{"commit":"7d5d530efbf4d2889913133318ad7e4875858b6b","rows":[{"n":"Causalean.DAG","k":"structure","m":"Causalean.Graph.DAG","a":"Graph","t":1,"d":"A Directed Acyclic Graph on a finite vertex type: a decidable edge relation together with the condition that no vertex is connected to itself by a directed path — the transitive closure of the edge relation is irreflexive. Irreflexivity of the transitive closure is exactly the statement that the graph has no directed cycle.","s":"(V : Type u_2) → [DecidableEq V] → [Fintype V] → Type u_2","l":48,"f":"Causalean/Graph/DAG.lean"},{"n":"Causalean.DAG.BBDir","k":"inductive","m":"Causalean.Graph.DSep.BayesBall","a":"Graph/DSep","t":1,"d":"Direction from which a node is visited in the Bayes Ball algorithm: arrival from a parent (fromParent) or arrival from a child (fromChild).","s":"Type","l":51,"f":"Causalean/Graph/DSep/BayesBall.lean"},{"n":"Causalean.DAG.BBState","k":"def","m":"Causalean.Graph.DSep.BayesBall","a":"Graph/DSep","t":1,"d":"A Bayes Ball state: a vertex paired with arrival direction.","s":"Type u_2 → Type (max u_2 0)","l":63,"f":"Causalean/Graph/DSep/BayesBall.lean"},{"n":"Causalean.DAG.HasActivePath","k":"def","m":"Causalean.Graph.DSep.ActivePath","a":"Graph/DSep","t":1,"d":"There exists an active path from some vertex in X to some vertex in Y.","s":"{V : Type u_1} → [inst : DecidableEq V] → [inst_1 : Fintype V] → Causalean.DAG V → Finset V → Finset V → Finset V → Prop","l":90,"f":"Causalean/Graph/DSep/ActivePath.lean"},{"n":"Causalean.DAG.IsActivePath","k":"def","m":"Causalean.Graph.DSep.ActivePath","a":"Graph/DSep","t":1,"d":"A path (list of vertices) is active (unblocked) given conditioning set Z if: - consecutive vertices are undirected-adjacent - for every intermediate triple (pᵢ, pᵢ₊₁, pᵢ₊₂): - if pᵢ₊₁ is a collider: pᵢ₊₁ ∈ G.bbZAncestors Z - if pᵢ₊₁ is not a collider: pᵢ₊₁ ∉ Z","s":"{V : Type u_1} → [inst : DecidableEq V] → [inst_1 : Fintype V] → Causalean.DAG V → Finset V → List V → Prop","l":72,"f":"Causalean/Graph/DSep/ActivePath.lean"},{"n":"Causalean.DAG.IsCollider","k":"def","m":"Causalean.Graph.DSep.ActivePath","a":"Graph/DSep","t":1,"d":"Whether vertex m is a collider on the triple (l, m, r): both edges point toward m, i.e., G.edge l m ∧ G.edge r m.","s":"{V : Type u_1} → [inst : DecidableEq V] → [inst_1 : Fintype V] → Causalean.DAG V → V → V → V → Prop","l":64,"f":"Causalean/Graph/DSep/ActivePath.lean"},{"n":"Causalean.DAG.IsCoveredEdge","k":"def","m":"Causalean.Graph.MarkovEquiv.CoveredReversal","a":"Graph/MarkovEquiv","t":1,"d":"A directed edge a → b is covered when a and b share all parents apart from the edge a → b itself: every vertex c ≠ a is a parent of a iff it is a parent of b. Then pa(a) = pa(b) \\ {a}. Covered edges are exactly the reversible (unprotected) ones.","s":"{V : Type u_1} → [inst : DecidableEq V] → [inst_1 : Fintype V] → Causalean.DAG V → V → V → Prop","l":39,"f":"Causalean/Graph/MarkovEquiv/CoveredReversal.lean"},{"n":"Causalean.DAG.IsCoveredEdge.ne","k":"theorem","m":"Causalean.Graph.MarkovEquiv.CoveredReversal","a":"Graph/MarkovEquiv","t":2,"d":"A covered edge is genuinely an edge a → b, so a ≠ b.","s":"∀ {V : Type u_1} [inst : DecidableEq V] [inst_1 : Fintype V] {G : Causalean.DAG V} {a b : V}, G.IsCoveredEdge a b → a ≠ b","l":65,"f":"Causalean/Graph/MarkovEquiv/CoveredReversal.lean"},{"n":"Causalean.DAG.IsImmorality","k":"def","m":"Causalean.Graph.MarkovEquiv.Defs","a":"Graph/MarkovEquiv","t":1,"d":"A v-structure (immorality) at b: directed edges a → b and c → b whose tails a and c are distinct and not joined by any edge. Immoralities are the colliders whose parents are non-adjacent; they are exactly the part of the collider structure that is visible to conditional independence.","s":"{V : Type u_1} → [inst : DecidableEq V] → [inst_1 : Fintype V] → Causalean.DAG V → V → V → V → Prop","l":39,"f":"Causalean/Graph/MarkovEquiv/Defs.lean"},{"n":"Causalean.DAG.MoralAdj","k":"def","m":"Causalean.Graph.MarkovEquiv.Moralization","a":"Graph/MarkovEquiv","t":1,"d":"Moral adjacency within a ground set S: distinct vertices u, v ∈ S that are either skeleton-adjacent or share a common child inside S (\"married parents\"). This is the undirected edge relation of the moral graph of G restricted to S.","s":"{V : Type u_1} → [inst : DecidableEq V] → [inst_1 : Fintype V] → Causalean.DAG V → Finset V → V → V → Prop","l":40,"f":"Causalean/Graph/MarkovEquiv/Moralization.lean"},{"n":"Causalean.DAG.MoralConn","k":"def","m":"Causalean.Graph.MarkovEquiv.Moralization","a":"Graph/MarkovEquiv","t":1,"d":"Moral connectivity: u reaches v by a (possibly empty) sequence of moral steps inside S, every vertex of which avoids Z.","s":"{V : Type u_1} → [inst : DecidableEq V] → [inst_1 : Fintype V] → Causalean.DAG V → Finset V → Finset V → V → V → Prop","l":51,"f":"Causalean/Graph/MarkovEquiv/Moralization.lean"},{"n":"Causalean.DAG.MoralSep","k":"def","m":"Causalean.Graph.MarkovEquiv.Moralization","a":"Graph/MarkovEquiv","t":1,"d":"Moral separation: no vertex of X is moral-connected to a vertex of Y inside the ancestral set An(X ∪ Y ∪ Z) while avoiding Z.","s":"{V : Type u_1} → [inst : DecidableEq V] → [inst_1 : Fintype V] → Causalean.DAG V → Finset V → Finset V → Finset V → Prop","l":56,"f":"Causalean/Graph/MarkovEquiv/Moralization.lean"},{"n":"Causalean.DAG.MoralStep","k":"def","m":"Causalean.Graph.MarkovEquiv.Moralization","a":"Graph/MarkovEquiv","t":1,"d":"A single moral step inside S that avoids the conditioning set Z (both endpoints outside Z).","s":"{V : Type u_1} → [inst : DecidableEq V] → [inst_1 : Fintype V] → Causalean.DAG V → Finset V → Finset V → V → V → Prop","l":46,"f":"Causalean/Graph/MarkovEquiv/Moralization.lean"},{"n":"Causalean.DAG.OrderedLocalSG","k":"inductive","m":"Causalean.Graph.DSep.OrderedLocalSG","a":"Graph/DSep","t":1,"d":"Ordered-local semi-graphoid closure. A conditional-independence claim between two random-node blocks, given a third block, belongs to this closure when it can be derived from the DAG's ordered-local Markov basis using only the semi-graphoid rules.","s":"{V : Type u_1} → [inst : DecidableEq V] → [inst_1 : Fintype V] → Causalean.DAG V → Finset V → Finset V → Finset V → Finset V → Prop","l":37,"f":"Causalean/Graph/DSep/OrderedLocalSG.lean"},{"n":"Causalean.DAG.OrderedLocalSG.subset_random","k":"theorem","m":"Causalean.Graph.DSep.OrderedLocalSG","a":"Graph/DSep","t":2,"d":"Every triple appearing in an ordered-local derivation has all three sets contained in the ambient random set R. Used by the SCM interpretation to recover the ⊆ randomVars side-conditions of FullCondIndep.","s":"∀ {V : Type u_1} [inst : DecidableEq V] [inst_1 : Fintype V] {G : Causalean.DAG V} {R X Y Z : Finset V}, G.OrderedLocalSG R X Y Z → X ⊆ R ∧ Y ⊆ R ∧ Z ⊆ R","l":66,"f":"Causalean/Graph/DSep/OrderedLocalSG.lean"},{"n":"Causalean.DAG.UAdj","k":"def","m":"Causalean.Graph.DSep.ActivePath","a":"Graph/DSep","t":1,"d":"An undirected edge in the DAG: either G.edge u v or G.edge v u.","s":"{V : Type u_1} → [inst : DecidableEq V] → [inst_1 : Fintype V] → Causalean.DAG V → V → V → Prop","l":57,"f":"Causalean/Graph/DSep/ActivePath.lean"},{"n":"Causalean.DAG.UAdj_symm","k":"theorem","m":"Causalean.Graph.DSep.ActivePath","a":"Graph/DSep","t":2,"d":"Undirected adjacency is symmetric: if two vertices are adjacent, they remain adjacent in the opposite order.","s":"∀ {V : Type u_1} [inst : DecidableEq V] [inst_1 : Fintype V] (G : Causalean.DAG V) {u v : V}, G.UAdj u v → G.UAdj v u","l":97,"f":"Causalean/Graph/DSep/ActivePath.lean"},{"n":"Causalean.DAG.activePath_drop2","k":"theorem","m":"Causalean.Graph.MarkovEquiv.Moralization","a":"Graph/MarkovEquiv","t":2,"d":"Removing the first two vertices of an active path with at least three vertices leaves an active path.","s":"∀ {V : Type u_1} [inst : DecidableEq V] [inst_1 : Fintype V] (G : Causalean.DAG V) {Z : Finset V} {a b c : V} {r : List V}, G.IsActivePath Z (a :: b :: c :: r) → G.IsActivePath Z (c :: r)","l":119,"f":"Causalean/Graph/MarkovEquiv/Moralization.lean"},{"n":"Causalean.DAG.activePath_forwardRun_absurd","k":"theorem","m":"Causalean.SCM.Do.CutsetDSep","a":"SCM/Do","t":2,"d":"An active path that starts with an arrow flowing away from an ancestor cannot end at a root when that ancestor has no directed route to any activated conditioning ancestor. The result rules out a forward run that must either enter a root or create an activated collider.","s":"∀ {V : Type u_3} [inst : DecidableEq V] [inst_1 : Fintype V] (G : Causalean.DAG V) {C : Finset V} {s : V}, (∀ k ∈ G.bbZAncestors C, ¬G.isAncestor s k) → ∀ (prev m : V) (rest : List V), G.IsActivePath C (prev :: m :: rest) → G.edge prev m → s = prev ∨ G.isAncestor s prev → (∀ (p : V), ¬G.edge p ((prev :: m :: rest).getLast (of_eq_true (Eq.trans (congrArg Not (eq_false' fun h => False.elim (noConfusion_of_Nat List.ctorIdx h))) not_false_eq_true)))) → False","l":357,"f":"Causalean/SCM/Do/CutsetDSep.lean"},{"n":"Causalean.DAG.activePath_head_triple","k":"theorem","m":"Causalean.Graph.MarkovEquiv.Moralization","a":"Graph/MarkovEquiv","t":2,"d":"In an active path whose first three vertices are a, b, and c, the middle vertex obeys the active-path condition: a collider belongs to the ancestral closure of the conditioning set, and a non-collider is not conditioned on.","s":"∀ {V : Type u_1} [inst : DecidableEq V] [inst_1 : Fintype V] (G : Causalean.DAG V) {Z : Finset V} {a b c : V} {r : List V}, G.IsActivePath Z (a :: b :: c :: r) → if G.IsCollider a b c then b ∈ G.bbZAncestors Z else b ∉ Z","l":110,"f":"Causalean/Graph/MarkovEquiv/Moralization.lean"},{"n":"Causalean.DAG.activePath_head_uAdj","k":"theorem","m":"Causalean.Graph.MarkovEquiv.Moralization","a":"Graph/MarkovEquiv","t":2,"d":"In an active path whose first two vertices are a and b, those vertices are adjacent in the underlying undirected graph.","s":"∀ {V : Type u_1} [inst : DecidableEq V] [inst_1 : Fintype V] (G : Causalean.DAG V) {Z : Finset V} {a b : V} {r : List V}, G.IsActivePath Z (a :: b :: r) → G.UAdj a b","l":103,"f":"Causalean/Graph/MarkovEquiv/Moralization.lean"},{"n":"Causalean.DAG.activePath_join_at_collider","k":"theorem","m":"Causalean.Graph.DSep.OrderedLocalSG","a":"Graph/DSep","t":2,"d":"Two active paths that meet at a conditioned collider can be joined into an active path when both incident edges point into the joining vertex.","s":"∀ {V : Type u_1} [inst : DecidableEq V] [inst_1 : Fintype V] (G : Causalean.DAG V) {Z : Finset V} {n x y : V} {pa pb : List V} (hpa_len : pa.length ≥ 2), pa.head? = some x → pa.getLast? = some n → G.IsActivePath Z pa → ∀ (hpb_len : pb.length ≥ 2), pb.head? = some n → pb.getLast? = some y → G.IsActivePath Z pb → G.edge (pa.get ⟨pa.length - 2, Decidable.byContradiction fun a => Causalean.DAG.activePath_join_at_collider._proof_1 hpa_len a⟩) n → G.edge (pb.get ⟨1, Decidable.byContradiction fun a => Causalean.DAG.activePath_join_at_collider._proof_2 hpb_len a⟩) n → n ∈ Z → have p := pa ++ pb.tail; p.length ≥ 2 ∧ p.head? = some x ∧ p.getLast? = some y ∧ G.IsActivePath Z p","l":425,"f":"Causalean/Graph/DSep/OrderedLocalSG.lean"},{"n":"Causalean.DAG.activePath_nodes_are_ancestors","k":"theorem","m":"Causalean.Graph.DSep.Ancestral","a":"Graph/DSep","t":1,"d":"Active-path nodes lie in the ancestral set. For finite vertex sets X, Y, Z, consider a path p that is active given Z running from a node x in X to a node y in Y. Then every vertex on p lies in the ancestral set of X ∪ Y ∪ Z. This is the main classical lemma used to justify ancestral reduction of d-separation.","s":"∀ {V : Type u_1} [inst : DecidableEq V] [inst_1 : Fintype V] (G : Causalean.DAG V) {X Y Z : Finset V} {x y : V} {p : List V}, x ∈ X → y ∈ Y → G.IsActivePath Z p → p.head? = some x → p.getLast? = some y → ∀ v ∈ p, v ∈ G.ancestralSet (X ∪ Y ∪ Z)","l":319,"f":"Causalean/Graph/DSep/Ancestral.lean"},{"n":"Causalean.DAG.activePath_transfer_cond_to_source","k":"theorem","m":"Causalean.Graph.DSep.Separation","a":"Graph/DSep","t":2,"d":"An active path under an enlarged conditioning set can be replaced by an active path whose source may additionally come from the newly conditioned vertices.","s":"∀ {V : Type u_1} [inst : DecidableEq V] [inst_1 : Fintype V] (G : Causalean.DAG V) {X Z S : Finset V} {p : List V} {x w : V}, x ∈ X → p.length ≥ 2 → G.IsActivePath (Z ∪ S) p → p.head? = some x → p.getLast? = some w → ∃ x' p', x' ∈ X ∪ S ∧ p'.length ≥ 2 ∧ G.IsActivePath Z p' ∧ p'.head? = some x' ∧ p'.getLast? = some w","l":215,"f":"Causalean/Graph/DSep/Separation.lean"},{"n":"Causalean.DAG.activePath_witness_subset_ancestralSet","k":"theorem","m":"Causalean.Graph.DSep.Ancestral","a":"Graph/DSep","t":2,"d":"Active-path inner nodes lie in the ancestral set (informative form).","s":"∀ {V : Type u_1} [inst : DecidableEq V] [inst_1 : Fintype V] (G : Causalean.DAG V) {X Y Z : Finset V} {v : V}, v ∈ G.bbReachableVertices Z X → v ∈ Y → ∃ p, ∃ x ∈ X, p.length ≥ 2 ∧ G.IsActivePath Z p ∧ p.head? = some x ∧ p.getLast? = some v ∧ ∀ w ∈ p, w ∈ G.ancestralSet (X ∪ Y ∪ Z)","l":356,"f":"Causalean/Graph/DSep/Ancestral.lean"},{"n":"Causalean.DAG.active_triple_swap_outer","k":"theorem","m":"Causalean.Graph.DSep.ActivePath","a":"Graph/DSep","t":2,"d":"The active-path condition at a three-vertex segment is unchanged when its two outer vertices are swapped.","s":"∀ {V : Type u_1} [inst : DecidableEq V] [inst_1 : Fintype V] (G : Causalean.DAG V) {Z : Finset V} {u w z : V}, (if G.IsCollider u w z then w ∈ G.bbZAncestors Z else w ∉ Z) → if G.IsCollider z w u then w ∈ G.bbZAncestors Z else w ∉ Z","l":538,"f":"Causalean/Graph/DSep/ActivePath.lean"},{"n":"Causalean.DAG.acyclic_of_topoOrder","k":"theorem","m":"Causalean.Graph.DAG","a":"Graph","t":1,"d":"Acyclicity from a topological ranking. Given an edge relation e on V and a ranking function τ into a type equipped with a transitive, irreflexive relation r, if τ strictly increases (with respect to r) along every edge of e, then e has no directed cycle: no vertex is reachable from itself via the transitive closure of e.","s":"∀ {V : Type u_2} {W : Type u_3} {r : W → W → Prop} [IsTrans W r] [Std.Irrefl r] {e : V → V → Prop} {τ : V → W}, (∀ (u v : V), e u v → r (τ u) (τ v)) → ∀ (v : V), ¬Relation.TransGen e v v","l":450,"f":"Causalean/Graph/DAG.lean"},{"n":"Causalean.DAG.adjacent_iff_not_dSeparable","k":"theorem","m":"Causalean.Graph.MarkovEquiv.Readoff","a":"Graph/MarkovEquiv","t":1,"d":"Skeleton read-off. For two distinct vertices a and b, a and b are adjacent (in either direction) exactly when no conditioning set d-separates a from b.","s":"∀ {V : Type u_1} [inst : DecidableEq V] [inst_1 : Fintype V] (G : Causalean.DAG V) {a b : V}, a ≠ b → (G.UAdj a b ↔ ¬∃ Z, G.dSep {a} {b} Z)","l":178,"f":"Causalean/Graph/MarkovEquiv/Readoff.lean"},{"n":"Causalean.DAG.ancClosure","k":"def","m":"Causalean.Graph.DAG","a":"Graph","t":1,"d":"The strict ancestors of v: all vertices u with a directed path u ⇝ v, computed by iterating the backward-parent step |V| times starting from v's parents. |V| iterations suffice because the accumulating set is an increasing chain of subsets of a |V|-element type, hence reaches its fixpoint.","s":"{V : Type u_2} → [inst : DecidableEq V] → [inst_1 : Fintype V] → Causalean.DAG V → V → Finset V","l":165,"f":"Causalean/Graph/DAG.lean"},{"n":"Causalean.DAG.ancClosure_closed","k":"theorem","m":"Causalean.Graph.DAG","a":"Graph","t":2,"d":"Every parent of a vertex in its computed ancestor set also belongs to that ancestor set.","s":"∀ {V : Type u_2} [inst : DecidableEq V] [inst_1 : Fintype V] (G : Causalean.DAG V) (v : V) {x : V}, x ∈ G.ancClosure v → G.parents x ⊆ G.ancClosure v","l":264,"f":"Causalean/Graph/DAG.lean"},{"n":"Causalean.DAG.ancStep","k":"def","m":"Causalean.Graph.DAG","a":"Graph","t":1,"d":"One backward reachability step: enlarge S by the parents of every vertex in S. Iterating this from G.parents v accumulates all strict ancestors of v.","s":"{V : Type u_2} → [inst : DecidableEq V] → [inst_1 : Fintype V] → Causalean.DAG V → Finset V → Finset V","l":161,"f":"Causalean/Graph/DAG.lean"},{"n":"Causalean.DAG.ancestorRank","k":"def","m":"Causalean.Graph.DAG","a":"Graph","t":1,"d":"The rank of v: the number of strict ancestors of v. Along an edge the strict-ancestor set strictly grows, so the rank strictly increases; this makes it the basis of a topological numbering.","s":"{V : Type u_2} → [inst : DecidableEq V] → [inst_1 : Fintype V] → Causalean.DAG V → V → ℕ","l":360,"f":"Causalean/Graph/DAG.lean"},{"n":"Causalean.DAG.ancestorRank_lt_of_edge","k":"theorem","m":"Causalean.Graph.DAG","a":"Graph","t":2,"d":"Along an edge the strict-ancestor count strictly increases.","s":"∀ {V : Type u_2} [inst : DecidableEq V] [inst_1 : Fintype V] (G : Causalean.DAG V) {a b : V}, G.edge a b → G.ancestorRank a < G.ancestorRank b","l":365,"f":"Causalean/Graph/DAG.lean"},{"n":"Causalean.DAG.ancestors","k":"def","m":"Causalean.Graph.DAG","a":"Graph","t":1,"d":"The ancestors of v in G: all vertices u such that u is an ancestor of v.","s":"{V : Type u_2} → [inst : DecidableEq V] → [inst_1 : Fintype V] → Causalean.DAG V → V → Finset V","l":306,"f":"Causalean/Graph/DAG.lean"},{"n":"Causalean.DAG.ancestorsSet","k":"def","m":"Causalean.Graph.DAG","a":"Graph","t":1,"d":"The ancestors of a set S: all vertices that are ancestors of some vertex in S.","s":"{V : Type u_2} → [inst : DecidableEq V] → [inst_1 : Fintype V] → Causalean.DAG V → Finset V → Finset V","l":338,"f":"Causalean/Graph/DAG.lean"},{"n":"Causalean.DAG.ancestralSet","k":"def","m":"Causalean.Graph.DAG","a":"Graph","t":1,"d":"The ancestral set of S: the set S together with all its ancestors (S ∪ G.ancestorsSet S).","s":"{V : Type u_2} → [inst : DecidableEq V] → [inst_1 : Fintype V] → Causalean.DAG V → Finset V → Finset V","l":342,"f":"Causalean/Graph/DAG.lean"},{"n":"Causalean.DAG.ancestralSet_cases","k":"theorem","m":"Causalean.Graph.MarkovEquiv.Moralization","a":"Graph/MarkovEquiv","t":2,"d":"A vertex in the ancestral closure of a set either belongs to that set itself or is a strict ancestor of one of its elements.","s":"∀ {V : Type u_1} [inst : DecidableEq V] [inst_1 : Fintype V] (G : Causalean.DAG V) {A : Finset V} {a : V}, a ∈ G.ancestralSet A → a ∈ A ∨ ∃ w ∈ A, G.isAncestor a w","l":331,"f":"Causalean/Graph/MarkovEquiv/Moralization.lean"},{"n":"Causalean.DAG.ancestralSet_idem","k":"theorem","m":"Causalean.Graph.DSep.InduceTransport","a":"Graph/DSep","t":2,"d":"Taking the ancestors of an ancestral set gives back the same set.","s":"∀ {V : Type u_2} [inst : DecidableEq V] [inst_1 : Fintype V] (G : Causalean.DAG V) (S : Finset V), G.ancestralSet (G.ancestralSet S) = G.ancestralSet S","l":34,"f":"Causalean/Graph/DSep/InduceTransport.lean"},{"n":"Causalean.DAG.ancestralSet_inter_subset_ancestralSet_of_dSep","k":"theorem","m":"Causalean.Graph.DSep.Ancestral","a":"Graph/DSep","t":1,"d":"Ancestral intersection from reachability separation. If X and Y are disjoint and no vertex of Y is Bayes-Ball-reachable from X given Z, then any vertex lying in both the ancestral closure of X and the ancestral closure of Y also lies in the ancestral closure of Z.","s":"∀ {V : Type u_1} [inst : DecidableEq V] [inst_1 : Fintype V] (G : Causalean.DAG V) {X Y Z : Finset V}, Disjoint X Y → Disjoint (G.bbReachableVertices Z X) Y → G.ancestralSet X ∩ G.ancestralSet Y ⊆ G.ancestralSet Z","l":730,"f":"Causalean/Graph/DSep/Ancestral.lean"},{"n":"Causalean.DAG.ancestralSet_mono","k":"theorem","m":"Causalean.Graph.DSep.Ancestral","a":"Graph/DSep","t":2,"d":"ancestralSet is monotone in its argument.","s":"∀ {V : Type u_1} [inst : DecidableEq V] [inst_1 : Fintype V] (G : Causalean.DAG V) {S T : Finset V}, S ⊆ T → G.ancestralSet S ⊆ G.ancestralSet T","l":62,"f":"Causalean/Graph/DSep/Ancestral.lean"},{"n":"Causalean.DAG.ancestralSet_subset_of_subset_ancestralSet","k":"theorem","m":"Causalean.Graph.DSep.OrderedLocalSG","a":"Graph/DSep","t":2,"d":"The ancestral closure of a subset of an ancestral closure remains inside the original ancestral closure.","s":"∀ {V : Type u_1} [inst : DecidableEq V] [inst_1 : Fintype V] (G : Causalean.DAG V) {Q Q' : Finset V}, Q' ⊆ G.ancestralSet Q → G.ancestralSet Q' ⊆ G.ancestralSet Q","l":122,"f":"Causalean/Graph/DSep/OrderedLocalSG.lean"},{"n":"Causalean.DAG.asymm","k":"theorem","m":"Causalean.Graph.DAG","a":"Graph","t":2,"d":"If there is an edge from u to v, then there is no edge from v to u (a two-cycle is forbidden by acyclicity).","s":"∀ {V : Type u_2} [inst : DecidableEq V] [inst_1 : Fintype V] (G : Causalean.DAG V) {u v : V}, G.edge u v → ¬G.edge v u","l":123,"f":"Causalean/Graph/DAG.lean"},{"n":"Causalean.DAG.bbInit","k":"def","m":"Causalean.Graph.DSep.BayesBall","a":"Graph/DSep","t":1,"d":"Initial BFS frontier from a source set X: for each x ∈ X, include all children of x in direction fromParent and all parents of x in direction fromChild. These are the states a \"ball\" passing through x can occupy.","s":"{V : Type u_1} → [inst : DecidableEq V] → [inst_1 : Fintype V] → Causalean.DAG V → Finset V → Finset (Causalean.DAG.BBState V)","l":238,"f":"Causalean/Graph/DSep/BayesBall.lean"},{"n":"Causalean.DAG.bbReachable","k":"def","m":"Causalean.Graph.DSep.BayesBall","a":"Graph/DSep","t":1,"d":"The set of Bayes Ball states reachable from source vertices X given conditioning set Z.","s":"{V : Type u_1} → [inst : DecidableEq V] → [inst_1 : Fintype V] → Causalean.DAG V → Finset V → Finset V → Finset (Causalean.DAG.BBState V)","l":248,"f":"Causalean/Graph/DSep/BayesBall.lean"},{"n":"Causalean.DAG.bbReachableVertices","k":"def","m":"Causalean.Graph.DSep.BayesBall","a":"Graph/DSep","t":1,"d":"Vertices reachable from X via Bayes Ball (ignoring direction).","s":"{V : Type u_1} → [inst : DecidableEq V] → [inst_1 : Fintype V] → Causalean.DAG V → Finset V → Finset V → Finset V","l":327,"f":"Causalean/Graph/DSep/BayesBall.lean"},{"n":"Causalean.DAG.bbReachableVertices_iff_activePath","k":"theorem","m":"Causalean.Graph.DSep.ActivePath","a":"Graph/DSep","t":1,"d":"Bayes Ball correctness. For a source vertex set X and a conditioning vertex set Z and a vertex v, v lies in the breadth-first-search reachable set bbReachableVertices Z X if and only if there is an active path, given Z, from some vertex of X to v.","s":"∀ {V : Type u_1} [inst : DecidableEq V] [inst_1 : Fintype V] (G : Causalean.DAG V) (X Z : Finset V) (v : V), v ∈ G.bbReachableVertices Z X ↔ ∃ x ∈ X, ∃ p, p.length ≥ 2 ∧ G.IsActivePath Z p ∧ p.head? = some x ∧ p.getLast? = some v","l":698,"f":"Causalean/Graph/DSep/ActivePath.lean"},{"n":"Causalean.DAG.bbReachableVertices_mono_source","k":"theorem","m":"Causalean.Graph.DSep.BayesBall","a":"Graph/DSep","t":2,"d":"Bayes Ball reachable vertices are monotone in the source set.","s":"∀ {V : Type u_1} [inst : DecidableEq V] [inst_1 : Fintype V] (G : Causalean.DAG V) {Z X X' : Finset V}, X' ⊆ X → G.bbReachableVertices Z X' ⊆ G.bbReachableVertices Z X","l":345,"f":"Causalean/Graph/DSep/BayesBall.lean"},{"n":"Causalean.DAG.bbReachable_bbStep_subset","k":"theorem","m":"Causalean.Graph.DSep.BayesBall","a":"Graph/DSep","t":2,"d":"bbReachable Z X is closed under bbStep Z. If s ∈ bbReachable Z X, then every state produced by bbStep Z s is also in bbReachable Z X.","s":"∀ {V : Type u_1} [inst : DecidableEq V] [inst_1 : Fintype V] (G : Causalean.DAG V) (Z X : Finset V) {s : Causalean.DAG.BBState V}, s ∈ G.bbReachable Z X → G.bbStep Z s ⊆ G.bbReachable Z X","l":377,"f":"Causalean/Graph/DSep/BayesBall.lean"},{"n":"Causalean.DAG.bbReachable_extend_directed_arm","k":"theorem","m":"Causalean.Graph.DSep.OrderedLocalSG","a":"Graph/DSep","t":2,"d":"Extending an active path by a directed arm at a non-collider seam.","s":"∀ {V : Type u_1} [inst : DecidableEq V] [inst_1 : Fintype V] (G : Causalean.DAG V) {Z : Finset V} {a c b : V} {pa q : List V}, pa.length ≥ 2 → pa.head? = some a → pa.getLast? = some c → G.IsActivePath Z pa → q.length ≥ 2 → q.head? = some c → q.getLast? = some b → (∀ (i : ℕ) (hi : i + 1 < q.length), G.edge (q.get ⟨i, Decidable.byContradiction fun a => Causalean.DAG.bbReachable_extend_directed_arm._proof_1 i hi a⟩) (q.get ⟨i + 1, hi⟩)) → (∀ (i : ℕ) (hi : i + 2 < q.length), q.get ⟨i + 1, Decidable.byContradiction fun a => Causalean.DAG.bbReachable_extend_directed_arm._proof_2 i hi a⟩ ∉ Z) → c ∉ Z → b ∈ G.bbReachableVertices Z {a}","l":1689,"f":"Causalean/Graph/DSep/OrderedLocalSG.lean"},{"n":"Causalean.DAG.bbReachable_init_subset","k":"theorem","m":"Causalean.Graph.DSep.BayesBall","a":"Graph/DSep","t":2,"d":"The initial BFS frontier bbInit X is contained in bbReachable Z X.","s":"∀ {V : Type u_1} [inst : DecidableEq V] [inst_1 : Fintype V] (G : Causalean.DAG V) (Z X : Finset V), G.bbInit X ⊆ G.bbReachable Z X","l":355,"f":"Causalean/Graph/DSep/BayesBall.lean"},{"n":"Causalean.DAG.bbReachable_minimal","k":"theorem","m":"Causalean.Graph.DSep.BayesBall","a":"Graph/DSep","t":1,"d":"bbReachable Z X is the *least* superset of bbInit X closed under bbStep Z: for a conditioning set Z and source set X, if a candidate set S of Bayes-Ball states contains the initial frontier bbInit X and S is closed under the Bayes-Ball step relation bbStep Z, then S contains every state reachable via bbReachable Z X.","s":"∀ {V : Type u_1} [inst : DecidableEq V] [inst_1 : Fintype V] (G : Causalean.DAG V) (Z X : Finset V) (S : Finset (Causalean.DAG.BBState V)), G.bbInit X ⊆ S → (∀ s ∈ S, G.bbStep Z s ⊆ S) → G.bbReachable Z X ⊆ S","l":361,"f":"Causalean/Graph/DSep/BayesBall.lean"},{"n":"Causalean.DAG.bbReachable_mono_source","k":"theorem","m":"Causalean.Graph.DSep.BayesBall","a":"Graph/DSep","t":2,"d":"Bayes Ball reachability is monotone in the source set.","s":"∀ {V : Type u_1} [inst : DecidableEq V] [inst_1 : Fintype V] (G : Causalean.DAG V) {Z X X' : Finset V}, X' ⊆ X → G.bbReachable Z X' ⊆ G.bbReachable Z X","l":335,"f":"Causalean/Graph/DSep/BayesBall.lean"},{"n":"Causalean.DAG.bbStep","k":"def","m":"Causalean.Graph.DSep.BayesBall","a":"Graph/DSep","t":1,"d":"One step of the Bayes Ball algorithm from state (w, dir).","s":"{V : Type u_1} → [inst : DecidableEq V] → [inst_1 : Fintype V] → Causalean.DAG V → Finset V → Causalean.DAG.BBState V → Finset (Causalean.DAG.BBState V)","l":72,"f":"Causalean/Graph/DSep/BayesBall.lean"},{"n":"Causalean.DAG.bbZAncestors","k":"def","m":"Causalean.Graph.DSep.BayesBall","a":"Graph/DSep","t":1,"d":"The set of vertices in Z together with all ancestors of Z. A collider is \"activated\" iff it or a descendant is in Z.","s":"{V : Type u_1} → [inst : DecidableEq V] → [inst_1 : Fintype V] → Causalean.DAG V → Finset V → Finset V","l":68,"f":"Causalean/Graph/DSep/BayesBall.lean"},{"n":"Causalean.DAG.bbZAncestors_edge_congr","k":"theorem","m":"Causalean.Graph.MarkovEquiv.CoveredReversal","a":"Graph/MarkovEquiv","t":2,"d":"Graphs with the same directed edges have exactly the same sets of vertices that are ancestors of the conditioning set and can activate colliders.","s":"∀ {V : Type u_1} [inst : DecidableEq V] [inst_1 : Fintype V] {G₁ G₂ : Causalean.DAG V}, (∀ (u w : V), G₁.edge u w ↔ G₂.edge u w) → ∀ (Z : Finset V) (v : V), v ∈ G₁.bbZAncestors Z ↔ v ∈ G₂.bbZAncestors Z","l":239,"f":"Causalean/Graph/MarkovEquiv/CoveredReversal.lean"},{"n":"Causalean.DAG.bbZAncestors_flip_of_ne","k":"theorem","m":"Causalean.Graph.MarkovEquiv.CoveredReversal","a":"Graph/MarkovEquiv","t":2,"d":"If a vertex other than the reversed tail belongs to the Bayes-ball ancestor closure of a conditioning set before a covered reversal, it belongs to that closure after the reversal.","s":"∀ {V : Type u_1} [inst : DecidableEq V] [inst_1 : Fintype V] {G : Causalean.DAG V} {a b v : V} (hcov : G.IsCoveredEdge a b) (Z : Finset V), v ≠ a → v ∈ G.bbZAncestors Z → v ∈ (Causalean.DAG.flipEdge hcov).bbZAncestors Z","l":445,"f":"Causalean/Graph/MarkovEquiv/CoveredReversal.lean"},{"n":"Causalean.DAG.bbZAncestors_mono_edge","k":"theorem","m":"Causalean.Graph.DSep.Separation","a":"Graph/DSep","t":2,"d":"Adding directed edges can only enlarge the set of vertices that are ancestors of the conditioning set and can activate colliders.","s":"∀ {V : Type u_1} [inst : DecidableEq V] [inst_1 : Fintype V] (G G' : Causalean.DAG V), (∀ (u v : V), G'.edge u v → G.edge u v) → ∀ (Z : Finset V), G'.bbZAncestors Z ⊆ G.bbZAncestors Z","l":703,"f":"Causalean/Graph/DSep/Separation.lean"},{"n":"Causalean.DAG.bbZAncestors_subset_ancestralSet_of_subset","k":"theorem","m":"Causalean.Graph.DSep.Ancestral","a":"Graph/DSep","t":2,"d":"bbZAncestors Z ⊆ ancestralSet S whenever Z ⊆ S. Used to lift a collider-activation witness m ∈ bbZAncestors Z into the larger ancestralSet (X ∪ Y ∪ Z).","s":"∀ {V : Type u_1} [inst : DecidableEq V] [inst_1 : Fintype V] (G : Causalean.DAG V) {Z S : Finset V}, Z ⊆ S → G.bbZAncestors Z ⊆ G.ancestralSet S","l":82,"f":"Causalean/Graph/DSep/Ancestral.lean"},{"n":"Causalean.DAG.bbZAncestors_union_eq","k":"theorem","m":"Causalean.Graph.DSep.Separation","a":"Graph/DSep","t":2,"d":"Ancestral-set distributes over union.","s":"∀ {V : Type u_1} [inst : DecidableEq V] [inst_1 : Fintype V] (G : Causalean.DAG V) (Z S : Finset V), G.bbZAncestors (Z ∪ S) = G.bbZAncestors Z ∪ G.bbZAncestors S","l":86,"f":"Causalean/Graph/DSep/Separation.lean"},{"n":"Causalean.DAG.branch_seed_eq","k":"theorem","m":"Causalean.Graph.DSep.OrderedLocalSG","a":"Graph/DSep","t":2,"d":"If a finite set contains a node, removing and then reinserting that node while taking unions recovers the same union as using the original set.","s":"∀ {V : Type u_1} [inst : DecidableEq V] {a b Zr c : Finset V} {n : V}, n ∈ Zr → a ∪ {n} ∪ (Zr.erase n ∪ b) ∪ c = a ∪ b ∪ Zr ∪ c","l":1269,"f":"Causalean/Graph/DSep/OrderedLocalSG.lean"},{"n":"Causalean.DAG.chain_join_active","k":"theorem","m":"Causalean.Graph.DSep.OrderedLocalSG","a":"Graph/DSep","t":2,"d":"Concatenating two active paths at a non-collider. If pa is active given Z ending at m, pb is active given Z starting at m, the first edge of pb points *out of* m (so m is a chain/fork point, not a collider), and m ∉ Z, then the glued path pa ++ pb.tail is active given Z.","s":"∀ {V : Type u_1} [inst : DecidableEq V] [inst_1 : Fintype V] (G : Causalean.DAG V) {Z : Finset V} {m x y : V} {pa pb : List V}, pa.length ≥ 2 → pa.head? = some x → pa.getLast? = some m → G.IsActivePath Z pa → ∀ (hpb_len : pb.length ≥ 2), pb.head? = some m → pb.getLast? = some y → G.IsActivePath Z pb → G.edge m (pb.get ⟨1, Decidable.byContradiction fun a => Causalean.DAG.activePath_join_at_collider._proof_2 hpb_len a⟩) → m ∉ Z → have p := pa ++ pb.tail; p.length ≥ 2 ∧ p.head? = some x ∧ p.getLast? = some y ∧ G.IsActivePath Z p","l":621,"f":"Causalean/Graph/DSep/OrderedLocalSG.lean"},{"n":"Causalean.DAG.children","k":"def","m":"Causalean.Graph.DAG","a":"Graph","t":1,"d":"The children of v in G: all vertices w such that (v, w) ∈ E.","s":"{V : Type u_2} → [inst : DecidableEq V] → [inst_1 : Fintype V] → Causalean.DAG V → V → Finset V","l":82,"f":"Causalean/Graph/DAG.lean"},{"n":"Causalean.DAG.children_subset_descendants","k":"theorem","m":"Causalean.Graph.DAG","a":"Graph","t":2,"d":"Children are a subset of descendants.","s":"∀ {V : Type u_2} [inst : DecidableEq V] [inst_1 : Fintype V] (G : Causalean.DAG V) (v : V), G.children v ⊆ G.descendants v","l":328,"f":"Causalean/Graph/DAG.lean"},{"n":"Causalean.DAG.dSep","k":"def","m":"Causalean.Graph.DSep.Separation","a":"Graph/DSep","t":1,"d":"The source, target, and conditioning sets are d-separated when they are pairwise disjoint and no target vertex is Bayes-Ball-reachable from the source set after conditioning.","s":"{V : Type u_1} → [inst : DecidableEq V] → [inst_1 : Fintype V] → Causalean.DAG V → Finset V → Finset V → Finset V → Prop","l":52,"f":"Causalean/Graph/DSep/Separation.lean"},{"n":"Causalean.DAG.dSep_iff_moralSep","k":"theorem","m":"Causalean.Graph.MarkovEquiv.Moralization","a":"Graph/MarkovEquiv","t":1,"d":"The moralization criterion. For pairwise-disjoint X, Y, Z, X and Y are d-separated by Z exactly when they are moral-separated: no moral path inside the ancestral set An(X ∪ Y ∪ Z) connects them while avoiding Z. (Lauritzen–Dawid–Larsen–Speed.)","s":"∀ {V : Type u_1} [inst : DecidableEq V] [inst_1 : Fintype V] (G : Causalean.DAG V) {X Y Z : Finset V}, Disjoint X Y → Disjoint X Z → Disjoint Y Z → (G.dSep X Y Z ↔ G.MoralSep X Y Z)","l":505,"f":"Causalean/Graph/MarkovEquiv/Moralization.lean"},{"n":"Causalean.DAG.dSep_mono_conditioningSet","k":"theorem","m":"Causalean.Graph.DSep.Separation","a":"Graph/DSep","t":1,"d":"d-separation transfers from a supergraph to a subgraph. For DAGs G and G' on the same vertex set, if every edge of G' is also an edge of G — G' is obtained from G by removing edges — and X and Y are d-separated by Z in G, then X and Y are also d-separated by Z in G'.","s":"∀ {V : Type u_1} [inst : DecidableEq V] [inst_1 : Fintype V] (G : Causalean.DAG V) {X Y Z : Finset V} (G' : Causalean.DAG V), (∀ (u v : V), G'.edge u v → G.edge u v) → G.dSep X Y Z → G'.dSep X Y Z","l":769,"f":"Causalean/Graph/DSep/Separation.lean"},{"n":"Causalean.DAG.dSep_parents_of_topoOrder_lt","k":"theorem","m":"Causalean.Graph.MarkovEquiv.Readoff","a":"Graph/MarkovEquiv","t":2,"d":"If one endpoint precedes a nonadjacent later endpoint topologically, conditioning on the later endpoint's parents d-separates the pair.","s":"∀ {V : Type u_1} [inst : DecidableEq V] [inst_1 : Fintype V] (G : Causalean.DAG V) {x y : V}, G.topoOrder x < G.topoOrder y → ¬G.UAdj x y → G.dSep {x} {y} (G.parents y)","l":57,"f":"Causalean/Graph/MarkovEquiv/Readoff.lean"},{"n":"Causalean.DAG.dSep_source_to_cond","k":"theorem","m":"Causalean.Graph.DSep.Separation","a":"Graph/DSep","t":1,"d":"Source-to-conditioning transfer for d-separation. If X and S are disjoint and X ∪ S and Y are d-separated by Z, then X and Y remain d-separated once S is moved into the conditioning set: G.dSep X Y (Z ∪ S). Equivalently: bbReachableVertices (Z ∪ S) X is a subset of bbReachableVertices Z (X ∪ S).","s":"∀ {V : Type u_1} [inst : DecidableEq V] [inst_1 : Fintype V] (G : Causalean.DAG V) {X Y Z S : Finset V}, Disjoint X S → G.dSep (X ∪ S) Y Z → G.dSep X Y (Z ∪ S)","l":656,"f":"Causalean/Graph/DSep/Separation.lean"},{"n":"Causalean.DAG.dSep_subset_left","k":"theorem","m":"Causalean.Graph.DSep.Separation","a":"Graph/DSep","t":2,"d":"d-separation is monotone in X: smaller source sets preserve d-separation.","s":"∀ {V : Type u_1} [inst : DecidableEq V] [inst_1 : Fintype V] (G : Causalean.DAG V) {X X' Y Z : Finset V}, X' ⊆ X → G.dSep X Y Z → G.dSep X' Y Z","l":72,"f":"Causalean/Graph/DSep/Separation.lean"},{"n":"Causalean.DAG.dSep_subset_right","k":"theorem","m":"Causalean.Graph.DSep.Separation","a":"Graph/DSep","t":2,"d":"d-separation is monotone in Y: shrinking the target set preserves d-separation.","s":"∀ {V : Type u_1} [inst : DecidableEq V] [inst_1 : Fintype V] (G : Causalean.DAG V) {X Y Y' Z : Finset V}, Y' ⊆ Y → G.dSep X Y Z → G.dSep X Y' Z","l":79,"f":"Causalean/Graph/DSep/Separation.lean"},{"n":"Causalean.DAG.dSep_symm","k":"theorem","m":"Causalean.Graph.DSep.Separation","a":"Graph/DSep","t":1,"d":"d-separation is symmetric. For three finite vertex sets X, Y, Z, if X is d-separated from Y given Z then Y is d-separated from X given Z.","s":"∀ {V : Type u_1} [inst : DecidableEq V] [inst_1 : Fintype V] (G : Causalean.DAG V) (X Y Z : Finset V), G.dSep X Y Z → G.dSep Y X Z","l":800,"f":"Causalean/Graph/DSep/Separation.lean"},{"n":"Causalean.DAG.dSep_union_roots_right","k":"theorem","m":"Causalean.Graph.DSep.BackdoorBridges","a":"Graph/DSep","t":1,"d":"Adding root nodes (no incoming edges) to the conditioning set preserves d-separation. In a DAG G, suppose X and Y are d-separated by Z, every vertex of R has no incoming edge in G, i.e. R consists of root vertices, and R is disjoint from X and from Y. Then X and Y remain d-separated once the root vertices R are added to the conditioning set: G.dSep X Y (Z ∪ R).","s":"∀ {V : Type u_1} [inst : DecidableEq V] [inst_1 : Fintype V] (G : Causalean.DAG V) {X Y Z R : Finset V}, G.dSep X Y Z → (∀ r ∈ R, ∀ (u : V), ¬G.edge u r) → Disjoint R X → Disjoint R Y → G.dSep X Y (Z ∪ R)","l":51,"f":"Causalean/Graph/DSep/BackdoorBridges.lean"},{"n":"Causalean.DAG.dSeparable_disjoint_of_not_uAdj","k":"theorem","m":"Causalean.Graph.MarkovEquiv.Readoff","a":"Graph/MarkovEquiv","t":2,"d":"Disjoint skeleton read-off witness. Two distinct non-adjacent vertices are d-separated by a conditioning set disjoint from both endpoints (the parents of the topologically later one, which contain neither a nor b).","s":"∀ {V : Type u_1} [inst : DecidableEq V] [inst_1 : Fintype V] (G : Causalean.DAG V) {a b : V}, a ≠ b → ¬G.UAdj a b → ∃ Z, a ∉ Z ∧ b ∉ Z ∧ G.dSep {a} {b} Z","l":286,"f":"Causalean/Graph/MarkovEquiv/Readoff.lean"},{"n":"Causalean.DAG.dSeparable_of_not_uAdj","k":"theorem","m":"Causalean.Graph.MarkovEquiv.Readoff","a":"Graph/MarkovEquiv","t":2,"d":"Two distinct non-adjacent vertices can always be d-separated: conditioning on the parents of the topologically later vertex blocks every path between them.","s":"∀ {V : Type u_1} [inst : DecidableEq V] [inst_1 : Fintype V] (G : Causalean.DAG V) {a b : V}, a ≠ b → ¬G.UAdj a b → ∃ Z, G.dSep {a} {b} Z","l":164,"f":"Causalean/Graph/MarkovEquiv/Readoff.lean"},{"n":"Causalean.DAG.decDSep","k":"instance","m":"Causalean.Graph.DSep.Separation","a":"Graph/DSep","t":2,"d":"D-separation is decidable by computing Bayes Ball reachability and checking disjointness.","s":"{V : Type u_1} → [inst : DecidableEq V] → [inst_1 : Fintype V] → (G : Causalean.DAG V) → (X Y Z : Finset V) → Decidable (G.dSep X Y Z)","l":62,"f":"Causalean/Graph/DSep/Separation.lean"},{"n":"Causalean.DAG.decIsAncestor","k":"instance","m":"Causalean.Graph.DAG","a":"Graph","t":2,"d":"Decidability of the ancestor relation, computed from the edge relation alone via the backward-reachability fixpoint ancClosure.","s":"{V : Type u_2} → [inst : DecidableEq V] → [inst_1 : Fintype V] → (G : Causalean.DAG V) → DecidableRel G.isAncestor","l":301,"f":"Causalean/Graph/DAG.lean"},{"n":"Causalean.DAG.decIsCollider","k":"instance","m":"Causalean.Graph.DSep.ActivePath","a":"Graph/DSep","t":2,"d":"Collider status of a triple is decidable whenever the directed edge relation is decidable.","s":"{V : Type u_1} → [inst : DecidableEq V] → [inst_1 : Fintype V] → (G : Causalean.DAG V) → (l m r : V) → Decidable (G.IsCollider l m r)","l":68,"f":"Causalean/Graph/DSep/ActivePath.lean"},{"n":"Causalean.DAG.decIsRoot","k":"instance","m":"Causalean.Graph.DAG","a":"Graph","t":2,"d":"Decidability of isRoot v (reduces to deciding G.parents v = ∅).","s":"{V : Type u_2} → [inst : DecidableEq V] → [inst_1 : Fintype V] → (G : Causalean.DAG V) → (v : V) → Decidable (G.isRoot v)","l":437,"f":"Causalean/Graph/DAG.lean"},{"n":"Causalean.DAG.decUAdj","k":"instance","m":"Causalean.Graph.DSep.ActivePath","a":"Graph/DSep","t":2,"d":"Undirected adjacency is decidable whenever the directed edge relation is decidable.","s":"{V : Type u_1} → [inst : DecidableEq V] → [inst_1 : Fintype V] → (G : Causalean.DAG V) → (u v : V) → Decidable (G.UAdj u v)","l":60,"f":"Causalean/Graph/DSep/ActivePath.lean"},{"n":"Causalean.DAG.descendants","k":"def","m":"Causalean.Graph.DAG","a":"Graph","t":1,"d":"The descendants of v in G: all vertices w such that v is an ancestor of w.","s":"{V : Type u_2} → [inst : DecidableEq V] → [inst_1 : Fintype V] → Causalean.DAG V → V → Finset V","l":310,"f":"Causalean/Graph/DAG.lean"},{"n":"Causalean.DAG.descendantsSet","k":"def","m":"Causalean.Graph.DAG","a":"Graph","t":1,"d":"The descendants of a set S: all vertices that are descendants of some vertex in S.","s":"{V : Type u_2} → [inst : DecidableEq V] → [inst_1 : Fintype V] → Causalean.DAG V → Finset V → Finset V","l":347,"f":"Causalean/Graph/DAG.lean"},{"n":"Causalean.DAG.edge_to_later_of_nonCollider","k":"theorem","m":"Causalean.Graph.MarkovEquiv.Readoff","a":"Graph/MarkovEquiv","t":2,"d":"On an unshielded two-edge triple, a non-collider middle vertex points into the topologically later endpoint.","s":"∀ {V : Type u_1} [inst : DecidableEq V] [inst_1 : Fintype V] (G : Causalean.DAG V) {x y b : V}, G.topoOrder x ≤ G.topoOrder y → G.UAdj x b → G.UAdj y b → ¬G.IsCollider x b y → G.edge b y","l":190,"f":"Causalean/Graph/MarkovEquiv/Readoff.lean"},{"n":"Causalean.DAG.exists_activePath_of_ancestor_avoiding","k":"theorem","m":"Causalean.Graph.DSep.Ancestral","a":"Graph/DSep","t":2,"d":"An ancestor outside the conditioning set's ancestral closure has an active path to its descendant.","s":"∀ {V : Type u_1} [inst : DecidableEq V] [inst_1 : Fintype V] (G : Causalean.DAG V) {Z : Finset V} {u v : V}, G.isAncestor u v → u ∉ G.ancestralSet Z → ∃ p, p.length ≥ 2 ∧ p.head? = some u ∧ p.getLast? = some v ∧ G.IsActivePath Z p","l":404,"f":"Causalean/Graph/DSep/Ancestral.lean"},{"n":"Causalean.DAG.exists_directedPath_avoiding","k":"theorem","m":"Causalean.Graph.DSep.Separation","a":"Graph/DSep","t":2,"d":"If u is an ancestor of v and u ∉ ancestralSet Z, there is a list u = p₀, p₁, …, pₖ = v (k ≥ 1) of directed edges all avoiding Z.","s":"∀ {V : Type u_1} [inst : DecidableEq V] [inst_1 : Fintype V] (G : Causalean.DAG V) {u v : V}, G.isAncestor u v → ∀ {Z : Finset V}, u ∉ G.ancestralSet Z → ∃ p, p.length ≥ 2 ∧ p.head? = some u ∧ p.getLast? = some v ∧ (∀ (i : ℕ) (hi : i + 1 < p.length), G.edge (p.get ⟨i, Decidable.byContradiction fun a => Causalean.DAG.exists_directedPath_avoiding._proof_1 p i hi a⟩) (p.get ⟨i + 1, hi⟩)) ∧ ∀ x ∈ p, x ∉ Z","l":110,"f":"Causalean/Graph/DSep/Separation.lean"},{"n":"Causalean.DAG.flipEdge","k":"def","m":"Causalean.Graph.MarkovEquiv.CoveredReversal","a":"Graph/MarkovEquiv","t":1,"d":"The directed acyclic graph obtained from G by reversing the covered edge a → b.","s":"{V : Type u_1} → [inst : DecidableEq V] → [inst_1 : Fintype V] → {G : Causalean.DAG V} → {a b : V} → G.IsCoveredEdge a b → Causalean.DAG V","l":124,"f":"Causalean/Graph/MarkovEquiv/CoveredReversal.lean"},{"n":"Causalean.DAG.flipEdge_edge","k":"theorem","m":"Causalean.Graph.MarkovEquiv.CoveredReversal","a":"Graph/MarkovEquiv","t":2,"d":"In the graph obtained by reversing a covered edge, the edges are exactly the old edges except for deleting a → b and adding b → a.","s":"∀ {V : Type u_1} [inst : DecidableEq V] [inst_1 : Fintype V] {G : Causalean.DAG V} {a b : V} (hcov : G.IsCoveredEdge a b) (u w : V), (Causalean.DAG.flipEdge hcov).edge u w ↔ G.edge u w ∧ ¬(u = a ∧ w = b) ∨ u = b ∧ w = a","l":128,"f":"Causalean/Graph/MarkovEquiv/CoveredReversal.lean"},{"n":"Causalean.DAG.flipEdge_edge_or_deleted","k":"theorem","m":"Causalean.Graph.MarkovEquiv.CoveredReversal","a":"Graph/MarkovEquiv","t":2,"d":"Under a covered reversal of the edge from a to b, every directed edge of the original graph either remains an edge or is the deleted edge from a to b.","s":"∀ {V : Type u_1} [inst : DecidableEq V] [inst_1 : Fintype V] {G : Causalean.DAG V} {a b u w : V} (hcov : G.IsCoveredEdge a b), G.edge u w → (Causalean.DAG.flipEdge hcov).edge u w ∨ u = a ∧ w = b","l":410,"f":"Causalean/Graph/MarkovEquiv/CoveredReversal.lean"},{"n":"Causalean.DAG.flipEdge_flipEdge_edge","k":"theorem","m":"Causalean.Graph.MarkovEquiv.CoveredReversal","a":"Graph/MarkovEquiv","t":2,"d":"Flipping the reversed covered edge restores the original edge relation.","s":"∀ {V : Type u_1} [inst : DecidableEq V] [inst_1 : Fintype V] {G : Causalean.DAG V} {a b : V} (hcov : G.IsCoveredEdge a b) (u w : V), (Causalean.DAG.flipEdge (Causalean.DAG.flipEdge_isCoveredEdge_back hcov)).edge u w ↔ G.edge u w","l":354,"f":"Causalean/Graph/MarkovEquiv/CoveredReversal.lean"},{"n":"Causalean.DAG.flipEdge_isCoveredEdge_back","k":"theorem","m":"Causalean.Graph.MarkovEquiv.CoveredReversal","a":"Graph/MarkovEquiv","t":2,"d":"In the flipped graph, the reversed edge b → a is covered.","s":"∀ {V : Type u_1} [inst : DecidableEq V] [inst_1 : Fintype V] {G : Causalean.DAG V} {a b : V} (hcov : G.IsCoveredEdge a b), (Causalean.DAG.flipEdge hcov).IsCoveredEdge b a","l":322,"f":"Causalean/Graph/MarkovEquiv/CoveredReversal.lean"},{"n":"Causalean.DAG.flipEdge_sameImmoralities","k":"theorem","m":"Causalean.Graph.MarkovEquiv.CoveredReversal","a":"Graph/MarkovEquiv","t":2,"d":"Reversing a covered edge preserves the immoralities. Because a and b share all other parents, no v-structure is created or destroyed by the single reversal.","s":"∀ {V : Type u_1} [inst : DecidableEq V] [inst_1 : Fintype V] {G : Causalean.DAG V} {a b : V} (hcov : G.IsCoveredEdge a b), Causalean.SameImmoralities G (Causalean.DAG.flipEdge hcov)","l":154,"f":"Causalean/Graph/MarkovEquiv/CoveredReversal.lean"},{"n":"Causalean.DAG.flipEdge_sameSkeleton","k":"theorem","m":"Causalean.Graph.MarkovEquiv.CoveredReversal","a":"Graph/MarkovEquiv","t":2,"d":"Reversing a covered edge preserves the skeleton. The undirected adjacency is unchanged: only the orientation of the single edge a — b flips.","s":"∀ {V : Type u_1} [inst : DecidableEq V] [inst_1 : Fintype V] {G : Causalean.DAG V} {a b : V} (hcov : G.IsCoveredEdge a b), Causalean.SameSkeleton G (Causalean.DAG.flipEdge hcov)","l":134,"f":"Causalean/Graph/MarkovEquiv/CoveredReversal.lean"},{"n":"Causalean.DAG.flipMinus","k":"def","m":"Causalean.Graph.MarkovEquiv.CoveredReversal","a":"Graph/MarkovEquiv","t":1,"d":"The edge relation of G with the single edge a → b deleted.","s":"{V : Type u_1} → [inst : DecidableEq V] → [inst_1 : Fintype V] → Causalean.DAG V → V → V → V → V → Prop","l":45,"f":"Causalean/Graph/MarkovEquiv/CoveredReversal.lean"},{"n":"Causalean.DAG.flipMinus_le","k":"theorem","m":"Causalean.Graph.MarkovEquiv.CoveredReversal","a":"Graph/MarkovEquiv","t":2,"d":"A flipMinus-edge is in particular a G-edge.","s":"∀ {V : Type u_1} [inst : DecidableEq V] [inst_1 : Fintype V] {G : Causalean.DAG V} {a b u w : V}, G.flipMinus a b u w → G.edge u w","l":55,"f":"Causalean/Graph/MarkovEquiv/CoveredReversal.lean"},{"n":"Causalean.DAG.flipRel","k":"def","m":"Causalean.Graph.MarkovEquiv.CoveredReversal","a":"Graph/MarkovEquiv","t":1,"d":"The edge relation of G with the single edge a → b reversed to b → a.","s":"{V : Type u_1} → [inst : DecidableEq V] → [inst_1 : Fintype V] → Causalean.DAG V → V → V → V → V → Prop","l":49,"f":"Causalean/Graph/MarkovEquiv/CoveredReversal.lean"},{"n":"Causalean.DAG.flipRel_acyclic","k":"theorem","m":"Causalean.Graph.MarkovEquiv.CoveredReversal","a":"Graph/MarkovEquiv","t":2,"d":"Reversing a covered edge keeps the graph acyclic. The transitive closure of the flipped relation is irreflexive. Key step (AMP Lemma 3.1): a directed G-path a ⇝ b of length ≥ 2 would end at a parent c ≠ a of b, hence (covered) a parent of a, closing a G-cycle; so no such detour exists and the single reversal introduces no cycle.","s":"∀ {V : Type u_1} [inst : DecidableEq V] [inst_1 : Fintype V] {G : Causalean.DAG V} {a b : V}, G.IsCoveredEdge a b → ∀ (v : V), ¬Relation.TransGen (G.flipRel a b) v v","l":69,"f":"Causalean/Graph/MarkovEquiv/CoveredReversal.lean"},{"n":"Causalean.DAG.fork_isActivePath","k":"theorem","m":"Causalean.Graph.DSep.Ancestral","a":"Graph/DSep","t":2,"d":"Two directed paths from one unconditioned vertex can be joined through that fork.","s":"∀ {V : Type u_1} [inst : DecidableEq V] [inst_1 : Fintype V] (G : Causalean.DAG V) {Z : Finset V} {u x y : V} {xp yp : List V}, xp.length ≥ 2 → xp.head? = some u → xp.getLast? = some x → (∀ (i : ℕ) (hi : i + 1 < xp.length), G.edge (xp.get ⟨i, Decidable.byContradiction fun a => Causalean.DAG.walk_right_witness._proof_1✝ i hi a⟩) (xp.get ⟨i + 1, hi⟩)) → (∀ z ∈ xp, z ∉ Z) → yp.length ≥ 2 → yp.head? = some u → yp.getLast? = some y → (∀ (i : ℕ) (hi : i + 1 < yp.length), G.edge (yp.get ⟨i, Decidable.byContradiction fun a => Causalean.DAG.fork_isActivePath._proof_1 i hi a⟩) (yp.get ⟨i + 1, hi⟩)) → (∀ z ∈ yp, z ∉ Z) → have p := xp.reverse ++ yp.tail; p.length ≥ 2 ∧ p.head? = some x ∧ p.getLast? = some y ∧ G.IsActivePath Z p","l":420,"f":"Causalean/Graph/DSep/Ancestral.lean"},{"n":"Causalean.DAG.hasActivePath_edge_congr","k":"theorem","m":"Causalean.Graph.MarkovEquiv.CoveredReversal","a":"Graph/MarkovEquiv","t":2,"d":"Equal directed-edge relations preserve existence of active paths.","s":"∀ {V : Type u_1} [inst : DecidableEq V] [inst_1 : Fintype V] {G₁ G₂ : Causalean.DAG V}, (∀ (u w : V), G₁.edge u w ↔ G₂.edge u w) → ∀ (X Y Z : Finset V), G₁.HasActivePath X Y Z ↔ G₂.HasActivePath X Y Z","l":312,"f":"Causalean/Graph/MarkovEquiv/CoveredReversal.lean"},{"n":"Causalean.DAG.hasActivePath_symm","k":"theorem","m":"Causalean.Graph.DSep.ActivePath","a":"Graph/DSep","t":2,"d":"HasActivePath is symmetric in X and Y.","s":"∀ {V : Type u_1} [inst : DecidableEq V] [inst_1 : Fintype V] (G : Causalean.DAG V) (X Y Z : Finset V), G.HasActivePath X Y Z → G.HasActivePath Y X Z","l":372,"f":"Causalean/Graph/DSep/ActivePath.lean"},{"n":"Causalean.DAG.immorality_iff_colliderSep","k":"theorem","m":"Causalean.Graph.MarkovEquiv.Readoff","a":"Graph/MarkovEquiv","t":1,"d":"Collider read-off. For vertices a, b, c forming an unshielded triple: a and b adjacent, c and b adjacent, and a, c non-adjacent and distinct, the middle vertex b is a collider a → b ← c — equivalently a → b ← c is a v-structure — exactly when b belongs to no conditioning set that d-separates a and c.","s":"∀ {V : Type u_1} [inst : DecidableEq V] [inst_1 : Fintype V] (G : Causalean.DAG V) {a b c : V}, G.UAdj a b → G.UAdj c b → ¬G.UAdj a c → a ≠ c → (G.IsImmorality a b c ↔ ∀ (Z : Finset V), b ∈ Z → ¬G.dSep {a} {c} Z)","l":212,"f":"Causalean/Graph/MarkovEquiv/Readoff.lean"},{"n":"Causalean.DAG.immorality_iff_colliderSep_disjoint","k":"theorem","m":"Causalean.Graph.MarkovEquiv.Readoff","a":"Graph/MarkovEquiv","t":2,"d":"Disjoint collider read-off. For an unshielded triple a — b — c, b is the collider a → b ← c iff every separator of a, c that contains b and excludes a, c fails to d-separate — equivalently, b lies in no such separator. (The endpoint-disjoint form used to transport immoralities across Markov-equivalent graphs.)","s":"∀ {V : Type u_1} [inst : DecidableEq V] [inst_1 : Fintype V] (G : Causalean.DAG V) {a b c : V}, G.UAdj a b → G.UAdj c b → ¬G.UAdj a c → a ≠ c → (G.IsImmorality a b c ↔ ∀ (Z : Finset V), b ∈ Z → a ∉ Z → c ∉ Z → ¬G.dSep {a} {c} Z)","l":301,"f":"Causalean/Graph/MarkovEquiv/Readoff.lean"},{"n":"Causalean.DAG.instDecidableEqBBDir","k":"instance","m":"Causalean.Graph.DSep.BayesBall","a":"Graph/DSep","t":2,"d":"","s":"DecidableEq Causalean.DAG.BBDir","l":56,"f":"Causalean/Graph/DSep/BayesBall.lean"},{"n":"Causalean.DAG.instDecidableIsImmorality","k":"instance","m":"Causalean.Graph.MarkovEquiv.Defs","a":"Graph/MarkovEquiv","t":2,"d":"Whether a proposed v-structure in a finite DAG is decidable.","s":"{V : Type u_1} → [inst : DecidableEq V] → [inst_1 : Fintype V] → (G : Causalean.DAG V) → (a b c : V) → Decidable (G.IsImmorality a b c)","l":46,"f":"Causalean/Graph/MarkovEquiv/Defs.lean"},{"n":"Causalean.DAG.instFintypeBBDir","k":"instance","m":"Causalean.Graph.DSep.BayesBall","a":"Graph/DSep","t":2,"d":"The two Bayes Ball arrival directions form a finite type.","s":"Fintype Causalean.DAG.BBDir","l":58,"f":"Causalean/Graph/DSep/BayesBall.lean"},{"n":"Causalean.DAG.instReprBBDir","k":"instance","m":"Causalean.Graph.DSep.BayesBall","a":"Graph/DSep","t":2,"d":"","s":"Repr Causalean.DAG.BBDir","l":56,"f":"Causalean/Graph/DSep/BayesBall.lean"},{"n":"Causalean.DAG.irrefl","k":"theorem","m":"Causalean.Graph.DAG","a":"Graph","t":2,"d":"No vertex has an edge to itself (a directed self-loop would be a length-one cycle).","s":"∀ {V : Type u_2} [inst : DecidableEq V] [inst_1 : Fintype V] (G : Causalean.DAG V) (v : V), ¬G.edge v v","l":118,"f":"Causalean/Graph/DAG.lean"},{"n":"Causalean.DAG.isActivePath_Z_of_no_S_only_collider","k":"theorem","m":"Causalean.Graph.DSep.ActivePath","a":"Graph/DSep","t":2,"d":"If q is active given Z ∪ S and every interior collider of q is Z-activated (not merely S-activated), then q is active given Z alone.","s":"∀ {V : Type u_1} [inst : DecidableEq V] [inst_1 : Fintype V] (G : Causalean.DAG V) {Z S : Finset V} {q : List V}, G.IsActivePath (Z ∪ S) q → (∀ (i : ℕ) (hi : i + 2 < q.length), G.IsCollider (q.get ⟨i, Decidable.byContradiction fun a => Causalean.DAG.IsActivePath._proof_2 q i hi a⟩) (q.get ⟨i + 1, Decidable.byContradiction fun a => Causalean.DAG.IsActivePath._proof_3 q i hi a⟩) (q.get ⟨i + 2, hi⟩) → q.get ⟨i + 1, Decidable.byContradiction fun a => Causalean.DAG.IsActivePath._proof_3 q i hi a⟩ ∈ G.bbZAncestors Z) → G.IsActivePath Z q","l":345,"f":"Causalean/Graph/DSep/ActivePath.lean"},{"n":"Causalean.DAG.isActivePath_cons_of_active_triple","k":"theorem","m":"Causalean.Graph.DSep.ActivePath","a":"Graph/DSep","t":2,"d":"Prepending an active triple to an active path keeps the path active.","s":"∀ {V : Type u_1} [inst : DecidableEq V] [inst_1 : Fintype V] (G : Causalean.DAG V) {Z : Finset V} {z w u : V} {r : List V}, G.UAdj z w → (if G.IsCollider z w u then w ∈ G.bbZAncestors Z else w ∉ Z) → G.IsActivePath Z (w :: u :: r) → G.IsActivePath Z (z :: w :: u :: r)","l":514,"f":"Causalean/Graph/DSep/ActivePath.lean"},{"n":"Causalean.DAG.isActivePath_cons_tail","k":"theorem","m":"Causalean.Graph.DSep.ActivePath","a":"Graph/DSep","t":2,"d":"Removing the first vertex from an active path leaves an active path.","s":"∀ {V : Type u_1} [inst : DecidableEq V] [inst_1 : Fintype V] (G : Causalean.DAG V) {Z : Finset V} {u w : V} {r : List V}, G.IsActivePath Z (u :: w :: r) → G.IsActivePath Z (w :: r)","l":633,"f":"Causalean/Graph/DSep/ActivePath.lean"},{"n":"Causalean.DAG.isActivePath_drop","k":"theorem","m":"Causalean.SCM.Do.CutsetDSep","a":"SCM/Do","t":2,"d":"Removing any initial segment of an active graph path leaves a path that is still active under the same conditioning set.","s":"∀ {V : Type u_3} [inst : DecidableEq V] [inst_1 : Fintype V] (G : Causalean.DAG V) {C : Finset V} {p : List V} (j : ℕ), G.IsActivePath C p → G.IsActivePath C (List.drop j p)","l":409,"f":"Causalean/SCM/Do/CutsetDSep.lean"},{"n":"Causalean.DAG.isActivePath_edge_congr","k":"theorem","m":"Causalean.Graph.MarkovEquiv.CoveredReversal","a":"Graph/MarkovEquiv","t":2,"d":"Graphs with the same directed edges have exactly the same active paths for every conditioning set.","s":"∀ {V : Type u_1} [inst : DecidableEq V] [inst_1 : Fintype V] {G₁ G₂ : Causalean.DAG V}, (∀ (u w : V), G₁.edge u w ↔ G₂.edge u w) → ∀ (Z : Finset V) (p : List V), G₁.IsActivePath Z p ↔ G₂.IsActivePath Z p","l":268,"f":"Causalean/Graph/MarkovEquiv/CoveredReversal.lean"},{"n":"Causalean.DAG.isActivePath_flip_of_not_mem","k":"theorem","m":"Causalean.Graph.MarkovEquiv.CoveredReversal","a":"Graph/MarkovEquiv","t":2,"d":"A path that is active relative to a conditioning set and does not contain the tail of a covered reversed edge remains active relative to the same set after the reversal.","s":"∀ {V : Type u_1} [inst : DecidableEq V] [inst_1 : Fintype V] {G : Causalean.DAG V} {a b : V} (hcov : G.IsCoveredEdge a b) {Z : Finset V} {p : List V}, G.IsActivePath Z p → a ∉ p → (Causalean.DAG.flipEdge hcov).IsActivePath Z p","l":526,"f":"Causalean/Graph/MarkovEquiv/CoveredReversal.lean"},{"n":"Causalean.DAG.isActivePath_insert_cond","k":"theorem","m":"Causalean.Graph.DSep.OrderedLocalSG","a":"Graph/DSep","t":2,"d":"Adding a conditioning vertex preserves activity when that vertex appears internally only as a collider.","s":"∀ {V : Type u_1} [inst : DecidableEq V] [inst_1 : Fintype V] (G : Causalean.DAG V) {D : Finset V} {p : List V} {n : V}, G.IsActivePath D p → (∀ (i : ℕ) (hi : i + 2 < p.length), p.get ⟨i + 1, Decidable.byContradiction fun a => Causalean.DAG.isActivePath_insert_cond._proof_1 i hi a⟩ = n → G.IsCollider (p.get ⟨i, Decidable.byContradiction fun a => Causalean.DAG.isActivePath_insert_cond._proof_2 i hi a⟩) (p.get ⟨i + 1, Decidable.byContradiction fun a => Causalean.DAG.isActivePath_insert_cond._proof_1 i hi a⟩) (p.get ⟨i + 2, hi⟩)) → G.IsActivePath (insert n D) p","l":256,"f":"Causalean/Graph/DSep/OrderedLocalSG.lean"},{"n":"Causalean.DAG.isActivePath_mono_edge","k":"theorem","m":"Causalean.Graph.DSep.Separation","a":"Graph/DSep","t":2,"d":"An active path in a graph with fewer edges remains active when those edges are restored, so path witnesses transfer across graph transformations.","s":"∀ {V : Type u_1} [inst : DecidableEq V] [inst_1 : Fintype V] (G G' : Causalean.DAG V), (∀ (u v : V), G'.edge u v → G.edge u v) → ∀ {Z : Finset V} {p : List V}, G'.IsActivePath Z p → G.IsActivePath Z p","l":741,"f":"Causalean/Graph/DSep/Separation.lean"},{"n":"Causalean.DAG.isActivePath_of_directed","k":"theorem","m":"Causalean.Graph.DSep.Ancestral","a":"Graph/DSep","t":2,"d":"A *forward*-directed path whose interior vertices avoid Z is an active path given Z. \"Forward-directed\" means each edge points from the *earlier* index to the *later* index. Every interior vertex is then a non-collider (incoming + outgoing), and by the avoidance hypothesis none is in Z; endpoints may lie in Z.","s":"∀ {V : Type u_1} [inst : DecidableEq V] [inst_1 : Fintype V] (G : Causalean.DAG V) {Z : Finset V} {p : List V}, (∀ (i : ℕ) (hi : i + 1 < p.length), G.edge (p.get ⟨i, Decidable.byContradiction fun a => Causalean.DAG.walk_right_witness._proof_1✝ i hi a⟩) (p.get ⟨i + 1, hi⟩)) → (∀ (i : ℕ) (hi : i + 2 < p.length), p.get ⟨i + 1, Decidable.byContradiction fun a => Causalean.DAG.isActivePath_of_directed._proof_1 i hi a⟩ ∉ Z) → G.IsActivePath Z p","l":379,"f":"Causalean/Graph/DSep/Ancestral.lean"},{"n":"Causalean.DAG.isActivePath_of_reversed_directed","k":"theorem","m":"Causalean.Graph.DSep.ActivePath","a":"Graph/DSep","t":2,"d":"A backward-directed path whose interior vertices avoid the conditioning set is active.","s":"∀ {V : Type u_1} [inst : DecidableEq V] [inst_1 : Fintype V] (G : Causalean.DAG V) {Z : Finset V} {p : List V}, (∀ (i : ℕ) (hi : i + 1 < p.length), G.edge (p.get ⟨i + 1, hi⟩) (p.get ⟨i, Decidable.byContradiction fun a => Causalean.DAG.IsActivePath._proof_1 p i hi a⟩)) → (∀ (i : ℕ) (hi : i + 2 < p.length), p.get ⟨i + 1, Decidable.byContradiction fun a => Causalean.DAG.IsActivePath._proof_3 p i hi a⟩ ∉ Z) → G.IsActivePath Z p","l":166,"f":"Causalean/Graph/DSep/ActivePath.lean"},{"n":"Causalean.DAG.isActivePath_pair","k":"theorem","m":"Causalean.Graph.DSep.ActivePath","a":"Graph/DSep","t":2,"d":"Any two adjacent vertices form an active path of length one for every conditioning set.","s":"∀ {V : Type u_1} [inst : DecidableEq V] [inst_1 : Fintype V] (G : Causalean.DAG V) {Z : Finset V} {a b : V}, G.UAdj a b → G.IsActivePath Z [a, b]","l":558,"f":"Causalean/Graph/DSep/ActivePath.lean"},{"n":"Causalean.DAG.isActivePath_reverse","k":"theorem","m":"Causalean.Graph.DSep.ActivePath","a":"Graph/DSep","t":2,"d":"Active paths are symmetric: reversing an active path is also active.","s":"∀ {V : Type u_1} [inst : DecidableEq V] [inst_1 : Fintype V] (G : Causalean.DAG V) {Z : Finset V} {p : List V}, G.IsActivePath Z p → G.IsActivePath Z p.reverse","l":108,"f":"Causalean/Graph/DSep/ActivePath.lean"},{"n":"Causalean.DAG.isActivePath_take","k":"theorem","m":"Causalean.Graph.DSep.OrderedLocalSG","a":"Graph/DSep","t":2,"d":"Every prefix of an active path is active.","s":"∀ {V : Type u_1} [inst : DecidableEq V] [inst_1 : Fintype V] (G : Causalean.DAG V) {Z : Finset V} {p : List V} {k : ℕ}, G.IsActivePath Z p → G.IsActivePath Z (List.take k p)","l":875,"f":"Causalean/Graph/DSep/OrderedLocalSG.lean"},{"n":"Causalean.DAG.isAncestor","k":"inductive","m":"Causalean.Graph.DAG","a":"Graph","t":1,"d":"isAncestor G u v means u is an ancestor of v: there is a directed path from u to v. Defined inductively as the transitive closure of the edge relation.","s":"{V : Type u_2} → [inst : DecidableEq V] → [inst_1 : Fintype V] → Causalean.DAG V → V → V → Prop","l":98,"f":"Causalean/Graph/DAG.lean"},{"n":"Causalean.DAG.isAncestorAvoiding","k":"inductive","m":"Causalean.SCM.Model.CutsetLatent","a":"SCM/Model","t":1,"d":"Avoiding ancestry means there is a directed path whose strictly interior nodes avoid a forbidden set.","s":"{V : Type u_1} → [inst : DecidableEq V] → [inst_1 : Fintype V] → Causalean.DAG V → Finset V → V → V → Prop","l":43,"f":"Causalean/SCM/Model/CutsetLatent.lean"},{"n":"Causalean.DAG.isAncestorAvoiding.cons","k":"theorem","m":"Causalean.SCM.Model.CutsetLatent","a":"SCM/Model","t":2,"d":"An avoiding path can be extended backward by one edge when the new interior node avoids the forbidden set.","s":"∀ {V : Type u_1} [inst : DecidableEq V] [inst_1 : Fintype V] (G : Causalean.DAG V) {C : Finset V} {u w v : V}, G.edge u w → w ∉ C → G.isAncestorAvoiding C w v → G.isAncestorAvoiding C u v","l":64,"f":"Causalean/SCM/Model/CutsetLatent.lean"},{"n":"Causalean.DAG.isAncestorAvoiding.exists_path","k":"theorem","m":"Causalean.SCM.Model.CutsetLatent","a":"SCM/Model","t":2,"d":"Avoiding ancestry can be materialized as a concrete directed path with all strictly interior nodes avoiding the forbidden set.","s":"∀ {V : Type u_1} [inst : DecidableEq V] [inst_1 : Fintype V] (G : Causalean.DAG V) {C : Finset V} {u v : V}, G.isAncestorAvoiding C u v → ∃ q, q.length ≥ 2 ∧ q.head? = some u ∧ q.getLast? = some v ∧ (∀ (i : ℕ) (hi : i + 1 < q.length), G.edge (q.get ⟨i, Decidable.byContradiction fun a => Causalean.DAG.isAncestorAvoiding.exists_path._proof_1 q i hi a⟩) (q.get ⟨i + 1, hi⟩)) ∧ ∀ (i : ℕ) (hi : i + 2 < q.length), q.get ⟨i + 1, Decidable.byContradiction fun a => Causalean.DAG.isAncestorAvoiding.exists_path._proof_2 q i hi a⟩ ∉ C","l":76,"f":"Causalean/SCM/Model/CutsetLatent.lean"},{"n":"Causalean.DAG.isAncestorAvoiding.toIsAncestor","k":"theorem","m":"Causalean.SCM.Model.CutsetLatent","a":"SCM/Model","t":2,"d":"Every avoiding ancestry relation is also ordinary directed ancestry.","s":"∀ {V : Type u_1} [inst : DecidableEq V] [inst_1 : Fintype V] (G : Causalean.DAG V) {C : Finset V} {u v : V}, G.isAncestorAvoiding C u v → G.isAncestor u v","l":54,"f":"Causalean/SCM/Model/CutsetLatent.lean"},{"n":"Causalean.DAG.isAncestor_child","k":"theorem","m":"Causalean.Graph.DAG","a":"Graph","t":2,"d":"First-step decomposition: if u is an ancestor of v, then either edge u v or there exists a child c of u such that c is an ancestor of v.","s":"∀ {V : Type u_2} [inst : DecidableEq V] [inst_1 : Fintype V] (G : Causalean.DAG V) {u v : V}, G.isAncestor u v → G.edge u v ∨ ∃ c, G.edge u c ∧ G.isAncestor c v","l":142,"f":"Causalean/Graph/DAG.lean"},{"n":"Causalean.DAG.isAncestor_edge_congr","k":"theorem","m":"Causalean.Graph.MarkovEquiv.CoveredReversal","a":"Graph/MarkovEquiv","t":2,"d":"Graphs with the same directed edges have exactly the same ancestor relations.","s":"∀ {V : Type u_1} [inst : DecidableEq V] [inst_1 : Fintype V] {G₁ G₂ : Causalean.DAG V}, (∀ (u w : V), G₁.edge u w ↔ G₂.edge u w) → ∀ {u v : V}, G₁.isAncestor u v ↔ G₂.isAncestor u v","l":225,"f":"Causalean/Graph/MarkovEquiv/CoveredReversal.lean"},{"n":"Causalean.DAG.isAncestor_has_parent","k":"theorem","m":"Causalean.Graph.DAG","a":"Graph","t":2,"d":"Every vertex reached by a nonempty directed path has an incoming edge, namely the final edge of that path.","s":"∀ {V : Type u_2} [inst : DecidableEq V] [inst_1 : Fintype V] (G : Causalean.DAG V) {u v : V}, G.isAncestor u v → G.parents v ≠ ∅","l":475,"f":"Causalean/Graph/DAG.lean"},{"n":"Causalean.DAG.isAncestor_iff_transGen","k":"theorem","m":"Causalean.Graph.DAG","a":"Graph","t":2,"d":"The inductive ancestor relation coincides with Relation.TransGen of the edge relation: both are the transitive closure of the edge relation.","s":"∀ {V : Type u_2} [inst : DecidableEq V] [inst_1 : Fintype V] (G : Causalean.DAG V) {u v : V}, G.isAncestor u v ↔ Relation.TransGen G.edge u v","l":104,"f":"Causalean/Graph/DAG.lean"},{"n":"Causalean.DAG.isAncestor_irrefl","k":"theorem","m":"Causalean.Graph.DAG","a":"Graph","t":2,"d":"Ancestor relation is irreflexive: no vertex is its own ancestor (this is acyclicity, restated for the inductive ancestor relation).","s":"∀ {V : Type u_2} [inst : DecidableEq V] [inst_1 : Fintype V] (G : Causalean.DAG V) (v : V), ¬G.isAncestor v v","l":129,"f":"Causalean/Graph/DAG.lean"},{"n":"Causalean.DAG.isAncestor_last","k":"theorem","m":"Causalean.Graph.MarkovEquiv.CoveredReversal","a":"Graph/MarkovEquiv","t":2,"d":"If one node is a strict ancestor of another in a directed acyclic graph, then it either has a direct edge to the latter or is a strict ancestor of a node that has a direct edge to the latter.","s":"∀ {V : Type u_1} [inst : DecidableEq V] [inst_1 : Fintype V] {H : Causalean.DAG V} {u v : V}, H.isAncestor u v → H.edge u v ∨ ∃ w, H.isAncestor u w ∧ H.edge w v","l":377,"f":"Causalean/Graph/MarkovEquiv/CoveredReversal.lean"},{"n":"Causalean.DAG.isAncestor_mem_of_closed","k":"theorem","m":"Causalean.Graph.DAG","a":"Graph","t":2,"d":"A finite set that contains every parent of each of its vertices contains every ancestor of each vertex it contains.","s":"∀ {V : Type u_2} [inst : DecidableEq V] [inst_1 : Fintype V] (G : Causalean.DAG V) {T : Finset V}, (∀ x ∈ T, G.parents x ⊆ T) → ∀ {u w : V}, G.isAncestor u w → w ∈ T → u ∈ T","l":273,"f":"Causalean/Graph/DAG.lean"},{"n":"Causalean.DAG.isAncestor_mono_edge","k":"theorem","m":"Causalean.Graph.DSep.Separation","a":"Graph/DSep","t":2,"d":"If every directed edge of one graph is also an edge of another graph, every ancestor relation in the first graph also holds in the second graph.","s":"∀ {V : Type u_1} [inst : DecidableEq V] [inst_1 : Fintype V] (G G' : Causalean.DAG V), (∀ (u v : V), G'.edge u v → G.edge u v) → ∀ {u v : V}, G'.isAncestor u v → G.isAncestor u v","l":694,"f":"Causalean/Graph/DSep/Separation.lean"},{"n":"Causalean.DAG.isAncestor_topoOrder_lt","k":"theorem","m":"Causalean.Graph.DAG","a":"Graph","t":2,"d":"Ancestors respect the topological order: if u is an ancestor of v then G.topoOrder u < G.topoOrder v, so ancestor pairs are strictly ordered by topoOrder.","s":"∀ {V : Type u_2} [inst : DecidableEq V] [inst_1 : Fintype V] (G : Causalean.DAG V) {u v : V}, G.isAncestor u v → G.topoOrder u < G.topoOrder v","l":422,"f":"Causalean/Graph/DAG.lean"},{"n":"Causalean.DAG.isAncestor_trans","k":"theorem","m":"Causalean.Graph.DAG","a":"Graph","t":2,"d":"Ancestor relation is transitive.","s":"∀ {V : Type u_2} [inst : DecidableEq V] [inst_1 : Fintype V] (G : Causalean.DAG V) {u v w : V}, G.isAncestor u v → G.isAncestor v w → G.isAncestor u w","l":135,"f":"Causalean/Graph/DAG.lean"},{"n":"Causalean.DAG.isCollider_edge_congr","k":"theorem","m":"Causalean.Graph.MarkovEquiv.CoveredReversal","a":"Graph/MarkovEquiv","t":2,"d":"Graphs with the same directed edges have exactly the same collider triples.","s":"∀ {V : Type u_1} [inst : DecidableEq V] [inst_1 : Fintype V] {G₁ G₂ : Causalean.DAG V}, (∀ (u w : V), G₁.edge u w ↔ G₂.edge u w) → ∀ {l m r : V}, G₁.IsCollider l m r ↔ G₂.IsCollider l m r","l":261,"f":"Causalean/Graph/MarkovEquiv/CoveredReversal.lean"},{"n":"Causalean.DAG.isCollider_of_supergraph","k":"theorem","m":"Causalean.Graph.DSep.Separation","a":"Graph/DSep","t":2,"d":"If every edge of one graph is also an edge of another, a collider in the larger graph remains a collider in the smaller graph whenever its two adjacent pairs are present there.","s":"∀ {V : Type u_1} [inst : DecidableEq V] [inst_1 : Fintype V] (G G' : Causalean.DAG V), (∀ (u v : V), G'.edge u v → G.edge u v) → ∀ {l m r : V}, G'.UAdj l m → G'.UAdj m r → G.IsCollider l m r → G'.IsCollider l m r","l":725,"f":"Causalean/Graph/DSep/Separation.lean"},{"n":"Causalean.DAG.isDescendant","k":"def","m":"Causalean.Graph.DAG","a":"Graph","t":1,"d":"u is a descendant of v in the DAG exactly when there is a directed path from v to u, equivalently when v is an ancestor of u.","s":"{V : Type u_2} → [inst : DecidableEq V] → [inst_1 : Fintype V] → Causalean.DAG V → V → V → Prop","l":153,"f":"Causalean/Graph/DAG.lean"},{"n":"Causalean.DAG.isRoot","k":"def","m":"Causalean.Graph.DAG","a":"Graph","t":1,"d":"A vertex is a root if it has no parents (G.parents v = ∅).","s":"{V : Type u_2} → [inst : DecidableEq V] → [inst_1 : Fintype V] → Causalean.DAG V → V → Prop","l":434,"f":"Causalean/Graph/DAG.lean"},{"n":"Causalean.DAG.lastIdxLt","k":"def","m":"Causalean.Graph.DSep.ActivePath","a":"Graph/DSep","t":1,"d":"Largest i < n satisfying a decidable predicate P, if any.","s":"ℕ → (P : ℕ → Prop) → [DecidablePred P] → Option ℕ","l":203,"f":"Causalean/Graph/DSep/ActivePath.lean"},{"n":"Causalean.DAG.lastIdxLt_eq_none","k":"theorem","m":"Causalean.Graph.DSep.ActivePath","a":"Graph/DSep","t":2,"d":"If there is no last index below a bound satisfying a predicate, then no index below the bound satisfies the predicate.","s":"∀ {n : ℕ} {P : ℕ → Prop} [inst : DecidablePred P], Causalean.DAG.lastIdxLt n P = none → ∀ i < n, ¬P i","l":229,"f":"Causalean/Graph/DSep/ActivePath.lean"},{"n":"Causalean.DAG.lastIdxLt_eq_some","k":"theorem","m":"Causalean.Graph.DSep.ActivePath","a":"Graph/DSep","t":2,"d":"If the last index below a bound exists, it is below the bound, satisfies the predicate, and no larger index below the bound satisfies it.","s":"∀ {n : ℕ} {P : ℕ → Prop} [inst : DecidablePred P] {i : ℕ}, Causalean.DAG.lastIdxLt n P = some i → i < n ∧ P i ∧ ∀ (j : ℕ), i < j → j < n → ¬P j","l":210,"f":"Causalean/Graph/DSep/ActivePath.lean"},{"n":"Causalean.DAG.last_edge_into_max","k":"theorem","m":"Causalean.Graph.DSep.OrderedLocalSG","a":"Graph/DSep","t":2,"d":"The final edge of a path to a topological maximum points into it. On an active path whose nodes all lie in an ancestral set with topological maximum n, if the path ends at n then its last edge points into n: a child of n on the path would be a strict descendant, hence a strictly larger node.","s":"∀ {V : Type u_1} [inst : DecidableEq V] [inst_1 : Fintype V] (G : Causalean.DAG V) {D bigQ : Finset V} {p : List V} {n : V} (hp_len : p.length ≥ 2), G.IsActivePath D p → p.getLast? = some n → (∀ v ∈ p, v ∈ G.ancestralSet bigQ) → (∀ m ∈ G.ancestralSet bigQ, G.topoOrder m ≤ G.topoOrder n) → G.edge (p.get ⟨p.length - 2, Decidable.byContradiction fun a => Causalean.DAG.last_edge_into_max._proof_1 hp_len a⟩) n","l":395,"f":"Causalean/Graph/DSep/OrderedLocalSG.lean"},{"n":"Causalean.DAG.le_card_iterate","k":"theorem","m":"Causalean.Graph.DAG","a":"Graph","t":2,"d":"If each of a specified number of successive applications of a function on finite sets strictly increases cardinality, the final set has grown by at least that number.","s":"∀ {V : Type u_2} (f : Finset V → Finset V) (S₀ : Finset V) (k : ℕ), (∀ j < k, (f^[j] S₀).card < (f^[j + 1] S₀).card) → (f^[0] S₀).card + k ≤ (f^[k] S₀).card","l":215,"f":"Causalean/Graph/DAG.lean"},{"n":"Causalean.DAG.markovEquiv_flipEdge","k":"theorem","m":"Causalean.Graph.MarkovEquiv.CoveredReversal","a":"Graph/MarkovEquiv","t":1,"d":"The analytic core (AMP, per-step invariance). In a DAG G, if the edge a → b is covered — every other parent of b is also a parent of a, and vice versa, then the DAG obtained by reversing that edge to b → a is Markov equivalent to G: the two graphs license exactly the same d-separation statements. This is the single-edge kernel used by the covered-reversal proof of the Verma--Pearl hard direction.","s":"∀ {V : Type u_1} [inst : DecidableEq V] [inst_1 : Fintype V] {G : Causalean.DAG V} {a b : V} (hcov : G.IsCoveredEdge a b), Causalean.MarkovEquiv G (Causalean.DAG.flipEdge hcov)","l":1713,"f":"Causalean/Graph/MarkovEquiv/CoveredReversal.lean"},{"n":"Causalean.DAG.mem_ancClosure","k":"theorem","m":"Causalean.Graph.DAG","a":"Graph","t":2,"d":"Membership in the backward-reachability fixpoint is exactly ancestry: a vertex lies in G.ancClosure v iff it is an ancestor of v. This makes the ancestor relation decidable using only the (decidable) edge relation, with no reference to any topological order.","s":"∀ {V : Type u_2} [inst : DecidableEq V] [inst_1 : Fintype V] (G : Causalean.DAG V) {u v : V}, u ∈ G.ancClosure v ↔ G.isAncestor u v","l":282,"f":"Causalean/Graph/DAG.lean"},{"n":"Causalean.DAG.mem_ancestors","k":"theorem","m":"Causalean.Graph.DAG","a":"Graph","t":2,"d":"Membership characterization for ancestors: u ∈ G.ancestors v ↔ G.isAncestor u v.","s":"∀ {V : Type u_2} [inst : DecidableEq V] [inst_1 : Fintype V] (G : Causalean.DAG V) {v u : V}, u ∈ G.ancestors v ↔ G.isAncestor u v","l":314,"f":"Causalean/Graph/DAG.lean"},{"n":"Causalean.DAG.mem_ancestralSet_of_edge_to_mem","k":"theorem","m":"Causalean.Graph.DSep.Ancestral","a":"Graph/DSep","t":2,"d":"When a directed edge points to a vertex in the ancestral closure of a set, its source vertex is also in that ancestral closure.","s":"∀ {V : Type u_1} [inst : DecidableEq V] [inst_1 : Fintype V] (G : Causalean.DAG V) {u v : V} {S : Finset V}, G.edge u v → v ∈ G.ancestralSet S → u ∈ G.ancestralSet S","l":114,"f":"Causalean/Graph/DSep/Ancestral.lean"},{"n":"Causalean.DAG.mem_ancestralSet_of_isAncestor","k":"theorem","m":"Causalean.Graph.DSep.Ancestral","a":"Graph/DSep","t":2,"d":"If u is an ancestor of some vertex w ∈ S, then u ∈ ancestralSet S.","s":"∀ {V : Type u_1} [inst : DecidableEq V] [inst_1 : Fintype V] (G : Causalean.DAG V) {u w : V} {S : Finset V}, w ∈ S → G.isAncestor u w → u ∈ G.ancestralSet S","l":74,"f":"Causalean/Graph/DSep/Ancestral.lean"},{"n":"Causalean.DAG.mem_children","k":"theorem","m":"Causalean.Graph.DAG","a":"Graph","t":2,"d":"Membership characterization for children: w ∈ G.children v ↔ G.edge v w.","s":"∀ {V : Type u_2} [inst : DecidableEq V] [inst_1 : Fintype V] (G : Causalean.DAG V) {v w : V}, w ∈ G.children v ↔ G.edge v w","l":90,"f":"Causalean/Graph/DAG.lean"},{"n":"Causalean.DAG.mem_descendants","k":"theorem","m":"Causalean.Graph.DAG","a":"Graph","t":2,"d":"Membership characterization for descendants: w ∈ G.descendants v ↔ G.isAncestor v w.","s":"∀ {V : Type u_2} [inst : DecidableEq V] [inst_1 : Fintype V] (G : Causalean.DAG V) {v w : V}, w ∈ G.descendants v ↔ G.isAncestor v w","l":318,"f":"Causalean/Graph/DAG.lean"},{"n":"Causalean.DAG.mem_parents","k":"theorem","m":"Causalean.Graph.DAG","a":"Graph","t":2,"d":"Membership characterization for parents: u ∈ G.parents v ↔ G.edge u v.","s":"∀ {V : Type u_2} [inst : DecidableEq V] [inst_1 : Fintype V] (G : Causalean.DAG V) {v u : V}, u ∈ G.parents v ↔ G.edge u v","l":86,"f":"Causalean/Graph/DAG.lean"},{"n":"Causalean.DAG.moralAdj_symm","k":"theorem","m":"Causalean.Graph.MarkovEquiv.Moralization","a":"Graph/MarkovEquiv","t":2,"d":"Moral adjacency is symmetric: an undirected moral edge from u to v is also one from v to u.","s":"∀ {V : Type u_1} [inst : DecidableEq V] [inst_1 : Fintype V] (G : Causalean.DAG V) {S : Finset V} {u v : V}, G.MoralAdj S u v → G.MoralAdj S v u","l":61,"f":"Causalean/Graph/MarkovEquiv/Moralization.lean"},{"n":"Causalean.DAG.moralConn_of_activePath","k":"theorem","m":"Causalean.Graph.MarkovEquiv.Moralization","a":"Graph/MarkovEquiv","t":2,"d":"An active path inside a ground set, with endpoints outside the conditioning set, induces a connection in the corresponding moral graph.","s":"∀ {V : Type u_1} [inst : DecidableEq V] [inst_1 : Fintype V] (G : Causalean.DAG V) {S Z : Finset V} (n : ℕ) {x y : V} {p : List V}, p.length ≤ n → G.IsActivePath Z p → p.length ≥ 2 → (∀ v ∈ p, v ∈ S) → p.head? = some x → p.getLast? = some y → x ∉ Z → y ∉ Z → G.MoralConn S Z x y","l":129,"f":"Causalean/Graph/MarkovEquiv/Moralization.lean"},{"n":"Causalean.DAG.moralConn_of_step","k":"theorem","m":"Causalean.Graph.MarkovEquiv.Moralization","a":"Graph/MarkovEquiv","t":2,"d":"A single moral step yields moral connectivity.","s":"∀ {V : Type u_1} [inst : DecidableEq V] [inst_1 : Fintype V] (G : Causalean.DAG V) {S Z : Finset V} {u v : V}, G.MoralStep S Z u v → G.MoralConn S Z u v","l":75,"f":"Causalean/Graph/MarkovEquiv/Moralization.lean"},{"n":"Causalean.DAG.moralConn_trans","k":"theorem","m":"Causalean.Graph.MarkovEquiv.Moralization","a":"Graph/MarkovEquiv","t":2,"d":"Moral connectivity is transitive.","s":"∀ {V : Type u_1} [inst : DecidableEq V] [inst_1 : Fintype V] (G : Causalean.DAG V) {S Z : Finset V} {u v w : V}, G.MoralConn S Z u v → G.MoralConn S Z v w → G.MoralConn S Z u w","l":80,"f":"Causalean/Graph/MarkovEquiv/Moralization.lean"},{"n":"Causalean.DAG.moralStep_of_married","k":"theorem","m":"Causalean.Graph.MarkovEquiv.Moralization","a":"Graph/MarkovEquiv","t":2,"d":"A married pair (common child c ∈ S) of distinct non-Z vertices of S is a moral step.","s":"∀ {V : Type u_1} [inst : DecidableEq V] [inst_1 : Fintype V] (G : Causalean.DAG V) {S Z : Finset V} {u v c : V}, u ≠ v → u ∈ S → v ∈ S → c ∈ S → G.edge u c → G.edge v c → u ∉ Z → v ∉ Z → G.MoralStep S Z u v","l":91,"f":"Causalean/Graph/MarkovEquiv/Moralization.lean"},{"n":"Causalean.DAG.moralStep_of_uAdj","k":"theorem","m":"Causalean.Graph.MarkovEquiv.Moralization","a":"Graph/MarkovEquiv","t":2,"d":"A skeleton edge between two non-Z vertices of S is a moral step.","s":"∀ {V : Type u_1} [inst : DecidableEq V] [inst_1 : Fintype V] (G : Causalean.DAG V) {S Z : Finset V} {u v : V}, u ≠ v → u ∈ S → v ∈ S → G.UAdj u v → u ∉ Z → v ∉ Z → G.MoralStep S Z u v","l":85,"f":"Causalean/Graph/MarkovEquiv/Moralization.lean"},{"n":"Causalean.DAG.node_isAncestor_last_of_directed","k":"theorem","m":"Causalean.Graph.DSep.OrderedLocalSG","a":"Graph/DSep","t":2,"d":"Every vertex on a forward directed path is either the path's endpoint or a strict ancestor of that endpoint.","s":"∀ {V : Type u_1} [inst : DecidableEq V] [inst_1 : Fintype V] (G : Causalean.DAG V) {q : List V} {w : V}, q.length ≥ 2 → q.getLast? = some w → (∀ (i : ℕ) (hi : i + 1 < q.length), G.edge (q.get ⟨i, Decidable.byContradiction fun a => Causalean.DAG.node_isAncestor_last_of_directed._proof_1 i hi a⟩) (q.get ⟨i + 1, hi⟩)) → ∀ (j : ℕ) (hj : j < q.length), G.isAncestor (q.get ⟨j, hj⟩) w ∨ q.get ⟨j, hj⟩ = w","l":804,"f":"Causalean/Graph/DSep/OrderedLocalSG.lean"},{"n":"Causalean.DAG.nonCollider_has_outgoing","k":"theorem","m":"Causalean.Graph.DSep.Ancestral","a":"Graph/DSep","t":2,"d":"A non-collider triple (l, m, r) has at least one outgoing edge from m.","s":"∀ {V : Type u_1} [inst : DecidableEq V] [inst_1 : Fintype V] (G : Causalean.DAG V) {l m r : V}, G.UAdj l m → G.UAdj m r → ¬G.IsCollider l m r → G.edge m l ∨ G.edge m r","l":95,"f":"Causalean/Graph/DSep/Ancestral.lean"},{"n":"Causalean.DAG.nonDescendants","k":"def","m":"Causalean.Graph.DAG","a":"Graph","t":1,"d":"The non-descendants of v: all vertices that are NOT descendants of v (and not v itself).","s":"{V : Type u_2} → [inst : DecidableEq V] → [inst_1 : Fintype V] → Causalean.DAG V → V → Finset V","l":351,"f":"Causalean/Graph/DAG.lean"},{"n":"Causalean.DAG.not_dSep_iff_hasActivePath","k":"theorem","m":"Causalean.Graph.MarkovEquiv.CoveredReversal","a":"Graph/MarkovEquiv","t":2,"d":"Bridge: failure of d-separation is exactly an active path. For pairwise-disjoint query sets, X and Y are *not* d-separated by Z iff there is an active path from X to Y given Z. Assembled from bbReachableVertices_iff_activePath. Reduces the covered-flip invariance to a pure active-path statement.","s":"∀ {V : Type u_1} [inst : DecidableEq V] [inst_1 : Fintype V] (H : Causalean.DAG V) (X Y Z : Finset V), Disjoint X Y → Disjoint X Z → Disjoint Y Z → (¬H.dSep X Y Z ↔ H.HasActivePath X Y Z)","l":198,"f":"Causalean/Graph/MarkovEquiv/CoveredReversal.lean"},{"n":"Causalean.DAG.not_dSeparable_of_uAdj","k":"theorem","m":"Causalean.Graph.MarkovEquiv.Readoff","a":"Graph/MarkovEquiv","t":2,"d":"An edge makes its endpoints inseparable: if a and b are adjacent then no conditioning set d-separates them, because the single edge [a, b] is an active path regardless of the conditioning set.","s":"∀ {V : Type u_1} [inst : DecidableEq V] [inst_1 : Fintype V] (G : Causalean.DAG V) {a b : V}, G.UAdj a b → ∀ (Z : Finset V), ¬G.dSep {a} {b} Z","l":37,"f":"Causalean/Graph/MarkovEquiv/Readoff.lean"},{"n":"Causalean.DAG.not_edge_self","k":"theorem","m":"Causalean.Graph.MarkovEquiv.Readoff","a":"Graph/MarkovEquiv","t":2,"d":"No self-loops in a DAG.","s":"∀ {V : Type u_1} [inst : DecidableEq V] [inst_1 : Fintype V] (G : Causalean.DAG V) (a : V), ¬G.edge a a","l":278,"f":"Causalean/Graph/MarkovEquiv/Readoff.lean"},{"n":"Causalean.DAG.not_isAncestor_of_root'","k":"theorem","m":"Causalean.Graph.DSep.BackdoorBridges","a":"Graph/DSep","t":2,"d":"A vertex with no incoming edges has no proper ancestors.","s":"∀ {V : Type u_1} [inst : DecidableEq V] [inst_1 : Fintype V] (G : Causalean.DAG V) {r : V}, (∀ (u : V), ¬G.edge u r) → ∀ (u : V), ¬G.isAncestor u r","l":39,"f":"Causalean/Graph/DSep/BackdoorBridges.lean"},{"n":"Causalean.DAG.not_uAdj_self","k":"theorem","m":"Causalean.Graph.MarkovEquiv.Readoff","a":"Graph/MarkovEquiv","t":2,"d":"No self-adjacency in a DAG.","s":"∀ {V : Type u_1} [inst : DecidableEq V] [inst_1 : Fintype V] (G : Causalean.DAG V) (a : V), ¬G.UAdj a a","l":282,"f":"Causalean/Graph/MarkovEquiv/Readoff.lean"},{"n":"Causalean.DAG.ofAcyclic","k":"def","m":"Causalean.Graph.AcyclicConstruct","a":"Graph","t":1,"d":"Build a DAG from an acyclic edge relation. Given e : V → V → Prop whose transitive closure is irreflexive (hac, i.e. e has no directed cycle), this is the directed acyclic graph with edge relation e.","s":"{V : Type u_1} → [inst : DecidableEq V] → [inst_1 : Fintype V] → (e : V → V → Prop) → (∀ (v : V), ¬Relation.TransGen e v v) → Causalean.DAG V","l":32,"f":"Causalean/Graph/AcyclicConstruct.lean"},{"n":"Causalean.DAG.ofAcyclic_edge","k":"theorem","m":"Causalean.Graph.AcyclicConstruct","a":"Graph","t":1,"d":"The directed acyclic graph ofAcyclic e hac, built from an edge relation e together with a proof that e has no directed cycle, has exactly e as its edge relation.","s":"∀ {V : Type u_1} [inst : DecidableEq V] [inst_1 : Fintype V] (e : V → V → Prop) (hac : ∀ (v : V), ¬Relation.TransGen e v v), (Causalean.DAG.ofAcyclic e hac).edge = e","l":41,"f":"Causalean/Graph/AcyclicConstruct.lean"},{"n":"Causalean.DAG.orderedLocalSG_of_dSep_with_fixed","k":"theorem","m":"Causalean.Graph.DSep.OrderedLocalSG","a":"Graph/DSep","t":1,"d":"d-separation yields an ordered-local derivation. In a DAG G, suppose every vertex of Zf has no parents, i.e. Zf consists of fixed roots, Zf is disjoint from R, and X, Y, and Zr are each contained in R. If X and Y are d-separated by Zr ∪ Zf, then the conditional-independence triple \"X ⊥ Y given Zr\" is derivable from the ordered-local basis on R via the semi-graphoid axioms.","s":"∀ {V : Type u_1} [inst : DecidableEq V] [inst_1 : Fintype V] (G : Causalean.DAG V) (R X Y Zr Zf : Finset V), (∀ f ∈ Zf, G.parents f = ∅) → Disjoint Zf R → X ⊆ R → Y ⊆ R → Zr ⊆ R → G.dSep X Y (Zr ∪ Zf) → G.OrderedLocalSG R X Y Zr","l":1293,"f":"Causalean/Graph/DSep/OrderedLocalSG.lean"},{"n":"Causalean.DAG.parents","k":"def","m":"Causalean.Graph.DAG","a":"Graph","t":1,"d":"The parents of v in G: all vertices u such that (u, v) ∈ E.","s":"{V : Type u_2} → [inst : DecidableEq V] → [inst_1 : Fintype V] → Causalean.DAG V → V → Finset V","l":78,"f":"Causalean/Graph/DAG.lean"},{"n":"Causalean.DAG.parents_subset_ancestors","k":"theorem","m":"Causalean.Graph.DAG","a":"Graph","t":2,"d":"Parents are a subset of ancestors.","s":"∀ {V : Type u_2} [inst : DecidableEq V] [inst_1 : Fintype V] (G : Causalean.DAG V) (v : V), G.parents v ⊆ G.ancestors v","l":322,"f":"Causalean/Graph/DAG.lean"},{"n":"Causalean.DAG.roots","k":"def","m":"Causalean.Graph.DAG","a":"Graph","t":1,"d":"The set of all root nodes.","s":"{V : Type u_2} → [inst : DecidableEq V] → [inst_1 : Fintype V] → Causalean.DAG V → Finset V","l":441,"f":"Causalean/Graph/DAG.lean"},{"n":"Causalean.DAG.subset_ancestralSet","k":"theorem","m":"Causalean.Graph.DSep.Ancestral","a":"Graph/DSep","t":2,"d":"S ⊆ ancestralSet S: every member is in its own ancestral set.","s":"∀ {V : Type u_1} [inst : DecidableEq V] [inst_1 : Fintype V] (G : Causalean.DAG V) (S : Finset V), S ⊆ G.ancestralSet S","l":57,"f":"Causalean/Graph/DSep/Ancestral.lean"},{"n":"Causalean.DAG.subset_ancestralSet_of_subset","k":"theorem","m":"Causalean.Graph.DSep.Ancestral","a":"Graph/DSep","t":2,"d":"Every element of a subset belongs to the ancestral closure of its ambient set.","s":"∀ {V : Type u_1} [inst : DecidableEq V] [inst_1 : Fintype V] (G : Causalean.DAG V) {Y S : Finset V}, Y ⊆ S → Y ⊆ G.ancestralSet S","l":343,"f":"Causalean/Graph/DSep/Ancestral.lean"},{"n":"Causalean.DAG.subset_iterate_ancStep","k":"theorem","m":"Causalean.Graph.DAG","a":"Graph","t":2,"d":"Any finite set of graph nodes remains contained after applying the graph's ancestor-step operation any number of times.","s":"∀ {V : Type u_2} [inst : DecidableEq V] [inst_1 : Fintype V] (G : Causalean.DAG V) (S : Finset V) (k : ℕ), S ⊆ G.ancStep^[k] S","l":197,"f":"Causalean/Graph/DAG.lean"},{"n":"Causalean.DAG.take_suffix_at_last_S","k":"theorem","m":"Causalean.Graph.DSep.ActivePath","a":"Graph/DSep","t":2,"d":"Suffix-step for source-to-cond transfer.","s":"∀ {V : Type u_1} [inst : DecidableEq V] [inst_1 : Fintype V] (G : Causalean.DAG V) {X Z S : Finset V} {p : List V} {x w : V}, x ∈ X → p.length ≥ 2 → G.IsActivePath (Z ∪ S) p → p.head? = some x → p.getLast? = some w → ∃ x' q, x' ∈ X ∪ S ∧ q.length ≥ 2 ∧ G.IsActivePath (Z ∪ S) q ∧ q.head? = some x' ∧ q.getLast? = some w ∧ ∀ (k : ℕ), 0 < k → ∀ (hk2 : k + 1 < q.length), q.get ⟨k, Decidable.byContradiction fun a => Causalean.DAG.take_suffix_at_last_S._proof_1 q k hk2 a⟩ ∉ S","l":242,"f":"Causalean/Graph/DSep/ActivePath.lean"},{"n":"Causalean.DAG.topoMax_mem_seed","k":"theorem","m":"Causalean.Graph.DSep.OrderedLocalSG","a":"Graph/DSep","t":2,"d":"A vertex with maximal topological order in the ancestral closure of a seed set must itself belong to that seed set.","s":"∀ {V : Type u_1} [inst : DecidableEq V] [inst_1 : Fintype V] (G : Causalean.DAG V) {Q : Finset V} {n : V}, n ∈ G.ancestralSet Q → (∀ m ∈ G.ancestralSet Q, G.topoOrder m ≤ G.topoOrder n) → n ∈ Q","l":103,"f":"Causalean/Graph/DSep/OrderedLocalSG.lean"},{"n":"Causalean.DAG.topoOrder","k":"def","m":"Causalean.Graph.DAG","a":"Graph","t":1,"d":"The topological order derived from the DAG: assign each vertex the value rank v * |V| + enum v, where rank v counts the strict ancestors of v (the computable ancestorRank) and enum : V ↪ Fin |V| breaks ties. The result is a natural number that strictly increases along edges (topoOrder_lt) and is injective (topoOrder_injective).","s":"{V : Type u_2} → [inst : DecidableEq V] → [inst_1 : Fintype V] → Causalean.DAG V → V → ℕ","l":378,"f":"Causalean/Graph/DAG.lean"},{"n":"Causalean.DAG.topoOrder_injective","k":"theorem","m":"Causalean.Graph.DAG","a":"Graph","t":2,"d":"The derived topological order is injective, so it provides a canonical total order on the finite vertex type.","s":"∀ {V : Type u_2} [inst : DecidableEq V] [inst_1 : Fintype V] (G : Causalean.DAG V), Function.Injective G.topoOrder","l":393,"f":"Causalean/Graph/DAG.lean"},{"n":"Causalean.DAG.topoOrder_lt","k":"theorem","m":"Causalean.Graph.DAG","a":"Graph","t":2,"d":"The derived topological order is edge-consistent: if there is an edge from u to v, then topoOrder u < topoOrder v. This witnesses acyclicity.","s":"∀ {V : Type u_2} [inst : DecidableEq V] [inst_1 : Fintype V] (G : Causalean.DAG V) (u v : V), G.edge u v → G.topoOrder u < G.topoOrder v","l":406,"f":"Causalean/Graph/DAG.lean"},{"n":"Causalean.DAG.topoOrder_lt_of_flipMinus_transGen","k":"theorem","m":"Causalean.Graph.MarkovEquiv.CoveredReversal","a":"Graph/MarkovEquiv","t":2,"d":"A directed flipMinus-path strictly increases the topological order.","s":"∀ {V : Type u_1} [inst : DecidableEq V] [inst_1 : Fintype V] {G : Causalean.DAG V} {a b u w : V}, Relation.TransGen (G.flipMinus a b) u w → G.topoOrder u < G.topoOrder w","l":58,"f":"Causalean/Graph/MarkovEquiv/CoveredReversal.lean"},{"n":"Causalean.DAG.uAdj_edge_congr","k":"theorem","m":"Causalean.Graph.MarkovEquiv.CoveredReversal","a":"Graph/MarkovEquiv","t":2,"d":"Graphs with the same directed edges have exactly the same undirected adjacencies.","s":"∀ {V : Type u_1} [inst : DecidableEq V] [inst_1 : Fintype V] {G₁ G₂ : Causalean.DAG V}, (∀ (u w : V), G₁.edge u w ↔ G₂.edge u w) → ∀ {u v : V}, G₁.UAdj u v ↔ G₂.UAdj u v","l":254,"f":"Causalean/Graph/MarkovEquiv/CoveredReversal.lean"},{"n":"Causalean.DAG.uAdj_mono_edge","k":"theorem","m":"Causalean.Graph.DSep.Separation","a":"Graph/DSep","t":2,"d":"If every directed edge of one graph is also an edge of another graph, vertices adjacent in the first graph are also adjacent in the second graph.","s":"∀ {V : Type u_1} [inst : DecidableEq V] [inst_1 : Fintype V] (G G' : Causalean.DAG V), (∀ (u v : V), G'.edge u v → G.edge u v) → ∀ {u v : V}, G'.UAdj u v → G.UAdj u v","l":716,"f":"Causalean/Graph/DSep/Separation.lean"},{"n":"Causalean.Discovery.InvariantPrediction.EnvFamily","k":"structure","m":"Causalean.Discovery.InvariantPrediction.Model","a":"Discovery/InvariantPrediction","t":1,"d":"An environment family for Invariant Causal Prediction (Peters, Bühlmann & Meinshausen 2016) bundles a finite, Fintype-indexed collection of structural causal models sharing a common target variable that every environment observes, a common observed-variable set, and a common latent-variable set. The environments never intervene on the target: they share the target's parent set, the target's structural mechanism, and the latent-noise law, while each environment carries the values assigned to its own intervened coordinates. It further packages the regularity needed to disintegrate the joint law into the target's conditional law given any predictor set — a standard Borel and nonempty target value space, and countable generation of the relevant kernels, and states the exogeneity assumption that in every environment the target's exogenous (latent) parents are independent of its observed parents under the joint law.","s":"(N : Type u_3) → [DecidableEq N] → [Fintype N] → (Ω : N → Type u_4) → [(n : N) → MeasurableSpace (Ω n)] → (ι : Type u_5) → [Fintype ι] → Type (max (max u_3 u_4) u_5)","l":40,"f":"Causalean/Discovery/InvariantPrediction/Model.lean"},{"n":"Causalean.Discovery.InvariantPrediction.EnvFamily.Invariant","k":"def","m":"Causalean.Discovery.InvariantPrediction.Invariance","a":"Discovery/InvariantPrediction","t":1,"d":"S is invariant across the environment family F.","s":"{N : Type u_1} → [inst : DecidableEq N] → [inst_1 : Fintype N] → {Ω : N → Type u_2} → [inst_2 : (n : N) → MeasurableSpace (Ω n)] → {ι : Type u_3} → [inst_3 : Fintype ι] → (F : Causalean.Discovery.InvariantPrediction.EnvFamily N Ω ι) → (S : Finset (Causalean.SWIGNode N)) → (∀ (i : ι), S ⊆ (F.M i).observed) → Prop","l":42,"f":"Causalean/Discovery/InvariantPrediction/Invariance.lean"},{"n":"Causalean.Discovery.InvariantPrediction.EnvFamily.condDistrib_eq_mechanismKernel_of_indep","k":"theorem","m":"Causalean.Discovery.InvariantPrediction.Helpers.MechanismFactor","a":"Discovery/InvariantPrediction/Helpers","t":1,"d":"Let $X$, $L$, and $Y$ be measurable random elements of a probability space, with $Y$'s value space standard Borel and nonempty, and let $Φ$ be a measurable map such that $Y$ equals $Φ(X,L)$ almost everywhere. If $L$ is independent of $X$, then the conditional distribution of $Y$ given $X$ agrees, for almost every pushed-forward value of $X$, with the mechanism kernel obtained by pushing the law of $L$ forward through $Φ$ paired with that value of $X$.","s":"∀ {α : Type u_4} {β : Type u_5} {γ : Type u_6} {δ : Type u_7} [inst : MeasurableSpace α] [inst_1 : MeasurableSpace β] [inst_2 : MeasurableSpace γ] [inst_3 : MeasurableSpace δ] [inst_4 : StandardBorelSpace δ] [inst_5 : Nonempty δ] (μ : MeasureTheory.Measure α) [inst_6 : MeasureTheory.IsProbabilityMeasure μ] {X : α → β} {L : α → γ} {Y : α → δ} {Φ : β × γ → δ}, Measurable X → Measurable L → Measurable Y → Measurable Φ → ProbabilityTheory.IndepFun L X μ → (Y =ᵐ[μ] fun ω => Φ (X ω, L ω)) → (fun x => (ProbabilityTheory.condDistrib Y X μ) x) =ᵐ[MeasureTheory.Measure.map X μ] ⇑(Causalean.Mathlib.GraphMapProd.mechanismKernel (MeasureTheory.Measure.map L μ) Φ)","l":308,"f":"Causalean/Discovery/InvariantPrediction/Helpers/MechanismFactor.lean"},{"n":"Causalean.Discovery.InvariantPrediction.EnvFamily.condDistrib_eq_mechanismKernel_of_indep_of_pair_map","k":"theorem","m":"Causalean.Discovery.InvariantPrediction.Helpers.MechanismFactor","a":"Discovery/InvariantPrediction/Helpers","t":2,"d":"Variant of condDistrib_eq_mechanismKernel_of_indep that takes equality of the joint (X,Y) push-forward measures directly. This avoids needing a MeasurableEq instance for the target when the equality is proved upstream by unfolding a map representation of μ.","s":"∀ {α : Type u_4} {β : Type u_5} {γ : Type u_6} {δ : Type u_7} [inst : MeasurableSpace α] [inst_1 : MeasurableSpace β] [inst_2 : MeasurableSpace γ] [inst_3 : MeasurableSpace δ] [inst_4 : StandardBorelSpace δ] [inst_5 : Nonempty δ] (μ : MeasureTheory.Measure α) [inst_6 : MeasureTheory.IsProbabilityMeasure μ] {X : α → β} {L : α → γ} {Y : α → δ} {Φ : β × γ → δ}, Measurable X → Measurable L → Measurable Y → Measurable Φ → ProbabilityTheory.IndepFun L X μ → MeasureTheory.Measure.map (fun ω => (X ω, Y ω)) μ = MeasureTheory.Measure.map (fun ω => (X ω, Φ (X ω, L ω))) μ → (fun x => (ProbabilityTheory.condDistrib Y X μ) x) =ᵐ[MeasureTheory.Measure.map X μ] ⇑(Causalean.Mathlib.GraphMapProd.mechanismKernel (MeasureTheory.Measure.map L μ) Φ)","l":365,"f":"Causalean/Discovery/InvariantPrediction/Helpers/MechanismFactor.lean"},{"n":"Causalean.Discovery.InvariantPrediction.EnvFamily.condDistrib_target_eq_mechanismKernel","k":"theorem","m":"Causalean.Discovery.InvariantPrediction.Helpers.MechanismFactor","a":"Discovery/InvariantPrediction/Helpers","t":1,"d":"For an invariant-prediction environment family and a reference index i₀ together with an environment i, in environment i the conditional law of the target given its observed parents, under the joint kernel restricted to that environment, agrees with the mechanism kernel formed by pushing the latent-parent law of environment i forward through the target's structural mechanism function, paired with the parent value.","s":"∀ {N : Type u_1} [inst : DecidableEq N] [inst_1 : Fintype N] {Ω : N → Type u_2} [inst_2 : (n : N) → MeasurableSpace (Ω n)] {ι : Type u_3} [inst_3 : Fintype ι] (F : Causalean.Discovery.InvariantPrediction.EnvFamily N Ω ι) (i₀ i : ι), have hYrv := fun ⦃w⦄ hw => Eq.symm (Eq.mp (congrArg (fun _a => _a) (propext Finset.mem_singleton)) hw) ▸ Finset.mem_union_left (F.M i).unobserved (F.hYobs i); have hPrv := Eq.mpr (id (congrArg (fun _a => _a ⊆ (F.M i).randomVars) (Causalean.Discovery.InvariantPrediction.EnvFamily.paObs_eq F i₀ i))) (LE.le.trans Finset.inter_subset_right (id Finset.subset_union_left)); have hLrv := LE.le.trans Finset.inter_subset_right (id Finset.subset_union_right); (fun c => (ProbabilityTheory.condDistrib (Causalean.valuesProjection hYrv) (Causalean.valuesProjection hPrv) ((F.M i).jointKernel (F.s i))) c) =ᵐ[MeasureTheory.Measure.map (Causalean.valuesProjection hPrv) ((F.M i).jointKernel (F.s i))] ⇑(Causalean.Mathlib.GraphMapProd.mechanismKernel (MeasureTheory.Measure.map (Causalean.valuesProjection hLrv) ((F.M i).jointKernel (F.s i))) (F.mechanismFun i₀ i))","l":410,"f":"Causalean/Discovery/InvariantPrediction/Helpers/MechanismFactor.lean"},{"n":"Causalean.Discovery.InvariantPrediction.EnvFamily.fixedParentVals","k":"def","m":"Causalean.Discovery.InvariantPrediction.Model","a":"Discovery/InvariantPrediction","t":1,"d":"The fixed-parent values assigned by environment i's intervention, projected from s i.","s":"{N : Type u_1} → [inst : DecidableEq N] → [inst_1 : Fintype N] → {Ω : N → Type u_2} → [inst_2 : (n : N) → MeasurableSpace (Ω n)] → {ι : Type u_3} → [inst_3 : Fintype ι] → (F : Causalean.Discovery.InvariantPrediction.EnvFamily N Ω ι) → (i : ι) → Causalean.ValuesOn (F.paFix i) (Causalean.swigΩ Ω)","l":188,"f":"Causalean/Discovery/InvariantPrediction/Model.lean"},{"n":"Causalean.Discovery.InvariantPrediction.EnvFamily.fixed_parent_mem_fixed_of_mem","k":"theorem","m":"Causalean.Discovery.InvariantPrediction.Model","a":"Discovery/InvariantPrediction","t":2,"d":"A target parent that is fixed in one environment is fixed in every environment. The point is not that EnvFamily shares the whole fixed set, but that it shares the target parent set (E4), and a fixed-form SWIG node cannot be an observed or unobserved random-form node in the other environment, while a parent of Y is always classified as fixed/observed/unobserved (dag_edges_classified).","s":"∀ {N : Type u_1} [inst : DecidableEq N] [inst_1 : Fintype N] {Ω : N → Type u_2} [inst_2 : (n : N) → MeasurableSpace (Ω n)] {ι : Type u_3} [inst_3 : Fintype ι] (F : Causalean.Discovery.InvariantPrediction.EnvFamily N Ω ι) {i j : ι} {d : Causalean.SWIGNode N}, d ∈ (F.M i).dag.parents F.yNode ∩ (F.M i).fixed → d ∈ (F.M j).fixed","l":137,"f":"Causalean/Discovery/InvariantPrediction/Model.lean"},{"n":"Causalean.Discovery.InvariantPrediction.EnvFamily.icp_sound","k":"theorem","m":"Causalean.Discovery.InvariantPrediction.Soundness","a":"Discovery/InvariantPrediction","t":1,"d":"Theorem 1 (soundness). For an environment family and an index i₀ selecting the target's observed-parent set, the identified set is contained in the target's observed parents: every node selected by ICP is a genuine direct cause.","s":"∀ {N : Type u_1} [inst : DecidableEq N] [inst_1 : Fintype N] {Ω : N → Type u_2} [inst_2 : (n : N) → MeasurableSpace (Ω n)] {ι : Type u_3} [inst_3 : Fintype ι] (F : Causalean.Discovery.InvariantPrediction.EnvFamily N Ω ι) (i₀ : ι), F.idSet ⊆ ↑(F.paObs i₀)","l":34,"f":"Causalean/Discovery/InvariantPrediction/Soundness.lean"},{"n":"Causalean.Discovery.InvariantPrediction.EnvFamily.idSet","k":"def","m":"Causalean.Discovery.InvariantPrediction.IdentifiedSet","a":"Discovery/InvariantPrediction","t":1,"d":"The identified set S(E): the intersection of all invariant predictor sets across the environment family.","s":"{N : Type u_1} → [inst : DecidableEq N] → [inst_1 : Fintype N] → {Ω : N → Type u_2} → [inst_2 : (n : N) → MeasurableSpace (Ω n)] → {ι : Type u_3} → [inst_3 : Fintype ι] → Causalean.Discovery.InvariantPrediction.EnvFamily N Ω ι → Set (Causalean.SWIGNode N)","l":34,"f":"Causalean/Discovery/InvariantPrediction/IdentifiedSet.lean"},{"n":"Causalean.Discovery.InvariantPrediction.EnvFamily.idSet_subset_of_mem","k":"theorem","m":"Causalean.Discovery.InvariantPrediction.IdentifiedSet","a":"Discovery/InvariantPrediction","t":2,"d":"The identified set is contained in every invariant set.","s":"∀ {N : Type u_1} [inst : DecidableEq N] [inst_1 : Fintype N] {Ω : N → Type u_2} [inst_2 : (n : N) → MeasurableSpace (Ω n)] {ι : Type u_3} [inst_3 : Fintype ι] (F : Causalean.Discovery.InvariantPrediction.EnvFamily N Ω ι) {T : Set (Causalean.SWIGNode N)}, T ∈ F.invariantSets → F.idSet ⊆ T","l":38,"f":"Causalean/Discovery/InvariantPrediction/IdentifiedSet.lean"},{"n":"Causalean.Discovery.InvariantPrediction.EnvFamily.invariantSets","k":"def","m":"Causalean.Discovery.InvariantPrediction.IdentifiedSet","a":"Discovery/InvariantPrediction","t":1,"d":"The collection of invariant predictor sets, viewed as sets of nodes.","s":"{N : Type u_1} → [inst : DecidableEq N] → [inst_1 : Fintype N] → {Ω : N → Type u_2} → [inst_2 : (n : N) → MeasurableSpace (Ω n)] → {ι : Type u_3} → [inst_3 : Fintype ι] → Causalean.Discovery.InvariantPrediction.EnvFamily N Ω ι → Set (Set (Causalean.SWIGNode N))","l":29,"f":"Causalean/Discovery/InvariantPrediction/IdentifiedSet.lean"},{"n":"Causalean.Discovery.InvariantPrediction.EnvFamily.jointKernel_map_paLat_eq_latentProduct_map","k":"theorem","m":"Causalean.Discovery.InvariantPrediction.Helpers.MechanismFactor","a":"Discovery/InvariantPrediction/Helpers","t":2,"d":"Projecting the joint kernel to latent parents is the same as projecting the latent product directly to those latent coordinates.","s":"∀ {N : Type u_1} [inst : DecidableEq N] [inst_1 : Fintype N] {Ω : N → Type u_2} [inst_2 : (n : N) → MeasurableSpace (Ω n)] {ι : Type u_3} [inst_3 : Fintype ι] (F : Causalean.Discovery.InvariantPrediction.EnvFamily N Ω ι) (i : ι) (hLrv : F.paLat i ⊆ (F.M i).randomVars) (hLun : F.paLat i ⊆ (F.M i).unobserved), MeasureTheory.Measure.map (Causalean.valuesProjection hLrv) ((F.M i).jointKernel (F.s i)) = MeasureTheory.Measure.map (Causalean.valuesProjection hLun) (F.M i).latentProduct","l":238,"f":"Causalean/Discovery/InvariantPrediction/Helpers/MechanismFactor.lean"},{"n":"Causalean.Discovery.InvariantPrediction.EnvFamily.latentProduct_heq","k":"theorem","m":"Causalean.Discovery.InvariantPrediction.Helpers.MechanismFactor","a":"Discovery/InvariantPrediction/Helpers","t":2,"d":"Cross-environment heterogeneous equality of latent product measures.","s":"∀ {N : Type u_1} [inst : DecidableEq N] [inst_1 : Fintype N] {Ω : N → Type u_2} [inst_2 : (n : N) → MeasurableSpace (Ω n)] {ι : Type u_3} [inst_3 : Fintype ι] (F : Causalean.Discovery.InvariantPrediction.EnvFamily N Ω ι) (i j : ι), (F.M i).latentProduct ≍ (F.M j).latentProduct","l":302,"f":"Causalean/Discovery/InvariantPrediction/Helpers/MechanismFactor.lean"},{"n":"Causalean.Discovery.InvariantPrediction.EnvFamily.latentValues_measurableSpace_heq","k":"theorem","m":"Causalean.Discovery.InvariantPrediction.Helpers.MechanismFactor","a":"Discovery/InvariantPrediction/Helpers","t":2,"d":"Cross-environment measurable-space transport for latent value spaces.","s":"∀ {N : Type u_1} [inst : DecidableEq N] [inst_1 : Fintype N] {Ω : N → Type u_2} [inst_2 : (n : N) → MeasurableSpace (Ω n)] {ι : Type u_3} [inst_3 : Fintype ι] (F : Causalean.Discovery.InvariantPrediction.EnvFamily N Ω ι) (i j : ι), inferInstance ≍ inferInstance","l":296,"f":"Causalean/Discovery/InvariantPrediction/Helpers/MechanismFactor.lean"},{"n":"Causalean.Discovery.InvariantPrediction.EnvFamily.map_heq_transport","k":"theorem","m":"Causalean.Discovery.InvariantPrediction.Helpers.MechanismFactor","a":"Discovery/InvariantPrediction/Helpers","t":2,"d":"Transport a push-forward μ.map g across a propositional equality of the source type together with HEq of the measurable structure, the measure, and the map. Used to identify latent-parent push-forwards across environments whose latent index sets coincide propositionally (via hUnobs) but whose latentDist families and value-space typing match only up to HEq.","s":"∀ {α₁ α₂ δ : Type u} [mα₁ : MeasurableSpace α₁] [mα₂ : MeasurableSpace α₂] [inst : MeasurableSpace δ] {μ₁ : MeasureTheory.Measure α₁} {μ₂ : MeasureTheory.Measure α₂} {g₁ : α₁ → δ} {g₂ : α₂ → δ}, α₁ = α₂ → mα₁ ≍ mα₂ → μ₁ ≍ μ₂ → g₁ ≍ g₂ → MeasureTheory.Measure.map g₁ μ₁ = MeasureTheory.Measure.map g₂ μ₂","l":260,"f":"Causalean/Discovery/InvariantPrediction/Helpers/MechanismFactor.lean"},{"n":"Causalean.Discovery.InvariantPrediction.EnvFamily.measurable_mechanismFun","k":"theorem","m":"Causalean.Discovery.InvariantPrediction.Helpers.MechanismFactor","a":"Discovery/InvariantPrediction/Helpers","t":2,"d":"mechanismFun is measurable.","s":"∀ {N : Type u_1} [inst : DecidableEq N] [inst_1 : Fintype N] {Ω : N → Type u_2} [inst_2 : (n : N) → MeasurableSpace (Ω n)] {ι : Type u_3} [inst_3 : Fintype ι] (F : Causalean.Discovery.InvariantPrediction.EnvFamily N Ω ι) (i₀ i : ι), Measurable (F.mechanismFun i₀ i)","l":169,"f":"Causalean/Discovery/InvariantPrediction/Helpers/MechanismFactor.lean"},{"n":"Causalean.Discovery.InvariantPrediction.EnvFamily.measurable_mechanismFunCf","k":"theorem","m":"Causalean.Discovery.InvariantPrediction.Helpers.MechanismFactor","a":"Discovery/InvariantPrediction/Helpers","t":2,"d":"mechanismFunCf is measurable.","s":"∀ {N : Type u_1} [inst : DecidableEq N] [inst_1 : Fintype N] {Ω : N → Type u_2} [inst_2 : (n : N) → MeasurableSpace (Ω n)] {ι : Type u_3} [inst_3 : Fintype ι] (F : Causalean.Discovery.InvariantPrediction.EnvFamily N Ω ι) (i₀ i : ι) (cf : Causalean.ValuesOn (F.paFix i) (Causalean.swigΩ Ω)), Measurable (F.mechanismFunCf i₀ i cf)","l":148,"f":"Causalean/Discovery/InvariantPrediction/Helpers/MechanismFactor.lean"},{"n":"Causalean.Discovery.InvariantPrediction.EnvFamily.measure_pi_heq","k":"theorem","m":"Causalean.Discovery.InvariantPrediction.Helpers.MechanismFactor","a":"Discovery/InvariantPrediction/Helpers","t":2,"d":"Measure.pi transports heterogeneously across equality of the finite index sets and heterogeneous equality of the coordinate measure families.","s":"∀ {M : Type u_4} {Ω' : M → Type u_5} [inst : (n : M) → MeasurableSpace (Ω' n)] {I J : Finset M} {μI : (i : ↥I) → MeasureTheory.Measure (Ω' ↑i)} {μJ : (j : ↥J) → MeasureTheory.Measure (Ω' ↑j)}, I = J → μI ≍ μJ → MeasureTheory.Measure.pi μI ≍ MeasureTheory.Measure.pi μJ","l":282,"f":"Causalean/Discovery/InvariantPrediction/Helpers/MechanismFactor.lean"},{"n":"Causalean.Discovery.InvariantPrediction.EnvFamily.mechanismFun","k":"def","m":"Causalean.Discovery.InvariantPrediction.Helpers.MechanismFactor","a":"Discovery/InvariantPrediction/Helpers","t":1,"d":"The target mechanism as a measurable map of observed and latent parents.","s":"{N : Type u_1} → [inst : DecidableEq N] → [inst_1 : Fintype N] → {Ω : N → Type u_2} → [inst_2 : (n : N) → MeasurableSpace (Ω n)] → {ι : Type u_3} → [inst_3 : Fintype ι] → (F : Causalean.Discovery.InvariantPrediction.EnvFamily N Ω ι) → (i₀ i : ι) → Causalean.ValuesOn (F.paObs i₀) (Causalean.swigΩ Ω) × Causalean.ValuesOn (F.paLat i) (Causalean.swigΩ Ω) → Causalean.ValuesOn {F.yNode} (Causalean.swigΩ Ω)","l":80,"f":"Causalean/Discovery/InvariantPrediction/Helpers/MechanismFactor.lean"},{"n":"Causalean.Discovery.InvariantPrediction.EnvFamily.mechanismFunCf","k":"def","m":"Causalean.Discovery.InvariantPrediction.Helpers.MechanismFactor","a":"Discovery/InvariantPrediction/Helpers","t":1,"d":"The target mechanism as a measurable map of observed and latent parents, parameterized by an explicit fixed-parent value argument cf. Identical to mechanismFun i₀ i except that the fixed-parent branch reads its value from cf instead of from the environment's intervention assignment s i. This lets the mechanism factor be keyed on paFix-values (used by the redesigned Invariant witness κ), so cross-environment agreement no longer needs the environments to assign the same value to a fixed parent.","s":"{N : Type u_1} → [inst : DecidableEq N] → [inst_1 : Fintype N] → {Ω : N → Type u_2} → [inst_2 : (n : N) → MeasurableSpace (Ω n)] → {ι : Type u_3} → [inst_3 : Fintype ι] → (F : Causalean.Discovery.InvariantPrediction.EnvFamily N Ω ι) → (i₀ i : ι) → Causalean.ValuesOn (F.paFix i) (Causalean.swigΩ Ω) → Causalean.ValuesOn (F.paObs i₀) (Causalean.swigΩ Ω) × Causalean.ValuesOn (F.paLat i) (Causalean.swigΩ Ω) → Causalean.ValuesOn {F.yNode} (Causalean.swigΩ Ω)","l":107,"f":"Causalean/Discovery/InvariantPrediction/Helpers/MechanismFactor.lean"},{"n":"Causalean.Discovery.InvariantPrediction.EnvFamily.mechanismFunCf_fixedParentVals","k":"theorem","m":"Causalean.Discovery.InvariantPrediction.Helpers.MechanismFactor","a":"Discovery/InvariantPrediction/Helpers","t":2,"d":"At the environment's own fixed-parent values fixedParentVals i, the parameterized mechanism mechanismFunCf coincides with the version that reads fixed-parent values from the environment assignment.","s":"∀ {N : Type u_1} [inst : DecidableEq N] [inst_1 : Fintype N] {Ω : N → Type u_2} [inst_2 : (n : N) → MeasurableSpace (Ω n)] {ι : Type u_3} [inst_3 : Fintype ι] (F : Causalean.Discovery.InvariantPrediction.EnvFamily N Ω ι) (i₀ i : ι), F.mechanismFunCf i₀ i (F.fixedParentVals i) = F.mechanismFun i₀ i","l":141,"f":"Causalean/Discovery/InvariantPrediction/Helpers/MechanismFactor.lean"},{"n":"Causalean.Discovery.InvariantPrediction.EnvFamily.mechanismKernel_cf_env_eq","k":"theorem","m":"Causalean.Discovery.InvariantPrediction.Helpers.MechanismFactor","a":"Discovery/InvariantPrediction/Helpers","t":1,"d":"Fix an environment family, a base environment i₀, two environments i and j, and a fixed-parent value assignment cf for environment i. If the latent parents of the target in environment i are among its random variables and likewise for environment j, then the target-mechanism kernel built from environment i at cf equals the target-mechanism kernel built from environment j at the value obtained by transporting cf through the shared fixed-parent set.","s":"∀ {N : Type u_1} [inst : DecidableEq N] [inst_1 : Fintype N] {Ω : N → Type u_2} [inst_2 : (n : N) → MeasurableSpace (Ω n)] {ι : Type u_3} [inst_3 : Fintype ι] (F : Causalean.Discovery.InvariantPrediction.EnvFamily N Ω ι) (i₀ i j : ι) (cf : Causalean.ValuesOn (F.paFix i) (Causalean.swigΩ Ω)) (hLrv : F.paLat i ⊆ (F.M i).randomVars) (hLrv' : F.paLat j ⊆ (F.M j).randomVars), Causalean.Mathlib.GraphMapProd.mechanismKernel (MeasureTheory.Measure.map (Causalean.valuesProjection hLrv) ((F.M i).jointKernel (F.s i))) (F.mechanismFunCf i₀ i cf) = Causalean.Mathlib.GraphMapProd.mechanismKernel (MeasureTheory.Measure.map (Causalean.valuesProjection hLrv') ((F.M j).jointKernel (F.s j))) (F.mechanismFunCf i₀ j (Causalean.valuesProjection (le_of_eq (Causalean.Discovery.InvariantPrediction.EnvFamily.paFix_eq F j i)) cf))","l":551,"f":"Causalean/Discovery/InvariantPrediction/Helpers/MechanismFactor.lean"},{"n":"Causalean.Discovery.InvariantPrediction.EnvFamily.mechanism_invariant","k":"theorem","m":"Causalean.Discovery.InvariantPrediction.Invariance","a":"Discovery/InvariantPrediction","t":1,"d":"Mechanism invariance (the heart of soundness). For an environment family and a target index i₀, the target's observed parents paObs form an invariant set: in every environment the conditional law of the target given its observed parents equals one fixed structural factor, so ICP never rejects the parents.","s":"∀ {N : Type u_1} [inst : DecidableEq N] [inst_1 : Fintype N] {Ω : N → Type u_2} [inst_2 : (n : N) → MeasurableSpace (Ω n)] {ι : Type u_3} [inst_3 : Fintype ι] (F : Causalean.Discovery.InvariantPrediction.EnvFamily N Ω ι) (i₀ : ι), F.Invariant (F.paObs i₀) fun j => Causalean.Discovery.InvariantPrediction.EnvFamily.paObs_subset_observed F i₀ j","l":178,"f":"Causalean/Discovery/InvariantPrediction/Invariance.lean"},{"n":"Causalean.Discovery.InvariantPrediction.EnvFamily.obsKernel_map_valuesProjection_eq_jointKernel_map","k":"theorem","m":"Causalean.Discovery.InvariantPrediction.Helpers.MechanismFactor","a":"Discovery/InvariantPrediction/Helpers","t":2,"d":"The observed marginal of obsKernel is the corresponding random marginal of jointKernel.","s":"∀ {N : Type u_1} [inst : DecidableEq N] [inst_1 : Fintype N] {Ω : N → Type u_2} [inst_2 : (n : N) → MeasurableSpace (Ω n)] (M : Causalean.SCM N Ω) (s : M.FixedValues) (S : Finset (Causalean.SWIGNode N)) (hSobs : S ⊆ M.observed) (hSrv : S ⊆ M.randomVars), MeasureTheory.Measure.map (Causalean.valuesProjection hSobs) (M.obsKernel s) = MeasureTheory.Measure.map (Causalean.valuesProjection hSrv) (M.jointKernel s)","l":66,"f":"Causalean/Discovery/InvariantPrediction/Helpers/MechanismFactor.lean"},{"n":"Causalean.Discovery.InvariantPrediction.EnvFamily.paFix","k":"def","m":"Causalean.Discovery.InvariantPrediction.Model","a":"Discovery/InvariantPrediction","t":1,"d":"The fixed parents of the target in environment i — the parents of Y that are intervened on (fixed) in that environment. These are the coordinates of Y's mechanism whose values live in s i and may *legitimately differ* across environments; the redesigned Invariant predicate conditions on them.","s":"{N : Type u_1} → [inst : DecidableEq N] → [inst_1 : Fintype N] → {Ω : N → Type u_2} → [inst_2 : (n : N) → MeasurableSpace (Ω n)] → {ι : Type u_3} → [inst_3 : Fintype ι] → Causalean.Discovery.InvariantPrediction.EnvFamily N Ω ι → ι → Finset (Causalean.SWIGNode N)","l":130,"f":"Causalean/Discovery/InvariantPrediction/Model.lean"},{"n":"Causalean.Discovery.InvariantPrediction.EnvFamily.paFix_eq","k":"theorem","m":"Causalean.Discovery.InvariantPrediction.Model","a":"Discovery/InvariantPrediction","t":1,"d":"For any two environments i and j, the fixed parents of the target coincide between environment i and environment j.","s":"∀ {N : Type u_1} [inst : DecidableEq N] [inst_1 : Fintype N] {Ω : N → Type u_2} [inst_2 : (n : N) → MeasurableSpace (Ω n)] {ι : Type u_3} [inst_3 : Fintype ι] (F : Causalean.Discovery.InvariantPrediction.EnvFamily N Ω ι) (i j : ι), F.paFix i = F.paFix j","l":164,"f":"Causalean/Discovery/InvariantPrediction/Model.lean"},{"n":"Causalean.Discovery.InvariantPrediction.EnvFamily.paFix_subset_fixed","k":"theorem","m":"Causalean.Discovery.InvariantPrediction.Model","a":"Discovery/InvariantPrediction","t":2,"d":"The fixed parents are a subset of the fixed coordinates, so they can be read off the environment's intervention assignment s i.","s":"∀ {N : Type u_1} [inst : DecidableEq N] [inst_1 : Fintype N] {Ω : N → Type u_2} [inst_2 : (n : N) → MeasurableSpace (Ω n)] {ι : Type u_3} [inst_3 : Fintype ι] (F : Causalean.Discovery.InvariantPrediction.EnvFamily N Ω ι) (i : ι), F.paFix i ⊆ (F.M i).fixed","l":183,"f":"Causalean/Discovery/InvariantPrediction/Model.lean"},{"n":"Causalean.Discovery.InvariantPrediction.EnvFamily.paLat","k":"def","m":"Causalean.Discovery.InvariantPrediction.Helpers.MechanismFactor","a":"Discovery/InvariantPrediction/Helpers","t":1,"d":"The latent parents of the target in environment i.","s":"{N : Type u_1} → [inst : DecidableEq N] → [inst_1 : Fintype N] → {Ω : N → Type u_2} → [inst_2 : (n : N) → MeasurableSpace (Ω n)] → {ι : Type u_3} → [inst_3 : Fintype ι] → Causalean.Discovery.InvariantPrediction.EnvFamily N Ω ι → ι → Finset (Causalean.SWIGNode N)","l":47,"f":"Causalean/Discovery/InvariantPrediction/Helpers/MechanismFactor.lean"},{"n":"Causalean.Discovery.InvariantPrediction.EnvFamily.paLat_eq","k":"theorem","m":"Causalean.Discovery.InvariantPrediction.Helpers.MechanismFactor","a":"Discovery/InvariantPrediction/Helpers","t":2,"d":"The latent-parent set is environment-independent.","s":"∀ {N : Type u_1} [inst : DecidableEq N] [inst_1 : Fintype N] {Ω : N → Type u_2} [inst_2 : (n : N) → MeasurableSpace (Ω n)] {ι : Type u_3} [inst_3 : Fintype ι] (F : Causalean.Discovery.InvariantPrediction.EnvFamily N Ω ι) (i j : ι), F.paLat i = F.paLat j","l":51,"f":"Causalean/Discovery/InvariantPrediction/Helpers/MechanismFactor.lean"},{"n":"Causalean.Discovery.InvariantPrediction.EnvFamily.paObs","k":"def","m":"Causalean.Discovery.InvariantPrediction.Model","a":"Discovery/InvariantPrediction","t":1,"d":"The observed parents of the target in environment i — the conditioning candidates ICP ranges over. Index-independent by hParents and hObs (see paObs_eq).","s":"{N : Type u_1} → [inst : DecidableEq N] → [inst_1 : Fintype N] → {Ω : N → Type u_2} → [inst_2 : (n : N) → MeasurableSpace (Ω n)] → {ι : Type u_3} → [inst_3 : Fintype ι] → Causalean.Discovery.InvariantPrediction.EnvFamily N Ω ι → ι → Finset (Causalean.SWIGNode N)","l":119,"f":"Causalean/Discovery/InvariantPrediction/Model.lean"},{"n":"Causalean.Discovery.InvariantPrediction.EnvFamily.paObs_eq","k":"theorem","m":"Causalean.Discovery.InvariantPrediction.Model","a":"Discovery/InvariantPrediction","t":2,"d":"paObs does not depend on the chosen environment.","s":"∀ {N : Type u_1} [inst : DecidableEq N] [inst_1 : Fintype N] {Ω : N → Type u_2} [inst_2 : (n : N) → MeasurableSpace (Ω n)] {ι : Type u_3} [inst_3 : Fintype ι] (F : Causalean.Discovery.InvariantPrediction.EnvFamily N Ω ι) (i j : ι), F.paObs i = F.paObs j","l":125,"f":"Causalean/Discovery/InvariantPrediction/Model.lean"},{"n":"Causalean.Discovery.InvariantPrediction.EnvFamily.paObs_mem_invariantSets","k":"theorem","m":"Causalean.Discovery.InvariantPrediction.IdentifiedSet","a":"Discovery/InvariantPrediction","t":1,"d":"For an invariant-prediction environment family and an index i₀ selecting the target's observed-parent set, the target's observed parents form an invariant predictor set — the membership form of mechanism_invariant.","s":"∀ {N : Type u_1} [inst : DecidableEq N] [inst_1 : Fintype N] {Ω : N → Type u_2} [inst_2 : (n : N) → MeasurableSpace (Ω n)] {ι : Type u_3} [inst_3 : Fintype ι] (F : Causalean.Discovery.InvariantPrediction.EnvFamily N Ω ι) (i₀ : ι), ↑(F.paObs i₀) ∈ F.invariantSets","l":43,"f":"Causalean/Discovery/InvariantPrediction/IdentifiedSet.lean"},{"n":"Causalean.Discovery.InvariantPrediction.EnvFamily.paObs_subset_observed","k":"theorem","m":"Causalean.Discovery.InvariantPrediction.Invariance","a":"Discovery/InvariantPrediction","t":2,"d":"The observed parents form a valid conditioning set in every environment.","s":"∀ {N : Type u_1} [inst : DecidableEq N] [inst_1 : Fintype N] {Ω : N → Type u_2} [inst_2 : (n : N) → MeasurableSpace (Ω n)] {ι : Type u_3} [inst_3 : Fintype ι] (F : Causalean.Discovery.InvariantPrediction.EnvFamily N Ω ι) (i j : ι), F.paObs i ⊆ (F.M j).observed","l":79,"f":"Causalean/Discovery/InvariantPrediction/Invariance.lean"},{"n":"Causalean.Discovery.InvariantPrediction.EnvFamily.structFun_yNode_apply_eq","k":"theorem","m":"Causalean.Discovery.InvariantPrediction.Helpers.MechanismFactor","a":"Discovery/InvariantPrediction/Helpers","t":2,"d":"Cross-environment equality of the target structural function applied to coordinatewise-equal target-parent tuples.","s":"∀ {N : Type u_1} [inst : DecidableEq N] [inst_1 : Fintype N] {Ω : N → Type u_2} [inst_2 : (n : N) → MeasurableSpace (Ω n)] {ι : Type u_3} [inst_3 : Fintype ι] (F : Causalean.Discovery.InvariantPrediction.EnvFamily N Ω ι) (i j : ι) {ξi : (w : ↥((F.M i).dag.parents F.yNode)) → Causalean.swigΩ Ω ↑w} {ξj : (w : ↥((F.M j).dag.parents F.yNode)) → Causalean.swigΩ Ω ↑w}, (∀ (d : Causalean.SWIGNode N) (hdi : d ∈ (F.M i).dag.parents F.yNode) (hdj : d ∈ (F.M j).dag.parents F.yNode), ξi ⟨d, hdi⟩ = ξj ⟨d, hdj⟩) → (F.M i).structFun ⟨F.yNode, F.hYobs i⟩ ξi = (F.M j).structFun ⟨F.yNode, F.hYobs j⟩ ξj","l":528,"f":"Causalean/Discovery/InvariantPrediction/Helpers/MechanismFactor.lean"},{"n":"Causalean.Discovery.InvariantPrediction.EnvFamily.targetSet","k":"def","m":"Causalean.Discovery.InvariantPrediction.Invariance","a":"Discovery/InvariantPrediction","t":1,"d":"The singleton node set {Y} used as the conditioned target.","s":"{N : Type u_1} → [inst : DecidableEq N] → [inst_1 : Fintype N] → {Ω : N → Type u_2} → [inst_2 : (n : N) → MeasurableSpace (Ω n)] → {ι : Type u_3} → [inst_3 : Fintype ι] → Causalean.Discovery.InvariantPrediction.EnvFamily N Ω ι → Finset (Causalean.SWIGNode N)","l":39,"f":"Causalean/Discovery/InvariantPrediction/Invariance.lean"},{"n":"Causalean.Discovery.InvariantPrediction.EnvFamily.target_projection_evalMap_eq_mechanismFun","k":"theorem","m":"Causalean.Discovery.InvariantPrediction.Helpers.MechanismFactor","a":"Discovery/InvariantPrediction/Helpers","t":2,"d":"Pointwise structural equation for the target, expressed through mechanismFun on observed and latent parents.","s":"∀ {N : Type u_1} [inst : DecidableEq N] [inst_1 : Fintype N] {Ω : N → Type u_2} [inst_2 : (n : N) → MeasurableSpace (Ω n)] {ι : Type u_3} [inst_3 : Fintype ι] (F : Causalean.Discovery.InvariantPrediction.EnvFamily N Ω ι) (i₀ i : ι) (ℓ : (F.M i).LatentValues), Causalean.valuesProjection (have this := fun ⦃w⦄ hw => Eq.symm (Eq.mp (congrArg (fun _a => _a) (propext Finset.mem_singleton)) hw) ▸ Finset.mem_union_left (F.M i).unobserved (F.hYobs i); this) ((F.M i).evalMap (F.s i) ℓ) = F.mechanismFun i₀ i (Causalean.valuesProjection (have this := Eq.mpr (id (congrArg (fun _a => _a ⊆ (F.M i).randomVars) (Causalean.Discovery.InvariantPrediction.EnvFamily.paObs_eq F i₀ i))) (LE.le.trans Finset.inter_subset_right (id Finset.subset_union_left)); this) ((F.M i).evalMap (F.s i) ℓ), Causalean.valuesProjection (have this := LE.le.trans Finset.inter_subset_right (id Finset.subset_union_right); this) ((F.M i).evalMap (F.s i) ℓ))","l":189,"f":"Causalean/Discovery/InvariantPrediction/Helpers/MechanismFactor.lean"},{"n":"Causalean.Discovery.InvariantPrediction.EnvFamily.valuesOn_measurableSpace_heq","k":"theorem","m":"Causalean.Discovery.InvariantPrediction.Helpers.MechanismFactor","a":"Discovery/InvariantPrediction/Helpers","t":2,"d":"ValuesOn measurable spaces transport heterogeneously across equality of the finite index sets.","s":"∀ {M : Type u_4} [Fintype M] {Ω' : M → Type u_5} [inst : (n : M) → MeasurableSpace (Ω' n)] {I J : Finset M}, I = J → inferInstance ≍ inferInstance","l":272,"f":"Causalean/Discovery/InvariantPrediction/Helpers/MechanismFactor.lean"},{"n":"Causalean.Discovery.InvariantPrediction.EnvFamily.valuesProjection_randomToObserved_eq","k":"theorem","m":"Causalean.Discovery.InvariantPrediction.Helpers.MechanismFactor","a":"Discovery/InvariantPrediction/Helpers","t":2,"d":"Observed-coordinate projection after randomToObserved is the same as the corresponding random-coordinate projection.","s":"∀ {N : Type u_1} [inst : DecidableEq N] [inst_1 : Fintype N] {Ω : N → Type u_2} [inst_2 : (n : N) → MeasurableSpace (Ω n)] (M : Causalean.SCM N Ω) (S : Finset (Causalean.SWIGNode N)) (hSobs : S ⊆ M.observed) (hSrv : S ⊆ M.randomVars), Causalean.valuesProjection hSobs ∘ M.randomToObserved = Causalean.valuesProjection hSrv","l":56,"f":"Causalean/Discovery/InvariantPrediction/Helpers/MechanismFactor.lean"},{"n":"Causalean.Discovery.InvariantPrediction.EnvFamily.yNode","k":"def","m":"Causalean.Discovery.InvariantPrediction.Model","a":"Discovery/InvariantPrediction","t":1,"d":"The target node SWIGNode.random Y.","s":"{N : Type u_1} → [inst : DecidableEq N] → [inst_1 : Fintype N] → {Ω : N → Type u_2} → [inst_2 : (n : N) → MeasurableSpace (Ω n)] → {ι : Type u_3} → [inst_3 : Fintype ι] → Causalean.Discovery.InvariantPrediction.EnvFamily N Ω ι → Causalean.SWIGNode N","l":116,"f":"Causalean/Discovery/InvariantPrediction/Model.lean"},{"n":"Causalean.Discovery.InvariantPrediction.LinearGaussian.Env","k":"structure","m":"Causalean.Discovery.InvariantPrediction.LinearGaussian.Model","a":"Discovery/InvariantPrediction/LinearGaussian","t":1,"d":"A single do-intervention environment for the observational SEM M (sec:idfirst) bundles an intervention set that never includes the target node, the constant values assigned to each intervened coordinate, and measurable post-intervention coordinates that equal their assigned constant, almost surely, on the intervened set, satisfy the same structural equation as M — with M's coefficients and noises — on every coordinate left un-intervened, and for which the target's noise remains independent of each parent coordinate of the target.","s":"{p : ℕ} → Causalean.Discovery.InvariantPrediction.LinearGaussian.ObsSEM p → Type u_1","l":161,"f":"Causalean/Discovery/InvariantPrediction/LinearGaussian/Model.lean"},{"n":"Causalean.Discovery.InvariantPrediction.LinearGaussian.EnvFamily","k":"structure","m":"Causalean.Discovery.InvariantPrediction.LinearGaussian.Model","a":"Discovery/InvariantPrediction/LinearGaussian","t":1,"d":"An environment family for the linear-Gaussian ICP problem bundles an observational linear-Gaussian structural equation model together with a finite index set of interventional environments over that model.","s":"ℕ → Type (u_1 + 1)","l":208,"f":"Causalean/Discovery/InvariantPrediction/LinearGaussian/Model.lean"},{"n":"Causalean.Discovery.InvariantPrediction.LinearGaussian.EnvFamily.HasShiftedSingleInterventions","k":"def","m":"Causalean.Discovery.InvariantPrediction.LinearGaussian.Completeness","a":"Discovery/InvariantPrediction/LinearGaussian","t":1,"d":"Do-intervention single-intervention hypothesis of prop:1(i): for every predictor j, some environment performs a single shifted do-intervention on j, i.e. A i = {j} and the assigned value differs from the observational mean a i j ≠ E[X¹_j].","s":"{p : ℕ} → Causalean.Discovery.InvariantPrediction.LinearGaussian.EnvFamily p → Prop","l":82,"f":"Causalean/Discovery/InvariantPrediction/LinearGaussian/Completeness.lean"},{"n":"Causalean.Discovery.InvariantPrediction.LinearGaussian.EnvFamily.InvarianceNull","k":"def","m":"Causalean.Discovery.InvariantPrediction.LinearGaussian.Regression","a":"Discovery/InvariantPrediction/LinearGaussian","t":1,"d":"Pairwise regression-invariance null.","s":"{p : ℕ} → Causalean.Discovery.InvariantPrediction.LinearGaussian.EnvFamily p → Finset (Fin (p + 1)) → Prop","l":80,"f":"Causalean/Discovery/InvariantPrediction/LinearGaussian/Regression.lean"},{"n":"Causalean.Discovery.InvariantPrediction.LinearGaussian.EnvFamily.ObsIntegrable","k":"def","m":"Causalean.Discovery.InvariantPrediction.LinearGaussian.Completeness","a":"Discovery/InvariantPrediction/LinearGaussian","t":1,"d":"Integrability of the observational coordinates (all Gaussian moments exist); carried as an explicit hypothesis so the observational means E[X¹_j] used by the shifted-intervention condition are available.","s":"{p : ℕ} → Causalean.Discovery.InvariantPrediction.LinearGaussian.EnvFamily p → Prop","l":90,"f":"Causalean/Discovery/InvariantPrediction/LinearGaussian/Completeness.lean"},{"n":"Causalean.Discovery.InvariantPrediction.LinearGaussian.EnvFamily.exists_youngest_nonzero","k":"theorem","m":"Causalean.Discovery.InvariantPrediction.LinearGaussian.Completeness","a":"Discovery/InvariantPrediction/LinearGaussian","t":1,"d":"Youngest-node selection (the \"youngest node X_{k₀}\" step). Given a nonempty set of coordinate indices $T$, there is an index $k_0 \\in T$ such that no other element of $T$ is a descendant of $k_0$ along the observational DAG's directed edges — i.e. $k_0$ is a sink of the subgraph induced by $T$. This is the \"youngest\" node with non-zero α of the paper's proof.","s":"∀ {p : ℕ} (F : Causalean.Discovery.InvariantPrediction.LinearGaussian.EnvFamily p) (T : Finset (Fin (p + 1))), T.Nonempty → ∃ k₀ ∈ T, ∀ k ∈ T, k ≠ k₀ → ¬F.obs.dag.isAncestor k₀ k","l":133,"f":"Causalean/Discovery/InvariantPrediction/LinearGaussian/Completeness.lean"},{"n":"Causalean.Discovery.InvariantPrediction.LinearGaussian.EnvFamily.icp_complete_linearGaussian","k":"theorem","m":"Causalean.Discovery.InvariantPrediction.LinearGaussian.Completeness","a":"Discovery/InvariantPrediction/LinearGaussian","t":1,"d":"Completeness for do-interventions — Theorem prop:1(i). For an observational linear-Gaussian SEM with integrable regressors such that every predictor $j$ receives at least one environment with a single do-intervention $A^e = \\{j\\}$ whose shifted value $a^e_j$ differs from its observational mean $E[X^1_j]$, the ICP identified set — the intersection of all invariant predictor sets — equals exactly the parent set of the target node, $S(E) = PA(Y)$.","s":"∀ {p : ℕ} (F : Causalean.Discovery.InvariantPrediction.LinearGaussian.EnvFamily p), F.ObsIntegrable → F.HasShiftedSingleInterventions → F.identifiedSet = F.paY","l":276,"f":"Causalean/Discovery/InvariantPrediction/LinearGaussian/Completeness.lean"},{"n":"Causalean.Discovery.InvariantPrediction.LinearGaussian.EnvFamily.icp_sound_linearGaussian","k":"theorem","m":"Causalean.Discovery.InvariantPrediction.LinearGaussian.Completeness","a":"Discovery/InvariantPrediction/LinearGaussian","t":1,"d":"Soundness (propos:sem). Assuming the observational predictor coordinates are integrable, so their means E[X¹_j] are well defined, the identified set is contained in the target's parents, S(E) ⊆ PA(Y).","s":"∀ {p : ℕ} (F : Causalean.Discovery.InvariantPrediction.LinearGaussian.EnvFamily p), F.ObsIntegrable → F.identifiedSet ⊆ F.paY","l":95,"f":"Causalean/Discovery/InvariantPrediction/LinearGaussian/Completeness.lean"},{"n":"Causalean.Discovery.InvariantPrediction.LinearGaussian.EnvFamily.identifiedSet","k":"def","m":"Causalean.Discovery.InvariantPrediction.LinearGaussian.Regression","a":"Discovery/InvariantPrediction/LinearGaussian","t":1,"d":"The identified set S(E) := ⋂ {S : H_{0,S} holds} (eq:ident).","s":"{p : ℕ} → Causalean.Discovery.InvariantPrediction.LinearGaussian.EnvFamily p → Finset (Fin (p + 1))","l":116,"f":"Causalean/Discovery/InvariantPrediction/LinearGaussian/Regression.lean"},{"n":"Causalean.Discovery.InvariantPrediction.LinearGaussian.EnvFamily.instDecidablePredFinsetFinHAddNatOfNatInvarianceNull","k":"instance","m":"Causalean.Discovery.InvariantPrediction.LinearGaussian.Regression","a":"Discovery/InvariantPrediction/LinearGaussian","t":2,"d":"Decidability of the invariance null, needed for the filter in invariantSets. The predicate is genuinely a Prop over measures, so this is supplied classically.","s":"{p : ℕ} → (F : Causalean.Discovery.InvariantPrediction.LinearGaussian.EnvFamily p) → DecidablePred F.InvarianceNull","l":106,"f":"Causalean/Discovery/InvariantPrediction/LinearGaussian/Regression.lean"},{"n":"Causalean.Discovery.InvariantPrediction.LinearGaussian.EnvFamily.invariantSets","k":"def","m":"Causalean.Discovery.InvariantPrediction.LinearGaussian.Regression","a":"Discovery/InvariantPrediction/LinearGaussian","t":1,"d":"The collection of predictor subsets S ⊆ {1,…,p} whose invariance null holds — the index set of the S(E) intersection (eq:ident).","s":"{p : ℕ} → Causalean.Discovery.InvariantPrediction.LinearGaussian.EnvFamily p → Finset (Finset (Fin (p + 1)))","l":111,"f":"Causalean/Discovery/InvariantPrediction/LinearGaussian/Regression.lean"},{"n":"Causalean.Discovery.InvariantPrediction.LinearGaussian.EnvFamily.mem_identifiedSet","k":"theorem","m":"Causalean.Discovery.InvariantPrediction.LinearGaussian.Regression","a":"Discovery/InvariantPrediction/LinearGaussian","t":1,"d":"For a predictor index k, k lies in the identified set S(E) exactly when k belongs to every predictor subset whose invariance null holds.","s":"∀ {p : ℕ} (F : Causalean.Discovery.InvariantPrediction.LinearGaussian.EnvFamily p) {k : Fin (p + 1)}, k ∈ F.identifiedSet ↔ ∀ S ∈ F.invariantSets, k ∈ S","l":125,"f":"Causalean/Discovery/InvariantPrediction/LinearGaussian/Regression.lean"},{"n":"Causalean.Discovery.InvariantPrediction.LinearGaussian.EnvFamily.mem_invariantSets","k":"theorem","m":"Causalean.Discovery.InvariantPrediction.LinearGaussian.Regression","a":"Discovery/InvariantPrediction/LinearGaussian","t":2,"d":"Membership in invariantSets.","s":"∀ {p : ℕ} (F : Causalean.Discovery.InvariantPrediction.LinearGaussian.EnvFamily p) {S : Finset (Fin (p + 1))}, S ∈ F.invariantSets ↔ S ⊆ Causalean.Discovery.InvariantPrediction.LinearGaussian.predictors p ∧ F.InvarianceNull S","l":132,"f":"Causalean/Discovery/InvariantPrediction/LinearGaussian/Regression.lean"},{"n":"Causalean.Discovery.InvariantPrediction.LinearGaussian.EnvFamily.obsMean","k":"def","m":"Causalean.Discovery.InvariantPrediction.LinearGaussian.Completeness","a":"Discovery/InvariantPrediction/LinearGaussian","t":1,"d":"The observational mean of coordinate j: E[X¹_j] = ∫ X_j dP¹.","s":"{p : ℕ} → Causalean.Discovery.InvariantPrediction.LinearGaussian.EnvFamily p → Fin (p + 1) → ℝ","l":79,"f":"Causalean/Discovery/InvariantPrediction/LinearGaussian/Completeness.lean"},{"n":"Causalean.Discovery.InvariantPrediction.LinearGaussian.EnvFamily.paY","k":"def","m":"Causalean.Discovery.InvariantPrediction.LinearGaussian.Model","a":"Discovery/InvariantPrediction/LinearGaussian","t":1,"d":"The observational parent set PA(Y) of the family.","s":"{p : ℕ} → Causalean.Discovery.InvariantPrediction.LinearGaussian.EnvFamily p → Finset (Fin (p + 1))","l":232,"f":"Causalean/Discovery/InvariantPrediction/LinearGaussian/Model.lean"},{"n":"Causalean.Discovery.InvariantPrediction.LinearGaussian.EnvFamily.residual_mean_shift_of_doIntervention","k":"theorem","m":"Causalean.Discovery.InvariantPrediction.LinearGaussian.Completeness","a":"Discovery/InvariantPrediction/LinearGaussian","t":1,"d":"Residual mean-shift (eq:help1/eq:help2). Fix an observational SEM with integrable regressors, a candidate coefficient vector γ, and an index k₀ such that the gap α_{k₀} = β₀,k₀ − γ_{k₀} between the causal coefficient and γ at k₀ is nonzero. Suppose k₀ is a youngest such index: every other index with a nonzero coefficient gap is not a descendant of k₀ in the observational DAG, and let i be an environment consisting of a single do-intervention pinning coordinate k₀ to a value different from its observational mean. Then the residual R^i = Y^i − Σ γ_k X_k^i computed in environment i and the observational residual R¹ computed the same way do not have the same distribution, since they have different means.","s":"∀ {p : ℕ} (F : Causalean.Discovery.InvariantPrediction.LinearGaussian.EnvFamily p), F.ObsIntegrable → ∀ (γ : Fin (p + 1) → ℝ) (k₀ : Fin (p + 1)), F.obs.β (Causalean.Discovery.InvariantPrediction.LinearGaussian.target p) k₀ - γ k₀ ≠ 0 → (∀ (k : Fin (p + 1)), k ≠ k₀ → F.obs.β (Causalean.Discovery.InvariantPrediction.LinearGaussian.target p) k - γ k ≠ 0 → ¬F.obs.dag.isAncestor k₀ k) → ∀ (i : F.ι), (F.env i).A = {k₀} → (F.env i).a k₀ ≠ F.obsMean k₀ → ¬ProbabilityTheory.IdentDistrib (Causalean.Discovery.InvariantPrediction.LinearGaussian.envResidual (F.env i) γ) (Causalean.Discovery.InvariantPrediction.LinearGaussian.obsResidual F.obs γ) F.obs.P F.obs.P","l":151,"f":"Causalean/Discovery/InvariantPrediction/LinearGaussian/Completeness.lean"},{"n":"Causalean.Discovery.InvariantPrediction.LinearGaussian.ObsSEM","k":"structure","m":"Causalean.Discovery.InvariantPrediction.LinearGaussian.Model","a":"Discovery/InvariantPrediction/LinearGaussian","t":1,"d":"The observational linear-Gaussian structural equation model (eq:semmmmm, the e = 1 block): it bundles a sample space with its σ-algebra carrying a probability measure that is a genuine probability measure, together with structural coefficients β obeying Xⱼ = Σ_{k≠j} βⱼₖ Xₖ + εⱼ for measurable observed coordinates X, an acyclic graph whose edge k → j holds exactly when βⱼₖ ≠ 0, and the absence of self-loops in β. The noises ε are declared to be the structural residuals εⱼ = Xⱼ − Σ_{k≠j} βⱼₖ Xₖ, jointly independent, and each centered Gaussian with a positive variance σⱼ², and the target's noise is independent of every parent coordinate of the target.","s":"ℕ → Type (u_1 + 1)","l":75,"f":"Causalean/Discovery/InvariantPrediction/LinearGaussian/Model.lean"},{"n":"Causalean.Discovery.InvariantPrediction.LinearGaussian.ObsSEM.mem_paY","k":"theorem","m":"Causalean.Discovery.InvariantPrediction.LinearGaussian.Model","a":"Discovery/InvariantPrediction/LinearGaussian","t":1,"d":"For a predictor index k, k belongs to the target's parent set PA(Y) exactly when the target's structural coefficient on k is nonzero.","s":"∀ {p : ℕ} (M : Causalean.Discovery.InvariantPrediction.LinearGaussian.ObsSEM p) {k : Fin (p + 1)}, k ∈ M.paY ↔ M.β (Causalean.Discovery.InvariantPrediction.LinearGaussian.target p) k ≠ 0","l":154,"f":"Causalean/Discovery/InvariantPrediction/LinearGaussian/Model.lean"},{"n":"Causalean.Discovery.InvariantPrediction.LinearGaussian.ObsSEM.paY","k":"def","m":"Causalean.Discovery.InvariantPrediction.LinearGaussian.Model","a":"Discovery/InvariantPrediction/LinearGaussian","t":1,"d":"The parents of the target Y in the observational graph: PA(Y) = {k : β 0 k ≠ 0}. This is the set the completeness theorem recovers.","s":"{p : ℕ} → Causalean.Discovery.InvariantPrediction.LinearGaussian.ObsSEM p → Finset (Fin (p + 1))","l":142,"f":"Causalean/Discovery/InvariantPrediction/LinearGaussian/Model.lean"},{"n":"Causalean.Discovery.InvariantPrediction.LinearGaussian.ObsSEM.paY_subset_predictors","k":"theorem","m":"Causalean.Discovery.InvariantPrediction.LinearGaussian.Model","a":"Discovery/InvariantPrediction/LinearGaussian","t":2,"d":"PA(Y) consists of predictors only (0 ∉ PA(Y)): the target is acyclic, so it is not its own parent.","s":"∀ {p : ℕ} (M : Causalean.Discovery.InvariantPrediction.LinearGaussian.ObsSEM p), M.paY ⊆ Causalean.Discovery.InvariantPrediction.LinearGaussian.predictors p","l":146,"f":"Causalean/Discovery/InvariantPrediction/LinearGaussian/Model.lean"},{"n":"Causalean.Discovery.InvariantPrediction.LinearGaussian.SupportedOn","k":"def","m":"Causalean.Discovery.InvariantPrediction.LinearGaussian.Regression","a":"Discovery/InvariantPrediction/LinearGaussian","t":1,"d":"γ is supported on S: every nonzero coordinate of γ lies in S (the paper's β_k = 0 if k ∉ S).","s":"{p : ℕ} → (Fin (p + 1) → ℝ) → Finset (Fin (p + 1)) → Prop","l":61,"f":"Causalean/Discovery/InvariantPrediction/LinearGaussian/Regression.lean"},{"n":"Causalean.Discovery.InvariantPrediction.LinearGaussian.causalCoeff","k":"def","m":"Causalean.Discovery.InvariantPrediction.LinearGaussian.Helpers.Residual","a":"Discovery/InvariantPrediction/LinearGaussian/Helpers","t":1,"d":"The causal coefficient γ* = β₀,· (row 0 of β).","s":"{p : ℕ} → Causalean.Discovery.InvariantPrediction.LinearGaussian.ObsSEM p → Fin (p + 1) → ℝ","l":31,"f":"Causalean/Discovery/InvariantPrediction/LinearGaussian/Helpers/Residual.lean"},{"n":"Causalean.Discovery.InvariantPrediction.LinearGaussian.envResidual","k":"def","m":"Causalean.Discovery.InvariantPrediction.LinearGaussian.Regression","a":"Discovery/InvariantPrediction/LinearGaussian","t":1,"d":"The regression residual in an interventional environment, on M's space: R^e = Y^e − Σ_k γ k · X_k^e.","s":"{p : ℕ} → {M : Causalean.Discovery.InvariantPrediction.LinearGaussian.ObsSEM p} → Causalean.Discovery.InvariantPrediction.LinearGaussian.Env M → (Fin (p + 1) → ℝ) → M.Ω → ℝ","l":71,"f":"Causalean/Discovery/InvariantPrediction/LinearGaussian/Regression.lean"},{"n":"Causalean.Discovery.InvariantPrediction.LinearGaussian.envResidual_eq_eps","k":"theorem","m":"Causalean.Discovery.InvariantPrediction.LinearGaussian.Helpers.Residual","a":"Discovery/InvariantPrediction/LinearGaussian/Helpers","t":1,"d":"Interventional residual is the target noise. For an observational SEM and a do-intervention environment built on it — where the target is never itself intervened on, so it keeps its structural equation — the environment residual Yᵉ − Σ_k β₀ₖ Xₖᵉ, evaluated at the causal coefficient, equals the target's structural noise ε₀ almost everywhere.","s":"∀ {p : ℕ} (M : Causalean.Discovery.InvariantPrediction.LinearGaussian.ObsSEM p) (e : Causalean.Discovery.InvariantPrediction.LinearGaussian.Env M), ∀ᵐ (ω : M.Ω) ∂M.P, Causalean.Discovery.InvariantPrediction.LinearGaussian.envResidual e (Causalean.Discovery.InvariantPrediction.LinearGaussian.causalCoeff M) ω = M.ε ω (Causalean.Discovery.InvariantPrediction.LinearGaussian.target p)","l":50,"f":"Causalean/Discovery/InvariantPrediction/LinearGaussian/Helpers/Residual.lean"},{"n":"Causalean.Discovery.InvariantPrediction.LinearGaussian.eps_aemeasurable","k":"theorem","m":"Causalean.Discovery.InvariantPrediction.LinearGaussian.Helpers.Moments","a":"Discovery/InvariantPrediction/LinearGaussian/Helpers","t":2,"d":"εⱼ is a.e.-measurable: by the structural-residual identity hε it agrees a.e. with the measurable map Xⱼ − Σ_{k≠j} βⱼₖ Xₖ.","s":"∀ {p : ℕ} (M : Causalean.Discovery.InvariantPrediction.LinearGaussian.ObsSEM p) (j : Fin (p + 1)), AEMeasurable (fun ω => M.ε ω j) M.P","l":27,"f":"Causalean/Discovery/InvariantPrediction/LinearGaussian/Helpers/Moments.lean"},{"n":"Causalean.Discovery.InvariantPrediction.LinearGaussian.eps_integrable","k":"theorem","m":"Causalean.Discovery.InvariantPrediction.LinearGaussian.Helpers.Moments","a":"Discovery/InvariantPrediction/LinearGaussian/Helpers","t":2,"d":"The noise εⱼ is M.P-integrable (Gaussian marginal has a first moment).","s":"∀ {p : ℕ} (M : Causalean.Discovery.InvariantPrediction.LinearGaussian.ObsSEM p) (j : Fin (p + 1)), MeasureTheory.Integrable (fun ω => M.ε ω j) M.P","l":37,"f":"Causalean/Discovery/InvariantPrediction/LinearGaussian/Helpers/Moments.lean"},{"n":"Causalean.Discovery.InvariantPrediction.LinearGaussian.eps_integral_zero","k":"theorem","m":"Causalean.Discovery.InvariantPrediction.LinearGaussian.Helpers.Moments","a":"Discovery/InvariantPrediction/LinearGaussian/Helpers","t":1,"d":"For a linear-Gaussian observational SEM and coordinate index j, the structural noise εⱼ is centered: its expectation under M's probability measure is zero.","s":"∀ {p : ℕ} (M : Causalean.Discovery.InvariantPrediction.LinearGaussian.ObsSEM p) (j : Fin (p + 1)), ∫ (ω : M.Ω), M.ε ω j ∂M.P = 0","l":53,"f":"Causalean/Discovery/InvariantPrediction/LinearGaussian/Helpers/Moments.lean"},{"n":"Causalean.Discovery.InvariantPrediction.LinearGaussian.mem_predictors","k":"theorem","m":"Causalean.Discovery.InvariantPrediction.LinearGaussian.Model","a":"Discovery/InvariantPrediction/LinearGaussian","t":2,"d":"A node is a predictor exactly when it is not the target node.","s":"∀ {p : ℕ} {k : Fin (p + 1)}, k ∈ Causalean.Discovery.InvariantPrediction.LinearGaussian.predictors p ↔ k ≠ 0","l":71,"f":"Causalean/Discovery/InvariantPrediction/LinearGaussian/Model.lean"},{"n":"Causalean.Discovery.InvariantPrediction.LinearGaussian.nonDescendant_invariance","k":"theorem","m":"Causalean.Discovery.InvariantPrediction.LinearGaussian.Helpers.Invariance","a":"Discovery/InvariantPrediction/LinearGaussian/Helpers","t":1,"d":"Non-descendant invariance. Let e be an environment of the observational SEM M in which the single intervened coordinate is k₀. Then every coordinate k that is neither k₀ nor a descendant of k₀ in the observational DAG keeps its observational value almost surely: Xₖᵉ = Xₖ¹. (The node k₀ itself is pinned to the assigned constant, hence excluded.)","s":"∀ {p : ℕ} (M : Causalean.Discovery.InvariantPrediction.LinearGaussian.ObsSEM p) (e : Causalean.Discovery.InvariantPrediction.LinearGaussian.Env M) (k₀ : Fin (p + 1)), e.A = {k₀} → ∀ᵐ (ω : M.Ω) ∂M.P, ∀ (k : Fin (p + 1)), k ≠ k₀ → ¬M.dag.isAncestor k₀ k → e.X ω k = M.X ω k","l":36,"f":"Causalean/Discovery/InvariantPrediction/LinearGaussian/Helpers/Invariance.lean"},{"n":"Causalean.Discovery.InvariantPrediction.LinearGaussian.obsResidual","k":"def","m":"Causalean.Discovery.InvariantPrediction.LinearGaussian.Regression","a":"Discovery/InvariantPrediction/LinearGaussian","t":1,"d":"The regression residual in the observational SEM: R = Y − Σ_k γ k · X_k (with γ supported on S, so only k ∈ S matter).","s":"{p : ℕ} → (M : Causalean.Discovery.InvariantPrediction.LinearGaussian.ObsSEM p) → (Fin (p + 1) → ℝ) → M.Ω → ℝ","l":66,"f":"Causalean/Discovery/InvariantPrediction/LinearGaussian/Regression.lean"},{"n":"Causalean.Discovery.InvariantPrediction.LinearGaussian.obsResidual_eq_eps","k":"theorem","m":"Causalean.Discovery.InvariantPrediction.LinearGaussian.Helpers.Residual","a":"Discovery/InvariantPrediction/LinearGaussian/Helpers","t":1,"d":"Observational residual is the target noise. For an observational SEM, evaluated at the causal coefficient γ* = β₀,·, the observational residual Y − Σ_k β₀ₖ X_k equals the target's structural noise ε₀ almost everywhere.","s":"∀ {p : ℕ} (M : Causalean.Discovery.InvariantPrediction.LinearGaussian.ObsSEM p), ∀ᵐ (ω : M.Ω) ∂M.P, Causalean.Discovery.InvariantPrediction.LinearGaussian.obsResidual M (Causalean.Discovery.InvariantPrediction.LinearGaussian.causalCoeff M) ω = M.ε ω (Causalean.Discovery.InvariantPrediction.LinearGaussian.target p)","l":42,"f":"Causalean/Discovery/InvariantPrediction/LinearGaussian/Helpers/Residual.lean"},{"n":"Causalean.Discovery.InvariantPrediction.LinearGaussian.predictors","k":"def","m":"Causalean.Discovery.InvariantPrediction.LinearGaussian.Model","a":"Discovery/InvariantPrediction/LinearGaussian","t":1,"d":"The predictor index set {X₂,…,X_{p+1}}, encoded as {k : k ≠ 0}.","s":"(p : ℕ) → Finset (Fin (p + 1))","l":68,"f":"Causalean/Discovery/InvariantPrediction/LinearGaussian/Model.lean"},{"n":"Causalean.Discovery.InvariantPrediction.LinearGaussian.sum_causalCoeff_eq","k":"theorem","m":"Causalean.Discovery.InvariantPrediction.LinearGaussian.Helpers.Residual","a":"Discovery/InvariantPrediction/LinearGaussian/Helpers","t":2,"d":"With the causal coefficient, the full-sum Σ_k β₀ₖ X_k equals the structural-equation sum Σ_{k≠0} β₀ₖ X_k, since β₀₀ = 0.","s":"∀ {p : ℕ} (M : Causalean.Discovery.InvariantPrediction.LinearGaussian.ObsSEM p) (x : Fin (p + 1) → ℝ), ∑ k, M.β (Causalean.Discovery.InvariantPrediction.LinearGaussian.target p) k * x k = ∑ k ∈ Finset.univ.erase (Causalean.Discovery.InvariantPrediction.LinearGaussian.target p), M.β (Causalean.Discovery.InvariantPrediction.LinearGaussian.target p) k * x k","l":34,"f":"Causalean/Discovery/InvariantPrediction/LinearGaussian/Helpers/Residual.lean"},{"n":"Causalean.Discovery.InvariantPrediction.LinearGaussian.target","k":"def","m":"Causalean.Discovery.InvariantPrediction.LinearGaussian.Model","a":"Discovery/InvariantPrediction/LinearGaussian","t":1,"d":"The target node Y = X₁, encoded at index 0 of Fin (p+1).","s":"(p : ℕ) → Fin (p + 1)","l":65,"f":"Causalean/Discovery/InvariantPrediction/LinearGaussian/Model.lean"},{"n":"Causalean.Discovery.LiNGAM.colSupport_of_kurtosis","k":"theorem","m":"Causalean.Discovery.LiNGAM.Kurtosis","a":"Discovery/LiNGAM","t":1,"d":"Kurtosis-based column support. Let e be a family of real sources on a probability space such that each coordinate eⱼ is measurable, the coordinates are mutually independent, each has finite fourth moment, each is centered, and the fourth cumulant (excess kurtosis) of every coordinate is nonzero and of one common sign, all positive or all negative. For two distinct row indices i ≠ k of a mixing matrix W, if the linear forms Σⱼ Wᵢⱼ eⱼ and Σⱼ Wₖⱼ eⱼ are independent, then every column j satisfies Wᵢⱼ · Wₖⱼ = 0. This is the input required by genPerm_of_det_ne_zero_of_colSupport.","s":"∀ {Ω : Type u_1} [inst : MeasurableSpace Ω] {P : MeasureTheory.Measure Ω} [MeasureTheory.IsProbabilityMeasure P] {n : ℕ} {e : Ω → Fin n → ℝ} {W : Matrix (Fin n) (Fin n) ℝ}, (∀ (j : Fin n), Measurable fun ω => e ω j) → ProbabilityTheory.iIndepFun (fun j ω => e ω j) P → (∀ (j : Fin n), MeasureTheory.MemLp (fun ω => e ω j) 4 P) → (∀ (j : Fin n), ∫ (ω : Ω), e ω j ∂P = 0) → ((∀ (j : Fin n), 0 < Causalean.Discovery.LiNGAM.kurt (fun ω => e ω j) P) ∨ ∀ (j : Fin n), Causalean.Discovery.LiNGAM.kurt (fun ω => e ω j) P < 0) → ∀ {i k : Fin n}, i ≠ k → ProbabilityTheory.IndepFun (fun ω => ∑ j, W i j * e ω j) (fun ω => ∑ j, W k j * e ω j) P → ∀ (j : Fin n), W i j * W k j = 0","l":611,"f":"Causalean/Discovery/LiNGAM/Kurtosis.lean"},{"n":"Causalean.Discovery.LiNGAM.cross_fourth_cumulant_eq_sum","k":"theorem","m":"Causalean.Discovery.LiNGAM.Kurtosis","a":"Discovery/LiNGAM","t":1,"d":"Fourth cross-cumulant identity (Isserlis / cumulant multilinearity). Let e be a family of real sources on a probability space such that each coordinate eⱼ is measurable, the coordinates are mutually independent, each has finite fourth moment, and each is centered. Then for any coefficient vectors a, b and the linear forms yₐ = Σⱼ aⱼ eⱼ, y_b = Σⱼ bⱼ eⱼ, the joint fourth cumulant cum(yₐ,yₐ,y_b,y_b) equals Σⱼ aⱼ² bⱼ² κ₄(eⱼ).","s":"∀ {Ω : Type u_1} [inst : MeasurableSpace Ω] {P : MeasureTheory.Measure Ω} [MeasureTheory.IsProbabilityMeasure P] {n : ℕ} {e : Ω → Fin n → ℝ} (a b : Fin n → ℝ), (∀ (j : Fin n), Measurable fun ω => e ω j) → ProbabilityTheory.iIndepFun (fun j ω => e ω j) P → (∀ (j : Fin n), MeasureTheory.MemLp (fun ω => e ω j) 4 P) → (∀ (j : Fin n), ∫ (ω : Ω), e ω j ∂P = 0) → ∫ (ω : Ω), (∑ j, a j * e ω j) ^ 2 * (∑ j, b j * e ω j) ^ 2 ∂P - (∫ (ω : Ω), (∑ j, a j * e ω j) ^ 2 ∂P) * ∫ (ω : Ω), (∑ j, b j * e ω j) ^ 2 ∂P - 2 * (∫ (ω : Ω), (∑ j, a j * e ω j) * ∑ j, b j * e ω j ∂P) ^ 2 = ∑ j, a j ^ 2 * b j ^ 2 * Causalean.Discovery.LiNGAM.kurt (fun ω => e ω j) P","l":42,"f":"Causalean/Discovery/LiNGAM/Kurtosis.lean"},{"n":"Causalean.Discovery.LiNGAM.ica_genPerm_relation","k":"theorem","m":"Causalean.Discovery.LiNGAM.LiNGAMKurtosis","a":"Discovery/LiNGAM","t":1,"d":"Let A, A' be n × n real matrices such that A is invertible and A' is invertible. If the matrix A'⁻¹ A has at most one non-zero entry per column — the column-support fact — then there exist a permutation τ and a nowhere-zero scale vector d such that A'⁻¹ and A⁻¹ agree up to that generalized permutation: A'⁻¹ i j = d i · A⁻¹ (τ i) j for every i, j.","s":"∀ {n : ℕ} {A A' : Matrix (Fin n) (Fin n) ℝ}, IsUnit A.det → IsUnit A'.det → (∀ (j i k : Fin n), i ≠ k → (A'⁻¹ * A) i j = 0 ∨ (A'⁻¹ * A) k j = 0) → ∃ τ d, ∀ (i j : Fin n), A'⁻¹ i j = d i * A⁻¹ (τ i) j","l":35,"f":"Causalean/Discovery/LiNGAM/LiNGAMKurtosis.lean"},{"n":"Causalean.Discovery.LiNGAM.kurt","k":"def","m":"Causalean.Discovery.LiNGAM.Kurtosis","a":"Discovery/LiNGAM","t":1,"d":"Excess kurtosis, the fourth cumulant of a real random variable. For a centered variable X it is E[X⁴] − 3 · (E[X²])²; Gaussian variables have zero excess kurtosis, while LiNGAM's kurtosis route assumes every source has nonzero excess kurtosis of one common sign.","s":"{Ω : Type u_1} → [inst : MeasurableSpace Ω] → (Ω → ℝ) → MeasureTheory.Measure Ω → ℝ","l":35,"f":"Causalean/Discovery/LiNGAM/Kurtosis.lean"},{"n":"Causalean.Discovery.LiNGAM.lingam_identifiability_kurtosis","k":"theorem","m":"Causalean.Discovery.LiNGAM.LiNGAMKurtosis","a":"Discovery/LiNGAM","t":1,"d":"LiNGAM identification theorem (kurtosis route, Marcinkiewicz-free). Let A, A' be n × n real mixing matrices, both invertible, with coefficient matrices A⁻¹ = I − B, A'⁻¹ = I − B' both having unit diagonal, and with A⁻¹ acyclic with respect to a causal order σ: (A⁻¹) i j = 0 whenever σ i < σ j. Let e, e' be families of real disturbances on a probability space, each coordinate measurable, each family's coordinates mutually independent, each coordinate of e with finite fourth moment, each coordinate of e centered, and the fourth cumulant of every coordinate of e nonzero and of one common sign. If the structural equations A·e and A'·e' produce the same observed law, then the coefficient matrices coincide, A⁻¹ = A'⁻¹ (i.e. B = B').","s":"∀ {Ω : Type u_1} [inst : MeasurableSpace Ω] {P : MeasureTheory.Measure Ω} [MeasureTheory.IsProbabilityMeasure P] {n : ℕ} {A A' : Matrix (Fin n) (Fin n) ℝ}, IsUnit A.det → IsUnit A'.det → (∀ (i : Fin n), A⁻¹ i i = 1) → (∀ (i : Fin n), A'⁻¹ i i = 1) → ∀ {σ : Equiv.Perm (Fin n)}, (∀ (i j : Fin n), σ i < σ j → A⁻¹ i j = 0) → ∀ {e e' : Ω → Fin n → ℝ}, (∀ (i : Fin n), Measurable fun ω => e ω i) → (∀ (i : Fin n), Measurable fun ω => e' ω i) → ProbabilityTheory.iIndepFun (fun i ω => e ω i) P → ProbabilityTheory.iIndepFun (fun i ω => e' ω i) P → (∀ (i : Fin n), MeasureTheory.MemLp (fun ω => e ω i) 4 P) → (∀ (i : Fin n), ∫ (ω : Ω), e ω i ∂P = 0) → ((∀ (j : Fin n), 0 < Causalean.Discovery.LiNGAM.kurt (fun ω => e ω j) P) ∨ ∀ (j : Fin n), Causalean.Discovery.LiNGAM.kurt (fun ω => e ω j) P < 0) → MeasureTheory.Measure.map (fun ω => A.mulVec (e ω)) P = MeasureTheory.Measure.map (fun ω => A'.mulVec (e' ω)) P → A⁻¹ = A'⁻¹","l":73,"f":"Causalean/Discovery/LiNGAM/LiNGAMKurtosis.lean"},{"n":"Causalean.Discovery.LiNGAM.lingam_identifiable","k":"theorem","m":"Causalean.Discovery.LiNGAM.Identifiability","a":"Discovery/LiNGAM","t":1,"d":"LiNGAM coefficient identifiability (modulo ICA). Let B and B' be two n × n real coefficient matrices such that B has zero diagonal, B' has zero diagonal, B is acyclic with respect to a causal order σ: B i j = 0 whenever σ i < σ j, and the matrices I − B and I − B' are related by a generalized permutation — a permutation τ composed with a nonzero diagonal scale d, so (I − B') i j = d i · (I − B) (τ i) j (as delivered by ICA identifiability). Then B equals B'. This packages eq_of_genPerm_triangular_unitDiag for C = I − B.","s":"∀ {n : ℕ} {B B' : Matrix (Fin n) (Fin n) ℝ}, (∀ (i : Fin n), B i i = 0) → (∀ (i : Fin n), B' i i = 0) → ∀ {σ : Equiv.Perm (Fin n)}, (∀ (i j : Fin n), σ i < σ j → B i j = 0) → ∀ {τ : Equiv.Perm (Fin n)} {d : Fin n → ℝ}, (∀ (i j : Fin n), (1 - B') i j = d i * (1 - B) (τ i) j) → B = B'","l":36,"f":"Causalean/Discovery/LiNGAM/Identifiability.lean"},{"n":"Causalean.Discovery.LinearDisentanglement.B0H_gram_invertible","k":"instance","m":"Causalean.Discovery.LinearDisentanglement.Uniqueness","a":"Discovery/LinearDisentanglement","t":2,"d":"The Gram matrix (B₀H)(B₀H)ᵀ = B₀ (H Hᵀ) B₀ᵀ is invertible (product of invertibles).","s":"{d p K : ℕ} → (S : Causalean.Discovery.LinearDisentanglement.Solution d p K) → Invertible (S.B0 * S.H * (S.B0 * S.H).transpose)","l":132,"f":"Causalean/Discovery/LinearDisentanglement/Uniqueness.lean"},{"n":"Causalean.Discovery.LinearDisentanglement.B0_blockTriangular","k":"theorem","m":"Causalean.Discovery.LinearDisentanglement.Uniqueness","a":"Discovery/LinearDisentanglement","t":2,"d":"B0 is upper triangular in the BlockTriangular id sense.","s":"∀ {d p K : ℕ} (S : Causalean.Discovery.LinearDisentanglement.Solution d p K), S.B0.BlockTriangular id","l":63,"f":"Causalean/Discovery/LinearDisentanglement/Uniqueness.lean"},{"n":"Causalean.Discovery.LinearDisentanglement.B0_det_pos","k":"theorem","m":"Causalean.Discovery.LinearDisentanglement.Uniqueness","a":"Discovery/LinearDisentanglement","t":2,"d":"det B0 = ∏ᵢ (B0)ᵢᵢ > 0.","s":"∀ {d p K : ℕ} (S : Causalean.Discovery.LinearDisentanglement.Solution d p K), 0 < S.B0.det","l":82,"f":"Causalean/Discovery/LinearDisentanglement/Uniqueness.lean"},{"n":"Causalean.Discovery.LinearDisentanglement.B0_invertible","k":"instance","m":"Causalean.Discovery.LinearDisentanglement.Uniqueness","a":"Discovery/LinearDisentanglement","t":2,"d":"B0 is invertible.","s":"{d p K : ℕ} → (S : Causalean.Discovery.LinearDisentanglement.Solution d p K) → Invertible S.B0","l":105,"f":"Causalean/Discovery/LinearDisentanglement/Uniqueness.lean"},{"n":"Causalean.Discovery.LinearDisentanglement.B0_source_row","k":"theorem","m":"Causalean.Discovery.LinearDisentanglement.Uniqueness","a":"Discovery/LinearDisentanglement","t":2,"d":"(R3 ingredient) The iₖ-th row of B₀ at a source. If iₖ is a *source* node (no parents: (B₀)_{iₖ,j} = 0 for every j ≠ iₖ), then wₖ = B₀ᵀ *ᵥ e_{iₖ} = β e_{iₖ} with β = (B₀)_{iₖ,iₖ} > 0. This collapses the latent key identity (C) to the rank-one Δₖ = (λₖ² − β²) e_{iₖ} e_{iₖ}ᵀ, the base case of the monomial induction (R3): the transported equation then forces the i'ₖ-th row of M to be a multiple of e_{iₖ}.","s":"∀ {d p K : ℕ} (S : Causalean.Discovery.LinearDisentanglement.Solution d p K) (i : Fin d), (∀ (j : Fin d), j ≠ i → S.B0 i j = 0) → S.B0.transpose.mulVec (Causalean.Discovery.LinearDisentanglement.stdVec d i) = S.B0 i i • Causalean.Discovery.LinearDisentanglement.stdVec d i","l":404,"f":"Causalean/Discovery/LinearDisentanglement/Uniqueness.lean"},{"n":"Causalean.Discovery.LinearDisentanglement.Bint_blockTriangular","k":"theorem","m":"Causalean.Discovery.LinearDisentanglement.Uniqueness","a":"Discovery/LinearDisentanglement","t":2,"d":"Bint k is upper triangular: the perfect intervention only rewrites the target row (which keeps the diagonal at λₖ and zeroes the strictly-lower entries already zero in B0).","s":"∀ {d p K : ℕ} (S : Causalean.Discovery.LinearDisentanglement.Solution d p K) (k : Fin K), (S.Bint k).BlockTriangular id","l":66,"f":"Causalean/Discovery/LinearDisentanglement/Uniqueness.lean"},{"n":"Causalean.Discovery.LinearDisentanglement.Bint_det_pos","k":"theorem","m":"Causalean.Discovery.LinearDisentanglement.Uniqueness","a":"Discovery/LinearDisentanglement","t":2,"d":"det (Bint k) = ∏ᵢ (Bint k)ᵢᵢ > 0.","s":"∀ {d p K : ℕ} (S : Causalean.Discovery.LinearDisentanglement.Solution d p K) (k : Fin K), 0 < (S.Bint k).det","l":100,"f":"Causalean/Discovery/LinearDisentanglement/Uniqueness.lean"},{"n":"Causalean.Discovery.LinearDisentanglement.Bint_diag_pos","k":"theorem","m":"Causalean.Discovery.LinearDisentanglement.Uniqueness","a":"Discovery/LinearDisentanglement","t":2,"d":"(Bint k)ᵢᵢ = (B0)ᵢᵢ off the target, and = λₖ on the target — in both cases strictly positive.","s":"∀ {d p K : ℕ} (S : Causalean.Discovery.LinearDisentanglement.Solution d p K) (k : Fin K) (i : Fin d), 0 < S.Bint k i i","l":87,"f":"Causalean/Discovery/LinearDisentanglement/Uniqueness.lean"},{"n":"Causalean.Discovery.LinearDisentanglement.Bint_invertible","k":"instance","m":"Causalean.Discovery.LinearDisentanglement.Uniqueness","a":"Discovery/LinearDisentanglement","t":2,"d":"Bint k is invertible.","s":"{d p K : ℕ} → (S : Causalean.Discovery.LinearDisentanglement.Solution d p K) → (k : Fin K) → Invertible (S.Bint k)","l":109,"f":"Causalean/Discovery/LinearDisentanglement/Uniqueness.lean"},{"n":"Causalean.Discovery.LinearDisentanglement.Bint_sub_B0","k":"theorem","m":"Causalean.Discovery.LinearDisentanglement.Uniqueness","a":"Discovery/LinearDisentanglement","t":2,"d":"Rank-one perturbation. Bₖ − B₀ = e_{iₖ} cₖᵀ (Matrix.vecMulVec): a perfect single-node intervention rewrites only the target row of B₀.","s":"∀ {d p K : ℕ} (S : Causalean.Discovery.LinearDisentanglement.Solution d p K) (k : Fin K), S.Bint k - S.B0 = Matrix.vecMulVec (Causalean.Discovery.LinearDisentanglement.stdVec d (S.target k)) (Causalean.Discovery.LinearDisentanglement.cvec S k)","l":253,"f":"Causalean/Discovery/LinearDisentanglement/Uniqueness.lean"},{"n":"Causalean.Discovery.LinearDisentanglement.Bint_transpose_mulVec_target","k":"theorem","m":"Causalean.Discovery.LinearDisentanglement.Uniqueness","a":"Discovery/LinearDisentanglement","t":2,"d":"(B) Target row of Bₖ. (Bₖ)ᵀ *ᵥ e_{iₖ} = λₖ • e_{iₖ}: the iₖ-th row of a perfect-intervention matrix is λₖ e_{iₖ}ᵀ (the intervention zeroes the parent entries and sets the diagonal to λₖ).","s":"∀ {d p K : ℕ} (S : Causalean.Discovery.LinearDisentanglement.Solution d p K) (k : Fin K), (S.Bint k).transpose.mulVec (Causalean.Discovery.LinearDisentanglement.stdVec d (S.target k)) = S.lam k • Causalean.Discovery.LinearDisentanglement.stdVec d (S.target k)","l":272,"f":"Causalean/Discovery/LinearDisentanglement/Uniqueness.lean"},{"n":"Causalean.Discovery.LinearDisentanglement.HHt_invertible","k":"instance","m":"Causalean.Discovery.LinearDisentanglement.Uniqueness","a":"Discovery/LinearDisentanglement","t":2,"d":"H Hᵀ is invertible.","s":"{d p K : ℕ} → (S : Causalean.Discovery.LinearDisentanglement.Solution d p K) → Invertible (S.H * S.H.transpose)","l":126,"f":"Causalean/Discovery/LinearDisentanglement/Uniqueness.lean"},{"n":"Causalean.Discovery.LinearDisentanglement.HHt_posDef","k":"theorem","m":"Causalean.Discovery.LinearDisentanglement.Uniqueness","a":"Discovery/LinearDisentanglement","t":2,"d":"The latent Gram matrix H Hᵀ is positive definite.","s":"∀ {d p K : ℕ} (S : Causalean.Discovery.LinearDisentanglement.Solution d p K), (S.H * S.H.transpose).PosDef","l":121,"f":"Causalean/Discovery/LinearDisentanglement/Uniqueness.lean"},{"n":"Causalean.Discovery.LinearDisentanglement.H_eq_recover_mul_Theta0","k":"theorem","m":"Causalean.Discovery.LinearDisentanglement.Uniqueness","a":"Discovery/LinearDisentanglement","t":2,"d":"The recovery identity H = W Θ₀. With C = B₀ H and Θ₀ = Cᵀ C, the matrix W = B₀⁻¹ (C Cᵀ)⁻¹ C satisfies W Θ₀ = H.","s":"∀ {d p K : ℕ} (S : Causalean.Discovery.LinearDisentanglement.Solution d p K), S.B0⁻¹ * (S.B0 * S.H * (S.B0 * S.H).transpose)⁻¹ * (S.B0 * S.H) * S.Theta0 = S.H","l":157,"f":"Causalean/Discovery/LinearDisentanglement/Uniqueness.lean"},{"n":"Causalean.Discovery.LinearDisentanglement.H_mul_rightInv","k":"theorem","m":"Causalean.Discovery.LinearDisentanglement.Uniqueness","a":"Discovery/LinearDisentanglement","t":2,"d":"R := Hᵀ (H Hᵀ)⁻¹ is a right inverse of H: H R = 1.","s":"∀ {d p K : ℕ} (S : Causalean.Discovery.LinearDisentanglement.Solution d p K), S.H * (S.H.transpose * (S.H * S.H.transpose)⁻¹) = 1","l":139,"f":"Causalean/Discovery/LinearDisentanglement/Uniqueness.lean"},{"n":"Causalean.Discovery.LinearDisentanglement.IsPORQ","k":"structure","m":"Causalean.Discovery.LinearDisentanglement.PartialOrderRQ","a":"Discovery/LinearDisentanglement","t":1,"d":"Definition 1 (partial order RQ decomposition). A witness that a full-row-rank matrix H factors as H = R Q with R having a non-negative diagonal and support confined to the reflexive partial order — Rᵢⱼ = 0 unless i ⪯ j, and the rows of Q each of unit Euclidean norm and pairwise orthogonal along the strict order.","s":"{d p K : ℕ} → Causalean.Discovery.LinearDisentanglement.Solution d p K → Matrix (Fin d) (Fin d) ℝ → Matrix (Fin d) (Fin p) ℝ → Prop","l":137,"f":"Causalean/Discovery/LinearDisentanglement/PartialOrderRQ.lean"},{"n":"Causalean.Discovery.LinearDisentanglement.Solution","k":"structure","m":"Causalean.Discovery.LinearDisentanglement.Model","a":"Discovery/LinearDisentanglement","t":1,"d":"A linear causal disentanglement model with d latent variables, p observed variables, and K interventional contexts (Squires, Seigal, Bhate & Uhler 2023) bundles a full-row-rank mixing pseudoinverse from the observed to the latent space, a latent edge relation that respects the node order, and an observational structural matrix that is upper triangular, has a positive diagonal, and has off-diagonal support exactly equal to the edge set. For each interventional context it further carries a structural matrix, an intervention target node, and a positive perfect-intervention scaling, tied to the observational matrix by the perfect-single-node-intervention formula.","s":"ℕ → ℕ → ℕ → Type","l":63,"f":"Causalean/Discovery/LinearDisentanglement/Model.lean"},{"n":"Causalean.Discovery.LinearDisentanglement.Solution.An","k":"def","m":"Causalean.Discovery.LinearDisentanglement.PartialOrderRQ","a":"Discovery/LinearDisentanglement","t":1,"d":"An(i) = an(i) ∪ {i}.","s":"{d p K : ℕ} → Causalean.Discovery.LinearDisentanglement.Solution d p K → Fin d → Set (Fin d)","l":45,"f":"Causalean/Discovery/LinearDisentanglement/PartialOrderRQ.lean"},{"n":"Causalean.Discovery.LinearDisentanglement.Solution.InSG","k":"def","m":"Causalean.Discovery.LinearDisentanglement.Model","a":"Discovery/LinearDisentanglement","t":1,"d":"The order-preserving relabelings S(𝒢): permutations σ with σ i < σ j for every edge j → i.","s":"{d p K : ℕ} → Causalean.Discovery.LinearDisentanglement.Solution d p K → Equiv.Perm (Fin d) → Prop","l":116,"f":"Causalean/Discovery/LinearDisentanglement/Model.lean"},{"n":"Causalean.Discovery.LinearDisentanglement.Solution.Pa","k":"def","m":"Causalean.Discovery.LinearDisentanglement.PartialOrderRQ","a":"Discovery/LinearDisentanglement","t":1,"d":"Pa(i) = pa(i) ∪ {i}.","s":"{d p K : ℕ} → Causalean.Discovery.LinearDisentanglement.Solution d p K → Fin d → Set (Fin d)","l":38,"f":"Causalean/Discovery/LinearDisentanglement/PartialOrderRQ.lean"},{"n":"Causalean.Discovery.LinearDisentanglement.Solution.Theta","k":"def","m":"Causalean.Discovery.LinearDisentanglement.Model","a":"Discovery/LinearDisentanglement","t":1,"d":"The interventional precision matrix Θ_k = Hᵀ Bₖᵀ Bₖ H ∈ ℝ^{p×p}.","s":"{d p K : ℕ} → Causalean.Discovery.LinearDisentanglement.Solution d p K → Fin K → Matrix (Fin p) (Fin p) ℝ","l":112,"f":"Causalean/Discovery/LinearDisentanglement/Model.lean"},{"n":"Causalean.Discovery.LinearDisentanglement.Solution.Theta0","k":"def","m":"Causalean.Discovery.LinearDisentanglement.Model","a":"Discovery/LinearDisentanglement","t":1,"d":"The observational precision matrix Θ₀ = Hᵀ B₀ᵀ B₀ H ∈ ℝ^{p×p}.","s":"{d p K : ℕ} → Causalean.Discovery.LinearDisentanglement.Solution d p K → Matrix (Fin p) (Fin p) ℝ","l":108,"f":"Causalean/Discovery/LinearDisentanglement/Model.lean"},{"n":"Causalean.Discovery.LinearDisentanglement.Solution.anc","k":"def","m":"Causalean.Discovery.LinearDisentanglement.PartialOrderRQ","a":"Discovery/LinearDisentanglement","t":1,"d":"Ancestors of i: nodes j with a directed path j ⇝ i (transitive closure of the edge relation).","s":"{d p K : ℕ} → Causalean.Discovery.LinearDisentanglement.Solution d p K → Fin d → Set (Fin d)","l":41,"f":"Causalean/Discovery/LinearDisentanglement/PartialOrderRQ.lean"},{"n":"Causalean.Discovery.LinearDisentanglement.Solution.not_prec_self","k":"theorem","m":"Causalean.Discovery.LinearDisentanglement.PartialOrderRQ","a":"Discovery/LinearDisentanglement","t":2,"d":"≺ is irreflexive: a node is never its own strict ancestor.","s":"∀ {d p K : ℕ} (S : Causalean.Discovery.LinearDisentanglement.Solution d p K) (i : Fin d), ¬S.prec i i","l":63,"f":"Causalean/Discovery/LinearDisentanglement/PartialOrderRQ.lean"},{"n":"Causalean.Discovery.LinearDisentanglement.Solution.pa","k":"def","m":"Causalean.Discovery.LinearDisentanglement.PartialOrderRQ","a":"Discovery/LinearDisentanglement","t":1,"d":"Parents of i: nodes j with an edge j → i.","s":"{d p K : ℕ} → Causalean.Discovery.LinearDisentanglement.Solution d p K → Fin d → Set (Fin d)","l":35,"f":"Causalean/Discovery/LinearDisentanglement/PartialOrderRQ.lean"},{"n":"Causalean.Discovery.LinearDisentanglement.Solution.permute","k":"def","m":"Causalean.Discovery.LinearDisentanglement.SigmaSolutions","a":"Discovery/LinearDisentanglement","t":1,"d":"Relabeling a solution by an order-preserving permutation gives another valid solution.","s":"{d p K : ℕ} → (S : Causalean.Discovery.LinearDisentanglement.Solution d p K) → (σ : Equiv.Perm (Fin d)) → S.InSG σ → Causalean.Discovery.LinearDisentanglement.Solution d p K","l":84,"f":"Causalean/Discovery/LinearDisentanglement/SigmaSolutions.lean"},{"n":"Causalean.Discovery.LinearDisentanglement.Solution.prec","k":"def","m":"Causalean.Discovery.LinearDisentanglement.PartialOrderRQ","a":"Discovery/LinearDisentanglement","t":1,"d":"The partial order ≺_𝒢: i ≺ j iff j is an ancestor of i.","s":"{d p K : ℕ} → Causalean.Discovery.LinearDisentanglement.Solution d p K → Fin d → Fin d → Prop","l":48,"f":"Causalean/Discovery/LinearDisentanglement/PartialOrderRQ.lean"},{"n":"Causalean.Discovery.LinearDisentanglement.Solution.prec_lt","k":"theorem","m":"Causalean.Discovery.LinearDisentanglement.PartialOrderRQ","a":"Discovery/LinearDisentanglement","t":2,"d":"The strict partial order embeds into the Fin d linear order: if i ≺ j (j is a strict ancestor of i) then i < j. This is the acyclicity of 𝒢 lifted through the transitive closure, and the engine behind every \"process the nodes in a topological order\" argument below.","s":"∀ {d p K : ℕ} (S : Causalean.Discovery.LinearDisentanglement.Solution d p K) {i j : Fin d}, S.prec i j → i < j","l":54,"f":"Causalean/Discovery/LinearDisentanglement/PartialOrderRQ.lean"},{"n":"Causalean.Discovery.LinearDisentanglement.Solution.preceq","k":"def","m":"Causalean.Discovery.LinearDisentanglement.PartialOrderRQ","a":"Discovery/LinearDisentanglement","t":1,"d":"The reflexive closure ⪯_𝒢 of ≺_𝒢.","s":"{d p K : ℕ} → Causalean.Discovery.LinearDisentanglement.Solution d p K → Fin d → Fin d → Prop","l":51,"f":"Causalean/Discovery/LinearDisentanglement/PartialOrderRQ.lean"},{"n":"Causalean.Discovery.LinearDisentanglement.cancel_Ht_H","k":"theorem","m":"Causalean.Discovery.LinearDisentanglement.Uniqueness","a":"Discovery/LinearDisentanglement","t":2,"d":"Left-cancel Hᵀ and right-cancel H in Hᵀ X H = Hᵀ Y H. H has full row rank, so Hᵀ has a left inverse and H a right inverse.","s":"∀ {d p K : ℕ} (S : Causalean.Discovery.LinearDisentanglement.Solution d p K) {X Y : Matrix (Fin d) (Fin d) ℝ}, S.H.transpose * X * S.H = S.H.transpose * Y * S.H → X = Y","l":209,"f":"Causalean/Discovery/LinearDisentanglement/Uniqueness.lean"},{"n":"Causalean.Discovery.LinearDisentanglement.central_rank2_eq","k":"theorem","m":"Causalean.Discovery.LinearDisentanglement.Uniqueness","a":"Discovery/LinearDisentanglement","t":2,"d":"(R1+C) The central rank-≤2 equation. Combining the Gram-difference transport Δₖ = Mᵀ Δ'ₖ M (gram_diff_transport) with the latent key identity (C) on both sides, Δₖ = λₖ²(eₖeₖᵀ) − wₖwₖᵀ and Δ'ₖ = λ'ₖ²(e'ₖe'ₖᵀ) − w'ₖw'ₖᵀ, gives the rank-≤2 identity in fully outer-product form, with the primed outer products transported by Mᵀ: λₖ²(eₖeₖᵀ) − wₖwₖᵀ = λ'ₖ²((Mᵀe'ₖ)(Mᵀe'ₖ)ᵀ) − (Mᵀw'ₖ)(Mᵀw'ₖ)ᵀ.","s":"∀ {d p K : ℕ} (S S' : Causalean.Discovery.LinearDisentanglement.Solution d p K) {M : Matrix (Fin d) (Fin d) ℝ}, S'.H = M * S.H → S.Theta0 = S'.Theta0 → (∀ (k : Fin K), S.Theta k = S'.Theta k) → ∀ (k : Fin K), S.lam k ^ 2 • Matrix.vecMulVec (Causalean.Discovery.LinearDisentanglement.stdVec d (S.target k)) (Causalean.Discovery.LinearDisentanglement.stdVec d (S.target k)) - Matrix.vecMulVec (S.B0.transpose.mulVec (Causalean.Discovery.LinearDisentanglement.stdVec d (S.target k))) (S.B0.transpose.mulVec (Causalean.Discovery.LinearDisentanglement.stdVec d (S.target k))) = S'.lam k ^ 2 • Matrix.vecMulVec (M.transpose.mulVec (Causalean.Discovery.LinearDisentanglement.stdVec d (S'.target k))) (M.transpose.mulVec (Causalean.Discovery.LinearDisentanglement.stdVec d (S'.target k))) - Matrix.vecMulVec (M.transpose.mulVec (S'.B0.transpose.mulVec (Causalean.Discovery.LinearDisentanglement.stdVec d (S'.target k)))) (M.transpose.mulVec (S'.B0.transpose.mulVec (Causalean.Discovery.LinearDisentanglement.stdVec d (S'.target k))))","l":376,"f":"Causalean/Discovery/LinearDisentanglement/Uniqueness.lean"},{"n":"Causalean.Discovery.LinearDisentanglement.conj_vecMulVec_transpose","k":"theorem","m":"Causalean.Discovery.LinearDisentanglement.Uniqueness","a":"Discovery/LinearDisentanglement","t":2,"d":"Mᵀ (u vᵀ) M = (Mᵀ u)(Mᵀ v)ᵀ (conjugation of a rank-one matrix by Mᵀ · M).","s":"∀ {q : ℕ} (M : Matrix (Fin q) (Fin q) ℝ) (u v : Fin q → ℝ), M.transpose * Matrix.vecMulVec u v * M = Matrix.vecMulVec (M.transpose.mulVec u) (M.transpose.mulVec v)","l":370,"f":"Causalean/Discovery/LinearDisentanglement/Uniqueness.lean"},{"n":"Causalean.Discovery.LinearDisentanglement.cvec","k":"def","m":"Causalean.Discovery.LinearDisentanglement.Uniqueness","a":"Discovery/LinearDisentanglement","t":1,"d":"The perturbation row cₖ : Fin d → ℝ of a perfect intervention: cₖ j = λₖ (eᵢₖ)ⱼ − (B₀)_{iₖ,j}, so that Bₖ = B₀ + e_{iₖ} cₖᵀ.","s":"{d p K : ℕ} → Causalean.Discovery.LinearDisentanglement.Solution d p K → Fin K → Fin d → ℝ","l":248,"f":"Causalean/Discovery/LinearDisentanglement/Uniqueness.lean"},{"n":"Causalean.Discovery.LinearDisentanglement.cvec_ne_zero","k":"theorem","m":"Causalean.Discovery.LinearDisentanglement.Uniqueness","a":"Discovery/LinearDisentanglement","t":2,"d":"(C, target-row form / sub-lemma (B)). The perturbation row is non-degenerate exactly when the intervention changes the precision matrix. Under Θₖ ≠ Θ₀, cₖ ≠ 0 (equivalently Bₖ ≠ B₀): if cₖ = 0 then Bₖ = B₀, hence Θₖ = Θ₀.","s":"∀ {d p K : ℕ} (S : Causalean.Discovery.LinearDisentanglement.Solution d p K) (k : Fin K), S.Theta k ≠ S.Theta0 → Causalean.Discovery.LinearDisentanglement.cvec S k ≠ 0","l":259,"f":"Causalean/Discovery/LinearDisentanglement/Uniqueness.lean"},{"n":"Causalean.Discovery.LinearDisentanglement.diag_permMat_mulVec_stdVec","k":"theorem","m":"Causalean.Discovery.LinearDisentanglement.Uniqueness","a":"Discovery/LinearDisentanglement","t":2,"d":"A signed permutation matrix sends eᵢ to the signed basis vector ν (σ i) • e_{σ i}.","s":"∀ {d : ℕ} (σ : Equiv.Perm (Fin d)) (ν : Fin d → ℝ) (i : Fin d), (Matrix.diagonal ν * Causalean.Discovery.LinearDisentanglement.permMat σ).mulVec (Causalean.Discovery.LinearDisentanglement.stdVec d i) = ν (σ i) • Causalean.Discovery.LinearDisentanglement.stdVec d (σ i)","l":438,"f":"Causalean/Discovery/LinearDisentanglement/Uniqueness.lean"},{"n":"Causalean.Discovery.LinearDisentanglement.disentanglement_identifiability","k":"theorem","m":"Causalean.Discovery.LinearDisentanglement.Identifiability","a":"Discovery/LinearDisentanglement","t":1,"d":"Linear causal disentanglement identifiability (Theorem 2). Let S and S' be two solutions of the linear causal disentanglement model such that each solution's intervention-target map is a bijection onto the latent coordinates, i.e. one intervention per latent node and S's interventions are non-degenerate (the paper's genericity / Assumption 1(b)): every intervened precision matrix Θ_k differs from the observational precision matrix Θ_0. If S and S' share the same observational precision matrix and agree, context by context, on every interventional precision matrix, then S and S' are related by a single order-preserving relabeling σ of the latent coordinates, a nonzero scaling vector μ, and a ±1 sign vector ν: σ, μ, ν transport S's latent-direction matrix and structural coefficient matrices onto S''s, and σ carries S's intervention targets onto S''s.","s":"∀ {d p K : ℕ} (S S' : Causalean.Discovery.LinearDisentanglement.Solution d p K), Function.Bijective S.target → Function.Bijective S'.target → (∀ (k : Fin K), S.Theta k ≠ S.Theta0) → S.Theta0 = S'.Theta0 → (∀ (k : Fin K), S.Theta k = S'.Theta k) → ∃ σ μ ν, S.InSG σ ∧ (∀ (i : Fin d), μ i ≠ 0) ∧ (∀ (i : Fin d), ν i = 1 ∨ ν i = -1) ∧ S'.H = Matrix.diagonal μ * Causalean.Discovery.LinearDisentanglement.permMat σ * S.H ∧ S'.B0 * (Matrix.diagonal μ * Causalean.Discovery.LinearDisentanglement.permMat σ) = Matrix.diagonal ν * Causalean.Discovery.LinearDisentanglement.permMat σ * S.B0 ∧ (∀ (k : Fin K), S'.Bint k * (Matrix.diagonal μ * Causalean.Discovery.LinearDisentanglement.permMat σ) = Matrix.diagonal ν * Causalean.Discovery.LinearDisentanglement.permMat σ * S.Bint k) ∧ ∀ (k : Fin K), S'.target k = σ (S.target k)","l":35,"f":"Causalean/Discovery/LinearDisentanglement/Identifiability.lean"},{"n":"Causalean.Discovery.LinearDisentanglement.disentanglement_uniqueness","k":"theorem","m":"Causalean.Discovery.LinearDisentanglement.Uniqueness","a":"Discovery/LinearDisentanglement","t":1,"d":"Uniqueness (⊆ direction of Theorem 2). Let S and S' be two solutions of the linear causal disentanglement model such that each solution's intervention-target map is a bijection onto the latent coordinates, i.e. one intervention per latent node and S's interventions are non-degenerate (the paper's genericity / Assumption 1(b)): every intervened precision matrix Θ_k differs from the observational precision matrix Θ_0. If S and S' share the same observational precision matrix and agree, context by context, on every interventional precision matrix, then S and S' are related by a single order-preserving relabeling σ of the latent coordinates, a nonzero scaling vector μ, and a ±1 sign vector ν, transporting S's latent-direction and structural coefficient matrices onto S''s, with σ carrying S's intervention targets onto S''s.","s":"∀ {d p K : ℕ} (S S' : Causalean.Discovery.LinearDisentanglement.Solution d p K), Function.Bijective S.target → Function.Bijective S'.target → (∀ (k : Fin K), S.Theta k ≠ S.Theta0) → S.Theta0 = S'.Theta0 → (∀ (k : Fin K), S.Theta k = S'.Theta k) → ∃ σ μ ν, S.InSG σ ∧ (∀ (i : Fin d), μ i ≠ 0) ∧ (∀ (i : Fin d), ν i = 1 ∨ ν i = -1) ∧ S'.H = Matrix.diagonal μ * Causalean.Discovery.LinearDisentanglement.permMat σ * S.H ∧ S'.B0 * (Matrix.diagonal μ * Causalean.Discovery.LinearDisentanglement.permMat σ) = Matrix.diagonal ν * Causalean.Discovery.LinearDisentanglement.permMat σ * S.B0 ∧ (∀ (k : Fin K), S'.Bint k * (Matrix.diagonal μ * Causalean.Discovery.LinearDisentanglement.permMat σ) = Matrix.diagonal ν * Causalean.Discovery.LinearDisentanglement.permMat σ * S.Bint k) ∧ ∀ (k : Fin K), S'.target k = σ (S.target k)","l":1466,"f":"Causalean/Discovery/LinearDisentanglement/Uniqueness.lean"},{"n":"Causalean.Discovery.LinearDisentanglement.dotProduct_self_pos","k":"theorem","m":"Causalean.Discovery.LinearDisentanglement.PartialOrderRQ","a":"Discovery/LinearDisentanglement","t":2,"d":"The self dotProduct of a nonzero real vector is strictly positive (positive-definiteness of ⬝ᵥ). Equivalently, x ⬝ᵥ x = 0 ↔ x = 0; this is what makes \"normalize the projection residual\" well defined in the construction.","s":"∀ {p : ℕ} {x : Fin p → ℝ}, x ≠ 0 → 0 < x ⬝ᵥ x","l":88,"f":"Causalean/Discovery/LinearDisentanglement/PartialOrderRQ.lean"},{"n":"Causalean.Discovery.LinearDisentanglement.eq_of_mul_eq_mul_row_indep","k":"theorem","m":"Causalean.Discovery.LinearDisentanglement.PartialOrderRQ","a":"Discovery/LinearDisentanglement","t":2,"d":"Right cancellation by a full-row-rank Q. If the rows of Q are linearly independent and R Q = R' Q, then R = R'. This pins the R factor once Q is fixed, and is the algebraic half of the uniqueness proof.","s":"∀ {d p : ℕ} {R R' : Matrix (Fin d) (Fin d) ℝ} {Q : Matrix (Fin d) (Fin p) ℝ}, LinearIndependent ℝ Q.row → R * Q = R' * Q → R = R'","l":101,"f":"Causalean/Discovery/LinearDisentanglement/PartialOrderRQ.lean"},{"n":"Causalean.Discovery.LinearDisentanglement.exists_change_of_basis","k":"theorem","m":"Causalean.Discovery.LinearDisentanglement.Uniqueness","a":"Discovery/LinearDisentanglement","t":2,"d":"(L1) The change-of-basis matrix. There is an invertible M with H' = M H. M is built from the recovery identity for H' (H' = W' Θ₀') by substituting Θ₀' = Θ₀ = Hᵀ B₀ᵀ B₀ H, giving H' = (W' Hᵀ B₀ᵀ B₀) H. Invertibility follows from the symmetric matrix N with H = N H' by right-cancelling the full-row-rank H/H'.","s":"∀ {d p K : ℕ} (S S' : Causalean.Discovery.LinearDisentanglement.Solution d p K), S.Theta0 = S'.Theta0 → ∃ M, IsUnit M ∧ S'.H = M * S.H","l":169,"f":"Causalean/Discovery/LinearDisentanglement/Uniqueness.lean"},{"n":"Causalean.Discovery.LinearDisentanglement.exists_orderPerm","k":"theorem","m":"Causalean.Discovery.LinearDisentanglement.Uniqueness","a":"Discovery/LinearDisentanglement","t":2,"d":"(L4) Orthogonal correctness (full statement).","s":"∀ {d p K : ℕ} (S S' : Causalean.Discovery.LinearDisentanglement.Solution d p K), Function.Bijective S.target → Function.Bijective S'.target → (∀ (k : Fin K), S.Theta k ≠ S.Theta0) → S.Theta0 = S'.Theta0 → (∀ (k : Fin K), S.Theta k = S'.Theta k) → ∀ {M : Matrix (Fin d) (Fin d) ℝ}, S'.H = M * S.H → ∃ σ μ ν, S.InSG σ ∧ (∀ (i : Fin d), μ i ≠ 0) ∧ (∀ (i : Fin d), ν i = 1 ∨ ν i = -1) ∧ M = Matrix.diagonal μ * Causalean.Discovery.LinearDisentanglement.permMat σ ∧ S'.B0 * M = Matrix.diagonal ν * Causalean.Discovery.LinearDisentanglement.permMat σ * S.B0 ∧ (∀ (k : Fin K), S'.Bint k * M = Matrix.diagonal ν * Causalean.Discovery.LinearDisentanglement.permMat σ * S.Bint k) ∧ ∀ (k : Fin K), S'.target k = σ (S.target k)","l":1437,"f":"Causalean/Discovery/LinearDisentanglement/Uniqueness.lean"},{"n":"Causalean.Discovery.LinearDisentanglement.fact_transpose_mul","k":"theorem","m":"Causalean.Discovery.LinearDisentanglement.KeyIdentity","a":"Discovery/LinearDisentanglement","t":2,"d":"Fact (rank-one decomposition). For any square B, BᵀB = Σᵢ (Bᵀeᵢ)⊗², where v⊗² = v vᵀ (Matrix.vecMulVec v v).","s":"∀ {d : ℕ} (B : Matrix (Fin d) (Fin d) ℝ), B.transpose * B = ∑ i, Matrix.vecMulVec (B.transpose.mulVec (Causalean.Discovery.LinearDisentanglement.stdVec d i)) (B.transpose.mulVec (Causalean.Discovery.LinearDisentanglement.stdVec d i))","l":33,"f":"Causalean/Discovery/LinearDisentanglement/KeyIdentity.lean"},{"n":"Causalean.Discovery.LinearDisentanglement.gram_diff_transport","k":"theorem","m":"Causalean.Discovery.LinearDisentanglement.Uniqueness","a":"Discovery/LinearDisentanglement","t":2,"d":"(R1) Latent Gram-difference transport. Bₖᵀ Bₖ − B₀ᵀ B₀ = Mᵀ (B'ₖᵀ B'ₖ − B'₀ᵀ B'₀) M. Subtracting the observational conjugation identity from the k-th one.","s":"∀ {d p K : ℕ} (S S' : Causalean.Discovery.LinearDisentanglement.Solution d p K) {M : Matrix (Fin d) (Fin d) ℝ}, S'.H = M * S.H → S.Theta0 = S'.Theta0 → (∀ (k : Fin K), S.Theta k = S'.Theta k) → ∀ (k : Fin K), (S.Bint k).transpose * S.Bint k - S.B0.transpose * S.B0 = M.transpose * ((S'.Bint k).transpose * S'.Bint k - S'.B0.transpose * S'.B0) * M","l":350,"f":"Causalean/Discovery/LinearDisentanglement/Uniqueness.lean"},{"n":"Causalean.Discovery.LinearDisentanglement.gram_identity","k":"theorem","m":"Causalean.Discovery.LinearDisentanglement.Uniqueness","a":"Discovery/LinearDisentanglement","t":2,"d":"(L2) The per-context Gram identity. Given H' = M H and Θₖ = Θₖ' for the observational (B₀/B'₀) and interventional (Bₖ/B'ₖ) matrices, the d × d Gram identity BᵀB = (B' M)ᵀ (B' M) holds.","s":"∀ {d p K : ℕ} (S S' : Causalean.Discovery.LinearDisentanglement.Solution d p K) {M : Matrix (Fin d) (Fin d) ℝ}, S'.H = M * S.H → ∀ {B B' : Matrix (Fin d) (Fin d) ℝ}, S.H.transpose * B.transpose * B * S.H = S'.H.transpose * B'.transpose * B' * S'.H → B.transpose * B = (B' * M).transpose * (B' * M)","l":225,"f":"Causalean/Discovery/LinearDisentanglement/Uniqueness.lean"},{"n":"Causalean.Discovery.LinearDisentanglement.gram_identity_conj","k":"theorem","m":"Causalean.Discovery.LinearDisentanglement.Uniqueness","a":"Discovery/LinearDisentanglement","t":2,"d":"The per-context Gram identity in the symmetric form BᵀB = Mᵀ B'ᵀ B' M (the conjugation form of gram_identity).","s":"∀ {d p K : ℕ} (S S' : Causalean.Discovery.LinearDisentanglement.Solution d p K) {M : Matrix (Fin d) (Fin d) ℝ}, S'.H = M * S.H → ∀ {B B' : Matrix (Fin d) (Fin d) ℝ}, S.H.transpose * B.transpose * B * S.H = S'.H.transpose * B'.transpose * B' * S'.H → B.transpose * B = M.transpose * (B'.transpose * B') * M","l":340,"f":"Causalean/Discovery/LinearDisentanglement/Uniqueness.lean"},{"n":"Causalean.Discovery.LinearDisentanglement.gram_to_orthogonal","k":"theorem","m":"Causalean.Discovery.LinearDisentanglement.Uniqueness","a":"Discovery/LinearDisentanglement","t":2,"d":"(L3) Gram ⟹ orthogonal. If XᵀX = YᵀY with X, Y invertible, then the transition matrix O = Y X⁻¹ is orthogonal: Oᵀ O = 1. This is the algebraic step that turns the equality of Gram matrices BᵀB = (B' M)ᵀ(B' M) into an orthogonality statement about O = B' M B⁻¹.","s":"∀ {q : ℕ} {X Y : Matrix (Fin q) (Fin q) ℝ} [Invertible X] [Invertible Y], X.transpose * X = Y.transpose * Y → (Y * X⁻¹).transpose * (Y * X⁻¹) = 1","l":47,"f":"Causalean/Discovery/LinearDisentanglement/Uniqueness.lean"},{"n":"Causalean.Discovery.LinearDisentanglement.hSpan","k":"def","m":"Causalean.Discovery.LinearDisentanglement.Rowspan","a":"Discovery/LinearDisentanglement","t":1,"d":"The span of the rows of H indexed by a set 𝓘 ⊆ [d].","s":"{d p K : ℕ} → Causalean.Discovery.LinearDisentanglement.Solution d p K → Set (Fin d) → Submodule ℝ (Fin p → ℝ)","l":40,"f":"Causalean/Discovery/LinearDisentanglement/Rowspan.lean"},{"n":"Causalean.Discovery.LinearDisentanglement.inner_toLp_eq_dotProduct","k":"theorem","m":"Causalean.Discovery.LinearDisentanglement.PartialOrderRQ","a":"Discovery/LinearDisentanglement","t":2,"d":"The EuclideanSpace ℝ (Fin p) inner product of toLp x and toLp y is exactly the dotProduct x ⬝ᵥ y. This is the bridge used to import all of Mathlib's inner-product-space machinery into the dotProduct world.","s":"∀ {p : ℕ} (x y : Fin p → ℝ), inner ℝ (WithLp.toLp 2 x) (WithLp.toLp 2 y) = x ⬝ᵥ y","l":76,"f":"Causalean/Discovery/LinearDisentanglement/PartialOrderRQ.lean"},{"n":"Causalean.Discovery.LinearDisentanglement.key_identity","k":"theorem","m":"Causalean.Discovery.LinearDisentanglement.KeyIdentity","a":"Discovery/LinearDisentanglement","t":1,"d":"Key identity (Proposition). For a linear causal disentanglement solution and an interventional context k with target iₖ, the difference of precision matrices Θ_k − Θ₀ equals exactly the difference between two outer products: one built from the target row of the interventional structural matrix, and one built from the target row of the observational structural matrix.","s":"∀ {d p K : ℕ} (S : Causalean.Discovery.LinearDisentanglement.Solution d p K) (k : Fin K), S.Theta k - S.Theta0 = Matrix.vecMulVec (S.H.transpose.mulVec ((S.Bint k).transpose.mulVec (Causalean.Discovery.LinearDisentanglement.stdVec d (S.target k)))) (S.H.transpose.mulVec ((S.Bint k).transpose.mulVec (Causalean.Discovery.LinearDisentanglement.stdVec d (S.target k)))) - Matrix.vecMulVec (S.H.transpose.mulVec (S.B0.transpose.mulVec (Causalean.Discovery.LinearDisentanglement.stdVec d (S.target k)))) (S.H.transpose.mulVec (S.B0.transpose.mulVec (Causalean.Discovery.LinearDisentanglement.stdVec d (S.target k))))","l":42,"f":"Causalean/Discovery/LinearDisentanglement/KeyIdentity.lean"},{"n":"Causalean.Discovery.LinearDisentanglement.latent_key_identity","k":"theorem","m":"Causalean.Discovery.LinearDisentanglement.Uniqueness","a":"Discovery/LinearDisentanglement","t":2,"d":"(C) Latent key identity. At the d × d (latent) level the difference of Gram matrices is a difference of two rank-one (outer-product) matrices: BₖᵀBₖ − B₀ᵀB₀ = λₖ² (e_{iₖ} e_{iₖ}ᵀ) − wₖ wₖᵀ where wₖ = B₀ᵀ *ᵥ e_{iₖ} is the iₖ-th row of B₀ (supported on Pa(iₖ)). This is the latent analogue of key_identity, and the structural engine of the orthogonal-correctness core.","s":"∀ {d p K : ℕ} (S : Causalean.Discovery.LinearDisentanglement.Solution d p K) (k : Fin K), (S.Bint k).transpose * S.Bint k - S.B0.transpose * S.B0 = S.lam k ^ 2 • Matrix.vecMulVec (Causalean.Discovery.LinearDisentanglement.stdVec d (S.target k)) (Causalean.Discovery.LinearDisentanglement.stdVec d (S.target k)) - Matrix.vecMulVec (S.B0.transpose.mulVec (Causalean.Discovery.LinearDisentanglement.stdVec d (S.target k))) (S.B0.transpose.mulVec (Causalean.Discovery.LinearDisentanglement.stdVec d (S.target k)))","l":288,"f":"Causalean/Discovery/LinearDisentanglement/Uniqueness.lean"},{"n":"Causalean.Discovery.LinearDisentanglement.leftInv_mul_Ht","k":"theorem","m":"Causalean.Discovery.LinearDisentanglement.Uniqueness","a":"Discovery/LinearDisentanglement","t":2,"d":"L := (H Hᵀ)⁻¹ H is a left inverse of Hᵀ: L Hᵀ = 1.","s":"∀ {d p K : ℕ} (S : Causalean.Discovery.LinearDisentanglement.Solution d p K), (S.H * S.H.transpose)⁻¹ * S.H * S.H.transpose = 1","l":144,"f":"Causalean/Discovery/LinearDisentanglement/Uniqueness.lean"},{"n":"Causalean.Discovery.LinearDisentanglement.orthogonal_of_diag_conj","k":"theorem","m":"Causalean.Discovery.LinearDisentanglement.Uniqueness","a":"Discovery/LinearDisentanglement","t":2,"d":"(S2) Diagonal conjugation ⟹ orthogonal rescaling. From Mᵀ diag(d') M = diag(d) with d, d' strictly positive, the matrix O = diag(fun i => √(d' i)) * M * diag(fun i => (√(d i))⁻¹) is orthogonal: Oᵀ O = 1.","s":"∀ {q : ℕ} {M : Matrix (Fin q) (Fin q) ℝ} {dv dv' : Fin q → ℝ}, (∀ (i : Fin q), 0 < dv i) → (∀ (i : Fin q), 0 < dv' i) → M.transpose * Matrix.diagonal dv' * M = Matrix.diagonal dv → ((Matrix.diagonal fun i => √(dv' i)) * M * Matrix.diagonal fun i => (√(dv i))⁻¹).transpose * ((Matrix.diagonal fun i => √(dv' i)) * M * Matrix.diagonal fun i => (√(dv i))⁻¹) = 1","l":651,"f":"Causalean/Discovery/LinearDisentanglement/Uniqueness.lean"},{"n":"Causalean.Discovery.LinearDisentanglement.permMat","k":"def","m":"Causalean.Discovery.LinearDisentanglement.Model","a":"Discovery/LinearDisentanglement","t":1,"d":"The permutation matrix of σ, with (permMat σ) i j = 1 if i = σ j and 0 otherwise — the convention (P_σ)_{ij} = ⟦i = σ(j)⟧ of the paper.","s":"{d : ℕ} → Equiv.Perm (Fin d) → Matrix (Fin d) (Fin d) ℝ","l":58,"f":"Causalean/Discovery/LinearDisentanglement/Model.lean"},{"n":"Causalean.Discovery.LinearDisentanglement.permMat_apply_symm","k":"theorem","m":"Causalean.Discovery.LinearDisentanglement.Uniqueness","a":"Discovery/LinearDisentanglement","t":2,"d":"(permMat σ) a c = 1 iff c = σ.symm a, else 0 (column read-off of permMat).","s":"∀ {d : ℕ} (σ : Equiv.Perm (Fin d)) (a c : Fin d), Causalean.Discovery.LinearDisentanglement.permMat σ a c = if c = (Equiv.symm σ) a then 1 else 0","l":741,"f":"Causalean/Discovery/LinearDisentanglement/Uniqueness.lean"},{"n":"Causalean.Discovery.LinearDisentanglement.permMat_conj_apply","k":"theorem","m":"Causalean.Discovery.LinearDisentanglement.Uniqueness","a":"Discovery/LinearDisentanglement","t":2,"d":"Permutation conjugation entry. (permMat σ * X * (permMat σ)ᵀ) a b = X (σ.symm a) (σ.symm b): conjugating X by permMat σ relabels rows and columns by σ.symm.","s":"∀ {d : ℕ} (σ : Equiv.Perm (Fin d)) (X : Matrix (Fin d) (Fin d) ℝ) (a b : Fin d), (Causalean.Discovery.LinearDisentanglement.permMat σ * X * (Causalean.Discovery.LinearDisentanglement.permMat σ).transpose) a b = X ((Equiv.symm σ) a) ((Equiv.symm σ) b)","l":750,"f":"Causalean/Discovery/LinearDisentanglement/Uniqueness.lean"},{"n":"Causalean.Discovery.LinearDisentanglement.permMat_mulVec_stdVec","k":"theorem","m":"Causalean.Discovery.LinearDisentanglement.Uniqueness","a":"Discovery/LinearDisentanglement","t":2,"d":"permMat σ *ᵥ eᵢ = e_{σ i}: the permutation matrix sends the i-th basis vector to the σ(i)-th.","s":"∀ {d : ℕ} (σ : Equiv.Perm (Fin d)) (i : Fin d), (Causalean.Discovery.LinearDisentanglement.permMat σ).mulVec (Causalean.Discovery.LinearDisentanglement.stdVec d i) = Causalean.Discovery.LinearDisentanglement.stdVec d (σ i)","l":430,"f":"Causalean/Discovery/LinearDisentanglement/Uniqueness.lean"},{"n":"Causalean.Discovery.LinearDisentanglement.permMat_mul_transpose","k":"theorem","m":"Causalean.Discovery.LinearDisentanglement.Model","a":"Discovery/LinearDisentanglement","t":1,"d":"For a dimension d and a permutation σ of Fin d, the permutation matrix permMat σ is orthogonal: its product with its own transpose is the identity matrix.","s":"∀ {d : ℕ} (σ : Equiv.Perm (Fin d)), Causalean.Discovery.LinearDisentanglement.permMat σ * (Causalean.Discovery.LinearDisentanglement.permMat σ).transpose = 1","l":125,"f":"Causalean/Discovery/LinearDisentanglement/Model.lean"},{"n":"Causalean.Discovery.LinearDisentanglement.permMat_transpose_mul","k":"theorem","m":"Causalean.Discovery.LinearDisentanglement.Model","a":"Discovery/LinearDisentanglement","t":2,"d":"permMat σ is orthogonal: (permMat σ)ᵀ * permMat σ = 1.","s":"∀ {d : ℕ} (σ : Equiv.Perm (Fin d)), (Causalean.Discovery.LinearDisentanglement.permMat σ).transpose * Causalean.Discovery.LinearDisentanglement.permMat σ = 1","l":143,"f":"Causalean/Discovery/LinearDisentanglement/Model.lean"},{"n":"Causalean.Discovery.LinearDisentanglement.porqAncSpan","k":"def","m":"Causalean.Discovery.LinearDisentanglement.PartialOrderRQ","a":"Discovery/LinearDisentanglement","t":1,"d":"The span Wᵢ = span {porqResidual k : i ≺ k} of the already-built strict-ancestor residuals, the subspace the i-th residual is orthogonalized against.","s":"{d p K : ℕ} → Causalean.Discovery.LinearDisentanglement.Solution d p K → Fin d → Submodule ℝ (EuclideanSpace ℝ (Fin p))","l":229,"f":"Causalean/Discovery/LinearDisentanglement/PartialOrderRQ.lean"},{"n":"Causalean.Discovery.LinearDisentanglement.porqResidual","k":"def","m":"Causalean.Discovery.LinearDisentanglement.PartialOrderRQ","a":"Discovery/LinearDisentanglement","t":1,"d":"The partial-order Gram–Schmidt residuals, working in EuclideanSpace ℝ (Fin p): porqResidual i = (toLp (hᵢ)) − proj_{Wᵢ} (toLp hᵢ) where Wᵢ = span {porqResidual k : i ≺ k} is the span of the already-built strict-ancestor residuals. The recursion terminates because every recursive call is to a strict ancestor k with i < k (Solution.prec_lt), so the measure d − i strictly decreases.","s":"{d p K : ℕ} → Causalean.Discovery.LinearDisentanglement.Solution d p K → Fin d → EuclideanSpace ℝ (Fin p)","l":213,"f":"Causalean/Discovery/LinearDisentanglement/PartialOrderRQ.lean"},{"n":"Causalean.Discovery.LinearDisentanglement.porqResidual_eq","k":"theorem","m":"Causalean.Discovery.LinearDisentanglement.PartialOrderRQ","a":"Discovery/LinearDisentanglement","t":2,"d":"Unfolding equation for porqResidual: it is toLp hᵢ minus its orthogonal projection onto the ancestor span porqAncSpan i.","s":"∀ {d p K : ℕ} (S : Causalean.Discovery.LinearDisentanglement.Solution d p K) (i : Fin d), Causalean.Discovery.LinearDisentanglement.porqResidual S i = WithLp.toLp 2 (S.H i) - (Causalean.Discovery.LinearDisentanglement.porqAncSpan S i).starProjection (WithLp.toLp 2 (S.H i))","l":236,"f":"Causalean/Discovery/LinearDisentanglement/PartialOrderRQ.lean"},{"n":"Causalean.Discovery.LinearDisentanglement.porqResidual_mem_HspanLE","k":"theorem","m":"Causalean.Discovery.LinearDisentanglement.PartialOrderRQ","a":"Discovery/LinearDisentanglement","t":2,"d":"The i-th residual lies in the span of {toLp hₘ : i ⪯ m} (the row itself and its strict ancestors). Proved by strong induction on the reversed Fin d order: the diagonal term is toLp hᵢ, and the projection lives in Wᵢ, whose generators porqResidual k (i ≺ k) lie, by induction, in span {toLp hₘ : k ⪯ m} ⊆ span {toLp hₘ : i ⪯ m} (by transitivity of ⪯).","s":"∀ {d p K : ℕ} (S : Causalean.Discovery.LinearDisentanglement.Solution d p K) (i : Fin d), Causalean.Discovery.LinearDisentanglement.porqResidual S i ∈ Submodule.span ℝ {x | ∃ m, S.preceq i m ∧ x = WithLp.toLp 2 (S.H m)}","l":277,"f":"Causalean/Discovery/LinearDisentanglement/PartialOrderRQ.lean"},{"n":"Causalean.Discovery.LinearDisentanglement.porqResidual_mem_ancSpan","k":"theorem","m":"Causalean.Discovery.LinearDisentanglement.PartialOrderRQ","a":"Discovery/LinearDisentanglement","t":2,"d":"Each strict-ancestor residual lies in the ancestor span porqAncSpan i.","s":"∀ {d p K : ℕ} (S : Causalean.Discovery.LinearDisentanglement.Solution d p K) {i k : Fin d}, S.prec i k → Causalean.Discovery.LinearDisentanglement.porqResidual S k ∈ Causalean.Discovery.LinearDisentanglement.porqAncSpan S i","l":243,"f":"Causalean/Discovery/LinearDisentanglement/PartialOrderRQ.lean"},{"n":"Causalean.Discovery.LinearDisentanglement.porqResidual_mem_orthogonal","k":"theorem","m":"Causalean.Discovery.LinearDisentanglement.PartialOrderRQ","a":"Discovery/LinearDisentanglement","t":2,"d":"The i-th residual is orthogonal to its ancestor span: porqResidual i ∈ (Wᵢ)ᗮ.","s":"∀ {d p K : ℕ} (S : Causalean.Discovery.LinearDisentanglement.Solution d p K) (i : Fin d), Causalean.Discovery.LinearDisentanglement.porqResidual S i ∈ (Causalean.Discovery.LinearDisentanglement.porqAncSpan S i)ᗮ","l":250,"f":"Causalean/Discovery/LinearDisentanglement/PartialOrderRQ.lean"},{"n":"Causalean.Discovery.LinearDisentanglement.porqResidual_ne_zero","k":"theorem","m":"Causalean.Discovery.LinearDisentanglement.PartialOrderRQ","a":"Discovery/LinearDisentanglement","t":2,"d":"Non-vanishing of the residual. porqResidual i ≠ 0: if it were zero then toLp hᵢ would lie in Wᵢ ⊆ span {toLp hₘ : i ≺ m}, contradicting the linear independence of {toLp hⱼ} (a member is never in the span of the strictly-others).","s":"∀ {d p K : ℕ} (S : Causalean.Discovery.LinearDisentanglement.Solution d p K) (i : Fin d), Causalean.Discovery.LinearDisentanglement.porqResidual S i ≠ 0","l":307,"f":"Causalean/Discovery/LinearDisentanglement/PartialOrderRQ.lean"},{"n":"Causalean.Discovery.LinearDisentanglement.porqResidual_orthogonal","k":"theorem","m":"Causalean.Discovery.LinearDisentanglement.PartialOrderRQ","a":"Discovery/LinearDisentanglement","t":2,"d":"Orthogonality along the order. The i-th residual is orthogonal to every strict-ancestor residual porqResidual k (i ≺ k), since the latter lies in Wᵢ and the former in (Wᵢ)ᗮ.","s":"∀ {d p K : ℕ} (S : Causalean.Discovery.LinearDisentanglement.Solution d p K) {i k : Fin d}, S.prec i k → inner ℝ (Causalean.Discovery.LinearDisentanglement.porqResidual S i) (Causalean.Discovery.LinearDisentanglement.porqResidual S k) = 0","l":256,"f":"Causalean/Discovery/LinearDisentanglement/PartialOrderRQ.lean"},{"n":"Causalean.Discovery.LinearDisentanglement.porq_diag_pos","k":"theorem","m":"Causalean.Discovery.LinearDisentanglement.PartialOrderRQ","a":"Discovery/LinearDisentanglement","t":2,"d":"The diagonal of any PORQ factor is strictly positive.","s":"∀ {d p K : ℕ} (S : Causalean.Discovery.LinearDisentanglement.Solution d p K) {R : Matrix (Fin d) (Fin d) ℝ} {Q : Matrix (Fin d) (Fin p) ℝ}, Causalean.Discovery.LinearDisentanglement.IsPORQ S R Q → ∀ (i : Fin d), 0 < R i i","l":155,"f":"Causalean/Discovery/LinearDisentanglement/PartialOrderRQ.lean"},{"n":"Causalean.Discovery.LinearDisentanglement.porq_exists","k":"theorem","m":"Causalean.Discovery.LinearDisentanglement.PartialOrderRQ","a":"Discovery/LinearDisentanglement","t":1,"d":"Existence of the partial order RQ decomposition (Appendix B Proposition). For a linear causal disentanglement solution, there exist matrices R and Q realizing a partial order RQ decomposition of S's factor loading matrix H — that is, H = R·Q with R's diagonal entries nonnegative and R's support confined to the partial order on nodes, and with Q's rows unit-norm and mutually orthogonal along that order.","s":"∀ {d p K : ℕ} (S : Causalean.Discovery.LinearDisentanglement.Solution d p K), ∃ R Q, Causalean.Discovery.LinearDisentanglement.IsPORQ S R Q","l":428,"f":"Causalean/Discovery/LinearDisentanglement/PartialOrderRQ.lean"},{"n":"Causalean.Discovery.LinearDisentanglement.porq_rowCoeffs","k":"theorem","m":"Causalean.Discovery.LinearDisentanglement.PartialOrderRQ","a":"Discovery/LinearDisentanglement","t":2,"d":"Row coefficients of the existence factorization. Writing qₘ = ‖rₘ‖⁻¹ • rₘ for the normalized residuals, the row toLp hᵢ decomposes as ‖rᵢ‖ • qᵢ + ∑_{i ≺ k} (coefficient) • qₖ: the diagonal coefficient is the residual norm ‖rᵢ‖ ≥ 0, the off-diagonal coefficients are supported on the strict ancestors i ≺ k, and there are none off ⪯. This packages factor, diag_nonneg and supp into one existence statement (the i-th row of R).","s":"∀ {d p K : ℕ} (S : Causalean.Discovery.LinearDisentanglement.Solution d p K) (i : Fin d), ∃ c, (∀ (j : Fin d), ¬S.preceq i j → c j = 0) ∧ c i = ‖Causalean.Discovery.LinearDisentanglement.porqResidual S i‖ ∧ WithLp.toLp 2 (S.H i) = ∑ m, c m • ‖Causalean.Discovery.LinearDisentanglement.porqResidual S m‖⁻¹ • Causalean.Discovery.LinearDisentanglement.porqResidual S m","l":345,"f":"Causalean/Discovery/LinearDisentanglement/PartialOrderRQ.lean"},{"n":"Causalean.Discovery.LinearDisentanglement.porq_row_split","k":"theorem","m":"Causalean.Discovery.LinearDisentanglement.PartialOrderRQ","a":"Discovery/LinearDisentanglement","t":2,"d":"Row-split of a PORQ factorization. Splitting off the diagonal term, the i-th row of H is Rᵢᵢ • qᵢ plus a combination of the *other* rows qₖ (k ≠ i); by supp only the strict ancestors i ≺ k contribute, so this is exactly the triangular equation hᵢ = Rᵢᵢ qᵢ + ∑_{i ≺ k} Rᵢₖ qₖ. It is the membership half of the orthogonal-decomposition uniqueness argument.","s":"∀ {d p K : ℕ} (S : Causalean.Discovery.LinearDisentanglement.Solution d p K) {R : Matrix (Fin d) (Fin d) ℝ} {Q : Matrix (Fin d) (Fin p) ℝ}, Causalean.Discovery.LinearDisentanglement.IsPORQ S R Q → ∀ (i : Fin d), S.H i = R i i • Q i + ∑ k ∈ Finset.univ.erase i, R i k • Q k","l":199,"f":"Causalean/Discovery/LinearDisentanglement/PartialOrderRQ.lean"},{"n":"Causalean.Discovery.LinearDisentanglement.porq_unique","k":"theorem","m":"Causalean.Discovery.LinearDisentanglement.PartialOrderRQ","a":"Discovery/LinearDisentanglement","t":1,"d":"Uniqueness of the partial order RQ decomposition (Appendix B Proposition). Given a solution S, if both (R,Q) and (R',Q') are partial order RQ decompositions of S's latent-direction matrix H — factoring H as a lower- triangular-along-the-order matrix R times a row-orthonormal-along-the-order matrix Q, then the two decompositions coincide: R = R' and Q = Q'.","s":"∀ {d p K : ℕ} (S : Causalean.Discovery.LinearDisentanglement.Solution d p K) {R R' : Matrix (Fin d) (Fin d) ℝ} {Q Q' : Matrix (Fin d) (Fin p) ℝ}, Causalean.Discovery.LinearDisentanglement.IsPORQ S R Q → Causalean.Discovery.LinearDisentanglement.IsPORQ S R' Q' → R = R' ∧ Q = Q'","l":497,"f":"Causalean/Discovery/LinearDisentanglement/PartialOrderRQ.lean"},{"n":"Causalean.Discovery.LinearDisentanglement.qSpan","k":"def","m":"Causalean.Discovery.LinearDisentanglement.Rowspan","a":"Discovery/LinearDisentanglement","t":1,"d":"The span of the rows of Q (from a partial order RQ decomposition) indexed by 𝓘.","s":"{d p : ℕ} → Matrix (Fin d) (Fin p) ℝ → Set (Fin d) → Submodule ℝ (Fin p → ℝ)","l":44,"f":"Causalean/Discovery/LinearDisentanglement/Rowspan.lean"},{"n":"Causalean.Discovery.LinearDisentanglement.rowSpan","k":"def","m":"Causalean.Discovery.LinearDisentanglement.Rowspan","a":"Discovery/LinearDisentanglement","t":1,"d":"The row span of a p × p matrix: the subspace of Fin p → ℝ spanned by its rows.","s":"{p : ℕ} → Matrix (Fin p) (Fin p) ℝ → Submodule ℝ (Fin p → ℝ)","l":36,"f":"Causalean/Discovery/LinearDisentanglement/Rowspan.lean"},{"n":"Causalean.Discovery.LinearDisentanglement.row_indep_of_mul_row_indep","k":"theorem","m":"Causalean.Discovery.LinearDisentanglement.PartialOrderRQ","a":"Discovery/LinearDisentanglement","t":2,"d":"A full-row-rank product forces a full-row-rank right factor. If the rows of R Q are linearly independent (d of them), then so are the rows of Q. The rows of R Q lie in the span of the rows of Q, which has at most d of them, so a dimension count forces the d rows of Q to be independent. Applied to H = R Q with H of full row rank, this shows the rows qᵢ of any PORQ factor are independent — the hypothesis eq_of_mul_eq_mul_row_indep needs.","s":"∀ {d p : ℕ} (R : Matrix (Fin d) (Fin d) ℝ) (Q : Matrix (Fin d) (Fin p) ℝ), LinearIndependent ℝ (R * Q).row → LinearIndependent ℝ Q.row","l":114,"f":"Causalean/Discovery/LinearDisentanglement/PartialOrderRQ.lean"},{"n":"Causalean.Discovery.LinearDisentanglement.rowspan_inclusion_a","k":"theorem","m":"Causalean.Discovery.LinearDisentanglement.Rowspan","a":"Discovery/LinearDisentanglement","t":1,"d":"Lemma 1(a). For a solution S, an intervention k, and a set of latent indices I, provided the intervention actually changes the observational precision matrix, Θₖ ≠ Θ₀ (the paper's genericity / Assumption (b)), the row span of the precision difference Θₖ − Θ₀ is contained in the span of the rows of the mixing pseudoinverse H indexed by I if and only if the parent set of the intervened target node is contained in I.","s":"∀ {d p K : ℕ} (S : Causalean.Discovery.LinearDisentanglement.Solution d p K) (k : Fin K) (I : Set (Fin d)), S.Theta k ≠ S.Theta0 → (Causalean.Discovery.LinearDisentanglement.rowSpan (S.Theta k - S.Theta0) ≤ Causalean.Discovery.LinearDisentanglement.hSpan S I ↔ S.Pa (S.target k) ⊆ I)","l":205,"f":"Causalean/Discovery/LinearDisentanglement/Rowspan.lean"},{"n":"Causalean.Discovery.LinearDisentanglement.rowspan_inclusion_b","k":"theorem","m":"Causalean.Discovery.LinearDisentanglement.Rowspan","a":"Discovery/LinearDisentanglement","t":1,"d":"Lemma 1(b). For a solution S, an intervention k, and any partial order RQ decomposition of S's latent-direction matrix H, witnessed by R and Q, the row span of the precision difference Θₖ − Θ₀ is contained in the span of the rows of Q indexed by the ancestor set of the intervened target node.","s":"∀ {d p K : ℕ} (S : Causalean.Discovery.LinearDisentanglement.Solution d p K) {R : Matrix (Fin d) (Fin d) ℝ} {Q : Matrix (Fin d) (Fin p) ℝ}, Causalean.Discovery.LinearDisentanglement.IsPORQ S R Q → ∀ (k : Fin K), Causalean.Discovery.LinearDisentanglement.rowSpan (S.Theta k - S.Theta0) ≤ Causalean.Discovery.LinearDisentanglement.qSpan Q (S.An (S.target k))","l":344,"f":"Causalean/Discovery/LinearDisentanglement/Rowspan.lean"},{"n":"Causalean.Discovery.LinearDisentanglement.rowspan_inclusion_c","k":"theorem","m":"Causalean.Discovery.LinearDisentanglement.Rowspan","a":"Discovery/LinearDisentanglement","t":1,"d":"Lemma 1(c). For a solution S, an intervention k, and a set of latent indices I, given a partial order RQ decomposition of S's latent-direction matrix H, witnessed by R and Q, the non-degeneracy condition Θₖ ≠ Θ₀ (the paper's genericity / Assumption (b)), that I is ancestor-closed, and that the parent set of the intervened target node is not contained in I, then the row span of the precision difference Θₖ − Θ₀ is not contained in the span of the rows of Q indexed by I.","s":"∀ {d p K : ℕ} (S : Causalean.Discovery.LinearDisentanglement.Solution d p K) {R : Matrix (Fin d) (Fin d) ℝ} {Q : Matrix (Fin d) (Fin p) ℝ}, Causalean.Discovery.LinearDisentanglement.IsPORQ S R Q → ∀ (k : Fin K) (I : Set (Fin d)), S.Theta k ≠ S.Theta0 → (∀ i ∈ I, S.An i ⊆ I) → ¬S.Pa (S.target k) ⊆ I → ¬Causalean.Discovery.LinearDisentanglement.rowSpan (S.Theta k - S.Theta0) ≤ Causalean.Discovery.LinearDisentanglement.qSpan Q I","l":522,"f":"Causalean/Discovery/LinearDisentanglement/Rowspan.lean"},{"n":"Causalean.Discovery.LinearDisentanglement.sigma_solutions","k":"theorem","m":"Causalean.Discovery.LinearDisentanglement.SigmaSolutions","a":"Discovery/LinearDisentanglement","t":1,"d":"For a solution S and a permutation σ of the latent coordinates such that σ is order-preserving, i.e. lies in the paper's group S(𝒢), the relabeled solution obtained by applying σ to S produces exactly the same observational precision matrix Θ₀, and for every intervention k the same interventional precision matrix Θₖ, as S itself.","s":"∀ {d p K : ℕ} (S : Causalean.Discovery.LinearDisentanglement.Solution d p K) (σ : Equiv.Perm (Fin d)) (hσ : S.InSG σ), (S.permute σ hσ).Theta0 = S.Theta0 ∧ ∀ (k : Fin K), (S.permute σ hσ).Theta k = S.Theta k","l":143,"f":"Causalean/Discovery/LinearDisentanglement/SigmaSolutions.lean"},{"n":"Causalean.Discovery.LinearDisentanglement.stdVec","k":"def","m":"Causalean.Discovery.LinearDisentanglement.Model","a":"Discovery/LinearDisentanglement","t":1,"d":"Standard basis (column) vector e_i : Fin d → ℝ, e_i j = ⟦j = i⟧.","s":"(d : ℕ) → Fin d → Fin d → ℝ","l":55,"f":"Causalean/Discovery/LinearDisentanglement/Model.lean"},{"n":"Causalean.Discovery.LinearDisentanglement.sum_latent_diff_unprimed","k":"theorem","m":"Causalean.Discovery.LinearDisentanglement.Uniqueness","a":"Discovery/LinearDisentanglement","t":2,"d":"(S1) The unprimed SUM TRICK. Let kof be a section of the surjective target map (S.target (kof n) = n). Summing the latent key identity over kof gives ∑ₙ (B_{kof n}ᵀ B_{kof n} − B₀ᵀB₀) = diagonal(d) − B₀ᵀB₀ with d n = (λ_{kof n})².","s":"∀ {d p K : ℕ} (S : Causalean.Discovery.LinearDisentanglement.Solution d p K) (kof : Fin d → Fin K), (∀ (n : Fin d), S.target (kof n) = n) → ∑ n, ((S.Bint (kof n)).transpose * S.Bint (kof n) - S.B0.transpose * S.B0) = (Matrix.diagonal fun n => S.lam (kof n) ^ 2) - S.B0.transpose * S.B0","l":520,"f":"Causalean/Discovery/LinearDisentanglement/Uniqueness.lean"},{"n":"Causalean.Discovery.LinearDisentanglement.sum_smul_vecMulVec_stdVec_eq_diagonal","k":"theorem","m":"Causalean.Discovery.LinearDisentanglement.Uniqueness","a":"Discovery/LinearDisentanglement","t":2,"d":"∑ n, (c n) • (e_n e_nᵀ) = diagonal c (the rank-one standard-basis outer products sum to a diagonal matrix).","s":"∀ {n : ℕ} (c : Fin n → ℝ), ∑ i, c i • Matrix.vecMulVec (Causalean.Discovery.LinearDisentanglement.stdVec n i) (Causalean.Discovery.LinearDisentanglement.stdVec n i) = Matrix.diagonal c","l":506,"f":"Causalean/Discovery/LinearDisentanglement/Uniqueness.lean"},{"n":"Causalean.Discovery.LinearDisentanglement.target_readoff","k":"theorem","m":"Causalean.Discovery.LinearDisentanglement.Uniqueness","a":"Discovery/LinearDisentanglement","t":2,"d":"Signed target read-off. Suppose the collapse holds for context k and the observational context: B'ₖ M = diagonal ν permMat σ Bₖ and B'₀ M = diagonal ν permMat σ B₀, with Θₖ ≠ Θ₀. Then the primed target equals the relabelled target: i'ₖ = σ(iₖ).","s":"∀ {d p K : ℕ} (S S' : Causalean.Discovery.LinearDisentanglement.Solution d p K) {M : Matrix (Fin d) (Fin d) ℝ} {σ : Equiv.Perm (Fin d)} {ν : Fin d → ℝ} (k : Fin K), S.Theta k ≠ S.Theta0 → (∀ (i : Fin d), ν i = 1 ∨ ν i = -1) → S'.Bint k * M = Matrix.diagonal ν * Causalean.Discovery.LinearDisentanglement.permMat σ * S.Bint k → S'.B0 * M = Matrix.diagonal ν * Causalean.Discovery.LinearDisentanglement.permMat σ * S.B0 → S'.target k = σ (S.target k)","l":462,"f":"Causalean/Discovery/LinearDisentanglement/Uniqueness.lean"},{"n":"Causalean.Discovery.LinearDisentanglement.toLp_H_linearIndependent","k":"theorem","m":"Causalean.Discovery.LinearDisentanglement.PartialOrderRQ","a":"Discovery/LinearDisentanglement","t":2,"d":"The images of the rows of H under toLp are linearly independent (full row rank of H transported through the linear isomorphism toLp).","s":"∀ {d p K : ℕ} (S : Causalean.Discovery.LinearDisentanglement.Solution d p K), LinearIndependent ℝ fun j => WithLp.toLp 2 (S.H j)","l":266,"f":"Causalean/Discovery/LinearDisentanglement/PartialOrderRQ.lean"},{"n":"Causalean.Discovery.LinearDisentanglement.transition_orthogonal","k":"theorem","m":"Causalean.Discovery.LinearDisentanglement.Uniqueness","a":"Discovery/LinearDisentanglement","t":2,"d":"(D) The transition factor is orthogonal. From the per-context Gram identity BₖᵀBₖ = (B'ₖ M)ᵀ(B'ₖ M) and invertibility, Oₖ = B'ₖ M Bₖ⁻¹ satisfies Oₖᵀ Oₖ = 1.","s":"∀ {d p K : ℕ} (S S' : Causalean.Discovery.LinearDisentanglement.Solution d p K) {M : Matrix (Fin d) (Fin d) ℝ}, S'.H = M * S.H → ∀ {B B' : Matrix (Fin d) (Fin d) ℝ} [Invertible B] [Invertible (B' * M)], S.H.transpose * B.transpose * B * S.H = S'.H.transpose * B'.transpose * B' * S'.H → (B' * M * B⁻¹).transpose * (B' * M * B⁻¹) = 1","l":323,"f":"Causalean/Discovery/LinearDisentanglement/Uniqueness.lean"},{"n":"Causalean.Discovery.LinearDisentanglement.transpose_mulVec_stdVec","k":"theorem","m":"Causalean.Discovery.LinearDisentanglement.Uniqueness","a":"Discovery/LinearDisentanglement","t":2,"d":"Mᵀ *ᵥ eᵢ is the i-th row of M, viewed as a vector: (Mᵀ *ᵥ eᵢ) j = Mᵢⱼ. This is the bridge that turns the transported outer products of central_rank2_eq into statements about the rows of M.","s":"∀ {n : ℕ} (M : Matrix (Fin n) (Fin n) ℝ) (i : Fin n), M.transpose.mulVec (Causalean.Discovery.LinearDisentanglement.stdVec n i) = fun j => M i j","l":396,"f":"Causalean/Discovery/LinearDisentanglement/Uniqueness.lean"},{"n":"Causalean.Discovery.LinearDisentanglement.vecMul_H_injective","k":"theorem","m":"Causalean.Discovery.LinearDisentanglement.Uniqueness","a":"Discovery/LinearDisentanglement","t":2,"d":"vecMul · H is injective (full row rank of H).","s":"∀ {d p K : ℕ} (S : Causalean.Discovery.LinearDisentanglement.Solution d p K), Function.Injective fun v => Matrix.vecMul v S.H","l":116,"f":"Causalean/Discovery/LinearDisentanglement/Uniqueness.lean"},{"n":"Causalean.EdgeType","k":"inductive","m":"Causalean.SCM.Model.EdgeType","a":"SCM/Model","t":1,"d":"An edge type records whether an edge is nonparametric, monotonic, linear, or parametric.","s":"Type","l":46,"f":"Causalean/SCM/Model/EdgeType.lean"},{"n":"Causalean.EdgeType.decRefines","k":"instance","m":"Causalean.SCM.Model.EdgeType","a":"SCM/Model","t":2,"d":"Refinement between two edge-type assumptions is decidable.","s":"(e₁ e₂ : Causalean.EdgeType) → Decidable (e₁.refines e₂)","l":90,"f":"Causalean/SCM/Model/EdgeType.lean"},{"n":"Causalean.EdgeType.refines","k":"def","m":"Causalean.SCM.Model.EdgeType","a":"SCM/Model","t":1,"d":"The refinement relation says that one edge-type assumption is at least as specific as another.","s":"Causalean.EdgeType → Causalean.EdgeType → Prop","l":85,"f":"Causalean/SCM/Model/EdgeType.lean"},{"n":"Causalean.EdgeType.refinesBool","k":"def","m":"Causalean.SCM.Model.EdgeType","a":"SCM/Model","t":1,"d":"The Boolean refinement check decides whether one edge-type assumption is at least as specific as another.","s":"Causalean.EdgeType → Causalean.EdgeType → Bool","l":70,"f":"Causalean/SCM/Model/EdgeType.lean"},{"n":"Causalean.EdgeType.refines_nonparametric","k":"theorem","m":"Causalean.SCM.Model.EdgeType","a":"SCM/Model","t":1,"d":"Every edge-type functional-form assumption e refines the nonparametric assumption: nonparametric is the weakest assumption in the refinement order, so every other assumption is at least as specific as it.","s":"∀ (e : Causalean.EdgeType), e.refines Causalean.EdgeType.nonparametric","l":101,"f":"Causalean/SCM/Model/EdgeType.lean"},{"n":"Causalean.EdgeType.refines_refl","k":"theorem","m":"Causalean.SCM.Model.EdgeType","a":"SCM/Model","t":2,"d":"Every edge-type assumption refines itself.","s":"∀ (e : Causalean.EdgeType), e.refines e","l":94,"f":"Causalean/SCM/Model/EdgeType.lean"},{"n":"Causalean.EdgeTypeAssignment","k":"structure","m":"Causalean.SCM.Model.EdgeType","a":"SCM/Model","t":1,"d":"An edge type assignment attaches a functional-assumption label to each directed edge of a graph.","s":"{V : Type u_1} → [inst : DecidableEq V] → [inst_1 : Fintype V] → Causalean.DAG V → Type u_1","l":111,"f":"Causalean/SCM/Model/EdgeType.lean"},{"n":"Causalean.EdgeTypeAssignment.allNonparametric","k":"def","m":"Causalean.SCM.Model.EdgeType","a":"SCM/Model","t":1,"d":"The default edge-type assignment labels every edge as nonparametric.","s":"{V : Type u_2} → [inst : DecidableEq V] → [inst_1 : Fintype V] → (G : Causalean.DAG V) → Causalean.EdgeTypeAssignment G","l":130,"f":"Causalean/SCM/Model/EdgeType.lean"},{"n":"Causalean.EdgeTypeAssignment.decIsFullyNonparametric","k":"instance","m":"Causalean.SCM.Model.EdgeType","a":"SCM/Model","t":2,"d":"Full nonparametricity of an edge-type assignment is decidable.","s":"{V : Type u_2} → [inst : DecidableEq V] → [inst_1 : Fintype V] → {G : Causalean.DAG V} → (a : Causalean.EdgeTypeAssignment G) → Decidable a.isFullyNonparametric","l":143,"f":"Causalean/SCM/Model/EdgeType.lean"},{"n":"Causalean.EdgeTypeAssignment.incomingTypes","k":"def","m":"Causalean.SCM.Model.EdgeType","a":"SCM/Model","t":1,"d":"The incoming edge-type set collects the labels on all edges pointing into a vertex.","s":"{V : Type u_2} → [inst : DecidableEq V] → [inst_1 : Fintype V] → {G : Causalean.DAG V} → Causalean.EdgeTypeAssignment G → V → Finset Causalean.EdgeType","l":134,"f":"Causalean/SCM/Model/EdgeType.lean"},{"n":"Causalean.EdgeTypeAssignment.isFullyNonparametric","k":"def","m":"Causalean.SCM.Model.EdgeType","a":"SCM/Model","t":1,"d":"A graph is fully nonparametric under an assignment when every directed edge is labeled nonparametric.","s":"{V : Type u_2} → [inst : DecidableEq V] → [inst_1 : Fintype V] → {G : Causalean.DAG V} → Causalean.EdgeTypeAssignment G → Prop","l":138,"f":"Causalean/SCM/Model/EdgeType.lean"},{"n":"Causalean.Estimation.ATE.BackdoorEstimationSystem","k":"structure","m":"Causalean.Estimation.ATE.Setup","a":"Estimation/ATE","t":1,"d":"A back-door estimation system extends a potential-outcome back-door system with value-space representatives of the nuisance parameters used by AIPW-style estimators: an outcome regression μ(a,x) and a propensity score e(x) that is bounded away from 0 and away from 1, together with the outcome regression's agreement, almost everywhere, with the σ(X)-measurable observable regression adjustedCE, composed with the factual covariate and the analogous agreement of the propensity score with the observable propensity propScore.","s":"(P : Causalean.PO.POSystem) → (γ : Type u_1) → [MeasurableSpace γ] → [StandardBorelSpace P.Ω] → [MeasureTheory.IsFiniteMeasure P.μ] → Type (max (max u_1 u_2) u_3)","l":51,"f":"Causalean/Estimation/ATE/Setup.lean"},{"n":"Causalean.Estimation.ATE.BackdoorEstimationSystem.H_ε","k":"def","m":"Causalean.Estimation.ATE.Score.AIPWMoment","a":"Estimation/ATE/Score","t":1,"d":"The legacy overlap-bounded realization set consists of nuisance vectors whose propensity component lies between ε and 1 - ε at every covariate value.","s":"{γ : Type u_1} → [inst : MeasurableSpace γ] → ℝ → Set (Causalean.Estimation.ATE.NuisanceVec γ)","l":258,"f":"Causalean/Estimation/ATE/Score/AIPWMoment.lean"},{"n":"Causalean.Estimation.ATE.BackdoorEstimationSystem.H_ε_aeL2","k":"def","m":"Causalean.Estimation.ATE.Score.AIPWMoment","a":"Estimation/ATE/Score","t":1,"d":"Source-shaped AIPW nuisance class: the propensity is overlap-bounded P_X-almost everywhere, both outcome regressions are square-integrable under the covariate law, and the propensity belongs to L∞(P_X).","s":"{P : Causalean.PO.POSystem} → {γ : Type u_1} → [inst : MeasurableSpace γ] → [inst_1 : StandardBorelSpace P.Ω] → [inst_2 : MeasureTheory.IsFiniteMeasure P.μ] → Causalean.Estimation.ATE.BackdoorEstimationSystem P γ → ℝ → Set (Causalean.Estimation.ATE.NuisanceVec γ)","l":266,"f":"Causalean/Estimation/ATE/Score/AIPWMoment.lean"},{"n":"Causalean.Estimation.ATE.BackdoorEstimationSystem.H_ε_aeL2_e_memLp_top","k":"theorem","m":"Causalean.Estimation.ATE.Score.AIPWMoment","a":"Estimation/ATE/Score","t":2,"d":"The propensity component of an H_ε_aeL2 nuisance is in L∞(P_X).","s":"∀ {P : Causalean.PO.POSystem} {γ : Type u_1} [inst : MeasurableSpace γ] [inst_1 : StandardBorelSpace P.Ω] [inst_2 : MeasureTheory.IsFiniteMeasure P.μ] (S : Causalean.Estimation.ATE.BackdoorEstimationSystem P γ) {ε : ℝ} {η : Causalean.Estimation.ATE.NuisanceVec γ}, η ∈ S.H_ε_aeL2 ε → MeasureTheory.MemLp η.e_fn ⊤ S.P_X","l":318,"f":"Causalean/Estimation/ATE/Score/AIPWMoment.lean"},{"n":"Causalean.Estimation.ATE.BackdoorEstimationSystem.H_ε_aeL2_mu_memLp","k":"theorem","m":"Causalean.Estimation.ATE.Score.AIPWMoment","a":"Estimation/ATE/Score","t":1,"d":"For an estimation system S, overlap level ε, and treatment arm d, if η is a member of the ε-overlap L² nuisance class H_ε_aeL2, then η's outcome-regression component at arm d lies in L²(P_X).","s":"∀ {P : Causalean.PO.POSystem} {γ : Type u_1} [inst : MeasurableSpace γ] [inst_1 : StandardBorelSpace P.Ω] [inst_2 : MeasureTheory.IsFiniteMeasure P.μ] (S : Causalean.Estimation.ATE.BackdoorEstimationSystem P γ) {ε : ℝ} {η : Causalean.Estimation.ATE.NuisanceVec γ}, η ∈ S.H_ε_aeL2 ε → ∀ (d : Bool), MeasureTheory.MemLp (η.μ_fn d) 2 S.P_X","l":309,"f":"Causalean/Estimation/ATE/Score/AIPWMoment.lean"},{"n":"Causalean.Estimation.ATE.BackdoorEstimationSystem.H_ε_aeL2_overlap_P_Z","k":"theorem","m":"Causalean.Estimation.ATE.Score.AIPWMoment","a":"Estimation/ATE/Score","t":2,"d":"Membership in H_ε_aeL2 transports its a.e. overlap condition from the covariate law to the observed-data law along the covariate projection.","s":"∀ {P : Causalean.PO.POSystem} {γ : Type u_1} [inst : MeasurableSpace γ] [inst_1 : StandardBorelSpace P.Ω] [inst_2 : MeasureTheory.IsFiniteMeasure P.μ] (S : Causalean.Estimation.ATE.BackdoorEstimationSystem P γ) {ε : ℝ} {η : Causalean.Estimation.ATE.NuisanceVec γ}, η ∈ S.H_ε_aeL2 ε → ∀ᵐ (z : γ × Bool × ℝ) ∂S.P_Z, ε ≤ η.e_fn (Causalean.Estimation.ATE.BackdoorEstimationSystem.projX z) ∧ η.e_fn (Causalean.Estimation.ATE.BackdoorEstimationSystem.projX z) ≤ 1 - ε","l":294,"f":"Causalean/Estimation/ATE/Score/AIPWMoment.lean"},{"n":"Causalean.Estimation.ATE.BackdoorEstimationSystem.H_ε_aeL2_overlap_factualX","k":"theorem","m":"Causalean.Estimation.ATE.Score.AIPWMoment","a":"Estimation/ATE/Score","t":2,"d":"Membership in H_ε_aeL2 transports its a.e. overlap condition from the covariate law to the original probability space along the observed covariate.","s":"∀ {P : Causalean.PO.POSystem} {γ : Type u_1} [inst : MeasurableSpace γ] [inst_1 : StandardBorelSpace P.Ω] [inst_2 : MeasureTheory.IsFiniteMeasure P.μ] (S : Causalean.Estimation.ATE.BackdoorEstimationSystem P γ) {ε : ℝ} {η : Causalean.Estimation.ATE.NuisanceVec γ}, η ∈ S.H_ε_aeL2 ε → ∀ᵐ (ω : P.Ω) ∂P.μ, ε ≤ η.e_fn (S.factualX ω) ∧ η.e_fn (S.factualX ω) ≤ 1 - ε","l":279,"f":"Causalean/Estimation/ATE/Score/AIPWMoment.lean"},{"n":"Causalean.Estimation.ATE.BackdoorEstimationSystem.K_AIPW","k":"def","m":"Causalean.Estimation.ATE.Score.AIPWScoreL2","a":"Estimation/ATE/Score","t":1,"d":"Pointwise Lipschitz constant for the AIPW moment in η on H_ε: K_AIPW ε := 1 + 2/ε + 2/ε². Tracks the quadratic blow-up of the inverse weights 1/ê, 1/(1−ê) and the cross terms (ê − e)/(ê·e).","s":"ℝ → ℝ","l":72,"f":"Causalean/Estimation/ATE/Score/AIPWScoreL2.lean"},{"n":"Causalean.Estimation.ATE.BackdoorEstimationSystem.P_X","k":"def","m":"Causalean.Estimation.ATE.Setup","a":"Estimation/ATE","t":1,"d":"Covariate marginal: P_X := μ.map factualX.","s":"{P : Causalean.PO.POSystem} → {γ : Type u_1} → [inst : MeasurableSpace γ] → [inst_1 : StandardBorelSpace P.Ω] → [inst_2 : MeasureTheory.IsFiniteMeasure P.μ] → Causalean.Estimation.ATE.BackdoorEstimationSystem P γ → MeasureTheory.Measure γ","l":121,"f":"Causalean/Estimation/ATE/Setup.lean"},{"n":"Causalean.Estimation.ATE.BackdoorEstimationSystem.P_Z","k":"def","m":"Causalean.Estimation.ATE.Setup","a":"Estimation/ATE","t":1,"d":"Joint data law P_Z := μ.map (X, A, Y).","s":"{P : Causalean.PO.POSystem} → {γ : Type u_1} → [inst : MeasurableSpace γ] → [inst_1 : StandardBorelSpace P.Ω] → [inst_2 : MeasureTheory.IsFiniteMeasure P.μ] → Causalean.Estimation.ATE.BackdoorEstimationSystem P γ → MeasureTheory.Measure (γ × Bool × ℝ)","l":139,"f":"Causalean/Estimation/ATE/Setup.lean"},{"n":"Causalean.Estimation.ATE.BackdoorEstimationSystem.P_Z_map_projX_eq_P_X","k":"theorem","m":"Causalean.Estimation.ATE.Setup","a":"Estimation/ATE","t":2,"d":"The covariate marginal P_X is the pushforward of P_Z along the projection (x, a, y) ↦ x. Used to bridge integrals/eLpNorm between P_X (covariates only) and P_Z (full data triple).","s":"∀ {P : Causalean.PO.POSystem} {γ : Type u_1} [inst : MeasurableSpace γ] [inst_1 : StandardBorelSpace P.Ω] [inst_2 : MeasureTheory.IsFiniteMeasure P.μ] (S : Causalean.Estimation.ATE.BackdoorEstimationSystem P γ), MeasureTheory.Measure.map (fun z => z.1) S.P_Z = S.P_X","l":144,"f":"Causalean/Estimation/ATE/Setup.lean"},{"n":"Causalean.Estimation.ATE.BackdoorEstimationSystem.StrictOverlap","k":"def","m":"Causalean.Estimation.ATE.Setup","a":"Estimation/ATE","t":1,"d":"Strict-overlap predicate ε ≤ propScore true ω ≤ 1 − ε a.s., with ε ∈ (0, 1/2]. Restated to the value-space propensity via e_compat.","s":"{P : Causalean.PO.POSystem} → {γ : Type u_1} → [inst : MeasurableSpace γ] → [inst_1 : StandardBorelSpace P.Ω] → [inst_2 : MeasureTheory.IsFiniteMeasure P.μ] → Causalean.Estimation.ATE.BackdoorEstimationSystem P γ → ℝ → Prop","l":112,"f":"Causalean/Estimation/ATE/Setup.lean"},{"n":"Causalean.Estimation.ATE.BackdoorEstimationSystem.Tfull","k":"def","m":"Causalean.Estimation.Efficiency.ATETangent","a":"Estimation/Efficiency","t":1,"d":"This is the mean-zero tangent space for the backdoor average treatment effect.","s":"{P : Causalean.PO.POSystem} → {γ : Type u_1} → [inst : MeasurableSpace γ] → [inst_1 : StandardBorelSpace P.Ω] → [inst_2 : MeasureTheory.IsFiniteMeasure P.μ] → (S : Causalean.Estimation.ATE.BackdoorEstimationSystem P γ) → Submodule ℝ ↥(MeasureTheory.Lp ℝ 2 S.P_Z)","l":455,"f":"Causalean/Estimation/Efficiency/ATETangent.lean"},{"n":"Causalean.Estimation.ATE.BackdoorEstimationSystem.YMuVal_residual_sq","k":"def","m":"Causalean.Estimation.ATE.Score.AIPWScoreL2","a":"Estimation/ATE/Score","t":1,"d":"The \"tilted\" cross-term integrand (|y − μ_val(1, x)| + |y − μ_val(0, x)|)², viewed as a fixed L¹(P_Z) function (witness via h_y2 + h_yd2 + Cauchy–Schwarz).","s":"{P : Causalean.PO.POSystem} → {γ : Type u_1} → [inst : MeasurableSpace γ] → [inst_1 : StandardBorelSpace P.Ω] → [inst_2 : MeasureTheory.IsFiniteMeasure P.μ] → Causalean.Estimation.ATE.BackdoorEstimationSystem P γ → γ × Bool × ℝ → ℝ","l":351,"f":"Causalean/Estimation/ATE/Score/AIPWScoreL2.lean"},{"n":"Causalean.Estimation.ATE.BackdoorEstimationSystem.aipwLp","k":"def","m":"Causalean.Estimation.Efficiency.ATETangent","a":"Estimation/Efficiency","t":1,"d":"This is the AIPW influence function represented as a point of the square-integrable Hilbert space.","s":"{P : Causalean.PO.POSystem} → {γ : Type u_1} → [inst : MeasurableSpace γ] → [inst_1 : StandardBorelSpace P.Ω] → [inst_2 : MeasureTheory.IsFiniteMeasure P.μ] → (S : Causalean.Estimation.ATE.BackdoorEstimationSystem P γ) → {ε : ℝ} → S.StrictOverlap ε → S.Assumptions → MeasureTheory.Integrable (fun ω => S.factualY ω ^ 2) P.μ → (∀ (d : Bool), MeasureTheory.Integrable (fun ω => S.YofD d ω ^ 2) P.μ) → ↥(MeasureTheory.Lp ℝ 2 S.P_Z)","l":440,"f":"Causalean/Estimation/Efficiency/ATETangent.lean"},{"n":"Causalean.Estimation.ATE.BackdoorEstimationSystem.aipwLp_mem_tangent","k":"theorem","m":"Causalean.Estimation.Efficiency.ATETangent","a":"Estimation/Efficiency","t":2,"d":"The AIPW influence function lies in the mean-zero tangent space.","s":"∀ {P : Causalean.PO.POSystem} {γ : Type u_1} [inst : MeasurableSpace γ] [inst_1 : StandardBorelSpace P.Ω] [inst_2 : MeasureTheory.IsFiniteMeasure P.μ] (S : Causalean.Estimation.ATE.BackdoorEstimationSystem P γ) {ε : ℝ} (h_overlap : S.StrictOverlap ε) (hA : S.Assumptions) (h_y2 : MeasureTheory.Integrable (fun ω => S.factualY ω ^ 2) P.μ) (h_yd2 : ∀ (d : Bool), MeasureTheory.Integrable (fun ω => S.YofD d ω ^ 2) P.μ), S.aipwLp h_overlap hA h_y2 h_yd2 ∈ S.Tfull","l":485,"f":"Causalean/Estimation/Efficiency/ATETangent.lean"},{"n":"Causalean.Estimation.ATE.BackdoorEstimationSystem.aipwMoment","k":"def","m":"Causalean.Estimation.ATE.Score.AIPWMoment","a":"Estimation/ATE/Score","t":1,"d":"The AIPW moment m_AIPW(η, z, θ) from def:est-aipw-moment:","s":"{γ : Type u_1} → γ × Bool × ℝ → (Bool → γ → ℝ) → (γ → ℝ) → ℝ → ℝ","l":61,"f":"Causalean/Estimation/ATE/Score/AIPWMoment.lean"},{"n":"Causalean.Estimation.ATE.BackdoorEstimationSystem.aipwMomentFunctional","k":"def","m":"Causalean.Estimation.ATE.Score.AIPWMoment","a":"Estimation/ATE/Score","t":1,"d":"The AIPW moment as a moment functional NuisanceVec γ → X → ℝ → ℝ, suitable for feeding NeymanOrthogonal.","s":"{γ : Type u_1} → [inst : MeasurableSpace γ] → Causalean.Estimation.ATE.NuisanceVec γ → γ × Bool × ℝ → ℝ → ℝ","l":325,"f":"Causalean/Estimation/ATE/Score/AIPWMoment.lean"},{"n":"Causalean.Estimation.ATE.BackdoorEstimationSystem.aipw_finite_var","k":"theorem","m":"Causalean.Estimation.ATE.Score.FiniteVar","a":"Estimation/ATE/Score","t":1,"d":"Finite variance of ψ_AIPW — lem:est-aipw-finite-var. For an estimation system S, if the true propensity satisfies strict overlap at some level in (0, 1/2], the observed outcome has finite second moment, and each treatment arm's outcome regression, evaluated at the covariate, is square-integrable, then the AIPW influence function is square-integrable against the observed-data law P_Z.","s":"∀ {P : Causalean.PO.POSystem} {γ : Type u_1} [inst : MeasurableSpace γ] [inst_1 : StandardBorelSpace P.Ω] [inst_2 : MeasureTheory.IsFiniteMeasure P.μ] (S : Causalean.Estimation.ATE.BackdoorEstimationSystem P γ) {ε : ℝ}, S.StrictOverlap ε → MeasureTheory.Integrable (fun ω => S.factualY ω ^ 2) P.μ → (∀ (d : Bool), MeasureTheory.MemLp (fun ω => S.μ_val d (S.factualX ω)) 2 P.μ) → MeasureTheory.Integrable (fun z => S.ψ_AIPW z ^ 2) S.P_Z","l":39,"f":"Causalean/Estimation/ATE/Score/FiniteVar.lean"},{"n":"Causalean.Estimation.ATE.BackdoorEstimationSystem.aipw_finite_var_of_counterfactual_sq","k":"theorem","m":"Causalean.Estimation.ATE.Score.FiniteVar","a":"Estimation/ATE/Score","t":2,"d":"Counterfactual second moments are a stronger sufficient condition for aipw_finite_var.","s":"∀ {P : Causalean.PO.POSystem} {γ : Type u_1} [inst : MeasurableSpace γ] [inst_1 : StandardBorelSpace P.Ω] [inst_2 : MeasureTheory.IsFiniteMeasure P.μ] (S : Causalean.Estimation.ATE.BackdoorEstimationSystem P γ) {ε : ℝ}, S.StrictOverlap ε → S.Assumptions → MeasureTheory.Integrable (fun ω => S.factualY ω ^ 2) P.μ → (∀ (d : Bool), MeasureTheory.Integrable (fun ω => S.YofD d ω ^ 2) P.μ) → MeasureTheory.Integrable (fun z => S.ψ_AIPW z ^ 2) S.P_Z","l":174,"f":"Causalean/Estimation/ATE/Score/FiniteVar.lean"},{"n":"Causalean.Estimation.ATE.BackdoorEstimationSystem.aipw_isPathwiseGradient_ATE","k":"theorem","m":"Causalean.Estimation.Efficiency.ATEEfficientIF","a":"Estimation/Efficiency","t":2,"d":"Hahn's pathwise-derivative identity makes the AIPW score a pathwise gradient of the backdoor ATE functional along every regular submodel.","s":"∀ {P : Causalean.PO.POSystem} {γ : Type u_1} [inst : MeasurableSpace γ] [inst_1 : StandardBorelSpace P.Ω] [inst_2 : MeasureTheory.IsFiniteMeasure P.μ] (S : Causalean.Estimation.ATE.BackdoorEstimationSystem P γ) {ε : ℝ} (h_overlap : S.StrictOverlap ε) (hA : S.Assumptions) (h_y2 : MeasureTheory.Integrable (fun ω => S.factualY ω ^ 2) P.μ) (h_yd2 : ∀ (d : Bool), MeasureTheory.Integrable (fun ω => S.YofD d ω ^ 2) P.μ) (ψ : MeasureTheory.Measure (γ × Bool × ℝ) → ℝ), (∀ (m : Causalean.Estimation.Efficiency.RegularSubmodel S.oneLp S.P_Z), HasDerivAt (fun t => ψ (m.path t)) (∫ (z : γ × Bool × ℝ), S.ψ_AIPW z * ↑↑m.score z ∂S.P_Z) 0) → Causalean.Estimation.Efficiency.IsPathwiseGradient S.oneLp S.P_Z ψ (S.aipwLp h_overlap hA h_y2 h_yd2)","l":82,"f":"Causalean/Estimation/Efficiency/ATEEfficientIF.lean"},{"n":"Causalean.Estimation.ATE.BackdoorEstimationSystem.aipw_is_efficientInfluenceFunction","k":"theorem","m":"Causalean.Estimation.Efficiency.ATEEfficientIF","a":"Estimation/Efficiency","t":1,"d":"AIPW is the efficient influence function for the backdoor ATE. Let S be a backdoor average-treatment-effect estimation system with strict overlap at level ε, satisfying the system's core identification assumptions, in which the observed outcome has finite second moment and each potential outcome under treatment level d has finite second moment. Suppose a functional ψ of the observed-data law satisfies Hahn's pathwise-derivative identity — along every regular submodel, ψ's derivative at the true law equals the covariance of the AIPW score with the submodel's score, and that the full mean-zero space is a genuine nonparametric tangent space, i.e. it is contained in the closed span of regular-submodel scores. Then the AIPW influence function is a pathwise gradient of ψ, lies in the full mean-zero tangent space, and is the canonical efficient influence function: for every other pathwise gradient g of ψ, projecting g onto the tangent space returns the AIPW influence function.","s":"∀ {P : Causalean.PO.POSystem} {γ : Type u_1} [inst : MeasurableSpace γ] [inst_1 : StandardBorelSpace P.Ω] [inst_2 : MeasureTheory.IsFiniteMeasure P.μ] (S : Causalean.Estimation.ATE.BackdoorEstimationSystem P γ) {ε : ℝ} (h_overlap : S.StrictOverlap ε) (hA : S.Assumptions) (h_y2 : MeasureTheory.Integrable (fun ω => S.factualY ω ^ 2) P.μ) (h_yd2 : ∀ (d : Bool), MeasureTheory.Integrable (fun ω => S.YofD d ω ^ 2) P.μ) (ψ : MeasureTheory.Measure (γ × Bool × ℝ) → ℝ), (∀ (m : Causalean.Estimation.Efficiency.RegularSubmodel S.oneLp S.P_Z), HasDerivAt (fun t => ψ (m.path t)) (∫ (z : γ × Bool × ℝ), S.ψ_AIPW z * ↑↑m.score z ∂S.P_Z) 0) → S.Tfull ≤ Causalean.Estimation.Efficiency.tangentSpace S.oneLp S.P_Z → Causalean.Estimation.Efficiency.IsPathwiseGradient S.oneLp S.P_Z ψ (S.aipwLp h_overlap hA h_y2 h_yd2) ∧ S.aipwLp h_overlap hA h_y2 h_yd2 ∈ S.Tfull ∧ ∀ (g : ↥(MeasureTheory.Lp ℝ 2 S.P_Z)), Causalean.Estimation.Efficiency.IsPathwiseGradient S.oneLp S.P_Z ψ g → Causalean.Estimation.Efficiency.efficientIF S.Tfull g = S.aipwLp h_overlap hA h_y2 h_yd2","l":104,"f":"Causalean/Estimation/Efficiency/ATEEfficientIF.lean"},{"n":"Causalean.Estimation.ATE.BackdoorEstimationSystem.aipw_mean_zero","k":"theorem","m":"Causalean.Estimation.ATE.Score.MeanZero","a":"Estimation/ATE/Score","t":1,"d":"Mean zero of the AIPW influence function. Under the back-door identification assumptions, if the inverse-propensity-weighted residual correction on the treated arm, 1{D=1}/e(X) · (Y − μ(1,X)), is integrable and the analogous correction on the control arm, 1{D=0}/(1 − e(X)) · (Y − μ(0,X)), is integrable, then the AIPW influence function ψ_AIPW has mean zero under the joint law of the covariates, treatment indicator, and outcome.","s":"∀ {P : Causalean.PO.POSystem} {γ : Type u_1} [inst : MeasurableSpace γ] [inst_1 : StandardBorelSpace P.Ω] [inst_2 : MeasureTheory.IsFiniteMeasure P.μ] (S : Causalean.Estimation.ATE.BackdoorEstimationSystem P γ), S.Assumptions → MeasureTheory.Integrable (fun ω => 1 / S.e_val (S.factualX ω) * (S.dVar.indicator true ω * (S.factualY ω - S.μ_val true (S.factualX ω)))) P.μ → MeasureTheory.Integrable (fun ω => 1 / (1 - S.e_val (S.factualX ω)) * (S.dVar.indicator false ω * (S.factualY ω - S.μ_val false (S.factualX ω)))) P.μ → ∫ (z : γ × Bool × ℝ), S.ψ_AIPW z ∂S.P_Z = 0","l":636,"f":"Causalean/Estimation/ATE/Score/MeanZero.lean"},{"n":"Causalean.Estimation.ATE.BackdoorEstimationSystem.aipw_mean_zero_of_square_integrable","k":"theorem","m":"Causalean.Estimation.ATE.Score.MeanZero","a":"Estimation/ATE/Score","t":2,"d":"A stronger sufficient-condition corollary for aipw_mean_zero.","s":"∀ {P : Causalean.PO.POSystem} {γ : Type u_1} [inst : MeasurableSpace γ] [inst_1 : StandardBorelSpace P.Ω] [inst_2 : MeasureTheory.IsFiniteMeasure P.μ] (S : Causalean.Estimation.ATE.BackdoorEstimationSystem P γ) {ε : ℝ}, S.StrictOverlap ε → S.Assumptions → MeasureTheory.Integrable (fun ω => S.factualY ω ^ 2) P.μ → (∀ (d : Bool), MeasureTheory.Integrable (fun ω => S.YofD d ω ^ 2) P.μ) → ∫ (z : γ × Bool × ℝ), S.ψ_AIPW z ∂S.P_Z = 0","l":664,"f":"Causalean/Estimation/ATE/Score/MeanZero.lean"},{"n":"Causalean.Estimation.ATE.BackdoorEstimationSystem.aipw_memLp","k":"theorem","m":"Causalean.Estimation.Efficiency.ATETangent","a":"Estimation/Efficiency","t":2,"d":"The AIPW influence function is square-integrable under the observed data law.","s":"∀ {P : Causalean.PO.POSystem} {γ : Type u_1} [inst : MeasurableSpace γ] [inst_1 : StandardBorelSpace P.Ω] [inst_2 : MeasureTheory.IsFiniteMeasure P.μ] (S : Causalean.Estimation.ATE.BackdoorEstimationSystem P γ) {ε : ℝ}, S.StrictOverlap ε → S.Assumptions → MeasureTheory.Integrable (fun ω => S.factualY ω ^ 2) P.μ → (∀ (d : Bool), MeasureTheory.Integrable (fun ω => S.YofD d ω ^ 2) P.μ) → MeasureTheory.MemLp S.ψ_AIPW 2 S.P_Z","l":429,"f":"Causalean/Estimation/Efficiency/ATETangent.lean"},{"n":"Causalean.Estimation.ATE.BackdoorEstimationSystem.aipw_rem_const","k":"def","m":"Causalean.Estimation.ATE.Remainder.Identity","a":"Estimation/ATE/Remainder","t":1,"d":"The AIPW remainder constant is the uniform strict-overlap weight bound used in the second-order remainder estimate.","s":"ℝ → ℝ","l":93,"f":"Causalean/Estimation/ATE/Remainder/Identity.lean"},{"n":"Causalean.Estimation.ATE.BackdoorEstimationSystem.aipw_remainder_bound","k":"theorem","m":"Causalean.Estimation.ATE.Remainder.Bound","a":"Estimation/ATE/Remainder","t":1,"d":"Fix strict overlap for the true propensity at level ε, the back-door identification assumptions, and finite second moments of the observed and potential outcomes. For a candidate nuisance vector η such that η lies in the ε-overlap L² nuisance class H_ε_aeL2, with each treatment-arm outcome-regression error in L²(P_X) and the propensity error in L²(P_X), the population AIPW moment functional at η and the true ATE θ₀ is bounded in absolute value by an overlap-dependent constant times the sum, over treatment arms, of the product of the outcome-regression and propensity L² errors.","s":"∀ {P : Causalean.PO.POSystem} {γ : Type u_1} [inst : MeasurableSpace γ] [inst_1 : StandardBorelSpace P.Ω] [inst_2 : MeasureTheory.IsFiniteMeasure P.μ] (S : Causalean.Estimation.ATE.BackdoorEstimationSystem P γ) {ε : ℝ}, S.StrictOverlap ε → S.Assumptions → MeasureTheory.Integrable (fun ω => S.factualY ω ^ 2) P.μ → (∀ (d : Bool), MeasureTheory.Integrable (fun ω => S.YofD d ω ^ 2) P.μ) → ∀ η ∈ S.H_ε_aeL2 ε, (∀ (a : Bool), MeasureTheory.MemLp (fun x => η.μ_fn a x - S.μ_val a x) 2 S.P_X) → MeasureTheory.MemLp (fun x => η.e_fn x - S.e_val x) 2 S.P_X → |∫ (z : γ × Bool × ℝ), Causalean.Estimation.ATE.BackdoorEstimationSystem.aipwMomentFunctional η z S.θ₀ ∂S.P_Z| ≤ Causalean.Estimation.ATE.BackdoorEstimationSystem.aipw_rem_const ε * ∑ a, (MeasureTheory.eLpNorm (fun x => η.μ_fn a x - S.μ_val a x) 2 S.P_X).toReal * (MeasureTheory.eLpNorm (fun x => η.e_fn x - S.e_val x) 2 S.P_X).toReal","l":44,"f":"Causalean/Estimation/ATE/Remainder/Bound.lean"},{"n":"Causalean.Estimation.ATE.BackdoorEstimationSystem.aipw_remainder_identity","k":"theorem","m":"Causalean.Estimation.ATE.Remainder.Identity","a":"Estimation/ATE/Remainder","t":1,"d":"Integrated AIPW remainder identity. Fix strict overlap for the true propensity at level ε, the back-door identification assumptions, and finite second moments of the observed and potential outcomes. For a nuisance vector η such that η lies in the ε-overlap L² nuisance class H_ε_aeL2 with each treatment-arm outcome-regression error in L²(P_X), the population AIPW moment functional at η and the true ATE θ₀, integrated over the observed-data law P_Z, equals the covariate-law integral of η's propensity error (its propensity estimate minus the truth) times the sum of each treatment-arm outcome-regression error divided by the corresponding true or complementary propensity.","s":"∀ {P : Causalean.PO.POSystem} {γ : Type u_1} [inst : MeasurableSpace γ] [inst_1 : StandardBorelSpace P.Ω] [inst_2 : MeasureTheory.IsFiniteMeasure P.μ] (S : Causalean.Estimation.ATE.BackdoorEstimationSystem P γ) {ε : ℝ}, S.StrictOverlap ε → S.Assumptions → MeasureTheory.Integrable (fun ω => S.factualY ω ^ 2) P.μ → (∀ (d : Bool), MeasureTheory.Integrable (fun ω => S.YofD d ω ^ 2) P.μ) → ∀ η ∈ S.H_ε_aeL2 ε, (∀ (a : Bool), MeasureTheory.MemLp (fun x => η.μ_fn a x - S.μ_val a x) 2 S.P_X) → ∫ (z : γ × Bool × ℝ), Causalean.Estimation.ATE.BackdoorEstimationSystem.aipwMomentFunctional η z S.θ₀ ∂S.P_Z = ∫ (x : γ), (η.e_fn x - S.e_val x) * ((η.μ_fn true x - S.μ_val true x) / η.e_fn x + (η.μ_fn false x - S.μ_val false x) / (1 - η.e_fn x)) ∂S.P_X","l":120,"f":"Causalean/Estimation/ATE/Remainder/Identity.lean"},{"n":"Causalean.Estimation.ATE.BackdoorEstimationSystem.aipw_remainder_op","k":"theorem","m":"Causalean.Estimation.ATE.Remainder.Bound","a":"Estimation/ATE/Remainder","t":1,"d":"AIPW remainder is o_p(n^{-1/2}) under the product rate. Fix strict overlap at level ε, the back-door identification assumptions, and finite second moments of the observed and potential outcomes. For a sequence of nuisance estimators η̂ such that every realization η̂(n,ω) lies in the ε-overlap L² nuisance class, with outcome-regression errors in L²(P_X) at every horizon and realization and propensity errors in L²(P_X) at every horizon and realization, and whose L² outcome-regression and propensity errors have product rate o_p(n^{-1/2}) for each treatment arm, the population AIPW moment functional at the random nuisance η̂(n) and θ₀ is o_p(n^{-1/2}) under μ. Direct consequence of aipw_remainder_bound plus closure of IsLittleOp under finite sums and constant scaling.","s":"∀ {P : Causalean.PO.POSystem} {γ : Type u_1} [inst : MeasurableSpace γ] [inst_1 : StandardBorelSpace P.Ω] [inst_2 : MeasureTheory.IsFiniteMeasure P.μ] (S : Causalean.Estimation.ATE.BackdoorEstimationSystem P γ) {ε : ℝ}, S.StrictOverlap ε → S.Assumptions → MeasureTheory.Integrable (fun ω => S.factualY ω ^ 2) P.μ → (∀ (d : Bool), MeasureTheory.Integrable (fun ω => S.YofD d ω ^ 2) P.μ) → ∀ (η_hat : ℕ → P.Ω → Causalean.Estimation.ATE.NuisanceVec γ), (∀ (n : ℕ) (ω : P.Ω), η_hat n ω ∈ S.H_ε_aeL2 ε) → (∀ (n : ℕ) (ω : P.Ω) (a : Bool), MeasureTheory.MemLp (fun x => (η_hat n ω).μ_fn a x - S.μ_val a x) 2 S.P_X) → (∀ (n : ℕ) (ω : P.Ω), MeasureTheory.MemLp (fun x => (η_hat n ω).e_fn x - S.e_val x) 2 S.P_X) → (∀ (a : Bool), Causalean.Stat.IsLittleOp (fun n ω => (MeasureTheory.eLpNorm (fun x => (η_hat n ω).μ_fn a x - S.μ_val a x) 2 S.P_X).toReal * (MeasureTheory.eLpNorm (fun x => (η_hat n ω).e_fn x - S.e_val x) 2 S.P_X).toReal) (fun n => ↑n ^ (-(1 / 2))) P.μ) → Causalean.Stat.IsLittleOp (fun n ω => ∫ (z : γ × Bool × ℝ), Causalean.Estimation.ATE.BackdoorEstimationSystem.aipwMomentFunctional (η_hat n ω) z S.θ₀ ∂S.P_Z) (fun n => ↑n ^ (-(1 / 2))) P.μ","l":231,"f":"Causalean/Estimation/ATE/Remainder/Bound.lean"},{"n":"Causalean.Estimation.ATE.BackdoorEstimationSystem.aipw_score_diff_isLittleOp_one","k":"theorem","m":"Causalean.Estimation.ATE.Score.AIPWScoreL2","a":"Estimation/ATE/Score","t":1,"d":"Fix strict overlap at level ε, the back-door identification assumptions, and finite second moments of the observed and potential outcomes. For a sequence of nuisance estimators η̂ such that every realization lies in the ε-overlap L² nuisance class, with outcome-regression errors in L²(P_X) at every horizon and realization and propensity errors in L²(P_X) at every horizon and realization, if the outcome-regression error converges to zero in L²(P_X) in probability and the propensity error converges to zero in L²(P_X) in probability, then the L²(P_Z) norm of the AIPW score difference between the estimated and true nuisance converges to zero in probability.","s":"∀ {P : Causalean.PO.POSystem} {γ : Type u_1} [inst : MeasurableSpace γ] [inst_1 : StandardBorelSpace P.Ω] [inst_2 : MeasureTheory.IsFiniteMeasure P.μ] (S : Causalean.Estimation.ATE.BackdoorEstimationSystem P γ) {ε : ℝ}, S.StrictOverlap ε → S.Assumptions → MeasureTheory.Integrable (fun ω => S.factualY ω ^ 2) P.μ → (∀ (d : Bool), MeasureTheory.Integrable (fun ω => S.YofD d ω ^ 2) P.μ) → ∀ (η_hat : ℕ → P.Ω → Causalean.Estimation.ATE.NuisanceVec γ), (∀ (n : ℕ) (ω : P.Ω), η_hat n ω ∈ S.H_ε_aeL2 ε) → (∀ (n : ℕ) (ω : P.Ω) (a : Bool), MeasureTheory.MemLp (fun x => (η_hat n ω).μ_fn a x - S.μ_val a x) 2 S.P_X) → (∀ (n : ℕ) (ω : P.Ω), MeasureTheory.MemLp (fun x => (η_hat n ω).e_fn x - S.e_val x) 2 S.P_X) → (∀ (a : Bool), Causalean.Stat.IsLittleOp (fun n ω => (MeasureTheory.eLpNorm (fun x => (η_hat n ω).μ_fn a x - S.μ_val a x) 2 S.P_X).toReal) (fun x => 1) P.μ) → Causalean.Stat.IsLittleOp (fun n ω => (MeasureTheory.eLpNorm (fun x => (η_hat n ω).e_fn x - S.e_val x) 2 S.P_X).toReal) (fun x => 1) P.μ → Causalean.Stat.IsLittleOp (fun n ω => (MeasureTheory.eLpNorm (fun z => Causalean.Estimation.ATE.BackdoorEstimationSystem.aipwMomentFunctional (η_hat n ω) z S.θ₀ - Causalean.Estimation.ATE.BackdoorEstimationSystem.aipwMomentFunctional S.η₀ z S.θ₀) 2 S.P_Z).toReal) (fun x => 1) P.μ","l":940,"f":"Causalean/Estimation/ATE/Score/AIPWScoreL2.lean"},{"n":"Causalean.Estimation.ATE.BackdoorEstimationSystem.aipw_score_diff_pointwise_bound","k":"theorem","m":"Causalean.Estimation.ATE.Score.AIPWScoreL2","a":"Estimation/ATE/Score","t":2,"d":"AIPW score Lipschitz bound on H_ε, P_Z-a.e.","s":"∀ {P : Causalean.PO.POSystem} {γ : Type u_1} [inst : MeasurableSpace γ] [inst_1 : StandardBorelSpace P.Ω] [inst_2 : MeasureTheory.IsFiniteMeasure P.μ] (S : Causalean.Estimation.ATE.BackdoorEstimationSystem P γ) {ε : ℝ}, S.StrictOverlap ε → ∀ η ∈ S.H_ε_aeL2 ε, ∀ᵐ (z : γ × Bool × ℝ) ∂S.P_Z, |Causalean.Estimation.ATE.BackdoorEstimationSystem.aipwMomentFunctional η z S.θ₀ - Causalean.Estimation.ATE.BackdoorEstimationSystem.aipwMomentFunctional S.η₀ z S.θ₀| ≤ Causalean.Estimation.ATE.BackdoorEstimationSystem.K_AIPW ε * (|η.μ_fn true (Causalean.Estimation.ATE.BackdoorEstimationSystem.projX z) - S.μ_val true (Causalean.Estimation.ATE.BackdoorEstimationSystem.projX z)| + |η.μ_fn false (Causalean.Estimation.ATE.BackdoorEstimationSystem.projX z) - S.μ_val false (Causalean.Estimation.ATE.BackdoorEstimationSystem.projX z)| + (|Causalean.Estimation.ATE.BackdoorEstimationSystem.projY z - S.μ_val true (Causalean.Estimation.ATE.BackdoorEstimationSystem.projX z)| + |Causalean.Estimation.ATE.BackdoorEstimationSystem.projY z - S.μ_val false (Causalean.Estimation.ATE.BackdoorEstimationSystem.projX z)|) * |η.e_fn (Causalean.Estimation.ATE.BackdoorEstimationSystem.projX z) - S.e_val (Causalean.Estimation.ATE.BackdoorEstimationSystem.projX z)|)","l":270,"f":"Causalean/Estimation/ATE/Score/AIPWScoreL2.lean"},{"n":"Causalean.Estimation.ATE.BackdoorEstimationSystem.aipw_score_meanZero_projection_eq","k":"theorem","m":"Causalean.Estimation.Efficiency.ATETangent","a":"Estimation/Efficiency","t":2,"d":"The square-integrable AIPW score is already mean-zero, so projecting it onto the full mean-zero tangent space leaves it unchanged.","s":"∀ {P : Causalean.PO.POSystem} {γ : Type u_1} [inst : MeasurableSpace γ] [inst_1 : StandardBorelSpace P.Ω] [inst_2 : MeasureTheory.IsFiniteMeasure P.μ] (S : Causalean.Estimation.ATE.BackdoorEstimationSystem P γ) {ε : ℝ} (h_overlap : S.StrictOverlap ε) (hA : S.Assumptions) (h_y2 : MeasureTheory.Integrable (fun ω => S.factualY ω ^ 2) P.μ) (h_yd2 : ∀ (d : Bool), MeasureTheory.Integrable (fun ω => S.YofD d ω ^ 2) P.μ), Causalean.Estimation.Efficiency.efficientIF S.Tfull (S.aipwLp h_overlap hA h_y2 h_yd2) = S.aipwLp h_overlap hA h_y2 h_yd2","l":505,"f":"Causalean/Estimation/Efficiency/ATETangent.lean"},{"n":"Causalean.Estimation.ATE.BackdoorEstimationSystem.aipw_variance_hahn_decomposition","k":"theorem","m":"Causalean.Estimation.Efficiency.ATEVariance","a":"Estimation/Efficiency","t":1,"d":"Hahn (1998) variance decomposition of the AIPW influence function. Assume the back-door identifying assumptions and strict overlap of the true propensity score with margin ε, and suppose the factual outcome is square-integrable and every potential outcome Y(d) is square-integrable. Then the variance of the augmented inverse-propensity-weighted (AIPW) influence function decomposes as the variance of the conditional treatment-effect function μ₁ − μ₀ − θ₀ plus two inverse-propensity-weighted conditional-outcome-variance terms — the Hahn semiparametric efficiency bound V_H.","s":"∀ {P : Causalean.PO.POSystem} {γ : Type u_1} [inst : MeasurableSpace γ] [inst_1 : StandardBorelSpace P.Ω] [inst_2 : MeasureTheory.IsFiniteMeasure P.μ] (S : Causalean.Estimation.ATE.BackdoorEstimationSystem P γ) {ε : ℝ}, S.StrictOverlap ε → S.Assumptions → MeasureTheory.Integrable (fun ω => S.factualY ω ^ 2) P.μ → (∀ (d : Bool), MeasureTheory.Integrable (fun ω => S.YofD d ω ^ 2) P.μ) → ∫ (z : γ × Bool × ℝ), S.ψ_AIPW z ^ 2 ∂S.P_Z = ∫ (x : γ), (S.μ_val true x - S.μ_val false x - S.θ₀) ^ 2 ∂S.P_X + ∫ (z : γ × Bool × ℝ), Causalean.Estimation.ATE.BackdoorEstimationSystem.indA z / S.e_val (Causalean.Estimation.ATE.BackdoorEstimationSystem.projX z) ^ 2 * (Causalean.Estimation.ATE.BackdoorEstimationSystem.projY z - S.μ_val true (Causalean.Estimation.ATE.BackdoorEstimationSystem.projX z)) ^ 2 ∂S.P_Z + ∫ (z : γ × Bool × ℝ), (1 - Causalean.Estimation.ATE.BackdoorEstimationSystem.indA z) / (1 - S.e_val (Causalean.Estimation.ATE.BackdoorEstimationSystem.projX z)) ^ 2 * (Causalean.Estimation.ATE.BackdoorEstimationSystem.projY z - S.μ_val false (Causalean.Estimation.ATE.BackdoorEstimationSystem.projX z)) ^ 2 ∂S.P_Z","l":153,"f":"Causalean/Estimation/Efficiency/ATEVariance.lean"},{"n":"Causalean.Estimation.ATE.BackdoorEstimationSystem.cond_exp_residual_zero","k":"theorem","m":"Causalean.Estimation.ATE.Score.MeanZero","a":"Estimation/ATE/Score","t":2,"d":"Conditional expectation of the residual ind_d · (Y − μ_val d X) given σ(X) is zero a.s. — the σ(X)-cleared form of lem:est-aipw-mean-zero.","s":"∀ {P : Causalean.PO.POSystem} {γ : Type u_1} [inst : MeasurableSpace γ] [inst_1 : StandardBorelSpace P.Ω] [inst_2 : MeasureTheory.IsFiniteMeasure P.μ] (S : Causalean.Estimation.ATE.BackdoorEstimationSystem P γ), S.Assumptions → ∀ (d : Bool), P.μ[fun ω => S.dVar.indicator d ω * (S.factualY ω - S.μ_val d (S.factualX ω)) | S.sigmaX] =ᵐ[P.μ] fun x => 0","l":123,"f":"Causalean/Estimation/ATE/Score/MeanZero.lean"},{"n":"Causalean.Estimation.ATE.BackdoorEstimationSystem.e_val_label","k":"def","m":"Causalean.Estimation.ATE.Score.ScorePullout","a":"Estimation/ATE/Score","t":1,"d":"Value-space propensity for label d: e_val for d = true, 1 − e_val for d = false.","s":"{P : Causalean.PO.POSystem} → {γ : Type u_1} → [inst : MeasurableSpace γ] → [inst_1 : StandardBorelSpace P.Ω] → [inst_2 : MeasureTheory.IsFiniteMeasure P.μ] → Causalean.Estimation.ATE.BackdoorEstimationSystem P γ → Bool → γ → ℝ","l":40,"f":"Causalean/Estimation/ATE/Score/ScorePullout.lean"},{"n":"Causalean.Estimation.ATE.BackdoorEstimationSystem.effBound_eq_hahn","k":"theorem","m":"Causalean.Estimation.Efficiency.ATETangent","a":"Estimation/Efficiency","t":1,"d":"Semiparametric efficiency bound equals Hahn's three-term variance formula. Let S be a backdoor average-treatment-effect estimation system with strict overlap at level ε, satisfying the system's core identification assumptions, in which the observed outcome has finite second moment and each potential outcome under treatment level d has finite second moment. Then the semiparametric efficiency bound for the backdoor ATE equals the sum of the between-arms regression-contrast variance ∫(μ(1,X)-μ(0,X)-θ₀)²dP_X, the treated-arm weighted residual variance ∫(A/e(X)²)(Y-μ(1,X))²dP_Z, and the control-arm weighted residual variance ∫((1-A)/(1-e(X))²)(Y-μ(0,X))²dP_Z.","s":"∀ {P : Causalean.PO.POSystem} {γ : Type u_1} [inst : MeasurableSpace γ] [inst_1 : StandardBorelSpace P.Ω] [inst_2 : MeasureTheory.IsFiniteMeasure P.μ] (S : Causalean.Estimation.ATE.BackdoorEstimationSystem P γ) {ε : ℝ} (h_overlap : S.StrictOverlap ε) (hA : S.Assumptions) (h_y2 : MeasureTheory.Integrable (fun ω => S.factualY ω ^ 2) P.μ) (h_yd2 : ∀ (d : Bool), MeasureTheory.Integrable (fun ω => S.YofD d ω ^ 2) P.μ), Causalean.Estimation.Efficiency.effBound S.Tfull (S.aipwLp h_overlap hA h_y2 h_yd2) = ∫ (x : γ), (S.μ_val true x - S.μ_val false x - S.θ₀) ^ 2 ∂S.P_X + ∫ (z : γ × Bool × ℝ), Causalean.Estimation.ATE.BackdoorEstimationSystem.indA z / S.e_val (Causalean.Estimation.ATE.BackdoorEstimationSystem.projX z) ^ 2 * (Causalean.Estimation.ATE.BackdoorEstimationSystem.projY z - S.μ_val true (Causalean.Estimation.ATE.BackdoorEstimationSystem.projX z)) ^ 2 ∂S.P_Z + ∫ (z : γ × Bool × ℝ), (1 - Causalean.Estimation.ATE.BackdoorEstimationSystem.indA z) / (1 - S.e_val (Causalean.Estimation.ATE.BackdoorEstimationSystem.projX z)) ^ 2 * (Causalean.Estimation.ATE.BackdoorEstimationSystem.projY z - S.μ_val false (Causalean.Estimation.ATE.BackdoorEstimationSystem.projX z)) ^ 2 ∂S.P_Z","l":548,"f":"Causalean/Estimation/Efficiency/ATETangent.lean"},{"n":"Causalean.Estimation.ATE.BackdoorEstimationSystem.effBound_eq_of_smaller_tangent_containing_aipw","k":"theorem","m":"Causalean.Estimation.Efficiency.ATETangent","a":"Estimation/Efficiency","t":2,"d":"For any supplied smaller tangent space that is contained in the mean-zero tangent space and still contains the AIPW score, the abstract squared-norm efficiency bound is unchanged.","s":"∀ {P : Causalean.PO.POSystem} {γ : Type u_1} [inst : MeasurableSpace γ] [inst_1 : StandardBorelSpace P.Ω] [inst_2 : MeasureTheory.IsFiniteMeasure P.μ] (S : Causalean.Estimation.ATE.BackdoorEstimationSystem P γ) {ε : ℝ} (h_overlap : S.StrictOverlap ε) (hA : S.Assumptions) (h_y2 : MeasureTheory.Integrable (fun ω => S.factualY ω ^ 2) P.μ) (h_yd2 : ∀ (d : Bool), MeasureTheory.Integrable (fun ω => S.YofD d ω ^ 2) P.μ) (T' : Submodule ℝ ↥(MeasureTheory.Lp ℝ 2 S.P_Z)) [inst_3 : T'.HasOrthogonalProjection], T' ≤ S.Tfull → S.aipwLp h_overlap hA h_y2 h_yd2 ∈ T' → Causalean.Estimation.Efficiency.effBound T' (S.aipwLp h_overlap hA h_y2 h_yd2) = Causalean.Estimation.Efficiency.effBound S.Tfull (S.aipwLp h_overlap hA h_y2 h_yd2)","l":603,"f":"Causalean/Estimation/Efficiency/ATETangent.lean"},{"n":"Causalean.Estimation.ATE.BackdoorEstimationSystem.effBound_eq_variance","k":"theorem","m":"Causalean.Estimation.Efficiency.ATETangent","a":"Estimation/Efficiency","t":1,"d":"Semiparametric efficiency bound equals the AIPW variance. Let S be a backdoor average-treatment-effect estimation system with strict overlap at level ε, satisfying the system's core identification assumptions, in which the observed outcome has finite second moment and each potential outcome under treatment level d has finite second moment. Then the semiparametric efficiency bound for the backdoor ATE, computed against the full mean-zero tangent space, equals the second moment of the AIPW influence function ψ_AIPW under the observed-data law P_Z.","s":"∀ {P : Causalean.PO.POSystem} {γ : Type u_1} [inst : MeasurableSpace γ] [inst_1 : StandardBorelSpace P.Ω] [inst_2 : MeasureTheory.IsFiniteMeasure P.μ] (S : Causalean.Estimation.ATE.BackdoorEstimationSystem P γ) {ε : ℝ} (h_overlap : S.StrictOverlap ε) (hA : S.Assumptions) (h_y2 : MeasureTheory.Integrable (fun ω => S.factualY ω ^ 2) P.μ) (h_yd2 : ∀ (d : Bool), MeasureTheory.Integrable (fun ω => S.YofD d ω ^ 2) P.μ), Causalean.Estimation.Efficiency.effBound S.Tfull (S.aipwLp h_overlap hA h_y2 h_yd2) = ∫ (z : γ × Bool × ℝ), S.ψ_AIPW z ^ 2 ∂S.P_Z","l":523,"f":"Causalean/Estimation/Efficiency/ATETangent.lean"},{"n":"Causalean.Estimation.ATE.BackdoorEstimationSystem.efficiency_bound_optimal","k":"theorem","m":"Causalean.Estimation.Efficiency.ATETangent","a":"Estimation/Efficiency","t":1,"d":"The efficiency bound lower-bounds every gradient's squared norm. Let S be a backdoor average-treatment-effect estimation system with strict overlap at level ε, satisfying the system's core identification assumptions, in which the observed outcome has finite second moment and each potential outcome under treatment level d has finite second moment. If ψ is a square-integrable element of L²(P_Z) that is a gradient for the AIPW influence function relative to the full mean-zero tangent space, then the semiparametric efficiency bound is at most the squared L² norm of ψ.","s":"∀ {P : Causalean.PO.POSystem} {γ : Type u_1} [inst : MeasurableSpace γ] [inst_1 : StandardBorelSpace P.Ω] [inst_2 : MeasureTheory.IsFiniteMeasure P.μ] (S : Causalean.Estimation.ATE.BackdoorEstimationSystem P γ) {ε : ℝ} (h_overlap : S.StrictOverlap ε) (hA : S.Assumptions) (h_y2 : MeasureTheory.Integrable (fun ω => S.factualY ω ^ 2) P.μ) (h_yd2 : ∀ (d : Bool), MeasureTheory.Integrable (fun ω => S.YofD d ω ^ 2) P.μ) (ψ : ↥(MeasureTheory.Lp ℝ 2 S.P_Z)), Causalean.Estimation.Efficiency.IsGradient S.Tfull (S.aipwLp h_overlap hA h_y2 h_yd2) ψ → Causalean.Estimation.Efficiency.effBound S.Tfull (S.aipwLp h_overlap hA h_y2 h_yd2) ≤ ‖ψ‖ ^ 2","l":578,"f":"Causalean/Estimation/Efficiency/ATETangent.lean"},{"n":"Causalean.Estimation.ATE.BackdoorEstimationSystem.factualZ","k":"def","m":"Causalean.Estimation.ATE.Setup","a":"Estimation/ATE","t":1,"d":"Data triple (X, A, Y) : Ω → γ × Bool × ℝ.","s":"{P : Causalean.PO.POSystem} → {γ : Type u_1} → [inst : MeasurableSpace γ] → [inst_1 : StandardBorelSpace P.Ω] → [inst_2 : MeasureTheory.IsFiniteMeasure P.μ] → Causalean.Estimation.ATE.BackdoorEstimationSystem P γ → P.Ω → γ × Bool × ℝ","l":125,"f":"Causalean/Estimation/ATE/Setup.lean"},{"n":"Causalean.Estimation.ATE.BackdoorEstimationSystem.indA","k":"def","m":"Causalean.Estimation.ATE.Score.AIPWMoment","a":"Estimation/ATE/Score","t":1,"d":"Real-valued indicator of {a = true} viewed as a function of the data.","s":"{γ : Type u_1} → γ × Bool × ℝ → ℝ","l":57,"f":"Causalean/Estimation/ATE/Score/AIPWMoment.lean"},{"n":"Causalean.Estimation.ATE.BackdoorEstimationSystem.indicator_to_propScore_integral","k":"theorem","m":"Causalean.Estimation.ATE.Score.ScorePullout","a":"Estimation/ATE/Score","t":1,"d":"Propensity-score pull-out for the treatment indicator. Fix a treatment label $d$, under the back-door identification assumptions. If f : γ → ℝ is measurable and the product f(X) · 1{D=d} is integrable, then replacing the treatment indicator 1{D=d} by the value-space propensity e_val_label d inside the integral leaves the integral unchanged: ∫ f(X) · 1{D=d} dμ = ∫ f(X) · e_val_label d(X) dμ.","s":"∀ {P : Causalean.PO.POSystem} {γ : Type u_1} [inst : MeasurableSpace γ] [inst_1 : StandardBorelSpace P.Ω] [inst_2 : MeasureTheory.IsFiniteMeasure P.μ] (S : Causalean.Estimation.ATE.BackdoorEstimationSystem P γ), S.Assumptions → ∀ (d : Bool) (f : γ → ℝ), Measurable f → MeasureTheory.Integrable (fun ω => f (S.factualX ω) * S.dVar.indicator d ω) P.μ → ∫ (ω : P.Ω), f (S.factualX ω) * S.dVar.indicator d ω ∂P.μ = ∫ (ω : P.Ω), f (S.factualX ω) * S.e_val_label d (S.factualX ω) ∂P.μ","l":225,"f":"Causalean/Estimation/ATE/Score/ScorePullout.lean"},{"n":"Causalean.Estimation.ATE.BackdoorEstimationSystem.inner_aipwLp_eq_integral","k":"theorem","m":"Causalean.Estimation.Efficiency.ATEEfficientIF","a":"Estimation/Efficiency","t":2,"d":"The covariance pairing between the AIPW influence function and any square-integrable score equals the Hilbert-space inner product against the AIPW element in the observed-data square-integrable space.","s":"∀ {P : Causalean.PO.POSystem} {γ : Type u_1} [inst : MeasurableSpace γ] [inst_1 : StandardBorelSpace P.Ω] [inst_2 : MeasureTheory.IsFiniteMeasure P.μ] (S : Causalean.Estimation.ATE.BackdoorEstimationSystem P γ) {ε : ℝ} (h_overlap : S.StrictOverlap ε) (hA : S.Assumptions) (h_y2 : MeasureTheory.Integrable (fun ω => S.factualY ω ^ 2) P.μ) (h_yd2 : ∀ (d : Bool), MeasureTheory.Integrable (fun ω => S.YofD d ω ^ 2) P.μ) (f : ↥(MeasureTheory.Lp ℝ 2 S.P_Z)), inner ℝ (S.aipwLp h_overlap hA h_y2 h_yd2) f = ∫ (z : γ × Bool × ℝ), S.ψ_AIPW z * ↑↑f z ∂S.P_Z","l":35,"f":"Causalean/Estimation/Efficiency/ATEEfficientIF.lean"},{"n":"Causalean.Estimation.ATE.BackdoorEstimationSystem.inner_oneLp","k":"theorem","m":"Causalean.Estimation.Efficiency.ATETangent","a":"Estimation/Efficiency","t":2,"d":"Inner product against the constant-one function equals integration under the observed data law.","s":"∀ {P : Causalean.PO.POSystem} {γ : Type u_1} [inst : MeasurableSpace γ] [inst_1 : StandardBorelSpace P.Ω] [inst_2 : MeasureTheory.IsFiniteMeasure P.μ] (S : Causalean.Estimation.ATE.BackdoorEstimationSystem P γ) (f : ↥(MeasureTheory.Lp ℝ 2 S.P_Z)), inner ℝ f S.oneLp = ∫ (z : γ × Bool × ℝ), ↑↑f z ∂S.P_Z","l":476,"f":"Causalean/Estimation/Efficiency/ATETangent.lean"},{"n":"Causalean.Estimation.ATE.BackdoorEstimationSystem.instHasOrthogonalProjection_Tfull","k":"instance","m":"Causalean.Estimation.Efficiency.ATETangent","a":"Estimation/Efficiency","t":2,"d":"The mean-zero tangent space has an orthogonal projection.","s":"∀ {P : Causalean.PO.POSystem} {γ : Type u_1} [inst : MeasurableSpace γ] [inst_1 : StandardBorelSpace P.Ω] [inst_2 : MeasureTheory.IsFiniteMeasure P.μ] (S : Causalean.Estimation.ATE.BackdoorEstimationSystem P γ), S.Tfull.HasOrthogonalProjection","l":469,"f":"Causalean/Estimation/Efficiency/ATETangent.lean"},{"n":"Causalean.Estimation.ATE.BackdoorEstimationSystem.instHasOrthogonalProjection_span_oneLp","k":"instance","m":"Causalean.Estimation.Efficiency.ATETangent","a":"Estimation/Efficiency","t":2,"d":"The span of the constant-one function has an orthogonal projection.","s":"∀ {P : Causalean.PO.POSystem} {γ : Type u_1} [inst : MeasurableSpace γ] [inst_1 : StandardBorelSpace P.Ω] [inst_2 : MeasureTheory.IsFiniteMeasure P.μ] (S : Causalean.Estimation.ATE.BackdoorEstimationSystem P γ), (ℝ ∙ S.oneLp).HasOrthogonalProjection","l":462,"f":"Causalean/Estimation/Efficiency/ATETangent.lean"},{"n":"Causalean.Estimation.ATE.BackdoorEstimationSystem.instIsProbabilityMeasure_P_Z","k":"instance","m":"Causalean.Estimation.Efficiency.ATETangent","a":"Estimation/Efficiency","t":2,"d":"The observed data law of a backdoor estimation system is a probability measure.","s":"∀ {P : Causalean.PO.POSystem} {γ : Type u_1} [inst : MeasurableSpace γ] [inst_1 : StandardBorelSpace P.Ω] [inst_2 : MeasureTheory.IsFiniteMeasure P.μ] (S : Causalean.Estimation.ATE.BackdoorEstimationSystem P γ), MeasureTheory.IsProbabilityMeasure S.P_Z","l":422,"f":"Causalean/Estimation/Efficiency/ATETangent.lean"},{"n":"Causalean.Estimation.ATE.BackdoorEstimationSystem.isTangentSpace_Tfull","k":"theorem","m":"Causalean.Estimation.Efficiency.ATEEfficientIF","a":"Estimation/Efficiency","t":2,"d":"The full mean-zero space is a genuine tangent space once the supplied nonparametric-model hypothesis says it is contained in the closed span of regular-submodel scores.","s":"∀ {P : Causalean.PO.POSystem} {γ : Type u_1} [inst : MeasurableSpace γ] [inst_1 : StandardBorelSpace P.Ω] [inst_2 : MeasureTheory.IsFiniteMeasure P.μ] (S : Causalean.Estimation.ATE.BackdoorEstimationSystem P γ), S.Tfull ≤ Causalean.Estimation.Efficiency.tangentSpace S.oneLp S.P_Z → Causalean.Estimation.Efficiency.IsTangentSpace S.oneLp S.P_Z S.Tfull","l":70,"f":"Causalean/Estimation/Efficiency/ATEEfficientIF.lean"},{"n":"Causalean.Estimation.ATE.BackdoorEstimationSystem.measurable_aipwMomentFunctional","k":"theorem","m":"Causalean.Estimation.ATE.Remainder.Identity","a":"Estimation/ATE/Remainder","t":2,"d":"The AIPW moment functional is measurable in the observed data triple for any fixed nuisance vector and target value.","s":"∀ {γ : Type u_1} [inst : MeasurableSpace γ] (η : Causalean.Estimation.ATE.NuisanceVec γ) (θ : ℝ), Measurable fun z => Causalean.Estimation.ATE.BackdoorEstimationSystem.aipwMomentFunctional η z θ","l":97,"f":"Causalean/Estimation/ATE/Remainder/Identity.lean"},{"n":"Causalean.Estimation.ATE.BackdoorEstimationSystem.measurable_e_val_label","k":"theorem","m":"Causalean.Estimation.ATE.Score.ScorePullout","a":"Estimation/ATE/Score","t":2,"d":"The value-space propensity for any treatment label is measurable.","s":"∀ {P : Causalean.PO.POSystem} {γ : Type u_1} [inst : MeasurableSpace γ] [inst_1 : StandardBorelSpace P.Ω] [inst_2 : MeasureTheory.IsFiniteMeasure P.μ] (S : Causalean.Estimation.ATE.BackdoorEstimationSystem P γ) (d : Bool), Measurable (S.e_val_label d)","l":46,"f":"Causalean/Estimation/ATE/Score/ScorePullout.lean"},{"n":"Causalean.Estimation.ATE.BackdoorEstimationSystem.measurable_factualZ","k":"theorem","m":"Causalean.Estimation.ATE.Setup","a":"Estimation/ATE","t":2,"d":"The observed covariate, treatment, and outcome triple is measurable.","s":"∀ {P : Causalean.PO.POSystem} {γ : Type u_1} [inst : MeasurableSpace γ] [inst_1 : StandardBorelSpace P.Ω] [inst_2 : MeasureTheory.IsFiniteMeasure P.μ] (S : Causalean.Estimation.ATE.BackdoorEstimationSystem P γ), Measurable S.factualZ","l":132,"f":"Causalean/Estimation/ATE/Setup.lean"},{"n":"Causalean.Estimation.ATE.BackdoorEstimationSystem.measurable_ψ_AIPW","k":"theorem","m":"Causalean.Estimation.ATE.Score.MeanZero","a":"Estimation/ATE/Score","t":2,"d":"The AIPW influence function is measurable as a function of the observed data triple.","s":"∀ {P : Causalean.PO.POSystem} {γ : Type u_1} [inst : MeasurableSpace γ] [inst_1 : StandardBorelSpace P.Ω] [inst_2 : MeasureTheory.IsFiniteMeasure P.μ] (S : Causalean.Estimation.ATE.BackdoorEstimationSystem P γ), Measurable S.ψ_AIPW","l":47,"f":"Causalean/Estimation/ATE/Score/MeanZero.lean"},{"n":"Causalean.Estimation.ATE.BackdoorEstimationSystem.oneLp","k":"def","m":"Causalean.Estimation.Efficiency.ATETangent","a":"Estimation/Efficiency","t":1,"d":"This is the constant-one function represented in the square-integrable Hilbert space.","s":"{P : Causalean.PO.POSystem} → {γ : Type u_1} → [inst : MeasurableSpace γ] → [inst_1 : StandardBorelSpace P.Ω] → [inst_2 : MeasureTheory.IsFiniteMeasure P.μ] → (S : Causalean.Estimation.ATE.BackdoorEstimationSystem P γ) → ↥(MeasureTheory.Lp ℝ 2 S.P_Z)","l":450,"f":"Causalean/Estimation/Efficiency/ATETangent.lean"},{"n":"Causalean.Estimation.ATE.BackdoorEstimationSystem.plugin_bias_le_eLpNorm","k":"theorem","m":"Causalean.Estimation.ATE.Remainder.Identity","a":"Estimation/ATE/Remainder","t":2,"d":"Plug-in bias bound. The integrated plug-in bias in component a satisfies |∫ (μ̂(a,·) − μ_val(a,·)) dP_X| ≤ ‖μ̂(a,·) − μ_val(a,·)‖_{L²(P_X)}.","s":"∀ {P : Causalean.PO.POSystem} {γ : Type u_1} [inst : MeasurableSpace γ] [inst_1 : StandardBorelSpace P.Ω] [inst_2 : MeasureTheory.IsFiniteMeasure P.μ] (S : Causalean.Estimation.ATE.BackdoorEstimationSystem P γ) [MeasureTheory.IsProbabilityMeasure P.μ] {μ_fn : Bool → γ → ℝ}, (∀ (a : Bool), Measurable (μ_fn a)) → (∀ (a : Bool), MeasureTheory.MemLp (fun x => μ_fn a x - S.μ_val a x) 2 S.P_X) → ∀ (a : Bool), |∫ (x : γ), μ_fn a x - S.μ_val a x ∂S.P_X| ≤ (MeasureTheory.eLpNorm (fun x => μ_fn a x - S.μ_val a x) 2 S.P_X).toReal","l":59,"f":"Causalean/Estimation/ATE/Remainder/Identity.lean"},{"n":"Causalean.Estimation.ATE.BackdoorEstimationSystem.projA","k":"def","m":"Causalean.Estimation.ATE.Score.AIPWMoment","a":"Estimation/ATE/Score","t":1,"d":"Projection (x, a, y) ↦ a.","s":"{γ : Type u_1} → γ × Bool × ℝ → Bool","l":51,"f":"Causalean/Estimation/ATE/Score/AIPWMoment.lean"},{"n":"Causalean.Estimation.ATE.BackdoorEstimationSystem.projX","k":"def","m":"Causalean.Estimation.ATE.Score.AIPWMoment","a":"Estimation/ATE/Score","t":1,"d":"Projection (x, a, y) ↦ x.","s":"{γ : Type u_1} → γ × Bool × ℝ → γ","l":48,"f":"Causalean/Estimation/ATE/Score/AIPWMoment.lean"},{"n":"Causalean.Estimation.ATE.BackdoorEstimationSystem.projY","k":"def","m":"Causalean.Estimation.ATE.Score.AIPWMoment","a":"Estimation/ATE/Score","t":1,"d":"Projection (x, a, y) ↦ y.","s":"{γ : Type u_1} → γ × Bool × ℝ → ℝ","l":54,"f":"Causalean/Estimation/ATE/Score/AIPWMoment.lean"},{"n":"Causalean.Estimation.ATE.BackdoorEstimationSystem.propScore_eq_e_val_label_ae","k":"theorem","m":"Causalean.Estimation.ATE.Score.ScorePullout","a":"Estimation/ATE/Score","t":2,"d":"The σ(X)-conditional expectation of 1_{D=d} is e_val_label d (X) a.s.","s":"∀ {P : Causalean.PO.POSystem} {γ : Type u_1} [inst : MeasurableSpace γ] [inst_1 : StandardBorelSpace P.Ω] [inst_2 : MeasureTheory.IsFiniteMeasure P.μ] (S : Causalean.Estimation.ATE.BackdoorEstimationSystem P γ), S.Assumptions → ∀ (d : Bool), S.propScore d =ᵐ[P.μ] fun ω => S.e_val_label d (S.factualX ω)","l":99,"f":"Causalean/Estimation/ATE/Score/ScorePullout.lean"},{"n":"Causalean.Estimation.ATE.BackdoorEstimationSystem.propScore_false_ae","k":"theorem","m":"Causalean.Estimation.ATE.Score.ScorePullout","a":"Estimation/ATE/Score","t":2,"d":"propScore false =ᵐ 1 - propScore true under back-door assumptions. The indicator-pair sums to one pointwise, conditional expectation is linear, and preserves constants.","s":"∀ {P : Causalean.PO.POSystem} {γ : Type u_1} [inst : MeasurableSpace γ] [inst_1 : StandardBorelSpace P.Ω] [inst_2 : MeasureTheory.IsFiniteMeasure P.μ] (S : Causalean.Estimation.ATE.BackdoorEstimationSystem P γ), S.Assumptions → S.propScore false =ᵐ[P.μ] fun ω => 1 - S.propScore true ω","l":53,"f":"Causalean/Estimation/ATE/Score/ScorePullout.lean"},{"n":"Causalean.Estimation.ATE.BackdoorEstimationSystem.propScore_ne_zero","k":"theorem","m":"Causalean.Estimation.ATE.Score.MeanZero","a":"Estimation/ATE/Score","t":2,"d":"The conditional treatment probability for either treatment label is nonzero almost surely under the back-door assumptions.","s":"∀ {P : Causalean.PO.POSystem} {γ : Type u_1} [inst : MeasurableSpace γ] [inst_1 : StandardBorelSpace P.Ω] [inst_2 : MeasureTheory.IsFiniteMeasure P.μ] (S : Causalean.Estimation.ATE.BackdoorEstimationSystem P γ), S.Assumptions → ∀ (d : Bool), ∀ᵐ (ω : P.Ω) ∂P.μ, S.propScore d ω ≠ 0","l":70,"f":"Causalean/Estimation/ATE/Score/MeanZero.lean"},{"n":"Causalean.Estimation.ATE.BackdoorEstimationSystem.score_mem_Tfull","k":"theorem","m":"Causalean.Estimation.Efficiency.ATEEfficientIF","a":"Estimation/Efficiency","t":2,"d":"Every regular submodel score is mean-zero, so every such score belongs to the full nonparametric tangent space for the observed-data law.","s":"∀ {P : Causalean.PO.POSystem} {γ : Type u_1} [inst : MeasurableSpace γ] [inst_1 : StandardBorelSpace P.Ω] [inst_2 : MeasureTheory.IsFiniteMeasure P.μ] (S : Causalean.Estimation.ATE.BackdoorEstimationSystem P γ) (m : Causalean.Estimation.Efficiency.RegularSubmodel S.oneLp S.P_Z), m.score ∈ S.Tfull","l":59,"f":"Causalean/Estimation/Efficiency/ATEEfficientIF.lean"},{"n":"Causalean.Estimation.ATE.BackdoorEstimationSystem.theta_zero_factualX_integral","k":"theorem","m":"Causalean.Estimation.ATE.Score.MeanZero","a":"Estimation/ATE/Score","t":2,"d":"The estimand θ₀ = ∫ x, (μ_val(1,x) − μ_val(0,x)) ∂P_X lifts back to the ambient measure via factualX.","s":"∀ {P : Causalean.PO.POSystem} {γ : Type u_1} [inst : MeasurableSpace γ] [inst_1 : StandardBorelSpace P.Ω] [inst_2 : MeasureTheory.IsFiniteMeasure P.μ] (S : Causalean.Estimation.ATE.BackdoorEstimationSystem P γ), S.θ₀ = ∫ (ω : P.Ω), S.μ_val true (S.factualX ω) - S.μ_val false (S.factualX ω) ∂P.μ","l":259,"f":"Causalean/Estimation/ATE/Score/MeanZero.lean"},{"n":"Causalean.Estimation.ATE.BackdoorEstimationSystem.weighted_residual_integral_zero","k":"theorem","m":"Causalean.Estimation.ATE.Score.ScorePullout","a":"Estimation/ATE/Score","t":1,"d":"Weighted-residual mean-zero identity (pull-out lemma). Fix a treatment label $d$ and a measurable weight function g : γ → ℝ on the covariates, under the back-door identification assumptions. If the product g(X) · 1{D=d} · (Y − μ(d,X)) is integrable and the σ(X)-conditional expectation of the treatment-d residual 1{D=d} · (Y − μ(d,X)) vanishes almost surely, then the integral of the weighted residual against the observed-data law vanishes: ∫ g(X) · 1{D=d} · (Y − μ(d,X)) dμ = 0.","s":"∀ {P : Causalean.PO.POSystem} {γ : Type u_1} [inst : MeasurableSpace γ] [inst_1 : StandardBorelSpace P.Ω] [inst_2 : MeasureTheory.IsFiniteMeasure P.μ] (S : Causalean.Estimation.ATE.BackdoorEstimationSystem P γ), S.Assumptions → ∀ (d : Bool) (g : γ → ℝ), Measurable g → MeasureTheory.Integrable (fun ω => g (S.factualX ω) * (S.dVar.indicator d ω * (S.factualY ω - S.μ_val d (S.factualX ω)))) P.μ → (P.μ[fun ω => S.dVar.indicator d ω * (S.factualY ω - S.μ_val d (S.factualX ω)) | S.sigmaX] =ᵐ[P.μ] fun x => 0) → ∫ (ω : P.Ω), g (S.factualX ω) * (S.dVar.indicator d ω * (S.factualY ω - S.μ_val d (S.factualX ω))) ∂P.μ = 0","l":114,"f":"Causalean/Estimation/ATE/Score/ScorePullout.lean"},{"n":"Causalean.Estimation.ATE.BackdoorEstimationSystem.yMuVal_residual_sq_integrable","k":"theorem","m":"Causalean.Estimation.ATE.Score.AIPWScoreL2","a":"Estimation/ATE/Score","t":2,"d":"The cross-term integrand is P_Z-integrable, with the bound coming from (a+b)² ≤ 2(a² + b²) and Y² ∈ L¹(P_Z), μ_val(d, X)² ∈ L¹(P_Z).","s":"∀ {P : Causalean.PO.POSystem} {γ : Type u_1} [inst : MeasurableSpace γ] [inst_1 : StandardBorelSpace P.Ω] [inst_2 : MeasureTheory.IsFiniteMeasure P.μ] (S : Causalean.Estimation.ATE.BackdoorEstimationSystem P γ), S.Assumptions → MeasureTheory.Integrable (fun ω => S.factualY ω ^ 2) P.μ → (∀ (d : Bool), MeasureTheory.Integrable (fun ω => S.YofD d ω ^ 2) P.μ) → MeasureTheory.Integrable S.YMuVal_residual_sq S.P_Z","l":375,"f":"Causalean/Estimation/ATE/Score/AIPWScoreL2.lean"},{"n":"Causalean.Estimation.ATE.BackdoorEstimationSystem.η₀","k":"def","m":"Causalean.Estimation.ATE.Score.AIPWMoment","a":"Estimation/ATE/Score","t":1,"d":"The truth η₀ : NuisanceVec γ.","s":"{P : Causalean.PO.POSystem} → {γ : Type u_1} → [inst : MeasurableSpace γ] → [inst_1 : StandardBorelSpace P.Ω] → [inst_2 : MeasureTheory.IsFiniteMeasure P.μ] → Causalean.Estimation.ATE.BackdoorEstimationSystem P γ → Causalean.Estimation.ATE.NuisanceVec γ","l":254,"f":"Causalean/Estimation/ATE/Score/AIPWMoment.lean"},{"n":"Causalean.Estimation.ATE.BackdoorEstimationSystem.θ₀","k":"def","m":"Causalean.Estimation.ATE.Setup","a":"Estimation/ATE","t":1,"d":"Value-space ATE: θ₀ = ∫ (μ(1, x) - μ(0, x)) dP_X.","s":"{P : Causalean.PO.POSystem} → {γ : Type u_1} → [inst : MeasurableSpace γ] → [inst_1 : StandardBorelSpace P.Ω] → [inst_2 : MeasureTheory.IsFiniteMeasure P.μ] → Causalean.Estimation.ATE.BackdoorEstimationSystem P γ → ℝ","l":156,"f":"Causalean/Estimation/ATE/Setup.lean"},{"n":"Causalean.Estimation.ATE.BackdoorEstimationSystem.θ₀_eq_ATE","k":"theorem","m":"Causalean.Estimation.ATE.Setup","a":"Estimation/ATE","t":1,"d":"Value-space estimand equals the potential-outcome ATE. Under the back-door identification assumptions, the value-space estimand θ₀ = ∫ (μ(1,x) − μ(0,x)) dP_X, built from the outcome-regression nuisance, coincides with the average treatment effect defined on potential outcomes.","s":"∀ {P : Causalean.PO.POSystem} {γ : Type u_1} [inst : MeasurableSpace γ] [inst_1 : StandardBorelSpace P.Ω] [inst_2 : MeasureTheory.IsFiniteMeasure P.μ] (S : Causalean.Estimation.ATE.BackdoorEstimationSystem P γ), S.Assumptions → S.θ₀ = S.ATE","l":160,"f":"Causalean/Estimation/ATE/Setup.lean"},{"n":"Causalean.Estimation.ATE.BackdoorEstimationSystem.μ_compat","k":"theorem","m":"Causalean.Estimation.ATE.Setup","a":"Estimation/ATE","t":2,"d":"Counterfactual compatibility (derived, under identification). Recovers the former μ_compat field — the *counterfactual* reading μ[Y(d) | σ(X)] =ᵐ μ_val d ∘ factualX — now as a theorem rather than an assumption. It is the observable μ_reg_compat (μ_val d ∘ factualX =ᵐ adjustedCE d) composed with back-door identification (cate_backdoor : μ[Y(d)|σX] =ᵐ adjustedCE d), so the counterfactual binding is NOT part of the estimation system's data: it holds only under Assumptions. Every downstream proof that used the old field calls this with the ambient hA.","s":"∀ {P : Causalean.PO.POSystem} {γ : Type u_1} [inst : MeasurableSpace γ] [inst_1 : StandardBorelSpace P.Ω] [inst_2 : MeasureTheory.IsFiniteMeasure P.μ] (S : Causalean.Estimation.ATE.BackdoorEstimationSystem P γ), S.Assumptions → ∀ (d : Bool), P.μ[S.YofD d | S.sigmaX] =ᵐ[P.μ] fun ω => S.μ_val d (S.factualX ω)","l":98,"f":"Causalean/Estimation/ATE/Setup.lean"},{"n":"Causalean.Estimation.ATE.BackdoorEstimationSystem.ψ_AIPW","k":"def","m":"Causalean.Estimation.ATE.Score.AIPWMoment","a":"Estimation/ATE/Score","t":1,"d":"The AIPW influence function at the truth: ψ_AIPW(z) := m_AIPW(η₀, z, θ₀) with η₀ = (μ_val, e_val).","s":"{P : Causalean.PO.POSystem} → {γ : Type u_1} → [inst : MeasurableSpace γ] → [inst_1 : StandardBorelSpace P.Ω] → [inst_2 : MeasureTheory.IsFiniteMeasure P.μ] → Causalean.Estimation.ATE.BackdoorEstimationSystem P γ → γ × Bool × ℝ → ℝ","l":71,"f":"Causalean/Estimation/ATE/Score/AIPWMoment.lean"},{"n":"Causalean.Estimation.ATE.NuisanceVec","k":"structure","m":"Causalean.Estimation.ATE.Score.AIPWMoment","a":"Estimation/ATE/Score","t":1,"d":"A pair of value-space AIPW nuisance functions, used as the abstract nuisance space for the AIPW moment functional: a treatment-arm-indexed outcome regression and a propensity score, together with their measurability.","s":"(γ : Type u_1) → [MeasurableSpace γ] → Type u_1","l":86,"f":"Causalean/Estimation/ATE/Score/AIPWMoment.lean"},{"n":"Causalean.Estimation.ATE.NuisanceVec.ext","k":"theorem","m":"Causalean.Estimation.ATE.Score.AIPWMoment","a":"Estimation/ATE/Score","t":2,"d":"Two AIPW nuisance vectors are equal when their outcome regressions and propensity functions agree everywhere.","s":"∀ {γ : Type u_1} [inst : MeasurableSpace γ] {η η' : Causalean.Estimation.ATE.NuisanceVec γ}, (∀ (b : Bool) (x : γ), η.μ_fn b x = η'.μ_fn b x) → (∀ (x : γ), η.e_fn x = η'.e_fn x) → η = η'","l":136,"f":"Causalean/Estimation/ATE/Score/AIPWMoment.lean"},{"n":"Causalean.Estimation.ATE.NuisanceVec.ext_iff","k":"theorem","m":"Causalean.Estimation.ATE.Score.AIPWMoment","a":"Estimation/ATE/Score","t":2,"d":"","s":"∀ {γ : Type u_1} [inst : MeasurableSpace γ] {η η' : Causalean.Estimation.ATE.NuisanceVec γ}, η = η' ↔ (∀ (b : Bool) (x : γ), η.μ_fn b x = η'.μ_fn b x) ∧ ∀ (x : γ), η.e_fn x = η'.e_fn x","l":138,"f":"Causalean/Estimation/ATE/Score/AIPWMoment.lean"},{"n":"Causalean.Estimation.ATE.NuisanceVec.instAdd","k":"instance","m":"Causalean.Estimation.ATE.Score.AIPWMoment","a":"Estimation/ATE/Score","t":2,"d":"Addition of nuisance vectors is componentwise addition of the outcome regressions and propensity function.","s":"{γ : Type u_1} → [inst : MeasurableSpace γ] → Add (Causalean.Estimation.ATE.NuisanceVec γ)","l":104,"f":"Causalean/Estimation/ATE/Score/AIPWMoment.lean"},{"n":"Causalean.Estimation.ATE.NuisanceVec.instAddCommGroup","k":"instance","m":"Causalean.Estimation.ATE.Score.AIPWMoment","a":"Estimation/ATE/Score","t":2,"d":"Nuisance vectors form an additive commutative group under componentwise operations.","s":"{γ : Type u_1} → [inst : MeasurableSpace γ] → AddCommGroup (Causalean.Estimation.ATE.NuisanceVec γ)","l":151,"f":"Causalean/Estimation/ATE/Score/AIPWMoment.lean"},{"n":"Causalean.Estimation.ATE.NuisanceVec.instModuleReal","k":"instance","m":"Causalean.Estimation.ATE.Score.AIPWMoment","a":"Estimation/ATE/Score","t":2,"d":"Nuisance vectors form a real vector space under componentwise scalar multiplication.","s":"{γ : Type u_1} → [inst : MeasurableSpace γ] → Module ℝ (Causalean.Estimation.ATE.NuisanceVec γ)","l":200,"f":"Causalean/Estimation/ATE/Score/AIPWMoment.lean"},{"n":"Causalean.Estimation.ATE.NuisanceVec.instNeg","k":"instance","m":"Causalean.Estimation.ATE.Score.AIPWMoment","a":"Estimation/ATE/Score","t":2,"d":"Negation of a nuisance vector negates each outcome regression and the propensity function.","s":"{γ : Type u_1} → [inst : MeasurableSpace γ] → Neg (Causalean.Estimation.ATE.NuisanceVec γ)","l":113,"f":"Causalean/Estimation/ATE/Score/AIPWMoment.lean"},{"n":"Causalean.Estimation.ATE.NuisanceVec.instSMulReal","k":"instance","m":"Causalean.Estimation.ATE.Score.AIPWMoment","a":"Estimation/ATE/Score","t":2,"d":"Scalar multiplication of a nuisance vector scales each outcome regression and the propensity function.","s":"{γ : Type u_1} → [inst : MeasurableSpace γ] → SMul ℝ (Causalean.Estimation.ATE.NuisanceVec γ)","l":128,"f":"Causalean/Estimation/ATE/Score/AIPWMoment.lean"},{"n":"Causalean.Estimation.ATE.NuisanceVec.instSub","k":"instance","m":"Causalean.Estimation.ATE.Score.AIPWMoment","a":"Estimation/ATE/Score","t":2,"d":"Subtraction of nuisance vectors is componentwise subtraction of the outcome regressions and propensity function.","s":"{γ : Type u_1} → [inst : MeasurableSpace γ] → Sub (Causalean.Estimation.ATE.NuisanceVec γ)","l":119,"f":"Causalean/Estimation/ATE/Score/AIPWMoment.lean"},{"n":"Causalean.Estimation.ATE.NuisanceVec.instZero","k":"instance","m":"Causalean.Estimation.ATE.Score.AIPWMoment","a":"Estimation/ATE/Score","t":2,"d":"The zero nuisance sets both outcome regressions and the propensity function to zero.","s":"{γ : Type u_1} → [inst : MeasurableSpace γ] → Zero (Causalean.Estimation.ATE.NuisanceVec γ)","l":99,"f":"Causalean/Estimation/ATE/Score/AIPWMoment.lean"},{"n":"Causalean.Estimation.ATE.aipwGeneralMoment","k":"def","m":"Causalean.Estimation.OrthogonalMoments.AIPWInstance","a":"Estimation/OrthogonalMoments","t":1,"d":"AIPW instance of the abstract GeneralMoment. The bilinear seminorms are the L²(P_X) norms of the μ_fn and e_fn differences; ρ₁ aggregates both treatment arms of μ_fn (matching the Σ_a ‖Δμ_a‖ factor produced by aipw_remainder_bound).","s":"{P : Causalean.PO.POSystem} → {γ : Type u_1} → [inst : MeasurableSpace γ] → [inst_1 : StandardBorelSpace P.Ω] → [inst_2 : MeasureTheory.IsFiniteMeasure P.μ] → (S : Causalean.Estimation.ATE.BackdoorEstimationSystem P γ) → {ε : ℝ} → S.η₀ ∈ S.H_ε_aeL2 ε → Causalean.Estimation.OrthogonalMoments.GeneralMoment P.Ω P.μ (γ × Bool × ℝ) S.P_Z (Causalean.Estimation.ATE.NuisanceVec γ)","l":56,"f":"Causalean/Estimation/OrthogonalMoments/AIPWInstance.lean"},{"n":"Causalean.Estimation.ATE.aipw_bilinearRem","k":"theorem","m":"Causalean.Estimation.OrthogonalMoments.AIPWInstance","a":"Estimation/OrthogonalMoments","t":2,"d":"AIPW satisfies BilinearRemainder with constant aipw_rem_const ε.","s":"∀ {P : Causalean.PO.POSystem} {γ : Type u_1} [inst : MeasurableSpace γ] [inst_1 : StandardBorelSpace P.Ω] [inst_2 : MeasureTheory.IsFiniteMeasure P.μ] (S : Causalean.Estimation.ATE.BackdoorEstimationSystem P γ) {ε : ℝ} (hη₀_mem : S.η₀ ∈ S.H_ε_aeL2 ε), S.StrictOverlap ε → S.Assumptions → MeasureTheory.Integrable (fun ω => S.factualY ω ^ 2) P.μ → (∀ (d : Bool), MeasureTheory.Integrable (fun ω => S.YofD d ω ^ 2) P.μ) → (∀ η ∈ S.H_ε_aeL2 ε, (∀ (d : Bool), MeasureTheory.MemLp (fun x => η.μ_fn d x - S.μ_val d x) 2 S.P_X) ∧ MeasureTheory.MemLp (fun x => η.e_fn x - S.e_val x) 2 S.P_X) → ∃ C, Causalean.Estimation.OrthogonalMoments.BilinearRemainder (Causalean.Estimation.ATE.aipwGeneralMoment S hη₀_mem) C","l":96,"f":"Causalean/Estimation/OrthogonalMoments/AIPWInstance.lean"},{"n":"Causalean.Estimation.ATE.aipw_dml_isAsymLinear","k":"theorem","m":"Causalean.Estimation.OrthogonalMoments.AIPWInstance","a":"Estimation/OrthogonalMoments","t":1,"d":"Headline AIPW DML asymptotic-linearity theorem, derived from the abstract dml_chernozhukov_asymptoticLinear in Estimation/OrthogonalMoments/DMLChernozhukov.lean. For the back-door AIPW estimator with true nuisance η₀ known to lie in the ε-ball H_ε_aeL2 S ε, assume the propensity score has ε-strict overlap, that the identification assumptions of the back-door system hold, and that the factual and potential outcomes are square-integrable. Given an i.i.d. sample with a one-shot fold split whose fold-B fraction converges to a strictly positive limit c > 0 along card (foldB n) / n → c, and a sequence of cross-fitted nuisance estimators η̂ that stay in the ε-ball at every fold and sample point with outcome-regression and propensity-score differences from the truth square-integrable in S.P_X. Assume the technical regularity package that the AIPW moment functional at η̂ is jointly measurable, fold-A-measurable in ω, and, at every fold and sample point, integrable and square-integrable under S.P_Z. Finally suppose the two nuisance-error rates are individually negligible ρ₁(η̂, η₀) = o_P(1), ρ₂(η̂, η₀) = o_P(1), and their product decays at the parametric rate ρ₁(η̂, η₀) · ρ₂(η̂, η₀) = o_P(n^{-1/2}). Then the Chernozhukov one-step AIPW-DML estimator is asymptotically linear at the true parameter S.θ₀ with the standard AIPW influence function ψ(z) = −J₀⁻¹ · ψ_AIPW(η₀, z), indexed over the fold-B subsample.","s":"∀ {P : Causalean.PO.POSystem} {γ : Type u_1} [inst : MeasurableSpace γ] [inst_1 : StandardBorelSpace P.Ω] [inst_2 : MeasureTheory.IsFiniteMeasure P.μ] (S : Causalean.Estimation.ATE.BackdoorEstimationSystem P γ) {ε : ℝ} (hη₀_mem : S.η₀ ∈ S.H_ε_aeL2 ε), S.StrictOverlap ε → S.Assumptions → MeasureTheory.Integrable (fun ω => S.factualY ω ^ 2) P.μ → (∀ (d : Bool), MeasureTheory.Integrable (fun ω => S.YofD d ω ^ 2) P.μ) → ∀ (sample : Causalean.Stat.IIDSample P.Ω (γ × Bool × ℝ) P.μ S.P_Z) (split : Causalean.Stat.OneShotSplit sample) {c : ℝ}, 0 < c → Filter.Tendsto (fun n => ↑(split.foldB n).card / ↑n) Filter.atTop (nhds c) → ∀ (η_hat : ℕ → P.Ω → Causalean.Estimation.ATE.NuisanceVec γ), (∀ (n : ℕ) (ω : P.Ω), η_hat n ω ∈ S.H_ε_aeL2 ε) → (∀ (n : ℕ) (ω : P.Ω) (a : Bool), MeasureTheory.MemLp (fun x => (η_hat n ω).μ_fn a x - S.μ_val a x) 2 S.P_X) → (∀ (n : ℕ) (ω : P.Ω), MeasureTheory.MemLp (fun x => (η_hat n ω).e_fn x - S.e_val x) 2 S.P_X) → (∀ (n : ℕ), Measurable fun p => Causalean.Estimation.ATE.BackdoorEstimationSystem.aipwMomentFunctional (η_hat n p.1) p.2 S.θ₀) → (∀ (n : ℕ), Measurable fun ω z => Causalean.Estimation.ATE.BackdoorEstimationSystem.aipwMomentFunctional (η_hat n ω) z S.θ₀) → (∀ (n : ℕ), Measurable fun p => Causalean.Estimation.ATE.BackdoorEstimationSystem.aipwMomentFunctional (η_hat n p.1) p.2 S.θ₀) → (∀ (n : ℕ) (ω : P.Ω), MeasureTheory.Integrable (fun z => Causalean.Estimation.ATE.BackdoorEstimationSystem.aipwMomentFunctional (η_hat n ω) z S.θ₀) S.P_Z) → (∀ (n : ℕ) (ω : P.Ω), MeasureTheory.Integrable (fun z => Causalean.Estimation.ATE.BackdoorEstimationSystem.aipwMomentFunctional (η_hat n ω) z S.θ₀ ^ 2) S.P_Z) → Causalean.Stat.IsLittleOp (fun n ω => ↑((Causalean.Estimation.ATE.aipwGeneralMoment S hη₀_mem).ρ₁ (η_hat n ω) S.η₀)) (fun x => 1) P.μ → Causalean.Stat.IsLittleOp (fun n ω => ↑((Causalean.Estimation.ATE.aipwGeneralMoment S hη₀_mem).ρ₂ (η_hat n ω) S.η₀)) (fun x => 1) P.μ → Causalean.Stat.IsLittleOp (fun n ω => ↑((Causalean.Estimation.ATE.aipwGeneralMoment S hη₀_mem).ρ₁ (η_hat n ω) S.η₀) * ↑((Causalean.Estimation.ATE.aipwGeneralMoment S hη₀_mem).ρ₂ (η_hat n ω) S.η₀)) (fun n => ↑n ^ (-(1 / 2))) P.μ → Causalean.Stat.IsAsymLinear (Causalean.Estimation.OrthogonalMoments.dmlChernozhukovEstimator (Causalean.Estimation.ATE.aipwGeneralMoment S hη₀_mem) sample split η_hat) S.θ₀ (fun z => -(Causalean.Estimation.ATE.aipwGeneralMoment S hη₀_mem).J₀_inv * Causalean.Estimation.ATE.BackdoorEstimationSystem.aipwMomentFunctional S.η₀ z S.θ₀) sample split.foldB","l":132,"f":"Causalean/Estimation/OrthogonalMoments/AIPWInstance.lean"},{"n":"Causalean.Estimation.ATE.aipw_meanZero","k":"theorem","m":"Causalean.Estimation.OrthogonalMoments.AIPWInstance","a":"Estimation/OrthogonalMoments","t":2,"d":"AIPW satisfies MeanZero.","s":"∀ {P : Causalean.PO.POSystem} {γ : Type u_1} [inst : MeasurableSpace γ] [inst_1 : StandardBorelSpace P.Ω] [inst_2 : MeasureTheory.IsFiniteMeasure P.μ] (S : Causalean.Estimation.ATE.BackdoorEstimationSystem P γ) {ε : ℝ} (hη₀_mem : S.η₀ ∈ S.H_ε_aeL2 ε), S.StrictOverlap ε → S.Assumptions → MeasureTheory.Integrable (fun ω => S.factualY ω ^ 2) P.μ → (∀ (d : Bool), MeasureTheory.Integrable (fun ω => S.YofD d ω ^ 2) P.μ) → Causalean.Estimation.OrthogonalMoments.MeanZero (Causalean.Estimation.ATE.aipwGeneralMoment S hη₀_mem)","l":83,"f":"Causalean/Estimation/OrthogonalMoments/AIPWInstance.lean"},{"n":"Causalean.Estimation.ATE.dmlEstimator","k":"def","m":"Causalean.Estimation.ATE.DML","a":"Estimation/ATE","t":1,"d":"One-shot DML / AIPW estimator of the back-door ATE (def:est-dml-ate).","s":"{P : Causalean.PO.POSystem} → {γ : Type u_1} → [inst : MeasurableSpace γ] → [inst_1 : StandardBorelSpace P.Ω] → [inst_2 : MeasureTheory.IsFiniteMeasure P.μ] → (S : Causalean.Estimation.ATE.BackdoorEstimationSystem P γ) → (sample : Causalean.Stat.IIDSample P.Ω (γ × Bool × ℝ) P.μ S.P_Z) → Causalean.Stat.OneShotSplit sample → (ℕ → P.Ω → Bool → γ → ℝ) → (ℕ → P.Ω → γ → ℝ) → ℕ → P.Ω → ℝ","l":62,"f":"Causalean/Estimation/ATE/DML.lean"},{"n":"Causalean.Estimation.ATE.dml_ATE_attains_hahn_bound","k":"theorem","m":"Causalean.Estimation.Efficiency.ATEVariance","a":"Estimation/Efficiency","t":1,"d":"The one-shot DML ATE attains the Hahn efficiency bound. Assume the back-door identifying assumptions and strict overlap of the true propensity score with margin ε, with the factual outcome and every potential outcome Y(d) square-integrable. Let sample be an i.i.d. draw of the observed data and split a one-shot fold split whose training-fold share converges to a limit c strictly between 0 and 1. For the nuisance estimators μ_hat, e_hat fit on the training fold, assume each is jointly measurable in the training data and the evaluation point and square-integrable against P_X, and each is a measurable function of the training fold alone, that the estimated propensity score also satisfies overlap with margin ε, that each nuisance estimator is L²(P_X)-consistent, and that the product of the two L² estimation errors is o_P(n^{-1/2}) — the Neyman-orthogonality rate condition. Assume finally the AIPW influence function, the rescaled estimator, and the normalized influence-function sum are all measurable. Then the rescaled one-shot DML ATE estimator converges in distribution to the mean-zero Gaussian law whose variance is the Hahn semiparametric efficiency bound.","s":"∀ {P : Causalean.PO.POSystem} {γ : Type u_1} [inst : MeasurableSpace γ] [inst_1 : StandardBorelSpace P.Ω] [inst_2 : MeasureTheory.IsFiniteMeasure P.μ] (S : Causalean.Estimation.ATE.BackdoorEstimationSystem P γ) {ε : ℝ}, S.Assumptions → S.StrictOverlap ε → MeasureTheory.Integrable (fun ω => S.factualY ω ^ 2) P.μ → (∀ (d : Bool), MeasureTheory.Integrable (fun ω => S.YofD d ω ^ 2) P.μ) → ∀ (sample : Causalean.Stat.IIDSample P.Ω (γ × Bool × ℝ) P.μ S.P_Z) (split : Causalean.Stat.OneShotSplit sample) {c : ℝ}, 0 < c → c < 1 → Filter.Tendsto (fun n => ↑(split.foldB n).card / ↑n) Filter.atTop (nhds c) → ∀ (μ_hat : ℕ → P.Ω → Bool → γ → ℝ) (e_hat : ℕ → P.Ω → γ → ℝ), (∀ (n : ℕ) (a : Bool), Measurable fun p => μ_hat n p.1 a p.2) → (∀ (n : ℕ), Measurable fun p => e_hat n p.1 p.2) → (∀ (n : ℕ) (ω : P.Ω) (a : Bool), MeasureTheory.MemLp (fun x => μ_hat n ω a x) 2 S.P_X) → (∀ (n : ℕ) (ω : P.Ω), MeasureTheory.MemLp (fun x => e_hat n ω x) 2 S.P_X) → (∀ (n : ℕ) (ω : P.Ω), ∀ᵐ (x : γ) ∂S.P_X, ε ≤ e_hat n ω x ∧ e_hat n ω x ≤ 1 - ε) → (∀ (n : ℕ), Measurable (μ_hat n)) → (∀ (n : ℕ), Measurable (e_hat n)) → (∀ (n : ℕ) (a : Bool), Measurable fun p => μ_hat n p.1 a p.2) → (∀ (n : ℕ), Measurable fun p => e_hat n p.1 p.2) → (∀ (a : Bool), Causalean.Stat.IsLittleOp (fun n ω => (MeasureTheory.eLpNorm (fun x => μ_hat n ω a x - S.μ_val a x) 2 S.P_X).toReal) (fun x => 1) P.μ) → Causalean.Stat.IsLittleOp (fun n ω => (MeasureTheory.eLpNorm (fun x => e_hat n ω x - S.e_val x) 2 S.P_X).toReal) (fun x => 1) P.μ → (∀ (a : Bool), Causalean.Stat.IsLittleOp (fun n ω => (MeasureTheory.eLpNorm (fun x => μ_hat n ω a x - S.μ_val a x) 2 S.P_X).toReal * (MeasureTheory.eLpNorm (fun x => e_hat n ω x - S.e_val x) 2 S.P_X).toReal) (fun n => ↑n ^ (-(1 / 2))) P.μ) → Measurable S.ψ_AIPW → ∀ (hθn_meas : ∀ (n : ℕ), AEMeasurable (Causalean.Stat.IsAsymLinear.rescaledEstimator (Causalean.Estimation.ATE.dmlEstimator S sample split μ_hat e_hat) S.θ₀ split.foldB n) P.μ), (∀ (n : ℕ), AEMeasurable (Causalean.Stat.IsAsymLinear.normalizedSum sample S.ψ_AIPW split.foldB n) P.μ) → Causalean.Stat.Tendsto_dist (Causalean.Stat.IsAsymLinear.rescaledEstimator (Causalean.Estimation.ATE.dmlEstimator S sample split μ_hat e_hat) S.θ₀ split.foldB) (Causalean.Stat.gaussianMeasure 0 (∫ (x : γ), (S.μ_val true x - S.μ_val false x - S.θ₀) ^ 2 ∂S.P_X + ∫ (z : γ × Bool × ℝ), Causalean.Estimation.ATE.BackdoorEstimationSystem.indA z / S.e_val (Causalean.Estimation.ATE.BackdoorEstimationSystem.projX z) ^ 2 * (Causalean.Estimation.ATE.BackdoorEstimationSystem.projY z - S.μ_val true (Causalean.Estimation.ATE.BackdoorEstimationSystem.projX z)) ^ 2 ∂S.P_Z + ∫ (z : γ × Bool × ℝ), (1 - Causalean.Estimation.ATE.BackdoorEstimationSystem.indA z) / (1 - S.e_val (Causalean.Estimation.ATE.BackdoorEstimationSystem.projX z)) ^ 2 * (Causalean.Estimation.ATE.BackdoorEstimationSystem.projY z - S.μ_val false (Causalean.Estimation.ATE.BackdoorEstimationSystem.projX z)) ^ 2 ∂S.P_Z)) P.μ hθn_meas","l":426,"f":"Causalean/Estimation/Efficiency/ATEVariance.lean"},{"n":"Causalean.Estimation.ATE.dml_ATE_isAsymLinear","k":"theorem","m":"Causalean.Estimation.ATE.DML","a":"Estimation/ATE","t":1,"d":"Asymptotic linearity of the one-shot DML ATE — thm:est-dml-ate-al. Fix the back-door identification assumptions for the estimation system S, with strict overlap $\\varepsilon \\le e(X) \\le 1-\\varepsilon$ for the true propensity and a.e. overlap at the same $\\varepsilon$ for every learner realization $\\hat e(n,\\omega)$, a finite second moment for the observed outcome, and a finite second moment for each potential outcome. Take a one-shot sample split whose training-fold size fraction converges to a limit $c$ with $0 < c < 1$. Suppose the outcome-regression and propensity learners $\\hat\\mu, \\hat e$ are measurable, lie in $L^2(P_X)$ at every realization, depend only on the nuisance-training fold $A(n)$, both as functions of that fold alone and jointly with the covariate, and converge individually to the truth in $L^2(P_X)$ at rate $o_p(1)$ with a product rate of $o_p(n^{-1/2})$. Then the one-shot DML/AIPW estimator of the back-door ATE is asymptotically linear at the true ATE $\\theta_0$ with influence function $\\psi_{AIPW}$ along the training folds.","s":"∀ {P : Causalean.PO.POSystem} {γ : Type u_1} [inst : MeasurableSpace γ] [inst_1 : StandardBorelSpace P.Ω] [inst_2 : MeasureTheory.IsFiniteMeasure P.μ] (S : Causalean.Estimation.ATE.BackdoorEstimationSystem P γ) {ε : ℝ}, S.Assumptions → S.StrictOverlap ε → MeasureTheory.Integrable (fun ω => S.factualY ω ^ 2) P.μ → (∀ (d : Bool), MeasureTheory.Integrable (fun ω => S.YofD d ω ^ 2) P.μ) → ∀ (sample : Causalean.Stat.IIDSample P.Ω (γ × Bool × ℝ) P.μ S.P_Z) (split : Causalean.Stat.OneShotSplit sample) {c : ℝ}, 0 < c → c < 1 → Filter.Tendsto (fun n => ↑(split.foldB n).card / ↑n) Filter.atTop (nhds c) → ∀ (μ_hat : ℕ → P.Ω → Bool → γ → ℝ) (e_hat : ℕ → P.Ω → γ → ℝ), (∀ (n : ℕ) (a : Bool), Measurable fun p => μ_hat n p.1 a p.2) → (∀ (n : ℕ), Measurable fun p => e_hat n p.1 p.2) → (∀ (n : ℕ) (ω : P.Ω) (a : Bool), MeasureTheory.MemLp (fun x => μ_hat n ω a x) 2 S.P_X) → (∀ (n : ℕ) (ω : P.Ω), MeasureTheory.MemLp (fun x => e_hat n ω x) 2 S.P_X) → (∀ (n : ℕ) (ω : P.Ω), ∀ᵐ (x : γ) ∂S.P_X, ε ≤ e_hat n ω x ∧ e_hat n ω x ≤ 1 - ε) → (∀ (n : ℕ), Measurable (μ_hat n)) → (∀ (n : ℕ), Measurable (e_hat n)) → (∀ (n : ℕ) (a : Bool), Measurable fun p => μ_hat n p.1 a p.2) → (∀ (n : ℕ), Measurable fun p => e_hat n p.1 p.2) → (∀ (a : Bool), Causalean.Stat.IsLittleOp (fun n ω => (MeasureTheory.eLpNorm (fun x => μ_hat n ω a x - S.μ_val a x) 2 S.P_X).toReal) (fun x => 1) P.μ) → Causalean.Stat.IsLittleOp (fun n ω => (MeasureTheory.eLpNorm (fun x => e_hat n ω x - S.e_val x) 2 S.P_X).toReal) (fun x => 1) P.μ → (∀ (a : Bool), Causalean.Stat.IsLittleOp (fun n ω => (MeasureTheory.eLpNorm (fun x => μ_hat n ω a x - S.μ_val a x) 2 S.P_X).toReal * (MeasureTheory.eLpNorm (fun x => e_hat n ω x - S.e_val x) 2 S.P_X).toReal) (fun n => ↑n ^ (-(1 / 2))) P.μ) → Causalean.Stat.IsAsymLinear (Causalean.Estimation.ATE.dmlEstimator S sample split μ_hat e_hat) S.θ₀ S.ψ_AIPW sample split.foldB","l":92,"f":"Causalean/Estimation/ATE/DML.lean"},{"n":"Causalean.Estimation.ATE.dml_ATE_tendstoNormal","k":"theorem","m":"Causalean.Estimation.ATE.DML","a":"Estimation/ATE","t":1,"d":"Asymptotic normality of the one-shot DML ATE (thm:est-dml-ate-al, \"In particular ...\" clause). Under the back-door identification assumptions for S, with strict overlap for the true propensity and a.e. overlap for every learner realization, finite second moments of the observed and potential outcomes, and a one-shot sample split whose training-fold fraction converges to some c with 0 < c < 1: suppose the learners μ̂, ê are measurable, in L²(P_X) at every realization, depend only on the nuisance-training fold, marginally and jointly with the covariate, and converge individually at rate o_p(1) with product rate o_p(n^{-1/2}) — the same hypotheses as dml_ATE_isAsymLinear. Given in addition measurability of the AIPW influence function, a.e. measurability of the rescaled estimator at every horizon, and a.e. measurability of the normalized influence-sum at every horizon, then the rescaled estimator √|B(n)| (θ̂ⁿ − θ₀) converges in distribution to N(0, ∫ ψ_AIPW² dP_Z).","s":"∀ {P : Causalean.PO.POSystem} {γ : Type u_1} [inst : MeasurableSpace γ] [inst_1 : StandardBorelSpace P.Ω] [inst_2 : MeasureTheory.IsFiniteMeasure P.μ] (S : Causalean.Estimation.ATE.BackdoorEstimationSystem P γ) {ε : ℝ}, S.Assumptions → S.StrictOverlap ε → MeasureTheory.Integrable (fun ω => S.factualY ω ^ 2) P.μ → (∀ (d : Bool), MeasureTheory.Integrable (fun ω => S.YofD d ω ^ 2) P.μ) → ∀ (sample : Causalean.Stat.IIDSample P.Ω (γ × Bool × ℝ) P.μ S.P_Z) (split : Causalean.Stat.OneShotSplit sample) {c : ℝ}, 0 < c → c < 1 → Filter.Tendsto (fun n => ↑(split.foldB n).card / ↑n) Filter.atTop (nhds c) → ∀ (μ_hat : ℕ → P.Ω → Bool → γ → ℝ) (e_hat : ℕ → P.Ω → γ → ℝ), (∀ (n : ℕ) (a : Bool), Measurable fun p => μ_hat n p.1 a p.2) → (∀ (n : ℕ), Measurable fun p => e_hat n p.1 p.2) → (∀ (n : ℕ) (ω : P.Ω) (a : Bool), MeasureTheory.MemLp (fun x => μ_hat n ω a x) 2 S.P_X) → (∀ (n : ℕ) (ω : P.Ω), MeasureTheory.MemLp (fun x => e_hat n ω x) 2 S.P_X) → (∀ (n : ℕ) (ω : P.Ω), ∀ᵐ (x : γ) ∂S.P_X, ε ≤ e_hat n ω x ∧ e_hat n ω x ≤ 1 - ε) → (∀ (n : ℕ), Measurable (μ_hat n)) → (∀ (n : ℕ), Measurable (e_hat n)) → (∀ (n : ℕ) (a : Bool), Measurable fun p => μ_hat n p.1 a p.2) → (∀ (n : ℕ), Measurable fun p => e_hat n p.1 p.2) → (∀ (a : Bool), Causalean.Stat.IsLittleOp (fun n ω => (MeasureTheory.eLpNorm (fun x => μ_hat n ω a x - S.μ_val a x) 2 S.P_X).toReal) (fun x => 1) P.μ) → Causalean.Stat.IsLittleOp (fun n ω => (MeasureTheory.eLpNorm (fun x => e_hat n ω x - S.e_val x) 2 S.P_X).toReal) (fun x => 1) P.μ → (∀ (a : Bool), Causalean.Stat.IsLittleOp (fun n ω => (MeasureTheory.eLpNorm (fun x => μ_hat n ω a x - S.μ_val a x) 2 S.P_X).toReal * (MeasureTheory.eLpNorm (fun x => e_hat n ω x - S.e_val x) 2 S.P_X).toReal) (fun n => ↑n ^ (-(1 / 2))) P.μ) → Measurable S.ψ_AIPW → ∀ (hθn_meas : ∀ (n : ℕ), AEMeasurable (Causalean.Stat.IsAsymLinear.rescaledEstimator (Causalean.Estimation.ATE.dmlEstimator S sample split μ_hat e_hat) S.θ₀ split.foldB n) P.μ), (∀ (n : ℕ), AEMeasurable (Causalean.Stat.IsAsymLinear.normalizedSum sample S.ψ_AIPW split.foldB n) P.μ) → Causalean.Stat.Tendsto_dist (Causalean.Stat.IsAsymLinear.rescaledEstimator (Causalean.Estimation.ATE.dmlEstimator S sample split μ_hat e_hat) S.θ₀ split.foldB) (Causalean.Stat.gaussianMeasure 0 (∫ (x : γ × Bool × ℝ), S.ψ_AIPW x ^ 2 ∂S.P_Z)) P.μ hθn_meas","l":927,"f":"Causalean/Estimation/ATE/DML.lean"},{"n":"Causalean.Estimation.ATE.plugInEstimator","k":"def","m":"Causalean.Estimation.ATE.PlugIn","a":"Estimation/ATE","t":1,"d":"Sample-split plug-in estimator of the back-door ATE (def:est-plug-in-ate).","s":"{P : Causalean.PO.POSystem} → {γ : Type u_1} → [inst : MeasurableSpace γ] → [inst_1 : StandardBorelSpace P.Ω] → [inst_2 : MeasureTheory.IsFiniteMeasure P.μ] → (S : Causalean.Estimation.ATE.BackdoorEstimationSystem P γ) → (sample : Causalean.Stat.IIDSample P.Ω (γ × Bool × ℝ) P.μ S.P_Z) → Causalean.Stat.OneShotSplit sample → (ℕ → P.Ω → Bool → γ → ℝ) → ℕ → P.Ω → ℝ","l":63,"f":"Causalean/Estimation/ATE/PlugIn.lean"},{"n":"Causalean.Estimation.ATE.plugIn_isAsymLinear","k":"theorem","m":"Causalean.Estimation.ATE.PlugIn","a":"Estimation/ATE","t":1,"d":"Asymptotic linearity of the plug-in ATE — thm:est-plug-in-ate-al. Fix the back-door identification assumptions for S, square-integrability of both potential outcomes, and a one-shot sample split whose training-fold fraction converges to some c with 0 < c < 1. Suppose the outcome-regression learner μ̂ is measurable, lies in L²(P_X) at every realization, and depends only on the nuisance-training fold, marginally and jointly with the covariate, with joint L²(P_X) estimation error at rate o_p(n^{-1/2}). Then the sample-split plug-in estimator of the back-door ATE is asymptotically linear at the true ATE θ₀ with influence function ψ_plugin along the training folds.","s":"∀ {P : Causalean.PO.POSystem} {γ : Type u_1} [inst : MeasurableSpace γ] [inst_1 : StandardBorelSpace P.Ω] [inst_2 : MeasureTheory.IsFiniteMeasure P.μ] (S : Causalean.Estimation.ATE.BackdoorEstimationSystem P γ), S.Assumptions → (∀ (d : Bool), MeasureTheory.Integrable (fun ω => S.YofD d ω ^ 2) P.μ) → ∀ (sample : Causalean.Stat.IIDSample P.Ω (γ × Bool × ℝ) P.μ S.P_Z) (split : Causalean.Stat.OneShotSplit sample) {c : ℝ}, 0 < c → c < 1 → Filter.Tendsto (fun n => ↑(split.foldB n).card / ↑n) Filter.atTop (nhds c) → ∀ (μ_hat : ℕ → P.Ω → Bool → γ → ℝ), (∀ (n : ℕ) (a : Bool), Measurable fun p => μ_hat n p.1 a p.2) → (∀ (n : ℕ) (ω : P.Ω) (a : Bool), MeasureTheory.MemLp (fun x => μ_hat n ω a x) 2 S.P_X) → (∀ (n : ℕ), Measurable (μ_hat n)) → (∀ (n : ℕ) (a : Bool), Measurable fun p => μ_hat n p.1 a p.2) → Causalean.Stat.IsLittleOp (fun n ω => √(∑ a, (MeasureTheory.eLpNorm (fun x => μ_hat n ω a x - S.μ_val a x) 2 S.P_X).toReal ^ 2)) (fun n => ↑n ^ (-(1 / 2))) P.μ → Causalean.Stat.IsAsymLinear (Causalean.Estimation.ATE.plugInEstimator S sample split μ_hat) S.θ₀ (Causalean.Estimation.ATE.ψ_plugin S) sample split.foldB","l":94,"f":"Causalean/Estimation/ATE/PlugIn.lean"},{"n":"Causalean.Estimation.ATE.plugIn_tendstoNormal","k":"theorem","m":"Causalean.Estimation.ATE.PlugIn","a":"Estimation/ATE","t":1,"d":"Asymptotic normality of the plug-in ATE (thm:est-plug-in-ate-al, \"In particular ...\" clause). Under the back-door identification assumptions for S, square-integrability of both potential outcomes, and a one-shot sample split whose training-fold fraction converges to some c with 0 < c < 1: suppose the learner μ̂ is measurable, in L²(P_X) at every realization, depends only on the nuisance-training fold, marginally and jointly with the covariate, with joint L²(P_X) estimation error at rate o_p(n^{-1/2}) — the same hypotheses as plugIn_isAsymLinear. Given in addition a.e. measurability of the rescaled estimator at every horizon and a.e. measurability of the normalized influence-sum at every horizon, then the rescaled estimator √|B(n)| (θ̂ⁿ − θ₀) converges in distribution to N(0, ∫ ψ_plugin² dP_Z).","s":"∀ {P : Causalean.PO.POSystem} {γ : Type u_1} [inst : MeasurableSpace γ] [inst_1 : StandardBorelSpace P.Ω] [inst_2 : MeasureTheory.IsFiniteMeasure P.μ] (S : Causalean.Estimation.ATE.BackdoorEstimationSystem P γ), S.Assumptions → (∀ (d : Bool), MeasureTheory.Integrable (fun ω => S.YofD d ω ^ 2) P.μ) → ∀ (sample : Causalean.Stat.IIDSample P.Ω (γ × Bool × ℝ) P.μ S.P_Z) (split : Causalean.Stat.OneShotSplit sample) {c : ℝ}, 0 < c → c < 1 → Filter.Tendsto (fun n => ↑(split.foldB n).card / ↑n) Filter.atTop (nhds c) → ∀ (μ_hat : ℕ → P.Ω → Bool → γ → ℝ), (∀ (n : ℕ) (a : Bool), Measurable fun p => μ_hat n p.1 a p.2) → (∀ (n : ℕ) (ω : P.Ω) (a : Bool), MeasureTheory.MemLp (fun x => μ_hat n ω a x) 2 S.P_X) → (∀ (n : ℕ), Measurable (μ_hat n)) → (∀ (n : ℕ) (a : Bool), Measurable fun p => μ_hat n p.1 a p.2) → Causalean.Stat.IsLittleOp (fun n ω => √(∑ a, (MeasureTheory.eLpNorm (fun x => μ_hat n ω a x - S.μ_val a x) 2 S.P_X).toReal ^ 2)) (fun n => ↑n ^ (-(1 / 2))) P.μ → ∀ (hθn_meas : ∀ (n : ℕ), AEMeasurable (Causalean.Stat.IsAsymLinear.rescaledEstimator (Causalean.Estimation.ATE.plugInEstimator S sample split μ_hat) S.θ₀ split.foldB n) P.μ), (∀ (n : ℕ), AEMeasurable (Causalean.Stat.IsAsymLinear.normalizedSum sample (Causalean.Estimation.ATE.ψ_plugin S) split.foldB n) P.μ) → Causalean.Stat.Tendsto_dist (Causalean.Stat.IsAsymLinear.rescaledEstimator (Causalean.Estimation.ATE.plugInEstimator S sample split μ_hat) S.θ₀ split.foldB) (Causalean.Stat.gaussianMeasure 0 (∫ (x : γ × Bool × ℝ), Causalean.Estimation.ATE.ψ_plugin S x ^ 2 ∂S.P_Z)) P.μ hθn_meas","l":765,"f":"Causalean/Estimation/ATE/PlugIn.lean"},{"n":"Causalean.Estimation.ATE.ψ_plugin","k":"def","m":"Causalean.Estimation.ATE.PlugIn","a":"Estimation/ATE","t":1,"d":"Plug-in influence function ψ_plugin(z) := μ(1, x) − μ(0, x) − θ₀ from thm:est-plug-in-ate-al.","s":"{P : Causalean.PO.POSystem} → {γ : Type u_1} → [inst : MeasurableSpace γ] → [inst_1 : StandardBorelSpace P.Ω] → [inst_2 : MeasureTheory.IsFiniteMeasure P.μ] → Causalean.Estimation.ATE.BackdoorEstimationSystem P γ → γ × Bool × ℝ → ℝ","l":88,"f":"Causalean/Estimation/ATE/PlugIn.lean"},{"n":"Causalean.Estimation.ATT.TreatedEstimationSystem","k":"structure","m":"Causalean.Estimation.ATT.Setup","a":"Estimation/ATT","t":1,"d":"A treated estimation system extends a potential-outcome back-door system with the value-space nuisance representatives needed for ATT AIPW estimation: the control-arm outcome regression μ₀(x) and a propensity score e(x) that is bounded away from 1, together with the control regression's agreement, almost everywhere, with the σ(X)-measurable observable control regression adjustedCE false, composed with the factual covariate and the analogous agreement of the propensity score with the observable propensity propScore.","s":"(P : Causalean.PO.POSystem) → (γ : Type u_1) → [MeasurableSpace γ] → [StandardBorelSpace P.Ω] → [MeasureTheory.IsFiniteMeasure P.μ] → Type (max (max u_1 u_2) u_3)","l":56,"f":"Causalean/Estimation/ATT/Setup.lean"},{"n":"Causalean.Estimation.ATT.TreatedEstimationSystem.H_ε","k":"def","m":"Causalean.Estimation.ATT.Score.AIPWMoment","a":"Estimation/ATT/Score","t":1,"d":"The ATT nuisance class carries one-sided upper overlap P_X-a.e., an L²(P_X) outcome-regression component, and an L∞(P_X) propensity component.","s":"{P : Causalean.PO.POSystem} → {γ : Type u_1} → [inst : MeasurableSpace γ] → [inst_1 : StandardBorelSpace P.Ω] → [inst_2 : MeasureTheory.IsFiniteMeasure P.μ] → Causalean.Estimation.ATT.TreatedEstimationSystem P γ → ℝ → Set (Causalean.Estimation.ATT.TreatedNuisanceVec γ)","l":247,"f":"Causalean/Estimation/ATT/Score/AIPWMoment.lean"},{"n":"Causalean.Estimation.ATT.TreatedEstimationSystem.H_ε_overlap_P_Z","k":"theorem","m":"Causalean.Estimation.ATT.Score.AIPWMoment","a":"Estimation/ATT/Score","t":1,"d":"Overlap transports to the joint data law. For a candidate nuisance pair lying in the ATT overlap-bounded realization set H_ε, in particular satisfying the one-sided upper overlap bound η.e_fn ≤ 1 − ε almost everywhere on the covariates, the same bound holds almost everywhere under the joint law of the covariates, treatment, and outcome, once the candidate propensity is evaluated at the covariate projection of the data triple.","s":"∀ {P : Causalean.PO.POSystem} {γ : Type u_1} [inst : MeasurableSpace γ] [inst_1 : StandardBorelSpace P.Ω] [inst_2 : MeasureTheory.IsFiniteMeasure P.μ] (S : Causalean.Estimation.ATT.TreatedEstimationSystem P γ) {ε : ℝ} {η : Causalean.Estimation.ATT.TreatedNuisanceVec γ}, η ∈ S.H_ε ε → ∀ᵐ (z : γ × Bool × ℝ) ∂S.P_Z, η.e_fn (Causalean.Estimation.ATE.BackdoorEstimationSystem.projX z) ≤ 1 - ε","l":272,"f":"Causalean/Estimation/ATT/Score/AIPWMoment.lean"},{"n":"Causalean.Estimation.ATT.TreatedEstimationSystem.H_ε_overlap_factualX","k":"theorem","m":"Causalean.Estimation.ATT.Score.AIPWMoment","a":"Estimation/ATT/Score","t":2,"d":"Membership in the ATT H_ε transports the one-sided a.e. overlap bound to the original probability space along the observed covariate.","s":"∀ {P : Causalean.PO.POSystem} {γ : Type u_1} [inst : MeasurableSpace γ] [inst_1 : StandardBorelSpace P.Ω] [inst_2 : MeasureTheory.IsFiniteMeasure P.μ] (S : Causalean.Estimation.ATT.TreatedEstimationSystem P γ) {ε : ℝ} {η : Causalean.Estimation.ATT.TreatedNuisanceVec γ}, η ∈ S.H_ε ε → ∀ᵐ (ω : P.Ω) ∂P.μ, η.e_fn (S.factualX ω) ≤ 1 - ε","l":259,"f":"Causalean/Estimation/ATT/Score/AIPWMoment.lean"},{"n":"Causalean.Estimation.ATT.TreatedEstimationSystem.K_AIPW_ATT","k":"def","m":"Causalean.Estimation.ATT.Score.AIPWScoreL2","a":"Estimation/ATT/Score","t":1,"d":"Pointwise Lipschitz constant for the ATT AIPW moment in η on H_ε: K_AIPW_ATT ε := 1 + 2/ε + 2/ε². Tracks the quadratic blow-up of the inverse weight 1/(1−ê) and the cross term (ê − e)/((1−ê)·(1−e)) arising from the single IPW factor e/(1−e) in the ATT AIPW form. Same shape as the ATE constant K_AIPW, which is fine — both are dominated by the worst-case 1/ε² term.","s":"ℝ → ℝ","l":76,"f":"Causalean/Estimation/ATT/Score/AIPWScoreL2.lean"},{"n":"Causalean.Estimation.ATT.TreatedEstimationSystem.OneSidedOverlap","k":"def","m":"Causalean.Estimation.ATT.Setup","a":"Estimation/ATT","t":1,"d":"One-sided overlap predicate propScore true ω ≤ 1 − ε a.s., with ε ∈ (0, 1/2]. The 0 < propScore true half is implied at the PO level by Assumptions.overlap; for ATT only the upper bound matters because the IPW correction divides by 1 − e(X).","s":"{P : Causalean.PO.POSystem} → {γ : Type u_1} → [inst : MeasurableSpace γ] → [inst_1 : StandardBorelSpace P.Ω] → [inst_2 : MeasureTheory.IsFiniteMeasure P.μ] → Causalean.Estimation.ATT.TreatedEstimationSystem P γ → ℝ → Prop","l":143,"f":"Causalean/Estimation/ATT/Setup.lean"},{"n":"Causalean.Estimation.ATT.TreatedEstimationSystem.P_X","k":"def","m":"Causalean.Estimation.ATT.Setup","a":"Estimation/ATT","t":1,"d":"Covariate marginal: P_X := μ.map factualX.","s":"{P : Causalean.PO.POSystem} → {γ : Type u_1} → [inst : MeasurableSpace γ] → [inst_1 : StandardBorelSpace P.Ω] → [inst_2 : MeasureTheory.IsFiniteMeasure P.μ] → Causalean.Estimation.ATT.TreatedEstimationSystem P γ → MeasureTheory.Measure γ","l":153,"f":"Causalean/Estimation/ATT/Setup.lean"},{"n":"Causalean.Estimation.ATT.TreatedEstimationSystem.P_Z","k":"def","m":"Causalean.Estimation.ATT.Setup","a":"Estimation/ATT","t":1,"d":"Joint data law P_Z := μ.map (X, A, Y).","s":"{P : Causalean.PO.POSystem} → {γ : Type u_1} → [inst : MeasurableSpace γ] → [inst_1 : StandardBorelSpace P.Ω] → [inst_2 : MeasureTheory.IsFiniteMeasure P.μ] → Causalean.Estimation.ATT.TreatedEstimationSystem P γ → MeasureTheory.Measure (γ × Bool × ℝ)","l":171,"f":"Causalean/Estimation/ATT/Setup.lean"},{"n":"Causalean.Estimation.ATT.TreatedEstimationSystem.P_Z_map_projX_eq_P_X","k":"theorem","m":"Causalean.Estimation.ATT.Setup","a":"Estimation/ATT","t":2,"d":"The covariate marginal P_X is the pushforward of P_Z along the projection (x, a, y) ↦ x. Used to bridge integrals/eLpNorm between P_X (covariates only) and P_Z (full data triple).","s":"∀ {P : Causalean.PO.POSystem} {γ : Type u_1} [inst : MeasurableSpace γ] [inst_1 : StandardBorelSpace P.Ω] [inst_2 : MeasureTheory.IsFiniteMeasure P.μ] (S : Causalean.Estimation.ATT.TreatedEstimationSystem P γ), MeasureTheory.Measure.map (fun z => z.1) S.P_Z = S.P_X","l":176,"f":"Causalean/Estimation/ATT/Setup.lean"},{"n":"Causalean.Estimation.ATT.TreatedEstimationSystem.YMuVal_residual_sq_ATT","k":"def","m":"Causalean.Estimation.ATT.Score.AIPWScoreL2","a":"Estimation/ATT/Score","t":1,"d":"The single ATT residual integrand |Y − μ₀(X)|, viewed on the data space.","s":"{P : Causalean.PO.POSystem} → {γ : Type u_1} → [inst : MeasurableSpace γ] → [inst_1 : StandardBorelSpace P.Ω] → [inst_2 : MeasureTheory.IsFiniteMeasure P.μ] → Causalean.Estimation.ATT.TreatedEstimationSystem P γ → γ × Bool × ℝ → ℝ","l":281,"f":"Causalean/Estimation/ATT/Score/AIPWScoreL2.lean"},{"n":"Causalean.Estimation.ATT.TreatedEstimationSystem.aipwMomentATT","k":"def","m":"Causalean.Estimation.ATT.Score.AIPWMoment","a":"Estimation/ATT/Score","t":1,"d":"The ATT AIPW moment","s":"{γ : Type u_1} → γ × Bool × ℝ → (γ → ℝ) → (γ → ℝ) → ℝ → ℝ","l":52,"f":"Causalean/Estimation/ATT/Score/AIPWMoment.lean"},{"n":"Causalean.Estimation.ATT.TreatedEstimationSystem.aipwMomentATTFunctional","k":"def","m":"Causalean.Estimation.ATT.Score.AIPWMoment","a":"Estimation/ATT/Score","t":1,"d":"The ATT AIPW moment as a moment functional TreatedNuisanceVec γ → X → ℝ → ℝ, suitable for feeding NeymanOrthogonal.","s":"{γ : Type u_1} → [inst : MeasurableSpace γ] → Causalean.Estimation.ATT.TreatedNuisanceVec γ → γ × Bool × ℝ → ℝ → ℝ","l":292,"f":"Causalean/Estimation/ATT/Score/AIPWMoment.lean"},{"n":"Causalean.Estimation.ATT.TreatedEstimationSystem.aipw_finite_var_ATT","k":"theorem","m":"Causalean.Estimation.ATT.Score.FiniteVar","a":"Estimation/ATT/Score","t":1,"d":"Finite variance of the ATT AIPW moment. Under one-sided overlap: the true treatment propensity is bounded above by 1 − ε for some ε ∈ (0, 1/2] almost surely, the one-sided back-door ATT assumptions, and square-integrability of the factual outcome together with square-integrability of the untreated potential outcome Y(0), the ATT AIPW moment evaluated at the truth nuisance is square-integrable under the joint law of the covariates, treatment, and outcome.","s":"∀ {P : Causalean.PO.POSystem} {γ : Type u_1} [inst : MeasurableSpace γ] [inst_1 : StandardBorelSpace P.Ω] [inst_2 : MeasureTheory.IsFiniteMeasure P.μ] (S : Causalean.Estimation.ATT.TreatedEstimationSystem P γ) {ε : ℝ}, S.OneSidedOverlap ε → S.ATTAssumptions → MeasureTheory.Integrable (fun ω => S.factualY ω ^ 2) P.μ → MeasureTheory.Integrable (fun ω => S.YofD false ω ^ 2) P.μ → MeasureTheory.Integrable (fun z => Causalean.Estimation.ATT.TreatedEstimationSystem.aipwMomentATT z S.μ₀_val S.e_val S.θ₀ ^ 2) S.P_Z","l":49,"f":"Causalean/Estimation/ATT/Score/FiniteVar.lean"},{"n":"Causalean.Estimation.ATT.TreatedEstimationSystem.aipw_mean_zero_ATT","k":"theorem","m":"Causalean.Estimation.ATT.Score.MeanZero","a":"Estimation/ATT/Score","t":1,"d":"Mean zero of the ATT AIPW moment. Under the one-sided back-door ATT assumptions, a strictly positive marginal treatment probability, and integrability of the control-arm inverse-propensity-weighted residual correction (1 − 1{D=1}) · (e(X)/(1 − e(X))) · (Y − μ₀(X)), the ATT AIPW moment evaluated at the truth has mean zero under the joint law of the covariates, treatment, and outcome.","s":"∀ {P : Causalean.PO.POSystem} {γ : Type u_1} [inst : MeasurableSpace γ] [inst_1 : StandardBorelSpace P.Ω] [inst_2 : MeasureTheory.IsFiniteMeasure P.μ] (S : Causalean.Estimation.ATT.TreatedEstimationSystem P γ), S.ATTAssumptions → 0 < S.π_val → MeasureTheory.Integrable (fun ω => (1 - S.dVar.indicator true ω) * (S.propScore true ω / (1 - S.propScore true ω)) * (S.factualY ω - S.adjustedCE false ω)) P.μ → ∫ (z : γ × Bool × ℝ), Causalean.Estimation.ATT.TreatedEstimationSystem.aipwMomentATT z S.μ₀_val S.e_val S.θ₀ ∂S.P_Z = 0","l":108,"f":"Causalean/Estimation/ATT/Score/MeanZero.lean"},{"n":"Causalean.Estimation.ATT.TreatedEstimationSystem.aipw_rem_const_ATT","k":"def","m":"Causalean.Estimation.ATT.Remainder.Bound","a":"Estimation/ATT/Remainder","t":1,"d":"AIPW remainder constant for one-sided overlap ε (ATT version).","s":"ℝ → ℝ","l":49,"f":"Causalean/Estimation/ATT/Remainder/Bound.lean"},{"n":"Causalean.Estimation.ATT.TreatedEstimationSystem.aipw_remainder_bound_ATT","k":"theorem","m":"Causalean.Estimation.ATT.Remainder.Bound","a":"Estimation/ATT/Remainder","t":1,"d":"AIPW remainder bound (ATT). Fix a candidate nuisance pair η. Under one-sided overlap ε on the true propensity, the one-sided back-door ATT assumptions, a strictly positive marginal treatment probability, square-integrability of the factual outcome, and square-integrability of the untreated potential outcome Y(0): if η lies in the overlap-bounded candidate realization set H_ε, its control-regression error μ̂₀ − μ₀ is square-integrable against the covariate law, its propensity error ê − e is square-integrable against the covariate law, and its IPW correction is integrable against the observed data law, then the absolute value of the population AIPW moment at η is bounded by (1/ε) · ‖μ̂₀ − μ₀‖_{L²(P_X)} · ‖ê − e‖_{L²(P_X)}.","s":"∀ {P : Causalean.PO.POSystem} {γ : Type u_1} [inst : MeasurableSpace γ] [inst_1 : StandardBorelSpace P.Ω] [inst_2 : MeasureTheory.IsFiniteMeasure P.μ] (S : Causalean.Estimation.ATT.TreatedEstimationSystem P γ) {ε : ℝ}, S.OneSidedOverlap ε → S.ATTAssumptions → 0 < S.π_val → MeasureTheory.Integrable (fun ω => S.factualY ω ^ 2) P.μ → MeasureTheory.Integrable (fun ω => S.YofD false ω ^ 2) P.μ → ∀ η ∈ S.H_ε ε, MeasureTheory.MemLp (fun x => η.μ₀_fn x - S.μ₀_val x) 2 S.P_X → MeasureTheory.MemLp (fun x => η.e_fn x - S.e_val x) 2 S.P_X → MeasureTheory.Integrable (fun z => (1 - Causalean.Estimation.ATE.BackdoorEstimationSystem.indA z) * (η.e_fn (Causalean.Estimation.ATE.BackdoorEstimationSystem.projX z) / (1 - η.e_fn (Causalean.Estimation.ATE.BackdoorEstimationSystem.projX z))) * (Causalean.Estimation.ATE.BackdoorEstimationSystem.projY z - η.μ₀_fn (Causalean.Estimation.ATE.BackdoorEstimationSystem.projX z))) S.P_Z → |∫ (z : γ × Bool × ℝ), Causalean.Estimation.ATT.TreatedEstimationSystem.aipwMomentATTFunctional η z S.θ₀ ∂S.P_Z| ≤ Causalean.Estimation.ATT.TreatedEstimationSystem.aipw_rem_const_ATT ε * (MeasureTheory.eLpNorm (fun x => η.μ₀_fn x - S.μ₀_val x) 2 S.P_X).toReal * (MeasureTheory.eLpNorm (fun x => η.e_fn x - S.e_val x) 2 S.P_X).toReal","l":58,"f":"Causalean/Estimation/ATT/Remainder/Bound.lean"},{"n":"Causalean.Estimation.ATT.TreatedEstimationSystem.aipw_remainder_identity_ATT","k":"theorem","m":"Causalean.Estimation.ATT.Remainder.Identity","a":"Estimation/ATT/Remainder","t":1,"d":"AIPW remainder identity (ATT). Fix a candidate nuisance pair η. Under one-sided overlap ε on the true propensity, the one-sided back-door ATT assumptions, a strictly positive marginal treatment probability, square-integrability of the factual outcome, square-integrability of the untreated potential outcome Y(0): if η lies in the overlap-bounded candidate realization set H_ε, its control-regression error μ̂₀ − μ₀ is square-integrable against the covariate law, its propensity error ê − e is square-integrable against the covariate law, and its IPW correction is integrable against the observed data law, then the population AIPW moment at η equals the single cross-product ∫ ((ê(x) − e(x))/(1 − ê(x))) · (μ̂₀(x) − μ₀(x)) dP_X.","s":"∀ {P : Causalean.PO.POSystem} {γ : Type u_1} [inst : MeasurableSpace γ] [inst_1 : StandardBorelSpace P.Ω] [inst_2 : MeasureTheory.IsFiniteMeasure P.μ] (S : Causalean.Estimation.ATT.TreatedEstimationSystem P γ) {ε : ℝ}, S.OneSidedOverlap ε → S.ATTAssumptions → 0 < S.π_val → MeasureTheory.Integrable (fun ω => S.factualY ω ^ 2) P.μ → MeasureTheory.Integrable (fun ω => S.YofD false ω ^ 2) P.μ → ∀ η ∈ S.H_ε ε, MeasureTheory.MemLp (fun x => η.μ₀_fn x - S.μ₀_val x) 2 S.P_X → MeasureTheory.MemLp (fun x => η.e_fn x - S.e_val x) 2 S.P_X → MeasureTheory.Integrable (fun z => (1 - Causalean.Estimation.ATE.BackdoorEstimationSystem.indA z) * (η.e_fn (Causalean.Estimation.ATE.BackdoorEstimationSystem.projX z) / (1 - η.e_fn (Causalean.Estimation.ATE.BackdoorEstimationSystem.projX z))) * (Causalean.Estimation.ATE.BackdoorEstimationSystem.projY z - η.μ₀_fn (Causalean.Estimation.ATE.BackdoorEstimationSystem.projX z))) S.P_Z → ∫ (z : γ × Bool × ℝ), Causalean.Estimation.ATT.TreatedEstimationSystem.aipwMomentATTFunctional η z S.θ₀ ∂S.P_Z = ∫ (x : γ), (η.e_fn x - S.e_val x) / (1 - η.e_fn x) * (η.μ₀_fn x - S.μ₀_val x) ∂S.P_X","l":66,"f":"Causalean/Estimation/ATT/Remainder/Identity.lean"},{"n":"Causalean.Estimation.ATT.TreatedEstimationSystem.aipw_remainder_op_ATT","k":"theorem","m":"Causalean.Estimation.ATT.Remainder.Bound","a":"Estimation/ATT/Remainder","t":1,"d":"AIPW remainder is o_p(n^{-1/2}) under the ATT product rate. Fix a sequence of random candidate nuisance pairs indexed by sample size. Under one-sided overlap ε on the true propensity, the one-sided back-door ATT assumptions, a strictly positive marginal treatment probability, square-integrability of the factual outcome, and square-integrability of the untreated potential outcome Y(0): if every draw of the candidate lies in the overlap-bounded realization set H_ε, each control-regression error is square-integrable against the covariate law, each propensity error is square-integrable against the covariate law, each candidate IPW correction is integrable against the observed data law, and the product of the two L²(P_X) error norms is o_p(n^{-1/2}), then the population AIPW moment evaluated at the random candidate nuisance is itself o_p(n^{-1/2}).","s":"∀ {P : Causalean.PO.POSystem} {γ : Type u_1} [inst : MeasurableSpace γ] [inst_1 : StandardBorelSpace P.Ω] [inst_2 : MeasureTheory.IsFiniteMeasure P.μ] (S : Causalean.Estimation.ATT.TreatedEstimationSystem P γ) {ε : ℝ}, S.OneSidedOverlap ε → S.ATTAssumptions → 0 < S.π_val → MeasureTheory.Integrable (fun ω => S.factualY ω ^ 2) P.μ → MeasureTheory.Integrable (fun ω => S.YofD false ω ^ 2) P.μ → ∀ (η_hat : ℕ → P.Ω → Causalean.Estimation.ATT.TreatedNuisanceVec γ), (∀ (n : ℕ) (ω : P.Ω), η_hat n ω ∈ S.H_ε ε) → (∀ (n : ℕ) (ω : P.Ω), MeasureTheory.MemLp (fun x => (η_hat n ω).μ₀_fn x - S.μ₀_val x) 2 S.P_X) → (∀ (n : ℕ) (ω : P.Ω), MeasureTheory.MemLp (fun x => (η_hat n ω).e_fn x - S.e_val x) 2 S.P_X) → (∀ (n : ℕ) (ω : P.Ω), MeasureTheory.Integrable (fun z => (1 - Causalean.Estimation.ATE.BackdoorEstimationSystem.indA z) * ((η_hat n ω).e_fn (Causalean.Estimation.ATE.BackdoorEstimationSystem.projX z) / (1 - (η_hat n ω).e_fn (Causalean.Estimation.ATE.BackdoorEstimationSystem.projX z))) * (Causalean.Estimation.ATE.BackdoorEstimationSystem.projY z - (η_hat n ω).μ₀_fn (Causalean.Estimation.ATE.BackdoorEstimationSystem.projX z))) S.P_Z) → Causalean.Stat.IsLittleOp (fun n ω => (MeasureTheory.eLpNorm (fun x => (η_hat n ω).μ₀_fn x - S.μ₀_val x) 2 S.P_X).toReal * (MeasureTheory.eLpNorm (fun x => (η_hat n ω).e_fn x - S.e_val x) 2 S.P_X).toReal) (fun n => ↑n ^ (-(1 / 2))) P.μ → Causalean.Stat.IsLittleOp (fun n ω => ∫ (z : γ × Bool × ℝ), Causalean.Estimation.ATT.TreatedEstimationSystem.aipwMomentATTFunctional (η_hat n ω) z S.θ₀ ∂S.P_Z) (fun n => ↑n ^ (-(1 / 2))) P.μ","l":187,"f":"Causalean/Estimation/ATT/Remainder/Bound.lean"},{"n":"Causalean.Estimation.ATT.TreatedEstimationSystem.aipw_score_diff_isLittleOp_one_ATT","k":"theorem","m":"Causalean.Estimation.ATT.Score.AIPWScoreL2","a":"Estimation/ATT/Score","t":1,"d":"Headline L²(P_Z) o_p(1) continuity bound for the ATT AIPW score. Fix a sequence of random candidate nuisance pairs indexed by sample size. Under one-sided overlap ε on the true propensity, membership of the truth nuisance in the overlap-bounded realization set H_ε, nonnegativity of the true propensity, the one-sided back-door ATT assumptions, square-integrability of the factual outcome and of the untreated potential outcome Y(0): if every draw of the candidate lies in the overlap-bounded realization set H_ε, every candidate propensity is nonnegative, each candidate control-regression error admits an L²(P_X) witness, each candidate propensity error admits an L²(P_X) witness, the control-regression error rate is o_p(1) in L²(P_X), and the propensity error rate is o_p(1) in L²(P_X), then the L²(P_Z) distance between the ATT AIPW score evaluated at the random candidate nuisance and at the truth nuisance is o_p(1).","s":"∀ {P : Causalean.PO.POSystem} {γ : Type u_1} [inst : MeasurableSpace γ] [inst_1 : StandardBorelSpace P.Ω] [inst_2 : MeasureTheory.IsFiniteMeasure P.μ] (S : Causalean.Estimation.ATT.TreatedEstimationSystem P γ) {ε : ℝ}, S.OneSidedOverlap ε → S.η₀ ∈ S.H_ε ε → (∀ (x : γ), 0 ≤ S.e_val x) → S.ATTAssumptions → MeasureTheory.Integrable (fun ω => S.factualY ω ^ 2) P.μ → MeasureTheory.Integrable (fun ω => S.YofD false ω ^ 2) P.μ → ∀ (η_hat : ℕ → P.Ω → Causalean.Estimation.ATT.TreatedNuisanceVec γ), (∀ (n : ℕ) (ω : P.Ω), η_hat n ω ∈ S.H_ε ε) → (∀ (n : ℕ) (ω : P.Ω) (x : γ), 0 ≤ (η_hat n ω).e_fn x) → (∀ (n : ℕ) (ω : P.Ω), MeasureTheory.MemLp (fun x => (η_hat n ω).μ₀_fn x - S.μ₀_val x) 2 S.P_X) → (∀ (n : ℕ) (ω : P.Ω), MeasureTheory.MemLp (fun x => (η_hat n ω).e_fn x - S.e_val x) 2 S.P_X) → Causalean.Stat.IsLittleOp (fun n ω => (MeasureTheory.eLpNorm (fun x => (η_hat n ω).μ₀_fn x - S.μ₀_val x) 2 S.P_X).toReal) (fun x => 1) P.μ → Causalean.Stat.IsLittleOp (fun n ω => (MeasureTheory.eLpNorm (fun x => (η_hat n ω).e_fn x - S.e_val x) 2 S.P_X).toReal) (fun x => 1) P.μ → Causalean.Stat.IsLittleOp (fun n ω => (MeasureTheory.eLpNorm (fun z => Causalean.Estimation.ATT.TreatedEstimationSystem.aipwMomentATTFunctional (η_hat n ω) z S.θ₀ - Causalean.Estimation.ATT.TreatedEstimationSystem.aipwMomentATTFunctional S.η₀ z S.θ₀) 2 S.P_Z).toReal) (fun x => 1) P.μ","l":598,"f":"Causalean/Estimation/ATT/Score/AIPWScoreL2.lean"},{"n":"Causalean.Estimation.ATT.TreatedEstimationSystem.aipw_score_lipschitz_ATT","k":"theorem","m":"Causalean.Estimation.ATT.Score.AIPWScoreL2","a":"Estimation/ATT/Score","t":2,"d":"ATT AIPW score Lipschitz bound on the a.e. one-sided-overlap set H_ε.","s":"∀ {P : Causalean.PO.POSystem} {γ : Type u_1} [inst : MeasurableSpace γ] [inst_1 : StandardBorelSpace P.Ω] [inst_2 : MeasureTheory.IsFiniteMeasure P.μ] (S : Causalean.Estimation.ATT.TreatedEstimationSystem P γ) {ε : ℝ}, S.OneSidedOverlap ε → S.η₀ ∈ S.H_ε ε → ∀ η ∈ S.H_ε ε, ∀ᵐ (z : γ × Bool × ℝ) ∂S.P_Z, |Causalean.Estimation.ATT.TreatedEstimationSystem.aipwMomentATTFunctional η z S.θ₀ - Causalean.Estimation.ATT.TreatedEstimationSystem.aipwMomentATTFunctional S.η₀ z S.θ₀| ≤ Causalean.Estimation.ATT.TreatedEstimationSystem.K_AIPW_ATT ε * |η.μ₀_fn (Causalean.Estimation.ATE.BackdoorEstimationSystem.projX z) - S.μ₀_val (Causalean.Estimation.ATE.BackdoorEstimationSystem.projX z)| + Causalean.Estimation.ATT.TreatedEstimationSystem.K_AIPW_ATT ε * |Causalean.Estimation.ATE.BackdoorEstimationSystem.projY z - S.μ₀_val (Causalean.Estimation.ATE.BackdoorEstimationSystem.projX z)| * |η.e_fn (Causalean.Estimation.ATE.BackdoorEstimationSystem.projX z) - S.e_val (Causalean.Estimation.ATE.BackdoorEstimationSystem.projX z)|","l":237,"f":"Causalean/Estimation/ATT/Score/AIPWScoreL2.lean"},{"n":"Causalean.Estimation.ATT.TreatedEstimationSystem.control_cate_backdoor","k":"theorem","m":"Causalean.Estimation.ATT.Setup","a":"Estimation/ATT","t":2,"d":"Control-arm backdoor CATE under ATT assumptions. The conditional mean of the untreated potential outcome given the covariates equals the observable control regression when consistency, conditional ignorability, integrability, and one-sided control overlap hold. No treated-arm overlap is used.","s":"∀ {P : Causalean.PO.POSystem} {γ : Type u_1} [inst : MeasurableSpace γ] [inst_1 : StandardBorelSpace P.Ω] [inst_2 : MeasureTheory.IsFiniteMeasure P.μ] (S : Causalean.Estimation.ATT.TreatedEstimationSystem P γ), S.ATTAssumptions → S.CATE false =ᵐ[P.μ] S.adjustedCE false","l":107,"f":"Causalean/Estimation/ATT/Setup.lean"},{"n":"Causalean.Estimation.ATT.TreatedEstimationSystem.factualZ","k":"def","m":"Causalean.Estimation.ATT.Setup","a":"Estimation/ATT","t":1,"d":"Data triple (X, A, Y) : Ω → γ × Bool × ℝ.","s":"{P : Causalean.PO.POSystem} → {γ : Type u_1} → [inst : MeasurableSpace γ] → [inst_1 : StandardBorelSpace P.Ω] → [inst_2 : MeasureTheory.IsFiniteMeasure P.μ] → Causalean.Estimation.ATT.TreatedEstimationSystem P γ → P.Ω → γ × Bool × ℝ","l":157,"f":"Causalean/Estimation/ATT/Setup.lean"},{"n":"Causalean.Estimation.ATT.TreatedEstimationSystem.indicator_to_propScore_integral","k":"theorem","m":"Causalean.Estimation.ATT.Score.ScorePullout","a":"Estimation/ATT/Score","t":1,"d":"Propensity-score pull-out for the treatment indicator (ATT). Fix a treatment label d, under the one-sided back-door ATT assumptions. If f : γ → ℝ is measurable and the product f(X) · 1{D=d} is integrable, then replacing the treatment indicator 1{D=d} by the value-space propensity — e_val when d is true, 1 − e_val when d is false — inside the integral leaves the integral unchanged.","s":"∀ {P : Causalean.PO.POSystem} {γ : Type u_1} [inst : MeasurableSpace γ] [inst_1 : StandardBorelSpace P.Ω] [inst_2 : MeasureTheory.IsFiniteMeasure P.μ] (S : Causalean.Estimation.ATT.TreatedEstimationSystem P γ), S.ATTAssumptions → ∀ (d : Bool) (f : γ → ℝ), Measurable f → MeasureTheory.Integrable (fun ω => f (S.factualX ω) * S.dVar.indicator d ω) P.μ → ∫ (ω : P.Ω), f (S.factualX ω) * S.dVar.indicator d ω ∂P.μ = ∫ (ω : P.Ω), f (S.factualX ω) * if d = true then S.e_val (S.factualX ω) else 1 - S.e_val (S.factualX ω) ∂P.μ","l":340,"f":"Causalean/Estimation/ATT/Score/ScorePullout.lean"},{"n":"Causalean.Estimation.ATT.TreatedEstimationSystem.ipwWeight_false","k":"def","m":"Causalean.Estimation.ATT.Score.ScorePullout","a":"Estimation/ATT/Score","t":1,"d":"Value-space IPW weight e(x) / (1 − e(x)) used in the ATT correction.","s":"{P : Causalean.PO.POSystem} → {γ : Type u_1} → [inst : MeasurableSpace γ] → [inst_1 : StandardBorelSpace P.Ω] → [inst_2 : MeasureTheory.IsFiniteMeasure P.μ] → Causalean.Estimation.ATT.TreatedEstimationSystem P γ → γ → ℝ","l":45,"f":"Causalean/Estimation/ATT/Score/ScorePullout.lean"},{"n":"Causalean.Estimation.ATT.TreatedEstimationSystem.ipw_estimated_integrable","k":"theorem","m":"Causalean.Estimation.ATT.Score.FiniteVar","a":"Estimation/ATT/Score","t":2,"d":"Integrability of an estimated-nuisance ATT IPW correction (over P_Z).","s":"∀ {P : Causalean.PO.POSystem} {γ : Type u_1} [inst : MeasurableSpace γ] [inst_1 : StandardBorelSpace P.Ω] [inst_2 : MeasureTheory.IsFiniteMeasure P.μ] (S : Causalean.Estimation.ATT.TreatedEstimationSystem P γ) {ε : ℝ}, 0 < ε → ∀ (η : Causalean.Estimation.ATT.TreatedNuisanceVec γ), (∀ᵐ (x : γ) ∂S.P_X, 0 ≤ η.e_fn x) → (∀ᵐ (x : γ) ∂S.P_X, η.e_fn x ≤ 1 - ε) → MeasureTheory.MemLp η.μ₀_fn 2 S.P_X → MeasureTheory.Integrable (fun ω => S.factualY ω ^ 2) P.μ → MeasureTheory.Integrable (fun z => (1 - Causalean.Estimation.ATE.BackdoorEstimationSystem.indA z) * (η.e_fn (Causalean.Estimation.ATE.BackdoorEstimationSystem.projX z) / (1 - η.e_fn (Causalean.Estimation.ATE.BackdoorEstimationSystem.projX z))) * (Causalean.Estimation.ATE.BackdoorEstimationSystem.projY z - η.μ₀_fn (Causalean.Estimation.ATE.BackdoorEstimationSystem.projX z))) S.P_Z","l":229,"f":"Causalean/Estimation/ATT/Score/FiniteVar.lean"},{"n":"Causalean.Estimation.ATT.TreatedEstimationSystem.ipw_truth_integrable","k":"theorem","m":"Causalean.Estimation.ATT.Score.FiniteVar","a":"Estimation/ATT/Score","t":2,"d":"Integrability of the truth-nuisance ATT IPW correction (over P.μ).","s":"∀ {P : Causalean.PO.POSystem} {γ : Type u_1} [inst : MeasurableSpace γ] [inst_1 : StandardBorelSpace P.Ω] [inst_2 : MeasureTheory.IsFiniteMeasure P.μ] (S : Causalean.Estimation.ATT.TreatedEstimationSystem P γ) {ε : ℝ}, S.OneSidedOverlap ε → S.ATTAssumptions → MeasureTheory.Integrable (fun ω => S.factualY ω ^ 2) P.μ → MeasureTheory.Integrable (fun ω => S.YofD false ω ^ 2) P.μ → MeasureTheory.Integrable (fun ω => (1 - S.dVar.indicator true ω) * (S.propScore true ω / (1 - S.propScore true ω)) * (S.factualY ω - S.adjustedCE false ω)) P.μ","l":311,"f":"Causalean/Estimation/ATT/Score/FiniteVar.lean"},{"n":"Causalean.Estimation.ATT.TreatedEstimationSystem.measurable_aipwMomentATTFunctional","k":"theorem","m":"Causalean.Estimation.ATT.Score.AIPWMoment","a":"Estimation/ATT/Score","t":2,"d":"Measurability of the ATT AIPW moment functional in the data variable z.","s":"∀ {γ : Type u_1} [inst : MeasurableSpace γ] (η : Causalean.Estimation.ATT.TreatedNuisanceVec γ) (θ : ℝ), Measurable fun z => Causalean.Estimation.ATT.TreatedEstimationSystem.aipwMomentATTFunctional η z θ","l":298,"f":"Causalean/Estimation/ATT/Score/AIPWMoment.lean"},{"n":"Causalean.Estimation.ATT.TreatedEstimationSystem.measurable_aipwMomentATT_at_θ₀","k":"theorem","m":"Causalean.Estimation.ATT.Score.MeanZero","a":"Estimation/ATT/Score","t":2,"d":"Measurability of aipwMomentATT z η₀ θ₀ as a function of z.","s":"∀ {P : Causalean.PO.POSystem} {γ : Type u_1} [inst : MeasurableSpace γ] [inst_1 : StandardBorelSpace P.Ω] [inst_2 : MeasureTheory.IsFiniteMeasure P.μ] (S : Causalean.Estimation.ATT.TreatedEstimationSystem P γ), Measurable fun z => Causalean.Estimation.ATT.TreatedEstimationSystem.aipwMomentATT z S.μ₀_val S.e_val S.θ₀","l":69,"f":"Causalean/Estimation/ATT/Score/MeanZero.lean"},{"n":"Causalean.Estimation.ATT.TreatedEstimationSystem.measurable_factualZ","k":"theorem","m":"Causalean.Estimation.ATT.Setup","a":"Estimation/ATT","t":2,"d":"Measurability of the data triple.","s":"∀ {P : Causalean.PO.POSystem} {γ : Type u_1} [inst : MeasurableSpace γ] [inst_1 : StandardBorelSpace P.Ω] [inst_2 : MeasureTheory.IsFiniteMeasure P.μ] (S : Causalean.Estimation.ATT.TreatedEstimationSystem P γ), Measurable S.factualZ","l":164,"f":"Causalean/Estimation/ATT/Setup.lean"},{"n":"Causalean.Estimation.ATT.TreatedEstimationSystem.measurable_ipwWeight_false","k":"theorem","m":"Causalean.Estimation.ATT.Score.ScorePullout","a":"Estimation/ATT/Score","t":2,"d":"Measurability of the value-space IPW weight e/(1−e).","s":"∀ {P : Causalean.PO.POSystem} {γ : Type u_1} [inst : MeasurableSpace γ] [inst_1 : StandardBorelSpace P.Ω] [inst_2 : MeasureTheory.IsFiniteMeasure P.μ] (S : Causalean.Estimation.ATT.TreatedEstimationSystem P γ), Measurable S.ipwWeight_false","l":50,"f":"Causalean/Estimation/ATT/Score/ScorePullout.lean"},{"n":"Causalean.Estimation.ATT.TreatedEstimationSystem.measurable_ψ_ATT","k":"theorem","m":"Causalean.Estimation.ATT.Score.MeanZero","a":"Estimation/ATT/Score","t":2,"d":"Measurability of the ATT influence function ψ_ATT.","s":"∀ {P : Causalean.PO.POSystem} {γ : Type u_1} [inst : MeasurableSpace γ] [inst_1 : StandardBorelSpace P.Ω] [inst_2 : MeasureTheory.IsFiniteMeasure P.μ] (S : Causalean.Estimation.ATT.TreatedEstimationSystem P γ), Measurable S.ψ_ATT","l":58,"f":"Causalean/Estimation/ATT/Score/MeanZero.lean"},{"n":"Causalean.Estimation.ATT.TreatedEstimationSystem.propScore_eq_e_val_ae","k":"theorem","m":"Causalean.Estimation.ATT.Score.ScorePullout","a":"Estimation/ATT/Score","t":2,"d":"The treated-arm propensity propScore true factors through factualX via the value-space e_val. Direct restatement of S.e_compat.","s":"∀ {P : Causalean.PO.POSystem} {γ : Type u_1} [inst : MeasurableSpace γ] [inst_1 : StandardBorelSpace P.Ω] [inst_2 : MeasureTheory.IsFiniteMeasure P.μ] (S : Causalean.Estimation.ATT.TreatedEstimationSystem P γ), S.ATTAssumptions → S.propScore true =ᵐ[P.μ] fun ω => S.e_val (S.factualX ω)","l":103,"f":"Causalean/Estimation/ATT/Score/ScorePullout.lean"},{"n":"Causalean.Estimation.ATT.TreatedEstimationSystem.propScore_false_ae","k":"theorem","m":"Causalean.Estimation.ATT.Score.ScorePullout","a":"Estimation/ATT/Score","t":2,"d":"propScore false =ᵐ 1 − propScore true. The indicator pair sums to one pointwise, conditional expectation is linear and preserves constants.","s":"∀ {P : Causalean.PO.POSystem} {γ : Type u_1} [inst : MeasurableSpace γ] [inst_1 : StandardBorelSpace P.Ω] [inst_2 : MeasureTheory.IsFiniteMeasure P.μ] (S : Causalean.Estimation.ATT.TreatedEstimationSystem P γ), S.ATTAssumptions → S.propScore false =ᵐ[P.μ] fun ω => 1 - S.propScore true ω","l":55,"f":"Causalean/Estimation/ATT/Score/ScorePullout.lean"},{"n":"Causalean.Estimation.ATT.TreatedEstimationSystem.propScore_false_eq_one_minus_e_val_ae","k":"theorem","m":"Causalean.Estimation.ATT.Score.ScorePullout","a":"Estimation/ATT/Score","t":2,"d":"Control-arm propensity factors through factualX as 1 − e_val.","s":"∀ {P : Causalean.PO.POSystem} {γ : Type u_1} [inst : MeasurableSpace γ] [inst_1 : StandardBorelSpace P.Ω] [inst_2 : MeasureTheory.IsFiniteMeasure P.μ] (S : Causalean.Estimation.ATT.TreatedEstimationSystem P γ), S.ATTAssumptions → S.propScore false =ᵐ[P.μ] fun ω => 1 - S.e_val (S.factualX ω)","l":112,"f":"Causalean/Estimation/ATT/Score/ScorePullout.lean"},{"n":"Causalean.Estimation.ATT.TreatedEstimationSystem.propScore_true_nonneg_ae","k":"theorem","m":"Causalean.Estimation.ATT.Setup","a":"Estimation/ATT","t":2,"d":"Treated propensity nonnegativity. The conditional treatment probability P[D=1 | X] is nonnegative almost surely because it is the conditional expectation of a nonnegative treatment indicator.","s":"∀ {P : Causalean.PO.POSystem} {γ : Type u_1} [inst : MeasurableSpace γ] [inst_1 : StandardBorelSpace P.Ω] [inst_2 : MeasureTheory.IsFiniteMeasure P.μ] (S : Causalean.Estimation.ATT.TreatedEstimationSystem P γ), ∀ᵐ (ω : P.Ω) ∂P.μ, 0 ≤ S.propScore true ω","l":119,"f":"Causalean/Estimation/ATT/Setup.lean"},{"n":"Causalean.Estimation.ATT.TreatedEstimationSystem.residual_false_condExp_zero","k":"theorem","m":"Causalean.Estimation.ATT.Score.ScorePullout","a":"Estimation/ATT/Score","t":2,"d":"The σ(X)-conditional expectation of 1_{D=false}·(Y − μ₀(X)) vanishes a.s. The witness Y(false) =ᵐ μ₀_val ∘ factualX comes from μ₀_compat together with consistency on {D = false}.","s":"∀ {P : Causalean.PO.POSystem} {γ : Type u_1} [inst : MeasurableSpace γ] [inst_1 : StandardBorelSpace P.Ω] [inst_2 : MeasureTheory.IsFiniteMeasure P.μ] (S : Causalean.Estimation.ATT.TreatedEstimationSystem P γ), S.ATTAssumptions → P.μ[fun ω => S.dVar.indicator false ω * (S.factualY ω - S.μ₀_val (S.factualX ω)) | S.sigmaX] =ᵐ[P.μ] fun x => 0","l":129,"f":"Causalean/Estimation/ATT/Score/ScorePullout.lean"},{"n":"Causalean.Estimation.ATT.TreatedEstimationSystem.weighted_residual_false_integral_zero","k":"theorem","m":"Causalean.Estimation.ATT.Score.ScorePullout","a":"Estimation/ATT/Score","t":1,"d":"Weighted-residual mean-zero identity, control arm (ATT). Under the one-sided back-door ATT assumptions, if g : γ → ℝ is measurable and the product g(X) · 1{D=false} · (Y − μ₀(X)) is integrable, then the integral of the weighted control-arm residual against the observed-data law vanishes: ∫ g(X) · 1{D=false} · (Y − μ₀(X)) dμ = 0.","s":"∀ {P : Causalean.PO.POSystem} {γ : Type u_1} [inst : MeasurableSpace γ] [inst_1 : StandardBorelSpace P.Ω] [inst_2 : MeasureTheory.IsFiniteMeasure P.μ] (S : Causalean.Estimation.ATT.TreatedEstimationSystem P γ), S.ATTAssumptions → ∀ (g : γ → ℝ), Measurable g → MeasureTheory.Integrable (fun ω => g (S.factualX ω) * (S.dVar.indicator false ω * (S.factualY ω - S.μ₀_val (S.factualX ω)))) P.μ → ∫ (ω : P.Ω), g (S.factualX ω) * (S.dVar.indicator false ω * (S.factualY ω - S.μ₀_val (S.factualX ω))) ∂P.μ = 0","l":230,"f":"Causalean/Estimation/ATT/Score/ScorePullout.lean"},{"n":"Causalean.Estimation.ATT.TreatedEstimationSystem.η₀","k":"def","m":"Causalean.Estimation.ATT.Score.AIPWMoment","a":"Estimation/ATT/Score","t":1,"d":"The truth η₀ : TreatedNuisanceVec γ.","s":"{P : Causalean.PO.POSystem} → {γ : Type u_1} → [inst : MeasurableSpace γ] → [inst_1 : StandardBorelSpace P.Ω] → [inst_2 : MeasureTheory.IsFiniteMeasure P.μ] → Causalean.Estimation.ATT.TreatedEstimationSystem P γ → Causalean.Estimation.ATT.TreatedNuisanceVec γ","l":243,"f":"Causalean/Estimation/ATT/Score/AIPWMoment.lean"},{"n":"Causalean.Estimation.ATT.TreatedEstimationSystem.θ₀","k":"def","m":"Causalean.Estimation.ATT.Setup","a":"Estimation/ATT","t":1,"d":"Value-space ATT estimand: delegates to the PO-level adjusted form POBackdoorSystem.adjustedATT.","s":"{P : Causalean.PO.POSystem} → {γ : Type u_1} → [inst : MeasurableSpace γ] → [inst_1 : StandardBorelSpace P.Ω] → [inst_2 : MeasureTheory.IsFiniteMeasure P.μ] → Causalean.Estimation.ATT.TreatedEstimationSystem P γ → ℝ","l":193,"f":"Causalean/Estimation/ATT/Setup.lean"},{"n":"Causalean.Estimation.ATT.TreatedEstimationSystem.θ₀_eq_ATT","k":"theorem","m":"Causalean.Estimation.ATT.Setup","a":"Estimation/ATT","t":1,"d":"Value-space estimand equals the potential-outcome ATT. Under the one-sided back-door ATT assumptions, the value-space ATT estimand θ₀ (the adjusted control-regression functional) coincides with the average treatment effect on the treated defined on potential outcomes.","s":"∀ {P : Causalean.PO.POSystem} {γ : Type u_1} [inst : MeasurableSpace γ] [inst_1 : StandardBorelSpace P.Ω] [inst_2 : MeasureTheory.IsFiniteMeasure P.μ] (S : Causalean.Estimation.ATT.TreatedEstimationSystem P γ), S.ATTAssumptions → S.θ₀ = S.ATT","l":198,"f":"Causalean/Estimation/ATT/Setup.lean"},{"n":"Causalean.Estimation.ATT.TreatedEstimationSystem.μ₀_compat","k":"theorem","m":"Causalean.Estimation.ATT.Setup","a":"Estimation/ATT","t":2,"d":"Counterfactual compatibility (derived, under ATT identification). The control-arm outcome-regression representative equals the conditional mean of the untreated potential outcome given the covariates, almost surely.","s":"∀ {P : Causalean.PO.POSystem} {γ : Type u_1} [inst : MeasurableSpace γ] [inst_1 : StandardBorelSpace P.Ω] [inst_2 : MeasureTheory.IsFiniteMeasure P.μ] (S : Causalean.Estimation.ATT.TreatedEstimationSystem P γ), S.ATTAssumptions → P.μ[S.YofD false | S.sigmaX] =ᵐ[P.μ] fun ω => S.μ₀_val (S.factualX ω)","l":129,"f":"Causalean/Estimation/ATT/Setup.lean"},{"n":"Causalean.Estimation.ATT.TreatedEstimationSystem.π_val","k":"def","m":"Causalean.Estimation.ATT.Setup","a":"Estimation/ATT","t":1,"d":"Marginal treatment probability π = P[A = 1], viewed at the value-space layer. Delegates to the PO-level definition POBackdoorSystem.propTreated.","s":"{P : Causalean.PO.POSystem} → {γ : Type u_1} → [inst : MeasurableSpace γ] → [inst_1 : StandardBorelSpace P.Ω] → [inst_2 : MeasureTheory.IsFiniteMeasure P.μ] → Causalean.Estimation.ATT.TreatedEstimationSystem P γ → ℝ","l":188,"f":"Causalean/Estimation/ATT/Setup.lean"},{"n":"Causalean.Estimation.ATT.TreatedEstimationSystem.ψ_ATT","k":"def","m":"Causalean.Estimation.ATT.Score.AIPWMoment","a":"Estimation/ATT/Score","t":1,"d":"The ATT AIPW influence function at the truth for the population-π_T one-shot estimator: ψ_ATT(z) := (1/π) · m_AIPW(η₀, z, 0) − θ₀.","s":"{P : Causalean.PO.POSystem} → {γ : Type u_1} → [inst : MeasurableSpace γ] → [inst_1 : StandardBorelSpace P.Ω] → [inst_2 : MeasureTheory.IsFiniteMeasure P.μ] → Causalean.Estimation.ATT.TreatedEstimationSystem P γ → γ × Bool × ℝ → ℝ","l":68,"f":"Causalean/Estimation/ATT/Score/AIPWMoment.lean"},{"n":"Causalean.Estimation.ATT.TreatedNuisanceVec","k":"structure","m":"Causalean.Estimation.ATT.Score.AIPWMoment","a":"Estimation/ATT/Score","t":1,"d":"A pair of value-space ATT nuisance functions, used as the abstract nuisance space for the ATT AIPW moment functional: the control-arm outcome regression and a propensity score, together with their measurability.","s":"(γ : Type u_1) → [MeasurableSpace γ] → Type u_1","l":86,"f":"Causalean/Estimation/ATT/Score/AIPWMoment.lean"},{"n":"Causalean.Estimation.ATT.TreatedNuisanceVec.ext","k":"theorem","m":"Causalean.Estimation.ATT.Score.AIPWMoment","a":"Estimation/ATT/Score","t":2,"d":"Two treated-nuisance vectors are equal when their control regressions and propensity functions agree everywhere.","s":"∀ {γ : Type u_1} [inst : MeasurableSpace γ] {η η' : Causalean.Estimation.ATT.TreatedNuisanceVec γ}, (∀ (x : γ), η.μ₀_fn x = η'.μ₀_fn x) → (∀ (x : γ), η.e_fn x = η'.e_fn x) → η = η'","l":132,"f":"Causalean/Estimation/ATT/Score/AIPWMoment.lean"},{"n":"Causalean.Estimation.ATT.TreatedNuisanceVec.ext_iff","k":"theorem","m":"Causalean.Estimation.ATT.Score.AIPWMoment","a":"Estimation/ATT/Score","t":2,"d":"","s":"∀ {γ : Type u_1} [inst : MeasurableSpace γ] {η η' : Causalean.Estimation.ATT.TreatedNuisanceVec γ}, η = η' ↔ (∀ (x : γ), η.μ₀_fn x = η'.μ₀_fn x) ∧ ∀ (x : γ), η.e_fn x = η'.e_fn x","l":133,"f":"Causalean/Estimation/ATT/Score/AIPWMoment.lean"},{"n":"Causalean.Estimation.ATT.TreatedNuisanceVec.instAdd","k":"instance","m":"Causalean.Estimation.ATT.Score.AIPWMoment","a":"Estimation/ATT/Score","t":2,"d":"Addition of treated-nuisance vectors is componentwise addition of the control regression and propensity function.","s":"{γ : Type u_1} → [inst : MeasurableSpace γ] → Add (Causalean.Estimation.ATT.TreatedNuisanceVec γ)","l":103,"f":"Causalean/Estimation/ATT/Score/AIPWMoment.lean"},{"n":"Causalean.Estimation.ATT.TreatedNuisanceVec.instAddCommGroup","k":"instance","m":"Causalean.Estimation.ATT.Score.AIPWMoment","a":"Estimation/ATT/Score","t":2,"d":"Treated-nuisance vectors form an additive commutative group under componentwise operations.","s":"{γ : Type u_1} → [inst : MeasurableSpace γ] → AddCommGroup (Causalean.Estimation.ATT.TreatedNuisanceVec γ)","l":146,"f":"Causalean/Estimation/ATT/Score/AIPWMoment.lean"},{"n":"Causalean.Estimation.ATT.TreatedNuisanceVec.instModuleReal","k":"instance","m":"Causalean.Estimation.ATT.Score.AIPWMoment","a":"Estimation/ATT/Score","t":2,"d":"Treated-nuisance vectors form a real vector space under componentwise scalar multiplication.","s":"{γ : Type u_1} → [inst : MeasurableSpace γ] → Module ℝ (Causalean.Estimation.ATT.TreatedNuisanceVec γ)","l":184,"f":"Causalean/Estimation/ATT/Score/AIPWMoment.lean"},{"n":"Causalean.Estimation.ATT.TreatedNuisanceVec.instNeg","k":"instance","m":"Causalean.Estimation.ATT.Score.AIPWMoment","a":"Estimation/ATT/Score","t":2,"d":"Negation of a treated-nuisance vector negates the control regression and propensity function.","s":"{γ : Type u_1} → [inst : MeasurableSpace γ] → Neg (Causalean.Estimation.ATT.TreatedNuisanceVec γ)","l":111,"f":"Causalean/Estimation/ATT/Score/AIPWMoment.lean"},{"n":"Causalean.Estimation.ATT.TreatedNuisanceVec.instSMulReal","k":"instance","m":"Causalean.Estimation.ATT.Score.AIPWMoment","a":"Estimation/ATT/Score","t":2,"d":"Scalar multiplication of a treated-nuisance vector scales the control regression and propensity function.","s":"{γ : Type u_1} → [inst : MeasurableSpace γ] → SMul ℝ (Causalean.Estimation.ATT.TreatedNuisanceVec γ)","l":125,"f":"Causalean/Estimation/ATT/Score/AIPWMoment.lean"},{"n":"Causalean.Estimation.ATT.TreatedNuisanceVec.instSub","k":"instance","m":"Causalean.Estimation.ATT.Score.AIPWMoment","a":"Estimation/ATT/Score","t":2,"d":"Subtraction of treated-nuisance vectors is componentwise subtraction of the control regression and propensity function.","s":"{γ : Type u_1} → [inst : MeasurableSpace γ] → Sub (Causalean.Estimation.ATT.TreatedNuisanceVec γ)","l":117,"f":"Causalean/Estimation/ATT/Score/AIPWMoment.lean"},{"n":"Causalean.Estimation.ATT.TreatedNuisanceVec.instZero","k":"instance","m":"Causalean.Estimation.ATT.Score.AIPWMoment","a":"Estimation/ATT/Score","t":2,"d":"The zero treated-nuisance vector sets the control regression and propensity function to zero.","s":"{γ : Type u_1} → [inst : MeasurableSpace γ] → Zero (Causalean.Estimation.ATT.TreatedNuisanceVec γ)","l":99,"f":"Causalean/Estimation/ATT/Score/AIPWMoment.lean"},{"n":"Causalean.Estimation.ATT.attGeneralMoment","k":"def","m":"Causalean.Estimation.ATT.ATTInstance","a":"Estimation/ATT","t":1,"d":"ATT AIPW instance of the abstract GeneralMoment.","s":"{P : Causalean.PO.POSystem} → {γ : Type u_1} → [inst : MeasurableSpace γ] → [inst_1 : StandardBorelSpace P.Ω] → [inst_2 : MeasureTheory.IsFiniteMeasure P.μ] → (S : Causalean.Estimation.ATT.TreatedEstimationSystem P γ) → {ε : ℝ} → S.η₀ ∈ S.H_ε ε → 0 < S.π_val → Causalean.Estimation.OrthogonalMoments.GeneralMoment P.Ω P.μ (γ × Bool × ℝ) S.P_Z (Causalean.Estimation.ATT.TreatedNuisanceVec γ)","l":73,"f":"Causalean/Estimation/ATT/ATTInstance.lean"},{"n":"Causalean.Estimation.ATT.att_bilinearRem","k":"theorem","m":"Causalean.Estimation.ATT.ATTInstance","a":"Estimation/ATT","t":2,"d":"ATT AIPW satisfies BilinearRemainder with constant aipw_rem_const_ATT ε.","s":"∀ {P : Causalean.PO.POSystem} {γ : Type u_1} [inst : MeasurableSpace γ] [inst_1 : StandardBorelSpace P.Ω] [inst_2 : MeasureTheory.IsFiniteMeasure P.μ] (S : Causalean.Estimation.ATT.TreatedEstimationSystem P γ) {ε : ℝ} (hη₀_mem : S.η₀ ∈ S.H_ε ε), S.OneSidedOverlap ε → S.ATTAssumptions → ∀ (hπ_pos : 0 < S.π_val), MeasureTheory.Integrable (fun ω => S.factualY ω ^ 2) P.μ → MeasureTheory.Integrable (fun ω => S.YofD false ω ^ 2) P.μ → (∀ η ∈ S.H_ε ε, MeasureTheory.MemLp (fun x => η.μ₀_fn x - S.μ₀_val x) 2 S.P_X ∧ MeasureTheory.MemLp (fun x => η.e_fn x - S.e_val x) 2 S.P_X) → (∀ η ∈ S.H_ε ε, MeasureTheory.Integrable (fun z => (1 - Causalean.Estimation.ATE.BackdoorEstimationSystem.indA z) * (η.e_fn (Causalean.Estimation.ATE.BackdoorEstimationSystem.projX z) / (1 - η.e_fn (Causalean.Estimation.ATE.BackdoorEstimationSystem.projX z))) * (Causalean.Estimation.ATE.BackdoorEstimationSystem.projY z - η.μ₀_fn (Causalean.Estimation.ATE.BackdoorEstimationSystem.projX z))) S.P_Z) → ∃ C, Causalean.Estimation.OrthogonalMoments.BilinearRemainder (Causalean.Estimation.ATT.attGeneralMoment S hη₀_mem hπ_pos) C","l":130,"f":"Causalean/Estimation/ATT/ATTInstance.lean"},{"n":"Causalean.Estimation.ATT.att_dml_isAsymLinear","k":"theorem","m":"Causalean.Estimation.ATT.ATTInstance","a":"Estimation/ATT","t":1,"d":"Headline ATT AIPW DML asymptotic-linearity theorem, derived from the abstract dml_chernozhukov_asymptoticLinear in Estimation/OrthogonalMoments/DMLChernozhukov.lean. Fix an estimated-nuisance sequence η_hat, an i.i.d. sample of the data triple, and a one-shot cross-fitting split of that sample. Under membership of the truth nuisance in the overlap-bounded realization set H_ε, nonnegativity of the true propensity, one-sided overlap ε on the true propensity, the one-sided back-door ATT assumptions, a strictly positive marginal treatment probability, square-integrability of the factual outcome and of the untreated potential outcome Y(0), integrability of the truth-side control-arm IPW correction, and a limiting fold-size fraction c strictly between 0 and 1 with the treated-fold cardinality fraction converging to c: if every candidate draw η_hat n ω lies in the overlap-bounded realization set H_ε, every candidate propensity is nonnegative, each candidate control-regression and propensity error admits an L²(P_X) witness, each candidate IPW correction is integrable, the AIPW moment functional is measurable jointly in the probability-space and data arguments, and on each cross-fitting fold, both singly and jointly, the moment at every candidate nuisance is integrable and square-integrable against the observed data law, the control-regression and propensity error rates are individually o_p(1) in L²(P_X), and their product is o_p(n^{-1/2}), then the Chernozhukov one-step DML estimator built from the ATT AIPW moment, the sample, the split, and the candidate nuisance sequence is asymptotically linear at the true ATT θ₀, with influence function ψ(z) = (1/π_T) · aipwMomentATTFunctional η₀ z θ₀.","s":"∀ {P : Causalean.PO.POSystem} {γ : Type u_1} [inst : MeasurableSpace γ] [inst_1 : StandardBorelSpace P.Ω] [inst_2 : MeasureTheory.IsFiniteMeasure P.μ] (S : Causalean.Estimation.ATT.TreatedEstimationSystem P γ) {ε : ℝ} (hη₀_mem : S.η₀ ∈ S.H_ε ε), (∀ (x : γ), 0 ≤ S.e_val x) → S.OneSidedOverlap ε → S.ATTAssumptions → ∀ (hπ_pos : 0 < S.π_val), MeasureTheory.Integrable (fun ω => S.factualY ω ^ 2) P.μ → MeasureTheory.Integrable (fun ω => S.YofD false ω ^ 2) P.μ → MeasureTheory.Integrable (fun ω => (1 - S.dVar.indicator true ω) * (S.propScore true ω / (1 - S.propScore true ω)) * (S.factualY ω - S.adjustedCE false ω)) P.μ → ∀ (sample : Causalean.Stat.IIDSample P.Ω (γ × Bool × ℝ) P.μ S.P_Z) (split : Causalean.Stat.OneShotSplit sample) {c : ℝ}, 0 < c → c < 1 → Filter.Tendsto (fun n => ↑(split.foldB n).card / ↑n) Filter.atTop (nhds c) → ∀ (η_hat : ℕ → P.Ω → Causalean.Estimation.ATT.TreatedNuisanceVec γ), (∀ (n : ℕ) (ω : P.Ω), η_hat n ω ∈ S.H_ε ε) → (∀ (n : ℕ) (ω : P.Ω) (x : γ), 0 ≤ (η_hat n ω).e_fn x) → (∀ (n : ℕ) (ω : P.Ω), MeasureTheory.MemLp (fun x => (η_hat n ω).μ₀_fn x - S.μ₀_val x) 2 S.P_X) → (∀ (n : ℕ) (ω : P.Ω), MeasureTheory.MemLp (fun x => (η_hat n ω).e_fn x - S.e_val x) 2 S.P_X) → (∀ (n : ℕ) (ω : P.Ω), MeasureTheory.Integrable (fun z => (1 - Causalean.Estimation.ATE.BackdoorEstimationSystem.indA z) * ((η_hat n ω).e_fn (Causalean.Estimation.ATE.BackdoorEstimationSystem.projX z) / (1 - (η_hat n ω).e_fn (Causalean.Estimation.ATE.BackdoorEstimationSystem.projX z))) * (Causalean.Estimation.ATE.BackdoorEstimationSystem.projY z - (η_hat n ω).μ₀_fn (Causalean.Estimation.ATE.BackdoorEstimationSystem.projX z))) S.P_Z) → (∀ (n : ℕ), Measurable fun p => Causalean.Estimation.ATT.TreatedEstimationSystem.aipwMomentATTFunctional (η_hat n p.1) p.2 S.θ₀) → (∀ (n : ℕ), Measurable fun ω z => Causalean.Estimation.ATT.TreatedEstimationSystem.aipwMomentATTFunctional (η_hat n ω) z S.θ₀) → (∀ (n : ℕ), Measurable fun p => Causalean.Estimation.ATT.TreatedEstimationSystem.aipwMomentATTFunctional (η_hat n p.1) p.2 S.θ₀) → (∀ (n : ℕ) (ω : P.Ω), MeasureTheory.Integrable (fun z => Causalean.Estimation.ATT.TreatedEstimationSystem.aipwMomentATTFunctional (η_hat n ω) z S.θ₀) S.P_Z) → (∀ (n : ℕ) (ω : P.Ω), MeasureTheory.Integrable (fun z => Causalean.Estimation.ATT.TreatedEstimationSystem.aipwMomentATTFunctional (η_hat n ω) z S.θ₀ ^ 2) S.P_Z) → Causalean.Stat.IsLittleOp (fun n ω => ↑((Causalean.Estimation.ATT.attGeneralMoment S hη₀_mem hπ_pos).ρ₁ (η_hat n ω) S.η₀)) (fun x => 1) P.μ → Causalean.Stat.IsLittleOp (fun n ω => ↑((Causalean.Estimation.ATT.attGeneralMoment S hη₀_mem hπ_pos).ρ₂ (η_hat n ω) S.η₀)) (fun x => 1) P.μ → Causalean.Stat.IsLittleOp (fun n ω => ↑((Causalean.Estimation.ATT.attGeneralMoment S hη₀_mem hπ_pos).ρ₁ (η_hat n ω) S.η₀) * ↑((Causalean.Estimation.ATT.attGeneralMoment S hη₀_mem hπ_pos).ρ₂ (η_hat n ω) S.η₀)) (fun n => ↑n ^ (-(1 / 2))) P.μ → Causalean.Stat.IsAsymLinear (Causalean.Estimation.OrthogonalMoments.dmlChernozhukovEstimator (Causalean.Estimation.ATT.attGeneralMoment S hη₀_mem hπ_pos) sample split η_hat) S.θ₀ (fun z => -(Causalean.Estimation.ATT.attGeneralMoment S hη₀_mem hπ_pos).J₀_inv * Causalean.Estimation.ATT.TreatedEstimationSystem.aipwMomentATTFunctional S.η₀ z S.θ₀) sample split.foldB","l":168,"f":"Causalean/Estimation/ATT/ATTInstance.lean"},{"n":"Causalean.Estimation.ATT.att_meanZero","k":"theorem","m":"Causalean.Estimation.ATT.ATTInstance","a":"Estimation/ATT","t":2,"d":"ATT AIPW satisfies MeanZero. Direct repackaging of aipw_mean_zero_ATT from Estimation/ATT/Score/MeanZero.lean.","s":"∀ {P : Causalean.PO.POSystem} {γ : Type u_1} [inst : MeasurableSpace γ] [inst_1 : StandardBorelSpace P.Ω] [inst_2 : MeasureTheory.IsFiniteMeasure P.μ] (S : Causalean.Estimation.ATT.TreatedEstimationSystem P γ) {ε : ℝ} (hη₀_mem : S.η₀ ∈ S.H_ε ε), S.OneSidedOverlap ε → S.ATTAssumptions → ∀ (hπ_pos : 0 < S.π_val), MeasureTheory.Integrable (fun ω => S.factualY ω ^ 2) P.μ → MeasureTheory.Integrable (fun ω => S.YofD false ω ^ 2) P.μ → MeasureTheory.Integrable (fun ω => (1 - S.dVar.indicator true ω) * (S.propScore true ω / (1 - S.propScore true ω)) * (S.factualY ω - S.adjustedCE false ω)) P.μ → Causalean.Estimation.OrthogonalMoments.MeanZero (Causalean.Estimation.ATT.attGeneralMoment S hη₀_mem hπ_pos)","l":110,"f":"Causalean/Estimation/ATT/ATTInstance.lean"},{"n":"Causalean.Estimation.ATT.dmlEstimator_ATT","k":"def","m":"Causalean.Estimation.ATT.DML","a":"Estimation/ATT","t":1,"d":"One-shot DML / AIPW estimator of the back-door ATT (def:est-dml-att).","s":"{P : Causalean.PO.POSystem} → {γ : Type u_1} → [inst : MeasurableSpace γ] → [inst_1 : StandardBorelSpace P.Ω] → [inst_2 : MeasureTheory.IsFiniteMeasure P.μ] → (S : Causalean.Estimation.ATT.TreatedEstimationSystem P γ) → (sample : Causalean.Stat.IIDSample P.Ω (γ × Bool × ℝ) P.μ S.P_Z) → Causalean.Stat.OneShotSplit sample → (ℕ → P.Ω → γ → ℝ) → (ℕ → P.Ω → γ → ℝ) → ℕ → P.Ω → ℝ","l":81,"f":"Causalean/Estimation/ATT/DML.lean"},{"n":"Causalean.Estimation.ATT.dml_ATT_isAsymLinear","k":"theorem","m":"Causalean.Estimation.ATT.DML","a":"Estimation/ATT","t":1,"d":"Asymptotic linearity of the one-shot DML ATT — thm:est-dml-att-al. Fix candidate control-regression and propensity estimator sequences μ₀_hat and e_hat, an i.i.d. sample of the data triple, and a one-shot cross-fitting split of that sample. Under the true propensity bounded above by 1 − ε almost everywhere, nonnegativity of the true propensity, one-sided overlap ε on the treated-arm propensity, the one-sided back-door ATT assumptions, a strictly positive marginal treatment probability, square-integrability of the factual outcome and of the untreated potential outcome Y(0), and a limiting fold-size fraction c strictly between 0 and 1 with the treated-fold cardinality fraction converging to c: if the candidate propensity is bounded above by 1 − ε almost everywhere, for every n, ω, the candidate propensity is nonnegative everywhere, the candidate regressions are jointly measurable in the probability-space and covariate arguments, each candidate regression, at every n, ω, is square-integrable against the covariate law, each candidate regression depends only on its own cross-fitting fold, singly and jointly with the covariate, the ATT AIPW moment at every candidate regression pair is integrable and square-integrable against the observed data law, the control-regression and propensity error rates are individually o_p(1) in L²(P_X), and their product is o_p(n^{-1/2}), then the population-π one-shot DML/AIPW ATT estimator is asymptotically linear at the true ATT θ₀, with influence function ψ_ATT, along the sample and the cross-fitting folds.","s":"∀ {P : Causalean.PO.POSystem} {γ : Type u_1} [inst : MeasurableSpace γ] [inst_1 : StandardBorelSpace P.Ω] [inst_2 : MeasureTheory.IsFiniteMeasure P.μ] (S : Causalean.Estimation.ATT.TreatedEstimationSystem P γ) {ε : ℝ}, (∀ᵐ (x : γ) ∂S.P_X, S.e_val x ≤ 1 - ε) → (∀ (x : γ), 0 ≤ S.e_val x) → S.OneSidedOverlap ε → S.ATTAssumptions → 0 < S.π_val → MeasureTheory.Integrable (fun ω => S.factualY ω ^ 2) P.μ → MeasureTheory.Integrable (fun ω => S.YofD false ω ^ 2) P.μ → ∀ (sample : Causalean.Stat.IIDSample P.Ω (γ × Bool × ℝ) P.μ S.P_Z) (split : Causalean.Stat.OneShotSplit sample) {c : ℝ}, 0 < c → c < 1 → Filter.Tendsto (fun n => ↑(split.foldB n).card / ↑n) Filter.atTop (nhds c) → ∀ (μ₀_hat e_hat : ℕ → P.Ω → γ → ℝ), (∀ (n : ℕ) (ω : P.Ω), ∀ᵐ (x : γ) ∂S.P_X, e_hat n ω x ≤ 1 - ε) → (∀ (n : ℕ) (ω : P.Ω) (x : γ), 0 ≤ e_hat n ω x) → (∀ (n : ℕ), Measurable fun p => μ₀_hat n p.1 p.2) → (∀ (n : ℕ), Measurable fun p => e_hat n p.1 p.2) → (∀ (n : ℕ) (ω : P.Ω), MeasureTheory.MemLp (fun x => μ₀_hat n ω x) 2 S.P_X) → (∀ (n : ℕ) (ω : P.Ω), MeasureTheory.MemLp (fun x => e_hat n ω x) 2 S.P_X) → (∀ (n : ℕ), Measurable (μ₀_hat n)) → (∀ (n : ℕ), Measurable (e_hat n)) → (∀ (n : ℕ), Measurable fun p => μ₀_hat n p.1 p.2) → (∀ (n : ℕ), Measurable fun p => e_hat n p.1 p.2) → (∀ (n : ℕ) (ω : P.Ω), MeasureTheory.Integrable (fun z => Causalean.Estimation.ATT.TreatedEstimationSystem.aipwMomentATT z (μ₀_hat n ω) (e_hat n ω) S.θ₀) S.P_Z) → (∀ (n : ℕ) (ω : P.Ω), MeasureTheory.Integrable (fun z => Causalean.Estimation.ATT.TreatedEstimationSystem.aipwMomentATT z (μ₀_hat n ω) (e_hat n ω) S.θ₀ ^ 2) S.P_Z) → Causalean.Stat.IsLittleOp (fun n ω => (MeasureTheory.eLpNorm (fun x => μ₀_hat n ω x - S.μ₀_val x) 2 S.P_X).toReal) (fun x => 1) P.μ → Causalean.Stat.IsLittleOp (fun n ω => (MeasureTheory.eLpNorm (fun x => e_hat n ω x - S.e_val x) 2 S.P_X).toReal) (fun x => 1) P.μ → Causalean.Stat.IsLittleOp (fun n ω => (MeasureTheory.eLpNorm (fun x => μ₀_hat n ω x - S.μ₀_val x) 2 S.P_X).toReal * (MeasureTheory.eLpNorm (fun x => e_hat n ω x - S.e_val x) 2 S.P_X).toReal) (fun n => ↑n ^ (-(1 / 2))) P.μ → Causalean.Stat.IsAsymLinear (Causalean.Estimation.ATT.dmlEstimator_ATT S sample split μ₀_hat e_hat) S.θ₀ S.ψ_ATT sample split.foldB","l":237,"f":"Causalean/Estimation/ATT/DML.lean"},{"n":"Causalean.Estimation.ATT.ψ_ATT_finite_var","k":"theorem","m":"Causalean.Estimation.ATT.DML","a":"Estimation/ATT","t":2,"d":"Finite variance of the centered population-π ATT influence function. Integrable ψ_ATT² against P_Z. Derived (not assumed) from aipw_finite_var_ATT (square-integrability of the moment at θ₀) and boundedness of the treatment indicator, via ψ_ATT = (1/π_T)·(m(η₀,·,θ₀) + θ₀·A) − θ₀ ∈ L²(P_Z).","s":"∀ {P : Causalean.PO.POSystem} {γ : Type u_1} [inst : MeasurableSpace γ] [inst_1 : StandardBorelSpace P.Ω] [inst_2 : MeasureTheory.IsFiniteMeasure P.μ] (S : Causalean.Estimation.ATT.TreatedEstimationSystem P γ) {ε : ℝ}, S.OneSidedOverlap ε → S.ATTAssumptions → MeasureTheory.Integrable (fun ω => S.factualY ω ^ 2) P.μ → MeasureTheory.Integrable (fun ω => S.YofD false ω ^ 2) P.μ → MeasureTheory.Integrable (fun z => S.ψ_ATT z ^ 2) S.P_Z","l":189,"f":"Causalean/Estimation/ATT/DML.lean"},{"n":"Causalean.Estimation.ATT.ψ_ATT_integral_zero","k":"theorem","m":"Causalean.Estimation.ATT.DML","a":"Estimation/ATT","t":2,"d":"Mean zero of the centered population-π ATT influence function ψ_ATT.","s":"∀ {P : Causalean.PO.POSystem} {γ : Type u_1} [inst : MeasurableSpace γ] [inst_1 : StandardBorelSpace P.Ω] [inst_2 : MeasureTheory.IsFiniteMeasure P.μ] (S : Causalean.Estimation.ATT.TreatedEstimationSystem P γ) {ε : ℝ}, S.OneSidedOverlap ε → S.ATTAssumptions → 0 < S.π_val → MeasureTheory.Integrable (fun ω => S.factualY ω ^ 2) P.μ → MeasureTheory.Integrable (fun ω => S.YofD false ω ^ 2) P.μ → MeasureTheory.Integrable (fun ω => (1 - S.dVar.indicator true ω) * (S.propScore true ω / (1 - S.propScore true ω)) * (S.factualY ω - S.adjustedCE false ω)) P.μ → ∫ (z : γ × Bool × ℝ), S.ψ_ATT z ∂S.P_Z = 0","l":111,"f":"Causalean/Estimation/ATT/DML.lean"},{"n":"Causalean.Estimation.CATE.CATEEstimationSystem","k":"structure","m":"Causalean.Estimation.CATE.Setup","a":"Estimation/CATE","t":1,"d":"Wrapper around BackdoorEstimationSystem whose semantic role is to expose the CATE target τ_val x = μ_val 1 x − μ_val 0 x as a derived function-valued estimand. No new fields are introduced; the underlying BackdoorEstimationSystem carries consistency, conditional exchangeability, overlap, integrability, the value-space μ_val / e_val, observable outcome-regression compatibility μ_reg_compat, and propensity compatibility e_compat. Counterfactual outcome-regression compatibility is derived later from the back-door causal assumptions, not inherited as a field.","s":"(P : Causalean.PO.POSystem) → (γ : Type u_1) → [MeasurableSpace γ] → [StandardBorelSpace P.Ω] → [MeasureTheory.IsFiniteMeasure P.μ] → Type (max (max u_1 u_2) u_3)","l":48,"f":"Causalean/Estimation/CATE/Setup.lean"},{"n":"Causalean.Estimation.CATE.CATEEstimationSystem.measurable_τ_val","k":"theorem","m":"Causalean.Estimation.CATE.Setup","a":"Estimation/CATE","t":2,"d":"The value-space CATE is measurable, since both arms of μ_val are.","s":"∀ {P : Causalean.PO.POSystem} {γ : Type u_1} [inst : MeasurableSpace γ] [inst_1 : StandardBorelSpace P.Ω] [inst_2 : MeasureTheory.IsFiniteMeasure P.μ] (S : Causalean.Estimation.CATE.CATEEstimationSystem P γ), Measurable S.τ_val","l":70,"f":"Causalean/Estimation/CATE/Setup.lean"},{"n":"Causalean.Estimation.CATE.CATEEstimationSystem.phi_eta_cond_exp_eq_tau","k":"theorem","m":"Causalean.Estimation.CATE.Core.PseudoOutcomeMean","a":"Estimation/CATE/Core","t":1,"d":"Value-space form of the DR pseudo-outcome mean-CATE identity. Under the back-door identification assumptions, two-sided strict overlap: the treatment propensity lies in [ε, 1 − ε] for some ε ∈ (0, 1/2] almost surely, and integrability of the true DR pseudo-outcome φ₀ evaluated at the factual data triple, for covariate-law-almost-every x, the mean of φ₀ under the regular conditional distribution of the data triple given the covariate value x equals the value-space CATE τ_val x.","s":"∀ {P : Causalean.PO.POSystem} {γ : Type u_1} [inst : MeasurableSpace γ] [inst_1 : StandardBorelSpace P.Ω] [inst_2 : MeasureTheory.IsFiniteMeasure P.μ] [inst_3 : StandardBorelSpace γ] [inst_4 : Nonempty γ] (S : Causalean.Estimation.CATE.CATEEstimationSystem P γ), S.Assumptions → ∀ {ε : ℝ}, S.StrictOverlap ε → MeasureTheory.Integrable (fun ω => Causalean.Estimation.CATE.phi₀ S (S.factualZ ω)) P.μ → ∀ᵐ (x : γ) ∂S.P_X, ∫ (z : γ × Bool × ℝ), Causalean.Estimation.CATE.phi₀ S z ∂(ProbabilityTheory.condDistrib S.factualZ S.factualX P.μ) x = S.τ_val x","l":341,"f":"Causalean/Estimation/CATE/Core/PseudoOutcomeMean.lean"},{"n":"Causalean.Estimation.CATE.CATEEstimationSystem.phi₀_factualZ_cond_exp","k":"theorem","m":"Causalean.Estimation.CATE.Core.PseudoOutcomeMean","a":"Estimation/CATE/Core","t":1,"d":"σ(X)-conditional mean of the true DR pseudo-outcome equals the CATE. Under the back-door identification assumptions and two-sided strict overlap: the treatment propensity lies in [ε, 1 − ε] for some ε ∈ (0, 1/2] almost surely, the σ(X)-conditional expectation of the true DR pseudo-outcome φ₀, evaluated at the factual data triple, equals the value-space CATE τ_val pulled back along the covariate, almost surely.","s":"∀ {P : Causalean.PO.POSystem} {γ : Type u_1} [inst : MeasurableSpace γ] [inst_1 : StandardBorelSpace P.Ω] [inst_2 : MeasureTheory.IsFiniteMeasure P.μ] (S : Causalean.Estimation.CATE.CATEEstimationSystem P γ), S.Assumptions → ∀ {ε : ℝ}, S.StrictOverlap ε → P.μ[fun ω => Causalean.Estimation.CATE.phi₀ S (S.factualZ ω) | S.sigmaX] =ᵐ[P.μ] fun ω => S.τ_val (S.factualX ω)","l":134,"f":"Causalean/Estimation/CATE/Core/PseudoOutcomeMean.lean"},{"n":"Causalean.Estimation.CATE.CATEEstimationSystem.tau_val_eq_CATE","k":"theorem","m":"Causalean.Estimation.CATE.Setup","a":"Estimation/CATE","t":1,"d":"Value-space CATE recovers the conditional average treatment effect. Under the back-door causal assumptions — consistency of observed and potential outcomes, treatment ignorability given the covariates, two-sided overlap, and integrability of both potential outcomes, the value-space CATE, evaluated at the observed covariate and viewed as a random variable on the sample space, agrees almost surely with the σ(X)-conditional expectation of the potential-outcome contrast Y(1) − Y(0). This is the last sentence of def:est-cate-causal-assumptions:","s":"∀ {P : Causalean.PO.POSystem} {γ : Type u_1} [inst : MeasurableSpace γ] [inst_1 : StandardBorelSpace P.Ω] [inst_2 : MeasureTheory.IsFiniteMeasure P.μ] (S : Causalean.Estimation.CATE.CATEEstimationSystem P γ), S.Assumptions → (fun ω => S.τ_val (S.factualX ω)) =ᵐ[P.μ] P.μ[fun ω => S.YofD true ω - S.YofD false ω | S.sigmaX]","l":75,"f":"Causalean/Estimation/CATE/Setup.lean"},{"n":"Causalean.Estimation.CATE.CATEEstimationSystem.τ_val","k":"def","m":"Causalean.Estimation.CATE.Setup","a":"Estimation/CATE","t":1,"d":"Value-space CATE: τ_val x := μ_val 1 x − μ_val 0 x. This is the \"observable CATE regression\" τ_0 of def:est-cate-system.","s":"{P : Causalean.PO.POSystem} → {γ : Type u_1} → [inst : MeasurableSpace γ] → [inst_1 : StandardBorelSpace P.Ω] → [inst_2 : MeasureTheory.IsFiniteMeasure P.μ] → Causalean.Estimation.CATE.CATEEstimationSystem P γ → γ → ℝ","l":65,"f":"Causalean/Estimation/CATE/Setup.lean"},{"n":"Causalean.Estimation.CATE.abs_integral_phiDiff_mul_le_product","k":"theorem","m":"Causalean.Estimation.CATE.Core.SecondOrderBias","a":"Estimation/CATE/Core","t":1,"d":"Second-order product bias bound for the DR-Learner. Fix a candidate nuisance pair η and a test function w. Under ε strictly positive, the back-door identification assumptions, if η's propensity is uniformly bounded in [ε, 1 − ε] for every covariate value, the truth nuisance likewise has propensity uniformly bounded in [ε, 1 − ε], each candidate outcome-regression arm, composed with the covariate, is integrable, w is measurable, w is dominated by a nonnegative bound B, the w-weighted pseudo-outcome bias is integrable, each candidate outcome-regression error is square-integrable against the covariate law, and the propensity error is square-integrable against the covariate law, then the absolute value of the population integral of the w-weighted pseudo-outcome bias (phi_eta η − φ₀) · w is bounded by B/ε times the sum, over the two treatment arms, of ‖η.μ_fn a − μ_val a‖_{L²(P_X)} · ‖η.e_fn − e_val‖_{L²(P_X)}.","s":"∀ {P : Causalean.PO.POSystem} {γ : Type u_1} [inst : MeasurableSpace γ] [inst_1 : StandardBorelSpace P.Ω] [inst_2 : MeasureTheory.IsFiniteMeasure P.μ] (S : Causalean.Estimation.CATE.CATEEstimationSystem P γ) {ε : ℝ}, 0 < ε → S.Assumptions → ∀ η ∈ Causalean.Estimation.ATE.BackdoorEstimationSystem.H_ε ε, S.η₀ ∈ Causalean.Estimation.ATE.BackdoorEstimationSystem.H_ε ε → (∀ (a : Bool), MeasureTheory.Integrable (fun ω => η.μ_fn a (S.factualX ω)) P.μ) → ∀ (w : γ → ℝ), Measurable w → ∀ {B : ℝ}, 0 ≤ B → (∀ (x : γ), |w x| ≤ B) → MeasureTheory.Integrable (fun ω => Causalean.Estimation.CATE.phi_eta (S.factualZ ω) η - Causalean.Estimation.CATE.phi₀ S (S.factualZ ω)) P.μ → MeasureTheory.Integrable (fun ω => (Causalean.Estimation.CATE.phi_eta (S.factualZ ω) η - Causalean.Estimation.CATE.phi₀ S (S.factualZ ω)) * w (S.factualX ω)) P.μ → (∀ (a : Bool), MeasureTheory.MemLp (fun x => η.μ_fn a x - S.μ_val a x) 2 S.P_X) → MeasureTheory.MemLp (fun x => η.e_fn x - S.e_val x) 2 S.P_X → |∫ (z : γ × Bool × ℝ), (Causalean.Estimation.CATE.phi_eta z η - Causalean.Estimation.CATE.phi₀ S z) * w z.1 ∂S.P_Z| ≤ B / ε * ∑ a, (MeasureTheory.eLpNorm (fun x => η.μ_fn a x - S.μ_val a x) 2 S.P_X).toReal * (MeasureTheory.eLpNorm (fun x => η.e_fn x - S.e_val x) 2 S.P_X).toReal","l":147,"f":"Causalean/Estimation/CATE/Core/SecondOrderBias.lean"},{"n":"Causalean.Estimation.CATE.cate_dr_oracle_efficient_linear","k":"theorem","m":"Causalean.Estimation.CATE.Kennedy.LinearSmootherCATE","a":"Estimation/CATE/Kennedy","t":1,"d":"Oracle efficiency for the DR-Learner with a linear-smoother second stage. Fix a CATE estimation system, a linear-smoother second-stage operator op, an estimated nuisance sequence η_hat, an evaluation point x, a centering-rate sequence d_n, and a bias-identity relation BiasIdent. Under the back-door identification assumptions and two-sided strict overlap for the truth, if the smoothed oracle estimator is stable at (τ_val, d_n, x) relative to BiasIdent, the centering sequence d_n converges to 0 in probability, the pseudo-outcome bias, the true pseudo-outcome, and the smoothed conditional bias jointly satisfy the identity relation BiasIdent, and the smoothed conditional-bias evaluation is o_p of the oracle risk scale, then the DR-Learner CATE estimator and the oracle estimator, both built from the linear-smoother second-stage operator, differ by o_p of the oracle risk scale.","s":"∀ {P : Causalean.PO.POSystem} {γ : Type u_1} [inst : MeasurableSpace γ] [inst_1 : StandardBorelSpace P.Ω] [inst_2 : MeasureTheory.IsFiniteMeasure P.μ] (S : Causalean.Estimation.CATE.CATEEstimationSystem P γ), S.Assumptions → ∀ {ε : ℝ}, S.StrictOverlap ε → ∀ (op : Causalean.Estimation.OrthogonalMoments.LinearSmootherOp P.Ω P.μ γ) (η_hat : ℕ → P.Ω → Causalean.Estimation.ATE.NuisanceVec γ) (x : γ) (d_n : ℕ → P.Ω → ℝ) (BiasIdent : (ℕ → P.Ω → γ × Bool × ℝ → ℝ) → (γ × Bool × ℝ → ℝ) → (ℕ → P.Ω → γ → ℝ) → Prop), Causalean.Estimation.OrthogonalMoments.Stable op.toSecondStageOperator S.τ_val d_n x BiasIdent → Causalean.Stat.Tendsto_inProb d_n (fun x => 0) P.μ → (BiasIdent (fun n ω z => Causalean.Estimation.CATE.phi_eta z (η_hat n ω)) (fun z => Causalean.Estimation.CATE.phi₀ S z) fun n ω u => Causalean.Estimation.CATE.condBias (η_hat n ω) S.η₀ u) → Causalean.Stat.IsLittleOp (fun n ω => op.evalAt n ω (fun z => Causalean.Estimation.CATE.condBias (η_hat n ω) S.η₀ z.1) x) (fun n => Causalean.Estimation.CATE.drOracleRiskScale S op.toSecondStageOperator x n) P.μ → Causalean.Stat.IsLittleOp (fun n ω => Causalean.Estimation.CATE.drLearnerEstimator S op.toSecondStageOperator η_hat n ω x - Causalean.Estimation.CATE.drOracleEstimator S op.toSecondStageOperator n ω x) (fun n => Causalean.Estimation.CATE.drOracleRiskScale S op.toSecondStageOperator x n) P.μ","l":227,"f":"Causalean/Estimation/CATE/Kennedy/LinearSmootherCATE.lean"},{"n":"Causalean.Estimation.CATE.cate_linear_smoother_bias_bound","k":"theorem","m":"Causalean.Estimation.CATE.Kennedy.LinearSmootherCATE","a":"Estimation/CATE/Kennedy","t":1,"d":"Linear-smoother bias bound for the DR-Learner CATE estimator. Fix a candidate nuisance sequence η_hat, a linear-smoother operator op, a sample index n, a realization ω, an evaluation point x, a data enumeration xs over an index set B with weights w, and constants c_n, p, q. Under two-sided strict overlap for the truth, if the estimated nuisance η_hat n ω has propensity uniformly bounded in [ε, 1 − ε], op realizes the linear smoother Σ_i w_i · f(xs i) at (n, ω, x) over B, the weights satisfy the absolute-value envelope Σ |w_i| ≤ c_n, and p and q are Hölder-conjugate exponents, then the smoothed conditional-bias evaluation op.evalAt n ω (condBias η_hat η₀ ∘ proj₁) x is bounded in absolute value by aipw_rem_const ε · c_n times the weighted-p-norm of the propensity error Δπ times the sum over treatment arms of the weighted-q-norm of the outcome-regression error Δμ_a.","s":"∀ {P : Causalean.PO.POSystem} {γ : Type u_1} [inst : MeasurableSpace γ] [inst_1 : StandardBorelSpace P.Ω] [inst_2 : MeasureTheory.IsFiniteMeasure P.μ] (S : Causalean.Estimation.CATE.CATEEstimationSystem P γ) {ε : ℝ}, S.StrictOverlap ε → ∀ (op : Causalean.Estimation.OrthogonalMoments.LinearSmootherOp P.Ω P.μ γ) (η_hat : ℕ → P.Ω → Causalean.Estimation.ATE.NuisanceVec γ) (n : ℕ) (ω : P.Ω) (x : γ) {ι : Type u_2} (B : Finset ι) (w : ι → ℝ) (xs : ι → γ × Bool × ℝ) (c_n p q : ℝ), η_hat n ω ∈ Causalean.Estimation.ATE.BackdoorEstimationSystem.H_ε ε → op.IsLinearSmoother n ω x B w xs → ∑ i ∈ B, |w i| ≤ c_n → p.HolderConjugate q → |op.evalAt n ω (fun z => Causalean.Estimation.CATE.condBias (η_hat n ω) S.η₀ z.1) x| ≤ Causalean.Estimation.ATE.BackdoorEstimationSystem.aipw_rem_const ε * c_n * Causalean.Estimation.OrthogonalMoments.WeightedNorm B w (fun i => (η_hat n ω).e_fn (xs i).1 - S.e_val (xs i).1) p * ∑ a, Causalean.Estimation.OrthogonalMoments.WeightedNorm B w (fun i => (η_hat n ω).μ_fn a (xs i).1 - S.μ_val a (xs i).1) q","l":46,"f":"Causalean/Estimation/CATE/Kennedy/LinearSmootherCATE.lean"},{"n":"Causalean.Estimation.CATE.condBias","k":"def","m":"Causalean.Estimation.CATE.Core.ConditionalBias","a":"Estimation/CATE/Core","t":1,"d":"The closed-form σ(X)-conditional bias of the DR pseudo-outcome contrast φ_η − φ_0, summed over the two treatment arms:","s":"{γ : Type u_1} → [inst : MeasurableSpace γ] → Causalean.Estimation.ATE.NuisanceVec γ → Causalean.Estimation.ATE.NuisanceVec γ → γ → ℝ","l":65,"f":"Causalean/Estimation/CATE/Core/ConditionalBias.lean"},{"n":"Causalean.Estimation.CATE.condBias_zero_of_outcome_match","k":"theorem","m":"Causalean.Estimation.CATE.Core.ConditionalBias","a":"Estimation/CATE/Core","t":2,"d":"If both outcome arms match the truth at x, then condBias η η₀ x = 0.","s":"∀ {γ : Type u_1} [inst : MeasurableSpace γ] (η η₀ : Causalean.Estimation.ATE.NuisanceVec γ) (x : γ), (∀ (a : Bool), η.μ_fn a x = η₀.μ_fn a x) → Causalean.Estimation.CATE.condBias η η₀ x = 0","l":740,"f":"Causalean/Estimation/CATE/Core/ConditionalBias.lean"},{"n":"Causalean.Estimation.CATE.condBias_zero_of_propensity_match","k":"theorem","m":"Causalean.Estimation.CATE.Core.ConditionalBias","a":"Estimation/CATE/Core","t":2,"d":"If the propensity matches the truth at x, then condBias η η₀ x = 0.","s":"∀ {γ : Type u_1} [inst : MeasurableSpace γ] (η η₀ : Causalean.Estimation.ATE.NuisanceVec γ) (x : γ), η.e_fn x = η₀.e_fn x → Causalean.Estimation.CATE.condBias η η₀ x = 0","l":726,"f":"Causalean/Estimation/CATE/Core/ConditionalBias.lean"},{"n":"Causalean.Estimation.CATE.cond_exp_phi_eta_dir_deriv_at_truth_zero","k":"theorem","m":"Causalean.Estimation.CATE.Core.ConditionalBias","a":"Estimation/CATE/Core","t":2,"d":"The σ(X)-conditional expectation of the directional derivative phi_eta_dir_deriv at the truth g₀ = η₀ is zero a.e.","s":"∀ {γ : Type u_1} [inst : MeasurableSpace γ] {P : Causalean.PO.POSystem} [inst_1 : StandardBorelSpace P.Ω] [inst_2 : MeasureTheory.IsFiniteMeasure P.μ] (S : Causalean.Estimation.CATE.CATEEstimationSystem P γ), S.Assumptions → ∀ {ε : ℝ}, 0 < ε → S.η₀ ∈ Causalean.Estimation.ATE.BackdoorEstimationSystem.H_ε ε → ∀ (v : Causalean.Estimation.ATE.NuisanceVec γ), (∃ Cμ, ∀ (b : Bool) (x : γ), |v.μ_fn b x| ≤ Cμ) → (∃ Ce, ∀ (x : γ), |v.e_fn x| ≤ Ce) → P.μ[fun ω => Causalean.Estimation.CATE.phi_eta_dir_deriv S.η₀ v (S.factualZ ω) | S.sigmaX] =ᵐ[P.μ] fun x => 0","l":758,"f":"Causalean/Estimation/CATE/Core/ConditionalBias.lean"},{"n":"Causalean.Estimation.CATE.drLearnerEstimator","k":"def","m":"Causalean.Estimation.CATE.Kennedy.DRLearner","a":"Estimation/CATE/Kennedy","t":1,"d":"DR-Learner CATE estimator at x (Def def:est-cate-dr-learner, τ̂^{DR}_n(x)).","s":"{P : Causalean.PO.POSystem} → {γ : Type u_1} → [inst : MeasurableSpace γ] → [inst_1 : StandardBorelSpace P.Ω] → [inst_2 : MeasureTheory.IsFiniteMeasure P.μ] → Causalean.Estimation.CATE.CATEEstimationSystem P γ → Causalean.Estimation.OrthogonalMoments.SecondStageOperator P.Ω P.μ γ → (ℕ → P.Ω → Causalean.Estimation.ATE.NuisanceVec γ) → ℕ → P.Ω → γ → ℝ","l":47,"f":"Causalean/Estimation/CATE/Kennedy/DRLearner.lean"},{"n":"Causalean.Estimation.CATE.drOracleEstimator","k":"def","m":"Causalean.Estimation.CATE.Kennedy.DRLearner","a":"Estimation/CATE/Kennedy","t":1,"d":"Oracle DR-Learner at x (Def def:est-cate-dr-learner, τ̃_n(x)).","s":"{P : Causalean.PO.POSystem} → {γ : Type u_1} → [inst : MeasurableSpace γ] → [inst_1 : StandardBorelSpace P.Ω] → [inst_2 : MeasureTheory.IsFiniteMeasure P.μ] → Causalean.Estimation.CATE.CATEEstimationSystem P γ → Causalean.Estimation.OrthogonalMoments.SecondStageOperator P.Ω P.μ γ → ℕ → P.Ω → γ → ℝ","l":67,"f":"Causalean/Estimation/CATE/Kennedy/DRLearner.lean"},{"n":"Causalean.Estimation.CATE.drOracleEstimator_eq_oracleEstimator","k":"theorem","m":"Causalean.Estimation.CATE.Kennedy.DRLearner","a":"Estimation/CATE/Kennedy","t":1,"d":"For a CATE estimation system and a second-stage operator, the oracle DR-learner estimator built from S and op equals the abstract oracle estimator of op applied to the true pseudo-outcome φ_0.","s":"∀ {P : Causalean.PO.POSystem} {γ : Type u_1} [inst : MeasurableSpace γ] [inst_1 : StandardBorelSpace P.Ω] [inst_2 : MeasureTheory.IsFiniteMeasure P.μ] (S : Causalean.Estimation.CATE.CATEEstimationSystem P γ) (op : Causalean.Estimation.OrthogonalMoments.SecondStageOperator P.Ω P.μ γ), Causalean.Estimation.CATE.drOracleEstimator S op = op.oracleEstimator fun z => Causalean.Estimation.CATE.phi₀ S z","l":93,"f":"Causalean/Estimation/CATE/Kennedy/DRLearner.lean"},{"n":"Causalean.Estimation.CATE.drOracleRiskScale","k":"def","m":"Causalean.Estimation.CATE.Kennedy.DRLearner","a":"Estimation/CATE/Kennedy","t":1,"d":"Oracle pointwise risk scale R^*_n(x) (Def def:est-cate-dr-learner).","s":"{P : Causalean.PO.POSystem} → {γ : Type u_1} → [inst : MeasurableSpace γ] → [inst_1 : StandardBorelSpace P.Ω] → [inst_2 : MeasureTheory.IsFiniteMeasure P.μ] → Causalean.Estimation.CATE.CATEEstimationSystem P γ → Causalean.Estimation.OrthogonalMoments.SecondStageOperator P.Ω P.μ γ → γ → ℕ → ℝ","l":78,"f":"Causalean/Estimation/CATE/Kennedy/DRLearner.lean"},{"n":"Causalean.Estimation.CATE.dr_oracle_efficient","k":"theorem","m":"Causalean.Estimation.CATE.Kennedy.OracleExpansion","a":"Estimation/CATE/Kennedy","t":1,"d":"Oracle efficiency for the DR-Learner CATE estimator, at a query point x — corollary of dr_oracle_expansion. Under the back-door causal assumptions and the same operator-stability, consistency, and bias-identification hypotheses as dr_oracle_expansion — op is stable at x w.r.t. a distance d_n, via the bias-identification predicate BiasIdent, d_n converges to zero in probability, and the AIPW pseudo-outcome contrast is identified with the closed-form conditional bias condBias(η_hat, η₀) — if in addition that smoothed conditional-bias term, the operator applied to condBias(η_hat, η₀) at x, is itself o_p(R*_n(x)), then the DR-Learner CATE estimator is oracle-efficient at x: it differs from its oracle counterpart by o_p(R*_n(x)).","s":"∀ {P : Causalean.PO.POSystem} {γ : Type u_1} [inst : MeasurableSpace γ] [inst_1 : StandardBorelSpace P.Ω] [inst_2 : MeasureTheory.IsFiniteMeasure P.μ] (S : Causalean.Estimation.CATE.CATEEstimationSystem P γ), S.Assumptions → ∀ (op : Causalean.Estimation.OrthogonalMoments.SecondStageOperator P.Ω P.μ γ) (η_hat : ℕ → P.Ω → Causalean.Estimation.ATE.NuisanceVec γ) (x : γ) (d_n : ℕ → P.Ω → ℝ) (BiasIdent : (ℕ → P.Ω → γ × Bool × ℝ → ℝ) → (γ × Bool × ℝ → ℝ) → (ℕ → P.Ω → γ → ℝ) → Prop), Causalean.Estimation.OrthogonalMoments.Stable op S.τ_val d_n x BiasIdent → Causalean.Stat.Tendsto_inProb d_n (fun x => 0) P.μ → (BiasIdent (fun n ω z => Causalean.Estimation.CATE.phi_eta z (η_hat n ω)) (fun z => Causalean.Estimation.CATE.phi₀ S z) fun n ω u => Causalean.Estimation.CATE.condBias (η_hat n ω) S.η₀ u) → Causalean.Stat.IsLittleOp (fun n ω => op.evalAt n ω (fun z => Causalean.Estimation.CATE.condBias (η_hat n ω) S.η₀ z.1) x) (fun n => Causalean.Estimation.CATE.drOracleRiskScale S op x n) P.μ → Causalean.Stat.IsLittleOp (fun n ω => Causalean.Estimation.CATE.drLearnerEstimator S op η_hat n ω x - Causalean.Estimation.CATE.drOracleEstimator S op n ω x) (fun n => Causalean.Estimation.CATE.drOracleRiskScale S op x n) P.μ","l":98,"f":"Causalean/Estimation/CATE/Kennedy/OracleExpansion.lean"},{"n":"Causalean.Estimation.CATE.dr_oracle_expansion","k":"theorem","m":"Causalean.Estimation.CATE.Kennedy.OracleExpansion","a":"Estimation/CATE/Kennedy","t":1,"d":"Oracle expansion for the DR-Learner CATE estimator. Fix a CATE estimation system under the back-door causal assumptions, a query point x, and a sequence of estimated nuisance vectors η_hat. Suppose the abstract second-stage regression operator op is stable at x with respect to a distance d_n, meaning the caller-supplied bias-identification predicate BiasIdent correctly separates the operator-level discrepancy between the estimated-nuisance and true pseudo-outcomes into a bias term plus a negligible remainder; d_n converges to zero in probability; and the contrast between the AIPW pseudo-outcome built from η_hat and the true pseudo-outcome is identified, through BiasIdent, with the closed-form conditional bias condBias(η_hat, η₀). Then the DR-Learner CATE estimator, minus its oracle counterpart, minus the operator applied to that closed-form bias, equals o_p(R*_n(x)), where R*_n(x) is the oracle risk scale at x.","s":"∀ {P : Causalean.PO.POSystem} {γ : Type u_1} [inst : MeasurableSpace γ] [inst_1 : StandardBorelSpace P.Ω] [inst_2 : MeasureTheory.IsFiniteMeasure P.μ] (S : Causalean.Estimation.CATE.CATEEstimationSystem P γ), S.Assumptions → ∀ (op : Causalean.Estimation.OrthogonalMoments.SecondStageOperator P.Ω P.μ γ) (η_hat : ℕ → P.Ω → Causalean.Estimation.ATE.NuisanceVec γ) (x : γ) (d_n : ℕ → P.Ω → ℝ) (BiasIdent : (ℕ → P.Ω → γ × Bool × ℝ → ℝ) → (γ × Bool × ℝ → ℝ) → (ℕ → P.Ω → γ → ℝ) → Prop), Causalean.Estimation.OrthogonalMoments.Stable op S.τ_val d_n x BiasIdent → Causalean.Stat.Tendsto_inProb d_n (fun x => 0) P.μ → (BiasIdent (fun n ω z => Causalean.Estimation.CATE.phi_eta z (η_hat n ω)) (fun z => Causalean.Estimation.CATE.phi₀ S z) fun n ω u => Causalean.Estimation.CATE.condBias (η_hat n ω) S.η₀ u) → Causalean.Stat.IsLittleOp (fun n ω => Causalean.Estimation.CATE.drLearnerEstimator S op η_hat n ω x - Causalean.Estimation.CATE.drOracleEstimator S op n ω x - op.evalAt n ω (fun z => Causalean.Estimation.CATE.condBias (η_hat n ω) S.η₀ z.1) x) (fun n => Causalean.Estimation.CATE.drOracleRiskScale S op x n) P.μ","l":49,"f":"Causalean/Estimation/CATE/Kennedy/OracleExpansion.lean"},{"n":"Causalean.Estimation.CATE.integral_phiDiff_mul_eq_condBias","k":"theorem","m":"Causalean.Estimation.CATE.Core.SecondOrderBias","a":"Estimation/CATE/Core","t":2,"d":"Conditioning identity for the weighted pseudo-outcome bias.","s":"∀ {P : Causalean.PO.POSystem} {γ : Type u_1} [inst : MeasurableSpace γ] [inst_1 : StandardBorelSpace P.Ω] [inst_2 : MeasureTheory.IsFiniteMeasure P.μ] (S : Causalean.Estimation.CATE.CATEEstimationSystem P γ), S.Assumptions → ∀ (η : Causalean.Estimation.ATE.NuisanceVec γ) {ε : ℝ}, 0 < ε → η ∈ Causalean.Estimation.ATE.BackdoorEstimationSystem.H_ε ε → S.η₀ ∈ Causalean.Estimation.ATE.BackdoorEstimationSystem.H_ε ε → (∀ (a : Bool), MeasureTheory.Integrable (fun ω => η.μ_fn a (S.factualX ω)) P.μ) → ∀ (w : γ → ℝ), Measurable w → MeasureTheory.Integrable (fun ω => Causalean.Estimation.CATE.phi_eta (S.factualZ ω) η - Causalean.Estimation.CATE.phi₀ S (S.factualZ ω)) P.μ → MeasureTheory.Integrable (fun ω => (Causalean.Estimation.CATE.phi_eta (S.factualZ ω) η - Causalean.Estimation.CATE.phi₀ S (S.factualZ ω)) * w (S.factualX ω)) P.μ → ∫ (z : γ × Bool × ℝ), (Causalean.Estimation.CATE.phi_eta z η - Causalean.Estimation.CATE.phi₀ S z) * w z.1 ∂S.P_Z = ∫ (x : γ), Causalean.Estimation.CATE.condBias η S.η₀ x * w x ∂S.P_X","l":66,"f":"Causalean/Estimation/CATE/Core/SecondOrderBias.lean"},{"n":"Causalean.Estimation.CATE.measurable_condBias","k":"theorem","m":"Causalean.Estimation.CATE.Core.ConditionalBias","a":"Estimation/CATE/Core","t":2,"d":"condBias η η₀ is measurable in x.","s":"∀ {γ : Type u_1} [inst : MeasurableSpace γ] (η η₀ : Causalean.Estimation.ATE.NuisanceVec γ), Measurable fun x => Causalean.Estimation.CATE.condBias η η₀ x","l":82,"f":"Causalean/Estimation/CATE/Core/ConditionalBias.lean"},{"n":"Causalean.Estimation.CATE.measurable_phi_eta","k":"theorem","m":"Causalean.Estimation.CATE.Core.PseudoOutcome","a":"Estimation/CATE/Core","t":2,"d":"The uncentered AIPW pseudo-outcome is measurable in the data argument.","s":"∀ {γ : Type u_1} [inst : MeasurableSpace γ] (η : Causalean.Estimation.ATE.NuisanceVec γ), Measurable fun z => Causalean.Estimation.CATE.phi_eta z η","l":63,"f":"Causalean/Estimation/CATE/Core/PseudoOutcome.lean"},{"n":"Causalean.Estimation.CATE.measurable_phi_eta_dir_deriv","k":"theorem","m":"Causalean.Estimation.CATE.Core.PhiEtaDeriv","a":"Estimation/CATE/Core","t":2,"d":"phi_eta_dir_deriv g₀ v is measurable in z.","s":"∀ {γ : Type u_1} [inst : MeasurableSpace γ] (g₀ v : Causalean.Estimation.ATE.NuisanceVec γ), Measurable fun z => Causalean.Estimation.CATE.phi_eta_dir_deriv g₀ v z","l":92,"f":"Causalean/Estimation/CATE/Core/PhiEtaDeriv.lean"},{"n":"Causalean.Estimation.CATE.measurable_phi₀","k":"theorem","m":"Causalean.Estimation.CATE.Core.PseudoOutcome","a":"Estimation/CATE/Core","t":1,"d":"For a CATE estimation system, the true doubly-robust pseudo-outcome φ_0 is measurable as a function of the observed data.","s":"∀ {γ : Type u_1} [inst : MeasurableSpace γ] {P : Causalean.PO.POSystem} [inst_1 : StandardBorelSpace P.Ω] [inst_2 : MeasureTheory.IsFiniteMeasure P.μ] (S : Causalean.Estimation.CATE.CATEEstimationSystem P γ), Measurable fun z => Causalean.Estimation.CATE.phi₀ S z","l":92,"f":"Causalean/Estimation/CATE/Core/PseudoOutcome.lean"},{"n":"Causalean.Estimation.CATE.phi_eta","k":"def","m":"Causalean.Estimation.CATE.Core.PseudoOutcome","a":"Estimation/CATE/Core","t":1,"d":"Uncentered AIPW pseudo-outcome (Kennedy's φ_η):","s":"{γ : Type u_1} → [inst : MeasurableSpace γ] → γ × Bool × ℝ → Causalean.Estimation.ATE.NuisanceVec γ → ℝ","l":44,"f":"Causalean/Estimation/CATE/Core/PseudoOutcome.lean"},{"n":"Causalean.Estimation.CATE.phi_eta_dir_deriv","k":"def","m":"Causalean.Estimation.CATE.Core.PhiEtaDeriv","a":"Estimation/CATE/Core","t":1,"d":"Closed-form directional derivative of phi_eta z η in η at g₀ along the direction v : NuisanceVec γ.","s":"{γ : Type u_1} → [inst : MeasurableSpace γ] → Causalean.Estimation.ATE.NuisanceVec γ → Causalean.Estimation.ATE.NuisanceVec γ → γ × Bool × ℝ → ℝ","l":76,"f":"Causalean/Estimation/CATE/Core/PhiEtaDeriv.lean"},{"n":"Causalean.Estimation.CATE.phi_eta_dir_deriv_tendsto","k":"theorem","m":"Causalean.Estimation.CATE.Core.PhiEtaDeriv","a":"Estimation/CATE/Core","t":1,"d":"Pointwise convergence of the DR pseudo-outcome difference quotient. Fix an anchor nuisance pair g₀, a perturbation direction v, and a data point z. If ε is strictly positive and the anchor g₀ has propensity uniformly bounded in [ε, 1 − ε] for every covariate value, then as the step size t tends to 0 along nonzero values, the difference quotient (phi_eta z (g₀ + t•v) − phi_eta z g₀) / t converges to the closed-form directional derivative phi_eta_dir_deriv g₀ v z.","s":"∀ {γ : Type u_1} [inst : MeasurableSpace γ] (g₀ v : Causalean.Estimation.ATE.NuisanceVec γ) {ε : ℝ}, 0 < ε → g₀ ∈ Causalean.Estimation.ATE.BackdoorEstimationSystem.H_ε ε → ∀ (z : γ × Bool × ℝ), Filter.Tendsto (fun t => (Causalean.Estimation.CATE.phi_eta z (g₀ + t • v) - Causalean.Estimation.CATE.phi_eta z g₀) / t) (nhdsWithin 0 {0}ᶜ) (nhds (Causalean.Estimation.CATE.phi_eta_dir_deriv g₀ v z))","l":213,"f":"Causalean/Estimation/CATE/Core/PhiEtaDeriv.lean"},{"n":"Causalean.Estimation.CATE.phi_eta_minus_phi₀_at_x","k":"theorem","m":"Causalean.Estimation.CATE.Core.ConditionalBias","a":"Estimation/CATE/Core","t":1,"d":"Value-space form of the DR pseudo-outcome bias identity. Fix a candidate nuisance pair η. Under the back-door identification assumptions, if η's propensity is uniformly bounded in [ε, 1 − ε] for every covariate value, the truth nuisance likewise has propensity uniformly bounded in [ε, 1 − ε], ε is strictly positive, each candidate outcome-regression arm, composed with the covariate, is integrable, and the DR pseudo-outcome contrast φ_η − φ_0 is integrable, then for covariate-law-almost-every x, the mean of φ_η − φ_0 under the regular conditional distribution of the data triple given the covariate value x equals the closed-form cross-product remainder condBias η η₀ x.","s":"∀ {γ : Type u_1} [inst : MeasurableSpace γ] {P : Causalean.PO.POSystem} [inst_1 : StandardBorelSpace P.Ω] [inst_2 : MeasureTheory.IsFiniteMeasure P.μ] [inst_3 : StandardBorelSpace γ] [inst_4 : Nonempty γ] (S : Causalean.Estimation.CATE.CATEEstimationSystem P γ), S.Assumptions → ∀ (η : Causalean.Estimation.ATE.NuisanceVec γ) {ε : ℝ}, η ∈ Causalean.Estimation.ATE.BackdoorEstimationSystem.H_ε ε → S.η₀ ∈ Causalean.Estimation.ATE.BackdoorEstimationSystem.H_ε ε → 0 < ε → (∀ (a : Bool), MeasureTheory.Integrable (fun ω => η.μ_fn a (S.factualX ω)) P.μ) → MeasureTheory.Integrable (fun ω => Causalean.Estimation.CATE.phi_eta (S.factualZ ω) η - Causalean.Estimation.CATE.phi₀ S (S.factualZ ω)) P.μ → ∀ᵐ (x : γ) ∂S.P_X, ∫ (z : γ × Bool × ℝ), Causalean.Estimation.CATE.phi_eta z η - Causalean.Estimation.CATE.phi₀ S z ∂(ProbabilityTheory.condDistrib S.factualZ S.factualX P.μ) x = Causalean.Estimation.CATE.condBias η S.η₀ x","l":616,"f":"Causalean/Estimation/CATE/Core/ConditionalBias.lean"},{"n":"Causalean.Estimation.CATE.phi_eta_minus_phi₀_cond_exp","k":"theorem","m":"Causalean.Estimation.CATE.Core.ConditionalBias","a":"Estimation/CATE/Core","t":1,"d":"σ(X)-conditional bias identity for the DR pseudo-outcome (Kennedy / DR-Learner Proposition 2). Fix a candidate nuisance pair η. Under the back-door identification assumptions, if η's propensity is uniformly bounded in [ε, 1 − ε] for every covariate value, the truth nuisance likewise has propensity uniformly bounded in [ε, 1 − ε], ε is strictly positive, and each candidate outcome-regression arm, composed with the covariate, is integrable, then the σ(X)-conditional expectation of the DR pseudo-outcome contrast φ_η − φ_0 equals the closed-form cross-product remainder condBias η η₀, evaluated at the covariate, almost surely.","s":"∀ {γ : Type u_1} [inst : MeasurableSpace γ] {P : Causalean.PO.POSystem} [inst_1 : StandardBorelSpace P.Ω] [inst_2 : MeasureTheory.IsFiniteMeasure P.μ] (S : Causalean.Estimation.CATE.CATEEstimationSystem P γ), S.Assumptions → ∀ (η : Causalean.Estimation.ATE.NuisanceVec γ) {ε : ℝ}, η ∈ Causalean.Estimation.ATE.BackdoorEstimationSystem.H_ε ε → S.η₀ ∈ Causalean.Estimation.ATE.BackdoorEstimationSystem.H_ε ε → 0 < ε → (∀ (a : Bool), MeasureTheory.Integrable (fun ω => η.μ_fn a (S.factualX ω)) P.μ) → P.μ[fun ω => Causalean.Estimation.CATE.phi_eta (S.factualZ ω) η - Causalean.Estimation.CATE.phi₀ S (S.factualZ ω) | S.sigmaX] =ᵐ[P.μ] fun ω => Causalean.Estimation.CATE.condBias η S.η₀ (S.factualX ω)","l":217,"f":"Causalean/Estimation/CATE/Core/ConditionalBias.lean"},{"n":"Causalean.Estimation.CATE.phi₀","k":"def","m":"Causalean.Estimation.CATE.Core.PseudoOutcome","a":"Estimation/CATE/Core","t":1,"d":"True DR pseudo-outcome: φ_0(z) := φ_{η_0}(z) where η_0 is the truth nuisance vector carried by the back-door substrate of S.","s":"{γ : Type u_1} → [inst : MeasurableSpace γ] → {P : Causalean.PO.POSystem} → [inst_1 : StandardBorelSpace P.Ω] → [inst_2 : MeasureTheory.IsFiniteMeasure P.μ] → Causalean.Estimation.CATE.CATEEstimationSystem P γ → γ × Bool × ℝ → ℝ","l":56,"f":"Causalean/Estimation/CATE/Core/PseudoOutcome.lean"},{"n":"Causalean.Estimation.DTR.DTREstimationSystem","k":"structure","m":"Causalean.Estimation.DTR.Setup","a":"Estimation/DTR","t":1,"d":"This structure extends a two-stage potential-outcome dynamic-treatment-regime system with, at a fixed target regime, measurable value-space representatives of the stage-0 and stage-1 outcome regressions and of the propensities at both stages, bounded away from zero and one, each required to agree almost surely with the corresponding observable conditional regression or propensity built from the factual treatment and covariate history.","s":"(P : Causalean.PO.POSystem) → (δ : Type) → (γ : Fin 2 → Type) → [inst : MeasurableSpace δ] → [MeasurableSingletonClass δ] → [(k : Fin 2) → MeasurableSpace (γ k)] → [StandardBorelSpace P.Ω] → [MeasureTheory.IsFiniteMeasure P.μ] → Type (max u_1 u_2)","l":61,"f":"Causalean/Estimation/DTR/Setup.lean"},{"n":"Causalean.Estimation.DTR.DTREstimationSystem.H_ε","k":"def","m":"Causalean.Estimation.DTR.SeqDRMoment","a":"Estimation/DTR","t":1,"d":"This set contains nuisance candidates whose two propensity components are uniformly overlap-bounded.","s":"{δ : Type} → [inst : MeasurableSpace δ] → [inst_1 : MeasurableSingletonClass δ] → {γ : Fin 2 → Type} → [inst_2 : (k : Fin 2) → MeasurableSpace (γ k)] → ℝ → Set (Causalean.Estimation.DTR.DTRNuisanceVec₂ δ γ)","l":357,"f":"Causalean/Estimation/DTR/SeqDRMoment.lean"},{"n":"Causalean.Estimation.DTR.DTREstimationSystem.K_seqDR","k":"def","m":"Causalean.Estimation.DTR.ScoreL2.Helpers","a":"Estimation/DTR/ScoreL2","t":1,"d":"The sequential doubly robust Lipschitz constant combines the stagewise inverse-overlap bounds used in score continuity.","s":"ℝ → ℝ","l":51,"f":"Causalean/Estimation/DTR/ScoreL2/Helpers.lean"},{"n":"Causalean.Estimation.DTR.DTREstimationSystem.K_seqDR_inv_cubed_le","k":"theorem","m":"Causalean.Estimation.DTR.ScoreL2.Helpers","a":"Estimation/DTR/ScoreL2","t":2,"d":"The sequential doubly robust Lipschitz constant dominates the cubed inverse-overlap bound.","s":"∀ {ε : ℝ}, 0 < ε → 1 / ε ^ 3 ≤ Causalean.Estimation.DTR.DTREstimationSystem.K_seqDR ε","l":100,"f":"Causalean/Estimation/DTR/ScoreL2/Helpers.lean"},{"n":"Causalean.Estimation.DTR.DTREstimationSystem.K_seqDR_inv_sq_le","k":"theorem","m":"Causalean.Estimation.DTR.ScoreL2.Helpers","a":"Estimation/DTR/ScoreL2","t":2,"d":"The sequential doubly robust Lipschitz constant dominates the squared inverse-overlap bound.","s":"∀ {ε : ℝ}, 0 < ε → 1 / ε ^ 2 ≤ Causalean.Estimation.DTR.DTREstimationSystem.K_seqDR ε","l":93,"f":"Causalean/Estimation/DTR/ScoreL2/Helpers.lean"},{"n":"Causalean.Estimation.DTR.DTREstimationSystem.K_seqDR_mu0_le","k":"theorem","m":"Causalean.Estimation.DTR.ScoreL2.Helpers","a":"Estimation/DTR/ScoreL2","t":2,"d":"The sequential doubly robust Lipschitz constant dominates the stage-zero regression coefficient bound.","s":"∀ {ε : ℝ}, 0 < ε → 1 + 1 / ε ≤ Causalean.Estimation.DTR.DTREstimationSystem.K_seqDR ε","l":79,"f":"Causalean/Estimation/DTR/ScoreL2/Helpers.lean"},{"n":"Causalean.Estimation.DTR.DTREstimationSystem.K_seqDR_mu1_le","k":"theorem","m":"Causalean.Estimation.DTR.ScoreL2.Helpers","a":"Estimation/DTR/ScoreL2","t":2,"d":"The sequential doubly robust Lipschitz constant dominates the stage-one regression coefficient bound.","s":"∀ {ε : ℝ}, 0 < ε → 1 / ε + 1 / ε ^ 2 ≤ Causalean.Estimation.DTR.DTREstimationSystem.K_seqDR ε","l":86,"f":"Causalean/Estimation/DTR/ScoreL2/Helpers.lean"},{"n":"Causalean.Estimation.DTR.DTREstimationSystem.K_seqDR_one_le","k":"theorem","m":"Causalean.Estimation.DTR.ScoreL2.Helpers","a":"Estimation/DTR/ScoreL2","t":2,"d":"The sequential doubly robust Lipschitz constant dominates one under positive overlap.","s":"∀ {ε : ℝ}, 0 < ε → 1 ≤ Causalean.Estimation.DTR.DTREstimationSystem.K_seqDR ε","l":72,"f":"Causalean/Estimation/DTR/ScoreL2/Helpers.lean"},{"n":"Causalean.Estimation.DTR.DTREstimationSystem.P_H₀","k":"def","m":"Causalean.Estimation.DTR.Setup","a":"Estimation/DTR","t":1,"d":"This is the marginal law of the stage-0 history.","s":"{P : Causalean.PO.POSystem} → {δ : Type} → {γ : Fin 2 → Type} → [inst : MeasurableSpace δ] → [inst_1 : MeasurableSingletonClass δ] → [inst_2 : (k : Fin 2) → MeasurableSpace (γ k)] → [inst_3 : StandardBorelSpace P.Ω] → [inst_4 : MeasureTheory.IsFiniteMeasure P.μ] → Causalean.Estimation.DTR.DTREstimationSystem P δ γ → MeasureTheory.Measure (γ 0)","l":422,"f":"Causalean/Estimation/DTR/Setup.lean"},{"n":"Causalean.Estimation.DTR.DTREstimationSystem.P_H₁","k":"def","m":"Causalean.Estimation.DTR.Setup","a":"Estimation/DTR","t":1,"d":"This is the marginal law of the stage-1 history in the DTR history ordering.","s":"{P : Causalean.PO.POSystem} → {δ : Type} → {γ : Fin 2 → Type} → [inst : MeasurableSpace δ] → [inst_1 : MeasurableSingletonClass δ] → [inst_2 : (k : Fin 2) → MeasurableSpace (γ k)] → [inst_3 : StandardBorelSpace P.Ω] → [inst_4 : MeasureTheory.IsFiniteMeasure P.μ] → Causalean.Estimation.DTR.DTREstimationSystem P δ γ → MeasureTheory.Measure (γ 1 × δ × γ 0)","l":426,"f":"Causalean/Estimation/DTR/Setup.lean"},{"n":"Causalean.Estimation.DTR.DTREstimationSystem.P_Z","k":"def","m":"Causalean.Estimation.DTR.Setup","a":"Estimation/DTR","t":1,"d":"This is the joint law of the full observed two-stage data tuple.","s":"{P : Causalean.PO.POSystem} → {δ : Type} → {γ : Fin 2 → Type} → [inst : MeasurableSpace δ] → [inst_1 : MeasurableSingletonClass δ] → [inst_2 : (k : Fin 2) → MeasurableSpace (γ k)] → [inst_3 : StandardBorelSpace P.Ω] → [inst_4 : MeasureTheory.IsFiniteMeasure P.μ] → Causalean.Estimation.DTR.DTREstimationSystem P δ γ → MeasureTheory.Measure (γ 0 × δ × γ 1 × δ × ℝ)","l":453,"f":"Causalean/Estimation/DTR/Setup.lean"},{"n":"Causalean.Estimation.DTR.DTREstimationSystem.P_Z_map_histH₁_eq_P_H₁","k":"theorem","m":"Causalean.Estimation.DTR.SeqDRMoment","a":"Estimation/DTR","t":1,"d":"For a dynamic-treatment-regime estimation system, pushing the full observed-data law forward through the stage-1 history projection yields exactly the stage-1 history marginal law.","s":"∀ {δ : Type} [inst : MeasurableSpace δ] [inst_1 : MeasurableSingletonClass δ] {γ : Fin 2 → Type} [inst_2 : (k : Fin 2) → MeasurableSpace (γ k)] {P : Causalean.PO.POSystem} [inst_3 : StandardBorelSpace P.Ω] [inst_4 : MeasureTheory.IsFiniteMeasure P.μ] (S : Causalean.Estimation.DTR.DTREstimationSystem P δ γ), MeasureTheory.Measure.map (fun z => Causalean.Estimation.DTR.histH₁ z) S.P_Z = S.P_H₁","l":450,"f":"Causalean/Estimation/DTR/SeqDRMoment.lean"},{"n":"Causalean.Estimation.DTR.DTREstimationSystem.P_Z_map_projS₀_eq_P_H₀","k":"theorem","m":"Causalean.Estimation.DTR.SeqDRMoment","a":"Estimation/DTR","t":1,"d":"For a dynamic-treatment-regime estimation system, pushing the full observed-data law forward through the initial-state projection yields exactly the stage-0 history marginal law.","s":"∀ {δ : Type} [inst : MeasurableSpace δ] [inst_1 : MeasurableSingletonClass δ] {γ : Fin 2 → Type} [inst_2 : (k : Fin 2) → MeasurableSpace (γ k)] {P : Causalean.PO.POSystem} [inst_3 : StandardBorelSpace P.Ω] [inst_4 : MeasureTheory.IsFiniteMeasure P.μ] (S : Causalean.Estimation.DTR.DTREstimationSystem P δ γ), MeasureTheory.Measure.map (fun z => Causalean.Estimation.DTR.projS₀ z) S.P_Z = S.P_H₀","l":442,"f":"Causalean/Estimation/DTR/SeqDRMoment.lean"},{"n":"Causalean.Estimation.DTR.DTREstimationSystem.StrictOverlap","k":"def","m":"Causalean.Estimation.DTR.Setup","a":"Estimation/DTR","t":1,"d":"This predicate requires both stagewise target-regime propensities to stay uniformly away from zero and one.","s":"{P : Causalean.PO.POSystem} → {δ : Type} → {γ : Fin 2 → Type} → [inst : MeasurableSpace δ] → [inst_1 : MeasurableSingletonClass δ] → [inst_2 : (k : Fin 2) → MeasurableSpace (γ k)] → [inst_3 : StandardBorelSpace P.Ω] → [inst_4 : MeasureTheory.IsFiniteMeasure P.μ] → Causalean.Estimation.DTR.DTREstimationSystem P δ γ → ℝ → Prop","l":350,"f":"Causalean/Estimation/DTR/Setup.lean"},{"n":"Causalean.Estimation.DTR.DTREstimationSystem.cond_exp_residual_zero_stage0","k":"theorem","m":"Causalean.Estimation.DTR.MeanZero","a":"Estimation/DTR","t":2,"d":"Stage-0 residual conditional expectation is zero a.s.: under DTR assumptions, μ[1{D₀=dbar 0}·(μ₁_val(history₁) − μ₀_val(S₀)) | σ(historyBundle 0)] =ᵐ 0.","s":"∀ {P : Causalean.PO.POSystem} {δ : Type} {γ : Fin 2 → Type} [inst : MeasurableSpace δ] [inst_1 : MeasurableSingletonClass δ] [inst_2 : (k : Fin 2) → MeasurableSpace (γ k)] [inst_3 : StandardBorelSpace P.Ω] [inst_4 : MeasureTheory.IsFiniteMeasure P.μ] (S : Causalean.Estimation.DTR.DTREstimationSystem P δ γ) {ε : ℝ}, S.StrictOverlap ε → S.Assumptions → MeasureTheory.Integrable (fun ω => S.factualY ω ^ 2) P.μ → ((S.historyBundle 0 (of_decide_eq_true (id (Eq.refl true)))).condExpGiven fun ω => (S.dVar ⟨0, of_decide_eq_true (id (Eq.refl true))⟩).indicator (S.dbar ⟨0, of_decide_eq_true (id (Eq.refl true))⟩) ω * (S.μ₁_val (S.factualS ⟨1, of_decide_eq_true (id (Eq.refl true))⟩ ω, S.factualD ⟨0, of_decide_eq_true (id (Eq.refl true))⟩ ω, S.factualS ⟨0, of_decide_eq_true (id (Eq.refl true))⟩ ω) - S.μ₀_val (S.factualS ⟨0, of_decide_eq_true (id (Eq.refl true))⟩ ω))) =ᵐ[P.μ] fun x => 0","l":90,"f":"Causalean/Estimation/DTR/MeanZero.lean"},{"n":"Causalean.Estimation.DTR.DTREstimationSystem.cond_exp_residual_zero_stage1","k":"theorem","m":"Causalean.Estimation.DTR.MeanZero","a":"Estimation/DTR","t":2,"d":"Stage-1 residual conditional expectation is zero a.s.: under DTR assumptions, the σ(historyBundle 1)-conditional expectation of 1{D₀ = dbar 0} · 1{D₁ = dbar 1} · (factualY − μ₁_val(S₁,D₀,S₀)) is zero a.s.","s":"∀ {P : Causalean.PO.POSystem} {δ : Type} {γ : Fin 2 → Type} [inst : MeasurableSpace δ] [inst_1 : MeasurableSingletonClass δ] [inst_2 : (k : Fin 2) → MeasurableSpace (γ k)] [inst_3 : StandardBorelSpace P.Ω] [inst_4 : MeasureTheory.IsFiniteMeasure P.μ] (S : Causalean.Estimation.DTR.DTREstimationSystem P δ γ) {ε : ℝ}, S.StrictOverlap ε → S.Assumptions → MeasureTheory.Integrable (fun ω => S.factualY ω ^ 2) P.μ → ((S.historyBundle 1 (of_decide_eq_true (id (Eq.refl true)))).condExpGiven fun ω => (S.dVar ⟨0, of_decide_eq_true (id (Eq.refl true))⟩).indicator (S.dbar ⟨0, of_decide_eq_true (id (Eq.refl true))⟩) ω * ((S.dVar ⟨1, of_decide_eq_true (id (Eq.refl true))⟩).indicator (S.dbar ⟨1, of_decide_eq_true (id (Eq.refl true))⟩) ω * (S.factualY ω - S.μ₁_val (S.factualS ⟨1, of_decide_eq_true (id (Eq.refl true))⟩ ω, S.factualD ⟨0, of_decide_eq_true (id (Eq.refl true))⟩ ω, S.factualS ⟨0, of_decide_eq_true (id (Eq.refl true))⟩ ω)))) =ᵐ[P.μ] fun x => 0","l":343,"f":"Causalean/Estimation/DTR/MeanZero.lean"},{"n":"Causalean.Estimation.DTR.DTREstimationSystem.eLpNorm_comp_histH₁_eq","k":"theorem","m":"Causalean.Estimation.DTR.ScoreL2.Helpers","a":"Estimation/DTR/ScoreL2","t":2,"d":"Pulling a stage-one history function back along the full DTR data law preserves its L² norm.","s":"∀ {P : Causalean.PO.POSystem} {δ : Type} {γ : Fin 2 → Type} [inst : MeasurableSpace δ] [inst_1 : MeasurableSingletonClass δ] [inst_2 : (k : Fin 2) → MeasurableSpace (γ k)] [inst_3 : StandardBorelSpace P.Ω] [inst_4 : MeasureTheory.IsFiniteMeasure P.μ] (S : Causalean.Estimation.DTR.DTREstimationSystem P δ γ) {f : γ 1 × δ × γ 0 → ℝ}, MeasureTheory.AEStronglyMeasurable f S.P_H₁ → MeasureTheory.eLpNorm (fun z => f (Causalean.Estimation.DTR.histH₁ z)) 2 S.P_Z = MeasureTheory.eLpNorm f 2 S.P_H₁","l":122,"f":"Causalean/Estimation/DTR/ScoreL2/Helpers.lean"},{"n":"Causalean.Estimation.DTR.DTREstimationSystem.eLpNorm_comp_projS₀_eq","k":"theorem","m":"Causalean.Estimation.DTR.ScoreL2.Helpers","a":"Estimation/DTR/ScoreL2","t":2,"d":"Pulling a stage-zero function back along the full DTR data law preserves its L² norm.","s":"∀ {P : Causalean.PO.POSystem} {δ : Type} {γ : Fin 2 → Type} [inst : MeasurableSpace δ] [inst_1 : MeasurableSingletonClass δ] [inst_2 : (k : Fin 2) → MeasurableSpace (γ k)] [inst_3 : StandardBorelSpace P.Ω] [inst_4 : MeasureTheory.IsFiniteMeasure P.μ] (S : Causalean.Estimation.DTR.DTREstimationSystem P δ γ) {f : γ 0 → ℝ}, MeasureTheory.AEStronglyMeasurable f S.P_H₀ → MeasureTheory.eLpNorm (fun z => f (Causalean.Estimation.DTR.projS₀ z)) 2 S.P_Z = MeasureTheory.eLpNorm f 2 S.P_H₀","l":107,"f":"Causalean/Estimation/DTR/ScoreL2/Helpers.lean"},{"n":"Causalean.Estimation.DTR.DTREstimationSystem.eta_e0_pos_of_mem_Hε","k":"theorem","m":"Causalean.Estimation.DTR.RemainderIdentity.Helpers","a":"Estimation/DTR/RemainderIdentity","t":2,"d":"Any nuisance vector in the overlap-bounded set has a positive stage-zero propensity.","s":"∀ {δ : Type} {γ : Fin 2 → Type} [inst : MeasurableSpace δ] [inst_1 : MeasurableSingletonClass δ] [inst_2 : (k : Fin 2) → MeasurableSpace (γ k)] {η : Causalean.Estimation.DTR.DTRNuisanceVec₂ δ γ} {ε : ℝ}, 0 < ε → η ∈ Causalean.Estimation.DTR.DTREstimationSystem.H_ε ε → ∀ (s₀ : γ 0), 0 < η.e₀_fn s₀","l":97,"f":"Causalean/Estimation/DTR/RemainderIdentity/Helpers.lean"},{"n":"Causalean.Estimation.DTR.DTREstimationSystem.eta_e1_pos_of_mem_Hε","k":"theorem","m":"Causalean.Estimation.DTR.RemainderIdentity.Helpers","a":"Estimation/DTR/RemainderIdentity","t":2,"d":"Any nuisance vector in the overlap-bounded set has a positive stage-one propensity.","s":"∀ {δ : Type} {γ : Fin 2 → Type} [inst : MeasurableSpace δ] [inst_1 : MeasurableSingletonClass δ] [inst_2 : (k : Fin 2) → MeasurableSpace (γ k)] {η : Causalean.Estimation.DTR.DTRNuisanceVec₂ δ γ} {ε : ℝ}, 0 < ε → η ∈ Causalean.Estimation.DTR.DTREstimationSystem.H_ε ε → ∀ (h : γ 1 × δ × γ 0), 0 < η.e₁_fn h","l":105,"f":"Causalean/Estimation/DTR/RemainderIdentity/Helpers.lean"},{"n":"Causalean.Estimation.DTR.DTREstimationSystem.factualZ","k":"def","m":"Causalean.Estimation.DTR.Setup","a":"Estimation/DTR","t":1,"d":"This map collects the full observed two-stage data tuple from the underlying outcome system.","s":"{P : Causalean.PO.POSystem} → {δ : Type} → {γ : Fin 2 → Type} → [inst : MeasurableSpace δ] → [inst_1 : MeasurableSingletonClass δ] → [inst_2 : (k : Fin 2) → MeasurableSpace (γ k)] → [inst_3 : StandardBorelSpace P.Ω] → [inst_4 : MeasureTheory.IsFiniteMeasure P.μ] → Causalean.Estimation.DTR.DTREstimationSystem P δ γ → P.Ω → γ 0 × δ × γ 1 × δ × ℝ","l":434,"f":"Causalean/Estimation/DTR/Setup.lean"},{"n":"Causalean.Estimation.DTR.DTREstimationSystem.indD_mul_μ₁_val_comp_eq","k":"theorem","m":"Causalean.Estimation.DTR.Setup","a":"Estimation/DTR","t":2,"d":"Same as μ₁_val_comp_mul_indD_eq, with the partial regime indicator written on the left.","s":"∀ {P : Causalean.PO.POSystem} {δ : Type} {γ : Fin 2 → Type} [inst : MeasurableSpace δ] [inst_1 : MeasurableSingletonClass δ] [inst_2 : (k : Fin 2) → MeasurableSpace (γ k)] [inst_3 : StandardBorelSpace P.Ω] [inst_4 : MeasureTheory.IsFiniteMeasure P.μ] (S : Causalean.Estimation.DTR.DTREstimationSystem P δ γ), S.Assumptions → (fun ω => S.indD S.dbar 1 ω * S.μ₁_val (S.factualS ⟨1, of_decide_eq_true (id (Eq.refl true))⟩ ω, S.factualD ⟨0, of_decide_eq_true (id (Eq.refl true))⟩ ω, S.factualS ⟨0, of_decide_eq_true (id (Eq.refl true))⟩ ω)) =ᵐ[P.μ] fun ω => S.indD S.dbar 1 ω * (S.historyBundle 1 (of_decide_eq_true (id (Eq.refl true)))).condExpGiven (S.Y_of S.dbar) P.μ ω","l":332,"f":"Causalean/Estimation/DTR/Setup.lean"},{"n":"Causalean.Estimation.DTR.DTREstimationSystem.indEq_factualD0_eq_indicator","k":"theorem","m":"Causalean.Estimation.DTR.RemainderIdentity.Helpers","a":"Estimation/DTR/RemainderIdentity","t":2,"d":"The stage-zero equality indicator agrees with the stage-zero treatment indicator for the target regime.","s":"∀ {P : Causalean.PO.POSystem} {δ : Type} {γ : Fin 2 → Type} [inst : MeasurableSpace δ] [inst_1 : MeasurableSingletonClass δ] [inst_2 : (k : Fin 2) → MeasurableSpace (γ k)] [inst_3 : StandardBorelSpace P.Ω] [inst_4 : MeasureTheory.IsFiniteMeasure P.μ] (S : Causalean.Estimation.DTR.DTREstimationSystem P δ γ) (ω : P.Ω), Causalean.Estimation.DTR.indEq (S.factualD ⟨0, of_decide_eq_true (id (Eq.refl true))⟩ ω) (S.dbar ⟨0, of_decide_eq_true (id (Eq.refl true))⟩) = (S.dVar ⟨0, of_decide_eq_true (id (Eq.refl true))⟩).indicator (S.dbar ⟨0, of_decide_eq_true (id (Eq.refl true))⟩) ω","l":41,"f":"Causalean/Estimation/DTR/RemainderIdentity/Helpers.lean"},{"n":"Causalean.Estimation.DTR.DTREstimationSystem.indEq_factualD1_eq_indicator","k":"theorem","m":"Causalean.Estimation.DTR.RemainderIdentity.Helpers","a":"Estimation/DTR/RemainderIdentity","t":2,"d":"The stage-one equality indicator agrees with the stage-one treatment indicator for the target regime.","s":"∀ {P : Causalean.PO.POSystem} {δ : Type} {γ : Fin 2 → Type} [inst : MeasurableSpace δ] [inst_1 : MeasurableSingletonClass δ] [inst_2 : (k : Fin 2) → MeasurableSpace (γ k)] [inst_3 : StandardBorelSpace P.Ω] [inst_4 : MeasureTheory.IsFiniteMeasure P.μ] (S : Causalean.Estimation.DTR.DTREstimationSystem P δ γ) (ω : P.Ω), Causalean.Estimation.DTR.indEq (S.factualD ⟨1, of_decide_eq_true (id (Eq.refl true))⟩ ω) (S.dbar ⟨1, of_decide_eq_true (id (Eq.refl true))⟩) = (S.dVar ⟨1, of_decide_eq_true (id (Eq.refl true))⟩).indicator (S.dbar ⟨1, of_decide_eq_true (id (Eq.refl true))⟩) ω","l":69,"f":"Causalean/Estimation/DTR/RemainderIdentity/Helpers.lean"},{"n":"Causalean.Estimation.DTR.DTREstimationSystem.indEq_le_one","k":"theorem","m":"Causalean.Estimation.DTR.ScoreL2.Helpers","a":"Estimation/DTR/ScoreL2","t":2,"d":"The real-valued equality indicator is always bounded above by one.","s":"∀ {δ : Type} (d d' : δ), Causalean.Estimation.DTR.indEq d d' ≤ 1","l":67,"f":"Causalean/Estimation/DTR/ScoreL2/Helpers.lean"},{"n":"Causalean.Estimation.DTR.DTREstimationSystem.indEq_nonneg","k":"theorem","m":"Causalean.Estimation.DTR.ScoreL2.Helpers","a":"Estimation/DTR/ScoreL2","t":2,"d":"The real-valued equality indicator is always nonnegative.","s":"∀ {δ : Type} (d d' : δ), 0 ≤ Causalean.Estimation.DTR.indEq d d'","l":61,"f":"Causalean/Estimation/DTR/ScoreL2/Helpers.lean"},{"n":"Causalean.Estimation.DTR.DTREstimationSystem.indicator_to_propScore_integral_stage0","k":"theorem","m":"Causalean.Estimation.DTR.ScorePullout","a":"Estimation/DTR","t":1,"d":"Stage-0 indicator-to-propensity rewrite. Let S be a two-stage dynamic-treatment-regime estimation system and let f be a real-valued function of the stage-0 history that is measurable and for which the product f(S₀) · 1{D₀ = dbar 0} is integrable. Then the expectation of f(S₀) times the indicator of following the target stage-0 treatment dbar 0 equals the expectation of f(S₀) times the true stage-0 propensity score e₀_val(S₀).","s":"∀ {P : Causalean.PO.POSystem} {δ : Type} {γ : Fin 2 → Type} [inst : MeasurableSpace δ] [inst_1 : MeasurableSingletonClass δ] [inst_2 : (k : Fin 2) → MeasurableSpace (γ k)] [inst_3 : StandardBorelSpace P.Ω] [inst_4 : MeasureTheory.IsFiniteMeasure P.μ] (S : Causalean.Estimation.DTR.DTREstimationSystem P δ γ) (f : γ 0 → ℝ), Measurable f → MeasureTheory.Integrable (fun ω => f (S.factualS ⟨0, of_decide_eq_true (id (Eq.refl true))⟩ ω) * (S.dVar ⟨0, of_decide_eq_true (id (Eq.refl true))⟩).indicator (S.dbar ⟨0, of_decide_eq_true (id (Eq.refl true))⟩) ω) P.μ → ∫ (ω : P.Ω), f (S.factualS ⟨0, of_decide_eq_true (id (Eq.refl true))⟩ ω) * (S.dVar ⟨0, of_decide_eq_true (id (Eq.refl true))⟩).indicator (S.dbar ⟨0, of_decide_eq_true (id (Eq.refl true))⟩) ω ∂P.μ = ∫ (ω : P.Ω), f (S.factualS ⟨0, of_decide_eq_true (id (Eq.refl true))⟩ ω) * S.e₀_val (S.factualS ⟨0, of_decide_eq_true (id (Eq.refl true))⟩ ω) ∂P.μ","l":189,"f":"Causalean/Estimation/DTR/ScorePullout.lean"},{"n":"Causalean.Estimation.DTR.DTREstimationSystem.indicator_to_propScore_integral_stage1","k":"theorem","m":"Causalean.Estimation.DTR.ScorePullout","a":"Estimation/DTR","t":1,"d":"Stage-1 indicator-to-propensity rewrite. Let S be a two-stage dynamic-treatment-regime estimation system and let f be a real-valued function of the stage-1 history (S₁,D₀,S₀) that is measurable and for which the product f(S₁,D₀,S₀) · 1{D₁ = dbar 1} is integrable. Then the expectation of f(S₁,D₀,S₀) times the indicator of following the target stage-1 treatment dbar 1 equals the expectation of f(S₁,D₀,S₀) times the true stage-1 propensity score e₁_val(S₁,D₀,S₀).","s":"∀ {P : Causalean.PO.POSystem} {δ : Type} {γ : Fin 2 → Type} [inst : MeasurableSpace δ] [inst_1 : MeasurableSingletonClass δ] [inst_2 : (k : Fin 2) → MeasurableSpace (γ k)] [inst_3 : StandardBorelSpace P.Ω] [inst_4 : MeasureTheory.IsFiniteMeasure P.μ] (S : Causalean.Estimation.DTR.DTREstimationSystem P δ γ) (f : γ 1 × δ × γ 0 → ℝ), Measurable f → MeasureTheory.Integrable (fun ω => f (S.factualS ⟨1, of_decide_eq_true (id (Eq.refl true))⟩ ω, S.factualD ⟨0, of_decide_eq_true (id (Eq.refl true))⟩ ω, S.factualS ⟨0, of_decide_eq_true (id (Eq.refl true))⟩ ω) * (S.dVar ⟨1, of_decide_eq_true (id (Eq.refl true))⟩).indicator (S.dbar ⟨1, of_decide_eq_true (id (Eq.refl true))⟩) ω) P.μ → ∫ (ω : P.Ω), f (S.factualS ⟨1, of_decide_eq_true (id (Eq.refl true))⟩ ω, S.factualD ⟨0, of_decide_eq_true (id (Eq.refl true))⟩ ω, S.factualS ⟨0, of_decide_eq_true (id (Eq.refl true))⟩ ω) * (S.dVar ⟨1, of_decide_eq_true (id (Eq.refl true))⟩).indicator (S.dbar ⟨1, of_decide_eq_true (id (Eq.refl true))⟩) ω ∂P.μ = ∫ (ω : P.Ω), f (S.factualS ⟨1, of_decide_eq_true (id (Eq.refl true))⟩ ω, S.factualD ⟨0, of_decide_eq_true (id (Eq.refl true))⟩ ω, S.factualS ⟨0, of_decide_eq_true (id (Eq.refl true))⟩ ω) * S.e₁_val (S.factualS ⟨1, of_decide_eq_true (id (Eq.refl true))⟩ ω, S.factualD ⟨0, of_decide_eq_true (id (Eq.refl true))⟩ ω, S.factualS ⟨0, of_decide_eq_true (id (Eq.refl true))⟩ ω) ∂P.μ","l":406,"f":"Causalean/Estimation/DTR/ScorePullout.lean"},{"n":"Causalean.Estimation.DTR.DTREstimationSystem.indicator_weighted_delta_mu0_integrable","k":"theorem","m":"Causalean.Estimation.DTR.RemainderIdentity.Helpers","a":"Estimation/DTR/RemainderIdentity","t":2,"d":"The stage-zero indicator-weighted stage-zero outcome-regression error is integrable.","s":"∀ {P : Causalean.PO.POSystem} {δ : Type} {γ : Fin 2 → Type} [inst : MeasurableSpace δ] [inst_1 : MeasurableSingletonClass δ] [inst_2 : (k : Fin 2) → MeasurableSpace (γ k)] [inst_3 : StandardBorelSpace P.Ω] [inst_4 : MeasureTheory.IsFiniteMeasure P.μ] (S : Causalean.Estimation.DTR.DTREstimationSystem P δ γ) {ε : ℝ}, 0 < ε → ∀ η ∈ Causalean.Estimation.DTR.DTREstimationSystem.H_ε ε, MeasureTheory.MemLp (fun s₀ => η.μ₀_fn s₀ - S.μ₀_val s₀) 2 S.P_H₀ → MeasureTheory.Integrable (fun ω => (S.dVar ⟨0, of_decide_eq_true (id (Eq.refl true))⟩).indicator (S.dbar ⟨0, of_decide_eq_true (id (Eq.refl true))⟩) ω * (1 / η.e₀_fn (S.factualS ⟨0, of_decide_eq_true (id (Eq.refl true))⟩ ω)) * (η.μ₀_fn (S.factualS ⟨0, of_decide_eq_true (id (Eq.refl true))⟩ ω) - S.μ₀_val (S.factualS ⟨0, of_decide_eq_true (id (Eq.refl true))⟩ ω))) P.μ","l":113,"f":"Causalean/Estimation/DTR/RemainderIdentity/Helpers.lean"},{"n":"Causalean.Estimation.DTR.DTREstimationSystem.indicator_weighted_delta_mu1_stage0_integrable","k":"theorem","m":"Causalean.Estimation.DTR.RemainderIdentity.Helpers","a":"Estimation/DTR/RemainderIdentity","t":2,"d":"The stage-zero indicator-weighted stage-one outcome-regression error is integrable.","s":"∀ {P : Causalean.PO.POSystem} {δ : Type} {γ : Fin 2 → Type} [inst : MeasurableSpace δ] [inst_1 : MeasurableSingletonClass δ] [inst_2 : (k : Fin 2) → MeasurableSpace (γ k)] [inst_3 : StandardBorelSpace P.Ω] [inst_4 : MeasureTheory.IsFiniteMeasure P.μ] (S : Causalean.Estimation.DTR.DTREstimationSystem P δ γ) {ε : ℝ}, 0 < ε → ∀ η ∈ Causalean.Estimation.DTR.DTREstimationSystem.H_ε ε, MeasureTheory.MemLp (fun h => η.μ₁_fn h - S.μ₁_val h) 2 S.P_H₁ → MeasureTheory.Integrable (fun ω => (S.dVar ⟨0, of_decide_eq_true (id (Eq.refl true))⟩).indicator (S.dbar ⟨0, of_decide_eq_true (id (Eq.refl true))⟩) ω * (1 / η.e₀_fn (S.factualS ⟨0, of_decide_eq_true (id (Eq.refl true))⟩ ω)) * (η.μ₁_fn (S.factualS ⟨1, of_decide_eq_true (id (Eq.refl true))⟩ ω, S.factualD ⟨0, of_decide_eq_true (id (Eq.refl true))⟩ ω, S.factualS ⟨0, of_decide_eq_true (id (Eq.refl true))⟩ ω) - S.μ₁_val (S.factualS ⟨1, of_decide_eq_true (id (Eq.refl true))⟩ ω, S.factualD ⟨0, of_decide_eq_true (id (Eq.refl true))⟩ ω, S.factualS ⟨0, of_decide_eq_true (id (Eq.refl true))⟩ ω))) P.μ","l":174,"f":"Causalean/Estimation/DTR/RemainderIdentity/Helpers.lean"},{"n":"Causalean.Estimation.DTR.DTREstimationSystem.indicator_weighted_delta_mu1_stage1_integrable","k":"theorem","m":"Causalean.Estimation.DTR.RemainderIdentity.Helpers","a":"Estimation/DTR/RemainderIdentity","t":2,"d":"The double-indicator-weighted stage-one outcome-regression error is integrable.","s":"∀ {P : Causalean.PO.POSystem} {δ : Type} {γ : Fin 2 → Type} [inst : MeasurableSpace δ] [inst_1 : MeasurableSingletonClass δ] [inst_2 : (k : Fin 2) → MeasurableSpace (γ k)] [inst_3 : StandardBorelSpace P.Ω] [inst_4 : MeasureTheory.IsFiniteMeasure P.μ] (S : Causalean.Estimation.DTR.DTREstimationSystem P δ γ) {ε : ℝ}, 0 < ε → ∀ η ∈ Causalean.Estimation.DTR.DTREstimationSystem.H_ε ε, MeasureTheory.MemLp (fun h => η.μ₁_fn h - S.μ₁_val h) 2 S.P_H₁ → MeasureTheory.Integrable (fun ω => (S.dVar ⟨0, of_decide_eq_true (id (Eq.refl true))⟩).indicator (S.dbar ⟨0, of_decide_eq_true (id (Eq.refl true))⟩) ω * (S.dVar ⟨1, of_decide_eq_true (id (Eq.refl true))⟩).indicator (S.dbar ⟨1, of_decide_eq_true (id (Eq.refl true))⟩) ω * (1 / (η.e₀_fn (S.factualS ⟨0, of_decide_eq_true (id (Eq.refl true))⟩ ω) * η.e₁_fn (S.factualS ⟨1, of_decide_eq_true (id (Eq.refl true))⟩ ω, S.factualD ⟨0, of_decide_eq_true (id (Eq.refl true))⟩ ω, S.factualS ⟨0, of_decide_eq_true (id (Eq.refl true))⟩ ω))) * (η.μ₁_fn (S.factualS ⟨1, of_decide_eq_true (id (Eq.refl true))⟩ ω, S.factualD ⟨0, of_decide_eq_true (id (Eq.refl true))⟩ ω, S.factualS ⟨0, of_decide_eq_true (id (Eq.refl true))⟩ ω) - S.μ₁_val (S.factualS ⟨1, of_decide_eq_true (id (Eq.refl true))⟩ ω, S.factualD ⟨0, of_decide_eq_true (id (Eq.refl true))⟩ ω, S.factualS ⟨0, of_decide_eq_true (id (Eq.refl true))⟩ ω))) P.μ","l":246,"f":"Causalean/Estimation/DTR/RemainderIdentity/Helpers.lean"},{"n":"Causalean.Estimation.DTR.DTREstimationSystem.measurable_factualZ","k":"theorem","m":"Causalean.Estimation.DTR.Setup","a":"Estimation/DTR","t":2,"d":"The full observed two-stage data tuple is measurable.","s":"∀ {P : Causalean.PO.POSystem} {δ : Type} {γ : Fin 2 → Type} [inst : MeasurableSpace δ] [inst_1 : MeasurableSingletonClass δ] [inst_2 : (k : Fin 2) → MeasurableSpace (γ k)] [inst_3 : StandardBorelSpace P.Ω] [inst_4 : MeasureTheory.IsFiniteMeasure P.μ] (S : Causalean.Estimation.DTR.DTREstimationSystem P δ γ), Measurable S.factualZ","l":444,"f":"Causalean/Estimation/DTR/Setup.lean"},{"n":"Causalean.Estimation.DTR.DTREstimationSystem.measurable_histH₁","k":"theorem","m":"Causalean.Estimation.DTR.SeqDRMoment","a":"Estimation/DTR","t":2,"d":"The cons-ordered stage-1 history projection is measurable.","s":"∀ {δ : Type} [inst : MeasurableSpace δ] {γ : Fin 2 → Type} [inst_1 : (k : Fin 2) → MeasurableSpace (γ k)], Measurable fun z => Causalean.Estimation.DTR.histH₁ z","l":437,"f":"Causalean/Estimation/DTR/SeqDRMoment.lean"},{"n":"Causalean.Estimation.DTR.DTREstimationSystem.measurable_projS₀","k":"theorem","m":"Causalean.Estimation.DTR.SeqDRMoment","a":"Estimation/DTR","t":2,"d":"The initial-state projection is measurable.","s":"∀ {δ : Type} [inst : MeasurableSpace δ] {γ : Fin 2 → Type} [inst_1 : (k : Fin 2) → MeasurableSpace (γ k)], Measurable fun z => Causalean.Estimation.DTR.projS₀ z","l":431,"f":"Causalean/Estimation/DTR/SeqDRMoment.lean"},{"n":"Causalean.Estimation.DTR.DTREstimationSystem.measurable_seqDRMomentFunctional","k":"theorem","m":"Causalean.Estimation.DTR.SeqDRMoment","a":"Estimation/DTR","t":2,"d":"The sequential doubly robust moment functional is measurable as a function of the observed data tuple.","s":"∀ {δ : Type} [inst : MeasurableSpace δ] [inst_1 : MeasurableSingletonClass δ] {γ : Fin 2 → Type} [inst_2 : (k : Fin 2) → MeasurableSpace (γ k)] {P : Causalean.PO.POSystem} [inst_3 : StandardBorelSpace P.Ω] [inst_4 : MeasureTheory.IsFiniteMeasure P.μ] (S : Causalean.Estimation.DTR.DTREstimationSystem P δ γ) (η : Causalean.Estimation.DTR.DTRNuisanceVec₂ δ γ) (θ : ℝ), Measurable fun z => S.seqDRMomentFunctional η z θ","l":369,"f":"Causalean/Estimation/DTR/SeqDRMoment.lean"},{"n":"Causalean.Estimation.DTR.DTREstimationSystem.measurable_ψ_seqDR","k":"theorem","m":"Causalean.Estimation.DTR.MeanZero","a":"Estimation/DTR","t":2,"d":"Measurability of the sequential DR influence function on the data tuple (s₀, d₀, s₁, d₁, y) : γ 0 × δ × γ 1 × δ × ℝ. Decomposes into Measurable.add/Measurable.mul/Measurable.div chained against the projections, the indicator functions indEq, and the value-space nuisance functions stored in S.η₀.","s":"∀ {P : Causalean.PO.POSystem} {δ : Type} {γ : Fin 2 → Type} [inst : MeasurableSpace δ] [inst_1 : MeasurableSingletonClass δ] [inst_2 : (k : Fin 2) → MeasurableSpace (γ k)] [inst_3 : StandardBorelSpace P.Ω] [inst_4 : MeasureTheory.IsFiniteMeasure P.μ] (S : Causalean.Estimation.DTR.DTREstimationSystem P δ γ), Measurable S.ψ_seqDR","l":48,"f":"Causalean/Estimation/DTR/MeanZero.lean"},{"n":"Causalean.Estimation.DTR.DTREstimationSystem.measurable_ψ_seqDR_squared","k":"theorem","m":"Causalean.Estimation.DTR.FiniteVar","a":"Estimation/DTR","t":2,"d":"Measurability helper: the squared sequential DR influence function on the data tuple is measurable. Used in the L² bookkeeping for seqDR_finite_var.","s":"∀ {P : Causalean.PO.POSystem} {δ : Type} {γ : Fin 2 → Type} [inst : MeasurableSpace δ] [inst_1 : MeasurableSingletonClass δ] [inst_2 : (k : Fin 2) → MeasurableSpace (γ k)] [inst_3 : StandardBorelSpace P.Ω] [inst_4 : MeasureTheory.IsFiniteMeasure P.μ] (S : Causalean.Estimation.DTR.DTREstimationSystem P δ γ), Measurable fun z => S.ψ_seqDR z ^ 2","l":42,"f":"Causalean/Estimation/DTR/FiniteVar.lean"},{"n":"Causalean.Estimation.DTR.DTREstimationSystem.propScore_ne_zero_stage0","k":"theorem","m":"Causalean.Estimation.DTR.MeanZero","a":"Estimation/DTR","t":2,"d":"Stage-0 propensity is a.e. nonzero under the DTR backdoor assumptions. The conditional indicator μ[1{D₀ = dbar 0} | σ(historyBundle 0)] is identified via e₀_compat with e₀_val ∘ factualS 0, and e₀_val > 0 pointwise on γ 0.","s":"∀ {P : Causalean.PO.POSystem} {δ : Type} {γ : Fin 2 → Type} [inst : MeasurableSpace δ] [inst_1 : MeasurableSingletonClass δ] [inst_2 : (k : Fin 2) → MeasurableSpace (γ k)] [inst_3 : StandardBorelSpace P.Ω] [inst_4 : MeasureTheory.IsFiniteMeasure P.μ] (S : Causalean.Estimation.DTR.DTREstimationSystem P δ γ), S.Assumptions → ∀ᵐ (ω : P.Ω) ∂P.μ, (S.historyBundle 0 (of_decide_eq_true (id (Eq.refl true)))).condExpGiven ((S.dVar ⟨0, of_decide_eq_true (id (Eq.refl true))⟩).indicator (S.dbar ⟨0, of_decide_eq_true (id (Eq.refl true))⟩)) P.μ ω ≠ 0","l":59,"f":"Causalean/Estimation/DTR/MeanZero.lean"},{"n":"Causalean.Estimation.DTR.DTREstimationSystem.propScore_ne_zero_stage1","k":"theorem","m":"Causalean.Estimation.DTR.MeanZero","a":"Estimation/DTR","t":2,"d":"Stage-1 propensity is a.e. nonzero under the DTR backdoor assumptions. Analogous to propScore_ne_zero_stage0 via e₁_compat and e₁_pos.","s":"∀ {P : Causalean.PO.POSystem} {δ : Type} {γ : Fin 2 → Type} [inst : MeasurableSpace δ] [inst_1 : MeasurableSingletonClass δ] [inst_2 : (k : Fin 2) → MeasurableSpace (γ k)] [inst_3 : StandardBorelSpace P.Ω] [inst_4 : MeasureTheory.IsFiniteMeasure P.μ] (S : Causalean.Estimation.DTR.DTREstimationSystem P δ γ), S.Assumptions → ∀ᵐ (ω : P.Ω) ∂P.μ, (S.historyBundle 1 (of_decide_eq_true (id (Eq.refl true)))).condExpGiven ((S.dVar ⟨1, of_decide_eq_true (id (Eq.refl true))⟩).indicator (S.dbar ⟨1, of_decide_eq_true (id (Eq.refl true))⟩)) P.μ ω ≠ 0","l":72,"f":"Causalean/Estimation/DTR/MeanZero.lean"},{"n":"Causalean.Estimation.DTR.DTREstimationSystem.residual_mul_error_isLittleOp_one","k":"theorem","m":"Causalean.Estimation.DTR.ScoreL2.Helpers","a":"Estimation/DTR/ScoreL2","t":2,"d":"Multiplying an L²-convergent error by a fixed square-integrable residual still gives a stochastic little-o L² norm.","s":"∀ {P : Causalean.PO.POSystem} {α : Type u_1} [inst : MeasurableSpace α] {ν : MeasureTheory.Measure α} [MeasureTheory.IsProbabilityMeasure ν] {R : α → ℝ}, Measurable R → (∀ (z : α), 0 ≤ R z) → MeasureTheory.MemLp R 2 ν → ∀ {deZ : ℕ → P.Ω → α → ℝ}, (∀ (n : ℕ) (ω : P.Ω), MeasureTheory.MemLp (deZ n ω) 2 ν) → (∀ (n : ℕ) (ω : P.Ω), Measurable (deZ n ω)) → (∀ (n : ℕ) (ω : P.Ω) (z : α), |deZ n ω z| ≤ 1) → Causalean.Stat.IsLittleOp (fun n ω => (MeasureTheory.eLpNorm (deZ n ω) 2 ν).toReal) (fun x => 1) P.μ → Causalean.Stat.IsLittleOp (fun n ω => (MeasureTheory.eLpNorm (fun z => R z * |deZ n ω z|) 2 ν).toReal) (fun x => 1) P.μ","l":140,"f":"Causalean/Estimation/DTR/ScoreL2/Helpers.lean"},{"n":"Causalean.Estimation.DTR.DTREstimationSystem.seqDRMoment","k":"def","m":"Causalean.Estimation.DTR.SeqDRMoment","a":"Estimation/DTR","t":1,"d":"This is the sequential doubly robust moment specialized to the system's target regime.","s":"{δ : Type} → [inst : MeasurableSpace δ] → [inst_1 : MeasurableSingletonClass δ] → {γ : Fin 2 → Type} → [inst_2 : (k : Fin 2) → MeasurableSpace (γ k)] → {P : Causalean.PO.POSystem} → [inst_3 : StandardBorelSpace P.Ω] → [inst_4 : MeasureTheory.IsFiniteMeasure P.μ] → Causalean.Estimation.DTR.DTREstimationSystem P δ γ → γ 0 × δ × γ 1 × δ × ℝ → Causalean.Estimation.DTR.DTRNuisanceVec₂ δ γ → ℝ → ℝ","l":347,"f":"Causalean/Estimation/DTR/SeqDRMoment.lean"},{"n":"Causalean.Estimation.DTR.DTREstimationSystem.seqDRMomentFunctional","k":"def","m":"Causalean.Estimation.DTR.SeqDRMoment","a":"Estimation/DTR","t":1,"d":"This is the sequential doubly robust moment packaged as a functional of nuisance, data, and target value.","s":"{δ : Type} → [inst : MeasurableSpace δ] → [inst_1 : MeasurableSingletonClass δ] → {γ : Fin 2 → Type} → [inst_2 : (k : Fin 2) → MeasurableSpace (γ k)] → {P : Causalean.PO.POSystem} → [inst_3 : StandardBorelSpace P.Ω] → [inst_4 : MeasureTheory.IsFiniteMeasure P.μ] → Causalean.Estimation.DTR.DTREstimationSystem P δ γ → Causalean.Estimation.DTR.DTRNuisanceVec₂ δ γ → γ 0 × δ × γ 1 × δ × ℝ → ℝ → ℝ","l":364,"f":"Causalean/Estimation/DTR/SeqDRMoment.lean"},{"n":"Causalean.Estimation.DTR.DTREstimationSystem.seqDR_finite_var","k":"theorem","m":"Causalean.Estimation.DTR.FiniteVar","a":"Estimation/DTR","t":1,"d":"Finite variance of the sequential doubly robust score — sequential DR (DTR) analogue of aipw_finite_var. Under the two-stage DTR backdoor assumptions — sequential exchangeability, consistency, stagewise positivity, and integrability of every counterfactual outcome, uniform two-stage strict overlap: the target-regime propensity at each stage lies almost surely in [ε, 1-ε] for some ε in (0, 1/2], a finite second moment for the observed factual outcome, and a finite second moment for every counterfactual outcome under a treatment sequence, the sequential doubly robust influence function ψ_seqDR is square-integrable against the joint law of the observed two-stage data tuple.","s":"∀ {P : Causalean.PO.POSystem} {δ : Type} {γ : Fin 2 → Type} [inst : MeasurableSpace δ] [inst_1 : MeasurableSingletonClass δ] [inst_2 : (k : Fin 2) → MeasurableSpace (γ k)] [inst_3 : StandardBorelSpace P.Ω] [inst_4 : MeasureTheory.IsFiniteMeasure P.μ] (S : Causalean.Estimation.DTR.DTREstimationSystem P δ γ) {ε : ℝ}, S.StrictOverlap ε → S.Assumptions → MeasureTheory.Integrable (fun ω => S.factualY ω ^ 2) P.μ → (∀ (dbar : Fin 2 → δ), MeasureTheory.Integrable (fun ω => S.Y_of dbar ω ^ 2) P.μ) → MeasureTheory.Integrable (fun z => S.ψ_seqDR z ^ 2) S.P_Z","l":56,"f":"Causalean/Estimation/DTR/FiniteVar.lean"},{"n":"Causalean.Estimation.DTR.DTREstimationSystem.seqDR_mean_zero","k":"theorem","m":"Causalean.Estimation.DTR.MeanZero","a":"Estimation/DTR","t":1,"d":"Mean zero of the sequential doubly robust score. Under the two-stage DTR backdoor assumptions — sequential exchangeability, consistency, stagewise positivity, and integrability of every counterfactual outcome, uniform two-stage strict overlap: the target-regime propensity at each stage lies almost surely in [ε, 1-ε] for some ε in (0, 1/2], and a finite second moment for the observed factual outcome, then the sequential doubly robust influence function ψ_seqDR has expectation zero under the observed two-stage data law.","s":"∀ {P : Causalean.PO.POSystem} {δ : Type} {γ : Fin 2 → Type} [inst : MeasurableSpace δ] [inst_1 : MeasurableSingletonClass δ] [inst_2 : (k : Fin 2) → MeasurableSpace (γ k)] [inst_3 : StandardBorelSpace P.Ω] [inst_4 : MeasureTheory.IsFiniteMeasure P.μ] (S : Causalean.Estimation.DTR.DTREstimationSystem P δ γ) {ε : ℝ}, S.StrictOverlap ε → S.Assumptions → MeasureTheory.Integrable (fun ω => S.factualY ω ^ 2) P.μ → ∫ (z : γ 0 × δ × γ 1 × δ × ℝ), S.ψ_seqDR z ∂S.P_Z = 0","l":1025,"f":"Causalean/Estimation/DTR/MeanZero.lean"},{"n":"Causalean.Estimation.DTR.DTREstimationSystem.seqDR_real_bound","k":"theorem","m":"Causalean.Estimation.DTR.ScoreL2.Helpers","a":"Estimation/DTR/ScoreL2","t":1,"d":"Pointwise Lipschitz bound for the two-stage sequential DR score (real-valued form). Consider a stage-0 propensity e0 and its estimate ê0, a stage-1 propensity e1 and its estimate ê1, true and estimated stage-0/stage-1 outcome-regression values μ0, μ0h, μ1, μ1h, an outcome value y, a target parameter θ, and treatment indicators I0, I1, where ε is strictly positive, both the true and estimated stage-0 propensities are at least ε, both the true and estimated stage-1 propensities are at least ε, and both treatment indicators have absolute value at most 1. Then the absolute difference between the sequential doubly-robust moment built from the estimated nuisances (ê0, ê1, μ0h, μ1h) and from the true nuisances (e0, e1, μ0, μ1) is bounded by the Lipschitz constant K_seqDR ε times the sum of the stage-0 regression error, the stage-1 regression error, and cross terms in which the stage-0 and stage-1 propensity errors are weighted by outcome/regression residuals.","s":"∀ {ε e0 ê0 e1 ê1 μ0 μ0h μ1 μ1h y θ I0 I1 : ℝ}, 0 < ε → ε ≤ e0 → ε ≤ ê0 → ε ≤ e1 → ε ≤ ê1 → |I0| ≤ 1 → |I1| ≤ 1 → |μ0h + I0 / ê0 * (μ1h - μ0h) + I0 * I1 / (ê0 * ê1) * (y - μ1h) - θ - (μ0 + I0 / e0 * (μ1 - μ0) + I0 * I1 / (e0 * e1) * (y - μ1) - θ)| ≤ Causalean.Estimation.DTR.DTREstimationSystem.K_seqDR ε * (|μ0h - μ0| + |μ1h - μ1| + (|μ1 - μ0| + |y - μ1|) * |ê0 - e0| + |y - μ1| * |ê1 - e1|)","l":413,"f":"Causalean/Estimation/DTR/ScoreL2/Helpers.lean"},{"n":"Causalean.Estimation.DTR.DTREstimationSystem.seqDR_rem_const","k":"def","m":"Causalean.Estimation.DTR.RemainderBound","a":"Estimation/DTR","t":1,"d":"Sequential DR (DTR, n = 2) remainder constant for strict overlap ε.","s":"ℝ → ℝ","l":71,"f":"Causalean/Estimation/DTR/RemainderBound.lean"},{"n":"Causalean.Estimation.DTR.DTREstimationSystem.seqDR_remainder_bound","k":"theorem","m":"Causalean.Estimation.DTR.RemainderBound","a":"Estimation/DTR","t":1,"d":"Sequential DR (DTR, n = 2) remainder bound. Consider a two-stage dynamic treatment-regime estimation system satisfying the sequential causal assumptions, for which the stage-0 and stage-1 propensity scores are bounded within a margin ε of 0 and 1 (strict overlap), and where the factual outcome and the potential outcome under every fixed treatment regime each have finite second moment. For any candidate nuisance vector η whose propensities likewise lie in this strict-overlap band, and whose stage-0 outcome-regression error, stage-1 outcome-regression error, stage-0 propensity error, and stage-1 propensity error are each square-integrable against the corresponding stage's history law, the absolute value of the population sequential doubly robust moment at η and the true target θ₀ is at most an explicit O(ε⁻²) constant times the sum of the two stagewise outcome-regression L² errors, times the sum of the two stagewise propensity L² errors.","s":"∀ {P : Causalean.PO.POSystem} {δ : Type} {γ : Fin 2 → Type} [inst : MeasurableSpace δ] [inst_1 : MeasurableSingletonClass δ] [inst_2 : (k : Fin 2) → MeasurableSpace (γ k)] [inst_3 : StandardBorelSpace P.Ω] [inst_4 : MeasureTheory.IsFiniteMeasure P.μ] (S : Causalean.Estimation.DTR.DTREstimationSystem P δ γ) {ε : ℝ}, S.StrictOverlap ε → S.Assumptions → MeasureTheory.Integrable (fun ω => S.factualY ω ^ 2) P.μ → (∀ (dbar : Fin 2 → δ), MeasureTheory.Integrable (fun ω => S.Y_of dbar ω ^ 2) P.μ) → ∀ η ∈ Causalean.Estimation.DTR.DTREstimationSystem.H_ε ε, MeasureTheory.MemLp (fun s₀ => η.μ₀_fn s₀ - S.μ₀_val s₀) 2 S.P_H₀ → MeasureTheory.MemLp (fun h => η.μ₁_fn h - S.μ₁_val h) 2 S.P_H₁ → MeasureTheory.MemLp (fun s₀ => η.e₀_fn s₀ - S.e₀_val s₀) 2 S.P_H₀ → MeasureTheory.MemLp (fun h => η.e₁_fn h - S.e₁_val h) 2 S.P_H₁ → |∫ (z : γ 0 × δ × γ 1 × δ × ℝ), S.seqDRMomentFunctional η z S.θ₀ ∂S.P_Z| ≤ Causalean.Estimation.DTR.DTREstimationSystem.seqDR_rem_const ε * ((MeasureTheory.eLpNorm (fun s₀ => η.μ₀_fn s₀ - S.μ₀_val s₀) 2 S.P_H₀).toReal + (MeasureTheory.eLpNorm (fun h => η.μ₁_fn h - S.μ₁_val h) 2 S.P_H₁).toReal) * ((MeasureTheory.eLpNorm (fun s₀ => η.e₀_fn s₀ - S.e₀_val s₀) 2 S.P_H₀).toReal + (MeasureTheory.eLpNorm (fun h => η.e₁_fn h - S.e₁_val h) 2 S.P_H₁).toReal)","l":78,"f":"Causalean/Estimation/DTR/RemainderBound.lean"},{"n":"Causalean.Estimation.DTR.DTREstimationSystem.seqDR_remainder_identity","k":"theorem","m":"Causalean.Estimation.DTR.RemainderIdentity","a":"Estimation/DTR/RemainderIdentity","t":1,"d":"Sequential DR (DTR, n = 2) remainder identity. Consider a two-stage dynamic treatment-regime estimation system satisfying the sequential causal assumptions (consistency and sequential ignorability), for which the stage-0 and stage-1 propensity scores are bounded within a margin ε of 0 and 1 (strict overlap), and where the factual outcome and the potential outcome under every fixed treatment regime each have finite second moment. For any candidate nuisance vector η whose stage-0 and stage-1 propensities likewise lie in this strict-overlap band, and whose stage-0 outcome-regression error, stage-1 outcome-regression error, stage-0 propensity error, and stage-1 propensity error are each square-integrable against the corresponding stage's history law, the population sequential doubly robust moment at η and the true target θ₀ equals the sum of two stagewise cross-product integrals — propensity error times inverse-propensity weight times outcome-regression error, at stage 0 against the stage-0 history law and at stage 1 against the stage-1 history law.","s":"∀ {P : Causalean.PO.POSystem} {δ : Type} {γ : Fin 2 → Type} [inst : MeasurableSpace δ] [inst_1 : MeasurableSingletonClass δ] [inst_2 : (k : Fin 2) → MeasurableSpace (γ k)] [inst_3 : StandardBorelSpace P.Ω] [inst_4 : MeasureTheory.IsFiniteMeasure P.μ] (S : Causalean.Estimation.DTR.DTREstimationSystem P δ γ) {ε : ℝ}, S.StrictOverlap ε → S.Assumptions → MeasureTheory.Integrable (fun ω => S.factualY ω ^ 2) P.μ → (∀ (dbar : Fin 2 → δ), MeasureTheory.Integrable (fun ω => S.Y_of dbar ω ^ 2) P.μ) → ∀ η ∈ Causalean.Estimation.DTR.DTREstimationSystem.H_ε ε, MeasureTheory.MemLp (fun s₀ => η.μ₀_fn s₀ - S.μ₀_val s₀) 2 S.P_H₀ → MeasureTheory.MemLp (fun h => η.μ₁_fn h - S.μ₁_val h) 2 S.P_H₁ → MeasureTheory.MemLp (fun s₀ => η.e₀_fn s₀ - S.e₀_val s₀) 2 S.P_H₀ → MeasureTheory.MemLp (fun h => η.e₁_fn h - S.e₁_val h) 2 S.P_H₁ → ∫ (z : γ 0 × δ × γ 1 × δ × ℝ), S.seqDRMomentFunctional η z S.θ₀ ∂S.P_Z = ∫ (s₀ : γ 0), (η.e₀_fn s₀ - S.e₀_val s₀) * (1 / η.e₀_fn s₀) * (η.μ₀_fn s₀ - S.μ₀_val s₀) ∂S.P_H₀ + ∫ (h : γ 1 × δ × γ 0), Causalean.Estimation.DTR.indEq h.2.1 (S.dbar 0) * (η.e₁_fn h - S.e₁_val h) * (1 / (η.e₀_fn h.2.2 * η.e₁_fn h)) * (η.μ₁_fn h - S.μ₁_val h) ∂S.P_H₁","l":96,"f":"Causalean/Estimation/DTR/RemainderIdentity.lean"},{"n":"Causalean.Estimation.DTR.DTREstimationSystem.seqDR_remainder_op","k":"theorem","m":"Causalean.Estimation.DTR.RemainderBound","a":"Estimation/DTR","t":1,"d":"Sequential DR remainder is o_p(n^{-1/2}) under the two-stage product rate. Consider a two-stage dynamic treatment-regime estimation system satisfying the sequential causal assumptions, for which the stage-0 and stage-1 propensity scores are bounded within a margin ε of 0 and 1 (strict overlap), and where the factual outcome and the potential outcome under every fixed treatment regime each have finite second moment. Let η̂ₙ be a sequence of sample-size-indexed, possibly random, candidate nuisance vectors that always land in the strict-overlap band, for every sample size and every outcome of the underlying randomness, with stage-0 outcome-regression error, stage-1 outcome-regression error, stage-0 propensity error, and stage-1 propensity error each square-integrable against the corresponding stage's history law at every sample size and outcome. If the four stagewise L² products of outcome-regression and propensity error — own-stage at stage 0, own-stage at stage 1, stage-0 outcome-regression with stage-1 propensity, and stage-1 outcome-regression with stage-0 propensity — are each o_p(n^{-1/2}), then the population sequential doubly robust moment evaluated at the random nuisance η̂ₙ is itself o_p(n^{-1/2}).","s":"∀ {P : Causalean.PO.POSystem} {δ : Type} {γ : Fin 2 → Type} [inst : MeasurableSpace δ] [inst_1 : MeasurableSingletonClass δ] [inst_2 : (k : Fin 2) → MeasurableSpace (γ k)] [inst_3 : StandardBorelSpace P.Ω] [inst_4 : MeasureTheory.IsFiniteMeasure P.μ] (S : Causalean.Estimation.DTR.DTREstimationSystem P δ γ) {ε : ℝ}, S.StrictOverlap ε → S.Assumptions → MeasureTheory.Integrable (fun ω => S.factualY ω ^ 2) P.μ → (∀ (dbar : Fin 2 → δ), MeasureTheory.Integrable (fun ω => S.Y_of dbar ω ^ 2) P.μ) → ∀ (η_hat : ℕ → P.Ω → Causalean.Estimation.DTR.DTRNuisanceVec₂ δ γ), (∀ (n : ℕ) (ω : P.Ω), η_hat n ω ∈ Causalean.Estimation.DTR.DTREstimationSystem.H_ε ε) → (∀ (n : ℕ) (ω : P.Ω), MeasureTheory.MemLp (fun s₀ => (η_hat n ω).μ₀_fn s₀ - S.μ₀_val s₀) 2 S.P_H₀) → (∀ (n : ℕ) (ω : P.Ω), MeasureTheory.MemLp (fun h => (η_hat n ω).μ₁_fn h - S.μ₁_val h) 2 S.P_H₁) → (∀ (n : ℕ) (ω : P.Ω), MeasureTheory.MemLp (fun s₀ => (η_hat n ω).e₀_fn s₀ - S.e₀_val s₀) 2 S.P_H₀) → (∀ (n : ℕ) (ω : P.Ω), MeasureTheory.MemLp (fun h => (η_hat n ω).e₁_fn h - S.e₁_val h) 2 S.P_H₁) → Causalean.Stat.IsLittleOp (fun n ω => (MeasureTheory.eLpNorm (fun s₀ => (η_hat n ω).μ₀_fn s₀ - S.μ₀_val s₀) 2 S.P_H₀).toReal * (MeasureTheory.eLpNorm (fun s₀ => (η_hat n ω).e₀_fn s₀ - S.e₀_val s₀) 2 S.P_H₀).toReal) (fun n => ↑n ^ (-(1 / 2))) P.μ → Causalean.Stat.IsLittleOp (fun n ω => (MeasureTheory.eLpNorm (fun h => (η_hat n ω).μ₁_fn h - S.μ₁_val h) 2 S.P_H₁).toReal * (MeasureTheory.eLpNorm (fun h => (η_hat n ω).e₁_fn h - S.e₁_val h) 2 S.P_H₁).toReal) (fun n => ↑n ^ (-(1 / 2))) P.μ → Causalean.Stat.IsLittleOp (fun n ω => (MeasureTheory.eLpNorm (fun s₀ => (η_hat n ω).μ₀_fn s₀ - S.μ₀_val s₀) 2 S.P_H₀).toReal * (MeasureTheory.eLpNorm (fun h => (η_hat n ω).e₁_fn h - S.e₁_val h) 2 S.P_H₁).toReal) (fun n => ↑n ^ (-(1 / 2))) P.μ → Causalean.Stat.IsLittleOp (fun n ω => (MeasureTheory.eLpNorm (fun h => (η_hat n ω).μ₁_fn h - S.μ₁_val h) 2 S.P_H₁).toReal * (MeasureTheory.eLpNorm (fun s₀ => (η_hat n ω).e₀_fn s₀ - S.e₀_val s₀) 2 S.P_H₀).toReal) (fun n => ↑n ^ (-(1 / 2))) P.μ → Causalean.Stat.IsLittleOp (fun n ω => ∫ (z : γ 0 × δ × γ 1 × δ × ℝ), S.seqDRMomentFunctional (η_hat n ω) z S.θ₀ ∂S.P_Z) (fun n => ↑n ^ (-(1 / 2))) P.μ","l":307,"f":"Causalean/Estimation/DTR/RemainderBound.lean"},{"n":"Causalean.Estimation.DTR.DTREstimationSystem.seqDR_score_diff_isLittleOp_one","k":"theorem","m":"Causalean.Estimation.DTR.ScoreL2","a":"Estimation/DTR/ScoreL2","t":1,"d":"Sequential DR score L²(P_Z) continuity. Consider a two-stage dynamic-treatment-regime estimation system in which the estimated propensity scores stay strictly between ε and 1-ε, the system's identification assumptions (consistency, sequential exchangeability, positivity) hold, the observed outcome has finite second moment, and every potential outcome under a fixed two-stage treatment regime has finite second moment. Given a sequence of nuisance-estimator draws η̂(n) that always land inside the ε-overlap-bounded nuisance set, whose stage-0 outcome-regression, stage-1 outcome-regression, stage-0 propensity-score, and stage-1 propensity-score estimation errors are each square-integrable against the corresponding history law, for every draw and every sample point, and whose four stagewise L² estimation-error rates each vanish in probability (are $o_p(1)$), then the L²(P_Z) distance between the sequential doubly-robust score evaluated at η̂(n) and at the true nuisance η₀ is itself $o_p(1)$.","s":"∀ {P : Causalean.PO.POSystem} {δ : Type} {γ : Fin 2 → Type} [inst : MeasurableSpace δ] [inst_1 : MeasurableSingletonClass δ] [inst_2 : (k : Fin 2) → MeasurableSpace (γ k)] [inst_3 : StandardBorelSpace P.Ω] [inst_4 : MeasureTheory.IsFiniteMeasure P.μ] (S : Causalean.Estimation.DTR.DTREstimationSystem P δ γ) {ε : ℝ}, S.StrictOverlap ε → S.Assumptions → MeasureTheory.Integrable (fun ω => S.factualY ω ^ 2) P.μ → (∀ (dbar : Fin 2 → δ), MeasureTheory.Integrable (fun ω => S.Y_of dbar ω ^ 2) P.μ) → ∀ (η_hat : ℕ → P.Ω → Causalean.Estimation.DTR.DTRNuisanceVec₂ δ γ), (∀ (n : ℕ) (ω : P.Ω), η_hat n ω ∈ Causalean.Estimation.DTR.DTREstimationSystem.H_ε ε) → (∀ (n : ℕ) (ω : P.Ω), MeasureTheory.MemLp (fun s₀ => (η_hat n ω).μ₀_fn s₀ - S.μ₀_val s₀) 2 S.P_H₀) → (∀ (n : ℕ) (ω : P.Ω), MeasureTheory.MemLp (fun h => (η_hat n ω).μ₁_fn h - S.μ₁_val h) 2 S.P_H₁) → (∀ (n : ℕ) (ω : P.Ω), MeasureTheory.MemLp (fun s₀ => (η_hat n ω).e₀_fn s₀ - S.e₀_val s₀) 2 S.P_H₀) → (∀ (n : ℕ) (ω : P.Ω), MeasureTheory.MemLp (fun h => (η_hat n ω).e₁_fn h - S.e₁_val h) 2 S.P_H₁) → Causalean.Stat.IsLittleOp (fun n ω => (MeasureTheory.eLpNorm (fun s₀ => (η_hat n ω).μ₀_fn s₀ - S.μ₀_val s₀) 2 S.P_H₀).toReal) (fun x => 1) P.μ → Causalean.Stat.IsLittleOp (fun n ω => (MeasureTheory.eLpNorm (fun h => (η_hat n ω).μ₁_fn h - S.μ₁_val h) 2 S.P_H₁).toReal) (fun x => 1) P.μ → Causalean.Stat.IsLittleOp (fun n ω => (MeasureTheory.eLpNorm (fun s₀ => (η_hat n ω).e₀_fn s₀ - S.e₀_val s₀) 2 S.P_H₀).toReal) (fun x => 1) P.μ → Causalean.Stat.IsLittleOp (fun n ω => (MeasureTheory.eLpNorm (fun h => (η_hat n ω).e₁_fn h - S.e₁_val h) 2 S.P_H₁).toReal) (fun x => 1) P.μ → Causalean.Stat.IsLittleOp (fun n ω => (MeasureTheory.eLpNorm (fun z => S.seqDRMomentFunctional (η_hat n ω) z S.θ₀ - S.seqDRMomentFunctional S.η₀ z S.θ₀) 2 S.P_Z).toReal) (fun x => 1) P.μ","l":72,"f":"Causalean/Estimation/DTR/ScoreL2.lean"},{"n":"Causalean.Estimation.DTR.DTREstimationSystem.seqDR_score_diff_pointwise_bound","k":"theorem","m":"Causalean.Estimation.DTR.ScoreL2.Helpers","a":"Estimation/DTR/ScoreL2","t":1,"d":"Pointwise Lipschitz bound for the sequential DR score difference. Let S be a two-stage dynamic-treatment-regime estimation system with strict overlap at level ε (the true propensity scores lie in [ε, 1-ε]), and let η be a candidate nuisance vector whose estimated propensity components are likewise confined to [ε, 1-ε]. Then, for almost every observation z under the observed-data law S.P_Z, the absolute difference between the sequential doubly-robust moment evaluated at η and at the true nuisance vector η₀ is bounded by the Lipschitz constant K_seqDR ε times the sum of the stage-0 regression error, the stage-1 regression error, and cross terms in which the stage-0 and stage-1 propensity-score errors are weighted by the corresponding outcome/regression residuals.","s":"∀ {P : Causalean.PO.POSystem} {δ : Type} {γ : Fin 2 → Type} [inst : MeasurableSpace δ] [inst_1 : MeasurableSingletonClass δ] [inst_2 : (k : Fin 2) → MeasurableSpace (γ k)] [inst_3 : StandardBorelSpace P.Ω] [inst_4 : MeasureTheory.IsFiniteMeasure P.μ] (S : Causalean.Estimation.DTR.DTREstimationSystem P δ γ) {ε : ℝ}, S.StrictOverlap ε → ∀ η ∈ Causalean.Estimation.DTR.DTREstimationSystem.H_ε ε, ∀ᵐ (z : γ 0 × δ × γ 1 × δ × ℝ) ∂S.P_Z, |S.seqDRMomentFunctional η z S.θ₀ - S.seqDRMomentFunctional S.η₀ z S.θ₀| ≤ Causalean.Estimation.DTR.DTREstimationSystem.K_seqDR ε * (|η.μ₀_fn (Causalean.Estimation.DTR.projS₀ z) - S.μ₀_val (Causalean.Estimation.DTR.projS₀ z)| + |η.μ₁_fn (Causalean.Estimation.DTR.histH₁ z) - S.μ₁_val (Causalean.Estimation.DTR.histH₁ z)| + (|S.μ₁_val (Causalean.Estimation.DTR.histH₁ z) - S.μ₀_val (Causalean.Estimation.DTR.projS₀ z)| + |Causalean.Estimation.DTR.projY z - S.μ₁_val (Causalean.Estimation.DTR.histH₁ z)|) * |η.e₀_fn (Causalean.Estimation.DTR.projS₀ z) - S.e₀_val (Causalean.Estimation.DTR.projS₀ z)| + |Causalean.Estimation.DTR.projY z - S.μ₁_val (Causalean.Estimation.DTR.histH₁ z)| * |η.e₁_fn (Causalean.Estimation.DTR.histH₁ z) - S.e₁_val (Causalean.Estimation.DTR.histH₁ z)|)","l":605,"f":"Causalean/Estimation/DTR/ScoreL2/Helpers.lean"},{"n":"Causalean.Estimation.DTR.DTREstimationSystem.seqDR_stage0_ratio_bound","k":"theorem","m":"Causalean.Estimation.DTR.ScoreL2.Helpers","a":"Estimation/DTR/ScoreL2","t":2,"d":"The stage-zero weighted regression contrast is Lipschitz in the stage-zero nuisance errors under overlap.","s":"∀ {ε e ê μ0 μ0h μ1 μ1h : ℝ}, 0 < ε → ε ≤ e → ε ≤ ê → |(μ1h - μ0h) / ê - (μ1 - μ0) / e| ≤ (|μ1h - μ1| + |μ0h - μ0|) / ε + |μ1 - μ0| * |ê - e| / ε ^ 2","l":286,"f":"Causalean/Estimation/DTR/ScoreL2/Helpers.lean"},{"n":"Causalean.Estimation.DTR.DTREstimationSystem.seqDR_stage1_ratio_bound","k":"theorem","m":"Causalean.Estimation.DTR.ScoreL2.Helpers","a":"Estimation/DTR/ScoreL2","t":2,"d":"The stage-one weighted residual contrast is Lipschitz in the stage-one regression error and both propensity errors under overlap.","s":"∀ {ε e0 ê0 e1 ê1 y μ1 μ1h : ℝ}, 0 < ε → ε ≤ e0 → ε ≤ ê0 → ε ≤ e1 → ε ≤ ê1 → |(y - μ1h) / (ê0 * ê1) - (y - μ1) / (e0 * e1)| ≤ |μ1h - μ1| / ε ^ 2 + |y - μ1| * |ê0 - e0| / ε ^ 3 + |y - μ1| * |ê1 - e1| / ε ^ 3","l":333,"f":"Causalean/Estimation/DTR/ScoreL2/Helpers.lean"},{"n":"Causalean.Estimation.DTR.DTREstimationSystem.split_stage_history_integral","k":"theorem","m":"Causalean.Estimation.DTR.RemainderIdentity.Helpers","a":"Estimation/DTR/RemainderIdentity","t":1,"d":"Additivity of stage-history integrals under the joint DTR data law. Consider a real-valued function f₀ of the stage-0 state that is measurable and integrable against the stage-0 history marginal law and a real-valued function f₁ of the stage-1 history (current state, previous treatment, previous state) that is measurable and integrable against the stage-1 history marginal law. Then the integral, against the joint law of the observed two-stage data tuple, of the sum of f₀ and f₁ each pulled back through its respective projection out of the data tuple equals the sum of the separate integrals of f₀ against the stage-0 marginal law and f₁ against the stage-1 marginal law.","s":"∀ {P : Causalean.PO.POSystem} {δ : Type} {γ : Fin 2 → Type} [inst : MeasurableSpace δ] [inst_1 : MeasurableSingletonClass δ] [inst_2 : (k : Fin 2) → MeasurableSpace (γ k)] [inst_3 : StandardBorelSpace P.Ω] [inst_4 : MeasureTheory.IsFiniteMeasure P.μ] (S : Causalean.Estimation.DTR.DTREstimationSystem P δ γ) (f₀ : γ 0 → ℝ) (f₁ : γ 1 × δ × γ 0 → ℝ), Measurable f₀ → Measurable f₁ → MeasureTheory.Integrable f₀ S.P_H₀ → MeasureTheory.Integrable f₁ S.P_H₁ → ∫ (z : γ 0 × δ × γ 1 × δ × ℝ), f₀ (Causalean.Estimation.DTR.projS₀ z) + f₁ (Causalean.Estimation.DTR.histH₁ z) ∂S.P_Z = ∫ (s₀ : γ 0), f₀ s₀ ∂S.P_H₀ + ∫ (h : γ 1 × δ × γ 0), f₁ h ∂S.P_H₁","l":349,"f":"Causalean/Estimation/DTR/RemainderIdentity/Helpers.lean"},{"n":"Causalean.Estimation.DTR.DTREstimationSystem.stageOneReg","k":"def","m":"Causalean.Estimation.DTR.Setup","a":"Estimation/DTR","t":1,"d":"This is the observable stage-1 regression of the outcome within the target final treatment arm.","s":"{P : Causalean.PO.POSystem} → {δ : Type} → {γ : Fin 2 → Type} → [inst : MeasurableSpace δ] → [inst_1 : MeasurableSingletonClass δ] → [inst_2 : (k : Fin 2) → MeasurableSpace (γ k)] → [inst_3 : StandardBorelSpace P.Ω] → [inst_4 : MeasureTheory.IsFiniteMeasure P.μ] → Causalean.Estimation.DTR.DTREstimationSystem P δ γ → P.Ω → ℝ","l":176,"f":"Causalean/Estimation/DTR/Setup.lean"},{"n":"Causalean.Estimation.DTR.DTREstimationSystem.stageOneReg_indD_eq","k":"theorem","m":"Causalean.Estimation.DTR.Setup","a":"Estimation/DTR","t":2,"d":"On the regime-consistent path, the observable stage-1 regression agrees with the counterfactual stage-1 regression.","s":"∀ {P : Causalean.PO.POSystem} {δ : Type} {γ : Fin 2 → Type} [inst : MeasurableSpace δ] [inst_1 : MeasurableSingletonClass δ] [inst_2 : (k : Fin 2) → MeasurableSpace (γ k)] [inst_3 : StandardBorelSpace P.Ω] [inst_4 : MeasureTheory.IsFiniteMeasure P.μ] (S : Causalean.Estimation.DTR.DTREstimationSystem P δ γ), S.Assumptions → (fun ω => S.stageOneReg ω * S.indD S.dbar 1 ω) =ᵐ[P.μ] fun ω => S.indD S.dbar 1 ω * (S.historyBundle 1 (of_decide_eq_true (id (Eq.refl true)))).condExpGiven (S.Y_of S.dbar) P.μ ω","l":197,"f":"Causalean/Estimation/DTR/Setup.lean"},{"n":"Causalean.Estimation.DTR.DTREstimationSystem.stageOneReg_memLp","k":"theorem","m":"Causalean.Estimation.DTR.Setup","a":"Estimation/DTR","t":2,"d":"The observable stage-1 regression is square-integrable under strict overlap and a factual second moment.","s":"∀ {P : Causalean.PO.POSystem} {δ : Type} {γ : Fin 2 → Type} [inst : MeasurableSpace δ] [inst_1 : MeasurableSingletonClass δ] [inst_2 : (k : Fin 2) → MeasurableSpace (γ k)] [inst_3 : StandardBorelSpace P.Ω] [inst_4 : MeasureTheory.IsFiniteMeasure P.μ] (S : Causalean.Estimation.DTR.DTREstimationSystem P δ γ) {ε : ℝ}, S.StrictOverlap ε → MeasureTheory.Integrable (fun ω => S.factualY ω ^ 2) P.μ → MeasureTheory.MemLp S.stageOneReg 2 P.μ","l":367,"f":"Causalean/Estimation/DTR/Setup.lean"},{"n":"Causalean.Estimation.DTR.DTREstimationSystem.theta_zero_factualS₀_integral","k":"theorem","m":"Causalean.Estimation.DTR.MeanZero","a":"Estimation/DTR","t":2,"d":"The DTR estimand θ₀ = E[Y(dbar)] lifts to an integral against P.μ: under DTR backdoor assumptions, θ₀ = ∫ ω, μ₀_val(factualS 0 ω) ∂P.μ, since μ₀_val ∘ factualS 0 is the σ(historyBundle 0)-conditional expectation of Y_of dbar and P.μ is a probability measure.","s":"∀ {P : Causalean.PO.POSystem} {δ : Type} {γ : Fin 2 → Type} [inst : MeasurableSpace δ] [inst_1 : MeasurableSingletonClass δ] [inst_2 : (k : Fin 2) → MeasurableSpace (γ k)] [inst_3 : StandardBorelSpace P.Ω] [inst_4 : MeasureTheory.IsFiniteMeasure P.μ] (S : Causalean.Estimation.DTR.DTREstimationSystem P δ γ), S.Assumptions → S.θ₀ = ∫ (ω : P.Ω), S.μ₀_val (S.factualS ⟨0, of_decide_eq_true (id (Eq.refl true))⟩ ω) ∂P.μ","l":716,"f":"Causalean/Estimation/DTR/MeanZero.lean"},{"n":"Causalean.Estimation.DTR.DTREstimationSystem.weighted_residual_integral_zero_stage0","k":"theorem","m":"Causalean.Estimation.DTR.ScorePullout","a":"Estimation/DTR","t":1,"d":"Stage-0 weighted-residual integral vanishes. Let S be a two-stage dynamic-treatment-regime estimation system with strict overlap at level ε, satisfying the system's core identification assumptions (consistency, sequential exchangeability, positivity), and in which the observed outcome has finite second moment. For any weight function g on the stage-0 history that is measurable and for which the product g(S₀) · 1{D₀ = dbar 0} · (μ₁_val(S₁,D₀,S₀) − μ₀_val(S₀)) is integrable, then its expectation under P.μ is zero: the stage-0-weighted, treatment-indicator-gated gap between the stage-1 and stage-0 regression functions has zero mean.","s":"∀ {P : Causalean.PO.POSystem} {δ : Type} {γ : Fin 2 → Type} [inst : MeasurableSpace δ] [inst_1 : MeasurableSingletonClass δ] [inst_2 : (k : Fin 2) → MeasurableSpace (γ k)] [inst_3 : StandardBorelSpace P.Ω] [inst_4 : MeasureTheory.IsFiniteMeasure P.μ] (S : Causalean.Estimation.DTR.DTREstimationSystem P δ γ) {ε : ℝ}, S.StrictOverlap ε → S.Assumptions → MeasureTheory.Integrable (fun ω => S.factualY ω ^ 2) P.μ → ∀ (g : γ 0 → ℝ), Measurable g → MeasureTheory.Integrable (fun ω => g (S.factualS ⟨0, of_decide_eq_true (id (Eq.refl true))⟩ ω) * ((S.dVar ⟨0, of_decide_eq_true (id (Eq.refl true))⟩).indicator (S.dbar ⟨0, of_decide_eq_true (id (Eq.refl true))⟩) ω * (S.μ₁_val (S.factualS ⟨1, of_decide_eq_true (id (Eq.refl true))⟩ ω, S.factualD ⟨0, of_decide_eq_true (id (Eq.refl true))⟩ ω, S.factualS ⟨0, of_decide_eq_true (id (Eq.refl true))⟩ ω) - S.μ₀_val (S.factualS ⟨0, of_decide_eq_true (id (Eq.refl true))⟩ ω)))) P.μ → ∫ (ω : P.Ω), g (S.factualS ⟨0, of_decide_eq_true (id (Eq.refl true))⟩ ω) * ((S.dVar ⟨0, of_decide_eq_true (id (Eq.refl true))⟩).indicator (S.dbar ⟨0, of_decide_eq_true (id (Eq.refl true))⟩) ω * (S.μ₁_val (S.factualS ⟨1, of_decide_eq_true (id (Eq.refl true))⟩ ω, S.factualD ⟨0, of_decide_eq_true (id (Eq.refl true))⟩ ω, S.factualS ⟨0, of_decide_eq_true (id (Eq.refl true))⟩ ω) - S.μ₀_val (S.factualS ⟨0, of_decide_eq_true (id (Eq.refl true))⟩ ω))) ∂P.μ = 0","l":62,"f":"Causalean/Estimation/DTR/ScorePullout.lean"},{"n":"Causalean.Estimation.DTR.DTREstimationSystem.weighted_residual_integral_zero_stage1","k":"theorem","m":"Causalean.Estimation.DTR.ScorePullout","a":"Estimation/DTR","t":1,"d":"Stage-1 weighted-residual integral vanishes. Let S be a two-stage dynamic-treatment-regime estimation system with strict overlap at level ε, satisfying the system's core identification assumptions (consistency, sequential exchangeability, positivity), and in which the observed outcome has finite second moment. For any weight function g on the stage-1 history (S₁,D₀,S₀) that is measurable and for which the product g(S₁,D₀,S₀) · 1{D₀ = dbar 0} · 1{D₁ = dbar 1} · (Y − μ₁_val(S₁,D₀,S₀)) is integrable, then its expectation under P.μ is zero: the doubly indicator-gated stage-1 outcome residual, weighted by g, has zero mean.","s":"∀ {P : Causalean.PO.POSystem} {δ : Type} {γ : Fin 2 → Type} [inst : MeasurableSpace δ] [inst_1 : MeasurableSingletonClass δ] [inst_2 : (k : Fin 2) → MeasurableSpace (γ k)] [inst_3 : StandardBorelSpace P.Ω] [inst_4 : MeasureTheory.IsFiniteMeasure P.μ] (S : Causalean.Estimation.DTR.DTREstimationSystem P δ γ) {ε : ℝ}, S.StrictOverlap ε → S.Assumptions → MeasureTheory.Integrable (fun ω => S.factualY ω ^ 2) P.μ → ∀ (g : γ 1 × δ × γ 0 → ℝ), Measurable g → MeasureTheory.Integrable (fun ω => g (S.factualS ⟨1, of_decide_eq_true (id (Eq.refl true))⟩ ω, S.factualD ⟨0, of_decide_eq_true (id (Eq.refl true))⟩ ω, S.factualS ⟨0, of_decide_eq_true (id (Eq.refl true))⟩ ω) * ((S.dVar ⟨0, of_decide_eq_true (id (Eq.refl true))⟩).indicator (S.dbar ⟨0, of_decide_eq_true (id (Eq.refl true))⟩) ω * ((S.dVar ⟨1, of_decide_eq_true (id (Eq.refl true))⟩).indicator (S.dbar ⟨1, of_decide_eq_true (id (Eq.refl true))⟩) ω * (S.factualY ω - S.μ₁_val (S.factualS ⟨1, of_decide_eq_true (id (Eq.refl true))⟩ ω, S.factualD ⟨0, of_decide_eq_true (id (Eq.refl true))⟩ ω, S.factualS ⟨0, of_decide_eq_true (id (Eq.refl true))⟩ ω))))) P.μ → ∫ (ω : P.Ω), g (S.factualS ⟨1, of_decide_eq_true (id (Eq.refl true))⟩ ω, S.factualD ⟨0, of_decide_eq_true (id (Eq.refl true))⟩ ω, S.factualS ⟨0, of_decide_eq_true (id (Eq.refl true))⟩ ω) * ((S.dVar ⟨0, of_decide_eq_true (id (Eq.refl true))⟩).indicator (S.dbar ⟨0, of_decide_eq_true (id (Eq.refl true))⟩) ω * ((S.dVar ⟨1, of_decide_eq_true (id (Eq.refl true))⟩).indicator (S.dbar ⟨1, of_decide_eq_true (id (Eq.refl true))⟩) ω * (S.factualY ω - S.μ₁_val (S.factualS ⟨1, of_decide_eq_true (id (Eq.refl true))⟩ ω, S.factualD ⟨0, of_decide_eq_true (id (Eq.refl true))⟩ ω, S.factualS ⟨0, of_decide_eq_true (id (Eq.refl true))⟩ ω)))) ∂P.μ = 0","l":260,"f":"Causalean/Estimation/DTR/ScorePullout.lean"},{"n":"Causalean.Estimation.DTR.DTREstimationSystem.η₀","k":"def","m":"Causalean.Estimation.DTR.SeqDRMoment","a":"Estimation/DTR","t":1,"d":"This is the true nuisance vector extracted from a two-stage DTR estimation system.","s":"{δ : Type} → [inst : MeasurableSpace δ] → [inst_1 : MeasurableSingletonClass δ] → {γ : Fin 2 → Type} → [inst_2 : (k : Fin 2) → MeasurableSpace (γ k)] → {P : Causalean.PO.POSystem} → [inst_3 : StandardBorelSpace P.Ω] → [inst_4 : MeasureTheory.IsFiniteMeasure P.μ] → Causalean.Estimation.DTR.DTREstimationSystem P δ γ → Causalean.Estimation.DTR.DTRNuisanceVec₂ δ γ","l":341,"f":"Causalean/Estimation/DTR/SeqDRMoment.lean"},{"n":"Causalean.Estimation.DTR.DTREstimationSystem.θ₀","k":"def","m":"Causalean.Estimation.DTR.Setup","a":"Estimation/DTR","t":1,"d":"This is the fixed-regime mean outcome targeted by the two-stage DTR estimation system.","s":"{P : Causalean.PO.POSystem} → {δ : Type} → {γ : Fin 2 → Type} → [inst : MeasurableSpace δ] → [inst_1 : MeasurableSingletonClass δ] → [inst_2 : (k : Fin 2) → MeasurableSpace (γ k)] → [inst_3 : StandardBorelSpace P.Ω] → [inst_4 : MeasureTheory.IsFiniteMeasure P.μ] → Causalean.Estimation.DTR.DTREstimationSystem P δ γ → ℝ","l":460,"f":"Causalean/Estimation/DTR/Setup.lean"},{"n":"Causalean.Estimation.DTR.DTREstimationSystem.θ₀_eq_dtrEffect","k":"theorem","m":"Causalean.Estimation.DTR.Setup","a":"Estimation/DTR","t":1,"d":"For a two-stage dynamic treatment regime estimation system, its value-space DTR estimand θ₀ equals the potential-outcome DTR effect evaluated at the chosen regime dbar.","s":"∀ {P : Causalean.PO.POSystem} {δ : Type} {γ : Fin 2 → Type} [inst : MeasurableSpace δ] [inst_1 : MeasurableSingletonClass δ] [inst_2 : (k : Fin 2) → MeasurableSpace (γ k)] [inst_3 : StandardBorelSpace P.Ω] [inst_4 : MeasureTheory.IsFiniteMeasure P.μ] (S : Causalean.Estimation.DTR.DTREstimationSystem P δ γ), S.θ₀ = S.dtrEffect S.dbar","l":464,"f":"Causalean/Estimation/DTR/Setup.lean"},{"n":"Causalean.Estimation.DTR.DTREstimationSystem.μ₀_compat","k":"theorem","m":"Causalean.Estimation.DTR.Setup","a":"Estimation/DTR","t":2,"d":"The stage-0 value-space regression equals the counterfactual stage-0 regression under identification.","s":"∀ {P : Causalean.PO.POSystem} {δ : Type} {γ : Fin 2 → Type} [inst : MeasurableSpace δ] [inst_1 : MeasurableSingletonClass δ] [inst_2 : (k : Fin 2) → MeasurableSpace (γ k)] [inst_3 : StandardBorelSpace P.Ω] [inst_4 : MeasureTheory.IsFiniteMeasure P.μ] (S : Causalean.Estimation.DTR.DTREstimationSystem P δ γ), S.Assumptions → (S.historyBundle 0 (of_decide_eq_true (id (Eq.refl true)))).condExpGiven (S.Y_of S.dbar) =ᵐ[P.μ] fun ω => S.μ₀_val (S.factualS ⟨0, of_decide_eq_true (id (Eq.refl true))⟩ ω)","l":163,"f":"Causalean/Estimation/DTR/Setup.lean"},{"n":"Causalean.Estimation.DTR.DTREstimationSystem.μ₀_val_memLp","k":"theorem","m":"Causalean.Estimation.DTR.ScoreL2.Helpers","a":"Estimation/DTR/ScoreL2","t":2,"d":"The true stage-zero regression representative is square-integrable under the stage-zero history marginal.","s":"∀ {P : Causalean.PO.POSystem} {δ : Type} {γ : Fin 2 → Type} [inst : MeasurableSpace δ] [inst_1 : MeasurableSingletonClass δ] [inst_2 : (k : Fin 2) → MeasurableSpace (γ k)] [inst_3 : StandardBorelSpace P.Ω] [inst_4 : MeasureTheory.IsFiniteMeasure P.μ] (S : Causalean.Estimation.DTR.DTREstimationSystem P δ γ), S.Assumptions → (∀ (dbar : Fin 2 → δ), MeasureTheory.Integrable (fun ω => S.Y_of dbar ω ^ 2) P.μ) → MeasureTheory.MemLp S.μ₀_val 2 S.P_H₀","l":558,"f":"Causalean/Estimation/DTR/ScoreL2/Helpers.lean"},{"n":"Causalean.Estimation.DTR.DTREstimationSystem.μ₁_val_comp_eq_stageOneReg","k":"theorem","m":"Causalean.Estimation.DTR.Setup","a":"Estimation/DTR","t":2,"d":"The stage-1 value-space regression agrees almost everywhere with the observable stage-1 regression.","s":"∀ {P : Causalean.PO.POSystem} {δ : Type} {γ : Fin 2 → Type} [inst : MeasurableSpace δ] [inst_1 : MeasurableSingletonClass δ] [inst_2 : (k : Fin 2) → MeasurableSpace (γ k)] [inst_3 : StandardBorelSpace P.Ω] [inst_4 : MeasureTheory.IsFiniteMeasure P.μ] (S : Causalean.Estimation.DTR.DTREstimationSystem P δ γ), (fun ω => S.μ₁_val (S.factualS ⟨1, of_decide_eq_true (id (Eq.refl true))⟩ ω, S.factualD ⟨0, of_decide_eq_true (id (Eq.refl true))⟩ ω, S.factualS ⟨0, of_decide_eq_true (id (Eq.refl true))⟩ ω)) =ᵐ[P.μ] S.stageOneReg","l":185,"f":"Causalean/Estimation/DTR/Setup.lean"},{"n":"Causalean.Estimation.DTR.DTREstimationSystem.μ₁_val_comp_mul_indD_eq","k":"theorem","m":"Causalean.Estimation.DTR.Setup","a":"Estimation/DTR","t":2,"d":"Composed observable μ₁_val, multiplied by the partial regime indicator indD dbar 1, agrees with the corresponding counterfactual conditional expectation. This is the consumer-facing form of stageOneReg_indD_eq.","s":"∀ {P : Causalean.PO.POSystem} {δ : Type} {γ : Fin 2 → Type} [inst : MeasurableSpace δ] [inst_1 : MeasurableSingletonClass δ] [inst_2 : (k : Fin 2) → MeasurableSpace (γ k)] [inst_3 : StandardBorelSpace P.Ω] [inst_4 : MeasureTheory.IsFiniteMeasure P.μ] (S : Causalean.Estimation.DTR.DTREstimationSystem P δ γ), S.Assumptions → (fun ω => S.μ₁_val (S.factualS ⟨1, of_decide_eq_true (id (Eq.refl true))⟩ ω, S.factualD ⟨0, of_decide_eq_true (id (Eq.refl true))⟩ ω, S.factualS ⟨0, of_decide_eq_true (id (Eq.refl true))⟩ ω) * S.indD S.dbar 1 ω) =ᵐ[P.μ] fun ω => S.indD S.dbar 1 ω * (S.historyBundle 1 (of_decide_eq_true (id (Eq.refl true)))).condExpGiven (S.Y_of S.dbar) P.μ ω","l":314,"f":"Causalean/Estimation/DTR/Setup.lean"},{"n":"Causalean.Estimation.DTR.DTREstimationSystem.μ₁_val_memLp","k":"theorem","m":"Causalean.Estimation.DTR.ScoreL2.Helpers","a":"Estimation/DTR/ScoreL2","t":2,"d":"The true stage-one regression representative is square-integrable under the stage-one history marginal.","s":"∀ {P : Causalean.PO.POSystem} {δ : Type} {γ : Fin 2 → Type} [inst : MeasurableSpace δ] [inst_1 : MeasurableSingletonClass δ] [inst_2 : (k : Fin 2) → MeasurableSpace (γ k)] [inst_3 : StandardBorelSpace P.Ω] [inst_4 : MeasureTheory.IsFiniteMeasure P.μ] (S : Causalean.Estimation.DTR.DTREstimationSystem P δ γ) {ε : ℝ}, S.StrictOverlap ε → MeasureTheory.Integrable (fun ω => S.factualY ω ^ 2) P.μ → MeasureTheory.MemLp S.μ₁_val 2 S.P_H₁","l":581,"f":"Causalean/Estimation/DTR/ScoreL2/Helpers.lean"},{"n":"Causalean.Estimation.DTR.DTREstimationSystem.ψ_seqDR","k":"def","m":"Causalean.Estimation.DTR.SeqDRMoment","a":"Estimation/DTR","t":1,"d":"This is the sequential doubly robust influence function evaluated at the true nuisances and target.","s":"{δ : Type} → [inst : MeasurableSpace δ] → [inst_1 : MeasurableSingletonClass δ] → {γ : Fin 2 → Type} → [inst_2 : (k : Fin 2) → MeasurableSpace (γ k)] → {P : Causalean.PO.POSystem} → [inst_3 : StandardBorelSpace P.Ω] → [inst_4 : MeasureTheory.IsFiniteMeasure P.μ] → Causalean.Estimation.DTR.DTREstimationSystem P δ γ → γ 0 × δ × γ 1 × δ × ℝ → ℝ","l":352,"f":"Causalean/Estimation/DTR/SeqDRMoment.lean"},{"n":"Causalean.Estimation.DTR.DTRNuisanceVec₂","k":"structure","m":"Causalean.Estimation.DTR.SeqDRMoment","a":"Estimation/DTR","t":1,"d":"A stagewise nuisance vector for two-period dynamic-treatment-regime estimation: the baseline outcome regression and treatment propensity at the first stage, the second-stage outcome regression and treatment propensity given the second-period history, together with the measurability of all four nuisance functions.","s":"(δ : Type) → (γ : Fin 2 → Type) → [inst : MeasurableSpace δ] → [MeasurableSingletonClass δ] → [(k : Fin 2) → MeasurableSpace (γ k)] → Type","l":91,"f":"Causalean/Estimation/DTR/SeqDRMoment.lean"},{"n":"Causalean.Estimation.DTR.DTRNuisanceVec₂.ext","k":"theorem","m":"Causalean.Estimation.DTR.SeqDRMoment","a":"Estimation/DTR","t":2,"d":"Two nuisance vectors are equal when all four stagewise components agree pointwise.","s":"∀ {δ : Type} [inst : MeasurableSpace δ] [inst_1 : MeasurableSingletonClass δ] {γ : Fin 2 → Type} [inst_2 : (k : Fin 2) → MeasurableSpace (γ k)] {η η' : Causalean.Estimation.DTR.DTRNuisanceVec₂ δ γ}, (∀ (s : γ 0), η.μ₀_fn s = η'.μ₀_fn s) → (∀ (s : γ 0), η.e₀_fn s = η'.e₀_fn s) → (∀ (h : γ 1 × δ × γ 0), η.μ₁_fn h = η'.μ₁_fn h) → (∀ (h : γ 1 × δ × γ 0), η.e₁_fn h = η'.e₁_fn h) → η = η'","l":168,"f":"Causalean/Estimation/DTR/SeqDRMoment.lean"},{"n":"Causalean.Estimation.DTR.DTRNuisanceVec₂.ext_iff","k":"theorem","m":"Causalean.Estimation.DTR.SeqDRMoment","a":"Estimation/DTR","t":2,"d":"","s":"∀ {δ : Type} [inst : MeasurableSpace δ] [inst_1 : MeasurableSingletonClass δ] {γ : Fin 2 → Type} [inst_2 : (k : Fin 2) → MeasurableSpace (γ k)] {η η' : Causalean.Estimation.DTR.DTRNuisanceVec₂ δ γ}, η = η' ↔ (∀ (s : γ 0), η.μ₀_fn s = η'.μ₀_fn s) ∧ (∀ (s : γ 0), η.e₀_fn s = η'.e₀_fn s) ∧ (∀ (h : γ 1 × δ × γ 0), η.μ₁_fn h = η'.μ₁_fn h) ∧ ∀ (h : γ 1 × δ × γ 0), η.e₁_fn h = η'.e₁_fn h","l":169,"f":"Causalean/Estimation/DTR/SeqDRMoment.lean"},{"n":"Causalean.Estimation.DTR.DTRNuisanceVec₂.instAdd","k":"instance","m":"Causalean.Estimation.DTR.SeqDRMoment","a":"Estimation/DTR","t":2,"d":"Addition of nuisance vectors is performed component by component.","s":"{δ : Type} → [inst : MeasurableSpace δ] → [inst_1 : MeasurableSingletonClass δ] → {γ : Fin 2 → Type} → [inst_2 : (k : Fin 2) → MeasurableSpace (γ k)] → Add (Causalean.Estimation.DTR.DTRNuisanceVec₂ δ γ)","l":127,"f":"Causalean/Estimation/DTR/SeqDRMoment.lean"},{"n":"Causalean.Estimation.DTR.DTRNuisanceVec₂.instAddCommGroup","k":"instance","m":"Causalean.Estimation.DTR.SeqDRMoment","a":"Estimation/DTR","t":2,"d":"The nuisance vectors form an additive commutative group under componentwise operations.","s":"{δ : Type} → [inst : MeasurableSpace δ] → [inst_1 : MeasurableSingletonClass δ] → {γ : Fin 2 → Type} → [inst_2 : (k : Fin 2) → MeasurableSpace (γ k)] → AddCommGroup (Causalean.Estimation.DTR.DTRNuisanceVec₂ δ γ)","l":184,"f":"Causalean/Estimation/DTR/SeqDRMoment.lean"},{"n":"Causalean.Estimation.DTR.DTRNuisanceVec₂.instModuleReal","k":"instance","m":"Causalean.Estimation.DTR.SeqDRMoment","a":"Estimation/DTR","t":2,"d":"The nuisance vectors form a real module under componentwise scalar multiplication.","s":"{δ : Type} → [inst : MeasurableSpace δ] → [inst_1 : MeasurableSingletonClass δ] → {γ : Fin 2 → Type} → [inst_2 : (k : Fin 2) → MeasurableSpace (γ k)] → Module ℝ (Causalean.Estimation.DTR.DTRNuisanceVec₂ δ γ)","l":234,"f":"Causalean/Estimation/DTR/SeqDRMoment.lean"},{"n":"Causalean.Estimation.DTR.DTRNuisanceVec₂.instNeg","k":"instance","m":"Causalean.Estimation.DTR.SeqDRMoment","a":"Estimation/DTR","t":2,"d":"Negation of a nuisance vector is performed component by component.","s":"{δ : Type} → [inst : MeasurableSpace δ] → [inst_1 : MeasurableSingletonClass δ] → {γ : Fin 2 → Type} → [inst_2 : (k : Fin 2) → MeasurableSpace (γ k)] → Neg (Causalean.Estimation.DTR.DTRNuisanceVec₂ δ γ)","l":139,"f":"Causalean/Estimation/DTR/SeqDRMoment.lean"},{"n":"Causalean.Estimation.DTR.DTRNuisanceVec₂.instSMulReal","k":"instance","m":"Causalean.Estimation.DTR.SeqDRMoment","a":"Estimation/DTR","t":2,"d":"Real scalar multiplication of a nuisance vector is performed component by component.","s":"{δ : Type} → [inst : MeasurableSpace δ] → [inst_1 : MeasurableSingletonClass δ] → {γ : Fin 2 → Type} → [inst_2 : (k : Fin 2) → MeasurableSpace (γ k)] → SMul ℝ (Causalean.Estimation.DTR.DTRNuisanceVec₂ δ γ)","l":158,"f":"Causalean/Estimation/DTR/SeqDRMoment.lean"},{"n":"Causalean.Estimation.DTR.DTRNuisanceVec₂.instSub","k":"instance","m":"Causalean.Estimation.DTR.SeqDRMoment","a":"Estimation/DTR","t":2,"d":"Subtraction of nuisance vectors is performed component by component.","s":"{δ : Type} → [inst : MeasurableSpace δ] → [inst_1 : MeasurableSingletonClass δ] → {γ : Fin 2 → Type} → [inst_2 : (k : Fin 2) → MeasurableSpace (γ k)] → Sub (Causalean.Estimation.DTR.DTRNuisanceVec₂ δ γ)","l":146,"f":"Causalean/Estimation/DTR/SeqDRMoment.lean"},{"n":"Causalean.Estimation.DTR.DTRNuisanceVec₂.instZero","k":"instance","m":"Causalean.Estimation.DTR.SeqDRMoment","a":"Estimation/DTR","t":2,"d":"The zero nuisance vector sets every stagewise regression and propensity component to zero.","s":"{δ : Type} → [inst : MeasurableSpace δ] → [inst_1 : MeasurableSingletonClass δ] → {γ : Fin 2 → Type} → [inst_2 : (k : Fin 2) → MeasurableSpace (γ k)] → Zero (Causalean.Estimation.DTR.DTRNuisanceVec₂ δ γ)","l":121,"f":"Causalean/Estimation/DTR/SeqDRMoment.lean"},{"n":"Causalean.Estimation.DTR.dml_DTR_estimator","k":"def","m":"Causalean.Estimation.DTR.DML","a":"Estimation/DTR","t":1,"d":"One-shot DML / sequential DR estimator of the DTR effect (def:est-dml-dtr).","s":"{P : Causalean.PO.POSystem} → {δ : Type} → {γ : Fin 2 → Type} → [inst : MeasurableSpace δ] → [inst_1 : MeasurableSingletonClass δ] → [inst_2 : (k : Fin 2) → MeasurableSpace (γ k)] → [inst_3 : StandardBorelSpace P.Ω] → [inst_4 : MeasureTheory.IsFiniteMeasure P.μ] → (S : Causalean.Estimation.DTR.DTREstimationSystem P δ γ) → (sample : Causalean.Stat.IIDSample P.Ω (γ 0 × δ × γ 1 × δ × ℝ) P.μ S.P_Z) → Causalean.Stat.OneShotSplit sample → (ℕ → P.Ω → Causalean.Estimation.DTR.DTRNuisanceVec₂ δ γ) → ℕ → P.Ω → ℝ","l":358,"f":"Causalean/Estimation/DTR/DML.lean"},{"n":"Causalean.Estimation.DTR.dml_DTR_isAsymLinear","k":"theorem","m":"Causalean.Estimation.DTR.DML","a":"Estimation/DTR","t":1,"d":"Asymptotic linearity of the one-shot DML DTR (n = 2) estimator — thm:est-dml-dtr-al. Assuming the DTR backdoor identification conditions hold, the theorem shows that the one-shot double/debiased-machine-learning estimator of the two-period dynamic-treatment-regime effect is asymptotically linear, with influence function ψ_seqDR, around the true effect θ₀. The population-truth propensities obey strict overlap — some ε ∈ (0, 1/2] sandwiches them a.s. at both stages, restated pointwise on the value-space propensity functions e₀_val, e₁_val; the factual outcome and every counterfactual outcome under a fixed regime are square-integrable; and the one-shot sample split's auxiliary-fold fraction |B(n)|/n converges to some c ∈ (0, 1). For every horizon n, the stage-0 nuisance learners μ̂₀, ê₀ and the stage-1 learners μ̂₁, ê₁ are each jointly measurable in the sample outcome and the covariate history at stage 0 and likewise at stage 1; the fitted propensities satisfy the same strict-overlap bound ε pointwise at both stages; each learner lies in L² of the covariate-history distribution at stage 0 and likewise at stage 1; and, viewed as a function of the sample outcome alone, each learner is measurable with respect to the auxiliary training fold's σ-algebra at stage 0 and likewise at stage 1, and jointly with the covariate in uncurried form at stage 0 and at stage 1. Finally, each stagewise estimation error converges to zero in L² at rate o_p(1), and every cross-stage product of an outcome-regression error with a propensity error vanishes at the doubly-robust rate o_p(n^{-1/2}).","s":"∀ {P : Causalean.PO.POSystem} {δ : Type} {γ : Fin 2 → Type} [inst : MeasurableSpace δ] [inst_1 : MeasurableSingletonClass δ] [inst_2 : (k : Fin 2) → MeasurableSpace (γ k)] [inst_3 : StandardBorelSpace P.Ω] [inst_4 : MeasureTheory.IsFiniteMeasure P.μ] (S : Causalean.Estimation.DTR.DTREstimationSystem P δ γ) {ε : ℝ}, S.Assumptions → S.StrictOverlap ε → ((∀ (s₀ : γ 0), ε ≤ S.e₀_val s₀ ∧ S.e₀_val s₀ ≤ 1 - ε) ∧ ∀ (h : γ 1 × δ × γ 0), ε ≤ S.e₁_val h ∧ S.e₁_val h ≤ 1 - ε) → MeasureTheory.Integrable (fun ω => S.factualY ω ^ 2) P.μ → (∀ (dbar : Fin 2 → δ), MeasureTheory.Integrable (fun ω => S.Y_of dbar ω ^ 2) P.μ) → ∀ (sample : Causalean.Stat.IIDSample P.Ω (γ 0 × δ × γ 1 × δ × ℝ) P.μ S.P_Z) (split : Causalean.Stat.OneShotSplit sample) {c : ℝ}, 0 < c → c < 1 → Filter.Tendsto (fun n => ↑(split.foldB n).card / ↑n) Filter.atTop (nhds c) → ∀ (μ₀_hat e₀_hat : ℕ → P.Ω → γ 0 → ℝ) (μ₁_hat e₁_hat : ℕ → P.Ω → γ 1 × δ × γ 0 → ℝ) (h_mu0_meas : ∀ (n : ℕ), Measurable fun p => μ₀_hat n p.1 p.2) (h_e0_meas : ∀ (n : ℕ), Measurable fun p => e₀_hat n p.1 p.2) (h_mu1_meas : ∀ (n : ℕ), Measurable fun p => μ₁_hat n p.1 p.2) (h_e1_meas : ∀ (n : ℕ), Measurable fun p => e₁_hat n p.1 p.2), (∀ (n : ℕ) (ω : P.Ω), (∀ (s₀ : γ 0), ε ≤ e₀_hat n ω s₀ ∧ e₀_hat n ω s₀ ≤ 1 - ε) ∧ ∀ (h : γ 1 × δ × γ 0), ε ≤ e₁_hat n ω h ∧ e₁_hat n ω h ≤ 1 - ε) → (∀ (n : ℕ) (ω : P.Ω), MeasureTheory.MemLp (fun s₀ => μ₀_hat n ω s₀) 2 S.P_H₀) → (∀ (n : ℕ) (ω : P.Ω), MeasureTheory.MemLp (fun s₀ => e₀_hat n ω s₀) 2 S.P_H₀) → (∀ (n : ℕ) (ω : P.Ω), MeasureTheory.MemLp (fun h => μ₁_hat n ω h) 2 S.P_H₁) → (∀ (n : ℕ) (ω : P.Ω), MeasureTheory.MemLp (fun h => e₁_hat n ω h) 2 S.P_H₁) → (∀ (n : ℕ), Measurable (μ₀_hat n)) → (∀ (n : ℕ), Measurable (e₀_hat n)) → (∀ (n : ℕ), Measurable (μ₁_hat n)) → (∀ (n : ℕ), Measurable (e₁_hat n)) → (∀ (n : ℕ), Measurable fun p => μ₀_hat n p.1 p.2) → (∀ (n : ℕ), Measurable fun p => e₀_hat n p.1 p.2) → (∀ (n : ℕ), Measurable fun p => μ₁_hat n p.1 p.2) → (∀ (n : ℕ), Measurable fun p => e₁_hat n p.1 p.2) → Causalean.Stat.IsLittleOp (fun n ω => (MeasureTheory.eLpNorm (fun s₀ => μ₀_hat n ω s₀ - S.μ₀_val s₀) 2 S.P_H₀).toReal) (fun x => 1) P.μ → Causalean.Stat.IsLittleOp (fun n ω => (MeasureTheory.eLpNorm (fun h => μ₁_hat n ω h - S.μ₁_val h) 2 S.P_H₁).toReal) (fun x => 1) P.μ → Causalean.Stat.IsLittleOp (fun n ω => (MeasureTheory.eLpNorm (fun s₀ => e₀_hat n ω s₀ - S.e₀_val s₀) 2 S.P_H₀).toReal) (fun x => 1) P.μ → Causalean.Stat.IsLittleOp (fun n ω => (MeasureTheory.eLpNorm (fun h => e₁_hat n ω h - S.e₁_val h) 2 S.P_H₁).toReal) (fun x => 1) P.μ → Causalean.Stat.IsLittleOp (fun n ω => (MeasureTheory.eLpNorm (fun s₀ => μ₀_hat n ω s₀ - S.μ₀_val s₀) 2 S.P_H₀).toReal * (MeasureTheory.eLpNorm (fun s₀ => e₀_hat n ω s₀ - S.e₀_val s₀) 2 S.P_H₀).toReal) (fun n => ↑n ^ (-(1 / 2))) P.μ → Causalean.Stat.IsLittleOp (fun n ω => (MeasureTheory.eLpNorm (fun h => μ₁_hat n ω h - S.μ₁_val h) 2 S.P_H₁).toReal * (MeasureTheory.eLpNorm (fun h => e₁_hat n ω h - S.e₁_val h) 2 S.P_H₁).toReal) (fun n => ↑n ^ (-(1 / 2))) P.μ → Causalean.Stat.IsLittleOp (fun n ω => (MeasureTheory.eLpNorm (fun s₀ => μ₀_hat n ω s₀ - S.μ₀_val s₀) 2 S.P_H₀).toReal * (MeasureTheory.eLpNorm (fun h => e₁_hat n ω h - S.e₁_val h) 2 S.P_H₁).toReal) (fun n => ↑n ^ (-(1 / 2))) P.μ → Causalean.Stat.IsLittleOp (fun n ω => (MeasureTheory.eLpNorm (fun h => μ₁_hat n ω h - S.μ₁_val h) 2 S.P_H₁).toReal * (MeasureTheory.eLpNorm (fun s₀ => e₀_hat n ω s₀ - S.e₀_val s₀) 2 S.P_H₀).toReal) (fun n => ↑n ^ (-(1 / 2))) P.μ → Causalean.Stat.IsAsymLinear (Causalean.Estimation.DTR.dml_DTR_estimator S sample split fun n ω => { μ₀_fn := μ₀_hat n ω, e₀_fn := e₀_hat n ω, μ₁_fn := μ₁_hat n ω, e₁_fn := e₁_hat n ω, μ₀_meas := Measurable.comp (h_mu0_meas n) (Measurable.prodMk measurable_const measurable_id), e₀_meas := Measurable.comp (h_e0_meas n) (Measurable.prodMk measurable_const measurable_id), μ₁_meas := Measurable.comp (h_mu1_meas n) (Measurable.prodMk measurable_const measurable_id), e₁_meas := Measurable.comp (h_e1_meas n) (Measurable.prodMk measurable_const measurable_id) }) S.θ₀ S.ψ_seqDR sample split.foldB","l":391,"f":"Causalean/Estimation/DTR/DML.lean"},{"n":"Causalean.Estimation.DTR.exists_stage0_lift","k":"theorem","m":"Causalean.Estimation.DTR.Constructor","a":"Estimation/DTR","t":1,"d":"Doob–Dynkin factorization through the stage-0 state. Consider a two-stage dynamic-treatment-regime potential-outcome system and a real-valued function g on the sample space. If g is measurable with respect to the σ-algebra generated by the stage-0 history — here just the initial state S₀, then g factors through the stage-0 state: there is a measurable function f on the stage-0 state space such that g equals f composed with the stage-0 state map.","s":"∀ {P : Causalean.PO.POSystem} {δ : Type} {γ : Fin 2 → Type} [inst : MeasurableSpace δ] [inst_1 : MeasurableSingletonClass δ] [inst_2 : (k : Fin 2) → MeasurableSpace (γ k)] (S : Causalean.PO.PODTRSystem P 2 δ γ) (g : P.Ω → ℝ), Measurable g → ∃ f, Measurable f ∧ g = fun ω => f (S.factualS ⟨0, of_decide_eq_true (id (Eq.refl true))⟩ ω)","l":39,"f":"Causalean/Estimation/DTR/Constructor.lean"},{"n":"Causalean.Estimation.DTR.exists_stage1_lift","k":"theorem","m":"Causalean.Estimation.DTR.Constructor","a":"Estimation/DTR","t":1,"d":"Doob–Dynkin factorization through the stage-1 history. Consider a two-stage dynamic-treatment-regime potential-outcome system and a real-valued function g on the sample space. If g is measurable with respect to the σ-algebra generated by the stage-1 history — the current state, the previous treatment, and the previous state, then g factors through that history tuple: there is a measurable function f on the stage-1 history space such that g equals f composed with the map recording the current state, the previous treatment, and the previous state.","s":"∀ {P : Causalean.PO.POSystem} {δ : Type} {γ : Fin 2 → Type} [inst : MeasurableSpace δ] [inst_1 : MeasurableSingletonClass δ] [inst_2 : (k : Fin 2) → MeasurableSpace (γ k)] (S : Causalean.PO.PODTRSystem P 2 δ γ) (g : P.Ω → ℝ), Measurable g → ∃ f, Measurable f ∧ g = fun ω => f (S.factualS ⟨1, of_decide_eq_true (id (Eq.refl true))⟩ ω, S.factualD ⟨0, of_decide_eq_true (id (Eq.refl true))⟩ ω, S.factualS ⟨0, of_decide_eq_true (id (Eq.refl true))⟩ ω)","l":69,"f":"Causalean/Estimation/DTR/Constructor.lean"},{"n":"Causalean.Estimation.DTR.histH₁","k":"def","m":"Causalean.Estimation.DTR.SeqDRMoment","a":"Estimation/DTR","t":1,"d":"This projection returns the stage-1 history in the order used by the DTR history bundle.","s":"{δ : Type} → {γ : Fin 2 → Type} → γ 0 × δ × γ 1 × δ × ℝ → γ 1 × δ × γ 0","l":70,"f":"Causalean/Estimation/DTR/SeqDRMoment.lean"},{"n":"Causalean.Estimation.DTR.indEq","k":"def","m":"Causalean.Estimation.DTR.SeqDRMoment","a":"Estimation/DTR","t":1,"d":"This function is the real-valued indicator that two discrete treatments are equal.","s":"{δ : Type} → δ → δ → ℝ","l":76,"f":"Causalean/Estimation/DTR/SeqDRMoment.lean"},{"n":"Causalean.Estimation.DTR.projD₀","k":"def","m":"Causalean.Estimation.DTR.SeqDRMoment","a":"Estimation/DTR","t":1,"d":"This projection returns the first treatment from the observed two-stage data tuple.","s":"{δ : Type} → {γ : Fin 2 → Type} → γ 0 × δ × γ 1 × δ × ℝ → δ","l":58,"f":"Causalean/Estimation/DTR/SeqDRMoment.lean"},{"n":"Causalean.Estimation.DTR.projD₁","k":"def","m":"Causalean.Estimation.DTR.SeqDRMoment","a":"Estimation/DTR","t":1,"d":"This projection returns the second treatment from the observed two-stage data tuple.","s":"{δ : Type} → {γ : Fin 2 → Type} → γ 0 × δ × γ 1 × δ × ℝ → δ","l":64,"f":"Causalean/Estimation/DTR/SeqDRMoment.lean"},{"n":"Causalean.Estimation.DTR.projS₀","k":"def","m":"Causalean.Estimation.DTR.SeqDRMoment","a":"Estimation/DTR","t":1,"d":"This projection returns the initial state from the observed two-stage data tuple.","s":"{δ : Type} → {γ : Fin 2 → Type} → γ 0 × δ × γ 1 × δ × ℝ → γ 0","l":55,"f":"Causalean/Estimation/DTR/SeqDRMoment.lean"},{"n":"Causalean.Estimation.DTR.projS₁","k":"def","m":"Causalean.Estimation.DTR.SeqDRMoment","a":"Estimation/DTR","t":1,"d":"This projection returns the second state from the observed two-stage data tuple.","s":"{δ : Type} → {γ : Fin 2 → Type} → γ 0 × δ × γ 1 × δ × ℝ → γ 1","l":61,"f":"Causalean/Estimation/DTR/SeqDRMoment.lean"},{"n":"Causalean.Estimation.DTR.projY","k":"def","m":"Causalean.Estimation.DTR.SeqDRMoment","a":"Estimation/DTR","t":1,"d":"This projection returns the outcome from the observed two-stage data tuple.","s":"{δ : Type} → {γ : Fin 2 → Type} → γ 0 × δ × γ 1 × δ × ℝ → ℝ","l":67,"f":"Causalean/Estimation/DTR/SeqDRMoment.lean"},{"n":"Causalean.Estimation.DTR.seqDRGeneralMoment","k":"def","m":"Causalean.Estimation.DTR.DTRInstance","a":"Estimation/DTR","t":1,"d":"Sequential DR (DTR, n = 2) instance of the abstract GeneralMoment.","s":"{P : Causalean.PO.POSystem} → {δ : Type} → {γ : Fin 2 → Type} → [inst : MeasurableSpace δ] → [inst_1 : MeasurableSingletonClass δ] → [inst_2 : (k : Fin 2) → MeasurableSpace (γ k)] → [inst_3 : StandardBorelSpace P.Ω] → [inst_4 : MeasureTheory.IsFiniteMeasure P.μ] → (S : Causalean.Estimation.DTR.DTREstimationSystem P δ γ) → {ε : ℝ} → ((∀ (s₀ : γ 0), ε ≤ S.e₀_val s₀ ∧ S.e₀_val s₀ ≤ 1 - ε) ∧ ∀ (h : γ 1 × δ × γ 0), ε ≤ S.e₁_val h ∧ S.e₁_val h ≤ 1 - ε) → Causalean.Estimation.OrthogonalMoments.GeneralMoment P.Ω P.μ (γ 0 × δ × γ 1 × δ × ℝ) S.P_Z (Causalean.Estimation.DTR.DTRNuisanceVec₂ δ γ)","l":62,"f":"Causalean/Estimation/DTR/DTRInstance.lean"},{"n":"Causalean.Estimation.DTR.seqDRMoment","k":"def","m":"Causalean.Estimation.DTR.SeqDRMoment","a":"Estimation/DTR","t":1,"d":"This is the explicit two-stage sequential doubly robust moment for a fixed treatment regime.","s":"{δ : Type} → [inst : MeasurableSpace δ] → [inst_1 : MeasurableSingletonClass δ] → {γ : Fin 2 → Type} → [inst_2 : (k : Fin 2) → MeasurableSpace (γ k)] → (Fin 2 → δ) → γ 0 × δ × γ 1 × δ × ℝ → Causalean.Estimation.DTR.DTRNuisanceVec₂ δ γ → ℝ → ℝ","l":307,"f":"Causalean/Estimation/DTR/SeqDRMoment.lean"},{"n":"Causalean.Estimation.DTR.seqDR_bilinearRem","k":"theorem","m":"Causalean.Estimation.DTR.DTRInstance","a":"Estimation/DTR","t":2,"d":"Sequential DR (DTR) satisfies BilinearRemainder with constant seqDR_rem_const ε.","s":"∀ {P : Causalean.PO.POSystem} {δ : Type} {γ : Fin 2 → Type} [inst : MeasurableSpace δ] [inst_1 : MeasurableSingletonClass δ] [inst_2 : (k : Fin 2) → MeasurableSpace (γ k)] [inst_3 : StandardBorelSpace P.Ω] [inst_4 : MeasureTheory.IsFiniteMeasure P.μ] (S : Causalean.Estimation.DTR.DTREstimationSystem P δ γ) {ε : ℝ} (h_e_pointwise : (∀ (s₀ : γ 0), ε ≤ S.e₀_val s₀ ∧ S.e₀_val s₀ ≤ 1 - ε) ∧ ∀ (h : γ 1 × δ × γ 0), ε ≤ S.e₁_val h ∧ S.e₁_val h ≤ 1 - ε), S.StrictOverlap ε → S.Assumptions → MeasureTheory.Integrable (fun ω => S.factualY ω ^ 2) P.μ → (∀ (dbar : Fin 2 → δ), MeasureTheory.Integrable (fun ω => S.Y_of dbar ω ^ 2) P.μ) → (∀ η ∈ Causalean.Estimation.DTR.DTREstimationSystem.H_ε ε, MeasureTheory.MemLp (fun s₀ => η.μ₀_fn s₀ - S.μ₀_val s₀) 2 S.P_H₀ ∧ MeasureTheory.MemLp (fun h => η.μ₁_fn h - S.μ₁_val h) 2 S.P_H₁ ∧ MeasureTheory.MemLp (fun s₀ => η.e₀_fn s₀ - S.e₀_val s₀) 2 S.P_H₀ ∧ MeasureTheory.MemLp (fun h => η.e₁_fn h - S.e₁_val h) 2 S.P_H₁) → ∃ C, Causalean.Estimation.OrthogonalMoments.BilinearRemainder (Causalean.Estimation.DTR.seqDRGeneralMoment S h_e_pointwise) C","l":109,"f":"Causalean/Estimation/DTR/DTRInstance.lean"},{"n":"Causalean.Estimation.DTR.seqDR_dml_isAsymLinear","k":"theorem","m":"Causalean.Estimation.DTR.DTRInstance","a":"Estimation/DTR","t":1,"d":"Headline sequential DR (DTR) DML asymptotic-linearity theorem, derived from the abstract dml_chernozhukov_asymptoticLinear in Estimation/OrthogonalMoments/DMLChernozhukov.lean. Fix a dynamic-treatment-regime estimation system with strict two-stage propensity overlap and satisfying the DTR identification assumptions, and suppose the factual outcome and every counterfactual outcome under a fixed treatment history have finite second moment. Given an i.i.d. sample together with a one-shot cross-fitting split whose estimation-fold share converges to some constant strictly between $0$ and $1$, and a sequence of nuisance estimators η_hat that remain in the $ε$-overlap ball, with stagewise outcome-regression and propensity errors that are square-integrable, such that the resulting moment function is measurable against the sample and each cross-fitting fold, and is both integrable and square-integrable, and such that the individual L² nuisance-error rates vanish while their product is $o_P(n^{-1/2})$, then the resulting Chernozhukov one-step DML estimator is asymptotically linear at the true sequential-DR parameter, with influence function the sequential doubly-robust score evaluated at the truth.","s":"∀ {P : Causalean.PO.POSystem} {δ : Type} {γ : Fin 2 → Type} [inst : MeasurableSpace δ] [inst_1 : MeasurableSingletonClass δ] [inst_2 : (k : Fin 2) → MeasurableSpace (γ k)] [inst_3 : StandardBorelSpace P.Ω] [inst_4 : MeasureTheory.IsFiniteMeasure P.μ] (S : Causalean.Estimation.DTR.DTREstimationSystem P δ γ) {ε : ℝ} (h_e_pointwise : (∀ (s₀ : γ 0), ε ≤ S.e₀_val s₀ ∧ S.e₀_val s₀ ≤ 1 - ε) ∧ ∀ (h : γ 1 × δ × γ 0), ε ≤ S.e₁_val h ∧ S.e₁_val h ≤ 1 - ε), S.StrictOverlap ε → S.Assumptions → MeasureTheory.Integrable (fun ω => S.factualY ω ^ 2) P.μ → (∀ (dbar : Fin 2 → δ), MeasureTheory.Integrable (fun ω => S.Y_of dbar ω ^ 2) P.μ) → ∀ (sample : Causalean.Stat.IIDSample P.Ω (γ 0 × δ × γ 1 × δ × ℝ) P.μ S.P_Z) (split : Causalean.Stat.OneShotSplit sample) {c : ℝ}, 0 < c → c < 1 → Filter.Tendsto (fun n => ↑(split.foldB n).card / ↑n) Filter.atTop (nhds c) → ∀ (η_hat : ℕ → P.Ω → Causalean.Estimation.DTR.DTRNuisanceVec₂ δ γ), (∀ (n : ℕ) (ω : P.Ω), η_hat n ω ∈ Causalean.Estimation.DTR.DTREstimationSystem.H_ε ε) → (∀ (n : ℕ) (ω : P.Ω), MeasureTheory.MemLp (fun s₀ => (η_hat n ω).μ₀_fn s₀ - S.μ₀_val s₀) 2 S.P_H₀) → (∀ (n : ℕ) (ω : P.Ω), MeasureTheory.MemLp (fun h => (η_hat n ω).μ₁_fn h - S.μ₁_val h) 2 S.P_H₁) → (∀ (n : ℕ) (ω : P.Ω), MeasureTheory.MemLp (fun s₀ => (η_hat n ω).e₀_fn s₀ - S.e₀_val s₀) 2 S.P_H₀) → (∀ (n : ℕ) (ω : P.Ω), MeasureTheory.MemLp (fun h => (η_hat n ω).e₁_fn h - S.e₁_val h) 2 S.P_H₁) → (∀ (n : ℕ), Measurable fun p => S.seqDRMomentFunctional (η_hat n p.1) p.2 S.θ₀) → (∀ (n : ℕ), Measurable fun ω z => S.seqDRMomentFunctional (η_hat n ω) z S.θ₀) → (∀ (n : ℕ), Measurable fun p => S.seqDRMomentFunctional (η_hat n p.1) p.2 S.θ₀) → (∀ (n : ℕ) (ω : P.Ω), MeasureTheory.Integrable (fun z => S.seqDRMomentFunctional (η_hat n ω) z S.θ₀) S.P_Z) → (∀ (n : ℕ) (ω : P.Ω), MeasureTheory.Integrable (fun z => S.seqDRMomentFunctional (η_hat n ω) z S.θ₀ ^ 2) S.P_Z) → Causalean.Stat.IsLittleOp (fun n ω => ↑((Causalean.Estimation.DTR.seqDRGeneralMoment S h_e_pointwise).ρ₁ (η_hat n ω) S.η₀)) (fun x => 1) P.μ → Causalean.Stat.IsLittleOp (fun n ω => ↑((Causalean.Estimation.DTR.seqDRGeneralMoment S h_e_pointwise).ρ₂ (η_hat n ω) S.η₀)) (fun x => 1) P.μ → Causalean.Stat.IsLittleOp (fun n ω => ↑((Causalean.Estimation.DTR.seqDRGeneralMoment S h_e_pointwise).ρ₁ (η_hat n ω) S.η₀) * ↑((Causalean.Estimation.DTR.seqDRGeneralMoment S h_e_pointwise).ρ₂ (η_hat n ω) S.η₀)) (fun n => ↑n ^ (-(1 / 2))) P.μ → Causalean.Stat.IsAsymLinear (Causalean.Estimation.OrthogonalMoments.dmlChernozhukovEstimator (Causalean.Estimation.DTR.seqDRGeneralMoment S h_e_pointwise) sample split η_hat) S.θ₀ (fun z => -(Causalean.Estimation.DTR.seqDRGeneralMoment S h_e_pointwise).J₀_inv * S.seqDRMomentFunctional S.η₀ z S.θ₀) sample split.foldB","l":145,"f":"Causalean/Estimation/DTR/DTRInstance.lean"},{"n":"Causalean.Estimation.DTR.seqDR_meanZero","k":"theorem","m":"Causalean.Estimation.DTR.DTRInstance","a":"Estimation/DTR","t":2,"d":"Sequential DR (DTR) satisfies MeanZero.","s":"∀ {P : Causalean.PO.POSystem} {δ : Type} {γ : Fin 2 → Type} [inst : MeasurableSpace δ] [inst_1 : MeasurableSingletonClass δ] [inst_2 : (k : Fin 2) → MeasurableSpace (γ k)] [inst_3 : StandardBorelSpace P.Ω] [inst_4 : MeasureTheory.IsFiniteMeasure P.μ] (S : Causalean.Estimation.DTR.DTREstimationSystem P δ γ) {ε : ℝ} (h_e_pointwise : (∀ (s₀ : γ 0), ε ≤ S.e₀_val s₀ ∧ S.e₀_val s₀ ≤ 1 - ε) ∧ ∀ (h : γ 1 × δ × γ 0), ε ≤ S.e₁_val h ∧ S.e₁_val h ≤ 1 - ε), S.StrictOverlap ε → S.Assumptions → MeasureTheory.Integrable (fun ω => S.factualY ω ^ 2) P.μ → Causalean.Estimation.OrthogonalMoments.MeanZero (Causalean.Estimation.DTR.seqDRGeneralMoment S h_e_pointwise)","l":96,"f":"Causalean/Estimation/DTR/DTRInstance.lean"},{"n":"Causalean.Estimation.Efficiency.BackdoorEstimationSystem.aipw_orthogonal_propensity_score","k":"theorem","m":"Causalean.Estimation.Efficiency.ATETangent","a":"Estimation/Efficiency","t":2,"d":"The role of the propensity score (Hahn).","s":"∀ {P : Causalean.PO.POSystem} {γ : Type u_1} [inst : MeasurableSpace γ] [inst_1 : StandardBorelSpace P.Ω] [inst_2 : MeasureTheory.IsFiniteMeasure P.μ] (S : Causalean.Estimation.ATE.BackdoorEstimationSystem P γ), S.Assumptions → ∀ {ε : ℝ}, S.StrictOverlap ε → ∀ (α : γ → ℝ), Measurable α → MeasureTheory.Integrable (fun ω => (S.μ_val true (S.factualX ω) - S.μ_val false (S.factualX ω) - S.θ₀) * α (S.factualX ω) * (S.dVar.indicator true ω - S.e_val (S.factualX ω))) P.μ → MeasureTheory.Integrable (fun ω => (fun x => α x * (1 - S.e_val x) / S.e_val x) (S.factualX ω) * (S.dVar.indicator true ω * (S.factualY ω - S.μ_val true (S.factualX ω)))) P.μ → MeasureTheory.Integrable (fun ω => (fun x => α x * S.e_val x / (1 - S.e_val x)) (S.factualX ω) * (S.dVar.indicator false ω * (S.factualY ω - S.μ_val false (S.factualX ω)))) P.μ → ∫ (z : γ × Bool × ℝ), S.ψ_AIPW z * (α (Causalean.Estimation.ATE.BackdoorEstimationSystem.projX z) * (Causalean.Estimation.ATE.BackdoorEstimationSystem.indA z - S.e_val (Causalean.Estimation.ATE.BackdoorEstimationSystem.projX z))) ∂S.P_Z = 0","l":196,"f":"Causalean/Estimation/Efficiency/ATETangent.lean"},{"n":"Causalean.Estimation.Efficiency.BackdoorEstimationSystem.propensity_score_residual_integral_zero","k":"theorem","m":"Causalean.Estimation.Efficiency.ATETangent","a":"Estimation/Efficiency","t":2,"d":"Any covariate-measurable multiplier has zero integral against the propensity-score residual.","s":"∀ {P : Causalean.PO.POSystem} {γ : Type u_1} [inst : MeasurableSpace γ] [inst_1 : StandardBorelSpace P.Ω] [inst_2 : MeasureTheory.IsFiniteMeasure P.μ] (S : Causalean.Estimation.ATE.BackdoorEstimationSystem P γ) (h : γ → ℝ), Measurable h → MeasureTheory.Integrable (fun ω => h (S.factualX ω) * (S.dVar.indicator true ω - S.e_val (S.factualX ω))) P.μ → ∫ (ω : P.Ω), h (S.factualX ω) * (S.dVar.indicator true ω - S.e_val (S.factualX ω)) ∂P.μ = 0","l":82,"f":"Causalean/Estimation/Efficiency/ATETangent.lean"},{"n":"Causalean.Estimation.Efficiency.IsGradient","k":"def","m":"Causalean.Estimation.Efficiency.TangentProjection","a":"Estimation/Efficiency","t":1,"d":"ψ is a *gradient* of g relative to the tangent space T when its inner product against every tangent direction matches that of g. In semiparametric models g is a reference influence function and the gradients are exactly the influence functions of regular asymptotically linear estimators.","s":"{H : Type u_1} → [inst : NormedAddCommGroup H] → [inst_1 : InnerProductSpace ℝ H] → Submodule ℝ H → H → H → Prop","l":44,"f":"Causalean/Estimation/Efficiency/TangentProjection.lean"},{"n":"Causalean.Estimation.Efficiency.IsPathwiseGradient","k":"def","m":"Causalean.Estimation.Efficiency.PathwiseGradient","a":"Estimation/Efficiency","t":1,"d":"g : H is a pathwise gradient of the functional ψ : 𝓛 → ℝ at P₀: along every regular submodel, t ↦ ψ (path t) is differentiable at 0 with derivative ⟪g, score⟫. This is the abstract semiparametric characterization of an influence function as the pathwise derivative of the target functional.","s":"{𝓛 : Type u_1} → {H : Type u_2} → [inst : NormedAddCommGroup H] → [InnerProductSpace ℝ H] → H → 𝓛 → (𝓛 → ℝ) → H → Prop","l":94,"f":"Causalean/Estimation/Efficiency/PathwiseGradient.lean"},{"n":"Causalean.Estimation.Efficiency.IsTangentSpace","k":"structure","m":"Causalean.Estimation.Efficiency.PathwiseGradient","a":"Estimation/Efficiency","t":1,"d":"A candidate closed subspace T of the Hilbert space is a genuine tangent space for the regular submodels through the base law P₀ when every submodel's score lies in T and T is contained in the closed linear span of all submodel scores — together these force T to equal that closed span, the tangent space generated by the available parametric perturbations.","s":"{𝓛 : Type u_1} → {H : Type u_2} → [inst : NormedAddCommGroup H] → [inst_1 : InnerProductSpace ℝ H] → H → 𝓛 → Submodule ℝ H → Prop","l":80,"f":"Causalean/Estimation/Efficiency/PathwiseGradient.lean"},{"n":"Causalean.Estimation.Efficiency.RegularSubmodel","k":"structure","m":"Causalean.Estimation.Efficiency.PathwiseGradient","a":"Estimation/Efficiency","t":1,"d":"A regular submodel through the base law P₀ bundles a one-parameter path of laws, indexed by a real perturbation parameter, that passes through P₀ at parameter zero, together with its tangent direction, an L² score vector required to be mean-zero, i.e. orthogonal to the distinguished vector one.","s":"{𝓛 : Type u_1} → {H : Type u_2} → [inst : NormedAddCommGroup H] → [InnerProductSpace ℝ H] → H → 𝓛 → Type (max u_1 u_2)","l":54,"f":"Causalean/Estimation/Efficiency/PathwiseGradient.lean"},{"n":"Causalean.Estimation.Efficiency.effBound","k":"def","m":"Causalean.Estimation.Efficiency.TangentProjection","a":"Estimation/Efficiency","t":1,"d":"The semiparametric efficiency bound: the squared norm of the efficient influence function.","s":"{H : Type u_1} → [inst : NormedAddCommGroup H] → [inst_1 : InnerProductSpace ℝ H] → (T : Submodule ℝ H) → [T.HasOrthogonalProjection] → H → ℝ","l":57,"f":"Causalean/Estimation/Efficiency/TangentProjection.lean"},{"n":"Causalean.Estimation.Efficiency.effBound_eq_of_mem_sub","k":"theorem","m":"Causalean.Estimation.Efficiency.TangentProjection","a":"Estimation/Efficiency","t":2,"d":"Tangent-shrinking corollary. If the reference influence function lies in the smaller tangent space T' ≤ T, then both the smaller and larger tangent spaces leave it fixed, so the efficiency bound is unchanged. Interpretation: knowing the propensity score shrinks the tangent space, but if ψ_AIPW already lives in the smaller space the efficiency bound does not move.","s":"∀ {H : Type u_1} [inst : NormedAddCommGroup H] [inst_1 : InnerProductSpace ℝ H] (T T' : Submodule ℝ H) [inst_2 : T.HasOrthogonalProjection] [inst_3 : T'.HasOrthogonalProjection], T' ≤ T → ∀ {g : H}, g ∈ T' → Causalean.Estimation.Efficiency.efficientIF T' g = g ∧ Causalean.Estimation.Efficiency.efficientIF T g = g ∧ Causalean.Estimation.Efficiency.effBound T' g = Causalean.Estimation.Efficiency.effBound T g","l":166,"f":"Causalean/Estimation/Efficiency/TangentProjection.lean"},{"n":"Causalean.Estimation.Efficiency.effBound_le_normSq","k":"theorem","m":"Causalean.Estimation.Efficiency.TangentProjection","a":"Estimation/Efficiency","t":1,"d":"Efficiency lower bound. For any function ψ that is a gradient at g, the efficiency bound is at most the squared norm of ψ.","s":"∀ {H : Type u_1} [inst : NormedAddCommGroup H] [inst_1 : InnerProductSpace ℝ H] {T : Submodule ℝ H} [inst_2 : T.HasOrthogonalProjection] {g ψ : H}, Causalean.Estimation.Efficiency.IsGradient T g ψ → Causalean.Estimation.Efficiency.effBound T g ≤ ‖ψ‖ ^ 2","l":125,"f":"Causalean/Estimation/Efficiency/TangentProjection.lean"},{"n":"Causalean.Estimation.Efficiency.efficientIF","k":"def","m":"Causalean.Estimation.Efficiency.TangentProjection","a":"Estimation/Efficiency","t":1,"d":"The efficient influence function: the orthogonal projection of the reference gradient g onto the tangent space T, coerced back into H.","s":"{H : Type u_1} → [inst : NormedAddCommGroup H] → [inst_1 : InnerProductSpace ℝ H] → (T : Submodule ℝ H) → [T.HasOrthogonalProjection] → H → H","l":51,"f":"Causalean/Estimation/Efficiency/TangentProjection.lean"},{"n":"Causalean.Estimation.Efficiency.efficientIF_eq_of_isPathwiseGradient","k":"theorem","m":"Causalean.Estimation.Efficiency.PathwiseGradient","a":"Estimation/Efficiency","t":2,"d":"All pathwise gradients share the same projection onto a tangent space. Hence the efficient influence function efficientIF T g does not depend on the chosen pathwise gradient g: it is the canonical gradient.","s":"∀ {𝓛 : Type u_1} {H : Type u_2} [inst : NormedAddCommGroup H] [inst_1 : InnerProductSpace ℝ H] {T : Submodule ℝ H} [inst_2 : T.HasOrthogonalProjection] {one : H} {P₀ : 𝓛} {ψ : 𝓛 → ℝ} {g g' : H}, Causalean.Estimation.Efficiency.IsTangentSpace one P₀ T → Causalean.Estimation.Efficiency.IsPathwiseGradient one P₀ ψ g → Causalean.Estimation.Efficiency.IsPathwiseGradient one P₀ ψ g' → Causalean.Estimation.Efficiency.efficientIF T g = Causalean.Estimation.Efficiency.efficientIF T g'","l":187,"f":"Causalean/Estimation/Efficiency/PathwiseGradient.lean"},{"n":"Causalean.Estimation.Efficiency.efficientIF_eq_self_of_isPathwiseGradient_mem","k":"theorem","m":"Causalean.Estimation.Efficiency.PathwiseGradient","a":"Estimation/Efficiency","t":2,"d":"A pathwise gradient lying in T equals its own projection. This recovers efficientIF_eq_self_of_mem, now as a corollary of \"g is the efficient influence function\": a gradient already in the tangent space is the canonical gradient.","s":"∀ {𝓛 : Type u_1} {H : Type u_2} [inst : NormedAddCommGroup H] [inst_1 : InnerProductSpace ℝ H] {T : Submodule ℝ H} [inst_2 : T.HasOrthogonalProjection] {one : H} {P₀ : 𝓛} {ψ : 𝓛 → ℝ} {g : H}, Causalean.Estimation.Efficiency.IsTangentSpace one P₀ T → Causalean.Estimation.Efficiency.IsPathwiseGradient one P₀ ψ g → g ∈ T → Causalean.Estimation.Efficiency.efficientIF T g = g","l":212,"f":"Causalean/Estimation/Efficiency/PathwiseGradient.lean"},{"n":"Causalean.Estimation.Efficiency.efficientIF_eq_self_of_mem","k":"theorem","m":"Causalean.Estimation.Efficiency.TangentProjection","a":"Estimation/Efficiency","t":2,"d":"If the reference gradient already lies in the tangent space, the efficient influence function equals it.","s":"∀ {H : Type u_1} [inst : NormedAddCommGroup H] [inst_1 : InnerProductSpace ℝ H] (T : Submodule ℝ H) [inst_2 : T.HasOrthogonalProjection] {g : H}, g ∈ T → Causalean.Estimation.Efficiency.efficientIF T g = g","l":159,"f":"Causalean/Estimation/Efficiency/TangentProjection.lean"},{"n":"Causalean.Estimation.Efficiency.efficientIF_eq_starProjection","k":"theorem","m":"Causalean.Estimation.Efficiency.TangentProjection","a":"Estimation/Efficiency","t":2,"d":"efficientIF is the coerced star-projection (the projection seen as an endomorphism of H).","s":"∀ {H : Type u_1} [inst : NormedAddCommGroup H] [inst_1 : InnerProductSpace ℝ H] {T : Submodule ℝ H} [inst_2 : T.HasOrthogonalProjection] (g : H), Causalean.Estimation.Efficiency.efficientIF T g = T.starProjection g","l":65,"f":"Causalean/Estimation/Efficiency/TangentProjection.lean"},{"n":"Causalean.Estimation.Efficiency.efficientIF_isGradient","k":"theorem","m":"Causalean.Estimation.Efficiency.TangentProjection","a":"Estimation/Efficiency","t":2,"d":"The efficient influence function is itself a gradient of g.","s":"∀ {H : Type u_1} [inst : NormedAddCommGroup H] [inst_1 : InnerProductSpace ℝ H] {T : Submodule ℝ H} [inst_2 : T.HasOrthogonalProjection] (g : H), Causalean.Estimation.Efficiency.IsGradient T g (Causalean.Estimation.Efficiency.efficientIF T g)","l":90,"f":"Causalean/Estimation/Efficiency/TangentProjection.lean"},{"n":"Causalean.Estimation.Efficiency.efficientIF_mem","k":"theorem","m":"Causalean.Estimation.Efficiency.TangentProjection","a":"Estimation/Efficiency","t":2,"d":"The efficient influence function lies in the tangent space.","s":"∀ {H : Type u_1} [inst : NormedAddCommGroup H] [inst_1 : InnerProductSpace ℝ H] {T : Submodule ℝ H} [inst_2 : T.HasOrthogonalProjection] (g : H), Causalean.Estimation.Efficiency.efficientIF T g ∈ T","l":70,"f":"Causalean/Estimation/Efficiency/TangentProjection.lean"},{"n":"Causalean.Estimation.Efficiency.efficientIF_unique","k":"theorem","m":"Causalean.Estimation.Efficiency.TangentProjection","a":"Estimation/Efficiency","t":2,"d":"A gradient that lies in the tangent space is the efficient influence function.","s":"∀ {H : Type u_1} [inst : NormedAddCommGroup H] [inst_1 : InnerProductSpace ℝ H] {T : Submodule ℝ H} [inst_2 : T.HasOrthogonalProjection] {g ψ : H}, Causalean.Estimation.Efficiency.IsGradient T g ψ → ψ ∈ T → ψ = Causalean.Estimation.Efficiency.efficientIF T g","l":148,"f":"Causalean/Estimation/Efficiency/TangentProjection.lean"},{"n":"Causalean.Estimation.Efficiency.hasDerivAt_tiltExp","k":"theorem","m":"Causalean.Estimation.Efficiency.Submodel","a":"Estimation/Efficiency","t":1,"d":"The tilt-derivative lemma. Along the exponential tilt with a measurable score s that is bounded in absolute value by a constant M and has mean zero under P, the tilted expectation of a fixed function φ that is almost-everywhere strongly measurable and integrable against P is differentiable at t = 0, with derivative equal to ∫ φ·s dP. (Mean-zero s makes the normalizing-constant contribution vanish, so the derivative is the raw covariance ∫ φ·s dP.)","s":"∀ {Z : Type u_1} [inst : MeasurableSpace Z] {P : MeasureTheory.Measure Z} [MeasureTheory.IsProbabilityMeasure P] {s φ : Z → ℝ} {M : ℝ}, Measurable s → (∀ (z : Z), |s z| ≤ M) → ∫ (z : Z), s z ∂P = 0 → MeasureTheory.AEStronglyMeasurable φ P → MeasureTheory.Integrable φ P → HasDerivAt (Causalean.Estimation.Efficiency.tiltExp P s φ) (∫ (z : Z), φ z * s z ∂P) 0","l":153,"f":"Causalean/Estimation/Efficiency/Submodel.lean"},{"n":"Causalean.Estimation.Efficiency.hasDerivAt_tilt_numerator","k":"theorem","m":"Causalean.Estimation.Efficiency.Submodel","a":"Estimation/Efficiency","t":2,"d":"Numerator derivative. d/dt ∫ φ·e^{t·s} dP |₀ = ∫ φ·s dP, for bounded measurable score s and integrable φ.","s":"∀ {Z : Type u_1} [inst : MeasurableSpace Z] {P : MeasureTheory.Measure Z} [MeasureTheory.IsProbabilityMeasure P] {s φ : Z → ℝ} {M : ℝ}, Measurable s → (∀ (z : Z), |s z| ≤ M) → MeasureTheory.AEStronglyMeasurable φ P → MeasureTheory.Integrable φ P → HasDerivAt (fun t => ∫ (z : Z), φ z * Real.exp (t * s z) ∂P) (∫ (z : Z), φ z * s z ∂P) 0","l":89,"f":"Causalean/Estimation/Efficiency/Submodel.lean"},{"n":"Causalean.Estimation.Efficiency.inner_score_eq_of_isPathwiseGradient","k":"theorem","m":"Causalean.Estimation.Efficiency.PathwiseGradient","a":"Estimation/Efficiency","t":2,"d":"Two pathwise gradients of the same functional pair identically with every submodel score: ⟪g, s⟫ = ⟪g', s⟫. (Both equal the pathwise derivative d/dt ψ(path t)|₀, which is unique.)","s":"∀ {𝓛 : Type u_1} {H : Type u_2} [inst : NormedAddCommGroup H] [inst_1 : InnerProductSpace ℝ H] {one : H} {P₀ : 𝓛} {ψ : 𝓛 → ℝ} {g g' : H}, Causalean.Estimation.Efficiency.IsPathwiseGradient one P₀ ψ g → Causalean.Estimation.Efficiency.IsPathwiseGradient one P₀ ψ g' → ∀ (m : Causalean.Estimation.Efficiency.RegularSubmodel one P₀), inner ℝ g m.score = inner ℝ g' m.score","l":102,"f":"Causalean/Estimation/Efficiency/PathwiseGradient.lean"},{"n":"Causalean.Estimation.Efficiency.inner_sub_score_eq_zero_of_isPathwiseGradient","k":"theorem","m":"Causalean.Estimation.Efficiency.PathwiseGradient","a":"Estimation/Efficiency","t":2,"d":"The difference of two pathwise gradients is orthogonal to every submodel score.","s":"∀ {𝓛 : Type u_1} {H : Type u_2} [inst : NormedAddCommGroup H] [inst_1 : InnerProductSpace ℝ H] {one : H} {P₀ : 𝓛} {ψ : 𝓛 → ℝ} {g g' : H}, Causalean.Estimation.Efficiency.IsPathwiseGradient one P₀ ψ g → Causalean.Estimation.Efficiency.IsPathwiseGradient one P₀ ψ g' → ∀ (m : Causalean.Estimation.Efficiency.RegularSubmodel one P₀), inner ℝ (g - g') m.score = 0","l":111,"f":"Causalean/Estimation/Efficiency/PathwiseGradient.lean"},{"n":"Causalean.Estimation.Efficiency.integral_tiltMeasure","k":"theorem","m":"Causalean.Estimation.Efficiency.Submodel","a":"Estimation/Efficiency","t":2,"d":"The expectation of h_fn under the tilted law equals tiltExp P s h_fn t. (No integrability hypothesis is needed: both sides reduce to the same P-integral of h_fn · e^{t·s} scaled by 1/c(t), and a non-integrable h_fn makes both the integral and tiltExp's numerator the same junk default.)","s":"∀ {Z : Type u_1} [inst : MeasurableSpace Z] {P : MeasureTheory.Measure Z} [MeasureTheory.IsProbabilityMeasure P] {s : Z → ℝ} {M t : ℝ}, Measurable s → (∀ (z : Z), |s z| ≤ M) → ∀ {h_fn : Z → ℝ}, ∫ (z : Z), h_fn z ∂Causalean.Estimation.Efficiency.tiltMeasure P s t = Causalean.Estimation.Efficiency.tiltExp P s h_fn t","l":246,"f":"Causalean/Estimation/Efficiency/Submodel.lean"},{"n":"Causalean.Estimation.Efficiency.isGradient_iff_sub_mem_orthogonal","k":"theorem","m":"Causalean.Estimation.Efficiency.TangentProjection","a":"Estimation/Efficiency","t":2,"d":"A vector is a gradient of g iff it differs from g by an element of the orthogonal complement of the tangent space.","s":"∀ {H : Type u_1} [inst : NormedAddCommGroup H] [inst_1 : InnerProductSpace ℝ H] {T : Submodule ℝ H} (g ψ : H), Causalean.Estimation.Efficiency.IsGradient T g ψ ↔ ψ - g ∈ Tᗮ","l":75,"f":"Causalean/Estimation/Efficiency/TangentProjection.lean"},{"n":"Causalean.Estimation.Efficiency.isGradient_of_isPathwiseGradient","k":"theorem","m":"Causalean.Estimation.Efficiency.PathwiseGradient","a":"Estimation/Efficiency","t":2,"d":"Two pathwise gradients differ by an orthogonal-complement element, i.e. either is a TangentProjection-gradient of the other relative to any tangent space T. This is the bridge from the pathwise (HasDerivAt) layer to the inner product IsGradient algebra of TangentProjection.lean.","s":"∀ {𝓛 : Type u_1} {H : Type u_2} [inst : NormedAddCommGroup H] [inst_1 : InnerProductSpace ℝ H] {one : H} {P₀ : 𝓛} {ψ : 𝓛 → ℝ} {g g' : H} {T : Submodule ℝ H}, Causalean.Estimation.Efficiency.IsTangentSpace one P₀ T → Causalean.Estimation.Efficiency.IsPathwiseGradient one P₀ ψ g → Causalean.Estimation.Efficiency.IsPathwiseGradient one P₀ ψ g' → Causalean.Estimation.Efficiency.IsGradient T g g'","l":152,"f":"Causalean/Estimation/Efficiency/PathwiseGradient.lean"},{"n":"Causalean.Estimation.Efficiency.isPathwiseGradient_eq_efficientIF_of_mem","k":"theorem","m":"Causalean.Estimation.Efficiency.PathwiseGradient","a":"Estimation/Efficiency","t":2,"d":"The efficient influence function is the unique pathwise gradient in T. If g is any pathwise gradient and g' is a pathwise gradient lying in the tangent space T, then g' = efficientIF T g. Thus a pathwise gradient inside T is uniquely determined and equals the projection of every gradient — the efficient influence function / canonical gradient.","s":"∀ {𝓛 : Type u_1} {H : Type u_2} [inst : NormedAddCommGroup H] [inst_1 : InnerProductSpace ℝ H] {T : Submodule ℝ H} [inst_2 : T.HasOrthogonalProjection] {one : H} {P₀ : 𝓛} {ψ : 𝓛 → ℝ} {g g' : H}, Causalean.Estimation.Efficiency.IsTangentSpace one P₀ T → Causalean.Estimation.Efficiency.IsPathwiseGradient one P₀ ψ g → Causalean.Estimation.Efficiency.IsPathwiseGradient one P₀ ψ g' → g' ∈ T → g' = Causalean.Estimation.Efficiency.efficientIF T g","l":200,"f":"Causalean/Estimation/Efficiency/PathwiseGradient.lean"},{"n":"Causalean.Estimation.Efficiency.isProbabilityMeasure_tiltMeasure","k":"theorem","m":"Causalean.Estimation.Efficiency.Submodel","a":"Estimation/Efficiency","t":2,"d":"The tilted law tiltMeasure P s t is a probability measure.","s":"∀ {Z : Type u_1} [inst : MeasurableSpace Z] {P : MeasureTheory.Measure Z} [MeasureTheory.IsProbabilityMeasure P] {s : Z → ℝ} {M t : ℝ}, Measurable s → (∀ (z : Z), |s z| ≤ M) → MeasureTheory.IsProbabilityMeasure (Causalean.Estimation.Efficiency.tiltMeasure P s t)","l":227,"f":"Causalean/Estimation/Efficiency/Submodel.lean"},{"n":"Causalean.Estimation.Efficiency.normSq_gradient_decomp","k":"theorem","m":"Causalean.Estimation.Efficiency.TangentProjection","a":"Estimation/Efficiency","t":1,"d":"Pythagoras for gradients. For any function ψ that is a gradient at g, the squared norm of ψ splits into the efficiency bound plus the squared norm of the remainder orthogonal to the efficient influence function.","s":"∀ {H : Type u_1} [inst : NormedAddCommGroup H] [inst_1 : InnerProductSpace ℝ H] {T : Submodule ℝ H} [inst_2 : T.HasOrthogonalProjection] {g ψ : H}, Causalean.Estimation.Efficiency.IsGradient T g ψ → ‖ψ‖ ^ 2 = ‖Causalean.Estimation.Efficiency.efficientIF T g‖ ^ 2 + ‖ψ - Causalean.Estimation.Efficiency.efficientIF T g‖ ^ 2","l":107,"f":"Causalean/Estimation/Efficiency/TangentProjection.lean"},{"n":"Causalean.Estimation.Efficiency.norm_eq_iff_eq_efficientIF","k":"theorem","m":"Causalean.Estimation.Efficiency.TangentProjection","a":"Estimation/Efficiency","t":1,"d":"Sharpness. For any function ψ that is a gradient at g, ψ attains the efficiency bound if and only if it equals the efficient influence function.","s":"∀ {H : Type u_1} [inst : NormedAddCommGroup H] [inst_1 : InnerProductSpace ℝ H] {T : Submodule ℝ H} [inst_2 : T.HasOrthogonalProjection] {g ψ : H}, Causalean.Estimation.Efficiency.IsGradient T g ψ → (‖ψ‖ ^ 2 = Causalean.Estimation.Efficiency.effBound T g ↔ ψ = Causalean.Estimation.Efficiency.efficientIF T g)","l":133,"f":"Causalean/Estimation/Efficiency/TangentProjection.lean"},{"n":"Causalean.Estimation.Efficiency.orthogonalProjection_eq_of_isGradient","k":"theorem","m":"Causalean.Estimation.Efficiency.TangentProjection","a":"Estimation/Efficiency","t":2,"d":"A gradient ψ projects onto the same efficient influence function as g.","s":"∀ {H : Type u_1} [inst : NormedAddCommGroup H] [inst_1 : InnerProductSpace ℝ H] {T : Submodule ℝ H} [inst_2 : T.HasOrthogonalProjection] {g ψ : H}, Causalean.Estimation.Efficiency.IsGradient T g ψ → ↑(T.orthogonalProjection ψ) = Causalean.Estimation.Efficiency.efficientIF T g","l":95,"f":"Causalean/Estimation/Efficiency/TangentProjection.lean"},{"n":"Causalean.Estimation.Efficiency.orthogonalProjection_isPathwiseGradient","k":"theorem","m":"Causalean.Estimation.Efficiency.PathwiseGradient","a":"Estimation/Efficiency","t":1,"d":"The projection of a pathwise gradient is a pathwise gradient. Suppose T is a tangent space for the regular submodels through the base law P₀, and suppose g reproduces the pathwise derivative of the functional ψ along every such submodel — it is a pathwise gradient of ψ at P₀. Then the orthogonal projection of g onto T, written efficientIF T g, is again a pathwise gradient of ψ.","s":"∀ {𝓛 : Type u_1} {H : Type u_2} [inst : NormedAddCommGroup H] [inst_1 : InnerProductSpace ℝ H] {T : Submodule ℝ H} [inst_2 : T.HasOrthogonalProjection] {one : H} {P₀ : 𝓛} {ψ : 𝓛 → ℝ} {g : H}, Causalean.Estimation.Efficiency.IsTangentSpace one P₀ T → Causalean.Estimation.Efficiency.IsPathwiseGradient one P₀ ψ g → Causalean.Estimation.Efficiency.IsPathwiseGradient one P₀ ψ (Causalean.Estimation.Efficiency.efficientIF T g)","l":168,"f":"Causalean/Estimation/Efficiency/PathwiseGradient.lean"},{"n":"Causalean.Estimation.Efficiency.scoreSet","k":"def","m":"Causalean.Estimation.Efficiency.PathwiseGradient","a":"Estimation/Efficiency","t":1,"d":"The set of all submodel scores (tangent directions) through P₀.","s":"{𝓛 : Type u_1} → {H : Type u_2} → [inst : NormedAddCommGroup H] → [InnerProductSpace ℝ H] → H → 𝓛 → Set H","l":71,"f":"Causalean/Estimation/Efficiency/PathwiseGradient.lean"},{"n":"Causalean.Estimation.Efficiency.sub_mem_orthogonal_of_isPathwiseGradient","k":"theorem","m":"Causalean.Estimation.Efficiency.PathwiseGradient","a":"Estimation/Efficiency","t":2,"d":"The difference of two pathwise gradients lies in the orthogonal complement of any tangent space T.","s":"∀ {𝓛 : Type u_1} {H : Type u_2} [inst : NormedAddCommGroup H] [inst_1 : InnerProductSpace ℝ H] {one : H} {P₀ : 𝓛} {ψ : 𝓛 → ℝ} {g g' : H} {T : Submodule ℝ H}, Causalean.Estimation.Efficiency.IsTangentSpace one P₀ T → Causalean.Estimation.Efficiency.IsPathwiseGradient one P₀ ψ g → Causalean.Estimation.Efficiency.IsPathwiseGradient one P₀ ψ g' → g - g' ∈ Tᗮ","l":119,"f":"Causalean/Estimation/Efficiency/PathwiseGradient.lean"},{"n":"Causalean.Estimation.Efficiency.tangentSpace","k":"def","m":"Causalean.Estimation.Efficiency.PathwiseGradient","a":"Estimation/Efficiency","t":1,"d":"The tangent space generated by the regular submodels through P₀: the topological closure of the linear span of the score set.","s":"{𝓛 : Type u_1} → {H : Type u_2} → [inst : NormedAddCommGroup H] → [inst_1 : InnerProductSpace ℝ H] → H → 𝓛 → Submodule ℝ H","l":75,"f":"Causalean/Estimation/Efficiency/PathwiseGradient.lean"},{"n":"Causalean.Estimation.Efficiency.tiltExp","k":"def","m":"Causalean.Estimation.Efficiency.Submodel","a":"Estimation/Efficiency","t":1,"d":"Tilted expectation E_{P_t}[φ] = (∫ φ·e^{t·s} dP)/(∫ e^{t·s} dP). This is the expectation of φ under the exponentially tilted law P_t.","s":"{Z : Type u_1} → [inst : MeasurableSpace Z] → MeasureTheory.Measure Z → (Z → ℝ) → (Z → ℝ) → ℝ → ℝ","l":56,"f":"Causalean/Estimation/Efficiency/Submodel.lean"},{"n":"Causalean.Estimation.Efficiency.tiltMeasure","k":"def","m":"Causalean.Estimation.Efficiency.Submodel","a":"Estimation/Efficiency","t":1,"d":"The exponentially tilted measure P_t = (e^{t·s}/c(t)) · P.","s":"{Z : Type u_1} → [inst : MeasurableSpace Z] → MeasureTheory.Measure Z → (Z → ℝ) → ℝ → MeasureTheory.Measure Z","l":61,"f":"Causalean/Estimation/Efficiency/Submodel.lean"},{"n":"Causalean.Estimation.Efficiency.tiltNorm","k":"def","m":"Causalean.Estimation.Efficiency.Submodel","a":"Estimation/Efficiency","t":1,"d":"Normalizing constant c(t) = ∫ e^{t·s} dP of the exponential tilt.","s":"{Z : Type u_1} → [inst : MeasurableSpace Z] → MeasureTheory.Measure Z → (Z → ℝ) → ℝ → ℝ","l":52,"f":"Causalean/Estimation/Efficiency/Submodel.lean"},{"n":"Causalean.Estimation.Efficiency.tiltNorm_pos","k":"theorem","m":"Causalean.Estimation.Efficiency.Submodel","a":"Estimation/Efficiency","t":2,"d":"The normalizing constant c(t) = ∫ e^{t·s} dP is strictly positive.","s":"∀ {Z : Type u_1} [inst : MeasurableSpace Z] {P : MeasureTheory.Measure Z} [MeasureTheory.IsProbabilityMeasure P] {s : Z → ℝ} {M t : ℝ}, Measurable s → (∀ (z : Z), |s z| ≤ M) → 0 < Causalean.Estimation.Efficiency.tiltNorm P s t","l":209,"f":"Causalean/Estimation/Efficiency/Submodel.lean"},{"n":"Causalean.Estimation.Efficiency.tiltNorm_zero","k":"theorem","m":"Causalean.Estimation.Efficiency.Submodel","a":"Estimation/Efficiency","t":2,"d":"At t = 0 the normalizing constant is the total mass = 1.","s":"∀ {Z : Type u_1} [inst : MeasurableSpace Z] {P : MeasureTheory.Measure Z} [MeasureTheory.IsProbabilityMeasure P] {s : Z → ℝ}, Causalean.Estimation.Efficiency.tiltNorm P s 0 = 1","l":66,"f":"Causalean/Estimation/Efficiency/Submodel.lean"},{"n":"Causalean.Estimation.Efficiency.tilt_exp_integrable","k":"theorem","m":"Causalean.Estimation.Efficiency.Submodel","a":"Estimation/Efficiency","t":2,"d":"For a bounded measurable score, e^{t·s} is P-integrable (it is bounded by the constant e^{|t|·M} on the finite measure P).","s":"∀ {Z : Type u_1} [inst : MeasurableSpace Z] {P : MeasureTheory.Measure Z} [MeasureTheory.IsProbabilityMeasure P] {s : Z → ℝ} {M t : ℝ}, Measurable s → (∀ (z : Z), |s z| ≤ M) → MeasureTheory.Integrable (fun z => Real.exp (t * s z)) P","l":196,"f":"Causalean/Estimation/Efficiency/Submodel.lean"},{"n":"Causalean.Estimation.MinimaxATE.Causal.AEquiv","k":"def","m":"Causalean.Estimation.MinimaxATE.Causal.Construction","a":"Estimation/MinimaxATE/Causal","t":1,"d":"This identifies the treatment node's value space with Booleans.","s":"{C : Type} → [inst : Fintype C] → [inst_1 : Nonempty C] → [inst_2 : MeasurableSpace C] → [inst_3 : MeasurableSingletonClass C] → (m : C → ℝ) → (g : Bool → C → ℝ) → (Causalean.Estimation.MinimaxATE.Causal.dgpPO m g).X (Causalean.Estimation.MinimaxATE.Causal.AIdx m g) ≃ᵐ Bool","l":363,"f":"Causalean/Estimation/MinimaxATE/Causal/Construction.lean"},{"n":"Causalean.Estimation.MinimaxATE.Causal.AIdx","k":"def","m":"Causalean.Estimation.MinimaxATE.Causal.Construction","a":"Estimation/MinimaxATE/Causal","t":1,"d":"This is the observed-node index of the treatment in the induced potential-outcome system.","s":"{C : Type} → [inst : Fintype C] → [inst_1 : Nonempty C] → [inst_2 : MeasurableSpace C] → [inst_3 : MeasurableSingletonClass C] → (m : C → ℝ) → (g : Bool → C → ℝ) → (Causalean.Estimation.MinimaxATE.Causal.dgpPO m g).V","l":351,"f":"Causalean/Estimation/MinimaxATE/Causal/Construction.lean"},{"n":"Causalean.Estimation.MinimaxATE.Causal.WNode","k":"inductive","m":"Causalean.Estimation.MinimaxATE.Causal.Construction","a":"Estimation/MinimaxATE/Causal","t":1,"d":"These are the nodes of the finite backdoor witness graph.","s":"Type","l":55,"f":"Causalean/Estimation/MinimaxATE/Causal/Construction.lean"},{"n":"Causalean.Estimation.MinimaxATE.Causal.WNode.instFintype","k":"instance","m":"Causalean.Estimation.MinimaxATE.Causal.Construction","a":"Estimation/MinimaxATE/Causal","t":2,"d":"The witness graph nodes form a finite type.","s":"Fintype Causalean.Estimation.MinimaxATE.Causal.WNode","l":74,"f":"Causalean/Estimation/MinimaxATE/Causal/Construction.lean"},{"n":"Causalean.Estimation.MinimaxATE.Causal.WNode.instRepr","k":"instance","m":"Causalean.Estimation.MinimaxATE.Causal.Construction","a":"Estimation/MinimaxATE/Causal","t":2,"d":"The witness graph nodes can be rendered for debugging and generated instances.","s":"Repr Causalean.Estimation.MinimaxATE.Causal.WNode","l":71,"f":"Causalean/Estimation/MinimaxATE/Causal/Construction.lean"},{"n":"Causalean.Estimation.MinimaxATE.Causal.WNode.repr","k":"def","m":"Causalean.Estimation.MinimaxATE.Causal.Construction","a":"Estimation/MinimaxATE/Causal","t":1,"d":"This gives a printable representation for the witness graph nodes.","s":"Causalean.Estimation.MinimaxATE.Causal.WNode → ℕ → Std.Format","l":62,"f":"Causalean/Estimation/MinimaxATE/Causal/Construction.lean"},{"n":"Causalean.Estimation.MinimaxATE.Causal.WΩ","k":"def","m":"Causalean.Estimation.MinimaxATE.Causal.Construction","a":"Estimation/MinimaxATE/Causal","t":1,"d":"This assigns value spaces to the witness graph nodes.","s":"Type → Causalean.Estimation.MinimaxATE.Causal.WNode → Type","l":151,"f":"Causalean/Estimation/MinimaxATE/Causal/Construction.lean"},{"n":"Causalean.Estimation.MinimaxATE.Causal.WΩ_borel","k":"instance","m":"Causalean.Estimation.MinimaxATE.Causal.Construction","a":"Estimation/MinimaxATE/Causal","t":2,"d":"Each witness node value space is standard Borel.","s":"∀ (C : Type) [inst : MeasurableSpace C] [StandardBorelSpace C] (n : Causalean.Estimation.MinimaxATE.Causal.WNode), StandardBorelSpace (Causalean.Estimation.MinimaxATE.Causal.WΩ C n)","l":169,"f":"Causalean/Estimation/MinimaxATE/Causal/Construction.lean"},{"n":"Causalean.Estimation.MinimaxATE.Causal.WΩ_meas","k":"instance","m":"Causalean.Estimation.MinimaxATE.Causal.Construction","a":"Estimation/MinimaxATE/Causal","t":2,"d":"Each witness node value space has its measurable-space structure.","s":"(C : Type) → [MeasurableSpace C] → (n : Causalean.Estimation.MinimaxATE.Causal.WNode) → MeasurableSpace (Causalean.Estimation.MinimaxATE.Causal.WΩ C n)","l":160,"f":"Causalean/Estimation/MinimaxATE/Causal/Construction.lean"},{"n":"Causalean.Estimation.MinimaxATE.Causal.WΩ_nonempty","k":"instance","m":"Causalean.Estimation.MinimaxATE.Causal.Construction","a":"Estimation/MinimaxATE/Causal","t":2,"d":"Each witness node value space is nonempty.","s":"∀ (C : Type) [Nonempty C] (n : Causalean.Estimation.MinimaxATE.Causal.WNode), Nonempty (Causalean.Estimation.MinimaxATE.Causal.WΩ C n)","l":178,"f":"Causalean/Estimation/MinimaxATE/Causal/Construction.lean"},{"n":"Causalean.Estimation.MinimaxATE.Causal.XEquiv","k":"def","m":"Causalean.Estimation.MinimaxATE.Causal.Construction","a":"Estimation/MinimaxATE/Causal","t":1,"d":"This identifies the covariate node's value space with the finite covariate type.","s":"{C : Type} → [inst : Fintype C] → [inst_1 : Nonempty C] → [inst_2 : MeasurableSpace C] → [inst_3 : MeasurableSingletonClass C] → (m : C → ℝ) → (g : Bool → C → ℝ) → (Causalean.Estimation.MinimaxATE.Causal.dgpPO m g).X (Causalean.Estimation.MinimaxATE.Causal.XIdx m g) ≃ᵐ C","l":373,"f":"Causalean/Estimation/MinimaxATE/Causal/Construction.lean"},{"n":"Causalean.Estimation.MinimaxATE.Causal.XIdx","k":"def","m":"Causalean.Estimation.MinimaxATE.Causal.Construction","a":"Estimation/MinimaxATE/Causal","t":1,"d":"This is the observed-node index of the covariate in the induced potential-outcome system.","s":"{C : Type} → [inst : Fintype C] → [inst_1 : Nonempty C] → [inst_2 : MeasurableSpace C] → [inst_3 : MeasurableSingletonClass C] → (m : C → ℝ) → (g : Bool → C → ℝ) → (Causalean.Estimation.MinimaxATE.Causal.dgpPO m g).V","l":359,"f":"Causalean/Estimation/MinimaxATE/Causal/Construction.lean"},{"n":"Causalean.Estimation.MinimaxATE.Causal.YEquiv","k":"def","m":"Causalean.Estimation.MinimaxATE.Causal.Construction","a":"Estimation/MinimaxATE/Causal","t":1,"d":"This identifies the outcome node's value space with real numbers.","s":"{C : Type} → [inst : Fintype C] → [inst_1 : Nonempty C] → [inst_2 : MeasurableSpace C] → [inst_3 : MeasurableSingletonClass C] → (m : C → ℝ) → (g : Bool → C → ℝ) → (Causalean.Estimation.MinimaxATE.Causal.dgpPO m g).X (Causalean.Estimation.MinimaxATE.Causal.YIdx m g) ≃ᵐ ℝ","l":368,"f":"Causalean/Estimation/MinimaxATE/Causal/Construction.lean"},{"n":"Causalean.Estimation.MinimaxATE.Causal.YIdx","k":"def","m":"Causalean.Estimation.MinimaxATE.Causal.Construction","a":"Estimation/MinimaxATE/Causal","t":1,"d":"This is the observed-node index of the outcome in the induced potential-outcome system.","s":"{C : Type} → [inst : Fintype C] → [inst_1 : Nonempty C] → [inst_2 : MeasurableSpace C] → [inst_3 : MeasurableSingletonClass C] → (m : C → ℝ) → (g : Bool → C → ℝ) → (Causalean.Estimation.MinimaxATE.Causal.dgpPO m g).V","l":355,"f":"Causalean/Estimation/MinimaxATE/Causal/Construction.lean"},{"n":"Causalean.Estimation.MinimaxATE.Causal.causalATE","k":"def","m":"Causalean.Estimation.MinimaxATE.Causal.Bridge","a":"Estimation/MinimaxATE/Causal","t":1,"d":"The causal average treatment effect of the constructed finite backdoor system, namely (dgpBackdoor m g).ATE. The theorem causalATE_eq_ate later identifies this potential-outcome estimand with the finite observed-data contrast ate g under validity and strict overlap.","s":"{C : Type} → [Fintype C] → [Nonempty C] → [inst : MeasurableSpace C] → [MeasurableSingletonClass C] → (C → ℝ) → (Bool → C → ℝ) → ℝ","l":79,"f":"Causalean/Estimation/MinimaxATE/Causal/Bridge.lean"},{"n":"Causalean.Estimation.MinimaxATE.Causal.causalATE_eq_ate","k":"theorem","m":"Causalean.Estimation.MinimaxATE.Causal.Bridge","a":"Estimation/MinimaxATE/Causal","t":1,"d":"Causal identification bridge. Suppose the data-generating process (m, g) is valid and satisfies strict overlap: the propensity m lies strictly between 0 and 1 at every covariate value. Then the causal average treatment effect E[Y(1) − Y(0)] of the backdoor potential-outcome system built from (m, g) equals the finite observed-data contrast ate g = (1/|C|)·Σₓ(g(1,x) − g(0,x)).","s":"∀ {C : Type} [inst : Fintype C] [inst_1 : Nonempty C] [inst_2 : MeasurableSpace C] [inst_3 : MeasurableSingletonClass C] [StandardBorelSpace C] {m : C → ℝ} {g : Bool → C → ℝ}, Causalean.Estimation.MinimaxATE.ValidDGP m g → (∀ (x : C), m x ∈ Set.Ioo 0 1) → Causalean.Estimation.MinimaxATE.Causal.causalATE m g = Causalean.Estimation.MinimaxATE.ate g","l":1561,"f":"Causalean/Estimation/MinimaxATE/Causal/Bridge.lean"},{"n":"Causalean.Estimation.MinimaxATE.Causal.covLaw","k":"def","m":"Causalean.Estimation.MinimaxATE.Causal.Construction","a":"Estimation/MinimaxATE/Causal","t":1,"d":"This is the uniform law on the finite covariate space.","s":"(C : Type) → [Fintype C] → [Nonempty C] → [inst : MeasurableSpace C] → MeasureTheory.Measure C","l":221,"f":"Causalean/Estimation/MinimaxATE/Causal/Construction.lean"},{"n":"Causalean.Estimation.MinimaxATE.Causal.dgpBES","k":"def","m":"Causalean.Estimation.MinimaxATE.Causal.Bridge","a":"Estimation/MinimaxATE/Causal","t":1,"d":"This is the backdoor estimation system whose value-space regression and propensity are the supplied functions.","s":"{C : Type} → [inst : Fintype C] → [inst_1 : Nonempty C] → [inst_2 : MeasurableSpace C] → [inst_3 : MeasurableSingletonClass C] → [inst_4 : StandardBorelSpace C] → {m : C → ℝ} → {g : Bool → C → ℝ} → Causalean.Estimation.MinimaxATE.ValidDGP m g → (∀ (x : C), m x ∈ Set.Ioo 0 1) → Causalean.Estimation.ATE.BackdoorEstimationSystem (Causalean.Estimation.MinimaxATE.Causal.dgpPO m g) C","l":1543,"f":"Causalean/Estimation/MinimaxATE/Causal/Bridge.lean"},{"n":"Causalean.Estimation.MinimaxATE.Causal.dgpBackdoor","k":"def","m":"Causalean.Estimation.MinimaxATE.Causal.Construction","a":"Estimation/MinimaxATE/Causal","t":1,"d":"This is the backdoor potential-outcome system extracted from the finite witness construction.","s":"{C : Type} → [inst : Fintype C] → [inst_1 : Nonempty C] → [inst_2 : MeasurableSpace C] → [inst_3 : MeasurableSingletonClass C] → (m : C → ℝ) → (g : Bool → C → ℝ) → Causalean.PO.POBackdoorSystem (Causalean.Estimation.MinimaxATE.Causal.dgpPO m g) C","l":378,"f":"Causalean/Estimation/MinimaxATE/Causal/Construction.lean"},{"n":"Causalean.Estimation.MinimaxATE.Causal.dgpFixed","k":"def","m":"Causalean.Estimation.MinimaxATE.Causal.Construction","a":"Estimation/MinimaxATE/Causal","t":1,"d":"This is the empty background assignment for the witness SCM.","s":"{C : Type} → [inst : Fintype C] → [inst_1 : Nonempty C] → [inst_2 : MeasurableSpace C] → [inst_3 : MeasurableSingletonClass C] → (m : C → ℝ) → (g : Bool → C → ℝ) → (Causalean.Estimation.MinimaxATE.Causal.dgpSCM m g).FixedValues","l":342,"f":"Causalean/Estimation/MinimaxATE/Causal/Construction.lean"},{"n":"Causalean.Estimation.MinimaxATE.Causal.dgpPO","k":"def","m":"Causalean.Estimation.MinimaxATE.Causal.Construction","a":"Estimation/MinimaxATE/Causal","t":1,"d":"This is the potential-outcome system induced by the witness structural causal model.","s":"{C : Type} → [Fintype C] → [Nonempty C] → [inst : MeasurableSpace C] → [MeasurableSingletonClass C] → (C → ℝ) → (Bool → C → ℝ) → Causalean.PO.POSystem","l":347,"f":"Causalean/Estimation/MinimaxATE/Causal/Construction.lean"},{"n":"Causalean.Estimation.MinimaxATE.Causal.dgpPO_borel","k":"theorem","m":"Causalean.Estimation.MinimaxATE.Causal.Bridge","a":"Estimation/MinimaxATE/Causal","t":2,"d":"The constructed potential-outcome system has a standard Borel sample space.","s":"∀ {C : Type} [inst : Fintype C] [inst_1 : Nonempty C] [inst_2 : MeasurableSpace C] [inst_3 : MeasurableSingletonClass C] [StandardBorelSpace C] {m : C → ℝ} {g : Bool → C → ℝ}, StandardBorelSpace (Causalean.Estimation.MinimaxATE.Causal.dgpPO m g).Ω","l":65,"f":"Causalean/Estimation/MinimaxATE/Causal/Bridge.lean"},{"n":"Causalean.Estimation.MinimaxATE.Causal.dgpPO_isProb","k":"instance","m":"Causalean.Estimation.MinimaxATE.Causal.Bridge","a":"Estimation/MinimaxATE/Causal","t":2,"d":"The constructed potential-outcome system has a probability measure as its population law.","s":"∀ {C : Type} [inst : Fintype C] [inst_1 : Nonempty C] [inst_2 : MeasurableSpace C] [inst_3 : MeasurableSingletonClass C] {m : C → ℝ} {g : Bool → C → ℝ}, MeasureTheory.IsProbabilityMeasure (Causalean.Estimation.MinimaxATE.Causal.dgpPO m g).μ","l":60,"f":"Causalean/Estimation/MinimaxATE/Causal/Bridge.lean"},{"n":"Causalean.Estimation.MinimaxATE.Causal.dgpPO_standardBorel","k":"instance","m":"Causalean.Estimation.MinimaxATE.Causal.Bridge","a":"Estimation/MinimaxATE/Causal","t":2,"d":"The standard Borel structure for the constructed potential-outcome system is available as an instance.","s":"∀ {C : Type} [inst : Fintype C] [inst_1 : Nonempty C] [inst_2 : MeasurableSpace C] [inst_3 : MeasurableSingletonClass C] [StandardBorelSpace C] {m : C → ℝ} {g : Bool → C → ℝ}, StandardBorelSpace (Causalean.Estimation.MinimaxATE.Causal.dgpPO m g).Ω","l":72,"f":"Causalean/Estimation/MinimaxATE/Causal/Bridge.lean"},{"n":"Causalean.Estimation.MinimaxATE.Causal.dgpSCM","k":"def","m":"Causalean.Estimation.MinimaxATE.Causal.Construction","a":"Estimation/MinimaxATE/Causal","t":1,"d":"This is the concrete stochastic structural causal model for a finite propensity and outcome regression.","s":"{C : Type} → [Fintype C] → [Nonempty C] → [inst : MeasurableSpace C] → [MeasurableSingletonClass C] → (C → ℝ) → (Bool → C → ℝ) → Causalean.SCM Causalean.Estimation.MinimaxATE.Causal.WNode (Causalean.Estimation.MinimaxATE.Causal.WΩ C)","l":230,"f":"Causalean/Estimation/MinimaxATE/Causal/Construction.lean"},{"n":"Causalean.Estimation.MinimaxATE.Causal.dgp_P_X_eq_covLaw","k":"theorem","m":"Causalean.Estimation.MinimaxATE.Causal.Bridge","a":"Estimation/MinimaxATE/Causal","t":2,"d":"The factual covariate marginal of the constructed system is uniform on the finite covariate space.","s":"∀ {C : Type} [inst : Fintype C] [inst_1 : Nonempty C] [inst_2 : MeasurableSpace C] [inst_3 : MeasurableSingletonClass C] [StandardBorelSpace C] {m : C → ℝ} {g : Bool → C → ℝ}, MeasureTheory.Measure.map (Causalean.Estimation.MinimaxATE.Causal.dgpBackdoor m g).factualX (Causalean.Estimation.MinimaxATE.Causal.dgpPO m g).μ = Causalean.Estimation.MinimaxATE.Causal.covLaw C","l":1507,"f":"Causalean/Estimation/MinimaxATE/Causal/Bridge.lean"},{"n":"Causalean.Estimation.MinimaxATE.Causal.dgp_adjustedCE_eq_g","k":"theorem","m":"Causalean.Estimation.MinimaxATE.Causal.Bridge","a":"Estimation/MinimaxATE/Causal","t":2,"d":"The constructed adjusted conditional mean equals the supplied outcome regression in each treatment arm.","s":"∀ {C : Type} [inst : Fintype C] [inst_1 : Nonempty C] [inst_2 : MeasurableSpace C] [inst_3 : MeasurableSingletonClass C] [StandardBorelSpace C] {m : C → ℝ} {g : Bool → C → ℝ}, Causalean.Estimation.MinimaxATE.ValidDGP m g → (∀ (x : C), m x ∈ Set.Ioo 0 1) → ∀ (d : Bool), (Causalean.Estimation.MinimaxATE.Causal.dgpBackdoor m g).adjustedCE d =ᵐ[(Causalean.Estimation.MinimaxATE.Causal.dgpPO m g).μ] fun ω => g d ((Causalean.Estimation.MinimaxATE.Causal.dgpBackdoor m g).factualX ω)","l":1314,"f":"Causalean/Estimation/MinimaxATE/Causal/Bridge.lean"},{"n":"Causalean.Estimation.MinimaxATE.Causal.dgp_assumptions","k":"theorem","m":"Causalean.Estimation.MinimaxATE.Causal.Bridge","a":"Estimation/MinimaxATE/Causal","t":2,"d":"The constructed finite backdoor system satisfies the standard backdoor assumptions.","s":"∀ {C : Type} [inst : Fintype C] [inst_1 : Nonempty C] [inst_2 : MeasurableSpace C] [inst_3 : MeasurableSingletonClass C] [inst_4 : StandardBorelSpace C] {m : C → ℝ} {g : Bool → C → ℝ}, Causalean.Estimation.MinimaxATE.ValidDGP m g → (∀ (x : C), m x ∈ Set.Ioo 0 1) → (Causalean.Estimation.MinimaxATE.Causal.dgpBackdoor m g).Assumptions","l":1496,"f":"Causalean/Estimation/MinimaxATE/Causal/Bridge.lean"},{"n":"Causalean.Estimation.MinimaxATE.Causal.dgp_consistency","k":"theorem","m":"Causalean.Estimation.MinimaxATE.Causal.Bridge","a":"Estimation/MinimaxATE/Causal","t":2,"d":"The constructed potential-outcome system satisfies consistency.","s":"∀ {C : Type} [inst : Fintype C] [inst_1 : Nonempty C] [inst_2 : MeasurableSpace C] [inst_3 : MeasurableSingletonClass C] [StandardBorelSpace C] {m : C → ℝ} {g : Bool → C → ℝ}, (Causalean.Estimation.MinimaxATE.Causal.dgpPO m g).Consistency","l":88,"f":"Causalean/Estimation/MinimaxATE/Causal/Bridge.lean"},{"n":"Causalean.Estimation.MinimaxATE.Causal.dgp_integrable_Y0","k":"theorem","m":"Causalean.Estimation.MinimaxATE.Causal.Bridge","a":"Estimation/MinimaxATE/Causal","t":2,"d":"The control potential outcome is integrable because it is bounded Bernoulli-valued.","s":"∀ {C : Type} [inst : Fintype C] [inst_1 : Nonempty C] [inst_2 : MeasurableSpace C] [inst_3 : MeasurableSingletonClass C] [StandardBorelSpace C] {m : C → ℝ} {g : Bool → C → ℝ}, MeasureTheory.Integrable ((Causalean.Estimation.MinimaxATE.Causal.dgpBackdoor m g).YofD false) (Causalean.Estimation.MinimaxATE.Causal.dgpPO m g).μ","l":1491,"f":"Causalean/Estimation/MinimaxATE/Causal/Bridge.lean"},{"n":"Causalean.Estimation.MinimaxATE.Causal.dgp_integrable_Y1","k":"theorem","m":"Causalean.Estimation.MinimaxATE.Causal.Bridge","a":"Estimation/MinimaxATE/Causal","t":2,"d":"The treated potential outcome is integrable because it is bounded Bernoulli-valued.","s":"∀ {C : Type} [inst : Fintype C] [inst_1 : Nonempty C] [inst_2 : MeasurableSpace C] [inst_3 : MeasurableSingletonClass C] [StandardBorelSpace C] {m : C → ℝ} {g : Bool → C → ℝ}, MeasureTheory.Integrable ((Causalean.Estimation.MinimaxATE.Causal.dgpBackdoor m g).YofD true) (Causalean.Estimation.MinimaxATE.Causal.dgpPO m g).μ","l":1486,"f":"Causalean/Estimation/MinimaxATE/Causal/Bridge.lean"},{"n":"Causalean.Estimation.MinimaxATE.Causal.dgp_overlap","k":"theorem","m":"Causalean.Estimation.MinimaxATE.Causal.Bridge","a":"Estimation/MinimaxATE/Causal","t":2,"d":"The constructed propensity satisfies overlap whenever the supplied propensity is strictly between zero and one.","s":"∀ {C : Type} [inst : Fintype C] [inst_1 : Nonempty C] [inst_2 : MeasurableSpace C] [inst_3 : MeasurableSingletonClass C] [StandardBorelSpace C] {m : C → ℝ} {g : Bool → C → ℝ}, Causalean.Estimation.MinimaxATE.ValidDGP m g → (∀ (x : C), m x ∈ Set.Ioo 0 1) → ∀ᵐ (ω : (Causalean.Estimation.MinimaxATE.Causal.dgpPO m g).Ω) ∂(Causalean.Estimation.MinimaxATE.Causal.dgpPO m g).μ, 0 < (Causalean.Estimation.MinimaxATE.Causal.dgpBackdoor m g).propScore true ω ∧ (Causalean.Estimation.MinimaxATE.Causal.dgpBackdoor m g).propScore true ω < 1","l":1463,"f":"Causalean/Estimation/MinimaxATE/Causal/Bridge.lean"},{"n":"Causalean.Estimation.MinimaxATE.Causal.dgp_propScore_eq_m","k":"theorem","m":"Causalean.Estimation.MinimaxATE.Causal.Bridge","a":"Estimation/MinimaxATE/Causal","t":2,"d":"The constructed treatment propensity equals the supplied propensity function given the covariate.","s":"∀ {C : Type} [inst : Fintype C] [inst_1 : Nonempty C] [inst_2 : MeasurableSpace C] [inst_3 : MeasurableSingletonClass C] [StandardBorelSpace C] {m : C → ℝ} {g : Bool → C → ℝ}, Causalean.Estimation.MinimaxATE.ValidDGP m g → (Causalean.Estimation.MinimaxATE.Causal.dgpBackdoor m g).propScore true =ᵐ[(Causalean.Estimation.MinimaxATE.Causal.dgpPO m g).μ] fun ω => m ((Causalean.Estimation.MinimaxATE.Causal.dgpBackdoor m g).factualX ω)","l":1302,"f":"Causalean/Estimation/MinimaxATE/Causal/Bridge.lean"},{"n":"Causalean.Estimation.MinimaxATE.Causal.dgp_unconfoundedness","k":"theorem","m":"Causalean.Estimation.MinimaxATE.Causal.Bridge","a":"Estimation/MinimaxATE/Causal","t":2,"d":"Unconfoundedness A ⟂ (Y(1), Y(0)) | X. The treatment noise Ea and outcome noise Ey are independent latent roots given the covariate, so the realized treatment is conditionally independent of the potential-outcome bundle given X. Discharged via POSystem.ofSCM_condIndepCF_of_dSep (d-separation in the split graph + value correspondences).","s":"∀ {C : Type} [inst : Fintype C] [inst_1 : Nonempty C] [inst_2 : MeasurableSpace C] [inst_3 : MeasurableSingletonClass C] [inst_4 : StandardBorelSpace C] {m : C → ℝ} {g : Bool → C → ℝ}, (Causalean.Estimation.MinimaxATE.Causal.dgpPO m g).CondIndepCF (Causalean.PO.RegimedVar.ofFactual (Causalean.Estimation.MinimaxATE.Causal.dgpBackdoor m g).dVar) (Causalean.Estimation.MinimaxATE.Causal.dgpBackdoor m g).cfBundle (Causalean.PO.RegimedVar.ofFactual (Causalean.Estimation.MinimaxATE.Causal.dgpBackdoor m g).xVar) (Causalean.Estimation.MinimaxATE.Causal.dgpPO m g).μ","l":1088,"f":"Causalean/Estimation/MinimaxATE/Causal/Bridge.lean"},{"n":"Causalean.Estimation.MinimaxATE.Causal.instDecidableEqWNode","k":"instance","m":"Causalean.Estimation.MinimaxATE.Causal.Construction","a":"Estimation/MinimaxATE/Causal","t":2,"d":"","s":"DecidableEq Causalean.Estimation.MinimaxATE.Causal.WNode","l":58,"f":"Causalean/Estimation/MinimaxATE/Causal/Construction.lean"},{"n":"Causalean.Estimation.MinimaxATE.Causal.instDecidableRelWNodeWEdge","k":"instance","m":"Causalean.Estimation.MinimaxATE.Causal.Construction","a":"Estimation/MinimaxATE/Causal","t":2,"d":"The witness graph edge relation is decidable.","s":"DecidableRel Causalean.Estimation.MinimaxATE.Causal.wEdge","l":93,"f":"Causalean/Estimation/MinimaxATE/Causal/Construction.lean"},{"n":"Causalean.Estimation.MinimaxATE.Causal.instIsProbabilityMeasureCovLaw","k":"instance","m":"Causalean.Estimation.MinimaxATE.Causal.Construction","a":"Estimation/MinimaxATE/Causal","t":2,"d":"The finite covariate uniform law is a probability measure.","s":"∀ (C : Type) [inst : Fintype C] [inst_1 : Nonempty C] [inst_2 : MeasurableSpace C], MeasureTheory.IsProbabilityMeasure (Causalean.Estimation.MinimaxATE.Causal.covLaw C)","l":224,"f":"Causalean/Estimation/MinimaxATE/Causal/Construction.lean"},{"n":"Causalean.Estimation.MinimaxATE.Causal.instIsProbabilityMeasureUnifLaw","k":"instance","m":"Causalean.Estimation.MinimaxATE.Causal.Construction","a":"Estimation/MinimaxATE/Causal","t":2,"d":"The unit-interval uniform law is a probability measure.","s":"MeasureTheory.IsProbabilityMeasure Causalean.Estimation.MinimaxATE.Causal.unifLaw","l":215,"f":"Causalean/Estimation/MinimaxATE/Causal/Construction.lean"},{"n":"Causalean.Estimation.MinimaxATE.Causal.outFun","k":"def","m":"Causalean.Estimation.MinimaxATE.Causal.Construction","a":"Estimation/MinimaxATE/Causal","t":1,"d":"This structural function turns treatment, covariate, and outcome noise into a Bernoulli outcome.","s":"(C : Type) → (Bool → C → ℝ) → Bool → C → ℝ → ℝ","l":192,"f":"Causalean/Estimation/MinimaxATE/Causal/Construction.lean"},{"n":"Causalean.Estimation.MinimaxATE.Causal.parentVal","k":"def","m":"Causalean.Estimation.MinimaxATE.Causal.Construction","a":"Estimation/MinimaxATE/Causal","t":1,"d":"This extracts a parent node's value from the tuple of parent values supplied to a structural function.","s":"(C : Type) → {c : Causalean.Estimation.MinimaxATE.Causal.WNode} → ((w : ↥((Causalean.initialSWIG Causalean.Estimation.MinimaxATE.Causal.wDAG).parents (Causalean.SWIGNode.random c))) → Causalean.swigΩ (Causalean.Estimation.MinimaxATE.Causal.WΩ C) ↑w) → {p : Causalean.Estimation.MinimaxATE.Causal.WNode} → Causalean.Estimation.MinimaxATE.Causal.wEdge p c → Causalean.Estimation.MinimaxATE.Causal.WΩ C p","l":204,"f":"Causalean/Estimation/MinimaxATE/Causal/Construction.lean"},{"n":"Causalean.Estimation.MinimaxATE.Causal.treatFun","k":"def","m":"Causalean.Estimation.MinimaxATE.Causal.Construction","a":"Estimation/MinimaxATE/Causal","t":1,"d":"This structural function turns a uniform treatment noise draw into a Boolean treatment.","s":"ℝ → ℝ → Bool","l":189,"f":"Causalean/Estimation/MinimaxATE/Causal/Construction.lean"},{"n":"Causalean.Estimation.MinimaxATE.Causal.unifLaw","k":"def","m":"Causalean.Estimation.MinimaxATE.Causal.Construction","a":"Estimation/MinimaxATE/Causal","t":1,"d":"This is the uniform law on the unit interval used for the latent noise roots.","s":"MeasureTheory.Measure ℝ","l":211,"f":"Causalean/Estimation/MinimaxATE/Causal/Construction.lean"},{"n":"Causalean.Estimation.MinimaxATE.Causal.wDAG","k":"def","m":"Causalean.Estimation.MinimaxATE.Causal.Construction","a":"Estimation/MinimaxATE/Causal","t":1,"d":"This is the directed acyclic graph for the finite backdoor witness.","s":"Causalean.DAG Causalean.Estimation.MinimaxATE.Causal.WNode","l":112,"f":"Causalean/Estimation/MinimaxATE/Causal/Construction.lean"},{"n":"Causalean.Estimation.MinimaxATE.Causal.wEdge","k":"def","m":"Causalean.Estimation.MinimaxATE.Causal.Construction","a":"Estimation/MinimaxATE/Causal","t":1,"d":"This is the edge relation for the finite backdoor witness graph.","s":"Causalean.Estimation.MinimaxATE.Causal.WNode → Causalean.Estimation.MinimaxATE.Causal.WNode → Prop","l":83,"f":"Causalean/Estimation/MinimaxATE/Causal/Construction.lean"},{"n":"Causalean.Estimation.MinimaxATE.Causal.wParent_mem","k":"theorem","m":"Causalean.Estimation.MinimaxATE.Causal.Construction","a":"Estimation/MinimaxATE/Causal","t":1,"d":"For any edge from node p to node c in the finite backdoor witness graph, the SWIG node for p is a parent of the SWIG node for c in the initial SWIG built from that graph.","s":"∀ {p c : Causalean.Estimation.MinimaxATE.Causal.WNode}, Causalean.Estimation.MinimaxATE.Causal.wEdge p c → Causalean.SWIGNode.random p ∈ (Causalean.initialSWIG Causalean.Estimation.MinimaxATE.Causal.wDAG).parents (Causalean.SWIGNode.random c)","l":197,"f":"Causalean/Estimation/MinimaxATE/Causal/Construction.lean"},{"n":"Causalean.Estimation.MinimaxATE.Causal.wSWIGGraph","k":"def","m":"Causalean.Estimation.MinimaxATE.Causal.Construction","a":"Estimation/MinimaxATE/Causal","t":1,"d":"This is the SWIG graph with observed covariate, treatment, and outcome nodes and latent noise roots.","s":"Causalean.SWIGGraph Causalean.Estimation.MinimaxATE.Causal.WNode","l":118,"f":"Causalean/Estimation/MinimaxATE/Causal/Construction.lean"},{"n":"Causalean.Estimation.MinimaxATE.Causal.wTopo","k":"def","m":"Causalean.Estimation.MinimaxATE.Causal.Construction","a":"Estimation/MinimaxATE/Causal","t":1,"d":"This topological order places latent roots before the observed covariate, treatment, and outcome.","s":"Causalean.Estimation.MinimaxATE.Causal.WNode → ℕ","l":97,"f":"Causalean/Estimation/MinimaxATE/Causal/Construction.lean"},{"n":"Causalean.Estimation.MinimaxATE.Causal.wTopo_lt","k":"theorem","m":"Causalean.Estimation.MinimaxATE.Causal.Construction","a":"Estimation/MinimaxATE/Causal","t":1,"d":"Every edge of the finite backdoor witness graph points strictly forward in the chosen topological order on its nodes.","s":"∀ (u v : Causalean.Estimation.MinimaxATE.Causal.WNode), Causalean.Estimation.MinimaxATE.Causal.wEdge u v → Causalean.Estimation.MinimaxATE.Causal.wTopo u < Causalean.Estimation.MinimaxATE.Causal.wTopo v","l":107,"f":"Causalean/Estimation/MinimaxATE/Causal/Construction.lean"},{"n":"Causalean.Estimation.MinimaxATE.GenConstr","k":"structure","m":"Causalean.Estimation.MinimaxATE.ConstCenterGeneral.Construction","a":"Estimation/MinimaxATE/ConstCenterGeneral","t":1,"d":"General-constant-center construction data. This record packages two nonnegative Rademacher-bump magnitudes together with a constant nuisance center (m₀, g₀, g₁) with each coordinate strictly between zero and one, and four further inequalities — the propensity bump smaller than the treated center, the outcome bump at most the treated center, and two worst-case upper bounds keeping the perturbed propensity and treated-arm regression at most one — that together certify the perturbed propensity and outcome-regression functions built from this data stay in the unit interval.","s":"Type","l":51,"f":"Causalean/Estimation/MinimaxATE/ConstCenterGeneral/Construction.lean"},{"n":"Causalean.Estimation.MinimaxATE.GenConstr.QfalseG","k":"def","m":"Causalean.Estimation.MinimaxATE.ConstCenterGeneral.LowerBound","a":"Estimation/MinimaxATE/ConstCenterGeneral","t":1,"d":"The null n-sample law P̂^⊗n.","s":"Causalean.Estimation.MinimaxATE.GenConstr → (K n : ℕ) → [NeZero K] → MeasureTheory.Measure (Fin n → Causalean.Estimation.MinimaxATE.Obs (Fin K × Bool))","l":79,"f":"Causalean/Estimation/MinimaxATE/ConstCenterGeneral/LowerBound.lean"},{"n":"Causalean.Estimation.MinimaxATE.GenConstr.QfalseG_isProb","k":"theorem","m":"Causalean.Estimation.MinimaxATE.ConstCenterGeneral.LowerBound","a":"Estimation/MinimaxATE/ConstCenterGeneral","t":2,"d":"The null sample law is a probability measure.","s":"∀ (P : Causalean.Estimation.MinimaxATE.GenConstr) (K n : ℕ) [inst : NeZero K], MeasureTheory.IsProbabilityMeasure (P.QfalseG K n)","l":94,"f":"Causalean/Estimation/MinimaxATE/ConstCenterGeneral/LowerBound.lean"},{"n":"Causalean.Estimation.MinimaxATE.GenConstr.QfalseG_singleton_ne_zero","k":"theorem","m":"Causalean.Estimation.MinimaxATE.ConstCenterGeneral.LowerBound","a":"Estimation/MinimaxATE/ConstCenterGeneral","t":2,"d":"The null n-sample law charges every point.","s":"∀ (P : Causalean.Estimation.MinimaxATE.GenConstr) {K n : ℕ} [inst : NeZero K] (ω : Fin n → Causalean.Estimation.MinimaxATE.Obs (Fin K × Bool)), (P.QfalseG K n) {ω} ≠ 0","l":122,"f":"Causalean/Estimation/MinimaxATE/ConstCenterGeneral/LowerBound.lean"},{"n":"Causalean.Estimation.MinimaxATE.GenConstr.QpertG","k":"def","m":"Causalean.Estimation.MinimaxATE.ConstCenterGeneral.LowerBound","a":"Estimation/MinimaxATE/ConstCenterGeneral","t":1,"d":"The perturbed n-sample law Qλ^⊗n.","s":"Causalean.Estimation.MinimaxATE.GenConstr → (K n : ℕ) → [NeZero K] → (Fin K → Bool) → MeasureTheory.Measure (Fin n → Causalean.Estimation.MinimaxATE.Obs (Fin K × Bool))","l":84,"f":"Causalean/Estimation/MinimaxATE/ConstCenterGeneral/LowerBound.lean"},{"n":"Causalean.Estimation.MinimaxATE.GenConstr.QpertG_isProb","k":"theorem","m":"Causalean.Estimation.MinimaxATE.ConstCenterGeneral.LowerBound","a":"Estimation/MinimaxATE/ConstCenterGeneral","t":2,"d":"Each perturbed sample law is a probability measure.","s":"∀ (P : Causalean.Estimation.MinimaxATE.GenConstr) (K n : ℕ) [inst : NeZero K] (lam : Fin K → Bool), MeasureTheory.IsProbabilityMeasure (P.QpertG K n lam)","l":98,"f":"Causalean/Estimation/MinimaxATE/ConstCenterGeneral/LowerBound.lean"},{"n":"Causalean.Estimation.MinimaxATE.GenConstr.QtrueG","k":"def","m":"Causalean.Estimation.MinimaxATE.ConstCenterGeneral.LowerBound","a":"Estimation/MinimaxATE/ConstCenterGeneral","t":1,"d":"The alternative n-sample law: the uniform Rademacher mixture of the perturbed laws.","s":"Causalean.Estimation.MinimaxATE.GenConstr → (K n : ℕ) → [NeZero K] → MeasureTheory.Measure (Fin n → Causalean.Estimation.MinimaxATE.Obs (Fin K × Bool))","l":89,"f":"Causalean/Estimation/MinimaxATE/ConstCenterGeneral/LowerBound.lean"},{"n":"Causalean.Estimation.MinimaxATE.GenConstr.QtrueG_isProb","k":"theorem","m":"Causalean.Estimation.MinimaxATE.ConstCenterGeneral.LowerBound","a":"Estimation/MinimaxATE/ConstCenterGeneral","t":2,"d":"The Rademacher mixture of perturbed sample laws is a probability measure.","s":"∀ (P : Causalean.Estimation.MinimaxATE.GenConstr) (K n : ℕ) [inst : NeZero K], MeasureTheory.IsProbabilityMeasure (P.QtrueG K n)","l":102,"f":"Causalean/Estimation/MinimaxATE/ConstCenterGeneral/LowerBound.lean"},{"n":"Causalean.Estimation.MinimaxATE.GenConstr.QtrueG_real_singleton","k":"theorem","m":"Causalean.Estimation.MinimaxATE.ConstCenterGeneral.LowerBound","a":"Estimation/MinimaxATE/ConstCenterGeneral","t":2,"d":"The alternative law's .real point mass: a uniform mixture over sign vectors.","s":"∀ (P : Causalean.Estimation.MinimaxATE.GenConstr) {K n : ℕ} [inst : NeZero K] (ω : Fin n → Causalean.Estimation.MinimaxATE.Obs (Fin K × Bool)), (P.QtrueG K n).real {ω} = ∑ lam, (2 ^ K)⁻¹ * ∏ i, Causalean.Estimation.MinimaxATE.obsReal (P.mPertG lam) (P.gPertG lam) (ω i)","l":141,"f":"Causalean/Estimation/MinimaxATE/ConstCenterGeneral/LowerBound.lean"},{"n":"Causalean.Estimation.MinimaxATE.GenConstr.ate_gPertG","k":"theorem","m":"Causalean.Estimation.MinimaxATE.ConstCenterGeneral.Gap","a":"Estimation/MinimaxATE/ConstCenterGeneral","t":1,"d":"Exact ATE of the perturbed construction. For any Rademacher sign vector lam, the average treatment effect of the perturbed outcome regression equals (g₁ − g₀) + g₁β(α+β)/(g₁² − β²), independent of lam.","s":"∀ {K : ℕ} (P : Causalean.Estimation.MinimaxATE.GenConstr) [NeZero K] (lam : Fin K → Bool), Causalean.Estimation.MinimaxATE.ate (P.gPertG lam) = P.g₁ - P.g₀ + P.g₁ * P.β * (P.α + P.β) / (P.g₁ ^ 2 - P.β ^ 2)","l":99,"f":"Causalean/Estimation/MinimaxATE/ConstCenterGeneral/Gap.lean"},{"n":"Causalean.Estimation.MinimaxATE.GenConstr.ate_gapG","k":"theorem","m":"Causalean.Estimation.MinimaxATE.ConstCenterGeneral.Gap","a":"Estimation/MinimaxATE/ConstCenterGeneral","t":1,"d":"For any Rademacher sign vector lam, the gap between the perturbed construction's ATE and the null estimate's ATE equals g₁β(α+β)/(g₁² − β²).","s":"∀ {K : ℕ} (P : Causalean.Estimation.MinimaxATE.GenConstr) [NeZero K] (lam : Fin K → Bool), Causalean.Estimation.MinimaxATE.ate (P.gPertG lam) - Causalean.Estimation.MinimaxATE.ate P.ghatG = P.g₁ * P.β * (P.α + P.β) / (P.g₁ ^ 2 - P.β ^ 2)","l":124,"f":"Causalean/Estimation/MinimaxATE/ConstCenterGeneral/Gap.lean"},{"n":"Causalean.Estimation.MinimaxATE.GenConstr.ate_ghatG","k":"theorem","m":"Causalean.Estimation.MinimaxATE.ConstCenterGeneral.Gap","a":"Estimation/MinimaxATE/ConstCenterGeneral","t":2,"d":"The null estimate ĝ has ATE g₁ − g₀: every cell contributes g₁ − g₀.","s":"∀ {K : ℕ} (P : Causalean.Estimation.MinimaxATE.GenConstr) [NeZero K], Causalean.Estimation.MinimaxATE.ate P.ghatG = P.g₁ - P.g₀","l":64,"f":"Causalean/Estimation/MinimaxATE/ConstCenterGeneral/Gap.lean"},{"n":"Causalean.Estimation.MinimaxATE.GenConstr.chiSqDiv_QtrueG_QfalseG_le_one","k":"theorem","m":"Causalean.Estimation.MinimaxATE.ConstCenterGeneral.LowerBound","a":"Estimation/MinimaxATE/ConstCenterGeneral","t":2,"d":"The χ² indistinguishability bound (general center): Γ ≤ 1, 2n²(Γ/2)² ≤ K·log 2 imply χ²(QtrueG‖QfalseG) ≤ 1.","s":"∀ (P : Causalean.Estimation.MinimaxATE.GenConstr) {K n : ℕ} [inst : NeZero K], P.Γ ≤ 1 → 2 * ↑n ^ 2 * (P.Γ / 2) ^ 2 ≤ ↑K * Real.log 2 → Causalean.Stat.chiSqDiv (P.QtrueG K n) (P.QfalseG K n) ≤ 1","l":194,"f":"Causalean/Estimation/MinimaxATE/ConstCenterGeneral/LowerBound.lean"},{"n":"Causalean.Estimation.MinimaxATE.GenConstr.chiSqOverlapG","k":"def","m":"Causalean.Estimation.MinimaxATE.ConstCenterGeneral.ChiSqOverlap","a":"Estimation/MinimaxATE/ConstCenterGeneral","t":1,"d":"The single-observation χ² second-moment overlap of two perturbed laws relative to the null.","s":"{K : ℕ} → Causalean.Estimation.MinimaxATE.GenConstr → (Fin K → Bool) → (Fin K → Bool) → ℝ","l":56,"f":"Causalean/Estimation/MinimaxATE/ConstCenterGeneral/ChiSqOverlap.lean"},{"n":"Causalean.Estimation.MinimaxATE.GenConstr.chiSqOverlap_eqG","k":"theorem","m":"Causalean.Estimation.MinimaxATE.ConstCenterGeneral.ChiSqOverlap","a":"Estimation/MinimaxATE/ConstCenterGeneral","t":1,"d":"For any two Rademacher sign vectors lam and lam' indexing perturbed data-generating processes, the single-observation χ² overlap between them equals one plus the per-cell coefficient Γ/K times the sum of pairwise sign agreements between lam and lam'.","s":"∀ {K : ℕ} (P : Causalean.Estimation.MinimaxATE.GenConstr) [NeZero K] (lam lam' : Fin K → Bool), P.chiSqOverlapG lam lam' = 1 + P.Γ / ↑K * ∑ j, Causalean.Estimation.MinimaxATE.signOf (lam j) * Causalean.Estimation.MinimaxATE.signOf (lam' j)","l":84,"f":"Causalean/Estimation/MinimaxATE/ConstCenterGeneral/ChiSqOverlap.lean"},{"n":"Causalean.Estimation.MinimaxATE.GenConstr.denomG_pos","k":"theorem","m":"Causalean.Estimation.MinimaxATE.ConstCenterGeneral.Construction","a":"Estimation/MinimaxATE/ConstCenterGeneral","t":2,"d":"The denominator 1 − (β/g₁)·Δ is positive: Δ ≤ 1 gives 1 − (β/g₁)·Δ ≥ 1 − β/g₁ > 0.","s":"∀ {K : ℕ} (P : Causalean.Estimation.MinimaxATE.GenConstr) (lam : Fin K → Bool) (x : Fin K × Bool), 0 < 1 - P.β / P.g₁ * Causalean.Estimation.MinimaxATE.Δ lam x","l":113,"f":"Causalean/Estimation/MinimaxATE/ConstCenterGeneral/Construction.lean"},{"n":"Causalean.Estimation.MinimaxATE.GenConstr.g1sq_sub_betasq_pos","k":"theorem","m":"Causalean.Estimation.MinimaxATE.ConstCenterGeneral.Gap","a":"Estimation/MinimaxATE/ConstCenterGeneral","t":2,"d":"g₁² − β² > 0 since 0 ≤ β < g₁.","s":"∀ (P : Causalean.Estimation.MinimaxATE.GenConstr), 0 < P.g₁ ^ 2 - P.β ^ 2","l":46,"f":"Causalean/Estimation/MinimaxATE/ConstCenterGeneral/Gap.lean"},{"n":"Causalean.Estimation.MinimaxATE.GenConstr.gPertG","k":"def","m":"Causalean.Estimation.MinimaxATE.ConstCenterGeneral.Construction","a":"Estimation/MinimaxATE/ConstCenterGeneral","t":1,"d":"The perturbed outcome regression: control arm = g₀, treated arm gλ(1,·) = (g₁ + α·Δ)/(1 − (β/g₁)·Δ).","s":"{K : ℕ} → Causalean.Estimation.MinimaxATE.GenConstr → (Fin K → Bool) → Bool → Fin K × Bool → ℝ","l":108,"f":"Causalean/Estimation/MinimaxATE/ConstCenterGeneral/Construction.lean"},{"n":"Causalean.Estimation.MinimaxATE.GenConstr.gPertG_true_eq","k":"theorem","m":"Causalean.Estimation.MinimaxATE.ConstCenterGeneral.Gap","a":"Estimation/MinimaxATE/ConstCenterGeneral","t":2,"d":"Cleared treated arm. Multiplying numerator and denominator by g₁ turns the inner fraction β/g₁ into the polynomial denominator g₁ − β·Δ: gλ(1,x) = (g₁² + α·g₁·Δ)/(g₁ − β·Δ).","s":"∀ {K : ℕ} (P : Causalean.Estimation.MinimaxATE.GenConstr) (lam : Fin K → Bool) (x : Fin K × Bool), P.gPertG lam true x = (P.g₁ ^ 2 + P.α * P.g₁ * Causalean.Estimation.MinimaxATE.Δ lam x) / (P.g₁ - P.β * Causalean.Estimation.MinimaxATE.Δ lam x)","l":50,"f":"Causalean/Estimation/MinimaxATE/ConstCenterGeneral/Gap.lean"},{"n":"Causalean.Estimation.MinimaxATE.GenConstr.gPertG_true_sub","k":"theorem","m":"Causalean.Estimation.MinimaxATE.ConstCenterGeneral.Membership","a":"Estimation/MinimaxATE/ConstCenterGeneral","t":2,"d":"The treated outcome arm: its pointwise deviation from ĝ(1) = g₁ is g₁(α+β)·Δ/(g₁ − β·Δ).","s":"∀ {K : ℕ} (P : Causalean.Estimation.MinimaxATE.GenConstr) (lam : Fin K → Bool) (x : Fin K × Bool), P.gPertG lam true x - P.g₁ = P.g₁ * (P.α + P.β) * Causalean.Estimation.MinimaxATE.Δ lam x / (P.g₁ - P.β * Causalean.Estimation.MinimaxATE.Δ lam x)","l":71,"f":"Causalean/Estimation/MinimaxATE/ConstCenterGeneral/Membership.lean"},{"n":"Causalean.Estimation.MinimaxATE.GenConstr.ghatG","k":"def","m":"Causalean.Estimation.MinimaxATE.ConstCenterGeneral.Construction","a":"Estimation/MinimaxATE/ConstCenterGeneral","t":1,"d":"The constant outcome-regression center: g₁ on the treated arm, g₀ on control.","s":"{K : ℕ} → Causalean.Estimation.MinimaxATE.GenConstr → Bool → Fin K × Bool → ℝ","l":101,"f":"Causalean/Estimation/MinimaxATE/ConstCenterGeneral/Construction.lean"},{"n":"Causalean.Estimation.MinimaxATE.GenConstr.inClassG","k":"theorem","m":"Causalean.Estimation.MinimaxATE.ConstCenterGeneral.Membership","a":"Estimation/MinimaxATE/ConstCenterGeneral","t":1,"d":"Class membership. Given the squared propensity-perturbation size (m₀·(β/g₁))² within the budget εm and the squared treated-arm outcome-regression perturbation bound g₁²(α+β)²/(g₁ − β)² within the budget εg, the Rademacher-perturbed data-generating process (mλ, gλ) lies in the structure-agnostic nuisance class ℱ(εg, εm) around the constant center (m̂, ĝ).","s":"∀ {K : ℕ} (P : Causalean.Estimation.MinimaxATE.GenConstr) [NeZero K] {εg εm : ℝ}, (P.m₀ * (P.β / P.g₁)) ^ 2 ≤ εm → P.g₁ ^ 2 * (P.α + P.β) ^ 2 / (P.g₁ - P.β) ^ 2 ≤ εg → ∀ (lam : Fin K → Bool), Causalean.Estimation.MinimaxATE.InClass P.mhatG P.ghatG εg εm (P.mPertG lam) (P.gPertG lam)","l":130,"f":"Causalean/Estimation/MinimaxATE/ConstCenterGeneral/Membership.lean"},{"n":"Causalean.Estimation.MinimaxATE.GenConstr.inClass_nullG","k":"theorem","m":"Causalean.Estimation.MinimaxATE.ConstCenterGeneral.LowerBound","a":"Estimation/MinimaxATE/ConstCenterGeneral","t":2,"d":"The null estimate is itself in the class (zero nuisance error).","s":"∀ (P : Causalean.Estimation.MinimaxATE.GenConstr) {K : ℕ} {εg εm : ℝ}, 0 ≤ εg → 0 ≤ εm → Causalean.Estimation.MinimaxATE.InClass P.mhatG P.ghatG εg εm P.mhatG P.ghatG","l":72,"f":"Causalean/Estimation/MinimaxATE/ConstCenterGeneral/LowerBound.lean"},{"n":"Causalean.Estimation.MinimaxATE.GenConstr.instNonemptyFinBoolProd","k":"instance","m":"Causalean.Estimation.MinimaxATE.ConstCenterGeneral.LowerBound","a":"Estimation/MinimaxATE/ConstCenterGeneral","t":2,"d":"The paired-cell covariate is nonempty whenever K ≠ 0.","s":"∀ {K : ℕ} [NeZero K], Nonempty (Fin K × Bool)","l":55,"f":"Causalean/Estimation/MinimaxATE/ConstCenterGeneral/LowerBound.lean"},{"n":"Causalean.Estimation.MinimaxATE.GenConstr.l2sq_gPertG_false","k":"theorem","m":"Causalean.Estimation.MinimaxATE.ConstCenterGeneral.Membership","a":"Estimation/MinimaxATE/ConstCenterGeneral","t":2,"d":"The control outcome arm is unchanged (gλ(0,·) = g₀ = ĝ(0,·)), so its L² error is 0.","s":"∀ {K : ℕ} (P : Causalean.Estimation.MinimaxATE.GenConstr) (lam : Fin K → Bool), Causalean.Estimation.MinimaxATE.l2sq (P.gPertG lam false) (P.ghatG false) = 0","l":62,"f":"Causalean/Estimation/MinimaxATE/ConstCenterGeneral/Membership.lean"},{"n":"Causalean.Estimation.MinimaxATE.GenConstr.l2sq_gPertG_true_le","k":"theorem","m":"Causalean.Estimation.MinimaxATE.ConstCenterGeneral.Membership","a":"Estimation/MinimaxATE/ConstCenterGeneral","t":2,"d":"Uniform bound on the treated arm's squared L² error: every pointwise term is ≤ g₁²(α+β)²/(g₁ − β)², so the average is too.","s":"∀ {K : ℕ} (P : Causalean.Estimation.MinimaxATE.GenConstr) [NeZero K] (lam : Fin K → Bool), Causalean.Estimation.MinimaxATE.l2sq (P.gPertG lam true) (P.ghatG true) ≤ P.g₁ ^ 2 * (P.α + P.β) ^ 2 / (P.g₁ - P.β) ^ 2","l":81,"f":"Causalean/Estimation/MinimaxATE/ConstCenterGeneral/Membership.lean"},{"n":"Causalean.Estimation.MinimaxATE.GenConstr.l2sq_mPertG","k":"theorem","m":"Causalean.Estimation.MinimaxATE.ConstCenterGeneral.Membership","a":"Estimation/MinimaxATE/ConstCenterGeneral","t":2,"d":"The propensity perturbation has exact squared L² size (m₀·(β/g₁))²: it is a Rademacher bump of magnitude m₀·(β/g₁) on Δ.","s":"∀ {K : ℕ} (P : Causalean.Estimation.MinimaxATE.GenConstr) [NeZero K] (lam : Fin K → Bool), Causalean.Estimation.MinimaxATE.l2sq (P.mPertG lam) P.mhatG = (P.m₀ * (P.β / P.g₁)) ^ 2","l":47,"f":"Causalean/Estimation/MinimaxATE/ConstCenterGeneral/Membership.lean"},{"n":"Causalean.Estimation.MinimaxATE.GenConstr.mPertG","k":"def","m":"Causalean.Estimation.MinimaxATE.ConstCenterGeneral.Construction","a":"Estimation/MinimaxATE/ConstCenterGeneral","t":1,"d":"The perturbed propensity mλ = m₀·(1 − (β/g₁)·Δ).","s":"{K : ℕ} → Causalean.Estimation.MinimaxATE.GenConstr → (Fin K → Bool) → Fin K × Bool → ℝ","l":104,"f":"Causalean/Estimation/MinimaxATE/ConstCenterGeneral/Construction.lean"},{"n":"Causalean.Estimation.MinimaxATE.GenConstr.mhatG","k":"def","m":"Causalean.Estimation.MinimaxATE.ConstCenterGeneral.Construction","a":"Estimation/MinimaxATE/ConstCenterGeneral","t":1,"d":"The constant propensity center as a function of the covariate.","s":"{K : ℕ} → Causalean.Estimation.MinimaxATE.GenConstr → Fin K × Bool → ℝ","l":98,"f":"Causalean/Estimation/MinimaxATE/ConstCenterGeneral/Construction.lean"},{"n":"Causalean.Estimation.MinimaxATE.GenConstr.minimax_lower_bound_gen","k":"theorem","m":"Causalean.Estimation.MinimaxATE.ConstCenterGeneral.LowerBound","a":"Estimation/MinimaxATE/ConstCenterGeneral","t":1,"d":"Structure-agnostic minimax lower bound (general constant center). Fix a constant nuisance center (m₀, g₀, g₁) ∈ (0,1)³ and Rademacher bump magnitudes (α, β). Suppose the squared propensity-perturbation size (m₀·β/g₁)² is within the budget εm and the squared outcome-regression perturbation size g₁²(α+β)²/(g₁−β)² is within the budget εg, with both budgets nonnegative. If further the per-cell overlap coefficient Γ is at most 1 and the sample size obeys the regime 2n²(Γ/2)² ≤ K·log 2, then for every measurable estimator, there is a data-generating process in the structure-agnostic class around this center on which the estimator misses the true ATE by s = g₁β(α+β)/(2(g₁²−β²)) with probability at least 1/4.","s":"∀ (P : Causalean.Estimation.MinimaxATE.GenConstr) {K n : ℕ} [inst : NeZero K] {εg εm : ℝ}, (P.m₀ * (P.β / P.g₁)) ^ 2 ≤ εm → P.g₁ ^ 2 * (P.α + P.β) ^ 2 / (P.g₁ - P.β) ^ 2 ≤ εg → 0 ≤ εg → 0 ≤ εm → P.Γ ≤ 1 → 2 * ↑n ^ 2 * (P.Γ / 2) ^ 2 ≤ ↑K * Real.log 2 → ∀ {est : (Fin n → Causalean.Estimation.MinimaxATE.Obs (Fin K × Bool)) → ℝ}, Measurable est → 1 / 4 ≤ Causalean.Estimation.MinimaxATE.minimaxMiss P.mhatG P.ghatG εg εm n est (P.g₁ * P.β * (P.α + P.β) / (2 * (P.g₁ ^ 2 - P.β ^ 2)))","l":231,"f":"Causalean/Estimation/MinimaxATE/ConstCenterGeneral/LowerBound.lean"},{"n":"Causalean.Estimation.MinimaxATE.GenConstr.minimax_lower_bound_mse_gen","k":"theorem","m":"Causalean.Estimation.MinimaxATE.ConstCenterGeneral.MSE","a":"Estimation/MinimaxATE/ConstCenterGeneral","t":1,"d":"Structure-agnostic minimax lower bound, expected-risk (MSE) form (general center). Under the same budget and regularity hypotheses as minimax_lower_bound_gen — the squared propensity-perturbation size within the budget εm, the squared outcome-regression perturbation size within the budget εg, both budgets nonnegative, the per-cell overlap coefficient Γ at most 1, and the sample size in the regime 2n²(Γ/2)² ≤ K·log 2 — every measurable estimator has the weaker expected-risk consequence: there is a data-generating process in the class on which the estimator's mean-squared error is at least s²/4, where s = g₁β(α+β)/(2(g₁²−β²)) ≍ √(εg·εm), obtained from the quantile bound by a Chebyshev (1−γ)-factor conversion at γ = 3/4.","s":"∀ (P : Causalean.Estimation.MinimaxATE.GenConstr) {K n : ℕ} [inst : NeZero K] {εg εm : ℝ}, (P.m₀ * (P.β / P.g₁)) ^ 2 ≤ εm → P.g₁ ^ 2 * (P.α + P.β) ^ 2 / (P.g₁ - P.β) ^ 2 ≤ εg → 0 ≤ εg → 0 ≤ εm → P.Γ ≤ 1 → 2 * ↑n ^ 2 * (P.Γ / 2) ^ 2 ≤ ↑K * Real.log 2 → ∀ {est : (Fin n → Causalean.Estimation.MinimaxATE.Obs (Fin K × Bool)) → ℝ}, Measurable est → ∃ p, (P.g₁ * P.β * (P.α + P.β) / (2 * (P.g₁ ^ 2 - P.β ^ 2))) ^ 2 / 4 ≤ Causalean.Estimation.MinimaxATE.nMSE p.property.valid n est","l":41,"f":"Causalean/Estimation/MinimaxATE/ConstCenterGeneral/MSE.lean"},{"n":"Causalean.Estimation.MinimaxATE.GenConstr.obsReal_pertG_eq","k":"theorem","m":"Causalean.Estimation.MinimaxATE.ConstCenterGeneral.ChiSqOverlap","a":"Estimation/MinimaxATE/ConstCenterGeneral","t":2,"d":"Denominator-free value of the perturbed observed mass obsReal mλ gλ at the four (d, y) combinations. The treated arm's denominator 1 − (β/g₁)Δ cancels against mλ = m₀·(1 − (β/g₁)Δ).","s":"∀ {K : ℕ} (P : Causalean.Estimation.MinimaxATE.GenConstr) (lam : Fin K → Bool) (x : Fin K × Bool) (d y : Bool), Causalean.Estimation.MinimaxATE.obsReal (P.mPertG lam) (P.gPertG lam) (x, d, y) = (↑(Fintype.card (Fin K × Bool)))⁻¹ * match d, y with | true, true => P.m₀ * (P.g₁ + P.α * Causalean.Estimation.MinimaxATE.Δ lam x) | true, false => P.m₀ * (1 - P.g₁ - (P.α + P.β / P.g₁) * Causalean.Estimation.MinimaxATE.Δ lam x) | false, true => (1 - P.m₀ + P.m₀ * (P.β / P.g₁) * Causalean.Estimation.MinimaxATE.Δ lam x) * P.g₀ | false, false => (1 - P.m₀ + P.m₀ * (P.β / P.g₁) * Causalean.Estimation.MinimaxATE.Δ lam x) * (1 - P.g₀)","l":63,"f":"Causalean/Estimation/MinimaxATE/ConstCenterGeneral/ChiSqOverlap.lean"},{"n":"Causalean.Estimation.MinimaxATE.GenConstr.one_add_chiSqDiv_QtrueG_QfalseG","k":"theorem","m":"Causalean.Estimation.MinimaxATE.ConstCenterGeneral.LowerBound","a":"Estimation/MinimaxATE/ConstCenterGeneral","t":2,"d":"Mixture second-moment identity (general center).","s":"∀ (P : Causalean.Estimation.MinimaxATE.GenConstr) {K n : ℕ} [inst : NeZero K], 1 + Causalean.Stat.chiSqDiv (P.QtrueG K n) (P.QfalseG K n) = ∑ lam, ∑ lam', (2 ^ K)⁻¹ * (2 ^ K)⁻¹ * P.chiSqOverlapG lam lam' ^ n","l":158,"f":"Causalean/Estimation/MinimaxATE/ConstCenterGeneral/LowerBound.lean"},{"n":"Causalean.Estimation.MinimaxATE.GenConstr.ratio_lt_one","k":"theorem","m":"Causalean.Estimation.MinimaxATE.ConstCenterGeneral.Construction","a":"Estimation/MinimaxATE/ConstCenterGeneral","t":2,"d":"β/g₁ < 1 since β < g₁.","s":"∀ (P : Causalean.Estimation.MinimaxATE.GenConstr), P.β / P.g₁ < 1","l":95,"f":"Causalean/Estimation/MinimaxATE/ConstCenterGeneral/Construction.lean"},{"n":"Causalean.Estimation.MinimaxATE.GenConstr.ratio_nonneg","k":"theorem","m":"Causalean.Estimation.MinimaxATE.ConstCenterGeneral.Construction","a":"Estimation/MinimaxATE/ConstCenterGeneral","t":2,"d":"0 < β/g₁ is false in general (β may be 0); but 0 ≤ β/g₁ always.","s":"∀ (P : Causalean.Estimation.MinimaxATE.GenConstr), 0 ≤ P.β / P.g₁","l":92,"f":"Causalean/Estimation/MinimaxATE/ConstCenterGeneral/Construction.lean"},{"n":"Causalean.Estimation.MinimaxATE.GenConstr.realG_le_minimaxMiss","k":"theorem","m":"Causalean.Estimation.MinimaxATE.ConstCenterGeneral.LowerBound","a":"Estimation/MinimaxATE/ConstCenterGeneral","t":2,"d":"An in-class DGP's n-sample miss probability at its own ATE is dominated by the minimax miss probability.","s":"∀ (P : Causalean.Estimation.MinimaxATE.GenConstr) {K n : ℕ} [inst : NeZero K] {εg εm : ℝ} {m : Fin K × Bool → ℝ} {g : Bool → Fin K × Bool → ℝ} (hin : Causalean.Estimation.MinimaxATE.InClass P.mhatG P.ghatG εg εm m g) (est : (Fin n → Causalean.Estimation.MinimaxATE.Obs (Fin K × Bool)) → ℝ) (s : ℝ), (Causalean.Estimation.MinimaxATE.productLaw hin.valid n).real {x | s ≤ |est x - Causalean.Estimation.MinimaxATE.ate g|} ≤ Causalean.Estimation.MinimaxATE.minimaxMiss P.mhatG P.ghatG εg εm n est s","l":110,"f":"Causalean/Estimation/MinimaxATE/ConstCenterGeneral/LowerBound.lean"},{"n":"Causalean.Estimation.MinimaxATE.GenConstr.tvDist_QfalseG_QtrueG_le_half","k":"theorem","m":"Causalean.Estimation.MinimaxATE.ConstCenterGeneral.LowerBound","a":"Estimation/MinimaxATE/ConstCenterGeneral","t":2,"d":"Total-variation indistinguishability (general center).","s":"∀ (P : Causalean.Estimation.MinimaxATE.GenConstr) {K n : ℕ} [inst : NeZero K], P.Γ ≤ 1 → 2 * ↑n ^ 2 * (P.Γ / 2) ^ 2 ≤ ↑K * Real.log 2 → Causalean.Stat.tvDist (P.QfalseG K n) (P.QtrueG K n) ≤ 1 / 2","l":214,"f":"Causalean/Estimation/MinimaxATE/ConstCenterGeneral/LowerBound.lean"},{"n":"Causalean.Estimation.MinimaxATE.GenConstr.validDGP_hatG","k":"theorem","m":"Causalean.Estimation.MinimaxATE.ConstCenterGeneral.Construction","a":"Estimation/MinimaxATE/ConstCenterGeneral","t":1,"d":"The null constant-center data-generating process, with propensity m₀ and outcome regressions (g₀, g₁), is a valid finite observed-data model, i.e. all its component probabilities lie in [0,1].","s":"∀ {K : ℕ} (P : Causalean.Estimation.MinimaxATE.GenConstr), Causalean.Estimation.MinimaxATE.ValidDGP P.mhatG P.ghatG","l":123,"f":"Causalean/Estimation/MinimaxATE/ConstCenterGeneral/Construction.lean"},{"n":"Causalean.Estimation.MinimaxATE.GenConstr.validDGP_pertG","k":"theorem","m":"Causalean.Estimation.MinimaxATE.ConstCenterGeneral.Construction","a":"Estimation/MinimaxATE/ConstCenterGeneral","t":1,"d":"For any Rademacher sign vector lam indexing the perturbation, the perturbed propensity and outcome-regression functions define a valid finite observed-data model, i.e. take values in [0,1].","s":"∀ {K : ℕ} (P : Causalean.Estimation.MinimaxATE.GenConstr) (lam : Fin K → Bool), Causalean.Estimation.MinimaxATE.ValidDGP (P.mPertG lam) (P.gPertG lam)","l":133,"f":"Causalean/Estimation/MinimaxATE/ConstCenterGeneral/Construction.lean"},{"n":"Causalean.Estimation.MinimaxATE.GenConstr.Γ","k":"def","m":"Causalean.Estimation.MinimaxATE.ConstCenterGeneral.ChiSqOverlap","a":"Estimation/MinimaxATE/ConstCenterGeneral","t":1,"d":"The per-cell χ²-overlap coefficient.","s":"Causalean.Estimation.MinimaxATE.GenConstr → ℝ","l":50,"f":"Causalean/Estimation/MinimaxATE/ConstCenterGeneral/ChiSqOverlap.lean"},{"n":"Causalean.Estimation.MinimaxATE.GenConstr.Γ_nonneg","k":"theorem","m":"Causalean.Estimation.MinimaxATE.ConstCenterGeneral.LowerBound","a":"Estimation/MinimaxATE/ConstCenterGeneral","t":2,"d":"The per-cell overlap coefficient is nonnegative.","s":"∀ (P : Causalean.Estimation.MinimaxATE.GenConstr), 0 ≤ P.Γ","l":59,"f":"Causalean/Estimation/MinimaxATE/ConstCenterGeneral/LowerBound.lean"},{"n":"Causalean.Estimation.MinimaxATE.InClass","k":"structure","m":"Causalean.Estimation.MinimaxATE.Model","a":"Estimation/MinimaxATE","t":1,"d":"Structure-agnostic nuisance class. A candidate data-generating process (m, g) belongs to the class ℱ(εg, εm) around fixed nuisance estimates (mhat, ghat) when it is a valid DGP, with propensity and both outcome-regression arms taking values in [0,1], each outcome-regression arm lies within squared L²(P_X) distance εg of the corresponding estimated arm, and the propensity lies within squared L²(P_X) distance εm of the estimated propensity.","s":"{C : Type u_1} → [Fintype C] → (C → ℝ) → (Bool → C → ℝ) → ℝ → ℝ → (C → ℝ) → (Bool → C → ℝ) → Prop","l":146,"f":"Causalean/Estimation/MinimaxATE/Model.lean"},{"n":"Causalean.Estimation.MinimaxATE.InClassDGP","k":"def","m":"Causalean.Estimation.MinimaxATE.Model","a":"Estimation/MinimaxATE","t":1,"d":"A DGP packaged with a proof that it lies in the class ℱ(εg, εm).","s":"{C : Type u_1} → [Fintype C] → (C → ℝ) → (Bool → C → ℝ) → ℝ → ℝ → Type u_1","l":158,"f":"Causalean/Estimation/MinimaxATE/Model.lean"},{"n":"Causalean.Estimation.MinimaxATE.MinimaxRateThreshold","k":"structure","m":"Causalean.Estimation.MinimaxATE.Optimality","a":"Estimation/MinimaxATE","t":1,"d":"Minimax rate threshold (reusable vocabulary). An estimation problem over the finite model exhibits a rate threshold when a *single* estimator's worst-case miss is bounded above at one separation (sepUpper), while *every* estimator's worst-case miss is bounded below at a (smaller) separation (sepLower). When sepLower and sepUpper are of the same order, this certifies the minimax rate and that estimator attains it.","s":"{C : Type u_1} → [Fintype C] → [MeasurableSpace C] → [Nonempty C] → (C → ℝ) → (Bool → C → ℝ) → ℝ → ℝ → ℕ → Type u_1","l":155,"f":"Causalean/Estimation/MinimaxATE/Optimality.lean"},{"n":"Causalean.Estimation.MinimaxATE.Obs","k":"def","m":"Causalean.Estimation.MinimaxATE.Model","a":"Estimation/MinimaxATE","t":1,"d":"One observation: covariate X : C, treatment D : Bool, outcome Y : Bool.","s":"Type u_1 → Type (max u_1 0)","l":74,"f":"Causalean/Estimation/MinimaxATE/Model.lean"},{"n":"Causalean.Estimation.MinimaxATE.Parametric.absolutelyContinuous_of_singleton_pos","k":"theorem","m":"Causalean.Estimation.MinimaxATE.ConstCenterHalf.Parametric","a":"Estimation/MinimaxATE/ConstCenterHalf","t":2,"d":"Absolute continuity from full support. If ν charges every singleton, every measure is absolutely continuous w.r.t. ν.","s":"∀ {Ω : Type u_2} [inst : MeasurableSpace Ω] (μ ν : MeasureTheory.Measure Ω), (∀ (x : Ω), ν {x} ≠ 0) → μ.AbsolutelyContinuous ν","l":152,"f":"Causalean/Estimation/MinimaxATE/ConstCenterHalf/Parametric.lean"},{"n":"Causalean.Estimation.MinimaxATE.Parametric.ate_gNull","k":"theorem","m":"Causalean.Estimation.MinimaxATE.ConstCenterHalf.Parametric","a":"Estimation/MinimaxATE/ConstCenterHalf","t":2,"d":"The ATE of the null DGP is g₁ − g₀.","s":"∀ {C : Type u_1} [inst : Fintype C] [Nonempty C] {g₀ g₁ : ℝ}, Causalean.Estimation.MinimaxATE.ate (Causalean.Estimation.MinimaxATE.Parametric.gNull g₀ g₁) = g₁ - g₀","l":101,"f":"Causalean/Estimation/MinimaxATE/ConstCenterHalf/Parametric.lean"},{"n":"Causalean.Estimation.MinimaxATE.Parametric.ate_gPert","k":"theorem","m":"Causalean.Estimation.MinimaxATE.ConstCenterHalf.Parametric","a":"Estimation/MinimaxATE/ConstCenterHalf","t":2,"d":"The ATE of the perturbed DGP is (g₁ + δ) − g₀.","s":"∀ {C : Type u_1} [inst : Fintype C] [Nonempty C] {g₀ g₁ δ : ℝ}, Causalean.Estimation.MinimaxATE.ate (Causalean.Estimation.MinimaxATE.Parametric.gPert g₀ g₁ δ) = g₁ + δ - g₀","l":109,"f":"Causalean/Estimation/MinimaxATE/ConstCenterHalf/Parametric.lean"},{"n":"Causalean.Estimation.MinimaxATE.Parametric.chiSqDiv_obsPert_obsNull_eq","k":"theorem","m":"Causalean.Estimation.MinimaxATE.ConstCenterHalf.Parametric","a":"Estimation/MinimaxATE/ConstCenterHalf","t":2,"d":"The single-observation χ² is nonnegative and equals m₀ δ²/(g₁(1−g₁)).","s":"∀ {C : Type u_1} [inst : Fintype C] [inst_1 : Nonempty C] [inst_2 : MeasurableSpace C] [MeasurableSingletonClass C] {m₀ g₀ g₁ δ : ℝ} (hvN : Causalean.Estimation.MinimaxATE.ValidDGP (Causalean.Estimation.MinimaxATE.Parametric.mC m₀) (Causalean.Estimation.MinimaxATE.Parametric.gNull g₀ g₁)) (hvP : Causalean.Estimation.MinimaxATE.ValidDGP (Causalean.Estimation.MinimaxATE.Parametric.mC m₀) (Causalean.Estimation.MinimaxATE.Parametric.gPert g₀ g₁ δ)), 0 < m₀ → m₀ < 1 → 0 < g₀ → g₀ < 1 → 0 < g₁ → g₁ < 1 → Causalean.Stat.chiSqDiv (Causalean.Estimation.MinimaxATE.obsLaw hvP) (Causalean.Estimation.MinimaxATE.obsLaw hvN) = m₀ * δ ^ 2 / (g₁ * (1 - g₁))","l":227,"f":"Causalean/Estimation/MinimaxATE/ConstCenterHalf/Parametric.lean"},{"n":"Causalean.Estimation.MinimaxATE.Parametric.chiSqDiv_productLaw_le_one","k":"theorem","m":"Causalean.Estimation.MinimaxATE.ConstCenterHalf.Parametric","a":"Estimation/MinimaxATE/ConstCenterHalf","t":2,"d":"χ² indistinguishability of the n-sample laws. In the regime n · m₀ δ²/(g₁(1−g₁)) ≤ log 2, the χ²-divergence of the n-sample perturbed law from the null is at most 1. Proved by tensorization (one_add_chiSqDiv_pi_iid) and (1 + x)^n ≤ exp(n x) ≤ 2.","s":"∀ {C : Type u_1} [inst : Fintype C] [inst_1 : Nonempty C] [inst_2 : MeasurableSpace C] [MeasurableSingletonClass C] {m₀ g₀ g₁ δ : ℝ} {n : ℕ} (hvN : Causalean.Estimation.MinimaxATE.ValidDGP (Causalean.Estimation.MinimaxATE.Parametric.mC m₀) (Causalean.Estimation.MinimaxATE.Parametric.gNull g₀ g₁)) (hvP : Causalean.Estimation.MinimaxATE.ValidDGP (Causalean.Estimation.MinimaxATE.Parametric.mC m₀) (Causalean.Estimation.MinimaxATE.Parametric.gPert g₀ g₁ δ)), 0 < m₀ → m₀ < 1 → 0 < g₀ → g₀ < 1 → 0 < g₁ → g₁ < 1 → ↑n * (m₀ * δ ^ 2 / (g₁ * (1 - g₁))) ≤ Real.log 2 → Causalean.Stat.chiSqDiv (Causalean.Estimation.MinimaxATE.productLaw hvP n) (Causalean.Estimation.MinimaxATE.productLaw hvN n) ≤ 1","l":237,"f":"Causalean/Estimation/MinimaxATE/ConstCenterHalf/Parametric.lean"},{"n":"Causalean.Estimation.MinimaxATE.Parametric.gNull","k":"def","m":"Causalean.Estimation.MinimaxATE.ConstCenterHalf.Parametric","a":"Estimation/MinimaxATE/ConstCenterHalf","t":1,"d":"The null outcome regression: g₁ on the treated arm, g₀ on the control arm.","s":"{C : Type u_1} → ℝ → ℝ → Bool → C → ℝ","l":76,"f":"Causalean/Estimation/MinimaxATE/ConstCenterHalf/Parametric.lean"},{"n":"Causalean.Estimation.MinimaxATE.Parametric.gPert","k":"def","m":"Causalean.Estimation.MinimaxATE.ConstCenterHalf.Parametric","a":"Estimation/MinimaxATE/ConstCenterHalf","t":1,"d":"The perturbed outcome regression: the treated arm is shifted to g₁ + δ.","s":"{C : Type u_1} → ℝ → ℝ → ℝ → Bool → C → ℝ","l":79,"f":"Causalean/Estimation/MinimaxATE/ConstCenterHalf/Parametric.lean"},{"n":"Causalean.Estimation.MinimaxATE.Parametric.inClass_null","k":"theorem","m":"Causalean.Estimation.MinimaxATE.ConstCenterHalf.Parametric","a":"Estimation/MinimaxATE/ConstCenterHalf","t":2,"d":"The null DGP lies in the class around (mC, gNull) (zero nuisance error).","s":"∀ {C : Type u_1} [inst : Fintype C] {m₀ g₀ g₁ εg εm : ℝ}, Causalean.Estimation.MinimaxATE.ValidDGP (Causalean.Estimation.MinimaxATE.Parametric.mC m₀) (Causalean.Estimation.MinimaxATE.Parametric.gNull g₀ g₁) → 0 ≤ εg → 0 ≤ εm → Causalean.Estimation.MinimaxATE.InClass (Causalean.Estimation.MinimaxATE.Parametric.mC m₀) (Causalean.Estimation.MinimaxATE.Parametric.gNull g₀ g₁) εg εm (Causalean.Estimation.MinimaxATE.Parametric.mC m₀) (Causalean.Estimation.MinimaxATE.Parametric.gNull g₀ g₁)","l":131,"f":"Causalean/Estimation/MinimaxATE/ConstCenterHalf/Parametric.lean"},{"n":"Causalean.Estimation.MinimaxATE.Parametric.inClass_pert","k":"theorem","m":"Causalean.Estimation.MinimaxATE.ConstCenterHalf.Parametric","a":"Estimation/MinimaxATE/ConstCenterHalf","t":2,"d":"The perturbed DGP lies in the class around (mC, gNull) when δ² ≤ εg.","s":"∀ {C : Type u_1} [inst : Fintype C] [Nonempty C] {m₀ g₀ g₁ δ εg εm : ℝ}, Causalean.Estimation.MinimaxATE.ValidDGP (Causalean.Estimation.MinimaxATE.Parametric.mC m₀) (Causalean.Estimation.MinimaxATE.Parametric.gPert g₀ g₁ δ) → δ ^ 2 ≤ εg → 0 ≤ εm → Causalean.Estimation.MinimaxATE.InClass (Causalean.Estimation.MinimaxATE.Parametric.mC m₀) (Causalean.Estimation.MinimaxATE.Parametric.gNull g₀ g₁) εg εm (Causalean.Estimation.MinimaxATE.Parametric.mC m₀) (Causalean.Estimation.MinimaxATE.Parametric.gPert g₀ g₁ δ)","l":140,"f":"Causalean/Estimation/MinimaxATE/ConstCenterHalf/Parametric.lean"},{"n":"Causalean.Estimation.MinimaxATE.Parametric.l2sq_gPert_gNull","k":"theorem","m":"Causalean.Estimation.MinimaxATE.ConstCenterHalf.Parametric","a":"Estimation/MinimaxATE/ConstCenterHalf","t":2,"d":"The squared L² distance between the perturbed and null treated arms is δ²; control arms agree. Hence each arm's error from gNull is ≤ εg once δ² ≤ εg.","s":"∀ {C : Type u_1} [inst : Fintype C] [Nonempty C] {g₀ g₁ δ : ℝ} (d : Bool), Causalean.Estimation.MinimaxATE.l2sq (Causalean.Estimation.MinimaxATE.Parametric.gPert g₀ g₁ δ d) (Causalean.Estimation.MinimaxATE.Parametric.gNull g₀ g₁ d) = if d = true then δ ^ 2 else 0","l":117,"f":"Causalean/Estimation/MinimaxATE/ConstCenterHalf/Parametric.lean"},{"n":"Causalean.Estimation.MinimaxATE.Parametric.mC","k":"def","m":"Causalean.Estimation.MinimaxATE.ConstCenterHalf.Parametric","a":"Estimation/MinimaxATE/ConstCenterHalf","t":1,"d":"The constant propensity center m ≡ m₀.","s":"{C : Type u_1} → ℝ → C → ℝ","l":73,"f":"Causalean/Estimation/MinimaxATE/ConstCenterHalf/Parametric.lean"},{"n":"Causalean.Estimation.MinimaxATE.Parametric.obsLaw_null_singleton_ne_zero","k":"theorem","m":"Causalean.Estimation.MinimaxATE.ConstCenterHalf.Parametric","a":"Estimation/MinimaxATE/ConstCenterHalf","t":2,"d":"The null single-observation law charges every point (its mass is positive).","s":"∀ {C : Type u_1} [inst : Fintype C] [inst_1 : Nonempty C] [inst_2 : MeasurableSpace C] [MeasurableSingletonClass C] {m₀ g₀ g₁ : ℝ} (hv : Causalean.Estimation.MinimaxATE.ValidDGP (Causalean.Estimation.MinimaxATE.Parametric.mC m₀) (Causalean.Estimation.MinimaxATE.Parametric.gNull g₀ g₁)), 0 < m₀ → m₀ < 1 → 0 < g₀ → g₀ < 1 → 0 < g₁ → g₁ < 1 → ∀ (z : Causalean.Estimation.MinimaxATE.Obs C), (Causalean.Estimation.MinimaxATE.obsLaw hv) {z} ≠ 0","l":163,"f":"Causalean/Estimation/MinimaxATE/ConstCenterHalf/Parametric.lean"},{"n":"Causalean.Estimation.MinimaxATE.Parametric.one_add_chiSqDiv_obsPert_obsNull","k":"theorem","m":"Causalean.Estimation.MinimaxATE.ConstCenterHalf.Parametric","a":"Estimation/MinimaxATE/ConstCenterHalf","t":2,"d":"Single-observation χ² (closed form). With μ = obsLaw pert, ν = obsLaw null, the (shifted) χ²-divergence is 1 + m₀ δ²/(g₁(1−g₁)).","s":"∀ {C : Type u_1} [inst : Fintype C] [inst_1 : Nonempty C] [inst_2 : MeasurableSpace C] [MeasurableSingletonClass C] {m₀ g₀ g₁ δ : ℝ} (hvN : Causalean.Estimation.MinimaxATE.ValidDGP (Causalean.Estimation.MinimaxATE.Parametric.mC m₀) (Causalean.Estimation.MinimaxATE.Parametric.gNull g₀ g₁)) (hvP : Causalean.Estimation.MinimaxATE.ValidDGP (Causalean.Estimation.MinimaxATE.Parametric.mC m₀) (Causalean.Estimation.MinimaxATE.Parametric.gPert g₀ g₁ δ)), 0 < m₀ → m₀ < 1 → 0 < g₀ → g₀ < 1 → 0 < g₁ → g₁ < 1 → 1 + Causalean.Stat.chiSqDiv (Causalean.Estimation.MinimaxATE.obsLaw hvP) (Causalean.Estimation.MinimaxATE.obsLaw hvN) = 1 + m₀ * δ ^ 2 / (g₁ * (1 - g₁))","l":182,"f":"Causalean/Estimation/MinimaxATE/ConstCenterHalf/Parametric.lean"},{"n":"Causalean.Estimation.MinimaxATE.Parametric.tvDist_productLaw_le_half","k":"theorem","m":"Causalean.Estimation.MinimaxATE.ConstCenterHalf.Parametric","a":"Estimation/MinimaxATE/ConstCenterHalf","t":2,"d":"Total-variation indistinguishability of the n-sample laws.","s":"∀ {C : Type u_1} [inst : Fintype C] [inst_1 : Nonempty C] [inst_2 : MeasurableSpace C] [MeasurableSingletonClass C] {m₀ g₀ g₁ δ : ℝ} {n : ℕ} (hvN : Causalean.Estimation.MinimaxATE.ValidDGP (Causalean.Estimation.MinimaxATE.Parametric.mC m₀) (Causalean.Estimation.MinimaxATE.Parametric.gNull g₀ g₁)) (hvP : Causalean.Estimation.MinimaxATE.ValidDGP (Causalean.Estimation.MinimaxATE.Parametric.mC m₀) (Causalean.Estimation.MinimaxATE.Parametric.gPert g₀ g₁ δ)), 0 < m₀ → m₀ < 1 → 0 < g₀ → g₀ < 1 → 0 < g₁ → g₁ < 1 → ↑n * (m₀ * δ ^ 2 / (g₁ * (1 - g₁))) ≤ Real.log 2 → Causalean.Stat.tvDist (Causalean.Estimation.MinimaxATE.productLaw hvN n) (Causalean.Estimation.MinimaxATE.productLaw hvP n) ≤ 1 / 2","l":277,"f":"Causalean/Estimation/MinimaxATE/ConstCenterHalf/Parametric.lean"},{"n":"Causalean.Estimation.MinimaxATE.Parametric.validDGP_null","k":"theorem","m":"Causalean.Estimation.MinimaxATE.ConstCenterHalf.Parametric","a":"Estimation/MinimaxATE/ConstCenterHalf","t":2,"d":"Validity of the null DGP.","s":"∀ {C : Type u_1} {m₀ g₀ g₁ : ℝ}, 0 < m₀ → m₀ < 1 → 0 < g₀ → g₀ < 1 → 0 < g₁ → g₁ < 1 → Causalean.Estimation.MinimaxATE.ValidDGP (Causalean.Estimation.MinimaxATE.Parametric.mC m₀) (Causalean.Estimation.MinimaxATE.Parametric.gNull g₀ g₁)","l":85,"f":"Causalean/Estimation/MinimaxATE/ConstCenterHalf/Parametric.lean"},{"n":"Causalean.Estimation.MinimaxATE.Parametric.validDGP_pert","k":"theorem","m":"Causalean.Estimation.MinimaxATE.ConstCenterHalf.Parametric","a":"Estimation/MinimaxATE/ConstCenterHalf","t":2,"d":"Validity of the perturbed DGP (using 0 ≤ δ and g₁ + δ ≤ 1).","s":"∀ {C : Type u_1} {m₀ g₀ g₁ δ : ℝ}, 0 < m₀ → m₀ < 1 → 0 < g₀ → g₀ < 1 → 0 < g₁ → 0 ≤ δ → g₁ + δ ≤ 1 → Causalean.Estimation.MinimaxATE.ValidDGP (Causalean.Estimation.MinimaxATE.Parametric.mC m₀) (Causalean.Estimation.MinimaxATE.Parametric.gPert g₀ g₁ δ)","l":93,"f":"Causalean/Estimation/MinimaxATE/ConstCenterHalf/Parametric.lean"},{"n":"Causalean.Estimation.MinimaxATE.Qfalse","k":"def","m":"Causalean.Estimation.MinimaxATE.ConstCenterHalf.ExplicitWitness","a":"Estimation/MinimaxATE/ConstCenterHalf","t":1,"d":"The n-sample law of the centered null data-generating process.","s":"(K n : ℕ) → [NeZero K] → MeasureTheory.Measure (Fin n → Causalean.Estimation.MinimaxATE.Obs (Fin K × Bool))","l":65,"f":"Causalean/Estimation/MinimaxATE/ConstCenterHalf/ExplicitWitness.lean"},{"n":"Causalean.Estimation.MinimaxATE.QfalseCont","k":"def","m":"Causalean.Estimation.MinimaxATE.ConstCenterHalf.ContinuousX","a":"Estimation/MinimaxATE/ConstCenterHalf","t":1,"d":"Continuous-covariate null law, pairing the finite-cell null sample with ancillary positions.","s":"{W : Type u_1} → [inst : MeasurableSpace W] → (K n : ℕ) → [NeZero K] → MeasureTheory.Measure W → MeasureTheory.Measure ((Fin n → Causalean.Estimation.MinimaxATE.Obs (Fin K × Bool)) × (Fin n → W))","l":64,"f":"Causalean/Estimation/MinimaxATE/ConstCenterHalf/ContinuousX.lean"},{"n":"Causalean.Estimation.MinimaxATE.QfalseCont_isProb","k":"theorem","m":"Causalean.Estimation.MinimaxATE.ConstCenterHalf.ContinuousX","a":"Estimation/MinimaxATE/ConstCenterHalf","t":2,"d":"The continuous-covariate null law is a probability measure.","s":"∀ {K n : ℕ} {W : Type u_1} [inst : MeasurableSpace W] [inst_1 : NeZero K] (ρ : MeasureTheory.Measure W) [MeasureTheory.IsProbabilityMeasure ρ], MeasureTheory.IsProbabilityMeasure (Causalean.Estimation.MinimaxATE.QfalseCont K n ρ)","l":78,"f":"Causalean/Estimation/MinimaxATE/ConstCenterHalf/ContinuousX.lean"},{"n":"Causalean.Estimation.MinimaxATE.Qfalse_isProb","k":"theorem","m":"Causalean.Estimation.MinimaxATE.ConstCenterHalf.ExplicitWitness","a":"Estimation/MinimaxATE/ConstCenterHalf","t":2,"d":"The centered null n-sample law is a probability measure.","s":"∀ (K n : ℕ) [inst : NeZero K], MeasureTheory.IsProbabilityMeasure (Causalean.Estimation.MinimaxATE.Qfalse K n)","l":90,"f":"Causalean/Estimation/MinimaxATE/ConstCenterHalf/ExplicitWitness.lean"},{"n":"Causalean.Estimation.MinimaxATE.Qfalse_singleton_ne_zero","k":"theorem","m":"Causalean.Estimation.MinimaxATE.ConstCenterHalf.ChiSquaredCore","a":"Estimation/MinimaxATE/ConstCenterHalf","t":2,"d":"The null n-sample law charges every point (its mass is (8K)⁻ⁿ > 0).","s":"∀ {K n : ℕ} [inst : NeZero K] (ω : Fin n → Causalean.Estimation.MinimaxATE.Obs (Fin K × Bool)), (Causalean.Estimation.MinimaxATE.Qfalse K n) {ω} ≠ 0","l":74,"f":"Causalean/Estimation/MinimaxATE/ConstCenterHalf/ChiSquaredCore.lean"},{"n":"Causalean.Estimation.MinimaxATE.Qpert","k":"def","m":"Causalean.Estimation.MinimaxATE.ConstCenterHalf.ExplicitWitness","a":"Estimation/MinimaxATE/ConstCenterHalf","t":1,"d":"The n-sample law for one perturbed data-generating process indexed by a sign vector.","s":"{K : ℕ} → {α β : ℝ} → [NeZero K] → 0 ≤ α → 0 ≤ β → α + 2 * β ≤ 1 / 2 → (n : ℕ) → (Fin K → Bool) → MeasureTheory.Measure (Fin n → Causalean.Estimation.MinimaxATE.Obs (Fin K × Bool))","l":73,"f":"Causalean/Estimation/MinimaxATE/ConstCenterHalf/ExplicitWitness.lean"},{"n":"Causalean.Estimation.MinimaxATE.Qpert_isProb","k":"theorem","m":"Causalean.Estimation.MinimaxATE.ConstCenterHalf.ExplicitWitness","a":"Estimation/MinimaxATE/ConstCenterHalf","t":2,"d":"Each perturbed n-sample law is a probability measure in the valid parameter regime.","s":"∀ {K : ℕ} {α β : ℝ} [inst : NeZero K] (hα : 0 ≤ α) (hβ : 0 ≤ β) (hαβ : α + 2 * β ≤ 1 / 2) (n : ℕ) (lam : Fin K → Bool), MeasureTheory.IsProbabilityMeasure (Causalean.Estimation.MinimaxATE.Qpert hα hβ hαβ n lam)","l":94,"f":"Causalean/Estimation/MinimaxATE/ConstCenterHalf/ExplicitWitness.lean"},{"n":"Causalean.Estimation.MinimaxATE.Qtrue","k":"def","m":"Causalean.Estimation.MinimaxATE.ConstCenterHalf.ExplicitWitness","a":"Estimation/MinimaxATE/ConstCenterHalf","t":1,"d":"Alternative n-sample law formed by uniformly mixing over all perturbation signs.","s":"{K : ℕ} → {α β : ℝ} → [NeZero K] → 0 ≤ α → 0 ≤ β → α + 2 * β ≤ 1 / 2 → (n : ℕ) → MeasureTheory.Measure (Fin n → Causalean.Estimation.MinimaxATE.Obs (Fin K × Bool))","l":78,"f":"Causalean/Estimation/MinimaxATE/ConstCenterHalf/ExplicitWitness.lean"},{"n":"Causalean.Estimation.MinimaxATE.QtrueCont","k":"def","m":"Causalean.Estimation.MinimaxATE.ConstCenterHalf.ContinuousX","a":"Estimation/MinimaxATE/ConstCenterHalf","t":1,"d":"Continuous-covariate alternative law, pairing the finite-cell mixture with ancillary positions.","s":"{K : ℕ} → {α β : ℝ} → {W : Type u_1} → [inst : MeasurableSpace W] → [NeZero K] → 0 ≤ α → 0 ≤ β → α + 2 * β ≤ 1 / 2 → (n : ℕ) → MeasureTheory.Measure W → MeasureTheory.Measure ((Fin n → Causalean.Estimation.MinimaxATE.Obs (Fin K × Bool)) × (Fin n → W))","l":70,"f":"Causalean/Estimation/MinimaxATE/ConstCenterHalf/ContinuousX.lean"},{"n":"Causalean.Estimation.MinimaxATE.QtrueCont_isProb","k":"theorem","m":"Causalean.Estimation.MinimaxATE.ConstCenterHalf.ContinuousX","a":"Estimation/MinimaxATE/ConstCenterHalf","t":2,"d":"The continuous-covariate alternative law is a probability measure in the valid parameter regime.","s":"∀ {K n : ℕ} {α β : ℝ} {W : Type u_1} [inst : MeasurableSpace W] [inst_1 : NeZero K] (hα : 0 ≤ α) (hβ : 0 ≤ β) (hαβ : α + 2 * β ≤ 1 / 2) (ρ : MeasureTheory.Measure W) [MeasureTheory.IsProbabilityMeasure ρ], MeasureTheory.IsProbabilityMeasure (Causalean.Estimation.MinimaxATE.QtrueCont hα hβ hαβ n ρ)","l":84,"f":"Causalean/Estimation/MinimaxATE/ConstCenterHalf/ContinuousX.lean"},{"n":"Causalean.Estimation.MinimaxATE.Qtrue_isProb","k":"theorem","m":"Causalean.Estimation.MinimaxATE.ConstCenterHalf.ExplicitWitness","a":"Estimation/MinimaxATE/ConstCenterHalf","t":2,"d":"The uniformly mixed alternative n-sample law is a probability measure in the valid parameter regime.","s":"∀ {K : ℕ} {α β : ℝ} [inst : NeZero K] (hα : 0 ≤ α) (hβ : 0 ≤ β) (hαβ : α + 2 * β ≤ 1 / 2) (n : ℕ), MeasureTheory.IsProbabilityMeasure (Causalean.Estimation.MinimaxATE.Qtrue hα hβ hαβ n)","l":99,"f":"Causalean/Estimation/MinimaxATE/ConstCenterHalf/ExplicitWitness.lean"},{"n":"Causalean.Estimation.MinimaxATE.Qtrue_real_singleton","k":"theorem","m":"Causalean.Estimation.MinimaxATE.ConstCenterHalf.ChiSquaredCore","a":"Estimation/MinimaxATE/ConstCenterHalf","t":2,"d":"The alternative law's .real point mass: a uniform mixture over sign vectors.","s":"∀ {K n : ℕ} {α β : ℝ} [inst : NeZero K] (hα : 0 ≤ α) (hβ : 0 ≤ β) (hαβ : α + 2 * β ≤ 1 / 2) (ω : Fin n → Causalean.Estimation.MinimaxATE.Obs (Fin K × Bool)), (Causalean.Estimation.MinimaxATE.Qtrue hα hβ hαβ n).real {ω} = ∑ lam, (2 ^ K)⁻¹ * ∏ i, Causalean.Estimation.MinimaxATE.obsReal (Causalean.Estimation.MinimaxATE.mPerturbed β lam) (Causalean.Estimation.MinimaxATE.gPerturbed α β lam) (ω i)","l":99,"f":"Causalean/Estimation/MinimaxATE/ConstCenterHalf/ChiSquaredCore.lean"},{"n":"Causalean.Estimation.MinimaxATE.TwoPointWitness","k":"structure","m":"Causalean.Estimation.MinimaxATE.Reduction.Witness","a":"Estimation/MinimaxATE/Reduction","t":1,"d":"Two-point (Le Cam) witness for the structure-agnostic ATE lower bound: it bundles two n-sample data laws, indexed by a hypothesis label j : Bool, together with the data needed to run Le Cam's two-point method. For each hypothesis, Q j is a probability measure with true average treatment effect θ j; the two hypotheses' ATE values are separated by at least twice the half-scale s, while the total-variation distance between the two laws is bounded by c, and for every estimator the probability under Q j of missing θ j by s is dominated by the in-class minimax miss probability — the realizability condition that a mixture-of-in-class-DGPs construction discharges.","s":"(C : Type u_2) → [Fintype C] → [Nonempty C] → [MeasurableSpace C] → ℕ → (C → ℝ) → (Bool → C → ℝ) → ℝ → ℝ → Type u_2","l":60,"f":"Causalean/Estimation/MinimaxATE/Reduction/Witness.lean"},{"n":"Causalean.Estimation.MinimaxATE.TwoPointWitnessCausal","k":"structure","m":"Causalean.Estimation.MinimaxATE.Causal.Minimax","a":"Estimation/MinimaxATE/Causal","t":1,"d":"Two-point Le Cam witness whose realizability target is minimaxMissCausal.","s":"(C : Type) → [Fintype C] → [Nonempty C] → [inst : MeasurableSpace C] → [MeasurableSingletonClass C] → ℕ → (C → ℝ) → (Bool → C → ℝ) → ℝ → ℝ → Type","l":78,"f":"Causalean/Estimation/MinimaxATE/Causal/Minimax.lean"},{"n":"Causalean.Estimation.MinimaxATE.ValidDGP","k":"structure","m":"Causalean.Estimation.MinimaxATE.Model","a":"Estimation/MinimaxATE","t":1,"d":"A data-generating process (m, g) is valid when the propensity m takes values in the unit interval [0,1] and each outcome-regression arm g d also takes values in [0,1].","s":"{C : Type u_1} → (C → ℝ) → (Bool → C → ℝ) → Prop","l":79,"f":"Causalean/Estimation/MinimaxATE/Model.lean"},{"n":"Causalean.Estimation.MinimaxATE.VarConstr","k":"structure","m":"Causalean.Estimation.MinimaxATE.VaryingCenterCase1.Construction","a":"Estimation/MinimaxATE/VaryingCenterCase1","t":1,"d":"Cell-varying-center construction data for a Rademacher perturbation whose nuisance center may vary by paired cell (Jin–Syrgkanis 2024, Assumption 2, restricted to centers constant within a pair). It packages two bump-magnitude scalars, one on the treated outcome arm and one on the propensity, together with a nuisance center given by the pair-indexed functions m₀, g₀ and g₁ for the propensity and the two potential-outcome regressions, plus the inequalities certifying that both bump magnitudes are nonnegative, the center is pointwise strictly inside (0,1) for m₀, g₀ and g₁, the propensity bump is smaller than the treated-arm center, keeping the perturbation denominator positive, the treated-arm bump does not exceed the treated center, keeping the perturbed outcome regression nonnegative, and worst-case bounds forcing the perturbed outcome regression to stay at most one and the perturbed propensity to stay at most one.","s":"ℕ → Type","l":54,"f":"Causalean/Estimation/MinimaxATE/VaryingCenterCase1/Construction.lean"},{"n":"Causalean.Estimation.MinimaxATE.VarConstr.QfalseV","k":"def","m":"Causalean.Estimation.MinimaxATE.VaryingCenterCase1.LowerBound","a":"Estimation/MinimaxATE/VaryingCenterCase1","t":1,"d":"The null n-sample law P̂^⊗n.","s":"{K : ℕ} → Causalean.Estimation.MinimaxATE.VarConstr K → (n : ℕ) → [NeZero K] → MeasureTheory.Measure (Fin n → Causalean.Estimation.MinimaxATE.Obs (Fin K × Bool))","l":65,"f":"Causalean/Estimation/MinimaxATE/VaryingCenterCase1/LowerBound.lean"},{"n":"Causalean.Estimation.MinimaxATE.VarConstr.QfalseV_isProb","k":"theorem","m":"Causalean.Estimation.MinimaxATE.VaryingCenterCase1.LowerBound","a":"Estimation/MinimaxATE/VaryingCenterCase1","t":2,"d":"The null sample law is a probability measure.","s":"∀ {K : ℕ} (P : Causalean.Estimation.MinimaxATE.VarConstr K) (n : ℕ) [inst : NeZero K], MeasureTheory.IsProbabilityMeasure (P.QfalseV n)","l":80,"f":"Causalean/Estimation/MinimaxATE/VaryingCenterCase1/LowerBound.lean"},{"n":"Causalean.Estimation.MinimaxATE.VarConstr.QfalseV_singleton_ne_zero","k":"theorem","m":"Causalean.Estimation.MinimaxATE.VaryingCenterCase1.LowerBound","a":"Estimation/MinimaxATE/VaryingCenterCase1","t":2,"d":"The null n-sample law charges every point.","s":"∀ {K : ℕ} (P : Causalean.Estimation.MinimaxATE.VarConstr K) {n : ℕ} [inst : NeZero K] (ω : Fin n → Causalean.Estimation.MinimaxATE.Obs (Fin K × Bool)), (P.QfalseV n) {ω} ≠ 0","l":107,"f":"Causalean/Estimation/MinimaxATE/VaryingCenterCase1/LowerBound.lean"},{"n":"Causalean.Estimation.MinimaxATE.VarConstr.QpertV","k":"def","m":"Causalean.Estimation.MinimaxATE.VaryingCenterCase1.LowerBound","a":"Estimation/MinimaxATE/VaryingCenterCase1","t":1,"d":"The perturbed n-sample law Qλ^⊗n.","s":"{K : ℕ} → Causalean.Estimation.MinimaxATE.VarConstr K → (n : ℕ) → [NeZero K] → (Fin K → Bool) → MeasureTheory.Measure (Fin n → Causalean.Estimation.MinimaxATE.Obs (Fin K × Bool))","l":70,"f":"Causalean/Estimation/MinimaxATE/VaryingCenterCase1/LowerBound.lean"},{"n":"Causalean.Estimation.MinimaxATE.VarConstr.QpertV_isProb","k":"theorem","m":"Causalean.Estimation.MinimaxATE.VaryingCenterCase1.LowerBound","a":"Estimation/MinimaxATE/VaryingCenterCase1","t":2,"d":"Each perturbed sample law is a probability measure.","s":"∀ {K : ℕ} (P : Causalean.Estimation.MinimaxATE.VarConstr K) (n : ℕ) [inst : NeZero K] (lam : Fin K → Bool), MeasureTheory.IsProbabilityMeasure (P.QpertV n lam)","l":84,"f":"Causalean/Estimation/MinimaxATE/VaryingCenterCase1/LowerBound.lean"},{"n":"Causalean.Estimation.MinimaxATE.VarConstr.QtrueV","k":"def","m":"Causalean.Estimation.MinimaxATE.VaryingCenterCase1.LowerBound","a":"Estimation/MinimaxATE/VaryingCenterCase1","t":1,"d":"The alternative n-sample law: the uniform Rademacher mixture of the perturbed laws.","s":"{K : ℕ} → Causalean.Estimation.MinimaxATE.VarConstr K → (n : ℕ) → [NeZero K] → MeasureTheory.Measure (Fin n → Causalean.Estimation.MinimaxATE.Obs (Fin K × Bool))","l":75,"f":"Causalean/Estimation/MinimaxATE/VaryingCenterCase1/LowerBound.lean"},{"n":"Causalean.Estimation.MinimaxATE.VarConstr.QtrueV_isProb","k":"theorem","m":"Causalean.Estimation.MinimaxATE.VaryingCenterCase1.LowerBound","a":"Estimation/MinimaxATE/VaryingCenterCase1","t":2,"d":"The Rademacher mixture of perturbed sample laws is a probability measure.","s":"∀ {K : ℕ} (P : Causalean.Estimation.MinimaxATE.VarConstr K) (n : ℕ) [inst : NeZero K], MeasureTheory.IsProbabilityMeasure (P.QtrueV n)","l":88,"f":"Causalean/Estimation/MinimaxATE/VaryingCenterCase1/LowerBound.lean"},{"n":"Causalean.Estimation.MinimaxATE.VarConstr.QtrueV_real_singleton","k":"theorem","m":"Causalean.Estimation.MinimaxATE.VaryingCenterCase1.LowerBound","a":"Estimation/MinimaxATE/VaryingCenterCase1","t":2,"d":"The alternative law's .real point mass: a uniform mixture over sign vectors.","s":"∀ {K : ℕ} (P : Causalean.Estimation.MinimaxATE.VarConstr K) {n : ℕ} [inst : NeZero K] (ω : Fin n → Causalean.Estimation.MinimaxATE.Obs (Fin K × Bool)), (P.QtrueV n).real {ω} = ∑ lam, (2 ^ K)⁻¹ * ∏ i, Causalean.Estimation.MinimaxATE.obsReal (P.mPertV lam) (P.gPertV lam) (ω i)","l":130,"f":"Causalean/Estimation/MinimaxATE/VaryingCenterCase1/LowerBound.lean"},{"n":"Causalean.Estimation.MinimaxATE.VarConstr.ate_gPertV","k":"theorem","m":"Causalean.Estimation.MinimaxATE.VaryingCenterCase1.Gap","a":"Estimation/MinimaxATE/VaryingCenterCase1","t":1,"d":"Exact ATE of the perturbed construction. For any Rademacher sign vector lam, the average treatment effect of the perturbed outcome regression equals the average over pairs j of 2g₁ⱼ(g₁ⱼ²+αβ)/(g₁ⱼ²−β²) − 2g₀ⱼ, independent of lam.","s":"∀ {K : ℕ} (P : Causalean.Estimation.MinimaxATE.VarConstr K) [NeZero K] (lam : Fin K → Bool), Causalean.Estimation.MinimaxATE.ate (P.gPertV lam) = (↑(Fintype.card (Fin K × Bool)))⁻¹ * ∑ j, (2 * P.g₁ j * (P.g₁ j ^ 2 + P.α * P.β) / (P.g₁ j ^ 2 - P.β ^ 2) - 2 * P.g₀ j)","l":99,"f":"Causalean/Estimation/MinimaxATE/VaryingCenterCase1/Gap.lean"},{"n":"Causalean.Estimation.MinimaxATE.VarConstr.ate_gapV","k":"theorem","m":"Causalean.Estimation.MinimaxATE.VaryingCenterCase1.Gap","a":"Estimation/MinimaxATE/VaryingCenterCase1","t":1,"d":"For any Rademacher sign vector lam, the gap between the perturbed construction's ATE and the null estimate's ATE equals 2β(α+β)/(2K) times the sum over pairs j of g₁ⱼ/(g₁ⱼ² − β²).","s":"∀ {K : ℕ} (P : Causalean.Estimation.MinimaxATE.VarConstr K) [NeZero K] (lam : Fin K → Bool), Causalean.Estimation.MinimaxATE.ate (P.gPertV lam) - Causalean.Estimation.MinimaxATE.ate P.ghatV = (↑(Fintype.card (Fin K × Bool)))⁻¹ * (2 * P.β * (P.α + P.β)) * ∑ j, P.g₁ j / (P.g₁ j ^ 2 - P.β ^ 2)","l":114,"f":"Causalean/Estimation/MinimaxATE/VaryingCenterCase1/Gap.lean"},{"n":"Causalean.Estimation.MinimaxATE.VarConstr.ate_gap_nonneg","k":"theorem","m":"Causalean.Estimation.MinimaxATE.VaryingCenterCase1.Gap","a":"Estimation/MinimaxATE/VaryingCenterCase1","t":2,"d":"The ATE gap is nonnegative.","s":"∀ {K : ℕ} (P : Causalean.Estimation.MinimaxATE.VarConstr K) [NeZero K] (lam : Fin K → Bool), 0 ≤ Causalean.Estimation.MinimaxATE.ate (P.gPertV lam) - Causalean.Estimation.MinimaxATE.ate P.ghatV","l":130,"f":"Causalean/Estimation/MinimaxATE/VaryingCenterCase1/Gap.lean"},{"n":"Causalean.Estimation.MinimaxATE.VarConstr.ate_ghatV","k":"theorem","m":"Causalean.Estimation.MinimaxATE.VaryingCenterCase1.Gap","a":"Estimation/MinimaxATE/VaryingCenterCase1","t":2,"d":"The null estimate ĝ has ATE (1/2K) Σⱼ 2(g₁ⱼ − g₀ⱼ).","s":"∀ {K : ℕ} (P : Causalean.Estimation.MinimaxATE.VarConstr K), Causalean.Estimation.MinimaxATE.ate P.ghatV = (↑(Fintype.card (Fin K × Bool)))⁻¹ * ∑ j, 2 * (P.g₁ j - P.g₀ j)","l":63,"f":"Causalean/Estimation/MinimaxATE/VaryingCenterCase1/Gap.lean"},{"n":"Causalean.Estimation.MinimaxATE.VarConstr.chiSqDiv_QtrueV_QfalseV_le_one","k":"theorem","m":"Causalean.Estimation.MinimaxATE.VaryingCenterCase1.LowerBound","a":"Estimation/MinimaxATE/VaryingCenterCase1","t":2,"d":"The χ² indistinguishability bound (cell-varying center).","s":"∀ {K : ℕ} (P : Causalean.Estimation.MinimaxATE.VarConstr K) {n : ℕ} [inst : NeZero K], ∑ j, P.ΓV j / ↑K ≤ 1 → ↑n ^ 2 / 2 * ∑ j, (P.ΓV j / ↑K) ^ 2 ≤ Real.log 2 → Causalean.Stat.chiSqDiv (P.QtrueV n) (P.QfalseV n) ≤ 1","l":183,"f":"Causalean/Estimation/MinimaxATE/VaryingCenterCase1/LowerBound.lean"},{"n":"Causalean.Estimation.MinimaxATE.VarConstr.chiSqOverlapV","k":"def","m":"Causalean.Estimation.MinimaxATE.VaryingCenterCase1.ChiSqOverlap","a":"Estimation/MinimaxATE/VaryingCenterCase1","t":1,"d":"The single-observation χ² second-moment overlap of two perturbed laws relative to the null.","s":"{K : ℕ} → Causalean.Estimation.MinimaxATE.VarConstr K → (Fin K → Bool) → (Fin K → Bool) → ℝ","l":67,"f":"Causalean/Estimation/MinimaxATE/VaryingCenterCase1/ChiSqOverlap.lean"},{"n":"Causalean.Estimation.MinimaxATE.VarConstr.chiSqOverlap_eqV","k":"theorem","m":"Causalean.Estimation.MinimaxATE.VaryingCenterCase1.ChiSqOverlap","a":"Estimation/MinimaxATE/VaryingCenterCase1","t":1,"d":"For any two Rademacher sign vectors lam and lam' indexing perturbed data-generating processes, the single-observation χ² overlap between them equals one plus the sum over pairs j of the per-pair coefficient ΓV j / K times the sign agreement between lam and lam' at pair j.","s":"∀ {K : ℕ} (P : Causalean.Estimation.MinimaxATE.VarConstr K) [NeZero K] (lam lam' : Fin K → Bool), P.chiSqOverlapV lam lam' = 1 + ∑ j, P.ΓV j / ↑K * (Causalean.Estimation.MinimaxATE.signOf (lam j) * Causalean.Estimation.MinimaxATE.signOf (lam' j))","l":94,"f":"Causalean/Estimation/MinimaxATE/VaryingCenterCase1/ChiSqOverlap.lean"},{"n":"Causalean.Estimation.MinimaxATE.VarConstr.denomV_pos","k":"theorem","m":"Causalean.Estimation.MinimaxATE.VaryingCenterCase1.Construction","a":"Estimation/MinimaxATE/VaryingCenterCase1","t":2,"d":"The denominator 1 − (β/g₁ j)·Δ is positive.","s":"∀ {K : ℕ} (P : Causalean.Estimation.MinimaxATE.VarConstr K) (lam : Fin K → Bool) (x : Fin K × Bool), 0 < 1 - P.β / P.g₁ x.1 * Causalean.Estimation.MinimaxATE.Δ lam x","l":122,"f":"Causalean/Estimation/MinimaxATE/VaryingCenterCase1/Construction.lean"},{"n":"Causalean.Estimation.MinimaxATE.VarConstr.g1sq_sub_betasq_pos","k":"theorem","m":"Causalean.Estimation.MinimaxATE.VaryingCenterCase1.Gap","a":"Estimation/MinimaxATE/VaryingCenterCase1","t":2,"d":"g₁ⱼ² − β² > 0 since 0 ≤ β < g₁ⱼ.","s":"∀ {K : ℕ} (P : Causalean.Estimation.MinimaxATE.VarConstr K) (j : Fin K), 0 < P.g₁ j ^ 2 - P.β ^ 2","l":46,"f":"Causalean/Estimation/MinimaxATE/VaryingCenterCase1/Gap.lean"},{"n":"Causalean.Estimation.MinimaxATE.VarConstr.gPertV","k":"def","m":"Causalean.Estimation.MinimaxATE.VaryingCenterCase1.Construction","a":"Estimation/MinimaxATE/VaryingCenterCase1","t":1,"d":"The perturbed outcome regression: control arm = g₀ x.1, treated arm gλ(1,x) = (g₁ x.1 + α·Δ)/(1 − (β/g₁ x.1)·Δ).","s":"{K : ℕ} → Causalean.Estimation.MinimaxATE.VarConstr K → (Fin K → Bool) → Bool → Fin K × Bool → ℝ","l":117,"f":"Causalean/Estimation/MinimaxATE/VaryingCenterCase1/Construction.lean"},{"n":"Causalean.Estimation.MinimaxATE.VarConstr.gPertV_true_eq","k":"theorem","m":"Causalean.Estimation.MinimaxATE.VaryingCenterCase1.Gap","a":"Estimation/MinimaxATE/VaryingCenterCase1","t":2,"d":"Cleared treated arm. gλ(1,x) = (g₁ x.1² + α·g₁ x.1·Δ)/(g₁ x.1 − β·Δ).","s":"∀ {K : ℕ} (P : Causalean.Estimation.MinimaxATE.VarConstr K) (lam : Fin K → Bool) (x : Fin K × Bool), P.gPertV lam true x = (P.g₁ x.1 ^ 2 + P.α * P.g₁ x.1 * Causalean.Estimation.MinimaxATE.Δ lam x) / (P.g₁ x.1 - P.β * Causalean.Estimation.MinimaxATE.Δ lam x)","l":50,"f":"Causalean/Estimation/MinimaxATE/VaryingCenterCase1/Gap.lean"},{"n":"Causalean.Estimation.MinimaxATE.VarConstr.gPertV_true_sub","k":"theorem","m":"Causalean.Estimation.MinimaxATE.VaryingCenterCase1.Membership","a":"Estimation/MinimaxATE/VaryingCenterCase1","t":2,"d":"The treated arm's pointwise deviation from ĝ(1,x) = g₁ x.1 is g₁ x.1·(α+β)·Δ/(g₁ x.1 − β·Δ).","s":"∀ {K : ℕ} (P : Causalean.Estimation.MinimaxATE.VarConstr K) (lam : Fin K → Bool) (x : Fin K × Bool), P.gPertV lam true x - P.g₁ x.1 = P.g₁ x.1 * (P.α + P.β) * Causalean.Estimation.MinimaxATE.Δ lam x / (P.g₁ x.1 - P.β * Causalean.Estimation.MinimaxATE.Δ lam x)","l":75,"f":"Causalean/Estimation/MinimaxATE/VaryingCenterCase1/Membership.lean"},{"n":"Causalean.Estimation.MinimaxATE.VarConstr.ghatV","k":"def","m":"Causalean.Estimation.MinimaxATE.VaryingCenterCase1.Construction","a":"Estimation/MinimaxATE/VaryingCenterCase1","t":1,"d":"The cell-varying outcome-regression center: g₁ j on the treated arm, g₀ j on control, where j is the pair index of the cell.","s":"{K : ℕ} → Causalean.Estimation.MinimaxATE.VarConstr K → Bool → Fin K × Bool → ℝ","l":108,"f":"Causalean/Estimation/MinimaxATE/VaryingCenterCase1/Construction.lean"},{"n":"Causalean.Estimation.MinimaxATE.VarConstr.inClassV","k":"theorem","m":"Causalean.Estimation.MinimaxATE.VaryingCenterCase1.Membership","a":"Estimation/MinimaxATE/VaryingCenterCase1","t":1,"d":"Class membership. Suppose every pair's propensity-bump budget (m₀ⱼ·(β/g₁ⱼ))² ≤ εm holds and every pair's treated-arm budget g₁ⱼ²(α+β)²/(g₁ⱼ−β)² ≤ εg holds for a nonnegative outcome-error tolerance εg. Then the perturbed data-generating process (mλ, gλ) indexed by a Rademacher sign vector lam lies in the structure-agnostic nuisance class ℱ(εg, εm) around the cell-varying center (m̂, ĝ).","s":"∀ {K : ℕ} (P : Causalean.Estimation.MinimaxATE.VarConstr K) [NeZero K] {εg εm : ℝ}, (∀ (j : Fin K), (P.m₀ j * (P.β / P.g₁ j)) ^ 2 ≤ εm) → (∀ (j : Fin K), P.g₁ j ^ 2 * (P.α + P.β) ^ 2 / (P.g₁ j - P.β) ^ 2 ≤ εg) → 0 ≤ εg → ∀ (lam : Fin K → Bool), Causalean.Estimation.MinimaxATE.InClass P.mhatV P.ghatV εg εm (P.mPertV lam) (P.gPertV lam)","l":126,"f":"Causalean/Estimation/MinimaxATE/VaryingCenterCase1/Membership.lean"},{"n":"Causalean.Estimation.MinimaxATE.VarConstr.inClass_nullV","k":"theorem","m":"Causalean.Estimation.MinimaxATE.VaryingCenterCase1.LowerBound","a":"Estimation/MinimaxATE/VaryingCenterCase1","t":2,"d":"The null estimate is itself in the class (zero nuisance error).","s":"∀ {K : ℕ} (P : Causalean.Estimation.MinimaxATE.VarConstr K) {εg εm : ℝ}, 0 ≤ εg → 0 ≤ εm → Causalean.Estimation.MinimaxATE.InClass P.mhatV P.ghatV εg εm P.mhatV P.ghatV","l":58,"f":"Causalean/Estimation/MinimaxATE/VaryingCenterCase1/LowerBound.lean"},{"n":"Causalean.Estimation.MinimaxATE.VarConstr.instNonemptyFinBoolProd","k":"instance","m":"Causalean.Estimation.MinimaxATE.VaryingCenterCase1.LowerBound","a":"Estimation/MinimaxATE/VaryingCenterCase1","t":2,"d":"The paired-cell covariate is nonempty whenever K ≠ 0.","s":"∀ {K : ℕ} [NeZero K], Nonempty (Fin K × Bool)","l":52,"f":"Causalean/Estimation/MinimaxATE/VaryingCenterCase1/LowerBound.lean"},{"n":"Causalean.Estimation.MinimaxATE.VarConstr.l2sq_gPertV_false","k":"theorem","m":"Causalean.Estimation.MinimaxATE.VaryingCenterCase1.Membership","a":"Estimation/MinimaxATE/VaryingCenterCase1","t":2,"d":"The control outcome arm is unchanged, so its L² error is 0.","s":"∀ {K : ℕ} (P : Causalean.Estimation.MinimaxATE.VarConstr K) (lam : Fin K → Bool), Causalean.Estimation.MinimaxATE.l2sq (P.gPertV lam false) (P.ghatV false) = 0","l":68,"f":"Causalean/Estimation/MinimaxATE/VaryingCenterCase1/Membership.lean"},{"n":"Causalean.Estimation.MinimaxATE.VarConstr.l2sq_gPertV_true_le","k":"theorem","m":"Causalean.Estimation.MinimaxATE.VaryingCenterCase1.Membership","a":"Estimation/MinimaxATE/VaryingCenterCase1","t":2,"d":"The treated arm's squared L² error is ≤ εg whenever every pair's bound g₁ⱼ²(α+β)²/(g₁ⱼ − β)² is ≤ εg.","s":"∀ {K : ℕ} (P : Causalean.Estimation.MinimaxATE.VarConstr K) [NeZero K] {εg : ℝ}, (∀ (j : Fin K), P.g₁ j ^ 2 * (P.α + P.β) ^ 2 / (P.g₁ j - P.β) ^ 2 ≤ εg) → ∀ (lam : Fin K → Bool), Causalean.Estimation.MinimaxATE.l2sq (P.gPertV lam true) (P.ghatV true) ≤ εg","l":86,"f":"Causalean/Estimation/MinimaxATE/VaryingCenterCase1/Membership.lean"},{"n":"Causalean.Estimation.MinimaxATE.VarConstr.l2sq_mPertV_le","k":"theorem","m":"Causalean.Estimation.MinimaxATE.VaryingCenterCase1.Membership","a":"Estimation/MinimaxATE/VaryingCenterCase1","t":2,"d":"The propensity perturbation's squared L² error is ≤ εm whenever every pair's bump magnitude (m₀ⱼ·(β/g₁ⱼ))² is ≤ εm.","s":"∀ {K : ℕ} (P : Causalean.Estimation.MinimaxATE.VarConstr K) [NeZero K] {εm : ℝ}, (∀ (j : Fin K), (P.m₀ j * (P.β / P.g₁ j)) ^ 2 ≤ εm) → ∀ (lam : Fin K → Bool), Causalean.Estimation.MinimaxATE.l2sq (P.mPertV lam) P.mhatV ≤ εm","l":46,"f":"Causalean/Estimation/MinimaxATE/VaryingCenterCase1/Membership.lean"},{"n":"Causalean.Estimation.MinimaxATE.VarConstr.mPertV","k":"def","m":"Causalean.Estimation.MinimaxATE.VaryingCenterCase1.Construction","a":"Estimation/MinimaxATE/VaryingCenterCase1","t":1,"d":"The perturbed propensity mλ x = m₀ x.1·(1 − (β/g₁ x.1)·Δ).","s":"{K : ℕ} → Causalean.Estimation.MinimaxATE.VarConstr K → (Fin K → Bool) → Fin K × Bool → ℝ","l":113,"f":"Causalean/Estimation/MinimaxATE/VaryingCenterCase1/Construction.lean"},{"n":"Causalean.Estimation.MinimaxATE.VarConstr.mPertV_strictOverlap","k":"theorem","m":"Causalean.Estimation.MinimaxATE.Causal.Minimax","a":"Estimation/MinimaxATE/Causal","t":2,"d":"The perturbed VaryingCenterCase1 witness has strict propensity overlap.","s":"∀ {K : ℕ} (P : Causalean.Estimation.MinimaxATE.VarConstr K) [NeZero K], (∀ (j : Fin K), P.m₀ j * (1 + P.β / P.g₁ j) < 1) → ∀ (lam : Fin K → Bool) (x : Fin K × Bool), P.mPertV lam x ∈ Set.Ioo 0 1","l":130,"f":"Causalean/Estimation/MinimaxATE/Causal/Minimax.lean"},{"n":"Causalean.Estimation.MinimaxATE.VarConstr.mhatV","k":"def","m":"Causalean.Estimation.MinimaxATE.VaryingCenterCase1.Construction","a":"Estimation/MinimaxATE/VaryingCenterCase1","t":1,"d":"The cell-varying propensity center as a function of the covariate.","s":"{K : ℕ} → Causalean.Estimation.MinimaxATE.VarConstr K → Fin K × Bool → ℝ","l":105,"f":"Causalean/Estimation/MinimaxATE/VaryingCenterCase1/Construction.lean"},{"n":"Causalean.Estimation.MinimaxATE.VarConstr.mhatV_strictOverlap","k":"theorem","m":"Causalean.Estimation.MinimaxATE.Causal.Minimax","a":"Estimation/MinimaxATE/Causal","t":2,"d":"The null VaryingCenterCase1 witness has strict propensity overlap.","s":"∀ {K : ℕ} (P : Causalean.Estimation.MinimaxATE.VarConstr K) [NeZero K] (x : Fin K × Bool), P.mhatV x ∈ Set.Ioo 0 1","l":124,"f":"Causalean/Estimation/MinimaxATE/Causal/Minimax.lean"},{"n":"Causalean.Estimation.MinimaxATE.VarConstr.minimax_lower_bound_var","k":"theorem","m":"Causalean.Estimation.MinimaxATE.VaryingCenterCase1.LowerBound","a":"Estimation/MinimaxATE/VaryingCenterCase1","t":1,"d":"Structure-agnostic minimax lower bound (cell-varying center). Around a nuisance center P that varies across pairs (constant within each pair), suppose every pair's propensity-bump budget (m₀ⱼ·(β/g₁ⱼ))² ≤ εm holds and every pair's treated-arm budget g₁ⱼ²(α+β)²/(g₁ⱼ−β)² ≤ εg holds for nonnegative error tolerances εg, εm, and the per-pair overlap coefficients satisfy the total-mass bound Σⱼ Γⱼ/K ≤ 1 and the sample-size regularity budget (n²/2)·Σⱼ (Γⱼ/K)² ≤ log 2. Then for any measurable estimator of the average treatment effect, the worst-case-over-class probability that it misses the true ATE by half the cell-varying gap ate gλ − ate ĝ is at least 1/4.","s":"∀ {K : ℕ} (P : Causalean.Estimation.MinimaxATE.VarConstr K) {n : ℕ} [inst : NeZero K] {εg εm : ℝ}, (∀ (j : Fin K), (P.m₀ j * (P.β / P.g₁ j)) ^ 2 ≤ εm) → (∀ (j : Fin K), P.g₁ j ^ 2 * (P.α + P.β) ^ 2 / (P.g₁ j - P.β) ^ 2 ≤ εg) → 0 ≤ εg → 0 ≤ εm → ∑ j, P.ΓV j / ↑K ≤ 1 → ↑n ^ 2 / 2 * ∑ j, (P.ΓV j / ↑K) ^ 2 ≤ Real.log 2 → ∀ {est : (Fin n → Causalean.Estimation.MinimaxATE.Obs (Fin K × Bool)) → ℝ}, Measurable est → 1 / 4 ≤ Causalean.Estimation.MinimaxATE.minimaxMiss P.mhatV P.ghatV εg εm n est (((↑(Fintype.card (Fin K × Bool)))⁻¹ * (2 * P.β * (P.α + P.β)) * ∑ j, P.g₁ j / (P.g₁ j ^ 2 - P.β ^ 2)) / 2)","l":222,"f":"Causalean/Estimation/MinimaxATE/VaryingCenterCase1/LowerBound.lean"},{"n":"Causalean.Estimation.MinimaxATE.VarConstr.minimax_lower_bound_var_causal","k":"theorem","m":"Causalean.Estimation.MinimaxATE.Causal.Minimax","a":"Estimation/MinimaxATE/Causal","t":1,"d":"Causal-centered structure-agnostic minimax lower bound (Case 1). For the outcome-dominant cell-varying construction P, suppose the squared propensity perturbation stays within the budget εm, the squared outcome-regression perturbation stays within the budget εg, the perturbed propensity remains strictly below 1 in every cell, and both budgets are nonnegative. If in addition the aggregate separation budget across cells is at most 1 and the sample size satisfies the stated regularity regime relative to that budget, then for every measurable estimator, the causal-centered miss probability — of missing the true backdoor-identified ATE E[Y(1) − Y(0)] by at least half the displayed Case-1 separation gap — is at least 1/4 for some data-generating process in the class; the strict perturbed-overlap hypothesis is exactly what lets the observed-data Case-1 bound be re-centered onto the genuine causal estimand.","s":"∀ {K : ℕ} (P : Causalean.Estimation.MinimaxATE.VarConstr K) {n : ℕ} [inst : NeZero K] {εg εm : ℝ}, (∀ (j : Fin K), (P.m₀ j * (P.β / P.g₁ j)) ^ 2 ≤ εm) → (∀ (j : Fin K), P.g₁ j ^ 2 * (P.α + P.β) ^ 2 / (P.g₁ j - P.β) ^ 2 ≤ εg) → (∀ (j : Fin K), P.m₀ j * (1 + P.β / P.g₁ j) < 1) → 0 ≤ εg → 0 ≤ εm → ∑ j, P.ΓV j / ↑K ≤ 1 → ↑n ^ 2 / 2 * ∑ j, (P.ΓV j / ↑K) ^ 2 ≤ Real.log 2 → ∀ {est : (Fin n → Causalean.Estimation.MinimaxATE.Obs (Fin K × Bool)) → ℝ}, Measurable est → 1 / 4 ≤ Causalean.Estimation.MinimaxATE.minimaxMissCausal P.mhatV P.ghatV εg εm n est (((↑(Fintype.card (Fin K × Bool)))⁻¹ * (2 * P.β * (P.α + P.β)) * ∑ j, P.g₁ j / (P.g₁ j ^ 2 - P.β ^ 2)) / 2)","l":172,"f":"Causalean/Estimation/MinimaxATE/Causal/Minimax.lean"},{"n":"Causalean.Estimation.MinimaxATE.VarConstr.obsReal_pertV_eq","k":"theorem","m":"Causalean.Estimation.MinimaxATE.VaryingCenterCase1.ChiSqOverlap","a":"Estimation/MinimaxATE/VaryingCenterCase1","t":2,"d":"Denominator-free value of the perturbed observed mass at the four (d, y).","s":"∀ {K : ℕ} (P : Causalean.Estimation.MinimaxATE.VarConstr K) (lam : Fin K → Bool) (x : Fin K × Bool) (d y : Bool), Causalean.Estimation.MinimaxATE.obsReal (P.mPertV lam) (P.gPertV lam) (x, d, y) = (↑(Fintype.card (Fin K × Bool)))⁻¹ * match d, y with | true, true => P.m₀ x.1 * (P.g₁ x.1 + P.α * Causalean.Estimation.MinimaxATE.Δ lam x) | true, false => P.m₀ x.1 * (1 - P.g₁ x.1 - (P.α + P.β / P.g₁ x.1) * Causalean.Estimation.MinimaxATE.Δ lam x) | false, true => (1 - P.m₀ x.1 + P.m₀ x.1 * (P.β / P.g₁ x.1) * Causalean.Estimation.MinimaxATE.Δ lam x) * P.g₀ x.1 | false, false => (1 - P.m₀ x.1 + P.m₀ x.1 * (P.β / P.g₁ x.1) * Causalean.Estimation.MinimaxATE.Δ lam x) * (1 - P.g₀ x.1)","l":74,"f":"Causalean/Estimation/MinimaxATE/VaryingCenterCase1/ChiSqOverlap.lean"},{"n":"Causalean.Estimation.MinimaxATE.VarConstr.one_add_chiSqDiv_QtrueV_QfalseV","k":"theorem","m":"Causalean.Estimation.MinimaxATE.VaryingCenterCase1.LowerBound","a":"Estimation/MinimaxATE/VaryingCenterCase1","t":2,"d":"Mixture second-moment identity (cell-varying center).","s":"∀ {K : ℕ} (P : Causalean.Estimation.MinimaxATE.VarConstr K) {n : ℕ} [inst : NeZero K], 1 + Causalean.Stat.chiSqDiv (P.QtrueV n) (P.QfalseV n) = ∑ lam, ∑ lam', (2 ^ K)⁻¹ * (2 ^ K)⁻¹ * P.chiSqOverlapV lam lam' ^ n","l":147,"f":"Causalean/Estimation/MinimaxATE/VaryingCenterCase1/LowerBound.lean"},{"n":"Causalean.Estimation.MinimaxATE.VarConstr.ratio_lt_one","k":"theorem","m":"Causalean.Estimation.MinimaxATE.VaryingCenterCase1.Construction","a":"Estimation/MinimaxATE/VaryingCenterCase1","t":2,"d":"β/g₁ j < 1 since β < g₁ j.","s":"∀ {K : ℕ} (P : Causalean.Estimation.MinimaxATE.VarConstr K) (j : Fin K), P.β / P.g₁ j < 1","l":102,"f":"Causalean/Estimation/MinimaxATE/VaryingCenterCase1/Construction.lean"},{"n":"Causalean.Estimation.MinimaxATE.VarConstr.ratio_nonneg","k":"theorem","m":"Causalean.Estimation.MinimaxATE.VaryingCenterCase1.Construction","a":"Estimation/MinimaxATE/VaryingCenterCase1","t":2,"d":"0 ≤ β/g₁ j always.","s":"∀ {K : ℕ} (P : Causalean.Estimation.MinimaxATE.VarConstr K) (j : Fin K), 0 ≤ P.β / P.g₁ j","l":99,"f":"Causalean/Estimation/MinimaxATE/VaryingCenterCase1/Construction.lean"},{"n":"Causalean.Estimation.MinimaxATE.VarConstr.realV_le_minimaxMiss","k":"theorem","m":"Causalean.Estimation.MinimaxATE.VaryingCenterCase1.LowerBound","a":"Estimation/MinimaxATE/VaryingCenterCase1","t":2,"d":"An in-class DGP's miss probability at its own ATE is dominated by the minimax miss.","s":"∀ {K : ℕ} (P : Causalean.Estimation.MinimaxATE.VarConstr K) {n : ℕ} [inst : NeZero K] {εg εm : ℝ} {m : Fin K × Bool → ℝ} {g : Bool → Fin K × Bool → ℝ} (hin : Causalean.Estimation.MinimaxATE.InClass P.mhatV P.ghatV εg εm m g) (est : (Fin n → Causalean.Estimation.MinimaxATE.Obs (Fin K × Bool)) → ℝ) (s : ℝ), (Causalean.Estimation.MinimaxATE.productLaw hin.valid n).real {x | s ≤ |est x - Causalean.Estimation.MinimaxATE.ate g|} ≤ Causalean.Estimation.MinimaxATE.minimaxMiss P.mhatV P.ghatV εg εm n est s","l":96,"f":"Causalean/Estimation/MinimaxATE/VaryingCenterCase1/LowerBound.lean"},{"n":"Causalean.Estimation.MinimaxATE.VarConstr.realV_le_minimaxMissCausal","k":"theorem","m":"Causalean.Estimation.MinimaxATE.Causal.Minimax","a":"Estimation/MinimaxATE/Causal","t":2,"d":"An in-class DGP's causal-centered miss probability is dominated by minimaxMissCausal.","s":"∀ {K : ℕ} (P : Causalean.Estimation.MinimaxATE.VarConstr K) {n : ℕ} [inst : NeZero K] {εg εm : ℝ} {m : Fin K × Bool → ℝ} {g : Bool → Fin K × Bool → ℝ} (hin : Causalean.Estimation.MinimaxATE.InClass P.mhatV P.ghatV εg εm m g) (est : (Fin n → Causalean.Estimation.MinimaxATE.Obs (Fin K × Bool)) → ℝ) (s : ℝ), (Causalean.Estimation.MinimaxATE.productLaw hin.valid n).real {x | s ≤ |est x - Causalean.Estimation.MinimaxATE.Causal.causalATE m g|} ≤ Causalean.Estimation.MinimaxATE.minimaxMissCausal P.mhatV P.ghatV εg εm n est s","l":157,"f":"Causalean/Estimation/MinimaxATE/Causal/Minimax.lean"},{"n":"Causalean.Estimation.MinimaxATE.VarConstr.tvDist_QfalseV_QtrueV_le_half","k":"theorem","m":"Causalean.Estimation.MinimaxATE.VaryingCenterCase1.LowerBound","a":"Estimation/MinimaxATE/VaryingCenterCase1","t":2,"d":"Total-variation indistinguishability (cell-varying center).","s":"∀ {K : ℕ} (P : Causalean.Estimation.MinimaxATE.VarConstr K) {n : ℕ} [inst : NeZero K], ∑ j, P.ΓV j / ↑K ≤ 1 → ↑n ^ 2 / 2 * ∑ j, (P.ΓV j / ↑K) ^ 2 ≤ Real.log 2 → Causalean.Stat.tvDist (P.QfalseV n) (P.QtrueV n) ≤ 1 / 2","l":204,"f":"Causalean/Estimation/MinimaxATE/VaryingCenterCase1/LowerBound.lean"},{"n":"Causalean.Estimation.MinimaxATE.VarConstr.validDGP_hatV","k":"theorem","m":"Causalean.Estimation.MinimaxATE.VaryingCenterCase1.Construction","a":"Estimation/MinimaxATE/VaryingCenterCase1","t":1,"d":"The null cell-varying-center data-generating process, with propensity mhatV and outcome regressions ghatV, is a valid finite observed-data model, i.e. all its component probabilities lie in [0,1].","s":"∀ {K : ℕ} (P : Causalean.Estimation.MinimaxATE.VarConstr K), Causalean.Estimation.MinimaxATE.ValidDGP P.mhatV P.ghatV","l":131,"f":"Causalean/Estimation/MinimaxATE/VaryingCenterCase1/Construction.lean"},{"n":"Causalean.Estimation.MinimaxATE.VarConstr.validDGP_pertV","k":"theorem","m":"Causalean.Estimation.MinimaxATE.VaryingCenterCase1.Construction","a":"Estimation/MinimaxATE/VaryingCenterCase1","t":1,"d":"For any Rademacher sign vector lam indexing the perturbation, the perturbed propensity and outcome-regression functions define a valid finite observed-data model, i.e. take values in [0,1].","s":"∀ {K : ℕ} (P : Causalean.Estimation.MinimaxATE.VarConstr K) (lam : Fin K → Bool), Causalean.Estimation.MinimaxATE.ValidDGP (P.mPertV lam) (P.gPertV lam)","l":141,"f":"Causalean/Estimation/MinimaxATE/VaryingCenterCase1/Construction.lean"},{"n":"Causalean.Estimation.MinimaxATE.VarConstr.ΓV","k":"def","m":"Causalean.Estimation.MinimaxATE.VaryingCenterCase1.ChiSqOverlap","a":"Estimation/MinimaxATE/VaryingCenterCase1","t":1,"d":"The per-pair χ²-overlap coefficient.","s":"{K : ℕ} → Causalean.Estimation.MinimaxATE.VarConstr K → Fin K → ℝ","l":48,"f":"Causalean/Estimation/MinimaxATE/VaryingCenterCase1/ChiSqOverlap.lean"},{"n":"Causalean.Estimation.MinimaxATE.VarConstr.ΓV_nonneg","k":"theorem","m":"Causalean.Estimation.MinimaxATE.VaryingCenterCase1.ChiSqOverlap","a":"Estimation/MinimaxATE/VaryingCenterCase1","t":2,"d":"Γⱼ ≥ 0.","s":"∀ {K : ℕ} (P : Causalean.Estimation.MinimaxATE.VarConstr K) (j : Fin K), 0 ≤ P.ΓV j","l":54,"f":"Causalean/Estimation/MinimaxATE/VaryingCenterCase1/ChiSqOverlap.lean"},{"n":"Causalean.Estimation.MinimaxATE.VarConstr2","k":"structure","m":"Causalean.Estimation.MinimaxATE.VaryingCenterCase2.Construction","a":"Estimation/MinimaxATE/VaryingCenterCase2","t":1,"d":"Second (propensity-dominant) cell-varying construction data for the same style of Rademacher perturbation as VarConstr but with the roles of the two nuisances swapped, so the propensity carries the larger deviation. It packages two bump-magnitude scalars, the larger on the propensity and the smaller on the treated outcome arm, together with a nuisance center given by the pair-indexed functions m₀, g₀ and g₁ for the propensity and the two potential-outcome regressions, plus the inequalities certifying that both bump magnitudes are nonnegative, the center is pointwise strictly inside (0,1) for m₀, g₀ and g₁, a worst-case bound forcing the perturbed treated outcome regression to stay at most one, which also keeps the perturbation denominator positive, the propensity bump coefficient not to exceed one, keeping the perturbed propensity nonnegative, and the perturbed propensity to stay at most one.","s":"ℕ → Type","l":57,"f":"Causalean/Estimation/MinimaxATE/VaryingCenterCase2/Construction.lean"},{"n":"Causalean.Estimation.MinimaxATE.VarConstr2.D2","k":"def","m":"Causalean.Estimation.MinimaxATE.VaryingCenterCase2.Construction","a":"Estimation/MinimaxATE/VaryingCenterCase2","t":1,"d":"The treated-arm denominator D = 1 + (β/g₁)·Δ − α·β.","s":"{K : ℕ} → Causalean.Estimation.MinimaxATE.VarConstr2 K → (Fin K → Bool) → Fin K × Bool → ℝ","l":105,"f":"Causalean/Estimation/MinimaxATE/VaryingCenterCase2/Construction.lean"},{"n":"Causalean.Estimation.MinimaxATE.VarConstr2.D2_pos","k":"theorem","m":"Causalean.Estimation.MinimaxATE.VaryingCenterCase2.Construction","a":"Estimation/MinimaxATE/VaryingCenterCase2","t":2,"d":"The denominator D = 1 + (β/g₁)·Δ − αβ is positive (at least g₁ > 0).","s":"∀ {K : ℕ} (P : Causalean.Estimation.MinimaxATE.VarConstr2 K) (lam : Fin K → Bool) (x : Fin K × Bool), 0 < P.D2 lam x","l":128,"f":"Causalean/Estimation/MinimaxATE/VaryingCenterCase2/Construction.lean"},{"n":"Causalean.Estimation.MinimaxATE.VarConstr2.Qfalse2","k":"def","m":"Causalean.Estimation.MinimaxATE.VaryingCenterCase2.LowerBound","a":"Estimation/MinimaxATE/VaryingCenterCase2","t":1,"d":"The null n-sample law P̂^⊗n.","s":"{K : ℕ} → Causalean.Estimation.MinimaxATE.VarConstr2 K → (n : ℕ) → [NeZero K] → MeasureTheory.Measure (Fin n → Causalean.Estimation.MinimaxATE.Obs (Fin K × Bool))","l":59,"f":"Causalean/Estimation/MinimaxATE/VaryingCenterCase2/LowerBound.lean"},{"n":"Causalean.Estimation.MinimaxATE.VarConstr2.Qfalse2_isProb","k":"theorem","m":"Causalean.Estimation.MinimaxATE.VaryingCenterCase2.LowerBound","a":"Estimation/MinimaxATE/VaryingCenterCase2","t":2,"d":"The null sample law is a probability measure.","s":"∀ {K : ℕ} (P : Causalean.Estimation.MinimaxATE.VarConstr2 K) (n : ℕ) [inst : NeZero K], MeasureTheory.IsProbabilityMeasure (P.Qfalse2 n)","l":74,"f":"Causalean/Estimation/MinimaxATE/VaryingCenterCase2/LowerBound.lean"},{"n":"Causalean.Estimation.MinimaxATE.VarConstr2.Qfalse2_singleton_ne_zero","k":"theorem","m":"Causalean.Estimation.MinimaxATE.VaryingCenterCase2.LowerBound","a":"Estimation/MinimaxATE/VaryingCenterCase2","t":2,"d":"The null n-sample law charges every point.","s":"∀ {K : ℕ} (P : Causalean.Estimation.MinimaxATE.VarConstr2 K) {n : ℕ} [inst : NeZero K] (ω : Fin n → Causalean.Estimation.MinimaxATE.Obs (Fin K × Bool)), (P.Qfalse2 n) {ω} ≠ 0","l":101,"f":"Causalean/Estimation/MinimaxATE/VaryingCenterCase2/LowerBound.lean"},{"n":"Causalean.Estimation.MinimaxATE.VarConstr2.Qpert2","k":"def","m":"Causalean.Estimation.MinimaxATE.VaryingCenterCase2.LowerBound","a":"Estimation/MinimaxATE/VaryingCenterCase2","t":1,"d":"The perturbed n-sample law Qλ^⊗n.","s":"{K : ℕ} → Causalean.Estimation.MinimaxATE.VarConstr2 K → (n : ℕ) → [NeZero K] → (Fin K → Bool) → MeasureTheory.Measure (Fin n → Causalean.Estimation.MinimaxATE.Obs (Fin K × Bool))","l":64,"f":"Causalean/Estimation/MinimaxATE/VaryingCenterCase2/LowerBound.lean"},{"n":"Causalean.Estimation.MinimaxATE.VarConstr2.Qpert2_isProb","k":"theorem","m":"Causalean.Estimation.MinimaxATE.VaryingCenterCase2.LowerBound","a":"Estimation/MinimaxATE/VaryingCenterCase2","t":2,"d":"Each perturbed sample law is a probability measure.","s":"∀ {K : ℕ} (P : Causalean.Estimation.MinimaxATE.VarConstr2 K) (n : ℕ) [inst : NeZero K] (lam : Fin K → Bool), MeasureTheory.IsProbabilityMeasure (P.Qpert2 n lam)","l":78,"f":"Causalean/Estimation/MinimaxATE/VaryingCenterCase2/LowerBound.lean"},{"n":"Causalean.Estimation.MinimaxATE.VarConstr2.Qtrue2","k":"def","m":"Causalean.Estimation.MinimaxATE.VaryingCenterCase2.LowerBound","a":"Estimation/MinimaxATE/VaryingCenterCase2","t":1,"d":"The alternative n-sample law: the uniform Rademacher mixture of the perturbed laws.","s":"{K : ℕ} → Causalean.Estimation.MinimaxATE.VarConstr2 K → (n : ℕ) → [NeZero K] → MeasureTheory.Measure (Fin n → Causalean.Estimation.MinimaxATE.Obs (Fin K × Bool))","l":69,"f":"Causalean/Estimation/MinimaxATE/VaryingCenterCase2/LowerBound.lean"},{"n":"Causalean.Estimation.MinimaxATE.VarConstr2.Qtrue2_isProb","k":"theorem","m":"Causalean.Estimation.MinimaxATE.VaryingCenterCase2.LowerBound","a":"Estimation/MinimaxATE/VaryingCenterCase2","t":2,"d":"The Rademacher mixture of perturbed sample laws is a probability measure.","s":"∀ {K : ℕ} (P : Causalean.Estimation.MinimaxATE.VarConstr2 K) (n : ℕ) [inst : NeZero K], MeasureTheory.IsProbabilityMeasure (P.Qtrue2 n)","l":82,"f":"Causalean/Estimation/MinimaxATE/VaryingCenterCase2/LowerBound.lean"},{"n":"Causalean.Estimation.MinimaxATE.VarConstr2.Qtrue2_real_singleton","k":"theorem","m":"Causalean.Estimation.MinimaxATE.VaryingCenterCase2.LowerBound","a":"Estimation/MinimaxATE/VaryingCenterCase2","t":2,"d":"The alternative law's .real point mass: a uniform mixture over sign vectors.","s":"∀ {K : ℕ} (P : Causalean.Estimation.MinimaxATE.VarConstr2 K) {n : ℕ} [inst : NeZero K] (ω : Fin n → Causalean.Estimation.MinimaxATE.Obs (Fin K × Bool)), (P.Qtrue2 n).real {ω} = ∑ lam, (2 ^ K)⁻¹ * ∏ i, Causalean.Estimation.MinimaxATE.obsReal (P.mPert2 lam) (P.gPert2 lam) (ω i)","l":124,"f":"Causalean/Estimation/MinimaxATE/VaryingCenterCase2/LowerBound.lean"},{"n":"Causalean.Estimation.MinimaxATE.VarConstr2.alphabeta_le_one","k":"theorem","m":"Causalean.Estimation.MinimaxATE.VaryingCenterCase2.Gap","a":"Estimation/MinimaxATE/VaryingCenterCase2","t":2,"d":"αβ ≤ 1 (a consequence of hgU and g₁ⱼ > 0).","s":"∀ {K : ℕ} (P : Causalean.Estimation.MinimaxATE.VarConstr2 K) (j : Fin K), P.α * P.β ≤ 1","l":42,"f":"Causalean/Estimation/MinimaxATE/VaryingCenterCase2/Gap.lean"},{"n":"Causalean.Estimation.MinimaxATE.VarConstr2.ate_gPert2","k":"theorem","m":"Causalean.Estimation.MinimaxATE.VaryingCenterCase2.Gap","a":"Estimation/MinimaxATE/VaryingCenterCase2","t":1,"d":"Exact ATE of the perturbed construction. For any Rademacher sign vector lam, the average treatment effect of the perturbed outcome regression equals the average over pairs j of 2g₁ⱼ³(1−αβ)/(g₁ⱼ²(1−αβ)²−β²) − 2g₀ⱼ, independent of lam.","s":"∀ {K : ℕ} (P : Causalean.Estimation.MinimaxATE.VarConstr2 K) [NeZero K] (lam : Fin K → Bool), Causalean.Estimation.MinimaxATE.ate (P.gPert2 lam) = (↑(Fintype.card (Fin K × Bool)))⁻¹ * ∑ j, (2 * P.g₁ j ^ 3 * (1 - P.α * P.β) / (P.g₁ j ^ 2 * (1 - P.α * P.β) ^ 2 - P.β ^ 2) - 2 * P.g₀ j)","l":146,"f":"Causalean/Estimation/MinimaxATE/VaryingCenterCase2/Gap.lean"},{"n":"Causalean.Estimation.MinimaxATE.VarConstr2.ate_gap2","k":"theorem","m":"Causalean.Estimation.MinimaxATE.VaryingCenterCase2.Gap","a":"Estimation/MinimaxATE/VaryingCenterCase2","t":1,"d":"For any Rademacher sign vector lam, the gap between the perturbed construction's ATE and the null estimate's ATE equals 2β/(2K) times the sum over pairs j of g₁ⱼ(αg₁ⱼ²(1−αβ) + β)/(g₁ⱼ²(1−αβ)² − β²).","s":"∀ {K : ℕ} (P : Causalean.Estimation.MinimaxATE.VarConstr2 K) [NeZero K] (lam : Fin K → Bool), Causalean.Estimation.MinimaxATE.ate (P.gPert2 lam) - Causalean.Estimation.MinimaxATE.ate P.ghat2 = (↑(Fintype.card (Fin K × Bool)))⁻¹ * (2 * P.β) * ∑ j, P.g₁ j * (P.α * P.g₁ j ^ 2 * (1 - P.α * P.β) + P.β) / (P.g₁ j ^ 2 * (1 - P.α * P.β) ^ 2 - P.β ^ 2)","l":161,"f":"Causalean/Estimation/MinimaxATE/VaryingCenterCase2/Gap.lean"},{"n":"Causalean.Estimation.MinimaxATE.VarConstr2.ate_gap2_nonneg","k":"theorem","m":"Causalean.Estimation.MinimaxATE.VaryingCenterCase2.Gap","a":"Estimation/MinimaxATE/VaryingCenterCase2","t":2,"d":"The ATE gap is nonnegative.","s":"∀ {K : ℕ} (P : Causalean.Estimation.MinimaxATE.VarConstr2 K) [NeZero K] (lam : Fin K → Bool), 0 ≤ Causalean.Estimation.MinimaxATE.ate (P.gPert2 lam) - Causalean.Estimation.MinimaxATE.ate P.ghat2","l":179,"f":"Causalean/Estimation/MinimaxATE/VaryingCenterCase2/Gap.lean"},{"n":"Causalean.Estimation.MinimaxATE.VarConstr2.ate_ghat2","k":"theorem","m":"Causalean.Estimation.MinimaxATE.VaryingCenterCase2.Gap","a":"Estimation/MinimaxATE/VaryingCenterCase2","t":2,"d":"The null estimate ĝ has ATE (1/2K) Σⱼ 2(g₁ⱼ − g₀ⱼ).","s":"∀ {K : ℕ} (P : Causalean.Estimation.MinimaxATE.VarConstr2 K), Causalean.Estimation.MinimaxATE.ate P.ghat2 = (↑(Fintype.card (Fin K × Bool)))⁻¹ * ∑ j, 2 * (P.g₁ j - P.g₀ j)","l":91,"f":"Causalean/Estimation/MinimaxATE/VaryingCenterCase2/Gap.lean"},{"n":"Causalean.Estimation.MinimaxATE.VarConstr2.chiSqDiv_Qtrue2_Qfalse2_le_one","k":"theorem","m":"Causalean.Estimation.MinimaxATE.VaryingCenterCase2.LowerBound","a":"Estimation/MinimaxATE/VaryingCenterCase2","t":2,"d":"The χ² indistinguishability bound (second construction).","s":"∀ {K : ℕ} (P : Causalean.Estimation.MinimaxATE.VarConstr2 K) {n : ℕ} [inst : NeZero K], ∑ j, P.ΓV2 j / ↑K ≤ 1 → ↑n ^ 2 / 2 * ∑ j, (P.ΓV2 j / ↑K) ^ 2 ≤ Real.log 2 → Causalean.Stat.chiSqDiv (P.Qtrue2 n) (P.Qfalse2 n) ≤ 1","l":177,"f":"Causalean/Estimation/MinimaxATE/VaryingCenterCase2/LowerBound.lean"},{"n":"Causalean.Estimation.MinimaxATE.VarConstr2.chiSqOverlapV2","k":"def","m":"Causalean.Estimation.MinimaxATE.VaryingCenterCase2.ChiSqOverlap","a":"Estimation/MinimaxATE/VaryingCenterCase2","t":1,"d":"The single-observation χ² second-moment overlap of two perturbed laws relative to the null.","s":"{K : ℕ} → Causalean.Estimation.MinimaxATE.VarConstr2 K → (Fin K → Bool) → (Fin K → Bool) → ℝ","l":66,"f":"Causalean/Estimation/MinimaxATE/VaryingCenterCase2/ChiSqOverlap.lean"},{"n":"Causalean.Estimation.MinimaxATE.VarConstr2.chiSqOverlap_eq2","k":"theorem","m":"Causalean.Estimation.MinimaxATE.VaryingCenterCase2.ChiSqOverlap","a":"Estimation/MinimaxATE/VaryingCenterCase2","t":1,"d":"For any two Rademacher sign vectors lam and lam' indexing perturbed data-generating processes, the single-observation χ² overlap between them equals one plus the sum over pairs j of the propensity-dominant coefficient ΓV2 j / K times the sign agreement between lam and lam' at pair j.","s":"∀ {K : ℕ} (P : Causalean.Estimation.MinimaxATE.VarConstr2 K) [NeZero K] (lam lam' : Fin K → Bool), P.chiSqOverlapV2 lam lam' = 1 + ∑ j, P.ΓV2 j / ↑K * (Causalean.Estimation.MinimaxATE.signOf (lam j) * Causalean.Estimation.MinimaxATE.signOf (lam' j))","l":102,"f":"Causalean/Estimation/MinimaxATE/VaryingCenterCase2/ChiSqOverlap.lean"},{"n":"Causalean.Estimation.MinimaxATE.VarConstr2.clearedDenom_pos","k":"theorem","m":"Causalean.Estimation.MinimaxATE.VaryingCenterCase2.Gap","a":"Estimation/MinimaxATE/VaryingCenterCase2","t":2,"d":"The cleared denominator g₁ⱼ(1 − αβ) + β·σ is positive.","s":"∀ {K : ℕ} (P : Causalean.Estimation.MinimaxATE.VarConstr2 K) (j : Fin K) (σ : ℝ), σ = 1 ∨ σ = -1 → 0 < P.g₁ j * (1 - P.α * P.β) + P.β * σ","l":81,"f":"Causalean/Estimation/MinimaxATE/VaryingCenterCase2/Gap.lean"},{"n":"Causalean.Estimation.MinimaxATE.VarConstr2.denomE_pos","k":"theorem","m":"Causalean.Estimation.MinimaxATE.VaryingCenterCase2.Gap","a":"Estimation/MinimaxATE/VaryingCenterCase2","t":2,"d":"The pair denominator Eⱼ = g₁ⱼ²(1 − αβ)² − β² > 0.","s":"∀ {K : ℕ} (P : Causalean.Estimation.MinimaxATE.VarConstr2 K) (j : Fin K), 0 < P.g₁ j ^ 2 * (1 - P.α * P.β) ^ 2 - P.β ^ 2","l":59,"f":"Causalean/Estimation/MinimaxATE/VaryingCenterCase2/Gap.lean"},{"n":"Causalean.Estimation.MinimaxATE.VarConstr2.g1_one_sub_ab_sub_beta","k":"theorem","m":"Causalean.Estimation.MinimaxATE.VaryingCenterCase2.Gap","a":"Estimation/MinimaxATE/VaryingCenterCase2","t":2,"d":"g₁ⱼ(1 − αβ) − β ≥ g₁ⱼ² > 0: clearing hgU by g₁ⱼ.","s":"∀ {K : ℕ} (P : Causalean.Estimation.MinimaxATE.VarConstr2 K) (j : Fin K), P.g₁ j ^ 2 ≤ P.g₁ j * (1 - P.α * P.β) - P.β","l":46,"f":"Causalean/Estimation/MinimaxATE/VaryingCenterCase2/Gap.lean"},{"n":"Causalean.Estimation.MinimaxATE.VarConstr2.gPert2","k":"def","m":"Causalean.Estimation.MinimaxATE.VaryingCenterCase2.Construction","a":"Estimation/MinimaxATE/VaryingCenterCase2","t":1,"d":"The perturbed outcome regression: control arm = g₀ x.1, treated arm gλ(1,x) = g₁ x.1 / D.","s":"{K : ℕ} → Causalean.Estimation.MinimaxATE.VarConstr2 K → (Fin K → Bool) → Bool → Fin K × Bool → ℝ","l":150,"f":"Causalean/Estimation/MinimaxATE/VaryingCenterCase2/Construction.lean"},{"n":"Causalean.Estimation.MinimaxATE.VarConstr2.gPert2_true_eq2","k":"theorem","m":"Causalean.Estimation.MinimaxATE.VaryingCenterCase2.Gap","a":"Estimation/MinimaxATE/VaryingCenterCase2","t":2,"d":"Cleared treated arm. gλ(1,x) = g₁ x.1² / (g₁ x.1·(1 − αβ) + β·Δ) — no nested fractions, so field_simp can clear it given the (positive) denominator.","s":"∀ {K : ℕ} (P : Causalean.Estimation.MinimaxATE.VarConstr2 K) (lam : Fin K → Bool) (x : Fin K × Bool), P.gPert2 lam true x = P.g₁ x.1 ^ 2 / (P.g₁ x.1 * (1 - P.α * P.β) + P.β * Causalean.Estimation.MinimaxATE.Δ lam x)","l":69,"f":"Causalean/Estimation/MinimaxATE/VaryingCenterCase2/Gap.lean"},{"n":"Causalean.Estimation.MinimaxATE.VarConstr2.gPert2_true_sub","k":"theorem","m":"Causalean.Estimation.MinimaxATE.VaryingCenterCase2.Membership","a":"Estimation/MinimaxATE/VaryingCenterCase2","t":2,"d":"The treated arm's pointwise deviation from ĝ(1,x) = g₁ x.1 is β·(α g₁ x.1 − Δ)/D.","s":"∀ {K : ℕ} (P : Causalean.Estimation.MinimaxATE.VarConstr2 K) (lam : Fin K → Bool) (x : Fin K × Bool), P.gPert2 lam true x - P.g₁ x.1 = P.β * (P.α * P.g₁ x.1 - Causalean.Estimation.MinimaxATE.Δ lam x) / P.D2 lam x","l":71,"f":"Causalean/Estimation/MinimaxATE/VaryingCenterCase2/Membership.lean"},{"n":"Causalean.Estimation.MinimaxATE.VarConstr2.ghat2","k":"def","m":"Causalean.Estimation.MinimaxATE.VaryingCenterCase2.Construction","a":"Estimation/MinimaxATE/VaryingCenterCase2","t":1,"d":"The cell-varying outcome-regression center: g₁ j on the treated arm, g₀ j on control, where j is the pair index of the cell.","s":"{K : ℕ} → Causalean.Estimation.MinimaxATE.VarConstr2 K → Bool → Fin K × Bool → ℝ","l":141,"f":"Causalean/Estimation/MinimaxATE/VaryingCenterCase2/Construction.lean"},{"n":"Causalean.Estimation.MinimaxATE.VarConstr2.inClass2","k":"theorem","m":"Causalean.Estimation.MinimaxATE.VaryingCenterCase2.Membership","a":"Estimation/MinimaxATE/VaryingCenterCase2","t":1,"d":"Class membership. Fix a sign pattern lam and nuisance-error budgets εg, εm. If the squared quantity (m₀ⱼ·κⱼ)² is at most εm at every cell j, the worst-case squared deviation of the perturbed treated-arm outcome regression is at most εg at every cell j, and εg is nonnegative, then the perturbed propensity and outcome-regression pair (mλ, gλ) lies in the structure-agnostic nuisance class ℱ(εg, εm) around the cell-varying center (m̂, ĝ).","s":"∀ {K : ℕ} (P : Causalean.Estimation.MinimaxATE.VarConstr2 K) [NeZero K] {εg εm : ℝ}, (∀ (j : Fin K), (P.m₀ j * P.κ j) ^ 2 ≤ εm) → (∀ (j : Fin K), P.β ^ 2 * (P.α * P.g₁ j + 1) ^ 2 / (1 - P.β / P.g₁ j - P.α * P.β) ^ 2 ≤ εg) → 0 ≤ εg → ∀ (lam : Fin K → Bool), Causalean.Estimation.MinimaxATE.InClass P.mhat2 P.ghat2 εg εm (P.mPert2 lam) (P.gPert2 lam)","l":137,"f":"Causalean/Estimation/MinimaxATE/VaryingCenterCase2/Membership.lean"},{"n":"Causalean.Estimation.MinimaxATE.VarConstr2.inClass_null2","k":"theorem","m":"Causalean.Estimation.MinimaxATE.VaryingCenterCase2.LowerBound","a":"Estimation/MinimaxATE/VaryingCenterCase2","t":2,"d":"The null estimate is itself in the class (zero nuisance error).","s":"∀ {K : ℕ} (P : Causalean.Estimation.MinimaxATE.VarConstr2 K) {εg εm : ℝ}, 0 ≤ εg → 0 ≤ εm → Causalean.Estimation.MinimaxATE.InClass P.mhat2 P.ghat2 εg εm P.mhat2 P.ghat2","l":52,"f":"Causalean/Estimation/MinimaxATE/VaryingCenterCase2/LowerBound.lean"},{"n":"Causalean.Estimation.MinimaxATE.VarConstr2.instNonemptyFinBoolProd","k":"instance","m":"Causalean.Estimation.MinimaxATE.VaryingCenterCase2.LowerBound","a":"Estimation/MinimaxATE/VaryingCenterCase2","t":2,"d":"The paired-cell covariate is nonempty whenever K ≠ 0.","s":"∀ {K : ℕ} [NeZero K], Nonempty (Fin K × Bool)","l":46,"f":"Causalean/Estimation/MinimaxATE/VaryingCenterCase2/LowerBound.lean"},{"n":"Causalean.Estimation.MinimaxATE.VarConstr2.l2sq_gPert2_false","k":"theorem","m":"Causalean.Estimation.MinimaxATE.VaryingCenterCase2.Membership","a":"Estimation/MinimaxATE/VaryingCenterCase2","t":2,"d":"The control outcome arm is unchanged, so its L² error is 0.","s":"∀ {K : ℕ} (P : Causalean.Estimation.MinimaxATE.VarConstr2 K) (lam : Fin K → Bool), Causalean.Estimation.MinimaxATE.l2sq (P.gPert2 lam false) (P.ghat2 false) = 0","l":64,"f":"Causalean/Estimation/MinimaxATE/VaryingCenterCase2/Membership.lean"},{"n":"Causalean.Estimation.MinimaxATE.VarConstr2.l2sq_gPert2_true_le","k":"theorem","m":"Causalean.Estimation.MinimaxATE.VaryingCenterCase2.Membership","a":"Estimation/MinimaxATE/VaryingCenterCase2","t":2,"d":"The treated arm's squared L² error is ≤ εg whenever every pair satisfies β²·(α g₁ⱼ + 1)²/(1 − β/g₁ⱼ − αβ)² ≤ εg.","s":"∀ {K : ℕ} (P : Causalean.Estimation.MinimaxATE.VarConstr2 K) [NeZero K] {εg : ℝ}, (∀ (j : Fin K), P.β ^ 2 * (P.α * P.g₁ j + 1) ^ 2 / (1 - P.β / P.g₁ j - P.α * P.β) ^ 2 ≤ εg) → ∀ (lam : Fin K → Bool), Causalean.Estimation.MinimaxATE.l2sq (P.gPert2 lam true) (P.ghat2 true) ≤ εg","l":84,"f":"Causalean/Estimation/MinimaxATE/VaryingCenterCase2/Membership.lean"},{"n":"Causalean.Estimation.MinimaxATE.VarConstr2.l2sq_mPert2_le","k":"theorem","m":"Causalean.Estimation.MinimaxATE.VaryingCenterCase2.Membership","a":"Estimation/MinimaxATE/VaryingCenterCase2","t":2,"d":"The propensity perturbation's squared L² error is exactly the average of (m₀ⱼ·κⱼ)², hence ≤ εm whenever every pair satisfies (m₀ⱼ·κⱼ)² ≤ εm.","s":"∀ {K : ℕ} (P : Causalean.Estimation.MinimaxATE.VarConstr2 K) [NeZero K] {εm : ℝ}, (∀ (j : Fin K), (P.m₀ j * P.κ j) ^ 2 ≤ εm) → ∀ (lam : Fin K → Bool), Causalean.Estimation.MinimaxATE.l2sq (P.mPert2 lam) P.mhat2 ≤ εm","l":42,"f":"Causalean/Estimation/MinimaxATE/VaryingCenterCase2/Membership.lean"},{"n":"Causalean.Estimation.MinimaxATE.VarConstr2.mPert2","k":"def","m":"Causalean.Estimation.MinimaxATE.VaryingCenterCase2.Construction","a":"Estimation/MinimaxATE/VaryingCenterCase2","t":1,"d":"The perturbed propensity mλ x = m₀ x.1·(1 + α·g₁ x.1·Δ)·D.","s":"{K : ℕ} → Causalean.Estimation.MinimaxATE.VarConstr2 K → (Fin K → Bool) → Fin K × Bool → ℝ","l":146,"f":"Causalean/Estimation/MinimaxATE/VaryingCenterCase2/Construction.lean"},{"n":"Causalean.Estimation.MinimaxATE.VarConstr2.mPert2_eq","k":"theorem","m":"Causalean.Estimation.MinimaxATE.VaryingCenterCase2.Construction","a":"Estimation/MinimaxATE/VaryingCenterCase2","t":1,"d":"Affine collapse of the propensity. For any Rademacher sign vector lam and cell x, the perturbed propensity at x equals m₀(x.1)·(1 + κ(x.1)·Δ(lam,x)), an exactly affine function of the perturbation.","s":"∀ {K : ℕ} (P : Causalean.Estimation.MinimaxATE.VarConstr2 K) (lam : Fin K → Bool) (x : Fin K × Bool), P.mPert2 lam x = P.m₀ x.1 * (1 + P.κ x.1 * Causalean.Estimation.MinimaxATE.Δ lam x)","l":155,"f":"Causalean/Estimation/MinimaxATE/VaryingCenterCase2/Construction.lean"},{"n":"Causalean.Estimation.MinimaxATE.VarConstr2.mPert2_strictOverlap","k":"theorem","m":"Causalean.Estimation.MinimaxATE.Causal.Minimax","a":"Estimation/MinimaxATE/Causal","t":2,"d":"The perturbed VaryingCenterCase2 witness has strict propensity overlap.","s":"∀ {K : ℕ} (P : Causalean.Estimation.MinimaxATE.VarConstr2 K) [NeZero K], (∀ (j : Fin K), P.κ j < 1) → (∀ (j : Fin K), P.m₀ j * (1 + P.κ j) < 1) → ∀ (lam : Fin K → Bool) (x : Fin K × Bool), P.mPert2 lam x ∈ Set.Ioo 0 1","l":269,"f":"Causalean/Estimation/MinimaxATE/Causal/Minimax.lean"},{"n":"Causalean.Estimation.MinimaxATE.VarConstr2.mhat2","k":"def","m":"Causalean.Estimation.MinimaxATE.VaryingCenterCase2.Construction","a":"Estimation/MinimaxATE/VaryingCenterCase2","t":1,"d":"The cell-varying propensity center as a function of the covariate.","s":"{K : ℕ} → Causalean.Estimation.MinimaxATE.VarConstr2 K → Fin K × Bool → ℝ","l":138,"f":"Causalean/Estimation/MinimaxATE/VaryingCenterCase2/Construction.lean"},{"n":"Causalean.Estimation.MinimaxATE.VarConstr2.mhat2_strictOverlap","k":"theorem","m":"Causalean.Estimation.MinimaxATE.Causal.Minimax","a":"Estimation/MinimaxATE/Causal","t":2,"d":"The null VaryingCenterCase2 witness has strict propensity overlap.","s":"∀ {K : ℕ} (P : Causalean.Estimation.MinimaxATE.VarConstr2 K) [NeZero K] (x : Fin K × Bool), P.mhat2 x ∈ Set.Ioo 0 1","l":263,"f":"Causalean/Estimation/MinimaxATE/Causal/Minimax.lean"},{"n":"Causalean.Estimation.MinimaxATE.VarConstr2.minimax_lower_bound_var2","k":"theorem","m":"Causalean.Estimation.MinimaxATE.VaryingCenterCase2.LowerBound","a":"Estimation/MinimaxATE/VaryingCenterCase2","t":1,"d":"Structure-agnostic minimax lower bound (second / propensity-dominant construction). Fix a cell-varying nuisance center P, sample size n, and nuisance-error budgets εg, εm, and suppose the treated-arm outcome bump magnitude is strictly positive. If the perturbed propensity's squared deviation from its center is at most εm in every cell, the perturbed treated-arm outcome regression's squared deviation from its center is at most εg in every cell, εm is nonnegative, εg is nonnegative, the normalized per-pair χ² coefficients sum to at most 1, and n²/2 times the sum of their squares is at most log 2, then for every measurable estimator, the worst-case-over-class probability that the estimator's error exceeds half of the displayed strictly positive separation gap is at least 1/4.","s":"∀ {K : ℕ} (P : Causalean.Estimation.MinimaxATE.VarConstr2 K) {n : ℕ} [inst : NeZero K] {εg εm : ℝ}, 0 < P.β → (∀ (j : Fin K), (P.m₀ j * P.κ j) ^ 2 ≤ εm) → (∀ (j : Fin K), P.β ^ 2 * (P.α * P.g₁ j + 1) ^ 2 / (1 - P.β / P.g₁ j - P.α * P.β) ^ 2 ≤ εg) → 0 ≤ εg → 0 ≤ εm → ∑ j, P.ΓV2 j / ↑K ≤ 1 → ↑n ^ 2 / 2 * ∑ j, (P.ΓV2 j / ↑K) ^ 2 ≤ Real.log 2 → ∀ {est : (Fin n → Causalean.Estimation.MinimaxATE.Obs (Fin K × Bool)) → ℝ}, Measurable est → 1 / 4 ≤ Causalean.Estimation.MinimaxATE.minimaxMiss P.mhat2 P.ghat2 εg εm n est (((↑(Fintype.card (Fin K × Bool)))⁻¹ * (2 * P.β) * ∑ j, P.g₁ j * (P.α * P.g₁ j ^ 2 * (1 - P.α * P.β) + P.β) / (P.g₁ j ^ 2 * (1 - P.α * P.β) ^ 2 - P.β ^ 2)) / 2)","l":216,"f":"Causalean/Estimation/MinimaxATE/VaryingCenterCase2/LowerBound.lean"},{"n":"Causalean.Estimation.MinimaxATE.VarConstr2.minimax_lower_bound_var2_causal","k":"theorem","m":"Causalean.Estimation.MinimaxATE.Causal.Minimax","a":"Estimation/MinimaxATE/Causal","t":2,"d":"Causal-centered structure-agnostic minimax lower bound (Case 2). For the propensity-dominant cell-varying construction, a strictly positive treated-arm bump and strict perturbed-propensity overlap imply that every measurable estimator has causal-centered miss probability at least 1/4 at half of the displayed strictly positive ATE separation.","s":"∀ {K : ℕ} (P : Causalean.Estimation.MinimaxATE.VarConstr2 K) {n : ℕ} [inst : NeZero K] {εg εm : ℝ}, 0 < P.β → (∀ (j : Fin K), (P.m₀ j * P.κ j) ^ 2 ≤ εm) → (∀ (j : Fin K), P.β ^ 2 * (P.α * P.g₁ j + 1) ^ 2 / (1 - P.β / P.g₁ j - P.α * P.β) ^ 2 ≤ εg) → (∀ (j : Fin K), P.κ j < 1) → (∀ (j : Fin K), P.m₀ j * (1 + P.κ j) < 1) → 0 ≤ εg → 0 ≤ εm → ∑ j, P.ΓV2 j / ↑K ≤ 1 → ↑n ^ 2 / 2 * ∑ j, (P.ΓV2 j / ↑K) ^ 2 ≤ Real.log 2 → ∀ {est : (Fin n → Causalean.Estimation.MinimaxATE.Obs (Fin K × Bool)) → ℝ}, Measurable est → 1 / 4 ≤ Causalean.Estimation.MinimaxATE.minimaxMissCausal P.mhat2 P.ghat2 εg εm n est (((↑(Fintype.card (Fin K × Bool)))⁻¹ * (2 * P.β) * ∑ j, P.g₁ j * (P.α * P.g₁ j ^ 2 * (1 - P.α * P.β) + P.β) / (P.g₁ j ^ 2 * (1 - P.α * P.β) ^ 2 - P.β ^ 2)) / 2)","l":314,"f":"Causalean/Estimation/MinimaxATE/Causal/Minimax.lean"},{"n":"Causalean.Estimation.MinimaxATE.VarConstr2.obsReal_pert2_eq","k":"theorem","m":"Causalean.Estimation.MinimaxATE.VaryingCenterCase2.ChiSqOverlap","a":"Estimation/MinimaxATE/VaryingCenterCase2","t":2,"d":"Denominator-free value of the perturbed observed mass at the four (d, y).","s":"∀ {K : ℕ} (P : Causalean.Estimation.MinimaxATE.VarConstr2 K) (lam : Fin K → Bool) (x : Fin K × Bool) (d y : Bool), Causalean.Estimation.MinimaxATE.obsReal (P.mPert2 lam) (P.gPert2 lam) (x, d, y) = (↑(Fintype.card (Fin K × Bool)))⁻¹ * match d, y with | true, true => P.m₀ x.1 * (P.g₁ x.1 + P.α * P.g₁ x.1 ^ 2 * Causalean.Estimation.MinimaxATE.Δ lam x) | true, false => P.m₀ x.1 * (1 + P.α * P.g₁ x.1 * Causalean.Estimation.MinimaxATE.Δ lam x) * (P.D2 lam x - P.g₁ x.1) | false, true => (1 - P.m₀ x.1 - P.m₀ x.1 * P.κ x.1 * Causalean.Estimation.MinimaxATE.Δ lam x) * P.g₀ x.1 | false, false => (1 - P.m₀ x.1 - P.m₀ x.1 * P.κ x.1 * Causalean.Estimation.MinimaxATE.Δ lam x) * (1 - P.g₀ x.1)","l":73,"f":"Causalean/Estimation/MinimaxATE/VaryingCenterCase2/ChiSqOverlap.lean"},{"n":"Causalean.Estimation.MinimaxATE.VarConstr2.one_add_chiSqDiv_Qtrue2_Qfalse2","k":"theorem","m":"Causalean.Estimation.MinimaxATE.VaryingCenterCase2.LowerBound","a":"Estimation/MinimaxATE/VaryingCenterCase2","t":2,"d":"Mixture second-moment identity (second construction).","s":"∀ {K : ℕ} (P : Causalean.Estimation.MinimaxATE.VarConstr2 K) {n : ℕ} [inst : NeZero K], 1 + Causalean.Stat.chiSqDiv (P.Qtrue2 n) (P.Qfalse2 n) = ∑ lam, ∑ lam', (2 ^ K)⁻¹ * (2 ^ K)⁻¹ * P.chiSqOverlapV2 lam lam' ^ n","l":141,"f":"Causalean/Estimation/MinimaxATE/VaryingCenterCase2/LowerBound.lean"},{"n":"Causalean.Estimation.MinimaxATE.VarConstr2.ratio_nonneg","k":"theorem","m":"Causalean.Estimation.MinimaxATE.VaryingCenterCase2.Construction","a":"Estimation/MinimaxATE/VaryingCenterCase2","t":2,"d":"0 ≤ β/g₁ j.","s":"∀ {K : ℕ} (P : Causalean.Estimation.MinimaxATE.VarConstr2 K) (j : Fin K), 0 ≤ P.β / P.g₁ j","l":109,"f":"Causalean/Estimation/MinimaxATE/VaryingCenterCase2/Construction.lean"},{"n":"Causalean.Estimation.MinimaxATE.VarConstr2.real2_le_minimaxMiss","k":"theorem","m":"Causalean.Estimation.MinimaxATE.VaryingCenterCase2.LowerBound","a":"Estimation/MinimaxATE/VaryingCenterCase2","t":2,"d":"An in-class DGP's miss probability at its own ATE is dominated by the minimax miss.","s":"∀ {K : ℕ} (P : Causalean.Estimation.MinimaxATE.VarConstr2 K) {n : ℕ} [inst : NeZero K] {εg εm : ℝ} {m : Fin K × Bool → ℝ} {g : Bool → Fin K × Bool → ℝ} (hin : Causalean.Estimation.MinimaxATE.InClass P.mhat2 P.ghat2 εg εm m g) (est : (Fin n → Causalean.Estimation.MinimaxATE.Obs (Fin K × Bool)) → ℝ) (s : ℝ), (Causalean.Estimation.MinimaxATE.productLaw hin.valid n).real {x | s ≤ |est x - Causalean.Estimation.MinimaxATE.ate g|} ≤ Causalean.Estimation.MinimaxATE.minimaxMiss P.mhat2 P.ghat2 εg εm n est s","l":90,"f":"Causalean/Estimation/MinimaxATE/VaryingCenterCase2/LowerBound.lean"},{"n":"Causalean.Estimation.MinimaxATE.VarConstr2.real2_le_minimaxMissCausal","k":"theorem","m":"Causalean.Estimation.MinimaxATE.Causal.Minimax","a":"Estimation/MinimaxATE/Causal","t":2,"d":"An in-class Case-2 DGP's causal-centered miss probability is dominated by minimaxMissCausal.","s":"∀ {K : ℕ} (P : Causalean.Estimation.MinimaxATE.VarConstr2 K) {n : ℕ} [inst : NeZero K] {εg εm : ℝ} {m : Fin K × Bool → ℝ} {g : Bool → Fin K × Bool → ℝ} (hin : Causalean.Estimation.MinimaxATE.InClass P.mhat2 P.ghat2 εg εm m g) (est : (Fin n → Causalean.Estimation.MinimaxATE.Obs (Fin K × Bool)) → ℝ) (s : ℝ), (Causalean.Estimation.MinimaxATE.productLaw hin.valid n).real {x | s ≤ |est x - Causalean.Estimation.MinimaxATE.Causal.causalATE m g|} ≤ Causalean.Estimation.MinimaxATE.minimaxMissCausal P.mhat2 P.ghat2 εg εm n est s","l":299,"f":"Causalean/Estimation/MinimaxATE/Causal/Minimax.lean"},{"n":"Causalean.Estimation.MinimaxATE.VarConstr2.tvDist_Qfalse2_Qtrue2_le_half","k":"theorem","m":"Causalean.Estimation.MinimaxATE.VaryingCenterCase2.LowerBound","a":"Estimation/MinimaxATE/VaryingCenterCase2","t":2,"d":"Total-variation indistinguishability (second construction).","s":"∀ {K : ℕ} (P : Causalean.Estimation.MinimaxATE.VarConstr2 K) {n : ℕ} [inst : NeZero K], ∑ j, P.ΓV2 j / ↑K ≤ 1 → ↑n ^ 2 / 2 * ∑ j, (P.ΓV2 j / ↑K) ^ 2 ≤ Real.log 2 → Causalean.Stat.tvDist (P.Qfalse2 n) (P.Qtrue2 n) ≤ 1 / 2","l":198,"f":"Causalean/Estimation/MinimaxATE/VaryingCenterCase2/LowerBound.lean"},{"n":"Causalean.Estimation.MinimaxATE.VarConstr2.validDGP_hat2","k":"theorem","m":"Causalean.Estimation.MinimaxATE.VaryingCenterCase2.Construction","a":"Estimation/MinimaxATE/VaryingCenterCase2","t":2,"d":"The null DGP (m̂, ĝ) is valid.","s":"∀ {K : ℕ} (P : Causalean.Estimation.MinimaxATE.VarConstr2 K), Causalean.Estimation.MinimaxATE.ValidDGP P.mhat2 P.ghat2","l":169,"f":"Causalean/Estimation/MinimaxATE/VaryingCenterCase2/Construction.lean"},{"n":"Causalean.Estimation.MinimaxATE.VarConstr2.validDGP_pert2","k":"theorem","m":"Causalean.Estimation.MinimaxATE.VaryingCenterCase2.Construction","a":"Estimation/MinimaxATE/VaryingCenterCase2","t":1,"d":"For any Rademacher sign vector lam indexing the perturbation, the perturbed propensity and outcome-regression functions define a valid finite observed-data model, i.e. take values in [0,1].","s":"∀ {K : ℕ} (P : Causalean.Estimation.MinimaxATE.VarConstr2 K) (lam : Fin K → Bool), Causalean.Estimation.MinimaxATE.ValidDGP (P.mPert2 lam) (P.gPert2 lam)","l":177,"f":"Causalean/Estimation/MinimaxATE/VaryingCenterCase2/Construction.lean"},{"n":"Causalean.Estimation.MinimaxATE.VarConstr2.ΓV2","k":"def","m":"Causalean.Estimation.MinimaxATE.VaryingCenterCase2.ChiSqOverlap","a":"Estimation/MinimaxATE/VaryingCenterCase2","t":1,"d":"The per-pair χ²-overlap coefficient (second construction).","s":"{K : ℕ} → Causalean.Estimation.MinimaxATE.VarConstr2 K → Fin K → ℝ","l":47,"f":"Causalean/Estimation/MinimaxATE/VaryingCenterCase2/ChiSqOverlap.lean"},{"n":"Causalean.Estimation.MinimaxATE.VarConstr2.ΓV2_nonneg","k":"theorem","m":"Causalean.Estimation.MinimaxATE.VaryingCenterCase2.ChiSqOverlap","a":"Estimation/MinimaxATE/VaryingCenterCase2","t":2,"d":"Γⱼ ≥ 0.","s":"∀ {K : ℕ} (P : Causalean.Estimation.MinimaxATE.VarConstr2 K) (j : Fin K), 0 ≤ P.ΓV2 j","l":54,"f":"Causalean/Estimation/MinimaxATE/VaryingCenterCase2/ChiSqOverlap.lean"},{"n":"Causalean.Estimation.MinimaxATE.VarConstr2.κ","k":"def","m":"Causalean.Estimation.MinimaxATE.VaryingCenterCase2.Construction","a":"Estimation/MinimaxATE/VaryingCenterCase2","t":1,"d":"The propensity bump coefficient κⱼ = β/g₁ⱼ + α·g₁ⱼ − α²·β·g₁ⱼ.","s":"{K : ℕ} → Causalean.Estimation.MinimaxATE.VarConstr2 K → Fin K → ℝ","l":101,"f":"Causalean/Estimation/MinimaxATE/VaryingCenterCase2/Construction.lean"},{"n":"Causalean.Estimation.MinimaxATE.VarConstr2.κ_nonneg","k":"theorem","m":"Causalean.Estimation.MinimaxATE.VaryingCenterCase2.Construction","a":"Estimation/MinimaxATE/VaryingCenterCase2","t":2,"d":"0 ≤ κⱼ.","s":"∀ {K : ℕ} (P : Causalean.Estimation.MinimaxATE.VarConstr2 K) (j : Fin K), 0 ≤ P.κ j","l":112,"f":"Causalean/Estimation/MinimaxATE/VaryingCenterCase2/Construction.lean"},{"n":"Causalean.Estimation.MinimaxATE.absolutelyContinuous_of_singleton_pos","k":"theorem","m":"Causalean.Estimation.MinimaxATE.ConstCenterHalf.ChiSquaredCore","a":"Estimation/MinimaxATE/ConstCenterHalf","t":2,"d":"Absolute continuity from full support. If ν charges every singleton, every measure is absolutely continuous w.r.t. ν (a ν-null set must be empty).","s":"∀ {Ω : Type u_1} [inst : MeasurableSpace Ω] (μ ν : MeasureTheory.Measure Ω), (∀ (x : Ω), ν {x} ≠ 0) → μ.AbsolutelyContinuous ν","l":56,"f":"Causalean/Estimation/MinimaxATE/ConstCenterHalf/ChiSquaredCore.lean"},{"n":"Causalean.Estimation.MinimaxATE.aipwScoreFin","k":"def","m":"Causalean.Estimation.MinimaxATE.Achievability.AIPWEstimator","a":"Estimation/MinimaxATE/Achievability","t":1,"d":"The fixed-center augmented inverse-propensity score combines the fitted treated-versus-control outcome-regression contrast with the residual from the observed treatment arm, weighted by the fitted propensity at the covariate value.","s":"{C : Type u_1} → (C → ℝ) → (Bool → C → ℝ) → Causalean.Estimation.MinimaxATE.Obs C → ℝ","l":61,"f":"Causalean/Estimation/MinimaxATE/Achievability/AIPWEstimator.lean"},{"n":"Causalean.Estimation.MinimaxATE.aipwScore_bound","k":"theorem","m":"Causalean.Estimation.MinimaxATE.Achievability.AIPWEstimator","a":"Estimation/MinimaxATE/Achievability","t":2,"d":"Score bound. The AIPW score is bounded by B = 1 + 2/ε whenever the center nuisances are valid and mhat is ε-bounded off {0,1}.","s":"∀ {C : Type u_1} [Nonempty C] (mhat : C → ℝ) (ghat : Bool → C → ℝ), Causalean.Estimation.MinimaxATE.ValidDGP mhat ghat → ∀ {ε : ℝ}, 0 < ε → (∀ (x : C), ε ≤ mhat x ∧ ε ≤ 1 - mhat x) → ∀ (z : Causalean.Estimation.MinimaxATE.Obs C), |Causalean.Estimation.MinimaxATE.aipwScoreFin mhat ghat z| ≤ 1 + 2 / ε","l":253,"f":"Causalean/Estimation/MinimaxATE/Achievability/AIPWEstimator.lean"},{"n":"Causalean.Estimation.MinimaxATE.aipw_attains_minimax_rate","k":"def","m":"Causalean.Estimation.MinimaxATE.Optimality","a":"Estimation/MinimaxATE","t":1,"d":"Capstone — DR/AIPW attains the structure-agnostic minimax rate. For the cell-varying construction P with the per-pair budgets and regularity conditions of the lower bound, the problem exhibits a MinimaxRateThreshold:","s":"{K : ℕ} → (P : Causalean.Estimation.MinimaxATE.VarConstr K) → [inst : NeZero K] → {n : ℕ} → 0 < n → {εg εm : ℝ} → (∀ (j : Fin K), (P.m₀ j * (P.β / P.g₁ j)) ^ 2 ≤ εm) → (∀ (j : Fin K), P.g₁ j ^ 2 * (P.α + P.β) ^ 2 / (P.g₁ j - P.β) ^ 2 ≤ εg) → 0 ≤ εg → 0 ≤ εm → ∑ j, P.ΓV j / ↑K ≤ 1 → ↑n ^ 2 / 2 * ∑ j, (P.ΓV j / ↑K) ^ 2 ≤ Real.log 2 → {ε : ℝ} → 0 < ε → (∀ (x : Fin K × Bool), ε ≤ P.mhatV x ∧ ε ≤ 1 - P.mhatV x) → {s : ℝ} → ε⁻¹ * (2 * √εg * √εm) < s → Causalean.Estimation.MinimaxATE.MinimaxRateThreshold P.mhatV P.ghatV εg εm n","l":177,"f":"Causalean/Estimation/MinimaxATE/Optimality.lean"},{"n":"Causalean.Estimation.MinimaxATE.aipw_bias_bound","k":"theorem","m":"Causalean.Estimation.MinimaxATE.Achievability.AIPWEstimator","a":"Estimation/MinimaxATE/Achievability","t":1,"d":"Doubly-robust bias bound. Given a positive slack ε such that the fitted propensity mhat stays at least ε away from both 0 and 1 at every covariate value — the center has uniform overlap, the absolute bias of the population AIPW mean relative to the true ATE is at most ε⁻¹ times the product of the combined treated/control L²(P_X) outcome-regression error and the L²(P_X) propensity error.","s":"∀ {C : Type u_1} [inst : Fintype C] [Nonempty C] {m : C → ℝ} {g : Bool → C → ℝ} (mhat : C → ℝ) (ghat : Bool → C → ℝ) {ε : ℝ}, 0 < ε → (∀ (x : C), ε ≤ mhat x ∧ ε ≤ 1 - mhat x) → |∑ z, Causalean.Estimation.MinimaxATE.obsReal m g z * Causalean.Estimation.MinimaxATE.aipwScoreFin mhat ghat z - Causalean.Estimation.MinimaxATE.ate g| ≤ ε⁻¹ * ((√(Causalean.Estimation.MinimaxATE.l2sq (g true) (ghat true)) + √(Causalean.Estimation.MinimaxATE.l2sq (g false) (ghat false))) * √(Causalean.Estimation.MinimaxATE.l2sq m mhat))","l":148,"f":"Causalean/Estimation/MinimaxATE/Achievability/AIPWEstimator.lean"},{"n":"Causalean.Estimation.MinimaxATE.aipw_bias_identity","k":"theorem","m":"Causalean.Estimation.MinimaxATE.Achievability.AIPWEstimator","a":"Estimation/MinimaxATE/Achievability","t":1,"d":"Doubly-robust bias identity. Assume the fitted propensity mhat takes values strictly between 0 and 1 at every covariate value. Then the population mean of the AIPW score under the true data-generating process (m, g), minus the true average treatment effect, equals a finite doubly-robust remainder built from cell-by-cell products of the propensity error m − mhat and the outcome-regression errors g − ghat on each treatment arm.","s":"∀ {C : Type u_1} [inst : Fintype C] [Nonempty C] {m : C → ℝ} {g : Bool → C → ℝ} (mhat : C → ℝ) (ghat : Bool → C → ℝ), (∀ (x : C), 0 < mhat x) → (∀ (x : C), mhat x < 1) → ∑ z, Causalean.Estimation.MinimaxATE.obsReal m g z * Causalean.Estimation.MinimaxATE.aipwScoreFin mhat ghat z - Causalean.Estimation.MinimaxATE.ate g = (↑(Fintype.card C))⁻¹ * ∑ x, ((1 - m x / mhat x) * (ghat true x - g true x) - (1 - (1 - m x) / (1 - mhat x)) * (ghat false x - g false x))","l":94,"f":"Causalean/Estimation/MinimaxATE/Achievability/AIPWEstimator.lean"},{"n":"Causalean.Estimation.MinimaxATE.aipw_inclass_bias_bound","k":"theorem","m":"Causalean.Estimation.MinimaxATE.Optimality","a":"Estimation/MinimaxATE","t":1,"d":"In-class bias bound. For a positive overlap threshold ε such that the fixed propensity estimate mhat stays at least ε away from both 0 and 1 at every covariate value, and for any data-generating process (m, g) lying in the structure-agnostic nuisance class around the fixed estimates (mhat, ghat) with error budgets εg, εm, the plug-in bias of the fixed-center AIPW estimator's population score is at most ε⁻¹·2·√εg·√εm.","s":"∀ {C : Type u_1} [inst : Fintype C] [Nonempty C] {mhat : C → ℝ} {ghat : Bool → C → ℝ} {εg εm ε : ℝ}, 0 < ε → (∀ (x : C), ε ≤ mhat x ∧ ε ≤ 1 - mhat x) → ∀ {m : C → ℝ} {g : Bool → C → ℝ}, Causalean.Estimation.MinimaxATE.InClass mhat ghat εg εm m g → |∑ z, Causalean.Estimation.MinimaxATE.obsReal m g z * Causalean.Estimation.MinimaxATE.aipwScoreFin mhat ghat z - Causalean.Estimation.MinimaxATE.ate g| ≤ ε⁻¹ * (2 * √εg * √εm)","l":108,"f":"Causalean/Estimation/MinimaxATE/Optimality.lean"},{"n":"Causalean.Estimation.MinimaxATE.aipw_mean_eq","k":"theorem","m":"Causalean.Estimation.MinimaxATE.Optimality","a":"Estimation/MinimaxATE","t":2,"d":"The mean of the AIPW estimator equals the single-observation population mean of the score (mean of an i.i.d. average), for n > 0.","s":"∀ {C : Type u_1} [inst : Fintype C] [inst_1 : MeasurableSpace C] [MeasurableSingletonClass C] [inst_3 : Nonempty C] {m : C → ℝ} {g : Bool → C → ℝ} (hv : Causalean.Estimation.MinimaxATE.ValidDGP m g) (mhat : C → ℝ) (ghat : Bool → C → ℝ) {n : ℕ}, 0 < n → ∫ (sample : Fin n → Causalean.Estimation.MinimaxATE.Obs C), Causalean.Estimation.MinimaxATE.estAIPW mhat ghat n sample ∂Causalean.Estimation.MinimaxATE.productLaw hv n = ∑ z, Causalean.Estimation.MinimaxATE.obsReal m g z * Causalean.Estimation.MinimaxATE.aipwScoreFin mhat ghat z","l":51,"f":"Causalean/Estimation/MinimaxATE/Optimality.lean"},{"n":"Causalean.Estimation.MinimaxATE.aipw_minimaxMiss_le","k":"theorem","m":"Causalean.Estimation.MinimaxATE.Optimality","a":"Estimation/MinimaxATE","t":1,"d":"Worst-case (minimax) miss bound for AIPW. Suppose the fixed nuisance estimates (mhat, ghat) are themselves valid, for a positive overlap threshold ε such that mhat stays at least ε away from 0 and 1 everywhere, nonnegative error budgets εg, εm, a positive sample size n, and a separation s exceeding the uniform bias bound b = ε⁻¹·2·√εg·√εm. Then the worst-case-over-class miss probability of the fixed-center AIPW estimator at separation s is at most ((1+2/ε)²/n)/(s−b)².","s":"∀ {C : Type u_1} [inst : Fintype C] [inst_1 : MeasurableSpace C] [MeasurableSingletonClass C] [inst_3 : Nonempty C] {mhat : C → ℝ} {ghat : Bool → C → ℝ} {εg εm : ℝ}, Causalean.Estimation.MinimaxATE.ValidDGP mhat ghat → ∀ {ε : ℝ}, 0 < ε → (∀ (x : C), ε ≤ mhat x ∧ ε ≤ 1 - mhat x) → 0 ≤ εg → 0 ≤ εm → ∀ {n : ℕ}, 0 < n → ∀ {s : ℝ}, ε⁻¹ * (2 * √εg * √εm) < s → Causalean.Estimation.MinimaxATE.minimaxMiss mhat ghat εg εm n (Causalean.Estimation.MinimaxATE.estAIPW mhat ghat n) s ≤ (1 + 2 / ε) ^ 2 / ↑n / (s - ε⁻¹ * (2 * √εg * √εm)) ^ 2","l":133,"f":"Causalean/Estimation/MinimaxATE/Optimality.lean"},{"n":"Causalean.Estimation.MinimaxATE.aipw_nMiss_le","k":"theorem","m":"Causalean.Estimation.MinimaxATE.Optimality","a":"Estimation/MinimaxATE","t":2,"d":"Per-DGP Chebyshev miss bound. If the plug-in bias is ≤ b and the variance is ≤ V, then for any separation s > b the miss probability is ≤ V / (s − b)².","s":"∀ {C : Type u_1} [inst : Fintype C] [inst_1 : MeasurableSpace C] [MeasurableSingletonClass C] [inst_3 : Nonempty C] {m : C → ℝ} {g : Bool → C → ℝ} (hv : Causalean.Estimation.MinimaxATE.ValidDGP m g) (mhat : C → ℝ) (ghat : Bool → C → ℝ) {n : ℕ}, 0 < n → ∀ {b V s : ℝ}, |∑ z, Causalean.Estimation.MinimaxATE.obsReal m g z * Causalean.Estimation.MinimaxATE.aipwScoreFin mhat ghat z - Causalean.Estimation.MinimaxATE.ate g| ≤ b → ProbabilityTheory.variance (Causalean.Estimation.MinimaxATE.estAIPW mhat ghat n) (Causalean.Estimation.MinimaxATE.productLaw hv n) ≤ V → b < s → Causalean.Estimation.MinimaxATE.nMiss hv n (Causalean.Estimation.MinimaxATE.estAIPW mhat ghat n) s ≤ V / (s - b) ^ 2","l":73,"f":"Causalean/Estimation/MinimaxATE/Optimality.lean"},{"n":"Causalean.Estimation.MinimaxATE.aipw_pop_mean","k":"theorem","m":"Causalean.Estimation.MinimaxATE.Achievability.AIPWEstimator","a":"Estimation/MinimaxATE/Achievability","t":2,"d":"Population mean of the AIPW score under the DGP (m, g) is the finite sum Σ_z obsReal m g z · ψ_AIPW(z; m̂, ĝ).","s":"∀ {C : Type u_1} [inst : Fintype C] [inst_1 : MeasurableSpace C] [MeasurableSingletonClass C] [inst_3 : Nonempty C] {m : C → ℝ} {g : Bool → C → ℝ} (hv : Causalean.Estimation.MinimaxATE.ValidDGP m g) (mhat : C → ℝ) (ghat : Bool → C → ℝ), ∫ (z : Causalean.Estimation.MinimaxATE.Obs C), Causalean.Estimation.MinimaxATE.aipwScoreFin mhat ghat z ∂Causalean.Estimation.MinimaxATE.obsLaw hv = ∑ z, Causalean.Estimation.MinimaxATE.obsReal m g z * Causalean.Estimation.MinimaxATE.aipwScoreFin mhat ghat z","l":79,"f":"Causalean/Estimation/MinimaxATE/Achievability/AIPWEstimator.lean"},{"n":"Causalean.Estimation.MinimaxATE.aipw_var_bound","k":"theorem","m":"Causalean.Estimation.MinimaxATE.Achievability.AIPWEstimator","a":"Estimation/MinimaxATE/Achievability","t":1,"d":"Variance bound. Suppose the true data-generating process (m, g) is valid, the fixed nuisance center (mhat, ghat) is itself a valid data-generating process, and that center has uniform overlap: the fitted propensity mhat stays at least a positive constant ε away from both 0 and 1 at every covariate value. Then on an n-observation i.i.d. sample from the true single-observation law, the variance of the fixed-center AIPW sample-average estimator is at most (1 + 2/ε)²/n.","s":"∀ {C : Type u_1} [inst : Fintype C] [inst_1 : MeasurableSpace C] [MeasurableSingletonClass C] [inst_3 : Nonempty C] {m : C → ℝ} {g : Bool → C → ℝ} (hv : Causalean.Estimation.MinimaxATE.ValidDGP m g) (mhat : C → ℝ) (ghat : Bool → C → ℝ), Causalean.Estimation.MinimaxATE.ValidDGP mhat ghat → ∀ {ε : ℝ}, 0 < ε → (∀ (x : C), ε ≤ mhat x ∧ ε ≤ 1 - mhat x) → ∀ (n : ℕ), ProbabilityTheory.variance (Causalean.Estimation.MinimaxATE.estAIPW mhat ghat n) (Causalean.Estimation.MinimaxATE.productLaw hv n) ≤ (1 + 2 / ε) ^ 2 / ↑n","l":288,"f":"Causalean/Estimation/MinimaxATE/Achievability/AIPWEstimator.lean"},{"n":"Causalean.Estimation.MinimaxATE.ancillaryLaw","k":"def","m":"Causalean.Estimation.MinimaxATE.ConstCenterHalf.ContinuousX","a":"Estimation/MinimaxATE/ConstCenterHalf","t":1,"d":"Joint law of the ancillary within-cell positions in the n-sample experiment.","s":"{W : Type u_1} → [inst : MeasurableSpace W] → MeasureTheory.Measure W → (n : ℕ) → MeasureTheory.Measure (Fin n → W)","l":54,"f":"Causalean/Estimation/MinimaxATE/ConstCenterHalf/ContinuousX.lean"},{"n":"Causalean.Estimation.MinimaxATE.ancillaryLaw_isProb","k":"instance","m":"Causalean.Estimation.MinimaxATE.ConstCenterHalf.ContinuousX","a":"Estimation/MinimaxATE/ConstCenterHalf","t":2,"d":"The ancillary n-sample law is a probability measure when the one-draw law is a probability measure.","s":"∀ {W : Type u_1} [inst : MeasurableSpace W] (ρ : MeasureTheory.Measure W) [MeasureTheory.IsProbabilityMeasure ρ] (n : ℕ), MeasureTheory.IsProbabilityMeasure (Causalean.Estimation.MinimaxATE.ancillaryLaw ρ n)","l":58,"f":"Causalean/Estimation/MinimaxATE/ConstCenterHalf/ContinuousX.lean"},{"n":"Causalean.Estimation.MinimaxATE.ate","k":"def","m":"Causalean.Estimation.MinimaxATE.Model","a":"Estimation/MinimaxATE","t":1,"d":"The average treatment effect (1/card C) Σ_x (g true x − g false x). Depends only on the outcome regression g.","s":"{C : Type u_1} → [Fintype C] → (Bool → C → ℝ) → ℝ","l":136,"f":"Causalean/Estimation/MinimaxATE/Model.lean"},{"n":"Causalean.Estimation.MinimaxATE.ate_gPerturbed","k":"theorem","m":"Causalean.Estimation.MinimaxATE.ConstCenterHalf.Gap","a":"Estimation/MinimaxATE/ConstCenterHalf","t":1,"d":"Exact ATE of the perturbed construction. For nonnegative bump magnitudes α and β with α + 2β ≤ 1/2 and any Rademacher sign vector lam, the average treatment effect of the perturbed construction equals 2β(α+β)/(1−4β²), independent of lam.","s":"∀ {K : ℕ} {α β : ℝ} [NeZero K], 0 ≤ α → 0 ≤ β → α + 2 * β ≤ 1 / 2 → ∀ (lam : Fin K → Bool), Causalean.Estimation.MinimaxATE.ate (Causalean.Estimation.MinimaxATE.gPerturbed α β lam) = 2 * β * (α + β) / (1 - 4 * β ^ 2)","l":81,"f":"Causalean/Estimation/MinimaxATE/ConstCenterHalf/Gap.lean"},{"n":"Causalean.Estimation.MinimaxATE.ate_gap","k":"theorem","m":"Causalean.Estimation.MinimaxATE.ConstCenterHalf.Gap","a":"Estimation/MinimaxATE/ConstCenterHalf","t":1,"d":"For nonnegative bump magnitudes α and β with α + 2β ≤ 1/2 and any Rademacher sign vector lam, the gap between the perturbed construction's average treatment effect and the null estimate's average treatment effect (which is zero) equals 2β(α+β)/(1−4β²).","s":"∀ {K : ℕ} {α β : ℝ} [NeZero K], 0 ≤ α → 0 ≤ β → α + 2 * β ≤ 1 / 2 → ∀ (lam : Fin K → Bool), Causalean.Estimation.MinimaxATE.ate (Causalean.Estimation.MinimaxATE.gPerturbed α β lam) - Causalean.Estimation.MinimaxATE.ate Causalean.Estimation.MinimaxATE.ghat = 2 * β * (α + β) / (1 - 4 * β ^ 2)","l":110,"f":"Causalean/Estimation/MinimaxATE/ConstCenterHalf/Gap.lean"},{"n":"Causalean.Estimation.MinimaxATE.ate_ghat","k":"theorem","m":"Causalean.Estimation.MinimaxATE.ConstCenterHalf.Gap","a":"Estimation/MinimaxATE/ConstCenterHalf","t":2,"d":"The null estimate ĝ ≡ 1/2 has zero ATE: every cell contributes 1/2 − 1/2 = 0.","s":"∀ {K : ℕ}, Causalean.Estimation.MinimaxATE.ate Causalean.Estimation.MinimaxATE.ghat = 0","l":44,"f":"Causalean/Estimation/MinimaxATE/ConstCenterHalf/Gap.lean"},{"n":"Causalean.Estimation.MinimaxATE.bddAbove_nMissCausal_range","k":"theorem","m":"Causalean.Estimation.MinimaxATE.Causal.Minimax","a":"Estimation/MinimaxATE/Causal","t":2,"d":"Each in-class causal-centered miss probability is bounded above by 1.","s":"∀ {C : Type} [inst : Fintype C] [inst_1 : Nonempty C] [inst_2 : MeasurableSpace C] [inst_3 : MeasurableSingletonClass C] (mhat : C → ℝ) (ghat : Bool → C → ℝ) (εg εm : ℝ) (n : ℕ) (est : (Fin n → Causalean.Estimation.MinimaxATE.Obs C) → ℝ) (s : ℝ), BddAbove (Set.range fun p => (Causalean.Estimation.MinimaxATE.productLaw p.property.valid n).real {x | s ≤ |est x - Causalean.Estimation.MinimaxATE.Causal.causalATE (↑p).1 (↑p).2|})","l":52,"f":"Causalean/Estimation/MinimaxATE/Causal/Minimax.lean"},{"n":"Causalean.Estimation.MinimaxATE.bddAbove_nMiss_range","k":"theorem","m":"Causalean.Estimation.MinimaxATE.Model","a":"Estimation/MinimaxATE","t":2,"d":"Each in-class miss probability is bounded above by 1 (it is a probability), so the range of nMiss over the class is bounded above.","s":"∀ {C : Type u_1} [inst : Fintype C] [inst_1 : MeasurableSpace C] [inst_2 : Nonempty C] (mhat : C → ℝ) (ghat : Bool → C → ℝ) (εg εm : ℝ) (n : ℕ) (est : (Fin n → Causalean.Estimation.MinimaxATE.Obs C) → ℝ) (s : ℝ), BddAbove (Set.range fun p => Causalean.Estimation.MinimaxATE.nMiss p.property.valid n est s)","l":218,"f":"Causalean/Estimation/MinimaxATE/Model.lean"},{"n":"Causalean.Estimation.MinimaxATE.chiSqDiv_QtrueCont_QfalseCont_le_one","k":"theorem","m":"Causalean.Estimation.MinimaxATE.ConstCenterHalf.ContinuousX","a":"Estimation/MinimaxATE/ConstCenterHalf","t":2,"d":"The continuous-covariate alternative is chi-squared close to the null in the finite-cell regime.","s":"∀ {K n : ℕ} {α β : ℝ} {W : Type u_1} [inst : MeasurableSpace W] [inst_1 : NeZero K] (ρ : MeasureTheory.Measure W) [MeasureTheory.IsProbabilityMeasure ρ] (hα : 0 ≤ α) (hβ : 0 ≤ β) (hαβ : α + 2 * β ≤ 1 / 2), 2 * (α ^ 2 + 2 * α * β + 3 * β ^ 2) ≤ 1 → 2 * ↑n ^ 2 * (α ^ 2 + 2 * α * β + 3 * β ^ 2) ^ 2 ≤ ↑K * Real.log 2 → Causalean.Stat.chiSqDiv (Causalean.Estimation.MinimaxATE.QtrueCont hα hβ hαβ n ρ) (Causalean.Estimation.MinimaxATE.QfalseCont K n ρ) ≤ 1","l":93,"f":"Causalean/Estimation/MinimaxATE/ConstCenterHalf/ContinuousX.lean"},{"n":"Causalean.Estimation.MinimaxATE.chiSqDiv_Qtrue_Qfalse_le_one","k":"theorem","m":"Causalean.Estimation.MinimaxATE.ConstCenterHalf.ChiSquaredCore","a":"Estimation/MinimaxATE/ConstCenterHalf","t":2,"d":"The χ² indistinguishability bound. In the regime 2n²γ² ≤ K·log 2 (with γ = α²+2αβ+3β², 2γ ≤ 1), the χ²-divergence of the alternative mixture from the null is at most 1.","s":"∀ {K n : ℕ} {α β : ℝ} [inst : NeZero K] (hα : 0 ≤ α) (hβ : 0 ≤ β) (hαβ : α + 2 * β ≤ 1 / 2), 2 * (α ^ 2 + 2 * α * β + 3 * β ^ 2) ≤ 1 → 2 * ↑n ^ 2 * (α ^ 2 + 2 * α * β + 3 * β ^ 2) ^ 2 ≤ ↑K * Real.log 2 → Causalean.Stat.chiSqDiv (Causalean.Estimation.MinimaxATE.Qtrue hα hβ hαβ n) (Causalean.Estimation.MinimaxATE.Qfalse K n) ≤ 1","l":157,"f":"Causalean/Estimation/MinimaxATE/ConstCenterHalf/ChiSquaredCore.lean"},{"n":"Causalean.Estimation.MinimaxATE.chiSqOverlap","k":"def","m":"Causalean.Estimation.MinimaxATE.ConstCenterHalf.ChiSqOverlap","a":"Estimation/MinimaxATE/ConstCenterHalf","t":1,"d":"The one-observation χ² second-moment overlap of two perturbed laws relative to the null.","s":"{K : ℕ} → ℝ → ℝ → (Fin K → Bool) → (Fin K → Bool) → ℝ","l":49,"f":"Causalean/Estimation/MinimaxATE/ConstCenterHalf/ChiSqOverlap.lean"},{"n":"Causalean.Estimation.MinimaxATE.chiSqOverlap_eq","k":"theorem","m":"Causalean.Estimation.MinimaxATE.ConstCenterHalf.ChiSqOverlap","a":"Estimation/MinimaxATE/ConstCenterHalf","t":1,"d":"Closed form of the one-observation χ² overlap. For nonnegative bump magnitudes α and β with α + 2β ≤ 1/2, the regime keeping every perturbed nuisance in [0,1], and any two Rademacher sign vectors lam, lam' indexing perturbed laws, the one-observation χ² second-moment overlap of the two perturbed laws relative to the null equals 1 + (2(α²+2αβ+3β²)/K)·Σⱼ signOf(lam j)·signOf(lam' j).","s":"∀ {K : ℕ} {α β : ℝ} [NeZero K], 0 ≤ α → 0 ≤ β → α + 2 * β ≤ 1 / 2 → ∀ (lam lam' : Fin K → Bool), Causalean.Estimation.MinimaxATE.chiSqOverlap α β lam lam' = 1 + 2 * (α ^ 2 + 2 * α * β + 3 * β ^ 2) / ↑K * ∑ j, Causalean.Estimation.MinimaxATE.signOf (lam j) * Causalean.Estimation.MinimaxATE.signOf (lam' j)","l":78,"f":"Causalean/Estimation/MinimaxATE/ConstCenterHalf/ChiSqOverlap.lean"},{"n":"Causalean.Estimation.MinimaxATE.denom_pos","k":"theorem","m":"Causalean.Estimation.MinimaxATE.ConstCenterHalf.Construction","a":"Estimation/MinimaxATE/ConstCenterHalf","t":2,"d":"The treated-arm denominator in the perturbed outcome regression is positive in the valid parameter regime.","s":"∀ {K : ℕ} {α β : ℝ}, 0 ≤ β → α + 2 * β ≤ 1 / 2 → 0 ≤ α → ∀ (lam : Fin K → Bool) (x : Fin K × Bool), 0 < 1 - 2 * β * Causalean.Estimation.MinimaxATE.Δ lam x","l":123,"f":"Causalean/Estimation/MinimaxATE/ConstCenterHalf/Construction.lean"},{"n":"Causalean.Estimation.MinimaxATE.estAIPW","k":"def","m":"Causalean.Estimation.MinimaxATE.Achievability.AIPWEstimator","a":"Estimation/MinimaxATE/Achievability","t":1,"d":"The fixed-center AIPW estimator: the sample average of the AIPW score.","s":"{C : Type u_1} → (C → ℝ) → (Bool → C → ℝ) → (n : ℕ) → (Fin n → Causalean.Estimation.MinimaxATE.Obs C) → ℝ","l":74,"f":"Causalean/Estimation/MinimaxATE/Achievability/AIPWEstimator.lean"},{"n":"Causalean.Estimation.MinimaxATE.exists_center_overlap","k":"theorem","m":"Causalean.Estimation.MinimaxATE.Achievability.AIPWEstimator","a":"Estimation/MinimaxATE/Achievability","t":2,"d":"Uniform lower bound on the center weights. With mhat strictly inside (0,1) on the finite type C, there is ε > 0 with ε ≤ mhat x and ε ≤ 1 − mhat x for all x.","s":"∀ {C : Type u_1} [Fintype C] [Nonempty C] (mhat : C → ℝ), (∀ (x : C), 0 < mhat x) → (∀ (x : C), mhat x < 1) → ∃ ε > 0, ∀ (x : C), ε ≤ mhat x ∧ ε ≤ 1 - mhat x","l":132,"f":"Causalean/Estimation/MinimaxATE/Achievability/AIPWEstimator.lean"},{"n":"Causalean.Estimation.MinimaxATE.explicitWitness","k":"def","m":"Causalean.Estimation.MinimaxATE.ConstCenterHalf.ExplicitWitness","a":"Estimation/MinimaxATE/ConstCenterHalf","t":1,"d":"Assemble the explicit Le Cam witness from the centered null and the uniform perturbation mixture.","s":"{K n : ℕ} → {α β εg εm : ℝ} → [inst : NeZero K] → (hα : 0 ≤ α) → (hβ : 0 ≤ β) → (hαβ : α + 2 * β ≤ 1 / 2) → β ^ 2 ≤ εm → (α + β) ^ 2 / (1 - 2 * β) ^ 2 ≤ εg → 0 ≤ εg → 0 ≤ εm → Causalean.Stat.tvDist (Causalean.Estimation.MinimaxATE.Qfalse K n) (Causalean.Estimation.MinimaxATE.Qtrue hα hβ hαβ n) ≤ 1 / 2 → Causalean.Estimation.MinimaxATE.TwoPointWitness (Fin K × Bool) n Causalean.Estimation.MinimaxATE.mhat Causalean.Estimation.MinimaxATE.ghat εg εm","l":120,"f":"Causalean/Estimation/MinimaxATE/ConstCenterHalf/ExplicitWitness.lean"},{"n":"Causalean.Estimation.MinimaxATE.explicit_minimax_lower_bound","k":"theorem","m":"Causalean.Estimation.MinimaxATE.ConstCenterHalf.ExplicitWitness","a":"Estimation/MinimaxATE/ConstCenterHalf","t":1,"d":"Fix nonnegative bump magnitudes α and β with α + 2β ≤ 1/2 meeting the Rademacher perturbation budgets β² ≤ εm and (α+β)²/(1−2β)² ≤ εg for nonnegative error tolerances εg, εm, and suppose the centered null law and the uniform mixture of perturbed laws are statistically close, at total-variation distance at most 1/2. Then for any measurable estimator of the average treatment effect, the worst-case-over-class probability that it misses the true ATE by s = β(α+β)/(1−4β²) is at least 1/4.","s":"∀ {K n : ℕ} {α β εg εm : ℝ} [inst : NeZero K] (hα : 0 ≤ α) (hβ : 0 ≤ β) (hαβ : α + 2 * β ≤ 1 / 2), β ^ 2 ≤ εm → (α + β) ^ 2 / (1 - 2 * β) ^ 2 ≤ εg → 0 ≤ εg → 0 ≤ εm → Causalean.Stat.tvDist (Causalean.Estimation.MinimaxATE.Qfalse K n) (Causalean.Estimation.MinimaxATE.Qtrue hα hβ hαβ n) ≤ 1 / 2 → ∀ {est : (Fin n → Causalean.Estimation.MinimaxATE.Obs (Fin K × Bool)) → ℝ}, Measurable est → 1 / 4 ≤ Causalean.Estimation.MinimaxATE.minimaxMiss Causalean.Estimation.MinimaxATE.mhat Causalean.Estimation.MinimaxATE.ghat εg εm n est (β * (α + β) / (1 - 4 * β ^ 2))","l":173,"f":"Causalean/Estimation/MinimaxATE/ConstCenterHalf/ExplicitWitness.lean"},{"n":"Causalean.Estimation.MinimaxATE.gPerturbed","k":"def","m":"Causalean.Estimation.MinimaxATE.ConstCenterHalf.Construction","a":"Estimation/MinimaxATE/ConstCenterHalf","t":1,"d":"Perturbed outcome regression with an unchanged control arm and a nonlinear treated-arm shift.","s":"{K : ℕ} → ℝ → ℝ → (Fin K → Bool) → Bool → Fin K × Bool → ℝ","l":111,"f":"Causalean/Estimation/MinimaxATE/ConstCenterHalf/Construction.lean"},{"n":"Causalean.Estimation.MinimaxATE.gPerturbed_true_sub_half","k":"theorem","m":"Causalean.Estimation.MinimaxATE.ConstCenterHalf.Membership","a":"Estimation/MinimaxATE/ConstCenterHalf","t":2,"d":"The treated outcome arm: its pointwise deviation from ĝ(1) = 1/2 is (α+β)·Δ/(1 − 2β·Δ).","s":"∀ {K : ℕ} {α β : ℝ}, 0 ≤ α → 0 ≤ β → α + 2 * β ≤ 1 / 2 → ∀ (lam : Fin K → Bool) (x : Fin K × Bool), Causalean.Estimation.MinimaxATE.gPerturbed α β lam true x - 1 / 2 = (α + β) * Causalean.Estimation.MinimaxATE.Δ lam x / (1 - 2 * β * Causalean.Estimation.MinimaxATE.Δ lam x)","l":67,"f":"Causalean/Estimation/MinimaxATE/ConstCenterHalf/Membership.lean"},{"n":"Causalean.Estimation.MinimaxATE.ghat","k":"def","m":"Causalean.Estimation.MinimaxATE.ConstCenterHalf.Construction","a":"Estimation/MinimaxATE/ConstCenterHalf","t":1,"d":"Centered outcome-regression estimate that assigns one half in both treatment arms.","s":"{K : ℕ} → Bool → Fin K × Bool → ℝ","l":102,"f":"Causalean/Estimation/MinimaxATE/ConstCenterHalf/Construction.lean"},{"n":"Causalean.Estimation.MinimaxATE.inClass_null","k":"theorem","m":"Causalean.Estimation.MinimaxATE.ConstCenterHalf.ExplicitWitness","a":"Estimation/MinimaxATE/ConstCenterHalf","t":2,"d":"The centered null nuisance functions are in the nuisance class whenever the budgets are nonnegative.","s":"∀ {K : ℕ} {εg εm : ℝ}, 0 ≤ εg → 0 ≤ εm → Causalean.Estimation.MinimaxATE.InClass Causalean.Estimation.MinimaxATE.mhat Causalean.Estimation.MinimaxATE.ghat εg εm Causalean.Estimation.MinimaxATE.mhat Causalean.Estimation.MinimaxATE.ghat","l":57,"f":"Causalean/Estimation/MinimaxATE/ConstCenterHalf/ExplicitWitness.lean"},{"n":"Causalean.Estimation.MinimaxATE.inClass_perturbed","k":"theorem","m":"Causalean.Estimation.MinimaxATE.ConstCenterHalf.Membership","a":"Estimation/MinimaxATE/ConstCenterHalf","t":1,"d":"Class membership. For nonnegative bump magnitudes α and β with α + 2β ≤ 1/2 meeting the Rademacher perturbation budgets β² ≤ εm and (α+β)²/(1−2β)² ≤ εg, the perturbed data-generating process (mλ, gλ) indexed by a Rademacher sign vector lam lies in the structure-agnostic nuisance class ℱ(εg, εm) around the centered estimates (m̂, ĝ) = (1/2, 1/2).","s":"∀ {K : ℕ} {α β : ℝ} [NeZero K] {εg εm : ℝ}, 0 ≤ α → 0 ≤ β → α + 2 * β ≤ 1 / 2 → β ^ 2 ≤ εm → (α + β) ^ 2 / (1 - 2 * β) ^ 2 ≤ εg → ∀ (lam : Fin K → Bool), Causalean.Estimation.MinimaxATE.InClass Causalean.Estimation.MinimaxATE.mhat Causalean.Estimation.MinimaxATE.ghat εg εm (Causalean.Estimation.MinimaxATE.mPerturbed β lam) (Causalean.Estimation.MinimaxATE.gPerturbed α β lam)","l":128,"f":"Causalean/Estimation/MinimaxATE/ConstCenterHalf/Membership.lean"},{"n":"Causalean.Estimation.MinimaxATE.ingster_bound","k":"theorem","m":"Causalean.Estimation.MinimaxATE.ConstCenterHalf.Ingster","a":"Estimation/MinimaxATE/ConstCenterHalf","t":1,"d":"For a nonnegative coefficient γ with 2γ ≤ 1 satisfying the regularity budget 2n²γ² ≤ K·log 2, the uniform double average, over pairs of Rademacher sign vectors lam, lam' : Fin K → Bool, of (1 + (2γ/K)·Σⱼ signOf(lam j)·signOf(lam' j))^n is at most 2.","s":"∀ (K n : ℕ) [NeZero K] {γ : ℝ}, 0 ≤ γ → 2 * γ ≤ 1 → 2 * ↑n ^ 2 * γ ^ 2 ≤ ↑K * Real.log 2 → ∑ lam, ∑ lam', (2 ^ K)⁻¹ * (2 ^ K)⁻¹ * (1 + 2 * γ / ↑K * ∑ j, Causalean.Estimation.MinimaxATE.signOf (lam j) * Causalean.Estimation.MinimaxATE.signOf (lam' j)) ^ n ≤ 2","l":37,"f":"Causalean/Estimation/MinimaxATE/ConstCenterHalf/Ingster.lean"},{"n":"Causalean.Estimation.MinimaxATE.ingster_bound_general","k":"theorem","m":"Causalean.Estimation.MinimaxATE.VaryingCenterCase1.Ingster","a":"Estimation/MinimaxATE/VaryingCenterCase1","t":1,"d":"For nonnegative per-coordinate weights d j summing to at most 1 satisfying the regularity budget (n²/2)·Σⱼ (d j)² ≤ log 2, the uniform double average, over pairs of Rademacher sign vectors lam, lam' : Fin K → Bool, of (1 + Σⱼ d j·signOf(lam j)·signOf(lam' j))^n is at most 2.","s":"∀ (K n : ℕ) [NeZero K] {d : Fin K → ℝ}, (∀ (j : Fin K), 0 ≤ d j) → ∑ j, d j ≤ 1 → ↑n ^ 2 / 2 * ∑ j, d j ^ 2 ≤ Real.log 2 → ∑ lam, ∑ lam', (2 ^ K)⁻¹ * (2 ^ K)⁻¹ * (1 + ∑ j, d j * (Causalean.Estimation.MinimaxATE.signOf (lam j) * Causalean.Estimation.MinimaxATE.signOf (lam' j))) ^ n ≤ 2","l":42,"f":"Causalean/Estimation/MinimaxATE/VaryingCenterCase1/Ingster.lean"},{"n":"Causalean.Estimation.MinimaxATE.integrable_sq_rnDeriv_QtrueCont_QfalseCont","k":"theorem","m":"Causalean.Estimation.MinimaxATE.ConstCenterHalf.ContinuousX","a":"Estimation/MinimaxATE/ConstCenterHalf","t":2,"d":"The squared density deviation for the continuous experiment is integrable under the null.","s":"∀ {K n : ℕ} {α β : ℝ} {W : Type u_1} [inst : MeasurableSpace W] [inst_1 : NeZero K] (ρ : MeasureTheory.Measure W) [MeasureTheory.IsProbabilityMeasure ρ] (hα : 0 ≤ α) (hβ : 0 ≤ β) (hαβ : α + 2 * β ≤ 1 / 2), MeasureTheory.Integrable (fun z => (((Causalean.Estimation.MinimaxATE.QtrueCont hα hβ hαβ n ρ).rnDeriv (Causalean.Estimation.MinimaxATE.QfalseCont K n ρ) z).toReal - 1) ^ 2) (Causalean.Estimation.MinimaxATE.QfalseCont K n ρ)","l":111,"f":"Causalean/Estimation/MinimaxATE/ConstCenterHalf/ContinuousX.lean"},{"n":"Causalean.Estimation.MinimaxATE.l2sq","k":"def","m":"Causalean.Estimation.MinimaxATE.Model","a":"Estimation/MinimaxATE","t":1,"d":"The squared L²(P_X) distance (1/card C) Σ_x (a x − b x)² between two functions of the covariate.","s":"{C : Type u_1} → [Fintype C] → (C → ℝ) → (C → ℝ) → ℝ","l":141,"f":"Causalean/Estimation/MinimaxATE/Model.lean"},{"n":"Causalean.Estimation.MinimaxATE.l2sq_bump","k":"theorem","m":"Causalean.Estimation.MinimaxATE.Reduction.Bump","a":"Estimation/MinimaxATE/Reduction","t":1,"d":"Constant-magnitude (Rademacher) bump. If a sign function σ satisfies (σ x)² = 1 at every covariate value, then on a nonempty finite covariate space, the squared L²(P_X) distance between b shifted by δ·σ and b itself equals δ². This is exactly what makes a Rademacher-bump perturbation land on the boundary of the nuisance class ℱ(ε,·) when δ = √ε.","s":"∀ {C : Type u_1} [inst : Fintype C] [Nonempty C] (b : C → ℝ) (δ : ℝ) (σ : C → ℝ), (∀ (x : C), σ x ^ 2 = 1) → Causalean.Estimation.MinimaxATE.l2sq (fun x => b x + δ * σ x) b = δ ^ 2","l":51,"f":"Causalean/Estimation/MinimaxATE/Reduction/Bump.lean"},{"n":"Causalean.Estimation.MinimaxATE.l2sq_comm","k":"theorem","m":"Causalean.Estimation.MinimaxATE.Reduction.Bump","a":"Estimation/MinimaxATE/Reduction","t":2,"d":"l2sq is symmetric.","s":"∀ {C : Type u_1} [inst : Fintype C] (a b : C → ℝ), Causalean.Estimation.MinimaxATE.l2sq a b = Causalean.Estimation.MinimaxATE.l2sq b a","l":36,"f":"Causalean/Estimation/MinimaxATE/Reduction/Bump.lean"},{"n":"Causalean.Estimation.MinimaxATE.l2sq_gPerturbed_false","k":"theorem","m":"Causalean.Estimation.MinimaxATE.ConstCenterHalf.Membership","a":"Estimation/MinimaxATE/ConstCenterHalf","t":2,"d":"The control outcome arm is unchanged (gλ(0,·) = 1/2 = ĝ(0,·)), so its L² error is 0.","s":"∀ {K : ℕ} {α β : ℝ} (lam : Fin K → Bool), Causalean.Estimation.MinimaxATE.l2sq (Causalean.Estimation.MinimaxATE.gPerturbed α β lam false) (Causalean.Estimation.MinimaxATE.ghat false) = 0","l":58,"f":"Causalean/Estimation/MinimaxATE/ConstCenterHalf/Membership.lean"},{"n":"Causalean.Estimation.MinimaxATE.l2sq_gPerturbed_true_le","k":"theorem","m":"Causalean.Estimation.MinimaxATE.ConstCenterHalf.Membership","a":"Estimation/MinimaxATE/ConstCenterHalf","t":2,"d":"Uniform bound on the treated arm's squared L² error: every pointwise term is ≤ (α+β)²/(1 − 2β)², so the average is too.","s":"∀ {K : ℕ} {α β : ℝ} [NeZero K], 0 ≤ α → 0 ≤ β → α + 2 * β ≤ 1 / 2 → ∀ (lam : Fin K → Bool), Causalean.Estimation.MinimaxATE.l2sq (Causalean.Estimation.MinimaxATE.gPerturbed α β lam true) (Causalean.Estimation.MinimaxATE.ghat true) ≤ (α + β) ^ 2 / (1 - 2 * β) ^ 2","l":79,"f":"Causalean/Estimation/MinimaxATE/ConstCenterHalf/Membership.lean"},{"n":"Causalean.Estimation.MinimaxATE.l2sq_mPerturbed","k":"theorem","m":"Causalean.Estimation.MinimaxATE.ConstCenterHalf.Membership","a":"Estimation/MinimaxATE/ConstCenterHalf","t":2,"d":"The propensity perturbation has exact squared L² size β²: it is a Rademacher bump of magnitude β on Δ.","s":"∀ {K : ℕ} {β : ℝ} [NeZero K] (lam : Fin K → Bool), Causalean.Estimation.MinimaxATE.l2sq (Causalean.Estimation.MinimaxATE.mPerturbed β lam) Causalean.Estimation.MinimaxATE.mhat = β ^ 2","l":44,"f":"Causalean/Estimation/MinimaxATE/ConstCenterHalf/Membership.lean"},{"n":"Causalean.Estimation.MinimaxATE.l2sq_nonneg","k":"theorem","m":"Causalean.Estimation.MinimaxATE.Reduction.Bump","a":"Estimation/MinimaxATE/Reduction","t":2,"d":"l2sq is nonnegative.","s":"∀ {C : Type u_1} [inst : Fintype C] (a b : C → ℝ), 0 ≤ Causalean.Estimation.MinimaxATE.l2sq a b","l":44,"f":"Causalean/Estimation/MinimaxATE/Reduction/Bump.lean"},{"n":"Causalean.Estimation.MinimaxATE.l2sq_self","k":"theorem","m":"Causalean.Estimation.MinimaxATE.Reduction.Bump","a":"Estimation/MinimaxATE/Reduction","t":2,"d":"l2sq of a function with itself is zero.","s":"∀ {C : Type u_1} [inst : Fintype C] (a : C → ℝ), Causalean.Estimation.MinimaxATE.l2sq a a = 0","l":32,"f":"Causalean/Estimation/MinimaxATE/Reduction/Bump.lean"},{"n":"Causalean.Estimation.MinimaxATE.mPerturbed","k":"def","m":"Causalean.Estimation.MinimaxATE.ConstCenterHalf.Construction","a":"Estimation/MinimaxATE/ConstCenterHalf","t":1,"d":"Perturbed propensity obtained by shifting the centered propensity along the Rademacher bump.","s":"{K : ℕ} → ℝ → (Fin K → Bool) → Fin K × Bool → ℝ","l":106,"f":"Causalean/Estimation/MinimaxATE/ConstCenterHalf/Construction.lean"},{"n":"Causalean.Estimation.MinimaxATE.mhat","k":"def","m":"Causalean.Estimation.MinimaxATE.ConstCenterHalf.Construction","a":"Estimation/MinimaxATE/ConstCenterHalf","t":1,"d":"Centered propensity estimate that assigns one half to every covariate value.","s":"{K : ℕ} → Fin K × Bool → ℝ","l":98,"f":"Causalean/Estimation/MinimaxATE/ConstCenterHalf/Construction.lean"},{"n":"Causalean.Estimation.MinimaxATE.minimaxMiss","k":"def","m":"Causalean.Estimation.MinimaxATE.Model","a":"Estimation/MinimaxATE","t":1,"d":"Minimax (worst-case-over-class) miss probability. The supremum over all in-class DGPs of the probability that est misses that DGP's true ATE by s. A lower bound on this quantity is a minimax lower bound: no estimator can be within s of the truth with high probability uniformly over the class.","s":"{C : Type u_1} → [Fintype C] → [MeasurableSpace C] → [Nonempty C] → (C → ℝ) → (Bool → C → ℝ) → ℝ → ℝ → (n : ℕ) → ((Fin n → Causalean.Estimation.MinimaxATE.Obs C) → ℝ) → ℝ → ℝ","l":210,"f":"Causalean/Estimation/MinimaxATE/Model.lean"},{"n":"Causalean.Estimation.MinimaxATE.minimaxMissCausal","k":"def","m":"Causalean.Estimation.MinimaxATE.Causal.Minimax","a":"Estimation/MinimaxATE/Causal","t":1,"d":"Causal-centered minimax miss probability. This mirrors minimaxMiss, but centers each DGP's miss event at the potential-outcome estimand causalATE m g of the concrete backdoor system.","s":"{C : Type} → [Fintype C] → [Nonempty C] → [inst : MeasurableSpace C] → [MeasurableSingletonClass C] → (C → ℝ) → (Bool → C → ℝ) → ℝ → ℝ → (n : ℕ) → ((Fin n → Causalean.Estimation.MinimaxATE.Obs C) → ℝ) → ℝ → ℝ","l":44,"f":"Causalean/Estimation/MinimaxATE/Causal/Minimax.lean"},{"n":"Causalean.Estimation.MinimaxATE.minimax_lower_bound","k":"theorem","m":"Causalean.Estimation.MinimaxATE.ConstCenterHalf.ChiSquaredCore","a":"Estimation/MinimaxATE/ConstCenterHalf","t":1,"d":"Structure-agnostic minimax lower bound (unconditional). Fix nonnegative bump magnitudes α and β with α + 2β ≤ 1/2 meeting the Rademacher perturbation budgets β² ≤ εm and (α+β)²/(1−2β)² ≤ εg for nonnegative outcome- and propensity-error tolerances εg, εm, in the sample-size regime 2n²γ² ≤ K·log 2 with γ = α²+2αβ+3β² and 2γ ≤ 1. Then for any measurable estimator of the average treatment effect from n i.i.d. paired-cell observations, the worst-case-over-class probability that it misses the true ATE by s = β(α+β)/(1−4β²), over the structure-agnostic nuisance class centered at the constant estimates (m̂, ĝ) = (1/2, 1/2), is at least 1/4 — the doubly-robust product rate is unbeatable.","s":"∀ {K n : ℕ} {α β εg εm : ℝ} [inst : NeZero K], 0 ≤ α → 0 ≤ β → α + 2 * β ≤ 1 / 2 → β ^ 2 ≤ εm → (α + β) ^ 2 / (1 - 2 * β) ^ 2 ≤ εg → 0 ≤ εg → 0 ≤ εm → 2 * (α ^ 2 + 2 * α * β + 3 * β ^ 2) ≤ 1 → 2 * ↑n ^ 2 * (α ^ 2 + 2 * α * β + 3 * β ^ 2) ^ 2 ≤ ↑K * Real.log 2 → ∀ {est : (Fin n → Causalean.Estimation.MinimaxATE.Obs (Fin K × Bool)) → ℝ}, Measurable est → 1 / 4 ≤ Causalean.Estimation.MinimaxATE.minimaxMiss Causalean.Estimation.MinimaxATE.mhat Causalean.Estimation.MinimaxATE.ghat εg εm n est (β * (α + β) / (1 - 4 * β ^ 2))","l":204,"f":"Causalean/Estimation/MinimaxATE/ConstCenterHalf/ChiSquaredCore.lean"},{"n":"Causalean.Estimation.MinimaxATE.minimax_lower_bound_mse","k":"theorem","m":"Causalean.Estimation.MinimaxATE.ConstCenterHalf.MSE","a":"Estimation/MinimaxATE/ConstCenterHalf","t":1,"d":"Structure-agnostic minimax lower bound, expected-risk (MSE) form. Fix nonnegative bump magnitudes α and β with α + 2β ≤ 1/2 meeting the Rademacher perturbation budgets β² ≤ εm and (α+β)²/(1−2β)² ≤ εg for nonnegative error tolerances εg, εm, in the sample-size regime 2n²γ² ≤ K·log 2 with γ = α²+2αβ+3β² and 2γ ≤ 1. Then for any measurable estimator of the average treatment effect, there is a data-generating process in the structure-agnostic nuisance class on which its mean-squared error is at least s²/4, with s = β(α+β)/(1−4β²) ≍ √(εg·εm) — the doubly-robust product rate is an unbeatable expected-squared-error floor. This is the weaker, (1−γ)-factored consequence (γ = 3/4) of the quantile bound, in the form used by Balakrishnan et al.","s":"∀ {K n : ℕ} {α β εg εm : ℝ} [inst : NeZero K], 0 ≤ α → 0 ≤ β → α + 2 * β ≤ 1 / 2 → β ^ 2 ≤ εm → (α + β) ^ 2 / (1 - 2 * β) ^ 2 ≤ εg → 0 ≤ εg → 0 ≤ εm → 2 * (α ^ 2 + 2 * α * β + 3 * β ^ 2) ≤ 1 → 2 * ↑n ^ 2 * (α ^ 2 + 2 * α * β + 3 * β ^ 2) ^ 2 ≤ ↑K * Real.log 2 → ∀ {est : (Fin n → Causalean.Estimation.MinimaxATE.Obs (Fin K × Bool)) → ℝ}, Measurable est → ∃ p, (β * (α + β) / (1 - 4 * β ^ 2)) ^ 2 / 4 ≤ Causalean.Estimation.MinimaxATE.nMSE p.property.valid n est","l":46,"f":"Causalean/Estimation/MinimaxATE/ConstCenterHalf/MSE.lean"},{"n":"Causalean.Estimation.MinimaxATE.mixture_two_point_lower_bound","k":"theorem","m":"Causalean.Estimation.MinimaxATE.Reduction.WitnessMixture","a":"Estimation/MinimaxATE/Reduction","t":1,"d":"Mixture two-point lower bound (one-sided separation). Suppose a null data-generating process (m₀, g₀) lies in the structure-agnostic nuisance class and a finite family of alternative data-generating processes (mlam i, glam i) also lie in the class, with nonnegative mixture weights w summing to 1, where every alternative's average treatment effect is at least 2s away from the null's, and the null's n-sample law and the weighted mixture of the alternatives' n-sample laws are at total-variation distance at most c. Then for any measurable estimator of the average treatment effect, the worst-case-over-class probability that it misses the true ATE by s is at least (1 − c)/2. Unlike TwoPointWitness, the alternatives need not share a common ATE — only 2 s ≤ |ate g₀ − ate (glam i)| per component.","s":"∀ {C : Type u_1} [inst : Fintype C] [inst_1 : Nonempty C] [inst_2 : MeasurableSpace C] {n : ℕ} {mhat : C → ℝ} {ghat : Bool → C → ℝ} {εg εm : ℝ} {ι : Type u_2} [inst_3 : Fintype ι] {m₀ : C → ℝ} {g₀ : Bool → C → ℝ} (hnull : Causalean.Estimation.MinimaxATE.InClass mhat ghat εg εm m₀ g₀) {mlam : ι → C → ℝ} {glam : ι → Bool → C → ℝ} (hin : ∀ (i : ι), Causalean.Estimation.MinimaxATE.InClass mhat ghat εg εm (mlam i) (glam i)) (w : ι → ENNReal), ∑ i, w i = 1 → ∀ {s c : ℝ}, (∀ (i : ι), 2 * s ≤ |Causalean.Estimation.MinimaxATE.ate g₀ - Causalean.Estimation.MinimaxATE.ate (glam i)|) → Causalean.Stat.tvDist (Causalean.Estimation.MinimaxATE.productLaw hnull.valid n) (Causalean.Stat.mixture w fun i => Causalean.Estimation.MinimaxATE.productLaw (hin i).valid n) ≤ c → ∀ {est : (Fin n → Causalean.Estimation.MinimaxATE.Obs C) → ℝ}, Measurable est → (1 - c) / 2 ≤ Causalean.Estimation.MinimaxATE.minimaxMiss mhat ghat εg εm n est s","l":58,"f":"Causalean/Estimation/MinimaxATE/Reduction/WitnessMixture.lean"},{"n":"Causalean.Estimation.MinimaxATE.nMSE","k":"def","m":"Causalean.Estimation.MinimaxATE.Model","a":"Estimation/MinimaxATE","t":1,"d":"The mean-squared error of est on n samples from the DGP (m, g): the expected squared deviation from the true ATE ate g. This is the expected-risk functional whose minimax lower bound the paper (Jin–Syrgkanis 2024, eq. for 𝔐ⁿ,γ) deduces — as the weaker (1−γ)-factor consequence — from the quantile (probability-of-miss) form.","s":"{C : Type u_1} → [Fintype C] → [MeasurableSpace C] → [Nonempty C] → {m : C → ℝ} → {g : Bool → C → ℝ} → Causalean.Estimation.MinimaxATE.ValidDGP m g → (n : ℕ) → ((Fin n → Causalean.Estimation.MinimaxATE.Obs C) → ℝ) → ℝ","l":179,"f":"Causalean/Estimation/MinimaxATE/Model.lean"},{"n":"Causalean.Estimation.MinimaxATE.nMiss","k":"def","m":"Causalean.Estimation.MinimaxATE.Model","a":"Estimation/MinimaxATE","t":1,"d":"The probability that the estimator est, run on n samples from the DGP (m, g), misses the true ATE ate g by at least s.","s":"{C : Type u_1} → [Fintype C] → [MeasurableSpace C] → [Nonempty C] → {m : C → ℝ} → {g : Bool → C → ℝ} → Causalean.Estimation.MinimaxATE.ValidDGP m g → (n : ℕ) → ((Fin n → Causalean.Estimation.MinimaxATE.Obs C) → ℝ) → ℝ → ℝ","l":173,"f":"Causalean/Estimation/MinimaxATE/Model.lean"},{"n":"Causalean.Estimation.MinimaxATE.nMissCausal_le_minimaxMissCausal","k":"theorem","m":"Causalean.Estimation.MinimaxATE.Causal.Minimax","a":"Estimation/MinimaxATE/Causal","t":2,"d":"A specific in-class DGP's causal-centered miss probability is dominated by the causal-centered minimax miss.","s":"∀ {C : Type} [inst : Fintype C] [inst_1 : Nonempty C] [inst_2 : MeasurableSpace C] [inst_3 : MeasurableSingletonClass C] {mhat : C → ℝ} {ghat : Bool → C → ℝ} {εg εm : ℝ} {n : ℕ} {est : (Fin n → Causalean.Estimation.MinimaxATE.Obs C) → ℝ} {s : ℝ} (p : Causalean.Estimation.MinimaxATE.InClassDGP mhat ghat εg εm), (Causalean.Estimation.MinimaxATE.productLaw p.property.valid n).real {x | s ≤ |est x - Causalean.Estimation.MinimaxATE.Causal.causalATE (↑p).1 (↑p).2|} ≤ Causalean.Estimation.MinimaxATE.minimaxMissCausal mhat ghat εg εm n est s","l":66,"f":"Causalean/Estimation/MinimaxATE/Causal/Minimax.lean"},{"n":"Causalean.Estimation.MinimaxATE.nMiss_le_minimaxMiss","k":"theorem","m":"Causalean.Estimation.MinimaxATE.Model","a":"Estimation/MinimaxATE","t":2,"d":"A specific in-class DGP's miss probability is ≤ minimaxMiss.","s":"∀ {C : Type u_1} [inst : Fintype C] [inst_1 : MeasurableSpace C] [inst_2 : Nonempty C] {mhat : C → ℝ} {ghat : Bool → C → ℝ} {εg εm : ℝ} {n : ℕ} {est : (Fin n → Causalean.Estimation.MinimaxATE.Obs C) → ℝ} {s : ℝ} (p : Causalean.Estimation.MinimaxATE.InClassDGP mhat ghat εg εm), Causalean.Estimation.MinimaxATE.nMiss p.property.valid n est s ≤ Causalean.Estimation.MinimaxATE.minimaxMiss mhat ghat εg εm n est s","l":231,"f":"Causalean/Estimation/MinimaxATE/Model.lean"},{"n":"Causalean.Estimation.MinimaxATE.nMiss_sq_le_nMSE","k":"theorem","m":"Causalean.Estimation.MinimaxATE.Model","a":"Estimation/MinimaxATE","t":1,"d":"Chebyshev/Markov bridge. For a valid data-generating process (m, g) and a nonnegative separation threshold s, the squared threshold times the probability that an estimator misses the true average treatment effect by at least s, on n i.i.d. draws, is at most the estimator's mean-squared error. This is the quantitative form of the paper's observation that the quantile risk lower bound implies the expected-risk one (𝔐ₙ,γ ≥ ρ ⟹ minimax 𝔼-risk ≥ (1−γ)ρ`).","s":"∀ {C : Type u_1} [inst : Fintype C] [inst_1 : MeasurableSpace C] [inst_2 : Nonempty C] [MeasurableSingletonClass C] {m : C → ℝ} {g : Bool → C → ℝ} (hv : Causalean.Estimation.MinimaxATE.ValidDGP m g) (n : ℕ) {est : (Fin n → Causalean.Estimation.MinimaxATE.Obs C) → ℝ} {s : ℝ}, 0 ≤ s → s ^ 2 * Causalean.Estimation.MinimaxATE.nMiss hv n est s ≤ Causalean.Estimation.MinimaxATE.nMSE hv n est","l":188,"f":"Causalean/Estimation/MinimaxATE/Model.lean"},{"n":"Causalean.Estimation.MinimaxATE.neg_one_le_Δ","k":"theorem","m":"Causalean.Estimation.MinimaxATE.ConstCenterHalf.Construction","a":"Estimation/MinimaxATE/ConstCenterHalf","t":2,"d":"The Rademacher bump is always at least minus one.","s":"∀ {K : ℕ} (lam : Fin K → Bool) (x : Fin K × Bool), -1 ≤ Causalean.Estimation.MinimaxATE.Δ lam x","l":92,"f":"Causalean/Estimation/MinimaxATE/ConstCenterHalf/Construction.lean"},{"n":"Causalean.Estimation.MinimaxATE.obsLaw","k":"def","m":"Causalean.Estimation.MinimaxATE.Model","a":"Estimation/MinimaxATE","t":1,"d":"The single-observation law of the DGP (m, g) as a probability Measure.","s":"{C : Type u_1} → [Fintype C] → [inst : MeasurableSpace C] → [Nonempty C] → {m : C → ℝ} → {g : Bool → C → ℝ} → Causalean.Estimation.MinimaxATE.ValidDGP m g → MeasureTheory.Measure (Causalean.Estimation.MinimaxATE.Obs C)","l":126,"f":"Causalean/Estimation/MinimaxATE/Model.lean"},{"n":"Causalean.Estimation.MinimaxATE.obsLaw_isProb","k":"instance","m":"Causalean.Estimation.MinimaxATE.Model","a":"Estimation/MinimaxATE","t":2,"d":"The single-observation law of any valid data-generating process is a probability measure.","s":"∀ {C : Type u_1} [inst : Fintype C] [inst_1 : MeasurableSpace C] [inst_2 : Nonempty C] {m : C → ℝ} {g : Bool → C → ℝ} (hv : Causalean.Estimation.MinimaxATE.ValidDGP m g), MeasureTheory.IsProbabilityMeasure (Causalean.Estimation.MinimaxATE.obsLaw hv)","l":131,"f":"Causalean/Estimation/MinimaxATE/Model.lean"},{"n":"Causalean.Estimation.MinimaxATE.obsLaw_real_singleton","k":"theorem","m":"Causalean.Estimation.MinimaxATE.ConstCenterHalf.ChiSqOverlap","a":"Estimation/MinimaxATE/ConstCenterHalf","t":2,"d":"The one-observation law assigns each observed point exactly its finite observed-data mass.","s":"∀ {C : Type u_1} [inst : Fintype C] [inst_1 : Nonempty C] [inst_2 : MeasurableSpace C] [MeasurableSingletonClass C] {m : C → ℝ} {g : Bool → C → ℝ} (hv : Causalean.Estimation.MinimaxATE.ValidDGP m g) (z : Causalean.Estimation.MinimaxATE.Obs C), (Causalean.Estimation.MinimaxATE.obsLaw hv).real {z} = Causalean.Estimation.MinimaxATE.obsReal m g z","l":40,"f":"Causalean/Estimation/MinimaxATE/ConstCenterHalf/ChiSqOverlap.lean"},{"n":"Causalean.Estimation.MinimaxATE.obsPMF","k":"def","m":"Causalean.Estimation.MinimaxATE.Model","a":"Estimation/MinimaxATE","t":1,"d":"The single-observation law of the DGP (m, g) as a probability PMF.","s":"{C : Type u_1} → [Fintype C] → [Nonempty C] → {m : C → ℝ} → {g : Bool → C → ℝ} → Causalean.Estimation.MinimaxATE.ValidDGP m g → PMF (Causalean.Estimation.MinimaxATE.Obs C)","l":117,"f":"Causalean/Estimation/MinimaxATE/Model.lean"},{"n":"Causalean.Estimation.MinimaxATE.obsReal","k":"def","m":"Causalean.Estimation.MinimaxATE.Model","a":"Estimation/MinimaxATE","t":1,"d":"The real-valued mass the DGP (m, g) assigns to the observation z = (x, d, y): (1/card C) · P(D = d | x) · P(Y = y | d, x).","s":"{C : Type u_1} → [Fintype C] → (C → ℝ) → (Bool → C → ℝ) → Causalean.Estimation.MinimaxATE.Obs C → ℝ","l":86,"f":"Causalean/Estimation/MinimaxATE/Model.lean"},{"n":"Causalean.Estimation.MinimaxATE.obsReal_nonneg","k":"theorem","m":"Causalean.Estimation.MinimaxATE.Model","a":"Estimation/MinimaxATE","t":2,"d":"Nonnegativity of obsReal for a valid DGP.","s":"∀ {C : Type u_1} [inst : Fintype C] {m : C → ℝ} {g : Bool → C → ℝ}, Causalean.Estimation.MinimaxATE.ValidDGP m g → ∀ (z : Causalean.Estimation.MinimaxATE.Obs C), 0 ≤ Causalean.Estimation.MinimaxATE.obsReal m g z","l":107,"f":"Causalean/Estimation/MinimaxATE/Model.lean"},{"n":"Causalean.Estimation.MinimaxATE.obsReal_perturbed_eq","k":"theorem","m":"Causalean.Estimation.MinimaxATE.ConstCenterHalf.ChiSqOverlap","a":"Estimation/MinimaxATE/ConstCenterHalf","t":2,"d":"The perturbed observed-data mass has an explicit denominator-free value at every treatment-outcome cell.","s":"∀ {K : ℕ} {α β : ℝ}, 0 ≤ α → 0 ≤ β → α + 2 * β ≤ 1 / 2 → ∀ (lam : Fin K → Bool) (x : Fin K × Bool) (d y : Bool), Causalean.Estimation.MinimaxATE.obsReal (Causalean.Estimation.MinimaxATE.mPerturbed β lam) (Causalean.Estimation.MinimaxATE.gPerturbed α β lam) (x, d, y) = (↑(Fintype.card (Fin K × Bool)))⁻¹ * match d, y with | true, true => (1 / 2 + α * Causalean.Estimation.MinimaxATE.Δ lam x) / 2 | true, false => (1 / 2 - (α + 2 * β) * Causalean.Estimation.MinimaxATE.Δ lam x) / 2 | false, true => (1 / 2 + β * Causalean.Estimation.MinimaxATE.Δ lam x) / 2 | false, false => (1 / 2 + β * Causalean.Estimation.MinimaxATE.Δ lam x) / 2","l":56,"f":"Causalean/Estimation/MinimaxATE/ConstCenterHalf/ChiSqOverlap.lean"},{"n":"Causalean.Estimation.MinimaxATE.obsReal_sum","k":"theorem","m":"Causalean.Estimation.MinimaxATE.Model","a":"Estimation/MinimaxATE","t":2,"d":"The total mass of obsReal is 1 (for nonempty C): summing over Y gives 1, then over D gives 1, then over X gives card C · (1/card C) = 1.","s":"∀ {C : Type u_1} [inst : Fintype C] [Nonempty C] (m : C → ℝ) (g : Bool → C → ℝ), ∑ z, Causalean.Estimation.MinimaxATE.obsReal m g z = 1","l":92,"f":"Causalean/Estimation/MinimaxATE/Model.lean"},{"n":"Causalean.Estimation.MinimaxATE.one_add_chiSqDiv_Qtrue_Qfalse","k":"theorem","m":"Causalean.Estimation.MinimaxATE.ConstCenterHalf.ChiSquaredCore","a":"Estimation/MinimaxATE/ConstCenterHalf","t":2,"d":"Mixture second-moment identity. On the finite product space, 1 + χ²(Qtrue‖Qfalse) is the uniform average over Rademacher sign pairs of the single-observation overlap raised to the sample size n.","s":"∀ {K n : ℕ} {α β : ℝ} [inst : NeZero K] (hα : 0 ≤ α) (hβ : 0 ≤ β) (hαβ : α + 2 * β ≤ 1 / 2), 1 + Causalean.Stat.chiSqDiv (Causalean.Estimation.MinimaxATE.Qtrue hα hβ hαβ n) (Causalean.Estimation.MinimaxATE.Qfalse K n) = ∑ lam, ∑ lam', (2 ^ K)⁻¹ * (2 ^ K)⁻¹ * Causalean.Estimation.MinimaxATE.chiSqOverlap α β lam lam' ^ n","l":116,"f":"Causalean/Estimation/MinimaxATE/ConstCenterHalf/ChiSquaredCore.lean"},{"n":"Causalean.Estimation.MinimaxATE.parametric_lower_bound","k":"theorem","m":"Causalean.Estimation.MinimaxATE.ConstCenterHalf.Parametric","a":"Estimation/MinimaxATE/ConstCenterHalf","t":1,"d":"Parametric Ω(1/n) minimax lower bound for structure-agnostic ATE estimation. Around a constant nuisance center (m₀, g₀, g₁) with all three coordinates strictly between 0 and 1, with a nonnegative treated-arm shift δ satisfying g₁ + δ ≤ 1 that lies inside the outcome-error budget δ² ≤ εg for a nonnegative propensity-error tolerance εm, and in the sample-size regime n·m₀δ²/(g₁(1−g₁)) ≤ log 2, for any measurable estimator of the average treatment effect the worst-case-over-class probability that it misses the true ATE by s = δ/2 is at least 1/4.","s":"∀ {C : Type u_1} [inst : Fintype C] [inst_1 : Nonempty C] [inst_2 : MeasurableSpace C] [MeasurableSingletonClass C] {n : ℕ} {m₀ g₀ g₁ δ εg εm : ℝ}, 0 < m₀ → m₀ < 1 → 0 < g₀ → g₀ < 1 → 0 < g₁ → g₁ < 1 → 0 ≤ δ → g₁ + δ ≤ 1 → δ ^ 2 ≤ εg → 0 ≤ εm → ↑n * (m₀ * δ ^ 2 / (g₁ * (1 - g₁))) ≤ Real.log 2 → ∀ {est : (Fin n → Causalean.Estimation.MinimaxATE.Obs C) → ℝ}, Measurable est → 1 / 4 ≤ Causalean.Estimation.MinimaxATE.minimaxMiss (fun x => m₀) (fun d x => if d = true then g₁ else g₀) εg εm n est (δ / 2)","l":304,"f":"Causalean/Estimation/MinimaxATE/ConstCenterHalf/Parametric.lean"},{"n":"Causalean.Estimation.MinimaxATE.productLaw","k":"def","m":"Causalean.Estimation.MinimaxATE.Model","a":"Estimation/MinimaxATE","t":1,"d":"The n-sample data law: the product Measure.pi of n independent copies of the single-observation law.","s":"{C : Type u_1} → [Fintype C] → [inst : MeasurableSpace C] → [Nonempty C] → {m : C → ℝ} → {g : Bool → C → ℝ} → Causalean.Estimation.MinimaxATE.ValidDGP m g → (n : ℕ) → MeasureTheory.Measure (Fin n → Causalean.Estimation.MinimaxATE.Obs C)","l":162,"f":"Causalean/Estimation/MinimaxATE/Model.lean"},{"n":"Causalean.Estimation.MinimaxATE.productLaw_isProb","k":"instance","m":"Causalean.Estimation.MinimaxATE.Model","a":"Estimation/MinimaxATE","t":2,"d":"The independent sample law of any valid data-generating process is a probability measure.","s":"∀ {C : Type u_1} [inst : Fintype C] [inst_1 : MeasurableSpace C] [inst_2 : Nonempty C] {m : C → ℝ} {g : Bool → C → ℝ} (hv : Causalean.Estimation.MinimaxATE.ValidDGP m g) (n : ℕ), MeasureTheory.IsProbabilityMeasure (Causalean.Estimation.MinimaxATE.productLaw hv n)","l":168,"f":"Causalean/Estimation/MinimaxATE/Model.lean"},{"n":"Causalean.Estimation.MinimaxATE.productLaw_real_singleton","k":"theorem","m":"Causalean.Estimation.MinimaxATE.ConstCenterHalf.ChiSquaredCore","a":"Estimation/MinimaxATE/ConstCenterHalf","t":2,"d":"The .real product point mass of an n-sample DGP law factorizes over draws.","s":"∀ {K n : ℕ} [inst : NeZero K] {m : Fin K × Bool → ℝ} {g : Bool → Fin K × Bool → ℝ} (hv : Causalean.Estimation.MinimaxATE.ValidDGP m g) (ω : Fin n → Causalean.Estimation.MinimaxATE.Obs (Fin K × Bool)), (Causalean.Estimation.MinimaxATE.productLaw hv n).real {ω} = ∏ i, Causalean.Estimation.MinimaxATE.obsReal m g (ω i)","l":67,"f":"Causalean/Estimation/MinimaxATE/ConstCenterHalf/ChiSquaredCore.lean"},{"n":"Causalean.Estimation.MinimaxATE.real_le_minimaxMiss","k":"theorem","m":"Causalean.Estimation.MinimaxATE.ConstCenterHalf.ExplicitWitness","a":"Estimation/MinimaxATE/ConstCenterHalf","t":2,"d":"The miss probability of any in-class data-generating process is bounded by the minimax miss probability.","s":"∀ {K n : ℕ} {εg εm : ℝ} [inst : NeZero K] {m : Fin K × Bool → ℝ} {g : Bool → Fin K × Bool → ℝ} (hin : Causalean.Estimation.MinimaxATE.InClass Causalean.Estimation.MinimaxATE.mhat Causalean.Estimation.MinimaxATE.ghat εg εm m g) (est : (Fin n → Causalean.Estimation.MinimaxATE.Obs (Fin K × Bool)) → ℝ) (s : ℝ), (Causalean.Estimation.MinimaxATE.productLaw hin.valid n).real {x | s ≤ |est x - Causalean.Estimation.MinimaxATE.ate g|} ≤ Causalean.Estimation.MinimaxATE.minimaxMiss Causalean.Estimation.MinimaxATE.mhat Causalean.Estimation.MinimaxATE.ghat εg εm n est s","l":109,"f":"Causalean/Estimation/MinimaxATE/ConstCenterHalf/ExplicitWitness.lean"},{"n":"Causalean.Estimation.MinimaxATE.signOf","k":"def","m":"Causalean.Estimation.MinimaxATE.ConstCenterHalf.Construction","a":"Estimation/MinimaxATE/ConstCenterHalf","t":1,"d":"This maps a Boolean position to its plus-or-minus-one sign.","s":"Bool → ℝ","l":51,"f":"Causalean/Estimation/MinimaxATE/ConstCenterHalf/Construction.lean"},{"n":"Causalean.Estimation.MinimaxATE.signOf_false","k":"theorem","m":"Causalean.Estimation.MinimaxATE.ConstCenterHalf.Construction","a":"Estimation/MinimaxATE/ConstCenterHalf","t":2,"d":"The sign of the false Boolean value is minus one.","s":"Causalean.Estimation.MinimaxATE.signOf false = -1","l":56,"f":"Causalean/Estimation/MinimaxATE/ConstCenterHalf/Construction.lean"},{"n":"Causalean.Estimation.MinimaxATE.signOf_mem","k":"theorem","m":"Causalean.Estimation.MinimaxATE.ConstCenterHalf.Construction","a":"Estimation/MinimaxATE/ConstCenterHalf","t":2,"d":"Every Boolean sign is either one or minus one.","s":"∀ (b : Bool), Causalean.Estimation.MinimaxATE.signOf b = 1 ∨ Causalean.Estimation.MinimaxATE.signOf b = -1","l":63,"f":"Causalean/Estimation/MinimaxATE/ConstCenterHalf/Construction.lean"},{"n":"Causalean.Estimation.MinimaxATE.signOf_sq","k":"theorem","m":"Causalean.Estimation.MinimaxATE.ConstCenterHalf.Construction","a":"Estimation/MinimaxATE/ConstCenterHalf","t":2,"d":"The square of every Boolean sign is one.","s":"∀ (b : Bool), Causalean.Estimation.MinimaxATE.signOf b ^ 2 = 1","l":59,"f":"Causalean/Estimation/MinimaxATE/ConstCenterHalf/Construction.lean"},{"n":"Causalean.Estimation.MinimaxATE.signOf_true","k":"theorem","m":"Causalean.Estimation.MinimaxATE.ConstCenterHalf.Construction","a":"Estimation/MinimaxATE/ConstCenterHalf","t":2,"d":"The sign of the true Boolean value is one.","s":"Causalean.Estimation.MinimaxATE.signOf true = 1","l":54,"f":"Causalean/Estimation/MinimaxATE/ConstCenterHalf/Construction.lean"},{"n":"Causalean.Estimation.MinimaxATE.signWeight","k":"def","m":"Causalean.Estimation.MinimaxATE.ConstCenterHalf.ExplicitWitness","a":"Estimation/MinimaxATE/ConstCenterHalf","t":1,"d":"Equal mixture weight on every Rademacher sign vector.","s":"(K : ℕ) → (Fin K → Bool) → ENNReal","l":69,"f":"Causalean/Estimation/MinimaxATE/ConstCenterHalf/ExplicitWitness.lean"},{"n":"Causalean.Estimation.MinimaxATE.signWeight_sum","k":"theorem","m":"Causalean.Estimation.MinimaxATE.ConstCenterHalf.ExplicitWitness","a":"Estimation/MinimaxATE/ConstCenterHalf","t":2,"d":"The uniform weights over sign vectors have total mass one.","s":"∀ (K : ℕ), ∑ lam, Causalean.Estimation.MinimaxATE.signWeight K lam = 1","l":83,"f":"Causalean/Estimation/MinimaxATE/ConstCenterHalf/ExplicitWitness.lean"},{"n":"Causalean.Estimation.MinimaxATE.signWeight_toReal","k":"theorem","m":"Causalean.Estimation.MinimaxATE.ConstCenterHalf.ChiSquaredCore","a":"Estimation/MinimaxATE/ConstCenterHalf","t":2,"d":"The real-valued uniform sign weight (2^K)⁻¹.","s":"∀ {K : ℕ} (lam : Fin K → Bool), (Causalean.Estimation.MinimaxATE.signWeight K lam).toReal = (2 ^ K)⁻¹","l":91,"f":"Causalean/Estimation/MinimaxATE/ConstCenterHalf/ChiSquaredCore.lean"},{"n":"Causalean.Estimation.MinimaxATE.tvDist_QfalseCont_QtrueCont_le_half","k":"theorem","m":"Causalean.Estimation.MinimaxATE.ConstCenterHalf.ContinuousX","a":"Estimation/MinimaxATE/ConstCenterHalf","t":2,"d":"The continuous-covariate null and alternative are indistinguishable up to total variation one half.","s":"∀ {K n : ℕ} {α β : ℝ} {W : Type u_1} [inst : MeasurableSpace W] [inst_1 : NeZero K] (ρ : MeasureTheory.Measure W) [MeasureTheory.IsProbabilityMeasure ρ] (hα : 0 ≤ α) (hβ : 0 ≤ β) (hαβ : α + 2 * β ≤ 1 / 2), 2 * (α ^ 2 + 2 * α * β + 3 * β ^ 2) ≤ 1 → 2 * ↑n ^ 2 * (α ^ 2 + 2 * α * β + 3 * β ^ 2) ^ 2 ≤ ↑K * Real.log 2 → Causalean.Stat.tvDist (Causalean.Estimation.MinimaxATE.QfalseCont K n ρ) (Causalean.Estimation.MinimaxATE.QtrueCont hα hβ hαβ n ρ) ≤ 1 / 2","l":167,"f":"Causalean/Estimation/MinimaxATE/ConstCenterHalf/ContinuousX.lean"},{"n":"Causalean.Estimation.MinimaxATE.tvDist_Qfalse_Qtrue_le_half","k":"theorem","m":"Causalean.Estimation.MinimaxATE.ConstCenterHalf.ChiSquaredCore","a":"Estimation/MinimaxATE/ConstCenterHalf","t":2,"d":"Total-variation indistinguishability. In the same regime, the null and alternative n-sample laws are statistically 1/2-close in total variation — discharging the hypothesis carried abstractly in ExplicitWitness.lean.","s":"∀ {K n : ℕ} {α β : ℝ} [inst : NeZero K] (hα : 0 ≤ α) (hβ : 0 ≤ β) (hαβ : α + 2 * β ≤ 1 / 2), 2 * (α ^ 2 + 2 * α * β + 3 * β ^ 2) ≤ 1 → 2 * ↑n ^ 2 * (α ^ 2 + 2 * α * β + 3 * β ^ 2) ^ 2 ≤ ↑K * Real.log 2 → Causalean.Stat.tvDist (Causalean.Estimation.MinimaxATE.Qfalse K n) (Causalean.Estimation.MinimaxATE.Qtrue hα hβ hαβ n) ≤ 1 / 2","l":182,"f":"Causalean/Estimation/MinimaxATE/ConstCenterHalf/ChiSquaredCore.lean"},{"n":"Causalean.Estimation.MinimaxATE.twoPointWitnessCausal_lower_bound","k":"theorem","m":"Causalean.Estimation.MinimaxATE.Causal.Minimax","a":"Estimation/MinimaxATE/Causal","t":2,"d":"Le Cam lower bound for a causal-centered two-point witness.","s":"∀ {C : Type} [inst : Fintype C] [inst_1 : Nonempty C] [inst_2 : MeasurableSpace C] [inst_3 : MeasurableSingletonClass C] {n : ℕ} {mhat : C → ℝ} {ghat : Bool → C → ℝ} {εg εm : ℝ} (W : Causalean.Estimation.MinimaxATE.TwoPointWitnessCausal C n mhat ghat εg εm) {est : (Fin n → Causalean.Estimation.MinimaxATE.Obs C) → ℝ}, Measurable est → (1 - W.c) / 2 ≤ Causalean.Estimation.MinimaxATE.minimaxMissCausal mhat ghat εg εm n est W.s","l":94,"f":"Causalean/Estimation/MinimaxATE/Causal/Minimax.lean"},{"n":"Causalean.Estimation.MinimaxATE.twoPointWitnessCausal_quarter","k":"theorem","m":"Causalean.Estimation.MinimaxATE.Causal.Minimax","a":"Estimation/MinimaxATE/Causal","t":2,"d":"A causal-centered witness with c ≤ 1/2 yields a 1/4 minimax miss lower bound.","s":"∀ {C : Type} [inst : Fintype C] [inst_1 : Nonempty C] [inst_2 : MeasurableSpace C] [inst_3 : MeasurableSingletonClass C] {n : ℕ} {mhat : C → ℝ} {ghat : Bool → C → ℝ} {εg εm : ℝ} (W : Causalean.Estimation.MinimaxATE.TwoPointWitnessCausal C n mhat ghat εg εm), W.c ≤ 1 / 2 → ∀ {est : (Fin n → Causalean.Estimation.MinimaxATE.Obs C) → ℝ}, Measurable est → 1 / 4 ≤ Causalean.Estimation.MinimaxATE.minimaxMissCausal mhat ghat εg εm n est W.s","l":110,"f":"Causalean/Estimation/MinimaxATE/Causal/Minimax.lean"},{"n":"Causalean.Estimation.MinimaxATE.twoPointWitness_lower_bound","k":"theorem","m":"Causalean.Estimation.MinimaxATE.Reduction.Witness","a":"Estimation/MinimaxATE/Reduction","t":1,"d":"Structure-agnostic two-point lower bound. Given a two-point (Le Cam) witness W packaging two statistically close n-sample laws with separated true average-treatment-effect values, for any measurable estimator, the worst-case-over-class probability that it misses the true ATE by W.s is at least (1 − W.c)/2. The proof is two_point_lower_bound_of_tvDist_le applied to the two witness laws, followed by the realizability domination.","s":"∀ {C : Type u_1} [inst : Fintype C] [inst_1 : Nonempty C] [inst_2 : MeasurableSpace C] {n : ℕ} {mhat : C → ℝ} {ghat : Bool → C → ℝ} {εg εm : ℝ} (W : Causalean.Estimation.MinimaxATE.TwoPointWitness C n mhat ghat εg εm) {est : (Fin n → Causalean.Estimation.MinimaxATE.Obs C) → ℝ}, Measurable est → (1 - W.c) / 2 ≤ Causalean.Estimation.MinimaxATE.minimaxMiss mhat ghat εg εm n est W.s","l":94,"f":"Causalean/Estimation/MinimaxATE/Reduction/Witness.lean"},{"n":"Causalean.Estimation.MinimaxATE.twoPointWitness_quarter","k":"theorem","m":"Causalean.Estimation.MinimaxATE.Reduction.Witness","a":"Estimation/MinimaxATE/Reduction","t":2,"d":"A witness with total-variation budget c ≤ 1/2 forces every estimator to miss the true ATE by W.s with probability at least 1/4 somewhere in the class.","s":"∀ {C : Type u_1} [inst : Fintype C] [inst_1 : Nonempty C] [inst_2 : MeasurableSpace C] {n : ℕ} {mhat : C → ℝ} {ghat : Bool → C → ℝ} {εg εm : ℝ} (W : Causalean.Estimation.MinimaxATE.TwoPointWitness C n mhat ghat εg εm), W.c ≤ 1 / 2 → ∀ {est : (Fin n → Causalean.Estimation.MinimaxATE.Obs C) → ℝ}, Measurable est → 1 / 4 ≤ Causalean.Estimation.MinimaxATE.minimaxMiss mhat ghat εg εm n est W.s","l":114,"f":"Causalean/Estimation/MinimaxATE/Reduction/Witness.lean"},{"n":"Causalean.Estimation.MinimaxATE.two_point_lower_bound_continuous","k":"theorem","m":"Causalean.Estimation.MinimaxATE.ConstCenterHalf.ContinuousX","a":"Estimation/MinimaxATE/ConstCenterHalf","t":1,"d":"Fix nonnegative bump magnitudes α and β with α + 2β ≤ 1/2, in the sample-size regime 2n²γ² ≤ K·log 2 with γ = α²+2αβ+3β² and 2γ ≤ 1. For any measurable estimator est of the average treatment effect that observes both the finite paired-cell data and n independent ancillary continuous within-cell positions drawn from an arbitrary probability space, the larger of the two miss probabilities — under the mixed alternative law and under the null law — that est misses the corresponding true ATE (2β(α+β)/(1−4β²) or 0) by β(α+β)/(1−4β²) is at least 1/4.","s":"∀ {K n : ℕ} {α β : ℝ} {W : Type u_1} [inst : MeasurableSpace W] [inst_1 : NeZero K] (ρ : MeasureTheory.Measure W) [MeasureTheory.IsProbabilityMeasure ρ] (hα : 0 ≤ α) (hβ : 0 ≤ β) (hαβ : α + 2 * β ≤ 1 / 2), 2 * (α ^ 2 + 2 * α * β + 3 * β ^ 2) ≤ 1 → 2 * ↑n ^ 2 * (α ^ 2 + 2 * α * β + 3 * β ^ 2) ^ 2 ≤ ↑K * Real.log 2 → ∀ {est : (Fin n → Causalean.Estimation.MinimaxATE.Obs (Fin K × Bool)) × (Fin n → W) → ℝ}, Measurable est → 1 / 4 ≤ max ((Causalean.Estimation.MinimaxATE.QtrueCont hα hβ hαβ n ρ).real {x | β * (α + β) / (1 - 4 * β ^ 2) ≤ |est x - 2 * β * (α + β) / (1 - 4 * β ^ 2)|}) ((Causalean.Estimation.MinimaxATE.QfalseCont K n ρ).real {x | β * (α + β) / (1 - 4 * β ^ 2) ≤ |est x - 0|})","l":194,"f":"Causalean/Estimation/MinimaxATE/ConstCenterHalf/ContinuousX.lean"},{"n":"Causalean.Estimation.MinimaxATE.validDGP_hat","k":"theorem","m":"Causalean.Estimation.MinimaxATE.ConstCenterHalf.Construction","a":"Estimation/MinimaxATE/ConstCenterHalf","t":1,"d":"The null construction's centered propensity and outcome-regression functions, both fixed at one half, define a valid finite observed-data model, i.e. take values in [0,1].","s":"∀ {K : ℕ}, Causalean.Estimation.MinimaxATE.ValidDGP Causalean.Estimation.MinimaxATE.mhat Causalean.Estimation.MinimaxATE.ghat","l":132,"f":"Causalean/Estimation/MinimaxATE/ConstCenterHalf/Construction.lean"},{"n":"Causalean.Estimation.MinimaxATE.validDGP_perturbed","k":"theorem","m":"Causalean.Estimation.MinimaxATE.ConstCenterHalf.Construction","a":"Estimation/MinimaxATE/ConstCenterHalf","t":1,"d":"Given nonnegative bump magnitudes α and β with α + 2β ≤ 1/2, the perturbed propensity and outcome-regression functions indexed by a Rademacher sign vector lam define a valid finite observed-data model, i.e. take values in [0,1].","s":"∀ {K : ℕ} {α β : ℝ}, 0 ≤ α → 0 ≤ β → α + 2 * β ≤ 1 / 2 → ∀ (lam : Fin K → Bool), Causalean.Estimation.MinimaxATE.ValidDGP (Causalean.Estimation.MinimaxATE.mPerturbed β lam) (Causalean.Estimation.MinimaxATE.gPerturbed α β lam)","l":138,"f":"Causalean/Estimation/MinimaxATE/ConstCenterHalf/Construction.lean"},{"n":"Causalean.Estimation.MinimaxATE.yReal","k":"def","m":"Causalean.Estimation.MinimaxATE.Achievability.AIPWEstimator","a":"Estimation/MinimaxATE/Achievability","t":1,"d":"The outcome Y : Bool read as a real number {0,1}.","s":"Bool → ℝ","l":58,"f":"Causalean/Estimation/MinimaxATE/Achievability/AIPWEstimator.lean"},{"n":"Causalean.Estimation.MinimaxATE.Δ","k":"def","m":"Causalean.Estimation.MinimaxATE.ConstCenterHalf.Construction","a":"Estimation/MinimaxATE/ConstCenterHalf","t":1,"d":"Paired-cell Rademacher bump used to perturb the nuisance functions.","s":"{K : ℕ} → (Fin K → Bool) → Fin K × Bool → ℝ","l":69,"f":"Causalean/Estimation/MinimaxATE/ConstCenterHalf/Construction.lean"},{"n":"Causalean.Estimation.MinimaxATE.Δ_le_one","k":"theorem","m":"Causalean.Estimation.MinimaxATE.ConstCenterHalf.Construction","a":"Estimation/MinimaxATE/ConstCenterHalf","t":2,"d":"The Rademacher bump is always at most one.","s":"∀ {K : ℕ} (lam : Fin K → Bool) (x : Fin K × Bool), Causalean.Estimation.MinimaxATE.Δ lam x ≤ 1","l":86,"f":"Causalean/Estimation/MinimaxATE/ConstCenterHalf/Construction.lean"},{"n":"Causalean.Estimation.MinimaxATE.Δ_mem","k":"theorem","m":"Causalean.Estimation.MinimaxATE.ConstCenterHalf.Construction","a":"Estimation/MinimaxATE/ConstCenterHalf","t":2,"d":"The Rademacher bump only takes the values one and minus one.","s":"∀ {K : ℕ} (lam : Fin K → Bool) (x : Fin K × Bool), Causalean.Estimation.MinimaxATE.Δ lam x = 1 ∨ Causalean.Estimation.MinimaxATE.Δ lam x = -1","l":80,"f":"Causalean/Estimation/MinimaxATE/ConstCenterHalf/Construction.lean"},{"n":"Causalean.Estimation.MinimaxATE.Δ_sq","k":"theorem","m":"Causalean.Estimation.MinimaxATE.ConstCenterHalf.Construction","a":"Estimation/MinimaxATE/ConstCenterHalf","t":2,"d":"The Rademacher bump has square one at every paired cell.","s":"∀ {K : ℕ} (lam : Fin K → Bool) (x : Fin K × Bool), Causalean.Estimation.MinimaxATE.Δ lam x ^ 2 = 1","l":75,"f":"Causalean/Estimation/MinimaxATE/ConstCenterHalf/Construction.lean"},{"n":"Causalean.Estimation.NPIV.Complexification.complexLift","k":"def","m":"Causalean.Estimation.NPIV.Operator.Complexification","a":"Estimation/NPIV/Operator","t":1,"d":"Complex lift of a real CLM.","s":"{Ω : Type u_1} → [inst : MeasurableSpace Ω] → {μ : MeasureTheory.Measure Ω} → (↥(MeasureTheory.Lp ℝ 2 μ) →L[ℝ] ↥(MeasureTheory.Lp ℝ 2 μ)) → ↥(MeasureTheory.Lp ℂ 2 μ) →L[ℂ] ↥(MeasureTheory.Lp ℂ 2 μ)","l":353,"f":"Causalean/Estimation/NPIV/Operator/Complexification.lean"},{"n":"Causalean.Estimation.NPIV.Complexification.complexLiftFun","k":"def","m":"Causalean.Estimation.NPIV.Operator.Complexification","a":"Estimation/NPIV/Operator","t":1,"d":"The function-level definition of the complex lift. Given a real CLM A : Lp ℝ 2 μ →L[ℝ] Lp ℝ 2 μ and a complex L² class f, the lift sends f = (Re f) + i (Im f) to A(Re f) + i · A(Im f) (with the real outputs re-embedded via ιLp).","s":"{Ω : Type u_1} → [inst : MeasurableSpace Ω] → {μ : MeasureTheory.Measure Ω} → (↥(MeasureTheory.Lp ℝ 2 μ) →L[ℝ] ↥(MeasureTheory.Lp ℝ 2 μ)) → ↥(MeasureTheory.Lp ℂ 2 μ) → ↥(MeasureTheory.Lp ℂ 2 μ)","l":229,"f":"Causalean/Estimation/NPIV/Operator/Complexification.lean"},{"n":"Causalean.Estimation.NPIV.Complexification.complexLift_adjoint","k":"theorem","m":"Causalean.Estimation.NPIV.Operator.Complexification","a":"Estimation/NPIV/Operator","t":2,"d":"The complex lift commutes with taking adjoints.","s":"∀ {Ω : Type u_1} [inst : MeasurableSpace Ω] {μ : MeasureTheory.Measure Ω} (A : ↥(MeasureTheory.Lp ℝ 2 μ) →L[ℝ] ↥(MeasureTheory.Lp ℝ 2 μ)), ContinuousLinearMap.adjoint (Causalean.Estimation.NPIV.Complexification.complexLift A) = Causalean.Estimation.NPIV.Complexification.complexLift (ContinuousLinearMap.adjoint A)","l":480,"f":"Causalean/Estimation/NPIV/Operator/Complexification.lean"},{"n":"Causalean.Estimation.NPIV.Complexification.complexLift_apply","k":"theorem","m":"Causalean.Estimation.NPIV.Operator.Complexification","a":"Estimation/NPIV/Operator","t":2,"d":"The action equation for complexLift: it agrees with complexLiftFun A on every input.","s":"∀ {Ω : Type u_1} [inst : MeasurableSpace Ω] {μ : MeasureTheory.Measure Ω} (A : ↥(MeasureTheory.Lp ℝ 2 μ) →L[ℝ] ↥(MeasureTheory.Lp ℝ 2 μ)) (f : ↥(MeasureTheory.Lp ℂ 2 μ)), (Causalean.Estimation.NPIV.Complexification.complexLift A) f = Causalean.Estimation.NPIV.Complexification.ιLp (A (Causalean.Estimation.NPIV.Complexification.reLp f)) + Complex.I • Causalean.Estimation.NPIV.Complexification.ιLp (A (Causalean.Estimation.NPIV.Complexification.imLp f))","l":384,"f":"Causalean/Estimation/NPIV/Operator/Complexification.lean"},{"n":"Causalean.Estimation.NPIV.Complexification.complexLift_isSelfAdjoint","k":"theorem","m":"Causalean.Estimation.NPIV.Operator.Complexification","a":"Estimation/NPIV/Operator","t":2,"d":"Self-adjointness is preserved by the complex lift. Direct corollary of complexLift_adjoint (rewrite A.adjoint = A inside the RHS).","s":"∀ {Ω : Type u_1} [inst : MeasurableSpace Ω] {μ : MeasureTheory.Measure Ω} {A : ↥(MeasureTheory.Lp ℝ 2 μ) →L[ℝ] ↥(MeasureTheory.Lp ℝ 2 μ)}, IsSelfAdjoint A → IsSelfAdjoint (Causalean.Estimation.NPIV.Complexification.complexLift A)","l":497,"f":"Causalean/Estimation/NPIV/Operator/Complexification.lean"},{"n":"Causalean.Estimation.NPIV.Complexification.complexLift_real","k":"theorem","m":"Causalean.Estimation.NPIV.Operator.Complexification","a":"Estimation/NPIV/Operator","t":2,"d":"Restriction of the complex lift to the real subspace recovers A. reLp_comp_ιLp gives reLp (ιLp g) = g and imLp_comp_ιLp gives imLp (ιLp g) = 0, so complexLift_apply collapses to ιLp (A g) + I • ιLp (A 0) = ιLp (A g).","s":"∀ {Ω : Type u_1} [inst : MeasurableSpace Ω] {μ : MeasureTheory.Measure Ω} (A : ↥(MeasureTheory.Lp ℝ 2 μ) →L[ℝ] ↥(MeasureTheory.Lp ℝ 2 μ)) (g : ↥(MeasureTheory.Lp ℝ 2 μ)), (Causalean.Estimation.NPIV.Complexification.complexLift A) (Causalean.Estimation.NPIV.Complexification.ιLp g) = Causalean.Estimation.NPIV.Complexification.ιLp (A g)","l":395,"f":"Causalean/Estimation/NPIV/Operator/Complexification.lean"},{"n":"Causalean.Estimation.NPIV.Complexification.imLp","k":"def","m":"Causalean.Estimation.NPIV.Operator.Complexification","a":"Estimation/NPIV/Operator","t":1,"d":"Pointwise imaginary part on L²(Ω, ℂ), bundled as a continuous ℝ-linear map Lp ℂ 2 μ →L[ℝ] Lp ℝ 2 μ. Same construction as reLp but with RCLike.imCLM.","s":"{Ω : Type u_1} → [inst : MeasurableSpace Ω] → {μ : MeasureTheory.Measure Ω} → ↥(MeasureTheory.Lp ℂ 2 μ) →L[ℝ] ↥(MeasureTheory.Lp ℝ 2 μ)","l":113,"f":"Causalean/Estimation/NPIV/Operator/Complexification.lean"},{"n":"Causalean.Estimation.NPIV.Complexification.imLp_comp_ιLp","k":"theorem","m":"Causalean.Estimation.NPIV.Operator.Complexification","a":"Estimation/NPIV/Operator","t":2,"d":"The imaginary part vanishes on the image of ιLp (pointwise RCLike.im_ofReal : RCLike.im (r : ℂ) = 0).","s":"∀ {Ω : Type u_1} [inst : MeasurableSpace Ω] {μ : MeasureTheory.Measure Ω} (f : ↥(MeasureTheory.Lp ℝ 2 μ)), Causalean.Estimation.NPIV.Complexification.imLp (Causalean.Estimation.NPIV.Complexification.ιLp f) = 0","l":144,"f":"Causalean/Estimation/NPIV/Operator/Complexification.lean"},{"n":"Causalean.Estimation.NPIV.Complexification.instAlgebraRealLpCLM","k":"def","m":"Causalean.Estimation.NPIV.Operator.Complexification","a":"Estimation/NPIV/Operator","t":1,"d":"Local disambiguation of the real-scalar algebra structure on the complex operator algebra Lp ℂ 2 μ →L[ℂ] Lp ℂ 2 μ.","s":"{Ω : Type u_1} → [inst : MeasurableSpace Ω] → {μ : MeasureTheory.Measure Ω} → Algebra ℝ (↥(MeasureTheory.Lp ℂ 2 μ) →L[ℂ] ↥(MeasureTheory.Lp ℂ 2 μ))","l":83,"f":"Causalean/Estimation/NPIV/Operator/Complexification.lean"},{"n":"Causalean.Estimation.NPIV.Complexification.reLp","k":"def","m":"Causalean.Estimation.NPIV.Operator.Complexification","a":"Estimation/NPIV/Operator","t":1,"d":"Pointwise real part on L²(Ω, ℂ), bundled as a continuous ℝ-linear map Lp ℂ 2 μ →L[ℝ] Lp ℝ 2 μ.","s":"{Ω : Type u_1} → [inst : MeasurableSpace Ω] → {μ : MeasureTheory.Measure Ω} → ↥(MeasureTheory.Lp ℂ 2 μ) →L[ℝ] ↥(MeasureTheory.Lp ℝ 2 μ)","l":103,"f":"Causalean/Estimation/NPIV/Operator/Complexification.lean"},{"n":"Causalean.Estimation.NPIV.Complexification.reLp_add_smul_imLp","k":"theorem","m":"Causalean.Estimation.NPIV.Operator.Complexification","a":"Estimation/NPIV/Operator","t":2,"d":"Reconstruction identity: every complex L²-class is the complex combination of its real and imaginary parts re-embedded via ιLp. Pointwise this is RCLike.re_add_im : (r.re : ℂ) + I * r.im = r.","s":"∀ {Ω : Type u_1} [inst : MeasurableSpace Ω] {μ : MeasureTheory.Measure Ω} (f : ↥(MeasureTheory.Lp ℂ 2 μ)), Causalean.Estimation.NPIV.Complexification.ιLp (Causalean.Estimation.NPIV.Complexification.reLp f) + Complex.I • Causalean.Estimation.NPIV.Complexification.ιLp (Causalean.Estimation.NPIV.Complexification.imLp f) = f","l":189,"f":"Causalean/Estimation/NPIV/Operator/Complexification.lean"},{"n":"Causalean.Estimation.NPIV.Complexification.reLp_comp_ιLp","k":"theorem","m":"Causalean.Estimation.NPIV.Operator.Complexification","a":"Estimation/NPIV/Operator","t":2,"d":"reLp is a left-inverse of ιLp.","s":"∀ {Ω : Type u_1} [inst : MeasurableSpace Ω] {μ : MeasureTheory.Measure Ω} (f : ↥(MeasureTheory.Lp ℝ 2 μ)), Causalean.Estimation.NPIV.Complexification.reLp (Causalean.Estimation.NPIV.Complexification.ιLp f) = f","l":127,"f":"Causalean/Estimation/NPIV/Operator/Complexification.lean"},{"n":"Causalean.Estimation.NPIV.Complexification.reLp_complexLift_real","k":"theorem","m":"Causalean.Estimation.NPIV.Operator.Complexification","a":"Estimation/NPIV/Operator","t":2,"d":"Corollary of complexLift_real: projecting back via reLp gives exactly A g. Direct from reLp_comp_ιLp.","s":"∀ {Ω : Type u_1} [inst : MeasurableSpace Ω] {μ : MeasureTheory.Measure Ω} (A : ↥(MeasureTheory.Lp ℝ 2 μ) →L[ℝ] ↥(MeasureTheory.Lp ℝ 2 μ)) (g : ↥(MeasureTheory.Lp ℝ 2 μ)), Causalean.Estimation.NPIV.Complexification.reLp ((Causalean.Estimation.NPIV.Complexification.complexLift A) (Causalean.Estimation.NPIV.Complexification.ιLp g)) = A g","l":405,"f":"Causalean/Estimation/NPIV/Operator/Complexification.lean"},{"n":"Causalean.Estimation.NPIV.Complexification.realCFC","k":"def","m":"Causalean.Estimation.NPIV.Operator.Complexification","a":"Estimation/NPIV/Operator","t":1,"d":"Real CFC by complexification.","s":"{Ω : Type u_1} → [inst : MeasurableSpace Ω] → {μ : MeasureTheory.Measure Ω} → (↥(MeasureTheory.Lp ℝ 2 μ) →L[ℝ] ↥(MeasureTheory.Lp ℝ 2 μ)) → (ℝ → ℝ) → ↥(MeasureTheory.Lp ℝ 2 μ) →L[ℝ] ↥(MeasureTheory.Lp ℝ 2 μ)","l":541,"f":"Causalean/Estimation/NPIV/Operator/Complexification.lean"},{"n":"Causalean.Estimation.NPIV.Complexification.realCFC_apply","k":"theorem","m":"Causalean.Estimation.NPIV.Operator.Complexification","a":"Estimation/NPIV/Operator","t":2,"d":"Action equation for realCFC — the basic rewrite rule used by spectral proofs.","s":"∀ {Ω : Type u_1} [inst : MeasurableSpace Ω] {μ : MeasureTheory.Measure Ω} (A : ↥(MeasureTheory.Lp ℝ 2 μ) →L[ℝ] ↥(MeasureTheory.Lp ℝ 2 μ)), IsSelfAdjoint A → ∀ (f : ℝ → ℝ), Continuous f → ∀ (g : ↥(MeasureTheory.Lp ℝ 2 μ)), (Causalean.Estimation.NPIV.Complexification.realCFC A f) g = Causalean.Estimation.NPIV.Complexification.reLp ((cfc (fun z => ↑(f z.re)) (Causalean.Estimation.NPIV.Complexification.complexLift A)) (Causalean.Estimation.NPIV.Complexification.ιLp g))","l":590,"f":"Causalean/Estimation/NPIV/Operator/Complexification.lean"},{"n":"Causalean.Estimation.NPIV.Complexification.realCFC_id","k":"theorem","m":"Causalean.Estimation.NPIV.Operator.Complexification","a":"Estimation/NPIV/Operator","t":2,"d":"realCFC of the identity symbol recovers A.","s":"∀ {Ω : Type u_1} [inst : MeasurableSpace Ω] {μ : MeasureTheory.Measure Ω} (A : ↥(MeasureTheory.Lp ℝ 2 μ) →L[ℝ] ↥(MeasureTheory.Lp ℝ 2 μ)), IsSelfAdjoint A → ∀ (g : ↥(MeasureTheory.Lp ℝ 2 μ)), (Causalean.Estimation.NPIV.Complexification.realCFC A id) g = A g","l":605,"f":"Causalean/Estimation/NPIV/Operator/Complexification.lean"},{"n":"Causalean.Estimation.NPIV.Complexification.realCFC_isSelfAdjoint","k":"theorem","m":"Causalean.Estimation.NPIV.Operator.Complexification","a":"Estimation/NPIV/Operator","t":2,"d":"realCFC preserves self-adjointness.","s":"∀ {Ω : Type u_1} [inst : MeasurableSpace Ω] {μ : MeasureTheory.Measure Ω} (A : ↥(MeasureTheory.Lp ℝ 2 μ) →L[ℝ] ↥(MeasureTheory.Lp ℝ 2 μ)), IsSelfAdjoint A → ∀ (f : ℝ → ℝ), Continuous f → IsSelfAdjoint (Causalean.Estimation.NPIV.Complexification.realCFC A f)","l":637,"f":"Causalean/Estimation/NPIV/Operator/Complexification.lean"},{"n":"Causalean.Estimation.NPIV.Complexification.realCFC_mul","k":"theorem","m":"Causalean.Estimation.NPIV.Operator.Complexification","a":"Estimation/NPIV/Operator","t":2,"d":"Symbol-multiplication law for realCFC.","s":"∀ {Ω : Type u_1} [inst : MeasurableSpace Ω] {μ : MeasureTheory.Measure Ω} (A : ↥(MeasureTheory.Lp ℝ 2 μ) →L[ℝ] ↥(MeasureTheory.Lp ℝ 2 μ)), IsSelfAdjoint A → ∀ (f g : ℝ → ℝ), Continuous f → Continuous g → (Causalean.Estimation.NPIV.Complexification.realCFC A fun x => f x * g x) = Causalean.Estimation.NPIV.Complexification.realCFC A f ∘SL Causalean.Estimation.NPIV.Complexification.realCFC A g","l":829,"f":"Causalean/Estimation/NPIV/Operator/Complexification.lean"},{"n":"Causalean.Estimation.NPIV.Complexification.realCFC_norm_le","k":"theorem","m":"Causalean.Estimation.NPIV.Operator.Complexification","a":"Estimation/NPIV/Operator","t":1,"d":"Norm-via-spectrum bound for realCFC. For a self-adjoint bounded operator A and a continuous real-valued symbol f, if f is bounded in absolute value by a nonnegative constant c on the real spectrum of A, then the operator realCFC A f is a contraction up to c: for every vector g, ‖realCFC A f g‖ ≤ c · ‖g‖.","s":"∀ {Ω : Type u_1} [inst : MeasurableSpace Ω] {μ : MeasureTheory.Measure Ω} (A : ↥(MeasureTheory.Lp ℝ 2 μ) →L[ℝ] ↥(MeasureTheory.Lp ℝ 2 μ)), IsSelfAdjoint A → ∀ (f : ℝ → ℝ), Continuous f → ∀ (c : ℝ), 0 ≤ c → (∀ x ∈ spectrum ℝ A, |f x| ≤ c) → ∀ (g : ↥(MeasureTheory.Lp ℝ 2 μ)), ‖(Causalean.Estimation.NPIV.Complexification.realCFC A f) g‖ ≤ c * ‖g‖","l":1029,"f":"Causalean/Estimation/NPIV/Operator/Complexification.lean"},{"n":"Causalean.Estimation.NPIV.Complexification.realCFC_resolvent_mul_self","k":"theorem","m":"Causalean.Estimation.NPIV.Operator.Complexification","a":"Estimation/NPIV/Operator","t":1,"d":"Resolvent symbol for realCFC. For a self-adjoint bounded operator A whose real spectrum lies in the nonnegative reals, and for a strictly positive regularization parameter λ, composing the real-functional-calculus operator for the affine symbol x ↦ λ + x with that for the resolvent symbol x ↦ (λ + x)⁻¹ yields the identity operator on the ambient L² space.","s":"∀ {Ω : Type u_1} [inst : MeasurableSpace Ω] {μ : MeasureTheory.Measure Ω} (A : ↥(MeasureTheory.Lp ℝ 2 μ) →L[ℝ] ↥(MeasureTheory.Lp ℝ 2 μ)), IsSelfAdjoint A → (∀ x ∈ spectrum ℝ A, 0 ≤ x) → ∀ (lambda : ℝ), 0 < lambda → ((Causalean.Estimation.NPIV.Complexification.realCFC A fun x => lambda + x) ∘SL Causalean.Estimation.NPIV.Complexification.realCFC A fun x => (lambda + x)⁻¹) = ContinuousLinearMap.id ℝ ↥(MeasureTheory.Lp ℝ 2 μ)","l":878,"f":"Causalean/Estimation/NPIV/Operator/Complexification.lean"},{"n":"Causalean.Estimation.NPIV.Complexification.ιLp","k":"def","m":"Causalean.Estimation.NPIV.Operator.Complexification","a":"Estimation/NPIV/Operator","t":1,"d":"Pointwise embedding of real-valued L² into complex-valued L², bundled as a continuous ℝ-linear map Lp ℝ 2 μ →L[ℝ] Lp ℂ 2 μ.","s":"{Ω : Type u_1} → [inst : MeasurableSpace Ω] → {μ : MeasureTheory.Measure Ω} → ↥(MeasureTheory.Lp ℝ 2 μ) →L[ℝ] ↥(MeasureTheory.Lp ℂ 2 μ)","l":119,"f":"Causalean/Estimation/NPIV/Operator/Complexification.lean"},{"n":"Causalean.Estimation.NPIV.Complexification.ιLp_isometry","k":"theorem","m":"Causalean.Estimation.NPIV.Operator.Complexification","a":"Estimation/NPIV/Operator","t":2,"d":"ιLp is an isometric embedding. Follows from RCLike.ofRealLI being a LinearIsometry and the fact that compLpL of a norm-one CLM preserves norms on L²; concretely, ‖ιLp f‖ = ‖f‖ reduces pointwise to ‖(r : ℂ)‖ = ‖r‖ (RCLike.norm_ofReal).","s":"∀ {Ω : Type u_1} [inst : MeasurableSpace Ω] {μ : MeasureTheory.Measure Ω} (f : ↥(MeasureTheory.Lp ℝ 2 μ)), ‖Causalean.Estimation.NPIV.Complexification.ιLp f‖ = ‖f‖","l":167,"f":"Causalean/Estimation/NPIV/Operator/Complexification.lean"},{"n":"Causalean.Estimation.NPIV.DR.TRAEDRRemainderHyps","k":"structure","m":"Causalean.Estimation.NPIV.DR.AsymptoticLinear","a":"Estimation/NPIV/DR","t":1,"d":"The TRAE-DR remainder hypotheses bundle the sufficient conditions under which the doubly-robust TRAE estimator, built from primal nuisance estimators ĥ_n and dual nuisance estimators q̂_n over a cross-fitting split, is asymptotically linear: the fitted primal nuisance is L²-consistent for the truth in probability, likewise for the fitted dual nuisance, both fitted nuisances stay in their respective candidate classes at every sample size and outcome, the √-scaled mixed-bias integral between the two nuisance errors is asymptotically negligible, the centered empirical-process remainder from plugging the fitted nuisances into the oracle score vanishes at the √-rate, and the oracle score has finite variance under the observation law.","s":"{Ω : Type u_1} → [inst : MeasurableSpace Ω] → {μ : MeasureTheory.Measure Ω} → [MeasureTheory.IsProbabilityMeasure μ] → (S : Causalean.Estimation.NPIV.InverseProblemSystem Ω μ) → {q₀ : S.𝒵 → ℝ} → S.DualSolution q₀ → {P_W : MeasureTheory.Measure S.𝒲} → [MeasureTheory.IsProbabilityMeasure P_W] → (sample : Causalean.Stat.IIDSample Ω S.𝒲 μ P_W) → Causalean.Stat.OneShotSplit sample → (ℕ → Ω → S.𝒳 → ℝ) → (ℕ → Ω → S.𝒵 → ℝ) → Prop","l":117,"f":"Causalean/Estimation/NPIV/DR/AsymptoticLinear.lean"},{"n":"Causalean.Estimation.NPIV.DR.instMeasurableSpace_𝒲","k":"def","m":"Causalean.Estimation.NPIV.DR.Estimator","a":"Estimation/NPIV/DR","t":1,"d":"The observation space carries the measurable space stored in the inverse problem system.","s":"{Ω : Type u_1} → [inst : MeasurableSpace Ω] → {μ : MeasureTheory.Measure Ω} → (S : Causalean.Estimation.NPIV.InverseProblemSystem Ω μ) → MeasurableSpace S.𝒲","l":47,"f":"Causalean/Estimation/NPIV/DR/Estimator.lean"},{"n":"Causalean.Estimation.NPIV.DR.instMeasurableSpace_𝒳","k":"def","m":"Causalean.Estimation.NPIV.DR.Estimator","a":"Estimation/NPIV/DR","t":1,"d":"The covariate space carries the measurable space stored in the inverse problem system.","s":"{Ω : Type u_1} → [inst : MeasurableSpace Ω] → {μ : MeasureTheory.Measure Ω} → (S : Causalean.Estimation.NPIV.InverseProblemSystem Ω μ) → MeasurableSpace S.𝒳","l":53,"f":"Causalean/Estimation/NPIV/DR/Estimator.lean"},{"n":"Causalean.Estimation.NPIV.DR.instMeasurableSpace_𝒵","k":"def","m":"Causalean.Estimation.NPIV.DR.Estimator","a":"Estimation/NPIV/DR","t":1,"d":"The instrument space carries the measurable space stored in the inverse problem system.","s":"{Ω : Type u_1} → [inst : MeasurableSpace Ω] → {μ : MeasureTheory.Measure Ω} → (S : Causalean.Estimation.NPIV.InverseProblemSystem Ω μ) → MeasurableSpace S.𝒵","l":59,"f":"Causalean/Estimation/NPIV/DR/Estimator.lean"},{"n":"Causalean.Estimation.NPIV.DR.mean_zero_of_DualSolution","k":"theorem","m":"Causalean.Estimation.NPIV.DR.AsymptoticLinear","a":"Estimation/NPIV/DR","t":2,"d":"Mean-zero of the oracle score under the observation law P_W.","s":"∀ {Ω : Type u_1} [inst : MeasurableSpace Ω] {μ : MeasureTheory.Measure Ω} [MeasureTheory.IsProbabilityMeasure μ] (S : Causalean.Estimation.NPIV.InverseProblemSystem Ω μ) {q₀ : S.𝒵 → ℝ}, S.DualSolution q₀ → ∀ {P_W : MeasureTheory.Measure S.𝒲}, MeasureTheory.Measure.map S.W μ = P_W → Measurable (Causalean.Estimation.NPIV.DR.ρ₀ S q₀) → ∫ (w : S.𝒲), Causalean.Estimation.NPIV.DR.ρ₀ S q₀ w ∂P_W = 0","l":97,"f":"Causalean/Estimation/NPIV/DR/AsymptoticLinear.lean"},{"n":"Causalean.Estimation.NPIV.DR.mean_zero_of_DualSolution_mu","k":"theorem","m":"Causalean.Estimation.NPIV.DR.AsymptoticLinear","a":"Estimation/NPIV/DR","t":2,"d":"Mean-zero of the oracle score under the structural law μ.","s":"∀ {Ω : Type u_1} [inst : MeasurableSpace Ω] {μ : MeasureTheory.Measure Ω} [MeasureTheory.IsProbabilityMeasure μ] (S : Causalean.Estimation.NPIV.InverseProblemSystem Ω μ) {q₀ : S.𝒵 → ℝ}, S.DualSolution q₀ → ∫ (ω : Ω), Causalean.Estimation.NPIV.DR.ρ₀ S q₀ (S.W ω) ∂μ = 0","l":73,"f":"Causalean/Estimation/NPIV/DR/AsymptoticLinear.lean"},{"n":"Causalean.Estimation.NPIV.DR.mixed_bias_sufficient","k":"theorem","m":"Causalean.Estimation.NPIV.DR.AsymptoticLinear","a":"Estimation/NPIV/DR","t":2,"d":"Sufficient condition for the mixed-bias hypothesis using either operator side (the displayed min{·,·} form in thm:est-trae-dr-al-criterion):","s":"∀ {Ω : Type u_1} [inst : MeasurableSpace Ω] {μ : MeasureTheory.Measure Ω} [MeasureTheory.IsProbabilityMeasure μ] (S : Causalean.Estimation.NPIV.InverseProblemSystem Ω μ) {q₀ : S.𝒵 → ℝ} {P_W : MeasureTheory.Measure S.𝒲} [MeasureTheory.IsProbabilityMeasure P_W] (sample : Causalean.Stat.IIDSample Ω S.𝒲 μ P_W) (split : Causalean.Stat.OneShotSplit sample) (h_hat : ℕ → Ω → S.𝒳 → ℝ) (q_hat : ℕ → Ω → S.𝒵 → ℝ) (bnd : ℕ → Ω → ℝ), (∀ (n : ℕ) (ω : Ω), |∫ (ω' : Ω), (q₀ (S.zOf (S.W ω')) - q_hat n ω (S.zOf (S.W ω'))) * (h_hat n ω (S.xOf (S.W ω')) - S.h₀ (S.xOf (S.W ω'))) ∂μ| ≤ bnd n ω) → Causalean.Stat.IsLittleOp (fun n ω => √↑(split.foldB n).card * bnd n ω) (fun x => 1) μ → Causalean.Stat.IsLittleOp (fun n ω => √↑(split.foldB n).card * ∫ (ω' : Ω), (q₀ (S.zOf (S.W ω')) - q_hat n ω (S.zOf (S.W ω'))) * (h_hat n ω (S.xOf (S.W ω')) - S.h₀ (S.xOf (S.W ω'))) ∂μ) (fun x => 1) μ","l":395,"f":"Causalean/Estimation/NPIV/DR/AsymptoticLinear.lean"},{"n":"Causalean.Estimation.NPIV.DR.trae_dr_asymp_normal","k":"theorem","m":"Causalean.Estimation.NPIV.DR.AsymptoticNormal","a":"Estimation/NPIV/DR","t":1,"d":"TRAE-DR asymptotic normality — thm:est-trae-dr-asymp-normal. Under the dual-solution hypothesis on q₀, primal nuisance estimators ĥ_n, indexed by sample size and outcome, satisfying — together with the paired dual estimators — the bundled TRAE-DR remainder conditions, and the law bridge identifying the pushforward of μ along the observation map W with the observation law P_W, suppose further that the √n-rescaled estimator sequence is almost-everywhere measurable at every sample size and the normalized influence-function partial sum is almost-everywhere measurable at every sample size. Then the rescaled TRAE-DR estimator converges in distribution, along the estimation-fold sizes, to the centered Gaussian law with variance σ₀² := ∫ ρ₀(w)² dP_W.","s":"∀ {Ω : Type u_1} [inst : MeasurableSpace Ω] {μ : MeasureTheory.Measure Ω} [inst_1 : MeasureTheory.IsProbabilityMeasure μ] (S : Causalean.Estimation.NPIV.InverseProblemSystem Ω μ) {q₀ : S.𝒵 → ℝ} (hq₀ : S.DualSolution q₀) {P_W : MeasureTheory.Measure S.𝒲} [inst_2 : MeasureTheory.IsProbabilityMeasure P_W] (sample : Causalean.Stat.IIDSample Ω S.𝒲 μ P_W) (split : Causalean.Stat.OneShotSplit sample) (h_hat : ℕ → Ω → S.𝒳 → ℝ) (q_hat : ℕ → Ω → S.𝒵 → ℝ), Causalean.Estimation.NPIV.DR.TRAEDRRemainderHyps S hq₀ sample split h_hat q_hat → MeasureTheory.Measure.map S.W μ = P_W → Measurable (Causalean.Estimation.NPIV.DR.ρ₀ S q₀) → ∀ (h_meas_θ : ∀ (n : ℕ), AEMeasurable (Causalean.Stat.IsAsymLinear.rescaledEstimator (Causalean.Estimation.NPIV.DR.trae_dr_estimator S sample split h_hat q_hat) S.θ₀ split.foldB n) μ), (∀ (n : ℕ), AEMeasurable (Causalean.Stat.IsAsymLinear.normalizedSum sample (Causalean.Estimation.NPIV.DR.ρ₀ S q₀) split.foldB n) μ) → Causalean.Stat.Tendsto_dist (Causalean.Stat.IsAsymLinear.rescaledEstimator (Causalean.Estimation.NPIV.DR.trae_dr_estimator S sample split h_hat q_hat) S.θ₀ split.foldB) (Causalean.Stat.gaussianMeasure 0 (∫ (w : S.𝒲), Causalean.Estimation.NPIV.DR.ρ₀ S q₀ w ^ 2 ∂P_W)) μ h_meas_θ","l":51,"f":"Causalean/Estimation/NPIV/DR/AsymptoticNormal.lean"},{"n":"Causalean.Estimation.NPIV.DR.trae_dr_estimator","k":"def","m":"Causalean.Estimation.NPIV.DR.Estimator","a":"Estimation/NPIV/DR","t":1,"d":"One-shot TRAE doubly robust estimator (def:est-trae-dr-estimator):","s":"{Ω : Type u_1} → [inst : MeasurableSpace Ω] → {μ : MeasureTheory.Measure Ω} → (S : Causalean.Estimation.NPIV.InverseProblemSystem Ω μ) → {P_W : MeasureTheory.Measure S.𝒲} → (sample : Causalean.Stat.IIDSample Ω S.𝒲 μ P_W) → Causalean.Stat.OneShotSplit sample → (ℕ → Ω → S.𝒳 → ℝ) → (ℕ → Ω → S.𝒵 → ℝ) → ℕ → Ω → ℝ","l":65,"f":"Causalean/Estimation/NPIV/DR/Estimator.lean"},{"n":"Causalean.Estimation.NPIV.DR.trae_dr_estimator_eq_avg_phi","k":"theorem","m":"Causalean.Estimation.NPIV.DR.Estimator","a":"Estimation/NPIV/DR","t":1,"d":"Estimator unfolding. For primal nuisance estimators ĥ_n, indexed by sample size and by outcome, paired with dual nuisance estimators q̂_n, an inverse-problem system, an i.i.d. sample, and a one-shot cross-fitting split, the one-shot TRAE doubly robust estimator evaluated at sample size n and outcome ω equals the average, over the evaluation fold B(n), of the pointwise doubly-robust pseudo-outcome φ_{ĥ_n,q̂_n} computed at each fold member's observation.","s":"∀ {Ω : Type u_1} [inst : MeasurableSpace Ω] {μ : MeasureTheory.Measure Ω} (S : Causalean.Estimation.NPIV.InverseProblemSystem Ω μ) {P_W : MeasureTheory.Measure S.𝒲} (sample : Causalean.Stat.IIDSample Ω S.𝒲 μ P_W) (split : Causalean.Stat.OneShotSplit sample) (h_hat : ℕ → Ω → S.𝒳 → ℝ) (q_hat : ℕ → Ω → S.𝒵 → ℝ) (n : ℕ) (ω : Ω), Causalean.Estimation.NPIV.DR.trae_dr_estimator S sample split h_hat q_hat n ω = (↑(split.foldB n).card)⁻¹ * ∑ i ∈ split.foldB n, S.phiVal (h_hat n ω) (q_hat n ω) (sample.Z i ω)","l":85,"f":"Causalean/Estimation/NPIV/DR/Estimator.lean"},{"n":"Causalean.Estimation.NPIV.DR.trae_dr_isAsymLinear","k":"theorem","m":"Causalean.Estimation.NPIV.DR.AsymptoticLinear","a":"Estimation/NPIV/DR","t":1,"d":"TRAE-DR asymptotic-linearity criterion — thm:est-trae-dr-al-criterion. Fix a linear inverse-problem functional system S, an i.i.d. sample, and a one-shot cross-fitting split, and suppose q₀ solves the associated dual moment equation. Given primal nuisance estimators ĥ_n, indexed by sample size and outcome, paired with dual nuisance estimators satisfying the bundled L²-consistency, candidate-membership, mixed-bias, and empirical-process remainder conditions, together with the law bridge identifying the pushforward of μ along the observation map W with the observation law P_W and measurability of the oracle score ρ₀, the TRAE-DR estimator is asymptotically linear at the structural target θ₀, with mean-zero, finite-variance influence function ρ₀ := φ_{h₀,q₀} − θ₀ and vanishing √n-rescaled remainder, indexed along the estimation folds.","s":"∀ {Ω : Type u_1} [inst : MeasurableSpace Ω] {μ : MeasureTheory.Measure Ω} [inst_1 : MeasureTheory.IsProbabilityMeasure μ] (S : Causalean.Estimation.NPIV.InverseProblemSystem Ω μ) {q₀ : S.𝒵 → ℝ} (hq₀ : S.DualSolution q₀) {P_W : MeasureTheory.Measure S.𝒲} [inst_2 : MeasureTheory.IsProbabilityMeasure P_W] (sample : Causalean.Stat.IIDSample Ω S.𝒲 μ P_W) (split : Causalean.Stat.OneShotSplit sample) (h_hat : ℕ → Ω → S.𝒳 → ℝ) (q_hat : ℕ → Ω → S.𝒵 → ℝ), Causalean.Estimation.NPIV.DR.TRAEDRRemainderHyps S hq₀ sample split h_hat q_hat → MeasureTheory.Measure.map S.W μ = P_W → Measurable (Causalean.Estimation.NPIV.DR.ρ₀ S q₀) → Causalean.Stat.IsAsymLinear (Causalean.Estimation.NPIV.DR.trae_dr_estimator S sample split h_hat q_hat) S.θ₀ (Causalean.Estimation.NPIV.DR.ρ₀ S q₀) sample split.foldB","l":190,"f":"Causalean/Estimation/NPIV/DR/AsymptoticLinear.lean"},{"n":"Causalean.Estimation.NPIV.DR.trae_dr_studentized","k":"theorem","m":"Causalean.Estimation.NPIV.DR.AsymptoticNormal","a":"Estimation/NPIV/DR","t":1,"d":"Studentized TRAE-DR convergence. Fix a dual solution q₀ of the inverse-problem system S together with a first-stage nuisance-estimator sequence ĥ_n, and suppose the sample's W-marginal is identified as P_W. Assume the efficient influence function ρ₀ is measurable and the rescaled estimator sequence, its normalized-sum representation, and the studentized statistic itself are all almost-everywhere measurable at every sample size. If the asymptotic standard deviation σ₀ is strictly positive with σ₀² = ∫ ρ₀(w)² dP_W and a variance-estimator sequence σ̂_n converges to σ₀ in probability, then the studentized statistic √|B(n)| · (θ̂_n − θ₀) / σ̂_n converges in distribution to the standard normal law N(0, 1).","s":"∀ {Ω : Type u_1} [inst : MeasurableSpace Ω] {μ : MeasureTheory.Measure Ω} [inst_1 : MeasureTheory.IsProbabilityMeasure μ] (S : Causalean.Estimation.NPIV.InverseProblemSystem Ω μ) {q₀ : S.𝒵 → ℝ} (hq₀ : S.DualSolution q₀) {P_W : MeasureTheory.Measure S.𝒲} [inst_2 : MeasureTheory.IsProbabilityMeasure P_W] (sample : Causalean.Stat.IIDSample Ω S.𝒲 μ P_W) (split : Causalean.Stat.OneShotSplit sample) (h_hat : ℕ → Ω → S.𝒳 → ℝ) (q_hat : ℕ → Ω → S.𝒵 → ℝ), Causalean.Estimation.NPIV.DR.TRAEDRRemainderHyps S hq₀ sample split h_hat q_hat → MeasureTheory.Measure.map S.W μ = P_W → ∀ (σ_hat_n : ℕ → Ω → ℝ) (σ₀ : ℝ), 0 < σ₀ → σ₀ ^ 2 = ∫ (w : S.𝒲), Causalean.Estimation.NPIV.DR.ρ₀ S q₀ w ^ 2 ∂P_W → Causalean.Stat.Tendsto_inProb σ_hat_n (fun x => σ₀) μ → Measurable (Causalean.Estimation.NPIV.DR.ρ₀ S q₀) → (∀ (n : ℕ), AEMeasurable (Causalean.Stat.IsAsymLinear.rescaledEstimator (Causalean.Estimation.NPIV.DR.trae_dr_estimator S sample split h_hat q_hat) S.θ₀ split.foldB n) μ) → (∀ (n : ℕ), AEMeasurable (Causalean.Stat.IsAsymLinear.normalizedSum sample (Causalean.Estimation.NPIV.DR.ρ₀ S q₀) split.foldB n) μ) → ∀ (h_studentized_meas : ∀ (n : ℕ), AEMeasurable (fun ω => √↑(split.foldB n).card * (Causalean.Estimation.NPIV.DR.trae_dr_estimator S sample split h_hat q_hat n ω - S.θ₀) / σ_hat_n n ω) μ), Causalean.Stat.Tendsto_dist (fun n ω => √↑(split.foldB n).card * (Causalean.Estimation.NPIV.DR.trae_dr_estimator S sample split h_hat q_hat n ω - S.θ₀) / σ_hat_n n ω) (Causalean.Stat.gaussianMeasure 0 1) μ h_studentized_meas","l":106,"f":"Causalean/Estimation/NPIV/DR/AsymptoticNormal.lean"},{"n":"Causalean.Estimation.NPIV.DR.trae_dr_wald_coverage","k":"theorem","m":"Causalean.Estimation.NPIV.DR.AsymptoticNormal","a":"Estimation/NPIV/DR","t":1,"d":"Wald asymptotic coverage — thm:est-trae-dr-asymp-normal. Under the same setup as trae_dr_studentized — a dual solution q₀ of the inverse-problem system S, a first-stage nuisance-estimator sequence ĥ_n, the sample's W-marginal identified as P_W, measurability of ρ₀, of the rescaled estimator, of its normalized-sum representation, and of the studentized statistic, and a variance-estimator sequence σ̂_n that is consistent in probability for a strictly positive σ₀ satisfying σ₀² = ∫ ρ₀(w)² dP_W — fix any strictly positive real number z. Then provided the coverage probability of the Wald interval θ̂_n ± z · σ̂_n / √|B(n)| and the probability that the studentized statistic lands in [-z, z] become asymptotically equal, the Wald-interval coverage probability converges to the standard-normal mass 2 Φ(z) − 1 on [-z, z].","s":"∀ {Ω : Type u_1} [inst : MeasurableSpace Ω] {μ : MeasureTheory.Measure Ω} [inst_1 : MeasureTheory.IsProbabilityMeasure μ] (S : Causalean.Estimation.NPIV.InverseProblemSystem Ω μ) {q₀ : S.𝒵 → ℝ} (hq₀ : S.DualSolution q₀) {P_W : MeasureTheory.Measure S.𝒲} [inst_2 : MeasureTheory.IsProbabilityMeasure P_W] (sample : Causalean.Stat.IIDSample Ω S.𝒲 μ P_W) (split : Causalean.Stat.OneShotSplit sample) (h_hat : ℕ → Ω → S.𝒳 → ℝ) (q_hat : ℕ → Ω → S.𝒵 → ℝ), Causalean.Estimation.NPIV.DR.TRAEDRRemainderHyps S hq₀ sample split h_hat q_hat → MeasureTheory.Measure.map S.W μ = P_W → ∀ (σ_hat_n : ℕ → Ω → ℝ) (σ₀ : ℝ), 0 < σ₀ → σ₀ ^ 2 = ∫ (w : S.𝒲), Causalean.Estimation.NPIV.DR.ρ₀ S q₀ w ^ 2 ∂P_W → Causalean.Stat.Tendsto_inProb σ_hat_n (fun x => σ₀) μ → Measurable (Causalean.Estimation.NPIV.DR.ρ₀ S q₀) → (∀ (n : ℕ), AEMeasurable (Causalean.Stat.IsAsymLinear.rescaledEstimator (Causalean.Estimation.NPIV.DR.trae_dr_estimator S sample split h_hat q_hat) S.θ₀ split.foldB n) μ) → (∀ (n : ℕ), AEMeasurable (Causalean.Stat.IsAsymLinear.normalizedSum sample (Causalean.Estimation.NPIV.DR.ρ₀ S q₀) split.foldB n) μ) → (∀ (n : ℕ), AEMeasurable (fun ω => √↑(split.foldB n).card * (Causalean.Estimation.NPIV.DR.trae_dr_estimator S sample split h_hat q_hat n ω - S.θ₀) / σ_hat_n n ω) μ) → ∀ (z : ℝ), 0 < z → Filter.Tendsto (fun n => (μ {ω | |Causalean.Estimation.NPIV.DR.trae_dr_estimator S sample split h_hat q_hat n ω - S.θ₀| ≤ z * σ_hat_n n ω / √↑(split.foldB n).card}).toReal - (μ {ω | √↑(split.foldB n).card * (Causalean.Estimation.NPIV.DR.trae_dr_estimator S sample split h_hat q_hat n ω - S.θ₀) / σ_hat_n n ω ∈ Set.Icc (-z) z}).toReal) Filter.atTop (nhds 0) → Filter.Tendsto (fun n => (μ {ω | |Causalean.Estimation.NPIV.DR.trae_dr_estimator S sample split h_hat q_hat n ω - S.θ₀| ≤ z * σ_hat_n n ω / √↑(split.foldB n).card}).toReal) Filter.atTop (nhds ((Causalean.Stat.gaussianMeasure 0 1) (Set.Icc (-z) z)).toReal)","l":181,"f":"Causalean/Estimation/NPIV/DR/AsymptoticNormal.lean"},{"n":"Causalean.Estimation.NPIV.DR.ρ₀","k":"def","m":"Causalean.Estimation.NPIV.DR.AsymptoticLinear","a":"Estimation/NPIV/DR","t":1,"d":"Oracle score ρ₀(w) := φ_{h₀, q₀}(w) − θ₀.","s":"{Ω : Type u_1} → [inst : MeasurableSpace Ω] → {μ : MeasureTheory.Measure Ω} → (S : Causalean.Estimation.NPIV.InverseProblemSystem Ω μ) → (S.𝒵 → ℝ) → S.𝒲 → ℝ","l":65,"f":"Causalean/Estimation/NPIV/DR/AsymptoticLinear.lean"},{"n":"Causalean.Estimation.NPIV.InverseProblemSystem","k":"structure","m":"Causalean.Estimation.NPIV.Setup","a":"Estimation/NPIV","t":1,"d":"A linear inverse-problem functional system packages an observation space together with measurable treatment-side and instrument-side covariate projections of the observed data, measurable closed candidate sets for the primal and dual nuisances, each closed under subtraction, observation-level moment maps that are linear in their nuisance argument, jointly measurable and integrable against every candidate, whose cross product q(Z)h(X) is integrable as well, and a primal nuisance in the candidate set satisfying the inverse-problem moment equation against every dual candidate (def:est-trae-system).","s":"(Ω : Type u_1) → [inst : MeasurableSpace Ω] → MeasureTheory.Measure Ω → Type (max (max (max u_1 (u_2 + 1)) (u_3 + 1)) (u_4 + 1))","l":43,"f":"Causalean/Estimation/NPIV/Setup.lean"},{"n":"Causalean.Estimation.NPIV.InverseProblemSystem.DualSolution","k":"structure","m":"Causalean.Estimation.NPIV.Setup","a":"Estimation/NPIV","t":1,"d":"DualSolution S q₀ says q₀ ∈ Qbar and q₀ solves the dual moment identity E[m_e(W; h)] = E[q₀(Z) h(X)] for every h ∈ Hbar (def:est-trae-dual-solution).","s":"{Ω : Type u_1} → [inst : MeasurableSpace Ω] → {μ : MeasureTheory.Measure Ω} → (S : Causalean.Estimation.NPIV.InverseProblemSystem Ω μ) → (S.𝒵 → ℝ) → Prop","l":166,"f":"Causalean/Estimation/NPIV/Setup.lean"},{"n":"Causalean.Estimation.NPIV.InverseProblemSystem.X","k":"def","m":"Causalean.Estimation.NPIV.Setup","a":"Estimation/NPIV","t":1,"d":"Treatment-side covariate as a random variable X := xOf ∘ W.","s":"{Ω : Type u_1} → [inst : MeasurableSpace Ω] → {μ : MeasureTheory.Measure Ω} → (S : Causalean.Estimation.NPIV.InverseProblemSystem Ω μ) → Ω → S.𝒳","l":141,"f":"Causalean/Estimation/NPIV/Setup.lean"},{"n":"Causalean.Estimation.NPIV.InverseProblemSystem.Z","k":"def","m":"Causalean.Estimation.NPIV.Setup","a":"Estimation/NPIV","t":1,"d":"Instrument-side covariate as a random variable Z := zOf ∘ W.","s":"{Ω : Type u_1} → [inst : MeasurableSpace Ω] → {μ : MeasureTheory.Measure Ω} → (S : Causalean.Estimation.NPIV.InverseProblemSystem Ω μ) → Ω → S.𝒵","l":144,"f":"Causalean/Estimation/NPIV/Setup.lean"},{"n":"Causalean.Estimation.NPIV.InverseProblemSystem.phi","k":"def","m":"Causalean.Estimation.NPIV.Setup","a":"Estimation/NPIV","t":1,"d":"The pseudo-outcome along the random observation W: φ_{h,q}(ω) := φ_{h,q}(W ω).","s":"{Ω : Type u_1} → [inst : MeasurableSpace Ω] → {μ : MeasureTheory.Measure Ω} → (S : Causalean.Estimation.NPIV.InverseProblemSystem Ω μ) → (S.𝒳 → ℝ) → (S.𝒵 → ℝ) → Ω → ℝ","l":160,"f":"Causalean/Estimation/NPIV/Setup.lean"},{"n":"Causalean.Estimation.NPIV.InverseProblemSystem.phiVal","k":"def","m":"Causalean.Estimation.NPIV.Setup","a":"Estimation/NPIV","t":1,"d":"The pointwise pseudo-outcome at a sample value w : 𝒲: φ_{h,q}(w) := m_e(w; h) + m(w; q) − q(zOf w) h(xOf w).","s":"{Ω : Type u_1} → [inst : MeasurableSpace Ω] → {μ : MeasureTheory.Measure Ω} → (S : Causalean.Estimation.NPIV.InverseProblemSystem Ω μ) → (S.𝒳 → ℝ) → (S.𝒵 → ℝ) → S.𝒲 → ℝ","l":151,"f":"Causalean/Estimation/NPIV/Setup.lean"},{"n":"Causalean.Estimation.NPIV.InverseProblemSystem.Θ","k":"def","m":"Causalean.Estimation.NPIV.Setup","a":"Estimation/NPIV","t":1,"d":"The doubly-robust functional from def:est-trae-dr-functional: Θ(h, q) := E[m_e(W; h) + m(W; q) − q(Z) h(X)] = E[φ_{h,q}(W)].","s":"{Ω : Type u_1} → [inst : MeasurableSpace Ω] → {μ : MeasureTheory.Measure Ω} → (S : Causalean.Estimation.NPIV.InverseProblemSystem Ω μ) → (S.𝒳 → ℝ) → (S.𝒵 → ℝ) → ℝ","l":179,"f":"Causalean/Estimation/NPIV/Setup.lean"},{"n":"Causalean.Estimation.NPIV.InverseProblemSystem.θ₀","k":"def","m":"Causalean.Estimation.NPIV.Setup","a":"Estimation/NPIV","t":1,"d":"Scalar target θ₀ := E[m_e(W; h₀)] (Definition 4.5, item 6).","s":"{Ω : Type u_1} → [inst : MeasurableSpace Ω] → {μ : MeasureTheory.Measure Ω} → Causalean.Estimation.NPIV.InverseProblemSystem Ω μ → ℝ","l":147,"f":"Causalean/Estimation/NPIV/Setup.lean"},{"n":"Causalean.Estimation.NPIV.OperatorSystem","k":"structure","m":"Causalean.Estimation.NPIV.Operator","a":"Estimation/NPIV/Operator","t":1,"d":"An operator system: an InverseProblemSystem enriched with the L² data needed to define the concrete population projection operator T = Π_{Q̄} ∘ E[· | σ(Z)]. On top of the base system it packages the σ-algebra generated by the instrument Z, with the facts that it is coarser than the ambient σ-algebra on the sample space and coincides with the σ-algebra pulled back along Z, that the ambient probability measure is finite, and closed submodules of the ambient L²(μ) space realizing the primal and dual candidate function classes, the dual one further required to admit an orthogonal projection; finally, every candidate primal or dual function is square-integrable once composed with the covariate or instrument map, with its L² class coherently landing inside the matching submodule.","s":"(Ω : Type u_1) → [m₀ : MeasurableSpace Ω] → MeasureTheory.Measure Ω → Type (max (max (max u_1 (u_2 + 1)) (u_3 + 1)) (u_4 + 1))","l":65,"f":"Causalean/Estimation/NPIV/Operator.lean"},{"n":"Causalean.Estimation.NPIV.OperatorSystem.QbarAEMeasurableMZ","k":"def","m":"Causalean.Estimation.NPIV.Operator","a":"Estimation/NPIV/Operator","t":1,"d":"Hypothesis used by T_inner_eq_integral: q ∘ Z is m_Z-strongly measurable (a.e.). Derivable from m_Z_eq and meas_of_Qbar; see qbar_aestrongly_measurable_mZ below. Kept as a def so existing call sites (which take it as a separate hypothesis) continue to compile.","s":"{Ω : Type u_1} → [inst : MeasurableSpace Ω] → {μ : MeasureTheory.Measure Ω} → Causalean.Estimation.NPIV.OperatorSystem Ω μ → Prop","l":233,"f":"Causalean/Estimation/NPIV/Operator.lean"},{"n":"Causalean.Estimation.NPIV.OperatorSystem.T","k":"def","m":"Causalean.Estimation.NPIV.Operator","a":"Estimation/NPIV/Operator","t":1,"d":"The concrete projection operator T = Π_{Q̄} ∘ E[· | σ(Z)].","s":"{Ω : Type u_1} → [inst : MeasurableSpace Ω] → {μ : MeasureTheory.Measure Ω} → Causalean.Estimation.NPIV.OperatorSystem Ω μ → ↥(MeasureTheory.Lp ℝ 2 μ) → ↥(MeasureTheory.Lp ℝ 2 μ)","l":145,"f":"Causalean/Estimation/NPIV/Operator.lean"},{"n":"Causalean.Estimation.NPIV.OperatorSystem.T_add","k":"theorem","m":"Causalean.Estimation.NPIV.Operator","a":"Estimation/NPIV/Operator","t":2,"d":"T distributes over addition (companion to T_sub).","s":"∀ {Ω : Type u_1} [inst : MeasurableSpace Ω] {μ : MeasureTheory.Measure Ω} (S : Causalean.Estimation.NPIV.OperatorSystem Ω μ) (f g : ↥(MeasureTheory.Lp ℝ 2 μ)), S.T (f + g) = S.T f + S.T g","l":192,"f":"Causalean/Estimation/NPIV/Operator.lean"},{"n":"Causalean.Estimation.NPIV.OperatorSystem.T_inner_eq_integral","k":"theorem","m":"Causalean.Estimation.NPIV.Operator","a":"Estimation/NPIV/Operator","t":1,"d":"Population identity for T. Consider an operator system S — an inverse-problem system enriched with an L² realization of the candidate sets — and fix two primal candidate functions h, h' in the primal candidate set Hbar together with a dual candidate function f in the dual candidate set Qbar. Writing T for the operator that conditions on the σ-algebra generated by the instrument Z and then orthogonally projects onto the L² realization of Qbar, the L²(μ) inner product of T applied to the lifted residual h − h' against the lifted f equals the expectation of the pointwise product (h(X) − h'(X))·f(Z).","s":"∀ {Ω : Type u_1} [inst : MeasurableSpace Ω] {μ : MeasureTheory.Measure Ω} (S : Causalean.Estimation.NPIV.OperatorSystem Ω μ) {h h' : S.𝒳 → ℝ} (hh : h ∈ S.Hbar) (hh' : h' ∈ S.Hbar) {f : S.𝒵 → ℝ} (hf : f ∈ S.Qbar), inner ℝ (S.T (S.hL2 hh - S.hL2 hh')) (S.qL2 hf) = ∫ (ω : Ω), (h (S.xOf (S.W ω)) - h' (S.xOf (S.W ω))) * f (S.zOf (S.W ω)) ∂μ","l":260,"f":"Causalean/Estimation/NPIV/Operator.lean"},{"n":"Causalean.Estimation.NPIV.OperatorSystem.T_sub","k":"theorem","m":"Causalean.Estimation.NPIV.Operator","a":"Estimation/NPIV/Operator","t":2,"d":"T distributes over subtraction.","s":"∀ {Ω : Type u_1} [inst : MeasurableSpace Ω] {μ : MeasureTheory.Measure Ω} (S : Causalean.Estimation.NPIV.OperatorSystem Ω μ) (f g : ↥(MeasureTheory.Lp ℝ 2 μ)), S.T (f - g) = S.T f - S.T g","l":176,"f":"Causalean/Estimation/NPIV/Operator.lean"},{"n":"Causalean.Estimation.NPIV.OperatorSystem.Tadjoint","k":"def","m":"Causalean.Estimation.NPIV.Operator.Adjoint","a":"Estimation/NPIV/Operator","t":1,"d":"The adjoint operator T† : Lp ℝ 2 μ →L[ℝ] Lp ℝ 2 μ.","s":"{Ω : Type u_1} → [inst : MeasurableSpace Ω] → {μ : MeasureTheory.Measure Ω} → Causalean.Estimation.NPIV.OperatorSystem Ω μ → ↥(MeasureTheory.Lp ℝ 2 μ) →L[ℝ] ↥(MeasureTheory.Lp ℝ 2 μ)","l":83,"f":"Causalean/Estimation/NPIV/Operator/Adjoint.lean"},{"n":"Causalean.Estimation.NPIV.OperatorSystem.Tlin","k":"def","m":"Causalean.Estimation.NPIV.Operator.Adjoint","a":"Estimation/NPIV/Operator","t":1,"d":"The bundled continuous-linear-map version of OperatorSystem.T:","s":"{Ω : Type u_1} → [inst : MeasurableSpace Ω] → {μ : MeasureTheory.Measure Ω} → Causalean.Estimation.NPIV.OperatorSystem Ω μ → ↥(MeasureTheory.Lp ℝ 2 μ) →L[ℝ] ↥(MeasureTheory.Lp ℝ 2 μ)","l":61,"f":"Causalean/Estimation/NPIV/Operator/Adjoint.lean"},{"n":"Causalean.Estimation.NPIV.OperatorSystem.Tlin_apply","k":"theorem","m":"Causalean.Estimation.NPIV.Operator.Adjoint","a":"Estimation/NPIV/Operator","t":2,"d":"Coherence: applying the bundled CLM Tlin to f agrees with the function-level operator T from Causalean/Estimation/NPIV/Operator.lean. The proof unfolds both sides and uses Submodule.starProjection_apply = orthogonalProjectionFn.","s":"∀ {Ω : Type u_1} [inst : MeasurableSpace Ω] {μ : MeasureTheory.Measure Ω} (S : Causalean.Estimation.NPIV.OperatorSystem Ω μ) (f : ↥(MeasureTheory.Lp ℝ 2 μ)), S.Tlin f = S.T f","l":73,"f":"Causalean/Estimation/NPIV/Operator/Adjoint.lean"},{"n":"Causalean.Estimation.NPIV.OperatorSystem.Tstar_T","k":"def","m":"Causalean.Estimation.NPIV.Operator.Adjoint","a":"Estimation/NPIV/Operator","t":1,"d":"The self-adjoint, positive-semidefinite composite T† T. This is the operator the β-source condition refers to: h₀ = (T*T)^{β/2} w₀ is interpreted via Mathlib's continuous functional calculus on this CLM.","s":"{Ω : Type u_1} → [inst : MeasurableSpace Ω] → {μ : MeasureTheory.Measure Ω} → Causalean.Estimation.NPIV.OperatorSystem Ω μ → ↥(MeasureTheory.Lp ℝ 2 μ) →L[ℝ] ↥(MeasureTheory.Lp ℝ 2 μ)","l":92,"f":"Causalean/Estimation/NPIV/Operator/Adjoint.lean"},{"n":"Causalean.Estimation.NPIV.OperatorSystem.Tstar_T_isSelfAdjoint","k":"theorem","m":"Causalean.Estimation.NPIV.Operator.Adjoint","a":"Estimation/NPIV/Operator","t":1,"d":"For any NPIV operator system, the composite operator T† T, obtained by composing the second-stage conditional-expectation operator with its adjoint, is self-adjoint.","s":"∀ {Ω : Type u_1} [inst : MeasurableSpace Ω] {μ : MeasureTheory.Measure Ω} (S : Causalean.Estimation.NPIV.OperatorSystem Ω μ), IsSelfAdjoint S.Tstar_T","l":100,"f":"Causalean/Estimation/NPIV/Operator/Adjoint.lean"},{"n":"Causalean.Estimation.NPIV.OperatorSystem.condExpStep","k":"def","m":"Causalean.Estimation.NPIV.Operator","a":"Estimation/NPIV/Operator","t":1,"d":"The conditional-expectation step E[· | σ(Z)], viewed as a self-map of Lp ℝ 2 μ (via the canonical inclusion of the σ(Z)-measurable subspace).","s":"{Ω : Type u_1} → [inst : MeasurableSpace Ω] → {μ : MeasureTheory.Measure Ω} → Causalean.Estimation.NPIV.OperatorSystem Ω μ → ↥(MeasureTheory.Lp ℝ 2 μ) → ↥(MeasureTheory.Lp ℝ 2 μ)","l":132,"f":"Causalean/Estimation/NPIV/Operator.lean"},{"n":"Causalean.Estimation.NPIV.OperatorSystem.condExpStepL","k":"def","m":"Causalean.Estimation.NPIV.Operator.Adjoint","a":"Estimation/NPIV/Operator","t":1,"d":"The conditional-expectation step f ↦ E[f | σ(Z)], viewed as a continuous ℝ-linear endomorphism of Lp ℝ 2 μ.","s":"{Ω : Type u_1} → [inst : MeasurableSpace Ω] → {μ : MeasureTheory.Measure Ω} → Causalean.Estimation.NPIV.OperatorSystem Ω μ → ↥(MeasureTheory.Lp ℝ 2 μ) →L[ℝ] ↥(MeasureTheory.Lp ℝ 2 μ)","l":50,"f":"Causalean/Estimation/NPIV/Operator/Adjoint.lean"},{"n":"Causalean.Estimation.NPIV.OperatorSystem.condExpStep_sub","k":"theorem","m":"Causalean.Estimation.NPIV.Operator","a":"Estimation/NPIV/Operator","t":2,"d":"condExpStep distributes over subtraction (linearity of condExpL2).","s":"∀ {Ω : Type u_1} [inst : MeasurableSpace Ω] {μ : MeasureTheory.Measure Ω} (S : Causalean.Estimation.NPIV.OperatorSystem Ω μ) (f g : ↥(MeasureTheory.Lp ℝ 2 μ)), S.condExpStep (f - g) = S.condExpStep f - S.condExpStep g","l":168,"f":"Causalean/Estimation/NPIV/Operator.lean"},{"n":"Causalean.Estimation.NPIV.OperatorSystem.hL2","k":"def","m":"Causalean.Estimation.NPIV.Operator","a":"Estimation/NPIV/Operator","t":1,"d":"Lift h ∈ Hbar to its L² class (h ∘ X)_{Lp} ∈ Lp ℝ 2 μ. This is just a name.","s":"{Ω : Type u_1} → [inst : MeasurableSpace Ω] → {μ : MeasureTheory.Measure Ω} → (S : Causalean.Estimation.NPIV.OperatorSystem Ω μ) → {h : S.𝒳 → ℝ} → h ∈ S.Hbar → ↥(MeasureTheory.Lp ℝ 2 μ)","l":114,"f":"Causalean/Estimation/NPIV/Operator.lean"},{"n":"Causalean.Estimation.NPIV.OperatorSystem.qL2","k":"def","m":"Causalean.Estimation.NPIV.Operator","a":"Estimation/NPIV/Operator","t":1,"d":"Lift q ∈ Qbar to its L² class (q ∘ Z)_{Lp} ∈ Lp ℝ 2 μ.","s":"{Ω : Type u_1} → [inst : MeasurableSpace Ω] → {μ : MeasureTheory.Measure Ω} → (S : Causalean.Estimation.NPIV.OperatorSystem Ω μ) → {q : S.𝒵 → ℝ} → q ∈ S.Qbar → ↥(MeasureTheory.Lp ℝ 2 μ)","l":120,"f":"Causalean/Estimation/NPIV/Operator.lean"},{"n":"Causalean.Estimation.NPIV.OperatorSystem.qbar_aestrongly_measurable_mZ","k":"theorem","m":"Causalean.Estimation.NPIV.Operator","a":"Estimation/NPIV/Operator","t":2,"d":"q ∘ zOf ∘ W is m_Z-strongly measurable for every q ∈ Qbar. Direct consequence of m_Z_eq : m_Z = comap (zOf ∘ W) inst𝒵 and meas_of_Qbar: the canonical map (zOf ∘ W) : Ω → 𝒵 is measurable from comap (zOf ∘ W) inst𝒵 to inst𝒵 (comap_measurable), so its composition with the inst𝒵-measurable q : 𝒵 → ℝ is m_Z-measurable, hence m_Z-strongly measurable into ℝ.","s":"∀ {Ω : Type u_1} [inst : MeasurableSpace Ω] {μ : MeasureTheory.Measure Ω} (S : Causalean.Estimation.NPIV.OperatorSystem Ω μ), S.QbarAEMeasurableMZ","l":241,"f":"Causalean/Estimation/NPIV/Operator.lean"},{"n":"Causalean.Estimation.NPIV.OperatorSystem.strongNorm","k":"def","m":"Causalean.Estimation.NPIV.Operator","a":"Estimation/NPIV/Operator","t":1,"d":"Strong (L²) norm of an L² element.","s":"{Ω : Type u_1} → [inst : MeasurableSpace Ω] → {μ : MeasureTheory.Measure Ω} → Causalean.Estimation.NPIV.OperatorSystem Ω μ → ↥(MeasureTheory.Lp ℝ 2 μ) → ℝ","l":359,"f":"Causalean/Estimation/NPIV/Operator.lean"},{"n":"Causalean.Estimation.NPIV.OperatorSystem.strongNorm_sq_hL2_eq_integral","k":"theorem","m":"Causalean.Estimation.NPIV.Operator","a":"Estimation/NPIV/Operator","t":2,"d":"Squared strong norm of a lifted Hbar member, unfolded as the integral of the squared representative.","s":"∀ {Ω : Type u_1} [inst : MeasurableSpace Ω] {μ : MeasureTheory.Measure Ω} (S : Causalean.Estimation.NPIV.OperatorSystem Ω μ) {h : S.𝒳 → ℝ} (hh : h ∈ S.Hbar), S.strongNorm (S.hL2 hh) ^ 2 = ∫ (ω : Ω), h (S.xOf (S.W ω)) ^ 2 ∂μ","l":383,"f":"Causalean/Estimation/NPIV/Operator.lean"},{"n":"Causalean.Estimation.NPIV.OperatorSystem.tikhonovBilin","k":"def","m":"Causalean.Estimation.NPIV.Operator.Tikhonov","a":"Estimation/NPIV/Operator","t":1,"d":"The Tikhonov bilinear form on the ambient Lp ℝ 2 μ:","s":"{Ω : Type u_1} → [inst : MeasurableSpace Ω] → {μ : MeasureTheory.Measure Ω} → Causalean.Estimation.NPIV.OperatorSystem Ω μ → ℝ → ↥(MeasureTheory.Lp ℝ 2 μ) →L[ℝ] ↥(MeasureTheory.Lp ℝ 2 μ) →L[ℝ] ℝ","l":74,"f":"Causalean/Estimation/NPIV/Operator/Tikhonov.lean"},{"n":"Causalean.Estimation.NPIV.OperatorSystem.tikhonovBilinSub","k":"def","m":"Causalean.Estimation.NPIV.Operator.Tikhonov","a":"Estimation/NPIV/Operator","t":1,"d":"The bilinear form, as a bilinear form on the closed subspace Hbar_L2. We pre/post-compose tikhonovBilin with the subtype inclusion Hbar_L2 →L[ℝ] Lp ℝ 2 μ so the result lives on the Hilbert space Hbar_L2 itself. This is the form fed into Lax–Milgram.","s":"{Ω : Type u_1} → [inst : MeasurableSpace Ω] → {μ : MeasureTheory.Measure Ω} → (S : Causalean.Estimation.NPIV.OperatorSystem Ω μ) → ℝ → ↥S.Hbar_L2 →L[ℝ] ↥S.Hbar_L2 →L[ℝ] ℝ","l":103,"f":"Causalean/Estimation/NPIV/Operator/Tikhonov.lean"},{"n":"Causalean.Estimation.NPIV.OperatorSystem.tikhonovBilinSub_apply","k":"theorem","m":"Causalean.Estimation.NPIV.Operator.Tikhonov","a":"Estimation/NPIV/Operator","t":2,"d":"Pointwise formula for the restricted bilinear form.","s":"∀ {Ω : Type u_1} [inst : MeasurableSpace Ω] {μ : MeasureTheory.Measure Ω} (S : Causalean.Estimation.NPIV.OperatorSystem Ω μ) (lambda : ℝ) (u v : ↥S.Hbar_L2), ((S.tikhonovBilinSub lambda) u) v = ((S.tikhonovBilin lambda) ↑u) ↑v","l":111,"f":"Causalean/Estimation/NPIV/Operator/Tikhonov.lean"},{"n":"Causalean.Estimation.NPIV.OperatorSystem.tikhonovBilinSub_isCoercive","k":"theorem","m":"Causalean.Estimation.NPIV.Operator.Tikhonov","a":"Estimation/NPIV/Operator","t":2,"d":"Coercivity of tikhonovBilinSub for 0 < λ. Witness constant: λ. Proof: bilin u u = ‖T u‖² + λ ‖u‖² ≥ λ ‖u‖² since ‖T u‖² ≥ 0.","s":"∀ {Ω : Type u_1} [inst : MeasurableSpace Ω] {μ : MeasureTheory.Measure Ω} (S : Causalean.Estimation.NPIV.OperatorSystem Ω μ) [S.Hbar_L2.HasOrthogonalProjection] {lambda : ℝ}, 0 < lambda → IsCoercive (S.tikhonovBilinSub lambda)","l":118,"f":"Causalean/Estimation/NPIV/Operator/Tikhonov.lean"},{"n":"Causalean.Estimation.NPIV.OperatorSystem.tikhonovBilin_apply","k":"theorem","m":"Causalean.Estimation.NPIV.Operator.Tikhonov","a":"Estimation/NPIV/Operator","t":2,"d":"Pointwise formula for tikhonovBilin.","s":"∀ {Ω : Type u_1} [inst : MeasurableSpace Ω] {μ : MeasureTheory.Measure Ω} (S : Causalean.Estimation.NPIV.OperatorSystem Ω μ) (lambda : ℝ) (u v : ↥(MeasureTheory.Lp ℝ 2 μ)), ((S.tikhonovBilin lambda) u) v = inner ℝ (S.Tlin u) (S.Tlin v) + lambda * inner ℝ u v","l":87,"f":"Causalean/Estimation/NPIV/Operator/Tikhonov.lean"},{"n":"Causalean.Estimation.NPIV.OperatorSystem.tikhonovBilin_symm","k":"theorem","m":"Causalean.Estimation.NPIV.Operator.Tikhonov","a":"Estimation/NPIV/Operator","t":2,"d":"Symmetry of tikhonovBilin.","s":"∀ {Ω : Type u_1} [inst : MeasurableSpace Ω] {μ : MeasureTheory.Measure Ω} (S : Causalean.Estimation.NPIV.OperatorSystem Ω μ) (lambda : ℝ) (u v : ↥(MeasureTheory.Lp ℝ 2 μ)), ((S.tikhonovBilin lambda) u) v = ((S.tikhonovBilin lambda) v) u","l":97,"f":"Causalean/Estimation/NPIV/Operator/Tikhonov.lean"},{"n":"Causalean.Estimation.NPIV.OperatorSystem.tikhonovMinimiserL2","k":"def","m":"Causalean.Estimation.NPIV.Operator.Tikhonov","a":"Estimation/NPIV/Operator","t":1,"d":"The population Tikhonov minimiser at level λ (L² level).","s":"{Ω : Type u_1} → [inst : MeasurableSpace Ω] → {μ : MeasureTheory.Measure Ω} → (S : Causalean.Estimation.NPIV.OperatorSystem Ω μ) → [S.Hbar_L2.HasOrthogonalProjection] → ℝ → ↥(MeasureTheory.Lp ℝ 2 μ)","l":159,"f":"Causalean/Estimation/NPIV/Operator/Tikhonov.lean"},{"n":"Causalean.Estimation.NPIV.OperatorSystem.tikhonovMinimiserL2_mem","k":"theorem","m":"Causalean.Estimation.NPIV.Operator.Tikhonov","a":"Estimation/NPIV/Operator","t":2,"d":"The minimiser lies in Hbar_L2.","s":"∀ {Ω : Type u_1} [inst : MeasurableSpace Ω] {μ : MeasureTheory.Measure Ω} (S : Causalean.Estimation.NPIV.OperatorSystem Ω μ) [inst_1 : S.Hbar_L2.HasOrthogonalProjection] (lambda : ℝ), S.tikhonovMinimiserL2 lambda ∈ S.Hbar_L2","l":183,"f":"Causalean/Estimation/NPIV/Operator/Tikhonov.lean"},{"n":"Causalean.Estimation.NPIV.OperatorSystem.tikhonovMinimiserL2_optimality","k":"theorem","m":"Causalean.Estimation.NPIV.Operator.Tikhonov","a":"Estimation/NPIV/Operator","t":1,"d":"Variational identity for the Tikhonov minimiser. For a strictly positive Tikhonov regularization level λ and any function v in the closed primal candidate subspace Hbar_L2, the population Tikhonov minimiser h*_λ at level λ satisfies the identity ⟪T h*_λ, T v⟫ + λ · ⟪h*_λ, v⟫ = ⟪T h₀, T v⟫, where T is the projection-composed conditional-expectation operator and h₀ is the L² class of the structural function.","s":"∀ {Ω : Type u_1} [inst : MeasurableSpace Ω] {μ : MeasureTheory.Measure Ω} (S : Causalean.Estimation.NPIV.OperatorSystem Ω μ) [inst_1 : S.Hbar_L2.HasOrthogonalProjection] {lambda : ℝ}, 0 < lambda → ∀ {v : ↥(MeasureTheory.Lp ℝ 2 μ)}, v ∈ S.Hbar_L2 → inner ℝ (S.Tlin (S.tikhonovMinimiserL2 lambda)) (S.Tlin v) + lambda * inner ℝ (S.tikhonovMinimiserL2 lambda) v = inner ℝ (S.Tlin (S.hL2 S.h₀_mem)) (S.Tlin v)","l":192,"f":"Causalean/Estimation/NPIV/Operator/Tikhonov.lean"},{"n":"Causalean.Estimation.NPIV.OperatorSystem.tikhonovMinimiserL2_strong_convexity","k":"theorem","m":"Causalean.Estimation.NPIV.Operator.Tikhonov","a":"Estimation/NPIV/Operator","t":1,"d":"Population strong convexity at the Tikhonov minimiser (L² level). For a strictly positive Tikhonov regularization level λ and any function h in the closed primal candidate subspace Hbar_L2, the non-negative excess λ‖h − h*_λ‖² + ‖T(h − h*_λ)‖² — the amount by which the quadratic Tikhonov objective at h exceeds its value at the population minimiser h*_λ — is bounded above by ‖T(h − h₀)‖² − ‖T(h*_λ − h₀)‖² + λ(‖h‖² − ‖h*_λ‖²), where T is the projection-composed conditional-expectation operator and h₀ is the structural function's L² class.","s":"∀ {Ω : Type u_1} [inst : MeasurableSpace Ω] {μ : MeasureTheory.Measure Ω} (S : Causalean.Estimation.NPIV.OperatorSystem Ω μ) [inst_1 : S.Hbar_L2.HasOrthogonalProjection] {lambda : ℝ}, 0 < lambda → ∀ {h : ↥(MeasureTheory.Lp ℝ 2 μ)}, h ∈ S.Hbar_L2 → lambda * ‖h - S.tikhonovMinimiserL2 lambda‖ ^ 2 + ‖S.Tlin (h - S.tikhonovMinimiserL2 lambda)‖ ^ 2 ≤ ‖S.Tlin (h - S.hL2 S.h₀_mem)‖ ^ 2 - ‖S.Tlin (S.tikhonovMinimiserL2 lambda - S.hL2 S.h₀_mem)‖ ^ 2 + lambda * (‖h‖ ^ 2 - ‖S.tikhonovMinimiserL2 lambda‖ ^ 2)","l":236,"f":"Causalean/Estimation/NPIV/Operator/Tikhonov.lean"},{"n":"Causalean.Estimation.NPIV.OperatorSystem.tikhonovTargetSub","k":"def","m":"Causalean.Estimation.NPIV.Operator.Tikhonov","a":"Estimation/NPIV/Operator","t":1,"d":"The Tikhonov target functional evaluated on the closed subspace Hbar_L2:","s":"{Ω : Type u_1} → [inst : MeasurableSpace Ω] → {μ : MeasureTheory.Measure Ω} → (S : Causalean.Estimation.NPIV.OperatorSystem Ω μ) → ↥S.Hbar_L2 →L[ℝ] ℝ","l":133,"f":"Causalean/Estimation/NPIV/Operator/Tikhonov.lean"},{"n":"Causalean.Estimation.NPIV.OperatorSystem.tikhonovTargetSub_apply","k":"theorem","m":"Causalean.Estimation.NPIV.Operator.Tikhonov","a":"Estimation/NPIV/Operator","t":2,"d":"Pointwise formula for the target functional.","s":"∀ {Ω : Type u_1} [inst : MeasurableSpace Ω] {μ : MeasureTheory.Measure Ω} (S : Causalean.Estimation.NPIV.OperatorSystem Ω μ) (v : ↥S.Hbar_L2), S.tikhonovTargetSub v = inner ℝ (S.Tlin (S.hL2 S.h₀_mem)) (S.Tlin ↑v)","l":145,"f":"Causalean/Estimation/NPIV/Operator/Tikhonov.lean"},{"n":"Causalean.Estimation.NPIV.OperatorSystem.weakNorm","k":"def","m":"Causalean.Estimation.NPIV.Operator","a":"Estimation/NPIV/Operator","t":1,"d":"Weak norm associated to T: ‖T f‖_{L²(μ)}.","s":"{Ω : Type u_1} → [inst : MeasurableSpace Ω] → {μ : MeasureTheory.Measure Ω} → Causalean.Estimation.NPIV.OperatorSystem Ω μ → ↥(MeasureTheory.Lp ℝ 2 μ) → ℝ","l":363,"f":"Causalean/Estimation/NPIV/Operator.lean"},{"n":"Causalean.Estimation.NPIV.OperatorSystem.weakNorm_le_strongNorm","k":"theorem","m":"Causalean.Estimation.NPIV.Operator","a":"Estimation/NPIV/Operator","t":2,"d":"The projected conditional-expectation norm is bounded by the ambient L² norm.","s":"∀ {Ω : Type u_1} [inst : MeasurableSpace Ω] {μ : MeasureTheory.Measure Ω} (S : Causalean.Estimation.NPIV.OperatorSystem Ω μ) (f : ↥(MeasureTheory.Lp ℝ 2 μ)), S.weakNorm f ≤ S.strongNorm f","l":367,"f":"Causalean/Estimation/NPIV/Operator.lean"},{"n":"Causalean.Estimation.NPIV.Primal.IsTRAEPrimalEstimator","k":"structure","m":"Causalean.Estimation.NPIV.Primal.Estimator","a":"Estimation/NPIV/Primal","t":1,"d":"An estimator, indexed by sample size and randomness, is a TRAE primal estimator relative to a nuisance fold when, at every sample size and realization, it belongs to the statistical candidate class, attains an empirical sup-min objective on that fold no worse than at any other candidate in the class, and is jointly measurable in the randomness and the covariate argument.","s":"{Ω : Type u_1} → [inst : MeasurableSpace Ω] → {μ : MeasureTheory.Measure Ω} → (S : Causalean.Estimation.NPIV.OperatorSystem Ω μ) → Causalean.Estimation.NPIV.Primal.TRAEClasses S → {P_W : MeasureTheory.Measure S.𝒲} → (sample : Causalean.Stat.IIDSample Ω S.𝒲 μ P_W) → Causalean.Stat.OneShotSplit sample → ℝ → (ℕ → Ω → S.𝒳 → ℝ) → Prop","l":146,"f":"Causalean/Estimation/NPIV/Primal/Estimator.lean"},{"n":"Causalean.Estimation.NPIV.Primal.LocalizedRegimeBundle","k":"structure","m":"Causalean.Estimation.NPIV.Primal.EmpiricalProcessEvent.Regime","a":"Estimation/NPIV/Primal/EmpiricalProcessEvent","t":1,"d":"A localized regime bundle packages a countable index set and a measurable family of loss functions on a nonempty covariate space, together with a localization norm, nonnegative on every class member and a measurable sample embedding into that covariate space. It further records a localized empirical-process regime whose critical radius is at most the target localization scale δ_n, is strictly positive, and is a fixed point of the process's expected-supremum function at that scale. Finally, uniformly over every radius at least δ_n, it requires that the empirical Rademacher process on the star-hull is almost-surely bounded and that the corresponding empirical Rademacher complexity is integrable.","s":"(Ω : Type u_2) → [inst : MeasurableSpace Ω] → MeasureTheory.Measure Ω → ℕ → ℝ → Type (max 1 u_2)","l":32,"f":"Causalean/Estimation/NPIV/Primal/EmpiricalProcessEvent/Regime.lean"},{"n":"Causalean.Estimation.NPIV.Primal.LocalizedRegimes","k":"structure","m":"Causalean.Estimation.NPIV.Primal.EmpiricalProcessEvent.Regime","a":"Estimation/NPIV/Primal/EmpiricalProcessEvent","t":1,"d":"This structure bundles four localized regime witnesses, one for each function class entering the empirical-process and centred-regulariser arguments — the product, moment, critic, and candidate classes, with the deterministic hypotheses that the observation variable has the stated law, that the population Tikhonov solution is realizable in the statistical candidate class, that the moment, candidate, and critic maps are almost-surely uniformly bounded, and that every candidate admits a critic whose L² lift realizes the projected residual. It also records that the empirical sup-min objective is bounded above by, and attained at, the population sup objective; diameter caps on the candidate class and the critic class; and nonnegative proportionality constants together with a matching radius lower bound for the moment-class pair differences and for the product-class cross terms. For the candidate class it further supplies an interpretation triple: an index into the H-bundle whose norm controls the pairwise loss gap, and whose composed evaluation equals the difference of squared candidate losses. For the product class it supplies both a single-candidate interpretation triple bounded by the localization scale and a pair-difference interpretation triple scaling with the candidate gap. For the moment class it likewise supplies a single-critic interpretation triple and a pair-difference interpretation triple.","s":"{Ω : Type u_1} → [inst : MeasurableSpace Ω] → {μ : MeasureTheory.Measure Ω} → (S : Causalean.Estimation.NPIV.OperatorSystem Ω μ) → Causalean.Estimation.NPIV.Primal.TRAEClasses S → {P_W : MeasureTheory.Measure S.𝒲} → Causalean.Stat.IIDSample Ω S.𝒲 μ P_W → {β lambda : ℝ} → (sc : Causalean.Estimation.NPIV.SourceCondition S β) → Causalean.Estimation.NPIV.TikhonovBiasBound S β lambda sc → ℕ → ℝ → Type (max (max (max 1 u_2) u_3) u_4)","l":103,"f":"Causalean/Estimation/NPIV/Primal/EmpiricalProcessEvent/Regime.lean"},{"n":"Causalean.Estimation.NPIV.Primal.TRAEClasses","k":"structure","m":"Causalean.Estimation.NPIV.Primal.Estimator","a":"Estimation/NPIV/Primal","t":1,"d":"This structure packages a nonempty statistical candidate class for the primal nuisance that sits inside the ambient closed candidate set, together with a nonempty statistical critic class that sits inside the ambient closed critic set, for the primal Tikhonov-regularized adversarial optimization (def:est-trae-population-criterion, line 139).","s":"{Ω : Type u_1} → [inst : MeasurableSpace Ω] → {μ : MeasureTheory.Measure Ω} → Causalean.Estimation.NPIV.OperatorSystem Ω μ → Type (max u_2 u_3)","l":82,"f":"Causalean/Estimation/NPIV/Primal/Estimator.lean"},{"n":"Causalean.Estimation.NPIV.Primal.TRAERatePrimalAbstractHyps","k":"structure","m":"Causalean.Estimation.NPIV.Primal.Rate","a":"Estimation/NPIV/Primal","t":1,"d":"This structure augments the deterministic TRAE primal rate hypotheses with a discharged high-probability empirical-process / centred-regulariser event, bounding the right-hand side of the population strong-convexity inequality by an explicit deterministic-looking expression, abstracting away the localized empirical-process and centred-regulariser arguments that the underlying localized wrapper supplies.","s":"{Ω : Type u_1} → [inst : MeasurableSpace Ω] → {μ : MeasureTheory.Measure Ω} → (S : Causalean.Estimation.NPIV.OperatorSystem Ω μ) → (TC : Causalean.Estimation.NPIV.Primal.TRAEClasses S) → {P_W : MeasureTheory.Measure S.𝒲} → (sample : Causalean.Stat.IIDSample Ω S.𝒲 μ P_W) → (split : Causalean.Stat.OneShotSplit sample) → (lambda : ℝ) → ℝ → ℝ → (ℕ → ℝ) → (h_hat : ℕ → Ω → S.𝒳 → ℝ) → Causalean.Estimation.NPIV.Primal.IsTRAEPrimalEstimator S TC sample split lambda h_hat → Type u_3","l":158,"f":"Causalean/Estimation/NPIV/Primal/Rate.lean"},{"n":"Causalean.Estimation.NPIV.Primal.TRAERatePrimalHyps","k":"structure","m":"Causalean.Estimation.NPIV.Primal.Rate","a":"Estimation/NPIV/Primal","t":1,"d":"This structure bundles the deterministic hypotheses of the primal TRAE rate theorem: a β-source condition for the target nuisance, a Tikhonov bias certificate at the chosen regularization level, realizability of the resulting population Tikhonov solution in the statistical candidate class, closedness of the critic class for primal residuals, mean-square continuity of the moment map on critics, and almost-sure uniform boundedness of the moment map and the candidate and critic functions.","s":"{Ω : Type u_1} → [inst : MeasurableSpace Ω] → {μ : MeasureTheory.Measure Ω} → (S : Causalean.Estimation.NPIV.OperatorSystem Ω μ) → (TC : Causalean.Estimation.NPIV.Primal.TRAEClasses S) → {P_W : MeasureTheory.Measure S.𝒲} → (sample : Causalean.Stat.IIDSample Ω S.𝒲 μ P_W) → (split : Causalean.Stat.OneShotSplit sample) → (lambda : ℝ) → ℝ → ℝ → (ℕ → ℝ) → (h_hat : ℕ → Ω → S.𝒳 → ℝ) → Causalean.Estimation.NPIV.Primal.IsTRAEPrimalEstimator S TC sample split lambda h_hat → Type u_3","l":91,"f":"Causalean/Estimation/NPIV/Primal/Rate.lean"},{"n":"Causalean.Estimation.NPIV.Primal.centred_regulariser_bound_from_localized","k":"theorem","m":"Causalean.Estimation.NPIV.Primal.EmpiricalProcessEvent.Regulariser","a":"Estimation/NPIV/Primal/EmpiricalProcessEvent","t":1,"d":"The centred empirical regularizer gap is bounded uniformly over all sample sizes by the localized rate from the squared candidate-difference event. For every sample size n, suppose a localized-regimes witness at the corresponding fold size and localization rate delta n is available, the regularization weight satisfies lambda ≥ 0, and the confidence level satisfies 0 < ζ < 1. Then there is an event of probability at least 1 - ζ on which, for every split size with at least one fold-A observation, the empirical-vs- population regularizer discrepancy between the estimator and the population Tikhonov candidate is controlled by an explicit bound built from the fold's diameter, critical radius, and localization rate.","s":"∀ {Ω : Type u_1} [inst : MeasurableSpace Ω] {μ : MeasureTheory.Measure Ω} {S : Causalean.Estimation.NPIV.OperatorSystem Ω μ} {TC : Causalean.Estimation.NPIV.Primal.TRAEClasses S} {P_W : MeasureTheory.Measure S.𝒲} {sample : Causalean.Stat.IIDSample Ω S.𝒲 μ P_W} {split : Causalean.Stat.OneShotSplit sample} {lambda β : ℝ} {delta : ℕ → ℝ} {h_hat : ℕ → Ω → S.𝒳 → ℝ} {is_estimator : Causalean.Estimation.NPIV.Primal.IsTRAEPrimalEstimator S TC sample split lambda h_hat} (sc : Causalean.Estimation.NPIV.SourceCondition S β) (tb : Causalean.Estimation.NPIV.TikhonovBiasBound S β lambda sc) [MeasureTheory.IsProbabilityMeasure μ] (regimes : (n : ℕ) → Causalean.Estimation.NPIV.Primal.LocalizedRegimes S TC sample sc tb (split.n₁ n) (delta n)), 0 ≤ lambda → ∀ {ζ : ℝ}, 0 < ζ → ζ < 1 → ∃ Aζ_reg, MeasurableSet Aζ_reg ∧ μ Aζ_reg ≥ 1 - ENNReal.ofReal ζ ∧ ∀ ω ∈ Aζ_reg, ∀ (n : ℕ), 1 ≤ split.n₁ n → |lambda * ((↑(split.n₁ n))⁻¹ * ∑ k, tb.h_lambda_star_fun (S.xOf (sample.Z (↑k) ω)) ^ 2 - (↑(split.n₁ n))⁻¹ * ∑ k, h_hat n ω (S.xOf (sample.Z (↑k) ω)) ^ 2 - (S.strongNorm (S.hL2 tb.h_lambda_star_mem) ^ 2 - S.strongNorm (S.hL2 (TC.H_subset (is_estimator.mem_H n ω))) ^ 2))| ≤ lambda * (4 * ((regimes n).H_diameter + delta n) * Causalean.Stat.Concentration.criticalRadius ((regimes n).bundle_H.regime.ψ (split.n₁ n)) + (regimes n).bundle_H.regime.b * √(2 * Real.log (2 ^ (n + 1) / ζ) / ↑(split.n₁ n)))","l":55,"f":"Causalean/Estimation/NPIV/Primal/EmpiricalProcessEvent/Regulariser.lean"},{"n":"Causalean.Estimation.NPIV.Primal.combined_metric_rate","k":"theorem","m":"Causalean.Estimation.NPIV.Primal.Rate","a":"Estimation/NPIV/Primal","t":2,"d":"Combined-metric rate (helper, deterministic).","s":"∀ {Ω : Type u_1} [inst : MeasurableSpace Ω] {μ : MeasureTheory.Measure Ω} {S : Causalean.Estimation.NPIV.OperatorSystem Ω μ} {TC : Causalean.Estimation.NPIV.Primal.TRAEClasses S} {P_W : MeasureTheory.Measure S.𝒲} {sample : Causalean.Stat.IIDSample Ω S.𝒲 μ P_W} {split : Causalean.Stat.OneShotSplit sample} {lambda β ζ : ℝ} {delta : ℕ → ℝ} {h_hat : ℕ → Ω → S.𝒳 → ℝ} {is_estimator : Causalean.Estimation.NPIV.Primal.IsTRAEPrimalEstimator S TC sample split lambda h_hat} (hyps : Causalean.Estimation.NPIV.Primal.TRAERatePrimalAbstractHyps S TC sample split lambda β ζ delta h_hat is_estimator), 0 < lambda → lambda < 2 → 0 < ζ → ζ < 1 → ∃ Aζ K, MeasurableSet Aζ ∧ μ Aζ ≥ 1 - ENNReal.ofReal ζ ∧ 0 ≤ K ∧ ∀ ω ∈ Aζ, ∀ (n : ℕ), lambda * S.strongNorm (S.hL2 (TC.H_subset (is_estimator.mem_H n ω)) - S.hL2 hyps.tikhonov_bias.h_lambda_star_mem) ^ 2 + S.weakNorm (S.hL2 (TC.H_subset (is_estimator.mem_H n ω)) - S.hL2 hyps.tikhonov_bias.h_lambda_star_mem) ^ 2 ≤ K * (S.weakNorm (S.hL2 hyps.tikhonov_bias.h_lambda_star_mem - S.hL2 S.h₀_mem) ^ 2 + delta n ^ 2)","l":290,"f":"Causalean/Estimation/NPIV/Primal/Rate.lean"},{"n":"Causalean.Estimation.NPIV.Primal.empirical_critic_argmax_localized","k":"theorem","m":"Causalean.Estimation.NPIV.Primal.EmpiricalProcessEvent.EPMasterEvent","a":"Estimation/NPIV/Primal/EmpiricalProcessEvent","t":2,"d":"Deterministic argmax-localization bridge. If f_emp empirically beats the closedness witness for the same candidate h, and the population-vs-empirical loss difference is controlled by R, then the empirical critic is within squared L² distance R of the closedness witness.","s":"∀ {Ω : Type u_1} [inst : MeasurableSpace Ω] {μ : MeasureTheory.Measure Ω} {S : Causalean.Estimation.NPIV.OperatorSystem Ω μ} {TC : Causalean.Estimation.NPIV.Primal.TRAEClasses S} {P_W : MeasureTheory.Measure S.𝒲} {sample : Causalean.Stat.IIDSample Ω S.𝒲 μ P_W} {split : Causalean.Stat.OneShotSplit sample} [MeasureTheory.IsProbabilityMeasure μ] {lambda : ℝ} {h : S.𝒳 → ℝ} (hh : h ∈ TC.H) {f_closed : S.𝒵 → ℝ} (hf_closed : f_closed ∈ TC.F) {f_emp : S.𝒵 → ℝ} (hf_emp : f_emp ∈ TC.F) {n : ℕ} {ω : Ω} {R : ℝ}, S.T (S.hL2 S.h₀_mem - S.hL2 (TC.H_subset hh)) = S.qL2 (TC.F_subset hf_closed) → Causalean.Estimation.NPIV.Primal.innerObjective S sample split lambda h f_closed n ω ≤ Causalean.Estimation.NPIV.Primal.innerObjective S sample split lambda h f_emp n ω → 2 * ∫ (ω' : Ω), S.m (S.W ω') f_closed ∂μ - 2 * ∫ (ω' : Ω), h (S.xOf (S.W ω')) * f_closed (S.zOf (S.W ω')) ∂μ - ∫ (ω' : Ω), f_closed (S.zOf (S.W ω')) ^ 2 ∂μ - (2 * ∫ (ω' : Ω), S.m (S.W ω') f_emp ∂μ - 2 * ∫ (ω' : Ω), h (S.xOf (S.W ω')) * f_emp (S.zOf (S.W ω')) ∂μ - ∫ (ω' : Ω), f_emp (S.zOf (S.W ω')) ^ 2 ∂μ) ≤ Causalean.Estimation.NPIV.Primal.innerObjective S sample split lambda h f_closed n ω - Causalean.Estimation.NPIV.Primal.innerObjective S sample split lambda h f_emp n ω + R → ‖S.qL2 (TC.F_subset hf_closed) - S.qL2 (TC.F_subset hf_emp)‖ ^ 2 ≤ R","l":379,"f":"Causalean/Estimation/NPIV/Primal/EmpiricalProcessEvent/EPMasterEvent.lean"},{"n":"Causalean.Estimation.NPIV.Primal.empirical_critic_argmax_localized_from_components","k":"theorem","m":"Causalean.Estimation.NPIV.Primal.EmpiricalProcessEvent.EPMasterEvent","a":"Estimation/NPIV/Primal/EmpiricalProcessEvent","t":2,"d":"Componentwise version of empirical_critic_argmax_localized.","s":"∀ {Ω : Type u_1} [inst : MeasurableSpace Ω] {μ : MeasureTheory.Measure Ω} {S : Causalean.Estimation.NPIV.OperatorSystem Ω μ} {TC : Causalean.Estimation.NPIV.Primal.TRAEClasses S} {P_W : MeasureTheory.Measure S.𝒲} {sample : Causalean.Stat.IIDSample Ω S.𝒲 μ P_W} {split : Causalean.Stat.OneShotSplit sample} [MeasureTheory.IsProbabilityMeasure μ] {lambda : ℝ} {h : S.𝒳 → ℝ} (hh : h ∈ TC.H) {f_closed : S.𝒵 → ℝ} (hf_closed : f_closed ∈ TC.F) {f_emp : S.𝒵 → ℝ} (hf_emp : f_emp ∈ TC.F) {n : ℕ} {ω : Ω} {Rm_closed RHF_closed RF_closed Rm_emp RHF_emp RF_emp : ℝ}, S.T (S.hL2 S.h₀_mem - S.hL2 (TC.H_subset hh)) = S.qL2 (TC.F_subset hf_closed) → Causalean.Estimation.NPIV.Primal.innerObjective S sample split lambda h f_closed n ω ≤ Causalean.Estimation.NPIV.Primal.innerObjective S sample split lambda h f_emp n ω → |(↑(split.n₁ n))⁻¹ * ∑ k, S.m (sample.Z (↑k) ω) f_closed - ∫ (ω' : Ω), S.m (S.W ω') f_closed ∂μ| ≤ Rm_closed → |(↑(split.n₁ n))⁻¹ * ∑ k, h (S.xOf (sample.Z (↑k) ω)) * f_closed (S.zOf (sample.Z (↑k) ω)) - ∫ (ω' : Ω), h (S.xOf (S.W ω')) * f_closed (S.zOf (S.W ω')) ∂μ| ≤ RHF_closed → |(↑(split.n₁ n))⁻¹ * ∑ k, f_closed (S.zOf (sample.Z (↑k) ω)) ^ 2 - ∫ (ω' : Ω), f_closed (S.zOf (S.W ω')) ^ 2 ∂μ| ≤ RF_closed → |(↑(split.n₁ n))⁻¹ * ∑ k, S.m (sample.Z (↑k) ω) f_emp - ∫ (ω' : Ω), S.m (S.W ω') f_emp ∂μ| ≤ Rm_emp → |(↑(split.n₁ n))⁻¹ * ∑ k, h (S.xOf (sample.Z (↑k) ω)) * f_emp (S.zOf (sample.Z (↑k) ω)) - ∫ (ω' : Ω), h (S.xOf (S.W ω')) * f_emp (S.zOf (S.W ω')) ∂μ| ≤ RHF_emp → |(↑(split.n₁ n))⁻¹ * ∑ k, f_emp (S.zOf (sample.Z (↑k) ω)) ^ 2 - ∫ (ω' : Ω), f_emp (S.zOf (S.W ω')) ^ 2 ∂μ| ≤ RF_emp → ‖S.qL2 (TC.F_subset hf_closed) - S.qL2 (TC.F_subset hf_emp)‖ ^ 2 ≤ 2 * Rm_closed + 2 * RHF_closed + RF_closed + (2 * Rm_emp + 2 * RHF_emp + RF_emp)","l":639,"f":"Causalean/Estimation/NPIV/Primal/EmpiricalProcessEvent/EPMasterEvent.lean"},{"n":"Causalean.Estimation.NPIV.Primal.empirical_process_event_from_localized","k":"theorem","m":"Causalean.Estimation.NPIV.Primal.EmpiricalProcessEvent.EventAssembly","a":"Estimation/NPIV/Primal/EmpiricalProcessEvent","t":1,"d":"Discharge of empirical_process_event from localized_uniform_deviation — explicit-rate form. Given a localized-regime bundle for the weak-norm, regularizer, and cross function classes at each fold-A sample size and a nonnegative Tikhonov regularization weight lambda, for every confidence level ζ strictly between 0 and 1 there is an event of probability at least 1 − ζ on which, simultaneously for every n with 1 ≤ split.n₁ n, the weak-norm estimation excess plus lambda times the strong-norm estimation excess is bounded by the sum of the explicit per-n empirical-process rate and the explicit per-n centred-regularizer rate — each an additive combination of critical radii and a √(log/n) deviation term.","s":"∀ {Ω : Type u_1} [inst : MeasurableSpace Ω] {μ : MeasureTheory.Measure Ω} {S : Causalean.Estimation.NPIV.OperatorSystem Ω μ} {TC : Causalean.Estimation.NPIV.Primal.TRAEClasses S} {P_W : MeasureTheory.Measure S.𝒲} {sample : Causalean.Stat.IIDSample Ω S.𝒲 μ P_W} {split : Causalean.Stat.OneShotSplit sample} {lambda β : ℝ} {delta : ℕ → ℝ} {h_hat : ℕ → Ω → S.𝒳 → ℝ} {is_estimator : Causalean.Estimation.NPIV.Primal.IsTRAEPrimalEstimator S TC sample split lambda h_hat} (sc : Causalean.Estimation.NPIV.SourceCondition S β) (tb : Causalean.Estimation.NPIV.TikhonovBiasBound S β lambda sc) [MeasureTheory.IsProbabilityMeasure μ] (regimes : (n : ℕ) → Causalean.Estimation.NPIV.Primal.LocalizedRegimes S TC sample sc tb (split.n₁ n) (delta n)), 0 ≤ lambda → ∀ (ζ : ℝ), 0 < ζ → ζ < 1 → ∃ Aζ, MeasurableSet Aζ ∧ μ Aζ ≥ 1 - ENNReal.ofReal ζ ∧ ∀ ω ∈ Aζ, ∀ (n : ℕ), 1 ≤ split.n₁ n → S.weakNorm (S.hL2 (TC.H_subset (is_estimator.mem_H n ω)) - S.hL2 S.h₀_mem) ^ 2 - S.weakNorm (S.hL2 tb.h_lambda_star_mem - S.hL2 S.h₀_mem) ^ 2 + lambda * (S.strongNorm (S.hL2 (TC.H_subset (is_estimator.mem_H n ω))) ^ 2 - S.strongNorm (S.hL2 tb.h_lambda_star_mem) ^ 2) ≤ 16 * delta n * Causalean.Stat.Concentration.criticalRadius ((regimes n).bundle_HF.regime.ψ (split.n₁ n)) + 16 * delta n * Causalean.Stat.Concentration.criticalRadius ((regimes n).bundle_mF.regime.ψ (split.n₁ n)) + 8 * delta n * Causalean.Stat.Concentration.criticalRadius ((regimes n).bundle_F.regime.ψ (split.n₁ n)) + (4 * (regimes n).bundle_HF.regime.b + 4 * (regimes n).bundle_mF.regime.b + 2 * (regimes n).bundle_F.regime.b) * √(2 * Real.log (8 * 2 ^ (n + 1) / ζ) / ↑(split.n₁ n)) + lambda * (4 * ((regimes n).H_diameter + delta n) * Causalean.Stat.Concentration.criticalRadius ((regimes n).bundle_H.regime.ψ (split.n₁ n)) + (regimes n).bundle_H.regime.b * √(2 * Real.log (2 ^ (n + 2) / ζ) / ↑(split.n₁ n)))","l":83,"f":"Causalean/Estimation/NPIV/Primal/EmpiricalProcessEvent/EventAssembly.lean"},{"n":"Causalean.Estimation.NPIV.Primal.empirical_process_event_of_absorption","k":"theorem","m":"Causalean.Estimation.NPIV.Primal.EmpiricalProcessEvent.EventAssembly","a":"Estimation/NPIV/Primal/EmpiricalProcessEvent","t":1,"d":"Empirical-process event in the Rate.lean shape, conditional on absorption. Given a localized-regime bundle for the weak-norm, regularizer, and cross classes at each fold-A sample size, a nonnegative regularization weight lambda, an absorption hypothesis providing, for every confidence level ζ in (0, 1), a nonnegative constant K_ep bounding the explicit per-n empirical-process rate by K_ep times the population shape R² + δ_n·y + δ_n² + λ·δ_n·x + λ·δ_n², for every ω and every n with 1 ≤ split.n₁ n, and a small-n slack hypothesis extending the same K_ep-domination, for every set Aζ and every nonnegative K_ep, to every n with split.n₁ n = 0 and every ω ∈ Aζ, then for every confidence level ζ in (0, 1) there is an event of probability at least 1 − ζ and a nonnegative constant K_ep such that, for every ω in the event and every n, the weak-norm estimation excess plus lambda times the strong-norm estimation excess is bounded by K_ep times the population shape.","s":"∀ {Ω : Type u_1} [inst : MeasurableSpace Ω] {μ : MeasureTheory.Measure Ω} {S : Causalean.Estimation.NPIV.OperatorSystem Ω μ} {TC : Causalean.Estimation.NPIV.Primal.TRAEClasses S} {P_W : MeasureTheory.Measure S.𝒲} {sample : Causalean.Stat.IIDSample Ω S.𝒲 μ P_W} {split : Causalean.Stat.OneShotSplit sample} {lambda β : ℝ} {delta : ℕ → ℝ} {h_hat : ℕ → Ω → S.𝒳 → ℝ} {is_estimator : Causalean.Estimation.NPIV.Primal.IsTRAEPrimalEstimator S TC sample split lambda h_hat} (sc : Causalean.Estimation.NPIV.SourceCondition S β) (tb : Causalean.Estimation.NPIV.TikhonovBiasBound S β lambda sc) [MeasureTheory.IsProbabilityMeasure μ] (regimes : (n : ℕ) → Causalean.Estimation.NPIV.Primal.LocalizedRegimes S TC sample sc tb (split.n₁ n) (delta n)), 0 ≤ lambda → (∀ (ζ : ℝ), 0 < ζ → ζ < 1 → ∃ K_ep, 0 ≤ K_ep ∧ ∀ (ω : Ω) (n : ℕ), 1 ≤ split.n₁ n → Causalean.Estimation.NPIV.Primal.explicitRate regimes is_estimator ω n ζ ≤ K_ep * Causalean.Estimation.NPIV.Primal.populationShape is_estimator ω n) → (∀ (ζ : ℝ), 0 < ζ → ζ < 1 → ∀ (Aζ : Set Ω) (K_ep : ℝ), 0 ≤ K_ep → ∀ ω ∈ Aζ, ∀ (n : ℕ), ¬1 ≤ split.n₁ n → S.weakNorm (S.hL2 (TC.H_subset (is_estimator.mem_H n ω)) - S.hL2 S.h₀_mem) ^ 2 - S.weakNorm (S.hL2 tb.h_lambda_star_mem - S.hL2 S.h₀_mem) ^ 2 + lambda * (S.strongNorm (S.hL2 (TC.H_subset (is_estimator.mem_H n ω))) ^ 2 - S.strongNorm (S.hL2 tb.h_lambda_star_mem) ^ 2) ≤ K_ep * Causalean.Estimation.NPIV.Primal.populationShape is_estimator ω n) → ∀ (ζ : ℝ), 0 < ζ → ζ < 1 → ∃ Aζ K_ep, MeasurableSet Aζ ∧ μ Aζ ≥ 1 - ENNReal.ofReal ζ ∧ 0 ≤ K_ep ∧ ∀ ω ∈ Aζ, ∀ (n : ℕ), S.weakNorm (S.hL2 (TC.H_subset (is_estimator.mem_H n ω)) - S.hL2 S.h₀_mem) ^ 2 - S.weakNorm (S.hL2 tb.h_lambda_star_mem - S.hL2 S.h₀_mem) ^ 2 + lambda * (S.strongNorm (S.hL2 (TC.H_subset (is_estimator.mem_H n ω))) ^ 2 - S.strongNorm (S.hL2 tb.h_lambda_star_mem) ^ 2) ≤ K_ep * Causalean.Estimation.NPIV.Primal.populationShape is_estimator ω n","l":253,"f":"Causalean/Estimation/NPIV/Primal/EmpiricalProcessEvent/EventAssembly.lean"},{"n":"Causalean.Estimation.NPIV.Primal.ep_inequality_from_localized","k":"theorem","m":"Causalean.Estimation.NPIV.Primal.EmpiricalProcessEvent.EPPerN","a":"Estimation/NPIV/Primal/EmpiricalProcessEvent","t":1,"d":"Localized empirical-process inequality for the primal NPIV estimator. Given, for every n, a localized-regime bundle for the weak-norm, regularizer, and cross function classes at that fold-A sample size, and a confidence level ζ strictly between 0 and 1, there is an event of probability at least 1 − ζ on which, simultaneously for every n with 1 ≤ split.n₁ n, the weak-norm estimation excess ‖T(ĥ_n − h_0)‖² − ‖T(h*_λ − h_0)‖² is bounded by the empirical regularizer gap λ · (‖h*‖²_{A(n)} − ‖ĥ‖²_{A(n)}) plus a localized envelope built from the per-n critical radii scaled by δ_n and a √(log(1/ζ)/n) deviation term, where ‖h‖²_{A(n)} := (split.n₁ n)⁻¹ ∑_{k < split.n₁ n} h(X_k)² is the fold-A empirical second moment.","s":"∀ {Ω : Type u_1} [inst : MeasurableSpace Ω] {μ : MeasureTheory.Measure Ω} {S : Causalean.Estimation.NPIV.OperatorSystem Ω μ} {TC : Causalean.Estimation.NPIV.Primal.TRAEClasses S} {P_W : MeasureTheory.Measure S.𝒲} {sample : Causalean.Stat.IIDSample Ω S.𝒲 μ P_W} {split : Causalean.Stat.OneShotSplit sample} {lambda β : ℝ} {delta : ℕ → ℝ} {h_hat : ℕ → Ω → S.𝒳 → ℝ} {is_estimator : Causalean.Estimation.NPIV.Primal.IsTRAEPrimalEstimator S TC sample split lambda h_hat} (sc : Causalean.Estimation.NPIV.SourceCondition S β) (tb : Causalean.Estimation.NPIV.TikhonovBiasBound S β lambda sc) [MeasureTheory.IsProbabilityMeasure μ] (regimes : (n : ℕ) → Causalean.Estimation.NPIV.Primal.LocalizedRegimes S TC sample sc tb (split.n₁ n) (delta n)) {ζ : ℝ}, 0 < ζ → ζ < 1 → ∃ Aζ_ep, MeasurableSet Aζ_ep ∧ μ Aζ_ep ≥ 1 - ENNReal.ofReal ζ ∧ ∀ ω ∈ Aζ_ep, ∀ (n : ℕ), 1 ≤ split.n₁ n → S.weakNorm (S.hL2 (TC.H_subset (is_estimator.mem_H n ω)) - S.hL2 S.h₀_mem) ^ 2 - S.weakNorm (S.hL2 tb.h_lambda_star_mem - S.hL2 S.h₀_mem) ^ 2 ≤ lambda * ((↑(split.n₁ n))⁻¹ * ∑ k, tb.h_lambda_star_fun (S.xOf (sample.Z (↑k) ω)) ^ 2 - (↑(split.n₁ n))⁻¹ * ∑ k, h_hat n ω (S.xOf (sample.Z (↑k) ω)) ^ 2) + (16 * delta n * Causalean.Stat.Concentration.criticalRadius ((regimes n).bundle_HF.regime.ψ (split.n₁ n)) + 16 * delta n * Causalean.Stat.Concentration.criticalRadius ((regimes n).bundle_mF.regime.ψ (split.n₁ n)) + 8 * delta n * Causalean.Stat.Concentration.criticalRadius ((regimes n).bundle_F.regime.ψ (split.n₁ n)) + (4 * (regimes n).bundle_HF.regime.b + 4 * (regimes n).bundle_mF.regime.b + 2 * (regimes n).bundle_F.regime.b) * √(2 * Real.log (4 * 2 ^ (n + 1) / ζ) / ↑(split.n₁ n)))","l":227,"f":"Causalean/Estimation/NPIV/Primal/EmpiricalProcessEvent/EPPerN.lean"},{"n":"Causalean.Estimation.NPIV.Primal.ep_master_event_from_localized","k":"theorem","m":"Causalean.Estimation.NPIV.Primal.EmpiricalProcessEvent.EPMasterEvent","a":"Estimation/NPIV/Primal/EmpiricalProcessEvent","t":1,"d":"Master localized empirical-process event for the primal NPIV analysis. Given, for every sample size n, a localized-regime bundle for the weak-norm, regularizer, and cross function classes at that fold-A sample size, and a confidence level ζ strictly between 0 and 1, there is a single event of probability at least 1 − ζ on which, simultaneously for every n with 1 ≤ split.n₁ n, the population weak-objective excess plus the empirical regularizer excess is bounded by the empirical sup-objective excess plus a localized envelope built from the regimes' critical radii and a √(log(1/ζ)/n) term.","s":"∀ {Ω : Type u_1} [inst : MeasurableSpace Ω] {μ : MeasureTheory.Measure Ω} {S : Causalean.Estimation.NPIV.OperatorSystem Ω μ} {TC : Causalean.Estimation.NPIV.Primal.TRAEClasses S} {P_W : MeasureTheory.Measure S.𝒲} {sample : Causalean.Stat.IIDSample Ω S.𝒲 μ P_W} {split : Causalean.Stat.OneShotSplit sample} {lambda β : ℝ} {delta : ℕ → ℝ} {h_hat : ℕ → Ω → S.𝒳 → ℝ} (is_estimator : Causalean.Estimation.NPIV.Primal.IsTRAEPrimalEstimator S TC sample split lambda h_hat) (sc : Causalean.Estimation.NPIV.SourceCondition S β) (tb : Causalean.Estimation.NPIV.TikhonovBiasBound S β lambda sc) [MeasureTheory.IsProbabilityMeasure μ] (regimes : (n : ℕ) → Causalean.Estimation.NPIV.Primal.LocalizedRegimes S TC sample sc tb (split.n₁ n) (delta n)) {ζ : ℝ}, 0 < ζ → ζ < 1 → ∃ Aζ_master, MeasurableSet Aζ_master ∧ μ Aζ_master ≥ 1 - ENNReal.ofReal ζ ∧ ∀ ω ∈ Aζ_master, ∀ (n : ℕ), 1 ≤ split.n₁ n → S.weakNorm (S.hL2 (TC.H_subset (is_estimator.mem_H n ω)) - S.hL2 S.h₀_mem) ^ 2 + lambda * ((↑(split.n₁ n))⁻¹ * ∑ k, h_hat n ω (S.xOf (sample.Z (↑k) ω)) ^ 2) - (S.weakNorm (S.hL2 tb.h_lambda_star_mem - S.hL2 S.h₀_mem) ^ 2 + lambda * ((↑(split.n₁ n))⁻¹ * ∑ k, tb.h_lambda_star_fun (S.xOf (sample.Z (↑k) ω)) ^ 2)) ≤ Causalean.Estimation.NPIV.Primal.supObjective S TC sample split lambda (h_hat n ω) n ω - Causalean.Estimation.NPIV.Primal.supObjective S TC sample split lambda tb.h_lambda_star_fun n ω + (16 * delta n * Causalean.Stat.Concentration.criticalRadius ((regimes n).bundle_HF.regime.ψ (split.n₁ n)) + 16 * delta n * Causalean.Stat.Concentration.criticalRadius ((regimes n).bundle_mF.regime.ψ (split.n₁ n)) + 8 * delta n * Causalean.Stat.Concentration.criticalRadius ((regimes n).bundle_F.regime.ψ (split.n₁ n)) + (4 * (regimes n).bundle_HF.regime.b + 4 * (regimes n).bundle_mF.regime.b + 2 * (regimes n).bundle_F.regime.b) * √(2 * Real.log (4 * 2 ^ (n + 1) / ζ) / ↑(split.n₁ n)))","l":733,"f":"Causalean/Estimation/NPIV/Primal/EmpiricalProcessEvent/EPMasterEvent.lean"},{"n":"Causalean.Estimation.NPIV.Primal.ep_per_n_inequality_from_deviations","k":"theorem","m":"Causalean.Estimation.NPIV.Primal.EmpiricalProcessEvent.EPPerN","a":"Estimation/NPIV/Primal/EmpiricalProcessEvent","t":1,"d":"EP per-n inequality from an objective-level localized modulus (Helper B). Pure analytic per-n step, with no probability statement. Given a sample size n with 1 ≤ split.n₁ n, a localized-regime bundle at that fold-A sample size, a confidence level ζ strictly between 0 and 1, and the pointwise objective-level inequality, at sample point ω, bounding the population weak-objective-plus-regularizer excess by the empirical sup-objective excess plus the localized envelope, the weak-norm estimation excess is bounded by the empirical regularizer gap plus the same localized envelope, the empirical sup-objective excess having been eliminated using the optimality of ĥ_n (is_estimator.opt) against tb.h_lambda_star_fun. This statement is intentionally not vacuous: the hypothesis still contains the empirical sup-objective excess, which the proof cancels rather than assumes away.","s":"∀ {Ω : Type u_1} [inst : MeasurableSpace Ω] {μ : MeasureTheory.Measure Ω} {S : Causalean.Estimation.NPIV.OperatorSystem Ω μ} {TC : Causalean.Estimation.NPIV.Primal.TRAEClasses S} {P_W : MeasureTheory.Measure S.𝒲} {sample : Causalean.Stat.IIDSample Ω S.𝒲 μ P_W} {split : Causalean.Stat.OneShotSplit sample} {lambda β : ℝ} {delta : ℕ → ℝ} {h_hat : ℕ → Ω → S.𝒳 → ℝ} (is_estimator : Causalean.Estimation.NPIV.Primal.IsTRAEPrimalEstimator S TC sample split lambda h_hat) (sc : Causalean.Estimation.NPIV.SourceCondition S β) (tb : Causalean.Estimation.NPIV.TikhonovBiasBound S β lambda sc) [MeasureTheory.IsProbabilityMeasure μ] (regimes : (n : ℕ) → Causalean.Estimation.NPIV.Primal.LocalizedRegimes S TC sample sc tb (split.n₁ n) (delta n)) {ζ : ℝ}, 0 < ζ → ζ < 1 → ∀ (n : ℕ), 1 ≤ split.n₁ n → ∀ (ω : Ω), S.weakNorm (S.hL2 (TC.H_subset (is_estimator.mem_H n ω)) - S.hL2 S.h₀_mem) ^ 2 + lambda * ((↑(split.n₁ n))⁻¹ * ∑ k, h_hat n ω (S.xOf (sample.Z (↑k) ω)) ^ 2) - (S.weakNorm (S.hL2 tb.h_lambda_star_mem - S.hL2 S.h₀_mem) ^ 2 + lambda * ((↑(split.n₁ n))⁻¹ * ∑ k, tb.h_lambda_star_fun (S.xOf (sample.Z (↑k) ω)) ^ 2)) ≤ Causalean.Estimation.NPIV.Primal.supObjective S TC sample split lambda (h_hat n ω) n ω - Causalean.Estimation.NPIV.Primal.supObjective S TC sample split lambda tb.h_lambda_star_fun n ω + (16 * delta n * Causalean.Stat.Concentration.criticalRadius ((regimes n).bundle_HF.regime.ψ (split.n₁ n)) + 16 * delta n * Causalean.Stat.Concentration.criticalRadius ((regimes n).bundle_mF.regime.ψ (split.n₁ n)) + 8 * delta n * Causalean.Stat.Concentration.criticalRadius ((regimes n).bundle_F.regime.ψ (split.n₁ n)) + (4 * (regimes n).bundle_HF.regime.b + 4 * (regimes n).bundle_mF.regime.b + 2 * (regimes n).bundle_F.regime.b) * √(2 * Real.log (4 * 2 ^ (n + 1) / ζ) / ↑(split.n₁ n))) → S.weakNorm (S.hL2 (TC.H_subset (is_estimator.mem_H n ω)) - S.hL2 S.h₀_mem) ^ 2 - S.weakNorm (S.hL2 tb.h_lambda_star_mem - S.hL2 S.h₀_mem) ^ 2 ≤ lambda * ((↑(split.n₁ n))⁻¹ * ∑ k, tb.h_lambda_star_fun (S.xOf (sample.Z (↑k) ω)) ^ 2 - (↑(split.n₁ n))⁻¹ * ∑ k, h_hat n ω (S.xOf (sample.Z (↑k) ω)) ^ 2) + (16 * delta n * Causalean.Stat.Concentration.criticalRadius ((regimes n).bundle_HF.regime.ψ (split.n₁ n)) + 16 * delta n * Causalean.Stat.Concentration.criticalRadius ((regimes n).bundle_mF.regime.ψ (split.n₁ n)) + 8 * delta n * Causalean.Stat.Concentration.criticalRadius ((regimes n).bundle_F.regime.ψ (split.n₁ n)) + (4 * (regimes n).bundle_HF.regime.b + 4 * (regimes n).bundle_mF.regime.b + 2 * (regimes n).bundle_F.regime.b) * √(2 * Real.log (4 * 2 ^ (n + 1) / ζ) / ↑(split.n₁ n)))","l":146,"f":"Causalean/Estimation/NPIV/Primal/EmpiricalProcessEvent/EPPerN.lean"},{"n":"Causalean.Estimation.NPIV.Primal.ep_pop_inner_at_closedness_witness","k":"theorem","m":"Causalean.Estimation.NPIV.Primal.EmpiricalProcessEvent.EPPerN","a":"Estimation/NPIV/Primal/EmpiricalProcessEvent","t":2,"d":"EP closedness-witness population identity (Helper B1).","s":"∀ {Ω : Type u_1} [inst : MeasurableSpace Ω] {μ : MeasureTheory.Measure Ω} {S : Causalean.Estimation.NPIV.OperatorSystem Ω μ} {TC : Causalean.Estimation.NPIV.Primal.TRAEClasses S} [MeasureTheory.IsProbabilityMeasure μ] {h : S.𝒳 → ℝ} (hh : h ∈ TC.H) {f : S.𝒵 → ℝ} (hf : f ∈ TC.F), S.T (S.hL2 S.h₀_mem - S.hL2 (TC.H_subset hh)) = S.qL2 (TC.F_subset hf) → 2 * ∫ (ω : Ω), S.m (S.W ω) f ∂μ - 2 * ∫ (ω : Ω), h (S.xOf (S.W ω)) * f (S.zOf (S.W ω)) ∂μ - ∫ (ω : Ω), f (S.zOf (S.W ω)) ^ 2 ∂μ = S.weakNorm (S.hL2 (TC.H_subset hh) - S.hL2 S.h₀_mem) ^ 2","l":24,"f":"Causalean/Estimation/NPIV/Primal/EmpiricalProcessEvent/EPPerN.lean"},{"n":"Causalean.Estimation.NPIV.Primal.explicitRate","k":"def","m":"Causalean.Estimation.NPIV.Primal.EmpiricalProcessEvent.EventAssembly","a":"Estimation/NPIV/Primal/EmpiricalProcessEvent","t":1,"d":"Per-n explicit rate appearing on the RHS of empirical_process_event_from_localized, packaged as a function of (ω, n, ζ) for use as the LHS of the absorption hypothesis below.","s":"{Ω : Type u_1} → [inst : MeasurableSpace Ω] → {μ : MeasureTheory.Measure Ω} → {S : Causalean.Estimation.NPIV.OperatorSystem Ω μ} → {TC : Causalean.Estimation.NPIV.Primal.TRAEClasses S} → {P_W : MeasureTheory.Measure S.𝒲} → {sample : Causalean.Stat.IIDSample Ω S.𝒲 μ P_W} → {split : Causalean.Stat.OneShotSplit sample} → {lambda β : ℝ} → {delta : ℕ → ℝ} → {h_hat : ℕ → Ω → S.𝒳 → ℝ} → {sc : Causalean.Estimation.NPIV.SourceCondition S β} → {tb : Causalean.Estimation.NPIV.TikhonovBiasBound S β lambda sc} → ((n : ℕ) → Causalean.Estimation.NPIV.Primal.LocalizedRegimes S TC sample sc tb (split.n₁ n) (delta n)) → Causalean.Estimation.NPIV.Primal.IsTRAEPrimalEstimator S TC sample split lambda h_hat → Ω → ℕ → ℝ → ℝ","l":188,"f":"Causalean/Estimation/NPIV/Primal/EmpiricalProcessEvent/EventAssembly.lean"},{"n":"Causalean.Estimation.NPIV.Primal.innerIntegrand","k":"def","m":"Causalean.Estimation.NPIV.Primal.Estimator","a":"Estimation/NPIV/Primal","t":1,"d":"The fold-A empirical pointwise integrand at (h, f) and observation w : 𝒲:","s":"{Ω : Type u_1} → [inst : MeasurableSpace Ω] → {μ : MeasureTheory.Measure Ω} → (S : Causalean.Estimation.NPIV.OperatorSystem Ω μ) → ℝ → (S.𝒳 → ℝ) → (S.𝒵 → ℝ) → S.𝒲 → ℝ","l":103,"f":"Causalean/Estimation/NPIV/Primal/Estimator.lean"},{"n":"Causalean.Estimation.NPIV.Primal.innerObjective","k":"def","m":"Causalean.Estimation.NPIV.Primal.Estimator","a":"Estimation/NPIV/Primal","t":1,"d":"The fold-A empirical inner objective at a candidate h and critic f: the average of innerIntegrand over the nuisance fold A(n).","s":"{Ω : Type u_1} → [inst : MeasurableSpace Ω] → {μ : MeasureTheory.Measure Ω} → (S : Causalean.Estimation.NPIV.OperatorSystem Ω μ) → {P_W : MeasureTheory.Measure S.𝒲} → (sample : Causalean.Stat.IIDSample Ω S.𝒲 μ P_W) → Causalean.Stat.OneShotSplit sample → ℝ → (S.𝒳 → ℝ) → (S.𝒵 → ℝ) → ℕ → Ω → ℝ","l":115,"f":"Causalean/Estimation/NPIV/Primal/Estimator.lean"},{"n":"Causalean.Estimation.NPIV.Primal.instMeasurableSpace_𝒲","k":"def","m":"Causalean.Estimation.NPIV.Primal.Estimator","a":"Estimation/NPIV/Primal","t":1,"d":"The observation space carries the measurable space stored in the inverse problem system.","s":"{Ω : Type u_1} → [inst : MeasurableSpace Ω] → {μ : MeasureTheory.Measure Ω} → (S : Causalean.Estimation.NPIV.InverseProblemSystem Ω μ) → MeasurableSpace S.𝒲","l":60,"f":"Causalean/Estimation/NPIV/Primal/Estimator.lean"},{"n":"Causalean.Estimation.NPIV.Primal.instMeasurableSpace_𝒳","k":"def","m":"Causalean.Estimation.NPIV.Primal.Estimator","a":"Estimation/NPIV/Primal","t":1,"d":"The covariate space carries the measurable space stored in the inverse problem system.","s":"{Ω : Type u_1} → [inst : MeasurableSpace Ω] → {μ : MeasureTheory.Measure Ω} → (S : Causalean.Estimation.NPIV.InverseProblemSystem Ω μ) → MeasurableSpace S.𝒳","l":66,"f":"Causalean/Estimation/NPIV/Primal/Estimator.lean"},{"n":"Causalean.Estimation.NPIV.Primal.instMeasurableSpace_𝒵","k":"def","m":"Causalean.Estimation.NPIV.Primal.Estimator","a":"Estimation/NPIV/Primal","t":1,"d":"The instrument space carries the measurable space stored in the inverse problem system.","s":"{Ω : Type u_1} → [inst : MeasurableSpace Ω] → {μ : MeasureTheory.Measure Ω} → (S : Causalean.Estimation.NPIV.InverseProblemSystem Ω μ) → MeasurableSpace S.𝒵","l":72,"f":"Causalean/Estimation/NPIV/Primal/Estimator.lean"},{"n":"Causalean.Estimation.NPIV.Primal.integral_comp_law_W","k":"theorem","m":"Causalean.Estimation.NPIV.Primal.EmpiricalProcessEvent.LocalizedEventsBase","a":"Estimation/NPIV/Primal/EmpiricalProcessEvent","t":2,"d":"Integrating a measurable function under the law of the observed variable is the same as integrating its pullback over the ambient sample space.","s":"∀ {Ω : Type u_1} [inst : MeasurableSpace Ω] {μ : MeasureTheory.Measure Ω} {S : Causalean.Estimation.NPIV.OperatorSystem Ω μ} {P_W : MeasureTheory.Measure S.𝒲}, MeasureTheory.Measure.map S.W μ = P_W → ∀ {g : S.𝒲 → ℝ}, Measurable g → ∫ (w : S.𝒲), g w ∂P_W = ∫ (ω : Ω), g (S.W ω) ∂μ","l":51,"f":"Causalean/Estimation/NPIV/Primal/EmpiricalProcessEvent/LocalizedEventsBase.lean"},{"n":"Causalean.Estimation.NPIV.Primal.localized_omega_event_for_F","k":"theorem","m":"Causalean.Estimation.NPIV.Primal.EmpiricalProcessEvent.LocalizedEventF","a":"Estimation/NPIV/Primal/EmpiricalProcessEvent","t":2,"d":"Ω-side localized deviation event for the critic class star(F) — controls |(1/n) Σ f(Z_i)² − E[f(Z)²]| uniformly over f ∈ TC.F.","s":"∀ {Ω : Type u_1} [inst : MeasurableSpace Ω] {μ : MeasureTheory.Measure Ω} {S : Causalean.Estimation.NPIV.OperatorSystem Ω μ} {TC : Causalean.Estimation.NPIV.Primal.TRAEClasses S} {P_W : MeasureTheory.Measure S.𝒲} {sample : Causalean.Stat.IIDSample Ω S.𝒲 μ P_W} [MeasureTheory.IsProbabilityMeasure μ] {β lambda : ℝ} {sc : Causalean.Estimation.NPIV.SourceCondition S β} {tb : Causalean.Estimation.NPIV.TikhonovBiasBound S β lambda sc} {n : ℕ} {δ_n : ℝ} (regime : Causalean.Estimation.NPIV.Primal.LocalizedRegimes S TC sample sc tb n δ_n), 0 < n → ∀ {δ : ℝ}, 0 < δ → δ ≤ 1 → ∃ E, MeasurableSet E ∧ μ E ≥ 1 - ENNReal.ofReal δ ∧ ∀ ω ∈ E, ∀ f ∈ TC.F, |(↑n)⁻¹ * ∑ k, f (S.zOf (sample.Z (↑k) ω)) ^ 2 - ∫ (ω' : Ω), f (S.zOf (S.W ω')) ^ 2 ∂μ| ≤ 4 * δ_n * Causalean.Stat.Concentration.criticalRadius (regime.bundle_F.regime.ψ n) + regime.bundle_F.regime.b * √(2 * Real.log (1 / δ) / ↑n)","l":26,"f":"Causalean/Estimation/NPIV/Primal/EmpiricalProcessEvent/LocalizedEventF.lean"},{"n":"Causalean.Estimation.NPIV.Primal.localized_omega_event_for_F_pair","k":"theorem","m":"Causalean.Estimation.NPIV.Primal.EmpiricalProcessEvent.LocalizedEventF","a":"Estimation/NPIV/Primal/EmpiricalProcessEvent","t":2,"d":"Ω-side fixed-diameter pair-form localized deviation event for the squared critic class star(F). This is the non-peeled building block for the Foster pair-gap bridge.","s":"∀ {Ω : Type u_1} [inst : MeasurableSpace Ω] {μ : MeasureTheory.Measure Ω} {S : Causalean.Estimation.NPIV.OperatorSystem Ω μ} {TC : Causalean.Estimation.NPIV.Primal.TRAEClasses S} {P_W : MeasureTheory.Measure S.𝒲} {sample : Causalean.Stat.IIDSample Ω S.𝒲 μ P_W} [MeasureTheory.IsProbabilityMeasure μ] {β lambda : ℝ} {sc : Causalean.Estimation.NPIV.SourceCondition S β} {tb : Causalean.Estimation.NPIV.TikhonovBiasBound S β lambda sc} {n : ℕ} {δ_n : ℝ} (regime : Causalean.Estimation.NPIV.Primal.LocalizedRegimes S TC sample sc tb n δ_n), 0 < n → ∀ {δ : ℝ}, 0 < δ → δ ≤ 1 → ∃ E, MeasurableSet E ∧ μ E ≥ 1 - ENNReal.ofReal δ ∧ ∀ ω ∈ E, ∀ f₁ ∈ TC.F, ∀ f₂ ∈ TC.F, |(↑n)⁻¹ * ∑ k, (f₁ (S.zOf (sample.Z (↑k) ω)) ^ 2 - f₂ (S.zOf (sample.Z (↑k) ω)) ^ 2) - ∫ (ω' : Ω), f₁ (S.zOf (S.W ω')) ^ 2 - f₂ (S.zOf (S.W ω')) ^ 2 ∂μ| ≤ 4 * (regime.F_L2_const * regime.F_diameter) * Causalean.Stat.Concentration.criticalRadius (regime.bundle_F.regime.ψ n) + regime.bundle_F.regime.b * √(2 * Real.log (1 / δ) / ↑n)","l":88,"f":"Causalean/Estimation/NPIV/Primal/EmpiricalProcessEvent/LocalizedEventF.lean"},{"n":"Causalean.Estimation.NPIV.Primal.localized_omega_event_for_F_pair_peeled","k":"theorem","m":"Causalean.Estimation.NPIV.Primal.EmpiricalProcessEvent.LocalizedEventF","a":"Estimation/NPIV/Primal/EmpiricalProcessEvent","t":1,"d":"Peeled pair-form localized deviation event for the squared critic class star(F). Given a positive sample size n and a confidence level δ in (0, 1], there is a single event of probability at least 1 − δ, valid simultaneously for every pair f₁, f₂ in the critic class TC.F, on which the gap between the empirical and population second-moment differences of f₁ and f₂ is bounded by 8 · F_L2_const · ‖f₁ − f₂‖_strong · δ_n + 5 · δ_n², where the strong norm is taken in the critic's L² embedding and F_L2_const comes from the supplied localized-regime witness.","s":"∀ {Ω : Type u_1} [inst : MeasurableSpace Ω] {μ : MeasureTheory.Measure Ω} {S : Causalean.Estimation.NPIV.OperatorSystem Ω μ} {TC : Causalean.Estimation.NPIV.Primal.TRAEClasses S} {P_W : MeasureTheory.Measure S.𝒲} {sample : Causalean.Stat.IIDSample Ω S.𝒲 μ P_W} [MeasureTheory.IsProbabilityMeasure μ] {β lambda : ℝ} {sc : Causalean.Estimation.NPIV.SourceCondition S β} {tb : Causalean.Estimation.NPIV.TikhonovBiasBound S β lambda sc} {n : ℕ} {δ_n : ℝ} (regime : Causalean.Estimation.NPIV.Primal.LocalizedRegimes S TC sample sc tb n δ_n), 0 < n → ∀ {δ : ℝ}, 0 < δ → δ ≤ 1 → ∃ E, MeasurableSet E ∧ μ E ≥ 1 - ENNReal.ofReal δ ∧ ∀ ω ∈ E, ∀ (f₁ : S.𝒵 → ℝ) (hf₁ : f₁ ∈ TC.F) (f₂ : S.𝒵 → ℝ) (hf₂ : f₂ ∈ TC.F), |(↑n)⁻¹ * ∑ k, (f₁ (S.zOf (sample.Z (↑k) ω)) ^ 2 - f₂ (S.zOf (sample.Z (↑k) ω)) ^ 2) - ∫ (ω' : Ω), f₁ (S.zOf (S.W ω')) ^ 2 - f₂ (S.zOf (S.W ω')) ^ 2 ∂μ| ≤ 8 * regime.F_L2_const * S.strongNorm (S.qL2 (TC.F_subset hf₁) - S.qL2 (TC.F_subset hf₂)) * δ_n + 5 * δ_n ^ 2","l":171,"f":"Causalean/Estimation/NPIV/Primal/EmpiricalProcessEvent/LocalizedEventF.lean"},{"n":"Causalean.Estimation.NPIV.Primal.localized_omega_event_for_H","k":"theorem","m":"Causalean.Estimation.NPIV.Primal.EmpiricalProcessEvent.LocalizedEventH","a":"Estimation/NPIV/Primal/EmpiricalProcessEvent","t":1,"d":"Fixed-diameter localized deviation event for the candidate class star(H). Given a positive sample size n and a confidence level δ in (0, 1], there is a single event of probability at least 1 − δ, valid simultaneously for every pair h₁, h₂ in the primal hypothesis class TC.H, on which the gap between the fold's empirical second-moment difference (1/n) Σ_k h₁(X_k)² − (1/n) Σ_k h₂(X_k)² and its population counterpart E[h₁(X)²] − E[h₂(X)²] is bounded by 4 · (H_diameter + δ_n) · criticalRadius(ψ n) plus a √(2·log(1/δ)/n) deviation term, where H_diameter and the critical-radius regime come from the supplied localized-regime witness.","s":"∀ {Ω : Type u_1} [inst : MeasurableSpace Ω] {μ : MeasureTheory.Measure Ω} {S : Causalean.Estimation.NPIV.OperatorSystem Ω μ} {TC : Causalean.Estimation.NPIV.Primal.TRAEClasses S} {P_W : MeasureTheory.Measure S.𝒲} {sample : Causalean.Stat.IIDSample Ω S.𝒲 μ P_W} [MeasureTheory.IsProbabilityMeasure μ] {β lambda : ℝ} {sc : Causalean.Estimation.NPIV.SourceCondition S β} {tb : Causalean.Estimation.NPIV.TikhonovBiasBound S β lambda sc} {n : ℕ} {δ_n : ℝ} (regime : Causalean.Estimation.NPIV.Primal.LocalizedRegimes S TC sample sc tb n δ_n), 0 < n → ∀ {δ : ℝ}, 0 < δ → δ ≤ 1 → ∃ E, MeasurableSet E ∧ μ E ≥ 1 - ENNReal.ofReal δ ∧ ∀ ω ∈ E, ∀ h₁ ∈ TC.H, ∀ h₂ ∈ TC.H, |(↑n)⁻¹ * ∑ k, h₁ (S.xOf (sample.Z (↑k) ω)) ^ 2 - (↑n)⁻¹ * ∑ k, h₂ (S.xOf (sample.Z (↑k) ω)) ^ 2 - (∫ (ω' : Ω), h₁ (S.xOf (S.W ω')) ^ 2 ∂μ - ∫ (ω' : Ω), h₂ (S.xOf (S.W ω')) ^ 2 ∂μ)| ≤ 4 * (regime.H_diameter + δ_n) * Causalean.Stat.Concentration.criticalRadius (regime.bundle_H.regime.ψ n) + regime.bundle_H.regime.b * √(2 * Real.log (1 / δ) / ↑n)","l":26,"f":"Causalean/Estimation/NPIV/Primal/EmpiricalProcessEvent/LocalizedEventH.lean"},{"n":"Causalean.Estimation.NPIV.Primal.localized_omega_event_for_HF","k":"theorem","m":"Causalean.Estimation.NPIV.Primal.EmpiricalProcessEvent.LocalizedEventHF","a":"Estimation/NPIV/Primal/EmpiricalProcessEvent","t":2,"d":"Ω-side localized deviation event for the product class star(H · F) — controls |(1/n) Σ h(X_i) f(Z_i) − E[h(X) f(Z)]| uniformly over h ∈ TC.H, f ∈ TC.F.","s":"∀ {Ω : Type u_1} [inst : MeasurableSpace Ω] {μ : MeasureTheory.Measure Ω} {S : Causalean.Estimation.NPIV.OperatorSystem Ω μ} {TC : Causalean.Estimation.NPIV.Primal.TRAEClasses S} {P_W : MeasureTheory.Measure S.𝒲} {sample : Causalean.Stat.IIDSample Ω S.𝒲 μ P_W} [MeasureTheory.IsProbabilityMeasure μ] {β lambda : ℝ} {sc : Causalean.Estimation.NPIV.SourceCondition S β} {tb : Causalean.Estimation.NPIV.TikhonovBiasBound S β lambda sc} {n : ℕ} {δ_n : ℝ} (regime : Causalean.Estimation.NPIV.Primal.LocalizedRegimes S TC sample sc tb n δ_n), 0 < n → ∀ {δ : ℝ}, 0 < δ → δ ≤ 1 → ∃ E, MeasurableSet E ∧ μ E ≥ 1 - ENNReal.ofReal δ ∧ ∀ ω ∈ E, ∀ h ∈ TC.H, ∀ f ∈ TC.F, |(↑n)⁻¹ * ∑ k, h (S.xOf (sample.Z (↑k) ω)) * f (S.zOf (sample.Z (↑k) ω)) - ∫ (ω' : Ω), h (S.xOf (S.W ω')) * f (S.zOf (S.W ω')) ∂μ| ≤ 4 * δ_n * Causalean.Stat.Concentration.criticalRadius (regime.bundle_HF.regime.ψ n) + regime.bundle_HF.regime.b * √(2 * Real.log (1 / δ) / ↑n)","l":26,"f":"Causalean/Estimation/NPIV/Primal/EmpiricalProcessEvent/LocalizedEventHF.lean"},{"n":"Causalean.Estimation.NPIV.Primal.localized_omega_event_for_HF_pair","k":"theorem","m":"Causalean.Estimation.NPIV.Primal.EmpiricalProcessEvent.LocalizedEventHF","a":"Estimation/NPIV/Primal/EmpiricalProcessEvent","t":2,"d":"Ω-side fixed-diameter pair-form localized deviation event for the cross class star(H · F) — produces a *single* μ-event simultaneously valid for every triple (h₁, h₂, f) ∈ TC.H × TC.H × TC.F.","s":"∀ {Ω : Type u_1} [inst : MeasurableSpace Ω] {μ : MeasureTheory.Measure Ω} {S : Causalean.Estimation.NPIV.OperatorSystem Ω μ} {TC : Causalean.Estimation.NPIV.Primal.TRAEClasses S} {P_W : MeasureTheory.Measure S.𝒲} {sample : Causalean.Stat.IIDSample Ω S.𝒲 μ P_W} [MeasureTheory.IsProbabilityMeasure μ] {β lambda : ℝ} {sc : Causalean.Estimation.NPIV.SourceCondition S β} {tb : Causalean.Estimation.NPIV.TikhonovBiasBound S β lambda sc} {n : ℕ} {δ_n : ℝ} (regime : Causalean.Estimation.NPIV.Primal.LocalizedRegimes S TC sample sc tb n δ_n), 0 < n → ∀ {δ : ℝ}, 0 < δ → δ ≤ 1 → ∃ E, MeasurableSet E ∧ μ E ≥ 1 - ENNReal.ofReal δ ∧ ∀ ω ∈ E, ∀ h₁ ∈ TC.H, ∀ h₂ ∈ TC.H, ∀ f ∈ TC.F, |(↑n)⁻¹ * ∑ k, (h₁ (S.xOf (sample.Z (↑k) ω)) - h₂ (S.xOf (sample.Z (↑k) ω))) * f (S.zOf (sample.Z (↑k) ω)) - ∫ (ω' : Ω), (h₁ (S.xOf (S.W ω')) - h₂ (S.xOf (S.W ω'))) * f (S.zOf (S.W ω')) ∂μ| ≤ 4 * (regime.HF_pair_const * regime.H_diameter * δ_n + δ_n) * Causalean.Stat.Concentration.criticalRadius (regime.bundle_HF.regime.ψ n) + regime.bundle_HF.regime.b * √(2 * Real.log (1 / δ) / ↑n)","l":91,"f":"Causalean/Estimation/NPIV/Primal/EmpiricalProcessEvent/LocalizedEventHF.lean"},{"n":"Causalean.Estimation.NPIV.Primal.localized_omega_event_for_HF_pair_peeled","k":"theorem","m":"Causalean.Estimation.NPIV.Primal.EmpiricalProcessEvent.LocalizedEventHF","a":"Estimation/NPIV/Primal/EmpiricalProcessEvent","t":1,"d":"Peeled pair-form localized deviation event for the cross class star(H · F). Given a positive sample size n and a confidence level δ in (0, 1], there is a single event of probability at least 1 − δ, valid simultaneously for every pair h₁, h₂ in the primal class TC.H and every critic f in TC.F, on which the gap between the empirical and population means of (h₁ − h₂) · f is bounded by 8 · HF_pair_const · δ_n² · ‖h₁ − h₂‖_strong + 5 · δ_n², where the strong norm is taken in the primal L² embedding and HF_pair_const comes from the supplied localized-regime witness.","s":"∀ {Ω : Type u_1} [inst : MeasurableSpace Ω] {μ : MeasureTheory.Measure Ω} {S : Causalean.Estimation.NPIV.OperatorSystem Ω μ} {TC : Causalean.Estimation.NPIV.Primal.TRAEClasses S} {P_W : MeasureTheory.Measure S.𝒲} {sample : Causalean.Stat.IIDSample Ω S.𝒲 μ P_W} [MeasureTheory.IsProbabilityMeasure μ] {β lambda : ℝ} {sc : Causalean.Estimation.NPIV.SourceCondition S β} {tb : Causalean.Estimation.NPIV.TikhonovBiasBound S β lambda sc} {n : ℕ} {δ_n : ℝ} (regime : Causalean.Estimation.NPIV.Primal.LocalizedRegimes S TC sample sc tb n δ_n), 0 < n → ∀ {δ : ℝ}, 0 < δ → δ ≤ 1 → ∃ E, MeasurableSet E ∧ μ E ≥ 1 - ENNReal.ofReal δ ∧ ∀ ω ∈ E, ∀ (h₁ : S.𝒳 → ℝ) (hh₁ : h₁ ∈ TC.H) (h₂ : S.𝒳 → ℝ) (hh₂ : h₂ ∈ TC.H), ∀ f ∈ TC.F, |(↑n)⁻¹ * ∑ k, (h₁ (S.xOf (sample.Z (↑k) ω)) - h₂ (S.xOf (sample.Z (↑k) ω))) * f (S.zOf (sample.Z (↑k) ω)) - ∫ (ω' : Ω), (h₁ (S.xOf (S.W ω')) - h₂ (S.xOf (S.W ω'))) * f (S.zOf (S.W ω')) ∂μ| ≤ 8 * regime.HF_pair_const * δ_n ^ 2 * S.strongNorm (S.hL2 (TC.H_subset hh₁) - S.hL2 (TC.H_subset hh₂)) + 5 * δ_n ^ 2","l":218,"f":"Causalean/Estimation/NPIV/Primal/EmpiricalProcessEvent/LocalizedEventHF.lean"},{"n":"Causalean.Estimation.NPIV.Primal.localized_omega_event_for_mF","k":"theorem","m":"Causalean.Estimation.NPIV.Primal.EmpiricalProcessEvent.LocalizedEventMF","a":"Estimation/NPIV/Primal/EmpiricalProcessEvent","t":2,"d":"Ω-side localized deviation event for the moment class star(m ∘ F) — controls |(1/n) Σ m(W_i; f) − E[m(W; f)]| uniformly over f ∈ TC.F.","s":"∀ {Ω : Type u_1} [inst : MeasurableSpace Ω] {μ : MeasureTheory.Measure Ω} {S : Causalean.Estimation.NPIV.OperatorSystem Ω μ} {TC : Causalean.Estimation.NPIV.Primal.TRAEClasses S} {P_W : MeasureTheory.Measure S.𝒲} {sample : Causalean.Stat.IIDSample Ω S.𝒲 μ P_W} [MeasureTheory.IsProbabilityMeasure μ] {β lambda : ℝ} {sc : Causalean.Estimation.NPIV.SourceCondition S β} {tb : Causalean.Estimation.NPIV.TikhonovBiasBound S β lambda sc} {n : ℕ} {δ_n : ℝ} (regime : Causalean.Estimation.NPIV.Primal.LocalizedRegimes S TC sample sc tb n δ_n), 0 < n → ∀ {δ : ℝ}, 0 < δ → δ ≤ 1 → ∃ E, MeasurableSet E ∧ μ E ≥ 1 - ENNReal.ofReal δ ∧ ∀ ω ∈ E, ∀ f ∈ TC.F, |(↑n)⁻¹ * ∑ k, S.m (sample.Z (↑k) ω) f - ∫ (ω' : Ω), S.m (S.W ω') f ∂μ| ≤ 4 * δ_n * Causalean.Stat.Concentration.criticalRadius (regime.bundle_mF.regime.ψ n) + regime.bundle_mF.regime.b * √(2 * Real.log (1 / δ) / ↑n)","l":26,"f":"Causalean/Estimation/NPIV/Primal/EmpiricalProcessEvent/LocalizedEventMF.lean"},{"n":"Causalean.Estimation.NPIV.Primal.localized_omega_event_for_mF_pair","k":"theorem","m":"Causalean.Estimation.NPIV.Primal.EmpiricalProcessEvent.LocalizedEventMF","a":"Estimation/NPIV/Primal/EmpiricalProcessEvent","t":2,"d":"Ω-side fixed-diameter pair-form localized deviation event for the moment class star(m ∘ F). This is the non-peeled building block for the Foster pair-gap bridge; the peeled variant can reuse the same interpretation at dyadic radii below mF_L2_const * F_diameter.","s":"∀ {Ω : Type u_1} [inst : MeasurableSpace Ω] {μ : MeasureTheory.Measure Ω} {S : Causalean.Estimation.NPIV.OperatorSystem Ω μ} {TC : Causalean.Estimation.NPIV.Primal.TRAEClasses S} {P_W : MeasureTheory.Measure S.𝒲} {sample : Causalean.Stat.IIDSample Ω S.𝒲 μ P_W} [MeasureTheory.IsProbabilityMeasure μ] {β lambda : ℝ} {sc : Causalean.Estimation.NPIV.SourceCondition S β} {tb : Causalean.Estimation.NPIV.TikhonovBiasBound S β lambda sc} {n : ℕ} {δ_n : ℝ} (regime : Causalean.Estimation.NPIV.Primal.LocalizedRegimes S TC sample sc tb n δ_n), 0 < n → ∀ {δ : ℝ}, 0 < δ → δ ≤ 1 → ∃ E, MeasurableSet E ∧ μ E ≥ 1 - ENNReal.ofReal δ ∧ ∀ ω ∈ E, ∀ f₁ ∈ TC.F, ∀ f₂ ∈ TC.F, |(↑n)⁻¹ * ∑ k, (S.m (sample.Z (↑k) ω) f₁ - S.m (sample.Z (↑k) ω) f₂) - ∫ (ω' : Ω), S.m (S.W ω') f₁ - S.m (S.W ω') f₂ ∂μ| ≤ 4 * (regime.mF_L2_const * regime.F_diameter) * Causalean.Stat.Concentration.criticalRadius (regime.bundle_mF.regime.ψ n) + regime.bundle_mF.regime.b * √(2 * Real.log (1 / δ) / ↑n)","l":89,"f":"Causalean/Estimation/NPIV/Primal/EmpiricalProcessEvent/LocalizedEventMF.lean"},{"n":"Causalean.Estimation.NPIV.Primal.localized_omega_event_for_mF_pair_peeled","k":"theorem","m":"Causalean.Estimation.NPIV.Primal.EmpiricalProcessEvent.LocalizedEventMF","a":"Estimation/NPIV/Primal/EmpiricalProcessEvent","t":1,"d":"Peeled pair-form localized deviation event for the moment class star(m ∘ F). Given a positive sample size n and a confidence level δ in (0, 1], there is a single event of probability at least 1 − δ, valid simultaneously for every pair f₁, f₂ in the critic class TC.F, on which the gap between the empirical and population means of m(·; f₁) − m(·; f₂) is bounded by 8 · mF_L2_const · δ_n · ‖f₁ − f₂‖_strong + 5 · δ_n², where the strong norm is taken in the critic's L² embedding and mF_L2_const comes from the supplied localized-regime witness.","s":"∀ {Ω : Type u_1} [inst : MeasurableSpace Ω] {μ : MeasureTheory.Measure Ω} {S : Causalean.Estimation.NPIV.OperatorSystem Ω μ} {TC : Causalean.Estimation.NPIV.Primal.TRAEClasses S} {P_W : MeasureTheory.Measure S.𝒲} {sample : Causalean.Stat.IIDSample Ω S.𝒲 μ P_W} [MeasureTheory.IsProbabilityMeasure μ] {β lambda : ℝ} {sc : Causalean.Estimation.NPIV.SourceCondition S β} {tb : Causalean.Estimation.NPIV.TikhonovBiasBound S β lambda sc} {n : ℕ} {δ_n : ℝ} (regime : Causalean.Estimation.NPIV.Primal.LocalizedRegimes S TC sample sc tb n δ_n), 0 < n → ∀ {δ : ℝ}, 0 < δ → δ ≤ 1 → ∃ E, MeasurableSet E ∧ μ E ≥ 1 - ENNReal.ofReal δ ∧ ∀ ω ∈ E, ∀ (f₁ : S.𝒵 → ℝ) (_hf₁ : f₁ ∈ TC.F) (f₂ : S.𝒵 → ℝ) (_hf₂ : f₂ ∈ TC.F), |(↑n)⁻¹ * ∑ k, (S.m (sample.Z (↑k) ω) f₁ - S.m (sample.Z (↑k) ω) f₂) - ∫ (ω' : Ω), S.m (S.W ω') f₁ - S.m (S.W ω') f₂ ∂μ| ≤ 8 * regime.mF_L2_const * δ_n * S.strongNorm (S.qL2 (TC.F_subset _hf₁) - S.qL2 (TC.F_subset _hf₂)) + 5 * δ_n ^ 2","l":166,"f":"Causalean/Estimation/NPIV/Primal/EmpiricalProcessEvent/LocalizedEventMF.lean"},{"n":"Causalean.Estimation.NPIV.Primal.localized_omega_event_sharp_for_bundle","k":"theorem","m":"Causalean.Estimation.NPIV.Primal.EmpiricalProcessEvent.LocalizedEventsBase","a":"Estimation/NPIV/Primal/EmpiricalProcessEvent","t":1,"d":"Ω-side Foster–Syrgkanis sharp localized event for a single NPIV LocalizedRegimeBundle. Consider an IID sample drawn from a probability space, where the pushforward of the ambient measure under the observed-variable map equals the stated observation law $P_W$ and the sample size n is positive. Fix a confidence parameter ζ with $0 < ζ ≤ 1$, a localized rate $δ_n$ with $δ_n > 0$, and a radius Rmax with δ_n ≤ Rmax. If for every dyadic level K with Rmax ≤ δ_n · 2 ^ K, the corresponding complexity term stays below δ_n ^ 2, then there is an event of probability at least 1 - ζ on which, for every index in the regime bundle whose function norm is at most Rmax, the empirical average of that function over the first n sample draws deviates from its population mean by at most 8 * δ_n * norm + 5 * δ_n ^ 2.","s":"∀ {Ω : Type u_1} [inst : MeasurableSpace Ω] {μ : MeasureTheory.Measure Ω} {S : Causalean.Estimation.NPIV.OperatorSystem Ω μ} {P_W : MeasureTheory.Measure S.𝒲} {sample : Causalean.Stat.IIDSample Ω S.𝒲 μ P_W} [MeasureTheory.IsProbabilityMeasure μ] {n : ℕ} {δ_n Rmax : ℝ} (B : Causalean.Estimation.NPIV.Primal.LocalizedRegimeBundle S.𝒲 P_W n δ_n), MeasureTheory.Measure.map S.W μ = P_W → 0 < n → ∀ {ζ : ℝ}, 0 < ζ → ζ ≤ 1 → 0 < δ_n → δ_n ≤ Rmax → (∀ (K : ℕ), Rmax ≤ δ_n * 2 ^ K → B.regime.b * √(2 * Real.log (2 * (↑K + 1) / ζ) / ↑n) ≤ δ_n ^ 2) → ∃ E, MeasurableSet E ∧ μ E ≥ 1 - ENNReal.ofReal ζ ∧ ∀ ω ∈ E, ∀ (i : B.ι), B.norm (B.F i) ≤ Rmax → |(↑n)⁻¹ * ∑ k, B.F i (B.X (sample.Z (↑k) ω)) - ∫ (ω' : Ω), B.F i (B.X (S.W ω')) ∂μ| ≤ 8 * δ_n * B.norm (B.F i) + 5 * δ_n ^ 2","l":61,"f":"Causalean/Estimation/NPIV/Primal/EmpiricalProcessEvent/LocalizedEventsBase.lean"},{"n":"Causalean.Estimation.NPIV.Primal.measure_iInter_nat_ge_one_sub_tsum_of_ge","k":"theorem","m":"Causalean.Estimation.NPIV.Primal.EmpiricalProcessEvent.Regulariser","a":"Estimation/NPIV/Primal/EmpiricalProcessEvent","t":2,"d":"If each event in a countable family has probability at least one minus its assigned error, then their intersection has probability at least one minus the sum of those errors.","s":"∀ {Ω : Type u_1} [inst : MeasurableSpace Ω] {μ : MeasureTheory.Measure Ω} [MeasureTheory.IsProbabilityMeasure μ] {E : ℕ → Set Ω} {a : ℕ → ENNReal}, (∀ (n : ℕ), MeasurableSet (E n)) → (∀ (n : ℕ), μ (E n) ≥ 1 - a n) → μ (⋂ n, E n) ≥ 1 - ∑' (n : ℕ), a n","l":27,"f":"Causalean/Estimation/NPIV/Primal/EmpiricalProcessEvent/Regulariser.lean"},{"n":"Causalean.Estimation.NPIV.Primal.measure_inter_ge_one_sub_add_of_ge","k":"theorem","m":"Causalean.Estimation.NPIV.Primal.EmpiricalProcessEvent.EventAssembly","a":"Estimation/NPIV/Primal/EmpiricalProcessEvent","t":2,"d":"The intersection of two high-probability events has probability at least one minus the sum of their two failure probabilities.","s":"∀ {Ω : Type u_1} [inst : MeasurableSpace Ω] {μ : MeasureTheory.Measure Ω} [MeasureTheory.IsProbabilityMeasure μ] {A B : Set Ω} {a b : ENNReal}, MeasurableSet A → MeasurableSet B → μ A ≥ 1 - a → μ B ≥ 1 - b → μ (A ∩ B) ≥ 1 - (a + b)","l":58,"f":"Causalean/Estimation/NPIV/Primal/EmpiricalProcessEvent/EventAssembly.lean"},{"n":"Causalean.Estimation.NPIV.Primal.populationShape","k":"def","m":"Causalean.Estimation.NPIV.Primal.EmpiricalProcessEvent.EventAssembly","a":"Estimation/NPIV/Primal/EmpiricalProcessEvent","t":1,"d":"Population shape appearing on the RHS of TRAERatePrimalAbstractHyps.empirical_process_event (Rate.lean, line 167):","s":"{Ω : Type u_1} → [inst : MeasurableSpace Ω] → {μ : MeasureTheory.Measure Ω} → {S : Causalean.Estimation.NPIV.OperatorSystem Ω μ} → {TC : Causalean.Estimation.NPIV.Primal.TRAEClasses S} → {P_W : MeasureTheory.Measure S.𝒲} → {sample : Causalean.Stat.IIDSample Ω S.𝒲 μ P_W} → {split : Causalean.Stat.OneShotSplit sample} → {lambda β : ℝ} → {delta : ℕ → ℝ} → {h_hat : ℕ → Ω → S.𝒳 → ℝ} → {sc : Causalean.Estimation.NPIV.SourceCondition S β} → {tb : Causalean.Estimation.NPIV.TikhonovBiasBound S β lambda sc} → Causalean.Estimation.NPIV.Primal.IsTRAEPrimalEstimator S TC sample split lambda h_hat → Ω → ℕ → ℝ","l":222,"f":"Causalean/Estimation/NPIV/Primal/EmpiricalProcessEvent/EventAssembly.lean"},{"n":"Causalean.Estimation.NPIV.Primal.population_closedness_critic_gap_eq","k":"theorem","m":"Causalean.Estimation.NPIV.Primal.EmpiricalProcessEvent.EPMasterEvent","a":"Estimation/NPIV/Primal/EmpiricalProcessEvent","t":2,"d":"Population curvature at the closedness critic. For a fixed candidate h, the closedness witness f_closed is the population maximizer of the quadratic adversarial criterion, and the drop at any critic f is exactly the squared L² critic gap.","s":"∀ {Ω : Type u_1} [inst : MeasurableSpace Ω] {μ : MeasureTheory.Measure Ω} {S : Causalean.Estimation.NPIV.OperatorSystem Ω μ} {TC : Causalean.Estimation.NPIV.Primal.TRAEClasses S} [MeasureTheory.IsProbabilityMeasure μ] {h : S.𝒳 → ℝ} (hh : h ∈ TC.H) {f_closed : S.𝒵 → ℝ} (hf_closed : f_closed ∈ TC.F) {f : S.𝒵 → ℝ} (hf : f ∈ TC.F), S.T (S.hL2 S.h₀_mem - S.hL2 (TC.H_subset hh)) = S.qL2 (TC.F_subset hf_closed) → 2 * ∫ (ω : Ω), S.m (S.W ω) f_closed ∂μ - 2 * ∫ (ω : Ω), h (S.xOf (S.W ω)) * f_closed (S.zOf (S.W ω)) ∂μ - ∫ (ω : Ω), f_closed (S.zOf (S.W ω)) ^ 2 ∂μ - (2 * ∫ (ω : Ω), S.m (S.W ω) f ∂μ - 2 * ∫ (ω : Ω), h (S.xOf (S.W ω)) * f (S.zOf (S.W ω)) ∂μ - ∫ (ω : Ω), f (S.zOf (S.W ω)) ^ 2 ∂μ) = ‖S.qL2 (TC.F_subset hf_closed) - S.qL2 (TC.F_subset hf)‖ ^ 2","l":334,"f":"Causalean/Estimation/NPIV/Primal/EmpiricalProcessEvent/EPMasterEvent.lean"},{"n":"Causalean.Estimation.NPIV.Primal.strong_rate_from_combined","k":"theorem","m":"Causalean.Estimation.NPIV.Primal.Rate","a":"Estimation/NPIV/Primal","t":2,"d":"Strong rate (helper, deterministic).","s":"∀ {Ω : Type u_1} [inst : MeasurableSpace Ω] {μ : MeasureTheory.Measure Ω} {S : Causalean.Estimation.NPIV.OperatorSystem Ω μ} {TC : Causalean.Estimation.NPIV.Primal.TRAEClasses S} {P_W : MeasureTheory.Measure S.𝒲} {sample : Causalean.Stat.IIDSample Ω S.𝒲 μ P_W} {split : Causalean.Stat.OneShotSplit sample} {lambda β ζ : ℝ} {delta : ℕ → ℝ} {h_hat : ℕ → Ω → S.𝒳 → ℝ} {is_estimator : Causalean.Estimation.NPIV.Primal.IsTRAEPrimalEstimator S TC sample split lambda h_hat} (hyps : Causalean.Estimation.NPIV.Primal.TRAERatePrimalAbstractHyps S TC sample split lambda β ζ delta h_hat is_estimator), 0 < lambda → lambda < 2 → ∀ {Aζ : Set Ω} {K : ℝ}, 0 ≤ K → (∀ ω ∈ Aζ, ∀ (n : ℕ), lambda * S.strongNorm (S.hL2 (TC.H_subset (is_estimator.mem_H n ω)) - S.hL2 hyps.tikhonov_bias.h_lambda_star_mem) ^ 2 + S.weakNorm (S.hL2 (TC.H_subset (is_estimator.mem_H n ω)) - S.hL2 hyps.tikhonov_bias.h_lambda_star_mem) ^ 2 ≤ K * (S.weakNorm (S.hL2 hyps.tikhonov_bias.h_lambda_star_mem - S.hL2 S.h₀_mem) ^ 2 + delta n ^ 2)) → ∃ C_strong, 0 ≤ C_strong ∧ ∀ ω ∈ Aζ, ∀ (n : ℕ), S.strongNorm (S.hL2 (TC.H_subset (is_estimator.mem_H n ω)) - S.hL2 S.h₀_mem) ^ 2 ≤ C_strong * (delta n ^ 2 / lambda + S.strongNorm (S.hL2 hyps.source_condition.w₀_mem) * lambda ^ min β 1)","l":378,"f":"Causalean/Estimation/NPIV/Primal/Rate.lean"},{"n":"Causalean.Estimation.NPIV.Primal.supObjective","k":"def","m":"Causalean.Estimation.NPIV.Primal.Estimator","a":"Estimation/NPIV/Primal","t":1,"d":"The TRAE primal sup-min objective: sup_{f ∈ TC.F} P_{A(n)} [innerIntegrand λ h f W].","s":"{Ω : Type u_1} → [inst : MeasurableSpace Ω] → {μ : MeasureTheory.Measure Ω} → (S : Causalean.Estimation.NPIV.OperatorSystem Ω μ) → Causalean.Estimation.NPIV.Primal.TRAEClasses S → {P_W : MeasureTheory.Measure S.𝒲} → (sample : Causalean.Stat.IIDSample Ω S.𝒲 μ P_W) → Causalean.Stat.OneShotSplit sample → ℝ → (S.𝒳 → ℝ) → ℕ → Ω → ℝ","l":128,"f":"Causalean/Estimation/NPIV/Primal/Estimator.lean"},{"n":"Causalean.Estimation.NPIV.Primal.trae_primal_rate","k":"theorem","m":"Causalean.Estimation.NPIV.Primal.Rate","a":"Estimation/NPIV/Primal","t":1,"d":"TRAE primal rate from localized regimes. Assume the base deterministic TRAE primal rate hypotheses hold (source condition and Tikhonov bias bound), and that a localized-regimes witness is available at every fold-A sample size, with regularization parameter 0 < lambda < 2 and confidence level 0 < ζ < 1. Suppose further that the localized discharge rate is absorbed by a constant multiple of the population shape at every confidence level and every eligible fold size, and that the same absorption inequality also holds, with an explicit weak/strong-norm correction term, at every fold size too small for the localized regime. Then there exist nonnegative constants C_strong, C_weak and an event of probability at least 1 - ζ on which, for every fold-A sample size n, both the strong-norm rate ‖ĥ_n − h₀‖²_{L²(P_X)} ≤ C_strong · (δ_n²/λ + ‖w₀‖_{L²(P_X)} · λ^min(β,1)) and the weak-norm rate ‖T(ĥ_n − h₀)‖²_{L²(P_Z)} ≤ C_weak · (δ_n² + ‖w₀‖_{L²(P_X)} · λ^min(β+1,2)) hold simultaneously.","s":"∀ {Ω : Type u_1} [inst : MeasurableSpace Ω] {μ : MeasureTheory.Measure Ω} {S : Causalean.Estimation.NPIV.OperatorSystem Ω μ} {TC : Causalean.Estimation.NPIV.Primal.TRAEClasses S} {P_W : MeasureTheory.Measure S.𝒲} {sample : Causalean.Stat.IIDSample Ω S.𝒲 μ P_W} {split : Causalean.Stat.OneShotSplit sample} {lambda β ζ : ℝ} {delta : ℕ → ℝ} {h_hat : ℕ → Ω → S.𝒳 → ℝ} {is_estimator : Causalean.Estimation.NPIV.Primal.IsTRAEPrimalEstimator S TC sample split lambda h_hat} (base : Causalean.Estimation.NPIV.Primal.TRAERatePrimalHyps S TC sample split lambda β ζ delta h_hat is_estimator) [MeasureTheory.IsProbabilityMeasure μ] (regimes : (n : ℕ) → Causalean.Estimation.NPIV.Primal.LocalizedRegimes S TC sample base.source_condition base.tikhonov_bias (split.n₁ n) (delta n)), 0 < lambda → lambda < 2 → 0 < ζ → ζ < 1 → (∀ (ζ' : ℝ), 0 < ζ' → ζ' < 1 → ∃ K_ep, 0 ≤ K_ep ∧ ∀ (ω : Ω) (n : ℕ), 1 ≤ split.n₁ n → Causalean.Estimation.NPIV.Primal.explicitRate regimes is_estimator ω n ζ' ≤ K_ep * Causalean.Estimation.NPIV.Primal.populationShape is_estimator ω n) → (∀ (ζ' : ℝ), 0 < ζ' → ζ' < 1 → ∀ (Aζ : Set Ω) (K_ep : ℝ), 0 ≤ K_ep → ∀ ω ∈ Aζ, ∀ (n : ℕ), ¬1 ≤ split.n₁ n → S.weakNorm (S.hL2 (TC.H_subset (is_estimator.mem_H n ω)) - S.hL2 S.h₀_mem) ^ 2 - S.weakNorm (S.hL2 base.tikhonov_bias.h_lambda_star_mem - S.hL2 S.h₀_mem) ^ 2 + lambda * (S.strongNorm (S.hL2 (TC.H_subset (is_estimator.mem_H n ω))) ^ 2 - S.strongNorm (S.hL2 base.tikhonov_bias.h_lambda_star_mem) ^ 2) ≤ K_ep * Causalean.Estimation.NPIV.Primal.populationShape is_estimator ω n) → ∃ C_strong C_weak Aζ, MeasurableSet Aζ ∧ μ Aζ ≥ 1 - ENNReal.ofReal ζ ∧ 0 ≤ C_strong ∧ 0 ≤ C_weak ∧ ∀ ω ∈ Aζ, ∀ (n : ℕ), S.strongNorm (S.hL2 (TC.H_subset (is_estimator.mem_H n ω)) - S.hL2 S.h₀_mem) ^ 2 ≤ C_strong * (delta n ^ 2 / lambda + S.strongNorm (S.hL2 base.source_condition.w₀_mem) * lambda ^ min β 1) ∧ S.weakNorm (S.hL2 (TC.H_subset (is_estimator.mem_H n ω)) - S.hL2 S.h₀_mem) ^ 2 ≤ C_weak * (delta n ^ 2 + S.strongNorm (S.hL2 base.source_condition.w₀_mem) * lambda ^ min (β + 1) 2)","l":827,"f":"Causalean/Estimation/NPIV/Primal/Rate.lean"},{"n":"Causalean.Estimation.NPIV.Primal.trae_primal_rate_from_empirical_process","k":"theorem","m":"Causalean.Estimation.NPIV.Primal.Rate","a":"Estimation/NPIV/Primal","t":1,"d":"Non-iterated TRAE primal rate theorem (thm:est-trae-rate-theorem, lines 174–248 of trae_inverse_problems.tex). Assume the abstract deterministic TRAE primal rate hypotheses — source condition, Tikhonov bias bound, and localized empirical-process event — hold, the regularization parameter satisfies 0 < lambda < 2, and the confidence level satisfies 0 < ζ < 1. Then there exist nonnegative constants C_strong, C_weak and an event of probability at least 1 - ζ on which, for every fold-A sample size n, both the strong-norm rate ‖ĥ_n − h₀‖²_{L²(P_X)} ≤ C_strong · (δ_n²/λ + ‖w₀‖_{L²(P_X)} · λ^min(β,1)) and the weak-norm rate ‖T(ĥ_n − h₀)‖²_{L²(P_Z)} ≤ C_weak · (δ_n² + ‖w₀‖_{L²(P_X)} · λ^min(β+1,2)) hold simultaneously.","s":"∀ {Ω : Type u_1} [inst : MeasurableSpace Ω] {μ : MeasureTheory.Measure Ω} {S : Causalean.Estimation.NPIV.OperatorSystem Ω μ} {TC : Causalean.Estimation.NPIV.Primal.TRAEClasses S} {P_W : MeasureTheory.Measure S.𝒲} {sample : Causalean.Stat.IIDSample Ω S.𝒲 μ P_W} {split : Causalean.Stat.OneShotSplit sample} {lambda β ζ : ℝ} {delta : ℕ → ℝ} {h_hat : ℕ → Ω → S.𝒳 → ℝ} {is_estimator : Causalean.Estimation.NPIV.Primal.IsTRAEPrimalEstimator S TC sample split lambda h_hat} (hyps : Causalean.Estimation.NPIV.Primal.TRAERatePrimalAbstractHyps S TC sample split lambda β ζ delta h_hat is_estimator), 0 < lambda → lambda < 2 → 0 < ζ → ζ < 1 → ∃ C_strong C_weak Aζ, MeasurableSet Aζ ∧ μ Aζ ≥ 1 - ENNReal.ofReal ζ ∧ 0 ≤ C_strong ∧ 0 ≤ C_weak ∧ ∀ ω ∈ Aζ, ∀ (n : ℕ), S.strongNorm (S.hL2 (TC.H_subset (is_estimator.mem_H n ω)) - S.hL2 S.h₀_mem) ^ 2 ≤ C_strong * (delta n ^ 2 / lambda + S.strongNorm (S.hL2 hyps.source_condition.w₀_mem) * lambda ^ min β 1) ∧ S.weakNorm (S.hL2 (TC.H_subset (is_estimator.mem_H n ω)) - S.hL2 S.h₀_mem) ^ 2 ≤ C_weak * (delta n ^ 2 + S.strongNorm (S.hL2 hyps.source_condition.w₀_mem) * lambda ^ min (β + 1) 2)","l":767,"f":"Causalean/Estimation/NPIV/Primal/Rate.lean"},{"n":"Causalean.Estimation.NPIV.Primal.weak_rate_from_combined","k":"theorem","m":"Causalean.Estimation.NPIV.Primal.Rate","a":"Estimation/NPIV/Primal","t":2,"d":"Weak rate (helper, deterministic).","s":"∀ {Ω : Type u_1} [inst : MeasurableSpace Ω] {μ : MeasureTheory.Measure Ω} {S : Causalean.Estimation.NPIV.OperatorSystem Ω μ} {TC : Causalean.Estimation.NPIV.Primal.TRAEClasses S} {P_W : MeasureTheory.Measure S.𝒲} {sample : Causalean.Stat.IIDSample Ω S.𝒲 μ P_W} {split : Causalean.Stat.OneShotSplit sample} {lambda β ζ : ℝ} {delta : ℕ → ℝ} {h_hat : ℕ → Ω → S.𝒳 → ℝ} {is_estimator : Causalean.Estimation.NPIV.Primal.IsTRAEPrimalEstimator S TC sample split lambda h_hat} (hyps : Causalean.Estimation.NPIV.Primal.TRAERatePrimalAbstractHyps S TC sample split lambda β ζ delta h_hat is_estimator), 0 < lambda → ∀ {Aζ : Set Ω} {K : ℝ}, 0 ≤ K → (∀ ω ∈ Aζ, ∀ (n : ℕ), lambda * S.strongNorm (S.hL2 (TC.H_subset (is_estimator.mem_H n ω)) - S.hL2 hyps.tikhonov_bias.h_lambda_star_mem) ^ 2 + S.weakNorm (S.hL2 (TC.H_subset (is_estimator.mem_H n ω)) - S.hL2 hyps.tikhonov_bias.h_lambda_star_mem) ^ 2 ≤ K * (S.weakNorm (S.hL2 hyps.tikhonov_bias.h_lambda_star_mem - S.hL2 S.h₀_mem) ^ 2 + delta n ^ 2)) → ∃ C_weak, 0 ≤ C_weak ∧ ∀ ω ∈ Aζ, ∀ (n : ℕ), S.weakNorm (S.hL2 (TC.H_subset (is_estimator.mem_H n ω)) - S.hL2 S.h₀_mem) ^ 2 ≤ C_weak * (delta n ^ 2 + S.strongNorm (S.hL2 hyps.source_condition.w₀_mem) * lambda ^ min (β + 1) 2)","l":602,"f":"Causalean/Estimation/NPIV/Primal/Rate.lean"},{"n":"Causalean.Estimation.NPIV.Primal.young_cross_envelope","k":"theorem","m":"Causalean.Estimation.NPIV.Primal.EmpiricalProcessEvent.Algebra","a":"Estimation/NPIV/Primal/EmpiricalProcessEvent","t":1,"d":"Young / AM-GM cross-term envelope. For nonnegative reals R, δ, w, and κ, the mixed sum δ · (R + w + δ) + R · κ is bounded above by the quadratic envelope R² + δ · w + (3/2) · δ² + (1/2) · κ².","s":"∀ (R δ w κ : ℝ), 0 ≤ R → 0 ≤ δ → 0 ≤ w → 0 ≤ κ → δ * (R + w + δ) + R * κ ≤ R ^ 2 + δ * w + 3 / 2 * δ ^ 2 + 1 / 2 * κ ^ 2","l":58,"f":"Causalean/Estimation/NPIV/Primal/EmpiricalProcessEvent/Algebra.lean"},{"n":"Causalean.Estimation.NPIV.SourceCondition","k":"structure","m":"Causalean.Estimation.NPIV.SourceCondition","a":"Estimation/NPIV","t":1,"d":"The β-source condition represents the true primal nuisance as the image, under a nonnegative spectral power of the normal NPIV operator, of an admissible witness function in the primal candidate class, via the spectral identity expressing the nuisance as that power of the operator applied to the witness, inside the L² space.","s":"{Ω : Type u_1} → [inst : MeasurableSpace Ω] → {μ : MeasureTheory.Measure Ω} → Causalean.Estimation.NPIV.OperatorSystem Ω μ → ℝ → Type u_3","l":51,"f":"Causalean/Estimation/NPIV/SourceCondition.lean"},{"n":"Causalean.Estimation.NPIV.SpectralSourceCondition","k":"structure","m":"Causalean.Estimation.NPIV.Operator.SpectralCalculus","a":"Estimation/NPIV/Operator","t":1,"d":"Spectral β-source condition at the primal nuisance h₀: strengthens SourceCondition by additionally requiring that the primal candidate subspace coincides with the full ambient L² space, which is what lets the Lax–Milgram Tikhonov minimiser on the candidate class agree with the resolvent expression needed to run the spectral argument.","s":"{Ω : Type u_1} → [inst : MeasurableSpace Ω] → {μ : MeasureTheory.Measure Ω} → Causalean.Estimation.NPIV.OperatorSystem Ω μ → ℝ → Type u_3","l":98,"f":"Causalean/Estimation/NPIV/Operator/SpectralCalculus.lean"},{"n":"Causalean.Estimation.NPIV.SpectralSourceCondition.Hbar_L2_hasProj","k":"theorem","m":"Causalean.Estimation.NPIV.Operator.SpectralCalculus","a":"Estimation/NPIV/Operator","t":2,"d":"Closedness witness for the full primal candidate subspace.","s":"∀ {Ω : Type u_1} [inst : MeasurableSpace Ω] {μ : MeasureTheory.Measure Ω} {S : Causalean.Estimation.NPIV.OperatorSystem Ω μ} {β : ℝ} (sc : Causalean.Estimation.NPIV.SpectralSourceCondition S β), S.Hbar_L2.HasOrthogonalProjection","l":123,"f":"Causalean/Estimation/NPIV/Operator/SpectralCalculus.lean"},{"n":"Causalean.Estimation.NPIV.SpectralSourceCondition.Tstar_T_isPositive","k":"theorem","m":"Causalean.Estimation.NPIV.Operator.SpectralCalculus","a":"Estimation/NPIV/Operator","t":2,"d":"T†T is a positive operator.","s":"∀ {Ω : Type u_1} [inst : MeasurableSpace Ω] {μ : MeasureTheory.Measure Ω} (S : Causalean.Estimation.NPIV.OperatorSystem Ω μ), S.Tstar_T.IsPositive","l":129,"f":"Causalean/Estimation/NPIV/Operator/SpectralCalculus.lean"},{"n":"Causalean.Estimation.NPIV.SpectralSourceCondition.Tstar_T_spectrum_nonneg","k":"theorem","m":"Causalean.Estimation.NPIV.Operator.SpectralCalculus","a":"Estimation/NPIV/Operator","t":2,"d":"Positivity of T†T: its real spectrum lies in [0, ∞).","s":"∀ {Ω : Type u_1} [inst : MeasurableSpace Ω] {μ : MeasureTheory.Measure Ω} {S : Causalean.Estimation.NPIV.OperatorSystem Ω μ} {β : ℝ} (_sc : Causalean.Estimation.NPIV.SpectralSourceCondition S β), ∀ x ∈ spectrum ℝ S.Tstar_T, 0 ≤ x","l":136,"f":"Causalean/Estimation/NPIV/Operator/SpectralCalculus.lean"},{"n":"Causalean.Estimation.NPIV.SpectralSourceCondition.biasConst","k":"def","m":"Causalean.Estimation.NPIV.Operator.SpectralCalculus","a":"Estimation/NPIV/Operator","t":1,"d":"Uniform constant absorbing both regimes (β ≤ 2 and β > 2) of the sup-on-spectrum analysis.","s":"{Ω : Type u_1} → [inst : MeasurableSpace Ω] → {μ : MeasureTheory.Measure Ω} → {S : Causalean.Estimation.NPIV.OperatorSystem Ω μ} → {β : ℝ} → Causalean.Estimation.NPIV.SpectralSourceCondition S β → ℝ","l":172,"f":"Causalean/Estimation/NPIV/Operator/SpectralCalculus.lean"},{"n":"Causalean.Estimation.NPIV.SpectralSourceCondition.biasConst_nonneg","k":"theorem","m":"Causalean.Estimation.NPIV.Operator.SpectralCalculus","a":"Estimation/NPIV/Operator","t":2,"d":"The uniform Tikhonov bias constant is nonnegative.","s":"∀ {Ω : Type u_1} [inst : MeasurableSpace Ω] {μ : MeasureTheory.Measure Ω} {S : Causalean.Estimation.NPIV.OperatorSystem Ω μ} {β : ℝ} (sc : Causalean.Estimation.NPIV.SpectralSourceCondition S β), 0 ≤ sc.biasConst","l":182,"f":"Causalean/Estimation/NPIV/Operator/SpectralCalculus.lean"},{"n":"Causalean.Estimation.NPIV.SpectralSourceCondition.continuous_sourceSymbol","k":"theorem","m":"Causalean.Estimation.NPIV.Operator.SpectralCalculus","a":"Estimation/NPIV/Operator","t":2,"d":"The source symbol is continuous whenever the source exponent is nonnegative.","s":"∀ {β : ℝ}, 0 ≤ β → Continuous (Causalean.Estimation.NPIV.SpectralSourceCondition.sourceSymbol β)","l":152,"f":"Causalean/Estimation/NPIV/Operator/SpectralCalculus.lean"},{"n":"Causalean.Estimation.NPIV.SpectralSourceCondition.sourceSymbol","k":"def","m":"Causalean.Estimation.NPIV.Operator.SpectralCalculus","a":"Estimation/NPIV/Operator","t":1,"d":"The symbol x ↦ Real.rpow (max x 0) (β/2) used by spectralPower. Continuous on all of ℝ for β ≥ 0, and agrees with x^{β/2} on [0, ∞).","s":"ℝ → ℝ → ℝ","l":146,"f":"Causalean/Estimation/NPIV/Operator/SpectralCalculus.lean"},{"n":"Causalean.Estimation.NPIV.SpectralSourceCondition.spectralPower","k":"def","m":"Causalean.Estimation.NPIV.Operator.SpectralCalculus","a":"Estimation/NPIV/Operator","t":1,"d":"The operator (T†T)^{β/2}, defined as the real CFC of T†T applied to the continuous symbol x ↦ Real.rpow (max x 0) (β/2).","s":"{Ω : Type u_1} → [inst : MeasurableSpace Ω] → {μ : MeasureTheory.Measure Ω} → {S : Causalean.Estimation.NPIV.OperatorSystem Ω μ} → {β : ℝ} → Causalean.Estimation.NPIV.SpectralSourceCondition S β → ↥(MeasureTheory.Lp ℝ 2 μ) →L[ℝ] ↥(MeasureTheory.Lp ℝ 2 μ)","l":159,"f":"Causalean/Estimation/NPIV/Operator/SpectralCalculus.lean"},{"n":"Causalean.Estimation.NPIV.SpectralSourceCondition.spectral_identity_h₀","k":"theorem","m":"Causalean.Estimation.NPIV.Operator.SpectralCalculus","a":"Estimation/NPIV/Operator","t":2,"d":"Restated spectral identity using spectralPower.","s":"∀ {Ω : Type u_1} [inst : MeasurableSpace Ω] {μ : MeasureTheory.Measure Ω} {S : Causalean.Estimation.NPIV.OperatorSystem Ω μ} {β : ℝ} (sc : Causalean.Estimation.NPIV.SpectralSourceCondition S β), S.hL2 S.h₀_mem = sc.spectralPower (S.hL2 sc.w₀_mem)","l":165,"f":"Causalean/Estimation/NPIV/Operator/SpectralCalculus.lean"},{"n":"Causalean.Estimation.NPIV.SpectralSourceCondition.strong_bias","k":"theorem","m":"Causalean.Estimation.NPIV.Operator.SpectralCalculus","a":"Estimation/NPIV/Operator","t":1,"d":"Strong-metric Tikhonov bias bound. Given a spectral β-source condition sc linking the structural function h₀ to a coefficient w₀ through the spectral power operator (T†T)^{β/2}, for any strictly positive Tikhonov regularization level λ, the squared strong-metric ($L^2(P_X)$) distance between the Tikhonov minimiser h*_λ at level λ and h₀ is bounded by biasConst · ‖w₀‖² · λ^{min(β,2)}, where biasConst is a constant determined by T†T and β.","s":"∀ {Ω : Type u_1} [inst : MeasurableSpace Ω] {μ : MeasureTheory.Measure Ω} {S : Causalean.Estimation.NPIV.OperatorSystem Ω μ} {β : ℝ} (sc : Causalean.Estimation.NPIV.SpectralSourceCondition S β) {lambda : ℝ}, 0 < lambda → ‖S.tikhonovMinimiserL2 lambda - S.hL2 S.h₀_mem‖ ^ 2 ≤ sc.biasConst * ‖S.hL2 sc.w₀_mem‖ ^ 2 * lambda ^ min β 2","l":808,"f":"Causalean/Estimation/NPIV/Operator/SpectralCalculus.lean"},{"n":"Causalean.Estimation.NPIV.SpectralSourceCondition.tikhonovMinimiserL2_eq_resolvent","k":"theorem","m":"Causalean.Estimation.NPIV.Operator.SpectralCalculus","a":"Estimation/NPIV/Operator","t":1,"d":"Resolvent identification of the Lax–Milgram minimiser. Given a spectral β-source condition sc whose primal candidate set coincides with the whole ambient L² space, for any strictly positive Tikhonov regularization level λ, the population Tikhonov minimiser at level λ equals the resolvent expression obtained by applying the real functional calculus of T†T to the symbol x ↦ x/(λ+x), evaluated at the L² class of the structural function h₀.","s":"∀ {Ω : Type u_1} [inst : MeasurableSpace Ω] {μ : MeasureTheory.Measure Ω} {S : Causalean.Estimation.NPIV.OperatorSystem Ω μ} {β : ℝ} (sc : Causalean.Estimation.NPIV.SpectralSourceCondition S β) {lambda : ℝ}, 0 < lambda → S.tikhonovMinimiserL2 lambda = (Causalean.Estimation.NPIV.Complexification.realCFC S.Tstar_T fun x => x / (lambda + x)) (S.hL2 S.h₀_mem)","l":460,"f":"Causalean/Estimation/NPIV/Operator/SpectralCalculus.lean"},{"n":"Causalean.Estimation.NPIV.SpectralSourceCondition.weak_bias","k":"theorem","m":"Causalean.Estimation.NPIV.Operator.SpectralCalculus","a":"Estimation/NPIV/Operator","t":1,"d":"Weak-metric Tikhonov bias bound. Given the same spectral β-source condition sc linking h₀ to w₀ through (T†T)^{β/2}, for any strictly positive Tikhonov regularization level λ, the squared weak-metric ($L^2(P_Z)$) norm of the operator T applied to the Tikhonov-minimiser bias h*_λ − h₀ is bounded by biasConst · ‖w₀‖² · λ^{min(β+1,2)}.","s":"∀ {Ω : Type u_1} [inst : MeasurableSpace Ω] {μ : MeasureTheory.Measure Ω} {S : Causalean.Estimation.NPIV.OperatorSystem Ω μ} {β : ℝ} (sc : Causalean.Estimation.NPIV.SpectralSourceCondition S β) {lambda : ℝ}, 0 < lambda → ‖S.Tlin (S.tikhonovMinimiserL2 lambda - S.hL2 S.h₀_mem)‖ ^ 2 ≤ sc.biasConst * ‖S.hL2 sc.w₀_mem‖ ^ 2 * lambda ^ min (β + 1) 2","l":985,"f":"Causalean/Estimation/NPIV/Operator/SpectralCalculus.lean"},{"n":"Causalean.Estimation.NPIV.TikhonovBiasBound","k":"structure","m":"Causalean.Estimation.NPIV.SourceCondition","a":"Estimation/NPIV","t":1,"d":"The Tikhonov bias-bound bundle records, at a positive regularization level, the properties of the corresponding population Tikhonov solution: a nonnegative constant for which its squared distance to the true nuisance in the strong candidate-space norm and its squared distance to the true nuisance in the weak operator-image norm are each bounded by that constant times a power of the regularization level, together with the population strong-convexity inequality it satisfies relative to every other candidate.","s":"{Ω : Type u_1} → [inst : MeasurableSpace Ω] → {μ : MeasureTheory.Measure Ω} → (S : Causalean.Estimation.NPIV.OperatorSystem Ω μ) → (β : ℝ) → ℝ → Causalean.Estimation.NPIV.SourceCondition S β → Type u_3","l":84,"f":"Causalean/Estimation/NPIV/SourceCondition.lean"},{"n":"Causalean.Estimation.NPIV.TikhonovPullback","k":"structure","m":"Causalean.Estimation.NPIV.Operator.SpectralCalculus","a":"Estimation/NPIV/Operator","t":1,"d":"Function-level pullback datum for the discharge. The user provides a Hbar-element h_lambda_star_fun whose L² class equals the Lax–Milgram minimiser tikhonovMinimiserL2 S λ constructed in Operator/Tikhonov.lean. This single pullback is the only function-level commitment needed: the bias and convexity inequalities all live at the L² level and transport along this equation.","s":"{Ω : Type u_1} → [inst : MeasurableSpace Ω] → {μ : MeasureTheory.Measure Ω} → (S : Causalean.Estimation.NPIV.OperatorSystem Ω μ) → (β : ℝ) → ℝ → Causalean.Estimation.NPIV.SpectralSourceCondition S β → Type u_3","l":1199,"f":"Causalean/Estimation/NPIV/Operator/SpectralCalculus.lean"},{"n":"Causalean.Estimation.NPIV.instAlgebraRealLpCLM","k":"def","m":"Causalean.Estimation.NPIV.Operator.SpectralCalculus","a":"Estimation/NPIV/Operator","t":1,"d":"Local disambiguation of the real-scalar algebra structure on the complex operator algebra Lp ℂ 2 μ →L[ℂ] Lp ℂ 2 μ.","s":"{Ω : Type u_1} → [inst : MeasurableSpace Ω] → {μ : MeasureTheory.Measure Ω} → Algebra ℝ (↥(MeasureTheory.Lp ℂ 2 μ) →L[ℂ] ↥(MeasureTheory.Lp ℂ 2 μ))","l":76,"f":"Causalean/Estimation/NPIV/Operator/SpectralCalculus.lean"},{"n":"Causalean.Estimation.NPIV.mixed_bias_identity","k":"theorem","m":"Causalean.Estimation.NPIV.MixedBias","a":"Estimation/NPIV","t":1,"d":"Mixed-bias / DR identity — prop:est-trae-mixed-bias. Fix a linear inverse-problem functional system, and suppose q₀ solves the associated dual moment equation. Then for any primal candidate function h in the closed candidate set Hbar and any dual candidate function q in the closed candidate set Qbar, the bias of the doubly-robust functional Θ(h, q) relative to the true target θ₀ equals the expectation of the product (q₀(Z) − q(Z))·(h(X) − h₀(X)) of the dual and primal nuisance errors.","s":"∀ {Ω : Type u_1} [inst : MeasurableSpace Ω] {μ : MeasureTheory.Measure Ω} (S : Causalean.Estimation.NPIV.InverseProblemSystem Ω μ) {q₀ : S.𝒵 → ℝ}, S.DualSolution q₀ → ∀ {h : S.𝒳 → ℝ}, h ∈ S.Hbar → ∀ {q : S.𝒵 → ℝ}, q ∈ S.Qbar → S.Θ h q - S.θ₀ = ∫ (ω : Ω), (q₀ (S.Z ω) - q (S.Z ω)) * (h (S.X ω) - S.h₀ (S.X ω)) ∂μ","l":42,"f":"Causalean/Estimation/NPIV/MixedBias.lean"},{"n":"Causalean.Estimation.NPIV.tikhonov_bias_from_spectral","k":"def","m":"Causalean.Estimation.NPIV.Operator.SpectralCalculus","a":"Estimation/NPIV/Operator","t":1,"d":"Discharge of the Tikhonov bias bound from the spectral source condition.","s":"{Ω : Type u_1} → [inst : MeasurableSpace Ω] → {μ : MeasureTheory.Measure Ω} → (S : Causalean.Estimation.NPIV.OperatorSystem Ω μ) → (β lambda : ℝ) → 0 < lambda → (sc : Causalean.Estimation.NPIV.SpectralSourceCondition S β) → Causalean.Estimation.NPIV.TikhonovPullback S β lambda sc → Causalean.Estimation.NPIV.TikhonovBiasBound S β lambda sc.toSourceCondition","l":1218,"f":"Causalean/Estimation/NPIV/Operator/SpectralCalculus.lean"},{"n":"Causalean.Estimation.NPIV.Θ_h₀_eq_θ₀","k":"theorem","m":"Causalean.Estimation.NPIV.MixedBias","a":"Estimation/NPIV","t":2,"d":"DR cancellation at the truth on the primal side: Θ(h₀, q) = θ₀ for every q ∈ Qbar.","s":"∀ {Ω : Type u_1} [inst : MeasurableSpace Ω] {μ : MeasureTheory.Measure Ω} (S : Causalean.Estimation.NPIV.InverseProblemSystem Ω μ) {q₀ : S.𝒵 → ℝ}, S.DualSolution q₀ → ∀ {q : S.𝒵 → ℝ}, q ∈ S.Qbar → S.Θ S.h₀ q = S.θ₀","l":171,"f":"Causalean/Estimation/NPIV/MixedBias.lean"},{"n":"Causalean.Estimation.NPIV.Θ_q₀_eq_θ₀","k":"theorem","m":"Causalean.Estimation.NPIV.MixedBias","a":"Estimation/NPIV","t":2,"d":"DR cancellation at the truth on the dual side: Θ(h, q₀) = θ₀ for every h ∈ Hbar.","s":"∀ {Ω : Type u_1} [inst : MeasurableSpace Ω] {μ : MeasureTheory.Measure Ω} (S : Causalean.Estimation.NPIV.InverseProblemSystem Ω μ) {q₀ : S.𝒵 → ℝ}, S.DualSolution q₀ → ∀ {h : S.𝒳 → ℝ}, h ∈ S.Hbar → S.Θ h q₀ = S.θ₀","l":183,"f":"Causalean/Estimation/NPIV/MixedBias.lean"},{"n":"Causalean.Estimation.OrthogonalLearning.Bias_n","k":"def","m":"Causalean.Estimation.OrthogonalLearning.Population.SecondOrderBias","a":"Estimation/OrthogonalLearning/Population","t":1,"d":"Loss-gradient nuisance bias for a sample-split plug-in ERM.","s":"{Ω : Type u_1} → [inst : MeasurableSpace Ω] → {μ : MeasureTheory.Measure Ω} → {Z : Type u_2} → [inst_1 : MeasurableSpace Z] → {P_Z : MeasureTheory.Measure Z} → {Θ : Type u_3} → [inst_2 : NormedAddCommGroup Θ] → [inst_3 : InnerProductSpace ℝ Θ] → {G : Type u_4} → [inst_4 : AddCommGroup G] → [inst_5 : Module ℝ G] → (S : Causalean.Estimation.OrthogonalLearning.LearningSystem Ω μ Z P_Z Θ G) → Causalean.Estimation.OrthogonalLearning.HasDirDerivTheta S S.g₀ → {g : G} → Causalean.Estimation.OrthogonalLearning.HasDirDerivTheta S g → Θ → ℝ","l":57,"f":"Causalean/Estimation/OrthogonalLearning/Population/SecondOrderBias.lean"},{"n":"Causalean.Estimation.OrthogonalLearning.BoundedNuisanceDirs","k":"def","m":"Causalean.Estimation.CATE.OrthogonalLearning.DRLearner","a":"Estimation/CATE/OrthogonalLearning/DRLearner","t":1,"d":"Bounded-direction nuisance slice anchored at η₀.","s":"{γ : Type u_1} → [inst : MeasurableSpace γ] → Causalean.Estimation.ATE.NuisanceVec γ → Set (Causalean.Estimation.ATE.NuisanceVec γ)","l":59,"f":"Causalean/Estimation/CATE/OrthogonalLearning/DRLearner.lean"},{"n":"Causalean.Estimation.OrthogonalLearning.CenteredClampedThetaMinimizes","k":"def","m":"Causalean.Estimation.OrthogonalLearning.LocalEmpProcess.Localized","a":"Estimation/OrthogonalLearning/LocalEmpProcess","t":1,"d":"The target minimizes the auxiliary population risk whose centered excess loss has been clamped to radius b.","s":"{Ω : Type u_1} → [inst : MeasurableSpace Ω] → {μ : MeasureTheory.Measure Ω} → {Z : Type u_2} → [inst_1 : MeasurableSpace Z] → {P_Z : MeasureTheory.Measure Z} → {Θ : Type u_3} → [inst_2 : NormedAddCommGroup Θ] → [inst_3 : InnerProductSpace ℝ Θ] → {G : Type u_4} → [inst_4 : AddCommGroup G] → [inst_5 : Module ℝ G] → Causalean.Estimation.OrthogonalLearning.LearningSystem Ω μ Z P_Z Θ G → ℝ → Prop","l":411,"f":"Causalean/Estimation/OrthogonalLearning/LocalEmpProcess/Localized.lean"},{"n":"Causalean.Estimation.OrthogonalLearning.DRClampedThetaMinimizes","k":"def","m":"Causalean.Estimation.CATE.OrthogonalLearning.DRLearner","a":"Estimation/CATE/OrthogonalLearning/DRLearner","t":1,"d":"The DR-Learner target θ₀ minimizes the clamped true-nuisance squared loss used by almost-everywhere empirical-process reductions.","s":"{P : Causalean.PO.POSystem} → {γ : Type u_1} → [inst : MeasurableSpace γ] → [inst_1 : StandardBorelSpace P.Ω] → [inst_2 : MeasureTheory.IsFiniteMeasure P.μ] → Causalean.Estimation.CATE.CATEEstimationSystem P γ → {Θ : Type u_2} → Set Θ → Θ → (Θ → γ → ℝ) → ℝ → Prop","l":116,"f":"Causalean/Estimation/CATE/OrthogonalLearning/DRLearner.lean"},{"n":"Causalean.Estimation.OrthogonalLearning.DRCriticalRadius","k":"def","m":"Causalean.Estimation.CATE.OrthogonalLearning.LocalEmpProcess.LocalizedDRLearner","a":"Estimation/CATE/OrthogonalLearning/LocalEmpProcess","t":1,"d":"Critical-radius hypothesis for the DR-Learner.","s":"{P : Causalean.PO.POSystem} → {γ : Type u_1} → [inst : MeasurableSpace γ] → {Θ : Type u_2} → [inst_1 : StandardBorelSpace P.Ω] → [inst_2 : MeasureTheory.IsFiniteMeasure P.μ] → Causalean.Estimation.CATE.CATEEstimationSystem P γ → (Θ_set : Set Θ) → (Θ → γ → ℝ) → (ℕ → ↑Θ_set) → Causalean.Estimation.ATE.NuisanceVec γ → ((γ × Bool × ℝ → ℝ) → ℝ) → (ℕ → ℝ → ℝ) → Prop","l":47,"f":"Causalean/Estimation/CATE/OrthogonalLearning/LocalEmpProcess/LocalizedDRLearner.lean"},{"n":"Causalean.Estimation.OrthogonalLearning.DREvalBounded","k":"def","m":"Causalean.Estimation.CATE.OrthogonalLearning.LocalEmpProcess.DRLearner","a":"Estimation/CATE/OrthogonalLearning/LocalEmpProcess","t":1,"d":"Boundedness of the target candidate evaluation map: |eval θ x| ≤ M_Θ uniformly over θ ∈ Θ_set and x : γ.","s":"{γ : Type u_1} → {Θ : Type u_2} → Set Θ → (Θ → γ → ℝ) → ℝ → Prop","l":56,"f":"Causalean/Estimation/CATE/OrthogonalLearning/LocalEmpProcess/DRLearner.lean"},{"n":"Causalean.Estimation.OrthogonalLearning.DRNuisanceMuBounded","k":"def","m":"Causalean.Estimation.CATE.OrthogonalLearning.LocalEmpProcess.DRLearner","a":"Estimation/CATE/OrthogonalLearning/LocalEmpProcess","t":1,"d":"Uniform L∞-bound on the outcome-regression component of the realised nuisance h: |h.μ_fn b x| ≤ M_μ for both treatment arms.","s":"{γ : Type u_1} → [inst : MeasurableSpace γ] → Causalean.Estimation.ATE.NuisanceVec γ → ℝ → Prop","l":81,"f":"Causalean/Estimation/CATE/OrthogonalLearning/LocalEmpProcess/DRLearner.lean"},{"n":"Causalean.Estimation.OrthogonalLearning.DRNuisanceOverlap","k":"def","m":"Causalean.Estimation.CATE.OrthogonalLearning.LocalEmpProcess.DRLearner","a":"Estimation/CATE/OrthogonalLearning/LocalEmpProcess","t":1,"d":"Strict overlap floor on the realised nuisance h: the propensity component h.e_fn is bounded inside [ε, 1 − ε] uniformly in x.","s":"{P : Causalean.PO.POSystem} → {γ : Type u_1} → [inst : MeasurableSpace γ] → {Θ : Type u_2} → [inst_1 : StandardBorelSpace P.Ω] → [inst_2 : MeasureTheory.IsFiniteMeasure P.μ] → Causalean.Estimation.CATE.CATEEstimationSystem P γ → Set Θ → Causalean.Estimation.ATE.NuisanceVec γ → ℝ → Prop","l":72,"f":"Causalean/Estimation/CATE/OrthogonalLearning/LocalEmpProcess/DRLearner.lean"},{"n":"Causalean.Estimation.OrthogonalLearning.DROutcomeBounded","k":"def","m":"Causalean.Estimation.CATE.OrthogonalLearning.LocalEmpProcess.DRLearner","a":"Estimation/CATE/OrthogonalLearning/LocalEmpProcess","t":1,"d":"Bounded outcome assumption: the outcome coordinate is bounded by M_Y under the observed-data law P_Z, almost everywhere.","s":"{P : Causalean.PO.POSystem} → {γ : Type u_1} → [inst : MeasurableSpace γ] → [inst_1 : StandardBorelSpace P.Ω] → [inst_2 : MeasureTheory.IsFiniteMeasure P.μ] → Causalean.Estimation.CATE.CATEEstimationSystem P γ → ℝ → Prop","l":62,"f":"Causalean/Estimation/CATE/OrthogonalLearning/LocalEmpProcess/DRLearner.lean"},{"n":"Causalean.Estimation.OrthogonalLearning.DRThetaMinimizes","k":"def","m":"Causalean.Estimation.CATE.OrthogonalLearning.DRLearner","a":"Estimation/CATE/OrthogonalLearning/DRLearner","t":1,"d":"The DR-Learner target θ₀ minimizes the population squared-loss risk against the true nuisance over the candidate target class.","s":"{P : Causalean.PO.POSystem} → {γ : Type u_1} → [inst : MeasurableSpace γ] → [inst_1 : StandardBorelSpace P.Ω] → [inst_2 : MeasureTheory.IsFiniteMeasure P.μ] → Causalean.Estimation.CATE.CATEEstimationSystem P γ → {Θ : Type u_2} → Set Θ → Θ → (Θ → γ → ℝ) → Prop","l":98,"f":"Causalean/Estimation/CATE/OrthogonalLearning/DRLearner.lean"},{"n":"Causalean.Estimation.OrthogonalLearning.DiffQuotientEnvelopeG","k":"def","m":"Causalean.Estimation.OrthogonalLearning.Population.NeymanOrthogonal","a":"Estimation/OrthogonalLearning/Population","t":1,"d":"L¹(P_Z) envelope dominating the *nuisance-direction* difference quotient of the loss locally near t = 0, uniformly in g ∈ G_set. Mirrors Estimation.OrthogonalMoments.DiffQuotientEnvelope.","s":"{Ω : Type u_1} → [inst : MeasurableSpace Ω] → {μ : MeasureTheory.Measure Ω} → {Z : Type u_2} → [inst_1 : MeasurableSpace Z] → {P_Z : MeasureTheory.Measure Z} → {Θ : Type u_3} → [inst_2 : NormedAddCommGroup Θ] → [inst_3 : InnerProductSpace ℝ Θ] → {G : Type u_4} → [inst_4 : AddCommGroup G] → [inst_5 : Module ℝ G] → Causalean.Estimation.OrthogonalLearning.LearningSystem Ω μ Z P_Z Θ G → Prop","l":75,"f":"Causalean/Estimation/OrthogonalLearning/Population/NeymanOrthogonal.lean"},{"n":"Causalean.Estimation.OrthogonalLearning.DiffQuotientEnvelopeTheta","k":"def","m":"Causalean.Estimation.OrthogonalLearning.Population.NeymanOrthogonal","a":"Estimation/OrthogonalLearning/Population","t":1,"d":"L¹(P_Z) envelope dominating the *target-direction* difference quotient of the loss locally near t = 0, uniformly in θ ∈ Θ_set. Mirrors Estimation.OrthogonalMoments.DiffQuotientEnvelope.","s":"{Ω : Type u_1} → [inst : MeasurableSpace Ω] → {μ : MeasureTheory.Measure Ω} → {Z : Type u_2} → [inst_1 : MeasurableSpace Z] → {P_Z : MeasureTheory.Measure Z} → {Θ : Type u_3} → [inst_2 : NormedAddCommGroup Θ] → [inst_3 : InnerProductSpace ℝ Θ] → {G : Type u_4} → [inst_4 : AddCommGroup G] → [inst_5 : Module ℝ G] → Causalean.Estimation.OrthogonalLearning.LearningSystem Ω μ Z P_Z Θ G → G → Prop","l":65,"f":"Causalean/Estimation/OrthogonalLearning/Population/NeymanOrthogonal.lean"},{"n":"Causalean.Estimation.OrthogonalLearning.EvalDirDeriv","k":"structure","m":"Causalean.Estimation.CATE.OrthogonalLearning.DRLearner.Analytic","a":"Estimation/CATE/OrthogonalLearning/DRLearner","t":1,"d":"A pointwise directional-derivative bundle for a candidate evaluation map along the segment from a base point to a candidate parameter: it packages the derivative's value at each covariate together with the fact that the map's difference quotient along the segment converges to that value as the perturbation parameter vanishes, the derivative's measurability in the covariate, and a uniform bound on the derivative over the parameter set.","s":"{Θ : Type u_1} → [inst : NormedAddCommGroup Θ] → [InnerProductSpace ℝ Θ] → {γ : Type u_2} → [MeasurableSpace γ] → Set Θ → Θ → (Θ → γ → ℝ) → Type (max u_1 u_2)","l":59,"f":"Causalean/Estimation/CATE/OrthogonalLearning/DRLearner/Analytic.lean"},{"n":"Causalean.Estimation.OrthogonalLearning.FirstOrderInequality","k":"def","m":"Causalean.Estimation.OrthogonalLearning.Population.DirectionalDeriv","a":"Estimation/OrthogonalLearning/Population","t":1,"d":"Population first-order inequality at the truth: for every admissible target θ ∈ Θ_set, the integrated target directional derivative at the true nuisance is non-negative. This is the population KKT condition characterising θ₀ as a minimizer of L(·, g₀) over the convex set Θ_set.","s":"{Ω : Type u_1} → [inst : MeasurableSpace Ω] → {μ : MeasureTheory.Measure Ω} → {Z : Type u_2} → [inst_1 : MeasurableSpace Z] → {P_Z : MeasureTheory.Measure Z} → {Θ : Type u_3} → [inst_2 : NormedAddCommGroup Θ] → [inst_3 : InnerProductSpace ℝ Θ] → {G : Type u_4} → [inst_4 : AddCommGroup G] → [inst_5 : Module ℝ G] → (S : Causalean.Estimation.OrthogonalLearning.LearningSystem Ω μ Z P_Z Θ G) → Causalean.Estimation.OrthogonalLearning.HasDirDerivTheta S S.g₀ → Prop","l":122,"f":"Causalean/Estimation/OrthogonalLearning/Population/DirectionalDeriv.lean"},{"n":"Causalean.Estimation.OrthogonalLearning.G_PerturbClosed","k":"def","m":"Causalean.Estimation.OrthogonalLearning.Setup","a":"Estimation/OrthogonalLearning","t":1,"d":"Closure-under-perturbation predicate for G_set.","s":"{Ω : Type u_1} → [inst : MeasurableSpace Ω] → {μ : MeasureTheory.Measure Ω} → {Z : Type u_2} → [inst_1 : MeasurableSpace Z] → {P_Z : MeasureTheory.Measure Z} → {Θ : Type u_3} → [inst_2 : NormedAddCommGroup Θ] → [inst_3 : InnerProductSpace ℝ Θ] → {G : Type u_4} → [inst_4 : AddCommGroup G] → [inst_5 : Module ℝ G] → Causalean.Estimation.OrthogonalLearning.LearningSystem Ω μ Z P_Z Θ G → Prop","l":106,"f":"Causalean/Estimation/OrthogonalLearning/Setup.lean"},{"n":"Causalean.Estimation.OrthogonalLearning.HasDirDerivG","k":"structure","m":"Causalean.Estimation.OrthogonalLearning.Population.DirectionalDeriv","a":"Estimation/OrthogonalLearning/Population","t":1,"d":"Pointwise directional derivative of g ↦ ℓ z θ₀ g along the segment from g₀ to g, packaged with the pointwise tendsto witness and measurability.","s":"{Ω : Type u_1} → [inst : MeasurableSpace Ω] → {μ : MeasureTheory.Measure Ω} → {Z : Type u_2} → [inst_1 : MeasurableSpace Z] → {P_Z : MeasureTheory.Measure Z} → {Θ : Type u_3} → [inst_2 : NormedAddCommGroup Θ] → [inst_3 : InnerProductSpace ℝ Θ] → {G : Type u_4} → [inst_4 : AddCommGroup G] → [inst_5 : Module ℝ G] → Causalean.Estimation.OrthogonalLearning.LearningSystem Ω μ Z P_Z Θ G → Type (max u_2 u_4)","l":72,"f":"Causalean/Estimation/OrthogonalLearning/Population/DirectionalDeriv.lean"},{"n":"Causalean.Estimation.OrthogonalLearning.HasDirDerivTheta","k":"structure","m":"Causalean.Estimation.OrthogonalLearning.Population.DirectionalDeriv","a":"Estimation/OrthogonalLearning/Population","t":1,"d":"Directional derivative of the learning-system loss in the target coordinate, at a fixed nuisance g. Bundles a candidate directional-derivative function dℓ_θ, giving a real number for each target θ and observation z, the witness that for every target θ in the system's target class and every observation z, the loss's difference quotient along the segment from θ₀ to θ at nuisance g converges to dℓ_θ θ z as the step size shrinks to zero, and measurability of dℓ_θ θ in the observation for every target θ.","s":"{Ω : Type u_1} → [inst : MeasurableSpace Ω] → {μ : MeasureTheory.Measure Ω} → {Z : Type u_2} → [inst_1 : MeasurableSpace Z] → {P_Z : MeasureTheory.Measure Z} → {Θ : Type u_3} → [inst_2 : NormedAddCommGroup Θ] → [inst_3 : InnerProductSpace ℝ Θ] → {G : Type u_4} → [inst_4 : AddCommGroup G] → [inst_5 : Module ℝ G] → Causalean.Estimation.OrthogonalLearning.LearningSystem Ω μ Z P_Z Θ G → G → Type (max u_2 u_3)","l":52,"f":"Causalean/Estimation/OrthogonalLearning/Population/DirectionalDeriv.lean"},{"n":"Causalean.Estimation.OrthogonalLearning.HasMixedDirDeriv","k":"structure","m":"Causalean.Estimation.OrthogonalLearning.Population.DirectionalDeriv","a":"Estimation/OrthogonalLearning/Population","t":1,"d":"Mixed target-nuisance directional derivative of the learning-system loss, at the truth (θ₀, g₀). Bundles a target-direction directional-derivative bundle Dθ_at g anchored at every accessible nuisance value g, a real-valued mixed directional-derivative function dℓ_θg of the target, the nuisance, and the observation, the witness that for every target θ and nuisance g in the system's classes and every observation z, the target-direction derivative anchored at the perturbed nuisance g₀ + t(g - g₀) converges, as t → 0, to dℓ_θg θ g z, and measurability of dℓ_θg θ g in the observation for every target θ and nuisance g.","s":"{Ω : Type u_1} → [inst : MeasurableSpace Ω] → {μ : MeasureTheory.Measure Ω} → {Z : Type u_2} → [inst_1 : MeasurableSpace Z] → {P_Z : MeasureTheory.Measure Z} → {Θ : Type u_3} → [inst_2 : NormedAddCommGroup Θ] → [inst_3 : InnerProductSpace ℝ Θ] → {G : Type u_4} → [inst_4 : AddCommGroup G] → [inst_5 : Module ℝ G] → Causalean.Estimation.OrthogonalLearning.LearningSystem Ω μ Z P_Z Θ G → Type (max (max u_2 u_3) u_4)","l":87,"f":"Causalean/Estimation/OrthogonalLearning/Population/DirectionalDeriv.lean"},{"n":"Causalean.Estimation.OrthogonalLearning.LearningSystem","k":"structure","m":"Causalean.Estimation.OrthogonalLearning.Setup","a":"Estimation/OrthogonalLearning","t":1,"d":"Orthogonal statistical-learning system. Bundles a convex target class Θ_set containing a distinguished target θ₀, a nuisance class G_set containing a distinguished nuisance g₀, a jointly measurable loss ℓ, and the condition that θ₀ minimizes the population risk over the target class at the true nuisance g₀.","s":"(Ω : Type u_1) → [inst : MeasurableSpace Ω] → MeasureTheory.Measure Ω → (Z : Type u_2) → [inst : MeasurableSpace Z] → MeasureTheory.Measure Z → (Θ : Type u_3) → [inst : NormedAddCommGroup Θ] → [InnerProductSpace ℝ Θ] → (G : Type u_4) → [inst : AddCommGroup G] → [Module ℝ G] → Type (max (max u_2 u_3) u_4)","l":43,"f":"Causalean/Estimation/OrthogonalLearning/Setup.lean"},{"n":"Causalean.Estimation.OrthogonalLearning.LearningSystem.L","k":"def","m":"Causalean.Estimation.OrthogonalLearning.Setup","a":"Estimation/OrthogonalLearning","t":1,"d":"Population risk: L(θ, g) := ∫ ℓ(z; θ, g) dP_Z.","s":"{Ω : Type u_1} → [inst : MeasurableSpace Ω] → {μ : MeasureTheory.Measure Ω} → {Z : Type u_2} → [inst_1 : MeasurableSpace Z] → {P_Z : MeasureTheory.Measure Z} → {Θ : Type u_3} → [inst_2 : NormedAddCommGroup Θ] → [inst_3 : InnerProductSpace ℝ Θ] → {G : Type u_4} → [inst_4 : AddCommGroup G] → [inst_5 : Module ℝ G] → Causalean.Estimation.OrthogonalLearning.LearningSystem Ω μ Z P_Z Θ G → Θ → G → ℝ","l":88,"f":"Causalean/Estimation/OrthogonalLearning/Setup.lean"},{"n":"Causalean.Estimation.OrthogonalLearning.LocalEmpProcessModulus","k":"def","m":"Causalean.Estimation.OrthogonalLearning.LocalEmpProcess.Local","a":"Estimation/OrthogonalLearning/LocalEmpProcess","t":1,"d":"Local empirical-process modulus — the named high-probability regularity condition on the empirical excess risk.","s":"{Ω : Type u_1} → [inst : MeasurableSpace Ω] → {μ : MeasureTheory.Measure Ω} → {Z : Type u_2} → [inst_1 : MeasurableSpace Z] → {P_Z : MeasureTheory.Measure Z} → {Θ : Type u_3} → [inst_2 : NormedAddCommGroup Θ] → [inst_3 : InnerProductSpace ℝ Θ] → {G : Type u_4} → [inst_4 : AddCommGroup G] → [inst_5 : Module ℝ G] → Causalean.Estimation.OrthogonalLearning.LearningSystem Ω μ Z P_Z Θ G → (S_iid : Causalean.Stat.IIDSample Ω Z μ P_Z) → Causalean.Stat.OneShotSplit S_iid → (ℕ → ℝ) → ℝ → G → Prop","l":45,"f":"Causalean/Estimation/OrthogonalLearning/LocalEmpProcess/Local.lean"},{"n":"Causalean.Estimation.OrthogonalLearning.LocalEmpProcessModulusRandom","k":"def","m":"Causalean.Estimation.OrthogonalLearning.LocalEmpProcess.RandomNuisance","a":"Estimation/OrthogonalLearning/LocalEmpProcess","t":1,"d":"Random-nuisance local empirical-process modulus. The hMod shape: for each n there is a high-probability event on which the centred excess risk at the *random* nuisance ĥ n ω obeys the modulus inequality uniformly in θ.","s":"{Ω : Type u_1} → [inst : MeasurableSpace Ω] → {μ : MeasureTheory.Measure Ω} → {Z : Type u_2} → [inst_1 : MeasurableSpace Z] → {P_Z : MeasureTheory.Measure Z} → {Θ : Type u_3} → [inst_2 : NormedAddCommGroup Θ] → [inst_3 : InnerProductSpace ℝ Θ] → {G : Type u_4} → [inst_4 : AddCommGroup G] → [inst_5 : Module ℝ G] → Causalean.Estimation.OrthogonalLearning.LearningSystem Ω μ Z P_Z Θ G → (S_iid : Causalean.Stat.IIDSample Ω Z μ P_Z) → Causalean.Stat.OneShotSplit S_iid → (ℕ → ℝ) → ℝ → (ℕ → Ω → G) → Prop","l":60,"f":"Causalean/Estimation/OrthogonalLearning/LocalEmpProcess/RandomNuisance.lean"},{"n":"Causalean.Estimation.OrthogonalLearning.LocalizedRademacherRegime","k":"def","m":"Causalean.Estimation.OrthogonalLearning.LocalEmpProcess.Localized","a":"Estimation/OrthogonalLearning/LocalEmpProcess","t":1,"d":"Localized regime predicate for a LearningSystem.","s":"{Ω : Type u_1} → [inst : MeasurableSpace Ω] → {μ : MeasureTheory.Measure Ω} → {Z : Type u_2} → [inst_1 : MeasurableSpace Z] → {P_Z : MeasureTheory.Measure Z} → {Θ : Type u_3} → [inst_2 : NormedAddCommGroup Θ] → [inst_3 : InnerProductSpace ℝ Θ] → {G : Type u_4} → [inst_4 : AddCommGroup G] → [inst_5 : Module ℝ G] → (S : Causalean.Estimation.OrthogonalLearning.LearningSystem Ω μ Z P_Z Θ G) → (_S_iid : Causalean.Stat.IIDSample Ω Z μ P_Z) → Causalean.Stat.OneShotSplit _S_iid → G → (ℕ → ↑S.Θ_set) → ((Z → ℝ) → ℝ) → (ℕ → ℝ → ℝ) → ℝ → Prop","l":92,"f":"Causalean/Estimation/OrthogonalLearning/LocalEmpProcess/Localized.lean"},{"n":"Causalean.Estimation.OrthogonalLearning.LocalizedRademacherRegimeAE","k":"def","m":"Causalean.Estimation.OrthogonalLearning.LocalEmpProcess.Localized","a":"Estimation/OrthogonalLearning/LocalEmpProcess","t":1,"d":"Almost-everywhere localized regime predicate for a LearningSystem.","s":"{Ω : Type u_1} → [inst : MeasurableSpace Ω] → {μ : MeasureTheory.Measure Ω} → {Z : Type u_2} → [inst_1 : MeasurableSpace Z] → {P_Z : MeasureTheory.Measure Z} → {Θ : Type u_3} → [inst_2 : NormedAddCommGroup Θ] → [inst_3 : InnerProductSpace ℝ Θ] → {G : Type u_4} → [inst_4 : AddCommGroup G] → [inst_5 : Module ℝ G] → (S : Causalean.Estimation.OrthogonalLearning.LearningSystem Ω μ Z P_Z Θ G) → (_S_iid : Causalean.Stat.IIDSample Ω Z μ P_Z) → Causalean.Stat.OneShotSplit _S_iid → G → (ℕ → ↑S.Θ_set) → ((Z → ℝ) → ℝ) → (ℕ → ℝ → ℝ) → ℝ → Prop","l":135,"f":"Causalean/Estimation/OrthogonalLearning/LocalEmpProcess/Localized.lean"},{"n":"Causalean.Estimation.OrthogonalLearning.LossContinuousOnΘset","k":"def","m":"Causalean.Estimation.OrthogonalLearning.LocalEmpProcess.Rademacher","a":"Estimation/OrthogonalLearning/LocalEmpProcess","t":1,"d":"Loss continuous in θ on Θ_set (for each z and fixed nuisance g).","s":"{Ω : Type u_1} → [inst : MeasurableSpace Ω] → {μ : MeasureTheory.Measure Ω} → {Z : Type u_2} → [inst_1 : MeasurableSpace Z] → {P_Z : MeasureTheory.Measure Z} → {Θ : Type u_3} → [inst_2 : NormedAddCommGroup Θ] → [inst_3 : InnerProductSpace ℝ Θ] → {G : Type u_4} → [inst_4 : AddCommGroup G] → [inst_5 : Module ℝ G] → Causalean.Estimation.OrthogonalLearning.LearningSystem Ω μ Z P_Z Θ G → G → Prop","l":169,"f":"Causalean/Estimation/OrthogonalLearning/LocalEmpProcess/Rademacher.lean"},{"n":"Causalean.Estimation.OrthogonalLearning.MixedScoreDCTBridge","k":"def","m":"Causalean.Estimation.OrthogonalLearning.Population.NeymanOrthogonal","a":"Estimation/OrthogonalLearning/Population","t":1,"d":"DCT-bridge hypothesis for the score reformulation: for every admissible target direction ν_θ = θ - θ₀ and nuisance direction ν_g = g - g₀, the integrated centred difference quotient of the target dir derivatives along the nuisance perturbation tends to the integrated mixed dir derivative ∫ z, M.dℓ_θg θ g z ∂P_Z as t → 0 along 𝓝[≠] 0.","s":"{Ω : Type u_1} → [inst : MeasurableSpace Ω] → {μ : MeasureTheory.Measure Ω} → {Z : Type u_2} → [inst_1 : MeasurableSpace Z] → {P_Z : MeasureTheory.Measure Z} → {Θ : Type u_3} → [inst_2 : NormedAddCommGroup Θ] → [inst_3 : InnerProductSpace ℝ Θ] → {G : Type u_4} → [inst_4 : AddCommGroup G] → [inst_5 : Module ℝ G] → (S : Causalean.Estimation.OrthogonalLearning.LearningSystem Ω μ Z P_Z Θ G) → Causalean.Estimation.OrthogonalLearning.HasMixedDirDeriv S → Prop","l":85,"f":"Causalean/Estimation/OrthogonalLearning/Population/NeymanOrthogonal.lean"},{"n":"Causalean.Estimation.OrthogonalLearning.NeymanOrthogLoss","k":"def","m":"Causalean.Estimation.OrthogonalLearning.Population.NeymanOrthogonal","a":"Estimation/OrthogonalLearning/Population","t":1,"d":"Neyman orthogonality of the loss: for every admissible target and nuisance direction, the integrated mixed directional derivative at (θ₀, g₀) vanishes.","s":"{Ω : Type u_1} → [inst : MeasurableSpace Ω] → {μ : MeasureTheory.Measure Ω} → {Z : Type u_2} → [inst_1 : MeasurableSpace Z] → {P_Z : MeasureTheory.Measure Z} → {Θ : Type u_3} → [inst_2 : NormedAddCommGroup Θ] → [inst_3 : InnerProductSpace ℝ Θ] → {G : Type u_4} → [inst_4 : AddCommGroup G] → [inst_5 : Module ℝ G] → (S : Causalean.Estimation.OrthogonalLearning.LearningSystem Ω μ Z P_Z Θ G) → Causalean.Estimation.OrthogonalLearning.HasMixedDirDeriv S → Prop","l":58,"f":"Causalean/Estimation/OrthogonalLearning/Population/NeymanOrthogonal.lean"},{"n":"Causalean.Estimation.OrthogonalLearning.NuisanceDirDeriv","k":"structure","m":"Causalean.Estimation.CATE.OrthogonalLearning.DRLearner.Analytic","a":"Estimation/CATE/OrthogonalLearning/DRLearner","t":1,"d":"A nuisance derivative bundle records, for the doubly robust pseudo-outcome, its directional derivative in each nuisance direction as the limit of the corresponding difference quotient, together with that derivative's measurability.","s":"{γ : Type u_1} → [inst : MeasurableSpace γ] → Causalean.Estimation.ATE.NuisanceVec γ → Type u_1","l":90,"f":"Causalean/Estimation/CATE/OrthogonalLearning/DRLearner/Analytic.lean"},{"n":"Causalean.Estimation.OrthogonalLearning.RademacherBound","k":"def","m":"Causalean.Estimation.OrthogonalLearning.LocalEmpProcess.Rademacher","a":"Estimation/OrthogonalLearning/LocalEmpProcess","t":1,"d":"Rademacher-complexity bound on the centred loss class on fold B, indexed by a countable dense sequence in S.Θ_set.","s":"{Ω : Type u_1} → [inst : MeasurableSpace Ω] → {μ : MeasureTheory.Measure Ω} → {Z : Type u_2} → [inst_1 : MeasurableSpace Z] → {P_Z : MeasureTheory.Measure Z} → {Θ : Type u_3} → [inst_2 : NormedAddCommGroup Θ] → [inst_3 : InnerProductSpace ℝ Θ] → {G : Type u_4} → [inst_4 : AddCommGroup G] → [inst_5 : Module ℝ G] → (S : Causalean.Estimation.OrthogonalLearning.LearningSystem Ω μ Z P_Z Θ G) → (S_iid : Causalean.Stat.IIDSample Ω Z μ P_Z) → Causalean.Stat.OneShotSplit S_iid → G → (ℕ → ↑S.Θ_set) → (ℕ → ℝ) → Prop","l":116,"f":"Causalean/Estimation/OrthogonalLearning/LocalEmpProcess/Rademacher.lean"},{"n":"Causalean.Estimation.OrthogonalLearning.SampleSplitPluginERM","k":"structure","m":"Causalean.Estimation.OrthogonalLearning.PluginERM","a":"Estimation/OrthogonalLearning","t":1,"d":"This predicate says that a sample-size-indexed target estimator, evaluated against a plug-in nuisance estimate on the estimation fold, is an approximate empirical-risk minimizer: the optimization slack is nonnegative at every sample size, the estimator's value always lies in the target parameter set, and its fold-B empirical risk at the plug-in nuisance is within that slack of the risk at every other point in the target set.","s":"{Ω : Type u_1} → [inst : MeasurableSpace Ω] → {μ : MeasureTheory.Measure Ω} → {Z : Type u_2} → [inst_1 : MeasurableSpace Z] → {P_Z : MeasureTheory.Measure Z} → {Θ : Type u_3} → [inst_2 : NormedAddCommGroup Θ] → [inst_3 : InnerProductSpace ℝ Θ] → {G : Type u_4} → [inst_4 : AddCommGroup G] → [inst_5 : Module ℝ G] → Causalean.Estimation.OrthogonalLearning.LearningSystem Ω μ Z P_Z Θ G → (S_iid : Causalean.Stat.IIDSample Ω Z μ P_Z) → Causalean.Stat.OneShotSplit S_iid → (ℕ → Ω → Θ) → (ℕ → Ω → G) → (ℕ → ℝ) → Prop","l":55,"f":"Causalean/Estimation/OrthogonalLearning/PluginERM.lean"},{"n":"Causalean.Estimation.OrthogonalLearning.Sparse.LinftyDevTailBound","k":"def","m":"Causalean.Estimation.OrthogonalLearning.Sparse.PluginGuarantee","a":"Estimation/OrthogonalLearning/Sparse","t":1,"d":"High-probability ℓ∞ tail bound for an Ω-indexed deviation field dev : Ω → EuclideanSpace ℝ (Fin p): there is a measurable event E ⊆ Ω of μ-mass at least 1 - δ on which linftyDev hp (dev ω) ≤ ρ.","s":"{Ω : Type u_1} → [inst : MeasurableSpace Ω] → MeasureTheory.Measure Ω → {p : ℕ} → Finset.univ.Nonempty → (Ω → EuclideanSpace ℝ (Fin p)) → ℝ → ℝ → Prop","l":264,"f":"Causalean/Estimation/OrthogonalLearning/Sparse/PluginGuarantee.lean"},{"n":"Causalean.Estimation.OrthogonalLearning.Sparse.RestrictedCone","k":"def","m":"Causalean.Estimation.OrthogonalLearning.Sparse.Setup","a":"Estimation/OrthogonalLearning/Sparse","t":1,"d":"The restricted cone C(S₀) := {ν : ‖ν_{S₀ᶜ}‖₁ ≤ 3 ‖ν_{S₀}‖₁}. The complement is taken in Finset.univ.","s":"{p : ℕ} → Finset (Fin p) → Set (EuclideanSpace ℝ (Fin p))","l":73,"f":"Causalean/Estimation/OrthogonalLearning/Sparse/Setup.lean"},{"n":"Causalean.Estimation.OrthogonalLearning.Sparse.RestrictedStrongConvexity","k":"def","m":"Causalean.Estimation.OrthogonalLearning.Sparse.RSC","a":"Estimation/OrthogonalLearning/Sparse","t":1,"d":"σₙ-restricted strong convexity of empRiskFn at θ₀ over S₀.","s":"{p : ℕ} → (EuclideanSpace ℝ (Fin p) → ℝ) → (EuclideanSpace ℝ (Fin p) → EuclideanSpace ℝ (Fin p)) → EuclideanSpace ℝ (Fin p) → Finset (Fin p) → ℝ → Prop","l":43,"f":"Causalean/Estimation/OrthogonalLearning/Sparse/RSC.lean"},{"n":"Causalean.Estimation.OrthogonalLearning.Sparse.SparsePluginERM","k":"structure","m":"Causalean.Estimation.OrthogonalLearning.Sparse.Setup","a":"Estimation/OrthogonalLearning/Sparse","t":1,"d":"Sparse plug-in regularised ERM (predicate form). Given an empirical risk with the plug-in nuisance already absorbed and a candidate parameter estimate, this predicate records that the ℓ₁-penalty level is nonnegative and that the candidate minimises the ℓ₁-penalised empirical risk over the entire ambient parameter space, with no restriction to a support set.","s":"{p : ℕ} → (EuclideanSpace ℝ (Fin p) → ℝ) → EuclideanSpace ℝ (Fin p) → ℝ → Prop","l":137,"f":"Causalean/Estimation/OrthogonalLearning/Sparse/Setup.lean"},{"n":"Causalean.Estimation.OrthogonalLearning.Sparse.l1Full","k":"def","m":"Causalean.Estimation.OrthogonalLearning.Sparse.Setup","a":"Estimation/OrthogonalLearning/Sparse","t":1,"d":"Full coordinate ℓ¹ norm of ν: ∑ i, |ν i|.","s":"{p : ℕ} → EuclideanSpace ℝ (Fin p) → ℝ","l":59,"f":"Causalean/Estimation/OrthogonalLearning/Sparse/Setup.lean"},{"n":"Causalean.Estimation.OrthogonalLearning.Sparse.l1Full_eq","k":"theorem","m":"Causalean.Estimation.OrthogonalLearning.Sparse.Setup","a":"Estimation/OrthogonalLearning/Sparse","t":2,"d":"l1Full decomposes along any subset and its complement in Finset.univ.","s":"∀ {p : ℕ} (ν : EuclideanSpace ℝ (Fin p)) (S₀ : Finset (Fin p)), Causalean.Estimation.OrthogonalLearning.Sparse.l1Full ν = Causalean.Estimation.OrthogonalLearning.Sparse.l1Norm ν S₀ + Causalean.Estimation.OrthogonalLearning.Sparse.l1Norm ν (Finset.univ \\ S₀)","l":88,"f":"Causalean/Estimation/OrthogonalLearning/Sparse/Setup.lean"},{"n":"Causalean.Estimation.OrthogonalLearning.Sparse.l1Norm","k":"def","m":"Causalean.Estimation.OrthogonalLearning.Sparse.Setup","a":"Estimation/OrthogonalLearning/Sparse","t":1,"d":"Coordinate ℓ¹ norm of ν restricted to a subset T : Finset (Fin p): l1Norm ν T = ∑ i ∈ T, |ν i|.","s":"{p : ℕ} → EuclideanSpace ℝ (Fin p) → Finset (Fin p) → ℝ","l":53,"f":"Causalean/Estimation/OrthogonalLearning/Sparse/Setup.lean"},{"n":"Causalean.Estimation.OrthogonalLearning.Sparse.l1Norm_supp_le_card_sqrt_mul_l2norm","k":"theorem","m":"Causalean.Estimation.OrthogonalLearning.Sparse.Setup","a":"Estimation/OrthogonalLearning/Sparse","t":1,"d":"Cauchy–Schwarz on the support. The ℓ¹ norm of a vector ν restricted to a finite index set S₀ is bounded by √|S₀| times its full ℓ² norm.","s":"∀ {p : ℕ} (ν : EuclideanSpace ℝ (Fin p)) (S₀ : Finset (Fin p)), Causalean.Estimation.OrthogonalLearning.Sparse.l1Norm ν S₀ ≤ √↑S₀.card * ‖ν‖","l":96,"f":"Causalean/Estimation/OrthogonalLearning/Sparse/Setup.lean"},{"n":"Causalean.Estimation.OrthogonalLearning.Sparse.linftyDev","k":"def","m":"Causalean.Estimation.OrthogonalLearning.Sparse.Setup","a":"Estimation/OrthogonalLearning/Sparse","t":1,"d":"Coordinate ℓ∞ deviation of ν: max_i |ν i|.","s":"{p : ℕ} → Finset.univ.Nonempty → EuclideanSpace ℝ (Fin p) → ℝ","l":63,"f":"Causalean/Estimation/OrthogonalLearning/Sparse/Setup.lean"},{"n":"Causalean.Estimation.OrthogonalLearning.Sparse.mem_RestrictedCone_iff","k":"theorem","m":"Causalean.Estimation.OrthogonalLearning.Sparse.Setup","a":"Estimation/OrthogonalLearning/Sparse","t":1,"d":"Restricted cone membership unfolded. A vector ν lies in the restricted cone RestrictedCone S₀ exactly when its ℓ¹ norm off the support S₀ is at most three times its ℓ¹ norm on S₀.","s":"∀ {p : ℕ} (S₀ : Finset (Fin p)) (ν : EuclideanSpace ℝ (Fin p)), ν ∈ Causalean.Estimation.OrthogonalLearning.Sparse.RestrictedCone S₀ ↔ Causalean.Estimation.OrthogonalLearning.Sparse.l1Norm ν (Finset.univ \\ S₀) ≤ 3 * Causalean.Estimation.OrthogonalLearning.Sparse.l1Norm ν S₀","l":79,"f":"Causalean/Estimation/OrthogonalLearning/Sparse/Setup.lean"},{"n":"Causalean.Estimation.OrthogonalLearning.Sparse.sparse_plugin_guarantee","k":"theorem","m":"Causalean.Estimation.OrthogonalLearning.Sparse.PluginGuarantee","a":"Estimation/OrthogonalLearning/Sparse","t":1,"d":"Deterministic sparse plug-in ERM guarantee. Let empRiskFn be a convex empirical risk function on EuclideanSpace ℝ (Fin p), with population gradient popGrad and empirical gradient map gradEmp. Assume the support S₀ of the truth θ₀ has cardinality s, that θ₀ vanishes off S₀, that the restricted-strong-convexity modulus σn is strictly positive and empRiskFn is σn-restricted-strongly-convex at θ₀ over S₀, and that the empirical gradient at θ₀ satisfies the subgradient basic inequality for empRiskFn. Given a penalty level with lambda > 0, suppose the index set Fin p is nonempty so that lambda is at least twice the sup-norm deviation of the empirical gradient from the population gradient at θ₀, that the population gradient vanishes at the truth (the first-order condition), and that θhat is a sparse plug-in regularised empirical-risk minimizer at penalty level lambda. Then the estimation error θhat − θ₀ lies in the restricted cone around S₀, and its Euclidean norm is at most 12 · lambda · √s / σn.","s":"∀ {p : ℕ} (empRiskFn : EuclideanSpace ℝ (Fin p) → ℝ) (popGrad : EuclideanSpace ℝ (Fin p)) (gradEmp : EuclideanSpace ℝ (Fin p) → EuclideanSpace ℝ (Fin p)) (θ₀ θhat : EuclideanSpace ℝ (Fin p)) (S₀ : Finset (Fin p)) (s : ℕ), S₀.card = s → (∀ i ∉ S₀, θ₀.ofLp i = 0) → ConvexOn ℝ Set.univ empRiskFn → ∀ (σn : ℝ), 0 < σn → Causalean.Estimation.OrthogonalLearning.Sparse.RestrictedStrongConvexity empRiskFn gradEmp θ₀ S₀ σn → (∀ (θ : EuclideanSpace ℝ (Fin p)), empRiskFn θ - empRiskFn θ₀ ≥ inner ℝ (gradEmp θ₀) (θ - θ₀)) → ∀ (lambda : ℝ), 0 < lambda → ∀ (hp : Finset.univ.Nonempty), lambda ≥ 2 * Causalean.Estimation.OrthogonalLearning.Sparse.linftyDev hp (gradEmp θ₀ - popGrad) → popGrad = 0 → Causalean.Estimation.OrthogonalLearning.Sparse.SparsePluginERM empRiskFn θhat lambda → θhat - θ₀ ∈ Causalean.Estimation.OrthogonalLearning.Sparse.RestrictedCone S₀ ∧ ‖θhat - θ₀‖ ≤ 12 * lambda * √↑s / σn","l":49,"f":"Causalean/Estimation/OrthogonalLearning/Sparse/PluginGuarantee.lean"},{"n":"Causalean.Estimation.OrthogonalLearning.Sparse.sparse_plugin_guarantee_highProb","k":"theorem","m":"Causalean.Estimation.OrthogonalLearning.Sparse.PluginGuarantee","a":"Estimation/OrthogonalLearning/Sparse","t":1,"d":"High-probability sparse plug-in ERM guarantee. Let empRiskFn(ω) be a random empirical risk function on EuclideanSpace ℝ (Fin p) with random empirical gradient map gradEmp(ω), and popGrad the population gradient. Assume the support S₀ of the truth θ₀ has cardinality s, that θ₀ vanishes off S₀, and that empRiskFn(ω) is convex on the ambient space for every ω. Suppose the restricted-strong- convexity modulus σn is strictly positive and empRiskFn(ω) is σn-restricted-strongly-convex at θ₀ over S₀ for every ω, and that the empirical gradient at θ₀ satisfies the subgradient basic inequality for empRiskFn(ω), for every ω. Given a penalty level with lambda > 0, and assuming the index set Fin p is nonempty, that the population gradient vanishes at the truth, and that θhat(ω) is a sparse plug-in regularised empirical-risk minimizer of empRiskFn(ω) at penalty level lambda, for every ω. Suppose further that the deviation of the empirical gradient from popGrad at θ₀ obeys a sup-norm tail bound ρ at confidence level δ, with lambda at least twice that tail level, lambda ≥ 2ρ. Then there is an event of probability at least 1 - δ on which, for every ω in it, the estimation error θhat ω − θ₀ lies in the restricted cone around S₀ and its Euclidean norm is at most 12 · lambda · √s / σn.","s":"∀ {p : ℕ} {Ω : Type u_1} [inst : MeasurableSpace Ω] (μ : MeasureTheory.Measure Ω) (empRiskFn : Ω → EuclideanSpace ℝ (Fin p) → ℝ) (popGrad : EuclideanSpace ℝ (Fin p)) (gradEmp : Ω → EuclideanSpace ℝ (Fin p) → EuclideanSpace ℝ (Fin p)) (θ₀ : EuclideanSpace ℝ (Fin p)) (θhat : Ω → EuclideanSpace ℝ (Fin p)) (S₀ : Finset (Fin p)) (s : ℕ), S₀.card = s → (∀ i ∉ S₀, θ₀.ofLp i = 0) → (∀ (ω : Ω), ConvexOn ℝ Set.univ (empRiskFn ω)) → ∀ (σn : ℝ), 0 < σn → (∀ (ω : Ω), Causalean.Estimation.OrthogonalLearning.Sparse.RestrictedStrongConvexity (empRiskFn ω) (gradEmp ω) θ₀ S₀ σn) → (∀ (ω : Ω) (θ : EuclideanSpace ℝ (Fin p)), empRiskFn ω θ - empRiskFn ω θ₀ ≥ inner ℝ (gradEmp ω θ₀) (θ - θ₀)) → ∀ (lambda : ℝ), 0 < lambda → ∀ (hp : Finset.univ.Nonempty), popGrad = 0 → (∀ (ω : Ω), Causalean.Estimation.OrthogonalLearning.Sparse.SparsePluginERM (empRiskFn ω) (θhat ω) lambda) → ∀ (ρ δ : ℝ), Causalean.Estimation.OrthogonalLearning.Sparse.LinftyDevTailBound μ hp (fun ω => gradEmp ω θ₀ - popGrad) ρ δ → lambda ≥ 2 * ρ → ∃ E, MeasurableSet E ∧ μ E ≥ 1 - ENNReal.ofReal δ ∧ ∀ ω ∈ E, θhat ω - θ₀ ∈ Causalean.Estimation.OrthogonalLearning.Sparse.RestrictedCone S₀ ∧ ‖θhat ω - θ₀‖ ≤ 12 * lambda * √↑s / σn","l":280,"f":"Causalean/Estimation/OrthogonalLearning/Sparse/PluginGuarantee.lean"},{"n":"Causalean.Estimation.OrthogonalLearning.UniformlyBoundedLoss","k":"def","m":"Causalean.Estimation.OrthogonalLearning.LocalEmpProcess.Rademacher","a":"Estimation/OrthogonalLearning/LocalEmpProcess","t":1,"d":"Loss bounded uniformly over Θ_set at fixed nuisance g.","s":"{Ω : Type u_1} → [inst : MeasurableSpace Ω] → {μ : MeasureTheory.Measure Ω} → {Z : Type u_2} → [inst_1 : MeasurableSpace Z] → {P_Z : MeasureTheory.Measure Z} → {Θ : Type u_3} → [inst_2 : NormedAddCommGroup Θ] → [inst_3 : InnerProductSpace ℝ Θ] → {G : Type u_4} → [inst_4 : AddCommGroup G] → [inst_5 : Module ℝ G] → Causalean.Estimation.OrthogonalLearning.LearningSystem Ω μ Z P_Z Θ G → G → ℝ → Prop","l":156,"f":"Causalean/Estimation/OrthogonalLearning/LocalEmpProcess/Rademacher.lean"},{"n":"Causalean.Estimation.OrthogonalLearning.UniformlyBoundedLossAE","k":"def","m":"Causalean.Estimation.OrthogonalLearning.LocalEmpProcess.Rademacher","a":"Estimation/OrthogonalLearning/LocalEmpProcess","t":1,"d":"Loss bounded uniformly over Θ_set at fixed nuisance g, almost everywhere.","s":"{Ω : Type u_1} → [inst : MeasurableSpace Ω] → {μ : MeasureTheory.Measure Ω} → {Z : Type u_2} → [inst_1 : MeasurableSpace Z] → {P_Z : MeasureTheory.Measure Z} → {Θ : Type u_3} → [inst_2 : NormedAddCommGroup Θ] → [inst_3 : InnerProductSpace ℝ Θ] → {G : Type u_4} → [inst_4 : AddCommGroup G] → [inst_5 : Module ℝ G] → Causalean.Estimation.OrthogonalLearning.LearningSystem Ω μ Z P_Z Θ G → G → ℝ → Prop","l":161,"f":"Causalean/Estimation/OrthogonalLearning/LocalEmpProcess/Rademacher.lean"},{"n":"Causalean.Estimation.OrthogonalLearning.anchor_mem_boundedNuisanceDirs","k":"theorem","m":"Causalean.Estimation.CATE.OrthogonalLearning.DRLearner","a":"Estimation/CATE/OrthogonalLearning/DRLearner","t":2,"d":"The anchor η₀ belongs to BoundedNuisanceDirs η₀ (with Cμ = Ce = 0).","s":"∀ {γ : Type u_1} [inst : MeasurableSpace γ] (η₀ : Causalean.Estimation.ATE.NuisanceVec γ), η₀ ∈ Causalean.Estimation.OrthogonalLearning.BoundedNuisanceDirs η₀","l":81,"f":"Causalean/Estimation/CATE/OrthogonalLearning/DRLearner.lean"},{"n":"Causalean.Estimation.OrthogonalLearning.badDataSet","k":"def","m":"Causalean.Estimation.OrthogonalLearning.LocalEmpProcess.RandomNuisance","a":"Estimation/OrthogonalLearning/LocalEmpProcess","t":1,"d":"The product-sample bad event for nuisance g.","s":"{Ω : Type u_1} → [inst : MeasurableSpace Ω] → {μ : MeasureTheory.Measure Ω} → {Z : Type u_2} → [inst_1 : MeasurableSpace Z] → {P_Z : MeasureTheory.Measure Z} → {Θ : Type u_3} → [inst_2 : NormedAddCommGroup Θ] → [inst_3 : InnerProductSpace ℝ Θ] → {G : Type u_4} → [inst_4 : AddCommGroup G] → [inst_5 : Module ℝ G] → Causalean.Estimation.OrthogonalLearning.LearningSystem Ω μ Z P_Z Θ G → (S_iid : Causalean.Stat.IIDSample Ω Z μ P_Z) → (split : Causalean.Stat.OneShotSplit S_iid) → (n : ℕ) → ℝ → ℝ → (ℕ → ℝ) → G → Set (Fin (split.foldB n).card → Z)","l":85,"f":"Causalean/Estimation/OrthogonalLearning/LocalEmpProcess/RandomNuisance.lean"},{"n":"Causalean.Estimation.OrthogonalLearning.badDataSet_mass_le","k":"theorem","m":"Causalean.Estimation.OrthogonalLearning.LocalEmpProcess.RandomNuisance","a":"Estimation/OrthogonalLearning/LocalEmpProcess","t":2,"d":"Mass bound. The product-space bad set has Measure.pi P_Z-mass ≤ δ. The proof uses the deterministic fixed-nuisance construction from Rademacher.lean: McDiarmid concentration, symmetrization, and separable lifting.","s":"∀ {Ω : Type u_1} [inst : MeasurableSpace Ω] {μ : MeasureTheory.Measure Ω} {Z : Type u_2} [inst_1 : MeasurableSpace Z] {P_Z : MeasureTheory.Measure Z} {Θ : Type u_3} [inst_2 : NormedAddCommGroup Θ] [inst_3 : InnerProductSpace ℝ Θ] {G : Type u_4} [inst_4 : AddCommGroup G] [inst_5 : Module ℝ G] (S : Causalean.Estimation.OrthogonalLearning.LearningSystem Ω μ Z P_Z Θ G) [MeasureTheory.IsProbabilityMeasure μ] [MeasureTheory.IsProbabilityMeasure P_Z] (S_iid : Causalean.Stat.IIDSample Ω Z μ P_Z) (split : Causalean.Stat.OneShotSplit S_iid) {b : ℝ}, 0 < b → ∀ (g : G), Causalean.Estimation.OrthogonalLearning.UniformlyBoundedLoss S g b → Causalean.Estimation.OrthogonalLearning.LossContinuousOnΘset S g → ∀ (idx : ℕ → ↑S.Θ_set), DenseRange idx → ∀ (R : ℕ → ℝ), Causalean.Estimation.OrthogonalLearning.RademacherBound S S_iid split g idx R → ∀ {δ : ℝ}, 0 < δ → δ ≤ 1 → ∀ (n : ℕ), 0 < (split.foldB n).card → (MeasureTheory.Measure.pi fun x => P_Z) (Causalean.Estimation.OrthogonalLearning.badDataSet S S_iid split n b δ R g) ≤ ENNReal.ofReal δ","l":136,"f":"Causalean/Estimation/OrthogonalLearning/LocalEmpProcess/RandomNuisance.lean"},{"n":"Causalean.Estimation.OrthogonalLearning.drBias_le_product","k":"theorem","m":"Causalean.Estimation.CATE.OrthogonalLearning.SecondOrderBias","a":"Estimation/CATE/OrthogonalLearning","t":1,"d":"DR-Learner second-order product bias bound. For a CATE estimation system built on a potential-outcome model that satisfies the back-door identification assumptions, with the true nuisance η₀ lying in the strict-overlap band at some margin ε > 0, fix a convex candidate target class in an inner-product space with a real-valued evaluation map, and suppose θ₀ belongs to this class, every candidate's evaluation is measurable, and θ₀'s evaluation agrees pointwise with the true value-space CATE. Fix a candidate nuisance h that also lies in the strict-overlap band at the same margin, a candidate target θ̂, and a nonnegative constant B bounding the evaluation-map directional derivative at θ̂. Assume the arm-wise outcome-regression fit of h, the AIPW pseudo-outcome discrepancy between h and the truth, and that discrepancy weighted by the directional derivative are all integrable, the outcome-regression error of h and the propensity error of h are square-integrable arm by arm, and the loss-gradient integrand at the true nuisance and the loss-gradient integrand at the candidate nuisance h are each integrable against the observation law. Then the loss-gradient nuisance bias Bias_n, evaluated between the true and candidate nuisance directional derivatives at θ̂, is bounded in absolute value by (2B/ε) times the sum over treatment arms of the outcome-regression L² error times the propensity L² error.","s":"∀ {P : Causalean.PO.POSystem} {γ : Type u_1} [inst : MeasurableSpace γ] [inst_1 : StandardBorelSpace P.Ω] [inst_2 : MeasureTheory.IsFiniteMeasure P.μ] (S : Causalean.Estimation.CATE.CATEEstimationSystem P γ) (hA : S.Assumptions) {ε : ℝ} (hε_pos : 0 < ε) (h_overlap_η₀ : S.η₀ ∈ Causalean.Estimation.ATE.BackdoorEstimationSystem.H_ε ε) (Θ : Type u_2) [inst_3 : NormedAddCommGroup Θ] [inst_4 : InnerProductSpace ℝ Θ] (Θ_set : Set Θ) (Θ_convex : Convex ℝ Θ_set) (θ₀ : Θ) (θ₀_mem : θ₀ ∈ Θ_set) (eval : Θ → γ → ℝ) (eval_meas : ∀ (θ : Θ), Measurable (eval θ)) (eval_θ₀ : ∀ (x : γ), eval θ₀ x = S.τ_val x) (θ₀_minimizes : Causalean.Estimation.OrthogonalLearning.DRThetaMinimizes S Θ_set θ₀ eval) (D : Causalean.Estimation.OrthogonalLearning.EvalDirDeriv Θ_set θ₀ eval) (ND : Causalean.Estimation.OrthogonalLearning.NuisanceDirDeriv S.η₀), ∀ h ∈ Causalean.Estimation.ATE.BackdoorEstimationSystem.H_ε ε, ∀ (θhat : Θ) {B : ℝ}, 0 ≤ B → (∀ (x : γ), |D.dEval θhat x| ≤ B) → (∀ (a : Bool), MeasureTheory.Integrable (fun ω => h.μ_fn a (S.factualX ω)) P.μ) → MeasureTheory.Integrable (fun ω => Causalean.Estimation.CATE.phi_eta (S.factualZ ω) h - Causalean.Estimation.CATE.phi₀ S (S.factualZ ω)) P.μ → MeasureTheory.Integrable (fun ω => (Causalean.Estimation.CATE.phi_eta (S.factualZ ω) h - Causalean.Estimation.CATE.phi₀ S (S.factualZ ω)) * D.dEval θhat (S.factualX ω)) P.μ → (∀ (a : Bool), MeasureTheory.MemLp (fun x => h.μ_fn a x - S.μ_val a x) 2 S.P_X) → MeasureTheory.MemLp (fun x => h.e_fn x - S.e_val x) 2 S.P_X → MeasureTheory.Integrable (fun z => ((Causalean.Estimation.OrthogonalLearning.drMixedDirDeriv S hA hε_pos h_overlap_η₀ Θ Θ_set Θ_convex θ₀ θ₀_mem eval eval_meas eval_θ₀ θ₀_minimizes D ND).Dθ_at S.η₀).dℓ_θ θhat z) S.P_Z → MeasureTheory.Integrable (fun z => ((Causalean.Estimation.OrthogonalLearning.drMixedDirDeriv S hA hε_pos h_overlap_η₀ Θ Θ_set Θ_convex θ₀ θ₀_mem eval eval_meas eval_θ₀ θ₀_minimizes D ND).Dθ_at h).dℓ_θ θhat z) S.P_Z → |Causalean.Estimation.OrthogonalLearning.Bias_n (Causalean.Estimation.OrthogonalLearning.drLearningSystem S Θ Θ_set Θ_convex θ₀ θ₀_mem eval eval_meas eval_θ₀ θ₀_minimizes) ((Causalean.Estimation.OrthogonalLearning.drMixedDirDeriv S hA hε_pos h_overlap_η₀ Θ Θ_set Θ_convex θ₀ θ₀_mem eval eval_meas eval_θ₀ θ₀_minimizes D ND).Dθ_at S.η₀) ((Causalean.Estimation.OrthogonalLearning.drMixedDirDeriv S hA hε_pos h_overlap_η₀ Θ Θ_set Θ_convex θ₀ θ₀_mem eval eval_meas eval_θ₀ θ₀_minimizes D ND).Dθ_at h) θhat| ≤ 2 * B / ε * ∑ a, (MeasureTheory.eLpNorm (fun x => h.μ_fn a x - S.μ_val a x) 2 S.P_X).toReal * (MeasureTheory.eLpNorm (fun x => h.e_fn x - S.e_val x) 2 S.P_X).toReal","l":56,"f":"Causalean/Estimation/CATE/OrthogonalLearning/SecondOrderBias.lean"},{"n":"Causalean.Estimation.OrthogonalLearning.drLearningSystem","k":"def","m":"Causalean.Estimation.CATE.OrthogonalLearning.DRLearner","a":"Estimation/CATE/OrthogonalLearning/DRLearner","t":1,"d":"The DR-Learner orthogonal-learning system: a LearningSystem whose data law is the CATE observation triple's joint law P_Z, target space Θ is a user-provided convex subset of an inner-product space (with a candidate evaluation map eval : Θ → γ → ℝ), nuisance space G := NuisanceVec γ, and loss","s":"{P : Causalean.PO.POSystem} → {γ : Type u_1} → [inst : MeasurableSpace γ] → [inst_1 : StandardBorelSpace P.Ω] → [inst_2 : MeasureTheory.IsFiniteMeasure P.μ] → (S : Causalean.Estimation.CATE.CATEEstimationSystem P γ) → (Θ : Type u_2) → [inst_3 : NormedAddCommGroup Θ] → [inst_4 : InnerProductSpace ℝ Θ] → (Θ_set : Set Θ) → Convex ℝ Θ_set → (θ₀ : Θ) → θ₀ ∈ Θ_set → (eval : Θ → γ → ℝ) → (∀ (θ : Θ), Measurable (eval θ)) → (∀ (x : γ), eval θ₀ x = S.τ_val x) → Causalean.Estimation.OrthogonalLearning.DRThetaMinimizes S Θ_set θ₀ eval → Causalean.Estimation.OrthogonalLearning.LearningSystem P.Ω P.μ (γ × Bool × ℝ) S.P_Z Θ (Causalean.Estimation.ATE.NuisanceVec γ)","l":136,"f":"Causalean/Estimation/CATE/OrthogonalLearning/DRLearner.lean"},{"n":"Causalean.Estimation.OrthogonalLearning.drMixedDirDeriv","k":"def","m":"Causalean.Estimation.CATE.OrthogonalLearning.DRLearner.Analytic","a":"Estimation/CATE/OrthogonalLearning/DRLearner","t":1,"d":"The closed-form HasMixedDirDeriv bundle for the DR-Learner orthogonal-learning system.","s":"{P : Causalean.PO.POSystem} → {γ : Type u_1} → [inst : MeasurableSpace γ] → [inst_1 : StandardBorelSpace P.Ω] → [inst_2 : MeasureTheory.IsFiniteMeasure P.μ] → (S : Causalean.Estimation.CATE.CATEEstimationSystem P γ) → S.Assumptions → {ε : ℝ} → 0 < ε → S.η₀ ∈ Causalean.Estimation.ATE.BackdoorEstimationSystem.H_ε ε → (Θ : Type u_2) → [inst_3 : NormedAddCommGroup Θ] → [inst_4 : InnerProductSpace ℝ Θ] → (Θ_set : Set Θ) → (Θ_convex : Convex ℝ Θ_set) → (θ₀ : Θ) → (θ₀_mem : θ₀ ∈ Θ_set) → (eval : Θ → γ → ℝ) → (eval_meas : ∀ (θ : Θ), Measurable (eval θ)) → (eval_θ₀ : ∀ (x : γ), eval θ₀ x = S.τ_val x) → (θ₀_minimizes : Causalean.Estimation.OrthogonalLearning.DRThetaMinimizes S Θ_set θ₀ eval) → Causalean.Estimation.OrthogonalLearning.EvalDirDeriv Θ_set θ₀ eval → Causalean.Estimation.OrthogonalLearning.NuisanceDirDeriv S.η₀ → Causalean.Estimation.OrthogonalLearning.HasMixedDirDeriv (Causalean.Estimation.OrthogonalLearning.drLearningSystem S Θ Θ_set Θ_convex θ₀ θ₀_mem eval eval_meas eval_θ₀ θ₀_minimizes)","l":122,"f":"Causalean/Estimation/CATE/OrthogonalLearning/DRLearner/Analytic.lean"},{"n":"Causalean.Estimation.OrthogonalLearning.drNeymanOrthog","k":"theorem","m":"Causalean.Estimation.CATE.OrthogonalLearning.DRLearner.Analytic","a":"Estimation/CATE/OrthogonalLearning/DRLearner","t":1,"d":"DR-Learner loss orthogonality under a DCT bridge (prop:est-osl-dr-loss-orthogonal). For a CATE estimation system built on a potential-outcome model that satisfies the back-door identification assumptions, fix a margin ε > 0 such that the true nuisance η₀ lies in the strict-overlap slice at that margin, i.e. the propensity score is bounded away from 0 and 1. Fix a convex candidate target class inside an inner-product space together with a real-valued evaluation map on it, and suppose θ₀ belongs to this class, every candidate's evaluation is measurable, θ₀'s evaluation agrees pointwise with the true value-space CATE, and θ₀ minimizes the population AIPW pseudo-outcome squared-loss risk against the true nuisance over the candidate class. Given directional-derivative data for the evaluation map and for the doubly robust pseudo-outcome that together assemble a closed-form mixed directional derivative for the loss, if a dominated-convergence bridge licenses passing the limit defining the integrated mixed target/nuisance score through the integral, then this closed-form derivative witnesses that the DR-Learner squared loss is Neyman-orthogonal: its integrated mixed directional derivative vanishes at the truth (θ₀, η₀) for every admissible target and bounded nuisance direction.","s":"∀ {P : Causalean.PO.POSystem} {γ : Type u_1} [inst : MeasurableSpace γ] [inst_1 : StandardBorelSpace P.Ω] [inst_2 : MeasureTheory.IsFiniteMeasure P.μ] (S : Causalean.Estimation.CATE.CATEEstimationSystem P γ) (hA : S.Assumptions) {ε : ℝ} (hε_pos : 0 < ε) (h_overlap_η₀ : S.η₀ ∈ Causalean.Estimation.ATE.BackdoorEstimationSystem.H_ε ε) (Θ : Type u_2) [inst_3 : NormedAddCommGroup Θ] [inst_4 : InnerProductSpace ℝ Θ] (Θ_set : Set Θ) (Θ_convex : Convex ℝ Θ_set) (θ₀ : Θ) (θ₀_mem : θ₀ ∈ Θ_set) (eval : Θ → γ → ℝ) (eval_meas : ∀ (θ : Θ), Measurable (eval θ)) (eval_θ₀ : ∀ (x : γ), eval θ₀ x = S.τ_val x) (θ₀_minimizes : Causalean.Estimation.OrthogonalLearning.DRThetaMinimizes S Θ_set θ₀ eval) (D : Causalean.Estimation.OrthogonalLearning.EvalDirDeriv Θ_set θ₀ eval) (ND : Causalean.Estimation.OrthogonalLearning.NuisanceDirDeriv S.η₀), Causalean.Estimation.OrthogonalLearning.MixedScoreDCTBridge (Causalean.Estimation.OrthogonalLearning.drLearningSystem S Θ Θ_set Θ_convex θ₀ θ₀_mem eval eval_meas eval_θ₀ θ₀_minimizes) (Causalean.Estimation.OrthogonalLearning.drMixedDirDeriv S hA hε_pos h_overlap_η₀ Θ Θ_set Θ_convex θ₀ θ₀_mem eval eval_meas eval_θ₀ θ₀_minimizes D ND) → Causalean.Estimation.OrthogonalLearning.NeymanOrthogLoss (Causalean.Estimation.OrthogonalLearning.drLearningSystem S Θ Θ_set Θ_convex θ₀ θ₀_mem eval eval_meas eval_θ₀ θ₀_minimizes) (Causalean.Estimation.OrthogonalLearning.drMixedDirDeriv S hA hε_pos h_overlap_η₀ Θ Θ_set Θ_convex θ₀ θ₀_mem eval eval_meas eval_θ₀ θ₀_minimizes D ND)","l":723,"f":"Causalean/Estimation/CATE/OrthogonalLearning/DRLearner/Analytic.lean"},{"n":"Causalean.Estimation.OrthogonalLearning.drNeymanOrthog_witness","k":"theorem","m":"Causalean.Estimation.CATE.OrthogonalLearning.DRLearner","a":"Estimation/CATE/OrthogonalLearning/DRLearner","t":1,"d":"DR-Learner loss orthogonality (prop:est-osl-dr-loss-orthogonal). For a CATE estimation system built on a potential-outcome model that satisfies the back-door identification assumptions with the propensity score bounded away from 0 and 1 by some margin (strict overlap), fix a convex candidate target class inside an inner-product space together with a real-valued evaluation map on it, and suppose the candidate θ₀ belongs to this class, every candidate's evaluation is measurable, θ₀'s evaluation agrees pointwise with the true value-space CATE, and θ₀ minimizes the population AIPW pseudo-outcome squared-loss risk against the true nuisance over the candidate class. If, for the resulting DR-Learner learning system, a dominated-convergence bridge licenses passing the limit defining the mixed target/nuisance directional derivative through the integral and the integrated target-directional score has vanishing derivative, at every point of the bounded nuisance slice and for every candidate target, along the segment toward that point from the true nuisance, then the DR-Learner squared loss is Neyman-orthogonal: its integrated mixed directional derivative between target and nuisance directions vanishes at the truth (θ₀, η₀) for every admissible target and nuisance direction.","s":"∀ {P : Causalean.PO.POSystem} {γ : Type u_1} [inst : MeasurableSpace γ] [inst_1 : StandardBorelSpace P.Ω] [inst_2 : MeasureTheory.IsFiniteMeasure P.μ] (S : Causalean.Estimation.CATE.CATEEstimationSystem P γ), S.Assumptions → (∃ ε > 0, S.StrictOverlap ε) → ∀ (Θ : Type u_2) [inst_3 : NormedAddCommGroup Θ] [inst_4 : InnerProductSpace ℝ Θ] (Θ_set : Set Θ) (Θ_convex : Convex ℝ Θ_set) (θ₀ : Θ) (θ₀_mem : θ₀ ∈ Θ_set) (eval : Θ → γ → ℝ) (eval_meas : ∀ (θ : Θ), Measurable (eval θ)) (eval_θ₀ : ∀ (x : γ), eval θ₀ x = S.τ_val x) (θ₀_minimizes : Causalean.Estimation.OrthogonalLearning.DRThetaMinimizes S Θ_set θ₀ eval) (M : Causalean.Estimation.OrthogonalLearning.HasMixedDirDeriv (Causalean.Estimation.OrthogonalLearning.drLearningSystem S Θ Θ_set Θ_convex θ₀ θ₀_mem eval eval_meas eval_θ₀ θ₀_minimizes)), Causalean.Estimation.OrthogonalLearning.MixedScoreDCTBridge (Causalean.Estimation.OrthogonalLearning.drLearningSystem S Θ Θ_set Θ_convex θ₀ θ₀_mem eval eval_meas eval_θ₀ θ₀_minimizes) M → (∀ θ ∈ Θ_set, ∀ η ∈ Causalean.Estimation.OrthogonalLearning.BoundedNuisanceDirs S.η₀, Filter.Tendsto (fun t => (∫ (z : γ × Bool × ℝ), (M.Dθ_at (S.η₀ + t • (η - S.η₀))).dℓ_θ θ z ∂S.P_Z - ∫ (z : γ × Bool × ℝ), (M.Dθ_at S.η₀).dℓ_θ θ z ∂S.P_Z) / t) (nhdsWithin 0 {0}ᶜ) (nhds 0)) → Causalean.Estimation.OrthogonalLearning.NeymanOrthogLoss (Causalean.Estimation.OrthogonalLearning.drLearningSystem S Θ Θ_set Θ_convex θ₀ θ₀_mem eval eval_meas eval_θ₀ θ₀_minimizes) M","l":185,"f":"Causalean/Estimation/CATE/OrthogonalLearning/DRLearner.lean"},{"n":"Causalean.Estimation.OrthogonalLearning.dr_loss_uniformly_bounded","k":"theorem","m":"Causalean.Estimation.CATE.OrthogonalLearning.LocalEmpProcess.DRLearner","a":"Estimation/CATE/OrthogonalLearning/LocalEmpProcess","t":2,"d":"Bridge: DR-Learner squared loss is bounded under (M_Θ, M_Y, M_μ, ε).","s":"∀ {P : Causalean.PO.POSystem} {γ : Type u_1} [inst : MeasurableSpace γ] [inst_1 : StandardBorelSpace P.Ω] [inst_2 : MeasureTheory.IsFiniteMeasure P.μ] (S : Causalean.Estimation.CATE.CATEEstimationSystem P γ) (Θ : Type u_2) [inst_3 : NormedAddCommGroup Θ] [inst_4 : InnerProductSpace ℝ Θ] (Θ_set : Set Θ) (Θ_convex : Convex ℝ Θ_set) (θ₀ : Θ) (θ₀_mem : θ₀ ∈ Θ_set) (eval : Θ → γ → ℝ) (eval_meas : ∀ (θ : Θ), Measurable (eval θ)) (eval_θ₀ : ∀ (x : γ), eval θ₀ x = S.τ_val x) (θ₀_minimizes : Causalean.Estimation.OrthogonalLearning.DRThetaMinimizes S Θ_set θ₀ eval) {M_Θ M_Y M_μ ε : ℝ}, Causalean.Estimation.OrthogonalLearning.DREvalBounded Θ_set eval M_Θ → Causalean.Estimation.OrthogonalLearning.DROutcomeBounded S M_Y → ∀ (h : Causalean.Estimation.ATE.NuisanceVec γ), Causalean.Estimation.OrthogonalLearning.DRNuisanceMuBounded h M_μ → Causalean.Estimation.OrthogonalLearning.DRNuisanceOverlap S Θ_set h ε → Causalean.Estimation.OrthogonalLearning.UniformlyBoundedLossAE (Causalean.Estimation.OrthogonalLearning.drLearningSystem S Θ Θ_set Θ_convex θ₀ θ₀_mem eval eval_meas eval_θ₀ θ₀_minimizes) h ((M_Θ + 2 * M_μ + 2 * (M_Y + M_μ) / ε) ^ 2)","l":86,"f":"Causalean/Estimation/CATE/OrthogonalLearning/LocalEmpProcess/DRLearner.lean"},{"n":"Causalean.Estimation.OrthogonalLearning.dr_scoreFlat","k":"theorem","m":"Causalean.Estimation.CATE.OrthogonalLearning.DRLearner.Analytic","a":"Estimation/CATE/OrthogonalLearning/DRLearner","t":2,"d":"The score-flatness hypothesis for drMixedDirDeriv.","s":"∀ {P : Causalean.PO.POSystem} {γ : Type u_1} [inst : MeasurableSpace γ] [inst_1 : StandardBorelSpace P.Ω] [inst_2 : MeasureTheory.IsFiniteMeasure P.μ] (S : Causalean.Estimation.CATE.CATEEstimationSystem P γ) (hA : S.Assumptions) {ε : ℝ} (hε_pos : 0 < ε) (h_overlap_η₀ : S.η₀ ∈ Causalean.Estimation.ATE.BackdoorEstimationSystem.H_ε ε) (Θ : Type u_2) [inst_3 : NormedAddCommGroup Θ] [inst_4 : InnerProductSpace ℝ Θ] (Θ_set : Set Θ) (Θ_convex : Convex ℝ Θ_set) (θ₀ : Θ) (θ₀_mem : θ₀ ∈ Θ_set) (eval : Θ → γ → ℝ) (eval_meas : ∀ (θ : Θ), Measurable (eval θ)) (eval_θ₀ : ∀ (x : γ), eval θ₀ x = S.τ_val x) (θ₀_minimizes : Causalean.Estimation.OrthogonalLearning.DRThetaMinimizes S Θ_set θ₀ eval) (D : Causalean.Estimation.OrthogonalLearning.EvalDirDeriv Θ_set θ₀ eval) (ND : Causalean.Estimation.OrthogonalLearning.NuisanceDirDeriv S.η₀), Causalean.Estimation.OrthogonalLearning.MixedScoreDCTBridge (Causalean.Estimation.OrthogonalLearning.drLearningSystem S Θ Θ_set Θ_convex θ₀ θ₀_mem eval eval_meas eval_θ₀ θ₀_minimizes) (Causalean.Estimation.OrthogonalLearning.drMixedDirDeriv S hA hε_pos h_overlap_η₀ Θ Θ_set Θ_convex θ₀ θ₀_mem eval eval_meas eval_θ₀ θ₀_minimizes D ND) → ∀ θ ∈ Θ_set, ∀ η ∈ Causalean.Estimation.OrthogonalLearning.BoundedNuisanceDirs S.η₀, Filter.Tendsto (fun t => (∫ (z : γ × Bool × ℝ), ((Causalean.Estimation.OrthogonalLearning.drMixedDirDeriv S hA hε_pos h_overlap_η₀ Θ Θ_set Θ_convex θ₀ θ₀_mem eval eval_meas eval_θ₀ θ₀_minimizes D ND).Dθ_at (S.η₀ + t • (η - S.η₀))).dℓ_θ θ z ∂S.P_Z - ∫ (z : γ × Bool × ℝ), ((Causalean.Estimation.OrthogonalLearning.drMixedDirDeriv S hA hε_pos h_overlap_η₀ Θ Θ_set Θ_convex θ₀ θ₀_mem eval eval_meas eval_θ₀ θ₀_minimizes D ND).Dθ_at S.η₀).dℓ_θ θ z ∂S.P_Z) / t) (nhdsWithin 0 {0}ᶜ) (nhds 0)","l":633,"f":"Causalean/Estimation/CATE/OrthogonalLearning/DRLearner/Analytic.lean"},{"n":"Causalean.Estimation.OrthogonalLearning.dr_scoreZero_of_bounded","k":"theorem","m":"Causalean.Estimation.CATE.OrthogonalLearning.DRLearner.Analytic","a":"Estimation/CATE/OrthogonalLearning/DRLearner","t":2,"d":"Discharge of hScoreZero for bounded nuisance directions.","s":"∀ {P : Causalean.PO.POSystem} {γ : Type u_1} [inst : MeasurableSpace γ] [inst_1 : StandardBorelSpace P.Ω] [inst_2 : MeasureTheory.IsFiniteMeasure P.μ] (S : Causalean.Estimation.CATE.CATEEstimationSystem P γ), S.Assumptions → ∀ {ε : ℝ}, 0 < ε → S.η₀ ∈ Causalean.Estimation.ATE.BackdoorEstimationSystem.H_ε ε → ∀ (Θ : Type u_2) [inst_3 : NormedAddCommGroup Θ] [inst_4 : InnerProductSpace ℝ Θ] (Θ_set : Set Θ) (θ₀ : Θ) (eval : Θ → γ → ℝ) (D : Causalean.Estimation.OrthogonalLearning.EvalDirDeriv Θ_set θ₀ eval) (ND : Causalean.Estimation.OrthogonalLearning.NuisanceDirDeriv S.η₀), ∀ θ ∈ Θ_set, ∀ (η : Causalean.Estimation.ATE.NuisanceVec γ), (∃ Cμ, ∀ (b : Bool) (x : γ), |(η - S.η₀).μ_fn b x| ≤ Cμ) → (∃ Ce, ∀ (x : γ), |(η - S.η₀).e_fn x| ≤ Ce) → ∫ (z : γ × Bool × ℝ), ND.dPhi η z * D.dEval θ z.1 ∂S.P_Z = 0","l":470,"f":"Causalean/Estimation/CATE/OrthogonalLearning/DRLearner/Analytic.lean"},{"n":"Causalean.Estimation.OrthogonalLearning.empRiskFoldB","k":"def","m":"Causalean.Estimation.OrthogonalLearning.PluginERM","a":"Estimation/OrthogonalLearning","t":1,"d":"Empirical risk on the estimation fold B(n): (1 / |B(n)|) · Σ_{i ∈ B(n)} ℓ (Z_i ω) θ g.","s":"{Ω : Type u_1} → [inst : MeasurableSpace Ω] → {μ : MeasureTheory.Measure Ω} → {Z : Type u_2} → [inst_1 : MeasurableSpace Z] → {P_Z : MeasureTheory.Measure Z} → {Θ : Type u_3} → [inst_2 : NormedAddCommGroup Θ] → [inst_3 : InnerProductSpace ℝ Θ] → {G : Type u_4} → [inst_4 : AddCommGroup G] → [inst_5 : Module ℝ G] → Causalean.Estimation.OrthogonalLearning.LearningSystem Ω μ Z P_Z Θ G → (S_iid : Causalean.Stat.IIDSample Ω Z μ P_Z) → Causalean.Stat.OneShotSplit S_iid → ℕ → Ω → Θ → G → ℝ","l":45,"f":"Causalean/Estimation/OrthogonalLearning/PluginERM.lean"},{"n":"Causalean.Estimation.OrthogonalLearning.foldBCoord","k":"def","m":"Causalean.Estimation.OrthogonalLearning.LocalEmpProcess.RandomNuisance","a":"Estimation/OrthogonalLearning/LocalEmpProcess","t":1,"d":"The fold-B coordinate map Y : Ω → (Fin m → Z), m = card (foldB n), reindexing the fold-B subsample by the canonical order isomorphism.","s":"{Ω : Type u_1} → [inst : MeasurableSpace Ω] → {μ : MeasureTheory.Measure Ω} → {Z : Type u_2} → [inst_1 : MeasurableSpace Z] → {P_Z : MeasureTheory.Measure Z} → (S_iid : Causalean.Stat.IIDSample Ω Z μ P_Z) → (split : Causalean.Stat.OneShotSplit S_iid) → (n : ℕ) → Ω → Fin (split.foldB n).card → Z","l":77,"f":"Causalean/Estimation/OrthogonalLearning/LocalEmpProcess/RandomNuisance.lean"},{"n":"Causalean.Estimation.OrthogonalLearning.foldBCoord_law","k":"theorem","m":"Causalean.Estimation.OrthogonalLearning.LocalEmpProcess.RandomNuisance","a":"Estimation/OrthogonalLearning/LocalEmpProcess","t":2,"d":"Fold-B coordinate law. μ.map (foldBCoord n) = Measure.pi P_Z. This identifies the validation-fold coordinates as an i.i.d. product sample.","s":"∀ {Ω : Type u_1} [inst : MeasurableSpace Ω] {μ : MeasureTheory.Measure Ω} {Z : Type u_2} [inst_1 : MeasurableSpace Z] {P_Z : MeasureTheory.Measure Z} {S_iid : Causalean.Stat.IIDSample Ω Z μ P_Z} [MeasureTheory.IsProbabilityMeasure μ] [MeasureTheory.IsProbabilityMeasure P_Z] (split : Causalean.Stat.OneShotSplit S_iid) (n : ℕ), MeasureTheory.Measure.map (Causalean.Estimation.OrthogonalLearning.foldBCoord S_iid split n) μ = MeasureTheory.Measure.pi fun x => P_Z","l":112,"f":"Causalean/Estimation/OrthogonalLearning/LocalEmpProcess/RandomNuisance.lean"},{"n":"Causalean.Estimation.OrthogonalLearning.foldBCoord_meas","k":"theorem","m":"Causalean.Estimation.OrthogonalLearning.LocalEmpProcess.RandomNuisance","a":"Estimation/OrthogonalLearning/LocalEmpProcess","t":2,"d":"The fold-B coordinate map is measurable.","s":"∀ {Ω : Type u_1} [inst : MeasurableSpace Ω] {μ : MeasureTheory.Measure Ω} {Z : Type u_2} [inst_1 : MeasurableSpace Z] {P_Z : MeasureTheory.Measure Z} {S_iid : Causalean.Stat.IIDSample Ω Z μ P_Z} (split : Causalean.Stat.OneShotSplit S_iid) (n : ℕ), Measurable (Causalean.Estimation.OrthogonalLearning.foldBCoord S_iid split n)","l":541,"f":"Causalean/Estimation/OrthogonalLearning/LocalEmpProcess/RandomNuisance.lean"},{"n":"Causalean.Estimation.OrthogonalLearning.foldB_pi_law","k":"theorem","m":"Causalean.Estimation.OrthogonalLearning.LocalEmpProcess.Rademacher","a":"Estimation/OrthogonalLearning/LocalEmpProcess","t":1,"d":"Fold-B joint-law identification (public alias). For an IID sample S split via split, at any fold size n, the joint distribution of the fold-B subsample, mapped from μ via ω ↦ (i ↦ S.Z i ω) indexed by i ∈ split.foldB n, equals the product measure Measure.pi (fun _ ↦ P).","s":"∀ {Ω : Type u_1} [inst : MeasurableSpace Ω] {μ : MeasureTheory.Measure Ω} {X : Type u_2} [inst_1 : MeasurableSpace X] {P : MeasureTheory.Measure X} [MeasureTheory.IsProbabilityMeasure μ] [MeasureTheory.IsProbabilityMeasure P] (S : Causalean.Stat.IIDSample Ω X μ P) (split : Causalean.Stat.OneShotSplit S) (n : ℕ), MeasureTheory.Measure.map (fun ω i => S.Z (↑i) ω) μ = MeasureTheory.Measure.pi fun x => P","l":96,"f":"Causalean/Estimation/OrthogonalLearning/LocalEmpProcess/Rademacher.lean"},{"n":"Causalean.Estimation.OrthogonalLearning.localEmpProcessModulus_drLearner","k":"theorem","m":"Causalean.Estimation.CATE.OrthogonalLearning.LocalEmpProcess.DRLearner","a":"Estimation/CATE/OrthogonalLearning/LocalEmpProcess","t":1,"d":"DR-Learner bounded-loss Rademacher modulus. Consider the doubly robust orthogonal-learning system for conditional treatment effects, cross-fitted via a one-shot sample split, whose truth-identifying candidate belongs to the target set, has a measurable evaluation map, and recovers the population CATE function through that evaluation map. Suppose the candidate evaluation maps are uniformly bounded, the observed outcome is almost-surely bounded, the realised nuisance outcome-regression is uniformly bounded on both treatment arms, and the realised propensity stays away from 0 and 1 by a fixed margin; suppose also that the resulting doubly robust loss is continuous in the candidate over the target set at that realised nuisance, a sequence R n bounds the Rademacher complexity of the centred loss class on the estimation fold, the truth-identifying candidate still minimizes the clamped population loss at the realised nuisance, and the confidence level δ lies strictly above 0 and at most 1. Then there is a nonnegative constant b for which the system obeys the local empirical-process modulus condition at the explicit rate ρ n = √(2 R n + 2b · √(2 log(1/δ) / |B(n)|)) on nonempty estimation folds, with boundary value ρ n = √(2b) on empty folds.","s":"∀ {P : Causalean.PO.POSystem} {γ : Type u_1} [inst : MeasurableSpace γ] [inst_1 : StandardBorelSpace P.Ω] [inst_2 : MeasureTheory.IsFiniteMeasure P.μ] [MeasureTheory.IsProbabilityMeasure P.μ] (S : Causalean.Estimation.CATE.CATEEstimationSystem P γ) (Θ : Type u_2) [inst_4 : NormedAddCommGroup Θ] [inst_5 : InnerProductSpace ℝ Θ] (Θ_set : Set Θ) (Θ_convex : Convex ℝ Θ_set) (θ₀ : Θ) (θ₀_mem : θ₀ ∈ Θ_set) (eval : Θ → γ → ℝ) (eval_meas : ∀ (θ : Θ), Measurable (eval θ)) (eval_θ₀ : ∀ (x : γ), eval θ₀ x = S.τ_val x) (θ₀_minimizes : Causalean.Estimation.OrthogonalLearning.DRThetaMinimizes S Θ_set θ₀ eval) (S_iid : Causalean.Stat.IIDSample P.Ω (γ × Bool × ℝ) P.μ S.P_Z) (split : Causalean.Stat.OneShotSplit S_iid) {M_Θ M_Y M_μ ε : ℝ}, Causalean.Estimation.OrthogonalLearning.DREvalBounded Θ_set eval M_Θ → Causalean.Estimation.OrthogonalLearning.DROutcomeBounded S M_Y → ∀ (h : Causalean.Estimation.ATE.NuisanceVec γ), Causalean.Estimation.OrthogonalLearning.DRNuisanceMuBounded h M_μ → Causalean.Estimation.OrthogonalLearning.DRNuisanceOverlap S Θ_set h ε → Causalean.Estimation.OrthogonalLearning.LossContinuousOnΘset (Causalean.Estimation.OrthogonalLearning.drLearningSystem S Θ Θ_set Θ_convex θ₀ θ₀_mem eval eval_meas eval_θ₀ θ₀_minimizes) h → ∀ (idx : ℕ → ↑(Causalean.Estimation.OrthogonalLearning.drLearningSystem S Θ Θ_set Θ_convex θ₀ θ₀_mem eval eval_meas eval_θ₀ θ₀_minimizes).Θ_set), DenseRange idx → ∀ (R : ℕ → ℝ), Causalean.Estimation.OrthogonalLearning.RademacherBound (Causalean.Estimation.OrthogonalLearning.drLearningSystem S Θ Θ_set Θ_convex θ₀ θ₀_mem eval eval_meas eval_θ₀ θ₀_minimizes) S_iid split h idx R → Causalean.Estimation.OrthogonalLearning.DRClampedThetaMinimizes S Θ_set θ₀ eval ((M_Θ + 2 * M_μ + 2 * (M_Y + M_μ) / ε) ^ 2) → ∀ {δ : ℝ}, 0 < δ → δ ≤ 1 → ∃ b, 0 ≤ b ∧ Causalean.Estimation.OrthogonalLearning.LocalEmpProcessModulus (Causalean.Estimation.OrthogonalLearning.drLearningSystem S Θ Θ_set Θ_convex θ₀ θ₀_mem eval eval_meas eval_θ₀ θ₀_minimizes) S_iid split (fun n => √(if (split.foldB n).card = 0 then 2 * b else 2 * R n + 2 * b * √(2 * Real.log (1 / δ) / ↑(split.foldB n).card))) δ h","l":250,"f":"Causalean/Estimation/CATE/OrthogonalLearning/LocalEmpProcess/DRLearner.lean"},{"n":"Causalean.Estimation.OrthogonalLearning.localEmpProcessModulus_localized_drLearner","k":"theorem","m":"Causalean.Estimation.CATE.OrthogonalLearning.LocalEmpProcess.LocalizedDRLearner","a":"Estimation/CATE/OrthogonalLearning/LocalEmpProcess","t":1,"d":"DR-Learner localized modulus, sharp Foster–Syrgkanis form. For the DR-Learner CATE estimation system, suppose the evaluation functional is measurable in its parameter and recovers the true CATE at a parameter θ₀ in the constraint set, and that the evaluation functional, outcome, and fixed nuisance conditional-mean are uniformly bounded while the nuisance's propensity score satisfies ε-overlap. Assume the centred DR-loss is continuous in θ, a clamped version of θ₀ minimizes it, its population Rademacher complexity along a dense index sequence is controlled by a sub-root envelope ψ with respect to a seminorm that is invariant under almost-everywhere modification, and the same Rademacher upper bound extends to loss differences across the whole constraint set, together with Lipschitz and diameter control of the centred loss increments — nonnegative Lipschitz constant L, a diameter bound Rmax dominating every critical radius criticalRadius (ψ m), and the sub-root fixed-point property, plus boundedness and integrability of the empirical star-hull Rademacher process needed by the localization bridge and a confidence level in $(0,1]$ together with the Foster–Syrgkanis critical-radius domination inequality across dyadic shell counts. Then there is a nonnegative envelope b such that the DR-Learner system satisfies the localized empirical-process modulus predicate at rate ρ n = (8L+3)·criticalRadius (ψ |B(n)|) on nonempty validation folds, falling back to √(2b) when the fold is empty.","s":"∀ {P : Causalean.PO.POSystem} {γ : Type u_1} [inst : MeasurableSpace γ] [inst_1 : StandardBorelSpace P.Ω] [inst_2 : MeasureTheory.IsFiniteMeasure P.μ] [MeasureTheory.IsProbabilityMeasure P.μ] (S : Causalean.Estimation.CATE.CATEEstimationSystem P γ) (Θ : Type u_2) [inst_4 : NormedAddCommGroup Θ] [inst_5 : InnerProductSpace ℝ Θ] (Θ_set : Set Θ) (Θ_convex : Convex ℝ Θ_set) [Nonempty ↑Θ_set] [Countable ↑Θ_set] (θ₀ : Θ) (θ₀_mem : θ₀ ∈ Θ_set) (eval : Θ → γ → ℝ) (eval_meas : ∀ (θ : Θ), Measurable (eval θ)) (eval_θ₀ : ∀ (x : γ), eval θ₀ x = S.τ_val x) (θ₀_minimizes : Causalean.Estimation.OrthogonalLearning.DRThetaMinimizes S Θ_set θ₀ eval) (S_iid : Causalean.Stat.IIDSample P.Ω (γ × Bool × ℝ) P.μ S.P_Z) (split : Causalean.Stat.OneShotSplit S_iid) {M_Θ M_Y M_μ ε : ℝ}, Causalean.Estimation.OrthogonalLearning.DREvalBounded Θ_set eval M_Θ → Causalean.Estimation.OrthogonalLearning.DROutcomeBounded S M_Y → ∀ (h : Causalean.Estimation.ATE.NuisanceVec γ), Causalean.Estimation.OrthogonalLearning.DRNuisanceMuBounded h M_μ → Causalean.Estimation.OrthogonalLearning.DRNuisanceOverlap S Θ_set h ε → (∀ (z : γ × Bool × ℝ), Continuous fun θ => (Causalean.Estimation.OrthogonalLearning.drLearningSystem S Θ Θ_set Θ_convex θ₀ θ₀_mem eval eval_meas eval_θ₀ θ₀_minimizes).ℓ z (↑θ) h) → ∀ (idx : ℕ → ↑(Causalean.Estimation.OrthogonalLearning.drLearningSystem S Θ Θ_set Θ_convex θ₀ θ₀_mem eval eval_meas eval_θ₀ θ₀_minimizes).Θ_set), DenseRange idx → ∀ {norm : (γ × Bool × ℝ → ℝ) → ℝ} {ψ : ℕ → ℝ → ℝ}, Causalean.Estimation.OrthogonalLearning.DRCriticalRadius S Θ_set eval (fun k => ⟨↑(idx k), (idx k).property⟩) h norm ψ → (∀ (F F' : γ × Bool × ℝ → ℝ), F =ᵐ[S.P_Z] F' → norm F = norm F') → ∀ {Rmax L : ℝ}, 0 ≤ L → (∀ θ ∈ (Causalean.Estimation.OrthogonalLearning.drLearningSystem S Θ Θ_set Θ_convex θ₀ θ₀_mem eval eval_meas eval_θ₀ θ₀_minimizes).Θ_set, (norm fun z => (Causalean.Estimation.OrthogonalLearning.drLearningSystem S Θ Θ_set Θ_convex θ₀ θ₀_mem eval eval_meas eval_θ₀ θ₀_minimizes).ℓ z θ h - (Causalean.Estimation.OrthogonalLearning.drLearningSystem S Θ Θ_set Θ_convex θ₀ θ₀_mem eval eval_meas eval_θ₀ θ₀_minimizes).ℓ z (Causalean.Estimation.OrthogonalLearning.drLearningSystem S Θ Θ_set Θ_convex θ₀ θ₀_mem eval eval_meas eval_θ₀ θ₀_minimizes).θ₀ h) ≤ L * ‖θ - (Causalean.Estimation.OrthogonalLearning.drLearningSystem S Θ Θ_set Θ_convex θ₀ θ₀_mem eval eval_meas eval_θ₀ θ₀_minimizes).θ₀‖) → (∀ θ ∈ (Causalean.Estimation.OrthogonalLearning.drLearningSystem S Θ Θ_set Θ_convex θ₀ θ₀_mem eval eval_meas eval_θ₀ θ₀_minimizes).Θ_set, (norm fun z => (Causalean.Estimation.OrthogonalLearning.drLearningSystem S Θ Θ_set Θ_convex θ₀ θ₀_mem eval eval_meas eval_θ₀ θ₀_minimizes).ℓ z θ h - (Causalean.Estimation.OrthogonalLearning.drLearningSystem S Θ Θ_set Θ_convex θ₀ θ₀_mem eval eval_meas eval_θ₀ θ₀_minimizes).ℓ z (Causalean.Estimation.OrthogonalLearning.drLearningSystem S Θ Θ_set Θ_convex θ₀ θ₀_mem eval eval_meas eval_θ₀ θ₀_minimizes).θ₀ h) ≤ Rmax) → (∀ (m : ℕ), Causalean.Stat.Concentration.criticalRadius (ψ m) ≤ Rmax) → (∀ (m : ℕ), 0 < Causalean.Stat.Concentration.criticalRadius (ψ m)) → (∀ (m : ℕ), ψ m (Causalean.Stat.Concentration.criticalRadius (ψ m)) ≤ Causalean.Stat.Concentration.criticalRadius (ψ m) ^ 2) → (∀ (m : ℕ), Causalean.Stat.Concentration.RademacherUpperBound (fun θ z => (Causalean.Estimation.OrthogonalLearning.drLearningSystem S Θ Θ_set Θ_convex θ₀ θ₀_mem eval eval_meas eval_θ₀ θ₀_minimizes).ℓ z (↑θ) h - (Causalean.Estimation.OrthogonalLearning.drLearningSystem S Θ Θ_set Θ_convex θ₀ θ₀_mem eval eval_meas eval_θ₀ θ₀_minimizes).ℓ z (Causalean.Estimation.OrthogonalLearning.drLearningSystem S Θ Θ_set Θ_convex θ₀ θ₀_mem eval eval_meas eval_θ₀ θ₀_minimizes).θ₀ h) norm S.P_Z id m (ψ m)) → (∀ (m : ℕ) (r : ℝ) (S_fin : Fin m → γ × Bool × ℝ) (σ : Signs m), BddAbove (Set.range fun p => |(↑m)⁻¹ * ∑ k, ↑↑(σ k) * Causalean.Stat.Concentration.starHullZeroOut (fun θ z => (Causalean.Estimation.OrthogonalLearning.drLearningSystem S Θ Θ_set Θ_convex θ₀ θ₀_mem eval eval_meas eval_θ₀ θ₀_minimizes).ℓ z (↑θ) h - (Causalean.Estimation.OrthogonalLearning.drLearningSystem S Θ Θ_set Θ_convex θ₀ θ₀_mem eval eval_meas eval_θ₀ θ₀_minimizes).ℓ z (Causalean.Estimation.OrthogonalLearning.drLearningSystem S Θ Θ_set Θ_convex θ₀ θ₀_mem eval eval_meas eval_θ₀ θ₀_minimizes).θ₀ h) norm r p (S_fin k)|)) → (∀ (m : ℕ) (r : ℝ), MeasureTheory.Integrable (fun ω => empiricalRademacherComplexity m (Causalean.Stat.Concentration.starHullZeroOut (fun θ z => (Causalean.Estimation.OrthogonalLearning.drLearningSystem S Θ Θ_set Θ_convex θ₀ θ₀_mem eval eval_meas eval_θ₀ θ₀_minimizes).ℓ z (↑θ) h - (Causalean.Estimation.OrthogonalLearning.drLearningSystem S Θ Θ_set Θ_convex θ₀ θ₀_mem eval eval_meas eval_θ₀ θ₀_minimizes).ℓ z (Causalean.Estimation.OrthogonalLearning.drLearningSystem S Θ Θ_set Θ_convex θ₀ θ₀_mem eval eval_meas eval_θ₀ θ₀_minimizes).θ₀ h) norm r) (id ∘ ω)) (MeasureTheory.Measure.pi fun x => S.P_Z)) → Causalean.Estimation.OrthogonalLearning.CenteredClampedThetaMinimizes (Causalean.Estimation.OrthogonalLearning.drLearningSystem S Θ Θ_set Θ_convex θ₀ θ₀_mem eval eval_meas eval_θ₀ θ₀_minimizes) (2 * (M_Θ + 2 * M_μ + 2 * (M_Y + M_μ) / ε) ^ 2) → ∀ {δ : ℝ}, 0 < δ → δ ≤ 1 → (∀ (n K : ℕ), 0 < (split.foldB n).card → Rmax ≤ Causalean.Stat.Concentration.criticalRadius (ψ (split.foldB n).card) * 2 ^ K → 2 * (M_Θ + 2 * M_μ + 2 * (M_Y + M_μ) / ε) ^ 2 * √(2 * Real.log (2 * (↑K + 1) / δ) / ↑(split.foldB n).card) ≤ Causalean.Stat.Concentration.criticalRadius (ψ (split.foldB n).card) ^ 2) → ∃ b, 0 ≤ b ∧ Causalean.Estimation.OrthogonalLearning.LocalEmpProcessModulus (Causalean.Estimation.OrthogonalLearning.drLearningSystem S Θ Θ_set Θ_convex θ₀ θ₀_mem eval eval_meas eval_θ₀ θ₀_minimizes) S_iid split (fun n => if (split.foldB n).card = 0 then √(2 * b) else (8 * L + 3) * Causalean.Stat.Concentration.criticalRadius (ψ (split.foldB n).card)) δ h","l":82,"f":"Causalean/Estimation/CATE/OrthogonalLearning/LocalEmpProcess/LocalizedDRLearner.lean"},{"n":"Causalean.Estimation.OrthogonalLearning.localEmpProcessModulus_localized_drLearner_bounded","k":"theorem","m":"Causalean.Estimation.CATE.OrthogonalLearning.LocalEmpProcess.LocalizedDRLearner","a":"Estimation/CATE/OrthogonalLearning/LocalEmpProcess","t":2,"d":"DR-Learner bounded-loss localized modulus.","s":"∀ {P : Causalean.PO.POSystem} {γ : Type u_1} [inst : MeasurableSpace γ] [inst_1 : StandardBorelSpace P.Ω] [inst_2 : MeasureTheory.IsFiniteMeasure P.μ] [MeasureTheory.IsProbabilityMeasure P.μ] (S : Causalean.Estimation.CATE.CATEEstimationSystem P γ) (Θ : Type u_2) [inst_4 : NormedAddCommGroup Θ] [inst_5 : InnerProductSpace ℝ Θ] (Θ_set : Set Θ) (Θ_convex : Convex ℝ Θ_set) [Nonempty ↑Θ_set] (θ₀ : Θ) (θ₀_mem : θ₀ ∈ Θ_set) (eval : Θ → γ → ℝ) (eval_meas : ∀ (θ : Θ), Measurable (eval θ)) (eval_θ₀ : ∀ (x : γ), eval θ₀ x = S.τ_val x) (θ₀_minimizes : Causalean.Estimation.OrthogonalLearning.DRThetaMinimizes S Θ_set θ₀ eval) (S_iid : Causalean.Stat.IIDSample P.Ω (γ × Bool × ℝ) P.μ S.P_Z) (split : Causalean.Stat.OneShotSplit S_iid) {M_Θ M_Y M_μ ε : ℝ}, Causalean.Estimation.OrthogonalLearning.DREvalBounded Θ_set eval M_Θ → Causalean.Estimation.OrthogonalLearning.DROutcomeBounded S M_Y → ∀ (h : Causalean.Estimation.ATE.NuisanceVec γ), Causalean.Estimation.OrthogonalLearning.DRNuisanceMuBounded h M_μ → Causalean.Estimation.OrthogonalLearning.DRNuisanceOverlap S Θ_set h ε → (∀ (z : γ × Bool × ℝ), Continuous fun θ => (Causalean.Estimation.OrthogonalLearning.drLearningSystem S Θ Θ_set Θ_convex θ₀ θ₀_mem eval eval_meas eval_θ₀ θ₀_minimizes).ℓ z (↑θ) h) → ∀ (idx : ℕ → ↑(Causalean.Estimation.OrthogonalLearning.drLearningSystem S Θ Θ_set Θ_convex θ₀ θ₀_mem eval eval_meas eval_θ₀ θ₀_minimizes).Θ_set), DenseRange idx → ∀ {norm : (γ × Bool × ℝ → ℝ) → ℝ} {ψ : ℕ → ℝ → ℝ}, Causalean.Estimation.OrthogonalLearning.DRCriticalRadius S Θ_set eval (fun k => ⟨↑(idx k), (idx k).property⟩) h norm ψ → ∀ {δ : ℝ}, 0 < δ → δ ≤ 1 → ∃ b, 0 ≤ b ∧ Causalean.Estimation.OrthogonalLearning.LocalEmpProcessModulus (Causalean.Estimation.OrthogonalLearning.drLearningSystem S Θ Θ_set Θ_convex θ₀ θ₀_mem eval eval_meas eval_θ₀ θ₀_minimizes) S_iid split (fun _n => √(2 * (2 * b))) δ h","l":306,"f":"Causalean/Estimation/CATE/OrthogonalLearning/LocalEmpProcess/LocalizedDRLearner.lean"},{"n":"Causalean.Estimation.OrthogonalLearning.localEmpProcessModulus_of_bounded_rademacher","k":"theorem","m":"Causalean.Estimation.OrthogonalLearning.LocalEmpProcess.Rademacher","a":"Estimation/OrthogonalLearning/LocalEmpProcess","t":1,"d":"Bounded-loss Rademacher bridge theorem. Assume b is nonnegative, that the loss magnitude is uniformly bounded by b over the parameter set, and that the loss is continuous in the parameter on the parameter set. Given a sequence R n that is nonnegative and upper-bounds the population Rademacher complexity of the centred loss class on the fold-B sample at every sample size, then for any confidence level 0 < δ ≤ 1 the local empirical-process modulus condition holds, with rate ρ n := √(2 · b) when the fold-B sample is empty and ρ n := √(2 · R n + 2 · b · √(2 · log(1/δ) / |foldB n|)) otherwise.","s":"∀ {Ω : Type u_1} [inst : MeasurableSpace Ω] {μ : MeasureTheory.Measure Ω} {Z : Type u_2} [inst_1 : MeasurableSpace Z] {P_Z : MeasureTheory.Measure Z} {Θ : Type u_3} [inst_2 : NormedAddCommGroup Θ] [inst_3 : InnerProductSpace ℝ Θ] {G : Type u_4} [inst_4 : AddCommGroup G] [inst_5 : Module ℝ G] (S : Causalean.Estimation.OrthogonalLearning.LearningSystem Ω μ Z P_Z Θ G) [MeasureTheory.IsProbabilityMeasure μ] (S_iid : Causalean.Stat.IIDSample Ω Z μ P_Z) (split : Causalean.Stat.OneShotSplit S_iid) {b : ℝ}, 0 ≤ b → ∀ (g : G), Causalean.Estimation.OrthogonalLearning.UniformlyBoundedLoss S g b → Causalean.Estimation.OrthogonalLearning.LossContinuousOnΘset S g → ∀ (idx : ℕ → ↑S.Θ_set), DenseRange idx → ∀ (R : ℕ → ℝ), Causalean.Estimation.OrthogonalLearning.RademacherBound S S_iid split g idx R → ∀ {δ : ℝ}, 0 < δ → δ ≤ 1 → Causalean.Estimation.OrthogonalLearning.LocalEmpProcessModulus S S_iid split (fun n => √(if (split.foldB n).card = 0 then 2 * b else 2 * R n + 2 * b * √(2 * Real.log (1 / δ) / ↑(split.foldB n).card))) δ g","l":389,"f":"Causalean/Estimation/OrthogonalLearning/LocalEmpProcess/Rademacher.lean"},{"n":"Causalean.Estimation.OrthogonalLearning.localEmpProcessModulus_of_bounded_rademacher_ae","k":"theorem","m":"Causalean.Estimation.OrthogonalLearning.LocalEmpProcess.Rademacher","a":"Estimation/OrthogonalLearning/LocalEmpProcess","t":1,"d":"Almost-everywhere bounded-loss bridge. Assume b is nonnegative, that the loss magnitude is bounded by b at P_Z-almost-every observation, over the parameter set, and that the loss is continuous in the parameter on the parameter set. Given a sequence R n that is nonnegative and upper-bounds the population Rademacher complexity of the centred loss class on the fold-B sample at every sample size, and that the target parameter minimizes, over the parameter set, the population risk of the loss — evaluated at the model's baseline nuisance and clamped to [-b, b], then for any confidence level 0 < δ ≤ 1 the local empirical-process modulus condition holds, with rate ρ n := √(2 · b) when the fold-B sample is empty and ρ n := √(2 · R n + 2 · b · √(2 · log(1/δ) / |foldB n|)) otherwise.","s":"∀ {Ω : Type u_1} [inst : MeasurableSpace Ω] {μ : MeasureTheory.Measure Ω} {Z : Type u_2} [inst_1 : MeasurableSpace Z] {P_Z : MeasureTheory.Measure Z} {Θ : Type u_3} [inst_2 : NormedAddCommGroup Θ] [inst_3 : InnerProductSpace ℝ Θ] {G : Type u_4} [inst_4 : AddCommGroup G] [inst_5 : Module ℝ G] (S : Causalean.Estimation.OrthogonalLearning.LearningSystem Ω μ Z P_Z Θ G) [MeasureTheory.IsProbabilityMeasure μ] (S_iid : Causalean.Stat.IIDSample Ω Z μ P_Z) (split : Causalean.Stat.OneShotSplit S_iid) {b : ℝ}, 0 ≤ b → ∀ (g : G), Causalean.Estimation.OrthogonalLearning.UniformlyBoundedLossAE S g b → Causalean.Estimation.OrthogonalLearning.LossContinuousOnΘset S g → ∀ (idx : ℕ → ↑S.Θ_set), DenseRange idx → ∀ (R : ℕ → ℝ), Causalean.Estimation.OrthogonalLearning.RademacherBound S S_iid split g idx R → (∀ θ ∈ S.Θ_set, ∫ (z : Z), max (-b) (min b (S.ℓ z S.θ₀ S.g₀)) ∂P_Z ≤ ∫ (z : Z), max (-b) (min b (S.ℓ z θ S.g₀)) ∂P_Z) → ∀ {δ : ℝ}, 0 < δ → δ ≤ 1 → Causalean.Estimation.OrthogonalLearning.LocalEmpProcessModulus S S_iid split (fun n => √(if (split.foldB n).card = 0 then 2 * b else 2 * R n + 2 * b * √(2 * Real.log (1 / δ) / ↑(split.foldB n).card))) δ g","l":877,"f":"Causalean/Estimation/OrthogonalLearning/LocalEmpProcess/Rademacher.lean"},{"n":"Causalean.Estimation.OrthogonalLearning.localEmpProcessModulus_of_localized_bounded","k":"theorem","m":"Causalean.Estimation.OrthogonalLearning.LocalEmpProcess.Localized","a":"Estimation/OrthogonalLearning/LocalEmpProcess","t":1,"d":"Uniform-bound fallback bridge. Under a localized Rademacher regime — a uniform bound b on the centred pointwise loss difference together with a sub-root Rademacher- complexity envelope ψ, and given that the population excess risk at every admissible parameter is likewise bounded by b, then for any confidence level 0 < δ ≤ 1 the local empirical-process modulus condition holds, with the constant envelope ρ n := √(2 · b).","s":"∀ {Ω : Type u_1} [inst : MeasurableSpace Ω] {μ : MeasureTheory.Measure Ω} {Z : Type u_2} [inst_1 : MeasurableSpace Z] {P_Z : MeasureTheory.Measure Z} {Θ : Type u_3} [inst_2 : NormedAddCommGroup Θ] [inst_3 : InnerProductSpace ℝ Θ] {G : Type u_4} [inst_4 : AddCommGroup G] [inst_5 : Module ℝ G] (S : Causalean.Estimation.OrthogonalLearning.LearningSystem Ω μ Z P_Z Θ G) [MeasureTheory.IsProbabilityMeasure μ] (S_iid : Causalean.Stat.IIDSample Ω Z μ P_Z) (split : Causalean.Stat.OneShotSplit S_iid) [Nonempty ↑S.Θ_set] (g : G) (idx : ℕ → ↑S.Θ_set) {norm : (Z → ℝ) → ℝ} {ψ : ℕ → ℝ → ℝ} {b : ℝ}, Causalean.Estimation.OrthogonalLearning.LocalizedRademacherRegime S S_iid split g idx norm ψ b → (∀ θ ∈ S.Θ_set, |S.L θ g - S.L S.θ₀ g| ≤ b) → ∀ {δ : ℝ}, 0 < δ → δ ≤ 1 → Causalean.Estimation.OrthogonalLearning.LocalEmpProcessModulus S S_iid split (fun _n => √(2 * b)) δ g","l":240,"f":"Causalean/Estimation/OrthogonalLearning/LocalEmpProcess/Localized.lean"},{"n":"Causalean.Estimation.OrthogonalLearning.localEmpProcessModulus_of_localized_bounded_ae","k":"theorem","m":"Causalean.Estimation.OrthogonalLearning.LocalEmpProcess.Localized","a":"Estimation/OrthogonalLearning/LocalEmpProcess","t":1,"d":"Almost-everywhere uniform-bound fallback bridge. Under an almost-everywhere localized Rademacher regime — a P_Z-a.e. bound b on the centred pointwise loss difference together with a sub-root Rademacher-complexity envelope ψ, and given that the population excess risk at every admissible parameter is likewise bounded by b, then for any confidence level 0 < δ ≤ 1 the local empirical-process modulus condition holds, with the constant envelope ρ n := √(2 · b).","s":"∀ {Ω : Type u_1} [inst : MeasurableSpace Ω] {μ : MeasureTheory.Measure Ω} {Z : Type u_2} [inst_1 : MeasurableSpace Z] {P_Z : MeasureTheory.Measure Z} {Θ : Type u_3} [inst_2 : NormedAddCommGroup Θ] [inst_3 : InnerProductSpace ℝ Θ] {G : Type u_4} [inst_4 : AddCommGroup G] [inst_5 : Module ℝ G] (S : Causalean.Estimation.OrthogonalLearning.LearningSystem Ω μ Z P_Z Θ G) [MeasureTheory.IsProbabilityMeasure μ] (S_iid : Causalean.Stat.IIDSample Ω Z μ P_Z) (split : Causalean.Stat.OneShotSplit S_iid) [Nonempty ↑S.Θ_set] (g : G) (idx : ℕ → ↑S.Θ_set) {norm : (Z → ℝ) → ℝ} {ψ : ℕ → ℝ → ℝ} {b : ℝ}, Causalean.Estimation.OrthogonalLearning.LocalizedRademacherRegimeAE S S_iid split g idx norm ψ b → (∀ θ ∈ S.Θ_set, |S.L θ g - S.L S.θ₀ g| ≤ b) → ∀ {δ : ℝ}, 0 < δ → δ ≤ 1 → Causalean.Estimation.OrthogonalLearning.LocalEmpProcessModulus S S_iid split (fun _n => √(2 * b)) δ g","l":321,"f":"Causalean/Estimation/OrthogonalLearning/LocalEmpProcess/Localized.lean"},{"n":"Causalean.Estimation.OrthogonalLearning.localEmpProcessModulus_of_localized_sharp","k":"theorem","m":"Causalean.Estimation.OrthogonalLearning.LocalEmpProcess.Localized","a":"Estimation/OrthogonalLearning/LocalEmpProcess","t":1,"d":"Localized critical-radius modulus, countable-class Foster–Syrgkanis form. Fix a countable, densely-indexed target class on which the loss is continuous in the parameter for every observation and a localized Rademacher regime holds — a uniform centred-loss bound and a sub-root Rademacher-complexity envelope ψ. Suppose the Lipschitz constant L is nonnegative, that the chosen function-space norm of the centred loss difference at each parameter is bounded by L · ‖θ − θ₀‖, that the loss at each parameter is measurable and integrable under P_Z, and that the same centred loss difference has diameter at most Rmax in that norm. Assume the critical radius of ψ at every fold-B sample size satisfies criticalRadius (ψ m) ≤ Rmax and criticalRadius (ψ m) > 0, together with the sub-root fixed-point inequality ψ m (criticalRadius (ψ m)) ≤ criticalRadius (ψ m) ^ 2; that ψ upper-bounds the population Rademacher complexity of the star-hulled centred loss class on a fold-B-sized sample; and the technical regularity conditions that the empirical Rademacher supremum is bounded above and the upper empirical Rademacher complexity process is integrable. For any confidence level 0 < δ ≤ 1, assume the Foster–Syrgkanis peeling-absorption condition that at every dyadic shell count covering Rmax, the McDiarmid concentration slack at the union-bound-adjusted confidence is dominated by the squared critical radius. Then the local empirical-process modulus condition holds, with envelope ρ n := (8 · L + 3) · criticalRadius (ψ |foldB n|) (falling back to √(2 · b) when the fold-B sample is empty).","s":"∀ {Ω : Type u_1} [inst : MeasurableSpace Ω] {μ : MeasureTheory.Measure Ω} {Z : Type u_2} [inst_1 : MeasurableSpace Z] {P_Z : MeasureTheory.Measure Z} {Θ : Type u_3} [inst_2 : NormedAddCommGroup Θ] [inst_3 : InnerProductSpace ℝ Θ] {G : Type u_4} [inst_4 : AddCommGroup G] [inst_5 : Module ℝ G] (S : Causalean.Estimation.OrthogonalLearning.LearningSystem Ω μ Z P_Z Θ G) [MeasureTheory.IsProbabilityMeasure μ] (S_iid : Causalean.Stat.IIDSample Ω Z μ P_Z) (split : Causalean.Stat.OneShotSplit S_iid) [Nonempty ↑S.Θ_set] [Countable ↑S.Θ_set] (g : G), (∀ (z : Z), Continuous fun θ => S.ℓ z (↑θ) g) → ∀ (idx : ℕ → ↑S.Θ_set), DenseRange idx → ∀ {norm : (Z → ℝ) → ℝ} {ψ : ℕ → ℝ → ℝ} {b L Rmax : ℝ}, Causalean.Estimation.OrthogonalLearning.LocalizedRademacherRegime S S_iid split g idx norm ψ b → 0 ≤ L → (∀ θ ∈ S.Θ_set, (norm fun z => S.ℓ z θ g - S.ℓ z S.θ₀ g) ≤ L * ‖θ - S.θ₀‖) → (∀ θ ∈ S.Θ_set, Measurable fun z => S.ℓ z θ g) → (∀ θ ∈ S.Θ_set, MeasureTheory.Integrable (fun z => S.ℓ z θ g) P_Z) → (∀ θ ∈ S.Θ_set, (norm fun z => S.ℓ z θ g - S.ℓ z S.θ₀ g) ≤ Rmax) → (∀ (m : ℕ), Causalean.Stat.Concentration.criticalRadius (ψ m) ≤ Rmax) → (∀ (m : ℕ), 0 < Causalean.Stat.Concentration.criticalRadius (ψ m)) → (∀ (m : ℕ), ψ m (Causalean.Stat.Concentration.criticalRadius (ψ m)) ≤ Causalean.Stat.Concentration.criticalRadius (ψ m) ^ 2) → (∀ (m : ℕ), Causalean.Stat.Concentration.RademacherUpperBound (fun θ z => S.ℓ z (↑θ) g - S.ℓ z S.θ₀ g) norm P_Z id m (ψ m)) → (∀ (m : ℕ) (r : ℝ) (S_fin : Fin m → Z) (σ : Signs m), BddAbove (Set.range fun p => |(↑m)⁻¹ * ∑ k, ↑↑(σ k) * Causalean.Stat.Concentration.starHullZeroOut (fun θ z => S.ℓ z (↑θ) g - S.ℓ z S.θ₀ g) norm r p (S_fin k)|)) → (∀ (m : ℕ) (r : ℝ), MeasureTheory.Integrable (fun ω => empiricalRademacherComplexity m (Causalean.Stat.Concentration.starHullZeroOut (fun θ z => S.ℓ z (↑θ) g - S.ℓ z S.θ₀ g) norm r) (id ∘ ω)) (MeasureTheory.Measure.pi fun x => P_Z)) → ∀ {δ : ℝ}, 0 < δ → δ ≤ 1 → (∀ (n K : ℕ), 0 < (split.foldB n).card → Rmax ≤ Causalean.Stat.Concentration.criticalRadius (ψ (split.foldB n).card) * 2 ^ K → b * √(2 * Real.log (2 * (↑K + 1) / δ) / ↑(split.foldB n).card) ≤ Causalean.Stat.Concentration.criticalRadius (ψ (split.foldB n).card) ^ 2) → Causalean.Estimation.OrthogonalLearning.LocalEmpProcessModulus S S_iid split (fun n => if (split.foldB n).card = 0 then √(2 * b) else (8 * L + 3) * Causalean.Stat.Concentration.criticalRadius (ψ (split.foldB n).card)) δ g","l":426,"f":"Causalean/Estimation/OrthogonalLearning/LocalEmpProcess/Localized.lean"},{"n":"Causalean.Estimation.OrthogonalLearning.localEmpProcessModulus_of_localized_sharp_ae","k":"theorem","m":"Causalean.Estimation.OrthogonalLearning.LocalEmpProcess.Localized","a":"Estimation/OrthogonalLearning/LocalEmpProcess","t":1,"d":"Almost-everywhere countable-class localized bridge. Fix a countable, densely-indexed target class on which the loss is continuous in the parameter for every observation and the chosen function-space norm is invariant under P_Z-a.e. equality of its argument. Assume an almost-everywhere localized Rademacher regime holds — a uniform centred-loss bound and a sub-root Rademacher-complexity envelope ψ, that the Lipschitz constant L is nonnegative, that the norm of the centred loss difference at each parameter is bounded by L · ‖θ − θ₀‖, that the loss at each parameter is measurable and integrable under P_Z, and that the same centred loss difference has diameter at most Rmax in that norm. Assume the critical radius of ψ at every fold-B sample size satisfies criticalRadius (ψ m) ≤ Rmax and criticalRadius (ψ m) > 0, together with the sub-root fixed-point inequality ψ m (criticalRadius (ψ m)) ≤ criticalRadius (ψ m) ^ 2; that ψ upper-bounds the population Rademacher complexity of the star-hulled centred loss class on a fold-B-sized sample; the technical regularity conditions that the empirical Rademacher supremum is bounded above and the upper empirical Rademacher complexity process is integrable; and that the population-optimal parameter minimizes the auxiliary population risk with the centred excess loss clamped to [-b, b]. For any confidence level 0 < δ ≤ 1, assume the Foster–Syrgkanis peeling-absorption condition that at every dyadic shell count covering Rmax, the McDiarmid concentration slack at the union-bound-adjusted confidence is dominated by the squared critical radius. Then the local empirical-process modulus condition holds, with envelope ρ n := (8 · L + 3) · criticalRadius (ψ |foldB n|) (falling back to √(2 · b) when the fold-B sample is empty).","s":"∀ {Ω : Type u_1} [inst : MeasurableSpace Ω] {μ : MeasureTheory.Measure Ω} {Z : Type u_2} [inst_1 : MeasurableSpace Z] {P_Z : MeasureTheory.Measure Z} {Θ : Type u_3} [inst_2 : NormedAddCommGroup Θ] [inst_3 : InnerProductSpace ℝ Θ] {G : Type u_4} [inst_4 : AddCommGroup G] [inst_5 : Module ℝ G] (S : Causalean.Estimation.OrthogonalLearning.LearningSystem Ω μ Z P_Z Θ G) [MeasureTheory.IsProbabilityMeasure μ] (S_iid : Causalean.Stat.IIDSample Ω Z μ P_Z) (split : Causalean.Stat.OneShotSplit S_iid) [Nonempty ↑S.Θ_set] [Countable ↑S.Θ_set] (g : G), (∀ (z : Z), Continuous fun θ => S.ℓ z (↑θ) g) → ∀ (idx : ℕ → ↑S.Θ_set), DenseRange idx → ∀ {norm : (Z → ℝ) → ℝ}, (∀ (F F' : Z → ℝ), F =ᵐ[P_Z] F' → norm F = norm F') → ∀ {ψ : ℕ → ℝ → ℝ} {b L Rmax : ℝ}, Causalean.Estimation.OrthogonalLearning.LocalizedRademacherRegimeAE S S_iid split g idx norm ψ b → 0 ≤ L → (∀ θ ∈ S.Θ_set, (norm fun z => S.ℓ z θ g - S.ℓ z S.θ₀ g) ≤ L * ‖θ - S.θ₀‖) → (∀ θ ∈ S.Θ_set, Measurable fun z => S.ℓ z θ g) → (∀ θ ∈ S.Θ_set, MeasureTheory.Integrable (fun z => S.ℓ z θ g) P_Z) → (∀ θ ∈ S.Θ_set, (norm fun z => S.ℓ z θ g - S.ℓ z S.θ₀ g) ≤ Rmax) → (∀ (m : ℕ), Causalean.Stat.Concentration.criticalRadius (ψ m) ≤ Rmax) → (∀ (m : ℕ), 0 < Causalean.Stat.Concentration.criticalRadius (ψ m)) → (∀ (m : ℕ), ψ m (Causalean.Stat.Concentration.criticalRadius (ψ m)) ≤ Causalean.Stat.Concentration.criticalRadius (ψ m) ^ 2) → (∀ (m : ℕ), Causalean.Stat.Concentration.RademacherUpperBound (fun θ z => S.ℓ z (↑θ) g - S.ℓ z S.θ₀ g) norm P_Z id m (ψ m)) → (∀ (m : ℕ) (r : ℝ) (S_fin : Fin m → Z) (σ : Signs m), BddAbove (Set.range fun p => |(↑m)⁻¹ * ∑ k, ↑↑(σ k) * Causalean.Stat.Concentration.starHullZeroOut (fun θ z => S.ℓ z (↑θ) g - S.ℓ z S.θ₀ g) norm r p (S_fin k)|)) → (∀ (m : ℕ) (r : ℝ), MeasureTheory.Integrable (fun ω => empiricalRademacherComplexity m (Causalean.Stat.Concentration.starHullZeroOut (fun θ z => S.ℓ z (↑θ) g - S.ℓ z S.θ₀ g) norm r) (id ∘ ω)) (MeasureTheory.Measure.pi fun x => P_Z)) → Causalean.Estimation.OrthogonalLearning.CenteredClampedThetaMinimizes S b → ∀ {δ : ℝ}, 0 < δ → δ ≤ 1 → (∀ (n K : ℕ), 0 < (split.foldB n).card → Rmax ≤ Causalean.Stat.Concentration.criticalRadius (ψ (split.foldB n).card) * 2 ^ K → b * √(2 * Real.log (2 * (↑K + 1) / δ) / ↑(split.foldB n).card) ≤ Causalean.Stat.Concentration.criticalRadius (ψ (split.foldB n).card) ^ 2) → Causalean.Estimation.OrthogonalLearning.LocalEmpProcessModulus S S_iid split (fun n => if (split.foldB n).card = 0 then √(2 * b) else (8 * L + 3) * Causalean.Stat.Concentration.criticalRadius (ψ (split.foldB n).card)) δ g","l":970,"f":"Causalean/Estimation/OrthogonalLearning/LocalEmpProcess/Localized.lean"},{"n":"Causalean.Estimation.OrthogonalLearning.localEmpProcessModulus_of_localized_singleton","k":"theorem","m":"Causalean.Estimation.OrthogonalLearning.LocalEmpProcess.Localized","a":"Estimation/OrthogonalLearning/LocalEmpProcess","t":2,"d":"Trivial finite class. When Θ_set = {θ₀} the modulus inequality holds with ρ n := 0 (the centred excess risk is identically zero). Mirrors localEmpProcessModulus_singleton in the global-Rademacher bridge.","s":"∀ {Ω : Type u_1} [inst : MeasurableSpace Ω] {μ : MeasureTheory.Measure Ω} {Z : Type u_2} [inst_1 : MeasurableSpace Z] {P_Z : MeasureTheory.Measure Z} {Θ : Type u_3} [inst_2 : NormedAddCommGroup Θ] [inst_3 : InnerProductSpace ℝ Θ] {G : Type u_4} [inst_4 : AddCommGroup G] [inst_5 : Module ℝ G] (S : Causalean.Estimation.OrthogonalLearning.LearningSystem Ω μ Z P_Z Θ G) (S_iid : Causalean.Stat.IIDSample Ω Z μ P_Z) (split : Causalean.Stat.OneShotSplit S_iid) [MeasureTheory.IsProbabilityMeasure μ] (g : G), S.Θ_set = {S.θ₀} → ∀ {δ : ℝ}, 0 < δ → Causalean.Estimation.OrthogonalLearning.LocalEmpProcessModulus S S_iid split (fun x => 0) δ g","l":1456,"f":"Causalean/Estimation/OrthogonalLearning/LocalEmpProcess/Localized.lean"},{"n":"Causalean.Estimation.OrthogonalLearning.localEmpProcessModulus_random_of_bounded_rademacher","k":"theorem","m":"Causalean.Estimation.OrthogonalLearning.LocalEmpProcess.RandomNuisance","a":"Estimation/OrthogonalLearning/LocalEmpProcess","t":1,"d":"Cross-fit lift of the empirical-process modulus to a random nuisance. Let ĥ be a fold-A-measurable random nuisance sequence. Assume b is strictly positive, that at every sample size and sample point the loss magnitude at the realised nuisance ĥ n ω is uniformly bounded by b, and that the loss at that realised nuisance is continuous in the parameter. Given a sequence R n that Rademacher-controls the centred loss class at every realisation ĥ n ω, and a confidence level 0 < δ ≤ 1. Suppose further that the σ-algebra m_A n generating the fold-A information is coarser than the ambient σ-algebra, that fold A is independent of the fold-B coordinate block under this σ-algebra, and that the nuisance-indexed bad event depends jointly measurably on the fold-A outcome and the fold-B sample. Then the random-nuisance local empirical-process modulus condition holds, at rate ρ n := √(2·b) on empty folds and ρ n := √(2·R n + 2·b·√(2·log(1/δ)/|foldB n|)) otherwise, evaluated at the random nuisance ĥ.","s":"∀ {Ω : Type u_1} [inst : MeasurableSpace Ω] {μ : MeasureTheory.Measure Ω} {Z : Type u_2} [inst_1 : MeasurableSpace Z] {P_Z : MeasureTheory.Measure Z} {Θ : Type u_3} [inst_2 : NormedAddCommGroup Θ] [inst_3 : InnerProductSpace ℝ Θ] {G : Type u_4} [inst_4 : AddCommGroup G] [inst_5 : Module ℝ G] (S : Causalean.Estimation.OrthogonalLearning.LearningSystem Ω μ Z P_Z Θ G) [MeasureTheory.IsProbabilityMeasure μ] [MeasureTheory.IsProbabilityMeasure P_Z] (S_iid : Causalean.Stat.IIDSample Ω Z μ P_Z) (split : Causalean.Stat.OneShotSplit S_iid) {b : ℝ}, 0 < b → ∀ (ĥ : ℕ → Ω → G), (∀ (n : ℕ) (ω : Ω), Causalean.Estimation.OrthogonalLearning.UniformlyBoundedLoss S (ĥ n ω) b) → (∀ (n : ℕ) (ω : Ω), Causalean.Estimation.OrthogonalLearning.LossContinuousOnΘset S (ĥ n ω)) → ∀ (idx : ℕ → ↑S.Θ_set), DenseRange idx → ∀ (R : ℕ → ℝ), (∀ (n : ℕ) (ω : Ω), Causalean.Estimation.OrthogonalLearning.RademacherBound S S_iid split (ĥ n ω) idx R) → ∀ {δ : ℝ}, 0 < δ → δ ≤ 1 → ∀ (m_A : ℕ → MeasurableSpace Ω), (∀ (n : ℕ), m_A n ≤ inferInstance) → (∀ (n : ℕ), ProbabilityTheory.Indep (m_A n) (MeasurableSpace.comap (Causalean.Estimation.OrthogonalLearning.foldBCoord S_iid split n) inferInstance) μ) → (∀ (n : ℕ), MeasurableSet {p | p.2 ∈ Causalean.Estimation.OrthogonalLearning.badDataSet S S_iid split n b δ R (ĥ n p.1)}) → Causalean.Estimation.OrthogonalLearning.LocalEmpProcessModulusRandom S S_iid split (Causalean.Estimation.OrthogonalLearning.modulusRadius split b δ R) δ ĥ","l":550,"f":"Causalean/Estimation/OrthogonalLearning/LocalEmpProcess/RandomNuisance.lean"},{"n":"Causalean.Estimation.OrthogonalLearning.localEmpProcessModulus_singleton","k":"theorem","m":"Causalean.Estimation.OrthogonalLearning.LocalEmpProcess.Rademacher","a":"Estimation/OrthogonalLearning/LocalEmpProcess","t":2,"d":"Trivial finite class. When Θ_set = {θ₀} (the class collapses to the truth), the modulus inequality holds with ρ n := 0.","s":"∀ {Ω : Type u_1} [inst : MeasurableSpace Ω] {μ : MeasureTheory.Measure Ω} {Z : Type u_2} [inst_1 : MeasurableSpace Z] {P_Z : MeasureTheory.Measure Z} {Θ : Type u_3} [inst_2 : NormedAddCommGroup Θ] [inst_3 : InnerProductSpace ℝ Θ] {G : Type u_4} [inst_4 : AddCommGroup G] [inst_5 : Module ℝ G] (S : Causalean.Estimation.OrthogonalLearning.LearningSystem Ω μ Z P_Z Θ G) (S_iid : Causalean.Stat.IIDSample Ω Z μ P_Z) (split : Causalean.Stat.OneShotSplit S_iid) [MeasureTheory.IsProbabilityMeasure μ] (g : G), S.Θ_set = {S.θ₀} → ∀ {δ : ℝ}, 0 < δ → Causalean.Estimation.OrthogonalLearning.LocalEmpProcessModulus S S_iid split (fun x => 0) δ g","l":1138,"f":"Causalean/Estimation/OrthogonalLearning/LocalEmpProcess/Rademacher.lean"},{"n":"Causalean.Estimation.OrthogonalLearning.modulusRadius","k":"def","m":"Causalean.Estimation.OrthogonalLearning.LocalEmpProcess.RandomNuisance","a":"Estimation/OrthogonalLearning/LocalEmpProcess","t":1,"d":"The deterministic modulus radius ρ n.","s":"{Ω : Type u_1} → [inst : MeasurableSpace Ω] → {μ : MeasureTheory.Measure Ω} → {Z : Type u_2} → [inst_1 : MeasurableSpace Z] → {P_Z : MeasureTheory.Measure Z} → {S_iid : Causalean.Stat.IIDSample Ω Z μ P_Z} → Causalean.Stat.OneShotSplit S_iid → ℝ → ℝ → (ℕ → ℝ) → ℕ → ℝ","l":103,"f":"Causalean/Estimation/OrthogonalLearning/LocalEmpProcess/RandomNuisance.lean"},{"n":"Causalean.Estimation.OrthogonalLearning.modulus_of_not_badData","k":"theorem","m":"Causalean.Estimation.OrthogonalLearning.LocalEmpProcess.RandomNuisance","a":"Estimation/OrthogonalLearning/LocalEmpProcess","t":2,"d":"On-event modulus. Off the bad set, the modulus inequality holds uniformly over Θ_set for a fixed nuisance g, using the same bounded-loss Rademacher radius as the deterministic bridge in Rademacher.lean.","s":"∀ {Ω : Type u_1} [inst : MeasurableSpace Ω] {μ : MeasureTheory.Measure Ω} {Z : Type u_2} [inst_1 : MeasurableSpace Z] {P_Z : MeasureTheory.Measure Z} {Θ : Type u_3} [inst_2 : NormedAddCommGroup Θ] [inst_3 : InnerProductSpace ℝ Θ] {G : Type u_4} [inst_4 : AddCommGroup G] [inst_5 : Module ℝ G] (S : Causalean.Estimation.OrthogonalLearning.LearningSystem Ω μ Z P_Z Θ G) [MeasureTheory.IsProbabilityMeasure μ] (S_iid : Causalean.Stat.IIDSample Ω Z μ P_Z) (split : Causalean.Stat.OneShotSplit S_iid) {b : ℝ}, 0 < b → ∀ (g : G), Causalean.Estimation.OrthogonalLearning.UniformlyBoundedLoss S g b → Causalean.Estimation.OrthogonalLearning.LossContinuousOnΘset S g → ∀ (idx : ℕ → ↑S.Θ_set), DenseRange idx → ∀ (R : ℕ → ℝ), Causalean.Estimation.OrthogonalLearning.RademacherBound S S_iid split g idx R → ∀ {δ : ℝ}, 0 < δ → δ ≤ 1 → ∀ (n : ℕ) (ω : Ω), 0 < (split.foldB n).card → Causalean.Estimation.OrthogonalLearning.foldBCoord S_iid split n ω ∉ Causalean.Estimation.OrthogonalLearning.badDataSet S S_iid split n b δ R g → ∀ θ ∈ S.Θ_set, S.L θ g - S.L S.θ₀ g - (Causalean.Estimation.OrthogonalLearning.empRiskFoldB S S_iid split n ω θ g - Causalean.Estimation.OrthogonalLearning.empRiskFoldB S S_iid split n ω S.θ₀ g) ≤ Causalean.Estimation.OrthogonalLearning.modulusRadius split b δ R n * ‖θ - S.θ₀‖ + Causalean.Estimation.OrthogonalLearning.modulusRadius split b δ R n ^ 2","l":218,"f":"Causalean/Estimation/OrthogonalLearning/LocalEmpProcess/RandomNuisance.lean"},{"n":"Causalean.Estimation.OrthogonalLearning.neymanOrthog_iff_score_deriv_zero","k":"theorem","m":"Causalean.Estimation.OrthogonalLearning.Population.NeymanOrthogonal","a":"Estimation/OrthogonalLearning/Population","t":1,"d":"Score reformulation of Neyman orthogonality. Assume the DCT-bridge condition — the integrated centred target-direction difference quotient along a nuisance perturbation converges to the integrated mixed directional derivative as the perturbation size tends to zero. Then Neyman orthogonality of the loss is equivalent to the score map g' ↦ D_θ L(θ₀, g')[ν_θ] having zero first derivative at g₀, for every admissible target direction ν_θ.","s":"∀ {Ω : Type u_1} [inst : MeasurableSpace Ω] {μ : MeasureTheory.Measure Ω} {Z : Type u_2} [inst_1 : MeasurableSpace Z] {P_Z : MeasureTheory.Measure Z} {Θ : Type u_3} [inst_2 : NormedAddCommGroup Θ] [inst_3 : InnerProductSpace ℝ Θ] {G : Type u_4} [inst_4 : AddCommGroup G] [inst_5 : Module ℝ G] (S : Causalean.Estimation.OrthogonalLearning.LearningSystem Ω μ Z P_Z Θ G) (M : Causalean.Estimation.OrthogonalLearning.HasMixedDirDeriv S), Causalean.Estimation.OrthogonalLearning.MixedScoreDCTBridge S M → (Causalean.Estimation.OrthogonalLearning.NeymanOrthogLoss S M ↔ ∀ θ ∈ S.Θ_set, ∀ g ∈ S.G_set, Filter.Tendsto (fun t => (∫ (z : Z), (M.Dθ_at (S.g₀ + t • (g - S.g₀))).dℓ_θ θ z ∂P_Z - ∫ (z : Z), (M.Dθ_at S.g₀).dℓ_θ θ z ∂P_Z) / t) (nhdsWithin 0 {0}ᶜ) (nhds 0))","l":103,"f":"Causalean/Estimation/OrthogonalLearning/Population/NeymanOrthogonal.lean"},{"n":"Causalean.Estimation.OrthogonalLearning.oracle_inequality_drLearner_highProb","k":"theorem","m":"Causalean.Estimation.CATE.OrthogonalLearning.LocalEmpProcess.DRLearnerEndToEnd","a":"Estimation/CATE/OrthogonalLearning/LocalEmpProcess","t":1,"d":"DR-Learner end-to-end high-probability oracle inequality. Consider the same doubly robust orthogonal-learning system, one-shot sample split, and boundedness / overlap / continuity / Rademacher package as in localEmpProcessModulus_drLearner — truth-identifying admissibility and evaluation correctness, uniform bounds on the candidate evaluations, the outcome, and the realised nuisance regression together with propensity overlap, and loss continuity, a Rademacher bound R n, the clamped-loss minimizer property, and a confidence level δ in (0, 1] . Assume in addition that a plug-in empirical-risk-minimisation estimator sequence τhat, evaluated against the same realised nuisance on every cross-fitting fold, attains the empirical risk up to an optimization slack r_opt n, that the population risk is strongly convex at the realised nuisance with modulus σ > 0, and that the truth-identifying candidate satisfies the first-order optimality inequality for the population risk's directional derivative at the truth. Then there is a nonnegative constant b such that, for every sample size n, with P.μ-probability at least 1 − δ the estimation error obeys ‖τhat n ω − θ₀‖² ≤ (4(1+σ)/σ²) · ρ_{n,δ}² + (4/σ) · Bias_n + (4/σ) · r_opt n, where ρ_{n,δ} is the modulus rate realised by localEmpProcessModulus_drLearner.","s":"∀ {P : Causalean.PO.POSystem} {γ : Type u_1} [inst : MeasurableSpace γ] [inst_1 : StandardBorelSpace P.Ω] [inst_2 : MeasureTheory.IsFiniteMeasure P.μ] [MeasureTheory.IsProbabilityMeasure P.μ] (S : Causalean.Estimation.CATE.CATEEstimationSystem P γ) (Θ : Type u_2) [inst_4 : NormedAddCommGroup Θ] [inst_5 : InnerProductSpace ℝ Θ] (Θ_set : Set Θ) (Θ_convex : Convex ℝ Θ_set) (θ₀ : Θ) (θ₀_mem : θ₀ ∈ Θ_set) (eval : Θ → γ → ℝ) (eval_meas : ∀ (θ : Θ), Measurable (eval θ)) (eval_θ₀ : ∀ (x : γ), eval θ₀ x = S.τ_val x) (θ₀_minimizes : Causalean.Estimation.OrthogonalLearning.DRThetaMinimizes S Θ_set θ₀ eval) (S_iid : Causalean.Stat.IIDSample P.Ω (γ × Bool × ℝ) P.μ S.P_Z) (split : Causalean.Stat.OneShotSplit S_iid) {M_Θ M_Y M_μ ε : ℝ}, Causalean.Estimation.OrthogonalLearning.DREvalBounded Θ_set eval M_Θ → Causalean.Estimation.OrthogonalLearning.DROutcomeBounded S M_Y → ∀ (h : Causalean.Estimation.ATE.NuisanceVec γ), Causalean.Estimation.OrthogonalLearning.DRNuisanceMuBounded h M_μ → Causalean.Estimation.OrthogonalLearning.DRNuisanceOverlap S Θ_set h ε → Causalean.Estimation.OrthogonalLearning.LossContinuousOnΘset (Causalean.Estimation.OrthogonalLearning.drLearningSystem S Θ Θ_set Θ_convex θ₀ θ₀_mem eval eval_meas eval_θ₀ θ₀_minimizes) h → ∀ (idx : ℕ → ↑(Causalean.Estimation.OrthogonalLearning.drLearningSystem S Θ Θ_set Θ_convex θ₀ θ₀_mem eval eval_meas eval_θ₀ θ₀_minimizes).Θ_set), DenseRange idx → ∀ (R : ℕ → ℝ), Causalean.Estimation.OrthogonalLearning.RademacherBound (Causalean.Estimation.OrthogonalLearning.drLearningSystem S Θ Θ_set Θ_convex θ₀ θ₀_mem eval eval_meas eval_θ₀ θ₀_minimizes) S_iid split h idx R → Causalean.Estimation.OrthogonalLearning.DRClampedThetaMinimizes S Θ_set θ₀ eval ((M_Θ + 2 * M_μ + 2 * (M_Y + M_μ) / ε) ^ 2) → ∀ {δ : ℝ}, 0 < δ → δ ≤ 1 → ∀ (Dθ_truth : Causalean.Estimation.OrthogonalLearning.HasDirDerivTheta (Causalean.Estimation.OrthogonalLearning.drLearningSystem S Θ Θ_set Θ_convex θ₀ θ₀_mem eval eval_meas eval_θ₀ θ₀_minimizes) (Causalean.Estimation.OrthogonalLearning.drLearningSystem S Θ Θ_set Θ_convex θ₀ θ₀_mem eval eval_meas eval_θ₀ θ₀_minimizes).g₀) (Dθ_at_h : Causalean.Estimation.OrthogonalLearning.HasDirDerivTheta (Causalean.Estimation.OrthogonalLearning.drLearningSystem S Θ Θ_set Θ_convex θ₀ θ₀_mem eval eval_meas eval_θ₀ θ₀_minimizes) h) (τhat : ℕ → P.Ω → Θ) (r_opt : ℕ → ℝ), Causalean.Estimation.OrthogonalLearning.SampleSplitPluginERM (Causalean.Estimation.OrthogonalLearning.drLearningSystem S Θ Θ_set Θ_convex θ₀ θ₀_mem eval eval_meas eval_θ₀ θ₀_minimizes) S_iid split τhat (fun x x_1 => h) r_opt → ∀ (σ : ℝ), 0 < σ → (∀ θ ∈ Θ_set, (Causalean.Estimation.OrthogonalLearning.drLearningSystem S Θ Θ_set Θ_convex θ₀ θ₀_mem eval eval_meas eval_θ₀ θ₀_minimizes).L θ h - (Causalean.Estimation.OrthogonalLearning.drLearningSystem S Θ Θ_set Θ_convex θ₀ θ₀_mem eval eval_meas eval_θ₀ θ₀_minimizes).L θ₀ h ≥ ∫ (z : γ × Bool × ℝ), Dθ_at_h.dℓ_θ θ z ∂S.P_Z + σ / 2 * ‖θ - θ₀‖ ^ 2) → Causalean.Estimation.OrthogonalLearning.FirstOrderInequality (Causalean.Estimation.OrthogonalLearning.drLearningSystem S Θ Θ_set Θ_convex θ₀ θ₀_mem eval eval_meas eval_θ₀ θ₀_minimizes) Dθ_truth → ∃ b, 0 ≤ b ∧ ∀ (n : ℕ), ∃ E, MeasurableSet E ∧ P.μ E ≥ 1 - ENNReal.ofReal δ ∧ ∀ ω ∈ E, ‖τhat n ω - θ₀‖ ^ 2 ≤ 4 * (1 + σ) / σ ^ 2 * √(if (split.foldB n).card = 0 then 2 * b else 2 * R n + 2 * b * √(2 * Real.log (1 / δ) / ↑(split.foldB n).card)) ^ 2 + 4 / σ * Causalean.Estimation.OrthogonalLearning.Bias_n (Causalean.Estimation.OrthogonalLearning.drLearningSystem S Θ Θ_set Θ_convex θ₀ θ₀_mem eval eval_meas eval_θ₀ θ₀_minimizes) Dθ_truth Dθ_at_h (τhat n ω) + 4 / σ * r_opt n","l":49,"f":"Causalean/Estimation/CATE/OrthogonalLearning/LocalEmpProcess/DRLearnerEndToEnd.lean"},{"n":"Causalean.Estimation.OrthogonalLearning.oracle_inequality_drLearner_highProb_honest","k":"theorem","m":"Causalean.Estimation.CATE.OrthogonalLearning.LocalEmpProcess.DRLearnerHonest","a":"Estimation/CATE/OrthogonalLearning/LocalEmpProcess","t":1,"d":"Product-bias DR-Learner oracle inequality. Consider a CATE estimation system satisfying the underlying backdoor identification assumptions, with an evaluation functional that is measurable in its parameter and recovers the true CATE at a parameter θ₀ belonging to the constraint set, and suppose the evaluation functional, the outcome, and a fixed plug-in nuisance's conditional-mean function are uniformly bounded while that nuisance's propensity score satisfies ε-overlap. Assume further the DR-loss is continuous in the parameter, its localized empirical process admits a Rademacher complexity bound, and a clamped version of θ₀ still minimizes the loss, the confidence level lies in $(0,1]$, the overlap parameter is positive, and both the true and the plug-in nuisance lie in the ε-overlap slice, and that the estimator sequence stays within the constraint set as an approximate sample-split plug-in empirical-risk minimizer (slack r_opt) whose excess population risk satisfies a strong-convexity-type lower bound with constant σ>0 and a first-order orthogonality inequality at the true nuisance's directional derivative. Finally, a battery of boundedness and integrability conditions bounds the directional-derivative envelope and controls the two nuisance-error terms entering the second-order bias decomposition. Then for every sample fold n, with probability at least 1-δ the squared estimation error of the DR-Learner is bounded by an oracle/Rademacher term plus an explicit second-order product-bias term (2B/ε)·Σ_a ‖Δμ_a‖₂·‖Δe‖₂ in the two nuisance L²-errors, plus an optimization slack.","s":"∀ {P : Causalean.PO.POSystem} {γ : Type u_1} [inst : MeasurableSpace γ] [inst_1 : StandardBorelSpace P.Ω] [inst_2 : MeasureTheory.IsFiniteMeasure P.μ] [MeasureTheory.IsProbabilityMeasure P.μ] (S : Causalean.Estimation.CATE.CATEEstimationSystem P γ) (hA : S.Assumptions) (Θ : Type u_2) [inst_4 : NormedAddCommGroup Θ] [inst_5 : InnerProductSpace ℝ Θ] (Θ_set : Set Θ) (Θ_convex : Convex ℝ Θ_set) (θ₀ : Θ) (θ₀_mem : θ₀ ∈ Θ_set) (eval : Θ → γ → ℝ) (eval_meas : ∀ (θ : Θ), Measurable (eval θ)) (eval_θ₀ : ∀ (x : γ), eval θ₀ x = S.τ_val x) (θ₀_minimizes : Causalean.Estimation.OrthogonalLearning.DRThetaMinimizes S Θ_set θ₀ eval) (S_iid : Causalean.Stat.IIDSample P.Ω (γ × Bool × ℝ) P.μ S.P_Z) (split : Causalean.Stat.OneShotSplit S_iid) {M_Θ M_Y M_μ ε : ℝ}, Causalean.Estimation.OrthogonalLearning.DREvalBounded Θ_set eval M_Θ → Causalean.Estimation.OrthogonalLearning.DROutcomeBounded S M_Y → ∀ (h : Causalean.Estimation.ATE.NuisanceVec γ), Causalean.Estimation.OrthogonalLearning.DRNuisanceMuBounded h M_μ → Causalean.Estimation.OrthogonalLearning.DRNuisanceOverlap S Θ_set h ε → Causalean.Estimation.OrthogonalLearning.LossContinuousOnΘset (Causalean.Estimation.OrthogonalLearning.drLearningSystem S Θ Θ_set Θ_convex θ₀ θ₀_mem eval eval_meas eval_θ₀ θ₀_minimizes) h → ∀ (idx : ℕ → ↑(Causalean.Estimation.OrthogonalLearning.drLearningSystem S Θ Θ_set Θ_convex θ₀ θ₀_mem eval eval_meas eval_θ₀ θ₀_minimizes).Θ_set), DenseRange idx → ∀ (R : ℕ → ℝ), Causalean.Estimation.OrthogonalLearning.RademacherBound (Causalean.Estimation.OrthogonalLearning.drLearningSystem S Θ Θ_set Θ_convex θ₀ θ₀_mem eval eval_meas eval_θ₀ θ₀_minimizes) S_iid split h idx R → Causalean.Estimation.OrthogonalLearning.DRClampedThetaMinimizes S Θ_set θ₀ eval ((M_Θ + 2 * M_μ + 2 * (M_Y + M_μ) / ε) ^ 2) → ∀ {δ : ℝ}, 0 < δ → δ ≤ 1 → ∀ (hε_pos : 0 < ε) (h_overlap_η₀ : S.η₀ ∈ Causalean.Estimation.ATE.BackdoorEstimationSystem.H_ε ε), h ∈ Causalean.Estimation.ATE.BackdoorEstimationSystem.H_ε ε → ∀ (D : Causalean.Estimation.OrthogonalLearning.EvalDirDeriv Θ_set θ₀ eval) (ND : Causalean.Estimation.OrthogonalLearning.NuisanceDirDeriv S.η₀) (τhat : ℕ → P.Ω → Θ), (∀ (n : ℕ) (ω : P.Ω), τhat n ω ∈ Θ_set) → ∀ (r_opt : ℕ → ℝ), Causalean.Estimation.OrthogonalLearning.SampleSplitPluginERM (Causalean.Estimation.OrthogonalLearning.drLearningSystem S Θ Θ_set Θ_convex θ₀ θ₀_mem eval eval_meas eval_θ₀ θ₀_minimizes) S_iid split τhat (fun x x_1 => h) r_opt → ∀ (σ : ℝ), 0 < σ → (∀ θ ∈ Θ_set, (Causalean.Estimation.OrthogonalLearning.drLearningSystem S Θ Θ_set Θ_convex θ₀ θ₀_mem eval eval_meas eval_θ₀ θ₀_minimizes).L θ h - (Causalean.Estimation.OrthogonalLearning.drLearningSystem S Θ Θ_set Θ_convex θ₀ θ₀_mem eval eval_meas eval_θ₀ θ₀_minimizes).L θ₀ h ≥ ∫ (z : γ × Bool × ℝ), ((Causalean.Estimation.OrthogonalLearning.drMixedDirDeriv S hA hε_pos h_overlap_η₀ Θ Θ_set Θ_convex θ₀ θ₀_mem eval eval_meas eval_θ₀ θ₀_minimizes D ND).Dθ_at h).dℓ_θ θ z ∂S.P_Z + σ / 2 * ‖θ - θ₀‖ ^ 2) → Causalean.Estimation.OrthogonalLearning.FirstOrderInequality (Causalean.Estimation.OrthogonalLearning.drLearningSystem S Θ Θ_set Θ_convex θ₀ θ₀_mem eval eval_meas eval_θ₀ θ₀_minimizes) ((Causalean.Estimation.OrthogonalLearning.drMixedDirDeriv S hA hε_pos h_overlap_η₀ Θ Θ_set Θ_convex θ₀ θ₀_mem eval eval_meas eval_θ₀ θ₀_minimizes D ND).Dθ_at S.η₀) → ∀ {B : ℝ}, 0 ≤ B → (∀ θ ∈ Θ_set, ∀ (x : γ), |D.dEval θ x| ≤ B) → (∀ (a : Bool), MeasureTheory.Integrable (fun ω => h.μ_fn a (S.factualX ω)) P.μ) → MeasureTheory.Integrable (fun ω => Causalean.Estimation.CATE.phi_eta (S.factualZ ω) h - Causalean.Estimation.CATE.phi₀ S (S.factualZ ω)) P.μ → (∀ (n : ℕ) (ω : P.Ω), MeasureTheory.Integrable (fun ω' => (Causalean.Estimation.CATE.phi_eta (S.factualZ ω') h - Causalean.Estimation.CATE.phi₀ S (S.factualZ ω')) * D.dEval (τhat n ω) (S.factualX ω')) P.μ) → (∀ (a : Bool), MeasureTheory.MemLp (fun x => h.μ_fn a x - S.μ_val a x) 2 S.P_X) → MeasureTheory.MemLp (fun x => h.e_fn x - S.e_val x) 2 S.P_X → (∀ (n : ℕ) (ω : P.Ω), MeasureTheory.Integrable (fun z => ((Causalean.Estimation.OrthogonalLearning.drMixedDirDeriv S hA hε_pos h_overlap_η₀ Θ Θ_set Θ_convex θ₀ θ₀_mem eval eval_meas eval_θ₀ θ₀_minimizes D ND).Dθ_at S.η₀).dℓ_θ (τhat n ω) z) S.P_Z) → (∀ (n : ℕ) (ω : P.Ω), MeasureTheory.Integrable (fun z => ((Causalean.Estimation.OrthogonalLearning.drMixedDirDeriv S hA hε_pos h_overlap_η₀ Θ Θ_set Θ_convex θ₀ θ₀_mem eval eval_meas eval_θ₀ θ₀_minimizes D ND).Dθ_at h).dℓ_θ (τhat n ω) z) S.P_Z) → ∃ b, 0 ≤ b ∧ ∀ (n : ℕ), ∃ E, MeasurableSet E ∧ P.μ E ≥ 1 - ENNReal.ofReal δ ∧ ∀ ω ∈ E, ‖τhat n ω - θ₀‖ ^ 2 ≤ 4 * (1 + σ) / σ ^ 2 * √(if (split.foldB n).card = 0 then 2 * b else 2 * R n + 2 * b * √(2 * Real.log (1 / δ) / ↑(split.foldB n).card)) ^ 2 + 4 / σ * (2 * B / ε * ∑ a, (MeasureTheory.eLpNorm (fun x => h.μ_fn a x - S.μ_val a x) 2 S.P_X).toReal * (MeasureTheory.eLpNorm (fun x => h.e_fn x - S.e_val x) 2 S.P_X).toReal) + 4 / σ * r_opt n","l":53,"f":"Causalean/Estimation/CATE/OrthogonalLearning/LocalEmpProcess/DRLearnerHonest.lean"},{"n":"Causalean.Estimation.OrthogonalLearning.oracle_inequality_drLearner_random_nuisance","k":"theorem","m":"Causalean.Estimation.CATE.OrthogonalLearning.LocalEmpProcess.DRLearnerRandomNuisance","a":"Estimation/CATE/OrthogonalLearning/LocalEmpProcess","t":1,"d":"DR-Learner oracle inequality at a random cross-fitted nuisance. Consider a CATE estimation system satisfying the backdoor identification assumptions, with an evaluation functional measurable in its parameter and recovering the true CATE at a parameter θ₀ in the constraint set, and suppose the overlap parameter is positive while the true nuisance and every realization of the random, fold-A-estimated nuisance ĥ n ω lie in the ε-overlap slice. Assume the estimator sequence stays in the constraint set as an approximate sample-split plug-in empirical-risk minimizer with respect to the random nuisance (slack r_opt), whose excess population risk at each realized ĥ n ω obeys a strong-convexity-type lower bound with constant σ>0, together with a first-order orthogonality inequality at the true nuisance's directional derivative, and that on a high-probability event the centred excess empirical risk at the realized estimator and nuisance is controlled by a rate ρ n. Finally, a battery of boundedness and integrability conditions governs the directional-derivative envelope and bounds the two random nuisance-error terms. Then for every n, with probability at least 1-δ the squared estimation error is bounded by the oracle/Rademacher term (4(1+σ)/σ²)·(ρ n)² plus a random second-order product-bias term (4/σ)·(2B/ε)·Σ_a ‖(ĥ n ω).μ_fn a − μ_val a‖₂·‖(ĥ n ω).e_fn − e_val‖₂ evaluated at the realized nuisance, plus an optimization slack (4/σ)·r_opt n.","s":"∀ {P : Causalean.PO.POSystem} {γ : Type u_1} [inst : MeasurableSpace γ] [inst_1 : StandardBorelSpace P.Ω] [inst_2 : MeasureTheory.IsFiniteMeasure P.μ] (S : Causalean.Estimation.CATE.CATEEstimationSystem P γ) (hA : S.Assumptions) (Θ : Type u_2) [inst_3 : NormedAddCommGroup Θ] [inst_4 : InnerProductSpace ℝ Θ] (Θ_set : Set Θ) (Θ_convex : Convex ℝ Θ_set) (θ₀ : Θ) (θ₀_mem : θ₀ ∈ Θ_set) (eval : Θ → γ → ℝ) (eval_meas : ∀ (θ : Θ), Measurable (eval θ)) (eval_θ₀ : ∀ (x : γ), eval θ₀ x = S.τ_val x) (θ₀_minimizes : Causalean.Estimation.OrthogonalLearning.DRThetaMinimizes S Θ_set θ₀ eval) (S_iid : Causalean.Stat.IIDSample P.Ω (γ × Bool × ℝ) P.μ S.P_Z) (split : Causalean.Stat.OneShotSplit S_iid) {ε : ℝ} (hε_pos : 0 < ε) (h_overlap_η₀ : S.η₀ ∈ Causalean.Estimation.ATE.BackdoorEstimationSystem.H_ε ε) (D : Causalean.Estimation.OrthogonalLearning.EvalDirDeriv Θ_set θ₀ eval) (ND : Causalean.Estimation.OrthogonalLearning.NuisanceDirDeriv S.η₀) (ĥ : ℕ → P.Ω → Causalean.Estimation.ATE.NuisanceVec γ), (∀ (n : ℕ) (ω : P.Ω), ĥ n ω ∈ Causalean.Estimation.ATE.BackdoorEstimationSystem.H_ε ε) → ∀ (τhat : ℕ → P.Ω → Θ), (∀ (n : ℕ) (ω : P.Ω), τhat n ω ∈ Θ_set) → ∀ (r_opt : ℕ → ℝ), Causalean.Estimation.OrthogonalLearning.SampleSplitPluginERM (Causalean.Estimation.OrthogonalLearning.drLearningSystem S Θ Θ_set Θ_convex θ₀ θ₀_mem eval eval_meas eval_θ₀ θ₀_minimizes) S_iid split τhat ĥ r_opt → ∀ (σ : ℝ), 0 < σ → (∀ (n : ℕ) (ω : P.Ω), ∀ θ ∈ Θ_set, (Causalean.Estimation.OrthogonalLearning.drLearningSystem S Θ Θ_set Θ_convex θ₀ θ₀_mem eval eval_meas eval_θ₀ θ₀_minimizes).L θ (ĥ n ω) - (Causalean.Estimation.OrthogonalLearning.drLearningSystem S Θ Θ_set Θ_convex θ₀ θ₀_mem eval eval_meas eval_θ₀ θ₀_minimizes).L θ₀ (ĥ n ω) ≥ ∫ (z : γ × Bool × ℝ), ((Causalean.Estimation.OrthogonalLearning.drMixedDirDeriv S hA hε_pos h_overlap_η₀ Θ Θ_set Θ_convex θ₀ θ₀_mem eval eval_meas eval_θ₀ θ₀_minimizes D ND).Dθ_at (ĥ n ω)).dℓ_θ θ z ∂S.P_Z + σ / 2 * ‖θ - θ₀‖ ^ 2) → Causalean.Estimation.OrthogonalLearning.FirstOrderInequality (Causalean.Estimation.OrthogonalLearning.drLearningSystem S Θ Θ_set Θ_convex θ₀ θ₀_mem eval eval_meas eval_θ₀ θ₀_minimizes) ((Causalean.Estimation.OrthogonalLearning.drMixedDirDeriv S hA hε_pos h_overlap_η₀ Θ Θ_set Θ_convex θ₀ θ₀_mem eval eval_meas eval_θ₀ θ₀_minimizes D ND).Dθ_at S.η₀) → ∀ (ρ : ℕ → ℝ) {δ : ℝ}, (∀ (n : ℕ), ∃ E, MeasurableSet E ∧ P.μ E ≥ 1 - ENNReal.ofReal δ ∧ ∀ ω ∈ E, (Causalean.Estimation.OrthogonalLearning.drLearningSystem S Θ Θ_set Θ_convex θ₀ θ₀_mem eval eval_meas eval_θ₀ θ₀_minimizes).L (τhat n ω) (ĥ n ω) - (Causalean.Estimation.OrthogonalLearning.drLearningSystem S Θ Θ_set Θ_convex θ₀ θ₀_mem eval eval_meas eval_θ₀ θ₀_minimizes).L θ₀ (ĥ n ω) - (Causalean.Estimation.OrthogonalLearning.empRiskFoldB (Causalean.Estimation.OrthogonalLearning.drLearningSystem S Θ Θ_set Θ_convex θ₀ θ₀_mem eval eval_meas eval_θ₀ θ₀_minimizes) S_iid split n ω (τhat n ω) (ĥ n ω) - Causalean.Estimation.OrthogonalLearning.empRiskFoldB (Causalean.Estimation.OrthogonalLearning.drLearningSystem S Θ Θ_set Θ_convex θ₀ θ₀_mem eval eval_meas eval_θ₀ θ₀_minimizes) S_iid split n ω θ₀ (ĥ n ω)) ≤ ρ n * ‖τhat n ω - θ₀‖ + ρ n ^ 2) → ∀ {B : ℝ}, 0 ≤ B → (∀ θ ∈ Θ_set, ∀ (x : γ), |D.dEval θ x| ≤ B) → (∀ (n : ℕ) (ω : P.Ω) (a : Bool), MeasureTheory.Integrable (fun ω' => (ĥ n ω).μ_fn a (S.factualX ω')) P.μ) → (∀ (n : ℕ) (ω : P.Ω), MeasureTheory.Integrable (fun ω' => Causalean.Estimation.CATE.phi_eta (S.factualZ ω') (ĥ n ω) - Causalean.Estimation.CATE.phi₀ S (S.factualZ ω')) P.μ) → (∀ (n : ℕ) (ω : P.Ω), MeasureTheory.Integrable (fun ω' => (Causalean.Estimation.CATE.phi_eta (S.factualZ ω') (ĥ n ω) - Causalean.Estimation.CATE.phi₀ S (S.factualZ ω')) * D.dEval (τhat n ω) (S.factualX ω')) P.μ) → (∀ (n : ℕ) (ω : P.Ω) (a : Bool), MeasureTheory.MemLp (fun x => (ĥ n ω).μ_fn a x - S.μ_val a x) 2 S.P_X) → (∀ (n : ℕ) (ω : P.Ω), MeasureTheory.MemLp (fun x => (ĥ n ω).e_fn x - S.e_val x) 2 S.P_X) → (∀ (n : ℕ) (ω : P.Ω), MeasureTheory.Integrable (fun z => ((Causalean.Estimation.OrthogonalLearning.drMixedDirDeriv S hA hε_pos h_overlap_η₀ Θ Θ_set Θ_convex θ₀ θ₀_mem eval eval_meas eval_θ₀ θ₀_minimizes D ND).Dθ_at S.η₀).dℓ_θ (τhat n ω) z) S.P_Z) → (∀ (n : ℕ) (ω : P.Ω), MeasureTheory.Integrable (fun z => ((Causalean.Estimation.OrthogonalLearning.drMixedDirDeriv S hA hε_pos h_overlap_η₀ Θ Θ_set Θ_convex θ₀ θ₀_mem eval eval_meas eval_θ₀ θ₀_minimizes D ND).Dθ_at (ĥ n ω)).dℓ_θ (τhat n ω) z) S.P_Z) → ∀ (n : ℕ), ∃ E, MeasurableSet E ∧ P.μ E ≥ 1 - ENNReal.ofReal δ ∧ ∀ ω ∈ E, ‖τhat n ω - θ₀‖ ^ 2 ≤ 4 * (1 + σ) / σ ^ 2 * ρ n ^ 2 + 4 / σ * (2 * B / ε * ∑ a, (MeasureTheory.eLpNorm (fun x => (ĥ n ω).μ_fn a x - S.μ_val a x) 2 S.P_X).toReal * (MeasureTheory.eLpNorm (fun x => (ĥ n ω).e_fn x - S.e_val x) 2 S.P_X).toReal) + 4 / σ * r_opt n","l":57,"f":"Causalean/Estimation/CATE/OrthogonalLearning/LocalEmpProcess/DRLearnerRandomNuisance.lean"},{"n":"Causalean.Estimation.OrthogonalLearning.oracle_inequality_localized_drLearner_highProb","k":"theorem","m":"Causalean.Estimation.CATE.OrthogonalLearning.LocalEmpProcess.LocalizedDRLearnerEndToEnd","a":"Estimation/CATE/OrthogonalLearning/LocalEmpProcess","t":1,"d":"DR-Learner localized high-probability oracle inequality. For the DR-Learner CATE estimation system, suppose the evaluation functional is measurable in its parameter and recovers the true CATE at a parameter θ₀ in the constraint set, and that the evaluation functional, outcome, and fixed nuisance conditional-mean are uniformly bounded while the nuisance's propensity score satisfies ε-overlap. Assume the centred DR-loss is continuous in θ, a clamped version of θ₀ minimizes it, its population Rademacher complexity along a dense index sequence is controlled by a sub-root envelope ψ with respect to a seminorm invariant under almost-everywhere modification, and the same Rademacher upper bound extends to loss differences across the whole constraint set, together with Lipschitz and diameter control of the centred loss increments — nonnegative Lipschitz constant L, a diameter bound Rmax dominating every critical radius, and the sub-root fixed-point property, plus boundedness and integrability of the empirical star-hull Rademacher process needed by the localization bridge and a confidence level in $(0,1]$ together with the Foster–Syrgkanis critical-radius domination inequality across dyadic shell counts. Finally, suppose the estimator sequence is an approximate sample-split plug-in empirical-risk minimizer at the fixed nuisance h (slack r_opt), whose excess population risk obeys a strong-convexity-type lower bound with constant σ>0, together with a first-order orthogonality inequality at the truth's directional derivative. Then there is a nonnegative envelope b such that, for every n, with probability at least 1-δ the squared estimation error is bounded by the oracle term (4(1+σ)/σ²)·(ρ n)² at the sharp localized Foster–Syrgkanis rate ρ n = (8L+3)·criticalRadius (ψ |B(n)|) (falling back to √(2b) on an empty validation fold), plus the nuisance bias term (4/σ)·Bias_n and an optimization slack (4/σ)·r_opt n.","s":"∀ {P : Causalean.PO.POSystem} {γ : Type u_1} [inst : MeasurableSpace γ] [inst_1 : StandardBorelSpace P.Ω] [inst_2 : MeasureTheory.IsFiniteMeasure P.μ] [MeasureTheory.IsProbabilityMeasure P.μ] (S : Causalean.Estimation.CATE.CATEEstimationSystem P γ) (Θ : Type u_2) [inst_4 : NormedAddCommGroup Θ] [inst_5 : InnerProductSpace ℝ Θ] (Θ_set : Set Θ) (Θ_convex : Convex ℝ Θ_set) [Nonempty ↑Θ_set] [Countable ↑Θ_set] (θ₀ : Θ) (θ₀_mem : θ₀ ∈ Θ_set) (eval : Θ → γ → ℝ) (eval_meas : ∀ (θ : Θ), Measurable (eval θ)) (eval_θ₀ : ∀ (x : γ), eval θ₀ x = S.τ_val x) (θ₀_minimizes : Causalean.Estimation.OrthogonalLearning.DRThetaMinimizes S Θ_set θ₀ eval) (S_iid : Causalean.Stat.IIDSample P.Ω (γ × Bool × ℝ) P.μ S.P_Z) (split : Causalean.Stat.OneShotSplit S_iid) {M_Θ M_Y M_μ ε : ℝ}, Causalean.Estimation.OrthogonalLearning.DREvalBounded Θ_set eval M_Θ → Causalean.Estimation.OrthogonalLearning.DROutcomeBounded S M_Y → ∀ (h : Causalean.Estimation.ATE.NuisanceVec γ), Causalean.Estimation.OrthogonalLearning.DRNuisanceMuBounded h M_μ → Causalean.Estimation.OrthogonalLearning.DRNuisanceOverlap S Θ_set h ε → Causalean.Estimation.OrthogonalLearning.CenteredClampedThetaMinimizes (Causalean.Estimation.OrthogonalLearning.drLearningSystem S Θ Θ_set Θ_convex θ₀ θ₀_mem eval eval_meas eval_θ₀ θ₀_minimizes) (2 * (M_Θ + 2 * M_μ + 2 * (M_Y + M_μ) / ε) ^ 2) → (∀ (z : γ × Bool × ℝ), Continuous fun θ => (Causalean.Estimation.OrthogonalLearning.drLearningSystem S Θ Θ_set Θ_convex θ₀ θ₀_mem eval eval_meas eval_θ₀ θ₀_minimizes).ℓ z (↑θ) h) → ∀ (idx : ℕ → ↑(Causalean.Estimation.OrthogonalLearning.drLearningSystem S Θ Θ_set Θ_convex θ₀ θ₀_mem eval eval_meas eval_θ₀ θ₀_minimizes).Θ_set), DenseRange idx → ∀ {norm : (γ × Bool × ℝ → ℝ) → ℝ} {ψ : ℕ → ℝ → ℝ}, Causalean.Estimation.OrthogonalLearning.DRCriticalRadius S Θ_set eval (fun k => ⟨↑(idx k), (idx k).property⟩) h norm ψ → (∀ (F F' : γ × Bool × ℝ → ℝ), F =ᵐ[S.P_Z] F' → norm F = norm F') → ∀ {Rmax L : ℝ}, 0 ≤ L → (∀ θ ∈ (Causalean.Estimation.OrthogonalLearning.drLearningSystem S Θ Θ_set Θ_convex θ₀ θ₀_mem eval eval_meas eval_θ₀ θ₀_minimizes).Θ_set, (norm fun z => (Causalean.Estimation.OrthogonalLearning.drLearningSystem S Θ Θ_set Θ_convex θ₀ θ₀_mem eval eval_meas eval_θ₀ θ₀_minimizes).ℓ z θ h - (Causalean.Estimation.OrthogonalLearning.drLearningSystem S Θ Θ_set Θ_convex θ₀ θ₀_mem eval eval_meas eval_θ₀ θ₀_minimizes).ℓ z (Causalean.Estimation.OrthogonalLearning.drLearningSystem S Θ Θ_set Θ_convex θ₀ θ₀_mem eval eval_meas eval_θ₀ θ₀_minimizes).θ₀ h) ≤ L * ‖θ - (Causalean.Estimation.OrthogonalLearning.drLearningSystem S Θ Θ_set Θ_convex θ₀ θ₀_mem eval eval_meas eval_θ₀ θ₀_minimizes).θ₀‖) → (∀ θ ∈ (Causalean.Estimation.OrthogonalLearning.drLearningSystem S Θ Θ_set Θ_convex θ₀ θ₀_mem eval eval_meas eval_θ₀ θ₀_minimizes).Θ_set, (norm fun z => (Causalean.Estimation.OrthogonalLearning.drLearningSystem S Θ Θ_set Θ_convex θ₀ θ₀_mem eval eval_meas eval_θ₀ θ₀_minimizes).ℓ z θ h - (Causalean.Estimation.OrthogonalLearning.drLearningSystem S Θ Θ_set Θ_convex θ₀ θ₀_mem eval eval_meas eval_θ₀ θ₀_minimizes).ℓ z (Causalean.Estimation.OrthogonalLearning.drLearningSystem S Θ Θ_set Θ_convex θ₀ θ₀_mem eval eval_meas eval_θ₀ θ₀_minimizes).θ₀ h) ≤ Rmax) → (∀ (m : ℕ), Causalean.Stat.Concentration.criticalRadius (ψ m) ≤ Rmax) → (∀ (m : ℕ), 0 < Causalean.Stat.Concentration.criticalRadius (ψ m)) → (∀ (m : ℕ), ψ m (Causalean.Stat.Concentration.criticalRadius (ψ m)) ≤ Causalean.Stat.Concentration.criticalRadius (ψ m) ^ 2) → (∀ (m : ℕ), Causalean.Stat.Concentration.RademacherUpperBound (fun θ z => (Causalean.Estimation.OrthogonalLearning.drLearningSystem S Θ Θ_set Θ_convex θ₀ θ₀_mem eval eval_meas eval_θ₀ θ₀_minimizes).ℓ z (↑θ) h - (Causalean.Estimation.OrthogonalLearning.drLearningSystem S Θ Θ_set Θ_convex θ₀ θ₀_mem eval eval_meas eval_θ₀ θ₀_minimizes).ℓ z (Causalean.Estimation.OrthogonalLearning.drLearningSystem S Θ Θ_set Θ_convex θ₀ θ₀_mem eval eval_meas eval_θ₀ θ₀_minimizes).θ₀ h) norm S.P_Z id m (ψ m)) → (∀ (m : ℕ) (r : ℝ) (S_fin : Fin m → γ × Bool × ℝ) (σ : Signs m), BddAbove (Set.range fun p => |(↑m)⁻¹ * ∑ k, ↑↑(σ k) * Causalean.Stat.Concentration.starHullZeroOut (fun θ z => (Causalean.Estimation.OrthogonalLearning.drLearningSystem S Θ Θ_set Θ_convex θ₀ θ₀_mem eval eval_meas eval_θ₀ θ₀_minimizes).ℓ z (↑θ) h - (Causalean.Estimation.OrthogonalLearning.drLearningSystem S Θ Θ_set Θ_convex θ₀ θ₀_mem eval eval_meas eval_θ₀ θ₀_minimizes).ℓ z (Causalean.Estimation.OrthogonalLearning.drLearningSystem S Θ Θ_set Θ_convex θ₀ θ₀_mem eval eval_meas eval_θ₀ θ₀_minimizes).θ₀ h) norm r p (S_fin k)|)) → (∀ (m : ℕ) (r : ℝ), MeasureTheory.Integrable (fun ω => empiricalRademacherComplexity m (Causalean.Stat.Concentration.starHullZeroOut (fun θ z => (Causalean.Estimation.OrthogonalLearning.drLearningSystem S Θ Θ_set Θ_convex θ₀ θ₀_mem eval eval_meas eval_θ₀ θ₀_minimizes).ℓ z (↑θ) h - (Causalean.Estimation.OrthogonalLearning.drLearningSystem S Θ Θ_set Θ_convex θ₀ θ₀_mem eval eval_meas eval_θ₀ θ₀_minimizes).ℓ z (Causalean.Estimation.OrthogonalLearning.drLearningSystem S Θ Θ_set Θ_convex θ₀ θ₀_mem eval eval_meas eval_θ₀ θ₀_minimizes).θ₀ h) norm r) (id ∘ ω)) (MeasureTheory.Measure.pi fun x => S.P_Z)) → ∀ {δ : ℝ}, 0 < δ → δ ≤ 1 → (∀ (n K : ℕ), 0 < (split.foldB n).card → Rmax ≤ Causalean.Stat.Concentration.criticalRadius (ψ (split.foldB n).card) * 2 ^ K → 2 * (M_Θ + 2 * M_μ + 2 * (M_Y + M_μ) / ε) ^ 2 * √(2 * Real.log (2 * (↑K + 1) / δ) / ↑(split.foldB n).card) ≤ Causalean.Stat.Concentration.criticalRadius (ψ (split.foldB n).card) ^ 2) → ∀ (Dθ_truth : Causalean.Estimation.OrthogonalLearning.HasDirDerivTheta (Causalean.Estimation.OrthogonalLearning.drLearningSystem S Θ Θ_set Θ_convex θ₀ θ₀_mem eval eval_meas eval_θ₀ θ₀_minimizes) (Causalean.Estimation.OrthogonalLearning.drLearningSystem S Θ Θ_set Θ_convex θ₀ θ₀_mem eval eval_meas eval_θ₀ θ₀_minimizes).g₀) (Dθ_at_h : Causalean.Estimation.OrthogonalLearning.HasDirDerivTheta (Causalean.Estimation.OrthogonalLearning.drLearningSystem S Θ Θ_set Θ_convex θ₀ θ₀_mem eval eval_meas eval_θ₀ θ₀_minimizes) h) (τhat : ℕ → P.Ω → Θ) (r_opt : ℕ → ℝ), Causalean.Estimation.OrthogonalLearning.SampleSplitPluginERM (Causalean.Estimation.OrthogonalLearning.drLearningSystem S Θ Θ_set Θ_convex θ₀ θ₀_mem eval eval_meas eval_θ₀ θ₀_minimizes) S_iid split τhat (fun x x_1 => h) r_opt → ∀ (σ : ℝ), 0 < σ → (∀ θ ∈ Θ_set, (Causalean.Estimation.OrthogonalLearning.drLearningSystem S Θ Θ_set Θ_convex θ₀ θ₀_mem eval eval_meas eval_θ₀ θ₀_minimizes).L θ h - (Causalean.Estimation.OrthogonalLearning.drLearningSystem S Θ Θ_set Θ_convex θ₀ θ₀_mem eval eval_meas eval_θ₀ θ₀_minimizes).L θ₀ h ≥ ∫ (z : γ × Bool × ℝ), Dθ_at_h.dℓ_θ θ z ∂S.P_Z + σ / 2 * ‖θ - θ₀‖ ^ 2) → Causalean.Estimation.OrthogonalLearning.FirstOrderInequality (Causalean.Estimation.OrthogonalLearning.drLearningSystem S Θ Θ_set Θ_convex θ₀ θ₀_mem eval eval_meas eval_θ₀ θ₀_minimizes) Dθ_truth → ∃ b, 0 ≤ b ∧ ∀ (n : ℕ), ∃ E, MeasurableSet E ∧ P.μ E ≥ 1 - ENNReal.ofReal δ ∧ ∀ ω ∈ E, ‖τhat n ω - θ₀‖ ^ 2 ≤ 4 * (1 + σ) / σ ^ 2 * (if (split.foldB n).card = 0 then √(2 * b) else (8 * L + 3) * Causalean.Stat.Concentration.criticalRadius (ψ (split.foldB n).card)) ^ 2 + 4 / σ * Causalean.Estimation.OrthogonalLearning.Bias_n (Causalean.Estimation.OrthogonalLearning.drLearningSystem S Θ Θ_set Θ_convex θ₀ θ₀_mem eval eval_meas eval_θ₀ θ₀_minimizes) Dθ_truth Dθ_at_h (τhat n ω) + 4 / σ * r_opt n","l":39,"f":"Causalean/Estimation/CATE/OrthogonalLearning/LocalEmpProcess/LocalizedDRLearnerEndToEnd.lean"},{"n":"Causalean.Estimation.OrthogonalLearning.oracle_inequality_plugin_ERM","k":"theorem","m":"Causalean.Estimation.OrthogonalLearning.OracleInequality","a":"Estimation/OrthogonalLearning","t":1,"d":"Conservative oracle inequality for the orthogonal sample-split plug-in ERM. Assume the realised nuisance at every sample point admits the directional-derivative structure needed for the first-order expansion of the loss, that the estimator is a sample-split plug-in ERM with optimization slack r_opt, and that the strong-convexity modulus σ is strictly positive. Suppose the population risk at the realised nuisance is σ-strongly convex around θ₀ in the chosen norm, and that the first-order inequality holds at the truth (θ₀, g₀). If, at the specific sample point (n, ω), the local empirical-process modulus inequality is realised for the estimator's target value against the realised nuisance, then the squared target-space estimation error is bounded by 4(1+σ)/σ² · (ρ n)² + (4/σ) · Bias_n + (4/σ) · r_opt n.","s":"∀ {Ω : Type u_1} [inst : MeasurableSpace Ω] {μ : MeasureTheory.Measure Ω} {Z : Type u_2} [inst_1 : MeasurableSpace Z] {P_Z : MeasureTheory.Measure Z} {Θ : Type u_3} [inst_2 : NormedAddCommGroup Θ] [inst_3 : InnerProductSpace ℝ Θ] {G : Type u_4} [inst_4 : AddCommGroup G] [inst_5 : Module ℝ G] (S : Causalean.Estimation.OrthogonalLearning.LearningSystem Ω μ Z P_Z Θ G) (S_iid : Causalean.Stat.IIDSample Ω Z μ P_Z) (split : Causalean.Stat.OneShotSplit S_iid) (Dθ_truth : Causalean.Estimation.OrthogonalLearning.HasDirDerivTheta S S.g₀) (ghat : ℕ → Ω → G) (Dθ_hat : (n : ℕ) → (ω : Ω) → Causalean.Estimation.OrthogonalLearning.HasDirDerivTheta S (ghat n ω)) (θhat : ℕ → Ω → Θ) (r_opt : ℕ → ℝ), Causalean.Estimation.OrthogonalLearning.SampleSplitPluginERM S S_iid split θhat ghat r_opt → ∀ (σ : ℝ), 0 < σ → (∀ (n : ℕ) (ω : Ω), ∀ θ ∈ S.Θ_set, S.L θ (ghat n ω) - S.L S.θ₀ (ghat n ω) ≥ ∫ (z : Z), (Dθ_hat n ω).dℓ_θ θ z ∂P_Z + σ / 2 * ‖θ - S.θ₀‖ ^ 2) → Causalean.Estimation.OrthogonalLearning.FirstOrderInequality S Dθ_truth → ∀ (ρ : ℕ → ℝ) (n : ℕ) (ω : Ω), S.L (θhat n ω) (ghat n ω) - S.L S.θ₀ (ghat n ω) - (Causalean.Estimation.OrthogonalLearning.empRiskFoldB S S_iid split n ω (θhat n ω) (ghat n ω) - Causalean.Estimation.OrthogonalLearning.empRiskFoldB S S_iid split n ω S.θ₀ (ghat n ω)) ≤ ρ n * ‖θhat n ω - S.θ₀‖ + ρ n ^ 2 → ‖θhat n ω - S.θ₀‖ ^ 2 ≤ 4 * (1 + σ) / σ ^ 2 * ρ n ^ 2 + 4 / σ * Causalean.Estimation.OrthogonalLearning.Bias_n S Dθ_truth (Dθ_hat n ω) (θhat n ω) + 4 / σ * r_opt n","l":69,"f":"Causalean/Estimation/OrthogonalLearning/OracleInequality.lean"},{"n":"Causalean.Estimation.OrthogonalLearning.oracle_inequality_plugin_ERM_highProb","k":"theorem","m":"Causalean.Estimation.OrthogonalLearning.OracleInequality","a":"Estimation/OrthogonalLearning","t":1,"d":"High-probability conservative oracle inequality for the orthogonal sample-split plug-in ERM. Fix a single nuisance value g. Assume the estimator is a sample-split plug-in ERM at this fixed nuisance, with optimization slack r_opt, and that the strong-convexity modulus σ is strictly positive. Suppose the population risk at g is σ-strongly convex around θ₀ in the chosen norm, and that the first-order inequality holds at the truth (θ₀, g₀). If the local empirical-process modulus condition holds at rate ρ and confidence level δ for the fixed nuisance g, then for every sample size n there is an event of probability at least 1 - δ on which the squared target-space estimation error is bounded by 4(1+σ)/σ² · (ρ n)² + (4/σ) · Bias_n + (4/σ) · r_opt n.","s":"∀ {Ω : Type u_1} [inst : MeasurableSpace Ω] {μ : MeasureTheory.Measure Ω} {Z : Type u_2} [inst_1 : MeasurableSpace Z] {P_Z : MeasureTheory.Measure Z} {Θ : Type u_3} [inst_2 : NormedAddCommGroup Θ] [inst_3 : InnerProductSpace ℝ Θ] {G : Type u_4} [inst_4 : AddCommGroup G] [inst_5 : Module ℝ G] (S : Causalean.Estimation.OrthogonalLearning.LearningSystem Ω μ Z P_Z Θ G) (S_iid : Causalean.Stat.IIDSample Ω Z μ P_Z) (split : Causalean.Stat.OneShotSplit S_iid) (Dθ_truth : Causalean.Estimation.OrthogonalLearning.HasDirDerivTheta S S.g₀) (g : G) (Dθ_at_g : Causalean.Estimation.OrthogonalLearning.HasDirDerivTheta S g) (θhat : ℕ → Ω → Θ) (r_opt : ℕ → ℝ), Causalean.Estimation.OrthogonalLearning.SampleSplitPluginERM S S_iid split θhat (fun x x_1 => g) r_opt → ∀ (σ : ℝ), 0 < σ → (∀ θ ∈ S.Θ_set, S.L θ g - S.L S.θ₀ g ≥ ∫ (z : Z), Dθ_at_g.dℓ_θ θ z ∂P_Z + σ / 2 * ‖θ - S.θ₀‖ ^ 2) → Causalean.Estimation.OrthogonalLearning.FirstOrderInequality S Dθ_truth → ∀ (ρ : ℕ → ℝ) (δ : ℝ), Causalean.Estimation.OrthogonalLearning.LocalEmpProcessModulus S S_iid split ρ δ g → ∀ (n : ℕ), ∃ E, MeasurableSet E ∧ μ E ≥ 1 - ENNReal.ofReal δ ∧ ∀ ω ∈ E, ‖θhat n ω - S.θ₀‖ ^ 2 ≤ 4 * (1 + σ) / σ ^ 2 * ρ n ^ 2 + 4 / σ * Causalean.Estimation.OrthogonalLearning.Bias_n S Dθ_truth Dθ_at_g (θhat n ω) + 4 / σ * r_opt n","l":180,"f":"Causalean/Estimation/OrthogonalLearning/OracleInequality.lean"},{"n":"Causalean.Estimation.OrthogonalLearning.randomParam_event_le","k":"theorem","m":"Causalean.Estimation.OrthogonalLearning.LocalEmpProcess.RandomParam","a":"Estimation/OrthogonalLearning/LocalEmpProcess","t":1,"d":"Cross-fit conditioning keystone. Let Y be measurable with pushforward law μ.map Y = ν, and let m_A be a sub-σ-algebra with m_A ≤ mΩ that is independent of the σ-algebra generated by Y. Suppose the family of \"bad\" subsets of β indexed by ω has an m_A-measurable joint dependence on (ω, ·) and each individual bad set has ν-mass at most δ. Then the random event where Y ω falls in the ω-dependent bad set also has μ-mass at most δ.","s":"∀ {Ω : Type u_1} {β : Type u_2} [mΩ : MeasurableSpace Ω] [mβ : MeasurableSpace β] {μ : MeasureTheory.Measure Ω} [MeasureTheory.IsProbabilityMeasure μ] {Y : Ω → β}, Measurable Y → ∀ {ν : MeasureTheory.Measure β} [MeasureTheory.IsProbabilityMeasure ν], MeasureTheory.Measure.map Y μ = ν → ∀ m_A ≤ mΩ, ProbabilityTheory.Indep m_A (MeasurableSpace.comap Y mβ) μ → ∀ {δ : ℝ} (Bad : Ω → Set β), MeasurableSet {p | p.2 ∈ Bad p.1} → (∀ (ω : Ω), ν (Bad ω) ≤ ENNReal.ofReal δ) → μ {ω | Y ω ∈ Bad ω} ≤ ENNReal.ofReal δ","l":54,"f":"Causalean/Estimation/OrthogonalLearning/LocalEmpProcess/RandomParam.lean"},{"n":"Causalean.Estimation.OrthogonalLearning.Θ_PerturbClosed","k":"def","m":"Causalean.Estimation.OrthogonalLearning.Setup","a":"Estimation/OrthogonalLearning","t":1,"d":"Closure-under-perturbation predicate for Θ_set: every line segment between θ₀ and θ ∈ Θ_set stays in Θ_set. Strictly weaker than convexity; useful for stating directional-derivative hypotheses without requiring the whole set to be convex.","s":"{Ω : Type u_1} → [inst : MeasurableSpace Ω] → {μ : MeasureTheory.Measure Ω} → {Z : Type u_2} → [inst_1 : MeasurableSpace Z] → {P_Z : MeasureTheory.Measure Z} → {Θ : Type u_3} → [inst_2 : NormedAddCommGroup Θ] → [inst_3 : InnerProductSpace ℝ Θ] → {G : Type u_4} → [inst_4 : AddCommGroup G] → [inst_5 : Module ℝ G] → Causalean.Estimation.OrthogonalLearning.LearningSystem Ω μ Z P_Z Θ G → Prop","l":99,"f":"Causalean/Estimation/OrthogonalLearning/Setup.lean"},{"n":"Causalean.Estimation.OrthogonalMoments.AutoDebias.AutoDebiasRepresenter","k":"structure","m":"Causalean.Estimation.OrthogonalMoments.AutoDebias.General","a":"Estimation/OrthogonalMoments/AutoDebias","t":1,"d":"Automatic debiasing representer. For a regression-nuisance moment system, a function α₀ on the covariate space that is measurable and integrable against the covariate measure, and that represents the Gateaux derivative of the population moment at the truth, in every direction, as the L²-inner product of α₀ against the regression target evaluated in that direction.","s":"Causalean.Estimation.OrthogonalMoments.AutoDebias.RegNuisanceMomentSys → Type u_2","l":109,"f":"Causalean/Estimation/OrthogonalMoments/AutoDebias/General.lean"},{"n":"Causalean.Estimation.OrthogonalMoments.AutoDebias.L_of_m","k":"def","m":"Causalean.Estimation.OrthogonalMoments.AutoDebias.Linear","a":"Estimation/OrthogonalMoments/AutoDebias","t":1,"d":"Population linear functional L(γ) := ∫ m_lin(z, γ) dP_Z.","s":"(S : Causalean.Estimation.OrthogonalMoments.AutoDebias.LinRegFnSys) → S.H_γ → ℝ","l":101,"f":"Causalean/Estimation/OrthogonalMoments/AutoDebias/Linear.lean"},{"n":"Causalean.Estimation.OrthogonalMoments.AutoDebias.L_of_m_add","k":"theorem","m":"Causalean.Estimation.OrthogonalMoments.AutoDebias.Linear","a":"Estimation/OrthogonalMoments/AutoDebias","t":2,"d":"Additivity of L_of_m on integrable summands.","s":"∀ (S : Causalean.Estimation.OrthogonalMoments.AutoDebias.LinRegFnSys) (γ₁ γ₂ : S.H_γ), MeasureTheory.Integrable (fun z => S.m_lin z γ₁) S.P_Z → MeasureTheory.Integrable (fun z => S.m_lin z γ₂) S.P_Z → Causalean.Estimation.OrthogonalMoments.AutoDebias.L_of_m S (γ₁ + γ₂) = Causalean.Estimation.OrthogonalMoments.AutoDebias.L_of_m S γ₁ + Causalean.Estimation.OrthogonalMoments.AutoDebias.L_of_m S γ₂","l":105,"f":"Causalean/Estimation/OrthogonalMoments/AutoDebias/Linear.lean"},{"n":"Causalean.Estimation.OrthogonalMoments.AutoDebias.L_of_m_smul","k":"theorem","m":"Causalean.Estimation.OrthogonalMoments.AutoDebias.Linear","a":"Estimation/OrthogonalMoments/AutoDebias","t":2,"d":"ℝ-homogeneity of L_of_m.","s":"∀ (S : Causalean.Estimation.OrthogonalMoments.AutoDebias.LinRegFnSys) (c : ℝ) (γ : S.H_γ), Causalean.Estimation.OrthogonalMoments.AutoDebias.L_of_m S (c • γ) = c * Causalean.Estimation.OrthogonalMoments.AutoDebias.L_of_m S γ","l":115,"f":"Causalean/Estimation/OrthogonalMoments/AutoDebias/Linear.lean"},{"n":"Causalean.Estimation.OrthogonalMoments.AutoDebias.LinRegFnSys","k":"structure","m":"Causalean.Estimation.OrthogonalMoments.AutoDebias.Linear","a":"Estimation/OrthogonalMoments/AutoDebias","t":1,"d":"Linear regression-functional system. Bundles an observation space with an induced regression-argument space, a regression class (a real vector space of nuisance parameters) paired with an evaluation map into the argument space, an observation-level functional linear in the nuisance argument, and a regression truth; the evaluation map is required additive and homogeneous in the nuisance argument, the observation-level functional is likewise additive and homogeneous in that argument and measurable in the observation for every fixed nuisance value, the regression-argument law is the pushforward of the observation law under the projection, and the regression residual at the truth, weighted by any measurable integrable function of the regression argument, integrates to zero.","s":"Type (max (max (u_1 + 1) (u_2 + 1)) (u_3 + 1))","l":39,"f":"Causalean/Estimation/OrthogonalMoments/AutoDebias/Linear.lean"},{"n":"Causalean.Estimation.OrthogonalMoments.AutoDebias.RegNuisanceMomentSys","k":"structure","m":"Causalean.Estimation.OrthogonalMoments.AutoDebias.General","a":"Estimation/OrthogonalMoments/AutoDebias","t":1,"d":"Regression-nuisance moment system. Bundles an observation space with an induced covariate space, an outcome variable, a regression target functional on a normed nuisance class, the true nuisance and true scalar parameter, a population moment functional of the parameter and nuisance, and an observation-level moment kernel that averages to it; the regression target is required additive and homogeneous in its nuisance argument, the population moment vanishes at the truth, the supplied Gateaux derivative of the moment at the truth is itself additive and homogeneous, the kernel is measurable and integrates against the observation measure to the population moment, the covariate measure is the pushforward of the observation measure under the projection, and the regression residual at the truth, weighted by any measurable integrable function of the covariates, integrates to zero.","s":"Type (max (max (u_1 + 1) (u_2 + 1)) (u_3 + 1))","l":42,"f":"Causalean/Estimation/OrthogonalMoments/AutoDebias/General.lean"},{"n":"Causalean.Estimation.OrthogonalMoments.AutoDebias.autoDebiasedMoment","k":"def","m":"Causalean.Estimation.OrthogonalMoments.AutoDebias.General","a":"Estimation/OrthogonalMoments/AutoDebias","t":1,"d":"Automatically debiased moment (population form):","s":"(S : Causalean.Estimation.OrthogonalMoments.AutoDebias.RegNuisanceMomentSys) → S.H → (S.X → ℝ) → ℝ → ℝ","l":132,"f":"Causalean/Estimation/OrthogonalMoments/AutoDebias/General.lean"},{"n":"Causalean.Estimation.OrthogonalMoments.AutoDebias.autoDebiasedMoment_directional_g_zero","k":"theorem","m":"Causalean.Estimation.OrthogonalMoments.AutoDebias.General","a":"Estimation/OrthogonalMoments/AutoDebias","t":1,"d":"Directional zero in the regression direction. For a regression-nuisance moment system with Riesz representer and any perturbation ν_g of the regression nuisance, the Gateaux derivative of the population debiased moment in the g-direction at the truth vanishes — equivalently, this is the representer identity for the perturbation ν_g.","s":"∀ (S : Causalean.Estimation.OrthogonalMoments.AutoDebias.RegNuisanceMomentSys) (rep : Causalean.Estimation.OrthogonalMoments.AutoDebias.AutoDebiasRepresenter S) (ν_g : S.H), S.D_g_M ν_g - ∫ (x : S.X), rep.α₀ x * S.γ_target ν_g x ∂S.P_X = 0","l":162,"f":"Causalean/Estimation/OrthogonalMoments/AutoDebias/General.lean"},{"n":"Causalean.Estimation.OrthogonalMoments.AutoDebias.autoDebiasedMoment_directional_α_zero","k":"theorem","m":"Causalean.Estimation.OrthogonalMoments.AutoDebias.General","a":"Estimation/OrthogonalMoments/AutoDebias","t":1,"d":"Directional zero in the representer direction. For any perturbation ν_α of the representer, assume ν_α is measurable and the ν_α-weighted regression-residual product at the truth is integrable. Then the directional derivative of the population debiased moment in the α-direction at the truth vanishes: the integral of the weighted regression residual is zero.","s":"∀ (S : Causalean.Estimation.OrthogonalMoments.AutoDebias.RegNuisanceMomentSys) (ν_α : S.X → ℝ), Measurable ν_α → MeasureTheory.Integrable (fun z => ν_α (S.proj_X z) * (S.Y_obs z - S.γ_target S.g₀ (S.proj_X z))) S.P_Z → ∫ (z : S.Z), ν_α (S.proj_X z) * (S.Y_obs z - S.γ_target S.g₀ (S.proj_X z)) ∂S.P_Z = 0","l":173,"f":"Causalean/Estimation/OrthogonalMoments/AutoDebias/General.lean"},{"n":"Causalean.Estimation.OrthogonalMoments.AutoDebias.autoDebiasedMoment_meanZero_at_truth","k":"theorem","m":"Causalean.Estimation.OrthogonalMoments.AutoDebias.General","a":"Estimation/OrthogonalMoments/AutoDebias","t":1,"d":"Mean-zero of the debiased moment at the truth. Given a regression-nuisance moment system with representer rep, assume the α₀-weighted regression-residual product at the truth is integrable and the baseline moment integrand at the truth is integrable. Then the automatically debiased population moment, evaluated at the true nuisance and true parameter, equals zero.","s":"∀ (S : Causalean.Estimation.OrthogonalMoments.AutoDebias.RegNuisanceMomentSys) (rep : Causalean.Estimation.OrthogonalMoments.AutoDebias.AutoDebiasRepresenter S), MeasureTheory.Integrable (fun z => rep.α₀ (S.proj_X z) * (S.Y_obs z - S.γ_target S.g₀ (S.proj_X z))) S.P_Z → MeasureTheory.Integrable (fun z => S.m S.g₀ z S.θ₀) S.P_Z → Causalean.Estimation.OrthogonalMoments.AutoDebias.autoDebiasedMoment S S.g₀ rep.α₀ S.θ₀ = 0","l":141,"f":"Causalean/Estimation/OrthogonalMoments/AutoDebias/General.lean"},{"n":"Causalean.Estimation.OrthogonalMoments.AutoDebias.autoDebiasedScore","k":"def","m":"Causalean.Estimation.OrthogonalMoments.AutoDebias.General","a":"Estimation/OrthogonalMoments/AutoDebias","t":1,"d":"Automatically debiased score (observation-level form):","s":"(S : Causalean.Estimation.OrthogonalMoments.AutoDebias.RegNuisanceMomentSys) → S.H → (S.X → ℝ) → ℝ → S.Z → ℝ","l":125,"f":"Causalean/Estimation/OrthogonalMoments/AutoDebias/General.lean"},{"n":"Causalean.Estimation.OrthogonalMoments.AutoDebias.finDiffRieszLoss","k":"def","m":"Causalean.Estimation.OrthogonalMoments.AutoDebias.General","a":"Estimation/OrthogonalMoments/AutoDebias","t":1,"d":"Finite-difference representer loss.","s":"(S : Causalean.Estimation.OrthogonalMoments.AutoDebias.RegNuisanceMomentSys) → {Ω : Type u_1} → (ℕ → Ω → S.Z) → (ℕ → Ω → ℝ) → (ℕ → Ω → S.H) → (ℕ → ℝ) → S.H → (ℕ → Finset ℕ) → ℕ → Ω → ℝ","l":198,"f":"Causalean/Estimation/OrthogonalMoments/AutoDebias/General.lean"},{"n":"Causalean.Estimation.OrthogonalMoments.AutoDebias.genRieszLoss","k":"def","m":"Causalean.Estimation.OrthogonalMoments.AutoDebias.General","a":"Estimation/OrthogonalMoments/AutoDebias","t":1,"d":"General Riesz loss.","s":"(S : Causalean.Estimation.OrthogonalMoments.AutoDebias.RegNuisanceMomentSys) → S.H → ℝ","l":187,"f":"Causalean/Estimation/OrthogonalMoments/AutoDebias/General.lean"},{"n":"Causalean.Estimation.OrthogonalMoments.AutoDebias.linAutoDMLEstimator","k":"def","m":"Causalean.Estimation.OrthogonalMoments.AutoDebias.DML","a":"Estimation/OrthogonalMoments/AutoDebias","t":1,"d":"One-shot linear Auto-DML estimator.","s":"{Ω : Type u_1} → [inst : MeasurableSpace Ω] → {μ : MeasureTheory.Measure Ω} → (S : Causalean.Estimation.OrthogonalMoments.AutoDebias.LinRegFnSys) → Causalean.Estimation.OrthogonalMoments.RieszRepresentation S.H_γ S.γ_target (Causalean.Estimation.OrthogonalMoments.AutoDebias.L_of_m S) S.P_X → (ε : ℝ) → 0 ≤ ε → (∀ (η : Causalean.Estimation.OrthogonalMoments.AutoDebias.linAutoNuisance S) (θ : ℝ), Measurable fun z => Causalean.Estimation.OrthogonalMoments.AutoDebias.linRieszScore S η.1 η.2 θ z) → (sample : Causalean.Stat.IIDSample Ω S.Z μ S.P_Z) → Causalean.Stat.OneShotSplit sample → (ℕ → Ω → Causalean.Estimation.OrthogonalMoments.AutoDebias.linAutoNuisance S) → ℕ → Ω → ℝ","l":220,"f":"Causalean/Estimation/OrthogonalMoments/AutoDebias/DML.lean"},{"n":"Causalean.Estimation.OrthogonalMoments.AutoDebias.linAutoDML_asymptoticLinear","k":"theorem","m":"Causalean.Estimation.OrthogonalMoments.AutoDebias.DML","a":"Estimation/OrthogonalMoments/AutoDebias","t":1,"d":"Linear Auto-DML asymptotic-linearity wrapper. Assume ε is nonnegative and that the linear Riesz score is measurable in the observation for every nuisance and target value. Given an i.i.d. sample with a one-shot fold split whose fold-B fraction converges to a strictly positive limit c > 0 along card (foldB n) / n → c, and a sequence of cross-fitted nuisance estimators η̂, suppose the Auto-DML moment has mean zero at the truth, the baseline score is square-integrable (finite variance), and the population moment at η̂ is bounded by a constant times the product of the two bilinear-remainder seminorms, at every fold and sample point. Assume the technical regularity package that the moment at η̂ is jointly measurable, fold-A-measurable in ω, and, at every fold and sample point, integrable and square-integrable. Finally suppose the L² score difference between the estimated and true nuisance is o_P(1), and the product of the two nuisance-error rates decays at the parametric rate o_P(n^{-1/2}). Then the one-shot linear Auto-DML estimator is asymptotically linear at the target value L_of_m S S.g₀, with influence function the baseline linear Riesz score scaled by the inverse Jacobian factor, indexed over the fold-B subsample.","s":"∀ {Ω : Type u_1} [inst : MeasurableSpace Ω] {μ : MeasureTheory.Measure Ω} [StandardBorelSpace Ω] [MeasureTheory.IsFiniteMeasure μ] [MeasureTheory.IsProbabilityMeasure μ] (S : Causalean.Estimation.OrthogonalMoments.AutoDebias.LinRegFnSys) (rep : Causalean.Estimation.OrthogonalMoments.RieszRepresentation S.H_γ S.γ_target (Causalean.Estimation.OrthogonalMoments.AutoDebias.L_of_m S) S.P_X) (ε : ℝ) (hε_nn : 0 ≤ ε) (h_score_meas : ∀ (η : Causalean.Estimation.OrthogonalMoments.AutoDebias.linAutoNuisance S) (θ : ℝ), Measurable fun z => Causalean.Estimation.OrthogonalMoments.AutoDebias.linRieszScore S η.1 η.2 θ z) (sample : Causalean.Stat.IIDSample Ω S.Z μ S.P_Z) (split : Causalean.Stat.OneShotSplit sample) {c : ℝ}, 0 < c → Filter.Tendsto (fun n => ↑(split.foldB n).card / ↑n) Filter.atTop (nhds c) → ∀ (η_hat : ℕ → Ω → Causalean.Estimation.OrthogonalMoments.AutoDebias.linAutoNuisance S), Causalean.Estimation.OrthogonalMoments.MeanZero (Causalean.Estimation.OrthogonalMoments.AutoDebias.linAutoGeneralMoment μ S rep ε hε_nn h_score_meas) → MeasureTheory.Integrable (fun z => Causalean.Estimation.OrthogonalMoments.AutoDebias.linRieszScore S S.g₀ rep.α₀ (Causalean.Estimation.OrthogonalMoments.AutoDebias.L_of_m S S.g₀) z ^ 2) S.P_Z → ∀ {Crem : ℝ}, (∀ (n : ℕ) (ω : Ω), |∫ (z : S.Z), (Causalean.Estimation.OrthogonalMoments.AutoDebias.linAutoGeneralMoment μ S rep ε hε_nn h_score_meas).m (η_hat n ω) z (Causalean.Estimation.OrthogonalMoments.AutoDebias.linAutoGeneralMoment μ S rep ε hε_nn h_score_meas).θ₀ ∂S.P_Z| ≤ Crem * ↑((Causalean.Estimation.OrthogonalMoments.AutoDebias.linAutoGeneralMoment μ S rep ε hε_nn h_score_meas).ρ₁ (η_hat n ω) (Causalean.Estimation.OrthogonalMoments.AutoDebias.linAutoGeneralMoment μ S rep ε hε_nn h_score_meas).η₀) * ↑((Causalean.Estimation.OrthogonalMoments.AutoDebias.linAutoGeneralMoment μ S rep ε hε_nn h_score_meas).ρ₂ (η_hat n ω) (Causalean.Estimation.OrthogonalMoments.AutoDebias.linAutoGeneralMoment μ S rep ε hε_nn h_score_meas).η₀)) → (∀ (n : ℕ), Measurable fun p => (Causalean.Estimation.OrthogonalMoments.AutoDebias.linAutoGeneralMoment μ S rep ε hε_nn h_score_meas).m (η_hat n p.1) p.2 (Causalean.Estimation.OrthogonalMoments.AutoDebias.linAutoGeneralMoment μ S rep ε hε_nn h_score_meas).θ₀) → (∀ (n : ℕ), Measurable fun ω z => (Causalean.Estimation.OrthogonalMoments.AutoDebias.linAutoGeneralMoment μ S rep ε hε_nn h_score_meas).m (η_hat n ω) z (Causalean.Estimation.OrthogonalMoments.AutoDebias.linAutoGeneralMoment μ S rep ε hε_nn h_score_meas).θ₀) → (∀ (n : ℕ), Measurable fun p => (Causalean.Estimation.OrthogonalMoments.AutoDebias.linAutoGeneralMoment μ S rep ε hε_nn h_score_meas).m (η_hat n p.1) p.2 (Causalean.Estimation.OrthogonalMoments.AutoDebias.linAutoGeneralMoment μ S rep ε hε_nn h_score_meas).θ₀) → (∀ (n : ℕ) (ω : Ω), MeasureTheory.Integrable (fun z => (Causalean.Estimation.OrthogonalMoments.AutoDebias.linAutoGeneralMoment μ S rep ε hε_nn h_score_meas).m (η_hat n ω) z (Causalean.Estimation.OrthogonalMoments.AutoDebias.linAutoGeneralMoment μ S rep ε hε_nn h_score_meas).θ₀) S.P_Z) → (∀ (n : ℕ) (ω : Ω), MeasureTheory.Integrable (fun z => (Causalean.Estimation.OrthogonalMoments.AutoDebias.linAutoGeneralMoment μ S rep ε hε_nn h_score_meas).m (η_hat n ω) z (Causalean.Estimation.OrthogonalMoments.AutoDebias.linAutoGeneralMoment μ S rep ε hε_nn h_score_meas).θ₀ ^ 2) S.P_Z) → Causalean.Stat.IsLittleOp (fun n ω => (MeasureTheory.eLpNorm (fun z => (Causalean.Estimation.OrthogonalMoments.AutoDebias.linAutoGeneralMoment μ S rep ε hε_nn h_score_meas).m (η_hat n ω) z (Causalean.Estimation.OrthogonalMoments.AutoDebias.linAutoGeneralMoment μ S rep ε hε_nn h_score_meas).θ₀ - (Causalean.Estimation.OrthogonalMoments.AutoDebias.linAutoGeneralMoment μ S rep ε hε_nn h_score_meas).m (Causalean.Estimation.OrthogonalMoments.AutoDebias.linAutoGeneralMoment μ S rep ε hε_nn h_score_meas).η₀ z (Causalean.Estimation.OrthogonalMoments.AutoDebias.linAutoGeneralMoment μ S rep ε hε_nn h_score_meas).θ₀) 2 S.P_Z).toReal) (fun x => 1) μ → Causalean.Stat.IsLittleOp (fun n ω => ↑((Causalean.Estimation.OrthogonalMoments.AutoDebias.linAutoGeneralMoment μ S rep ε hε_nn h_score_meas).ρ₁ (η_hat n ω) (Causalean.Estimation.OrthogonalMoments.AutoDebias.linAutoGeneralMoment μ S rep ε hε_nn h_score_meas).η₀) * ↑((Causalean.Estimation.OrthogonalMoments.AutoDebias.linAutoGeneralMoment μ S rep ε hε_nn h_score_meas).ρ₂ (η_hat n ω) (Causalean.Estimation.OrthogonalMoments.AutoDebias.linAutoGeneralMoment μ S rep ε hε_nn h_score_meas).η₀)) (fun n => ↑n ^ (-(1 / 2))) μ → Causalean.Stat.IsAsymLinear (Causalean.Estimation.OrthogonalMoments.AutoDebias.linAutoDMLEstimator S rep ε hε_nn h_score_meas sample split η_hat) (Causalean.Estimation.OrthogonalMoments.AutoDebias.L_of_m S S.g₀) (fun z => -(Causalean.Estimation.OrthogonalMoments.AutoDebias.linAutoGeneralMoment μ S rep ε hε_nn h_score_meas).J₀_inv * Causalean.Estimation.OrthogonalMoments.AutoDebias.linRieszScore S S.g₀ rep.α₀ (Causalean.Estimation.OrthogonalMoments.AutoDebias.L_of_m S S.g₀) z) sample split.foldB","l":240,"f":"Causalean/Estimation/OrthogonalMoments/AutoDebias/DML.lean"},{"n":"Causalean.Estimation.OrthogonalMoments.AutoDebias.linAutoGeneralMoment","k":"def","m":"Causalean.Estimation.OrthogonalMoments.AutoDebias.DML","a":"Estimation/OrthogonalMoments/AutoDebias","t":1,"d":"Linear Auto-DML moment as a GeneralMoment instance.","s":"{Ω : Type u_1} → [inst : MeasurableSpace Ω] → (μ : MeasureTheory.Measure Ω) → (S : Causalean.Estimation.OrthogonalMoments.AutoDebias.LinRegFnSys) → Causalean.Estimation.OrthogonalMoments.RieszRepresentation S.H_γ S.γ_target (Causalean.Estimation.OrthogonalMoments.AutoDebias.L_of_m S) S.P_X → (ε : ℝ) → 0 ≤ ε → (∀ (η : Causalean.Estimation.OrthogonalMoments.AutoDebias.linAutoNuisance S) (θ : ℝ), Measurable fun z => Causalean.Estimation.OrthogonalMoments.AutoDebias.linRieszScore S η.1 η.2 θ z) → Causalean.Estimation.OrthogonalMoments.GeneralMoment Ω μ S.Z S.P_Z (Causalean.Estimation.OrthogonalMoments.AutoDebias.linAutoNuisance S)","l":61,"f":"Causalean/Estimation/OrthogonalMoments/AutoDebias/DML.lean"},{"n":"Causalean.Estimation.OrthogonalMoments.AutoDebias.linAutoNuisance","k":"def","m":"Causalean.Estimation.OrthogonalMoments.AutoDebias.DML","a":"Estimation/OrthogonalMoments/AutoDebias","t":1,"d":"Joint nuisance type for the linear Auto-DML moment: a regression function γ ∈ H_γ paired with a Riesz-representer candidate α : X → ℝ. Carries componentwise AddCommGroup / Module ℝ.","s":"Causalean.Estimation.OrthogonalMoments.AutoDebias.LinRegFnSys → Type (max u_2 u_3)","l":46,"f":"Causalean/Estimation/OrthogonalMoments/AutoDebias/DML.lean"},{"n":"Causalean.Estimation.OrthogonalMoments.AutoDebias.linAuto_bilinearRem","k":"theorem","m":"Causalean.Estimation.OrthogonalMoments.AutoDebias.DML","a":"Estimation/OrthogonalMoments/AutoDebias","t":2,"d":"Bilinear remainder bound for the linear Auto-DML moment.","s":"∀ {Ω : Type u_1} [inst : MeasurableSpace Ω] (μ : MeasureTheory.Measure Ω) (S : Causalean.Estimation.OrthogonalMoments.AutoDebias.LinRegFnSys) (rep : Causalean.Estimation.OrthogonalMoments.RieszRepresentation S.H_γ S.γ_target (Causalean.Estimation.OrthogonalMoments.AutoDebias.L_of_m S) S.P_X) (ε : ℝ) (hε_nn : 0 ≤ ε) (h_score_meas : ∀ (η : Causalean.Estimation.OrthogonalMoments.AutoDebias.linAutoNuisance S) (θ : ℝ), Measurable fun z => Causalean.Estimation.OrthogonalMoments.AutoDebias.linRieszScore S η.1 η.2 θ z), ∃ C, ∀ η ∈ (Causalean.Estimation.OrthogonalMoments.AutoDebias.linAutoGeneralMoment μ S rep ε hε_nn h_score_meas).H_ε, Causalean.Estimation.OrthogonalMoments.AutoDebias.linAuto_int_pred✝ S rep η → MeasureTheory.MemLp (fun x => S.γ_target η.1 x - S.γ_target S.g₀ x) 2 S.P_X → MeasureTheory.MemLp (fun x => η.2 x - rep.α₀ x) 2 S.P_X → |∫ (z : S.Z), (Causalean.Estimation.OrthogonalMoments.AutoDebias.linAutoGeneralMoment μ S rep ε hε_nn h_score_meas).m η z (Causalean.Estimation.OrthogonalMoments.AutoDebias.linAutoGeneralMoment μ S rep ε hε_nn h_score_meas).θ₀ ∂S.P_Z| ≤ C * ↑((Causalean.Estimation.OrthogonalMoments.AutoDebias.linAutoGeneralMoment μ S rep ε hε_nn h_score_meas).ρ₁ η (Causalean.Estimation.OrthogonalMoments.AutoDebias.linAutoGeneralMoment μ S rep ε hε_nn h_score_meas).η₀) * ↑((Causalean.Estimation.OrthogonalMoments.AutoDebias.linAutoGeneralMoment μ S rep ε hε_nn h_score_meas).ρ₂ η (Causalean.Estimation.OrthogonalMoments.AutoDebias.linAutoGeneralMoment μ S rep ε hε_nn h_score_meas).η₀)","l":146,"f":"Causalean/Estimation/OrthogonalMoments/AutoDebias/DML.lean"},{"n":"Causalean.Estimation.OrthogonalMoments.AutoDebias.linAuto_meanZero","k":"theorem","m":"Causalean.Estimation.OrthogonalMoments.AutoDebias.DML","a":"Estimation/OrthogonalMoments/AutoDebias","t":2,"d":"Mean-zero of the linear Auto-DML moment at the truth. Specialises linRieszScore_meanZero.","s":"∀ {Ω : Type u_1} [inst : MeasurableSpace Ω] (μ : MeasureTheory.Measure Ω) (S : Causalean.Estimation.OrthogonalMoments.AutoDebias.LinRegFnSys) (rep : Causalean.Estimation.OrthogonalMoments.RieszRepresentation S.H_γ S.γ_target (Causalean.Estimation.OrthogonalMoments.AutoDebias.L_of_m S) S.P_X) (ε : ℝ) (hε_nn : 0 ≤ ε) (h_score_meas : ∀ (η : Causalean.Estimation.OrthogonalMoments.AutoDebias.linAutoNuisance S) (θ : ℝ), Measurable fun z => Causalean.Estimation.OrthogonalMoments.AutoDebias.linRieszScore S η.1 η.2 θ z), MeasureTheory.Integrable (fun z => rep.α₀ (S.proj_X z) * (S.Y_obs z - S.γ_target S.g₀ (S.proj_X z))) S.P_Z → Causalean.Estimation.OrthogonalMoments.MeanZero (Causalean.Estimation.OrthogonalMoments.AutoDebias.linAutoGeneralMoment μ S rep ε hε_nn h_score_meas)","l":102,"f":"Causalean/Estimation/OrthogonalMoments/AutoDebias/DML.lean"},{"n":"Causalean.Estimation.OrthogonalMoments.AutoDebias.linRieszLoss","k":"def","m":"Causalean.Estimation.OrthogonalMoments.AutoDebias.Linear","a":"Estimation/OrthogonalMoments/AutoDebias","t":1,"d":"Linear Riesz loss:","s":"(S : Causalean.Estimation.OrthogonalMoments.AutoDebias.LinRegFnSys) → S.H_γ → ℝ","l":172,"f":"Causalean/Estimation/OrthogonalMoments/AutoDebias/Linear.lean"},{"n":"Causalean.Estimation.OrthogonalMoments.AutoDebias.linRieszLoss_FOC_iff_representer","k":"theorem","m":"Causalean.Estimation.OrthogonalMoments.AutoDebias.Linear","a":"Estimation/OrthogonalMoments/AutoDebias","t":1,"d":"First-order condition for Riesz loss minimizers (Prop 4, first half). Assume the squared regression functional along every perturbed line α₀_idx + t • ν is integrable, that the linear moment integrand m_lin(·, ν) is integrable for every direction ν, and that the product γ_target α₀_idx · γ_target ν is integrable for every direction ν. Then α₀_idx is a directional minimizer of the Riesz loss along every line through it if and only if it indexes a Riesz representer, i.e. L_of_m S ν = ∫ γ_target α₀_idx · γ_target ν dP_X for every ν.","s":"∀ (S : Causalean.Estimation.OrthogonalMoments.AutoDebias.LinRegFnSys) (α₀_idx : S.H_γ), (∀ (ν : S.H_γ) (t : ℝ), MeasureTheory.Integrable (fun x => S.γ_target (α₀_idx + t • ν) x ^ 2) S.P_X) → (∀ (ν : S.H_γ), MeasureTheory.Integrable (fun z => S.m_lin z ν) S.P_Z) → (∀ (ν : S.H_γ), MeasureTheory.Integrable (fun x => S.γ_target α₀_idx x * S.γ_target ν x) S.P_X) → ((∀ (ν : S.H_γ) (t : ℝ), Causalean.Estimation.OrthogonalMoments.AutoDebias.linRieszLoss S α₀_idx ≤ Causalean.Estimation.OrthogonalMoments.AutoDebias.linRieszLoss S (α₀_idx + t • ν)) ↔ ∀ (ν : S.H_γ), Causalean.Estimation.OrthogonalMoments.AutoDebias.L_of_m S ν = ∫ (x : S.X), S.γ_target α₀_idx x * S.γ_target ν x ∂S.P_X)","l":244,"f":"Causalean/Estimation/OrthogonalMoments/AutoDebias/Linear.lean"},{"n":"Causalean.Estimation.OrthogonalMoments.AutoDebias.linRieszLoss_excess_eq_l2dist","k":"theorem","m":"Causalean.Estimation.OrthogonalMoments.AutoDebias.Linear","a":"Estimation/OrthogonalMoments/AutoDebias","t":1,"d":"Excess Riesz loss equals the squared L²(P_X) distance to the representer. Let α₀_idx index the Riesz representer via rep.α₀ = γ_target α₀_idx pointwise. Assume (γ_target α) ^ 2 is integrable, (γ_target α₀_idx) ^ 2 is integrable, the product γ_target α · γ_target α₀_idx is integrable, and the squared difference (γ_target α − γ_target α₀_idx) ^ 2 is integrable. Then the excess linear Riesz loss of α over α₀_idx equals the squared L²(P_X) distance between γ_target α and γ_target α₀_idx.","s":"∀ (S : Causalean.Estimation.OrthogonalMoments.AutoDebias.LinRegFnSys) (α α₀_idx : S.H_γ) (rep : Causalean.Estimation.OrthogonalMoments.RieszRepresentation S.H_γ S.γ_target (Causalean.Estimation.OrthogonalMoments.AutoDebias.L_of_m S) S.P_X), (∀ (x : S.X), rep.α₀ x = S.γ_target α₀_idx x) → MeasureTheory.Integrable (fun x => S.γ_target α x ^ 2) S.P_X → MeasureTheory.Integrable (fun x => S.γ_target α₀_idx x ^ 2) S.P_X → MeasureTheory.Integrable (fun x => S.γ_target α x * S.γ_target α₀_idx x) S.P_X → MeasureTheory.Integrable (fun x => (S.γ_target α x - S.γ_target α₀_idx x) ^ 2) S.P_X → Causalean.Estimation.OrthogonalMoments.AutoDebias.linRieszLoss S α - Causalean.Estimation.OrthogonalMoments.AutoDebias.linRieszLoss S α₀_idx = ∫ (x : S.X), (S.γ_target α x - S.γ_target α₀_idx x) ^ 2 ∂S.P_X","l":181,"f":"Causalean/Estimation/OrthogonalMoments/AutoDebias/Linear.lean"},{"n":"Causalean.Estimation.OrthogonalMoments.AutoDebias.linRieszScore","k":"def","m":"Causalean.Estimation.OrthogonalMoments.AutoDebias.Linear","a":"Estimation/OrthogonalMoments/AutoDebias","t":1,"d":"Linear Riesz score: alias of the generic rieszScore applied to the linear regression-functional system's data.","s":"(S : Causalean.Estimation.OrthogonalMoments.AutoDebias.LinRegFnSys) → S.H_γ → (S.X → ℝ) → ℝ → S.Z → ℝ","l":123,"f":"Causalean/Estimation/OrthogonalMoments/AutoDebias/Linear.lean"},{"n":"Causalean.Estimation.OrthogonalMoments.AutoDebias.linRieszScore_directional_g_zero","k":"theorem","m":"Causalean.Estimation.OrthogonalMoments.AutoDebias.Linear","a":"Estimation/OrthogonalMoments/AutoDebias","t":1,"d":"Directional zero in the regression direction. For a linear regression-functional system with Riesz representer and any perturbation ν_g of the regression nuisance, the Gateaux derivative of the population debiased moment in the g-direction at the truth vanishes — equivalently, this is the representer identity for the perturbation ν_g.","s":"∀ (S : Causalean.Estimation.OrthogonalMoments.AutoDebias.LinRegFnSys) (rep : Causalean.Estimation.OrthogonalMoments.RieszRepresentation S.H_γ S.γ_target (Causalean.Estimation.OrthogonalMoments.AutoDebias.L_of_m S) S.P_X) (ν_g : S.H_γ), ∫ (z : S.Z), S.m_lin z ν_g ∂S.P_Z - ∫ (x : S.X), rep.α₀ x * S.γ_target ν_g x ∂S.P_X = 0","l":146,"f":"Causalean/Estimation/OrthogonalMoments/AutoDebias/Linear.lean"},{"n":"Causalean.Estimation.OrthogonalMoments.AutoDebias.linRieszScore_directional_α_zero","k":"theorem","m":"Causalean.Estimation.OrthogonalMoments.AutoDebias.Linear","a":"Estimation/OrthogonalMoments/AutoDebias","t":1,"d":"Directional zero in the representer direction. For any perturbation ν_α of the representer, assume ν_α is measurable and the ν_α-weighted regression-residual product at the truth is integrable. Then the population mean of the ν_α-weighted regression residual at the truth is zero.","s":"∀ (S : Causalean.Estimation.OrthogonalMoments.AutoDebias.LinRegFnSys) (ν_α : S.X → ℝ), Measurable ν_α → MeasureTheory.Integrable (fun z => ν_α (S.proj_X z) * (S.Y_obs z - S.γ_target S.g₀ (S.proj_X z))) S.P_Z → ∫ (z : S.Z), ν_α (S.proj_X z) * (S.Y_obs z - S.γ_target S.g₀ (S.proj_X z)) ∂S.P_Z = 0","l":159,"f":"Causalean/Estimation/OrthogonalMoments/AutoDebias/Linear.lean"},{"n":"Causalean.Estimation.OrthogonalMoments.AutoDebias.linRieszScore_meanZero","k":"theorem","m":"Causalean.Estimation.OrthogonalMoments.AutoDebias.Linear","a":"Estimation/OrthogonalMoments/AutoDebias","t":1,"d":"Mean-zero of the debiased linear score at the truth. Given the linear regression-functional system with Riesz representer rep, assume the α₀-weighted regression-residual product at the truth is integrable. Then the population mean of the linear Riesz score, evaluated at the true regression function and the representer's α₀, equals zero.","s":"∀ (S : Causalean.Estimation.OrthogonalMoments.AutoDebias.LinRegFnSys) (rep : Causalean.Estimation.OrthogonalMoments.RieszRepresentation S.H_γ S.γ_target (Causalean.Estimation.OrthogonalMoments.AutoDebias.L_of_m S) S.P_X), MeasureTheory.Integrable (fun z => rep.α₀ (S.proj_X z) * (S.Y_obs z - S.γ_target S.g₀ (S.proj_X z))) S.P_Z → ∫ (z : S.Z), Causalean.Estimation.OrthogonalMoments.AutoDebias.linRieszScore S S.g₀ rep.α₀ (Causalean.Estimation.OrthogonalMoments.AutoDebias.L_of_m S S.g₀) z ∂S.P_Z = 0","l":129,"f":"Causalean/Estimation/OrthogonalMoments/AutoDebias/Linear.lean"},{"n":"Causalean.Estimation.OrthogonalMoments.BilinearRemainder","k":"def","m":"Causalean.Estimation.OrthogonalMoments.RemainderBound","a":"Estimation/OrthogonalMoments","t":1,"d":"Bilinear remainder predicate: the population moment at any η ∈ H_ε is bounded by C · ρ₁(η, η₀) · ρ₂(η, η₀).","s":"{Ω : Type u_1} → [inst : MeasurableSpace Ω] → {μ : MeasureTheory.Measure Ω} → {Z : Type u_2} → [inst_1 : MeasurableSpace Z] → {P_Z : MeasureTheory.Measure Z} → {H : Type u_3} → [inst_2 : AddCommGroup H] → [inst_3 : Module ℝ H] → Causalean.Estimation.OrthogonalMoments.GeneralMoment Ω μ Z P_Z H → ℝ → Prop","l":36,"f":"Causalean/Estimation/OrthogonalMoments/RemainderBound.lean"},{"n":"Causalean.Estimation.OrthogonalMoments.DiffQuotientEnvelope","k":"def","m":"Causalean.Estimation.OrthogonalMoments.NeymanOrthogonal","a":"Estimation/OrthogonalMoments","t":1,"d":"DiffQuotientEnvelope M asserts that, locally near t = 0, the difference quotient of m along the segment η₀ → η is dominated by a fixed L¹(P_Z) function.","s":"{Ω : Type u_1} → [inst : MeasurableSpace Ω] → {μ : MeasureTheory.Measure Ω} → {Z : Type u_2} → [inst_1 : MeasurableSpace Z] → {P_Z : MeasureTheory.Measure Z} → {H : Type u_3} → [inst_2 : AddCommGroup H] → [inst_3 : Module ℝ H] → Causalean.Estimation.OrthogonalMoments.GeneralMoment Ω μ Z P_Z H → Prop","l":44,"f":"Causalean/Estimation/OrthogonalMoments/NeymanOrthogonal.lean"},{"n":"Causalean.Estimation.OrthogonalMoments.GeneralMoment","k":"structure","m":"Causalean.Estimation.OrthogonalMoments.MomentFunctional","a":"Estimation/OrthogonalMoments","t":1,"d":"A general moment bundles a score function of a nuisance, an observation, and a scalar parameter, a truth nuisance η₀ and truth parameter θ₀, a set of admissible nuisance perturbations, and a pair of bilinear seminorms used to bound product-rate remainders, subject to: the score is jointly measurable in the observation for every nuisance and parameter value, the truth nuisance belongs to the perturbation set, and the population moment's parameter-derivative at the truth (its Jacobian) is nonzero, so that its inverse is well-defined.","s":"(Ω : Type u_1) → [inst : MeasurableSpace Ω] → MeasureTheory.Measure Ω → (Z : Type u_2) → [inst : MeasurableSpace Z] → MeasureTheory.Measure Z → (H : Type u_3) → [inst : AddCommGroup H] → [Module ℝ H] → Type (max u_2 u_3)","l":36,"f":"Causalean/Estimation/OrthogonalMoments/MomentFunctional.lean"},{"n":"Causalean.Estimation.OrthogonalMoments.GeneralMoment.J₀_inv","k":"def","m":"Causalean.Estimation.OrthogonalMoments.MomentFunctional","a":"Estimation/OrthogonalMoments","t":1,"d":"The inverse Jacobian is the reciprocal of the nonzero population Jacobian.","s":"{Ω : Type u_1} → [inst : MeasurableSpace Ω] → {μ : MeasureTheory.Measure Ω} → {Z : Type u_2} → [inst_1 : MeasurableSpace Z] → {P_Z : MeasureTheory.Measure Z} → {H : Type u_3} → [inst_2 : AddCommGroup H] → [inst_3 : Module ℝ H] → Causalean.Estimation.OrthogonalMoments.GeneralMoment Ω μ Z P_Z H → ℝ","l":83,"f":"Causalean/Estimation/OrthogonalMoments/MomentFunctional.lean"},{"n":"Causalean.Estimation.OrthogonalMoments.GeneralMoment.J₀_mul_J₀_inv","k":"theorem","m":"Causalean.Estimation.OrthogonalMoments.MomentFunctional","a":"Estimation/OrthogonalMoments","t":1,"d":"Jacobian times its inverse is one. For a general orthogonal-moment system, the population Jacobian times its inverse equals one.","s":"∀ {Ω : Type u_1} [inst : MeasurableSpace Ω] {μ : MeasureTheory.Measure Ω} {Z : Type u_2} [inst_1 : MeasurableSpace Z] {P_Z : MeasureTheory.Measure Z} {H : Type u_3} [inst_2 : AddCommGroup H] [inst_3 : Module ℝ H] (M : Causalean.Estimation.OrthogonalMoments.GeneralMoment Ω μ Z P_Z H), M.J₀ * M.J₀_inv = 1","l":89,"f":"Causalean/Estimation/OrthogonalMoments/MomentFunctional.lean"},{"n":"Causalean.Estimation.OrthogonalMoments.H_ε_PerturbClosed","k":"def","m":"Causalean.Estimation.OrthogonalMoments.MomentFunctional","a":"Estimation/OrthogonalMoments","t":1,"d":"The perturbation set is closed under line segments from the true nuisance to any nuisance already in the set.","s":"{Ω : Type u_1} → [inst : MeasurableSpace Ω] → {μ : MeasureTheory.Measure Ω} → {Z : Type u_2} → [inst_1 : MeasurableSpace Z] → {P_Z : MeasureTheory.Measure Z} → {H : Type u_3} → [inst_2 : AddCommGroup H] → [inst_3 : Module ℝ H] → Causalean.Estimation.OrthogonalMoments.GeneralMoment Ω μ Z P_Z H → Prop","l":106,"f":"Causalean/Estimation/OrthogonalMoments/MomentFunctional.lean"},{"n":"Causalean.Estimation.OrthogonalMoments.HasDirDeriv","k":"structure","m":"Causalean.Estimation.OrthogonalMoments.DirectionalDeriv","a":"Estimation/OrthogonalMoments","t":1,"d":"For a general moment, a candidate pointwise directional-derivative function dM of the score along the line segment from the truth to a perturbed nuisance, evaluated at each observation, together with the witnesses that for every perturbation in the admissible set and every observation, the score's difference quotient along that segment tends to dM's value there as the step size shrinks to zero, and that dM at each perturbation is measurable in the observation.","s":"{Ω : Type u_1} → [inst : MeasurableSpace Ω] → {μ : MeasureTheory.Measure Ω} → {Z : Type u_2} → [inst_1 : MeasurableSpace Z] → {P_Z : MeasureTheory.Measure Z} → {H : Type u_3} → [inst_2 : AddCommGroup H] → [inst_3 : Module ℝ H] → Causalean.Estimation.OrthogonalMoments.GeneralMoment Ω μ Z P_Z H → Type (max u_2 u_3)","l":35,"f":"Causalean/Estimation/OrthogonalMoments/DirectionalDeriv.lean"},{"n":"Causalean.Estimation.OrthogonalMoments.LinearMoment","k":"structure","m":"Causalean.Estimation.OrthogonalMoments.MomentFunctional","a":"Estimation/OrthogonalMoments","t":1,"d":"A linear moment is a general moment whose score is affine in the scalar target parameter.","s":"(Ω : Type u_4) → [inst : MeasurableSpace Ω] → MeasureTheory.Measure Ω → (Z : Type u_5) → [inst : MeasurableSpace Z] → MeasureTheory.Measure Z → (H : Type u_6) → [inst : AddCommGroup H] → [Module ℝ H] → Type (max u_5 u_6)","l":111,"f":"Causalean/Estimation/OrthogonalMoments/MomentFunctional.lean"},{"n":"Causalean.Estimation.OrthogonalMoments.LinearSmootherOp","k":"structure","m":"Causalean.Estimation.OrthogonalMoments.LinearSmoother","a":"Estimation/OrthogonalMoments","t":1,"d":"Linear-smoother operator (Def def:est-cate-second-stage, smoother form). A second-stage regression operator extended with an abstract array of smoothing weights indexed by sample size, randomness scope, query point, and data tuple, from which the weighted-sum representation of the operator's output can be built; the linear-combination identity itself is not required here but recorded separately as a predicate below.","s":"(Ω : Type u_1) → [inst : MeasurableSpace Ω] → MeasureTheory.Measure Ω → (γ : Type u_2) → [MeasurableSpace γ] → Type (max u_1 u_2)","l":43,"f":"Causalean/Estimation/OrthogonalMoments/LinearSmoother.lean"},{"n":"Causalean.Estimation.OrthogonalMoments.LinearSmootherOp.IsLinearSmoother","k":"def","m":"Causalean.Estimation.OrthogonalMoments.LinearSmoother","a":"Estimation/OrthogonalMoments","t":1,"d":"Predicate witnessing that the operator is genuinely a linear smoother: the value of evalAt n ω f x is the weighted sum Σ_{i ∈ B} w i · f (xs i), where B : Finset ι enumerates the data fold, w : ι → ℝ provides the weights, and xs : ι → γ × Bool × ℝ provides the data tuples. The exact relationship between w and op.weights is left to the caller.","s":"{Ω : Type u_1} → [inst : MeasurableSpace Ω] → {μ : MeasureTheory.Measure Ω} → {γ : Type u_2} → [inst_1 : MeasurableSpace γ] → {ι : Type u_3} → Causalean.Estimation.OrthogonalMoments.LinearSmootherOp Ω μ γ → ℕ → Ω → γ → Finset ι → (ι → ℝ) → (ι → γ × Bool × ℝ) → Prop","l":72,"f":"Causalean/Estimation/OrthogonalMoments/LinearSmoother.lean"},{"n":"Causalean.Estimation.OrthogonalMoments.MeanZero","k":"def","m":"Causalean.Estimation.OrthogonalMoments.MomentFunctional","a":"Estimation/OrthogonalMoments","t":1,"d":"The moment has zero population mean at the true nuisance and target.","s":"{Ω : Type u_1} → [inst : MeasurableSpace Ω] → {μ : MeasureTheory.Measure Ω} → {Z : Type u_2} → [inst_1 : MeasurableSpace Z] → {P_Z : MeasureTheory.Measure Z} → {H : Type u_3} → [inst_2 : AddCommGroup H] → [inst_3 : Module ℝ H] → Causalean.Estimation.OrthogonalMoments.GeneralMoment Ω μ Z P_Z H → Prop","l":102,"f":"Causalean/Estimation/OrthogonalMoments/MomentFunctional.lean"},{"n":"Causalean.Estimation.OrthogonalMoments.NeymanOrthogonal","k":"def","m":"Causalean.Estimation.OrthogonalMoments.NeymanOrthogonal","a":"Estimation/OrthogonalMoments","t":1,"d":"Neyman orthogonality for the (M, D) pair: the population integral of the directional derivative vanishes at every nuisance perturbation.","s":"{Ω : Type u_1} → [inst : MeasurableSpace Ω] → {μ : MeasureTheory.Measure Ω} → {Z : Type u_2} → [inst_1 : MeasurableSpace Z] → {P_Z : MeasureTheory.Measure Z} → {H : Type u_3} → [inst_2 : AddCommGroup H] → [inst_3 : Module ℝ H] → (M : Causalean.Estimation.OrthogonalMoments.GeneralMoment Ω μ Z P_Z H) → Causalean.Estimation.OrthogonalMoments.HasDirDeriv M → Prop","l":38,"f":"Causalean/Estimation/OrthogonalMoments/NeymanOrthogonal.lean"},{"n":"Causalean.Estimation.OrthogonalMoments.RieszRepresentation","k":"structure","m":"Causalean.Estimation.OrthogonalMoments.Riesz","a":"Estimation/OrthogonalMoments","t":1,"d":"A Riesz representation for a continuous linear functional on a regression class under a covariate measure: a function α₀ on the covariate space that is measurable, integrable against the covariate measure, and that represents the functional's value at every element of the regression class as the covariate-measure integral of α₀ against that element's evaluation.","s":"{X : Type u_3} → [inst : MeasurableSpace X] → (H_γ : Type u_4) → [inst_1 : AddCommGroup H_γ] → [Module ℝ H_γ] → (H_γ → X → ℝ) → (H_γ → ℝ) → MeasureTheory.Measure X → Type u_3","l":48,"f":"Causalean/Estimation/OrthogonalMoments/Riesz.lean"},{"n":"Causalean.Estimation.OrthogonalMoments.SecondStageOperator","k":"structure","m":"Causalean.Estimation.OrthogonalMoments.SecondStageOperator","a":"Estimation/OrthogonalMoments","t":1,"d":"Abstract bundle for a second-stage regression operator (Def def:est-cate-second-stage): an operator mapping a sample size, a randomness scope, a real-valued pseudo-outcome function of a data tuple, and a query point to a real-valued estimate, together with the minimal requirement that for every sample size and constant pseudo-outcome, the map from randomness scope and query point to the operator's value is jointly measurable; stronger measurability, and any linearity of the operator in its function input, are deferred to concrete instances or the separate IsLinearInInput predicate.","s":"(Ω : Type u_1) → [inst : MeasurableSpace Ω] → MeasureTheory.Measure Ω → (γ : Type u_2) → [MeasurableSpace γ] → Type (max u_1 u_2)","l":47,"f":"Causalean/Estimation/OrthogonalMoments/SecondStageOperator.lean"},{"n":"Causalean.Estimation.OrthogonalMoments.SecondStageOperator.IsLinearInInput","k":"def","m":"Causalean.Estimation.OrthogonalMoments.SecondStageOperator","a":"Estimation/OrthogonalMoments","t":1,"d":"Linearity of the operator in its pseudo-outcome input. A second-stage operator is *linear in input* iff evalAt n ω (f + g) x = evalAt n ω f x + evalAt n ω g x for all sample sizes, randomness, pseudo-outcomes, and query points. Linear smoothers satisfy this predicate; kernel-or-tree mean estimators with random splits need not satisfy it.","s":"{Ω : Type u_1} → [inst : MeasurableSpace Ω] → {μ : MeasureTheory.Measure Ω} → {γ : Type u_2} → [inst_1 : MeasurableSpace γ] → Causalean.Estimation.OrthogonalMoments.SecondStageOperator Ω μ γ → Prop","l":81,"f":"Causalean/Estimation/OrthogonalMoments/SecondStageOperator.lean"},{"n":"Causalean.Estimation.OrthogonalMoments.SecondStageOperator.oracleEstimator","k":"def","m":"Causalean.Estimation.OrthogonalMoments.SecondStageOperator","a":"Estimation/OrthogonalMoments","t":1,"d":"Oracle estimator: the operator applied to a fixed \"true\" pseudo-outcome f (Def def:est-cate-dr-learner, \\tilde\\tau_n).","s":"{Ω : Type u_1} → [inst : MeasurableSpace Ω] → {μ : MeasureTheory.Measure Ω} → {γ : Type u_2} → [inst_1 : MeasurableSpace γ] → Causalean.Estimation.OrthogonalMoments.SecondStageOperator Ω μ γ → (γ × Bool × ℝ → ℝ) → ℕ → Ω → γ → ℝ","l":92,"f":"Causalean/Estimation/OrthogonalMoments/SecondStageOperator.lean"},{"n":"Causalean.Estimation.OrthogonalMoments.SecondStageOperator.oracleRiskScale","k":"def","m":"Causalean.Estimation.OrthogonalMoments.SecondStageOperator","a":"Estimation/OrthogonalMoments","t":1,"d":"Oracle pointwise risk scale R^*_n(x) from def:est-cate-dr-learner:","s":"{Ω : Type u_1} → [inst : MeasurableSpace Ω] → {μ : MeasureTheory.Measure Ω} → {γ : Type u_2} → [inst_1 : MeasurableSpace γ] → Causalean.Estimation.OrthogonalMoments.SecondStageOperator Ω μ γ → (γ × Bool × ℝ → ℝ) → (γ → ℝ) → γ → ℕ → ℝ","l":98,"f":"Causalean/Estimation/OrthogonalMoments/SecondStageOperator.lean"},{"n":"Causalean.Estimation.OrthogonalMoments.Stable","k":"def","m":"Causalean.Estimation.OrthogonalMoments.SecondStageOperator","a":"Estimation/OrthogonalMoments","t":1,"d":"Stability of a second-stage regression operator at a query point x with respect to a distance d_n between pseudo-outcomes (Def def:est-cate-stability).","s":"{Ω : Type u_1} → [inst : MeasurableSpace Ω] → {μ : MeasureTheory.Measure Ω} → {γ : Type u_2} → [inst_1 : MeasurableSpace γ] → Causalean.Estimation.OrthogonalMoments.SecondStageOperator Ω μ γ → (γ → ℝ) → (ℕ → Ω → ℝ) → γ → ((ℕ → Ω → γ × Bool × ℝ → ℝ) → (γ × Bool × ℝ → ℝ) → (ℕ → Ω → γ → ℝ) → Prop) → Prop","l":114,"f":"Causalean/Estimation/OrthogonalMoments/SecondStageOperator.lean"},{"n":"Causalean.Estimation.OrthogonalMoments.WeightedNorm","k":"def","m":"Causalean.Estimation.OrthogonalMoments.LinearSmoother","a":"Estimation/OrthogonalMoments","t":1,"d":"The weighted norm is the normalized absolute-weight empirical norm of a real-valued function on the sample indices.","s":"{ι : Type u_1} → Finset ι → (ι → ℝ) → (ι → ℝ) → ℝ → ℝ","l":100,"f":"Causalean/Estimation/OrthogonalMoments/LinearSmoother.lean"},{"n":"Causalean.Estimation.OrthogonalMoments.WeightedNorm_nonneg","k":"theorem","m":"Causalean.Estimation.OrthogonalMoments.LinearSmoother","a":"Estimation/OrthogonalMoments","t":2,"d":"The weighted norm is non-negative.","s":"∀ {ι : Type u_1} (B : Finset ι) (w g : ι → ℝ) (p : ℝ), 0 ≤ Causalean.Estimation.OrthogonalMoments.WeightedNorm B w g p","l":115,"f":"Causalean/Estimation/OrthogonalMoments/LinearSmoother.lean"},{"n":"Causalean.Estimation.OrthogonalMoments.bilinear_remainder_of_smoothness","k":"theorem","m":"Causalean.Estimation.OrthogonalMoments.RemainderBound","a":"Estimation/OrthogonalMoments","t":1,"d":"Smoothness bridge to a bilinear remainder bound. For a general moment M with directional derivative D satisfying Neyman orthogonality, given an envelope g that is P_Z-integrable, suppose the linearization residual obeys a uniform second-order envelope with constant K: for every perturbation η in the nuisance neighborhood, |m(η,·,θ₀) − m(η₀,·,θ₀) − dM(η,·)| ≤ K·ρ₁(η,η₀)·ρ₂(η,η₀)·g almost everywhere, with the directional derivative integrable at every such η, the moment integrable at the baseline η₀, the moment integrable at every perturbed η, and the moment having population mean zero. Then some constant C makes the population moment obey the bilinear remainder bound C·ρ₁(η,η₀)·ρ₂(η,η₀) uniformly over the nuisance neighborhood.","s":"∀ {Ω : Type u_1} [inst : MeasurableSpace Ω] {μ : MeasureTheory.Measure Ω} {Z : Type u_2} [inst_1 : MeasurableSpace Z] {P_Z : MeasureTheory.Measure Z} {H : Type u_3} [inst_2 : AddCommGroup H] [inst_3 : Module ℝ H] (M : Causalean.Estimation.OrthogonalMoments.GeneralMoment Ω μ Z P_Z H) (D : Causalean.Estimation.OrthogonalMoments.HasDirDeriv M), Causalean.Estimation.OrthogonalMoments.NeymanOrthogonal M D → ∀ {K : ℝ} {g : Z → ℝ}, MeasureTheory.Integrable g P_Z → (∀ η ∈ M.H_ε, ∀ᵐ (z : Z) ∂P_Z, |M.m η z M.θ₀ - M.m M.η₀ z M.θ₀ - D.dM η z| ≤ K * ↑(M.ρ₁ η M.η₀) * ↑(M.ρ₂ η M.η₀) * g z) → (∀ η ∈ M.H_ε, MeasureTheory.Integrable (fun z => D.dM η z) P_Z) → MeasureTheory.Integrable (fun z => M.m M.η₀ z M.θ₀) P_Z → (∀ η ∈ M.H_ε, MeasureTheory.Integrable (fun z => M.m η z M.θ₀) P_Z) → Causalean.Estimation.OrthogonalMoments.MeanZero M → ∃ C, Causalean.Estimation.OrthogonalMoments.BilinearRemainder M C","l":43,"f":"Causalean/Estimation/OrthogonalMoments/RemainderBound.lean"},{"n":"Causalean.Estimation.OrthogonalMoments.dmlChernozhukovEstimator","k":"def","m":"Causalean.Estimation.OrthogonalMoments.DMLChernozhukov","a":"Estimation/OrthogonalMoments","t":1,"d":"Chernozhukov one-step DML estimator. Evaluates the score at the truth M.θ₀ and rescales by the Jacobian inverse:","s":"{Ω : Type u_1} → [inst : MeasurableSpace Ω] → {μ : MeasureTheory.Measure Ω} → {Z : Type u_2} → [inst_1 : MeasurableSpace Z] → {P_Z : MeasureTheory.Measure Z} → {H : Type u_3} → [inst_2 : AddCommGroup H] → [inst_3 : Module ℝ H] → Causalean.Estimation.OrthogonalMoments.GeneralMoment Ω μ Z P_Z H → (sample : Causalean.Stat.IIDSample Ω Z μ P_Z) → Causalean.Stat.OneShotSplit sample → (ℕ → Ω → H) → ℕ → Ω → ℝ","l":57,"f":"Causalean/Estimation/OrthogonalMoments/DMLChernozhukov.lean"},{"n":"Causalean.Estimation.OrthogonalMoments.dmlCrossFitEstimator","k":"def","m":"Causalean.Estimation.OrthogonalMoments.DMLCrossFit","a":"Estimation/OrthogonalMoments","t":1,"d":"K-fold cross-fitted Chernozhukov DML estimator. At each fold k, the nuisance estimator η_hat n k ω : H is trained on the complement of fold k. The fold-k score is the empirical mean of m(η̂^{(-k)}, ·, θ₀) over fold k, rescaled by −J₀⁻¹ and shifted by θ₀. The final estimator averages these K fold scores.","s":"{Ω : Type u_1} → [inst : MeasurableSpace Ω] → {μ : MeasureTheory.Measure Ω} → {Z : Type u_2} → [inst_1 : MeasurableSpace Z] → {P_Z : MeasureTheory.Measure Z} → {H : Type u_3} → [inst_2 : AddCommGroup H] → [inst_3 : Module ℝ H] → Causalean.Estimation.OrthogonalMoments.GeneralMoment Ω μ Z P_Z H → (sample : Causalean.Stat.IIDSample Ω Z μ P_Z) → {K : ℕ} → Causalean.Stat.KFoldSplit sample K → (ℕ → Fin K → Ω → H) → ℕ → Ω → ℝ","l":440,"f":"Causalean/Estimation/OrthogonalMoments/DMLCrossFit.lean"},{"n":"Causalean.Estimation.OrthogonalMoments.dml_chernozhukov_asymptoticLinear","k":"theorem","m":"Causalean.Estimation.OrthogonalMoments.DMLChernozhukov","a":"Estimation/OrthogonalMoments","t":1,"d":"Asymptotic linearity of the Chernozhukov DML estimator. Drops the zero-centering hypothesis hθ_zero from dml_asymptoticLinear. Given a general moment M with mean zero at the truth, assume the truth-evaluated score is square-integrable (finite variance). Given an i.i.d. sample with a one-shot fold split whose fold-B fraction converges to a strictly positive limit c > 0 along card (foldB n) / n → c, and a sequence of cross-fitted nuisance estimators η̂, suppose the population moment at η̂ is bounded by a constant times the product of the two bilinear-remainder seminorms, at every fold and sample point. Assume the technical regularity package that the moment at η̂ is jointly measurable, fold-A-measurable in ω, and, at every fold and sample point, integrable and square-integrable. Finally suppose the L² score difference between the estimated and true nuisance is o_P(1), and the product of the two nuisance-error rates decays at the parametric rate o_P(n^{-1/2}). Then the Chernozhukov one-step estimator is asymptotically linear at the truth M.θ₀, with influence function −J₀⁻¹ · m(η₀, ·, θ₀) and asymptotic variance J₀⁻¹ Σ J₀⁻ᵀ where Σ := ∫ m(η₀, z, θ₀)² dP_Z, indexed over split.foldB.","s":"∀ {Ω : Type u_1} [inst : MeasurableSpace Ω] {μ : MeasureTheory.Measure Ω} {Z : Type u_2} [inst_1 : MeasurableSpace Z] {P_Z : MeasureTheory.Measure Z} {H : Type u_3} [inst_2 : AddCommGroup H] [inst_3 : Module ℝ H] [StandardBorelSpace Ω] [MeasureTheory.IsFiniteMeasure μ] [MeasureTheory.IsProbabilityMeasure μ] (M : Causalean.Estimation.OrthogonalMoments.GeneralMoment Ω μ Z P_Z H), Causalean.Estimation.OrthogonalMoments.MeanZero M → MeasureTheory.Integrable (fun z => M.m M.η₀ z M.θ₀ ^ 2) P_Z → ∀ (sample : Causalean.Stat.IIDSample Ω Z μ P_Z) (split : Causalean.Stat.OneShotSplit sample) {c : ℝ}, 0 < c → Filter.Tendsto (fun n => ↑(split.foldB n).card / ↑n) Filter.atTop (nhds c) → ∀ (η_hat : ℕ → Ω → H) {Crem : ℝ}, (∀ (n : ℕ) (ω : Ω), |∫ (z : Z), M.m (η_hat n ω) z M.θ₀ ∂P_Z| ≤ Crem * ↑(M.ρ₁ (η_hat n ω) M.η₀) * ↑(M.ρ₂ (η_hat n ω) M.η₀)) → (∀ (n : ℕ), Measurable fun p => M.m (η_hat n p.1) p.2 M.θ₀) → (∀ (n : ℕ), Measurable fun ω z => M.m (η_hat n ω) z M.θ₀) → (∀ (n : ℕ), Measurable fun p => M.m (η_hat n p.1) p.2 M.θ₀) → (∀ (n : ℕ) (ω : Ω), MeasureTheory.Integrable (fun z => M.m (η_hat n ω) z M.θ₀) P_Z) → (∀ (n : ℕ) (ω : Ω), MeasureTheory.Integrable (fun z => M.m (η_hat n ω) z M.θ₀ ^ 2) P_Z) → Causalean.Stat.IsLittleOp (fun n ω => (MeasureTheory.eLpNorm (fun z => M.m (η_hat n ω) z M.θ₀ - M.m M.η₀ z M.θ₀) 2 P_Z).toReal) (fun x => 1) μ → Causalean.Stat.IsLittleOp (fun n ω => ↑(M.ρ₁ (η_hat n ω) M.η₀) * ↑(M.ρ₂ (η_hat n ω) M.η₀)) (fun n => ↑n ^ (-(1 / 2))) μ → Causalean.Stat.IsAsymLinear (Causalean.Estimation.OrthogonalMoments.dmlChernozhukovEstimator M sample split η_hat) M.θ₀ (fun z => -M.J₀_inv * M.m M.η₀ z M.θ₀) sample split.foldB","l":77,"f":"Causalean/Estimation/OrthogonalMoments/DMLChernozhukov.lean"},{"n":"Causalean.Estimation.OrthogonalMoments.dml_crossFit_asymptoticLinear","k":"theorem","m":"Causalean.Estimation.OrthogonalMoments.DMLCrossFit","a":"Estimation/OrthogonalMoments","t":1,"d":"Asymptotic linearity of the K-fold cross-fitted Chernozhukov DML estimator. Same Chernozhukov form as dml_chernozhukov_asymptoticLinear, but with K folds. Given a general moment M with mean zero at the truth, assume the truth-evaluated score is square-integrable (finite variance), that there are at least two folds, K > 1, and a sequence of per-fold cross-fitted nuisance estimators η̂. Suppose the population moment at η̂, trained on each fold's complement, is bounded by a constant times the product of the two bilinear-remainder seminorms, at every fold count, fold index, and sample point. Assume the technical regularity package that the moment at η̂ is jointly measurable at every fold, measurable with respect to the fold's training complement, and, at every fold count, fold index, and sample point, integrable and square-integrable. Finally suppose that, at every fold, the L² score difference between the fold's estimated and true nuisance is o_P(1), each of the two nuisance-error rates is individually o_P(1), and their product decays at the parametric rate o_P(n^{-1/2}). Then the K-fold cross-fitted estimator is asymptotically linear at the truth M.θ₀ with influence function −J₀⁻¹ · m(η₀, ·, θ₀), indexed over the full sample (the fold-level sub-aggregations sum to a full-sample average asymptotically).","s":"∀ {Ω : Type u_1} [inst : MeasurableSpace Ω] {μ : MeasureTheory.Measure Ω} {Z : Type u_2} [inst_1 : MeasurableSpace Z] {P_Z : MeasureTheory.Measure Z} {H : Type u_3} [inst_2 : AddCommGroup H] [inst_3 : Module ℝ H] [StandardBorelSpace Ω] [MeasureTheory.IsFiniteMeasure μ] [MeasureTheory.IsProbabilityMeasure μ] (M : Causalean.Estimation.OrthogonalMoments.GeneralMoment Ω μ Z P_Z H), Causalean.Estimation.OrthogonalMoments.MeanZero M → MeasureTheory.Integrable (fun z => M.m M.η₀ z M.θ₀ ^ 2) P_Z → ∀ (sample : Causalean.Stat.IIDSample Ω Z μ P_Z) {K : ℕ}, 1 < K → ∀ (split : Causalean.Stat.KFoldSplit sample K) (η_hat : ℕ → Fin K → Ω → H) {Crem : ℝ}, (∀ (n : ℕ) (k : Fin K) (ω : Ω), |∫ (z : Z), M.m (η_hat n k ω) z M.θ₀ ∂P_Z| ≤ Crem * ↑(M.ρ₁ (η_hat n k ω) M.η₀) * ↑(M.ρ₂ (η_hat n k ω) M.η₀)) → (∀ (n : ℕ) (k : Fin K), Measurable fun p => M.m (η_hat n k p.1) p.2 M.θ₀) → (∀ (n : ℕ) (k : Fin K), Measurable fun ω z => M.m (η_hat n k ω) z M.θ₀) → (∀ (n : ℕ) (k : Fin K), Measurable fun p => M.m (η_hat n k p.1) p.2 M.θ₀) → (∀ (n : ℕ) (k : Fin K) (ω : Ω), MeasureTheory.Integrable (fun z => M.m (η_hat n k ω) z M.θ₀) P_Z) → (∀ (n : ℕ) (k : Fin K) (ω : Ω), MeasureTheory.Integrable (fun z => M.m (η_hat n k ω) z M.θ₀ ^ 2) P_Z) → (∀ (k : Fin K), Causalean.Stat.IsLittleOp (fun n ω => (MeasureTheory.eLpNorm (fun z => M.m (η_hat n k ω) z M.θ₀ - M.m M.η₀ z M.θ₀) 2 P_Z).toReal) (fun x => 1) μ) → (∀ (k : Fin K), Causalean.Stat.IsLittleOp (fun n ω => ↑(M.ρ₁ (η_hat n k ω) M.η₀)) (fun x => 1) μ) → (∀ (k : Fin K), Causalean.Stat.IsLittleOp (fun n ω => ↑(M.ρ₂ (η_hat n k ω) M.η₀)) (fun x => 1) μ) → (∀ (k : Fin K), Causalean.Stat.IsLittleOp (fun n ω => ↑(M.ρ₁ (η_hat n k ω) M.η₀) * ↑(M.ρ₂ (η_hat n k ω) M.η₀)) (fun n => ↑n ^ (-(1 / 2))) μ) → Causalean.Stat.IsAsymLinear (Causalean.Estimation.OrthogonalMoments.dmlCrossFitEstimator M sample split η_hat) M.θ₀ (fun z => -M.J₀_inv * M.m M.η₀ z M.θ₀) sample fun n => Finset.range n","l":458,"f":"Causalean/Estimation/OrthogonalMoments/DMLCrossFit.lean"},{"n":"Causalean.Estimation.OrthogonalMoments.integratedMoment_diffQuotient_tendsto_zero","k":"theorem","m":"Causalean.Estimation.OrthogonalMoments.NeymanOrthogonal","a":"Estimation/OrthogonalMoments","t":1,"d":"Abstract DCT bridge. Given a general moment M with directional-derivative structure D, assume Neyman orthogonality — the population directional derivative vanishes at every admissible nuisance perturbation, and an L¹(P_Z) envelope dominating the difference quotient of the moment near t = 0. Suppose η lies in the admissible nuisance neighborhood M.H_ε, that the moment along the segment from η₀ to η is integrable at every nonzero t, and that the moment at η₀ is integrable. Then the integrated difference quotient (∫ m(η₀ + t · (η − η₀)) dP_Z − ∫ m(η₀) dP_Z) / t tends to zero as t → 0 along the punctured neighborhood.","s":"∀ {Ω : Type u_1} [inst : MeasurableSpace Ω] {μ : MeasureTheory.Measure Ω} {Z : Type u_2} [inst_1 : MeasurableSpace Z] {P_Z : MeasureTheory.Measure Z} {H : Type u_3} [inst_2 : AddCommGroup H] [inst_3 : Module ℝ H] (M : Causalean.Estimation.OrthogonalMoments.GeneralMoment Ω μ Z P_Z H) (D : Causalean.Estimation.OrthogonalMoments.HasDirDeriv M), Causalean.Estimation.OrthogonalMoments.NeymanOrthogonal M D → Causalean.Estimation.OrthogonalMoments.DiffQuotientEnvelope M → ∀ {η : H}, η ∈ M.H_ε → (∀ (t : ℝ), t ≠ 0 → MeasureTheory.Integrable (fun z => M.m (M.η₀ + t • (η - M.η₀)) z M.θ₀) P_Z) → MeasureTheory.Integrable (fun z => M.m M.η₀ z M.θ₀) P_Z → Filter.Tendsto (fun t => (∫ (z : Z), M.m (M.η₀ + t • (η - M.η₀)) z M.θ₀ ∂P_Z - ∫ (z : Z), M.m M.η₀ z M.θ₀ ∂P_Z) / t) (nhdsWithin 0 {0}ᶜ) (nhds 0)","l":53,"f":"Causalean/Estimation/OrthogonalMoments/NeymanOrthogonal.lean"},{"n":"Causalean.Estimation.OrthogonalMoments.oracle_expansion","k":"theorem","m":"Causalean.Estimation.OrthogonalMoments.SecondStageOperator","a":"Estimation/OrthogonalMoments","t":1,"d":"Oracle expansion for the DR-Learner (Thm thm:est-cate-dr-oracle, abstract operator-level form). Given a second-stage regression operator op that is stable at the query point x for target function target, with respect to a distance d_n between pseudo-outcomes and a caller-supplied conditional-bias identification predicate BiasIdent, suppose d_n converges to zero in probability under μ, i.e. the first-stage pseudo-outcome estimate is consistent, and suppose the estimated pseudo-outcome fHat_n, the true pseudo-outcome f, and the claimed conditional bias bHat_n satisfy BiasIdent. Then the discrepancy between the operator applied to fHat_n and to f, minus the operator applied to bHat_n, is o_p of the oracle risk scale under μ: the operator-level oracle expansion holds modulo o_p(R^*_n(x)).","s":"∀ {Ω : Type u_1} [inst : MeasurableSpace Ω] {μ : MeasureTheory.Measure Ω} {γ : Type u_2} [inst_1 : MeasurableSpace γ] (op : Causalean.Estimation.OrthogonalMoments.SecondStageOperator Ω μ γ) (target : γ → ℝ) (x : γ) (d_n : ℕ → Ω → ℝ) (fHat_n : ℕ → Ω → γ × Bool × ℝ → ℝ) (f : γ × Bool × ℝ → ℝ) (bHat_n : ℕ → Ω → γ → ℝ) (BiasIdent : (ℕ → Ω → γ × Bool × ℝ → ℝ) → (γ × Bool × ℝ → ℝ) → (ℕ → Ω → γ → ℝ) → Prop), Causalean.Estimation.OrthogonalMoments.Stable op target d_n x BiasIdent → Causalean.Stat.Tendsto_inProb d_n (fun x => 0) μ → BiasIdent fHat_n f bHat_n → Causalean.Stat.IsLittleOp (fun n ω => op.evalAt n ω (fHat_n n ω) x - op.evalAt n ω f x - op.evalAt n ω (fun z => bHat_n n ω z.1) x) (fun n => op.oracleRiskScale f target x n) μ","l":146,"f":"Causalean/Estimation/OrthogonalMoments/SecondStageOperator.lean"},{"n":"Causalean.Estimation.OrthogonalMoments.parametricMoment","k":"def","m":"Causalean.Estimation.OrthogonalMoments.Parametric","a":"Estimation/OrthogonalMoments","t":1,"d":"Parametric moment: the moment depends only on θ (no nuisance). m_par θ z is the user-supplied moment; J₀ : ℝ is its scalar Jacobian at θ₀.","s":"{Ω : Type u_1} → [inst : MeasurableSpace Ω] → {μ : MeasureTheory.Measure Ω} → {Z : Type u_2} → [inst_1 : MeasurableSpace Z] → {P_Z : MeasureTheory.Measure Z} → (m_par : ℝ → Z → ℝ) → ℝ → (J₀ : ℝ) → J₀ ≠ 0 → (∀ (θ : ℝ), Measurable (m_par θ)) → Causalean.Estimation.OrthogonalMoments.GeneralMoment Ω μ Z P_Z Unit","l":40,"f":"Causalean/Estimation/OrthogonalMoments/Parametric.lean"},{"n":"Causalean.Estimation.OrthogonalMoments.parametric_asymptoticLinear","k":"theorem","m":"Causalean.Estimation.OrthogonalMoments.Parametric","a":"Estimation/OrthogonalMoments","t":1,"d":"Asymptotic linearity of the parametric one-step estimator. Fix a score m_par with a nonzero scalar Jacobian J₀ at the true parameter θ₀, where m_par θ is measurable for every θ. If the score m_par θ₀ has population mean zero and it has finite second moment, then for an i.i.d. sample and a one-shot fold split of it, the parametric one-step estimator built from m_par and J₀ is asymptotically linear at θ₀ with influence function ψ(z) := −J₀⁻¹·m_par(θ₀, z).","s":"∀ {Ω : Type u_1} [inst : MeasurableSpace Ω] {μ : MeasureTheory.Measure Ω} {Z : Type u_2} [inst_1 : MeasurableSpace Z] {P_Z : MeasureTheory.Measure Z} (m_par : ℝ → Z → ℝ) (θ₀ J₀ : ℝ) (hJ : J₀ ≠ 0) (m_meas : ∀ (θ : ℝ), Measurable (m_par θ)), ∫ (z : Z), m_par θ₀ z ∂P_Z = 0 → MeasureTheory.Integrable (fun z => m_par θ₀ z ^ 2) P_Z → ∀ (sample : Causalean.Stat.IIDSample Ω Z μ P_Z) (split : Causalean.Stat.OneShotSplit sample), Causalean.Stat.IsAsymLinear (Causalean.Estimation.OrthogonalMoments.dmlChernozhukovEstimator (Causalean.Estimation.OrthogonalMoments.parametricMoment m_par θ₀ J₀ hJ m_meas) sample split fun x x_1 => ()) θ₀ (fun z => -J₀⁻¹ * m_par θ₀ z) sample split.foldB","l":58,"f":"Causalean/Estimation/OrthogonalMoments/Parametric.lean"},{"n":"Causalean.Estimation.OrthogonalMoments.rieszScore","k":"def","m":"Causalean.Estimation.OrthogonalMoments.Riesz","a":"Estimation/OrthogonalMoments","t":1,"d":"Generic Riesz orthogonal score. Given a Riesz representation, the orthogonal moment for the target θ(P) := L(γ_0) is","s":"{Z : Type u_2} → {X : Type u_3} → {H_γ : Type u_4} → [inst : AddCommGroup H_γ] → [Module ℝ H_γ] → (H_γ → X → ℝ) → (H_γ → ℝ) → (Z → X) → (Z → ℝ) → H_γ → (X → ℝ) → ℝ → Z → ℝ","l":66,"f":"Causalean/Estimation/OrthogonalMoments/Riesz.lean"},{"n":"Causalean.Estimation.OrthogonalMoments.rieszScore_bilinearRem","k":"theorem","m":"Causalean.Estimation.OrthogonalMoments.Riesz","a":"Estimation/OrthogonalMoments","t":2,"d":"Bilinear remainder identity for the Riesz score.","s":"∀ {Z : Type u_2} [inst : MeasurableSpace Z] {P_Z : MeasureTheory.Measure Z} {X : Type u_3} [inst_1 : MeasurableSpace X] {H_γ : Type u_4} [inst_2 : AddCommGroup H_γ] [inst_3 : Module ℝ H_γ] {γ_target : H_γ → X → ℝ} {L : H_γ → ℝ} {P_X : MeasureTheory.Measure X} [MeasureTheory.IsProbabilityMeasure P_Z] (rep : Causalean.Estimation.OrthogonalMoments.RieszRepresentation H_γ γ_target L P_X) (γ₀ γ : H_γ) (α : X → ℝ) (proj_X : Z → X) (Y_obs : Z → ℝ), P_X = MeasureTheory.Measure.map proj_X P_Z → Measurable proj_X → ∫ (z : Z), rep.α₀ (proj_X z) * (Y_obs z - γ_target γ₀ (proj_X z)) ∂P_Z = 0 → ∫ (z : Z), α (proj_X z) * (Y_obs z - γ_target γ₀ (proj_X z)) ∂P_Z = 0 → MeasureTheory.Integrable (fun z => α (proj_X z) * (Y_obs z - γ_target γ₀ (proj_X z))) P_Z → MeasureTheory.Integrable (fun z => α (proj_X z) * γ_target γ (proj_X z)) P_Z → MeasureTheory.Integrable (fun z => α (proj_X z) * γ_target γ₀ (proj_X z)) P_Z → MeasureTheory.Integrable (fun x => rep.α₀ x * γ_target γ x) P_X → MeasureTheory.Integrable (fun x => rep.α₀ x * γ_target γ₀ x) P_X → MeasureTheory.Integrable α P_X → MeasureTheory.Integrable (γ_target γ) P_X → MeasureTheory.Integrable (γ_target γ₀) P_X → ∫ (z : Z), Causalean.Estimation.OrthogonalMoments.rieszScore γ_target L proj_X Y_obs γ α (L γ₀) z ∂P_Z - ∫ (z : Z), Causalean.Estimation.OrthogonalMoments.rieszScore γ_target L proj_X Y_obs γ₀ rep.α₀ (L γ₀) z ∂P_Z = -∫ (x : X), (α x - rep.α₀ x) * (γ_target γ x - γ_target γ₀ x) ∂P_X","l":181,"f":"Causalean/Estimation/OrthogonalMoments/Riesz.lean"},{"n":"Causalean.Estimation.OrthogonalMoments.rieszScore_meanZero","k":"theorem","m":"Causalean.Estimation.OrthogonalMoments.Riesz","a":"Estimation/OrthogonalMoments","t":1,"d":"Mean-zero of the Riesz score at the truth. Given a Riesz representation rep of a linear functional L on the regression class H_γ under P_X, with true regression function γ₀, and observed data given by the covariate projection proj_X and the outcome Y_obs, suppose the representer residual α₀(proj_X z)·(Y_obs z − γ_target γ₀ (proj_X z)) has population mean zero under P_Z — which holds when γ_target γ₀ is the conditional expectation of Y_obs given proj_X, since residuals are orthogonal to all square-integrable functions of proj_X. Then the orthogonal score rieszScore evaluated at the truth (γ₀, α₀, L γ₀) integrates to zero under P_Z:","s":"∀ {Z : Type u_2} [inst : MeasurableSpace Z] {P_Z : MeasureTheory.Measure Z} {X : Type u_3} [inst_1 : MeasurableSpace X] {H_γ : Type u_4} [inst_2 : AddCommGroup H_γ] [inst_3 : Module ℝ H_γ] {γ_target : H_γ → X → ℝ} {L : H_γ → ℝ} {P_X : MeasureTheory.Measure X} [MeasureTheory.IsProbabilityMeasure P_Z] (rep : Causalean.Estimation.OrthogonalMoments.RieszRepresentation H_γ γ_target L P_X) (γ₀ : H_γ) (proj_X : Z → X) (Y_obs : Z → ℝ), ∫ (z : Z), rep.α₀ (proj_X z) * (Y_obs z - γ_target γ₀ (proj_X z)) ∂P_Z = 0 → ∫ (z : Z), Causalean.Estimation.OrthogonalMoments.rieszScore γ_target L proj_X Y_obs γ₀ rep.α₀ (L γ₀) z ∂P_Z = 0","l":82,"f":"Causalean/Estimation/OrthogonalMoments/Riesz.lean"},{"n":"Causalean.Estimation.OrthogonalMoments.smoother_bias_holder","k":"theorem","m":"Causalean.Estimation.OrthogonalMoments.LinearSmoother","a":"Estimation/OrthogonalMoments","t":1,"d":"Single-function Hölder bound for a linear smoother. Assume op realises a linear smoother at sample size n, randomness ω, and query point x — its evaluation of any function on the fold B is the weighted sum Σ w_i · f(xs_i), and that the absolute weights sum to at most c_n. Then the absolute smoothed bias of g (evaluated on the γ-component of the data) at x is bounded by c_n times the weighted L¹ norm of g ∘ xs.1.","s":"∀ {Ω : Type u_1} [inst : MeasurableSpace Ω] {μ : MeasureTheory.Measure Ω} {γ : Type u_2} [inst_1 : MeasurableSpace γ] {ι : Type u_3} (op : Causalean.Estimation.OrthogonalMoments.LinearSmootherOp Ω μ γ) (n : ℕ) (ω : Ω) (x : γ) (g : γ → ℝ) (B : Finset ι) (w : ι → ℝ) (xs : ι → γ × Bool × ℝ) (c_n : ℝ), op.IsLinearSmoother n ω x B w xs → ∑ i ∈ B, |w i| ≤ c_n → |op.evalAt n ω (fun z => g z.1) x| ≤ c_n * Causalean.Estimation.OrthogonalMoments.WeightedNorm B w (fun i => g (xs i).1) 1","l":130,"f":"Causalean/Estimation/OrthogonalMoments/LinearSmoother.lean"},{"n":"Causalean.Estimation.OrthogonalMoments.smoother_bias_product_holder","k":"theorem","m":"Causalean.Estimation.OrthogonalMoments.LinearSmoother","a":"Estimation/OrthogonalMoments","t":1,"d":"Product Hölder bound for a linear smoother (Prop prop:est-cate-linear-smoother-bound). Assume op realises a linear smoother at sample size n, randomness ω, and query point x, that the absolute weights sum to at most c_n, and that p and q are Hölder-conjugate exponents, 1/p + 1/q = 1. Then the absolute smoothed bias of the product g₁ · g₂ (evaluated on the γ-component of the data) at x is bounded by c_n times the weighted L^p norm of g₁ ∘ xs.1 times the weighted L^q norm of g₂ ∘ xs.1.","s":"∀ {Ω : Type u_1} [inst : MeasurableSpace Ω] {μ : MeasureTheory.Measure Ω} {γ : Type u_2} [inst_1 : MeasurableSpace γ] {ι : Type u_3} (op : Causalean.Estimation.OrthogonalMoments.LinearSmootherOp Ω μ γ) (n : ℕ) (ω : Ω) (x : γ) (g₁ g₂ : γ → ℝ) (B : Finset ι) (w : ι → ℝ) (xs : ι → γ × Bool × ℝ) (c_n p q : ℝ), op.IsLinearSmoother n ω x B w xs → ∑ i ∈ B, |w i| ≤ c_n → p.HolderConjugate q → |op.evalAt n ω (fun z => g₁ z.1 * g₂ z.1) x| ≤ c_n * Causalean.Estimation.OrthogonalMoments.WeightedNorm B w (fun i => g₁ (xs i).1) p * Causalean.Estimation.OrthogonalMoments.WeightedNorm B w (fun i => g₂ (xs i).1) q","l":209,"f":"Causalean/Estimation/OrthogonalMoments/LinearSmoother.lean"},{"n":"Causalean.Estimation.PLR.PLRNuisance","k":"structure","m":"Causalean.Estimation.PLR.Nuisance","a":"Estimation/PLR","t":1,"d":"A partially linear nuisance bundles an outcome regression and a treatment regression on the covariates, requiring each to be measurable.","s":"(γ : Type u_1) → [MeasurableSpace γ] → Type u_1","l":35,"f":"Causalean/Estimation/PLR/Nuisance.lean"},{"n":"Causalean.Estimation.PLR.PLRNuisance.ext","k":"theorem","m":"Causalean.Estimation.PLR.Nuisance","a":"Estimation/PLR","t":1,"d":"Two nuisance pairs η and η' are equal whenever their outcome-regression components agree at every covariate value and their treatment-regression components agree at every covariate value, so the two pairs are equal.","s":"∀ {γ : Type u_1} [inst : MeasurableSpace γ] {η η' : Causalean.Estimation.PLR.PLRNuisance γ}, (∀ (x : γ), η.lFn x = η'.lFn x) → (∀ (x : γ), η.mFn x = η'.mFn x) → η = η'","l":89,"f":"Causalean/Estimation/PLR/Nuisance.lean"},{"n":"Causalean.Estimation.PLR.PLRNuisance.ext_iff","k":"theorem","m":"Causalean.Estimation.PLR.Nuisance","a":"Estimation/PLR","t":2,"d":"","s":"∀ {γ : Type u_1} [inst : MeasurableSpace γ] {η η' : Causalean.Estimation.PLR.PLRNuisance γ}, η = η' ↔ (∀ (x : γ), η.lFn x = η'.lFn x) ∧ ∀ (x : γ), η.mFn x = η'.mFn x","l":93,"f":"Causalean/Estimation/PLR/Nuisance.lean"},{"n":"Causalean.Estimation.PLR.PLRNuisance.instAdd","k":"instance","m":"Causalean.Estimation.PLR.Nuisance","a":"Estimation/PLR","t":2,"d":"Addition of two nuisance pairs, performed separately on the outcome regression and on the treatment regression.","s":"{γ : Type u_1} → [inst : MeasurableSpace γ] → Add (Causalean.Estimation.PLR.PLRNuisance γ)","l":56,"f":"Causalean/Estimation/PLR/Nuisance.lean"},{"n":"Causalean.Estimation.PLR.PLRNuisance.instAddCommGroup","k":"instance","m":"Causalean.Estimation.PLR.Nuisance","a":"Estimation/PLR","t":2,"d":"The nuisance pairs form an additive commutative group under componentwise operations, inheriting all the group laws from the reals applied at each covariate value.","s":"{γ : Type u_1} → [inst : MeasurableSpace γ] → AddCommGroup (Causalean.Estimation.PLR.PLRNuisance γ)","l":106,"f":"Causalean/Estimation/PLR/Nuisance.lean"},{"n":"Causalean.Estimation.PLR.PLRNuisance.instModuleReal","k":"instance","m":"Causalean.Estimation.PLR.Nuisance","a":"Estimation/PLR","t":2,"d":"The nuisance pairs form a real vector space, with scalar multiplication acting pointwise on both the outcome regression and the treatment regression.","s":"{γ : Type u_1} → [inst : MeasurableSpace γ] → Module ℝ (Causalean.Estimation.PLR.PLRNuisance γ)","l":151,"f":"Causalean/Estimation/PLR/Nuisance.lean"},{"n":"Causalean.Estimation.PLR.PLRNuisance.instNeg","k":"instance","m":"Causalean.Estimation.PLR.Nuisance","a":"Estimation/PLR","t":2,"d":"Negation of a nuisance pair, negating both the outcome regression and the treatment regression.","s":"{γ : Type u_1} → [inst : MeasurableSpace γ] → Neg (Causalean.Estimation.PLR.PLRNuisance γ)","l":65,"f":"Causalean/Estimation/PLR/Nuisance.lean"},{"n":"Causalean.Estimation.PLR.PLRNuisance.instSMulReal","k":"instance","m":"Causalean.Estimation.PLR.Nuisance","a":"Estimation/PLR","t":2,"d":"Scaling a nuisance pair by a real number, multiplying both the outcome regression and the treatment regression pointwise by that scalar.","s":"{γ : Type u_1} → [inst : MeasurableSpace γ] → SMul ℝ (Causalean.Estimation.PLR.PLRNuisance γ)","l":81,"f":"Causalean/Estimation/PLR/Nuisance.lean"},{"n":"Causalean.Estimation.PLR.PLRNuisance.instSub","k":"instance","m":"Causalean.Estimation.PLR.Nuisance","a":"Estimation/PLR","t":2,"d":"Subtraction of two nuisance pairs, performed separately on the outcome regression and on the treatment regression.","s":"{γ : Type u_1} → [inst : MeasurableSpace γ] → Sub (Causalean.Estimation.PLR.PLRNuisance γ)","l":72,"f":"Causalean/Estimation/PLR/Nuisance.lean"},{"n":"Causalean.Estimation.PLR.PLRNuisance.instZero","k":"instance","m":"Causalean.Estimation.PLR.Nuisance","a":"Estimation/PLR","t":2,"d":"The zero nuisance pair: both the outcome regression and the treatment regression are the constant function zero.","s":"{γ : Type u_1} → [inst : MeasurableSpace γ] → Zero (Causalean.Estimation.PLR.PLRNuisance γ)","l":51,"f":"Causalean/Estimation/PLR/Nuisance.lean"},{"n":"Causalean.Estimation.PLR.PLRSystem","k":"structure","m":"Causalean.Estimation.PLR.Setup","a":"Estimation/PLR","t":1,"d":"A partially linear estimation system extends a partially linear causal model with value-space regression representatives — an outcome-regression representative and a treatment-regression representative on the covariates — subject to: each representative is measurable; the outcome representative agrees almost surely with the conditional mean of the outcome given the covariates and the treatment representative agrees almost surely with the conditional mean of the treatment given the covariates; and the treatment retains nonzero variation after partialling out the covariate, which is what makes the partialling-out Jacobian invertible.","s":"(P : Causalean.PO.POSystem) → (γ : Type u_1) → [MeasurableSpace γ] → [MeasureTheory.IsFiniteMeasure P.μ] → Type (max (max (max u_1 u_2) u_3) u_4)","l":53,"f":"Causalean/Estimation/PLR/Setup.lean"},{"n":"Causalean.Estimation.PLR.PLRSystem.P_X","k":"def","m":"Causalean.Estimation.PLR.Setup","a":"Estimation/PLR","t":1,"d":"The covariate marginal is the distribution of the observed covariate induced by the population measure.","s":"{P : Causalean.PO.POSystem} → {γ : Type u_1} → [inst : MeasurableSpace γ] → [inst_1 : MeasureTheory.IsFiniteMeasure P.μ] → Causalean.Estimation.PLR.PLRSystem P γ → MeasureTheory.Measure γ","l":111,"f":"Causalean/Estimation/PLR/Setup.lean"},{"n":"Causalean.Estimation.PLR.PLRSystem.P_Z","k":"def","m":"Causalean.Estimation.PLR.Setup","a":"Estimation/PLR","t":1,"d":"The joint observed-data law is the distribution of covariate, treatment, and outcome induced by the population measure.","s":"{P : Causalean.PO.POSystem} → {γ : Type u_1} → [inst : MeasurableSpace γ] → [inst_1 : MeasureTheory.IsFiniteMeasure P.μ] → Causalean.Estimation.PLR.PLRSystem P γ → MeasureTheory.Measure (γ × ℝ × ℝ)","l":107,"f":"Causalean/Estimation/PLR/Setup.lean"},{"n":"Causalean.Estimation.PLR.PLRSystem.factualZ","k":"def","m":"Causalean.Estimation.PLR.Setup","a":"Estimation/PLR","t":1,"d":"The observed-data map returns the covariate, treatment, and outcome for each unit in the population space.","s":"{P : Causalean.PO.POSystem} → {γ : Type u_1} → [inst : MeasurableSpace γ] → [inst_1 : MeasureTheory.IsFiniteMeasure P.μ] → Causalean.Estimation.PLR.PLRSystem P γ → P.Ω → γ × ℝ × ℝ","l":98,"f":"Causalean/Estimation/PLR/Setup.lean"},{"n":"Causalean.Estimation.PLR.PLRSystem.instIsProbabilityMeasureP_Z","k":"instance","m":"Causalean.Estimation.PLR.JacobianConsistency","a":"Estimation/PLR","t":2,"d":"The joint observed-data law P_Z is a probability measure: it is the pushforward of the probability measure P.μ along the measurable observation map (X, D, Y).","s":"∀ {P : Causalean.PO.POSystem} {γ : Type u_1} [inst : MeasurableSpace γ] [inst_1 : MeasureTheory.IsFiniteMeasure P.μ] [MeasureTheory.IsProbabilityMeasure P.μ] (S : Causalean.Estimation.PLR.PLRSystem P γ), MeasureTheory.IsProbabilityMeasure S.P_Z","l":87,"f":"Causalean/Estimation/PLR/JacobianConsistency.lean"},{"n":"Causalean.Estimation.PLR.PLRSystem.integral_P_Z","k":"theorem","m":"Causalean.Estimation.PLR.Setup","a":"Estimation/PLR","t":2,"d":"Integrating a measurable function under the joint observed-data law equals integrating its pullback under the population measure.","s":"∀ {P : Causalean.PO.POSystem} {γ : Type u_1} [inst : MeasurableSpace γ] [inst_1 : MeasureTheory.IsFiniteMeasure P.μ] (S : Causalean.Estimation.PLR.PLRSystem P γ) {f : γ × ℝ × ℝ → ℝ}, Measurable f → ∫ (z : γ × ℝ × ℝ), f z ∂S.P_Z = ∫ (ω : P.Ω), f (S.factualZ ω) ∂P.μ","l":155,"f":"Causalean/Estimation/PLR/Setup.lean"},{"n":"Causalean.Estimation.PLR.PLRSystem.integral_condExpZero_mul_comp_factualX","k":"theorem","m":"Causalean.Estimation.PLR.RemainderBound","a":"Estimation/PLR","t":2,"d":"Orthogonality of a σ(X)-conditionally-mean-zero variable w to any covariate function h(X): E[w·h(X)] = 0. Mirrors integral_U_resid, but the σ-algebra is σ(X) (so h(X) pulls out of the conditional expectation) instead of σ(X,D). Used to kill the three orthogonal cross terms U·Δm, Δℓ·V, V·Δm in the doubly-robust remainder.","s":"∀ {P : Causalean.PO.POSystem} {γ : Type u_1} [inst : MeasurableSpace γ] [inst_1 : MeasureTheory.IsFiniteMeasure P.μ] (S : Causalean.Estimation.PLR.PLRSystem P γ) {w : P.Ω → ℝ} {h : γ → ℝ}, Measurable h → P.μ[w | S.sigmaX] =ᵐ[P.μ] 0 → MeasureTheory.Integrable w P.μ → MeasureTheory.Integrable (fun ω => w ω * h (S.factualX ω)) P.μ → ∫ (ω : P.Ω), w ω * h (S.factualX ω) ∂P.μ = 0","l":47,"f":"Causalean/Estimation/PLR/RemainderBound.lean"},{"n":"Causalean.Estimation.PLR.PLRSystem.integral_plrMomentA_diff_eq","k":"theorem","m":"Causalean.Estimation.PLR.JacobianConsistency","a":"Estimation/PLR","t":2,"d":"Bias of the empirical-Jacobian increment. The population integral of the increment Δa := mₐ(η̂, ·) − mₐ(η₀, ·) is minus the squared L²(P_X) magnitude of the treatment-regression error Δm := m_val − m̂:","s":"∀ {P : Causalean.PO.POSystem} {γ : Type u_1} [inst : MeasurableSpace γ] [StandardBorelSpace P.Ω] [inst_2 : MeasureTheory.IsFiniteMeasure P.μ] [MeasureTheory.IsProbabilityMeasure P.μ] (S : Causalean.Estimation.PLR.PLRSystem P γ) (η : Causalean.Estimation.PLR.PLRNuisance γ), MeasureTheory.Integrable S.factualD P.μ → MeasureTheory.Integrable (fun ω => (S.factualD ω - S.mVal (S.factualX ω)) ^ 2) P.μ → MeasureTheory.Integrable (fun ω => (S.mVal (S.factualX ω) - η.mFn (S.factualX ω)) ^ 2) P.μ → MeasureTheory.Integrable (fun ω => (S.factualD ω - S.mVal (S.factualX ω)) * (S.mVal (S.factualX ω) - η.mFn (S.factualX ω))) P.μ → ∫ (z : γ × ℝ × ℝ), Causalean.Estimation.PLR.plrMomentA η z - Causalean.Estimation.PLR.plrMomentA S.η₀ z ∂S.P_Z = -∫ (ω : P.Ω), (S.mVal (S.factualX ω) - η.mFn (S.factualX ω)) ^ 2 ∂P.μ","l":93,"f":"Causalean/Estimation/PLR/JacobianConsistency.lean"},{"n":"Causalean.Estimation.PLR.PLRSystem.integral_plrMomentA_η₀_eq_J₀","k":"theorem","m":"Causalean.Estimation.PLR.JacobianConsistency","a":"Estimation/PLR","t":2,"d":"The population integral of the partialling-out moment at the truth equals the Jacobian: ∫ mₐ(η₀, ·) dP_Z = J₀ = −E[(D − m_val(X))²].","s":"∀ {P : Causalean.PO.POSystem} {γ : Type u_1} [inst : MeasurableSpace γ] [StandardBorelSpace P.Ω] [inst_2 : MeasureTheory.IsFiniteMeasure P.μ] [MeasureTheory.IsProbabilityMeasure P.μ] (S : Causalean.Estimation.PLR.PLRSystem P γ), ∫ (z : γ × ℝ × ℝ), Causalean.Estimation.PLR.plrMomentA S.η₀ z ∂S.P_Z = S.plrGeneralMoment.J₀","l":72,"f":"Causalean/Estimation/PLR/JacobianConsistency.lean"},{"n":"Causalean.Estimation.PLR.PLRSystem.measurable_factualZ","k":"theorem","m":"Causalean.Estimation.PLR.Setup","a":"Estimation/PLR","t":2,"d":"The observed-data map is measurable.","s":"∀ {P : Causalean.PO.POSystem} {γ : Type u_1} [inst : MeasurableSpace γ] [inst_1 : MeasureTheory.IsFiniteMeasure P.μ] (S : Causalean.Estimation.PLR.PLRSystem P γ), Measurable S.factualZ","l":103,"f":"Causalean/Estimation/PLR/Setup.lean"},{"n":"Causalean.Estimation.PLR.PLRSystem.plrFeasibleEstimator","k":"def","m":"Causalean.Estimation.PLR.Feasible","a":"Estimation/PLR","t":1,"d":"Feasible (solved) partially linear DML estimator. Solving the empirical Robinson partialling-out moment equation Pₙ ψ(η̂, ·, θ) = 0 for θ — which, because the score is affine in θ, is the explicit ratio of the empirical covariance of the residualized outcome and residualized treatment to the empirical second moment of the residualized treatment:","s":"{P : Causalean.PO.POSystem} → {γ : Type u_1} → [inst : MeasurableSpace γ] → [inst_1 : MeasureTheory.IsFiniteMeasure P.μ] → (S : Causalean.Estimation.PLR.PLRSystem P γ) → (sample : Causalean.Stat.IIDSample P.Ω (γ × ℝ × ℝ) P.μ S.P_Z) → Causalean.Stat.OneShotSplit sample → (ℕ → P.Ω → Causalean.Estimation.PLR.PLRNuisance γ) → ℕ → P.Ω → ℝ","l":69,"f":"Causalean/Estimation/PLR/Feasible.lean"},{"n":"Causalean.Estimation.PLR.PLRSystem.plrGeneralMoment","k":"def","m":"Causalean.Estimation.PLR.Setup","a":"Estimation/PLR","t":1,"d":"The partially linear moment instance plugs the Robinson partialling-out score into the abstract double-machine-learning framework.","s":"{P : Causalean.PO.POSystem} → {γ : Type u_1} → [inst : MeasurableSpace γ] → [inst_1 : MeasureTheory.IsFiniteMeasure P.μ] → (S : Causalean.Estimation.PLR.PLRSystem P γ) → Causalean.Estimation.OrthogonalMoments.GeneralMoment P.Ω P.μ (γ × ℝ × ℝ) S.P_Z (Causalean.Estimation.PLR.PLRNuisance γ)","l":130,"f":"Causalean/Estimation/PLR/Setup.lean"},{"n":"Causalean.Estimation.PLR.PLRSystem.plrMomentFunctional_factualZ","k":"theorem","m":"Causalean.Estimation.PLR.Setup","a":"Estimation/PLR","t":1,"d":"Robinson score at the true nuisance, pulled back to the population space. For the observed data generated from a population outcome ω, the Robinson partialling-out score evaluated at the true nuisance pair, true data, and true parameter equals the true residualized outcome times the true treatment residual.","s":"∀ {P : Causalean.PO.POSystem} {γ : Type u_1} [inst : MeasurableSpace γ] [inst_1 : MeasureTheory.IsFiniteMeasure P.μ] (S : Causalean.Estimation.PLR.PLRSystem P γ) (ω : P.Ω), Causalean.Estimation.PLR.plrMomentFunctional S.η₀ (S.factualZ ω) S.θ₀ = (S.factualY ω - S.lVal (S.factualX ω) - S.θ * (S.factualD ω - S.mVal (S.factualX ω))) * (S.factualD ω - S.mVal (S.factualX ω))","l":165,"f":"Causalean/Estimation/PLR/Setup.lean"},{"n":"Causalean.Estimation.PLR.PLRSystem.plr_dml_feasible_tendstoNormal","k":"theorem","m":"Causalean.Estimation.PLR.Feasible","a":"Estimation/PLR","t":1,"d":"Feasible partially linear DML asymptotic-normality theorem. Fix a partially linear estimation system, an i.i.d. sample of covariate-treatment-outcome triples, and a sample split whose evaluation-fold share converges to a fixed positive limit. Suppose the structural error, its product with the treatment residual, the baseline-covariate term, and the treatment are integrable, the true treatment residual is square-integrable, and the true score has finite second moment; for the estimated nuisance sequence η_hat, the outcome- and treatment-regression errors are square-integrable in the covariate law at every fold and draw, with the resulting cross terms against the structural error and the treatment residual integrable; the estimated score is jointly measurable and measurable as a function of the nuisance-training fold alone and jointly with the observation, and integrable and square-integrable at every fold and draw; the estimated score converges to the true score in L²(P_Z) at rate $o_p(1)$, and the product of the two nuisance-error seminorms is $o_p(n^{-1/2})$; and the influence function, the one-step rescaled estimator at each n, and the normalized influence sum at each n are all measurable. Suppose in addition the empirical partialling-out Jacobian over fold B converges in probability to its population value J₀, and the rescaled feasible estimator is measurable at each n. Then the rescaled *feasible* estimator — the solved Robinson partialling-out estimator, recentered at the true slope and scaled by the square root of the fold-B sample size — converges in distribution to the same centered Gaussian as the one-step estimator: a normal law whose variance is the population second moment of the inverse-Jacobian-scaled partialling-out score at the true regressions.","s":"∀ {P : Causalean.PO.POSystem} {γ : Type u_1} [inst : MeasurableSpace γ] [StandardBorelSpace P.Ω] [inst_2 : MeasureTheory.IsFiniteMeasure P.μ] [inst_3 : MeasureTheory.IsProbabilityMeasure P.μ] (S : Causalean.Estimation.PLR.PLRSystem P γ) (sample : Causalean.Stat.IIDSample P.Ω (γ × ℝ × ℝ) P.μ S.P_Z) (split : Causalean.Stat.OneShotSplit sample) {c : ℝ}, 0 < c → Filter.Tendsto (fun n => ↑(split.foldB n).card / ↑n) Filter.atTop (nhds c) → ∀ (η_hat : ℕ → P.Ω → Causalean.Estimation.PLR.PLRNuisance γ), MeasureTheory.Integrable S.U P.μ → MeasureTheory.Integrable (fun ω => S.U ω * S.resid ω) P.μ → MeasureTheory.Integrable (fun ω => S.b (S.factualX ω)) P.μ → MeasureTheory.Integrable S.factualD P.μ → MeasureTheory.MemLp S.resid 2 P.μ → MeasureTheory.Integrable (fun ω => Causalean.Estimation.PLR.plrMomentFunctional S.η₀ (S.factualZ ω) S.θ₀ ^ 2) P.μ → (∀ (n : ℕ) (ω : P.Ω), MeasureTheory.MemLp (fun x => (η_hat n ω).lFn x - S.lVal x) 2 S.P_X) → (∀ (n : ℕ) (ω : P.Ω), MeasureTheory.MemLp (fun x => (η_hat n ω).mFn x - S.mVal x) 2 S.P_X) → (∀ (n : ℕ) (ω : P.Ω), MeasureTheory.Integrable (fun ω' => S.U ω' * ((η_hat n ω).mFn (S.factualX ω') - S.mVal (S.factualX ω'))) P.μ) → (∀ (n : ℕ) (ω : P.Ω), MeasureTheory.Integrable (fun ω' => ((η_hat n ω).lFn (S.factualX ω') - S.lVal (S.factualX ω')) * S.resid ω') P.μ) → (∀ (n : ℕ) (ω : P.Ω), MeasureTheory.Integrable (fun ω' => S.resid ω' * ((η_hat n ω).mFn (S.factualX ω') - S.mVal (S.factualX ω'))) P.μ) → (∀ (n : ℕ), Measurable fun p => S.plrGeneralMoment.m (η_hat n p.1) p.2 S.plrGeneralMoment.θ₀) → (∀ (n : ℕ), Measurable fun ω z => S.plrGeneralMoment.m (η_hat n ω) z S.plrGeneralMoment.θ₀) → (∀ (n : ℕ), Measurable fun p => S.plrGeneralMoment.m (η_hat n p.1) p.2 S.plrGeneralMoment.θ₀) → (∀ (n : ℕ) (ω : P.Ω), MeasureTheory.Integrable (fun z => S.plrGeneralMoment.m (η_hat n ω) z S.plrGeneralMoment.θ₀) S.P_Z) → (∀ (n : ℕ) (ω : P.Ω), MeasureTheory.Integrable (fun z => S.plrGeneralMoment.m (η_hat n ω) z S.plrGeneralMoment.θ₀ ^ 2) S.P_Z) → Causalean.Stat.IsLittleOp (fun n ω => (MeasureTheory.eLpNorm (fun z => S.plrGeneralMoment.m (η_hat n ω) z S.plrGeneralMoment.θ₀ - S.plrGeneralMoment.m S.plrGeneralMoment.η₀ z S.plrGeneralMoment.θ₀) 2 S.P_Z).toReal) (fun x => 1) P.μ → Causalean.Stat.IsLittleOp (fun n ω => ↑(S.plrGeneralMoment.ρ₁ (η_hat n ω) S.plrGeneralMoment.η₀) * ↑(S.plrGeneralMoment.ρ₂ (η_hat n ω) S.plrGeneralMoment.η₀)) (fun n => ↑n ^ (-(1 / 2))) P.μ → (Measurable fun z => -S.plrGeneralMoment.J₀_inv * Causalean.Estimation.PLR.plrMomentFunctional S.η₀ z S.θ₀) → (∀ (n : ℕ), AEMeasurable (Causalean.Stat.IsAsymLinear.rescaledEstimator (Causalean.Estimation.OrthogonalMoments.dmlChernozhukovEstimator S.plrGeneralMoment sample split η_hat) S.θ₀ split.foldB n) P.μ) → (∀ (n : ℕ), AEMeasurable (Causalean.Stat.IsAsymLinear.normalizedSum sample (fun z => -S.plrGeneralMoment.J₀_inv * Causalean.Estimation.PLR.plrMomentFunctional S.η₀ z S.θ₀) split.foldB n) P.μ) → Causalean.Stat.Tendsto_inProb (fun n ω => (↑(split.foldB n).card)⁻¹ * ∑ i ∈ split.foldB n, Causalean.Estimation.PLR.plrMomentA (η_hat n ω) (sample.Z i ω)) (fun x => S.plrGeneralMoment.J₀) P.μ → ∀ (hθn_feas : ∀ (n : ℕ), AEMeasurable (Causalean.Stat.IsAsymLinear.rescaledEstimator (S.plrFeasibleEstimator sample split η_hat) S.θ₀ split.foldB n) P.μ), Causalean.Stat.Tendsto_dist (Causalean.Stat.IsAsymLinear.rescaledEstimator (S.plrFeasibleEstimator sample split η_hat) S.θ₀ split.foldB) (Causalean.Stat.gaussianMeasure 0 (∫ (z : γ × ℝ × ℝ), (-S.plrGeneralMoment.J₀_inv * Causalean.Estimation.PLR.plrMomentFunctional S.η₀ z S.θ₀) ^ 2 ∂S.P_Z)) P.μ hθn_feas","l":89,"f":"Causalean/Estimation/PLR/Feasible.lean"},{"n":"Causalean.Estimation.PLR.PLRSystem.plr_dml_isAsymLinear","k":"theorem","m":"Causalean.Estimation.PLR.DML","a":"Estimation/PLR","t":1,"d":"Headline partially linear DML asymptotic-linearity theorem. Fix a partially linear estimation system, an i.i.d. sample of covariate-treatment-outcome triples, and a sample split whose evaluation-fold share converges to a fixed positive limit. If the model's integrability and square-integrability conditions hold — the structural error, its product with the treatment residual, the baseline-covariate term, and the treatment are integrable, the true treatment residual is square-integrable, and the true score has finite second moment; for the estimated nuisance sequence η_hat, at every fold and draw the outcome- and treatment-regression errors are square-integrable in the covariate law, with the resulting cross terms against the structural error and the treatment residual integrable; the estimated score is jointly measurable, fold-A measurable, and integrable/square-integrable at every fold; the L²(P_Z) distance between the estimated and true score is $o_p(1)$; and the product of the two nuisance-error seminorms is $o_p(n^{-1/2})$; then the one-step double-machine-learning estimator of the structural slope is asymptotically linear at the true slope, with influence function $-J_0^{-1}\\psi(\\eta_0,\\cdot,\\theta_0)$ — the inverse-Jacobian-scaled Robinson partialling-out score at the truth.","s":"∀ {P : Causalean.PO.POSystem} {γ : Type u_1} [inst : MeasurableSpace γ] [StandardBorelSpace P.Ω] [inst_2 : MeasureTheory.IsFiniteMeasure P.μ] [MeasureTheory.IsProbabilityMeasure P.μ] (S : Causalean.Estimation.PLR.PLRSystem P γ) (sample : Causalean.Stat.IIDSample P.Ω (γ × ℝ × ℝ) P.μ S.P_Z) (split : Causalean.Stat.OneShotSplit sample) {c : ℝ}, 0 < c → Filter.Tendsto (fun n => ↑(split.foldB n).card / ↑n) Filter.atTop (nhds c) → ∀ (η_hat : ℕ → P.Ω → Causalean.Estimation.PLR.PLRNuisance γ), MeasureTheory.Integrable S.U P.μ → MeasureTheory.Integrable (fun ω => S.U ω * S.resid ω) P.μ → MeasureTheory.Integrable (fun ω => S.b (S.factualX ω)) P.μ → MeasureTheory.Integrable S.factualD P.μ → MeasureTheory.MemLp S.resid 2 P.μ → MeasureTheory.Integrable (fun ω => Causalean.Estimation.PLR.plrMomentFunctional S.η₀ (S.factualZ ω) S.θ₀ ^ 2) P.μ → (∀ (n : ℕ) (ω : P.Ω), MeasureTheory.MemLp (fun x => (η_hat n ω).lFn x - S.lVal x) 2 S.P_X) → (∀ (n : ℕ) (ω : P.Ω), MeasureTheory.MemLp (fun x => (η_hat n ω).mFn x - S.mVal x) 2 S.P_X) → (∀ (n : ℕ) (ω : P.Ω), MeasureTheory.Integrable (fun ω' => S.U ω' * ((η_hat n ω).mFn (S.factualX ω') - S.mVal (S.factualX ω'))) P.μ) → (∀ (n : ℕ) (ω : P.Ω), MeasureTheory.Integrable (fun ω' => ((η_hat n ω).lFn (S.factualX ω') - S.lVal (S.factualX ω')) * S.resid ω') P.μ) → (∀ (n : ℕ) (ω : P.Ω), MeasureTheory.Integrable (fun ω' => S.resid ω' * ((η_hat n ω).mFn (S.factualX ω') - S.mVal (S.factualX ω'))) P.μ) → (∀ (n : ℕ), Measurable fun p => S.plrGeneralMoment.m (η_hat n p.1) p.2 S.plrGeneralMoment.θ₀) → (∀ (n : ℕ), Measurable fun ω z => S.plrGeneralMoment.m (η_hat n ω) z S.plrGeneralMoment.θ₀) → (∀ (n : ℕ), Measurable fun p => S.plrGeneralMoment.m (η_hat n p.1) p.2 S.plrGeneralMoment.θ₀) → (∀ (n : ℕ) (ω : P.Ω), MeasureTheory.Integrable (fun z => S.plrGeneralMoment.m (η_hat n ω) z S.plrGeneralMoment.θ₀) S.P_Z) → (∀ (n : ℕ) (ω : P.Ω), MeasureTheory.Integrable (fun z => S.plrGeneralMoment.m (η_hat n ω) z S.plrGeneralMoment.θ₀ ^ 2) S.P_Z) → Causalean.Stat.IsLittleOp (fun n ω => (MeasureTheory.eLpNorm (fun z => S.plrGeneralMoment.m (η_hat n ω) z S.plrGeneralMoment.θ₀ - S.plrGeneralMoment.m S.plrGeneralMoment.η₀ z S.plrGeneralMoment.θ₀) 2 S.P_Z).toReal) (fun x => 1) P.μ → Causalean.Stat.IsLittleOp (fun n ω => ↑(S.plrGeneralMoment.ρ₁ (η_hat n ω) S.plrGeneralMoment.η₀) * ↑(S.plrGeneralMoment.ρ₂ (η_hat n ω) S.plrGeneralMoment.η₀)) (fun n => ↑n ^ (-(1 / 2))) P.μ → Causalean.Stat.IsAsymLinear (Causalean.Estimation.OrthogonalMoments.dmlChernozhukovEstimator S.plrGeneralMoment sample split η_hat) S.θ₀ (fun z => -S.plrGeneralMoment.J₀_inv * Causalean.Estimation.PLR.plrMomentFunctional S.η₀ z S.θ₀) sample split.foldB","l":53,"f":"Causalean/Estimation/PLR/DML.lean"},{"n":"Causalean.Estimation.PLR.PLRSystem.plr_dml_tendstoNormal","k":"theorem","m":"Causalean.Estimation.PLR.DML","a":"Estimation/PLR","t":1,"d":"Headline partially linear DML asymptotic-normality theorem. Fix a partially linear estimation system, an i.i.d. sample of covariate-treatment-outcome triples, and a sample split whose evaluation-fold share converges to a fixed positive limit. Suppose the structural error, its product with the treatment residual, the baseline-covariate term, and the treatment are integrable, the true treatment residual is square-integrable, and the true score has finite second moment; for the estimated nuisance sequence η_hat, the outcome- and treatment-regression errors are square-integrable in the covariate law at every fold and draw, with the resulting cross terms against the structural error and the treatment residual integrable; the estimated score is jointly measurable and measurable as a function of the nuisance-training fold alone and jointly with the observation, and integrable and square-integrable at every fold and draw; the estimated score converges to the true score in L²(P_Z) at rate $o_p(1)$, and the product of the two nuisance-error seminorms is $o_p(n^{-1/2})$; and the influence function, the rescaled estimator at each n, and the normalized influence sum at each n are all measurable. Then the rescaled double-machine-learning estimator of the structural slope, recentered at the true slope and scaled by the square root of the fold-B sample size, converges in distribution to a centered Gaussian whose variance is the second moment of the inverse-Jacobian-scaled partialling-out score at the true regressions.","s":"∀ {P : Causalean.PO.POSystem} {γ : Type u_1} [inst : MeasurableSpace γ] [StandardBorelSpace P.Ω] [inst_2 : MeasureTheory.IsFiniteMeasure P.μ] [inst_3 : MeasureTheory.IsProbabilityMeasure P.μ] (S : Causalean.Estimation.PLR.PLRSystem P γ) (sample : Causalean.Stat.IIDSample P.Ω (γ × ℝ × ℝ) P.μ S.P_Z) (split : Causalean.Stat.OneShotSplit sample) {c : ℝ}, 0 < c → Filter.Tendsto (fun n => ↑(split.foldB n).card / ↑n) Filter.atTop (nhds c) → ∀ (η_hat : ℕ → P.Ω → Causalean.Estimation.PLR.PLRNuisance γ), MeasureTheory.Integrable S.U P.μ → MeasureTheory.Integrable (fun ω => S.U ω * S.resid ω) P.μ → MeasureTheory.Integrable (fun ω => S.b (S.factualX ω)) P.μ → MeasureTheory.Integrable S.factualD P.μ → MeasureTheory.MemLp S.resid 2 P.μ → MeasureTheory.Integrable (fun ω => Causalean.Estimation.PLR.plrMomentFunctional S.η₀ (S.factualZ ω) S.θ₀ ^ 2) P.μ → (∀ (n : ℕ) (ω : P.Ω), MeasureTheory.MemLp (fun x => (η_hat n ω).lFn x - S.lVal x) 2 S.P_X) → (∀ (n : ℕ) (ω : P.Ω), MeasureTheory.MemLp (fun x => (η_hat n ω).mFn x - S.mVal x) 2 S.P_X) → (∀ (n : ℕ) (ω : P.Ω), MeasureTheory.Integrable (fun ω' => S.U ω' * ((η_hat n ω).mFn (S.factualX ω') - S.mVal (S.factualX ω'))) P.μ) → (∀ (n : ℕ) (ω : P.Ω), MeasureTheory.Integrable (fun ω' => ((η_hat n ω).lFn (S.factualX ω') - S.lVal (S.factualX ω')) * S.resid ω') P.μ) → (∀ (n : ℕ) (ω : P.Ω), MeasureTheory.Integrable (fun ω' => S.resid ω' * ((η_hat n ω).mFn (S.factualX ω') - S.mVal (S.factualX ω'))) P.μ) → (∀ (n : ℕ), Measurable fun p => S.plrGeneralMoment.m (η_hat n p.1) p.2 S.plrGeneralMoment.θ₀) → (∀ (n : ℕ), Measurable fun ω z => S.plrGeneralMoment.m (η_hat n ω) z S.plrGeneralMoment.θ₀) → (∀ (n : ℕ), Measurable fun p => S.plrGeneralMoment.m (η_hat n p.1) p.2 S.plrGeneralMoment.θ₀) → (∀ (n : ℕ) (ω : P.Ω), MeasureTheory.Integrable (fun z => S.plrGeneralMoment.m (η_hat n ω) z S.plrGeneralMoment.θ₀) S.P_Z) → (∀ (n : ℕ) (ω : P.Ω), MeasureTheory.Integrable (fun z => S.plrGeneralMoment.m (η_hat n ω) z S.plrGeneralMoment.θ₀ ^ 2) S.P_Z) → Causalean.Stat.IsLittleOp (fun n ω => (MeasureTheory.eLpNorm (fun z => S.plrGeneralMoment.m (η_hat n ω) z S.plrGeneralMoment.θ₀ - S.plrGeneralMoment.m S.plrGeneralMoment.η₀ z S.plrGeneralMoment.θ₀) 2 S.P_Z).toReal) (fun x => 1) P.μ → Causalean.Stat.IsLittleOp (fun n ω => ↑(S.plrGeneralMoment.ρ₁ (η_hat n ω) S.plrGeneralMoment.η₀) * ↑(S.plrGeneralMoment.ρ₂ (η_hat n ω) S.plrGeneralMoment.η₀)) (fun n => ↑n ^ (-(1 / 2))) P.μ → (Measurable fun z => -S.plrGeneralMoment.J₀_inv * Causalean.Estimation.PLR.plrMomentFunctional S.η₀ z S.θ₀) → ∀ (hθn_meas : ∀ (n : ℕ), AEMeasurable (Causalean.Stat.IsAsymLinear.rescaledEstimator (Causalean.Estimation.OrthogonalMoments.dmlChernozhukovEstimator S.plrGeneralMoment sample split η_hat) S.θ₀ split.foldB n) P.μ), (∀ (n : ℕ), AEMeasurable (Causalean.Stat.IsAsymLinear.normalizedSum sample (fun z => -S.plrGeneralMoment.J₀_inv * Causalean.Estimation.PLR.plrMomentFunctional S.η₀ z S.θ₀) split.foldB n) P.μ) → Causalean.Stat.Tendsto_dist (Causalean.Stat.IsAsymLinear.rescaledEstimator (Causalean.Estimation.OrthogonalMoments.dmlChernozhukovEstimator S.plrGeneralMoment sample split η_hat) S.θ₀ split.foldB) (Causalean.Stat.gaussianMeasure 0 (∫ (z : γ × ℝ × ℝ), (-S.plrGeneralMoment.J₀_inv * Causalean.Estimation.PLR.plrMomentFunctional S.η₀ z S.θ₀) ^ 2 ∂S.P_Z)) P.μ hθn_meas","l":171,"f":"Causalean/Estimation/PLR/DML.lean"},{"n":"Causalean.Estimation.PLR.PLRSystem.plr_finite_var","k":"theorem","m":"Causalean.Estimation.PLR.MeanZero","a":"Estimation/PLR","t":1,"d":"If the squared Robinson partialling-out score — evaluated at the true outcome and treatment regressions and the true structural slope, pulled back to the population space — is integrable, then the squared score is integrable under the observed-data law $P_Z$, i.e. the score has finite variance at the truth.","s":"∀ {P : Causalean.PO.POSystem} {γ : Type u_1} [inst : MeasurableSpace γ] [inst_1 : MeasureTheory.IsFiniteMeasure P.μ] (S : Causalean.Estimation.PLR.PLRSystem P γ), MeasureTheory.Integrable (fun ω => Causalean.Estimation.PLR.plrMomentFunctional S.η₀ (S.factualZ ω) S.θ₀ ^ 2) P.μ → MeasureTheory.Integrable (fun z => Causalean.Estimation.PLR.plrMomentFunctional S.η₀ z S.θ₀ ^ 2) S.P_Z","l":63,"f":"Causalean/Estimation/PLR/MeanZero.lean"},{"n":"Causalean.Estimation.PLR.PLRSystem.plr_jacobian_consistency","k":"theorem","m":"Causalean.Estimation.PLR.JacobianConsistency","a":"Estimation/PLR","t":1,"d":"Fold-B Jacobian consistency. Assume the treatment is integrable; the true partialling-out moment at the truth is square-integrable under the observed-data law, and the squared true treatment residual is integrable; that for the estimated nuisance sequence η_hat, at every fold and draw, the treatment-regression error is square-integrable in the covariate law, both marginally and against the true treatment residual; that the treatment-regression error converges to zero in L²(P_X) at rate $o_p(1)$; and that the partialling-out-moment increment mₐ(η̂,·) − mₐ(η₀,·) is jointly measurable, fold-A measurable, square-integrable under the observed-data law, and converges to zero in L²(P_Z) at rate $o_p(1)$. Then the empirical partialling-out Jacobian at the estimated nuisance, averaged over the estimation fold, converges in probability to its population value $J_0=-E[(D-m_{val}(X))^2]$:","s":"∀ {P : Causalean.PO.POSystem} {γ : Type u_1} [inst : MeasurableSpace γ] [StandardBorelSpace P.Ω] [inst_2 : MeasureTheory.IsFiniteMeasure P.μ] [MeasureTheory.IsProbabilityMeasure P.μ] (S : Causalean.Estimation.PLR.PLRSystem P γ) (sample : Causalean.Stat.IIDSample P.Ω (γ × ℝ × ℝ) P.μ S.P_Z) (split : Causalean.Stat.OneShotSplit sample) (η_hat : ℕ → P.Ω → Causalean.Estimation.PLR.PLRNuisance γ), MeasureTheory.Integrable S.factualD P.μ → MeasureTheory.MemLp (Causalean.Estimation.PLR.plrMomentA S.η₀) 2 S.P_Z → MeasureTheory.Integrable (fun ω => (S.factualD ω - S.mVal (S.factualX ω)) ^ 2) P.μ → (∀ (n : ℕ) (ω : P.Ω), MeasureTheory.Integrable (fun ω' => (S.mVal (S.factualX ω') - (η_hat n ω).mFn (S.factualX ω')) ^ 2) P.μ) → (∀ (n : ℕ) (ω : P.Ω), MeasureTheory.Integrable (fun ω' => (S.factualD ω' - S.mVal (S.factualX ω')) * (S.mVal (S.factualX ω') - (η_hat n ω).mFn (S.factualX ω'))) P.μ) → (∀ (n : ℕ) (ω : P.Ω), MeasureTheory.MemLp (fun x => S.mVal x - (η_hat n ω).mFn x) 2 S.P_X) → Causalean.Stat.IsLittleOp (fun n ω => (MeasureTheory.eLpNorm (fun x => S.mVal x - (η_hat n ω).mFn x) 2 S.P_X).toReal) (fun x => 1) P.μ → (∀ (n : ℕ), Measurable (Function.uncurry fun ω z => Causalean.Estimation.PLR.plrMomentA (η_hat n ω) z - Causalean.Estimation.PLR.plrMomentA S.η₀ z)) → (∀ (n : ℕ), Measurable fun ω z => Causalean.Estimation.PLR.plrMomentA (η_hat n ω) z - Causalean.Estimation.PLR.plrMomentA S.η₀ z) → (∀ (n : ℕ), Measurable (Function.uncurry fun ω z => Causalean.Estimation.PLR.plrMomentA (η_hat n ω) z - Causalean.Estimation.PLR.plrMomentA S.η₀ z)) → (∀ (n : ℕ) (ω : P.Ω), MeasureTheory.MemLp (fun z => Causalean.Estimation.PLR.plrMomentA (η_hat n ω) z - Causalean.Estimation.PLR.plrMomentA S.η₀ z) 2 S.P_Z) → Causalean.Stat.IsLittleOp (fun n ω => (MeasureTheory.eLpNorm (fun z => Causalean.Estimation.PLR.plrMomentA (η_hat n ω) z - Causalean.Estimation.PLR.plrMomentA S.η₀ z) 2 S.P_Z).toReal) (fun x => 1) P.μ → Causalean.Stat.Tendsto_inProb (fun n ω => (↑(split.foldB n).card)⁻¹ * ∑ i ∈ split.foldB n, Causalean.Estimation.PLR.plrMomentA (η_hat n ω) (sample.Z i ω)) (fun x => S.plrGeneralMoment.J₀) P.μ","l":220,"f":"Causalean/Estimation/PLR/JacobianConsistency.lean"},{"n":"Causalean.Estimation.PLR.PLRSystem.plr_meanZero","k":"theorem","m":"Causalean.Estimation.PLR.MeanZero","a":"Estimation/PLR","t":1,"d":"If the structural error is integrable, the product of the structural error and the true treatment residual is integrable, the baseline covariate function is integrable, and the treatment is integrable, then the Robinson partialling-out score, evaluated at the true outcome and treatment regressions and the true structural slope, has zero mean under the observed-data law.","s":"∀ {P : Causalean.PO.POSystem} {γ : Type u_1} [inst : MeasurableSpace γ] [inst_1 : MeasureTheory.IsFiniteMeasure P.μ] (S : Causalean.Estimation.PLR.PLRSystem P γ), MeasureTheory.Integrable S.U P.μ → MeasureTheory.Integrable (fun ω => S.U ω * S.resid ω) P.μ → MeasureTheory.Integrable (fun ω => S.b (S.factualX ω)) P.μ → MeasureTheory.Integrable S.factualD P.μ → Causalean.Estimation.OrthogonalMoments.MeanZero S.plrGeneralMoment","l":38,"f":"Causalean/Estimation/PLR/MeanZero.lean"},{"n":"Causalean.Estimation.PLR.PLRSystem.plr_remainder_bound","k":"theorem","m":"Causalean.Estimation.PLR.RemainderBound","a":"Estimation/PLR","t":1,"d":"Doubly-robust bilinear remainder. Fix any candidate nuisance pair η of outcome and treatment regressions. Assume the treatment, the baseline covariate function, and the structural error are integrable; that the outcome- and treatment-regression errors of η and the true treatment residual are square-integrable; and that the resulting orthogonal cross terms and the true structural-error/treatment-residual product are all integrable. Then the population Robinson partialling-out score at η and the true structural slope is bounded in absolute value by $(1+|\\theta_0|)$ times the product of the two L²(P_X) nuisance-error seminorms:","s":"∀ {P : Causalean.PO.POSystem} {γ : Type u_1} [inst : MeasurableSpace γ] [inst_1 : MeasureTheory.IsFiniteMeasure P.μ] (S : Causalean.Estimation.PLR.PLRSystem P γ) (η : Causalean.Estimation.PLR.PLRNuisance γ), MeasureTheory.Integrable S.factualD P.μ → MeasureTheory.Integrable (fun ω => S.b (S.factualX ω)) P.μ → MeasureTheory.Integrable S.U P.μ → MeasureTheory.MemLp (fun x => η.lFn x - S.lVal x) 2 S.P_X → MeasureTheory.MemLp (fun x => η.mFn x - S.mVal x) 2 S.P_X → MeasureTheory.MemLp S.resid 2 P.μ → MeasureTheory.Integrable (fun ω => S.U ω * (η.mFn (S.factualX ω) - S.mVal (S.factualX ω))) P.μ → MeasureTheory.Integrable (fun ω => (η.lFn (S.factualX ω) - S.lVal (S.factualX ω)) * S.resid ω) P.μ → MeasureTheory.Integrable (fun ω => S.resid ω * (η.mFn (S.factualX ω) - S.mVal (S.factualX ω))) P.μ → MeasureTheory.Integrable (fun ω => S.U ω * S.resid ω) P.μ → |∫ (z : γ × ℝ × ℝ), Causalean.Estimation.PLR.plrMomentFunctional η z S.θ₀ ∂S.P_Z| ≤ (1 + |S.θ₀|) * ↑(S.plrGeneralMoment.ρ₁ η S.η₀) * ↑(S.plrGeneralMoment.ρ₂ η S.η₀)","l":84,"f":"Causalean/Estimation/PLR/RemainderBound.lean"},{"n":"Causalean.Estimation.PLR.PLRSystem.plr_score_diff_isLittleOp_one","k":"theorem","m":"Causalean.Estimation.PLR.ScoreL2","a":"Estimation/PLR","t":1,"d":"Score-difference L²(P_Z) o_p(1) for the partially linear model. Suppose three nonnegative bounding constants Ca, Cv, Cm satisfy: the residualized outcome $A = Y-\\ell_{val}(X)-\\theta_0(D-m_{val}(X))$ is almost surely bounded by Ca under the observed-data law; the true treatment residual $D-m_{val}(X)$ is almost surely bounded by Cv; and the treatment-regression estimation error of η_hat is uniformly bounded by Cm at every fold, draw, and covariate value. Suppose further that for the estimated nuisance sequence η_hat on the system S, the outcome- and treatment-regression estimation errors are square-integrable over the covariate law at every fold and draw, and that both regression errors converge to zero in L²(P_X) at rate $o_p(1)$. Then the L²(P_Z)-seminorm of the Robinson partialling-out score difference between the estimated and the true nuisance is itself $o_p(1)$.","s":"∀ {P : Causalean.PO.POSystem} {γ : Type u_1} [inst : MeasurableSpace γ] [inst_1 : MeasureTheory.IsFiniteMeasure P.μ] (S : Causalean.Estimation.PLR.PLRSystem P γ) (η_hat : ℕ → P.Ω → Causalean.Estimation.PLR.PLRNuisance γ) {Ca Cv Cm : ℝ}, 0 ≤ Ca → 0 ≤ Cv → 0 ≤ Cm → (∀ᵐ (z : γ × ℝ × ℝ) ∂S.P_Z, |z.2.2 - S.lVal z.1 - S.θ₀ * (z.2.1 - S.mVal z.1)| ≤ Ca) → (∀ᵐ (z : γ × ℝ × ℝ) ∂S.P_Z, |z.2.1 - S.mVal z.1| ≤ Cv) → (∀ (n : ℕ) (ω : P.Ω) (x : γ), |S.mVal x - (η_hat n ω).mFn x| ≤ Cm) → (∀ (n : ℕ) (ω : P.Ω), MeasureTheory.MemLp (fun x => (η_hat n ω).lFn x - S.lVal x) 2 S.P_X) → (∀ (n : ℕ) (ω : P.Ω), MeasureTheory.MemLp (fun x => (η_hat n ω).mFn x - S.mVal x) 2 S.P_X) → Causalean.Stat.IsLittleOp (fun n ω => (MeasureTheory.eLpNorm (fun x => (η_hat n ω).lFn x - S.lVal x) 2 S.P_X).toReal) (fun x => 1) P.μ → Causalean.Stat.IsLittleOp (fun n ω => (MeasureTheory.eLpNorm (fun x => (η_hat n ω).mFn x - S.mVal x) 2 S.P_X).toReal) (fun x => 1) P.μ → Causalean.Stat.IsLittleOp (fun n ω => (MeasureTheory.eLpNorm (fun z => Causalean.Estimation.PLR.plrMomentFunctional (η_hat n ω) z S.θ₀ - Causalean.Estimation.PLR.plrMomentFunctional S.η₀ z S.θ₀) 2 S.P_Z).toReal) (fun x => 1) P.μ","l":334,"f":"Causalean/Estimation/PLR/ScoreL2.lean"},{"n":"Causalean.Estimation.PLR.PLRSystem.residSecondMoment","k":"def","m":"Causalean.Estimation.PLR.Setup","a":"Estimation/PLR","t":1,"d":"The residual second moment measures treatment variation left after partialling out the covariate.","s":"{P : Causalean.PO.POSystem} → {γ : Type u_1} → [inst : MeasurableSpace γ] → [inst_1 : MeasureTheory.IsFiniteMeasure P.μ] → Causalean.Estimation.PLR.PLRSystem P γ → ℝ","l":123,"f":"Causalean/Estimation/PLR/Setup.lean"},{"n":"Causalean.Estimation.PLR.PLRSystem.η₀","k":"def","m":"Causalean.Estimation.PLR.Setup","a":"Estimation/PLR","t":1,"d":"The true nuisance is the pair of value-space outcome and treatment regressions.","s":"{P : Causalean.PO.POSystem} → {γ : Type u_1} → [inst : MeasurableSpace γ] → [inst_1 : MeasureTheory.IsFiniteMeasure P.μ] → Causalean.Estimation.PLR.PLRSystem P γ → Causalean.Estimation.PLR.PLRNuisance γ","l":115,"f":"Causalean/Estimation/PLR/Setup.lean"},{"n":"Causalean.Estimation.PLR.PLRSystem.θ₀","k":"def","m":"Causalean.Estimation.PLR.Setup","a":"Estimation/PLR","t":1,"d":"The target parameter is the structural slope in the partially linear model.","s":"{P : Causalean.PO.POSystem} → {γ : Type u_1} → [inst : MeasurableSpace γ] → [inst_1 : MeasureTheory.IsFiniteMeasure P.μ] → Causalean.Estimation.PLR.PLRSystem P γ → ℝ","l":120,"f":"Causalean/Estimation/PLR/Setup.lean"},{"n":"Causalean.Estimation.PLR.measurable_plrMomentA","k":"theorem","m":"Causalean.Estimation.PLR.Moment","a":"Estimation/PLR","t":2,"d":"The linear-score coefficient is measurable in the observed data.","s":"∀ {γ : Type u_1} [inst : MeasurableSpace γ] (η : Causalean.Estimation.PLR.PLRNuisance γ), Measurable fun z => Causalean.Estimation.PLR.plrMomentA η z","l":83,"f":"Causalean/Estimation/PLR/Moment.lean"},{"n":"Causalean.Estimation.PLR.measurable_plrMomentB","k":"theorem","m":"Causalean.Estimation.PLR.Moment","a":"Estimation/PLR","t":2,"d":"The linear-score constant term is measurable in the observed data.","s":"∀ {γ : Type u_1} [inst : MeasurableSpace γ] (η : Causalean.Estimation.PLR.PLRNuisance γ), Measurable fun z => Causalean.Estimation.PLR.plrMomentB η z","l":89,"f":"Causalean/Estimation/PLR/Moment.lean"},{"n":"Causalean.Estimation.PLR.measurable_plrMomentFunctional","k":"theorem","m":"Causalean.Estimation.PLR.Moment","a":"Estimation/PLR","t":2,"d":"The Robinson partialling-out score is measurable in the observed data.","s":"∀ {γ : Type u_1} [inst : MeasurableSpace γ] (η : Causalean.Estimation.PLR.PLRNuisance γ) (θ : ℝ), Measurable fun z => Causalean.Estimation.PLR.plrMomentFunctional η z θ","l":75,"f":"Causalean/Estimation/PLR/Moment.lean"},{"n":"Causalean.Estimation.PLR.measurable_plrResidual","k":"theorem","m":"Causalean.Estimation.PLR.Moment","a":"Estimation/PLR","t":2,"d":"The treatment residual is measurable as a function of the observed data.","s":"∀ {γ : Type u_1} [inst : MeasurableSpace γ] (η : Causalean.Estimation.PLR.PLRNuisance γ), Measurable fun z => Causalean.Estimation.PLR.plrResidual η z","l":70,"f":"Causalean/Estimation/PLR/Moment.lean"},{"n":"Causalean.Estimation.PLR.plrMomentA","k":"def","m":"Causalean.Estimation.PLR.Moment","a":"Estimation/PLR","t":1,"d":"The linear-score coefficient is minus the squared treatment residual.","s":"{γ : Type u_1} → [inst : MeasurableSpace γ] → Causalean.Estimation.PLR.PLRNuisance γ → γ × ℝ × ℝ → ℝ","l":54,"f":"Causalean/Estimation/PLR/Moment.lean"},{"n":"Causalean.Estimation.PLR.plrMomentB","k":"def","m":"Causalean.Estimation.PLR.Moment","a":"Estimation/PLR","t":1,"d":"The linear-score constant term is the residualized outcome times the treatment residual.","s":"{γ : Type u_1} → [inst : MeasurableSpace γ] → Causalean.Estimation.PLR.PLRNuisance γ → γ × ℝ × ℝ → ℝ","l":58,"f":"Causalean/Estimation/PLR/Moment.lean"},{"n":"Causalean.Estimation.PLR.plrMomentFunctional","k":"def","m":"Causalean.Estimation.PLR.Moment","a":"Estimation/PLR","t":1,"d":"The Robinson partialling-out score multiplies the structural residualized outcome by the treatment residual.","s":"{γ : Type u_1} → [inst : MeasurableSpace γ] → Causalean.Estimation.PLR.PLRNuisance γ → γ × ℝ × ℝ → ℝ → ℝ","l":49,"f":"Causalean/Estimation/PLR/Moment.lean"},{"n":"Causalean.Estimation.PLR.plrMoment_decomp","k":"theorem","m":"Causalean.Estimation.PLR.Moment","a":"Estimation/PLR","t":1,"d":"Robinson score decomposition. The partialling-out score decomposes into its linear coefficient in θ times the target parameter θ plus a constant term.","s":"∀ {γ : Type u_1} [inst : MeasurableSpace γ] (η : Causalean.Estimation.PLR.PLRNuisance γ) (z : γ × ℝ × ℝ) (θ : ℝ), Causalean.Estimation.PLR.plrMomentFunctional η z θ = Causalean.Estimation.PLR.plrMomentA η z * θ + Causalean.Estimation.PLR.plrMomentB η z","l":63,"f":"Causalean/Estimation/PLR/Moment.lean"},{"n":"Causalean.Estimation.PLR.plrResidual","k":"def","m":"Causalean.Estimation.PLR.Moment","a":"Estimation/PLR","t":1,"d":"The treatment residual subtracts the nuisance treatment regression from the observed treatment value.","s":"{γ : Type u_1} → [inst : MeasurableSpace γ] → Causalean.Estimation.PLR.PLRNuisance γ → γ × ℝ × ℝ → ℝ","l":45,"f":"Causalean/Estimation/PLR/Moment.lean"},{"n":"Causalean.Experimentation.BettingMean.capital","k":"def","m":"Causalean.Experimentation.BettingMean","a":"Experimentation","t":1,"d":"The betting capital process for testing that the mean is m, with observations X and predictable betting fractions lam: K₀ = 1 and Kₙ₊₁ = Kₙ · (1 + lamₙ·(Xₙ − m)).","s":"{Ω : Type u_1} → (ℕ → Ω → ℝ) → (ℕ → Ω → ℝ) → ℝ → ℕ → Ω → ℝ","l":36,"f":"Causalean/Experimentation/BettingMean.lean"},{"n":"Causalean.Experimentation.BettingMean.capital_nonneg","k":"theorem","m":"Causalean.Experimentation.BettingMean","a":"Experimentation","t":2,"d":"The capital process stays nonnegative provided each bet keeps the per-step factor nonnegative (0 ≤ 1 + lamₙ(Xₙ − m)) — the admissibility constraint on the betting fractions.","s":"∀ {Ω : Type u_1} {X lam : ℕ → Ω → ℝ} {m : ℝ}, (∀ (n : ℕ) (ω : Ω), 0 ≤ 1 + lam n ω * (X n ω - m)) → ∀ (n : ℕ), 0 ≤ Causalean.Experimentation.BettingMean.capital X lam m n","l":50,"f":"Causalean/Experimentation/BettingMean.lean"},{"n":"Causalean.Experimentation.BettingMean.capital_succ","k":"theorem","m":"Causalean.Experimentation.BettingMean","a":"Experimentation","t":2,"d":"One step of the betting capital multiplies current wealth by the return 1 + lam n · (X n − m).","s":"∀ {Ω : Type u_1} (X lam : ℕ → Ω → ℝ) (m : ℝ) (n : ℕ), Causalean.Experimentation.BettingMean.capital X lam m (n + 1) = fun ω => Causalean.Experimentation.BettingMean.capital X lam m n ω * (1 + lam n ω * (X n ω - m))","l":45,"f":"Causalean/Experimentation/BettingMean.lean"},{"n":"Causalean.Experimentation.BettingMean.capital_zero","k":"theorem","m":"Causalean.Experimentation.BettingMean","a":"Experimentation","t":2,"d":"The betting capital starts at one before any observations are processed.","s":"∀ {Ω : Type u_1} (X lam : ℕ → Ω → ℝ) (m : ℝ), Causalean.Experimentation.BettingMean.capital X lam m 0 = fun x => 1","l":42,"f":"Causalean/Experimentation/BettingMean.lean"},{"n":"Causalean.Experimentation.BettingMean.isAnytimeValid_betting","k":"theorem","m":"Causalean.Experimentation.BettingMean","a":"Experimentation","t":1,"d":"Anytime-valid test by betting. If the betting capital process built from the data stream X, betting fractions lam, and candidate mean m is a test supermartingale under μ and the target error level α is strictly positive, then the rejection region that declares significance once the capital ever reaches 1/α is an anytime-valid level-α test.","s":"∀ {Ω : Type u_1} {m0 : MeasurableSpace Ω} {μ : MeasureTheory.Measure Ω} {ℱ : MeasureTheory.Filtration ℕ m0} [MeasureTheory.IsFiniteMeasure μ] {X lam : ℕ → Ω → ℝ} {m : ℝ}, Causalean.Experimentation.Sequential.IsTestSupermartingale (Causalean.Experimentation.BettingMean.capital X lam m) ℱ μ → ∀ {α : ℝ}, 0 < α → Causalean.Experimentation.Sequential.IsAnytimeValid (Causalean.Experimentation.Sequential.rejectionRegion (Causalean.Experimentation.BettingMean.capital X lam m) α) μ α","l":131,"f":"Causalean/Experimentation/BettingMean.lean"},{"n":"Causalean.Experimentation.BettingMean.isConfidenceSequence_bettingCI","k":"theorem","m":"Causalean.Experimentation.BettingMean","a":"Experimentation","t":1,"d":"Betting confidence sequence (coverage). If the betting capital process built from the data stream X, betting fractions lam, and candidate mean m is a test supermartingale under μ and the target error level α is strictly positive, then the sequence of confidence sets that retains a candidate mean m only while its capital has not yet reached 1/α is a valid confidence sequence with time-uniform coverage 1 − α.","s":"∀ {Ω : Type u_1} {m0 : MeasurableSpace Ω} {μ : MeasureTheory.Measure Ω} {ℱ : MeasureTheory.Filtration ℕ m0} [MeasureTheory.IsFiniteMeasure μ] {X lam : ℕ → Ω → ℝ} {m : ℝ}, Causalean.Experimentation.Sequential.IsTestSupermartingale (Causalean.Experimentation.BettingMean.capital X lam m) ℱ μ → ∀ {α : ℝ}, 0 < α → Causalean.Experimentation.Sequential.IsConfidenceSequence (Causalean.Experimentation.Sequential.confSeqOfWealth (Causalean.Experimentation.BettingMean.capital X lam m) α) μ α","l":121,"f":"Causalean/Experimentation/BettingMean.lean"},{"n":"Causalean.Experimentation.BettingMean.isTestSupermartingale_capital","k":"theorem","m":"Causalean.Experimentation.BettingMean","a":"Experimentation","t":1,"d":"The capital process is a test supermartingale (the Waudby-Smith–Ramdas construction). Consider a data stream X tested against a candidate mean m via betting fractions lam, all adapted to a filtration ℱ. If the capital process itself is adapted to ℱ, each bet lam n is measurable with respect to the time-n information, the capital process, the centered increment X n − m, and the bet-scaled increment lam n · (X n − m) are all integrable at every time n, every per-step return factor 1 + lam n (X n − m) stays nonnegative, and the conditional mean of X n − m given the time-n information is zero, i.e. the bet is conditionally fair, then the betting capital process is a nonnegative test supermartingale for ℱ under μ, with E[K₀] ≤ 1.","s":"∀ {Ω : Type u_1} {m0 : MeasurableSpace Ω} {μ : MeasureTheory.Measure Ω} {ℱ : MeasureTheory.Filtration ℕ m0} [MeasureTheory.IsProbabilityMeasure μ] {X lam : ℕ → Ω → ℝ} {m : ℝ}, MeasureTheory.Adapted ℱ (Causalean.Experimentation.BettingMean.capital X lam m) → (∀ (n : ℕ), MeasureTheory.StronglyMeasurable (lam n)) → (∀ (n : ℕ), MeasureTheory.Integrable (Causalean.Experimentation.BettingMean.capital X lam m n) μ) → (∀ (n : ℕ), MeasureTheory.Integrable (fun ω => X n ω - m) μ) → (∀ (n : ℕ), MeasureTheory.Integrable (fun ω => lam n ω * (X n ω - m)) μ) → (∀ (n : ℕ) (ω : Ω), 0 ≤ 1 + lam n ω * (X n ω - m)) → (∀ (n : ℕ), μ[fun ω => X n ω - m | ↑ℱ n] =ᵐ[μ] 0) → Causalean.Experimentation.Sequential.IsTestSupermartingale (Causalean.Experimentation.BettingMean.capital X lam m) ℱ μ","l":65,"f":"Causalean/Experimentation/BettingMean.lean"},{"n":"Causalean.Experimentation.ClusterRandomizedHT.E_htClusterEffect","k":"theorem","m":"Causalean.Experimentation.ClusterRandomizedHT","a":"Experimentation","t":1,"d":"Unbiasedness for the finite-population average treatment effect. Consider cluster randomization where each cluster c is assigned treatment with probability p c; suppose p c lies between 0 and 1 and, more sharply, p c lies strictly between 0 and 1, and the total number of units across clusters is positive. Then the Middleton–Aronow Horvitz–Thompson estimator of the effect — the inverse-probability-weighted treated-minus-control cluster-total contrast, normalized by the total unit count — is unbiased for the finite-population average treatment effect, the analogous unweighted contrast of the cluster-total potential outcomes normalized by the same unit count.","s":"∀ {C : Type u_1} [inst : Fintype C] [inst_1 : DecidableEq C] (p : C → ℝ) (n : C → ℕ) (y1 y0 : C → ℝ) (hp0 : ∀ (c : C), 0 ≤ p c) (hp1 : ∀ (c : C), p c ≤ 1), (∀ (c : C), 0 < p c) → (∀ (c : C), p c < 1) → 0 < Causalean.Experimentation.ClusterRandomizedHT.totalUnits n → (Causalean.Experimentation.DesignBased.clusterDesign p hp0 hp1).E (Causalean.Experimentation.ClusterRandomizedHT.htClusterEffect p n y1 y0) = Causalean.Experimentation.ClusterRandomizedHT.totalEffect n y1 y0","l":111,"f":"Causalean/Experimentation/ClusterRandomizedHT.lean"},{"n":"Causalean.Experimentation.ClusterRandomizedHT.E_htControlTotal","k":"theorem","m":"Causalean.Experimentation.ClusterRandomizedHT","a":"Experimentation","t":2,"d":"The HT control total is unbiased for the control-arm population total ∑ y0 c: each cluster is a control with probability 1 − p c, undone by the weight 1 / (1 − p c).","s":"∀ {C : Type u_1} [inst : Fintype C] [inst_1 : DecidableEq C] (p y0 : C → ℝ) (hp0 : ∀ (c : C), 0 ≤ p c) (hp1 : ∀ (c : C), p c ≤ 1), (∀ (c : C), p c < 1) → (Causalean.Experimentation.DesignBased.clusterDesign p hp0 hp1).E (Causalean.Experimentation.ClusterRandomizedHT.htControlTotal p y0) = ∑ c, y0 c","l":82,"f":"Causalean/Experimentation/ClusterRandomizedHT.lean"},{"n":"Causalean.Experimentation.ClusterRandomizedHT.E_htTreatedTotal","k":"theorem","m":"Causalean.Experimentation.ClusterRandomizedHT","a":"Experimentation","t":2,"d":"The HT treated total is unbiased for the treated-arm population total ∑ y1 c: each cluster is treated with probability p c, which the inverse weight 1 / p c exactly undoes.","s":"∀ {C : Type u_1} [inst : Fintype C] [inst_1 : DecidableEq C] (p y1 : C → ℝ) (hp0 : ∀ (c : C), 0 ≤ p c) (hp1 : ∀ (c : C), p c ≤ 1), (∀ (c : C), 0 < p c) → (Causalean.Experimentation.DesignBased.clusterDesign p hp0 hp1).E (Causalean.Experimentation.ClusterRandomizedHT.htTreatedTotal p y1) = ∑ c, y1 c","l":59,"f":"Causalean/Experimentation/ClusterRandomizedHT.lean"},{"n":"Causalean.Experimentation.ClusterRandomizedHT.htClusterEffect","k":"def","m":"Causalean.Experimentation.ClusterRandomizedHT","a":"Experimentation","t":1,"d":"The Middleton-Aronow Horvitz-Thompson ATE estimator divides the inverse-probability weighted treated-minus-control cluster-total estimator by the total number of units.","s":"{C : Type u_1} → [Fintype C] → (C → ℝ) → (C → ℕ) → (C → ℝ) → (C → ℝ) → (C → Bool) → ℝ","l":48,"f":"Causalean/Experimentation/ClusterRandomizedHT.lean"},{"n":"Causalean.Experimentation.ClusterRandomizedHT.htControlTotal","k":"def","m":"Causalean.Experimentation.ClusterRandomizedHT","a":"Experimentation","t":1,"d":"The Horvitz–Thompson control total: each control cluster's total outcome y0 c, weighted by the inverse control probability 1 / (1 − p c).","s":"{C : Type u_1} → [Fintype C] → (C → ℝ) → (C → ℝ) → (C → Bool) → ℝ","l":40,"f":"Causalean/Experimentation/ClusterRandomizedHT.lean"},{"n":"Causalean.Experimentation.ClusterRandomizedHT.htTreatedTotal","k":"def","m":"Causalean.Experimentation.ClusterRandomizedHT","a":"Experimentation","t":1,"d":"The Horvitz–Thompson treated total: each treated cluster's total outcome y1 c, weighted by the inverse assignment probability 1 / p c.","s":"{C : Type u_1} → [Fintype C] → (C → ℝ) → (C → ℝ) → (C → Bool) → ℝ","l":35,"f":"Causalean/Experimentation/ClusterRandomizedHT.lean"},{"n":"Causalean.Experimentation.ClusterRandomizedHT.totalEffect","k":"def","m":"Causalean.Experimentation.ClusterRandomizedHT","a":"Experimentation","t":1,"d":"The finite-population average treatment effect is the all-unit treated-minus-control potential-outcome total, aggregated through cluster totals, divided by the total number of units.","s":"{C : Type u_1} → [Fintype C] → (C → ℕ) → (C → ℝ) → (C → ℝ) → ℝ","l":54,"f":"Causalean/Experimentation/ClusterRandomizedHT.lean"},{"n":"Causalean.Experimentation.ClusterRandomizedHT.totalUnits","k":"def","m":"Causalean.Experimentation.ClusterRandomizedHT","a":"Experimentation","t":1,"d":"The total number of experimental units represented by the cluster counts n c.","s":"{C : Type u_1} → [Fintype C] → (C → ℕ) → ℝ","l":45,"f":"Causalean/Experimentation/ClusterRandomizedHT.lean"},{"n":"Causalean.Experimentation.DesignBased.BlockDegreeLE","k":"def","m":"Causalean.Experimentation.DesignBased.LocalDependenceVariance","a":"Experimentation/DesignBased","t":1,"d":"A family of dependence blocks has degree at most d if no single unit's treatment is read by more than d of the blocks. When the blocks are the in-neighbourhoods of a dependency graph this is the bound on out-degree.","s":"{U : Type u_1} → [Fintype U] → [DecidableEq U] → (U → Finset U) → ℕ → Prop","l":46,"f":"Causalean/Experimentation/DesignBased/LocalDependenceVariance.lean"},{"n":"Causalean.Experimentation.DesignBased.Cov_expoInd_cross","k":"theorem","m":"Causalean.Experimentation.DesignBased.Exposure","a":"Experimentation/DesignBased","t":1,"d":"Covariance of cross-exposure indicators. For a design, an exposure mapping, unit traits, units i and j, and exposure levels d and d', the design covariance of unit i's d-exposure indicator and unit j's d'-exposure indicator equals the cross joint exposure probability π_ij(d,d') minus the product of their marginal exposure probabilities.","s":"∀ {Ω : Type u_1} [inst : Fintype Ω] {ι : Type u_2} {Θ : Type u_3} {Δ : Type u_4} [inst_1 : DecidableEq Δ] (D : Causalean.Experimentation.DesignBased.FiniteDesign Ω) (f : Ω → Θ → Δ) (θ : ι → Θ) (i j : ι) (d d' : Δ), D.Cov (Causalean.Experimentation.DesignBased.expoInd f θ i d) (Causalean.Experimentation.DesignBased.expoInd f θ j d') = Causalean.Experimentation.DesignBased.propPairCross D f θ i j d d' - Causalean.Experimentation.DesignBased.prop D f θ i d * Causalean.Experimentation.DesignBased.prop D f θ j d'","l":84,"f":"Causalean/Experimentation/DesignBased/Exposure.lean"},{"n":"Causalean.Experimentation.DesignBased.Cov_expoInd_same","k":"theorem","m":"Causalean.Experimentation.DesignBased.Exposure","a":"Experimentation/DesignBased","t":1,"d":"Covariance of same-exposure indicators. For a design, an exposure mapping, unit traits, units i and j, and an exposure level d, the design covariance of the two units' d-exposure indicators equals the joint exposure probability π_ij(d) minus the product of their marginal exposure probabilities.","s":"∀ {Ω : Type u_1} [inst : Fintype Ω] {ι : Type u_2} {Θ : Type u_3} {Δ : Type u_4} [inst_1 : DecidableEq Δ] (D : Causalean.Experimentation.DesignBased.FiniteDesign Ω) (f : Ω → Θ → Δ) (θ : ι → Θ) (i j : ι) (d : Δ), D.Cov (Causalean.Experimentation.DesignBased.expoInd f θ i d) (Causalean.Experimentation.DesignBased.expoInd f θ j d) = Causalean.Experimentation.DesignBased.propPairSame D f θ i j d - Causalean.Experimentation.DesignBased.prop D f θ i d * Causalean.Experimentation.DesignBased.prop D f θ j d","l":73,"f":"Causalean/Experimentation/DesignBased/Exposure.lean"},{"n":"Causalean.Experimentation.DesignBased.Cov_htTotal","k":"theorem","m":"Causalean.Experimentation.DesignBased.HT.Variance","a":"Experimentation/DesignBased/HT","t":1,"d":"Proposition 4.4 (covariance), expanded form eq:totals_covariance. Given a finite design, an outcome function y, an exposure map f, and an assignment θ, suppose the two exposures dk and dl are distinct, every unit has nonzero probability of realizing exposure dk, and every unit has nonzero probability of realizing exposure dl. Then the covariance of the two Horvitz–Thompson totals equals the off-diagonal double sum of inverse-probability-weighted joint-exposure covariance terms, minus the diagonal term ∑ᵢ y_i(dk)·y_i(dl) — a unit cannot occupy two distinct exposures at once, so the diagonal cross-indicator vanishes and leaves this unidentified term.","s":"∀ {Ω : Type u_1} [inst : Fintype Ω] {ι : Type u_2} {Θ : Type u_3} {Δ : Type u_4} [inst_1 : Fintype ι] [inst_2 : DecidableEq ι] [inst_3 : DecidableEq Δ] (D : Causalean.Experimentation.DesignBased.FiniteDesign Ω) (y : ι → Δ → ℝ) (f : Ω → Θ → Δ) (θ : ι → Θ) (dk dl : Δ), dk ≠ dl → (∀ (i : ι), Causalean.Experimentation.DesignBased.prop D f θ i dk ≠ 0) → (∀ (i : ι), Causalean.Experimentation.DesignBased.prop D f θ i dl ≠ 0) → D.Cov (Causalean.Experimentation.DesignBased.htTotal D y f θ dk) (Causalean.Experimentation.DesignBased.htTotal D y f θ dl) = ∑ i, ∑ j ∈ Finset.univ.erase i, (Causalean.Experimentation.DesignBased.propPairCross D f θ i j dk dl - Causalean.Experimentation.DesignBased.prop D f θ i dk * Causalean.Experimentation.DesignBased.prop D f θ j dl) * (y i dk / Causalean.Experimentation.DesignBased.prop D f θ i dk * (y j dl / Causalean.Experimentation.DesignBased.prop D f θ j dl)) - ∑ i, y i dk * y i dl","l":102,"f":"Causalean/Experimentation/DesignBased/HT/Variance.lean"},{"n":"Causalean.Experimentation.DesignBased.Cov_htTotal_cov","k":"theorem","m":"Causalean.Experimentation.DesignBased.HT.Variance","a":"Experimentation/DesignBased/HT","t":1,"d":"For a finite design, outcome function y, exposure map f, and assignment θ, the covariance between two Horvitz-Thompson exposure totals equals the double sum, over ordered unit pairs, of the two inverse-probability outcome weights times the covariance of the corresponding exposure indicators.","s":"∀ {Ω : Type u_1} [inst : Fintype Ω] {ι : Type u_2} {Θ : Type u_3} {Δ : Type u_4} [inst_1 : Fintype ι] [inst_2 : DecidableEq Δ] (D : Causalean.Experimentation.DesignBased.FiniteDesign Ω) (y : ι → Δ → ℝ) (f : Ω → Θ → Δ) (θ : ι → Θ) (dk dl : Δ), D.Cov (Causalean.Experimentation.DesignBased.htTotal D y f θ dk) (Causalean.Experimentation.DesignBased.htTotal D y f θ dl) = ∑ i, ∑ j, y i dk / Causalean.Experimentation.DesignBased.prop D f θ i dk * (y j dl / Causalean.Experimentation.DesignBased.prop D f θ j dl) * D.Cov (Causalean.Experimentation.DesignBased.expoInd f θ i dk) (Causalean.Experimentation.DesignBased.expoInd f θ j dl)","l":86,"f":"Causalean/Experimentation/DesignBased/HT/Variance.lean"},{"n":"Causalean.Experimentation.DesignBased.DependsOnBlock","k":"def","m":"Causalean.Experimentation.DesignBased.Designs.BernoulliMoments","a":"Experimentation/DesignBased/Designs","t":1,"d":"A statistic depends only on the coordinate block N if changing the assignment of any unit outside N leaves its value unchanged.","s":"{U : Type u_1} → Finset U → ((U → Bool) → ℝ) → Prop","l":373,"f":"Causalean/Experimentation/DesignBased/Designs/BernoulliMoments.lean"},{"n":"Causalean.Experimentation.DesignBased.DesignFamily","k":"def","m":"Causalean.Experimentation.DesignBased.Optimality","a":"Experimentation/DesignBased/Optimality","t":1,"d":"A design family: a set of candidate randomization designs on a common assignment space Ω, among which the experimenter chooses.","s":"(Ω : Type u_2) → [Fintype Ω] → Type u_2","l":41,"f":"Causalean/Experimentation/DesignBased/Optimality.lean"},{"n":"Causalean.Experimentation.DesignBased.Dominates","k":"def","m":"Causalean.Experimentation.DesignBased.Optimality","a":"Experimentation/DesignBased/Optimality","t":1,"d":"D₁ weakly dominates D₂ under the risk criterion R when it carries no larger risk: R D₁ ≤ R D₂.","s":"{Ω : Type u_1} → [inst : Fintype Ω] → {α : Type u_2} → [Preorder α] → (Causalean.Experimentation.DesignBased.FiniteDesign Ω → α) → Causalean.Experimentation.DesignBased.FiniteDesign Ω → Causalean.Experimentation.DesignBased.FiniteDesign Ω → Prop","l":45,"f":"Causalean/Experimentation/DesignBased/Optimality.lean"},{"n":"Causalean.Experimentation.DesignBased.Dominates.refl","k":"theorem","m":"Causalean.Experimentation.DesignBased.Optimality","a":"Experimentation/DesignBased/Optimality","t":2,"d":"Domination is reflexive: every design dominates itself.","s":"∀ {Ω : Type u_1} [inst : Fintype Ω] {α : Type u_2} [inst_1 : Preorder α] (R : Causalean.Experimentation.DesignBased.FiniteDesign Ω → α) (D : Causalean.Experimentation.DesignBased.FiniteDesign Ω), Causalean.Experimentation.DesignBased.Dominates R D D","l":50,"f":"Causalean/Experimentation/DesignBased/Optimality.lean"},{"n":"Causalean.Experimentation.DesignBased.Dominates.trans","k":"theorem","m":"Causalean.Experimentation.DesignBased.Optimality","a":"Experimentation/DesignBased/Optimality","t":2,"d":"Domination is transitive.","s":"∀ {Ω : Type u_1} [inst : Fintype Ω] {α : Type u_2} [inst_1 : Preorder α] {R : Causalean.Experimentation.DesignBased.FiniteDesign Ω → α} {D₁ D₂ D₃ : Causalean.Experimentation.DesignBased.FiniteDesign Ω}, Causalean.Experimentation.DesignBased.Dominates R D₁ D₂ → Causalean.Experimentation.DesignBased.Dominates R D₂ D₃ → Causalean.Experimentation.DesignBased.Dominates R D₁ D₃","l":55,"f":"Causalean/Experimentation/DesignBased/Optimality.lean"},{"n":"Causalean.Experimentation.DesignBased.E_controlMean","k":"theorem","m":"Causalean.Experimentation.DesignBased.Estimators.DifferenceInMeans","a":"Experimentation/DesignBased/Estimators","t":2,"d":"The control-arm mean is unbiased for the control population mean (1/N) ∑ Y0. Each unit enters the control arm with probability n₀ / N.","s":"∀ {U : Type u_1} [inst : Fintype U] [inst_1 : DecidableEq U] (n₁ : ℕ) (hn0 : n₁ < Fintype.card U) (Y0 : U → ℝ), (Causalean.Experimentation.DesignBased.completeRandomization n₁ (Nat.le_of_lt hn0)).E (Causalean.Experimentation.DesignBased.controlMean n₁ Y0) = (∑ i, Y0 i) / ↑(Fintype.card U)","l":111,"f":"Causalean/Experimentation/DesignBased/Estimators/DifferenceInMeans.lean"},{"n":"Causalean.Experimentation.DesignBased.E_diffInMeans_eq_sate","k":"theorem","m":"Causalean.Experimentation.DesignBased.Estimators.DifferenceInMeans","a":"Experimentation/DesignBased/Estimators","t":1,"d":"Unbiasedness of difference in means. Under complete randomization in which at least one unit is treated and the treated count n₁ is strictly below the population size N, so at least one unit remains a control, the difference-in-means estimator, computed from the potential outcomes Y1 and Y0, has expectation exactly equal to the sample average treatment effect.","s":"∀ {U : Type u_1} [inst : Fintype U] [inst_1 : DecidableEq U] (n₁ : ℕ), 0 < n₁ → ∀ (hn0 : n₁ < Fintype.card U) (Y1 Y0 : U → ℝ), (Causalean.Experimentation.DesignBased.completeRandomization n₁ (Nat.le_of_lt hn0)).E (Causalean.Experimentation.DesignBased.diffInMeans n₁ Y1 Y0) = Causalean.Experimentation.DesignBased.sateEstimand Y1 Y0","l":179,"f":"Causalean/Experimentation/DesignBased/Estimators/DifferenceInMeans.lean"},{"n":"Causalean.Experimentation.DesignBased.E_expoInd","k":"theorem","m":"Causalean.Experimentation.DesignBased.Exposure","a":"Experimentation/DesignBased","t":1,"d":"Exposure probability as an expectation. For a design, an exposure mapping, unit traits, a unit i, and an exposure level d, the design expectation of the exposure indicator equals the generalized probability of exposure π_i(d).","s":"∀ {Ω : Type u_1} [inst : Fintype Ω] {ι : Type u_2} {Θ : Type u_3} {Δ : Type u_4} [inst_1 : DecidableEq Δ] (D : Causalean.Experimentation.DesignBased.FiniteDesign Ω) (f : Ω → Θ → Δ) (θ : ι → Θ) (i : ι) (d : Δ), D.E (Causalean.Experimentation.DesignBased.expoInd f θ i d) = Causalean.Experimentation.DesignBased.prop D f θ i d","l":66,"f":"Causalean/Experimentation/DesignBased/Exposure.lean"},{"n":"Causalean.Experimentation.DesignBased.E_htEffect","k":"theorem","m":"Causalean.Experimentation.DesignBased.HT.Unbiased","a":"Experimentation/DesignBased/HT","t":1,"d":"Proposition 4.4 (expectation). Given a finite design, an outcome function y, an exposure map f, and an assignment θ, suppose every unit has nonzero probability of realizing exposure dk and every unit has nonzero probability of realizing exposure dl. Then the Horvitz–Thompson effect estimator contrasting dk against dl has expectation exactly the average causal effect τ(dk,dl).","s":"∀ {Ω : Type u_1} [inst : Fintype Ω] {ι : Type u_2} {Θ : Type u_3} {Δ : Type u_4} [inst_1 : Fintype ι] [inst_2 : DecidableEq Δ] (D : Causalean.Experimentation.DesignBased.FiniteDesign Ω) (y : ι → Δ → ℝ) (f : Ω → Θ → Δ) (θ : ι → Θ) (dk dl : Δ), (∀ (i : ι), Causalean.Experimentation.DesignBased.prop D f θ i dk ≠ 0) → (∀ (i : ι), Causalean.Experimentation.DesignBased.prop D f θ i dl ≠ 0) → D.E (Causalean.Experimentation.DesignBased.htEffect D y f θ dk dl) = Causalean.Experimentation.DesignBased.tauTrue y dk dl","l":65,"f":"Causalean/Experimentation/DesignBased/HT/Unbiased.lean"},{"n":"Causalean.Experimentation.DesignBased.E_htMean","k":"theorem","m":"Causalean.Experimentation.DesignBased.HT.Unbiased","a":"Experimentation/DesignBased/HT","t":1,"d":"Given a finite design, an outcome function y, an exposure map f, and an assignment θ, suppose every unit has nonzero probability of realizing exposure d. Then the Horvitz–Thompson mean estimator has expectation exactly the population mean μ(d).","s":"∀ {Ω : Type u_1} [inst : Fintype Ω] {ι : Type u_2} {Θ : Type u_3} {Δ : Type u_4} [inst_1 : Fintype ι] [inst_2 : DecidableEq Δ] (D : Causalean.Experimentation.DesignBased.FiniteDesign Ω) (y : ι → Δ → ℝ) (f : Ω → Θ → Δ) (θ : ι → Θ) (d : Δ), (∀ (i : ι), Causalean.Experimentation.DesignBased.prop D f θ i d ≠ 0) → D.E (Causalean.Experimentation.DesignBased.htMean D y f θ d) = Causalean.Experimentation.DesignBased.muTrue y d","l":54,"f":"Causalean/Experimentation/DesignBased/HT/Unbiased.lean"},{"n":"Causalean.Experimentation.DesignBased.E_htTotal","k":"theorem","m":"Causalean.Experimentation.DesignBased.HT.Unbiased","a":"Experimentation/DesignBased/HT","t":1,"d":"Lemma 4.1 (expectation). Given a finite design, an outcome function y, an exposure map f, and an assignment θ, suppose every unit has nonzero probability of realizing exposure d. Then the Horvitz–Thompson total estimator has expectation exactly the population total ∑ᵢ y i d.","s":"∀ {Ω : Type u_1} [inst : Fintype Ω] {ι : Type u_2} {Θ : Type u_3} {Δ : Type u_4} [inst_1 : Fintype ι] [inst_2 : DecidableEq Δ] (D : Causalean.Experimentation.DesignBased.FiniteDesign Ω) (y : ι → Δ → ℝ) (f : Ω → Θ → Δ) (θ : ι → Θ) (d : Δ), (∀ (i : ι), Causalean.Experimentation.DesignBased.prop D f θ i d ≠ 0) → D.E (Causalean.Experimentation.DesignBased.htTotal D y f θ d) = ∑ i, y i d","l":38,"f":"Causalean/Experimentation/DesignBased/HT/Unbiased.lean"},{"n":"Causalean.Experimentation.DesignBased.E_treatedMean","k":"theorem","m":"Causalean.Experimentation.DesignBased.Estimators.DifferenceInMeans","a":"Experimentation/DesignBased/Estimators","t":2,"d":"The treated-arm mean is unbiased for the treated population mean (1/N) ∑ Y1. Each unit enters the treated arm with probability n₁ / N, so its inverse-n₁ weight averages to 1/N.","s":"∀ {U : Type u_1} [inst : Fintype U] [inst_1 : DecidableEq U] (n₁ : ℕ) (hn : n₁ ≤ Fintype.card U), 0 < n₁ → ∀ (Y1 : U → ℝ), (Causalean.Experimentation.DesignBased.completeRandomization n₁ hn).E (Causalean.Experimentation.DesignBased.treatedMean n₁ Y1) = (∑ i, Y1 i) / ↑(Fintype.card U)","l":63,"f":"Causalean/Experimentation/DesignBased/Estimators/DifferenceInMeans.lean"},{"n":"Causalean.Experimentation.DesignBased.FiniteDesign","k":"structure","m":"Causalean.Experimentation.DesignBased.DesignCore","a":"Experimentation/DesignBased","t":1,"d":"A randomization design: a probability mass function p on a finite assignment space Ω whose values are nonnegative and sum to one.","s":"(Ω : Type u_1) → [Fintype Ω] → Type u_1","l":43,"f":"Causalean/Experimentation/DesignBased/DesignCore.lean"},{"n":"Causalean.Experimentation.DesignBased.FiniteDesign.BoundedInProb","k":"def","m":"Causalean.Experimentation.DesignBased.InProb","a":"Experimentation/DesignBased","t":1,"d":"Uniform tightness (bounded in probability). The statistics X m are uniformly tight if, for every tolerance η, there is a single threshold M past which the deviation probability Pr(M ≤ |X m|) is at most η for every index m. This is the O_p(1) counterpart of TendstoInProb, satisfied by a bounded (non-vanishing) factor of a delta-method remainder.","s":"{Ω : ℕ → Type u_1} → [inst : (m : ℕ) → Fintype (Ω m)] → ((m : ℕ) → Causalean.Experimentation.DesignBased.FiniteDesign (Ω m)) → ((m : ℕ) → Ω m → ℝ) → Prop","l":261,"f":"Causalean/Experimentation/DesignBased/InProb.lean"},{"n":"Causalean.Experimentation.DesignBased.FiniteDesign.Cov","k":"def","m":"Causalean.Experimentation.DesignBased.DesignCore","a":"Experimentation/DesignBased","t":1,"d":"Covariance of X and Y under the design.","s":"{Ω : Type u_1} → [inst : Fintype Ω] → Causalean.Experimentation.DesignBased.FiniteDesign Ω → (Ω → ℝ) → (Ω → ℝ) → ℝ","l":63,"f":"Causalean/Experimentation/DesignBased/DesignCore.lean"},{"n":"Causalean.Experimentation.DesignBased.FiniteDesign.Cov_comm","k":"theorem","m":"Causalean.Experimentation.DesignBased.DesignCore","a":"Experimentation/DesignBased","t":2,"d":"Covariance is symmetric in its two arguments.","s":"∀ {Ω : Type u_1} [inst : Fintype Ω] (D : Causalean.Experimentation.DesignBased.FiniteDesign Ω) (X Y : Ω → ℝ), D.Cov X Y = D.Cov Y X","l":149,"f":"Causalean/Experimentation/DesignBased/DesignCore.lean"},{"n":"Causalean.Experimentation.DesignBased.FiniteDesign.Cov_congr","k":"theorem","m":"Causalean.Experimentation.DesignBased.DesignCore","a":"Experimentation/DesignBased","t":2,"d":"Congruence: pointwise-equal random variables have equal covariance.","s":"∀ {Ω : Type u_1} [inst : Fintype Ω] (D : Causalean.Experimentation.DesignBased.FiniteDesign Ω) {X Y X' Y' : Ω → ℝ}, (∀ (z : Ω), X z = X' z) → (∀ (z : Ω), Y z = Y' z) → D.Cov X Y = D.Cov X' Y'","l":118,"f":"Causalean/Experimentation/DesignBased/DesignCore.lean"},{"n":"Causalean.Experimentation.DesignBased.FiniteDesign.Cov_const_mul_left","k":"theorem","m":"Causalean.Experimentation.DesignBased.DesignCore","a":"Experimentation/DesignBased","t":2,"d":"Multiplying the left random variable by a constant multiplies covariance by that constant.","s":"∀ {Ω : Type u_1} [inst : Fintype Ω] (D : Causalean.Experimentation.DesignBased.FiniteDesign Ω) (c : ℝ) (X Y : Ω → ℝ), D.Cov (fun z => c * X z) Y = c * D.Cov X Y","l":155,"f":"Causalean/Experimentation/DesignBased/DesignCore.lean"},{"n":"Causalean.Experimentation.DesignBased.FiniteDesign.Cov_const_mul_right","k":"theorem","m":"Causalean.Experimentation.DesignBased.DesignCore","a":"Experimentation/DesignBased","t":2,"d":"Multiplying the right random variable by a constant multiplies covariance by that constant.","s":"∀ {Ω : Type u_1} [inst : Fintype Ω] (D : Causalean.Experimentation.DesignBased.FiniteDesign Ω) (c : ℝ) (X Y : Ω → ℝ), (D.Cov X fun z => c * Y z) = c * D.Cov X Y","l":178,"f":"Causalean/Experimentation/DesignBased/DesignCore.lean"},{"n":"Causalean.Experimentation.DesignBased.FiniteDesign.Cov_eq","k":"theorem","m":"Causalean.Experimentation.DesignBased.DesignCore","a":"Experimentation/DesignBased","t":1,"d":"Covariance identity. For any two statistics X and Y under the design, their design covariance equals the design expectation of X·Y minus the product of their design expectations.","s":"∀ {Ω : Type u_1} [inst : Fintype Ω] (D : Causalean.Experimentation.DesignBased.FiniteDesign Ω) (X Y : Ω → ℝ), D.Cov X Y = (D.E fun z => X z * Y z) - D.E X * D.E Y","l":135,"f":"Causalean/Experimentation/DesignBased/DesignCore.lean"},{"n":"Causalean.Experimentation.DesignBased.FiniteDesign.Cov_eq_covariance","k":"theorem","m":"Causalean.Experimentation.DesignBased.MeasureBridge","a":"Experimentation/DesignBased","t":2,"d":"Reverse rewrite. The design covariance is the measure-theoretic covariance under the design measure.","s":"∀ {Ω : Type u_1} [inst : Fintype Ω] [inst_1 : MeasurableSpace Ω] [MeasurableSingletonClass Ω] (D : Causalean.Experimentation.DesignBased.FiniteDesign Ω) (X Y : Ω → ℝ), D.Cov X Y = ProbabilityTheory.covariance X Y D.toMeasure","l":83,"f":"Causalean/Experimentation/DesignBased/MeasureBridge.lean"},{"n":"Causalean.Experimentation.DesignBased.FiniteDesign.Cov_linear_comb","k":"theorem","m":"Causalean.Experimentation.DesignBased.DesignCore","a":"Experimentation/DesignBased","t":1,"d":"Bilinear expansion of covariance for linear combinations. For finite weighted sums ∑ᵢ∈s cᵢXᵢ and ∑ⱼ∈t eⱼYⱼ, built from index sets s ⊆ ι, t ⊆ κ, weights c, e, and random-variable families X, Y, their design covariance equals the double sum over s×t of the weighted covariances cᵢeⱼ·Cov(Xᵢ,Yⱼ).","s":"∀ {Ω : Type u_1} [inst : Fintype Ω] (D : Causalean.Experimentation.DesignBased.FiniteDesign Ω) {ι : Type u_2} {κ : Type u_3} (s : Finset ι) (t : Finset κ) (c : ι → ℝ) (e : κ → ℝ) (X : ι → Ω → ℝ) (Y : κ → Ω → ℝ), (D.Cov (fun z => ∑ i ∈ s, c i * X i z) fun z => ∑ j ∈ t, e j * Y j z) = ∑ i ∈ s, ∑ j ∈ t, c i * e j * D.Cov (X i) (Y j)","l":207,"f":"Causalean/Experimentation/DesignBased/DesignCore.lean"},{"n":"Causalean.Experimentation.DesignBased.FiniteDesign.Cov_prod_apply_of_ne","k":"theorem","m":"Causalean.Experimentation.DesignBased.ProductVariance","a":"Experimentation/DesignBased","t":2,"d":"Cross-coordinate independence: under the product design, functions of two distinct coordinates have zero covariance.","s":"∀ {ι : Type u_1} [inst : Fintype ι] [inst_1 : DecidableEq ι] {α : ι → Type u_2} [inst_2 : (i : ι) → Fintype (α i)] (D : (i : ι) → Causalean.Experimentation.DesignBased.FiniteDesign (α i)) {i j : ι}, i ≠ j → ∀ (g : α i → ℝ) (hfun : α j → ℝ), ((Causalean.Experimentation.DesignBased.prodDesign D).Cov (fun w => g (w i)) fun w => hfun (w j)) = 0","l":75,"f":"Causalean/Experimentation/DesignBased/ProductVariance.lean"},{"n":"Causalean.Experimentation.DesignBased.FiniteDesign.Cov_prod_block_zero","k":"theorem","m":"Causalean.Experimentation.DesignBased.ProductBlock","a":"Experimentation/DesignBased","t":1,"d":"Disjoint-block covariance vanishing (complement form). For a family of independent coordinate designs D and real-valued functions f and g, suppose f depends only on the coordinates in a block A and g depends only on the coordinates outside A. Then their covariance under the product design is zero.","s":"∀ {ι : Type u_1} [inst : Fintype ι] [inst_1 : DecidableEq ι] {α : ι → Type u_2} [inst_2 : (i : ι) → Fintype (α i)] (D : (i : ι) → Causalean.Experimentation.DesignBased.FiniteDesign (α i)) (A : Finset ι) (f g : ((i : ι) → α i) → ℝ), (∀ (w w' : (i : ι) → α i), (∀ i ∈ A, w i = w' i) → f w = f w') → (∀ (w w' : (i : ι) → α i), (∀ i ∉ A, w i = w' i) → g w = g w') → (Causalean.Experimentation.DesignBased.prodDesign D).Cov f g = 0","l":106,"f":"Causalean/Experimentation/DesignBased/ProductBlock.lean"},{"n":"Causalean.Experimentation.DesignBased.FiniteDesign.Cov_prod_disjoint_zero","k":"theorem","m":"Causalean.Experimentation.DesignBased.ProductBlock","a":"Experimentation/DesignBased","t":1,"d":"Disjoint-block covariance vanishing (two-block form). For a family of independent coordinate designs D and real-valued functions f and g, suppose the index sets S and T are disjoint, f depends only on the coordinates in S, and g depends only on the coordinates in T. Then their covariance under the product design is zero.","s":"∀ {ι : Type u_1} [inst : Fintype ι] [inst_1 : DecidableEq ι] {α : ι → Type u_2} [inst_2 : (i : ι) → Fintype (α i)] (D : (i : ι) → Causalean.Experimentation.DesignBased.FiniteDesign (α i)) (S T : Finset ι), Disjoint S T → ∀ (f g : ((i : ι) → α i) → ℝ), (∀ (w w' : (i : ι) → α i), (∀ i ∈ S, w i = w' i) → f w = f w') → (∀ (w w' : (i : ι) → α i), (∀ i ∈ T, w i = w' i) → g w = g w') → (Causalean.Experimentation.DesignBased.prodDesign D).Cov f g = 0","l":117,"f":"Causalean/Experimentation/DesignBased/ProductBlock.lean"},{"n":"Causalean.Experimentation.DesignBased.FiniteDesign.Cov_self","k":"theorem","m":"Causalean.Experimentation.DesignBased.DesignCore","a":"Experimentation/DesignBased","t":2,"d":"The covariance of a random variable with itself is its variance.","s":"∀ {Ω : Type u_1} [inst : Fintype Ω] (D : Causalean.Experimentation.DesignBased.FiniteDesign Ω) (X : Ω → ℝ), D.Cov X X = D.Var X","l":145,"f":"Causalean/Experimentation/DesignBased/DesignCore.lean"},{"n":"Causalean.Experimentation.DesignBased.FiniteDesign.Cov_sub_left","k":"theorem","m":"Causalean.Experimentation.DesignBased.DesignCore","a":"Experimentation/DesignBased","t":2,"d":"Covariance is linear over differences in its left argument.","s":"∀ {Ω : Type u_1} [inst : Fintype Ω] (D : Causalean.Experimentation.DesignBased.FiniteDesign Ω) (X Y Z : Ω → ℝ), D.Cov (fun z => X z - Y z) Z = D.Cov X Z - D.Cov Y Z","l":183,"f":"Causalean/Experimentation/DesignBased/DesignCore.lean"},{"n":"Causalean.Experimentation.DesignBased.FiniteDesign.Cov_sub_right","k":"theorem","m":"Causalean.Experimentation.DesignBased.DesignCore","a":"Experimentation/DesignBased","t":2,"d":"Covariance is linear over differences in its right argument.","s":"∀ {Ω : Type u_1} [inst : Fintype Ω] (D : Causalean.Experimentation.DesignBased.FiniteDesign Ω) (X Y Z : Ω → ℝ), (D.Cov X fun z => Y z - Z z) = D.Cov X Y - D.Cov X Z","l":191,"f":"Causalean/Experimentation/DesignBased/DesignCore.lean"},{"n":"Causalean.Experimentation.DesignBased.FiniteDesign.Cov_sum_left","k":"theorem","m":"Causalean.Experimentation.DesignBased.DesignCore","a":"Experimentation/DesignBased","t":2,"d":"Covariance with a finite sum in the left argument is the finite sum of covariances.","s":"∀ {Ω : Type u_1} [inst : Fintype Ω] (D : Causalean.Experimentation.DesignBased.FiniteDesign Ω) {ι : Type u_2} (s : Finset ι) (f : ι → Ω → ℝ) (Y : Ω → ℝ), D.Cov (fun z => ∑ i ∈ s, f i z) Y = ∑ i ∈ s, D.Cov (f i) Y","l":163,"f":"Causalean/Experimentation/DesignBased/DesignCore.lean"},{"n":"Causalean.Experimentation.DesignBased.FiniteDesign.Cov_sum_right","k":"theorem","m":"Causalean.Experimentation.DesignBased.DesignCore","a":"Experimentation/DesignBased","t":2,"d":"Covariance with a finite sum in the right argument is the finite sum of covariances.","s":"∀ {Ω : Type u_1} [inst : Fintype Ω] (D : Causalean.Experimentation.DesignBased.FiniteDesign Ω) {ι : Type u_2} (s : Finset ι) (X : Ω → ℝ) (g : ι → Ω → ℝ), (D.Cov X fun z => ∑ i ∈ s, g i z) = ∑ i ∈ s, D.Cov X (g i)","l":172,"f":"Causalean/Experimentation/DesignBased/DesignCore.lean"},{"n":"Causalean.Experimentation.DesignBased.FiniteDesign.Cov_zero_left","k":"theorem","m":"Causalean.Experimentation.DesignBased.EdgeVarianceBound","a":"Experimentation/DesignBased","t":2,"d":"The covariance of the identically-zero random variable with anything is zero.","s":"∀ {Ω : Type u_1} [inst : Fintype Ω] (D : Causalean.Experimentation.DesignBased.FiniteDesign Ω) (Y : Ω → ℝ), D.Cov (fun x => 0) Y = 0","l":83,"f":"Causalean/Experimentation/DesignBased/EdgeVarianceBound.lean"},{"n":"Causalean.Experimentation.DesignBased.FiniteDesign.E","k":"def","m":"Causalean.Experimentation.DesignBased.DesignCore","a":"Experimentation/DesignBased","t":1,"d":"Expectation of a random variable X : Ω → ℝ under the design.","s":"{Ω : Type u_1} → [inst : Fintype Ω] → Causalean.Experimentation.DesignBased.FiniteDesign Ω → (Ω → ℝ) → ℝ","l":57,"f":"Causalean/Experimentation/DesignBased/DesignCore.lean"},{"n":"Causalean.Experimentation.DesignBased.FiniteDesign.E_add","k":"theorem","m":"Causalean.Experimentation.DesignBased.DesignCore","a":"Experimentation/DesignBased","t":2,"d":"The expectation of a sum is the sum of expectations.","s":"∀ {Ω : Type u_1} [inst : Fintype Ω] (D : Causalean.Experimentation.DesignBased.FiniteDesign Ω) (X Y : Ω → ℝ), (D.E fun z => X z + Y z) = D.E X + D.E Y","l":78,"f":"Causalean/Experimentation/DesignBased/DesignCore.lean"},{"n":"Causalean.Experimentation.DesignBased.FiniteDesign.E_centered_ratio","k":"theorem","m":"Causalean.Experimentation.DesignBased.RatioLinearization","a":"Experimentation/DesignBased","t":2,"d":"Centering of a plug-in ratio. In a finite design, the mean-normalized ratio X / E[X] − 1 has design mean zero whenever the mean E[X] is nonzero — the first-order (linearization) fact that a ratio statistic is exactly centered at its plug-in point.","s":"∀ {Ω : Type u_1} [inst : Fintype Ω] (D : Causalean.Experimentation.DesignBased.FiniteDesign Ω) (X : Ω → ℝ) (a : ℝ), a ≠ 0 → D.E X = a → (D.E fun z => X z / a - 1) = 0","l":74,"f":"Causalean/Experimentation/DesignBased/RatioLinearization.lean"},{"n":"Causalean.Experimentation.DesignBased.FiniteDesign.E_centered_ratio_mul","k":"theorem","m":"Causalean.Experimentation.DesignBased.RatioLinearization","a":"Experimentation/DesignBased","t":2,"d":"Covariance of two normalized ratios (the ratio linearization kernel). In a finite design, the mean of the product of two mean-normalized ratios (X/E[X] − 1)(Y/E[Y] − 1) equals E[XY] / (E[X]·E[Y]) − 1. This is the exact second cross-moment at the heart of every Horvitz–Thompson / Hájek ratio-variance linearization.","s":"∀ {Ω : Type u_1} [inst : Fintype Ω] (D : Causalean.Experimentation.DesignBased.FiniteDesign Ω) (X Y : Ω → ℝ) (a b c : ℝ), a ≠ 0 → b ≠ 0 → D.E X = a → D.E Y = b → (D.E fun z => X z * Y z) = c → (D.E fun z => (X z / a - 1) * (Y z / b - 1)) = c / (a * b) - 1","l":87,"f":"Causalean/Experimentation/DesignBased/RatioLinearization.lean"},{"n":"Causalean.Experimentation.DesignBased.FiniteDesign.E_compound","k":"theorem","m":"Causalean.Experimentation.DesignBased.TwoStage","a":"Experimentation/DesignBased","t":1,"d":"For the stage-2 design map assigning each stage-1 outcome s a per-coordinate design D₂ s and any real-valued function X of the joint outcome, built on top of a stage-1 design D₁, the expectation of X under the compound (two-stage) design equals the double sum over stage-1 outcomes s and stage-2 profiles w of the compound probability D₁.p(s)·∏ᵢ(D₂ s i).p(wᵢ) times X(s,w).","s":"∀ {Ω₁ : Type u_1} [inst : Fintype Ω₁] {ι : Type u_2} [inst_1 : Fintype ι] [inst_2 : DecidableEq ι] {α : ι → Type u_3} [inst_3 : (i : ι) → Fintype (α i)] (D₁ : Causalean.Experimentation.DesignBased.FiniteDesign Ω₁) (D₂ : Ω₁ → (i : ι) → Causalean.Experimentation.DesignBased.FiniteDesign (α i)) (X : Ω₁ × ((i : ι) → α i) → ℝ), (Causalean.Experimentation.DesignBased.compound D₁ D₂).E X = ∑ s, ∑ w, (D₁.p s * ∏ i, (D₂ s i).p (w i)) * X (s, w)","l":69,"f":"Causalean/Experimentation/DesignBased/TwoStage.lean"},{"n":"Causalean.Experimentation.DesignBased.FiniteDesign.E_compound_factor","k":"theorem","m":"Causalean.Experimentation.DesignBased.TwoStage","a":"Experimentation/DesignBased","t":1,"d":"Stage-2 collapse. For the stage-2 design map assigning each stage-1 outcome s a per-coordinate design D₂ s, the compound-design expectation of the product of a stage-1 quantity h(s) and a function g of a single group j's within-assignment wⱼ factors through the marginal expectation of group j's conditional design: E[h(s)·g(wⱼ)] = E_s[h(s)·E_{D₂ s j}[g]].","s":"∀ {Ω₁ : Type u_1} [inst : Fintype Ω₁] {ι : Type u_2} [inst_1 : Fintype ι] [inst_2 : DecidableEq ι] {α : ι → Type u_3} [inst_3 : (i : ι) → Fintype (α i)] (D₁ : Causalean.Experimentation.DesignBased.FiniteDesign Ω₁) (D₂ : Ω₁ → (i : ι) → Causalean.Experimentation.DesignBased.FiniteDesign (α i)) (h : Ω₁ → ℝ) (j : ι) (g : α j → ℝ), ((Causalean.Experimentation.DesignBased.compound D₁ D₂).E fun sw => h sw.1 * g (sw.2 j)) = D₁.E fun s => h s * (D₂ s j).E g","l":81,"f":"Causalean/Experimentation/DesignBased/TwoStage.lean"},{"n":"Causalean.Experimentation.DesignBased.FiniteDesign.E_compound_tower","k":"theorem","m":"Causalean.Experimentation.DesignBased.CompoundVariance","a":"Experimentation/DesignBased","t":2,"d":"Tower property of expectation for the compound design. The compound expectation iterates as the stage-1 expectation of the stage-2 product-design conditional expectation, E[F] = E_s[E_{D₂ s}[F(s, ·)]].","s":"∀ {Ω₁ : Type u_1} [inst : Fintype Ω₁] {ι : Type u_2} [inst_1 : Fintype ι] [inst_2 : DecidableEq ι] {α : ι → Type u_3} [inst_3 : (i : ι) → Fintype (α i)] (D₁ : Causalean.Experimentation.DesignBased.FiniteDesign Ω₁) (D₂ : Ω₁ → (i : ι) → Causalean.Experimentation.DesignBased.FiniteDesign (α i)) (F : Ω₁ × ((i : ι) → α i) → ℝ), (Causalean.Experimentation.DesignBased.compound D₁ D₂).E F = D₁.E fun s => (Causalean.Experimentation.DesignBased.prodDesign (D₂ s)).E fun w => F (s, w)","l":48,"f":"Causalean/Experimentation/DesignBased/CompoundVariance.lean"},{"n":"Causalean.Experimentation.DesignBased.FiniteDesign.E_congr","k":"theorem","m":"Causalean.Experimentation.DesignBased.DesignCore","a":"Experimentation/DesignBased","t":2,"d":"Congruence: pointwise-equal random variables have equal expectation.","s":"∀ {Ω : Type u_1} [inst : Fintype Ω] (D : Causalean.Experimentation.DesignBased.FiniteDesign Ω) {X Y : Ω → ℝ}, (∀ (z : Ω), X z = Y z) → D.E X = D.E Y","l":109,"f":"Causalean/Experimentation/DesignBased/DesignCore.lean"},{"n":"Causalean.Experimentation.DesignBased.FiniteDesign.E_const","k":"theorem","m":"Causalean.Experimentation.DesignBased.DesignCore","a":"Experimentation/DesignBased","t":2,"d":"The expectation of a constant random variable is that constant.","s":"∀ {Ω : Type u_1} [inst : Fintype Ω] (D : Causalean.Experimentation.DesignBased.FiniteDesign Ω) (c : ℝ), (D.E fun x => c) = c","l":74,"f":"Causalean/Experimentation/DesignBased/DesignCore.lean"},{"n":"Causalean.Experimentation.DesignBased.FiniteDesign.E_const_mul","k":"theorem","m":"Causalean.Experimentation.DesignBased.DesignCore","a":"Experimentation/DesignBased","t":2,"d":"Multiplying a random variable by a constant on the left multiplies its expectation by that constant.","s":"∀ {Ω : Type u_1} [inst : Fintype Ω] (D : Causalean.Experimentation.DesignBased.FiniteDesign Ω) (c : ℝ) (X : Ω → ℝ), (D.E fun z => c * X z) = c * D.E X","l":86,"f":"Causalean/Experimentation/DesignBased/DesignCore.lean"},{"n":"Causalean.Experimentation.DesignBased.FiniteDesign.E_eq_integral","k":"theorem","m":"Causalean.Experimentation.DesignBased.MeasureBridge","a":"Experimentation/DesignBased","t":2,"d":"Reverse rewrite. The design expectation is the integral against the design measure.","s":"∀ {Ω : Type u_1} [inst : Fintype Ω] [inst_1 : MeasurableSpace Ω] [MeasurableSingletonClass Ω] (D : Causalean.Experimentation.DesignBased.FiniteDesign Ω) (g : Ω → ℝ), D.E g = ∫ (x : Ω), g x ∂D.toMeasure","l":68,"f":"Causalean/Experimentation/DesignBased/MeasureBridge.lean"},{"n":"Causalean.Experimentation.DesignBased.FiniteDesign.E_ind","k":"theorem","m":"Causalean.Experimentation.DesignBased.DesignCore","a":"Experimentation/DesignBased","t":1,"d":"Expectation of an indicator. For any event A, the design expectation of its indicator equals the design probability of A.","s":"∀ {Ω : Type u_1} [inst : Fintype Ω] (D : Causalean.Experimentation.DesignBased.FiniteDesign Ω) (A : Ω → Prop) [inst_1 : DecidablePred A], D.E (Causalean.Experimentation.DesignBased.FiniteDesign.ind A) = D.Pr A","l":232,"f":"Causalean/Experimentation/DesignBased/DesignCore.lean"},{"n":"Causalean.Experimentation.DesignBased.FiniteDesign.E_le_one","k":"theorem","m":"Causalean.Experimentation.DesignBased.DesignCore","a":"Experimentation/DesignBased","t":2,"d":"Expectation of a random variable bounded above by one is at most one.","s":"∀ {Ω : Type u_1} [inst : Fintype Ω] (D : Causalean.Experimentation.DesignBased.FiniteDesign Ω) {X : Ω → ℝ}, (∀ (z : Ω), X z ≤ 1) → D.E X ≤ 1","l":254,"f":"Causalean/Experimentation/DesignBased/DesignCore.lean"},{"n":"Causalean.Experimentation.DesignBased.FiniteDesign.E_lin_expand","k":"theorem","m":"Causalean.Experimentation.DesignBased.RatioLinearization","a":"Experimentation/DesignBased","t":1,"d":"Bilinear expansion of a design mean. In a finite design, suppose the mean of A·C equals AA, the mean of A·F equals AB, the mean of B·C equals BA, and the mean of B·F equals BB — the four pairwise cross-moments of the two-term linear forms (A·ai − B·bi) and (C·aj − F·bj). Then the mean of the product (A·ai − B·bi)(C·aj − F·bj) equals AA·ai·aj − AB·ai·bj − BA·bi·aj + BB·bi·bj. This is the generic bookkeeping step assembling a two-arm (e.g. treated/control) linearized variance from its component moments.","s":"∀ {Ω : Type u_1} [inst : Fintype Ω] (D : Causalean.Experimentation.DesignBased.FiniteDesign Ω) (A B C F : Ω → ℝ) (ai bi aj bj AA AB BA BB : ℝ), (D.E fun z => A z * C z) = AA → (D.E fun z => A z * F z) = AB → (D.E fun z => B z * C z) = BA → (D.E fun z => B z * F z) = BB → (D.E fun z => (A z * ai - B z * bi) * (C z * aj - F z * bj)) = AA * ai * aj - AB * ai * bj - BA * bi * aj + BB * bi * bj","l":107,"f":"Causalean/Experimentation/DesignBased/RatioLinearization.lean"},{"n":"Causalean.Experimentation.DesignBased.FiniteDesign.E_map","k":"theorem","m":"Causalean.Experimentation.DesignBased.DesignCore","a":"Experimentation/DesignBased","t":2,"d":"Transfer of expectation across a pushforward. The expectation of g under the pushforward f_* D equals the expectation of the composite g ∘ f under D.","s":"∀ {Ω : Type u_1} [inst : Fintype Ω] (D : Causalean.Experimentation.DesignBased.FiniteDesign Ω) {Ω' : Type u_2} [inst_1 : Fintype Ω'] (f : Ω → Ω') (g : Ω' → ℝ), (D.map f).E g = D.E fun z => g (f z)","l":327,"f":"Causalean/Experimentation/DesignBased/DesignCore.lean"},{"n":"Causalean.Experimentation.DesignBased.FiniteDesign.E_mul_const","k":"theorem","m":"Causalean.Experimentation.DesignBased.DesignCore","a":"Experimentation/DesignBased","t":2,"d":"Multiplying a random variable by a constant on the right multiplies its expectation by that constant.","s":"∀ {Ω : Type u_1} [inst : Fintype Ω] (D : Causalean.Experimentation.DesignBased.FiniteDesign Ω) (X : Ω → ℝ) (c : ℝ), (D.E fun z => X z * c) = D.E X * c","l":91,"f":"Causalean/Experimentation/DesignBased/DesignCore.lean"},{"n":"Causalean.Experimentation.DesignBased.FiniteDesign.E_neg","k":"theorem","m":"Causalean.Experimentation.DesignBased.DesignCore","a":"Experimentation/DesignBased","t":2,"d":"The expectation of a negated random variable is the negated expectation.","s":"∀ {Ω : Type u_1} [inst : Fintype Ω] (D : Causalean.Experimentation.DesignBased.FiniteDesign Ω) (X : Ω → ℝ), (D.E fun z => -X z) = -D.E X","l":96,"f":"Causalean/Experimentation/DesignBased/DesignCore.lean"},{"n":"Causalean.Experimentation.DesignBased.FiniteDesign.E_nonneg","k":"theorem","m":"Causalean.Experimentation.DesignBased.DesignCore","a":"Experimentation/DesignBased","t":2,"d":"Expectation of a nonnegative random variable is nonnegative.","s":"∀ {Ω : Type u_1} [inst : Fintype Ω] (D : Causalean.Experimentation.DesignBased.FiniteDesign Ω) {X : Ω → ℝ}, (∀ (z : Ω), 0 ≤ X z) → 0 ≤ D.E X","l":250,"f":"Causalean/Experimentation/DesignBased/DesignCore.lean"},{"n":"Causalean.Experimentation.DesignBased.FiniteDesign.E_prod_apply","k":"theorem","m":"Causalean.Experimentation.DesignBased.Product","a":"Experimentation/DesignBased","t":1,"d":"For a family of independent coordinate designs D, the expectation under the product design of a function of a single coordinate j equals the expectation of that same function under coordinate j's own marginal design.","s":"∀ {ι : Type u_1} [inst : Fintype ι] [inst_1 : DecidableEq ι] {α : ι → Type u_2} [inst_2 : (i : ι) → Fintype (α i)] (D : (i : ι) → Causalean.Experimentation.DesignBased.FiniteDesign (α i)) (j : ι) (g : α j → ℝ), ((Causalean.Experimentation.DesignBased.prodDesign D).E fun w => g (w j)) = (D j).E g","l":74,"f":"Causalean/Experimentation/DesignBased/Product.lean"},{"n":"Causalean.Experimentation.DesignBased.FiniteDesign.E_prod_apply₂","k":"theorem","m":"Causalean.Experimentation.DesignBased.ProductVariance","a":"Experimentation/DesignBased","t":2,"d":"Two-coordinate factorization: under the product design, the expectation of a product of a function of one coordinate and a function of a distinct coordinate factors into the product of the two marginal expectations.","s":"∀ {ι : Type u_1} [inst : Fintype ι] [inst_1 : DecidableEq ι] {α : ι → Type u_2} [inst_2 : (i : ι) → Fintype (α i)] (D : (i : ι) → Causalean.Experimentation.DesignBased.FiniteDesign (α i)) {i j : ι}, i ≠ j → ∀ (g : α i → ℝ) (hfun : α j → ℝ), ((Causalean.Experimentation.DesignBased.prodDesign D).E fun w => g (w i) * hfun (w j)) = (D i).E g * (D j).E hfun","l":49,"f":"Causalean/Experimentation/DesignBased/ProductVariance.lean"},{"n":"Causalean.Experimentation.DesignBased.FiniteDesign.E_prod_block_mul","k":"theorem","m":"Causalean.Experimentation.DesignBased.ProductBlock","a":"Experimentation/DesignBased","t":1,"d":"Disjoint-block factorization of expectation. For a family of independent coordinate designs D and real-valued functions f and g on the joint assignment space, suppose f is invariant under any change to the coordinates outside a block A — it depends only on the coordinates in A, and g depends only on the coordinates outside A. Then under the product design, the expectation of the product f·g factors as E[f·g] = E[f]·E[g]. The general (block-level) form of E_prod_apply₂.","s":"∀ {ι : Type u_1} [inst : Fintype ι] [inst_1 : DecidableEq ι] {α : ι → Type u_2} [inst_2 : (i : ι) → Fintype (α i)] (D : (i : ι) → Causalean.Experimentation.DesignBased.FiniteDesign (α i)) (A : Finset ι) (f g : ((i : ι) → α i) → ℝ), (∀ (w w' : (i : ι) → α i), (∀ i ∈ A, w i = w' i) → f w = f w') → (∀ (w w' : (i : ι) → α i), (∀ i ∉ A, w i = w' i) → g w = g w') → ((Causalean.Experimentation.DesignBased.prodDesign D).E fun w => f w * g w) = (Causalean.Experimentation.DesignBased.prodDesign D).E f * (Causalean.Experimentation.DesignBased.prodDesign D).E g","l":59,"f":"Causalean/Experimentation/DesignBased/ProductBlock.lean"},{"n":"Causalean.Experimentation.DesignBased.FiniteDesign.E_prod_prod","k":"theorem","m":"Causalean.Experimentation.DesignBased.Product","a":"Experimentation/DesignBased","t":1,"d":"For a family of independent coordinate designs D and single-coordinate functions g, the expectation under the product design of the product ∏ᵢ g i (w i) factors into the product of the individual coordinate expectations ∏ᵢ (D i).E (g i).","s":"∀ {ι : Type u_1} [inst : Fintype ι] [inst_1 : DecidableEq ι] {α : ι → Type u_2} [inst_2 : (i : ι) → Fintype (α i)] (D : (i : ι) → Causalean.Experimentation.DesignBased.FiniteDesign (α i)) (g : (i : ι) → α i → ℝ), ((Causalean.Experimentation.DesignBased.prodDesign D).E fun w => ∏ i, g i (w i)) = ∏ i, (D i).E (g i)","l":63,"f":"Causalean/Experimentation/DesignBased/Product.lean"},{"n":"Causalean.Experimentation.DesignBased.FiniteDesign.E_sub","k":"theorem","m":"Causalean.Experimentation.DesignBased.DesignCore","a":"Experimentation/DesignBased","t":2,"d":"The expectation of a difference is the difference of expectations.","s":"∀ {Ω : Type u_1} [inst : Fintype Ω] (D : Causalean.Experimentation.DesignBased.FiniteDesign Ω) (X Y : Ω → ℝ), (D.E fun z => X z - Y z) = D.E X - D.E Y","l":82,"f":"Causalean/Experimentation/DesignBased/DesignCore.lean"},{"n":"Causalean.Experimentation.DesignBased.FiniteDesign.E_sum","k":"theorem","m":"Causalean.Experimentation.DesignBased.DesignCore","a":"Experimentation/DesignBased","t":1,"d":"Linearity of expectation over a finite sum. For a finite index set s over an index type ι and a family of random variables f, the design expectation of the finite sum ∑ᵢ∈s fᵢ equals the finite sum of the design expectations ∑ᵢ∈s E[fᵢ].","s":"∀ {Ω : Type u_1} [inst : Fintype Ω] (D : Causalean.Experimentation.DesignBased.FiniteDesign Ω) {ι : Type u_2} (s : Finset ι) (f : ι → Ω → ℝ), (D.E fun z => ∑ i ∈ s, f i z) = ∑ i ∈ s, D.E (f i)","l":100,"f":"Causalean/Experimentation/DesignBased/DesignCore.lean"},{"n":"Causalean.Experimentation.DesignBased.FiniteDesign.IsConservativeVarEst","k":"def","m":"Causalean.Experimentation.DesignBased.Variance.Conservative","a":"Experimentation/DesignBased/Variance","t":1,"d":"A variance estimator Vhat is conservative for X when its expectation is at least the randomization variance of X.","s":"{Ω : Type u_1} → [inst : Fintype Ω] → Causalean.Experimentation.DesignBased.FiniteDesign Ω → (Ω → ℝ) → (Ω → ℝ) → Prop","l":41,"f":"Causalean/Experimentation/DesignBased/Variance/Conservative.lean"},{"n":"Causalean.Experimentation.DesignBased.FiniteDesign.Pr","k":"def","m":"Causalean.Experimentation.DesignBased.DesignCore","a":"Experimentation/DesignBased","t":1,"d":"Probability of an event A under the design.","s":"{Ω : Type u_1} → [inst : Fintype Ω] → Causalean.Experimentation.DesignBased.FiniteDesign Ω → (A : Ω → Prop) → [DecidablePred A] → ℝ","l":69,"f":"Causalean/Experimentation/DesignBased/DesignCore.lean"},{"n":"Causalean.Experimentation.DesignBased.FiniteDesign.Pr_compound_eq_E_condPr","k":"theorem","m":"Causalean.Experimentation.TwoStageInterference.Asymptotic.CLT","a":"Experimentation/TwoStageInterference/Asymptotic","t":2,"d":"Tower bridge for probabilities. Under the two-stage compound design, the unconditional probability of an event P equals the stage-1 expectation of its stage-2 conditional probability:","s":"∀ {Ω₁ : Type u_1} {ι : Type u_2} [inst : Fintype Ω₁] [inst_1 : Fintype ι] [inst_2 : DecidableEq ι] {α : ι → Type u_3} [inst_3 : (i : ι) → Fintype (α i)] (D₁ : Causalean.Experimentation.DesignBased.FiniteDesign Ω₁) (D₂ : Ω₁ → (i : ι) → Causalean.Experimentation.DesignBased.FiniteDesign (α i)) (P : Ω₁ × ((i : ι) → α i) → Prop) [inst_4 : DecidablePred P], (Causalean.Experimentation.DesignBased.compound D₁ D₂).Pr P = D₁.E fun s => (Causalean.Experimentation.DesignBased.prodDesign (D₂ s)).Pr fun w => P (s, w)","l":85,"f":"Causalean/Experimentation/TwoStageInterference/Asymptotic/CLT.lean"},{"n":"Causalean.Experimentation.DesignBased.FiniteDesign.Pr_congr","k":"theorem","m":"Causalean.Experimentation.DesignBased.DesignCore","a":"Experimentation/DesignBased","t":2,"d":"Probability respects pointwise-equivalent events.","s":"∀ {Ω : Type u_1} [inst : Fintype Ω] (D : Causalean.Experimentation.DesignBased.FiniteDesign Ω) (A B : Ω → Prop) [inst_1 : DecidablePred A] [inst_2 : DecidablePred B], (∀ (z : Ω), A z ↔ B z) → D.Pr A = D.Pr B","l":279,"f":"Causalean/Experimentation/DesignBased/DesignCore.lean"},{"n":"Causalean.Experimentation.DesignBased.FiniteDesign.Pr_eq_measureReal","k":"theorem","m":"Causalean.Experimentation.DesignBased.MeasureBridge","a":"Experimentation/DesignBased","t":2,"d":"Reverse rewrite. The design probability of an event is the real-valued measure of the event under the design measure.","s":"∀ {Ω : Type u_1} [inst : Fintype Ω] [inst_1 : MeasurableSpace Ω] [MeasurableSingletonClass Ω] (D : Causalean.Experimentation.DesignBased.FiniteDesign Ω) (A : Ω → Prop) [inst_3 : DecidablePred A], D.Pr A = D.toMeasure.real {z | A z}","l":77,"f":"Causalean/Experimentation/DesignBased/MeasureBridge.lean"},{"n":"Causalean.Experimentation.DesignBased.FiniteDesign.Pr_eq_zero_le","k":"theorem","m":"Causalean.Experimentation.DesignBased.Chebyshev","a":"Experimentation/DesignBased","t":2,"d":"Zero-hitting bound for a nonzero-mean statistic. In a finite design, a statistic whose design mean is nonzero equals zero with probability at most its design variance divided by the square of its mean. This is the Chebyshev corollary that controls degeneracy of a random Horvitz–Thompson / Hájek denominator: a zero value is exactly a deviation from the mean of size equal to the mean, so it is Var / mean²-rare.","s":"∀ {Ω : Type u_1} [inst : Fintype Ω] (D : Causalean.Experimentation.DesignBased.FiniteDesign Ω) (X : Ω → ℝ), D.E X ≠ 0 → (D.Pr fun z => X z = 0) ≤ D.Var X / D.E X ^ 2","l":77,"f":"Causalean/Experimentation/DesignBased/Chebyshev.lean"},{"n":"Causalean.Experimentation.DesignBased.FiniteDesign.Pr_le_one","k":"theorem","m":"Causalean.Experimentation.DesignBased.DesignCore","a":"Experimentation/DesignBased","t":2,"d":"A probability lies in [0,1]: at most one.","s":"∀ {Ω : Type u_1} [inst : Fintype Ω] (D : Causalean.Experimentation.DesignBased.FiniteDesign Ω) (A : Ω → Prop) [inst_1 : DecidablePred A], D.Pr A ≤ 1","l":275,"f":"Causalean/Experimentation/DesignBased/DesignCore.lean"},{"n":"Causalean.Experimentation.DesignBased.FiniteDesign.Pr_map","k":"theorem","m":"Causalean.Experimentation.DesignBased.DesignCore","a":"Experimentation/DesignBased","t":2,"d":"Transfer of probability across a pushforward. The probability of an event A under the pushforward f_* D equals the probability of its preimage under D.","s":"∀ {Ω : Type u_1} [inst : Fintype Ω] (D : Causalean.Experimentation.DesignBased.FiniteDesign Ω) {Ω' : Type u_2} [inst_1 : Fintype Ω'] (f : Ω → Ω') (A : Ω' → Prop) [inst_2 : DecidablePred A], (D.map f).Pr A = D.Pr fun z => A (f z)","l":341,"f":"Causalean/Experimentation/DesignBased/DesignCore.lean"},{"n":"Causalean.Experimentation.DesignBased.FiniteDesign.Pr_mono","k":"theorem","m":"Causalean.Experimentation.DesignBased.DesignCore","a":"Experimentation/DesignBased","t":2,"d":"Monotonicity of probability: a smaller event has smaller probability.","s":"∀ {Ω : Type u_1} [inst : Fintype Ω] (D : Causalean.Experimentation.DesignBased.FiniteDesign Ω) (A B : Ω → Prop) [inst_1 : DecidablePred A] [inst_2 : DecidablePred B], (∀ (z : Ω), A z → B z) → D.Pr A ≤ D.Pr B","l":284,"f":"Causalean/Experimentation/DesignBased/DesignCore.lean"},{"n":"Causalean.Experimentation.DesignBased.FiniteDesign.Pr_nonneg","k":"theorem","m":"Causalean.Experimentation.DesignBased.DesignCore","a":"Experimentation/DesignBased","t":2,"d":"A probability lies in [0,1]: nonnegativity.","s":"∀ {Ω : Type u_1} [inst : Fintype Ω] (D : Causalean.Experimentation.DesignBased.FiniteDesign Ω) (A : Ω → Prop) [inst_1 : DecidablePred A], 0 ≤ D.Pr A","l":271,"f":"Causalean/Experimentation/DesignBased/DesignCore.lean"},{"n":"Causalean.Experimentation.DesignBased.FiniteDesign.Pr_or_le","k":"theorem","m":"Causalean.Experimentation.DesignBased.InProb","a":"Experimentation/DesignBased","t":2,"d":"Union bound for two events: Pr(P ∨ Q) ≤ Pr P + Pr Q.","s":"∀ {Ω' : Type u_2} [inst : Fintype Ω'] (D : Causalean.Experimentation.DesignBased.FiniteDesign Ω') (P Q : Ω' → Prop) [inst_1 : DecidablePred P] [inst_2 : DecidablePred Q], (D.Pr fun z => P z ∨ Q z) ≤ D.Pr P + D.Pr Q","l":71,"f":"Causalean/Experimentation/DesignBased/InProb.lean"},{"n":"Causalean.Experimentation.DesignBased.FiniteDesign.Pr_split","k":"theorem","m":"Causalean.Experimentation.DesignBased.DesignCore","a":"Experimentation/DesignBased","t":2,"d":"Finite additivity: splitting an event by a second event.","s":"∀ {Ω : Type u_1} [inst : Fintype Ω] (D : Causalean.Experimentation.DesignBased.FiniteDesign Ω) (B A : Ω → Prop) [inst_1 : DecidablePred A] [inst_2 : DecidablePred B], D.Pr B = (D.Pr fun z => B z ∧ A z) + D.Pr fun z => B z ∧ ¬A z","l":295,"f":"Causalean/Experimentation/DesignBased/DesignCore.lean"},{"n":"Causalean.Experimentation.DesignBased.FiniteDesign.TendstoInProb","k":"def","m":"Causalean.Experimentation.DesignBased.InProb","a":"Experimentation/DesignBased","t":1,"d":"Convergence in probability along a sequence of finite designs: X m converges in probability to the target c m if Pr(|X m − c m| ≥ ε) → 0 for every ε > 0.","s":"{Ω : ℕ → Type u_1} → [inst : (m : ℕ) → Fintype (Ω m)] → ((m : ℕ) → Causalean.Experimentation.DesignBased.FiniteDesign (Ω m)) → ((m : ℕ) → Ω m → ℝ) → (ℕ → ℝ) → Prop","l":51,"f":"Causalean/Experimentation/DesignBased/InProb.lean"},{"n":"Causalean.Experimentation.DesignBased.FiniteDesign.TendstoInProb.abs","k":"theorem","m":"Causalean.Experimentation.DesignBased.InProb","a":"Experimentation/DesignBased","t":2,"d":"If X m → 0 in probability then |X m| → 0 in probability.","s":"∀ {Ω : ℕ → Type u_1} [inst : (m : ℕ) → Fintype (Ω m)] {D : (m : ℕ) → Causalean.Experimentation.DesignBased.FiniteDesign (Ω m)} {X : (m : ℕ) → Ω m → ℝ}, (Causalean.Experimentation.DesignBased.FiniteDesign.TendstoInProb D X fun x => 0) → Causalean.Experimentation.DesignBased.FiniteDesign.TendstoInProb D (fun m z => |X m z|) fun x => 0","l":149,"f":"Causalean/Experimentation/DesignBased/InProb.lean"},{"n":"Causalean.Experimentation.DesignBased.FiniteDesign.TendstoInProb.add","k":"theorem","m":"Causalean.Experimentation.DesignBased.InProb","a":"Experimentation/DesignBased","t":2,"d":"Convergence in probability is closed under sums.","s":"∀ {Ω : ℕ → Type u_1} [inst : (m : ℕ) → Fintype (Ω m)] {D : (m : ℕ) → Causalean.Experimentation.DesignBased.FiniteDesign (Ω m)} {X Y : (m : ℕ) → Ω m → ℝ} {a b : ℕ → ℝ}, Causalean.Experimentation.DesignBased.FiniteDesign.TendstoInProb D X a → Causalean.Experimentation.DesignBased.FiniteDesign.TendstoInProb D Y b → Causalean.Experimentation.DesignBased.FiniteDesign.TendstoInProb D (fun m z => X m z + Y m z) fun m => a m + b m","l":118,"f":"Causalean/Experimentation/DesignBased/InProb.lean"},{"n":"Causalean.Experimentation.DesignBased.FiniteDesign.TendstoInProb.const_mul","k":"theorem","m":"Causalean.Experimentation.DesignBased.InProb","a":"Experimentation/DesignBased","t":2,"d":"Scaling a convergent sequence by a constant scales its probability limit: if X m → a m in probability then c · X m → c · a m in probability.","s":"∀ {Ω : ℕ → Type u_1} [inst : (m : ℕ) → Fintype (Ω m)] {D : (m : ℕ) → Causalean.Experimentation.DesignBased.FiniteDesign (Ω m)} {X : (m : ℕ) → Ω m → ℝ} {a : ℕ → ℝ} (c : ℝ), Causalean.Experimentation.DesignBased.FiniteDesign.TendstoInProb D X a → Causalean.Experimentation.DesignBased.FiniteDesign.TendstoInProb D (fun m z => c * X m z) fun m => c * a m","l":304,"f":"Causalean/Experimentation/DesignBased/InProb.lean"},{"n":"Causalean.Experimentation.DesignBased.FiniteDesign.TendstoInProb.mul_boundedInProb","k":"theorem","m":"Causalean.Experimentation.DesignBased.InProb","a":"Experimentation/DesignBased","t":2,"d":"Product-tightness (o_p × O_p = o_p). If U m converges to zero in probability and V m is uniformly tight (bounded in probability), then the product U m · V m converges to zero in probability. This is the engine that turns a delta-method remainder — a vanishing factor times a bounded factor — into an o_p(1) term.","s":"∀ {Ω : ℕ → Type u_1} [inst : (m : ℕ) → Fintype (Ω m)] {D : (m : ℕ) → Causalean.Experimentation.DesignBased.FiniteDesign (Ω m)} {U V : (m : ℕ) → Ω m → ℝ}, (Causalean.Experimentation.DesignBased.FiniteDesign.TendstoInProb D U fun x => 0) → Causalean.Experimentation.DesignBased.FiniteDesign.BoundedInProb D V → Causalean.Experimentation.DesignBased.FiniteDesign.TendstoInProb D (fun m z => U m z * V m z) fun x => 0","l":324,"f":"Causalean/Experimentation/DesignBased/InProb.lean"},{"n":"Causalean.Experimentation.DesignBased.FiniteDesign.TendstoInProb.sub","k":"theorem","m":"Causalean.Experimentation.DesignBased.InProb","a":"Experimentation/DesignBased","t":1,"d":"Along a sequence of finite designs, if the statistics X m converge in probability to a m and the statistics Y m converge in probability to b m, then the difference X m − Y m converges in probability to a m − b m: convergence in probability is closed under differences.","s":"∀ {Ω : ℕ → Type u_1} [inst : (m : ℕ) → Fintype (Ω m)] {D : (m : ℕ) → Causalean.Experimentation.DesignBased.FiniteDesign (Ω m)} {X Y : (m : ℕ) → Ω m → ℝ} {a b : ℕ → ℝ}, Causalean.Experimentation.DesignBased.FiniteDesign.TendstoInProb D X a → Causalean.Experimentation.DesignBased.FiniteDesign.TendstoInProb D Y b → Causalean.Experimentation.DesignBased.FiniteDesign.TendstoInProb D (fun m z => X m z - Y m z) fun m => a m - b m","l":83,"f":"Causalean/Experimentation/DesignBased/InProb.lean"},{"n":"Causalean.Experimentation.DesignBased.FiniteDesign.Unbiased","k":"def","m":"Causalean.Experimentation.DesignBased.Risk","a":"Experimentation/DesignBased","t":1,"d":"An estimator is unbiased for μ under the design when its expectation equals μ.","s":"{Ω : Type u_1} → [inst : Fintype Ω] → Causalean.Experimentation.DesignBased.FiniteDesign Ω → (Ω → ℝ) → ℝ → Prop","l":47,"f":"Causalean/Experimentation/DesignBased/Risk.lean"},{"n":"Causalean.Experimentation.DesignBased.FiniteDesign.Var","k":"def","m":"Causalean.Experimentation.DesignBased.DesignCore","a":"Experimentation/DesignBased","t":1,"d":"Variance of X under the design.","s":"{Ω : Type u_1} → [inst : Fintype Ω] → Causalean.Experimentation.DesignBased.FiniteDesign Ω → (Ω → ℝ) → ℝ","l":60,"f":"Causalean/Experimentation/DesignBased/DesignCore.lean"},{"n":"Causalean.Experimentation.DesignBased.FiniteDesign.Var_compound_eq_tower","k":"theorem","m":"Causalean.Experimentation.DesignBased.CompoundVariance","a":"Experimentation/DesignBased","t":1,"d":"Law of total variance for the two-stage compound design. For a stage-1 design D₁ paired with a stage-2 design D₂ s at every stage-1 outcome s, applied to any statistic X on the joint outcome space, the total variance of X under the compound design decomposes as the stage-1 expectation of the stage-2 conditional variance of X, plus the stage-1 variance of the stage-2 conditional mean of X:","s":"∀ {Ω₁ : Type u_1} [inst : Fintype Ω₁] {ι : Type u_2} [inst_1 : Fintype ι] [inst_2 : DecidableEq ι] {α : ι → Type u_3} [inst_3 : (i : ι) → Fintype (α i)] (D₁ : Causalean.Experimentation.DesignBased.FiniteDesign Ω₁) (D₂ : Ω₁ → (i : ι) → Causalean.Experimentation.DesignBased.FiniteDesign (α i)) (X : Ω₁ × ((i : ι) → α i) → ℝ), (Causalean.Experimentation.DesignBased.compound D₁ D₂).Var X = (D₁.E fun s => (Causalean.Experimentation.DesignBased.prodDesign (D₂ s)).Var fun w => X (s, w)) + D₁.Var fun s => (Causalean.Experimentation.DesignBased.prodDesign (D₂ s)).E fun w => X (s, w)","l":64,"f":"Causalean/Experimentation/DesignBased/CompoundVariance.lean"},{"n":"Causalean.Experimentation.DesignBased.FiniteDesign.Var_congr","k":"theorem","m":"Causalean.Experimentation.DesignBased.DesignCore","a":"Experimentation/DesignBased","t":2,"d":"Congruence: pointwise-equal random variables have equal variance.","s":"∀ {Ω : Type u_1} [inst : Fintype Ω] (D : Causalean.Experimentation.DesignBased.FiniteDesign Ω) {X Y : Ω → ℝ}, (∀ (z : Ω), X z = Y z) → D.Var X = D.Var Y","l":113,"f":"Causalean/Experimentation/DesignBased/DesignCore.lean"},{"n":"Causalean.Experimentation.DesignBased.FiniteDesign.Var_const_mul","k":"theorem","m":"Causalean.Experimentation.DesignBased.DesignCore","a":"Experimentation/DesignBased","t":2,"d":"Variance scales by the square of a constant: Var(c·X) = c²·Var X.","s":"∀ {Ω : Type u_1} [inst : Fintype Ω] (D : Causalean.Experimentation.DesignBased.FiniteDesign Ω) (c : ℝ) (X : Ω → ℝ), (D.Var fun z => c * X z) = c ^ 2 * D.Var X","l":202,"f":"Causalean/Experimentation/DesignBased/DesignCore.lean"},{"n":"Causalean.Experimentation.DesignBased.FiniteDesign.Var_eq","k":"theorem","m":"Causalean.Experimentation.DesignBased.DesignCore","a":"Experimentation/DesignBased","t":1,"d":"Variance identity. For any statistic X, its design variance equals the design expectation of X² minus the square of its design expectation.","s":"∀ {Ω : Type u_1} [inst : Fintype Ω] (D : Causalean.Experimentation.DesignBased.FiniteDesign Ω) (X : Ω → ℝ), D.Var X = (D.E fun z => X z ^ 2) - D.E X ^ 2","l":127,"f":"Causalean/Experimentation/DesignBased/DesignCore.lean"},{"n":"Causalean.Experimentation.DesignBased.FiniteDesign.Var_eq_variance","k":"theorem","m":"Causalean.Experimentation.DesignBased.MeasureBridge","a":"Experimentation/DesignBased","t":2,"d":"Reverse rewrite. The design variance is the measure-theoretic variance under the design measure.","s":"∀ {Ω : Type u_1} [inst : Fintype Ω] [inst_1 : MeasurableSpace Ω] [MeasurableSingletonClass Ω] (D : Causalean.Experimentation.DesignBased.FiniteDesign Ω) (g : Ω → ℝ), D.Var g = ProbabilityTheory.variance g D.toMeasure","l":72,"f":"Causalean/Experimentation/DesignBased/MeasureBridge.lean"},{"n":"Causalean.Experimentation.DesignBased.FiniteDesign.Var_ind","k":"theorem","m":"Causalean.Experimentation.DesignBased.DesignCore","a":"Experimentation/DesignBased","t":1,"d":"Variance of an indicator. For any event A, the design variance of its indicator equals the design probability of A times one minus that probability.","s":"∀ {Ω : Type u_1} [inst : Fintype Ω] (D : Causalean.Experimentation.DesignBased.FiniteDesign Ω) (A : Ω → Prop) [inst_1 : DecidablePred A], D.Var (Causalean.Experimentation.DesignBased.FiniteDesign.ind A) = D.Pr A * (1 - D.Pr A)","l":242,"f":"Causalean/Experimentation/DesignBased/DesignCore.lean"},{"n":"Causalean.Experimentation.DesignBased.FiniteDesign.Var_linear_comb","k":"theorem","m":"Causalean.Experimentation.DesignBased.DesignCore","a":"Experimentation/DesignBased","t":1,"d":"Variance of a finite linear combination. For a finite weighted sum ∑ᵢ∈s cᵢXᵢ, built from an index set s ⊆ ι, weights c, and a random-variable family X, its design variance equals the double sum of weighted covariances ∑ᵢ∈s∑ⱼ∈s cᵢcⱼ·Cov(Xᵢ,Xⱼ).","s":"∀ {Ω : Type u_1} [inst : Fintype Ω] (D : Causalean.Experimentation.DesignBased.FiniteDesign Ω) {ι : Type u_2} (s : Finset ι) (c : ι → ℝ) (X : ι → Ω → ℝ), (D.Var fun z => ∑ i ∈ s, c i * X i z) = ∑ i ∈ s, ∑ j ∈ s, c i * c j * D.Cov (X i) (X j)","l":221,"f":"Causalean/Experimentation/DesignBased/DesignCore.lean"},{"n":"Causalean.Experimentation.DesignBased.FiniteDesign.Var_nonneg","k":"theorem","m":"Causalean.Experimentation.DesignBased.Risk","a":"Experimentation/DesignBased","t":2,"d":"Variance is nonnegative (it is the mean squared error of the centered estimator).","s":"∀ {Ω : Type u_1} [inst : Fintype Ω] (D : Causalean.Experimentation.DesignBased.FiniteDesign Ω) (X : Ω → ℝ), 0 ≤ D.Var X","l":77,"f":"Causalean/Experimentation/DesignBased/Risk.lean"},{"n":"Causalean.Experimentation.DesignBased.FiniteDesign.Var_prod_apply","k":"theorem","m":"Causalean.Experimentation.DesignBased.ProductVariance","a":"Experimentation/DesignBased","t":2,"d":"Single-coordinate variance: under the product design, the variance of a function of one coordinate equals the variance of that function under the coordinate's own design.","s":"∀ {ι : Type u_1} [inst : Fintype ι] [inst_1 : DecidableEq ι] {α : ι → Type u_2} [inst_2 : (i : ι) → Fintype (α i)] (D : (i : ι) → Causalean.Experimentation.DesignBased.FiniteDesign (α i)) (j : ι) (g : α j → ℝ), ((Causalean.Experimentation.DesignBased.prodDesign D).Var fun w => g (w j)) = (D j).Var g","l":63,"f":"Causalean/Experimentation/DesignBased/ProductVariance.lean"},{"n":"Causalean.Experimentation.DesignBased.FiniteDesign.Var_prod_linear_comb","k":"theorem","m":"Causalean.Experimentation.DesignBased.ProductVariance","a":"Experimentation/DesignBased","t":1,"d":"The payload. Under the product design formed from a family of per-coordinate finite designs, the variance of a linear combination ∑ᵢ cᵢ·gᵢ(wᵢ) of single-coordinate functions gᵢ, with coefficients c, equals the sum over coordinates of the squared coefficient times the coordinate's own variance — cross-coordinate covariances vanish.","s":"∀ {ι : Type u_1} [inst : Fintype ι] [inst_1 : DecidableEq ι] {α : ι → Type u_2} [inst_2 : (i : ι) → Fintype (α i)] (D : (i : ι) → Causalean.Experimentation.DesignBased.FiniteDesign (α i)) (c : ι → ℝ) (g : (i : ι) → α i → ℝ), ((Causalean.Experimentation.DesignBased.prodDesign D).Var fun w => ∑ i, c i * g i (w i)) = ∑ i, c i ^ 2 * (D i).Var (g i)","l":86,"f":"Causalean/Experimentation/DesignBased/ProductVariance.lean"},{"n":"Causalean.Experimentation.DesignBased.FiniteDesign.Var_sub","k":"theorem","m":"Causalean.Experimentation.DesignBased.DesignCore","a":"Experimentation/DesignBased","t":2,"d":"Variance of a difference: Var(X−Y) = Var X + Var Y − 2 Cov(X,Y).","s":"∀ {Ω : Type u_1} [inst : Fintype Ω] (D : Causalean.Experimentation.DesignBased.FiniteDesign Ω) (X Y : Ω → ℝ), (D.Var fun z => X z - Y z) = D.Var X + D.Var Y - 2 * D.Cov X Y","l":196,"f":"Causalean/Experimentation/DesignBased/DesignCore.lean"},{"n":"Causalean.Experimentation.DesignBased.FiniteDesign.Var_sub_const","k":"theorem","m":"Causalean.Experimentation.TwoStageInterference.Asymptotic.CLTDischarge","a":"Experimentation/TwoStageInterference/Asymptotic","t":2,"d":"Shifting a random variable by a constant leaves its variance unchanged.","s":"∀ {Ω : Type u_1} [inst : Fintype Ω] (D : Causalean.Experimentation.DesignBased.FiniteDesign Ω) (X : Ω → ℝ) (c : ℝ), (D.Var fun z => X z - c) = D.Var X","l":75,"f":"Causalean/Experimentation/TwoStageInterference/Asymptotic/CLTDischarge.lean"},{"n":"Causalean.Experimentation.DesignBased.FiniteDesign.abs_Cov_le","k":"theorem","m":"Causalean.Experimentation.DesignBased.MeasureBridge","a":"Experimentation/DesignBased","t":1,"d":"Cauchy–Schwarz for the design covariance. In any finite design, the absolute covariance of two statistics X and Y is at most the product of their design standard deviations, |Cov(X,Y)| ≤ √(Var X) · √(Var Y).","s":"∀ {Ω : Type u_1} [inst : Fintype Ω] [inst_1 : MeasurableSpace Ω] [MeasurableSingletonClass Ω] (D : Causalean.Experimentation.DesignBased.FiniteDesign Ω) (X Y : Ω → ℝ), |D.Cov X Y| ≤ √(D.Var X) * √(D.Var Y)","l":90,"f":"Causalean/Experimentation/DesignBased/MeasureBridge.lean"},{"n":"Causalean.Experimentation.DesignBased.FiniteDesign.abs_Cov_le_two_sq","k":"theorem","m":"Causalean.Experimentation.DesignBased.EdgeVarianceBound","a":"Experimentation/DesignBased","t":2,"d":"The design covariance of random variables bounded by MX and MY in absolute value pointwise is bounded by 2·MX·MY in absolute value.","s":"∀ {Ω : Type u_1} [inst : Fintype Ω] (D : Causalean.Experimentation.DesignBased.FiniteDesign Ω) {X Y : Ω → ℝ} {MX MY : ℝ}, 0 ≤ MX → (∀ (z : Ω), |X z| ≤ MX) → (∀ (z : Ω), |Y z| ≤ MY) → |D.Cov X Y| ≤ 2 * MX * MY","l":62,"f":"Causalean/Experimentation/DesignBased/EdgeVarianceBound.lean"},{"n":"Causalean.Experimentation.DesignBased.FiniteDesign.abs_E_le","k":"theorem","m":"Causalean.Experimentation.DesignBased.EdgeVarianceBound","a":"Experimentation/DesignBased","t":2,"d":"The expectation of a random variable whose absolute value is bounded by M pointwise is itself bounded by M in absolute value.","s":"∀ {Ω : Type u_1} [inst : Fintype Ω] (D : Causalean.Experimentation.DesignBased.FiniteDesign Ω) {X : Ω → ℝ} {M : ℝ}, (∀ (z : Ω), |X z| ≤ M) → |D.E X| ≤ M","l":42,"f":"Causalean/Experimentation/DesignBased/EdgeVarianceBound.lean"},{"n":"Causalean.Experimentation.DesignBased.FiniteDesign.aemeasurable_toMeasure","k":"theorem","m":"Causalean.Experimentation.DesignBased.MeasureBridge","a":"Experimentation/DesignBased","t":2,"d":"On a finite assignment space every statistic with a measurable codomain is a.e.-measurable under the design measure.","s":"∀ {Ω : Type u_1} [inst : Fintype Ω] [inst_1 : MeasurableSpace Ω] [MeasurableSingletonClass Ω] (D : Causalean.Experimentation.DesignBased.FiniteDesign Ω) {β : Type u_2} [inst_3 : MeasurableSpace β] (g : Ω → β), AEMeasurable g D.toMeasure","l":51,"f":"Causalean/Experimentation/DesignBased/MeasureBridge.lean"},{"n":"Causalean.Experimentation.DesignBased.FiniteDesign.aestronglyMeasurable_toMeasure","k":"theorem","m":"Causalean.Experimentation.DesignBased.MeasureBridge","a":"Experimentation/DesignBased","t":2,"d":"On a finite assignment space every statistic is strongly measurable, because singletons — hence all sets — are measurable.","s":"∀ {Ω : Type u_1} [inst : Fintype Ω] [inst_1 : MeasurableSpace Ω] [MeasurableSingletonClass Ω] (D : Causalean.Experimentation.DesignBased.FiniteDesign Ω) {β : Type u_2} [inst_3 : MeasurableSpace β] [inst_4 : TopologicalSpace β] [TopologicalSpace.PseudoMetrizableSpace β] [SecondCountableTopology β] [OpensMeasurableSpace β] (g : Ω → β), MeasureTheory.AEStronglyMeasurable g D.toMeasure","l":43,"f":"Causalean/Experimentation/DesignBased/MeasureBridge.lean"},{"n":"Causalean.Experimentation.DesignBased.FiniteDesign.bernstein_abs_ge","k":"theorem","m":"Causalean.Experimentation.DesignBased.Concentration","a":"Experimentation/DesignBased","t":1,"d":"Bernstein tail for a bounded design statistic (two-sided). For a statistic X on a finite design, suppose the bound c is nonnegative, X has design mean 0 , X is bounded in absolute value by c everywhere, the design variance of X is at most v, and the threshold ε is nonnegative. Then the design probability that X deviates from 0 by at least ε in absolute value is at most 2·exp(−ε²/(2(2v + cε))), twice the corresponding one-sided Bernstein bound.","s":"∀ {Ω : Type u_1} [inst : Fintype Ω] [inst_1 : MeasurableSpace Ω] [MeasurableSingletonClass Ω] (D : Causalean.Experimentation.DesignBased.FiniteDesign Ω) (X : Ω → ℝ) {c v ε : ℝ}, 0 ≤ c → D.E X = 0 → (∀ (z : Ω), |X z| ≤ c) → D.Var X ≤ v → 0 ≤ ε → (D.Pr fun z => ε ≤ |X z|) ≤ 2 * Real.exp (-ε ^ 2 / (2 * (2 * v + c * ε)))","l":71,"f":"Causalean/Experimentation/DesignBased/Concentration.lean"},{"n":"Causalean.Experimentation.DesignBased.FiniteDesign.bernstein_ge","k":"theorem","m":"Causalean.Experimentation.DesignBased.Concentration","a":"Experimentation/DesignBased","t":2,"d":"Bernstein tail for a bounded design statistic (one-sided). If a statistic is bounded by c, has design mean 0, and has design variance at most v, then it exceeds a nonnegative threshold ε with probability at most exp(−ε² / (2(2v + cε))), far sharper than Chebyshev.","s":"∀ {Ω : Type u_1} [inst : Fintype Ω] [inst_1 : MeasurableSpace Ω] [MeasurableSingletonClass Ω] (D : Causalean.Experimentation.DesignBased.FiniteDesign Ω) (X : Ω → ℝ) {c v ε : ℝ}, 0 ≤ c → D.E X = 0 → (∀ (z : Ω), |X z| ≤ c) → D.Var X ≤ v → 0 ≤ ε → (D.Pr fun z => ε ≤ X z) ≤ Real.exp (-ε ^ 2 / (2 * (2 * v + c * ε)))","l":58,"f":"Causalean/Experimentation/DesignBased/Concentration.lean"},{"n":"Causalean.Experimentation.DesignBased.FiniteDesign.bias","k":"def","m":"Causalean.Experimentation.DesignBased.Risk","a":"Experimentation/DesignBased","t":1,"d":"Bias of an estimator est for the target μ under the design: E[est] − μ.","s":"{Ω : Type u_1} → [inst : Fintype Ω] → Causalean.Experimentation.DesignBased.FiniteDesign Ω → (Ω → ℝ) → ℝ → ℝ","l":41,"f":"Causalean/Experimentation/DesignBased/Risk.lean"},{"n":"Causalean.Experimentation.DesignBased.FiniteDesign.bias_of_unbiased","k":"theorem","m":"Causalean.Experimentation.DesignBased.Risk","a":"Experimentation/DesignBased","t":2,"d":"The bias of an unbiased estimator is zero.","s":"∀ {Ω : Type u_1} [inst : Fintype Ω] (D : Causalean.Experimentation.DesignBased.FiniteDesign Ω) {est : Ω → ℝ} {μ : ℝ}, D.Unbiased est μ → D.bias est μ = 0","l":68,"f":"Causalean/Experimentation/DesignBased/Risk.lean"},{"n":"Causalean.Experimentation.DesignBased.FiniteDesign.boundedInProb_of_var_bound","k":"theorem","m":"Causalean.Experimentation.DesignBased.InProb","a":"Experimentation/DesignBased","t":2,"d":"An eventual variance bound together with an eventual mean bound makes a sequence bounded in probability: if the design variances Var(X m) are eventually at most V and the means E(X m) eventually lie within c of zero, then X is uniformly tight.","s":"∀ {Ω : ℕ → Type u_1} [inst : (m : ℕ) → Fintype (Ω m)] (D : (m : ℕ) → Causalean.Experimentation.DesignBased.FiniteDesign (Ω m)) (X : (m : ℕ) → Ω m → ℝ) {V c : ℝ}, (∀ᶠ (m : ℕ) in Filter.atTop, (D m).Var (X m) ≤ V) → (∀ᶠ (m : ℕ) in Filter.atTop, |(D m).E (X m)| ≤ c) → Causalean.Experimentation.DesignBased.FiniteDesign.BoundedInProb D X","l":268,"f":"Causalean/Experimentation/DesignBased/InProb.lean"},{"n":"Causalean.Experimentation.DesignBased.FiniteDesign.chebyshev","k":"theorem","m":"Causalean.Experimentation.DesignBased.Chebyshev","a":"Experimentation/DesignBased","t":1,"d":"Finite-design Chebyshev inequality. In a finite design, for any statistic X and a strictly positive threshold ε, the design probability that X differs from its design mean by at least ε is at most the design variance of X divided by ε².","s":"∀ {Ω : Type u_1} [inst : Fintype Ω] (D : Causalean.Experimentation.DesignBased.FiniteDesign Ω) (X : Ω → ℝ) {ε : ℝ}, 0 < ε → (D.Pr fun z => ε ≤ |X z - D.E X|) ≤ D.Var X / ε ^ 2","l":37,"f":"Causalean/Experimentation/DesignBased/Chebyshev.lean"},{"n":"Causalean.Experimentation.DesignBased.FiniteDesign.chebyshev_conservative","k":"theorem","m":"Causalean.Experimentation.DesignBased.Variance.Conservative","a":"Experimentation/DesignBased/Variance","t":1,"d":"Conservative Chebyshev bound. If the design expectation of a proposed variance estimator V̂ is at least the randomization variance of X, then for any positive threshold ε, the probability that X deviates from its mean by at least ε is bounded by that expected estimator divided by ε².","s":"∀ {Ω : Type u_1} [inst : Fintype Ω] (D : Causalean.Experimentation.DesignBased.FiniteDesign Ω) {Vhat X : Ω → ℝ}, D.IsConservativeVarEst Vhat X → ∀ {ε : ℝ}, 0 < ε → (D.Pr fun z => ε ≤ |X z - D.E X|) ≤ D.E Vhat / ε ^ 2","l":56,"f":"Causalean/Experimentation/DesignBased/Variance/Conservative.lean"},{"n":"Causalean.Experimentation.DesignBased.FiniteDesign.hasSubexponentialMGF_of_bounded","k":"theorem","m":"Causalean.Experimentation.DesignBased.Concentration","a":"Experimentation/DesignBased","t":2,"d":"A bounded, mean-zero statistic on a finite design is sub-exponential under the design measure, with variance-proxy 2v and scale c. This is the finite-design instance of the Bernstein sub-exponential lemma, obtained through the measure bridge.","s":"∀ {Ω : Type u_1} [inst : Fintype Ω] [inst_1 : MeasurableSpace Ω] [MeasurableSingletonClass Ω] (D : Causalean.Experimentation.DesignBased.FiniteDesign Ω) (X : Ω → ℝ) {c v : ℝ} (hc : 0 ≤ c), D.E X = 0 → (∀ (z : Ω), |X z| ≤ c) → ∀ (hvar : D.Var X ≤ v), Causalean.Stat.Concentration.HasSubexponentialMGF X ⟨2 * v, mul_nonneg (Mathlib.Meta.NormNum.isNat_le_true (Mathlib.Meta.NormNum.isNat_ofNat ℝ Nat.cast_zero) (Mathlib.Meta.NormNum.isNat_ofNat ℝ (Eq.refl 2)) (Eq.refl true)) (le_trans (Causalean.Experimentation.DesignBased.FiniteDesign.E_nonneg D fun x => sq_nonneg (X x - D.E X)) hvar)⟩ ⟨c, hc⟩ D.toMeasure","l":40,"f":"Causalean/Experimentation/DesignBased/Concentration.lean"},{"n":"Causalean.Experimentation.DesignBased.FiniteDesign.ind","k":"def","m":"Causalean.Experimentation.DesignBased.DesignCore","a":"Experimentation/DesignBased","t":1,"d":"Indicator of an event A, as a {0,1}-valued random variable.","s":"{Ω : Type u_1} → (A : Ω → Prop) → [DecidablePred A] → Ω → ℝ","l":66,"f":"Causalean/Experimentation/DesignBased/DesignCore.lean"},{"n":"Causalean.Experimentation.DesignBased.FiniteDesign.ind_le_one","k":"theorem","m":"Causalean.Experimentation.DesignBased.DesignCore","a":"Experimentation/DesignBased","t":2,"d":"An event indicator is always at most one.","s":"∀ {Ω : Type u_1} (A : Ω → Prop) [inst : DecidablePred A] (z : Ω), Causalean.Experimentation.DesignBased.FiniteDesign.ind A z ≤ 1","l":267,"f":"Causalean/Experimentation/DesignBased/DesignCore.lean"},{"n":"Causalean.Experimentation.DesignBased.FiniteDesign.ind_nonneg","k":"theorem","m":"Causalean.Experimentation.DesignBased.DesignCore","a":"Experimentation/DesignBased","t":2,"d":"An event indicator is always nonnegative.","s":"∀ {Ω : Type u_1} (A : Ω → Prop) [inst : DecidablePred A] (z : Ω), 0 ≤ Causalean.Experimentation.DesignBased.FiniteDesign.ind A z","l":262,"f":"Causalean/Experimentation/DesignBased/DesignCore.lean"},{"n":"Causalean.Experimentation.DesignBased.FiniteDesign.ind_sq","k":"theorem","m":"Causalean.Experimentation.DesignBased.DesignCore","a":"Experimentation/DesignBased","t":2,"d":"1_A ^ 2 = 1_A, the idempotence of an indicator.","s":"∀ {Ω : Type u_1} (A : Ω → Prop) [inst : DecidablePred A], (fun z => Causalean.Experimentation.DesignBased.FiniteDesign.ind A z ^ 2) = Causalean.Experimentation.DesignBased.FiniteDesign.ind A","l":237,"f":"Causalean/Experimentation/DesignBased/DesignCore.lean"},{"n":"Causalean.Experimentation.DesignBased.FiniteDesign.instIsProbabilityMeasureToMeasure","k":"instance","m":"Causalean.Experimentation.DesignBased.FiniteDesignMeasure","a":"Experimentation/DesignBased","t":2,"d":"The measure induced by a finite randomization design has total mass one.","s":"∀ {Ω : Type u_1} [inst : Fintype Ω] [inst_1 : MeasurableSpace Ω] (D : Causalean.Experimentation.DesignBased.FiniteDesign Ω), MeasureTheory.IsProbabilityMeasure D.toMeasure","l":60,"f":"Causalean/Experimentation/DesignBased/FiniteDesignMeasure.lean"},{"n":"Causalean.Experimentation.DesignBased.FiniteDesign.integral_toMeasure","k":"theorem","m":"Causalean.Experimentation.DesignBased.FiniteDesignMeasure","a":"Experimentation/DesignBased","t":1,"d":"Integral against the induced measure is the design expectation. For any statistic g, the measure-theoretic integral of g against the design's induced probability measure equals its design expectation.","s":"∀ {Ω : Type u_1} [inst : Fintype Ω] [inst_1 : MeasurableSpace Ω] [MeasurableSingletonClass Ω] (D : Causalean.Experimentation.DesignBased.FiniteDesign Ω) (g : Ω → ℝ), ∫ (x : Ω), g x ∂D.toMeasure = D.E g","l":70,"f":"Causalean/Experimentation/DesignBased/FiniteDesignMeasure.lean"},{"n":"Causalean.Experimentation.DesignBased.FiniteDesign.isConservativeVarEst_of_E_eq_add","k":"theorem","m":"Causalean.Experimentation.DesignBased.Variance.Conservative","a":"Experimentation/DesignBased/Variance","t":2,"d":"A variance estimator whose expectation equals the variance plus a nonnegative bias is conservative.","s":"∀ {Ω : Type u_1} [inst : Fintype Ω] (D : Causalean.Experimentation.DesignBased.FiniteDesign Ω) {Vhat X : Ω → ℝ} {B : ℝ}, 0 ≤ B → D.E Vhat = D.Var X + B → D.IsConservativeVarEst Vhat X","l":45,"f":"Causalean/Experimentation/DesignBased/Variance/Conservative.lean"},{"n":"Causalean.Experimentation.DesignBased.FiniteDesign.isConservativeVarEst_of_unbiased","k":"theorem","m":"Causalean.Experimentation.DesignBased.Variance.Conservative","a":"Experimentation/DesignBased/Variance","t":2,"d":"An unbiased variance estimator (its expectation equals the variance) is conservative.","s":"∀ {Ω : Type u_1} [inst : Fintype Ω] (D : Causalean.Experimentation.DesignBased.FiniteDesign Ω) {Vhat X : Ω → ℝ}, D.E Vhat = D.Var X → D.IsConservativeVarEst Vhat X","l":51,"f":"Causalean/Experimentation/DesignBased/Variance/Conservative.lean"},{"n":"Causalean.Experimentation.DesignBased.FiniteDesign.map","k":"def","m":"Causalean.Experimentation.DesignBased.DesignCore","a":"Experimentation/DesignBased","t":1,"d":"The pushforward design f_* D of D along f : Ω → Ω': the law of the transformed assignment f z when z is drawn from D. Its weight on y is the total design weight of the fiber f⁻¹{y}.","s":"{Ω : Type u_1} → [inst : Fintype Ω] → Causalean.Experimentation.DesignBased.FiniteDesign Ω → {Ω' : Type u_2} → [inst : Fintype Ω'] → (Ω → Ω') → Causalean.Experimentation.DesignBased.FiniteDesign Ω'","l":306,"f":"Causalean/Experimentation/DesignBased/DesignCore.lean"},{"n":"Causalean.Experimentation.DesignBased.FiniteDesign.map_p","k":"theorem","m":"Causalean.Experimentation.DesignBased.DesignCore","a":"Experimentation/DesignBased","t":2,"d":"The pushforward weight of y is the fiber sum ∑_{z} 1[f z = y] · D.p z.","s":"∀ {Ω : Type u_1} [inst : Fintype Ω] (D : Causalean.Experimentation.DesignBased.FiniteDesign Ω) {Ω' : Type u_2} [inst_1 : Fintype Ω'] (f : Ω → Ω') (y : Ω'), (D.map f).p y = ∑ z, if f z = y then D.p z else 0","l":321,"f":"Causalean/Experimentation/DesignBased/DesignCore.lean"},{"n":"Causalean.Experimentation.DesignBased.FiniteDesign.memLp_toMeasure","k":"theorem","m":"Causalean.Experimentation.DesignBased.MeasureBridge","a":"Experimentation/DesignBased","t":2,"d":"Every normed-additive statistic with a suitable measurable second-countable codomain on a finite assignment space is Lᵖ under the design measure: the space is finite so every statistic is bounded, and the design measure is a probability measure, so every power is integrable.","s":"∀ {Ω : Type u_1} [inst : Fintype Ω] [inst_1 : MeasurableSpace Ω] [MeasurableSingletonClass Ω] (D : Causalean.Experimentation.DesignBased.FiniteDesign Ω) {β : Type u_2} [inst_3 : NormedAddCommGroup β] [inst_4 : MeasurableSpace β] [SecondCountableTopology β] [OpensMeasurableSpace β] (g : Ω → β) (p : ENNReal), MeasureTheory.MemLp g p D.toMeasure","l":57,"f":"Causalean/Experimentation/DesignBased/MeasureBridge.lean"},{"n":"Causalean.Experimentation.DesignBased.FiniteDesign.mse","k":"def","m":"Causalean.Experimentation.DesignBased.Risk","a":"Experimentation/DesignBased","t":1,"d":"Mean squared error of est for the target μ under the design: E[(est − μ)²].","s":"{Ω : Type u_1} → [inst : Fintype Ω] → Causalean.Experimentation.DesignBased.FiniteDesign Ω → (Ω → ℝ) → ℝ → ℝ","l":44,"f":"Causalean/Experimentation/DesignBased/Risk.lean"},{"n":"Causalean.Experimentation.DesignBased.FiniteDesign.mse_congr","k":"theorem","m":"Causalean.Experimentation.DesignBased.Risk","a":"Experimentation/DesignBased","t":2,"d":"Congruence: pointwise-equal estimators have equal mean squared error.","s":"∀ {Ω : Type u_1} [inst : Fintype Ω] (D : Causalean.Experimentation.DesignBased.FiniteDesign Ω) {est est' : Ω → ℝ} {μ : ℝ}, (∀ (z : Ω), est z = est' z) → D.mse est μ = D.mse est' μ","l":87,"f":"Causalean/Experimentation/DesignBased/Risk.lean"},{"n":"Causalean.Experimentation.DesignBased.FiniteDesign.mse_eq_var_add_bias_sq","k":"theorem","m":"Causalean.Experimentation.DesignBased.Risk","a":"Experimentation/DesignBased","t":1,"d":"Bias–variance decomposition. The mean squared error mse of estimator est for target μ equals its randomization variance plus the square of its bias: mse = Var + bias².","s":"∀ {Ω : Type u_1} [inst : Fintype Ω] (D : Causalean.Experimentation.DesignBased.FiniteDesign Ω) (est : Ω → ℝ) (μ : ℝ), D.mse est μ = D.Var est + D.bias est μ ^ 2","l":50,"f":"Causalean/Experimentation/DesignBased/Risk.lean"},{"n":"Causalean.Experimentation.DesignBased.FiniteDesign.mse_eq_var_of_unbiased","k":"theorem","m":"Causalean.Experimentation.DesignBased.Risk","a":"Experimentation/DesignBased","t":2,"d":"For an unbiased estimator the mean squared error equals the variance.","s":"∀ {Ω : Type u_1} [inst : Fintype Ω] (D : Causalean.Experimentation.DesignBased.FiniteDesign Ω) {est : Ω → ℝ} {μ : ℝ}, D.Unbiased est μ → D.mse est μ = D.Var est","l":61,"f":"Causalean/Experimentation/DesignBased/Risk.lean"},{"n":"Causalean.Experimentation.DesignBased.FiniteDesign.mse_nonneg","k":"theorem","m":"Causalean.Experimentation.DesignBased.Risk","a":"Experimentation/DesignBased","t":2,"d":"Mean squared error is nonnegative.","s":"∀ {Ω : Type u_1} [inst : Fintype Ω] (D : Causalean.Experimentation.DesignBased.FiniteDesign Ω) (est : Ω → ℝ) (μ : ℝ), 0 ≤ D.mse est μ","l":73,"f":"Causalean/Experimentation/DesignBased/Risk.lean"},{"n":"Causalean.Experimentation.DesignBased.FiniteDesign.nonempty_of_design","k":"theorem","m":"Causalean.Experimentation.DesignBased.ProductBlock","a":"Experimentation/DesignBased","t":2,"d":"Every coordinate type carrying a finite design is nonempty.","s":"∀ {ι : Type u_1} {α : ι → Type u_2} [inst : (i : ι) → Fintype (α i)] (D : (i : ι) → Causalean.Experimentation.DesignBased.FiniteDesign (α i)) (i : ι), Nonempty (α i)","l":50,"f":"Causalean/Experimentation/DesignBased/ProductBlock.lean"},{"n":"Causalean.Experimentation.DesignBased.FiniteDesign.prodDesign_Pr_reindex","k":"theorem","m":"Causalean.Experimentation.DesignBased.ProductReindex","a":"Experimentation/DesignBased","t":1,"d":"Relabeling a product design by a coordinate permutation. For a family of designs D over a common coordinate space W, indexed by ι, and a permutation σ : ι ≃ ι, the probability under prodDesign D of the predicate w ↦ P (w ∘ σ) equals the probability under the permuted product prodDesign (D ∘ σ) of P.","s":"∀ {ι : Type u_1} [inst : Fintype ι] [inst_1 : DecidableEq ι] {W : Type u_2} [inst_2 : Fintype W] (σ : ι ≃ ι) (D : ι → Causalean.Experimentation.DesignBased.FiniteDesign W) (P : (ι → W) → Prop) [inst_3 : DecidablePred P], ((Causalean.Experimentation.DesignBased.prodDesign D).Pr fun w => P fun i => w (σ i)) = (Causalean.Experimentation.DesignBased.prodDesign fun i => D (σ i)).Pr P","l":40,"f":"Causalean/Experimentation/DesignBased/ProductReindex.lean"},{"n":"Causalean.Experimentation.DesignBased.FiniteDesign.tendstoInProb_div_one","k":"theorem","m":"Causalean.Experimentation.DesignBased.InProb","a":"Experimentation/DesignBased","t":1,"d":"Slutsky ratio step. Along a sequence of finite designs D, fix statistics X and Y. If X m converges in probability to a m, the denominator Y m converges in probability to 1, and the limit sequence a m is uniformly bounded by a constant M, then the ratio X m / Y m converges in probability to a m. (The realized normalizer tends to one, so dividing by it does not change the probability limit.)","s":"∀ {Ω : ℕ → Type u_1} [inst : (m : ℕ) → Fintype (Ω m)] (D : (m : ℕ) → Causalean.Experimentation.DesignBased.FiniteDesign (Ω m)) (X Y : (m : ℕ) → Ω m → ℝ) (a : ℕ → ℝ) (M : ℝ), (∀ (m : ℕ), |a m| ≤ M) → Causalean.Experimentation.DesignBased.FiniteDesign.TendstoInProb D X a → (Causalean.Experimentation.DesignBased.FiniteDesign.TendstoInProb D Y fun x => 1) → Causalean.Experimentation.DesignBased.FiniteDesign.TendstoInProb D (fun m z => X m z / Y m z) a","l":156,"f":"Causalean/Experimentation/DesignBased/InProb.lean"},{"n":"Causalean.Experimentation.DesignBased.FiniteDesign.tendstoInProb_of_var","k":"theorem","m":"Causalean.Experimentation.DesignBased.InProb","a":"Experimentation/DesignBased","t":1,"d":"Chebyshev consistency engine. Along a sequence of finite designs D, for statistics X, if the design variance of X m tends to zero as m → ∞, then X m converges in probability to its design mean E[X m].","s":"∀ {Ω : ℕ → Type u_1} [inst : (m : ℕ) → Fintype (Ω m)] (D : (m : ℕ) → Causalean.Experimentation.DesignBased.FiniteDesign (Ω m)) (X : (m : ℕ) → Ω m → ℝ), Filter.Tendsto (fun m => (D m).Var (X m)) Filter.atTop (nhds 0) → Causalean.Experimentation.DesignBased.FiniteDesign.TendstoInProb D X fun m => (D m).E (X m)","l":57,"f":"Causalean/Experimentation/DesignBased/InProb.lean"},{"n":"Causalean.Experimentation.DesignBased.FiniteDesign.tendsto_E_of_uniformBound","k":"theorem","m":"Causalean.Experimentation.TwoStageInterference.Asymptotic.CLT","a":"Experimentation/TwoStageInterference/Asymptotic","t":2,"d":"Mixture-lifting lemma. If a sequence of design random variables F n : Ω n → ℝ converges to a constant L *uniformly* over the assignment space — |F n s − L| ≤ B n for every s, with B n → 0 — then their expectations converge to L: (D n).E (F n) → L. Averaging a uniformly convergent family preserves the limit, since the expectation of a B n-bounded deviation is itself B n-bounded. This is the abstract content of the two-stage mixture-lifting argument: the joint law is the stage-1 average of conditional laws, and a uniform conditional limit lifts to the average.","s":"∀ {Ω : ℕ → Type u_1} [inst : (n : ℕ) → Fintype (Ω n)] (D : (n : ℕ) → Causalean.Experimentation.DesignBased.FiniteDesign (Ω n)) (F : (n : ℕ) → Ω n → ℝ) (L : ℝ) (B : ℕ → ℝ), (∀ (n : ℕ) (s : Ω n), |F n s - L| ≤ B n) → Filter.Tendsto B Filter.atTop (nhds 0) → Filter.Tendsto (fun n => (D n).E (F n)) Filter.atTop (nhds L)","l":63,"f":"Causalean/Experimentation/TwoStageInterference/Asymptotic/CLT.lean"},{"n":"Causalean.Experimentation.DesignBased.FiniteDesign.tendsto_E_of_uniformBound_ae","k":"theorem","m":"Causalean.Experimentation.TwoStageInterference.Asymptotic.CLTDischarge","a":"Experimentation/TwoStageInterference/Asymptotic","t":2,"d":"Design expectations converge to a constant when the random variables converge uniformly on the support of each design.","s":"∀ {Ω : ℕ → Type u_2} [inst : (n : ℕ) → Fintype (Ω n)] (D : (n : ℕ) → Causalean.Experimentation.DesignBased.FiniteDesign (Ω n)) (F : (n : ℕ) → Ω n → ℝ) (L : ℝ) (B : ℕ → ℝ), (∀ (n : ℕ) (s : Ω n), (D n).p s ≠ 0 → |F n s - L| ≤ B n) → Filter.Tendsto B Filter.atTop (nhds 0) → Filter.Tendsto (fun n => (D n).E (F n)) Filter.atTop (nhds L)","l":83,"f":"Causalean/Experimentation/TwoStageInterference/Asymptotic/CLTDischarge.lean"},{"n":"Causalean.Experimentation.DesignBased.FiniteDesign.toMeasure","k":"def","m":"Causalean.Experimentation.DesignBased.FiniteDesignMeasure","a":"Experimentation/DesignBased","t":1,"d":"The probability measure on Ω induced by the design: ∑_z (p z)·δ_z.","s":"{Ω : Type u_1} → [inst : Fintype Ω] → [inst_1 : MeasurableSpace Ω] → Causalean.Experimentation.DesignBased.FiniteDesign Ω → MeasureTheory.Measure Ω","l":48,"f":"Causalean/Experimentation/DesignBased/FiniteDesignMeasure.lean"},{"n":"Causalean.Experimentation.DesignBased.FiniteDesign.toMeasure_apply","k":"theorem","m":"Causalean.Experimentation.DesignBased.FiniteDesignMeasure","a":"Experimentation/DesignBased","t":2,"d":"The measure induced by a finite randomization design assigns each set the sum, over assignments, of each assignment's probability times the point mass of that assignment on the set.","s":"∀ {Ω : Type u_1} [inst : Fintype Ω] [inst_1 : MeasurableSpace Ω] (D : Causalean.Experimentation.DesignBased.FiniteDesign Ω) (s : Set Ω), D.toMeasure s = ∑ z, ENNReal.ofReal (D.p z) * (MeasureTheory.Measure.dirac z) s","l":52,"f":"Causalean/Experimentation/DesignBased/FiniteDesignMeasure.lean"},{"n":"Causalean.Experimentation.DesignBased.FiniteDesign.toMeasure_real_setOf","k":"theorem","m":"Causalean.Experimentation.DesignBased.FiniteDesignMeasure","a":"Experimentation/DesignBased","t":1,"d":"Induced measure matches the design probability. For any event A, the induced measure's probability of A equals the design probability of A.","s":"∀ {Ω : Type u_1} [inst : Fintype Ω] [inst_1 : MeasurableSpace Ω] [MeasurableSingletonClass Ω] (D : Causalean.Experimentation.DesignBased.FiniteDesign Ω) (A : Ω → Prop) [inst_3 : DecidablePred A], D.toMeasure.real {z | A z} = D.Pr A","l":85,"f":"Causalean/Experimentation/DesignBased/FiniteDesignMeasure.lean"},{"n":"Causalean.Experimentation.DesignBased.FiniteDesign.toMeasure_singleton","k":"theorem","m":"Causalean.Experimentation.DesignBased.ProductMeasure","a":"Experimentation/DesignBased","t":2,"d":"The design measure of a singleton equals that singleton's design probability: (D i).toMeasure {a} = (D i).p a. (The dirac sum collapses to the single term z = a.)","s":"∀ {β : Type u_3} [inst : Fintype β] [inst_1 : MeasurableSpace β] [MeasurableSingletonClass β] (D : Causalean.Experimentation.DesignBased.FiniteDesign β) (a : β), D.toMeasure {a} = ENNReal.ofReal (D.p a)","l":52,"f":"Causalean/Experimentation/DesignBased/ProductMeasure.lean"},{"n":"Causalean.Experimentation.DesignBased.FiniteDesign.var_edge_sum_le","k":"theorem","m":"Causalean.Experimentation.DesignBased.EdgeVarianceBound","a":"Experimentation/DesignBased","t":1,"d":"Variance of an edge-sum over a bounded-degree dependency graph. Consider real-valued statistics b i j attached to ordered pairs of units, all defined on the same finite design. Suppose every statistic attached to an edge of a graph G is bounded in absolute value by a nonnegative constant M, G is symmetric, every unit has at most m neighbours in G, every statistic attached to a pair that is not an edge of G is identically zero, and two statistics are uncorrelated whenever no edge of G connects an endpoint of one pair to an endpoint of the other. Then the variance of the double sum ∑ᵢ∑ⱼ b i j is at most 8·M²·m³·N, where N is the number of units.","s":"∀ {Ω : Type u_1} [inst : Fintype Ω] (D : Causalean.Experimentation.DesignBased.FiniteDesign Ω) {ι : Type u_2} [inst_1 : Fintype ι] (b : ι → ι → Ω → ℝ) (G : ι → ι → Prop) [inst_2 : DecidableRel G] {M : ℝ}, 0 ≤ M → (∀ (i j : ι), G i j → ∀ (z : Ω), |b i j z| ≤ M) → (∀ (i j : ι), G i j → G j i) → ∀ {m : ℕ}, (∀ (i : ι), {j | G i j}.card ≤ m) → (∀ (i j : ι), ¬G i j → b i j = fun x => 0) → (∀ (i j k l : ι), ¬(G i k ∨ G i l ∨ G j k ∨ G j l) → D.Cov (b i j) (b k l) = 0) → (D.Var fun z => ∑ i, ∑ j, b i j z) ≤ 8 * M ^ 2 * (↑m ^ 3 * ↑(Fintype.card ι))","l":95,"f":"Causalean/Experimentation/DesignBased/EdgeVarianceBound.lean"},{"n":"Causalean.Experimentation.DesignBased.FiniteDesign.var_le_mse","k":"theorem","m":"Causalean.Experimentation.DesignBased.Risk","a":"Experimentation/DesignBased","t":2,"d":"Mean squared error is bounded below by the variance.","s":"∀ {Ω : Type u_1} [inst : Fintype Ω] (D : Causalean.Experimentation.DesignBased.FiniteDesign Ω) (est : Ω → ℝ) (μ : ℝ), D.Var est ≤ D.mse est μ","l":82,"f":"Causalean/Experimentation/DesignBased/Risk.lean"},{"n":"Causalean.Experimentation.DesignBased.FiniteDesign.variance_toMeasure","k":"theorem","m":"Causalean.Experimentation.DesignBased.FiniteDesignMeasure","a":"Experimentation/DesignBased","t":1,"d":"Measure-theoretic variance equals the design variance. For any statistic g, the measure-theoretic variance of g under the design's induced measure equals its design variance.","s":"∀ {Ω : Type u_1} [inst : Fintype Ω] [inst_1 : MeasurableSpace Ω] [MeasurableSingletonClass Ω] (D : Causalean.Experimentation.DesignBased.FiniteDesign Ω) (g : Ω → ℝ), ProbabilityTheory.variance g D.toMeasure = D.Var g","l":95,"f":"Causalean/Experimentation/DesignBased/FiniteDesignMeasure.lean"},{"n":"Causalean.Experimentation.DesignBased.IsMinimaxOn","k":"def","m":"Causalean.Experimentation.DesignBased.Optimality.Minimax","a":"Experimentation/DesignBased/Optimality","t":1,"d":"A design is minimax in the family 𝒟 over the states s when it belongs to 𝒟 and has the least worst-case risk among all members.","s":"{Ω : Type u_1} → [inst : Fintype Ω] → {Y : Type u_2} → Causalean.Experimentation.DesignBased.DesignFamily Ω → (s : Finset Y) → s.Nonempty → {α : Type u_3} → [SemilatticeSup α] → (Y → Causalean.Experimentation.DesignBased.FiniteDesign Ω → α) → Causalean.Experimentation.DesignBased.FiniteDesign Ω → Prop","l":50,"f":"Causalean/Experimentation/DesignBased/Optimality/Minimax.lean"},{"n":"Causalean.Experimentation.DesignBased.IsOptimalOn","k":"def","m":"Causalean.Experimentation.DesignBased.Optimality","a":"Experimentation/DesignBased/Optimality","t":1,"d":"A design D₀ is optimal in the family 𝒟 under risk R when it belongs to 𝒟 and carries the least risk among all members.","s":"{Ω : Type u_1} → [inst : Fintype Ω] → {α : Type u_2} → [Preorder α] → Causalean.Experimentation.DesignBased.DesignFamily Ω → (Causalean.Experimentation.DesignBased.FiniteDesign Ω → α) → Causalean.Experimentation.DesignBased.FiniteDesign Ω → Prop","l":60,"f":"Causalean/Experimentation/DesignBased/Optimality.lean"},{"n":"Causalean.Experimentation.DesignBased.IsOptimalOn.dominates","k":"theorem","m":"Causalean.Experimentation.DesignBased.Optimality","a":"Experimentation/DesignBased/Optimality","t":2,"d":"An optimal design dominates every member of its family.","s":"∀ {Ω : Type u_1} [inst : Fintype Ω] {α : Type u_2} [inst_1 : Preorder α] {𝒟 : Causalean.Experimentation.DesignBased.DesignFamily Ω} {R : Causalean.Experimentation.DesignBased.FiniteDesign Ω → α} {D₀ : Causalean.Experimentation.DesignBased.FiniteDesign Ω}, Causalean.Experimentation.DesignBased.IsOptimalOn 𝒟 R D₀ → ∀ {D : Causalean.Experimentation.DesignBased.FiniteDesign Ω}, D ∈ 𝒟 → Causalean.Experimentation.DesignBased.Dominates R D₀ D","l":66,"f":"Causalean/Experimentation/DesignBased/Optimality.lean"},{"n":"Causalean.Experimentation.DesignBased.ProperlySpecified","k":"def","m":"Causalean.Experimentation.DesignBased.PotentialOutcome","a":"Experimentation/DesignBased","t":1,"d":"A properly specified exposure mapping: the randomization potential outcome yr factors through the exposure via exposure-indexed potential outcomes y.","s":"{Ω : Type u_1} → {ι : Type u_2} → {Θ : Type u_3} → {Δ : Type u_4} → (ι → Δ → ℝ) → (ι → Ω → ℝ) → (Ω → Θ → Δ) → (ι → Θ) → Prop","l":43,"f":"Causalean/Experimentation/DesignBased/PotentialOutcome.lean"},{"n":"Causalean.Experimentation.DesignBased.Var_htTotal","k":"theorem","m":"Causalean.Experimentation.DesignBased.HT.Variance","a":"Experimentation/DesignBased/HT","t":1,"d":"Lemma 4.1 (variance), expanded form eq:total_variance. For a finite design, outcome function y, exposure map f, and assignment θ, the randomization variance of the Horvitz–Thompson total under exposure d equals the diagonal sum of inverse-probability-weighted indicator variances plus the off-diagonal sum of inverse-probability-weighted indicator covariances.","s":"∀ {Ω : Type u_1} [inst : Fintype Ω] {ι : Type u_2} {Θ : Type u_3} {Δ : Type u_4} [inst_1 : Fintype ι] [inst_2 : DecidableEq ι] [inst_3 : DecidableEq Δ] (D : Causalean.Experimentation.DesignBased.FiniteDesign Ω) (y : ι → Δ → ℝ) (f : Ω → Θ → Δ) (θ : ι → Θ) (d : Δ), D.Var (Causalean.Experimentation.DesignBased.htTotal D y f θ d) = ∑ i, Causalean.Experimentation.DesignBased.prop D f θ i d * (1 - Causalean.Experimentation.DesignBased.prop D f θ i d) * (y i d / Causalean.Experimentation.DesignBased.prop D f θ i d) ^ 2 + ∑ i, ∑ j ∈ Finset.univ.erase i, (Causalean.Experimentation.DesignBased.propPairSame D f θ i j d - Causalean.Experimentation.DesignBased.prop D f θ i d * Causalean.Experimentation.DesignBased.prop D f θ j d) * (y i d / Causalean.Experimentation.DesignBased.prop D f θ i d * (y j d / Causalean.Experimentation.DesignBased.prop D f θ j d))","l":57,"f":"Causalean/Experimentation/DesignBased/HT/Variance.lean"},{"n":"Causalean.Experimentation.DesignBased.Var_htTotal_cov","k":"theorem","m":"Causalean.Experimentation.DesignBased.HT.Variance","a":"Experimentation/DesignBased/HT","t":1,"d":"Lemma 4.1 (variance), covariance form. For a finite design, outcome function y, exposure map f, and assignment θ, the randomization variance of the Horvitz–Thompson total under exposure d equals the double sum, over unit pairs, of inverse-probability-weighted indicator covariances.","s":"∀ {Ω : Type u_1} [inst : Fintype Ω] {ι : Type u_2} {Θ : Type u_3} {Δ : Type u_4} [inst_1 : Fintype ι] [inst_2 : DecidableEq Δ] (D : Causalean.Experimentation.DesignBased.FiniteDesign Ω) (y : ι → Δ → ℝ) (f : Ω → Θ → Δ) (θ : ι → Θ) (d : Δ), D.Var (Causalean.Experimentation.DesignBased.htTotal D y f θ d) = ∑ i, ∑ j, y i d / Causalean.Experimentation.DesignBased.prop D f θ i d * (y j d / Causalean.Experimentation.DesignBased.prop D f θ j d) * D.Cov (Causalean.Experimentation.DesignBased.expoInd f θ i d) (Causalean.Experimentation.DesignBased.expoInd f θ j d)","l":42,"f":"Causalean/Experimentation/DesignBased/HT/Variance.lean"},{"n":"Causalean.Experimentation.DesignBased.Yobs","k":"def","m":"Causalean.Experimentation.DesignBased.PotentialOutcome","a":"Experimentation/DesignBased","t":1,"d":"Observed outcome of unit i under assignment z: y i (expo i z).","s":"{Ω : Type u_1} → {ι : Type u_2} → {Θ : Type u_3} → {Δ : Type u_4} → (ι → Δ → ℝ) → (Ω → Θ → Δ) → (ι → Θ) → ι → Ω → ℝ","l":48,"f":"Causalean/Experimentation/DesignBased/PotentialOutcome.lean"},{"n":"Causalean.Experimentation.DesignBased.Yobs_eq_sum","k":"theorem","m":"Causalean.Experimentation.DesignBased.PotentialOutcome","a":"Experimentation/DesignBased","t":1,"d":"Condition 2 (consistency). The observed outcome of unit i under assignment z equals the sum, over every exposure level, of the exposure indicator times the exposure-indexed potential outcome.","s":"∀ {Ω : Type u_1} {ι : Type u_2} {Θ : Type u_3} {Δ : Type u_4} [inst : DecidableEq Δ] [inst_1 : Fintype Δ] (y : ι → Δ → ℝ) (f : Ω → Θ → Δ) (θ : ι → Θ) (i : ι) (z : Ω), Causalean.Experimentation.DesignBased.Yobs y f θ i z = ∑ d, Causalean.Experimentation.DesignBased.expoInd f θ i d z * y i d","l":55,"f":"Causalean/Experimentation/DesignBased/PotentialOutcome.lean"},{"n":"Causalean.Experimentation.DesignBased.bernoulliDesign","k":"def","m":"Causalean.Experimentation.DesignBased.Designs.Bernoulli","a":"Experimentation/DesignBased/Designs","t":1,"d":"The Bernoulli randomization design: each unit i is independently assigned treatment with probability p i. Built as the product of the per-unit coin designs.","s":"{U : Type u_1} → [inst : Fintype U] → [inst_1 : DecidableEq U] → (p : U → ℝ) → (∀ (i : U), 0 ≤ p i) → (∀ (i : U), p i ≤ 1) → Causalean.Experimentation.DesignBased.FiniteDesign (U → Bool)","l":57,"f":"Causalean/Experimentation/DesignBased/Designs/Bernoulli.lean"},{"n":"Causalean.Experimentation.DesignBased.bernoulliDesign_Cov_treatInd","k":"theorem","m":"Causalean.Experimentation.DesignBased.Designs.Bernoulli","a":"Experimentation/DesignBased/Designs","t":2,"d":"For distinct units i ≠ j, the two treatment indicators are uncorrelated under the Bernoulli design.","s":"∀ {U : Type u_1} [inst : Fintype U] [inst_1 : DecidableEq U] (p : U → ℝ) (hp0 : ∀ (i : U), 0 ≤ p i) (hp1 : ∀ (i : U), p i ≤ 1) {i j : U}, i ≠ j → (Causalean.Experimentation.DesignBased.bernoulliDesign p hp0 hp1).Cov (Causalean.Experimentation.DesignBased.treatInd i) (Causalean.Experimentation.DesignBased.treatInd j) = 0","l":99,"f":"Causalean/Experimentation/DesignBased/Designs/Bernoulli.lean"},{"n":"Causalean.Experimentation.DesignBased.bernoulliDesign_E_centeredMonomial","k":"theorem","m":"Causalean.Experimentation.DesignBased.Designs.BernoulliMoments","a":"Experimentation/DesignBased/Designs","t":2,"d":"A centered monomial has mean zero unless it is the constant one: the design expectation of the S-centered monomial is 1 when S is empty and 0 otherwise.","s":"∀ {U : Type u_1} [inst : Fintype U] [inst_1 : DecidableEq U] (p : U → ℝ) (hp0 : ∀ (i : U), 0 ≤ p i) (hp1 : ∀ (i : U), p i ≤ 1) (S : Finset U), (Causalean.Experimentation.DesignBased.bernoulliDesign p hp0 hp1).E (Causalean.Experimentation.DesignBased.centeredMonomial p S) = if S = ∅ then 1 else 0","l":217,"f":"Causalean/Experimentation/DesignBased/Designs/BernoulliMoments.lean"},{"n":"Causalean.Experimentation.DesignBased.bernoulliDesign_E_centeredMonomial_mul","k":"theorem","m":"Causalean.Experimentation.DesignBased.Designs.BernoulliMoments","a":"Experimentation/DesignBased/Designs","t":1,"d":"Orthogonality of the centered monomials. Under a Bernoulli randomization in which every unit's treatment probability p i lies in [0,1], the centered monomials attached to two finite sets of units S and T are uncorrelated whenever S ≠ T, while the monomial attached to S itself has second moment ∏_{j ∈ S} p j · (1 − p j) — the product of the per-unit assignment variances — when S = T. This is the design-based analogue of the orthonormality of a Fourier basis.","s":"∀ {U : Type u_1} [inst : Fintype U] [inst_1 : DecidableEq U] (p : U → ℝ) (hp0 : ∀ (i : U), 0 ≤ p i) (hp1 : ∀ (i : U), p i ≤ 1) (S T : Finset U), ((Causalean.Experimentation.DesignBased.bernoulliDesign p hp0 hp1).E fun z => Causalean.Experimentation.DesignBased.centeredMonomial p S z * Causalean.Experimentation.DesignBased.centeredMonomial p T z) = if S = T then ∏ j ∈ S, p j * (1 - p j) else 0","l":176,"f":"Causalean/Experimentation/DesignBased/Designs/BernoulliMoments.lean"},{"n":"Causalean.Experimentation.DesignBased.bernoulliDesign_E_centeredMonomial_mul_treatInd_prod","k":"theorem","m":"Causalean.Experimentation.DesignBased.Designs.BernoulliMoments","a":"Experimentation/DesignBased/Designs","t":2,"d":"Pairing a centered monomial with a raw monomial. The expectation of the S-centered monomial times the product of the treatment indicators of a set T vanishes unless every unit of S also lies in T; when it does, it equals the product of the assignment variances p j (1 − p j) over S times the treatment probabilities of the units of T outside S.","s":"∀ {U : Type u_1} [inst : Fintype U] [inst_1 : DecidableEq U] (p : U → ℝ) (hp0 : ∀ (i : U), 0 ≤ p i) (hp1 : ∀ (i : U), p i ≤ 1) (S T : Finset U), ((Causalean.Experimentation.DesignBased.bernoulliDesign p hp0 hp1).E fun z => Causalean.Experimentation.DesignBased.centeredMonomial p S z * ∏ j ∈ T, Causalean.Experimentation.DesignBased.treatInd j z) = if S ⊆ T then (∏ j ∈ S, p j * (1 - p j)) * ∏ j ∈ T \\ S, p j else 0","l":234,"f":"Causalean/Experimentation/DesignBased/Designs/BernoulliMoments.lean"},{"n":"Causalean.Experimentation.DesignBased.bernoulliDesign_E_ctrlInd_prod","k":"theorem","m":"Causalean.Experimentation.DesignBased.Designs.BernoulliMoments","a":"Experimentation/DesignBased/Designs","t":2,"d":"The units in a finite set S are all left untreated together with probability ∏_{j ∈ S} (1 − p j): the expectation of the product of their control indicators is the product of their control probabilities.","s":"∀ {U : Type u_1} [inst : Fintype U] [inst_1 : DecidableEq U] (p : U → ℝ) (hp0 : ∀ (i : U), 0 ≤ p i) (hp1 : ∀ (i : U), p i ≤ 1) (S : Finset U), ((Causalean.Experimentation.DesignBased.bernoulliDesign p hp0 hp1).E fun z => ∏ j ∈ S, (1 - Causalean.Experimentation.DesignBased.treatInd j z)) = ∏ j ∈ S, (1 - p j)","l":91,"f":"Causalean/Experimentation/DesignBased/Designs/BernoulliMoments.lean"},{"n":"Causalean.Experimentation.DesignBased.bernoulliDesign_E_ctrlInd_prod_mul_prod","k":"theorem","m":"Causalean.Experimentation.DesignBased.Designs.BernoulliMoments","a":"Experimentation/DesignBased/Designs","t":2,"d":"Two overlapping sets of units are simultaneously left untreated with probability ∏_{j ∈ S ∪ T} (1 − p j): the shared units are counted once, not twice.","s":"∀ {U : Type u_1} [inst : Fintype U] [inst_1 : DecidableEq U] (p : U → ℝ) (hp0 : ∀ (i : U), 0 ≤ p i) (hp1 : ∀ (i : U), p i ≤ 1) (S T : Finset U), ((Causalean.Experimentation.DesignBased.bernoulliDesign p hp0 hp1).E fun z => (∏ j ∈ S, (1 - Causalean.Experimentation.DesignBased.treatInd j z)) * ∏ j ∈ T, (1 - Causalean.Experimentation.DesignBased.treatInd j z)) = ∏ j ∈ S ∪ T, (1 - p j)","l":152,"f":"Causalean/Experimentation/DesignBased/Designs/BernoulliMoments.lean"},{"n":"Causalean.Experimentation.DesignBased.bernoulliDesign_E_prod","k":"theorem","m":"Causalean.Experimentation.DesignBased.Designs.BernoulliMoments","a":"Experimentation/DesignBased/Designs","t":2,"d":"Independence across units turns the expectation of a product into a product of expectations: if every unit contributes a factor that depends only on its own coin, the design expectation of the whole product is the product over units of p i · g i (treated) + (1 − p i) · g i (control).","s":"∀ {U : Type u_1} [inst : Fintype U] [inst_1 : DecidableEq U] (p : U → ℝ) (hp0 : ∀ (i : U), 0 ≤ p i) (hp1 : ∀ (i : U), p i ≤ 1) (g : U → Bool → ℝ), ((Causalean.Experimentation.DesignBased.bernoulliDesign p hp0 hp1).E fun z => ∏ i, g i (z i)) = ∏ i, (p i * g i true + (1 - p i) * g i false)","l":51,"f":"Causalean/Experimentation/DesignBased/Designs/BernoulliMoments.lean"},{"n":"Causalean.Experimentation.DesignBased.bernoulliDesign_E_prod_finset","k":"theorem","m":"Causalean.Experimentation.DesignBased.Designs.BernoulliMoments","a":"Experimentation/DesignBased/Designs","t":2,"d":"The same factorization when only the units in a finite set S contribute a factor: the units outside S are simply averaged out and leave no trace.","s":"∀ {U : Type u_1} [inst : Fintype U] [inst_1 : DecidableEq U] (p : U → ℝ) (hp0 : ∀ (i : U), 0 ≤ p i) (hp1 : ∀ (i : U), p i ≤ 1) (S : Finset U) (g : U → Bool → ℝ), ((Causalean.Experimentation.DesignBased.bernoulliDesign p hp0 hp1).E fun z => ∏ k ∈ S, g k (z k)) = ∏ k ∈ S, (p k * g k true + (1 - p k) * g k false)","l":62,"f":"Causalean/Experimentation/DesignBased/Designs/BernoulliMoments.lean"},{"n":"Causalean.Experimentation.DesignBased.bernoulliDesign_E_sq_of_expansion","k":"theorem","m":"Causalean.Experimentation.DesignBased.Designs.BernoulliMoments","a":"Experimentation/DesignBased/Designs","t":2,"d":"Parseval's identity for the Bernoulli design. Once a statistic is written in the centered-monomial basis, its second moment is the sum over subsets of the squared coefficient times the product of the per-unit assignment variances on that subset — the basis is orthogonal, so no cross terms survive.","s":"∀ {U : Type u_1} [inst : Fintype U] [inst_1 : DecidableEq U] (p : U → ℝ) (hp0 : ∀ (i : U), 0 ≤ p i) (hp1 : ∀ (i : U), p i ≤ 1) (F : (U → Bool) → ℝ) (a : Finset U → ℝ), (∀ (z : U → Bool), F z = ∑ S ∈ Finset.univ.powerset, a S * Causalean.Experimentation.DesignBased.centeredMonomial p S z) → ((Causalean.Experimentation.DesignBased.bernoulliDesign p hp0 hp1).E fun z => F z ^ 2) = ∑ S ∈ Finset.univ.powerset, a S ^ 2 * ∏ j ∈ S, p j * (1 - p j)","l":327,"f":"Causalean/Experimentation/DesignBased/Designs/BernoulliMoments.lean"},{"n":"Causalean.Experimentation.DesignBased.bernoulliDesign_E_sum_sq_le_blockDegree_mul_sum_sq","k":"theorem","m":"Causalean.Experimentation.DesignBased.LocalDependenceVariance","a":"Experimentation/DesignBased","t":1,"d":"Local-dependence variance bound. Under a Bernoulli design in which every unit's treatment probability lies strictly between 0 and 1, suppose each unit i contributes a term F i — attached to a dependence block N i — such that no unit's treatment is read by more than d of these blocks, every term has mean zero under the Bernoulli design, and each term F i depends only on the treatments of its own block N i. Then the second moment of the total ∑ᵢ F i is at most d times the total of the individual second moments, d · ∑ᵢ E[(F i)²]. The bound is scale free: it needs no uniform bound on the terms, no explicit decorrelation assumption, and no dependence on the population size.","s":"∀ {U : Type u_1} [inst : Fintype U] [inst_1 : DecidableEq U] (p : U → ℝ) (hp0 : ∀ (i : U), 0 < p i) (hp1 : ∀ (i : U), p i < 1) (N : U → Finset U) (d : ℕ), Causalean.Experimentation.DesignBased.BlockDegreeLE N d → ∀ (F : U → (U → Bool) → ℝ), (∀ (i : U), (Causalean.Experimentation.DesignBased.bernoulliDesign p (fun k => LT.lt.le (hp0 k)) fun k => LT.lt.le (hp1 k)).E (F i) = 0) → (∀ (i : U), Causalean.Experimentation.DesignBased.DependsOnBlock (N i) (F i)) → ((Causalean.Experimentation.DesignBased.bernoulliDesign p (fun k => LT.lt.le (hp0 k)) fun k => LT.lt.le (hp1 k)).E fun z => (∑ i, F i z) ^ 2) ≤ ↑d * ∑ i, (Causalean.Experimentation.DesignBased.bernoulliDesign p (fun k => LT.lt.le (hp0 k)) fun k => LT.lt.le (hp1 k)).E fun z => F i z ^ 2","l":52,"f":"Causalean/Experimentation/DesignBased/LocalDependenceVariance.lean"},{"n":"Causalean.Experimentation.DesignBased.bernoulliDesign_E_treatInd","k":"theorem","m":"Causalean.Experimentation.DesignBased.Designs.Bernoulli","a":"Experimentation/DesignBased/Designs","t":2,"d":"A quantity depending only on unit i's treatment coin has expectation equal to its value under treatment weighted by p i plus its value under control weighted by 1 - p i.","s":"∀ {U : Type u_1} [inst : Fintype U] [inst_1 : DecidableEq U] (p : U → ℝ) (hp0 : ∀ (i : U), 0 ≤ p i) (hp1 : ∀ (i : U), p i ≤ 1) (i : U) (g : Bool → ℝ), ((Causalean.Experimentation.DesignBased.bernoulliDesign p hp0 hp1).E fun z => g (z i)) = p i * g true + (1 - p i) * g false","l":66,"f":"Causalean/Experimentation/DesignBased/Designs/Bernoulli.lean"},{"n":"Causalean.Experimentation.DesignBased.bernoulliDesign_E_treatInd_pair","k":"theorem","m":"Causalean.Experimentation.DesignBased.Designs.Bernoulli","a":"Experimentation/DesignBased/Designs","t":1,"d":"Second-order inclusion probability. Under the Bernoulli randomization design with per-unit treatment probabilities p i lying between 0 and 1, for two distinct units i ≠ j, the joint expectation of their treatment indicators factors as p i · p j — the units are independently assigned.","s":"∀ {U : Type u_1} [inst : Fintype U] [inst_1 : DecidableEq U] (p : U → ℝ) (hp0 : ∀ (i : U), 0 ≤ p i) (hp1 : ∀ (i : U), p i ≤ 1) {i j : U}, i ≠ j → ((Causalean.Experimentation.DesignBased.bernoulliDesign p hp0 hp1).E fun z => Causalean.Experimentation.DesignBased.treatInd i z * Causalean.Experimentation.DesignBased.treatInd j z) = p i * p j","l":84,"f":"Causalean/Experimentation/DesignBased/Designs/Bernoulli.lean"},{"n":"Causalean.Experimentation.DesignBased.bernoulliDesign_E_treatInd_prod","k":"theorem","m":"Causalean.Experimentation.DesignBased.Designs.BernoulliMoments","a":"Experimentation/DesignBased/Designs","t":2,"d":"Higher-order inclusion probability. The units in a finite set S are all treated together with probability ∏_{j ∈ S} p j; equivalently, the expectation of the product of their treatment indicators is the product of their treatment probabilities.","s":"∀ {U : Type u_1} [inst : Fintype U] [inst_1 : DecidableEq U] (p : U → ℝ) (hp0 : ∀ (i : U), 0 ≤ p i) (hp1 : ∀ (i : U), p i ≤ 1) (S : Finset U), ((Causalean.Experimentation.DesignBased.bernoulliDesign p hp0 hp1).E fun z => ∏ j ∈ S, Causalean.Experimentation.DesignBased.treatInd j z) = ∏ j ∈ S, p j","l":80,"f":"Causalean/Experimentation/DesignBased/Designs/BernoulliMoments.lean"},{"n":"Causalean.Experimentation.DesignBased.bernoulliDesign_E_treatInd_prod_mul_prod","k":"theorem","m":"Causalean.Experimentation.DesignBased.Designs.BernoulliMoments","a":"Experimentation/DesignBased/Designs","t":2,"d":"Two overlapping sets of units are simultaneously treated with probability ∏_{j ∈ S ∪ T} p j: the shared units are counted once, not twice.","s":"∀ {U : Type u_1} [inst : Fintype U] [inst_1 : DecidableEq U] (p : U → ℝ) (hp0 : ∀ (i : U), 0 ≤ p i) (hp1 : ∀ (i : U), p i ≤ 1) (S T : Finset U), ((Causalean.Experimentation.DesignBased.bernoulliDesign p hp0 hp1).E fun z => (∏ j ∈ S, Causalean.Experimentation.DesignBased.treatInd j z) * ∏ j ∈ T, Causalean.Experimentation.DesignBased.treatInd j z) = ∏ j ∈ S ∪ T, p j","l":142,"f":"Causalean/Experimentation/DesignBased/Designs/BernoulliMoments.lean"},{"n":"Causalean.Experimentation.DesignBased.bernoulliDesign_Var_treatInd","k":"theorem","m":"Causalean.Experimentation.DesignBased.Designs.Bernoulli","a":"Experimentation/DesignBased/Designs","t":2,"d":"The treatment indicator of unit i has variance p i (1 − p i) under the Bernoulli design.","s":"∀ {U : Type u_1} [inst : Fintype U] [inst_1 : DecidableEq U] (p : U → ℝ) (hp0 : ∀ (i : U), 0 ≤ p i) (hp1 : ∀ (i : U), p i ≤ 1) (i : U), (Causalean.Experimentation.DesignBased.bernoulliDesign p hp0 hp1).Var (Causalean.Experimentation.DesignBased.treatInd i) = p i * (1 - p i)","l":76,"f":"Causalean/Experimentation/DesignBased/Designs/Bernoulli.lean"},{"n":"Causalean.Experimentation.DesignBased.bestRisk","k":"def","m":"Causalean.Experimentation.DesignBased.Optimality.Minimax","a":"Experimentation/DesignBased/Optimality","t":1,"d":"The best achievable risk at state y over the finite design family 𝒟: the least risk R y D as D ranges over 𝒟. (Defined via a chosen minimizer, which exists by exists_isOptimalOn; bestRisk_le and le_bestRisk characterize it.)","s":"{Ω : Type u_1} → [inst : Fintype Ω] → {Y : Type u_2} → {α : Type u_3} → [LinearOrder α] → (𝒟 : Causalean.Experimentation.DesignBased.DesignFamily Ω) → Set.Finite 𝒟 → Set.Nonempty 𝒟 → (Y → Causalean.Experimentation.DesignBased.FiniteDesign Ω → α) → Y → α","l":68,"f":"Causalean/Experimentation/DesignBased/Optimality/Minimax.lean"},{"n":"Causalean.Experimentation.DesignBased.bestRisk_le","k":"theorem","m":"Causalean.Experimentation.DesignBased.Optimality.Minimax","a":"Experimentation/DesignBased/Optimality","t":2,"d":"The best achievable risk is attained, hence no larger than the risk of any family member.","s":"∀ {Ω : Type u_1} [inst : Fintype Ω] {Y : Type u_2} {α : Type u_3} [inst_1 : LinearOrder α] (𝒟 : Causalean.Experimentation.DesignBased.DesignFamily Ω) (hfin : Set.Finite 𝒟) (hne : Set.Nonempty 𝒟) (R : Y → Causalean.Experimentation.DesignBased.FiniteDesign Ω → α) (y : Y) {D : Causalean.Experimentation.DesignBased.FiniteDesign Ω}, D ∈ 𝒟 → Causalean.Experimentation.DesignBased.bestRisk 𝒟 hfin hne R y ≤ R y D","l":75,"f":"Causalean/Experimentation/DesignBased/Optimality/Minimax.lean"},{"n":"Causalean.Experimentation.DesignBased.card_design_mem_pair","k":"theorem","m":"Causalean.Experimentation.DesignBased.Designs.CompleteRandomization","a":"Experimentation/DesignBased/Designs","t":2,"d":"The number of subsets of a finite population with k + 2 units that contain two distinct specified units equals the number of ways to choose k units from the remaining population.","s":"∀ {U : Type u_1} [inst : Fintype U] [inst_1 : DecidableEq U] (i j : U), i ≠ j → ∀ (k : ℕ), Fintype.card { S // i ∈ ↑S ∧ j ∈ ↑S } = (Fintype.card U - 2).choose k","l":214,"f":"Causalean/Experimentation/DesignBased/Designs/CompleteRandomization.lean"},{"n":"Causalean.Experimentation.DesignBased.card_design_mem_succ","k":"theorem","m":"Causalean.Experimentation.DesignBased.Designs.CompleteRandomization","a":"Experimentation/DesignBased/Designs","t":2,"d":"The number of subsets of a finite population with k + 1 units that contain one specified unit equals the number of ways to choose k units from the remaining population.","s":"∀ {U : Type u_1} [inst : Fintype U] [inst_1 : DecidableEq U] (i : U) (k : ℕ), Fintype.card { S // i ∈ ↑S } = (Fintype.card U - 1).choose k","l":115,"f":"Causalean/Experimentation/DesignBased/Designs/CompleteRandomization.lean"},{"n":"Causalean.Experimentation.DesignBased.card_powersetCard_filter_mem_pair","k":"theorem","m":"Causalean.Experimentation.DesignBased.Designs.CompleteRandomization","a":"Experimentation/DesignBased/Designs","t":2,"d":"Among the fixed-size subsets of a finite set that contain two distinct specified elements, the count equals the number of subsets of the remaining set after those two elements are removed. This is the finite-population counting identity behind pairwise inclusion probabilities.","s":"∀ {α : Type u_2} [inst : DecidableEq α] (s : Finset α) (i j : α), i ∈ s → j ∈ s → i ≠ j → ∀ (k : ℕ), {t ∈ Finset.powersetCard (k + 2) s | i ∈ t ∧ j ∈ t}.card = (Finset.powersetCard k ((s.erase i).erase j)).card","l":137,"f":"Causalean/Experimentation/DesignBased/Designs/CompleteRandomization.lean"},{"n":"Causalean.Experimentation.DesignBased.card_powersetCard_filter_mem_succ","k":"theorem","m":"Causalean.Experimentation.DesignBased.Designs.CompleteRandomization","a":"Experimentation/DesignBased/Designs","t":2,"d":"Among subsets containing a specified unit and of size k + 1, there are exactly as many as there are size-k subsets after that unit is removed.","s":"∀ {α : Type u_2} [inst : DecidableEq α] (s : Finset α), ∀ i ∈ s, ∀ (k : ℕ), {t ∈ Finset.powersetCard (k + 1) s | i ∈ t}.card = (Finset.powersetCard k (s.erase i)).card","l":73,"f":"Causalean/Experimentation/DesignBased/Designs/CompleteRandomization.lean"},{"n":"Causalean.Experimentation.DesignBased.centeredMonomial","k":"def","m":"Causalean.Experimentation.DesignBased.Designs.BernoulliMoments","a":"Experimentation/DesignBased/Designs","t":1,"d":"The centered monomial attached to a finite set of units S: the product over j ∈ S of unit j's treatment indicator minus its treatment probability. These are the Walsh/Fourier basis functions of the biased Boolean cube; the empty set gives the constant 1.","s":"{U : Type u_1} → (U → ℝ) → Finset U → (U → Bool) → ℝ","l":165,"f":"Causalean/Experimentation/DesignBased/Designs/BernoulliMoments.lean"},{"n":"Causalean.Experimentation.DesignBased.centeredMonomial_coef_empty_eq_zero","k":"theorem","m":"Causalean.Experimentation.DesignBased.Designs.BernoulliMoments","a":"Experimentation/DesignBased/Designs","t":2,"d":"A statistic with design mean zero has no constant term: the coefficient of the empty subset in any centered-monomial expansion of it vanishes.","s":"∀ {U : Type u_1} [inst : Fintype U] [inst_1 : DecidableEq U] (p : U → ℝ) (hp0 : ∀ (i : U), 0 ≤ p i) (hp1 : ∀ (i : U), p i ≤ 1) (F : (U → Bool) → ℝ) (a : Finset U → ℝ), (∀ (z : U → Bool), F z = ∑ S ∈ Finset.univ.powerset, a S * Causalean.Experimentation.DesignBased.centeredMonomial p S z) → (Causalean.Experimentation.DesignBased.bernoulliDesign p hp0 hp1).E F = 0 → a ∅ = 0","l":356,"f":"Causalean/Experimentation/DesignBased/Designs/BernoulliMoments.lean"},{"n":"Causalean.Experimentation.DesignBased.centeredMonomial_coef_eq_zero_of_not_subset","k":"theorem","m":"Causalean.Experimentation.DesignBased.Designs.BernoulliMoments","a":"Experimentation/DesignBased/Designs","t":2,"d":"Locality of the Fourier support. If a statistic depends only on the units of a block N, every centered-monomial coefficient attached to a subset that is not contained in N is zero, so its expansion is supported on the subsets of N. (Only treatment probabilities on S must be strictly between zero and one, so those selected coordinates are nondegenerate.)","s":"∀ {U : Type u_1} [inst : Fintype U] (p : U → ℝ), (∀ (i : U), 0 ≤ p i) → (∀ (i : U), p i ≤ 1) → ∀ (N : Finset U) (F : (U → Bool) → ℝ), Causalean.Experimentation.DesignBased.DependsOnBlock N F → ∀ (a : Finset U → ℝ), (∀ (z : U → Bool), F z = ∑ S ∈ Finset.univ.powerset, a S * Causalean.Experimentation.DesignBased.centeredMonomial p S z) → ∀ (S : Finset U), ¬S ⊆ N → (∀ i ∈ S, 0 < p i) → (∀ i ∈ S, p i < 1) → a S = 0","l":379,"f":"Causalean/Experimentation/DesignBased/Designs/BernoulliMoments.lean"},{"n":"Causalean.Experimentation.DesignBased.centeredMonomial_empty","k":"theorem","m":"Causalean.Experimentation.DesignBased.Designs.BernoulliMoments","a":"Experimentation/DesignBased/Designs","t":2,"d":"The centered monomial of the empty set of units is the constant function 1.","s":"∀ {U : Type u_1} (p : U → ℝ), Causalean.Experimentation.DesignBased.centeredMonomial p ∅ = fun x => 1","l":172,"f":"Causalean/Experimentation/DesignBased/Designs/BernoulliMoments.lean"},{"n":"Causalean.Experimentation.DesignBased.clusterDesign","k":"def","m":"Causalean.Experimentation.DesignBased.Designs.ClusterRandomization","a":"Experimentation/DesignBased/Designs","t":1,"d":"The cluster-randomization design: each cluster c is independently assigned treatment with probability p c (a Bernoulli design over the cluster labels). A unit is treated iff its cluster is.","s":"{C : Type u_2} → [inst : Fintype C] → [inst_1 : DecidableEq C] → (p : C → ℝ) → (∀ (c : C), 0 ≤ p c) → (∀ (c : C), p c ≤ 1) → Causalean.Experimentation.DesignBased.FiniteDesign (C → Bool)","l":35,"f":"Causalean/Experimentation/DesignBased/Designs/ClusterRandomization.lean"},{"n":"Causalean.Experimentation.DesignBased.clusterDesign_E_unitTreatInd","k":"theorem","m":"Causalean.Experimentation.DesignBased.Designs.ClusterRandomization","a":"Experimentation/DesignBased/Designs","t":2,"d":"First-order inclusion probability. A unit i is treated with probability p (clus i), the treatment rate of its own cluster.","s":"∀ {U : Type u_1} {C : Type u_2} [inst : Fintype C] [inst_1 : DecidableEq C] (p : C → ℝ) (hp0 : ∀ (c : C), 0 ≤ p c) (hp1 : ∀ (c : C), p c ≤ 1) (clus : U → C) (i : U), (Causalean.Experimentation.DesignBased.clusterDesign p hp0 hp1).E (Causalean.Experimentation.DesignBased.unitTreatInd clus i) = p (clus i)","l":46,"f":"Causalean/Experimentation/DesignBased/Designs/ClusterRandomization.lean"},{"n":"Causalean.Experimentation.DesignBased.clusterDesign_E_unitTreatInd_pair_diff","k":"theorem","m":"Causalean.Experimentation.DesignBased.Designs.ClusterRandomization","a":"Experimentation/DesignBased/Designs","t":2,"d":"Different-cluster joint treatment. Two units in distinct clusters are jointly treated with the product of their cluster rates — distinct clusters are randomized independently.","s":"∀ {U : Type u_1} {C : Type u_2} [inst : Fintype C] [inst_1 : DecidableEq C] (p : C → ℝ) (hp0 : ∀ (c : C), 0 ≤ p c) (hp1 : ∀ (c : C), p c ≤ 1) (clus : U → C) {i j : U}, clus i ≠ clus j → ((Causalean.Experimentation.DesignBased.clusterDesign p hp0 hp1).E fun z => Causalean.Experimentation.DesignBased.unitTreatInd clus i z * Causalean.Experimentation.DesignBased.unitTreatInd clus j z) = p (clus i) * p (clus j)","l":74,"f":"Causalean/Experimentation/DesignBased/Designs/ClusterRandomization.lean"},{"n":"Causalean.Experimentation.DesignBased.clusterDesign_E_unitTreatInd_pair_same","k":"theorem","m":"Causalean.Experimentation.DesignBased.Designs.ClusterRandomization","a":"Experimentation/DesignBased/Designs","t":1,"d":"Same-cluster joint treatment. For a cluster-randomization design in which each cluster's treatment probability p c lies in [0,1], if two units i and j belong to the same cluster, then they are jointly treated with probability exactly their shared cluster's rate p (clus i) — because being in the same cluster makes their treatment indicators identical.","s":"∀ {U : Type u_1} {C : Type u_2} [inst : Fintype C] [inst_1 : DecidableEq C] (p : C → ℝ) (hp0 : ∀ (c : C), 0 ≤ p c) (hp1 : ∀ (c : C), p c ≤ 1) (clus : U → C) {i j : U}, clus i = clus j → ((Causalean.Experimentation.DesignBased.clusterDesign p hp0 hp1).E fun z => Causalean.Experimentation.DesignBased.unitTreatInd clus i z * Causalean.Experimentation.DesignBased.unitTreatInd clus j z) = p (clus i)","l":56,"f":"Causalean/Experimentation/DesignBased/Designs/ClusterRandomization.lean"},{"n":"Causalean.Experimentation.DesignBased.coinDesign","k":"def","m":"Causalean.Experimentation.DesignBased.Designs.Coin","a":"Experimentation/DesignBased/Designs","t":1,"d":"The single-unit coin design on Bool: treatment (true) with probability p, control (false) with probability 1 − p.","s":"(p : ℝ) → 0 ≤ p → p ≤ 1 → Causalean.Experimentation.DesignBased.FiniteDesign Bool","l":23,"f":"Causalean/Experimentation/DesignBased/Designs/Coin.lean"},{"n":"Causalean.Experimentation.DesignBased.coinDesign_E","k":"theorem","m":"Causalean.Experimentation.DesignBased.Designs.Coin","a":"Experimentation/DesignBased/Designs","t":1,"d":"For a probability p lying in [0,1], the expectation of a real-valued function g of a single coin flip that lands heads with probability p equals the probability-weighted two-point sum p · g(true) + (1 − p) · g(false).","s":"∀ (p : ℝ) (hp0 : 0 ≤ p) (hp1 : p ≤ 1) (g : Bool → ℝ), (Causalean.Experimentation.DesignBased.coinDesign p hp0 hp1).E g = p * g true + (1 - p) * g false","l":37,"f":"Causalean/Experimentation/DesignBased/Designs/Coin.lean"},{"n":"Causalean.Experimentation.DesignBased.coinDesign_E_treat","k":"theorem","m":"Causalean.Experimentation.DesignBased.Designs.Bernoulli","a":"Experimentation/DesignBased/Designs","t":2,"d":"The treatment indicator of a single coin, 1(true), has expectation p.","s":"∀ (p : ℝ) (hp0 : 0 ≤ p) (hp1 : p ≤ 1), ((Causalean.Experimentation.DesignBased.coinDesign p hp0 hp1).E fun b => if b = true then 1 else 0) = p","l":40,"f":"Causalean/Experimentation/DesignBased/Designs/Bernoulli.lean"},{"n":"Causalean.Experimentation.DesignBased.coinDesign_Var_treat","k":"theorem","m":"Causalean.Experimentation.DesignBased.Designs.Bernoulli","a":"Experimentation/DesignBased/Designs","t":2,"d":"The treatment indicator of a single coin has variance p(1 − p).","s":"∀ (p : ℝ) (hp0 : 0 ≤ p) (hp1 : p ≤ 1), ((Causalean.Experimentation.DesignBased.coinDesign p hp0 hp1).Var fun b => if b = true then 1 else 0) = p * (1 - p)","l":45,"f":"Causalean/Experimentation/DesignBased/Designs/Bernoulli.lean"},{"n":"Causalean.Experimentation.DesignBased.completeRandomization","k":"def","m":"Causalean.Experimentation.DesignBased.Designs.CompleteRandomization","a":"Experimentation/DesignBased/Designs","t":1,"d":"The complete-randomization design: the uniform law on size-n₁ subsets of the population, i.e. exactly n₁ of the N = card U units are treated, with every such treated set equally likely. Requires n₁ ≤ N so that the design space is nonempty.","s":"{V : Type u_2} → [inst : Fintype V] → (n₁ : ℕ) → n₁ ≤ Fintype.card V → Causalean.Experimentation.DesignBased.FiniteDesign { S // S.card = n₁ }","l":54,"f":"Causalean/Experimentation/DesignBased/Designs/CompleteRandomization.lean"},{"n":"Causalean.Experimentation.DesignBased.completeRandomization_card","k":"theorem","m":"Causalean.Experimentation.DesignBased.Designs.CompleteRandomization","a":"Experimentation/DesignBased/Designs","t":2,"d":"The number of possible treated sets in complete randomization is the binomial coefficient (Fintype.card U).choose n₁.","s":"∀ {V : Type u_2} [inst : Fintype V] (n₁ : ℕ), Fintype.card { S // S.card = n₁ } = (Fintype.card V).choose n₁","l":42,"f":"Causalean/Experimentation/DesignBased/Designs/CompleteRandomization.lean"},{"n":"Causalean.Experimentation.DesignBased.completeRandomization_incl","k":"theorem","m":"Causalean.Experimentation.DesignBased.Designs.CompleteRandomization","a":"Experimentation/DesignBased/Designs","t":2,"d":"First-order inclusion probability. Under complete randomization, a unit is treated with probability n₁ / N.","s":"∀ {U : Type u_1} [inst : Fintype U] [inst_1 : DecidableEq U] (n₁ : ℕ) (hn : n₁ ≤ Fintype.card U) (i : U), ((Causalean.Experimentation.DesignBased.completeRandomization n₁ hn).Pr fun S => i ∈ ↑S) = ↑n₁ / ↑(Fintype.card U)","l":243,"f":"Causalean/Experimentation/DesignBased/Designs/CompleteRandomization.lean"},{"n":"Causalean.Experimentation.DesignBased.completeRandomization_incl_pair","k":"theorem","m":"Causalean.Experimentation.DesignBased.Designs.CompleteRandomization","a":"Experimentation/DesignBased/Designs","t":1,"d":"Second-order inclusion probability. Under complete randomization treating exactly n₁ of the N units, where the treated count n₁ does not exceed the population size N, two distinct units are jointly treated with probability n₁(n₁−1) / (N(N−1)).","s":"∀ {U : Type u_1} [inst : Fintype U] [inst_1 : DecidableEq U] (n₁ : ℕ) (hn : n₁ ≤ Fintype.card U) {i j : U}, i ≠ j → ((Causalean.Experimentation.DesignBased.completeRandomization n₁ hn).Pr fun S => i ∈ ↑S ∧ j ∈ ↑S) = ↑n₁ * (↑n₁ - 1) / (↑(Fintype.card U) * (↑(Fintype.card U) - 1))","l":283,"f":"Causalean/Experimentation/DesignBased/Designs/CompleteRandomization.lean"},{"n":"Causalean.Experimentation.DesignBased.compound","k":"def","m":"Causalean.Experimentation.DesignBased.TwoStage","a":"Experimentation/DesignBased","t":1,"d":"The compound (two-stage) design: stage-1 design D₁ on Ω₁, then, conditionally on the stage-1 outcome s, the independent within-coordinate designs D₂ s i.","s":"{Ω₁ : Type u_1} → [inst : Fintype Ω₁] → {ι : Type u_2} → [inst_1 : Fintype ι] → [inst_2 : DecidableEq ι] → {α : ι → Type u_3} → [inst_3 : (i : ι) → Fintype (α i)] → Causalean.Experimentation.DesignBased.FiniteDesign Ω₁ → (Ω₁ → (i : ι) → Causalean.Experimentation.DesignBased.FiniteDesign (α i)) → Causalean.Experimentation.DesignBased.FiniteDesign (Ω₁ × ((i : ι) → α i))","l":61,"f":"Causalean/Experimentation/DesignBased/TwoStage.lean"},{"n":"Causalean.Experimentation.DesignBased.compoundCore","k":"def","m":"Causalean.Experimentation.DesignBased.TwoStage","a":"Experimentation/DesignBased","t":1,"d":"Generic conditional composition of a first-stage finite design with a finite second-stage design chosen after observing the first-stage outcome.","s":"{Ω₁ : Type u_1} → [inst : Fintype Ω₁] → {Ω₂ : Type u_4} → [inst_1 : Fintype Ω₂] → Causalean.Experimentation.DesignBased.FiniteDesign Ω₁ → (Ω₁ → Causalean.Experimentation.DesignBased.FiniteDesign Ω₂) → Causalean.Experimentation.DesignBased.FiniteDesign (Ω₁ × Ω₂)","l":47,"f":"Causalean/Experimentation/DesignBased/TwoStage.lean"},{"n":"Causalean.Experimentation.DesignBased.conservative_wald_liminf_of_studentized_cdf","k":"theorem","m":"Causalean.Experimentation.DesignBased.WaldCoverage","a":"Experimentation/DesignBased","t":1,"d":"Conservative Wald-interval liminf coverage. Consider a sequence of finite designs with an estimator est n of a target θ n. Suppose the normalization size m n is eventually positive, the true variance scale v n is eventually positive, and a deterministic conservative variance v̂ n eventually dominates v n. Suppose further that the studentized statistic √(m n)·(est n − θ n)/√(v n) has design-probability CDF converging to the standard-normal CDF at z and at −z, where z is nonnegative and satisfies Φ(z) = 1 − α/2. Then the two-sided interval |θ n − est n| ≤ z·√(v̂ n / m n) has asymptotic (liminf) coverage at least 1 − α.","s":"∀ {Ω : ℕ → Type u_1} [inst : (n : ℕ) → Fintype (Ω n)] (D : (n : ℕ) → Causalean.Experimentation.DesignBased.FiniteDesign (Ω n)) (est : (n : ℕ) → Ω n → ℝ) (θ v vhat m : ℕ → ℝ), (∀ᶠ (n : ℕ) in Filter.atTop, 0 < m n) → (∀ᶠ (n : ℕ) in Filter.atTop, 0 < v n) → (∀ᶠ (n : ℕ) in Filter.atTop, v n ≤ vhat n) → ∀ (α z : ℝ), Filter.Tendsto (fun n => (D n).Pr fun zz => √(m n) * (est n zz - θ n) / √(v n) ≤ z) Filter.atTop (nhds (Causalean.Experimentation.DesignBased.stdNormalCdf z)) → Filter.Tendsto (fun n => (D n).Pr fun zz => √(m n) * (est n zz - θ n) / √(v n) ≤ -z) Filter.atTop (nhds (Causalean.Experimentation.DesignBased.stdNormalCdf (-z))) → 0 ≤ z → Causalean.Experimentation.DesignBased.stdNormalCdf z = 1 - α / 2 → 1 - α ≤ Filter.liminf (fun n => (D n).Pr fun zz => |θ n - est n zz| ≤ z * √(vhat n / m n)) Filter.atTop","l":32,"f":"Causalean/Experimentation/DesignBased/WaldCoverage.lean"},{"n":"Causalean.Experimentation.DesignBased.continuous_stdNormalCdf","k":"theorem","m":"Causalean.Experimentation.DesignBased.GaussianCDF","a":"Experimentation/DesignBased","t":1,"d":"Continuity of the standard normal CDF. The standard normal cumulative distribution function Φ is continuous.","s":"Continuous Causalean.Experimentation.DesignBased.stdNormalCdf","l":65,"f":"Causalean/Experimentation/DesignBased/GaussianCDF.lean"},{"n":"Causalean.Experimentation.DesignBased.controlMean","k":"def","m":"Causalean.Experimentation.DesignBased.Estimators.DifferenceInMeans","a":"Experimentation/DesignBased/Estimators","t":1,"d":"The control-arm mean of Y0 under a treated set S: the average of Y0 over the n₀ = N − n₁ control units.","s":"{U : Type u_1} → [Fintype U] → [DecidableEq U] → (n₁ : ℕ) → (U → ℝ) → { S // S.card = n₁ } → ℝ","l":53,"f":"Causalean/Experimentation/DesignBased/Estimators/DifferenceInMeans.lean"},{"n":"Causalean.Experimentation.DesignBased.ctrlInd_prod_mul_prod","k":"theorem","m":"Causalean.Experimentation.DesignBased.Designs.BernoulliMoments","a":"Experimentation/DesignBased/Designs","t":2,"d":"Control indicators take only the values zero and one, so two overlapping products of them collapse into a single product over the union of the two sets of units.","s":"∀ {U : Type u_1} [inst : DecidableEq U] (S T : Finset U) (z : U → Bool), (∏ j ∈ S, (1 - Causalean.Experimentation.DesignBased.treatInd j z)) * ∏ j ∈ T, (1 - Causalean.Experimentation.DesignBased.treatInd j z) = ∏ j ∈ S ∪ T, (1 - Causalean.Experimentation.DesignBased.treatInd j z)","l":123,"f":"Causalean/Experimentation/DesignBased/Designs/BernoulliMoments.lean"},{"n":"Causalean.Experimentation.DesignBased.dependency_studentized_cdf","k":"theorem","m":"Causalean.Experimentation.DesignBased.DependencyCLT","a":"Experimentation/DesignBased","t":1,"d":"Design-based dependency-graph CLT (studentized CDF form). Consider a sequence of finite designs D n together with a triangular array of unit contributions X n i and, for each n, a dependency-graph structure Dep n on the units. Suppose every neighbourhood in the dependency graph has size at most Dmax, the contributions are uniformly bounded by a nonnegative constant M, each contribution has design mean 0, the design second moment v n of the aggregate depSum(X n) is recorded exactly, a positive constant c eventually lower-bounds v n by c times the number of units, and the number of units diverges to infinity. Then, at every point s, the studentized aggregate depSum(X n)/√(v n) has, under the design D n, a limiting CDF equal to the standard normal CDF at s.","s":"∀ {Ω : ℕ → Type u_1} [inst : (n : ℕ) → Fintype (Ω n)] [inst_1 : (n : ℕ) → MeasurableSpace (Ω n)] [∀ (n : ℕ), MeasurableSingletonClass (Ω n)] (D : (n : ℕ) → Causalean.Experimentation.DesignBased.FiniteDesign (Ω n)) {ι : ℕ → Type u_2} [inst_3 : (n : ℕ) → Fintype (ι n)] (X : (n : ℕ) → ι n → Ω n → ℝ) (Dep : (n : ℕ) → Causalean.SteinMethod.DepGraph (X n) (D n).toMeasure) (Dmax : ℕ), (∀ (n : ℕ) (i : ι n), ((Dep n).nbhd i).card ≤ Dmax) → ∀ (M : ℝ), 0 ≤ M → (∀ (n : ℕ) (i : ι n) (ω : Ω n), |X n i ω| ≤ M) → (∀ (n : ℕ) (i : ι n), (D n).E (X n i) = 0) → ∀ (v : ℕ → ℝ), (∀ (n : ℕ), ((D n).E fun ω => Causalean.SteinMethod.depSum (X n) ω ^ 2) = v n) → ∀ (c : ℝ), 0 < c → (∀ᶠ (n : ℕ) in Filter.atTop, c * ↑(Fintype.card (ι n)) ≤ v n) → Filter.Tendsto (fun n => Fintype.card (ι n)) Filter.atTop Filter.atTop → ∀ (s : ℝ), Filter.Tendsto (fun n => (D n).Pr fun ω => Causalean.SteinMethod.depSum (X n) ω / √(v n) ≤ s) Filter.atTop (nhds (Causalean.Experimentation.DesignBased.stdNormalCdf s))","l":43,"f":"Causalean/Experimentation/DesignBased/DependencyCLT.lean"},{"n":"Causalean.Experimentation.DesignBased.dependency_wald_coverage","k":"theorem","m":"Causalean.Experimentation.DesignBased.WaldPipeline","a":"Experimentation/DesignBased","t":1,"d":"Design-based Wald-interval coverage from dependency-graph primitives. Consider a sequence of finite designs together with per-unit contributions X n i governed by a dependency graph Dep n on those contributions whose neighbourhoods have cardinality at most Dmax, where the contributions are uniformly bounded by a nonnegative constant M and have design mean zero. Suppose the standardizing quantity v n equals the design second moment of the contributions' sum depSum(X n), is bounded below by a positive constant c times the number of units eventually, and the number of units diverges. Suppose further that the standardized sum eventually equals √(m n)·(est n − θ n) for an estimator est n of a target θ n, the normalization m n is eventually positive, and a deterministic conservative variance v̂ n eventually dominates v n, with z the nonnegative quantile satisfying Φ(z) = 1 − α/2. Then the two-sided Wald interval |θ n − est n| ≤ z·√(v̂ n / m n) has asymptotic (liminf) coverage at least 1 − α.","s":"∀ {Ω : ℕ → Type u_1} [inst : (n : ℕ) → Fintype (Ω n)] [inst_1 : (n : ℕ) → MeasurableSpace (Ω n)] [∀ (n : ℕ), MeasurableSingletonClass (Ω n)] (D : (n : ℕ) → Causalean.Experimentation.DesignBased.FiniteDesign (Ω n)) {ι : ℕ → Type u_2} [inst_3 : (n : ℕ) → Fintype (ι n)] (X : (n : ℕ) → ι n → Ω n → ℝ) (Dep : (n : ℕ) → Causalean.SteinMethod.DepGraph (X n) (D n).toMeasure) (Dmax : ℕ), (∀ (n : ℕ) (i : ι n), ((Dep n).nbhd i).card ≤ Dmax) → ∀ (M : ℝ), 0 ≤ M → (∀ (n : ℕ) (i : ι n) (ω : Ω n), |X n i ω| ≤ M) → (∀ (n : ℕ) (i : ι n), (D n).E (X n i) = 0) → ∀ (v : ℕ → ℝ), (∀ (n : ℕ), ((D n).E fun ω => Causalean.SteinMethod.depSum (X n) ω ^ 2) = v n) → ∀ (c : ℝ), 0 < c → (∀ᶠ (n : ℕ) in Filter.atTop, c * ↑(Fintype.card (ι n)) ≤ v n) → Filter.Tendsto (fun n => Fintype.card (ι n)) Filter.atTop Filter.atTop → ∀ (est : (n : ℕ) → Ω n → ℝ) (θ m : ℕ → ℝ), (∀ᶠ (n : ℕ) in Filter.atTop, ∀ (ω : Ω n), Causalean.SteinMethod.depSum (X n) ω = √(m n) * (est n ω - θ n)) → (∀ᶠ (n : ℕ) in Filter.atTop, 0 < m n) → ∀ (vhat : ℕ → ℝ), (∀ᶠ (n : ℕ) in Filter.atTop, v n ≤ vhat n) → ∀ (α z : ℝ), 0 ≤ z → Causalean.Experimentation.DesignBased.stdNormalCdf z = 1 - α / 2 → 1 - α ≤ Filter.liminf (fun n => (D n).Pr fun ω => |θ n - est n ω| ≤ z * √(vhat n / m n)) Filter.atTop","l":39,"f":"Causalean/Experimentation/DesignBased/WaldPipeline.lean"},{"n":"Causalean.Experimentation.DesignBased.diagDepGraph","k":"def","m":"Causalean.Experimentation.DesignBased.IndepSummandsCLT","a":"Experimentation/DesignBased","t":1,"d":"The diagonal dependency graph for the per-coordinate summands X n i = fun w => g n i (w i) under the product-design measure: each coordinate depends only on itself (G a b := a = b). The independence field is the product-measure block independence indepFun_prodDesign_apply_blocks, since distinct coordinate-index sets are disjoint.","s":"{ι : ℕ → Type u_1} → [inst : (n : ℕ) → Fintype (ι n)] → [inst_1 : (n : ℕ) → DecidableEq (ι n)] → {α : (n : ℕ) → ι n → Type u_2} → [inst_2 : (n : ℕ) → (i : ι n) → Fintype (α n i)] → [inst_3 : (n : ℕ) → (i : ι n) → MeasurableSpace (α n i)] → [∀ (n : ℕ) (i : ι n), MeasurableSingletonClass (α n i)] → (D : (n : ℕ) → (i : ι n) → Causalean.Experimentation.DesignBased.FiniteDesign (α n i)) → (g : (n : ℕ) → (i : ι n) → α n i → ℝ) → (n : ℕ) → Causalean.SteinMethod.DepGraph (fun i w => g n i (w i)) (Causalean.Experimentation.DesignBased.prodDesign (D n)).toMeasure","l":51,"f":"Causalean/Experimentation/DesignBased/IndepSummandsCLT.lean"},{"n":"Causalean.Experimentation.DesignBased.diffInMeans","k":"def","m":"Causalean.Experimentation.DesignBased.Estimators.DifferenceInMeans","a":"Experimentation/DesignBased/Estimators","t":1,"d":"The difference-in-means estimator: treated-arm mean of Y1 minus control-arm mean of Y0.","s":"{U : Type u_1} → [Fintype U] → [DecidableEq U] → (n₁ : ℕ) → (U → ℝ) → (U → ℝ) → { S // S.card = n₁ } → ℝ","l":58,"f":"Causalean/Experimentation/DesignBased/Estimators/DifferenceInMeans.lean"},{"n":"Causalean.Experimentation.DesignBased.exists_centeredMonomial_expansion","k":"theorem","m":"Causalean.Experimentation.DesignBased.Designs.BernoulliMoments","a":"Experimentation/DesignBased/Designs","t":2,"d":"Completeness of the centered-monomial basis. Every real-valued statistic of a Bernoulli assignment can be written as a linear combination of the centered monomials indexed by the subsets of the population — the design-based Fourier (Walsh) expansion of the statistic.","s":"∀ {U : Type u_1} [inst : Fintype U] (p : U → ℝ) (F : (U → Bool) → ℝ), ∃ a, ∀ (z : U → Bool), F z = ∑ S ∈ Finset.univ.powerset, a S * Causalean.Experimentation.DesignBased.centeredMonomial p S z","l":278,"f":"Causalean/Experimentation/DesignBased/Designs/BernoulliMoments.lean"},{"n":"Causalean.Experimentation.DesignBased.exists_isMinimaxOn","k":"theorem","m":"Causalean.Experimentation.DesignBased.Optimality.Minimax","a":"Experimentation/DesignBased/Optimality","t":1,"d":"Existence of a minimax design. For a design family 𝒟 and a risk criterion R indexed by states of nature, if the state set s is nonempty, the design family 𝒟 is finite, and 𝒟 is nonempty, then there exists a design in 𝒟 that is minimax — it minimizes the worst-case risk over s among all members of 𝒟. Immediate from exists_isOptimalOn applied to the worst-case-risk criterion.","s":"∀ {Ω : Type u_1} [inst : Fintype Ω] {Y : Type u_2} (𝒟 : Causalean.Experimentation.DesignBased.DesignFamily Ω) (s : Finset Y) (hs : s.Nonempty) {α : Type u_3} [inst_1 : LinearOrder α] (R : Y → Causalean.Experimentation.DesignBased.FiniteDesign Ω → α), Set.Finite 𝒟 → Set.Nonempty 𝒟 → ∃ D₀, Causalean.Experimentation.DesignBased.IsMinimaxOn 𝒟 s hs R D₀","l":57,"f":"Causalean/Experimentation/DesignBased/Optimality/Minimax.lean"},{"n":"Causalean.Experimentation.DesignBased.exists_isOptimalOn","k":"theorem","m":"Causalean.Experimentation.DesignBased.Optimality","a":"Experimentation/DesignBased/Optimality","t":1,"d":"Existence of an optimal design. For a design family 𝒟 and a risk criterion R valued in a linear order, if 𝒟 is finite and 𝒟 is nonempty, then there exists a design in 𝒟 that is optimal — it dominates every other member of the family under R.","s":"∀ {Ω : Type u_1} [inst : Fintype Ω] {α : Type u_2} [inst_1 : LinearOrder α] (𝒟 : Causalean.Experimentation.DesignBased.DesignFamily Ω) (R : Causalean.Experimentation.DesignBased.FiniteDesign Ω → α), Set.Finite 𝒟 → Set.Nonempty 𝒟 → ∃ D₀, Causalean.Experimentation.DesignBased.IsOptimalOn 𝒟 R D₀","l":72,"f":"Causalean/Experimentation/DesignBased/Optimality.lean"},{"n":"Causalean.Experimentation.DesignBased.expo","k":"def","m":"Causalean.Experimentation.DesignBased.Exposure","a":"Experimentation/DesignBased","t":1,"d":"The exposure unit i receives under assignment z: f z (θ i).","s":"{Ω : Type u_1} → {ι : Type u_2} → {Θ : Type u_3} → {Δ : Type u_4} → (Ω → Θ → Δ) → (ι → Θ) → ι → Ω → Δ","l":45,"f":"Causalean/Experimentation/DesignBased/Exposure.lean"},{"n":"Causalean.Experimentation.DesignBased.expoInd","k":"def","m":"Causalean.Experimentation.DesignBased.Exposure","a":"Experimentation/DesignBased","t":1,"d":"Indicator that unit i is in exposure condition d under assignment z.","s":"{Ω : Type u_1} → {ι : Type u_2} → {Θ : Type u_3} → {Δ : Type u_4} → [DecidableEq Δ] → (Ω → Θ → Δ) → (ι → Θ) → ι → Δ → Ω → ℝ","l":50,"f":"Causalean/Experimentation/DesignBased/Exposure.lean"},{"n":"Causalean.Experimentation.DesignBased.expoInd_mul_Yobs","k":"theorem","m":"Causalean.Experimentation.DesignBased.PotentialOutcome","a":"Experimentation/DesignBased","t":2,"d":"On the event expo i = d, the observed outcome agrees with the potential outcome y i d; hence 1(expo i = d)·Yobs i = 1(expo i = d)·y i d.","s":"∀ {Ω : Type u_1} {ι : Type u_2} {Θ : Type u_3} {Δ : Type u_4} [inst : DecidableEq Δ] (y : ι → Δ → ℝ) (f : Ω → Θ → Δ) (θ : ι → Θ) (i : ι) (d : Δ) (z : Ω), Causalean.Experimentation.DesignBased.expoInd f θ i d z * Causalean.Experimentation.DesignBased.Yobs y f θ i z = Causalean.Experimentation.DesignBased.expoInd f θ i d z * y i d","l":67,"f":"Causalean/Experimentation/DesignBased/PotentialOutcome.lean"},{"n":"Causalean.Experimentation.DesignBased.expoInd_mul_Yobs_sq","k":"theorem","m":"Causalean.Experimentation.DesignBased.PotentialOutcome","a":"Experimentation/DesignBased","t":2,"d":"Squared on-event substitution: 1(expo i = d)·(Yobs i)² = 1(expo i = d)·(y i d)².","s":"∀ {Ω : Type u_1} {ι : Type u_2} {Θ : Type u_3} {Δ : Type u_4} [inst : DecidableEq Δ] (y : ι → Δ → ℝ) (f : Ω → Θ → Δ) (θ : ι → Θ) (i : ι) (d : Δ) (z : Ω), Causalean.Experimentation.DesignBased.expoInd f θ i d z * Causalean.Experimentation.DesignBased.Yobs y f θ i z ^ 2 = Causalean.Experimentation.DesignBased.expoInd f θ i d z * y i d ^ 2","l":75,"f":"Causalean/Experimentation/DesignBased/PotentialOutcome.lean"},{"n":"Causalean.Experimentation.DesignBased.expoInd_mul_self_of_ne","k":"theorem","m":"Causalean.Experimentation.DesignBased.Exposure","a":"Experimentation/DesignBased","t":2,"d":"A unit cannot be in two distinct exposures at once: the indicators are pointwise disjoint.","s":"∀ {Ω : Type u_1} {ι : Type u_2} {Θ : Type u_3} {Δ : Type u_4} [inst : DecidableEq Δ] (f : Ω → Θ → Δ) (θ : ι → Θ) (i : ι) {d d' : Δ}, d ≠ d' → ∀ (z : Ω), Causalean.Experimentation.DesignBased.expoInd f θ i d z * Causalean.Experimentation.DesignBased.expoInd f θ i d' z = 0","l":96,"f":"Causalean/Experimentation/DesignBased/Exposure.lean"},{"n":"Causalean.Experimentation.DesignBased.expoInd₂_mul_Yobs","k":"theorem","m":"Causalean.Experimentation.DesignBased.PotentialOutcome","a":"Experimentation/DesignBased","t":2,"d":"On the event expo i = di, multiplying the observed outcome for unit i by any real quantity leaves it valid to replace that outcome by potential outcome y i di.","s":"∀ {Ω : Type u_1} {ι : Type u_2} {Θ : Type u_3} {Δ : Type u_4} [inst : DecidableEq Δ] (y : ι → Δ → ℝ) (f : Ω → Θ → Δ) (θ : ι → Θ) (i : ι) (di : Δ) (q : Ω → ℝ) (z : Ω), Causalean.Experimentation.DesignBased.expoInd f θ i di z * q z * Causalean.Experimentation.DesignBased.Yobs y f θ i z = Causalean.Experimentation.DesignBased.expoInd f θ i di z * q z * y i di","l":82,"f":"Causalean/Experimentation/DesignBased/PotentialOutcome.lean"},{"n":"Causalean.Experimentation.DesignBased.finiteDesign_cdf_converging_together","k":"theorem","m":"Causalean.Experimentation.DesignBased.Slutsky","a":"Experimentation/DesignBased","t":1,"d":"Finite-design CDF converging-together. Along a sequence of finite designs on two sequences of real-valued statistics Sₙ and Tₙ, suppose Sₙ is asymptotically indistinguishable from Tₙ in probability: for every η > 0, Pr(η ≤ |Sₙ − Tₙ|) → 0, the CDFs of Tₙ converge pointwise to a limit function Φ, and Φ is continuous. Then the CDFs of Sₙ converge pointwise to the same Φ.","s":"∀ {Ω : ℕ → Type u_1} [inst : (n : ℕ) → Fintype (Ω n)] (D : (n : ℕ) → Causalean.Experimentation.DesignBased.FiniteDesign (Ω n)) (S T : (n : ℕ) → Ω n → ℝ) (Φ : ℝ → ℝ), (∀ (η : ℝ), 0 < η → Filter.Tendsto (fun n => (D n).Pr fun z => η ≤ |S n z - T n z|) Filter.atTop (nhds 0)) → (∀ (x : ℝ), Filter.Tendsto (fun n => (D n).Pr fun z => T n z ≤ x) Filter.atTop (nhds (Φ x))) → Continuous Φ → ∀ (x : ℝ), Filter.Tendsto (fun n => (D n).Pr fun z => S n z ≤ x) Filter.atTop (nhds (Φ x))","l":29,"f":"Causalean/Experimentation/DesignBased/Slutsky.lean"},{"n":"Causalean.Experimentation.DesignBased.htEffect","k":"def","m":"Causalean.Experimentation.DesignBased.HT.Estimator","a":"Experimentation/DesignBased/HT","t":1,"d":"Horvitz–Thompson estimator of the average causal effect of dk versus dl.","s":"{Ω : Type u_1} → [inst : Fintype Ω] → {ι : Type u_2} → {Θ : Type u_3} → {Δ : Type u_4} → [Fintype ι] → [DecidableEq Δ] → Causalean.Experimentation.DesignBased.FiniteDesign Ω → (ι → Δ → ℝ) → (Ω → Θ → Δ) → (ι → Θ) → Δ → Δ → Ω → ℝ","l":63,"f":"Causalean/Experimentation/DesignBased/HT/Estimator.lean"},{"n":"Causalean.Experimentation.DesignBased.htMean","k":"def","m":"Causalean.Experimentation.DesignBased.HT.Estimator","a":"Experimentation/DesignBased/HT","t":1,"d":"Horvitz–Thompson estimator of the mean potential outcome under exposure d.","s":"{Ω : Type u_1} → [inst : Fintype Ω] → {ι : Type u_2} → {Θ : Type u_3} → {Δ : Type u_4} → [Fintype ι] → [DecidableEq Δ] → Causalean.Experimentation.DesignBased.FiniteDesign Ω → (ι → Δ → ℝ) → (Ω → Θ → Δ) → (ι → Θ) → Δ → Ω → ℝ","l":58,"f":"Causalean/Experimentation/DesignBased/HT/Estimator.lean"},{"n":"Causalean.Experimentation.DesignBased.htTotal","k":"def","m":"Causalean.Experimentation.DesignBased.HT.Estimator","a":"Experimentation/DesignBased/HT","t":1,"d":"Totalized Horvitz–Thompson estimator of the total potential outcome under exposure d.","s":"{Ω : Type u_1} → [inst : Fintype Ω] → {ι : Type u_2} → {Θ : Type u_3} → {Δ : Type u_4} → [Fintype ι] → [DecidableEq Δ] → Causalean.Experimentation.DesignBased.FiniteDesign Ω → (ι → Δ → ℝ) → (Ω → Θ → Δ) → (ι → Θ) → Δ → Ω → ℝ","l":45,"f":"Causalean/Experimentation/DesignBased/HT/Estimator.lean"},{"n":"Causalean.Experimentation.DesignBased.htTotal_eq","k":"theorem","m":"Causalean.Experimentation.DesignBased.HT.Estimator","a":"Experimentation/DesignBased/HT","t":1,"d":"For a finite design D, potential outcomes y, exposure map f, assignment θ, exposure level d, and realized assignment z, the Horvitz–Thompson total equals the same sum with the observed outcome replaced termwise by the potential outcome y i d, because on each unit's exposure-indicator term the observed outcome coincides with y i d.","s":"∀ {Ω : Type u_1} [inst : Fintype Ω] {ι : Type u_2} {Θ : Type u_3} {Δ : Type u_4} [inst_1 : Fintype ι] [inst_2 : DecidableEq Δ] (D : Causalean.Experimentation.DesignBased.FiniteDesign Ω) (y : ι → Δ → ℝ) (f : Ω → Θ → Δ) (θ : ι → Θ) (d : Δ) (z : Ω), Causalean.Experimentation.DesignBased.htTotal D y f θ d z = ∑ i, Causalean.Experimentation.DesignBased.expoInd f θ i d z * y i d / Causalean.Experimentation.DesignBased.prop D f θ i d","l":68,"f":"Causalean/Experimentation/DesignBased/HT/Estimator.lean"},{"n":"Causalean.Experimentation.DesignBased.iIndepFun_prodDesign_eval","k":"theorem","m":"Causalean.Experimentation.DesignBased.ProductMeasure","a":"Experimentation/DesignBased","t":2,"d":"Coordinate evaluations are mutually independent under the product-design measure: viewing fun (i) (w) => w i as the family of coordinate projections, this family is iIndepFun under (prodDesign D).toMeasure. (The push-forward of iIndepFun_pi along prodDesign_toMeasure_eq_pi.)","s":"∀ {ι : Type u_1} [inst : Fintype ι] [inst_1 : DecidableEq ι] {α : ι → Type u_2} [inst_2 : (i : ι) → Fintype (α i)] [inst_3 : (i : ι) → MeasurableSpace (α i)] [∀ (i : ι), MeasurableSingletonClass (α i)] (D : (i : ι) → Causalean.Experimentation.DesignBased.FiniteDesign (α i)), ProbabilityTheory.iIndepFun (fun i w => w i) (Causalean.Experimentation.DesignBased.prodDesign D).toMeasure","l":86,"f":"Causalean/Experimentation/DesignBased/ProductMeasure.lean"},{"n":"Causalean.Experimentation.DesignBased.indepFun_prodDesign_apply_blocks","k":"theorem","m":"Causalean.Experimentation.DesignBased.ProductMeasure","a":"Experimentation/DesignBased","t":2,"d":"Functions of disjoint coordinate blocks are independent. Applying separate measurable function families to the coordinates in two disjoint blocks preserves their independence, even when the two blocks have different coordinatewise output spaces.","s":"∀ {ι : Type u_1} [inst : Fintype ι] [inst_1 : DecidableEq ι] {α : ι → Type u_2} [inst_2 : (i : ι) → Fintype (α i)] [inst_3 : (i : ι) → MeasurableSpace (α i)] [∀ (i : ι), MeasurableSingletonClass (α i)] (D : (i : ι) → Causalean.Experimentation.DesignBased.FiniteDesign (α i)) {β : ι → Type u_3} {γ : ι → Type u_4} [inst_5 : (i : ι) → MeasurableSpace (β i)] [inst_6 : (i : ι) → MeasurableSpace (γ i)] {g : (i : ι) → α i → β i} {h : (i : ι) → α i → γ i}, (∀ (i : ι), Measurable (g i)) → (∀ (i : ι), Measurable (h i)) → ∀ {A B : Finset ι}, Disjoint A B → ProbabilityTheory.IndepFun (fun w k => g (↑k) (w ↑k)) (fun w k => h (↑k) (w ↑k)) (Causalean.Experimentation.DesignBased.prodDesign D).toMeasure","l":113,"f":"Causalean/Experimentation/DesignBased/ProductMeasure.lean"},{"n":"Causalean.Experimentation.DesignBased.indepFun_prodDesign_blocks","k":"theorem","m":"Causalean.Experimentation.DesignBased.ProductMeasure","a":"Experimentation/DesignBased","t":1,"d":"Disjoint-block independence. For a family of independent coordinate designs D, suppose the finite index sets A and B are disjoint. Then the tuple of coordinates indexed by A is probabilistically independent of the tuple of coordinates indexed by B, under the measure induced by the product design. This is the form a diagonal dependency graph G a b := a = b consumes.","s":"∀ {ι : Type u_1} [inst : Fintype ι] [inst_1 : DecidableEq ι] {α : ι → Type u_2} [inst_2 : (i : ι) → Fintype (α i)] [inst_3 : (i : ι) → MeasurableSpace (α i)] [∀ (i : ι), MeasurableSingletonClass (α i)] (D : (i : ι) → Causalean.Experimentation.DesignBased.FiniteDesign (α i)) {A B : Finset ι}, Disjoint A B → ProbabilityTheory.IndepFun (fun w k => w ↑k) (fun w k => w ↑k) (Causalean.Experimentation.DesignBased.prodDesign D).toMeasure","l":101,"f":"Causalean/Experimentation/DesignBased/ProductMeasure.lean"},{"n":"Causalean.Experimentation.DesignBased.indepFun_prodDesign_eval","k":"theorem","m":"Causalean.Experimentation.DesignBased.ProductMeasure","a":"Experimentation/DesignBased","t":2,"d":"For i ≠ j, the two coordinate evaluations w ↦ w i and w ↦ w j are independent under the product-design measure.","s":"∀ {ι : Type u_1} [inst : Fintype ι] [inst_1 : DecidableEq ι] {α : ι → Type u_2} [inst_2 : (i : ι) → Fintype (α i)] [inst_3 : (i : ι) → MeasurableSpace (α i)] [∀ (i : ι), MeasurableSingletonClass (α i)] (D : (i : ι) → Causalean.Experimentation.DesignBased.FiniteDesign (α i)) {i j : ι}, i ≠ j → ProbabilityTheory.IndepFun (fun w => w i) (fun w => w j) (Causalean.Experimentation.DesignBased.prodDesign D).toMeasure","l":95,"f":"Causalean/Experimentation/DesignBased/ProductMeasure.lean"},{"n":"Causalean.Experimentation.DesignBased.indepFun_prodDesign_of_depends_on_disjoint_blocks","k":"theorem","m":"Causalean.Experimentation.DesignBased.ProductMeasure","a":"Experimentation/DesignBased","t":2,"d":"General functions of disjoint coordinate blocks are independent. The strict generalization of indepFun_prodDesign_apply_blocks from coordinatewise-separable maps (g k (w k))_k to *arbitrary* measurable functions that each depend only on a block: if F is unchanged by any assignment that agrees on S, and G is unchanged by any assignment that agrees on the disjoint set T, then F and G are independent under the product-design measure. This is the hypothesis a dependency-graph construction actually consumes, where each summand depends jointly (not coordinatewise) on a neighbourhood block.","s":"∀ {ι : Type u_1} [inst : Fintype ι] [inst_1 : DecidableEq ι] {α : ι → Type u_2} [inst_2 : (i : ι) → Fintype (α i)] [inst_3 : (i : ι) → MeasurableSpace (α i)] [∀ (i : ι), MeasurableSingletonClass (α i)] {β : Type u_3} {γ : Type u_4} [inst_5 : MeasurableSpace β] [inst_6 : MeasurableSpace γ] (D : (i : ι) → Causalean.Experimentation.DesignBased.FiniteDesign (α i)) {S T : Finset ι}, Disjoint S T → ∀ (F : ((i : ι) → α i) → β) (G : ((i : ι) → α i) → γ), (∀ (w w' : (i : ι) → α i), (∀ i ∈ S, w i = w' i) → F w = F w') → (∀ (w w' : (i : ι) → α i), (∀ i ∈ T, w i = w' i) → G w = G w') → ProbabilityTheory.IndepFun F G (Causalean.Experimentation.DesignBased.prodDesign D).toMeasure","l":128,"f":"Causalean/Experimentation/DesignBased/ProductMeasure.lean"},{"n":"Causalean.Experimentation.DesignBased.le_worstRisk","k":"theorem","m":"Causalean.Experimentation.DesignBased.Optimality.Minimax","a":"Experimentation/DesignBased/Optimality","t":2,"d":"The worst-case risk dominates the risk at every state in the family.","s":"∀ {Ω : Type u_1} [inst : Fintype Ω] {Y : Type u_2} {α : Type u_3} [inst_1 : SemilatticeSup α] (s : Finset Y) (hs : s.Nonempty) (R : Y → Causalean.Experimentation.DesignBased.FiniteDesign Ω → α) (D : Causalean.Experimentation.DesignBased.FiniteDesign Ω) {y : Y}, y ∈ s → R y D ≤ Causalean.Experimentation.DesignBased.worstRisk s hs R D","l":44,"f":"Causalean/Experimentation/DesignBased/Optimality/Minimax.lean"},{"n":"Causalean.Experimentation.DesignBased.monotone_stdNormalCdf","k":"theorem","m":"Causalean.Experimentation.DesignBased.GaussianCDF","a":"Experimentation/DesignBased","t":2,"d":"The standard-normal cumulative distribution function is monotone in its threshold.","s":"Monotone Causalean.Experimentation.DesignBased.stdNormalCdf","l":55,"f":"Causalean/Experimentation/DesignBased/GaussianCDF.lean"},{"n":"Causalean.Experimentation.DesignBased.mseRisk","k":"def","m":"Causalean.Experimentation.DesignBased.Optimality","a":"Experimentation/DesignBased/Optimality","t":1,"d":"The mean-squared-error risk of a design-indexed estimator est for target μ: the risk assigned to a design D is est D's mean squared error under D.","s":"{Ω : Type u_1} → [inst : Fintype Ω] → (Causalean.Experimentation.DesignBased.FiniteDesign Ω → Ω → ℝ) → ℝ → Causalean.Experimentation.DesignBased.FiniteDesign Ω → ℝ","l":91,"f":"Causalean/Experimentation/DesignBased/Optimality.lean"},{"n":"Causalean.Experimentation.DesignBased.mseRisk_eq_var_of_unbiased","k":"theorem","m":"Causalean.Experimentation.DesignBased.Optimality","a":"Experimentation/DesignBased/Optimality","t":2,"d":"For an estimator that is unbiased under every design in the family, the mean-squared-error risk coincides with the variance there.","s":"∀ {Ω : Type u_1} [inst : Fintype Ω] {est : Causalean.Experimentation.DesignBased.FiniteDesign Ω → Ω → ℝ} {μ : ℝ} {D : Causalean.Experimentation.DesignBased.FiniteDesign Ω}, D.Unbiased (est D) μ → Causalean.Experimentation.DesignBased.mseRisk est μ D = D.Var (est D)","l":101,"f":"Causalean/Experimentation/DesignBased/Optimality.lean"},{"n":"Causalean.Experimentation.DesignBased.mseRisk_nonneg","k":"theorem","m":"Causalean.Experimentation.DesignBased.Optimality","a":"Experimentation/DesignBased/Optimality","t":2,"d":"Mean-squared-error risk is nonnegative.","s":"∀ {Ω : Type u_1} [inst : Fintype Ω] (est : Causalean.Experimentation.DesignBased.FiniteDesign Ω → Ω → ℝ) (μ : ℝ) (D : Causalean.Experimentation.DesignBased.FiniteDesign Ω), 0 ≤ Causalean.Experimentation.DesignBased.mseRisk est μ D","l":96,"f":"Causalean/Experimentation/DesignBased/Optimality.lean"},{"n":"Causalean.Experimentation.DesignBased.muTrue","k":"def","m":"Causalean.Experimentation.DesignBased.HT.Estimator","a":"Experimentation/DesignBased/HT","t":1,"d":"Population mean potential outcome under exposure d: μ(d) = (1/N)∑ᵢ y i d.","s":"{ι : Type u_2} → {Δ : Type u_4} → [Fintype ι] → (ι → Δ → ℝ) → Δ → ℝ","l":50,"f":"Causalean/Experimentation/DesignBased/HT/Estimator.lean"},{"n":"Causalean.Experimentation.DesignBased.neymanFraction","k":"def","m":"Causalean.Experimentation.DesignBased.Optimality.Neyman","a":"Experimentation/DesignBased/Optimality","t":1,"d":"The Neyman-fraction formula: √A / (√A + √B), the treatment fraction that minimizes the two-arm variance A/x + B/(1−x) when the arm variances are positive.","s":"ℝ → ℝ → ℝ","l":37,"f":"Causalean/Experimentation/DesignBased/Optimality/Neyman.lean"},{"n":"Causalean.Experimentation.DesignBased.neymanFraction_mem_Ioo","k":"theorem","m":"Causalean.Experimentation.DesignBased.Optimality.Neyman","a":"Experimentation/DesignBased/Optimality","t":2,"d":"The Neyman fraction lies in (0,1).","s":"∀ {A B : ℝ}, 0 < A → 0 < B → 0 < Causalean.Experimentation.DesignBased.neymanFraction A B ∧ Causalean.Experimentation.DesignBased.neymanFraction A B < 1","l":65,"f":"Causalean/Experimentation/DesignBased/Optimality/Neyman.lean"},{"n":"Causalean.Experimentation.DesignBased.neymanOptimalValue","k":"def","m":"Causalean.Experimentation.DesignBased.Optimality.Neyman","a":"Experimentation/DesignBased/Optimality","t":1,"d":"The Neyman optimal-value formula: (√A + √B)², the minimized two-arm variance when the arm variances are positive.","s":"ℝ → ℝ → ℝ","l":41,"f":"Causalean/Experimentation/DesignBased/Optimality/Neyman.lean"},{"n":"Causalean.Experimentation.DesignBased.neyman_allocation_eq_at_fraction","k":"theorem","m":"Causalean.Experimentation.DesignBased.Optimality.Neyman","a":"Experimentation/DesignBased/Optimality","t":1,"d":"Neyman allocation optimum. For nonnegative treated-arm and control-arm outcome variances A and B, evaluating the two-arm variance A/x + B/(1−x) at the Neyman fraction x* = √A/(√A+√B) yields exactly its lower bound (√A + √B)²; hence the Neyman fraction minimizes the variance over (0,1).","s":"∀ {A B : ℝ}, 0 ≤ A → 0 ≤ B → A / Causalean.Experimentation.DesignBased.neymanFraction A B + B / (1 - Causalean.Experimentation.DesignBased.neymanFraction A B) = Causalean.Experimentation.DesignBased.neymanOptimalValue A B","l":76,"f":"Causalean/Experimentation/DesignBased/Optimality/Neyman.lean"},{"n":"Causalean.Experimentation.DesignBased.neyman_allocation_isMinimizer","k":"theorem","m":"Causalean.Experimentation.DesignBased.Optimality.Neyman","a":"Experimentation/DesignBased/Optimality","t":2,"d":"The Neyman fraction is a minimizer: its two-arm variance is no larger than the variance at any treatment fraction in (0,1).","s":"∀ {A B : ℝ}, 0 ≤ A → 0 ≤ B → ∀ (x : ℝ), 0 < x → x < 1 → A / Causalean.Experimentation.DesignBased.neymanFraction A B + B / (1 - Causalean.Experimentation.DesignBased.neymanFraction A B) ≤ A / x + B / (1 - x)","l":103,"f":"Causalean/Experimentation/DesignBased/Optimality/Neyman.lean"},{"n":"Causalean.Experimentation.DesignBased.neyman_allocation_lower_bound","k":"theorem","m":"Causalean.Experimentation.DesignBased.Optimality.Neyman","a":"Experimentation/DesignBased/Optimality","t":2,"d":"Neyman allocation lower bound. For nonnegative arm variances A, B and any treatment fraction x ∈ (0,1), the two-arm variance is at least (√A + √B)².","s":"∀ {A B x : ℝ}, 0 ≤ A → 0 ≤ B → 0 < x → x < 1 → Causalean.Experimentation.DesignBased.neymanOptimalValue A B ≤ A / x + B / (1 - x)","l":45,"f":"Causalean/Experimentation/DesignBased/Optimality/Neyman.lean"},{"n":"Causalean.Experimentation.DesignBased.prodDesign","k":"def","m":"Causalean.Experimentation.DesignBased.Product","a":"Experimentation/DesignBased","t":1,"d":"The product design of a finite family D i : FiniteDesign (α i): each coordinate is randomized independently, with joint pmf w ↦ ∏ i, (D i).p (w i).","s":"{ι : Type u_1} → [inst : Fintype ι] → [inst_1 : DecidableEq ι] → {α : ι → Type u_2} → [inst_2 : (i : ι) → Fintype (α i)] → ((i : ι) → Causalean.Experimentation.DesignBased.FiniteDesign (α i)) → Causalean.Experimentation.DesignBased.FiniteDesign ((i : ι) → α i)","l":45,"f":"Causalean/Experimentation/DesignBased/Product.lean"},{"n":"Causalean.Experimentation.DesignBased.prodDesign_clt","k":"theorem","m":"Causalean.Experimentation.DesignBased.IndepSummandsCLT","a":"Experimentation/DesignBased","t":1,"d":"Independent-summands CLT over product designs. Fix a family of coordinate designs D n i and real-valued per-coordinate summands g n i, one pair per stage n and coordinate i. Suppose there is a sequence of nonnegative bounds B n tending to zero, every summand g n i a is bounded in absolute value by B n, the number of coordinates at stage n times B n cubed tends to zero, each summand has mean zero under its own coordinate design, and the total sum ∑ᵢ g n i (w i) has design variance exactly one under the product design at every stage n. Then the design probability that the sum is at most any fixed threshold s converges, as n → ∞, to the standard normal cumulative distribution function Φ(s).","s":"∀ {ι : ℕ → Type u_1} [inst : (n : ℕ) → Fintype (ι n)] [inst_1 : (n : ℕ) → DecidableEq (ι n)] {α : (n : ℕ) → ι n → Type u_2} [inst_2 : (n : ℕ) → (i : ι n) → Fintype (α n i)] [inst_3 : (n : ℕ) → (i : ι n) → MeasurableSpace (α n i)] [∀ (n : ℕ) (i : ι n), MeasurableSingletonClass (α n i)] (D : (n : ℕ) → (i : ι n) → Causalean.Experimentation.DesignBased.FiniteDesign (α n i)) (g : (n : ℕ) → (i : ι n) → α n i → ℝ) (B : ℕ → ℝ), (∀ (n : ℕ), 0 ≤ B n) → Filter.Tendsto B Filter.atTop (nhds 0) → (∀ (n : ℕ) (i : ι n) (a : α n i), |g n i a| ≤ B n) → Filter.Tendsto (fun n => ↑(Fintype.card (ι n)) * B n ^ 3) Filter.atTop (nhds 0) → (∀ (n : ℕ) (i : ι n), (D n i).E (g n i) = 0) → (∀ (n : ℕ), ((Causalean.Experimentation.DesignBased.prodDesign (D n)).Var fun w => ∑ i, g n i (w i)) = 1) → ∀ (s : ℝ), Filter.Tendsto (fun n => (Causalean.Experimentation.DesignBased.prodDesign (D n)).Pr fun w => ∑ i, g n i (w i) ≤ s) Filter.atTop (nhds (Causalean.Experimentation.DesignBased.stdNormalCdf s))","l":72,"f":"Causalean/Experimentation/DesignBased/IndepSummandsCLT.lean"},{"n":"Causalean.Experimentation.DesignBased.prodDesign_p","k":"theorem","m":"Causalean.Experimentation.DesignBased.Product","a":"Experimentation/DesignBased","t":1,"d":"For a family of independent coordinate designs D and an assignment w, the product-design probability of w equals the product of the coordinate probabilities ∏ᵢ (D i).p (w i).","s":"∀ {ι : Type u_1} [inst : Fintype ι] [inst_1 : DecidableEq ι] {α : ι → Type u_2} [inst_2 : (i : ι) → Fintype (α i)] (D : (i : ι) → Causalean.Experimentation.DesignBased.FiniteDesign (α i)) (w : (i : ι) → α i), (Causalean.Experimentation.DesignBased.prodDesign D).p w = ∏ i, (D i).p (w i)","l":55,"f":"Causalean/Experimentation/DesignBased/Product.lean"},{"n":"Causalean.Experimentation.DesignBased.prodDesign_toMeasure_eq_pi","k":"theorem","m":"Causalean.Experimentation.DesignBased.ProductMeasure","a":"Experimentation/DesignBased","t":1,"d":"The product design IS Mathlib's product measure. For a family of independent coordinate designs D, the probability measure induced by the product design prodDesign D coincides exactly with Mathlib's product measure of the coordinate design measures. Both are finite measures on the Fintype ∀ i, α i, so they agree iff they agree on every singleton {w}; on a singleton both sides evaluate to ∏ i, (D i).p (w i).","s":"∀ {ι : Type u_1} [inst : Fintype ι] [inst_1 : DecidableEq ι] {α : ι → Type u_2} [inst_2 : (i : ι) → Fintype (α i)] [inst_3 : (i : ι) → MeasurableSpace (α i)] [∀ (i : ι), MeasurableSingletonClass (α i)] (D : (i : ι) → Causalean.Experimentation.DesignBased.FiniteDesign (α i)), (Causalean.Experimentation.DesignBased.prodDesign D).toMeasure = MeasureTheory.Measure.pi fun i => (D i).toMeasure","l":69,"f":"Causalean/Experimentation/DesignBased/ProductMeasure.lean"},{"n":"Causalean.Experimentation.DesignBased.prop","k":"def","m":"Causalean.Experimentation.DesignBased.Exposure","a":"Experimentation/DesignBased","t":1,"d":"Generalized probability of exposure: π_i(d) = Pr[expo i = d].","s":"{Ω : Type u_1} → [inst : Fintype Ω] → {ι : Type u_2} → {Θ : Type u_3} → {Δ : Type u_4} → [DecidableEq Δ] → Causalean.Experimentation.DesignBased.FiniteDesign Ω → (Ω → Θ → Δ) → (ι → Θ) → ι → Δ → ℝ","l":54,"f":"Causalean/Experimentation/DesignBased/Exposure.lean"},{"n":"Causalean.Experimentation.DesignBased.propPairCross","k":"def","m":"Causalean.Experimentation.DesignBased.Exposure","a":"Experimentation/DesignBased","t":1,"d":"Cross joint exposure probability π_{ij}(d,d') = E[1(expo i = d)·1(expo j = d')].","s":"{Ω : Type u_1} → [inst : Fintype Ω] → {ι : Type u_2} → {Θ : Type u_3} → {Δ : Type u_4} → [DecidableEq Δ] → Causalean.Experimentation.DesignBased.FiniteDesign Ω → (Ω → Θ → Δ) → (ι → Θ) → ι → ι → Δ → Δ → ℝ","l":62,"f":"Causalean/Experimentation/DesignBased/Exposure.lean"},{"n":"Causalean.Experimentation.DesignBased.propPairCross_self_of_ne","k":"theorem","m":"Causalean.Experimentation.DesignBased.Exposure","a":"Experimentation/DesignBased","t":2,"d":"The self cross-exposure joint probability vanishes for distinct exposures.","s":"∀ {Ω : Type u_1} [inst : Fintype Ω] {ι : Type u_2} {Θ : Type u_3} {Δ : Type u_4} [inst_1 : DecidableEq Δ] (D : Causalean.Experimentation.DesignBased.FiniteDesign Ω) (f : Ω → Θ → Δ) (θ : ι → Θ) (i : ι) {d d' : Δ}, d ≠ d' → Causalean.Experimentation.DesignBased.propPairCross D f θ i i d d' = 0","l":105,"f":"Causalean/Experimentation/DesignBased/Exposure.lean"},{"n":"Causalean.Experimentation.DesignBased.propPairSame","k":"def","m":"Causalean.Experimentation.DesignBased.Exposure","a":"Experimentation/DesignBased","t":1,"d":"Joint exposure probability π_{ij}(d) = E[1(expo i = d)·1(expo j = d)].","s":"{Ω : Type u_1} → [inst : Fintype Ω] → {ι : Type u_2} → {Θ : Type u_3} → {Δ : Type u_4} → [DecidableEq Δ] → Causalean.Experimentation.DesignBased.FiniteDesign Ω → (Ω → Θ → Δ) → (ι → Θ) → ι → ι → Δ → ℝ","l":58,"f":"Causalean/Experimentation/DesignBased/Exposure.lean"},{"n":"Causalean.Experimentation.DesignBased.ratio_remainder_capped_bound","k":"theorem","m":"Causalean.Experimentation.DesignBased.RatioLinearization","a":"Experimentation/DesignBased","t":2,"d":"Capped ratio-remainder bound. Consider a ratio estimator A / D of a target μ, scaled by √n (here sn with sn² = n), and let G = A − μ·D be the centered numerator. The scaled second-order remainder √n·(A/D − μ) − √n⁻¹·G equals √n⁻¹·G·(n/D − 1), and on the event where the denominator is at least half its target (equivalently n/D ≤ 2) it is bounded by 2·|√n⁻¹·G|·|n⁻¹·D − 1|. This is the per-arm remainder control a design-based ratio / Hájek CLT uses to reduce the studentized estimator to its linear score.","s":"∀ {n sn A D μ G : ℝ}, 0 < n → sn ^ 2 = n → 0 < D → n / D ≤ 2 → G = A - μ * D → |sn * (A / D - μ) - sn⁻¹ * G| ≤ 2 * |sn⁻¹ * G| * |n⁻¹ * D - 1|","l":34,"f":"Causalean/Experimentation/DesignBased/RatioLinearization.lean"},{"n":"Causalean.Experimentation.DesignBased.regret","k":"def","m":"Causalean.Experimentation.DesignBased.Optimality.Minimax","a":"Experimentation/DesignBased/Optimality","t":1,"d":"The regret of a design D at state y, relative to the finite family 𝒟: how much worse its risk is than the best risk achievable in 𝒟 at that state.","s":"{Ω : Type u_1} → [inst : Fintype Ω] → {Y : Type u_2} → (𝒟 : Causalean.Experimentation.DesignBased.DesignFamily Ω) → Set.Finite 𝒟 → Set.Nonempty 𝒟 → {α : Type u_3} → [inst_1 : AddGroup α] → [inst_2 : LinearOrder α] → [AddRightMono α] → (Y → Causalean.Experimentation.DesignBased.FiniteDesign Ω → α) → Y → Causalean.Experimentation.DesignBased.FiniteDesign Ω → α","l":82,"f":"Causalean/Experimentation/DesignBased/Optimality/Minimax.lean"},{"n":"Causalean.Experimentation.DesignBased.regret_nonneg","k":"theorem","m":"Causalean.Experimentation.DesignBased.Optimality.Minimax","a":"Experimentation/DesignBased/Optimality","t":2,"d":"Regret is nonnegative for every member of the family.","s":"∀ {Ω : Type u_1} [inst : Fintype Ω] {Y : Type u_2} (𝒟 : Causalean.Experimentation.DesignBased.DesignFamily Ω) (hfin : Set.Finite 𝒟) (hne : Set.Nonempty 𝒟) {α : Type u_3} [inst_1 : AddGroup α] [inst_2 : LinearOrder α] [inst_3 : AddRightMono α] (R : Y → Causalean.Experimentation.DesignBased.FiniteDesign Ω → α) (y : Y) {D : Causalean.Experimentation.DesignBased.FiniteDesign Ω}, D ∈ 𝒟 → 0 ≤ Causalean.Experimentation.DesignBased.regret 𝒟 hfin hne R y D","l":89,"f":"Causalean/Experimentation/DesignBased/Optimality/Minimax.lean"},{"n":"Causalean.Experimentation.DesignBased.sateEstimand","k":"def","m":"Causalean.Experimentation.DesignBased.Estimators.DifferenceInMeans","a":"Experimentation/DesignBased/Estimators","t":1,"d":"The sample average treatment effect (SATE): the population mean of the unit-level treatment effects Y1 i − Y0 i.","s":"{U : Type u_1} → [Fintype U] → (U → ℝ) → (U → ℝ) → ℝ","l":43,"f":"Causalean/Experimentation/DesignBased/Estimators/DifferenceInMeans.lean"},{"n":"Causalean.Experimentation.DesignBased.stdNormalCdf","k":"def","m":"Causalean.Experimentation.DesignBased.GaussianCDF","a":"Experimentation/DesignBased","t":1,"d":"Standard normal cumulative distribution function Φ(t) = P[Z ≤ t], Z ∼ 𝒩(0,1), in the (gaussianReal 0 1).real (Iic t) presentation used by the design-based interval theorems. Definitionally the canonical Causalean.Mathlib.stdNormalCDF (see stdNormalCdf_eq).","s":"ℝ → ℝ","l":37,"f":"Causalean/Experimentation/DesignBased/GaussianCDF.lean"},{"n":"Causalean.Experimentation.DesignBased.stdNormalCdf_eq","k":"theorem","m":"Causalean.Experimentation.DesignBased.GaussianCDF","a":"Experimentation/DesignBased","t":2,"d":"The design-based .real (Iic) presentation agrees with the canonical stdNormalCDF.","s":"∀ (t : ℝ), Causalean.Experimentation.DesignBased.stdNormalCdf t = Causalean.Mathlib.stdNormalCDF t","l":43,"f":"Causalean/Experimentation/DesignBased/GaussianCDF.lean"},{"n":"Causalean.Experimentation.DesignBased.stdNormalCdf_le_one","k":"theorem","m":"Causalean.Experimentation.DesignBased.GaussianCDF","a":"Experimentation/DesignBased","t":2,"d":"The standard-normal cumulative probability is at most one.","s":"∀ (t : ℝ), Causalean.Experimentation.DesignBased.stdNormalCdf t ≤ 1","l":51,"f":"Causalean/Experimentation/DesignBased/GaussianCDF.lean"},{"n":"Causalean.Experimentation.DesignBased.stdNormalCdf_neg","k":"theorem","m":"Causalean.Experimentation.DesignBased.GaussianCDF","a":"Experimentation/DesignBased","t":1,"d":"Symmetry of the standard normal CDF. For any threshold t, the standard normal CDF satisfies Φ(−t) = 1 − Φ(t).","s":"∀ (t : ℝ), Causalean.Experimentation.DesignBased.stdNormalCdf (-t) = 1 - Causalean.Experimentation.DesignBased.stdNormalCdf t","l":60,"f":"Causalean/Experimentation/DesignBased/GaussianCDF.lean"},{"n":"Causalean.Experimentation.DesignBased.stdNormalCdf_nonneg","k":"theorem","m":"Causalean.Experimentation.DesignBased.GaussianCDF","a":"Experimentation/DesignBased","t":2,"d":"The standard-normal cumulative probability is nonnegative.","s":"∀ (t : ℝ), 0 ≤ Causalean.Experimentation.DesignBased.stdNormalCdf t","l":47,"f":"Causalean/Experimentation/DesignBased/GaussianCDF.lean"},{"n":"Causalean.Experimentation.DesignBased.stratifiedDesign","k":"def","m":"Causalean.Experimentation.DesignBased.Designs.Stratified","a":"Experimentation/DesignBased/Designs","t":1,"d":"The stratified randomization design: an independent complete randomization within each stratum k, treating exactly n₁ k of that stratum's units. Built as the product of the per-stratum complete-randomization designs.","s":"{K : Type u_1} → [inst : Fintype K] → [inst_1 : DecidableEq K] → {V : K → Type u_2} → [inst_2 : (k : K) → Fintype (V k)] → (n₁ : K → ℕ) → (∀ (k : K), n₁ k ≤ Fintype.card (V k)) → Causalean.Experimentation.DesignBased.FiniteDesign ((k : K) → { S // S.card = n₁ k })","l":39,"f":"Causalean/Experimentation/DesignBased/Designs/Stratified.lean"},{"n":"Causalean.Experimentation.DesignBased.stratifiedDesign_incl","k":"theorem","m":"Causalean.Experimentation.DesignBased.Designs.Stratified","a":"Experimentation/DesignBased/Designs","t":2,"d":"First-order inclusion probability. A unit i in stratum k is treated with probability n₁ k / N k, the complete-randomization rate of its own stratum.","s":"∀ {K : Type u_1} [inst : Fintype K] [inst_1 : DecidableEq K] {V : K → Type u_2} [inst_2 : (k : K) → Fintype (V k)] [inst_3 : (k : K) → DecidableEq (V k)] (n₁ : K → ℕ) (hn : ∀ (k : K), n₁ k ≤ Fintype.card (V k)) (k : K) (i : V k), ((Causalean.Experimentation.DesignBased.stratifiedDesign n₁ hn).Pr fun z => i ∈ ↑(z k)) = ↑(n₁ k) / ↑(Fintype.card (V k))","l":46,"f":"Causalean/Experimentation/DesignBased/Designs/Stratified.lean"},{"n":"Causalean.Experimentation.DesignBased.stratifiedDesign_incl_pair_across","k":"theorem","m":"Causalean.Experimentation.DesignBased.Designs.Stratified","a":"Experimentation/DesignBased/Designs","t":1,"d":"Second-order inclusion probability across strata. For a stratified design in which each stratum k treats exactly n₁ k of its N k units, with n₁ k never exceeding N k, if k and k' are distinct strata, then a unit i in stratum k and a unit i' in stratum k' are jointly treated with probability the product of their two strata's treatment rates, (n₁ k / N k) · (n₁ k' / N k') — the strata are randomized independently.","s":"∀ {K : Type u_1} [inst : Fintype K] [inst_1 : DecidableEq K] {V : K → Type u_2} [inst_2 : (k : K) → Fintype (V k)] [inst_3 : (k : K) → DecidableEq (V k)] (n₁ : K → ℕ) (hn : ∀ (k : K), n₁ k ≤ Fintype.card (V k)) {k k' : K}, k ≠ k' → ∀ (i : V k) (i' : V k'), ((Causalean.Experimentation.DesignBased.stratifiedDesign n₁ hn).Pr fun z => i ∈ ↑(z k) ∧ i' ∈ ↑(z k')) = ↑(n₁ k) / ↑(Fintype.card (V k)) * (↑(n₁ k') / ↑(Fintype.card (V k')))","l":86,"f":"Causalean/Experimentation/DesignBased/Designs/Stratified.lean"},{"n":"Causalean.Experimentation.DesignBased.stratifiedDesign_incl_pair_within","k":"theorem","m":"Causalean.Experimentation.DesignBased.Designs.Stratified","a":"Experimentation/DesignBased/Designs","t":2,"d":"Second-order inclusion probability within one stratum. Two distinct units in the same stratum k are jointly treated with the complete-randomization second-order inclusion probability for that stratum.","s":"∀ {K : Type u_1} [inst : Fintype K] [inst_1 : DecidableEq K] {V : K → Type u_2} [inst_2 : (k : K) → Fintype (V k)] [inst_3 : (k : K) → DecidableEq (V k)] (n₁ : K → ℕ) (hn : ∀ (k : K), n₁ k ≤ Fintype.card (V k)) (k : K) {i j : V k}, i ≠ j → ((Causalean.Experimentation.DesignBased.stratifiedDesign n₁ hn).Pr fun z => i ∈ ↑(z k) ∧ j ∈ ↑(z k)) = ↑(n₁ k) * (↑(n₁ k) - 1) / (↑(Fintype.card (V k)) * (↑(Fintype.card (V k)) - 1))","l":64,"f":"Causalean/Experimentation/DesignBased/Designs/Stratified.lean"},{"n":"Causalean.Experimentation.DesignBased.sum_prop_eq_one","k":"theorem","m":"Causalean.Experimentation.DesignBased.Exposure","a":"Experimentation/DesignBased","t":1,"d":"Exposure probabilities sum to one. For a design, an exposure mapping, unit traits, and a unit i, the generalized exposure probabilities of unit i, summed over all exposure levels, equal one.","s":"∀ {Ω : Type u_1} [inst : Fintype Ω] {ι : Type u_2} {Θ : Type u_3} {Δ : Type u_4} [inst_1 : DecidableEq Δ] [inst_2 : Fintype Δ] (D : Causalean.Experimentation.DesignBased.FiniteDesign Ω) (f : Ω → Θ → Δ) (θ : ι → Θ) (i : ι), ∑ d, Causalean.Experimentation.DesignBased.prop D f θ i d = 1","l":115,"f":"Causalean/Experimentation/DesignBased/Exposure.lean"},{"n":"Causalean.Experimentation.DesignBased.tauTrue","k":"def","m":"Causalean.Experimentation.DesignBased.HT.Estimator","a":"Experimentation/DesignBased/HT","t":1,"d":"Average causal effect of exposure dk versus dl: τ = μ(dk) − μ(dl).","s":"{ι : Type u_2} → {Δ : Type u_4} → [Fintype ι] → (ι → Δ → ℝ) → Δ → Δ → ℝ","l":54,"f":"Causalean/Experimentation/DesignBased/HT/Estimator.lean"},{"n":"Causalean.Experimentation.DesignBased.treatInd","k":"def","m":"Causalean.Experimentation.DesignBased.Designs.Bernoulli","a":"Experimentation/DesignBased/Designs","t":1,"d":"The treatment indicator of unit i under an assignment z: 1 if i is treated, else 0.","s":"{U : Type u_1} → U → (U → Bool) → ℝ","l":63,"f":"Causalean/Experimentation/DesignBased/Designs/Bernoulli.lean"},{"n":"Causalean.Experimentation.DesignBased.treatInd_prod_mul_prod","k":"theorem","m":"Causalean.Experimentation.DesignBased.Designs.BernoulliMoments","a":"Experimentation/DesignBased/Designs","t":2,"d":"Treatment indicators take only the values zero and one, so two overlapping products of them collapse into a single product over the union of the two sets of units.","s":"∀ {U : Type u_1} [inst : DecidableEq U] (S T : Finset U) (z : U → Bool), (∏ j ∈ S, Causalean.Experimentation.DesignBased.treatInd j z) * ∏ j ∈ T, Causalean.Experimentation.DesignBased.treatInd j z = ∏ j ∈ S ∪ T, Causalean.Experimentation.DesignBased.treatInd j z","l":105,"f":"Causalean/Experimentation/DesignBased/Designs/BernoulliMoments.lean"},{"n":"Causalean.Experimentation.DesignBased.treatedMean","k":"def","m":"Causalean.Experimentation.DesignBased.Estimators.DifferenceInMeans","a":"Experimentation/DesignBased/Estimators","t":1,"d":"The treated-arm mean of Y1 under a treated set S: the average of Y1 over the n₁ treated units.","s":"{U : Type u_1} → [Fintype U] → [DecidableEq U] → (n₁ : ℕ) → (U → ℝ) → { S // S.card = n₁ } → ℝ","l":48,"f":"Causalean/Experimentation/DesignBased/Estimators/DifferenceInMeans.lean"},{"n":"Causalean.Experimentation.DesignBased.unbiased_diffInMeans","k":"theorem","m":"Causalean.Experimentation.DesignBased.Estimators.DifferenceInMeans","a":"Experimentation/DesignBased/Estimators","t":2,"d":"The difference in means is unbiased for the SATE, in the Unbiased predicate form.","s":"∀ {U : Type u_1} [inst : Fintype U] [inst_1 : DecidableEq U] (n₁ : ℕ), 0 < n₁ → ∀ (hn0 : n₁ < Fintype.card U) (Y1 Y0 : U → ℝ), (Causalean.Experimentation.DesignBased.completeRandomization n₁ (Nat.le_of_lt hn0)).Unbiased (Causalean.Experimentation.DesignBased.diffInMeans n₁ Y1 Y0) (Causalean.Experimentation.DesignBased.sateEstimand Y1 Y0)","l":195,"f":"Causalean/Experimentation/DesignBased/Estimators/DifferenceInMeans.lean"},{"n":"Causalean.Experimentation.DesignBased.unitTreatInd","k":"def","m":"Causalean.Experimentation.DesignBased.Designs.ClusterRandomization","a":"Experimentation/DesignBased/Designs","t":1,"d":"The treatment indicator of unit i under cluster assignment z: 1 if i's cluster is treated, else 0.","s":"{U : Type u_1} → {C : Type u_2} → (U → C) → U → (C → Bool) → ℝ","l":42,"f":"Causalean/Experimentation/DesignBased/Designs/ClusterRandomization.lean"},{"n":"Causalean.Experimentation.DesignBased.worstRisk","k":"def","m":"Causalean.Experimentation.DesignBased.Optimality.Minimax","a":"Experimentation/DesignBased/Optimality","t":1,"d":"The worst-case risk of a design D over a nonempty finite set s of states of nature: the largest risk R y D incurred as the state y ranges over s.","s":"{Ω : Type u_1} → [inst : Fintype Ω] → {Y : Type u_2} → {α : Type u_3} → [SemilatticeSup α] → (s : Finset Y) → s.Nonempty → (Y → Causalean.Experimentation.DesignBased.FiniteDesign Ω → α) → Causalean.Experimentation.DesignBased.FiniteDesign Ω → α","l":38,"f":"Causalean/Experimentation/DesignBased/Optimality/Minimax.lean"},{"n":"Causalean.Experimentation.ExposureMappingInterference.E_htCovEstA_le","k":"theorem","m":"Causalean.Experimentation.ExposureMappingInterference.Variance.Conservative","a":"Experimentation/ExposureMappingInterference/Variance","t":1,"d":"Proposition 5.6 (cova). Suppose only the two treatment arms dk, dl are distinct and every unit has nonzero exposure propensity under dk and under dl — with NO positive-cross-joint assumption. Then the general Horvitz–Thompson covariance estimator Ĉov_A is nonpositively biased for the true design covariance of the two HT totals: zero-cross-joint off-diagonal pairs drop from the first sum, while the Young correction (summing over every j with π_{ij}(dk,dl)=0, including the diagonal) dominates the corresponding −y_i(dk)y_j(dl) covariance contributions termwise via y_i(dk)y_j(dl) ≤ y_i(dk)²/2 + y_j(dl)²/2.","s":"∀ {Ω : Type u_1} [inst : Fintype Ω] {ι : Type u_2} {Θ : Type u_3} {Δ : Type u_4} [inst_1 : Fintype ι] [inst_2 : DecidableEq ι] [inst_3 : DecidableEq Δ] (D : Causalean.Experimentation.DesignBased.FiniteDesign Ω) (y : ι → Δ → ℝ) (f : Ω → Θ → Δ) (θ : ι → Θ) (dk dl : Δ), dk ≠ dl → (∀ (i : ι), Causalean.Experimentation.DesignBased.prop D f θ i dk ≠ 0) → (∀ (i : ι), Causalean.Experimentation.DesignBased.prop D f θ i dl ≠ 0) → D.E (Causalean.Experimentation.ExposureMappingInterference.htCovEstA D y f θ dk dl) ≤ D.Cov (Causalean.Experimentation.DesignBased.htTotal D y f θ dk) (Causalean.Experimentation.DesignBased.htTotal D y f θ dl)","l":570,"f":"Causalean/Experimentation/ExposureMappingInterference/Variance/Conservative.lean"},{"n":"Causalean.Experimentation.ExposureMappingInterference.E_htCovEst_eq_of_noEffect","k":"theorem","m":"Causalean.Experimentation.ExposureMappingInterference.Variance.Conservative","a":"Experimentation/ExposureMappingInterference/Variance","t":1,"d":"Proposition 5.5 (no_bias_cov). Suppose the two treatment arms dk, dl are distinct, every unit has nonzero exposure propensity under dk and under dl, every off-diagonal pair has nonzero cross-arm joint exposure propensity — the positive marginal and positive cross-joint regime — and the two exposures share the same potential outcomes: y_i(dk) = y_i(dl) for every unit i. Then the Horvitz–Thompson covariance estimator is exactly unbiased for the true design covariance of the two HT totals: the Young diagonal correction (y_i²/2 + y_i²/2) = y_i² = y_i(dk)y_i(dl) is exact, so the nonpositive bias of Proposition 5.4 vanishes.","s":"∀ {Ω : Type u_1} [inst : Fintype Ω] {ι : Type u_2} {Θ : Type u_3} {Δ : Type u_4} [inst_1 : Fintype ι] [inst_2 : DecidableEq ι] [inst_3 : DecidableEq Δ] (D : Causalean.Experimentation.DesignBased.FiniteDesign Ω) (y : ι → Δ → ℝ) (f : Ω → Θ → Δ) (θ : ι → Θ) (dk dl : Δ), dk ≠ dl → (∀ (i : ι), Causalean.Experimentation.DesignBased.prop D f θ i dk ≠ 0) → (∀ (i : ι), Causalean.Experimentation.DesignBased.prop D f θ i dl ≠ 0) → (∀ (i j : ι), i ≠ j → Causalean.Experimentation.DesignBased.propPairCross D f θ i j dk dl ≠ 0) → (∀ (i : ι), y i dk = y i dl) → D.E (Causalean.Experimentation.ExposureMappingInterference.htCovEst D y f θ dk dl) = D.Cov (Causalean.Experimentation.DesignBased.htTotal D y f θ dk) (Causalean.Experimentation.DesignBased.htTotal D y f θ dl)","l":453,"f":"Causalean/Experimentation/ExposureMappingInterference/Variance/Conservative.lean"},{"n":"Causalean.Experimentation.ExposureMappingInterference.E_htCovEst_le","k":"theorem","m":"Causalean.Experimentation.ExposureMappingInterference.Variance.Conservative","a":"Experimentation/ExposureMappingInterference/Variance","t":1,"d":"Proposition 5.4 (ncov). Suppose the two treatment arms dk, dl are distinct, every unit has nonzero exposure propensity under dk and under dl, and every off-diagonal pair has nonzero cross-arm joint exposure propensity — the positive marginal and positive cross-joint regime. Then the Horvitz–Thompson covariance estimator is nonpositively biased for the true design covariance of the two HT totals.","s":"∀ {Ω : Type u_1} [inst : Fintype Ω] {ι : Type u_2} {Θ : Type u_3} {Δ : Type u_4} [inst_1 : Fintype ι] [inst_2 : DecidableEq ι] [inst_3 : DecidableEq Δ] (D : Causalean.Experimentation.DesignBased.FiniteDesign Ω) (y : ι → Δ → ℝ) (f : Ω → Θ → Δ) (θ : ι → Θ) (dk dl : Δ), dk ≠ dl → (∀ (i : ι), Causalean.Experimentation.DesignBased.prop D f θ i dk ≠ 0) → (∀ (i : ι), Causalean.Experimentation.DesignBased.prop D f θ i dl ≠ 0) → (∀ (i j : ι), i ≠ j → Causalean.Experimentation.DesignBased.propPairCross D f θ i j dk dl ≠ 0) → D.E (Causalean.Experimentation.ExposureMappingInterference.htCovEst D y f θ dk dl) ≤ D.Cov (Causalean.Experimentation.DesignBased.htTotal D y f θ dk) (Causalean.Experimentation.DesignBased.htTotal D y f θ dl)","l":147,"f":"Causalean/Experimentation/ExposureMappingInterference/Variance/Conservative.lean"},{"n":"Causalean.Experimentation.ExposureMappingInterference.E_htEffectVarEstA_ge","k":"theorem","m":"Causalean.Experimentation.ExposureMappingInterference.Variance.Conservative","a":"Experimentation/ExposureMappingInterference/Variance","t":1,"d":"Proposition 5.7, general form (consvar). Suppose only the two treatment arms dk, dl are distinct and every unit has nonzero exposure propensity under dk and under dl — without any positive-joint assumption. Then the assembled general Horvitz–Thompson effect-variance estimator has nonnegative bias: its expectation is at least the true design variance of the effect estimator τ̂.","s":"∀ {Ω : Type u_1} [inst : Fintype Ω] {ι : Type u_2} {Θ : Type u_3} {Δ : Type u_4} [inst_1 : Fintype ι] [inst_2 : DecidableEq ι] [inst_3 : DecidableEq Δ] (D : Causalean.Experimentation.DesignBased.FiniteDesign Ω) (y : ι → Δ → ℝ) (f : Ω → Θ → Δ) (θ : ι → Θ) (dk dl : Δ), dk ≠ dl → (∀ (i : ι), Causalean.Experimentation.DesignBased.prop D f θ i dk ≠ 0) → (∀ (i : ι), Causalean.Experimentation.DesignBased.prop D f θ i dl ≠ 0) → D.Var (Causalean.Experimentation.DesignBased.htEffect D y f θ dk dl) ≤ D.E (Causalean.Experimentation.ExposureMappingInterference.htEffectVarEstA D y f θ dk dl)","l":758,"f":"Causalean/Experimentation/ExposureMappingInterference/Variance/Conservative.lean"},{"n":"Causalean.Experimentation.ExposureMappingInterference.E_htEffectVarEst_ge","k":"theorem","m":"Causalean.Experimentation.ExposureMappingInterference.Variance.Conservative","a":"Experimentation/ExposureMappingInterference/Variance","t":1,"d":"Proposition 5.7 (consvar). Suppose the two treatment arms dk, dl are distinct, every unit has nonzero exposure propensity under dk and under dl, and every off-diagonal pair has nonzero same-arm joint exposure propensity under dk and under dl, as well as nonzero cross-arm joint exposure propensity — the positive-joint regime. Then the assembled Horvitz–Thompson effect-variance estimator has nonnegative bias: its expectation is at least the true design variance of the effect estimator τ̂.","s":"∀ {Ω : Type u_1} [inst : Fintype Ω] {ι : Type u_2} {Θ : Type u_3} {Δ : Type u_4} [inst_1 : Fintype ι] [inst_2 : DecidableEq ι] [inst_3 : DecidableEq Δ] (D : Causalean.Experimentation.DesignBased.FiniteDesign Ω) (y : ι → Δ → ℝ) (f : Ω → Θ → Δ) (θ : ι → Θ) (dk dl : Δ), dk ≠ dl → (∀ (i : ι), Causalean.Experimentation.DesignBased.prop D f θ i dk ≠ 0) → (∀ (i : ι), Causalean.Experimentation.DesignBased.prop D f θ i dl ≠ 0) → (∀ (i j : ι), i ≠ j → Causalean.Experimentation.DesignBased.propPairSame D f θ i j dk ≠ 0) → (∀ (i j : ι), i ≠ j → Causalean.Experimentation.DesignBased.propPairSame D f θ i j dl ≠ 0) → (∀ (i j : ι), i ≠ j → Causalean.Experimentation.DesignBased.propPairCross D f θ i j dk dl ≠ 0) → D.Var (Causalean.Experimentation.DesignBased.htEffect D y f θ dk dl) ≤ D.E (Causalean.Experimentation.ExposureMappingInterference.htEffectVarEst D y f θ dk dl)","l":250,"f":"Causalean/Experimentation/ExposureMappingInterference/Variance/Conservative.lean"},{"n":"Causalean.Experimentation.ExposureMappingInterference.E_htVarEst","k":"theorem","m":"Causalean.Experimentation.ExposureMappingInterference.Variance.Conservative","a":"Experimentation/ExposureMappingInterference/Variance","t":1,"d":"Lemma 5.1 (varun). In the positive-joint regime — every unit has nonzero exposure propensity under d and every off-diagonal pair has nonzero same-arm joint exposure propensity under d — the Horvitz–Thompson variance estimator is exactly unbiased for the true design variance of the HT total.","s":"∀ {Ω : Type u_1} [inst : Fintype Ω] {ι : Type u_2} {Θ : Type u_3} {Δ : Type u_4} [inst_1 : Fintype ι] [inst_2 : DecidableEq ι] [inst_3 : DecidableEq Δ] (D : Causalean.Experimentation.DesignBased.FiniteDesign Ω) (y : ι → Δ → ℝ) (f : Ω → Θ → Δ) (θ : ι → Θ) (d : Δ), (∀ (i : ι), Causalean.Experimentation.DesignBased.prop D f θ i d ≠ 0) → (∀ (i j : ι), i ≠ j → Causalean.Experimentation.DesignBased.propPairSame D f θ i j d ≠ 0) → D.E (Causalean.Experimentation.ExposureMappingInterference.htVarEst D y f θ d) = D.Var (Causalean.Experimentation.DesignBased.htTotal D y f θ d)","l":78,"f":"Causalean/Experimentation/ExposureMappingInterference/Variance/Conservative.lean"},{"n":"Causalean.Experimentation.ExposureMappingInterference.E_htVarEst_add_htA2_ge","k":"theorem","m":"Causalean.Experimentation.ExposureMappingInterference.Variance.Conservative","a":"Experimentation/ExposureMappingInterference/Variance","t":1,"d":"Proposition 5.3 (a2). Assuming only every unit has nonzero exposure propensity under d, adding the Young correction Â₂ to the Horvitz–Thompson variance estimator makes it conservative for the true design variance of the HT total: Var[ŷᵀ(d)] ≤ E[V̂ + Â₂].","s":"∀ {Ω : Type u_1} [inst : Fintype Ω] {ι : Type u_2} {Θ : Type u_3} {Δ : Type u_4} [inst_1 : Fintype ι] [inst_2 : DecidableEq ι] [inst_3 : DecidableEq Δ] (D : Causalean.Experimentation.DesignBased.FiniteDesign Ω) (y : ι → Δ → ℝ) (f : Ω → Θ → Δ) (θ : ι → Θ) (d : Δ), (∀ (i : ι), Causalean.Experimentation.DesignBased.prop D f θ i d ≠ 0) → D.Var (Causalean.Experimentation.DesignBased.htTotal D y f θ d) ≤ D.E fun z => Causalean.Experimentation.ExposureMappingInterference.htVarEst D y f θ d z + Causalean.Experimentation.ExposureMappingInterference.htA2 D y f θ d z","l":393,"f":"Causalean/Experimentation/ExposureMappingInterference/Variance/Conservative.lean"},{"n":"Causalean.Experimentation.ExposureMappingInterference.E_htVarEst_eq_addBias","k":"theorem","m":"Causalean.Experimentation.ExposureMappingInterference.Variance.Conservative","a":"Experimentation/ExposureMappingInterference/Variance","t":1,"d":"Proposition 5.2 (varbias). Assuming only every unit has nonzero exposure propensity under d — without the positive-joint assumption — the expectation of the Horvitz–Thompson variance estimator equals the true design variance of the HT total plus the signed zero-joint correction A = ∑_{π_{ij}(d)=0} y_i(d)·y_j(d).","s":"∀ {Ω : Type u_1} [inst : Fintype Ω] {ι : Type u_2} {Θ : Type u_3} {Δ : Type u_4} [inst_1 : Fintype ι] [inst_2 : DecidableEq ι] [inst_3 : DecidableEq Δ] (D : Causalean.Experimentation.DesignBased.FiniteDesign Ω) (y : ι → Δ → ℝ) (f : Ω → Θ → Δ) (θ : ι → Θ) (d : Δ), (∀ (i : ι), Causalean.Experimentation.DesignBased.prop D f θ i d ≠ 0) → D.E (Causalean.Experimentation.ExposureMappingInterference.htVarEst D y f θ d) = D.Var (Causalean.Experimentation.DesignBased.htTotal D y f θ d) + ∑ i, ∑ j ∈ Finset.univ.erase i, (if Causalean.Experimentation.DesignBased.propPairSame D f θ i j d = 0 then 1 else 0) * (y i d * y j d)","l":295,"f":"Causalean/Experimentation/ExposureMappingInterference/Variance/Conservative.lean"},{"n":"Causalean.Experimentation.ExposureMappingInterference.Experiment","k":"structure","m":"Causalean.Experimentation.ExposureMappingInterference.Asymptotics.Consistency","a":"Experimentation/ExposureMappingInterference/Asymptotics","t":1,"d":"A design-based experiment on a finite population: a finite assignment space Ω equipped with a randomization design D over it, a finite population of units ι each assigned a trait in a trait space Θ, exposure-indexed potential outcomes y, one real value per unit and exposure in an exposure space Δ, and an exposure mapping f sending each assignment and unit trait to a realized exposure.","s":"Type 1","l":37,"f":"Causalean/Experimentation/ExposureMappingInterference/Asymptotics/Consistency.lean"},{"n":"Causalean.Experimentation.ExposureMappingInterference.Experiment.N","k":"def","m":"Causalean.Experimentation.ExposureMappingInterference.Asymptotics.Consistency","a":"Experimentation/ExposureMappingInterference/Asymptotics","t":1,"d":"Population size.","s":"Causalean.Experimentation.ExposureMappingInterference.Experiment → ℕ","l":77,"f":"Causalean/Experimentation/ExposureMappingInterference/Asymptotics/Consistency.lean"},{"n":"Causalean.Experimentation.ExposureMappingInterference.Experiment.Var_htMean_le","k":"theorem","m":"Causalean.Experimentation.ExposureMappingInterference.Asymptotics.Consistency","a":"Experimentation/ExposureMappingInterference/Asymptotics","t":2,"d":"Variance bound. Under a uniform bound c on |y_i(d)/π_i(d)|, the variance of the HT mean estimator is O((N + ∑_{i,j} g_{ij})/N²).","s":"∀ (E : Causalean.Experimentation.ExposureMappingInterference.Experiment) (d : E.Δ) {c : ℝ}, 0 ≤ c → (∀ (i : E.ι), |E.y i d / Causalean.Experimentation.DesignBased.prop E.D E.f E.θ i d| ≤ c) → E.D.Var (Causalean.Experimentation.DesignBased.htMean E.D E.y E.f E.θ d) ≤ c ^ 2 * (↑E.N + ∑ i, ∑ j ∈ Finset.univ.erase i, E.gdep d i j) / ↑E.N ^ 2","l":97,"f":"Causalean/Experimentation/ExposureMappingInterference/Asymptotics/Consistency.lean"},{"n":"Causalean.Experimentation.ExposureMappingInterference.Experiment.effRaw","k":"def","m":"Causalean.Experimentation.ExposureMappingInterference.Asymptotics.SteinInstance","a":"Experimentation/ExposureMappingInterference/Asymptotics","t":1,"d":"This is one unit's inverse-propensity-weighted contribution to the unnormalized Horvitz-Thompson effect estimator.","s":"(E : Causalean.Experimentation.ExposureMappingInterference.Experiment) → E.Δ → E.Δ → E.ι → E.Ω → ℝ","l":49,"f":"Causalean/Experimentation/ExposureMappingInterference/Asymptotics/SteinInstance.lean"},{"n":"Causalean.Experimentation.ExposureMappingInterference.Experiment.effSummand","k":"def","m":"Causalean.Experimentation.ExposureMappingInterference.Asymptotics.SteinInstance","a":"Experimentation/ExposureMappingInterference/Asymptotics","t":1,"d":"This is the centered and standardized per-unit summand used to form the studentized Horvitz-Thompson effect statistic.","s":"(E : Causalean.Experimentation.ExposureMappingInterference.Experiment) → E.Δ → E.Δ → E.ι → E.Ω → ℝ","l":58,"f":"Causalean/Experimentation/ExposureMappingInterference/Asymptotics/SteinInstance.lean"},{"n":"Causalean.Experimentation.ExposureMappingInterference.Experiment.gdep","k":"def","m":"Causalean.Experimentation.ExposureMappingInterference.Asymptotics.Consistency","a":"Experimentation/ExposureMappingInterference/Asymptotics","t":1,"d":"Pairwise dependency indicator: 0 when the exposure indicators of i and j for exposure d are uncorrelated, 1 otherwise.","s":"(E : Causalean.Experimentation.ExposureMappingInterference.Experiment) → E.Δ → E.ι → E.ι → ℝ","l":71,"f":"Causalean/Experimentation/ExposureMappingInterference/Asymptotics/Consistency.lean"},{"n":"Causalean.Experimentation.ExposureMappingInterference.Experiment.instMeasurableSingletonΩ","k":"instance","m":"Causalean.Experimentation.ExposureMappingInterference.Asymptotics.SteinInstance","a":"Experimentation/ExposureMappingInterference/Asymptotics","t":2,"d":"The top sigma-algebra on a finite assignment space makes every singleton measurable.","s":"∀ (E : Causalean.Experimentation.ExposureMappingInterference.Experiment), MeasurableSingletonClass E.Ω","l":170,"f":"Causalean/Experimentation/ExposureMappingInterference/Asymptotics/SteinInstance.lean"},{"n":"Causalean.Experimentation.ExposureMappingInterference.Experiment.instMeasurableSpaceΩ","k":"instance","m":"Causalean.Experimentation.ExposureMappingInterference.Asymptotics.SteinInstance","a":"Experimentation/ExposureMappingInterference/Asymptotics","t":2,"d":"Equip a finite assignment space with the top σ-algebra (every set measurable), so the design measure and the measure-theoretic CLT machinery apply with no extra hypotheses.","s":"(E : Causalean.Experimentation.ExposureMappingInterference.Experiment) → MeasurableSpace E.Ω","l":166,"f":"Causalean/Experimentation/ExposureMappingInterference/Asymptotics/SteinInstance.lean"},{"n":"Causalean.Experimentation.ExposureMappingInterference.Experiment.studentizedEffect","k":"def","m":"Causalean.Experimentation.ExposureMappingInterference.Asymptotics.SteinCLT","a":"Experimentation/ExposureMappingInterference/Asymptotics","t":1,"d":"The studentized HT effect statistic: centered at τ, scaled by the true standard error √Var[τ̂].","s":"(E : Causalean.Experimentation.ExposureMappingInterference.Experiment) → E.Δ → E.Δ → E.Ω → ℝ","l":36,"f":"Causalean/Experimentation/ExposureMappingInterference/Asymptotics/SteinCLT.lean"},{"n":"Causalean.Experimentation.ExposureMappingInterference.LocalDependenceCLT","k":"structure","m":"Causalean.Experimentation.ExposureMappingInterference.Asymptotics.SteinCLT","a":"Experimentation/ExposureMappingInterference/Asymptotics","t":1,"d":"Chen-Shao local-dependence CLT interface, bounded-neighborhood case. Along a sequence of experiments Exp with a chosen pair of exposure levels dk, dl at each sample size, this records that the design probability that the studentized Horvitz–Thompson effect statistic falls below any fixed threshold converges, as the sample size grows, to the standard normal CDF at that threshold.","s":"(Exp : ℕ → Causalean.Experimentation.ExposureMappingInterference.Experiment) → ((n : ℕ) → (Exp n).Δ) → ((n : ℕ) → (Exp n).Δ) → Prop","l":44,"f":"Causalean/Experimentation/ExposureMappingInterference/Asymptotics/SteinCLT.lean"},{"n":"Causalean.Experimentation.ExposureMappingInterference.abs_vb_le","k":"theorem","m":"Causalean.Experimentation.ExposureMappingInterference.Asymptotics.VarEstQuadBound","a":"Experimentation/ExposureMappingInterference/Asymptotics","t":2,"d":"Boundedness of the edge-function (Conditions 1 + 1'): |vb i j z| ≤ vbBound c₁ c₂ c₃.","s":"∀ {Ω : Type u_1} [inst : Fintype Ω] {ι : Type u_2} {Θ : Type u_3} {Δ : Type u_4} [inst_1 : DecidableEq ι] [inst_2 : DecidableEq Δ] (D : Causalean.Experimentation.DesignBased.FiniteDesign Ω) (y : ι → Δ → ℝ) (f : Ω → Θ → Δ) (θ : ι → Θ) (dk dl : Δ), dk ≠ dl → ∀ {c₁ c₂ c₃ : ℝ}, 0 ≤ c₁ → 0 ≤ c₂ → 0 ≤ c₃ → (∀ (i : ι) (d : Δ), |y i d| ≤ c₁) → (∀ (i : ι), 0 < Causalean.Experimentation.DesignBased.prop D f θ i dk) → (∀ (i : ι), 0 < Causalean.Experimentation.DesignBased.prop D f θ i dl) → (∀ (i : ι), 1 / Causalean.Experimentation.DesignBased.prop D f θ i dk ≤ c₂) → (∀ (i : ι), 1 / Causalean.Experimentation.DesignBased.prop D f θ i dl ≤ c₂) → (∀ (i j : ι), 1 / Causalean.Experimentation.DesignBased.propPairSame D f θ i j dk ≤ c₃) → (∀ (i j : ι), 1 / Causalean.Experimentation.DesignBased.propPairSame D f θ i j dl ≤ c₃) → (∀ (i j : ι), 1 / Causalean.Experimentation.DesignBased.propPairCross D f θ i j dk dl ≤ c₃) → ∀ (i j : ι) (z : Ω), |Causalean.Experimentation.ExposureMappingInterference.vb D y f θ dk dl i j z| ≤ Causalean.Experimentation.ExposureMappingInterference.vbBound c₁ c₂ c₃","l":304,"f":"Causalean/Experimentation/ExposureMappingInterference/Asymptotics/VarEstQuadBound.lean"},{"n":"Causalean.Experimentation.ExposureMappingInterference.diagCov","k":"def","m":"Causalean.Experimentation.ExposureMappingInterference.Asymptotics.VarEstQuadBound","a":"Experimentation/ExposureMappingInterference/Asymptotics","t":1,"d":"Diagonal (Young) correction summand of Ĉov.","s":"{Ω : Type u_1} → [inst : Fintype Ω] → {ι : Type u_2} → {Θ : Type u_3} → {Δ : Type u_4} → [DecidableEq Δ] → Causalean.Experimentation.DesignBased.FiniteDesign Ω → (ι → Δ → ℝ) → (Ω → Θ → Δ) → (ι → Θ) → Δ → Δ → ι → Ω → ℝ","l":67,"f":"Causalean/Experimentation/ExposureMappingInterference/Asymptotics/VarEstQuadBound.lean"},{"n":"Causalean.Experimentation.ExposureMappingInterference.diagVar","k":"def","m":"Causalean.Experimentation.ExposureMappingInterference.Asymptotics.VarEstQuadBound","a":"Experimentation/ExposureMappingInterference/Asymptotics","t":1,"d":"Diagonal (variance) summand of ŷVar(d): 1ᵢ(d)·(1−πᵢ)·(Yᵢ/πᵢ)².","s":"{Ω : Type u_1} → [inst : Fintype Ω] → {ι : Type u_2} → {Θ : Type u_3} → {Δ : Type u_4} → [DecidableEq Δ] → Causalean.Experimentation.DesignBased.FiniteDesign Ω → (ι → Δ → ℝ) → (Ω → Θ → Δ) → (ι → Θ) → Δ → ι → Ω → ℝ","l":47,"f":"Causalean/Experimentation/ExposureMappingInterference/Asymptotics/VarEstQuadBound.lean"},{"n":"Causalean.Experimentation.ExposureMappingInterference.htA2","k":"def","m":"Causalean.Experimentation.ExposureMappingInterference.Variance.Conservative","a":"Experimentation/ExposureMappingInterference/Variance","t":1,"d":"Young-inequality correction Â₂ (eq for Prop 5.3): for each zero-joint pair (i,j) with π_{ij}(d)=0, it adds the diagonal Young terms 1(expo i=d)·(Yobs i)²/(2π_i) + 1(expo j=d)·(Yobs j)²/(2π_j), whose expectation is y_i(d)²/2 + y_j(d)²/2. Added to htVarEst, it makes the estimator conservative even when some joint exposure probabilities vanish.","s":"{Ω : Type u_1} → [inst : Fintype Ω] → {ι : Type u_2} → {Θ : Type u_3} → {Δ : Type u_4} → [Fintype ι] → [DecidableEq ι] → [DecidableEq Δ] → Causalean.Experimentation.DesignBased.FiniteDesign Ω → (ι → Δ → ℝ) → (Ω → Θ → Δ) → (ι → Θ) → Δ → Ω → ℝ","l":381,"f":"Causalean/Experimentation/ExposureMappingInterference/Variance/Conservative.lean"},{"n":"Causalean.Experimentation.ExposureMappingInterference.htCovEst","k":"def","m":"Causalean.Experimentation.ExposureMappingInterference.Variance.Conservative","a":"Experimentation/ExposureMappingInterference/Variance","t":1,"d":"Horvitz–Thompson-type estimator of Cov[ŷᵀ(dk),ŷᵀ(dl)] (eq:ht_cov_estimator), positive-joint regime. Conservative (nonpositively biased) via the Young-inequality diagonal correction.","s":"{Ω : Type u_1} → [inst : Fintype Ω] → {ι : Type u_2} → {Θ : Type u_3} → {Δ : Type u_4} → [Fintype ι] → [DecidableEq ι] → [DecidableEq Δ] → Causalean.Experimentation.DesignBased.FiniteDesign Ω → (ι → Δ → ℝ) → (Ω → Θ → Δ) → (ι → Θ) → Δ → Δ → Ω → ℝ","l":58,"f":"Causalean/Experimentation/ExposureMappingInterference/Variance/Conservative.lean"},{"n":"Causalean.Experimentation.ExposureMappingInterference.htCovEstA","k":"def","m":"Causalean.Experimentation.ExposureMappingInterference.Variance.Conservative","a":"Experimentation/ExposureMappingInterference/Variance","t":1,"d":"General Horvitz–Thompson covariance estimator (eq:ht_cov_general_estimator), handling zero cross-joint exposure probabilities. The first double sum (over j ≠ i) keeps π_{ij}(d_k,d_l) > 0 pairs via /π_{ij} (zero-joint pairs drop, x/0 = 0); the subtracted Young correction ranges over *all* j ∈ U with π_{ij}(d_k,d_l) = 0, including the diagonal j = i (always zero since d_k ≠ d_l), which recovers the positive-joint estimator's diagonal correction.","s":"{Ω : Type u_1} → [inst : Fintype Ω] → {ι : Type u_2} → {Θ : Type u_3} → {Δ : Type u_4} → [Fintype ι] → [DecidableEq ι] → [DecidableEq Δ] → Causalean.Experimentation.DesignBased.FiniteDesign Ω → (ι → Δ → ℝ) → (Ω → Θ → Δ) → (ι → Θ) → Δ → Δ → Ω → ℝ","l":553,"f":"Causalean/Experimentation/ExposureMappingInterference/Variance/Conservative.lean"},{"n":"Causalean.Experimentation.ExposureMappingInterference.htEffectVarEst","k":"def","m":"Causalean.Experimentation.ExposureMappingInterference.Variance.Conservative","a":"Experimentation/ExposureMappingInterference/Variance","t":1,"d":"Conservative variance estimator for Var[τ̂(dk,dl)] (eq:ate_var_estimator), positive-joint regime.","s":"{Ω : Type u_1} → [inst : Fintype Ω] → {ι : Type u_2} → {Θ : Type u_3} → {Δ : Type u_4} → [Fintype ι] → [DecidableEq ι] → [DecidableEq Δ] → Causalean.Experimentation.DesignBased.FiniteDesign Ω → (ι → Δ → ℝ) → (Ω → Θ → Δ) → (ι → Θ) → Δ → Δ → Ω → ℝ","l":71,"f":"Causalean/Experimentation/ExposureMappingInterference/Variance/Conservative.lean"},{"n":"Causalean.Experimentation.ExposureMappingInterference.htEffectVarEstA","k":"def","m":"Causalean.Experimentation.ExposureMappingInterference.Variance.Conservative","a":"Experimentation/ExposureMappingInterference/Variance","t":1,"d":"General conservative variance estimator for Var[τ̂(dk,dl)] (eq:ate_var_estimator), assembling the zero-joint-robust variance corrections Â₂ and the general covariance estimator Ĉov_A.","s":"{Ω : Type u_1} → [inst : Fintype Ω] → {ι : Type u_2} → {Θ : Type u_3} → {Δ : Type u_4} → [Fintype ι] → [DecidableEq ι] → [DecidableEq Δ] → Causalean.Experimentation.DesignBased.FiniteDesign Ω → (ι → Δ → ℝ) → (Ω → Θ → Δ) → (ι → Θ) → Δ → Δ → Ω → ℝ","l":749,"f":"Causalean/Experimentation/ExposureMappingInterference/Variance/Conservative.lean"},{"n":"Causalean.Experimentation.ExposureMappingInterference.htEffectVarEst_undershoot_tendsto_zero","k":"theorem","m":"Causalean.Experimentation.ExposureMappingInterference.Asymptotics.VarianceConsistency","a":"Experimentation/ExposureMappingInterference/Asymptotics","t":1,"d":"Reduction of feasible-interval variance consistency to L²-relative consistency. For a pair of treatment sequences dk, dl with dk n ≠ dl n for every n, suppose every unit has nonzero exposure propensity under dk and under dl, and every off-diagonal pair has nonzero same-arm and nonzero cross-arm joint exposure propensities — the standing overlap conditions giving the conservativeness Var[τ̂] ≤ E[V̂]. If the true effect-estimator variance is everywhere positive and the L²-relative-consistency limit Var[V̂]/Var[τ̂]² → 0 holds, then the conservative variance estimator undershoots the true variance only with vanishing probability: for every ε > 0, Pr[Vh < (1−ε)·Var[τ̂]] → 0.","s":"∀ (Exp : ℕ → Causalean.Experimentation.ExposureMappingInterference.Experiment) (dk dl : (n : ℕ) → (Exp n).Δ), (∀ (n : ℕ), dk n ≠ dl n) → (∀ (n : ℕ) (i : (Exp n).ι), Causalean.Experimentation.DesignBased.prop (Exp n).D (Exp n).f (Exp n).θ i (dk n) ≠ 0) → (∀ (n : ℕ) (i : (Exp n).ι), Causalean.Experimentation.DesignBased.prop (Exp n).D (Exp n).f (Exp n).θ i (dl n) ≠ 0) → (∀ (n : ℕ) (i j : (Exp n).ι), i ≠ j → Causalean.Experimentation.DesignBased.propPairSame (Exp n).D (Exp n).f (Exp n).θ i j (dk n) ≠ 0) → (∀ (n : ℕ) (i j : (Exp n).ι), i ≠ j → Causalean.Experimentation.DesignBased.propPairSame (Exp n).D (Exp n).f (Exp n).θ i j (dl n) ≠ 0) → (∀ (n : ℕ) (i j : (Exp n).ι), i ≠ j → Causalean.Experimentation.DesignBased.propPairCross (Exp n).D (Exp n).f (Exp n).θ i j (dk n) (dl n) ≠ 0) → (∀ (n : ℕ), 0 < (Exp n).D.Var (Causalean.Experimentation.DesignBased.htEffect (Exp n).D (Exp n).y (Exp n).f (Exp n).θ (dk n) (dl n))) → Filter.Tendsto (fun n => (Exp n).D.Var (Causalean.Experimentation.ExposureMappingInterference.htEffectVarEst (Exp n).D (Exp n).y (Exp n).f (Exp n).θ (dk n) (dl n)) / (Exp n).D.Var (Causalean.Experimentation.DesignBased.htEffect (Exp n).D (Exp n).y (Exp n).f (Exp n).θ (dk n) (dl n)) ^ 2) Filter.atTop (nhds 0) → ∀ (ε : ℝ), 0 < ε → Filter.Tendsto (fun n => (Exp n).D.Pr fun z => Causalean.Experimentation.ExposureMappingInterference.htEffectVarEst (Exp n).D (Exp n).y (Exp n).f (Exp n).θ (dk n) (dl n) z < (1 - ε) * (Exp n).D.Var (Causalean.Experimentation.DesignBased.htEffect (Exp n).D (Exp n).y (Exp n).f (Exp n).θ (dk n) (dl n))) Filter.atTop (nhds 0)","l":48,"f":"Causalean/Experimentation/ExposureMappingInterference/Asymptotics/VarianceConsistency.lean"},{"n":"Causalean.Experimentation.ExposureMappingInterference.htMean_consistent_of_var","k":"theorem","m":"Causalean.Experimentation.ExposureMappingInterference.Asymptotics.Consistency","a":"Experimentation/ExposureMappingInterference/Asymptotics","t":1,"d":"Chebyshev consistency. Along a sequence of experiments with a sequence of treatment assignments d such that every unit's exposure probability under d is nonzero, if the design variance of the Horvitz–Thompson mean estimator tends to 0, then for any positive threshold ε, the estimator is consistent: Pr[|μ̂ − μ| ≥ ε] → 0.","s":"∀ (Exp : ℕ → Causalean.Experimentation.ExposureMappingInterference.Experiment) (d : (n : ℕ) → (Exp n).Δ), (∀ (n : ℕ) (i : (Exp n).ι), Causalean.Experimentation.DesignBased.prop (Exp n).D (Exp n).f (Exp n).θ i (d n) ≠ 0) → Filter.Tendsto (fun n => (Exp n).D.Var (Causalean.Experimentation.DesignBased.htMean (Exp n).D (Exp n).y (Exp n).f (Exp n).θ (d n))) Filter.atTop (nhds 0) → ∀ {ε : ℝ}, 0 < ε → Filter.Tendsto (fun n => (Exp n).D.Pr fun z => ε ≤ |Causalean.Experimentation.DesignBased.htMean (Exp n).D (Exp n).y (Exp n).f (Exp n).θ (d n) z - Causalean.Experimentation.DesignBased.muTrue (Exp n).y (d n)|) Filter.atTop (nhds 0)","l":184,"f":"Causalean/Experimentation/ExposureMappingInterference/Asymptotics/Consistency.lean"},{"n":"Causalean.Experimentation.ExposureMappingInterference.htVarEst","k":"def","m":"Causalean.Experimentation.ExposureMappingInterference.Variance.Conservative","a":"Experimentation/ExposureMappingInterference/Variance","t":1,"d":"Horvitz–Thompson estimator of Var[ŷᵀ(d)] (eq:ht_variance_estimator), positive-joint regime. Uses observed outcomes Yobs.","s":"{Ω : Type u_1} → [inst : Fintype Ω] → {ι : Type u_2} → {Θ : Type u_3} → {Δ : Type u_4} → [Fintype ι] → [DecidableEq ι] → [DecidableEq Δ] → Causalean.Experimentation.DesignBased.FiniteDesign Ω → (ι → Δ → ℝ) → (Ω → Θ → Δ) → (ι → Θ) → Δ → Ω → ℝ","l":47,"f":"Causalean/Experimentation/ExposureMappingInterference/Variance/Conservative.lean"},{"n":"Causalean.Experimentation.ExposureMappingInterference.htVarEst_add_sub_eq_edgeSum","k":"theorem","m":"Causalean.Experimentation.ExposureMappingInterference.Asymptotics.VarEstQuadBound","a":"Experimentation/ExposureMappingInterference/Asymptotics","t":2,"d":"Edge-sum identity. V̂_raw = ∑_{i,j} vb i j: the three single-index sums of ŷVar(dk), ŷVar(dl), Ĉov fold into the diagonal i = j, and the three pairwise sums into the off-diagonal i ≠ j.","s":"∀ {Ω : Type u_1} [inst : Fintype Ω] {ι : Type u_2} {Θ : Type u_3} {Δ : Type u_4} [inst_1 : Fintype ι] [inst_2 : DecidableEq ι] [inst_3 : DecidableEq Δ] (D : Causalean.Experimentation.DesignBased.FiniteDesign Ω) (y : ι → Δ → ℝ) (f : Ω → Θ → Δ) (θ : ι → Θ) (dk dl : Δ), (fun z => Causalean.Experimentation.ExposureMappingInterference.htVarEst D y f θ dk z + Causalean.Experimentation.ExposureMappingInterference.htVarEst D y f θ dl z - 2 * Causalean.Experimentation.ExposureMappingInterference.htCovEst D y f θ dk dl z) = fun z => ∑ i, ∑ j, Causalean.Experimentation.ExposureMappingInterference.vb D y f θ dk dl i j z","l":96,"f":"Causalean/Experimentation/ExposureMappingInterference/Asymptotics/VarEstQuadBound.lean"},{"n":"Causalean.Experimentation.ExposureMappingInterference.localDependenceCLT_of_conditions","k":"theorem","m":"Causalean.Experimentation.ExposureMappingInterference.Asymptotics.SteinInstance","a":"Experimentation/ExposureMappingInterference/Asymptotics","t":1,"d":"Local-dependence CLT from a bounded-degree dependency graph. For a pair of treatment sequences dk, dl, suppose the per-unit effect summands admit a dependency graph Dg whose neighbourhoods have cardinality at most m, the design variance of the Horvitz–Thompson effect estimator is everywhere positive, every unit has nonzero exposure probability under dk and under dl, and the summands are pointwise bounded by a nonnegative sequence B n with B n → 0 and population size times B n cubed tending to 0. Then the studentized Horvitz–Thompson effect statistic satisfies the local-dependence central limit theorem.","s":"∀ (Exp : ℕ → Causalean.Experimentation.ExposureMappingInterference.Experiment) (dk dl : (n : ℕ) → (Exp n).Δ) (Dg : (n : ℕ) → Causalean.SteinMethod.DepGraph (fun i => (Exp n).effSummand (dk n) (dl n) i) (Exp n).D.toMeasure) (m : ℕ), (∀ (n : ℕ) (i : (Exp n).ι), ((Dg n).nbhd i).card ≤ m) → (∀ (n : ℕ), 0 < (Exp n).D.Var (Causalean.Experimentation.DesignBased.htEffect (Exp n).D (Exp n).y (Exp n).f (Exp n).θ (dk n) (dl n))) → (∀ (n : ℕ) (i : (Exp n).ι), Causalean.Experimentation.DesignBased.prop (Exp n).D (Exp n).f (Exp n).θ i (dk n) ≠ 0) → (∀ (n : ℕ) (i : (Exp n).ι), Causalean.Experimentation.DesignBased.prop (Exp n).D (Exp n).f (Exp n).θ i (dl n) ≠ 0) → ∀ (B : ℕ → ℝ), (∀ (n : ℕ), 0 ≤ B n) → (∀ (n : ℕ) (i : (Exp n).ι) (z : (Exp n).Ω), |(Exp n).effSummand (dk n) (dl n) i z| ≤ B n) → Filter.Tendsto B Filter.atTop (nhds 0) → Filter.Tendsto (fun n => ↑(Fintype.card (Exp n).ι) * B n ^ 3) Filter.atTop (nhds 0) → Causalean.Experimentation.ExposureMappingInterference.LocalDependenceCLT Exp dk dl","l":304,"f":"Causalean/Experimentation/ExposureMappingInterference/Asymptotics/SteinInstance.lean"},{"n":"Causalean.Experimentation.ExposureMappingInterference.localDependenceCLT_of_paper_conditions","k":"theorem","m":"Causalean.Experimentation.ExposureMappingInterference.Asymptotics.SteinInstance","a":"Experimentation/ExposureMappingInterference/Asymptotics","t":1,"d":"Local-dependence CLT from the literal Aronow–Samii conditions. For a pair of treatment sequences dk, dl, suppose the per-unit effect summands admit a bounded-degree dependency graph Dg, potential outcomes under dk and under dl are uniformly bounded by a constant c1, and the exposure propensities under dk and under dl are bounded away from 0 by a positive constant c2. If the population size diverges, the design variance of the Horvitz–Thompson effect estimator is everywhere positive, and population size times that variance converges to a positive limit cVar, then the studentized Horvitz–Thompson effect statistic satisfies the local-dependence central limit theorem.","s":"∀ (Exp : ℕ → Causalean.Experimentation.ExposureMappingInterference.Experiment) (dk dl : (n : ℕ) → (Exp n).Δ) (Dg : (n : ℕ) → Causalean.SteinMethod.DepGraph (fun i => (Exp n).effSummand (dk n) (dl n) i) (Exp n).D.toMeasure) (m : ℕ), (∀ (n : ℕ) (i : (Exp n).ι), ((Dg n).nbhd i).card ≤ m) → ∀ (c1 : ℝ), (∀ (n : ℕ) (i : (Exp n).ι), |(Exp n).y i (dk n)| ≤ c1) → (∀ (n : ℕ) (i : (Exp n).ι), |(Exp n).y i (dl n)| ≤ c1) → ∀ (c2 : ℝ), 0 < c2 → (∀ (n : ℕ) (i : (Exp n).ι), 1 / c2 ≤ Causalean.Experimentation.DesignBased.prop (Exp n).D (Exp n).f (Exp n).θ i (dk n)) → (∀ (n : ℕ) (i : (Exp n).ι), 1 / c2 ≤ Causalean.Experimentation.DesignBased.prop (Exp n).D (Exp n).f (Exp n).θ i (dl n)) → Filter.Tendsto (fun n => ↑(Fintype.card (Exp n).ι)) Filter.atTop Filter.atTop → (∀ (n : ℕ), 0 < (Exp n).D.Var (Causalean.Experimentation.DesignBased.htEffect (Exp n).D (Exp n).y (Exp n).f (Exp n).θ (dk n) (dl n))) → ∀ (cVar : ℝ), 0 < cVar → Filter.Tendsto (fun n => ↑(Fintype.card (Exp n).ι) * (Exp n).D.Var (Causalean.Experimentation.DesignBased.htEffect (Exp n).D (Exp n).y (Exp n).f (Exp n).θ (dk n) (dl n))) Filter.atTop (nhds cVar) → Causalean.Experimentation.ExposureMappingInterference.LocalDependenceCLT Exp dk dl","l":454,"f":"Causalean/Experimentation/ExposureMappingInterference/Asymptotics/SteinInstance.lean"},{"n":"Causalean.Experimentation.ExposureMappingInterference.localDependenceCLT_of_stein","k":"theorem","m":"Causalean.Experimentation.ExposureMappingInterference.Asymptotics.SteinInstance","a":"Experimentation/ExposureMappingInterference/Asymptotics","t":1,"d":"Local-dependence CLT from the abstract Stein conditions. For a pair of treatment sequences dk, dl and a choice of dependency neighbourhoods N n i, suppose the design variance of the Horvitz–Thompson effect estimator is everywhere positive, every unit has nonzero exposure probability under dk and under dl, and the per-unit effect summands are pointwise bounded by a nonnegative sequence B n. If each summand is independent of the sum of summands outside its neighbourhood, and the two Stein negligibility limits hold — the design variance of the neighbourhood cross-term sum tends to 0 and the summed third-moment-type error term tends to 0 — then the studentized Horvitz–Thompson effect statistic converges in distribution to a standard normal.","s":"∀ (Exp : ℕ → Causalean.Experimentation.ExposureMappingInterference.Experiment) (dk dl : (n : ℕ) → (Exp n).Δ) (N : (n : ℕ) → (Exp n).ι → Finset (Exp n).ι), (∀ (n : ℕ), 0 < (Exp n).D.Var (Causalean.Experimentation.DesignBased.htEffect (Exp n).D (Exp n).y (Exp n).f (Exp n).θ (dk n) (dl n))) → (∀ (n : ℕ) (i : (Exp n).ι), Causalean.Experimentation.DesignBased.prop (Exp n).D (Exp n).f (Exp n).θ i (dk n) ≠ 0) → (∀ (n : ℕ) (i : (Exp n).ι), Causalean.Experimentation.DesignBased.prop (Exp n).D (Exp n).f (Exp n).θ i (dl n) ≠ 0) → ∀ (B : ℕ → ℝ), (∀ (n : ℕ), 0 ≤ B n) → (∀ (n : ℕ) (i : (Exp n).ι) (z : (Exp n).Ω), |(Exp n).effSummand (dk n) (dl n) i z| ≤ B n) → (∀ (n : ℕ) (i : (Exp n).ι), ProbabilityTheory.IndepFun ((Exp n).effSummand (dk n) (dl n) i) (fun z => ∑ j ∈ Finset.univ \\ N n i, (Exp n).effSummand (dk n) (dl n) j z) (Exp n).D.toMeasure) → Filter.Tendsto (fun n => (Exp n).D.Var fun z => ∑ i, (Exp n).effSummand (dk n) (dl n) i z * ∑ j ∈ N n i, (Exp n).effSummand (dk n) (dl n) j z) Filter.atTop (nhds 0) → Filter.Tendsto (fun n => ∑ i, (Exp n).D.E fun z => |(Exp n).effSummand (dk n) (dl n) i z| * (∑ j ∈ N n i, (Exp n).effSummand (dk n) (dl n) j z) ^ 2) Filter.atTop (nhds 0) → Causalean.Experimentation.ExposureMappingInterference.LocalDependenceCLT Exp dk dl","l":176,"f":"Causalean/Experimentation/ExposureMappingInterference/Asymptotics/SteinInstance.lean"},{"n":"Causalean.Experimentation.ExposureMappingInterference.offCov","k":"def","m":"Causalean.Experimentation.ExposureMappingInterference.Asymptotics.VarEstQuadBound","a":"Experimentation/ExposureMappingInterference/Asymptotics","t":1,"d":"Off-diagonal summand of the covariance estimator Ĉov.","s":"{Ω : Type u_1} → [inst : Fintype Ω] → {ι : Type u_2} → {Θ : Type u_3} → {Δ : Type u_4} → [DecidableEq Δ] → Causalean.Experimentation.DesignBased.FiniteDesign Ω → (ι → Δ → ℝ) → (Ω → Θ → Δ) → (ι → Θ) → Δ → Δ → ι → ι → Ω → ℝ","l":59,"f":"Causalean/Experimentation/ExposureMappingInterference/Asymptotics/VarEstQuadBound.lean"},{"n":"Causalean.Experimentation.ExposureMappingInterference.offVar","k":"def","m":"Causalean.Experimentation.ExposureMappingInterference.Asymptotics.VarEstQuadBound","a":"Experimentation/ExposureMappingInterference/Asymptotics","t":1,"d":"Off-diagonal (variance) summand of ŷVar(d).","s":"{Ω : Type u_1} → [inst : Fintype Ω] → {ι : Type u_2} → {Θ : Type u_3} → {Δ : Type u_4} → [DecidableEq Δ] → Causalean.Experimentation.DesignBased.FiniteDesign Ω → (ι → Δ → ℝ) → (Ω → Θ → Δ) → (ι → Θ) → Δ → ι → ι → Ω → ℝ","l":52,"f":"Causalean/Experimentation/ExposureMappingInterference/Asymptotics/VarEstQuadBound.lean"},{"n":"Causalean.Experimentation.ExposureMappingInterference.relVar_of_NsqVar_tendsto","k":"theorem","m":"Causalean.Experimentation.ExposureMappingInterference.Asymptotics.VarianceConsistency","a":"Experimentation/ExposureMappingInterference/Asymptotics","t":2,"d":"Bridge from Var[N·V̂] → 0 to L²-relative consistency. The paper's appendix establishes Var[N·V̂_n] → 0 (here as N²·Var[V̂_n] → 0); together with Condition 4 (N·Var[τ̂_n] → c > 0) this yields the relative limit Var[V̂_n]/Var[τ̂_n]² → 0, via the identity Var[V̂]/Var² = (N²·Var[V̂])/(N·Var)² and Tendsto.div (numerator → 0, denominator → c² > 0).","s":"∀ (Exp : ℕ → Causalean.Experimentation.ExposureMappingInterference.Experiment) (dk dl : (n : ℕ) → (Exp n).Δ) {c : ℝ}, 0 < c → Filter.Tendsto (fun n => ↑(Fintype.card (Exp n).ι) * (Exp n).D.Var (Causalean.Experimentation.DesignBased.htEffect (Exp n).D (Exp n).y (Exp n).f (Exp n).θ (dk n) (dl n))) Filter.atTop (nhds c) → Filter.Tendsto (fun n => ↑(Fintype.card (Exp n).ι) ^ 2 * (Exp n).D.Var (Causalean.Experimentation.ExposureMappingInterference.htEffectVarEst (Exp n).D (Exp n).y (Exp n).f (Exp n).θ (dk n) (dl n))) Filter.atTop (nhds 0) → Filter.Tendsto (fun n => (Exp n).D.Var (Causalean.Experimentation.ExposureMappingInterference.htEffectVarEst (Exp n).D (Exp n).y (Exp n).f (Exp n).θ (dk n) (dl n)) / (Exp n).D.Var (Causalean.Experimentation.DesignBased.htEffect (Exp n).D (Exp n).y (Exp n).f (Exp n).θ (dk n) (dl n)) ^ 2) Filter.atTop (nhds 0)","l":171,"f":"Causalean/Experimentation/ExposureMappingInterference/Asymptotics/VarianceConsistency.lean"},{"n":"Causalean.Experimentation.ExposureMappingInterference.var_NsqVhat_tendsto_zero_of_conditions","k":"theorem","m":"Causalean.Experimentation.ExposureMappingInterference.Asymptotics.VarEstConsistencyConditions","a":"Experimentation/ExposureMappingInterference/Asymptotics","t":1,"d":"Scaled variance-estimator variance vanishes (Aronow–Samii Prop 6.6 core). Along a sequence of experiments with distinct treatment arms dk ≠ dl whose outcomes are uniformly bounded by a nonnegative constant c₁ (Condition 1), suppose the marginal exposure propensities under dk and dl are positive with inverses uniformly bounded by a nonnegative constant c₂, and the same-arm and cross-arm pairwise joint exposure propensities have inverses uniformly bounded by a nonnegative constant c₃ (Condition 1'). Suppose further a symmetric, reflexive dependency relation G of degree at most m such that every non-adjacent pair's joint exposure propensities factor as if independent and every quadruple with no adjacent index pair has zero covariance between the corresponding variance-estimator kernel terms (Condition 3), and the population size diverges. Then the scaled variance of the conservative variance estimator vanishes: N²·Var[V̂_n] → 0.","s":"∀ (Exp : ℕ → Causalean.Experimentation.ExposureMappingInterference.Experiment) (dk dl : (n : ℕ) → (Exp n).Δ), (∀ (n : ℕ), dk n ≠ dl n) → ∀ {c₁ c₂ c₃ : ℝ}, 0 ≤ c₁ → 0 ≤ c₂ → 0 ≤ c₃ → (∀ (n : ℕ) (i : (Exp n).ι) (d : (Exp n).Δ), |(Exp n).y i d| ≤ c₁) → (∀ (n : ℕ) (i : (Exp n).ι), 0 < Causalean.Experimentation.DesignBased.prop (Exp n).D (Exp n).f (Exp n).θ i (dk n)) → (∀ (n : ℕ) (i : (Exp n).ι), 0 < Causalean.Experimentation.DesignBased.prop (Exp n).D (Exp n).f (Exp n).θ i (dl n)) → (∀ (n : ℕ) (i : (Exp n).ι), 1 / Causalean.Experimentation.DesignBased.prop (Exp n).D (Exp n).f (Exp n).θ i (dk n) ≤ c₂) → (∀ (n : ℕ) (i : (Exp n).ι), 1 / Causalean.Experimentation.DesignBased.prop (Exp n).D (Exp n).f (Exp n).θ i (dl n) ≤ c₂) → (∀ (n : ℕ) (i j : (Exp n).ι), 1 / Causalean.Experimentation.DesignBased.propPairSame (Exp n).D (Exp n).f (Exp n).θ i j (dk n) ≤ c₃) → (∀ (n : ℕ) (i j : (Exp n).ι), 1 / Causalean.Experimentation.DesignBased.propPairSame (Exp n).D (Exp n).f (Exp n).θ i j (dl n) ≤ c₃) → (∀ (n : ℕ) (i j : (Exp n).ι), 1 / Causalean.Experimentation.DesignBased.propPairCross (Exp n).D (Exp n).f (Exp n).θ i j (dk n) (dl n) ≤ c₃) → ∀ (G : (n : ℕ) → (Exp n).ι → (Exp n).ι → Prop) (decG : (n : ℕ) → DecidableRel (G n)), (∀ (n : ℕ) (i : (Exp n).ι), G n i i) → (∀ (n : ℕ) (i j : (Exp n).ι), G n i j → G n j i) → ∀ {m : ℕ}, (∀ (n : ℕ) (i : (Exp n).ι), {j | G n i j}.card ≤ m) → (∀ (n : ℕ) (i j : (Exp n).ι), ¬G n i j → Causalean.Experimentation.DesignBased.propPairSame (Exp n).D (Exp n).f (Exp n).θ i j (dk n) = Causalean.Experimentation.DesignBased.prop (Exp n).D (Exp n).f (Exp n).θ i (dk n) * Causalean.Experimentation.DesignBased.prop (Exp n).D (Exp n).f (Exp n).θ j (dk n) ∧ Causalean.Experimentation.DesignBased.propPairSame (Exp n).D (Exp n).f (Exp n).θ i j (dl n) = Causalean.Experimentation.DesignBased.prop (Exp n).D (Exp n).f (Exp n).θ i (dl n) * Causalean.Experimentation.DesignBased.prop (Exp n).D (Exp n).f (Exp n).θ j (dl n) ∧ Causalean.Experimentation.DesignBased.propPairCross (Exp n).D (Exp n).f (Exp n).θ i j (dk n) (dl n) = Causalean.Experimentation.DesignBased.prop (Exp n).D (Exp n).f (Exp n).θ i (dk n) * Causalean.Experimentation.DesignBased.prop (Exp n).D (Exp n).f (Exp n).θ j (dl n)) → (∀ (n : ℕ) (i j k l : (Exp n).ι), ¬(G n i k ∨ G n i l ∨ G n j k ∨ G n j l) → (Exp n).D.Cov (Causalean.Experimentation.ExposureMappingInterference.vb (Exp n).D (Exp n).y (Exp n).f (Exp n).θ (dk n) (dl n) i j) (Causalean.Experimentation.ExposureMappingInterference.vb (Exp n).D (Exp n).y (Exp n).f (Exp n).θ (dk n) (dl n) k l) = 0) → Filter.Tendsto (fun n => ↑(Fintype.card (Exp n).ι)) Filter.atTop Filter.atTop → Filter.Tendsto (fun n => ↑(Fintype.card (Exp n).ι) ^ 2 * (Exp n).D.Var (Causalean.Experimentation.ExposureMappingInterference.htEffectVarEst (Exp n).D (Exp n).y (Exp n).f (Exp n).θ (dk n) (dl n))) Filter.atTop (nhds 0)","l":94,"f":"Causalean/Experimentation/ExposureMappingInterference/Asymptotics/VarEstConsistencyConditions.lean"},{"n":"Causalean.Experimentation.ExposureMappingInterference.var_htEdgeStat_le","k":"theorem","m":"Causalean.Experimentation.ExposureMappingInterference.Asymptotics.VarEstQuadBound","a":"Experimentation/ExposureMappingInterference/Asymptotics","t":1,"d":"Per-population quadruple-sum bound (Aronow–Samii appendix, Prop 6.6). Suppose the two treatment arms are distinct and outcomes are uniformly bounded by a nonnegative constant c₁ (Condition 1). Suppose the marginal exposure propensities under each arm are positive with inverses uniformly bounded by a nonnegative constant c₂, and the same-arm and cross-arm pairwise joint exposure propensities have inverses uniformly bounded by a nonnegative constant c₃ (Condition 1'). Suppose further a symmetric, reflexive dependency relation G of degree at most m makes every non-adjacent pair's joint exposure propensities factor as if independent and gives every quadruple with no adjacent index pair zero covariance between the variance-estimator kernel terms (Condition 3). Then the variance of the raw conservative variance estimator is linear in the population size: Var[ŷVar(dk)+ŷVar(dl)−2Ĉov] ≤ 8·(vbBound c₁ c₂ c₃)²·m³·N.","s":"∀ {Ω : Type u_1} [inst : Fintype Ω] {ι : Type u_2} {Θ : Type u_3} {Δ : Type u_4} [inst_1 : Fintype ι] [inst_2 : DecidableEq ι] [inst_3 : DecidableEq Δ] (D : Causalean.Experimentation.DesignBased.FiniteDesign Ω) (y : ι → Δ → ℝ) (f : Ω → Θ → Δ) (θ : ι → Θ) (dk dl : Δ), dk ≠ dl → ∀ {c₁ c₂ c₃ : ℝ}, 0 ≤ c₁ → 0 ≤ c₂ → 0 ≤ c₃ → (∀ (i : ι) (d : Δ), |y i d| ≤ c₁) → (∀ (i : ι), 0 < Causalean.Experimentation.DesignBased.prop D f θ i dk) → (∀ (i : ι), 0 < Causalean.Experimentation.DesignBased.prop D f θ i dl) → (∀ (i : ι), 1 / Causalean.Experimentation.DesignBased.prop D f θ i dk ≤ c₂) → (∀ (i : ι), 1 / Causalean.Experimentation.DesignBased.prop D f θ i dl ≤ c₂) → (∀ (i j : ι), 1 / Causalean.Experimentation.DesignBased.propPairSame D f θ i j dk ≤ c₃) → (∀ (i j : ι), 1 / Causalean.Experimentation.DesignBased.propPairSame D f θ i j dl ≤ c₃) → (∀ (i j : ι), 1 / Causalean.Experimentation.DesignBased.propPairCross D f θ i j dk dl ≤ c₃) → ∀ (G : ι → ι → Prop) [inst_4 : DecidableRel G], (∀ (i : ι), G i i) → (∀ (i j : ι), G i j → G j i) → ∀ {m : ℕ}, (∀ (i : ι), {j | G i j}.card ≤ m) → (∀ (i j : ι), ¬G i j → Causalean.Experimentation.DesignBased.propPairSame D f θ i j dk = Causalean.Experimentation.DesignBased.prop D f θ i dk * Causalean.Experimentation.DesignBased.prop D f θ j dk ∧ Causalean.Experimentation.DesignBased.propPairSame D f θ i j dl = Causalean.Experimentation.DesignBased.prop D f θ i dl * Causalean.Experimentation.DesignBased.prop D f θ j dl ∧ Causalean.Experimentation.DesignBased.propPairCross D f θ i j dk dl = Causalean.Experimentation.DesignBased.prop D f θ i dk * Causalean.Experimentation.DesignBased.prop D f θ j dl) → (∀ (i j k l : ι), ¬(G i k ∨ G i l ∨ G j k ∨ G j l) → D.Cov (Causalean.Experimentation.ExposureMappingInterference.vb D y f θ dk dl i j) (Causalean.Experimentation.ExposureMappingInterference.vb D y f θ dk dl k l) = 0) → (D.Var fun z => Causalean.Experimentation.ExposureMappingInterference.htVarEst D y f θ dk z + Causalean.Experimentation.ExposureMappingInterference.htVarEst D y f θ dl z - 2 * Causalean.Experimentation.ExposureMappingInterference.htCovEst D y f θ dk dl z) ≤ 8 * Causalean.Experimentation.ExposureMappingInterference.vbBound c₁ c₂ c₃ ^ 2 * (↑m ^ 3 * ↑(Fintype.card ι))","l":387,"f":"Causalean/Experimentation/ExposureMappingInterference/Asymptotics/VarEstQuadBound.lean"},{"n":"Causalean.Experimentation.ExposureMappingInterference.vb","k":"def","m":"Causalean.Experimentation.ExposureMappingInterference.Asymptotics.VarEstQuadBound","a":"Experimentation/ExposureMappingInterference/Asymptotics","t":1,"d":"The edge-function for V̂_raw: diagonal term on i = j, off-diagonal term otherwise.","s":"{Ω : Type u_1} → [inst : Fintype Ω] → {ι : Type u_2} → {Θ : Type u_3} → {Δ : Type u_4} → [DecidableEq ι] → [DecidableEq Δ] → Causalean.Experimentation.DesignBased.FiniteDesign Ω → (ι → Δ → ℝ) → (Ω → Θ → Δ) → (ι → Θ) → Δ → Δ → ι → ι → Ω → ℝ","l":83,"f":"Causalean/Experimentation/ExposureMappingInterference/Asymptotics/VarEstQuadBound.lean"},{"n":"Causalean.Experimentation.ExposureMappingInterference.vbBound","k":"def","m":"Causalean.Experimentation.ExposureMappingInterference.Asymptotics.VarEstQuadBound","a":"Experimentation/ExposureMappingInterference/Asymptotics","t":1,"d":"Uniform pointwise bound on the edge-function, from Conditions 1 and 1'. Chosen large enough to dominate both the squared inverse-propensity terms (∝ c₂²) and the Young-correction terms (∝ c₂), uniformly in the joint-overlap constant c₃.","s":"ℝ → ℝ → ℝ → ℝ","l":88,"f":"Causalean/Experimentation/ExposureMappingInterference/Asymptotics/VarEstQuadBound.lean"},{"n":"Causalean.Experimentation.ExposureMappingInterference.vbDiag","k":"def","m":"Causalean.Experimentation.ExposureMappingInterference.Asymptotics.VarEstQuadBound","a":"Experimentation/ExposureMappingInterference/Asymptotics","t":1,"d":"Diagonal edge term: diagVar(dk) + diagVar(dl) + 2·diagCov.","s":"{Ω : Type u_1} → [inst : Fintype Ω] → {ι : Type u_2} → {Θ : Type u_3} → {Δ : Type u_4} → [DecidableEq Δ] → Causalean.Experimentation.DesignBased.FiniteDesign Ω → (ι → Δ → ℝ) → (Ω → Θ → Δ) → (ι → Θ) → Δ → Δ → ι → Ω → ℝ","l":73,"f":"Causalean/Experimentation/ExposureMappingInterference/Asymptotics/VarEstQuadBound.lean"},{"n":"Causalean.Experimentation.ExposureMappingInterference.vbOff","k":"def","m":"Causalean.Experimentation.ExposureMappingInterference.Asymptotics.VarEstQuadBound","a":"Experimentation/ExposureMappingInterference/Asymptotics","t":1,"d":"Off-diagonal edge term: offVar(dk) + offVar(dl) − 2·offCov.","s":"{Ω : Type u_1} → [inst : Fintype Ω] → {ι : Type u_2} → {Θ : Type u_3} → {Δ : Type u_4} → [DecidableEq Δ] → Causalean.Experimentation.DesignBased.FiniteDesign Ω → (ι → Δ → ℝ) → (Ω → Θ → Δ) → (ι → Θ) → Δ → Δ → ι → ι → Ω → ℝ","l":78,"f":"Causalean/Experimentation/ExposureMappingInterference/Asymptotics/VarEstQuadBound.lean"},{"n":"Causalean.Experimentation.ExposureMappingInterference.vb_eq_zero_of_indep","k":"theorem","m":"Causalean.Experimentation.ExposureMappingInterference.Asymptotics.VarEstQuadBound","a":"Experimentation/ExposureMappingInterference/Asymptotics","t":2,"d":"Off-edge vanishing. If the same-exposure and cross-exposure pair propensities factor as products for an off-diagonal pair i ≠ j, then the centered-cross factors vanish and vb i j = 0; Condition 3 supplies these factorization hypotheses for non-adjacent pairs.","s":"∀ {Ω : Type u_1} [inst : Fintype Ω] {ι : Type u_2} {Θ : Type u_3} {Δ : Type u_4} [inst_1 : DecidableEq ι] [inst_2 : DecidableEq Δ] (D : Causalean.Experimentation.DesignBased.FiniteDesign Ω) (y : ι → Δ → ℝ) (f : Ω → Θ → Δ) (θ : ι → Θ) (dk dl : Δ) (i j : ι), i ≠ j → Causalean.Experimentation.DesignBased.propPairSame D f θ i j dk = Causalean.Experimentation.DesignBased.prop D f θ i dk * Causalean.Experimentation.DesignBased.prop D f θ j dk → Causalean.Experimentation.DesignBased.propPairSame D f θ i j dl = Causalean.Experimentation.DesignBased.prop D f θ i dl * Causalean.Experimentation.DesignBased.prop D f θ j dl → Causalean.Experimentation.DesignBased.propPairCross D f θ i j dk dl = Causalean.Experimentation.DesignBased.prop D f θ i dk * Causalean.Experimentation.DesignBased.prop D f θ j dl → Causalean.Experimentation.ExposureMappingInterference.vb D y f θ dk dl i j = fun x => 0","l":374,"f":"Causalean/Experimentation/ExposureMappingInterference/Asymptotics/VarEstQuadBound.lean"},{"n":"Causalean.Experimentation.ExposureMappingInterference.wald_coverage","k":"theorem","m":"Causalean.Experimentation.ExposureMappingInterference.Asymptotics.Intervals","a":"Experimentation/ExposureMappingInterference/Asymptotics","t":1,"d":"Proposition 6.5 (asymptotic coverage, true standard error). For a pair of treatment sequences dk, dl, suppose the studentized Horvitz–Thompson effect statistic satisfies the local-dependence central limit theorem and the design variance of the effect estimator is everywhere positive — the operative form of Condition 4 (nonzero limiting variance). Then with zq a nonnegative quantile satisfying Φ(zq) = 1 − α/2, the Wald interval centered at the Horvitz–Thompson effect estimator, using its true standard error, attains asymptotic (liminf) coverage at least 1 − α.","s":"∀ (Exp : ℕ → Causalean.Experimentation.ExposureMappingInterference.Experiment) (dk dl : (n : ℕ) → (Exp n).Δ), Causalean.Experimentation.ExposureMappingInterference.LocalDependenceCLT Exp dk dl → (∀ (n : ℕ), 0 < (Exp n).D.Var (Causalean.Experimentation.DesignBased.htEffect (Exp n).D (Exp n).y (Exp n).f (Exp n).θ (dk n) (dl n))) → ∀ {α : ℝ} (zq : ℝ), 0 ≤ zq → Causalean.Experimentation.DesignBased.stdNormalCdf zq = 1 - α / 2 → 1 - α ≤ Filter.liminf (fun n => (Exp n).D.Pr fun z => |Causalean.Experimentation.DesignBased.htEffect (Exp n).D (Exp n).y (Exp n).f (Exp n).θ (dk n) (dl n) z - Causalean.Experimentation.DesignBased.tauTrue (Exp n).y (dk n) (dl n)| ≤ zq * √((Exp n).D.Var (Causalean.Experimentation.DesignBased.htEffect (Exp n).D (Exp n).y (Exp n).f (Exp n).θ (dk n) (dl n)))) Filter.atTop","l":35,"f":"Causalean/Experimentation/ExposureMappingInterference/Asymptotics/Intervals.lean"},{"n":"Causalean.Experimentation.ExposureMappingInterference.wald_coverage_feasible","k":"theorem","m":"Causalean.Experimentation.ExposureMappingInterference.Asymptotics.Intervals","a":"Experimentation/ExposureMappingInterference/Asymptotics","t":1,"d":"Proposition 6.5 (asymptotic coverage, feasible/estimated standard error — the paper's actual interval). For a pair of treatment sequences dk, dl, suppose the studentized effect statistic satisfies the local-dependence central limit theorem, the design variance of the effect estimator is everywhere positive, and the conservative variance estimator V̂ undershoots the true variance only with vanishing probability. Then with zq a nonnegative quantile satisfying Φ(zq) = 1 − α/2, the Wald interval τ̂ ± zq·√(V̂[τ̂]), using the estimated variance htEffectVarEst exactly as in Aronow–Samii Prop 6.5, attains asymptotic (liminf) coverage at least 1 − α.","s":"∀ (Exp : ℕ → Causalean.Experimentation.ExposureMappingInterference.Experiment) (dk dl : (n : ℕ) → (Exp n).Δ), Causalean.Experimentation.ExposureMappingInterference.LocalDependenceCLT Exp dk dl → (∀ (n : ℕ), 0 < (Exp n).D.Var (Causalean.Experimentation.DesignBased.htEffect (Exp n).D (Exp n).y (Exp n).f (Exp n).θ (dk n) (dl n))) → (∀ (ε : ℝ), 0 < ε → Filter.Tendsto (fun n => (Exp n).D.Pr fun z => Causalean.Experimentation.ExposureMappingInterference.htEffectVarEst (Exp n).D (Exp n).y (Exp n).f (Exp n).θ (dk n) (dl n) z < (1 - ε) * (Exp n).D.Var (Causalean.Experimentation.DesignBased.htEffect (Exp n).D (Exp n).y (Exp n).f (Exp n).θ (dk n) (dl n))) Filter.atTop (nhds 0)) → ∀ {α : ℝ} (zq : ℝ), 0 ≤ zq → Causalean.Experimentation.DesignBased.stdNormalCdf zq = 1 - α / 2 → 1 - α ≤ Filter.liminf (fun n => (Exp n).D.Pr fun z => |Causalean.Experimentation.DesignBased.htEffect (Exp n).D (Exp n).y (Exp n).f (Exp n).θ (dk n) (dl n) z - Causalean.Experimentation.DesignBased.tauTrue (Exp n).y (dk n) (dl n)| ≤ zq * √(Causalean.Experimentation.ExposureMappingInterference.htEffectVarEst (Exp n).D (Exp n).y (Exp n).f (Exp n).θ (dk n) (dl n) z)) Filter.atTop","l":176,"f":"Causalean/Experimentation/ExposureMappingInterference/Asymptotics/Intervals.lean"},{"n":"Causalean.Experimentation.ExposureMappingInterference.wald_coverage_feasible_of_conditions","k":"theorem","m":"Causalean.Experimentation.ExposureMappingInterference.Asymptotics.VarEstConsistencyConditions","a":"Experimentation/ExposureMappingInterference/Asymptotics","t":1,"d":"Feasible Wald coverage from primitive boundedness, overlap, dependency-graph, and variance-growth conditions. Along a sequence of experiments with distinct treatment arms dk ≠ dl whose outcomes are uniformly bounded by a nonnegative constant c₁, suppose the marginal exposure propensities under dk and dl are positive with inverses uniformly bounded by a nonnegative constant c₂ — taken strictly positive — and the same-arm and cross-arm pairwise joint exposure propensities have inverses uniformly bounded by a nonnegative constant c₃, with all off-diagonal joint propensities strictly positive. Suppose a symmetric, reflexive dependency relation G of degree at most m makes every non-adjacent pair's joint exposure propensities factor as if independent and gives every quadruple with no adjacent index pair zero covariance between the variance-estimator kernel terms; suppose also the per-unit effect summands admit a Stein dependency graph Dg of degree at most m, that the population size diverges while population size times the true effect-estimator variance converges to a positive constant cVar and that variance is everywhere positive, and that zq is a nonnegative quantile satisfying Φ(zq) = 1 − α/2. Then the feasible Wald interval τ̂ ± zq·√V̂ attains asymptotic (liminf) coverage at least 1 − α.","s":"∀ (Exp : ℕ → Causalean.Experimentation.ExposureMappingInterference.Experiment) (dk dl : (n : ℕ) → (Exp n).Δ) {m : ℕ} (Dg : (n : ℕ) → Causalean.SteinMethod.DepGraph (fun i => (Exp n).effSummand (dk n) (dl n) i) (Exp n).D.toMeasure), (∀ (n : ℕ) (i : (Exp n).ι), ((Dg n).nbhd i).card ≤ m) → (∀ (n : ℕ), dk n ≠ dl n) → ∀ {c₁ c₂ c₃ : ℝ}, 0 ≤ c₁ → 0 ≤ c₂ → 0 ≤ c₃ → 0 < c₂ → (∀ (n : ℕ) (i : (Exp n).ι) (d : (Exp n).Δ), |(Exp n).y i d| ≤ c₁) → (∀ (n : ℕ) (i : (Exp n).ι), 0 < Causalean.Experimentation.DesignBased.prop (Exp n).D (Exp n).f (Exp n).θ i (dk n)) → (∀ (n : ℕ) (i : (Exp n).ι), 0 < Causalean.Experimentation.DesignBased.prop (Exp n).D (Exp n).f (Exp n).θ i (dl n)) → (∀ (n : ℕ) (i : (Exp n).ι), 1 / Causalean.Experimentation.DesignBased.prop (Exp n).D (Exp n).f (Exp n).θ i (dk n) ≤ c₂) → (∀ (n : ℕ) (i : (Exp n).ι), 1 / Causalean.Experimentation.DesignBased.prop (Exp n).D (Exp n).f (Exp n).θ i (dl n) ≤ c₂) → (∀ (n : ℕ) (i j : (Exp n).ι), 1 / Causalean.Experimentation.DesignBased.propPairSame (Exp n).D (Exp n).f (Exp n).θ i j (dk n) ≤ c₃) → (∀ (n : ℕ) (i j : (Exp n).ι), 1 / Causalean.Experimentation.DesignBased.propPairSame (Exp n).D (Exp n).f (Exp n).θ i j (dl n) ≤ c₃) → (∀ (n : ℕ) (i j : (Exp n).ι), 1 / Causalean.Experimentation.DesignBased.propPairCross (Exp n).D (Exp n).f (Exp n).θ i j (dk n) (dl n) ≤ c₃) → ∀ (G : (n : ℕ) → (Exp n).ι → (Exp n).ι → Prop) (decG : (n : ℕ) → DecidableRel (G n)), (∀ (n : ℕ) (i : (Exp n).ι), G n i i) → (∀ (n : ℕ) (i j : (Exp n).ι), G n i j → G n j i) → (∀ (n : ℕ) (i : (Exp n).ι), {j | G n i j}.card ≤ m) → (∀ (n : ℕ) (i j : (Exp n).ι), ¬G n i j → Causalean.Experimentation.DesignBased.propPairSame (Exp n).D (Exp n).f (Exp n).θ i j (dk n) = Causalean.Experimentation.DesignBased.prop (Exp n).D (Exp n).f (Exp n).θ i (dk n) * Causalean.Experimentation.DesignBased.prop (Exp n).D (Exp n).f (Exp n).θ j (dk n) ∧ Causalean.Experimentation.DesignBased.propPairSame (Exp n).D (Exp n).f (Exp n).θ i j (dl n) = Causalean.Experimentation.DesignBased.prop (Exp n).D (Exp n).f (Exp n).θ i (dl n) * Causalean.Experimentation.DesignBased.prop (Exp n).D (Exp n).f (Exp n).θ j (dl n) ∧ Causalean.Experimentation.DesignBased.propPairCross (Exp n).D (Exp n).f (Exp n).θ i j (dk n) (dl n) = Causalean.Experimentation.DesignBased.prop (Exp n).D (Exp n).f (Exp n).θ i (dk n) * Causalean.Experimentation.DesignBased.prop (Exp n).D (Exp n).f (Exp n).θ j (dl n)) → (∀ (n : ℕ) (i j k l : (Exp n).ι), ¬(G n i k ∨ G n i l ∨ G n j k ∨ G n j l) → (Exp n).D.Cov (Causalean.Experimentation.ExposureMappingInterference.vb (Exp n).D (Exp n).y (Exp n).f (Exp n).θ (dk n) (dl n) i j) (Causalean.Experimentation.ExposureMappingInterference.vb (Exp n).D (Exp n).y (Exp n).f (Exp n).θ (dk n) (dl n) k l) = 0) → Filter.Tendsto (fun n => ↑(Fintype.card (Exp n).ι)) Filter.atTop Filter.atTop → (∀ (n : ℕ) (i j : (Exp n).ι), i ≠ j → 0 < Causalean.Experimentation.DesignBased.propPairSame (Exp n).D (Exp n).f (Exp n).θ i j (dk n)) → (∀ (n : ℕ) (i j : (Exp n).ι), i ≠ j → 0 < Causalean.Experimentation.DesignBased.propPairSame (Exp n).D (Exp n).f (Exp n).θ i j (dl n)) → (∀ (n : ℕ) (i j : (Exp n).ι), i ≠ j → 0 < Causalean.Experimentation.DesignBased.propPairCross (Exp n).D (Exp n).f (Exp n).θ i j (dk n) (dl n)) → ∀ {cVar : ℝ}, 0 < cVar → Filter.Tendsto (fun n => ↑(Fintype.card (Exp n).ι) * (Exp n).D.Var (Causalean.Experimentation.DesignBased.htEffect (Exp n).D (Exp n).y (Exp n).f (Exp n).θ (dk n) (dl n))) Filter.atTop (nhds cVar) → (∀ (n : ℕ), 0 < (Exp n).D.Var (Causalean.Experimentation.DesignBased.htEffect (Exp n).D (Exp n).y (Exp n).f (Exp n).θ (dk n) (dl n))) → ∀ {α : ℝ} (zq : ℝ), 0 ≤ zq → Causalean.Experimentation.DesignBased.stdNormalCdf zq = 1 - α / 2 → 1 - α ≤ Filter.liminf (fun n => (Exp n).D.Pr fun z => |Causalean.Experimentation.DesignBased.htEffect (Exp n).D (Exp n).y (Exp n).f (Exp n).θ (dk n) (dl n) z - Causalean.Experimentation.DesignBased.tauTrue (Exp n).y (dk n) (dl n)| ≤ zq * √(Causalean.Experimentation.ExposureMappingInterference.htEffectVarEst (Exp n).D (Exp n).y (Exp n).f (Exp n).θ (dk n) (dl n) z)) Filter.atTop","l":151,"f":"Causalean/Experimentation/ExposureMappingInterference/Asymptotics/VarEstConsistencyConditions.lean"},{"n":"Causalean.Experimentation.ExposureMappingInterference.wald_coverage_feasible_of_relVar","k":"theorem","m":"Causalean.Experimentation.ExposureMappingInterference.Asymptotics.VarianceConsistency","a":"Experimentation/ExposureMappingInterference/Asymptotics","t":1,"d":"Feasible Wald coverage from L²-relative variance consistency (capstone). For a pair of treatment sequences dk, dl with dk n ≠ dl n for every n, suppose the studentized effect statistic satisfies the local-dependence central limit theorem, every unit has nonzero exposure propensity under dk and under dl, and every off-diagonal pair has nonzero same-arm and nonzero cross-arm joint exposure propensities. If the true effect-estimator variance is everywhere positive and the L²-relative-consistency limit Var[V̂]/Var[τ̂]² → 0 holds, with zq a nonnegative quantile satisfying Φ(zq) = 1 − α/2, then the paper's actual interval τ̂ ± zq·√V̂ attains asymptotic (liminf) coverage at least 1 − α.","s":"∀ (Exp : ℕ → Causalean.Experimentation.ExposureMappingInterference.Experiment) (dk dl : (n : ℕ) → (Exp n).Δ), Causalean.Experimentation.ExposureMappingInterference.LocalDependenceCLT Exp dk dl → (∀ (n : ℕ), dk n ≠ dl n) → (∀ (n : ℕ) (i : (Exp n).ι), Causalean.Experimentation.DesignBased.prop (Exp n).D (Exp n).f (Exp n).θ i (dk n) ≠ 0) → (∀ (n : ℕ) (i : (Exp n).ι), Causalean.Experimentation.DesignBased.prop (Exp n).D (Exp n).f (Exp n).θ i (dl n) ≠ 0) → (∀ (n : ℕ) (i j : (Exp n).ι), i ≠ j → Causalean.Experimentation.DesignBased.propPairSame (Exp n).D (Exp n).f (Exp n).θ i j (dk n) ≠ 0) → (∀ (n : ℕ) (i j : (Exp n).ι), i ≠ j → Causalean.Experimentation.DesignBased.propPairSame (Exp n).D (Exp n).f (Exp n).θ i j (dl n) ≠ 0) → (∀ (n : ℕ) (i j : (Exp n).ι), i ≠ j → Causalean.Experimentation.DesignBased.propPairCross (Exp n).D (Exp n).f (Exp n).θ i j (dk n) (dl n) ≠ 0) → (∀ (n : ℕ), 0 < (Exp n).D.Var (Causalean.Experimentation.DesignBased.htEffect (Exp n).D (Exp n).y (Exp n).f (Exp n).θ (dk n) (dl n))) → Filter.Tendsto (fun n => (Exp n).D.Var (Causalean.Experimentation.ExposureMappingInterference.htEffectVarEst (Exp n).D (Exp n).y (Exp n).f (Exp n).θ (dk n) (dl n)) / (Exp n).D.Var (Causalean.Experimentation.DesignBased.htEffect (Exp n).D (Exp n).y (Exp n).f (Exp n).θ (dk n) (dl n)) ^ 2) Filter.atTop (nhds 0) → ∀ {α : ℝ} (zq : ℝ), 0 ≤ zq → Causalean.Experimentation.DesignBased.stdNormalCdf zq = 1 - α / 2 → 1 - α ≤ Filter.liminf (fun n => (Exp n).D.Pr fun z => |Causalean.Experimentation.DesignBased.htEffect (Exp n).D (Exp n).y (Exp n).f (Exp n).θ (dk n) (dl n) z - Causalean.Experimentation.DesignBased.tauTrue (Exp n).y (dk n) (dl n)| ≤ zq * √(Causalean.Experimentation.ExposureMappingInterference.htEffectVarEst (Exp n).D (Exp n).y (Exp n).f (Exp n).θ (dk n) (dl n) z)) Filter.atTop","l":126,"f":"Causalean/Experimentation/ExposureMappingInterference/Asymptotics/VarianceConsistency.lean"},{"n":"Causalean.Experimentation.ExposureMappingInterference.wald_coverage_of_conditions","k":"theorem","m":"Causalean.Experimentation.ExposureMappingInterference.Asymptotics.SteinInstance","a":"Experimentation/ExposureMappingInterference/Asymptotics","t":1,"d":"Aronow–Samii oracle Wald coverage from bounded-degree primitive conditions. For a pair of treatment sequences dk, dl, suppose the per-unit effect summands admit a dependency graph Dg whose neighbourhoods have cardinality at most m, the design variance of the Horvitz–Thompson effect estimator is everywhere positive, and every unit has nonzero exposure probability under dk and under dl. If the summands are pointwise bounded by a nonnegative sequence B n with B n → 0 and population size times B n cubed tending to 0, and zq is a nonnegative quantile satisfying Φ(zq) = 1 − α/2, then the oracle Wald interval τ̂ ± zq·√Var[τ̂] attains asymptotic (liminf) coverage at least 1 − α.","s":"∀ (Exp : ℕ → Causalean.Experimentation.ExposureMappingInterference.Experiment) (dk dl : (n : ℕ) → (Exp n).Δ) (Dg : (n : ℕ) → Causalean.SteinMethod.DepGraph (fun i => (Exp n).effSummand (dk n) (dl n) i) (Exp n).D.toMeasure) (m : ℕ), (∀ (n : ℕ) (i : (Exp n).ι), ((Dg n).nbhd i).card ≤ m) → (∀ (n : ℕ), 0 < (Exp n).D.Var (Causalean.Experimentation.DesignBased.htEffect (Exp n).D (Exp n).y (Exp n).f (Exp n).θ (dk n) (dl n))) → (∀ (n : ℕ) (i : (Exp n).ι), Causalean.Experimentation.DesignBased.prop (Exp n).D (Exp n).f (Exp n).θ i (dk n) ≠ 0) → (∀ (n : ℕ) (i : (Exp n).ι), Causalean.Experimentation.DesignBased.prop (Exp n).D (Exp n).f (Exp n).θ i (dl n) ≠ 0) → ∀ (B : ℕ → ℝ), (∀ (n : ℕ), 0 ≤ B n) → (∀ (n : ℕ) (i : (Exp n).ι) (z : (Exp n).Ω), |(Exp n).effSummand (dk n) (dl n) i z| ≤ B n) → Filter.Tendsto B Filter.atTop (nhds 0) → Filter.Tendsto (fun n => ↑(Fintype.card (Exp n).ι) * B n ^ 3) Filter.atTop (nhds 0) → ∀ {α : ℝ} (zq : ℝ), 0 ≤ zq → Causalean.Experimentation.DesignBased.stdNormalCdf zq = 1 - α / 2 → 1 - α ≤ Filter.liminf (fun n => (Exp n).D.Pr fun z => |Causalean.Experimentation.DesignBased.htEffect (Exp n).D (Exp n).y (Exp n).f (Exp n).θ (dk n) (dl n) z - Causalean.Experimentation.DesignBased.tauTrue (Exp n).y (dk n) (dl n)| ≤ zq * √((Exp n).D.Var (Causalean.Experimentation.DesignBased.htEffect (Exp n).D (Exp n).y (Exp n).f (Exp n).θ (dk n) (dl n)))) Filter.atTop","l":365,"f":"Causalean/Experimentation/ExposureMappingInterference/Asymptotics/SteinInstance.lean"},{"n":"Causalean.Experimentation.ExposureMappingInterference.wald_coverage_of_stein","k":"theorem","m":"Causalean.Experimentation.ExposureMappingInterference.Asymptotics.SteinInstance","a":"Experimentation/ExposureMappingInterference/Asymptotics","t":1,"d":"Aronow–Samii oracle Wald coverage from primitive Stein-discharge conditions. For a pair of treatment sequences dk, dl and a choice of dependency neighbourhoods N n i, suppose the design variance of the Horvitz–Thompson effect estimator is everywhere positive, every unit has nonzero exposure probability under dk and under dl, and the per-unit effect summands are pointwise bounded by a nonnegative sequence B n. If each summand is independent of the sum of summands outside its neighbourhood and the two Stein negligibility limits hold — the design variance of the neighbourhood cross-term sum tends to 0 and the summed third-moment-type error term tends to 0 — and zq is a nonnegative quantile satisfying Φ(zq) = 1 − α/2, then the oracle Wald interval τ̂ ± zq·√Var[τ̂] attains asymptotic (liminf) coverage at least 1 − α.","s":"∀ (Exp : ℕ → Causalean.Experimentation.ExposureMappingInterference.Experiment) (dk dl : (n : ℕ) → (Exp n).Δ) (N : (n : ℕ) → (Exp n).ι → Finset (Exp n).ι), (∀ (n : ℕ), 0 < (Exp n).D.Var (Causalean.Experimentation.DesignBased.htEffect (Exp n).D (Exp n).y (Exp n).f (Exp n).θ (dk n) (dl n))) → (∀ (n : ℕ) (i : (Exp n).ι), Causalean.Experimentation.DesignBased.prop (Exp n).D (Exp n).f (Exp n).θ i (dk n) ≠ 0) → (∀ (n : ℕ) (i : (Exp n).ι), Causalean.Experimentation.DesignBased.prop (Exp n).D (Exp n).f (Exp n).θ i (dl n) ≠ 0) → ∀ (B : ℕ → ℝ), (∀ (n : ℕ), 0 ≤ B n) → (∀ (n : ℕ) (i : (Exp n).ι) (z : (Exp n).Ω), |(Exp n).effSummand (dk n) (dl n) i z| ≤ B n) → (∀ (n : ℕ) (i : (Exp n).ι), ProbabilityTheory.IndepFun ((Exp n).effSummand (dk n) (dl n) i) (fun z => ∑ j ∈ Finset.univ \\ N n i, (Exp n).effSummand (dk n) (dl n) j z) (Exp n).D.toMeasure) → Filter.Tendsto (fun n => (Exp n).D.Var fun z => ∑ i, (Exp n).effSummand (dk n) (dl n) i z * ∑ j ∈ N n i, (Exp n).effSummand (dk n) (dl n) j z) Filter.atTop (nhds 0) → Filter.Tendsto (fun n => ∑ i, (Exp n).D.E fun z => |(Exp n).effSummand (dk n) (dl n) i z| * (∑ j ∈ N n i, (Exp n).effSummand (dk n) (dl n) j z) ^ 2) Filter.atTop (nhds 0) → ∀ {α : ℝ} (zq : ℝ), 0 ≤ zq → Causalean.Experimentation.DesignBased.stdNormalCdf zq = 1 - α / 2 → 1 - α ≤ Filter.liminf (fun n => (Exp n).D.Pr fun z => |Causalean.Experimentation.DesignBased.htEffect (Exp n).D (Exp n).y (Exp n).f (Exp n).θ (dk n) (dl n) z - Causalean.Experimentation.DesignBased.tauTrue (Exp n).y (dk n) (dl n)| ≤ zq * √((Exp n).D.Var (Causalean.Experimentation.DesignBased.htEffect (Exp n).D (Exp n).y (Exp n).f (Exp n).θ (dk n) (dl n)))) Filter.atTop","l":259,"f":"Causalean/Experimentation/ExposureMappingInterference/Asymptotics/SteinInstance.lean"},{"n":"Causalean.Experimentation.FinitePopulationMoments.E_sampleMean","k":"theorem","m":"Causalean.Experimentation.FinitePopulationMoments","a":"Experimentation","t":1,"d":"Unbiasedness of the sample mean (Li & Ding 2017, Thm 1 moments). For a sample size that is positive and at most the population size, the mean of a simple random sample of that size, drawn without replacement from the finite population of outcomes y, is unbiased for the population mean: each unit is sampled with probability n/N, which the 1/n weight averages to 1/N.","s":"∀ {U : Type u_1} [inst : Fintype U] [inst_1 : DecidableEq U] (n : ℕ) (hn : n ≤ Fintype.card U), 0 < n → ∀ (y : U → ℝ), (Causalean.Experimentation.DesignBased.completeRandomization n hn).E (Causalean.Experimentation.FinitePopulationMoments.sampleMean n y) = Causalean.Experimentation.FinitePopulationMoments.popMean y","l":59,"f":"Causalean/Experimentation/FinitePopulationMoments.lean"},{"n":"Causalean.Experimentation.FinitePopulationMoments.Var_sampleMean","k":"theorem","m":"Causalean.Experimentation.FinitePopulationMoments","a":"Experimentation","t":1,"d":"Variance of the sample mean (Li & Ding 2017, Thm 1 / Cochran). For a sample size that is positive and at most the population size, when the population contains at least two units, the randomization variance of the simple-random-sample mean equals (1/n − 1/N)·v_N, the sampling-without-replacement variance.","s":"∀ {U : Type u_1} [inst : Fintype U] [inst_1 : DecidableEq U] (n : ℕ) (hn : n ≤ Fintype.card U), 0 < n → 2 ≤ Fintype.card U → ∀ (y : U → ℝ), (Causalean.Experimentation.DesignBased.completeRandomization n hn).Var (Causalean.Experimentation.FinitePopulationMoments.sampleMean n y) = (1 / ↑n - 1 / ↑(Fintype.card U)) * Causalean.Experimentation.FinitePopulationMoments.popVar y","l":69,"f":"Causalean/Experimentation/FinitePopulationMoments.lean"},{"n":"Causalean.Experimentation.FinitePopulationMoments.popMaxSqDev","k":"def","m":"Causalean.Experimentation.FinitePopulationMoments","a":"Experimentation","t":1,"d":"The maximum squared deviation m_N = maxᵢ (y_i − ȳ_N)² driving the Hájek CLT condition.","s":"{U : Type u_1} → [Fintype U] → [Nonempty U] → (U → ℝ) → ℝ","l":51,"f":"Causalean/Experimentation/FinitePopulationMoments.lean"},{"n":"Causalean.Experimentation.FinitePopulationMoments.popMean","k":"def","m":"Causalean.Experimentation.FinitePopulationMoments","a":"Experimentation","t":1,"d":"The finite-population mean ȳ_N = (1/N) ∑ y_i.","s":"{U : Type u_1} → [Fintype U] → (U → ℝ) → ℝ","l":44,"f":"Causalean/Experimentation/FinitePopulationMoments.lean"},{"n":"Causalean.Experimentation.FinitePopulationMoments.popVar","k":"def","m":"Causalean.Experimentation.FinitePopulationMoments","a":"Experimentation","t":1,"d":"The finite-population variance v_N = (1/(N−1)) ∑ (y_i − ȳ_N)².","s":"{U : Type u_1} → [Fintype U] → (U → ℝ) → ℝ","l":47,"f":"Causalean/Experimentation/FinitePopulationMoments.lean"},{"n":"Causalean.Experimentation.FinitePopulationMoments.sampleMean","k":"def","m":"Causalean.Experimentation.FinitePopulationMoments","a":"Experimentation","t":1,"d":"The simple-random-sample mean ȳ_S = (1/n) ∑_{i∈S} y_i of a size-n sample S.","s":"{U : Type u_1} → [Fintype U] → [DecidableEq U] → (n : ℕ) → (U → ℝ) → { S // S.card = n } → ℝ","l":55,"f":"Causalean/Experimentation/FinitePopulationMoments.lean"},{"n":"Causalean.Experimentation.MatchedPairDesign.E_matchedPairEstimator","k":"theorem","m":"Causalean.Experimentation.MatchedPairDesign.Estimator","a":"Experimentation/MatchedPairDesign","t":1,"d":"Unbiasedness of the matched-pair estimator. Provided there is at least one pair, under the matched-pair design the difference-in-means estimator is unbiased for the sample average treatment effect.","s":"∀ {P : Type u_1} [inst : Fintype P] [inst_1 : DecidableEq P] (y1 y0 : P → Bool → ℝ), 0 < Fintype.card P → Causalean.Experimentation.MatchedPairDesign.matchedPairDesign.E (Causalean.Experimentation.MatchedPairDesign.matchedPairEstimator y1 y0) = Causalean.Experimentation.MatchedPairDesign.sate y1 y0","l":80,"f":"Causalean/Experimentation/MatchedPairDesign/Estimator.lean"},{"n":"Causalean.Experimentation.MatchedPairDesign.E_pairContribution","k":"theorem","m":"Causalean.Experimentation.MatchedPairDesign.Estimator","a":"Experimentation/MatchedPairDesign","t":2,"d":"The expected per-pair contribution is pair p's average treatment effect ½ ∑_b (y1 p b − y0 p b) — the fair coin gives each position probability ½ of being treated.","s":"∀ {P : Type u_1} [inst : Fintype P] [inst_1 : DecidableEq P] (y1 y0 : P → Bool → ℝ) (p : P), Causalean.Experimentation.MatchedPairDesign.matchedPairDesign.E (Causalean.Experimentation.MatchedPairDesign.pairContribution y1 y0 p) = (∑ b, (y1 p b - y0 p b)) / 2","l":61,"f":"Causalean/Experimentation/MatchedPairDesign/Estimator.lean"},{"n":"Causalean.Experimentation.MatchedPairDesign.Var_matchedPairEstimator","k":"theorem","m":"Causalean.Experimentation.MatchedPairDesign.Variance","a":"Experimentation/MatchedPairDesign","t":1,"d":"Variance of the matched-pair estimator. Under the matched-pair design, the difference-in-means estimator built from potential outcomes y1 and y0 has randomization variance equal to 1/(4N²) times the sum of squared within-pair imbalances.","s":"∀ {P : Type u_1} [inst : Fintype P] [inst_1 : DecidableEq P] (y1 y0 : P → Bool → ℝ), Causalean.Experimentation.MatchedPairDesign.matchedPairDesign.Var (Causalean.Experimentation.MatchedPairDesign.matchedPairEstimator y1 y0) = (∑ p, Causalean.Experimentation.MatchedPairDesign.pairImbalance y1 y0 p ^ 2) / (4 * ↑(Fintype.card P) ^ 2)","l":73,"f":"Causalean/Experimentation/MatchedPairDesign/Variance.lean"},{"n":"Causalean.Experimentation.MatchedPairDesign.Var_pairContribution","k":"theorem","m":"Causalean.Experimentation.MatchedPairDesign.Variance","a":"Experimentation/MatchedPairDesign","t":2,"d":"The variance of a single pair's contribution is ¼ times its squared within-pair imbalance — the variance of the fair two-point random variable y1 p (z p) − y0 p (¬ z p).","s":"∀ {P : Type u_1} [inst : Fintype P] [inst_1 : DecidableEq P] (y1 y0 : P → Bool → ℝ) (p : P), Causalean.Experimentation.MatchedPairDesign.matchedPairDesign.Var (Causalean.Experimentation.MatchedPairDesign.pairContribution y1 y0 p) = Causalean.Experimentation.MatchedPairDesign.pairImbalance y1 y0 p ^ 2 / 4","l":52,"f":"Causalean/Experimentation/MatchedPairDesign/Variance.lean"},{"n":"Causalean.Experimentation.MatchedPairDesign.matchedPairDesign","k":"def","m":"Causalean.Experimentation.MatchedPairDesign.MatchedPair","a":"Experimentation/MatchedPairDesign","t":1,"d":"The matched-pair design: each pair p independently treats one of its two positions by a fair coin. The assignment z p : Bool is the treated position in pair p; the other position is control. This is a size-two stratified design with one treated unit per stratum, represented in the smaller assignment space of treated positions rather than as independent unit-level assignments.","s":"{P : Type u_1} → [inst : Fintype P] → [inst_1 : DecidableEq P] → Causalean.Experimentation.DesignBased.FiniteDesign (P → Bool)","l":53,"f":"Causalean/Experimentation/MatchedPairDesign/MatchedPair.lean"},{"n":"Causalean.Experimentation.MatchedPairDesign.matchedPairDesign_E_mpTreatInd","k":"theorem","m":"Causalean.Experimentation.MatchedPairDesign.MatchedPair","a":"Experimentation/MatchedPairDesign","t":1,"d":"First-order inclusion probability. Under the matched-pair design, the unit occupying position b of pair p is treated with probability 1/2.","s":"∀ {P : Type u_1} [inst : Fintype P] [inst_1 : DecidableEq P] (p : P) (b : Bool), Causalean.Experimentation.MatchedPairDesign.matchedPairDesign.E (Causalean.Experimentation.MatchedPairDesign.mpTreatInd p b) = 1 / 2","l":76,"f":"Causalean/Experimentation/MatchedPairDesign/MatchedPair.lean"},{"n":"Causalean.Experimentation.MatchedPairDesign.matchedPairDesign_E_mpTreatInd_cross","k":"theorem","m":"Causalean.Experimentation.MatchedPairDesign.MatchedPair","a":"Experimentation/MatchedPairDesign","t":1,"d":"Cross-pair independence. For pairs p and p' that are distinct, the probability that position b of p and position b' of p' are treated simultaneously equals ¼ — units in distinct pairs are treated independently, and each position is treated with probability ½, so the joint probability factors as ½ · ½.","s":"∀ {P : Type u_1} [inst : Fintype P] [inst_1 : DecidableEq P] (p p' : P), p ≠ p' → ∀ (b b' : Bool), (Causalean.Experimentation.MatchedPairDesign.matchedPairDesign.E fun z => Causalean.Experimentation.MatchedPairDesign.mpTreatInd p b z * Causalean.Experimentation.MatchedPairDesign.mpTreatInd p' b' z) = 1 / 4","l":100,"f":"Causalean/Experimentation/MatchedPairDesign/MatchedPair.lean"},{"n":"Causalean.Experimentation.MatchedPairDesign.matchedPairDesign_E_mpTreatInd_within","k":"theorem","m":"Causalean.Experimentation.MatchedPairDesign.MatchedPair","a":"Experimentation/MatchedPairDesign","t":1,"d":"Within-pair negative dependence. Under the matched-pair design, the two units of pair p are never treated together, so the expectation of the product of their treatment indicators is zero — perfect negative dependence.","s":"∀ {P : Type u_1} [inst : Fintype P] [inst_1 : DecidableEq P] (p : P), (Causalean.Experimentation.MatchedPairDesign.matchedPairDesign.E fun z => Causalean.Experimentation.MatchedPairDesign.mpTreatInd p true z * Causalean.Experimentation.MatchedPairDesign.mpTreatInd p false z) = 0","l":88,"f":"Causalean/Experimentation/MatchedPairDesign/MatchedPair.lean"},{"n":"Causalean.Experimentation.MatchedPairDesign.matchedPairDesign_eq_prod_pairCoin","k":"theorem","m":"Causalean.Experimentation.MatchedPairDesign.MatchedPair","a":"Experimentation/MatchedPairDesign","t":2,"d":"The matched-pair design is the product of the independent pair-level fair coins.","s":"∀ {P : Type u_1} [inst : Fintype P] [inst_1 : DecidableEq P], Causalean.Experimentation.MatchedPairDesign.matchedPairDesign = Causalean.Experimentation.DesignBased.prodDesign fun x => Causalean.Experimentation.MatchedPairDesign.pairCoinDesign","l":60,"f":"Causalean/Experimentation/MatchedPairDesign/MatchedPair.lean"},{"n":"Causalean.Experimentation.MatchedPairDesign.matchedPairEstimator","k":"def","m":"Causalean.Experimentation.MatchedPairDesign.Estimator","a":"Experimentation/MatchedPairDesign","t":1,"d":"The matched-pair difference-in-means estimator: the average over pairs of the per-pair treated-minus-control contributions.","s":"{P : Type u_1} → [Fintype P] → (P → Bool → ℝ) → (P → Bool → ℝ) → (P → Bool) → ℝ","l":56,"f":"Causalean/Experimentation/MatchedPairDesign/Estimator.lean"},{"n":"Causalean.Experimentation.MatchedPairDesign.mpTreatInd","k":"def","m":"Causalean.Experimentation.MatchedPairDesign.MatchedPair","a":"Experimentation/MatchedPairDesign","t":1,"d":"The treatment indicator of the unit at position b of pair p under assignment z: 1 if the coin selected position b, else 0.","s":"{P : Type u_1} → P → Bool → (P → Bool) → ℝ","l":64,"f":"Causalean/Experimentation/MatchedPairDesign/MatchedPair.lean"},{"n":"Causalean.Experimentation.MatchedPairDesign.mpTreatInd_within","k":"theorem","m":"Causalean.Experimentation.MatchedPairDesign.MatchedPair","a":"Experimentation/MatchedPairDesign","t":2,"d":"Within-pair exclusivity. Exactly one position of each pair is treated: the two units' indicators sum to one on every assignment.","s":"∀ {P : Type u_1} (p : P) (z : P → Bool), Causalean.Experimentation.MatchedPairDesign.mpTreatInd p true z + Causalean.Experimentation.MatchedPairDesign.mpTreatInd p false z = 1","l":69,"f":"Causalean/Experimentation/MatchedPairDesign/MatchedPair.lean"},{"n":"Causalean.Experimentation.MatchedPairDesign.pairCoinDesign","k":"def","m":"Causalean.Experimentation.MatchedPairDesign.MatchedPair","a":"Experimentation/MatchedPairDesign","t":1,"d":"The per-pair fair coin: true means the true position is treated, and false means the false position is treated.","s":"Causalean.Experimentation.DesignBased.FiniteDesign Bool","l":48,"f":"Causalean/Experimentation/MatchedPairDesign/MatchedPair.lean"},{"n":"Causalean.Experimentation.MatchedPairDesign.pairContribution","k":"def","m":"Causalean.Experimentation.MatchedPairDesign.Estimator","a":"Experimentation/MatchedPairDesign","t":1,"d":"The observed treated-minus-control contribution of pair p under assignment z: the treated position is z p (outcome y1 p (z p)), the control position is ¬ z p (outcome y0 p (¬ z p)).","s":"{P : Type u_1} → (P → Bool → ℝ) → (P → Bool → ℝ) → P → (P → Bool) → ℝ","l":50,"f":"Causalean/Experimentation/MatchedPairDesign/Estimator.lean"},{"n":"Causalean.Experimentation.MatchedPairDesign.pairImbalance","k":"def","m":"Causalean.Experimentation.MatchedPairDesign.Variance","a":"Experimentation/MatchedPairDesign","t":1,"d":"The within-pair imbalance of pair p: the difference, between its two positions, of the unit totals y1 + y0. Good matching makes this small.","s":"{P : Type u_1} → (P → Bool → ℝ) → (P → Bool → ℝ) → P → ℝ","l":47,"f":"Causalean/Experimentation/MatchedPairDesign/Variance.lean"},{"n":"Causalean.Experimentation.MatchedPairDesign.sate","k":"def","m":"Causalean.Experimentation.MatchedPairDesign.Estimator","a":"Experimentation/MatchedPairDesign","t":1,"d":"The sample average treatment effect over the 2|P| units (both positions of every pair).","s":"{P : Type u_1} → [Fintype P] → (P → Bool → ℝ) → (P → Bool → ℝ) → ℝ","l":46,"f":"Causalean/Experimentation/MatchedPairDesign/Estimator.lean"},{"n":"Causalean.Experimentation.Sequential.AdaptiveExperiment","k":"structure","m":"Causalean.Experimentation.Sequential.AdaptiveDesign","a":"Experimentation/Sequential","t":1,"d":"An adaptive sequential experiment: a time-indexed information flow ℱ on the outcome space together with a propensity process propensity recording the treatment probability of the unit arriving at each time, subject to three conditions — the process is predictable: the time-0 propensity is measurable with respect to the initial information, and the time-(t+1) propensity depends only on the information available at time t, every propensity is nonnegative, and every propensity is at most one.","s":"(Ω : Type u_1) → MeasurableSpace Ω → Type u_1","l":41,"f":"Causalean/Experimentation/Sequential/AdaptiveDesign.lean"},{"n":"Causalean.Experimentation.Sequential.AdaptiveExperiment.HasOverlap","k":"def","m":"Causalean.Experimentation.Sequential.AdaptiveDesign","a":"Experimentation/Sequential","t":1,"d":"The experiment satisfies overlap with margin δ when every propensity stays at least δ away from the boundary: δ ≤ propensity t ω ≤ 1 − δ. This is the positivity condition that makes inverse-propensity weighting (and hence the martingale construction underlying valid inference) well-behaved.","s":"{Ω : Type u_1} → {m0 : MeasurableSpace Ω} → Causalean.Experimentation.Sequential.AdaptiveExperiment Ω m0 → ℝ → Prop","l":67,"f":"Causalean/Experimentation/Sequential/AdaptiveDesign.lean"},{"n":"Causalean.Experimentation.Sequential.AdaptiveExperiment.propensity_pos_of_overlap","k":"theorem","m":"Causalean.Experimentation.Sequential.AdaptiveDesign","a":"Experimentation/Sequential","t":1,"d":"Under overlap with margin δ, every propensity is at least δ, hence strictly positive — so inverse-propensity weights are finite.","s":"∀ {Ω : Type u_1} {m0 : MeasurableSpace Ω} {E : Causalean.Experimentation.Sequential.AdaptiveExperiment Ω m0} {δ : ℝ}, E.HasOverlap δ → ∀ (t : ℕ) (ω : Ω), 0 < E.propensity t ω","l":74,"f":"Causalean/Experimentation/Sequential/AdaptiveDesign.lean"},{"n":"Causalean.Experimentation.Sequential.IsAnytimeValid","k":"def","m":"Causalean.Experimentation.Sequential.AnytimeValid","a":"Experimentation/Sequential","t":1,"d":"A rejection region is anytime-valid at level α under μ when its probability is at most α.","s":"{Ω : Type u_1} → {m0 : MeasurableSpace Ω} → Set Ω → MeasureTheory.Measure Ω → ℝ → Prop","l":45,"f":"Causalean/Experimentation/Sequential/AnytimeValid.lean"},{"n":"Causalean.Experimentation.Sequential.IsConfidenceSequence","k":"def","m":"Causalean.Experimentation.Sequential.AnytimeValid","a":"Experimentation/Sequential","t":1,"d":"A predicate cover : ℕ → Ω → Prop (with cover n ω meaning \"the target lies in the time-n set on outcome ω\") is a confidence sequence at level α when the miscoverage probability — that the cover ever fails — is at most α.","s":"{Ω : Type u_1} → {m0 : MeasurableSpace Ω} → (ℕ → Ω → Prop) → MeasureTheory.Measure Ω → ℝ → Prop","l":59,"f":"Causalean/Experimentation/Sequential/AnytimeValid.lean"},{"n":"Causalean.Experimentation.Sequential.IsTestSupermartingale","k":"def","m":"Causalean.Experimentation.Sequential.Ville","a":"Experimentation/Sequential","t":1,"d":"A test supermartingale for a null hypothesis: a nonnegative supermartingale adapted to the data filtration whose initial expected value is at most one. It is the wealth process of a bet against the null that cannot grow in expectation under it.","s":"{Ω : Type u_1} → {m0 : MeasurableSpace Ω} → (ℕ → Ω → ℝ) → MeasureTheory.Filtration ℕ m0 → MeasureTheory.Measure Ω → Prop","l":45,"f":"Causalean/Experimentation/Sequential/Ville.lean"},{"n":"Causalean.Experimentation.Sequential.confSeqOfWealth","k":"def","m":"Causalean.Experimentation.Sequential.AnytimeValid","a":"Experimentation/Sequential","t":1,"d":"The wealth-based cover predicate obtained by inversion. At time n, the cover holds exactly when the wealth M n has not yet reached 1/α.","s":"{Ω : Type u_1} → (ℕ → Ω → ℝ) → ℝ → ℕ → Ω → Prop","l":65,"f":"Causalean/Experimentation/Sequential/AnytimeValid.lean"},{"n":"Causalean.Experimentation.Sequential.isAnytimeValid_rejectionRegion","k":"theorem","m":"Causalean.Experimentation.Sequential.AnytimeValid","a":"Experimentation/Sequential","t":1,"d":"Anytime-valid type-I error control. If the wealth process M is a test supermartingale for the filtration ℱ under μ and the level α is positive, then the event that M ever reaches 1/α has probability at most α.","s":"∀ {Ω : Type u_1} {m0 : MeasurableSpace Ω} {μ : MeasureTheory.Measure Ω} {ℱ : MeasureTheory.Filtration ℕ m0} [MeasureTheory.IsFiniteMeasure μ] {M : ℕ → Ω → ℝ}, Causalean.Experimentation.Sequential.IsTestSupermartingale M ℱ μ → ∀ {α : ℝ}, 0 < α → Causalean.Experimentation.Sequential.IsAnytimeValid (Causalean.Experimentation.Sequential.rejectionRegion M α) μ α","l":49,"f":"Causalean/Experimentation/Sequential/AnytimeValid.lean"},{"n":"Causalean.Experimentation.Sequential.isConfidenceSequence_confSeqOfWealth","k":"theorem","m":"Causalean.Experimentation.Sequential.AnytimeValid","a":"Experimentation/Sequential","t":1,"d":"Confidence-sequence coverage. If M is a test supermartingale for the filtration ℱ under μ and the level α is positive, then the cover obtained by requiring M's wealth to stay below 1/α fails at some time with probability at most α.","s":"∀ {Ω : Type u_1} {m0 : MeasurableSpace Ω} {μ : MeasureTheory.Measure Ω} {ℱ : MeasureTheory.Filtration ℕ m0} [MeasureTheory.IsFiniteMeasure μ] {M : ℕ → Ω → ℝ}, Causalean.Experimentation.Sequential.IsTestSupermartingale M ℱ μ → ∀ {α : ℝ}, 0 < α → Causalean.Experimentation.Sequential.IsConfidenceSequence (Causalean.Experimentation.Sequential.confSeqOfWealth M α) μ α","l":69,"f":"Causalean/Experimentation/Sequential/AnytimeValid.lean"},{"n":"Causalean.Experimentation.Sequential.rejectionRegion","k":"def","m":"Causalean.Experimentation.Sequential.AnytimeValid","a":"Experimentation/Sequential","t":1,"d":"The rejection region of the sequential test driven by wealth M at level α: the test rejects on the event that M ever reaches 1/α.","s":"{Ω : Type u_1} → (ℕ → Ω → ℝ) → ℝ → Set Ω","l":41,"f":"Causalean/Experimentation/Sequential/AnytimeValid.lean"},{"n":"Causalean.Experimentation.Sequential.supermartingale_maximal_ineq","k":"theorem","m":"Causalean.Experimentation.Sequential.Ville","a":"Experimentation/Sequential","t":2,"d":"Finite-horizon supermartingale maximal inequality. For a nonnegative supermartingale M and level λ > 0, the probability that M reaches λ by time n is at most E[M₀] / λ.","s":"∀ {Ω : Type u_1} {m0 : MeasurableSpace Ω} {μ : MeasureTheory.Measure Ω} {ℱ : MeasureTheory.Filtration ℕ m0} [MeasureTheory.IsFiniteMeasure μ] {M : ℕ → Ω → ℝ}, MeasureTheory.Supermartingale M ℱ μ → (∀ (n : ℕ), 0 ≤ M n) → ∀ {lam : ℝ}, 0 < lam → ∀ (n : ℕ), μ {ω | lam ≤ (Finset.range (n + 1)).sup' Finset.nonempty_range_add_one fun k => M k ω} ≤ ENNReal.ofReal ((∫ (x : Ω), M 0 x ∂μ) / lam)","l":51,"f":"Causalean/Experimentation/Sequential/Ville.lean"},{"n":"Causalean.Experimentation.Sequential.ville_inequality","k":"theorem","m":"Causalean.Experimentation.Sequential.Ville","a":"Experimentation/Sequential","t":1,"d":"Ville's inequality (time-uniform maximal inequality). If M is a supermartingale adapted to the filtration ℱ under the finite measure μ, M is everywhere nonnegative, and the level λ is positive, then the probability that M ever reaches λ is at most E[M₀] / λ, the bound taken over the event of reaching the boundary at some finite time.","s":"∀ {Ω : Type u_1} {m0 : MeasurableSpace Ω} {μ : MeasureTheory.Measure Ω} {ℱ : MeasureTheory.Filtration ℕ m0} [MeasureTheory.IsFiniteMeasure μ] {M : ℕ → Ω → ℝ}, MeasureTheory.Supermartingale M ℱ μ → (∀ (n : ℕ), 0 ≤ M n) → ∀ {lam : ℝ}, 0 < lam → μ {ω | ∃ n, lam ≤ M n ω} ≤ ENNReal.ofReal ((∫ (x : Ω), M 0 x ∂μ) / lam)","l":128,"f":"Causalean/Experimentation/Sequential/Ville.lean"},{"n":"Causalean.Experimentation.Sequential.ville_test","k":"theorem","m":"Causalean.Experimentation.Sequential.Ville","a":"Experimentation/Sequential","t":2,"d":"Ville's inequality for a test supermartingale: the chance of ever reaching 1/α is at most α.","s":"∀ {Ω : Type u_1} {m0 : MeasurableSpace Ω} {μ : MeasureTheory.Measure Ω} {ℱ : MeasureTheory.Filtration ℕ m0} [MeasureTheory.IsFiniteMeasure μ] {M : ℕ → Ω → ℝ}, Causalean.Experimentation.Sequential.IsTestSupermartingale M ℱ μ → ∀ {α : ℝ}, 0 < α → μ {ω | ∃ n, 1 / α ≤ M n ω} ≤ ENNReal.ofReal α","l":159,"f":"Causalean/Experimentation/Sequential/Ville.lean"},{"n":"Causalean.Experimentation.SuperPopulation.HACConsistency.netHACVarEst_eq_locProd","k":"theorem","m":"Causalean.Experimentation.SuperPopulation.HACConsistency.VarianceBound","a":"Experimentation/SuperPopulation/HACConsistency","t":2,"d":"The network-HAC estimator equals the localized double sum ∑ᵢ Xᵢ · (∑_{j ∈ N i} Xⱼ): the pointwise identity that lets the dependency-graph variance bound apply to V̂ verbatim.","s":"∀ {V : Type u_1} {Ω : Type u_2} [inst : Fintype V] [inst_1 : DecidableEq V] [inst_2 : MeasurableSpace Ω] {μ : MeasureTheory.Measure Ω} (F : Causalean.Experimentation.SuperPopulation.NetworkDependence V Ω μ) (ω : Ω), F.netHACVarEst ω = ∑ i, F.X i ω * ∑ k ∈ F.toDepGraph.nbhd i, F.X k ω","l":44,"f":"Causalean/Experimentation/SuperPopulation/HACConsistency/VarianceBound.lean"},{"n":"Causalean.Experimentation.SuperPopulation.HACConsistency.netHACVarEst_memLp","k":"theorem","m":"Causalean.Experimentation.SuperPopulation.HACConsistency.Consistency","a":"Experimentation/SuperPopulation/HACConsistency","t":2,"d":"The network-HAC estimator is in L². With summands bounded by B and degree ≤ m, the estimator V̂ is pointwise bounded by card(V)·m·B², hence square-integrable; this is the moment hypothesis Chebyshev's inequality needs.","s":"∀ {V : Type u_1} {Ω : Type u_2} [inst : Fintype V] [inst_1 : DecidableEq V] [inst_2 : MeasurableSpace Ω] {μ : MeasureTheory.Measure Ω} (F : Causalean.Experimentation.SuperPopulation.NetworkDependence V Ω μ) [MeasureTheory.IsProbabilityMeasure μ] {B : ℝ}, 0 ≤ B → (∀ (i : V) (ω : Ω), |F.X i ω| ≤ B) → ∀ {m : ℕ}, (∀ (i : V), (F.toDepGraph.nbhd i).card ≤ m) → MeasureTheory.MemLp (fun ω => F.netHACVarEst ω) 2 μ","l":42,"f":"Causalean/Experimentation/SuperPopulation/HACConsistency/Consistency.lean"},{"n":"Causalean.Experimentation.SuperPopulation.HACConsistency.netHACVarEst_variance_le","k":"theorem","m":"Causalean.Experimentation.SuperPopulation.HACConsistency.VarianceBound","a":"Experimentation/SuperPopulation/HACConsistency","t":1,"d":"Estimator-variance bound. For a super-population network field F, given a nonnegative bound B such that every summand is bounded in absolute value by B and every network neighborhood has size at most m, the variance of the network-HAC estimator V̂ = ∑ᵢ ∑_{j∈N i} Xᵢ Xⱼ is at most 2·m⁵·card(V)·B⁴.","s":"∀ {V : Type u_1} {Ω : Type u_2} [inst : Fintype V] [inst_1 : DecidableEq V] [inst_2 : MeasurableSpace Ω] {μ : MeasureTheory.Measure Ω} (F : Causalean.Experimentation.SuperPopulation.NetworkDependence V Ω μ) [MeasureTheory.IsProbabilityMeasure μ] {B : ℝ}, 0 ≤ B → (∀ (i : V) (ω : Ω), |F.X i ω| ≤ B) → ∀ {m : ℕ}, (∀ (i : V), (F.toDepGraph.nbhd i).card ≤ m) → ProbabilityTheory.variance (fun ω => F.netHACVarEst ω) μ ≤ 2 * ↑m ^ 5 * ↑(Fintype.card V) * B ^ 4","l":50,"f":"Causalean/Experimentation/SuperPopulation/HACConsistency/VarianceBound.lean"},{"n":"Causalean.Experimentation.SuperPopulation.HACConsistency.netHACVarEst_variance_tendsto_zero","k":"theorem","m":"Causalean.Experimentation.SuperPopulation.HACConsistency.VarianceBound","a":"Experimentation/SuperPopulation/HACConsistency","t":2,"d":"Estimator variance tends to zero. Along a sequence of super-population network fields F n in the CLT regime — bounded degree m, summands bounded by B n with B n → 0, and card(V n)·(B n)³ → 0 — the variance of the network-HAC estimator tends to zero.","s":"∀ {V : ℕ → Type u_3} [inst : (n : ℕ) → Fintype (V n)] [inst_1 : (n : ℕ) → DecidableEq (V n)] {Ω : ℕ → Type u_4} [inst_2 : (n : ℕ) → MeasurableSpace (Ω n)] (μ : (n : ℕ) → MeasureTheory.Measure (Ω n)) [∀ (n : ℕ), MeasureTheory.IsProbabilityMeasure (μ n)] (F : (n : ℕ) → Causalean.Experimentation.SuperPopulation.NetworkDependence (V n) (Ω n) (μ n)) (m : ℕ), (∀ (n : ℕ) (i : V n), ((F n).toDepGraph.nbhd i).card ≤ m) → ∀ (B : ℕ → ℝ), (∀ (n : ℕ), 0 ≤ B n) → (∀ (n : ℕ) (i : V n) (ω : Ω n), |(F n).X i ω| ≤ B n) → Filter.Tendsto B Filter.atTop (nhds 0) → Filter.Tendsto (fun n => ↑(Fintype.card (V n)) * B n ^ 3) Filter.atTop (nhds 0) → Filter.Tendsto (fun n => ProbabilityTheory.variance (fun ω => (F n).netHACVarEst ω) (μ n)) Filter.atTop (nhds 0)","l":70,"f":"Causalean/Experimentation/SuperPopulation/HACConsistency/VarianceBound.lean"},{"n":"Causalean.Experimentation.SuperPopulation.HACConsistency.netHAC_consistent","k":"theorem","m":"Causalean.Experimentation.SuperPopulation.HACConsistency.Consistency","a":"Experimentation/SuperPopulation/HACConsistency","t":1,"d":"HAC consistency (convergence in probability). Consider a sequence of super-population network fields F n over probability spaces with measures μ n, each in the CLT regime — dependency-graph degree at most m, summands uniformly bounded by a sequence B n tending to zero with card(Vₙ)·(Bₙ)³ → 0, and square-integrable, mean-zero summands — and an arbitrary error tolerance ε > 0. Then the network-HAC variance estimator converges in probability to the variance of the network sum: the probability that the estimator deviates from that variance by at least ε tends to zero as n → ∞.","s":"∀ {V : ℕ → Type u_3} [inst : (n : ℕ) → Fintype (V n)] [inst_1 : (n : ℕ) → DecidableEq (V n)] {Ω : ℕ → Type u_4} [inst_2 : (n : ℕ) → MeasurableSpace (Ω n)] (μ : (n : ℕ) → MeasureTheory.Measure (Ω n)) [∀ (n : ℕ), MeasureTheory.IsProbabilityMeasure (μ n)] (F : (n : ℕ) → Causalean.Experimentation.SuperPopulation.NetworkDependence (V n) (Ω n) (μ n)) (m : ℕ), (∀ (n : ℕ) (i : V n), ((F n).toDepGraph.nbhd i).card ≤ m) → ∀ (B : ℕ → ℝ), (∀ (n : ℕ), 0 ≤ B n) → (∀ (n : ℕ) (i : V n) (ω : Ω n), |(F n).X i ω| ≤ B n) → Filter.Tendsto B Filter.atTop (nhds 0) → Filter.Tendsto (fun n => ↑(Fintype.card (V n)) * B n ^ 3) Filter.atTop (nhds 0) → (∀ (n : ℕ) (i : V n), MeasureTheory.MemLp ((F n).X i) 2 (μ n)) → (∀ (n : ℕ) (i : V n), ∫ (ω : Ω n), (F n).X i ω ∂μ n = 0) → ∀ (ε : ℝ), 0 < ε → Filter.Tendsto (fun n => (μ n).real {ω | ε ≤ |(F n).netHACVarEst ω - ProbabilityTheory.variance (Causalean.SteinMethod.depSum (F n).X) (μ n)|}) Filter.atTop (nhds 0)","l":85,"f":"Causalean/Experimentation/SuperPopulation/HACConsistency/Consistency.lean"},{"n":"Causalean.Experimentation.SuperPopulation.MeanCLT.centeredNormalizedField","k":"def","m":"Causalean.Experimentation.SuperPopulation.MeanCLT.Field","a":"Experimentation/SuperPopulation/MeanCLT","t":1,"d":"The centered/normalized network field for a super-population mean. From raw outcomes Y i : Ω → ℝ carrying a reflexive/symmetric measurable m-dependent network and a normalizer s (intended s = √Var(∑ᵢ Yᵢ)), build the NetworkDependence whose summand is the standardized contribution Xᵢ = (Yᵢ − E[Yᵢ]) / s.","s":"{V : Type u_1} → {Ω : Type u_2} → [inst : Fintype V] → [inst_1 : DecidableEq V] → [inst_2 : MeasurableSpace Ω] → {μ : MeasureTheory.Measure Ω} → (Y : V → Ω → ℝ) → (adj : V → V → Prop) → [DecidableRel adj] → (∀ (i : V), adj i i) → (∀ (i j : V), adj i j → adj j i) → (∀ (i : V), Measurable (Y i)) → (∀ (A B : Finset V), (∀ a ∈ A, ∀ b ∈ B, ¬adj a b) → ProbabilityTheory.IndepFun (fun ω k => Y (↑k) ω) (fun ω k => Y (↑k) ω) μ) → ℝ → Causalean.Experimentation.SuperPopulation.NetworkDependence V Ω μ","l":41,"f":"Causalean/Experimentation/SuperPopulation/MeanCLT/Field.lean"},{"n":"Causalean.Experimentation.SuperPopulation.MeanCLT.centeredNormalizedField_X","k":"theorem","m":"Causalean.Experimentation.SuperPopulation.MeanCLT.Field","a":"Experimentation/SuperPopulation/MeanCLT","t":1,"d":"The standardized summand for unit i at sample point ω is the centered outcome Yᵢ ω − E[Yᵢ] divided by the normalizing constant s.","s":"∀ {V : Type u_1} {Ω : Type u_2} [inst : Fintype V] [inst_1 : DecidableEq V] [inst_2 : MeasurableSpace Ω] {μ : MeasureTheory.Measure Ω} (Y : V → Ω → ℝ) (adj : V → V → Prop) [inst_3 : DecidableRel adj] (hrefl : ∀ (i : V), adj i i) (hsymm : ∀ (i j : V), adj i j → adj j i) (hmeasY : ∀ (i : V), Measurable (Y i)) (hindepY : ∀ (A B : Finset V), (∀ a ∈ A, ∀ b ∈ B, ¬adj a b) → ProbabilityTheory.IndepFun (fun ω k => Y (↑k) ω) (fun ω k => Y (↑k) ω) μ) (s : ℝ) (i : V) (ω : Ω), (Causalean.Experimentation.SuperPopulation.MeanCLT.centeredNormalizedField Y adj hrefl hsymm hmeasY hindepY s).X i ω = (Y i ω - ∫ (x : Ω), Y i x ∂μ) / s","l":78,"f":"Causalean/Experimentation/SuperPopulation/MeanCLT/Field.lean"},{"n":"Causalean.Experimentation.SuperPopulation.MeanCLT.centeredNormalizedField_abs_le","k":"theorem","m":"Causalean.Experimentation.SuperPopulation.MeanCLT.Hypotheses","a":"Experimentation/SuperPopulation/MeanCLT","t":2,"d":"Uniform summand bound. If the centered outcomes are bounded, |Yᵢ − E[Yᵢ]| ≤ c, then each standardized summand satisfies |Xᵢ| ≤ 2c/s (with s > 0). The tight bound is c/s, weakened to 2c/s to match the engine's card·Bₙ³ → 0 smallness with Bₙ = 2c/s.","s":"∀ {V : Type u_1} {Ω : Type u_2} [inst : Fintype V] [inst_1 : DecidableEq V] [inst_2 : MeasurableSpace Ω] {μ : MeasureTheory.Measure Ω} (Y : V → Ω → ℝ) (adj : V → V → Prop) [inst_3 : DecidableRel adj] (hrefl : ∀ (i : V), adj i i) (hsymm : ∀ (i j : V), adj i j → adj j i) (hmeasY : ∀ (i : V), Measurable (Y i)) (hindepY : ∀ (A B : Finset V), (∀ a ∈ A, ∀ b ∈ B, ¬adj a b) → ProbabilityTheory.IndepFun (fun ω k => Y (↑k) ω) (fun ω k => Y (↑k) ω) μ) (s : ℝ), 0 < s → ∀ (c : ℝ), (∀ (i : V) (ω : Ω), |Y i ω - ∫ (x : Ω), Y i x ∂μ| ≤ c) → ∀ (i : V) (ω : Ω), |(Causalean.Experimentation.SuperPopulation.MeanCLT.centeredNormalizedField Y adj hrefl hsymm hmeasY hindepY s).X i ω| ≤ 2 * c / s","l":124,"f":"Causalean/Experimentation/SuperPopulation/MeanCLT/Hypotheses.lean"},{"n":"Causalean.Experimentation.SuperPopulation.MeanCLT.centeredNormalizedField_adj","k":"theorem","m":"Causalean.Experimentation.SuperPopulation.MeanCLT.Field","a":"Experimentation/SuperPopulation/MeanCLT","t":1,"d":"The interference network underlying the standardized field is exactly the supplied network adj.","s":"∀ {V : Type u_1} {Ω : Type u_2} [inst : Fintype V] [inst_1 : DecidableEq V] [inst_2 : MeasurableSpace Ω] {μ : MeasureTheory.Measure Ω} (Y : V → Ω → ℝ) (adj : V → V → Prop) [inst_3 : DecidableRel adj] (hrefl : ∀ (i : V), adj i i) (hsymm : ∀ (i j : V), adj i j → adj j i) (hmeasY : ∀ (i : V), Measurable (Y i)) (hindepY : ∀ (A B : Finset V), (∀ a ∈ A, ∀ b ∈ B, ¬adj a b) → ProbabilityTheory.IndepFun (fun ω k => Y (↑k) ω) (fun ω k => Y (↑k) ω) μ) (s : ℝ), (Causalean.Experimentation.SuperPopulation.MeanCLT.centeredNormalizedField Y adj hrefl hsymm hmeasY hindepY s).adj = adj","l":84,"f":"Causalean/Experimentation/SuperPopulation/MeanCLT/Field.lean"},{"n":"Causalean.Experimentation.SuperPopulation.MeanCLT.centeredNormalizedField_integral_eq_zero","k":"theorem","m":"Causalean.Experimentation.SuperPopulation.MeanCLT.Hypotheses","a":"Experimentation/SuperPopulation/MeanCLT","t":2,"d":"Mean-zero summands. Each standardized summand has integral zero: E[Xᵢ] = (E[Yᵢ] − E[Yᵢ]) / s = 0. (Uses integrability of Yᵢ, from MemLp Yᵢ 2.)","s":"∀ {V : Type u_1} {Ω : Type u_2} [inst : Fintype V] [inst_1 : DecidableEq V] [inst_2 : MeasurableSpace Ω] {μ : MeasureTheory.Measure Ω} [MeasureTheory.IsProbabilityMeasure μ] (Y : V → Ω → ℝ) (adj : V → V → Prop) [inst_4 : DecidableRel adj] (hrefl : ∀ (i : V), adj i i) (hsymm : ∀ (i j : V), adj i j → adj j i) (hmeasY : ∀ (i : V), Measurable (Y i)) (hindepY : ∀ (A B : Finset V), (∀ a ∈ A, ∀ b ∈ B, ¬adj a b) → ProbabilityTheory.IndepFun (fun ω k => Y (↑k) ω) (fun ω k => Y (↑k) ω) μ), (∀ (i : V), MeasureTheory.MemLp (Y i) 2 μ) → ∀ (s : ℝ) (i : V), ∫ (ω : Ω), (Causalean.Experimentation.SuperPopulation.MeanCLT.centeredNormalizedField Y adj hrefl hsymm hmeasY hindepY s).X i ω ∂μ = 0","l":39,"f":"Causalean/Experimentation/SuperPopulation/MeanCLT/Hypotheses.lean"},{"n":"Causalean.Experimentation.SuperPopulation.MeanCLT.centeredNormalizedField_sq_integral","k":"theorem","m":"Causalean.Experimentation.SuperPopulation.MeanCLT.Hypotheses","a":"Experimentation/SuperPopulation/MeanCLT","t":1,"d":"Unit total variance. If each outcome is square-integrable, the normalizing constant s is positive, and s² equals the variance of the network sum of outcomes, s² = Var(∑ᵢ Yᵢ), then the standardized network sum ∑ᵢ Xᵢ = (∑ᵢ Yᵢ − ∑ᵢ E[Yᵢ]) / s has unit total variance: ∫ (∑ᵢ Xᵢ)² = 1. This is the field-variance hypothesis of networkSum_clt (∫ (depSum X)² = 1), derived from the outcome sum-variance.","s":"∀ {V : Type u_1} {Ω : Type u_2} [inst : Fintype V] [inst_1 : DecidableEq V] [inst_2 : MeasurableSpace Ω] {μ : MeasureTheory.Measure Ω} [MeasureTheory.IsProbabilityMeasure μ] (Y : V → Ω → ℝ) (adj : V → V → Prop) [inst_4 : DecidableRel adj] (hrefl : ∀ (i : V), adj i i) (hsymm : ∀ (i j : V), adj i j → adj j i) (hmeasY : ∀ (i : V), Measurable (Y i)) (hindepY : ∀ (A B : Finset V), (∀ a ∈ A, ∀ b ∈ B, ¬adj a b) → ProbabilityTheory.IndepFun (fun ω k => Y (↑k) ω) (fun ω k => Y (↑k) ω) μ), (∀ (i : V), MeasureTheory.MemLp (Y i) 2 μ) → ∀ (s : ℝ), 0 < s → s ^ 2 = ProbabilityTheory.variance (fun ω => ∑ i, Y i ω) μ → ∫ (ω : Ω), Causalean.SteinMethod.depSum (Causalean.Experimentation.SuperPopulation.MeanCLT.centeredNormalizedField Y adj hrefl hsymm hmeasY hindepY s).X ω ^ 2 ∂μ = 1","l":49,"f":"Causalean/Experimentation/SuperPopulation/MeanCLT/Hypotheses.lean"},{"n":"Causalean.Experimentation.SuperPopulation.MeanCLT.networkMean_clt","k":"theorem","m":"Causalean.Experimentation.SuperPopulation.MeanCLT.MeanCLT","a":"Experimentation/SuperPopulation/MeanCLT","t":1,"d":"Asymptotic normality of a centered super-population network sum. Fix a family of outcomes Y n over probability spaces with measures μ n and a reflexive, symmetric adjacency relation adj n recording which units interfere, with every outcome measurable, outcome tuples on non-adjacent unit sets independent — i.e. m-dependence — and adjacency degree bounded by m. Assume every outcome is square-integrable, a positive normalizing constant s n with (s n)² = Var(∑ᵢ Yₙᵢ), outcomes uniformly bounded around their means by a sequence c n, and the negligibility rate card(Vₙ)·(cₙ/sₙ)³ → 0. Then the network sum, centered by subtracting the sum of the individual outcome means and divided by s n, converges in distribution to the standard normal.","s":"∀ {V : ℕ → Type u_1} [inst : (n : ℕ) → Fintype (V n)] {Ω : ℕ → Type u_2} [inst_1 : (n : ℕ) → MeasurableSpace (Ω n)] (μ : (n : ℕ) → MeasureTheory.Measure (Ω n)) [∀ (n : ℕ), MeasureTheory.IsProbabilityMeasure (μ n)] (Y : (n : ℕ) → V n → Ω n → ℝ) (adj : (n : ℕ) → V n → V n → Prop) [inst_3 : (n : ℕ) → DecidableRel (adj n)], (∀ (n : ℕ) (i : V n), adj n i i) → (∀ (n : ℕ) (i j : V n), adj n i j → adj n j i) → (∀ (n : ℕ) (i : V n), Measurable (Y n i)) → (∀ (n : ℕ) (A B : Finset (V n)), (∀ a ∈ A, ∀ b ∈ B, ¬adj n a b) → ProbabilityTheory.IndepFun (fun ω k => Y n (↑k) ω) (fun ω k => Y n (↑k) ω) (μ n)) → ∀ (m : ℕ), (∀ (n : ℕ) (i : V n), {j | adj n i j}.card ≤ m) → (∀ (n : ℕ) (i : V n), MeasureTheory.MemLp (Y n i) 2 (μ n)) → ∀ (s : ℕ → ℝ), (∀ (n : ℕ), 0 < s n) → (∀ (n : ℕ), s n ^ 2 = ProbabilityTheory.variance (fun ω => ∑ i, Y n i ω) (μ n)) → ∀ (c : ℕ → ℝ), (∀ (n : ℕ) (i : V n) (ω : Ω n), |Y n i ω - ∫ (x : Ω n), Y n i x ∂μ n| ≤ c n) → Filter.Tendsto (fun n => ↑(Fintype.card (V n)) * (c n / s n) ^ 3) Filter.atTop (nhds 0) → ∀ (t : ℝ), Filter.Tendsto (fun n => (μ n).real {ω | (∑ i, Y n i ω - ∑ i, ∫ (x : Ω n), Y n i x ∂μ n) / s n ≤ t}) Filter.atTop (nhds ((ProbabilityTheory.gaussianReal 0 1).real (Set.Iic t)))","l":36,"f":"Causalean/Experimentation/SuperPopulation/MeanCLT/MeanCLT.lean"},{"n":"Causalean.Experimentation.SuperPopulation.NetworkDependence","k":"structure","m":"Causalean.Experimentation.SuperPopulation.Basic","a":"Experimentation/SuperPopulation","t":1,"d":"A super-population locally-dependent network field: bundles node-level random summands on a common ambient probability space together with a network relation between units that is reflexive and symmetric, requires every summand to be measurable, and requires any two collections of units joined by no edge to carry independent summand tuples (exact m-dependence beyond the network). This is the model-based counterpart of the finite design — the randomness is the population draw, not the assignment — and the m-dependence sibling of an i.i.d. sample.","s":"(V : Type u_3) → (Ω : Type u_4) → [Fintype V] → [DecidableEq V] → [inst : MeasurableSpace Ω] → MeasureTheory.Measure Ω → Type (max u_3 u_4)","l":39,"f":"Causalean/Experimentation/SuperPopulation/Basic.lean"},{"n":"Causalean.Experimentation.SuperPopulation.NetworkDependence.mem_nbhd_iff","k":"theorem","m":"Causalean.Experimentation.SuperPopulation.Basic","a":"Experimentation/SuperPopulation","t":1,"d":"Unit j lies in unit i's network neighborhood if and only if i and j are adjacent in the underlying interference network.","s":"∀ {V : Type u_1} {Ω : Type u_2} [inst : Fintype V] [inst_1 : DecidableEq V] [inst_2 : MeasurableSpace Ω] {μ : MeasureTheory.Measure Ω} (F : Causalean.Experimentation.SuperPopulation.NetworkDependence V Ω μ) {i j : V}, j ∈ F.nbhd i ↔ F.adj i j","l":82,"f":"Causalean/Experimentation/SuperPopulation/Basic.lean"},{"n":"Causalean.Experimentation.SuperPopulation.NetworkDependence.nbhd","k":"def","m":"Causalean.Experimentation.SuperPopulation.Basic","a":"Experimentation/SuperPopulation","t":1,"d":"The closed network neighborhood N i = {j | adj i j} of a unit.","s":"{V : Type u_1} → {Ω : Type u_2} → [inst : Fintype V] → [inst_1 : DecidableEq V] → [inst_2 : MeasurableSpace Ω] → {μ : MeasureTheory.Measure Ω} → Causalean.Experimentation.SuperPopulation.NetworkDependence V Ω μ → V → Finset V","l":78,"f":"Causalean/Experimentation/SuperPopulation/Basic.lean"},{"n":"Causalean.Experimentation.SuperPopulation.NetworkDependence.netHACVarEst","k":"def","m":"Causalean.Experimentation.SuperPopulation.HAC","a":"Experimentation/SuperPopulation","t":1,"d":"The network-HAC variance estimator for a network field: the empirical sum of summand products over network-adjacent pairs, V̂(ω) = ∑ᵢ ∑_{j ∈ N i} Xᵢ(ω)·Xⱼ(ω). It targets the variance of the network sum, whose cross-terms vanish outside the network neighborhoods.","s":"{V : Type u_1} → {Ω : Type u_2} → [inst : Fintype V] → [inst_1 : DecidableEq V] → [inst_2 : MeasurableSpace Ω] → {μ : MeasureTheory.Measure Ω} → Causalean.Experimentation.SuperPopulation.NetworkDependence V Ω μ → Ω → ℝ","l":33,"f":"Causalean/Experimentation/SuperPopulation/HAC.lean"},{"n":"Causalean.Experimentation.SuperPopulation.NetworkDependence.netHACVarEst_integral_eq_variance","k":"theorem","m":"Causalean.Experimentation.SuperPopulation.HAC","a":"Experimentation/SuperPopulation","t":1,"d":"Unbiasedness of the network-HAC estimator. Under a probability measure with square- integrable summands that are mean zero, the expectation of the network-HAC estimator equals the variance of the network sum: E[V̂] = Var(∑ᵢ Xᵢ).","s":"∀ {V : Type u_1} {Ω : Type u_2} [inst : Fintype V] [inst_1 : DecidableEq V] [inst_2 : MeasurableSpace Ω] {μ : MeasureTheory.Measure Ω} (F : Causalean.Experimentation.SuperPopulation.NetworkDependence V Ω μ) [MeasureTheory.IsProbabilityMeasure μ], (∀ (i : V), MeasureTheory.MemLp (F.X i) 2 μ) → (∀ (i : V), ∫ (ω : Ω), F.X i ω ∂μ = 0) → ∫ (ω : Ω), F.netHACVarEst ω ∂μ = ProbabilityTheory.variance (Causalean.SteinMethod.depSum F.X) μ","l":39,"f":"Causalean/Experimentation/SuperPopulation/HAC.lean"},{"n":"Causalean.Experimentation.SuperPopulation.NetworkDependence.self_mem_nbhd","k":"theorem","m":"Causalean.Experimentation.SuperPopulation.Basic","a":"Experimentation/SuperPopulation","t":1,"d":"Every unit lies in its own network neighborhood.","s":"∀ {V : Type u_1} {Ω : Type u_2} [inst : Fintype V] [inst_1 : DecidableEq V] [inst_2 : MeasurableSpace Ω] {μ : MeasureTheory.Measure Ω} (F : Causalean.Experimentation.SuperPopulation.NetworkDependence V Ω μ) (i : V), i ∈ F.nbhd i","l":87,"f":"Causalean/Experimentation/SuperPopulation/Basic.lean"},{"n":"Causalean.Experimentation.SuperPopulation.NetworkDependence.toDepGraph","k":"def","m":"Causalean.Experimentation.SuperPopulation.Basic","a":"Experimentation/SuperPopulation","t":1,"d":"The network field, viewed as a Stein dependency graph for its summands — a pure field rename, so the proved dependency-graph CLT applies verbatim.","s":"{V : Type u_1} → {Ω : Type u_2} → [inst : Fintype V] → [inst_1 : DecidableEq V] → [inst_2 : MeasurableSpace Ω] → {μ : MeasureTheory.Measure Ω} → (F : Causalean.Experimentation.SuperPopulation.NetworkDependence V Ω μ) → Causalean.SteinMethod.DepGraph F.X μ","l":68,"f":"Causalean/Experimentation/SuperPopulation/Basic.lean"},{"n":"Causalean.Experimentation.SuperPopulation.networkSum_clt","k":"theorem","m":"Causalean.Experimentation.SuperPopulation.CLT","a":"Experimentation/SuperPopulation","t":1,"d":"m-dependent network CLT (super-population). Consider a sequence of super-population network fields F n, each on a probability space with measure μ n, whose dependency graph has degree at most m, whose summands are uniformly bounded in absolute value by a sequence B n tending to zero fast enough that card(Vₙ)·(Bₙ)³ → 0, and whose summands are mean zero with the network sum having unit total variance for every n. Then the network sum's cumulative distribution function converges pointwise to the standard normal CDF: P[∑ᵢ (F n).X i ≤ s] → Φ(s).","s":"∀ {V : ℕ → Type u_1} [inst : (n : ℕ) → Fintype (V n)] [inst_1 : (n : ℕ) → DecidableEq (V n)] {Ω : ℕ → Type u_2} [inst_2 : (n : ℕ) → MeasurableSpace (Ω n)] (μ : (n : ℕ) → MeasureTheory.Measure (Ω n)) [∀ (n : ℕ), MeasureTheory.IsProbabilityMeasure (μ n)] (F : (n : ℕ) → Causalean.Experimentation.SuperPopulation.NetworkDependence (V n) (Ω n) (μ n)) (m : ℕ), (∀ (n : ℕ) (i : V n), ((F n).toDepGraph.nbhd i).card ≤ m) → ∀ (B : ℕ → ℝ), (∀ (n : ℕ), 0 ≤ B n) → (∀ (n : ℕ) (i : V n) (ω : Ω n), |(F n).X i ω| ≤ B n) → Filter.Tendsto B Filter.atTop (nhds 0) → Filter.Tendsto (fun n => ↑(Fintype.card (V n)) * B n ^ 3) Filter.atTop (nhds 0) → (∀ (n : ℕ) (i : V n), ∫ (ω : Ω n), (F n).X i ω ∂μ n = 0) → (∀ (n : ℕ), ∫ (ω : Ω n), Causalean.SteinMethod.depSum (F n).X ω ^ 2 ∂μ n = 1) → ∀ (s : ℝ), Filter.Tendsto (fun n => (MeasureTheory.Measure.map (Causalean.SteinMethod.depSum (F n).X) (μ n)).real (Set.Iic s)) Filter.atTop (nhds ((ProbabilityTheory.gaussianReal 0 1).real (Set.Iic s)))","l":29,"f":"Causalean/Experimentation/SuperPopulation/CLT.lean"},{"n":"Causalean.Experimentation.TwoStageInterference.CE_direct","k":"def","m":"Causalean.Experimentation.TwoStageInterference.Basic","a":"Experimentation/TwoStageInterference","t":1,"d":"Hudgens-Halloran direct-effect contrast. This is the population average outcome under treatment minus the population average outcome under control, evaluated under the strategy ψ.","s":"{ι : Type u_1} → [Fintype ι] → {n : ι → ℕ} → ((i : ι) → Causalean.Experimentation.DesignBased.FiniteDesign (Causalean.Experimentation.TwoStageInterference.WAssign n i)) → ((i : ι) → Fin (n i) → Causalean.Experimentation.TwoStageInterference.WAssign n i → ℝ) → ℝ","l":82,"f":"Causalean/Experimentation/TwoStageInterference/Basic.lean"},{"n":"Causalean.Experimentation.TwoStageInterference.CE_indirect","k":"def","m":"Causalean.Experimentation.TwoStageInterference.Basic","a":"Experimentation/TwoStageInterference","t":1,"d":"Indirect (spillover) causal effect C̄E^I(φ,ψ) = ȳ(0;φ) − ȳ(0;ψ).","s":"{ι : Type u_1} → [Fintype ι] → {n : ι → ℕ} → ((i : ι) → Causalean.Experimentation.DesignBased.FiniteDesign (Causalean.Experimentation.TwoStageInterference.WAssign n i)) → ((i : ι) → Causalean.Experimentation.DesignBased.FiniteDesign (Causalean.Experimentation.TwoStageInterference.WAssign n i)) → ((i : ι) → Fin (n i) → Causalean.Experimentation.TwoStageInterference.WAssign n i → ℝ) → ℝ","l":88,"f":"Causalean/Experimentation/TwoStageInterference/Basic.lean"},{"n":"Causalean.Experimentation.TwoStageInterference.CE_overall","k":"def","m":"Causalean.Experimentation.TwoStageInterference.Basic","a":"Experimentation/TwoStageInterference","t":1,"d":"Overall causal effect C̄E^O(φ,ψ) = ȳ(φ) − ȳ(ψ).","s":"{ι : Type u_1} → [Fintype ι] → {n : ι → ℕ} → ((i : ι) → Causalean.Experimentation.DesignBased.FiniteDesign (Causalean.Experimentation.TwoStageInterference.WAssign n i)) → ((i : ι) → Causalean.Experimentation.DesignBased.FiniteDesign (Causalean.Experimentation.TwoStageInterference.WAssign n i)) → ((i : ι) → Fin (n i) → Causalean.Experimentation.TwoStageInterference.WAssign n i → ℝ) → ℝ","l":101,"f":"Causalean/Experimentation/TwoStageInterference/Basic.lean"},{"n":"Causalean.Experimentation.TwoStageInterference.CE_total","k":"def","m":"Causalean.Experimentation.TwoStageInterference.Basic","a":"Experimentation/TwoStageInterference","t":1,"d":"Total causal effect C̄E^T(φ,ψ) = ȳ(0;φ) − ȳ(1;ψ).","s":"{ι : Type u_1} → [Fintype ι] → {n : ι → ℕ} → ((i : ι) → Causalean.Experimentation.DesignBased.FiniteDesign (Causalean.Experimentation.TwoStageInterference.WAssign n i)) → ((i : ι) → Causalean.Experimentation.DesignBased.FiniteDesign (Causalean.Experimentation.TwoStageInterference.WAssign n i)) → ((i : ι) → Fin (n i) → Causalean.Experimentation.TwoStageInterference.WAssign n i → ℝ) → ℝ","l":93,"f":"Causalean/Experimentation/TwoStageInterference/Basic.lean"},{"n":"Causalean.Experimentation.TwoStageInterference.CE_total_decomp","k":"theorem","m":"Causalean.Experimentation.TwoStageInterference.Unbiased","a":"Experimentation/TwoStageInterference","t":1,"d":"Mixed-orientation decomposition identity. For per-group two-stage designs ψ (treatment strategy) and φ (control strategy) governing each group's within-group assignment and an outcome recorded for every group, unit, and realized within-group assignment, the total contrast — the population control-state mean under φ minus the population treated-state mean under ψ — equals the indirect contrast — the population control-state mean under φ minus the population control-state mean under ψ — minus the direct contrast — the population treated-state mean under ψ minus the population control-state mean under ψ.","s":"∀ {ι : Type u_1} [inst : Fintype ι] {n : ι → ℕ} (ψ φ : (i : ι) → Causalean.Experimentation.DesignBased.FiniteDesign (Causalean.Experimentation.TwoStageInterference.WAssign n i)) (Y : (i : ι) → Fin (n i) → Causalean.Experimentation.TwoStageInterference.WAssign n i → ℝ), Causalean.Experimentation.TwoStageInterference.CE_total ψ φ Y = Causalean.Experimentation.TwoStageInterference.CE_indirect ψ φ Y - Causalean.Experimentation.TwoStageInterference.CE_direct ψ Y","l":136,"f":"Causalean/Experimentation/TwoStageInterference/Unbiased.lean"},{"n":"Causalean.Experimentation.TwoStageInterference.DEbar_eq_of_homogeneous","k":"theorem","m":"Causalean.Experimentation.TwoStageInterference.Asymptotic.CLTDischarge","a":"Experimentation/TwoStageInterference/Asymptotic","t":1,"d":"Estimand reduction. Under the homogeneity and regularity bundle, the population average treatment-minus-control direct-effect contrast collapses to the common group-level contrast δ.","s":"∀ {Exp : ℕ → Causalean.Experimentation.TwoStageInterference.LHExperiment} {t δ M : ℝ} {v : ℕ → ℝ} {stud : (n : ℕ) → Causalean.Experimentation.TwoStageInterference.StratAssign (Exp n).ι × ((i : (Exp n).ι) → Fin ((Exp n).gsize i) → Bool) → ℝ}, Causalean.Experimentation.TwoStageInterference.Homogeneous Exp t stud δ M v → ∀ (n : ℕ), (Exp n).DEbar = δ","l":190,"f":"Causalean/Experimentation/TwoStageInterference/Asymptotic/CLTDischarge.lean"},{"n":"Causalean.Experimentation.TwoStageInterference.E_Shat","k":"theorem","m":"Causalean.Experimentation.TwoStageInterference.VarianceMoments","a":"Experimentation/TwoStageInterference","t":1,"d":"Expectation of an observed sample variance. Let U be a {0,1}-valued selection family over a design ρ on length-n binary assignments, so that every Uⱼ is idempotent, taking only the values 0 and 1, and suppose the group size n and n − 1 are both nonzero, as are the real-valued selection count M and M − 1. If each Uⱼ has first moment M/n, every two distinct units j and k have second moment M(M−1)/(n(n−1)) for the product Uⱼ·Uₖ, and exactly M units are selected on every assignment the design gives positive weight, then the expectation of the realized sample variance of x over the selected units — (1/(M−1))∑ⱼ Uⱼ(xⱼ − x̄_U)² with x̄_U = (∑ Uⱼxⱼ)/M — equals the population sample variance (1/(n−1))∑ⱼ(xⱼ − x̄)².","s":"∀ {n : ℕ} (ρ : Causalean.Experimentation.DesignBased.FiniteDesign (Fin n → Bool)) (M : ℝ) (x : Fin n → ℝ) (U : Fin n → (Fin n → Bool) → ℝ), ↑n ≠ 0 → ↑n - 1 ≠ 0 → M ≠ 0 → M - 1 ≠ 0 → (∀ (j : Fin n) (w : Fin n → Bool), U j w * U j w = U j w) → (∀ (j : Fin n), ρ.E (U j) = M / ↑n) → (∀ (j k : Fin n), j ≠ k → (ρ.E fun w => U j w * U k w) = M * (M - 1) / (↑n * (↑n - 1))) → (∀ (w : Fin n → Bool), ρ.p w ≠ 0 → ∑ j, U j w = M) → (ρ.E fun w => (∑ j, U j w * (x j - (∑ i, U i w * x i) / M) ^ 2) / (M - 1)) = (∑ j, (x j - (∑ i, x i) / ↑n) ^ 2) / (↑n - 1)","l":64,"f":"Causalean/Experimentation/TwoStageInterference/VarianceMoments.lean"},{"n":"Causalean.Experimentation.TwoStageInterference.E_ShatControl","k":"theorem","m":"Causalean.Experimentation.TwoStageInterference.VarianceConservative","a":"Experimentation/TwoStageInterference","t":1,"d":"Expected observed control sample variance. The control analogue: for any design satisfying the same completely-randomized moment hypotheses, the expectation of the realized untreated-state sample variance among the n−K control units equals the population untreated-state sample variance S₀: E[Ŝ₀] = S₀.","s":"∀ {n : ℕ} (K : ℕ) (b : Fin n → ℝ) (ρ : Causalean.Experimentation.DesignBased.FiniteDesign (Fin n → Bool)), 2 ≤ K → K + 2 ≤ n → (∀ (j : Fin n), ρ.E (Causalean.Experimentation.TwoStageInterference.T j) = ↑K / ↑n) → (∀ (j k : Fin n), j ≠ k → (ρ.E fun w => Causalean.Experimentation.TwoStageInterference.T j w * Causalean.Experimentation.TwoStageInterference.T k w) = ↑K * (↑K - 1) / (↑n * (↑n - 1))) → (∀ (w : Fin n → Bool), ρ.p w ≠ 0 → ∑ j, Causalean.Experimentation.TwoStageInterference.T j w = ↑K) → ρ.E (Causalean.Experimentation.TwoStageInterference.ShatControl K b) = Causalean.Experimentation.TwoStageInterference.S0 b","l":168,"f":"Causalean/Experimentation/TwoStageInterference/VarianceConservative.lean"},{"n":"Causalean.Experimentation.TwoStageInterference.E_ShatTreated","k":"theorem","m":"Causalean.Experimentation.TwoStageInterference.VarianceConservative","a":"Experimentation/TwoStageInterference","t":1,"d":"Expected observed treated sample variance. For any within-group design whose treatment indicators satisfy the completely-randomized moment hypotheses — first moment K/n (hmean), pairwise second moment K(K−1)/(n(n−1)) (hpair), and a deterministic treated count K on the support (hsupp) — the expectation of the realized treated-state sample variance equals the population treated-state sample variance S₁: E[Ŝ₁] = S₁.","s":"∀ {n : ℕ} (K : ℕ) (a : Fin n → ℝ) (ρ : Causalean.Experimentation.DesignBased.FiniteDesign (Fin n → Bool)), 2 ≤ K → K + 2 ≤ n → (∀ (j : Fin n), ρ.E (Causalean.Experimentation.TwoStageInterference.T j) = ↑K / ↑n) → (∀ (j k : Fin n), j ≠ k → (ρ.E fun w => Causalean.Experimentation.TwoStageInterference.T j w * Causalean.Experimentation.TwoStageInterference.T k w) = ↑K * (↑K - 1) / (↑n * (↑n - 1))) → (∀ (w : Fin n → Bool), ρ.p w ≠ 0 → ∑ j, Causalean.Experimentation.TwoStageInterference.T j w = ↑K) → ρ.E (Causalean.Experimentation.TwoStageInterference.ShatTreated K a) = Causalean.Experimentation.TwoStageInterference.S1 a","l":146,"f":"Causalean/Experimentation/TwoStageInterference/VarianceConservative.lean"},{"n":"Causalean.Experimentation.TwoStageInterference.E_congr_supp","k":"theorem","m":"Causalean.Experimentation.TwoStageInterference.VarianceMoments","a":"Experimentation/TwoStageInterference","t":2,"d":"Support congruence for expectation. Two random variables that agree on every assignment the design gives positive weight have equal expectation, since the off-support assignments contribute 0 = p w · _ to the finite sum either way.","s":"∀ {n : ℕ} (ρ : Causalean.Experimentation.DesignBased.FiniteDesign (Fin n → Bool)) {f g : (Fin n → Bool) → ℝ}, (∀ (w : Fin n → Bool), ρ.p w ≠ 0 → f w = g w) → ρ.E f = ρ.E g","l":53,"f":"Causalean/Experimentation/TwoStageInterference/VarianceMoments.lean"},{"n":"Causalean.Experimentation.TwoStageInterference.E_estDirect","k":"theorem","m":"Causalean.Experimentation.TwoStageInterference.Effects","a":"Experimentation/TwoStageInterference","t":1,"d":"Direct-contrast unbiasedness (Theorem 1 contrast). For the two-stage design that allocates groups to strategy ψ or strategy φ and records outcomes via the potential-outcome function Y, assume the target sample size C of ψ-selected groups is nonzero, every group's control-arm unit count m0 is nonzero, every group's treatment-arm unit count m1 is nonzero, and every group's size n is nonzero. Suppose that within each group randomized by ψ each unit's control propensity is m0/n and each unit's treatment propensity is m1/n, and that the stage-1 design selects each group into the ψ arm with probability C/N. Then the Horvitz–Thompson estimator built from the ψ-selected groups is unbiased for the direct-effect contrast — the population average outcome under treatment minus under control, both evaluated under strategy ψ.","s":"∀ {ι : Type u_1} [inst : Fintype ι] [inst_1 : DecidableEq ι] {n : ι → ℕ} (D₁ : Causalean.Experimentation.DesignBased.FiniteDesign (Causalean.Experimentation.TwoStageInterference.StratAssign ι)) (ψ φ : (i : ι) → Causalean.Experimentation.DesignBased.FiniteDesign (Causalean.Experimentation.TwoStageInterference.WAssign n i)) (Y : (i : ι) → Fin (n i) → Causalean.Experimentation.TwoStageInterference.WAssign n i → ℝ) (m0 m1 : ι → ℝ) (C : ℝ), C ≠ 0 → (∀ (i : ι), m0 i ≠ 0) → (∀ (i : ι), m1 i ≠ 0) → (∀ (i : ι), ↑(n i) ≠ 0) → (∀ (i : ι) (j : Fin (n i)), ((ψ i).Pr fun w => w j = false) = m0 i / ↑(n i)) → (∀ (i : ι) (j : Fin (n i)), ((ψ i).Pr fun w => w j = true) = m1 i / ↑(n i)) → (∀ (i : ι), (D₁.Pr fun s => s i = true) = C / ↑(Fintype.card ι)) → (Causalean.Experimentation.TwoStageInterference.jointDesign D₁ ψ φ).E (Causalean.Experimentation.TwoStageInterference.estDirect Y m0 m1 C) = Causalean.Experimentation.TwoStageInterference.CE_direct ψ Y","l":137,"f":"Causalean/Experimentation/TwoStageInterference/Effects.lean"},{"n":"Causalean.Experimentation.TwoStageInterference.E_estIndirect","k":"theorem","m":"Causalean.Experimentation.TwoStageInterference.Effects","a":"Experimentation/TwoStageInterference","t":1,"d":"Indirect-effect unbiasedness (Theorem 2 contrast). For the two-stage design that allocates groups to strategy ψ or strategy φ and records outcomes via the potential-outcome function Y, assume the target sample size dφ of φ-selected groups is nonzero, the target sample size dψ of ψ-selected groups is nonzero, every group's φ-arm control unit count m0φ is nonzero, every group's ψ-arm control unit count m0ψ is nonzero, and every group's size n is nonzero. Suppose that within each group randomized by φ each unit's control propensity is m0φ/n, that within each group randomized by ψ each unit's control propensity is m0ψ/n, that the stage-1 design selects each group into the φ arm with probability dφ/N, and that it selects each group into the ψ arm with probability dψ/N. Then the indirect-effect estimator is unbiased for the spillover contrast: the population average control outcome under φ minus the population average control outcome under ψ.","s":"∀ {ι : Type u_1} [inst : Fintype ι] [inst_1 : DecidableEq ι] {n : ι → ℕ} (D₁ : Causalean.Experimentation.DesignBased.FiniteDesign (Causalean.Experimentation.TwoStageInterference.StratAssign ι)) (ψ φ : (i : ι) → Causalean.Experimentation.DesignBased.FiniteDesign (Causalean.Experimentation.TwoStageInterference.WAssign n i)) (Y : (i : ι) → Fin (n i) → Causalean.Experimentation.TwoStageInterference.WAssign n i → ℝ) (m0φ m0ψ : ι → ℝ) (dφ dψ : ℝ), dφ ≠ 0 → dψ ≠ 0 → (∀ (i : ι), m0φ i ≠ 0) → (∀ (i : ι), m0ψ i ≠ 0) → (∀ (i : ι), ↑(n i) ≠ 0) → (∀ (i : ι) (j : Fin (n i)), ((φ i).Pr fun w => w j = false) = m0φ i / ↑(n i)) → (∀ (i : ι) (j : Fin (n i)), ((ψ i).Pr fun w => w j = false) = m0ψ i / ↑(n i)) → (∀ (i : ι), (D₁.Pr fun s => s i = false) = dφ / ↑(Fintype.card ι)) → (∀ (i : ι), (D₁.Pr fun s => s i = true) = dψ / ↑(Fintype.card ι)) → (Causalean.Experimentation.TwoStageInterference.jointDesign D₁ ψ φ).E (Causalean.Experimentation.TwoStageInterference.estIndirect Y m0φ m0ψ dφ dψ) = Causalean.Experimentation.TwoStageInterference.CE_indirect ψ φ Y","l":162,"f":"Causalean/Experimentation/TwoStageInterference/Effects.lean"},{"n":"Causalean.Experimentation.TwoStageInterference.E_estTotal","k":"theorem","m":"Causalean.Experimentation.TwoStageInterference.Effects","a":"Experimentation/TwoStageInterference","t":1,"d":"Total-effect unbiasedness (Theorem 3 contrast). For the two-stage design that allocates groups to strategy ψ or strategy φ and records outcomes via the potential-outcome function Y, assume the target sample size dφ of φ-selected groups is nonzero, the target sample size dψ of ψ-selected groups is nonzero, every group's φ-arm control unit count m0φ is nonzero, every group's ψ-arm treatment unit count m1ψ is nonzero, and every group's size n is nonzero. Suppose that within each group randomized by φ each unit's control propensity is m0φ/n, that within each group randomized by ψ each unit's treatment propensity is m1ψ/n, that the stage-1 design selects each group into the φ arm with probability dφ/N, and that it selects each group into the ψ arm with probability dψ/N. Then the total-effect estimator is unbiased for the contrast between the population average control outcome under φ and the population average treatment outcome under ψ.","s":"∀ {ι : Type u_1} [inst : Fintype ι] [inst_1 : DecidableEq ι] {n : ι → ℕ} (D₁ : Causalean.Experimentation.DesignBased.FiniteDesign (Causalean.Experimentation.TwoStageInterference.StratAssign ι)) (ψ φ : (i : ι) → Causalean.Experimentation.DesignBased.FiniteDesign (Causalean.Experimentation.TwoStageInterference.WAssign n i)) (Y : (i : ι) → Fin (n i) → Causalean.Experimentation.TwoStageInterference.WAssign n i → ℝ) (m0φ m1ψ : ι → ℝ) (dφ dψ : ℝ), dφ ≠ 0 → dψ ≠ 0 → (∀ (i : ι), m0φ i ≠ 0) → (∀ (i : ι), m1ψ i ≠ 0) → (∀ (i : ι), ↑(n i) ≠ 0) → (∀ (i : ι) (j : Fin (n i)), ((φ i).Pr fun w => w j = false) = m0φ i / ↑(n i)) → (∀ (i : ι) (j : Fin (n i)), ((ψ i).Pr fun w => w j = true) = m1ψ i / ↑(n i)) → (∀ (i : ι), (D₁.Pr fun s => s i = false) = dφ / ↑(Fintype.card ι)) → (∀ (i : ι), (D₁.Pr fun s => s i = true) = dψ / ↑(Fintype.card ι)) → (Causalean.Experimentation.TwoStageInterference.jointDesign D₁ ψ φ).E (Causalean.Experimentation.TwoStageInterference.estTotal Y m0φ m1ψ dφ dψ) = Causalean.Experimentation.TwoStageInterference.CE_total ψ φ Y","l":191,"f":"Causalean/Experimentation/TwoStageInterference/Effects.lean"},{"n":"Causalean.Experimentation.TwoStageInterference.E_groupDiff_eq_of_homogeneous","k":"theorem","m":"Causalean.Experimentation.TwoStageInterference.Asymptotic.CLTDischarge","a":"Experimentation/TwoStageInterference/Asymptotic","t":2,"d":"The conditional mean of the per-group contrast estimator under the ψ-design is the common group-level contrast: (ψ i).E (dᵢ) = δ. Immediate from E_groupEst (twice) and homogeneity.","s":"∀ {Exp : ℕ → Causalean.Experimentation.TwoStageInterference.LHExperiment} {t δ M : ℝ} {v : ℕ → ℝ} {stud : (n : ℕ) → Causalean.Experimentation.TwoStageInterference.StratAssign (Exp n).ι × ((i : (Exp n).ι) → Fin ((Exp n).gsize i) → Bool) → ℝ}, Causalean.Experimentation.TwoStageInterference.Homogeneous Exp t stud δ M v → ∀ (n : ℕ) (i : (Exp n).ι), ((Exp n).ψ i).E (Causalean.Experimentation.TwoStageInterference.groupDiff (Exp n) i) = δ","l":282,"f":"Causalean/Experimentation/TwoStageInterference/Asymptotic/CLTDischarge.lean"},{"n":"Causalean.Experimentation.TwoStageInterference.E_groupEst","k":"theorem","m":"Causalean.Experimentation.TwoStageInterference.Unbiased","a":"Experimentation/TwoStageInterference","t":1,"d":"Within-group unbiasedness. Fix a per-group design ρ governing the within-group treatment randomization and an outcome recorded for every group, unit, and realized within-group assignment, together with a group i, a treatment state z, and a nonzero real number m used as the treated-count denominator. Assume group i has a nonzero number of units and that every unit of group i receives treatment state z with the same probability m divided by the group's size. Then the expected value, under ρ, of the empirical mean outcome among the z-treated units of group i equals the group's average potential outcome under z.","s":"∀ {ι : Type u_1} {n : ι → ℕ} (ρ : (i : ι) → Causalean.Experimentation.DesignBased.FiniteDesign (Causalean.Experimentation.TwoStageInterference.WAssign n i)) (Y : (i : ι) → Fin (n i) → Causalean.Experimentation.TwoStageInterference.WAssign n i → ℝ) (i : ι) (z : Bool) (m : ℝ), m ≠ 0 → ↑(n i) ≠ 0 → (∀ (j : Fin (n i)), ((ρ i).Pr fun w => w j = z) = m / ↑(n i)) → (ρ i).E (Causalean.Experimentation.TwoStageInterference.groupEst Y i z m) = Causalean.Experimentation.TwoStageInterference.groupMean ρ Y i z","l":45,"f":"Causalean/Experimentation/TwoStageInterference/Unbiased.lean"},{"n":"Causalean.Experimentation.TwoStageInterference.E_popEst","k":"theorem","m":"Causalean.Experimentation.TwoStageInterference.Unbiased","a":"Experimentation/TwoStageInterference","t":1,"d":"Population unbiasedness (Theorem 1). Consider the two per-group designs ψ and φ governing the within-group randomization when a group is respectively assigned the ψ-strategy or the φ-strategy at stage 1 and an outcome recorded for every group, unit, and realized within-group assignment. Fix a treatment state z, a nonzero real number C used as the population-estimator denominator, and a family m with every group's value m i nonzero; suppose every group has a nonzero number of units, within every ψ-assigned group every unit receives treatment state z with the same probability m i divided by the group's size, and the marginal probability of each group being assigned the ψ-strategy at stage 1 equals C divided by the number of groups. Then the expected value, under the compound two-stage design, of the population estimator restricted to the ψ-assigned groups equals the population average potential outcome under z computed from the ψ-design.","s":"∀ {ι : Type u_1} [inst : Fintype ι] [inst_1 : DecidableEq ι] {n : ι → ℕ} (D₁ : Causalean.Experimentation.DesignBased.FiniteDesign (Causalean.Experimentation.TwoStageInterference.StratAssign ι)) (ψ φ : (i : ι) → Causalean.Experimentation.DesignBased.FiniteDesign (Causalean.Experimentation.TwoStageInterference.WAssign n i)) (Y : (i : ι) → Fin (n i) → Causalean.Experimentation.TwoStageInterference.WAssign n i → ℝ) (z : Bool) (m : ι → ℝ) (C : ℝ), C ≠ 0 → (∀ (i : ι), m i ≠ 0) → (∀ (i : ι), ↑(n i) ≠ 0) → (∀ (i : ι) (j : Fin (n i)), ((ψ i).Pr fun w => w j = z) = m i / ↑(n i)) → (∀ (i : ι), (D₁.Pr fun s => s i = true) = C / ↑(Fintype.card ι)) → (Causalean.Experimentation.TwoStageInterference.jointDesign D₁ ψ φ).E (Causalean.Experimentation.TwoStageInterference.popEst Y z true m C) = Causalean.Experimentation.TwoStageInterference.popMean ψ Y z","l":83,"f":"Causalean/Experimentation/TwoStageInterference/Unbiased.lean"},{"n":"Causalean.Experimentation.TwoStageInterference.E_popEst_pick","k":"theorem","m":"Causalean.Experimentation.TwoStageInterference.Effects","a":"Experimentation/TwoStageInterference","t":1,"d":"Population unbiasedness, either selection (generalizing Theorem 1). For the two-stage design that allocates groups to strategy ψ or strategy φ and records outcomes via the potential-outcome function Y, fix an arbitrary selection flag pick together with a within-group design ρ meant to govern every group whose stage-1 flag equals pick, where on the event that a group's stage-1 flag equals pick, its conditional within-group design (ψ if flagged true, φ if flagged false) actually equals ρ. Assume the normalizing group count denom is nonzero, every group's unit count m at treatment level z is nonzero, and every group's size n is nonzero. Suppose that within each group governed by ρ each unit's propensity of being assigned treatment level z is m/n, and that the stage-1 design selects each group flagged pick with probability denom/N. Then the population estimator on the groups selected by pick is unbiased for the population average potential outcome at level z computed under design ρ.","s":"∀ {ι : Type u_1} [inst : Fintype ι] [inst_1 : DecidableEq ι] {n : ι → ℕ} (D₁ : Causalean.Experimentation.DesignBased.FiniteDesign (Causalean.Experimentation.TwoStageInterference.StratAssign ι)) (ψ φ : (i : ι) → Causalean.Experimentation.DesignBased.FiniteDesign (Causalean.Experimentation.TwoStageInterference.WAssign n i)) (Y : (i : ι) → Fin (n i) → Causalean.Experimentation.TwoStageInterference.WAssign n i → ℝ) (z : Bool) (m : ι → ℝ) (pick : Bool) (denom : ℝ) (ρ : (i : ι) → Causalean.Experimentation.DesignBased.FiniteDesign (Causalean.Experimentation.TwoStageInterference.WAssign n i)), denom ≠ 0 → (∀ (i : ι), m i ≠ 0) → (∀ (i : ι), ↑(n i) ≠ 0) → (∀ (s : Causalean.Experimentation.TwoStageInterference.StratAssign ι) (i : ι), s i = pick → (if s i = true then ψ i else φ i) = ρ i) → (∀ (i : ι) (j : Fin (n i)), ((ρ i).Pr fun w => w j = z) = m i / ↑(n i)) → (∀ (i : ι), (D₁.Pr fun s => s i = pick) = denom / ↑(Fintype.card ι)) → (Causalean.Experimentation.TwoStageInterference.jointDesign D₁ ψ φ).E (Causalean.Experimentation.TwoStageInterference.popEst Y z pick m denom) = Causalean.Experimentation.TwoStageInterference.popMean ρ Y z","l":64,"f":"Causalean/Experimentation/TwoStageInterference/Effects.lean"},{"n":"Causalean.Experimentation.TwoStageInterference.E_varHat_conservative","k":"theorem","m":"Causalean.Experimentation.TwoStageInterference.VarianceConservative","a":"Experimentation/TwoStageInterference","t":1,"d":"Conservativeness of the within-group variance estimator (Hudgens–Halloran 2008, Eq. 9). For any design satisfying the completely-randomized moment hypotheses (hmean/hpair/hsupp) and treating exactly K of n units (with 2 ≤ K and K + 2 ≤ n, so both sample variances are well defined), the conservative estimator overstates the randomization variance, Var(τ̂) ≤ E[v̂ar].","s":"∀ {n : ℕ} (K : ℕ) (a b : Fin n → ℝ) (ρ : Causalean.Experimentation.DesignBased.FiniteDesign (Fin n → Bool)), 2 ≤ K → K + 2 ≤ n → (∀ (j : Fin n), ρ.E (Causalean.Experimentation.TwoStageInterference.T j) = ↑K / ↑n) → (∀ (j k : Fin n), j ≠ k → (ρ.E fun w => Causalean.Experimentation.TwoStageInterference.T j w * Causalean.Experimentation.TwoStageInterference.T k w) = ↑K * (↑K - 1) / (↑n * (↑n - 1))) → (∀ (w : Fin n → Bool), ρ.p w ≠ 0 → ∑ j, Causalean.Experimentation.TwoStageInterference.T j w = ↑K) → ρ.Var (Causalean.Experimentation.TwoStageInterference.tauHat K a b) ≤ ρ.E (Causalean.Experimentation.TwoStageInterference.varHat K a b)","l":228,"f":"Causalean/Experimentation/TwoStageInterference/VarianceConservative.lean"},{"n":"Causalean.Experimentation.TwoStageInterference.E_varHat_conservative_CRD","k":"theorem","m":"Causalean.Experimentation.TwoStageInterference.VarianceConservative","a":"Experimentation/TwoStageInterference","t":1,"d":"Conservativeness of the within-group variance estimator, for the completely randomized design (Hudgens–Halloran 2008, Eq. 9). For a group of n units with potential outcomes a (treated state) and b (untreated state), consider the completely randomized within-group design that treats exactly K units uniformly at random, where at least two units are treated and at least two units remain in control (K + 2 ≤ n) — so both within-group sample variances are well defined. Then the randomization variance of the control-minus-treatment difference-in-means estimator under this design is at most the expectation, under the same design, of the conservative variance estimator v̂ar.","s":"∀ {n : ℕ} (K : ℕ) (a b : Fin n → ℝ), 2 ≤ K → ∀ (hKn2 : K + 2 ≤ n), (Causalean.Experimentation.TwoStageInterference.crd K (le_trans (Nat.le_add_right K 2) hKn2)).Var (Causalean.Experimentation.TwoStageInterference.tauHat K a b) ≤ (Causalean.Experimentation.TwoStageInterference.crd K (le_trans (Nat.le_add_right K 2) hKn2)).E (Causalean.Experimentation.TwoStageInterference.varHat K a b)","l":271,"f":"Causalean/Experimentation/TwoStageInterference/VarianceConservative.lean"},{"n":"Causalean.Experimentation.TwoStageInterference.FiniteDesign.Var_add_const","k":"theorem","m":"Causalean.Experimentation.TwoStageInterference.Variance","a":"Experimentation/TwoStageInterference","t":2,"d":"Variance is invariant under adding a constant: Var(X + c) = Var X.","s":"∀ {Ω : Type u_1} [inst : Fintype Ω] (D : Causalean.Experimentation.DesignBased.FiniteDesign Ω) (X : Ω → ℝ) (c : ℝ), (D.Var fun z => X z + c) = D.Var X","l":57,"f":"Causalean/Experimentation/TwoStageInterference/Variance.lean"},{"n":"Causalean.Experimentation.TwoStageInterference.Homogeneous","k":"structure","m":"Causalean.Experimentation.TwoStageInterference.Asymptotic.CLTDischarge","a":"Experimentation/TwoStageInterference/Asymptotic","t":1,"d":"Homogeneity and regularity bundle for a sequence of Liu–Hudgens experiments. Faithfully encodes the hypotheses of Proposition 5.1 for the Hudgens-Halloran orientation: the studentized statistic is the standardized contrast estimator; every group-level direct-effect contrast equals a common value δ (homogeneity); every within-group contrast-estimator variance equals a common value v(n) that is positive; the centered per-group contrast estimator is uniformly bounded; every stage-1 selection supported by the design flags exactly C groups; the resulting rate sequence tends to zero together with its cubed Lyapunov rate (the many-groups asymptotic regime); and the conditional distribution of the studentized statistic does not depend on which stage-1 selection occurred, the analytic form of homogeneity that lifts the conditional CLT to the average.","s":"(Exp : ℕ → Causalean.Experimentation.TwoStageInterference.LHExperiment) → ℝ → ((n : ℕ) → Causalean.Experimentation.TwoStageInterference.StratAssign (Exp n).ι × ((i : (Exp n).ι) → Fin ((Exp n).gsize i) → Bool) → ℝ) → ℝ → ℝ → (ℕ → ℝ) → Prop","l":147,"f":"Causalean/Experimentation/TwoStageInterference/Asymptotic/CLTDischarge.lean"},{"n":"Causalean.Experimentation.TwoStageInterference.IdenticalRef","k":"structure","m":"Causalean.Experimentation.TwoStageInterference.Asymptotic.Identical","a":"Experimentation/TwoStageInterference/Asymptotic","t":1,"d":"Reference data for one identical-groups experiment. All groups in the population share a common size K, a common pair of within-group allocation strategies ψ₀ and φ₀, common potential outcomes Y₀, and common control and treatment unit counts, each assumed nonzero, as is the group size K. A stage-1 design D₁ assigns each group a strategy, selecting a nonzero number C of groups for ψ, out of a population of at least two groups; every unit's within-group control propensity equals m0₀/K and its treatment propensity equals m1₀/K, every group's stage-1 selection propensity equals C/N, and every pair's joint selection propensity equals C(C−1)/(N(N−1)). Together these assemble the constant LHExperiment toExp.","s":"Type 1","l":107,"f":"Causalean/Experimentation/TwoStageInterference/Asymptotic/Identical.lean"},{"n":"Causalean.Experimentation.TwoStageInterference.IdenticalRef.condDesign_toExp","k":"theorem","m":"Causalean.Experimentation.TwoStageInterference.Asymptotic.Identical","a":"Experimentation/TwoStageInterference/Asymptotic","t":2,"d":"The conditional design of the constant experiment is the genuine product, over the non-dependent space R.ι → (Fin K → Bool), of the per-group strategy designs.","s":"∀ (R : Causalean.Experimentation.TwoStageInterference.IdenticalRef) (s : Causalean.Experimentation.TwoStageInterference.StratAssign R.ι), Causalean.Experimentation.TwoStageInterference.condDesign R.toExp s = Causalean.Experimentation.DesignBased.prodDesign fun i => if s i = true then R.ψ₀ else R.φ₀","l":203,"f":"Causalean/Experimentation/TwoStageInterference/Asymptotic/Identical.lean"},{"n":"Causalean.Experimentation.TwoStageInterference.IdenticalRef.estD_equivariant","k":"theorem","m":"Causalean.Experimentation.TwoStageInterference.Asymptotic.Identical","a":"Experimentation/TwoStageInterference/Asymptotic","t":2,"d":"Estimator equivariance. Because all groups are identical, the aggregate contrast estimator sees a selection only through its pattern: if σ aligns the patterns of s' and s (s' i = s (σ i)), then evaluating at s' and the relabeled assignment w ∘ σ equals the value at s and w.","s":"∀ (R : Causalean.Experimentation.TwoStageInterference.IdenticalRef) (s s' : Causalean.Experimentation.TwoStageInterference.StratAssign R.ι) (σ : R.ι ≃ R.ι), (∀ (i : R.ι), s' i = s (σ i)) → ∀ (w : R.ι → Fin R.K → Bool), R.toExp.estD (s', fun i => w (σ i)) = R.toExp.estD (s, w)","l":213,"f":"Causalean/Experimentation/TwoStageInterference/Asymptotic/Identical.lean"},{"n":"Causalean.Experimentation.TwoStageInterference.IdenticalRef.groupDiff_toExp","k":"theorem","m":"Causalean.Experimentation.TwoStageInterference.Asymptotic.Identical","a":"Experimentation/TwoStageInterference/Asymptotic","t":2,"d":"In the constant experiment, every group's contrast estimator equals the common groupDiff₀.","s":"∀ (R : Causalean.Experimentation.TwoStageInterference.IdenticalRef) (i : R.ι), Causalean.Experimentation.TwoStageInterference.groupDiff R.toExp i = R.groupDiff₀","l":199,"f":"Causalean/Experimentation/TwoStageInterference/Asymptotic/Identical.lean"},{"n":"Causalean.Experimentation.TwoStageInterference.IdenticalRef.groupDiff₀","k":"def","m":"Causalean.Experimentation.TwoStageInterference.Asymptotic.Identical","a":"Experimentation/TwoStageInterference/Asymptotic","t":1,"d":"The common per-group treatment-minus-control contrast estimator of the identical groups; it does not depend on the group index.","s":"(R : Causalean.Experimentation.TwoStageInterference.IdenticalRef) → (Fin R.K → Bool) → ℝ","l":193,"f":"Causalean/Experimentation/TwoStageInterference/Asymptotic/Identical.lean"},{"n":"Causalean.Experimentation.TwoStageInterference.IdenticalRef.hhom_of_identical","k":"theorem","m":"Causalean.Experimentation.TwoStageInterference.Asymptotic.Identical","a":"Experimentation/TwoStageInterference/Asymptotic","t":2,"d":"Derived homogeneity hypothesis under identical groups. For two stage-1 selections each flagging exactly C groups, the conditional studentized CDF is the same. This is exactly the analytic homogeneity hypothesis hhom, now a theorem: the selection-symmetry permutation σ (exists_equiv_selection) relabels the conditional product design (prodDesign_Pr_reindex) while carrying the studentized statistic (studId_equivariant), so the two CDFs coincide.","s":"∀ (R : Causalean.Experimentation.TwoStageInterference.IdenticalRef) (t : ℝ) (s s' : Causalean.Experimentation.TwoStageInterference.StratAssign R.ι), (∑ i, if s i = true then 1 else 0) = R.C → (∑ i, if s' i = true then 1 else 0) = R.C → ((Causalean.Experimentation.TwoStageInterference.condDesign R.toExp s).Pr fun w => R.studId (s, w) ≤ t) = (Causalean.Experimentation.TwoStageInterference.condDesign R.toExp s').Pr fun w => R.studId (s', w) ≤ t","l":237,"f":"Causalean/Experimentation/TwoStageInterference/Asymptotic/Identical.lean"},{"n":"Causalean.Experimentation.TwoStageInterference.IdenticalRef.hv_toExp","k":"theorem","m":"Causalean.Experimentation.TwoStageInterference.Asymptotic.Identical","a":"Experimentation/TwoStageInterference/Asymptotic","t":2,"d":"In the constant experiment every group's within-group variance equals the common refVar.","s":"∀ (R : Causalean.Experimentation.TwoStageInterference.IdenticalRef) (i : R.ι), (R.toExp.ψ i).Var (Causalean.Experimentation.TwoStageInterference.groupDiff R.toExp i) = R.refVar","l":299,"f":"Causalean/Experimentation/TwoStageInterference/Asymptotic/Identical.lean"},{"n":"Causalean.Experimentation.TwoStageInterference.IdenticalRef.hδ_toExp","k":"theorem","m":"Causalean.Experimentation.TwoStageInterference.Asymptotic.Identical","a":"Experimentation/TwoStageInterference/Asymptotic","t":2,"d":"In the constant experiment every group's level contrast equals the common refDelta.","s":"∀ (R : Causalean.Experimentation.TwoStageInterference.IdenticalRef) (i : R.ι), Causalean.Experimentation.TwoStageInterference.groupMean R.toExp.ψ R.toExp.Y i true - Causalean.Experimentation.TwoStageInterference.groupMean R.toExp.ψ R.toExp.Y i false = R.refDelta","l":294,"f":"Causalean/Experimentation/TwoStageInterference/Asymptotic/Identical.lean"},{"n":"Causalean.Experimentation.TwoStageInterference.IdenticalRef.refDelta","k":"def","m":"Causalean.Experimentation.TwoStageInterference.Asymptotic.Identical","a":"Experimentation/TwoStageInterference/Asymptotic","t":1,"d":"The common group-level treatment-minus-control direct-effect contrast δ of the identical groups; it does not depend on any group index.","s":"Causalean.Experimentation.TwoStageInterference.IdenticalRef → ℝ","l":287,"f":"Causalean/Experimentation/TwoStageInterference/Asymptotic/Identical.lean"},{"n":"Causalean.Experimentation.TwoStageInterference.IdenticalRef.refGroupMean","k":"def","m":"Causalean.Experimentation.TwoStageInterference.Asymptotic.Identical","a":"Experimentation/TwoStageInterference/Asymptotic","t":1,"d":"The common group-average potential outcome ȳ(z) of the identical groups under treatment status z, computed from the reference allocation strategy and outcomes — index-free.","s":"Causalean.Experimentation.TwoStageInterference.IdenticalRef → Bool → ℝ","l":281,"f":"Causalean/Experimentation/TwoStageInterference/Asymptotic/Identical.lean"},{"n":"Causalean.Experimentation.TwoStageInterference.IdenticalRef.refVar","k":"def","m":"Causalean.Experimentation.TwoStageInterference.Asymptotic.Identical","a":"Experimentation/TwoStageInterference/Asymptotic","t":1,"d":"The common within-group contrast-estimator variance v of the identical groups.","s":"Causalean.Experimentation.TwoStageInterference.IdenticalRef → ℝ","l":291,"f":"Causalean/Experimentation/TwoStageInterference/Asymptotic/Identical.lean"},{"n":"Causalean.Experimentation.TwoStageInterference.IdenticalRef.studId","k":"def","m":"Causalean.Experimentation.TwoStageInterference.Asymptotic.Identical","a":"Experimentation/TwoStageInterference/Asymptotic","t":1,"d":"The concrete studentized treatment-minus-control contrast statistic of the constant experiment.","s":"(R : Causalean.Experimentation.TwoStageInterference.IdenticalRef) → Causalean.Experimentation.TwoStageInterference.StratAssign R.ι × (R.ι → Fin R.K → Bool) → ℝ","l":208,"f":"Causalean/Experimentation/TwoStageInterference/Asymptotic/Identical.lean"},{"n":"Causalean.Experimentation.TwoStageInterference.IdenticalRef.studId_equivariant","k":"theorem","m":"Causalean.Experimentation.TwoStageInterference.Asymptotic.Identical","a":"Experimentation/TwoStageInterference/Asymptotic","t":2,"d":"Studentized equivariance. The studentized statistic is likewise selection-pattern equivariant, since DEbar and directVar are scalars and the estimator is equivariant.","s":"∀ (R : Causalean.Experimentation.TwoStageInterference.IdenticalRef) (s s' : Causalean.Experimentation.TwoStageInterference.StratAssign R.ι) (σ : R.ι ≃ R.ι), (∀ (i : R.ι), s' i = s (σ i)) → ∀ (w : R.ι → Fin R.K → Bool), R.studId (s', fun i => w (σ i)) = R.studId (s, w)","l":227,"f":"Causalean/Experimentation/TwoStageInterference/Asymptotic/Identical.lean"},{"n":"Causalean.Experimentation.TwoStageInterference.IdenticalRef.toExp","k":"def","m":"Causalean.Experimentation.TwoStageInterference.Asymptotic.Identical","a":"Experimentation/TwoStageInterference/Asymptotic","t":1,"d":"The constant LHExperiment assembled from identical-groups reference data: every group has gsize := K, strategies ψ₀/φ₀, outcomes Y₀, and counts m0₀/m1₀. Because gsize is the literal constant fun _ => K, the within-group assignment space is the non-dependent space Fin K → Bool and the conditional design is a genuine product over ι → (Fin K → Bool).","s":"Causalean.Experimentation.TwoStageInterference.IdenticalRef → Causalean.Experimentation.TwoStageInterference.LHExperiment","l":168,"f":"Causalean/Experimentation/TwoStageInterference/Asymptotic/Identical.lean"},{"n":"Causalean.Experimentation.TwoStageInterference.LHExperiment","k":"structure","m":"Causalean.Experimentation.TwoStageInterference.Asymptotic.Setup","a":"Experimentation/TwoStageInterference/Asymptotic","t":1,"d":"A single Liu–Hudgens (2014) two-stage experiment, packaged so that a sequence of them can be studied for large-sample inference. Carries a finite population of groups with per-group sizes, a stage-1 design assigning each group a strategy, the per-group allocation strategies ψ and φ, partial-interference potential outcomes, and design-fixed control and treatment unit counts per group, together with the regularity conditions that the number C of ψ-selected groups is nonzero, every group has nonzero control and treatment counts and nonzero size, the population has at least one group and at least two groups, every unit's within-group control propensity equals m0 i / nᵢ and its treatment propensity equals m1 i / nᵢ, every group's stage-1 selection propensity equals C/N, and every pair's joint selection propensity equals C(C−1)/(N(N−1)) — exactly the hypothesis lists of E_estDirect and Var_estDirect.","s":"Type 1","l":52,"f":"Causalean/Experimentation/TwoStageInterference/Asymptotic/Setup.lean"},{"n":"Causalean.Experimentation.TwoStageInterference.LHExperiment.DEbar","k":"def","m":"Causalean.Experimentation.TwoStageInterference.Asymptotic.Setup","a":"Experimentation/TwoStageInterference/Asymptotic","t":1,"d":"The population average treatment-minus-control direct-effect contrast: the treatment mean under ψ minus the control mean under ψ.","s":"Causalean.Experimentation.TwoStageInterference.LHExperiment → ℝ","l":127,"f":"Causalean/Experimentation/TwoStageInterference/Asymptotic/Setup.lean"},{"n":"Causalean.Experimentation.TwoStageInterference.LHExperiment.E_estD","k":"theorem","m":"Causalean.Experimentation.TwoStageInterference.Asymptotic.Setup","a":"Experimentation/TwoStageInterference/Asymptotic","t":1,"d":"Unbiasedness bridge. The Horvitz–Thompson estimator is unbiased for the population average treatment-minus-control direct-effect contrast.","s":"∀ (E : Causalean.Experimentation.TwoStageInterference.LHExperiment), E.jointD.E E.estD = E.DEbar","l":141,"f":"Causalean/Experimentation/TwoStageInterference/Asymptotic/Setup.lean"},{"n":"Causalean.Experimentation.TwoStageInterference.LHExperiment.directVar","k":"def","m":"Causalean.Experimentation.TwoStageInterference.Asymptotic.Setup","a":"Experimentation/TwoStageInterference/Asymptotic","t":1,"d":"The closed-form two-stage design variance of the treatment-minus-control direct-effect contrast estimator (Theorem 6): a between-group SRS term over the group-level contrasts plus a within-group term averaging the per-group contrast-estimator variances.","s":"Causalean.Experimentation.TwoStageInterference.LHExperiment → ℝ","l":131,"f":"Causalean/Experimentation/TwoStageInterference/Asymptotic/Setup.lean"},{"n":"Causalean.Experimentation.TwoStageInterference.LHExperiment.estD","k":"def","m":"Causalean.Experimentation.TwoStageInterference.Asymptotic.Setup","a":"Experimentation/TwoStageInterference/Asymptotic","t":1,"d":"The Horvitz-Thompson estimator of the experiment's treatment-minus-control direct-effect contrast.","s":"(E : Causalean.Experimentation.TwoStageInterference.LHExperiment) → Causalean.Experimentation.TwoStageInterference.StratAssign E.ι × ((i : E.ι) → Fin (E.gsize i) → Bool) → ℝ","l":122,"f":"Causalean/Experimentation/TwoStageInterference/Asymptotic/Setup.lean"},{"n":"Causalean.Experimentation.TwoStageInterference.LHExperiment.jointD","k":"def","m":"Causalean.Experimentation.TwoStageInterference.Asymptotic.Setup","a":"Experimentation/TwoStageInterference/Asymptotic","t":1,"d":"The joint two-stage design of the experiment.","s":"(E : Causalean.Experimentation.TwoStageInterference.LHExperiment) → Causalean.Experimentation.DesignBased.FiniteDesign (Causalean.Experimentation.TwoStageInterference.StratAssign E.ι × ((i : E.ι) → Fin (E.gsize i) → Bool))","l":118,"f":"Causalean/Experimentation/TwoStageInterference/Asymptotic/Setup.lean"},{"n":"Causalean.Experimentation.TwoStageInterference.LHExperiment.var_estD","k":"theorem","m":"Causalean.Experimentation.TwoStageInterference.Asymptotic.Setup","a":"Experimentation/TwoStageInterference/Asymptotic","t":1,"d":"Variance bridge. The design variance of the treatment-minus-control direct-effect contrast estimator equals the closed-form two-stage variance directVar.","s":"∀ (E : Causalean.Experimentation.TwoStageInterference.LHExperiment), E.jointD.Var E.estD = E.directVar","l":149,"f":"Causalean/Experimentation/TwoStageInterference/Asymptotic/Setup.lean"},{"n":"Causalean.Experimentation.TwoStageInterference.M_nonneg_of_homogeneous","k":"theorem","m":"Causalean.Experimentation.TwoStageInterference.Asymptotic.CLTDischarge","a":"Experimentation/TwoStageInterference/Asymptotic","t":2,"d":"Nonnegativity of the bound constant M, available whenever the experiment has at least one group.","s":"∀ {Exp : ℕ → Causalean.Experimentation.TwoStageInterference.LHExperiment} {t δ M : ℝ} {v : ℕ → ℝ} {stud : (n : ℕ) → Causalean.Experimentation.TwoStageInterference.StratAssign (Exp n).ι × ((i : (Exp n).ι) → Fin ((Exp n).gsize i) → Bool) → ℝ}, Causalean.Experimentation.TwoStageInterference.Homogeneous Exp t stud δ M v → ∀ (n : ℕ) [Nonempty (Exp n).ι], 0 ≤ M","l":294,"f":"Causalean/Experimentation/TwoStageInterference/Asymptotic/CLTDischarge.lean"},{"n":"Causalean.Experimentation.TwoStageInterference.S0","k":"def","m":"Causalean.Experimentation.TwoStageInterference.Variance","a":"Experimentation/TwoStageInterference","t":1,"d":"Population sample variance of the untreated-state outcomes b, S₀ = (1/(n−1))∑ⱼ(b j − b̄)².","s":"{n : ℕ} → (Fin n → ℝ) → ℝ","l":126,"f":"Causalean/Experimentation/TwoStageInterference/Variance.lean"},{"n":"Causalean.Experimentation.TwoStageInterference.S1","k":"def","m":"Causalean.Experimentation.TwoStageInterference.Variance","a":"Experimentation/TwoStageInterference","t":1,"d":"Population sample variance of the treated-state outcomes a, S₁ = (1/(n−1))∑ⱼ(a j − ā)².","s":"{n : ℕ} → (Fin n → ℝ) → ℝ","l":123,"f":"Causalean/Experimentation/TwoStageInterference/Variance.lean"},{"n":"Causalean.Experimentation.TwoStageInterference.ShatControl","k":"def","m":"Causalean.Experimentation.TwoStageInterference.VarianceConservative","a":"Experimentation/TwoStageInterference","t":1,"d":"Observed sample variance among the n−K control units, Ŝ₀, with n−K−1 denominator: (1/(n−K−1))∑_{j: wⱼ=0}(b j − b̄_obs)².","s":"{n : ℕ} → ℕ → (Fin n → ℝ) → (Fin n → Bool) → ℝ","l":71,"f":"Causalean/Experimentation/TwoStageInterference/VarianceConservative.lean"},{"n":"Causalean.Experimentation.TwoStageInterference.ShatControl_nonneg","k":"theorem","m":"Causalean.Experimentation.TwoStageInterference.VarianceConservative","a":"Experimentation/TwoStageInterference","t":2,"d":"The observed control sample variance is nonnegative, provided K + 1 ≤ n (so the denominator n−K−1 ≥ 0).","s":"∀ {n : ℕ} (K : ℕ) (b : Fin n → ℝ), K + 1 ≤ n → ∀ (w : Fin n → Bool), 0 ≤ Causalean.Experimentation.TwoStageInterference.ShatControl K b w","l":98,"f":"Causalean/Experimentation/TwoStageInterference/VarianceConservative.lean"},{"n":"Causalean.Experimentation.TwoStageInterference.ShatTreated","k":"def","m":"Causalean.Experimentation.TwoStageInterference.VarianceConservative","a":"Experimentation/TwoStageInterference","t":1,"d":"Observed sample variance among the K treated units, Ŝ₁, with K−1 denominator: (1/(K−1))∑_{j: wⱼ=1}(a j − ā_obs)².","s":"{n : ℕ} → ℕ → (Fin n → ℝ) → (Fin n → Bool) → ℝ","l":66,"f":"Causalean/Experimentation/TwoStageInterference/VarianceConservative.lean"},{"n":"Causalean.Experimentation.TwoStageInterference.ShatTreated_nonneg","k":"theorem","m":"Causalean.Experimentation.TwoStageInterference.VarianceConservative","a":"Experimentation/TwoStageInterference","t":2,"d":"The observed treated sample variance is nonnegative, provided 1 ≤ K (so the denominator K−1 ≥ 0): it is a nonnegative-weighted sum of squares divided by a nonnegative number.","s":"∀ {n : ℕ} (K : ℕ) (a : Fin n → ℝ), 1 ≤ K → ∀ (w : Fin n → Bool), 0 ≤ Causalean.Experimentation.TwoStageInterference.ShatTreated K a w","l":89,"f":"Causalean/Experimentation/TwoStageInterference/VarianceConservative.lean"},{"n":"Causalean.Experimentation.TwoStageInterference.SmuVar","k":"def","m":"Causalean.Experimentation.TwoStageInterference.StageOne","a":"Experimentation/TwoStageInterference","t":1,"d":"The population sample variance (Neyman N−1 denominator) of the group-level quantities μ, Sμ² = (∑ᵢ(μᵢ − μ̄)²)/(N−1), where N := card ι and μ̄ = (∑ μ)/N.","s":"{ι : Type u_1} → [Fintype ι] → (ι → ℝ) → ℝ","l":46,"f":"Causalean/Experimentation/TwoStageInterference/StageOne.lean"},{"n":"Causalean.Experimentation.TwoStageInterference.Stau","k":"def","m":"Causalean.Experimentation.TwoStageInterference.Variance","a":"Experimentation/TwoStageInterference","t":1,"d":"Population sample variance of the unit-level treatment effects a j − b j, Sτ = (1/(n−1))∑ⱼ((a j − b j) − (ā − b̄))².","s":"{n : ℕ} → (Fin n → ℝ) → (Fin n → ℝ) → ℝ","l":130,"f":"Causalean/Experimentation/TwoStageInterference/Variance.lean"},{"n":"Causalean.Experimentation.TwoStageInterference.StratAssign","k":"def","m":"Causalean.Experimentation.TwoStageInterference.Basic","a":"Experimentation/TwoStageInterference","t":1,"d":"Stage-1 strategy assignment space: each group is flagged ψ (true) or φ (false).","s":"Type u_2 → Type u_2","l":43,"f":"Causalean/Experimentation/TwoStageInterference/Basic.lean"},{"n":"Causalean.Experimentation.TwoStageInterference.StratifiedInterference","k":"def","m":"Causalean.Experimentation.TwoStageInterference.Stratified","a":"Experimentation/TwoStageInterference","t":1,"d":"Assumption 2 (stratified interference). A population of potential outcomes Y satisfies stratified interference when each unit's outcome depends on the within-group assignment only through its stratified exposure: whenever two assignments give a unit the same own treatment and the same count of treated others, the unit's outcome agrees.","s":"{ι : Type u_1} → {n : ι → ℕ} → ((i : ι) → Fin (n i) → Causalean.Experimentation.TwoStageInterference.WAssign n i → ℝ) → Prop","l":60,"f":"Causalean/Experimentation/TwoStageInterference/Stratified.lean"},{"n":"Causalean.Experimentation.TwoStageInterference.StratifiedInterference.elim","k":"theorem","m":"Causalean.Experimentation.TwoStageInterference.Stratified","a":"Experimentation/TwoStageInterference","t":2,"d":"Restatement of stratified interference in terms of the two raw summaries: equal own treatment and equal count of treated others force equal outcomes.","s":"∀ {ι : Type u_1} {n : ι → ℕ} {Y : (i : ι) → Fin (n i) → Causalean.Experimentation.TwoStageInterference.WAssign n i → ℝ}, Causalean.Experimentation.TwoStageInterference.StratifiedInterference Y → ∀ (i : ι) (j : Fin (n i)) (w w' : Causalean.Experimentation.TwoStageInterference.WAssign n i), w j = w' j → Causalean.Experimentation.TwoStageInterference.numTreatedOthers i j w = Causalean.Experimentation.TwoStageInterference.numTreatedOthers i j w' → Y i j w = Y i j w'","l":69,"f":"Causalean/Experimentation/TwoStageInterference/Stratified.lean"},{"n":"Causalean.Experimentation.TwoStageInterference.T","k":"def","m":"Causalean.Experimentation.TwoStageInterference.Variance","a":"Experimentation/TwoStageInterference","t":1,"d":"Treatment indicator of unit j: 1 on within-group assignments that treat j, 0 otherwise. (The indicator does not depend on the design.)","s":"{n : ℕ} → Fin n → (Fin n → Bool) → ℝ","l":104,"f":"Causalean/Experimentation/TwoStageInterference/Variance.lean"},{"n":"Causalean.Experimentation.TwoStageInterference.T_nonneg","k":"theorem","m":"Causalean.Experimentation.TwoStageInterference.VarianceConservative","a":"Experimentation/TwoStageInterference","t":2,"d":"The treatment indicator is nonnegative.","s":"∀ {n : ℕ} (j : Fin n) (w : Fin n → Bool), 0 ≤ Causalean.Experimentation.TwoStageInterference.T j w","l":81,"f":"Causalean/Experimentation/TwoStageInterference/VarianceConservative.lean"},{"n":"Causalean.Experimentation.TwoStageInterference.Var_estDirect","k":"theorem","m":"Causalean.Experimentation.TwoStageInterference.BetweenGroupEffect","a":"Experimentation/TwoStageInterference","t":1,"d":"Hudgens–Halloran (2008), Theorem 6 (two-stage variance of the direct-effect contrast estimator). Consider the two-stage design that first allocates each group to strategy ψ or strategy φ and then randomizes the group's units accordingly, with outcomes recorded by the potential-outcome function Y. Assume the target sample size C of selected groups is nonzero, the number N of groups is nonzero, N minus one is nonzero, every group's control-arm unit count m0 is nonzero, every group's treatment-arm unit count m1 is nonzero, and every group's size n is nonzero. Suppose the stage-1 design draws a simple random sample of C of the N groups, so that each group is selected with probability C/N and each pair of distinct groups is jointly selected with probability C(C−1)/(N(N−1)), and that within a selected ψ-group each unit's control propensity is m0/n and each unit's treatment propensity is m1/n. Then the randomization variance of the treatment-minus-control estimator decomposes into a between-group term — the finite-population-corrected sample variance of the group-level treatment-minus-control contrasts, scaled by (1 − C/N)/C — plus a within-group term averaging, over the N groups and scaled by 1/(C·N), the conditional variance of each group's within-group contrast estimator.","s":"∀ {ι : Type u_1} [inst : Fintype ι] [inst_1 : DecidableEq ι] {n : ι → ℕ} (D₁ : Causalean.Experimentation.DesignBased.FiniteDesign (Causalean.Experimentation.TwoStageInterference.StratAssign ι)) (ψ φ : (i : ι) → Causalean.Experimentation.DesignBased.FiniteDesign (Causalean.Experimentation.TwoStageInterference.WAssign n i)) (Y : (i : ι) → Fin (n i) → Causalean.Experimentation.TwoStageInterference.WAssign n i → ℝ) (m0 m1 : ι → ℝ) (C : ℝ), C ≠ 0 → ↑(Fintype.card ι) ≠ 0 → ↑(Fintype.card ι) - 1 ≠ 0 → (∀ (i : ι), m0 i ≠ 0) → (∀ (i : ι), m1 i ≠ 0) → (∀ (i : ι), ↑(n i) ≠ 0) → (∀ (i : ι) (j : Fin (n i)), ((ψ i).Pr fun w => w j = false) = m0 i / ↑(n i)) → (∀ (i : ι) (j : Fin (n i)), ((ψ i).Pr fun w => w j = true) = m1 i / ↑(n i)) → (∀ (i : ι), (D₁.Pr fun s => s i = true) = C / ↑(Fintype.card ι)) → (∀ (i j : ι), i ≠ j → (D₁.E fun s => Causalean.Experimentation.DesignBased.FiniteDesign.ind (fun s => s i = true) s * Causalean.Experimentation.DesignBased.FiniteDesign.ind (fun s => s j = true) s) = C * (C - 1) / (↑(Fintype.card ι) * (↑(Fintype.card ι) - 1))) → (Causalean.Experimentation.TwoStageInterference.jointDesign D₁ ψ φ).Var (Causalean.Experimentation.TwoStageInterference.estDirect Y m0 m1 C) = ((1 - C / ↑(Fintype.card ι)) / C * Causalean.Experimentation.TwoStageInterference.SmuVar fun i => Causalean.Experimentation.TwoStageInterference.groupMean ψ Y i true - Causalean.Experimentation.TwoStageInterference.groupMean ψ Y i false) + 1 / (C * ↑(Fintype.card ι)) * ∑ i, (ψ i).Var fun w => Causalean.Experimentation.TwoStageInterference.groupEst Y i true (m1 i) w - Causalean.Experimentation.TwoStageInterference.groupEst Y i false (m0 i) w","l":49,"f":"Causalean/Experimentation/TwoStageInterference/BetweenGroupEffect.lean"},{"n":"Causalean.Experimentation.TwoStageInterference.Var_estDirect_CRD","k":"theorem","m":"Causalean.Experimentation.TwoStageInterference.BetweenGroupEffect","a":"Experimentation/TwoStageInterference","t":1,"d":"Hudgens–Halloran (2008), Theorem 6, for the mixed two-stage design. Consider the completely randomized two-stage design in which stage 1 draws a simple random sample of C groups out of the population of N groups, and each drawn group is completely randomized by treating K of its n units, with outcomes recorded by the potential-outcome function Y. Assume the sample size C is strictly positive, C is strictly less than the number N of groups, every group's treated-unit count K is strictly positive, and every group's treated count K is strictly less than its size n. Then the randomization variance of the treatment-minus-control estimator on this design equals the same between-group/within-group decomposition as Var_estDirect, with control count n−K and treatment count K in each group.","s":"∀ {ι : Type u_1} [inst : Fintype ι] [inst_1 : DecidableEq ι] {n : ι → ℕ} (Y : (i : ι) → Fin (n i) → Causalean.Experimentation.TwoStageInterference.WAssign n i → ℝ) (K : ι → ℕ) (C : ℕ), 0 < C → ∀ (hCN : C < Fintype.card ι), (∀ (i : ι), 0 < K i) → ∀ (hKn : ∀ (i : ι), K i < n i), (Causalean.Experimentation.TwoStageInterference.jointDesign (Causalean.Experimentation.TwoStageInterference.crdOn C (LT.lt.le hCN)) (fun i => Causalean.Experimentation.TwoStageInterference.crd (K i) (LT.lt.le (hKn i))) fun i => Causalean.Experimentation.TwoStageInterference.crd (K i) (LT.lt.le (hKn i))).Var (Causalean.Experimentation.TwoStageInterference.estDirect Y (fun i => ↑(n i) - ↑(K i)) (fun i => ↑(K i)) ↑C) = ((1 - ↑C / ↑(Fintype.card ι)) / ↑C * Causalean.Experimentation.TwoStageInterference.SmuVar fun i => Causalean.Experimentation.TwoStageInterference.groupMean (fun i => Causalean.Experimentation.TwoStageInterference.crd (K i) (LT.lt.le (hKn i))) Y i true - Causalean.Experimentation.TwoStageInterference.groupMean (fun i => Causalean.Experimentation.TwoStageInterference.crd (K i) (LT.lt.le (hKn i))) Y i false) + 1 / (↑C * ↑(Fintype.card ι)) * ∑ i, (Causalean.Experimentation.TwoStageInterference.crd (K i) (LT.lt.le (hKn i))).Var fun w => Causalean.Experimentation.TwoStageInterference.groupEst Y i true (↑(K i)) w - Causalean.Experimentation.TwoStageInterference.groupEst Y i false (↑(n i) - ↑(K i)) w","l":112,"f":"Causalean/Experimentation/TwoStageInterference/BetweenGroupEffect.lean"},{"n":"Causalean.Experimentation.TwoStageInterference.Var_groupAgg","k":"theorem","m":"Causalean.Experimentation.TwoStageInterference.BetweenGroup","a":"Experimentation/TwoStageInterference","t":1,"d":"Two-stage variance decomposition, abstract per-group statistic. For an arbitrary within-group allocation design ψ and comparison design φ and an arbitrary per-group statistic g, assume the selected-group count C is nonzero, the population has nonzero size and at least two groups, and that the stage-1 selection satisfies first-order inclusion probability C/N for every group and second-order (pairwise) inclusion probability C(C−1)/(N(N−1)) — the moments of a simple random sample of C of the N groups. Then the randomization variance of the aggregate (∑ᵢ 1(Sᵢ=ψ)·g i(wᵢ))/C under the two-stage design jointDesign D₁ ψ φ splits into a between-group SRS term with finite-population correction (1 − C/N)/C applied to the population sample variance of the conditional means (ψ i).E (g i), plus a within-group term (1/(C·N)) times the sum of the conditional variances (ψ i).Var (g i).","s":"∀ {ι : Type u_1} [inst : Fintype ι] [inst_1 : DecidableEq ι] {n : ι → ℕ} (D₁ : Causalean.Experimentation.DesignBased.FiniteDesign (Causalean.Experimentation.TwoStageInterference.StratAssign ι)) (ψ φ : (i : ι) → Causalean.Experimentation.DesignBased.FiniteDesign (Causalean.Experimentation.TwoStageInterference.WAssign n i)) (g : (i : ι) → Causalean.Experimentation.TwoStageInterference.WAssign n i → ℝ) (C : ℝ), C ≠ 0 → ↑(Fintype.card ι) ≠ 0 → ↑(Fintype.card ι) - 1 ≠ 0 → (∀ (i : ι), (D₁.Pr fun s => s i = true) = C / ↑(Fintype.card ι)) → (∀ (i j : ι), i ≠ j → (D₁.E fun s => Causalean.Experimentation.DesignBased.FiniteDesign.ind (fun s => s i = true) s * Causalean.Experimentation.DesignBased.FiniteDesign.ind (fun s => s j = true) s) = C * (C - 1) / (↑(Fintype.card ι) * (↑(Fintype.card ι) - 1))) → ((Causalean.Experimentation.TwoStageInterference.jointDesign D₁ ψ φ).Var fun sw => (∑ i, if sw.1 i = true then g i (sw.2 i) else 0) / C) = ((1 - C / ↑(Fintype.card ι)) / C * Causalean.Experimentation.TwoStageInterference.SmuVar fun i => (ψ i).E (g i)) + 1 / (C * ↑(Fintype.card ι)) * ∑ i, (ψ i).Var (g i)","l":145,"f":"Causalean/Experimentation/TwoStageInterference/BetweenGroup.lean"},{"n":"Causalean.Experimentation.TwoStageInterference.Var_popEst","k":"theorem","m":"Causalean.Experimentation.TwoStageInterference.BetweenGroup","a":"Experimentation/TwoStageInterference","t":1,"d":"Hudgens–Halloran (2008), Theorem 4 (two-stage variance decomposition of Ŷ(z;ψ)). For arbitrary within-group allocation designs ψ and comparison designs φ and potential outcomes Y, assume the treated/control unit counts m i are nonzero, the group sizes n i are nonzero, the population has nonzero size and at least two groups, the selected-group count C is nonzero, that the within-group z-propensity of every unit equals m i / n i, and that the stage-1 selection satisfies first-order inclusion probability C/N and second-order (pairwise) inclusion probability C(C−1)/(N(N−1)) — the moments of a simple random sample of C of the N groups, each then randomized by its within-group design. Then the randomization variance of the population estimator Ŷ(z;ψ) under the two-stage design jointDesign D₁ ψ φ splits into a between-group SRS term with finite-population correction (1 − C/N)/C applied to the population sample variance of the group-level potential outcomes ȳ_i(z;ψ), plus a within-group term (1/(C·N)) times the sum of the per-group conditional variances of the within-group estimator.","s":"∀ {ι : Type u_1} [inst : Fintype ι] [inst_1 : DecidableEq ι] {n : ι → ℕ} (D₁ : Causalean.Experimentation.DesignBased.FiniteDesign (Causalean.Experimentation.TwoStageInterference.StratAssign ι)) (ψ φ : (i : ι) → Causalean.Experimentation.DesignBased.FiniteDesign (Causalean.Experimentation.TwoStageInterference.WAssign n i)) (Y : (i : ι) → Fin (n i) → Causalean.Experimentation.TwoStageInterference.WAssign n i → ℝ) (z : Bool) (m : ι → ℝ) (C : ℝ), C ≠ 0 → (∀ (i : ι), m i ≠ 0) → (∀ (i : ι), ↑(n i) ≠ 0) → ↑(Fintype.card ι) ≠ 0 → ↑(Fintype.card ι) - 1 ≠ 0 → (∀ (i : ι) (j : Fin (n i)), ((ψ i).Pr fun w => w j = z) = m i / ↑(n i)) → (∀ (i : ι), (D₁.Pr fun s => s i = true) = C / ↑(Fintype.card ι)) → (∀ (i j : ι), i ≠ j → (D₁.E fun s => Causalean.Experimentation.DesignBased.FiniteDesign.ind (fun s => s i = true) s * Causalean.Experimentation.DesignBased.FiniteDesign.ind (fun s => s j = true) s) = C * (C - 1) / (↑(Fintype.card ι) * (↑(Fintype.card ι) - 1))) → (Causalean.Experimentation.TwoStageInterference.jointDesign D₁ ψ φ).Var (Causalean.Experimentation.TwoStageInterference.popEst Y z true m C) = ((1 - C / ↑(Fintype.card ι)) / C * Causalean.Experimentation.TwoStageInterference.SmuVar fun i => Causalean.Experimentation.TwoStageInterference.groupMean ψ Y i z) + 1 / (C * ↑(Fintype.card ι)) * ∑ i, (ψ i).Var (Causalean.Experimentation.TwoStageInterference.groupEst Y i z (m i))","l":187,"f":"Causalean/Experimentation/TwoStageInterference/BetweenGroup.lean"},{"n":"Causalean.Experimentation.TwoStageInterference.Var_popEst_CRD","k":"theorem","m":"Causalean.Experimentation.TwoStageInterference.BetweenGroup","a":"Experimentation/TwoStageInterference","t":1,"d":"Hudgens–Halloran (2008), Theorem 4, for the mixed two-stage design. For arbitrary potential outcomes Y, assume the selected-group count C is positive and strictly less than the number of groups, and that every group's treated-unit count K i is positive and strictly less than the group's size n i. Then, under the mixed two-stage design — stage 1 a simple random sample of C of the groups, each selected group i completely randomized to treat K i of its n i units — Var_popEst's between/within variance decomposition holds for the population estimator Ŷ(z; ·), with all of Var_popEst's design-moment hypotheses discharged by the completely-randomized design's inclusion and propensity facts.","s":"∀ {ι : Type u_1} [inst : Fintype ι] [inst_1 : DecidableEq ι] {n : ι → ℕ} (Y : (i : ι) → Fin (n i) → Causalean.Experimentation.TwoStageInterference.WAssign n i → ℝ) (z : Bool) (K : ι → ℕ) (C : ℕ), 0 < C → ∀ (hCN : C < Fintype.card ι), (∀ (i : ι), 0 < K i) → ∀ (hKn : ∀ (i : ι), K i < n i), (Causalean.Experimentation.TwoStageInterference.jointDesign (Causalean.Experimentation.TwoStageInterference.crdOn C (LT.lt.le hCN)) (fun i => Causalean.Experimentation.TwoStageInterference.crd (K i) (LT.lt.le (hKn i))) fun i => Causalean.Experimentation.TwoStageInterference.crd (K i) (LT.lt.le (hKn i))).Var (Causalean.Experimentation.TwoStageInterference.popEst Y z true (fun i => bif z then ↑(K i) else ↑(n i) - ↑(K i)) ↑C) = ((1 - ↑C / ↑(Fintype.card ι)) / ↑C * Causalean.Experimentation.TwoStageInterference.SmuVar fun i => Causalean.Experimentation.TwoStageInterference.groupMean (fun i => Causalean.Experimentation.TwoStageInterference.crd (K i) (LT.lt.le (hKn i))) Y i z) + 1 / (↑C * ↑(Fintype.card ι)) * ∑ i, (Causalean.Experimentation.TwoStageInterference.crd (K i) (LT.lt.le (hKn i))).Var (Causalean.Experimentation.TwoStageInterference.groupEst Y i z (bif z then ↑(K i) else ↑(n i) - ↑(K i)))","l":225,"f":"Causalean/Experimentation/TwoStageInterference/BetweenGroup.lean"},{"n":"Causalean.Experimentation.TwoStageInterference.Var_srs_mean","k":"theorem","m":"Causalean.Experimentation.TwoStageInterference.StageOne","a":"Experimentation/TwoStageInterference","t":1,"d":"Stage-1 / between-group SRS variance term (Hudgens–Halloran 2008, the between-group term of Theorems 4 and 6). Under simple random sampling of m of the N := card ι groups, with {0,1} selection indicators U satisfying the SRS first- and second-order selection moments (hmean, hpair) and the Bernoulli diagonal variance (hvar), the sampling variance of the sample mean (∑ᵢ Uᵢ·μᵢ)/m of the group-level quantities μ equals (1 − m/N)/m times the population sample variance SmuVar μ.","s":"∀ {ι : Type u_1} [inst : Fintype ι] [DecidableEq ι] {Ω₁ : Type u_2} [inst_2 : Fintype Ω₁] (D₁ : Causalean.Experimentation.DesignBased.FiniteDesign Ω₁) (U : ι → Ω₁ → ℝ) (μ : ι → ℝ) (m : ℝ), (∀ (i : ι), D₁.E (U i) = m / ↑(Fintype.card ι)) → (∀ (i j : ι), i ≠ j → (D₁.E fun s => U i s * U j s) = m * (m - 1) / (↑(Fintype.card ι) * (↑(Fintype.card ι) - 1))) → (∀ (i : ι), D₁.Var (U i) = m / ↑(Fintype.card ι) * (1 - m / ↑(Fintype.card ι))) → m ≠ 0 → ↑(Fintype.card ι) - 1 ≠ 0 → ↑(Fintype.card ι) ≠ 0 → (D₁.Var fun s => (∑ i, U i s * μ i) / m) = (1 - m / ↑(Fintype.card ι)) / m * Causalean.Experimentation.TwoStageInterference.SmuVar μ","l":62,"f":"Causalean/Experimentation/TwoStageInterference/StageOne.lean"},{"n":"Causalean.Experimentation.TwoStageInterference.Var_tauHat","k":"theorem","m":"Causalean.Experimentation.TwoStageInterference.Variance","a":"Experimentation/TwoStageInterference","t":1,"d":"Hudgens–Halloran (2008), Theorem 5 (within-group / Neyman form). For any within-group design whose treatment indicators have first moment K/n (hmean) and pairwise second moment K(K−1)/(n(n−1)) (hpair) — the moments of the completely randomized (mixed) design of Assumption 1, which treats exactly K of n units — with the two-valued potential outcomes a (treated state) and b (control state), the randomization variance of the difference-in-means estimator is S₁/K + S₀/(n−K) − Sτ/n.","s":"∀ {n : ℕ} (K : ℕ) (a b : Fin n → ℝ) (ρ : Causalean.Experimentation.DesignBased.FiniteDesign (Fin n → Bool)), 0 < K → K < n → (∀ (j : Fin n), ρ.E (Causalean.Experimentation.TwoStageInterference.T j) = ↑K / ↑n) → (∀ (j k : Fin n), j ≠ k → (ρ.E fun w => Causalean.Experimentation.TwoStageInterference.T j w * Causalean.Experimentation.TwoStageInterference.T k w) = ↑K * (↑K - 1) / (↑n * (↑n - 1))) → ρ.Var (Causalean.Experimentation.TwoStageInterference.tauHat K a b) = Causalean.Experimentation.TwoStageInterference.S1 a / ↑K + Causalean.Experimentation.TwoStageInterference.S0 b / (↑n - ↑K) - Causalean.Experimentation.TwoStageInterference.Stau a b / ↑n","l":174,"f":"Causalean/Experimentation/TwoStageInterference/Variance.lean"},{"n":"Causalean.Experimentation.TwoStageInterference.Var_tauHat_CRD","k":"theorem","m":"Causalean.Experimentation.TwoStageInterference.Variance","a":"Experimentation/TwoStageInterference","t":1,"d":"Hudgens–Halloran (2008), Theorem 5, for the completely randomized design. For a group of n units with potential outcomes a (treated state) and b (untreated state), consider the completely randomized within-group design that treats exactly K units uniformly at random, where K is positive and strictly less than the group size n. Then the randomization variance of the control-minus-treatment difference-in-means estimator under this design equals S₁/K + S₀/(n−K) − Sτ/n, where S₁ and S₀ are the population sample variances of the treated-state and untreated-state outcomes and Sτ is the population sample variance of the unit-level treatment effects.","s":"∀ {n : ℕ} (K : ℕ) (a b : Fin n → ℝ), 0 < K → ∀ (hKn : K < n), (Causalean.Experimentation.TwoStageInterference.crd K (LT.lt.le hKn)).Var (Causalean.Experimentation.TwoStageInterference.tauHat K a b) = Causalean.Experimentation.TwoStageInterference.S1 a / ↑K + Causalean.Experimentation.TwoStageInterference.S0 b / (↑n - ↑K) - Causalean.Experimentation.TwoStageInterference.Stau a b / ↑n","l":262,"f":"Causalean/Experimentation/TwoStageInterference/Variance.lean"},{"n":"Causalean.Experimentation.TwoStageInterference.WAssign","k":"def","m":"Causalean.Experimentation.TwoStageInterference.Basic","a":"Experimentation/TwoStageInterference","t":1,"d":"Within-group assignment space for group i: a {treated/untreated} flag per unit.","s":"{ι : Type u_1} → (ι → ℕ) → ι → Type","l":40,"f":"Causalean/Experimentation/TwoStageInterference/Basic.lean"},{"n":"Causalean.Experimentation.TwoStageInterference.cltSummand","k":"def","m":"Causalean.Experimentation.TwoStageInterference.Asymptotic.CLTDischarge","a":"Experimentation/TwoStageInterference/Asymptotic","t":1,"d":"This is the mean-zero scaled per-coordinate summand of the conditional studentized statistic under homogeneity.","s":"{Exp : ℕ → Causalean.Experimentation.TwoStageInterference.LHExperiment} → (n : ℕ) → ℝ → (ℕ → ℝ) → Causalean.Experimentation.TwoStageInterference.StratAssign (Exp n).ι → (i : (Exp n).ι) → (Fin ((Exp n).gsize i) → Bool) → ℝ","l":233,"f":"Causalean/Experimentation/TwoStageInterference/Asymptotic/CLTDischarge.lean"},{"n":"Causalean.Experimentation.TwoStageInterference.condCLT_ref","k":"theorem","m":"Causalean.Experimentation.TwoStageInterference.Asymptotic.CLTDischargeMain","a":"Experimentation/TwoStageInterference/Asymptotic","t":2,"d":"Conditional CLT for the reference selection. Applying the independent-summands CLT prodDesign_clt to the per-coordinate summands gₛ₀,ᵢ over the conditional product design gives the conditional studentized CDF at the reference selection refSel Exp n converging to Φ(t).","s":"∀ {Exp : ℕ → Causalean.Experimentation.TwoStageInterference.LHExperiment} {t δ M : ℝ} {v : ℕ → ℝ} {stud : (n : ℕ) → Causalean.Experimentation.TwoStageInterference.StratAssign (Exp n).ι × ((i : (Exp n).ι) → Fin ((Exp n).gsize i) → Bool) → ℝ}, Causalean.Experimentation.TwoStageInterference.Homogeneous Exp t stud δ M v → Filter.Tendsto (fun n => (Causalean.Experimentation.TwoStageInterference.condDesign (Exp n) (Causalean.Experimentation.TwoStageInterference.refSel Exp n)).Pr fun w => stud n (Causalean.Experimentation.TwoStageInterference.refSel Exp n, w) ≤ t) Filter.atTop (nhds (Causalean.Experimentation.DesignBased.stdNormalCdf t))","l":45,"f":"Causalean/Experimentation/TwoStageInterference/Asymptotic/CLTDischargeMain.lean"},{"n":"Causalean.Experimentation.TwoStageInterference.condDesign","k":"def","m":"Causalean.Experimentation.TwoStageInterference.Asymptotic.CLTDischarge","a":"Experimentation/TwoStageInterference/Asymptotic","t":1,"d":"This is the stage-two product design conditional on a stage-one strategy assignment.","s":"(E : Causalean.Experimentation.TwoStageInterference.LHExperiment) → Causalean.Experimentation.TwoStageInterference.StratAssign E.ι → Causalean.Experimentation.DesignBased.FiniteDesign ((i : E.ι) → Fin (E.gsize i) → Bool)","l":126,"f":"Causalean/Experimentation/TwoStageInterference/Asymptotic/CLTDischarge.lean"},{"n":"Causalean.Experimentation.TwoStageInterference.cov_diag","k":"theorem","m":"Causalean.Experimentation.TwoStageInterference.Variance","a":"Experimentation/TwoStageInterference","t":2,"d":"The diagonal covariance Cov(Tⱼ, Tⱼ) = Var(Tⱼ) = (K/n)(1 − K/n).","s":"∀ {n : ℕ} (K : ℕ) (ρ : Causalean.Experimentation.DesignBased.FiniteDesign (Fin n → Bool)), (∀ (j : Fin n), ρ.E (Causalean.Experimentation.TwoStageInterference.T j) = ↑K / ↑n) → ∀ (j : Fin n), ρ.Cov (Causalean.Experimentation.TwoStageInterference.T j) (Causalean.Experimentation.TwoStageInterference.T j) = ↑K / ↑n * (1 - ↑K / ↑n)","l":145,"f":"Causalean/Experimentation/TwoStageInterference/Variance.lean"},{"n":"Causalean.Experimentation.TwoStageInterference.cov_offdiag","k":"theorem","m":"Causalean.Experimentation.TwoStageInterference.Variance","a":"Experimentation/TwoStageInterference","t":2,"d":"The off-diagonal covariance Cov(Tⱼ, Tₖ) = K(K−1)/(n(n−1)) − (K/n)² for j ≠ k.","s":"∀ {n : ℕ} (K : ℕ) (ρ : Causalean.Experimentation.DesignBased.FiniteDesign (Fin n → Bool)), (∀ (j : Fin n), ρ.E (Causalean.Experimentation.TwoStageInterference.T j) = ↑K / ↑n) → (∀ (j k : Fin n), j ≠ k → (ρ.E fun w => Causalean.Experimentation.TwoStageInterference.T j w * Causalean.Experimentation.TwoStageInterference.T k w) = ↑K * (↑K - 1) / (↑n * (↑n - 1))) → ∀ (j k : Fin n), j ≠ k → ρ.Cov (Causalean.Experimentation.TwoStageInterference.T j) (Causalean.Experimentation.TwoStageInterference.T k) = ↑K * (↑K - 1) / (↑n * (↑n - 1)) - ↑K / ↑n * (↑K / ↑n)","l":155,"f":"Causalean/Experimentation/TwoStageInterference/Variance.lean"},{"n":"Causalean.Experimentation.TwoStageInterference.crd","k":"def","m":"Causalean.Experimentation.TwoStageInterference.CompleteRandomization","a":"Experimentation/TwoStageInterference","t":1,"d":"The within-group completely randomized design on Fin n → Bool. Exactly K of the n units are treated, uniformly over the (n choose K) treated sets — the U = Fin n specialization of crdOn.","s":"{n : ℕ} → (K : ℕ) → K ≤ n → Causalean.Experimentation.DesignBased.FiniteDesign (Fin n → Bool)","l":143,"f":"Causalean/Experimentation/TwoStageInterference/CompleteRandomization.lean"},{"n":"Causalean.Experimentation.TwoStageInterference.crdOn","k":"def","m":"Causalean.Experimentation.TwoStageInterference.CompleteRandomization","a":"Experimentation/TwoStageInterference","t":1,"d":"The completely randomized design on U → Bool. Exactly K of the N = card U units are treated, with every one of the (N choose K) treated sets equally likely. Realized as the pushforward of completeRandomization K (on size-K treated sets) along the indicator-vector map crdToBoolOn.","s":"{U : Type u_1} → [inst : Fintype U] → [inst_1 : DecidableEq U] → (K : ℕ) → K ≤ Fintype.card U → Causalean.Experimentation.DesignBased.FiniteDesign (U → Bool)","l":64,"f":"Causalean/Experimentation/TwoStageInterference/CompleteRandomization.lean"},{"n":"Causalean.Experimentation.TwoStageInterference.crdOn_mean","k":"theorem","m":"Causalean.Experimentation.TwoStageInterference.CompleteRandomization","a":"Experimentation/TwoStageInterference","t":2,"d":"First-order inclusion probability: each unit i is treated with probability K/N, i.e. E[Tᵢ] = K/N. Derived from completeRandomization_incl.","s":"∀ {U : Type u_1} [inst : Fintype U] [inst_1 : DecidableEq U] (K : ℕ) (hK : K ≤ Fintype.card U) (i : U), (Causalean.Experimentation.TwoStageInterference.crdOn K hK).E (Causalean.Experimentation.DesignBased.FiniteDesign.ind fun w => w i = true) = ↑K / ↑(Fintype.card U)","l":71,"f":"Causalean/Experimentation/TwoStageInterference/CompleteRandomization.lean"},{"n":"Causalean.Experimentation.TwoStageInterference.crdOn_mean_compl","k":"theorem","m":"Causalean.Experimentation.TwoStageInterference.CompleteRandomization","a":"Experimentation/TwoStageInterference","t":2,"d":"Complementary first moment: each unit i is untreated with probability (N−K)/N, i.e. E[1−Tᵢ] = (N−K)/N. The control-arm propensity, from the indicator complement 1 − Tᵢ.","s":"∀ {U : Type u_1} [inst : Fintype U] [inst_1 : DecidableEq U] (K : ℕ) (hK : K ≤ Fintype.card U) (i : U), ((Causalean.Experimentation.TwoStageInterference.crdOn K hK).E fun w => 1 - Causalean.Experimentation.DesignBased.FiniteDesign.ind (fun w => w i = true) w) = (↑(Fintype.card U) - ↑K) / ↑(Fintype.card U)","l":81,"f":"Causalean/Experimentation/TwoStageInterference/CompleteRandomization.lean"},{"n":"Causalean.Experimentation.TwoStageInterference.crdOn_pair","k":"theorem","m":"Causalean.Experimentation.TwoStageInterference.CompleteRandomization","a":"Experimentation/TwoStageInterference","t":1,"d":"Second-order inclusion probability of the completely randomized design. Under the completely randomized design that treats exactly K of the N units in the population uniformly over all size-K treated subsets, if i and j are two distinct units then the probability that both i and j are treated equals K(K−1)/(N(N−1)).","s":"∀ {U : Type u_1} [inst : Fintype U] [inst_1 : DecidableEq U] (K : ℕ) (hK : K ≤ Fintype.card U) (i j : U), i ≠ j → ((Causalean.Experimentation.TwoStageInterference.crdOn K hK).E fun w => Causalean.Experimentation.DesignBased.FiniteDesign.ind (fun w => w i = true) w * Causalean.Experimentation.DesignBased.FiniteDesign.ind (fun w => w j = true) w) = ↑K * (↑K - 1) / (↑(Fintype.card U) * (↑(Fintype.card U) - 1))","l":91,"f":"Causalean/Experimentation/TwoStageInterference/CompleteRandomization.lean"},{"n":"Causalean.Experimentation.TwoStageInterference.crdOn_supp","k":"theorem","m":"Causalean.Experimentation.TwoStageInterference.CompleteRandomization","a":"Experimentation/TwoStageInterference","t":2,"d":"Deterministic treated count on the design's support: any assignment with positive design weight treats exactly K units, i.e. ∑ᵢ Tᵢ = K.","s":"∀ {U : Type u_1} [inst : Fintype U] [inst_1 : DecidableEq U] (K : ℕ) (hK : K ≤ Fintype.card U) (w : U → Bool), (Causalean.Experimentation.TwoStageInterference.crdOn K hK).p w ≠ 0 → ∑ i, Causalean.Experimentation.DesignBased.FiniteDesign.ind (fun w => w i = true) w = ↑K","l":111,"f":"Causalean/Experimentation/TwoStageInterference/CompleteRandomization.lean"},{"n":"Causalean.Experimentation.TwoStageInterference.crdToBool","k":"def","m":"Causalean.Experimentation.TwoStageInterference.CompleteRandomization","a":"Experimentation/TwoStageInterference","t":1,"d":"The within-group treated-set-to-indicator map (specialization of crdToBoolOn to Fin n).","s":"{n : ℕ} → (K : ℕ) → { S // S.card = K } → Fin n → Bool","l":140,"f":"Causalean/Experimentation/TwoStageInterference/CompleteRandomization.lean"},{"n":"Causalean.Experimentation.TwoStageInterference.crdToBoolOn","k":"def","m":"Causalean.Experimentation.TwoStageInterference.CompleteRandomization","a":"Experimentation/TwoStageInterference","t":1,"d":"The treated-set-to-indicator map: a size-K treated set S becomes the Boolean assignment that flags unit i true exactly when i ∈ S.","s":"{U : Type u_1} → [DecidableEq U] → (K : ℕ) → { S // S.card = K } → U → Bool","l":59,"f":"Causalean/Experimentation/TwoStageInterference/CompleteRandomization.lean"},{"n":"Causalean.Experimentation.TwoStageInterference.crd_mean","k":"theorem","m":"Causalean.Experimentation.TwoStageInterference.CompleteRandomization","a":"Experimentation/TwoStageInterference","t":2,"d":"First-order inclusion probability of the within-group design: E[Tⱼ] = K/n.","s":"∀ {n : ℕ} (K : ℕ) (hK : K ≤ n) (j : Fin n), (Causalean.Experimentation.TwoStageInterference.crd K hK).E (Causalean.Experimentation.DesignBased.FiniteDesign.ind fun w => w j = true) = ↑K / ↑n","l":149,"f":"Causalean/Experimentation/TwoStageInterference/CompleteRandomization.lean"},{"n":"Causalean.Experimentation.TwoStageInterference.crd_pair","k":"theorem","m":"Causalean.Experimentation.TwoStageInterference.CompleteRandomization","a":"Experimentation/TwoStageInterference","t":2,"d":"Second-order inclusion probability of the within-group design: E[Tⱼ Tₖ] = K(K−1)/(n(n−1)) for j ≠ k.","s":"∀ {n : ℕ} (K : ℕ) (hK : K ≤ n) (j k : Fin n), j ≠ k → ((Causalean.Experimentation.TwoStageInterference.crd K hK).E fun w => Causalean.Experimentation.DesignBased.FiniteDesign.ind (fun w => w j = true) w * Causalean.Experimentation.DesignBased.FiniteDesign.ind (fun w => w k = true) w) = ↑K * (↑K - 1) / (↑n * (↑n - 1))","l":154,"f":"Causalean/Experimentation/TwoStageInterference/CompleteRandomization.lean"},{"n":"Causalean.Experimentation.TwoStageInterference.crd_prop_false","k":"theorem","m":"Causalean.Experimentation.TwoStageInterference.CompleteRandomization","a":"Experimentation/TwoStageInterference","t":2,"d":"Control propensity of the within-group design: each unit j is in control with probability (n−K)/n, i.e. Pr[wⱼ = false] = (n−K)/n.","s":"∀ {n : ℕ} (K : ℕ) (hK : K ≤ n) (j : Fin n), ((Causalean.Experimentation.TwoStageInterference.crd K hK).Pr fun w => w j = false) = (↑n - ↑K) / ↑n","l":174,"f":"Causalean/Experimentation/TwoStageInterference/CompleteRandomization.lean"},{"n":"Causalean.Experimentation.TwoStageInterference.crd_prop_true","k":"theorem","m":"Causalean.Experimentation.TwoStageInterference.CompleteRandomization","a":"Experimentation/TwoStageInterference","t":2,"d":"Treatment propensity of the within-group design: each unit j is treated with probability K/n, i.e. Pr[wⱼ = true] = K/n.","s":"∀ {n : ℕ} (K : ℕ) (hK : K ≤ n) (j : Fin n), ((Causalean.Experimentation.TwoStageInterference.crd K hK).Pr fun w => w j = true) = ↑K / ↑n","l":168,"f":"Causalean/Experimentation/TwoStageInterference/CompleteRandomization.lean"},{"n":"Causalean.Experimentation.TwoStageInterference.crd_supp","k":"theorem","m":"Causalean.Experimentation.TwoStageInterference.CompleteRandomization","a":"Experimentation/TwoStageInterference","t":2,"d":"Deterministic treated count on the within-group design's support: ∑ⱼ Tⱼ = K.","s":"∀ {n : ℕ} (K : ℕ) (hK : K ≤ n) (w : Fin n → Bool), (Causalean.Experimentation.TwoStageInterference.crd K hK).p w ≠ 0 → ∑ j, Causalean.Experimentation.DesignBased.FiniteDesign.ind (fun w => w j = true) w = ↑K","l":162,"f":"Causalean/Experimentation/TwoStageInterference/CompleteRandomization.lean"},{"n":"Causalean.Experimentation.TwoStageInterference.directEffect_clt","k":"theorem","m":"Causalean.Experimentation.TwoStageInterference.Asymptotic.CLT","a":"Experimentation/TwoStageInterference/Asymptotic","t":1,"d":"Proposition 5.1 (Liu–Hudgens 2014), asymptotic normality of the treatment-minus-control direct-effect contrast — conditional on the uniform conditional CLT. Along a sequence of two-stage Hudgens–Halloran experiments Exp : ℕ → LHExperiment, let stud n be the studentized statistic (D̂E − DE̅)/√directVar for the treatment-minus-control direct-effect contrast and cond n s the within-selection (stage-2 product) design when stage 1 selects the strategy assignment s. Assume the conditional studentized CDFs at t converge to the standard normal CDF Φ(t) uniformly over selections s, with a vanishing uniform bound. Then the unconditional (joint-design) studentized CDF at t converges to Φ(t):","s":"∀ (Exp : ℕ → Causalean.Experimentation.TwoStageInterference.LHExperiment) (t : ℝ) (stud : (n : ℕ) → Causalean.Experimentation.TwoStageInterference.StratAssign (Exp n).ι × ((i : (Exp n).ι) → Fin ((Exp n).gsize i) → Bool) → ℝ), (∀ (n : ℕ) (sw : Causalean.Experimentation.TwoStageInterference.StratAssign (Exp n).ι × ((i : (Exp n).ι) → Fin ((Exp n).gsize i) → Bool)), stud n sw = ((Exp n).estD sw - (Exp n).DEbar) / √(Exp n).directVar) → ∀ (cond : (n : ℕ) → Causalean.Experimentation.TwoStageInterference.StratAssign (Exp n).ι → Causalean.Experimentation.DesignBased.FiniteDesign ((i : (Exp n).ι) → Fin ((Exp n).gsize i) → Bool)), (∀ (n : ℕ) (s : Causalean.Experimentation.TwoStageInterference.StratAssign (Exp n).ι), cond n s = Causalean.Experimentation.DesignBased.prodDesign fun i => if s i = true then (Exp n).ψ i else (Exp n).φ i) → (∃ B, Filter.Tendsto B Filter.atTop (nhds 0) ∧ ∀ (n : ℕ) (s : Causalean.Experimentation.TwoStageInterference.StratAssign (Exp n).ι), |((cond n s).Pr fun w => stud n (s, w) ≤ t) - Causalean.Experimentation.DesignBased.stdNormalCdf t| ≤ B n) → Filter.Tendsto (fun n => (Exp n).jointD.Pr fun sw => stud n sw ≤ t) Filter.atTop (nhds (Causalean.Experimentation.DesignBased.stdNormalCdf t))","l":118,"f":"Causalean/Experimentation/TwoStageInterference/Asymptotic/CLT.lean"},{"n":"Causalean.Experimentation.TwoStageInterference.directEffect_clt_homogeneous","k":"theorem","m":"Causalean.Experimentation.TwoStageInterference.Asymptotic.CLTDischargeMain","a":"Experimentation/TwoStageInterference/Asymptotic","t":1,"d":"Proposition 5.1 (Liu–Hudgens 2014), fully primitive. Under the homogeneity and regularity bundle, the studentized treatment-minus-control direct-effect contrast is asymptotically standard normal: its joint-design CDF at t converges to Φ(t).","s":"∀ {Exp : ℕ → Causalean.Experimentation.TwoStageInterference.LHExperiment} {t δ M : ℝ} {v : ℕ → ℝ} {stud : (n : ℕ) → Causalean.Experimentation.TwoStageInterference.StratAssign (Exp n).ι × ((i : (Exp n).ι) → Fin ((Exp n).gsize i) → Bool) → ℝ}, Causalean.Experimentation.TwoStageInterference.Homogeneous Exp t stud δ M v → Filter.Tendsto (fun n => (Exp n).jointD.Pr fun sw => stud n sw ≤ t) Filter.atTop (nhds (Causalean.Experimentation.DesignBased.stdNormalCdf t))","l":146,"f":"Causalean/Experimentation/TwoStageInterference/Asymptotic/CLTDischargeMain.lean"},{"n":"Causalean.Experimentation.TwoStageInterference.directEffect_clt_identical","k":"theorem","m":"Causalean.Experimentation.TwoStageInterference.Asymptotic.Identical","a":"Experimentation/TwoStageInterference/Asymptotic","t":1,"d":"Proposition 5.1 under literally identical groups. Along a sequence of identical-groups experiments R — common size, allocation strategies, potential outcomes, and unit counts — sharing one group-level treatment-minus-control direct-effect contrast δ with a positive common within-group variance, a uniform bound M on the centered per-group contrast estimator, every supported stage-1 selection flagging exactly C groups, and the many-groups rate M/√(C·v) → 0 together with its Lyapunov cube card·(M/√(C·v))³ → 0 — the studentized contrast statistic is asymptotically standard normal. No analytic homogeneity hypothesis is assumed: it is derived from the concrete identical-groups structure via hhom_of_identical.","s":"∀ (R : ℕ → Causalean.Experimentation.TwoStageInterference.IdenticalRef) (t δ M : ℝ), (∀ (n : ℕ), (R n).refDelta = δ) → (∀ (n : ℕ), 0 < (R n).refVar) → (∀ (n : ℕ) (i : (R n).toExp.ι) (w : Fin ((R n).toExp.gsize i) → Bool), |Causalean.Experimentation.TwoStageInterference.groupDiff (R n).toExp i w - δ| ≤ M) → (∀ (n : ℕ) (s : Causalean.Experimentation.TwoStageInterference.StratAssign (R n).toExp.ι), (R n).toExp.D₁.p s ≠ 0 → (∑ i, if s i = true then 1 else 0) = (R n).toExp.C) → Filter.Tendsto (fun n => M / √((R n).toExp.C * (R n).refVar)) Filter.atTop (nhds 0) → Filter.Tendsto (fun n => ↑(Fintype.card (R n).toExp.ι) * (M / √((R n).toExp.C * (R n).refVar)) ^ 3) Filter.atTop (nhds 0) → Filter.Tendsto (fun n => (R n).toExp.jointD.Pr fun sw => (R n).studId sw ≤ t) Filter.atTop (nhds (Causalean.Experimentation.DesignBased.stdNormalCdf t))","l":336,"f":"Causalean/Experimentation/TwoStageInterference/Asymptotic/Identical.lean"},{"n":"Causalean.Experimentation.TwoStageInterference.directVar_eq_of_homogeneous","k":"theorem","m":"Causalean.Experimentation.TwoStageInterference.Asymptotic.CLTDischarge","a":"Experimentation/TwoStageInterference/Asymptotic","t":1,"d":"Variance reduction. Under the homogeneity and regularity bundle, the two-stage design variance of the direct-effect contrast collapses to v n / C: the between-group term vanishes since the population variance of a constant is zero, and the within-group term averages to v n / C.","s":"∀ {Exp : ℕ → Causalean.Experimentation.TwoStageInterference.LHExperiment} {t δ M : ℝ} {v : ℕ → ℝ} {stud : (n : ℕ) → Causalean.Experimentation.TwoStageInterference.StratAssign (Exp n).ι × ((i : (Exp n).ι) → Fin ((Exp n).gsize i) → Bool) → ℝ}, Causalean.Experimentation.TwoStageInterference.Homogeneous Exp t stud δ M v → ∀ (n : ℕ), (Exp n).directVar = v n / (Exp n).C","l":200,"f":"Causalean/Experimentation/TwoStageInterference/Asymptotic/CLTDischarge.lean"},{"n":"Causalean.Experimentation.TwoStageInterference.estD_eq_agg","k":"theorem","m":"Causalean.Experimentation.TwoStageInterference.Asymptotic.CLTDischarge","a":"Experimentation/TwoStageInterference/Asymptotic","t":2,"d":"The experiment-level contrast estimator is the mean of the selected groups' per-group treatment-minus-control contrast estimators.","s":"∀ (E : Causalean.Experimentation.TwoStageInterference.LHExperiment) (s : Causalean.Experimentation.TwoStageInterference.StratAssign E.ι) (w : (i : E.ι) → Fin (E.gsize i) → Bool), E.estD (s, w) = (∑ i, (if s i = true then 1 else 0) * Causalean.Experimentation.TwoStageInterference.groupDiff E i (w i)) / E.C","l":131,"f":"Causalean/Experimentation/TwoStageInterference/Asymptotic/CLTDischarge.lean"},{"n":"Causalean.Experimentation.TwoStageInterference.estDirect","k":"def","m":"Causalean.Experimentation.TwoStageInterference.Basic","a":"Experimentation/TwoStageInterference","t":1,"d":"The Horvitz-Thompson estimator of the treatment-minus-control direct-effect contrast on the ψ-groups: the estimated treatment mean minus the estimated control mean.","s":"{ι : Type u_1} → [Fintype ι] → {n : ι → ℕ} → ((i : ι) → Fin (n i) → Causalean.Experimentation.TwoStageInterference.WAssign n i → ℝ) → (ι → ℝ) → (ι → ℝ) → ℝ → Causalean.Experimentation.TwoStageInterference.StratAssign ι × ((i : ι) → Causalean.Experimentation.TwoStageInterference.WAssign n i) → ℝ","l":126,"f":"Causalean/Experimentation/TwoStageInterference/Basic.lean"},{"n":"Causalean.Experimentation.TwoStageInterference.estDirect_consistent","k":"theorem","m":"Causalean.Experimentation.TwoStageInterference.Asymptotic.Consistency","a":"Experimentation/TwoStageInterference/Asymptotic","t":1,"d":"Consistency of the direct-effect contrast estimator (Liu–Hudgens 2014). Along a sequence of two-stage experiments, if the closed-form design variance directVar tends to zero as the number of groups grows, then for every positive tolerance ε, the Horvitz-Thompson estimator is consistent for the population average treatment-minus-control direct-effect contrast: the probability that the deviation exceeds ε tends to zero.","s":"∀ (Exp : ℕ → Causalean.Experimentation.TwoStageInterference.LHExperiment), Filter.Tendsto (fun n => (Exp n).directVar) Filter.atTop (nhds 0) → ∀ {ε : ℝ}, 0 < ε → Filter.Tendsto (fun n => (Exp n).jointD.Pr fun sw => ε ≤ |(Exp n).estD sw - (Exp n).DEbar|) Filter.atTop (nhds 0)","l":41,"f":"Causalean/Experimentation/TwoStageInterference/Asymptotic/Consistency.lean"},{"n":"Causalean.Experimentation.TwoStageInterference.estIndirect","k":"def","m":"Causalean.Experimentation.TwoStageInterference.Effects","a":"Experimentation/TwoStageInterference","t":1,"d":"The HT-effect estimator of the indirect (spillover) effect: Ŷ(0;φ) − Ŷ(0;ψ), the control-treatment population estimators on the φ-groups minus on the ψ-groups.","s":"{ι : Type u_1} → [Fintype ι] → {n : ι → ℕ} → ((i : ι) → Fin (n i) → Causalean.Experimentation.TwoStageInterference.WAssign n i → ℝ) → (ι → ℝ) → (ι → ℝ) → ℝ → ℝ → Causalean.Experimentation.TwoStageInterference.StratAssign ι × ((i : ι) → Causalean.Experimentation.TwoStageInterference.WAssign n i) → ℝ","l":121,"f":"Causalean/Experimentation/TwoStageInterference/Effects.lean"},{"n":"Causalean.Experimentation.TwoStageInterference.estTotal","k":"def","m":"Causalean.Experimentation.TwoStageInterference.Effects","a":"Experimentation/TwoStageInterference","t":1,"d":"The HT-effect estimator of the total effect: Ŷ(0;φ) − Ŷ(1;ψ), the control-treatment population estimator on the φ-groups minus the treatment population estimator on the ψ-groups.","s":"{ι : Type u_1} → [Fintype ι] → {n : ι → ℕ} → ((i : ι) → Fin (n i) → Causalean.Experimentation.TwoStageInterference.WAssign n i → ℝ) → (ι → ℝ) → (ι → ℝ) → ℝ → ℝ → Causalean.Experimentation.TwoStageInterference.StratAssign ι × ((i : ι) → Causalean.Experimentation.TwoStageInterference.WAssign n i) → ℝ","l":128,"f":"Causalean/Experimentation/TwoStageInterference/Effects.lean"},{"n":"Causalean.Experimentation.TwoStageInterference.exists_equiv_selection","k":"theorem","m":"Causalean.Experimentation.TwoStageInterference.Asymptotic.Identical","a":"Experimentation/TwoStageInterference/Asymptotic","t":2,"d":"Selection-symmetry permutation. Given two Boolean selections s, s' on a finite type ι that flag the same number of indices (card {i | s i} = card {i | s' i}), there is a permutation σ : ι ≃ ι aligning their patterns: s' i = s (σ i) for every i. Built by gluing a bijection of the selected sets (equal cardinality) with a bijection of their complements (equal cardinality, since the total is fixed) through Equiv.sumCompl.","s":"∀ {ι : Type u_1} [inst : Fintype ι] (s s' : ι → Bool), {i | s i = true}.card = {i | s' i = true}.card → ∃ σ, ∀ (i : ι), s' i = s (σ i)","l":63,"f":"Causalean/Experimentation/TwoStageInterference/Asymptotic/Identical.lean"},{"n":"Causalean.Experimentation.TwoStageInterference.exists_strat_factor","k":"theorem","m":"Causalean.Experimentation.TwoStageInterference.Stratified","a":"Experimentation/TwoStageInterference","t":1,"d":"Factorization through the exposure. Under stratified interference of the potential outcomes Y, there is a family g i j : Bool × ℕ → ℝ of exposure-indexed potential outcomes such that every outcome factors as Y i j w = g i j (stratExpo i j w).","s":"∀ {ι : Type u_1} {n : ι → ℕ} {Y : (i : ι) → Fin (n i) → Causalean.Experimentation.TwoStageInterference.WAssign n i → ℝ}, Causalean.Experimentation.TwoStageInterference.StratifiedInterference Y → ∃ g, ∀ (i : ι) (j : Fin (n i)) (w : Causalean.Experimentation.TwoStageInterference.WAssign n i), Y i j w = g i j (Causalean.Experimentation.TwoStageInterference.stratExpo i j w)","l":85,"f":"Causalean/Experimentation/TwoStageInterference/Stratified.lean"},{"n":"Causalean.Experimentation.TwoStageInterference.exists_support_selection","k":"theorem","m":"Causalean.Experimentation.TwoStageInterference.Asymptotic.CLTDischarge","a":"Experimentation/TwoStageInterference/Asymptotic","t":2,"d":"Some stage-1 selection lies in the support of D₁ (its probabilities sum to one).","s":"∀ (E : Causalean.Experimentation.TwoStageInterference.LHExperiment), ∃ s, E.D₁.p s ≠ 0","l":301,"f":"Causalean/Experimentation/TwoStageInterference/Asymptotic/CLTDischarge.lean"},{"n":"Causalean.Experimentation.TwoStageInterference.groupDiff","k":"def","m":"Causalean.Experimentation.TwoStageInterference.Asymptotic.CLTDischarge","a":"Experimentation/TwoStageInterference/Asymptotic","t":1,"d":"This is the per-group estimator for the treatment-minus-control direct-effect contrast in one group of a Liu-Hudgens experiment.","s":"(E : Causalean.Experimentation.TwoStageInterference.LHExperiment) → (i : E.ι) → (Fin (E.gsize i) → Bool) → ℝ","l":117,"f":"Causalean/Experimentation/TwoStageInterference/Asymptotic/CLTDischarge.lean"},{"n":"Causalean.Experimentation.TwoStageInterference.groupEst","k":"def","m":"Causalean.Experimentation.TwoStageInterference.Basic","a":"Experimentation/TwoStageInterference","t":1,"d":"Within-group estimator Ŷ_i(z;ρ): the empirical mean outcome among the m units of group i whose own treatment equals z, as a function of the realized within-group assignment w. The denominator m is the (design-fixed) number of such units.","s":"{ι : Type u_1} → {n : ι → ℕ} → ((i : ι) → Fin (n i) → Causalean.Experimentation.TwoStageInterference.WAssign n i → ℝ) → (i : ι) → Bool → ℝ → Causalean.Experimentation.TwoStageInterference.WAssign n i → ℝ","l":111,"f":"Causalean/Experimentation/TwoStageInterference/Basic.lean"},{"n":"Causalean.Experimentation.TwoStageInterference.groupMean","k":"def","m":"Causalean.Experimentation.TwoStageInterference.Basic","a":"Experimentation/TwoStageInterference","t":1,"d":"Group average potential outcome ȳ_i(z;ρ) = (1/nᵢ)∑ⱼ ȳ_ij(z;ρ).","s":"{ι : Type u_1} → {n : ι → ℕ} → ((i : ι) → Causalean.Experimentation.DesignBased.FiniteDesign (Causalean.Experimentation.TwoStageInterference.WAssign n i)) → ((i : ι) → Fin (n i) → Causalean.Experimentation.TwoStageInterference.WAssign n i → ℝ) → ι → Bool → ℝ","l":61,"f":"Causalean/Experimentation/TwoStageInterference/Basic.lean"},{"n":"Causalean.Experimentation.TwoStageInterference.homogeneous_of_identical","k":"def","m":"Causalean.Experimentation.TwoStageInterference.Asymptotic.Identical","a":"Experimentation/TwoStageInterference/Asymptotic","t":1,"d":"Homogeneity bundle from identical groups. From a sequence of identical-groups experiments sharing one group-level treatment-minus-control direct-effect contrast δ, with a uniform bound M on the centered per-group estimator, the many-groups rate, and the exact-C selection support, the full Homogeneous bundle is assembled — with its hhom field discharged by hhom_of_identical. The within-group variances v n := refVar (R n) may vary across n.","s":"∀ (R : ℕ → Causalean.Experimentation.TwoStageInterference.IdenticalRef) (t δ M : ℝ), (∀ (n : ℕ), (R n).refDelta = δ) → (∀ (n : ℕ), 0 < (R n).refVar) → (∀ (n : ℕ) (i : (R n).toExp.ι) (w : Fin ((R n).toExp.gsize i) → Bool), |Causalean.Experimentation.TwoStageInterference.groupDiff (R n).toExp i w - δ| ≤ M) → (∀ (n : ℕ) (s : Causalean.Experimentation.TwoStageInterference.StratAssign (R n).toExp.ι), (R n).toExp.D₁.p s ≠ 0 → (∑ i, if s i = true then 1 else 0) = (R n).toExp.C) → Filter.Tendsto (fun n => M / √((R n).toExp.C * (R n).refVar)) Filter.atTop (nhds 0) → Filter.Tendsto (fun n => ↑(Fintype.card (R n).toExp.ι) * (M / √((R n).toExp.C * (R n).refVar)) ^ 3) Filter.atTop (nhds 0) → Causalean.Experimentation.TwoStageInterference.Homogeneous (fun n => (R n).toExp) t (fun n => (R n).studId) δ M fun n => (R n).refVar","l":308,"f":"Causalean/Experimentation/TwoStageInterference/Asymptotic/Identical.lean"},{"n":"Causalean.Experimentation.TwoStageInterference.indMarg","k":"def","m":"Causalean.Experimentation.TwoStageInterference.Basic","a":"Experimentation/TwoStageInterference","t":1,"d":"Marginal individual average potential outcome ȳ_ij(ρ): expected outcome of (i,j) under design ρ, averaging over its own treatment as well (the overall-effect target).","s":"{ι : Type u_1} → {n : ι → ℕ} → ((i : ι) → Causalean.Experimentation.DesignBased.FiniteDesign (Causalean.Experimentation.TwoStageInterference.WAssign n i)) → ((i : ι) → Fin (n i) → Causalean.Experimentation.TwoStageInterference.WAssign n i → ℝ) → (i : ι) → Fin (n i) → ℝ","l":71,"f":"Causalean/Experimentation/TwoStageInterference/Basic.lean"},{"n":"Causalean.Experimentation.TwoStageInterference.indMean","k":"def","m":"Causalean.Experimentation.TwoStageInterference.Basic","a":"Experimentation/TwoStageInterference","t":1,"d":"Individual average potential outcome ȳ_ij(z;ρ): unit (i,j)'s expected outcome under within-group design ρ, conditional on its own treatment being z.","s":"{ι : Type u_1} → {n : ι → ℕ} → ((i : ι) → Causalean.Experimentation.DesignBased.FiniteDesign (Causalean.Experimentation.TwoStageInterference.WAssign n i)) → ((i : ι) → Fin (n i) → Causalean.Experimentation.TwoStageInterference.WAssign n i → ℝ) → (i : ι) → Fin (n i) → Bool → ℝ","l":55,"f":"Causalean/Experimentation/TwoStageInterference/Basic.lean"},{"n":"Causalean.Experimentation.TwoStageInterference.jointDesign","k":"def","m":"Causalean.Experimentation.TwoStageInterference.Basic","a":"Experimentation/TwoStageInterference","t":1,"d":"The joint two-stage design: stage-1 design D₁ over strategy assignments, then, conditionally, group i is randomized by ψ i if assigned ψ and by φ i otherwise.","s":"{ι : Type u_1} → [inst : Fintype ι] → [inst_1 : DecidableEq ι] → {n : ι → ℕ} → Causalean.Experimentation.DesignBased.FiniteDesign (Causalean.Experimentation.TwoStageInterference.StratAssign ι) → ((i : ι) → Causalean.Experimentation.DesignBased.FiniteDesign (Causalean.Experimentation.TwoStageInterference.WAssign n i)) → ((i : ι) → Causalean.Experimentation.DesignBased.FiniteDesign (Causalean.Experimentation.TwoStageInterference.WAssign n i)) → Causalean.Experimentation.DesignBased.FiniteDesign (Causalean.Experimentation.TwoStageInterference.StratAssign ι × ((i : ι) → Causalean.Experimentation.TwoStageInterference.WAssign n i))","l":46,"f":"Causalean/Experimentation/TwoStageInterference/Basic.lean"},{"n":"Causalean.Experimentation.TwoStageInterference.nonempty_of_refSel","k":"theorem","m":"Causalean.Experimentation.TwoStageInterference.Asymptotic.CLTDischarge","a":"Experimentation/TwoStageInterference/Asymptotic","t":2,"d":"The reference selection flags at least one group, so the group index type is nonempty.","s":"∀ {Exp : ℕ → Causalean.Experimentation.TwoStageInterference.LHExperiment} {t δ M : ℝ} {v : ℕ → ℝ} {stud : (n : ℕ) → Causalean.Experimentation.TwoStageInterference.StratAssign (Exp n).ι × ((i : (Exp n).ι) → Fin ((Exp n).gsize i) → Bool) → ℝ}, Causalean.Experimentation.TwoStageInterference.Homogeneous Exp t stud δ M v → ∀ (n : ℕ), Nonempty (Exp n).ι","l":319,"f":"Causalean/Experimentation/TwoStageInterference/Asymptotic/CLTDischarge.lean"},{"n":"Causalean.Experimentation.TwoStageInterference.numTreatedOthers","k":"def","m":"Causalean.Experimentation.TwoStageInterference.Stratified","a":"Experimentation/TwoStageInterference","t":1,"d":"The number of units k ≠ j in group i that are treated under within-group assignment w (the count of *other* treated units relevant to unit j).","s":"{ι : Type u_1} → {n : ι → ℕ} → (i : ι) → Fin (n i) → Causalean.Experimentation.TwoStageInterference.WAssign n i → ℕ","l":49,"f":"Causalean/Experimentation/TwoStageInterference/Stratified.lean"},{"n":"Causalean.Experimentation.TwoStageInterference.obsMeanControl","k":"def","m":"Causalean.Experimentation.TwoStageInterference.VarianceConservative","a":"Experimentation/TwoStageInterference","t":1,"d":"Empirical mean of the untreated-state outcomes among the units observed in control under the realized assignment w: (1/(n−K))∑_{j: wⱼ=0} b j.","s":"{n : ℕ} → ℕ → (Fin n → ℝ) → (Fin n → Bool) → ℝ","l":61,"f":"Causalean/Experimentation/TwoStageInterference/VarianceConservative.lean"},{"n":"Causalean.Experimentation.TwoStageInterference.obsMeanTreated","k":"def","m":"Causalean.Experimentation.TwoStageInterference.VarianceConservative","a":"Experimentation/TwoStageInterference","t":1,"d":"Empirical mean of the treated-state outcomes among the units observed treated under the realized assignment w: (1/K)∑_{j: wⱼ=1} a j, written with the treatment indicators.","s":"{n : ℕ} → ℕ → (Fin n → ℝ) → (Fin n → Bool) → ℝ","l":56,"f":"Causalean/Experimentation/TwoStageInterference/VarianceConservative.lean"},{"n":"Causalean.Experimentation.TwoStageInterference.one_sub_T_nonneg","k":"theorem","m":"Causalean.Experimentation.TwoStageInterference.VarianceConservative","a":"Experimentation/TwoStageInterference","t":2,"d":"1 − Tⱼ (the control indicator) is nonnegative.","s":"∀ {n : ℕ} (j : Fin n) (w : Fin n → Bool), 0 ≤ 1 - Causalean.Experimentation.TwoStageInterference.T j w","l":85,"f":"Causalean/Experimentation/TwoStageInterference/VarianceConservative.lean"},{"n":"Causalean.Experimentation.TwoStageInterference.popEst","k":"def","m":"Causalean.Experimentation.TwoStageInterference.Basic","a":"Experimentation/TwoStageInterference","t":1,"d":"Population estimator Ŷ(z;ρ) on the groups assigned strategy pick: the average of the within-group estimators over the denom groups with s i = pick, as a function of the realized joint assignment (s, w). m i is the design-fixed count of z-units in group i.","s":"{ι : Type u_1} → [Fintype ι] → {n : ι → ℕ} → ((i : ι) → Fin (n i) → Causalean.Experimentation.TwoStageInterference.WAssign n i → ℝ) → Bool → Bool → (ι → ℝ) → ℝ → Causalean.Experimentation.TwoStageInterference.StratAssign ι × ((i : ι) → Causalean.Experimentation.TwoStageInterference.WAssign n i) → ℝ","l":118,"f":"Causalean/Experimentation/TwoStageInterference/Basic.lean"},{"n":"Causalean.Experimentation.TwoStageInterference.popMarg","k":"def","m":"Causalean.Experimentation.TwoStageInterference.Basic","a":"Experimentation/TwoStageInterference","t":1,"d":"Population marginal average potential outcome ȳ(ρ) = (1/N)∑ᵢ(1/nᵢ)∑ⱼ ȳ_ij(ρ).","s":"{ι : Type u_1} → [Fintype ι] → {n : ι → ℕ} → ((i : ι) → Causalean.Experimentation.DesignBased.FiniteDesign (Causalean.Experimentation.TwoStageInterference.WAssign n i)) → ((i : ι) → Fin (n i) → Causalean.Experimentation.TwoStageInterference.WAssign n i → ℝ) → ℝ","l":77,"f":"Causalean/Experimentation/TwoStageInterference/Basic.lean"},{"n":"Causalean.Experimentation.TwoStageInterference.popMean","k":"def","m":"Causalean.Experimentation.TwoStageInterference.Basic","a":"Experimentation/TwoStageInterference","t":1,"d":"Population average potential outcome ȳ(z;ρ) = (1/N)∑ᵢ ȳ_i(z;ρ).","s":"{ι : Type u_1} → [Fintype ι] → {n : ι → ℕ} → ((i : ι) → Causalean.Experimentation.DesignBased.FiniteDesign (Causalean.Experimentation.TwoStageInterference.WAssign n i)) → ((i : ι) → Fin (n i) → Causalean.Experimentation.TwoStageInterference.WAssign n i → ℝ) → Bool → ℝ","l":66,"f":"Causalean/Experimentation/TwoStageInterference/Basic.lean"},{"n":"Causalean.Experimentation.TwoStageInterference.popMeanV","k":"def","m":"Causalean.Experimentation.TwoStageInterference.Variance","a":"Experimentation/TwoStageInterference","t":1,"d":"Population mean of a unit-indexed quantity: (1/n)∑ⱼ x j.","s":"{n : ℕ} → (Fin n → ℝ) → ℝ","l":120,"f":"Causalean/Experimentation/TwoStageInterference/Variance.lean"},{"n":"Causalean.Experimentation.TwoStageInterference.refSel","k":"def","m":"Causalean.Experimentation.TwoStageInterference.Asymptotic.CLTDischarge","a":"Experimentation/TwoStageInterference/Asymptotic","t":1,"d":"A fixed reference selection in the support of stage 1.","s":"(Exp : ℕ → Causalean.Experimentation.TwoStageInterference.LHExperiment) → (n : ℕ) → Causalean.Experimentation.TwoStageInterference.StratAssign (Exp n).ι","l":310,"f":"Causalean/Experimentation/TwoStageInterference/Asymptotic/CLTDischarge.lean"},{"n":"Causalean.Experimentation.TwoStageInterference.refSel_mem","k":"theorem","m":"Causalean.Experimentation.TwoStageInterference.Asymptotic.CLTDischarge","a":"Experimentation/TwoStageInterference/Asymptotic","t":2,"d":"The fixed reference selection has positive stage-one design probability.","s":"∀ (Exp : ℕ → Causalean.Experimentation.TwoStageInterference.LHExperiment) (n : ℕ), (Exp n).D₁.p (Causalean.Experimentation.TwoStageInterference.refSel Exp n) ≠ 0","l":314,"f":"Causalean/Experimentation/TwoStageInterference/Asymptotic/CLTDischarge.lean"},{"n":"Causalean.Experimentation.TwoStageInterference.stratExpo","k":"def","m":"Causalean.Experimentation.TwoStageInterference.Stratified","a":"Experimentation/TwoStageInterference","t":1,"d":"The stratified-interference exposure of unit (i,j) under within-group assignment w: the pair of its own treatment w j and the number of other treated units in its group. This is the exposure summary through which outcomes are allowed to depend on the assignment.","s":"{ι : Type u_1} → {n : ι → ℕ} → (i : ι) → Fin (n i) → Causalean.Experimentation.TwoStageInterference.WAssign n i → Bool × ℕ","l":54,"f":"Causalean/Experimentation/TwoStageInterference/Stratified.lean"},{"n":"Causalean.Experimentation.TwoStageInterference.stratExpo_exists","k":"theorem","m":"Causalean.Experimentation.TwoStageInterference.Stratified","a":"Experimentation/TwoStageInterference","t":2,"d":"The exposure summary of unit (i,j) is always realized — namely by w itself; so every stratified exposure value reachable from some assignment has a witnessing assignment.","s":"∀ {ι : Type u_1} {n : ι → ℕ} (i : ι) (j : Fin (n i)) (w : Causalean.Experimentation.TwoStageInterference.WAssign n i), ∃ w', Causalean.Experimentation.TwoStageInterference.stratExpo i j w' = Causalean.Experimentation.TwoStageInterference.stratExpo i j w","l":78,"f":"Causalean/Experimentation/TwoStageInterference/Stratified.lean"},{"n":"Causalean.Experimentation.TwoStageInterference.stud_eq_sum_of_homogeneous","k":"theorem","m":"Causalean.Experimentation.TwoStageInterference.Asymptotic.CLTDischarge","a":"Experimentation/TwoStageInterference/Asymptotic","t":1,"d":"Studentized = independent sum. For a stage-1 stratified assignment s that the design supports with positive probability, then, under homogeneity, for every within-group assignment pattern w, the conditional studentized statistic decomposes as the normalized independent sum stud(s,w) = ∑ᵢ cltSummand n δ v s i (w i).","s":"∀ {Exp : ℕ → Causalean.Experimentation.TwoStageInterference.LHExperiment} {t δ M : ℝ} {v : ℕ → ℝ} {stud : (n : ℕ) → Causalean.Experimentation.TwoStageInterference.StratAssign (Exp n).ι × ((i : (Exp n).ι) → Fin ((Exp n).gsize i) → Bool) → ℝ}, Causalean.Experimentation.TwoStageInterference.Homogeneous Exp t stud δ M v → ∀ (n : ℕ) (s : Causalean.Experimentation.TwoStageInterference.StratAssign (Exp n).ι), (Exp n).D₁.p s ≠ 0 → ∀ (w : (i : (Exp n).ι) → Fin ((Exp n).gsize i) → Bool), stud n (s, w) = ∑ i, Causalean.Experimentation.TwoStageInterference.cltSummand n δ v s i (w i)","l":240,"f":"Causalean/Experimentation/TwoStageInterference/Asymptotic/CLTDischarge.lean"},{"n":"Causalean.Experimentation.TwoStageInterference.sum_sub_mean_sq","k":"theorem","m":"Causalean.Experimentation.TwoStageInterference.Variance","a":"Experimentation/TwoStageInterference","t":2,"d":"The sum of squared deviations equals the raw second moment minus the squared first moment divided by m: ∑ⱼ (xⱼ − x̄)² = ∑ⱼ xⱼ² − (∑ⱼ xⱼ)²/m, where x̄ = (∑ x)/m.","s":"∀ {m : ℕ}, 0 < m → ∀ (x : Fin m → ℝ), ∑ j, (x j - (∑ i, x i) / ↑m) ^ 2 = ∑ j, x j ^ 2 - (∑ i, x i) ^ 2 / ↑m","l":85,"f":"Causalean/Experimentation/TwoStageInterference/Variance.lean"},{"n":"Causalean.Experimentation.TwoStageInterference.sum_sum_ite_quadratic","k":"theorem","m":"Causalean.Experimentation.TwoStageInterference.Variance","a":"Experimentation/TwoStageInterference","t":2,"d":"A double sum of cⱼ cₖ weighted by a two-valued kernel (vd on the diagonal, vo off it) collapses to vo·(∑ c)² + (vd − vo)·∑ c². This is the algebraic core that turns the Var_linear_comb double sum into the Neyman split form.","s":"∀ {α : Type u_1} [inst : DecidableEq α] (s : Finset α) (c : α → ℝ) (vd vo : ℝ), (∑ j ∈ s, ∑ k ∈ s, c j * c k * if j = k then vd else vo) = vo * (∑ j ∈ s, c j) ^ 2 + (vd - vo) * ∑ j ∈ s, c j ^ 2","l":67,"f":"Causalean/Experimentation/TwoStageInterference/Variance.lean"},{"n":"Causalean.Experimentation.TwoStageInterference.tauHat","k":"def","m":"Causalean.Experimentation.TwoStageInterference.Variance","a":"Experimentation/TwoStageInterference","t":1,"d":"The difference-in-means estimator ȳ(0) − ȳ(1): the mean untreated-state outcome among the n − K control units minus the mean treated-state outcome among the K treated units, as a function of the realized assignment. On the design's support the treated units realize a and the control units realize b, so this is linear in the treatment indicators.","s":"{n : ℕ} → ℕ → (Fin n → ℝ) → (Fin n → ℝ) → (Fin n → Bool) → ℝ","l":111,"f":"Causalean/Experimentation/TwoStageInterference/Variance.lean"},{"n":"Causalean.Experimentation.TwoStageInterference.varHat","k":"def","m":"Causalean.Experimentation.TwoStageInterference.VarianceConservative","a":"Experimentation/TwoStageInterference","t":1,"d":"Conservative variance estimator v̂ar = Ŝ₁/K + Ŝ₀/(n−K), the empirical analogue of the first two terms of the Neyman variance. Computable from a single realized assignment.","s":"{n : ℕ} → ℕ → (Fin n → ℝ) → (Fin n → ℝ) → (Fin n → Bool) → ℝ","l":76,"f":"Causalean/Experimentation/TwoStageInterference/VarianceConservative.lean"},{"n":"Causalean.Experimentation.TwoStageInterference.varHat_nonneg","k":"theorem","m":"Causalean.Experimentation.TwoStageInterference.VarianceConservative","a":"Experimentation/TwoStageInterference","t":1,"d":"Pointwise nonnegativity of the conservative variance estimator. For a group of n units with potential outcomes a and b, provided at least one unit is treated and at least one unit remains in control (K + 1 ≤ n) — so both within-group sample-variance denominators K and n − K are positive — then for every realized assignment w, the conservative variance estimator v̂ar is nonnegative, being the sum of two nonnegative sample variances each divided by a positive count.","s":"∀ {n : ℕ} (K : ℕ) (a b : Fin n → ℝ), 1 ≤ K → K + 1 ≤ n → ∀ (w : Fin n → Bool), 0 ≤ Causalean.Experimentation.TwoStageInterference.varHat K a b w","l":107,"f":"Causalean/Experimentation/TwoStageInterference/VarianceConservative.lean"},{"n":"Causalean.Experimentation.TwoStageInterference.wald_coverage_feasible","k":"theorem","m":"Causalean.Experimentation.TwoStageInterference.Asymptotic.WaldFeasible","a":"Experimentation/TwoStageInterference/Asymptotic","t":1,"d":"Asymptotic feasible Wald coverage (Liu–Hudgens 2014). Along a sequence of two-stage Hudgens–Halloran experiments Exp, let stud n be the studentized statistic (D̂E − DE̅)/√directVar for the treatment-minus-control direct-effect contrast, and assume its per-threshold CDF converges to the standard normal CDF at every threshold and the design variance is everywhere positive. Let Vh n be an arbitrary variance estimator that is conservative-consistent — for every slack ε > 0 the probability it undershoots (1−ε) times the true variance tends to zero, and let zq ≥ 0 be the standard-normal upper quantile at level γ, Φ(zq) = 1 − γ/2. Then the feasible Wald interval D̂E ± zq·√(Vh) attains asymptotic coverage of DE̅ at least 1 − γ.","s":"∀ (Exp : ℕ → Causalean.Experimentation.TwoStageInterference.LHExperiment) (stud : (n : ℕ) → Causalean.Experimentation.TwoStageInterference.StratAssign (Exp n).ι × ((i : (Exp n).ι) → Fin ((Exp n).gsize i) → Bool) → ℝ), (∀ (n : ℕ) (sw : Causalean.Experimentation.TwoStageInterference.StratAssign (Exp n).ι × ((i : (Exp n).ι) → Fin ((Exp n).gsize i) → Bool)), stud n sw = ((Exp n).estD sw - (Exp n).DEbar) / √(Exp n).directVar) → ∀ (Vh : (n : ℕ) → Causalean.Experimentation.TwoStageInterference.StratAssign (Exp n).ι × ((i : (Exp n).ι) → Fin ((Exp n).gsize i) → Bool) → ℝ), (∀ (t : ℝ), Filter.Tendsto (fun n => (Exp n).jointD.Pr fun sw => stud n sw ≤ t) Filter.atTop (nhds (Causalean.Experimentation.DesignBased.stdNormalCdf t))) → (∀ (n : ℕ), 0 < (Exp n).directVar) → (∀ (ε : ℝ), 0 < ε → Filter.Tendsto (fun n => (Exp n).jointD.Pr fun sw => Vh n sw < (1 - ε) * (Exp n).directVar) Filter.atTop (nhds 0)) → ∀ {γ : ℝ} (zq : ℝ), 0 ≤ zq → Causalean.Experimentation.DesignBased.stdNormalCdf zq = 1 - γ / 2 → 1 - γ ≤ Filter.liminf (fun n => (Exp n).jointD.Pr fun sw => |(Exp n).estD sw - (Exp n).DEbar| ≤ zq * √(Vh n sw)) Filter.atTop","l":123,"f":"Causalean/Experimentation/TwoStageInterference/Asymptotic/WaldFeasible.lean"},{"n":"Causalean.Experimentation.TwoStageInterference.wald_coverage_oracle","k":"theorem","m":"Causalean.Experimentation.TwoStageInterference.Asymptotic.Wald","a":"Experimentation/TwoStageInterference/Asymptotic","t":1,"d":"Asymptotic oracle Wald coverage (Liu–Hudgens 2014). Along a sequence of two-stage Hudgens–Halloran experiments Exp, let stud n be the studentized statistic (D̂E − DE̅)/√directVar for the treatment-minus-control direct-effect contrast, and assume the design variance is everywhere positive. Let zq ≥ 0 be the standard-normal upper quantile at level γ, i.e. Φ(zq) = 1 − γ/2, and assume the two studentized-CDF limits of stud at zq and −zq converge to Φ(zq) and Φ(−zq) — exactly the conclusion of the direct-contrast CLT. Then the oracle Wald interval D̂E ± zq·√directVar attains asymptotic coverage of DE̅ at least 1 − γ.","s":"∀ (Exp : ℕ → Causalean.Experimentation.TwoStageInterference.LHExperiment) (stud : (n : ℕ) → Causalean.Experimentation.TwoStageInterference.StratAssign (Exp n).ι × ((i : (Exp n).ι) → Fin ((Exp n).gsize i) → Bool) → ℝ), (∀ (n : ℕ) (sw : Causalean.Experimentation.TwoStageInterference.StratAssign (Exp n).ι × ((i : (Exp n).ι) → Fin ((Exp n).gsize i) → Bool)), stud n sw = ((Exp n).estD sw - (Exp n).DEbar) / √(Exp n).directVar) → (∀ (n : ℕ), 0 < (Exp n).directVar) → ∀ {γ : ℝ} (zq : ℝ), 0 ≤ zq → Causalean.Experimentation.DesignBased.stdNormalCdf zq = 1 - γ / 2 → Filter.Tendsto (fun n => (Exp n).jointD.Pr fun sw => stud n sw ≤ zq) Filter.atTop (nhds (Causalean.Experimentation.DesignBased.stdNormalCdf zq)) → Filter.Tendsto (fun n => (Exp n).jointD.Pr fun sw => stud n sw ≤ -zq) Filter.atTop (nhds (Causalean.Experimentation.DesignBased.stdNormalCdf (-zq))) → 1 - γ ≤ Filter.liminf (fun n => (Exp n).jointD.Pr fun sw => |(Exp n).estD sw - (Exp n).DEbar| ≤ zq * √(Exp n).directVar) Filter.atTop","l":54,"f":"Causalean/Experimentation/TwoStageInterference/Asymptotic/Wald.lean"},{"n":"Causalean.Experimentation.UnknownInterference.ACATE","k":"def","m":"Causalean.Experimentation.UnknownInterference.Basic","a":"Experimentation/UnknownInterference","t":1,"d":"Assignment-conditional average treatment effect ACATE(z) = n⁻¹ ∑ᵢ τ_i(z_{-i}).","s":"{U : Type u_1} → [Fintype U] → [DecidableEq U] → (U → (U → Bool) → ℝ) → (U → Bool) → ℝ","l":81,"f":"Causalean/Experimentation/UnknownInterference/Basic.lean"},{"n":"Causalean.Experimentation.UnknownInterference.AhatCtrl","k":"def","m":"Causalean.Experimentation.UnknownInterference.Hajek","a":"Experimentation/UnknownInterference","t":1,"d":"The control numerator average Â₀ = n⁻¹ ∑ᵢ (1−Zᵢ) Yᵢ / (1−pᵢ).","s":"{U : Type u_1} → [Fintype U] → (U → ℝ) → (U → (U → Bool) → ℝ) → (U → Bool) → ℝ","l":76,"f":"Causalean/Experimentation/UnknownInterference/Hajek.lean"},{"n":"Causalean.Experimentation.UnknownInterference.AhatTreat","k":"def","m":"Causalean.Experimentation.UnknownInterference.Hajek","a":"Experimentation/UnknownInterference","t":1,"d":"The treated numerator average Â₁ = n⁻¹ ∑ᵢ Zᵢ Yᵢ / pᵢ.","s":"{U : Type u_1} → [Fintype U] → (U → ℝ) → (U → (U → Bool) → ℝ) → (U → Bool) → ℝ","l":72,"f":"Causalean/Experimentation/UnknownInterference/Hajek.lean"},{"n":"Causalean.Experimentation.UnknownInterference.BhatCtrl","k":"def","m":"Causalean.Experimentation.UnknownInterference.Hajek","a":"Experimentation/UnknownInterference","t":1,"d":"The control weight average B̂₀ = n⁻¹ ∑ᵢ (1−Zᵢ) / (1−pᵢ).","s":"{U : Type u_1} → [Fintype U] → (U → ℝ) → (U → Bool) → ℝ","l":84,"f":"Causalean/Experimentation/UnknownInterference/Hajek.lean"},{"n":"Causalean.Experimentation.UnknownInterference.BhatTreat","k":"def","m":"Causalean.Experimentation.UnknownInterference.Hajek","a":"Experimentation/UnknownInterference","t":1,"d":"The treated weight average B̂₁ = n⁻¹ ∑ᵢ Zᵢ / pᵢ.","s":"{U : Type u_1} → [Fintype U] → (U → ℝ) → (U → Bool) → ℝ","l":80,"f":"Causalean/Experimentation/UnknownInterference/Hajek.lean"},{"n":"Causalean.Experimentation.UnknownInterference.EATE","k":"def","m":"Causalean.Experimentation.UnknownInterference.Basic","a":"Experimentation/UnknownInterference","t":1,"d":"Expected average treatment effect (Definition, Sävje–Aronow–Hudgens 2021): EATE = E[ACATE(Z)], the design average of the assignment-conditional ATE. It generalizes the conventional ATE — under no interference ACATE(z) is constant in z, so the marginalization is inconsequential and EATE = ATE.","s":"{U : Type u_1} → [inst : Fintype U] → [inst_1 : DecidableEq U] → Causalean.Experimentation.DesignBased.FiniteDesign (U → Bool) → (U → (U → Bool) → ℝ) → ℝ","l":85,"f":"Causalean/Experimentation/UnknownInterference/Basic.lean"},{"n":"Causalean.Experimentation.UnknownInterference.E_AhatTreat_sub_E_AhatCtrl","k":"theorem","m":"Causalean.Experimentation.UnknownInterference.Hajek","a":"Experimentation/UnknownInterference","t":2,"d":"The difference of the numerator means is EATE: E[Â₁] − E[Â₀] = EATE (the HT estimator is Â₁ − Â₀, and it is unbiased).","s":"∀ {U : Type u_1} [inst : Fintype U] [inst_1 : DecidableEq U] (p : U → ℝ) (hp0 : ∀ (i : U), 0 ≤ p i) (hp1 : ∀ (i : U), p i ≤ 1), (∀ (i : U), p i ≠ 0) → (∀ (i : U), 1 - p i ≠ 0) → ∀ (y : U → (U → Bool) → ℝ), (Causalean.Experimentation.UnknownInterference.bernoulliDesign p hp0 hp1).E (Causalean.Experimentation.UnknownInterference.AhatTreat p y) - (Causalean.Experimentation.UnknownInterference.bernoulliDesign p hp0 hp1).E (Causalean.Experimentation.UnknownInterference.AhatCtrl p y) = Causalean.Experimentation.UnknownInterference.EATE (Causalean.Experimentation.UnknownInterference.bernoulliDesign p hp0 hp1) y","l":663,"f":"Causalean/Experimentation/UnknownInterference/Hajek.lean"},{"n":"Causalean.Experimentation.UnknownInterference.E_BhatCtrl","k":"theorem","m":"Causalean.Experimentation.UnknownInterference.Hajek","a":"Experimentation/UnknownInterference","t":2,"d":"With nonzero control propensities and a nonempty population, the control weight average has mean exactly one: E[B̂₀] = 1.","s":"∀ {U : Type u_1} [inst : Fintype U] [inst_1 : DecidableEq U] (p : U → ℝ) (hp0 : ∀ (i : U), 0 ≤ p i) (hp1 : ∀ (i : U), p i ≤ 1), (∀ (i : U), 1 - p i ≠ 0) → 1 ≤ Fintype.card U → (Causalean.Experimentation.UnknownInterference.bernoulliDesign p hp0 hp1).E (Causalean.Experimentation.UnknownInterference.BhatCtrl p) = 1","l":120,"f":"Causalean/Experimentation/UnknownInterference/Hajek.lean"},{"n":"Causalean.Experimentation.UnknownInterference.E_BhatTreat","k":"theorem","m":"Causalean.Experimentation.UnknownInterference.Hajek","a":"Experimentation/UnknownInterference","t":2,"d":"With nonzero treatment propensities and a nonempty population, the treated weight average has mean exactly one: E[B̂₁] = 1.","s":"∀ {U : Type u_1} [inst : Fintype U] [inst_1 : DecidableEq U] (p : U → ℝ) (hp0 : ∀ (i : U), 0 ≤ p i) (hp1 : ∀ (i : U), p i ≤ 1), (∀ (i : U), p i ≠ 0) → 1 ≤ Fintype.card U → (Causalean.Experimentation.UnknownInterference.bernoulliDesign p hp0 hp1).E (Causalean.Experimentation.UnknownInterference.BhatTreat p) = 1","l":95,"f":"Causalean/Experimentation/UnknownInterference/Hajek.lean"},{"n":"Causalean.Experimentation.UnknownInterference.E_VhatBer_bias","k":"theorem","m":"Causalean.Experimentation.UnknownInterference.Confidence","a":"Experimentation/UnknownInterference","t":1,"d":"The anti-conservativeness mechanism (Sävje–Aronow–Hudgens 2021). Under the Bernoulli design with per-unit treatment probabilities p taking values in [0, 1], the conventional variance estimator's expected value equals Var(ĤT) + n⁻²·(∑ᵢ(E ĤTᵢ)² − ∑ᵢ∑_{j≠i} Cov(ĤTᵢ,ĤTⱼ)): the true sampling variance of the Horvitz–Thompson estimator, plus the average squared per-unit mean, minus the off-diagonal covariances between units' Horvitz–Thompson summands.","s":"∀ {U : Type u_1} [inst : Fintype U] [inst_1 : DecidableEq U] (p : U → ℝ) (hp0 : ∀ (i : U), 0 ≤ p i) (hp1 : ∀ (i : U), p i ≤ 1) (y : U → (U → Bool) → ℝ), (Causalean.Experimentation.UnknownInterference.bernoulliDesign p hp0 hp1).E (Causalean.Experimentation.UnknownInterference.VhatBer p y) = (Causalean.Experimentation.UnknownInterference.bernoulliDesign p hp0 hp1).Var (Causalean.Experimentation.UnknownInterference.htEst p y) + (∑ i, (Causalean.Experimentation.UnknownInterference.bernoulliDesign p hp0 hp1).E (Causalean.Experimentation.UnknownInterference.htSummand p y i) ^ 2 - ∑ i, ∑ j ∈ Finset.univ.erase i, (Causalean.Experimentation.UnknownInterference.bernoulliDesign p hp0 hp1).Cov (Causalean.Experimentation.UnknownInterference.htSummand p y i) (Causalean.Experimentation.UnknownInterference.htSummand p y j)) / ↑(Fintype.card U) ^ 2","l":108,"f":"Causalean/Experimentation/UnknownInterference/Confidence.lean"},{"n":"Causalean.Experimentation.UnknownInterference.E_htSummand","k":"theorem","m":"Causalean.Experimentation.UnknownInterference.Unbiased","a":"Experimentation/UnknownInterference","t":1,"d":"Per-unit unbiasedness. Under a Bernoulli design in which every unit's treatment probability lies in [0,1] and is neither exactly zero nor exactly one, so both the treatment and control propensities are nonzero, the i-th Horvitz–Thompson summand has the same expectation as unit i's assignment-conditional treatment effect τ_i.","s":"∀ {U : Type u_1} [inst : Fintype U] [inst_1 : DecidableEq U] (p : U → ℝ) (hp0 : ∀ (i : U), 0 ≤ p i) (hp1 : ∀ (i : U), p i ≤ 1), (∀ (i : U), p i ≠ 0) → (∀ (i : U), 1 - p i ≠ 0) → ∀ (y : U → (U → Bool) → ℝ) (i : U), (Causalean.Experimentation.UnknownInterference.bernoulliDesign p hp0 hp1).E (Causalean.Experimentation.UnknownInterference.htSummand p y i) = (Causalean.Experimentation.UnknownInterference.bernoulliDesign p hp0 hp1).E (Causalean.Experimentation.UnknownInterference.tau y i)","l":49,"f":"Causalean/Experimentation/UnknownInterference/Unbiased.lean"},{"n":"Causalean.Experimentation.UnknownInterference.InterfDep","k":"def","m":"Causalean.Experimentation.UnknownInterference.Basic","a":"Experimentation/UnknownInterference","t":1,"d":"Interference dependence. Units i and j are interference dependent if some unit ℓ interferes with both — i.e. they may be affected by a common treatment. The paper's d̄_{ij}.","s":"{U : Type u_1} → [DecidableEq U] → (U → (U → Bool) → ℝ) → U → U → Prop","l":56,"f":"Causalean/Experimentation/UnknownInterference/Basic.lean"},{"n":"Causalean.Experimentation.UnknownInterference.Interferes","k":"def","m":"Causalean.Experimentation.UnknownInterference.Basic","a":"Experimentation/UnknownInterference","t":1,"d":"Interference indicator. Unit ℓ *interferes with* unit i if changing ℓ's treatment changes i's outcome under some assignment, or if ℓ = i (a unit always interferes with itself). y i z is unit i's outcome under the full assignment z.","s":"{U : Type u_1} → [DecidableEq U] → (U → (U → Bool) → ℝ) → U → U → Prop","l":50,"f":"Causalean/Experimentation/UnknownInterference/Basic.lean"},{"n":"Causalean.Experimentation.UnknownInterference.SAHExperiment","k":"structure","m":"Causalean.Experimentation.UnknownInterference.Consistency","a":"Experimentation/UnknownInterference","t":1,"d":"A single Sävje–Aronow–Hudgens Bernoulli experiment, packaged so that a sequence of them models the growing-sample regime. Carries a finite population of units, marginal treatment probabilities, and potential outcomes indexed by the full assignment vector, together with an overlap regularity constant k at least one; the conditions that there is at least one unit, every treatment probability lies between 0 and 1, is bounded below by 1/k and above by 1 − 1/k (overlap); and every unit's potential outcome has second moment at most k² under the resulting Bernoulli design.","s":"Type 1","l":52,"f":"Causalean/Experimentation/UnknownInterference/Consistency.lean"},{"n":"Causalean.Experimentation.UnknownInterference.SAHExperiment.D","k":"def","m":"Causalean.Experimentation.UnknownInterference.Consistency","a":"Experimentation/UnknownInterference","t":1,"d":"The Bernoulli design of the experiment.","s":"(E : Causalean.Experimentation.UnknownInterference.SAHExperiment) → Causalean.Experimentation.DesignBased.FiniteDesign (E.U → Bool)","l":92,"f":"Causalean/Experimentation/UnknownInterference/Consistency.lean"},{"n":"Causalean.Experimentation.UnknownInterference.SAHExperiment.D_E_htEst","k":"theorem","m":"Causalean.Experimentation.UnknownInterference.Consistency","a":"Experimentation/UnknownInterference","t":1,"d":"Unbiasedness (bundle form). The Horvitz–Thompson estimator's expectation under the experiment's Bernoulli design equals the experiment's EATE estimand.","s":"∀ (E : Causalean.Experimentation.UnknownInterference.SAHExperiment), E.D.E (Causalean.Experimentation.UnknownInterference.htEst E.p E.y) = E.eate","l":113,"f":"Causalean/Experimentation/UnknownInterference/Consistency.lean"},{"n":"Causalean.Experimentation.UnknownInterference.SAHExperiment.D_Var_htEst_le","k":"theorem","m":"Causalean.Experimentation.UnknownInterference.Consistency","a":"Experimentation/UnknownInterference","t":1,"d":"Variance bound (bundle form). The Horvitz–Thompson estimator's variance is at most the fourth power of the regularity constant times the average interference degree, divided by the population size.","s":"∀ (E : Causalean.Experimentation.UnknownInterference.SAHExperiment), E.D.Var (Causalean.Experimentation.UnknownInterference.htEst E.p E.y) ≤ E.k ^ 4 * Causalean.Experimentation.UnknownInterference.dbar E.y / ↑(Fintype.card E.U)","l":120,"f":"Causalean/Experimentation/UnknownInterference/Consistency.lean"},{"n":"Causalean.Experimentation.UnknownInterference.SAHExperiment.chebyshev_eate","k":"theorem","m":"Causalean.Experimentation.UnknownInterference.Consistency","a":"Experimentation/UnknownInterference","t":1,"d":"Chebyshev tail bound. For any deviation threshold ε > 0, the probability that the Horvitz–Thompson estimator deviates from the EATE by at least ε is at most (k⁴·d̄/n)/ε², where k is the experiment's regularity constant, d̄ is the average interference degree, and n is the population size.","s":"∀ (E : Causalean.Experimentation.UnknownInterference.SAHExperiment) {ε : ℝ}, 0 < ε → (E.D.Pr fun z => ε ≤ |Causalean.Experimentation.UnknownInterference.htEst E.p E.y z - E.eate|) ≤ E.k ^ 4 * Causalean.Experimentation.UnknownInterference.dbar E.y / ↑(Fintype.card E.U) / ε ^ 2","l":128,"f":"Causalean/Experimentation/UnknownInterference/Consistency.lean"},{"n":"Causalean.Experimentation.UnknownInterference.SAHExperiment.eate","k":"def","m":"Causalean.Experimentation.UnknownInterference.Consistency","a":"Experimentation/UnknownInterference","t":1,"d":"The EATE estimand of the experiment.","s":"Causalean.Experimentation.UnknownInterference.SAHExperiment → ℝ","l":95,"f":"Causalean/Experimentation/UnknownInterference/Consistency.lean"},{"n":"Causalean.Experimentation.UnknownInterference.SAHExperiment.one_sub_p_ne_zero","k":"theorem","m":"Causalean.Experimentation.UnknownInterference.Consistency","a":"Experimentation/UnknownInterference","t":2,"d":"One minus the treatment probability is nonzero (from upper overlap).","s":"∀ (E : Causalean.Experimentation.UnknownInterference.SAHExperiment) (i : E.U), 1 - E.p i ≠ 0","l":104,"f":"Causalean/Experimentation/UnknownInterference/Consistency.lean"},{"n":"Causalean.Experimentation.UnknownInterference.SAHExperiment.p_ne_zero","k":"theorem","m":"Causalean.Experimentation.UnknownInterference.Consistency","a":"Experimentation/UnknownInterference","t":2,"d":"The treatment probabilities are nonzero (from lower overlap).","s":"∀ (E : Causalean.Experimentation.UnknownInterference.SAHExperiment) (i : E.U), E.p i ≠ 0","l":98,"f":"Causalean/Experimentation/UnknownInterference/Consistency.lean"},{"n":"Causalean.Experimentation.UnknownInterference.VhatBer","k":"def","m":"Causalean.Experimentation.UnknownInterference.Confidence","a":"Experimentation/UnknownInterference","t":1,"d":"The conventional Horvitz–Thompson variance estimator V̂_Ber = n⁻² ∑ᵢ ĤTᵢ². Pointwise this equals the paper's n⁻²[∑ᵢ ZᵢYᵢ²/pᵢ² + ∑ᵢ(1−Zᵢ)Yᵢ²/(1−pᵢ)²] (the cross term vanishes since Zᵢ(1−Zᵢ)=0).","s":"{U : Type u_1} → [Fintype U] → (U → ℝ) → (U → (U → Bool) → ℝ) → (U → Bool) → ℝ","l":63,"f":"Causalean/Experimentation/UnknownInterference/Confidence.lean"},{"n":"Causalean.Experimentation.UnknownInterference.abs_E_AhatCtrl_le","k":"theorem","m":"Causalean.Experimentation.UnknownInterference.Hajek","a":"Experimentation/UnknownInterference","t":2,"d":"The control numerator mean is the average control potential outcome, uniformly bounded by k.","s":"∀ {U : Type u_1} [inst : Fintype U] [inst_1 : DecidableEq U] (p : U → ℝ) (hp0 : ∀ (i : U), 0 ≤ p i) (hp1 : ∀ (i : U), p i ≤ 1), (∀ (i : U), 1 - p i ≠ 0) → ∀ (y : U → (U → Bool) → ℝ) (k : ℝ), 1 ≤ k → 1 ≤ Fintype.card U → (∀ (i : U), ((Causalean.Experimentation.UnknownInterference.bernoulliDesign p hp0 hp1).E fun z => |y i (Function.update z i false)|) ≤ k) → |(Causalean.Experimentation.UnknownInterference.bernoulliDesign p hp0 hp1).E (Causalean.Experimentation.UnknownInterference.AhatCtrl p y)| ≤ k","l":632,"f":"Causalean/Experimentation/UnknownInterference/Hajek.lean"},{"n":"Causalean.Experimentation.UnknownInterference.abs_E_AhatTreat_le","k":"theorem","m":"Causalean.Experimentation.UnknownInterference.Hajek","a":"Experimentation/UnknownInterference","t":2,"d":"The treated numerator mean is the average treated potential outcome, uniformly bounded by k via the potential-outcome moment bound (Assumption C).","s":"∀ {U : Type u_1} [inst : Fintype U] [inst_1 : DecidableEq U] (p : U → ℝ) (hp0 : ∀ (i : U), 0 ≤ p i) (hp1 : ∀ (i : U), p i ≤ 1), (∀ (i : U), p i ≠ 0) → ∀ (y : U → (U → Bool) → ℝ) (k : ℝ), 1 ≤ k → 1 ≤ Fintype.card U → (∀ (i : U), ((Causalean.Experimentation.UnknownInterference.bernoulliDesign p hp0 hp1).E fun z => |y i (Function.update z i true)|) ≤ k) → |(Causalean.Experimentation.UnknownInterference.bernoulliDesign p hp0 hp1).E (Causalean.Experimentation.UnknownInterference.AhatTreat p y)| ≤ k","l":599,"f":"Causalean/Experimentation/UnknownInterference/Hajek.lean"},{"n":"Causalean.Experimentation.UnknownInterference.bernoulliDesign","k":"def","m":"Causalean.Experimentation.UnknownInterference.Bernoulli","a":"Experimentation/UnknownInterference","t":1,"d":"The Bernoulli randomization design: each unit i is independently assigned treatment with probability p i. Built as the product of the per-unit coin designs.","s":"{U : Type u_1} → [inst : Fintype U] → [inst_1 : DecidableEq U] → (p : U → ℝ) → (∀ (i : U), 0 ≤ p i) → (∀ (i : U), p i ≤ 1) → Causalean.Experimentation.DesignBased.FiniteDesign (U → Bool)","l":44,"f":"Causalean/Experimentation/UnknownInterference/Bernoulli.lean"},{"n":"Causalean.Experimentation.UnknownInterference.bernoulliDesign_E_ctrl","k":"theorem","m":"Causalean.Experimentation.UnknownInterference.Bernoulli","a":"Experimentation/UnknownInterference","t":1,"d":"Marginal control-indicator expectation under the Bernoulli design. Under the Bernoulli design with per-unit treatment probabilities p taking values in [0, 1], the expected value of the indicator that unit i is untreated equals 1 - p i.","s":"∀ {U : Type u_1} [inst : Fintype U] [inst_1 : DecidableEq U] (p : U → ℝ) (hp0 : ∀ (i : U), 0 ≤ p i) (hp1 : ∀ (i : U), p i ≤ 1) (i : U), ((Causalean.Experimentation.UnknownInterference.bernoulliDesign p hp0 hp1).E fun z => if z i = true then 0 else 1) = 1 - p i","l":69,"f":"Causalean/Experimentation/UnknownInterference/Bernoulli.lean"},{"n":"Causalean.Experimentation.UnknownInterference.bernoulliDesign_E_eval","k":"theorem","m":"Causalean.Experimentation.UnknownInterference.Bernoulli","a":"Experimentation/UnknownInterference","t":1,"d":"Marginalizing the Bernoulli design to a single coordinate. Under the Bernoulli design with per-unit treatment probabilities p taking values in [0, 1], the expected value of any function g of a single unit i's treatment status coincides with its expectation under that unit's own coin design with success probability p i — the Zᵢ ⊥ Z₋ᵢ marginalization.","s":"∀ {U : Type u_1} [inst : Fintype U] [inst_1 : DecidableEq U] (p : U → ℝ) (hp0 : ∀ (i : U), 0 ≤ p i) (hp1 : ∀ (i : U), p i ≤ 1) (i : U) (g : Bool → ℝ), ((Causalean.Experimentation.UnknownInterference.bernoulliDesign p hp0 hp1).E fun z => g (z i)) = (Causalean.Experimentation.DesignBased.coinDesign (p i) (hp0 i) (hp1 i)).E g","l":50,"f":"Causalean/Experimentation/UnknownInterference/Bernoulli.lean"},{"n":"Causalean.Experimentation.UnknownInterference.bernoulliDesign_E_treat","k":"theorem","m":"Causalean.Experimentation.UnknownInterference.Bernoulli","a":"Experimentation/UnknownInterference","t":1,"d":"Marginal treatment-indicator expectation under the Bernoulli design. Under the Bernoulli design with per-unit treatment probabilities p taking values in [0, 1], the expected value of the indicator that unit i is treated equals p i.","s":"∀ {U : Type u_1} [inst : Fintype U] [inst_1 : DecidableEq U] (p : U → ℝ) (hp0 : ∀ (i : U), 0 ≤ p i) (hp1 : ∀ (i : U), p i ≤ 1) (i : U), ((Causalean.Experimentation.UnknownInterference.bernoulliDesign p hp0 hp1).E fun z => if z i = true then 1 else 0) = p i","l":61,"f":"Causalean/Experimentation/UnknownInterference/Bernoulli.lean"},{"n":"Causalean.Experimentation.UnknownInterference.chebyshev_ci_eate","k":"theorem","m":"Causalean.Experimentation.UnknownInterference.Confidence","a":"Experimentation/UnknownInterference","t":1,"d":"Chebyshev confidence interval for EATE. For the Bernoulli design with per-unit treatment probabilities p that take values in [0, 1] and are never exactly zero or one, for any value V that is positive and bounds the Horvitz–Thompson estimator's true sampling variance from above, and for any positive significance level α, the interval ĤT ± √(V/α) covers the EATE estimand with probability at least 1 − α.","s":"∀ {U : Type u_1} [inst : Fintype U] [inst_1 : DecidableEq U] (p : U → ℝ) (hp0 : ∀ (i : U), 0 ≤ p i) (hp1 : ∀ (i : U), p i ≤ 1), (∀ (i : U), p i ≠ 0) → (∀ (i : U), 1 - p i ≠ 0) → ∀ (y : U → (U → Bool) → ℝ) {V α : ℝ}, 0 < V → (Causalean.Experimentation.UnknownInterference.bernoulliDesign p hp0 hp1).Var (Causalean.Experimentation.UnknownInterference.htEst p y) ≤ V → 0 < α → 1 - α ≤ (Causalean.Experimentation.UnknownInterference.bernoulliDesign p hp0 hp1).Pr fun z => |Causalean.Experimentation.UnknownInterference.htEst p y z - Causalean.Experimentation.UnknownInterference.EATE (Causalean.Experimentation.UnknownInterference.bernoulliDesign p hp0 hp1) y| ≤ √(V / α)","l":262,"f":"Causalean/Experimentation/UnknownInterference/Confidence.lean"},{"n":"Causalean.Experimentation.UnknownInterference.cov_htSummand_zero","k":"theorem","m":"Causalean.Experimentation.UnknownInterference.VarianceBound","a":"Experimentation/UnknownInterference","t":1,"d":"No covariance off the interference-dependence graph. Under a Bernoulli design in which each unit i is treated independently with probability p i, where every p i lies in the unit interval, if units i and j are not interference dependent — no unit's treatment affects both units' outcomes — then their Horvitz–Thompson summands have zero covariance under this design.","s":"∀ {U : Type u_1} [inst : Fintype U] [inst_1 : DecidableEq U] (p : U → ℝ) (hp0 : ∀ (i : U), 0 ≤ p i) (hp1 : ∀ (i : U), p i ≤ 1) (y : U → (U → Bool) → ℝ) {i j : U}, ¬Causalean.Experimentation.UnknownInterference.InterfDep y i j → (Causalean.Experimentation.UnknownInterference.bernoulliDesign p hp0 hp1).Cov (Causalean.Experimentation.UnknownInterference.htSummand p y i) (Causalean.Experimentation.UnknownInterference.htSummand p y j) = 0","l":89,"f":"Causalean/Experimentation/UnknownInterference/VarianceBound.lean"},{"n":"Causalean.Experimentation.UnknownInterference.dbar","k":"def","m":"Causalean.Experimentation.UnknownInterference.Basic","a":"Experimentation/UnknownInterference","t":1,"d":"Average interference dependence d̄ = n⁻¹ ∑ᵢ ∑ⱼ 1[InterfDep i j] — the paper's basic measure of the amount of interference. For a nonempty population, d̄ = 1 under no interference; d̄ = n when every pair is interference dependent. \"Restricted interference\" is the assumption d̄ = o(n).","s":"{U : Type u_1} → [Fintype U] → [DecidableEq U] → (U → (U → Bool) → ℝ) → ℝ","l":66,"f":"Causalean/Experimentation/UnknownInterference/Basic.lean"},{"n":"Causalean.Experimentation.UnknownInterference.dbarCount","k":"def","m":"Causalean.Experimentation.UnknownInterference.Basic","a":"Experimentation/UnknownInterference","t":1,"d":"The (unnormalized) count ∑ᵢ ∑ⱼ 1[InterfDep i j] of interference-dependent ordered pairs.","s":"{U : Type u_1} → [Fintype U] → [DecidableEq U] → (U → (U → Bool) → ℝ) → ℝ","l":62,"f":"Causalean/Experimentation/UnknownInterference/Basic.lean"},{"n":"Causalean.Experimentation.UnknownInterference.degDep","k":"def","m":"Causalean.Experimentation.UnknownInterference.Confidence","a":"Experimentation/UnknownInterference","t":1,"d":"The interference degree of unit i: the number of units interference-dependent with i (d̄ᵢ = ∑ⱼ 1[InterfDep i j]).","s":"{U : Type u_1} → [Fintype U] → [DecidableEq U] → (U → (U → Bool) → ℝ) → U → ℝ","l":70,"f":"Causalean/Experimentation/UnknownInterference/Confidence.lean"},{"n":"Causalean.Experimentation.UnknownInterference.disjoint_interferers_of_not_interfDep","k":"theorem","m":"Causalean.Experimentation.UnknownInterference.VarianceBound","a":"Experimentation/UnknownInterference","t":2,"d":"If i and j are not interference dependent, their interferer blocks are disjoint.","s":"∀ {U : Type u_1} [inst : Fintype U] [inst_1 : DecidableEq U] (y : U → (U → Bool) → ℝ) {i j : U}, ¬Causalean.Experimentation.UnknownInterference.InterfDep y i j → Disjoint (Causalean.Experimentation.UnknownInterference.interferers y i) (Causalean.Experimentation.UnknownInterference.interferers y j)","l":78,"f":"Causalean/Experimentation/UnknownInterference/VarianceBound.lean"},{"n":"Causalean.Experimentation.UnknownInterference.eate_ci_kbound","k":"theorem","m":"Causalean.Experimentation.UnknownInterference.Confidence","a":"Experimentation/UnknownInterference","t":1,"d":"A concrete finite-sample confidence interval for EATE (Sävje–Aronow–Hudgens 2021). Suppose the Bernoulli design has treatment probabilities p that lie in [0, 1], are never exactly zero or one, and in fact stay within [1/k, 1 - 1/k] for some regularity constant k ≥ 1; suppose also that the population is nonempty, every unit's outcome has second moment at most k², and the significance level α is positive. Then the Chebyshev interval ĤT ± √(k⁴·d̄/(n·α)), where d̄ is the average interference degree and n the population size, covers the EATE estimand with probability at least 1 − α — a valid (conservative) interval that needs only the regularity constant k and the interference measure d̄.","s":"∀ {U : Type u_1} [inst : Fintype U] [inst_1 : DecidableEq U] (p : U → ℝ) (hp0 : ∀ (i : U), 0 ≤ p i) (hp1 : ∀ (i : U), p i ≤ 1), (∀ (i : U), p i ≠ 0) → (∀ (i : U), 1 - p i ≠ 0) → ∀ (y : U → (U → Bool) → ℝ) (k : ℝ), 1 ≤ k → 1 ≤ Fintype.card U → (∀ (i : U), k⁻¹ ≤ p i) → (∀ (i : U), p i ≤ 1 - k⁻¹) → (∀ (i : U), ((Causalean.Experimentation.UnknownInterference.bernoulliDesign p hp0 hp1).E fun z => y i z ^ 2) ≤ k ^ 2) → ∀ {α : ℝ}, 0 < α → 1 - α ≤ (Causalean.Experimentation.UnknownInterference.bernoulliDesign p hp0 hp1).Pr fun z => |Causalean.Experimentation.UnknownInterference.htEst p y z - Causalean.Experimentation.UnknownInterference.EATE (Causalean.Experimentation.UnknownInterference.bernoulliDesign p hp0 hp1) y| ≤ √(k ^ 4 * Causalean.Experimentation.UnknownInterference.dbar y / (↑(Fintype.card U) * α))","l":339,"f":"Causalean/Experimentation/UnknownInterference/Confidence.lean"},{"n":"Causalean.Experimentation.UnknownInterference.hajekEst","k":"def","m":"Causalean.Experimentation.UnknownInterference.Hajek","a":"Experimentation/UnknownInterference","t":1,"d":"The Hájek estimator ĤA = Â₁/B̂₁ − Â₀/B̂₀, the realized-weight-normalized inverse-probability-weighted estimator.","s":"{U : Type u_1} → [Fintype U] → (U → ℝ) → (U → (U → Bool) → ℝ) → (U → Bool) → ℝ","l":88,"f":"Causalean/Experimentation/UnknownInterference/Hajek.lean"},{"n":"Causalean.Experimentation.UnknownInterference.hajek_consistent_eate","k":"theorem","m":"Causalean.Experimentation.UnknownInterference.Hajek","a":"Experimentation/UnknownInterference","t":1,"d":"Consistency of the Hájek estimator for EATE (Sävje–Aronow–Hudgens 2021). Along a sequence of Bernoulli experiments for which the regularity constants are uniformly bounded above by a constant M, k⁴·d̄/n → 0 along the sequence — restricted interference, and the mean absolute treated and control potential outcomes are each bounded by that experiment's own regularity constant k (the paper's Assumption C), the Hájek estimator converges in probability to the EATE.","s":"∀ (Exp : ℕ → Causalean.Experimentation.UnknownInterference.SAHExperiment) (M : ℝ), (∀ (m : ℕ), (Exp m).k ≤ M) → Filter.Tendsto (fun m => (Exp m).k ^ 4 * Causalean.Experimentation.UnknownInterference.dbar (Exp m).y / ↑(Fintype.card (Exp m).U)) Filter.atTop (nhds 0) → (∀ (m : ℕ) (i : (Exp m).U), ((Exp m).D.E fun z => |(Exp m).y i (Function.update z i true)|) ≤ (Exp m).k) → (∀ (m : ℕ) (i : (Exp m).U), ((Exp m).D.E fun z => |(Exp m).y i (Function.update z i false)|) ≤ (Exp m).k) → Causalean.Experimentation.DesignBased.FiniteDesign.TendstoInProb (fun m => (Exp m).D) (fun m => Causalean.Experimentation.UnknownInterference.hajekEst (Exp m).p (Exp m).y) fun m => (Exp m).eate","l":706,"f":"Causalean/Experimentation/UnknownInterference/Hajek.lean"},{"n":"Causalean.Experimentation.UnknownInterference.htCtrlSummand","k":"def","m":"Causalean.Experimentation.UnknownInterference.Hajek","a":"Experimentation/UnknownInterference","t":1,"d":"Control Horvitz–Thompson summand (1 − Zᵢ) Yᵢ / (1 − pᵢ).","s":"{U : Type u_1} → (U → ℝ) → (U → (U → Bool) → ℝ) → U → (U → Bool) → ℝ","l":60,"f":"Causalean/Experimentation/UnknownInterference/Hajek.lean"},{"n":"Causalean.Experimentation.UnknownInterference.htEst","k":"def","m":"Causalean.Experimentation.UnknownInterference.Basic","a":"Experimentation/UnknownInterference","t":1,"d":"Horvitz–Thompson estimator htEst = n⁻¹ ∑ᵢ [Z_i Y_i / p_i − (1 − Z_i) Y_i / (1 − p_i)], with p i the marginal treatment probability of unit i.","s":"{U : Type u_1} → [Fintype U] → (U → ℝ) → (U → (U → Bool) → ℝ) → (U → Bool) → ℝ","l":100,"f":"Causalean/Experimentation/UnknownInterference/Basic.lean"},{"n":"Causalean.Experimentation.UnknownInterference.htEst_consistent_eate","k":"theorem","m":"Causalean.Experimentation.UnknownInterference.Consistency","a":"Experimentation/UnknownInterference","t":1,"d":"Consistency of Horvitz–Thompson for EATE under unknown interference (Sävje–Aronow–Hudgens 2021). Along a sequence of Bernoulli experiments along which k⁴·d̄/n → 0 — restricted interference with a controlled regularity constant, for every fixed deviation threshold ε > 0, the probability that the Horvitz–Thompson estimator deviates from the EATE by at least ε tends to zero along the sequence, i.e. the HT estimator converges in probability to the EATE.","s":"∀ (Exp : ℕ → Causalean.Experimentation.UnknownInterference.SAHExperiment), Filter.Tendsto (fun m => (Exp m).k ^ 4 * Causalean.Experimentation.UnknownInterference.dbar (Exp m).y / ↑(Fintype.card (Exp m).U)) Filter.atTop (nhds 0) → ∀ {ε : ℝ}, 0 < ε → Filter.Tendsto (fun m => (Exp m).D.Pr fun z => ε ≤ |Causalean.Experimentation.UnknownInterference.htEst (Exp m).p (Exp m).y z - (Exp m).eate|) Filter.atTop (nhds 0)","l":143,"f":"Causalean/Experimentation/UnknownInterference/Consistency.lean"},{"n":"Causalean.Experimentation.UnknownInterference.htEst_unbiased","k":"theorem","m":"Causalean.Experimentation.UnknownInterference.Unbiased","a":"Experimentation/UnknownInterference","t":1,"d":"Horvitz–Thompson unbiasedness for EATE (Sävje–Aronow–Hudgens 2021). Under a Bernoulli design in which every unit's treatment probability lies in [0,1] and is neither exactly zero nor exactly one, so both the treatment and control propensities are nonzero, the Horvitz–Thompson estimator is exactly unbiased for the expected average treatment effect.","s":"∀ {U : Type u_1} [inst : Fintype U] [inst_1 : DecidableEq U] (p : U → ℝ) (hp0 : ∀ (i : U), 0 ≤ p i) (hp1 : ∀ (i : U), p i ≤ 1), (∀ (i : U), p i ≠ 0) → (∀ (i : U), 1 - p i ≠ 0) → ∀ (y : U → (U → Bool) → ℝ), (Causalean.Experimentation.UnknownInterference.bernoulliDesign p hp0 hp1).E (Causalean.Experimentation.UnknownInterference.htEst p y) = Causalean.Experimentation.UnknownInterference.EATE (Causalean.Experimentation.UnknownInterference.bernoulliDesign p hp0 hp1) y","l":150,"f":"Causalean/Experimentation/UnknownInterference/Unbiased.lean"},{"n":"Causalean.Experimentation.UnknownInterference.htSummand","k":"def","m":"Causalean.Experimentation.UnknownInterference.Basic","a":"Experimentation/UnknownInterference","t":1,"d":"The iᵗʰ Horvitz–Thompson summand Z_i Y_i / p_i − (1 − Z_i) Y_i / (1 − p_i), with Z_i = 1[z i] and Y_i = y i z.","s":"{U : Type u_1} → (U → ℝ) → (U → (U → Bool) → ℝ) → U → (U → Bool) → ℝ","l":94,"f":"Causalean/Experimentation/UnknownInterference/Basic.lean"},{"n":"Causalean.Experimentation.UnknownInterference.htSummand_depends_on_interferers","k":"theorem","m":"Causalean.Experimentation.UnknownInterference.VarianceBound","a":"Experimentation/UnknownInterference","t":2,"d":"The iᵗʰ HT summand depends only on the treatments of the units interfering with i.","s":"∀ {U : Type u_1} [inst : Fintype U] [inst_1 : DecidableEq U] (p : U → ℝ) (y : U → (U → Bool) → ℝ) (i : U) (z z' : U → Bool), (∀ ℓ ∈ Causalean.Experimentation.UnknownInterference.interferers y i, z ℓ = z' ℓ) → Causalean.Experimentation.UnknownInterference.htSummand p y i z = Causalean.Experimentation.UnknownInterference.htSummand p y i z'","l":62,"f":"Causalean/Experimentation/UnknownInterference/VarianceBound.lean"},{"n":"Causalean.Experimentation.UnknownInterference.htTreatSummand","k":"def","m":"Causalean.Experimentation.UnknownInterference.Hajek","a":"Experimentation/UnknownInterference","t":1,"d":"Treated Horvitz–Thompson summand Zᵢ Yᵢ / pᵢ.","s":"{U : Type u_1} → (U → ℝ) → (U → (U → Bool) → ℝ) → U → (U → Bool) → ℝ","l":56,"f":"Causalean/Experimentation/UnknownInterference/Hajek.lean"},{"n":"Causalean.Experimentation.UnknownInterference.interferers","k":"def","m":"Causalean.Experimentation.UnknownInterference.VarianceBound","a":"Experimentation/UnknownInterference","t":1,"d":"The block of units that interfere with i (its own treatment included): the coordinate support of the iᵗʰ HT summand.","s":"{U : Type u_1} → [Fintype U] → [DecidableEq U] → (U → (U → Bool) → ℝ) → U → Finset U","l":57,"f":"Causalean/Experimentation/UnknownInterference/VarianceBound.lean"},{"n":"Causalean.Experimentation.UnknownInterference.root_n_var","k":"theorem","m":"Causalean.Experimentation.UnknownInterference.Consistency","a":"Experimentation/UnknownInterference","t":1,"d":"Root-n variance scaling under bounded interference (Sävje–Aronow–Hudgens 2021). In one bundled experiment, if the average interference degree d̄ is bounded above by a constant C, then the sample size n times the variance of the Horvitz–Thompson estimator is at most k⁴·C, where k is the experiment's regularity constant — the finite-sample variance inequality that supports a root-n rate in bounded-interference sequences.","s":"∀ (E : Causalean.Experimentation.UnknownInterference.SAHExperiment) (C : ℝ), Causalean.Experimentation.UnknownInterference.dbar E.y ≤ C → ↑(Fintype.card E.U) * E.D.Var (Causalean.Experimentation.UnknownInterference.htEst E.p E.y) ≤ E.k ^ 4 * C","l":163,"f":"Causalean/Experimentation/UnknownInterference/Consistency.lean"},{"n":"Causalean.Experimentation.UnknownInterference.tau","k":"def","m":"Causalean.Experimentation.UnknownInterference.Basic","a":"Experimentation/UnknownInterference","t":1,"d":"Assignment-conditional unit-level treatment effect. τ_i(z_{-i}) = y_i(1; z_{-i}) − y_i(0; z_{-i}), the effect of changing unit i's own treatment with all others held at z. Encoded on the full assignment via Function.update; it does not depend on z i.","s":"{U : Type u_1} → [DecidableEq U] → (U → (U → Bool) → ℝ) → U → (U → Bool) → ℝ","l":75,"f":"Causalean/Experimentation/UnknownInterference/Basic.lean"},{"n":"Causalean.Experimentation.UnknownInterference.var_AhatCtrl_le","k":"theorem","m":"Causalean.Experimentation.UnknownInterference.Hajek","a":"Experimentation/UnknownInterference","t":2,"d":"Variance bound for the control numerator average: Var(Â₀) ≤ k⁴·d̄/n.","s":"∀ {U : Type u_1} [inst : Fintype U] [inst_1 : DecidableEq U] (p : U → ℝ) (hp0 : ∀ (i : U), 0 ≤ p i) (hp1 : ∀ (i : U), p i ≤ 1) (y : U → (U → Bool) → ℝ) (k : ℝ), 1 ≤ k → 1 ≤ Fintype.card U → (∀ (i : U), p i ≤ 1 - k⁻¹) → (∀ (i : U), ((Causalean.Experimentation.UnknownInterference.bernoulliDesign p hp0 hp1).E fun z => y i z ^ 2) ≤ k ^ 2) → (Causalean.Experimentation.UnknownInterference.bernoulliDesign p hp0 hp1).Var (Causalean.Experimentation.UnknownInterference.AhatCtrl p y) ≤ k ^ 4 * Causalean.Experimentation.UnknownInterference.dbar y / ↑(Fintype.card U)","l":453,"f":"Causalean/Experimentation/UnknownInterference/Hajek.lean"},{"n":"Causalean.Experimentation.UnknownInterference.var_AhatTreat_le","k":"theorem","m":"Causalean.Experimentation.UnknownInterference.Hajek","a":"Experimentation/UnknownInterference","t":2,"d":"Variance bound for the treated numerator average: Var(Â₁) ≤ k⁴·d̄/n (same disjoint-block argument as the HT estimator).","s":"∀ {U : Type u_1} [inst : Fintype U] [inst_1 : DecidableEq U] (p : U → ℝ) (hp0 : ∀ (i : U), 0 ≤ p i) (hp1 : ∀ (i : U), p i ≤ 1) (y : U → (U → Bool) → ℝ) (k : ℝ), 1 ≤ k → 1 ≤ Fintype.card U → (∀ (i : U), k⁻¹ ≤ p i) → (∀ (i : U), ((Causalean.Experimentation.UnknownInterference.bernoulliDesign p hp0 hp1).E fun z => y i z ^ 2) ≤ k ^ 2) → (Causalean.Experimentation.UnknownInterference.bernoulliDesign p hp0 hp1).Var (Causalean.Experimentation.UnknownInterference.AhatTreat p y) ≤ k ^ 4 * Causalean.Experimentation.UnknownInterference.dbar y / ↑(Fintype.card U)","l":399,"f":"Causalean/Experimentation/UnknownInterference/Hajek.lean"},{"n":"Causalean.Experimentation.UnknownInterference.var_BhatCtrl_le","k":"theorem","m":"Causalean.Experimentation.UnknownInterference.Hajek","a":"Experimentation/UnknownInterference","t":2,"d":"Variance bound for the control weight average: Var(B̂₀) ≤ k²/n.","s":"∀ {U : Type u_1} [inst : Fintype U] [inst_1 : DecidableEq U] (p : U → ℝ) (hp0 : ∀ (i : U), 0 ≤ p i) (hp1 : ∀ (i : U), p i ≤ 1) (k : ℝ), 1 ≤ k → 1 ≤ Fintype.card U → (∀ (i : U), p i ≤ 1 - k⁻¹) → (Causalean.Experimentation.UnknownInterference.bernoulliDesign p hp0 hp1).Var (Causalean.Experimentation.UnknownInterference.BhatCtrl p) ≤ k ^ 2 / ↑(Fintype.card U)","l":554,"f":"Causalean/Experimentation/UnknownInterference/Hajek.lean"},{"n":"Causalean.Experimentation.UnknownInterference.var_BhatTreat_le","k":"theorem","m":"Causalean.Experimentation.UnknownInterference.Hajek","a":"Experimentation/UnknownInterference","t":2,"d":"Variance bound for the treated weight average: Var(B̂₁) ≤ k²/n (single-coordinate independence — the weight summands Zᵢ/pᵢ depend on disjoint singletons).","s":"∀ {U : Type u_1} [inst : Fintype U] [inst_1 : DecidableEq U] (p : U → ℝ) (hp0 : ∀ (i : U), 0 ≤ p i) (hp1 : ∀ (i : U), p i ≤ 1) (k : ℝ), 1 ≤ k → 1 ≤ Fintype.card U → (∀ (i : U), k⁻¹ ≤ p i) → (Causalean.Experimentation.UnknownInterference.bernoulliDesign p hp0 hp1).Var (Causalean.Experimentation.UnknownInterference.BhatTreat p) ≤ k ^ 2 / ↑(Fintype.card U)","l":505,"f":"Causalean/Experimentation/UnknownInterference/Hajek.lean"},{"n":"Causalean.Experimentation.UnknownInterference.var_htEst_le","k":"theorem","m":"Causalean.Experimentation.UnknownInterference.VarianceBound","a":"Experimentation/UnknownInterference","t":1,"d":"The Horvitz–Thompson variance bound (Sävje–Aronow–Hudgens 2021). Fix a regularity constant k at least 1 over a nonempty finite unit population, with a Bernoulli design whose per-unit treatment probabilities lie in the unit interval and further satisfy the overlap bounds k⁻¹ ≤ p i ≤ 1 − k⁻¹ for every unit. If the outcome function's second moment E[(y i)²] is at most k² for every unit under this design, then the Horvitz–Thompson estimator's variance is at most k⁴ · d̄ / n, where d̄ is the average interference-dependence degree and n the number of units.","s":"∀ {U : Type u_1} [inst : Fintype U] [inst_1 : DecidableEq U] (p : U → ℝ) (hp0 : ∀ (i : U), 0 ≤ p i) (hp1 : ∀ (i : U), p i ≤ 1) (y : U → (U → Bool) → ℝ) (k : ℝ), 1 ≤ k → 1 ≤ Fintype.card U → (∀ (i : U), k⁻¹ ≤ p i) → (∀ (i : U), p i ≤ 1 - k⁻¹) → (∀ (i : U), ((Causalean.Experimentation.UnknownInterference.bernoulliDesign p hp0 hp1).E fun z => y i z ^ 2) ≤ k ^ 2) → (Causalean.Experimentation.UnknownInterference.bernoulliDesign p hp0 hp1).Var (Causalean.Experimentation.UnknownInterference.htEst p y) ≤ k ^ 4 * Causalean.Experimentation.UnknownInterference.dbar y / ↑(Fintype.card U)","l":178,"f":"Causalean/Experimentation/UnknownInterference/VarianceBound.lean"},{"n":"Causalean.Experimentation.UnknownInterference.var_htEst_le_inflated","k":"theorem","m":"Causalean.Experimentation.UnknownInterference.Confidence","a":"Experimentation/UnknownInterference","t":1,"d":"Conservative inflation (Sävje–Aronow–Hudgens 2021). Under the Bernoulli design with per-unit treatment probabilities p taking values in [0, 1], if every unit's interference degree — the number of units it is interference-dependent with — is at most a bound D, then inflating the conventional variance estimator's expectation by a factor 1 + D gives a conservative bound on the Horvitz–Thompson estimator's true variance: Var(ĤT) ≤ (1 + D)·E[V̂_Ber].","s":"∀ {U : Type u_1} [inst : Fintype U] [inst_1 : DecidableEq U] (p : U → ℝ) (hp0 : ∀ (i : U), 0 ≤ p i) (hp1 : ∀ (i : U), p i ≤ 1) (y : U → (U → Bool) → ℝ) (D : ℝ), (∀ (i : U), Causalean.Experimentation.UnknownInterference.degDep y i ≤ D) → (Causalean.Experimentation.UnknownInterference.bernoulliDesign p hp0 hp1).Var (Causalean.Experimentation.UnknownInterference.htEst p y) ≤ (1 + D) * (Causalean.Experimentation.UnknownInterference.bernoulliDesign p hp0 hp1).E (Causalean.Experimentation.UnknownInterference.VhatBer p y)","l":148,"f":"Causalean/Experimentation/UnknownInterference/Confidence.lean"},{"n":"Causalean.Experimentation.UnknownInterference.var_htSummand_le","k":"theorem","m":"Causalean.Experimentation.UnknownInterference.VarianceBound","a":"Experimentation/UnknownInterference","t":1,"d":"Per-summand variance bound. Fix a regularity constant k at least 1 and a Bernoulli design with per-unit treatment probabilities lying in the unit interval and further satisfying the overlap bounds k⁻¹ ≤ p i ≤ 1 − k⁻¹ for every unit. If the outcome function's second moment E[(y i)²] is at most k² for every unit under this design, then the variance of the iᵗʰ Horvitz–Thompson summand is at most k⁴.","s":"∀ {U : Type u_1} [inst : Fintype U] [inst_1 : DecidableEq U] (p : U → ℝ) (hp0 : ∀ (i : U), 0 ≤ p i) (hp1 : ∀ (i : U), p i ≤ 1) (y : U → (U → Bool) → ℝ) (k : ℝ), 1 ≤ k → (∀ (i : U), k⁻¹ ≤ p i) → (∀ (i : U), p i ≤ 1 - k⁻¹) → (∀ (i : U), ((Causalean.Experimentation.UnknownInterference.bernoulliDesign p hp0 hp1).E fun z => y i z ^ 2) ≤ k ^ 2) → ∀ (i : U), (Causalean.Experimentation.UnknownInterference.bernoulliDesign p hp0 hp1).Var (Causalean.Experimentation.UnknownInterference.htSummand p y i) ≤ k ^ 4","l":116,"f":"Causalean/Experimentation/UnknownInterference/VarianceBound.lean"},{"n":"Causalean.Experimentation.UnknownInterference.weightCtrlSummand","k":"def","m":"Causalean.Experimentation.UnknownInterference.Hajek","a":"Experimentation/UnknownInterference","t":1,"d":"Control weight summand (1 − Zᵢ) / (1 − pᵢ).","s":"{U : Type u_1} → (U → ℝ) → U → (U → Bool) → ℝ","l":68,"f":"Causalean/Experimentation/UnknownInterference/Hajek.lean"},{"n":"Causalean.Experimentation.UnknownInterference.weightTreatSummand","k":"def","m":"Causalean.Experimentation.UnknownInterference.Hajek","a":"Experimentation/UnknownInterference","t":1,"d":"Treated weight summand Zᵢ / pᵢ.","s":"{U : Type u_1} → (U → ℝ) → U → (U → Bool) → ℝ","l":64,"f":"Causalean/Experimentation/UnknownInterference/Hajek.lean"},{"n":"Causalean.Experimentation.UnknownInterference.y_eq_of_agree_on_interferers","k":"theorem","m":"Causalean.Experimentation.UnknownInterference.Basic","a":"Experimentation/UnknownInterference","t":1,"d":"A unit's outcome depends only on the units that interfere with it. If two treatment assignments z and z' agree on every unit that interferes with unit i, then unit i's outcome is the same under both assignments: y i z = y i z'.","s":"∀ {U : Type u_1} [inst : DecidableEq U] [Finite U] (y : U → (U → Bool) → ℝ) (i : U) (z z' : U → Bool), (∀ (ℓ : U), Causalean.Experimentation.UnknownInterference.Interferes y ℓ i → z ℓ = z' ℓ) → y i z = y i z'","l":129,"f":"Causalean/Experimentation/UnknownInterference/Basic.lean"},{"n":"Causalean.Experimentation.UnknownInterference.y_update_eq_of_not_interferes","k":"theorem","m":"Causalean.Experimentation.UnknownInterference.Basic","a":"Experimentation/UnknownInterference","t":1,"d":"When unit ℓ does not interfere with unit i, flipping ℓ's treatment while holding every other unit's assignment fixed never changes i's realized outcome.","s":"∀ {U : Type u_1} [inst : DecidableEq U] {y : U → (U → Bool) → ℝ} {ℓ i : U}, ¬Causalean.Experimentation.UnknownInterference.Interferes y ℓ i → ∀ (z : U → Bool), y i z = y i (Function.update z ℓ !z ℓ)","l":111,"f":"Causalean/Experimentation/UnknownInterference/Basic.lean"},{"n":"Causalean.GaussMarkov.SphericalErrors","k":"def","m":"Causalean.Estimation.GaussMarkov.QuadForm","a":"Estimation/GaussMarkov","t":1,"d":"This condition says the covariance matrix has homoskedastic uncorrelated errors.","s":"{Obs : Type u_1} → [DecidableEq Obs] → Matrix Obs Obs ℝ → ℝ → Prop","l":66,"f":"Causalean/Estimation/GaussMarkov/QuadForm.lean"},{"n":"Causalean.GaussMarkov.SphericalFamily","k":"def","m":"Causalean.Estimation.GaussMarkov.Variance","a":"Estimation/GaussMarkov","t":1,"d":"A random family has *spherical errors* with scale σ: distinct cells are uncorrelated and each has variance σ².","s":"{Ω : Type u_1} → {Obs : Type u_2} → {mΩ : MeasurableSpace Ω} → (Obs → Ω → ℝ) → MeasureTheory.Measure Ω → ℝ → Prop","l":71,"f":"Causalean/Estimation/GaussMarkov/Variance.lean"},{"n":"Causalean.GaussMarkov.colSpan_dotProduct_leftNull","k":"theorem","m":"Causalean.Estimation.GaussMarkov.LeastNorm","a":"Estimation/GaussMarkov","t":2,"d":"Column span is Euclidean-orthogonal to the left null space: if z lies in the left null space of X (z ᵥ* X = 0), then any column-span vector X *ᵥ g is orthogonal to z.","s":"∀ {Obs : Type u_1} {Param : Type u_2} [inst : Fintype Obs] [inst_1 : Fintype Param] {X : Matrix Obs Param ℝ} {g : Param → ℝ} {z : Obs → ℝ}, Matrix.vecMul z X = 0 → X.mulVec g ⬝ᵥ z = 0","l":51,"f":"Causalean/Estimation/GaussMarkov/LeastNorm.lean"},{"n":"Causalean.GaussMarkov.covMatrix","k":"def","m":"Causalean.Estimation.GaussMarkov.Variance","a":"Estimation/GaussMarkov","t":1,"d":"Covariance matrix of a finite family of random variables Y i.","s":"{Ω : Type u_1} → {Obs : Type u_2} → {mΩ : MeasurableSpace Ω} → (Obs → Ω → ℝ) → MeasureTheory.Measure Ω → Matrix Obs Obs ℝ","l":37,"f":"Causalean/Estimation/GaussMarkov/Variance.lean"},{"n":"Causalean.GaussMarkov.covMatrix_posSemidef","k":"theorem","m":"Causalean.Estimation.GaussMarkov.BLUE","a":"Estimation/GaussMarkov","t":2,"d":"A covariance matrix of an L² random family is positive semidefinite: it is symmetric (cov is symmetric) and its quadratic form is a genuine variance, hence nonnegative. (Fintype Obs appears only under the PosSemidef definition, which the unusedFintypeInType linter cannot see; it is genuinely required.)","s":"∀ {Ω : Type u_1} {Obs : Type u_2} [Fintype Obs] {mΩ : MeasurableSpace Ω} {μ : MeasureTheory.Measure Ω} [MeasureTheory.IsProbabilityMeasure μ] (Y : Obs → Ω → ℝ), (∀ (i : Obs), MeasureTheory.MemLp (Y i) 2 μ) → (Causalean.GaussMarkov.covMatrix Y μ).PosSemidef","l":37,"f":"Causalean/Estimation/GaussMarkov/BLUE.lean"},{"n":"Causalean.GaussMarkov.dotProduct_self_nonneg'","k":"theorem","m":"Causalean.Estimation.GaussMarkov.LeastNorm","a":"Estimation/GaussMarkov","t":2,"d":"Nonnegativity of the Euclidean self dot product.","s":"∀ {Obs : Type u_1} [inst : Fintype Obs] (v : Obs → ℝ), 0 ≤ v ⬝ᵥ v","l":58,"f":"Causalean/Estimation/GaussMarkov/LeastNorm.lean"},{"n":"Causalean.GaussMarkov.gauss_markov_gls","k":"theorem","m":"Causalean.Estimation.GaussMarkov.LeastNorm","a":"Estimation/GaussMarkov","t":1,"d":"Finite Gauss-Markov theorem, general covariance (GLS). Fix a design matrix X, a target combination c, and a positive-semidefinite covariance matrix S. Among all weight vectors w satisfying the same unbiasedness constraint w ᵥ* X = c, if wStar is a GLS weight — meaning S *ᵥ wStar lies in the column span of X, then the quadratic form wStarᵀ S wStar is no larger than wᵀ S w. Specializes to gauss_markov_spherical when S = σ² I.","s":"∀ {Obs : Type u_1} {Param : Type u_2} [inst : Fintype Obs] [inst_1 : Fintype Param] {X : Matrix Obs Param ℝ} {c : Param → ℝ} {S : Matrix Obs Obs ℝ}, S.PosSemidef → ∀ {w wStar : Obs → ℝ} {g : Param → ℝ}, S.mulVec wStar = X.mulVec g → Matrix.vecMul wStar X = c → Matrix.vecMul w X = c → Causalean.GaussMarkov.quadVar S wStar ≤ Causalean.GaussMarkov.quadVar S w","l":93,"f":"Causalean/Estimation/GaussMarkov/LeastNorm.lean"},{"n":"Causalean.GaussMarkov.gauss_markov_spherical","k":"theorem","m":"Causalean.Estimation.GaussMarkov.LeastNorm","a":"Estimation/GaussMarkov","t":1,"d":"Finite Gauss-Markov theorem, spherical case. Fix a design matrix X, a target combination c, and suppose the covariance matrix S is spherical: S = σ² I for some scale σ. Among all weight vectors w satisfying the same unbiasedness constraint w ᵥ* X = c, if wStar lies in the column span of X, wStar = X *ᵥ g, then the quadratic form wStarᵀ S wStar is no larger than wᵀ S w.","s":"∀ {Obs : Type u_1} {Param : Type u_2} [inst : Fintype Obs] [inst_1 : Fintype Param] [inst_2 : DecidableEq Obs] {X : Matrix Obs Param ℝ} {c : Param → ℝ} {S : Matrix Obs Obs ℝ} {σ : ℝ}, Causalean.GaussMarkov.SphericalErrors S σ → ∀ {w wStar : Obs → ℝ} {g : Param → ℝ}, wStar = X.mulVec g → Matrix.vecMul wStar X = c → Matrix.vecMul w X = c → Causalean.GaussMarkov.quadVar S wStar ≤ Causalean.GaussMarkov.quadVar S w","l":63,"f":"Causalean/Estimation/GaussMarkov/LeastNorm.lean"},{"n":"Causalean.GaussMarkov.olsWeight","k":"def","m":"Causalean.Estimation.GaussMarkov.OLS","a":"Estimation/GaussMarkov","t":1,"d":"OLS weight via the normal-equations inverse: wStar = X (XᵀX)⁻¹ c.","s":"{Obs : Type u_1} → {Param : Type u_2} → [Fintype Obs] → [Fintype Param] → [DecidableEq Param] → Matrix Obs Param ℝ → (Param → ℝ) → Obs → ℝ","l":36,"f":"Causalean/Estimation/GaussMarkov/OLS.lean"},{"n":"Causalean.GaussMarkov.olsWeight_blue_spherical","k":"theorem","m":"Causalean.Estimation.GaussMarkov.OLS","a":"Estimation/GaussMarkov","t":1,"d":"OLS is BLUE under spherical errors. For an invertible normal-equations matrix Xᵀ X — full column rank — the ordinary-least-squares weight olsWeight X c estimating the target combination c is well defined. Under spherical errors S = σ² I, among all weight vectors w satisfying the same unbiasedness constraint w ᵥ* X = c, the quadratic form at the OLS weight is no larger than at any other unbiased weight w.","s":"∀ {Obs : Type u_1} {Param : Type u_2} [inst : Fintype Obs] [inst_1 : Fintype Param] [inst_2 : DecidableEq Param] [inst_3 : DecidableEq Obs] {X : Matrix Obs Param ℝ} (c : Param → ℝ), IsUnit (X.transpose * X).det → ∀ {S : Matrix Obs Obs ℝ} {σ : ℝ}, Causalean.GaussMarkov.SphericalErrors S σ → ∀ {w : Obs → ℝ}, Matrix.vecMul w X = c → Causalean.GaussMarkov.quadVar S (Causalean.GaussMarkov.olsWeight X c) ≤ Causalean.GaussMarkov.quadVar S w","l":61,"f":"Causalean/Estimation/GaussMarkov/OLS.lean"},{"n":"Causalean.GaussMarkov.olsWeight_mem_colSpan","k":"theorem","m":"Causalean.Estimation.GaussMarkov.OLS","a":"Estimation/GaussMarkov","t":2,"d":"The OLS weight lies in the column span of X.","s":"∀ {Obs : Type u_1} {Param : Type u_2} [inst : Fintype Obs] [inst_1 : Fintype Param] [inst_2 : DecidableEq Param] (X : Matrix Obs Param ℝ) (c : Param → ℝ), Causalean.GaussMarkov.olsWeight X c = X.mulVec ((X.transpose * X)⁻¹.mulVec c)","l":44,"f":"Causalean/Estimation/GaussMarkov/OLS.lean"},{"n":"Causalean.GaussMarkov.olsWeight_unbiased","k":"theorem","m":"Causalean.Estimation.GaussMarkov.OLS","a":"Estimation/GaussMarkov","t":2,"d":"The OLS weight satisfies the unbiasedness constraint wStar ᵥ* X = c, provided XᵀX is invertible (full column rank).","s":"∀ {Obs : Type u_1} {Param : Type u_2} [inst : Fintype Obs] [inst_1 : Fintype Param] [inst_2 : DecidableEq Param] {X : Matrix Obs Param ℝ} (c : Param → ℝ), IsUnit (X.transpose * X).det → Matrix.vecMul (Causalean.GaussMarkov.olsWeight X c) X = c","l":48,"f":"Causalean/Estimation/GaussMarkov/OLS.lean"},{"n":"Causalean.GaussMarkov.quadVar","k":"def","m":"Causalean.Estimation.GaussMarkov.QuadForm","a":"Estimation/GaussMarkov","t":1,"d":"This is the covariance quadratic form that gives the variance of a linear estimator.","s":"{Obs : Type u_1} → [Fintype Obs] → Matrix Obs Obs ℝ → (Obs → ℝ) → ℝ","l":46,"f":"Causalean/Estimation/GaussMarkov/QuadForm.lean"},{"n":"Causalean.GaussMarkov.quadVar_def","k":"theorem","m":"Causalean.Estimation.GaussMarkov.QuadForm","a":"Estimation/GaussMarkov","t":2,"d":"The quadratic variance form unfolds to the weight-covariance-weight product.","s":"∀ {Obs : Type u_1} [inst : Fintype Obs] (S : Matrix Obs Obs ℝ) (w : Obs → ℝ), Causalean.GaussMarkov.quadVar S w = w ⬝ᵥ S.mulVec w","l":53,"f":"Causalean/Estimation/GaussMarkov/QuadForm.lean"},{"n":"Causalean.GaussMarkov.quadVar_nonneg","k":"theorem","m":"Causalean.Estimation.GaussMarkov.QuadForm","a":"Estimation/GaussMarkov","t":2,"d":"A positive-semidefinite covariance matrix gives a nonnegative variance for every linear weight.","s":"∀ {Obs : Type u_1} [inst : Fintype Obs] {S : Matrix Obs Obs ℝ}, S.PosSemidef → ∀ (w : Obs → ℝ), 0 ≤ Causalean.GaussMarkov.quadVar S w","l":57,"f":"Causalean/Estimation/GaussMarkov/QuadForm.lean"},{"n":"Causalean.GaussMarkov.quadVar_spherical","k":"theorem","m":"Causalean.Estimation.GaussMarkov.QuadForm","a":"Estimation/GaussMarkov","t":1,"d":"Under spherical errors, i.e. when the error covariance matrix S equals σ² times the identity, the linear-estimator variance functional at weight vector w reduces to the common variance σ² times the squared Euclidean length of w.","s":"∀ {Obs : Type u_1} [inst : Fintype Obs] [inst_1 : DecidableEq Obs] {S : Matrix Obs Obs ℝ} {σ : ℝ}, Causalean.GaussMarkov.SphericalErrors S σ → ∀ (w : Obs → ℝ), Causalean.GaussMarkov.quadVar S w = σ ^ 2 * w ⬝ᵥ w","l":70,"f":"Causalean/Estimation/GaussMarkov/QuadForm.lean"},{"n":"Causalean.GaussMarkov.sphericalFamily_covMatrix","k":"theorem","m":"Causalean.Estimation.GaussMarkov.Variance","a":"Estimation/GaussMarkov","t":2,"d":"A spherical random family has a spherical (scalar-identity) covariance matrix.","s":"∀ {Ω : Type u_1} {Obs : Type u_2} {mΩ : MeasurableSpace Ω} {μ : MeasureTheory.Measure Ω} [inst : DecidableEq Obs] {Y : Obs → Ω → ℝ} {σ : ℝ}, (∀ (i : Obs), AEMeasurable (Y i) μ) → Causalean.GaussMarkov.SphericalFamily Y μ σ → Causalean.GaussMarkov.SphericalErrors (Causalean.GaussMarkov.covMatrix Y μ) σ","l":77,"f":"Causalean/Estimation/GaussMarkov/Variance.lean"},{"n":"Causalean.GaussMarkov.variance_blue_gls","k":"theorem","m":"Causalean.Estimation.GaussMarkov.BLUE","a":"Estimation/GaussMarkov","t":1,"d":"BLUE for known covariance (GLS, variance form). For a finite family of random variables Y i, each square-integrable, consider linear combinations ∑ i, w i * Y i whose weights satisfy the same unbiasedness constraint w ᵥ* X = c relative to a design X and target combination c. If the weight wStar is a generalized-least-squares (GLS) weight — i.e. the family's covariance matrix applied to wStar lies in the column span of X, then the linear combination built from wStar has variance no larger than that of any other unbiased linear combination.","s":"∀ {Ω : Type u_1} {Obs : Type u_2} [inst : Fintype Obs] {mΩ : MeasurableSpace Ω} {μ : MeasureTheory.Measure Ω} {Param : Type u_3} [inst_1 : Fintype Param] [MeasureTheory.IsProbabilityMeasure μ] {X : Matrix Obs Param ℝ} {c : Param → ℝ} (Y : Obs → Ω → ℝ), (∀ (i : Obs), MeasureTheory.MemLp (Y i) 2 μ) → ∀ {w wStar : Obs → ℝ} {g : Param → ℝ}, (Causalean.GaussMarkov.covMatrix Y μ).mulVec wStar = X.mulVec g → Matrix.vecMul wStar X = c → Matrix.vecMul w X = c → ProbabilityTheory.variance (fun ω => ∑ i, wStar i * Y i ω) μ ≤ ProbabilityTheory.variance (fun ω => ∑ i, w i * Y i ω) μ","l":79,"f":"Causalean/Estimation/GaussMarkov/BLUE.lean"},{"n":"Causalean.GaussMarkov.variance_blue_spherical","k":"theorem","m":"Causalean.Estimation.GaussMarkov.BLUE","a":"Estimation/GaussMarkov","t":1,"d":"BLUE under spherical errors (variance form). For a finite family of random variables Y i, each square-integrable, suppose the family is spherical: distinct cells are uncorrelated and every cell has the same variance σ². Among all linear combinations ∑ i, w i * Y i whose weights satisfy the same unbiasedness constraint w ᵥ* X = c relative to a design X and target combination c, if wStar lies in the column span of X, wStar = X *ᵥ g, then the linear combination built from wStar (the OLS weight) has variance no larger than that of any other unbiased linear combination.","s":"∀ {Ω : Type u_1} {Obs : Type u_2} [inst : Fintype Obs] {mΩ : MeasurableSpace Ω} {μ : MeasureTheory.Measure Ω} {Param : Type u_3} [inst_1 : Fintype Param] [MeasureTheory.IsProbabilityMeasure μ] {X : Matrix Obs Param ℝ} {c : Param → ℝ} (Y : Obs → Ω → ℝ), (∀ (i : Obs), MeasureTheory.MemLp (Y i) 2 μ) → ∀ {σ : ℝ}, Causalean.GaussMarkov.SphericalFamily Y μ σ → ∀ {w wStar : Obs → ℝ} {g : Param → ℝ}, wStar = X.mulVec g → Matrix.vecMul wStar X = c → Matrix.vecMul w X = c → ProbabilityTheory.variance (fun ω => ∑ i, wStar i * Y i ω) μ ≤ ProbabilityTheory.variance (fun ω => ∑ i, w i * Y i ω) μ","l":57,"f":"Causalean/Estimation/GaussMarkov/BLUE.lean"},{"n":"Causalean.GaussMarkov.variance_linearCombination","k":"theorem","m":"Causalean.Estimation.GaussMarkov.Variance","a":"Estimation/GaussMarkov","t":1,"d":"Variance of a linear combination = covariance quadratic form. For a finite family of random variables Y i, each square-integrable, and any weight vector w, the variance of the random linear combination ∑ i, w i * Y i equals the quadratic form wᵀ Σ w, where Σ is the family's covariance matrix.","s":"∀ {Ω : Type u_1} {Obs : Type u_2} [inst : Fintype Obs] {mΩ : MeasurableSpace Ω} {μ : MeasureTheory.Measure Ω} [MeasureTheory.IsProbabilityMeasure μ] (Y : Obs → Ω → ℝ), (∀ (i : Obs), MeasureTheory.MemLp (Y i) 2 μ) → ∀ (w : Obs → ℝ), ProbabilityTheory.variance (fun ω => ∑ i, w i * Y i ω) μ = Causalean.GaussMarkov.quadVar (Causalean.GaussMarkov.covMatrix Y μ) w","l":41,"f":"Causalean/Estimation/GaussMarkov/Variance.lean"},{"n":"Causalean.InducedFrom","k":"def","m":"Causalean.SCM.ID.GraphicalThms.InducedSubgraph","a":"SCM/ID/GraphicalThms","t":1,"d":"The induced subgraph restricts a SWIG graph to the selected observed nodes.","s":"{N : Type u_1} → [inst : DecidableEq N] → [inst_1 : Fintype N] → Causalean.SWIGGraph N → Finset (Causalean.SWIGNode N) → Causalean.SWIGGraph N","l":61,"f":"Causalean/SCM/ID/GraphicalThms/InducedSubgraph.lean"},{"n":"Causalean.ML.AchievesL2Rate","k":"def","m":"Causalean.ML.Core.Rate","a":"ML/Core","t":1,"d":"The L² estimation error has a finite seminorm at every sample size and experiment outcome, and that finite seminorm is stochastically bounded at the supplied rate under the experiment law. The L² seminorm is computed using the covariate marginal of the joint law, so the real-valued stochastic-order claim never comes from converting an infinite extended norm to zero.","s":"{Ω : Type u_1} → [inst : MeasurableSpace Ω] → {X : Type u_2} → [inst_1 : MeasurableSpace X] → (ℕ → Ω → X → ℝ) → (X → ℝ) → MeasureTheory.Measure (X × ℝ) → (ℕ → ℝ) → MeasureTheory.Measure Ω → Prop","l":196,"f":"Causalean/ML/Core/Rate.lean"},{"n":"Causalean.ML.Activation","k":"structure","m":"Causalean.ML.NeuralNet.Layer","a":"ML/NeuralNet","t":1,"d":"An activation function bundles a scalar map together with a Lipschitz constant and the certificate that the map is Lipschitz with that constant (e.g. ReLU, sigmoid, and tanh are all 1-Lipschitz).","s":"Type","l":33,"f":"Causalean/ML/NeuralNet/Layer.lean"},{"n":"Causalean.ML.Activation.applyVec","k":"def","m":"Causalean.ML.NeuralNet.Layer","a":"ML/NeuralNet","t":1,"d":"Apply an activation coordinatewise to a vector.","s":"{n : ℕ} → Causalean.ML.Activation → (Fin n → ℝ) → Fin n → ℝ","l":44,"f":"Causalean/ML/NeuralNet/Layer.lean"},{"n":"Causalean.ML.Causal.aipw_mlNuisance_meanZero_of_wellSpecified","k":"theorem","m":"Causalean.ML.CausalApplication.AIPWExample","a":"ML/CausalApplication","t":1,"d":"End-to-end identification with ML nuisances. In a backdoor estimation system S, suppose the propensity is bounded away from 0 and 1 by ε (strict overlap), the backdoor identification assumptions hold, the squared factual outcome is integrable, and the squared potential outcome under each treatment arm is integrable. If the ML-learned outcome-regression functions and propensity function are measurable and agree pointwise with the true outcome regression and true propensity, respectively (correct specification), then the AIPW moment functional built from the ML-learned nuisance vector integrates to zero at the true average treatment effect S.θ₀ — the doubly-robust estimating equation identifies the ATE with machine-learned nuisances.","s":"∀ {P : Causalean.PO.POSystem} {γ : Type u_1} [inst : MeasurableSpace γ] [inst_1 : StandardBorelSpace P.Ω] [inst_2 : MeasureTheory.IsFiniteMeasure P.μ] (S : Causalean.Estimation.ATE.BackdoorEstimationSystem P γ) {ε : ℝ}, S.StrictOverlap ε → S.Assumptions → MeasureTheory.Integrable (fun ω => S.factualY ω ^ 2) P.μ → (∀ (d : Bool), MeasureTheory.Integrable (fun ω => S.YofD d ω ^ 2) P.μ) → ∀ {mhat : Bool → γ → ℝ} {ehat : γ → ℝ} (hmhat : ∀ (b : Bool), Measurable (mhat b)) (hehat : Measurable ehat), (∀ (b : Bool) (x : γ), mhat b x = S.μ_val b x) → (∀ (x : γ), ehat x = S.e_val x) → ∫ (z : γ × Bool × ℝ), Causalean.Estimation.ATE.BackdoorEstimationSystem.aipwMomentFunctional (Causalean.ML.Causal.mlNuisanceVec mhat ehat hmhat hehat) z S.θ₀ ∂S.P_Z = 0","l":27,"f":"Causalean/ML/CausalApplication/AIPWExample.lean"},{"n":"Causalean.ML.Causal.condExp_of_isL2Projection","k":"theorem","m":"Causalean.ML.CausalApplication.RegressionBridge","a":"ML/CausalApplication","t":1,"d":"On a finite joint covariate–response law P, if the candidate function m is measurable, the response coordinate is integrable, the composed function z ↦ m(z.1) is integrable, and m is an L² projection of the response onto functions of the covariate — its residual is uncorrelated with every measurable, integrable function of the covariate, then m agrees P-almost everywhere with the conditional expectation of the response given the covariate σ-algebra.","s":"∀ {X : Type u_1} [inst : MeasurableSpace X] (P : MeasureTheory.Measure (X × ℝ)) [MeasureTheory.IsFiniteMeasure P] {m : X → ℝ}, Measurable m → MeasureTheory.Integrable (fun z => z.2) P → MeasureTheory.Integrable (fun z => m z.1) P → Causalean.ML.IsL2Projection P m → (fun z => m z.1) =ᵐ[P] P[fun z => z.2 | Causalean.ML.Causal.covarSigma]","l":116,"f":"Causalean/ML/CausalApplication/RegressionBridge.lean"},{"n":"Causalean.ML.Causal.covarSigma","k":"def","m":"Causalean.ML.CausalApplication.RegressionBridge","a":"ML/CausalApplication","t":1,"d":"The covariate sigma-algebra records exactly the information carried by the covariate in a joint covariate-response observation.","s":"{X : Type u_1} → [MeasurableSpace X] → MeasurableSpace (X × ℝ)","l":31,"f":"Causalean/ML/CausalApplication/RegressionBridge.lean"},{"n":"Causalean.ML.Causal.covarSigma_le","k":"theorem","m":"Causalean.ML.CausalApplication.RegressionBridge","a":"ML/CausalApplication","t":2,"d":"The covariate sigma-algebra is a sub-sigma-algebra of the full joint observation sigma-algebra.","s":"∀ {X : Type u_1} [inst : MeasurableSpace X], Causalean.ML.Causal.covarSigma ≤ inferInstance","l":35,"f":"Causalean/ML/CausalApplication/RegressionBridge.lean"},{"n":"Causalean.ML.Causal.dml_rate_conditions_of_quarter_rates","k":"theorem","m":"Causalean.ML.CausalApplication.RateAssembly","a":"ML/CausalApplication","t":1,"d":"Assembly. If, for every treatment arm, the outcome-regression estimation error is o_p(n^{-1/4}), and the propensity estimation error is o_p(n^{-1/4}), then each error is separately o_p(1), and for every arm the pointwise product of the outcome and propensity errors is o_p(n^{-1/2}) — together these are DML's three nuisance-rate conditions.","s":"∀ {Ω : Type u_1} [inst : MeasurableSpace Ω] {μ : MeasureTheory.Measure Ω} {μErr : Bool → ℕ → Ω → ℝ} {eErr : ℕ → Ω → ℝ}, (∀ (a : Bool), Causalean.Stat.IsLittleOp (μErr a) (fun n => ↑n ^ (-(1 / 4))) μ) → Causalean.Stat.IsLittleOp eErr (fun n => ↑n ^ (-(1 / 4))) μ → (∀ (a : Bool), Causalean.Stat.IsLittleOp (μErr a) (fun x => 1) μ) ∧ Causalean.Stat.IsLittleOp eErr (fun x => 1) μ ∧ ∀ (a : Bool), Causalean.Stat.IsLittleOp (fun n ω => μErr a n ω * eErr n ω) (fun n => ↑n ^ (-(1 / 2))) μ","l":36,"f":"Causalean/ML/CausalApplication/RateAssembly.lean"},{"n":"Causalean.ML.Causal.isL2Projection_of_condExp","k":"theorem","m":"Causalean.ML.CausalApplication.RegressionBridge","a":"ML/CausalApplication","t":1,"d":"On a finite joint covariate–response law P, if the candidate function m is measurable, the response coordinate is integrable, and m agrees P-almost everywhere with the conditional expectation of the response given the covariate σ-algebra, then m is an L² projection of the response onto functions of the covariate: its residual is uncorrelated with every measurable, integrable function of the covariate.","s":"∀ {X : Type u_1} [inst : MeasurableSpace X] (P : MeasureTheory.Measure (X × ℝ)) [MeasureTheory.IsFiniteMeasure P] {m : X → ℝ}, Measurable m → MeasureTheory.Integrable (fun z => z.2) P → (fun z => m z.1) =ᵐ[P] P[fun z => z.2 | Causalean.ML.Causal.covarSigma] → Causalean.ML.IsL2Projection P m","l":40,"f":"Causalean/ML/CausalApplication/RegressionBridge.lean"},{"n":"Causalean.ML.Causal.mlNuisanceVec","k":"def","m":"Causalean.ML.CausalApplication.Nuisance","a":"ML/CausalApplication","t":1,"d":"Package ML-learned outcome arms μ_fn and propensity e_fn into the AIPW nuisance vector consumed by Estimation.ATE.","s":"{γ : Type u_1} → [inst : MeasurableSpace γ] → (μ_fn : Bool → γ → ℝ) → (e_fn : γ → ℝ) → (∀ (b : Bool), Measurable (μ_fn b)) → Measurable e_fn → Causalean.Estimation.ATE.NuisanceVec γ","l":55,"f":"Causalean/ML/CausalApplication/Nuisance.lean"},{"n":"Causalean.ML.Causal.mlOutcomeRegression_ae_eq","k":"theorem","m":"Causalean.ML.CausalApplication.Nuisance","a":"ML/CausalApplication","t":1,"d":"Outcome-regression recovery. On a finite covariate–outcome law Pd, if the candidate regression function m is measurable, the outcome coordinate is integrable, the composed function z ↦ m(z.1) is integrable, and m is the squared-loss population target — its residual Y − m(X) is uncorrelated with every measurable, integrable function of the covariate, then m agrees Pd-almost everywhere with the conditional expectation of the outcome given the covariate σ-algebra, i.e. m is the outcome regression μ(d, x) = E[Y ∣ X = x, D = d].","s":"∀ {γ : Type u_1} [inst : MeasurableSpace γ] (Pd : MeasureTheory.Measure (γ × ℝ)) [MeasureTheory.IsFiniteMeasure Pd] {m : γ → ℝ}, Measurable m → MeasureTheory.Integrable (fun z => z.2) Pd → MeasureTheory.Integrable (fun z => m z.1) Pd → Causalean.ML.IsL2Projection Pd m → (fun z => m z.1) =ᵐ[Pd] Pd[fun z => z.2 | Causalean.ML.Causal.covarSigma]","l":26,"f":"Causalean/ML/CausalApplication/Nuisance.lean"},{"n":"Causalean.ML.Causal.mlPropensity_ae_eq","k":"theorem","m":"Causalean.ML.CausalApplication.Nuisance","a":"ML/CausalApplication","t":1,"d":"Propensity recovery. On a finite covariate–treatment-indicator law Pe, if the candidate propensity function e is measurable, the treatment indicator is integrable, the composed function z ↦ e(z.1) is integrable, and e is the squared-loss population target — its residual is uncorrelated with every measurable, integrable function of the covariate, then e agrees Pe-almost everywhere with the conditional expectation of the treatment indicator given the covariate σ-algebra, i.e. e is the propensity P(D = 1 ∣ X).","s":"∀ {γ : Type u_1} [inst : MeasurableSpace γ] (Pe : MeasureTheory.Measure (γ × ℝ)) [MeasureTheory.IsFiniteMeasure Pe] {e : γ → ℝ}, Measurable e → MeasureTheory.Integrable (fun z => z.2) Pe → MeasureTheory.Integrable (fun z => e z.1) Pe → Causalean.ML.IsL2Projection Pe e → (fun z => e z.1) =ᵐ[Pe] Pe[fun z => z.2 | Causalean.ML.Causal.covarSigma]","l":41,"f":"Causalean/ML/CausalApplication/Nuisance.lean"},{"n":"Causalean.ML.Causal.rpow_quarter_le_one","k":"theorem","m":"Causalean.ML.CausalApplication.RateAssembly","a":"ML/CausalApplication","t":2,"d":"The n^{-1/4} rate is bounded by 1 (so it weakens to the o_p(1) rate).","s":"∀ (n : ℕ), ↑n ^ (-(1 / 4)) ≤ 1","l":28,"f":"Causalean/ML/CausalApplication/RateAssembly.lean"},{"n":"Causalean.ML.DenseLayer","k":"structure","m":"Causalean.ML.NeuralNet.Layer","a":"ML/NeuralNet","t":1,"d":"A dense affine layer bundles a weight matrix and a bias vector determining the map $x \\mapsto Wx + b$ from Fin m inputs to Fin n outputs.","s":"ℕ → ℕ → Type","l":21,"f":"Causalean/ML/NeuralNet/Layer.lean"},{"n":"Causalean.ML.DenseLayer.eval","k":"def","m":"Causalean.ML.NeuralNet.Layer","a":"ML/NeuralNet","t":1,"d":"The affine map computed by a dense layer.","s":"{m n : ℕ} → Causalean.ML.DenseLayer m n → (Fin m → ℝ) → Fin n → ℝ","l":29,"f":"Causalean/ML/NeuralNet/Layer.lean"},{"n":"Causalean.ML.FeatureMap","k":"structure","m":"Causalean.ML.Core.Hypothesis","a":"ML/Core","t":1,"d":"A finite feature map bundles a transform sending each input to its vector of K feature values. Linear-in-features predictors use x ↦ ⟪β, φ x⟫; the identity feature map recovers ordinary linear regression, and other choices of the transform recover polynomial, spline, or Fourier (sieve) regression.","s":"Type u_1 → (K : Type u_2) → [Fintype K] → Type (max u_1 u_2)","l":45,"f":"Causalean/ML/Core/Hypothesis.lean"},{"n":"Causalean.ML.FinitePartitionPredictor","k":"structure","m":"Causalean.ML.Tree.FinitePartitionPredictor","a":"ML/Tree","t":1,"d":"A piecewise-constant predictor on a finite partition: a finite index type cell of partition cells, a map chooseCell assigning each input to its cell, and a constant predicted value value on each cell.","s":"Type u_1 → Type (max 1 u_1)","l":18,"f":"Causalean/ML/Tree/FinitePartitionPredictor.lean"},{"n":"Causalean.ML.FinitePartitionPredictor.eval","k":"def","m":"Causalean.ML.Tree.FinitePartitionPredictor","a":"ML/Tree","t":1,"d":"The tree prediction: the value of the cell the input falls into.","s":"{X : Type u_1} → Causalean.ML.FinitePartitionPredictor X → X → ℝ","l":33,"f":"Causalean/ML/Tree/FinitePartitionPredictor.lean"},{"n":"Causalean.ML.FinitePartitionPredictor.eval_eqOn_cell","k":"theorem","m":"Causalean.ML.Tree.FinitePartitionPredictor","a":"ML/Tree","t":1,"d":"For a finite-partition predictor T, on the set of inputs mapped to a given cell c, the predictor's output equals the constant value assigned to that cell.","s":"∀ {X : Type u_1} (T : Causalean.ML.FinitePartitionPredictor X) (c : T.cell), Set.EqOn T.eval (fun x => T.value c) {x | T.chooseCell x = c}","l":37,"f":"Causalean/ML/Tree/FinitePartitionPredictor.lean"},{"n":"Causalean.ML.HasFinitePopulationRisk","k":"def","m":"Causalean.ML.Core.Risk","a":"ML/Core","t":1,"d":"The loss-integrand of h is integrable, so the population risk is finite.","s":"{X : Type u_1} → {Y : Type u_2} → [inst : MeasurableSpace X] → [inst_1 : MeasurableSpace Y] → Causalean.ML.Loss Y → MeasureTheory.Measure (X × Y) → (X → Y) → Prop","l":65,"f":"Causalean/ML/Core/Risk.lean"},{"n":"Causalean.ML.HypothesisClass","k":"structure","m":"Causalean.ML.Core.Hypothesis","a":"ML/Core","t":1,"d":"A hypothesis class packaged extensionally as a set of admissible prediction functions from the covariates to the outcome, subject to the requirement that every function admitted to the class is measurable.","s":"(X : Type u_1) → (Y : Type u_2) → [MeasurableSpace X] → [MeasurableSpace Y] → Type (max u_1 u_2)","l":36,"f":"Causalean/ML/Core/Hypothesis.lean"},{"n":"Causalean.ML.IsApproxERMP","k":"structure","m":"Causalean.ML.Core.ERM","a":"ML/Core","t":1,"d":"θhat minimizes a parametric objective up to slack ε ≥ 0.","s":"{Θ : Type u_1} → (Θ → ℝ) → Set Θ → Θ → ℝ → Prop","l":34,"f":"Causalean/ML/Core/ERM.lean"},{"n":"Causalean.ML.IsERM","k":"structure","m":"Causalean.ML.Core.ERM","a":"ML/Core","t":1,"d":"An empirical-risk minimizer is an admissible prediction rule whose finite-sample risk criterion is no larger than that of any other admissible rule. The criterion is the inverse-cardinality-scaled loss sum over a nonempty finite sample.","s":"{ι : Type u_1} → {X : Type u_2} → {Y : Type u_3} → [Fintype ι] → [Nonempty ι] → [inst : MeasurableSpace X] → [inst_1 : MeasurableSpace Y] → Causalean.ML.HypothesisClass X Y → Causalean.ML.Loss Y → (ι → X × Y) → (X → Y) → Prop","l":55,"f":"Causalean/ML/Core/ERM.lean"},{"n":"Causalean.ML.IsERMP","k":"structure","m":"Causalean.ML.Core.ERM","a":"ML/Core","t":1,"d":"θhat exactly minimizes a parametric objective over the admissible set.","s":"{Θ : Type u_1} → (Θ → ℝ) → Set Θ → Θ → Prop","l":27,"f":"Causalean/ML/Core/ERM.lean"},{"n":"Causalean.ML.IsL2Projection","k":"def","m":"Causalean.ML.Core.PopulationTarget","a":"ML/Core","t":1,"d":"The residual of m is orthogonal to every (integrable, measurable) function of the covariate. This is the defining property of the L²-projection onto the σ-algebra generated by the covariate, i.e. of the regression function x ↦ E[Y ∣ X = x]. Measurability of the test function g is required because orthogonality of the residual is only meaningful for measurable functions of X (it is what links this to condExp; see ML/CausalApplication/RegressionBridge.lean).","s":"{X : Type u_1} → [inst : MeasurableSpace X] → MeasureTheory.Measure (X × ℝ) → (X → ℝ) → Prop","l":47,"f":"Causalean/ML/Core/PopulationTarget.lean"},{"n":"Causalean.ML.IsPSDkernel","k":"def","m":"Causalean.ML.Kernel.Gram","a":"ML/Kernel","t":1,"d":"A kernel is positive semidefinite when every Gram matrix is PSD.","s":"{X : Type u_1} → (X → X → ℝ) → Prop","l":27,"f":"Causalean/ML/Kernel/Gram.lean"},{"n":"Causalean.ML.IsPopulationOLS","k":"def","m":"Causalean.ML.Linear.Population","a":"ML/Linear","t":1,"d":"Population normal equations: the residual of the linear predictor with coefficients βstar is uncorrelated with every feature φ·ₖ.","s":"{X' : Type u_1} → {K : Type u_2} → [inst : MeasurableSpace X'] → [inst_1 : Fintype K] → MeasureTheory.Measure (X' × ℝ) → Causalean.ML.FeatureMap X' K → (K → ℝ) → Prop","l":25,"f":"Causalean/ML/Linear/Population.lean"},{"n":"Causalean.ML.IsPopulationRegLogistic","k":"def","m":"Causalean.ML.Binary.Rate","a":"ML/Binary","t":1,"d":"The penalized population first-order condition at β⋆: E[(σ⟨β⋆,φ⟩ − Y) φₖ] + 2λ β⋆ₖ = 0 for every feature k.","s":"{γ : Type u_2} → {K : Type u_3} → [inst : MeasurableSpace γ] → [inst_1 : Fintype K] → MeasureTheory.Measure (γ × ℝ) → Causalean.ML.FeatureMap γ K → ℝ → (K → ℝ) → Prop","l":43,"f":"Causalean/ML/Binary/Rate.lean"},{"n":"Causalean.ML.IsPopulationRidge","k":"def","m":"Causalean.ML.Ridge.Population","a":"ML/Ridge","t":1,"d":"The regularized population normal equations: E[(Y − ⟪βstar,φ⟫) φₖ] = λ · βstarₖ for every feature k.","s":"{X' : Type u_1} → {K : Type u_2} → [inst : MeasurableSpace X'] → [inst_1 : Fintype K] → MeasureTheory.Measure (X' × ℝ) → Causalean.ML.FeatureMap X' K → ℝ → (K → ℝ) → Prop","l":33,"f":"Causalean/ML/Ridge/Population.lean"},{"n":"Causalean.ML.IsPopulationRiskMinimizer","k":"structure","m":"Causalean.ML.Core.ERM","a":"ML/Core","t":1,"d":"A population-risk minimizer bundles the claim that a prediction rule belongs to the hypothesis class, that it has finite expected loss under the population measure, that every other rule admissible in the class also has finite expected loss, and that its population risk is no larger than that of any other rule in the class.","s":"{X : Type u_1} → {Y : Type u_2} → [inst : MeasurableSpace X] → [inst_1 : MeasurableSpace Y] → Causalean.ML.HypothesisClass X Y → Causalean.ML.Loss Y → MeasureTheory.Measure (X × Y) → (X → Y) → Prop","l":67,"f":"Causalean/ML/Core/ERM.lean"},{"n":"Causalean.ML.IsRKHS","k":"structure","m":"Causalean.ML.Kernel.RKHS","a":"ML/Kernel","t":1,"d":"The candidate reproducing-kernel Hilbert space structure on a real inner-product space H of functions on X, given an evaluation map and a representer map, satisfies the reproducing property: evaluating any function of H at a point equals its inner product with the representer of that point.","s":"(X : Type u_1) → (H : Type u_2) → [inst : NormedAddCommGroup H] → [InnerProductSpace ℝ H] → (H → X → ℝ) → (X → H) → Prop","l":18,"f":"Causalean/ML/Kernel/RKHS.lean"},{"n":"Causalean.ML.KFeat","k":"def","m":"Causalean.ML.Kernel.SquaredLoss","a":"ML/Kernel","t":1,"d":"Feature × response data space: Xb-ball features paired with [-Yb, Yb] responses.","s":"ℕ → ℝ → ℝ → Type","l":34,"f":"Causalean/ML/Kernel/SquaredLoss.lean"},{"n":"Causalean.ML.KWeight","k":"def","m":"Causalean.ML.Kernel.SquaredLoss","a":"ML/Kernel","t":1,"d":"The W-ball of weight vectors.","s":"ℕ → ℝ → Type","l":38,"f":"Causalean/ML/Kernel/SquaredLoss.lean"},{"n":"Causalean.ML.LassoFeat","k":"def","m":"Causalean.ML.Lasso.SquaredLoss","a":"ML/Lasso","t":1,"d":"Feature × response data space: Xinf-bounded coordinates paired with [-Yb,Yb] responses.","s":"ℕ → ℝ → ℝ → Type","l":34,"f":"Causalean/ML/Lasso/SquaredLoss.lean"},{"n":"Causalean.ML.LassoWeight","k":"def","m":"Causalean.ML.Lasso.SquaredLoss","a":"ML/Lasso","t":1,"d":"The W-ball of weight vectors in coordinate ℓ¹ norm.","s":"ℕ → ℝ → Type","l":39,"f":"Causalean/ML/Lasso/SquaredLoss.lean"},{"n":"Causalean.ML.Loss","k":"def","m":"Causalean.ML.Core.Risk","a":"ML/Core","t":1,"d":"A pointwise loss: loss ŷ y compares a prediction ŷ to a label y.","s":"Type u_1 → Type u_1","l":25,"f":"Causalean/ML/Core/Risk.lean"},{"n":"Causalean.ML.Predictor","k":"structure","m":"Causalean.ML.Core.Hypothesis","a":"ML/Core","t":1,"d":"A parametrized family of predictors bundles an admissible set of parameter values together with a map sending each parameter to a prediction function from the covariates to the outcome.","s":"Type u_1 → Type u_2 → Type u_3 → Type (max (max u_1 u_2) u_3)","l":27,"f":"Causalean/ML/Core/Hypothesis.lean"},{"n":"Causalean.ML.ProperBinaryLoss","k":"def","m":"Causalean.ML.Surrogate.GenericERM","a":"ML/Surrogate","t":1,"d":"A binary loss L : ℝ → Bool → ℝ is proper: for every true probability η ∈ [0,1], the conditional expected loss q ↦ η·L q true + (1−η)·L q false is minimized over [0,1] at q = η.","s":"(ℝ → Bool → ℝ) → Prop","l":28,"f":"Causalean/ML/Surrogate/GenericERM.lean"},{"n":"Causalean.ML.RandomForest","k":"structure","m":"Causalean.ML.Tree.Forest","a":"ML/Tree","t":1,"d":"A random forest: one finite-partition regression tree tree for each index in Fin T.","s":"Type u_1 → ℕ → Type (max 1 u_1)","l":25,"f":"Causalean/ML/Tree/Forest.lean"},{"n":"Causalean.ML.RandomForest.eval","k":"def","m":"Causalean.ML.Tree.Forest","a":"ML/Tree","t":1,"d":"The forest prediction: the average of the tree predictions.","s":"{X : Type u_1} → {T : ℕ} → Causalean.ML.RandomForest X T → X → ℝ","l":31,"f":"Causalean/ML/Tree/Forest.lean"},{"n":"Causalean.ML.RandomForest.eval_mem_Icc","k":"theorem","m":"Causalean.ML.Tree.Forest","a":"ML/Tree","t":1,"d":"For a random forest F of T finite-partition regression trees, if the forest is nonempty (T > 0) and every tree predicts within [a, b] at every input, then the forest's averaged prediction also lies in [a, b] at every input x.","s":"∀ {X : Type u_1} {T : ℕ} (F : Causalean.ML.RandomForest X T), 0 < T → ∀ {a b : ℝ}, (∀ (t : Fin T) (x : X), (F.tree t).eval x ∈ Set.Icc a b) → ∀ (x : X), F.eval x ∈ Set.Icc a b","l":35,"f":"Causalean/ML/Tree/Forest.lean"},{"n":"Causalean.ML.RegularizedERM","k":"structure","m":"Causalean.ML.Core.ERM","a":"ML/Core","t":1,"d":"θhat minimizes a penalized objective objective θ + lam · penalty θ with a nonnegative regularization weight lam.","s":"{Θ : Type u_1} → (Θ → ℝ) → (Θ → ℝ) → ℝ → Set Θ → Θ → Prop","l":44,"f":"Causalean/ML/Core/ERM.lean"},{"n":"Causalean.ML.StrictProperBinaryLoss","k":"def","m":"Causalean.ML.Surrogate.GenericERM","a":"ML/Surrogate","t":1,"d":"A binary loss is strictly proper when it is proper and the true probability is the unique [0,1] minimizer of the conditional expected loss.","s":"(ℝ → Bool → ℝ) → Prop","l":35,"f":"Causalean/ML/Surrogate/GenericERM.lean"},{"n":"Causalean.ML.SubgradientAt","k":"def","m":"Causalean.ML.Core.Convex","a":"ML/Core","t":1,"d":"g is a subgradient of f at x relative to s: the affine support inequality f x + ⟪g, y − x⟫ ≤ f y holds for all y ∈ s (and x ∈ s).","s":"{E : Type u_1} → [inst : NormedAddCommGroup E] → [InnerProductSpace ℝ E] → (E → ℝ) → Set E → E → E → Prop","l":121,"f":"Causalean/ML/Core/Convex.lean"},{"n":"Causalean.ML.bernoulliCE","k":"def","m":"Causalean.ML.Binary.FisherConsistency","a":"ML/Binary","t":1,"d":"The Bernoulli cross-entropy of predicting probability q when the true probability is η.","s":"ℝ → ℝ → ℝ","l":19,"f":"Causalean/ML/Binary/FisherConsistency.lean"},{"n":"Causalean.ML.bernoulliCE_isMinOn","k":"theorem","m":"Causalean.ML.Binary.FisherConsistency","a":"ML/Binary","t":1,"d":"For a true probability η strictly between 0 and 1, the Bernoulli cross-entropy q ↦ bernoulliCE η q attains its minimum over the open interval (0,1) exactly at the truth q = η.","s":"∀ {η : ℝ}, η ∈ Set.Ioo 0 1 → IsMinOn (fun q => Causalean.ML.bernoulliCE η q) (Set.Ioo 0 1) η","l":24,"f":"Causalean/ML/Binary/FisherConsistency.lean"},{"n":"Causalean.ML.bestLinearPredictor_minimizes_populationRisk","k":"theorem","m":"Causalean.ML.Linear.Population","a":"ML/Linear","t":1,"d":"For a probability measure P on features and outcome and a finite feature map φ, if the residual of the linear predictor with coefficients βstar is uncorrelated in expectation with every feature, the population squared-loss risks of the βstar- and β-predictors are both finite, and each feature is integrable against that residual, then the population squared risk of the βstar-predictor is at most that of any other linear-in-features predictor with coefficients β.","s":"∀ {X' : Type u_1} {K : Type u_2} [inst : MeasurableSpace X'] [inst_1 : Fintype K] (P : MeasureTheory.Measure (X' × ℝ)) (φ : Causalean.ML.FeatureMap X' K) {βstar : K → ℝ}, Causalean.ML.IsPopulationOLS P φ βstar → ∀ (β : K → ℝ), (Causalean.ML.HasFinitePopulationRisk Causalean.ML.squaredLoss P fun x => ∑ k, βstar k * φ.φ x k) → (Causalean.ML.HasFinitePopulationRisk Causalean.ML.squaredLoss P fun x => ∑ k, β k * φ.φ x k) → (∀ (k : K), MeasureTheory.Integrable (fun z => (z.2 - ∑ j, βstar j * φ.φ z.1 j) * φ.φ z.1 k) P) → (Causalean.ML.populationRisk Causalean.ML.squaredLoss P fun x => ∑ k, βstar k * φ.φ x k) ≤ Causalean.ML.populationRisk Causalean.ML.squaredLoss P fun x => ∑ k, β k * φ.φ x k","l":30,"f":"Causalean/ML/Linear/Population.lean"},{"n":"Causalean.ML.bool01","k":"def","m":"Causalean.ML.Core.Losses","a":"ML/Core","t":1,"d":"Boolean labels are encoded as real zero-one outcomes.","s":"Bool → ℝ","l":23,"f":"Causalean/ML/Core/Losses.lean"},{"n":"Causalean.ML.bool01_false","k":"theorem","m":"Causalean.ML.Core.Losses","a":"ML/Core","t":2,"d":"A false Boolean label is encoded as zero.","s":"Causalean.ML.bool01 false = 0","l":42,"f":"Causalean/ML/Core/Losses.lean"},{"n":"Causalean.ML.bool01_le_one","k":"theorem","m":"Causalean.ML.Core.Losses","a":"ML/Core","t":2,"d":"The zero-one encoding of a Boolean label is always at most one.","s":"∀ (b : Bool), Causalean.ML.bool01 b ≤ 1","l":49,"f":"Causalean/ML/Core/Losses.lean"},{"n":"Causalean.ML.bool01_nonneg","k":"theorem","m":"Causalean.ML.Core.Losses","a":"ML/Core","t":2,"d":"The zero-one encoding of a Boolean label is always nonnegative.","s":"∀ (b : Bool), 0 ≤ Causalean.ML.bool01 b","l":45,"f":"Causalean/ML/Core/Losses.lean"},{"n":"Causalean.ML.bool01_true","k":"theorem","m":"Causalean.ML.Core.Losses","a":"ML/Core","t":1,"d":"A true Boolean label is encoded as the real number one.","s":"Causalean.ML.bool01 true = 1","l":39,"f":"Causalean/ML/Core/Losses.lean"},{"n":"Causalean.ML.centered_score_mean_isBigOp","k":"theorem","m":"Causalean.ML.Ridge.Rate","a":"ML/Ridge","t":2,"d":"The centered empirical score mean Ŝₙ − λβ⋆ = (Ĉₙ − Ĝₙβ⋆) − λβ⋆ = n⁻¹ Σ (φᵢ(yᵢ−⟨β⋆,φᵢ⟩) − λβ⋆) is O_p(n^{-1/2}). The population normal equations make the summands centered.","s":"∀ {Ω : Type u_1} {γ : Type u_2} {K : Type u_3} [inst : MeasurableSpace Ω] [inst_1 : MeasurableSpace γ] [inst_2 : Fintype K] {μ : MeasureTheory.Measure Ω} (φ : Causalean.ML.FeatureMap γ K) (P : MeasureTheory.Measure (γ × ℝ)) [MeasureTheory.IsProbabilityMeasure P] (S : Causalean.Stat.IIDSample Ω (γ × ℝ) μ P) [MeasureTheory.IsProbabilityMeasure μ] {lam : ℝ} (βstar : K → ℝ), Causalean.ML.IsPopulationRidge P φ lam βstar → (∀ (k : K), Measurable fun x => φ.φ x k) → (∀ (k : K), MeasureTheory.MemLp (fun z => (z.2 - ∑ j, βstar j * φ.φ z.1 j) * φ.φ z.1 k) 2 P) → Causalean.Stat.IsBigOp (fun n ω => ‖Causalean.ML.empiricalCross φ S.Z n ω - (Causalean.ML.empiricalGram φ S.Z n ω).mulVec βstar - lam • βstar‖) (fun n => (√↑n)⁻¹) μ","l":226,"f":"Causalean/ML/Ridge/Rate.lean"},{"n":"Causalean.ML.clampedSq","k":"def","m":"Causalean.ML.Surrogate.ClampedSquare","a":"ML/Surrogate","t":1,"d":"The square of the projection of t onto [-c, c]: a globally Lipschitz surrogate that agrees with t ↦ t² on [-c, c].","s":"ℝ → ℝ → ℝ","l":31,"f":"Causalean/ML/Surrogate/ClampedSquare.lean"},{"n":"Causalean.ML.clampedSq_eq_sq","k":"theorem","m":"Causalean.ML.Surrogate.ClampedSquare","a":"ML/Surrogate","t":1,"d":"On the band where t lies within c in absolute value, the clamped square clampedSq c t equals the genuine square t².","s":"∀ {c t : ℝ}, |t| ≤ c → Causalean.ML.clampedSq c t = t ^ 2","l":39,"f":"Causalean/ML/Surrogate/ClampedSquare.lean"},{"n":"Causalean.ML.clampedSq_le_sq","k":"theorem","m":"Causalean.ML.Surrogate.ClampedSquare","a":"ML/Surrogate","t":1,"d":"For a nonnegative bound c, the clamped square clampedSq c t never exceeds c², for every real t.","s":"∀ {c : ℝ}, 0 ≤ c → ∀ (t : ℝ), Causalean.ML.clampedSq c t ≤ c ^ 2","l":47,"f":"Causalean/ML/Surrogate/ClampedSquare.lean"},{"n":"Causalean.ML.clampedSq_nonneg","k":"theorem","m":"Causalean.ML.Surrogate.ClampedSquare","a":"ML/Surrogate","t":2,"d":"The clamped square is nonnegative.","s":"∀ (c t : ℝ), 0 ≤ Causalean.ML.clampedSq c t","l":44,"f":"Causalean/ML/Surrogate/ClampedSquare.lean"},{"n":"Causalean.ML.continuousAt_matrixInv_entry","k":"theorem","m":"Causalean.ML.Ridge.Rate","a":"ML/Ridge","t":2,"d":"Continuity of an inverse-matrix entry. At any matrix with nonzero determinant, the map M ↦ (M⁻¹)ₖₗ is continuous.","s":"∀ {K : Type u_3} [inst : Fintype K] [inst_1 : DecidableEq K] {G₀ : Matrix K K ℝ}, G₀.det ≠ 0 → ∀ (k l : K), ContinuousAt (fun M => M⁻¹ k l) G₀","l":367,"f":"Causalean/ML/Ridge/Rate.lean"},{"n":"Causalean.ML.continuous_clampedSq","k":"theorem","m":"Causalean.ML.Surrogate.ClampedSquare","a":"ML/Surrogate","t":2,"d":"The clamped square is continuous.","s":"∀ (c : ℝ), Continuous (Causalean.ML.clampedSq c)","l":35,"f":"Causalean/ML/Surrogate/ClampedSquare.lean"},{"n":"Causalean.ML.continuous_logisticEmpRisk","k":"theorem","m":"Causalean.ML.Binary.Logistic","a":"ML/Binary","t":2,"d":"The empirical logistic risk is continuous.","s":"∀ {ι : Type u_1} {E : Type u_2} [inst : Fintype ι] [inst_1 : NormedAddCommGroup E] [inst_2 : InnerProductSpace ℝ E] (Z : ι → E × Bool), Continuous (Causalean.ML.logisticEmpRisk Z)","l":103,"f":"Causalean/ML/Binary/Logistic.lean"},{"n":"Causalean.ML.convexOn_comp_inner","k":"theorem","m":"Causalean.ML.Core.Convex","a":"ML/Core","t":2,"d":"A convex scalar function precomposed with the linear score θ ↦ ⟪θ, x⟫ is convex. This is the workhorse turning convexity of a margin loss into convexity of the parameter objective.","s":"∀ {E : Type u_1} [inst : NormedAddCommGroup E] [inst_1 : InnerProductSpace ℝ E] {φ : ℝ → ℝ}, ConvexOn ℝ Set.univ φ → ∀ (x : E), ConvexOn ℝ Set.univ fun θ => φ (inner ℝ θ x)","l":64,"f":"Causalean/ML/Core/Convex.lean"},{"n":"Causalean.ML.convexOn_l1penalty","k":"theorem","m":"Causalean.ML.Lasso.Finite","a":"ML/Lasso","t":1,"d":"The L1 penalty is convex as a function of the coefficient vector.","s":"∀ {Param : Type u_2} [inst : Fintype Param], ConvexOn ℝ Set.univ Causalean.ML.l1penalty","l":42,"f":"Causalean/ML/Lasso/Finite.lean"},{"n":"Causalean.ML.convexOn_lassoObjective","k":"theorem","m":"Causalean.ML.Lasso.Finite","a":"ML/Lasso","t":2,"d":"The lasso objective is convex for λ ≥ 0.","s":"∀ {Obs : Type u_1} {Param : Type u_2} [inst : Fintype Obs] [inst_1 : Fintype Param] (X : Matrix Obs Param ℝ) (y : Obs → ℝ) {lam : ℝ}, 0 ≤ lam → ConvexOn ℝ Set.univ (Causalean.ML.lassoObjective X y lam)","l":108,"f":"Causalean/ML/Lasso/Finite.lean"},{"n":"Causalean.ML.convexOn_logisticEmpRisk","k":"theorem","m":"Causalean.ML.Binary.Logistic","a":"ML/Binary","t":1,"d":"For any finite sample of feature–label pairs Z, the empirical logistic risk, as a function of the coefficient, is convex.","s":"∀ {ι : Type u_1} {E : Type u_2} [inst : Fintype ι] [inst_1 : NormedAddCommGroup E] [inst_2 : InnerProductSpace ℝ E] (Z : ι → E × Bool), ConvexOn ℝ Set.univ (Causalean.ML.logisticEmpRisk Z)","l":78,"f":"Causalean/ML/Binary/Logistic.lean"},{"n":"Causalean.ML.convexOn_logisticScoreLoss","k":"theorem","m":"Causalean.ML.Binary.Logistic","a":"ML/Binary","t":2,"d":"The score-space logistic loss is convex in the score.","s":"∀ (y : Bool), ConvexOn ℝ Set.univ fun t => Causalean.ML.logisticScoreLoss y t","l":59,"f":"Causalean/ML/Binary/Logistic.lean"},{"n":"Causalean.ML.convexOn_olsObjective","k":"theorem","m":"Causalean.ML.Lasso.Finite","a":"ML/Lasso","t":2,"d":"The least-squares objective is convex in the coefficients.","s":"∀ {Obs : Type u_1} {Param : Type u_2} [inst : Fintype Obs] [inst_1 : Fintype Param] (X : Matrix Obs Param ℝ) (y : Obs → ℝ), ConvexOn ℝ Set.univ (Causalean.ML.olsObjective X y)","l":66,"f":"Causalean/ML/Lasso/Finite.lean"},{"n":"Causalean.ML.convexOn_softplus","k":"theorem","m":"Causalean.ML.Binary.Logistic","a":"ML/Binary","t":2,"d":"Softplus is convex.","s":"ConvexOn ℝ Set.univ Causalean.ML.softplus","l":52,"f":"Causalean/ML/Binary/Logistic.lean"},{"n":"Causalean.ML.eLpNorm_predictor_sub_le","k":"theorem","m":"Causalean.ML.Ridge.Rate","a":"ML/Ridge","t":2,"d":"Deterministic predictor Lipschitz bound. The prediction L²(P_X) error is bounded by a finite constant times the coefficient error: ‖∑ₖ δₖ φ·ₖ‖_{L²} ≤ C ‖δ‖.","s":"∀ {γ : Type u_2} {K : Type u_3} [inst : MeasurableSpace γ] [inst_1 : Fintype K] (φ : Causalean.ML.FeatureMap γ K) (P : MeasureTheory.Measure (γ × ℝ)) [MeasureTheory.IsProbabilityMeasure P], (∀ (k : K), Measurable fun x => φ.φ x k) → MeasureTheory.Integrable (fun z => (∑ k, φ.φ z.1 k ^ 2) ^ 2) P → ∀ (βstar : K → ℝ), ∃ C, 0 ≤ C ∧ ∀ (β : K → ℝ), (MeasureTheory.eLpNorm (fun x => ∑ k, β k * φ.φ x k - ∑ k, βstar k * φ.φ x k) 2 (MeasureTheory.Measure.map Prod.fst P)).toReal ≤ C * ‖β - βstar‖","l":613,"f":"Causalean/ML/Ridge/Rate.lean"},{"n":"Causalean.ML.effectiveDimension","k":"def","m":"Causalean.ML.Kernel.EffectiveDimension","a":"ML/Kernel","t":1,"d":"The effective dimension N(λ) = ∑ᵢ μᵢ/(μᵢ+λ) of an operator with eigenvalue family μ at regularization level λ.","s":"{ι : Type u_1} → (ι → ℝ) → ℝ → ℝ","l":44,"f":"Causalean/ML/Kernel/EffectiveDimension.lean"},{"n":"Causalean.ML.effectiveDimension_antitone","k":"theorem","m":"Causalean.ML.Kernel.EffectiveDimension","a":"ML/Kernel","t":2,"d":"Monotonicity in the regularization level. More regularization shrinks the effective dimension: λ₁ ≤ λ₂ ⇒ N(λ₂) ≤ N(λ₁).","s":"∀ {ι : Type u_1} {μ : ι → ℝ} {lam₁ lam₂ : ℝ}, (∀ (i : ι), 0 ≤ μ i) → 0 < lam₁ → lam₁ ≤ lam₂ → Summable μ → Causalean.ML.effectiveDimension μ lam₂ ≤ Causalean.ML.effectiveDimension μ lam₁","l":102,"f":"Causalean/ML/Kernel/EffectiveDimension.lean"},{"n":"Causalean.ML.effectiveDimension_le_card","k":"theorem","m":"Causalean.ML.Kernel.EffectiveDimension","a":"ML/Kernel","t":2,"d":"Over finitely many eigenvalues, N(λ) never exceeds the ambient dimension #ι.","s":"∀ {ι : Type u_1} [inst : Fintype ι] {μ : ι → ℝ} {lam : ℝ}, (∀ (i : ι), 0 ≤ μ i) → 0 < lam → Causalean.ML.effectiveDimension μ lam ≤ ↑(Fintype.card ι)","l":92,"f":"Causalean/ML/Kernel/EffectiveDimension.lean"},{"n":"Causalean.ML.effectiveDimension_le_trace_div","k":"theorem","m":"Causalean.ML.Kernel.EffectiveDimension","a":"ML/Kernel","t":1,"d":"Dimension-free bound. For an eigenvalue family μ and regularization level lam, if every eigenvalue is nonnegative, the regularization level is strictly positive, and the eigenvalues are summable, i.e. the operator is trace-class, then the effective dimension N(lam) = ∑ᵢ μᵢ/(μᵢ+lam) is at most the trace ∑ᵢ μᵢ divided by lam. This is the bound that, with eigenvalue decay μᵢ ≍ i^{-b}, yields N(λ) = O(λ^{-1/b}).","s":"∀ {ι : Type u_1} {μ : ι → ℝ} {lam : ℝ}, (∀ (i : ι), 0 ≤ μ i) → 0 < lam → Summable μ → Causalean.ML.effectiveDimension μ lam ≤ (∑' (i : ι), μ i) / lam","l":78,"f":"Causalean/ML/Kernel/EffectiveDimension.lean"},{"n":"Causalean.ML.effectiveDimension_nonneg","k":"theorem","m":"Causalean.ML.Kernel.EffectiveDimension","a":"ML/Kernel","t":2,"d":"The effective dimension is nonnegative.","s":"∀ {ι : Type u_1} {μ : ι → ℝ} {lam : ℝ}, (∀ (i : ι), 0 ≤ μ i) → 0 < lam → 0 ≤ Causalean.ML.effectiveDimension μ lam","l":73,"f":"Causalean/ML/Kernel/EffectiveDimension.lean"},{"n":"Causalean.ML.effectiveDimension_term_le_div","k":"theorem","m":"Causalean.ML.Kernel.EffectiveDimension","a":"ML/Kernel","t":2,"d":"Each summand is dominated by μᵢ/λ.","s":"∀ {ι : Type u_1} {μ : ι → ℝ} {lam : ℝ}, (∀ (i : ι), 0 ≤ μ i) → 0 < lam → ∀ (i : ι), μ i / (μ i + lam) ≤ μ i / lam","l":54,"f":"Causalean/ML/Kernel/EffectiveDimension.lean"},{"n":"Causalean.ML.effectiveDimension_term_le_one","k":"theorem","m":"Causalean.ML.Kernel.EffectiveDimension","a":"ML/Kernel","t":2,"d":"Each summand is at most 1.","s":"∀ {ι : Type u_1} {μ : ι → ℝ} {lam : ℝ}, (∀ (i : ι), 0 ≤ μ i) → 0 < lam → ∀ (i : ι), μ i / (μ i + lam) ≤ 1","l":59,"f":"Causalean/ML/Kernel/EffectiveDimension.lean"},{"n":"Causalean.ML.effectiveDimension_term_nonneg","k":"theorem","m":"Causalean.ML.Kernel.EffectiveDimension","a":"ML/Kernel","t":2,"d":"Each summand μᵢ/(μᵢ+λ) is nonnegative (for nonnegative eigenvalues and λ > 0).","s":"∀ {ι : Type u_1} {μ : ι → ℝ} {lam : ℝ}, (∀ (i : ι), 0 ≤ μ i) → 0 < lam → ∀ (i : ι), 0 ≤ μ i / (μ i + lam)","l":49,"f":"Causalean/ML/Kernel/EffectiveDimension.lean"},{"n":"Causalean.ML.empiricalCross","k":"def","m":"Causalean.ML.Ridge.Rate","a":"ML/Ridge","t":1,"d":"Empirical feature–response cross moment: Ĉₙ = n⁻¹ Σ_{i<n} yᵢ φ(xᵢ).","s":"{Ω : Type u_1} → {γ : Type u_2} → {K : Type u_3} → [inst : Fintype K] → Causalean.ML.FeatureMap γ K → (ℕ → Ω → γ × ℝ) → ℕ → Ω → K → ℝ","l":49,"f":"Causalean/ML/Ridge/Rate.lean"},{"n":"Causalean.ML.empiricalGram","k":"def","m":"Causalean.ML.Ridge.Rate","a":"ML/Ridge","t":1,"d":"Empirical feature Gram over the first n sample points: Ĝₙ = n⁻¹ Σ_{i<n} φ(xᵢ) φ(xᵢ)ᵀ.","s":"{Ω : Type u_1} → {γ : Type u_2} → {K : Type u_3} → [inst : Fintype K] → Causalean.ML.FeatureMap γ K → (ℕ → Ω → γ × ℝ) → ℕ → Ω → Matrix K K ℝ","l":42,"f":"Causalean/ML/Ridge/Rate.lean"},{"n":"Causalean.ML.empiricalGram_posSemidef","k":"theorem","m":"Causalean.ML.Ridge.Rate","a":"ML/Ridge","t":2,"d":"The empirical Gram is positive semidefinite (average of rank-one φφᵀ).","s":"∀ {Ω : Type u_1} {γ : Type u_2} {K : Type u_3} [inst : Fintype K] (φ : Causalean.ML.FeatureMap γ K) (Z : ℕ → Ω → γ × ℝ) (n : ℕ) (ω : Ω), (Causalean.ML.empiricalGram φ Z n ω).PosSemidef","l":178,"f":"Causalean/ML/Ridge/Rate.lean"},{"n":"Causalean.ML.empiricalRademacherComplexity_innerBall_le","k":"theorem","m":"Causalean.ML.Kernel.RKHSRademacher","a":"ML/Kernel","t":2,"d":"For a sample of vectors bounded by κ and linear weights bounded by r, the empirical Rademacher complexity of the resulting linear class is at most κ times r divided by the square root of the sample size.","s":"∀ {E : Type u_1} [inst : NormedAddCommGroup E] [inst_1 : InnerProductSpace ℝ E] {ι : Type u_2} {n : ℕ} (κ r : ℝ), 0 ≤ κ → 0 ≤ r → ∀ (Y : Fin n → E), (∀ (k : Fin n), ‖Y k‖ ≤ κ) → ∀ (w : ι → E), (∀ (i : ι), ‖w i‖ ≤ r) → empiricalRademacherComplexity n (fun i a => inner ℝ (w i) a) Y ≤ κ * r / √↑n","l":442,"f":"Causalean/ML/Kernel/RKHSRademacher.lean"},{"n":"Causalean.ML.empiricalRisk","k":"def","m":"Causalean.ML.Core.Risk","a":"ML/Core","t":1,"d":"The empirical risk criterion of a prediction rule is the sample loss sum scaled by the inverse cardinality of a nonempty finite sample index. The nonemptiness assumption rules out the empty-sample convention where the average would collapse to zero without data.","s":"{ι : Type u_1} → {X : Type u_2} → {Y : Type u_3} → [Fintype ι] → [Nonempty ι] → Causalean.ML.Loss Y → (ι → X × Y) → (X → Y) → ℝ","l":28,"f":"Causalean/ML/Core/Risk.lean"},{"n":"Causalean.ML.empiricalRiskP","k":"def","m":"Causalean.ML.Core.Risk","a":"ML/Core","t":1,"d":"The empirical risk criterion of a model parameter is the finite-sample criterion over a nonempty sample applied to the prediction rule selected by that parameter.","s":"{ι : Type u_1} → {Θ : Type u_2} → {X : Type u_3} → {Y : Type u_4} → [Fintype ι] → [Nonempty ι] → Causalean.ML.Predictor Θ X Y → Causalean.ML.Loss Y → (ι → X × Y) → Θ → ℝ","l":36,"f":"Causalean/ML/Core/Risk.lean"},{"n":"Causalean.ML.empiricalRiskP_convexOn_of_loss_convex","k":"theorem","m":"Causalean.ML.Core.Convex","a":"ML/Core","t":2,"d":"If every per-sample parameter loss is convex on Θset, so is the empirical risk of the predictor.","s":"∀ {ι : Type u_1} {Θ : Type u_2} {X : Type u_3} {Y : Type u_4} [inst : Fintype ι] [inst_1 : Nonempty ι] [inst_2 : AddCommGroup Θ] [inst_3 : Module ℝ Θ] (M : Causalean.ML.Predictor Θ X Y) (loss : Causalean.ML.Loss Y) (S : ι → X × Y) (Θset : Set Θ), Convex ℝ Θset → (∀ (i : ι), ConvexOn ℝ Θset fun θ => loss (M.predict θ (S i).1) (S i).2) → ConvexOn ℝ Θset (Causalean.ML.empiricalRiskP M loss S)","l":27,"f":"Causalean/ML/Core/Convex.lean"},{"n":"Causalean.ML.empiricalRisk_squaredLoss_linear","k":"theorem","m":"Causalean.ML.Linear.Finite","a":"ML/Linear","t":2,"d":"Bridge to the spine: the empirical squared-loss risk of the linear-in-features predictor x ↦ ⟪β, φ x⟫ equals (card ι)⁻¹ times the OLS objective of the design matrix Xᵢₖ = (φ xᵢ)ₖ.","s":"∀ {ι : Type u_3} {K : Type u_4} {X' : Type u_5} [inst : Fintype ι] [inst_1 : Nonempty ι] [inst_2 : Fintype K] (φ : Causalean.ML.FeatureMap X' K) (S : ι → X' × ℝ) (β : K → ℝ), (Causalean.ML.empiricalRisk Causalean.ML.squaredLoss S fun x => ∑ k, β k * φ.φ x k) = (↑(Fintype.card ι))⁻¹ * Causalean.ML.olsObjective (fun i k => φ.φ (S i).1 k) (fun i => (S i).2) β","l":85,"f":"Causalean/ML/Linear/Finite.lean"},{"n":"Causalean.ML.evalLayers","k":"def","m":"Causalean.ML.NeuralNet.FeedForward","a":"ML/NeuralNet","t":1,"d":"Evaluate a uniform-width feedforward network (a list of layers), folding left-to-right.","s":"{n : ℕ} → Causalean.ML.Activation → List (Causalean.ML.DenseLayer n n) → (Fin n → ℝ) → Fin n → ℝ","l":25,"f":"Causalean/ML/NeuralNet/FeedForward.lean"},{"n":"Causalean.ML.evalLayers_append","k":"theorem","m":"Causalean.ML.NeuralNet.FeedForward","a":"ML/NeuralNet","t":2,"d":"Structure — composition. Evaluating concatenated layer lists is the composition of the two evaluations.","s":"∀ {n : ℕ} (σ : Causalean.ML.Activation) (Ls Ms : List (Causalean.ML.DenseLayer n n)) (x : Fin n → ℝ), Causalean.ML.evalLayers σ (Ls ++ Ms) x = Causalean.ML.evalLayers σ Ms (Causalean.ML.evalLayers σ Ls x)","l":31,"f":"Causalean/ML/NeuralNet/FeedForward.lean"},{"n":"Causalean.ML.evalLayers_lipschitz","k":"theorem","m":"Causalean.ML.NeuralNet.FeedForward","a":"ML/NeuralNet","t":1,"d":"Structure — Lipschitz. For a uniform-width feedforward network with activation σ and layer list Ls, if each layer's affine-then-activation map is Lipschitz with the constant assigned to it by k, then the whole network evaluation is Lipschitz with constant equal to the product of the per-layer constants.","s":"∀ {n : ℕ} (σ : Causalean.ML.Activation) (Ls : List (Causalean.ML.DenseLayer n n)) (k : Causalean.ML.DenseLayer n n → NNReal), (∀ L ∈ Ls, LipschitzWith (k L) (Causalean.ML.layerMap σ L)) → LipschitzWith (List.map k Ls).prod (Causalean.ML.evalLayers σ Ls)","l":40,"f":"Causalean/ML/NeuralNet/FeedForward.lean"},{"n":"Causalean.ML.exists_isMinOn_of_isCompact","k":"theorem","m":"Causalean.ML.Core.Convex","a":"ML/Core","t":2,"d":"Existence of a minimizer on a nonempty compact set (Weierstrass).","s":"∀ {Θ : Type u_1} [inst : TopologicalSpace Θ] {objective : Θ → ℝ} {Θset : Set Θ}, Θset.Nonempty → IsCompact Θset → ContinuousOn objective Θset → ∃ θhat ∈ Θset, IsMinOn objective Θset θhat","l":78,"f":"Causalean/ML/Core/Convex.lean"},{"n":"Causalean.ML.exists_isMinOn_univ_of_coercive","k":"theorem","m":"Causalean.ML.Core.Convex","a":"ML/Core","t":1,"d":"Existence via coercivity. On a proper normed real vector space, if the objective function is continuous and it tends to infinity along the cocompact filter, i.e. it is coercive — it grows without bound as the argument leaves every compact set, then a global minimizer of the objective over the whole space exists.","s":"∀ {E : Type u_1} [inst : NormedAddCommGroup E] [ProperSpace E] {objective : E → ℝ}, Continuous objective → Filter.Tendsto objective (Filter.cocompact E) Filter.atTop → ∃ a, IsMinOn objective Set.univ a","l":91,"f":"Causalean/ML/Core/Convex.lean"},{"n":"Causalean.ML.fderiv_eq_zero_of_isMinOn_univ","k":"theorem","m":"Causalean.ML.Core.Convex","a":"ML/Core","t":2,"d":"First-order optimality on the whole space: at an unconstrained minimizer the Fréchet derivative vanishes.","s":"∀ {E : Type u_1} [inst : NormedAddCommGroup E] [inst_1 : NormedSpace ℝ E] {f : E → ℝ} {f' : E →L[ℝ] ℝ} {a : E}, IsMinOn f Set.univ a → HasFDerivAt f f' a → f' = 0","l":109,"f":"Causalean/ML/Core/Convex.lean"},{"n":"Causalean.ML.gram","k":"def","m":"Causalean.ML.Kernel.Gram","a":"ML/Kernel","t":1,"d":"The Gram matrix Kᵢⱼ = k(xᵢ, xⱼ) of a kernel k on a finite sample x.","s":"{X : Type u_1} → {n : ℕ} → (X → X → ℝ) → (Fin n → X) → Matrix (Fin n) (Fin n) ℝ","l":22,"f":"Causalean/ML/Kernel/Gram.lean"},{"n":"Causalean.ML.gram_add_lambda_posDef","k":"theorem","m":"Causalean.ML.Kernel.Gram","a":"ML/Kernel","t":1,"d":"For a finite sample x and kernel k, if k is positive semidefinite, i.e. every Gram matrix it produces is PSD and the regularization level lam is strictly positive, then the regularized Gram matrix gram k x + lam·I is positive definite, hence invertible.","s":"∀ {X : Type u_1} {n : ℕ} {k : X → X → ℝ}, Causalean.ML.IsPSDkernel k → ∀ (x : Fin n → X) {lam : ℝ}, 0 < lam → (Causalean.ML.gram k x + lam • 1).PosDef","l":31,"f":"Causalean/ML/Kernel/Gram.lean"},{"n":"Causalean.ML.iidEmpiricalRisk","k":"def","m":"Causalean.ML.Core.Risk","a":"ML/Core","t":1,"d":"The empirical risk along the first n points of an i.i.d. sample, expressed through Stat.IIDSample.sampleMean.","s":"{Ω : Type u_1} → {X : Type u_2} → {Y : Type u_3} → [inst : MeasurableSpace Ω] → [inst_1 : MeasurableSpace (X × Y)] → {μ : MeasureTheory.Measure Ω} → {P : MeasureTheory.Measure (X × Y)} → Causalean.Stat.IIDSample Ω (X × Y) μ P → Causalean.ML.Loss Y → (X → Y) → ℕ → Ω → ℝ","l":56,"f":"Causalean/ML/Core/Risk.lean"},{"n":"Causalean.ML.imageClass","k":"def","m":"Causalean.ML.Core.Bridge","a":"ML/Core","t":1,"d":"The hypothesis class realized by a predictor: the image of its admissible parameter set under the prediction map.","s":"{Θ : Type u_1} → {X : Type u_2} → {Y : Type u_3} → [inst : MeasurableSpace X] → [inst_1 : MeasurableSpace Y] → (M : Causalean.ML.Predictor Θ X Y) → (∀ θ ∈ M.paramSet, Measurable (M.predict θ)) → Causalean.ML.HypothesisClass X Y","l":22,"f":"Causalean/ML/Core/Bridge.lean"},{"n":"Causalean.ML.isERMP_to_extensional","k":"theorem","m":"Causalean.ML.Core.Bridge","a":"ML/Core","t":2,"d":"A parametric empirical-risk minimizer pushes forward to an extensional empirical-risk minimizer over the realized hypothesis class.","s":"∀ {ι : Type u_1} {Θ : Type u_2} {X : Type u_3} {Y : Type u_4} [inst : Fintype ι] [inst_1 : Nonempty ι] [inst_2 : MeasurableSpace X] [inst_3 : MeasurableSpace Y] (M : Causalean.ML.Predictor Θ X Y) (loss : Causalean.ML.Loss Y) (S : ι → X × Y) (hmeas : ∀ θ ∈ M.paramSet, Measurable (M.predict θ)) {θhat : Θ}, Causalean.ML.IsERMP (Causalean.ML.empiricalRiskP M loss S) M.paramSet θhat → Causalean.ML.IsERM (Causalean.ML.imageClass M hmeas) loss S (M.predict θhat)","l":32,"f":"Causalean/ML/Core/Bridge.lean"},{"n":"Causalean.ML.isLittleOp_mul_quarter","k":"theorem","m":"Causalean.ML.Core.Rate","a":"ML/Core","t":1,"d":"The n^{-1/4} product rule. If the sequence Xn is o_p(n^{-1/4}) and the sequence Yn is o_p(n^{-1/4}) under the probability law μ, then their pointwise product Xn·Yn is o_p(n^{-1/2}) — the DML product-rate condition from per-nuisance n^{-1/4} rates.","s":"∀ {Ω : Type u_1} [inst : MeasurableSpace Ω] {μ : MeasureTheory.Measure Ω} {Xn Yn : ℕ → Ω → ℝ}, Causalean.Stat.IsLittleOp Xn (fun n => ↑n ^ (-(1 / 4))) μ → Causalean.Stat.IsLittleOp Yn (fun n => ↑n ^ (-(1 / 4))) μ → Causalean.Stat.IsLittleOp (fun n ω => Xn n ω * Yn n ω) (fun n => ↑n ^ (-(1 / 2))) μ","l":43,"f":"Causalean/ML/Core/Rate.lean"},{"n":"Causalean.ML.isLittleOp_one_of_le_one","k":"theorem","m":"Causalean.ML.Core.Rate","a":"ML/Core","t":2,"d":"Rate weakening: an o_p(rn) sequence with rn ≤ 1 is o_p(1).","s":"∀ {Ω : Type u_1} [inst : MeasurableSpace Ω] {μ : MeasureTheory.Measure Ω} {Xn : ℕ → Ω → ℝ} {rn : ℕ → ℝ}, (∀ (n : ℕ), rn n ≤ 1) → Causalean.Stat.IsLittleOp Xn rn μ → Causalean.Stat.IsLittleOp Xn (fun x => 1) μ","l":29,"f":"Causalean/ML/Core/Rate.lean"},{"n":"Causalean.ML.isLittleOp_quarter_of_isBigOp_sqrt","k":"theorem","m":"Causalean.ML.Core.Rate","a":"ML/Core","t":2,"d":"An O_p(n^{-1/2}) (root-n) rate is o_p(n^{-1/4}): a method's root-n estimation rate clears the DML o_p(n^{-1/4}) bar.","s":"∀ {Ω : Type u_1} [inst : MeasurableSpace Ω] {μ : MeasureTheory.Measure Ω} {Xn : ℕ → Ω → ℝ}, Causalean.Stat.IsBigOp Xn (fun n => (√↑n)⁻¹) μ → Causalean.Stat.IsLittleOp Xn (fun n => ↑n ^ (-(1 / 4))) μ","l":106,"f":"Causalean/ML/Core/Rate.lean"},{"n":"Causalean.ML.kernel_erm_excess_rate","k":"theorem","m":"Causalean.ML.Kernel.Rate","a":"ML/Kernel","t":1,"d":"Kernel ERM excess-risk rate over the L² ball. For features valued in the closed Xb-ball and linear predictors indexed by the closed W-ball, if the feature bound Xb is nonnegative, the weight bound W is nonnegative, the feature map X is measurable, the constant t satisfies the calibration t·(Xb·W)² ≤ 1/2, the tolerance ε is nonnegative, and the estimator ŵ attains empirical risk no larger than that of the comparator wstar, then the probability that the excess population risk of ŵ over wstar exceeds 4·Xb·W/√n + 2ε is at most exp(-ε²tn).","s":"∀ {d n : ℕ} {Ω : Type u_1} [inst : MeasurableSpace Ω] {μ : MeasureTheory.Measure Ω} [MeasureTheory.IsProbabilityMeasure μ] {Xb W : ℝ}, 0 ≤ Xb → 0 ≤ W → ∀ (X : Ω → ↑(Metric.closedBall 0 Xb)), Measurable X → ∀ {t : ℝ}, t * (Xb * W) ^ 2 ≤ 1 / 2 → ∀ {ε : ℝ}, 0 ≤ ε → ∀ (ŵ : (Fin n → Ω) → ↑(Metric.closedBall 0 W)) (wstar : ↑(Metric.closedBall 0 W)), (∀ (ω : Fin n → Ω), (↑n)⁻¹ * ∑ k, inner ℝ ↑(ŵ ω) ↑(X (ω k)) ≤ (↑n)⁻¹ * ∑ k, inner ℝ ↑wstar ↑(X (ω k))) → ((MeasureTheory.Measure.pi fun x => μ) fun ω => 4 * (Xb * W / √↑n) + 2 * ε < ∫ (x : Ω), (fun ω' => inner ℝ ↑(ŵ ω) ↑(X ω')) x ∂μ - ∫ (x : Ω), (fun ω' => inner ℝ ↑wstar ↑(X ω')) x ∂μ).toReal ≤ Real.exp (-ε ^ 2 * t * ↑n)","l":82,"f":"Causalean/ML/Kernel/Rate.lean"},{"n":"Causalean.ML.kernel_erm_squaredLoss_excess_rate","k":"theorem","m":"Causalean.ML.Kernel.SquaredLoss","a":"ML/Kernel","t":1,"d":"Kernel ERM squared-loss excess-risk rate over the L² ball. For data pairing features in the closed Xb-ball with responses in [-Yb, Yb], and linear predictors indexed by the closed W-ball, if Xb is nonnegative, Yb is nonnegative, W is nonnegative, the data map X is measurable, the constant t satisfies the calibration t·((Xb·W)² + 2·Yb·Xb·W)² ≤ 1/2, the tolerance ε is nonnegative, and the estimator ŵ attains empirical squared loss no larger than that of the comparator wstar, then for the squared loss (⟪w,x⟩ − y)², the probability that the excess population risk of ŵ over wstar exceeds 4·(4(XbW)² + 2·Yb·Xb·W)/√n + 2ε is at most exp(-ε²tn).","s":"∀ {d n : ℕ} {Ω : Type u_1} [inst : MeasurableSpace Ω] {μ : MeasureTheory.Measure Ω} [MeasureTheory.IsProbabilityMeasure μ] {Xb Yb W : ℝ}, 0 ≤ Xb → 0 ≤ Yb → 0 ≤ W → ∀ (X : Ω → Causalean.ML.KFeat d Xb Yb), Measurable X → ∀ {t : ℝ}, t * ((Xb * W) ^ 2 + 2 * Yb * (Xb * W)) ^ 2 ≤ 1 / 2 → ∀ {ε : ℝ}, 0 ≤ ε → ∀ (ŵ : (Fin n → Ω) → Causalean.ML.KWeight d W) (wstar : Causalean.ML.KWeight d W), (∀ (ω : Fin n → Ω), (↑n)⁻¹ * ∑ k, (inner ℝ ↑(ŵ ω) ↑(X (ω k)).1 - ↑(X (ω k)).2) ^ 2 ≤ (↑n)⁻¹ * ∑ k, (inner ℝ ↑wstar ↑(X (ω k)).1 - ↑(X (ω k)).2) ^ 2) → ((MeasureTheory.Measure.pi fun x => μ) fun ω => 4 * ((4 * (Xb * W) ^ 2 + 2 * Yb * Xb * W) / √↑n) + 2 * ε < ∫ (x : Ω), (fun ω' => (inner ℝ ↑(ŵ ω) ↑(X ω').1 - ↑(X ω').2) ^ 2) x ∂μ - ∫ (x : Ω), (fun ω' => (inner ℝ ↑wstar ↑(X ω').1 - ↑(X ω').2) ^ 2) x ∂μ).toReal ≤ Real.exp (-ε ^ 2 * t * ↑n)","l":42,"f":"Causalean/ML/Kernel/SquaredLoss.lean"},{"n":"Causalean.ML.krrRisk","k":"def","m":"Causalean.ML.Kernel.Ridge","a":"ML/Kernel","t":1,"d":"The kernel-ridge regularized empirical risk of an RKHS element f: (1/n) ∑ᵢ (f(xᵢ) − yᵢ)² + λ‖f‖².","s":"{X : Type u_1} → {H : Type u_2} → [inst : NormedAddCommGroup H] → [InnerProductSpace ℝ H] → (H → X → ℝ) → {n : ℕ} → (Fin n → X) → (Fin n → ℝ) → ℝ → H → ℝ","l":23,"f":"Causalean/ML/Kernel/Ridge.lean"},{"n":"Causalean.ML.l1penalty","k":"def","m":"Causalean.ML.Lasso.Finite","a":"ML/Lasso","t":1,"d":"The L1 penalty ‖β‖₁ = ∑ₖ |βₖ|.","s":"{Param : Type u_2} → [Fintype Param] → (Param → ℝ) → ℝ","l":26,"f":"Causalean/ML/Lasso/Finite.lean"},{"n":"Causalean.ML.l1penalty_nonneg","k":"theorem","m":"Causalean.ML.Lasso.Finite","a":"ML/Lasso","t":2,"d":"The L1 penalty is nonnegative.","s":"∀ {Param : Type u_2} [inst : Fintype Param] (β : Param → ℝ), 0 ≤ Causalean.ML.l1penalty β","l":38,"f":"Causalean/ML/Lasso/Finite.lean"},{"n":"Causalean.ML.lassoObjective","k":"def","m":"Causalean.ML.Lasso.Finite","a":"ML/Lasso","t":1,"d":"The lasso objective: least-squares error plus the L1 penalty λ‖β‖₁.","s":"{Obs : Type u_1} → {Param : Type u_2} → [Fintype Obs] → [Fintype Param] → Matrix Obs Param ℝ → (Obs → ℝ) → ℝ → (Param → ℝ) → ℝ","l":29,"f":"Causalean/ML/Lasso/Finite.lean"},{"n":"Causalean.ML.lasso_erm_excess_rate","k":"theorem","m":"Causalean.ML.Lasso.Rate","a":"ML/Lasso","t":1,"d":"Lasso ERM excess-risk rate over the L¹ ball. For linear predictors indexed by the L¹ ball, if the dimension d is positive, the sample size n is positive, the coordinatewise feature bound Xinf is nonnegative, the weight bound W is nonnegative, the feature map X is measurable, the constant t satisfies the calibration t·(Xinf·W)² ≤ 1/2, the tolerance ε is nonnegative, and the estimator ŵ attains empirical risk no larger than that of the comparator wstar, then the probability that the excess population risk of ŵ over wstar exceeds 4·(Xinf·W/√n)·√(2 log 2d) + 2ε is at most exp(-ε²tn).","s":"∀ {d n : ℕ}, 0 < d → 0 < n → ∀ {Ω : Type u_1} [inst : MeasurableSpace Ω] {μ : MeasureTheory.Measure Ω} [MeasureTheory.IsProbabilityMeasure μ] {Xinf W : ℝ}, 0 ≤ Xinf → 0 ≤ W → ∀ (X : Ω → LinftyBall Xinf), (Measurable fun ω => ↑(X ω)) → ∀ {t : ℝ}, t * (Xinf * W) ^ 2 ≤ 1 / 2 → ∀ {ε : ℝ}, 0 ≤ ε → ∀ (ŵ : (Fin n → Ω) → L1Ball W) (wstar : L1Ball W), (∀ (ω : Fin n → Ω), (↑n)⁻¹ * ∑ k, ∑ j, (↑(ŵ ω)).ofLp j * (↑(X (ω k))).ofLp j ≤ (↑n)⁻¹ * ∑ k, ∑ j, (↑wstar).ofLp j * (↑(X (ω k))).ofLp j) → ((MeasureTheory.Measure.pi fun x => μ) fun ω => 4 * (Xinf * W / √↑n * √(2 * Real.log (2 * ↑d))) + 2 * ε < ∫ (x : Ω), (fun ω' => ∑ j, (↑(ŵ ω)).ofLp j * (↑(X ω')).ofLp j) x ∂μ - ∫ (x : Ω), (fun ω' => ∑ j, (↑wstar).ofLp j * (↑(X ω')).ofLp j) x ∂μ).toReal ≤ Real.exp (-ε ^ 2 * t * ↑n)","l":73,"f":"Causalean/ML/Lasso/Rate.lean"},{"n":"Causalean.ML.lasso_erm_squaredLoss_excess_rate","k":"theorem","m":"Causalean.ML.Lasso.SquaredLoss","a":"ML/Lasso","t":1,"d":"Lasso ERM squared-loss excess-risk rate over the L¹ ball. For data pairing coordinatewise-bounded features with bounded responses, and predictors indexed by the coordinate L¹ ball of radius W, if the dimension d is positive, the sample size n is positive, the coordinatewise feature bound Xinf is nonnegative, the response bound Yb is nonnegative, the weight bound W is nonnegative, the feature coordinate map is measurable, the response coordinate map is measurable, the constant t satisfies the calibration t·((Xinf·W)² + 2·Yb·Xinf·W)² ≤ 1/2, the tolerance ε is nonnegative, and the estimator ŵ attains empirical squared loss no larger than that of the comparator wstar, then for the squared regression loss (∑ⱼ wⱼxⱼ - y)², the probability that the excess population risk of ŵ over wstar exceeds 4·((4(XinfW)² + 2·Yb·Xinf·W)/√n)·√(2 log 2d) + 2ε is at most exp(-ε²tn).","s":"∀ {d n : ℕ}, 0 < d → 0 < n → ∀ {Ω : Type u_1} [inst : MeasurableSpace Ω] {μ : MeasureTheory.Measure Ω} [MeasureTheory.IsProbabilityMeasure μ] {Xinf Yb W : ℝ}, 0 ≤ Xinf → 0 ≤ Yb → 0 ≤ W → ∀ (X : Ω → Causalean.ML.LassoFeat d Xinf Yb), (Measurable fun ω => ↑(X ω).1) → (Measurable fun ω => ↑(X ω).2) → ∀ {t : ℝ}, t * ((Xinf * W) ^ 2 + 2 * Yb * (Xinf * W)) ^ 2 ≤ 1 / 2 → ∀ {ε : ℝ}, 0 ≤ ε → ∀ (ŵ : (Fin n → Ω) → Causalean.ML.LassoWeight d W) (wstar : Causalean.ML.LassoWeight d W), (∀ (ω : Fin n → Ω), (↑n)⁻¹ * ∑ k, (∑ j, (↑(ŵ ω)).ofLp j * (↑(X (ω k)).1).ofLp j - ↑(X (ω k)).2) ^ 2 ≤ (↑n)⁻¹ * ∑ k, (∑ j, (↑wstar).ofLp j * (↑(X (ω k)).1).ofLp j - ↑(X (ω k)).2) ^ 2) → ((MeasureTheory.Measure.pi fun x => μ) fun ω => 4 * ((4 * (Xinf * W) ^ 2 + 2 * Yb * Xinf * W) / √↑n * √(2 * Real.log (2 * ↑d))) + 2 * ε < ∫ (x : Ω), (fun ω' => (∑ j, (↑(ŵ ω)).ofLp j * (↑(X ω').1).ofLp j - ↑(X ω').2) ^ 2) x ∂μ - ∫ (x : Ω), (fun ω' => (∑ j, (↑wstar).ofLp j * (↑(X ω').1).ofLp j - ↑(X ω').2) ^ 2) x ∂μ).toReal ≤ Real.exp (-ε ^ 2 * t * ↑n)","l":43,"f":"Causalean/ML/Lasso/SquaredLoss.lean"},{"n":"Causalean.ML.layerMap","k":"def","m":"Causalean.ML.NeuralNet.FeedForward","a":"ML/NeuralNet","t":1,"d":"One network layer: the affine map followed by coordinatewise activation.","s":"{n : ℕ} → Causalean.ML.Activation → Causalean.ML.DenseLayer n n → (Fin n → ℝ) → Fin n → ℝ","l":21,"f":"Causalean/ML/NeuralNet/FeedForward.lean"},{"n":"Causalean.ML.linearPredict","k":"def","m":"Causalean.ML.Linear.Finite","a":"ML/Linear","t":1,"d":"The linear prediction vector X β for a design matrix X and coefficients β.","s":"{Obs : Type u_1} → {Param : Type u_2} → [Fintype Param] → Matrix Obs Param ℝ → (Param → ℝ) → Obs → ℝ","l":26,"f":"Causalean/ML/Linear/Finite.lean"},{"n":"Causalean.ML.linear_predictor_sub_memLp_of_l4","k":"theorem","m":"Causalean.ML.Ridge.Rate","a":"ML/Ridge","t":2,"d":"A finite linear combination of features has finite L² norm under the covariate marginal whenever the feature vector has a finite fourth moment under the joint law.","s":"∀ {γ : Type u_2} {K : Type u_3} [inst : MeasurableSpace γ] [inst_1 : Fintype K] (φ : Causalean.ML.FeatureMap γ K) (P : MeasureTheory.Measure (γ × ℝ)) [MeasureTheory.IsProbabilityMeasure P], (∀ (k : K), Measurable fun x => φ.φ x k) → MeasureTheory.Integrable (fun z => (∑ k, φ.φ z.1 k ^ 2) ^ 2) P → ∀ (β βstar : K → ℝ), MeasureTheory.MemLp (fun x => ∑ k, β k * φ.φ x k - ∑ k, βstar k * φ.φ x k) 2 (MeasureTheory.Measure.map Prod.fst P)","l":561,"f":"Causalean/ML/Ridge/Rate.lean"},{"n":"Causalean.ML.lipschitzAt0_clampedSq","k":"theorem","m":"Causalean.ML.Surrogate.ClampedSquare","a":"ML/Surrogate","t":1,"d":"For a nonnegative bound c, the clamped square clampedSq c is Lipschitz at 0 with constant 2c: it fixes 0 and is globally 2c-Lipschitz.","s":"∀ {c : ℝ}, 0 ≤ c → Causalean.Stat.Concentration.LipschitzAt0 (Causalean.ML.clampedSq c) (2 * c)","l":54,"f":"Causalean/ML/Surrogate/ClampedSquare.lean"},{"n":"Causalean.ML.logisticEmpRisk","k":"def","m":"Causalean.ML.Binary.Logistic","a":"ML/Binary","t":1,"d":"The empirical logistic risk at coefficient β: average score-space log-loss over the sample Z.","s":"{ι : Type u_1} → {E : Type u_2} → [Fintype ι] → [inst : NormedAddCommGroup E] → [InnerProductSpace ℝ E] → (ι → E × Bool) → E → ℝ","l":47,"f":"Causalean/ML/Binary/Logistic.lean"},{"n":"Causalean.ML.logisticPredictor","k":"def","m":"Causalean.ML.Binary.Rate","a":"ML/Binary","t":1,"d":"The logistic predictor x ↦ σ(⟨β, φ(x)⟩) (the fitted conditional probability).","s":"{γ : Type u_2} → {K : Type u_3} → [inst : Fintype K] → Causalean.ML.FeatureMap γ K → (K → ℝ) → γ → ℝ","l":50,"f":"Causalean/ML/Binary/Rate.lean"},{"n":"Causalean.ML.logisticScoreLoss","k":"def","m":"Causalean.ML.Core.Losses","a":"ML/Core","t":1,"d":"Logistic score loss is the binary cross-entropy loss written as a function of the raw prediction score.","s":"Bool → ℝ → ℝ","l":34,"f":"Causalean/ML/Core/Losses.lean"},{"n":"Causalean.ML.logistic_exists_minimizer_on_compact","k":"theorem","m":"Causalean.ML.Binary.Logistic","a":"ML/Binary","t":1,"d":"For any labeled sample Z and a parameter set Θset that is nonempty and compact, the empirical logistic risk attains its minimum over Θset at some parameter βhat in Θset.","s":"∀ {ι : Type u_1} {E : Type u_2} [inst : Fintype ι] [inst_1 : NormedAddCommGroup E] [inst_2 : InnerProductSpace ℝ E] (Z : ι → E × Bool) {Θset : Set E}, Θset.Nonempty → IsCompact Θset → ∃ βhat ∈ Θset, IsMinOn (Causalean.ML.logisticEmpRisk Z) Θset βhat","l":124,"f":"Causalean/ML/Binary/Logistic.lean"},{"n":"Causalean.ML.margin_erm_surrogate_excess_rate","k":"theorem","m":"Causalean.ML.Margin.Rate","a":"ML/Margin","t":1,"d":"Margin surrogate ERM excess-risk rate. Fix nonnegative feature-, label-, and weight-ball radii Xb, Yb, W and a nonnegative Lipschitz constant L for a surrogate loss φ satisfying |φ(s) − φ(t)| ≤ L·|s − t|. For a measurable feature-label map X valued in the Xb-by-Yb ball, a concentration parameter t satisfying t·(L·Yb·Xb·W)² ≤ 1/2, and a nonnegative slack ε, if ŵ empirically minimizes the average surrogate loss over an i.i.d. sample of size n against any comparator wstar in the W-ball, then the probability that the excess population surrogate risk of ŵ over wstar exceeds 8·L·Yb·Xb·W/√n + 2ε is at most exp(-ε²·t·n).","s":"∀ {d n : ℕ} {Ω : Type u_1} [inst : MeasurableSpace Ω] {μ : MeasureTheory.Measure Ω} [MeasureTheory.IsProbabilityMeasure μ] {Xb Yb W L : ℝ}, 0 ≤ Xb → 0 ≤ Yb → 0 ≤ W → 0 ≤ L → ∀ (φ : ℝ → ℝ), (∀ (s t : ℝ), |φ s - φ t| ≤ L * |s - t|) → ∀ (X : Ω → Causalean.ML.KFeat d Xb Yb), Measurable X → ∀ {t : ℝ}, t * (L * (Yb * Xb * W)) ^ 2 ≤ 1 / 2 → ∀ {ε : ℝ}, 0 ≤ ε → ∀ (ŵ : (Fin n → Ω) → Causalean.ML.KWeight d W) (wstar : Causalean.ML.KWeight d W), (∀ (ω : Fin n → Ω), (↑n)⁻¹ * ∑ k, φ (↑(X (ω k)).2 * inner ℝ ↑(ŵ ω) ↑(X (ω k)).1) ≤ (↑n)⁻¹ * ∑ k, φ (↑(X (ω k)).2 * inner ℝ ↑wstar ↑(X (ω k)).1)) → ((MeasureTheory.Measure.pi fun x => μ) fun ω => 8 * L * (Yb * Xb * W) / √↑n + 2 * ε < ∫ (x : Ω), (fun ω' => φ (↑(X ω').2 * inner ℝ ↑(ŵ ω) ↑(X ω').1)) x ∂μ - ∫ (x : Ω), (fun ω' => φ (↑(X ω').2 * inner ℝ ↑wstar ↑(X ω').1)) x ∂μ).toReal ≤ Real.exp (-ε ^ 2 * t * ↑n)","l":28,"f":"Causalean/ML/Margin/Rate.lean"},{"n":"Causalean.ML.olsCoef","k":"def","m":"Causalean.ML.Linear.ClosedForm","a":"ML/Linear","t":1,"d":"The OLS closed-form coefficient (XᵀX)⁻¹ Xᵀy.","s":"{Obs : Type u_1} → {Param : Type u_2} → [Fintype Obs] → [Fintype Param] → [DecidableEq Param] → Matrix Obs Param ℝ → (Obs → ℝ) → Param → ℝ","l":22,"f":"Causalean/ML/Linear/ClosedForm.lean"},{"n":"Causalean.ML.olsCoef_normalEq","k":"theorem","m":"Causalean.ML.Linear.ClosedForm","a":"ML/Linear","t":1,"d":"For a design matrix X and response vector y, if XᵀX is invertible, i.e. its determinant is a unit, then the closed-form OLS coefficient (XᵀX)⁻¹Xᵀy solves the normal equations (XᵀX)β = Xᵀy.","s":"∀ {Obs : Type u_1} {Param : Type u_2} [inst : Fintype Obs] [inst_1 : Fintype Param] [inst_2 : DecidableEq Param] (X : Matrix Obs Param ℝ) (y : Obs → ℝ), IsUnit (X.transpose * X).det → (X.transpose * X).mulVec (Causalean.ML.olsCoef X y) = X.transpose.mulVec y","l":110,"f":"Causalean/ML/Linear/ClosedForm.lean"},{"n":"Causalean.ML.olsCoef_unique","k":"theorem","m":"Causalean.ML.Linear.ClosedForm","a":"ML/Linear","t":2,"d":"When XᵀX is invertible, every normal-equation solution equals the closed form OLS coefficient.","s":"∀ {Obs : Type u_1} {Param : Type u_2} [inst : Fintype Obs] [inst_1 : Fintype Param] [inst_2 : DecidableEq Param] (X : Matrix Obs Param ℝ) (y : Obs → ℝ), IsUnit (X.transpose * X).det → ∀ {β : Param → ℝ}, (X.transpose * X).mulVec β = X.transpose.mulVec y → β = Causalean.ML.olsCoef X y","l":119,"f":"Causalean/ML/Linear/ClosedForm.lean"},{"n":"Causalean.ML.olsObjective","k":"def","m":"Causalean.ML.Linear.Finite","a":"ML/Linear","t":1,"d":"The ordinary-least-squares objective: the sum of squared residuals ∑ᵢ (yᵢ − (Xβ)ᵢ)².","s":"{Obs : Type u_1} → {Param : Type u_2} → [Fintype Obs] → [Fintype Param] → Matrix Obs Param ℝ → (Obs → ℝ) → (Param → ℝ) → ℝ","l":29,"f":"Causalean/ML/Linear/Finite.lean"},{"n":"Causalean.ML.ols_is_squaredLoss_ERM_of_normalEq","k":"theorem","m":"Causalean.ML.Linear.Finite","a":"ML/Linear","t":1,"d":"For any coefficient vector β̂ satisfying the normal equations XᵀX β̂ = Xᵀy built from a finite design matrix X and outcome vector y, that vector minimizes the sum of squared residuals over every coefficient vector β.","s":"∀ {Obs : Type u_1} {Param : Type u_2} [inst : Fintype Obs] [inst_1 : Fintype Param] (X : Matrix Obs Param ℝ) (y : Obs → ℝ) (βhat : Param → ℝ), (X.transpose * X).mulVec βhat = X.transpose.mulVec y → ∀ (β : Param → ℝ), Causalean.ML.olsObjective X y βhat ≤ Causalean.ML.olsObjective X y β","l":34,"f":"Causalean/ML/Linear/Finite.lean"},{"n":"Causalean.ML.ols_normalEq_of_minimizer","k":"theorem","m":"Causalean.ML.Linear.ClosedForm","a":"ML/Linear","t":2,"d":"A minimizer of the least-squares objective solves the normal equations.","s":"∀ {Obs : Type u_1} {Param : Type u_2} [inst : Fintype Obs] [inst_1 : Fintype Param] (X : Matrix Obs Param ℝ) (y : Obs → ℝ) (βhat : Param → ℝ), (∀ (β : Param → ℝ), Causalean.ML.olsObjective X y βhat ≤ Causalean.ML.olsObjective X y β) → (X.transpose * X).mulVec βhat = X.transpose.mulVec y","l":27,"f":"Causalean/ML/Linear/ClosedForm.lean"},{"n":"Causalean.ML.populationGram","k":"def","m":"Causalean.ML.Ridge.Rate","a":"ML/Ridge","t":1,"d":"The population feature Gram, defined entrywise (avoids matrix-valued Bochner integration): Gₖₗ = ∫ φ(x)ₖ φ(x)ₗ dP.","s":"{γ : Type u_2} → {K : Type u_3} → [inst : MeasurableSpace γ] → [inst_1 : Fintype K] → Causalean.ML.FeatureMap γ K → MeasureTheory.Measure (γ × ℝ) → Matrix K K ℝ","l":64,"f":"Causalean/ML/Ridge/Rate.lean"},{"n":"Causalean.ML.populationGram_posSemidef","k":"theorem","m":"Causalean.ML.Ridge.Rate","a":"ML/Ridge","t":2,"d":"The population feature Gram is positive semidefinite whenever the feature coordinates are measurable and the fourth moment of the feature norm is finite.","s":"∀ {γ : Type u_2} {K : Type u_3} [inst : MeasurableSpace γ] [inst_1 : Fintype K] (φ : Causalean.ML.FeatureMap γ K) (P : MeasureTheory.Measure (γ × ℝ)) [MeasureTheory.IsFiniteMeasure P], (∀ (k : K), Measurable fun x => φ.φ x k) → MeasureTheory.Integrable (fun z => (∑ k, φ.φ z.1 k ^ 2) ^ 2) P → (Causalean.ML.populationGram φ P).PosSemidef","l":71,"f":"Causalean/ML/Ridge/Rate.lean"},{"n":"Causalean.ML.populationRidgeObjective","k":"def","m":"Causalean.ML.Ridge.Population","a":"ML/Ridge","t":1,"d":"The population ridge objective: population squared risk plus λ‖β‖².","s":"{X' : Type u_1} → {K : Type u_2} → [inst : MeasurableSpace X'] → [inst_1 : Fintype K] → MeasureTheory.Measure (X' × ℝ) → Causalean.ML.FeatureMap X' K → ℝ → (K → ℝ) → ℝ","l":28,"f":"Causalean/ML/Ridge/Population.lean"},{"n":"Causalean.ML.populationRidge_minimizes","k":"theorem","m":"Causalean.ML.Ridge.Population","a":"ML/Ridge","t":1,"d":"With a nonnegative ridge penalty λ, for a probability measure P on features and outcome and a finite feature map φ, if the coefficient vector βstar satisfies the regularized population normal equations E[(Y − ⟪βstar,φ⟫)φₖ] = λβstarₖ for every feature k, the population squared-loss risks of the βstar- and β-predictors are both finite, and each feature is integrable against that residual, then the population ridge objective at βstar is at most its value at any other coefficient vector β.","s":"∀ {X' : Type u_1} {K : Type u_2} [inst : MeasurableSpace X'] [inst_1 : Fintype K] (P : MeasureTheory.Measure (X' × ℝ)) (φ : Causalean.ML.FeatureMap X' K) {lam : ℝ}, 0 ≤ lam → ∀ {βstar : K → ℝ}, Causalean.ML.IsPopulationRidge P φ lam βstar → ∀ (β : K → ℝ), (Causalean.ML.HasFinitePopulationRisk Causalean.ML.squaredLoss P fun x => ∑ k, βstar k * φ.φ x k) → (Causalean.ML.HasFinitePopulationRisk Causalean.ML.squaredLoss P fun x => ∑ k, β k * φ.φ x k) → (∀ (k : K), MeasureTheory.Integrable (fun z => (z.2 - ∑ j, βstar j * φ.φ z.1 j) * φ.φ z.1 k) P) → Causalean.ML.populationRidgeObjective P φ lam βstar ≤ Causalean.ML.populationRidgeObjective P φ lam β","l":39,"f":"Causalean/ML/Ridge/Population.lean"},{"n":"Causalean.ML.populationRisk","k":"def","m":"Causalean.ML.Core.Risk","a":"ML/Core","t":1,"d":"The population risk criterion of a prediction rule is the Bochner integral of its pointwise loss under the joint law. This definition does not by itself assert integrability or finite expected loss.","s":"{X : Type u_1} → {Y : Type u_2} → [inst : MeasurableSpace X] → [inst_1 : MeasurableSpace Y] → Causalean.ML.Loss Y → MeasureTheory.Measure (X × Y) → (X → Y) → ℝ","l":43,"f":"Causalean/ML/Core/Risk.lean"},{"n":"Causalean.ML.populationRiskP","k":"def","m":"Causalean.ML.Core.Risk","a":"ML/Core","t":1,"d":"The population risk criterion of a model parameter is the Bochner-integral criterion applied to the prediction rule selected by that parameter.","s":"{Θ : Type u_1} → {X : Type u_2} → {Y : Type u_3} → [inst : MeasurableSpace X] → [inst_1 : MeasurableSpace Y] → Causalean.ML.Predictor Θ X Y → Causalean.ML.Loss Y → MeasureTheory.Measure (X × Y) → Θ → ℝ","l":50,"f":"Causalean/ML/Core/Risk.lean"},{"n":"Causalean.ML.populationRisk_eq_of_both_minimizers","k":"theorem","m":"Causalean.ML.Core.PopulationTarget","a":"ML/Core","t":2,"d":"Any two population-risk minimizers over the same class attain the same risk.","s":"∀ {X : Type u_1} {Y : Type u_2} [inst : MeasurableSpace X] [inst_1 : MeasurableSpace Y] {H : Causalean.ML.HypothesisClass X Y} {loss : Causalean.ML.Loss Y} {P : MeasureTheory.Measure (X × Y)} {h₁ h₂ : X → Y}, Causalean.ML.IsPopulationRiskMinimizer H loss P h₁ → Causalean.ML.IsPopulationRiskMinimizer H loss P h₂ → Causalean.ML.populationRisk loss P h₁ = Causalean.ML.populationRisk loss P h₂","l":28,"f":"Causalean/ML/Core/PopulationTarget.lean"},{"n":"Causalean.ML.populationRisk_minimizer_eq_target","k":"theorem","m":"Causalean.ML.Core.PopulationTarget","a":"ML/Core","t":2,"d":"If a target is known to be the unique population-risk minimizer, every population-risk minimizer equals it.","s":"∀ {X : Type u_1} {Y : Type u_2} [inst : MeasurableSpace X] [inst_1 : MeasurableSpace Y] {H : Causalean.ML.HypothesisClass X Y} {loss : Causalean.ML.Loss Y} {P : MeasureTheory.Measure (X × Y)} {target hstar : X → Y}, Causalean.ML.IsPopulationRiskMinimizer H loss P hstar → Causalean.ML.IsPopulationRiskMinimizer H loss P target → (∀ (h₁ h₂ : X → Y), Causalean.ML.IsPopulationRiskMinimizer H loss P h₁ → Causalean.ML.IsPopulationRiskMinimizer H loss P h₂ → h₁ = h₂) → hstar = target","l":36,"f":"Causalean/ML/Core/PopulationTarget.lean"},{"n":"Causalean.ML.populationTarget_pushforward","k":"theorem","m":"Causalean.ML.Core.Bridge","a":"ML/Core","t":1,"d":"Population-target pushforward. For a predictor M with loss loss and population law P, suppose every admissible parameter's prediction function is measurable, the parameter θhat is itself admissible, every admissible parameter attains finite population risk, and θhat minimizes the population risk over the admissible parameter set. Then the predictor M.predict θhat is a population-risk minimizer over the hypothesis class realized by M's image: a parametric population-risk minimizer pushes forward to an extensional minimizer over the realized function class.","s":"∀ {Θ : Type u_1} {X : Type u_2} {Y : Type u_3} [inst : MeasurableSpace X] [inst_1 : MeasurableSpace Y] (M : Causalean.ML.Predictor Θ X Y) (loss : Causalean.ML.Loss Y) (P : MeasureTheory.Measure (X × Y)) (hmeas : ∀ θ ∈ M.paramSet, Measurable (M.predict θ)) {θhat : Θ}, θhat ∈ M.paramSet → (∀ θ ∈ M.paramSet, Causalean.ML.HasFinitePopulationRisk loss P (M.predict θ)) → IsMinOn (fun θ => Causalean.ML.populationRiskP M loss P θ) M.paramSet θhat → Causalean.ML.IsPopulationRiskMinimizer (Causalean.ML.imageClass M hmeas) loss P (M.predict θhat)","l":46,"f":"Causalean/ML/Core/Bridge.lean"},{"n":"Causalean.ML.properLoss_population_minimizer_recovers_eta","k":"theorem","m":"Causalean.ML.Surrogate.GenericERM","a":"ML/Surrogate","t":1,"d":"Strictly proper population minimizers recover the regression function. For a strictly proper binary loss L on a measure space PX, with the true conditional probability η taking values in [0,1] and a candidate prediction rule q also taking values in [0,1], if q minimizes the population conditional risk over every [0,1]-valued prediction rule and the conditional risks of η and q are both integrable, then q equals η almost everywhere.","s":"∀ {X : Type u_1} [inst : MeasurableSpace X] {L : ℝ → Bool → ℝ}, Causalean.ML.StrictProperBinaryLoss L → ∀ {PX : MeasureTheory.Measure X} (η : X → ℝ), (∀ (x : X), η x ∈ Set.Icc 0 1) → ∀ (q : X → ℝ), (∀ (x : X), q x ∈ Set.Icc 0 1) → IsMinOn (fun r => ∫ (x : X), η x * L (r x) true + (1 - η x) * L (r x) false ∂PX) {r | ∀ (x : X), r x ∈ Set.Icc 0 1} q → MeasureTheory.Integrable (fun x => η x * L (η x) true + (1 - η x) * L (η x) false) PX → MeasureTheory.Integrable (fun x => η x * L (q x) true + (1 - η x) * L (q x) false) PX → q =ᵐ[PX] η","l":61,"f":"Causalean/ML/Surrogate/GenericERM.lean"},{"n":"Causalean.ML.properLoss_population_risk_le","k":"theorem","m":"Causalean.ML.Surrogate.GenericERM","a":"ML/Surrogate","t":2,"d":"Proper-loss integrated risk inequality. For a proper binary loss, the integral of the conditional risk obtained by predicting the true conditional probability is no larger than the corresponding integral for any pointwise [0,1]-valued comparator. The hypotheses keep both conditional-risk integrands integrable.","s":"∀ {X : Type u_1} [inst : MeasurableSpace X] {L : ℝ → Bool → ℝ}, Causalean.ML.ProperBinaryLoss L → ∀ {PX : MeasureTheory.Measure X} (η : X → ℝ), (∀ (x : X), η x ∈ Set.Icc 0 1) → ∀ (q : X → ℝ), (∀ (x : X), q x ∈ Set.Icc 0 1) → MeasureTheory.Integrable (fun x => η x * L (η x) true + (1 - η x) * L (η x) false) PX → MeasureTheory.Integrable (fun x => η x * L (q x) true + (1 - η x) * L (q x) false) PX → ∫ (x : X), η x * L (η x) true + (1 - η x) * L (η x) false ∂PX ≤ ∫ (x : X), η x * L (q x) true + (1 - η x) * L (q x) false ∂PX","l":45,"f":"Causalean/ML/Surrogate/GenericERM.lean"},{"n":"Causalean.ML.rademacherComplexity_l1_ball_le","k":"theorem","m":"Causalean.ML.Lasso.Rate","a":"ML/Lasso","t":1,"d":"Rademacher complexity of the L¹-ball linear class. If the dimension d is positive, the sample size n is positive, the coordinatewise feature bound Xinf is nonnegative, the weight bound W is nonnegative, and every feature coordinate is bounded in absolute value by Xinf, then the expected Rademacher complexity of the class of linear predictors with L¹ weight norm at most W is at most (Xinf·W/√n)·√(2 log 2d).","s":"∀ {d n : ℕ}, 0 < d → 0 < n → ∀ {Ω : Type u_1} [inst : MeasurableSpace Ω] {μ : MeasureTheory.Measure Ω} [MeasureTheory.IsProbabilityMeasure μ] {Xinf W : ℝ}, 0 ≤ Xinf → 0 ≤ W → ∀ (X : Ω → EuclideanSpace ℝ (Fin d)), (∀ (ω : Ω) (j : Fin d), |(X ω).ofLp j| ≤ Xinf) → rademacherComplexity n (fun w a => ∑ j, (↑w).ofLp j * a.ofLp j) μ X ≤ Xinf * W / √↑n * √(2 * Real.log (2 * ↑d))","l":25,"f":"Causalean/ML/Lasso/Rate.lean"},{"n":"Causalean.ML.rademacherComplexity_l2_ball_le","k":"theorem","m":"Causalean.ML.Kernel.Rate","a":"ML/Kernel","t":1,"d":"Rademacher complexity of the L²-ball linear class. If the feature bound Xb is nonnegative, the weight bound W is nonnegative, and every feature vector has Euclidean norm at most Xb, then the expected Rademacher complexity of the class of linear predictors with weight norm at most W, on a sample of size n, is at most Xb·W/√n.","s":"∀ {d n : ℕ} {Ω : Type u_1} [inst : MeasurableSpace Ω] {μ : MeasureTheory.Measure Ω} [MeasureTheory.IsProbabilityMeasure μ] {Xb W : ℝ}, 0 ≤ Xb → 0 ≤ W → ∀ (X : Ω → EuclideanSpace ℝ (Fin d)), (∀ (ω : Ω), ‖X ω‖ ≤ Xb) → rademacherComplexity n (fun w a => inner ℝ (↑w) a) μ X ≤ Xb * W / √↑n","l":26,"f":"Causalean/ML/Kernel/Rate.lean"},{"n":"Causalean.ML.regLogisticCoef_isBigOp","k":"theorem","m":"Causalean.ML.Binary.Rate","a":"ML/Binary","t":2,"d":"The regularized-logistic coefficient error is O_p(n^{-1/2}): the strong-convexity basic inequality converts the centered-gradient bound into a coefficient-error bound.","s":"∀ {Ω : Type u_1} {γ : Type u_2} {K : Type u_3} [inst : MeasurableSpace Ω] [inst_1 : MeasurableSpace γ] [inst_2 : Fintype K] {μ : MeasureTheory.Measure Ω} (φ : Causalean.ML.FeatureMap γ K) (P : MeasureTheory.Measure (γ × ℝ)) [MeasureTheory.IsProbabilityMeasure P] (S : Causalean.Stat.IIDSample Ω (γ × ℝ) μ P) [MeasureTheory.IsProbabilityMeasure μ] {lam : ℝ}, 0 < lam → ∀ (βstar : K → ℝ) (βhat : ℕ → Ω → K → ℝ), Causalean.ML.IsPopulationRegLogistic P φ lam βstar → (∀ (n : ℕ) (ω : Ω), Causalean.ML.regLogisticGrad φ S.Z lam n ω (βhat n ω) = 0) → (∀ (k : K), Measurable fun x => φ.φ x k) → (∀ (k : K), MeasureTheory.MemLp (fun z => ((∑ j, βstar j * φ.φ z.1 j).sigmoid - z.2) * φ.φ z.1 k) 2 P) → Causalean.Stat.IsBigOp (fun n ω => ‖βhat n ω - βstar‖) (fun n => (√↑n)⁻¹) μ","l":370,"f":"Causalean/ML/Binary/Rate.lean"},{"n":"Causalean.ML.regLogisticGrad","k":"def","m":"Causalean.ML.Binary.Rate","a":"ML/Binary","t":1,"d":"Empirical penalized-logistic gradient at β (coordinate form): ∇ₙ(β)ₖ = n⁻¹ Σ_{i<n} (σ⟨β,φᵢ⟩ − yᵢ) φᵢₖ + 2λ βₖ.","s":"{Ω : Type u_1} → {γ : Type u_2} → {K : Type u_3} → [inst : Fintype K] → Causalean.ML.FeatureMap γ K → (ℕ → Ω → γ × ℝ) → ℝ → ℕ → Ω → (K → ℝ) → K → ℝ","l":35,"f":"Causalean/ML/Binary/Rate.lean"},{"n":"Causalean.ML.regLogisticGrad_coord_isBigOp","k":"theorem","m":"Causalean.ML.Binary.Rate","a":"ML/Binary","t":2,"d":"Each coordinate of the empirical penalized gradient at the population target is O_p(n^{-1/2}); the population score equations make the i.i.d. summands centered.","s":"∀ {Ω : Type u_1} {γ : Type u_2} {K : Type u_3} [inst : MeasurableSpace Ω] [inst_1 : MeasurableSpace γ] [inst_2 : Fintype K] {μ : MeasureTheory.Measure Ω} (φ : Causalean.ML.FeatureMap γ K) (P : MeasureTheory.Measure (γ × ℝ)) [MeasureTheory.IsProbabilityMeasure P] (S : Causalean.Stat.IIDSample Ω (γ × ℝ) μ P) [MeasureTheory.IsProbabilityMeasure μ] {lam : ℝ} (βstar : K → ℝ), Causalean.ML.IsPopulationRegLogistic P φ lam βstar → (∀ (k : K), Measurable fun x => φ.φ x k) → (∀ (k : K), MeasureTheory.MemLp (fun z => ((∑ j, βstar j * φ.φ z.1 j).sigmoid - z.2) * φ.φ z.1 k) 2 P) → ∀ (k : K), Causalean.Stat.IsBigOp (fun n ω => Causalean.ML.regLogisticGrad φ S.Z lam n ω βstar k) (fun n => (√↑n)⁻¹) μ","l":306,"f":"Causalean/ML/Binary/Rate.lean"},{"n":"Causalean.ML.regLogistic_achievesL2Rate","k":"theorem","m":"Causalean.ML.Binary.Rate","a":"ML/Binary","t":1,"d":"Regularized-logistic root-n estimation rate. For a strictly positive regularization weight lam, suppose βstar solves the penalized population first-order condition for the logistic quasi-score under the feature map φ and law P, and that for every sample size n and outcome ω, the fitted coefficients βhat n ω solve the corresponding empirical penalized first-order condition on the i.i.d. sample S. Suppose further that every feature coordinate is measurable, that the fourth moment of the squared feature norm is integrable under P, and that each coordinate of the population logistic score at βstar is square-integrable under P. Then the fitted logistic predictor σ(⟨βhat n ω, φ⟩) achieves the L²-rate n^{-1/2} toward the population target predictor σ(⟨βstar, φ⟩), under P and the sampling law μ. Assembled from the coefficient rate, the 1/4-Lipschitz σ, and the shared linear-predictor L² bound.","s":"∀ {Ω : Type u_1} {γ : Type u_2} {K : Type u_3} [inst : MeasurableSpace Ω] [inst_1 : MeasurableSpace γ] [inst_2 : Fintype K] {μ : MeasureTheory.Measure Ω} (φ : Causalean.ML.FeatureMap γ K) (P : MeasureTheory.Measure (γ × ℝ)) [MeasureTheory.IsProbabilityMeasure P] (S : Causalean.Stat.IIDSample Ω (γ × ℝ) μ P) [MeasureTheory.IsProbabilityMeasure μ] {lam : ℝ}, 0 < lam → ∀ (βstar : K → ℝ) (βhat : ℕ → Ω → K → ℝ), Causalean.ML.IsPopulationRegLogistic P φ lam βstar → (∀ (n : ℕ) (ω : Ω), Causalean.ML.regLogisticGrad φ S.Z lam n ω (βhat n ω) = 0) → (∀ (k : K), Measurable fun x => φ.φ x k) → MeasureTheory.Integrable (fun z => (∑ k, φ.φ z.1 k ^ 2) ^ 2) P → (∀ (k : K), MeasureTheory.MemLp (fun z => ((∑ j, βstar j * φ.φ z.1 j).sigmoid - z.2) * φ.φ z.1 k) 2 P) → Causalean.ML.AchievesL2Rate (fun n ω => Causalean.ML.logisticPredictor φ (βhat n ω)) (Causalean.ML.logisticPredictor φ βstar) P (fun n => (√↑n)⁻¹) μ","l":425,"f":"Causalean/ML/Binary/Rate.lean"},{"n":"Causalean.ML.regLogistic_basic_inequality","k":"theorem","m":"Causalean.ML.Binary.Rate","a":"ML/Binary","t":2,"d":"Strong-convexity basic inequality (deterministic). If β̂ solves the empirical penalized FOC ∇ₙ(β̂) = 0 and λ > 0, the coefficient error is controlled by the empirical gradient at the target: 2λ‖β̂ − β⋆‖ ≤ ∑ₖ |∇ₙ(β⋆)ₖ|.","s":"∀ {Ω : Type u_1} {γ : Type u_2} {K : Type u_3} [inst : Fintype K] (φ : Causalean.ML.FeatureMap γ K) (Z : ℕ → Ω → γ × ℝ) {lam : ℝ}, 0 < lam → ∀ (βstar : K → ℝ) (n : ℕ) (ω : Ω) {βhat : K → ℝ}, Causalean.ML.regLogisticGrad φ Z lam n ω βhat = 0 → 2 * lam * ‖βhat - βstar‖ ≤ ∑ k, |Causalean.ML.regLogisticGrad φ Z lam n ω βstar k|","l":143,"f":"Causalean/ML/Binary/Rate.lean"},{"n":"Causalean.ML.representer_theorem","k":"theorem","m":"Causalean.ML.Kernel.Ridge","a":"ML/Kernel","t":1,"d":"For a complete reproducing-kernel Hilbert space H, if (feval, representer) realizes H as an RKHS on X, i.e. evaluation at each point equals the inner product with its representer, the regularization level lam is strictly positive, and fhat minimizes the regularized empirical risk krrRisk over all of H for the sample (x, y), then fhat lies in the span of the sample representers representer x₁, …, representer xₙ, i.e. it is a finite linear combination of them.","s":"∀ {X : Type u_1} {H : Type u_2} [inst : NormedAddCommGroup H] [inst_1 : InnerProductSpace ℝ H] [CompleteSpace H] {feval : H → X → ℝ} {representer : X → H}, Causalean.ML.IsRKHS X H feval representer → ∀ {n : ℕ} (x : Fin n → X) (y : Fin n → ℝ) {lam : ℝ}, 0 < lam → ∀ {fhat : H}, (∀ (g : H), Causalean.ML.krrRisk feval x y lam fhat ≤ Causalean.ML.krrRisk feval x y lam g) → ∃ α, fhat = ∑ i, α i • representer (x i)","l":29,"f":"Causalean/ML/Kernel/Ridge.lean"},{"n":"Causalean.ML.ridgeCoef","k":"def","m":"Causalean.ML.Ridge.ClosedForm","a":"ML/Ridge","t":1,"d":"The ridge closed-form coefficient (XᵀX + λI)⁻¹ Xᵀy.","s":"{Obs : Type u_1} → {Param : Type u_2} → [Fintype Obs] → [Fintype Param] → [DecidableEq Param] → Matrix Obs Param ℝ → (Obs → ℝ) → ℝ → Param → ℝ","l":23,"f":"Causalean/ML/Ridge/ClosedForm.lean"},{"n":"Causalean.ML.ridgeCoef_normalEq","k":"theorem","m":"Causalean.ML.Ridge.ClosedForm","a":"ML/Ridge","t":1,"d":"For a strictly positive ridge penalty λ, the closed-form ridge coefficient (XᵀX + λI)⁻¹Xᵀy, built from a finite design matrix X and outcome vector y, satisfies the ridge normal equations (XᵀX + λI)β̂ = Xᵀy.","s":"∀ {Obs : Type u_1} {Param : Type u_2} [inst : Fintype Obs] [inst_1 : Fintype Param] [inst_2 : DecidableEq Param] (X : Matrix Obs Param ℝ) (y : Obs → ℝ) {lam : ℝ}, 0 < lam → (X.transpose * X + lam • 1).mulVec (Causalean.ML.ridgeCoef X y lam) = X.transpose.mulVec y","l":41,"f":"Causalean/ML/Ridge/ClosedForm.lean"},{"n":"Causalean.ML.ridgeCoef_unique","k":"theorem","m":"Causalean.ML.Ridge.ClosedForm","a":"ML/Ridge","t":2,"d":"For λ > 0, every solution of the ridge normal equations is the closed-form ridge coefficient.","s":"∀ {Obs : Type u_1} {Param : Type u_2} [inst : Fintype Obs] [inst_1 : Fintype Param] [inst_2 : DecidableEq Param] (X : Matrix Obs Param ℝ) (y : Obs → ℝ) {lam : ℝ}, 0 < lam → ∀ {β : Param → ℝ}, (X.transpose * X + lam • 1).mulVec β = X.transpose.mulVec y → β = Causalean.ML.ridgeCoef X y lam","l":55,"f":"Causalean/ML/Ridge/ClosedForm.lean"},{"n":"Causalean.ML.ridgeGram_posDef","k":"theorem","m":"Causalean.ML.Ridge.ClosedForm","a":"ML/Ridge","t":1,"d":"For a strictly positive ridge penalty λ, the ridge Gram matrix XᵀX + λI built from a finite design matrix X is positive definite.","s":"∀ {Obs : Type u_1} {Param : Type u_2} [inst : Fintype Obs] [Fintype Param] [inst_2 : DecidableEq Param] (X : Matrix Obs Param ℝ) {lam : ℝ}, 0 < lam → (X.transpose * X + lam • 1).PosDef","l":29,"f":"Causalean/ML/Ridge/ClosedForm.lean"},{"n":"Causalean.ML.ridgeObjective","k":"def","m":"Causalean.ML.Ridge.Finite","a":"ML/Ridge","t":1,"d":"The ridge objective: least-squares error plus the L² penalty λ‖β‖².","s":"{Obs : Type u_1} → {Param : Type u_2} → [Fintype Obs] → [Fintype Param] → Matrix Obs Param ℝ → (Obs → ℝ) → ℝ → (Param → ℝ) → ℝ","l":23,"f":"Causalean/ML/Ridge/Finite.lean"},{"n":"Causalean.ML.ridge_achievesL2Rate","k":"theorem","m":"Causalean.ML.Ridge.Rate","a":"ML/Ridge","t":1,"d":"Ridge root-n estimation rate. For a strictly positive ridge penalty λ, a finite feature map φ, and an i.i.d. sample S from a distribution P on features and outcome, if the true coefficient vector βstar satisfies the regularized population ridge normal equations, each feature coordinate is measurable, the fourth moment of the squared feature norm is finite, and each per-coordinate score function is square-integrable, then the sample ridge predictor converges to the population ridge predictor at the root-n rate in the L²(P) sense.","s":"∀ {Ω : Type u_1} {γ : Type u_2} {K : Type u_3} [inst : MeasurableSpace Ω] [inst_1 : MeasurableSpace γ] [inst_2 : Fintype K] [inst_3 : DecidableEq K] {μ : MeasureTheory.Measure Ω} (φ : Causalean.ML.FeatureMap γ K) (P : MeasureTheory.Measure (γ × ℝ)) [MeasureTheory.IsProbabilityMeasure P] (S : Causalean.Stat.IIDSample Ω (γ × ℝ) μ P) [MeasureTheory.IsProbabilityMeasure μ] {lam : ℝ}, 0 < lam → ∀ (βstar : K → ℝ), Causalean.ML.IsPopulationRidge P φ lam βstar → (∀ (k : K), Measurable fun x => φ.φ x k) → MeasureTheory.Integrable (fun z => (∑ k, φ.φ z.1 k ^ 2) ^ 2) P → (∀ (k : K), MeasureTheory.MemLp (fun z => (z.2 - ∑ j, βstar j * φ.φ z.1 j) * φ.φ z.1 k) 2 P) → Causalean.ML.AchievesL2Rate (Causalean.ML.sampleRidgePredictor φ S.Z lam) (fun x => ∑ k, βstar k * φ.φ x k) P (fun n => (√↑n)⁻¹) μ","l":719,"f":"Causalean/ML/Ridge/Rate.lean"},{"n":"Causalean.ML.ridge_is_regularized_squaredLoss_ERM_of_normalEq","k":"theorem","m":"Causalean.ML.Ridge.Finite","a":"ML/Ridge","t":1,"d":"With a nonnegative ridge penalty λ, for any coefficient vector β̂ satisfying the ridge normal equations (XᵀX + λI)β̂ = Xᵀy built from a finite design matrix X and outcome vector y, that vector minimizes the ridge objective — the sum of squared residuals plus λ‖β‖² — over every coefficient vector β.","s":"∀ {Obs : Type u_1} {Param : Type u_2} [inst : Fintype Obs] [inst_1 : Fintype Param] [inst_2 : DecidableEq Param] (X : Matrix Obs Param ℝ) (y : Obs → ℝ) {lam : ℝ}, 0 ≤ lam → ∀ (βhat : Param → ℝ), (X.transpose * X + lam • 1).mulVec βhat = X.transpose.mulVec y → ∀ (β : Param → ℝ), Causalean.ML.ridgeObjective X y lam βhat ≤ Causalean.ML.ridgeObjective X y lam β","l":28,"f":"Causalean/ML/Ridge/Finite.lean"},{"n":"Causalean.ML.rkhs_ball_empiricalRademacher_le","k":"theorem","m":"Causalean.ML.Kernel.RKHSRademacher","a":"ML/Kernel","t":1,"d":"In a reproducing-kernel Hilbert space H, if (feval, representer) realizes H as an RKHS on X, κ is a nonnegative bound, r is a nonnegative radius, and every sampled kernel representer representer (xs k) has norm at most κ, then the empirical Rademacher complexity, on the sample xs, of the closed ball of radius r in H is at most κ·r/√n.","s":"∀ {X : Type u_1} {H : Type u_2} [inst : NormedAddCommGroup H] [inst_1 : InnerProductSpace ℝ H] {feval : H → X → ℝ} {representer : X → H}, Causalean.ML.IsRKHS X H feval representer → ∀ {n : ℕ} {κ r : ℝ}, 0 ≤ κ → 0 ≤ r → ∀ (xs : Fin n → X), (∀ (k : Fin n), ‖representer (xs k)‖ ≤ κ) → empiricalRademacherComplexity n (fun f x => feval (↑f) x) xs ≤ κ * r / √↑n","l":465,"f":"Causalean/ML/Kernel/RKHSRademacher.lean"},{"n":"Causalean.ML.sampleRidgeCoef","k":"def","m":"Causalean.ML.Ridge.Rate","a":"ML/Ridge","t":1,"d":"The sample ridge coefficient β̂ₙ = (Ĝₙ + λI)⁻¹ Ĉₙ.","s":"{Ω : Type u_1} → {γ : Type u_2} → {K : Type u_3} → [inst : Fintype K] → [DecidableEq K] → Causalean.ML.FeatureMap γ K → (ℕ → Ω → γ × ℝ) → ℝ → ℕ → Ω → K → ℝ","l":54,"f":"Causalean/ML/Ridge/Rate.lean"},{"n":"Causalean.ML.sampleRidgeCoef_isBigOp","k":"theorem","m":"Causalean.ML.Ridge.Rate","a":"ML/Ridge","t":2,"d":"The ridge coefficient error is O_p(n^{-1/2}): the regularized empirical-Gram inverse is O_p(1) (entrywise matrix LLN + inverse continuity at the PosDef limit) times the centered score mean O_p(n^{-1/2}).","s":"∀ {Ω : Type u_1} {γ : Type u_2} {K : Type u_3} [inst : MeasurableSpace Ω] [inst_1 : MeasurableSpace γ] [inst_2 : Fintype K] [inst_3 : DecidableEq K] {μ : MeasureTheory.Measure Ω} (φ : Causalean.ML.FeatureMap γ K) (P : MeasureTheory.Measure (γ × ℝ)) [MeasureTheory.IsProbabilityMeasure P] (S : Causalean.Stat.IIDSample Ω (γ × ℝ) μ P) [MeasureTheory.IsProbabilityMeasure μ] {lam : ℝ}, 0 < lam → ∀ (βstar : K → ℝ), Causalean.ML.IsPopulationRidge P φ lam βstar → (∀ (k : K), Measurable fun x => φ.φ x k) → MeasureTheory.Integrable (fun z => (∑ k, φ.φ z.1 k ^ 2) ^ 2) P → (∀ (k : K), MeasureTheory.MemLp (fun z => (z.2 - ∑ j, βstar j * φ.φ z.1 j) * φ.φ z.1 k) 2 P) → (Causalean.ML.populationGram φ P + lam • 1).PosDef → Causalean.Stat.IsBigOp (fun n ω => ‖Causalean.ML.sampleRidgeCoef φ S.Z lam n ω - βstar‖) (fun n => (√↑n)⁻¹) μ","l":383,"f":"Causalean/ML/Ridge/Rate.lean"},{"n":"Causalean.ML.sampleRidgeCoef_sub_eq","k":"theorem","m":"Causalean.ML.Ridge.Rate","a":"ML/Ridge","t":2,"d":"The ridge coefficient error factors through the regularized empirical Gram inverse and the centered empirical score Ŝₙ − λ β⋆ = (Ĉₙ − Ĝₙ β⋆) − λ β⋆.","s":"∀ {Ω : Type u_1} {γ : Type u_2} {K : Type u_3} [inst : Fintype K] [inst_1 : DecidableEq K] (φ : Causalean.ML.FeatureMap γ K) (Z : ℕ → Ω → γ × ℝ) {lam : ℝ}, 0 < lam → ∀ (βstar : K → ℝ) (n : ℕ) (ω : Ω), Causalean.ML.sampleRidgeCoef φ Z lam n ω - βstar = (Causalean.ML.empiricalGram φ Z n ω + lam • 1)⁻¹.mulVec (Causalean.ML.empiricalCross φ Z n ω - (Causalean.ML.empiricalGram φ Z n ω).mulVec βstar - lam • βstar)","l":194,"f":"Causalean/ML/Ridge/Rate.lean"},{"n":"Causalean.ML.sampleRidgePredictor","k":"def","m":"Causalean.ML.Ridge.Rate","a":"ML/Ridge","t":1,"d":"The sample ridge predictor x ↦ ⟨β̂ₙ, φ(x)⟩.","s":"{Ω : Type u_1} → {γ : Type u_2} → {K : Type u_3} → [inst : Fintype K] → [DecidableEq K] → Causalean.ML.FeatureMap γ K → (ℕ → Ω → γ × ℝ) → ℝ → ℕ → Ω → γ → ℝ","l":59,"f":"Causalean/ML/Ridge/Rate.lean"},{"n":"Causalean.ML.sigmoid","k":"def","m":"Causalean.ML.Core.Losses","a":"ML/Core","t":1,"d":"The logistic sigmoid maps any real score to a probability-like number between zero and one.","s":"ℝ → ℝ","l":26,"f":"Causalean/ML/Core/Losses.lean"},{"n":"Causalean.ML.sigmoid_lt_one","k":"theorem","m":"Causalean.ML.Core.Losses","a":"ML/Core","t":2,"d":"The logistic sigmoid is strictly below one at every score.","s":"∀ (t : ℝ), Causalean.ML.sigmoid t < 1","l":61,"f":"Causalean/ML/Core/Losses.lean"},{"n":"Causalean.ML.sigmoid_pos","k":"theorem","m":"Causalean.ML.Core.Losses","a":"ML/Core","t":2,"d":"The logistic sigmoid is strictly positive at every score.","s":"∀ (t : ℝ), 0 < Causalean.ML.sigmoid t","l":56,"f":"Causalean/ML/Core/Losses.lean"},{"n":"Causalean.ML.softThreshold","k":"def","m":"Causalean.ML.Lasso.Finite","a":"ML/Lasso","t":1,"d":"The soft-thresholding operator, in the sign-free form S_λ(z) = max(z − λ, 0) − max(−z − λ, 0) (equal to sign z · max(|z| − λ, 0)).","s":"ℝ → ℝ → ℝ","l":34,"f":"Causalean/ML/Lasso/Finite.lean"},{"n":"Causalean.ML.softThreshold_isMinOn","k":"theorem","m":"Causalean.ML.Lasso.Optimality","a":"ML/Lasso","t":1,"d":"For a real number a, if the regularization level lam is nonnegative, then the soft-thresholded value softThreshold lam a minimizes the one-dimensional penalized least-squares objective u ↦ (u − a)² + 2·lam·|u| over every real u.","s":"∀ {lam : ℝ}, 0 ≤ lam → ∀ (a u : ℝ), (Causalean.ML.softThreshold lam a - a) ^ 2 + 2 * lam * |Causalean.ML.softThreshold lam a| ≤ (u - a) ^ 2 + 2 * lam * |u|","l":20,"f":"Causalean/ML/Lasso/Optimality.lean"},{"n":"Causalean.ML.softplus","k":"def","m":"Causalean.ML.Core.Losses","a":"ML/Core","t":1,"d":"Softplus is a smooth positive transformation used to write logistic losses in score space.","s":"ℝ → ℝ","l":30,"f":"Causalean/ML/Core/Losses.lean"},{"n":"Causalean.ML.softplus_pos","k":"theorem","m":"Causalean.ML.Core.Losses","a":"ML/Core","t":2,"d":"Softplus is strictly positive at every score.","s":"∀ (t : ℝ), 0 < Causalean.ML.softplus t","l":70,"f":"Causalean/ML/Core/Losses.lean"},{"n":"Causalean.ML.square_loss_population_target_of_isL2Projection","k":"theorem","m":"Causalean.ML.Core.PopulationTarget","a":"ML/Core","t":1,"d":"For a joint covariate–response law P and a competitor function h, if the residual of m is orthogonal to every function of the covariate, i.e. m is an L² projection onto functions of the covariate, m is measurable, the competitor h is measurable, m attains finite squared population risk, h attains finite squared population risk, and the cross term (Y − m(X))·(m(X) − h(X)) is integrable, then the squared population risk of m is at most that of h: m minimizes squared population risk against any such competitor.","s":"∀ {X : Type u_1} [inst : MeasurableSpace X] {P : MeasureTheory.Measure (X × ℝ)} {m : X → ℝ}, Causalean.ML.IsL2Projection P m → ∀ (h : X → ℝ), Measurable m → Measurable h → Causalean.ML.HasFinitePopulationRisk Causalean.ML.squaredLoss P m → Causalean.ML.HasFinitePopulationRisk Causalean.ML.squaredLoss P h → MeasureTheory.Integrable (fun z => (z.2 - m z.1) * (m z.1 - h z.1)) P → Causalean.ML.populationRisk Causalean.ML.squaredLoss P m ≤ Causalean.ML.populationRisk Causalean.ML.squaredLoss P h","l":57,"f":"Causalean/ML/Core/PopulationTarget.lean"},{"n":"Causalean.ML.squaredLoss","k":"def","m":"Causalean.ML.Core.Losses","a":"ML/Core","t":1,"d":"Squared loss measures prediction error by squaring the difference between the observed label and the prediction.","s":"ℝ → ℝ → ℝ","l":19,"f":"Causalean/ML/Core/Losses.lean"},{"n":"Causalean.ML.squaredLoss_nonneg","k":"theorem","m":"Causalean.ML.Core.Losses","a":"ML/Core","t":2,"d":"Squared loss is always nonnegative.","s":"∀ (ŷ y : ℝ), 0 ≤ Causalean.ML.squaredLoss ŷ y","l":53,"f":"Causalean/ML/Core/Losses.lean"},{"n":"Causalean.ML.subgradientAt_zero_iff_isMinOn","k":"theorem","m":"Causalean.ML.Core.Convex","a":"ML/Core","t":2,"d":"Fermat's rule, subgradient form: 0 is a subgradient at x over s iff x minimizes f over s.","s":"∀ {E : Type u_1} [inst : NormedAddCommGroup E] [inst_1 : InnerProductSpace ℝ E] {f : E → ℝ} {s : Set E} {x : E}, x ∈ s → (Causalean.ML.SubgradientAt f s 0 x ↔ IsMinOn f s x)","l":126,"f":"Causalean/ML/Core/Convex.lean"},{"n":"Causalean.ML.summable_effectiveDimension_term","k":"theorem","m":"Causalean.ML.Kernel.EffectiveDimension","a":"ML/Kernel","t":2,"d":"For a trace-class operator (summable eigenvalues) the effective-dimension summands are summable, so effectiveDimension is a genuine real number.","s":"∀ {ι : Type u_1} {μ : ι → ℝ} {lam : ℝ}, (∀ (i : ι), 0 ≤ μ i) → 0 < lam → Summable μ → Summable fun i => μ i / (μ i + lam)","l":65,"f":"Causalean/ML/Kernel/EffectiveDimension.lean"},{"n":"Causalean.MarkovEquiv","k":"def","m":"Causalean.Graph.MarkovEquiv.Defs","a":"Graph/MarkovEquiv","t":1,"d":"Two DAGs are Markov equivalent when they entail exactly the same d-separations: for every triple of vertex sets X, Y, Z, X and Y are d-separated by Z in G₁ iff they are in G₂. Equivalently (via the global Markov property) the two graphs impose the same conditional-independence constraints on every distribution. Pairwise disjointness is already part of dSep, so it need not be repeated here.","s":"{V : Type u_1} → [inst : DecidableEq V] → [inst_1 : Fintype V] → Causalean.DAG V → Causalean.DAG V → Prop","l":61,"f":"Causalean/Graph/MarkovEquiv/Defs.lean"},{"n":"Causalean.MarkovEquiv.refl","k":"theorem","m":"Causalean.Graph.MarkovEquiv.Defs","a":"Graph/MarkovEquiv","t":1,"d":"For any DAG G, G is Markov equivalent to itself.","s":"∀ {V : Type u_1} [inst : DecidableEq V] [inst_1 : Fintype V] (G : Causalean.DAG V), Causalean.MarkovEquiv G G","l":77,"f":"Causalean/Graph/MarkovEquiv/Defs.lean"},{"n":"Causalean.MarkovEquiv.symm","k":"theorem","m":"Causalean.Graph.MarkovEquiv.Defs","a":"Graph/MarkovEquiv","t":2,"d":"Markov equivalence is symmetric.","s":"∀ {V : Type u_1} [inst : DecidableEq V] [inst_1 : Fintype V] {G₁ G₂ : Causalean.DAG V}, Causalean.MarkovEquiv G₁ G₂ → Causalean.MarkovEquiv G₂ G₁","l":81,"f":"Causalean/Graph/MarkovEquiv/Defs.lean"},{"n":"Causalean.MarkovEquiv.trans","k":"theorem","m":"Causalean.Graph.MarkovEquiv.Defs","a":"Graph/MarkovEquiv","t":2,"d":"Markov equivalence is transitive.","s":"∀ {V : Type u_1} [inst : DecidableEq V] [inst_1 : Fintype V] {G₁ G₂ G₃ : Causalean.DAG V}, Causalean.MarkovEquiv G₁ G₂ → Causalean.MarkovEquiv G₂ G₃ → Causalean.MarkovEquiv G₁ G₃","l":85,"f":"Causalean/Graph/MarkovEquiv/Defs.lean"},{"n":"Causalean.Mathlib.AlgebraicGeometry.PolynomialImageDimension.HasAffineCodimensionIn","k":"def","m":"Causalean.Mathlib.AlgebraicGeometry.PolynomialImageDimension.CodimensionOne","a":"Mathlib/AlgebraicGeometry/PolynomialImageDimension","t":1,"d":"A locus has affine codimension d in an ambient set when every irreducible component admits an endpoint-fixed chain of length d, and one component admits no such chain of length d + 1.","s":"{ι : Type u_1} → ℕ → Set (ι → ℂ) → Set (ι → ℂ) → Prop","l":31,"f":"Causalean/Mathlib/AlgebraicGeometry/PolynomialImageDimension/CodimensionOne.lean"},{"n":"Causalean.Mathlib.AlgebraicGeometry.PolynomialImageDimension.HasAffineZariskiDimension","k":"def","m":"Causalean.Mathlib.AlgebraicGeometry.PolynomialImageDimension.ChainDimension","a":"Mathlib/AlgebraicGeometry/PolynomialImageDimension","t":1,"d":"Exact affine Zariski dimension is the largest number of strict containments in a chain of nonempty irreducible polynomially closed subsets of a set.","s":"{ι : Type u_1} → ℕ → Set (ι → ℂ) → Prop","l":20,"f":"Causalean/Mathlib/AlgebraicGeometry/PolynomialImageDimension/ChainDimension.lean"},{"n":"Causalean.Mathlib.AlgebraicGeometry.PolynomialImageDimension.IsIrreducibleAffineClosed","k":"def","m":"Causalean.Mathlib.AlgebraicGeometry.PolynomialImageDimension.Irreducibility","a":"Mathlib/AlgebraicGeometry/PolynomialImageDimension","t":1,"d":"An irreducible affine-closed set is a nonempty polynomially closed set that cannot be expressed as the union of two smaller polynomially closed sets.","s":"{ι : Type u_1} → Set (ι → ℂ) → Prop","l":21,"f":"Causalean/Mathlib/AlgebraicGeometry/PolynomialImageDimension/Irreducibility.lean"},{"n":"Causalean.Mathlib.AlgebraicGeometry.PolynomialImageDimension.IsIrreducibleAffineComponent","k":"def","m":"Causalean.Mathlib.AlgebraicGeometry.PolynomialImageDimension.CodimensionOne","a":"Mathlib/AlgebraicGeometry/PolynomialImageDimension","t":1,"d":"An irreducible affine component is a maximal irreducible affine-closed subset of a prescribed locus.","s":"{ι : Type u_1} → Set (ι → ℂ) → Set (ι → ℂ) → Prop","l":24,"f":"Causalean/Mathlib/AlgebraicGeometry/PolynomialImageDimension/CodimensionOne.lean"},{"n":"Causalean.Mathlib.AlgebraicGeometry.PolynomialImageDimension.IsPolynomialMap","k":"def","m":"Causalean.Mathlib.AlgebraicGeometry.PolynomialImageDimension.PolynomialMap","a":"Mathlib/AlgebraicGeometry/PolynomialImageDimension","t":1,"d":"Every coordinate of f is a polynomial in the source coordinates.","s":"{ι : Type u_1} → {κ : Type u_2} → ((ι → ℂ) → κ → ℂ) → Prop","l":20,"f":"Causalean/Mathlib/AlgebraicGeometry/PolynomialImageDimension/PolynomialMap.lean"},{"n":"Causalean.Mathlib.AlgebraicGeometry.PolynomialImageDimension.IsPolynomialMap.comp","k":"theorem","m":"Causalean.Mathlib.AlgebraicGeometry.PolynomialImageDimension.PolynomialMap","a":"Mathlib/AlgebraicGeometry/PolynomialImageDimension","t":2,"d":"A composite of polynomial maps is polynomial.","s":"∀ {ι : Type u_1} {κ : Type u_2} {τ : Type u_3} {f : (ι → ℂ) → κ → ℂ} {g : (κ → ℂ) → τ → ℂ}, Causalean.Mathlib.AlgebraicGeometry.PolynomialImageDimension.IsPolynomialMap g → Causalean.Mathlib.AlgebraicGeometry.PolynomialImageDimension.IsPolynomialMap f → Causalean.Mathlib.AlgebraicGeometry.PolynomialImageDimension.IsPolynomialMap (g ∘ f)","l":30,"f":"Causalean/Mathlib/AlgebraicGeometry/PolynomialImageDimension/PolynomialMap.lean"},{"n":"Causalean.Mathlib.AlgebraicGeometry.PolynomialImageDimension.IsPolynomialMap.eval_comp","k":"theorem","m":"Causalean.Mathlib.AlgebraicGeometry.PolynomialImageDimension.PolynomialMap","a":"Mathlib/AlgebraicGeometry/PolynomialImageDimension","t":2,"d":"Substituting a polynomial map into a target polynomial yields a source polynomial.","s":"∀ {ι : Type u_1} {κ : Type u_2} {f : (ι → ℂ) → κ → ℂ}, Causalean.Mathlib.AlgebraicGeometry.PolynomialImageDimension.IsPolynomialMap f → ∀ (Q : MvPolynomial κ ℂ), ∃ P, ∀ (x : ι → ℂ), (MvPolynomial.eval x) P = (MvPolynomial.eval (f x)) Q","l":54,"f":"Causalean/Mathlib/AlgebraicGeometry/PolynomialImageDimension/PolynomialMap.lean"},{"n":"Causalean.Mathlib.AlgebraicGeometry.PolynomialImageDimension.affineLinearMap_isPolynomial","k":"theorem","m":"Causalean.Mathlib.AlgebraicGeometry.PolynomialImageDimension.AffineSubspaceDimension","a":"Mathlib/AlgebraicGeometry/PolynomialImageDimension","t":2,"d":"An affine-linear map from a finite-coordinate source is polynomial.","s":"∀ {ι : Type u_1} {κ : Type u_2} [Finite ι] (F : (ι → ℂ) →ₗ[ℂ] κ → ℂ) (c : κ → ℂ), Causalean.Mathlib.AlgebraicGeometry.PolynomialImageDimension.IsPolynomialMap fun x => c + F x","l":48,"f":"Causalean/Mathlib/AlgebraicGeometry/PolynomialImageDimension/AffineSubspaceDimension.lean"},{"n":"Causalean.Mathlib.AlgebraicGeometry.PolynomialImageDimension.affineSpace_hasAffineZariskiDimension","k":"theorem","m":"Causalean.Mathlib.AlgebraicGeometry.PolynomialImageDimension.AffineSpaceDimension","a":"Mathlib/AlgebraicGeometry/PolynomialImageDimension","t":1,"d":"For any natural number d, complex affine d-space has affine Zariski dimension exactly d, in the sense of maximal chains of irreducible affine closed subsets.","s":"∀ (d : ℕ), Causalean.Mathlib.AlgebraicGeometry.PolynomialImageDimension.HasAffineZariskiDimension d Set.univ","l":46,"f":"Causalean/Mathlib/AlgebraicGeometry/PolynomialImageDimension/AffineSpaceDimension.lean"},{"n":"Causalean.Mathlib.AlgebraicGeometry.PolynomialImageDimension.affineSubspace_hasAffineZariskiDimension","k":"theorem","m":"Causalean.Mathlib.AlgebraicGeometry.PolynomialImageDimension.AffineSubspaceDimension","a":"Mathlib/AlgebraicGeometry/PolynomialImageDimension","t":1,"d":"An affine translate of a d-dimensional linear subspace has exact irreducible-chain dimension d. For a complex linear subspace V of κ → ℂ and a base point x₀, if V has finite rank exactly d, then the affine translate {x | x - x₀ ∈ V} has irreducible-chain (Zariski) dimension exactly d.","s":"∀ {κ : Type u_1} [Finite κ] (V : Submodule ℂ (κ → ℂ)) (x₀ : κ → ℂ) (d : ℕ), Module.finrank ℂ ↥V = d → Causalean.Mathlib.AlgebraicGeometry.PolynomialImageDimension.HasAffineZariskiDimension d {x | x - x₀ ∈ V}","l":59,"f":"Causalean/Mathlib/AlgebraicGeometry/PolynomialImageDimension/AffineSubspaceDimension.lean"},{"n":"Causalean.Mathlib.AlgebraicGeometry.PolynomialImageDimension.affineZariskiClosure","k":"def","m":"Causalean.Mathlib.AlgebraicGeometry.PolynomialImageDimension.ZariskiClosure","a":"Mathlib/AlgebraicGeometry/PolynomialImageDimension","t":1,"d":"Algebraic closure in a complex affine space.","s":"{ι : Type u_1} → Set (ι → ℂ) → Set (ι → ℂ)","l":23,"f":"Causalean/Mathlib/AlgebraicGeometry/PolynomialImageDimension/ZariskiClosure.lean"},{"n":"Causalean.Mathlib.AlgebraicGeometry.PolynomialImageDimension.affineZariskiClosure_eq_zeroLocus","k":"theorem","m":"Causalean.Mathlib.AlgebraicGeometry.PolynomialImageDimension.ZariskiClosure","a":"Mathlib/AlgebraicGeometry/PolynomialImageDimension","t":2,"d":"A closed affine set is the zero locus of its vanishing ideal.","s":"∀ {ι : Type u_1} {Z : Set (ι → ℂ)}, Causalean.Mathlib.AlgebraicGeometry.PolynomialImageDimension.affineZariskiClosure Z = Z → Z = MvPolynomial.zeroLocus ℂ (MvPolynomial.vanishingIdeal ℂ Z)","l":46,"f":"Causalean/Mathlib/AlgebraicGeometry/PolynomialImageDimension/ZariskiClosure.lean"},{"n":"Causalean.Mathlib.AlgebraicGeometry.PolynomialImageDimension.affineZariskiClosure_extensive","k":"theorem","m":"Causalean.Mathlib.AlgebraicGeometry.PolynomialImageDimension.ZariskiClosure","a":"Mathlib/AlgebraicGeometry/PolynomialImageDimension","t":2,"d":"Every set is contained in its affine Zariski closure.","s":"∀ {ι : Type u_1} (A : Set (ι → ℂ)), A ⊆ Causalean.Mathlib.AlgebraicGeometry.PolynomialImageDimension.affineZariskiClosure A","l":27,"f":"Causalean/Mathlib/AlgebraicGeometry/PolynomialImageDimension/ZariskiClosure.lean"},{"n":"Causalean.Mathlib.AlgebraicGeometry.PolynomialImageDimension.affineZariskiClosure_idem","k":"theorem","m":"Causalean.Mathlib.AlgebraicGeometry.PolynomialImageDimension.ZariskiClosure","a":"Mathlib/AlgebraicGeometry/PolynomialImageDimension","t":2,"d":"Affine Zariski closure is idempotent.","s":"∀ {ι : Type u_1} (A : Set (ι → ℂ)), Causalean.Mathlib.AlgebraicGeometry.PolynomialImageDimension.affineZariskiClosure (Causalean.Mathlib.AlgebraicGeometry.PolynomialImageDimension.affineZariskiClosure A) = Causalean.Mathlib.AlgebraicGeometry.PolynomialImageDimension.affineZariskiClosure A","l":38,"f":"Causalean/Mathlib/AlgebraicGeometry/PolynomialImageDimension/ZariskiClosure.lean"},{"n":"Causalean.Mathlib.AlgebraicGeometry.PolynomialImageDimension.affineZariskiClosure_inter","k":"theorem","m":"Causalean.Mathlib.AlgebraicGeometry.PolynomialImageDimension.ZariskiClosure","a":"Mathlib/AlgebraicGeometry/PolynomialImageDimension","t":2,"d":"The intersection of two affine-closed sets is affine-closed.","s":"∀ {ι : Type u_1} {A B : Set (ι → ℂ)}, Causalean.Mathlib.AlgebraicGeometry.PolynomialImageDimension.affineZariskiClosure A = A → Causalean.Mathlib.AlgebraicGeometry.PolynomialImageDimension.affineZariskiClosure B = B → Causalean.Mathlib.AlgebraicGeometry.PolynomialImageDimension.affineZariskiClosure (A ∩ B) = A ∩ B","l":52,"f":"Causalean/Mathlib/AlgebraicGeometry/PolynomialImageDimension/ZariskiClosure.lean"},{"n":"Causalean.Mathlib.AlgebraicGeometry.PolynomialImageDimension.affineZariskiClosure_mono","k":"theorem","m":"Causalean.Mathlib.AlgebraicGeometry.PolynomialImageDimension.ZariskiClosure","a":"Mathlib/AlgebraicGeometry/PolynomialImageDimension","t":2,"d":"Affine Zariski closure is monotone.","s":"∀ {ι : Type u_1} {A B : Set (ι → ℂ)}, A ⊆ B → Causalean.Mathlib.AlgebraicGeometry.PolynomialImageDimension.affineZariskiClosure A ⊆ Causalean.Mathlib.AlgebraicGeometry.PolynomialImageDimension.affineZariskiClosure B","l":32,"f":"Causalean/Mathlib/AlgebraicGeometry/PolynomialImageDimension/ZariskiClosure.lean"},{"n":"Causalean.Mathlib.AlgebraicGeometry.PolynomialImageDimension.affineZariskiClosure_nonvanishing_eq_univ","k":"theorem","m":"Causalean.Mathlib.AlgebraicGeometry.PolynomialImageDimension.ZariskiClosure","a":"Mathlib/AlgebraicGeometry/PolynomialImageDimension","t":1,"d":"For a polynomial P in coordinates indexed by ι, if P is not the zero polynomial, then the set of points where P does not vanish is Zariski-dense in the whole affine space.","s":"∀ {ι : Type u_1} (P : MvPolynomial ι ℂ), P ≠ 0 → Causalean.Mathlib.AlgebraicGeometry.PolynomialImageDimension.affineZariskiClosure {x | (MvPolynomial.eval x) P ≠ 0} = Set.univ","l":99,"f":"Causalean/Mathlib/AlgebraicGeometry/PolynomialImageDimension/ZariskiClosure.lean"},{"n":"Causalean.Mathlib.AlgebraicGeometry.PolynomialImageDimension.affineZariskiClosure_polynomial_image_of_dense","k":"theorem","m":"Causalean.Mathlib.AlgebraicGeometry.PolynomialImageDimension.DenseImage","a":"Mathlib/AlgebraicGeometry/PolynomialImageDimension","t":2,"d":"The closure of the image of a dense set under a polynomial map equals the closure of the full range.","s":"∀ {ι : Type u_1} {κ : Type u_2} {f : (ι → ℂ) → κ → ℂ}, Causalean.Mathlib.AlgebraicGeometry.PolynomialImageDimension.IsPolynomialMap f → ∀ {A : Set (ι → ℂ)}, Causalean.Mathlib.AlgebraicGeometry.PolynomialImageDimension.affineZariskiClosure A = Set.univ → Causalean.Mathlib.AlgebraicGeometry.PolynomialImageDimension.affineZariskiClosure (f '' A) = Causalean.Mathlib.AlgebraicGeometry.PolynomialImageDimension.affineZariskiClosure (Set.range f)","l":21,"f":"Causalean/Mathlib/AlgebraicGeometry/PolynomialImageDimension/DenseImage.lean"},{"n":"Causalean.Mathlib.AlgebraicGeometry.PolynomialImageDimension.affineZariskiClosure_union","k":"theorem","m":"Causalean.Mathlib.AlgebraicGeometry.PolynomialImageDimension.ZariskiClosure","a":"Mathlib/AlgebraicGeometry/PolynomialImageDimension","t":2,"d":"The union of two affine-closed sets is affine-closed.","s":"∀ {ι : Type u_1} {A B : Set (ι → ℂ)}, Causalean.Mathlib.AlgebraicGeometry.PolynomialImageDimension.affineZariskiClosure A = A → Causalean.Mathlib.AlgebraicGeometry.PolynomialImageDimension.affineZariskiClosure B = B → Causalean.Mathlib.AlgebraicGeometry.PolynomialImageDimension.affineZariskiClosure (A ∪ B) = A ∪ B","l":73,"f":"Causalean/Mathlib/AlgebraicGeometry/PolynomialImageDimension/ZariskiClosure.lean"},{"n":"Causalean.Mathlib.AlgebraicGeometry.PolynomialImageDimension.affineZariskiClosure_zero_of_polynomial","k":"theorem","m":"Causalean.Mathlib.AlgebraicGeometry.PolynomialImageDimension.ZariskiClosure","a":"Mathlib/AlgebraicGeometry/PolynomialImageDimension","t":2,"d":"The zero set of one multivariate polynomial is affine-closed.","s":"∀ {ι : Type u_1} (P : MvPolynomial ι ℂ), Causalean.Mathlib.AlgebraicGeometry.PolynomialImageDimension.affineZariskiClosure {x | (MvPolynomial.eval x) P = 0} = {x | (MvPolynomial.eval x) P = 0}","l":63,"f":"Causalean/Mathlib/AlgebraicGeometry/PolynomialImageDimension/ZariskiClosure.lean"},{"n":"Causalean.Mathlib.AlgebraicGeometry.PolynomialImageDimension.affineZariskiDimension_iff_coordinateRingKrullDim","k":"theorem","m":"Causalean.Mathlib.AlgebraicGeometry.PolynomialImageDimension.CoordinateRing","a":"Mathlib/AlgebraicGeometry/PolynomialImageDimension","t":1,"d":"For a subset Z of ι → ℂ (with ι finite), if Z is Zariski-closed, then Z has irreducible-chain (Zariski) dimension d exactly when the Krull dimension of its coordinate ring — the polynomial ring on ι variables modulo Z's vanishing ideal — equals d.","s":"∀ {ι : Type u_1} [Finite ι] {d : ℕ} {Z : Set (ι → ℂ)}, Causalean.Mathlib.AlgebraicGeometry.PolynomialImageDimension.affineZariskiClosure Z = Z → (Causalean.Mathlib.AlgebraicGeometry.PolynomialImageDimension.HasAffineZariskiDimension d Z ↔ ringKrullDim (MvPolynomial ι ℂ ⧸ MvPolynomial.vanishingIdeal ℂ Z) = ↑d)","l":58,"f":"Causalean/Mathlib/AlgebraicGeometry/PolynomialImageDimension/CoordinateRing.lean"},{"n":"Causalean.Mathlib.AlgebraicGeometry.PolynomialImageDimension.algebraicIndependent_of_polynomialJacobianMinor_ne_zero","k":"theorem","m":"Causalean.Mathlib.AlgebraicGeometry.PolynomialImageDimension.Jacobian","a":"Mathlib/AlgebraicGeometry/PolynomialImageDimension","t":2,"d":"Over the complex numbers, a nonzero square Jacobian minor proves that the corresponding coordinate polynomials satisfy no nontrivial algebraic relation.","s":"∀ {ι : Type u_1} {κ : Type u_2} {d : ℕ} (f : κ → MvPolynomial ι ℂ) (rows : Fin d → κ) (cols : Fin d → ι), Causalean.Mathlib.AlgebraicGeometry.PolynomialImageDimension.polynomialJacobianMinor f rows cols ≠ 0 → AlgebraicIndependent ℂ fun a => f (rows a)","l":187,"f":"Causalean/Mathlib/AlgebraicGeometry/PolynomialImageDimension/Jacobian.lean"},{"n":"Causalean.Mathlib.AlgebraicGeometry.PolynomialImageDimension.coordinateSubalgebraAlgebra","k":"def","m":"Causalean.Mathlib.AlgebraicGeometry.PolynomialImageDimension.Transcendence","a":"Mathlib/AlgebraicGeometry/PolynomialImageDimension","t":1,"d":"The polynomial coordinate subalgebra inherits its usual complex-algebra structure, allowing its transcendence degree to be formed locally.","s":"{ι : Type u_1} → {κ : Type u_2} → (f : κ → MvPolynomial ι ℂ) → Algebra ℂ ↥(Causalean.Mathlib.AlgebraicGeometry.PolynomialImageDimension.polynomialCoordinateSubalgebra f)","l":23,"f":"Causalean/Mathlib/AlgebraicGeometry/PolynomialImageDimension/Transcendence.lean"},{"n":"Causalean.Mathlib.AlgebraicGeometry.PolynomialImageDimension.coordinateSubalgebra_trdeg_le_of_polynomial_factorization","k":"theorem","m":"Causalean.Mathlib.AlgebraicGeometry.PolynomialImageDimension.Transcendence","a":"Mathlib/AlgebraicGeometry/PolynomialImageDimension","t":2,"d":"The algebraic dimension (transcendence degree) of the coordinate algebra generated by a polynomial map is at most k whenever each of its coordinates can be written as a polynomial expression in a fixed family of k polynomials. This is the factorization-based companion to a Jacobian-minor bound: it certifies an upper bound on dimension from an explicit polynomial reparametrization rather than from a derivative rank.","s":"∀ {ι κ τ : Type} [Finite τ] (f : κ → MvPolynomial ι ℂ) (g : τ → MvPolynomial ι ℂ) (h : κ → MvPolynomial τ ℂ), (∀ (k : κ), (MvPolynomial.bind₁ g) (h k) = f k) → Algebra.trdeg ℂ ↥(Causalean.Mathlib.AlgebraicGeometry.PolynomialImageDimension.polynomialCoordinateSubalgebra f) ≤ ↑(Nat.card τ)","l":207,"f":"Causalean/Mathlib/AlgebraicGeometry/PolynomialImageDimension/Transcendence.lean"},{"n":"Causalean.Mathlib.AlgebraicGeometry.PolynomialImageDimension.coordinateSubalgebra_trdeg_le_of_surjection","k":"theorem","m":"Causalean.Mathlib.AlgebraicGeometry.PolynomialImageDimension.Transcendence","a":"Mathlib/AlgebraicGeometry/PolynomialImageDimension","t":2,"d":"A surjective presentation by a polynomial algebra in a given number of generators certifies that the coordinate subalgebra has no larger transcendence degree.","s":"∀ {ι : Type u_1} {κ : Type u_2} {τ : Type u_3} [Finite τ] (f : κ → MvPolynomial ι ℂ) (present : MvPolynomial τ ℂ →ₐ[ℂ] ↥(Causalean.Mathlib.AlgebraicGeometry.PolynomialImageDimension.polynomialCoordinateSubalgebra f)), Function.Surjective ⇑present → Algebra.trdeg ℂ ↥(Causalean.Mathlib.AlgebraicGeometry.PolynomialImageDimension.polynomialCoordinateSubalgebra f) ≤ ↑(Nat.card τ)","l":188,"f":"Causalean/Mathlib/AlgebraicGeometry/PolynomialImageDimension/Transcendence.lean"},{"n":"Causalean.Mathlib.AlgebraicGeometry.PolynomialImageDimension.eq_C_of_forall_pderiv_eq_zero","k":"theorem","m":"Causalean.Mathlib.AlgebraicGeometry.PolynomialImageDimension.Jacobian","a":"Mathlib/AlgebraicGeometry/PolynomialImageDimension","t":2,"d":"A polynomial over a characteristic-zero field in finitely many variables whose partial derivatives all vanish equals its constant coefficient.","s":"∀ {σ : Type u_1} {K : Type u_2} [Finite σ] [inst : Field K] [CharZero K] (P : MvPolynomial σ K), (∀ (i : σ), (MvPolynomial.pderiv i) P = 0) → P = MvPolynomial.C (MvPolynomial.coeff 0 P)","l":137,"f":"Causalean/Mathlib/AlgebraicGeometry/PolynomialImageDimension/Jacobian.lean"},{"n":"Causalean.Mathlib.AlgebraicGeometry.PolynomialImageDimension.eval_polynomialPullback","k":"theorem","m":"Causalean.Mathlib.AlgebraicGeometry.PolynomialImageDimension.Definitions","a":"Mathlib/AlgebraicGeometry/PolynomialImageDimension","t":1,"d":"For index types ι, κ, a polynomial parameterization f, a point x, and a polynomial P, substituting the coordinate polynomials f into P and evaluating at x agrees with evaluating P directly at the point obtained by evaluating f's coordinates at x.","s":"∀ {ι : Type u_1} {κ : Type u_2} (f : κ → MvPolynomial ι ℂ) (x : ι → ℂ) (P : MvPolynomial κ ℂ), (MvPolynomial.eval x) ((Causalean.Mathlib.AlgebraicGeometry.PolynomialImageDimension.polynomialPullback f) P) = (MvPolynomial.eval (Causalean.Mathlib.AlgebraicGeometry.PolynomialImageDimension.polynomialCoordinateMap f x)) P","l":40,"f":"Causalean/Mathlib/AlgebraicGeometry/PolynomialImageDimension/Definitions.lean"},{"n":"Causalean.Mathlib.AlgebraicGeometry.PolynomialImageDimension.finiteCoordinateRange_closed","k":"theorem","m":"Causalean.Mathlib.AlgebraicGeometry.PolynomialImageDimension.IrreducibleFiniteRange","a":"Mathlib/AlgebraicGeometry/PolynomialImageDimension","t":2,"d":"The set of complex coordinate vectors whose chosen coordinate belongs to a fixed finite set is closed in the affine Zariski topology.","s":"∀ {κ : Type u_1} (c : κ) (S : Finset ℂ), Causalean.Mathlib.AlgebraicGeometry.PolynomialImageDimension.affineZariskiClosure {x | x c ∈ S} = {x | x c ∈ S}","l":30,"f":"Causalean/Mathlib/AlgebraicGeometry/PolynomialImageDimension/IrreducibleFiniteRange.lean"},{"n":"Causalean.Mathlib.AlgebraicGeometry.PolynomialImageDimension.finiteTypeDomain_ringKrullDim_eq_trdeg","k":"theorem","m":"Causalean.Mathlib.AlgebraicGeometry.PolynomialImageDimension.Transcendence","a":"Mathlib/AlgebraicGeometry/PolynomialImageDimension","t":2,"d":"A finitely generated integral algebra over a field has Krull dimension equal to its finite transcendence degree, the number of algebraically independent coordinates it contains.","s":"∀ (K : Type u_1) (A : Type u_2) [inst : Field K] [inst_1 : CommRing A] [inst_2 : Algebra K A] [IsDomain A] [Algebra.FiniteType K A] (d : ℕ), Algebra.trdeg K A = ↑d → ringKrullDim A = ↑d","l":30,"f":"Causalean/Mathlib/AlgebraicGeometry/PolynomialImageDimension/Transcendence.lean"},{"n":"Causalean.Mathlib.AlgebraicGeometry.PolynomialImageDimension.hasAffineCodimensionIn_one_of_minimalPrime_span","k":"theorem","m":"Causalean.Mathlib.AlgebraicGeometry.PolynomialImageDimension.CodimensionOne","a":"Mathlib/AlgebraicGeometry/PolynomialImageDimension","t":1,"d":"For subsets C, Z, X of ι → ℂ (with ι finite), suppose X is irreducible Zariski-closed, Z is a proper subset of X, C is an irreducible component of Z, and the vanishing ideal of C is a minimal prime over the vanishing ideal of X joined with the principal ideal generated by some polynomial P. Then Z has affine codimension exactly 1 in X.","s":"∀ {ι : Type u_1} [Finite ι] {C Z X : Set (ι → ℂ)}, Causalean.Mathlib.AlgebraicGeometry.PolynomialImageDimension.IsIrreducibleAffineClosed X → Z ⊆ X → Z ≠ X → Causalean.Mathlib.AlgebraicGeometry.PolynomialImageDimension.IsIrreducibleAffineComponent C Z → ∀ (P : MvPolynomial ι ℂ), MvPolynomial.vanishingIdeal ℂ C ∈ (MvPolynomial.vanishingIdeal ℂ X ⊔ Ideal.span {P}).minimalPrimes → Causalean.Mathlib.AlgebraicGeometry.PolynomialImageDimension.HasAffineCodimensionIn 1 Z X","l":280,"f":"Causalean/Mathlib/AlgebraicGeometry/PolynomialImageDimension/CodimensionOne.lean"},{"n":"Causalean.Mathlib.AlgebraicGeometry.PolynomialImageDimension.image_strictMono_on","k":"theorem","m":"Causalean.Mathlib.AlgebraicGeometry.PolynomialImageDimension.PolynomialRetractDimension","a":"Mathlib/AlgebraicGeometry/PolynomialImageDimension","t":2,"d":"If a function is injective on a set containing every member of a strictly increasing chain of sets, then the images of those sets form a strictly increasing chain.","s":"∀ {α : Type u_1} {β : Type u_2} {ι : Type u_3} [inst : Preorder ι] (f : α → β) {S : Set α}, Set.InjOn f S → ∀ {chain : ι → Set α}, StrictMono chain → (∀ (i : ι), chain i ⊆ S) → StrictMono fun i => f '' chain i","l":46,"f":"Causalean/Mathlib/AlgebraicGeometry/PolynomialImageDimension/PolynomialRetractDimension.lean"},{"n":"Causalean.Mathlib.AlgebraicGeometry.PolynomialImageDimension.irreducibleAffineClosed_iff_isPrime","k":"theorem","m":"Causalean.Mathlib.AlgebraicGeometry.PolynomialImageDimension.Irreducibility","a":"Mathlib/AlgebraicGeometry/PolynomialImageDimension","t":1,"d":"A nonempty polynomially closed complex set Z is irreducible exactly when the polynomial equations that vanish on it form a prime ideal.","s":"∀ {ι : Type u_1} [Finite ι] {Z : Set (ι → ℂ)}, Causalean.Mathlib.AlgebraicGeometry.PolynomialImageDimension.affineZariskiClosure Z = Z → Z.Nonempty → (Causalean.Mathlib.AlgebraicGeometry.PolynomialImageDimension.IsIrreducibleAffineClosed Z ↔ (MvPolynomial.vanishingIdeal ℂ Z).IsPrime)","l":50,"f":"Causalean/Mathlib/AlgebraicGeometry/PolynomialImageDimension/Irreducibility.lean"},{"n":"Causalean.Mathlib.AlgebraicGeometry.PolynomialImageDimension.irreducibleAffineComponent_of_no_intermediate","k":"theorem","m":"Causalean.Mathlib.AlgebraicGeometry.PolynomialImageDimension.CodimensionOne","a":"Mathlib/AlgebraicGeometry/PolynomialImageDimension","t":2,"d":"An irreducible closed subset with no irreducible closed set strictly between it and an ambient irreducible variety is a component of every proper closed locus lying between the two.","s":"∀ {ι : Type u_1} {C Z X : Set (ι → ℂ)}, Causalean.Mathlib.AlgebraicGeometry.PolynomialImageDimension.IsIrreducibleAffineClosed C → C ⊆ Z → Z ⊆ X → Z ≠ X → (¬∃ Y, Causalean.Mathlib.AlgebraicGeometry.PolynomialImageDimension.IsIrreducibleAffineClosed Y ∧ C ⊂ Y ∧ Y ⊂ X) → Causalean.Mathlib.AlgebraicGeometry.PolynomialImageDimension.IsIrreducibleAffineComponent C Z","l":196,"f":"Causalean/Mathlib/AlgebraicGeometry/PolynomialImageDimension/CodimensionOne.lean"},{"n":"Causalean.Mathlib.AlgebraicGeometry.PolynomialImageDimension.irreducible_affineClosure_polynomial_image_of_dense","k":"theorem","m":"Causalean.Mathlib.AlgebraicGeometry.PolynomialImageDimension.DenseImage","a":"Mathlib/AlgebraicGeometry/PolynomialImageDimension","t":2,"d":"The closure of a polynomial image of a nonempty dense subset of finite complex affine space is irreducible.","s":"∀ {ι : Type u_1} {κ : Type u_2} [Finite ι] {f : (ι → ℂ) → κ → ℂ}, Causalean.Mathlib.AlgebraicGeometry.PolynomialImageDimension.IsPolynomialMap f → ∀ {A : Set (ι → ℂ)}, Causalean.Mathlib.AlgebraicGeometry.PolynomialImageDimension.affineZariskiClosure A = Set.univ → A.Nonempty → Causalean.Mathlib.AlgebraicGeometry.PolynomialImageDimension.IsIrreducibleAffineClosed (Causalean.Mathlib.AlgebraicGeometry.PolynomialImageDimension.affineZariskiClosure (f '' A))","l":45,"f":"Causalean/Mathlib/AlgebraicGeometry/PolynomialImageDimension/DenseImage.lean"},{"n":"Causalean.Mathlib.AlgebraicGeometry.PolynomialImageDimension.irreducible_coordinate_constant_of_finite_range","k":"theorem","m":"Causalean.Mathlib.AlgebraicGeometry.PolynomialImageDimension.IrreducibleFiniteRange","a":"Mathlib/AlgebraicGeometry/PolynomialImageDimension","t":1,"d":"On an irreducible affine-closed set Z, if a fixed coordinate c takes values only within a fixed finite set S at every point of Z, then that coordinate is in fact constant on Z, equal to some single value in S.","s":"∀ {κ : Type u_1} {Z : Set (κ → ℂ)}, Causalean.Mathlib.AlgebraicGeometry.PolynomialImageDimension.IsIrreducibleAffineClosed Z → ∀ (c : κ) (S : Finset ℂ), (∀ x ∈ Z, x c ∈ S) → ∃ z ∈ S, ∀ x ∈ Z, x c = z","l":41,"f":"Causalean/Mathlib/AlgebraicGeometry/PolynomialImageDimension/IrreducibleFiniteRange.lean"},{"n":"Causalean.Mathlib.AlgebraicGeometry.PolynomialImageDimension.irreducible_image_polynomial_retract","k":"theorem","m":"Causalean.Mathlib.AlgebraicGeometry.PolynomialImageDimension.PolynomialRetractDimension","a":"Mathlib/AlgebraicGeometry/PolynomialImageDimension","t":2,"d":"A polynomial embedding with a polynomial retraction sends irreducible affine-closed subsets to irreducible affine-closed images.","s":"∀ {ι : Type u_1} {κ : Type u_2} {f : (ι → ℂ) → κ → ℂ} {g : (κ → ℂ) → ι → ℂ}, Causalean.Mathlib.AlgebraicGeometry.PolynomialImageDimension.IsPolynomialMap f → Causalean.Mathlib.AlgebraicGeometry.PolynomialImageDimension.IsPolynomialMap g → Function.LeftInverse g f → ∀ {A : Set (ι → ℂ)}, Causalean.Mathlib.AlgebraicGeometry.PolynomialImageDimension.IsIrreducibleAffineClosed A → Causalean.Mathlib.AlgebraicGeometry.PolynomialImageDimension.IsIrreducibleAffineClosed (f '' A)","l":74,"f":"Causalean/Mathlib/AlgebraicGeometry/PolynomialImageDimension/PolynomialRetractDimension.lean"},{"n":"Causalean.Mathlib.AlgebraicGeometry.PolynomialImageDimension.irreducible_zeroLocus_of_prime","k":"theorem","m":"Causalean.Mathlib.AlgebraicGeometry.PolynomialImageDimension.Irreducibility","a":"Mathlib/AlgebraicGeometry/PolynomialImageDimension","t":2,"d":"The zero locus of a prime ideal is irreducible affine-closed.","s":"∀ {ι : Type u_1} [Finite ι] (P : Ideal (MvPolynomial ι ℂ)) [P.IsPrime], Causalean.Mathlib.AlgebraicGeometry.PolynomialImageDimension.IsIrreducibleAffineClosed (MvPolynomial.zeroLocus ℂ P)","l":122,"f":"Causalean/Mathlib/AlgebraicGeometry/PolynomialImageDimension/Irreducibility.lean"},{"n":"Causalean.Mathlib.AlgebraicGeometry.PolynomialImageDimension.isPolynomialMap_id","k":"theorem","m":"Causalean.Mathlib.AlgebraicGeometry.PolynomialImageDimension.PolynomialMap","a":"Mathlib/AlgebraicGeometry/PolynomialImageDimension","t":2,"d":"The identity map is polynomial.","s":"∀ {ι : Type u_1}, Causalean.Mathlib.AlgebraicGeometry.PolynomialImageDimension.IsPolynomialMap id","l":24,"f":"Causalean/Mathlib/AlgebraicGeometry/PolynomialImageDimension/PolynomialMap.lean"},{"n":"Causalean.Mathlib.AlgebraicGeometry.PolynomialImageDimension.jacobianCoordinateSubalgebraAlgebra","k":"def","m":"Causalean.Mathlib.AlgebraicGeometry.PolynomialImageDimension.Jacobian","a":"Mathlib/AlgebraicGeometry/PolynomialImageDimension","t":1,"d":"The polynomial coordinate subalgebra inherits its usual complex-algebra structure, allowing the Jacobian argument to use its transcendence degree.","s":"{ι : Type u_1} → {κ : Type u_2} → (f : κ → MvPolynomial ι ℂ) → Algebra ℂ ↥(Causalean.Mathlib.AlgebraicGeometry.PolynomialImageDimension.polynomialCoordinateSubalgebra f)","l":21,"f":"Causalean/Mathlib/AlgebraicGeometry/PolynomialImageDimension/Jacobian.lean"},{"n":"Causalean.Mathlib.AlgebraicGeometry.PolynomialImageDimension.linearMap_isPolynomial","k":"theorem","m":"Causalean.Mathlib.AlgebraicGeometry.PolynomialImageDimension.AffineSubspaceDimension","a":"Mathlib/AlgebraicGeometry/PolynomialImageDimension","t":2,"d":"A linear map from a finite-coordinate source is polynomial.","s":"∀ {ι : Type u_1} {κ : Type u_2} [Finite ι] (F : (ι → ℂ) →ₗ[ℂ] κ → ℂ), Causalean.Mathlib.AlgebraicGeometry.PolynomialImageDimension.IsPolynomialMap ⇑F","l":22,"f":"Causalean/Mathlib/AlgebraicGeometry/PolynomialImageDimension/AffineSubspaceDimension.lean"},{"n":"Causalean.Mathlib.AlgebraicGeometry.PolynomialImageDimension.map_quotient_strictMono","k":"theorem","m":"Causalean.Mathlib.AlgebraicGeometry.PolynomialImageDimension.CodimensionOne","a":"Mathlib/AlgebraicGeometry/PolynomialImageDimension","t":2,"d":"If one ideal is strictly contained in another and both contain a third ideal, their images in the quotient by that third ideal remain strictly ordered.","s":"∀ {R : Type u_1} [inst : CommRing R] (I A B : Ideal R), I ≤ A → A < B → Ideal.map (Ideal.Quotient.mk I) A < Ideal.map (Ideal.Quotient.mk I) B","l":45,"f":"Causalean/Mathlib/AlgebraicGeometry/PolynomialImageDimension/CodimensionOne.lean"},{"n":"Causalean.Mathlib.AlgebraicGeometry.PolynomialImageDimension.no_intermediate_of_exact_affine_dimensions","k":"theorem","m":"Causalean.Mathlib.AlgebraicGeometry.PolynomialImageDimension.CodimensionOne","a":"Mathlib/AlgebraicGeometry/PolynomialImageDimension","t":2,"d":"Exact dimensions differing by one exclude an intermediate irreducible closed set. The proof appends Y and X to a maximal-length chain inside C, contradicting the upper bound for X.","s":"∀ {ι : Type u_1} {d : ℕ} {C X : Set (ι → ℂ)}, Causalean.Mathlib.AlgebraicGeometry.PolynomialImageDimension.IsIrreducibleAffineClosed X → Causalean.Mathlib.AlgebraicGeometry.PolynomialImageDimension.HasAffineZariskiDimension d C → Causalean.Mathlib.AlgebraicGeometry.PolynomialImageDimension.HasAffineZariskiDimension (d + 1) X → ¬∃ Y, Causalean.Mathlib.AlgebraicGeometry.PolynomialImageDimension.IsIrreducibleAffineClosed Y ∧ C ⊂ Y ∧ Y ⊂ X","l":168,"f":"Causalean/Mathlib/AlgebraicGeometry/PolynomialImageDimension/CodimensionOne.lean"},{"n":"Causalean.Mathlib.AlgebraicGeometry.PolynomialImageDimension.no_three_chain_of_minimalPrime_span","k":"theorem","m":"Causalean.Mathlib.AlgebraicGeometry.PolynomialImageDimension.CodimensionOne","a":"Mathlib/AlgebraicGeometry/PolynomialImageDimension","t":2,"d":"A prime affine subvariety whose prime ideal is minimal over one additional equation in the coordinate ring of X admits no intermediate irreducible closed set. This is the endpoint-fixed form of Krull's principal ideal theorem needed by HasAffineCodimensionIn 1.","s":"∀ {ι : Type u_1} [Finite ι] {C X : Set (ι → ℂ)}, Causalean.Mathlib.AlgebraicGeometry.PolynomialImageDimension.IsIrreducibleAffineClosed C → Causalean.Mathlib.AlgebraicGeometry.PolynomialImageDimension.IsIrreducibleAffineClosed X → ∀ (P : MvPolynomial ι ℂ), MvPolynomial.vanishingIdeal ℂ C ∈ (MvPolynomial.vanishingIdeal ℂ X ⊔ Ideal.span {P}).minimalPrimes → ¬∃ chain, StrictMono chain ∧ (∀ (i : Fin 3), Causalean.Mathlib.AlgebraicGeometry.PolynomialImageDimension.IsIrreducibleAffineClosed (chain i)) ∧ chain 0 = C ∧ chain (Fin.last 2) = X","l":56,"f":"Causalean/Mathlib/AlgebraicGeometry/PolynomialImageDimension/CodimensionOne.lean"},{"n":"Causalean.Mathlib.AlgebraicGeometry.PolynomialImageDimension.nonempty_zeroLocus_of_prime","k":"theorem","m":"Causalean.Mathlib.AlgebraicGeometry.PolynomialImageDimension.Irreducibility","a":"Mathlib/AlgebraicGeometry/PolynomialImageDimension","t":2,"d":"A prime ideal in a finite complex polynomial ring has a nonempty zero locus.","s":"∀ {ι : Type u_1} [Finite ι] (P : Ideal (MvPolynomial ι ℂ)) [P.IsPrime], (MvPolynomial.zeroLocus ℂ P).Nonempty","l":39,"f":"Causalean/Mathlib/AlgebraicGeometry/PolynomialImageDimension/Irreducibility.lean"},{"n":"Causalean.Mathlib.AlgebraicGeometry.PolynomialImageDimension.pderiv_polynomialPullback","k":"theorem","m":"Causalean.Mathlib.AlgebraicGeometry.PolynomialImageDimension.Jacobian","a":"Mathlib/AlgebraicGeometry/PolynomialImageDimension","t":2,"d":"The partial derivative of a polynomial after polynomial substitution equals the sum of the substituted partial derivatives, each weighted by the corresponding derivative of the substituted polynomial.","s":"∀ {ι : Type u_1} {κ : Type u_2} [inst : Fintype κ] (f : κ → MvPolynomial ι ℂ) (P : MvPolynomial κ ℂ) (i : ι), (MvPolynomial.pderiv i) ((Causalean.Mathlib.AlgebraicGeometry.PolynomialImageDimension.polynomialPullback f) P) = ∑ j, (Causalean.Mathlib.AlgebraicGeometry.PolynomialImageDimension.polynomialPullback f) ((MvPolynomial.pderiv j) P) * (MvPolynomial.pderiv i) (f j)","l":45,"f":"Causalean/Mathlib/AlgebraicGeometry/PolynomialImageDimension/Jacobian.lean"},{"n":"Causalean.Mathlib.AlgebraicGeometry.PolynomialImageDimension.polynomialCoordinateMap","k":"def","m":"Causalean.Mathlib.AlgebraicGeometry.PolynomialImageDimension.Definitions","a":"Mathlib/AlgebraicGeometry/PolynomialImageDimension","t":1,"d":"A polynomial coordinate map sends a complex parameter vector to the vector of values of a specified finite family of coordinate polynomials.","s":"{ι : Type u_1} → {κ : Type u_2} → (κ → MvPolynomial ι ℂ) → (ι → ℂ) → κ → ℂ","l":16,"f":"Causalean/Mathlib/AlgebraicGeometry/PolynomialImageDimension/Definitions.lean"},{"n":"Causalean.Mathlib.AlgebraicGeometry.PolynomialImageDimension.polynomialCoordinateSubalgebra","k":"def","m":"Causalean.Mathlib.AlgebraicGeometry.PolynomialImageDimension.Definitions","a":"Mathlib/AlgebraicGeometry/PolynomialImageDimension","t":1,"d":"The coordinate subalgebra is the complex algebra generated by the coordinate polynomials of a polynomial parameterization.","s":"{ι : Type u_1} → {κ : Type u_2} → (κ → MvPolynomial ι ℂ) → Subalgebra ℂ (MvPolynomial ι ℂ)","l":28,"f":"Causalean/Mathlib/AlgebraicGeometry/PolynomialImageDimension/Definitions.lean"},{"n":"Causalean.Mathlib.AlgebraicGeometry.PolynomialImageDimension.polynomialImageClosure","k":"def","m":"Causalean.Mathlib.AlgebraicGeometry.PolynomialImageDimension.Definitions","a":"Mathlib/AlgebraicGeometry/PolynomialImageDimension","t":1,"d":"The polynomial image closure is the smallest complex algebraic set containing all values attained by the polynomial parameterization.","s":"{ι : Type u_1} → {κ : Type u_2} → (κ → MvPolynomial ι ℂ) → Set (κ → ℂ)","l":34,"f":"Causalean/Mathlib/AlgebraicGeometry/PolynomialImageDimension/Definitions.lean"},{"n":"Causalean.Mathlib.AlgebraicGeometry.PolynomialImageDimension.polynomialImageClosure_coordinateRing","k":"def","m":"Causalean.Mathlib.AlgebraicGeometry.PolynomialImageDimension.CoordinateRing","a":"Mathlib/AlgebraicGeometry/PolynomialImageDimension","t":1,"d":"The coordinate ring of a polynomial image closure is canonically the same complex algebra as the source-polynomial subalgebra generated by the image coordinates.","s":"{ι : Type u_1} → {κ : Type u_2} → (f : κ → MvPolynomial ι ℂ) → (MvPolynomial κ ℂ ⧸ MvPolynomial.vanishingIdeal ℂ (Causalean.Mathlib.AlgebraicGeometry.PolynomialImageDimension.polynomialImageClosure f)) ≃ₐ[ℂ] ↥(Causalean.Mathlib.AlgebraicGeometry.PolynomialImageDimension.polynomialCoordinateSubalgebra f)","l":48,"f":"Causalean/Mathlib/AlgebraicGeometry/PolynomialImageDimension/CoordinateRing.lean"},{"n":"Causalean.Mathlib.AlgebraicGeometry.PolynomialImageDimension.polynomialImageClosure_dimension_of_jacobian","k":"theorem","m":"Causalean.Mathlib.AlgebraicGeometry.PolynomialImageDimension.Jacobian","a":"Mathlib/AlgebraicGeometry/PolynomialImageDimension","t":1,"d":"Given a finite polynomial parameterization f of κ-many coordinates by variables indexed by a finite set ι, together with d selected output coordinates rows and input variables cols, if the corresponding d × d Jacobian minor of f, formed from partial derivatives, is nonzero and the transcendence degree of the coordinate subalgebra generated by f is at most d, then the Zariski closure of the image of f has exact affine (irreducible-chain) dimension d.","s":"∀ {ι : Type u_1} {κ : Type u_2} [Finite ι] [Finite κ] {d : ℕ} (f : κ → MvPolynomial ι ℂ) (rows : Fin d → κ) (cols : Fin d → ι), Causalean.Mathlib.AlgebraicGeometry.PolynomialImageDimension.polynomialJacobianMinor f rows cols ≠ 0 → Algebra.trdeg ℂ ↥(Causalean.Mathlib.AlgebraicGeometry.PolynomialImageDimension.polynomialCoordinateSubalgebra f) ≤ ↑d → Causalean.Mathlib.AlgebraicGeometry.PolynomialImageDimension.HasAffineZariskiDimension d (Causalean.Mathlib.AlgebraicGeometry.PolynomialImageDimension.polynomialImageClosure f)","l":246,"f":"Causalean/Mathlib/AlgebraicGeometry/PolynomialImageDimension/Jacobian.lean"},{"n":"Causalean.Mathlib.AlgebraicGeometry.PolynomialImageDimension.polynomialImageClosure_dimension_of_jacobian_and_surjection","k":"theorem","m":"Causalean.Mathlib.AlgebraicGeometry.PolynomialImageDimension.Jacobian","a":"Mathlib/AlgebraicGeometry/PolynomialImageDimension","t":2,"d":"A nonzero Jacobian minor and a surjective presentation using the same number of generators identify the exact affine dimension of a polynomial image closure.","s":"∀ {ι : Type u_1} {κ : Type u_2} [Finite ι] [Finite κ] {d : ℕ} (f : κ → MvPolynomial ι ℂ) (rows : Fin d → κ) (cols : Fin d → ι), Causalean.Mathlib.AlgebraicGeometry.PolynomialImageDimension.polynomialJacobianMinor f rows cols ≠ 0 → ∀ (present : MvPolynomial (Fin d) ℂ →ₐ[ℂ] ↥(Causalean.Mathlib.AlgebraicGeometry.PolynomialImageDimension.polynomialCoordinateSubalgebra f)), Function.Surjective ⇑present → Causalean.Mathlib.AlgebraicGeometry.PolynomialImageDimension.HasAffineZariskiDimension d (Causalean.Mathlib.AlgebraicGeometry.PolynomialImageDimension.polynomialImageClosure f)","l":284,"f":"Causalean/Mathlib/AlgebraicGeometry/PolynomialImageDimension/Jacobian.lean"},{"n":"Causalean.Mathlib.AlgebraicGeometry.PolynomialImageDimension.polynomialImageClosure_dimension_of_trdeg","k":"theorem","m":"Causalean.Mathlib.AlgebraicGeometry.PolynomialImageDimension.Transcendence","a":"Mathlib/AlgebraicGeometry/PolynomialImageDimension","t":1,"d":"For a finite polynomial parameterization f of κ-many coordinates by variables indexed by a finite set ι, if the transcendence degree of the subalgebra of ℂ generated by the coordinate polynomials of f equals d, then the Zariski closure of the image of f has exact affine (irreducible-chain) dimension d.","s":"∀ {ι : Type u_1} {κ : Type u_2} [Finite ι] [Finite κ] (f : κ → MvPolynomial ι ℂ) (d : ℕ), Algebra.trdeg ℂ ↥(Causalean.Mathlib.AlgebraicGeometry.PolynomialImageDimension.polynomialCoordinateSubalgebra f) = ↑d → Causalean.Mathlib.AlgebraicGeometry.PolynomialImageDimension.HasAffineZariskiDimension d (Causalean.Mathlib.AlgebraicGeometry.PolynomialImageDimension.polynomialImageClosure f)","l":151,"f":"Causalean/Mathlib/AlgebraicGeometry/PolynomialImageDimension/Transcendence.lean"},{"n":"Causalean.Mathlib.AlgebraicGeometry.PolynomialImageDimension.polynomialImageClosure_isIrreducible","k":"theorem","m":"Causalean.Mathlib.AlgebraicGeometry.PolynomialImageDimension.DenseImage","a":"Mathlib/AlgebraicGeometry/PolynomialImageDimension","t":1,"d":"For a map f from the affine space ℂ^ι (with ι finite) to ℂ^κ that is coordinatewise polynomial, the Zariski closure of the range of f is irreducible as a polynomially closed set.","s":"∀ {ι : Type u_1} {κ : Type u_2} [Finite ι] {f : (ι → ℂ) → κ → ℂ}, Causalean.Mathlib.AlgebraicGeometry.PolynomialImageDimension.IsPolynomialMap f → Causalean.Mathlib.AlgebraicGeometry.PolynomialImageDimension.IsIrreducibleAffineClosed (Causalean.Mathlib.AlgebraicGeometry.PolynomialImageDimension.affineZariskiClosure (Set.range f))","l":109,"f":"Causalean/Mathlib/AlgebraicGeometry/PolynomialImageDimension/DenseImage.lean"},{"n":"Causalean.Mathlib.AlgebraicGeometry.PolynomialImageDimension.polynomialImageClosure_vanishingIdeal","k":"theorem","m":"Causalean.Mathlib.AlgebraicGeometry.PolynomialImageDimension.CoordinateRing","a":"Mathlib/AlgebraicGeometry/PolynomialImageDimension","t":2,"d":"A target polynomial relation holds throughout a polynomial image closure exactly when substituting the parameterization's coordinate polynomials makes the relation identically zero.","s":"∀ {ι : Type u_1} {κ : Type u_2} (f : κ → MvPolynomial ι ℂ), MvPolynomial.vanishingIdeal ℂ (Causalean.Mathlib.AlgebraicGeometry.PolynomialImageDimension.polynomialImageClosure f) = RingHom.ker (Causalean.Mathlib.AlgebraicGeometry.PolynomialImageDimension.polynomialPullback f)","l":17,"f":"Causalean/Mathlib/AlgebraicGeometry/PolynomialImageDimension/CoordinateRing.lean"},{"n":"Causalean.Mathlib.AlgebraicGeometry.PolynomialImageDimension.polynomialJacobianMinor","k":"def","m":"Causalean.Mathlib.AlgebraicGeometry.PolynomialImageDimension.Jacobian","a":"Mathlib/AlgebraicGeometry/PolynomialImageDimension","t":1,"d":"A polynomial Jacobian minor records the determinant of selected derivative coordinates of a polynomial parameterization before evaluating it at any point.","s":"{R : Type u_1} → {ι : Type u_2} → {κ : Type u_3} → [inst : CommRing R] → {d : ℕ} → (κ → MvPolynomial ι R) → (Fin d → κ) → (Fin d → ι) → MvPolynomial ι R","l":28,"f":"Causalean/Mathlib/AlgebraicGeometry/PolynomialImageDimension/Jacobian.lean"},{"n":"Causalean.Mathlib.AlgebraicGeometry.PolynomialImageDimension.polynomialPullback","k":"def","m":"Causalean.Mathlib.AlgebraicGeometry.PolynomialImageDimension.Definitions","a":"Mathlib/AlgebraicGeometry/PolynomialImageDimension","t":1,"d":"Polynomial pullback replaces each target coordinate in a polynomial relation by its corresponding source-coordinate polynomial.","s":"{ι : Type u_1} → {κ : Type u_2} → (κ → MvPolynomial ι ℂ) → MvPolynomial κ ℂ →ₐ[ℂ] MvPolynomial ι ℂ","l":22,"f":"Causalean/Mathlib/AlgebraicGeometry/PolynomialImageDimension/Definitions.lean"},{"n":"Causalean.Mathlib.AlgebraicGeometry.PolynomialImageDimension.polynomialRetract_range_dimension","k":"theorem","m":"Causalean.Mathlib.AlgebraicGeometry.PolynomialImageDimension.PolynomialRetractDimension","a":"Mathlib/AlgebraicGeometry/PolynomialImageDimension","t":1,"d":"For maps between the affine space ℂ^d and ℂ^κ, if f is coordinatewise polynomial, an accompanying map g back to ℂ^d is coordinatewise polynomial, and g is a left inverse of f, i.e. g (f x) = x for every x, then the range of f has exact affine (irreducible-chain) dimension d, matching the dimension of its source space.","s":"∀ {d : ℕ} {κ : Type u_1} {f : (Fin d → ℂ) → κ → ℂ} {g : (κ → ℂ) → Fin d → ℂ}, Causalean.Mathlib.AlgebraicGeometry.PolynomialImageDimension.IsPolynomialMap f → Causalean.Mathlib.AlgebraicGeometry.PolynomialImageDimension.IsPolynomialMap g → Function.LeftInverse g f → Causalean.Mathlib.AlgebraicGeometry.PolynomialImageDimension.HasAffineZariskiDimension d (Set.range f)","l":114,"f":"Causalean/Mathlib/AlgebraicGeometry/PolynomialImageDimension/PolynomialRetractDimension.lean"},{"n":"Causalean.Mathlib.AlgebraicGeometry.PolynomialImageDimension.polynomialRing_dimension","k":"theorem","m":"Causalean.Mathlib.AlgebraicGeometry.PolynomialImageDimension.AffineSpaceDimension","a":"Mathlib/AlgebraicGeometry/PolynomialImageDimension","t":2,"d":"The Krull dimension of the ring of complex polynomials in d variables is d.","s":"∀ (d : ℕ), ringKrullDim (MvPolynomial (Fin d) ℂ) = ↑d","l":40,"f":"Causalean/Mathlib/AlgebraicGeometry/PolynomialImageDimension/AffineSpaceDimension.lean"},{"n":"Causalean.Mathlib.AlgebraicGeometry.PolynomialImageDimension.polynomial_fixedPoints_closed","k":"theorem","m":"Causalean.Mathlib.AlgebraicGeometry.PolynomialImageDimension.PolynomialMap","a":"Mathlib/AlgebraicGeometry/PolynomialImageDimension","t":2,"d":"The fixed-point set of a polynomial endomorphism is affine-closed.","s":"∀ {ι : Type u_1} {f : (ι → ℂ) → ι → ℂ}, Causalean.Mathlib.AlgebraicGeometry.PolynomialImageDimension.IsPolynomialMap f → Causalean.Mathlib.AlgebraicGeometry.PolynomialImageDimension.affineZariskiClosure {x | f x = x} = {x | f x = x}","l":92,"f":"Causalean/Mathlib/AlgebraicGeometry/PolynomialImageDimension/PolynomialMap.lean"},{"n":"Causalean.Mathlib.AlgebraicGeometry.PolynomialImageDimension.polynomial_image_closed_of_retract","k":"theorem","m":"Causalean.Mathlib.AlgebraicGeometry.PolynomialImageDimension.PolynomialMap","a":"Mathlib/AlgebraicGeometry/PolynomialImageDimension","t":2,"d":"A polynomial retract sends affine-closed source subsets to affine-closed images.","s":"∀ {ι : Type u_1} {κ : Type u_2} {f : (ι → ℂ) → κ → ℂ} {g : (κ → ℂ) → ι → ℂ}, Causalean.Mathlib.AlgebraicGeometry.PolynomialImageDimension.IsPolynomialMap f → Causalean.Mathlib.AlgebraicGeometry.PolynomialImageDimension.IsPolynomialMap g → Function.LeftInverse g f → ∀ {A : Set (ι → ℂ)}, Causalean.Mathlib.AlgebraicGeometry.PolynomialImageDimension.affineZariskiClosure A = A → Causalean.Mathlib.AlgebraicGeometry.PolynomialImageDimension.affineZariskiClosure (f '' A) = f '' A","l":128,"f":"Causalean/Mathlib/AlgebraicGeometry/PolynomialImageDimension/PolynomialMap.lean"},{"n":"Causalean.Mathlib.AlgebraicGeometry.PolynomialImageDimension.polynomial_preimage_closed","k":"theorem","m":"Causalean.Mathlib.AlgebraicGeometry.PolynomialImageDimension.PolynomialMap","a":"Mathlib/AlgebraicGeometry/PolynomialImageDimension","t":2,"d":"The preimage of an affine-closed set under a polynomial map is affine-closed.","s":"∀ {ι : Type u_1} {κ : Type u_2} {f : (ι → ℂ) → κ → ℂ}, Causalean.Mathlib.AlgebraicGeometry.PolynomialImageDimension.IsPolynomialMap f → ∀ {A : Set (κ → ℂ)}, Causalean.Mathlib.AlgebraicGeometry.PolynomialImageDimension.affineZariskiClosure A = A → Causalean.Mathlib.AlgebraicGeometry.PolynomialImageDimension.affineZariskiClosure (f ⁻¹' A) = f ⁻¹' A","l":73,"f":"Causalean/Mathlib/AlgebraicGeometry/PolynomialImageDimension/PolynomialMap.lean"},{"n":"Causalean.Mathlib.AlgebraicGeometry.PolynomialImageDimension.polynomial_range_closed_of_retract","k":"theorem","m":"Causalean.Mathlib.AlgebraicGeometry.PolynomialImageDimension.PolynomialMap","a":"Mathlib/AlgebraicGeometry/PolynomialImageDimension","t":1,"d":"For maps between the finite-dimensional complex affine spaces ℂ^ι and ℂ^κ, if f is coordinatewise polynomial, an accompanying map g back to ℂ^ι is coordinatewise polynomial, and g is a left inverse of f, i.e. g (f x) = x for every x, then the range of f is already Zariski-closed.","s":"∀ {ι : Type u_1} {κ : Type u_2} {f : (ι → ℂ) → κ → ℂ} {g : (κ → ℂ) → ι → ℂ}, Causalean.Mathlib.AlgebraicGeometry.PolynomialImageDimension.IsPolynomialMap f → Causalean.Mathlib.AlgebraicGeometry.PolynomialImageDimension.IsPolynomialMap g → Function.LeftInverse g f → Causalean.Mathlib.AlgebraicGeometry.PolynomialImageDimension.affineZariskiClosure (Set.range f) = Set.range f","l":108,"f":"Causalean/Mathlib/AlgebraicGeometry/PolynomialImageDimension/PolynomialMap.lean"},{"n":"Causalean.Mathlib.AlgebraicGeometry.PolynomialImageDimension.primeOfIrreducible","k":"def","m":"Causalean.Mathlib.AlgebraicGeometry.PolynomialImageDimension.AffineSpaceDimension","a":"Mathlib/AlgebraicGeometry/PolynomialImageDimension","t":1,"d":"An irreducible affine closed subset of finite-dimensional complex affine space determines the point of the polynomial ring's prime spectrum given by its vanishing ideal.","s":"{ι : Type u_1} → [Finite ι] → (Z : Set (ι → ℂ)) → Causalean.Mathlib.AlgebraicGeometry.PolynomialImageDimension.IsIrreducibleAffineClosed Z → PrimeSpectrum (MvPolynomial ι ℂ)","l":24,"f":"Causalean/Mathlib/AlgebraicGeometry/PolynomialImageDimension/AffineSpaceDimension.lean"},{"n":"Causalean.Mathlib.AlgebraicGeometry.PolynomialImageDimension.primeOfIrreducible_lt","k":"theorem","m":"Causalean.Mathlib.AlgebraicGeometry.PolynomialImageDimension.AffineSpaceDimension","a":"Mathlib/AlgebraicGeometry/PolynomialImageDimension","t":2,"d":"For two irreducible affine closed subsets of finite-dimensional complex affine space, strict inclusion reverses the strict order of their corresponding vanishing-ideal prime points.","s":"∀ {ι : Type u_1} [inst : Finite ι] {A B : Set (ι → ℂ)} (hA : Causalean.Mathlib.AlgebraicGeometry.PolynomialImageDimension.IsIrreducibleAffineClosed A) (hB : Causalean.Mathlib.AlgebraicGeometry.PolynomialImageDimension.IsIrreducibleAffineClosed B), A ⊂ B → Causalean.Mathlib.AlgebraicGeometry.PolynomialImageDimension.primeOfIrreducible B hB < Causalean.Mathlib.AlgebraicGeometry.PolynomialImageDimension.primeOfIrreducible A hA","l":32,"f":"Causalean/Mathlib/AlgebraicGeometry/PolynomialImageDimension/AffineSpaceDimension.lean"},{"n":"Causalean.Mathlib.AlgebraicGeometry.PolynomialImageDimension.sum_X_mul_pderiv_eq_sum_degree","k":"theorem","m":"Causalean.Mathlib.AlgebraicGeometry.PolynomialImageDimension.Jacobian","a":"Mathlib/AlgebraicGeometry/PolynomialImageDimension","t":2,"d":"For a polynomial over a commutative semiring in finitely many variables, the sum of each variable times its partial derivative equals the sum of its monomials weighted by their total degrees.","s":"∀ {σ : Type u_1} {R : Type u_2} [inst : Fintype σ] [inst_1 : CommSemiring R] (P : MvPolynomial σ R), ∑ i, MvPolynomial.X i * (MvPolynomial.pderiv i) P = ∑ m ∈ P.support, Finsupp.degree m • (MvPolynomial.monomial m) (MvPolynomial.coeff m P)","l":112,"f":"Causalean/Mathlib/AlgebraicGeometry/PolynomialImageDimension/Jacobian.lean"},{"n":"Causalean.Mathlib.AlgebraicGeometry.PolynomialImageDimension.totalDegree_pderiv_lt","k":"theorem","m":"Causalean.Mathlib.AlgebraicGeometry.PolynomialImageDimension.Jacobian","a":"Mathlib/AlgebraicGeometry/PolynomialImageDimension","t":2,"d":"A nonzero partial derivative of a polynomial has strictly smaller total degree than the original polynomial.","s":"∀ {σ : Type u_1} {R : Type u_2} [inst : CommSemiring R] {i : σ} {P : MvPolynomial σ R}, (MvPolynomial.pderiv i) P ≠ 0 → ((MvPolynomial.pderiv i) P).totalDegree < P.totalDegree","l":76,"f":"Causalean/Mathlib/AlgebraicGeometry/PolynomialImageDimension/Jacobian.lean"},{"n":"Causalean.Mathlib.AlgebraicGeometry.PolynomialImageDimension.vanishingIdeal_mem_minimalPrimes_span_of_no_intermediate","k":"theorem","m":"Causalean.Mathlib.AlgebraicGeometry.PolynomialImageDimension.CodimensionOne","a":"Mathlib/AlgebraicGeometry/PolynomialImageDimension","t":2,"d":"Geometric height one plus one equation through C identifies its prime ideal as a minimal prime over that equation in the coordinate ring of X. This is the Nullstellensatz converse used to manufacture the explicit minimal-prime certificates from the D0 dimension argument.","s":"∀ {ι : Type u_1} [Finite ι] {C X : Set (ι → ℂ)}, Causalean.Mathlib.AlgebraicGeometry.PolynomialImageDimension.IsIrreducibleAffineClosed C → Causalean.Mathlib.AlgebraicGeometry.PolynomialImageDimension.IsIrreducibleAffineClosed X → C ⊆ X → ∀ P ∈ MvPolynomial.vanishingIdeal ℂ C, P ∉ MvPolynomial.vanishingIdeal ℂ X → (¬∃ Y, Causalean.Mathlib.AlgebraicGeometry.PolynomialImageDimension.IsIrreducibleAffineClosed Y ∧ C ⊂ Y ∧ Y ⊂ X) → MvPolynomial.vanishingIdeal ℂ C ∈ (MvPolynomial.vanishingIdeal ℂ X ⊔ Ideal.span {P}).minimalPrimes","l":224,"f":"Causalean/Mathlib/AlgebraicGeometry/PolynomialImageDimension/CodimensionOne.lean"},{"n":"Causalean.Mathlib.AlgebraicGeometry.PolynomialImageDimension.vanishingIdeal_ne_top_of_nonempty","k":"theorem","m":"Causalean.Mathlib.AlgebraicGeometry.PolynomialImageDimension.Irreducibility","a":"Mathlib/AlgebraicGeometry/PolynomialImageDimension","t":2,"d":"A nonempty affine set has a proper vanishing ideal.","s":"∀ {ι : Type u_1} {Z : Set (ι → ℂ)}, Z.Nonempty → MvPolynomial.vanishingIdeal ℂ Z ≠ ⊤","l":29,"f":"Causalean/Mathlib/AlgebraicGeometry/PolynomialImageDimension/Irreducibility.lean"},{"n":"Causalean.Mathlib.AlgebraicGeometry.PolynomialImageDimension.vanishingIdeal_strict_anti","k":"theorem","m":"Causalean.Mathlib.AlgebraicGeometry.PolynomialImageDimension.Irreducibility","a":"Mathlib/AlgebraicGeometry/PolynomialImageDimension","t":2,"d":"Strict inclusion of affine-closed sets strictly reverses their vanishing ideals.","s":"∀ {ι : Type u_1} {A B : Set (ι → ℂ)}, Causalean.Mathlib.AlgebraicGeometry.PolynomialImageDimension.affineZariskiClosure A = A → Causalean.Mathlib.AlgebraicGeometry.PolynomialImageDimension.affineZariskiClosure B = B → A ⊂ B → MvPolynomial.vanishingIdeal ℂ B < MvPolynomial.vanishingIdeal ℂ A","l":135,"f":"Causalean/Mathlib/AlgebraicGeometry/PolynomialImageDimension/Irreducibility.lean"},{"n":"Causalean.Mathlib.AlgebraicGeometry.PolynomialImageDimension.zeroLocus_strict_anti","k":"theorem","m":"Causalean.Mathlib.AlgebraicGeometry.PolynomialImageDimension.Irreducibility","a":"Mathlib/AlgebraicGeometry/PolynomialImageDimension","t":2,"d":"Strict inclusion of prime ideals strictly reverses their complex zero loci.","s":"∀ {ι : Type u_1} [Finite ι] {P Q : Ideal (MvPolynomial ι ℂ)} [P.IsPrime] [Q.IsPrime], P < Q → MvPolynomial.zeroLocus ℂ Q ⊂ MvPolynomial.zeroLocus ℂ P","l":145,"f":"Causalean/Mathlib/AlgebraicGeometry/PolynomialImageDimension/Irreducibility.lean"},{"n":"Causalean.Mathlib.Analysis.ArgumentPrincipleCircle.argumentPrinciple_circle","k":"theorem","m":"Causalean.Mathlib.Analysis.ArgumentPrincipleCircle.ArgumentPrinciple","a":"Mathlib/Analysis/ArgumentPrincipleCircle","t":1,"d":"Argument principle for a circle. For a positive radius R, if the function f is complex-analytic on a neighborhood of the closed disk of radius R centered at c and f is nonzero on the boundary circle, then the normalized logarithmic-derivative integral of f around that circle equals the number of zeros of f strictly inside the disk, counted with analytic multiplicity.","s":"∀ {f : ℂ → ℂ} {c : ℂ} {R : ℝ}, 0 < R → AnalyticOnNhd ℂ f (Metric.closedBall c R) → (∀ z ∈ Metric.sphere c R, f z ≠ 0) → Causalean.Mathlib.Analysis.ArgumentPrincipleCircle.normalizedLogDerivCircleIntegral f c R = ↑(Causalean.Mathlib.Analysis.ArgumentPrincipleCircle.zeroMultiplicityCount f c R)","l":105,"f":"Causalean/Mathlib/Analysis/ArgumentPrincipleCircle/ArgumentPrinciple.lean"},{"n":"Causalean.Mathlib.Analysis.ArgumentPrincipleCircle.eventuallyEq_logDeriv_add_order_div_sub","k":"theorem","m":"Causalean.Mathlib.Analysis.ArgumentPrincipleCircle.Basic","a":"Mathlib/Analysis/ArgumentPrincipleCircle","t":2,"d":"Near a finite-order zero of an analytic complex function, its logarithmic derivative is the zero multiplicity divided by displacement from the zero plus the logarithmic derivative of an analytic factor that does not vanish there.","s":"∀ {f : ℂ → ℂ} {a : ℂ}, AnalyticAt ℂ f a → analyticOrderAt f a ≠ ⊤ → ∃ g, AnalyticAt ℂ g a ∧ g a ≠ 0 ∧ ∀ᶠ (z : ℂ) in nhdsWithin a {a}ᶜ, logDeriv f z = ↑(analyticOrderNatAt f a) / (z - a) + logDeriv g z","l":48,"f":"Causalean/Mathlib/Analysis/ArgumentPrincipleCircle/Basic.lean"},{"n":"Causalean.Mathlib.Analysis.ArgumentPrincipleCircle.finiteSupport_orderWithinBall","k":"theorem","m":"Causalean.Mathlib.Analysis.ArgumentPrincipleCircle.ArgumentPrinciple","a":"Mathlib/Analysis/ArgumentPrincipleCircle","t":2,"d":"Under disk analyticity and boundary nonvanishing, the function assigning each interior zero its analytic multiplicity and all other points zero has finite support.","s":"∀ {f : ℂ → ℂ} {c : ℂ} {R : ℝ}, 0 < R → AnalyticOnNhd ℂ f (Metric.closedBall c R) → (∀ z ∈ Metric.sphere c R, f z ≠ 0) → Function.HasFiniteSupport fun z => if z ∈ Metric.ball c R then analyticOrderNatAt f z else 0","l":58,"f":"Causalean/Mathlib/Analysis/ArgumentPrincipleCircle/ArgumentPrinciple.lean"},{"n":"Causalean.Mathlib.Analysis.ArgumentPrincipleCircle.finite_interiorZeros","k":"theorem","m":"Causalean.Mathlib.Analysis.ArgumentPrincipleCircle.ArgumentPrinciple","a":"Mathlib/Analysis/ArgumentPrincipleCircle","t":2,"d":"A complex function analytic on a neighborhood of a closed disk and nonzero on its boundary has only finitely many zeros strictly inside that disk.","s":"∀ {f : ℂ → ℂ} {c : ℂ} {R : ℝ}, 0 < R → AnalyticOnNhd ℂ f (Metric.closedBall c R) → (∀ z ∈ Metric.sphere c R, f z ≠ 0) → (Causalean.Mathlib.Analysis.ArgumentPrincipleCircle.interiorZeros f c R).Finite","l":28,"f":"Causalean/Mathlib/Analysis/ArgumentPrincipleCircle/ArgumentPrinciple.lean"},{"n":"Causalean.Mathlib.Analysis.ArgumentPrincipleCircle.interiorZeros","k":"def","m":"Causalean.Mathlib.Analysis.ArgumentPrincipleCircle.Basic","a":"Mathlib/Analysis/ArgumentPrincipleCircle","t":1,"d":"This set comprises exactly the zeros of a complex-valued function that lie strictly inside a given open disk.","s":"(ℂ → ℂ) → ℂ → ℝ → Set ℂ","l":43,"f":"Causalean/Mathlib/Analysis/ArgumentPrincipleCircle/Basic.lean"},{"n":"Causalean.Mathlib.Analysis.ArgumentPrincipleCircle.normalizedLogDerivCircleIntegral","k":"def","m":"Causalean.Mathlib.Analysis.ArgumentPrincipleCircle.Basic","a":"Mathlib/Analysis/ArgumentPrincipleCircle","t":1,"d":"This quantity is the winding count obtained by integrating the logarithmic derivative of a complex-valued function around a positively oriented circle and scaling so that one enclosed simple zero contributes one.","s":"(ℂ → ℂ) → ℂ → ℝ → ℂ","l":30,"f":"Causalean/Mathlib/Analysis/ArgumentPrincipleCircle/Basic.lean"},{"n":"Causalean.Mathlib.Analysis.ArgumentPrincipleCircle.normalizedLogDerivCircleIntegral_centeredMonomial","k":"theorem","m":"Causalean.Mathlib.Analysis.ArgumentPrincipleCircle.Basic","a":"Mathlib/Analysis/ArgumentPrincipleCircle","t":1,"d":"For a positive radius R such that the point a lies strictly inside the open disk of radius R centered at c, the normalized logarithmic-derivative integral around that circle of the monomial z ↦ (z - a)^n equals n.","s":"∀ {c a : ℂ} {R : ℝ} {n : ℕ}, 0 < R → a ∈ Metric.ball c R → Causalean.Mathlib.Analysis.ArgumentPrincipleCircle.normalizedLogDerivCircleIntegral (fun z => (z - a) ^ n) c R = ↑n","l":77,"f":"Causalean/Mathlib/Analysis/ArgumentPrincipleCircle/Basic.lean"},{"n":"Causalean.Mathlib.Analysis.ArgumentPrincipleCircle.normalizedLogDerivCircleIntegral_eq_of_rouche","k":"theorem","m":"Causalean.Mathlib.Analysis.ArgumentPrincipleCircle.Homotopy","a":"Mathlib/Analysis/ArgumentPrincipleCircle","t":2,"d":"Under Rouché's strict boundary inequality, two analytic complex functions have equal normalized logarithmic-derivative circle integrals.","s":"∀ {f g : ℂ → ℂ} {c : ℂ} {R : ℝ}, 0 < R → AnalyticOnNhd ℂ f (Metric.closedBall c R) → AnalyticOnNhd ℂ g (Metric.closedBall c R) → (∀ z ∈ Metric.sphere c R, ‖g z - f z‖ < ‖f z‖) → Causalean.Mathlib.Analysis.ArgumentPrincipleCircle.normalizedLogDerivCircleIntegral f c R = Causalean.Mathlib.Analysis.ArgumentPrincipleCircle.normalizedLogDerivCircleIntegral g c R","l":251,"f":"Causalean/Mathlib/Analysis/ArgumentPrincipleCircle/Homotopy.lean"},{"n":"Causalean.Mathlib.Analysis.ArgumentPrincipleCircle.normalizedLogDerivCircleIntegral_eq_of_straightLineHomotopy","k":"theorem","m":"Causalean.Mathlib.Analysis.ArgumentPrincipleCircle.Homotopy","a":"Mathlib/Analysis/ArgumentPrincipleCircle","t":2,"d":"Two analytic complex functions have equal normalized logarithmic-derivative circle integrals when every function on their straight-line interpolation is nonzero on the boundary circle.","s":"∀ {f g : ℂ → ℂ} {c : ℂ} {R : ℝ}, 0 < R → AnalyticOnNhd ℂ f (Metric.closedBall c R) → AnalyticOnNhd ℂ g (Metric.closedBall c R) → (∀ t ∈ Set.Icc 0 1, ∀ z ∈ Metric.sphere c R, Causalean.Mathlib.Analysis.ArgumentPrincipleCircle.straightLineHomotopy f g t z ≠ 0) → Causalean.Mathlib.Analysis.ArgumentPrincipleCircle.normalizedLogDerivCircleIntegral f c R = Causalean.Mathlib.Analysis.ArgumentPrincipleCircle.normalizedLogDerivCircleIntegral g c R","l":33,"f":"Causalean/Mathlib/Analysis/ArgumentPrincipleCircle/Homotopy.lean"},{"n":"Causalean.Mathlib.Analysis.ArgumentPrincipleCircle.normalizedLogDerivCircleIntegral_exists_nat","k":"theorem","m":"Causalean.Mathlib.Analysis.ArgumentPrincipleCircle.ArgumentPrinciple","a":"Mathlib/Analysis/ArgumentPrincipleCircle","t":2,"d":"The normalized logarithmic-derivative integral of a boundary-zero-free analytic function is a nonnegative whole number, viewed as a complex number.","s":"∀ {f : ℂ → ℂ} {c : ℂ} {R : ℝ}, 0 < R → AnalyticOnNhd ℂ f (Metric.closedBall c R) → (∀ z ∈ Metric.sphere c R, f z ≠ 0) → ∃ n, Causalean.Mathlib.Analysis.ArgumentPrincipleCircle.normalizedLogDerivCircleIntegral f c R = ↑n","l":281,"f":"Causalean/Mathlib/Analysis/ArgumentPrincipleCircle/ArgumentPrinciple.lean"},{"n":"Causalean.Mathlib.Analysis.ArgumentPrincipleCircle.normalizedLogDerivCircleIntegral_re_pos_of_exists_zero","k":"theorem","m":"Causalean.Mathlib.Analysis.ArgumentPrincipleCircle.ArgumentPrinciple","a":"Mathlib/Analysis/ArgumentPrincipleCircle","t":2,"d":"A boundary-zero-free analytic function with an interior zero has normalized logarithmic-derivative integral with strictly positive real part.","s":"∀ {f : ℂ → ℂ} {c : ℂ} {R : ℝ}, 0 < R → AnalyticOnNhd ℂ f (Metric.closedBall c R) → (∀ z ∈ Metric.sphere c R, f z ≠ 0) → (∃ z ∈ Metric.ball c R, f z = 0) → 0 < (Causalean.Mathlib.Analysis.ArgumentPrincipleCircle.normalizedLogDerivCircleIntegral f c R).re","l":324,"f":"Causalean/Mathlib/Analysis/ArgumentPrincipleCircle/ArgumentPrinciple.lean"},{"n":"Causalean.Mathlib.Analysis.ArgumentPrincipleCircle.rouche_circle","k":"theorem","m":"Causalean.Mathlib.Analysis.ArgumentPrincipleCircle.Homotopy","a":"Mathlib/Analysis/ArgumentPrincipleCircle","t":1,"d":"Rouché's theorem for a circle. For a positive radius R, if both f and g are complex-analytic on a neighborhood of the closed disk of radius R centered at c and on the boundary circle the discrepancy ‖g z - f z‖ is everywhere strictly smaller than ‖f z‖, then f and g have the same multiplicity-weighted number of zeros strictly inside the disk.","s":"∀ {f g : ℂ → ℂ} {c : ℂ} {R : ℝ}, 0 < R → AnalyticOnNhd ℂ f (Metric.closedBall c R) → AnalyticOnNhd ℂ g (Metric.closedBall c R) → (∀ z ∈ Metric.sphere c R, ‖g z - f z‖ < ‖f z‖) → Causalean.Mathlib.Analysis.ArgumentPrincipleCircle.zeroMultiplicityCount f c R = Causalean.Mathlib.Analysis.ArgumentPrincipleCircle.zeroMultiplicityCount g c R","l":220,"f":"Causalean/Mathlib/Analysis/ArgumentPrincipleCircle/Homotopy.lean"},{"n":"Causalean.Mathlib.Analysis.ArgumentPrincipleCircle.straightLineHomotopy","k":"def","m":"Causalean.Mathlib.Analysis.ArgumentPrincipleCircle.Homotopy","a":"Mathlib/Analysis/ArgumentPrincipleCircle","t":1,"d":"This is the straight-line interpolation between two complex-valued functions, indexed from the first endpoint at time zero to the second endpoint at time one.","s":"(ℂ → ℂ) → (ℂ → ℂ) → ℝ → ℂ → ℂ","l":28,"f":"Causalean/Mathlib/Analysis/ArgumentPrincipleCircle/Homotopy.lean"},{"n":"Causalean.Mathlib.Analysis.ArgumentPrincipleCircle.zeroMultiplicityCount","k":"def","m":"Causalean.Mathlib.Analysis.ArgumentPrincipleCircle.Basic","a":"Mathlib/Analysis/ArgumentPrincipleCircle","t":1,"d":"This count adds the analytic multiplicity of every zero strictly inside a given open disk; outside finite-support settings it uses the standard totalized finite sum.","s":"(ℂ → ℂ) → ℂ → ℝ → ℕ","l":37,"f":"Causalean/Mathlib/Analysis/ArgumentPrincipleCircle/Basic.lean"},{"n":"Causalean.Mathlib.Analysis.ArgumentPrincipleCircle.zeroMultiplicityCount_eq_of_straightLineHomotopy","k":"theorem","m":"Causalean.Mathlib.Analysis.ArgumentPrincipleCircle.Homotopy","a":"Mathlib/Analysis/ArgumentPrincipleCircle","t":2,"d":"Two analytic complex functions have the same multiplicity-weighted number of interior zeros when every function on their straight-line interpolation is nonzero on the boundary circle.","s":"∀ {f g : ℂ → ℂ} {c : ℂ} {R : ℝ}, 0 < R → AnalyticOnNhd ℂ f (Metric.closedBall c R) → AnalyticOnNhd ℂ g (Metric.closedBall c R) → (∀ t ∈ Set.Icc 0 1, ∀ z ∈ Metric.sphere c R, Causalean.Mathlib.Analysis.ArgumentPrincipleCircle.straightLineHomotopy f g t z ≠ 0) → Causalean.Mathlib.Analysis.ArgumentPrincipleCircle.zeroMultiplicityCount f c R = Causalean.Mathlib.Analysis.ArgumentPrincipleCircle.zeroMultiplicityCount g c R","l":198,"f":"Causalean/Mathlib/Analysis/ArgumentPrincipleCircle/Homotopy.lean"},{"n":"Causalean.Mathlib.Analysis.ArgumentPrincipleCircle.zeroMultiplicityCount_pos_of_exists_zero","k":"theorem","m":"Causalean.Mathlib.Analysis.ArgumentPrincipleCircle.ArgumentPrinciple","a":"Mathlib/Analysis/ArgumentPrincipleCircle","t":2,"d":"A boundary-zero-free analytic function that vanishes somewhere strictly inside the disk has a strictly positive multiplicity-weighted interior zero count.","s":"∀ {f : ℂ → ℂ} {c : ℂ} {R : ℝ}, 0 < R → AnalyticOnNhd ℂ f (Metric.closedBall c R) → (∀ z ∈ Metric.sphere c R, f z ≠ 0) → (∃ z ∈ Metric.ball c R, f z = 0) → 0 < Causalean.Mathlib.Analysis.ArgumentPrincipleCircle.zeroMultiplicityCount f c R","l":289,"f":"Causalean/Mathlib/Analysis/ArgumentPrincipleCircle/ArgumentPrinciple.lean"},{"n":"Causalean.Mathlib.Analysis.BernsteinSzegoTrig.IsTrigPolyLE","k":"def","m":"Causalean.Mathlib.Analysis.BernsteinSzegoTrig.TrigPoly","a":"Mathlib/Analysis/BernsteinSzegoTrig","t":1,"d":"IsTrigPolyLE n f means f is a real trigonometric polynomial of degree at most n, i.e. there are coefficient sequences a b : ℕ → ℝ with f t = ∑_{k=0}^{n} (a k · cos (k t) + b k · sin (k t)) for all t.","s":"ℕ → (ℝ → ℝ) → Prop","l":45,"f":"Causalean/Mathlib/Analysis/BernsteinSzegoTrig/TrigPoly.lean"},{"n":"Causalean.Mathlib.Analysis.BernsteinSzegoTrig.IsTrigPolyLE.add","k":"theorem","m":"Causalean.Mathlib.Analysis.BernsteinSzegoTrig.Szego","a":"Mathlib/Analysis/BernsteinSzegoTrig","t":2,"d":"The sum of two real trigonometric polynomials of degree at most n is again a real trigonometric polynomial of degree at most n.","s":"∀ {n : ℕ} {f g : ℝ → ℝ}, Causalean.Mathlib.Analysis.BernsteinSzegoTrig.IsTrigPolyLE n f → Causalean.Mathlib.Analysis.BernsteinSzegoTrig.IsTrigPolyLE n g → Causalean.Mathlib.Analysis.BernsteinSzegoTrig.IsTrigPolyLE n fun t => f t + g t","l":69,"f":"Causalean/Mathlib/Analysis/BernsteinSzegoTrig/Szego.lean"},{"n":"Causalean.Mathlib.Analysis.BernsteinSzegoTrig.IsTrigPolyLE.card_simple_add_double_le","k":"theorem","m":"Causalean.Mathlib.Analysis.BernsteinSzegoTrig.TrigPoly","a":"Mathlib/Analysis/BernsteinSzegoTrig","t":1,"d":"Multiplicity-refined zero-count lemma for real trigonometric polynomials (the sharp input to Szegő's inequality; not in Mathlib). Suppose f is a real trigonometric polynomial of degree at most n that is not identically zero, S is a finite set of zeros of f lying in the half-open period [c, c + 2π), and there is a further point t₀ in that same period and not belonging to S that is itself a zero of f at which the derivative of f also vanishes, i.e. t₀ is a zero of order at least 2. Then the cardinality of S, plus 2 for the double zero at t₀, is at most 2 n.","s":"∀ {n : ℕ} {f : ℝ → ℝ}, Causalean.Mathlib.Analysis.BernsteinSzegoTrig.IsTrigPolyLE n f → (∃ t, f t ≠ 0) → ∀ {c : ℝ} {S : Finset ℝ}, ↑S ⊆ Set.Ico c (c + 2 * Real.pi) → (∀ t ∈ S, f t = 0) → ∀ {t₀ : ℝ}, t₀ ∈ Set.Ico c (c + 2 * Real.pi) → t₀ ∉ S → f t₀ = 0 → HasDerivAt f 0 t₀ → S.card + 2 ≤ 2 * n","l":294,"f":"Causalean/Mathlib/Analysis/BernsteinSzegoTrig/TrigPoly.lean"},{"n":"Causalean.Mathlib.Analysis.BernsteinSzegoTrig.IsTrigPolyLE.card_zeros_le","k":"theorem","m":"Causalean.Mathlib.Analysis.BernsteinSzegoTrig.TrigPoly","a":"Mathlib/Analysis/BernsteinSzegoTrig","t":1,"d":"Zero-count lemma for real trigonometric polynomials (the load-bearing input to Szegő's inequality; not in Mathlib). If f is a real trigonometric polynomial of degree at most n that is not identically zero, then, for a finite set S of zeros of f contained in a half-open period [c, c + 2π), the cardinality of S is at most 2 n.","s":"∀ {n : ℕ} {f : ℝ → ℝ}, Causalean.Mathlib.Analysis.BernsteinSzegoTrig.IsTrigPolyLE n f → (∃ t, f t ≠ 0) → ∀ {c : ℝ} {S : Finset ℝ}, ↑S ⊆ Set.Ico c (c + 2 * Real.pi) → (∀ t ∈ S, f t = 0) → S.card ≤ 2 * n","l":243,"f":"Causalean/Mathlib/Analysis/BernsteinSzegoTrig/TrigPoly.lean"},{"n":"Causalean.Mathlib.Analysis.BernsteinSzegoTrig.IsTrigPolyLE.congr","k":"theorem","m":"Causalean.Mathlib.Analysis.BernsteinSzegoTrig.Szego","a":"Mathlib/Analysis/BernsteinSzegoTrig","t":2,"d":"A function pointwise equal to a real trigonometric polynomial of degree at most n is itself a real trigonometric polynomial of degree at most n.","s":"∀ {n : ℕ} {f g : ℝ → ℝ}, Causalean.Mathlib.Analysis.BernsteinSzegoTrig.IsTrigPolyLE n f → (∀ (t : ℝ), g t = f t) → Causalean.Mathlib.Analysis.BernsteinSzegoTrig.IsTrigPolyLE n g","l":100,"f":"Causalean/Mathlib/Analysis/BernsteinSzegoTrig/Szego.lean"},{"n":"Causalean.Mathlib.Analysis.BernsteinSzegoTrig.IsTrigPolyLE.const_mul","k":"theorem","m":"Causalean.Mathlib.Analysis.BernsteinSzegoTrig.Szego","a":"Mathlib/Analysis/BernsteinSzegoTrig","t":2,"d":"Multiplying a real trigonometric polynomial of degree at most n by a scalar preserves the degree bound.","s":"∀ {n : ℕ} {f : ℝ → ℝ} (c : ℝ), Causalean.Mathlib.Analysis.BernsteinSzegoTrig.IsTrigPolyLE n f → Causalean.Mathlib.Analysis.BernsteinSzegoTrig.IsTrigPolyLE n fun t => c * f t","l":85,"f":"Causalean/Mathlib/Analysis/BernsteinSzegoTrig/Szego.lean"},{"n":"Causalean.Mathlib.Analysis.BernsteinSzegoTrig.IsTrigPolyLE.mono","k":"theorem","m":"Causalean.Mathlib.Analysis.BernsteinSzegoTrig.TrigPoly","a":"Mathlib/Analysis/BernsteinSzegoTrig","t":2,"d":"A degree-≤n trig polynomial is also a degree-≤m trig polynomial when n ≤ m (pad the higher coefficients with zeros).","s":"∀ {n m : ℕ} {f : ℝ → ℝ}, Causalean.Mathlib.Analysis.BernsteinSzegoTrig.IsTrigPolyLE n f → n ≤ m → Causalean.Mathlib.Analysis.BernsteinSzegoTrig.IsTrigPolyLE m f","l":60,"f":"Causalean/Mathlib/Analysis/BernsteinSzegoTrig/TrigPoly.lean"},{"n":"Causalean.Mathlib.Analysis.BernsteinSzegoTrig.IsTrigPolyLE.sub","k":"theorem","m":"Causalean.Mathlib.Analysis.BernsteinSzegoTrig.TrigPoly","a":"Mathlib/Analysis/BernsteinSzegoTrig","t":2,"d":"The difference of two degree-≤n trig polynomials is a degree-≤n trig polynomial (subtract the coefficient sequences termwise).","s":"∀ {n : ℕ} {f g : ℝ → ℝ}, Causalean.Mathlib.Analysis.BernsteinSzegoTrig.IsTrigPolyLE n f → Causalean.Mathlib.Analysis.BernsteinSzegoTrig.IsTrigPolyLE n g → Causalean.Mathlib.Analysis.BernsteinSzegoTrig.IsTrigPolyLE n fun t => f t - g t","l":90,"f":"Causalean/Mathlib/Analysis/BernsteinSzegoTrig/TrigPoly.lean"},{"n":"Causalean.Mathlib.Analysis.BernsteinSzegoTrig.cosComp_isTrigPolyLE","k":"theorem","m":"Causalean.Mathlib.Analysis.BernsteinSzegoTrig.Szego","a":"Mathlib/Analysis/BernsteinSzegoTrig","t":1,"d":"For a real polynomial R, if its degree is at most β, then the even trigonometric transform t ↦ R(cos t) is a real trigonometric polynomial of degree at most β.","s":"∀ (R : Polynomial ℝ) (β : ℕ), R.natDegree ≤ β → Causalean.Mathlib.Analysis.BernsteinSzegoTrig.IsTrigPolyLE β fun t => Polynomial.eval (Real.cos t) R","l":121,"f":"Causalean/Mathlib/Analysis/BernsteinSzegoTrig/Szego.lean"},{"n":"Causalean.Mathlib.Analysis.BernsteinSzegoTrig.cos_nat_mul_isTrigPolyLE","k":"theorem","m":"Causalean.Mathlib.Analysis.BernsteinSzegoTrig.Szego","a":"Mathlib/Analysis/BernsteinSzegoTrig","t":2,"d":"The function t ↦ cos(n t) is a real trigonometric polynomial of degree at most n.","s":"∀ (n : ℕ), Causalean.Mathlib.Analysis.BernsteinSzegoTrig.IsTrigPolyLE n fun t => Real.cos (↑n * t)","l":107,"f":"Causalean/Mathlib/Analysis/BernsteinSzegoTrig/Szego.lean"},{"n":"Causalean.Mathlib.Analysis.BernsteinSzegoTrig.deriv_cosComp","k":"theorem","m":"Causalean.Mathlib.Analysis.BernsteinSzegoTrig.Basic","a":"Mathlib/Analysis/BernsteinSzegoTrig","t":1,"d":"For any real polynomial R and any point t, the derivative of s ↦ R(cos s) at t equals R'(cos t) · (-sin t).","s":"∀ (R : Polynomial ℝ) (t : ℝ), deriv (fun s => Polynomial.eval (Real.cos s) R) t = Polynomial.eval (Real.cos t) (Polynomial.derivative R) * -Real.sin t","l":48,"f":"Causalean/Mathlib/Analysis/BernsteinSzegoTrig/Basic.lean"},{"n":"Causalean.Mathlib.Analysis.BernsteinSzegoTrig.exp_mul_I_injOn_Ico","k":"theorem","m":"Causalean.Mathlib.Analysis.BernsteinSzegoTrig.TrigPoly","a":"Mathlib/Analysis/BernsteinSzegoTrig","t":2,"d":"On any half-open real interval spanning one full period, the complex exponential evaluated at an imaginary real input is injective.","s":"∀ {c s t : ℝ}, s ∈ Set.Ico c (c + 2 * Real.pi) → t ∈ Set.Ico c (c + 2 * Real.pi) → Complex.exp (↑s * Complex.I) = Complex.exp (↑t * Complex.I) → s = t","l":214,"f":"Causalean/Mathlib/Analysis/BernsteinSzegoTrig/TrigPoly.lean"},{"n":"Causalean.Mathlib.Analysis.BernsteinSzegoTrig.hasDerivAt_cosComp","k":"theorem","m":"Causalean.Mathlib.Analysis.BernsteinSzegoTrig.Basic","a":"Mathlib/Analysis/BernsteinSzegoTrig","t":1,"d":"For any real polynomial R and any point t, the map s ↦ R(cos s) has derivative R'(cos t) · (-sin t) at t.","s":"∀ (R : Polynomial ℝ) (t : ℝ), HasDerivAt (fun s => Polynomial.eval (Real.cos s) R) (Polynomial.eval (Real.cos t) (Polynomial.derivative R) * -Real.sin t) t","l":35,"f":"Causalean/Mathlib/Analysis/BernsteinSzegoTrig/Basic.lean"},{"n":"Causalean.Mathlib.Analysis.BernsteinSzegoTrig.isTrigPolyLE_const","k":"theorem","m":"Causalean.Mathlib.Analysis.BernsteinSzegoTrig.TrigPoly","a":"Mathlib/Analysis/BernsteinSzegoTrig","t":2,"d":"Constant functions are trigonometric polynomials of degree ≤ n (they use only the k = 0 term).","s":"∀ (n : ℕ) (c : ℝ), Causalean.Mathlib.Analysis.BernsteinSzegoTrig.IsTrigPolyLE n fun x => c","l":53,"f":"Causalean/Mathlib/Analysis/BernsteinSzegoTrig/TrigPoly.lean"},{"n":"Causalean.Mathlib.Analysis.BernsteinSzegoTrig.szegoInterp","k":"def","m":"Causalean.Mathlib.Analysis.BernsteinSzegoTrig.Interp","a":"Mathlib/Analysis/BernsteinSzegoTrig","t":1,"d":"The Szegő comparison interpolant at base point t₀ matching prescribed value Q₀ and derivative Q₁: S(t) = Q₀ · cos(β(t − t₀)) + (Q₁/β) · sin(β(t − t₀)). For β ≥ 1 it satisfies S(t₀) = Q₀, S'(t₀) = Q₁, and is a trigonometric wave with constant amplitude √(Q₀² + (Q₁/β)²). The degree-≤ β trigonometric-polynomial statement is proved later as szegoInterp_isTrigPolyLE in Szego.","s":"ℕ → ℝ → ℝ → ℝ → ℝ → ℝ","l":40,"f":"Causalean/Mathlib/Analysis/BernsteinSzegoTrig/Interp.lean"},{"n":"Causalean.Mathlib.Analysis.BernsteinSzegoTrig.szegoInterp_abs_le","k":"theorem","m":"Causalean.Mathlib.Analysis.BernsteinSzegoTrig.Interp","a":"Mathlib/Analysis/BernsteinSzegoTrig","t":1,"d":"For a degree parameter β and values Q₀, Q₁, base point t₀, and evaluation point t, the Szegő interpolant's value is bounded in absolute value by its amplitude A = √(Q₀² + (Q₁/β)²).","s":"∀ (β : ℕ) (Q₀ Q₁ t₀ t : ℝ), |Causalean.Mathlib.Analysis.BernsteinSzegoTrig.szegoInterp β Q₀ Q₁ t₀ t| ≤ √(Q₀ ^ 2 + (Q₁ / ↑β) ^ 2)","l":140,"f":"Causalean/Mathlib/Analysis/BernsteinSzegoTrig/Interp.lean"},{"n":"Causalean.Mathlib.Analysis.BernsteinSzegoTrig.szegoInterp_amplitude","k":"theorem","m":"Causalean.Mathlib.Analysis.BernsteinSzegoTrig.Interp","a":"Mathlib/Analysis/BernsteinSzegoTrig","t":1,"d":"For a degree parameter β and [values Q₀, Q₁ and base point t₀] (hyp:Q₀,Q₁,t₀), the Szegő interpolant is an elementary cosine wave of amplitude A = √(Q₀² + (Q₁/β)²): there is a phase φ with S(t) = A · cos(β(t − t₀) − φ) for every t.","s":"∀ (β : ℕ) (Q₀ Q₁ t₀ : ℝ), ∃ φ, ∀ (t : ℝ), Causalean.Mathlib.Analysis.BernsteinSzegoTrig.szegoInterp β Q₀ Q₁ t₀ t = √(Q₀ ^ 2 + (Q₁ / ↑β) ^ 2) * Real.cos (↑β * (t - t₀) - φ)","l":129,"f":"Causalean/Mathlib/Analysis/BernsteinSzegoTrig/Interp.lean"},{"n":"Causalean.Mathlib.Analysis.BernsteinSzegoTrig.szegoInterp_amplitude_core","k":"theorem","m":"Causalean.Mathlib.Analysis.BernsteinSzegoTrig.Interp","a":"Mathlib/Analysis/BernsteinSzegoTrig","t":2,"d":"A cosine-sine wave is an elementary wave of amplitude A = √(Q₀² + q²): for every real frequency ω, there is a phase φ with Q₀ cos(ω(t − t₀)) + q sin(ω(t − t₀)) = A cos(ω(t − t₀) − φ) for all t.","s":"∀ (ω Q₀ q t₀ : ℝ), ∃ φ, ∀ (t : ℝ), Q₀ * Real.cos (ω * (t - t₀)) + q * Real.sin (ω * (t - t₀)) = √(Q₀ ^ 2 + q ^ 2) * Real.cos (ω * (t - t₀) - φ)","l":95,"f":"Causalean/Mathlib/Analysis/BernsteinSzegoTrig/Interp.lean"},{"n":"Causalean.Mathlib.Analysis.BernsteinSzegoTrig.szegoInterp_continuous","k":"theorem","m":"Causalean.Mathlib.Analysis.BernsteinSzegoTrig.Szego","a":"Mathlib/Analysis/BernsteinSzegoTrig","t":2,"d":"The Szegő interpolant is continuous as a function of its argument.","s":"∀ (β : ℕ) (Q₀ Q₁ t₀ : ℝ), Continuous (Causalean.Mathlib.Analysis.BernsteinSzegoTrig.szegoInterp β Q₀ Q₁ t₀)","l":188,"f":"Causalean/Mathlib/Analysis/BernsteinSzegoTrig/Szego.lean"},{"n":"Causalean.Mathlib.Analysis.BernsteinSzegoTrig.szegoInterp_hasDerivAt","k":"theorem","m":"Causalean.Mathlib.Analysis.BernsteinSzegoTrig.Interp","a":"Mathlib/Analysis/BernsteinSzegoTrig","t":1,"d":"The Szegő comparison interpolant S built at base point t₀ from a prescribed value Q₀ and derivative Q₁ reproduces the prescribed derivative there: provided the degree parameter β is at least 1, the derivative of S at t₀ equals Q₁.","s":"∀ (β : ℕ), 1 ≤ β → ∀ (Q₀ Q₁ t₀ : ℝ), HasDerivAt (fun t => Causalean.Mathlib.Analysis.BernsteinSzegoTrig.szegoInterp β Q₀ Q₁ t₀ t) Q₁ t₀","l":57,"f":"Causalean/Mathlib/Analysis/BernsteinSzegoTrig/Interp.lean"},{"n":"Causalean.Mathlib.Analysis.BernsteinSzegoTrig.szegoInterp_isTrigPolyLE","k":"theorem","m":"Causalean.Mathlib.Analysis.BernsteinSzegoTrig.Szego","a":"Mathlib/Analysis/BernsteinSzegoTrig","t":1,"d":"For a degree parameter β and [values Q₀, Q₁ and base point t₀] (hyp:Q₀,Q₁,t₀), the Szegő interpolant is a real trigonometric polynomial of degree at most β.","s":"∀ (β : ℕ) (Q₀ Q₁ t₀ : ℝ), Causalean.Mathlib.Analysis.BernsteinSzegoTrig.IsTrigPolyLE β (Causalean.Mathlib.Analysis.BernsteinSzegoTrig.szegoInterp β Q₀ Q₁ t₀)","l":44,"f":"Causalean/Mathlib/Analysis/BernsteinSzegoTrig/Szego.lean"},{"n":"Causalean.Mathlib.Analysis.BernsteinSzegoTrig.szegoInterp_self","k":"theorem","m":"Causalean.Mathlib.Analysis.BernsteinSzegoTrig.Interp","a":"Mathlib/Analysis/BernsteinSzegoTrig","t":1,"d":"For a degree parameter β and [values Q₀, Q₁ and base point t₀] (hyp:Q₀,Q₁,t₀), the Szegő interpolant reproduces its prescribed value at the base point: S(t₀) = Q₀.","s":"∀ (β : ℕ) (Q₀ Q₁ t₀ : ℝ), Causalean.Mathlib.Analysis.BernsteinSzegoTrig.szegoInterp β Q₀ Q₁ t₀ t₀ = Q₀","l":50,"f":"Causalean/Mathlib/Analysis/BernsteinSzegoTrig/Interp.lean"},{"n":"Causalean.Mathlib.Analysis.BernsteinSzegoTrig.szego_deriv_sq_bound","k":"theorem","m":"Causalean.Mathlib.Analysis.BernsteinSzegoTrig.Szego","a":"Mathlib/Analysis/BernsteinSzegoTrig","t":1,"d":"The sharp Bernstein/Szegő differential inequality (the deliverable; not available in Mathlib). For a real polynomial R, if its degree is at most β and the even trigonometric transform t ↦ R(cos t) is bounded in absolute value by M at every t, then at every t the squared derivative of that transform plus β² times its squared value is at most β² · M², equivalently |d/dt R(cos t)| ≤ β · √(M² − R(cos t)²).","s":"∀ (R : Polynomial ℝ) (β : ℕ), R.natDegree ≤ β → ∀ (M : ℝ), (∀ (t : ℝ), |Polynomial.eval (Real.cos t) R| ≤ M) → ∀ (t : ℝ), deriv (fun s => Polynomial.eval (Real.cos s) R) t ^ 2 + ↑β ^ 2 * Polynomial.eval (Real.cos t) R ^ 2 ≤ ↑β ^ 2 * M ^ 2","l":243,"f":"Causalean/Mathlib/Analysis/BernsteinSzegoTrig/Szego.lean"},{"n":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.CertifiedIntervalArithmetic","k":"structure","m":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.API","a":"Mathlib/Analysis/CertifiedContourIntervalArithmetic","t":1,"d":"This interface packages sound rational interval arithmetic and total certified-real refinement behind a paper-independent interface.","s":"Type","l":14,"f":"Causalean/Mathlib/Analysis/CertifiedContourIntervalArithmetic/API.lean"},{"n":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.CertifiedReal","k":"structure","m":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.CertifiedReal","a":"Mathlib/Analysis/CertifiedContourIntervalArithmetic","t":1,"d":"A certified real name represents a real number by a sequence of rational interval enclosures indexed by precision that is nested — each successive enclosure a subinterval of the one before — and always contains the represented value, together with a computable rule selecting, for any requested positive rational error, a precision level whose enclosure is no wider than that error.","s":"Type","l":18,"f":"Causalean/Mathlib/Analysis/CertifiedContourIntervalArithmetic/CertifiedReal.lean"},{"n":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.CertifiedReal.add","k":"def","m":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.CertifiedReal","a":"Mathlib/Analysis/CertifiedContourIntervalArithmetic","t":1,"d":"Adding certified names adds equal-precision interval enclosures and uses half of the requested tolerance for each input.","s":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.CertifiedReal → Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.CertifiedReal → Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.CertifiedReal","l":100,"f":"Causalean/Mathlib/Analysis/CertifiedContourIntervalArithmetic/CertifiedReal.lean"},{"n":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.CertifiedReal.approx_mono","k":"theorem","m":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.CertifiedReal","a":"Mathlib/Analysis/CertifiedContourIntervalArithmetic","t":2,"d":"Nestedness extends from successive precisions to every pair of ordered precisions.","s":"∀ (x : Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.CertifiedReal) {m n : ℕ}, m ≤ n → (x.approx n).Subinterval (x.approx m)","l":40,"f":"Causalean/Mathlib/Analysis/CertifiedContourIntervalArithmetic/CertifiedReal.lean"},{"n":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.CertifiedReal.exists_refinement","k":"theorem","m":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.CertifiedReal","a":"Mathlib/Analysis/CertifiedContourIntervalArithmetic","t":2,"d":"Every positive rational tolerance admits an explicitly returned enclosing interval of at most that width.","s":"∀ (x : Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.CertifiedReal) (ε : ℚ), 0 < ε → ∃ n, (x.approx n).Contains x.value ∧ (x.approx n).width ≤ ε","l":64,"f":"Causalean/Mathlib/Analysis/CertifiedContourIntervalArithmetic/CertifiedReal.lean"},{"n":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.CertifiedReal.neg","k":"def","m":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.CertifiedReal","a":"Mathlib/Analysis/CertifiedContourIntervalArithmetic","t":1,"d":"Negating a certified name negates every rational enclosure and preserves its modulus.","s":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.CertifiedReal → Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.CertifiedReal","l":85,"f":"Causalean/Mathlib/Analysis/CertifiedContourIntervalArithmetic/CertifiedReal.lean"},{"n":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.CertifiedReal.ofRat","k":"def","m":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.CertifiedReal","a":"Mathlib/Analysis/CertifiedContourIntervalArithmetic","t":1,"d":"A rational number has the constant point interval as a certified real name.","s":"ℚ → Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.CertifiedReal","l":70,"f":"Causalean/Mathlib/Analysis/CertifiedContourIntervalArithmetic/CertifiedReal.lean"},{"n":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.CertifiedReal.refine","k":"def","m":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.CertifiedReal","a":"Mathlib/Analysis/CertifiedContourIntervalArithmetic","t":1,"d":"Refinement evaluates the certified modulus and returns its rational enclosure.","s":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.CertifiedReal → Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.PosRat → Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.RatInterval","l":48,"f":"Causalean/Mathlib/Analysis/CertifiedContourIntervalArithmetic/CertifiedReal.lean"},{"n":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.CertifiedReal.refine_contains","k":"theorem","m":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.CertifiedReal","a":"Mathlib/Analysis/CertifiedContourIntervalArithmetic","t":2,"d":"Refinement always encloses the real value named by the certificate.","s":"∀ (x : Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.CertifiedReal) (ε : Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.PosRat), (x.refine ε).Contains x.value","l":52,"f":"Causalean/Mathlib/Analysis/CertifiedContourIntervalArithmetic/CertifiedReal.lean"},{"n":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.CertifiedReal.refine_width","k":"theorem","m":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.CertifiedReal","a":"Mathlib/Analysis/CertifiedContourIntervalArithmetic","t":1,"d":"For a certified real number x and a requested positive rational tolerance ε, refining x to the precision selected by ε's modulus yields an enclosure whose width is at most ε.","s":"∀ (x : Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.CertifiedReal) (ε : Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.PosRat), (x.refine ε).width ≤ ↑ε","l":57,"f":"Causalean/Mathlib/Analysis/CertifiedContourIntervalArithmetic/CertifiedReal.lean"},{"n":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.CertifiedReal.sub","k":"def","m":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.CertifiedReal","a":"Mathlib/Analysis/CertifiedContourIntervalArithmetic","t":1,"d":"Subtracting certified names adds the first enclosure to the negation of the second.","s":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.CertifiedReal → Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.CertifiedReal → Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.CertifiedReal","l":127,"f":"Causalean/Mathlib/Analysis/CertifiedContourIntervalArithmetic/CertifiedReal.lean"},{"n":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.CertifiedReal.sub_value","k":"theorem","m":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.CertifiedReal","a":"Mathlib/Analysis/CertifiedContourIntervalArithmetic","t":2,"d":"Certified subtraction denotes the difference of the two named real values.","s":"∀ (x y : Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.CertifiedReal), (x.sub y).value = x.value - y.value","l":130,"f":"Causalean/Mathlib/Analysis/CertifiedContourIntervalArithmetic/CertifiedReal.lean"},{"n":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.CertifiedReal.value_eq_of_common_approximations","k":"theorem","m":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.CertifiedReal","a":"Mathlib/Analysis/CertifiedContourIntervalArithmetic","t":2,"d":"Two certified names whose every approximation is shared denote the same real value.","s":"∀ (x y : Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.CertifiedReal), (∀ (n : ℕ), x.approx n = y.approx n) → x.value = y.value","l":135,"f":"Causalean/Mathlib/Analysis/CertifiedContourIntervalArithmetic/CertifiedReal.lean"},{"n":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.CircleMesh.circleContourIntegral","k":"def","m":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Mesh","a":"Mathlib/Analysis/CertifiedContourIntervalArithmetic","t":1,"d":"The contour integral around a circle is represented deterministically as an interval integral over one unit-length parameter cycle.","s":"(ℂ → ℂ) → ℂ → ℝ → ℂ","l":95,"f":"Causalean/Mathlib/Analysis/CertifiedContourIntervalArithmetic/Mesh.lean"},{"n":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.CircleMesh.circleContourIntegral_enclosed","k":"theorem","m":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Quadrature","a":"Mathlib/Analysis/CertifiedContourIntervalArithmetic","t":2,"d":"Applying the complex mesh theorem to the parameterized circle integrand encloses the contour integral.","s":"∀ {f : ℂ → ℂ} {c : ℂ} {r : ℝ} {nodes : ℕ → Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.ComplexRatInterval} {L : ℚ} (hL : 0 ≤ L) {n : ℕ} (hn : 0 < n), (∀ s ∈ Set.Icc 0 1, ∀ t ∈ Set.Icc 0 1, ‖Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.CircleMesh.circleIntegrand f c r s - Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.CircleMesh.circleIntegrand f c r t‖ ≤ ↑L * |s - t|) → (∀ k ≤ n, (nodes k).Contains (Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.CircleMesh.circleIntegrand f c r (Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.CircleMesh.meshPoint n k))) → (Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.CircleMesh.integralEnclosure nodes L hL n hn).Contains (Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.CircleMesh.circleContourIntegral f c r)","l":295,"f":"Causalean/Mathlib/Analysis/CertifiedContourIntervalArithmetic/Quadrature.lean"},{"n":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.CircleMesh.circleIntegrand","k":"def","m":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Mesh","a":"Mathlib/Analysis/CertifiedContourIntervalArithmetic","t":1,"d":"The parameterized contour integrand is the function value times the circle tangent.","s":"(ℂ → ℂ) → ℂ → ℝ → ℝ → ℂ","l":91,"f":"Causalean/Mathlib/Analysis/CertifiedContourIntervalArithmetic/Mesh.lean"},{"n":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.CircleMesh.circleMap","k":"def","m":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Mesh","a":"Mathlib/Analysis/CertifiedContourIntervalArithmetic","t":1,"d":"The standard once-around parameterization of a circle with a specified center and nonnegative radius.","s":"ℂ → ℝ → ℝ → ℂ","l":82,"f":"Causalean/Mathlib/Analysis/CertifiedContourIntervalArithmetic/Mesh.lean"},{"n":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.CircleMesh.circleTangent","k":"def","m":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Mesh","a":"Mathlib/Analysis/CertifiedContourIntervalArithmetic","t":1,"d":"The derivative of the standard unit-interval circle parameterization.","s":"ℝ → ℝ → ℂ","l":87,"f":"Causalean/Mathlib/Analysis/CertifiedContourIntervalArithmetic/Mesh.lean"},{"n":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.CircleMesh.continuousOn_of_lipschitz_bound","k":"theorem","m":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Mesh","a":"Mathlib/Analysis/CertifiedContourIntervalArithmetic","t":2,"d":"A function satisfying a finite Lipschitz bound on the unit interval is continuous there.","s":"∀ {E : Type u_1} [inst : NormedAddCommGroup E] {g : ℝ → E} {C : ℝ}, 0 ≤ C → (∀ s ∈ Set.Icc 0 1, ∀ t ∈ Set.Icc 0 1, ‖g s - g t‖ ≤ C * |s - t|) → ContinuousOn g (Set.Icc 0 1)","l":254,"f":"Causalean/Mathlib/Analysis/CertifiedContourIntervalArithmetic/Mesh.lean"},{"n":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.CircleMesh.infEnclosure","k":"def","m":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Mesh","a":"Mathlib/Analysis/CertifiedContourIntervalArithmetic","t":1,"d":"The infimum enclosure widens the minimum node enclosure downward by one Lipschitz mesh step.","s":"(ℕ → Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.RatInterval) → (L : ℚ) → 0 ≤ L → (n : ℕ) → 0 < n → Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.RatInterval","l":278,"f":"Causalean/Mathlib/Analysis/CertifiedContourIntervalArithmetic/Mesh.lean"},{"n":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.CircleMesh.infEnclosure_sound","k":"theorem","m":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Mesh","a":"Mathlib/Analysis/CertifiedContourIntervalArithmetic","t":1,"d":"Certified infimum enclosure on the unit mesh. Fix a nonnegative Lipschitz constant and a positive number of mesh nodes, and suppose the real function is Lipschitz on [0, 1] with that constant and each rational node interval contains the function's value at the corresponding mesh point. Then the infimum enclosure built from those node intervals contains the true infimum of the function over [0, 1].","s":"∀ {f : ℝ → ℝ} {nodes : ℕ → Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.RatInterval} {L : ℚ} (hL : 0 ≤ L) {n : ℕ} (hn : 0 < n), (∀ s ∈ Set.Icc 0 1, ∀ t ∈ Set.Icc 0 1, |f s - f t| ≤ ↑L * |s - t|) → (∀ k ≤ n, (nodes k).Contains (f (Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.CircleMesh.meshPoint n k))) → (Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.CircleMesh.infEnclosure nodes L hL n hn).Contains (sInf (f '' Set.Icc 0 1))","l":292,"f":"Causalean/Mathlib/Analysis/CertifiedContourIntervalArithmetic/Mesh.lean"},{"n":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.CircleMesh.integralEnclosure","k":"def","m":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Quadrature","a":"Mathlib/Analysis/CertifiedContourIntervalArithmetic","t":1,"d":"The certified contour enclosure widens the rational trapezoidal enclosure by its Lipschitz discretization error.","s":"(ℕ → Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.ComplexRatInterval) → (L : ℚ) → 0 ≤ L → (n : ℕ) → 0 < n → Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.ComplexRatInterval","l":29,"f":"Causalean/Mathlib/Analysis/CertifiedContourIntervalArithmetic/Quadrature.lean"},{"n":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.CircleMesh.integralEnclosure_sound","k":"theorem","m":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Quadrature","a":"Mathlib/Analysis/CertifiedContourIntervalArithmetic","t":1,"d":"Certified trapezoidal enclosure of an integral. Fix a nonnegative Lipschitz constant and a positive node count, and suppose the complex-valued integrand is Lipschitz on [0, 1] with that constant and each complex rational rectangle contains the integrand's value at the corresponding mesh point. Then the trapezoidal enclosure built from those rectangles contains the true integral of the integrand over [0, 1].","s":"∀ {g : ℝ → ℂ} {nodes : ℕ → Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.ComplexRatInterval} {L : ℚ} (hL : 0 ≤ L) {n : ℕ} (hn : 0 < n), (∀ s ∈ Set.Icc 0 1, ∀ t ∈ Set.Icc 0 1, ‖g s - g t‖ ≤ ↑L * |s - t|) → (∀ k ≤ n, (nodes k).Contains (g (Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.CircleMesh.meshPoint n k))) → (Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.CircleMesh.integralEnclosure nodes L hL n hn).Contains (∫ (u : ℝ) in 0..1, g u)","l":238,"f":"Causalean/Mathlib/Analysis/CertifiedContourIntervalArithmetic/Quadrature.lean"},{"n":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.CircleMesh.integralEnclosure_width","k":"theorem","m":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Quadrature","a":"Mathlib/Analysis/CertifiedContourIntervalArithmetic","t":2,"d":"Each coordinate width of the certified integral is bounded by the uniform node width plus one mesh-scale Lipschitz allowance.","s":"∀ {nodes : ℕ → Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.ComplexRatInterval} {w L : ℚ}, 0 ≤ w → ∀ (hL : 0 ≤ L) {n : ℕ} (hn : 0 < n), (∀ k ≤ n, (nodes k).re.width ≤ w ∧ (nodes k).im.width ≤ w) → (Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.CircleMesh.integralEnclosure nodes L hL n hn).re.width ≤ w + L / ↑n ∧ (Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.CircleMesh.integralEnclosure nodes L hL n hn).im.width ≤ w + L / ↑n","l":263,"f":"Causalean/Mathlib/Analysis/CertifiedContourIntervalArithmetic/Quadrature.lean"},{"n":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.CircleMesh.maxHiUpTo","k":"def","m":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Mesh","a":"Mathlib/Analysis/CertifiedContourIntervalArithmetic","t":1,"d":"The maximum upper endpoint among the nodes from the initial node through the requested terminal node is computed by primitive recursion.","s":"(ℕ → Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.RatInterval) → ℕ → ℚ","l":121,"f":"Causalean/Mathlib/Analysis/CertifiedContourIntervalArithmetic/Mesh.lean"},{"n":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.CircleMesh.maxLoUpTo","k":"def","m":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Mesh","a":"Mathlib/Analysis/CertifiedContourIntervalArithmetic","t":1,"d":"The maximum lower endpoint among the nodes from the initial node through the requested terminal node is computed by primitive recursion.","s":"(ℕ → Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.RatInterval) → ℕ → ℚ","l":114,"f":"Causalean/Mathlib/Analysis/CertifiedContourIntervalArithmetic/Mesh.lean"},{"n":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.CircleMesh.meshPoint","k":"def","m":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Mesh","a":"Mathlib/Analysis/CertifiedContourIntervalArithmetic","t":1,"d":"A uniform mesh node divides its whole-number position by the positive mesh size on the unit parameter interval.","s":"ℕ → ℕ → ℝ","l":78,"f":"Causalean/Mathlib/Analysis/CertifiedContourIntervalArithmetic/Mesh.lean"},{"n":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.CircleMesh.meshPoint_mem","k":"theorem","m":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Mesh","a":"Mathlib/Analysis/CertifiedContourIntervalArithmetic","t":2,"d":"A node of a nonempty uniform mesh lies in the unit parameter interval.","s":"∀ {n k : ℕ}, 0 < n → k ≤ n → Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.CircleMesh.meshPoint n k ∈ Set.Icc 0 1","l":229,"f":"Causalean/Mathlib/Analysis/CertifiedContourIntervalArithmetic/Mesh.lean"},{"n":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.CircleMesh.minHiUpTo","k":"def","m":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Mesh","a":"Mathlib/Analysis/CertifiedContourIntervalArithmetic","t":1,"d":"The minimum upper endpoint among the nodes from the initial node through the requested terminal node is computed by primitive recursion.","s":"(ℕ → Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.RatInterval) → ℕ → ℚ","l":107,"f":"Causalean/Mathlib/Analysis/CertifiedContourIntervalArithmetic/Mesh.lean"},{"n":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.CircleMesh.minLoUpTo","k":"def","m":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Mesh","a":"Mathlib/Analysis/CertifiedContourIntervalArithmetic","t":1,"d":"The minimum lower endpoint among the nodes from the initial node through the requested terminal node is computed by primitive recursion.","s":"(ℕ → Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.RatInterval) → ℕ → ℚ","l":100,"f":"Causalean/Mathlib/Analysis/CertifiedContourIntervalArithmetic/Mesh.lean"},{"n":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.CircleMesh.norm_integral_sub_trapezoid_le","k":"theorem","m":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Quadrature","a":"Mathlib/Analysis/CertifiedContourIntervalArithmetic","t":2,"d":"A Lipschitz complex function on the unit parameter interval differs from its deterministic trapezoidal rule by at most half a Lipschitz mesh unit.","s":"∀ {g : ℝ → ℂ} {L : ℚ}, 0 ≤ L → ∀ {n : ℕ}, 0 < n → (∀ s ∈ Set.Icc 0 1, ∀ t ∈ Set.Icc 0 1, ‖g s - g t‖ ≤ ↑L * |s - t|) → ‖(∫ (u : ℝ) in 0..1, g u) - 1 / ↑(2 * ↑n) * ∑ k ∈ Finset.range n, (g (Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.CircleMesh.meshPoint n k) + g (Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.CircleMesh.meshPoint n (k + 1)))‖ ≤ ↑L / (2 * ↑n)","l":77,"f":"Causalean/Mathlib/Analysis/CertifiedContourIntervalArithmetic/Quadrature.lean"},{"n":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.CircleMesh.supEnclosure","k":"def","m":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Mesh","a":"Mathlib/Analysis/CertifiedContourIntervalArithmetic","t":1,"d":"The supremum enclosure widens the maximum node enclosure upward by one Lipschitz mesh step.","s":"(ℕ → Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.RatInterval) → (L : ℚ) → 0 ≤ L → (n : ℕ) → 0 < n → Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.RatInterval","l":285,"f":"Causalean/Mathlib/Analysis/CertifiedContourIntervalArithmetic/Mesh.lean"},{"n":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.CircleMesh.supEnclosure_sound","k":"theorem","m":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Mesh","a":"Mathlib/Analysis/CertifiedContourIntervalArithmetic","t":2,"d":"Rational node enclosures and a Lipschitz bound enclose the supremum of a real function on the unit mesh interval.","s":"∀ {f : ℝ → ℝ} {nodes : ℕ → Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.RatInterval} {L : ℚ} (hL : 0 ≤ L) {n : ℕ} (hn : 0 < n), (∀ s ∈ Set.Icc 0 1, ∀ t ∈ Set.Icc 0 1, |f s - f t| ≤ ↑L * |s - t|) → (∀ k ≤ n, (nodes k).Contains (f (Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.CircleMesh.meshPoint n k))) → (Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.CircleMesh.supEnclosure nodes L hL n hn).Contains (sSup (f '' Set.Icc 0 1))","l":337,"f":"Causalean/Mathlib/Analysis/CertifiedContourIntervalArithmetic/Mesh.lean"},{"n":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.CircleMesh.trapezoidEnclosure","k":"def","m":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Quadrature","a":"Mathlib/Analysis/CertifiedContourIntervalArithmetic","t":1,"d":"The rational complex trapezoidal enclosure applies the usual half-cell average and mesh scaling to the recursive endpoint sum.","s":"(ℕ → Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.ComplexRatInterval) → ℕ → Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.ComplexRatInterval","l":24,"f":"Causalean/Mathlib/Analysis/CertifiedContourIntervalArithmetic/Quadrature.lean"},{"n":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.CircleMesh.trapezoidSum","k":"def","m":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Quadrature","a":"Mathlib/Analysis/CertifiedContourIntervalArithmetic","t":1,"d":"The unscaled trapezoidal rectangle sum is computed by primitive recursion over mesh cells.","s":"(ℕ → Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.ComplexRatInterval) → ℕ → Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.ComplexRatInterval","l":18,"f":"Causalean/Mathlib/Analysis/CertifiedContourIntervalArithmetic/Quadrature.lean"},{"n":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.CircleMesh.width_infEnclosure","k":"theorem","m":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Mesh","a":"Mathlib/Analysis/CertifiedContourIntervalArithmetic","t":2,"d":"The infimum enclosure width is at most one mesh error plus the uniform node-enclosure width.","s":"∀ {nodes : ℕ → Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.RatInterval} {w L : ℚ}, 0 ≤ w → ∀ (hL : 0 ≤ L) {n : ℕ} (hn : 0 < n), (∀ k ≤ n, (nodes k).width ≤ w) → (Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.CircleMesh.infEnclosure nodes L hL n hn).width ≤ w + L / ↑n","l":374,"f":"Causalean/Mathlib/Analysis/CertifiedContourIntervalArithmetic/Mesh.lean"},{"n":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.CircleMesh.width_supEnclosure","k":"theorem","m":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Mesh","a":"Mathlib/Analysis/CertifiedContourIntervalArithmetic","t":2,"d":"The supremum enclosure width is at most one mesh error plus the uniform node-enclosure width.","s":"∀ {nodes : ℕ → Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.RatInterval} {w L : ℚ}, 0 ≤ w → ∀ (hL : 0 ≤ L) {n : ℕ} (hn : 0 < n), (∀ k ≤ n, (nodes k).width ≤ w) → (Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.CircleMesh.supEnclosure nodes L hL n hn).width ≤ w + L / ↑n","l":387,"f":"Causalean/Mathlib/Analysis/CertifiedContourIntervalArithmetic/Mesh.lean"},{"n":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.CertifiedComplex","k":"structure","m":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.Names","a":"Mathlib/Analysis/CertifiedContourIntervalArithmetic/Complex","t":1,"d":"A certified complex name represents a complex number by a sequence of rational rectangle enclosures indexed by fuel that is nested — each successive enclosure a subrectangle of the one before — and always contains the represented value, together with a computable rule selecting, for any requested positive rational error, a fuel level whose enclosure meets that coordinate-width tolerance.","s":"Type","l":13,"f":"Causalean/Mathlib/Analysis/CertifiedContourIntervalArithmetic/Complex/Names.lean"},{"n":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.CertifiedComplex.add","k":"def","m":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.Names","a":"Mathlib/Analysis/CertifiedContourIntervalArithmetic/Complex","t":1,"d":"Adding certified complex names uses equal-fuel rectangle addition and a half-tolerance refinement for each operand.","s":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.CertifiedComplex → Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.CertifiedComplex → Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.CertifiedComplex","l":69,"f":"Causalean/Mathlib/Analysis/CertifiedContourIntervalArithmetic/Complex/Names.lean"},{"n":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.CertifiedComplex.norm","k":"def","m":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.Names","a":"Mathlib/Analysis/CertifiedContourIntervalArithmetic/Complex","t":1,"d":"Complex modulus lifts a refining certified complex name to a certified real name.","s":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.CertifiedComplex → Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.CertifiedReal","l":307,"f":"Causalean/Mathlib/Analysis/CertifiedContourIntervalArithmetic/Complex/Names.lean"},{"n":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.CertifiedComplex.normApprox","k":"def","m":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.Names","a":"Mathlib/Analysis/CertifiedContourIntervalArithmetic/Complex","t":1,"d":"Modulus evaluation on a refining complex name recursively intersects the current Newton enclosure with every preceding output.","s":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.CertifiedComplex → ℕ → Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.RatInterval","l":102,"f":"Causalean/Mathlib/Analysis/CertifiedContourIntervalArithmetic/Complex/Names.lean"},{"n":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.CertifiedComplex.normApprox_spec","k":"theorem","m":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.Names","a":"Mathlib/Analysis/CertifiedContourIntervalArithmetic/Complex","t":2,"d":"Recursive modulus approximations contain the norm and are adjacent-fuel nested.","s":"∀ (z : Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.CertifiedComplex) (fuel : ℕ), (z.normApprox fuel).Contains ‖z.value‖ ∧ (z.normApprox (fuel + 1)).Subinterval (z.normApprox fuel)","l":120,"f":"Causalean/Mathlib/Analysis/CertifiedContourIntervalArithmetic/Complex/Names.lean"},{"n":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.CertifiedComplex.normPrecision","k":"def","m":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.Names","a":"Mathlib/Analysis/CertifiedContourIntervalArithmetic/Complex","t":1,"d":"A conservative executable modulus precision includes both input-name diameter and Newton iteration fuel, including the nondifferentiable origin.","s":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.CertifiedComplex → Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.PosRat → ℕ","l":109,"f":"Causalean/Mathlib/Analysis/CertifiedContourIntervalArithmetic/Complex/Names.lean"},{"n":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.CertifiedComplex.norm_width_at_precision","k":"theorem","m":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.Names","a":"Mathlib/Analysis/CertifiedContourIntervalArithmetic/Complex","t":1,"d":"For a certified complex input and a requested positive rational tolerance ε, evaluating the certified-modulus approximation at the precision level selected for that tolerance yields an output enclosure whose width is no larger than ε.","s":"∀ (z : Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.CertifiedComplex) (ε : Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.PosRat), (z.normApprox (z.normPrecision ε)).width ≤ ↑ε","l":164,"f":"Causalean/Mathlib/Analysis/CertifiedContourIntervalArithmetic/Complex/Names.lean"},{"n":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.CertifiedComplex.ofRatPair","k":"def","m":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.Names","a":"Mathlib/Analysis/CertifiedContourIntervalArithmetic/Complex","t":1,"d":"A rational complex point has the constant point rectangle as its certified name.","s":"ℚ → ℚ → Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.CertifiedComplex","l":55,"f":"Causalean/Mathlib/Analysis/CertifiedContourIntervalArithmetic/Complex/Names.lean"},{"n":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.CertifiedComplex.refine","k":"def","m":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.Names","a":"Mathlib/Analysis/CertifiedContourIntervalArithmetic/Complex","t":1,"d":"A certified complex name refines effectively to a requested rational width.","s":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.CertifiedComplex → Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.PosRat → Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.ComplexRatInterval","l":47,"f":"Causalean/Mathlib/Analysis/CertifiedContourIntervalArithmetic/Complex/Names.lean"},{"n":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.CertifiedComplex.refine_spec","k":"theorem","m":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.Names","a":"Mathlib/Analysis/CertifiedContourIntervalArithmetic/Complex","t":2,"d":"Effective refinement preserves containment and meets its requested width.","s":"∀ (z : Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.CertifiedComplex) (ε : Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.PosRat), (z.refine ε).Contains z.value ∧ (z.refine ε).width ≤ ↑ε","l":50,"f":"Causalean/Mathlib/Analysis/CertifiedContourIntervalArithmetic/Complex/Names.lean"},{"n":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.CertifiedComplexMap","k":"structure","m":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.Program","a":"Mathlib/Analysis/CertifiedContourIntervalArithmetic/Complex","t":1,"d":"A certified complex map packages an exact complex function with an executable interval-evaluation program over rational input rectangles and a fuel level whose primitive-operation count is recorded, proved sound — every executable result contains the exact value at every enclosed input. Increasing fuel refines the enclosure to a subrectangle of the previous one, and refining the input rectangle cannot enlarge the output enclosure. The map carries [a nonnegative Lipschitz bound on how input width amplifies into output width](hyp:amplification, amplification_nonneg) and a nonincreasing sequence of algorithmic-error bounds in the fuel, such that the evaluated width never exceeds the algorithmic error plus the amplified input width, together with a computable fuel rule meeting any requested positive algorithmic-error target.","s":"Type","l":16,"f":"Causalean/Mathlib/Analysis/CertifiedContourIntervalArithmetic/Complex/Program.lean"},{"n":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.CertifiedComplexMap.add","k":"def","m":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.Program","a":"Mathlib/Analysis/CertifiedContourIntervalArithmetic/Complex","t":1,"d":"Pointwise addition composes two certified interval programs, adds their width amplifications and algorithmic errors, and records one new operation.","s":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.CertifiedComplexMap → Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.CertifiedComplexMap → Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.CertifiedComplexMap","l":107,"f":"Causalean/Mathlib/Analysis/CertifiedContourIntervalArithmetic/Complex/Program.lean"},{"n":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.CertifiedComplexMap.algorithmError_le_of_modulus","k":"theorem","m":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.Program","a":"Mathlib/Analysis/CertifiedContourIntervalArithmetic/Complex","t":1,"d":"Fuel sufficiency bound. For a certified complex-map evaluation algorithm and a requested positive rational error tolerance, if the number of iterations supplied is at least the algorithm's certified error modulus at that tolerance, then running the algorithm for that many iterations yields an approximation whose error is at most the requested tolerance.","s":"∀ {f : Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.CertifiedComplexMap} {ε : Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.PosRat} {fuel : ℕ}, f.errorModulus ε ≤ fuel → f.algorithmError fuel ≤ ↑ε","l":341,"f":"Causalean/Mathlib/Analysis/CertifiedContourIntervalArithmetic/Complex/Program.lean"},{"n":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.CertifiedComplexMap.constant","k":"def","m":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.Program","a":"Mathlib/Analysis/CertifiedContourIntervalArithmetic/Complex","t":1,"d":"A rational complex constant is a zero-operation certified program returning the same point rectangle for every input and fuel.","s":"ℚ → ℚ → Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.CertifiedComplexMap","l":82,"f":"Causalean/Mathlib/Analysis/CertifiedContourIntervalArithmetic/Complex/Program.lean"},{"n":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.CertifiedComplexMap.identity","k":"def","m":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.Program","a":"Mathlib/Analysis/CertifiedContourIntervalArithmetic/Complex","t":1,"d":"The identity interval program performs no primitive arithmetic and has unit input amplification with zero algorithmic error.","s":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.CertifiedComplexMap","l":62,"f":"Causalean/Mathlib/Analysis/CertifiedContourIntervalArithmetic/Complex/Program.lean"},{"n":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.CertifiedComplexMap.mulWithBounds","k":"def","m":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.Program","a":"Mathlib/Analysis/CertifiedContourIntervalArithmetic/Complex","t":1,"d":"Multiplication composes two certified programs when uniform executable magnitude bounds are supplied; its error and amplification use the standard product-width propagation formula and its operation count is structural.","s":"(f g : Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.CertifiedComplexMap) → (Bf Bg : ℚ) → 0 ≤ Bf → 0 ≤ Bg → (∀ (I : Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.ComplexRatInterval) (fuel : ℕ), (f.eval I fuel).maxAbs ≤ Bf) → (∀ (I : Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.ComplexRatInterval) (fuel : ℕ), (g.eval I fuel).maxAbs ≤ Bg) → Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.CertifiedComplexMap","l":217,"f":"Causalean/Mathlib/Analysis/CertifiedContourIntervalArithmetic/Complex/Program.lean"},{"n":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.CertifiedComplexMap.sub","k":"def","m":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.Program","a":"Mathlib/Analysis/CertifiedContourIntervalArithmetic/Complex","t":1,"d":"Pointwise subtraction is a certified operation combinator with the same width accounting as addition and one newly counted primitive operation.","s":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.CertifiedComplexMap → Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.CertifiedComplexMap → Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.CertifiedComplexMap","l":161,"f":"Causalean/Mathlib/Analysis/CertifiedContourIntervalArithmetic/Complex/Program.lean"},{"n":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.CertifiedProgramSchedule","k":"structure","m":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.ContourProgram","a":"Mathlib/Analysis/CertifiedContourIntervalArithmetic/Complex","t":1,"d":"A certified program schedule packages an exact schedule for evaluating a given contour program to a stated separation bound, together with the guarantees that its operation count matches the program's structural operation count, a common positive error target selected canonically from the tolerance, value bounds, and separation, an input precision equal to the uniform circle precision for that target, and that its fuel dominates the circle-exponential fuel requirement, the numerator's algorithmic-error modulus, and the denominator's algorithmic-error modulus, while the compositional arithmetic propagation bound fits within the schedule's node budget.","s":"(program : Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.ContourProgram) → Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.ContourValueBounds program → Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.PosRat → Type","l":348,"f":"Causalean/Mathlib/Analysis/CertifiedContourIntervalArithmetic/Complex/ContourProgram.lean"},{"n":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.ContourProgram","k":"structure","m":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.ContourProgram","a":"Mathlib/Analysis/CertifiedContourIntervalArithmetic/Complex","t":1,"d":"A finite rational contour program consists of a certified interval extension of the numerator function and of the denominator function, together with a rational circle radius that is strictly positive.","s":"Type","l":20,"f":"Causalean/Mathlib/Analysis/CertifiedContourIntervalArithmetic/Complex/ContourProgram.lean"},{"n":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.ContourProgram.canonicalNodeTarget","k":"def","m":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.ContourProgram","a":"Mathlib/Analysis/CertifiedContourIntervalArithmetic/Complex","t":1,"d":"The canonical common target spends the node budget after accounting for all program operations, map amplification, magnitudes, and denominator separation.","s":"(program : Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.ContourProgram) → Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.ContourValueBounds program → Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.PosRat → Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.PosRat → Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.PosRat","l":111,"f":"Causalean/Mathlib/Analysis/CertifiedContourIntervalArithmetic/Complex/ContourProgram.lean"},{"n":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.ContourProgram.canonicalScheduled","k":"def","m":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.ContourProgram","a":"Mathlib/Analysis/CertifiedContourIntervalArithmetic/Complex","t":1,"d":"The canonical certified schedule derives operation count, input precision, fuel, mesh, and all three budgets from the program and rational bounds.","s":"(program : Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.ContourProgram) → (bounds : Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.ContourValueBounds program) → (separation : Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.PosRat) → Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.PosRat → (magnitude : ℚ) → 0 ≤ magnitude → Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.CertifiedProgramSchedule program bounds separation","l":408,"f":"Causalean/Mathlib/Analysis/CertifiedContourIntervalArithmetic/Complex/ContourProgram.lean"},{"n":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.ContourProgram.constantUnit","k":"def","m":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.ContourProgram","a":"Mathlib/Analysis/CertifiedContourIntervalArithmetic/Complex","t":1,"d":"A concrete canonical rational program uses a rational constant numerator, the unit denominator, and a positive rational circle radius.","s":"ℚ → ℚ → (radius : ℚ) → 0 < radius → Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.ContourProgram","l":38,"f":"Causalean/Mathlib/Analysis/CertifiedContourIntervalArithmetic/Complex/ContourProgram.lean"},{"n":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.ContourProgram.constantUnitBounds","k":"def","m":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.ContourProgram","a":"Mathlib/Analysis/CertifiedContourIntervalArithmetic/Complex","t":1,"d":"The canonical constant-over-unit program has explicit global coordinate magnitude bounds computed only from its rational numerator coordinates.","s":"(numeratorRe numeratorIm radius : ℚ) → (hradius : 0 < radius) → Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.ContourValueBounds (Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.ContourProgram.constantUnit numeratorRe numeratorIm radius hradius)","l":311,"f":"Causalean/Mathlib/Analysis/CertifiedContourIntervalArithmetic/Complex/ContourProgram.lean"},{"n":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.ContourProgram.constantUnitCertificate","k":"def","m":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.ContourProgram","a":"Mathlib/Analysis/CertifiedContourIntervalArithmetic/Complex","t":1,"d":"The concrete unit-denominator program has an executable separation certificate with squared-modulus lower bound one at every endpoint.","s":"(numeratorRe numeratorIm radius : ℚ) → (hradius : 0 < radius) → (schedule : Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.Schedule) → Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.DenominatorCertificate (Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.ContourProgram.constantUnit numeratorRe numeratorIm radius hradius) schedule","l":460,"f":"Causalean/Mathlib/Analysis/CertifiedContourIntervalArithmetic/Complex/ContourProgram.lean"},{"n":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.ContourProgram.evaluate","k":"def","m":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.ContourEvaluation","a":"Mathlib/Analysis/CertifiedContourIntervalArithmetic/Complex","t":1,"d":"The executable contour result applies promoted deterministic quadrature and then spends the schedule's final symmetric quadrature budget.","s":"(program : Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.ContourProgram) → (schedule : Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.Schedule) → Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.DenominatorCertificate program schedule → Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.ComplexRatInterval","l":290,"f":"Causalean/Mathlib/Analysis/CertifiedContourIntervalArithmetic/Complex/ContourEvaluation.lean"},{"n":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.ContourProgram.integrandNodeFin","k":"def","m":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.ContourProgram","a":"Mathlib/Analysis/CertifiedContourIntervalArithmetic/Complex","t":1,"d":"The node function with an explicit endpoint-bound argument avoids ever constructing a division guard outside the scheduled finite endpoint range.","s":"(program : Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.ContourProgram) → (schedule : Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.Schedule) → Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.DenominatorCertificate program schedule → Fin (schedule.mesh + 1) → Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.ComplexRatInterval","l":480,"f":"Causalean/Mathlib/Analysis/CertifiedContourIntervalArithmetic/Complex/ContourProgram.lean"},{"n":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.ContourProgram.integrandNodes","k":"def","m":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.ContourProgram","a":"Mathlib/Analysis/CertifiedContourIntervalArithmetic/Complex","t":1,"d":"A total natural-indexed node family agrees with certified endpoints and uses the terminal endpoint outside the range, which quadrature never queries.","s":"(program : Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.ContourProgram) → (schedule : Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.Schedule) → Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.DenominatorCertificate program schedule → ℕ → Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.ComplexRatInterval","l":490,"f":"Causalean/Mathlib/Analysis/CertifiedContourIntervalArithmetic/Complex/ContourProgram.lean"},{"n":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.ContourProgram.mapWidthBounds","k":"def","m":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.ContourProgram","a":"Mathlib/Analysis/CertifiedContourIntervalArithmetic/Complex","t":1,"d":"A common leaf-error and circle-input target gives these numerator and denominator rectangle width bounds by CertifiedComplexMap.width_le.","s":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.ContourProgram → ℚ → ℚ × ℚ","l":81,"f":"Causalean/Mathlib/Analysis/CertifiedContourIntervalArithmetic/Complex/ContourProgram.lean"},{"n":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.ContourProgram.nodeBox","k":"def","m":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.ContourProgram","a":"Mathlib/Analysis/CertifiedContourIntervalArithmetic/Complex","t":1,"d":"The scheduled rational rectangle fed to both function interval extensions.","s":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.ContourProgram → Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.Schedule → ℕ → Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.ComplexRatInterval","l":47,"f":"Causalean/Mathlib/Analysis/CertifiedContourIntervalArithmetic/Complex/ContourProgram.lean"},{"n":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.ContourProgram.nodePropagationBound","k":"def","m":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.ContourProgram","a":"Mathlib/Analysis/CertifiedContourIntervalArithmetic/Complex","t":1,"d":"The explicit arithmetic propagation bound for one guarded contour node combines map errors, input refinement, magnitude amplification, separation, division, and the final multiplication by the circle point.","s":"(program : Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.ContourProgram) → Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.ContourValueBounds program → ℚ → ℚ → ℚ","l":87,"f":"Causalean/Mathlib/Analysis/CertifiedContourIntervalArithmetic/Complex/ContourProgram.lean"},{"n":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.ContourProgram.nodeScale","k":"def","m":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.ContourProgram","a":"Mathlib/Analysis/CertifiedContourIntervalArithmetic/Complex","t":1,"d":"A conservative positive scale controls every coefficient in the node propagation polynomial on targets no larger than one.","s":"(program : Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.ContourProgram) → Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.ContourValueBounds program → Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.PosRat → ℚ","l":105,"f":"Causalean/Mathlib/Analysis/CertifiedContourIntervalArithmetic/Complex/ContourProgram.lean"},{"n":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.ContourProgram.nodeTrace","k":"def","m":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.ContourProgram","a":"Mathlib/Analysis/CertifiedContourIntervalArithmetic/Complex","t":1,"d":"The finite trace for one scheduled node has one event for each structural primitive operation and every event retains the exact execution schedule.","s":"(program : Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.ContourProgram) → {bounds : Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.ContourValueBounds program} → {separation : Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.PosRat} → Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.CertifiedProgramSchedule program bounds separation → ℕ → List Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.TraceEvent","l":380,"f":"Causalean/Mathlib/Analysis/CertifiedContourIntervalArithmetic/Complex/ContourProgram.lean"},{"n":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.ContourProgram.nodeTrace_spec","k":"theorem","m":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.ContourProgram","a":"Mathlib/Analysis/CertifiedContourIntervalArithmetic/Complex","t":1,"d":"For a contour program, a target error separation, a certified schedule for that program, and a stage index k, the executable node trace at stage k has exactly one event per structural operation of the program, and every event in it records the full schedule together with that schedule's operation count.","s":"∀ (program : Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.ContourProgram) {bounds : Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.ContourValueBounds program} {separation : Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.PosRat} (scheduled : Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.CertifiedProgramSchedule program bounds separation) (k : ℕ), (program.nodeTrace scheduled k).length = program.operationCount ∧ ∀ event ∈ program.nodeTrace scheduled k, event.schedule = scheduled.schedule ∧ event.schedule.operationCount = program.operationCount","l":389,"f":"Causalean/Mathlib/Analysis/CertifiedContourIntervalArithmetic/Complex/ContourProgram.lean"},{"n":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.ContourProgram.normalizedContourIntegral","k":"def","m":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.ContourProgram","a":"Mathlib/Analysis/CertifiedContourIntervalArithmetic/Complex","t":1,"d":"The normalized exact contour integral is the usual circle contour integral divided by 2 * π * i; it is used only as the semantic target.","s":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.ContourProgram → ℂ","l":511,"f":"Causalean/Mathlib/Analysis/CertifiedContourIntervalArithmetic/Complex/ContourProgram.lean"},{"n":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.ContourProgram.normalizedIntegrand","k":"def","m":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.ContourProgram","a":"Mathlib/Analysis/CertifiedContourIntervalArithmetic/Complex","t":1,"d":"The exact normalized integrand after cancellation of the circle tangent's 2 * π * i factor is the quotient times the circle point.","s":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.ContourProgram → Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.Schedule → ℝ → ℂ","l":503,"f":"Causalean/Mathlib/Analysis/CertifiedContourIntervalArithmetic/Complex/ContourProgram.lean"},{"n":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.ContourProgram.operationCount","k":"def","m":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.ContourProgram","a":"Mathlib/Analysis/CertifiedContourIntervalArithmetic/Complex","t":1,"d":"The structural operation count of one contour node includes both certified map programs, guarded division, and multiplication by the circle node.","s":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.ContourProgram → ℕ","l":33,"f":"Causalean/Mathlib/Analysis/CertifiedContourIntervalArithmetic/Complex/ContourProgram.lean"},{"n":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.ContourValueBounds","k":"structure","m":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.ContourProgram","a":"Mathlib/Analysis/CertifiedContourIntervalArithmetic/Complex","t":1,"d":"Uniform semantic magnitude data for one contour program: a nonnegative bound that covers every real and imaginary coordinate of the exact numerator function everywhere on the circle, and a nonnegative bound that covers the denominator function likewise.","s":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.ContourProgram → Type","l":52,"f":"Causalean/Mathlib/Analysis/CertifiedContourIntervalArithmetic/Complex/ContourProgram.lean"},{"n":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.DenominatorCertificate","k":"structure","m":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.ContourProgram","a":"Mathlib/Analysis/CertifiedContourIntervalArithmetic/Complex","t":1,"d":"A denominator certificate for a contour program under a fixed schedule provides a common squared-modulus separation bound that is strictly positive, together with the guarantees that every mesh endpoint's denominator interval evaluation is executable and bounded away from zero and that the same separation bound lies below every such endpoint's lower squared-modulus bound.","s":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.ContourProgram → Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.Schedule → Type","l":331,"f":"Causalean/Mathlib/Analysis/CertifiedContourIntervalArithmetic/Complex/ContourProgram.lean"},{"n":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.Schedule","k":"structure","m":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.CircleSchedule","a":"Mathlib/Analysis/CertifiedContourIntervalArithmetic/Complex","t":1,"d":"A contour schedule packages a requested rational error tolerance, an operation count, an input precision, a mesh size that is nonempty, a fuel budget for transcendental evaluation, and a nonnegative magnitude-amplification bound, together with three nonnegative error budgets — [for node evaluation](hyp:nodeBudget, nodeBudget_nonneg), for the uniform mesh, and for the final quadrature widening — such that the three budgets together fit within the tolerance and the mesh budget absorbs the magnitude-amplified discretization error.","s":"Type","l":16,"f":"Causalean/Mathlib/Analysis/CertifiedContourIntervalArithmetic/Complex/CircleSchedule.lean"},{"n":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.Schedule.canonical","k":"def","m":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.CircleSchedule","a":"Mathlib/Analysis/CertifiedContourIntervalArithmetic/Complex","t":1,"d":"A canonical schedule splits tolerance equally and chooses conservative integer fuel and mesh from the exact rational denominator and magnitude.","s":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.PosRat → ℕ → (magnitude : ℚ) → 0 ≤ magnitude → Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.Schedule","l":59,"f":"Causalean/Mathlib/Analysis/CertifiedContourIntervalArithmetic/Complex/CircleSchedule.lean"},{"n":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.TraceEvent","k":"structure","m":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.CircleSchedule","a":"Mathlib/Analysis/CertifiedContourIntervalArithmetic/Complex","t":1,"d":"A trace event retains the schedule object itself, so execution and semantic audit cannot silently disagree about fuel, mesh, or precision.","s":"Type","l":122,"f":"Causalean/Mathlib/Analysis/CertifiedContourIntervalArithmetic/Complex/CircleSchedule.lean"},{"n":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.Transcendental.abs_sub_intervalMid_le_radius","k":"theorem","m":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.Transcendental","a":"Mathlib/Analysis/CertifiedContourIntervalArithmetic/Complex","t":2,"d":"Every real point enclosed by a rational interval is at most half its width away from the interval's rational midpoint.","s":"∀ {I : Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.RatInterval} {x : ℝ}, I.Contains x → |x - ↑(Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.Transcendental.intervalMid I)| ≤ ↑(Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.Transcendental.intervalRadius I)","l":477,"f":"Causalean/Mathlib/Analysis/CertifiedContourIntervalArithmetic/Complex/Transcendental.lean"},{"n":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.Transcendental.atanError","k":"def","m":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.Transcendental","a":"Mathlib/Analysis/CertifiedContourIntervalArithmetic/Complex","t":1,"d":"The first omitted arctangent term is a rational absolute error bound when the argument has absolute value at most one.","s":"ℚ → ℕ → ℚ","l":36,"f":"Causalean/Mathlib/Analysis/CertifiedContourIntervalArithmetic/Complex/Transcendental.lean"},{"n":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.Transcendental.atanPartial","k":"def","m":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.Transcendental","a":"Mathlib/Analysis/CertifiedContourIntervalArithmetic/Complex","t":1,"d":"The rational alternating Taylor sum through degree 2 * fuel + 1 approximates arctangent on the unit interval.","s":"ℚ → ℕ → ℚ","l":30,"f":"Causalean/Mathlib/Analysis/CertifiedContourIntervalArithmetic/Complex/Transcendental.lean"},{"n":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.Transcendental.atanRaw","k":"def","m":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.Transcendental","a":"Mathlib/Analysis/CertifiedContourIntervalArithmetic/Complex","t":1,"d":"A rational arctangent Taylor sum widened by its first omitted term.","s":"ℚ → ℕ → Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.RatInterval","l":41,"f":"Causalean/Mathlib/Analysis/CertifiedContourIntervalArithmetic/Complex/Transcendental.lean"},{"n":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.Transcendental.complexExp","k":"def","m":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.ComplexExp","a":"Mathlib/Analysis/CertifiedContourIntervalArithmetic/Complex","t":1,"d":"Complex exponential outputs recursively intersect raw compositional bounds for one fixed input rectangle.","s":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.ComplexRatInterval → ℕ → Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.ComplexRatInterval","l":27,"f":"Causalean/Mathlib/Analysis/CertifiedContourIntervalArithmetic/Complex/ComplexExp.lean"},{"n":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.Transcendental.complexExpInnerTolerance","k":"def","m":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.ComplexExp","a":"Mathlib/Analysis/CertifiedContourIntervalArithmetic/Complex","t":1,"d":"The inner scalar-operation tolerance spends the stage budget after dividing by the certified magnitude amplification scale.","s":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.CertifiedComplex → ℕ → Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.PosRat","l":130,"f":"Causalean/Mathlib/Analysis/CertifiedContourIntervalArithmetic/Complex/ComplexExp.lean"},{"n":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.Transcendental.complexExpInputTolerance","k":"def","m":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.ComplexExp","a":"Mathlib/Analysis/CertifiedContourIntervalArithmetic/Complex","t":1,"d":"The input-name tolerance also pays for exponential Lipschitz amplification, so a large positive real part requests a proportionally narrower input box.","s":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.CertifiedComplex → ℕ → Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.PosRat","l":138,"f":"Causalean/Mathlib/Analysis/CertifiedContourIntervalArithmetic/Complex/ComplexExp.lean"},{"n":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.Transcendental.complexExpMagnitude","k":"def","m":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.ComplexExp","a":"Mathlib/Analysis/CertifiedContourIntervalArithmetic/Complex","t":1,"d":"The initial real-part enclosure gives a positive rational amplification scale for all later exponential stages.","s":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.CertifiedComplex → ℚ","l":120,"f":"Causalean/Mathlib/Analysis/CertifiedContourIntervalArithmetic/Complex/ComplexExp.lean"},{"n":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.Transcendental.complexExpName","k":"def","m":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.ComplexExp","a":"Mathlib/Analysis/CertifiedContourIntervalArithmetic/Complex","t":1,"d":"Complex exponential lifts a certified complex input to a certified complex output without storing exact values in any returned rectangle.","s":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.CertifiedComplex → Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.CertifiedComplex","l":573,"f":"Causalean/Mathlib/Analysis/CertifiedContourIntervalArithmetic/Complex/ComplexExp.lean"},{"n":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.Transcendental.complexExpNameApprox","k":"def","m":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.ComplexExp","a":"Mathlib/Analysis/CertifiedContourIntervalArithmetic/Complex","t":1,"d":"Certified complex-exponential name approximations recursively intersect every scheduled stage output seen so far.","s":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.CertifiedComplex → ℕ → Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.ComplexRatInterval","l":173,"f":"Causalean/Mathlib/Analysis/CertifiedContourIntervalArithmetic/Complex/ComplexExp.lean"},{"n":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.Transcendental.complexExpNameApprox_spec","k":"theorem","m":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.ComplexExp","a":"Mathlib/Analysis/CertifiedContourIntervalArithmetic/Complex","t":2,"d":"Recursive complex-exponential name approximations remain sound and adjacent stages are nested by their defining finite intersection.","s":"∀ (z : Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.CertifiedComplex) (n : ℕ), (Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.Transcendental.complexExpNameApprox z n).Contains (Complex.exp z.value) ∧ (Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.Transcendental.complexExpNameApprox z (n + 1)).Subinterval (Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.Transcendental.complexExpNameApprox z n)","l":183,"f":"Causalean/Mathlib/Analysis/CertifiedContourIntervalArithmetic/Complex/ComplexExp.lean"},{"n":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.Transcendental.complexExpPrecision","k":"def","m":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.ComplexExp","a":"Mathlib/Analysis/CertifiedContourIntervalArithmetic/Complex","t":1,"d":"The denominator of a requested rational tolerance selects a stage whose canonical reciprocal target is no larger than that tolerance.","s":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.CertifiedComplex → Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.PosRat → ℕ","l":546,"f":"Causalean/Mathlib/Analysis/CertifiedContourIntervalArithmetic/Complex/ComplexExp.lean"},{"n":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.Transcendental.complexExpRaw","k":"def","m":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.ComplexExp","a":"Mathlib/Analysis/CertifiedContourIntervalArithmetic/Complex","t":1,"d":"Raw complex exponential evaluation composes certified real exponential, cosine, and sine interval extensions coordinatewise.","s":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.ComplexRatInterval → ℕ → Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.ComplexRatInterval","l":21,"f":"Causalean/Mathlib/Analysis/CertifiedContourIntervalArithmetic/Complex/ComplexExp.lean"},{"n":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.Transcendental.complexExpStage","k":"def","m":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.ComplexExp","a":"Mathlib/Analysis/CertifiedContourIntervalArithmetic/Complex","t":1,"d":"One executable complex exponential name stage uses its scheduled input rectangle and the Taylor fuel computed from that same rectangle.","s":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.CertifiedComplex → ℕ → Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.ComplexRatInterval","l":168,"f":"Causalean/Mathlib/Analysis/CertifiedContourIntervalArithmetic/Complex/ComplexExp.lean"},{"n":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.Transcendental.complexExpStageFuel","k":"def","m":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.ComplexExp","a":"Mathlib/Analysis/CertifiedContourIntervalArithmetic/Complex","t":1,"d":"Taylor fuel for a stage is computed from the rational midpoint and magnitude of the input rectangle actually returned, not from a denominator ordering heuristic.","s":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.CertifiedComplex → ℕ → ℕ","l":151,"f":"Causalean/Mathlib/Analysis/CertifiedContourIntervalArithmetic/Complex/ComplexExp.lean"},{"n":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.Transcendental.complexExpStageInput","k":"def","m":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.ComplexExp","a":"Mathlib/Analysis/CertifiedContourIntervalArithmetic/Complex","t":1,"d":"A complex exponential stage refines its input using the magnitude-sensitive tolerance selected for that stage.","s":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.CertifiedComplex → ℕ → Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.ComplexRatInterval","l":146,"f":"Causalean/Mathlib/Analysis/CertifiedContourIntervalArithmetic/Complex/ComplexExp.lean"},{"n":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.Transcendental.complexExpStageTolerance","k":"def","m":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.ComplexExp","a":"Mathlib/Analysis/CertifiedContourIntervalArithmetic/Complex","t":1,"d":"The canonical error target at name stage n is the reciprocal of n + 1.","s":"ℕ → Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.PosRat","l":126,"f":"Causalean/Mathlib/Analysis/CertifiedContourIntervalArithmetic/Complex/ComplexExp.lean"},{"n":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.Transcendental.complexExpStage_width","k":"theorem","m":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.ComplexExp","a":"Mathlib/Analysis/CertifiedContourIntervalArithmetic/Complex","t":2,"d":"Every scheduled stage has width at most its reciprocal stage tolerance; the proof combines input refinement, scalar Taylor convergence, magnitude amplification, and explicit rectangle-product propagation.","s":"∀ (z : Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.CertifiedComplex) (n : ℕ), (Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.Transcendental.complexExpStage z n).width ≤ ↑(Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.Transcendental.complexExpStageTolerance n)","l":278,"f":"Causalean/Mathlib/Analysis/CertifiedContourIntervalArithmetic/Complex/ComplexExp.lean"},{"n":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.Transcendental.complexExp_nested","k":"theorem","m":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.ComplexExp","a":"Mathlib/Analysis/CertifiedContourIntervalArithmetic/Complex","t":2,"d":"Adjacent complex exponential outputs for one fixed input rectangle are nested by coordinatewise finite intersection.","s":"∀ (I : Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.ComplexRatInterval) (fuel : ℕ), (Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.Transcendental.complexExp I (fuel + 1)).Subinterval (Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.Transcendental.complexExp I fuel)","l":51,"f":"Causalean/Mathlib/Analysis/CertifiedContourIntervalArithmetic/Complex/ComplexExp.lean"},{"n":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.Transcendental.complexExp_sound","k":"theorem","m":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.ComplexExp","a":"Mathlib/Analysis/CertifiedContourIntervalArithmetic/Complex","t":2,"d":"Compositional complex exponential evaluation encloses the exact complex exponential.","s":"∀ {I : Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.ComplexRatInterval} {z : ℂ}, I.Contains z → ∀ (fuel : ℕ), (Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.Transcendental.complexExp I fuel).Contains (Complex.exp z)","l":42,"f":"Causalean/Mathlib/Analysis/CertifiedContourIntervalArithmetic/Complex/ComplexExp.lean"},{"n":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.Transcendental.complexExp_width","k":"theorem","m":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.ComplexExp","a":"Mathlib/Analysis/CertifiedContourIntervalArithmetic/Complex","t":2,"d":"Complex exponential width is bounded explicitly by the widths and endpoint magnitudes of its certified exponential and trigonometric factors.","s":"∀ (I : Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.ComplexRatInterval) (fuel : ℕ), (Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.Transcendental.complexExp I fuel).width ≤ 2 * (max (Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.Transcendental.expInterval I.re fuel).maxAbs 0 * max (Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.Transcendental.cosInterval I.im fuel).width (Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.Transcendental.sinInterval I.im fuel).width + max (Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.Transcendental.cosInterval I.im fuel).maxAbs (Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.Transcendental.sinInterval I.im fuel).maxAbs * max (Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.Transcendental.expInterval I.re fuel).width 0)","l":69,"f":"Causalean/Mathlib/Analysis/CertifiedContourIntervalArithmetic/Complex/ComplexExp.lean"},{"n":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.Transcendental.complexExp_width_at_precision","k":"theorem","m":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.ComplexExp","a":"Mathlib/Analysis/CertifiedContourIntervalArithmetic/Complex","t":1,"d":"For a certified complex input and a requested positive rational tolerance ε, evaluating the certified complex-exponential approximation at the precision level selected for that tolerance yields an output enclosure whose width is no larger than ε.","s":"∀ (z : Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.CertifiedComplex) (ε : Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.PosRat), (Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.Transcendental.complexExpNameApprox z (Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.Transcendental.complexExpPrecision z ε)).width ≤ ↑ε","l":550,"f":"Causalean/Mathlib/Analysis/CertifiedContourIntervalArithmetic/Complex/ComplexExp.lean"},{"n":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.Transcendental.cosError","k":"def","m":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.Transcendental","a":"Mathlib/Analysis/CertifiedContourIntervalArithmetic/Complex","t":1,"d":"The absolute first omitted cosine Taylor scale is an executable rational remainder bound.","s":"ℚ → ℕ → ℚ","l":228,"f":"Causalean/Mathlib/Analysis/CertifiedContourIntervalArithmetic/Complex/Transcendental.lean"},{"n":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.Transcendental.cosInterval","k":"def","m":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.Transcendental","a":"Mathlib/Analysis/CertifiedContourIntervalArithmetic/Complex","t":1,"d":"Cosine interval outputs recursively intersect all raw bounds seen so far.","s":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.RatInterval → ℕ → Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.RatInterval","l":273,"f":"Causalean/Mathlib/Analysis/CertifiedContourIntervalArithmetic/Complex/Transcendental.lean"},{"n":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.Transcendental.cosIntervalRaw","k":"def","m":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.Transcendental","a":"Mathlib/Analysis/CertifiedContourIntervalArithmetic/Complex","t":1,"d":"Raw cosine interval evaluation adds the input-radius Lipschitz error to a rational Taylor enclosure at the midpoint.","s":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.RatInterval → ℕ → Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.RatInterval","l":262,"f":"Causalean/Mathlib/Analysis/CertifiedContourIntervalArithmetic/Complex/Transcendental.lean"},{"n":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.Transcendental.cosInterval_nested","k":"theorem","m":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.Transcendental","a":"Mathlib/Analysis/CertifiedContourIntervalArithmetic/Complex","t":2,"d":"Adjacent cosine outputs are nested by recursive finite intersection.","s":"∀ (I : Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.RatInterval) (fuel : ℕ), (Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.Transcendental.cosInterval I (fuel + 1)).Subinterval (Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.Transcendental.cosInterval I fuel)","l":534,"f":"Causalean/Mathlib/Analysis/CertifiedContourIntervalArithmetic/Complex/Transcendental.lean"},{"n":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.Transcendental.cosInterval_sound","k":"theorem","m":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.Transcendental","a":"Mathlib/Analysis/CertifiedContourIntervalArithmetic/Complex","t":2,"d":"Cosine interval evaluation encloses cosine throughout the input interval.","s":"∀ {I : Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.RatInterval} {x : ℝ}, I.Contains x → ∀ (fuel : ℕ), (Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.Transcendental.cosInterval I fuel).Contains (Real.cos x)","l":517,"f":"Causalean/Mathlib/Analysis/CertifiedContourIntervalArithmetic/Complex/Transcendental.lean"},{"n":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.Transcendental.cosInterval_width","k":"theorem","m":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.Transcendental","a":"Mathlib/Analysis/CertifiedContourIntervalArithmetic/Complex","t":2,"d":"Cosine output width is the shrinking input diameter plus its explicit Taylor error.","s":"∀ (I : Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.RatInterval) (fuel : ℕ), (Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.Transcendental.cosInterval I fuel).width ≤ I.width + 2 * Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.Transcendental.cosError (Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.Transcendental.intervalMid I) fuel","l":563,"f":"Causalean/Mathlib/Analysis/CertifiedContourIntervalArithmetic/Complex/Transcendental.lean"},{"n":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.Transcendental.cosName","k":"def","m":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.TrigNames","a":"Mathlib/Analysis/CertifiedContourIntervalArithmetic/Complex","t":1,"d":"Cosine lifts a refining certified real input to a certified real output.","s":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.CertifiedReal → Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.CertifiedReal","l":260,"f":"Causalean/Mathlib/Analysis/CertifiedContourIntervalArithmetic/Complex/TrigNames.lean"},{"n":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.Transcendental.cosNameApprox","k":"def","m":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.TrigNames","a":"Mathlib/Analysis/CertifiedContourIntervalArithmetic/Complex","t":1,"d":"A cosine output on a refining real name recursively intersects the current input enclosure result with every preceding output.","s":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.CertifiedReal → ℕ → Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.RatInterval","l":22,"f":"Causalean/Mathlib/Analysis/CertifiedContourIntervalArithmetic/Complex/TrigNames.lean"},{"n":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.Transcendental.cosNameApprox_spec","k":"theorem","m":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.TrigNames","a":"Mathlib/Analysis/CertifiedContourIntervalArithmetic/Complex","t":2,"d":"Recursive cosine-name approximations remain sound and adjacent-fuel nested.","s":"∀ (x : Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.CertifiedReal) (fuel : ℕ), (Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.Transcendental.cosNameApprox x fuel).Contains (Real.cos x.value) ∧ (Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.Transcendental.cosNameApprox x (fuel + 1)).Subinterval (Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.Transcendental.cosNameApprox x fuel)","l":56,"f":"Causalean/Mathlib/Analysis/CertifiedContourIntervalArithmetic/Complex/TrigNames.lean"},{"n":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.Transcendental.cosName_width_at_precision","k":"theorem","m":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.TrigNames","a":"Mathlib/Analysis/CertifiedContourIntervalArithmetic/Complex","t":2,"d":"Refinement of the input name plus Taylor fuel makes cosine output width effective.","s":"∀ (x : Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.CertifiedReal) (ε : Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.PosRat), (Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.Transcendental.cosNameApprox x (Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.Transcendental.trigNamePrecision x ε)).width ≤ ↑ε","l":198,"f":"Causalean/Mathlib/Analysis/CertifiedContourIntervalArithmetic/Complex/TrigNames.lean"},{"n":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.Transcendental.cosPartial","k":"def","m":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.Transcendental","a":"Mathlib/Analysis/CertifiedContourIntervalArithmetic/Complex","t":1,"d":"The rational cosine Taylor polynomial retains terms through the requested fuel.","s":"ℚ → ℕ → ℚ","l":223,"f":"Causalean/Mathlib/Analysis/CertifiedContourIntervalArithmetic/Complex/Transcendental.lean"},{"n":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.Transcendental.cosRaw","k":"def","m":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.Transcendental","a":"Mathlib/Analysis/CertifiedContourIntervalArithmetic/Complex","t":1,"d":"A rational cosine Taylor value widened by its explicit remainder.","s":"ℚ → ℕ → Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.RatInterval","l":241,"f":"Causalean/Mathlib/Analysis/CertifiedContourIntervalArithmetic/Complex/Transcendental.lean"},{"n":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.Transcendental.cosRaw_sound","k":"theorem","m":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.Transcendental","a":"Mathlib/Analysis/CertifiedContourIntervalArithmetic/Complex","t":2,"d":"Raw rational cosine bounds contain the exact cosine at every rational center.","s":"∀ (q : ℚ) (fuel : ℕ), (Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.Transcendental.cosRaw q fuel).Contains (Real.cos ↑q)","l":463,"f":"Causalean/Mathlib/Analysis/CertifiedContourIntervalArithmetic/Complex/Transcendental.lean"},{"n":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.Transcendental.expInterval","k":"def","m":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.IntervalExp","a":"Mathlib/Analysis/CertifiedContourIntervalArithmetic/Complex","t":1,"d":"Exponential interval outputs recursively intersect all midpoint bounds seen so far.","s":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.RatInterval → ℕ → Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.RatInterval","l":34,"f":"Causalean/Mathlib/Analysis/CertifiedContourIntervalArithmetic/Complex/IntervalExp.lean"},{"n":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.Transcendental.expIntervalRaw","k":"def","m":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.IntervalExp","a":"Mathlib/Analysis/CertifiedContourIntervalArithmetic/Complex","t":1,"d":"The exponential midpoint extension uses the promoted scalar exponential and a magnitude-dependent Lipschitz expansion for the input radius.","s":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.RatInterval → ℕ → Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.RatInterval","l":15,"f":"Causalean/Mathlib/Analysis/CertifiedContourIntervalArithmetic/Complex/IntervalExp.lean"},{"n":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.Transcendental.expInterval_nested","k":"theorem","m":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.IntervalExp","a":"Mathlib/Analysis/CertifiedContourIntervalArithmetic/Complex","t":2,"d":"Adjacent midpoint exponential outputs are nested by finite intersection.","s":"∀ (I : Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.RatInterval) (fuel : ℕ), (Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.Transcendental.expInterval I (fuel + 1)).Subinterval (Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.Transcendental.expInterval I fuel)","l":97,"f":"Causalean/Mathlib/Analysis/CertifiedContourIntervalArithmetic/Complex/IntervalExp.lean"},{"n":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.Transcendental.expInterval_sound","k":"theorem","m":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.IntervalExp","a":"Mathlib/Analysis/CertifiedContourIntervalArithmetic/Complex","t":2,"d":"The magnitude-dependent exponential interval contains the exponential of every enclosed real input.","s":"∀ {I : Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.RatInterval} {x : ℝ}, I.Contains x → ∀ (fuel : ℕ), (Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.Transcendental.expInterval I fuel).Contains (Real.exp x)","l":89,"f":"Causalean/Mathlib/Analysis/CertifiedContourIntervalArithmetic/Complex/IntervalExp.lean"},{"n":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.Transcendental.expInterval_width","k":"theorem","m":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.IntervalExp","a":"Mathlib/Analysis/CertifiedContourIntervalArithmetic/Complex","t":1,"d":"For a rational interval I and a Taylor fuel/iteration count, the width of the certified interval exponential is bounded by the scalar Taylor truncation error at the interval's midpoint plus twice the interval's radius amplified by a certified bound on the exponential's magnitude over the interval.","s":"∀ (I : Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.RatInterval) (fuel : ℕ), (Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.Transcendental.expInterval I fuel).width ≤ (Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Transcendental.expScalar (Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.Transcendental.intervalMid I) fuel).width + 2 * (Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Transcendental.expScalar I.maxAbs fuel).hi * Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.Transcendental.intervalRadius I","l":107,"f":"Causalean/Mathlib/Analysis/CertifiedContourIntervalArithmetic/Complex/IntervalExp.lean"},{"n":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.Transcendental.intervalMid","k":"def","m":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.Transcendental","a":"Mathlib/Analysis/CertifiedContourIntervalArithmetic/Complex","t":1,"d":"The rational midpoint of an interval is used only as a Taylor expansion center.","s":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.RatInterval → ℚ","l":250,"f":"Causalean/Mathlib/Analysis/CertifiedContourIntervalArithmetic/Complex/Transcendental.lean"},{"n":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.Transcendental.intervalRadius","k":"def","m":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.Transcendental","a":"Mathlib/Analysis/CertifiedContourIntervalArithmetic/Complex","t":1,"d":"Half the rational interval width bounds distance from every enclosed point to its midpoint.","s":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.RatInterval → ℚ","l":253,"f":"Causalean/Mathlib/Analysis/CertifiedContourIntervalArithmetic/Complex/Transcendental.lean"},{"n":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.Transcendental.piInterval","k":"def","m":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.Transcendental","a":"Mathlib/Analysis/CertifiedContourIntervalArithmetic/Complex","t":1,"d":"Successive π approximations are recursive finite intersections of Machin bounds.","s":"ℕ → Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.RatInterval","l":57,"f":"Causalean/Mathlib/Analysis/CertifiedContourIntervalArithmetic/Complex/Transcendental.lean"},{"n":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.Transcendental.piInterval_nested","k":"theorem","m":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.Transcendental","a":"Mathlib/Analysis/CertifiedContourIntervalArithmetic/Complex","t":2,"d":"Adjacent π fuel values are nested by construction through finite intersection.","s":"∀ (fuel : ℕ), (Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.Transcendental.piInterval (fuel + 1)).Subinterval (Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.Transcendental.piInterval fuel)","l":133,"f":"Causalean/Mathlib/Analysis/CertifiedContourIntervalArithmetic/Complex/Transcendental.lean"},{"n":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.Transcendental.piInterval_sound","k":"theorem","m":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.Transcendental","a":"Mathlib/Analysis/CertifiedContourIntervalArithmetic/Complex","t":2,"d":"Every recursively intersected π rectangle contains π.","s":"∀ (fuel : ℕ), (Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.Transcendental.piInterval fuel).Contains Real.pi","l":126,"f":"Causalean/Mathlib/Analysis/CertifiedContourIntervalArithmetic/Complex/Transcendental.lean"},{"n":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.Transcendental.piInterval_width","k":"theorem","m":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.Transcendental","a":"Mathlib/Analysis/CertifiedContourIntervalArithmetic/Complex","t":1,"d":"For a requested positive rational tolerance ε, evaluating the certified π-enclosure at the precision level selected for that tolerance yields an enclosure whose width is no larger than ε.","s":"∀ (ε : Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.PosRat), (Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.Transcendental.piInterval (Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.Transcendental.piPrecision ε)).width ≤ ↑ε","l":162,"f":"Causalean/Mathlib/Analysis/CertifiedContourIntervalArithmetic/Complex/Transcendental.lean"},{"n":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.Transcendental.piName","k":"def","m":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.Transcendental","a":"Mathlib/Analysis/CertifiedContourIntervalArithmetic/Complex","t":1,"d":"The Machin enclosure sequence packages π as an effectively refining certified real name.","s":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.CertifiedReal","l":205,"f":"Causalean/Mathlib/Analysis/CertifiedContourIntervalArithmetic/Complex/Transcendental.lean"},{"n":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.Transcendental.piPrecision","k":"def","m":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.Transcendental","a":"Mathlib/Analysis/CertifiedContourIntervalArithmetic/Complex","t":1,"d":"A denominator-sensitive executable fuel for a requested π width.","s":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.PosRat → ℕ","l":62,"f":"Causalean/Mathlib/Analysis/CertifiedContourIntervalArithmetic/Complex/Transcendental.lean"},{"n":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.Transcendental.piRaw","k":"def","m":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.Transcendental","a":"Mathlib/Analysis/CertifiedContourIntervalArithmetic/Complex","t":1,"d":"Machin's formula combines two rational arctangent enclosures without using an exact π endpoint.","s":"ℕ → Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.RatInterval","l":50,"f":"Causalean/Mathlib/Analysis/CertifiedContourIntervalArithmetic/Complex/Transcendental.lean"},{"n":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.Transcendental.piRaw_sound","k":"theorem","m":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.Transcendental","a":"Mathlib/Analysis/CertifiedContourIntervalArithmetic/Complex","t":2,"d":"Every raw Machin rectangle contains the mathematical constant π.","s":"∀ (fuel : ℕ), (Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.Transcendental.piRaw fuel).Contains Real.pi","l":111,"f":"Causalean/Mathlib/Analysis/CertifiedContourIntervalArithmetic/Complex/Transcendental.lean"},{"n":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.Transcendental.power_div_factorial_le","k":"theorem","m":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.TrigNames","a":"Mathlib/Analysis/CertifiedContourIntervalArithmetic/Complex","t":2,"d":"If the Taylor fuel dominates a rational magnitude and denominator scale, the corresponding rational power-over-factorial remainder is explicitly small.","s":"∀ (q : ℚ) (A d N r : ℕ), |q| ≤ ↑A → 1 ≤ A → 1 ≤ d → 1 ≤ r → 32 * d * A ^ 2 ≤ N → |q| ^ (2 * N + r) / ↑(2 * N + r).factorial ≤ 1 / ↑(4 * d)","l":101,"f":"Causalean/Mathlib/Analysis/CertifiedContourIntervalArithmetic/Complex/TrigNames.lean"},{"n":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.Transcendental.sinError","k":"def","m":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.Transcendental","a":"Mathlib/Analysis/CertifiedContourIntervalArithmetic/Complex","t":1,"d":"The absolute first omitted sine Taylor scale is an executable rational remainder bound.","s":"ℚ → ℕ → ℚ","l":219,"f":"Causalean/Mathlib/Analysis/CertifiedContourIntervalArithmetic/Complex/Transcendental.lean"},{"n":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.Transcendental.sinInterval","k":"def","m":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.Transcendental","a":"Mathlib/Analysis/CertifiedContourIntervalArithmetic/Complex","t":1,"d":"Sine interval outputs recursively intersect all raw bounds seen so far.","s":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.RatInterval → ℕ → Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.RatInterval","l":268,"f":"Causalean/Mathlib/Analysis/CertifiedContourIntervalArithmetic/Complex/Transcendental.lean"},{"n":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.Transcendental.sinIntervalRaw","k":"def","m":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.Transcendental","a":"Mathlib/Analysis/CertifiedContourIntervalArithmetic/Complex","t":1,"d":"Raw sine interval evaluation adds the input-radius Lipschitz error to a rational Taylor enclosure at the midpoint.","s":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.RatInterval → ℕ → Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.RatInterval","l":256,"f":"Causalean/Mathlib/Analysis/CertifiedContourIntervalArithmetic/Complex/Transcendental.lean"},{"n":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.Transcendental.sinInterval_nested","k":"theorem","m":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.Transcendental","a":"Mathlib/Analysis/CertifiedContourIntervalArithmetic/Complex","t":2,"d":"Adjacent sine outputs are nested by recursive finite intersection.","s":"∀ (I : Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.RatInterval) (fuel : ℕ), (Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.Transcendental.sinInterval I (fuel + 1)).Subinterval (Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.Transcendental.sinInterval I fuel)","l":524,"f":"Causalean/Mathlib/Analysis/CertifiedContourIntervalArithmetic/Complex/Transcendental.lean"},{"n":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.Transcendental.sinInterval_sound","k":"theorem","m":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.Transcendental","a":"Mathlib/Analysis/CertifiedContourIntervalArithmetic/Complex","t":2,"d":"Sine interval evaluation encloses sine throughout the input interval.","s":"∀ {I : Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.RatInterval} {x : ℝ}, I.Contains x → ∀ (fuel : ℕ), (Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.Transcendental.sinInterval I fuel).Contains (Real.sin x)","l":510,"f":"Causalean/Mathlib/Analysis/CertifiedContourIntervalArithmetic/Complex/Transcendental.lean"},{"n":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.Transcendental.sinInterval_width","k":"theorem","m":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.Transcendental","a":"Mathlib/Analysis/CertifiedContourIntervalArithmetic/Complex","t":2,"d":"Sine output width is the shrinking input diameter plus its explicit Taylor error.","s":"∀ (I : Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.RatInterval) (fuel : ℕ), (Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.Transcendental.sinInterval I fuel).width ≤ I.width + 2 * Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.Transcendental.sinError (Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.Transcendental.intervalMid I) fuel","l":544,"f":"Causalean/Mathlib/Analysis/CertifiedContourIntervalArithmetic/Complex/Transcendental.lean"},{"n":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.Transcendental.sinName","k":"def","m":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.TrigNames","a":"Mathlib/Analysis/CertifiedContourIntervalArithmetic/Complex","t":1,"d":"Sine lifts a refining certified real input to a certified real output.","s":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.CertifiedReal → Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.CertifiedReal","l":251,"f":"Causalean/Mathlib/Analysis/CertifiedContourIntervalArithmetic/Complex/TrigNames.lean"},{"n":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.Transcendental.sinNameApprox","k":"def","m":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.TrigNames","a":"Mathlib/Analysis/CertifiedContourIntervalArithmetic/Complex","t":1,"d":"A sine output on a refining real name recursively intersects the current input enclosure result with every preceding output.","s":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.CertifiedReal → ℕ → Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.RatInterval","l":15,"f":"Causalean/Mathlib/Analysis/CertifiedContourIntervalArithmetic/Complex/TrigNames.lean"},{"n":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.Transcendental.sinNameApprox_spec","k":"theorem","m":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.TrigNames","a":"Mathlib/Analysis/CertifiedContourIntervalArithmetic/Complex","t":2,"d":"Recursive sine-name approximations remain sound and adjacent-fuel nested.","s":"∀ (x : Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.CertifiedReal) (fuel : ℕ), (Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.Transcendental.sinNameApprox x fuel).Contains (Real.sin x.value) ∧ (Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.Transcendental.sinNameApprox x (fuel + 1)).Subinterval (Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.Transcendental.sinNameApprox x fuel)","l":36,"f":"Causalean/Mathlib/Analysis/CertifiedContourIntervalArithmetic/Complex/TrigNames.lean"},{"n":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.Transcendental.sinName_width_at_precision","k":"theorem","m":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.TrigNames","a":"Mathlib/Analysis/CertifiedContourIntervalArithmetic/Complex","t":1,"d":"For a certified real input and a requested positive rational tolerance ε, evaluating the certified sine name-approximation at the precision level selected for that tolerance yields an output enclosure whose width is no larger than ε.","s":"∀ (x : Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.CertifiedReal) (ε : Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.PosRat), (Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.Transcendental.sinNameApprox x (Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.Transcendental.trigNamePrecision x ε)).width ≤ ↑ε","l":142,"f":"Causalean/Mathlib/Analysis/CertifiedContourIntervalArithmetic/Complex/TrigNames.lean"},{"n":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.Transcendental.sinPartial","k":"def","m":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.Transcendental","a":"Mathlib/Analysis/CertifiedContourIntervalArithmetic/Complex","t":1,"d":"The rational sine Taylor polynomial retains terms through the requested fuel.","s":"ℚ → ℕ → ℚ","l":214,"f":"Causalean/Mathlib/Analysis/CertifiedContourIntervalArithmetic/Complex/Transcendental.lean"},{"n":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.Transcendental.sinRaw","k":"def","m":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.Transcendental","a":"Mathlib/Analysis/CertifiedContourIntervalArithmetic/Complex","t":1,"d":"A rational sine Taylor value widened by its explicit remainder.","s":"ℚ → ℕ → Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.RatInterval","l":232,"f":"Causalean/Mathlib/Analysis/CertifiedContourIntervalArithmetic/Complex/Transcendental.lean"},{"n":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.Transcendental.sinRaw_sound","k":"theorem","m":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.Transcendental","a":"Mathlib/Analysis/CertifiedContourIntervalArithmetic/Complex","t":2,"d":"Raw rational sine bounds contain the exact sine at every rational center.","s":"∀ (q : ℚ) (fuel : ℕ), (Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.Transcendental.sinRaw q fuel).Contains (Real.sin ↑q)","l":449,"f":"Causalean/Mathlib/Analysis/CertifiedContourIntervalArithmetic/Complex/Transcendental.lean"},{"n":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.Transcendental.trigNamePrecision","k":"def","m":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.TrigNames","a":"Mathlib/Analysis/CertifiedContourIntervalArithmetic/Complex","t":1,"d":"Conservative trigonometric precision combines the input modulus, initial argument magnitude, and target denominator.","s":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.CertifiedReal → Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.PosRat → ℕ","l":29,"f":"Causalean/Mathlib/Analysis/CertifiedContourIntervalArithmetic/Complex/TrigNames.lean"},{"n":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.certified_contour_evaluation","k":"theorem","m":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.ContourEvaluation","a":"Mathlib/Analysis/CertifiedContourIntervalArithmetic/Complex","t":1,"d":"Certified finite-precision evaluation of a contour integral. Fix a contour-integral program, value bounds for it, a target separation, a certified mesh schedule built to meet that target, and a certificate that the denominator stays bounded away from zero, whose guaranteed separation covers the target one; assume also that the program's denominator never vanishes anywhere on the parametrized circle and the normalized integrand is Lipschitz in the circle parameter with the schedule's constant. Then the rational rectangle obtained by evaluating the program both contains the true normalized contour integral and has real-part width at most the schedule's requested tolerance.","s":"∀ (program : Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.ContourProgram) (bounds : Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.ContourValueBounds program) (separation : Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.PosRat) (scheduled : Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.CertifiedProgramSchedule program bounds separation) (certificate : Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.DenominatorCertificate program scheduled.schedule), ↑separation ≤ certificate.separation → (∀ u ∈ Set.Icc 0 1, program.denominator.value (↑program.radius * Complex.exp (↑2 * ↑Real.pi * ↑u * Complex.I)) ≠ 0) → (∀ s ∈ Set.Icc 0 1, ∀ t ∈ Set.Icc 0 1, ‖program.normalizedIntegrand scheduled.schedule s - program.normalizedIntegrand scheduled.schedule t‖ ≤ ↑scheduled.schedule.magnitude * |s - t|) → (program.evaluate scheduled.schedule certificate).Contains program.normalizedContourIntegral ∧ (program.evaluate scheduled.schedule certificate).re.width ≤ ↑scheduled.schedule.tolerance","l":359,"f":"Causalean/Mathlib/Analysis/CertifiedContourIntervalArithmetic/Complex/ContourEvaluation.lean"},{"n":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.certified_contour_evaluation_inverseMax","k":"theorem","m":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.ContourEvaluation","a":"Mathlib/Analysis/CertifiedContourIntervalArithmetic/Complex","t":2,"d":"Specializing the generic contour certificate to reciprocal-max tolerance gives the width required by canonical finite-rational statistics.","s":"∀ (n : ℕ) (program : Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.ContourProgram) (bounds : Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.ContourValueBounds program) (separation : Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.PosRat) (scheduled : Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.CertifiedProgramSchedule program bounds separation), scheduled.schedule.tolerance = Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.inverseMaxTolerance n → ∀ (certificate : Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.DenominatorCertificate program scheduled.schedule), ↑separation ≤ certificate.separation → (∀ u ∈ Set.Icc 0 1, program.denominator.value (↑program.radius * Complex.exp (↑2 * ↑Real.pi * ↑u * Complex.I)) ≠ 0) → (∀ s ∈ Set.Icc 0 1, ∀ t ∈ Set.Icc 0 1, ‖program.normalizedIntegrand scheduled.schedule s - program.normalizedIntegrand scheduled.schedule t‖ ≤ ↑scheduled.schedule.magnitude * |s - t|) → (program.evaluate scheduled.schedule certificate).re.width ≤ 1 / max (↑n) 1","l":393,"f":"Causalean/Mathlib/Analysis/CertifiedContourIntervalArithmetic/Complex/ContourEvaluation.lean"},{"n":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.circleAngle","k":"def","m":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.CircleSchedule","a":"Mathlib/Analysis/CertifiedContourIntervalArithmetic/Complex","t":1,"d":"The rational rectangle for the pure-imaginary angle 2 * π * k / mesh uses the π enclosure at the schedule's input precision.","s":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.Schedule → ℕ → Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.ComplexRatInterval","l":136,"f":"Causalean/Mathlib/Analysis/CertifiedContourIntervalArithmetic/Complex/CircleSchedule.lean"},{"n":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.circleAngle_sound","k":"theorem","m":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.CircleSchedule","a":"Mathlib/Analysis/CertifiedContourIntervalArithmetic/Complex","t":2,"d":"The angle rectangle contains the exact pure-imaginary angle at every endpoint from zero through the terminal mesh endpoint.","s":"∀ (schedule : Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.Schedule) {k : ℕ}, k ≤ schedule.mesh → (Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.circleAngle schedule k).Contains (↑2 * ↑Real.pi * (↑↑k / ↑schedule.mesh) * Complex.I)","l":388,"f":"Causalean/Mathlib/Analysis/CertifiedContourIntervalArithmetic/Complex/CircleSchedule.lean"},{"n":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.circleExpFuel","k":"def","m":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.CircleSchedule","a":"Mathlib/Analysis/CertifiedContourIntervalArithmetic/Complex","t":1,"d":"A uniform Taylor fuel selected for a requested circle-node width is an explicit rational-data bound valid for all endpoints of the finite mesh.","s":"ℚ → ℕ → Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.PosRat → ℕ","l":162,"f":"Causalean/Mathlib/Analysis/CertifiedContourIntervalArithmetic/Complex/CircleSchedule.lean"},{"n":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.circleInnerTolerance","k":"def","m":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.CircleSchedule","a":"Mathlib/Analysis/CertifiedContourIntervalArithmetic/Complex","t":1,"d":"The internal circle tolerance is capped by one so the elementary-factor magnitude estimates used in complex multiplication remain uniform.","s":"ℚ → Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.PosRat → Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.PosRat","l":148,"f":"Causalean/Mathlib/Analysis/CertifiedContourIntervalArithmetic/Complex/CircleSchedule.lean"},{"n":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.circleInputPrecision","k":"def","m":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.CircleSchedule","a":"Mathlib/Analysis/CertifiedContourIntervalArithmetic/Complex","t":1,"d":"The uniform π precision selected for a requested circle-node width uses only rational data and is shared by every endpoint of the mesh.","s":"ℚ → Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.PosRat → ℕ","l":157,"f":"Causalean/Mathlib/Analysis/CertifiedContourIntervalArithmetic/Complex/CircleSchedule.lean"},{"n":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.circleNode","k":"def","m":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.CircleSchedule","a":"Mathlib/Analysis/CertifiedContourIntervalArithmetic/Complex","t":1,"d":"The scheduled rational circle node computes radius * exp(2 * π * i * k / mesh) entirely through executable rational rectangle operations.","s":"ℚ → Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.Schedule → ℕ → Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.ComplexRatInterval","l":143,"f":"Causalean/Mathlib/Analysis/CertifiedContourIntervalArithmetic/Complex/CircleSchedule.lean"},{"n":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.circleNodeEvent","k":"def","m":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.CircleSchedule","a":"Mathlib/Analysis/CertifiedContourIntervalArithmetic/Complex","t":1,"d":"A circle-node trace event uses the schedule definitionally, not a copied fuel field.","s":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.Schedule → ℕ → ℕ → Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.TraceEvent","l":132,"f":"Causalean/Mathlib/Analysis/CertifiedContourIntervalArithmetic/Complex/CircleSchedule.lean"},{"n":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.circleNode_endpoint","k":"theorem","m":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.CircleSchedule","a":"Mathlib/Analysis/CertifiedContourIntervalArithmetic/Complex","t":2,"d":"The terminal endpoint k = mesh, which is explicitly used by the trapezoidal program, is certified and denotes the same point as endpoint zero.","s":"∀ (radius : ℚ) (schedule : Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.Schedule), (Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.circleNode radius schedule schedule.mesh).Contains ↑radius ∧ Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.exactCircleNode radius schedule schedule.mesh = ↑radius","l":416,"f":"Causalean/Mathlib/Analysis/CertifiedContourIntervalArithmetic/Complex/CircleSchedule.lean"},{"n":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.circleNode_schedule_correspondence","k":"theorem","m":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.CircleSchedule","a":"Mathlib/Analysis/CertifiedContourIntervalArithmetic/Complex","t":2,"d":"Reading operation count, input precision, fuel, and mesh from the trace and from execution gives definitionally the same data because both retain the shared schedule object.","s":"∀ (radius : ℚ) (schedule : Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.Schedule) (k operation : ℕ), (Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.circleNodeEvent schedule k operation).schedule.operationCount = schedule.operationCount ∧ (Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.circleNodeEvent schedule k operation).schedule.inputPrecision = schedule.inputPrecision ∧ (Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.circleNodeEvent schedule k operation).schedule.fuel = schedule.fuel ∧ (Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.circleNodeEvent schedule k operation).schedule.mesh = schedule.mesh ∧ Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.circleNode radius (Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.circleNodeEvent schedule k operation).schedule k = Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.circleNode radius schedule k","l":455,"f":"Causalean/Mathlib/Analysis/CertifiedContourIntervalArithmetic/Complex/CircleSchedule.lean"},{"n":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.circleNode_sound","k":"theorem","m":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.CircleSchedule","a":"Mathlib/Analysis/CertifiedContourIntervalArithmetic/Complex","t":1,"d":"For a rational radius, a quadrature schedule, and a mesh index k, if k does not exceed the schedule's mesh size, then the computed rational circle-node rectangle at index k contains the exact complex circle point radius · exp(2πi · k / mesh).","s":"∀ (radius : ℚ) (schedule : Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.Schedule) {k : ℕ}, k ≤ schedule.mesh → (Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.circleNode radius schedule k).Contains (Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.exactCircleNode radius schedule k)","l":406,"f":"Causalean/Mathlib/Analysis/CertifiedContourIntervalArithmetic/Complex/CircleSchedule.lean"},{"n":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.circleNode_width","k":"theorem","m":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.CircleSchedule","a":"Mathlib/Analysis/CertifiedContourIntervalArithmetic/Complex","t":2,"d":"Circle-node width propagation exposes radius scaling and the scheduled complex-exponential approximation width.","s":"∀ (radius : ℚ), 0 ≤ radius → ∀ (schedule : Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.Schedule) (k : ℕ), (Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.circleNode radius schedule k).width ≤ radius * (Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.Transcendental.complexExp (Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.circleAngle schedule k) schedule.fuel).width","l":432,"f":"Causalean/Mathlib/Analysis/CertifiedContourIntervalArithmetic/Complex/CircleSchedule.lean"},{"n":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.circleNode_width_at_selected_precision","k":"theorem","m":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.CircleSchedule","a":"Mathlib/Analysis/CertifiedContourIntervalArithmetic/Complex","t":2,"d":"The explicit circle input precision and exponential fuel make every actually evaluated endpoint rectangle no wider than the requested target.","s":"∀ (radius : ℚ), 0 ≤ radius → ∀ (schedule : Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.Schedule) (target : Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.PosRat), schedule.inputPrecision = Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.circleInputPrecision radius target → Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.circleExpFuel radius schedule.mesh target ≤ schedule.fuel → ∀ {k : ℕ}, k ≤ schedule.mesh → (Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.circleNode radius schedule k).width ≤ ↑target","l":219,"f":"Causalean/Mathlib/Analysis/CertifiedContourIntervalArithmetic/Complex/CircleSchedule.lean"},{"n":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.denominator_ne_zero_at_nodes","k":"theorem","m":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.ContourEvaluation","a":"Mathlib/Analysis/CertifiedContourIntervalArithmetic/Complex","t":2,"d":"Denominator separation at all certified nodes implies nonvanishing of the exact denominator at all mesh endpoints.","s":"∀ (program : Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.ContourProgram) (schedule : Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.Schedule) (certificate : Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.DenominatorCertificate program schedule) {k : ℕ}, k ≤ schedule.mesh → program.denominator.value (Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.exactCircleNode program.radius schedule k) ≠ 0","l":300,"f":"Causalean/Mathlib/Analysis/CertifiedContourIntervalArithmetic/Complex/ContourEvaluation.lean"},{"n":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.evaluate_contains","k":"theorem","m":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.ContourEvaluation","a":"Mathlib/Analysis/CertifiedContourIntervalArithmetic/Complex","t":2,"d":"Under a Lipschitz bound and denominator nonvanishing on the full circle, the returned rational rectangle contains the normalized exact contour integral.","s":"∀ (program : Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.ContourProgram) (schedule : Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.Schedule) (certificate : Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.DenominatorCertificate program schedule), (∀ u ∈ Set.Icc 0 1, program.denominator.value (↑program.radius * Complex.exp (↑2 * ↑Real.pi * ↑u * Complex.I)) ≠ 0) → (∀ s ∈ Set.Icc 0 1, ∀ t ∈ Set.Icc 0 1, ‖program.normalizedIntegrand schedule s - program.normalizedIntegrand schedule t‖ ≤ ↑schedule.magnitude * |s - t|) → (program.evaluate schedule certificate).Contains program.normalizedContourIntegral","l":318,"f":"Causalean/Mathlib/Analysis/CertifiedContourIntervalArithmetic/Complex/ContourEvaluation.lean"},{"n":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.evaluate_width","k":"theorem","m":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.ContourEvaluation","a":"Mathlib/Analysis/CertifiedContourIntervalArithmetic/Complex","t":2,"d":"Uniform scheduled node widths propagate through mesh and final widening, so both result coordinates fit within the three split budgets.","s":"∀ (program : Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.ContourProgram) (bounds : Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.ContourValueBounds program) (separation : Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.PosRat) (scheduled : Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.CertifiedProgramSchedule program bounds separation) (certificate : Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.DenominatorCertificate program scheduled.schedule), ↑separation ≤ certificate.separation → (program.evaluate scheduled.schedule certificate).width ≤ ↑scheduled.schedule.tolerance","l":334,"f":"Causalean/Mathlib/Analysis/CertifiedContourIntervalArithmetic/Complex/ContourEvaluation.lean"},{"n":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.exactCircleNode","k":"def","m":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.CircleSchedule","a":"Mathlib/Analysis/CertifiedContourIntervalArithmetic/Complex","t":1,"d":"The exact semantic circle node corresponding to the scheduled rational endpoint.","s":"ℚ → Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.Schedule → ℕ → ℂ","l":383,"f":"Causalean/Mathlib/Analysis/CertifiedContourIntervalArithmetic/Complex/CircleSchedule.lean"},{"n":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.integrandNodes_sound","k":"theorem","m":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.ContourEvaluation","a":"Mathlib/Analysis/CertifiedContourIntervalArithmetic/Complex","t":2,"d":"Every certified finite endpoint rectangle contains the exact normalized integrand.","s":"∀ (program : Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.ContourProgram) (schedule : Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.Schedule) (certificate : Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.DenominatorCertificate program schedule) {k : ℕ}, k ≤ schedule.mesh → (program.integrandNodes schedule certificate k).Contains (program.normalizedIntegrand schedule (Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.CircleMesh.meshPoint schedule.mesh k))","l":53,"f":"Causalean/Mathlib/Analysis/CertifiedContourIntervalArithmetic/Complex/ContourEvaluation.lean"},{"n":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.integrandNodes_width_le_propagation","k":"theorem","m":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.ContourEvaluation","a":"Mathlib/Analysis/CertifiedContourIntervalArithmetic/Complex","t":2,"d":"The scheduled node-width bound follows compositionally from circle input precision, both map error moduli, their amplification fields, rational magnitude bounds, guarded-division separation, and primitive width lemmas.","s":"∀ (program : Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.ContourProgram) (bounds : Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.ContourValueBounds program) (separation : Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.PosRat) (scheduled : Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.CertifiedProgramSchedule program bounds separation) (certificate : Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.DenominatorCertificate program scheduled.schedule), ↑separation ≤ certificate.separation → ∀ {k : ℕ}, k ≤ scheduled.schedule.mesh → (program.integrandNodes scheduled.schedule certificate k).width ≤ program.nodePropagationBound bounds ↑separation ↑scheduled.target","l":69,"f":"Causalean/Mathlib/Analysis/CertifiedContourIntervalArithmetic/Complex/ContourEvaluation.lean"},{"n":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.inverseMaxTolerance","k":"def","m":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.ContourEvaluation","a":"Mathlib/Analysis/CertifiedContourIntervalArithmetic/Complex","t":1,"d":"The canonical statistical tolerance is the positive rational reciprocal of max n 1.","s":"ℕ → Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.PosRat","l":389,"f":"Causalean/Mathlib/Analysis/CertifiedContourIntervalArithmetic/Complex/ContourEvaluation.lean"},{"n":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.midpoint_error_le_half_width","k":"theorem","m":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.ContourEvaluation","a":"Mathlib/Analysis/CertifiedContourIntervalArithmetic/Complex","t":2,"d":"Containment in a rational real interval bounds its midpoint error by half the interval width, turning the reciprocal-max width into a statistic error bound.","s":"∀ {I : Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.RatInterval} {x : ℝ}, I.Contains x → |↑((I.lo + I.hi) / 2) - x| ≤ ↑I.width / 2","l":415,"f":"Causalean/Mathlib/Analysis/CertifiedContourIntervalArithmetic/Complex/ContourEvaluation.lean"},{"n":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.normalizedContourIntegral_eq","k":"theorem","m":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.ContourEvaluation","a":"Mathlib/Analysis/CertifiedContourIntervalArithmetic/Complex","t":2,"d":"The canceled unit-parameter integral equals the normalized exact circle contour integral whenever the denominator is nonzero along the circle.","s":"∀ (program : Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.ContourProgram) (schedule : Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.Schedule), (∀ u ∈ Set.Icc 0 1, program.denominator.value (↑program.radius * Complex.exp (↑2 * ↑Real.pi * ↑u * Complex.I)) ≠ 0) → program.normalizedContourIntegral = ∫ (u : ℝ) in 0..1, program.normalizedIntegrand schedule u","l":16,"f":"Causalean/Mathlib/Analysis/CertifiedContourIntervalArithmetic/Complex/ContourEvaluation.lean"},{"n":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.ComplexRatInterval","k":"structure","m":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Mesh","a":"Mathlib/Analysis/CertifiedContourIntervalArithmetic","t":1,"d":"A rational rectangle in the complex plane consists of a rational interval enclosing the real coordinate and a rational interval enclosing the imaginary coordinate.","s":"Type","l":19,"f":"Causalean/Mathlib/Analysis/CertifiedContourIntervalArithmetic/Mesh.lean"},{"n":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.ComplexRatInterval.Contains","k":"def","m":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Mesh","a":"Mathlib/Analysis/CertifiedContourIntervalArithmetic","t":1,"d":"A complex number belongs to a rational rectangle when both coordinates belong to their intervals.","s":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.ComplexRatInterval → ℂ → Prop","l":30,"f":"Causalean/Mathlib/Analysis/CertifiedContourIntervalArithmetic/Mesh.lean"},{"n":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.ComplexRatInterval.Subinterval","k":"def","m":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Mesh","a":"Mathlib/Analysis/CertifiedContourIntervalArithmetic","t":1,"d":"Coordinatewise inclusion is the refinement relation for complex rational rectangles.","s":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.ComplexRatInterval → Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.ComplexRatInterval → Prop","l":35,"f":"Causalean/Mathlib/Analysis/CertifiedContourIntervalArithmetic/Mesh.lean"},{"n":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.ComplexRatInterval.add","k":"def","m":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Mesh","a":"Mathlib/Analysis/CertifiedContourIntervalArithmetic","t":1,"d":"Complex rectangle addition is rational interval addition in both coordinates.","s":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.ComplexRatInterval → Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.ComplexRatInterval → Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.ComplexRatInterval","l":39,"f":"Causalean/Mathlib/Analysis/CertifiedContourIntervalArithmetic/Mesh.lean"},{"n":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.ComplexRatInterval.add_sound","k":"theorem","m":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Mesh","a":"Mathlib/Analysis/CertifiedContourIntervalArithmetic","t":2,"d":"Complex rectangle addition encloses sums of enclosed complex numbers.","s":"∀ {I J : Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.ComplexRatInterval} {z w : ℂ}, I.Contains z → J.Contains w → (I.add J).Contains (z + w)","l":54,"f":"Causalean/Mathlib/Analysis/CertifiedContourIntervalArithmetic/Mesh.lean"},{"n":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.ComplexRatInterval.conj","k":"def","m":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.Rectangles","a":"Mathlib/Analysis/CertifiedContourIntervalArithmetic/Complex","t":1,"d":"Complex conjugation negates only the imaginary coordinate interval.","s":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.ComplexRatInterval → Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.ComplexRatInterval","l":35,"f":"Causalean/Mathlib/Analysis/CertifiedContourIntervalArithmetic/Complex/Rectangles.lean"},{"n":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.ComplexRatInterval.div","k":"def","m":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.Rectangles","a":"Mathlib/Analysis/CertifiedContourIntervalArithmetic/Complex","t":1,"d":"Guarded complex division multiplies by the conjugate and divides both resulting coordinates by the denominator squared-modulus interval.","s":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.ComplexRatInterval → (J : Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.ComplexRatInterval) → J.normSq.AwayFromZero → Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.ComplexRatInterval","l":41,"f":"Causalean/Mathlib/Analysis/CertifiedContourIntervalArithmetic/Complex/Rectangles.lean"},{"n":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.ComplexRatInterval.div_maxAbs","k":"theorem","m":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.Rectangles","a":"Mathlib/Analysis/CertifiedContourIntervalArithmetic/Complex","t":2,"d":"If the squared-modulus denominator has positive lower bound δ, guarded division has maximum coordinate magnitude at most the numerator magnitude divided by δ.","s":"∀ {I J : Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.ComplexRatInterval} (hJ : J.normSq.AwayFromZero) {δ : ℚ}, 0 < δ → δ ≤ J.normSq.lo → (I.div J hJ).maxAbs ≤ (I.mul J.conj).maxAbs / δ","l":373,"f":"Causalean/Mathlib/Analysis/CertifiedContourIntervalArithmetic/Complex/Rectangles.lean"},{"n":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.ComplexRatInterval.div_sound","k":"theorem","m":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.Rectangles","a":"Mathlib/Analysis/CertifiedContourIntervalArithmetic/Complex","t":1,"d":"Guarded division of complex rational rectangles. For two complex rational rectangles enclosing complex values z and w respectively, provided the denominator rectangle's squared-modulus interval is bounded away from zero, the rectangle obtained by dividing the first rectangle by the second under that guard contains the quotient z / w .","s":"∀ {I J : Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.ComplexRatInterval} {z w : ℂ} (hJ : J.normSq.AwayFromZero), I.Contains z → J.Contains w → (I.div J hJ).Contains (z / w)","l":102,"f":"Causalean/Mathlib/Analysis/CertifiedContourIntervalArithmetic/Complex/Rectangles.lean"},{"n":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.ComplexRatInterval.div_width","k":"theorem","m":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.Rectangles","a":"Mathlib/Analysis/CertifiedContourIntervalArithmetic/Complex","t":2,"d":"If the squared-modulus denominator has certified lower bound δ, guarded division propagates widths with the usual inverse-square factor.","s":"∀ {I J : Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.ComplexRatInterval} (hJ : J.normSq.AwayFromZero) {δ : ℚ}, 0 < δ → δ ≤ J.normSq.lo → (I.div J hJ).width ≤ (I.mul J.conj).maxAbs * J.normSq.width / δ ^ 2 + (I.mul J.conj).width / δ","l":412,"f":"Causalean/Mathlib/Analysis/CertifiedContourIntervalArithmetic/Complex/Rectangles.lean"},{"n":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.ComplexRatInterval.expand","k":"def","m":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Mesh","a":"Mathlib/Analysis/CertifiedContourIntervalArithmetic","t":1,"d":"Coordinatewise widening by a nonnegative rational error produces a larger rectangle.","s":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.ComplexRatInterval → (e : ℚ) → 0 ≤ e → Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.ComplexRatInterval","l":47,"f":"Causalean/Mathlib/Analysis/CertifiedContourIntervalArithmetic/Mesh.lean"},{"n":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.ComplexRatInterval.expand_contains","k":"theorem","m":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Mesh","a":"Mathlib/Analysis/CertifiedContourIntervalArithmetic","t":2,"d":"Coordinatewise widening preserves containment.","s":"∀ {I : Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.ComplexRatInterval} {z : ℂ} {e : ℚ} (he : 0 ≤ e), I.Contains z → (I.expand e he).Contains z","l":66,"f":"Causalean/Mathlib/Analysis/CertifiedContourIntervalArithmetic/Mesh.lean"},{"n":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.ComplexRatInterval.ext","k":"theorem","m":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Mesh","a":"Mathlib/Analysis/CertifiedContourIntervalArithmetic","t":2,"d":"","s":"∀ {x y : Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.ComplexRatInterval}, x.re = y.re → x.im = y.im → x = y","l":21,"f":"Causalean/Mathlib/Analysis/CertifiedContourIntervalArithmetic/Mesh.lean"},{"n":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.ComplexRatInterval.ext_iff","k":"theorem","m":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Mesh","a":"Mathlib/Analysis/CertifiedContourIntervalArithmetic","t":2,"d":"","s":"∀ {x y : Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.ComplexRatInterval}, x = y ↔ x.re = y.re ∧ x.im = y.im","l":21,"f":"Causalean/Mathlib/Analysis/CertifiedContourIntervalArithmetic/Mesh.lean"},{"n":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.ComplexRatInterval.maxAbs","k":"def","m":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.Rectangles","a":"Mathlib/Analysis/CertifiedContourIntervalArithmetic/Complex","t":1,"d":"The maximum coordinate endpoint magnitude controls arithmetic error amplification.","s":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.ComplexRatInterval → ℚ","l":16,"f":"Causalean/Mathlib/Analysis/CertifiedContourIntervalArithmetic/Complex/Rectangles.lean"},{"n":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.ComplexRatInterval.maxAbs_conj","k":"theorem","m":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.Rectangles","a":"Mathlib/Analysis/CertifiedContourIntervalArithmetic/Complex","t":2,"d":"Conjugation preserves the maximum coordinate endpoint magnitude.","s":"∀ (I : Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.ComplexRatInterval), I.conj.maxAbs = I.maxAbs","l":128,"f":"Causalean/Mathlib/Analysis/CertifiedContourIntervalArithmetic/Complex/Rectangles.lean"},{"n":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.ComplexRatInterval.maxAbs_le_of_contains_width","k":"theorem","m":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.Rectangles","a":"Mathlib/Analysis/CertifiedContourIntervalArithmetic/Complex","t":2,"d":"Semantic coordinate bounds and rectangle width give an executable maximum endpoint-magnitude bound.","s":"∀ {I : Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.ComplexRatInterval} {z : ℂ} {C w : ℚ}, I.Contains z → max |z.re| |z.im| ≤ ↑C → I.width ≤ w → I.maxAbs ≤ C + w","l":59,"f":"Causalean/Mathlib/Analysis/CertifiedContourIntervalArithmetic/Complex/Rectangles.lean"},{"n":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.ComplexRatInterval.mul","k":"def","m":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.Rectangles","a":"Mathlib/Analysis/CertifiedContourIntervalArithmetic/Complex","t":1,"d":"Complex rectangle multiplication evaluates the Cartesian formula with outward rational real interval operations.","s":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.ComplexRatInterval → Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.ComplexRatInterval → Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.ComplexRatInterval","l":29,"f":"Causalean/Mathlib/Analysis/CertifiedContourIntervalArithmetic/Complex/Rectangles.lean"},{"n":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.ComplexRatInterval.mul_maxAbs","k":"theorem","m":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.Rectangles","a":"Mathlib/Analysis/CertifiedContourIntervalArithmetic/Complex","t":2,"d":"Complex rectangle multiplication has twice the product of operand coordinate magnitude bounds as an executable magnitude bound.","s":"∀ (I J : Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.ComplexRatInterval), (I.mul J).maxAbs ≤ 2 * I.maxAbs * J.maxAbs","l":148,"f":"Causalean/Mathlib/Analysis/CertifiedContourIntervalArithmetic/Complex/Rectangles.lean"},{"n":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.ComplexRatInterval.mul_sound","k":"theorem","m":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.Rectangles","a":"Mathlib/Analysis/CertifiedContourIntervalArithmetic/Complex","t":2,"d":"Complex rectangle multiplication encloses products of enclosed values.","s":"∀ {I J : Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.ComplexRatInterval} {z w : ℂ}, I.Contains z → J.Contains w → (I.mul J).Contains (z * w)","l":84,"f":"Causalean/Mathlib/Analysis/CertifiedContourIntervalArithmetic/Complex/Rectangles.lean"},{"n":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.ComplexRatInterval.mul_width","k":"theorem","m":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.Rectangles","a":"Mathlib/Analysis/CertifiedContourIntervalArithmetic/Complex","t":2,"d":"Each coordinate width after complex multiplication is bounded explicitly by the operand widths and endpoint magnitudes.","s":"∀ (I J : Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.ComplexRatInterval), (I.mul J).width ≤ 2 * (I.maxAbs * J.width + J.maxAbs * I.width)","l":309,"f":"Causalean/Mathlib/Analysis/CertifiedContourIntervalArithmetic/Complex/Rectangles.lean"},{"n":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.ComplexRatInterval.neg","k":"def","m":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.Rectangles","a":"Mathlib/Analysis/CertifiedContourIntervalArithmetic/Complex","t":1,"d":"Complex rectangle negation reverses both coordinate intervals.","s":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.ComplexRatInterval → Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.ComplexRatInterval","l":23,"f":"Causalean/Mathlib/Analysis/CertifiedContourIntervalArithmetic/Complex/Rectangles.lean"},{"n":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.ComplexRatInterval.neg_sound","k":"theorem","m":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.Rectangles","a":"Mathlib/Analysis/CertifiedContourIntervalArithmetic/Complex","t":2,"d":"Complex rectangle negation encloses negated complex values.","s":"∀ {I : Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.ComplexRatInterval} {z : ℂ}, I.Contains z → I.neg.Contains (-z)","l":73,"f":"Causalean/Mathlib/Analysis/CertifiedContourIntervalArithmetic/Complex/Rectangles.lean"},{"n":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.ComplexRatInterval.normInterval","k":"def","m":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.Rectangles","a":"Mathlib/Analysis/CertifiedContourIntervalArithmetic/Complex","t":1,"d":"Applying rational square-root bounds to the squared-modulus interval produces an executable enclosure of the complex norm.","s":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.ComplexRatInterval → ℕ → Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.RatInterval","l":475,"f":"Causalean/Mathlib/Analysis/CertifiedContourIntervalArithmetic/Complex/Rectangles.lean"},{"n":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.ComplexRatInterval.normInterval_sound","k":"theorem","m":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.Rectangles","a":"Mathlib/Analysis/CertifiedContourIntervalArithmetic/Complex","t":2,"d":"The executable modulus interval contains the norm of every enclosed complex value.","s":"∀ {I : Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.ComplexRatInterval} {z : ℂ}, I.Contains z → ∀ (fuel : ℕ), (I.normInterval fuel).Contains ‖z‖","l":480,"f":"Causalean/Mathlib/Analysis/CertifiedContourIntervalArithmetic/Complex/Rectangles.lean"},{"n":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.ComplexRatInterval.normSq","k":"def","m":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.Rectangles","a":"Mathlib/Analysis/CertifiedContourIntervalArithmetic/Complex","t":1,"d":"The squared-modulus interval sums the sign-aware squares of both coordinates.","s":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.ComplexRatInterval → Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.RatInterval","l":38,"f":"Causalean/Mathlib/Analysis/CertifiedContourIntervalArithmetic/Complex/Rectangles.lean"},{"n":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.ComplexRatInterval.normSq_lo_nonneg","k":"theorem","m":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.Rectangles","a":"Mathlib/Analysis/CertifiedContourIntervalArithmetic/Complex","t":2,"d":"Applying rational square-root bounds to the squared-modulus interval produces an executable enclosure of the complex norm.","s":"∀ (I : Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.ComplexRatInterval), 0 ≤ I.normSq.lo","l":466,"f":"Causalean/Mathlib/Analysis/CertifiedContourIntervalArithmetic/Complex/Rectangles.lean"},{"n":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.ComplexRatInterval.normSq_sound","k":"theorem","m":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.Rectangles","a":"Mathlib/Analysis/CertifiedContourIntervalArithmetic/Complex","t":2,"d":"The squared-modulus interval contains the squared complex norm.","s":"∀ {I : Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.ComplexRatInterval} {z : ℂ}, I.Contains z → I.normSq.Contains (‖z‖ ^ 2)","l":95,"f":"Causalean/Mathlib/Analysis/CertifiedContourIntervalArithmetic/Complex/Rectangles.lean"},{"n":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.ComplexRatInterval.normSq_width","k":"theorem","m":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.Rectangles","a":"Mathlib/Analysis/CertifiedContourIntervalArithmetic/Complex","t":2,"d":"The squared-modulus interval width is controlled by rectangle magnitude and rectangle width.","s":"∀ (I : Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.ComplexRatInterval), I.normSq.width ≤ 4 * I.maxAbs * I.width","l":356,"f":"Causalean/Mathlib/Analysis/CertifiedContourIntervalArithmetic/Complex/Rectangles.lean"},{"n":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.ComplexRatInterval.point","k":"def","m":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.Rectangles","a":"Mathlib/Analysis/CertifiedContourIntervalArithmetic/Complex","t":1,"d":"A rational pair determines a point rectangle in the complex plane.","s":"ℚ → ℚ → Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.ComplexRatInterval","l":19,"f":"Causalean/Mathlib/Analysis/CertifiedContourIntervalArithmetic/Complex/Rectangles.lean"},{"n":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.ComplexRatInterval.point_sound","k":"theorem","m":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.Rectangles","a":"Mathlib/Analysis/CertifiedContourIntervalArithmetic/Complex","t":2,"d":"The rational point rectangle contains the corresponding complex number.","s":"∀ (x y : ℚ), (Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.ComplexRatInterval.point x y).Contains (↑↑x + ↑↑y * Complex.I)","l":52,"f":"Causalean/Mathlib/Analysis/CertifiedContourIntervalArithmetic/Complex/Rectangles.lean"},{"n":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.ComplexRatInterval.rat_maxAbs_mono","k":"theorem","m":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.Rectangles","a":"Mathlib/Analysis/CertifiedContourIntervalArithmetic/Complex","t":2,"d":"Refining an interval cannot increase its maximum endpoint magnitude.","s":"∀ {K L : Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.RatInterval}, K.Subinterval L → K.maxAbs ≤ L.maxAbs","l":302,"f":"Causalean/Mathlib/Analysis/CertifiedContourIntervalArithmetic/Complex/Rectangles.lean"},{"n":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.ComplexRatInterval.rat_sq_hi_le","k":"theorem","m":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.Rectangles","a":"Mathlib/Analysis/CertifiedContourIntervalArithmetic/Complex","t":2,"d":"The upper endpoint of a squared interval is bounded by the square of its maximum endpoint magnitude.","s":"∀ (K : Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.RatInterval), K.sq.hi ≤ K.maxAbs ^ 2","l":287,"f":"Causalean/Mathlib/Analysis/CertifiedContourIntervalArithmetic/Complex/Rectangles.lean"},{"n":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.ComplexRatInterval.rat_sq_width_le","k":"theorem","m":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.Rectangles","a":"Mathlib/Analysis/CertifiedContourIntervalArithmetic/Complex","t":2,"d":"Squaring an interval has width at most twice its maximum endpoint magnitude times its width.","s":"∀ (K : Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.RatInterval), K.sq.width ≤ 2 * K.maxAbs * K.width","l":281,"f":"Causalean/Mathlib/Analysis/CertifiedContourIntervalArithmetic/Complex/Rectangles.lean"},{"n":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.ComplexRatInterval.smulRat","k":"def","m":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Mesh","a":"Mathlib/Analysis/CertifiedContourIntervalArithmetic","t":1,"d":"Multiplication by a rational scalar uses real interval multiplication in both coordinates.","s":"ℚ → Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.ComplexRatInterval → Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.ComplexRatInterval","l":43,"f":"Causalean/Mathlib/Analysis/CertifiedContourIntervalArithmetic/Mesh.lean"},{"n":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.ComplexRatInterval.smulRat_sound","k":"theorem","m":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Mesh","a":"Mathlib/Analysis/CertifiedContourIntervalArithmetic","t":2,"d":"Rational scalar multiplication encloses the corresponding complex scalar multiple.","s":"∀ {I : Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.ComplexRatInterval} {z : ℂ} (q : ℚ), I.Contains z → (Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.ComplexRatInterval.smulRat q I).Contains (↑q * z)","l":59,"f":"Causalean/Mathlib/Analysis/CertifiedContourIntervalArithmetic/Mesh.lean"},{"n":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.ComplexRatInterval.sub","k":"def","m":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.Rectangles","a":"Mathlib/Analysis/CertifiedContourIntervalArithmetic/Complex","t":1,"d":"Complex rectangle subtraction adds the coordinatewise negation.","s":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.ComplexRatInterval → Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.ComplexRatInterval → Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.ComplexRatInterval","l":26,"f":"Causalean/Mathlib/Analysis/CertifiedContourIntervalArithmetic/Complex/Rectangles.lean"},{"n":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.ComplexRatInterval.sub_sound","k":"theorem","m":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.Rectangles","a":"Mathlib/Analysis/CertifiedContourIntervalArithmetic/Complex","t":2,"d":"Complex rectangle subtraction encloses differences of enclosed values.","s":"∀ {I J : Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.ComplexRatInterval} {z w : ℂ}, I.Contains z → J.Contains w → (I.sub J).Contains (z - w)","l":79,"f":"Causalean/Mathlib/Analysis/CertifiedContourIntervalArithmetic/Complex/Rectangles.lean"},{"n":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.ComplexRatInterval.tighten","k":"def","m":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.Rectangles","a":"Mathlib/Analysis/CertifiedContourIntervalArithmetic/Complex","t":1,"d":"Coordinatewise conditional intersection makes a rectangle no larger while retaining a common enclosed complex value.","s":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.ComplexRatInterval → Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.ComplexRatInterval → Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.ComplexRatInterval","l":47,"f":"Causalean/Mathlib/Analysis/CertifiedContourIntervalArithmetic/Complex/Rectangles.lean"},{"n":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.ComplexRatInterval.tighten_sound_left","k":"theorem","m":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.Rectangles","a":"Mathlib/Analysis/CertifiedContourIntervalArithmetic/Complex","t":2,"d":"Tightening two rectangles containing a common value preserves that value and produces a subrectangle of the first argument.","s":"∀ {I J : Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.ComplexRatInterval} {z : ℂ}, I.Contains z → J.Contains z → (I.tighten J).Contains z ∧ (I.tighten J).Subinterval I","l":132,"f":"Causalean/Mathlib/Analysis/CertifiedContourIntervalArithmetic/Complex/Rectangles.lean"},{"n":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.ComplexRatInterval.width","k":"def","m":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.Rectangles","a":"Mathlib/Analysis/CertifiedContourIntervalArithmetic/Complex","t":1,"d":"The width of a complex rectangle is the larger of its two coordinate widths.","s":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.ComplexRatInterval → ℚ","l":13,"f":"Causalean/Mathlib/Analysis/CertifiedContourIntervalArithmetic/Complex/Rectangles.lean"},{"n":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.ComplexRatInterval.width_add","k":"theorem","m":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.Rectangles","a":"Mathlib/Analysis/CertifiedContourIntervalArithmetic/Complex","t":2,"d":"Complex rectangle addition has exactly the maximum of the two summed coordinate widths.","s":"∀ (I J : Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.ComplexRatInterval), (I.add J).width = max (I.re.width + J.re.width) (I.im.width + J.im.width)","l":142,"f":"Causalean/Mathlib/Analysis/CertifiedContourIntervalArithmetic/Complex/Rectangles.lean"},{"n":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.ComplexRatInterval.width_conj","k":"theorem","m":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.Rectangles","a":"Mathlib/Analysis/CertifiedContourIntervalArithmetic/Complex","t":2,"d":"Conjugation preserves rectangle width.","s":"∀ (I : Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.ComplexRatInterval), I.conj.width = I.width","l":124,"f":"Causalean/Mathlib/Analysis/CertifiedContourIntervalArithmetic/Complex/Rectangles.lean"},{"n":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.ComplexRatInterval.zero","k":"def","m":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Mesh","a":"Mathlib/Analysis/CertifiedContourIntervalArithmetic","t":1,"d":"The point rectangle at complex zero is the neutral enclosure for finite recursive sums.","s":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.ComplexRatInterval","l":51,"f":"Causalean/Mathlib/Analysis/CertifiedContourIntervalArithmetic/Mesh.lean"},{"n":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.FiniteSearch.finiteRefine","k":"def","m":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.FiniteSearch","a":"Mathlib/Analysis/CertifiedContourIntervalArithmetic","t":1,"d":"Finite refinement returns the first candidate interval meeting a rational width tolerance, with a total fallback.","s":"{n : ℕ} → (Fin (n + 1) → Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.RatInterval) → Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.PosRat → Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.RatInterval","l":84,"f":"Causalean/Mathlib/Analysis/CertifiedContourIntervalArithmetic/FiniteSearch.lean"},{"n":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.FiniteSearch.finiteRefine_width","k":"theorem","m":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.FiniteSearch","a":"Mathlib/Analysis/CertifiedContourIntervalArithmetic","t":2,"d":"If a candidate meets the requested width, finite refinement returns the least such candidate and meets the width.","s":"∀ {n : ℕ} {candidates : Fin (n + 1) → Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.RatInterval} (ε : Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.PosRat), (∃ i, (candidates i).width ≤ ↑ε) → (Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.FiniteSearch.finiteRefine candidates ε).width ≤ ↑ε","l":90,"f":"Causalean/Mathlib/Analysis/CertifiedContourIntervalArithmetic/FiniteSearch.lean"},{"n":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.FiniteSearch.instMeasurableSpaceComplexRatInterval","k":"instance","m":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.FiniteSearch","a":"Mathlib/Analysis/CertifiedContourIntervalArithmetic","t":2,"d":"Complex rational rectangles carry the countable discrete measurable space used for certified outputs.","s":"MeasurableSpace Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.ComplexRatInterval","l":38,"f":"Causalean/Mathlib/Analysis/CertifiedContourIntervalArithmetic/FiniteSearch.lean"},{"n":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.FiniteSearch.instMeasurableSpaceRatInterval","k":"instance","m":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.FiniteSearch","a":"Mathlib/Analysis/CertifiedContourIntervalArithmetic","t":2,"d":"Rational intervals carry the countable discrete measurable space used for certified outputs.","s":"MeasurableSpace Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.RatInterval","l":35,"f":"Causalean/Mathlib/Analysis/CertifiedContourIntervalArithmetic/FiniteSearch.lean"},{"n":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.FiniteSearch.leastScoreIndex","k":"def","m":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.FiniteSearch","a":"Mathlib/Analysis/CertifiedContourIntervalArithmetic","t":1,"d":"The least-score index minimizes lexicographically by real score and then by the original finite index.","s":"{n : ℕ} → (Fin (n + 1) → ℝ) → Fin (n + 1)","l":110,"f":"Causalean/Mathlib/Analysis/CertifiedContourIntervalArithmetic/FiniteSearch.lean"},{"n":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.FiniteSearch.leastScoreIndex_minimal","k":"theorem","m":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.FiniteSearch","a":"Mathlib/Analysis/CertifiedContourIntervalArithmetic","t":2,"d":"The selected least-score index has score no larger than every competing score.","s":"∀ {n : ℕ} (score : Fin (n + 1) → ℝ) (i : Fin (n + 1)), score (Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.FiniteSearch.leastScoreIndex score) ≤ score i","l":121,"f":"Causalean/Mathlib/Analysis/CertifiedContourIntervalArithmetic/FiniteSearch.lean"},{"n":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.FiniteSearch.leastScoreIndex_tie","k":"theorem","m":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.FiniteSearch","a":"Mathlib/Analysis/CertifiedContourIntervalArithmetic","t":2,"d":"Equal minimal scores are resolved in favor of the smaller original index.","s":"∀ {n : ℕ} (score : Fin (n + 1) → ℝ) {i : Fin (n + 1)}, score i = score (Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.FiniteSearch.leastScoreIndex score) → Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.FiniteSearch.leastScoreIndex score ≤ i","l":146,"f":"Causalean/Mathlib/Analysis/CertifiedContourIntervalArithmetic/FiniteSearch.lean"},{"n":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.FiniteSearch.leastTrue","k":"def","m":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.FiniteSearch","a":"Mathlib/Analysis/CertifiedContourIntervalArithmetic","t":1,"d":"The least successful index of a nonempty finite Boolean family is returned, or zero if none succeeds.","s":"{n : ℕ} → (Fin (n + 1) → Bool) → Fin (n + 1)","l":42,"f":"Causalean/Mathlib/Analysis/CertifiedContourIntervalArithmetic/FiniteSearch.lean"},{"n":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.FiniteSearch.leastTrue_accepts","k":"theorem","m":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.FiniteSearch","a":"Mathlib/Analysis/CertifiedContourIntervalArithmetic","t":2,"d":"When at least one Boolean test succeeds, the returned index succeeds.","s":"∀ {n : ℕ} {accept : Fin (n + 1) → Bool}, (∃ i, accept i = true) → accept (Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.FiniteSearch.leastTrue accept) = true","l":49,"f":"Causalean/Mathlib/Analysis/CertifiedContourIntervalArithmetic/FiniteSearch.lean"},{"n":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.FiniteSearch.leastTrue_eq_zero","k":"theorem","m":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.FiniteSearch","a":"Mathlib/Analysis/CertifiedContourIntervalArithmetic","t":2,"d":"If no Boolean test succeeds, the total search returns index zero.","s":"∀ {n : ℕ} {accept : Fin (n + 1) → Bool}, (∀ (i : Fin (n + 1)), accept i = false) → Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.FiniteSearch.leastTrue accept = 0","l":73,"f":"Causalean/Mathlib/Analysis/CertifiedContourIntervalArithmetic/FiniteSearch.lean"},{"n":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.FiniteSearch.leastTrue_le","k":"theorem","m":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.FiniteSearch","a":"Mathlib/Analysis/CertifiedContourIntervalArithmetic","t":2,"d":"Every successful index is no smaller than the successful index returned by the search.","s":"∀ {n : ℕ} {accept : Fin (n + 1) → Bool} {i : Fin (n + 1)}, accept i = true → Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.FiniteSearch.leastTrue accept ≤ i","l":62,"f":"Causalean/Mathlib/Analysis/CertifiedContourIntervalArithmetic/FiniteSearch.lean"},{"n":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.FiniteSearch.measurable_finiteRefine","k":"theorem","m":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.FiniteSearch","a":"Mathlib/Analysis/CertifiedContourIntervalArithmetic","t":2,"d":"Finite refinement is measurable as a function of a finite vector of rational interval candidates.","s":"∀ {n : ℕ} (ε : Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.PosRat), Measurable fun candidates => Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.FiniteSearch.finiteRefine candidates ε","l":103,"f":"Causalean/Mathlib/Analysis/CertifiedContourIntervalArithmetic/FiniteSearch.lean"},{"n":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.FiniteSearch.measurable_infEnclosure","k":"theorem","m":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.FiniteSearch","a":"Mathlib/Analysis/CertifiedContourIntervalArithmetic","t":2,"d":"The rational infimum enclosure is measurable as a function of its finite rational node vector.","s":"∀ {n : ℕ} (hn : 0 < n) (L : ℚ) (hL : 0 ≤ L), Measurable fun nodes => Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.CircleMesh.infEnclosure (fun k => if hk : k < n + 1 then nodes ⟨k, hk⟩ else nodes 0) L hL n hn","l":235,"f":"Causalean/Mathlib/Analysis/CertifiedContourIntervalArithmetic/FiniteSearch.lean"},{"n":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.FiniteSearch.measurable_integralEnclosure","k":"theorem","m":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.FiniteSearch","a":"Mathlib/Analysis/CertifiedContourIntervalArithmetic","t":2,"d":"The rational complex integral enclosure is measurable as a function of its finite node rectangles.","s":"∀ {n : ℕ} (hn : 0 < n) (L : ℚ) (hL : 0 ≤ L), Measurable fun nodes => Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.CircleMesh.integralEnclosure (fun k => if hk : k < n + 1 then nodes ⟨k, hk⟩ else nodes 0) L hL n hn","l":253,"f":"Causalean/Mathlib/Analysis/CertifiedContourIntervalArithmetic/FiniteSearch.lean"},{"n":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.FiniteSearch.measurable_leastScoreIndex","k":"theorem","m":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.FiniteSearch","a":"Mathlib/Analysis/CertifiedContourIntervalArithmetic","t":1,"d":"For a sample-size index n, the function that selects, from a vector of n+1 real scores, the smallest index attaining the least score is measurable with respect to the Borel σ-algebra on the score vector space and the discrete σ-algebra on the finite index set.","s":"∀ {n : ℕ}, Measurable Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.FiniteSearch.leastScoreIndex","l":173,"f":"Causalean/Mathlib/Analysis/CertifiedContourIntervalArithmetic/FiniteSearch.lean"},{"n":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.FiniteSearch.measurable_leastTrue","k":"theorem","m":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.FiniteSearch","a":"Mathlib/Analysis/CertifiedContourIntervalArithmetic","t":2,"d":"A finite least-success search is measurable when every individual success event is measurable.","s":"∀ {Ω : Type u_1} [inst : MeasurableSpace Ω] {n : ℕ} (accept : Fin (n + 1) → Ω → Bool), (∀ (i : Fin (n + 1)), Measurable (accept i)) → Measurable fun ω => Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.FiniteSearch.leastTrue fun i => accept i ω","l":224,"f":"Causalean/Mathlib/Analysis/CertifiedContourIntervalArithmetic/FiniteSearch.lean"},{"n":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.FiniteSearch.measurable_supEnclosure","k":"theorem","m":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.FiniteSearch","a":"Mathlib/Analysis/CertifiedContourIntervalArithmetic","t":2,"d":"The rational supremum enclosure is measurable as a function of its finite rational node vector.","s":"∀ {n : ℕ} (hn : 0 < n) (L : ℚ) (hL : 0 ≤ L), Measurable fun nodes => Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.CircleMesh.supEnclosure (fun k => if hk : k < n + 1 then nodes ⟨k, hk⟩ else nodes 0) L hL n hn","l":244,"f":"Causalean/Mathlib/Analysis/CertifiedContourIntervalArithmetic/FiniteSearch.lean"},{"n":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.PosRat","k":"def","m":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.CertifiedReal","a":"Mathlib/Analysis/CertifiedContourIntervalArithmetic","t":1,"d":"A positive rational is a rational number bundled with a proof that it is strictly positive.","s":"Type","l":15,"f":"Causalean/Mathlib/Analysis/CertifiedContourIntervalArithmetic/CertifiedReal.lean"},{"n":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.RatInterval","k":"structure","m":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Basic","a":"Mathlib/Analysis/CertifiedContourIntervalArithmetic","t":1,"d":"A rational closed interval consists of a rational lower endpoint and a rational upper endpoint, with the lower endpoint no greater than the upper endpoint.","s":"Type","l":18,"f":"Causalean/Mathlib/Analysis/CertifiedContourIntervalArithmetic/Basic.lean"},{"n":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.RatInterval.AwayFromZero","k":"def","m":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Basic","a":"Mathlib/Analysis/CertifiedContourIntervalArithmetic","t":1,"d":"An interval is separated from zero when it lies strictly on one side of zero.","s":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.RatInterval → Prop","l":77,"f":"Causalean/Mathlib/Analysis/CertifiedContourIntervalArithmetic/Basic.lean"},{"n":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.RatInterval.Contains","k":"def","m":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Basic","a":"Mathlib/Analysis/CertifiedContourIntervalArithmetic","t":1,"d":"A real number is contained in a rational interval when it lies between the real casts of its endpoints.","s":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.RatInterval → ℝ → Prop","l":32,"f":"Causalean/Mathlib/Analysis/CertifiedContourIntervalArithmetic/Basic.lean"},{"n":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.RatInterval.Contains.mono","k":"theorem","m":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Basic","a":"Mathlib/Analysis/CertifiedContourIntervalArithmetic","t":2,"d":"A real point contained in a subinterval is contained in the enclosing interval.","s":"∀ {I J : Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.RatInterval} {x : ℝ}, I.Subinterval J → I.Contains x → J.Contains x","l":129,"f":"Causalean/Mathlib/Analysis/CertifiedContourIntervalArithmetic/Basic.lean"},{"n":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.RatInterval.Subinterval","k":"def","m":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Basic","a":"Mathlib/Analysis/CertifiedContourIntervalArithmetic","t":1,"d":"Interval inclusion means that every point in the first enclosure also lies in the second.","s":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.RatInterval → Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.RatInterval → Prop","l":44,"f":"Causalean/Mathlib/Analysis/CertifiedContourIntervalArithmetic/Basic.lean"},{"n":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.RatInterval.add","k":"def","m":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Basic","a":"Mathlib/Analysis/CertifiedContourIntervalArithmetic","t":1,"d":"Addition evaluates all possible sums by adding the two lower and two upper endpoints.","s":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.RatInterval → Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.RatInterval → Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.RatInterval","l":56,"f":"Causalean/Mathlib/Analysis/CertifiedContourIntervalArithmetic/Basic.lean"},{"n":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.RatInterval.add_mono","k":"theorem","m":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Basic","a":"Mathlib/Analysis/CertifiedContourIntervalArithmetic","t":2,"d":"Addition is inclusion-isotone in both interval arguments.","s":"∀ {I I' J J' : Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.RatInterval}, I.Subinterval I' → J.Subinterval J' → (I.add J).Subinterval (I'.add J')","l":309,"f":"Causalean/Mathlib/Analysis/CertifiedContourIntervalArithmetic/Basic.lean"},{"n":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.RatInterval.add_sound","k":"theorem","m":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Basic","a":"Mathlib/Analysis/CertifiedContourIntervalArithmetic","t":2,"d":"Rational interval addition encloses the sum of any enclosed real operands.","s":"∀ {I J : Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.RatInterval} {x y : ℝ}, I.Contains x → J.Contains y → (I.add J).Contains (x + y)","l":145,"f":"Causalean/Mathlib/Analysis/CertifiedContourIntervalArithmetic/Basic.lean"},{"n":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.RatInterval.contains_point_iff","k":"theorem","m":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Basic","a":"Mathlib/Analysis/CertifiedContourIntervalArithmetic","t":2,"d":"Containment in a point interval is equality with that rational point.","s":"∀ (q : ℚ) (x : ℝ), (Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.RatInterval.point q).Contains x ↔ x = ↑q","l":110,"f":"Causalean/Mathlib/Analysis/CertifiedContourIntervalArithmetic/Basic.lean"},{"n":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.RatInterval.div","k":"def","m":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Basic","a":"Mathlib/Analysis/CertifiedContourIntervalArithmetic","t":1,"d":"Division multiplies by the reciprocal of a denominator interval certified away from zero.","s":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.RatInterval → (J : Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.RatInterval) → J.AwayFromZero → Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.RatInterval","l":87,"f":"Causalean/Mathlib/Analysis/CertifiedContourIntervalArithmetic/Basic.lean"},{"n":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.RatInterval.div_mono","k":"theorem","m":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Basic","a":"Mathlib/Analysis/CertifiedContourIntervalArithmetic","t":2,"d":"Division is inclusion-isotone when both denominator intervals avoid zero.","s":"∀ {I I' J J' : Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.RatInterval}, I.Subinterval I' → J.Subinterval J' → ∀ (hJ0 : J.AwayFromZero) (hJ0' : J'.AwayFromZero), (I.div J hJ0).Subinterval (I'.div J' hJ0')","l":386,"f":"Causalean/Mathlib/Analysis/CertifiedContourIntervalArithmetic/Basic.lean"},{"n":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.RatInterval.div_sound","k":"theorem","m":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Basic","a":"Mathlib/Analysis/CertifiedContourIntervalArithmetic","t":2,"d":"Rational interval division encloses every quotient whose denominator interval avoids zero.","s":"∀ {I J : Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.RatInterval} {x y : ℝ} (hJ : J.AwayFromZero), I.Contains x → J.Contains y → (I.div J hJ).Contains (x / y)","l":268,"f":"Causalean/Mathlib/Analysis/CertifiedContourIntervalArithmetic/Basic.lean"},{"n":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.RatInterval.expand","k":"def","m":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Basic","a":"Mathlib/Analysis/CertifiedContourIntervalArithmetic","t":1,"d":"Widening by a nonnegative rational amount subtracts it below and adds it above.","s":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.RatInterval → (e : ℚ) → 0 ≤ e → Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.RatInterval","l":102,"f":"Causalean/Mathlib/Analysis/CertifiedContourIntervalArithmetic/Basic.lean"},{"n":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.RatInterval.ext","k":"theorem","m":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Basic","a":"Mathlib/Analysis/CertifiedContourIntervalArithmetic","t":2,"d":"","s":"∀ {x y : Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.RatInterval}, x.lo = y.lo → x.hi = y.hi → x = y","l":21,"f":"Causalean/Mathlib/Analysis/CertifiedContourIntervalArithmetic/Basic.lean"},{"n":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.RatInterval.ext_iff","k":"theorem","m":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Basic","a":"Mathlib/Analysis/CertifiedContourIntervalArithmetic","t":2,"d":"","s":"∀ {x y : Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.RatInterval}, x = y ↔ x.lo = y.lo ∧ x.hi = y.hi","l":21,"f":"Causalean/Mathlib/Analysis/CertifiedContourIntervalArithmetic/Basic.lean"},{"n":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.RatInterval.hull","k":"def","m":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Basic","a":"Mathlib/Analysis/CertifiedContourIntervalArithmetic","t":1,"d":"The convex hull of two rational intervals is their least endpoint-wise hull.","s":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.RatInterval → Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.RatInterval → Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.RatInterval","l":51,"f":"Causalean/Mathlib/Analysis/CertifiedContourIntervalArithmetic/Basic.lean"},{"n":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.RatInterval.inv","k":"def","m":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Basic","a":"Mathlib/Analysis/CertifiedContourIntervalArithmetic","t":1,"d":"Reciprocal reverses the endpoints of an interval certified away from zero.","s":"(I : Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.RatInterval) → I.AwayFromZero → Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.RatInterval","l":80,"f":"Causalean/Mathlib/Analysis/CertifiedContourIntervalArithmetic/Basic.lean"},{"n":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.RatInterval.inv_mono","k":"theorem","m":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Basic","a":"Mathlib/Analysis/CertifiedContourIntervalArithmetic","t":2,"d":"Reciprocal preserves inclusion when both intervals are certified away from zero.","s":"∀ {I J : Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.RatInterval}, I.Subinterval J → ∀ (hI : I.AwayFromZero) (hJ : J.AwayFromZero), (I.inv hI).Subinterval (J.inv hJ)","l":370,"f":"Causalean/Mathlib/Analysis/CertifiedContourIntervalArithmetic/Basic.lean"},{"n":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.RatInterval.inv_sound","k":"theorem","m":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Basic","a":"Mathlib/Analysis/CertifiedContourIntervalArithmetic","t":2,"d":"Rational interval reciprocal encloses the reciprocal of every enclosed real operand.","s":"∀ {I : Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.RatInterval} {x : ℝ} (hI : I.AwayFromZero), I.Contains x → (I.inv hI).Contains x⁻¹","l":251,"f":"Causalean/Mathlib/Analysis/CertifiedContourIntervalArithmetic/Basic.lean"},{"n":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.RatInterval.maxAbs","k":"def","m":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Basic","a":"Mathlib/Analysis/CertifiedContourIntervalArithmetic","t":1,"d":"The maximum endpoint magnitude is a rational bound for the absolute value of every real number in the interval.","s":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.RatInterval → ℚ","l":40,"f":"Causalean/Mathlib/Analysis/CertifiedContourIntervalArithmetic/Basic.lean"},{"n":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.RatInterval.maxAbs_add","k":"theorem","m":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.Basic","a":"Mathlib/Analysis/CertifiedContourIntervalArithmetic/Complex","t":2,"d":"Addition increases endpoint magnitude by at most the sum of operand bounds.","s":"∀ (I J : Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.RatInterval), (I.add J).maxAbs ≤ I.maxAbs + J.maxAbs","l":244,"f":"Causalean/Mathlib/Analysis/CertifiedContourIntervalArithmetic/Complex/Basic.lean"},{"n":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.RatInterval.maxAbs_le_of_contains_width","k":"theorem","m":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.Basic","a":"Mathlib/Analysis/CertifiedContourIntervalArithmetic/Complex","t":2,"d":"Containment of one real value together with interval width controls both endpoint magnitudes.","s":"∀ {I : Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.RatInterval} {x : ℝ} {C w : ℚ}, I.Contains x → |x| ≤ ↑C → I.width ≤ w → I.maxAbs ≤ C + w","l":213,"f":"Causalean/Mathlib/Analysis/CertifiedContourIntervalArithmetic/Complex/Basic.lean"},{"n":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.RatInterval.maxAbs_mul","k":"theorem","m":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.Basic","a":"Mathlib/Analysis/CertifiedContourIntervalArithmetic/Complex","t":2,"d":"Interval multiplication has the product of operand endpoint magnitudes as an executable magnitude bound.","s":"∀ (I J : Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.RatInterval), (I.mul J).maxAbs ≤ I.maxAbs * J.maxAbs","l":260,"f":"Causalean/Mathlib/Analysis/CertifiedContourIntervalArithmetic/Complex/Basic.lean"},{"n":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.RatInterval.maxAbs_neg","k":"theorem","m":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.Basic","a":"Mathlib/Analysis/CertifiedContourIntervalArithmetic/Complex","t":2,"d":"Negation preserves the maximum endpoint magnitude.","s":"∀ (I : Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.RatInterval), I.neg.maxAbs = I.maxAbs","l":240,"f":"Causalean/Mathlib/Analysis/CertifiedContourIntervalArithmetic/Complex/Basic.lean"},{"n":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.RatInterval.maxAbs_sub","k":"theorem","m":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.Basic","a":"Mathlib/Analysis/CertifiedContourIntervalArithmetic/Complex","t":2,"d":"Subtraction increases endpoint magnitude by at most the sum of operand bounds.","s":"∀ (I J : Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.RatInterval), (I.sub J).maxAbs ≤ I.maxAbs + J.maxAbs","l":255,"f":"Causalean/Mathlib/Analysis/CertifiedContourIntervalArithmetic/Complex/Basic.lean"},{"n":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.RatInterval.mul","k":"def","m":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Basic","a":"Mathlib/Analysis/CertifiedContourIntervalArithmetic","t":1,"d":"Multiplication takes the minimum and maximum of the four endpoint products.","s":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.RatInterval → Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.RatInterval → Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.RatInterval","l":67,"f":"Causalean/Mathlib/Analysis/CertifiedContourIntervalArithmetic/Basic.lean"},{"n":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.RatInterval.mul_mono","k":"theorem","m":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Basic","a":"Mathlib/Analysis/CertifiedContourIntervalArithmetic","t":2,"d":"Multiplication is inclusion-isotone in both interval arguments.","s":"∀ {I I' J J' : Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.RatInterval}, I.Subinterval I' → J.Subinterval J' → (I.mul J).Subinterval (I'.mul J')","l":326,"f":"Causalean/Mathlib/Analysis/CertifiedContourIntervalArithmetic/Basic.lean"},{"n":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.RatInterval.mul_sound","k":"theorem","m":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Basic","a":"Mathlib/Analysis/CertifiedContourIntervalArithmetic","t":1,"d":"For rational intervals I and J, if x is a real number contained in I and y is a real number contained in J, then the interval-multiplication enclosure mul I J contains the real product x * y.","s":"∀ {I J : Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.RatInterval} {x y : ℝ}, I.Contains x → J.Contains y → (I.mul J).Contains (x * y)","l":164,"f":"Causalean/Mathlib/Analysis/CertifiedContourIntervalArithmetic/Basic.lean"},{"n":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.RatInterval.ne_zero_of_contains","k":"theorem","m":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Basic","a":"Mathlib/Analysis/CertifiedContourIntervalArithmetic","t":2,"d":"A real number contained in an interval separated from zero is nonzero.","s":"∀ {I : Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.RatInterval} {x : ℝ}, I.AwayFromZero → I.Contains x → x ≠ 0","l":242,"f":"Causalean/Mathlib/Analysis/CertifiedContourIntervalArithmetic/Basic.lean"},{"n":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.RatInterval.neg","k":"def","m":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Basic","a":"Mathlib/Analysis/CertifiedContourIntervalArithmetic","t":1,"d":"Negation reverses and negates the endpoints.","s":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.RatInterval → Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.RatInterval","l":60,"f":"Causalean/Mathlib/Analysis/CertifiedContourIntervalArithmetic/Basic.lean"},{"n":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.RatInterval.neg_mono","k":"theorem","m":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Basic","a":"Mathlib/Analysis/CertifiedContourIntervalArithmetic","t":2,"d":"Negation preserves interval inclusion.","s":"∀ {I J : Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.RatInterval}, I.Subinterval J → I.neg.Subinterval J.neg","l":315,"f":"Causalean/Mathlib/Analysis/CertifiedContourIntervalArithmetic/Basic.lean"},{"n":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.RatInterval.neg_sound","k":"theorem","m":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Basic","a":"Mathlib/Analysis/CertifiedContourIntervalArithmetic","t":2,"d":"Rational interval negation encloses the negation of every enclosed real operand.","s":"∀ {I : Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.RatInterval} {x : ℝ}, I.Contains x → I.neg.Contains (-x)","l":152,"f":"Causalean/Mathlib/Analysis/CertifiedContourIntervalArithmetic/Basic.lean"},{"n":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.RatInterval.npow","k":"def","m":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Basic","a":"Mathlib/Analysis/CertifiedContourIntervalArithmetic","t":1,"d":"Natural powers are evaluated by repeated rational interval multiplication.","s":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.RatInterval → ℕ → Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.RatInterval","l":90,"f":"Causalean/Mathlib/Analysis/CertifiedContourIntervalArithmetic/Basic.lean"},{"n":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.RatInterval.npow_mono","k":"theorem","m":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Basic","a":"Mathlib/Analysis/CertifiedContourIntervalArithmetic","t":2,"d":"Natural interval powers preserve interval inclusion.","s":"∀ {I J : Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.RatInterval}, I.Subinterval J → ∀ (n : ℕ), (I.npow n).Subinterval (J.npow n)","l":363,"f":"Causalean/Mathlib/Analysis/CertifiedContourIntervalArithmetic/Basic.lean"},{"n":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.RatInterval.npow_sound","k":"theorem","m":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Basic","a":"Mathlib/Analysis/CertifiedContourIntervalArithmetic","t":2,"d":"Repeated interval multiplication encloses every natural power of an enclosed real number.","s":"∀ {I : Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.RatInterval} {x : ℝ}, I.Contains x → ∀ (n : ℕ), (I.npow n).Contains (x ^ n)","l":273,"f":"Causalean/Mathlib/Analysis/CertifiedContourIntervalArithmetic/Basic.lean"},{"n":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.RatInterval.point","k":"def","m":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Basic","a":"Mathlib/Analysis/CertifiedContourIntervalArithmetic","t":1,"d":"A point interval contains exactly one rational value.","s":"ℚ → Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.RatInterval","l":48,"f":"Causalean/Mathlib/Analysis/CertifiedContourIntervalArithmetic/Basic.lean"},{"n":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.RatInterval.point_sound","k":"theorem","m":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Basic","a":"Mathlib/Analysis/CertifiedContourIntervalArithmetic","t":2,"d":"The point interval soundly encloses its rational value viewed as a real.","s":"∀ (q : ℚ), (Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.RatInterval.point q).Contains ↑q","l":141,"f":"Causalean/Mathlib/Analysis/CertifiedContourIntervalArithmetic/Basic.lean"},{"n":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.RatInterval.sq","k":"def","m":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.Basic","a":"Mathlib/Analysis/CertifiedContourIntervalArithmetic/Complex","t":1,"d":"Squaring a rational interval uses zero as the lower bound exactly when the interval crosses zero and the larger squared endpoint as the upper bound.","s":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.RatInterval → Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.RatInterval","l":16,"f":"Causalean/Mathlib/Analysis/CertifiedContourIntervalArithmetic/Complex/Basic.lean"},{"n":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.RatInterval.sq_mono","k":"theorem","m":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.Basic","a":"Mathlib/Analysis/CertifiedContourIntervalArithmetic/Complex","t":2,"d":"Squaring is inclusion-isotone for rational intervals.","s":"∀ {I J : Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.RatInterval}, I.Subinterval J → I.sq.Subinterval J.sq","l":49,"f":"Causalean/Mathlib/Analysis/CertifiedContourIntervalArithmetic/Complex/Basic.lean"},{"n":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.RatInterval.sq_sound","k":"theorem","m":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.Basic","a":"Mathlib/Analysis/CertifiedContourIntervalArithmetic/Complex","t":2,"d":"The sign-aware rational square enclosure contains the square of every real number contained in the input interval.","s":"∀ {I : Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.RatInterval} {x : ℝ}, I.Contains x → I.sq.Contains (x ^ 2)","l":26,"f":"Causalean/Mathlib/Analysis/CertifiedContourIntervalArithmetic/Complex/Basic.lean"},{"n":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.RatInterval.sqrtInterval","k":"def","m":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.Basic","a":"Mathlib/Analysis/CertifiedContourIntervalArithmetic/Complex","t":1,"d":"A nonnegative rational interval is mapped to a rational square-root interval by applying lower and upper Newton bounds to its endpoints.","s":"(I : Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.RatInterval) → 0 ≤ I.lo → ℕ → Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.RatInterval","l":183,"f":"Causalean/Mathlib/Analysis/CertifiedContourIntervalArithmetic/Complex/Basic.lean"},{"n":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.RatInterval.sqrtInterval_sound","k":"theorem","m":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.Basic","a":"Mathlib/Analysis/CertifiedContourIntervalArithmetic/Complex","t":2,"d":"The executable square-root interval encloses the square root of every nonnegative real contained in its input.","s":"∀ {I : Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.RatInterval} (hI : 0 ≤ I.lo) {x : ℝ}, I.Contains x → ∀ (fuel : ℕ), (I.sqrtInterval hI fuel).Contains √x","l":194,"f":"Causalean/Mathlib/Analysis/CertifiedContourIntervalArithmetic/Complex/Basic.lean"},{"n":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.RatInterval.sqrtInterval_width","k":"theorem","m":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.Basic","a":"Mathlib/Analysis/CertifiedContourIntervalArithmetic/Complex","t":2,"d":"Square-root interval width is exactly the endpoint Newton gap, exposing both input-diameter and iteration-error contributions.","s":"∀ (I : Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.RatInterval) (hI : 0 ≤ I.lo) (fuel : ℕ), (I.sqrtInterval hI fuel).width = Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.RatInterval.sqrtUpper I.hi fuel - Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.RatInterval.sqrtLower I.lo fuel","l":206,"f":"Causalean/Mathlib/Analysis/CertifiedContourIntervalArithmetic/Complex/Basic.lean"},{"n":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.RatInterval.sqrtLower","k":"def","m":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.Basic","a":"Mathlib/Analysis/CertifiedContourIntervalArithmetic/Complex","t":1,"d":"Dividing a nonnegative rational by its Newton upper iterate gives a matching executable lower square-root bound.","s":"ℚ → ℕ → ℚ","l":85,"f":"Causalean/Mathlib/Analysis/CertifiedContourIntervalArithmetic/Complex/Basic.lean"},{"n":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.RatInterval.sqrtUpper","k":"def","m":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.Basic","a":"Mathlib/Analysis/CertifiedContourIntervalArithmetic/Complex","t":1,"d":"The Newton upper iterate is an executable rational approximation to the square root of a nonnegative rational.","s":"ℚ → ℕ → ℚ","l":77,"f":"Causalean/Mathlib/Analysis/CertifiedContourIntervalArithmetic/Complex/Basic.lean"},{"n":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.RatInterval.sqrt_gap_rate","k":"theorem","m":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.Basic","a":"Mathlib/Analysis/CertifiedContourIntervalArithmetic/Complex","t":2,"d":"The Newton upper-minus-lower gap is bounded by an explicit reciprocal fuel rate.","s":"∀ (q : ℚ), 0 ≤ q → ∀ (n : ℕ), Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.RatInterval.sqrtUpper q n - Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.RatInterval.sqrtLower q n ≤ (q + 1) / (↑n + 1)","l":141,"f":"Causalean/Mathlib/Analysis/CertifiedContourIntervalArithmetic/Complex/Basic.lean"},{"n":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.RatInterval.sqrt_iterates_converge","k":"theorem","m":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.Basic","a":"Mathlib/Analysis/CertifiedContourIntervalArithmetic/Complex","t":2,"d":"The gap between the Newton lower and upper square-root bounds converges effectively to zero.","s":"∀ (q : ℚ), 0 ≤ q → ∀ (ε : Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.PosRat), ∃ n, Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.RatInterval.sqrtUpper q n - Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.RatInterval.sqrtLower q n ≤ ↑ε","l":171,"f":"Causalean/Mathlib/Analysis/CertifiedContourIntervalArithmetic/Complex/Basic.lean"},{"n":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.RatInterval.sqrt_iterates_sound","k":"theorem","m":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.Basic","a":"Mathlib/Analysis/CertifiedContourIntervalArithmetic/Complex","t":1,"d":"For a nonnegative rational number q and any fuel count n, the Newton-iterate rational lower bound sqrtLower q n and upper bound sqrtUpper q n both enclose the real square root of q: sqrtLower q n ≤ √q ≤ sqrtUpper q n.","s":"∀ (q : ℚ), 0 ≤ q → ∀ (n : ℕ), ↑(Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.RatInterval.sqrtLower q n) ≤ √↑q ∧ √↑q ≤ ↑(Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.RatInterval.sqrtUpper q n)","l":106,"f":"Causalean/Mathlib/Analysis/CertifiedContourIntervalArithmetic/Complex/Basic.lean"},{"n":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.RatInterval.sub","k":"def","m":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Basic","a":"Mathlib/Analysis/CertifiedContourIntervalArithmetic","t":1,"d":"Subtraction is addition with the negated second interval.","s":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.RatInterval → Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.RatInterval → Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.RatInterval","l":64,"f":"Causalean/Mathlib/Analysis/CertifiedContourIntervalArithmetic/Basic.lean"},{"n":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.RatInterval.sub_mono","k":"theorem","m":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Basic","a":"Mathlib/Analysis/CertifiedContourIntervalArithmetic","t":2,"d":"Subtraction is inclusion-isotone in both interval arguments.","s":"∀ {I I' J J' : Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.RatInterval}, I.Subinterval I' → J.Subinterval J' → (I.sub J).Subinterval (I'.sub J')","l":320,"f":"Causalean/Mathlib/Analysis/CertifiedContourIntervalArithmetic/Basic.lean"},{"n":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.RatInterval.sub_sound","k":"theorem","m":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Basic","a":"Mathlib/Analysis/CertifiedContourIntervalArithmetic","t":2,"d":"Rational interval subtraction encloses the difference of any enclosed real operands.","s":"∀ {I J : Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.RatInterval} {x y : ℝ}, I.Contains x → J.Contains y → (I.sub J).Contains (x - y)","l":159,"f":"Causalean/Mathlib/Analysis/CertifiedContourIntervalArithmetic/Basic.lean"},{"n":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.RatInterval.subinterval_refl","k":"theorem","m":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Basic","a":"Mathlib/Analysis/CertifiedContourIntervalArithmetic","t":2,"d":"Subinterval is reflexive.","s":"∀ (I : Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.RatInterval), I.Subinterval I","l":119,"f":"Causalean/Mathlib/Analysis/CertifiedContourIntervalArithmetic/Basic.lean"},{"n":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.RatInterval.subinterval_trans","k":"theorem","m":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Basic","a":"Mathlib/Analysis/CertifiedContourIntervalArithmetic","t":2,"d":"Subinterval is transitive.","s":"∀ {I J K : Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.RatInterval}, I.Subinterval J → J.Subinterval K → I.Subinterval K","l":123,"f":"Causalean/Mathlib/Analysis/CertifiedContourIntervalArithmetic/Basic.lean"},{"n":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.RatInterval.tighten","k":"def","m":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Basic","a":"Mathlib/Analysis/CertifiedContourIntervalArithmetic","t":1,"d":"Conditional intersection keeps the exact intersection when its rational endpoints overlap, and otherwise returns the first interval.","s":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.RatInterval → Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.RatInterval → Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.RatInterval","l":95,"f":"Causalean/Mathlib/Analysis/CertifiedContourIntervalArithmetic/Basic.lean"},{"n":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.RatInterval.tighten_sound","k":"theorem","m":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Basic","a":"Mathlib/Analysis/CertifiedContourIntervalArithmetic","t":2,"d":"Tightening two intervals that share an enclosed real point returns their intersection and continues to enclose that point.","s":"∀ {I J : Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.RatInterval} {x : ℝ}, I.Contains x → J.Contains x → (I.tighten J).Contains x","l":280,"f":"Causalean/Mathlib/Analysis/CertifiedContourIntervalArithmetic/Basic.lean"},{"n":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.RatInterval.tighten_subinterval_left","k":"theorem","m":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Basic","a":"Mathlib/Analysis/CertifiedContourIntervalArithmetic","t":2,"d":"Tightening by another sound interval produces a subinterval of the first interval.","s":"∀ {I J : Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.RatInterval} {x : ℝ}, I.Contains x → J.Contains x → (I.tighten J).Subinterval I","l":295,"f":"Causalean/Mathlib/Analysis/CertifiedContourIntervalArithmetic/Basic.lean"},{"n":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.RatInterval.width","k":"def","m":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Basic","a":"Mathlib/Analysis/CertifiedContourIntervalArithmetic","t":1,"d":"The width of a rational interval is its upper endpoint minus its lower endpoint.","s":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.RatInterval → ℚ","l":37,"f":"Causalean/Mathlib/Analysis/CertifiedContourIntervalArithmetic/Basic.lean"},{"n":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.RatInterval.width_add","k":"theorem","m":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Basic","a":"Mathlib/Analysis/CertifiedContourIntervalArithmetic","t":2,"d":"Addition makes widths add exactly.","s":"∀ (I J : Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.RatInterval), (I.add J).width = I.width + J.width","l":393,"f":"Causalean/Mathlib/Analysis/CertifiedContourIntervalArithmetic/Basic.lean"},{"n":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.RatInterval.width_expand","k":"theorem","m":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Basic","a":"Mathlib/Analysis/CertifiedContourIntervalArithmetic","t":2,"d":"Widening an interval by a nonnegative amount on each side increases its width by twice that amount.","s":"∀ (I : Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.RatInterval) (e : ℚ) (he : 0 ≤ e), (I.expand e he).width = I.width + 2 * e","l":407,"f":"Causalean/Mathlib/Analysis/CertifiedContourIntervalArithmetic/Basic.lean"},{"n":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.RatInterval.width_mono","k":"theorem","m":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Basic","a":"Mathlib/Analysis/CertifiedContourIntervalArithmetic","t":2,"d":"Inclusion of rational intervals cannot increase their width.","s":"∀ {I J : Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.RatInterval}, I.Subinterval J → I.width ≤ J.width","l":136,"f":"Causalean/Mathlib/Analysis/CertifiedContourIntervalArithmetic/Basic.lean"},{"n":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.RatInterval.width_neg","k":"theorem","m":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Basic","a":"Mathlib/Analysis/CertifiedContourIntervalArithmetic","t":2,"d":"Negation preserves interval width exactly.","s":"∀ (I : Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.RatInterval), I.neg.width = I.width","l":398,"f":"Causalean/Mathlib/Analysis/CertifiedContourIntervalArithmetic/Basic.lean"},{"n":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.RatInterval.width_nonneg","k":"theorem","m":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Basic","a":"Mathlib/Analysis/CertifiedContourIntervalArithmetic","t":2,"d":"Every rational interval has nonnegative width.","s":"∀ (I : Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.RatInterval), 0 ≤ I.width","l":106,"f":"Causalean/Mathlib/Analysis/CertifiedContourIntervalArithmetic/Basic.lean"},{"n":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.RatInterval.width_sub","k":"theorem","m":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Basic","a":"Mathlib/Analysis/CertifiedContourIntervalArithmetic","t":2,"d":"Subtraction makes widths add exactly.","s":"∀ (I J : Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.RatInterval), (I.sub J).width = I.width + J.width","l":403,"f":"Causalean/Mathlib/Analysis/CertifiedContourIntervalArithmetic/Basic.lean"},{"n":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Transcendental.abs_expReduced_le_one","k":"theorem","m":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Exponential","a":"Mathlib/Analysis/CertifiedContourIntervalArithmetic","t":2,"d":"Range reduction puts the exponential Taylor argument in the closed unit interval.","s":"∀ (q : ℚ), |Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Transcendental.expReduced q| ≤ 1","l":64,"f":"Causalean/Mathlib/Analysis/CertifiedContourIntervalArithmetic/Exponential.lean"},{"n":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Transcendental.abs_logCoordinate_lt_one","k":"theorem","m":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Logarithm","a":"Mathlib/Analysis/CertifiedContourIntervalArithmetic","t":2,"d":"A positive rational logarithm argument has atanh coordinate of absolute value strictly below one.","s":"∀ (q : ℚ), 0 < q → |Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Transcendental.logCoordinate q| < 1","l":131,"f":"Causalean/Mathlib/Analysis/CertifiedContourIntervalArithmetic/Logarithm.lean"},{"n":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Transcendental.expInterval","k":"def","m":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Operations","a":"Mathlib/Analysis/CertifiedContourIntervalArithmetic","t":1,"d":"Exponential interval evaluation encloses the exponential of every real number contained in its input interval.","s":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.RatInterval → ℕ → Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.RatInterval","l":31,"f":"Causalean/Mathlib/Analysis/CertifiedContourIntervalArithmetic/Operations.lean"},{"n":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Transcendental.expInterval_nested","k":"theorem","m":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Operations","a":"Mathlib/Analysis/CertifiedContourIntervalArithmetic","t":2,"d":"Exponential interval evaluation is nested in its precision argument.","s":"∀ (I : Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.RatInterval) (n : ℕ), (Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Transcendental.expInterval I (n + 1)).Subinterval (Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Transcendental.expInterval I n)","l":115,"f":"Causalean/Mathlib/Analysis/CertifiedContourIntervalArithmetic/Operations.lean"},{"n":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Transcendental.expInterval_sound","k":"theorem","m":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Operations","a":"Mathlib/Analysis/CertifiedContourIntervalArithmetic","t":2,"d":"Exponential interval evaluation encloses the exponential of every enclosed real input.","s":"∀ {I : Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.RatInterval} {x : ℝ}, I.Contains x → ∀ (n : ℕ), (Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Transcendental.expInterval I n).Contains (Real.exp x)","l":74,"f":"Causalean/Mathlib/Analysis/CertifiedContourIntervalArithmetic/Operations.lean"},{"n":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Transcendental.expName","k":"def","m":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Exponential","a":"Mathlib/Analysis/CertifiedContourIntervalArithmetic","t":1,"d":"A rational exponential is a certified real with a fully rational endpoint algorithm.","s":"ℚ → Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.CertifiedReal","l":484,"f":"Causalean/Mathlib/Analysis/CertifiedContourIntervalArithmetic/Exponential.lean"},{"n":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Transcendental.expPartial","k":"def","m":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Exponential","a":"Mathlib/Analysis/CertifiedContourIntervalArithmetic","t":1,"d":"The rational Taylor polynomial for the exponential retains all terms through the requested degree.","s":"ℚ → ℕ → ℚ","l":28,"f":"Causalean/Mathlib/Analysis/CertifiedContourIntervalArithmetic/Exponential.lean"},{"n":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Transcendental.expPrecision","k":"def","m":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Exponential","a":"Mathlib/Analysis/CertifiedContourIntervalArithmetic","t":1,"d":"The explicit exponential precision is a natural function of the rational input and target width.","s":"ℚ → Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.PosRat → ℕ","l":59,"f":"Causalean/Mathlib/Analysis/CertifiedContourIntervalArithmetic/Exponential.lean"},{"n":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Transcendental.expRaw","k":"def","m":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Exponential","a":"Mathlib/Analysis/CertifiedContourIntervalArithmetic","t":1,"d":"The raw exponential enclosure raises the reduced-argument enclosure to the exact scaling power.","s":"ℚ → ℕ → Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.RatInterval","l":49,"f":"Causalean/Mathlib/Analysis/CertifiedContourIntervalArithmetic/Exponential.lean"},{"n":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Transcendental.expRaw_sound","k":"theorem","m":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Exponential","a":"Mathlib/Analysis/CertifiedContourIntervalArithmetic","t":2,"d":"The raw rational exponential interval encloses the real exponential of the rational input.","s":"∀ (q : ℚ) (n : ℕ), (Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Transcendental.expRaw q n).Contains (Real.exp ↑q)","l":86,"f":"Causalean/Mathlib/Analysis/CertifiedContourIntervalArithmetic/Exponential.lean"},{"n":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Transcendental.expReduced","k":"def","m":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Exponential","a":"Mathlib/Analysis/CertifiedContourIntervalArithmetic","t":1,"d":"The reduced exponential argument divides by its positive natural scaling factor.","s":"ℚ → ℚ","l":25,"f":"Causalean/Mathlib/Analysis/CertifiedContourIntervalArithmetic/Exponential.lean"},{"n":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Transcendental.expReducedRaw","k":"def","m":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Exponential","a":"Mathlib/Analysis/CertifiedContourIntervalArithmetic","t":1,"d":"A reduced rational argument is enclosed by its Taylor polynomial plus or minus the rational remainder radius.","s":"ℚ → ℕ → Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.RatInterval","l":39,"f":"Causalean/Mathlib/Analysis/CertifiedContourIntervalArithmetic/Exponential.lean"},{"n":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Transcendental.expRemainder","k":"def","m":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Exponential","a":"Mathlib/Analysis/CertifiedContourIntervalArithmetic","t":1,"d":"The rational remainder radius is the explicit exponential-series bound after the retained Taylor terms.","s":"ℚ → ℕ → ℚ","l":33,"f":"Causalean/Mathlib/Analysis/CertifiedContourIntervalArithmetic/Exponential.lean"},{"n":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Transcendental.expScalar","k":"def","m":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Exponential","a":"Mathlib/Analysis/CertifiedContourIntervalArithmetic","t":1,"d":"Successive exponential enclosures are intersected to make the returned sequence nested.","s":"ℚ → ℕ → Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.RatInterval","l":54,"f":"Causalean/Mathlib/Analysis/CertifiedContourIntervalArithmetic/Exponential.lean"},{"n":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Transcendental.expScalar_nested","k":"theorem","m":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Exponential","a":"Mathlib/Analysis/CertifiedContourIntervalArithmetic","t":2,"d":"Scalar exponential enclosures are nested as precision increases.","s":"∀ (q : ℚ) (n : ℕ), (Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Transcendental.expScalar q (n + 1)).Subinterval (Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Transcendental.expScalar q n)","l":127,"f":"Causalean/Mathlib/Analysis/CertifiedContourIntervalArithmetic/Exponential.lean"},{"n":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Transcendental.expScalar_sound","k":"theorem","m":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Exponential","a":"Mathlib/Analysis/CertifiedContourIntervalArithmetic","t":2,"d":"Every tightened scalar exponential interval encloses the real exponential.","s":"∀ (q : ℚ) (n : ℕ), (Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Transcendental.expScalar q n).Contains (Real.exp ↑q)","l":119,"f":"Causalean/Mathlib/Analysis/CertifiedContourIntervalArithmetic/Exponential.lean"},{"n":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Transcendental.expScalar_width","k":"theorem","m":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Exponential","a":"Mathlib/Analysis/CertifiedContourIntervalArithmetic","t":1,"d":"For a rational input q and a requested positive rational tolerance ε, evaluating the certified scalar exponential at the precision level selected for that tolerance yields an enclosure whose width is no larger than ε.","s":"∀ (q : ℚ) (ε : Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.PosRat), (Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Transcendental.expScalar q (Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Transcendental.expPrecision q ε)).width ≤ ↑ε","l":338,"f":"Causalean/Mathlib/Analysis/CertifiedContourIntervalArithmetic/Exponential.lean"},{"n":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Transcendental.expScale","k":"def","m":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Exponential","a":"Mathlib/Analysis/CertifiedContourIntervalArithmetic","t":1,"d":"The integer scaling factor reduces a rational exponential argument to absolute value at most one.","s":"ℚ → ℕ","l":21,"f":"Causalean/Mathlib/Analysis/CertifiedContourIntervalArithmetic/Exponential.lean"},{"n":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Transcendental.inv_den_le_of_pos","k":"theorem","m":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Exponential","a":"Mathlib/Analysis/CertifiedContourIntervalArithmetic","t":2,"d":"The reciprocal of a positive rational's denominator is no larger than the rational itself.","s":"∀ (u : ℚ), 0 < u → 1 / ↑u.den ≤ u","l":321,"f":"Causalean/Mathlib/Analysis/CertifiedContourIntervalArithmetic/Exponential.lean"},{"n":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Transcendental.logCoordinate","k":"def","m":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Logarithm","a":"Mathlib/Analysis/CertifiedContourIntervalArithmetic","t":1,"d":"The atanh coordinate sends every positive rational logarithm argument into the open interval from minus one to one.","s":"ℚ → ℚ","l":19,"f":"Causalean/Mathlib/Analysis/CertifiedContourIntervalArithmetic/Logarithm.lean"},{"n":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Transcendental.logInterval","k":"def","m":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Operations","a":"Mathlib/Analysis/CertifiedContourIntervalArithmetic","t":1,"d":"Positive logarithm interval evaluation encloses the logarithm of every real number contained in its certified positive input interval.","s":"(I : Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.RatInterval) → 0 < I.lo → ℕ → Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.RatInterval","l":49,"f":"Causalean/Mathlib/Analysis/CertifiedContourIntervalArithmetic/Operations.lean"},{"n":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Transcendental.logInterval_nested","k":"theorem","m":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Operations","a":"Mathlib/Analysis/CertifiedContourIntervalArithmetic","t":2,"d":"Positive logarithm interval evaluation is nested in its precision argument.","s":"∀ (I : Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.RatInterval) (hI : 0 < I.lo) (n : ℕ), (Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Transcendental.logInterval I hI (n + 1)).Subinterval (Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Transcendental.logInterval I hI n)","l":123,"f":"Causalean/Mathlib/Analysis/CertifiedContourIntervalArithmetic/Operations.lean"},{"n":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Transcendental.logInterval_sound","k":"theorem","m":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Operations","a":"Mathlib/Analysis/CertifiedContourIntervalArithmetic","t":2,"d":"Positive logarithm interval evaluation encloses the logarithm of every enclosed real input.","s":"∀ {I : Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.RatInterval} {x : ℝ} (hI : 0 < I.lo), I.Contains x → ∀ (n : ℕ), (Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Transcendental.logInterval I hI n).Contains (Real.log x)","l":81,"f":"Causalean/Mathlib/Analysis/CertifiedContourIntervalArithmetic/Operations.lean"},{"n":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Transcendental.logName","k":"def","m":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Logarithm","a":"Mathlib/Analysis/CertifiedContourIntervalArithmetic","t":1,"d":"The logarithm of a positive rational is a certified real with fully rational endpoints.","s":"(q : ℚ) → 0 < q → Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.CertifiedReal","l":293,"f":"Causalean/Mathlib/Analysis/CertifiedContourIntervalArithmetic/Logarithm.lean"},{"n":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Transcendental.logPartial","k":"def","m":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Logarithm","a":"Mathlib/Analysis/CertifiedContourIntervalArithmetic","t":1,"d":"The rational logarithm polynomial retains the first requested number of terms of twice the atanh series.","s":"ℚ → ℕ → ℚ","l":23,"f":"Causalean/Mathlib/Analysis/CertifiedContourIntervalArithmetic/Logarithm.lean"},{"n":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Transcendental.logPrecision","k":"def","m":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Logarithm","a":"Mathlib/Analysis/CertifiedContourIntervalArithmetic","t":1,"d":"The explicit logarithm precision is a natural function of the input numerator, denominator, and target width.","s":"ℚ → Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.PosRat → ℕ","l":126,"f":"Causalean/Mathlib/Analysis/CertifiedContourIntervalArithmetic/Logarithm.lean"},{"n":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Transcendental.logRaw","k":"def","m":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Logarithm","a":"Mathlib/Analysis/CertifiedContourIntervalArithmetic","t":1,"d":"A positive rational logarithm argument is enclosed by its atanh polynomial plus or minus its tail bound.","s":"(q : ℚ) → 0 < q → ℕ → Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.RatInterval","l":107,"f":"Causalean/Mathlib/Analysis/CertifiedContourIntervalArithmetic/Logarithm.lean"},{"n":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Transcendental.logRaw_sound","k":"theorem","m":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Logarithm","a":"Mathlib/Analysis/CertifiedContourIntervalArithmetic","t":2,"d":"The raw rational logarithm interval encloses the real logarithm of its positive rational input.","s":"∀ (q : ℚ) (hq : 0 < q) (n : ℕ), (Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Transcendental.logRaw q hq n).Contains (Real.log ↑q)","l":174,"f":"Causalean/Mathlib/Analysis/CertifiedContourIntervalArithmetic/Logarithm.lean"},{"n":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Transcendental.logRemainder","k":"def","m":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Logarithm","a":"Mathlib/Analysis/CertifiedContourIntervalArithmetic","t":1,"d":"The rational logarithm remainder bounds the tail of the atanh series geometrically.","s":"ℚ → ℕ → ℚ","l":29,"f":"Causalean/Mathlib/Analysis/CertifiedContourIntervalArithmetic/Logarithm.lean"},{"n":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Transcendental.logScalar","k":"def","m":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Logarithm","a":"Mathlib/Analysis/CertifiedContourIntervalArithmetic","t":1,"d":"Successive positive-logarithm enclosures are intersected to make the sequence nested.","s":"(q : ℚ) → 0 < q → ℕ → Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.RatInterval","l":121,"f":"Causalean/Mathlib/Analysis/CertifiedContourIntervalArithmetic/Logarithm.lean"},{"n":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Transcendental.logScalar_nested","k":"theorem","m":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Logarithm","a":"Mathlib/Analysis/CertifiedContourIntervalArithmetic","t":2,"d":"Scalar logarithm enclosures are nested as precision increases.","s":"∀ (q : ℚ) (hq : 0 < q) (n : ℕ), (Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Transcendental.logScalar q hq (n + 1)).Subinterval (Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Transcendental.logScalar q hq n)","l":218,"f":"Causalean/Mathlib/Analysis/CertifiedContourIntervalArithmetic/Logarithm.lean"},{"n":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Transcendental.logScalar_sound","k":"theorem","m":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Logarithm","a":"Mathlib/Analysis/CertifiedContourIntervalArithmetic","t":2,"d":"Every tightened scalar logarithm interval encloses the real logarithm.","s":"∀ (q : ℚ) (hq : 0 < q) (n : ℕ), (Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Transcendental.logScalar q hq n).Contains (Real.log ↑q)","l":210,"f":"Causalean/Mathlib/Analysis/CertifiedContourIntervalArithmetic/Logarithm.lean"},{"n":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Transcendental.logScalar_width","k":"theorem","m":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Logarithm","a":"Mathlib/Analysis/CertifiedContourIntervalArithmetic","t":1,"d":"Precision-driven logarithm enclosure width. For a positive rational number and a requested positive rational tolerance, evaluating the scalar logarithm enclosure at the precision level determined from those two inputs produces an interval no wider than the requested tolerance.","s":"∀ (q : ℚ) (hq : 0 < q) (ε : Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.PosRat), (Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Transcendental.logScalar q hq (Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Transcendental.logPrecision q ε)).width ≤ ↑ε","l":232,"f":"Causalean/Mathlib/Analysis/CertifiedContourIntervalArithmetic/Logarithm.lean"},{"n":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Transcendental.rpowInterval","k":"def","m":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Operations","a":"Mathlib/Analysis/CertifiedContourIntervalArithmetic","t":1,"d":"Real-power interval evaluation tightens all raw evaluations up to the requested precision.","s":"(base : Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.RatInterval) → Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.RatInterval → 0 < base.lo → ℕ → Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.RatInterval","l":61,"f":"Causalean/Mathlib/Analysis/CertifiedContourIntervalArithmetic/Operations.lean"},{"n":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Transcendental.rpowInterval_nested","k":"theorem","m":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Operations","a":"Mathlib/Analysis/CertifiedContourIntervalArithmetic","t":2,"d":"Real-power interval evaluation is nested in its precision argument.","s":"∀ (base exponent : Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.RatInterval) (hbase : 0 < base.lo) (n : ℕ), (Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Transcendental.rpowInterval base exponent hbase (n + 1)).Subinterval (Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Transcendental.rpowInterval base exponent hbase n)","l":132,"f":"Causalean/Mathlib/Analysis/CertifiedContourIntervalArithmetic/Operations.lean"},{"n":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Transcendental.rpowInterval_sound","k":"theorem","m":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Operations","a":"Mathlib/Analysis/CertifiedContourIntervalArithmetic","t":1,"d":"Certified real-power interval evaluation. For a base rational interval whose lower endpoint is strictly positive that encloses a real base value, and an exponent rational interval enclosing a real exponent value, the real-power interval evaluation of the base and exponent, at any precision level, contains the true value of the base raised to that exponent.","s":"∀ {base exponent : Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.RatInterval} {x y : ℝ} (hbase : 0 < base.lo), base.Contains x → exponent.Contains y → ∀ (n : ℕ), (Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Transcendental.rpowInterval base exponent hbase n).Contains (x ^ y)","l":103,"f":"Causalean/Mathlib/Analysis/CertifiedContourIntervalArithmetic/Operations.lean"},{"n":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Transcendental.rpowRaw","k":"def","m":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Operations","a":"Mathlib/Analysis/CertifiedContourIntervalArithmetic","t":1,"d":"Raw real-power evaluation on a strictly positive base composes logarithm, multiplication, and exponential interval evaluation.","s":"(base : Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.RatInterval) → Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.RatInterval → 0 < base.lo → ℕ → Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.RatInterval","l":56,"f":"Causalean/Mathlib/Analysis/CertifiedContourIntervalArithmetic/Operations.lean"},{"n":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Transcendental.tighten_subinterval_right","k":"theorem","m":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Exponential","a":"Mathlib/Analysis/CertifiedContourIntervalArithmetic","t":2,"d":"Tightening two enclosures of the same real value is contained in either input enclosure.","s":"∀ {I J : Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.RatInterval} {x : ℝ}, I.Contains x → J.Contains x → (I.tighten J).Subinterval J","l":133,"f":"Causalean/Mathlib/Analysis/CertifiedContourIntervalArithmetic/Exponential.lean"},{"n":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.certifiedIntervalArithmetic","k":"def","m":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.API","a":"Mathlib/Analysis/CertifiedContourIntervalArithmetic","t":1,"d":"The concrete certified interval arithmetic implementation uses exact rational primitives, rational Taylor/atanh enclosures, and stored effective moduli for certified real refinement.","s":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.CertifiedIntervalArithmetic","l":63,"f":"Causalean/Mathlib/Analysis/CertifiedContourIntervalArithmetic/API.lean"},{"n":"Causalean.Mathlib.Analysis.EhlichZellerMesh.abs_czTrig_le_czSup","k":"theorem","m":"Causalean.Mathlib.Analysis.EhlichZellerMesh.Bernstein","a":"Mathlib/Analysis/EhlichZellerMesh","t":2,"d":"Every trigonometric value is bounded by the sup-norm: |czTrig R t| ≤ czSup R for all t ∈ [0, π].","s":"∀ (R : Polynomial ℝ) {t : ℝ}, t ∈ Set.Icc 0 Real.pi → |Causalean.Mathlib.Analysis.EhlichZellerMesh.czTrig R t| ≤ Causalean.Mathlib.Analysis.EhlichZellerMesh.czSup R","l":81,"f":"Causalean/Mathlib/Analysis/EhlichZellerMesh/Bernstein.lean"},{"n":"Causalean.Mathlib.Analysis.EhlichZellerMesh.czMeshLHS_le_czSup","k":"theorem","m":"Causalean.Mathlib.Analysis.EhlichZellerMesh.Mesh","a":"Mathlib/Analysis/EhlichZellerMesh","t":2,"d":"Sup transfer. The [-1, 1] sup-norm of R equals the sup-norm of its trigonometric transform on [0, π]: since t ↦ -cos t maps [0, π] *onto* [-1, 1], sup_{x∈[-1,1]} |R x| ≤ czSup R (in fact with equality).","s":"∀ (R : Polynomial ℝ), sSup ((fun x => |Polynomial.eval x R|) '' Set.Icc (-1) 1) ≤ Causalean.Mathlib.Analysis.EhlichZellerMesh.czSup R","l":117,"f":"Causalean/Mathlib/Analysis/EhlichZellerMesh/Mesh.lean"},{"n":"Causalean.Mathlib.Analysis.EhlichZellerMesh.czMeshMax","k":"def","m":"Causalean.Mathlib.Analysis.EhlichZellerMesh.Mesh","a":"Mathlib/Analysis/EhlichZellerMesh","t":1,"d":"The mesh maximum: max_{0 ≤ j ≤ k} |R(x_j)|, written as a bounded supremum over Finset.range (k+1). Since each summand is ≥ 0 and the index set is finite and nonempty, this is the ordinary finite maximum.","s":"Polynomial ℝ → ℕ → ℝ","l":45,"f":"Causalean/Mathlib/Analysis/EhlichZellerMesh/Mesh.lean"},{"n":"Causalean.Mathlib.Analysis.EhlichZellerMesh.czMeshMax_neg","k":"theorem","m":"Causalean.Mathlib.Analysis.EhlichZellerMesh.Mesh","a":"Mathlib/Analysis/EhlichZellerMesh","t":2,"d":"Negating a real polynomial leaves unchanged its mesh maximum at the Chebyshev–Lobatto nodes.","s":"∀ (R : Polynomial ℝ) (k : ℕ), Causalean.Mathlib.Analysis.EhlichZellerMesh.czMeshMax (-R) k = Causalean.Mathlib.Analysis.EhlichZellerMesh.czMeshMax R k","l":179,"f":"Causalean/Mathlib/Analysis/EhlichZellerMesh/Mesh.lean"},{"n":"Causalean.Mathlib.Analysis.EhlichZellerMesh.czMeshMax_nonneg","k":"theorem","m":"Causalean.Mathlib.Analysis.EhlichZellerMesh.Mesh","a":"Mathlib/Analysis/EhlichZellerMesh","t":2,"d":"The mesh maximum of the absolute values of a real polynomial at the Chebyshev–Lobatto nodes is nonnegative.","s":"∀ (R : Polynomial ℝ) (k : ℕ), 0 ≤ Causalean.Mathlib.Analysis.EhlichZellerMesh.czMeshMax R k","l":155,"f":"Causalean/Mathlib/Analysis/EhlichZellerMesh/Mesh.lean"},{"n":"Causalean.Mathlib.Analysis.EhlichZellerMesh.czNode","k":"def","m":"Causalean.Mathlib.Analysis.EhlichZellerMesh.Mesh","a":"Mathlib/Analysis/EhlichZellerMesh","t":1,"d":"The j-th Chebyshev–Lobatto (extrema) node on [-1, 1] for a mesh of order k: czNode k j = -cos(π j / k). When 0 < k, the indices j = 0..k give k+1 points with endpoints czNode k 0 = -1 and czNode k k = 1.","s":"ℕ → ℕ → ℝ","l":40,"f":"Causalean/Mathlib/Analysis/EhlichZellerMesh/Mesh.lean"},{"n":"Causalean.Mathlib.Analysis.EhlichZellerMesh.czSup","k":"def","m":"Causalean.Mathlib.Analysis.EhlichZellerMesh.Bernstein","a":"Mathlib/Analysis/EhlichZellerMesh","t":1,"d":"The sup-norm of the trigonometric transform of R over the compact interval [0, π]. By surjectivity of t ↦ -cos t this equals sup_{x∈[-1,1]} |R x|.","s":"Polynomial ℝ → ℝ","l":51,"f":"Causalean/Mathlib/Analysis/EhlichZellerMesh/Bernstein.lean"},{"n":"Causalean.Mathlib.Analysis.EhlichZellerMesh.czSup_attained","k":"theorem","m":"Causalean.Mathlib.Analysis.EhlichZellerMesh.Bernstein","a":"Mathlib/Analysis/EhlichZellerMesh","t":2,"d":"The sup-norm czSup R is attained at some point of [0, π]: there is t₀ ∈ [0, π] with |czTrig R t₀| = czSup R. This is the extreme-value theorem applied to the continuous map |czTrig R| on the compact interval [0, π].","s":"∀ (R : Polynomial ℝ), ∃ t₀ ∈ Set.Icc 0 Real.pi, |Causalean.Mathlib.Analysis.EhlichZellerMesh.czTrig R t₀| = Causalean.Mathlib.Analysis.EhlichZellerMesh.czSup R","l":62,"f":"Causalean/Mathlib/Analysis/EhlichZellerMesh/Bernstein.lean"},{"n":"Causalean.Mathlib.Analysis.EhlichZellerMesh.czSup_neg","k":"theorem","m":"Causalean.Mathlib.Analysis.EhlichZellerMesh.Mesh","a":"Mathlib/Analysis/EhlichZellerMesh","t":2,"d":"Negating a real polynomial leaves unchanged the supremum of the absolute value of its trigonometric transform over the cosine interval.","s":"∀ (R : Polynomial ℝ), Causalean.Mathlib.Analysis.EhlichZellerMesh.czSup (-R) = Causalean.Mathlib.Analysis.EhlichZellerMesh.czSup R","l":167,"f":"Causalean/Mathlib/Analysis/EhlichZellerMesh/Mesh.lean"},{"n":"Causalean.Mathlib.Analysis.EhlichZellerMesh.czSup_nonneg","k":"theorem","m":"Causalean.Mathlib.Analysis.EhlichZellerMesh.Bernstein","a":"Mathlib/Analysis/EhlichZellerMesh","t":2,"d":"czSup R is nonnegative (it is a supremum of absolute values).","s":"∀ (R : Polynomial ℝ), 0 ≤ Causalean.Mathlib.Analysis.EhlichZellerMesh.czSup R","l":75,"f":"Causalean/Mathlib/Analysis/EhlichZellerMesh/Bernstein.lean"},{"n":"Causalean.Mathlib.Analysis.EhlichZellerMesh.czTrig","k":"def","m":"Causalean.Mathlib.Analysis.EhlichZellerMesh.Bernstein","a":"Mathlib/Analysis/EhlichZellerMesh","t":1,"d":"The trigonometric transform of a real polynomial R: czTrig R t = R(-cos t). For R of degree ≤ β this is an even trigonometric polynomial of degree ≤ β, and as t ranges over [0, π] the argument -cos t ranges over all of [-1, 1].","s":"Polynomial ℝ → ℝ → ℝ","l":46,"f":"Causalean/Mathlib/Analysis/EhlichZellerMesh/Bernstein.lean"},{"n":"Causalean.Mathlib.Analysis.EhlichZellerMesh.czTrig_arccos_lipschitz","k":"theorem","m":"Causalean.Mathlib.Analysis.EhlichZellerMesh.Bernstein","a":"Mathlib/Analysis/EhlichZellerMesh","t":1,"d":"Arccos-Lipschitz reformulation of Szegő's inequality. For a real polynomial of degree at most β with a strictly positive trigonometric sup-norm on [0, π], and for any two points in [0, π], the arccosine of the polynomial's normalized trigonometric transform is β-Lipschitz between those two points: the difference of the two arccosine values is bounded by β times the distance between the points.","s":"∀ (R : Polynomial ℝ) (β : ℕ), R.natDegree ≤ β → 0 < Causalean.Mathlib.Analysis.EhlichZellerMesh.czSup R → ∀ {s t : ℝ}, s ∈ Set.Icc 0 Real.pi → t ∈ Set.Icc 0 Real.pi → |Real.arccos (Causalean.Mathlib.Analysis.EhlichZellerMesh.czTrig R t / Causalean.Mathlib.Analysis.EhlichZellerMesh.czSup R) - Real.arccos (Causalean.Mathlib.Analysis.EhlichZellerMesh.czTrig R s / Causalean.Mathlib.Analysis.EhlichZellerMesh.czSup R)| ≤ ↑β * |t - s|","l":269,"f":"Causalean/Mathlib/Analysis/EhlichZellerMesh/Bernstein.lean"},{"n":"Causalean.Mathlib.Analysis.EhlichZellerMesh.czTrig_arccos_lipschitz_regularized","k":"theorem","m":"Causalean.Mathlib.Analysis.EhlichZellerMesh.Bernstein","a":"Mathlib/Analysis/EhlichZellerMesh","t":2,"d":"For a real polynomial of degree at most β, the arccosine of its cosine-polynomial transform normalized by its supremum plus a positive constant is Lipschitz on [0, π] with constant β.","s":"∀ (R : Polynomial ℝ) (β : ℕ), R.natDegree ≤ β → ∀ {δ : ℝ}, 0 < δ → ∀ {s t : ℝ}, s ∈ Set.Icc 0 Real.pi → t ∈ Set.Icc 0 Real.pi → |Real.arccos (Causalean.Mathlib.Analysis.EhlichZellerMesh.czTrig R t / (Causalean.Mathlib.Analysis.EhlichZellerMesh.czSup R + δ)) - Real.arccos (Causalean.Mathlib.Analysis.EhlichZellerMesh.czTrig R s / (Causalean.Mathlib.Analysis.EhlichZellerMesh.czSup R + δ))| ≤ ↑β * |t - s|","l":153,"f":"Causalean/Mathlib/Analysis/EhlichZellerMesh/Bernstein.lean"},{"n":"Causalean.Mathlib.Analysis.EhlichZellerMesh.czTrig_at_meshParam","k":"theorem","m":"Causalean.Mathlib.Analysis.EhlichZellerMesh.Mesh","a":"Mathlib/Analysis/EhlichZellerMesh","t":2,"d":"The trigonometric transform evaluated at the mesh parameter t = π j / k recovers the node value: czTrig R (π j / k) = R(czNode k j).","s":"∀ (R : Polynomial ℝ) (k j : ℕ), Causalean.Mathlib.Analysis.EhlichZellerMesh.czTrig R (Real.pi * ↑j / ↑k) = Polynomial.eval (Causalean.Mathlib.Analysis.EhlichZellerMesh.czNode k j) R","l":51,"f":"Causalean/Mathlib/Analysis/EhlichZellerMesh/Mesh.lean"},{"n":"Causalean.Mathlib.Analysis.EhlichZellerMesh.czTrig_continuous","k":"theorem","m":"Causalean.Mathlib.Analysis.EhlichZellerMesh.Bernstein","a":"Mathlib/Analysis/EhlichZellerMesh","t":2,"d":"czTrig R is continuous (composition of the polynomial evaluation with t ↦ -cos t).","s":"∀ (R : Polynomial ℝ), Continuous (Causalean.Mathlib.Analysis.EhlichZellerMesh.czTrig R)","l":56,"f":"Causalean/Mathlib/Analysis/EhlichZellerMesh/Bernstein.lean"},{"n":"Causalean.Mathlib.Analysis.EhlichZellerMesh.czTrig_maximizer_bound","k":"theorem","m":"Causalean.Mathlib.Analysis.EhlichZellerMesh.Bernstein","a":"Mathlib/Analysis/EhlichZellerMesh","t":1,"d":"Maximizer node bound (the packaged output consumed by the mesh file). For a real polynomial of degree at most β with a strictly positive trigonometric sup-norm, suppose a point t₀ in [0, π] is a maximizer where the trigonometric transform attains the sup-norm, another point s also lies in [0, π], and the two points are close enough that β times their distance is at most π/2. Then the trigonometric transform at s is bounded below by the sup-norm times the cosine of β times the distance between the two points.","s":"∀ (R : Polynomial ℝ) (β : ℕ), R.natDegree ≤ β → 0 < Causalean.Mathlib.Analysis.EhlichZellerMesh.czSup R → ∀ {t₀ s : ℝ}, t₀ ∈ Set.Icc 0 Real.pi → s ∈ Set.Icc 0 Real.pi → Causalean.Mathlib.Analysis.EhlichZellerMesh.czTrig R t₀ = Causalean.Mathlib.Analysis.EhlichZellerMesh.czSup R → ↑β * |t₀ - s| ≤ Real.pi / 2 → Causalean.Mathlib.Analysis.EhlichZellerMesh.czSup R * Real.cos (↑β * |t₀ - s|) ≤ Causalean.Mathlib.Analysis.EhlichZellerMesh.czTrig R s","l":325,"f":"Causalean/Mathlib/Analysis/EhlichZellerMesh/Bernstein.lean"},{"n":"Causalean.Mathlib.Analysis.EhlichZellerMesh.czTrig_neg","k":"theorem","m":"Causalean.Mathlib.Analysis.EhlichZellerMesh.Mesh","a":"Mathlib/Analysis/EhlichZellerMesh","t":2,"d":"Negating a real polynomial negates its trigonometric transform at every point.","s":"∀ (R : Polynomial ℝ) (t : ℝ), Causalean.Mathlib.Analysis.EhlichZellerMesh.czTrig (-R) t = -Causalean.Mathlib.Analysis.EhlichZellerMesh.czTrig R t","l":162,"f":"Causalean/Mathlib/Analysis/EhlichZellerMesh/Mesh.lean"},{"n":"Causalean.Mathlib.Analysis.EhlichZellerMesh.czTrig_szego_deriv","k":"theorem","m":"Causalean.Mathlib.Analysis.EhlichZellerMesh.Bernstein","a":"Mathlib/Analysis/EhlichZellerMesh","t":1,"d":"Szegő's differential inequality (the crux; not available in Mathlib). For a real polynomial of degree at most β and any real point t, the derivative of the polynomial's trigonometric transform at that point is bounded in absolute value by β times the square root of the sup-norm squared minus the transform's value at that point squared.","s":"∀ (R : Polynomial ℝ) (β : ℕ), R.natDegree ≤ β → ∀ (t : ℝ), |deriv (Causalean.Mathlib.Analysis.EhlichZellerMesh.czTrig R) t| ≤ ↑β * √(Causalean.Mathlib.Analysis.EhlichZellerMesh.czSup R ^ 2 - Causalean.Mathlib.Analysis.EhlichZellerMesh.czTrig R t ^ 2)","l":90,"f":"Causalean/Mathlib/Analysis/EhlichZellerMesh/Bernstein.lean"},{"n":"Causalean.Mathlib.Analysis.EhlichZellerMesh.ehlichZeller_mesh_bound","k":"theorem","m":"Causalean.Mathlib.Analysis.EhlichZellerMesh.Mesh","a":"Mathlib/Analysis/EhlichZellerMesh","t":1,"d":"Ehlich–Zeller mesh (norming) inequality. For a real polynomial of degree at most β and a mesh order k strictly exceeding β, with Chebyshev–Lobatto nodes x_j = -cos(π j / k) for j = 0, …, k, the polynomial's supremum absolute value on [-1, 1] is bounded by the secant of π β / (2k) times the maximum of its absolute values at the mesh nodes.","s":"∀ (R : Polynomial ℝ) (β k : ℕ), R.natDegree ≤ β → β < k → sSup ((fun x => |Polynomial.eval x R|) '' Set.Icc (-1) 1) ≤ 1 / Real.cos (Real.pi * ↑β / (2 * ↑k)) * Causalean.Mathlib.Analysis.EhlichZellerMesh.czMeshMax R k","l":267,"f":"Causalean/Mathlib/Analysis/EhlichZellerMesh/Mesh.lean"},{"n":"Causalean.Mathlib.Analysis.EhlichZellerMesh.eval_node_le_czMeshMax","k":"theorem","m":"Causalean.Mathlib.Analysis.EhlichZellerMesh.Mesh","a":"Mathlib/Analysis/EhlichZellerMesh","t":2,"d":"The mesh maximum lower-bounds every node value it ranges over.","s":"∀ (R : Polynomial ℝ) (k : ℕ) {j : ℕ}, j ∈ Finset.range (k + 1) → |Polynomial.eval (Causalean.Mathlib.Analysis.EhlichZellerMesh.czNode k j) R| ≤ Causalean.Mathlib.Analysis.EhlichZellerMesh.czMeshMax R k","l":133,"f":"Causalean/Mathlib/Analysis/EhlichZellerMesh/Mesh.lean"},{"n":"Causalean.Mathlib.Analysis.EhlichZellerMesh.exists_meshParam_close","k":"theorem","m":"Causalean.Mathlib.Analysis.EhlichZellerMesh.Mesh","a":"Mathlib/Analysis/EhlichZellerMesh","t":2,"d":"Node-selection lemma. For k ≥ 1 and any t₀ ∈ [0, π] there is a mesh index j ≤ k whose parameter π j / k is within π / (2k) of t₀: |t₀ − π j / k| ≤ π / (2k). (The k mesh parameters partition [0, π] into subintervals of length π / k, so every point is within a half-step of a node.)","s":"∀ (k : ℕ), 0 < k → ∀ {t₀ : ℝ}, t₀ ∈ Set.Icc 0 Real.pi → ∃ j ∈ Finset.range (k + 1), |t₀ - Real.pi * ↑j / ↑k| ≤ Real.pi / (2 * ↑k)","l":57,"f":"Causalean/Mathlib/Analysis/EhlichZellerMesh/Mesh.lean"},{"n":"Causalean.Mathlib.Analysis.EhlichZellerMesh.meshParam_mem_Icc","k":"theorem","m":"Causalean.Mathlib.Analysis.EhlichZellerMesh.Mesh","a":"Mathlib/Analysis/EhlichZellerMesh","t":2,"d":"For a positive mesh order, every valid Chebyshev–Lobatto mesh parameter lies in the interval from zero to π.","s":"∀ (k : ℕ), 0 < k → ∀ {j : ℕ}, j ∈ Finset.range (k + 1) → Real.pi * ↑j / ↑k ∈ Set.Icc 0 Real.pi","l":185,"f":"Causalean/Mathlib/Analysis/EhlichZellerMesh/Mesh.lean"},{"n":"Causalean.Mathlib.Analysis.EhlichZellerMesh.oversampled_norming","k":"theorem","m":"Causalean.Mathlib.Analysis.EhlichZellerMesh.Mesh","a":"Mathlib/Analysis/EhlichZellerMesh","t":1,"d":"Oversampled Chebyshev–Lobatto norming (constant depending only on the oversampling ratio). For an oversampling ratio strictly greater than one, a polynomial degree bound of at least one, a real polynomial whose degree does not exceed that bound, and a mesh order at least the oversampling ratio times the degree bound, the polynomial's supremum absolute value on [-1, 1] is bounded by the secant of π / (2c) times the maximum of its absolute values at the mesh nodes — a norming constant depending only on the oversampling ratio, uniformly in the degree bound.","s":"∀ (R : Polynomial ℝ) (β k : ℕ) (c : ℝ), 1 < c → 1 ≤ β → R.natDegree ≤ β → c * ↑β ≤ ↑k → sSup ((fun x => |Polynomial.eval x R|) '' Set.Icc (-1) 1) ≤ 1 / Real.cos (Real.pi / (2 * c)) * Causalean.Mathlib.Analysis.EhlichZellerMesh.czMeshMax R k","l":304,"f":"Causalean/Mathlib/Analysis/EhlichZellerMesh/Mesh.lean"},{"n":"Causalean.Mathlib.Analysis.FiniteDimL1LinfDuality.Ev","k":"def","m":"Causalean.Mathlib.Analysis.FiniteDimL1LinfDuality.HahnBanachSetup","a":"Mathlib/Analysis/FiniteDimL1LinfDuality","t":1,"d":"Node evaluation as a linear map: Ev p β b is the vector of values of the degree-≤ β polynomial with coefficients b, sampled at the nodes, (Ev p β b) j = ∑ i, b i * (p j)^i. This is the transpose of the moment map.","s":"{k : ℕ} → (Fin (k + 1) → ℝ) → (β : ℕ) → (Fin (β + 1) → ℝ) →ₗ[ℝ] Fin (k + 1) → ℝ","l":65,"f":"Causalean/Mathlib/Analysis/FiniteDimL1LinfDuality/HahnBanachSetup.lean"},{"n":"Causalean.Mathlib.Analysis.FiniteDimL1LinfDuality.Ev_apply","k":"theorem","m":"Causalean.Mathlib.Analysis.FiniteDimL1LinfDuality.HahnBanachSetup","a":"Mathlib/Analysis/FiniteDimL1LinfDuality","t":2,"d":"Unfolding lemma for Ev (definitional).","s":"∀ {k β : ℕ} {p : Fin (k + 1) → ℝ} (b : Fin (β + 1) → ℝ) (j : Fin (k + 1)), (Causalean.Mathlib.Analysis.FiniteDimL1LinfDuality.Ev p β) b j = ∑ i, b i * p j ^ ↑i","l":80,"f":"Causalean/Mathlib/Analysis/FiniteDimL1LinfDuality/HahnBanachSetup.lean"},{"n":"Causalean.Mathlib.Analysis.FiniteDimL1LinfDuality.Ev_injective","k":"theorem","m":"Causalean.Mathlib.Analysis.FiniteDimL1LinfDuality.HahnBanachSetup","a":"Mathlib/Analysis/FiniteDimL1LinfDuality","t":1,"d":"Node evaluation is injective. For k + 1 pairwise distinct real nodes and a degree bound β at most k, the linear map sending a degree-≤ β coefficient vector to its values at the nodes is injective.","s":"∀ {k β : ℕ} {p : Fin (k + 1) → ℝ}, Function.Injective p → β ≤ k → Function.Injective ⇑(Causalean.Mathlib.Analysis.FiniteDimL1LinfDuality.Ev p β)","l":122,"f":"Causalean/Mathlib/Analysis/FiniteDimL1LinfDuality/HahnBanachSetup.lean"},{"n":"Causalean.Mathlib.Analysis.FiniteDimL1LinfDuality.Ev_single","k":"theorem","m":"Causalean.Mathlib.Analysis.FiniteDimL1LinfDuality.HahnBanachSetup","a":"Mathlib/Analysis/FiniteDimL1LinfDuality","t":2,"d":"Node evaluation of the ℓ-th coordinate vector recovers the monomial column (p j)^ℓ.","s":"∀ {k β : ℕ} {p : Fin (k + 1) → ℝ} (ℓ : Fin (β + 1)) (j : Fin (k + 1)), (Causalean.Mathlib.Analysis.FiniteDimL1LinfDuality.Ev p β) (Pi.single ℓ 1) j = p j ^ ↑ℓ","l":167,"f":"Causalean/Mathlib/Analysis/FiniteDimL1LinfDuality/HahnBanachSetup.lean"},{"n":"Causalean.Mathlib.Analysis.FiniteDimL1LinfDuality.MomentSol","k":"def","m":"Causalean.Mathlib.Analysis.FiniteDimL1LinfDuality.Basic","a":"Mathlib/Analysis/FiniteDimL1LinfDuality","t":1,"d":"The moment system of admissible weight vectors. w ∈ MomentSol p β means that for every degree ℓ ≤ β the sampled weighted sum ∑ j, w j * (p j) ^ ℓ equals 1^ℓ - 0^ℓ, i.e. 0 when ℓ = 0 and 1 otherwise. Equivalently, w reproduces the endpoint contrast r ↦ r.eval 1 - r.eval 0 on every real polynomial r of degree ≤ β via the node values (r.eval (p j))ⱼ.","s":"{k : ℕ} → (Fin (k + 1) → ℝ) → ℕ → Set (Fin (k + 1) → ℝ)","l":51,"f":"Causalean/Mathlib/Analysis/FiniteDimL1LinfDuality/Basic.lean"},{"n":"Causalean.Mathlib.Analysis.FiniteDimL1LinfDuality.coeffPoly","k":"def","m":"Causalean.Mathlib.Analysis.FiniteDimL1LinfDuality.HahnBanachSetup","a":"Mathlib/Analysis/FiniteDimL1LinfDuality","t":1,"d":"The degree-≤ β polynomial ∑ i, b i * X^i associated to a coefficient vector b : Fin (β+1) → ℝ. Its node values are Ev p β b and its endpoint contrast is contrastL β b.","s":"{β : ℕ} → (Fin (β + 1) → ℝ) → Polynomial ℝ","l":43,"f":"Causalean/Mathlib/Analysis/FiniteDimL1LinfDuality/HahnBanachSetup.lean"},{"n":"Causalean.Mathlib.Analysis.FiniteDimL1LinfDuality.coeffPoly_contrast","k":"theorem","m":"Causalean.Mathlib.Analysis.FiniteDimL1LinfDuality.HahnBanachSetup","a":"Mathlib/Analysis/FiniteDimL1LinfDuality","t":2,"d":"The contrast functional computes the endpoint contrast of coeffPoly b.","s":"∀ {β : ℕ} (b : Fin (β + 1) → ℝ), Polynomial.eval 1 (Causalean.Mathlib.Analysis.FiniteDimL1LinfDuality.coeffPoly b) - Polynomial.eval 0 (Causalean.Mathlib.Analysis.FiniteDimL1LinfDuality.coeffPoly b) = (Causalean.Mathlib.Analysis.FiniteDimL1LinfDuality.contrastL β) b","l":111,"f":"Causalean/Mathlib/Analysis/FiniteDimL1LinfDuality/HahnBanachSetup.lean"},{"n":"Causalean.Mathlib.Analysis.FiniteDimL1LinfDuality.coeffPoly_eval","k":"theorem","m":"Causalean.Mathlib.Analysis.FiniteDimL1LinfDuality.HahnBanachSetup","a":"Mathlib/Analysis/FiniteDimL1LinfDuality","t":2,"d":"Evaluating coeffPoly b at t gives the polynomial value ∑ i, b i * t^i.","s":"∀ {β : ℕ} (b : Fin (β + 1) → ℝ) (t : ℝ), Polynomial.eval t (Causalean.Mathlib.Analysis.FiniteDimL1LinfDuality.coeffPoly b) = ∑ i, b i * t ^ ↑i","l":59,"f":"Causalean/Mathlib/Analysis/FiniteDimL1LinfDuality/HahnBanachSetup.lean"},{"n":"Causalean.Mathlib.Analysis.FiniteDimL1LinfDuality.coeffPoly_eval_node","k":"theorem","m":"Causalean.Mathlib.Analysis.FiniteDimL1LinfDuality.HahnBanachSetup","a":"Mathlib/Analysis/FiniteDimL1LinfDuality","t":2,"d":"Node values of coeffPoly b coincide with Ev p β b.","s":"∀ {k β : ℕ} {p : Fin (k + 1) → ℝ} (b : Fin (β + 1) → ℝ) (j : Fin (k + 1)), Polynomial.eval (p j) (Causalean.Mathlib.Analysis.FiniteDimL1LinfDuality.coeffPoly b) = (Causalean.Mathlib.Analysis.FiniteDimL1LinfDuality.Ev p β) b j","l":85,"f":"Causalean/Mathlib/Analysis/FiniteDimL1LinfDuality/HahnBanachSetup.lean"},{"n":"Causalean.Mathlib.Analysis.FiniteDimL1LinfDuality.coeffPoly_natDegree_le","k":"theorem","m":"Causalean.Mathlib.Analysis.FiniteDimL1LinfDuality.HahnBanachSetup","a":"Mathlib/Analysis/FiniteDimL1LinfDuality","t":2,"d":"coeffPoly b has degree at most β (each monomial X^i has i ≤ β).","s":"∀ {β : ℕ} (b : Fin (β + 1) → ℝ), (Causalean.Mathlib.Analysis.FiniteDimL1LinfDuality.coeffPoly b).natDegree ≤ β","l":49,"f":"Causalean/Mathlib/Analysis/FiniteDimL1LinfDuality/HahnBanachSetup.lean"},{"n":"Causalean.Mathlib.Analysis.FiniteDimL1LinfDuality.contrastL","k":"def","m":"Causalean.Mathlib.Analysis.FiniteDimL1LinfDuality.HahnBanachSetup","a":"Mathlib/Analysis/FiniteDimL1LinfDuality","t":1,"d":"Endpoint-contrast functional on coefficient vectors: contrastL β b = ∑ i, b i * (if i = 0 then 0 else 1) = r.eval 1 - r.eval 0 for r = coeffPoly b.","s":"(β : ℕ) → (Fin (β + 1) → ℝ) →ₗ[ℝ] ℝ","l":90,"f":"Causalean/Mathlib/Analysis/FiniteDimL1LinfDuality/HahnBanachSetup.lean"},{"n":"Causalean.Mathlib.Analysis.FiniteDimL1LinfDuality.contrastL_apply","k":"theorem","m":"Causalean.Mathlib.Analysis.FiniteDimL1LinfDuality.HahnBanachSetup","a":"Mathlib/Analysis/FiniteDimL1LinfDuality","t":2,"d":"Unfolding lemma for contrastL (definitional).","s":"∀ {β : ℕ} (b : Fin (β + 1) → ℝ), (Causalean.Mathlib.Analysis.FiniteDimL1LinfDuality.contrastL β) b = ∑ i, b i * if ↑i = 0 then 0 else 1","l":106,"f":"Causalean/Mathlib/Analysis/FiniteDimL1LinfDuality/HahnBanachSetup.lean"},{"n":"Causalean.Mathlib.Analysis.FiniteDimL1LinfDuality.contrastL_le_dual_mul_ninf","k":"theorem","m":"Causalean.Mathlib.Analysis.FiniteDimL1LinfDuality.HahnBanachSetup","a":"Mathlib/Analysis/FiniteDimL1LinfDuality","t":1,"d":"Boundedness estimate (the Hahn–Banach hypothesis). For k + 1 pairwise distinct real nodes, a degree bound β at most k, and any coefficient vector b, the endpoint-contrast functional evaluated at b is bounded above by the dual supremum times the sup-norm of b's node values.","s":"∀ {k β : ℕ} {p : Fin (k + 1) → ℝ}, Function.Injective p → β ≤ k → ∀ (b : Fin (β + 1) → ℝ), (Causalean.Mathlib.Analysis.FiniteDimL1LinfDuality.contrastL β) b ≤ sSup (Causalean.Mathlib.Analysis.FiniteDimL1LinfDuality.dualValSet p β) * Causalean.Mathlib.Analysis.FiniteDimL1LinfDuality.ninf ((Causalean.Mathlib.Analysis.FiniteDimL1LinfDuality.Ev p β) b)","l":253,"f":"Causalean/Mathlib/Analysis/FiniteDimL1LinfDuality/HahnBanachSetup.lean"},{"n":"Causalean.Mathlib.Analysis.FiniteDimL1LinfDuality.contrastL_le_dual_mul_ninf_of_momentSol_nonempty","k":"theorem","m":"Causalean.Mathlib.Analysis.FiniteDimL1LinfDuality.NonemptyDuality","a":"Mathlib/Analysis/FiniteDimL1LinfDuality","t":2,"d":"Hahn-Banach domination estimate on the node-value subspace, using feasibility instead of node-evaluation injectivity: contrastL β b ≤ sSup (dualValSet p β) * ninf (Ev p β b).","s":"∀ {k β : ℕ} {p : Fin (k + 1) → ℝ}, (Causalean.Mathlib.Analysis.FiniteDimL1LinfDuality.MomentSol p β).Nonempty → ∀ (b : Fin (β + 1) → ℝ), (Causalean.Mathlib.Analysis.FiniteDimL1LinfDuality.contrastL β) b ≤ sSup (Causalean.Mathlib.Analysis.FiniteDimL1LinfDuality.dualValSet p β) * Causalean.Mathlib.Analysis.FiniteDimL1LinfDuality.ninf ((Causalean.Mathlib.Analysis.FiniteDimL1LinfDuality.Ev p β) b)","l":83,"f":"Causalean/Mathlib/Analysis/FiniteDimL1LinfDuality/NonemptyDuality.lean"},{"n":"Causalean.Mathlib.Analysis.FiniteDimL1LinfDuality.contrastL_single","k":"theorem","m":"Causalean.Mathlib.Analysis.FiniteDimL1LinfDuality.HahnBanachSetup","a":"Mathlib/Analysis/FiniteDimL1LinfDuality","t":2,"d":"The contrast of the ℓ-th coordinate vector is 1^ℓ - 0^ℓ.","s":"∀ {β : ℕ} (ℓ : Fin (β + 1)), (Causalean.Mathlib.Analysis.FiniteDimL1LinfDuality.contrastL β) (Pi.single ℓ 1) = if ↑ℓ = 0 then 0 else 1","l":179,"f":"Causalean/Mathlib/Analysis/FiniteDimL1LinfDuality/HahnBanachSetup.lean"},{"n":"Causalean.Mathlib.Analysis.FiniteDimL1LinfDuality.dualValSet","k":"def","m":"Causalean.Mathlib.Analysis.FiniteDimL1LinfDuality.Basic","a":"Mathlib/Analysis/FiniteDimL1LinfDuality","t":1,"d":"The set of achievable endpoint contrasts of node-bounded polynomials: dualValSet p β = { |r.eval 1 - r.eval 0| : r.natDegree ≤ β, ∀ j, |r.eval (p j)| ≤ 1 }. Its supremum is the dual value.","s":"{k : ℕ} → (Fin (k + 1) → ℝ) → ℕ → Set ℝ","l":66,"f":"Causalean/Mathlib/Analysis/FiniteDimL1LinfDuality/Basic.lean"},{"n":"Causalean.Mathlib.Analysis.FiniteDimL1LinfDuality.dualValSet_bddAbove","k":"theorem","m":"Causalean.Mathlib.Analysis.FiniteDimL1LinfDuality.WeakDuality","a":"Mathlib/Analysis/FiniteDimL1LinfDuality","t":2,"d":"The dual set is bounded above (by any primal value; the primal set is nonempty for distinct nodes and β ≤ k).","s":"∀ {k β : ℕ} {p : Fin (k + 1) → ℝ}, (Causalean.Mathlib.Analysis.FiniteDimL1LinfDuality.primalNormSet p β).Nonempty → BddAbove (Causalean.Mathlib.Analysis.FiniteDimL1LinfDuality.dualValSet p β)","l":124,"f":"Causalean/Mathlib/Analysis/FiniteDimL1LinfDuality/WeakDuality.lean"},{"n":"Causalean.Mathlib.Analysis.FiniteDimL1LinfDuality.dualValSet_bddAbove_of_momentSol_nonempty","k":"theorem","m":"Causalean.Mathlib.Analysis.FiniteDimL1LinfDuality.NonemptyDuality","a":"Mathlib/Analysis/FiniteDimL1LinfDuality","t":2,"d":"Feasibility of the moment system bounds every dual value by any feasible primal norm. This is the weak-duality estimate: if w ∈ MomentSol p β, then every node-bounded degree-≤ β polynomial has endpoint contrast at most ∑ j, |w j|, so the dual value set is bounded above.","s":"∀ {k β : ℕ} {p : Fin (k + 1) → ℝ}, (Causalean.Mathlib.Analysis.FiniteDimL1LinfDuality.MomentSol p β).Nonempty → BddAbove (Causalean.Mathlib.Analysis.FiniteDimL1LinfDuality.dualValSet p β)","l":47,"f":"Causalean/Mathlib/Analysis/FiniteDimL1LinfDuality/NonemptyDuality.lean"},{"n":"Causalean.Mathlib.Analysis.FiniteDimL1LinfDuality.dualValSet_nonempty","k":"theorem","m":"Causalean.Mathlib.Analysis.FiniteDimL1LinfDuality.Basic","a":"Mathlib/Analysis/FiniteDimL1LinfDuality","t":2,"d":"The dual set is nonempty: the zero polynomial contributes the value 0 (degree 0 ≤ β, trivially node-bounded, contrast 0).","s":"∀ {k β : ℕ} {p : Fin (k + 1) → ℝ}, (Causalean.Mathlib.Analysis.FiniteDimL1LinfDuality.dualValSet p β).Nonempty","l":73,"f":"Causalean/Mathlib/Analysis/FiniteDimL1LinfDuality/Basic.lean"},{"n":"Causalean.Mathlib.Analysis.FiniteDimL1LinfDuality.dual_le_primal","k":"theorem","m":"Causalean.Mathlib.Analysis.FiniteDimL1LinfDuality.WeakDuality","a":"Mathlib/Analysis/FiniteDimL1LinfDuality","t":1,"d":"Weak duality. If a real number is the ℓ¹ norm of some feasible weight vector and another real number is the endpoint contrast attained by some node-bounded degree-≤ β polynomial, then the second number is no larger than the first: every dual value is bounded by every primal value.","s":"∀ {k β : ℕ} {p : Fin (k + 1) → ℝ} {s t : ℝ}, s ∈ Causalean.Mathlib.Analysis.FiniteDimL1LinfDuality.primalNormSet p β → t ∈ Causalean.Mathlib.Analysis.FiniteDimL1LinfDuality.dualValSet p β → t ≤ s","l":97,"f":"Causalean/Mathlib/Analysis/FiniteDimL1LinfDuality/WeakDuality.lean"},{"n":"Causalean.Mathlib.Analysis.FiniteDimL1LinfDuality.dual_nonneg","k":"theorem","m":"Causalean.Mathlib.Analysis.FiniteDimL1LinfDuality.HahnBanachSetup","a":"Mathlib/Analysis/FiniteDimL1LinfDuality","t":2,"d":"Dual value is nonnegative. 0 ∈ dualValSet p β (the zero polynomial) and the set is bounded above, so its supremum is ≥ 0.","s":"∀ {k β : ℕ} {p : Fin (k + 1) → ℝ}, BddAbove (Causalean.Mathlib.Analysis.FiniteDimL1LinfDuality.dualValSet p β) → 0 ≤ sSup (Causalean.Mathlib.Analysis.FiniteDimL1LinfDuality.dualValSet p β)","l":241,"f":"Causalean/Mathlib/Analysis/FiniteDimL1LinfDuality/HahnBanachSetup.lean"},{"n":"Causalean.Mathlib.Analysis.FiniteDimL1LinfDuality.dual_nonneg_of_momentSol_nonempty","k":"theorem","m":"Causalean.Mathlib.Analysis.FiniteDimL1LinfDuality.NonemptyDuality","a":"Mathlib/Analysis/FiniteDimL1LinfDuality","t":2,"d":"With a feasible primal system, the dual supremum is nonnegative because 0 is dual-feasible.","s":"∀ {k β : ℕ} {p : Fin (k + 1) → ℝ}, (Causalean.Mathlib.Analysis.FiniteDimL1LinfDuality.MomentSol p β).Nonempty → 0 ≤ sSup (Causalean.Mathlib.Analysis.FiniteDimL1LinfDuality.dualValSet p β)","l":57,"f":"Causalean/Mathlib/Analysis/FiniteDimL1LinfDuality/NonemptyDuality.lean"},{"n":"Causalean.Mathlib.Analysis.FiniteDimL1LinfDuality.exists_moment_le_dual","k":"theorem","m":"Causalean.Mathlib.Analysis.FiniteDimL1LinfDuality.StrongDuality","a":"Mathlib/Analysis/FiniteDimL1LinfDuality","t":1,"d":"Strong duality (existence of an optimal weight). For k + 1 pairwise distinct real nodes and a degree bound β at most k, there exists a feasible weight vector — one solving the moment system — whose ℓ¹ norm does not exceed the dual supremum.","s":"∀ {k β : ℕ} {p : Fin (k + 1) → ℝ}, Function.Injective p → β ≤ k → ∃ w ∈ Causalean.Mathlib.Analysis.FiniteDimL1LinfDuality.MomentSol p β, ∑ j, |w j| ≤ sSup (Causalean.Mathlib.Analysis.FiniteDimL1LinfDuality.dualValSet p β)","l":21,"f":"Causalean/Mathlib/Analysis/FiniteDimL1LinfDuality/StrongDuality.lean"},{"n":"Causalean.Mathlib.Analysis.FiniteDimL1LinfDuality.exists_moment_le_dual_of_momentSol_nonempty","k":"theorem","m":"Causalean.Mathlib.Analysis.FiniteDimL1LinfDuality.NonemptyDuality","a":"Mathlib/Analysis/FiniteDimL1LinfDuality","t":1,"d":"Strong duality from moment-system feasibility. If the moment system for the given nodes and degree bound is feasible, i.e. it has at least one solution, then there exists a feasible weight vector whose ℓ¹ norm is at most the dual supremum.","s":"∀ {k β : ℕ} {p : Fin (k + 1) → ℝ}, (Causalean.Mathlib.Analysis.FiniteDimL1LinfDuality.MomentSol p β).Nonempty → ∃ w ∈ Causalean.Mathlib.Analysis.FiniteDimL1LinfDuality.MomentSol p β, ∑ j, |w j| ≤ sSup (Causalean.Mathlib.Analysis.FiniteDimL1LinfDuality.dualValSet p β)","l":152,"f":"Causalean/Mathlib/Analysis/FiniteDimL1LinfDuality/NonemptyDuality.lean"},{"n":"Causalean.Mathlib.Analysis.FiniteDimL1LinfDuality.l1_repr_eq_sup_dual","k":"theorem","m":"Causalean.Mathlib.Analysis.FiniteDimL1LinfDuality.Duality","a":"Mathlib/Analysis/FiniteDimL1LinfDuality","t":1,"d":"Finite-dimensional ℓ¹/ℓ∞ duality. For k + 1 pairwise distinct real interpolation nodes and a degree bound β at most k, the least ℓ¹ norm of a weight vector that reproduces the endpoint contrast r ↦ r.eval 1 - r.eval 0 of every degree-≤ β polynomial through its node values equals the largest such contrast attained by a degree-≤ β polynomial bounded by 1 at every node.","s":"∀ {k β : ℕ} {p : Fin (k + 1) → ℝ}, Function.Injective p → β ≤ k → sInf (Causalean.Mathlib.Analysis.FiniteDimL1LinfDuality.primalNormSet p β) = sSup (Causalean.Mathlib.Analysis.FiniteDimL1LinfDuality.dualValSet p β)","l":22,"f":"Causalean/Mathlib/Analysis/FiniteDimL1LinfDuality/Duality.lean"},{"n":"Causalean.Mathlib.Analysis.FiniteDimL1LinfDuality.l1_repr_eq_sup_dual_of_momentSol_nonempty","k":"theorem","m":"Causalean.Mathlib.Analysis.FiniteDimL1LinfDuality.NonemptyDuality","a":"Mathlib/Analysis/FiniteDimL1LinfDuality","t":1,"d":"Finite-dimensional ℓ¹/ℓ∞ duality under feasibility alone. If the moment system for the given nodes and degree bound has at least one solution, then the least ℓ¹ norm among feasible weight vectors equals the largest endpoint contrast attained by a degree-bounded polynomial bounded by 1 at every node.","s":"∀ {k β : ℕ} {p : Fin (k + 1) → ℝ}, (Causalean.Mathlib.Analysis.FiniteDimL1LinfDuality.MomentSol p β).Nonempty → sInf (Causalean.Mathlib.Analysis.FiniteDimL1LinfDuality.primalNormSet p β) = sSup (Causalean.Mathlib.Analysis.FiniteDimL1LinfDuality.dualValSet p β)","l":303,"f":"Causalean/Mathlib/Analysis/FiniteDimL1LinfDuality/NonemptyDuality.lean"},{"n":"Causalean.Mathlib.Analysis.FiniteDimL1LinfDuality.l1_repr_sq_eq_sup_dual_sq","k":"theorem","m":"Causalean.Mathlib.Analysis.FiniteDimL1LinfDuality.Duality","a":"Mathlib/Analysis/FiniteDimL1LinfDuality","t":1,"d":"Squared form. Under the same hypotheses as l1_repr_eq_sup_dual — k + 1 pairwise distinct real interpolation nodes and a degree bound β at most k — the square of the min-norm-representation value equals the square of the dual sup.","s":"∀ {k β : ℕ} {p : Fin (k + 1) → ℝ}, Function.Injective p → β ≤ k → sInf (Causalean.Mathlib.Analysis.FiniteDimL1LinfDuality.primalNormSet p β) ^ 2 = sSup (Causalean.Mathlib.Analysis.FiniteDimL1LinfDuality.dualValSet p β) ^ 2","l":36,"f":"Causalean/Mathlib/Analysis/FiniteDimL1LinfDuality/Duality.lean"},{"n":"Causalean.Mathlib.Analysis.FiniteDimL1LinfDuality.le_ninf","k":"theorem","m":"Causalean.Mathlib.Analysis.FiniteDimL1LinfDuality.HahnBanachSetup","a":"Mathlib/Analysis/FiniteDimL1LinfDuality","t":2,"d":"Each coordinate is bounded by the sup norm: |x j| ≤ ninf x.","s":"∀ {k : ℕ} (x : Fin (k + 1) → ℝ) (j : Fin (k + 1)), |x j| ≤ Causalean.Mathlib.Analysis.FiniteDimL1LinfDuality.ninf x","l":195,"f":"Causalean/Mathlib/Analysis/FiniteDimL1LinfDuality/HahnBanachSetup.lean"},{"n":"Causalean.Mathlib.Analysis.FiniteDimL1LinfDuality.momentSol_contrast_eq_sum_eval","k":"theorem","m":"Causalean.Mathlib.Analysis.FiniteDimL1LinfDuality.NonemptyDuality","a":"Mathlib/Analysis/FiniteDimL1LinfDuality","t":2,"d":"A feasible moment vector represents the coefficient contrast on node-value vectors: for every coefficient vector b, contrastL β b equals the pairing of w with the node-evaluation vector Ev p β b.","s":"∀ {k β : ℕ} {p w : Fin (k + 1) → ℝ}, w ∈ Causalean.Mathlib.Analysis.FiniteDimL1LinfDuality.MomentSol p β → ∀ (b : Fin (β + 1) → ℝ), (Causalean.Mathlib.Analysis.FiniteDimL1LinfDuality.contrastL β) b = ∑ j, w j * (Causalean.Mathlib.Analysis.FiniteDimL1LinfDuality.Ev p β) b j","l":69,"f":"Causalean/Mathlib/Analysis/FiniteDimL1LinfDuality/NonemptyDuality.lean"},{"n":"Causalean.Mathlib.Analysis.FiniteDimL1LinfDuality.momentSol_nonempty","k":"theorem","m":"Causalean.Mathlib.Analysis.FiniteDimL1LinfDuality.Basic","a":"Mathlib/Analysis/FiniteDimL1LinfDuality","t":1,"d":"Feasibility of the moment system (Vandermonde). For k + 1 pairwise distinct real interpolation nodes and a degree bound β at most k, the moment system — the set of weight vectors that reproduce the endpoint contrast of every degree-≤ β polynomial through the node values — has a solution.","s":"∀ {k β : ℕ} {p : Fin (k + 1) → ℝ}, Function.Injective p → β ≤ k → (Causalean.Mathlib.Analysis.FiniteDimL1LinfDuality.MomentSol p β).Nonempty","l":95,"f":"Causalean/Mathlib/Analysis/FiniteDimL1LinfDuality/Basic.lean"},{"n":"Causalean.Mathlib.Analysis.FiniteDimL1LinfDuality.ninf","k":"def","m":"Causalean.Mathlib.Analysis.FiniteDimL1LinfDuality.HahnBanachSetup","a":"Mathlib/Analysis/FiniteDimL1LinfDuality","t":1,"d":"The sup norm maxⱼ |x j| on Fin (k+1) → ℝ (a Finset.sup' over the nonempty index set). Used as the majorant M · ninf in Hahn–Banach.","s":"{k : ℕ} → (Fin (k + 1) → ℝ) → ℝ","l":190,"f":"Causalean/Mathlib/Analysis/FiniteDimL1LinfDuality/HahnBanachSetup.lean"},{"n":"Causalean.Mathlib.Analysis.FiniteDimL1LinfDuality.ninf_add_le","k":"theorem","m":"Causalean.Mathlib.Analysis.FiniteDimL1LinfDuality.HahnBanachSetup","a":"Mathlib/Analysis/FiniteDimL1LinfDuality","t":2,"d":"Subadditivity (triangle inequality) of the sup norm.","s":"∀ {k : ℕ} (x y : Fin (k + 1) → ℝ), Causalean.Mathlib.Analysis.FiniteDimL1LinfDuality.ninf (x + y) ≤ Causalean.Mathlib.Analysis.FiniteDimL1LinfDuality.ninf x + Causalean.Mathlib.Analysis.FiniteDimL1LinfDuality.ninf y","l":225,"f":"Causalean/Mathlib/Analysis/FiniteDimL1LinfDuality/HahnBanachSetup.lean"},{"n":"Causalean.Mathlib.Analysis.FiniteDimL1LinfDuality.ninf_nonneg","k":"theorem","m":"Causalean.Mathlib.Analysis.FiniteDimL1LinfDuality.HahnBanachSetup","a":"Mathlib/Analysis/FiniteDimL1LinfDuality","t":2,"d":"The sup norm is nonnegative.","s":"∀ {k : ℕ} (x : Fin (k + 1) → ℝ), 0 ≤ Causalean.Mathlib.Analysis.FiniteDimL1LinfDuality.ninf x","l":199,"f":"Causalean/Mathlib/Analysis/FiniteDimL1LinfDuality/HahnBanachSetup.lean"},{"n":"Causalean.Mathlib.Analysis.FiniteDimL1LinfDuality.ninf_sign","k":"theorem","m":"Causalean.Mathlib.Analysis.FiniteDimL1LinfDuality.HahnBanachSetup","a":"Mathlib/Analysis/FiniteDimL1LinfDuality","t":2,"d":"The sup norm of a ±1 sign vector is 1.","s":"∀ {k : ℕ} (s : Fin (k + 1) → ℝ), (∀ (j : Fin (k + 1)), |s j| = 1) → Causalean.Mathlib.Analysis.FiniteDimL1LinfDuality.ninf s = 1","l":235,"f":"Causalean/Mathlib/Analysis/FiniteDimL1LinfDuality/HahnBanachSetup.lean"},{"n":"Causalean.Mathlib.Analysis.FiniteDimL1LinfDuality.ninf_smul","k":"theorem","m":"Causalean.Mathlib.Analysis.FiniteDimL1LinfDuality.HahnBanachSetup","a":"Mathlib/Analysis/FiniteDimL1LinfDuality","t":2,"d":"Positive homogeneity (with absolute value) of the sup norm.","s":"∀ {k : ℕ} (c : ℝ) (x : Fin (k + 1) → ℝ), Causalean.Mathlib.Analysis.FiniteDimL1LinfDuality.ninf (c • x) = |c| * Causalean.Mathlib.Analysis.FiniteDimL1LinfDuality.ninf x","l":203,"f":"Causalean/Mathlib/Analysis/FiniteDimL1LinfDuality/HahnBanachSetup.lean"},{"n":"Causalean.Mathlib.Analysis.FiniteDimL1LinfDuality.primalNormSet","k":"def","m":"Causalean.Mathlib.Analysis.FiniteDimL1LinfDuality.Basic","a":"Mathlib/Analysis/FiniteDimL1LinfDuality","t":1,"d":"The set of achievable ℓ¹ norms of admissible weights: primalNormSet p β = { ∑ j, |w j| : w ∈ MomentSol p β }. Its infimum is the primal (min-norm-representation) value.","s":"{k : ℕ} → (Fin (k + 1) → ℝ) → ℕ → Set ℝ","l":60,"f":"Causalean/Mathlib/Analysis/FiniteDimL1LinfDuality/Basic.lean"},{"n":"Causalean.Mathlib.Analysis.FiniteDimL1LinfDuality.primalNormSet_bddBelow","k":"theorem","m":"Causalean.Mathlib.Analysis.FiniteDimL1LinfDuality.Basic","a":"Mathlib/Analysis/FiniteDimL1LinfDuality","t":2,"d":"Every element of the primal set is ≥ 0 (a sum of absolute values), so the set is bounded below by 0.","s":"∀ {k β : ℕ} {p : Fin (k + 1) → ℝ}, BddBelow (Causalean.Mathlib.Analysis.FiniteDimL1LinfDuality.primalNormSet p β)","l":83,"f":"Causalean/Mathlib/Analysis/FiniteDimL1LinfDuality/Basic.lean"},{"n":"Causalean.Mathlib.Analysis.FiniteDimL1LinfDuality.primalNormSet_nonempty","k":"theorem","m":"Causalean.Mathlib.Analysis.FiniteDimL1LinfDuality.Basic","a":"Mathlib/Analysis/FiniteDimL1LinfDuality","t":2,"d":"The primal set is nonempty whenever the moment system is solvable.","s":"∀ {k β : ℕ} {p : Fin (k + 1) → ℝ}, Function.Injective p → β ≤ k → (Causalean.Mathlib.Analysis.FiniteDimL1LinfDuality.primalNormSet p β).Nonempty","l":140,"f":"Causalean/Mathlib/Analysis/FiniteDimL1LinfDuality/Basic.lean"},{"n":"Causalean.Mathlib.Analysis.FiniteDimL1LinfDuality.primalNormSet_nonempty_of_momentSol_nonempty","k":"theorem","m":"Causalean.Mathlib.Analysis.FiniteDimL1LinfDuality.NonemptyDuality","a":"Mathlib/Analysis/FiniteDimL1LinfDuality","t":2,"d":"If the moment system is feasible, the primal norm set is nonempty.","s":"∀ {k β : ℕ} {p : Fin (k + 1) → ℝ}, (Causalean.Mathlib.Analysis.FiniteDimL1LinfDuality.MomentSol p β).Nonempty → (Causalean.Mathlib.Analysis.FiniteDimL1LinfDuality.primalNormSet p β).Nonempty","l":40,"f":"Causalean/Mathlib/Analysis/FiniteDimL1LinfDuality/NonemptyDuality.lean"},{"n":"Causalean.Mathlib.Analysis.FiniteDimL1LinfDuality.primalNormSet_nonneg","k":"theorem","m":"Causalean.Mathlib.Analysis.FiniteDimL1LinfDuality.Basic","a":"Mathlib/Analysis/FiniteDimL1LinfDuality","t":2,"d":"0 is a lower bound for the primal set.","s":"∀ {k β : ℕ} {p : Fin (k + 1) → ℝ} {s : ℝ}, s ∈ Causalean.Mathlib.Analysis.FiniteDimL1LinfDuality.primalNormSet p β → 0 ≤ s","l":90,"f":"Causalean/Mathlib/Analysis/FiniteDimL1LinfDuality/Basic.lean"},{"n":"Causalean.Mathlib.Analysis.FiniteDimL1LinfDuality.repr_identity","k":"theorem","m":"Causalean.Mathlib.Analysis.FiniteDimL1LinfDuality.WeakDuality","a":"Mathlib/Analysis/FiniteDimL1LinfDuality","t":1,"d":"Representation identity. If a weight vector solves the moment system for the given nodes and degree bound and a real polynomial has degree at most β, then the polynomial's endpoint contrast, r.eval 1 - r.eval 0, is reproduced exactly by the weighted sum of its values at the nodes.","s":"∀ {k β : ℕ} {p w : Fin (k + 1) → ℝ}, w ∈ Causalean.Mathlib.Analysis.FiniteDimL1LinfDuality.MomentSol p β → ∀ {r : Polynomial ℝ}, r.natDegree ≤ β → Polynomial.eval 1 r - Polynomial.eval 0 r = ∑ j, w j * Polynomial.eval (p j) r","l":29,"f":"Causalean/Mathlib/Analysis/FiniteDimL1LinfDuality/WeakDuality.lean"},{"n":"Causalean.Mathlib.Analysis.FiniteDimL1LinfDuality.sInf_primal_le_sSup_dual","k":"theorem","m":"Causalean.Mathlib.Analysis.FiniteDimL1LinfDuality.StrongDuality","a":"Mathlib/Analysis/FiniteDimL1LinfDuality","t":1,"d":"Strong duality (inequality form). For k + 1 pairwise distinct real nodes and a degree bound β at most k, the least ℓ¹ norm among feasible weight vectors is at most the dual supremum.","s":"∀ {k β : ℕ} {p : Fin (k + 1) → ℝ}, Function.Injective p → β ≤ k → sInf (Causalean.Mathlib.Analysis.FiniteDimL1LinfDuality.primalNormSet p β) ≤ sSup (Causalean.Mathlib.Analysis.FiniteDimL1LinfDuality.dualValSet p β)","l":29,"f":"Causalean/Mathlib/Analysis/FiniteDimL1LinfDuality/StrongDuality.lean"},{"n":"Causalean.Mathlib.Analysis.FiniteDimL1LinfDuality.sInf_primal_le_sSup_dual_of_momentSol_nonempty","k":"theorem","m":"Causalean.Mathlib.Analysis.FiniteDimL1LinfDuality.NonemptyDuality","a":"Mathlib/Analysis/FiniteDimL1LinfDuality","t":2,"d":"Inequality form of strong duality under moment-system feasibility.","s":"∀ {k β : ℕ} {p : Fin (k + 1) → ℝ}, (Causalean.Mathlib.Analysis.FiniteDimL1LinfDuality.MomentSol p β).Nonempty → sInf (Causalean.Mathlib.Analysis.FiniteDimL1LinfDuality.primalNormSet p β) ≤ sSup (Causalean.Mathlib.Analysis.FiniteDimL1LinfDuality.dualValSet p β)","l":285,"f":"Causalean/Mathlib/Analysis/FiniteDimL1LinfDuality/NonemptyDuality.lean"},{"n":"Causalean.Mathlib.Analysis.FiniteDimL1LinfDuality.sSup_dual_le_sInf_primal","k":"theorem","m":"Causalean.Mathlib.Analysis.FiniteDimL1LinfDuality.WeakDuality","a":"Mathlib/Analysis/FiniteDimL1LinfDuality","t":1,"d":"Weak duality (inequality form). If the primal set of achievable ℓ¹ norms is nonempty, then the dual supremum is at most the primal infimum: every dual value is bounded by every primal value, hence so is the supremum of dual values by the infimum of primal values.","s":"∀ {k β : ℕ} {p : Fin (k + 1) → ℝ}, (Causalean.Mathlib.Analysis.FiniteDimL1LinfDuality.primalNormSet p β).Nonempty → sSup (Causalean.Mathlib.Analysis.FiniteDimL1LinfDuality.dualValSet p β) ≤ sInf (Causalean.Mathlib.Analysis.FiniteDimL1LinfDuality.primalNormSet p β)","l":131,"f":"Causalean/Mathlib/Analysis/FiniteDimL1LinfDuality/WeakDuality.lean"},{"n":"Causalean.Mathlib.Analysis.FiniteDimL1LinfDuality.sSup_dual_le_sInf_primal_of_momentSol_nonempty","k":"theorem","m":"Causalean.Mathlib.Analysis.FiniteDimL1LinfDuality.NonemptyDuality","a":"Mathlib/Analysis/FiniteDimL1LinfDuality","t":2,"d":"Weak duality as an sSup ≤ sInf inequality under moment-system feasibility.","s":"∀ {k β : ℕ} {p : Fin (k + 1) → ℝ}, (Causalean.Mathlib.Analysis.FiniteDimL1LinfDuality.MomentSol p β).Nonempty → sSup (Causalean.Mathlib.Analysis.FiniteDimL1LinfDuality.dualValSet p β) ≤ sInf (Causalean.Mathlib.Analysis.FiniteDimL1LinfDuality.primalNormSet p β)","l":294,"f":"Causalean/Mathlib/Analysis/FiniteDimL1LinfDuality/NonemptyDuality.lean"},{"n":"Causalean.Mathlib.Analysis.abs_clipIcc_neg_le","k":"theorem","m":"Causalean.Mathlib.Analysis.ClipInterval","a":"Mathlib/Analysis","t":2,"d":"Clipping to a symmetric band around zero produces a value whose magnitude is at most the half-width of the band.","s":"∀ {B : ℝ}, 0 ≤ B → ∀ (x : ℝ), |Causalean.Mathlib.Analysis.clipIcc (-B) B x| ≤ B","l":109,"f":"Causalean/Mathlib/Analysis/ClipInterval.lean"},{"n":"Causalean.Mathlib.Analysis.abs_clipIcc_sub_clipIcc_le","k":"theorem","m":"Causalean.Mathlib.Analysis.ClipInterval","a":"Mathlib/Analysis","t":2,"d":"Clipping two numbers to the same interval never increases the distance between them.","s":"∀ (a b x y : ℝ), |Causalean.Mathlib.Analysis.clipIcc a b x - Causalean.Mathlib.Analysis.clipIcc a b y| ≤ |x - y|","l":62,"f":"Causalean/Mathlib/Analysis/ClipInterval.lean"},{"n":"Causalean.Mathlib.Analysis.abs_clipIcc_sub_le","k":"theorem","m":"Causalean.Mathlib.Analysis.ClipInterval","a":"Mathlib/Analysis","t":1,"d":"Clipping toward an in-interval target does not increase distance. If a real number lies in the closed interval from a to b, then for any real x, clipping x into that interval moves it no farther from the target than x itself was.","s":"∀ {a b t : ℝ}, t ∈ Set.Icc a b → ∀ (x : ℝ), |Causalean.Mathlib.Analysis.clipIcc a b x - t| ≤ |x - t|","l":77,"f":"Causalean/Mathlib/Analysis/ClipInterval.lean"},{"n":"Causalean.Mathlib.Analysis.abs_min_max_sub_min_max_le","k":"theorem","m":"Causalean.Mathlib.Analysis.ClipInterval","a":"Mathlib/Analysis","t":2,"d":"Capping from above and then flooring from below is a contraction as well: two numbers put through that order of operations are never pushed further apart. Unlike clipIcc_eq_min_max, this needs no assumption on the two thresholds, which is what makes it usable for a propensity band [q, 1 - q] whose definition carries no q ≤ 1/2 hypothesis.","s":"∀ (a b x y : ℝ), |min b (max a x) - min b (max a y)| ≤ |x - y|","l":67,"f":"Causalean/Mathlib/Analysis/ClipInterval.lean"},{"n":"Causalean.Mathlib.Analysis.abs_weighted_inner_le","k":"theorem","m":"Causalean.Mathlib.Analysis.WeightedCauchySchwarz","a":"Mathlib/Analysis","t":1,"d":"Weighted Cauchy-Schwarz inequality on a finite support, square-root form. With nonnegative weights, the absolute value of the weighted inner product of two real-valued functions is at most the product of the square roots of their weighted sums of squares.","s":"∀ {ι : Type u_1} (s : Finset ι) (w f g : ι → ℝ), (∀ i ∈ s, 0 ≤ w i) → |∑ i ∈ s, w i * (f i * g i)| ≤ √(∑ i ∈ s, w i * f i ^ 2) * √(∑ i ∈ s, w i * g i ^ 2)","l":67,"f":"Causalean/Mathlib/Analysis/WeightedCauchySchwarz.lean"},{"n":"Causalean.Mathlib.Analysis.abs_weighted_mean_le_sqrt_weighted_sq","k":"theorem","m":"Causalean.Mathlib.Analysis.WeightedCauchySchwarz","a":"Mathlib/Analysis","t":2,"d":"Weighted mean is dominated by the weighted root-mean-square. For weights that are nonnegative and sum to at most one, the absolute value of the weighted average of a real-valued function is at most the square root of the weighted average of its square.","s":"∀ {ι : Type u_1} (s : Finset ι) (w f : ι → ℝ), (∀ i ∈ s, 0 ≤ w i) → ∀ (c : ℝ), ∑ i ∈ s, w i ≤ c → |∑ i ∈ s, w i * f i| ≤ √c * √(∑ i ∈ s, w i * f i ^ 2)","l":82,"f":"Causalean/Mathlib/Analysis/WeightedCauchySchwarz.lean"},{"n":"Causalean.Mathlib.Analysis.affine_of_continuous_of_second_diff_zero","k":"theorem","m":"Causalean.Mathlib.Analysis.FrechetFunctionalEquation","a":"Mathlib/Analysis","t":1,"d":"Fréchet functional equation, degree one. If f is a continuous function from the reals into a normed real vector space and its second forward difference vanishes: f(x+2s) + f(x) = 2·f(x+s) for all real x and s, then f is affine — there exist vectors a and b such that f(x) = a + x·b for every real x.","s":"∀ {E : Type u_1} [inst : NormedAddCommGroup E] [inst_1 : NormedSpace ℝ E] {f : ℝ → E}, Continuous f → (∀ (x s : ℝ), f (x + s + s) + f x = f (x + s) + f (x + s)) → ∃ a b, ∀ (x : ℝ), f x = a + x • b","l":27,"f":"Causalean/Mathlib/Analysis/FrechetFunctionalEquation.lean"},{"n":"Causalean.Mathlib.Analysis.bernD","k":"def","m":"Causalean.Mathlib.Analysis.BernoulliKL","a":"Mathlib/Analysis","t":1,"d":"The negative Bernoulli entropy at a probability is the sum of that probability times its logarithm and its complementary probability times the logarithm of the complement.","s":"ℝ → ℝ","l":61,"f":"Causalean/Mathlib/Analysis/BernoulliKL.lean"},{"n":"Causalean.Mathlib.Analysis.bernoulliKL_eq_bregman","k":"theorem","m":"Causalean.Mathlib.Analysis.BernoulliKL","a":"Mathlib/Analysis","t":2,"d":"Bernoulli Kullback–Leibler divergence equals the Bregman remainder of the negative entropy function at the second probability.","s":"∀ {p q : ℝ}, p ≠ 0 → 1 - p ≠ 0 → q ≠ 0 → 1 - q ≠ 0 → p * Real.log (p / q) + (1 - p) * Real.log ((1 - p) / (1 - q)) = Causalean.Mathlib.Analysis.bernD p - Causalean.Mathlib.Analysis.bernD q - (Real.log q - Real.log (1 - q)) * (p - q)","l":183,"f":"Causalean/Mathlib/Analysis/BernoulliKL.lean"},{"n":"Causalean.Mathlib.Analysis.bernoulli_kl_le_four_sq_sub_of_mem_quarter_band","k":"theorem","m":"Causalean.Mathlib.Analysis.BernoulliKL","a":"Mathlib/Analysis","t":1,"d":"For probabilities p and q both restricted to the band [1/4, 3/4], the Bernoulli Kullback–Leibler divergence p · log(p / q) + (1 - p) · log((1 - p) / (1 - q)) is bounded above by 4 · (p - q) ^ 2.","s":"∀ {p q : ℝ}, 1 / 4 ≤ p → p ≤ 3 / 4 → 1 / 4 ≤ q → q ≤ 3 / 4 → p * Real.log (p / q) + (1 - p) * Real.log ((1 - p) / (1 - q)) ≤ 4 * (p - q) ^ 2","l":193,"f":"Causalean/Mathlib/Analysis/BernoulliKL.lean"},{"n":"Causalean.Mathlib.Analysis.clipIcc","k":"def","m":"Causalean.Mathlib.Analysis.ClipInterval","a":"Mathlib/Analysis","t":1,"d":"Clip to an interval. clipIcc a b x is x pushed into the closed interval from a to b: it returns a when x falls below a, b when x exceeds b, and x itself otherwise. It is the plain real-valued form of the projection onto the interval.","s":"ℝ → ℝ → ℝ → ℝ","l":31,"f":"Causalean/Mathlib/Analysis/ClipInterval.lean"},{"n":"Causalean.Mathlib.Analysis.clipIcc_eq_min_max","k":"theorem","m":"Causalean.Mathlib.Analysis.ClipInterval","a":"Mathlib/Analysis","t":2,"d":"Clipping from below first and then from above gives the same answer as clipping from above first and then from below. This reconciles the two spellings of the same operation that occur in practice, max a (min b x) and min b (max a x).","s":"∀ {a b : ℝ}, a ≤ b → ∀ (x : ℝ), Causalean.Mathlib.Analysis.clipIcc a b x = min b (max a x)","l":51,"f":"Causalean/Mathlib/Analysis/ClipInterval.lean"},{"n":"Causalean.Mathlib.Analysis.clipIcc_eq_self","k":"theorem","m":"Causalean.Mathlib.Analysis.ClipInterval","a":"Mathlib/Analysis","t":2,"d":"Clipping leaves untouched any number that already lies inside the interval.","s":"∀ {a b x : ℝ}, x ∈ Set.Icc a b → Causalean.Mathlib.Analysis.clipIcc a b x = x","l":47,"f":"Causalean/Mathlib/Analysis/ClipInterval.lean"},{"n":"Causalean.Mathlib.Analysis.clipIcc_mem_Icc","k":"theorem","m":"Causalean.Mathlib.Analysis.ClipInterval","a":"Mathlib/Analysis","t":2,"d":"A clipped value always lies in the interval it was clipped to (provided the interval is nonempty, i.e. its left endpoint does not exceed its right endpoint).","s":"∀ {a b : ℝ}, a ≤ b → ∀ (x : ℝ), Causalean.Mathlib.Analysis.clipIcc a b x ∈ Set.Icc a b","l":42,"f":"Causalean/Mathlib/Analysis/ClipInterval.lean"},{"n":"Causalean.Mathlib.Analysis.clipIcc_neg_eq_self","k":"theorem","m":"Causalean.Mathlib.Analysis.ClipInterval","a":"Mathlib/Analysis","t":2,"d":"Clipping to a symmetric band around zero leaves untouched any number whose magnitude is already within the half-width of the band.","s":"∀ {x B : ℝ}, |x| ≤ B → Causalean.Mathlib.Analysis.clipIcc (-B) B x = x","l":114,"f":"Causalean/Mathlib/Analysis/ClipInterval.lean"},{"n":"Causalean.Mathlib.Analysis.clipIcc_sub_sq_le","k":"theorem","m":"Causalean.Mathlib.Analysis.ClipInterval","a":"Mathlib/Analysis","t":1,"d":"Squared version: clipping toward an in-interval target does not increase squared distance. If a real number lies in the closed interval from a to b, then for any real x, the squared distance from the clipped value of x to the target is at most the squared distance from x itself to the target.","s":"∀ {a b t : ℝ}, t ∈ Set.Icc a b → ∀ (x : ℝ), (Causalean.Mathlib.Analysis.clipIcc a b x - t) ^ 2 ≤ (x - t) ^ 2","l":87,"f":"Causalean/Mathlib/Analysis/ClipInterval.lean"},{"n":"Causalean.Mathlib.Analysis.closedHalfDisc_weighted_first_div_radius_cancellation","k":"theorem","m":"Causalean.Mathlib.Analysis.HalfDiscPolar","a":"Mathlib/Analysis","t":2,"d":"Replacing the open diameter of an upper half-disc by the closed diameter does not affect the Cartesian cosine integral.","s":"∀ (g : ℝ → ℝ) (r : ℝ), ∫ (z : ℝ × ℝ) in {z | 0 ≤ z.2 ∧ Causalean.Mathlib.Analysis.planarRadius z ≤ r}, g (Causalean.Mathlib.Analysis.planarRadius z) * (z.1 / Causalean.Mathlib.Analysis.planarRadius z) = 0","l":218,"f":"Causalean/Mathlib/Analysis/HalfDiscPolar.lean"},{"n":"Causalean.Mathlib.Analysis.coe_projIcc_eq_clipIcc","k":"theorem","m":"Causalean.Mathlib.Analysis.ClipInterval","a":"Mathlib/Analysis","t":2,"d":"Clipping into an interval is exactly Mathlib's projection onto that interval, read as a plain real number instead of as an element of the interval. This is the bridge that makes every fact about the projection available to a development that never leaves the reals.","s":"∀ {a b : ℝ} (hab : a ≤ b) (x : ℝ), ↑(Set.projIcc a b hab x) = Causalean.Mathlib.Analysis.clipIcc a b x","l":36,"f":"Causalean/Mathlib/Analysis/ClipInterval.lean"},{"n":"Causalean.Mathlib.Analysis.contDiff_one_fderiv_apply","k":"theorem","m":"Causalean.Mathlib.Analysis.LineSecondDeriv","a":"Mathlib/Analysis","t":2,"d":"x ↦ fderiv ℝ f x d is Cⁿ when f is Cⁿ⁺¹. It is the composition of the derivative map with the continuous linear evaluation L ↦ L d.","s":"∀ {E : Type u_1} [inst : NormedAddCommGroup E] [inst_1 : NormedSpace ℝ E] {n : ℕ} {f : E → ℝ}, ContDiff ℝ (↑n + 1) f → ∀ (d : E), ContDiff ℝ ↑n fun x => (fderiv ℝ f x) d","l":59,"f":"Causalean/Mathlib/Analysis/LineSecondDeriv.lean"},{"n":"Causalean.Mathlib.Analysis.continuous_clipIcc","k":"theorem","m":"Causalean.Mathlib.Analysis.ClipInterval","a":"Mathlib/Analysis","t":2,"d":"Clipping to a fixed interval is a continuous operation.","s":"∀ (a b : ℝ), Continuous (Causalean.Mathlib.Analysis.clipIcc a b)","l":100,"f":"Causalean/Mathlib/Analysis/ClipInterval.lean"},{"n":"Causalean.Mathlib.Analysis.continuous_convexProj","k":"theorem","m":"Causalean.Mathlib.Analysis.ConvexProjection","a":"Mathlib/Analysis","t":2,"d":"Metric projection onto a nonempty closed convex set varies continuously with the point being projected.","s":"∀ {E : Type u_1} [inst : NormedAddCommGroup E] [inst_1 : InnerProductSpace ℝ E] [inst_2 : CompleteSpace E] (K : Set E) (hne : K.Nonempty) (hc : IsClosed K) (hconv : Convex ℝ K), Continuous (Causalean.Mathlib.Analysis.convexProj K hne hc hconv)","l":109,"f":"Causalean/Mathlib/Analysis/ConvexProjection.lean"},{"n":"Causalean.Mathlib.Analysis.continuous_monomial","k":"theorem","m":"Causalean.Mathlib.Analysis.MonomialGram","a":"Mathlib/Analysis","t":2,"d":"A multivariate real monomial varies continuously with its coordinate vector.","s":"∀ {ι : Type u_1} [inst : Fintype ι] (e : ι → ℕ), Continuous (Causalean.Mathlib.Analysis.monomial e)","l":51,"f":"Causalean/Mathlib/Analysis/MonomialGram.lean"},{"n":"Causalean.Mathlib.Analysis.continuous_monomialCombination","k":"theorem","m":"Causalean.Mathlib.Analysis.MonomialGram","a":"Mathlib/Analysis","t":2,"d":"A finite linear combination of multivariate real monomials is continuous.","s":"∀ {d p : ℕ} (expo : Fin p → Fin d → ℕ) (z : Fin p → ℝ), Continuous fun u => ∑ k, z k * Causalean.Mathlib.Analysis.monomial (expo k) u","l":57,"f":"Causalean/Mathlib/Analysis/MonomialGram.lean"},{"n":"Causalean.Mathlib.Analysis.continuous_secondDirDeriv","k":"theorem","m":"Causalean.Mathlib.Analysis.LineSecondDeriv","a":"Mathlib/Analysis","t":2,"d":"The curvature modulus is continuous in the base point. Hence bounded on any compact set, which supplies the BddAbove hypothesis of le_ciSup.","s":"∀ {E : Type u_1} [inst : NormedAddCommGroup E] [inst_1 : NormedSpace ℝ E] {f : E → ℝ}, ContDiff ℝ 2 f → ∀ (d : E), Continuous (Causalean.Mathlib.Analysis.secondDirDeriv f d)","l":85,"f":"Causalean/Mathlib/Analysis/LineSecondDeriv.lean"},{"n":"Causalean.Mathlib.Analysis.convexOn_deriv2_nonneg","k":"theorem","m":"Causalean.Mathlib.Analysis.LineSecondDeriv","a":"Mathlib/Analysis","t":1,"d":"A convex function of one real variable has nonnegative second derivative at interior points of its domain. If g is convex on the set S, differentiable at every point of S, x lies in the interior of S, and the derivative of g is itself differentiable at x, then the second derivative of g at x is nonnegative.","s":"∀ {S : Set ℝ} {g : ℝ → ℝ} {x : ℝ}, ConvexOn ℝ S g → (∀ y ∈ S, DifferentiableAt ℝ g y) → x ∈ interior S → DifferentiableAt ℝ (deriv g) x → 0 ≤ deriv (deriv g) x","l":92,"f":"Causalean/Mathlib/Analysis/LineSecondDeriv.lean"},{"n":"Causalean.Mathlib.Analysis.convexProj","k":"def","m":"Causalean.Mathlib.Analysis.ConvexProjection","a":"Mathlib/Analysis","t":1,"d":"Metric projection assigns each point in a real Hilbert space its nearest point in a specified nonempty closed convex set.","s":"{E : Type u_1} → [inst : NormedAddCommGroup E] → [inst_1 : InnerProductSpace ℝ E] → [CompleteSpace E] → (K : Set E) → K.Nonempty → IsClosed K → Convex ℝ K → E → E","l":35,"f":"Causalean/Mathlib/Analysis/ConvexProjection.lean"},{"n":"Causalean.Mathlib.Analysis.convexProj_eq_self","k":"theorem","m":"Causalean.Mathlib.Analysis.ConvexProjection","a":"Mathlib/Analysis","t":2,"d":"Every point in a nonempty closed convex target set is unchanged by metric projection onto that set.","s":"∀ {E : Type u_1} [inst : NormedAddCommGroup E] [inst_1 : InnerProductSpace ℝ E] [inst_2 : CompleteSpace E] (K : Set E) (hne : K.Nonempty) (hc : IsClosed K) (hconv : Convex ℝ K), ∀ x ∈ K, Causalean.Mathlib.Analysis.convexProj K hne hc hconv x = x","l":76,"f":"Causalean/Mathlib/Analysis/ConvexProjection.lean"},{"n":"Causalean.Mathlib.Analysis.convexProj_le_of_mem","k":"theorem","m":"Causalean.Mathlib.Analysis.ConvexProjection","a":"Mathlib/Analysis","t":2,"d":"Projecting a point onto a nonempty closed convex set cannot increase its distance from any point already in that set.","s":"∀ {E : Type u_1} [inst : NormedAddCommGroup E] [inst_1 : InnerProductSpace ℝ E] [inst_2 : CompleteSpace E] (K : Set E) (hne : K.Nonempty) (hc : IsClosed K) (hconv : Convex ℝ K) (x y : E), y ∈ K → ‖Causalean.Mathlib.Analysis.convexProj K hne hc hconv x - y‖ ≤ ‖x - y‖","l":62,"f":"Causalean/Mathlib/Analysis/ConvexProjection.lean"},{"n":"Causalean.Mathlib.Analysis.convexProj_lipschitz","k":"theorem","m":"Causalean.Mathlib.Analysis.ConvexProjection","a":"Mathlib/Analysis","t":2,"d":"Metric projection onto a nonempty closed convex set is nonexpansive: projected distances never exceed the original distances.","s":"∀ {E : Type u_1} [inst : NormedAddCommGroup E] [inst_1 : InnerProductSpace ℝ E] [inst_2 : CompleteSpace E] (K : Set E) (hne : K.Nonempty) (hc : IsClosed K) (hconv : Convex ℝ K), LipschitzWith 1 (Causalean.Mathlib.Analysis.convexProj K hne hc hconv)","l":84,"f":"Causalean/Mathlib/Analysis/ConvexProjection.lean"},{"n":"Causalean.Mathlib.Analysis.convexProj_mem","k":"theorem","m":"Causalean.Mathlib.Analysis.ConvexProjection","a":"Mathlib/Analysis","t":2,"d":"The metric projection of every point belongs to its nonempty closed convex target set.","s":"∀ {E : Type u_1} [inst : NormedAddCommGroup E] [inst_1 : InnerProductSpace ℝ E] [inst_2 : CompleteSpace E] (K : Set E) (hne : K.Nonempty) (hc : IsClosed K) (hconv : Convex ℝ K) (x : E), Causalean.Mathlib.Analysis.convexProj K hne hc hconv x ∈ K","l":49,"f":"Causalean/Mathlib/Analysis/ConvexProjection.lean"},{"n":"Causalean.Mathlib.Analysis.convexProj_spec","k":"theorem","m":"Causalean.Mathlib.Analysis.ConvexProjection","a":"Mathlib/Analysis","t":2,"d":"A point's metric projection onto a nonempty closed convex set belongs to that set and attains the smallest possible distance from the point among all points in the set.","s":"∀ {E : Type u_1} [inst : NormedAddCommGroup E] [inst_1 : InnerProductSpace ℝ E] [inst_2 : CompleteSpace E] (K : Set E) (hne : K.Nonempty) (hc : IsClosed K) (hconv : Convex ℝ K) (x : E), Causalean.Mathlib.Analysis.convexProj K hne hc hconv x ∈ K ∧ ‖x - Causalean.Mathlib.Analysis.convexProj K hne hc hconv x‖ = ⨅ y, ‖x - ↑y‖","l":41,"f":"Causalean/Mathlib/Analysis/ConvexProjection.lean"},{"n":"Causalean.Mathlib.Analysis.convexProj_variational","k":"theorem","m":"Causalean.Mathlib.Analysis.ConvexProjection","a":"Mathlib/Analysis","t":2,"d":"The residual from a point to its metric projection has nonpositive inner product with every feasible direction based at the projection.","s":"∀ {E : Type u_1} [inst : NormedAddCommGroup E] [inst_1 : InnerProductSpace ℝ E] [inst_2 : CompleteSpace E] (K : Set E) (hne : K.Nonempty) (hc : IsClosed K) (hconv : Convex ℝ K) (x y : E), y ∈ K → inner ℝ (x - Causalean.Mathlib.Analysis.convexProj K hne hc hconv x) (y - Causalean.Mathlib.Analysis.convexProj K hne hc hconv x) ≤ 0","l":54,"f":"Causalean/Mathlib/Analysis/ConvexProjection.lean"},{"n":"Causalean.Mathlib.Analysis.deriv_deriv_line","k":"theorem","m":"Causalean.Mathlib.Analysis.LineSecondDeriv","a":"Mathlib/Analysis","t":2,"d":"Second directional derivative along a line. The second derivative of the line restriction at parameter t is the second directional derivative of f at the moving base point q + t • d. Note this is an identity of deriv (not derivWithin): it holds at every t because the line restriction is differentiable on all of ℝ.","s":"∀ {E : Type u_1} [inst : NormedAddCommGroup E] [inst_1 : NormedSpace ℝ E] {f : E → ℝ}, ContDiff ℝ 2 f → ∀ (q d : E) (t : ℝ), deriv (deriv fun s => f (q + s • d)) t = Causalean.Mathlib.Analysis.secondDirDeriv f d (q + t • d)","l":72,"f":"Causalean/Mathlib/Analysis/LineSecondDeriv.lean"},{"n":"Causalean.Mathlib.Analysis.deriv_line","k":"theorem","m":"Causalean.Mathlib.Analysis.LineSecondDeriv","a":"Mathlib/Analysis","t":2,"d":"First directional derivative along a line.","s":"∀ {E : Type u_1} [inst : NormedAddCommGroup E] [inst_1 : NormedSpace ℝ E] {f : E → ℝ} (q d : E) (t : ℝ), DifferentiableAt ℝ f (q + t • d) → deriv (fun s => f (q + s • d)) t = (fderiv ℝ f (q + t • d)) d","l":52,"f":"Causalean/Mathlib/Analysis/LineSecondDeriv.lean"},{"n":"Causalean.Mathlib.Analysis.descentStep","k":"def","m":"Causalean.Mathlib.Analysis.SecondOrderDescent","a":"Mathlib/Analysis","t":1,"d":"The quadratic-model step chooses the endpoint when curvature is zero and otherwise clips the unconstrained optimizer at the available interval length.","s":"ℝ → ℝ → ℝ → ℝ","l":24,"f":"Causalean/Mathlib/Analysis/SecondOrderDescent.lean"},{"n":"Causalean.Mathlib.Analysis.descentStep_mem_Icc","k":"theorem","m":"Causalean.Mathlib.Analysis.SecondOrderDescent","a":"Mathlib/Analysis","t":2,"d":"The clipped quadratic-model step is feasible whenever the interval length, slope margin, and curvature bound are nonnegative.","s":"∀ {M c T : ℝ}, 0 ≤ M → 0 ≤ c → 0 ≤ T → Causalean.Mathlib.Analysis.descentStep M c T ∈ Set.Icc 0 T","l":43,"f":"Causalean/Mathlib/Analysis/SecondOrderDescent.lean"},{"n":"Causalean.Mathlib.Analysis.descentStep_of_eq_zero","k":"theorem","m":"Causalean.Mathlib.Analysis.SecondOrderDescent","a":"Mathlib/Analysis","t":2,"d":"With zero curvature bound, the clipped quadratic-model step is the endpoint.","s":"∀ (c T : ℝ), Causalean.Mathlib.Analysis.descentStep 0 c T = T","l":32,"f":"Causalean/Mathlib/Analysis/SecondOrderDescent.lean"},{"n":"Causalean.Mathlib.Analysis.descentStep_of_pos","k":"theorem","m":"Causalean.Mathlib.Analysis.SecondOrderDescent","a":"Mathlib/Analysis","t":2,"d":"With a positive curvature bound, the clipped quadratic-model step is the smaller of the available interval length and the unconstrained quadratic-model optimizer.","s":"∀ {M : ℝ}, 0 < M → ∀ (c T : ℝ), Causalean.Mathlib.Analysis.descentStep M c T = min T (c / M)","l":37,"f":"Causalean/Mathlib/Analysis/SecondOrderDescent.lean"},{"n":"Causalean.Mathlib.Analysis.differentiable_fderiv_apply","k":"theorem","m":"Causalean.Mathlib.Analysis.LineSecondDeriv","a":"Mathlib/Analysis","t":2,"d":"x ↦ fderiv ℝ f x d is differentiable when f is C².","s":"∀ {E : Type u_1} [inst : NormedAddCommGroup E] [inst_1 : NormedSpace ℝ E] {f : E → ℝ}, ContDiff ℝ 2 f → ∀ (d : E), Differentiable ℝ fun x => (fderiv ℝ f x) d","l":67,"f":"Causalean/Mathlib/Analysis/LineSecondDeriv.lean"},{"n":"Causalean.Mathlib.Analysis.exists_monomialGram_coercive","k":"theorem","m":"Causalean.Mathlib.Analysis.MonomialGram","a":"Mathlib/Analysis","t":2,"d":"A fixed family of distinct monomials on a cube with positive radius admits a positive uniform lower bound: its Gram quadratic form dominates the squared Euclidean norm of the coefficients.","s":"∀ {d p : ℕ} (expo : Fin p → Fin d → ℕ) {r : ℝ}, 0 < r → Function.Injective expo → ∃ cmin, 0 < cmin ∧ ∀ (z : Fin p → ℝ), cmin * ∑ k, z k ^ 2 ≤ ∑ k, ∑ l, z k * Causalean.Mathlib.Analysis.monomialGram expo r k l * z l","l":241,"f":"Causalean/Mathlib/Analysis/MonomialGram.lean"},{"n":"Causalean.Mathlib.Analysis.first_order_descent_gap","k":"theorem","m":"Causalean.Mathlib.Analysis.SecondOrderDescent","a":"Mathlib/Analysis","t":2,"d":"With zero curvature, the endpoint step gives the full linear descent guaranteed by the negative initial slope.","s":"∀ {f : ℝ → ℝ} {c T : ℝ}, 0 ≤ T → ContDiffOn ℝ 2 f (Set.Icc 0 T) → DifferentiableAt ℝ f 0 → (∀ t ∈ Set.Ioo 0 T, deriv (deriv f) t ≤ 0) → deriv f 0 ≤ -c → f 0 - f T ≥ c * T","l":162,"f":"Causalean/Mathlib/Analysis/SecondOrderDescent.lean"},{"n":"Causalean.Mathlib.Analysis.first_order_upper_bound","k":"theorem","m":"Causalean.Mathlib.Analysis.SecondOrderDescent","a":"Mathlib/Analysis","t":2,"d":"A real function with nonpositive second derivative in an interval's interior lies below its tangent line at the left endpoint.","s":"∀ {f : ℝ → ℝ} {T : ℝ}, ContDiffOn ℝ 2 f (Set.Icc 0 T) → DifferentiableAt ℝ f 0 → (∀ t ∈ Set.Ioo 0 T, deriv (deriv f) t ≤ 0) → ∀ {t : ℝ}, t ∈ Set.Icc 0 T → f t ≤ f 0 + deriv f 0 * t","l":100,"f":"Causalean/Mathlib/Analysis/SecondOrderDescent.lean"},{"n":"Causalean.Mathlib.Analysis.frobDist","k":"def","m":"Causalean.Mathlib.Analysis.ConvexProjection","a":"Mathlib/Analysis","t":1,"d":"The Frobenius distance between two real matrices is the square root of the sum of the squared entrywise differences.","s":"{p : ℕ} → Matrix (Fin p) (Fin p) ℝ → Matrix (Fin p) (Fin p) ℝ → ℝ","l":134,"f":"Causalean/Mathlib/Analysis/ConvexProjection.lean"},{"n":"Causalean.Mathlib.Analysis.frobDist_comm","k":"theorem","m":"Causalean.Mathlib.Analysis.ConvexProjection","a":"Mathlib/Analysis","t":2,"d":"Frobenius distance is symmetric in its two matrix arguments.","s":"∀ {p : ℕ} (A B : Matrix (Fin p) (Fin p) ℝ), Causalean.Mathlib.Analysis.frobDist A B = Causalean.Mathlib.Analysis.frobDist B A","l":165,"f":"Causalean/Mathlib/Analysis/ConvexProjection.lean"},{"n":"Causalean.Mathlib.Analysis.frobDist_eq_norm","k":"theorem","m":"Causalean.Mathlib.Analysis.ConvexProjection","a":"Mathlib/Analysis","t":2,"d":"The Frobenius distance between two finite real matrices equals the Euclidean norm of their difference after vectorizing the entries.","s":"∀ {p : ℕ} (A B : Matrix (Fin p) (Fin p) ℝ), Causalean.Mathlib.Analysis.frobDist A B = ‖(Causalean.Mathlib.Analysis.mtx p) A - (Causalean.Mathlib.Analysis.mtx p) B‖","l":152,"f":"Causalean/Mathlib/Analysis/ConvexProjection.lean"},{"n":"Causalean.Mathlib.Analysis.frobDist_nonneg","k":"theorem","m":"Causalean.Mathlib.Analysis.ConvexProjection","a":"Mathlib/Analysis","t":2,"d":"Frobenius distance is nonnegative.","s":"∀ {p : ℕ} (A B : Matrix (Fin p) (Fin p) ℝ), 0 ≤ Causalean.Mathlib.Analysis.frobDist A B","l":161,"f":"Causalean/Mathlib/Analysis/ConvexProjection.lean"},{"n":"Causalean.Mathlib.Analysis.frobDist_triangle","k":"theorem","m":"Causalean.Mathlib.Analysis.ConvexProjection","a":"Mathlib/Analysis","t":2,"d":"Frobenius distance satisfies the triangle inequality.","s":"∀ {p : ℕ} (A B D : Matrix (Fin p) (Fin p) ℝ), Causalean.Mathlib.Analysis.frobDist A D ≤ Causalean.Mathlib.Analysis.frobDist A B + Causalean.Mathlib.Analysis.frobDist B D","l":175,"f":"Causalean/Mathlib/Analysis/ConvexProjection.lean"},{"n":"Causalean.Mathlib.Analysis.halfDisc_cos_radial_cancellation","k":"theorem","m":"Causalean.Mathlib.Analysis.HalfDiscPolar","a":"Mathlib/Analysis","t":2,"d":"Multiplication by cosine contributes zero after angular integration on every radial shell.","s":"∀ (g : ℝ → ℝ) (r : ℝ), ∫ (s : ℝ) in Set.Ioc 0 r, s * g s * ∫ (θ : ℝ) in Set.Ioc 0 Real.pi, Real.cos θ = 0","l":381,"f":"Causalean/Mathlib/Analysis/HalfDiscPolar.lean"},{"n":"Causalean.Mathlib.Analysis.halfDisc_radialSet_weighted_cos_cancellation","k":"theorem","m":"Causalean.Mathlib.Analysis.HalfDiscPolar","a":"Mathlib/Analysis","t":2,"d":"A cosine angular tilt has zero mass on every measurable radial subset of an upper half-disc. This is the setwise interface used to identify radial pushforwards, rather than merely their total masses.","s":"∀ (g : ℝ → ℝ) (r : ℝ) {A : Set ℝ}, MeasurableSet A → ∫ (z : ℝ × ℝ) in {z | 0 < z.2 ∧ Causalean.Mathlib.Analysis.planarRadius z ≤ r} ∩ Causalean.Mathlib.Analysis.planarRadius ⁻¹' A, g (Causalean.Mathlib.Analysis.planarRadius z) * Real.cos (Causalean.Mathlib.Analysis.planarAngle z) = 0","l":148,"f":"Causalean/Mathlib/Analysis/HalfDiscPolar.lean"},{"n":"Causalean.Mathlib.Analysis.halfDisc_radial_integral","k":"theorem","m":"Causalean.Mathlib.Analysis.HalfDiscPolar","a":"Mathlib/Analysis","t":2,"d":"A radial integrand on a half-disc admits the expected polar-coordinate decomposition.","s":"∀ (g : ℝ → ℝ) (r : ℝ), ∫ (z : ℝ × ℝ) in {z | 0 < z.2 ∧ Causalean.Mathlib.Analysis.planarRadius z ≤ r}, g (Causalean.Mathlib.Analysis.planarRadius z) = ∫ (s : ℝ) in Set.Ioc 0 r, Real.pi * s * g s","l":299,"f":"Causalean/Mathlib/Analysis/HalfDiscPolar.lean"},{"n":"Causalean.Mathlib.Analysis.halfDisc_weighted_cos_cancellation","k":"theorem","m":"Causalean.Mathlib.Analysis.HalfDiscPolar","a":"Mathlib/Analysis","t":2,"d":"A cosine angular tilt has zero integral against every radial weight on an upper half-disc.","s":"∀ (g : ℝ → ℝ) (r : ℝ), ∫ (z : ℝ × ℝ) in {z | 0 < z.2 ∧ Causalean.Mathlib.Analysis.planarRadius z ≤ r}, g (Causalean.Mathlib.Analysis.planarRadius z) * Real.cos (Causalean.Mathlib.Analysis.planarAngle z) = 0","l":136,"f":"Causalean/Mathlib/Analysis/HalfDiscPolar.lean"},{"n":"Causalean.Mathlib.Analysis.halfDisc_weighted_cos_sq","k":"theorem","m":"Causalean.Mathlib.Analysis.HalfDiscPolar","a":"Mathlib/Analysis","t":2,"d":"The cosine-squared angular moment converts a radial weight into the nonzero π/2 factor used to cancel the affine regression term.","s":"∀ (g : ℝ → ℝ) (r : ℝ), ∫ (z : ℝ × ℝ) in {z | 0 < z.2 ∧ Causalean.Mathlib.Analysis.planarRadius z ≤ r}, g (Causalean.Mathlib.Analysis.planarRadius z) * Real.cos (Causalean.Mathlib.Analysis.planarAngle z) ^ 2 = (∫ (s : ℝ) in Set.Ioc 0 r, s * g s) * (Real.pi / 2)","l":283,"f":"Causalean/Mathlib/Analysis/HalfDiscPolar.lean"},{"n":"Causalean.Mathlib.Analysis.halfDisc_weighted_first_div_radius_cancellation","k":"theorem","m":"Causalean.Mathlib.Analysis.HalfDiscPolar","a":"Mathlib/Analysis","t":2,"d":"A radial weight times the Cartesian direction cosine has zero integral on an open upper half-disc.","s":"∀ (g : ℝ → ℝ) (r : ℝ), ∫ (z : ℝ × ℝ) in {z | 0 < z.2 ∧ Causalean.Mathlib.Analysis.planarRadius z ≤ r}, g (Causalean.Mathlib.Analysis.planarRadius z) * (z.1 / Causalean.Mathlib.Analysis.planarRadius z) = 0","l":194,"f":"Causalean/Mathlib/Analysis/HalfDiscPolar.lean"},{"n":"Causalean.Mathlib.Analysis.halfDisc_weighted_polar_integral","k":"theorem","m":"Causalean.Mathlib.Analysis.HalfDiscPolar","a":"Mathlib/Analysis","t":1,"d":"For radial and angular weight functions g and h and a radius r, the integral of the product g(radius)·h(angle) over the open upper half-disc of radius r factors as the product of the radial integral ∫ s·g(s) ds over (0, r] and the angular integral ∫ h(θ) dθ over (0, π).","s":"∀ (g h : ℝ → ℝ) (r : ℝ), ∫ (z : ℝ × ℝ) in {z | 0 < z.2 ∧ Causalean.Mathlib.Analysis.planarRadius z ≤ r}, g (Causalean.Mathlib.Analysis.planarRadius z) * h (Causalean.Mathlib.Analysis.planarAngle z) = (∫ (s : ℝ) in Set.Ioc 0 r, s * g s) * ∫ (θ : ℝ) in Set.Ioo 0 Real.pi, h θ","l":52,"f":"Causalean/Mathlib/Analysis/HalfDiscPolar.lean"},{"n":"Causalean.Mathlib.Analysis.hasDerivAt_bernEntropy","k":"theorem","m":"Causalean.Mathlib.Analysis.BernoulliKL","a":"Mathlib/Analysis","t":2,"d":"Away from zero and one, the negative Bernoulli entropy has derivative equal to the log odds of its argument.","s":"∀ (x : ℝ), x ≠ 0 → 1 - x ≠ 0 → HasDerivAt (fun t => t * Real.log t + (1 - t) * Real.log (1 - t)) (Real.log x - Real.log (1 - x)) x","l":36,"f":"Causalean/Mathlib/Analysis/BernoulliKL.lean"},{"n":"Causalean.Mathlib.Analysis.hasDerivAt_line","k":"theorem","m":"Causalean.Mathlib.Analysis.LineSecondDeriv","a":"Mathlib/Analysis","t":2,"d":"The line s ↦ q + s • d is differentiable with derivative d, as a HasDerivAt.","s":"∀ {E : Type u_1} [inst : NormedAddCommGroup E] [inst_1 : NormedSpace ℝ E] (q d : E) (t : ℝ), HasDerivAt (fun s => q + s • d) d t","l":47,"f":"Causalean/Mathlib/Analysis/LineSecondDeriv.lean"},{"n":"Causalean.Mathlib.Analysis.integral_cos_sq_zero_to_pi","k":"theorem","m":"Causalean.Mathlib.Analysis.HalfDiscPolar","a":"Mathlib/Analysis","t":2,"d":"The quadratic cosine moment on a half-circle is π / 2.","s":"∫ (θ : ℝ) in Set.Ioc 0 Real.pi, Real.cos θ ^ 2 = Real.pi / 2","l":45,"f":"Causalean/Mathlib/Analysis/HalfDiscPolar.lean"},{"n":"Causalean.Mathlib.Analysis.integral_cos_zero_to_pi","k":"theorem","m":"Causalean.Mathlib.Analysis.HalfDiscPolar","a":"Mathlib/Analysis","t":2,"d":"The cosine has zero integral on a half-circle.","s":"∫ (θ : ℝ) in Set.Ioc 0 Real.pi, Real.cos θ = 0","l":36,"f":"Causalean/Mathlib/Analysis/HalfDiscPolar.lean"},{"n":"Causalean.Mathlib.Analysis.isClosed_posSemidef","k":"theorem","m":"Causalean.Mathlib.Analysis.ConvexProjection","a":"Mathlib/Analysis","t":2,"d":"Real positive-semidefinite matrices indexed by any finite set form a closed set in the coordinatewise topology.","s":"∀ {ι : Type u_1} [Finite ι], IsClosed {A | A.PosSemidef}","l":209,"f":"Causalean/Mathlib/Analysis/ConvexProjection.lean"},{"n":"Causalean.Mathlib.Analysis.lipschitzWith_clipIcc","k":"theorem","m":"Causalean.Mathlib.Analysis.ClipInterval","a":"Mathlib/Analysis","t":2,"d":"Clipping is a contraction: two numbers are never pushed further apart by being clipped to a common interval.","s":"∀ (a b : ℝ), LipschitzWith 1 (Causalean.Mathlib.Analysis.clipIcc a b)","l":57,"f":"Causalean/Mathlib/Analysis/ClipInterval.lean"},{"n":"Causalean.Mathlib.Analysis.loewnerProj","k":"def","m":"Causalean.Mathlib.Analysis.ConvexProjection","a":"Mathlib/Analysis","t":1,"d":"Loewner projection is the nearest-point projection in Frobenius geometry when the interval endpoints are ordered, and is the identity map when they are reversed.","s":"(p : ℕ) → ℝ → ℝ → Matrix (Fin p) (Fin p) ℝ → Matrix (Fin p) (Fin p) ℝ","l":264,"f":"Causalean/Mathlib/Analysis/ConvexProjection.lean"},{"n":"Causalean.Mathlib.Analysis.loewnerProj_eq_self","k":"theorem","m":"Causalean.Mathlib.Analysis.ConvexProjection","a":"Mathlib/Analysis","t":2,"d":"Every matrix already in a nonempty Loewner interval is unchanged by metric projection onto that interval.","s":"∀ {p : ℕ} {c C : ℝ}, c ≤ C → ∀ G ∈ Causalean.Mathlib.Analysis.loewnerSet p c C, Causalean.Mathlib.Analysis.loewnerProj p c C G = G","l":398,"f":"Causalean/Mathlib/Analysis/ConvexProjection.lean"},{"n":"Causalean.Mathlib.Analysis.loewnerProj_frobDist_le","k":"theorem","m":"Causalean.Mathlib.Analysis.ConvexProjection","a":"Mathlib/Analysis","t":1,"d":"Loewner projection is a nonexpansive (nearest-point) map. For ordered interval endpoints c ≤ C, if a target matrix already lies in the Loewner interval [cI, CI], then projecting an arbitrary matrix onto that interval, in Frobenius geometry, does not increase its Frobenius distance to the in-interval target matrix.","s":"∀ {p : ℕ} {c C : ℝ}, c ≤ C → ∀ (G S : Matrix (Fin p) (Fin p) ℝ), S ∈ Causalean.Mathlib.Analysis.loewnerSet p c C → Causalean.Mathlib.Analysis.frobDist (Causalean.Mathlib.Analysis.loewnerProj p c C G) S ≤ Causalean.Mathlib.Analysis.frobDist G S","l":282,"f":"Causalean/Mathlib/Analysis/ConvexProjection.lean"},{"n":"Causalean.Mathlib.Analysis.loewnerProj_mem","k":"theorem","m":"Causalean.Mathlib.Analysis.ConvexProjection","a":"Mathlib/Analysis","t":2,"d":"When its endpoints are ordered, Loewner projection sends every finite real matrix into the corresponding Loewner interval.","s":"∀ {p : ℕ} {c C : ℝ}, c ≤ C → ∀ (G : Matrix (Fin p) (Fin p) ℝ), Causalean.Mathlib.Analysis.loewnerProj p c C G ∈ Causalean.Mathlib.Analysis.loewnerSet p c C","l":270,"f":"Causalean/Mathlib/Analysis/ConvexProjection.lean"},{"n":"Causalean.Mathlib.Analysis.loewnerSet","k":"def","m":"Causalean.Mathlib.Analysis.ConvexProjection","a":"Mathlib/Analysis","t":1,"d":"A Loewner interval contains the real matrices bounded between two scalar multiples of the identity matrix in positive-semidefinite order.","s":"(p : ℕ) → ℝ → ℝ → Set (Matrix (Fin p) (Fin p) ℝ)","l":128,"f":"Causalean/Mathlib/Analysis/ConvexProjection.lean"},{"n":"Causalean.Mathlib.Analysis.loewnerSet_convex","k":"theorem","m":"Causalean.Mathlib.Analysis.ConvexProjection","a":"Mathlib/Analysis","t":2,"d":"Every Loewner interval of finite real matrices is convex.","s":"∀ (p : ℕ) (c C : ℝ), Convex ℝ (Causalean.Mathlib.Analysis.loewnerSet p c C)","l":194,"f":"Causalean/Mathlib/Analysis/ConvexProjection.lean"},{"n":"Causalean.Mathlib.Analysis.loewnerSet_inv_mulVec_norm_le","k":"theorem","m":"Causalean.Mathlib.Analysis.ConvexProjection","a":"Mathlib/Analysis","t":2,"d":"The inverse of a matrix in a Loewner interval with a positive lower endpoint expands Euclidean norm by at most the reciprocal of that endpoint.","s":"∀ {p : ℕ} {c : ℝ}, 0 < c → ∀ {G : Matrix (Fin p) (Fin p) ℝ}, (G - c • 1).PosSemidef → ∀ (v : Fin p → ℝ), √(∑ k, G⁻¹.mulVec v k ^ 2) ≤ √(∑ k, v k ^ 2) / c","l":334,"f":"Causalean/Mathlib/Analysis/ConvexProjection.lean"},{"n":"Causalean.Mathlib.Analysis.loewnerSet_isClosed","k":"theorem","m":"Causalean.Mathlib.Analysis.ConvexProjection","a":"Mathlib/Analysis","t":2,"d":"Every Loewner interval of finite real matrices is closed in the coordinatewise product topology.","s":"∀ (p : ℕ) (c C : ℝ), IsClosed (Causalean.Mathlib.Analysis.loewnerSet p c C)","l":236,"f":"Causalean/Mathlib/Analysis/ConvexProjection.lean"},{"n":"Causalean.Mathlib.Analysis.loewnerSet_nonempty","k":"theorem","m":"Causalean.Mathlib.Analysis.ConvexProjection","a":"Mathlib/Analysis","t":2,"d":"A Loewner interval with ordered scalar endpoints is nonempty.","s":"∀ (p : ℕ) (c C : ℝ), c ≤ C → (Causalean.Mathlib.Analysis.loewnerSet p c C).Nonempty","l":181,"f":"Causalean/Mathlib/Analysis/ConvexProjection.lean"},{"n":"Causalean.Mathlib.Analysis.loewnerSet_posDef","k":"theorem","m":"Causalean.Mathlib.Analysis.ConvexProjection","a":"Mathlib/Analysis","t":2,"d":"Every matrix in a Loewner interval with a strictly positive lower endpoint is positive definite.","s":"∀ {p : ℕ} {c : ℝ}, 0 < c → ∀ {G : Matrix (Fin p) (Fin p) ℝ}, (G - c • 1).PosSemidef → G.PosDef","l":242,"f":"Causalean/Mathlib/Analysis/ConvexProjection.lean"},{"n":"Causalean.Mathlib.Analysis.matrixBorelSpace","k":"instance","m":"Causalean.Mathlib.Analysis.ConvexProjection","a":"Mathlib/Analysis","t":2,"d":"The coordinatewise measurable structure on finite real matrices agrees with their Borel sigma-algebra.","s":"∀ (p : ℕ), BorelSpace (Matrix (Fin p) (Fin p) ℝ)","l":390,"f":"Causalean/Mathlib/Analysis/ConvexProjection.lean"},{"n":"Causalean.Mathlib.Analysis.matrixMeasurableSpace","k":"instance","m":"Causalean.Mathlib.Analysis.ConvexProjection","a":"Mathlib/Analysis","t":2,"d":"Finite real matrices carry the coordinatewise product sigma-algebra.","s":"(p : ℕ) → MeasurableSpace (Matrix (Fin p) (Fin p) ℝ)","l":385,"f":"Causalean/Mathlib/Analysis/ConvexProjection.lean"},{"n":"Causalean.Mathlib.Analysis.measurable_clipIcc","k":"theorem","m":"Causalean.Mathlib.Analysis.ClipInterval","a":"Mathlib/Analysis","t":2,"d":"Clipping to a fixed interval is measurable, so clipping a random variable again yields a random variable.","s":"∀ (a b : ℝ), Measurable (Causalean.Mathlib.Analysis.clipIcc a b)","l":104,"f":"Causalean/Mathlib/Analysis/ClipInterval.lean"},{"n":"Causalean.Mathlib.Analysis.measurable_convexProj","k":"theorem","m":"Causalean.Mathlib.Analysis.ConvexProjection","a":"Mathlib/Analysis","t":2,"d":"Metric projection onto a nonempty closed convex set is measurable under the Borel sigma-algebra of the Hilbert space.","s":"∀ {E : Type u_1} [inst : NormedAddCommGroup E] [inst_1 : InnerProductSpace ℝ E] [inst_2 : CompleteSpace E] [inst_3 : MeasurableSpace E] [BorelSpace E] (K : Set E) (hne : K.Nonempty) (hc : IsClosed K) (hconv : Convex ℝ K), Measurable (Causalean.Mathlib.Analysis.convexProj K hne hc hconv)","l":115,"f":"Causalean/Mathlib/Analysis/ConvexProjection.lean"},{"n":"Causalean.Mathlib.Analysis.measurable_loewnerProj","k":"theorem","m":"Causalean.Mathlib.Analysis.ConvexProjection","a":"Mathlib/Analysis","t":2,"d":"Loewner projection is measurable under the coordinatewise Borel sigma-algebra on finite real matrix spaces.","s":"∀ (p : ℕ) [inst : MeasurableSpace (Matrix (Fin p) (Fin p) ℝ)] [BorelSpace (Matrix (Fin p) (Fin p) ℝ)] (c C : ℝ), Measurable (Causalean.Mathlib.Analysis.loewnerProj p c C)","l":296,"f":"Causalean/Mathlib/Analysis/ConvexProjection.lean"},{"n":"Causalean.Mathlib.Analysis.monomial","k":"def","m":"Causalean.Mathlib.Analysis.MonomialGram","a":"Mathlib/Analysis","t":1,"d":"A multivariate monomial multiplies the coordinates of a vector after raising each coordinate to its assigned nonnegative integer exponent.","s":"{ι : Type u_1} → [Fintype ι] → (ι → ℕ) → (ι → ℝ) → ℝ","l":26,"f":"Causalean/Mathlib/Analysis/MonomialGram.lean"},{"n":"Causalean.Mathlib.Analysis.monomialGram","k":"def","m":"Causalean.Mathlib.Analysis.MonomialGram","a":"Mathlib/Analysis","t":1,"d":"The monomial Gram matrix records the integrals of every pairwise product of a finite family of multivariate monomials over a centered closed cube.","s":"{d p : ℕ} → (Fin p → Fin d → ℕ) → ℝ → Matrix (Fin p) (Fin p) ℝ","l":31,"f":"Causalean/Mathlib/Analysis/MonomialGram.lean"},{"n":"Causalean.Mathlib.Analysis.monomialGram_isHermitian","k":"theorem","m":"Causalean.Mathlib.Analysis.MonomialGram","a":"Mathlib/Analysis","t":2,"d":"The monomial Gram matrix is symmetric, and hence Hermitian over the real numbers.","s":"∀ {d p : ℕ} (expo : Fin p → Fin d → ℕ) (r : ℝ), (Causalean.Mathlib.Analysis.monomialGram expo r).IsHermitian","l":163,"f":"Causalean/Mathlib/Analysis/MonomialGram.lean"},{"n":"Causalean.Mathlib.Analysis.monomialGram_posDef","k":"theorem","m":"Causalean.Mathlib.Analysis.MonomialGram","a":"Mathlib/Analysis","t":1,"d":"Positive-definiteness of the monomial Gram matrix. For a finite family of multivariate monomials indexed by their exponent vectors, if the cube radius r is strictly positive and the exponent vectors are pairwise distinct, then the Gram matrix of pairwise integrals of the monomials over the cube of radius r is positive-definite.","s":"∀ {d p : ℕ} (expo : Fin p → Fin d → ℕ) {r : ℝ}, 0 < r → Function.Injective expo → (Causalean.Mathlib.Analysis.monomialGram expo r).PosDef","l":183,"f":"Causalean/Mathlib/Analysis/MonomialGram.lean"},{"n":"Causalean.Mathlib.Analysis.monomialGram_posSemidef","k":"theorem","m":"Causalean.Mathlib.Analysis.MonomialGram","a":"Mathlib/Analysis","t":2,"d":"Every coefficient vector gives a nonnegative quadratic form under the monomial Gram matrix.","s":"∀ {d p : ℕ} (expo : Fin p → Fin d → ℕ) {r : ℝ}, (Causalean.Mathlib.Analysis.monomialGram expo r).PosSemidef","l":171,"f":"Causalean/Mathlib/Analysis/MonomialGram.lean"},{"n":"Causalean.Mathlib.Analysis.monomialGram_quadForm","k":"theorem","m":"Causalean.Mathlib.Analysis.MonomialGram","a":"Mathlib/Analysis","t":2,"d":"Evaluating the monomial Gram quadratic form at a coefficient vector equals integrating the square of the corresponding monomial combination over the cube.","s":"∀ {d p : ℕ} (expo : Fin p → Fin d → ℕ) {r : ℝ} (z : Fin p → ℝ), ∑ k, ∑ l, z k * Causalean.Mathlib.Analysis.monomialGram expo r k l * z l = ∫ (u : Fin d → ℝ) in {u | ∀ (j : Fin d), |u j| ≤ r}, (∑ k, z k * Causalean.Mathlib.Analysis.monomial (expo k) u) ^ 2","l":123,"f":"Causalean/Mathlib/Analysis/MonomialGram.lean"},{"n":"Causalean.Mathlib.Analysis.monomial_linearIndependent_on_cube","k":"theorem","m":"Causalean.Mathlib.Analysis.MonomialGram","a":"Mathlib/Analysis","t":2,"d":"A linear combination of distinct multivariate monomials that vanishes throughout the interior of a cube with positive radius must have every coefficient equal to zero.","s":"∀ {d p : ℕ} {expo : Fin p → Fin d → ℕ} {r : ℝ}, 0 < r → Function.Injective expo → ∀ (z : Fin p → ℝ), (∀ (u : Fin d → ℝ), (∀ (j : Fin d), u j ∈ Set.Ioo (-r) r) → ∑ k, z k * Causalean.Mathlib.Analysis.monomial (expo k) u = 0) → z = 0","l":115,"f":"Causalean/Mathlib/Analysis/MonomialGram.lean"},{"n":"Causalean.Mathlib.Analysis.mtx","k":"def","m":"Causalean.Mathlib.Analysis.ConvexProjection","a":"Mathlib/Analysis","t":1,"d":"Finite real matrices are linearly equivalent to Euclidean vectors indexed by pairs of row and column coordinates.","s":"(p : ℕ) → Matrix (Fin p) (Fin p) ℝ ≃ₗ[ℝ] EuclideanSpace ℝ (Fin p × Fin p)","l":139,"f":"Causalean/Mathlib/Analysis/ConvexProjection.lean"},{"n":"Causalean.Mathlib.Analysis.mtxHomeo","k":"def","m":"Causalean.Mathlib.Analysis.ConvexProjection","a":"Mathlib/Analysis","t":1,"d":"Finite square real matrices of dimension p are homeomorphic to Euclidean vectors indexed by their row and column coordinates.","s":"(p : ℕ) → Matrix (Fin p) (Fin p) ℝ ≃ₜ EuclideanSpace ℝ (Fin p × Fin p)","l":146,"f":"Causalean/Mathlib/Analysis/ConvexProjection.lean"},{"n":"Causalean.Mathlib.Analysis.mulVec_sub_norm_le","k":"theorem","m":"Causalean.Mathlib.Analysis.ConvexProjection","a":"Mathlib/Analysis","t":2,"d":"Multiplying a vector by the difference of two finite real matrices has Euclidean norm at most their Frobenius distance times the vector's Euclidean norm.","s":"∀ {p : ℕ} (A B : Matrix (Fin p) (Fin p) ℝ) (v : Fin p → ℝ), √(∑ k, (A - B).mulVec v k ^ 2) ≤ Causalean.Mathlib.Analysis.frobDist A B * √(∑ k, v k ^ 2)","l":320,"f":"Causalean/Mathlib/Analysis/ConvexProjection.lean"},{"n":"Causalean.Mathlib.Analysis.planarAngle","k":"def","m":"Causalean.Mathlib.Analysis.HalfDiscPolar","a":"Mathlib/Analysis","t":1,"d":"The polar angle of a nonzero point, using Mathlib's polar-coordinate chart. Only its values on the open upper half-plane are used below.","s":"ℝ × ℝ → ℝ","l":30,"f":"Causalean/Mathlib/Analysis/HalfDiscPolar.lean"},{"n":"Causalean.Mathlib.Analysis.planarFirst_div_radius_eq_cos","k":"theorem","m":"Causalean.Mathlib.Analysis.HalfDiscPolar","a":"Mathlib/Analysis","t":2,"d":"On the open upper half-plane, the first coordinate divided by the radius is the cosine of the polar angle.","s":"∀ (z : ℝ × ℝ), 0 < z.2 → z.1 / Causalean.Mathlib.Analysis.planarRadius z = Real.cos (Causalean.Mathlib.Analysis.planarAngle z)","l":173,"f":"Causalean/Mathlib/Analysis/HalfDiscPolar.lean"},{"n":"Causalean.Mathlib.Analysis.planarRadius","k":"def","m":"Causalean.Mathlib.Analysis.HalfDiscPolar","a":"Mathlib/Analysis","t":1,"d":"The Euclidean radius on the real coordinate plane. This is explicit because the product type ℝ × ℝ carries the max product norm, not the Euclidean norm.","s":"ℝ × ℝ → ℝ","l":18,"f":"Causalean/Mathlib/Analysis/HalfDiscPolar.lean"},{"n":"Causalean.Mathlib.Analysis.planarRadius_measurable","k":"theorem","m":"Causalean.Mathlib.Analysis.HalfDiscPolar","a":"Mathlib/Analysis","t":2,"d":"The Euclidean radius on the coordinate plane is Borel measurable.","s":"Measurable Causalean.Mathlib.Analysis.planarRadius","l":24,"f":"Causalean/Mathlib/Analysis/HalfDiscPolar.lean"},{"n":"Causalean.Mathlib.Analysis.secondDirDeriv","k":"def","m":"Causalean.Mathlib.Analysis.LineSecondDeriv","a":"Mathlib/Analysis","t":1,"d":"The second directional derivative of f at q along d: the directional derivative, along d, of the map x ↦ fderiv ℝ f x d. For C² functions this is the value Hess f q (d, d) of the Hessian quadratic form.","s":"{E : Type u_1} → [inst : NormedAddCommGroup E] → [NormedSpace ℝ E] → (E → ℝ) → E → E → ℝ","l":41,"f":"Causalean/Mathlib/Analysis/LineSecondDeriv.lean"},{"n":"Causalean.Mathlib.Analysis.second_order_descent_gap","k":"theorem","m":"Causalean.Mathlib.Analysis.SecondOrderDescent","a":"Mathlib/Analysis","t":1,"d":"Quadratic-model descent gap. For a real function f on [0, T] where the interval length T, the slope margin c, and the curvature bound M are all nonnegative, f is twice continuously differentiable on [0, T], f is differentiable at 0, its second derivative is bounded above by M throughout the open interval, and its derivative at 0 is at most −c, then the clipped quadratic-model step lies in [0, T], and the resulting decrease f(0) − f(step) is at least c·step − (M/2)·step².","s":"∀ {f : ℝ → ℝ} {M c T : ℝ}, 0 ≤ T → 0 ≤ c → 0 ≤ M → ContDiffOn ℝ 2 f (Set.Icc 0 T) → DifferentiableAt ℝ f 0 → (∀ t ∈ Set.Ioo 0 T, deriv (deriv f) t ≤ M) → deriv f 0 ≤ -c → Causalean.Mathlib.Analysis.descentStep M c T ∈ Set.Icc 0 T ∧ f 0 - f (Causalean.Mathlib.Analysis.descentStep M c T) ≥ c * Causalean.Mathlib.Analysis.descentStep M c T - M / 2 * Causalean.Mathlib.Analysis.descentStep M c T ^ 2","l":115,"f":"Causalean/Mathlib/Analysis/SecondOrderDescent.lean"},{"n":"Causalean.Mathlib.Analysis.second_order_descent_gap_half","k":"theorem","m":"Causalean.Mathlib.Analysis.SecondOrderDescent","a":"Mathlib/Analysis","t":2,"d":"In both zero and positive curvature regimes, the clipped quadratic-model step gives a uniform half-linear decrease.","s":"∀ {f : ℝ → ℝ} {M c T : ℝ}, 0 ≤ T → 0 ≤ c → 0 ≤ M → ContDiffOn ℝ 2 f (Set.Icc 0 T) → DifferentiableAt ℝ f 0 → (∀ t ∈ Set.Ioo 0 T, deriv (deriv f) t ≤ M) → deriv f 0 ≤ -c → Causalean.Mathlib.Analysis.descentStep M c T ∈ Set.Icc 0 T ∧ f 0 - f (Causalean.Mathlib.Analysis.descentStep M c T) ≥ c / 2 * Causalean.Mathlib.Analysis.descentStep M c T","l":176,"f":"Causalean/Mathlib/Analysis/SecondOrderDescent.lean"},{"n":"Causalean.Mathlib.Analysis.second_order_descent_gap_min","k":"theorem","m":"Causalean.Mathlib.Analysis.SecondOrderDescent","a":"Mathlib/Analysis","t":2,"d":"With positive curvature, the clipped quadratic-model step gives at least half of the linear descent term.","s":"∀ {f : ℝ → ℝ} {M c T : ℝ}, 0 ≤ T → 0 ≤ c → 0 < M → ContDiffOn ℝ 2 f (Set.Icc 0 T) → DifferentiableAt ℝ f 0 → (∀ t ∈ Set.Ioo 0 T, deriv (deriv f) t ≤ M) → deriv f 0 ≤ -c → f 0 - f (min T (c / M)) ≥ c / 2 * min T (c / M)","l":142,"f":"Causalean/Mathlib/Analysis/SecondOrderDescent.lean"},{"n":"Causalean.Mathlib.Analysis.second_order_upper_bound","k":"theorem","m":"Causalean.Mathlib.Analysis.SecondOrderDescent","a":"Mathlib/Analysis","t":2,"d":"A twice differentiable real function whose second derivative is bounded above on an interval's interior is no larger than its tangent at the left endpoint plus the quadratic curvature allowance.","s":"∀ {f : ℝ → ℝ} {M T : ℝ}, ContDiffOn ℝ 2 f (Set.Icc 0 T) → DifferentiableAt ℝ f 0 → (∀ t ∈ Set.Ioo 0 T, deriv (deriv f) t ≤ M) → ∀ {t : ℝ}, t ∈ Set.Icc 0 T → f t ≤ f 0 + deriv f 0 * t + M / 2 * t ^ 2","l":55,"f":"Causalean/Mathlib/Analysis/SecondOrderDescent.lean"},{"n":"Causalean.Mathlib.Analysis.sum_sq_pos","k":"theorem","m":"Causalean.Mathlib.Analysis.MonomialGram","a":"Mathlib/Analysis","t":2,"d":"The sum of squared coordinates of a nonzero finite real-valued vector is strictly positive.","s":"∀ {ι : Type u_1} [inst : Fintype ι] (z : ι → ℝ), z ≠ 0 → 0 < ∑ k, z k ^ 2","l":228,"f":"Causalean/Mathlib/Analysis/MonomialGram.lean"},{"n":"Causalean.Mathlib.Analysis.supBall","k":"def","m":"Causalean.Mathlib.Analysis.MonomialGram","a":"Mathlib/Analysis","t":1,"d":"The closed sup-norm ball contains the vectors whose every coordinate lies within the stated radius of the corresponding coordinate of the center.","s":"{ι : Type u_1} → (ι → ℝ) → ℝ → Set (ι → ℝ)","l":21,"f":"Causalean/Mathlib/Analysis/MonomialGram.lean"},{"n":"Causalean.Mathlib.Analysis.translatedClosedHalfDisc_weighted_first_div_radius_cancellation","k":"theorem","m":"Causalean.Mathlib.Analysis.HalfDiscPolar","a":"Mathlib/Analysis","t":2,"d":"Translation preserves the zero Cartesian-cosine integral over a closed upper half-disc.","s":"∀ (g : ℝ → ℝ) (r : ℝ) (c : ℝ × ℝ), ∫ (z : ℝ × ℝ) in {z | 0 ≤ (z - c).2 ∧ Causalean.Mathlib.Analysis.planarRadius (z - c) ≤ r}, g (Causalean.Mathlib.Analysis.planarRadius (z - c)) * ((z - c).1 / Causalean.Mathlib.Analysis.planarRadius (z - c)) = 0","l":250,"f":"Causalean/Mathlib/Analysis/HalfDiscPolar.lean"},{"n":"Causalean.Mathlib.Analysis.translatedHalfDisc_radialSet_weighted_cos_cancellation","k":"theorem","m":"Causalean.Mathlib.Analysis.HalfDiscPolar","a":"Mathlib/Analysis","t":2,"d":"Translation preserves cosine cancellation on every measurable radial subset of a half-disc.","s":"∀ (g : ℝ → ℝ) (r : ℝ) (c : ℝ × ℝ) {A : Set ℝ}, MeasurableSet A → ∫ (z : ℝ × ℝ) in (fun u => c + u) '' {u | 0 < u.2 ∧ Causalean.Mathlib.Analysis.planarRadius u ≤ r} ∩ {z | Causalean.Mathlib.Analysis.planarRadius (z - c) ∈ A}, g (Causalean.Mathlib.Analysis.planarRadius (z - c)) * Real.cos (Causalean.Mathlib.Analysis.planarAngle (z - c)) = 0","l":415,"f":"Causalean/Mathlib/Analysis/HalfDiscPolar.lean"},{"n":"Causalean.Mathlib.Analysis.translatedHalfDisc_weighted_cos_cancellation","k":"theorem","m":"Causalean.Mathlib.Analysis.HalfDiscPolar","a":"Mathlib/Analysis","t":2,"d":"Translating an upper half-disc does not change the cosine cancellation.","s":"∀ (g : ℝ → ℝ) (r : ℝ) (c : ℝ × ℝ), ∫ (z : ℝ × ℝ) in (fun u => c + u) '' {u | 0 < u.2 ∧ Causalean.Mathlib.Analysis.planarRadius u ≤ r}, g (Causalean.Mathlib.Analysis.planarRadius (z - c)) * Real.cos (Causalean.Mathlib.Analysis.planarAngle (z - c)) = 0","l":390,"f":"Causalean/Mathlib/Analysis/HalfDiscPolar.lean"},{"n":"Causalean.Mathlib.Analysis.weighted_inner_sq_le","k":"theorem","m":"Causalean.Mathlib.Analysis.WeightedCauchySchwarz","a":"Mathlib/Analysis","t":1,"d":"Weighted Cauchy-Schwarz inequality on a finite support, squared form. Weighting a finite collection of index points by nonnegative weights, the square of the weighted inner product of two real-valued functions is at most the product of their weighted sums of squares.","s":"∀ {ι : Type u_1} (s : Finset ι) (w f g : ι → ℝ), (∀ i ∈ s, 0 ≤ w i) → (∑ i ∈ s, w i * (f i * g i)) ^ 2 ≤ (∑ i ∈ s, w i * f i ^ 2) * ∑ i ∈ s, w i * g i ^ 2","l":33,"f":"Causalean/Mathlib/Analysis/WeightedCauchySchwarz.lean"},{"n":"Causalean.Mathlib.Analysis.weighted_sqrt_le_sqrt_weighted","k":"theorem","m":"Causalean.Mathlib.Analysis.WeightedCauchySchwarz","a":"Mathlib/Analysis","t":2,"d":"Jensen's inequality for the square root. Averaging nonnegative values with weights that are nonnegative and sum to at most one, the weighted average of their square roots is at most the square root of their weighted average.","s":"∀ {ι : Type u_1} (s : Finset ι) (w q : ι → ℝ), (∀ i ∈ s, 0 ≤ w i) → ∀ (c : ℝ), ∑ i ∈ s, w i ≤ c → (∀ i ∈ s, 0 ≤ q i) → ∑ i ∈ s, w i * √(q i) ≤ √c * √(∑ i ∈ s, w i * q i)","l":100,"f":"Causalean/Mathlib/Analysis/WeightedCauchySchwarz.lean"},{"n":"Causalean.Mathlib.CompProdAssembly.compProd_eq_of_inner_ae","k":"theorem","m":"Causalean.Mathlib.Probability.Kernel.CompProdAssembly","a":"Mathlib/Probability/Kernel","t":1,"d":"CompProd assembly from an a.e. inner equality. Consider two Markov-style kernels KL, KR from a base space to an outer space, each built by mixing the section at a base point of an inner kernel — fL, respectively fR, on the product of the base space and an intermediate space — against a fixed intermediate measure μ: KL a equals the section fL(a,·) composed with μ and KR a equals the section fR(a,·) composed with μ. If the inner kernels fL and fR agree almost everywhere with respect to the composition product of the base measure ν with the constant-μ kernel, then the composition products ν ⊗ₘ KL and ν ⊗ₘ KR are equal.","s":"∀ {α : Type u_1} {γ : Type u_2} {β : Type u_3} [inst : MeasurableSpace α] [inst_1 : MeasurableSpace γ] [inst_2 : MeasurableSpace β] (ν : MeasureTheory.Measure α) [MeasureTheory.IsFiniteMeasure ν] (μ : MeasureTheory.Measure γ) [MeasureTheory.SFinite μ] (KL KR : ProbabilityTheory.Kernel α β) [ProbabilityTheory.IsFiniteKernel KL] [ProbabilityTheory.IsSFiniteKernel KR] (fL fR : ProbabilityTheory.Kernel (α × γ) β), (∀ (a : α), KL a = μ.bind ⇑(fL.sectR a)) → (∀ (a : α), KR a = μ.bind ⇑(fR.sectR a)) → (∀ᵐ (p : α × γ) ∂ν.compProd (ProbabilityTheory.Kernel.const α μ), fL p = fR p) → ν.compProd KL = ν.compProd KR","l":77,"f":"Causalean/Mathlib/Probability/Kernel/CompProdAssembly.lean"},{"n":"Causalean.Mathlib.CompProdAssembly.compProd_eq_of_inner_ae_kernel","k":"theorem","m":"Causalean.Mathlib.Probability.Kernel.CompProdAssembly","a":"Mathlib/Probability/Kernel","t":1,"d":"CompProd assembly from an a.e. inner equality, indexed-integrator form. Consider two Markov-style kernels KL, KR from a base space to an outer space, each built by mixing the section at a base point of an inner kernel — fL, respectively fR, on the product of the base space and an intermediate space — against a fixed intermediate kernel κ evaluated at that point: KL a equals the section fL(a,·) composed with κ a and KR a equals the section fR(a,·) composed with κ a. If the inner kernels fL and fR agree almost everywhere with respect to the composition product of the base measure ν and κ, then the composition products ν ⊗ₘ KL and ν ⊗ₘ KR are equal.","s":"∀ {α : Type u_1} {γ : Type u_2} {β : Type u_3} [inst : MeasurableSpace α] [inst_1 : MeasurableSpace γ] [inst_2 : MeasurableSpace β] (ν : MeasureTheory.Measure α) [MeasureTheory.IsFiniteMeasure ν] (κ : ProbabilityTheory.Kernel α γ) [ProbabilityTheory.IsSFiniteKernel κ] (KL KR : ProbabilityTheory.Kernel α β) [ProbabilityTheory.IsFiniteKernel KL] [ProbabilityTheory.IsSFiniteKernel KR] (fL fR : ProbabilityTheory.Kernel (α × γ) β), (∀ (a : α), KL a = (κ a).bind ⇑(fL.sectR a)) → (∀ (a : α), KR a = (κ a).bind ⇑(fR.sectR a)) → (∀ᵐ (p : α × γ) ∂ν.compProd κ, fL p = fR p) → ν.compProd KL = ν.compProd KR","l":33,"f":"Causalean/Mathlib/Probability/Kernel/CompProdAssembly.lean"},{"n":"Causalean.Mathlib.CompProdAssembly.compProd_map_snd_apply","k":"theorem","m":"Causalean.Mathlib.Probability.Kernel.CompProdAssembly","a":"Mathlib/Probability/Kernel","t":2,"d":"Snd-marginal of a composition product, pointwise (disintegration backbone).","s":"∀ {α : Type u_1} {β : Type u_2} {γ : Type u_3} [inst : MeasurableSpace α] [inst_1 : MeasurableSpace β] [inst_2 : MeasurableSpace γ] (κ₁ : ProbabilityTheory.Kernel α β) (κ₂ : ProbabilityTheory.Kernel (α × β) γ) [ProbabilityTheory.IsSFiniteKernel κ₁] [ProbabilityTheory.IsSFiniteKernel κ₂] (a : α), ((κ₁.compProd κ₂).map Prod.snd) a = (κ₁ a).bind ⇑(κ₂.sectR a)","l":98,"f":"Causalean/Mathlib/Probability/Kernel/CompProdAssembly.lean"},{"n":"Causalean.Mathlib.FWL.fittedValue","k":"def","m":"Causalean.Mathlib.FWL","a":"Mathlib","t":1,"d":"Fitted value for a finite tuple of regressors.","s":"{V : Type u_1} → [inst : NormedAddCommGroup V] → [InnerProductSpace ℝ V] → {K : ℕ} → (Fin K → V) → (Fin K → ℝ) → V","l":147,"f":"Causalean/Mathlib/FWL.lean"},{"n":"Causalean.Mathlib.FWL.fittedValue_sub","k":"theorem","m":"Causalean.Mathlib.FWL","a":"Mathlib","t":2,"d":"Fitted values are linear in the coefficient vector.","s":"∀ {V : Type u_1} [inst : NormedAddCommGroup V] [inst_1 : InnerProductSpace ℝ V] {K : ℕ} (X : Fin K → V) (β γ : Fin K → ℝ), (Causalean.Mathlib.FWL.fittedValue X fun j => β j - γ j) = Causalean.Mathlib.FWL.fittedValue X β - Causalean.Mathlib.FWL.fittedValue X γ","l":151,"f":"Causalean/Mathlib/FWL.lean"},{"n":"Causalean.Mathlib.FWL.fwlCoef","k":"def","m":"Causalean.Mathlib.FWL","a":"Mathlib","t":1,"d":"The FWL coefficient. Defined directly as Q_{XX}⁻¹ *ᵥ (fun j => ⟨X̃ j, Y⟩).","s":"{V : Type u_1} → [inst : NormedAddCommGroup V] → [inst_1 : InnerProductSpace ℝ V] → (H : Submodule ℝ V) → [H.HasOrthogonalProjection] → {K : ℕ} → (Fin K → V) → V → Fin K → ℝ","l":200,"f":"Causalean/Mathlib/FWL.lean"},{"n":"Causalean.Mathlib.FWL.fwlCoef_eq_of_original_minimizer","k":"theorem","m":"Causalean.Mathlib.FWL","a":"Mathlib","t":2,"d":"Any minimizer of the original least-squares problem has coefficient block equal to the FWL coefficient. This is the uniqueness part of the standard FWL statement for the X-block.","s":"∀ {V : Type u_1} [inst : NormedAddCommGroup V] [inst_1 : InnerProductSpace ℝ V] (H : Submodule ℝ V) [inst_2 : H.HasOrthogonalProjection] {K : ℕ} (X : Fin K → V) (Y : V), IsUnit (Causalean.Mathlib.FWL.gramResid H X).det → ∀ (β : Fin K → ℝ) {h : V}, h ∈ H → (∀ (γ : Fin K → ℝ) {g : V}, g ∈ H → Causalean.Mathlib.FWL.originalObjective X Y β h ≤ Causalean.Mathlib.FWL.originalObjective X Y γ g) → β = Causalean.Mathlib.FWL.fwlCoef H X Y","l":401,"f":"Causalean/Mathlib/FWL.lean"},{"n":"Causalean.Mathlib.FWL.fwlCoef_original_minimizes","k":"theorem","m":"Causalean.Mathlib.FWL","a":"Mathlib","t":1,"d":"Standard finite-dimensional Frisch–Waugh–Lovell theorem. On a real inner-product space with a nuisance subspace H admitting an orthogonal projection, and a finite tuple of regressors X, if the residualized Gram matrix of X has nonzero determinant, i.e. is invertible and a candidate nuisance vector h lies in H, then the original least-squares objective — evaluated at the FWL coefficient together with the orthogonal-projection nuisance term of its raw residual — is at most the original objective at any other coefficient vector β and nuisance term h.","s":"∀ {V : Type u_1} [inst : NormedAddCommGroup V] [inst_1 : InnerProductSpace ℝ V] (H : Submodule ℝ V) [inst_2 : H.HasOrthogonalProjection] {K : ℕ} (X : Fin K → V) (Y : V), IsUnit (Causalean.Mathlib.FWL.gramResid H X).det → ∀ (β : Fin K → ℝ) {h : V}, h ∈ H → Causalean.Mathlib.FWL.originalObjective X Y (Causalean.Mathlib.FWL.fwlCoef H X Y) (Submodule.orthogonalProjectionFn (Y - Causalean.Mathlib.FWL.fittedValue X (Causalean.Mathlib.FWL.fwlCoef H X Y))) ≤ Causalean.Mathlib.FWL.originalObjective X Y β h","l":378,"f":"Causalean/Mathlib/FWL.lean"},{"n":"Causalean.Mathlib.FWL.fwlCoef_residualized_minimizes","k":"theorem","m":"Causalean.Mathlib.FWL","a":"Mathlib","t":2,"d":"The FWL coefficient minimizes the residualized least-squares objective.","s":"∀ {V : Type u_1} [inst : NormedAddCommGroup V] [inst_1 : InnerProductSpace ℝ V] (H : Submodule ℝ V) [inst_2 : H.HasOrthogonalProjection] {K : ℕ} (X : Fin K → V) (Y : V), IsUnit (Causalean.Mathlib.FWL.gramResid H X).det → ∀ (β : Fin K → ℝ), Causalean.Mathlib.FWL.residualizedObjective H X Y (Causalean.Mathlib.FWL.fwlCoef H X Y) ≤ Causalean.Mathlib.FWL.residualizedObjective H X Y β","l":312,"f":"Causalean/Mathlib/FWL.lean"},{"n":"Causalean.Mathlib.FWL.gramResid","k":"def","m":"Causalean.Mathlib.FWL","a":"Mathlib","t":1,"d":"The residualized Gram matrix Q_{XX} with entries ⟨residualize H (X j), residualize H (X k)⟩.","s":"{V : Type u_1} → [inst : NormedAddCommGroup V] → [inst_1 : InnerProductSpace ℝ V] → (H : Submodule ℝ V) → [H.HasOrthogonalProjection] → {K : ℕ} → (Fin K → V) → Matrix (Fin K) (Fin K) ℝ","l":179,"f":"Causalean/Mathlib/FWL.lean"},{"n":"Causalean.Mathlib.FWL.gramResid_apply","k":"theorem","m":"Causalean.Mathlib.FWL","a":"Mathlib","t":2,"d":"Each entry of the residualized Gram matrix is the inner product of two residualized regressors.","s":"∀ {V : Type u_1} [inst : NormedAddCommGroup V] [inst_1 : InnerProductSpace ℝ V] (H : Submodule ℝ V) [inst_2 : H.HasOrthogonalProjection] {K : ℕ} (X : Fin K → V) (j k : Fin K), Causalean.Mathlib.FWL.gramResid H X j k = inner ℝ (Causalean.Mathlib.FWL.residualize H (X j)) (Causalean.Mathlib.FWL.residualize H (X k))","l":184,"f":"Causalean/Mathlib/FWL.lean"},{"n":"Causalean.Mathlib.FWL.gramResid_mulVec_fwlCoef","k":"theorem","m":"Causalean.Mathlib.FWL","a":"Mathlib","t":2,"d":"Residualized normal equations (FWL coefficient form). If Q_{XX} = gramResid H X is invertible (equivalently, the residualized regressors are linearly independent), the FWL coefficient fwlCoef H X Y = Q_{XX}⁻¹ *ᵥ (fun j => ⟨X̃ j, Y⟩) satisfies the residualized normal equations Q_{XX} *ᵥ fwlCoef H X Y = (fun j => ⟨X̃ j, Y⟩).","s":"∀ {V : Type u_1} [inst : NormedAddCommGroup V] [inst_1 : InnerProductSpace ℝ V] (H : Submodule ℝ V) [inst_2 : H.HasOrthogonalProjection] {K : ℕ} (X : Fin K → V) (Y : V), IsUnit (Causalean.Mathlib.FWL.gramResid H X).det → (Causalean.Mathlib.FWL.gramResid H X).mulVec (Causalean.Mathlib.FWL.fwlCoef H X Y) = Causalean.Mathlib.FWL.residInnerVec H X Y","l":205,"f":"Causalean/Mathlib/FWL.lean"},{"n":"Causalean.Mathlib.FWL.inner_residualize_of_mem","k":"theorem","m":"Causalean.Mathlib.FWL","a":"Mathlib","t":2,"d":"Residual is orthogonal to every element of H. This is the defining property of the orthogonal residual.","s":"∀ {V : Type u_1} [inst : NormedAddCommGroup V] [inst_1 : InnerProductSpace ℝ V] (H : Submodule ℝ V) [inst_2 : H.HasOrthogonalProjection] (v : V) {w : V}, w ∈ H → inner ℝ (Causalean.Mathlib.FWL.residualize H v) w = 0","l":113,"f":"Causalean/Mathlib/FWL.lean"},{"n":"Causalean.Mathlib.FWL.inner_residualizedRegressor_fwlResidual","k":"theorem","m":"Causalean.Mathlib.FWL","a":"Mathlib","t":2,"d":"The residual left by fwlCoef is orthogonal to every residualized regressor. This is the normal-equation form used in the completing-square argument.","s":"∀ {V : Type u_1} [inst : NormedAddCommGroup V] [inst_1 : InnerProductSpace ℝ V] (H : Submodule ℝ V) [inst_2 : H.HasOrthogonalProjection] {K : ℕ} (X : Fin K → V) (Y : V), IsUnit (Causalean.Mathlib.FWL.gramResid H X).det → ∀ (j : Fin K), inner ℝ (Causalean.Mathlib.FWL.residualize H (X j)) (Causalean.Mathlib.FWL.residualize H Y - Causalean.Mathlib.FWL.fittedValue (fun k => Causalean.Mathlib.FWL.residualize H (X k)) (Causalean.Mathlib.FWL.fwlCoef H X Y)) = 0","l":237,"f":"Causalean/Mathlib/FWL.lean"},{"n":"Causalean.Mathlib.FWL.originalObjective","k":"def","m":"Causalean.Mathlib.FWL","a":"Mathlib","t":1,"d":"Original least-squares objective with an explicit nuisance term h ∈ H.","s":"{V : Type u_1} → [inst : NormedAddCommGroup V] → [InnerProductSpace ℝ V] → {K : ℕ} → (Fin K → V) → V → (Fin K → ℝ) → V → ℝ","l":233,"f":"Causalean/Mathlib/FWL.lean"},{"n":"Causalean.Mathlib.FWL.originalObjective_eq_residualizedObjective_add_projectionError","k":"theorem","m":"Causalean.Mathlib.FWL","a":"Mathlib","t":2,"d":"Pythagorean split of the original objective into the residualized objective plus the squared distance from h to the projection of the raw regression residual onto H.","s":"∀ {V : Type u_1} [inst : NormedAddCommGroup V] [inst_1 : InnerProductSpace ℝ V] (H : Submodule ℝ V) [inst_2 : H.HasOrthogonalProjection] {K : ℕ} (X : Fin K → V) (Y : V) (β : Fin K → ℝ) {h : V}, h ∈ H → Causalean.Mathlib.FWL.originalObjective X Y β h = Causalean.Mathlib.FWL.residualizedObjective H X Y β + inner ℝ (Submodule.orthogonalProjectionFn (Y - Causalean.Mathlib.FWL.fittedValue X β) - h) (Submodule.orthogonalProjectionFn (Y - Causalean.Mathlib.FWL.fittedValue X β) - h)","l":320,"f":"Causalean/Mathlib/FWL.lean"},{"n":"Causalean.Mathlib.FWL.originalObjective_projection_eq_residualizedObjective","k":"theorem","m":"Causalean.Mathlib.FWL","a":"Mathlib","t":2,"d":"At the nuisance projection, the original objective equals the residualized objective.","s":"∀ {V : Type u_1} [inst : NormedAddCommGroup V] [inst_1 : InnerProductSpace ℝ V] (H : Submodule ℝ V) [inst_2 : H.HasOrthogonalProjection] {K : ℕ} (X : Fin K → V) (Y : V) (β : Fin K → ℝ), Causalean.Mathlib.FWL.originalObjective X Y β (Submodule.orthogonalProjectionFn (Y - Causalean.Mathlib.FWL.fittedValue X β)) = Causalean.Mathlib.FWL.residualizedObjective H X Y β","l":367,"f":"Causalean/Mathlib/FWL.lean"},{"n":"Causalean.Mathlib.FWL.original_minimizer_eq_fwlCoef_projection","k":"theorem","m":"Causalean.Mathlib.FWL","a":"Mathlib","t":1,"d":"Uniqueness of the original least-squares minimizer. If the residualized Gram matrix of X is invertible, a candidate nuisance vector h lies in H, and the pair (β, h) minimizes the original least-squares objective over all coefficient vectors and nuisance terms in H, then β equals the FWL coefficient and h equals the orthogonal-projection nuisance term of the FWL coefficient's raw residual.","s":"∀ {V : Type u_1} [inst : NormedAddCommGroup V] [inst_1 : InnerProductSpace ℝ V] (H : Submodule ℝ V) [inst_2 : H.HasOrthogonalProjection] {K : ℕ} (X : Fin K → V) (Y : V), IsUnit (Causalean.Mathlib.FWL.gramResid H X).det → ∀ (β : Fin K → ℝ) {h : V}, h ∈ H → (∀ (γ : Fin K → ℝ) {g : V}, g ∈ H → Causalean.Mathlib.FWL.originalObjective X Y β h ≤ Causalean.Mathlib.FWL.originalObjective X Y γ g) → β = Causalean.Mathlib.FWL.fwlCoef H X Y ∧ h = Submodule.orthogonalProjectionFn (Y - Causalean.Mathlib.FWL.fittedValue X (Causalean.Mathlib.FWL.fwlCoef H X Y))","l":452,"f":"Causalean/Mathlib/FWL.lean"},{"n":"Causalean.Mathlib.FWL.residInnerVec","k":"def","m":"Causalean.Mathlib.FWL","a":"Mathlib","t":1,"d":"The right-hand side of the residualized normal equations, r_j := ⟨residualize H (X j), Y⟩.","s":"{V : Type u_1} → [inst : NormedAddCommGroup V] → [inst_1 : InnerProductSpace ℝ V] → (H : Submodule ℝ V) → [H.HasOrthogonalProjection] → {K : ℕ} → (Fin K → V) → V → Fin K → ℝ","l":190,"f":"Causalean/Mathlib/FWL.lean"},{"n":"Causalean.Mathlib.FWL.residInnerVec_apply","k":"theorem","m":"Causalean.Mathlib.FWL","a":"Mathlib","t":2,"d":"Each entry of the residualized right-hand side is the inner product of a residualized regressor with the outcome.","s":"∀ {V : Type u_1} [inst : NormedAddCommGroup V] [inst_1 : InnerProductSpace ℝ V] (H : Submodule ℝ V) [inst_2 : H.HasOrthogonalProjection] {K : ℕ} (X : Fin K → V) (Y : V) (j : Fin K), Causalean.Mathlib.FWL.residInnerVec H X Y j = inner ℝ (Causalean.Mathlib.FWL.residualize H (X j)) Y","l":195,"f":"Causalean/Mathlib/FWL.lean"},{"n":"Causalean.Mathlib.FWL.residInnerVec_eq_residualize_right","k":"theorem","m":"Causalean.Mathlib.FWL","a":"Mathlib","t":2,"d":"Symmetric residualization for the FWL right-hand side. The residualized inner products ⟨X̃ j, Y⟩ agree with ⟨X̃ j, Ỹ⟩, so the FWL coefficient depends only on the residualized response.","s":"∀ {V : Type u_1} [inst : NormedAddCommGroup V] [inst_1 : InnerProductSpace ℝ V] (H : Submodule ℝ V) [inst_2 : H.HasOrthogonalProjection] {K : ℕ} (X : Fin K → V) (Y : V), Causalean.Mathlib.FWL.residInnerVec H X Y = Causalean.Mathlib.FWL.residInnerVec H X (Causalean.Mathlib.FWL.residualize H Y)","l":219,"f":"Causalean/Mathlib/FWL.lean"},{"n":"Causalean.Mathlib.FWL.residualize","k":"def","m":"Causalean.Mathlib.FWL","a":"Mathlib","t":1,"d":"Orthogonal residual operator M_H v := v - P_H v as a function V → V. We use Submodule.orthogonalProjectionFn (which lands in V, not in ↥H) so that residualize H v : V.","s":"{V : Type u_1} → [inst : NormedAddCommGroup V] → [inst_1 : InnerProductSpace ℝ V] → (H : Submodule ℝ V) → [H.HasOrthogonalProjection] → V → V","l":93,"f":"Causalean/Mathlib/FWL.lean"},{"n":"Causalean.Mathlib.FWL.residualizeL","k":"def","m":"Causalean.Mathlib.FWL","a":"Mathlib","t":1,"d":"The residual maker as a continuous linear map. This bundled form is used only to transfer linearity facts back to residualize.","s":"{V : Type u_1} → [inst : NormedAddCommGroup V] → [inst_1 : InnerProductSpace ℝ V] → (H : Submodule ℝ V) → [H.HasOrthogonalProjection] → V →L[ℝ] V","l":103,"f":"Causalean/Mathlib/FWL.lean"},{"n":"Causalean.Mathlib.FWL.residualizeL_apply","k":"theorem","m":"Causalean.Mathlib.FWL","a":"Mathlib","t":2,"d":"Applying the residual-maker linear map gives the residualized vector.","s":"∀ {V : Type u_1} [inst : NormedAddCommGroup V] [inst_1 : InnerProductSpace ℝ V] (H : Submodule ℝ V) [inst_2 : H.HasOrthogonalProjection] (v : V), (Causalean.Mathlib.FWL.residualizeL H) v = Causalean.Mathlib.FWL.residualize H v","l":108,"f":"Causalean/Mathlib/FWL.lean"},{"n":"Causalean.Mathlib.FWL.residualize_def","k":"theorem","m":"Causalean.Mathlib.FWL","a":"Mathlib","t":2,"d":"Residualization is the original vector minus its orthogonal projection onto the nuisance subspace.","s":"∀ {V : Type u_1} [inst : NormedAddCommGroup V] [inst_1 : InnerProductSpace ℝ V] (H : Submodule ℝ V) [inst_2 : H.HasOrthogonalProjection] (v : V), Causalean.Mathlib.FWL.residualize H v = v - Submodule.orthogonalProjectionFn v","l":98,"f":"Causalean/Mathlib/FWL.lean"},{"n":"Causalean.Mathlib.FWL.residualize_fittedValue","k":"theorem","m":"Causalean.Mathlib.FWL","a":"Mathlib","t":2,"d":"Residualization commutes with forming a fitted value.","s":"∀ {V : Type u_1} [inst : NormedAddCommGroup V] [inst_1 : InnerProductSpace ℝ V] (H : Submodule ℝ V) [inst_2 : H.HasOrthogonalProjection] {K : ℕ} (X : Fin K → V) (β : Fin K → ℝ), Causalean.Mathlib.FWL.residualize H (Causalean.Mathlib.FWL.fittedValue X β) = Causalean.Mathlib.FWL.fittedValue (fun j => Causalean.Mathlib.FWL.residualize H (X j)) β","l":157,"f":"Causalean/Mathlib/FWL.lean"},{"n":"Causalean.Mathlib.FWL.residualize_inner_swap_right","k":"theorem","m":"Causalean.Mathlib.FWL","a":"Mathlib","t":2,"d":"Symmetric residualization in the right slot. Since M_H is self-adjoint and idempotent, ⟨X̃, Y⟩ = ⟨X̃, Ỹ⟩.","s":"∀ {V : Type u_1} [inst : NormedAddCommGroup V] [inst_1 : InnerProductSpace ℝ V] (H : Submodule ℝ V) [inst_2 : H.HasOrthogonalProjection] (v w : V), inner ℝ (Causalean.Mathlib.FWL.residualize H v) w = inner ℝ (Causalean.Mathlib.FWL.residualize H v) (Causalean.Mathlib.FWL.residualize H w)","l":121,"f":"Causalean/Mathlib/FWL.lean"},{"n":"Causalean.Mathlib.FWL.residualize_regressionResidual","k":"theorem","m":"Causalean.Mathlib.FWL","a":"Mathlib","t":2,"d":"Residualization of the raw regression residual is the residualized outcome minus the fitted value of the residualized regressors.","s":"∀ {V : Type u_1} [inst : NormedAddCommGroup V] [inst_1 : InnerProductSpace ℝ V] (H : Submodule ℝ V) [inst_2 : H.HasOrthogonalProjection] {K : ℕ} (X : Fin K → V) (Y : V) (β : Fin K → ℝ), Causalean.Mathlib.FWL.residualize H (Y - Causalean.Mathlib.FWL.fittedValue X β) = Causalean.Mathlib.FWL.residualize H Y - Causalean.Mathlib.FWL.fittedValue (fun j => Causalean.Mathlib.FWL.residualize H (X j)) β","l":166,"f":"Causalean/Mathlib/FWL.lean"},{"n":"Causalean.Mathlib.FWL.residualizedObjective","k":"def","m":"Causalean.Mathlib.FWL","a":"Mathlib","t":1,"d":"Residualized least-squares objective.","s":"{V : Type u_1} → [inst : NormedAddCommGroup V] → [inst_1 : InnerProductSpace ℝ V] → (H : Submodule ℝ V) → [H.HasOrthogonalProjection] → {K : ℕ} → (Fin K → V) → V → (Fin K → ℝ) → ℝ","l":227,"f":"Causalean/Mathlib/FWL.lean"},{"n":"Causalean.Mathlib.FWL.residualizedObjective_eq_fwlCoef_add_error","k":"theorem","m":"Causalean.Mathlib.FWL","a":"Mathlib","t":2,"d":"Completing-square identity for the residualized objective around the FWL coefficient.","s":"∀ {V : Type u_1} [inst : NormedAddCommGroup V] [inst_1 : InnerProductSpace ℝ V] (H : Submodule ℝ V) [inst_2 : H.HasOrthogonalProjection] {K : ℕ} (X : Fin K → V) (Y : V), IsUnit (Causalean.Mathlib.FWL.gramResid H X).det → ∀ (β : Fin K → ℝ), Causalean.Mathlib.FWL.residualizedObjective H X Y β = Causalean.Mathlib.FWL.residualizedObjective H X Y (Causalean.Mathlib.FWL.fwlCoef H X Y) + inner ℝ (Causalean.Mathlib.FWL.fittedValue (fun j => Causalean.Mathlib.FWL.residualize H (X j)) fun j => Causalean.Mathlib.FWL.fwlCoef H X Y j - β j) (Causalean.Mathlib.FWL.fittedValue (fun j => Causalean.Mathlib.FWL.residualize H (X j)) fun j => Causalean.Mathlib.FWL.fwlCoef H X Y j - β j)","l":263,"f":"Causalean/Mathlib/FWL.lean"},{"n":"Causalean.Mathlib.FWL.residualizedObjective_le_originalObjective","k":"theorem","m":"Causalean.Mathlib.FWL","a":"Mathlib","t":2,"d":"For fixed β, the residualized objective is the minimum of the original objective over the nuisance subspace H.","s":"∀ {V : Type u_1} [inst : NormedAddCommGroup V] [inst_1 : InnerProductSpace ℝ V] (H : Submodule ℝ V) [inst_2 : H.HasOrthogonalProjection] {K : ℕ} (X : Fin K → V) (Y : V) (β : Fin K → ℝ) {h : V}, h ∈ H → Causalean.Mathlib.FWL.residualizedObjective H X Y β ≤ Causalean.Mathlib.FWL.originalObjective X Y β h","l":358,"f":"Causalean/Mathlib/FWL.lean"},{"n":"Causalean.Mathlib.GraphMapProd.instIsMarkovKernelMechanismKernel","k":"theorem","m":"Causalean.Mathlib.Probability.Kernel.GraphMapProd","a":"Mathlib/Probability/Kernel","t":2,"d":"The mechanism kernel is Markov when the exogenous-noise law is a probability measure and the mechanism is measurable.","s":"∀ {β : Type u_1} {γ : Type u_2} {δ : Type u_3} [inst : MeasurableSpace β] [inst_1 : MeasurableSpace γ] [inst_2 : MeasurableSpace δ] (νₗ : MeasureTheory.Measure γ) [MeasureTheory.IsProbabilityMeasure νₗ] {Φ : β × γ → δ}, Measurable Φ → ProbabilityTheory.IsMarkovKernel (Causalean.Mathlib.GraphMapProd.mechanismKernel νₗ Φ)","l":55,"f":"Causalean/Mathlib/Probability/Kernel/GraphMapProd.lean"},{"n":"Causalean.Mathlib.GraphMapProd.map_graph_prod_eq_compProd","k":"theorem","m":"Causalean.Mathlib.Probability.Kernel.GraphMapProd","a":"Mathlib/Probability/Kernel","t":1,"d":"Graph push-forward of a product measure is a composition product. For an s-finite measure α on a first factor, an s-finite measure νₗ on a second factor, and a measurable mechanism map Φ combining the two factors into a third space, pushing the product measure α.prod νₗ forward along the graph map (o, l) ↦ (o, Φ (o, l)) equals the composition product of α with the mechanism kernel that sends each value o of the first coordinate to the pushforward of νₗ through the slice l ↦ Φ (o, l).","s":"∀ {β : Type u_1} {γ : Type u_2} {δ : Type u_3} [inst : MeasurableSpace β] [inst_1 : MeasurableSpace γ] [inst_2 : MeasurableSpace δ] (α : MeasureTheory.Measure β) [MeasureTheory.SFinite α] (νₗ : MeasureTheory.Measure γ) [MeasureTheory.SFinite νₗ] {Φ : β × γ → δ}, Measurable Φ → MeasureTheory.Measure.map (fun p => (p.1, Φ p)) (α.prod νₗ) = α.compProd (Causalean.Mathlib.GraphMapProd.mechanismKernel νₗ Φ)","l":68,"f":"Causalean/Mathlib/Probability/Kernel/GraphMapProd.lean"},{"n":"Causalean.Mathlib.GraphMapProd.mechanismKernel","k":"def","m":"Causalean.Mathlib.Probability.Kernel.GraphMapProd","a":"Mathlib/Probability/Kernel","t":1,"d":"The structural-mechanism kernel κ o = νₗ.map (Φ (o, ·)): at each value o of the first coordinate, push the γ-marginal νₗ through the slice l ↦ Φ (o, l). Built as the deterministic-times-constant product kernel (Kernel.id ×ₖ Kernel.const β νₗ) mapped through Φ.","s":"{β : Type u_1} → {γ : Type u_2} → {δ : Type u_3} → [inst : MeasurableSpace β] → [inst_1 : MeasurableSpace γ] → [inst_2 : MeasurableSpace δ] → MeasureTheory.Measure γ → (β × γ → δ) → ProbabilityTheory.Kernel β δ","l":34,"f":"Causalean/Mathlib/Probability/Kernel/GraphMapProd.lean"},{"n":"Causalean.Mathlib.GraphMapProd.mechanismKernel_apply","k":"theorem","m":"Causalean.Mathlib.Probability.Kernel.GraphMapProd","a":"Mathlib/Probability/Kernel","t":2,"d":"Pointwise value of mechanismKernel: κ o = νₗ.map (fun l => Φ (o, l)).","s":"∀ {β : Type u_1} {γ : Type u_2} {δ : Type u_3} [inst : MeasurableSpace β] [inst_1 : MeasurableSpace γ] [inst_2 : MeasurableSpace δ] (νₗ : MeasureTheory.Measure γ) [MeasureTheory.SFinite νₗ] {Φ : β × γ → δ}, Measurable Φ → ∀ (o : β), (Causalean.Mathlib.GraphMapProd.mechanismKernel νₗ Φ) o = MeasureTheory.Measure.map (fun l => Φ (o, l)) νₗ","l":43,"f":"Causalean/Mathlib/Probability/Kernel/GraphMapProd.lean"},{"n":"Causalean.Mathlib.InformationTheory.KlDensityTiltExpansion.abs_klRemainder_le","k":"theorem","m":"Causalean.Mathlib.InformationTheory.KlDensityTiltExpansion.KLExpansion","a":"Mathlib/InformationTheory/KlDensityTiltExpansion","t":1,"d":"Dominated cubic remainder bound. For a probability measure μ and a measurable score function s that is bounded by a constant C and has mean zero under μ, if the tilt strength satisfies |h| · C ≤ 1, then the Kullback–Leibler divergence of the linearly tilted measure tiltMeasure μ s h from μ deviates from its second-order Taylor approximation (h²/2) · ∫ s² dμ by at most C³ · |h|³:","s":"∀ {Z : Type u_1} [inst : MeasurableSpace Z] {μ : MeasureTheory.Measure Z} [MeasureTheory.IsProbabilityMeasure μ] {s : Z → ℝ} {C h : ℝ}, Measurable s → (∀ (y : Z), |s y| ≤ C) → ∫ (y : Z), s y ∂μ = 0 → |h| * C ≤ 1 / 2 → |(InformationTheory.klDiv (Causalean.Mathlib.InformationTheory.KlDensityTiltExpansion.tiltMeasure μ s h) μ).toReal - h ^ 2 / 2 * ∫ (y : Z), s y ^ 2 ∂μ| ≤ C ^ 3 * |h| ^ 3","l":105,"f":"Causalean/Mathlib/InformationTheory/KlDensityTiltExpansion/KLExpansion.lean"},{"n":"Causalean.Mathlib.InformationTheory.KlDensityTiltExpansion.abs_tiltRemainder_le","k":"theorem","m":"Causalean.Mathlib.InformationTheory.KlDensityTiltExpansion.CubicRemainder","a":"Mathlib/InformationTheory/KlDensityTiltExpansion","t":1,"d":"Cubic remainder bound for (1 + x) · log (1 + x). For a real number x with |x| ≤ 1/2, the second-order Taylor remainder of (1 + x) · log (1 + x) at 0 is bounded by |x|³:","s":"∀ {x : ℝ}, |x| ≤ 1 / 2 → |(1 + x) * Real.log (1 + x) - x - x ^ 2 / 2| ≤ |x| ^ 3","l":131,"f":"Causalean/Mathlib/InformationTheory/KlDensityTiltExpansion/CubicRemainder.lean"},{"n":"Causalean.Mathlib.InformationTheory.KlDensityTiltExpansion.integral_tiltDensity","k":"theorem","m":"Causalean.Mathlib.InformationTheory.KlDensityTiltExpansion.Basic","a":"Mathlib/InformationTheory/KlDensityTiltExpansion","t":2,"d":"The tilt density integrates to 1: ∫ (1 + h · s) dμ = 1 + h · ∫ s = 1 for a probability measure μ and a mean-zero score s.","s":"∀ {Z : Type u_1} [inst : MeasurableSpace Z] {μ : MeasureTheory.Measure Z} [MeasureTheory.IsProbabilityMeasure μ] {s : Z → ℝ} {h : ℝ}, MeasureTheory.Integrable s μ → ∫ (y : Z), s y ∂μ = 0 → ∫ (y : Z), 1 + h * s y ∂μ = 1","l":75,"f":"Causalean/Mathlib/InformationTheory/KlDensityTiltExpansion/Basic.lean"},{"n":"Causalean.Mathlib.InformationTheory.KlDensityTiltExpansion.isProbabilityMeasure_tiltMeasure","k":"theorem","m":"Causalean.Mathlib.InformationTheory.KlDensityTiltExpansion.Basic","a":"Mathlib/InformationTheory/KlDensityTiltExpansion","t":1,"d":"The linear tilt is a probability measure for small |h|. For a probability measure μ and a measurable score function s that is bounded by a constant C (|s y| ≤ C for every y) and has mean zero under μ, if the tilt strength satisfies |h| · C ≤ 1, then the linearly tilted measure tiltMeasure μ s h is again a probability measure.","s":"∀ {Z : Type u_1} [inst : MeasurableSpace Z] {μ : MeasureTheory.Measure Z} [MeasureTheory.IsProbabilityMeasure μ] {s : Z → ℝ} {C h : ℝ}, Measurable s → (∀ (y : Z), |s y| ≤ C) → ∫ (y : Z), s y ∂μ = 0 → |h| * C ≤ 1 → MeasureTheory.IsProbabilityMeasure (Causalean.Mathlib.InformationTheory.KlDensityTiltExpansion.tiltMeasure μ s h)","l":94,"f":"Causalean/Mathlib/InformationTheory/KlDensityTiltExpansion/Basic.lean"},{"n":"Causalean.Mathlib.InformationTheory.KlDensityTiltExpansion.klDiv_tiltMeasure_toReal_eq","k":"theorem","m":"Causalean.Mathlib.InformationTheory.KlDensityTiltExpansion.KLExpansion","a":"Mathlib/InformationTheory/KlDensityTiltExpansion","t":1,"d":"KL of the linear tilt as an x log x density integral. For a probability measure μ and a measurable score function s that is bounded by a constant C and has mean zero under μ, if the tilt strength satisfies |h| · C ≤ 1, then the Kullback–Leibler divergence of the linearly tilted measure tiltMeasure μ s h from μ, as a real number, equals the μ-integral of (1 + h·s)·log(1 + h·s):","s":"∀ {Z : Type u_1} [inst : MeasurableSpace Z] {μ : MeasureTheory.Measure Z} [MeasureTheory.IsProbabilityMeasure μ] {s : Z → ℝ} {C h : ℝ}, Measurable s → (∀ (y : Z), |s y| ≤ C) → ∫ (y : Z), s y ∂μ = 0 → |h| * C ≤ 1 → (InformationTheory.klDiv (Causalean.Mathlib.InformationTheory.KlDensityTiltExpansion.tiltMeasure μ s h) μ).toReal = ∫ (y : Z), (1 + h * s y) * Real.log (1 + h * s y) ∂μ","l":55,"f":"Causalean/Mathlib/InformationTheory/KlDensityTiltExpansion/KLExpansion.lean"},{"n":"Causalean.Mathlib.InformationTheory.KlDensityTiltExpansion.klDiv_tilt_expansion","k":"theorem","m":"Causalean.Mathlib.InformationTheory.KlDensityTiltExpansion.KLExpansion","a":"Mathlib/InformationTheory/KlDensityTiltExpansion","t":1,"d":"Second-order KL expansion of the linear density tilt (headline result). For a probability measure μ and a measurable score function s that is bounded by a constant C and has mean zero under μ, the Kullback–Leibler divergence of the linearly tilted measure tiltMeasure μ s h from μ, as a function of the tilt strength h, agrees with the quadratic approximation (h²/2) · ∫ s² dμ up to an error that is little-o of h² as h → 0:","s":"∀ {Z : Type u_1} [inst : MeasurableSpace Z] {μ : MeasureTheory.Measure Z} [MeasureTheory.IsProbabilityMeasure μ] {s : Z → ℝ} {C : ℝ}, Measurable s → (∀ (y : Z), |s y| ≤ C) → ∫ (y : Z), s y ∂μ = 0 → (fun h => (InformationTheory.klDiv (Causalean.Mathlib.InformationTheory.KlDensityTiltExpansion.tiltMeasure μ s h) μ).toReal - h ^ 2 / 2 * ∫ (y : Z), s y ^ 2 ∂μ) =o[nhds 0] fun h => h ^ 2","l":210,"f":"Causalean/Mathlib/InformationTheory/KlDensityTiltExpansion/KLExpansion.lean"},{"n":"Causalean.Mathlib.InformationTheory.KlDensityTiltExpansion.tiltDensity_nonneg","k":"theorem","m":"Causalean.Mathlib.InformationTheory.KlDensityTiltExpansion.Basic","a":"Mathlib/InformationTheory/KlDensityTiltExpansion","t":2,"d":"For a bounded score |s y| ≤ C and tilt strength with |h| · C ≤ 1, the tilt density 1 + h · s y is nonnegative: |h · s y| ≤ |h| · C ≤ 1.","s":"∀ {Z : Type u_1} {s : Z → ℝ} {C h : ℝ}, (∀ (y : Z), |s y| ≤ C) → |h| * C ≤ 1 → ∀ (y : Z), 0 ≤ 1 + h * s y","l":56,"f":"Causalean/Mathlib/InformationTheory/KlDensityTiltExpansion/Basic.lean"},{"n":"Causalean.Mathlib.InformationTheory.KlDensityTiltExpansion.tiltMeasure","k":"def","m":"Causalean.Mathlib.InformationTheory.KlDensityTiltExpansion.Basic","a":"Mathlib/InformationTheory/KlDensityTiltExpansion","t":1,"d":"The linear density tilt of μ by the score s at strength h: tiltMeasure μ s h = μ.withDensity (y ↦ ENNReal.ofReal (1 + h · s y)). When 1 + h · s ≥ 0 a.e. and s is mean-zero this is again a probability measure (see isProbabilityMeasure_tiltMeasure).","s":"{Z : Type u_1} → [inst : MeasurableSpace Z] → MeasureTheory.Measure Z → (Z → ℝ) → ℝ → MeasureTheory.Measure Z","l":42,"f":"Causalean/Mathlib/InformationTheory/KlDensityTiltExpansion/Basic.lean"},{"n":"Causalean.Mathlib.InformationTheory.KlDensityTiltExpansion.tiltMeasure_absolutelyContinuous","k":"theorem","m":"Causalean.Mathlib.InformationTheory.KlDensityTiltExpansion.Basic","a":"Mathlib/InformationTheory/KlDensityTiltExpansion","t":2,"d":"The tilt is absolutely continuous with respect to the base measure μ (any withDensity measure is).","s":"∀ {Z : Type u_1} [inst : MeasurableSpace Z] (μ : MeasureTheory.Measure Z) (s : Z → ℝ) (h : ℝ), (Causalean.Mathlib.InformationTheory.KlDensityTiltExpansion.tiltMeasure μ s h).AbsolutelyContinuous μ","l":49,"f":"Causalean/Mathlib/InformationTheory/KlDensityTiltExpansion/Basic.lean"},{"n":"Causalean.Mathlib.InformationTheory.Measure.klDiv_bind_eq_of_base_recording","k":"theorem","m":"Causalean.Mathlib.InformationTheory.KLBind","a":"Mathlib/InformationTheory","t":1,"d":"For measurable spaces B and Ω, a finite base measure m, finite kernels κ, η from B to Ω, and a measurable projection proj : Ω → B whose graph {(b, ω) | b = proj ω} is a measurable subset of B × Ω, suppose κ-almost every output, for m-almost every base point b, lands in the fibre proj⁻¹{b}, likewise for η, and κ b is absolutely continuous with respect to η b for m-almost every b. Then the Kullback–Leibler divergence between the bind of m with κ and the bind of m with η equals the m-average, over the base point b, of the Kullback–Leibler divergence between κ b and η b.","s":"∀ {B : Type u_4} {Ω : Type u_5} [inst : MeasurableSpace B] [inst_1 : MeasurableSpace Ω] [MeasurableSpace.CountableOrCountablyGenerated B Ω] (m : MeasureTheory.Measure B) [MeasureTheory.IsFiniteMeasure m] (κ η : ProbabilityTheory.Kernel B Ω) [ProbabilityTheory.IsFiniteKernel κ] [ProbabilityTheory.IsFiniteKernel η] (proj : Ω → B), Measurable proj → MeasurableSet {p | p.1 = proj p.2} → (∀ᵐ (b : B) ∂m, (κ b) {ω | proj ω = b}ᶜ = 0) → (∀ᵐ (b : B) ∂m, (η b) {ω | proj ω = b}ᶜ = 0) → (∀ᵐ (b : B) ∂m, (κ b).AbsolutelyContinuous (η b)) → InformationTheory.klDiv (m.bind ⇑κ) (m.bind ⇑η) = ∫⁻ (b : B), InformationTheory.klDiv (κ b) (η b) ∂m","l":203,"f":"Causalean/Mathlib/InformationTheory/KLBind.lean"},{"n":"Causalean.Mathlib.InformationTheory.Measure.klDiv_bind_le","k":"theorem","m":"Causalean.Mathlib.InformationTheory.KLBind","a":"Mathlib/InformationTheory","t":1,"d":"Passing two finite input laws μ and ν through the same randomized observation channel κ, the Kullback–Leibler divergence between the channel's output laws is no larger than the divergence between the original input laws, including when the channel is non-injective or the original divergence is infinite.","s":"∀ {α : Type u_1} {β : Type u_2} [inst : MeasurableSpace α] [inst_1 : MeasurableSpace β] (μ ν : MeasureTheory.Measure α) [MeasureTheory.IsFiniteMeasure μ] [MeasureTheory.IsFiniteMeasure ν] (κ : ProbabilityTheory.Kernel α β) [ProbabilityTheory.IsMarkovKernel κ], InformationTheory.klDiv (μ.bind ⇑κ) (ν.bind ⇑κ) ≤ InformationTheory.klDiv μ ν","l":462,"f":"Causalean/Mathlib/InformationTheory/KLBind.lean"},{"n":"Causalean.Mathlib.InformationTheory.Measure.klDiv_bind_le_of_isProbabilityMeasure","k":"theorem","m":"Causalean.Mathlib.InformationTheory.KLBind","a":"Mathlib/InformationTheory","t":2,"d":"Passing two probability laws through a shared Markov channel cannot increase their Kullback--Leibler divergence; this is the probability-law specialization of the finite-measure data-processing inequality.","s":"∀ {α : Type u_1} {β : Type u_2} [inst : MeasurableSpace α] [inst_1 : MeasurableSpace β] (μ ν : MeasureTheory.Measure α) [MeasureTheory.IsProbabilityMeasure μ] [MeasureTheory.IsProbabilityMeasure ν] (κ : ProbabilityTheory.Kernel α β) [ProbabilityTheory.IsMarkovKernel κ], InformationTheory.klDiv (μ.bind ⇑κ) (ν.bind ⇑κ) ≤ InformationTheory.klDiv μ ν","l":474,"f":"Causalean/Mathlib/InformationTheory/KLBind.lean"},{"n":"Causalean.Mathlib.InformationTheory.Measure.klDiv_compProd_left","k":"theorem","m":"Causalean.Mathlib.InformationTheory.KLBind","a":"Mathlib/InformationTheory","t":2,"d":"Adding an output drawn from the same Markov kernel preserves the Kullback--Leibler divergence between two finite input laws because the joint observation still retains the input coordinate.","s":"∀ {α : Type u_1} {β : Type u_2} [inst : MeasurableSpace α] [inst_1 : MeasurableSpace β] (μ ν : MeasureTheory.Measure α) [MeasureTheory.IsFiniteMeasure μ] [MeasureTheory.IsFiniteMeasure ν] (κ : ProbabilityTheory.Kernel α β) [ProbabilityTheory.IsMarkovKernel κ], InformationTheory.klDiv (μ.compProd κ) (ν.compProd κ) = InformationTheory.klDiv μ ν","l":434,"f":"Causalean/Mathlib/InformationTheory/KLBind.lean"},{"n":"Causalean.Mathlib.InformationTheory.Measure.klDiv_compProd_right_of_forall_ac","k":"theorem","m":"Causalean.Mathlib.InformationTheory.KLBind","a":"Mathlib/InformationTheory","t":1,"d":"For a countably-generated pair of measurable spaces, a finite base measure μ, and finite kernels κ, η out of the base, if κ b is absolutely continuous with respect to η b for μ-almost every base point b, then the Kullback–Leibler divergence between the composition products μ ⊗ₘ κ and μ ⊗ₘ η equals the μ-average, over the base point, of the Kullback–Leibler divergence between κ and η at that base point.","s":"∀ {α : Type u_1} {β : Type u_2} [inst : MeasurableSpace α] [inst_1 : MeasurableSpace β] {μ : MeasureTheory.Measure α} {κ η : ProbabilityTheory.Kernel α β} [MeasurableSpace.CountableOrCountablyGenerated α β] [MeasureTheory.IsFiniteMeasure μ] [ProbabilityTheory.IsFiniteKernel κ] [ProbabilityTheory.IsFiniteKernel η], (∀ᵐ (a : α) ∂μ, (κ a).AbsolutelyContinuous (η a)) → InformationTheory.klDiv (μ.compProd κ) (μ.compProd η) = ∫⁻ (a : α), InformationTheory.klDiv (κ a) (η a) ∂μ","l":86,"f":"Causalean/Mathlib/InformationTheory/KLBind.lean"},{"n":"Causalean.Mathlib.InformationTheory.Measure.klDiv_map_le","k":"theorem","m":"Causalean.Mathlib.InformationTheory.KLBind","a":"Mathlib/InformationTheory","t":2,"d":"A common measurable observation rule cannot increase the Kullback--Leibler divergence between two finite input laws, even when the rule merges distinct inputs.","s":"∀ {α : Type u_1} {β : Type u_2} [inst : MeasurableSpace α] [inst_1 : MeasurableSpace β] {μ ν : MeasureTheory.Measure α} [MeasureTheory.IsFiniteMeasure μ] [MeasureTheory.IsFiniteMeasure ν] {f : α → β}, Measurable f → InformationTheory.klDiv (MeasureTheory.Measure.map f μ) (MeasureTheory.Measure.map f ν) ≤ InformationTheory.klDiv μ ν","l":342,"f":"Causalean/Mathlib/InformationTheory/KLBind.lean"},{"n":"Causalean.Mathlib.InformationTheory.Measure.klDiv_map_measurableEmbedding","k":"theorem","m":"Causalean.Mathlib.InformationTheory.KLBind","a":"Mathlib/InformationTheory","t":2,"d":"KL is invariant under a measurable embedding.","s":"∀ {α : Type u_1} {γ : Type u_3} [inst : MeasurableSpace α] [inst_1 : MeasurableSpace γ] {μ : MeasureTheory.Measure α} {f : α → γ}, MeasurableEmbedding f → ∀ [MeasureTheory.IsFiniteMeasure μ] {ν : MeasureTheory.Measure α} [MeasureTheory.IsFiniteMeasure ν], InformationTheory.klDiv (MeasureTheory.Measure.map f μ) (MeasureTheory.Measure.map f ν) = InformationTheory.klDiv μ ν","l":122,"f":"Causalean/Mathlib/InformationTheory/KLBind.lean"},{"n":"Causalean.Mathlib.InformationTheory.Measure.map_bind_eq_compProd_of_base_recording","k":"theorem","m":"Causalean.Mathlib.InformationTheory.KLBind","a":"Mathlib/InformationTheory","t":2,"d":"If a kernel is supported almost everywhere on outputs that record their base coordinate, then mapping its bound measure to the recorded base-output pair gives the corresponding composition-product measure.","s":"∀ {B : Type u_4} {Ω : Type u_5} [inst : MeasurableSpace B] [inst_1 : MeasurableSpace Ω] (m : MeasureTheory.Measure B) [MeasureTheory.SFinite m] (κ : ProbabilityTheory.Kernel B Ω) [ProbabilityTheory.IsSFiniteKernel κ] (proj : Ω → B), Measurable proj → (∀ᵐ (b : B) ∂m, (κ b) {ω | proj ω = b}ᶜ = 0) → MeasureTheory.Measure.map (fun ω => (proj ω, ω)) (m.bind ⇑κ) = m.compProd κ","l":173,"f":"Causalean/Mathlib/InformationTheory/KLBind.lean"},{"n":"Causalean.Mathlib.InformationTheory.Measure.measurableEmbedding_base_recording","k":"theorem","m":"Causalean.Mathlib.InformationTheory.KLBind","a":"Mathlib/InformationTheory","t":2,"d":"If a measurable map has a measurable graph, pairing each observation with its map value produces a measurable embedding into the corresponding product space.","s":"∀ {B : Type u_4} {Ω : Type u_5} [inst : MeasurableSpace B] [inst_1 : MeasurableSpace Ω] (proj : Ω → B), Measurable proj → MeasurableSet {p | p.1 = proj p.2} → MeasurableEmbedding fun ω => (proj ω, ω)","l":153,"f":"Causalean/Mathlib/InformationTheory/KLBind.lean"},{"n":"Causalean.Mathlib.InformationTheory.Measure.rnDeriv_compProd_right_of_forall_ac","k":"theorem","m":"Causalean.Mathlib.InformationTheory.KLBind","a":"Mathlib/InformationTheory","t":2,"d":"Radon--Nikodym derivative of a shared-base composition product.","s":"∀ {α : Type u_1} {β : Type u_2} [inst : MeasurableSpace α] [inst_1 : MeasurableSpace β] {μ : MeasureTheory.Measure α} {κ η : ProbabilityTheory.Kernel α β} [inst_2 : MeasurableSpace.CountableOrCountablyGenerated α β] [MeasureTheory.IsFiniteMeasure μ] [ProbabilityTheory.IsFiniteKernel κ] [ProbabilityTheory.IsFiniteKernel η], (∀ᵐ (a : α) ∂μ, (κ a).AbsolutelyContinuous (η a)) → (μ.compProd κ).rnDeriv (μ.compProd η) =ᵐ[μ.compProd η] fun p => κ.rnDeriv η p.1 p.2","l":52,"f":"Causalean/Mathlib/InformationTheory/KLBind.lean"},{"n":"Causalean.Mathlib.InformationTheory.ProductKLPrivate.ae_prod_fst_of_ae","k":"theorem","m":"Causalean.Mathlib.InformationTheory.ProductKLLeCam","a":"Mathlib/InformationTheory","t":2,"d":"A measurable property holding almost everywhere under a measure also holds for the first coordinate almost everywhere under its product with a probability measure.","s":"∀ {α : Type u_1} {β : Type u_2} [inst : MeasurableSpace α] [inst_1 : MeasurableSpace β] {μ : MeasureTheory.Measure α} {ν : MeasureTheory.Measure β} [MeasureTheory.IsProbabilityMeasure ν] {p : α → Prop}, MeasurableSet {x | p x} → (∀ᵐ (x : α) ∂μ, p x) → ∀ᵐ (z : α × β) ∂μ.prod ν, p z.1","l":59,"f":"Causalean/Mathlib/InformationTheory/ProductKLLeCam.lean"},{"n":"Causalean.Mathlib.InformationTheory.ProductKLPrivate.ae_prod_snd_of_ae","k":"theorem","m":"Causalean.Mathlib.InformationTheory.ProductKLLeCam","a":"Mathlib/InformationTheory","t":2,"d":"A measurable property holding almost everywhere under a measure also holds for the second coordinate almost everywhere under its product with a probability measure.","s":"∀ {α : Type u_1} {β : Type u_2} [inst : MeasurableSpace α] [inst_1 : MeasurableSpace β] {μ : MeasureTheory.Measure α} {ν : MeasureTheory.Measure β} [MeasureTheory.IsProbabilityMeasure μ] [MeasureTheory.SFinite ν] {p : β → Prop}, MeasurableSet {y | p y} → (∀ᵐ (y : β) ∂ν, p y) → ∀ᵐ (z : α × β) ∂μ.prod ν, p z.2","l":69,"f":"Causalean/Mathlib/InformationTheory/ProductKLLeCam.lean"},{"n":"Causalean.Mathlib.InformationTheory.ProductKLPrivate.klDiv_prod_toReal_add","k":"theorem","m":"Causalean.Mathlib.InformationTheory.ProductKLLeCam","a":"Mathlib/InformationTheory","t":2,"d":"For product probability laws with integrable component log-likelihood ratios, the real-valued KL divergence of the product equals the sum of the component KL divergences.","s":"∀ {α : Type u_1} {β : Type u_2} [inst : MeasurableSpace α] [inst_1 : MeasurableSpace β] (μ₁ ν₁ : MeasureTheory.Measure α) (μ₂ ν₂ : MeasureTheory.Measure β) [MeasureTheory.IsProbabilityMeasure μ₁] [MeasureTheory.IsProbabilityMeasure ν₁] [MeasureTheory.IsProbabilityMeasure μ₂] [MeasureTheory.IsProbabilityMeasure ν₂], μ₁.AbsolutelyContinuous ν₁ → μ₂.AbsolutelyContinuous ν₂ → MeasureTheory.Integrable (MeasureTheory.llr μ₁ ν₁) μ₁ → MeasureTheory.Integrable (MeasureTheory.llr μ₂ ν₂) μ₂ → (InformationTheory.klDiv (μ₁.prod μ₂) (ν₁.prod ν₂)).toReal = (InformationTheory.klDiv μ₁ ν₁).toReal + (InformationTheory.klDiv μ₂ ν₂).toReal","l":241,"f":"Causalean/Mathlib/InformationTheory/ProductKLLeCam.lean"},{"n":"Causalean.Mathlib.InformationTheory.ProductKLPrivate.llr_integrable_of_map_measurableEquiv","k":"theorem","m":"Causalean.Mathlib.InformationTheory.ProductKLLeCam","a":"Mathlib/InformationTheory","t":2,"d":"Pushing two measures through a measurable relabelling preserves integrability of their log-likelihood ratio, allowing KL side conditions to transfer between equivalent sample spaces.","s":"∀ {α : Type u_1} {β : Type u_2} [inst : MeasurableSpace α] [inst_1 : MeasurableSpace β] (e : α ≃ᵐ β) (μ ν : MeasureTheory.Measure α) [MeasureTheory.SigmaFinite μ] [MeasureTheory.SigmaFinite ν], μ.AbsolutelyContinuous ν → MeasureTheory.Integrable (MeasureTheory.llr (MeasureTheory.Measure.map (⇑e) μ) (MeasureTheory.Measure.map (⇑e) ν)) (MeasureTheory.Measure.map (⇑e) μ) → MeasureTheory.Integrable (MeasureTheory.llr μ ν) μ","l":145,"f":"Causalean/Mathlib/InformationTheory/ProductKLLeCam.lean"},{"n":"Causalean.Mathlib.InformationTheory.ProductKLPrivate.llr_prod_ae","k":"theorem","m":"Causalean.Mathlib.InformationTheory.ProductKLLeCam","a":"Mathlib/InformationTheory","t":2,"d":"When each component law is absolutely continuous with respect to its reference law, the log-likelihood ratio of their product laws is almost surely the sum of the two component log-likelihood ratios.","s":"∀ {α : Type u_1} {β : Type u_2} [inst : MeasurableSpace α] [inst_1 : MeasurableSpace β] (μ₁ ν₁ : MeasureTheory.Measure α) (μ₂ ν₂ : MeasureTheory.Measure β) [MeasureTheory.IsProbabilityMeasure μ₁] [MeasureTheory.SigmaFinite ν₁] [MeasureTheory.IsProbabilityMeasure μ₂] [MeasureTheory.SigmaFinite ν₂], μ₁.AbsolutelyContinuous ν₁ → μ₂.AbsolutelyContinuous ν₂ → MeasureTheory.llr (μ₁.prod μ₂) (ν₁.prod ν₂) =ᵐ[μ₁.prod μ₂] fun z => MeasureTheory.llr μ₁ ν₁ z.1 + MeasureTheory.llr μ₂ ν₂ z.2","l":79,"f":"Causalean/Mathlib/InformationTheory/ProductKLLeCam.lean"},{"n":"Causalean.Mathlib.InformationTheory.ProductKLPrivate.llr_prod_integrable","k":"theorem","m":"Causalean.Mathlib.InformationTheory.ProductKLLeCam","a":"Mathlib/InformationTheory","t":2,"d":"Integrable component log-likelihood ratios imply that the log-likelihood ratio of the corresponding product laws is integrable.","s":"∀ {α : Type u_1} {β : Type u_2} [inst : MeasurableSpace α] [inst_1 : MeasurableSpace β] (μ₁ ν₁ : MeasureTheory.Measure α) (μ₂ ν₂ : MeasureTheory.Measure β) [MeasureTheory.IsProbabilityMeasure μ₁] [MeasureTheory.SigmaFinite ν₁] [MeasureTheory.IsProbabilityMeasure μ₂] [MeasureTheory.SigmaFinite ν₂], μ₁.AbsolutelyContinuous ν₁ → μ₂.AbsolutelyContinuous ν₂ → MeasureTheory.Integrable (MeasureTheory.llr μ₁ ν₁) μ₁ → MeasureTheory.Integrable (MeasureTheory.llr μ₂ ν₂) μ₂ → MeasureTheory.Integrable (MeasureTheory.llr (μ₁.prod μ₂) (ν₁.prod ν₂)) (μ₁.prod μ₂)","l":124,"f":"Causalean/Mathlib/InformationTheory/ProductKLLeCam.lean"},{"n":"Causalean.Mathlib.InformationTheory.ProductKLPrivate.productKL_tensorization_toReal_eq","k":"theorem","m":"Causalean.Mathlib.InformationTheory.ProductKLLeCam","a":"Mathlib/InformationTheory","t":2,"d":"Under absolute-continuity and integrability conditions for every finite product, the real-valued Kullback–Leibler divergence of two n-fold product laws is n times the one-law divergence.","s":"∀ {α : Type u_1} [inst : MeasurableSpace α] (n : ℕ) (μ ν : MeasureTheory.Measure α) [MeasureTheory.IsProbabilityMeasure μ] [MeasureTheory.IsProbabilityMeasure ν], μ.AbsolutelyContinuous ν → MeasureTheory.Integrable (MeasureTheory.llr μ ν) μ → (∀ (k : ℕ), (MeasureTheory.Measure.pi fun x => μ).AbsolutelyContinuous (MeasureTheory.Measure.pi fun x => ν)) → (∀ (k : ℕ), MeasureTheory.Integrable (MeasureTheory.llr (MeasureTheory.Measure.pi fun x => μ) (MeasureTheory.Measure.pi fun x => ν)) (MeasureTheory.Measure.pi fun x => μ)) → (InformationTheory.klDiv (MeasureTheory.Measure.pi fun x => μ) (MeasureTheory.Measure.pi fun x => ν)).toReal = ↑n * (InformationTheory.klDiv μ ν).toReal","l":288,"f":"Causalean/Mathlib/InformationTheory/ProductKLLeCam.lean"},{"n":"Causalean.Mathlib.InformationTheory.ProductKLTensorizationBound","k":"def","m":"Causalean.Mathlib.InformationTheory.ProductKLLeCam","a":"Mathlib/InformationTheory","t":1,"d":"Product-KL tensorisation bound for an n-fold i.i.d. product pair.","s":"{α : Type u_1} → [inst : MeasurableSpace α] → ℕ → MeasureTheory.Measure α → MeasureTheory.Measure α → Prop","l":357,"f":"Causalean/Mathlib/InformationTheory/ProductKLLeCam.lean"},{"n":"Causalean.Mathlib.InformationTheory.ProductKLTensorizationBound.apply","k":"theorem","m":"Causalean.Mathlib.InformationTheory.ProductKLLeCam","a":"Mathlib/InformationTheory","t":2,"d":"Unpack a supplied product-KL tensorisation bound.","s":"∀ {α : Type u_1} [inst : MeasurableSpace α] {n : ℕ} {μ ν : MeasureTheory.Measure α}, Causalean.Mathlib.InformationTheory.ProductKLTensorizationBound n μ ν → (InformationTheory.klDiv (MeasureTheory.Measure.pi fun x => μ) (MeasureTheory.Measure.pi fun x => ν)).toReal ≤ ↑n * (InformationTheory.klDiv μ ν).toReal","l":414,"f":"Causalean/Mathlib/InformationTheory/ProductKLLeCam.lean"},{"n":"Causalean.Mathlib.InformationTheory.ProductKLTensorizationBound.one_ne_top","k":"theorem","m":"Causalean.Mathlib.InformationTheory.ProductKLLeCam","a":"Mathlib/InformationTheory","t":2,"d":"The one-observation KL divergence in a supplied tensorisation bound is finite.","s":"∀ {α : Type u_1} [inst : MeasurableSpace α] {n : ℕ} {μ ν : MeasureTheory.Measure α}, Causalean.Mathlib.InformationTheory.ProductKLTensorizationBound n μ ν → InformationTheory.klDiv μ ν ≠ ⊤","l":433,"f":"Causalean/Mathlib/InformationTheory/ProductKLLeCam.lean"},{"n":"Causalean.Mathlib.InformationTheory.ProductKLTensorizationBound.product_ne_top","k":"theorem","m":"Causalean.Mathlib.InformationTheory.ProductKLLeCam","a":"Mathlib/InformationTheory","t":2,"d":"The product KL divergence in a supplied tensorisation bound is finite.","s":"∀ {α : Type u_1} [inst : MeasurableSpace α] {n : ℕ} {μ ν : MeasureTheory.Measure α}, Causalean.Mathlib.InformationTheory.ProductKLTensorizationBound n μ ν → InformationTheory.klDiv (MeasureTheory.Measure.pi fun x => μ) (MeasureTheory.Measure.pi fun x => ν) ≠ ⊤","l":424,"f":"Causalean/Mathlib/InformationTheory/ProductKLLeCam.lean"},{"n":"Causalean.Mathlib.InformationTheory.clippedStatisticSuccessParameter","k":"def","m":"Causalean.Mathlib.InformationTheory.CommonStatisticBernoulli","a":"Mathlib/InformationTheory","t":1,"d":"Globally clip the conditional parameter to the middle half.","s":"{A : Type u_1} → [inst : MeasurableSpace A] → MeasureTheory.Measure A → (A → ℝ) → (A → ℝ) → ℝ → ℝ","l":313,"f":"Causalean/Mathlib/InformationTheory/CommonStatisticBernoulli.lean"},{"n":"Causalean.Mathlib.InformationTheory.clippedStatisticSuccessParameter_abs_sub_le_ae","k":"theorem","m":"Causalean.Mathlib.InformationTheory.CommonStatisticBernoulli","a":"Mathlib/InformationTheory","t":2,"d":"A localized setwise bound on success-weighted statistic masses yields the corresponding almost-everywhere bound on conditional Bernoulli parameters.","s":"∀ {A : Type u_1} [inst : MeasurableSpace A] (nu nu' : MeasureTheory.Measure A) [MeasureTheory.IsFiniteMeasure nu] [MeasureTheory.IsFiniteMeasure nu'] (p p' stat : A → ℝ), Measurable p → Measurable p' → Measurable stat → (∀ (x : A), 1 / 4 ≤ p x) → (∀ (x : A), p x ≤ 3 / 4) → (∀ (x : A), 1 / 4 ≤ p' x) → (∀ (x : A), p' x ≤ 3 / 4) → MeasureTheory.Measure.map stat nu = MeasureTheory.Measure.map stat nu' → ∀ {D : ℝ}, 0 ≤ D → ∀ {E : Set ℝ}, MeasurableSet E → (∀ (B : Set ℝ), MeasurableSet B → |∫ (x : A) in {x | stat x ∈ B}, p x ∂nu - ∫ (x : A) in {x | stat x ∈ B}, p' x ∂nu'| ≤ D * ((MeasureTheory.Measure.map stat nu) (B ∩ E)).toReal) → ∀ᵐ (r : ℝ) ∂MeasureTheory.Measure.map stat nu, |Causalean.Mathlib.InformationTheory.clippedStatisticSuccessParameter nu p stat r - Causalean.Mathlib.InformationTheory.clippedStatisticSuccessParameter nu' p' stat r| ≤ E.indicator (fun x => D) r","l":496,"f":"Causalean/Mathlib/InformationTheory/CommonStatisticBernoulli.lean"},{"n":"Causalean.Mathlib.InformationTheory.clippedStatisticSuccessParameter_ae_eq","k":"theorem","m":"Causalean.Mathlib.InformationTheory.CommonStatisticBernoulli","a":"Mathlib/InformationTheory","t":2,"d":"","s":"∀ {A : Type u_1} [inst : MeasurableSpace A] (nu : MeasureTheory.Measure A) [MeasureTheory.IsFiniteMeasure nu] (p stat : A → ℝ), Measurable p → Measurable stat → (∀ (x : A), 1 / 4 ≤ p x) → (∀ (x : A), p x ≤ 3 / 4) → Causalean.Mathlib.InformationTheory.clippedStatisticSuccessParameter nu p stat =ᵐ[MeasureTheory.Measure.map stat nu] Causalean.Mathlib.InformationTheory.statisticSuccessParameter nu p stat","l":339,"f":"Causalean/Mathlib/InformationTheory/CommonStatisticBernoulli.lean"},{"n":"Causalean.Mathlib.InformationTheory.clippedStatisticSuccessParameter_measurable","k":"theorem","m":"Causalean.Mathlib.InformationTheory.CommonStatisticBernoulli","a":"Mathlib/InformationTheory","t":2,"d":"","s":"∀ {A : Type u_1} [inst : MeasurableSpace A] (nu : MeasureTheory.Measure A) (p stat : A → ℝ), Measurable (Causalean.Mathlib.InformationTheory.clippedStatisticSuccessParameter nu p stat)","l":322,"f":"Causalean/Mathlib/InformationTheory/CommonStatisticBernoulli.lean"},{"n":"Causalean.Mathlib.InformationTheory.clippedStatisticSuccessParameter_mem_Icc","k":"theorem","m":"Causalean.Mathlib.InformationTheory.CommonStatisticBernoulli","a":"Mathlib/InformationTheory","t":2,"d":"","s":"∀ {A : Type u_1} [inst : MeasurableSpace A] (nu : MeasureTheory.Measure A) (p stat : A → ℝ) (r : ℝ), Causalean.Mathlib.InformationTheory.clippedStatisticSuccessParameter nu p stat r ∈ Set.Icc (1 / 4) (3 / 4)","l":330,"f":"Causalean/Mathlib/InformationTheory/CommonStatisticBernoulli.lean"},{"n":"Causalean.Mathlib.InformationTheory.commonStatisticBernoulliKernel","k":"def","m":"Causalean.Mathlib.InformationTheory.CommonStatisticBernoulli","a":"Mathlib/InformationTheory","t":1,"d":"A measurable success-probability function determines the Markov kernel that returns the corresponding real-valued Bernoulli law at each input.","s":"{S : Type u_1} → [inst : MeasurableSpace S] → (p : S → ℝ) → Measurable p → ProbabilityTheory.Kernel S ℝ","l":38,"f":"Causalean/Mathlib/InformationTheory/CommonStatisticBernoulli.lean"},{"n":"Causalean.Mathlib.InformationTheory.commonStatisticBernoulliKernel_isMarkovKernel","k":"theorem","m":"Causalean.Mathlib.InformationTheory.CommonStatisticBernoulli","a":"Mathlib/InformationTheory","t":2,"d":"Pointwise unit-interval parameters make the common-statistic Bernoulli kernel Markov.","s":"∀ {S : Type u_1} [inst : MeasurableSpace S] (p : S → ℝ) (hp : Measurable p), (∀ (r : S), 0 ≤ p r) → (∀ (r : S), p r ≤ 1) → ProbabilityTheory.IsMarkovKernel (Causalean.Mathlib.InformationTheory.commonStatisticBernoulliKernel p hp)","l":49,"f":"Causalean/Mathlib/InformationTheory/CommonStatisticBernoulli.lean"},{"n":"Causalean.Mathlib.InformationTheory.commonStatisticBernoulliKernel_setLIntegral_eq","k":"theorem","m":"Causalean.Mathlib.InformationTheory.CommonStatisticBernoulli","a":"Mathlib/InformationTheory","t":2,"d":"Integrating Bernoulli kernels over two base sets gives the same outcome measure when the base masses and success-weighted masses agree.","s":"∀ {A : Type u_1} {B : Type u_2} [inst : MeasurableSpace A] [inst_1 : MeasurableSpace B] (nu : MeasureTheory.Measure A) [MeasureTheory.IsFiniteMeasure nu] (nu' : MeasureTheory.Measure B) [MeasureTheory.IsFiniteMeasure nu'] (p : A → ℝ) (p' : B → ℝ) (hp : Measurable p) (hp' : Measurable p'), (∀ (x : A), 0 ≤ p x) → (∀ (x : A), p x ≤ 1) → (∀ (x : B), 0 ≤ p' x) → (∀ (x : B), p' x ≤ 1) → ∀ {D : Set A} {D' : Set B}, nu D = nu' D' → ∫ (x : A) in D, p x ∂nu = ∫ (x : B) in D', p' x ∂nu' → ∀ (E : Set ℝ), ∫⁻ (x : A) in D, ((Causalean.Mathlib.InformationTheory.commonStatisticBernoulliKernel p hp) x) E ∂nu = ∫⁻ (x : B) in D', ((Causalean.Mathlib.InformationTheory.commonStatisticBernoulliKernel p' hp') x) E ∂nu'","l":351,"f":"Causalean/Mathlib/InformationTheory/CommonStatisticBernoulli.lean"},{"n":"Causalean.Mathlib.InformationTheory.commonStatisticBernoulliOutcome_klDiv_le_of_localized_parameter","k":"theorem","m":"Causalean.Mathlib.InformationTheory.CommonStatisticBernoulli","a":"Mathlib/InformationTheory","t":2,"d":"Swapping a common statistic behind its Bernoulli outcome preserves the localized KL estimate, giving the (outcome, statistic) coordinate order used by signed observations.","s":"∀ (m : MeasureTheory.Measure ℝ) [MeasureTheory.IsFiniteMeasure m] (p q : ℝ → ℝ) (hp : Measurable p) (hq : Measurable q), (∀ (r : ℝ), 1 / 4 ≤ p r) → (∀ (r : ℝ), p r ≤ 3 / 4) → (∀ (r : ℝ), 1 / 4 ≤ q r) → (∀ (r : ℝ), q r ≤ 3 / 4) → ∀ {D : ℝ}, 0 ≤ D → ∀ {E : Set ℝ}, MeasurableSet E → (∀ᵐ (r : ℝ) ∂m, |p r - q r| ≤ E.indicator (fun x => D) r) → InformationTheory.klDiv (MeasureTheory.Measure.map Prod.swap (m.compProd (Causalean.Mathlib.InformationTheory.commonStatisticBernoulliKernel p hp))) (MeasureTheory.Measure.map Prod.swap (m.compProd (Causalean.Mathlib.InformationTheory.commonStatisticBernoulliKernel q hq))) ≤ ENNReal.ofReal (4 * D ^ 2) * m E","l":124,"f":"Causalean/Mathlib/InformationTheory/CommonStatisticBernoulli.lean"},{"n":"Causalean.Mathlib.InformationTheory.commonStatisticBernoulli_klDiv_le_of_localized_parameter","k":"theorem","m":"Causalean.Mathlib.InformationTheory.CommonStatisticBernoulli","a":"Mathlib/InformationTheory","t":2,"d":"A common statistic with conditionally Bernoulli outcomes has KL bounded by the squared change in its success parameter, integrated only over the statistic region where that parameter can change. This is the generic disintegration step used by the signed hard-cell comparison.","s":"∀ (m : MeasureTheory.Measure ℝ) [MeasureTheory.IsFiniteMeasure m] (p q : ℝ → ℝ) (hp : Measurable p) (hq : Measurable q), (∀ (r : ℝ), 1 / 4 ≤ p r) → (∀ (r : ℝ), p r ≤ 3 / 4) → (∀ (r : ℝ), 1 / 4 ≤ q r) → (∀ (r : ℝ), q r ≤ 3 / 4) → ∀ {D : ℝ}, 0 ≤ D → ∀ {E : Set ℝ}, MeasurableSet E → (∀ᵐ (r : ℝ) ∂m, |p r - q r| ≤ E.indicator (fun x => D) r) → InformationTheory.klDiv (m.compProd (Causalean.Mathlib.InformationTheory.commonStatisticBernoulliKernel p hp)) (m.compProd (Causalean.Mathlib.InformationTheory.commonStatisticBernoulliKernel q hq)) ≤ ENNReal.ofReal (4 * D ^ 2) * m E","l":61,"f":"Causalean/Mathlib/InformationTheory/CommonStatisticBernoulli.lean"},{"n":"Causalean.Mathlib.InformationTheory.condEntropy","k":"def","m":"Causalean.Mathlib.InformationTheory.ConditionalEntropy","a":"Mathlib/InformationTheory","t":1,"d":"Conditional Shannon entropy H(X ∣ Y) of a joint mass function p : α × β → ℝ, defined via the chain rule H(X ∣ Y) = H(X,Y) − H(Y), i.e. entropy p − entropy (yMarginal p).","s":"{α : Type u_1} → {β : Type u_2} → [Fintype α] → [Fintype β] → (α × β → ℝ) → ℝ","l":51,"f":"Causalean/Mathlib/InformationTheory/ConditionalEntropy.lean"},{"n":"Causalean.Mathlib.InformationTheory.condEntropy_def","k":"theorem","m":"Causalean.Mathlib.InformationTheory.ConditionalEntropy","a":"Mathlib/InformationTheory","t":2,"d":"Conditional entropy unfolds to total joint entropy minus the entropy of the conditioning marginal.","s":"∀ {α : Type u_1} {β : Type u_2} [inst : Fintype α] [inst_1 : Fintype β] (p : α × β → ℝ), Causalean.Mathlib.InformationTheory.condEntropy p = Causalean.Mathlib.InformationTheory.entropy p - Causalean.Mathlib.InformationTheory.entropy (Causalean.Mathlib.InformationTheory.yMarginal p)","l":72,"f":"Causalean/Mathlib/InformationTheory/ConditionalEntropy.lean"},{"n":"Causalean.Mathlib.InformationTheory.correctMass_eq","k":"theorem","m":"Causalean.Mathlib.InformationTheory.ConditionalEntropy","a":"Mathlib/InformationTheory","t":2,"d":"The correct-decision mass equals 1 − errorProb: splitting the total mass = 1 into the correct cells (x = decode y) and the error cells gives ∑_{x = decode y} p (x, y) = 1 − errorProb p decode.","s":"∀ {α : Type u_1} {β : Type u_2} [inst : Fintype α] [inst_1 : Fintype β] [inst_2 : DecidableEq α] {p : α × β → ℝ}, ∑ xy, p xy = 1 → ∀ (decode : β → α), (∑ xy, if xy.1 = decode xy.2 then p xy else 0) = 1 - Causalean.Mathlib.InformationTheory.errorProb p decode","l":192,"f":"Causalean/Mathlib/InformationTheory/ConditionalEntropy.lean"},{"n":"Causalean.Mathlib.InformationTheory.entropy","k":"def","m":"Causalean.Mathlib.InformationTheory.Entropy","a":"Mathlib/InformationTheory","t":1,"d":"Shannon entropy (in nats) of a real-valued mass function p : α → ℝ on a finite alphabet α, defined as ∑ i, Real.negMulLog (p i) = ∑ i, -(p i) * log (p i).","s":"{α : Type u_1} → [Fintype α] → (α → ℝ) → ℝ","l":41,"f":"Causalean/Mathlib/InformationTheory/Entropy.lean"},{"n":"Causalean.Mathlib.InformationTheory.entropy_const_eq_log_card","k":"theorem","m":"Causalean.Mathlib.InformationTheory.Entropy","a":"Mathlib/InformationTheory","t":1,"d":"Sharpness of the maximum-entropy bound. The uniform probability mass function on a nonempty finite alphabet — assigning every outcome probability equal to the reciprocal of the alphabet's cardinality — attains entropy exactly equal to the natural logarithm of that cardinality.","s":"∀ {α : Type u_1} [inst : Fintype α] [Nonempty α], (Causalean.Mathlib.InformationTheory.entropy fun x => (↑(Fintype.card α))⁻¹) = Real.log ↑(Fintype.card α)","l":144,"f":"Causalean/Mathlib/InformationTheory/Entropy.lean"},{"n":"Causalean.Mathlib.InformationTheory.entropy_def","k":"theorem","m":"Causalean.Mathlib.InformationTheory.Entropy","a":"Mathlib/InformationTheory","t":2,"d":"The finite-alphabet entropy definition unfolds to the sum of Real.negMulLog over the alphabet.","s":"∀ {α : Type u_1} [inst : Fintype α] (p : α → ℝ), Causalean.Mathlib.InformationTheory.entropy p = ∑ i, (p i).negMulLog","l":49,"f":"Causalean/Mathlib/InformationTheory/Entropy.lean"},{"n":"Causalean.Mathlib.InformationTheory.entropy_le_crossEntropy","k":"theorem","m":"Causalean.Mathlib.InformationTheory.ConditionalEntropy","a":"Mathlib/InformationTheory","t":1,"d":"Gibbs / cross-entropy inequality. For a nonnegative mass function p and a nonnegative reference mass function g on a finite type γ, if the total mass of g is at most the total mass of p (∑ g ≤ ∑ p) and g dominates the support of p (p i ≠ 0 → 0 < g i), then the Shannon entropy of p is bounded by the cross-entropy of p relative to g: entropy p ≤ −∑ i, p i * Real.log (g i).","s":"∀ {γ : Type u_3} [inst : Fintype γ] {p g : γ → ℝ}, (∀ (i : γ), 0 ≤ p i) → (∀ (i : γ), 0 ≤ g i) → ∑ i, g i ≤ ∑ i, p i → (∀ (i : γ), p i ≠ 0 → 0 < g i) → Causalean.Mathlib.InformationTheory.entropy p ≤ -∑ i, p i * Real.log (g i)","l":108,"f":"Causalean/Mathlib/InformationTheory/ConditionalEntropy.lean"},{"n":"Causalean.Mathlib.InformationTheory.entropy_le_log_card","k":"theorem","m":"Causalean.Mathlib.InformationTheory.Entropy","a":"Mathlib/InformationTheory","t":1,"d":"Maximum-entropy (Gibbs) bound. For a nonnegative function p on a finite alphabet α that sums to one (∑ i, p i = 1, i.e. p is a probability mass function) with n = Fintype.card α, the Shannon entropy of p is at most log n: entropy p ≤ Real.log (Fintype.card α).","s":"∀ {α : Type u_1} [inst : Fintype α] {p : α → ℝ}, (∀ (i : α), 0 ≤ p i) → ∑ i, p i = 1 → Causalean.Mathlib.InformationTheory.entropy p ≤ Real.log ↑(Fintype.card α)","l":91,"f":"Causalean/Mathlib/InformationTheory/Entropy.lean"},{"n":"Causalean.Mathlib.InformationTheory.entropy_nonneg","k":"theorem","m":"Causalean.Mathlib.InformationTheory.Entropy","a":"Mathlib/InformationTheory","t":2,"d":"Entropy is nonnegative for any sub-probability vector: if 0 ≤ p i ≤ 1 for every i, then 0 ≤ entropy p. Each summand Real.negMulLog (p i) is nonnegative on [0,1] (Real.negMulLog_nonneg), so the finite sum is nonnegative.","s":"∀ {α : Type u_1} [inst : Fintype α] {p : α → ℝ}, (∀ (i : α), 0 ≤ p i) → (∀ (i : α), p i ≤ 1) → 0 ≤ Causalean.Mathlib.InformationTheory.entropy p","l":53,"f":"Causalean/Mathlib/InformationTheory/Entropy.lean"},{"n":"Causalean.Mathlib.InformationTheory.errorProb","k":"def","m":"Causalean.Mathlib.InformationTheory.ConditionalEntropy","a":"Mathlib/InformationTheory","t":1,"d":"Error probability of a deterministic decoder decode : β → α under the joint mass function p: the total mass on cells where the decoder is wrong, ∑_{x ≠ decode y} p (x, y). Encoded with an if so the correct cells contribute 0.","s":"{α : Type u_1} → {β : Type u_2} → [Fintype α] → [Fintype β] → [DecidableEq α] → (α × β → ℝ) → (β → α) → ℝ","l":58,"f":"Causalean/Mathlib/InformationTheory/ConditionalEntropy.lean"},{"n":"Causalean.Mathlib.InformationTheory.errorProb_def","k":"theorem","m":"Causalean.Mathlib.InformationTheory.ConditionalEntropy","a":"Mathlib/InformationTheory","t":2,"d":"The decoder error probability unfolds to the sum of the joint masses on incorrect decoding cells.","s":"∀ {α : Type u_1} {β : Type u_2} [inst : Fintype α] [inst_1 : Fintype β] [inst_2 : DecidableEq α] (p : α × β → ℝ) (decode : β → α), Causalean.Mathlib.InformationTheory.errorProb p decode = ∑ xy, if xy.1 = decode xy.2 then 0 else p xy","l":77,"f":"Causalean/Mathlib/InformationTheory/ConditionalEntropy.lean"},{"n":"Causalean.Mathlib.InformationTheory.errorProb_le_one","k":"theorem","m":"Causalean.Mathlib.InformationTheory.ConditionalEntropy","a":"Mathlib/InformationTheory","t":2,"d":"The error probability is at most one (it is a sub-sum of the total mass = 1).","s":"∀ {α : Type u_1} {β : Type u_2} [inst : Fintype α] [inst_1 : Fintype β] [inst_2 : DecidableEq α] {p : α × β → ℝ}, (∀ (xy : α × β), 0 ≤ p xy) → ∑ xy, p xy = 1 → ∀ (decode : β → α), Causalean.Mathlib.InformationTheory.errorProb p decode ≤ 1","l":181,"f":"Causalean/Mathlib/InformationTheory/ConditionalEntropy.lean"},{"n":"Causalean.Mathlib.InformationTheory.errorProb_nonneg","k":"theorem","m":"Causalean.Mathlib.InformationTheory.ConditionalEntropy","a":"Mathlib/InformationTheory","t":2,"d":"The error probability is nonnegative.","s":"∀ {α : Type u_1} {β : Type u_2} [inst : Fintype α] [inst_1 : Fintype β] [inst_2 : DecidableEq α] {p : α × β → ℝ}, (∀ (xy : α × β), 0 ≤ p xy) → ∀ (decode : β → α), 0 ≤ Causalean.Mathlib.InformationTheory.errorProb p decode","l":171,"f":"Causalean/Mathlib/InformationTheory/ConditionalEntropy.lean"},{"n":"Causalean.Mathlib.InformationTheory.fanoRef","k":"def","m":"Causalean.Mathlib.InformationTheory.Fano","a":"Mathlib/InformationTheory","t":1,"d":"The Fano reference distribution on α × β. On each column y it places mass (1 − Pe) on the decoded symbol decode y and spreads the error mass Pe uniformly over the card α − 1 other symbols, then weights the column by the marginal yMarginal p y: fanoRef p decode (x, y) = yMarginal p y · (if x = decode y then 1 − Pe else Pe / (card α − 1)) where Pe = errorProb p decode. It is the worst-case posterior that makes Gibbs tight.","s":"{α : Type u_1} → {β : Type u_2} → [Fintype α] → [Fintype β] → [DecidableEq α] → (α × β → ℝ) → (β → α) → α × β → ℝ","l":41,"f":"Causalean/Mathlib/InformationTheory/Fano.lean"},{"n":"Causalean.Mathlib.InformationTheory.fanoRef_ac","k":"theorem","m":"Causalean.Mathlib.InformationTheory.Fano","a":"Mathlib/InformationTheory","t":2,"d":"The Fano reference distribution dominates the support of p: p xy ≠ 0 → 0 < fanoRef xy. A nonzero p (x, y) forces yMarginal p y > 0; on a correct cell the complementary mass 1 − Pe ≥ p (x,y) > 0 and on an error cell Pe ≥ p (x,y) > 0, so the inner weight is positive. This is the absolute-continuity hypothesis of the Gibbs inequality.","s":"∀ {α : Type u_1} {β : Type u_2} [inst : Fintype α] [inst_1 : Fintype β] [inst_2 : DecidableEq α] {p : α × β → ℝ}, (∀ (xy : α × β), 0 ≤ p xy) → ∑ xy, p xy = 1 → 2 ≤ Fintype.card α → ∀ (decode : β → α) (xy : α × β), p xy ≠ 0 → 0 < Causalean.Mathlib.InformationTheory.fanoRef p decode xy","l":124,"f":"Causalean/Mathlib/InformationTheory/Fano.lean"},{"n":"Causalean.Mathlib.InformationTheory.fanoRef_def","k":"theorem","m":"Causalean.Mathlib.InformationTheory.Fano","a":"Mathlib/InformationTheory","t":2,"d":"","s":"∀ {α : Type u_1} {β : Type u_2} [inst : Fintype α] [inst_1 : Fintype β] [inst_2 : DecidableEq α] (p : α × β → ℝ) (decode : β → α) (xy : α × β), Causalean.Mathlib.InformationTheory.fanoRef p decode xy = Causalean.Mathlib.InformationTheory.yMarginal p xy.2 * if xy.1 = decode xy.2 then 1 - Causalean.Mathlib.InformationTheory.errorProb p decode else Causalean.Mathlib.InformationTheory.errorProb p decode / (↑(Fintype.card α) - 1)","l":51,"f":"Causalean/Mathlib/InformationTheory/Fano.lean"},{"n":"Causalean.Mathlib.InformationTheory.fanoRef_nonneg","k":"theorem","m":"Causalean.Mathlib.InformationTheory.Fano","a":"Mathlib/InformationTheory","t":2,"d":"The Fano reference distribution is nonnegative.","s":"∀ {α : Type u_1} {β : Type u_2} [inst : Fintype α] [inst_1 : Fintype β] [inst_2 : DecidableEq α] {p : α × β → ℝ}, (∀ (xy : α × β), 0 ≤ p xy) → ∑ xy, p xy = 1 → ∀ (decode : β → α) (xy : α × β), 0 ≤ Causalean.Mathlib.InformationTheory.fanoRef p decode xy","l":57,"f":"Causalean/Mathlib/InformationTheory/Fano.lean"},{"n":"Causalean.Mathlib.InformationTheory.fanoRef_sum_eq_one","k":"theorem","m":"Causalean.Mathlib.InformationTheory.Fano","a":"Mathlib/InformationTheory","t":2,"d":"The Fano reference distribution is a probability mass function: ∑ xy, fanoRef = 1. On each column the inner weights sum to (1 − Pe) + (card α − 1) · Pe/(card α − 1) = 1, so the total is ∑ y, yMarginal p y = 1.","s":"∀ {α : Type u_1} {β : Type u_2} [inst : Fintype α] [inst_1 : Fintype β] [inst_2 : DecidableEq α] {p : α × β → ℝ}, ∑ xy, p xy = 1 → 2 ≤ Fintype.card α → ∀ (decode : β → α), ∑ xy, Causalean.Mathlib.InformationTheory.fanoRef p decode xy = 1","l":70,"f":"Causalean/Mathlib/InformationTheory/Fano.lean"},{"n":"Causalean.Mathlib.InformationTheory.fano_error_lower_bound","k":"theorem","m":"Causalean.Mathlib.InformationTheory.Fano","a":"Mathlib/InformationTheory","t":1,"d":"Fano error lower bound (the standard weakened corollary). For a nonnegative function p on α × β that sums to one, with at least two symbols in the alphabet α (2 ≤ Fintype.card α) and a decoder decode : β → α, the error probability Pe = errorProb p decode is bounded below: Pe ≥ (condEntropy p − Real.log 2) / Real.log (Fintype.card α).","s":"∀ {α : Type u_1} {β : Type u_2} [inst : Fintype α] [inst_1 : Fintype β] [inst_2 : DecidableEq α] {p : α × β → ℝ}, (∀ (xy : α × β), 0 ≤ p xy) → ∑ xy, p xy = 1 → 2 ≤ Fintype.card α → ∀ (decode : β → α), (Causalean.Mathlib.InformationTheory.condEntropy p - Real.log 2) / Real.log ↑(Fintype.card α) ≤ Causalean.Mathlib.InformationTheory.errorProb p decode","l":308,"f":"Causalean/Mathlib/InformationTheory/Fano.lean"},{"n":"Causalean.Mathlib.InformationTheory.fano_inequality","k":"theorem","m":"Causalean.Mathlib.InformationTheory.Fano","a":"Mathlib/InformationTheory","t":1,"d":"Fano's inequality (Cover & Thomas, Thm 2.10.1). For a nonnegative function p on α × β that sums to one, i.e. a joint probability mass function, with at least two symbols in the alphabet α (2 ≤ Fintype.card α), and a decoder decode : β → α, the conditional entropy of p is bounded by the binary entropy of the error probability Pe = errorProb p decode plus Pe times the log of one less than the alphabet size: condEntropy p ≤ Real.binEntropy Pe + Pe * Real.log (Fintype.card α − 1).","s":"∀ {α : Type u_1} {β : Type u_2} [inst : Fintype α] [inst_1 : Fintype β] [inst_2 : DecidableEq α] {p : α × β → ℝ}, (∀ (xy : α × β), 0 ≤ p xy) → ∑ xy, p xy = 1 → 2 ≤ Fintype.card α → ∀ (decode : β → α), Causalean.Mathlib.InformationTheory.condEntropy p ≤ Real.binEntropy (Causalean.Mathlib.InformationTheory.errorProb p decode) + Causalean.Mathlib.InformationTheory.errorProb p decode * Real.log (↑(Fintype.card α) - 1)","l":284,"f":"Causalean/Mathlib/InformationTheory/Fano.lean"},{"n":"Causalean.Mathlib.InformationTheory.gaussianKL_eq","k":"theorem","m":"Causalean.Mathlib.InformationTheory.GaussianKL","a":"Mathlib/InformationTheory","t":1,"d":"Equal-variance Gaussian KL divergence (closed form). For means m₀, m₁ and a strictly positive common variance v, the Kullback–Leibler divergence between two real Gaussians of equal variance v and respective means m₀ and m₁ equals (m₀ - m₁)² / (2v):","s":"∀ {v : NNReal} (m₀ m₁ : ℝ), 0 < v → InformationTheory.klDiv (ProbabilityTheory.gaussianReal m₀ v) (ProbabilityTheory.gaussianReal m₁ v) = ENNReal.ofReal ((m₀ - m₁) ^ 2 / (2 * ↑v))","l":175,"f":"Causalean/Mathlib/InformationTheory/GaussianKL.lean"},{"n":"Causalean.Mathlib.InformationTheory.gaussianReal_ac_gaussianReal","k":"theorem","m":"Causalean.Mathlib.InformationTheory.GaussianKL","a":"Mathlib/InformationTheory","t":2,"d":"Absolute continuity of nondegenerate Gaussians. For v₀ ≠ 0 and v₁ ≠ 0, gaussianReal m₀ v₀ is absolutely continuous with respect to gaussianReal m₁ v₁: both are volume.withDensity of a strictly positive density, so each is mutually absolutely continuous with Lebesgue measure, and absolute continuity is transitive.","s":"∀ (m₀ m₁ : ℝ) {v₀ v₁ : NNReal}, v₀ ≠ 0 → v₁ ≠ 0 → (ProbabilityTheory.gaussianReal m₀ v₀).AbsolutelyContinuous (ProbabilityTheory.gaussianReal m₁ v₁)","l":56,"f":"Causalean/Mathlib/InformationTheory/GaussianKL.lean"},{"n":"Causalean.Mathlib.InformationTheory.integrable_llr_gaussianReal","k":"theorem","m":"Causalean.Mathlib.InformationTheory.GaussianKL","a":"Mathlib/InformationTheory","t":2,"d":"Integrability of the Gaussian log-likelihood ratio. For v ≠ 0, the affine log-likelihood ratio x ↦ (m₀ - m₁) * (2*x - m₀ - m₁) / (2*v) is integrable with respect to gaussianReal m₀ v, because the Gaussian has a finite first moment.","s":"∀ {v : NNReal} (m₀ m₁ : ℝ), v ≠ 0 → MeasureTheory.Integrable (MeasureTheory.llr (ProbabilityTheory.gaussianReal m₀ v) (ProbabilityTheory.gaussianReal m₁ v)) (ProbabilityTheory.gaussianReal m₀ v)","l":123,"f":"Causalean/Mathlib/InformationTheory/GaussianKL.lean"},{"n":"Causalean.Mathlib.InformationTheory.integral_llr_gaussianReal","k":"theorem","m":"Causalean.Mathlib.InformationTheory.GaussianKL","a":"Mathlib/InformationTheory","t":2,"d":"Integral of the Gaussian log-likelihood ratio. For v ≠ 0, ∫ llr (gaussianReal m₀ v) (gaussianReal m₁ v) ∂(gaussianReal m₀ v) = (m₀ - m₁)^2 / (2*v), obtained by integrating the affine a.e. form against ∫ x ∂gaussianReal m₀ v = m₀.","s":"∀ {v : NNReal} (m₀ m₁ : ℝ), v ≠ 0 → ∫ (x : ℝ), MeasureTheory.llr (ProbabilityTheory.gaussianReal m₀ v) (ProbabilityTheory.gaussianReal m₁ v) x ∂ProbabilityTheory.gaussianReal m₀ v = (m₀ - m₁) ^ 2 / (2 * ↑v)","l":144,"f":"Causalean/Mathlib/InformationTheory/GaussianKL.lean"},{"n":"Causalean.Mathlib.InformationTheory.le_yMarginal","k":"theorem","m":"Causalean.Mathlib.InformationTheory.ConditionalEntropy","a":"Mathlib/InformationTheory","t":2,"d":"A joint mass is dominated by its β-marginal: p (x, y) ≤ yMarginal p y for nonnegative p.","s":"∀ {α : Type u_1} {β : Type u_2} [inst : Fintype α] {p : α × β → ℝ}, (∀ (xy : α × β), 0 ≤ p xy) → ∀ (x : α) (y : β), p (x, y) ≤ Causalean.Mathlib.InformationTheory.yMarginal p y","l":164,"f":"Causalean/Mathlib/InformationTheory/ConditionalEntropy.lean"},{"n":"Causalean.Mathlib.InformationTheory.llr_gaussianReal_ae","k":"theorem","m":"Causalean.Mathlib.InformationTheory.GaussianKL","a":"Mathlib/InformationTheory","t":2,"d":"Log-likelihood ratio of equal-variance Gaussians. For v ≠ 0, the log-likelihood ratio llr (gaussianReal m₀ v) (gaussianReal m₁ v) is a.e. (with respect to gaussianReal m₀ v) equal to the affine function x ↦ (m₀ - m₁) * (2*x - m₀ - m₁) / (2*v). At equal variance the (√(2πv))⁻¹ normalising constants cancel in the density ratio, so the log-ratio reduces to ((x - m₁)^2 - (x - m₀)^2) / (2*v) = (m₀ - m₁)*(2*x - m₀ - m₁)/(2*v).","s":"∀ {v : NNReal} (m₀ m₁ : ℝ), v ≠ 0 → MeasureTheory.llr (ProbabilityTheory.gaussianReal m₀ v) (ProbabilityTheory.gaussianReal m₁ v) =ᵐ[ProbabilityTheory.gaussianReal m₀ v] fun x => (m₀ - m₁) * (2 * x - m₀ - m₁) / (2 * ↑v)","l":93,"f":"Causalean/Mathlib/InformationTheory/GaussianKL.lean"},{"n":"Causalean.Mathlib.InformationTheory.negMulLog_add_mul_log_le","k":"theorem","m":"Causalean.Mathlib.InformationTheory.ConditionalEntropy","a":"Mathlib/InformationTheory","t":2,"d":"Per-coordinate Gibbs lever. For 0 ≤ x and 0 ≤ g, with g positive whenever x is nonzero (absolute continuity), the cross-entropy summand is controlled: Real.negMulLog x + x * Real.log g ≤ g - x.","s":"∀ {x g : ℝ}, 0 ≤ x → 0 ≤ g → (x ≠ 0 → 0 < g) → x.negMulLog + x * Real.log g ≤ g - x","l":82,"f":"Causalean/Mathlib/InformationTheory/ConditionalEntropy.lean"},{"n":"Causalean.Mathlib.InformationTheory.negMulLog_sub_mul_log_le","k":"theorem","m":"Causalean.Mathlib.InformationTheory.Entropy","a":"Mathlib/InformationTheory","t":2,"d":"Per-coordinate Gibbs lever. At any positive real scale c, the entropy summand minus x · log c is bounded by 1/c - x: Real.negMulLog x - x * Real.log c ≤ c⁻¹ - x.","s":"∀ {c : ℝ}, 0 < c → ∀ {x : ℝ}, 0 ≤ x → x.negMulLog - x * Real.log c ≤ c⁻¹ - x","l":63,"f":"Causalean/Mathlib/InformationTheory/Entropy.lean"},{"n":"Causalean.Mathlib.InformationTheory.neg_crossEntropy_fanoRef","k":"theorem","m":"Causalean.Mathlib.InformationTheory.Fano","a":"Mathlib/InformationTheory","t":2,"d":"The cross-entropy of p against the Fano reference splits, via the chain rule for log on the support of p, into the marginal entropy plus the binary-entropy/error terms: −∑ xy, p xy · log (fanoRef p decode xy) = entropy (yMarginal p) + Real.binEntropy Pe + Pe · Real.log (card α − 1), where Pe = errorProb p decode. This is the key algebraic computation behind Fano.","s":"∀ {α : Type u_1} {β : Type u_2} [inst : Fintype α] [inst_1 : Fintype β] [inst_2 : DecidableEq α] {p : α × β → ℝ}, (∀ (xy : α × β), 0 ≤ p xy) → ∑ xy, p xy = 1 → 2 ≤ Fintype.card α → ∀ (decode : β → α), -∑ xy, p xy * Real.log (Causalean.Mathlib.InformationTheory.fanoRef p decode xy) = Causalean.Mathlib.InformationTheory.entropy (Causalean.Mathlib.InformationTheory.yMarginal p) + Real.binEntropy (Causalean.Mathlib.InformationTheory.errorProb p decode) + Causalean.Mathlib.InformationTheory.errorProb p decode * Real.log (↑(Fintype.card α) - 1)","l":170,"f":"Causalean/Mathlib/InformationTheory/Fano.lean"},{"n":"Causalean.Mathlib.InformationTheory.one_add_mul_one_sub_mem_Icc","k":"theorem","m":"Causalean.Mathlib.InformationTheory.CommonStatisticBernoulli","a":"Mathlib/InformationTheory","t":2,"d":"A Bernoulli parameter in the middle half of the unit interval has variance between zero and one quarter, so adding unit noise gives variance between one and five quarters.","s":"∀ {p : ℝ}, p ∈ Set.Icc (1 / 4) (3 / 4) → 1 + p * (1 - p) ∈ Set.Icc 1 (5 / 4)","l":25,"f":"Causalean/Mathlib/InformationTheory/CommonStatisticBernoulli.lean"},{"n":"Causalean.Mathlib.InformationTheory.pi_iid_absolutelyContinuous","k":"theorem","m":"Causalean.Mathlib.InformationTheory.ProductKLLeCam","a":"Mathlib/InformationTheory","t":2,"d":"Public: absolute continuity of i.i.d. finite products from the one-sample hypothesis μ ≪ ν for sigma-finite laws. (Thin wrapper over the private induction.)","s":"∀ {α : Type u_1} [inst : MeasurableSpace α] (μ ν : MeasureTheory.Measure α) [MeasureTheory.SigmaFinite μ] [MeasureTheory.SigmaFinite ν], μ.AbsolutelyContinuous ν → ∀ (n : ℕ), (MeasureTheory.Measure.pi fun x => μ).AbsolutelyContinuous (MeasureTheory.Measure.pi fun x => ν)","l":440,"f":"Causalean/Mathlib/InformationTheory/ProductKLLeCam.lean"},{"n":"Causalean.Mathlib.InformationTheory.pi_iid_llr_integrable","k":"theorem","m":"Causalean.Mathlib.InformationTheory.ProductKLLeCam","a":"Mathlib/InformationTheory","t":2,"d":"Public: log-likelihood-ratio integrability for i.i.d. finite products from the one-sample hypotheses μ ≪ ν and Integrable (llr μ ν) μ. Combined with pi_iid_absolutelyContinuous this certifies klDiv (pi μ) (pi ν) ≠ ⊤ (via InformationTheory.klDiv_ne_top).","s":"∀ {α : Type u_1} [inst : MeasurableSpace α] (μ ν : MeasureTheory.Measure α) [MeasureTheory.IsProbabilityMeasure μ] [MeasureTheory.SigmaFinite ν], μ.AbsolutelyContinuous ν → MeasureTheory.Integrable (MeasureTheory.llr μ ν) μ → ∀ (n : ℕ), MeasureTheory.Integrable (MeasureTheory.llr (MeasureTheory.Measure.pi fun x => μ) (MeasureTheory.Measure.pi fun x => ν)) (MeasureTheory.Measure.pi fun x => μ)","l":448,"f":"Causalean/Mathlib/InformationTheory/ProductKLLeCam.lean"},{"n":"Causalean.Mathlib.InformationTheory.productKL_tensorization","k":"theorem","m":"Causalean.Mathlib.InformationTheory.ProductKLLeCam","a":"Mathlib/InformationTheory","t":1,"d":"For a sample size n and probability measures μ, ν on a measurable space α, if μ is absolutely continuous with respect to ν and the log-likelihood ratio of μ against ν is μ-integrable, then both the KL divergence between the n-fold product of μ and the n-fold product of ν, and the one-observation KL divergence between μ and ν, are finite, and the real-valued product KL divergence is at most n times the real-valued one-observation KL divergence: this is product-KL tensorisation packaged in the Le Cam interface.","s":"∀ {α : Type u_1} [inst : MeasurableSpace α] (n : ℕ) (μ ν : MeasureTheory.Measure α) [MeasureTheory.IsProbabilityMeasure μ] [MeasureTheory.IsProbabilityMeasure ν], μ.AbsolutelyContinuous ν → MeasureTheory.Integrable (MeasureTheory.llr μ ν) μ → Causalean.Mathlib.InformationTheory.ProductKLTensorizationBound n μ ν","l":391,"f":"Causalean/Mathlib/InformationTheory/ProductKLLeCam.lean"},{"n":"Causalean.Mathlib.InformationTheory.productKL_tensorization_of_finite","k":"theorem","m":"Causalean.Mathlib.InformationTheory.ProductKLLeCam","a":"Mathlib/InformationTheory","t":2,"d":"Finite-branch product-KL tensorisation for i.i.d. finite products.","s":"∀ {α : Type u_1} [inst : MeasurableSpace α] (n : ℕ) (μ ν : MeasureTheory.Measure α) [MeasureTheory.IsProbabilityMeasure μ] [MeasureTheory.IsProbabilityMeasure ν], μ.AbsolutelyContinuous ν → MeasureTheory.Integrable (MeasureTheory.llr μ ν) μ → (InformationTheory.klDiv (MeasureTheory.Measure.pi fun x => μ) (MeasureTheory.Measure.pi fun x => ν)).toReal = ↑n * (InformationTheory.klDiv μ ν).toReal","l":374,"f":"Causalean/Mathlib/InformationTheory/ProductKLLeCam.lean"},{"n":"Causalean.Mathlib.InformationTheory.rnDeriv_toReal_gaussianReal_ae","k":"theorem","m":"Causalean.Mathlib.InformationTheory.GaussianKL","a":"Mathlib/InformationTheory","t":2,"d":"Radon–Nikodym ratio of equal-variance Gaussians, as a real number. For v ≠ 0, the real part of the Radon–Nikodym derivative ∂(gaussianReal m₀ v)/∂(gaussianReal m₁ v) is a.e. (with respect to gaussianReal m₀ v) the pointwise ratio of the two Gaussian densities gaussianPDFReal m₀ v x / gaussianPDFReal m₁ v x.","s":"∀ {v : NNReal} (m₀ m₁ : ℝ), v ≠ 0 → (fun x => ((ProbabilityTheory.gaussianReal m₀ v).rnDeriv (ProbabilityTheory.gaussianReal m₁ v) x).toReal) =ᵐ[ProbabilityTheory.gaussianReal m₀ v] fun x => ProbabilityTheory.gaussianPDFReal m₀ v x / ProbabilityTheory.gaussianPDFReal m₁ v x","l":66,"f":"Causalean/Mathlib/InformationTheory/GaussianKL.lean"},{"n":"Causalean.Mathlib.InformationTheory.statisticBernoulliOutcomeLaw_eq_map_swap_compProd","k":"theorem","m":"Causalean.Mathlib.InformationTheory.CommonStatisticBernoulli","a":"Mathlib/InformationTheory","t":2,"d":"Compressing the base coordinate to a statistic turns a Bernoulli mixture into a Bernoulli composition product over the statistic marginal.","s":"∀ {A : Type u_1} [inst : MeasurableSpace A] (nu : MeasureTheory.Measure A) [MeasureTheory.IsFiniteMeasure nu] (p stat : A → ℝ) (hp : Measurable p), Measurable stat → (∀ (x : A), 1 / 4 ≤ p x) → (∀ (x : A), p x ≤ 3 / 4) → MeasureTheory.Measure.map (fun z => (z.2, stat z.1)) (nu.compProd (Causalean.Mathlib.InformationTheory.commonStatisticBernoulliKernel p hp)) = MeasureTheory.Measure.map Prod.swap ((MeasureTheory.Measure.map stat nu).compProd (Causalean.Mathlib.InformationTheory.commonStatisticBernoulliKernel (Causalean.Mathlib.InformationTheory.clippedStatisticSuccessParameter nu p stat) (Causalean.Mathlib.InformationTheory.clippedStatisticSuccessParameter_measurable nu p stat)))","l":416,"f":"Causalean/Mathlib/InformationTheory/CommonStatisticBernoulli.lean"},{"n":"Causalean.Mathlib.InformationTheory.statisticBernoulliOutcome_klDiv_le_of_localized_success_bound","k":"theorem","m":"Causalean.Mathlib.InformationTheory.CommonStatisticBernoulli","a":"Mathlib/InformationTheory","t":1,"d":"For a measurable space A, two finite measures nu and nu' on it, and success-probability functions p, p' on A, suppose p, p', and a statistic stat are all measurable, p takes values in [1/4, 3/4] and p' likewise takes values in [1/4, 3/4], and stat pushes nu and nu' forward to the same marginal law. Given a nonnegative discrepancy bound D and a measurable exceptional set E such that for every measurable set B of statistic values, the setwise success-mass discrepancy |∫_{stat∈B} p dnu − ∫_{stat∈B} p' dnu'| is at most D times the stat-pushforward mass of nu on B ∩ E, then the Kullback–Leibler divergence between the compressed Bernoulli-outcome laws obtained by pairing the outcome with stat under nu and under nu' is at most 4·D² times the stat-pushforward mass of E under nu.","s":"∀ {A : Type u_1} [inst : MeasurableSpace A] (nu nu' : MeasureTheory.Measure A) [MeasureTheory.IsFiniteMeasure nu] [MeasureTheory.IsFiniteMeasure nu'] (p p' stat : A → ℝ) (hp : Measurable p) (hp' : Measurable p'), Measurable stat → (∀ (x : A), 1 / 4 ≤ p x) → (∀ (x : A), p x ≤ 3 / 4) → (∀ (x : A), 1 / 4 ≤ p' x) → (∀ (x : A), p' x ≤ 3 / 4) → MeasureTheory.Measure.map stat nu = MeasureTheory.Measure.map stat nu' → ∀ {D : ℝ}, 0 ≤ D → ∀ {E : Set ℝ}, MeasurableSet E → (∀ (B : Set ℝ), MeasurableSet B → |∫ (x : A) in {x | stat x ∈ B}, p x ∂nu - ∫ (x : A) in {x | stat x ∈ B}, p' x ∂nu'| ≤ D * ((MeasureTheory.Measure.map stat nu) (B ∩ E)).toReal) → InformationTheory.klDiv (MeasureTheory.Measure.map (fun z => (z.2, stat z.1)) (nu.compProd (Causalean.Mathlib.InformationTheory.commonStatisticBernoulliKernel p hp))) (MeasureTheory.Measure.map (fun z => (z.2, stat z.1)) (nu'.compProd (Causalean.Mathlib.InformationTheory.commonStatisticBernoulliKernel p' hp'))) ≤ ENNReal.ofReal (4 * D ^ 2) * (MeasureTheory.Measure.map stat nu) E","l":603,"f":"Causalean/Mathlib/InformationTheory/CommonStatisticBernoulli.lean"},{"n":"Causalean.Mathlib.InformationTheory.statisticBernoulliOutcome_restrict_compl_eq_of_localized_success_bound","k":"theorem","m":"Causalean.Mathlib.InformationTheory.CommonStatisticBernoulli","a":"Mathlib/InformationTheory","t":2,"d":"Common statistic marginals and a localized setwise success-mass bound also imply exact agreement of the compressed outcome laws away from the exceptional statistic set.","s":"∀ {A : Type u_1} [inst : MeasurableSpace A] (nu nu' : MeasureTheory.Measure A) [MeasureTheory.IsFiniteMeasure nu] [MeasureTheory.IsFiniteMeasure nu'] (p p' stat : A → ℝ) (hp : Measurable p) (hp' : Measurable p'), Measurable stat → (∀ (x : A), 1 / 4 ≤ p x) → (∀ (x : A), p x ≤ 3 / 4) → (∀ (x : A), 1 / 4 ≤ p' x) → (∀ (x : A), p' x ≤ 3 / 4) → MeasureTheory.Measure.map stat nu = MeasureTheory.Measure.map stat nu' → ∀ {D : ℝ}, 0 ≤ D → ∀ {E : Set ℝ}, MeasurableSet E → (∀ (B : Set ℝ), MeasurableSet B → |∫ (x : A) in {x | stat x ∈ B}, p x ∂nu - ∫ (x : A) in {x | stat x ∈ B}, p' x ∂nu'| ≤ D * ((MeasureTheory.Measure.map stat nu) (B ∩ E)).toReal) → (MeasureTheory.Measure.map (fun z => (z.2, stat z.1)) (nu.compProd (Causalean.Mathlib.InformationTheory.commonStatisticBernoulliKernel p hp))).restrict {z | z.2 ∉ E} = (MeasureTheory.Measure.map (fun z => (z.2, stat z.1)) (nu'.compProd (Causalean.Mathlib.InformationTheory.commonStatisticBernoulliKernel p' hp'))).restrict {z | z.2 ∉ E}","l":647,"f":"Causalean/Mathlib/InformationTheory/CommonStatisticBernoulli.lean"},{"n":"Causalean.Mathlib.InformationTheory.statisticSuccessMeasure","k":"def","m":"Causalean.Mathlib.InformationTheory.CommonStatisticBernoulli","a":"Mathlib/InformationTheory","t":1,"d":"The success-weighted pushforward associated with a real statistic.","s":"{A : Type u_1} → [inst : MeasurableSpace A] → MeasureTheory.Measure A → (A → ℝ) → (A → ℝ) → MeasureTheory.Measure ℝ","l":155,"f":"Causalean/Mathlib/InformationTheory/CommonStatisticBernoulli.lean"},{"n":"Causalean.Mathlib.InformationTheory.statisticSuccessMeasure_absolutelyContinuous","k":"theorem","m":"Causalean.Mathlib.InformationTheory.CommonStatisticBernoulli","a":"Mathlib/InformationTheory","t":2,"d":"The success-weighted statistic law is dominated by the statistic marginal when the pointwise success probability is at most one.","s":"∀ {A : Type u_1} [inst : MeasurableSpace A] (nu : MeasureTheory.Measure A) [MeasureTheory.IsFiniteMeasure nu] (p stat : A → ℝ), Measurable stat → (∀ (x : A), p x ≤ 1) → (Causalean.Mathlib.InformationTheory.statisticSuccessMeasure nu p stat).AbsolutelyContinuous (MeasureTheory.Measure.map stat nu)","l":169,"f":"Causalean/Mathlib/InformationTheory/CommonStatisticBernoulli.lean"},{"n":"Causalean.Mathlib.InformationTheory.statisticSuccessParameter","k":"def","m":"Causalean.Mathlib.InformationTheory.CommonStatisticBernoulli","a":"Mathlib/InformationTheory","t":1,"d":"A measurable version of the Bernoulli success probability conditional on the statistic.","s":"{A : Type u_1} → [inst : MeasurableSpace A] → MeasureTheory.Measure A → (A → ℝ) → (A → ℝ) → ℝ → ℝ","l":161,"f":"Causalean/Mathlib/InformationTheory/CommonStatisticBernoulli.lean"},{"n":"Causalean.Mathlib.InformationTheory.statisticSuccessParameter_mem_Icc_ae","k":"theorem","m":"Causalean.Mathlib.InformationTheory.CommonStatisticBernoulli","a":"Mathlib/InformationTheory","t":2,"d":"Middle-half pointwise bounds pass to the conditional statistic parameter almost everywhere.","s":"∀ {A : Type u_1} [inst : MeasurableSpace A] (nu : MeasureTheory.Measure A) [MeasureTheory.IsFiniteMeasure nu] (p stat : A → ℝ), Measurable p → Measurable stat → (∀ (x : A), 1 / 4 ≤ p x) → (∀ (x : A), p x ≤ 3 / 4) → ∀ᵐ (r : ℝ) ∂MeasureTheory.Measure.map stat nu, Causalean.Mathlib.InformationTheory.statisticSuccessParameter nu p stat r ∈ Set.Icc (1 / 4) (3 / 4)","l":231,"f":"Causalean/Mathlib/InformationTheory/CommonStatisticBernoulli.lean"},{"n":"Causalean.Mathlib.InformationTheory.statisticSuccessParameter_setIntegral","k":"theorem","m":"Causalean.Mathlib.InformationTheory.CommonStatisticBernoulli","a":"Mathlib/InformationTheory","t":2,"d":"Set integrals of the conditional parameter recover success-weighted integrals on statistic preimages.","s":"∀ {A : Type u_1} [inst : MeasurableSpace A] (nu : MeasureTheory.Measure A) [MeasureTheory.IsFiniteMeasure nu] (p stat : A → ℝ), Measurable p → Measurable stat → (∀ (x : A), 0 ≤ p x) → (∀ (x : A), p x ≤ 1) → ∀ (B : Set ℝ), MeasurableSet B → ∫ (r : ℝ) in B, Causalean.Mathlib.InformationTheory.statisticSuccessParameter nu p stat r ∂MeasureTheory.Measure.map stat nu = ∫ (x : A) in {x | stat x ∈ B}, p x ∂nu","l":188,"f":"Causalean/Mathlib/InformationTheory/CommonStatisticBernoulli.lean"},{"n":"Causalean.Mathlib.InformationTheory.yMarginal","k":"def","m":"Causalean.Mathlib.InformationTheory.ConditionalEntropy","a":"Mathlib/InformationTheory","t":1,"d":"The β-marginal of a joint mass function p : α × β → ℝ: yMarginal p y = ∑ x, p (x, y). This is the mass of the conditioning variable Y.","s":"{α : Type u_1} → {β : Type u_2} → [Fintype α] → (α × β → ℝ) → β → ℝ","l":47,"f":"Causalean/Mathlib/InformationTheory/ConditionalEntropy.lean"},{"n":"Causalean.Mathlib.InformationTheory.yMarginal_def","k":"theorem","m":"Causalean.Mathlib.InformationTheory.ConditionalEntropy","a":"Mathlib/InformationTheory","t":2,"d":"The β-marginal is the finite sum of joint masses over the α coordinate at the chosen value of β.","s":"∀ {α : Type u_1} {β : Type u_2} [inst : Fintype α] (p : α × β → ℝ) (y : β), Causalean.Mathlib.InformationTheory.yMarginal p y = ∑ x, p (x, y)","l":66,"f":"Causalean/Mathlib/InformationTheory/ConditionalEntropy.lean"},{"n":"Causalean.Mathlib.InformationTheory.yMarginal_nonneg","k":"theorem","m":"Causalean.Mathlib.InformationTheory.ConditionalEntropy","a":"Mathlib/InformationTheory","t":2,"d":"The β-marginal of a nonnegative mass function is nonnegative.","s":"∀ {α : Type u_1} {β : Type u_2} [inst : Fintype α] {p : α × β → ℝ}, (∀ (xy : α × β), 0 ≤ p xy) → ∀ (y : β), 0 ≤ Causalean.Mathlib.InformationTheory.yMarginal p y","l":157,"f":"Causalean/Mathlib/InformationTheory/ConditionalEntropy.lean"},{"n":"Causalean.Mathlib.InformationTheory.yMarginal_sum","k":"theorem","m":"Causalean.Mathlib.InformationTheory.ConditionalEntropy","a":"Mathlib/InformationTheory","t":2,"d":"The β-marginal of a pmf is itself a pmf summing to one: if ∑ xy, p xy = 1 then ∑ y, yMarginal p y = 1.","s":"∀ {α : Type u_1} {β : Type u_2} [inst : Fintype α] [inst_1 : Fintype β] {p : α × β → ℝ}, ∑ xy, p xy = 1 → ∑ y, Causalean.Mathlib.InformationTheory.yMarginal p y = 1","l":148,"f":"Causalean/Mathlib/InformationTheory/ConditionalEntropy.lean"},{"n":"Causalean.Mathlib.LinearAlgebra.IsUpperTri","k":"def","m":"Causalean.Mathlib.LinearAlgebra.Cholesky","a":"Mathlib/LinearAlgebra","t":1,"d":"An upper-triangular matrix in the sense j < i → U i j = 0, i.e. all entries strictly below the diagonal vanish.","s":"{ι : Type u_1} → {K : Type u_2} → [LT ι] → [Zero K] → Matrix ι ι K → Prop","l":31,"f":"Causalean/Mathlib/LinearAlgebra/Cholesky.lean"},{"n":"Causalean.Mathlib.LinearAlgebra.affineBinaryPower","k":"def","m":"Causalean.Mathlib.LinearAlgebra.StackedVandermonde","a":"Mathlib/LinearAlgebra","t":1,"d":"The affine power vector whose r-th entry is z^r. Over a field, this is the coefficient vector of (X₀ + z X₁)^k after normalization by the corresponding nonzero binomial coefficient.","s":"{K : Type u_1} → [Monoid K] → K → (k : ℕ) → Fin (k + 1) → K","l":30,"f":"Causalean/Mathlib/LinearAlgebra/StackedVandermonde.lean"},{"n":"Causalean.Mathlib.LinearAlgebra.blockVandermondeWitnessWeights","k":"def","m":"Causalean.Mathlib.LinearAlgebra.StackedVandermonde","a":"Mathlib/LinearAlgebra","t":1,"d":"The two-block specialization from the block-Vandermonde argument: J₀ = {0} and J_{N-1} = {1, …, N}.","s":"{K : Type u_1} → [Zero K] → [One K] → (N : ℕ) → 1 ≤ N → Fin (N + 1) → Fin N → K","l":45,"f":"Causalean/Mathlib/LinearAlgebra/StackedVandermonde.lean"},{"n":"Causalean.Mathlib.LinearAlgebra.cholesky_exists","k":"theorem","m":"Causalean.Mathlib.LinearAlgebra.Cholesky","a":"Mathlib/LinearAlgebra","t":1,"d":"For a real positive-definite d × d matrix M, there exists an upper-triangular matrix U (zero below the diagonal) with strictly positive diagonal entries such that M factors as Uᵀ · U.","s":"∀ {d : ℕ} {M : Matrix (Fin d) (Fin d) ℝ}, M.PosDef → ∃ U, (∀ (i j : Fin d), j < i → U i j = 0) ∧ (∀ (i : Fin d), 0 < U i i) ∧ M = U.transpose * U","l":106,"f":"Causalean/Mathlib/LinearAlgebra/Cholesky.lean"},{"n":"Causalean.Mathlib.LinearAlgebra.cholesky_unique","k":"theorem","m":"Causalean.Mathlib.LinearAlgebra.Cholesky","a":"Mathlib/LinearAlgebra","t":1,"d":"For a linearly ordered finite index type ι and a linearly ordered field K, and two matrices U, V indexed by ι × ι and valued in K, if U is upper-triangular, i.e. zero below the diagonal, with strictly positive diagonal entries, V is likewise upper-triangular with strictly positive diagonal entries, and U and V have the same Gram matrix, Uᵀ · U = Vᵀ · V, then U equals V.","s":"∀ {ι : Type u_1} {K : Type u_2} [inst : Fintype ι] [inst_1 : LinearOrder ι] [inst_2 : Field K] [inst_3 : LinearOrder K] [IsStrictOrderedRing K] {U V : Matrix ι ι K}, (∀ (i j : ι), j < i → U i j = 0) → (∀ (i : ι), 0 < U i i) → (∀ (i j : ι), j < i → V i j = 0) → (∀ (i : ι), 0 < V i i) → U.transpose * U = V.transpose * V → U = V","l":197,"f":"Causalean/Mathlib/LinearAlgebra/Cholesky.lean"},{"n":"Causalean.Mathlib.LinearAlgebra.coeff_doubledCoefficientPolynomial","k":"theorem","m":"Causalean.Mathlib.LinearAlgebra.ConfluentVandermonde","a":"Mathlib/LinearAlgebra","t":2,"d":"Reading a doubled coefficient polynomial at an encoded exponent recovers the corresponding coefficient.","s":"∀ {n : ℕ} {K : Type u_1} [inst : Semiring K] (v : Fin n ⊕ Fin n → K) (i : Fin n ⊕ Fin n), (Causalean.Mathlib.LinearAlgebra.doubledCoefficientPolynomial v).coeff (Causalean.Mathlib.LinearAlgebra.doubledExponent i) = v i","l":65,"f":"Causalean/Mathlib/LinearAlgebra/ConfluentVandermonde.lean"},{"n":"Causalean.Mathlib.LinearAlgebra.coeff_pinnedCoefficientPolynomial","k":"theorem","m":"Causalean.Mathlib.LinearAlgebra.ConfluentVandermonde","a":"Mathlib/LinearAlgebra","t":2,"d":"Reading a pinned coefficient polynomial at an encoded exponent recovers the corresponding coefficient.","s":"∀ {n : ℕ} {K : Type u_1} [inst : Semiring K] (v : Fin n ⊕ Fin (n - 1) → K) (i : Fin n ⊕ Fin (n - 1)), (Causalean.Mathlib.LinearAlgebra.pinnedCoefficientPolynomial v).coeff (Causalean.Mathlib.LinearAlgebra.pinnedExponent i) = v i","l":268,"f":"Causalean/Mathlib/LinearAlgebra/ConfluentVandermonde.lean"},{"n":"Causalean.Mathlib.LinearAlgebra.confluentVandermonde","k":"def","m":"Causalean.Mathlib.LinearAlgebra.ConfluentVandermonde","a":"Mathlib/LinearAlgebra","t":1,"d":"The square Hermite-evaluation matrix with value and first-derivative columns at every node. Rows are monomials of degrees 0, ..., 2n-1.","s":"{n : ℕ} → {K : Type u_1} → [Semiring K] → (Fin n → K) → Matrix (Fin n ⊕ Fin n) (Fin n ⊕ Fin n) K","l":104,"f":"Causalean/Mathlib/LinearAlgebra/ConfluentVandermonde.lean"},{"n":"Causalean.Mathlib.LinearAlgebra.det_confluentVandermonde_ne_zero","k":"theorem","m":"Causalean.Mathlib.LinearAlgebra.ConfluentVandermonde","a":"Mathlib/LinearAlgebra","t":1,"d":"For a positive number of nodes n (1 ≤ n), a field K, and pairwise distinct nodes s : Fin n → K, the confluent Vandermonde determinant at s is nonzero.","s":"∀ {n : ℕ} {K : Type u_1} [inst : Field K], 1 ≤ n → ∀ (s : Fin n → K), Function.Injective s → (Causalean.Mathlib.LinearAlgebra.confluentVandermonde s).det ≠ 0","l":179,"f":"Causalean/Mathlib/LinearAlgebra/ConfluentVandermonde.lean"},{"n":"Causalean.Mathlib.LinearAlgebra.det_pinnedConfluentVandermonde_ne_zero","k":"theorem","m":"Causalean.Mathlib.LinearAlgebra.ConfluentVandermonde","a":"Mathlib/LinearAlgebra","t":1,"d":"For a positive number of nodes n (1 ≤ n), a field K, and pairwise distinct nodes s : Fin n → K, the pinned confluent Vandermonde determinant at s is nonzero: one node contributes only value evaluation and every other node contributes both value and first-derivative evaluation.","s":"∀ {n : ℕ} {K : Type u_1} [inst : Field K], 1 ≤ n → ∀ (s : Fin n → K), Function.Injective s → (Causalean.Mathlib.LinearAlgebra.pinnedConfluentVandermonde s).det ≠ 0","l":408,"f":"Causalean/Mathlib/LinearAlgebra/ConfluentVandermonde.lean"},{"n":"Causalean.Mathlib.LinearAlgebra.doubledCoefficientPolynomial","k":"def","m":"Causalean.Mathlib.LinearAlgebra.ConfluentVandermonde","a":"Mathlib/LinearAlgebra","t":1,"d":"Monomial coefficients encoded by the doubled finite index.","s":"{n : ℕ} → {K : Type u_1} → [inst : Semiring K] → (Fin n ⊕ Fin n → K) → Polynomial K","l":60,"f":"Causalean/Mathlib/LinearAlgebra/ConfluentVandermonde.lean"},{"n":"Causalean.Mathlib.LinearAlgebra.doubledCoefficientPolynomial_eq_zero_iff","k":"theorem","m":"Causalean.Mathlib.LinearAlgebra.ConfluentVandermonde","a":"Mathlib/LinearAlgebra","t":2,"d":"A doubled coefficient polynomial vanishes exactly when its coefficient vector vanishes.","s":"∀ {n : ℕ} {K : Type u_1} [inst : Semiring K] (v : Fin n ⊕ Fin n → K), Causalean.Mathlib.LinearAlgebra.doubledCoefficientPolynomial v = 0 ↔ v = 0","l":81,"f":"Causalean/Mathlib/LinearAlgebra/ConfluentVandermonde.lean"},{"n":"Causalean.Mathlib.LinearAlgebra.doubledCoefficientPolynomial_eq_zero_of_eval_derivative","k":"theorem","m":"Causalean.Mathlib.LinearAlgebra.ConfluentVandermonde","a":"Mathlib/LinearAlgebra","t":2,"d":"A polynomial of degree below 2n whose value and derivative vanish at n distinct points is zero.","s":"∀ {n : ℕ} {K : Type u_1} [inst : Field K], 1 ≤ n → ∀ (s : Fin n → K), Function.Injective s → ∀ (v : Fin n ⊕ Fin n → K), (∀ (i : Fin n), Polynomial.eval (s i) (Causalean.Mathlib.LinearAlgebra.doubledCoefficientPolynomial v) = 0) → (∀ (i : Fin n), Polynomial.eval (s i) (Polynomial.derivative (Causalean.Mathlib.LinearAlgebra.doubledCoefficientPolynomial v)) = 0) → v = 0","l":141,"f":"Causalean/Mathlib/LinearAlgebra/ConfluentVandermonde.lean"},{"n":"Causalean.Mathlib.LinearAlgebra.doubledExponent","k":"def","m":"Causalean.Mathlib.LinearAlgebra.ConfluentVandermonde","a":"Mathlib/LinearAlgebra","t":1,"d":"The monomial exponent attached to the two copies of Fin n.","s":"{n : ℕ} → Fin n ⊕ Fin n → ℕ","l":29,"f":"Causalean/Mathlib/LinearAlgebra/ConfluentVandermonde.lean"},{"n":"Causalean.Mathlib.LinearAlgebra.doubledExponent_injective","k":"theorem","m":"Causalean.Mathlib.LinearAlgebra.ConfluentVandermonde","a":"Mathlib/LinearAlgebra","t":2,"d":"The doubled exponent encoding is injective.","s":"∀ {n : ℕ}, Function.Injective Causalean.Mathlib.LinearAlgebra.doubledExponent","l":34,"f":"Causalean/Mathlib/LinearAlgebra/ConfluentVandermonde.lean"},{"n":"Causalean.Mathlib.LinearAlgebra.doubledExponent_lt","k":"theorem","m":"Causalean.Mathlib.LinearAlgebra.ConfluentVandermonde","a":"Mathlib/LinearAlgebra","t":2,"d":"Every doubled exponent is strictly below 2n.","s":"∀ {n : ℕ} (i : Fin n ⊕ Fin n), Causalean.Mathlib.LinearAlgebra.doubledExponent i < 2 * n","l":52,"f":"Causalean/Mathlib/LinearAlgebra/ConfluentVandermonde.lean"},{"n":"Causalean.Mathlib.LinearAlgebra.endpointOrderSynthesis","k":"def","m":"Causalean.Mathlib.LinearAlgebra.VandermondeSynthesis","a":"Mathlib/LinearAlgebra","t":1,"d":"Moment synthesis from n commutative-semiring-valued nodes and one terminal coordinate. The terminal coordinate contributes only to the highest requested moment.","s":"{n : ℕ} → {K : Type u_1} → [inst : CommSemiring K] → (Fin n → K) → (r : ℕ) → (Fin (n + 1) → K) →ₗ[K] Fin (r + 1) → K","l":28,"f":"Causalean/Mathlib/LinearAlgebra/VandermondeSynthesis.lean"},{"n":"Causalean.Mathlib.LinearAlgebra.endpointOrderSynthesis_injective","k":"theorem","m":"Causalean.Mathlib.LinearAlgebra.VandermondeSynthesis","a":"Mathlib/LinearAlgebra","t":2,"d":"With distinct node values, moments through an order at least as large as the number of nodes uniquely determine every coordinate supplied to the moment-synthesis map.","s":"∀ {n r : ℕ} {K : Type u_1} [inst : CommRing K] [IsDomain K] (s : Fin n → K), Function.Injective s → n ≤ r → Function.Injective ⇑(Causalean.Mathlib.LinearAlgebra.endpointOrderSynthesis s r)","l":101,"f":"Causalean/Mathlib/LinearAlgebra/VandermondeSynthesis.lean"},{"n":"Causalean.Mathlib.LinearAlgebra.endpointOrderSynthesis_ker_finrank","k":"theorem","m":"Causalean.Mathlib.LinearAlgebra.VandermondeSynthesis","a":"Mathlib/LinearAlgebra","t":1,"d":"For natural numbers n, r, a field K, and pairwise distinct nodes s : Fin n → K, the dimension of the kernel of the order-r endpoint synthesis map equals n - r: below the square threshold (r < n) the kernel has dimension n - r, and at or above it (r ≥ n) the kernel is zero, matching n - r = 0 under truncated subtraction.","s":"∀ {n r : ℕ} {K : Type u_1} [inst : Field K] (s : Fin n → K), Function.Injective s → Module.finrank K ↥(Causalean.Mathlib.LinearAlgebra.endpointOrderSynthesis s r).ker = n - r","l":124,"f":"Causalean/Mathlib/LinearAlgebra/VandermondeSynthesis.lean"},{"n":"Causalean.Mathlib.LinearAlgebra.endpointOrderSynthesis_single","k":"theorem","m":"Causalean.Mathlib.LinearAlgebra.VandermondeSynthesis","a":"Mathlib/LinearAlgebra","t":2,"d":"Applying the moment-synthesis map to the standard basis vector for a node produces the vector of that node's powers across the requested moment orders.","s":"∀ {n r : ℕ} {K : Type u_1} [inst : CommSemiring K] (s : Fin n → K) (j : Fin n), (Causalean.Mathlib.LinearAlgebra.endpointOrderSynthesis s r) (Pi.single j.castSucc 1) = fun a => s j ^ ↑a","l":44,"f":"Causalean/Mathlib/LinearAlgebra/VandermondeSynthesis.lean"},{"n":"Causalean.Mathlib.LinearAlgebra.endpointOrderSynthesis_surjective","k":"theorem","m":"Causalean.Mathlib.LinearAlgebra.VandermondeSynthesis","a":"Mathlib/LinearAlgebra","t":2,"d":"With distinct node values, the finite moment-synthesis map can reproduce every target vector of moments up through an order smaller than the number of nodes.","s":"∀ {n r : ℕ} {K : Type u_1} [inst : Field K] (s : Fin n → K), Function.Injective s → r < n → Function.Surjective ⇑(Causalean.Mathlib.LinearAlgebra.endpointOrderSynthesis s r)","l":59,"f":"Causalean/Mathlib/LinearAlgebra/VandermondeSynthesis.lean"},{"n":"Causalean.Mathlib.LinearAlgebra.eq_of_genPerm_triangular_unitDiag","k":"theorem","m":"Causalean.Mathlib.LinearAlgebra.MonomialMatrix","a":"Mathlib/LinearAlgebra","t":1,"d":"Generalized-permutation reduction. For matrices C, C' over Fin n × Fin n valued in K, if C has unit diagonal and C' has unit diagonal, C is lower triangular in some causal order σ (C i j = 0 when σ i < σ j), and C' is obtained from C by a generalized permutation with permutation τ and scalings d, i.e. C' i j = d i · C (τ i) j for all i, j, then C = C': the unit diagonal plus triangularity force the underlying permutation to be the identity and every scaling to be one. (Formerly Discovery.LiNGAM.lingam_reduction.)","s":"∀ {n : ℕ} {K : Type u_1} [inst : MulZeroOneClass K] [Nontrivial K] {C C' : Matrix (Fin n) (Fin n) K}, (∀ (i : Fin n), C i i = 1) → (∀ (i : Fin n), C' i i = 1) → ∀ {σ : Equiv.Perm (Fin n)}, (∀ (i j : Fin n), σ i < σ j → C i j = 0) → ∀ {τ : Equiv.Perm (Fin n)} {d : Fin n → K}, (∀ (i j : Fin n), C' i j = d i * C (τ i) j) → C = C'","l":125,"f":"Causalean/Mathlib/LinearAlgebra/MonomialMatrix.lean"},{"n":"Causalean.Mathlib.LinearAlgebra.eval_derivative_doubledCoefficientPolynomial","k":"theorem","m":"Causalean.Mathlib.LinearAlgebra.ConfluentVandermonde","a":"Mathlib/LinearAlgebra","t":2,"d":"Evaluating the derivative of a polynomial whose coefficients are indexed by two blocks of monomial powers gives the finite sum of coefficients weighted by their monomial exponents and the corresponding reduced powers.","s":"∀ {n : ℕ} {K : Type u_1} [inst : CommSemiring K] (v : Fin n ⊕ Fin n → K) (x : K), Polynomial.eval x (Polynomial.derivative (Causalean.Mathlib.LinearAlgebra.doubledCoefficientPolynomial v)) = ∑ a, ↑(Causalean.Mathlib.LinearAlgebra.doubledExponent a) * x ^ (Causalean.Mathlib.LinearAlgebra.doubledExponent a - 1) * v a","l":126,"f":"Causalean/Mathlib/LinearAlgebra/ConfluentVandermonde.lean"},{"n":"Causalean.Mathlib.LinearAlgebra.eval_derivative_pinnedCoefficientPolynomial","k":"theorem","m":"Causalean.Mathlib.LinearAlgebra.ConfluentVandermonde","a":"Mathlib/LinearAlgebra","t":2,"d":"Evaluating the derivative of a pinned coefficient polynomial at a point equals the sum of its coefficients weighted by the corresponding derivative monomial values at that point.","s":"∀ {n : ℕ} {K : Type u_1} [inst : Field K] (v : Fin n ⊕ Fin (n - 1) → K) (x : K), Polynomial.eval x (Polynomial.derivative (Causalean.Mathlib.LinearAlgebra.pinnedCoefficientPolynomial v)) = ∑ a, ↑(Causalean.Mathlib.LinearAlgebra.pinnedExponent a) * x ^ (Causalean.Mathlib.LinearAlgebra.pinnedExponent a - 1) * v a","l":331,"f":"Causalean/Mathlib/LinearAlgebra/ConfluentVandermonde.lean"},{"n":"Causalean.Mathlib.LinearAlgebra.eval_doubledCoefficientPolynomial","k":"theorem","m":"Causalean.Mathlib.LinearAlgebra.ConfluentVandermonde","a":"Mathlib/LinearAlgebra","t":2,"d":"Evaluating a polynomial whose coefficients are indexed by two blocks of monomial powers gives the finite sum of those coefficients weighted by the corresponding powers of the evaluation point.","s":"∀ {n : ℕ} {K : Type u_1} [inst : CommSemiring K] (v : Fin n ⊕ Fin n → K) (x : K), Polynomial.eval x (Causalean.Mathlib.LinearAlgebra.doubledCoefficientPolynomial v) = ∑ a, x ^ Causalean.Mathlib.LinearAlgebra.doubledExponent a * v a","l":112,"f":"Causalean/Mathlib/LinearAlgebra/ConfluentVandermonde.lean"},{"n":"Causalean.Mathlib.LinearAlgebra.eval_pinnedCoefficientPolynomial","k":"theorem","m":"Causalean.Mathlib.LinearAlgebra.ConfluentVandermonde","a":"Mathlib/LinearAlgebra","t":2,"d":"Evaluating a pinned coefficient polynomial at a point equals the sum of its coefficients, each weighted by that point raised to the coefficient's associated pinned exponent.","s":"∀ {n : ℕ} {K : Type u_1} [inst : Field K] (v : Fin n ⊕ Fin (n - 1) → K) (x : K), Polynomial.eval x (Causalean.Mathlib.LinearAlgebra.pinnedCoefficientPolynomial v) = ∑ a, x ^ Causalean.Mathlib.LinearAlgebra.pinnedExponent a * v a","l":318,"f":"Causalean/Mathlib/LinearAlgebra/ConfluentVandermonde.lean"},{"n":"Causalean.Mathlib.LinearAlgebra.genPerm_of_det_ne_zero_of_colSupport","k":"theorem","m":"Causalean.Mathlib.LinearAlgebra.MonomialMatrix","a":"Mathlib/LinearAlgebra","t":1,"d":"For a finite index type ι and a commutative ring K, and a square matrix W over ι × ι valued in K with nonzero determinant, if every column of W has at most one non-zero entry (for any two distinct rows i ≠ k, at least one of W i j, W k j vanishes at column j), then W is a generalized permutation matrix: there are a permutation τ of ι and non-zero scalings d with W i j = if j = τ i then d i else 0.","s":"∀ {ι : Type u_1} {K : Type u_2} [inst : Fintype ι] [inst_1 : DecidableEq ι] [inst_2 : CommRing K] {W : Matrix ι ι K}, W.det ≠ 0 → (∀ (j i k : ι), i ≠ k → W i j = 0 ∨ W k j = 0) → ∃ τ d, (∀ (i : ι), d i ≠ 0) ∧ ∀ (i j : ι), W i j = if j = τ i then d i else 0","l":32,"f":"Causalean/Mathlib/LinearAlgebra/MonomialMatrix.lean"},{"n":"Causalean.Mathlib.LinearAlgebra.isUpperTri_iff_blockTriangular","k":"theorem","m":"Causalean.Mathlib.LinearAlgebra.Cholesky","a":"Mathlib/LinearAlgebra","t":2,"d":"The entrywise upper-triangular predicate is exactly Mathlib's block-triangular predicate for the identity order.","s":"∀ {ι : Type u_1} {K : Type u_2} [inst : LT ι] [inst_1 : Zero K] {U : Matrix ι ι K}, Causalean.Mathlib.LinearAlgebra.IsUpperTri U ↔ U.BlockTriangular id","l":36,"f":"Causalean/Mathlib/LinearAlgebra/Cholesky.lean"},{"n":"Causalean.Mathlib.LinearAlgebra.ldl_diagEntries_pos","k":"theorem","m":"Causalean.Mathlib.LinearAlgebra.Cholesky","a":"Mathlib/LinearAlgebra","t":2,"d":"The diagonal entries of the LDL decomposition of a real positive-definite matrix are strictly positive.","s":"∀ {d : ℕ} {M : Matrix (Fin d) (Fin d) ℝ} (hM : M.PosDef) (i : Fin d), 0 < LDL.diagEntries hM i","l":65,"f":"Causalean/Mathlib/LinearAlgebra/Cholesky.lean"},{"n":"Causalean.Mathlib.LinearAlgebra.ldl_lowerInv_blockTriangular","k":"theorem","m":"Causalean.Mathlib.LinearAlgebra.Cholesky","a":"Mathlib/LinearAlgebra","t":2,"d":"LDL.lowerInv is lower-triangular in the BlockTriangular toDual sense.","s":"∀ {d : ℕ} {M : Matrix (Fin d) (Fin d) ℝ} (hM : M.PosDef), (LDL.lowerInv hM).BlockTriangular ⇑OrderDual.toDual","l":78,"f":"Causalean/Mathlib/LinearAlgebra/Cholesky.lean"},{"n":"Causalean.Mathlib.LinearAlgebra.ldl_lowerInv_diag_one","k":"theorem","m":"Causalean.Mathlib.LinearAlgebra.Cholesky","a":"Mathlib/LinearAlgebra","t":2,"d":"The Gram-Schmidt lower-inverse matrix is unitriangular: its diagonal entries are 1.","s":"∀ {d : ℕ} {M : Matrix (Fin d) (Fin d) ℝ} (hM : M.PosDef) (i : Fin d), LDL.lowerInv hM i i = 1","l":49,"f":"Causalean/Mathlib/LinearAlgebra/Cholesky.lean"},{"n":"Causalean.Mathlib.LinearAlgebra.ldl_lower_blockTriangular","k":"theorem","m":"Causalean.Mathlib.LinearAlgebra.Cholesky","a":"Mathlib/LinearAlgebra","t":2,"d":"LDL.lower (the inverse of LDL.lowerInv) is lower-triangular.","s":"∀ {d : ℕ} {M : Matrix (Fin d) (Fin d) ℝ} (hM : M.PosDef), (LDL.lower hM).BlockTriangular ⇑OrderDual.toDual","l":84,"f":"Causalean/Mathlib/LinearAlgebra/Cholesky.lean"},{"n":"Causalean.Mathlib.LinearAlgebra.ldl_lower_diag_one","k":"theorem","m":"Causalean.Mathlib.LinearAlgebra.Cholesky","a":"Mathlib/LinearAlgebra","t":2,"d":"LDL.lower is unitriangular: its diagonal entries are 1.","s":"∀ {d : ℕ} {M : Matrix (Fin d) (Fin d) ℝ} (hM : M.PosDef) (i : Fin d), LDL.lower hM i i = 1","l":90,"f":"Causalean/Mathlib/LinearAlgebra/Cholesky.lean"},{"n":"Causalean.Mathlib.LinearAlgebra.natDegree_doubledCoefficientPolynomial_lt","k":"theorem","m":"Causalean.Mathlib.LinearAlgebra.ConfluentVandermonde","a":"Mathlib/LinearAlgebra","t":2,"d":"A doubled coefficient polynomial has degree below 2n when n is positive.","s":"∀ {n : ℕ} {K : Type u_1} [inst : Semiring K], 1 ≤ n → ∀ {v : Fin n ⊕ Fin n → K}, (Causalean.Mathlib.LinearAlgebra.doubledCoefficientPolynomial v).natDegree < 2 * n","l":93,"f":"Causalean/Mathlib/LinearAlgebra/ConfluentVandermonde.lean"},{"n":"Causalean.Mathlib.LinearAlgebra.natDegree_pinnedCoefficientPolynomial_lt","k":"theorem","m":"Causalean.Mathlib.LinearAlgebra.ConfluentVandermonde","a":"Mathlib/LinearAlgebra","t":2,"d":"A pinned coefficient polynomial has degree below 2n - 1.","s":"∀ {n : ℕ} {K : Type u_1} [inst : Semiring K], 1 ≤ n → ∀ {v : Fin n ⊕ Fin (n - 1) → K}, (Causalean.Mathlib.LinearAlgebra.pinnedCoefficientPolynomial v).natDegree < 2 * n - 1","l":297,"f":"Causalean/Mathlib/LinearAlgebra/ConfluentVandermonde.lean"},{"n":"Causalean.Mathlib.LinearAlgebra.orthogonal_upperTri_pos_diag_eq_one","k":"theorem","m":"Causalean.Mathlib.LinearAlgebra.Cholesky","a":"Mathlib/LinearAlgebra","t":2,"d":"An orthogonal (Wᵀ * W = 1) upper-triangular matrix with strictly positive diagonal is the identity matrix.","s":"∀ {ι : Type u_1} {K : Type u_2} [inst : Fintype ι] [inst_1 : LinearOrder ι] [inst_2 : Field K] [inst_3 : LinearOrder K] [IsStrictOrderedRing K] {W : Matrix ι ι K}, W.transpose * W = 1 → (∀ (i j : ι), j < i → W i j = 0) → (∀ (i : ι), 0 < W i i) → W = 1","l":152,"f":"Causalean/Mathlib/LinearAlgebra/Cholesky.lean"},{"n":"Causalean.Mathlib.LinearAlgebra.perm_uniqueness","k":"theorem","m":"Causalean.Mathlib.LinearAlgebra.MonomialMatrix","a":"Mathlib/LinearAlgebra","t":1,"d":"Permutation uniqueness for lower-triangular matrices (LiNGAM Appendix A, Lemma 1). For a matrix M over Fin n × Fin n that is lower-triangular (M i j = 0 whenever i < j) with non-zero diagonal entries, and permutations σ, τ of Fin n, the row/column-permuted matrix (i ↦ M (σ i) (τ i)) has a non-zero diagonal at every i if and only if σ = τ.","s":"∀ {n : ℕ} {K : Type u_1} [inst : Zero K] {M : Matrix (Fin n) (Fin n) K}, (∀ (i j : Fin n), i < j → M i j = 0) → (∀ (i : Fin n), M i i ≠ 0) → ∀ {σ τ : Equiv.Perm (Fin n)}, (∀ (i : Fin n), M (σ i) (τ i) ≠ 0) ↔ σ = τ","l":99,"f":"Causalean/Mathlib/LinearAlgebra/MonomialMatrix.lean"},{"n":"Causalean.Mathlib.LinearAlgebra.pinnedCoefficientPolynomial","k":"def","m":"Causalean.Mathlib.LinearAlgebra.ConfluentVandermonde","a":"Mathlib/LinearAlgebra","t":1,"d":"Encode pinned Hermite coefficients as a polynomial.","s":"{n : ℕ} → {K : Type u_1} → [inst : Semiring K] → (Fin n ⊕ Fin (n - 1) → K) → Polynomial K","l":263,"f":"Causalean/Mathlib/LinearAlgebra/ConfluentVandermonde.lean"},{"n":"Causalean.Mathlib.LinearAlgebra.pinnedCoefficientPolynomial_eq_zero_iff","k":"theorem","m":"Causalean.Mathlib.LinearAlgebra.ConfluentVandermonde","a":"Mathlib/LinearAlgebra","t":2,"d":"A pinned coefficient polynomial vanishes exactly when its coefficient vector vanishes.","s":"∀ {n : ℕ} {K : Type u_1} [inst : Semiring K] (v : Fin n ⊕ Fin (n - 1) → K), Causalean.Mathlib.LinearAlgebra.pinnedCoefficientPolynomial v = 0 ↔ v = 0","l":283,"f":"Causalean/Mathlib/LinearAlgebra/ConfluentVandermonde.lean"},{"n":"Causalean.Mathlib.LinearAlgebra.pinnedCoefficientPolynomial_eq_zero_of_eval_derivative","k":"theorem","m":"Causalean.Mathlib.LinearAlgebra.ConfluentVandermonde","a":"Mathlib/LinearAlgebra","t":2,"d":"A polynomial in the pinned coefficient model is zero when it vanishes at all nodes and its derivative vanishes at every nondistinguished node.","s":"∀ {n : ℕ} {K : Type u_1} [inst : Field K], 1 ≤ n → ∀ (s : Fin n → K), Function.Injective s → ∀ (v : Fin n ⊕ Fin (n - 1) → K), (∀ (i : Fin n), Polynomial.eval (s i) (Causalean.Mathlib.LinearAlgebra.pinnedCoefficientPolynomial v) = 0) → (∀ (i : Fin (n - 1)), Polynomial.eval (s (Causalean.Mathlib.LinearAlgebra.pinnedSucc i)) (Polynomial.derivative (Causalean.Mathlib.LinearAlgebra.pinnedCoefficientPolynomial v)) = 0) → v = 0","l":345,"f":"Causalean/Mathlib/LinearAlgebra/ConfluentVandermonde.lean"},{"n":"Causalean.Mathlib.LinearAlgebra.pinnedConfluentVandermonde","k":"def","m":"Causalean.Mathlib.LinearAlgebra.ConfluentVandermonde","a":"Mathlib/LinearAlgebra","t":1,"d":"The pinned Hermite matrix records values at all nodes and derivatives at nodes 1, ..., n-1, with the derivative at node zero deleted.","s":"{n : ℕ} → {K : Type u_1} → [Semiring K] → (Fin n → K) → Matrix (Fin n ⊕ Fin (n - 1)) (Fin n ⊕ Fin (n - 1)) K","l":308,"f":"Causalean/Mathlib/LinearAlgebra/ConfluentVandermonde.lean"},{"n":"Causalean.Mathlib.LinearAlgebra.pinnedExponent","k":"def","m":"Causalean.Mathlib.LinearAlgebra.ConfluentVandermonde","a":"Mathlib/LinearAlgebra","t":1,"d":"Monomial degrees for one simple node and n - 1 doubled nodes.","s":"{n : ℕ} → Fin n ⊕ Fin (n - 1) → ℕ","l":212,"f":"Causalean/Mathlib/LinearAlgebra/ConfluentVandermonde.lean"},{"n":"Causalean.Mathlib.LinearAlgebra.pinnedExponent_injective","k":"theorem","m":"Causalean.Mathlib.LinearAlgebra.ConfluentVandermonde","a":"Mathlib/LinearAlgebra","t":2,"d":"The pinned exponent encoding is injective.","s":"∀ {n : ℕ}, Function.Injective Causalean.Mathlib.LinearAlgebra.pinnedExponent","l":237,"f":"Causalean/Mathlib/LinearAlgebra/ConfluentVandermonde.lean"},{"n":"Causalean.Mathlib.LinearAlgebra.pinnedExponent_lt","k":"theorem","m":"Causalean.Mathlib.LinearAlgebra.ConfluentVandermonde","a":"Mathlib/LinearAlgebra","t":2,"d":"Every pinned exponent is strictly below 2n - 1.","s":"∀ {n : ℕ} (i : Fin n ⊕ Fin (n - 1)), Causalean.Mathlib.LinearAlgebra.pinnedExponent i < 2 * n - 1","l":254,"f":"Causalean/Mathlib/LinearAlgebra/ConfluentVandermonde.lean"},{"n":"Causalean.Mathlib.LinearAlgebra.pinnedSucc","k":"def","m":"Causalean.Mathlib.LinearAlgebra.ConfluentVandermonde","a":"Mathlib/LinearAlgebra","t":1,"d":"Embed the doubled-node index as a positive node index.","s":"{n : ℕ} → Fin (n - 1) → Fin n","l":217,"f":"Causalean/Mathlib/LinearAlgebra/ConfluentVandermonde.lean"},{"n":"Causalean.Mathlib.LinearAlgebra.pinnedSucc_injective","k":"theorem","m":"Causalean.Mathlib.LinearAlgebra.ConfluentVandermonde","a":"Mathlib/LinearAlgebra","t":2,"d":"The positive-node embedding is injective.","s":"∀ {n : ℕ}, Function.Injective Causalean.Mathlib.LinearAlgebra.pinnedSucc","l":221,"f":"Causalean/Mathlib/LinearAlgebra/ConfluentVandermonde.lean"},{"n":"Causalean.Mathlib.LinearAlgebra.pinnedSucc_ne_zero","k":"theorem","m":"Causalean.Mathlib.LinearAlgebra.ConfluentVandermonde","a":"Mathlib/LinearAlgebra","t":2,"d":"A positive-node index never equals the distinguished zero node.","s":"∀ {n : ℕ} (hn : 1 ≤ n) (i : Fin (n - 1)), Causalean.Mathlib.LinearAlgebra.pinnedSucc i ≠ ⟨0, hn⟩","l":230,"f":"Causalean/Mathlib/LinearAlgebra/ConfluentVandermonde.lean"},{"n":"Causalean.Mathlib.LinearAlgebra.stackedContraction","k":"def","m":"Causalean.Mathlib.LinearAlgebra.StackedVandermonde","a":"Mathlib/LinearAlgebra","t":1,"d":"The coefficient-vector version of the stacked contraction map. Over a field in the affine chart ℓ j = X₀ + slopes j X₁, its k-th component agrees with the coefficient vector of ∑ j, weights j k * e j • (ℓ j)^k up to invertible binomial diagonal rescaling.","s":"{K : Type u_1} → [Semiring K] → (N : ℕ) → (Fin (N + 1) → K) → (Fin (N + 1) → Fin N → K) → (Fin (N + 1) → K) → (k : Fin N) → Fin (↑k + 1) → K","l":36,"f":"Causalean/Mathlib/LinearAlgebra/StackedVandermonde.lean"},{"n":"Causalean.Mathlib.LinearAlgebra.stacked_contraction_injective_of_generic_weights","k":"theorem","m":"Causalean.Mathlib.LinearAlgebra.StackedVandermonde","a":"Mathlib/LinearAlgebra","t":1,"d":"For a commutative integral domain K, at least three indices N + 1 (2 ≤ N), and pairwise distinct slope values slopes : Fin (N + 1) → K, there exist weights Fin (N + 1) → Fin N → K for which the stacked contraction is injective. The first block detects coordinate 0, while the last block is a square Vandermonde system on coordinates 1,…,N.","s":"∀ {K : Type u_1} [inst : CommRing K] [IsDomain K] {N : ℕ}, 2 ≤ N → ∀ (slopes : Fin (N + 1) → K), Function.Injective slopes → ∃ weights, Function.Injective (Causalean.Mathlib.LinearAlgebra.stackedContraction N slopes weights)","l":56,"f":"Causalean/Mathlib/LinearAlgebra/StackedVandermonde.lean"},{"n":"Causalean.Mathlib.MeasureTheory.IsExtremePoint","k":"def","m":"Causalean.Mathlib.MeasureTheory.MomentSliceSupport","a":"Mathlib/MeasureTheory","t":1,"d":"μ is an extreme point of a set C of measures: it lies in C and is not an interior point of any nondegenerate segment inside C.","s":"Set (MeasureTheory.Measure ℝ) → MeasureTheory.Measure ℝ → Prop","l":65,"f":"Causalean/Mathlib/MeasureTheory/MomentSliceSupport.lean"},{"n":"Causalean.Mathlib.MeasureTheory.MemLp.of_measurable_of_integral_sq_le","k":"theorem","m":"Causalean.Mathlib.MeasureTheory.MemLp","a":"Mathlib/MeasureTheory","t":1,"d":"If a normed-vector-space-valued function f is almost-everywhere strongly measurable with respect to Q and the squared norm ‖f x‖^2 is integrable against Q, then f belongs to $L^2(Q)$.","s":"∀ {X : Type u_1} {F : Type u_2} [inst : MeasurableSpace X] [inst_1 : NormedAddCommGroup F] {Q : MeasureTheory.Measure X} {f : X → F}, MeasureTheory.AEStronglyMeasurable f Q → MeasureTheory.Integrable (fun x => ‖f x‖ ^ 2) Q → MeasureTheory.MemLp f 2 Q","l":22,"f":"Causalean/Mathlib/MeasureTheory/MemLp.lean"},{"n":"Causalean.Mathlib.MeasureTheory.MomentSlice","k":"def","m":"Causalean.Mathlib.MeasureTheory.MomentSliceSupport","a":"Mathlib/MeasureTheory","t":1,"d":"The moment slice: probability measures supported on [a,b] with mean 0 and second moment s.","s":"ℝ → ℝ → ℝ → Set (MeasureTheory.Measure ℝ)","l":59,"f":"Causalean/Mathlib/MeasureTheory/MomentSliceSupport.lean"},{"n":"Causalean.Mathlib.MeasureTheory.abs_le_one_ae_of_setIntegral_le_measure","k":"theorem","m":"Causalean.Mathlib.MeasureTheory.SetIntegralRecovery","a":"Mathlib/MeasureTheory","t":2,"d":"A measurable function is bounded in absolute value by one almost everywhere if the absolute value of its integral over every measurable event is at most that event's measure.","s":"∀ {Ω : Type u_1} [inst : MeasurableSpace Ω] (mu : MeasureTheory.Measure Ω) [MeasureTheory.IsFiniteMeasure mu] (f : Ω → ℝ), Measurable f → (∀ (A : Set Ω), MeasurableSet A → |∫ (x : Ω) in A, f x ∂mu| ≤ (mu A).toReal) → ∀ᵐ (x : Ω) ∂mu, |f x| ≤ 1","l":48,"f":"Causalean/Mathlib/MeasureTheory/SetIntegralRecovery.lean"},{"n":"Causalean.Mathlib.MeasureTheory.bddAbove_image_of_bound","k":"theorem","m":"Causalean.Mathlib.MeasureTheory.SupCountableDense","a":"Mathlib/MeasureTheory","t":2,"d":"A uniform upper bound F ω π ≤ C over the index class S makes the image (F ω) '' S bounded above (for every ω). Companion BddAbove fact accompanying the integrability lemma.","s":"∀ {Ω : Type u_1} {ι : Type u_2} (S : Set ι) (F : Ω → ι → ℝ) (C : ℝ), (∀ (ω : Ω), ∀ π ∈ S, F ω π ≤ C) → ∀ (ω : Ω), BddAbove ((fun π => F ω π) '' S)","l":107,"f":"Causalean/Mathlib/MeasureTheory/SupCountableDense.lean"},{"n":"Causalean.Mathlib.MeasureTheory.card_le_three_of_isExtremePoint","k":"theorem","m":"Causalean.Mathlib.MeasureTheory.MomentSliceSupport","a":"Mathlib/MeasureTheory","t":1,"d":"Richter–Rogosinski support bound (finite-atom case). Consider the discrete probability measure μ = ∑_{x∈T} w x · δ_x carried by a finite set T ⊆ ℝ. If every atom weight w x is strictly positive for x ∈ T, every atom lies in the interval [a, b], and μ is an extreme point of the moment slice — the probability measures on [a, b] with mean 0 and second moment s, then T has at most three elements: μ is supported on at most three atoms.","s":"∀ {a b s : ℝ} {T : Finset ℝ} {w : ℝ → ℝ}, (∀ x ∈ T, 0 < w x) → (∀ x ∈ T, x ∈ Set.Icc a b) → Causalean.Mathlib.MeasureTheory.IsExtremePoint (Causalean.Mathlib.MeasureTheory.MomentSlice a b s) (Causalean.Mathlib.MeasureTheory.discreteMeasure T w) → T.card ≤ 3","l":177,"f":"Causalean/Mathlib/MeasureTheory/MomentSliceSupport.lean"},{"n":"Causalean.Mathlib.MeasureTheory.discreteMeasure","k":"def","m":"Causalean.Mathlib.MeasureTheory.MomentSliceSupport","a":"Mathlib/MeasureTheory","t":1,"d":"The discrete probability-measure builder: ∑_{x ∈ T} w x · δ_x, the weighted sum of Dirac masses at the points of T.","s":"{α : Type u_1} → [inst : MeasurableSpace α] → Finset α → (α → ℝ) → MeasureTheory.Measure α","l":53,"f":"Causalean/Mathlib/MeasureTheory/MomentSliceSupport.lean"},{"n":"Causalean.Mathlib.MeasureTheory.discreteMeasure_apply_compl_of_subset","k":"theorem","m":"Causalean.Mathlib.MeasureTheory.MomentSliceSupport","a":"Mathlib/MeasureTheory","t":2,"d":"A discrete measure whose atoms all lie in a set gives zero mass to that set's complement.","s":"∀ {α : Type u_1} [inst : MeasurableSpace α] [MeasurableSingletonClass α] {T : Finset α} {w : α → ℝ} {K : Set α}, (∀ x ∈ T, x ∈ K) → (Causalean.Mathlib.MeasureTheory.discreteMeasure T w) Kᶜ = 0","l":100,"f":"Causalean/Mathlib/MeasureTheory/MomentSliceSupport.lean"},{"n":"Causalean.Mathlib.MeasureTheory.discreteMeasure_midpoint","k":"theorem","m":"Causalean.Mathlib.MeasureTheory.MomentSliceSupport","a":"Mathlib/MeasureTheory","t":2,"d":"A discrete measure whose atom weights are the pointwise average of two nonnegative weightings is the midpoint of the two corresponding discrete measures.","s":"∀ {α : Type u_1} [inst : MeasurableSpace α] {T : Finset α} {w wp wm : α → ℝ}, (∀ x ∈ T, 0 ≤ wp x) → (∀ x ∈ T, 0 ≤ wm x) → (∀ x ∈ T, w x = 1 / 2 * wp x + 1 / 2 * wm x) → Causalean.Mathlib.MeasureTheory.discreteMeasure T w = (1 / 2) • Causalean.Mathlib.MeasureTheory.discreteMeasure T wp + (1 / 2) • Causalean.Mathlib.MeasureTheory.discreteMeasure T wm","l":155,"f":"Causalean/Mathlib/MeasureTheory/MomentSliceSupport.lean"},{"n":"Causalean.Mathlib.MeasureTheory.discreteMeasure_singleton","k":"theorem","m":"Causalean.Mathlib.MeasureTheory.MomentSliceSupport","a":"Mathlib/MeasureTheory","t":2,"d":"The mass assigned by a finite discrete measure to an atom in its support is the corresponding atom weight, coerced to ℝ≥0∞.","s":"∀ {α : Type u_1} [inst : MeasurableSpace α] [MeasurableSingletonClass α] {T : Finset α} {w : α → ℝ} {x₀ : α}, x₀ ∈ T → (Causalean.Mathlib.MeasureTheory.discreteMeasure T w) {x₀} = ENNReal.ofReal (w x₀)","l":140,"f":"Causalean/Mathlib/MeasureTheory/MomentSliceSupport.lean"},{"n":"Causalean.Mathlib.MeasureTheory.exists_cardSupportLe_three_of_isExtremePoint","k":"theorem","m":"Causalean.Mathlib.MeasureTheory.MomentSliceSupport","a":"Mathlib/MeasureTheory","t":2,"d":"An extreme probability measure in the two-moment slice is carried by a finite set of at most three points.","s":"∀ {a b s : ℝ} {μ : MeasureTheory.Measure ℝ}, Causalean.Mathlib.MeasureTheory.IsExtremePoint (Causalean.Mathlib.MeasureTheory.MomentSlice a b s) μ → ∃ T, T.card ≤ 3 ∧ μ (↑T)ᶜ = 0","l":711,"f":"Causalean/Mathlib/MeasureTheory/MomentSliceSupport.lean"},{"n":"Causalean.Mathlib.MeasureTheory.exists_isMinOn_momentSlice","k":"theorem","m":"Causalean.Mathlib.MeasureTheory.MomentSliceSupport","a":"Mathlib/MeasureTheory","t":2,"d":"Attainment of the minimum over a moment slice. On a compact Hausdorff space Ω, the set of probability measures pinned by two bounded-continuous moment constraints ∫ g₁ = c₁, ∫ g₂ = c₂ is weak-* compact, so any bounded-continuous objective ∫ f attains its minimum over that (nonempty) slice. The number of constraints is immaterial; the two-moment case is stated to match the mean/second-moment slice.","s":"∀ {Ω : Type u_1} [inst : MeasurableSpace Ω] [inst_1 : TopologicalSpace Ω] [T2Space Ω] [BorelSpace Ω] [CompactSpace Ω] (g₁ g₂ f : BoundedContinuousFunction Ω ℝ) (c₁ c₂ : ℝ), {μ | ∫ (x : Ω), g₁ x ∂↑μ = c₁ ∧ ∫ (x : Ω), g₂ x ∂↑μ = c₂}.Nonempty → ∃ μ ∈ {μ | ∫ (x : Ω), g₁ x ∂↑μ = c₁ ∧ ∫ (x : Ω), g₂ x ∂↑μ = c₂}, ∀ ν ∈ {μ | ∫ (x : Ω), g₁ x ∂↑μ = c₁ ∧ ∫ (x : Ω), g₂ x ∂↑μ = c₂}, ∫ (x : Ω), f x ∂↑μ ≤ ∫ (x : Ω), f x ∂↑ν","l":277,"f":"Causalean/Mathlib/MeasureTheory/MomentSliceSupport.lean"},{"n":"Causalean.Mathlib.MeasureTheory.exists_moment_perturbation","k":"theorem","m":"Causalean.Mathlib.MeasureTheory.MomentSliceSupport","a":"Mathlib/MeasureTheory","t":2,"d":"Three-moment perturbation. On any finite set T ⊆ ℝ of more than three points there is a nonzero real weighting δ whose total mass, first moment and second moment all vanish. This is rank–nullity: three linear functionals (∑ δ, ∑ δ·x, ∑ δ·x²) on a space of dimension > 3 have a nonzero common kernel.","s":"∀ {T : Finset ℝ}, 3 < T.card → ∃ δ, ∑ x ∈ T, δ x = 0 ∧ ∑ x ∈ T, δ x * x = 0 ∧ ∑ x ∈ T, δ x * x ^ 2 = 0 ∧ ∃ x ∈ T, δ x ≠ 0","l":73,"f":"Causalean/Mathlib/MeasureTheory/MomentSliceSupport.lean"},{"n":"Causalean.Mathlib.MeasureTheory.integrable_of_finite_atomic_support","k":"theorem","m":"Causalean.Mathlib.MeasureTheory.FiniteAtomicMeasure","a":"Mathlib/MeasureTheory","t":2,"d":"Every strongly measurable normed-vector-valued function is integrable under a finite measure concentrated on finitely many measurable points.","s":"∀ {𝒳 : Type u_1} [inst : MeasurableSpace 𝒳] (μ : MeasureTheory.Measure 𝒳) [MeasureTheory.IsFiniteMeasure μ] {ι : Type u_2} [Finite ι] (cell : ι ↪ 𝒳), (∀ (i : ι), MeasurableSet {cell i}) → μ (Set.range ⇑cell) = μ Set.univ → ∀ {E : Type u_3} [inst_3 : NormedAddCommGroup E] (f : 𝒳 → E), MeasureTheory.StronglyMeasurable f → MeasureTheory.Integrable f μ","l":91,"f":"Causalean/Mathlib/MeasureTheory/FiniteAtomicMeasure.lean"},{"n":"Causalean.Mathlib.MeasureTheory.integrable_sSup_image_of_countable_dense","k":"theorem","m":"Causalean.Mathlib.MeasureTheory.SupCountableDense","a":"Mathlib/MeasureTheory","t":1,"d":"On a finite measure μ, if the bound C is nonnegative, the skeleton index set D is countable, each coordinate map ω ↦ F ω π is measurable for π in the skeleton D, the pointwise supremum of F ω over the index class S equals its supremum over the countable skeleton D, for every ω, and F ω π is bounded in absolute value by C uniformly over π ∈ S, then the pointwise supremum ω ↦ sSup ((F ω) '' S) is integrable against μ. This is the entry point that discharges the Bochner side conditions of a localized empirical-process supremum in one call.","s":"∀ {Ω : Type u_1} {ι : Type u_2} [inst : MeasurableSpace Ω] (μ : MeasureTheory.Measure Ω) [MeasureTheory.IsFiniteMeasure μ] (S D : Set ι) (F : Ω → ι → ℝ) (C : ℝ), 0 ≤ C → D.Countable → (∀ π ∈ D, Measurable fun ω => F ω π) → (∀ (ω : Ω), sSup ((fun π => F ω π) '' S) = sSup ((fun π => F ω π) '' D)) → (∀ (ω : Ω), ∀ π ∈ S, |F ω π| ≤ C) → MeasureTheory.Integrable (fun ω => sSup ((fun π => F ω π) '' S)) μ","l":116,"f":"Causalean/Mathlib/MeasureTheory/SupCountableDense.lean"},{"n":"Causalean.Mathlib.MeasureTheory.integral_bind","k":"theorem","m":"Causalean.Mathlib.MeasureTheory.IntegralBind","a":"Mathlib/MeasureTheory","t":1,"d":"If κ is a measurable family of measures, one per point of the base space (a measurable kernel) and f is Bochner-integrable against the measure m.bind κ obtained by mixing κ over the base measure m, then the Bochner integral of f against m.bind κ equals the iterated integral: first integrate f against κ a for each base point a, then integrate the result against m. This is the Bochner analogue of MeasureTheory.Measure.lintegral_bind.","s":"∀ {α : Type u_1} {β : Type u_2} {E : Type u_3} [inst : MeasurableSpace α] [inst_1 : MeasurableSpace β] [inst_2 : NormedAddCommGroup E] [inst_3 : NormedSpace ℝ E] {m : MeasureTheory.Measure α} {κ : α → MeasureTheory.Measure β} {f : β → E}, Measurable κ → MeasureTheory.Integrable f (m.bind κ) → ∫ (x : β), f x ∂m.bind κ = ∫ (a : α), ∫ (x : β), f x ∂κ a ∂m","l":39,"f":"Causalean/Mathlib/MeasureTheory/IntegralBind.lean"},{"n":"Causalean.Mathlib.MeasureTheory.integral_bind_bind_map","k":"theorem","m":"Causalean.Mathlib.MeasureTheory.IntegralBind","a":"Mathlib/MeasureTheory","t":1,"d":"Suppose each map g a b is measurable, for every base point a the pushforward kernel b ↦ (κ₂ a b).map (g a b) is measurable, the resulting doubly-nested mixed kernel a ↦ (κ₁ a).bind (fun b => (κ₂ a b).map (g a b)) is itself measurable, and f is Bochner-integrable against the measure obtained by mixing κ₁ over the base measure m and, within each fibre, mixing the pushforward of κ₂ under g. Then the Bochner integral of f against that triply-nested mixed measure equals the threefold iterated integral of the pulled-back integrand (a, b, c) ↦ f (g a b c), integrated successively against κ₂ a b, κ₁ a, and m. This is the single bridge for a bind-then-bind-then-map integrand, which neither integral_bind nor integral_bind_map covers in one step.","s":"∀ {α : Type u_1} {β : Type u_2} {γ : Type u_3} {δ : Type u_4} {E : Type u_5} [inst : MeasurableSpace α] [inst_1 : MeasurableSpace β] [inst_2 : MeasurableSpace γ] [inst_3 : MeasurableSpace δ] [inst_4 : NormedAddCommGroup E] [inst_5 : NormedSpace ℝ E] {m : MeasureTheory.Measure α} {κ₁ : α → MeasureTheory.Measure β} {κ₂ : α → β → MeasureTheory.Measure γ} {g : α → β → γ → δ} {f : δ → E}, (∀ (a : α) (b : β), Measurable (g a b)) → (∀ (a : α), Measurable fun b => MeasureTheory.Measure.map (g a b) (κ₂ a b)) → (Measurable fun a => (κ₁ a).bind fun b => MeasureTheory.Measure.map (g a b) (κ₂ a b)) → MeasureTheory.Integrable f (m.bind fun a => (κ₁ a).bind fun b => MeasureTheory.Measure.map (g a b) (κ₂ a b)) → (∫ (z : δ), f z ∂m.bind fun a => (κ₁ a).bind fun b => MeasureTheory.Measure.map (g a b) (κ₂ a b)) = ∫ (a : α), ∫ (b : β), ∫ (c : γ), f (g a b c) ∂κ₂ a b ∂κ₁ a ∂m","l":98,"f":"Causalean/Mathlib/MeasureTheory/IntegralBind.lean"},{"n":"Causalean.Mathlib.MeasureTheory.integral_bind_bind_map_proj","k":"theorem","m":"Causalean.Mathlib.MeasureTheory.IntegralBind","a":"Mathlib/MeasureTheory","t":1,"d":"Suppose every fibre κ₁ a is a probability measure, every inner fibre κ₂ a b is a probability measure, each map g a b is measurable, for every base point a the pushforward kernel b ↦ (κ₂ a b).map (g a b) is measurable, the resulting doubly-nested mixed kernel is measurable, the projection π is measurable, π undoes g by recovering the base point: π (g a b c) = a for all a, b, c, and f is Bochner-integrable against the base measure m. Then integrating the pulled-back function f ∘ π against the triply-nested mixed measure equals integrating f directly against the base measure m. This is the one-step bridge for marginalising a nested Giry-monad construction back onto the coordinate that the innermost pushforward carries through; the fibrewise probability-mass-one hypotheses are what make the two inner integrals of the constant f a evaluate to f a (no assumption on m is needed).","s":"∀ {α : Type u_1} {β : Type u_2} {γ : Type u_3} {δ : Type u_4} {E : Type u_5} [inst : MeasurableSpace α] [inst_1 : MeasurableSpace β] [inst_2 : MeasurableSpace γ] [inst_3 : MeasurableSpace δ] [inst_4 : NormedAddCommGroup E] [inst_5 : NormedSpace ℝ E] {m : MeasureTheory.Measure α} {κ₁ : α → MeasureTheory.Measure β} {κ₂ : α → β → MeasureTheory.Measure γ} {g : α → β → γ → δ} {π : δ → α} {f : α → E}, (∀ (a : α), MeasureTheory.IsProbabilityMeasure (κ₁ a)) → (∀ (a : α) (b : β), MeasureTheory.IsProbabilityMeasure (κ₂ a b)) → (∀ (a : α) (b : β), Measurable (g a b)) → (∀ (a : α), Measurable fun b => MeasureTheory.Measure.map (g a b) (κ₂ a b)) → (Measurable fun a => (κ₁ a).bind fun b => MeasureTheory.Measure.map (g a b) (κ₂ a b)) → Measurable π → (∀ (a : α) (b : β) (c : γ), π (g a b c) = a) → MeasureTheory.Integrable f m → (∫ (z : δ), f (π z) ∂m.bind fun a => (κ₁ a).bind fun b => MeasureTheory.Measure.map (g a b) (κ₂ a b)) = ∫ (a : α), f a ∂m","l":205,"f":"Causalean/Mathlib/MeasureTheory/IntegralBind.lean"},{"n":"Causalean.Mathlib.MeasureTheory.integral_bind_map","k":"theorem","m":"Causalean.Mathlib.MeasureTheory.IntegralBind","a":"Mathlib/MeasureTheory","t":1,"d":"Suppose each map g a is measurable, the kernel sending a base point a to the pushforward measure (κ a).map (g a) is itself measurable, and f is Bochner-integrable against the mixed measure m.bind (fun a => (κ a).map (g a)). Then the Bochner integral of f against that mixed measure equals the iterated integral of the pulled-back integrand a ↦ ∫ x, f (g a x) ∂κ a against the base measure m. This packages a single application of integral_bind with the fibrewise MeasureTheory.integral_map, supplying the bridge needed to expand a nested bind-then-map Bochner integral.","s":"∀ {α : Type u_1} {β : Type u_2} {γ : Type u_3} {E : Type u_4} [inst : MeasurableSpace α] [inst_1 : MeasurableSpace β] [inst_2 : MeasurableSpace γ] [inst_3 : NormedAddCommGroup E] [inst_4 : NormedSpace ℝ E] {m : MeasureTheory.Measure α} {κ : α → MeasureTheory.Measure β} {g : α → β → γ} {f : γ → E}, (∀ (a : α), Measurable (g a)) → (Measurable fun a => MeasureTheory.Measure.map (g a) (κ a)) → MeasureTheory.Integrable f (m.bind fun a => MeasureTheory.Measure.map (g a) (κ a)) → (∫ (z : γ), f z ∂m.bind fun a => MeasureTheory.Measure.map (g a) (κ a)) = ∫ (a : α), ∫ (x : β), f (g a x) ∂κ a ∂m","l":61,"f":"Causalean/Mathlib/MeasureTheory/IntegralBind.lean"},{"n":"Causalean.Mathlib.MeasureTheory.integral_bind_of_ae_eq_const","k":"theorem","m":"Causalean.Mathlib.MeasureTheory.IntegralBind","a":"Mathlib/MeasureTheory","t":1,"d":"Suppose κ is a measurable family of measures, every fibre κ a is a probability measure, the integrand f agrees κ a-almost everywhere with a constant f' a on that fibre, for every base point a, and f is Bochner-integrable against the mixed measure m.bind κ. Then the Bochner integral of f against m.bind κ equals the integral of the fibrewise constant f' against the base measure m. This is the bridge for the situation where the integrand only depends on a coordinate that is constant within each inner kernel, so the inner integral evaluates to that constant and the bind reduces to ∫ a, f' a ∂m.","s":"∀ {α : Type u_1} {β : Type u_2} {E : Type u_3} [inst : MeasurableSpace α] [inst_1 : MeasurableSpace β] [inst_2 : NormedAddCommGroup E] [inst_3 : NormedSpace ℝ E] {m : MeasureTheory.Measure α} {κ : α → MeasureTheory.Measure β} {f : β → E} {f' : α → E}, Measurable κ → (∀ (a : α), MeasureTheory.IsProbabilityMeasure (κ a)) → (∀ (a : α), ∀ᵐ (y : β) ∂κ a, f y = f' a) → MeasureTheory.Integrable f (m.bind κ) → ∫ (y : β), f y ∂m.bind κ = ∫ (a : α), f' a ∂m","l":136,"f":"Causalean/Mathlib/MeasureTheory/IntegralBind.lean"},{"n":"Causalean.Mathlib.MeasureTheory.integral_cellConst","k":"theorem","m":"Causalean.Mathlib.MeasureTheory.PartitionIntegral","a":"Mathlib/MeasureTheory","t":2,"d":"Cell-weight aggregation. For a finite measure, the integral of a cell-constant function c (H ω) is the cell-weighted sum ∑ h, c h · μ(H⁻¹{h}). Special case of integral_cellConst_mul with f ≡ 1.","s":"∀ {Ω : Type u_1} [inst : MeasurableSpace Ω] {μ : MeasureTheory.Measure Ω} {ι : Type u_2} [inst_1 : Fintype ι] [MeasureTheory.IsFiniteMeasure μ] {H : Ω → ι}, (∀ (h : ι), MeasurableSet (H ⁻¹' {h})) → ∀ (c : ι → ℝ), ∫ (ω : Ω), c (H ω) ∂μ = ∑ h, c h * (μ (H ⁻¹' {h})).toReal","l":138,"f":"Causalean/Mathlib/MeasureTheory/PartitionIntegral.lean"},{"n":"Causalean.Mathlib.MeasureTheory.integral_cellConst_mul","k":"theorem","m":"Causalean.Mathlib.MeasureTheory.PartitionIntegral","a":"Mathlib/MeasureTheory","t":1,"d":"Cell-constant pull-out. If H is a finite-valued map on the sample space whose fibers H⁻¹{h} are all measurable and f is integrable with respect to the ambient measure, then, for any weighting c of the fiber values, the integral of the ω-dependent product c (H ω) • f ω decomposes as the sum, over the values h of H, of c h scaling the set integral of f over the fiber H⁻¹{h}. No measurability of c is needed: on each fiber c (H ω) is the constant c h.","s":"∀ {Ω : Type u_1} [inst : MeasurableSpace Ω] {μ : MeasureTheory.Measure Ω} {ι : Type u_2} [inst_1 : Fintype ι] {H : Ω → ι}, (∀ (h : ι), MeasurableSet (H ⁻¹' {h})) → ∀ (c : ι → ℝ) {E : Type u_3} [inst_2 : NormedAddCommGroup E] [inst_3 : NormedSpace ℝ E] {f : Ω → E}, MeasureTheory.Integrable f μ → ∫ (ω : Ω), c (H ω) • f ω ∂μ = ∑ h, c h • ∫ (ω : Ω) in H ⁻¹' {h}, f ω ∂μ","l":97,"f":"Causalean/Mathlib/MeasureTheory/PartitionIntegral.lean"},{"n":"Causalean.Mathlib.MeasureTheory.integral_discreteMeasure","k":"theorem","m":"Causalean.Mathlib.MeasureTheory.MomentSliceSupport","a":"Mathlib/MeasureTheory","t":2,"d":"Integral against a discrete measure is the weighted sum of the integrand over the atoms. Every function into a real normed vector space is integrable because the measure has finite support.","s":"∀ {α : Type u_1} {E : Type u_2} [inst : MeasurableSpace α] [MeasurableSingletonClass α] [inst_2 : NormedAddCommGroup E] [inst_3 : NormedSpace ℝ E] [CompleteSpace E] {T : Finset α} {w : α → ℝ}, (∀ x ∈ T, 0 ≤ w x) → ∀ (f : α → E), ∫ (x : α), f x ∂Causalean.Mathlib.MeasureTheory.discreteMeasure T w = ∑ x ∈ T, w x • f x","l":125,"f":"Causalean/Mathlib/MeasureTheory/MomentSliceSupport.lean"},{"n":"Causalean.Mathlib.MeasureTheory.integral_eq_sum_setIntegral_fiber","k":"theorem","m":"Causalean.Mathlib.MeasureTheory.PartitionIntegral","a":"Mathlib/MeasureTheory","t":1,"d":"If H is a finite-valued map on the sample space whose fibers H⁻¹{h} are all measurable and f is integrable with respect to the ambient measure, then the integral of f splits as the sum, over the finitely many values h of H, of the set integrals of f restricted to the fiber H⁻¹{h}.","s":"∀ {Ω : Type u_1} [inst : MeasurableSpace Ω] {μ : MeasureTheory.Measure Ω} {ι : Type u_2} [inst_1 : Fintype ι] {H : Ω → ι}, (∀ (h : ι), MeasurableSet (H ⁻¹' {h})) → ∀ {E : Type u_3} [inst_2 : NormedAddCommGroup E] [inst_3 : NormedSpace ℝ E] {f : Ω → E}, MeasureTheory.Integrable f μ → ∫ (ω : Ω), f ω ∂μ = ∑ h, ∫ (ω : Ω) in H ⁻¹' {h}, f ω ∂μ","l":79,"f":"Causalean/Mathlib/MeasureTheory/PartitionIntegral.lean"},{"n":"Causalean.Mathlib.MeasureTheory.isAtomic_le_three_of_isExtremePoint","k":"theorem","m":"Causalean.Mathlib.MeasureTheory.MomentSliceSupport","a":"Mathlib/MeasureTheory","t":2,"d":"An extreme probability measure in the two-moment slice is a positive discrete measure supported on at most three points of the interval.","s":"∀ {a b s : ℝ} {μ : MeasureTheory.Measure ℝ}, Causalean.Mathlib.MeasureTheory.IsExtremePoint (Causalean.Mathlib.MeasureTheory.MomentSlice a b s) μ → ∃ T w, (∀ x ∈ T, 0 < w x) ∧ (∀ x ∈ T, x ∈ Set.Icc a b) ∧ T.card ≤ 3 ∧ μ = Causalean.Mathlib.MeasureTheory.discreteMeasure T w","l":635,"f":"Causalean/Mathlib/MeasureTheory/MomentSliceSupport.lean"},{"n":"Causalean.Mathlib.MeasureTheory.isProbabilityMeasure_discreteMeasure","k":"theorem","m":"Causalean.Mathlib.MeasureTheory.MomentSliceSupport","a":"Mathlib/MeasureTheory","t":2,"d":"A finite discrete measure is a probability measure when all atom weights are nonnegative and their sum is one.","s":"∀ {α : Type u_1} [inst : MeasurableSpace α] {T : Finset α} {w : α → ℝ}, (∀ x ∈ T, 0 ≤ w x) → ∑ x ∈ T, w x = 1 → MeasureTheory.IsProbabilityMeasure (Causalean.Mathlib.MeasureTheory.discreteMeasure T w)","l":113,"f":"Causalean/Mathlib/MeasureTheory/MomentSliceSupport.lean"},{"n":"Causalean.Mathlib.MeasureTheory.map_bind_bind_map_proj","k":"theorem","m":"Causalean.Mathlib.MeasureTheory.IntegralBind","a":"Mathlib/MeasureTheory","t":1,"d":"Suppose every fibre κ₁ a is a probability measure, every inner fibre κ₂ a b is a probability measure, each map g a b is measurable, for every base point a the pushforward kernel b ↦ (κ₂ a b).map (g a b) is measurable, the resulting doubly-nested mixed kernel is measurable, the projection π is measurable, and π undoes g by recovering the base point: π (g a b c) = a for all a, b, c. Then pushing the triply-nested mixed measure forward along π returns exactly the base measure m. This is the underlying measure identity behind integral_bind_bind_map_proj, stated without any integrability or integrand hypotheses: the two inner probability fibres each contribute total mass one over a fixed base point, so transporting back along π returns m unchanged (no hypothesis on m is needed). It is the bridge for marginalising a nested Giry-monad construction onto its reattached coordinate when only measurability of the eventual integrand is available.","s":"∀ {α : Type u_1} {β : Type u_2} {γ : Type u_3} {δ : Type u_4} [inst : MeasurableSpace α] [inst_1 : MeasurableSpace β] [inst_2 : MeasurableSpace γ] [inst_3 : MeasurableSpace δ] {m : MeasureTheory.Measure α} {κ₁ : α → MeasureTheory.Measure β} {κ₂ : α → β → MeasureTheory.Measure γ} {g : α → β → γ → δ} {π : δ → α}, (∀ (a : α), MeasureTheory.IsProbabilityMeasure (κ₁ a)) → (∀ (a : α) (b : β), MeasureTheory.IsProbabilityMeasure (κ₂ a b)) → (∀ (a : α) (b : β), Measurable (g a b)) → (∀ (a : α), Measurable fun b => MeasureTheory.Measure.map (g a b) (κ₂ a b)) → (Measurable fun a => (κ₁ a).bind fun b => MeasureTheory.Measure.map (g a b) (κ₂ a b)) → Measurable π → (∀ (a : α) (b : β) (c : γ), π (g a b c) = a) → MeasureTheory.Measure.map π (m.bind fun a => (κ₁ a).bind fun b => MeasureTheory.Measure.map (g a b) (κ₂ a b)) = m","l":163,"f":"Causalean/Mathlib/MeasureTheory/IntegralBind.lean"},{"n":"Causalean.Mathlib.MeasureTheory.measurable_sSup_image_of_countable_dense","k":"theorem","m":"Causalean.Mathlib.MeasureTheory.SupCountableDense","a":"Mathlib/MeasureTheory","t":2,"d":"The pointwise supremum ω ↦ sSup ((F ω) '' S) of a real process over an index class S is measurable, provided S has a countable skeleton D on which the process is measurable in ω, and the supremum over S agrees pointwise with the supremum over D. Paper-agnostic generalization of the policy-class skeleton measurability step.","s":"∀ {Ω : Type u_1} {ι : Type u_2} [inst : MeasurableSpace Ω] (S D : Set ι) (F : Ω → ι → ℝ), D.Countable → (∀ π ∈ D, Measurable fun ω => F ω π) → (∀ (ω : Ω), sSup ((fun π => F ω π) '' S) = sSup ((fun π => F ω π) '' D)) → Measurable fun ω => sSup ((fun π => F ω π) '' S)","l":68,"f":"Causalean/Mathlib/MeasureTheory/SupCountableDense.lean"},{"n":"Causalean.Mathlib.MeasureTheory.measure_eq_fin_sum_smul_dirac_of_range","k":"theorem","m":"Causalean.Mathlib.MeasureTheory.FiniteAtomicMeasure","a":"Mathlib/MeasureTheory","t":1,"d":"Let μ be a finite measure and cell an injective family of finitely many points of the sample space, indexed by a finite type ι. If every singleton {cell i} is measurable and μ assigns its full mass to the range of cell, i.e. μ puts no mass outside these finitely many points, then μ equals the sum, over the index i, of the point mass μ {cell i} scaling the Dirac measure at cell i.","s":"∀ {𝒳 : Type u_1} [inst : MeasurableSpace 𝒳] (μ : MeasureTheory.Measure 𝒳) [MeasureTheory.IsFiniteMeasure μ] {ι : Type u_2} [inst_2 : Fintype ι] (cell : ι ↪ 𝒳), (∀ (i : ι), MeasurableSet {cell i}) → μ (Set.range ⇑cell) = μ Set.univ → μ = ∑ i, μ {cell i} • MeasureTheory.Measure.dirac (cell i)","l":23,"f":"Causalean/Mathlib/MeasureTheory/FiniteAtomicMeasure.lean"},{"n":"Causalean.Mathlib.MeasureTheory.measure_eq_withDensity_of_toReal_setIntegral","k":"theorem","m":"Causalean.Mathlib.MeasureTheory.SetIntegralRecovery","a":"Mathlib/MeasureTheory","t":1,"d":"If a weight function w is integrable with respect to a reference measure μ and a finite measure ν's mass on every measurable event A equals, as a real number, the integral of w over A against μ, then ν is obtained from μ by weighting with the nonnegative part of w: ν = μ.withDensity (fun x => ENNReal.ofReal (w x)).","s":"∀ {α : Type u_1} [inst : MeasurableSpace α] {μ ν : MeasureTheory.Measure α} [MeasureTheory.IsFiniteMeasure ν] {w : α → ℝ}, MeasureTheory.Integrable w μ → (∀ (A : Set α), MeasurableSet A → (ν A).toReal = ∫ (x : α) in A, w x ∂μ) → ν = μ.withDensity fun x => ENNReal.ofReal (w x)","l":27,"f":"Causalean/Mathlib/MeasureTheory/SetIntegralRecovery.lean"},{"n":"Causalean.Mathlib.MeasureTheory.not_four_distinct_in_support","k":"theorem","m":"Causalean.Mathlib.MeasureTheory.MomentSliceSupport","a":"Mathlib/MeasureTheory","t":2,"d":"An extreme probability measure in the two-moment slice cannot have four distinct support points. Four separated neighborhoods would give a nonzero signed perturbation preserving mass, mean, and second moment, so the measure would be the midpoint of two different slice members.","s":"∀ {a b s : ℝ} {μ : MeasureTheory.Measure ℝ}, Causalean.Mathlib.MeasureTheory.IsExtremePoint (Causalean.Mathlib.MeasureTheory.MomentSlice a b s) μ → ∀ (x : Fin 4 → ℝ), Function.Injective x → (∀ (i : Fin 4), x i ∈ μ.support) → False","l":306,"f":"Causalean/Mathlib/MeasureTheory/MomentSliceSupport.lean"},{"n":"Causalean.Mathlib.MeasureTheory.partition_restrict_absolutelyContinuous","k":"theorem","m":"Causalean.Mathlib.MeasureTheory.PartitionRnDeriv","a":"Mathlib/MeasureTheory","t":1,"d":"Absolute continuity from a finite proportional partition. Suppose each cell s i of a finite family is measurable, the cells are pairwise disjoint, and the cells cover the whole ambient space — together, (s i) is a finite measurable partition — and suppose on every cell the numerator measure μ restricted to that cell equals the denominator measure ν restricted to the same cell, scaled by the constant r i. Then μ is absolutely continuous with respect to ν. The global density is the simple function whose value on cell i is r i, so μ = ν.withDensity d.","s":"∀ {Ω : Type u_1} [inst : MeasurableSpace Ω] {ι : Type u_2} [Finite ι] (μ ν : MeasureTheory.Measure Ω) (s : ι → Set Ω) (r : ι → ENNReal), (∀ (i : ι), MeasurableSet (s i)) → Pairwise (Function.onFun Disjoint s) → ⋃ i, s i = Set.univ → (∀ (i : ι), μ.restrict (s i) = r i • ν.restrict (s i)) → μ.AbsolutelyContinuous ν","l":28,"f":"Causalean/Mathlib/MeasureTheory/PartitionRnDeriv.lean"},{"n":"Causalean.Mathlib.MeasureTheory.partition_restrict_integrable_pow_rnDeriv","k":"theorem","m":"Causalean.Mathlib.MeasureTheory.PartitionRnDeriv","a":"Mathlib/MeasureTheory","t":1,"d":"Power-deviation integrability from a finite proportional partition. Under the same partition hypotheses as partition_restrict_absolutelyContinuous — the cells s i are measurable, pairwise disjoint, and cover the ambient space — and again assuming on every cell the numerator measure μ restricted to that cell equals the denominator measure ν restricted to the same cell, scaled by the constant r i, with ν finite, then for any natural number n, the n-th power of the deviation of the Radon–Nikodym derivative dμ/dν from 1 is integrable against ν. On each cell the derivative equals r i, so the function is a finite simple function and the integral is a finite sum of per-cell constants.","s":"∀ {Ω : Type u_1} [inst : MeasurableSpace Ω] {ι : Type u_2} [Finite ι] (μ ν : MeasureTheory.Measure Ω) [MeasureTheory.IsFiniteMeasure ν] (s : ι → Set Ω) (r : ι → ENNReal) (n : ℕ), (∀ (i : ι), MeasurableSet (s i)) → Pairwise (Function.onFun Disjoint s) → ⋃ i, s i = Set.univ → (∀ (i : ι), μ.restrict (s i) = r i • ν.restrict (s i)) → MeasureTheory.Integrable (fun x => ((μ.rnDeriv ν x).toReal - 1) ^ n) ν","l":83,"f":"Causalean/Mathlib/MeasureTheory/PartitionRnDeriv.lean"},{"n":"Causalean.Mathlib.MeasureTheory.property_at_of_ae_of_singleton_pos","k":"theorem","m":"Causalean.Mathlib.MeasureTheory.FiniteAtomicMeasure","a":"Mathlib/MeasureTheory","t":2,"d":"An almost-sure property holds at every point to which the measure assigns nonzero mass.","s":"∀ {𝒳 : Type u_1} [inst : MeasurableSpace 𝒳] (μ : MeasureTheory.Measure 𝒳) (p : 𝒳 → Prop) {x : 𝒳}, (∀ᵐ (y : 𝒳) ∂μ, p y) → μ {x} ≠ 0 → p x","l":78,"f":"Causalean/Mathlib/MeasureTheory/FiniteAtomicMeasure.lean"},{"n":"Causalean.Mathlib.MeasureTheory.sSup_image_eq_of_dense_tendsto","k":"theorem","m":"Causalean.Mathlib.MeasureTheory.SupCountableDense","a":"Mathlib/MeasureTheory","t":2,"d":"If F is bounded above on S, D ⊆ S, and every x ∈ S is the limit along a D-valued sequence of F-values (F (seq j) → F x), then the supremum of F over S coincides with the supremum over the countable skeleton D. This is the density-side input that turns a supremum over an uncountable class into a supremum over a countable skeleton (used both for measurability and for evaluating the supremum).","s":"∀ {ι : Type u_1} (F : ι → ℝ) (S D : Set ι), D ⊆ S → BddAbove (F '' S) → (∀ x ∈ S, ∃ seq, (∀ (j : ℕ), seq j ∈ D) ∧ Filter.Tendsto (fun j => F (seq j)) Filter.atTop (nhds (F x))) → sSup (F '' S) = sSup (F '' D)","l":41,"f":"Causalean/Mathlib/MeasureTheory/SupCountableDense.lean"},{"n":"Causalean.Mathlib.MeasureTheory.support_finite_ncard_le_three_of_isExtremePoint","k":"theorem","m":"Causalean.Mathlib.MeasureTheory.MomentSliceSupport","a":"Mathlib/MeasureTheory","t":2,"d":"An extreme probability measure in the two-moment slice has a finite topological support with at most three points.","s":"∀ {a b s : ℝ} {μ : MeasureTheory.Measure ℝ}, Causalean.Mathlib.MeasureTheory.IsExtremePoint (Causalean.Mathlib.MeasureTheory.MomentSlice a b s) μ → μ.support.Finite ∧ μ.support.ncard ≤ 3","l":599,"f":"Causalean/Mathlib/MeasureTheory/MomentSliceSupport.lean"},{"n":"Causalean.Mathlib.OffsetPeeling.max_two_split","k":"theorem","m":"Causalean.Mathlib.Analysis.OffsetPeeling","a":"Mathlib/Analysis","t":2,"d":"Split a clipped two-term offset across its summands: max 0 (a − 2r) ≤ max 0 (b − r) + max 0 (c − r) when a ≤ b + c.","s":"∀ (a b c r : ℝ), a ≤ b + c → max 0 (a - 2 * r) ≤ max 0 (b - r) + max 0 (c - r)","l":151,"f":"Causalean/Mathlib/Analysis/OffsetPeeling.lean"},{"n":"Causalean.Mathlib.OffsetPeeling.offsetPeelingConstant","k":"def","m":"Causalean.Mathlib.Analysis.OffsetPeeling","a":"Mathlib/Analysis","t":1,"d":"Offset-peeling constant for the c = 1/4 offset: (1 − θ)·(4θ)^{θ/(1−θ)}.","s":"ℝ → ℝ","l":22,"f":"Causalean/Mathlib/Analysis/OffsetPeeling.lean"},{"n":"Causalean.Mathlib.OffsetPeeling.offsetPeelingConstantC","k":"def","m":"Causalean.Mathlib.Analysis.OffsetPeeling","a":"Mathlib/Analysis","t":1,"d":"Peeling constant with a general offset coefficient c: (1 − θ)·(θ/c)^{θ/(1−θ)} (the c = 1/4 case is offsetPeelingConstant).","s":"ℝ → ℝ → ℝ","l":27,"f":"Causalean/Mathlib/Analysis/OffsetPeeling.lean"},{"n":"Causalean.Mathlib.OffsetPeeling.offsetPeelingConstantC_nonneg","k":"theorem","m":"Causalean.Mathlib.Analysis.OffsetPeeling","a":"Mathlib/Analysis","t":2,"d":"The general offset-peeling constant is nonnegative when the offset is positive and the exponent lies between zero and one.","s":"∀ (c θ : ℝ), 0 ≤ c → 0 ≤ θ → θ ≤ 1 → 0 ≤ Causalean.Mathlib.OffsetPeeling.offsetPeelingConstantC c θ","l":88,"f":"Causalean/Mathlib/Analysis/OffsetPeeling.lean"},{"n":"Causalean.Mathlib.OffsetPeeling.offsetPeelingConstant_nonneg","k":"theorem","m":"Causalean.Mathlib.Analysis.OffsetPeeling","a":"Mathlib/Analysis","t":2,"d":"The offset-peeling constant for the quarter-offset case is nonnegative when the exponent lies between zero and one.","s":"∀ (θ : ℝ), 0 ≤ θ → θ ≤ 1 → 0 ≤ Causalean.Mathlib.OffsetPeeling.offsetPeelingConstant θ","l":117,"f":"Causalean/Mathlib/Analysis/OffsetPeeling.lean"},{"n":"Causalean.Mathlib.OffsetPeeling.offset_peeling","k":"theorem","m":"Causalean.Mathlib.Analysis.OffsetPeeling","a":"Mathlib/Analysis","t":1,"d":"Deterministic Young/AM-GM peeling for the quarter-offset. For an exponent strictly between zero and one and nonnegative coefficients a and t, the positive part of a·t^θ − t/4 is at most the offset-peeling constant times a raised to the power 1/(1−θ).","s":"∀ (θ a t : ℝ), 0 < θ → θ < 1 → 0 ≤ a → 0 ≤ t → max 0 (a * t ^ θ - 1 / 4 * t) ≤ Causalean.Mathlib.OffsetPeeling.offsetPeelingConstant θ * a ^ (1 / (1 - θ))","l":125,"f":"Causalean/Mathlib/Analysis/OffsetPeeling.lean"},{"n":"Causalean.Mathlib.OffsetPeeling.offset_peeling_coeff","k":"theorem","m":"Causalean.Mathlib.Analysis.OffsetPeeling","a":"Mathlib/Analysis","t":2,"d":"Generalized offset peeling at coefficient c: for 0 < c, 0 < θ < 1, max 0 (a·t^θ − c·t) ≤ offsetPeelingConstantC c θ · a^{1/(1−θ)} (weighted AM-GM).","s":"∀ (c θ a t : ℝ), 0 < c → 0 < θ → θ < 1 → 0 ≤ a → 0 ≤ t → max 0 (a * t ^ θ - c * t) ≤ Causalean.Mathlib.OffsetPeeling.offsetPeelingConstantC c θ * a ^ (1 / (1 - θ))","l":32,"f":"Causalean/Mathlib/Analysis/OffsetPeeling.lean"},{"n":"Causalean.Mathlib.OffsetPeeling.offset_peeling_coeff_nonneg_theta","k":"theorem","m":"Causalean.Mathlib.Analysis.OffsetPeeling","a":"Mathlib/Analysis","t":2,"d":"The general offset-peeling bound remains valid at exponent zero by the same constant formula.","s":"∀ (c θ a t : ℝ), 0 < c → 0 ≤ θ → θ < 1 → 0 ≤ a → 0 ≤ t → max 0 (a * t ^ θ - c * t) ≤ Causalean.Mathlib.OffsetPeeling.offsetPeelingConstantC c θ * a ^ (1 / (1 - θ))","l":98,"f":"Causalean/Mathlib/Analysis/OffsetPeeling.lean"},{"n":"Causalean.Mathlib.OffsetPeeling.offset_peeling_nonneg_theta","k":"theorem","m":"Causalean.Mathlib.Analysis.OffsetPeeling","a":"Mathlib/Analysis","t":2,"d":"The quarter-offset peeling bound remains valid at exponent zero by continuity of the constant formula.","s":"∀ (θ a t : ℝ), 0 ≤ θ → θ < 1 → 0 ≤ a → 0 ≤ t → max 0 (a * t ^ θ - 1 / 4 * t) ≤ Causalean.Mathlib.OffsetPeeling.offsetPeelingConstant θ * a ^ (1 / (1 - θ))","l":140,"f":"Causalean/Mathlib/Analysis/OffsetPeeling.lean"},{"n":"Causalean.Mathlib.Optimization.InSimplex","k":"def","m":"Causalean.Mathlib.Optimization.SimplexActiveSetDefs","a":"Mathlib/Optimization","t":1,"d":"Membership in the scaled 3-point simplex Δ_M = {t ≥ 0 : Σ tᵢ = M}.","s":"ℝ → (Fin 3 → ℝ) → Prop","l":28,"f":"Causalean/Mathlib/Optimization/SimplexActiveSetDefs.lean"},{"n":"Causalean.Mathlib.Optimization.InTruncSimplex","k":"def","m":"Causalean.Mathlib.Optimization.SimplexTruncationDefs","a":"Mathlib/Optimization","t":1,"d":"Membership in the parity-truncated simplex K_d = {t ∈ Δ_M : t_y + t_z ≥ d}.","s":"ℝ → ℝ → (Fin 3 → ℝ) → Prop","l":20,"f":"Causalean/Mathlib/Optimization/SimplexTruncationDefs.lean"},{"n":"Causalean.Mathlib.Optimization.IsAdmissibleSupport","k":"def","m":"Causalean.Mathlib.Optimization.SimplexActiveSetDefs","a":"Mathlib/Optimization","t":1,"d":"KKT-admissible support/multiplier data for the SOCP: a nonempty support S ⊆ {x,y,z} and multiplier λ with Σ_{i∈S} (λ−αᵢ)²/βᵢ = κ², the strict activity λ > αᵢ on S, and the inactivity λ ≤ αⱼ off S.","s":"(Fin 3 → ℝ) → (Fin 3 → ℝ) → ℝ → Finset (Fin 3) → ℝ → Prop","l":32,"f":"Causalean/Mathlib/Optimization/SimplexActiveSetDefs.lean"},{"n":"Causalean.Mathlib.Optimization.activeSetPoint","k":"def","m":"Causalean.Mathlib.Optimization.SimplexActiveSetDefs","a":"Mathlib/Optimization","t":1,"d":"The active-set coordinate formula tᵢ = M ((λ−αᵢ)/βᵢ) / Σ_{h∈S} ((λ−αₕ)/βₕ) on S, and tᵢ = 0 off S.","s":"ℝ → (Fin 3 → ℝ) → (Fin 3 → ℝ) → Finset (Fin 3) → ℝ → Fin 3 → ℝ","l":42,"f":"Causalean/Mathlib/Optimization/SimplexActiveSetDefs.lean"},{"n":"Causalean.Mathlib.Optimization.activeSetPoint_isMinimizer","k":"theorem","m":"Causalean.Mathlib.Optimization.SimplexTruncationMinimizers","a":"Mathlib/Optimization","t":1,"d":"KKT admissible ⟹ global minimizer (κ > 0). For a positive total mass M, positive coordinate weights β, a positive SOCP scale κ, and a KKT-admissible support/multiplier pair (S, λ): a nonempty index set S with $\\sum_{i\\in S}(\\lambda-\\alpha_i)^2/\\beta_i = \\kappa^2$, strict activity $\\lambda>\\alpha_i$ on S, and inactivity $\\lambda\\le\\alpha_j$ off S, the induced active-set point lies in the simplex $\\{t\\ge0:\\sum t_i=M\\}$ and globally minimizes the objective $\\sum\\alpha_it_i+\\kappa\\sqrt{\\sum\\beta_it_i^2}$ over that simplex.","s":"∀ (M : ℝ), 0 < M → ∀ (α β : Fin 3 → ℝ) (kappa : ℝ), (∀ (i : Fin 3), 0 < β i) → 0 < kappa → ∀ (S : Finset (Fin 3)) (lam : ℝ), Causalean.Mathlib.Optimization.IsAdmissibleSupport α β kappa S lam → Causalean.Mathlib.Optimization.InSimplex M (Causalean.Mathlib.Optimization.activeSetPoint M α β S lam) ∧ ∀ (s : Fin 3 → ℝ), Causalean.Mathlib.Optimization.InSimplex M s → Causalean.Mathlib.Optimization.wsObj α β kappa (Causalean.Mathlib.Optimization.activeSetPoint M α β S lam) ≤ Causalean.Mathlib.Optimization.wsObj α β kappa s","l":30,"f":"Causalean/Mathlib/Optimization/SimplexTruncationMinimizers.lean"},{"n":"Causalean.Mathlib.Optimization.activeSetPoint_mem","k":"theorem","m":"Causalean.Mathlib.Optimization.WeightedSimplexKKT","a":"Mathlib/Optimization","t":2,"d":"The active-set point lies in the simplex Δ_M.","s":"∀ (M : ℝ) (α β : Fin 3 → ℝ) (S : Finset (Fin 3)) (lam : ℝ), 0 ≤ M → (∀ i ∈ S, 0 < β i) → S.Nonempty → (∀ i ∈ S, α i < lam) → Causalean.Mathlib.Optimization.InSimplex M (Causalean.Mathlib.Optimization.activeSetPoint M α β S lam)","l":45,"f":"Causalean/Mathlib/Optimization/WeightedSimplexKKT.lean"},{"n":"Causalean.Mathlib.Optimization.activeSetPoint_normSq","k":"theorem","m":"Causalean.Mathlib.Optimization.WeightedSimplexKKT","a":"Mathlib/Optimization","t":2,"d":"Weighted squared norm of the active-set point: Σ βᵢ tᵢ² = (Mκ)² / D², using the admissibility identity Σ_{i∈S}(λ−αᵢ)²/βᵢ = κ².","s":"∀ (M : ℝ) (α β : Fin 3 → ℝ) (kappa : ℝ) (S : Finset (Fin 3)) (lam : ℝ), (∀ i ∈ S, 0 < β i) → S.Nonempty → (∀ i ∈ S, α i < lam) → ∑ i ∈ S, (lam - α i) ^ 2 / β i = kappa ^ 2 → ∑ i, β i * Causalean.Mathlib.Optimization.activeSetPoint M α β S lam i ^ 2 = (M * kappa) ^ 2 / (∑ h ∈ S, (lam - α h) / β h) ^ 2","l":70,"f":"Causalean/Mathlib/Optimization/WeightedSimplexKKT.lean"},{"n":"Causalean.Mathlib.Optimization.activeSetPoint_pos_iff","k":"theorem","m":"Causalean.Mathlib.Optimization.WeightedSimplexKKT","a":"Mathlib/Optimization","t":2,"d":"The active-set point has support exactly S: on S its coordinate is positive, off S it is 0.","s":"∀ (M : ℝ) (α β : Fin 3 → ℝ) (S : Finset (Fin 3)) (lam : ℝ), 0 < M → (∀ i ∈ S, 0 < β i) → S.Nonempty → (∀ i ∈ S, α i < lam) → ∀ (i : Fin 3), 0 < Causalean.Mathlib.Optimization.activeSetPoint M α β S lam i ↔ i ∈ S","l":30,"f":"Causalean/Mathlib/Optimization/WeightedSimplexKKT.lean"},{"n":"Causalean.Mathlib.Optimization.activeSetPoint_strict_min","k":"theorem","m":"Causalean.Mathlib.Optimization.WeightedSimplexKKT","a":"Mathlib/Optimization","t":1,"d":"Strict minimality of the active-set point (κ > 0). For a positive total mass M, positive coordinate weights β, a positive SOCP scale κ, and a KKT-admissible support/multiplier pair (S, λ), every other point of the simplex $\\{t \\ge 0 : \\sum t_i = M\\}$ has strictly larger objective value than the induced active-set point.","s":"∀ (M : ℝ) (α β : Fin 3 → ℝ) (kappa : ℝ) (S : Finset (Fin 3)) (lam : ℝ), 0 < M → (∀ (i : Fin 3), 0 < β i) → 0 < kappa → Causalean.Mathlib.Optimization.IsAdmissibleSupport α β kappa S lam → ∀ (s : Fin 3 → ℝ), Causalean.Mathlib.Optimization.InSimplex M s → s ≠ Causalean.Mathlib.Optimization.activeSetPoint M α β S lam → Causalean.Mathlib.Optimization.wsObj α β kappa (Causalean.Mathlib.Optimization.activeSetPoint M α β S lam) < Causalean.Mathlib.Optimization.wsObj α β kappa s","l":156,"f":"Causalean/Mathlib/Optimization/WeightedSimplexKKT.lean"},{"n":"Causalean.Mathlib.Optimization.activeSetPoint_value","k":"theorem","m":"Causalean.Mathlib.Optimization.WeightedSimplexKKT","a":"Mathlib/Optimization","t":1,"d":"Objective value of the active-set point. For a nonnegative total mass M, positive weights on the support S, a positive SOCP scale κ, a nonempty support S, strict activity $\\lambda > \\alpha_i$ on S, and the admissibility identity $\\sum_{i\\in S}(\\lambda-\\alpha_i)^2/\\beta_i=\\kappa^2$, the objective wsObj evaluated at the induced active-set point equals the closed form $M\\lambda$.","s":"∀ (M : ℝ) (α β : Fin 3 → ℝ) (kappa : ℝ) (S : Finset (Fin 3)) (lam : ℝ), 0 ≤ M → (∀ i ∈ S, 0 < β i) → 0 < kappa → S.Nonempty → (∀ i ∈ S, α i < lam) → ∑ i ∈ S, (lam - α i) ^ 2 / β i = kappa ^ 2 → Causalean.Mathlib.Optimization.wsObj α β kappa (Causalean.Mathlib.Optimization.activeSetPoint M α β S lam) = M * lam","l":94,"f":"Causalean/Mathlib/Optimization/WeightedSimplexKKT.lean"},{"n":"Causalean.Mathlib.Optimization.activeSet_denom_pos","k":"theorem","m":"Causalean.Mathlib.Optimization.WeightedSimplexKKT","a":"Mathlib/Optimization","t":2,"d":"The active-set denominator D = Σ_{h∈S} (λ−αₕ)/βₕ is strictly positive: S is nonempty and each summand is positive (λ > αₕ on S, βₕ > 0).","s":"∀ (α β : Fin 3 → ℝ) (S : Finset (Fin 3)) (lam : ℝ), (∀ i ∈ S, 0 < β i) → S.Nonempty → (∀ i ∈ S, α i < lam) → 0 < ∑ h ∈ S, (lam - α h) / β h","l":22,"f":"Causalean/Mathlib/Optimization/WeightedSimplexKKT.lean"},{"n":"Causalean.Mathlib.Optimization.continuous_kktThreshold","k":"theorem","m":"Causalean.Mathlib.Optimization.WeightedSimplexExists","a":"Mathlib/Optimization","t":2,"d":"The active-set threshold function varies continuously with the multiplier because the weights are fixed real coefficients.","s":"∀ (α β : Fin 3 → ℝ), Continuous (Causalean.Mathlib.Optimization.kktThreshold α β)","l":26,"f":"Causalean/Mathlib/Optimization/WeightedSimplexExists.lean"},{"n":"Causalean.Mathlib.Optimization.eq_truncSegPoint_of_simplex_face","k":"theorem","m":"Causalean.Mathlib.Optimization.SimplexTruncationConvex","a":"Mathlib/Optimization","t":2,"d":"A three-coordinate vector with total mass M whose last two coordinates sum to d is the corresponding point on the truncation segment, indexed by its second coordinate.","s":"∀ (M d : ℝ) (x : Fin 3 → ℝ), Causalean.Mathlib.Optimization.InSimplex M x → x 1 + x 2 = d → x = Causalean.Mathlib.Optimization.truncSegPoint M d (x 1)","l":59,"f":"Causalean/Mathlib/Optimization/SimplexTruncationConvex.lean"},{"n":"Causalean.Mathlib.Optimization.exists_admissible","k":"theorem","m":"Causalean.Mathlib.Optimization.WeightedSimplexExists","a":"Mathlib/Optimization","t":1,"d":"Existence of an admissible support/multiplier pair (κ > 0). For positive coordinate weights β and a positive SOCP scale κ, there exists a nonempty index set S and a multiplier λ such that $\\sum_{i\\in S}(\\lambda-\\alpha_i)^2/\\beta_i=\\kappa^2$, with $\\lambda>\\alpha_i$ on S and $\\lambda\\le\\alpha_j$ off S.","s":"∀ (α β : Fin 3 → ℝ) (kappa : ℝ), (∀ (i : Fin 3), 0 < β i) → 0 < kappa → ∃ S lam, Causalean.Mathlib.Optimization.IsAdmissibleSupport α β kappa S lam","l":76,"f":"Causalean/Mathlib/Optimization/WeightedSimplexExists.lean"},{"n":"Causalean.Mathlib.Optimization.exposedMinFace","k":"def","m":"Causalean.Mathlib.Optimization.SimplexActiveSetDefs","a":"Mathlib/Optimization","t":1,"d":"The κ = 0 exposed face conv{M eᵢ : αᵢ = minⱼ αⱼ} of Δ_M: the simplex points supported only on the α-minimizing coordinates.","s":"ℝ → (Fin 3 → ℝ) → Set (Fin 3 → ℝ)","l":49,"f":"Causalean/Mathlib/Optimization/SimplexActiveSetDefs.lean"},{"n":"Causalean.Mathlib.Optimization.exposedMinFace_isMinimizer","k":"theorem","m":"Causalean.Mathlib.Optimization.SimplexTruncationMinimizers","a":"Mathlib/Optimization","t":1,"d":"Exposed face ⟹ global minimizer (κ = 0). If t_rel lies in the simplex $\\{t \\ge 0 : \\sum t_i = M\\}$ and every coordinate at which it is nonzero attains the minimum value of α, then t_rel globally minimizes the purely linear objective $\\sum \\alpha_i t_i$ — wsObj at $\\kappa = 0$ — over the whole simplex.","s":"∀ (M : ℝ) (α β t_rel : Fin 3 → ℝ), t_rel ∈ Causalean.Mathlib.Optimization.exposedMinFace M α → ∀ (s : Fin 3 → ℝ), Causalean.Mathlib.Optimization.InSimplex M s → Causalean.Mathlib.Optimization.wsObj α β 0 t_rel ≤ Causalean.Mathlib.Optimization.wsObj α β 0 s","l":275,"f":"Causalean/Mathlib/Optimization/SimplexTruncationMinimizers.lean"},{"n":"Causalean.Mathlib.Optimization.kappa_zero_face","k":"theorem","m":"Causalean.Mathlib.Optimization.WeightedSimplexFace","a":"Mathlib/Optimization","t":1,"d":"The κ = 0 minimizer set is the exposed α-minimizing face. For a simplex budget M, a linear weight vector α, and a candidate point t, t lies in the simplex of budget M and minimizes the linear objective Σᵢ αᵢtᵢ over that simplex if and only if t belongs to the exposed α-minimizing face of the simplex.","s":"∀ (M : ℝ) (α t : Fin 3 → ℝ), (Causalean.Mathlib.Optimization.InSimplex M t ∧ ∀ (s : Fin 3 → ℝ), Causalean.Mathlib.Optimization.InSimplex M s → ∑ i, α i * t i ≤ ∑ i, α i * s i) ↔ t ∈ Causalean.Mathlib.Optimization.exposedMinFace M α","l":23,"f":"Causalean/Mathlib/Optimization/WeightedSimplexFace.lean"},{"n":"Causalean.Mathlib.Optimization.kktThreshold","k":"def","m":"Causalean.Mathlib.Optimization.WeightedSimplexExists","a":"Mathlib/Optimization","t":1,"d":"The threshold function G(λ) = Σᵢ (max(λ − αᵢ, 0))² / βᵢ whose level set G(λ) = κ² selects the active support.","s":"(Fin 3 → ℝ) → (Fin 3 → ℝ) → ℝ → ℝ","l":21,"f":"Causalean/Mathlib/Optimization/WeightedSimplexExists.lean"},{"n":"Causalean.Mathlib.Optimization.kktThreshold_eq_zero_of_le","k":"theorem","m":"Causalean.Mathlib.Optimization.WeightedSimplexExists","a":"Mathlib/Optimization","t":2,"d":"The active-set threshold is zero when the multiplier is no larger than every coefficient.","s":"∀ (α β : Fin 3 → ℝ) (lam : ℝ), (∀ (i : Fin 3), lam ≤ α i) → Causalean.Mathlib.Optimization.kktThreshold α β lam = 0","l":42,"f":"Causalean/Mathlib/Optimization/WeightedSimplexExists.lean"},{"n":"Causalean.Mathlib.Optimization.kktThreshold_nonneg","k":"theorem","m":"Causalean.Mathlib.Optimization.WeightedSimplexExists","a":"Mathlib/Optimization","t":2,"d":"The active-set threshold is nonnegative when every coordinate has a nonnegative weight, because it sums squared multiplier gaps divided by those weights.","s":"∀ (α β : Fin 3 → ℝ), (∀ (i : Fin 3), 0 ≤ β i) → ∀ (lam : ℝ), 0 ≤ Causalean.Mathlib.Optimization.kktThreshold α β lam","l":33,"f":"Causalean/Mathlib/Optimization/WeightedSimplexExists.lean"},{"n":"Causalean.Mathlib.Optimization.support_sum_eq_kktThreshold","k":"theorem","m":"Causalean.Mathlib.Optimization.WeightedSimplexExists","a":"Mathlib/Optimization","t":2,"d":"Summing squared multiplier gaps only over coordinates whose coefficients lie below the multiplier gives exactly the active-set threshold.","s":"∀ (α β : Fin 3 → ℝ) (lam : ℝ), ∑ i with α i < lam, (lam - α i) ^ 2 / β i = Causalean.Mathlib.Optimization.kktThreshold α β lam","l":52,"f":"Causalean/Mathlib/Optimization/WeightedSimplexExists.lean"},{"n":"Causalean.Mathlib.Optimization.truncSegPoint","k":"def","m":"Causalean.Mathlib.Optimization.SimplexTruncationDefs","a":"Mathlib/Optimization","t":1,"d":"The boundary segment H_d: the point with t_x = M − d, t_y = s, t_z = d − s.","s":"ℝ → ℝ → ℝ → Fin 3 → ℝ","l":24,"f":"Causalean/Mathlib/Optimization/SimplexTruncationDefs.lean"},{"n":"Causalean.Mathlib.Optimization.truncSeg_cs_sqrt","k":"theorem","m":"Causalean.Mathlib.Optimization.SimplexTruncationSlice","a":"Mathlib/Optimization","t":2,"d":"For a nonnegative baseline component, the inner product of two three-dimensional boundary vectors is no greater than the product of their Euclidean norms.","s":"∀ {A d s σ : ℝ}, 0 ≤ A → A + s * σ + (d - s) * (d - σ) ≤ √(A + s ^ 2 + (d - s) ^ 2) * √(A + σ ^ 2 + (d - σ) ^ 2)","l":29,"f":"Causalean/Mathlib/Optimization/SimplexTruncationSlice.lean"},{"n":"Causalean.Mathlib.Optimization.truncSeg_reduction","k":"theorem","m":"Causalean.Mathlib.Optimization.SimplexTruncationConvex","a":"Mathlib/Optimization","t":1,"d":"Boundary reduction onto the truncation face. Fix nonnegative coordinate weights β and a nonnegative SOCP scale κ. If t_rel lies in the three-coordinate simplex $\\{t \\ge 0 : \\sum t_i = M\\}$ and globally minimizes the objective $\\sum \\alpha_i t_i + \\kappa\\sqrt{\\sum \\beta_i t_i^2}$ over that simplex, while t_rel fails the parity cut defining the truncated sub-simplex, since its last two coordinates sum to strictly less than $d$, then for every point t of the truncated sub-simplex $\\{t \\in \\Delta_M : t_y + t_z \\ge d\\}$, there is some $\\sigma$ between $0$ and $d$ such that the boundary point $(M-d,\\sigma,d-\\sigma)$ of the truncation face attains an objective value no larger than the objective at t.","s":"∀ (M d kappa : ℝ) (α β : Fin 3 → ℝ), (∀ (i : Fin 3), 0 ≤ β i) → 0 ≤ kappa → ∀ (t_rel : Fin 3 → ℝ), Causalean.Mathlib.Optimization.InSimplex M t_rel → (∀ (s : Fin 3 → ℝ), Causalean.Mathlib.Optimization.InSimplex M s → Causalean.Mathlib.Optimization.wsObj α β kappa t_rel ≤ Causalean.Mathlib.Optimization.wsObj α β kappa s) → t_rel 1 + t_rel 2 < d → ∀ (t : Fin 3 → ℝ), Causalean.Mathlib.Optimization.InTruncSimplex M d t → ∃ σ, 0 ≤ σ ∧ σ ≤ d ∧ Causalean.Mathlib.Optimization.wsObj α β kappa (Causalean.Mathlib.Optimization.truncSegPoint M d σ) ≤ Causalean.Mathlib.Optimization.wsObj α β kappa t","l":109,"f":"Causalean/Mathlib/Optimization/SimplexTruncationConvex.lean"},{"n":"Causalean.Mathlib.Optimization.truncSeg_selector_le","k":"theorem","m":"Causalean.Mathlib.Optimization.SimplexTruncationSlice","a":"Mathlib/Optimization","t":1,"d":"The selector minimizes the boundary slice. Suppose the truncation width $d$ is positive, the first-coordinate weight $\\beta_0$ is nonnegative while the other two weights both equal 1, and the SOCP scale $\\kappa$ is nonnegative. Then for every offset $\\sigma$ between $0$ and $d$, the objective wsObj evaluated at the boundary point truncSegPoint M d applied to the selector truncSelector M d α β κ is at most its value at the boundary point for $\\sigma$.","s":"∀ (M d : ℝ) (α β : Fin 3 → ℝ) (kappa : ℝ), 0 < d → 0 ≤ β 0 → β 1 = 1 → β 2 = 1 → 0 ≤ kappa → ∀ (σ : ℝ), 0 ≤ σ → σ ≤ d → Causalean.Mathlib.Optimization.wsObj α β kappa (Causalean.Mathlib.Optimization.truncSegPoint M d (Causalean.Mathlib.Optimization.truncSelector M d α β kappa)) ≤ Causalean.Mathlib.Optimization.wsObj α β kappa (Causalean.Mathlib.Optimization.truncSegPoint M d σ)","l":295,"f":"Causalean/Mathlib/Optimization/SimplexTruncationSlice.lean"},{"n":"Causalean.Mathlib.Optimization.truncSelector","k":"def","m":"Causalean.Mathlib.Optimization.SimplexTruncationDefs","a":"Mathlib/Optimization","t":1,"d":"The endpoint/interior selector s⋆ on H_d, with δ = α_y − α_z and A = β_x (M − d)²: s⋆ = 0 if δ ≥ κ d / √(A + d²), s⋆ = d if δ ≤ − κ d / √(A + d²), and otherwise the interior root s⋆ = (d − δ √((A + d²/2)/(κ² − δ²/2)))/2. At κ = 0 the two guards collapse to 0 ≤ δ and δ ≤ 0, recovering the κ = 0 endpoint rule s⋆ = 0 if δ ≥ 0, s⋆ = d if δ ≤ 0.","s":"ℝ → ℝ → (Fin 3 → ℝ) → (Fin 3 → ℝ) → ℝ → ℝ","l":27,"f":"Causalean/Mathlib/Optimization/SimplexTruncationDefs.lean"},{"n":"Causalean.Mathlib.Optimization.truncSelector_interior_den_pos","k":"theorem","m":"Causalean.Mathlib.Optimization.SimplexTruncationSlice","a":"Mathlib/Optimization","t":2,"d":"Under the selector's strict interior guard inequalities, a positive interval width, and nonnegative baseline and scale, the squared-scale denominator minus half the squared offset is positive.","s":"∀ {A d kappa δ : ℝ}, 0 < d → 0 ≤ A → 0 ≤ kappa → ¬kappa * d / √(A + d ^ 2) ≤ δ → ¬δ ≤ -(kappa * d / √(A + d ^ 2)) → 0 < kappa ^ 2 - δ ^ 2 / 2","l":95,"f":"Causalean/Mathlib/Optimization/SimplexTruncationSlice.lean"},{"n":"Causalean.Mathlib.Optimization.truncSelector_interior_sq_bound","k":"theorem","m":"Causalean.Mathlib.Optimization.SimplexTruncationSlice","a":"Mathlib/Optimization","t":2,"d":"If the truncation offset lies strictly between the two interior guard bounds, then its squared size times the radicand is strictly smaller than the squared truncation scale times the squared interval width.","s":"∀ {A d kappa δ : ℝ}, 0 < d → 0 ≤ A → 0 ≤ kappa → ¬kappa * d / √(A + d ^ 2) ≤ δ → ¬δ ≤ -(kappa * d / √(A + d ^ 2)) → δ ^ 2 * (A + d ^ 2) < kappa ^ 2 * d ^ 2","l":67,"f":"Causalean/Mathlib/Optimization/SimplexTruncationSlice.lean"},{"n":"Causalean.Mathlib.Optimization.truncSelector_mem","k":"theorem","m":"Causalean.Mathlib.Optimization.SimplexTruncationSlice","a":"Mathlib/Optimization","t":2,"d":"The selector lands in [0,d]. The endpoint/interior selector truncSelector satisfies 0 ≤ s⋆ ≤ d: the two endpoint branches give 0 and d directly, and in the interior branch the guard failures |δ| < κ d / √(A + d²) force δ²(A + d²) < κ² d², whence |s⋆ − d/2| < d/2.","s":"∀ (M d : ℝ) (α β : Fin 3 → ℝ) (kappa : ℝ), 0 < d → 0 ≤ β 0 → 0 ≤ kappa → 0 ≤ Causalean.Mathlib.Optimization.truncSelector M d α β kappa ∧ Causalean.Mathlib.Optimization.truncSelector M d α β kappa ≤ d","l":276,"f":"Causalean/Mathlib/Optimization/SimplexTruncationSlice.lean"},{"n":"Causalean.Mathlib.Optimization.trunc_from_minimizer","k":"theorem","m":"Causalean.Mathlib.Optimization.SimplexTruncation","a":"Mathlib/Optimization","t":1,"d":"Truncation dichotomy from a relaxed minimizer. Fix a truncation threshold d at most the total simplex mass M, a linear weighting α, coordinate weights β with its zeroth entry nonnegative and its remaining two entries fixed equal to 1, and a nonnegative regularization parameter κ. Given a point t_rel of the simplex Δ_M that globally minimizes the weighted objective wsObj over Δ_M, then the constrained problem over the truncated simplex K_d = {t ∈ Δ_M : t₁ + t₂ ≥ d} splits into two cases: if t_rel already satisfies the truncation constraint, it remains a global minimizer over K_d; otherwise, the face selector point truncSegPoint M d sStar is feasible for K_d and is a global minimizer over K_d. This is the κ-agnostic core shared by the κ > 0 and κ = 0 branches of the headline lemma.","s":"∀ (M d : ℝ), d ≤ M → ∀ (α β : Fin 3 → ℝ) (kappa : ℝ), 0 ≤ β 0 → β 1 = 1 → β 2 = 1 → 0 ≤ kappa → ∀ (t_rel : Fin 3 → ℝ), Causalean.Mathlib.Optimization.InSimplex M t_rel → (∀ (s : Fin 3 → ℝ), Causalean.Mathlib.Optimization.InSimplex M s → Causalean.Mathlib.Optimization.wsObj α β kappa t_rel ≤ Causalean.Mathlib.Optimization.wsObj α β kappa s) → (Causalean.Mathlib.Optimization.InTruncSimplex M d t_rel → ∀ (s : Fin 3 → ℝ), Causalean.Mathlib.Optimization.InTruncSimplex M d s → Causalean.Mathlib.Optimization.wsObj α β kappa t_rel ≤ Causalean.Mathlib.Optimization.wsObj α β kappa s) ∧ (¬d ≤ t_rel 1 + t_rel 2 → Causalean.Mathlib.Optimization.InTruncSimplex M d (Causalean.Mathlib.Optimization.truncSegPoint M d (Causalean.Mathlib.Optimization.truncSelector M d α β kappa)) ∧ ∀ (s : Fin 3 → ℝ), Causalean.Mathlib.Optimization.InTruncSimplex M d s → Causalean.Mathlib.Optimization.wsObj α β kappa (Causalean.Mathlib.Optimization.truncSegPoint M d (Causalean.Mathlib.Optimization.truncSelector M d α β kappa)) ≤ Causalean.Mathlib.Optimization.wsObj α β kappa s)","l":31,"f":"Causalean/Mathlib/Optimization/SimplexTruncation.lean"},{"n":"Causalean.Mathlib.Optimization.weighted_cs_simplex_strict","k":"theorem","m":"Causalean.Mathlib.Optimization.WeightedSimplexCS","a":"Mathlib/Optimization","t":1,"d":"Strict weighted Cauchy–Schwarz on the simplex slice. For a nonzero total mass M and positive coordinate weights β, if the coordinates of $s$ sum to $M$, the coordinates of $t$ sum to $M$, and $s$ and $t$ are distinct vectors, then the weighted inner product $\\sum \\beta_i s_i t_i$ is strictly less than the product of the weighted Euclidean norms $\\sqrt{\\sum \\beta_i s_i^2} \\cdot \\sqrt{\\sum \\beta_i t_i^2}$.","s":"∀ (M : ℝ), M ≠ 0 → ∀ (β s t : Fin 3 → ℝ), (∀ (i : Fin 3), 0 < β i) → ∑ i, s i = M → ∑ i, t i = M → s ≠ t → ∑ i, β i * (s i * t i) < √(∑ i, β i * s i ^ 2) * √(∑ i, β i * t i ^ 2)","l":39,"f":"Causalean/Mathlib/Optimization/WeightedSimplexCS.lean"},{"n":"Causalean.Mathlib.Optimization.weighted_cs_sq","k":"theorem","m":"Causalean.Mathlib.Optimization.WeightedSimplexCS","a":"Mathlib/Optimization","t":1,"d":"Weighted Cauchy–Schwarz (squared form). For finitely many coordinates weighted by nonnegative masses β, the square of the weighted inner product $\\sum \\beta_i s_i t_i$ of two vectors $s$ and $t$ is at most the product of their weighted sums of squares $(\\sum \\beta_i s_i^2)(\\sum \\beta_i t_i^2)$.","s":"∀ {ι : Type u_1} [inst : Fintype ι] (β s t : ι → ℝ), (∀ (i : ι), 0 ≤ β i) → (∑ i, β i * (s i * t i)) ^ 2 ≤ (∑ i, β i * s i ^ 2) * ∑ i, β i * t i ^ 2","l":26,"f":"Causalean/Mathlib/Optimization/WeightedSimplexCS.lean"},{"n":"Causalean.Mathlib.Optimization.weighted_simplex_active_set","k":"theorem","m":"Causalean.Mathlib.Optimization.SimplexActiveSet","a":"Mathlib/Optimization","t":1,"d":"Weighted-simplex active-set solution. Fix a positive total simplex mass M, linear weights α, coordinate weights β that are all strictly positive, and a nonnegative regularization parameter κ, and consider minimizing the second-order-cone objective Σ αᵢtᵢ + κ·√(Σβᵢtᵢ²) over the three-point simplex Δ_M of total mass M. Then the following two facts hold: whenever κ is strictly positive, there is a *unique* admissible support/multiplier pair (S, λ), its induced active-set point lies in Δ_M and is the *unique* global minimizer of the objective, with optimal value the closed form M · λ; and whenever κ equals zero, a point of Δ_M minimizes the objective exactly when it lies on the exposed α-minimizing face. This encodes the displayed KKT coordinate formula, the uniqueness of the admissible support, the value formula, and the κ = 0 face clause.","s":"∀ (M : ℝ), 0 < M → ∀ (α β : Fin 3 → ℝ) (kappa : ℝ), (∀ (i : Fin 3), 0 < β i) → 0 ≤ kappa → (0 < kappa → ∃! p, Causalean.Mathlib.Optimization.IsAdmissibleSupport α β kappa p.1 p.2 ∧ Causalean.Mathlib.Optimization.InSimplex M (Causalean.Mathlib.Optimization.activeSetPoint M α β p.1 p.2) ∧ (∀ (s : Fin 3 → ℝ), Causalean.Mathlib.Optimization.InSimplex M s → s ≠ Causalean.Mathlib.Optimization.activeSetPoint M α β p.1 p.2 → Causalean.Mathlib.Optimization.wsObj α β kappa (Causalean.Mathlib.Optimization.activeSetPoint M α β p.1 p.2) < Causalean.Mathlib.Optimization.wsObj α β kappa s) ∧ Causalean.Mathlib.Optimization.wsObj α β kappa (Causalean.Mathlib.Optimization.activeSetPoint M α β p.1 p.2) = M * p.2) ∧ (kappa = 0 → ∀ (t : Fin 3 → ℝ), (Causalean.Mathlib.Optimization.InSimplex M t ∧ ∀ (s : Fin 3 → ℝ), Causalean.Mathlib.Optimization.InSimplex M s → Causalean.Mathlib.Optimization.wsObj α β kappa t ≤ Causalean.Mathlib.Optimization.wsObj α β kappa s) ↔ t ∈ Causalean.Mathlib.Optimization.exposedMinFace M α)","l":29,"f":"Causalean/Mathlib/Optimization/SimplexActiveSet.lean"},{"n":"Causalean.Mathlib.Optimization.weighted_simplex_truncation","k":"theorem","m":"Causalean.Mathlib.Optimization.SimplexTruncation","a":"Mathlib/Optimization","t":1,"d":"Weighted-simplex truncation. In the notation of weighted_simplex_active_set, fix a positive total simplex mass M, a truncation threshold d at most M, a linear weighting α, coordinate weights β that are everywhere strictly positive with its last two entries fixed equal to 1, and a nonnegative regularization parameter κ. Then the global minimizer of the weighted-simplex objective over the truncated simplex K_d = {t ∈ Δ_M : t₁ + t₂ ≥ d} is given, case by case on the sign of κ, by the active-set/face-selector construction below.","s":"∀ (M d : ℝ), 0 < M → d ≤ M → ∀ (α β : Fin 3 → ℝ) (kappa : ℝ), (∀ (i : Fin 3), 0 < β i) → β 1 = 1 → β 2 = 1 → 0 ≤ kappa → (0 < kappa → ∀ (S : Finset (Fin 3)) (lam : ℝ), Causalean.Mathlib.Optimization.IsAdmissibleSupport α β kappa S lam → (Causalean.Mathlib.Optimization.InTruncSimplex M d (Causalean.Mathlib.Optimization.activeSetPoint M α β S lam) → ∀ (s : Fin 3 → ℝ), Causalean.Mathlib.Optimization.InTruncSimplex M d s → Causalean.Mathlib.Optimization.wsObj α β kappa (Causalean.Mathlib.Optimization.activeSetPoint M α β S lam) ≤ Causalean.Mathlib.Optimization.wsObj α β kappa s) ∧ (¬d ≤ Causalean.Mathlib.Optimization.activeSetPoint M α β S lam 1 + Causalean.Mathlib.Optimization.activeSetPoint M α β S lam 2 → Causalean.Mathlib.Optimization.InTruncSimplex M d (Causalean.Mathlib.Optimization.truncSegPoint M d (Causalean.Mathlib.Optimization.truncSelector M d α β kappa)) ∧ ∀ (s : Fin 3 → ℝ), Causalean.Mathlib.Optimization.InTruncSimplex M d s → Causalean.Mathlib.Optimization.wsObj α β kappa (Causalean.Mathlib.Optimization.truncSegPoint M d (Causalean.Mathlib.Optimization.truncSelector M d α β kappa)) ≤ Causalean.Mathlib.Optimization.wsObj α β kappa s)) ∧ (kappa = 0 → ∀ t_rel ∈ Causalean.Mathlib.Optimization.exposedMinFace M α, (Causalean.Mathlib.Optimization.InTruncSimplex M d t_rel → ∀ (s : Fin 3 → ℝ), Causalean.Mathlib.Optimization.InTruncSimplex M d s → Causalean.Mathlib.Optimization.wsObj α β kappa t_rel ≤ Causalean.Mathlib.Optimization.wsObj α β kappa s) ∧ (¬d ≤ t_rel 1 + t_rel 2 → Causalean.Mathlib.Optimization.InTruncSimplex M d (Causalean.Mathlib.Optimization.truncSegPoint M d (Causalean.Mathlib.Optimization.truncSelector M d α β kappa)) ∧ ∀ (s : Fin 3 → ℝ), Causalean.Mathlib.Optimization.InTruncSimplex M d s → Causalean.Mathlib.Optimization.wsObj α β kappa (Causalean.Mathlib.Optimization.truncSegPoint M d (Causalean.Mathlib.Optimization.truncSelector M d α β kappa)) ≤ Causalean.Mathlib.Optimization.wsObj α β kappa s))","l":95,"f":"Causalean/Mathlib/Optimization/SimplexTruncation.lean"},{"n":"Causalean.Mathlib.Optimization.weighted_sqrt_segment_le","k":"theorem","m":"Causalean.Mathlib.Optimization.SimplexTruncationConvex","a":"Mathlib/Optimization","t":2,"d":"The weighted Euclidean root-mean-square of a convex combination is no larger than the same convex combination of the two weighted root-mean-squares.","s":"∀ (β : Fin 3 → ℝ), (∀ (i : Fin 3), 0 ≤ β i) → ∀ (u v : Fin 3 → ℝ) (θ : ℝ), 0 ≤ θ → θ ≤ 1 → √(∑ i, β i * ((1 - θ) * u i + θ * v i) ^ 2) ≤ (1 - θ) * √(∑ i, β i * u i ^ 2) + θ * √(∑ i, β i * v i ^ 2)","l":33,"f":"Causalean/Mathlib/Optimization/SimplexTruncationConvex.lean"},{"n":"Causalean.Mathlib.Optimization.wsObj","k":"def","m":"Causalean.Mathlib.Optimization.SimplexActiveSetDefs","a":"Mathlib/Optimization","t":1,"d":"The weighted-simplex objective Σ αᵢ tᵢ + κ √(Σ βᵢ tᵢ²) in three coordinates (t x, t y, t z) indexed by Fin 3.","s":"(Fin 3 → ℝ) → (Fin 3 → ℝ) → ℝ → (Fin 3 → ℝ) → ℝ","l":23,"f":"Causalean/Mathlib/Optimization/SimplexActiveSetDefs.lean"},{"n":"Causalean.Mathlib.Optimization.wsObj_kappa_zero","k":"theorem","m":"Causalean.Mathlib.Optimization.WeightedSimplexFace","a":"Mathlib/Optimization","t":2,"d":"At κ = 0 the weighted-simplex objective collapses to the linear form Σ αᵢ tᵢ.","s":"∀ (α β t : Fin 3 → ℝ), Causalean.Mathlib.Optimization.wsObj α β 0 t = ∑ i, α i * t i","l":17,"f":"Causalean/Mathlib/Optimization/WeightedSimplexFace.lean"},{"n":"Causalean.Mathlib.Optimization.wsObj_segment_le","k":"theorem","m":"Causalean.Mathlib.Optimization.SimplexTruncationConvex","a":"Mathlib/Optimization","t":1,"d":"Two-point convexity of the weighted-simplex objective. For nonnegative coordinate weights β and a nonnegative SOCP scale κ, if θ lies between 0 and 1 then the objective $\\sum \\alpha_i t_i + \\kappa\\sqrt{\\sum \\beta_i t_i^2}$ evaluated at the convex combination $(1-\\theta)u + \\theta v$ of two points $u,v$ is at most the same convex combination of the objective values at $u$ and at $v$.","s":"∀ (α β : Fin 3 → ℝ) (kappa : ℝ), (∀ (i : Fin 3), 0 ≤ β i) → 0 ≤ kappa → ∀ (u v : Fin 3 → ℝ) (θ : ℝ), 0 ≤ θ → θ ≤ 1 → (Causalean.Mathlib.Optimization.wsObj α β kappa fun i => (1 - θ) * u i + θ * v i) ≤ (1 - θ) * Causalean.Mathlib.Optimization.wsObj α β kappa u + θ * Causalean.Mathlib.Optimization.wsObj α β kappa v","l":74,"f":"Causalean/Mathlib/Optimization/SimplexTruncationConvex.lean"},{"n":"Causalean.Mathlib.Optimization.wsObj_truncSeg_eq","k":"theorem","m":"Causalean.Mathlib.Optimization.SimplexTruncationSlice","a":"Mathlib/Optimization","t":1,"d":"Objective on the truncation face. For a total budget M and a truncation level d, a linear weight vector α and a quadratic weight vector β, and a curvature coefficient kappa and a face coordinate σ, evaluating the weighted-simplex objective at the truncation-face point (M−d, σ, d−σ) gives the explicit one-dimensional form α₀(M−d) + α₁σ + α₂(d−σ) + κ√(β₀(M−d)² + β₁σ² + β₂(d−σ)²).","s":"∀ (M d : ℝ) (α β : Fin 3 → ℝ) (kappa σ : ℝ), Causalean.Mathlib.Optimization.wsObj α β kappa (Causalean.Mathlib.Optimization.truncSegPoint M d σ) = α 0 * (M - d) + α 1 * σ + α 2 * (d - σ) + kappa * √(β 0 * (M - d) ^ 2 + β 1 * σ ^ 2 + β 2 * (d - σ) ^ 2)","l":261,"f":"Causalean/Mathlib/Optimization/SimplexTruncationSlice.lean"},{"n":"Causalean.Mathlib.Probability.ConvergingTogether.clt_of_l2_approx","k":"theorem","m":"Causalean.Mathlib.Probability.ConvergingTogether.Diagonal","a":"Mathlib/Probability/ConvergingTogether","t":1,"d":"Converging-together corollary for the standard normal (the CLT diagonal step). The specialization of tendsto_inDistribution_of_l2_approx to the standard normal limit law: let $(S_n)$ be a sequence of real random variables, one on each probability space in a sequence, and for every row index $m$ let $(T_{m,n})_n$ be the $m$-th approximating triangular row on the same spaces, with every $S_n$ square-integrable and every $T_{m,n}$ square-integrable. If for every fixed row $m$ the law of $T_{m,n}$ converges weakly to the standard normal as $n \\to \\infty$, i.e. row $m$ obeys its own standard-normal central limit theorem, and for every tolerance $\\varepsilon > 0$ some row $M$ makes the limit superior over $n$ of $E[(S_n - T_{M,n})^2]$ at most $\\varepsilon$, so row $M$ approximates $S_n$ in $L^2$ in the iterated-limsup sense, then the law of $S_n$ converges weakly to the standard normal as $n \\to \\infty$.","s":"∀ {Ω : ℕ → Type u_1} [inst : (n : ℕ) → MeasurableSpace (Ω n)] (μ : (n : ℕ) → MeasureTheory.Measure (Ω n)) [inst_1 : ∀ (n : ℕ), MeasureTheory.IsProbabilityMeasure (μ n)] (S : (n : ℕ) → Ω n → ℝ) (T : ℕ → (n : ℕ) → Ω n → ℝ) (hS_sq : ∀ (n : ℕ), MeasureTheory.MemLp (S n) 2 (μ n)) (hT_sq : ∀ (m n : ℕ), MeasureTheory.MemLp (T m n) 2 (μ n)), (∀ (m : ℕ), Filter.Tendsto (fun n => Causalean.Mathlib.Probability.ConvergingTogether.lawPM (MeasureTheory.Measure.map (T m n) (μ n)) (MeasureTheory.Measure.isProbabilityMeasure_map (MeasureTheory.AEStronglyMeasurable.aemeasurable (MeasureTheory.MemLp.aestronglyMeasurable (hT_sq m n))))) Filter.atTop (nhds (Causalean.Mathlib.Probability.ConvergingTogether.lawPM (ProbabilityTheory.gaussianReal 0 1) inferInstance))) → (∀ (ε : ℝ), 0 < ε → ∃ M, Filter.limsup (fun n => ENNReal.ofReal (∫ (ω : Ω n), (S n ω - T M n ω) ^ 2 ∂μ n)) Filter.atTop ≤ ENNReal.ofReal ε) → Filter.Tendsto (fun n => Causalean.Mathlib.Probability.ConvergingTogether.lawPM (MeasureTheory.Measure.map (S n) (μ n)) (MeasureTheory.Measure.isProbabilityMeasure_map (MeasureTheory.AEStronglyMeasurable.aemeasurable (MeasureTheory.MemLp.aestronglyMeasurable (hS_sq n))))) Filter.atTop (nhds (Causalean.Mathlib.Probability.ConvergingTogether.lawPM (ProbabilityTheory.gaussianReal 0 1) inferInstance))","l":172,"f":"Causalean/Mathlib/Probability/ConvergingTogether/Diagonal.lean"},{"n":"Causalean.Mathlib.Probability.ConvergingTogether.integral_abs_le_sqrt_integral_sq","k":"theorem","m":"Causalean.Mathlib.Probability.ConvergingTogether.CharFunBound","a":"Mathlib/Probability/ConvergingTogether","t":2,"d":"On a probability space, the expected norm of a square-integrable variable is at most the square root of its expected squared norm.","s":"∀ {Ω : Type u_1} [inst : MeasurableSpace Ω] (μ : MeasureTheory.Measure Ω) [MeasureTheory.IsProbabilityMeasure μ] {E : Type u_2} [inst_2 : NormedAddCommGroup E] (f : Ω → E), MeasureTheory.MemLp f 2 μ → ∫ (ω : Ω), ‖f ω‖ ∂μ ≤ √(∫ (ω : Ω), ‖f ω‖ ^ 2 ∂μ)","l":131,"f":"Causalean/Mathlib/Probability/ConvergingTogether/CharFunBound.lean"},{"n":"Causalean.Mathlib.Probability.ConvergingTogether.lawPM","k":"def","m":"Causalean.Mathlib.Probability.ConvergingTogether.Diagonal","a":"Mathlib/Probability/ConvergingTogether","t":1,"d":"Bundle a probability measure on ℝ (with an explicit IsProbabilityMeasure proof) as a ProbabilityMeasure ℝ. A thin wrapper around the subtype constructor whose declared return type keeps the bundled ProbabilityMeasure topology in scope (avoiding the raw-subtype unfolding that breaks 𝓝).","s":"(m : MeasureTheory.Measure ℝ) → MeasureTheory.IsProbabilityMeasure m → MeasureTheory.ProbabilityMeasure ℝ","l":47,"f":"Causalean/Mathlib/Probability/ConvergingTogether/Diagonal.lean"},{"n":"Causalean.Mathlib.Probability.ConvergingTogether.lawPM_coe","k":"theorem","m":"Causalean.Mathlib.Probability.ConvergingTogether.Diagonal","a":"Mathlib/Probability/ConvergingTogether","t":2,"d":"","s":"∀ (m : MeasureTheory.Measure ℝ) (h : MeasureTheory.IsProbabilityMeasure m), ↑(Causalean.Mathlib.Probability.ConvergingTogether.lawPM m h) = m","l":53,"f":"Causalean/Mathlib/Probability/ConvergingTogether/Diagonal.lean"},{"n":"Causalean.Mathlib.Probability.ConvergingTogether.norm_cexp_mul_I_sub_cexp_mul_I_le","k":"theorem","m":"Causalean.Mathlib.Probability.ConvergingTogether.CharFunBound","a":"Mathlib/Probability/ConvergingTogether","t":2,"d":"Pointwise Lipschitz bound for the unit-circle exponential. For real arguments a b, the chord between the points e^{ia} and e^{ib} on the unit circle is no longer than the arc, i.e. ‖exp (a·I) − exp (b·I)‖ ≤ |a − b|. This is the elementary input to the characteristic-function approximation bound.","s":"∀ (a b : ℝ), ‖Complex.exp (↑a * Complex.I) - Complex.exp (↑b * Complex.I)‖ ≤ |a - b|","l":38,"f":"Causalean/Mathlib/Probability/ConvergingTogether/CharFunBound.lean"},{"n":"Causalean.Mathlib.Probability.ConvergingTogether.norm_charFun_sub_le","k":"theorem","m":"Causalean.Mathlib.Probability.ConvergingTogether.CharFunBound","a":"Mathlib/Probability/ConvergingTogether","t":2,"d":"Almost-everywhere measurable real variables with an integrable difference have characteristic functions whose distance is at most the frequency magnitude times their expected absolute difference.","s":"∀ {Ω : Type u_1} [inst : MeasurableSpace Ω] (μ : MeasureTheory.Measure Ω) [MeasureTheory.IsFiniteMeasure μ] {S T : Ω → ℝ}, AEMeasurable S μ → AEMeasurable T μ → MeasureTheory.Integrable (fun ω => S ω - T ω) μ → ∀ (t : ℝ), ‖MeasureTheory.charFun (MeasureTheory.Measure.map S μ) t - MeasureTheory.charFun (MeasureTheory.Measure.map T μ) t‖ ≤ |t| * ∫ (ω : Ω), |S ω - T ω| ∂μ","l":65,"f":"Causalean/Mathlib/Probability/ConvergingTogether/CharFunBound.lean"},{"n":"Causalean.Mathlib.Probability.ConvergingTogether.norm_charFun_sub_le_L2","k":"theorem","m":"Causalean.Mathlib.Probability.ConvergingTogether.CharFunBound","a":"Mathlib/Probability/ConvergingTogether","t":1,"d":"Characteristic-function approximation bound (L² form). Let $\\mu$ be a probability measure and let $S, T$ be real random variables on the same space, with $S$ almost-everywhere measurable and $T$ almost-everywhere measurable; assume further that their difference $S - T$ is square-integrable under $\\mu$. Then for every real frequency $t$, the characteristic functions of the laws of $S$ and $T$ differ at $t$ by at most $|t|$ times the $L^2$ norm of $S - T$: $\\|{\\rm charFun}(\\mu \\circ S^{-1})(t) - {\\rm charFun}(\\mu \\circ T^{-1})(t)\\| \\le |t| \\cdot \\sqrt{\\int (S - T)^2 \\, d\\mu}$.","s":"∀ {Ω : Type u_1} [inst : MeasurableSpace Ω] (μ : MeasureTheory.Measure Ω) [MeasureTheory.IsProbabilityMeasure μ] {S T : Ω → ℝ}, AEMeasurable S μ → AEMeasurable T μ → MeasureTheory.MemLp (fun ω => S ω - T ω) 2 μ → ∀ (t : ℝ), ‖MeasureTheory.charFun (MeasureTheory.Measure.map S μ) t - MeasureTheory.charFun (MeasureTheory.Measure.map T μ) t‖ ≤ |t| * √(∫ (ω : Ω), (S ω - T ω) ^ 2 ∂μ)","l":166,"f":"Causalean/Mathlib/Probability/ConvergingTogether/CharFunBound.lean"},{"n":"Causalean.Mathlib.Probability.ConvergingTogether.tendsto_inDistribution_of_l2_approx","k":"theorem","m":"Causalean.Mathlib.Probability.ConvergingTogether.Diagonal","a":"Mathlib/Probability/ConvergingTogether","t":1,"d":"The converging-together theorem (Billingsley Thm 3.2). Let $G$ be a limit probability law on the reals, let $(S_n)$ be a sequence of real random variables, one on each probability space in a sequence, and for every row index $m$ let $(T_{m,n})_n$ be the $m$-th approximating triangular row of real random variables on the same spaces, with every $S_n$ square-integrable and every $T_{m,n}$ square-integrable. Suppose for every fixed row $m$ the law of $T_{m,n}$ converges weakly to $G$ as $n \\to \\infty$, and for every tolerance $\\varepsilon > 0$ some row $M$ makes the limit superior over $n$ of $E[(S_n - T_{M,n})^2]$ at most $\\varepsilon$, i.e. row $M$ approximates $S_n$ in $L^2$ uniformly enough in the iterated-limsup sense. Then the law of $S_n$ converges weakly to $G$ as $n \\to \\infty$.","s":"∀ {Ω : ℕ → Type u_1} [inst : (n : ℕ) → MeasurableSpace (Ω n)] (μ : (n : ℕ) → MeasureTheory.Measure (Ω n)) [inst_1 : ∀ (n : ℕ), MeasureTheory.IsProbabilityMeasure (μ n)] (G : MeasureTheory.ProbabilityMeasure ℝ) (S : (n : ℕ) → Ω n → ℝ) (T : ℕ → (n : ℕ) → Ω n → ℝ) (hS_sq : ∀ (n : ℕ), MeasureTheory.MemLp (S n) 2 (μ n)) (hT_sq : ∀ (m n : ℕ), MeasureTheory.MemLp (T m n) 2 (μ n)), (∀ (m : ℕ), Filter.Tendsto (fun n => Causalean.Mathlib.Probability.ConvergingTogether.lawPM (MeasureTheory.Measure.map (T m n) (μ n)) (MeasureTheory.Measure.isProbabilityMeasure_map (MeasureTheory.AEStronglyMeasurable.aemeasurable (MeasureTheory.MemLp.aestronglyMeasurable (hT_sq m n))))) Filter.atTop (nhds G)) → (∀ (ε : ℝ), 0 < ε → ∃ M, Filter.limsup (fun n => ENNReal.ofReal (∫ (ω : Ω n), (S n ω - T M n ω) ^ 2 ∂μ n)) Filter.atTop ≤ ENNReal.ofReal ε) → Filter.Tendsto (fun n => Causalean.Mathlib.Probability.ConvergingTogether.lawPM (MeasureTheory.Measure.map (S n) (μ n)) (MeasureTheory.Measure.isProbabilityMeasure_map (MeasureTheory.AEStronglyMeasurable.aemeasurable (MeasureTheory.MemLp.aestronglyMeasurable (hS_sq n))))) Filter.atTop (nhds G)","l":56,"f":"Causalean/Mathlib/Probability/ConvergingTogether/Diagonal.lean"},{"n":"Causalean.Mathlib.Probability.FiniteMarkedPoissonPartition.FiniteMeasurablePartition","k":"structure","m":"Causalean.Mathlib.Probability.FiniteMarkedPoissonPartition.Partition.Basic","a":"Mathlib/Probability/FiniteMarkedPoissonPartition/Partition","t":1,"d":"A finite measurable partition of the sample space X into cells indexed by ι, represented by a classifier assigning each observation to its cell — the cells are the fibres of this map — where that classifier is measurable.","s":"(X : Type u_3) → (ι : Type u_4) → [MeasurableSpace X] → [MeasurableSpace ι] → Type (max u_3 u_4)","l":22,"f":"Causalean/Mathlib/Probability/FiniteMarkedPoissonPartition/Partition/Basic.lean"},{"n":"Causalean.Mathlib.Probability.FiniteMarkedPoissonPartition.FiniteMeasurablePartition.cellIndices","k":"def","m":"Causalean.Mathlib.Probability.FiniteMarkedPoissonPartition.Partition.Basic","a":"Mathlib/Probability/FiniteMarkedPoissonPartition/Partition","t":1,"d":"Indices of the marked observations belonging to cell j.","s":"{X : Type u_1} → [inst : MeasurableSpace X] → {ι : Type u_2} → [inst_1 : MeasurableSpace ι] → Causalean.Mathlib.Probability.FiniteMarkedPoissonPartition.FiniteMeasurablePartition X ι → ι → (s : Causalean.Mathlib.Probability.FiniteMarkedPoissonPartition.FiniteSample (X × ℝ)) → Finset (Fin s.count)","l":161,"f":"Causalean/Mathlib/Probability/FiniteMarkedPoissonPartition/Partition/Basic.lean"},{"n":"Causalean.Mathlib.Probability.FiniteMarkedPoissonPartition.FiniteMeasurablePartition.cellLaw_restrict_count_eq","k":"theorem","m":"Causalean.Mathlib.Probability.FiniteMarkedPoissonPartition.Partition.CellLaws","a":"Mathlib/Probability/FiniteMarkedPoissonPartition/Partition","t":2,"d":"On its count-n fibre, the jth cell experiment is the Poisson mass at n times n independent draws from the normalised restriction and mark law.","s":"∀ {X : Type u_1} [inst : MeasurableSpace X] {ι : Type u_2} [Fintype ι] [inst_2 : MeasurableSpace ι] [MeasurableSingletonClass ι] (p : Causalean.Mathlib.Probability.FiniteMarkedPoissonPartition.FiniteMeasurablePartition X ι) (P : MeasureTheory.Measure X) [inst_4 : MeasureTheory.IsProbabilityMeasure P] (R : MeasureTheory.Measure ℝ) [inst_5 : MeasureTheory.IsProbabilityMeasure R] (lam : NNReal) (j : ι) (n : ℕ), (Causalean.Mathlib.Probability.FiniteMarkedPoissonPartition.finiteMarkedPoissonSampleLaw (p.cellObservationLaw P j) R (lam * p.cellMass P j)).restrict (Causalean.Mathlib.Probability.FiniteMarkedPoissonPartition.FiniteSample.count ⁻¹' {n}) = (ProbabilityTheory.poissonMeasure (lam * p.cellMass P j)) {n} • MeasureTheory.Measure.map (Causalean.Mathlib.Probability.FiniteMarkedPoissonPartition.fixedSizeEmbed n) (MeasureTheory.Measure.pi fun x => (p.cellObservationLaw P j).prod R)","l":64,"f":"Causalean/Mathlib/Probability/FiniteMarkedPoissonPartition/Partition/CellLaws.lean"},{"n":"Causalean.Mathlib.Probability.FiniteMarkedPoissonPartition.FiniteMeasurablePartition.cellMass","k":"def","m":"Causalean.Mathlib.Probability.FiniteMarkedPoissonPartition.Partition.Basic","a":"Mathlib/Probability/FiniteMarkedPoissonPartition/Partition","t":1,"d":"The probability mass of a cell, represented as a nonnegative real.","s":"{X : Type u_1} → [inst : MeasurableSpace X] → {ι : Type u_2} → [inst_1 : MeasurableSpace ι] → Causalean.Mathlib.Probability.FiniteMarkedPoissonPartition.FiniteMeasurablePartition X ι → MeasureTheory.Measure X → ι → NNReal","l":107,"f":"Causalean/Mathlib/Probability/FiniteMarkedPoissonPartition/Partition/Basic.lean"},{"n":"Causalean.Mathlib.Probability.FiniteMarkedPoissonPartition.FiniteMeasurablePartition.cellObservationLaw","k":"def","m":"Causalean.Mathlib.Probability.FiniteMarkedPoissonPartition.Partition.Basic","a":"Mathlib/Probability/FiniteMarkedPoissonPartition/Partition","t":1,"d":"The within-cell observation law is the normalised restriction when the cell has positive mass and the ambient probability law when its mass is zero.","s":"{X : Type u_1} → [inst : MeasurableSpace X] → {ι : Type u_2} → [inst_1 : MeasurableSpace ι] → Causalean.Mathlib.Probability.FiniteMarkedPoissonPartition.FiniteMeasurablePartition X ι → (P : MeasureTheory.Measure X) → [MeasureTheory.IsProbabilityMeasure P] → ι → MeasureTheory.Measure X","l":131,"f":"Causalean/Mathlib/Probability/FiniteMarkedPoissonPartition/Partition/Basic.lean"},{"n":"Causalean.Mathlib.Probability.FiniteMarkedPoissonPartition.FiniteMeasurablePartition.cellObservationLaw_apply_cellSet","k":"theorem","m":"Causalean.Mathlib.Probability.FiniteMarkedPoissonPartition.Partition.Basic","a":"Mathlib/Probability/FiniteMarkedPoissonPartition/Partition","t":2,"d":"A positive-mass within-cell law assigns probability one to its own cell.","s":"∀ {X : Type u_1} [inst : MeasurableSpace X] {ι : Type u_2} [Fintype ι] [inst_2 : MeasurableSpace ι] [MeasurableSingletonClass ι] (p : Causalean.Mathlib.Probability.FiniteMarkedPoissonPartition.FiniteMeasurablePartition X ι) (P : MeasureTheory.Measure X) [inst_4 : MeasureTheory.IsProbabilityMeasure P] (j : ι), P (p.cellSet j) ≠ 0 → (p.cellObservationLaw P j) (p.cellSet j) = 1","l":151,"f":"Causalean/Mathlib/Probability/FiniteMarkedPoissonPartition/Partition/Basic.lean"},{"n":"Causalean.Mathlib.Probability.FiniteMarkedPoissonPartition.FiniteMeasurablePartition.cellObservationLaw_isProbabilityMeasure","k":"instance","m":"Causalean.Mathlib.Probability.FiniteMarkedPoissonPartition.Partition.Basic","a":"Mathlib/Probability/FiniteMarkedPoissonPartition/Partition","t":2,"d":"The within-cell observation law is a probability measure, including the zero-mass fallback branch.","s":"∀ {X : Type u_1} [inst : MeasurableSpace X] {ι : Type u_2} [inst_1 : MeasurableSpace ι] (p : Causalean.Mathlib.Probability.FiniteMarkedPoissonPartition.FiniteMeasurablePartition X ι) (P : MeasureTheory.Measure X) [inst_2 : MeasureTheory.IsProbabilityMeasure P] (j : ι), MeasureTheory.IsProbabilityMeasure (p.cellObservationLaw P j)","l":138,"f":"Causalean/Mathlib/Probability/FiniteMarkedPoissonPartition/Partition/Basic.lean"},{"n":"Causalean.Mathlib.Probability.FiniteMarkedPoissonPartition.FiniteMeasurablePartition.cellSet","k":"def","m":"Causalean.Mathlib.Probability.FiniteMarkedPoissonPartition.Partition.Basic","a":"Mathlib/Probability/FiniteMarkedPoissonPartition/Partition","t":1,"d":"The measurable set forming cell j of a classifier partition.","s":"{X : Type u_1} → [inst : MeasurableSpace X] → {ι : Type u_2} → [inst_1 : MeasurableSpace ι] → Causalean.Mathlib.Probability.FiniteMarkedPoissonPartition.FiniteMeasurablePartition X ι → ι → Set X","l":85,"f":"Causalean/Mathlib/Probability/FiniteMarkedPoissonPartition/Partition/Basic.lean"},{"n":"Causalean.Mathlib.Probability.FiniteMarkedPoissonPartition.FiniteMeasurablePartition.disjoint_cellSet","k":"theorem","m":"Causalean.Mathlib.Probability.FiniteMarkedPoissonPartition.Partition.Basic","a":"Mathlib/Probability/FiniteMarkedPoissonPartition/Partition","t":2,"d":"Distinct classifier cells are disjoint.","s":"∀ {X : Type u_1} [inst : MeasurableSpace X] {ι : Type u_2} [Fintype ι] [inst_2 : MeasurableSpace ι] [MeasurableSingletonClass ι] (p : Causalean.Mathlib.Probability.FiniteMarkedPoissonPartition.FiniteMeasurablePartition X ι) {i j : ι}, i ≠ j → Disjoint (p.cellSet i) (p.cellSet j)","l":94,"f":"Causalean/Mathlib/Probability/FiniteMarkedPoissonPartition/Partition/Basic.lean"},{"n":"Causalean.Mathlib.Probability.FiniteMarkedPoissonPartition.FiniteMeasurablePartition.iUnion_cellSet","k":"theorem","m":"Causalean.Mathlib.Probability.FiniteMarkedPoissonPartition.Partition.Basic","a":"Mathlib/Probability/FiniteMarkedPoissonPartition/Partition","t":2,"d":"The union of all classifier cells is the whole observation space.","s":"∀ {X : Type u_1} [inst : MeasurableSpace X] {ι : Type u_2} [Fintype ι] [inst_2 : MeasurableSpace ι] [MeasurableSingletonClass ι] (p : Causalean.Mathlib.Probability.FiniteMarkedPoissonPartition.FiniteMeasurablePartition X ι), ⋃ j, p.cellSet j = Set.univ","l":101,"f":"Causalean/Mathlib/Probability/FiniteMarkedPoissonPartition/Partition/Basic.lean"},{"n":"Causalean.Mathlib.Probability.FiniteMarkedPoissonPartition.FiniteMeasurablePartition.map_restrictCell_count_finiteMarkedPoissonSampleLaw","k":"theorem","m":"Causalean.Mathlib.Probability.FiniteMarkedPoissonPartition.Partition.CellLaws","a":"Mathlib/Probability/FiniteMarkedPoissonPartition/Partition","t":1,"d":"Cell counts are Poisson. Under the marked Poisson sample law with base probability measure P, mark distribution R, and nonnegative intensity lam, the number of marked observations landing in cell j of the finite measurable partition p, viewed as a random variable, is Poisson distributed with mean equal to lam times the P-mass of cell j.","s":"∀ {X : Type u_1} [inst : MeasurableSpace X] {ι : Type u_2} [Fintype ι] [inst_2 : MeasurableSpace ι] [MeasurableSingletonClass ι] [StandardBorelSpace X] (p : Causalean.Mathlib.Probability.FiniteMarkedPoissonPartition.FiniteMeasurablePartition X ι) (P : MeasureTheory.Measure X) [inst_5 : MeasureTheory.IsProbabilityMeasure P] (R : MeasureTheory.Measure ℝ) [inst_6 : MeasureTheory.IsProbabilityMeasure R] (lam : NNReal) (j : ι), MeasureTheory.Measure.map (fun s => (p.restrictCell j s).count) (Causalean.Mathlib.Probability.FiniteMarkedPoissonPartition.finiteMarkedPoissonSampleLaw P R lam) = ProbabilityTheory.poissonMeasure (lam * p.cellMass P j)","l":20,"f":"Causalean/Mathlib/Probability/FiniteMarkedPoissonPartition/Partition/CellLaws.lean"},{"n":"Causalean.Mathlib.Probability.FiniteMarkedPoissonPartition.FiniteMeasurablePartition.map_restrictPartition_finiteMarkedPoissonSampleLaw","k":"theorem","m":"Causalean.Mathlib.Probability.FiniteMarkedPoissonPartition.Partition.Splitting","a":"Mathlib/Probability/FiniteMarkedPoissonPartition/Partition","t":1,"d":"Partition splitting. Under the marked Poisson sample law with base probability measure P, mark distribution R, and nonnegative intensity lam, restricting the sample to each cell of the finite measurable partition p yields, jointly across cells, the product of independent marked Poisson sample laws, one per cell j, each with base measure p.cellObservationLaw P j, mark distribution R, and intensity lam times the P-mass of cell j.","s":"∀ {X : Type u_1} [inst : MeasurableSpace X] {ι : Type u_2} [inst_1 : Fintype ι] [inst_2 : MeasurableSpace ι] [MeasurableSingletonClass ι] [StandardBorelSpace X] (p : Causalean.Mathlib.Probability.FiniteMarkedPoissonPartition.FiniteMeasurablePartition X ι) (P : MeasureTheory.Measure X) [inst_5 : MeasureTheory.IsProbabilityMeasure P] (R : MeasureTheory.Measure ℝ) [inst_6 : MeasureTheory.IsProbabilityMeasure R] (lam : NNReal), MeasureTheory.Measure.map p.restrictPartition (Causalean.Mathlib.Probability.FiniteMarkedPoissonPartition.finiteMarkedPoissonSampleLaw P R lam) = MeasureTheory.Measure.pi fun j => Causalean.Mathlib.Probability.FiniteMarkedPoissonPartition.finiteMarkedPoissonSampleLaw (p.cellObservationLaw P j) R (lam * p.cellMass P j)","l":749,"f":"Causalean/Mathlib/Probability/FiniteMarkedPoissonPartition/Partition/Splitting.lean"},{"n":"Causalean.Mathlib.Probability.FiniteMarkedPoissonPartition.FiniteMeasurablePartition.measurableSet_cellSet","k":"theorem","m":"Causalean.Mathlib.Probability.FiniteMarkedPoissonPartition.Partition.Basic","a":"Mathlib/Probability/FiniteMarkedPoissonPartition/Partition","t":2,"d":"Every classifier cell is measurable.","s":"∀ {X : Type u_1} [inst : MeasurableSpace X] {ι : Type u_2} [Fintype ι] [inst_2 : MeasurableSpace ι] [MeasurableSingletonClass ι] (p : Causalean.Mathlib.Probability.FiniteMarkedPoissonPartition.FiniteMeasurablePartition X ι) (j : ι), MeasurableSet (p.cellSet j)","l":89,"f":"Causalean/Mathlib/Probability/FiniteMarkedPoissonPartition/Partition/Basic.lean"},{"n":"Causalean.Mathlib.Probability.FiniteMarkedPoissonPartition.FiniteMeasurablePartition.measurable_restrictCell","k":"theorem","m":"Causalean.Mathlib.Probability.FiniteMarkedPoissonPartition.Partition.Basic","a":"Mathlib/Probability/FiniteMarkedPoissonPartition/Partition","t":2,"d":"Restriction to one measurable cell is a measurable map on finite marked sequences.","s":"∀ {X : Type u_1} [inst : MeasurableSpace X] {ι : Type u_2} [Fintype ι] [inst_2 : MeasurableSpace ι] [MeasurableSingletonClass ι] (p : Causalean.Mathlib.Probability.FiniteMarkedPoissonPartition.FiniteMeasurablePartition X ι) (j : ι), Measurable (p.restrictCell j)","l":180,"f":"Causalean/Mathlib/Probability/FiniteMarkedPoissonPartition/Partition/Basic.lean"},{"n":"Causalean.Mathlib.Probability.FiniteMarkedPoissonPartition.FiniteMeasurablePartition.measurable_restrictPartition","k":"theorem","m":"Causalean.Mathlib.Probability.FiniteMarkedPoissonPartition.Partition.Basic","a":"Mathlib/Probability/FiniteMarkedPoissonPartition/Partition","t":1,"d":"For a finite measurable partition p of the sample space into cells indexed by ι, the map sending a finite marked sequence to its family of restrictions to every cell simultaneously is measurable.","s":"∀ {X : Type u_1} [inst : MeasurableSpace X] {ι : Type u_2} [Fintype ι] [inst_2 : MeasurableSpace ι] [MeasurableSingletonClass ι] (p : Causalean.Mathlib.Probability.FiniteMarkedPoissonPartition.FiniteMeasurablePartition X ι), Measurable p.restrictPartition","l":254,"f":"Causalean/Mathlib/Probability/FiniteMarkedPoissonPartition/Partition/Basic.lean"},{"n":"Causalean.Mathlib.Probability.FiniteMarkedPoissonPartition.FiniteMeasurablePartition.ofSets","k":"def","m":"Causalean.Mathlib.Probability.FiniteMarkedPoissonPartition.Partition.Basic","a":"Mathlib/Probability/FiniteMarkedPoissonPartition/Partition","t":1,"d":"A finite family of measurable, pairwise disjoint sets covering the whole space determines its unique measurable classifier partition.","s":"{X : Type u_1} → [inst : MeasurableSpace X] → {ι : Type u_2} → [Fintype ι] → [inst_2 : MeasurableSpace ι] → (A : ι → Set X) → (∀ (j : ι), MeasurableSet (A j)) → (Pairwise fun i j => Disjoint (A i) (A j)) → ⋃ j, A j = Set.univ → Causalean.Mathlib.Probability.FiniteMarkedPoissonPartition.FiniteMeasurablePartition X ι","l":34,"f":"Causalean/Mathlib/Probability/FiniteMarkedPoissonPartition/Partition/Basic.lean"},{"n":"Causalean.Mathlib.Probability.FiniteMarkedPoissonPartition.FiniteMeasurablePartition.ofSets_cellSet","k":"theorem","m":"Causalean.Mathlib.Probability.FiniteMarkedPoissonPartition.Partition.Basic","a":"Mathlib/Probability/FiniteMarkedPoissonPartition/Partition","t":2,"d":"The classifier constructed from a disjoint measurable cover has exactly the supplied sets as its fibres.","s":"∀ {X : Type u_1} [inst : MeasurableSpace X] {ι : Type u_2} [inst_1 : Fintype ι] [inst_2 : MeasurableSpace ι] [MeasurableSingletonClass ι] (A : ι → Set X) (hA : ∀ (j : ι), MeasurableSet (A j)) (hdis : Pairwise fun i j => Disjoint (A i) (A j)) (hcover : ⋃ j, A j = Set.univ) (j : ι), (Causalean.Mathlib.Probability.FiniteMarkedPoissonPartition.FiniteMeasurablePartition.ofSets A hA hdis hcover).cell ⁻¹' {j} = A j","l":59,"f":"Causalean/Mathlib/Probability/FiniteMarkedPoissonPartition/Partition/Basic.lean"},{"n":"Causalean.Mathlib.Probability.FiniteMarkedPoissonPartition.FiniteMeasurablePartition.restrictCell","k":"def","m":"Causalean.Mathlib.Probability.FiniteMarkedPoissonPartition.Partition.Basic","a":"Mathlib/Probability/FiniteMarkedPoissonPartition/Partition","t":1,"d":"Restrict a finite marked sequence to one cell, preserving the original relative order of all points that lie in that cell.","s":"{X : Type u_1} → [inst : MeasurableSpace X] → {ι : Type u_2} → [inst_1 : MeasurableSpace ι] → Causalean.Mathlib.Probability.FiniteMarkedPoissonPartition.FiniteMeasurablePartition X ι → ι → Causalean.Mathlib.Probability.FiniteMarkedPoissonPartition.FiniteSample (X × ℝ) → Causalean.Mathlib.Probability.FiniteMarkedPoissonPartition.FiniteSample (X × ℝ)","l":167,"f":"Causalean/Mathlib/Probability/FiniteMarkedPoissonPartition/Partition/Basic.lean"},{"n":"Causalean.Mathlib.Probability.FiniteMarkedPoissonPartition.FiniteMeasurablePartition.restrictPartition","k":"def","m":"Causalean.Mathlib.Probability.FiniteMarkedPoissonPartition.Partition.Basic","a":"Mathlib/Probability/FiniteMarkedPoissonPartition/Partition","t":1,"d":"Restrict a finite marked sequence simultaneously to every partition cell.","s":"{X : Type u_1} → [inst : MeasurableSpace X] → {ι : Type u_2} → [inst_1 : MeasurableSpace ι] → Causalean.Mathlib.Probability.FiniteMarkedPoissonPartition.FiniteMeasurablePartition X ι → Causalean.Mathlib.Probability.FiniteMarkedPoissonPartition.FiniteSample (X × ℝ) → ι → Causalean.Mathlib.Probability.FiniteMarkedPoissonPartition.FiniteSample (X × ℝ)","l":175,"f":"Causalean/Mathlib/Probability/FiniteMarkedPoissonPartition/Partition/Basic.lean"},{"n":"Causalean.Mathlib.Probability.FiniteMarkedPoissonPartition.FiniteMeasurablePartition.sum_cellMass","k":"theorem","m":"Causalean.Mathlib.Probability.FiniteMarkedPoissonPartition.Partition.Basic","a":"Mathlib/Probability/FiniteMarkedPoissonPartition/Partition","t":2,"d":"Cell masses sum to one under a probability law.","s":"∀ {X : Type u_1} [inst : MeasurableSpace X] {ι : Type u_2} [inst_1 : Fintype ι] [inst_2 : MeasurableSpace ι] [MeasurableSingletonClass ι] (p : Causalean.Mathlib.Probability.FiniteMarkedPoissonPartition.FiniteMeasurablePartition X ι) (P : MeasureTheory.Measure X) [MeasureTheory.IsProbabilityMeasure P], ∑ j, p.cellMass P j = 1","l":112,"f":"Causalean/Mathlib/Probability/FiniteMarkedPoissonPartition/Partition/Basic.lean"},{"n":"Causalean.Mathlib.Probability.FiniteMarkedPoissonPartition.FiniteSample","k":"def","m":"Causalean.Mathlib.Probability.FiniteMarkedPoissonPartition.Basic","a":"Mathlib/Probability/FiniteMarkedPoissonPartition","t":1,"d":"A finite sequence is a length together with coordinates indexed by that length.","s":"(X : Type u_2) → [MeasurableSpace X] → Type (max 0 u_2)","l":34,"f":"Causalean/Mathlib/Probability/FiniteMarkedPoissonPartition/Basic.lean"},{"n":"Causalean.Mathlib.Probability.FiniteMarkedPoissonPartition.canonicalMarkedPoissonSampleLaw","k":"def","m":"Causalean.Mathlib.Probability.FiniteMarkedPoissonPartition.Superposition.Canonical","a":"Mathlib/Probability/FiniteMarkedPoissonPartition/Superposition","t":1,"d":"The canonical finite marked Poisson configuration law stores the atoms in increasing mark order. It is the measurable image of the raw conditionally i.i.d. marked sequence law.","s":"{X : Type u_1} → [inst : MeasurableSpace X] → (P : MeasureTheory.Measure X) → [MeasureTheory.IsProbabilityMeasure P] → (R : MeasureTheory.Measure ℝ) → [MeasureTheory.IsProbabilityMeasure R] → NNReal → MeasureTheory.Measure (Causalean.Mathlib.Probability.FiniteMarkedPoissonPartition.FiniteSample (X × ℝ))","l":298,"f":"Causalean/Mathlib/Probability/FiniteMarkedPoissonPartition/Superposition/Canonical.lean"},{"n":"Causalean.Mathlib.Probability.FiniteMarkedPoissonPartition.canonicalMarkedPoissonSampleLaw_isProbabilityMeasure","k":"instance","m":"Causalean.Mathlib.Probability.FiniteMarkedPoissonPartition.Superposition.Canonical","a":"Mathlib/Probability/FiniteMarkedPoissonPartition/Superposition","t":2,"d":"The canonical mark-ordered configuration law is a probability measure.","s":"∀ {X : Type u_1} [inst : MeasurableSpace X] (P : MeasureTheory.Measure X) [inst_1 : MeasureTheory.IsProbabilityMeasure P] (R : MeasureTheory.Measure ℝ) [inst_2 : MeasureTheory.IsProbabilityMeasure R] (lam : NNReal), MeasureTheory.IsProbabilityMeasure (Causalean.Mathlib.Probability.FiniteMarkedPoissonPartition.canonicalMarkedPoissonSampleLaw P R lam)","l":307,"f":"Causalean/Mathlib/Probability/FiniteMarkedPoissonPartition/Superposition/Canonical.lean"},{"n":"Causalean.Mathlib.Probability.FiniteMarkedPoissonPartition.canonicalMarkedPoissonSampleLaw_map_count","k":"theorem","m":"Causalean.Mathlib.Probability.FiniteMarkedPoissonPartition.Depoissonization","a":"Mathlib/Probability/FiniteMarkedPoissonPartition","t":2,"d":"","s":"∀ {X : Type u_1} [inst : MeasurableSpace X] (P : MeasureTheory.Measure X) [inst_1 : MeasureTheory.IsProbabilityMeasure P] (R : MeasureTheory.Measure ℝ) [inst_2 : MeasureTheory.IsProbabilityMeasure R] (lam : NNReal), MeasureTheory.Measure.map Causalean.Mathlib.Probability.FiniteMarkedPoissonPartition.FiniteSample.count (Causalean.Mathlib.Probability.FiniteMarkedPoissonPartition.canonicalMarkedPoissonSampleLaw P R lam) = ProbabilityTheory.poissonMeasure lam","l":372,"f":"Causalean/Mathlib/Probability/FiniteMarkedPoissonPartition/Depoissonization.lean"},{"n":"Causalean.Mathlib.Probability.FiniteMarkedPoissonPartition.canonicalPrefixObservations","k":"def","m":"Causalean.Mathlib.Probability.FiniteMarkedPoissonPartition.Depoissonization","a":"Mathlib/Probability/FiniteMarkedPoissonPartition","t":1,"d":"A marked finite sample supplies its first n observation coordinates when it is long enough, and otherwise supplies a fixed fallback tuple.","s":"{X : Type u_1} → [inst : MeasurableSpace X] → X → (n : ℕ) → Causalean.Mathlib.Probability.FiniteMarkedPoissonPartition.FiniteSample (X × ℝ) → Fin n → X","l":131,"f":"Causalean/Mathlib/Probability/FiniteMarkedPoissonPartition/Depoissonization.lean"},{"n":"Causalean.Mathlib.Probability.FiniteMarkedPoissonPartition.cellObservationLaw_eq_of_restrict_eq","k":"theorem","m":"Causalean.Mathlib.Probability.FiniteMarkedPoissonPartition.Depoissonization","a":"Mathlib/Probability/FiniteMarkedPoissonPartition","t":2,"d":"Equal restrictions and equal cell masses give equal normalized cell laws.","s":"∀ {X : Type u_1} {ι : Type u_2} [inst : MeasurableSpace X] [Fintype ι] [inst_2 : MeasurableSpace ι] [MeasurableSingletonClass ι] (p : Causalean.Mathlib.Probability.FiniteMarkedPoissonPartition.FiniteMeasurablePartition X ι) (μ ν : MeasureTheory.Measure X) [inst_4 : MeasureTheory.IsProbabilityMeasure μ] [inst_5 : MeasureTheory.IsProbabilityMeasure ν] (j : ι), μ (p.cellSet j) ≠ 0 → μ (p.cellSet j) = ν (p.cellSet j) → μ.restrict (p.cellSet j) = ν.restrict (p.cellSet j) → p.cellObservationLaw μ j = p.cellObservationLaw ν j","l":57,"f":"Causalean/Mathlib/Probability/FiniteMarkedPoissonPartition/Depoissonization.lean"},{"n":"Causalean.Mathlib.Probability.FiniteMarkedPoissonPartition.finiteMarkedPoissonSampleLaw","k":"def","m":"Causalean.Mathlib.Probability.FiniteMarkedPoissonPartition.Basic","a":"Mathlib/Probability/FiniteMarkedPoissonPartition","t":1,"d":"A finite marked Poisson sample has i.i.d. observation-mark pairs, with observation law P, independent mark law R, and Poisson mean lam.","s":"{X : Type u_1} → [inst : MeasurableSpace X] → (P : MeasureTheory.Measure X) → [MeasureTheory.IsProbabilityMeasure P] → (R : MeasureTheory.Measure ℝ) → [MeasureTheory.IsProbabilityMeasure R] → NNReal → MeasureTheory.Measure (Causalean.Mathlib.Probability.FiniteMarkedPoissonPartition.FiniteSample (X × ℝ))","l":143,"f":"Causalean/Mathlib/Probability/FiniteMarkedPoissonPartition/Basic.lean"},{"n":"Causalean.Mathlib.Probability.FiniteMarkedPoissonPartition.finiteMarkedPoissonSampleLaw_isProbabilityMeasure","k":"instance","m":"Causalean.Mathlib.Probability.FiniteMarkedPoissonPartition.Basic","a":"Mathlib/Probability/FiniteMarkedPoissonPartition","t":2,"d":"The finite marked Poisson sample law is a probability measure.","s":"∀ {X : Type u_1} [inst : MeasurableSpace X] (P : MeasureTheory.Measure X) [inst_1 : MeasureTheory.IsProbabilityMeasure P] (R : MeasureTheory.Measure ℝ) [inst_2 : MeasureTheory.IsProbabilityMeasure R] (lam : NNReal), MeasureTheory.IsProbabilityMeasure (Causalean.Mathlib.Probability.FiniteMarkedPoissonPartition.finiteMarkedPoissonSampleLaw P R lam)","l":150,"f":"Causalean/Mathlib/Probability/FiniteMarkedPoissonPartition/Basic.lean"},{"n":"Causalean.Mathlib.Probability.FiniteMarkedPoissonPartition.finiteMarkedPoissonSampleLaw_map_count","k":"theorem","m":"Causalean.Mathlib.Probability.FiniteMarkedPoissonPartition.Basic","a":"Mathlib/Probability/FiniteMarkedPoissonPartition","t":2,"d":"The count of a finite marked Poisson sample has scalar Poisson law with mean lam.","s":"∀ {X : Type u_1} [inst : MeasurableSpace X] (P : MeasureTheory.Measure X) [inst_1 : MeasureTheory.IsProbabilityMeasure P] (R : MeasureTheory.Measure ℝ) [inst_2 : MeasureTheory.IsProbabilityMeasure R] (lam : NNReal), MeasureTheory.Measure.map Causalean.Mathlib.Probability.FiniteMarkedPoissonPartition.FiniteSample.count (Causalean.Mathlib.Probability.FiniteMarkedPoissonPartition.finiteMarkedPoissonSampleLaw P R lam) = ProbabilityTheory.poissonMeasure lam","l":158,"f":"Causalean/Mathlib/Probability/FiniteMarkedPoissonPartition/Basic.lean"},{"n":"Causalean.Mathlib.Probability.FiniteMarkedPoissonPartition.finiteMarkedPoissonSampleLaw_marks_pairwise_distinct","k":"theorem","m":"Causalean.Mathlib.Probability.FiniteMarkedPoissonPartition.Superposition.Canonical","a":"Mathlib/Probability/FiniteMarkedPoissonPartition/Superposition","t":2,"d":"Atomless independent marks are pairwise distinct with probability one in the finite marked Poisson sample.","s":"∀ {X : Type u_1} [inst : MeasurableSpace X] (P : MeasureTheory.Measure X) [inst_1 : MeasureTheory.IsProbabilityMeasure P] (R : MeasureTheory.Measure ℝ) [inst_2 : MeasureTheory.IsProbabilityMeasure R] [MeasureTheory.NullSingletonClass R] (lam : NNReal), (Causalean.Mathlib.Probability.FiniteMarkedPoissonPartition.finiteMarkedPoissonSampleLaw P R lam) {s | ∀ (i j : Fin s.count), i ≠ j → (s.points i).2 ≠ (s.points j).2} = 1","l":752,"f":"Causalean/Mathlib/Probability/FiniteMarkedPoissonPartition/Superposition/Canonical.lean"},{"n":"Causalean.Mathlib.Probability.FiniteMarkedPoissonPartition.finiteMarkedPoissonSampleLaw_restrict_count_eq","k":"theorem","m":"Causalean.Mathlib.Probability.FiniteMarkedPoissonPartition.Basic","a":"Mathlib/Probability/FiniteMarkedPoissonPartition","t":1,"d":"Fix a nonnegative Poisson rate lam. For an observation law P and an independent mark law R, restricting the finite marked Poisson sample law to the event that the observed count equals n yields exactly poissonMeasure lam {n} times the pushforward, under the fixed-size embedding, of n independent draws from the product measure P.prod R.","s":"∀ {X : Type u_1} [inst : MeasurableSpace X] (P : MeasureTheory.Measure X) [inst_1 : MeasureTheory.IsProbabilityMeasure P] (R : MeasureTheory.Measure ℝ) [inst_2 : MeasureTheory.IsProbabilityMeasure R] (lam : NNReal) (n : ℕ), (Causalean.Mathlib.Probability.FiniteMarkedPoissonPartition.finiteMarkedPoissonSampleLaw P R lam).restrict (Causalean.Mathlib.Probability.FiniteMarkedPoissonPartition.FiniteSample.count ⁻¹' {n}) = (ProbabilityTheory.poissonMeasure lam) {n} • MeasureTheory.Measure.map (Causalean.Mathlib.Probability.FiniteMarkedPoissonPartition.fixedSizeEmbed n) (MeasureTheory.Measure.pi fun x => P.prod R)","l":167,"f":"Causalean/Mathlib/Probability/FiniteMarkedPoissonPartition/Basic.lean"},{"n":"Causalean.Mathlib.Probability.FiniteMarkedPoissonPartition.finiteMeasureMarkedPoissonLaw","k":"def","m":"Causalean.Mathlib.Probability.FiniteMarkedPoissonPartition.KL","a":"Mathlib/Probability/FiniteMarkedPoissonPartition","t":1,"d":"The marked Poisson experiment generated by finite intensity measure ν has count mean lam * ν(univ) and conditionally i.i.d. points from the normalisation of ν, with independent marks from R.","s":"{X : Type u_1} → [inst : MeasurableSpace X] → (ν : MeasureTheory.Measure X) → [MeasureTheory.IsFiniteMeasure ν] → (P₀ : MeasureTheory.Measure X) → [MeasureTheory.IsProbabilityMeasure P₀] → (R : MeasureTheory.Measure ℝ) → [MeasureTheory.IsProbabilityMeasure R] → NNReal → MeasureTheory.Measure (Causalean.Mathlib.Probability.FiniteMarkedPoissonPartition.FiniteSample (X × ℝ))","l":369,"f":"Causalean/Mathlib/Probability/FiniteMarkedPoissonPartition/KL.lean"},{"n":"Causalean.Mathlib.Probability.FiniteMarkedPoissonPartition.finiteMeasureMarkedPoissonLaw_isProbabilityMeasure","k":"instance","m":"Causalean.Mathlib.Probability.FiniteMarkedPoissonPartition.KL","a":"Mathlib/Probability/FiniteMarkedPoissonPartition","t":2,"d":"A finite-measure marked Poisson experiment is a probability measure.","s":"∀ {X : Type u_1} [inst : MeasurableSpace X] (ν : MeasureTheory.Measure X) [inst_1 : MeasureTheory.IsFiniteMeasure ν] (P₀ : MeasureTheory.Measure X) [inst_2 : MeasureTheory.IsProbabilityMeasure P₀] (R : MeasureTheory.Measure ℝ) [inst_3 : MeasureTheory.IsProbabilityMeasure R] (lam : NNReal), MeasureTheory.IsProbabilityMeasure (Causalean.Mathlib.Probability.FiniteMarkedPoissonPartition.finiteMeasureMarkedPoissonLaw ν P₀ R lam)","l":380,"f":"Causalean/Mathlib/Probability/FiniteMarkedPoissonPartition/KL.lean"},{"n":"Causalean.Mathlib.Probability.FiniteMarkedPoissonPartition.finiteMeasureMarkedPoissonLaw_map_count","k":"theorem","m":"Causalean.Mathlib.Probability.FiniteMarkedPoissonPartition.KL","a":"Mathlib/Probability/FiniteMarkedPoissonPartition","t":2,"d":"The count in the finite-measure marked Poisson experiment is Poisson with mean lam times the total mass of the intensity measure.","s":"∀ {X : Type u_1} [inst : MeasurableSpace X] (ν : MeasureTheory.Measure X) [inst_1 : MeasureTheory.IsFiniteMeasure ν] (P₀ : MeasureTheory.Measure X) [inst_2 : MeasureTheory.IsProbabilityMeasure P₀] (R : MeasureTheory.Measure ℝ) [inst_3 : MeasureTheory.IsProbabilityMeasure R] (lam : NNReal), MeasureTheory.Measure.map Causalean.Mathlib.Probability.FiniteMarkedPoissonPartition.FiniteSample.count (Causalean.Mathlib.Probability.FiniteMarkedPoissonPartition.finiteMeasureMarkedPoissonLaw ν P₀ R lam) = ProbabilityTheory.poissonMeasure (lam * Causalean.Mathlib.Probability.FiniteMarkedPoissonPartition.finiteMeasureMass ν)","l":389,"f":"Causalean/Mathlib/Probability/FiniteMarkedPoissonPartition/KL.lean"},{"n":"Causalean.Mathlib.Probability.FiniteMarkedPoissonPartition.finiteMeasureMarkedPoissonLaw_probability_eq","k":"theorem","m":"Causalean.Mathlib.Probability.FiniteMarkedPoissonPartition.Depoissonization","a":"Mathlib/Probability/FiniteMarkedPoissonPartition","t":2,"d":"When the intensity measure is already a probability law, the finite-measure Poisson wrapper agrees with the ordinary marked-Poisson sample law.","s":"∀ {X : Type u_1} [inst : MeasurableSpace X] (P : MeasureTheory.Measure X) [inst_1 : MeasureTheory.IsProbabilityMeasure P] (P0 : MeasureTheory.Measure X) [inst_2 : MeasureTheory.IsProbabilityMeasure P0] (R : MeasureTheory.Measure ℝ) [inst_3 : MeasureTheory.IsProbabilityMeasure R] (lam : NNReal), Causalean.Mathlib.Probability.FiniteMarkedPoissonPartition.finiteMeasureMarkedPoissonLaw P P0 R lam = Causalean.Mathlib.Probability.FiniteMarkedPoissonPartition.finiteMarkedPoissonSampleLaw P R lam","l":383,"f":"Causalean/Mathlib/Probability/FiniteMarkedPoissonPartition/Depoissonization.lean"},{"n":"Causalean.Mathlib.Probability.FiniteMarkedPoissonPartition.finiteMeasureMass","k":"def","m":"Causalean.Mathlib.Probability.FiniteMarkedPoissonPartition.KL","a":"Mathlib/Probability/FiniteMarkedPoissonPartition","t":1,"d":"The total mass of a finite measure as a nonnegative real.","s":"{X : Type u_1} → [inst : MeasurableSpace X] → (ν : MeasureTheory.Measure X) → [MeasureTheory.IsFiniteMeasure ν] → NNReal","l":329,"f":"Causalean/Mathlib/Probability/FiniteMarkedPoissonPartition/KL.lean"},{"n":"Causalean.Mathlib.Probability.FiniteMarkedPoissonPartition.finitePoissonSampleLaw","k":"def","m":"Causalean.Mathlib.Probability.FiniteMarkedPoissonPartition.Basic","a":"Mathlib/Probability/FiniteMarkedPoissonPartition","t":1,"d":"The finite Poisson sample law is obtained by drawing an independent scalar Poisson count and i.i.d. stream and retaining exactly the selected prefix.","s":"{X : Type u_1} → [inst : MeasurableSpace X] → (P : MeasureTheory.Measure X) → [MeasureTheory.IsProbabilityMeasure P] → NNReal → MeasureTheory.Measure (Causalean.Mathlib.Probability.FiniteMarkedPoissonPartition.FiniteSample X)","l":96,"f":"Causalean/Mathlib/Probability/FiniteMarkedPoissonPartition/Basic.lean"},{"n":"Causalean.Mathlib.Probability.FiniteMarkedPoissonPartition.finitePoissonSampleLaw_isProbabilityMeasure","k":"instance","m":"Causalean.Mathlib.Probability.FiniteMarkedPoissonPartition.Basic","a":"Mathlib/Probability/FiniteMarkedPoissonPartition","t":2,"d":"The finite Poisson sample law is a probability measure.","s":"∀ {X : Type u_1} [inst : MeasurableSpace X] (P : MeasureTheory.Measure X) [inst_1 : MeasureTheory.IsProbabilityMeasure P] (lam : NNReal), MeasureTheory.IsProbabilityMeasure (Causalean.Mathlib.Probability.FiniteMarkedPoissonPartition.finitePoissonSampleLaw P lam)","l":102,"f":"Causalean/Mathlib/Probability/FiniteMarkedPoissonPartition/Basic.lean"},{"n":"Causalean.Mathlib.Probability.FiniteMarkedPoissonPartition.finitePoissonSampleLaw_map_count","k":"theorem","m":"Causalean.Mathlib.Probability.FiniteMarkedPoissonPartition.Basic","a":"Mathlib/Probability/FiniteMarkedPoissonPartition","t":2,"d":"The count of a finite Poisson sample has scalar Poisson law with the requested mean.","s":"∀ {X : Type u_1} [inst : MeasurableSpace X] (P : MeasureTheory.Measure X) [inst_1 : MeasureTheory.IsProbabilityMeasure P] (lam : NNReal), MeasureTheory.Measure.map Causalean.Mathlib.Probability.FiniteMarkedPoissonPartition.FiniteSample.count (Causalean.Mathlib.Probability.FiniteMarkedPoissonPartition.finitePoissonSampleLaw P lam) = ProbabilityTheory.poissonMeasure lam","l":108,"f":"Causalean/Mathlib/Probability/FiniteMarkedPoissonPartition/Basic.lean"},{"n":"Causalean.Mathlib.Probability.FiniteMarkedPoissonPartition.finitePoissonSampleLaw_restrict_count_eq","k":"theorem","m":"Causalean.Mathlib.Probability.FiniteMarkedPoissonPartition.Basic","a":"Mathlib/Probability/FiniteMarkedPoissonPartition","t":2,"d":"On the fibre where the count equals n, the finite Poisson law is the Poisson mass at n times the embedded n-fold product law.","s":"∀ {X : Type u_1} [inst : MeasurableSpace X] (P : MeasureTheory.Measure X) [inst_1 : MeasureTheory.IsProbabilityMeasure P] (lam : NNReal) (n : ℕ), (Causalean.Mathlib.Probability.FiniteMarkedPoissonPartition.finitePoissonSampleLaw P lam).restrict (Causalean.Mathlib.Probability.FiniteMarkedPoissonPartition.FiniteSample.count ⁻¹' {n}) = (ProbabilityTheory.poissonMeasure lam) {n} • MeasureTheory.Measure.map (Causalean.Mathlib.Probability.FiniteMarkedPoissonPartition.fixedSizeEmbed n) (MeasureTheory.Measure.pi fun x => P)","l":118,"f":"Causalean/Mathlib/Probability/FiniteMarkedPoissonPartition/Basic.lean"},{"n":"Causalean.Mathlib.Probability.FiniteMarkedPoissonPartition.finiteSampleMap","k":"def","m":"Causalean.Mathlib.Probability.FiniteMarkedPoissonPartition.Depoissonization","a":"Mathlib/Probability/FiniteMarkedPoissonPartition","t":1,"d":"Mapping every point of a finite sample preserves its random count.","s":"{X : Type u_1} → {Y : Type u_2} → [inst : MeasurableSpace X] → [inst_1 : MeasurableSpace Y] → (X → Y) → Causalean.Mathlib.Probability.FiniteMarkedPoissonPartition.FiniteSample X → Causalean.Mathlib.Probability.FiniteMarkedPoissonPartition.FiniteSample Y","l":27,"f":"Causalean/Mathlib/Probability/FiniteMarkedPoissonPartition/Depoissonization.lean"},{"n":"Causalean.Mathlib.Probability.FiniteMarkedPoissonPartition.finiteSampleMap_fixedSizeEmbed","k":"theorem","m":"Causalean.Mathlib.Probability.FiniteMarkedPoissonPartition.Depoissonization","a":"Mathlib/Probability/FiniteMarkedPoissonPartition","t":2,"d":"Mapping commutes with fixed-size embedding.","s":"∀ {X : Type u_1} {Y : Type u_2} [inst : MeasurableSpace X] [inst_1 : MeasurableSpace Y] (f : X → Y) (n : ℕ) (x : Fin n → X), Causalean.Mathlib.Probability.FiniteMarkedPoissonPartition.finiteSampleMap f (Causalean.Mathlib.Probability.FiniteMarkedPoissonPartition.fixedSizeEmbed n x) = Causalean.Mathlib.Probability.FiniteMarkedPoissonPartition.fixedSizeEmbed n fun i => f (x i)","l":50,"f":"Causalean/Mathlib/Probability/FiniteMarkedPoissonPartition/Depoissonization.lean"},{"n":"Causalean.Mathlib.Probability.FiniteMarkedPoissonPartition.finiteSamplePaddedStream","k":"def","m":"Causalean.Mathlib.Probability.FiniteMarkedPoissonPartition.Depoissonization","a":"Mathlib/Probability/FiniteMarkedPoissonPartition","t":1,"d":"A finite sample embeds into a count-and-stream representation by padding all coordinates beyond its count with a fixed fallback observation.","s":"{X : Type u_1} → [inst : MeasurableSpace X] → X → Causalean.Mathlib.Probability.FiniteMarkedPoissonPartition.FiniteSample X → ℕ × (ℕ → X)","l":258,"f":"Causalean/Mathlib/Probability/FiniteMarkedPoissonPartition/Depoissonization.lean"},{"n":"Causalean.Mathlib.Probability.FiniteMarkedPoissonPartition.finiteSamplePaddedStream_measurable","k":"theorem","m":"Causalean.Mathlib.Probability.FiniteMarkedPoissonPartition.Depoissonization","a":"Mathlib/Probability/FiniteMarkedPoissonPartition","t":2,"d":"","s":"∀ {X : Type u_1} [inst : MeasurableSpace X] (x0 : X), Measurable (Causalean.Mathlib.Probability.FiniteMarkedPoissonPartition.finiteSamplePaddedStream x0)","l":266,"f":"Causalean/Mathlib/Probability/FiniteMarkedPoissonPartition/Depoissonization.lean"},{"n":"Causalean.Mathlib.Probability.FiniteMarkedPoissonPartition.finiteSamplePaddedStream_range","k":"theorem","m":"Causalean.Mathlib.Probability.FiniteMarkedPoissonPartition.Depoissonization","a":"Mathlib/Probability/FiniteMarkedPoissonPartition","t":2,"d":"","s":"∀ {X : Type u_1} [inst : MeasurableSpace X] (x0 : X), Set.range (Causalean.Mathlib.Probability.FiniteMarkedPoissonPartition.finiteSamplePaddedStream x0) = {z | ∀ (k : ℕ), z.1 ≤ k → z.2 k = x0}","l":296,"f":"Causalean/Mathlib/Probability/FiniteMarkedPoissonPartition/Depoissonization.lean"},{"n":"Causalean.Mathlib.Probability.FiniteMarkedPoissonPartition.finiteSample_standardBorelSpace","k":"instance","m":"Causalean.Mathlib.Probability.FiniteMarkedPoissonPartition.Depoissonization","a":"Mathlib/Probability/FiniteMarkedPoissonPartition","t":2,"d":"Finite samples over a nonempty standard Borel space are standard Borel. The explicit padded-stream presentation supplies the compatible Polish topology missing from the generic dependent-sum instance.","s":"∀ {X : Type u_1} [inst : MeasurableSpace X] [StandardBorelSpace X] [Nonempty X], StandardBorelSpace (Causalean.Mathlib.Probability.FiniteMarkedPoissonPartition.FiniteSample X)","l":316,"f":"Causalean/Mathlib/Probability/FiniteMarkedPoissonPartition/Depoissonization.lean"},{"n":"Causalean.Mathlib.Probability.FiniteMarkedPoissonPartition.fixedSizeEmbed","k":"def","m":"Causalean.Mathlib.Probability.FiniteMarkedPoissonPartition.Basic","a":"Mathlib/Probability/FiniteMarkedPoissonPartition","t":1,"d":"Embed a fixed-length tuple into the space of all finite sequences.","s":"{X : Type u_1} → [inst : MeasurableSpace X] → (n : ℕ) → (Fin n → X) → Causalean.Mathlib.Probability.FiniteMarkedPoissonPartition.FiniteSample X","l":44,"f":"Causalean/Mathlib/Probability/FiniteMarkedPoissonPartition/Basic.lean"},{"n":"Causalean.Mathlib.Probability.FiniteMarkedPoissonPartition.iidStreamLaw","k":"def","m":"Causalean.Mathlib.Probability.FiniteMarkedPoissonPartition.IIDPoisson","a":"Mathlib/Probability/FiniteMarkedPoissonPartition","t":1,"d":"The law of an infinite i.i.d. stream with one-coordinate law P.","s":"{X : Type u_1} → [inst : MeasurableSpace X] → (P : MeasureTheory.Measure X) → [MeasureTheory.IsProbabilityMeasure P] → MeasureTheory.Measure (ℕ → X)","l":20,"f":"Causalean/Mathlib/Probability/FiniteMarkedPoissonPartition/IIDPoisson.lean"},{"n":"Causalean.Mathlib.Probability.FiniteMarkedPoissonPartition.iidStreamLaw_isProbabilityMeasure","k":"instance","m":"Causalean.Mathlib.Probability.FiniteMarkedPoissonPartition.IIDPoisson","a":"Mathlib/Probability/FiniteMarkedPoissonPartition","t":2,"d":"The infinite i.i.d. stream law is a probability measure.","s":"∀ {X : Type u_1} [inst : MeasurableSpace X] (P : MeasureTheory.Measure X) [inst_1 : MeasureTheory.IsProbabilityMeasure P], MeasureTheory.IsProbabilityMeasure (Causalean.Mathlib.Probability.FiniteMarkedPoissonPartition.iidStreamLaw P)","l":25,"f":"Causalean/Mathlib/Probability/FiniteMarkedPoissonPartition/IIDPoisson.lean"},{"n":"Causalean.Mathlib.Probability.FiniteMarkedPoissonPartition.iidStreamLaw_map_finPrefix","k":"theorem","m":"Causalean.Mathlib.Probability.FiniteMarkedPoissonPartition.IIDPoisson","a":"Mathlib/Probability/FiniteMarkedPoissonPartition","t":1,"d":"Every finite prefix of length n of an infinite stream whose coordinates are i.i.d. with common law P has exactly the corresponding n-fold product law.","s":"∀ {X : Type u_1} [inst : MeasurableSpace X] (P : MeasureTheory.Measure X) [inst_1 : MeasureTheory.IsProbabilityMeasure P] (n : ℕ), MeasureTheory.Measure.map (fun z i => z ↑i) (Causalean.Mathlib.Probability.FiniteMarkedPoissonPartition.iidStreamLaw P) = MeasureTheory.Measure.pi fun x => P","l":31,"f":"Causalean/Mathlib/Probability/FiniteMarkedPoissonPartition/IIDPoisson.lean"},{"n":"Causalean.Mathlib.Probability.FiniteMarkedPoissonPartition.klDiv_finiteMeasureMarkedPoissonLaw","k":"theorem","m":"Causalean.Mathlib.Probability.FiniteMarkedPoissonPartition.KL","a":"Mathlib/Probability/FiniteMarkedPoissonPartition","t":1,"d":"Fix a nonnegative Poisson rate lam and suppose the finite intensity measures ν₀ and ν₁ have equal total mass. Then, for a common baseline probability measure P₀ and mark law R, the KL divergence between the finite-measure marked Poisson experiments generated by ν₀ and by ν₁ equals lam times the KL divergence between ν₀ and ν₁.","s":"∀ {X : Type u_1} [inst : MeasurableSpace X] [StandardBorelSpace X] (ν₀ ν₁ : MeasureTheory.Measure X) [inst_2 : MeasureTheory.IsFiniteMeasure ν₀] [inst_3 : MeasureTheory.IsFiniteMeasure ν₁] (P₀ : MeasureTheory.Measure X) [inst_4 : MeasureTheory.IsProbabilityMeasure P₀] (R : MeasureTheory.Measure ℝ) [inst_5 : MeasureTheory.IsProbabilityMeasure R] (lam : NNReal), ν₀ Set.univ = ν₁ Set.univ → InformationTheory.klDiv (Causalean.Mathlib.Probability.FiniteMarkedPoissonPartition.finiteMeasureMarkedPoissonLaw ν₀ P₀ R lam) (Causalean.Mathlib.Probability.FiniteMarkedPoissonPartition.finiteMeasureMarkedPoissonLaw ν₁ P₀ R lam) = ↑lam * InformationTheory.klDiv ν₀ ν₁","l":400,"f":"Causalean/Mathlib/Probability/FiniteMarkedPoissonPartition/KL.lean"},{"n":"Causalean.Mathlib.Probability.FiniteMarkedPoissonPartition.klDiv_finiteMeasureMarkedPoissonLaw_le","k":"theorem","m":"Causalean.Mathlib.Probability.FiniteMarkedPoissonPartition.KL","a":"Mathlib/Probability/FiniteMarkedPoissonPartition","t":2,"d":"A one-point finite-measure KL bound transfers directly to the corresponding marked Poisson experiments after multiplication by the scalar intensity.","s":"∀ {X : Type u_1} [inst : MeasurableSpace X] [StandardBorelSpace X] (ν₀ ν₁ : MeasureTheory.Measure X) [inst_2 : MeasureTheory.IsFiniteMeasure ν₀] [inst_3 : MeasureTheory.IsFiniteMeasure ν₁] (P₀ : MeasureTheory.Measure X) [inst_4 : MeasureTheory.IsProbabilityMeasure P₀] (R : MeasureTheory.Measure ℝ) [inst_5 : MeasureTheory.IsProbabilityMeasure R] (lam : NNReal) (B : ENNReal), ν₀ Set.univ = ν₁ Set.univ → InformationTheory.klDiv ν₀ ν₁ ≤ B → InformationTheory.klDiv (Causalean.Mathlib.Probability.FiniteMarkedPoissonPartition.finiteMeasureMarkedPoissonLaw ν₀ P₀ R lam) (Causalean.Mathlib.Probability.FiniteMarkedPoissonPartition.finiteMeasureMarkedPoissonLaw ν₁ P₀ R lam) ≤ ↑lam * B","l":459,"f":"Causalean/Mathlib/Probability/FiniteMarkedPoissonPartition/KL.lean"},{"n":"Causalean.Mathlib.Probability.FiniteMarkedPoissonPartition.map_canonicalPrefixObservations_restrict_count_ge","k":"theorem","m":"Causalean.Mathlib.Probability.FiniteMarkedPoissonPartition.Depoissonization","a":"Mathlib/Probability/FiniteMarkedPoissonPartition","t":2,"d":"On the successful-count event, the canonical marked-Poisson configuration's first n observations have the unnormalised product law.","s":"∀ {X : Type u_1} [inst : MeasurableSpace X] (P : MeasureTheory.Measure X) [inst_1 : MeasureTheory.IsProbabilityMeasure P] (R : MeasureTheory.Measure ℝ) [inst_2 : MeasureTheory.IsProbabilityMeasure R] [MeasureTheory.NullSingletonClass R] (lam : NNReal) (x₀ : X) (n : ℕ), MeasureTheory.Measure.map (Causalean.Mathlib.Probability.FiniteMarkedPoissonPartition.canonicalPrefixObservations x₀ n) ((Causalean.Mathlib.Probability.FiniteMarkedPoissonPartition.canonicalMarkedPoissonSampleLaw P R lam).restrict (Causalean.Mathlib.Probability.FiniteMarkedPoissonPartition.FiniteSample.count ⁻¹' Set.Ici n)) = (ProbabilityTheory.poissonMeasure lam) (Set.Ici n) • MeasureTheory.Measure.pi fun x => P","l":158,"f":"Causalean/Mathlib/Probability/FiniteMarkedPoissonPartition/Depoissonization.lean"},{"n":"Causalean.Mathlib.Probability.FiniteMarkedPoissonPartition.map_restrictPartition_canonicalMarkedPoissonSampleLaw","k":"theorem","m":"Causalean.Mathlib.Probability.FiniteMarkedPoissonPartition.Superposition.Canonical","a":"Mathlib/Probability/FiniteMarkedPoissonPartition/Superposition","t":2,"d":"Restricting the canonical global configuration gives exactly the product of independent canonical cell configurations.","s":"∀ {X : Type u_1} [inst : MeasurableSpace X] {ι : Type u_2} [inst_1 : Fintype ι] [inst_2 : MeasurableSpace ι] [MeasurableSingletonClass ι] [StandardBorelSpace X] (p : Causalean.Mathlib.Probability.FiniteMarkedPoissonPartition.FiniteMeasurablePartition X ι) (P : MeasureTheory.Measure X) [inst_5 : MeasureTheory.IsProbabilityMeasure P] (R : MeasureTheory.Measure ℝ) [inst_6 : MeasureTheory.IsProbabilityMeasure R] [MeasureTheory.NullSingletonClass R] (lam : NNReal), MeasureTheory.Measure.map p.restrictPartition (Causalean.Mathlib.Probability.FiniteMarkedPoissonPartition.canonicalMarkedPoissonSampleLaw P R lam) = MeasureTheory.Measure.pi fun j => Causalean.Mathlib.Probability.FiniteMarkedPoissonPartition.canonicalMarkedPoissonSampleLaw (p.cellObservationLaw P j) R (lam * p.cellMass P j)","l":772,"f":"Causalean/Mathlib/Probability/FiniteMarkedPoissonPartition/Superposition/Canonical.lean"},{"n":"Causalean.Mathlib.Probability.FiniteMarkedPoissonPartition.map_retainedObservations_restrict_count_ge","k":"theorem","m":"Causalean.Mathlib.Probability.FiniteMarkedPoissonPartition.Superposition.Retention","a":"Mathlib/Probability/FiniteMarkedPoissonPartition/Superposition","t":2,"d":"On the event that at least n points exist, retaining the n smallest atomless independent marks and forgetting marks gives the event probability times the exact product law P^n.","s":"∀ {X : Type u_1} [inst : MeasurableSpace X] (P : MeasureTheory.Measure X) [inst_1 : MeasureTheory.IsProbabilityMeasure P] (R : MeasureTheory.Measure ℝ) [inst_2 : MeasureTheory.IsProbabilityMeasure R] [MeasureTheory.NullSingletonClass R] (lam : NNReal) (x₀ : X) (n : ℕ), MeasureTheory.Measure.map (Causalean.Mathlib.Probability.FiniteMarkedPoissonPartition.retainedObservations x₀ n) ((Causalean.Mathlib.Probability.FiniteMarkedPoissonPartition.finiteMarkedPoissonSampleLaw P R lam).restrict (Causalean.Mathlib.Probability.FiniteMarkedPoissonPartition.FiniteSample.count ⁻¹' Set.Ici n)) = (ProbabilityTheory.poissonMeasure lam) (Set.Ici n) • MeasureTheory.Measure.pi fun x => P","l":195,"f":"Causalean/Mathlib/Probability/FiniteMarkedPoissonPartition/Superposition/Retention.lean"},{"n":"Causalean.Mathlib.Probability.FiniteMarkedPoissonPartition.map_superposeByMarks_canonicalCellLaws","k":"theorem","m":"Causalean.Mathlib.Probability.FiniteMarkedPoissonPartition.Superposition.Canonical","a":"Mathlib/Probability/FiniteMarkedPoissonPartition/Superposition","t":1,"d":"Superposition recovers the canonical global law. Under nonnegative intensity lam, and independently for each cell j of the finite measurable partition p, draw a canonical marked Poisson configuration with base measure p.cellObservationLaw P j, mark distribution R, and intensity lam times the P-mass of cell j; merging these per-cell configurations by increasing mark (mark-ordered superposition) has exactly the law of the canonical marked Poisson configuration with base measure P, mark distribution R, and intensity lam.","s":"∀ {X : Type u_1} [inst : MeasurableSpace X] {ι : Type u_2} [inst_1 : Fintype ι] [inst_2 : MeasurableSpace ι] [MeasurableSingletonClass ι] [StandardBorelSpace X] (p : Causalean.Mathlib.Probability.FiniteMarkedPoissonPartition.FiniteMeasurablePartition X ι) (P : MeasureTheory.Measure X) [inst_5 : MeasureTheory.IsProbabilityMeasure P] (R : MeasureTheory.Measure ℝ) [inst_6 : MeasureTheory.IsProbabilityMeasure R] [MeasureTheory.NullSingletonClass R] (lam : NNReal), MeasureTheory.Measure.map Causalean.Mathlib.Probability.FiniteMarkedPoissonPartition.superposeByMarks (MeasureTheory.Measure.pi fun j => Causalean.Mathlib.Probability.FiniteMarkedPoissonPartition.canonicalMarkedPoissonSampleLaw (p.cellObservationLaw P j) R (lam * p.cellMass P j)) = Causalean.Mathlib.Probability.FiniteMarkedPoissonPartition.canonicalMarkedPoissonSampleLaw P R lam","l":834,"f":"Causalean/Mathlib/Probability/FiniteMarkedPoissonPartition/Superposition/Canonical.lean"},{"n":"Causalean.Mathlib.Probability.FiniteMarkedPoissonPartition.map_superposeByMarks_map_restrictPartition","k":"theorem","m":"Causalean.Mathlib.Probability.FiniteMarkedPoissonPartition.Superposition.Canonical","a":"Mathlib/Probability/FiniteMarkedPoissonPartition/Superposition","t":2,"d":"Restriction followed by mark-ordered superposition is the identity in law on the canonical marked Poisson configuration.","s":"∀ {X : Type u_1} [inst : MeasurableSpace X] {ι : Type u_2} [inst_1 : Fintype ι] [inst_2 : MeasurableSpace ι] [MeasurableSingletonClass ι] [StandardBorelSpace X] (p : Causalean.Mathlib.Probability.FiniteMarkedPoissonPartition.FiniteMeasurablePartition X ι) (P : MeasureTheory.Measure X) [inst_5 : MeasureTheory.IsProbabilityMeasure P] (R : MeasureTheory.Measure ℝ) [inst_6 : MeasureTheory.IsProbabilityMeasure R] [MeasureTheory.NullSingletonClass R] (lam : NNReal), MeasureTheory.Measure.map Causalean.Mathlib.Probability.FiniteMarkedPoissonPartition.superposeByMarks (MeasureTheory.Measure.map p.restrictPartition (Causalean.Mathlib.Probability.FiniteMarkedPoissonPartition.canonicalMarkedPoissonSampleLaw P R lam)) = Causalean.Mathlib.Probability.FiniteMarkedPoissonPartition.canonicalMarkedPoissonSampleLaw P R lam","l":809,"f":"Causalean/Mathlib/Probability/FiniteMarkedPoissonPartition/Superposition/Canonical.lean"},{"n":"Causalean.Mathlib.Probability.FiniteMarkedPoissonPartition.markedKey_decode","k":"theorem","m":"Causalean.Mathlib.Probability.FiniteMarkedPoissonPartition.Superposition.Canonical","a":"Mathlib/Probability/FiniteMarkedPoissonPartition/Superposition","t":2,"d":"A key belonging to the mark-ordering set is recovered by decoding its stored original index and rebuilding its mark-and-index pair.","s":"∀ {X : Type u_1} [inst : MeasurableSpace X] (s : Causalean.Mathlib.Probability.FiniteMarkedPoissonPartition.FiniteSample (X × ℝ)) {z : Lex (ℝ × Fin s.count)}, z ∈ Causalean.Mathlib.Probability.FiniteMarkedPoissonPartition.markedKeys s → toLex ((s.points (ofLex z).2).2, (ofLex z).2) = z","l":188,"f":"Causalean/Mathlib/Probability/FiniteMarkedPoissonPartition/Superposition/Canonical.lean"},{"n":"Causalean.Mathlib.Probability.FiniteMarkedPoissonPartition.markedKeys","k":"def","m":"Causalean.Mathlib.Probability.FiniteMarkedPoissonPartition.Superposition.Canonical","a":"Mathlib/Probability/FiniteMarkedPoissonPartition/Superposition","t":1,"d":"The finite set of (mark, originalIndex) keys used to order a marked sample; the index makes all keys distinct even on the tie event.","s":"{X : Type u_1} → [inst : MeasurableSpace X] → (s : Causalean.Mathlib.Probability.FiniteMarkedPoissonPartition.FiniteSample (X × ℝ)) → Finset (Lex (ℝ × Fin s.count))","l":171,"f":"Causalean/Mathlib/Probability/FiniteMarkedPoissonPartition/Superposition/Canonical.lean"},{"n":"Causalean.Mathlib.Probability.FiniteMarkedPoissonPartition.markedKeys_card","k":"theorem","m":"Causalean.Mathlib.Probability.FiniteMarkedPoissonPartition.Superposition.Canonical","a":"Mathlib/Probability/FiniteMarkedPoissonPartition/Superposition","t":2,"d":"The number of lexicographic mark-and-index keys equals the number of marked observations in the finite sample.","s":"∀ {X : Type u_1} [inst : MeasurableSpace X] (s : Causalean.Mathlib.Probability.FiniteMarkedPoissonPartition.FiniteSample (X × ℝ)), (Causalean.Mathlib.Probability.FiniteMarkedPoissonPartition.markedKeys s).card = s.count","l":178,"f":"Causalean/Mathlib/Probability/FiniteMarkedPoissonPartition/Superposition/Canonical.lean"},{"n":"Causalean.Mathlib.Probability.FiniteMarkedPoissonPartition.markedPoissonKL_le_two_mul_of_piKL","k":"theorem","m":"Causalean.Mathlib.Probability.FiniteMarkedPoissonPartition.Depoissonization","a":"Mathlib/Probability/FiniteMarkedPoissonPartition","t":1,"d":"Consider a sample size n that is at least 1 and a nonnegative KL budget B, and suppose the KL divergence between n independent identically distributed draws from P and from Q is at most B. Then the KL divergence between the marked Poisson experiments with mean count 2n, mark law R, and intensity measures P and Q respectively (both built over the same baseline P) is at most 2B.","s":"∀ {X : Type u_1} [inst : MeasurableSpace X] [StandardBorelSpace X] (P Q : MeasureTheory.Measure X) [inst_2 : MeasureTheory.IsProbabilityMeasure P] [inst_3 : MeasureTheory.IsProbabilityMeasure Q] (R : MeasureTheory.Measure ℝ) [inst_4 : MeasureTheory.IsProbabilityMeasure R] (n : ℕ), 1 ≤ n → ∀ {B : ℝ}, 0 ≤ B → InformationTheory.klDiv (MeasureTheory.Measure.pi fun x => P) (MeasureTheory.Measure.pi fun x => Q) ≤ ENNReal.ofReal B → InformationTheory.klDiv (Causalean.Mathlib.Probability.FiniteMarkedPoissonPartition.finiteMeasureMarkedPoissonLaw P P R (2 * ↑n)) (Causalean.Mathlib.Probability.FiniteMarkedPoissonPartition.finiteMeasureMarkedPoissonLaw Q P R (2 * ↑n)) ≤ ENNReal.ofReal (2 * B)","l":72,"f":"Causalean/Mathlib/Probability/FiniteMarkedPoissonPartition/Depoissonization.lean"},{"n":"Causalean.Mathlib.Probability.FiniteMarkedPoissonPartition.measurable_canonicalPrefixObservations","k":"theorem","m":"Causalean.Mathlib.Probability.FiniteMarkedPoissonPartition.Depoissonization","a":"Mathlib/Probability/FiniteMarkedPoissonPartition","t":2,"d":"Reading a fixed prefix from a canonical finite configuration is measurable.","s":"∀ {X : Type u_1} [inst : MeasurableSpace X] (x₀ : X) (n : ℕ), Measurable (Causalean.Mathlib.Probability.FiniteMarkedPoissonPartition.canonicalPrefixObservations x₀ n)","l":139,"f":"Causalean/Mathlib/Probability/FiniteMarkedPoissonPartition/Depoissonization.lean"},{"n":"Causalean.Mathlib.Probability.FiniteMarkedPoissonPartition.measurable_finiteSampleMap","k":"theorem","m":"Causalean.Mathlib.Probability.FiniteMarkedPoissonPartition.Depoissonization","a":"Mathlib/Probability/FiniteMarkedPoissonPartition","t":2,"d":"Pointwise mapping of dependent finite samples is measurable.","s":"∀ {X : Type u_1} {Y : Type u_2} [inst : MeasurableSpace X] [inst_1 : MeasurableSpace Y] (f : X → Y), Measurable f → Measurable (Causalean.Mathlib.Probability.FiniteMarkedPoissonPartition.finiteSampleMap f)","l":33,"f":"Causalean/Mathlib/Probability/FiniteMarkedPoissonPartition/Depoissonization.lean"},{"n":"Causalean.Mathlib.Probability.FiniteMarkedPoissonPartition.measurable_finiteSample_count","k":"theorem","m":"Causalean.Mathlib.Probability.FiniteMarkedPoissonPartition.Basic","a":"Mathlib/Probability/FiniteMarkedPoissonPartition","t":2,"d":"Reading the count of a finite sequence is measurable.","s":"∀ {X : Type u_1} [inst : MeasurableSpace X], Measurable Causalean.Mathlib.Probability.FiniteMarkedPoissonPartition.FiniteSample.count","l":57,"f":"Causalean/Mathlib/Probability/FiniteMarkedPoissonPartition/Basic.lean"},{"n":"Causalean.Mathlib.Probability.FiniteMarkedPoissonPartition.measurable_fixedSizeEmbed","k":"theorem","m":"Causalean.Mathlib.Probability.FiniteMarkedPoissonPartition.Basic","a":"Mathlib/Probability/FiniteMarkedPoissonPartition","t":2,"d":"Embedding a fixed-size tuple into the finite-sequence space is measurable.","s":"∀ {X : Type u_1} [inst : MeasurableSpace X] (n : ℕ), Measurable (Causalean.Mathlib.Probability.FiniteMarkedPoissonPartition.fixedSizeEmbed n)","l":51,"f":"Causalean/Mathlib/Probability/FiniteMarkedPoissonPartition/Basic.lean"},{"n":"Causalean.Mathlib.Probability.FiniteMarkedPoissonPartition.measurable_orderByMarks","k":"theorem","m":"Causalean.Mathlib.Probability.FiniteMarkedPoissonPartition.Superposition.Canonical","a":"Mathlib/Probability/FiniteMarkedPoissonPartition/Superposition","t":2,"d":"The mark-ordering map is measurable.","s":"∀ {X : Type u_1} [inst : MeasurableSpace X], Measurable Causalean.Mathlib.Probability.FiniteMarkedPoissonPartition.orderByMarks","l":215,"f":"Causalean/Mathlib/Probability/FiniteMarkedPoissonPartition/Superposition/Canonical.lean"},{"n":"Causalean.Mathlib.Probability.FiniteMarkedPoissonPartition.measurable_retainedObservations","k":"theorem","m":"Causalean.Mathlib.Probability.FiniteMarkedPoissonPartition.Superposition.Retention","a":"Mathlib/Probability/FiniteMarkedPoissonPartition/Superposition","t":2,"d":"Retaining and forgetting marks is a measurable map to an n-tuple.","s":"∀ {X : Type u_1} [inst : MeasurableSpace X] (x₀ : X) (n : ℕ), Measurable (Causalean.Mathlib.Probability.FiniteMarkedPoissonPartition.retainedObservations x₀ n)","l":46,"f":"Causalean/Mathlib/Probability/FiniteMarkedPoissonPartition/Superposition/Retention.lean"},{"n":"Causalean.Mathlib.Probability.FiniteMarkedPoissonPartition.measurable_streamToFiniteSample","k":"theorem","m":"Causalean.Mathlib.Probability.FiniteMarkedPoissonPartition.Basic","a":"Mathlib/Probability/FiniteMarkedPoissonPartition","t":2,"d":"Truncating a count-and-stream outcome to its selected prefix is measurable.","s":"∀ {X : Type u_1} [inst : MeasurableSpace X], Measurable Causalean.Mathlib.Probability.FiniteMarkedPoissonPartition.streamToFiniteSample","l":72,"f":"Causalean/Mathlib/Probability/FiniteMarkedPoissonPartition/Basic.lean"},{"n":"Causalean.Mathlib.Probability.FiniteMarkedPoissonPartition.measurable_superpose","k":"theorem","m":"Causalean.Mathlib.Probability.FiniteMarkedPoissonPartition.Superposition.Canonical","a":"Mathlib/Probability/FiniteMarkedPoissonPartition/Superposition","t":2,"d":"Finite superposition is measurable.","s":"∀ {X : Type u_1} [inst : MeasurableSpace X] {ι : Type u_2} [inst_1 : Fintype ι] [inst_2 : MeasurableSpace ι] [MeasurableSingletonClass ι], Measurable Causalean.Mathlib.Probability.FiniteMarkedPoissonPartition.superpose","l":146,"f":"Causalean/Mathlib/Probability/FiniteMarkedPoissonPartition/Superposition/Canonical.lean"},{"n":"Causalean.Mathlib.Probability.FiniteMarkedPoissonPartition.measurable_superposeByMarks","k":"theorem","m":"Causalean.Mathlib.Probability.FiniteMarkedPoissonPartition.Superposition.Canonical","a":"Mathlib/Probability/FiniteMarkedPoissonPartition/Superposition","t":2,"d":"Mark-ordered finite superposition is measurable.","s":"∀ {X : Type u_1} [inst : MeasurableSpace X] {ι : Type u_2} [inst_1 : Fintype ι] [inst_2 : MeasurableSpace ι] [MeasurableSingletonClass ι], Measurable Causalean.Mathlib.Probability.FiniteMarkedPoissonPartition.superposeByMarks","l":321,"f":"Causalean/Mathlib/Probability/FiniteMarkedPoissonPartition/Superposition/Canonical.lean"},{"n":"Causalean.Mathlib.Probability.FiniteMarkedPoissonPartition.normalizedFiniteMeasure","k":"def","m":"Causalean.Mathlib.Probability.FiniteMarkedPoissonPartition.KL","a":"Mathlib/Probability/FiniteMarkedPoissonPartition","t":1,"d":"Normalise a nonzero finite measure to a probability measure, using the supplied probability law only on the zero-measure branch.","s":"{X : Type u_1} → [inst : MeasurableSpace X] → (ν : MeasureTheory.Measure X) → [MeasureTheory.IsFiniteMeasure ν] → (P₀ : MeasureTheory.Measure X) → [MeasureTheory.IsProbabilityMeasure P₀] → MeasureTheory.Measure X","l":333,"f":"Causalean/Mathlib/Probability/FiniteMarkedPoissonPartition/KL.lean"},{"n":"Causalean.Mathlib.Probability.FiniteMarkedPoissonPartition.normalizedFiniteMeasure_isProbabilityMeasure","k":"instance","m":"Causalean.Mathlib.Probability.FiniteMarkedPoissonPartition.KL","a":"Mathlib/Probability/FiniteMarkedPoissonPartition","t":2,"d":"The normalisation of a finite measure is a probability measure, including the explicit fallback on the zero branch.","s":"∀ {X : Type u_1} [inst : MeasurableSpace X] (ν : MeasureTheory.Measure X) [inst_1 : MeasureTheory.IsFiniteMeasure ν] (P₀ : MeasureTheory.Measure X) [inst_2 : MeasureTheory.IsProbabilityMeasure P₀], MeasureTheory.IsProbabilityMeasure (Causalean.Mathlib.Probability.FiniteMarkedPoissonPartition.normalizedFiniteMeasure ν P₀)","l":340,"f":"Causalean/Mathlib/Probability/FiniteMarkedPoissonPartition/KL.lean"},{"n":"Causalean.Mathlib.Probability.FiniteMarkedPoissonPartition.normalized_map_retainedObservations_restrict_count_ge","k":"theorem","m":"Causalean.Mathlib.Probability.FiniteMarkedPoissonPartition.Superposition.Retention","a":"Mathlib/Probability/FiniteMarkedPoissonPartition/Superposition","t":1,"d":"Conditioning on enough points gives i.i.d. draws. Fix a nonnegative intensity lam and suppose the Poisson(lam) probability of observing at least n points is nonzero. Under the marked Poisson sample law with base probability measure P, mark distribution R, and intensity lam, condition on the event that the sample count is at least n, retain the n mark-smallest points and forget their marks (x₀ is an irrelevant filler value used only outside this event); the resulting normalised law equals the product of n independent copies of P.","s":"∀ {X : Type u_1} [inst : MeasurableSpace X] (P : MeasureTheory.Measure X) [inst_1 : MeasureTheory.IsProbabilityMeasure P] (R : MeasureTheory.Measure ℝ) [inst_2 : MeasureTheory.IsProbabilityMeasure R] [MeasureTheory.NullSingletonClass R] (lam : NNReal) (x₀ : X) (n : ℕ), (ProbabilityTheory.poissonMeasure lam) (Set.Ici n) ≠ 0 → ((ProbabilityTheory.poissonMeasure lam) (Set.Ici n))⁻¹ • MeasureTheory.Measure.map (Causalean.Mathlib.Probability.FiniteMarkedPoissonPartition.retainedObservations x₀ n) ((Causalean.Mathlib.Probability.FiniteMarkedPoissonPartition.finiteMarkedPoissonSampleLaw P R lam).restrict (Causalean.Mathlib.Probability.FiniteMarkedPoissonPartition.FiniteSample.count ⁻¹' Set.Ici n)) = MeasureTheory.Measure.pi fun x => P","l":235,"f":"Causalean/Mathlib/Probability/FiniteMarkedPoissonPartition/Superposition/Retention.lean"},{"n":"Causalean.Mathlib.Probability.FiniteMarkedPoissonPartition.orderByMarks","k":"def","m":"Causalean.Mathlib.Probability.FiniteMarkedPoissonPartition.Superposition.Canonical","a":"Mathlib/Probability/FiniteMarkedPoissonPartition/Superposition","t":1,"d":"Order a finite marked sequence increasingly by mark, breaking mark ties by the original coordinate index.","s":"{X : Type u_1} → [inst : MeasurableSpace X] → Causalean.Mathlib.Probability.FiniteMarkedPoissonPartition.FiniteSample (X × ℝ) → Causalean.Mathlib.Probability.FiniteMarkedPoissonPartition.FiniteSample (X × ℝ)","l":200,"f":"Causalean/Mathlib/Probability/FiniteMarkedPoissonPartition/Superposition/Canonical.lean"},{"n":"Causalean.Mathlib.Probability.FiniteMarkedPoissonPartition.orderByMarks_count","k":"theorem","m":"Causalean.Mathlib.Probability.FiniteMarkedPoissonPartition.Superposition.Canonical","a":"Mathlib/Probability/FiniteMarkedPoissonPartition/Superposition","t":2,"d":"Ordering by marks preserves the sample count.","s":"∀ {X : Type u_1} [inst : MeasurableSpace X] (s : Causalean.Mathlib.Probability.FiniteMarkedPoissonPartition.FiniteSample (X × ℝ)), (Causalean.Mathlib.Probability.FiniteMarkedPoissonPartition.orderByMarks s).count = s.count","l":210,"f":"Causalean/Mathlib/Probability/FiniteMarkedPoissonPartition/Superposition/Canonical.lean"},{"n":"Causalean.Mathlib.Probability.FiniteMarkedPoissonPartition.orderByMarks_monotone_marks","k":"theorem","m":"Causalean.Mathlib.Probability.FiniteMarkedPoissonPartition.Superposition.Canonical","a":"Mathlib/Probability/FiniteMarkedPoissonPartition/Superposition","t":2,"d":"Marks in the ordered sequence are nondecreasing.","s":"∀ {X : Type u_1} [inst : MeasurableSpace X] (s : Causalean.Mathlib.Probability.FiniteMarkedPoissonPartition.FiniteSample (X × ℝ)) {a b : Fin s.count}, a ≤ b → ((Causalean.Mathlib.Probability.FiniteMarkedPoissonPartition.orderByMarks s).points (Fin.cast (Eq.symm (Causalean.Mathlib.Probability.FiniteMarkedPoissonPartition.orderByMarks_count s)) a)).2 ≤ ((Causalean.Mathlib.Probability.FiniteMarkedPoissonPartition.orderByMarks s).points (Fin.cast (Eq.symm (Causalean.Mathlib.Probability.FiniteMarkedPoissonPartition.orderByMarks_count s)) b)).2","l":854,"f":"Causalean/Mathlib/Probability/FiniteMarkedPoissonPartition/Superposition/Canonical.lean"},{"n":"Causalean.Mathlib.Probability.FiniteMarkedPoissonPartition.poissonIIDStreamLaw","k":"def","m":"Causalean.Mathlib.Probability.FiniteMarkedPoissonPartition.IIDPoisson","a":"Mathlib/Probability/FiniteMarkedPoissonPartition","t":1,"d":"An independent scalar Poisson count and infinite i.i.d. stream.","s":"{X : Type u_1} → [inst : MeasurableSpace X] → (P : MeasureTheory.Measure X) → [MeasureTheory.IsProbabilityMeasure P] → NNReal → MeasureTheory.Measure (ℕ × (ℕ → X))","l":66,"f":"Causalean/Mathlib/Probability/FiniteMarkedPoissonPartition/IIDPoisson.lean"},{"n":"Causalean.Mathlib.Probability.FiniteMarkedPoissonPartition.poissonIIDStreamLaw_isProbabilityMeasure","k":"instance","m":"Causalean.Mathlib.Probability.FiniteMarkedPoissonPartition.IIDPoisson","a":"Mathlib/Probability/FiniteMarkedPoissonPartition","t":2,"d":"The independent Poisson-count and i.i.d.-stream law is a probability measure.","s":"∀ {X : Type u_1} [inst : MeasurableSpace X] (P : MeasureTheory.Measure X) [inst_1 : MeasureTheory.IsProbabilityMeasure P] (lam : NNReal), MeasureTheory.IsProbabilityMeasure (Causalean.Mathlib.Probability.FiniteMarkedPoissonPartition.poissonIIDStreamLaw P lam)","l":71,"f":"Causalean/Mathlib/Probability/FiniteMarkedPoissonPartition/IIDPoisson.lean"},{"n":"Causalean.Mathlib.Probability.FiniteMarkedPoissonPartition.poissonIIDStreamLaw_map_count","k":"theorem","m":"Causalean.Mathlib.Probability.FiniteMarkedPoissonPartition.IIDPoisson","a":"Mathlib/Probability/FiniteMarkedPoissonPartition","t":2,"d":"The count coordinate has the requested scalar Poisson law.","s":"∀ {X : Type u_1} [inst : MeasurableSpace X] (P : MeasureTheory.Measure X) [inst_1 : MeasureTheory.IsProbabilityMeasure P] (lam : NNReal), MeasureTheory.Measure.map Prod.fst (Causalean.Mathlib.Probability.FiniteMarkedPoissonPartition.poissonIIDStreamLaw P lam) = ProbabilityTheory.poissonMeasure lam","l":78,"f":"Causalean/Mathlib/Probability/FiniteMarkedPoissonPartition/IIDPoisson.lean"},{"n":"Causalean.Mathlib.Probability.FiniteMarkedPoissonPartition.poissonIIDStreamLaw_map_finPrefix","k":"theorem","m":"Causalean.Mathlib.Probability.FiniteMarkedPoissonPartition.IIDPoisson","a":"Mathlib/Probability/FiniteMarkedPoissonPartition","t":2,"d":"Every finite stream prefix remains an exact product sample after pairing the stream with an independent Poisson count.","s":"∀ {X : Type u_1} [inst : MeasurableSpace X] (P : MeasureTheory.Measure X) [inst_1 : MeasureTheory.IsProbabilityMeasure P] (lam : NNReal) (n : ℕ), MeasureTheory.Measure.map (fun z i => z.2 ↑i) (Causalean.Mathlib.Probability.FiniteMarkedPoissonPartition.poissonIIDStreamLaw P lam) = MeasureTheory.Measure.pi fun x => P","l":85,"f":"Causalean/Mathlib/Probability/FiniteMarkedPoissonPartition/IIDPoisson.lean"},{"n":"Causalean.Mathlib.Probability.FiniteMarkedPoissonPartition.poissonMeasure_singleton_eq_poissonPMF","k":"theorem","m":"Causalean.Mathlib.Probability.FiniteMarkedPoissonPartition.Basic","a":"Mathlib/Probability/FiniteMarkedPoissonPartition","t":2,"d":"The Poisson law with mean r gives the single count n exactly the Poisson probability mass at n.","s":"∀ (r : NNReal) (n : ℕ), (ProbabilityTheory.poissonMeasure r) {n} = (ProbabilityTheory.poissonPMF r) n","l":18,"f":"Causalean/Mathlib/Probability/FiniteMarkedPoissonPartition/Basic.lean"},{"n":"Causalean.Mathlib.Probability.FiniteMarkedPoissonPartition.poisson_two_n_lower_tail","k":"theorem","m":"Causalean.Mathlib.Probability.FiniteMarkedPoissonPartition.Depoissonization","a":"Mathlib/Probability/FiniteMarkedPoissonPartition","t":2,"d":"The lower tail used in de-Poissonization is exponentially small.","s":"∀ (n : ℕ), (ProbabilityTheory.poissonMeasure (2 * ↑n)) {k | k < n} ≤ ENNReal.ofReal (Real.exp (-↑n * (1 - Real.log 2)))","l":195,"f":"Causalean/Mathlib/Probability/FiniteMarkedPoissonPartition/Depoissonization.lean"},{"n":"Causalean.Mathlib.Probability.FiniteMarkedPoissonPartition.retainedObservations","k":"def","m":"Causalean.Mathlib.Probability.FiniteMarkedPoissonPartition.Superposition.Retention","a":"Mathlib/Probability/FiniteMarkedPoissonPartition/Superposition","t":1,"d":"Retain the first n observations after sorting by marks and forget their marks; x₀ supplies an irrelevant value outside the event n ≤ count.","s":"{X : Type u_1} → [inst : MeasurableSpace X] → X → (n : ℕ) → Causalean.Mathlib.Probability.FiniteMarkedPoissonPartition.FiniteSample (X × ℝ) → Fin n → X","l":19,"f":"Causalean/Mathlib/Probability/FiniteMarkedPoissonPartition/Superposition/Retention.lean"},{"n":"Causalean.Mathlib.Probability.FiniteMarkedPoissonPartition.streamToFiniteSample","k":"def","m":"Causalean.Mathlib.Probability.FiniteMarkedPoissonPartition.Basic","a":"Mathlib/Probability/FiniteMarkedPoissonPartition","t":1,"d":"Truncate a count-and-stream outcome to the prefix selected by its count.","s":"{X : Type u_1} → [inst : MeasurableSpace X] → ℕ × (ℕ → X) → Causalean.Mathlib.Probability.FiniteMarkedPoissonPartition.FiniteSample X","l":47,"f":"Causalean/Mathlib/Probability/FiniteMarkedPoissonPartition/Basic.lean"},{"n":"Causalean.Mathlib.Probability.FiniteMarkedPoissonPartition.streamToFiniteSample_paddedStream","k":"theorem","m":"Causalean.Mathlib.Probability.FiniteMarkedPoissonPartition.Depoissonization","a":"Mathlib/Probability/FiniteMarkedPoissonPartition","t":2,"d":"","s":"∀ {X : Type u_1} [inst : MeasurableSpace X] (x0 : X) (s : Causalean.Mathlib.Probability.FiniteMarkedPoissonPartition.FiniteSample X), Causalean.Mathlib.Probability.FiniteMarkedPoissonPartition.streamToFiniteSample (Causalean.Mathlib.Probability.FiniteMarkedPoissonPartition.finiteSamplePaddedStream x0 s) = s","l":284,"f":"Causalean/Mathlib/Probability/FiniteMarkedPoissonPartition/Depoissonization.lean"},{"n":"Causalean.Mathlib.Probability.FiniteMarkedPoissonPartition.superpose","k":"def","m":"Causalean.Mathlib.Probability.FiniteMarkedPoissonPartition.Superposition.Canonical","a":"Mathlib/Probability/FiniteMarkedPoissonPartition/Superposition","t":1,"d":"Superpose finitely many finite sequences by enumerating their dependent disjoint union of coordinates.","s":"{X : Type u_1} → [inst : MeasurableSpace X] → {ι : Type u_2} → [Fintype ι] → (ι → Causalean.Mathlib.Probability.FiniteMarkedPoissonPartition.FiniteSample (X × ℝ)) → Causalean.Mathlib.Probability.FiniteMarkedPoissonPartition.FiniteSample (X × ℝ)","l":23,"f":"Causalean/Mathlib/Probability/FiniteMarkedPoissonPartition/Superposition/Canonical.lean"},{"n":"Causalean.Mathlib.Probability.FiniteMarkedPoissonPartition.superposeByMarks","k":"def","m":"Causalean.Mathlib.Probability.FiniteMarkedPoissonPartition.Superposition.Canonical","a":"Mathlib/Probability/FiniteMarkedPoissonPartition/Superposition","t":1,"d":"Superpose finitely many cell configurations and put the resulting atoms in their canonical increasing-mark order.","s":"{X : Type u_1} → [inst : MeasurableSpace X] → {ι : Type u_2} → [Fintype ι] → (ι → Causalean.Mathlib.Probability.FiniteMarkedPoissonPartition.FiniteSample (X × ℝ)) → Causalean.Mathlib.Probability.FiniteMarkedPoissonPartition.FiniteSample (X × ℝ)","l":315,"f":"Causalean/Mathlib/Probability/FiniteMarkedPoissonPartition/Superposition/Canonical.lean"},{"n":"Causalean.Mathlib.Probability.FiniteMarkedPoissonPartition.superpose_count","k":"theorem","m":"Causalean.Mathlib.Probability.FiniteMarkedPoissonPartition.Superposition.Canonical","a":"Mathlib/Probability/FiniteMarkedPoissonPartition/Superposition","t":2,"d":"The count after finite superposition is the sum of the cell counts.","s":"∀ {X : Type u_1} [inst : MeasurableSpace X] {ι : Type u_2} [inst_1 : Fintype ι] [inst_2 : MeasurableSpace ι] [MeasurableSingletonClass ι] (q : ι → Causalean.Mathlib.Probability.FiniteMarkedPoissonPartition.FiniteSample (X × ℝ)), (Causalean.Mathlib.Probability.FiniteMarkedPoissonPartition.superpose q).count = ∑ j, (q j).count","l":34,"f":"Causalean/Mathlib/Probability/FiniteMarkedPoissonPartition/Superposition/Canonical.lean"},{"n":"Causalean.Mathlib.Probability.bernoulliBool","k":"def","m":"Causalean.Mathlib.Probability.BernoulliMeasure","a":"Mathlib/Probability","t":1,"d":"The Boolean Bernoulli distribution assigns success probability to truth and failure probability to falsehood. It is the same two-point law as the existing real-valued Bernoulli distribution on zero and one, but its Boolean values make it usable as a Markov kernel into a Boolean coordinate of a potential-outcome tuple.","s":"ℝ → MeasureTheory.Measure Bool","l":271,"f":"Causalean/Mathlib/Probability/BernoulliMeasure.lean"},{"n":"Causalean.Mathlib.Probability.bernoulliBool_bind","k":"theorem","m":"Causalean.Mathlib.Probability.BernoulliMeasure","a":"Mathlib/Probability","t":2,"d":"Drawing a Bool-valued Bernoulli variable and then selecting a distribution according to its value produces the corresponding success-probability mixture of the two distributions.","s":"∀ {β : Type u_1} [inst : MeasurableSpace β] (p : ℝ) (K : Bool → MeasureTheory.Measure β), (Causalean.Mathlib.Probability.bernoulliBool p).bind K = ENNReal.ofReal p • K true + ENNReal.ofReal (1 - p) • K false","l":319,"f":"Causalean/Mathlib/Probability/BernoulliMeasure.lean"},{"n":"Causalean.Mathlib.Probability.bernoulliBool_integral","k":"theorem","m":"Causalean.Mathlib.Probability.BernoulliMeasure","a":"Mathlib/Probability","t":2,"d":"The expectation of a real-valued function of a Bool-valued Bernoulli draw is its value at success times the success probability plus its value at failure times the failure probability.","s":"∀ {p : ℝ}, 0 ≤ p → p ≤ 1 → ∀ (f : Bool → ℝ), ∫ (z : Bool), f z ∂Causalean.Mathlib.Probability.bernoulliBool p = p * f true + (1 - p) * f false","l":301,"f":"Causalean/Mathlib/Probability/BernoulliMeasure.lean"},{"n":"Causalean.Mathlib.Probability.bernoulliBool_isProbabilityMeasure","k":"theorem","m":"Causalean.Mathlib.Probability.BernoulliMeasure","a":"Mathlib/Probability","t":2,"d":"A Bool-valued Bernoulli distribution is a probability distribution whenever its success probability lies between zero and one.","s":"∀ {p : ℝ}, 0 ≤ p → p ≤ 1 → MeasureTheory.IsProbabilityMeasure (Causalean.Mathlib.Probability.bernoulliBool p)","l":287,"f":"Causalean/Mathlib/Probability/BernoulliMeasure.lean"},{"n":"Causalean.Mathlib.Probability.bernoulliBool_map","k":"theorem","m":"Causalean.Mathlib.Probability.BernoulliMeasure","a":"Mathlib/Probability","t":2,"d":"Transforming a Bool-valued Bernoulli draw produces a two-point distribution concentrated on the transformed success and failure values with their original probabilities.","s":"∀ {β : Type u_1} [inst : MeasurableSpace β] (p : ℝ) (f : Bool → β), MeasureTheory.Measure.map f (Causalean.Mathlib.Probability.bernoulliBool p) = ENNReal.ofReal p • MeasureTheory.Measure.dirac (f true) + ENNReal.ofReal (1 - p) • MeasureTheory.Measure.dirac (f false)","l":331,"f":"Causalean/Mathlib/Probability/BernoulliMeasure.lean"},{"n":"Causalean.Mathlib.Probability.bernoulliLaw","k":"def","m":"Causalean.Mathlib.Probability.BernoulliMeasure","a":"Mathlib/Probability","t":1,"d":"The Bernoulli law on ℝ with success probability p. Concentrates ENNReal.ofReal p on 1 and ENNReal.ofReal (1 − p) on 0.","s":"ℝ → MeasureTheory.Measure ℝ","l":34,"f":"Causalean/Mathlib/Probability/BernoulliMeasure.lean"},{"n":"Causalean.Mathlib.Probability.bernoulliLaw_ac_of_reference_interior","k":"theorem","m":"Causalean.Mathlib.Probability.BernoulliMeasure","a":"Mathlib/Probability","t":2,"d":"A Bernoulli law is absolutely continuous with respect to any Bernoulli law whose success probability is strictly between zero and one.","s":"∀ {p q : ℝ}, 0 < q → q < 1 → (Causalean.Mathlib.Probability.bernoulliLaw p).AbsolutelyContinuous (Causalean.Mathlib.Probability.bernoulliLaw q)","l":107,"f":"Causalean/Mathlib/Probability/BernoulliMeasure.lean"},{"n":"Causalean.Mathlib.Probability.bernoulliLaw_ae_le_one","k":"theorem","m":"Causalean.Mathlib.Probability.BernoulliMeasure","a":"Mathlib/Probability","t":2,"d":"The Bernoulli law on the real line is almost surely at most one, since all of its mass is placed at 0 and 1.","s":"∀ {p : ℝ}, (fun y => y) ≤ᵐ[Causalean.Mathlib.Probability.bernoulliLaw p] fun x => 1","l":85,"f":"Causalean/Mathlib/Probability/BernoulliMeasure.lean"},{"n":"Causalean.Mathlib.Probability.bernoulliLaw_ae_nonneg","k":"theorem","m":"Causalean.Mathlib.Probability.BernoulliMeasure","a":"Mathlib/Probability","t":2,"d":"The Bernoulli law on the real line is almost surely nonnegative, since all of its mass is placed at 0 and 1.","s":"∀ {p : ℝ}, 0 ≤ᵐ[Causalean.Mathlib.Probability.bernoulliLaw p] fun y => y","l":72,"f":"Causalean/Mathlib/Probability/BernoulliMeasure.lean"},{"n":"Causalean.Mathlib.Probability.bernoulliLaw_ae_zero_or_one","k":"theorem","m":"Causalean.Mathlib.Probability.BernoulliMeasure","a":"Mathlib/Probability","t":2,"d":"A draw from the real-valued Bernoulli law is almost surely either 0 or 1.","s":"∀ {p : ℝ}, ∀ᵐ (y : ℝ) ∂Causalean.Mathlib.Probability.bernoulliLaw p, y = 0 ∨ y = 1","l":98,"f":"Causalean/Mathlib/Probability/BernoulliMeasure.lean"},{"n":"Causalean.Mathlib.Probability.bernoulliLaw_integral","k":"theorem","m":"Causalean.Mathlib.Probability.BernoulliMeasure","a":"Mathlib/Probability","t":2,"d":"Two-point integral for the custom bernoulliLaw: since bernoulliLaw p = ENNReal.ofReal p • dirac 1 + ENNReal.ofReal (1-p) • dirac 0, its integral splits via integral_add_measure / integral_smul_measure / integral_dirac into the two-point weighted sum, with 0 ≤ p ≤ 1 collapsing the ENNReal → ℝ coercions to p and 1 - p.","s":"∀ {p : ℝ}, 0 ≤ p → p ≤ 1 → ∀ (f : ℝ → ℝ), ∫ (y : ℝ), f y ∂Causalean.Mathlib.Probability.bernoulliLaw p = p * f 1 + (1 - p) * f 0","l":40,"f":"Causalean/Mathlib/Probability/BernoulliMeasure.lean"},{"n":"Causalean.Mathlib.Probability.bernoulliLaw_isProbabilityMeasure","k":"theorem","m":"Causalean.Mathlib.Probability.BernoulliMeasure","a":"Mathlib/Probability","t":2,"d":"The custom bernoulliLaw is a probability measure for 0 ≤ p ≤ 1: its total mass is p + (1 - p) = 1.","s":"∀ {p : ℝ}, 0 ≤ p → p ≤ 1 → MeasureTheory.IsProbabilityMeasure (Causalean.Mathlib.Probability.bernoulliLaw p)","l":59,"f":"Causalean/Mathlib/Probability/BernoulliMeasure.lean"},{"n":"Causalean.Mathlib.Probability.bernoulliLaw_klDiv_le_four_sq_sub","k":"theorem","m":"Causalean.Mathlib.Probability.BernoulliMeasure","a":"Mathlib/Probability","t":1,"d":"If both success probabilities $p$ and $q$ lie in the middle half of the unit interval, $1/4 \\le p \\le 3/4$ and $1/4 \\le q \\le 3/4$, then the Kullback–Leibler divergence from the Bernoulli($p$) law to the Bernoulli($q$) law is at most four times the squared difference of the probabilities, $4(p-q)^2$.","s":"∀ {p q : ℝ}, 1 / 4 ≤ p → p ≤ 3 / 4 → 1 / 4 ≤ q → q ≤ 3 / 4 → InformationTheory.klDiv (Causalean.Mathlib.Probability.bernoulliLaw p) (Causalean.Mathlib.Probability.bernoulliLaw q) ≤ ENNReal.ofReal (4 * (p - q) ^ 2)","l":236,"f":"Causalean/Mathlib/Probability/BernoulliMeasure.lean"},{"n":"Causalean.Mathlib.Probability.bernoulliLaw_klDiv_toReal","k":"theorem","m":"Causalean.Mathlib.Probability.BernoulliMeasure","a":"Mathlib/Probability","t":1,"d":"Let $p$ be a Bernoulli success probability with $0 \\le p \\le 1$, and let $q$ be a reference success probability with $0 < q < 1$. Then the Kullback–Leibler divergence from the Bernoulli($p$) law to the Bernoulli($q$) law equals $p \\log(p/q) + (1-p)\\log((1-p)/(1-q))$, the usual two-point KL formula: a success contribution plus a failure contribution.","s":"∀ {p q : ℝ}, 0 ≤ p → p ≤ 1 → 0 < q → q < 1 → (InformationTheory.klDiv (Causalean.Mathlib.Probability.bernoulliLaw p) (Causalean.Mathlib.Probability.bernoulliLaw q)).toReal = p * Real.log (p / q) + (1 - p) * Real.log ((1 - p) / (1 - q))","l":154,"f":"Causalean/Mathlib/Probability/BernoulliMeasure.lean"},{"n":"Causalean.Mathlib.Probability.bernoulliLaw_lintegral_ofReal","k":"theorem","m":"Causalean.Mathlib.Probability.BernoulliMeasure","a":"Mathlib/Probability","t":2,"d":"ℝ≥0∞/lintegral analogue of bernoulliLaw_integral: a Bernoulli law integrates an ℝ≥0∞-valued function as the two-point weighted sum.","s":"∀ {p : ℝ} (f : ℝ → ENNReal), ∫⁻ (y : ℝ), f y ∂Causalean.Mathlib.Probability.bernoulliLaw p = ENNReal.ofReal p * f 1 + ENNReal.ofReal (1 - p) * f 0","l":262,"f":"Causalean/Mathlib/Probability/BernoulliMeasure.lean"},{"n":"Causalean.Mathlib.Probability.bernoulliLaw_llr_integrable","k":"theorem","m":"Causalean.Mathlib.Probability.BernoulliMeasure","a":"Mathlib/Probability","t":2,"d":"The log-likelihood ratio between any two real-parameter Bernoulli laws is integrable under the first law.","s":"∀ {p q : ℝ}, MeasureTheory.Integrable (MeasureTheory.llr (Causalean.Mathlib.Probability.bernoulliLaw p) (Causalean.Mathlib.Probability.bernoulliLaw q)) (Causalean.Mathlib.Probability.bernoulliLaw p)","l":134,"f":"Causalean/Mathlib/Probability/BernoulliMeasure.lean"},{"n":"Causalean.Mathlib.Probability.bernoulli_mean_channel_kl","k":"theorem","m":"Causalean.Mathlib.Probability.SignedTwoPoint","a":"Mathlib/Probability","t":1,"d":"KL band for the signed two-point mean channel. For a strictly positive spread parameter B and two channel means u and v each confined to the interval [-B/2, B/2], the Kullback–Leibler divergence between the two-point channels with means u and v is bounded by the quadratic (u − v)²/B². This is the affine transport of the {0,1} Bernoulli KL band onto the {−B, B} mean parametrization.","s":"∀ (B u v : ℝ), 0 < B → |u| ≤ B / 2 → |v| ≤ B / 2 → InformationTheory.klDiv (Causalean.Mathlib.Probability.twoPointMean B u) (Causalean.Mathlib.Probability.twoPointMean B v) ≤ ENNReal.ofReal ((u - v) ^ 2 / B ^ 2)","l":170,"f":"Causalean/Mathlib/Probability/SignedTwoPoint.lean"},{"n":"Causalean.Mathlib.Probability.bernstein","k":"theorem","m":"Causalean.Mathlib.Probability.DarmoisSkitovich","a":"Mathlib/Probability","t":1,"d":"Bernstein's theorem (finite-variance Darmois–Skitovich, two variables). Let X and Y be real random variables that are independent and each have finite second moment. If their sum X + Y and difference X − Y are independent, then both X and Y have Gaussian laws.","s":"∀ {Ω : Type u_1} [inst : MeasurableSpace Ω] {P : MeasureTheory.Measure Ω} [MeasureTheory.IsProbabilityMeasure P] {X Y : Ω → ℝ}, Measurable X → Measurable Y → MeasureTheory.MemLp X 2 P → MeasureTheory.MemLp Y 2 P → ProbabilityTheory.IndepFun X Y P → ProbabilityTheory.IndepFun (X + Y) (X - Y) P → ProbabilityTheory.IsGaussian (MeasureTheory.Measure.map X P) ∧ ProbabilityTheory.IsGaussian (MeasureTheory.Measure.map Y P)","l":621,"f":"Causalean/Mathlib/Probability/DarmoisSkitovich.lean"},{"n":"Causalean.Mathlib.Probability.bernstein_charFun_funeq","k":"theorem","m":"Causalean.Mathlib.Probability.DarmoisSkitovich","a":"Mathlib/Probability","t":2,"d":"Functional equation for Bernstein's theorem. If X and Y are independent and the pair (X + Y, X − Y) is independent, then the characteristic functions f = charFun (P.map X) and g = charFun (P.map Y) satisfy f (u+v) · g (u−v) = f u · g u · f v · g (−v) for all u, v.","s":"∀ {Ω : Type u_1} [inst : MeasurableSpace Ω] {P : MeasureTheory.Measure Ω} [MeasureTheory.IsProbabilityMeasure P] {X Y : Ω → ℝ}, Measurable X → Measurable Y → ProbabilityTheory.IndepFun X Y P → ProbabilityTheory.IndepFun (X + Y) (X - Y) P → ∀ (u v : ℝ), MeasureTheory.charFun (MeasureTheory.Measure.map X P) (u + v) * MeasureTheory.charFun (MeasureTheory.Measure.map Y P) (u - v) = MeasureTheory.charFun (MeasureTheory.Measure.map X P) u * MeasureTheory.charFun (MeasureTheory.Measure.map Y P) u * (MeasureTheory.charFun (MeasureTheory.Measure.map X P) v * MeasureTheory.charFun (MeasureTheory.Measure.map Y P) (-v))","l":58,"f":"Causalean/Mathlib/Probability/DarmoisSkitovich.lean"},{"n":"Causalean.Mathlib.Probability.bernstein_charFun_gaussian_nhds_zero","k":"theorem","m":"Causalean.Mathlib.Probability.DarmoisSkitovich","a":"Mathlib/Probability","t":2,"d":"Analytic core of Bernstein's theorem. Under the Bernstein hypotheses with finite second moments, the characteristic functions of X and Y coincide, on a neighbourhood of 0, with Gaussian characteristic functions that share one variance σ² ≥ 0 (with means mf, mg). The shared σ² — a consequence of the single constant c from gaussianForm_of_funeq — is exactly what the doubling bootstrap in bernstein needs to extend the Gaussian form to all of ℝ.","s":"∀ {Ω : Type u_1} [inst : MeasurableSpace Ω] {P : MeasureTheory.Measure Ω} [MeasureTheory.IsProbabilityMeasure P] {X Y : Ω → ℝ}, Measurable X → Measurable Y → MeasureTheory.MemLp X 2 P → MeasureTheory.MemLp Y 2 P → ProbabilityTheory.IndepFun X Y P → ProbabilityTheory.IndepFun (X + Y) (X - Y) P → ∃ mf mg σ2, 0 ≤ σ2 ∧ (∀ᶠ (t : ℝ) in nhds 0, MeasureTheory.charFun (MeasureTheory.Measure.map X P) t = Complex.exp (↑mf * ↑t * Complex.I - ↑σ2 * ↑t ^ 2 / 2)) ∧ ∀ᶠ (t : ℝ) in nhds 0, MeasureTheory.charFun (MeasureTheory.Measure.map Y P) t = Complex.exp (↑mg * ↑t * Complex.I - ↑σ2 * ↑t ^ 2 / 2)","l":486,"f":"Causalean/Mathlib/Probability/DarmoisSkitovich.lean"},{"n":"Causalean.Mathlib.Probability.binomialWeight","k":"def","m":"Causalean.Mathlib.Probability.BernoulliMeasure","a":"Mathlib/Probability","t":1,"d":"The real binomial weight of count j among m trials with success parameter p.","s":"ℕ → ℝ → ℕ → ℝ","l":348,"f":"Causalean/Mathlib/Probability/BernoulliMeasure.lean"},{"n":"Causalean.Mathlib.Probability.binomial_inverse_two_arms_interior_le","k":"theorem","m":"Causalean.Mathlib.Probability.BernoulliMeasure","a":"Mathlib/Probability","t":2,"d":"If the overlap margin is positive and the success probability lies between that margin and one minus the margin, the binomial expectation of the two inverse arm counts on the interior event is at most four divided by the trial count plus one times the margin.","s":"∀ (m : ℕ) (p epsilon : ℝ), 0 < epsilon → epsilon ≤ p → p ≤ 1 - epsilon → (∑ j ∈ Finset.range (m + 1), Causalean.Mathlib.Probability.binomialWeight m p j * if 0 < j ∧ j < m then (↑j)⁻¹ + (↑(m - j))⁻¹ else 0) ≤ 4 / (↑(m + 1) * epsilon)","l":524,"f":"Causalean/Mathlib/Probability/BernoulliMeasure.lean"},{"n":"Causalean.Mathlib.Probability.binomial_totalized_inverse_count_le","k":"theorem","m":"Causalean.Mathlib.Probability.BernoulliMeasure","a":"Mathlib/Probability","t":2,"d":"When the success probability is positive and at most one, the binomial expectation of the zero-safe inverse success count is at most twice the reciprocal of the trial count plus one times that probability.","s":"∀ (m : ℕ) (p : ℝ), 0 < p → p ≤ 1 → (∑ j ∈ Finset.range (m + 1), Causalean.Mathlib.Probability.binomialWeight m p j * if 0 < j then (↑j)⁻¹ else 0) ≤ 2 / (↑(m + 1) * p)","l":441,"f":"Causalean/Mathlib/Probability/BernoulliMeasure.lean"},{"n":"Causalean.Mathlib.Probability.charFun_contDiff_two","k":"theorem","m":"Causalean.Mathlib.Probability.DarmoisSkitovich","a":"Mathlib/Probability","t":2,"d":"charFun of a pushforward with a finite second moment is C². This is a specialisation of contDiff_charFun (the characteristic function is Cⁿ whenever the n-th moment is finite), transferring MemLp Z 2 P to MemLp id 2 (P.map Z) along the pushforward map.","s":"∀ {Ω : Type u_1} [inst : MeasurableSpace Ω] {P : MeasureTheory.Measure Ω} {Z : Ω → ℝ} [MeasureTheory.IsFiniteMeasure P], MeasureTheory.MemLp Z 2 P → ContDiff ℝ 2 (MeasureTheory.charFun (MeasureTheory.Measure.map Z P))","l":147,"f":"Causalean/Mathlib/Probability/DarmoisSkitovich.lean"},{"n":"Causalean.Mathlib.Probability.charFun_eventually_ne_zero","k":"theorem","m":"Causalean.Mathlib.Probability.DarmoisSkitovich","a":"Mathlib/Probability","t":2,"d":"The characteristic function of a probability measure on ℝ is nonzero on a neighbourhood of 0: it is continuous and equals 1 at 0. This is what makes the logarithmic derivative (charFun μ)' / charFun μ well defined near 0.","s":"∀ (μ : MeasureTheory.Measure ℝ) [MeasureTheory.IsProbabilityMeasure μ], ∀ᶠ (t : ℝ) in nhds 0, MeasureTheory.charFun μ t ≠ 0","l":138,"f":"Causalean/Mathlib/Probability/DarmoisSkitovich.lean"},{"n":"Causalean.Mathlib.Probability.gaussianForm_of_funeq","k":"theorem","m":"Causalean.Mathlib.Probability.DarmoisSkitovich","a":"Mathlib/Probability","t":1,"d":"Pure analytic core: the Bernstein functional equation forces a Gaussian form. Let f, g : ℝ → ℂ be functions that are twice continuously differentiable and equal to 1 at the origin, and suppose they satisfy the Bernstein functional equation f (u+v) · g (u−v) = f u · g u · (f v · g (−v)) for all real u, v. Then there is a single constant c such that, on a neighbourhood of 0, f t = exp (f'(0)·t + c·t²/2) and g t = exp (g'(0)·t + c·t²/2), with the same c in both formulas.","s":"∀ {f g : ℝ → ℂ}, ContDiff ℝ 2 f → ContDiff ℝ 2 g → f 0 = 1 → g 0 = 1 → (∀ (u v : ℝ), f (u + v) * g (u - v) = f u * g u * (f v * g (-v))) → ∃ c, (∀ᶠ (t : ℝ) in nhds 0, f t = Complex.exp (deriv f 0 * ↑t + c * ↑t ^ 2 / 2)) ∧ ∀ᶠ (t : ℝ) in nhds 0, g t = Complex.exp (deriv g 0 * ↑t + c * ↑t ^ 2 / 2)","l":157,"f":"Causalean/Mathlib/Probability/DarmoisSkitovich.lean"},{"n":"Causalean.Mathlib.Probability.iid_average_integral","k":"theorem","m":"Causalean.Mathlib.Probability.IidMeanVariance","a":"Mathlib/Probability","t":2,"d":"An average of n independent, identically distributed observations has the same expectation as the population statistic being averaged.","s":"∀ {Ω : Type u_1} [inst : MeasurableSpace Ω] (μ : MeasureTheory.Measure Ω) [MeasureTheory.IsProbabilityMeasure μ] (m : ℕ), 0 < m → ∀ (F : Ω → ℝ), MeasureTheory.Integrable F μ → (∫ (sample : Fin m → Ω), (↑m)⁻¹ * ∑ i, F (sample i) ∂MeasureTheory.Measure.pi fun x => μ) = ∫ (o : Ω), F o ∂μ","l":362,"f":"Causalean/Mathlib/Probability/IidMeanVariance.lean"},{"n":"Causalean.Mathlib.Probability.iid_average_integral_fintype","k":"theorem","m":"Causalean.Mathlib.Probability.IidMeanVariance","a":"Mathlib/Probability","t":2,"d":"An average of independent, identically distributed observations has the same expectation as the population statistic being averaged.","s":"∀ {Ω : Type u_1} {ι : Type u_2} [inst : MeasurableSpace Ω] [inst_1 : Fintype ι] [Nonempty ι] (μ : MeasureTheory.Measure Ω) [MeasureTheory.IsProbabilityMeasure μ] (F : Ω → ℝ), MeasureTheory.Integrable F μ → (∫ (sample : ι → Ω), (↑(Fintype.card ι))⁻¹ * ∑ i, F (sample i) ∂MeasureTheory.Measure.pi fun x => μ) = ∫ (o : Ω), F o ∂μ","l":227,"f":"Causalean/Mathlib/Probability/IidMeanVariance.lean"},{"n":"Causalean.Mathlib.Probability.iid_average_variance","k":"theorem","m":"Causalean.Mathlib.Probability.IidMeanVariance","a":"Mathlib/Probability","t":2,"d":"The variance of an average of independent, identically distributed observations is the population variance divided by the sample size (including the zero-length case).","s":"∀ {Ω : Type u_1} [inst : MeasurableSpace Ω] (μ : MeasureTheory.Measure Ω) [MeasureTheory.IsProbabilityMeasure μ] (m : ℕ) (F : Ω → ℝ), MeasureTheory.MemLp F 2 μ → ProbabilityTheory.variance (fun sample => (↑m)⁻¹ * ∑ i, F (sample i)) (MeasureTheory.Measure.pi fun x => μ) = (↑m)⁻¹ * ProbabilityTheory.variance F μ","l":375,"f":"Causalean/Mathlib/Probability/IidMeanVariance.lean"},{"n":"Causalean.Mathlib.Probability.iid_average_variance_fintype","k":"theorem","m":"Causalean.Mathlib.Probability.IidMeanVariance","a":"Mathlib/Probability","t":2,"d":"The variance of an average of independent, identically distributed observations is the population variance divided by the sample size.","s":"∀ {Ω : Type u_1} {ι : Type u_2} [inst : MeasurableSpace Ω] [inst_1 : Fintype ι] (μ : MeasureTheory.Measure Ω) [MeasureTheory.IsProbabilityMeasure μ] (F : Ω → ℝ), MeasureTheory.MemLp F 2 μ → ProbabilityTheory.variance (fun sample => (↑(Fintype.card ι))⁻¹ * ∑ i, F (sample i)) (MeasureTheory.Measure.pi fun x => μ) = (↑(Fintype.card ι))⁻¹ * ProbabilityTheory.variance F μ","l":250,"f":"Causalean/Mathlib/Probability/IidMeanVariance.lean"},{"n":"Causalean.Mathlib.Probability.iid_mean_abs_le","k":"theorem","m":"Causalean.Mathlib.Probability.IidMeanVariance","a":"Mathlib/Probability","t":2,"d":"The mean absolute error of a square-integrable scalar sample average of n independent, identically distributed observations is at most the square root of the population second moment divided by n.","s":"∀ {Ω : Type u_1} [inst : MeasurableSpace Ω] (μ : MeasureTheory.Measure Ω) [MeasureTheory.IsProbabilityMeasure μ] {n : ℕ}, 0 < n → ∀ (ξ : Ω → ℝ), MeasureTheory.MemLp ξ 2 μ → (∫ (s : Fin n → Ω), |(↑n)⁻¹ * ∑ i, ξ (s i) - ∫ (ω : Ω), ξ ω ∂μ| ∂MeasureTheory.Measure.pi fun x => μ) ≤ √((∫ (ω : Ω), ξ ω ^ 2 ∂μ) / ↑n)","l":301,"f":"Causalean/Mathlib/Probability/IidMeanVariance.lean"},{"n":"Causalean.Mathlib.Probability.iid_mean_abs_le_fintype","k":"theorem","m":"Causalean.Mathlib.Probability.IidMeanVariance","a":"Mathlib/Probability","t":2,"d":"The mean absolute error of a square-integrable scalar sample average from independent, identically distributed observations is at most the square root of the population second moment divided by the sample size.","s":"∀ {Ω : Type u_1} {ι : Type u_2} [inst : MeasurableSpace Ω] [inst_1 : Fintype ι] [Nonempty ι] (μ : MeasureTheory.Measure Ω) [MeasureTheory.IsProbabilityMeasure μ] (ξ : Ω → ℝ), MeasureTheory.MemLp ξ 2 μ → (∫ (s : ι → Ω), |(↑(Fintype.card ι))⁻¹ * ∑ i, ξ (s i) - ∫ (ω : Ω), ξ ω ∂μ| ∂MeasureTheory.Measure.pi fun x => μ) ≤ √((∫ (ω : Ω), ξ ω ^ 2 ∂μ) / ↑(Fintype.card ι))","l":140,"f":"Causalean/Mathlib/Probability/IidMeanVariance.lean"},{"n":"Causalean.Mathlib.Probability.iid_mean_euclidean_abs_le","k":"theorem","m":"Causalean.Mathlib.Probability.IidMeanVariance","a":"Mathlib/Probability","t":1,"d":"For a strictly positive sample size n and finitely many square-integrable real-valued statistics indexed by k, each observed on the same n-point independent, identically distributed sample, the expected Euclidean norm of the vector of centered sample averages — one coordinate per statistic — is at most the square root of the sum of the population second moments divided by n.","s":"∀ {Ω : Type u_1} {ι : Type u_2} [inst : MeasurableSpace Ω] [inst_1 : Fintype ι] (μ : MeasureTheory.Measure Ω) [MeasureTheory.IsProbabilityMeasure μ] {n : ℕ}, 0 < n → ∀ (ξ : ι → Ω → ℝ), (∀ (k : ι), MeasureTheory.MemLp (ξ k) 2 μ) → (∫ (s : Fin n → Ω), √(∑ k, ((↑n)⁻¹ * ∑ i, ξ k (s i) - ∫ (ω : Ω), ξ k ω ∂μ) ^ 2) ∂MeasureTheory.Measure.pi fun x => μ) ≤ √((∑ k, ∫ (ω : Ω), ξ k ω ^ 2 ∂μ) / ↑n)","l":313,"f":"Causalean/Mathlib/Probability/IidMeanVariance.lean"},{"n":"Causalean.Mathlib.Probability.iid_mean_euclidean_abs_le_fintype","k":"theorem","m":"Causalean.Mathlib.Probability.IidMeanVariance","a":"Mathlib/Probability","t":2,"d":"The expected Euclidean error of finitely many square-integrable sample averages from the same independent, identically distributed sample is controlled by their summed population second moments and the sample size.","s":"∀ {Ω : Type u_1} {ι : Type u_2} {κ : Type u_3} [inst : MeasurableSpace Ω] [inst_1 : Fintype ι] [Nonempty ι] [inst_3 : Fintype κ] (μ : MeasureTheory.Measure Ω) [MeasureTheory.IsProbabilityMeasure μ] (ξ : κ → Ω → ℝ), (∀ (k : κ), MeasureTheory.MemLp (ξ k) 2 μ) → (∫ (s : ι → Ω), √(∑ k, ((↑(Fintype.card ι))⁻¹ * ∑ i, ξ k (s i) - ∫ (ω : Ω), ξ k ω ∂μ) ^ 2) ∂MeasureTheory.Measure.pi fun x => μ) ≤ √((∑ k, ∫ (ω : Ω), ξ k ω ^ 2 ∂μ) / ↑(Fintype.card ι))","l":182,"f":"Causalean/Mathlib/Probability/IidMeanVariance.lean"},{"n":"Causalean.Mathlib.Probability.iid_mean_sq_le","k":"theorem","m":"Causalean.Mathlib.Probability.IidMeanVariance","a":"Mathlib/Probability","t":2,"d":"The mean squared error of a square-integrable scalar sample average of n independent, identically distributed observations is at most the population second moment divided by n.","s":"∀ {Ω : Type u_1} [inst : MeasurableSpace Ω] (μ : MeasureTheory.Measure Ω) [MeasureTheory.IsProbabilityMeasure μ] {n : ℕ}, 0 < n → ∀ (ξ : Ω → ℝ), MeasureTheory.MemLp ξ 2 μ → (∫ (s : Fin n → Ω), ((↑n)⁻¹ * ∑ i, ξ (s i) - ∫ (ω : Ω), ξ ω ∂μ) ^ 2 ∂MeasureTheory.Measure.pi fun x => μ) ≤ (∫ (ω : Ω), ξ ω ^ 2 ∂μ) / ↑n","l":290,"f":"Causalean/Mathlib/Probability/IidMeanVariance.lean"},{"n":"Causalean.Mathlib.Probability.iid_mean_sq_le_fintype","k":"theorem","m":"Causalean.Mathlib.Probability.IidMeanVariance","a":"Mathlib/Probability","t":2,"d":"The mean squared error of a square-integrable scalar sample average from independent, identically distributed observations is at most the population second moment divided by the sample size.","s":"∀ {Ω : Type u_1} {ι : Type u_2} [inst : MeasurableSpace Ω] [inst_1 : Fintype ι] [Nonempty ι] (μ : MeasureTheory.Measure Ω) [MeasureTheory.IsProbabilityMeasure μ] (ξ : Ω → ℝ), MeasureTheory.MemLp ξ 2 μ → (∫ (s : ι → Ω), ((↑(Fintype.card ι))⁻¹ * ∑ i, ξ (s i) - ∫ (ω : Ω), ξ ω ∂μ) ^ 2 ∂MeasureTheory.Measure.pi fun x => μ) ≤ (∫ (ω : Ω), ξ ω ^ 2 ∂μ) / ↑(Fintype.card ι)","l":77,"f":"Causalean/Mathlib/Probability/IidMeanVariance.lean"},{"n":"Causalean.Mathlib.Probability.integrable_euclidean_of_integrable","k":"theorem","m":"Causalean.Mathlib.Probability.IidMeanVariance","a":"Mathlib/Probability","t":2,"d":"The Euclidean length of a finite family of integrable real functions is itself integrable.","s":"∀ {Ω : Type u_1} {ι : Type u_2} [inst : MeasurableSpace Ω] [inst_1 : Fintype ι] {μ : MeasureTheory.Measure Ω} (v : Ω → ι → ℝ), (∀ (i : ι), MeasureTheory.Integrable (fun ω => v ω i) μ) → MeasureTheory.Integrable (fun ω => √(∑ i, v ω i ^ 2)) μ","l":44,"f":"Causalean/Mathlib/Probability/IidMeanVariance.lean"},{"n":"Causalean.Mathlib.Probability.integrable_iid_mean_euclidean","k":"theorem","m":"Causalean.Mathlib.Probability.IidMeanVariance","a":"Mathlib/Probability","t":2,"d":"The centred coordinatewise sample averages of finitely many integrable statistics over an n-point independent, identically distributed sample have integrable Euclidean length.","s":"∀ {Ω : Type u_1} {ι : Type u_2} [inst : MeasurableSpace Ω] [inst_1 : Fintype ι] (μ : MeasureTheory.Measure Ω) [MeasureTheory.IsProbabilityMeasure μ] {n : ℕ} (ξ : ι → Ω → ℝ), (∀ (k : ι), MeasureTheory.Integrable (ξ k) μ) → MeasureTheory.Integrable (fun s => √(∑ k, ((↑n)⁻¹ * ∑ i, ξ k (s i) - ∫ (ω : Ω), ξ k ω ∂μ) ^ 2)) (MeasureTheory.Measure.pi fun x => μ)","l":343,"f":"Causalean/Mathlib/Probability/IidMeanVariance.lean"},{"n":"Causalean.Mathlib.Probability.klDiv_map_measurableEquiv","k":"theorem","m":"Causalean.Mathlib.Probability.SignedTwoPoint","a":"Mathlib/Probability","t":2,"d":"KL-divergence is invariant under a measurable equivalence. Pushing both finite measures μ, ν forward through a measurable equivalence e leaves their Kullback–Leibler divergence unchanged: KL(e_* μ, e_* ν) = KL(μ, ν).","s":"∀ {α : Type u_1} {β : Type u_2} [inst : MeasurableSpace α] [inst_1 : MeasurableSpace β] (e : α ≃ᵐ β) (μ ν : MeasureTheory.Measure α) [MeasureTheory.IsFiniteMeasure μ] [MeasureTheory.IsFiniteMeasure ν], InformationTheory.klDiv (MeasureTheory.Measure.map (⇑e) μ) (MeasureTheory.Measure.map (⇑e) ν) = InformationTheory.klDiv μ ν","l":124,"f":"Causalean/Mathlib/Probability/SignedTwoPoint.lean"},{"n":"Causalean.Mathlib.Probability.measurable_bernoulliBool","k":"theorem","m":"Causalean.Mathlib.Probability.BernoulliMeasure","a":"Mathlib/Probability","t":2,"d":"The Bool-valued Bernoulli distribution varies measurably with its success probability, so a measurable probability parameter can be used to construct a measurable kernel.","s":"Measurable Causalean.Mathlib.Probability.bernoulliBool","l":281,"f":"Causalean/Mathlib/Probability/BernoulliMeasure.lean"},{"n":"Causalean.Mathlib.Probability.measurable_twoPointMean","k":"theorem","m":"Causalean.Mathlib.Probability.SignedTwoPoint","a":"Mathlib/Probability","t":2,"d":"For a fixed scale B, the signed two-point mean channel is measurable as a function of the target mean u.","s":"∀ (B : ℝ), Measurable fun u => Causalean.Mathlib.Probability.twoPointMean B u","l":44,"f":"Causalean/Mathlib/Probability/SignedTwoPoint.lean"},{"n":"Causalean.Mathlib.Probability.memLp_two_iid_mean_euclidean","k":"theorem","m":"Causalean.Mathlib.Probability.IidMeanVariance","a":"Mathlib/Probability","t":2,"d":"The centred coordinatewise sample averages of finitely many square-integrable statistics over an n-point independent, identically distributed sample form a square-integrable Euclidean length.","s":"∀ {Ω : Type u_1} {ι : Type u_2} [inst : MeasurableSpace Ω] [inst_1 : Fintype ι] (μ : MeasureTheory.Measure Ω) [MeasureTheory.IsProbabilityMeasure μ] {n : ℕ} (ξ : ι → Ω → ℝ), (∀ (k : ι), MeasureTheory.MemLp (ξ k) 2 μ) → MeasureTheory.MemLp (fun s => √(∑ k, ((↑n)⁻¹ * ∑ i, ξ k (s i) - ∫ (ω : Ω), ξ k ω ∂μ) ^ 2)) 2 (MeasureTheory.Measure.pi fun x => μ)","l":329,"f":"Causalean/Mathlib/Probability/IidMeanVariance.lean"},{"n":"Causalean.Mathlib.Probability.memLp_two_iid_mean_euclidean_fintype","k":"theorem","m":"Causalean.Mathlib.Probability.IidMeanVariance","a":"Mathlib/Probability","t":2,"d":"The centred coordinatewise sample averages of finitely many square-integrable statistics form a square-integrable Euclidean length on the product sample space.","s":"∀ {Ω : Type u_1} {ι : Type u_2} {κ : Type u_3} [inst : MeasurableSpace Ω] [inst_1 : Fintype ι] [inst_2 : Fintype κ] (μ : MeasureTheory.Measure Ω) [MeasureTheory.IsProbabilityMeasure μ] (ξ : κ → Ω → ℝ), (∀ (k : κ), MeasureTheory.MemLp (ξ k) 2 μ) → MeasureTheory.MemLp (fun s => √(∑ k, ((↑(Fintype.card ι))⁻¹ * ∑ i, ξ k (s i) - ∫ (ω : Ω), ξ k ω ∂μ) ^ 2)) 2 (MeasureTheory.Measure.pi fun x => μ)","l":164,"f":"Causalean/Mathlib/Probability/IidMeanVariance.lean"},{"n":"Causalean.Mathlib.Probability.memLp_two_sqrt_sum_sq","k":"theorem","m":"Causalean.Mathlib.Probability.IidMeanVariance","a":"Mathlib/Probability","t":2,"d":"The Euclidean length of a finite family of square-integrable real functions is itself square-integrable.","s":"∀ {Ω : Type u_1} {ι : Type u_2} [inst : MeasurableSpace Ω] [inst_1 : Fintype ι] {μ : MeasureTheory.Measure Ω} {Y : ι → Ω → ℝ}, (∀ (k : ι), MeasureTheory.MemLp (Y k) 2 μ) → MeasureTheory.MemLp (fun ω => √(∑ k, Y k ω ^ 2)) 2 μ","l":60,"f":"Causalean/Mathlib/Probability/IidMeanVariance.lean"},{"n":"Causalean.Mathlib.Probability.sqrt_sum_sq_le_sum_abs","k":"theorem","m":"Causalean.Mathlib.Probability.IidMeanVariance","a":"Mathlib/Probability","t":2,"d":"The Euclidean length of a finite real-valued vector is no greater than the sum of the absolute values of its components.","s":"∀ {ι : Type u_1} [inst : Fintype ι] (v : ι → ℝ), √(∑ i, v i ^ 2) ≤ ∑ i, |v i|","l":33,"f":"Causalean/Mathlib/Probability/IidMeanVariance.lean"},{"n":"Causalean.Mathlib.Probability.sum_bernoulli_eq_binomial","k":"theorem","m":"Causalean.Mathlib.Probability.BernoulliMeasure","a":"Mathlib/Probability","t":2,"d":"A Bernoulli-weighted sum of any function of the success count equals the corresponding sum against the binomial mass function.","s":"∀ {ι : Type u_1} [inst : Fintype ι] [inst_1 : DecidableEq ι] (p : ℝ) (F : ℕ → ℝ), ∑ b, (∏ i, if b i = true then p else 1 - p) * F {i | b i = true}.card = ∑ j ∈ Finset.range (Fintype.card ι + 1), Causalean.Mathlib.Probability.binomialWeight (Fintype.card ι) p j * F j","l":396,"f":"Causalean/Mathlib/Probability/BernoulliMeasure.lean"},{"n":"Causalean.Mathlib.Probability.totalized_inverse_count_le","k":"theorem","m":"Causalean.Mathlib.Probability.BernoulliMeasure","a":"Mathlib/Probability","t":2,"d":"The reciprocal of a positive integer, totalized to zero at the origin, is at most twice the reciprocal of its successor.","s":"∀ (j : ℕ), (if 0 < j then (↑j)⁻¹ else 0) ≤ 2 * (↑j + 1)⁻¹","l":421,"f":"Causalean/Mathlib/Probability/BernoulliMeasure.lean"},{"n":"Causalean.Mathlib.Probability.twoPointMean","k":"def","m":"Causalean.Mathlib.Probability.SignedTwoPoint","a":"Mathlib/Probability","t":1,"d":"Symmetric two-point mean channel. For a scale B and a target mean u, the law twoPointMean B u on {−B, B} puts mass (1 + u/B)/2 on B and (1 − u/B)/2 on −B; its mean is u.","s":"ℝ → ℝ → MeasureTheory.Measure ℝ","l":37,"f":"Causalean/Mathlib/Probability/SignedTwoPoint.lean"},{"n":"Causalean.Mathlib.Probability.twoPointMean_bad_support_zero","k":"theorem","m":"Causalean.Mathlib.Probability.SignedTwoPoint","a":"Mathlib/Probability","t":2,"d":"The channel is supported in [−M, M] whenever |B| ≤ M. The mass that any nonnegative mixture of point masses at B and −B places outside the interval [−M, M] is 0.","s":"∀ {B M : ℝ} (wplus wminus : ENNReal), |B| ≤ M → (wplus • MeasureTheory.Measure.dirac B + wminus • MeasureTheory.Measure.dirac (-B)) {y | y ∉ Set.Icc (-M) M} = 0","l":98,"f":"Causalean/Mathlib/Probability/SignedTwoPoint.lean"},{"n":"Causalean.Mathlib.Probability.twoPointMean_coef_nonneg","k":"theorem","m":"Causalean.Mathlib.Probability.SignedTwoPoint","a":"Mathlib/Probability","t":2,"d":"When the scale is positive and the target mean lies within that scale, both weights in the signed two-point distribution are nonnegative.","s":"∀ {B u : ℝ}, 0 < B → |u| ≤ B → 0 ≤ (1 + u / B) / 2 ∧ 0 ≤ (1 - u / B) / 2","l":51,"f":"Causalean/Mathlib/Probability/SignedTwoPoint.lean"},{"n":"Causalean.Mathlib.Probability.twoPointMean_eq_map_bernoulli","k":"theorem","m":"Causalean.Mathlib.Probability.SignedTwoPoint","a":"Mathlib/Probability","t":2,"d":"Affine-image representation. The {−B, B} mean channel is the pushforward of the {0,1} Bernoulli law bernoulliLaw ((1 + u/B)/2) under the affine map x ↦ 2Bx − B (for B ≠ 0).","s":"∀ (B u : ℝ) (hB : B ≠ 0), Causalean.Mathlib.Probability.twoPointMean B u = MeasureTheory.Measure.map (⇑(affineHomeomorph (2 * B) (-B) (mul_ne_zero (Mathlib.Meta.NormNum.isNat_eq_false (Mathlib.Meta.NormNum.isNat_ofNat ℝ (Eq.refl 2)) (Mathlib.Meta.NormNum.isNat_ofNat ℝ Nat.cast_zero) (Eq.refl false)) hB)).toMeasurableEquiv) (Causalean.Mathlib.Probability.bernoulliLaw ((1 + u / B) / 2))","l":150,"f":"Causalean/Mathlib/Probability/SignedTwoPoint.lean"},{"n":"Causalean.Mathlib.Probability.twoPointMean_integral","k":"theorem","m":"Causalean.Mathlib.Probability.SignedTwoPoint","a":"Mathlib/Probability","t":2,"d":"Two-point integral. For 0 < B and |u| ≤ B, integrating f against twoPointMean B u returns the two-point weighted average ((1 + u/B)/2)·f(B) + ((1 − u/B)/2)·f(−B).","s":"∀ {B u : ℝ}, 0 < B → |u| ≤ B → ∀ (f : ℝ → ℝ), ∫ (y : ℝ), f y ∂Causalean.Mathlib.Probability.twoPointMean B u = (1 + u / B) / 2 * f B + (1 - u / B) / 2 * f (-B)","l":72,"f":"Causalean/Mathlib/Probability/SignedTwoPoint.lean"},{"n":"Causalean.Mathlib.Probability.twoPointMean_isProbabilityMeasure","k":"theorem","m":"Causalean.Mathlib.Probability.SignedTwoPoint","a":"Mathlib/Probability","t":2,"d":"The signed two-point channel is a probability measure. For 0 < B and |u| ≤ B the total mass of twoPointMean B u is 1.","s":"∀ {B u : ℝ}, 0 < B → |u| ≤ B → MeasureTheory.IsProbabilityMeasure (Causalean.Mathlib.Probability.twoPointMean B u)","l":58,"f":"Causalean/Mathlib/Probability/SignedTwoPoint.lean"},{"n":"Causalean.Mathlib.Probability.twoPointMean_mean","k":"theorem","m":"Causalean.Mathlib.Probability.SignedTwoPoint","a":"Mathlib/Probability","t":2,"d":"The mean is u. For 0 < B and |u| ≤ B, the expectation of the identity under twoPointMean B u is exactly u.","s":"∀ {B u : ℝ}, 0 < B → |u| ≤ B → ∫ (y : ℝ), y ∂Causalean.Mathlib.Probability.twoPointMean B u = u","l":90,"f":"Causalean/Mathlib/Probability/SignedTwoPoint.lean"},{"n":"Causalean.Mathlib.Probability.variance_prod_eq_integral_variance_add","k":"theorem","m":"Causalean.Mathlib.Probability.VarianceProd","a":"Mathlib/Probability","t":1,"d":"For independent coordinates with laws μ and ν, and a real-valued statistic F on the product that is square-integrable under the product law, suppose that almost every slice of F along the first coordinate, at a fixed value of the second coordinate, is square-integrable, that the map m of slice means is square-integrable under ν, and that m records the correct slice means: the integral of F over the first coordinate at almost every value of the second coordinate equals m there. Then the variance of F under the product measure equals the ν-average of the within-slice variances of F plus the variance, across the second coordinate, of the slice-mean map m.","s":"∀ {Ω : Type u_1} {T : Type u_2} [inst : MeasurableSpace Ω] [inst_1 : MeasurableSpace T] (μ : MeasureTheory.Measure Ω) (ν : MeasureTheory.Measure T) [MeasureTheory.IsProbabilityMeasure μ] [MeasureTheory.IsProbabilityMeasure ν] (F : Ω × T → ℝ) (m : T → ℝ), MeasureTheory.MemLp F 2 (μ.prod ν) → (∀ᵐ (t : T) ∂ν, MeasureTheory.MemLp (fun s => F (s, t)) 2 μ) → MeasureTheory.MemLp m 2 ν → (∀ᵐ (t : T) ∂ν, ∫ (s : Ω), F (s, t) ∂μ = m t) → ProbabilityTheory.variance F (μ.prod ν) = ∫ (t : T), ProbabilityTheory.variance (fun s => F (s, t)) μ ∂ν + ProbabilityTheory.variance m ν","l":23,"f":"Causalean/Mathlib/Probability/VarianceProd.lean"},{"n":"Causalean.Mathlib.ProbabilityTheory.ProductCondDistrib.condDistrib_fst_of_compProd_prod","k":"theorem","m":"Causalean.Mathlib.Probability.Kernel.ProductCondDistrib","a":"Mathlib/Probability/Kernel","t":1,"d":"For the joint law formed by combining a finite base measure ρ with the product of two Markov kernels κ and η, the conditional distribution of the first product-kernel coordinate given the base coordinate equals κ, for ρ-almost every base point.","s":"∀ {α : Type u_1} {β : Type u_2} {γ : Type u_3} [inst : MeasurableSpace α] [inst_1 : MeasurableSpace β] [inst_2 : MeasurableSpace γ] [inst_3 : StandardBorelSpace β] [inst_4 : Nonempty β] {ρ : MeasureTheory.Measure α} [inst_5 : MeasureTheory.IsFiniteMeasure ρ] (κ : ProbabilityTheory.Kernel α β) (η : ProbabilityTheory.Kernel α γ) [inst_6 : ProbabilityTheory.IsMarkovKernel κ] [inst_7 : ProbabilityTheory.IsMarkovKernel η], ⇑(ProbabilityTheory.condDistrib (fun z => z.2.1) (fun z => z.1) (ρ.compProd (κ.prod η))) =ᵐ[ρ] ⇑κ","l":31,"f":"Causalean/Mathlib/Probability/Kernel/ProductCondDistrib.lean"},{"n":"Causalean.Mathlib.ProbabilityTheory.ProductCondDistrib.condDistrib_snd_of_compProd_prod","k":"theorem","m":"Causalean.Mathlib.Probability.Kernel.ProductCondDistrib","a":"Mathlib/Probability/Kernel","t":1,"d":"For the joint law formed by combining a finite base measure ρ with the product of two Markov kernels κ and η, the conditional distribution of the second product-kernel coordinate given the base coordinate equals η, for ρ-almost every base point.","s":"∀ {α : Type u_1} {β : Type u_2} {γ : Type u_3} [inst : MeasurableSpace α] [inst_1 : MeasurableSpace β] [inst_2 : MeasurableSpace γ] [inst_3 : StandardBorelSpace γ] [inst_4 : Nonempty γ] {ρ : MeasureTheory.Measure α} [inst_5 : MeasureTheory.IsFiniteMeasure ρ] (κ : ProbabilityTheory.Kernel α β) (η : ProbabilityTheory.Kernel α γ) [inst_6 : ProbabilityTheory.IsMarkovKernel κ] [inst_7 : ProbabilityTheory.IsMarkovKernel η], ⇑(ProbabilityTheory.condDistrib (fun z => z.2.2) (fun z => z.1) (ρ.compProd (κ.prod η))) =ᵐ[ρ] ⇑η","l":74,"f":"Causalean/Mathlib/Probability/Kernel/ProductCondDistrib.lean"},{"n":"Causalean.Mathlib.RpowArith.div_rpow_of_nonneg_of_pos","k":"theorem","m":"Causalean.Mathlib.Analysis.RpowArith","a":"Mathlib/Analysis","t":1,"d":"Factoring a real power of a quotient. For a nonnegative numerator A and a strictly positive denominator n, the real power (A/n)^p equals A^p times n raised to the power −p, for any real exponent p.","s":"∀ (A p n : ℝ), 0 ≤ A → 0 < n → (A / n) ^ p = A ^ p * n ^ (-p)","l":29,"f":"Causalean/Mathlib/Analysis/RpowArith.lean"},{"n":"Causalean.Mathlib.RpowArith.inv_eq_rpow_neg_one","k":"theorem","m":"Causalean.Mathlib.Analysis.RpowArith","a":"Mathlib/Analysis","t":1,"d":"For any real number x, its reciprocal equals its real power raised to the exponent −1.","s":"∀ (x : ℝ), x⁻¹ = x ^ (-1)","l":24,"f":"Causalean/Mathlib/Analysis/RpowArith.lean"},{"n":"Causalean.Mathlib.RpowArith.inv_mul_sqrt_eq_rpow_neg_half","k":"theorem","m":"Causalean.Mathlib.Analysis.RpowArith","a":"Mathlib/Analysis","t":1,"d":"Reciprocal times square root as a negative-half power. For a nonnegative real number q, the reciprocal of q times the square root of q equals q raised to the power −1/2.","s":"∀ (q : ℝ), 0 ≤ q → q⁻¹ * √q = q ^ (-(1 / 2))","l":66,"f":"Causalean/Mathlib/Analysis/RpowArith.lean"},{"n":"Causalean.Mathlib.RpowArith.rpow_natCast_nonpos_le_one","k":"theorem","m":"Causalean.Mathlib.Analysis.RpowArith","a":"Mathlib/Analysis","t":1,"d":"Nonpositive real power of a natural number is at most one. For a nonpositive real exponent e, the real power of any natural-number cast raised to e is at most 1.","s":"∀ (n : ℕ), ∀ e ≤ 0, ↑n ^ e ≤ 1","l":51,"f":"Causalean/Mathlib/Analysis/RpowArith.lean"},{"n":"Causalean.Mathlib.TwoStateMarkov.one_minus_a_b_abs_lt_one","k":"theorem","m":"Causalean.Mathlib.TwoStateMarkov","a":"Mathlib","t":2,"d":"Pointwise spectral gap: |1 - a - b| < 1 when a + b lies strictly between zero and two.","s":"∀ {K : Type u_1} [inst : CommRing K] [inst_1 : LinearOrder K] [IsStrictOrderedRing K] {a b : K}, 0 < a + b → a + b < 2 → |1 - a - b| < 1","l":119,"f":"Causalean/Mathlib/TwoStateMarkov.lean"},{"n":"Causalean.Mathlib.TwoStateMarkov.one_minus_a_b_uniform_gap_on_compact","k":"theorem","m":"Causalean.Mathlib.TwoStateMarkov","a":"Mathlib","t":2,"d":"Uniform spectral gap on compact subsets of the open strip: the continuous function (a,b) ↦ |1 - a - b| attains its supremum when 0 < a + b < 2 at some point of K, and that supremum is strictly less than 1 by the pointwise bound.","s":"∀ (K : Set (ℝ × ℝ)), IsCompact K → K ⊆ {p | 0 < p.1 + p.2 ∧ p.1 + p.2 < 2} → ∃ ρ < 1, ∀ p ∈ K, |1 - p.1 - p.2| ≤ ρ","l":133,"f":"Causalean/Mathlib/TwoStateMarkov.lean"},{"n":"Causalean.Mathlib.TwoStateMarkov.stationaryProjection","k":"def","m":"Causalean.Mathlib.TwoStateMarkov","a":"Mathlib","t":1,"d":"Stationary projection with both rows equal to (b/(a+b), a/(a+b)).","s":"{K : Type u_1} → [Field K] → K → K → Matrix (Fin 2) (Fin 2) K","l":55,"f":"Causalean/Mathlib/TwoStateMarkov.lean"},{"n":"Causalean.Mathlib.TwoStateMarkov.transitionMatrix","k":"def","m":"Causalean.Mathlib.TwoStateMarkov","a":"Mathlib","t":1,"d":"Transition matrix with rows (1 - a, a) and (b, 1 - b).","s":"{K : Type u_1} → [Field K] → K → K → Matrix (Fin 2) (Fin 2) K","l":46,"f":"Causalean/Mathlib/TwoStateMarkov.lean"},{"n":"Causalean.Mathlib.TwoStateMarkov.transitionMatrix_pow_eq_spectral","k":"theorem","m":"Causalean.Mathlib.TwoStateMarkov","a":"Mathlib","t":1,"d":"For parameters a, b in a field with a + b nonzero, the k-th power of the two-state transition matrix transitionMatrix a b decomposes as the stationary projection stationaryProjection a b plus (1 - a - b)^k times its complement, for every k.","s":"∀ {K : Type u_1} [inst : Field K] (a b : K), a + b ≠ 0 → ∀ (k : ℕ), Causalean.Mathlib.TwoStateMarkov.transitionMatrix a b ^ k = Causalean.Mathlib.TwoStateMarkov.stationaryProjection a b + (1 - a - b) ^ k • (1 - Causalean.Mathlib.TwoStateMarkov.stationaryProjection a b)","l":60,"f":"Causalean/Mathlib/TwoStateMarkov.lean"},{"n":"Causalean.Mathlib.TwoStateMarkov.transitionMatrix_pow_tendsto_stationary_uniform","k":"theorem","m":"Causalean.Mathlib.TwoStateMarkov","a":"Mathlib","t":1,"d":"For a set K of transition-parameter pairs (a,b) that is compact and contained in the open unit square, the k-th power of the transition matrix converges to the stationary projection entrywise, uniformly over K: for every ε > 0 there is a threshold N such that every entry of (transitionMatrix a b)^k - stationaryProjection a b has absolute value at most ε once k ≥ N, for every (a,b) in K.","s":"∀ (K : Set (ℝ × ℝ)), IsCompact K → K ⊆ {p | 0 < p.1 ∧ p.1 < 1 ∧ 0 < p.2 ∧ p.2 < 1} → ∀ ε > 0, ∃ N, ∀ (k : ℕ), N ≤ k → ∀ p ∈ K, ∀ (i j : Fin 2), |(Causalean.Mathlib.TwoStateMarkov.transitionMatrix p.1 p.2 ^ k - Causalean.Mathlib.TwoStateMarkov.stationaryProjection p.1 p.2) i j| ≤ ε","l":161,"f":"Causalean/Mathlib/TwoStateMarkov.lean"},{"n":"Causalean.Mathlib.abs_covariance_le_sqrt_mul","k":"theorem","m":"Causalean.Mathlib.Probability.CovarianceCauchySchwarz","a":"Mathlib/Probability","t":1,"d":"Cauchy–Schwarz for covariance. Under a finite measure, if $X$ is square-integrable and $Y$ is square-integrable, then the absolute value of their covariance is at most the product of their standard deviations, $|\\mathrm{Cov}(X,Y)| \\le \\sqrt{\\mathrm{Var}(X)} \\cdot \\sqrt{\\mathrm{Var}(Y)}$.","s":"∀ {Ω : Type u_1} {mΩ : MeasurableSpace Ω} {μ : MeasureTheory.Measure Ω} [MeasureTheory.IsFiniteMeasure μ] {X Y : Ω → ℝ}, MeasureTheory.MemLp X 2 μ → MeasureTheory.MemLp Y 2 μ → |ProbabilityTheory.covariance X Y μ| ≤ √(ProbabilityTheory.variance X μ) * √(ProbabilityTheory.variance Y μ)","l":69,"f":"Causalean/Mathlib/Probability/CovarianceCauchySchwarz.lean"},{"n":"Causalean.Mathlib.affine_preimage_Ioi","k":"theorem","m":"Causalean.Mathlib.Probability.GaussianMoments","a":"Mathlib/Probability","t":2,"d":"The inverse image of a right-hand tail under a positive affine transformation is a right-hand tail whose threshold is transformed by the inverse affine formula.","s":"∀ {α : Type u_1} [inst : Field α] [inst_1 : LinearOrder α] [IsStrictOrderedRing α] {s c m : α}, 0 < s → (fun z => s * z + m) ⁻¹' Set.Ioi c = Set.Ioi ((c - m) / s)","l":53,"f":"Causalean/Mathlib/Probability/GaussianMoments.lean"},{"n":"Causalean.Mathlib.bddBelow_condQuantileSet","k":"theorem","m":"Causalean.Mathlib.Probability.MeasurableCondQuantile","a":"Mathlib/Probability","t":2,"d":"At a strictly positive conditional-quantile level, the response values whose conditional cumulative distribution function reaches that level are bounded below.","s":"∀ {α : Type u_1} [inst : MeasurableSpace α] {ρ : MeasureTheory.Measure (α × ℝ)} {τ : α → ℝ} {a : α}, 0 < τ a → BddBelow {x | τ a ≤ ↑(ProbabilityTheory.condCDF ρ a) x}","l":43,"f":"Causalean/Mathlib/Probability/MeasurableCondQuantile.lean"},{"n":"Causalean.Mathlib.cdf_continuous_of_noAtoms","k":"theorem","m":"Causalean.Mathlib.Probability.StdNormalCDF","a":"Mathlib/Probability","t":2,"d":"The CDF of an atomless real probability measure is continuous.","s":"∀ (μ : MeasureTheory.Measure ℝ) [MeasureTheory.IsProbabilityMeasure μ] [MeasureTheory.NullSingletonClass μ], Continuous ↑(ProbabilityTheory.cdf μ)","l":90,"f":"Causalean/Mathlib/Probability/StdNormalCDF.lean"},{"n":"Causalean.Mathlib.cdf_neg_of_map_neg","k":"theorem","m":"Causalean.Mathlib.Probability.StdNormalCDF","a":"Mathlib/Probability","t":2,"d":"The CDF of an atomless probability measure invariant under reflection is symmetric.","s":"∀ {μ : MeasureTheory.Measure ℝ} [MeasureTheory.IsProbabilityMeasure μ] [MeasureTheory.NullSingletonClass μ], MeasureTheory.Measure.map (fun x => -x) μ = μ → ∀ (t : ℝ), ↑(ProbabilityTheory.cdf μ) (-t) = 1 - ↑(ProbabilityTheory.cdf μ) t","l":60,"f":"Causalean/Mathlib/Probability/StdNormalCDF.lean"},{"n":"Causalean.Mathlib.centered_sq_lintegral_le_eLpNorm_two_sq","k":"theorem","m":"Causalean.Mathlib.IIDCenteredSum","a":"Mathlib","t":2,"d":"Centering a square-integrable real variable cannot increase its second moment beyond its uncentered L2 energy under a probability distribution.","s":"∀ {X : Type u_1} [inst : MeasurableSpace X] {P : MeasureTheory.Measure X} [MeasureTheory.IsProbabilityMeasure P] {f : X → ℝ}, MeasureTheory.MemLp f 2 P → ∫⁻ (x : X), ENNReal.ofReal ((f x - ∫ (y : X), f y ∂P) ^ 2) ∂P ≤ ENNReal.ofReal ((MeasureTheory.eLpNorm f 2 P).toReal ^ 2)","l":153,"f":"Causalean/Mathlib/IIDCenteredSum.lean"},{"n":"Causalean.Mathlib.condQuantile","k":"def","m":"Causalean.Mathlib.Probability.MeasurableCondQuantile","a":"Mathlib/Probability","t":1,"d":"The conditional quantile at level τ: the generalized inverse of the conditional CDF, q(a) = inf { x : ℝ | τ(a) ≤ condCDF ρ a x }.","s":"{α : Type u_1} → [inst : MeasurableSpace α] → MeasureTheory.Measure (α × ℝ) → (α → ℝ) → α → ℝ","l":31,"f":"Causalean/Mathlib/Probability/MeasurableCondQuantile.lean"},{"n":"Causalean.Mathlib.condQuantileSet_up_closed","k":"theorem","m":"Causalean.Mathlib.Probability.MeasurableCondQuantile","a":"Mathlib/Probability","t":2,"d":"If a response value's conditional cumulative distribution function reaches a target level, then every larger response value also reaches that level.","s":"∀ {α : Type u_1} [inst : MeasurableSpace α] {ρ : MeasureTheory.Measure (α × ℝ)} {τ : α → ℝ} {a : α} {x x' : ℝ}, x ∈ {y | τ a ≤ ↑(ProbabilityTheory.condCDF ρ a) y} → x ≤ x' → x' ∈ {y | τ a ≤ ↑(ProbabilityTheory.condCDF ρ a) y}","l":36,"f":"Causalean/Mathlib/Probability/MeasurableCondQuantile.lean"},{"n":"Causalean.Mathlib.condQuantile_le_iff","k":"theorem","m":"Causalean.Mathlib.Probability.MeasurableCondQuantile","a":"Mathlib/Probability","t":2,"d":"At an interior target level, a point is above the conditional generalized inverse exactly when its conditional cumulative distribution function has reached that level.","s":"∀ {α : Type u_1} [inst : MeasurableSpace α] {ρ : MeasureTheory.Measure (α × ℝ)} {τ : α → ℝ} {a : α} {x : ℝ}, 0 < τ a → τ a < 1 → (Causalean.Mathlib.condQuantile ρ τ a ≤ x ↔ τ a ≤ ↑(ProbabilityTheory.condCDF ρ a) x)","l":93,"f":"Causalean/Mathlib/Probability/MeasurableCondQuantile.lean"},{"n":"Causalean.Mathlib.condQuantile_le_of_le_condCDF","k":"theorem","m":"Causalean.Mathlib.Probability.MeasurableCondQuantile","a":"Mathlib/Probability","t":2,"d":"When the conditional cumulative distribution function has reached a positive target level at a point, the conditional generalized inverse is no larger than that point.","s":"∀ {α : Type u_1} [inst : MeasurableSpace α] {ρ : MeasureTheory.Measure (α × ℝ)} {τ : α → ℝ} {a : α} {x : ℝ}, 0 < τ a → τ a ≤ ↑(ProbabilityTheory.condCDF ρ a) x → Causalean.Mathlib.condQuantile ρ τ a ≤ x","l":86,"f":"Causalean/Mathlib/Probability/MeasurableCondQuantile.lean"},{"n":"Causalean.Mathlib.covarianceBilin_stdGaussian","k":"theorem","m":"Causalean.Mathlib.StandardGaussian","a":"Mathlib","t":2,"d":"The covariance bilinear form of the standard Gaussian measure is the ambient inner product.","s":"∀ {E : Type u_1} [inst : NormedAddCommGroup E] [inst_1 : InnerProductSpace ℝ E] [inst_2 : FiniteDimensional ℝ E] [inst_3 : MeasurableSpace E] [inst_4 : BorelSpace E] (u v : E), ((ProbabilityTheory.covarianceBilin (Causalean.Mathlib.stdGaussian E)) u) v = inner ℝ u v","l":236,"f":"Causalean/Mathlib/StandardGaussian.lean"},{"n":"Causalean.Mathlib.covariance_sq_le_variance_mul","k":"theorem","m":"Causalean.Mathlib.Probability.CovarianceCauchySchwarz","a":"Mathlib/Probability","t":2,"d":"Cauchy–Schwarz for covariance (squared form). For square-integrable statistics under a finite measure, the squared covariance is at most the product of the two variances.","s":"∀ {Ω : Type u_1} {mΩ : MeasurableSpace Ω} {μ : MeasureTheory.Measure Ω} [MeasureTheory.IsFiniteMeasure μ] {X Y : Ω → ℝ}, MeasureTheory.MemLp X 2 μ → MeasureTheory.MemLp Y 2 μ → ProbabilityTheory.covariance X Y μ ^ 2 ≤ ProbabilityTheory.variance X μ * ProbabilityTheory.variance Y μ","l":31,"f":"Causalean/Mathlib/Probability/CovarianceCauchySchwarz.lean"},{"n":"Causalean.Mathlib.eLpNorm_two_sq_toReal_eq_integral_sq","k":"theorem","m":"Causalean.Mathlib.IIDCenteredSum","a":"Mathlib","t":2,"d":"The square of a square-integrable variable's L2 norm equals the integral of its squared norm, for any measure.","s":"∀ {X : Type u_1} [inst : MeasurableSpace X] {P : MeasureTheory.Measure X} {E : Type u_2} [inst_1 : NormedAddCommGroup E] {f : X → E}, MeasureTheory.MemLp f 2 P → ENNReal.ofReal ((MeasureTheory.eLpNorm f 2 P).toReal ^ 2) = ENNReal.ofReal (∫ (x : X), ‖f x‖ ^ 2 ∂P)","l":123,"f":"Causalean/Mathlib/IIDCenteredSum.lean"},{"n":"Causalean.Mathlib.exists_orthogonalProjection_of_posSemidef","k":"theorem","m":"Causalean.Mathlib.SemiInnerProjection","a":"Mathlib","t":1,"d":"In a vector space over a linearly ordered field equipped with a bilinear form B that is symmetric and positive-semidefinite, every finite-dimensional linear subspace H admits a linear self-map P of the ambient space, valued in H, whose residual X - P X is B-orthogonal to every vector of H, for every X.","s":"∀ {K : Type u_1} {V : Type u_2} [inst : Field K] [inst_1 : LinearOrder K] [IsStrictOrderedRing K] [inst_3 : AddCommGroup V] [inst_4 : Module K V] (B : LinearMap.BilinForm K V), (∀ (x y : V), (B x) y = (B y) x) → (∀ (x : V), 0 ≤ (B x) x) → ∀ (H : Submodule K V) [FiniteDimensional K ↥H], ∃ P, (∀ (X : V), P X ∈ H) ∧ ∀ (X h : V), h ∈ H → (B (X - P X)) h = 0","l":42,"f":"Causalean/Mathlib/SemiInnerProjection.lean"},{"n":"Causalean.Mathlib.gaussianReal_Ioi_eq","k":"theorem","m":"Causalean.Mathlib.Probability.GaussianMoments","a":"Mathlib/Probability","t":1,"d":"Gaussian survival. For a normal law with mean m and a nonnegative variance parameter v that is nonzero, the probability mass above a threshold c equals one minus the standard-normal CDF evaluated at the standardized threshold (c − m)/√v.","s":"∀ (m : ℝ) (v : NNReal), v ≠ 0 → ∀ (c : ℝ), ((ProbabilityTheory.gaussianReal m v) (Set.Ioi c)).toReal = 1 - Causalean.Mathlib.stdNormalCDF ((c - m) / √↑v)","l":143,"f":"Causalean/Mathlib/Probability/GaussianMoments.lean"},{"n":"Causalean.Mathlib.gaussianReal_eq_map_std","k":"theorem","m":"Causalean.Mathlib.Probability.GaussianMoments","a":"Mathlib/Probability","t":2,"d":"A real Gaussian distribution with nonnegative variance is the standard normal distribution after scaling by the standard deviation and shifting by the mean.","s":"∀ (m : ℝ) (v : NNReal), ProbabilityTheory.gaussianReal m v = MeasureTheory.Measure.map (fun z => √↑v * z + m) (ProbabilityTheory.gaussianReal 0 1)","l":31,"f":"Causalean/Mathlib/Probability/GaussianMoments.lean"},{"n":"Causalean.Mathlib.gradient_coord_eq_deriv","k":"theorem","m":"Causalean.Mathlib.Analysis.GradientCoord","a":"Mathlib/Analysis","t":1,"d":"Gradient coordinate as a directional derivative. For a real-valued function f on a finite-dimensional Euclidean space that is differentiable at the point x, the a-th coordinate of its gradient at x equals the derivative at 0 of the one-variable restriction t ↦ f(x + t·eₐ) along the a-th standard basis direction.","s":"∀ {ι : Type u_1} [inst : Fintype ι] [inst_1 : DecidableEq ι] (f : EuclideanSpace ℝ ι → ℝ) (x : EuclideanSpace ℝ ι) (a : ι), DifferentiableAt ℝ f x → (gradient f x).ofLp a = deriv (fun t => f (x + t • EuclideanSpace.single a 1)) 0","l":26,"f":"Causalean/Mathlib/Analysis/GradientCoord.lean"},{"n":"Causalean.Mathlib.iid_centered_sum_sq_lintegral_le","k":"theorem","m":"Causalean.Mathlib.IIDCenteredSum","a":"Mathlib","t":1,"d":"Conditional second-moment bound for a centered i.i.d. sum. For a nonempty finite index set s, a sub-σ-algebra m_A with m_A coarser than the ambient σ-algebra, and a finite family of measurable sample coordinates W i that is jointly independent of m_A and jointly distributed as the i.i.d. product law P^s, if the parametric integrand g is jointly measurable with respect to m_A and the σ-algebra on the sample space and g ω lies in L²(P) for every ω, then the second moment of the normalized centered sum (1/√|s|)·Σᵢ(g ω (W i ω) − ∫ g ω dP) is at most the integrated squared L²(P)-norm of g ω, stated in lintegral form via ENNReal.ofReal.","s":"∀ {Ω : Type u_1} {X : Type u_2} [mΩ : MeasurableSpace Ω] [mX : MeasurableSpace X] {μ : MeasureTheory.Measure Ω} {P : MeasureTheory.Measure X} [MeasureTheory.IsFiniteMeasure μ] [MeasureTheory.IsProbabilityMeasure P] {ι : Type u_3} (s : Finset ι), 0 < s.card → ∀ (W : ι → Ω → X), (∀ i ∈ s, Measurable (W i)) → ∀ m_A ≤ mΩ, ProbabilityTheory.Indep m_A (MeasurableSpace.comap (fun ω i => W (↑i) ω) inferInstance) μ → (MeasureTheory.Measure.map (fun ω i => W (↑i) ω) μ = MeasureTheory.Measure.pi fun x => P) → ∀ (g : Ω → X → ℝ), Measurable (Function.uncurry g) → (∀ (ω : Ω), MeasureTheory.MemLp (g ω) 2 P) → ∫⁻ (ω : Ω), ENNReal.ofReal (((√↑s.card)⁻¹ * ∑ i ∈ s, (g ω (W i ω) - ∫ (x : X), g ω x ∂P)) ^ 2) ∂μ ≤ ∫⁻ (ω : Ω), ENNReal.ofReal ((MeasureTheory.eLpNorm (g ω) 2 P).toReal ^ 2) ∂μ","l":467,"f":"Causalean/Mathlib/IIDCenteredSum.lean"},{"n":"Causalean.Mathlib.iid_centered_sum_sq_lintegral_unscaled_le","k":"theorem","m":"Causalean.Mathlib.IIDCenteredSum","a":"Mathlib","t":2,"d":"A finite conditionally independent identically distributed family has integrated squared centered sum bounded by its size times the integrated squared L² norm.","s":"∀ {Ω : Type u_1} {X : Type u_2} [mΩ : MeasurableSpace Ω] [mX : MeasurableSpace X] {μ : MeasureTheory.Measure Ω} {P : MeasureTheory.Measure X} [MeasureTheory.IsFiniteMeasure μ] [MeasureTheory.IsProbabilityMeasure P] {ι : Type u_3} (s : Finset ι) (W : ι → Ω → X), (∀ i ∈ s, Measurable (W i)) → ∀ m_A ≤ mΩ, ProbabilityTheory.Indep m_A (MeasurableSpace.comap (fun ω i => W (↑i) ω) inferInstance) μ → (MeasureTheory.Measure.map (fun ω i => W (↑i) ω) μ = MeasureTheory.Measure.pi fun x => P) → ∀ (g : Ω → X → ℝ), Measurable (Function.uncurry g) → (∀ (ω : Ω), MeasureTheory.MemLp (g ω) 2 P) → ∫⁻ (ω : Ω), ENNReal.ofReal ((∑ i ∈ s, (g ω (W i ω) - ∫ (x : X), g ω x ∂P)) ^ 2) ∂μ ≤ ↑s.card * ∫⁻ (ω : Ω), ENNReal.ofReal ((MeasureTheory.eLpNorm (g ω) 2 P).toReal ^ 2) ∂μ","l":408,"f":"Causalean/Mathlib/IIDCenteredSum.lean"},{"n":"Causalean.Mathlib.iid_centered_sum_sq_lintegral_unscaled_le_of_joined_law","k":"theorem","m":"Causalean.Mathlib.IIDCenteredSum","a":"Mathlib","t":2,"d":"When a finite family jointly has the product law of a base population measure and independent identically distributed draws, the integrated squared centered sum is bounded by the number of draws times the integrated squared L² norm.","s":"∀ {Ω : Type u_1} {X : Type u_2} [mΩ : MeasurableSpace Ω] [mX : MeasurableSpace X] {μ : MeasureTheory.Measure Ω} {P : MeasureTheory.Measure X} [MeasureTheory.IsFiniteMeasure μ] [MeasureTheory.IsProbabilityMeasure P] {ι : Type u_3} (s : Finset ι) (W : ι → Ω → X), (∀ i ∈ s, Measurable (W i)) → ∀ (m_A : MeasurableSpace Ω) (hm_A_le : m_A ≤ mΩ), MeasureTheory.Measure.map (fun ω => (ω, fun i => W (↑i) ω)) μ = (μ.trim hm_A_le).prod (MeasureTheory.Measure.pi fun x => P) → ∀ (g : Ω → X → ℝ), Measurable (Function.uncurry g) → (∀ (ω : Ω), MeasureTheory.MemLp (g ω) 2 P) → ∫⁻ (ω : Ω), ENNReal.ofReal ((∑ i ∈ s, (g ω (W i ω) - ∫ (x : X), g ω x ∂P)) ^ 2) ∂μ ≤ ↑s.card * ∫⁻ (ω : Ω), ENNReal.ofReal ((MeasureTheory.eLpNorm (g ω) 2 P).toReal ^ 2) ∂μ","l":284,"f":"Causalean/Mathlib/IIDCenteredSum.lean"},{"n":"Causalean.Mathlib.indep_trim_prod_map_eq","k":"theorem","m":"Causalean.Mathlib.IIDCenteredSum","a":"Mathlib","t":2,"d":"Convert independence of a sub-σ-algebra and a random element into the product law of the joined map, with the first marginal trimmed to the sub-σ-algebra.","s":"∀ {Ω : Type u_1} {β : Type u_2} [mΩ : MeasurableSpace Ω] [mβ : MeasurableSpace β] {μ : MeasureTheory.Measure Ω} [MeasureTheory.IsFiniteMeasure μ] (m_A : MeasurableSpace Ω) (hm_A_le : m_A ≤ mΩ) {Z : Ω → β}, AEMeasurable Z μ → ProbabilityTheory.Indep m_A (MeasurableSpace.comap Z mβ) μ → MeasureTheory.Measure.map (fun ω => (ω, Z ω)) μ = (μ.trim hm_A_le).prod (MeasureTheory.Measure.map Z μ)","l":259,"f":"Causalean/Mathlib/IIDCenteredSum.lean"},{"n":"Causalean.Mathlib.integrable_id_mul_stdNormalPDF","k":"theorem","m":"Causalean.Mathlib.Probability.GaussianMoments","a":"Mathlib/Probability","t":2,"d":"The standard normal density has a finite first absolute moment.","s":"MeasureTheory.Integrable (fun x => x * Causalean.Mathlib.stdNormalPDF x) MeasureTheory.volume","l":81,"f":"Causalean/Mathlib/Probability/GaussianMoments.lean"},{"n":"Causalean.Mathlib.integral_Ioi_affine_gaussianReal_eq_density","k":"theorem","m":"Causalean.Mathlib.Probability.GaussianMoments","a":"Mathlib/Probability","t":2,"d":"Above a threshold, the integral of an affine function under the standard-normal law equals the integral of that function against its density.","s":"∀ (s m t : ℝ), ∫ (z : ℝ) in Set.Ioi t, s * z + m ∂ProbabilityTheory.gaussianReal 0 1 = ∫ (z : ℝ) in Set.Ioi t, ProbabilityTheory.gaussianPDFReal 0 1 z * (s * z + m)","l":128,"f":"Causalean/Mathlib/Probability/GaussianMoments.lean"},{"n":"Causalean.Mathlib.integral_Ioi_affine_stdNormal","k":"theorem","m":"Causalean.Mathlib.Probability.GaussianMoments","a":"Mathlib/Probability","t":2,"d":"Above a threshold, integrating an affine function against the standard-normal density equals its slope times the truncated first moment plus its intercept times the tail mass.","s":"∀ (s m t : ℝ), ∫ (z : ℝ) in Set.Ioi t, ProbabilityTheory.gaussianPDFReal 0 1 z * (s * z + m) = (s * ∫ (z : ℝ) in Set.Ioi t, z * Causalean.Mathlib.stdNormalPDF z) + m * ∫ (z : ℝ) in Set.Ioi t, Causalean.Mathlib.stdNormalPDF z","l":99,"f":"Causalean/Mathlib/Probability/GaussianMoments.lean"},{"n":"Causalean.Mathlib.integral_Ioi_id_gaussianReal","k":"theorem","m":"Causalean.Mathlib.Probability.GaussianMoments","a":"Mathlib/Probability","t":1,"d":"Gaussian truncated first moment. For a normal law with mean m and a nonnegative variance parameter v that is nonzero, the first moment integrated over the tail above a threshold c equals m·(1 − Φ(t)) + √v·φ(t), where t = (c − m)/√v — the affine image of the standard-normal truncated moment ∫_{t}^∞ z φ(z) dz = φ(t).","s":"∀ (m : ℝ) (v : NNReal), v ≠ 0 → ∀ (c : ℝ), ∫ (y : ℝ) in Set.Ioi c, y ∂ProbabilityTheory.gaussianReal m v = m * (1 - Causalean.Mathlib.stdNormalCDF ((c - m) / √↑v)) + √↑v * Causalean.Mathlib.stdNormalPDF ((c - m) / √↑v)","l":173,"f":"Causalean/Mathlib/Probability/GaussianMoments.lean"},{"n":"Causalean.Mathlib.integral_Ioi_id_mul_stdNormalPDF","k":"theorem","m":"Causalean.Mathlib.Probability.StdNormalMoments","a":"Mathlib/Probability","t":1,"d":"The standard-normal truncated first moment. For any real cutoff c, the tail integral of t times the standard normal density over (c, ∞) equals the density value at c.","s":"∀ (c : ℝ), ∫ (t : ℝ) in Set.Ioi c, t * Causalean.Mathlib.stdNormalPDF t = Causalean.Mathlib.stdNormalPDF c","l":53,"f":"Causalean/Mathlib/Probability/StdNormalMoments.lean"},{"n":"Causalean.Mathlib.integral_Ioi_stdNormalPDF","k":"theorem","m":"Causalean.Mathlib.Probability.StdNormalMoments","a":"Mathlib/Probability","t":2,"d":"The standard-normal survival integral: ∫_{c}^∞ φ(t) dt = 1 − Φ(c).","s":"∀ (c : ℝ), ∫ (t : ℝ) in Set.Ioi c, Causalean.Mathlib.stdNormalPDF t = 1 - Causalean.Mathlib.stdNormalCDF c","l":27,"f":"Causalean/Mathlib/Probability/StdNormalMoments.lean"},{"n":"Causalean.Mathlib.integral_id_map_equiv","k":"theorem","m":"Causalean.Mathlib.StandardGaussian","a":"Mathlib","t":2,"d":"Transporting a measure through a continuous linear equivalence transports its vector integral through the same equivalence.","s":"∀ {F : Type u_2} {G : Type u_3} [inst : NormedAddCommGroup F] [inst_1 : NormedSpace ℝ F] [inst_2 : MeasurableSpace F] [BorelSpace F] [inst_4 : NormedAddCommGroup G] [inst_5 : NormedSpace ℝ G] [inst_6 : MeasurableSpace G] [BorelSpace G] [SecondCountableTopology G] {μ : MeasureTheory.Measure F} (L : F ≃L[ℝ] G), ∫ (x : G), x ∂MeasureTheory.Measure.map (⇑L) μ = L (∫ (x : F), x ∂μ)","l":135,"f":"Causalean/Mathlib/StandardGaussian.lean"},{"n":"Causalean.Mathlib.isGaussian_euclideanStdGaussian","k":"instance","m":"Causalean.Mathlib.StandardGaussian","a":"Mathlib","t":2,"d":"The product standard Gaussian on EuclideanSpace ℝ (Fin n) is Gaussian.","s":"∀ {E : Type u_1} [inst : NormedAddCommGroup E] [inst_1 : InnerProductSpace ℝ E], ProbabilityTheory.IsGaussian (Causalean.Mathlib.euclideanStdGaussian✝ E)","l":125,"f":"Causalean/Mathlib/StandardGaussian.lean"},{"n":"Causalean.Mathlib.isGaussian_piGaussian","k":"instance","m":"Causalean.Mathlib.StandardGaussian","a":"Mathlib","t":2,"d":"The product standard Gaussian on Fin n → ℝ is Gaussian.","s":"∀ {E : Type u_1} [inst : NormedAddCommGroup E] [inst_1 : InnerProductSpace ℝ E], ProbabilityTheory.IsGaussian (Causalean.Mathlib.piGaussian E)","l":92,"f":"Causalean/Mathlib/StandardGaussian.lean"},{"n":"Causalean.Mathlib.isGaussian_stdGaussian","k":"instance","m":"Causalean.Mathlib.StandardGaussian","a":"Mathlib","t":2,"d":"The standard Gaussian measure on the inner-product space is a Gaussian law.","s":"∀ {E : Type u_1} [inst : NormedAddCommGroup E] [inst_1 : InnerProductSpace ℝ E] [inst_2 : FiniteDimensional ℝ E] [inst_3 : MeasurableSpace E] [BorelSpace E], ProbabilityTheory.IsGaussian (Causalean.Mathlib.stdGaussian E)","l":130,"f":"Causalean/Mathlib/StandardGaussian.lean"},{"n":"Causalean.Mathlib.isProbabilityMeasure_euclideanStdGaussian","k":"instance","m":"Causalean.Mathlib.StandardGaussian","a":"Mathlib","t":2,"d":"The transported product standard Gaussian on Euclidean space is a probability measure.","s":"∀ {E : Type u_1} [inst : NormedAddCommGroup E] [inst_1 : InnerProductSpace ℝ E], MeasureTheory.IsProbabilityMeasure (Causalean.Mathlib.euclideanStdGaussian✝ E)","l":114,"f":"Causalean/Mathlib/StandardGaussian.lean"},{"n":"Causalean.Mathlib.isProbabilityMeasure_piGaussian","k":"instance","m":"Causalean.Mathlib.StandardGaussian","a":"Mathlib","t":2,"d":"The product of independent one-dimensional standard Gaussian laws is a probability measure.","s":"∀ {E : Type u_1} [inst : NormedAddCommGroup E] [inst_1 : InnerProductSpace ℝ E], MeasureTheory.IsProbabilityMeasure (Causalean.Mathlib.piGaussian E)","l":80,"f":"Causalean/Mathlib/StandardGaussian.lean"},{"n":"Causalean.Mathlib.isProbabilityMeasure_stdGaussian","k":"instance","m":"Causalean.Mathlib.StandardGaussian","a":"Mathlib","t":2,"d":"The standard Gaussian measure on the inner-product space is a probability measure.","s":"∀ {E : Type u_1} [inst : NormedAddCommGroup E] [inst_1 : InnerProductSpace ℝ E] [inst_2 : FiniteDimensional ℝ E] [inst_3 : MeasurableSpace E] [BorelSpace E], MeasureTheory.IsProbabilityMeasure (Causalean.Mathlib.stdGaussian E)","l":120,"f":"Causalean/Mathlib/StandardGaussian.lean"},{"n":"Causalean.Mathlib.le_condCDF_condQuantile","k":"theorem","m":"Causalean.Mathlib.Probability.MeasurableCondQuantile","a":"Mathlib/Probability","t":2,"d":"At a target level below one, the conditional generalized inverse reaches at least that level in the conditional cumulative distribution function.","s":"∀ {α : Type u_1} [inst : MeasurableSpace α] {ρ : MeasureTheory.Measure (α × ℝ)} {τ : α → ℝ} {a : α}, τ a < 1 → τ a ≤ ↑(ProbabilityTheory.condCDF ρ a) (Causalean.Mathlib.condQuantile ρ τ a)","l":62,"f":"Causalean/Mathlib/Probability/MeasurableCondQuantile.lean"},{"n":"Causalean.Mathlib.le_condCDF_of_condQuantile_le","k":"theorem","m":"Causalean.Mathlib.Probability.MeasurableCondQuantile","a":"Mathlib/Probability","t":2,"d":"A conditional quantile below a value makes the conditional CDF reach any level below one.","s":"∀ {α : Type u_1} [inst : MeasurableSpace α] {ρ : MeasureTheory.Measure (α × ℝ)} {τ : α → ℝ} {a : α} {x : ℝ}, τ a < 1 → Causalean.Mathlib.condQuantile ρ τ a ≤ x → τ a ≤ ↑(ProbabilityTheory.condCDF ρ a) x","l":79,"f":"Causalean/Mathlib/Probability/MeasurableCondQuantile.lean"},{"n":"Causalean.Mathlib.lintegral_ofReal_inv_sqrt_smul_sum_sq_le","k":"theorem","m":"Causalean.Mathlib.IIDCenteredSum","a":"Mathlib","t":2,"d":"A finite sum whose squared integral is at most its number of terms times a nonnegative benchmark remains bounded by that benchmark after division by the square root of the number of terms.","s":"∀ {Ω : Type u_1} [inst : MeasurableSpace Ω] {μ : MeasureTheory.Measure Ω} {ι : Type u_2} (s : Finset ι), 0 < s.card → ∀ (Y : ι → Ω → ℝ) (B : Ω → ENNReal), ∫⁻ (ω : Ω), ENNReal.ofReal ((∑ i ∈ s, Y i ω) ^ 2) ∂μ ≤ ↑s.card * ∫⁻ (ω : Ω), B ω ∂μ → ∫⁻ (ω : Ω), ENNReal.ofReal (((√↑s.card)⁻¹ * ∑ i ∈ s, Y i ω) ^ 2) ∂μ ≤ ∫⁻ (ω : Ω), B ω ∂μ","l":65,"f":"Causalean/Mathlib/IIDCenteredSum.lean"},{"n":"Causalean.Mathlib.measurable_condQuantile_and_attains","k":"theorem","m":"Causalean.Mathlib.Probability.MeasurableCondQuantile","a":"Mathlib/Probability","t":1,"d":"Measurable conditional quantile (selection). For a measure ρ on the product of a parameter space and the reals, and a measurable target level function τ that is everywhere strictly positive and everywhere strictly below one, if the conditional cumulative distribution function of ρ is continuous at the selected conditional quantile, for every parameter value, then the conditional quantile map condQuantile ρ τ is measurable and attains the target level — the conditional CDF at the selected quantile equals τ a for every a.","s":"∀ {α : Type u_1} [inst : MeasurableSpace α] (ρ : MeasureTheory.Measure (α × ℝ)) (τ : α → ℝ), Measurable τ → (∀ (a : α), 0 < τ a) → (∀ (a : α), τ a < 1) → (∀ (a : α), ContinuousAt (↑(ProbabilityTheory.condCDF ρ a)) (Causalean.Mathlib.condQuantile ρ τ a)) → Measurable (Causalean.Mathlib.condQuantile ρ τ) ∧ ∀ (a : α), ↑(ProbabilityTheory.condCDF ρ a) (Causalean.Mathlib.condQuantile ρ τ a) = τ a","l":101,"f":"Causalean/Mathlib/Probability/MeasurableCondQuantile.lean"},{"n":"Causalean.Mathlib.measurable_eLpNorm_toReal_of_uncurry","k":"theorem","m":"Causalean.Mathlib.ELpNormMeasurable","a":"Mathlib","t":2,"d":"Alias for measurable_eLpNorm_two_toReal_of_uncurry whose name reflects that the exponent may be any finite nonzero value.","s":"∀ {Ω : Type u_1} {X : Type u_2} [inst : MeasurableSpace X] [inst_1 : MeasurableSpace Ω] {P : MeasureTheory.Measure X} [MeasureTheory.SFinite P] {E : Type u_3} [inst_3 : MeasurableSpace E] [inst_4 : TopologicalSpace E] [inst_5 : ContinuousENorm E] [OpensMeasurableSpace E] {g : Ω → X → E} {p : ENNReal}, p ≠ 0 → p ≠ ⊤ → Measurable (Function.uncurry g) → Measurable fun ω => (MeasureTheory.eLpNorm (g ω) p P).toReal","l":77,"f":"Causalean/Mathlib/ELpNormMeasurable.lean"},{"n":"Causalean.Mathlib.measurable_eLpNorm_toReal_of_uncurry_of_factor","k":"theorem","m":"Causalean.Mathlib.ELpNormMeasurable","a":"Mathlib","t":2,"d":"Alias for measurable_eLpNorm_two_toReal_of_uncurry_of_factor whose name reflects that the exponent may be any finite nonzero value.","s":"∀ {Ω : Type u_1} {X : Type u_2} [inst : MeasurableSpace X] {mΩ : MeasurableSpace Ω} {P : MeasureTheory.Measure X} [MeasureTheory.SFinite P] {E : Type u_3} [inst_2 : MeasurableSpace E] [inst_3 : TopologicalSpace E] [inst_4 : ContinuousENorm E] [OpensMeasurableSpace E] {g : Ω → X → E} {p : ENNReal}, p ≠ 0 → p ≠ ⊤ → Measurable (Function.uncurry g) → Measurable fun ω => (MeasureTheory.eLpNorm (g ω) p P).toReal","l":117,"f":"Causalean/Mathlib/ELpNormMeasurable.lean"},{"n":"Causalean.Mathlib.measurable_eLpNorm_two_toReal_of_uncurry","k":"theorem","m":"Causalean.Mathlib.ELpNormMeasurable","a":"Mathlib","t":1,"d":"Lp-norm measurable from joint measurability (top σ-algebra). For a σ-finite base measure P and an exponent p that is neither zero nor infinite, if the map (ω, x) ↦ g ω x is jointly measurable on the product of Ω and X, then the real-valued Lp norm ω ↦ ‖g ω‖_{Lp(P)} is measurable as a function of ω.","s":"∀ {Ω : Type u_1} {X : Type u_2} [inst : MeasurableSpace X] [inst_1 : MeasurableSpace Ω] {P : MeasureTheory.Measure X} [MeasureTheory.SFinite P] {E : Type u_3} [inst_3 : MeasurableSpace E] [inst_4 : TopologicalSpace E] [inst_5 : ContinuousENorm E] [OpensMeasurableSpace E] {g : Ω → X → E} {p : ENNReal}, p ≠ 0 → p ≠ ⊤ → Measurable (Function.uncurry g) → Measurable fun ω => (MeasureTheory.eLpNorm (g ω) p P).toReal","l":50,"f":"Causalean/Mathlib/ELpNormMeasurable.lean"},{"n":"Causalean.Mathlib.measurable_eLpNorm_two_toReal_of_uncurry_of_factor","k":"theorem","m":"Causalean.Mathlib.ELpNormMeasurable","a":"Mathlib","t":1,"d":"Lp-norm measurable with respect to a sub-σ-algebra. For a σ-finite base measure P and an exponent p that is neither zero nor infinite, if the map (ω, x) ↦ g ω x is jointly measurable with respect to the product of a sub-σ-algebra mΩ on Ω and the σ-algebra on X, then the real-valued Lp norm ω ↦ ‖g ω‖_{Lp(P)} is measurable with respect to mΩ.","s":"∀ {Ω : Type u_1} {X : Type u_2} [inst : MeasurableSpace X] {mΩ : MeasurableSpace Ω} {P : MeasureTheory.Measure X} [MeasureTheory.SFinite P] {E : Type u_3} [inst_2 : MeasurableSpace E] [inst_3 : TopologicalSpace E] [inst_4 : ContinuousENorm E] [OpensMeasurableSpace E] {g : Ω → X → E} {p : ENNReal}, p ≠ 0 → p ≠ ⊤ → Measurable (Function.uncurry g) → Measurable fun ω => (MeasureTheory.eLpNorm (g ω) p P).toReal","l":88,"f":"Causalean/Mathlib/ELpNormMeasurable.lean"},{"n":"Causalean.Mathlib.neg_log_coord_convexOn","k":"theorem","m":"Causalean.Mathlib.Analysis.Convex.ReciprocalProduct","a":"Mathlib/Analysis/Convex","t":2,"d":"-log (p k) is convex on a convex set s whose k-th coordinate is positive throughout.","s":"∀ {ι : Type u_1} {s : Set (ι → ℝ)}, Convex ℝ s → ∀ (k : ι), (∀ p ∈ s, 0 < p k) → ConvexOn ℝ s fun p => -Real.log (p k)","l":30,"f":"Causalean/Mathlib/Analysis/Convex/ReciprocalProduct.lean"},{"n":"Causalean.Mathlib.neg_log_one_sub_coord_convexOn","k":"theorem","m":"Causalean.Mathlib.Analysis.Convex.ReciprocalProduct","a":"Mathlib/Analysis/Convex","t":2,"d":"-log (1 - p k) is convex on a convex set s where the k-th coordinate stays below 1.","s":"∀ {ι : Type u_1} {s : Set (ι → ℝ)}, Convex ℝ s → ∀ (k : ι), (∀ p ∈ s, p k < 1) → ConvexOn ℝ s fun p => -Real.log (1 - p k)","l":103,"f":"Causalean/Mathlib/Analysis/Convex/ReciprocalProduct.lean"},{"n":"Causalean.Mathlib.neg_log_one_sub_sum_convexOn","k":"theorem","m":"Causalean.Mathlib.Analysis.Convex.ReciprocalProduct","a":"Mathlib/Analysis/Convex","t":2,"d":"∑ k ∈ S, -log (1 - p k) is convex on a convex set s where the coordinates indexed by S stay below 1.","s":"∀ {ι : Type u_1} {s : Set (ι → ℝ)}, Convex ℝ s → ∀ (S : Finset ι), (∀ p ∈ s, ∀ k ∈ S, p k < 1) → ConvexOn ℝ s fun p => ∑ k ∈ S, -Real.log (1 - p k)","l":123,"f":"Causalean/Mathlib/Analysis/Convex/ReciprocalProduct.lean"},{"n":"Causalean.Mathlib.neg_log_sum_convexOn","k":"theorem","m":"Causalean.Mathlib.Analysis.Convex.ReciprocalProduct","a":"Mathlib/Analysis/Convex","t":2,"d":"∑ k ∈ S, -log (p k) is convex on a convex set s whose coordinates indexed by S are positive throughout.","s":"∀ {ι : Type u_1} {s : Set (ι → ℝ)}, Convex ℝ s → ∀ (S : Finset ι), (∀ p ∈ s, ∀ k ∈ S, 0 < p k) → ConvexOn ℝ s fun p => ∑ k ∈ S, -Real.log (p k)","l":45,"f":"Causalean/Mathlib/Analysis/Convex/ReciprocalProduct.lean"},{"n":"Causalean.Mathlib.nonempty_condQuantileSet","k":"theorem","m":"Causalean.Mathlib.Probability.MeasurableCondQuantile","a":"Mathlib/Probability","t":2,"d":"At a conditional-quantile level strictly below one, some response value has a conditional cumulative distribution function that reaches that level.","s":"∀ {α : Type u_1} [inst : MeasurableSpace α] {ρ : MeasureTheory.Measure (α × ℝ)} {τ : α → ℝ} {a : α}, τ a < 1 → {x | τ a ≤ ↑(ProbabilityTheory.condCDF ρ a) x}.Nonempty","l":54,"f":"Causalean/Mathlib/Probability/MeasurableCondQuantile.lean"},{"n":"Causalean.Mathlib.piGaussian","k":"def","m":"Causalean.Mathlib.StandardGaussian","a":"Mathlib","t":1,"d":"The product measure of independent standard normal laws, with one real-valued coordinate for each dimension of a finite-dimensional real inner-product space.","s":"(E : Type u_1) → [inst : NormedAddCommGroup E] → [inst_1 : InnerProductSpace ℝ E] → MeasureTheory.Measure (Fin (Module.finrank ℝ E) → ℝ)","l":59,"f":"Causalean/Mathlib/StandardGaussian.lean"},{"n":"Causalean.Mathlib.pi_centered_sum_sq_lintegral_le","k":"theorem","m":"Causalean.Mathlib.IIDCenteredSum","a":"Mathlib","t":2,"d":"Finitely many independent coordinates, each with its own square-integrable real-valued law, have a centered sum whose second moment is bounded by the sum of their individual squared L² norms.","s":"∀ {ι : Type u_1} [inst : Fintype ι] {X : ι → Type u_2} [inst_1 : (i : ι) → MeasurableSpace (X i)] {P : (i : ι) → MeasureTheory.Measure (X i)} [∀ (i : ι), MeasureTheory.IsProbabilityMeasure (P i)] {f : (i : ι) → X i → ℝ}, (∀ (i : ι), MeasureTheory.MemLp (f i) 2 (P i)) → ∫⁻ (v : (i : ι) → X i), ENNReal.ofReal ((∑ i, (f i (v i) - ∫ (x : X i), f i x ∂P i)) ^ 2) ∂MeasureTheory.Measure.pi P ≤ ∑ i, ENNReal.ofReal ((MeasureTheory.eLpNorm (f i) 2 (P i)).toReal ^ 2)","l":179,"f":"Causalean/Mathlib/IIDCenteredSum.lean"},{"n":"Causalean.Mathlib.probit","k":"def","m":"Causalean.Mathlib.Probability.StdNormalCDF","a":"Mathlib/Probability","t":1,"d":"The probit Φ⁻¹(p): the standard-normal quantile, the generalized inverse of Φ.","s":"ℝ → ℝ","l":150,"f":"Causalean/Mathlib/Probability/StdNormalCDF.lean"},{"n":"Causalean.Mathlib.probit_stdNormalCDF","k":"theorem","m":"Causalean.Mathlib.Probability.StdNormalCDF","a":"Mathlib/Probability","t":1,"d":"For any real score x, applying the probit transform to Φ(x) recovers x: the probit function Φ⁻¹ is a left inverse of the standard normal CDF Φ.","s":"∀ (x : ℝ), Causalean.Mathlib.probit (Causalean.Mathlib.stdNormalCDF x) = x","l":179,"f":"Causalean/Mathlib/Probability/StdNormalCDF.lean"},{"n":"Causalean.Mathlib.prod_inv_convexOn","k":"theorem","m":"Causalean.Mathlib.Analysis.Convex.ReciprocalProduct","a":"Mathlib/Analysis/Convex","t":1,"d":"Convexity of a reciprocal coordinate product. On a convex subset of the coordinate space, if every coordinate indexed by a fixed finite index set stays strictly positive throughout the set, then the map sending a point to the reciprocal of the product of its coordinates over that index set is convex on the set.","s":"∀ {ι : Type u_1} {s : Set (ι → ℝ)}, Convex ℝ s → ∀ (S : Finset ι), (∀ p ∈ s, ∀ k ∈ S, 0 < p k) → ConvexOn ℝ s fun p => (∏ k ∈ S, p k)⁻¹","l":68,"f":"Causalean/Mathlib/Analysis/Convex/ReciprocalProduct.lean"},{"n":"Causalean.Mathlib.prod_one_sub_inv_convexOn","k":"theorem","m":"Causalean.Mathlib.Analysis.Convex.ReciprocalProduct","a":"Mathlib/Analysis/Convex","t":2,"d":"Convexity of a reciprocal product of complements. On a convex set s whose coordinates indexed by S stay below 1, p ↦ (∏ k ∈ S, (1 - p k))⁻¹ is convex.","s":"∀ {ι : Type u_1} {s : Set (ι → ℝ)}, Convex ℝ s → ∀ (S : Finset ι), (∀ p ∈ s, ∀ k ∈ S, p k < 1) → ConvexOn ℝ s fun p => (∏ k ∈ S, (1 - p k))⁻¹","l":146,"f":"Causalean/Mathlib/Analysis/Convex/ReciprocalProduct.lean"},{"n":"Causalean.Mathlib.recipC","k":"def","m":"Causalean.Mathlib.Analysis.SmoothReciprocal","a":"Mathlib/Analysis","t":1,"d":"Floored reciprocal. The reciprocal x⁻¹ multiplied by a smooth cutoff that is 1 once x is a definite distance above 0 and 0 near and below 0. It coincides with x⁻¹ on x ≥ ε/2 (see recipC_eq_inv) yet is globally C¹ (see recipC_contDiff).","s":"ℝ → ℝ → ℝ","l":20,"f":"Causalean/Mathlib/Analysis/SmoothReciprocal.lean"},{"n":"Causalean.Mathlib.recipC_contDiff","k":"theorem","m":"Causalean.Mathlib.Analysis.SmoothReciprocal","a":"Mathlib/Analysis","t":1,"d":"Global continuous differentiability of the floored reciprocal. For a strictly positive threshold ε, the floored reciprocal recipC ε is continuously differentiable on all of the reals.","s":"∀ (ε : ℝ), 0 < ε → ContDiff ℝ 1 (Causalean.Mathlib.recipC ε)","l":75,"f":"Causalean/Mathlib/Analysis/SmoothReciprocal.lean"},{"n":"Causalean.Mathlib.recipC_contDiff_of","k":"theorem","m":"Causalean.Mathlib.Analysis.SmoothReciprocal","a":"Mathlib/Analysis","t":2,"d":"The floored reciprocal is globally Cⁿ on all of ℝ, for EVERY smoothness order n: near the singularity the smooth cutoff vanishes to infinite order, absorbing the blow-up of x⁻¹, and away from the singularity it is a product of Cⁿ functions.","s":"∀ (n : ℕ∞) (ε : ℝ), 0 < ε → ContDiff ℝ (↑n) (Causalean.Mathlib.recipC ε)","l":38,"f":"Causalean/Mathlib/Analysis/SmoothReciprocal.lean"},{"n":"Causalean.Mathlib.recipC_contDiff_two","k":"theorem","m":"Causalean.Mathlib.Analysis.SmoothReciprocal","a":"Mathlib/Analysis","t":2,"d":"recipC ε is globally C². Specialization of recipC_contDiff_of; this is the order the reciprocal-product envelope's directional curvature modulus needs.","s":"∀ (ε : ℝ), 0 < ε → ContDiff ℝ 2 (Causalean.Mathlib.recipC ε)","l":84,"f":"Causalean/Mathlib/Analysis/SmoothReciprocal.lean"},{"n":"Causalean.Mathlib.recipC_eq_inv","k":"theorem","m":"Causalean.Mathlib.Analysis.SmoothReciprocal","a":"Mathlib/Analysis","t":2,"d":"On the region x ≥ ε/2 (with ε > 0) the floored reciprocal is exactly x⁻¹, because the smooth cutoff has already saturated to 1 there.","s":"∀ {ε x : ℝ}, 0 < ε → ε / 2 ≤ x → Causalean.Mathlib.recipC ε x = x⁻¹","l":26,"f":"Causalean/Mathlib/Analysis/SmoothReciprocal.lean"},{"n":"Causalean.Mathlib.stdGaussian","k":"def","m":"Causalean.Mathlib.StandardGaussian","a":"Mathlib","t":1,"d":"The standard Gaussian measure on a finite-dimensional real inner-product space E: covariance equal to the identity (inner product), mean zero.","s":"(E : Type u_1) → [inst : NormedAddCommGroup E] → [inst_1 : InnerProductSpace ℝ E] → [FiniteDimensional ℝ E] → [inst : MeasurableSpace E] → MeasureTheory.Measure E","l":70,"f":"Causalean/Mathlib/StandardGaussian.lean"},{"n":"Causalean.Mathlib.stdGaussian_map_normSq_eq_pi","k":"theorem","m":"Causalean.Mathlib.StandardGaussian","a":"Mathlib","t":1,"d":"Product-of-1-D-Gaussians model for the squared norm. The law of the squared norm under the standard Gaussian measure on E equals the law of the sum of squared coordinates under a product of finrank ℝ E independent standard real Gaussians.","s":"∀ {E : Type u_1} [inst : NormedAddCommGroup E] [inst_1 : InnerProductSpace ℝ E] [inst_2 : FiniteDimensional ℝ E] [inst_3 : MeasurableSpace E] [BorelSpace E], MeasureTheory.Measure.map (fun x => ‖x‖ ^ 2) (Causalean.Mathlib.stdGaussian E) = MeasureTheory.Measure.map (fun w => ∑ i, w i ^ 2) (MeasureTheory.Measure.pi fun x => ProbabilityTheory.gaussianReal 0 1)","l":254,"f":"Causalean/Mathlib/StandardGaussian.lean"},{"n":"Causalean.Mathlib.stdGaussian_mean","k":"theorem","m":"Causalean.Mathlib.StandardGaussian","a":"Mathlib","t":2,"d":"The standard Gaussian measure on a finite-dimensional real inner-product space has mean zero.","s":"∀ {E : Type u_1} [inst : NormedAddCommGroup E] [inst_1 : InnerProductSpace ℝ E] [inst_2 : FiniteDimensional ℝ E] [inst_3 : MeasurableSpace E] [BorelSpace E], ∫ (x : E), x ∂Causalean.Mathlib.stdGaussian E = 0","l":176,"f":"Causalean/Mathlib/StandardGaussian.lean"},{"n":"Causalean.Mathlib.stdNormalCDF","k":"def","m":"Causalean.Mathlib.Probability.StdNormalCDF","a":"Mathlib/Probability","t":1,"d":"The standard-normal CDF Φ(x) = P(N(0,1) ≤ x), packaged from Mathlib's cdf.","s":"ℝ → ℝ","l":39,"f":"Causalean/Mathlib/Probability/StdNormalCDF.lean"},{"n":"Causalean.Mathlib.stdNormalCDF_continuous","k":"theorem","m":"Causalean.Mathlib.Probability.StdNormalCDF","a":"Mathlib/Probability","t":1,"d":"The standard normal CDF Φ is continuous: the standard normal has no atoms.","s":"Continuous Causalean.Mathlib.stdNormalCDF","l":107,"f":"Causalean/Mathlib/Probability/StdNormalCDF.lean"},{"n":"Causalean.Mathlib.stdNormalCDF_def","k":"theorem","m":"Causalean.Mathlib.Probability.StdNormalCDF","a":"Mathlib/Probability","t":2,"d":"The named standard-normal CDF is Mathlib's CDF for the standard real Gaussian law.","s":"∀ (x : ℝ), Causalean.Mathlib.stdNormalCDF x = ↑(ProbabilityTheory.cdf (ProbabilityTheory.gaussianReal 0 1)) x","l":46,"f":"Causalean/Mathlib/Probability/StdNormalCDF.lean"},{"n":"Causalean.Mathlib.stdNormalCDF_le_one","k":"theorem","m":"Causalean.Mathlib.Probability.StdNormalCDF","a":"Mathlib/Probability","t":2,"d":"The standard-normal CDF is at most one at every real point.","s":"∀ (x : ℝ), Causalean.Mathlib.stdNormalCDF x ≤ 1","l":57,"f":"Causalean/Mathlib/Probability/StdNormalCDF.lean"},{"n":"Causalean.Mathlib.stdNormalCDF_lt_one","k":"theorem","m":"Causalean.Mathlib.Probability.StdNormalCDF","a":"Mathlib/Probability","t":2,"d":"Φ(x) < 1 for every real x.","s":"∀ (x : ℝ), Causalean.Mathlib.stdNormalCDF x < 1","l":144,"f":"Causalean/Mathlib/Probability/StdNormalCDF.lean"},{"n":"Causalean.Mathlib.stdNormalCDF_monotone","k":"theorem","m":"Causalean.Mathlib.Probability.StdNormalCDF","a":"Mathlib/Probability","t":2,"d":"Φ is monotone (inherited from the StieltjesFunction structure of cdf).","s":"Monotone Causalean.Mathlib.stdNormalCDF","l":49,"f":"Causalean/Mathlib/Probability/StdNormalCDF.lean"},{"n":"Causalean.Mathlib.stdNormalCDF_neg","k":"theorem","m":"Causalean.Mathlib.Probability.StdNormalCDF","a":"Mathlib/Probability","t":2,"d":"Symmetry of the standard normal CDF: Φ(−t) = 1 − Φ(t), from the reflection symmetry of the Gaussian law and its atomlessness.","s":"∀ (t : ℝ), Causalean.Mathlib.stdNormalCDF (-t) = 1 - Causalean.Mathlib.stdNormalCDF t","l":75,"f":"Causalean/Mathlib/Probability/StdNormalCDF.lean"},{"n":"Causalean.Mathlib.stdNormalCDF_nonneg","k":"theorem","m":"Causalean.Mathlib.Probability.StdNormalCDF","a":"Mathlib/Probability","t":2,"d":"Φ(x) ∈ [0,1].","s":"∀ (x : ℝ), 0 ≤ Causalean.Mathlib.stdNormalCDF x","l":54,"f":"Causalean/Mathlib/Probability/StdNormalCDF.lean"},{"n":"Causalean.Mathlib.stdNormalCDF_pos","k":"theorem","m":"Causalean.Mathlib.Probability.StdNormalCDF","a":"Mathlib/Probability","t":2,"d":"0 < Φ(x) for every real x (full support of the Gaussian).","s":"∀ (x : ℝ), 0 < Causalean.Mathlib.stdNormalCDF x","l":138,"f":"Causalean/Mathlib/Probability/StdNormalCDF.lean"},{"n":"Causalean.Mathlib.stdNormalCDF_probit","k":"theorem","m":"Causalean.Mathlib.Probability.StdNormalCDF","a":"Mathlib/Probability","t":2,"d":"Φ(Φ⁻¹(p)) = p for p ∈ (0,1) (exact inversion, using continuity + strict monotonicity).","s":"∀ {p : ℝ}, 0 < p → p < 1 → Causalean.Mathlib.stdNormalCDF (Causalean.Mathlib.probit p) = p","l":153,"f":"Causalean/Mathlib/Probability/StdNormalCDF.lean"},{"n":"Causalean.Mathlib.stdNormalCDF_strictMono","k":"theorem","m":"Causalean.Mathlib.Probability.StdNormalCDF","a":"Mathlib/Probability","t":2,"d":"Φ is strictly monotone: the standard normal has full support.","s":"StrictMono Causalean.Mathlib.stdNormalCDF","l":113,"f":"Causalean/Mathlib/Probability/StdNormalCDF.lean"},{"n":"Causalean.Mathlib.stdNormalCDF_tendsto_atBot","k":"theorem","m":"Causalean.Mathlib.Probability.StdNormalCDF","a":"Mathlib/Probability","t":2,"d":"Φ → 0 at -∞.","s":"Filter.Tendsto Causalean.Mathlib.stdNormalCDF Filter.atBot (nhds 0)","l":84,"f":"Causalean/Mathlib/Probability/StdNormalCDF.lean"},{"n":"Causalean.Mathlib.stdNormalCDF_tendsto_atTop","k":"theorem","m":"Causalean.Mathlib.Probability.StdNormalCDF","a":"Mathlib/Probability","t":2,"d":"Φ → 1 at +∞.","s":"Filter.Tendsto Causalean.Mathlib.stdNormalCDF Filter.atTop (nhds 1)","l":87,"f":"Causalean/Mathlib/Probability/StdNormalCDF.lean"},{"n":"Causalean.Mathlib.stdNormalMeasure_Ioi_toReal_eq_integral","k":"theorem","m":"Causalean.Mathlib.Probability.GaussianMoments","a":"Mathlib/Probability","t":2,"d":"The standard normal probability of values above t equals the integral of the standard-normal density over that upper-tail region.","s":"∀ (t : ℝ), ((ProbabilityTheory.gaussianReal 0 1) (Set.Ioi t)).toReal = ∫ (x : ℝ) in Set.Ioi t, Causalean.Mathlib.stdNormalPDF x","l":69,"f":"Causalean/Mathlib/Probability/GaussianMoments.lean"},{"n":"Causalean.Mathlib.stdNormalPDF","k":"def","m":"Causalean.Mathlib.Probability.StdNormalCDF","a":"Mathlib/Probability","t":1,"d":"The standard-normal density φ(x) = exp(-x²/2)/√(2π) (= gaussianPDFReal 0 1).","s":"ℝ → ℝ","l":36,"f":"Causalean/Mathlib/Probability/StdNormalCDF.lean"},{"n":"Causalean.Mathlib.stdNormalPDF_def","k":"theorem","m":"Causalean.Mathlib.Probability.StdNormalCDF","a":"Mathlib/Probability","t":2,"d":"The named standard-normal density is Mathlib's real Gaussian density with mean zero and variance one.","s":"∀ (x : ℝ), Causalean.Mathlib.stdNormalPDF x = ProbabilityTheory.gaussianPDFReal 0 1 x","l":42,"f":"Causalean/Mathlib/Probability/StdNormalCDF.lean"},{"n":"Causalean.MonotonicityKind","k":"inductive","m":"Causalean.SCM.Model.EdgeType","a":"SCM/Model","t":1,"d":"A monotonicity kind records whether an edge is nondecreasing, nonincreasing, strictly increasing, or strictly decreasing.","s":"Type","l":30,"f":"Causalean/SCM/Model/EdgeType.lean"},{"n":"Causalean.PO.ID.Exact.MultipleInstrumentIV.Adj","k":"def","m":"Causalean.PO.ID.Exact.MultipleInstrumentIV.FiniteIndex","a":"PO/ID/Exact/MultipleInstrumentIV","t":1,"d":"Adjacent threshold for the ordered support. The value j : Adj K represents the source-paper threshold between support positions j-1 and j in zero-based Lean indexing, i.e. source notation 2, ..., K.","s":"ℕ → Type","l":44,"f":"Causalean/PO/ID/Exact/MultipleInstrumentIV/FiniteIndex.lean"},{"n":"Causalean.PO.ID.Exact.MultipleInstrumentIV.FiniteIndex","k":"structure","m":"Causalean.PO.ID.Exact.MultipleInstrumentIV.FiniteIndex","a":"PO/ID/Exact/MultipleInstrumentIV","t":1,"d":"A finite ordered first-stage index for the saturated multiple-instrument characterization: instrument support masses that are nonnegative and sum to one, together with a first-stage index that is weakly increasing in the displayed support order.","s":"ℕ → Type","l":61,"f":"Causalean/PO/ID/Exact/MultipleInstrumentIV/FiniteIndex.lean"},{"n":"Causalean.PO.ID.Exact.MultipleInstrumentIV.FiniteIndex.centeredIndex","k":"def","m":"Causalean.PO.ID.Exact.MultipleInstrumentIV.FiniteIndex","a":"PO/ID/Exact/MultipleInstrumentIV","t":1,"d":"Centered first-stage index a_k = dhat_k - dbar.","s":"{K : ℕ} → Causalean.PO.ID.Exact.MultipleInstrumentIV.FiniteIndex K → Fin K → ℝ","l":137,"f":"Causalean/PO/ID/Exact/MultipleInstrumentIV/FiniteIndex.lean"},{"n":"Causalean.PO.ID.Exact.MultipleInstrumentIV.FiniteIndex.centered_weight_sum_zero","k":"theorem","m":"Causalean.PO.ID.Exact.MultipleInstrumentIV.FiniteIndex","a":"PO/ID/Exact/MultipleInstrumentIV","t":2,"d":"Centering identity Σ_k ρ_k (dhat_k - dbar) = 0.","s":"∀ {K : ℕ} (I : Causalean.PO.ID.Exact.MultipleInstrumentIV.FiniteIndex K), ∑ k, I.rho k * I.centeredIndex k = 0","l":149,"f":"Causalean/PO/ID/Exact/MultipleInstrumentIV/FiniteIndex.lean"},{"n":"Causalean.PO.ID.Exact.MultipleInstrumentIV.FiniteIndex.fromMeasureScore","k":"def","m":"Causalean.PO.ID.Exact.MultipleInstrumentIV.FiniteIndex","a":"PO/ID/Exact/MultipleInstrumentIV","t":1,"d":"Construct the ordered finite first-stage index from a probability-space instrument Z : Ω → Fin K and a finite support score dhat. The support masses are the actual measure masses P(Z = k); the only remaining first-stage input is the displayed-order monotonicity of the score.","s":"{Ω : Type u_1} → [inst : MeasurableSpace Ω] → {K : ℕ} → (μ : MeasureTheory.Measure Ω) → [MeasureTheory.IsProbabilityMeasure μ] → (Z : Ω → Fin K) → Measurable Z → (dhat : Fin K → ℝ) → (∀ {k l : Fin K}, ↑k ≤ ↑l → dhat k ≤ dhat l) → Causalean.PO.ID.Exact.MultipleInstrumentIV.FiniteIndex K","l":113,"f":"Causalean/PO/ID/Exact/MultipleInstrumentIV/FiniteIndex.lean"},{"n":"Causalean.PO.ID.Exact.MultipleInstrumentIV.FiniteIndex.meanIndex","k":"def","m":"Causalean.PO.ID.Exact.MultipleInstrumentIV.FiniteIndex","a":"PO/ID/Exact/MultipleInstrumentIV","t":1,"d":"Mean first-stage index dbar = Σ_k ρ_k dhat_k.","s":"{K : ℕ} → Causalean.PO.ID.Exact.MultipleInstrumentIV.FiniteIndex K → ℝ","l":133,"f":"Causalean/PO/ID/Exact/MultipleInstrumentIV/FiniteIndex.lean"},{"n":"Causalean.PO.ID.Exact.MultipleInstrumentIV.FiniteIndex.tailCoeff","k":"def","m":"Causalean.PO.ID.Exact.MultipleInstrumentIV.FiniteIndex","a":"PO/ID/Exact/MultipleInstrumentIV","t":1,"d":"MTW tail coefficient B_j = Σ_{k ≥ j} ρ_k a_k.","s":"{K : ℕ} → Causalean.PO.ID.Exact.MultipleInstrumentIV.FiniteIndex K → Causalean.PO.ID.Exact.MultipleInstrumentIV.Adj K → ℝ","l":145,"f":"Causalean/PO/ID/Exact/MultipleInstrumentIV/FiniteIndex.lean"},{"n":"Causalean.PO.ID.Exact.MultipleInstrumentIV.FiniteIndex.tailCoeff_eq_cov","k":"theorem","m":"Causalean.PO.ID.Exact.MultipleInstrumentIV.FiniteIndex","a":"PO/ID/Exact/MultipleInstrumentIV","t":1,"d":"Covariance identity. For an adjacent threshold j, the MTW tail coefficient B_j equals the finite-support covariance between the first-stage index dhat(Z) and the upper-tail indicator 1_{Z ∈ T_j} under the instrument distribution ρ.","s":"∀ {K : ℕ} (I : Causalean.PO.ID.Exact.MultipleInstrumentIV.FiniteIndex K) (j : Causalean.PO.ID.Exact.MultipleInstrumentIV.Adj K), I.tailCoeff j = ∑ k, I.rho k * (I.dhat k - I.meanIndex) * ((if k ∈ Causalean.PO.ID.Exact.MultipleInstrumentIV.FiniteIndex.upperTail j then 1 else 0) - ∑ l, I.rho l * if l ∈ Causalean.PO.ID.Exact.MultipleInstrumentIV.FiniteIndex.upperTail j then 1 else 0)","l":269,"f":"Causalean/PO/ID/Exact/MultipleInstrumentIV/FiniteIndex.lean"},{"n":"Causalean.PO.ID.Exact.MultipleInstrumentIV.FiniteIndex.tailCoeff_nonneg","k":"theorem","m":"Causalean.PO.ID.Exact.MultipleInstrumentIV.FiniteIndex","a":"PO/ID/Exact/MultipleInstrumentIV","t":1,"d":"For an adjacent threshold j with the support ordered by the first-stage index, the MTW tail coefficient B_j is nonnegative (def:po-estimand-mtw-tail-coefficients).","s":"∀ {K : ℕ} (I : Causalean.PO.ID.Exact.MultipleInstrumentIV.FiniteIndex K) (j : Causalean.PO.ID.Exact.MultipleInstrumentIV.Adj K), 0 ≤ I.tailCoeff j","l":166,"f":"Causalean/PO/ID/Exact/MultipleInstrumentIV/FiniteIndex.lean"},{"n":"Causalean.PO.ID.Exact.MultipleInstrumentIV.FiniteIndex.tail_sum_interchange","k":"theorem","m":"Causalean.PO.ID.Exact.MultipleInstrumentIV.FiniteIndex","a":"PO/ID/Exact/MultipleInstrumentIV","t":2,"d":"Finite upper-tail interchange identity used by the signed decomposition. It is the algebraic form of moving from Σ_k ρ_k a_k Σ_{j≤k} x_j to Σ_j B_j x_j.","s":"∀ {K : ℕ} (I : Causalean.PO.ID.Exact.MultipleInstrumentIV.FiniteIndex K) (x : Causalean.PO.ID.Exact.MultipleInstrumentIV.Adj K → ℝ), (∑ k, I.rho k * I.centeredIndex k * ∑ j, if ↑↑j ≤ ↑k then x j else 0) = ∑ j, I.tailCoeff j * x j","l":235,"f":"Causalean/PO/ID/Exact/MultipleInstrumentIV/FiniteIndex.lean"},{"n":"Causalean.PO.ID.Exact.MultipleInstrumentIV.FiniteIndex.upperTail","k":"def","m":"Causalean.PO.ID.Exact.MultipleInstrumentIV.FiniteIndex","a":"PO/ID/Exact/MultipleInstrumentIV","t":1,"d":"Ordered upper tail T_j = {j, ..., K} for an adjacent threshold.","s":"{K : ℕ} → Causalean.PO.ID.Exact.MultipleInstrumentIV.Adj K → Finset (Fin K)","l":141,"f":"Causalean/PO/ID/Exact/MultipleInstrumentIV/FiniteIndex.lean"},{"n":"Causalean.PO.ID.Exact.MultipleInstrumentIV.MatrixFirstStage","k":"structure","m":"Causalean.PO.ID.Exact.MultipleInstrumentIV.FiniteIndex","a":"PO/ID/Exact/MultipleInstrumentIV","t":1,"d":"Finite-support matrix first stage for the source population-2SLS definition. The score vector q, support masses rho, reduced-form first-stage moments firstStageMoment, and invertible Gram matrix determine the projection coefficient by the displayed matrix inverse. The resulting fitted values are required to be in the displayed weakly increasing order so they can feed the MTW tail-coefficient algebra.","s":"ℕ → ℕ → Type","l":359,"f":"Causalean/PO/ID/Exact/MultipleInstrumentIV/FiniteIndex.lean"},{"n":"Causalean.PO.ID.Exact.MultipleInstrumentIV.MatrixFirstStage.fittedValue","k":"def","m":"Causalean.PO.ID.Exact.MultipleInstrumentIV.FiniteIndex","a":"PO/ID/Exact/MultipleInstrumentIV","t":1,"d":"Fitted saturated first-stage value dhat_k = q(zᵏ)'Π.","s":"{K L : ℕ} → Causalean.PO.ID.Exact.MultipleInstrumentIV.MatrixFirstStage K L → Fin K → ℝ","l":412,"f":"Causalean/PO/ID/Exact/MultipleInstrumentIV/FiniteIndex.lean"},{"n":"Causalean.PO.ID.Exact.MultipleInstrumentIV.MatrixFirstStage.gram","k":"def","m":"Causalean.PO.ID.Exact.MultipleInstrumentIV.FiniteIndex","a":"PO/ID/Exact/MultipleInstrumentIV","t":1,"d":"Finite second-moment matrix E[q(Z)q(Z)ᵀ].","s":"{K L : ℕ} → Causalean.PO.ID.Exact.MultipleInstrumentIV.MatrixFirstStage K L → Matrix (Fin L) (Fin L) ℝ","l":397,"f":"Causalean/PO/ID/Exact/MultipleInstrumentIV/FiniteIndex.lean"},{"n":"Causalean.PO.ID.Exact.MultipleInstrumentIV.MatrixFirstStage.instInvertibleGram","k":"instance","m":"Causalean.PO.ID.Exact.MultipleInstrumentIV.FiniteIndex","a":"PO/ID/Exact/MultipleInstrumentIV","t":2,"d":"The finite second-moment matrix is invertible by the matrix first-stage assumption.","s":"{K L : ℕ} → (S : Causalean.PO.ID.Exact.MultipleInstrumentIV.MatrixFirstStage K L) → Invertible S.gram","l":401,"f":"Causalean/PO/ID/Exact/MultipleInstrumentIV/FiniteIndex.lean"},{"n":"Causalean.PO.ID.Exact.MultipleInstrumentIV.MatrixFirstStage.projectionCoeff","k":"def","m":"Causalean.PO.ID.Exact.MultipleInstrumentIV.FiniteIndex","a":"PO/ID/Exact/MultipleInstrumentIV","t":1,"d":"Population first-stage projection coefficient (E[q(Z)q(Z)ᵀ])⁻¹ E[q(Z)D].","s":"{K L : ℕ} → Causalean.PO.ID.Exact.MultipleInstrumentIV.MatrixFirstStage K L → Fin L → ℝ","l":407,"f":"Causalean/PO/ID/Exact/MultipleInstrumentIV/FiniteIndex.lean"},{"n":"Causalean.PO.ID.Exact.MultipleInstrumentIV.MatrixFirstStage.toFiniteIndex","k":"def","m":"Causalean.PO.ID.Exact.MultipleInstrumentIV.FiniteIndex","a":"PO/ID/Exact/MultipleInstrumentIV","t":1,"d":"The matrix first stage induces the ordered finite index consumed by the MTW tail-coefficient and response-type algebra.","s":"{K L : ℕ} → Causalean.PO.ID.Exact.MultipleInstrumentIV.MatrixFirstStage K L → Causalean.PO.ID.Exact.MultipleInstrumentIV.FiniteIndex K","l":416,"f":"Causalean/PO/ID/Exact/MultipleInstrumentIV/FiniteIndex.lean"},{"n":"Causalean.PO.ID.Exact.MultipleInstrumentIV.POMultipleIVSystem","k":"structure","m":"Causalean.PO.ID.Exact.MultipleInstrumentIV.POBridge","a":"PO/ID/Exact/MultipleInstrumentIV","t":1,"d":"Multiple-instrument IV potential-outcome subsystem. Inside a potential-outcome system, this bundles a node serving as the instrument, taking values in a finite support of size K, a node serving as the binary treatment, and a node serving as the real-valued outcome, subject to the instrument, treatment, and outcome being pairwise distinct nodes. This is the Fin K generalization of PO.POIVSystem.","s":"Causalean.PO.POSystem → ℕ → Type (max u_1 u_2)","l":59,"f":"Causalean/PO/ID/Exact/MultipleInstrumentIV/POBridge.lean"},{"n":"Causalean.PO.ID.Exact.MultipleInstrumentIV.POMultipleIVSystem.Assumptions","k":"structure","m":"Causalean.PO.ID.Exact.MultipleInstrumentIV.POBridge","a":"PO/ID/Exact/MultipleInstrumentIV","t":1,"d":"Classical multi-instrument IV assumption bundle (ass:po-estimand-mtw-iv-validity). For a multiple-instrument potential-outcome subsystem, this packages the observed treatment and outcome equaling the potential treatment and outcome realized under the actual instrument value (SUTVA), and at each instrument support point, the instrument being independent of the counterfactual triple consisting of the potential treatment under that instrument value together with the two treatment-arm potential outcomes (IV exogeneity).","s":"{P : Causalean.PO.POSystem} → {K : ℕ} → Causalean.PO.ID.Exact.MultipleInstrumentIV.POMultipleIVSystem P K → Prop","l":185,"f":"Causalean/PO/ID/Exact/MultipleInstrumentIV/POBridge.lean"},{"n":"Causalean.PO.ID.Exact.MultipleInstrumentIV.POMultipleIVSystem.DofZ","k":"def","m":"Causalean.PO.ID.Exact.MultipleInstrumentIV.POBridge","a":"PO/ID/Exact/MultipleInstrumentIV","t":1,"d":"Potential treatment D(zᵏ) under fixing the instrument to support point k.","s":"{P : Causalean.PO.POSystem} → {K : ℕ} → Causalean.PO.ID.Exact.MultipleInstrumentIV.POMultipleIVSystem P K → Fin K → P.Ω → Bool","l":98,"f":"Causalean/PO/ID/Exact/MultipleInstrumentIV/POBridge.lean"},{"n":"Causalean.PO.ID.Exact.MultipleInstrumentIV.POMultipleIVSystem.DofZ_eq_factualD_on_zEvent","k":"theorem","m":"Causalean.PO.ID.Exact.MultipleInstrumentIV.POBridge","a":"PO/ID/Exact/MultipleInstrumentIV","t":2,"d":"On {Z = zᵏ}, the counterfactual treatment D(zᵏ) equals the factual D.","s":"∀ {P : Causalean.PO.POSystem} {K : ℕ} (S : Causalean.PO.ID.Exact.MultipleInstrumentIV.POMultipleIVSystem P K), S.Assumptions → ∀ (k : Fin K) {ω : P.Ω}, ω ∈ S.zEvent k → S.DofZ k ω = S.factualD ω","l":210,"f":"Causalean/PO/ID/Exact/MultipleInstrumentIV/POBridge.lean"},{"n":"Causalean.PO.ID.Exact.MultipleInstrumentIV.POMultipleIVSystem.DofZ_eq_on_gEvent","k":"theorem","m":"Causalean.PO.ID.Exact.MultipleInstrumentIV.POBridge","a":"PO/ID/Exact/MultipleInstrumentIV","t":2,"d":"On {G = g}, the potential treatment D(zᵏ) is the constant g k.","s":"∀ {P : Causalean.PO.POSystem} {K : ℕ} (S : Causalean.PO.ID.Exact.MultipleInstrumentIV.POMultipleIVSystem P K) (g : Causalean.PO.ID.Exact.MultipleInstrumentIV.ResponseType K) (k : Fin K) {ω : P.Ω}, ω ∈ S.gEvent g → S.DofZ k ω = g k","l":221,"f":"Causalean/PO/ID/Exact/MultipleInstrumentIV/POBridge.lean"},{"n":"Causalean.PO.ID.Exact.MultipleInstrumentIV.POMultipleIVSystem.YofD","k":"def","m":"Causalean.PO.ID.Exact.MultipleInstrumentIV.POBridge","a":"PO/ID/Exact/MultipleInstrumentIV","t":1,"d":"Potential outcome Y(d) under fixing the treatment to d.","s":"{P : Causalean.PO.POSystem} → {K : ℕ} → Causalean.PO.ID.Exact.MultipleInstrumentIV.POMultipleIVSystem P K → Bool → P.Ω → ℝ","l":102,"f":"Causalean/PO/ID/Exact/MultipleInstrumentIV/POBridge.lean"},{"n":"Causalean.PO.ID.Exact.MultipleInstrumentIV.POMultipleIVSystem.YofDofZ","k":"def","m":"Causalean.PO.ID.Exact.MultipleInstrumentIV.POBridge","a":"PO/ID/Exact/MultipleInstrumentIV","t":1,"d":"Potential outcome under the treatment that instrument value zᵏ induces, Y(D(zᵏ)) = 1_{D(zᵏ)=1} Y(1) + 1_{D(zᵏ)=0} Y(0).","s":"{P : Causalean.PO.POSystem} → {K : ℕ} → Causalean.PO.ID.Exact.MultipleInstrumentIV.POMultipleIVSystem P K → Fin K → P.Ω → ℝ","l":124,"f":"Causalean/PO/ID/Exact/MultipleInstrumentIV/POBridge.lean"},{"n":"Causalean.PO.ID.Exact.MultipleInstrumentIV.POMultipleIVSystem.baseOutcome","k":"def","m":"Causalean.PO.ID.Exact.MultipleInstrumentIV.POBridge","a":"PO/ID/Exact/MultipleInstrumentIV","t":1,"d":"Baseline response-type outcome E[Y(D(z⁰)) | G = g], the term the centered first-stage index cancels.","s":"{P : Causalean.PO.POSystem} → {K : ℕ} → Causalean.PO.ID.Exact.MultipleInstrumentIV.POMultipleIVSystem P K → 0 < K → Causalean.PO.ID.Exact.MultipleInstrumentIV.ResponseType K → ℝ","l":372,"f":"Causalean/PO/ID/Exact/MultipleInstrumentIV/POBridge.lean"},{"n":"Causalean.PO.ID.Exact.MultipleInstrumentIV.POMultipleIVSystem.baseTreatment","k":"def","m":"Causalean.PO.ID.Exact.MultipleInstrumentIV.POBridge","a":"PO/ID/Exact/MultipleInstrumentIV","t":1,"d":"Baseline treatment mean E[D(z⁰)].","s":"{P : Causalean.PO.POSystem} → {K : ℕ} → Causalean.PO.ID.Exact.MultipleInstrumentIV.POMultipleIVSystem P K → 0 < K → ℝ","l":377,"f":"Causalean/PO/ID/Exact/MultipleInstrumentIV/POBridge.lean"},{"n":"Causalean.PO.ID.Exact.MultipleInstrumentIV.POMultipleIVSystem.cfCell","k":"def","m":"Causalean.PO.ID.Exact.MultipleInstrumentIV.POBridge","a":"PO/ID/Exact/MultipleInstrumentIV","t":1,"d":"The three-element counterfactual cell (D(zᵏ), Y(1), Y(0)) relevant to identifying the conditional means on {Z = zᵏ}. Coordinates: 0 ↦ D(zᵏ), 1 ↦ Y(1), 2 ↦ Y(0).","s":"{P : Causalean.PO.POSystem} → {K : ℕ} → Causalean.PO.ID.Exact.MultipleInstrumentIV.POMultipleIVSystem P K → Fin K → Causalean.PO.POCFBundle P","l":137,"f":"Causalean/PO/ID/Exact/MultipleInstrumentIV/POBridge.lean"},{"n":"Causalean.PO.ID.Exact.MultipleInstrumentIV.POMultipleIVSystem.dUnderZ","k":"def","m":"Causalean.PO.ID.Exact.MultipleInstrumentIV.POBridge","a":"PO/ID/Exact/MultipleInstrumentIV","t":1,"d":"Regimed variable D(zᵏ) (treatment under instrument set to zᵏ).","s":"{P : Causalean.PO.POSystem} → {K : ℕ} → Causalean.PO.ID.Exact.MultipleInstrumentIV.POMultipleIVSystem P K → Fin K → Causalean.PO.RegimedVar P Bool","l":129,"f":"Causalean/PO/ID/Exact/MultipleInstrumentIV/POBridge.lean"},{"n":"Causalean.PO.ID.Exact.MultipleInstrumentIV.POMultipleIVSystem.dVar","k":"def","m":"Causalean.PO.ID.Exact.MultipleInstrumentIV.POBridge","a":"PO/ID/Exact/MultipleInstrumentIV","t":1,"d":"Treatment packaged as a POVar valued in Bool.","s":"{P : Causalean.PO.POSystem} → {K : ℕ} → Causalean.PO.ID.Exact.MultipleInstrumentIV.POMultipleIVSystem P K → Causalean.PO.POVar P Bool","l":92,"f":"Causalean/PO/ID/Exact/MultipleInstrumentIV/POBridge.lean"},{"n":"Causalean.PO.ID.Exact.MultipleInstrumentIV.POMultipleIVSystem.effect","k":"def","m":"Causalean.PO.ID.Exact.MultipleInstrumentIV.POBridge","a":"PO/ID/Exact/MultipleInstrumentIV","t":1,"d":"Within-type conditional causal effect Δ_g = E[Y(1) - Y(0) | G = g].","s":"{P : Causalean.PO.POSystem} → {K : ℕ} → Causalean.PO.ID.Exact.MultipleInstrumentIV.POMultipleIVSystem P K → Causalean.PO.ID.Exact.MultipleInstrumentIV.ResponseType K → ℝ","l":365,"f":"Causalean/PO/ID/Exact/MultipleInstrumentIV/POBridge.lean"},{"n":"Causalean.PO.ID.Exact.MultipleInstrumentIV.POMultipleIVSystem.factualD","k":"def","m":"Causalean.PO.ID.Exact.MultipleInstrumentIV.POBridge","a":"PO/ID/Exact/MultipleInstrumentIV","t":1,"d":"Factual treatment D.","s":"{P : Causalean.PO.POSystem} → {K : ℕ} → Causalean.PO.ID.Exact.MultipleInstrumentIV.POMultipleIVSystem P K → P.Ω → Bool","l":108,"f":"Causalean/PO/ID/Exact/MultipleInstrumentIV/POBridge.lean"},{"n":"Causalean.PO.ID.Exact.MultipleInstrumentIV.POMultipleIVSystem.factualY","k":"def","m":"Causalean.PO.ID.Exact.MultipleInstrumentIV.POBridge","a":"PO/ID/Exact/MultipleInstrumentIV","t":1,"d":"Factual outcome Y.","s":"{P : Causalean.PO.POSystem} → {K : ℕ} → Causalean.PO.ID.Exact.MultipleInstrumentIV.POMultipleIVSystem P K → P.Ω → ℝ","l":111,"f":"Causalean/PO/ID/Exact/MultipleInstrumentIV/POBridge.lean"},{"n":"Causalean.PO.ID.Exact.MultipleInstrumentIV.POMultipleIVSystem.factualY_eq_YofD_factualD","k":"theorem","m":"Causalean.PO.ID.Exact.MultipleInstrumentIV.POBridge","a":"PO/ID/Exact/MultipleInstrumentIV","t":2,"d":"Factual Y equals Y(factualD) pointwise.","s":"∀ {P : Causalean.PO.POSystem} {K : ℕ} (S : Causalean.PO.ID.Exact.MultipleInstrumentIV.POMultipleIVSystem P K), S.Assumptions → ∀ (ω : P.Ω), S.factualY ω = S.YofD (S.factualD ω) ω","l":216,"f":"Causalean/PO/ID/Exact/MultipleInstrumentIV/POBridge.lean"},{"n":"Causalean.PO.ID.Exact.MultipleInstrumentIV.POMultipleIVSystem.factualZ","k":"def","m":"Causalean.PO.ID.Exact.MultipleInstrumentIV.POBridge","a":"PO/ID/Exact/MultipleInstrumentIV","t":1,"d":"Factual instrument Z.","s":"{P : Causalean.PO.POSystem} → {K : ℕ} → Causalean.PO.ID.Exact.MultipleInstrumentIV.POMultipleIVSystem P K → P.Ω → Fin K","l":105,"f":"Causalean/PO/ID/Exact/MultipleInstrumentIV/POBridge.lean"},{"n":"Causalean.PO.ID.Exact.MultipleInstrumentIV.POMultipleIVSystem.gEvent","k":"def","m":"Causalean.PO.ID.Exact.MultipleInstrumentIV.POBridge","a":"PO/ID/Exact/MultipleInstrumentIV","t":1,"d":"The event {G = g} that a unit has response type g.","s":"{P : Causalean.PO.POSystem} → {K : ℕ} → Causalean.PO.ID.Exact.MultipleInstrumentIV.POMultipleIVSystem P K → Causalean.PO.ID.Exact.MultipleInstrumentIV.ResponseType K → Set P.Ω","l":118,"f":"Causalean/PO/ID/Exact/MultipleInstrumentIV/POBridge.lean"},{"n":"Causalean.PO.ID.Exact.MultipleInstrumentIV.POMultipleIVSystem.gEvent_iUnion","k":"theorem","m":"Causalean.PO.ID.Exact.MultipleInstrumentIV.POBridge","a":"PO/ID/Exact/MultipleInstrumentIV","t":2,"d":"The response-type events cover the sample space.","s":"∀ {P : Causalean.PO.POSystem} {K : ℕ} (S : Causalean.PO.ID.Exact.MultipleInstrumentIV.POMultipleIVSystem P K), ⋃ g, S.gEvent g = Set.univ","l":349,"f":"Causalean/PO/ID/Exact/MultipleInstrumentIV/POBridge.lean"},{"n":"Causalean.PO.ID.Exact.MultipleInstrumentIV.POMultipleIVSystem.gEvent_pairwise_disjoint","k":"theorem","m":"Causalean.PO.ID.Exact.MultipleInstrumentIV.POBridge","a":"PO/ID/Exact/MultipleInstrumentIV","t":2,"d":"The response-type events are pairwise disjoint.","s":"∀ {P : Causalean.PO.POSystem} {K : ℕ} (S : Causalean.PO.ID.Exact.MultipleInstrumentIV.POMultipleIVSystem P K), Pairwise (Function.onFun Disjoint S.gEvent)","l":341,"f":"Causalean/PO/ID/Exact/MultipleInstrumentIV/POBridge.lean"},{"n":"Causalean.PO.ID.Exact.MultipleInstrumentIV.POMultipleIVSystem.integral_partition","k":"theorem","m":"Causalean.PO.ID.Exact.MultipleInstrumentIV.POBridge","a":"PO/ID/Exact/MultipleInstrumentIV","t":2,"d":"Total law over response types for an integrable integrand.","s":"∀ {P : Causalean.PO.POSystem} {K : ℕ} (S : Causalean.PO.ID.Exact.MultipleInstrumentIV.POMultipleIVSystem P K) [MeasureTheory.IsFiniteMeasure P.μ] {f : P.Ω → ℝ}, MeasureTheory.Integrable f P.μ → ∫ (ω : P.Ω), f ω ∂P.μ = ∑ g, (P.μ (S.gEvent g)).toReal * Causalean.PO.eventCondExp P.μ (S.gEvent g) f","l":353,"f":"Causalean/PO/ID/Exact/MultipleInstrumentIV/POBridge.lean"},{"n":"Causalean.PO.ID.Exact.MultipleInstrumentIV.POMultipleIVSystem.mass","k":"def","m":"Causalean.PO.ID.Exact.MultipleInstrumentIV.POBridge","a":"PO/ID/Exact/MultipleInstrumentIV","t":1,"d":"Response-type mass π_g = P(G = g).","s":"{P : Causalean.PO.POSystem} → {K : ℕ} → Causalean.PO.ID.Exact.MultipleInstrumentIV.POMultipleIVSystem P K → Causalean.PO.ID.Exact.MultipleInstrumentIV.ResponseType K → ℝ","l":362,"f":"Causalean/PO/ID/Exact/MultipleInstrumentIV/POBridge.lean"},{"n":"Causalean.PO.ID.Exact.MultipleInstrumentIV.POMultipleIVSystem.mass_nonneg","k":"theorem","m":"Causalean.PO.ID.Exact.MultipleInstrumentIV.POBridge","a":"PO/ID/Exact/MultipleInstrumentIV","t":2,"d":"Response-type masses are nonnegative because they are real-valued probabilities.","s":"∀ {P : Causalean.PO.POSystem} {K : ℕ} (S : Causalean.PO.ID.Exact.MultipleInstrumentIV.POMultipleIVSystem P K) (g : Causalean.PO.ID.Exact.MultipleInstrumentIV.ResponseType K), 0 ≤ S.mass g","l":381,"f":"Causalean/PO/ID/Exact/MultipleInstrumentIV/POBridge.lean"},{"n":"Causalean.PO.ID.Exact.MultipleInstrumentIV.POMultipleIVSystem.mass_sum_one","k":"theorem","m":"Causalean.PO.ID.Exact.MultipleInstrumentIV.POBridge","a":"PO/ID/Exact/MultipleInstrumentIV","t":2,"d":"Response-type masses sum to one.","s":"∀ {P : Causalean.PO.POSystem} {K : ℕ} (S : Causalean.PO.ID.Exact.MultipleInstrumentIV.POMultipleIVSystem P K), ∑ g, S.mass g = 1","l":384,"f":"Causalean/PO/ID/Exact/MultipleInstrumentIV/POBridge.lean"},{"n":"Causalean.PO.ID.Exact.MultipleInstrumentIV.POMultipleIVSystem.measurableSet_gEvent","k":"theorem","m":"Causalean.PO.ID.Exact.MultipleInstrumentIV.POBridge","a":"PO/ID/Exact/MultipleInstrumentIV","t":2,"d":"Each response-type cell is measurable.","s":"∀ {P : Causalean.PO.POSystem} {K : ℕ} (S : Causalean.PO.ID.Exact.MultipleInstrumentIV.POMultipleIVSystem P K) (g : Causalean.PO.ID.Exact.MultipleInstrumentIV.ResponseType K), MeasurableSet (S.gEvent g)","l":169,"f":"Causalean/PO/ID/Exact/MultipleInstrumentIV/POBridge.lean"},{"n":"Causalean.PO.ID.Exact.MultipleInstrumentIV.POMultipleIVSystem.measurableSet_zEvent","k":"theorem","m":"Causalean.PO.ID.Exact.MultipleInstrumentIV.POBridge","a":"PO/ID/Exact/MultipleInstrumentIV","t":2,"d":"Each instrument support cell is measurable.","s":"∀ {P : Causalean.PO.POSystem} {K : ℕ} (S : Causalean.PO.ID.Exact.MultipleInstrumentIV.POMultipleIVSystem P K) (k : Fin K), MeasurableSet (S.zEvent k)","l":173,"f":"Causalean/PO/ID/Exact/MultipleInstrumentIV/POBridge.lean"},{"n":"Causalean.PO.ID.Exact.MultipleInstrumentIV.POMultipleIVSystem.measurable_DofZ","k":"theorem","m":"Causalean.PO.ID.Exact.MultipleInstrumentIV.POBridge","a":"PO/ID/Exact/MultipleInstrumentIV","t":2,"d":"The potential treatment under any instrument support point is measurable.","s":"∀ {P : Causalean.PO.POSystem} {K : ℕ} (S : Causalean.PO.ID.Exact.MultipleInstrumentIV.POMultipleIVSystem P K) (k : Fin K), Measurable (S.DofZ k)","l":148,"f":"Causalean/PO/ID/Exact/MultipleInstrumentIV/POBridge.lean"},{"n":"Causalean.PO.ID.Exact.MultipleInstrumentIV.POMultipleIVSystem.measurable_YofD","k":"theorem","m":"Causalean.PO.ID.Exact.MultipleInstrumentIV.POBridge","a":"PO/ID/Exact/MultipleInstrumentIV","t":2,"d":"The potential outcome under either treatment arm is measurable.","s":"∀ {P : Causalean.PO.POSystem} {K : ℕ} (S : Causalean.PO.ID.Exact.MultipleInstrumentIV.POMultipleIVSystem P K) (d : Bool), Measurable (S.YofD d)","l":152,"f":"Causalean/PO/ID/Exact/MultipleInstrumentIV/POBridge.lean"},{"n":"Causalean.PO.ID.Exact.MultipleInstrumentIV.POMultipleIVSystem.measurable_YofDofZ","k":"theorem","m":"Causalean.PO.ID.Exact.MultipleInstrumentIV.POBridge","a":"PO/ID/Exact/MultipleInstrumentIV","t":2,"d":"The outcome under the treatment induced by an instrument support point is measurable.","s":"∀ {P : Causalean.PO.POSystem} {K : ℕ} (S : Causalean.PO.ID.Exact.MultipleInstrumentIV.POMultipleIVSystem P K) (k : Fin K), Measurable (S.YofDofZ k)","l":177,"f":"Causalean/PO/ID/Exact/MultipleInstrumentIV/POBridge.lean"},{"n":"Causalean.PO.ID.Exact.MultipleInstrumentIV.POMultipleIVSystem.measurable_factualD","k":"theorem","m":"Causalean.PO.ID.Exact.MultipleInstrumentIV.POBridge","a":"PO/ID/Exact/MultipleInstrumentIV","t":2,"d":"The factual treatment value is measurable.","s":"∀ {P : Causalean.PO.POSystem} {K : ℕ} (S : Causalean.PO.ID.Exact.MultipleInstrumentIV.POMultipleIVSystem P K), Measurable S.factualD","l":159,"f":"Causalean/PO/ID/Exact/MultipleInstrumentIV/POBridge.lean"},{"n":"Causalean.PO.ID.Exact.MultipleInstrumentIV.POMultipleIVSystem.measurable_factualY","k":"theorem","m":"Causalean.PO.ID.Exact.MultipleInstrumentIV.POBridge","a":"PO/ID/Exact/MultipleInstrumentIV","t":2,"d":"The factual outcome value is measurable.","s":"∀ {P : Causalean.PO.POSystem} {K : ℕ} (S : Causalean.PO.ID.Exact.MultipleInstrumentIV.POMultipleIVSystem P K), Measurable S.factualY","l":162,"f":"Causalean/PO/ID/Exact/MultipleInstrumentIV/POBridge.lean"},{"n":"Causalean.PO.ID.Exact.MultipleInstrumentIV.POMultipleIVSystem.measurable_factualZ","k":"theorem","m":"Causalean.PO.ID.Exact.MultipleInstrumentIV.POBridge","a":"PO/ID/Exact/MultipleInstrumentIV","t":2,"d":"The factual instrument value is measurable.","s":"∀ {P : Causalean.PO.POSystem} {K : ℕ} (S : Causalean.PO.ID.Exact.MultipleInstrumentIV.POMultipleIVSystem P K), Measurable S.factualZ","l":156,"f":"Causalean/PO/ID/Exact/MultipleInstrumentIV/POBridge.lean"},{"n":"Causalean.PO.ID.Exact.MultipleInstrumentIV.POMultipleIVSystem.measurable_responseType","k":"theorem","m":"Causalean.PO.ID.Exact.MultipleInstrumentIV.POBridge","a":"PO/ID/Exact/MultipleInstrumentIV","t":2,"d":"The response-type map collecting all treatment responses is measurable.","s":"∀ {P : Causalean.PO.POSystem} {K : ℕ} (S : Causalean.PO.ID.Exact.MultipleInstrumentIV.POMultipleIVSystem P K), Measurable S.responseType","l":165,"f":"Causalean/PO/ID/Exact/MultipleInstrumentIV/POBridge.lean"},{"n":"Causalean.PO.ID.Exact.MultipleInstrumentIV.POMultipleIVSystem.observedBeta2SLS_eq_responseTypeWeightedSum","k":"theorem","m":"Causalean.PO.ID.Exact.MultipleInstrumentIV.POBridge","a":"PO/ID/Exact/MultipleInstrumentIV","t":1,"d":"Potential-outcome multiple-IV 2SLS characterization (prop:po-estimand-mtw-response-type-form, PO level). Consider a finite instrument support of positive size K together with weights dhat that are weakly increasing in the support order, under the potential-outcome identifying assumption bundle. If every instrument-support cell has positive probability, the potential outcome under treatment and under control are integrable, the centered-instrument-weighted outcome and treatment are integrable, and the observed first-stage moment is nonzero, then the observed population 2SLS ratio E[h(Z)Y] / E[h(Z)D] equals the response-type weighted sum Σ_g ω_g Δ_g of within-type conditional causal effects Δ_g = E[Y(1) - Y(0) | G = g].","s":"∀ {P : Causalean.PO.POSystem} {K : ℕ} (S : Causalean.PO.ID.Exact.MultipleInstrumentIV.POMultipleIVSystem P K) [MeasureTheory.IsFiniteMeasure P.μ] (hK : 0 < K), S.Assumptions → ∀ (dhat : Fin K → ℝ) (hmono : ∀ {k l : Fin K}, ↑k ≤ ↑l → dhat k ≤ dhat l), (∀ (k : Fin K), P.μ (S.zEvent k) ≠ 0) → MeasureTheory.Integrable (S.YofD true) P.μ → MeasureTheory.Integrable (S.YofD false) P.μ → MeasureTheory.Integrable (fun ω => (S.toFiniteIndex dhat fun {k l} => hmono).centeredIndex (S.factualZ ω) * S.factualY ω) P.μ → MeasureTheory.Integrable (fun ω => (S.toFiniteIndex dhat fun {k l} => hmono).centeredIndex (S.factualZ ω) * Causalean.PO.ID.Exact.MultipleInstrumentIV.boolToReal (S.factualD ω)) P.μ → Causalean.PO.ID.Exact.MultipleInstrumentIV.ResponseTypeStats.PopulationBridge.observedFirstStageMoment P.μ S.factualZ S.factualD (S.toFiniteIndex dhat fun {k l} => hmono) ≠ 0 → Causalean.PO.ID.Exact.MultipleInstrumentIV.ResponseTypeStats.PopulationBridge.observedBeta2SLS P.μ S.factualZ S.factualD S.factualY (S.toFiniteIndex dhat fun {k l} => hmono) = Causalean.PO.ID.Exact.MultipleInstrumentIV.ResponseTypeStats.responseTypeEstimand (S.toFiniteIndex dhat fun {k l} => hmono) (S.toPopulationBridge hK).stats","l":643,"f":"Causalean/PO/ID/Exact/MultipleInstrumentIV/POBridge.lean"},{"n":"Causalean.PO.ID.Exact.MultipleInstrumentIV.POMultipleIVSystem.outcomeDrop","k":"theorem","m":"Causalean.PO.ID.Exact.MultipleInstrumentIV.POBridge","a":"PO/ID/Exact/MultipleInstrumentIV","t":2,"d":"E[Y | Z = zᵏ] = E[Y(D(zᵏ))].","s":"∀ {P : Causalean.PO.POSystem} {K : ℕ} (S : Causalean.PO.ID.Exact.MultipleInstrumentIV.POMultipleIVSystem P K) [MeasureTheory.IsFiniteMeasure P.μ], S.Assumptions → ∀ (k : Fin K), P.μ (S.zEvent k) ≠ 0 → MeasureTheory.Integrable (S.YofD true) P.μ → MeasureTheory.Integrable (S.YofD false) P.μ → Causalean.PO.eventCondExp P.μ (S.zEvent k) S.factualY = ∫ (ω : P.Ω), S.YofDofZ k ω ∂P.μ","l":272,"f":"Causalean/PO/ID/Exact/MultipleInstrumentIV/POBridge.lean"},{"n":"Causalean.PO.ID.Exact.MultipleInstrumentIV.POMultipleIVSystem.outcome_cell_eq","k":"theorem","m":"Causalean.PO.ID.Exact.MultipleInstrumentIV.POBridge","a":"PO/ID/Exact/MultipleInstrumentIV","t":2,"d":"Outcome cell identity: E[Y | Z = zᵏ] = outcomeAtSupport k.","s":"∀ {P : Causalean.PO.POSystem} {K : ℕ} (S : Causalean.PO.ID.Exact.MultipleInstrumentIV.POMultipleIVSystem P K) [MeasureTheory.IsFiniteMeasure P.μ] (hK : 0 < K), S.Assumptions → ∀ (k : Fin K), P.μ (S.zEvent k) ≠ 0 → MeasureTheory.Integrable (S.YofD true) P.μ → MeasureTheory.Integrable (S.YofD false) P.μ → Causalean.PO.eventCondExp P.μ (S.zEvent k) S.factualY = (S.toPopulationBridge hK).outcomeAtSupport k","l":545,"f":"Causalean/PO/ID/Exact/MultipleInstrumentIV/POBridge.lean"},{"n":"Causalean.PO.ID.Exact.MultipleInstrumentIV.POMultipleIVSystem.responseType","k":"def","m":"Causalean.PO.ID.Exact.MultipleInstrumentIV.POBridge","a":"PO/ID/Exact/MultipleInstrumentIV","t":1,"d":"Response type G(ω) = (D(z⁰)(ω), …, D(z^{K-1})(ω)), the vector of potential treatments across the instrument support.","s":"{P : Causalean.PO.POSystem} → {K : ℕ} → Causalean.PO.ID.Exact.MultipleInstrumentIV.POMultipleIVSystem P K → P.Ω → Causalean.PO.ID.Exact.MultipleInstrumentIV.ResponseType K","l":114,"f":"Causalean/PO/ID/Exact/MultipleInstrumentIV/POBridge.lean"},{"n":"Causalean.PO.ID.Exact.MultipleInstrumentIV.POMultipleIVSystem.rho_eq_zMass","k":"theorem","m":"Causalean.PO.ID.Exact.MultipleInstrumentIV.POBridge","a":"PO/ID/Exact/MultipleInstrumentIV","t":2,"d":"The instrument masses ρ_k from toFiniteIndex are the probabilities of the observed instrument cells.","s":"∀ {P : Causalean.PO.POSystem} {K : ℕ} (S : Causalean.PO.ID.Exact.MultipleInstrumentIV.POMultipleIVSystem P K) (dhat : Fin K → ℝ) (hmono : ∀ {k l : Fin K}, ↑k ≤ ↑l → dhat k ≤ dhat l) (k : Fin K), (S.toFiniteIndex dhat fun {k l} => hmono).rho k = (P.μ (Causalean.PO.ID.Exact.MultipleInstrumentIV.ResponseTypeStats.PopulationBridge.zEvent S.factualZ k)).toReal","l":618,"f":"Causalean/PO/ID/Exact/MultipleInstrumentIV/POBridge.lean"},{"n":"Causalean.PO.ID.Exact.MultipleInstrumentIV.POMultipleIVSystem.telescoped_eq","k":"theorem","m":"Causalean.PO.ID.Exact.MultipleInstrumentIV.POBridge","a":"PO/ID/Exact/MultipleInstrumentIV","t":2,"d":"The telescoped adjacent step at support point k collapses to the difference of endpoint treatments: Σ_{j≤k}(d_j - d_{j-1}) = g k - g 0.","s":"∀ {P : Causalean.PO.POSystem} {K : ℕ} (S : Causalean.PO.ID.Exact.MultipleInstrumentIV.POMultipleIVSystem P K) (hK : 0 < K) (g : Causalean.PO.ID.Exact.MultipleInstrumentIV.ResponseType K) (k : Fin K), Causalean.PO.ID.Exact.MultipleInstrumentIV.ResponseTypeStats.PopulationBridge.telescopedTypeStep g k = Causalean.PO.ID.Exact.MultipleInstrumentIV.boolToReal (g k) - Causalean.PO.ID.Exact.MultipleInstrumentIV.boolToReal (g (S.z0 hK))","l":424,"f":"Causalean/PO/ID/Exact/MultipleInstrumentIV/POBridge.lean"},{"n":"Causalean.PO.ID.Exact.MultipleInstrumentIV.POMultipleIVSystem.toFiniteIndex","k":"def","m":"Causalean.PO.ID.Exact.MultipleInstrumentIV.POBridge","a":"PO/ID/Exact/MultipleInstrumentIV","t":1,"d":"Ordered finite index built from the instrument masses and a supplied saturated first-stage score.","s":"{P : Causalean.PO.POSystem} → {K : ℕ} → Causalean.PO.ID.Exact.MultipleInstrumentIV.POMultipleIVSystem P K → (dhat : Fin K → ℝ) → (∀ {k l : Fin K}, ↑k ≤ ↑l → dhat k ≤ dhat l) → Causalean.PO.ID.Exact.MultipleInstrumentIV.FiniteIndex K","l":415,"f":"Causalean/PO/ID/Exact/MultipleInstrumentIV/POBridge.lean"},{"n":"Causalean.PO.ID.Exact.MultipleInstrumentIV.POMultipleIVSystem.toObservedBridge","k":"def","m":"Causalean.PO.ID.Exact.MultipleInstrumentIV.POBridge","a":"PO/ID/Exact/MultipleInstrumentIV","t":1,"d":"The observed bridge assembled from the PO system, with the two conditional-mean fields *derived* (not assumed) from consistency and instrument independence.","s":"{P : Causalean.PO.POSystem} → {K : ℕ} → (S : Causalean.PO.ID.Exact.MultipleInstrumentIV.POMultipleIVSystem P K) → (hK : 0 < K) → S.Assumptions → (dhat : Fin K → ℝ) → (hmono : ∀ {k l : Fin K}, ↑k ≤ ↑l → dhat k ≤ dhat l) → (∀ (k : Fin K), P.μ (S.zEvent k) ≠ 0) → MeasureTheory.Integrable (S.YofD true) P.μ → MeasureTheory.Integrable (S.YofD false) P.μ → Causalean.PO.ID.Exact.MultipleInstrumentIV.ResponseTypeStats.PopulationBridge.ObservedBridge P.μ S.factualZ S.factualD S.factualY (S.toFiniteIndex dhat fun {k l} => hmono) (S.toPopulationBridge hK)","l":626,"f":"Causalean/PO/ID/Exact/MultipleInstrumentIV/POBridge.lean"},{"n":"Causalean.PO.ID.Exact.MultipleInstrumentIV.POMultipleIVSystem.toPopulationBridge","k":"def","m":"Causalean.PO.ID.Exact.MultipleInstrumentIV.POBridge","a":"PO/ID/Exact/MultipleInstrumentIV","t":1,"d":"Saturated finite-support population bridge assembled from the PO system.","s":"{P : Causalean.PO.POSystem} → {K : ℕ} → Causalean.PO.ID.Exact.MultipleInstrumentIV.POMultipleIVSystem P K → 0 < K → Causalean.PO.ID.Exact.MultipleInstrumentIV.ResponseTypeStats.PopulationBridge K","l":410,"f":"Causalean/PO/ID/Exact/MultipleInstrumentIV/POBridge.lean"},{"n":"Causalean.PO.ID.Exact.MultipleInstrumentIV.POMultipleIVSystem.toStats","k":"def","m":"Causalean.PO.ID.Exact.MultipleInstrumentIV.POBridge","a":"PO/ID/Exact/MultipleInstrumentIV","t":1,"d":"Finite response-type statistics assembled from the PO system.","s":"{P : Causalean.PO.POSystem} → {K : ℕ} → Causalean.PO.ID.Exact.MultipleInstrumentIV.POMultipleIVSystem P K → Causalean.PO.ID.Exact.MultipleInstrumentIV.ResponseTypeStats K","l":403,"f":"Causalean/PO/ID/Exact/MultipleInstrumentIV/POBridge.lean"},{"n":"Causalean.PO.ID.Exact.MultipleInstrumentIV.POMultipleIVSystem.treatmentDrop","k":"theorem","m":"Causalean.PO.ID.Exact.MultipleInstrumentIV.POBridge","a":"PO/ID/Exact/MultipleInstrumentIV","t":2,"d":"E[D | Z = zᵏ] = E[D(zᵏ)] (0/1-coded).","s":"∀ {P : Causalean.PO.POSystem} {K : ℕ} (S : Causalean.PO.ID.Exact.MultipleInstrumentIV.POMultipleIVSystem P K) [MeasureTheory.IsFiniteMeasure P.μ], S.Assumptions → ∀ (k : Fin K), P.μ (S.zEvent k) ≠ 0 → (Causalean.PO.eventCondExp P.μ (S.zEvent k) fun ω => Causalean.PO.ID.Exact.MultipleInstrumentIV.boolToReal (S.factualD ω)) = ∫ (ω : P.Ω), Causalean.PO.ID.Exact.MultipleInstrumentIV.boolToReal (S.DofZ k ω) ∂P.μ","l":236,"f":"Causalean/PO/ID/Exact/MultipleInstrumentIV/POBridge.lean"},{"n":"Causalean.PO.ID.Exact.MultipleInstrumentIV.POMultipleIVSystem.treatment_cell_eq","k":"theorem","m":"Causalean.PO.ID.Exact.MultipleInstrumentIV.POBridge","a":"PO/ID/Exact/MultipleInstrumentIV","t":2,"d":"Treatment cell identity: E[D | Z = zᵏ] = baseTreatment + treatmentAtSupport k.","s":"∀ {P : Causalean.PO.POSystem} {K : ℕ} (S : Causalean.PO.ID.Exact.MultipleInstrumentIV.POMultipleIVSystem P K) [MeasureTheory.IsFiniteMeasure P.μ] (hK : 0 < K), S.Assumptions → ∀ (k : Fin K), P.μ (S.zEvent k) ≠ 0 → (Causalean.PO.eventCondExp P.μ (S.zEvent k) fun ω => Causalean.PO.ID.Exact.MultipleInstrumentIV.boolToReal (S.factualD ω)) = S.baseTreatment hK + (S.toPopulationBridge hK).treatmentAtSupport k","l":485,"f":"Causalean/PO/ID/Exact/MultipleInstrumentIV/POBridge.lean"},{"n":"Causalean.PO.ID.Exact.MultipleInstrumentIV.POMultipleIVSystem.yUnderD","k":"def","m":"Causalean.PO.ID.Exact.MultipleInstrumentIV.POBridge","a":"PO/ID/Exact/MultipleInstrumentIV","t":1,"d":"Regimed variable Y(d) (outcome under treatment set to d).","s":"{P : Causalean.PO.POSystem} → {K : ℕ} → Causalean.PO.ID.Exact.MultipleInstrumentIV.POMultipleIVSystem P K → Bool → Causalean.PO.RegimedVar P ℝ","l":133,"f":"Causalean/PO/ID/Exact/MultipleInstrumentIV/POBridge.lean"},{"n":"Causalean.PO.ID.Exact.MultipleInstrumentIV.POMultipleIVSystem.yVar","k":"def","m":"Causalean.PO.ID.Exact.MultipleInstrumentIV.POBridge","a":"PO/ID/Exact/MultipleInstrumentIV","t":1,"d":"Outcome packaged as a POVar valued in ℝ.","s":"{P : Causalean.PO.POSystem} → {K : ℕ} → Causalean.PO.ID.Exact.MultipleInstrumentIV.POMultipleIVSystem P K → Causalean.PO.POVar P ℝ","l":95,"f":"Causalean/PO/ID/Exact/MultipleInstrumentIV/POBridge.lean"},{"n":"Causalean.PO.ID.Exact.MultipleInstrumentIV.POMultipleIVSystem.z0","k":"def","m":"Causalean.PO.ID.Exact.MultipleInstrumentIV.POBridge","a":"PO/ID/Exact/MultipleInstrumentIV","t":1,"d":"Reference support point z⁰ (requires 0 < K).","s":"{P : Causalean.PO.POSystem} → {K : ℕ} → Causalean.PO.ID.Exact.MultipleInstrumentIV.POMultipleIVSystem P K → 0 < K → Fin K","l":369,"f":"Causalean/PO/ID/Exact/MultipleInstrumentIV/POBridge.lean"},{"n":"Causalean.PO.ID.Exact.MultipleInstrumentIV.POMultipleIVSystem.zEvent","k":"def","m":"Causalean.PO.ID.Exact.MultipleInstrumentIV.POBridge","a":"PO/ID/Exact/MultipleInstrumentIV","t":1,"d":"The instrument cell {Z = zᵏ}.","s":"{P : Causalean.PO.POSystem} → {K : ℕ} → Causalean.PO.ID.Exact.MultipleInstrumentIV.POMultipleIVSystem P K → Fin K → Set P.Ω","l":121,"f":"Causalean/PO/ID/Exact/MultipleInstrumentIV/POBridge.lean"},{"n":"Causalean.PO.ID.Exact.MultipleInstrumentIV.POMultipleIVSystem.zVar","k":"def","m":"Causalean.PO.ID.Exact.MultipleInstrumentIV.POBridge","a":"PO/ID/Exact/MultipleInstrumentIV","t":1,"d":"Instrument packaged as a POVar valued in Fin K.","s":"{P : Causalean.PO.POSystem} → {K : ℕ} → Causalean.PO.ID.Exact.MultipleInstrumentIV.POMultipleIVSystem P K → Causalean.PO.POVar P (Fin K)","l":89,"f":"Causalean/PO/ID/Exact/MultipleInstrumentIV/POBridge.lean"},{"n":"Causalean.PO.ID.Exact.MultipleInstrumentIV.ResponseType","k":"def","m":"Causalean.PO.ID.Exact.MultipleInstrumentIV.ResponseTypes","a":"PO/ID/Exact/MultipleInstrumentIV","t":1,"d":"A response type is the binary vector g = (D(z^1), ..., D(z^K)).","s":"ℕ → Type","l":57,"f":"Causalean/PO/ID/Exact/MultipleInstrumentIV/ResponseTypes.lean"},{"n":"Causalean.PO.ID.Exact.MultipleInstrumentIV.ResponseTypeStats","k":"structure","m":"Causalean.PO.ID.Exact.MultipleInstrumentIV.ResponseTypes","a":"PO/ID/Exact/MultipleInstrumentIV","t":1,"d":"Finite response-type statistics. For a finite family of latent response types, this records the probability mass of each type and the type-specific average causal effect Δ_g, subject to every mass being nonnegative and the masses summing to one, so together they form a probability vector over response types.","s":"ℕ → Type","l":69,"f":"Causalean/PO/ID/Exact/MultipleInstrumentIV/ResponseTypes.lean"},{"n":"Causalean.PO.ID.Exact.MultipleInstrumentIV.ResponseTypeStats.ComponentwiseMonotoneRestriction","k":"structure","m":"Causalean.PO.ID.Exact.MultipleInstrumentIV.ResponseTypes","a":"PO/ID/Exact/MultipleInstrumentIV","t":1,"d":"Opaque response-type restriction interface.","s":"{K : ℕ} → Type","l":305,"f":"Causalean/PO/ID/Exact/MultipleInstrumentIV/ResponseTypes.lean"},{"n":"Causalean.PO.ID.Exact.MultipleInstrumentIV.ResponseTypeStats.PopulationBridge","k":"structure","m":"Causalean.PO.ID.Exact.MultipleInstrumentIV.ResponseTypes","a":"PO/ID/Exact/MultipleInstrumentIV","t":1,"d":"Saturated finite-support population bridge for the MTW identification step. Bundles a finite response-type statistics record supplying the type masses and type-specific effects together with a response-type-specific baseline outcome mean, the term the centered first-stage index cancels, leaving the telescoped adjacent treatment increments used by the finite algebra.","s":"ℕ → Type","l":115,"f":"Causalean/PO/ID/Exact/MultipleInstrumentIV/ResponseTypes.lean"},{"n":"Causalean.PO.ID.Exact.MultipleInstrumentIV.ResponseTypeStats.PopulationBridge.ObservedBridge","k":"structure","m":"Causalean.PO.ID.Exact.MultipleInstrumentIV.Population","a":"PO/ID/Exact/MultipleInstrumentIV","t":1,"d":"Bridge from an observed finite-instrument population to the response-type algebra. Given an instrument, a binary treatment, an outcome, a finite index, and a saturated response-type population P, this bundles the underlying law being a probability measure, the finite-index masses reproducing the probabilities of the observed instrument cells, the observed conditional mean of the outcome on each instrument cell equaling the response-type expansion P.outcomeAtSupport, a baseline treatment mean common across cells after exogeneity, and the observed conditional mean of the treatment on each instrument cell equaling that common baseline plus the response-type expansion P.treatmentAtSupport.","s":"{Ω : Type u_1} → [inst : MeasurableSpace Ω] → {K : ℕ} → MeasureTheory.Measure Ω → (Ω → Fin K) → (Ω → Bool) → (Ω → ℝ) → Causalean.PO.ID.Exact.MultipleInstrumentIV.FiniteIndex K → Causalean.PO.ID.Exact.MultipleInstrumentIV.ResponseTypeStats.PopulationBridge K → Type","l":134,"f":"Causalean/PO/ID/Exact/MultipleInstrumentIV/Population.lean"},{"n":"Causalean.PO.ID.Exact.MultipleInstrumentIV.ResponseTypeStats.PopulationBridge.ObservedBridge.observedBeta2SLS_eq_beta2SLSFiniteAlgebra","k":"theorem","m":"Causalean.PO.ID.Exact.MultipleInstrumentIV.Population","a":"PO/ID/Exact/MultipleInstrumentIV","t":1,"d":"End-to-end bridge to the finite response-type algebra. Given a bridge B linking the observed data to the response-type population, provided the instrument is measurable, the centered-instrument-weighted outcome is integrable, and the centered-instrument-weighted treatment is integrable, the observed population 2SLS ratio E[h(Z)Y] / E[h(Z)D] equals the finite response-type algebra estimand beta2SLSFiniteAlgebra.","s":"∀ {Ω : Type u_1} [inst : MeasurableSpace Ω] {K : ℕ} {μ : MeasureTheory.Measure Ω} {Z : Ω → Fin K} {D : Ω → Bool} {Y : Ω → ℝ} {I : Causalean.PO.ID.Exact.MultipleInstrumentIV.FiniteIndex K} {P : Causalean.PO.ID.Exact.MultipleInstrumentIV.ResponseTypeStats.PopulationBridge K} (B : Causalean.PO.ID.Exact.MultipleInstrumentIV.ResponseTypeStats.PopulationBridge.ObservedBridge μ Z D Y I P) [MeasureTheory.IsFiniteMeasure μ], Measurable Z → MeasureTheory.Integrable (fun ω => I.centeredIndex (Z ω) * Y ω) μ → MeasureTheory.Integrable (fun ω => I.centeredIndex (Z ω) * Causalean.PO.ID.Exact.MultipleInstrumentIV.boolToReal (D ω)) μ → Causalean.PO.ID.Exact.MultipleInstrumentIV.ResponseTypeStats.PopulationBridge.observedBeta2SLS μ Z D Y I = Causalean.PO.ID.Exact.MultipleInstrumentIV.ResponseTypeStats.beta2SLSFiniteAlgebra I P.stats","l":253,"f":"Causalean/PO/ID/Exact/MultipleInstrumentIV/Population.lean"},{"n":"Causalean.PO.ID.Exact.MultipleInstrumentIV.ResponseTypeStats.PopulationBridge.ObservedBridge.observedBeta2SLS_eq_beta2SLSPopulationBridge","k":"theorem","m":"Causalean.PO.ID.Exact.MultipleInstrumentIV.Population","a":"PO/ID/Exact/MultipleInstrumentIV","t":2,"d":"Observed population 2SLS, written as E[h(Z)Y]/E[h(Z)D], equals the saturated finite-support population bridge ratio.","s":"∀ {Ω : Type u_1} [inst : MeasurableSpace Ω] {K : ℕ} {μ : MeasureTheory.Measure Ω} {Z : Ω → Fin K} {D : Ω → Bool} {Y : Ω → ℝ} {I : Causalean.PO.ID.Exact.MultipleInstrumentIV.FiniteIndex K} {P : Causalean.PO.ID.Exact.MultipleInstrumentIV.ResponseTypeStats.PopulationBridge K} (B : Causalean.PO.ID.Exact.MultipleInstrumentIV.ResponseTypeStats.PopulationBridge.ObservedBridge μ Z D Y I P) [MeasureTheory.IsFiniteMeasure μ], Measurable Z → MeasureTheory.Integrable (fun ω => I.centeredIndex (Z ω) * Y ω) μ → MeasureTheory.Integrable (fun ω => I.centeredIndex (Z ω) * Causalean.PO.ID.Exact.MultipleInstrumentIV.boolToReal (D ω)) μ → Causalean.PO.ID.Exact.MultipleInstrumentIV.ResponseTypeStats.PopulationBridge.observedBeta2SLS μ Z D Y I = P.beta2SLSPopulationBridge I","l":242,"f":"Causalean/PO/ID/Exact/MultipleInstrumentIV/Population.lean"},{"n":"Causalean.PO.ID.Exact.MultipleInstrumentIV.ResponseTypeStats.PopulationBridge.ObservedBridge.observedBeta2SLS_eq_responseTypeWeightedSum","k":"theorem","m":"Causalean.PO.ID.Exact.MultipleInstrumentIV.Main","a":"PO/ID/Exact/MultipleInstrumentIV","t":1,"d":"End-to-end corollary (prop:po-estimand-mtw-response-type-form, observed level). Given a bridge B linking the data-generating instrument, treatment, and outcome to the response-type population, provided the instrument is measurable, the centered-instrument-weighted outcome is integrable, the centered-instrument-weighted treatment is integrable, and the observed first-stage moment observedFirstStageMoment μ Z D I is nonzero, the observed population 2SLS ratio equals the response-type weighted sum of within-type causal effects.","s":"∀ {Ω : Type u_1} [inst : MeasurableSpace Ω] {K : ℕ} {μ : MeasureTheory.Measure Ω} {Z : Ω → Fin K} {D : Ω → Bool} {Y : Ω → ℝ} {I : Causalean.PO.ID.Exact.MultipleInstrumentIV.FiniteIndex K} {P : Causalean.PO.ID.Exact.MultipleInstrumentIV.ResponseTypeStats.PopulationBridge K} (B : Causalean.PO.ID.Exact.MultipleInstrumentIV.ResponseTypeStats.PopulationBridge.ObservedBridge μ Z D Y I P) [MeasureTheory.IsFiniteMeasure μ], Measurable Z → MeasureTheory.Integrable (fun ω => I.centeredIndex (Z ω) * Y ω) μ → MeasureTheory.Integrable (fun ω => I.centeredIndex (Z ω) * Causalean.PO.ID.Exact.MultipleInstrumentIV.boolToReal (D ω)) μ → Causalean.PO.ID.Exact.MultipleInstrumentIV.ResponseTypeStats.PopulationBridge.observedFirstStageMoment μ Z D I ≠ 0 → Causalean.PO.ID.Exact.MultipleInstrumentIV.ResponseTypeStats.PopulationBridge.observedBeta2SLS μ Z D Y I = Causalean.PO.ID.Exact.MultipleInstrumentIV.ResponseTypeStats.responseTypeEstimand I P.stats","l":133,"f":"Causalean/PO/ID/Exact/MultipleInstrumentIV/Main.lean"},{"n":"Causalean.PO.ID.Exact.MultipleInstrumentIV.ResponseTypeStats.PopulationBridge.ObservedBridge.observedFirstStageMoment_eq_firstStageMoment","k":"theorem","m":"Causalean.PO.ID.Exact.MultipleInstrumentIV.Population","a":"PO/ID/Exact/MultipleInstrumentIV","t":2,"d":"The observed first-stage moment equals the finite response-type first-stage moment.","s":"∀ {Ω : Type u_1} [inst : MeasurableSpace Ω] {K : ℕ} {μ : MeasureTheory.Measure Ω} {Z : Ω → Fin K} {D : Ω → Bool} {Y : Ω → ℝ} {I : Causalean.PO.ID.Exact.MultipleInstrumentIV.FiniteIndex K} {P : Causalean.PO.ID.Exact.MultipleInstrumentIV.ResponseTypeStats.PopulationBridge K} (B : Causalean.PO.ID.Exact.MultipleInstrumentIV.ResponseTypeStats.PopulationBridge.ObservedBridge μ Z D Y I P) [MeasureTheory.IsFiniteMeasure μ], Measurable Z → MeasureTheory.Integrable (fun ω => I.centeredIndex (Z ω) * Causalean.PO.ID.Exact.MultipleInstrumentIV.boolToReal (D ω)) μ → Causalean.PO.ID.Exact.MultipleInstrumentIV.ResponseTypeStats.PopulationBridge.observedFirstStageMoment μ Z D I = P.firstStageMoment I","l":203,"f":"Causalean/PO/ID/Exact/MultipleInstrumentIV/Population.lean"},{"n":"Causalean.PO.ID.Exact.MultipleInstrumentIV.ResponseTypeStats.PopulationBridge.ObservedBridge.observedReducedFormMoment_eq_reducedFormMoment","k":"theorem","m":"Causalean.PO.ID.Exact.MultipleInstrumentIV.Population","a":"PO/ID/Exact/MultipleInstrumentIV","t":2,"d":"The observed reduced-form moment equals the finite response-type reduced-form moment.","s":"∀ {Ω : Type u_1} [inst : MeasurableSpace Ω] {K : ℕ} {μ : MeasureTheory.Measure Ω} {Z : Ω → Fin K} {D : Ω → Bool} {Y : Ω → ℝ} {I : Causalean.PO.ID.Exact.MultipleInstrumentIV.FiniteIndex K} {P : Causalean.PO.ID.Exact.MultipleInstrumentIV.ResponseTypeStats.PopulationBridge K} (B : Causalean.PO.ID.Exact.MultipleInstrumentIV.ResponseTypeStats.PopulationBridge.ObservedBridge μ Z D Y I P) [MeasureTheory.IsFiniteMeasure μ], Measurable Z → MeasureTheory.Integrable (fun ω => I.centeredIndex (Z ω) * Y ω) μ → Causalean.PO.ID.Exact.MultipleInstrumentIV.ResponseTypeStats.PopulationBridge.observedReducedFormMoment μ Z Y I = P.reducedFormMoment I","l":191,"f":"Causalean/PO/ID/Exact/MultipleInstrumentIV/Population.lean"},{"n":"Causalean.PO.ID.Exact.MultipleInstrumentIV.ResponseTypeStats.PopulationBridge.baselineMoment_eq_zero","k":"theorem","m":"Causalean.PO.ID.Exact.MultipleInstrumentIV.ResponseTypes","a":"PO/ID/Exact/MultipleInstrumentIV","t":2,"d":"The baseline outcome component vanishes because the first-stage index is centered.","s":"∀ {K : ℕ} (I : Causalean.PO.ID.Exact.MultipleInstrumentIV.FiniteIndex K) (P : Causalean.PO.ID.Exact.MultipleInstrumentIV.ResponseTypeStats.PopulationBridge K), ∑ k, I.rho k * I.centeredIndex k * ∑ g, P.stats.mass g * P.baseOutcome g = 0","l":164,"f":"Causalean/PO/ID/Exact/MultipleInstrumentIV/ResponseTypes.lean"},{"n":"Causalean.PO.ID.Exact.MultipleInstrumentIV.ResponseTypeStats.PopulationBridge.beta2SLSPopulationBridge","k":"def","m":"Causalean.PO.ID.Exact.MultipleInstrumentIV.ResponseTypes","a":"PO/ID/Exact/MultipleInstrumentIV","t":1,"d":"Population multiple-IV 2SLS ratio in the saturated finite-support bridge.","s":"{K : ℕ} → Causalean.PO.ID.Exact.MultipleInstrumentIV.ResponseTypeStats.PopulationBridge K → Causalean.PO.ID.Exact.MultipleInstrumentIV.FiniteIndex K → ℝ","l":160,"f":"Causalean/PO/ID/Exact/MultipleInstrumentIV/ResponseTypes.lean"},{"n":"Causalean.PO.ID.Exact.MultipleInstrumentIV.ResponseTypeStats.PopulationBridge.beta2SLSPopulationBridge_eq_beta2SLSFiniteAlgebra","k":"theorem","m":"Causalean.PO.ID.Exact.MultipleInstrumentIV.ResponseTypes","a":"PO/ID/Exact/MultipleInstrumentIV","t":1,"d":"Saturated finite-support MTW identification. The population 2SLS ratio E[h(Z)Y] / E[h(Z)D], after consistency, exogeneity, exclusion, and telescoping, is exactly the finite response-type ratio.","s":"∀ {K : ℕ} (I : Causalean.PO.ID.Exact.MultipleInstrumentIV.FiniteIndex K) (P : Causalean.PO.ID.Exact.MultipleInstrumentIV.ResponseTypeStats.PopulationBridge K), P.beta2SLSPopulationBridge I = Causalean.PO.ID.Exact.MultipleInstrumentIV.ResponseTypeStats.beta2SLSFiniteAlgebra I P.stats","l":288,"f":"Causalean/PO/ID/Exact/MultipleInstrumentIV/ResponseTypes.lean"},{"n":"Causalean.PO.ID.Exact.MultipleInstrumentIV.ResponseTypeStats.PopulationBridge.beta2SLSPopulationBridge_eq_positiveResponseTypeAverage","k":"theorem","m":"Causalean.PO.ID.Exact.MultipleInstrumentIV.Main","a":"PO/ID/Exact/MultipleInstrumentIV","t":1,"d":"Positive response-type average (population bridge). When the response types are sign-aligned with the instrument order and the first-stage type-weight denominator is strictly positive, the population-bridge 2SLS estimand beta2SLSPopulationBridge equals the response-type estimand, its normalized response-type weights are all nonnegative, and those weights sum to one.","s":"∀ {K : ℕ} (I : Causalean.PO.ID.Exact.MultipleInstrumentIV.FiniteIndex K) (P : Causalean.PO.ID.Exact.MultipleInstrumentIV.ResponseTypeStats.PopulationBridge K), Causalean.PO.ID.Exact.MultipleInstrumentIV.ResponseTypeStats.SignAligned I P.stats → 0 < Causalean.PO.ID.Exact.MultipleInstrumentIV.ResponseTypeStats.typeWeightDenom I P.stats → P.beta2SLSPopulationBridge I = Causalean.PO.ID.Exact.MultipleInstrumentIV.ResponseTypeStats.responseTypeEstimand I P.stats ∧ (∀ (g : Causalean.PO.ID.Exact.MultipleInstrumentIV.ResponseType K), 0 ≤ Causalean.PO.ID.Exact.MultipleInstrumentIV.ResponseTypeStats.normalizedTypeWeight I P.stats g) ∧ ∑ g, Causalean.PO.ID.Exact.MultipleInstrumentIV.ResponseTypeStats.normalizedTypeWeight I P.stats g = 1","l":107,"f":"Causalean/PO/ID/Exact/MultipleInstrumentIV/Main.lean"},{"n":"Causalean.PO.ID.Exact.MultipleInstrumentIV.ResponseTypeStats.PopulationBridge.beta2SLSPopulationBridge_eq_responseTypeWeightedSum","k":"theorem","m":"Causalean.PO.ID.Exact.MultipleInstrumentIV.Main","a":"PO/ID/Exact/MultipleInstrumentIV","t":1,"d":"Response-type weighted-sum form (population bridge). Provided the first-stage type-weight denominator is nonzero, the population- bridge 2SLS estimand equals the response-type-weighted sum Σ_g ω_g Δ_g of within-type causal effects.","s":"∀ {K : ℕ} (I : Causalean.PO.ID.Exact.MultipleInstrumentIV.FiniteIndex K) (P : Causalean.PO.ID.Exact.MultipleInstrumentIV.ResponseTypeStats.PopulationBridge K), Causalean.PO.ID.Exact.MultipleInstrumentIV.ResponseTypeStats.typeWeightDenom I P.stats ≠ 0 → P.beta2SLSPopulationBridge I = Causalean.PO.ID.Exact.MultipleInstrumentIV.ResponseTypeStats.responseTypeEstimand I P.stats","l":97,"f":"Causalean/PO/ID/Exact/MultipleInstrumentIV/Main.lean"},{"n":"Causalean.PO.ID.Exact.MultipleInstrumentIV.ResponseTypeStats.PopulationBridge.beta2SLSPopulationBridge_eq_signedAdjacentRatio","k":"theorem","m":"Causalean.PO.ID.Exact.MultipleInstrumentIV.Main","a":"PO/ID/Exact/MultipleInstrumentIV","t":1,"d":"Signed adjacent ratio form (population bridge). Provided the signed adjacent first-stage denominator is nonzero, the population- bridge 2SLS estimand equals the signed adjacent numerator divided by the signed adjacent denominator.","s":"∀ {K : ℕ} (I : Causalean.PO.ID.Exact.MultipleInstrumentIV.FiniteIndex K) (P : Causalean.PO.ID.Exact.MultipleInstrumentIV.ResponseTypeStats.PopulationBridge K), Causalean.PO.ID.Exact.MultipleInstrumentIV.ResponseTypeStats.signedAdjacentDenominator I P.stats ≠ 0 → P.beta2SLSPopulationBridge I = Causalean.PO.ID.Exact.MultipleInstrumentIV.ResponseTypeStats.signedAdjacentNumerator I P.stats / Causalean.PO.ID.Exact.MultipleInstrumentIV.ResponseTypeStats.signedAdjacentDenominator I P.stats","l":86,"f":"Causalean/PO/ID/Exact/MultipleInstrumentIV/Main.lean"},{"n":"Causalean.PO.ID.Exact.MultipleInstrumentIV.ResponseTypeStats.PopulationBridge.firstStageMoment","k":"def","m":"Causalean.PO.ID.Exact.MultipleInstrumentIV.ResponseTypes","a":"PO/ID/Exact/MultipleInstrumentIV","t":1,"d":"Population first-stage moment E[h(Z)D] after the same finite-support identification reductions.","s":"{K : ℕ} → Causalean.PO.ID.Exact.MultipleInstrumentIV.ResponseTypeStats.PopulationBridge K → Causalean.PO.ID.Exact.MultipleInstrumentIV.FiniteIndex K → ℝ","l":155,"f":"Causalean/PO/ID/Exact/MultipleInstrumentIV/ResponseTypes.lean"},{"n":"Causalean.PO.ID.Exact.MultipleInstrumentIV.ResponseTypeStats.PopulationBridge.firstStageMoment_eq_typeWeightDenom","k":"theorem","m":"Causalean.PO.ID.Exact.MultipleInstrumentIV.ResponseTypes","a":"PO/ID/Exact/MultipleInstrumentIV","t":2,"d":"The finite-support first-stage moment is the response-type denominator. This is the denominator half of the MTW identification step.","s":"∀ {K : ℕ} (I : Causalean.PO.ID.Exact.MultipleInstrumentIV.FiniteIndex K) (P : Causalean.PO.ID.Exact.MultipleInstrumentIV.ResponseTypeStats.PopulationBridge K), P.firstStageMoment I = Causalean.PO.ID.Exact.MultipleInstrumentIV.ResponseTypeStats.typeWeightDenom I P.stats","l":181,"f":"Causalean/PO/ID/Exact/MultipleInstrumentIV/ResponseTypes.lean"},{"n":"Causalean.PO.ID.Exact.MultipleInstrumentIV.ResponseTypeStats.PopulationBridge.observedBeta2SLS","k":"def","m":"Causalean.PO.ID.Exact.MultipleInstrumentIV.Population","a":"PO/ID/Exact/MultipleInstrumentIV","t":1,"d":"Source population 2SLS ratio E[h(Z)Y] / E[h(Z)D].","s":"{Ω : Type u_1} → [inst : MeasurableSpace Ω] → {K : ℕ} → MeasureTheory.Measure Ω → (Ω → Fin K) → (Ω → Bool) → (Ω → ℝ) → Causalean.PO.ID.Exact.MultipleInstrumentIV.FiniteIndex K → ℝ","l":59,"f":"Causalean/PO/ID/Exact/MultipleInstrumentIV/Population.lean"},{"n":"Causalean.PO.ID.Exact.MultipleInstrumentIV.ResponseTypeStats.PopulationBridge.observedFirstStageMoment","k":"def","m":"Causalean.PO.ID.Exact.MultipleInstrumentIV.Population","a":"PO/ID/Exact/MultipleInstrumentIV","t":1,"d":"Observed first-stage moment E[h(Z)D], with binary treatment coerced to the real values 0 and 1.","s":"{Ω : Type u_1} → [inst : MeasurableSpace Ω] → {K : ℕ} → MeasureTheory.Measure Ω → (Ω → Fin K) → (Ω → Bool) → Causalean.PO.ID.Exact.MultipleInstrumentIV.FiniteIndex K → ℝ","l":53,"f":"Causalean/PO/ID/Exact/MultipleInstrumentIV/Population.lean"},{"n":"Causalean.PO.ID.Exact.MultipleInstrumentIV.ResponseTypeStats.PopulationBridge.observedFirstStageMoment_eq_sum_eventCondExp","k":"theorem","m":"Causalean.PO.ID.Exact.MultipleInstrumentIV.Population","a":"PO/ID/Exact/MultipleInstrumentIV","t":2,"d":"Finite-support total-law rewrite of E[h(Z)D] into instrument-cell conditional expectations.","s":"∀ {Ω : Type u_1} [inst : MeasurableSpace Ω] {K : ℕ} (μ : MeasureTheory.Measure Ω) [MeasureTheory.IsFiniteMeasure μ] (Z : Ω → Fin K), Measurable Z → ∀ (D : Ω → Bool) (I : Causalean.PO.ID.Exact.MultipleInstrumentIV.FiniteIndex K), MeasureTheory.Integrable (fun ω => I.centeredIndex (Z ω) * Causalean.PO.ID.Exact.MultipleInstrumentIV.boolToReal (D ω)) μ → Causalean.PO.ID.Exact.MultipleInstrumentIV.ResponseTypeStats.PopulationBridge.observedFirstStageMoment μ Z D I = ∑ k, (μ (Causalean.PO.ID.Exact.MultipleInstrumentIV.ResponseTypeStats.PopulationBridge.zEvent Z k)).toReal * I.centeredIndex k * Causalean.PO.eventCondExp μ (Causalean.PO.ID.Exact.MultipleInstrumentIV.ResponseTypeStats.PopulationBridge.zEvent Z k) fun ω => Causalean.PO.ID.Exact.MultipleInstrumentIV.boolToReal (D ω)","l":121,"f":"Causalean/PO/ID/Exact/MultipleInstrumentIV/Population.lean"},{"n":"Causalean.PO.ID.Exact.MultipleInstrumentIV.ResponseTypeStats.PopulationBridge.observedReducedFormMoment","k":"def","m":"Causalean.PO.ID.Exact.MultipleInstrumentIV.Population","a":"PO/ID/Exact/MultipleInstrumentIV","t":1,"d":"Observed reduced-form moment E[h(Z)Y], where h(zᵏ) = dhat_k − Σ_l ρ_l dhat_l.","s":"{Ω : Type u_1} → [inst : MeasurableSpace Ω] → {K : ℕ} → MeasureTheory.Measure Ω → (Ω → Fin K) → (Ω → ℝ) → Causalean.PO.ID.Exact.MultipleInstrumentIV.FiniteIndex K → ℝ","l":47,"f":"Causalean/PO/ID/Exact/MultipleInstrumentIV/Population.lean"},{"n":"Causalean.PO.ID.Exact.MultipleInstrumentIV.ResponseTypeStats.PopulationBridge.observedReducedFormMoment_eq_sum_eventCondExp","k":"theorem","m":"Causalean.PO.ID.Exact.MultipleInstrumentIV.Population","a":"PO/ID/Exact/MultipleInstrumentIV","t":2,"d":"Finite-support total-law rewrite of E[h(Z)Y] into instrument-cell conditional expectations.","s":"∀ {Ω : Type u_1} [inst : MeasurableSpace Ω] {K : ℕ} (μ : MeasureTheory.Measure Ω) [MeasureTheory.IsFiniteMeasure μ] (Z : Ω → Fin K), Measurable Z → ∀ (Y : Ω → ℝ) (I : Causalean.PO.ID.Exact.MultipleInstrumentIV.FiniteIndex K), MeasureTheory.Integrable (fun ω => I.centeredIndex (Z ω) * Y ω) μ → Causalean.PO.ID.Exact.MultipleInstrumentIV.ResponseTypeStats.PopulationBridge.observedReducedFormMoment μ Z Y I = ∑ k, (μ (Causalean.PO.ID.Exact.MultipleInstrumentIV.ResponseTypeStats.PopulationBridge.zEvent Z k)).toReal * I.centeredIndex k * Causalean.PO.eventCondExp μ (Causalean.PO.ID.Exact.MultipleInstrumentIV.ResponseTypeStats.PopulationBridge.zEvent Z k) Y","l":99,"f":"Causalean/PO/ID/Exact/MultipleInstrumentIV/Population.lean"},{"n":"Causalean.PO.ID.Exact.MultipleInstrumentIV.ResponseTypeStats.PopulationBridge.outcomeAtSupport","k":"def","m":"Causalean.PO.ID.Exact.MultipleInstrumentIV.ResponseTypes","a":"PO/ID/Exact/MultipleInstrumentIV","t":1,"d":"Response-type expansion of the potential outcome at support point k after consistency, exogeneity, and exclusion have replaced conditioning on Z = zᵏ by response-type averages.","s":"{K : ℕ} → Causalean.PO.ID.Exact.MultipleInstrumentIV.ResponseTypeStats.PopulationBridge K → Fin K → ℝ","l":136,"f":"Causalean/PO/ID/Exact/MultipleInstrumentIV/ResponseTypes.lean"},{"n":"Causalean.PO.ID.Exact.MultipleInstrumentIV.ResponseTypeStats.PopulationBridge.reducedFormMoment","k":"def","m":"Causalean.PO.ID.Exact.MultipleInstrumentIV.ResponseTypes","a":"PO/ID/Exact/MultipleInstrumentIV","t":1,"d":"Population reduced-form moment E[h(Z)Y] after the finite-support identification reductions.","s":"{K : ℕ} → Causalean.PO.ID.Exact.MultipleInstrumentIV.ResponseTypeStats.PopulationBridge K → Causalean.PO.ID.Exact.MultipleInstrumentIV.FiniteIndex K → ℝ","l":150,"f":"Causalean/PO/ID/Exact/MultipleInstrumentIV/ResponseTypes.lean"},{"n":"Causalean.PO.ID.Exact.MultipleInstrumentIV.ResponseTypeStats.PopulationBridge.reducedFormMoment_eq_typeWeightNumerator","k":"theorem","m":"Causalean.PO.ID.Exact.MultipleInstrumentIV.ResponseTypes","a":"PO/ID/Exact/MultipleInstrumentIV","t":2,"d":"The finite-support reduced-form moment is the response-type numerator. This is the numerator half of the MTW identification step.","s":"∀ {K : ℕ} (I : Causalean.PO.ID.Exact.MultipleInstrumentIV.FiniteIndex K) (P : Causalean.PO.ID.Exact.MultipleInstrumentIV.ResponseTypeStats.PopulationBridge K), P.reducedFormMoment I = ∑ g, Causalean.PO.ID.Exact.MultipleInstrumentIV.ResponseTypeStats.unnormTypeWeight I P.stats g * P.stats.effect g","l":223,"f":"Causalean/PO/ID/Exact/MultipleInstrumentIV/ResponseTypes.lean"},{"n":"Causalean.PO.ID.Exact.MultipleInstrumentIV.ResponseTypeStats.PopulationBridge.telescopedTypeStep","k":"def","m":"Causalean.PO.ID.Exact.MultipleInstrumentIV.ResponseTypes","a":"PO/ID/Exact/MultipleInstrumentIV","t":1,"d":"Telescoped adjacent treatment response for support point k, i.e. Σ_{j≤k} (d_j - d_{j-1}) in zero-based Lean indexing.","s":"{K : ℕ} → Causalean.PO.ID.Exact.MultipleInstrumentIV.ResponseType K → Fin K → ℝ","l":131,"f":"Causalean/PO/ID/Exact/MultipleInstrumentIV/ResponseTypes.lean"},{"n":"Causalean.PO.ID.Exact.MultipleInstrumentIV.ResponseTypeStats.PopulationBridge.treatmentAtSupport","k":"def","m":"Causalean.PO.ID.Exact.MultipleInstrumentIV.ResponseTypes","a":"PO/ID/Exact/MultipleInstrumentIV","t":1,"d":"Response-type expansion of the potential treatment at support point k, written in the baseline-subtracted telescoped form used by the centered-index argument.","s":"{K : ℕ} → Causalean.PO.ID.Exact.MultipleInstrumentIV.ResponseTypeStats.PopulationBridge K → Fin K → ℝ","l":144,"f":"Causalean/PO/ID/Exact/MultipleInstrumentIV/ResponseTypes.lean"},{"n":"Causalean.PO.ID.Exact.MultipleInstrumentIV.ResponseTypeStats.PopulationBridge.zEvent","k":"def","m":"Causalean.PO.ID.Exact.MultipleInstrumentIV.Population","a":"PO/ID/Exact/MultipleInstrumentIV","t":1,"d":"Instrument cell {ω | Z ω = k}.","s":"{Ω : Type u_1} → {K : ℕ} → (Ω → Fin K) → Fin K → Set Ω","l":43,"f":"Causalean/PO/ID/Exact/MultipleInstrumentIV/Population.lean"},{"n":"Causalean.PO.ID.Exact.MultipleInstrumentIV.ResponseTypeStats.SignAligned","k":"def","m":"Causalean.PO.ID.Exact.MultipleInstrumentIV.ResponseTypes","a":"PO/ID/Exact/MultipleInstrumentIV","t":1,"d":"MTW sign alignment: every positive-mass response type has a nonnegative tail-weighted step contrast.","s":"{K : ℕ} → Causalean.PO.ID.Exact.MultipleInstrumentIV.FiniteIndex K → Causalean.PO.ID.Exact.MultipleInstrumentIV.ResponseTypeStats K → Prop","l":299,"f":"Causalean/PO/ID/Exact/MultipleInstrumentIV/ResponseTypes.lean"},{"n":"Causalean.PO.ID.Exact.MultipleInstrumentIV.ResponseTypeStats.beta2SLSFiniteAlgebra","k":"def","m":"Causalean.PO.ID.Exact.MultipleInstrumentIV.ResponseTypes","a":"PO/ID/Exact/MultipleInstrumentIV","t":1,"d":"Finite-algebra ratio after the MTW response-type partition: numerator Σ_g λ_g Δ_g, denominator Σ_g λ_g. The saturated finite-support population bridge below proves when the population 2SLS moment ratio reduces to this finite algebraic ratio.","s":"{K : ℕ} → Causalean.PO.ID.Exact.MultipleInstrumentIV.FiniteIndex K → Causalean.PO.ID.Exact.MultipleInstrumentIV.ResponseTypeStats K → ℝ","l":107,"f":"Causalean/PO/ID/Exact/MultipleInstrumentIV/ResponseTypes.lean"},{"n":"Causalean.PO.ID.Exact.MultipleInstrumentIV.ResponseTypeStats.beta2SLSFiniteAlgebra_eq_positiveResponseTypeAverage","k":"theorem","m":"Causalean.PO.ID.Exact.MultipleInstrumentIV.ResponseTypes","a":"PO/ID/Exact/MultipleInstrumentIV","t":1,"d":"Positive-weight response-type characterization. When the response types are sign-aligned with the instrument order and the first-stage type-weight denominator is strictly positive, the finite-algebra 2SLS estimand equals the response-type estimand, every normalized response-type weight is nonnegative, and the weights sum to one.","s":"∀ {K : ℕ} (I : Causalean.PO.ID.Exact.MultipleInstrumentIV.FiniteIndex K) (R : Causalean.PO.ID.Exact.MultipleInstrumentIV.ResponseTypeStats K), Causalean.PO.ID.Exact.MultipleInstrumentIV.ResponseTypeStats.SignAligned I R → 0 < Causalean.PO.ID.Exact.MultipleInstrumentIV.ResponseTypeStats.typeWeightDenom I R → Causalean.PO.ID.Exact.MultipleInstrumentIV.ResponseTypeStats.beta2SLSFiniteAlgebra I R = Causalean.PO.ID.Exact.MultipleInstrumentIV.ResponseTypeStats.responseTypeEstimand I R ∧ (∀ (g : Causalean.PO.ID.Exact.MultipleInstrumentIV.ResponseType K), 0 ≤ Causalean.PO.ID.Exact.MultipleInstrumentIV.ResponseTypeStats.normalizedTypeWeight I R g) ∧ ∑ g, Causalean.PO.ID.Exact.MultipleInstrumentIV.ResponseTypeStats.normalizedTypeWeight I R g = 1","l":389,"f":"Causalean/PO/ID/Exact/MultipleInstrumentIV/ResponseTypes.lean"},{"n":"Causalean.PO.ID.Exact.MultipleInstrumentIV.ResponseTypeStats.beta2SLSFiniteAlgebra_eq_positiveResponseTypeAverage'","k":"theorem","m":"Causalean.PO.ID.Exact.MultipleInstrumentIV.Main","a":"PO/ID/Exact/MultipleInstrumentIV","t":2,"d":"Positive response-type average under MTW sign alignment and a positive finite first-stage denominator (prop:po-estimand-mtw-positive-weights, finite algebra layer).","s":"∀ {K : ℕ} (I : Causalean.PO.ID.Exact.MultipleInstrumentIV.FiniteIndex K) (R : Causalean.PO.ID.Exact.MultipleInstrumentIV.ResponseTypeStats K), Causalean.PO.ID.Exact.MultipleInstrumentIV.ResponseTypeStats.SignAligned I R → 0 < Causalean.PO.ID.Exact.MultipleInstrumentIV.ResponseTypeStats.typeWeightDenom I R → Causalean.PO.ID.Exact.MultipleInstrumentIV.ResponseTypeStats.beta2SLSFiniteAlgebra I R = Causalean.PO.ID.Exact.MultipleInstrumentIV.ResponseTypeStats.responseTypeEstimand I R ∧ (∀ (g : Causalean.PO.ID.Exact.MultipleInstrumentIV.ResponseType K), 0 ≤ Causalean.PO.ID.Exact.MultipleInstrumentIV.ResponseTypeStats.normalizedTypeWeight I R g) ∧ ∑ g, Causalean.PO.ID.Exact.MultipleInstrumentIV.ResponseTypeStats.normalizedTypeWeight I R g = 1","l":69,"f":"Causalean/PO/ID/Exact/MultipleInstrumentIV/Main.lean"},{"n":"Causalean.PO.ID.Exact.MultipleInstrumentIV.ResponseTypeStats.beta2SLSFiniteAlgebra_eq_responseTypeWeightedSum","k":"theorem","m":"Causalean.PO.ID.Exact.MultipleInstrumentIV.ResponseTypes","a":"PO/ID/Exact/MultipleInstrumentIV","t":1,"d":"Response-type form of the finite MTW ratio (prop:po-estimand-mtw-response-type-form). Provided the first-stage type-weight denominator is nonzero, the finite-algebra 2SLS estimand equals the response-type-weighted sum of within-type causal effects.","s":"∀ {K : ℕ} (I : Causalean.PO.ID.Exact.MultipleInstrumentIV.FiniteIndex K) (R : Causalean.PO.ID.Exact.MultipleInstrumentIV.ResponseTypeStats K), Causalean.PO.ID.Exact.MultipleInstrumentIV.ResponseTypeStats.typeWeightDenom I R ≠ 0 → Causalean.PO.ID.Exact.MultipleInstrumentIV.ResponseTypeStats.beta2SLSFiniteAlgebra I R = Causalean.PO.ID.Exact.MultipleInstrumentIV.ResponseTypeStats.responseTypeEstimand I R","l":370,"f":"Causalean/PO/ID/Exact/MultipleInstrumentIV/ResponseTypes.lean"},{"n":"Causalean.PO.ID.Exact.MultipleInstrumentIV.ResponseTypeStats.beta2SLSFiniteAlgebra_eq_responseTypeWeightedSum'","k":"theorem","m":"Causalean.PO.ID.Exact.MultipleInstrumentIV.Main","a":"PO/ID/Exact/MultipleInstrumentIV","t":2,"d":"Response-type weighted-sum form of the finite MTW algebra (prop:po-estimand-mtw-response-type-form, finite algebra layer).","s":"∀ {K : ℕ} (I : Causalean.PO.ID.Exact.MultipleInstrumentIV.FiniteIndex K) (R : Causalean.PO.ID.Exact.MultipleInstrumentIV.ResponseTypeStats K), Causalean.PO.ID.Exact.MultipleInstrumentIV.ResponseTypeStats.typeWeightDenom I R ≠ 0 → Causalean.PO.ID.Exact.MultipleInstrumentIV.ResponseTypeStats.beta2SLSFiniteAlgebra I R = Causalean.PO.ID.Exact.MultipleInstrumentIV.ResponseTypeStats.responseTypeEstimand I R","l":62,"f":"Causalean/PO/ID/Exact/MultipleInstrumentIV/Main.lean"},{"n":"Causalean.PO.ID.Exact.MultipleInstrumentIV.ResponseTypeStats.beta2SLSFiniteAlgebra_eq_signedAdjacentRatio","k":"theorem","m":"Causalean.PO.ID.Exact.MultipleInstrumentIV.Main","a":"PO/ID/Exact/MultipleInstrumentIV","t":1,"d":"Finite response-type ratio as signed adjacent ratio. Provided the signed adjacent first-stage denominator is nonzero, the finite-algebra MTW estimand beta2SLSFiniteAlgebra equals the signed adjacent numerator Σ_g λ_g Δ_g divided by the signed adjacent denominator Σ_g λ_g.","s":"∀ {K : ℕ} (I : Causalean.PO.ID.Exact.MultipleInstrumentIV.FiniteIndex K) (R : Causalean.PO.ID.Exact.MultipleInstrumentIV.ResponseTypeStats K), Causalean.PO.ID.Exact.MultipleInstrumentIV.ResponseTypeStats.signedAdjacentDenominator I R ≠ 0 → Causalean.PO.ID.Exact.MultipleInstrumentIV.ResponseTypeStats.beta2SLSFiniteAlgebra I R = Causalean.PO.ID.Exact.MultipleInstrumentIV.ResponseTypeStats.signedAdjacentNumerator I R / Causalean.PO.ID.Exact.MultipleInstrumentIV.ResponseTypeStats.signedAdjacentDenominator I R","l":48,"f":"Causalean/PO/ID/Exact/MultipleInstrumentIV/Main.lean"},{"n":"Causalean.PO.ID.Exact.MultipleInstrumentIV.ResponseTypeStats.normalizedTypeWeight","k":"def","m":"Causalean.PO.ID.Exact.MultipleInstrumentIV.ResponseTypes","a":"PO/ID/Exact/MultipleInstrumentIV","t":1,"d":"Normalized response-type weight ω_g = λ_g / Σ_g' λ_g'.","s":"{K : ℕ} → Causalean.PO.ID.Exact.MultipleInstrumentIV.FiniteIndex K → Causalean.PO.ID.Exact.MultipleInstrumentIV.ResponseTypeStats K → Causalean.PO.ID.Exact.MultipleInstrumentIV.ResponseType K → ℝ","l":98,"f":"Causalean/PO/ID/Exact/MultipleInstrumentIV/ResponseTypes.lean"},{"n":"Causalean.PO.ID.Exact.MultipleInstrumentIV.ResponseTypeStats.normalizedTypeWeight_nonneg_of_signAligned","k":"theorem","m":"Causalean.PO.ID.Exact.MultipleInstrumentIV.ResponseTypes","a":"PO/ID/Exact/MultipleInstrumentIV","t":2,"d":"Normalized response-type weights are nonnegative when sign alignment holds and the denominator is positive (prop:po-estimand-mtw-positive-weights).","s":"∀ {K : ℕ} (I : Causalean.PO.ID.Exact.MultipleInstrumentIV.FiniteIndex K) (R : Causalean.PO.ID.Exact.MultipleInstrumentIV.ResponseTypeStats K), Causalean.PO.ID.Exact.MultipleInstrumentIV.ResponseTypeStats.SignAligned I R → 0 < Causalean.PO.ID.Exact.MultipleInstrumentIV.ResponseTypeStats.typeWeightDenom I R → ∀ (g : Causalean.PO.ID.Exact.MultipleInstrumentIV.ResponseType K), 0 ≤ Causalean.PO.ID.Exact.MultipleInstrumentIV.ResponseTypeStats.normalizedTypeWeight I R g","l":353,"f":"Causalean/PO/ID/Exact/MultipleInstrumentIV/ResponseTypes.lean"},{"n":"Causalean.PO.ID.Exact.MultipleInstrumentIV.ResponseTypeStats.normalizedTypeWeight_sum_eq_one_of_pos","k":"theorem","m":"Causalean.PO.ID.Exact.MultipleInstrumentIV.ResponseTypes","a":"PO/ID/Exact/MultipleInstrumentIV","t":2,"d":"Normalized response-type weights sum to one when the first-stage denominator is positive.","s":"∀ {K : ℕ} (I : Causalean.PO.ID.Exact.MultipleInstrumentIV.FiniteIndex K) (R : Causalean.PO.ID.Exact.MultipleInstrumentIV.ResponseTypeStats K), 0 < Causalean.PO.ID.Exact.MultipleInstrumentIV.ResponseTypeStats.typeWeightDenom I R → ∑ g, Causalean.PO.ID.Exact.MultipleInstrumentIV.ResponseTypeStats.normalizedTypeWeight I R g = 1","l":362,"f":"Causalean/PO/ID/Exact/MultipleInstrumentIV/ResponseTypes.lean"},{"n":"Causalean.PO.ID.Exact.MultipleInstrumentIV.ResponseTypeStats.responseTypeEstimand","k":"def","m":"Causalean.PO.ID.Exact.MultipleInstrumentIV.ResponseTypes","a":"PO/ID/Exact/MultipleInstrumentIV","t":1,"d":"Signed response-type weighted estimand Σ_g ω_g Δ_g.","s":"{K : ℕ} → Causalean.PO.ID.Exact.MultipleInstrumentIV.FiniteIndex K → Causalean.PO.ID.Exact.MultipleInstrumentIV.ResponseTypeStats K → ℝ","l":102,"f":"Causalean/PO/ID/Exact/MultipleInstrumentIV/ResponseTypes.lean"},{"n":"Causalean.PO.ID.Exact.MultipleInstrumentIV.ResponseTypeStats.signedAdjacentDenominator","k":"def","m":"Causalean.PO.ID.Exact.MultipleInstrumentIV.Main","a":"PO/ID/Exact/MultipleInstrumentIV","t":1,"d":"Signed adjacent first-stage denominator in the finite response-type algebra: Σ_g λ_g, corresponding to Σ_j B_j E[Δ_jD].","s":"{K : ℕ} → Causalean.PO.ID.Exact.MultipleInstrumentIV.FiniteIndex K → Causalean.PO.ID.Exact.MultipleInstrumentIV.ResponseTypeStats K → ℝ","l":42,"f":"Causalean/PO/ID/Exact/MultipleInstrumentIV/Main.lean"},{"n":"Causalean.PO.ID.Exact.MultipleInstrumentIV.ResponseTypeStats.signedAdjacentNumerator","k":"def","m":"Causalean.PO.ID.Exact.MultipleInstrumentIV.Main","a":"PO/ID/Exact/MultipleInstrumentIV","t":1,"d":"Signed adjacent reduced-form numerator in the finite response-type algebra: Σ_g λ_g Δ_g. This is the response-type counterpart of Σ_j B_j E[(Y(1)-Y(0)) Δ_jD].","s":"{K : ℕ} → Causalean.PO.ID.Exact.MultipleInstrumentIV.FiniteIndex K → Causalean.PO.ID.Exact.MultipleInstrumentIV.ResponseTypeStats K → ℝ","l":36,"f":"Causalean/PO/ID/Exact/MultipleInstrumentIV/Main.lean"},{"n":"Causalean.PO.ID.Exact.MultipleInstrumentIV.ResponseTypeStats.typeWeightDenom","k":"def","m":"Causalean.PO.ID.Exact.MultipleInstrumentIV.ResponseTypes","a":"PO/ID/Exact/MultipleInstrumentIV","t":1,"d":"First-stage denominator in response-type form, Σ_g λ_g.","s":"{K : ℕ} → Causalean.PO.ID.Exact.MultipleInstrumentIV.FiniteIndex K → Causalean.PO.ID.Exact.MultipleInstrumentIV.ResponseTypeStats K → ℝ","l":94,"f":"Causalean/PO/ID/Exact/MultipleInstrumentIV/ResponseTypes.lean"},{"n":"Causalean.PO.ID.Exact.MultipleInstrumentIV.ResponseTypeStats.unnormTypeWeight","k":"def","m":"Causalean.PO.ID.Exact.MultipleInstrumentIV.ResponseTypes","a":"PO/ID/Exact/MultipleInstrumentIV","t":1,"d":"Unnormalized MTW response-type weight λ_g = π_g Σ_j B_j (d_j - d_{j-1}).","s":"{K : ℕ} → Causalean.PO.ID.Exact.MultipleInstrumentIV.FiniteIndex K → Causalean.PO.ID.Exact.MultipleInstrumentIV.ResponseTypeStats K → Causalean.PO.ID.Exact.MultipleInstrumentIV.ResponseType K → ℝ","l":89,"f":"Causalean/PO/ID/Exact/MultipleInstrumentIV/ResponseTypes.lean"},{"n":"Causalean.PO.ID.Exact.MultipleInstrumentIV.ResponseTypeStats.unnormTypeWeight_nonneg_of_signAligned","k":"theorem","m":"Causalean.PO.ID.Exact.MultipleInstrumentIV.ResponseTypes","a":"PO/ID/Exact/MultipleInstrumentIV","t":2,"d":"Sign alignment and nonnegative type masses imply nonnegative unnormalized response-type weights.","s":"∀ {K : ℕ} (I : Causalean.PO.ID.Exact.MultipleInstrumentIV.FiniteIndex K) (R : Causalean.PO.ID.Exact.MultipleInstrumentIV.ResponseTypeStats K), Causalean.PO.ID.Exact.MultipleInstrumentIV.ResponseTypeStats.SignAligned I R → ∀ (g : Causalean.PO.ID.Exact.MultipleInstrumentIV.ResponseType K), 0 ≤ Causalean.PO.ID.Exact.MultipleInstrumentIV.ResponseTypeStats.unnormTypeWeight I R g","l":341,"f":"Causalean/PO/ID/Exact/MultipleInstrumentIV/ResponseTypes.lean"},{"n":"Causalean.PO.ID.Exact.MultipleInstrumentIV.boolToReal","k":"def","m":"Causalean.PO.ID.Exact.MultipleInstrumentIV.ResponseTypes","a":"PO/ID/Exact/MultipleInstrumentIV","t":1,"d":"Coerce a binary treatment indicator to the real values 0 and 1.","s":"Bool → ℝ","l":61,"f":"Causalean/PO/ID/Exact/MultipleInstrumentIV/ResponseTypes.lean"},{"n":"Causalean.PO.ID.Exact.MultipleInstrumentIV.exists_negativeNormalizedTypeWeight","k":"theorem","m":"Causalean.PO.ID.Exact.MultipleInstrumentIV.ResponseTypes","a":"PO/ID/Exact/MultipleInstrumentIV","t":1,"d":"Negative-weights theorem. There exists a finite-support instrument index, a response-type population, and a response type such that, with two support points and a 3/4-defier population, that type has positive mass yet a negative normalized response-type weight (equal to −1/2).","s":"∃ I R g, 0 < R.mass g ∧ Causalean.PO.ID.Exact.MultipleInstrumentIV.ResponseTypeStats.normalizedTypeWeight I R g < 0","l":594,"f":"Causalean/PO/ID/Exact/MultipleInstrumentIV/ResponseTypes.lean"},{"n":"Causalean.PO.ID.Exact.MultipleInstrumentIV.firstStageGram","k":"def","m":"Causalean.PO.ID.Exact.MultipleInstrumentIV.FiniteIndex","a":"PO/ID/Exact/MultipleInstrumentIV","t":1,"d":"Finite second-moment matrix E[q(Z)q(Z)ᵀ] for a score vector on finite instrument support.","s":"{K L : ℕ} → (Fin K → ℝ) → (Fin K → Fin L → ℝ) → Matrix (Fin L) (Fin L) ℝ","l":352,"f":"Causalean/PO/ID/Exact/MultipleInstrumentIV/FiniteIndex.lean"},{"n":"Causalean.PO.ID.Exact.MultipleInstrumentIV.supportMass","k":"def","m":"Causalean.PO.ID.Exact.MultipleInstrumentIV.FiniteIndex","a":"PO/ID/Exact/MultipleInstrumentIV","t":1,"d":"Measure-backed finite support mass P(Z = k) for a Fin K-valued instrument.","s":"{Ω : Type u_1} → [inst : MeasurableSpace Ω] → {K : ℕ} → MeasureTheory.Measure Ω → (Ω → Fin K) → Fin K → ℝ","l":77,"f":"Causalean/PO/ID/Exact/MultipleInstrumentIV/FiniteIndex.lean"},{"n":"Causalean.PO.ID.Exact.MultipleInstrumentIV.supportMass_nonneg","k":"theorem","m":"Causalean.PO.ID.Exact.MultipleInstrumentIV.FiniteIndex","a":"PO/ID/Exact/MultipleInstrumentIV","t":2,"d":"Support masses obtained from a measure are nonnegative.","s":"∀ {Ω : Type u_1} [inst : MeasurableSpace Ω] {K : ℕ} (μ : MeasureTheory.Measure Ω) (Z : Ω → Fin K) (k : Fin K), 0 ≤ Causalean.PO.ID.Exact.MultipleInstrumentIV.supportMass μ Z k","l":83,"f":"Causalean/PO/ID/Exact/MultipleInstrumentIV/FiniteIndex.lean"},{"n":"Causalean.PO.ID.Exact.MultipleInstrumentIV.supportMass_sum_eq_one","k":"theorem","m":"Causalean.PO.ID.Exact.MultipleInstrumentIV.FiniteIndex","a":"PO/ID/Exact/MultipleInstrumentIV","t":2,"d":"For a probability measure, the finite support masses induced by Z sum to one.","s":"∀ {Ω : Type u_1} [inst : MeasurableSpace Ω] {K : ℕ} (μ : MeasureTheory.Measure Ω) [MeasureTheory.IsProbabilityMeasure μ] (Z : Ω → Fin K), Measurable Z → ∑ k, Causalean.PO.ID.Exact.MultipleInstrumentIV.supportMass μ Z k = 1","l":89,"f":"Causalean/PO/ID/Exact/MultipleInstrumentIV/FiniteIndex.lean"},{"n":"Causalean.PO.ID.Exact.MultipleInstrumentIV.typeStep","k":"def","m":"Causalean.PO.ID.Exact.MultipleInstrumentIV.ResponseTypes","a":"PO/ID/Exact/MultipleInstrumentIV","t":1,"d":"Adjacent response increment d_j - d_{j-1} for a response type.","s":"{K : ℕ} → Causalean.PO.ID.Exact.MultipleInstrumentIV.ResponseType K → Causalean.PO.ID.Exact.MultipleInstrumentIV.Adj K → ℝ","l":65,"f":"Causalean/PO/ID/Exact/MultipleInstrumentIV/ResponseTypes.lean"},{"n":"Causalean.PO.ID.Exact.VariableIntensityIV.OrderedTreatment.Crossing","k":"def","m":"Causalean.PO.ID.Exact.VariableIntensityIV.OrderedTreatment","a":"PO/ID/Exact/VariableIntensityIV","t":1,"d":"A movement from a to b crosses margin j → j+1.","s":"{J : ℕ} → Fin (J + 1) → Fin (J + 1) → Fin J → Prop","l":51,"f":"Causalean/PO/ID/Exact/VariableIntensityIV/OrderedTreatment.lean"},{"n":"Causalean.PO.ID.Exact.VariableIntensityIV.OrderedTreatment.crossingIndicator","k":"def","m":"Causalean.PO.ID.Exact.VariableIntensityIV.OrderedTreatment","a":"PO/ID/Exact/VariableIntensityIV","t":1,"d":"Real-valued indicator of Crossing a b j.","s":"{J : ℕ} → Fin (J + 1) → Fin (J + 1) → Fin J → ℝ","l":55,"f":"Causalean/PO/ID/Exact/VariableIntensityIV/OrderedTreatment.lean"},{"n":"Causalean.PO.ID.Exact.VariableIntensityIV.OrderedTreatment.intensityValue","k":"def","m":"Causalean.PO.ID.Exact.VariableIntensityIV.OrderedTreatment","a":"PO/ID/Exact/VariableIntensityIV","t":1,"d":"Numeric value of an ordered treatment level.","s":"{J : ℕ} → Fin (J + 1) → ℝ","l":43,"f":"Causalean/PO/ID/Exact/VariableIntensityIV/OrderedTreatment.lean"},{"n":"Causalean.PO.ID.Exact.VariableIntensityIV.OrderedTreatment.lowerLevel","k":"def","m":"Causalean.PO.ID.Exact.VariableIntensityIV.OrderedTreatment","a":"PO/ID/Exact/VariableIntensityIV","t":1,"d":"Treatment intensity level j, the lower endpoint of margin j → j+1.","s":"{J : ℕ} → Fin J → Fin (J + 1)","l":35,"f":"Causalean/PO/ID/Exact/VariableIntensityIV/OrderedTreatment.lean"},{"n":"Causalean.PO.ID.Exact.VariableIntensityIV.OrderedTreatment.marginIncrement","k":"def","m":"Causalean.PO.ID.Exact.VariableIntensityIV.OrderedTreatment","a":"PO/ID/Exact/VariableIntensityIV","t":1,"d":"The margin increment f(j+1) - f(j) for j : Fin J.","s":"{J : ℕ} → (Fin (J + 1) → ℝ) → Fin J → ℝ","l":47,"f":"Causalean/PO/ID/Exact/VariableIntensityIV/OrderedTreatment.lean"},{"n":"Causalean.PO.ID.Exact.VariableIntensityIV.OrderedTreatment.normalizedWeight","k":"def","m":"Causalean.PO.ID.Exact.VariableIntensityIV.OrderedTreatment","a":"PO/ID/Exact/VariableIntensityIV","t":1,"d":"Generic normalized finite weight a_i / Σ_k a_k.","s":"{ι : Type u_1} → [Fintype ι] → (ι → ℝ) → ι → ℝ","l":117,"f":"Causalean/PO/ID/Exact/VariableIntensityIV/OrderedTreatment.lean"},{"n":"Causalean.PO.ID.Exact.VariableIntensityIV.OrderedTreatment.normalizedWeight_nonneg","k":"theorem","m":"Causalean.PO.ID.Exact.VariableIntensityIV.OrderedTreatment","a":"PO/ID/Exact/VariableIntensityIV","t":2,"d":"Nonnegativity of normalized weights from nonnegative raw weights and a positive normalizing sum.","s":"∀ {ι : Type u_1} [inst : Fintype ι] (a : ι → ℝ), (∀ (i : ι), 0 ≤ a i) → 0 < ∑ i, a i → ∀ (i : ι), 0 ≤ Causalean.PO.ID.Exact.VariableIntensityIV.OrderedTreatment.normalizedWeight a i","l":121,"f":"Causalean/PO/ID/Exact/VariableIntensityIV/OrderedTreatment.lean"},{"n":"Causalean.PO.ID.Exact.VariableIntensityIV.OrderedTreatment.ordered_telescope_identity","k":"theorem","m":"Causalean.PO.ID.Exact.VariableIntensityIV.OrderedTreatment","a":"PO/ID/Exact/VariableIntensityIV","t":1,"d":"Ordered telescoping for the identity intensity map. For an ordered treatment level a no larger than b among J + 1 ordered intensity levels, the numeric gap b − a equals the number of unit margins j → j+1 that the movement from a to b crosses.","s":"∀ {J : ℕ} {a b : Fin (J + 1)}, a ≤ b → Causalean.PO.ID.Exact.VariableIntensityIV.OrderedTreatment.intensityValue b - Causalean.PO.ID.Exact.VariableIntensityIV.OrderedTreatment.intensityValue a = ∑ j, Causalean.PO.ID.Exact.VariableIntensityIV.OrderedTreatment.crossingIndicator a b j","l":108,"f":"Causalean/PO/ID/Exact/VariableIntensityIV/OrderedTreatment.lean"},{"n":"Causalean.PO.ID.Exact.VariableIntensityIV.OrderedTreatment.ordered_telescope_indicator","k":"theorem","m":"Causalean.PO.ID.Exact.VariableIntensityIV.OrderedTreatment","a":"PO/ID/Exact/VariableIntensityIV","t":2,"d":"Ordered telescoping across crossed margins for an arbitrary real-valued function on finite ordered levels.","s":"∀ {J : ℕ} (f : Fin (J + 1) → ℝ) {a b : Fin (J + 1)}, a ≤ b → f b - f a = ∑ j, Causalean.PO.ID.Exact.VariableIntensityIV.OrderedTreatment.marginIncrement f j * Causalean.PO.ID.Exact.VariableIntensityIV.OrderedTreatment.crossingIndicator a b j","l":78,"f":"Causalean/PO/ID/Exact/VariableIntensityIV/OrderedTreatment.lean"},{"n":"Causalean.PO.ID.Exact.VariableIntensityIV.OrderedTreatment.sum_normalizedWeight_eq_one","k":"theorem","m":"Causalean.PO.ID.Exact.VariableIntensityIV.OrderedTreatment","a":"PO/ID/Exact/VariableIntensityIV","t":2,"d":"Normalized finite weights sum to one when the normalizing sum is positive.","s":"∀ {ι : Type u_1} [inst : Fintype ι] (a : ι → ℝ), 0 < ∑ i, a i → ∑ i, Causalean.PO.ID.Exact.VariableIntensityIV.OrderedTreatment.normalizedWeight a i = 1","l":128,"f":"Causalean/PO/ID/Exact/VariableIntensityIV/OrderedTreatment.lean"},{"n":"Causalean.PO.ID.Exact.VariableIntensityIV.OrderedTreatment.upperLevel","k":"def","m":"Causalean.PO.ID.Exact.VariableIntensityIV.OrderedTreatment","a":"PO/ID/Exact/VariableIntensityIV","t":1,"d":"Treatment intensity level j+1, the upper endpoint of margin j → j+1.","s":"{J : ℕ} → Fin J → Fin (J + 1)","l":39,"f":"Causalean/PO/ID/Exact/VariableIntensityIV/OrderedTreatment.lean"},{"n":"Causalean.PO.ID.Exact.VariableIntensityIV.VariableIntensityIVSystem","k":"structure","m":"Causalean.PO.ID.Exact.VariableIntensityIV.VariableIntensity","a":"PO/ID/Exact/VariableIntensityIV","t":1,"d":"A variable-intensity IV system records an instrument, an ordered treatment intensity, and an outcome inside a potential-outcome system, with the treatment taking at least one margin.","s":"Causalean.PO.POSystem → (𝒵 : Type u_1) → [inst : MeasurableSpace 𝒵] → [Fintype 𝒵] → [MeasurableSingletonClass 𝒵] → ℕ → Type (max (max u_1 u_2) u_3)","l":43,"f":"Causalean/PO/ID/Exact/VariableIntensityIV/VariableIntensity.lean"},{"n":"Causalean.PO.ID.Exact.VariableIntensityIV.VariableIntensityIVSystem.DofZ","k":"def","m":"Causalean.PO.ID.Exact.VariableIntensityIV.VariableIntensity","a":"PO/ID/Exact/VariableIntensityIV","t":1,"d":"Potential treatment intensity D(z): the intensity that would be observed if the instrument were fixed to z. A genuine single-intervention counterfactual.","s":"{P : Causalean.PO.POSystem} → {𝒵 : Type u_1} → [inst : MeasurableSpace 𝒵] → [inst_1 : Fintype 𝒵] → [inst_2 : MeasurableSingletonClass 𝒵] → {J : ℕ} → Causalean.PO.ID.Exact.VariableIntensityIV.VariableIntensityIVSystem P 𝒵 J → 𝒵 → P.Ω → Fin (J + 1)","l":90,"f":"Causalean/PO/ID/Exact/VariableIntensityIV/VariableIntensity.lean"},{"n":"Causalean.PO.ID.Exact.VariableIntensityIV.VariableIntensityIVSystem.DofZ_eq_factualD_on_zEvent","k":"theorem","m":"Causalean.PO.ID.Exact.VariableIntensityIV.VariableIntensity","a":"PO/ID/Exact/VariableIntensityIV","t":2,"d":"On zEvent z, the counterfactual intensity D(z) equals the factual D.","s":"∀ {P : Causalean.PO.POSystem} {𝒵 : Type u_1} [inst : MeasurableSpace 𝒵] [inst_1 : Fintype 𝒵] [inst_2 : MeasurableSingletonClass 𝒵] {J : ℕ} (S : Causalean.PO.ID.Exact.VariableIntensityIV.VariableIntensityIVSystem P 𝒵 J), P.Consistency → ∀ (z : 𝒵) {ω : P.Ω}, ω ∈ S.zEvent z → S.DofZ z ω = S.factualD ω","l":377,"f":"Causalean/PO/ID/Exact/VariableIntensityIV/VariableIntensity.lean"},{"n":"Causalean.PO.ID.Exact.VariableIntensityIV.VariableIntensityIVSystem.PopulationTwoSLSScore","k":"structure","m":"Causalean.PO.ID.Exact.VariableIntensityIV.VariableIntensity","a":"PO/ID/Exact/VariableIntensityIV","t":1,"d":"Centered finite instrument score used to define an interface-only population 2SLS estimand for multivalued instruments.","s":"{P : Causalean.PO.POSystem} → {𝒵 : Type u_1} → [inst : MeasurableSpace 𝒵] → [inst_1 : Fintype 𝒵] → [inst_2 : MeasurableSingletonClass 𝒵] → {J : ℕ} → Causalean.PO.ID.Exact.VariableIntensityIV.VariableIntensityIVSystem P 𝒵 J → ℕ → Type u_1","l":881,"f":"Causalean/PO/ID/Exact/VariableIntensityIV/VariableIntensity.lean"},{"n":"Causalean.PO.ID.Exact.VariableIntensityIV.VariableIntensityIVSystem.PopulationTwoSLSScore.beta2SLS","k":"def","m":"Causalean.PO.ID.Exact.VariableIntensityIV.VariableIntensity","a":"PO/ID/Exact/VariableIntensityIV","t":1,"d":"Population 2SLS estimand β_2SLS(S) = E[D_S Y] / E[D_S D]. This declaration is only the population ratio interface: the file does not derive the binary-instrument bridge beta2SLS T = wald z0 z1. Such a bridge would require centering algebra and the scalar FWL identity for a score of the form S(Z) = Z - E[Z].","s":"{k : ℕ} → {P : Causalean.PO.POSystem} → {𝒵 : Type u_1} → [inst : MeasurableSpace 𝒵] → [inst_1 : Fintype 𝒵] → [inst_2 : MeasurableSingletonClass 𝒵] → {J : ℕ} → {S : Causalean.PO.ID.Exact.VariableIntensityIV.VariableIntensityIVSystem P 𝒵 J} → S.PopulationTwoSLSScore k → ℝ","l":904,"f":"Causalean/PO/ID/Exact/VariableIntensityIV/VariableIntensity.lean"},{"n":"Causalean.PO.ID.Exact.VariableIntensityIV.VariableIntensityIVSystem.PopulationTwoSLSScore.fittedTreatment","k":"def","m":"Causalean.PO.ID.Exact.VariableIntensityIV.VariableIntensity","a":"PO/ID/Exact/VariableIntensityIV","t":1,"d":"First-stage fitted treatment D_S(ω) = γ_D^T S(Z(ω)); the linear projection of D onto the instrument score.","s":"{k : ℕ} → {P : Causalean.PO.POSystem} → {𝒵 : Type u_1} → [inst : MeasurableSpace 𝒵] → [inst_1 : Fintype 𝒵] → [inst_2 : MeasurableSingletonClass 𝒵] → {J : ℕ} → {S : Causalean.PO.ID.Exact.VariableIntensityIV.VariableIntensityIVSystem P 𝒵 J} → S.PopulationTwoSLSScore k → P.Ω → ℝ","l":899,"f":"Causalean/PO/ID/Exact/VariableIntensityIV/VariableIntensity.lean"},{"n":"Causalean.PO.ID.Exact.VariableIntensityIV.VariableIntensityIVSystem.SpecialCases.binaryMargin","k":"def","m":"Causalean.PO.ID.Exact.VariableIntensityIV.VariableIntensity","a":"PO/ID/Exact/VariableIntensityIV","t":1,"d":"Canonical single treatment margin transported from Fin 1 under J = 1.","s":"{J : ℕ} → J = 1 → Fin J","l":781,"f":"Causalean/PO/ID/Exact/VariableIntensityIV/VariableIntensity.lean"},{"n":"Causalean.PO.ID.Exact.VariableIntensityIV.VariableIntensityIVSystem.SpecialCases.crossingEvent_eq_complianceEvent","k":"theorem","m":"Causalean.PO.ID.Exact.VariableIntensityIV.VariableIntensity","a":"PO/ID/Exact/VariableIntensityIV","t":2,"d":"Under J = 1, the unique crossing event coincides with the complier event {D(z1) = Fin.last J ∧ D(z0) = 0}, i.e. the unit jump from level 0 to the maximum level. When J = 1, Fin.last 1 = 1 : Fin 2, so this recovers exactly the binary-treatment LATE complier event {D(z1) = 1 ∧ D(z0) = 0} of Imbens-Angrist (1994): β_Wald(z0,z1) = E[Y(1) − Y(0) | D(z1) = Fin.last J ∧ D(z0) = 0].","s":"∀ {P : Causalean.PO.POSystem} {𝒵 : Type u_1} [inst : MeasurableSpace 𝒵] [inst_1 : Fintype 𝒵] [inst_2 : MeasurableSingletonClass 𝒵] {J : ℕ} (S : Causalean.PO.ID.Exact.VariableIntensityIV.VariableIntensityIVSystem P 𝒵 J) (z0 z1 : 𝒵) (hJ : J = 1), S.crossingEvent z0 z1 (Causalean.PO.ID.Exact.VariableIntensityIV.VariableIntensityIVSystem.SpecialCases.binaryMargin hJ) = {ω | S.DofZ z1 ω = Fin.last J ∧ S.DofZ z0 ω = 0}","l":805,"f":"Causalean/PO/ID/Exact/VariableIntensityIV/VariableIntensity.lean"},{"n":"Causalean.PO.ID.Exact.VariableIntensityIV.VariableIntensityIVSystem.SpecialCases.wald_eq_constantResponse","k":"theorem","m":"Causalean.PO.ID.Exact.VariableIntensityIV.VariableIntensity","a":"PO/ID/Exact/VariableIntensityIV","t":2,"d":"Constant marginal response specialization.","s":"∀ {P : Causalean.PO.POSystem} {𝒵 : Type u_1} [inst : MeasurableSpace 𝒵] [inst_1 : Fintype 𝒵] [inst_2 : MeasurableSingletonClass 𝒵] {J : ℕ} (S : Causalean.PO.ID.Exact.VariableIntensityIV.VariableIntensityIVSystem P 𝒵 J) {z0 z1 : 𝒵} {τ : ℝ}, S.ValidContrastAssumptions z0 z1 → 0 < (P.μ (S.zEvent z0)).toReal → 0 < (P.μ (S.zEvent z1)).toReal → (∀ (j : Fin J), S.marginResponse j =ᵐ[P.μ] fun x => τ) → S.wald z0 z1 = τ","l":837,"f":"Causalean/PO/ID/Exact/VariableIntensityIV/VariableIntensity.lean"},{"n":"Causalean.PO.ID.Exact.VariableIntensityIV.VariableIntensityIVSystem.SpecialCases.wald_eq_late_of_binaryIntensity","k":"theorem","m":"Causalean.PO.ID.Exact.VariableIntensityIV.VariableIntensity","a":"PO/ID/Exact/VariableIntensityIV","t":1,"d":"Binary-intensity specialization: Wald recovers LATE. Under the variable-intensity IV validity assumptions, with positive probability of the instrument cell Z = z0, positive probability of the instrument cell Z = z1, and a single treatment margin, J = 1, the directed Wald estimand equals the conditional mean unit causal response given the unique crossing event — the classical binary-treatment local average treatment effect.","s":"∀ {P : Causalean.PO.POSystem} {𝒵 : Type u_1} [inst : MeasurableSpace 𝒵] [inst_1 : Fintype 𝒵] [inst_2 : MeasurableSingletonClass 𝒵] {J : ℕ} (S : Causalean.PO.ID.Exact.VariableIntensityIV.VariableIntensityIVSystem P 𝒵 J) {z0 z1 : 𝒵}, S.ValidContrastAssumptions z0 z1 → 0 < (P.μ (S.zEvent z0)).toReal → 0 < (P.μ (S.zEvent z1)).toReal → ∀ (hBinaryIntensity : J = 1), S.wald z0 z1 = Causalean.PO.eventCondExp P.μ (S.crossingEvent z0 z1 (Causalean.PO.ID.Exact.VariableIntensityIV.VariableIntensityIVSystem.SpecialCases.binaryMargin hBinaryIntensity)) (S.marginResponse (Causalean.PO.ID.Exact.VariableIntensityIV.VariableIntensityIVSystem.SpecialCases.binaryMargin hBinaryIntensity))","l":785,"f":"Causalean/PO/ID/Exact/VariableIntensityIV/VariableIntensity.lean"},{"n":"Causalean.PO.ID.Exact.VariableIntensityIV.VariableIntensityIVSystem.SpecialCases.wald_eq_marginResponseAverage","k":"theorem","m":"Causalean.PO.ID.Exact.VariableIntensityIV.VariableIntensity","a":"PO/ID/Exact/VariableIntensityIV","t":1,"d":"Margin-specific response average specialization. Under the variable-intensity IV validity assumptions, with positive probability of the instrument cell Z = z0, positive probability of the instrument cell Z = z1, and a candidate margin-response function m that agrees, on each treatment-intensity margin, with the conditional mean causal response given that margin's crossing event, the directed Wald estimand equals the crossing-probability-weighted average of m across margins.","s":"∀ {P : Causalean.PO.POSystem} {𝒵 : Type u_1} [inst : MeasurableSpace 𝒵] [inst_1 : Fintype 𝒵] [inst_2 : MeasurableSingletonClass 𝒵] {J : ℕ} (S : Causalean.PO.ID.Exact.VariableIntensityIV.VariableIntensityIVSystem P 𝒵 J) {z0 z1 : 𝒵} (m : Fin J → ℝ), S.ValidContrastAssumptions z0 z1 → 0 < (P.μ (S.zEvent z0)).toReal → 0 < (P.μ (S.zEvent z1)).toReal → (∀ (j : Fin J), m j = S.conditionalMarginResponse z0 z1 j) → S.wald z0 z1 = ∑ j, S.crossingWeight z0 z1 j * m j","l":863,"f":"Causalean/PO/ID/Exact/VariableIntensityIV/VariableIntensity.lean"},{"n":"Causalean.PO.ID.Exact.VariableIntensityIV.VariableIntensityIVSystem.TwoSLSContrastDecomposition","k":"structure","m":"Causalean.PO.ID.Exact.VariableIntensityIV.VariableIntensity","a":"PO/ID/Exact/VariableIntensityIV","t":1,"d":"Deferred interface for expanding a multivalued-instrument 2SLS estimand into finite directed contrasts. A later theorem can add sign-alignment assumptions to turn this signed decomposition into a convex ACR average.","s":"{P : Causalean.PO.POSystem} → {𝒵 : Type u_1} → [inst : MeasurableSpace 𝒵] → [inst_1 : Fintype 𝒵] → [inst_2 : MeasurableSingletonClass 𝒵] → {J : ℕ} → (S : Causalean.PO.ID.Exact.VariableIntensityIV.VariableIntensityIVSystem P 𝒵 J) → {k : ℕ} → S.PopulationTwoSLSScore k → Type u_1","l":915,"f":"Causalean/PO/ID/Exact/VariableIntensityIV/VariableIntensity.lean"},{"n":"Causalean.PO.ID.Exact.VariableIntensityIV.VariableIntensityIVSystem.ValidContrastAssumptions","k":"structure","m":"Causalean.PO.ID.Exact.VariableIntensityIV.VariableIntensity","a":"PO/ID/Exact/VariableIntensityIV","t":1,"d":"IV-validity assumptions for a fixed directed contrast (z0,z1).","s":"{P : Causalean.PO.POSystem} → {𝒵 : Type u_1} → [inst : MeasurableSpace 𝒵] → [inst_1 : Fintype 𝒵] → [inst_2 : MeasurableSingletonClass 𝒵] → {J : ℕ} → Causalean.PO.ID.Exact.VariableIntensityIV.VariableIntensityIVSystem P 𝒵 J → 𝒵 → 𝒵 → Prop","l":319,"f":"Causalean/PO/ID/Exact/VariableIntensityIV/VariableIntensity.lean"},{"n":"Causalean.PO.ID.Exact.VariableIntensityIV.VariableIntensityIVSystem.YofD","k":"def","m":"Causalean.PO.ID.Exact.VariableIntensityIV.VariableIntensity","a":"PO/ID/Exact/VariableIntensityIV","t":1,"d":"Treatment-indexed potential outcome Y(d): the outcome that would be observed if the treatment intensity were fixed to d. No instrument argument enters, so exclusion is structural.","s":"{P : Causalean.PO.POSystem} → {𝒵 : Type u_1} → [inst : MeasurableSpace 𝒵] → [inst_1 : Fintype 𝒵] → [inst_2 : MeasurableSingletonClass 𝒵] → {J : ℕ} → Causalean.PO.ID.Exact.VariableIntensityIV.VariableIntensityIVSystem P 𝒵 J → Fin (J + 1) → P.Ω → ℝ","l":94,"f":"Causalean/PO/ID/Exact/VariableIntensityIV/VariableIntensity.lean"},{"n":"Causalean.PO.ID.Exact.VariableIntensityIV.VariableIntensityIVSystem.YofDofZ","k":"def","m":"Causalean.PO.ID.Exact.VariableIntensityIV.VariableIntensity","a":"PO/ID/Exact/VariableIntensityIV","t":1,"d":"Potential outcome Y(D(z)): the outcome if the instrument were set to z. Equals Y(d) at d = D(z)(ω). No direct instrument effect enters because YofD fixes only the treatment intensity d, not z (structural exclusion).","s":"{P : Causalean.PO.POSystem} → {𝒵 : Type u_1} → [inst : MeasurableSpace 𝒵] → [inst_1 : Fintype 𝒵] → [inst_2 : MeasurableSingletonClass 𝒵] → {J : ℕ} → Causalean.PO.ID.Exact.VariableIntensityIV.VariableIntensityIVSystem P 𝒵 J → 𝒵 → P.Ω → ℝ","l":267,"f":"Causalean/PO/ID/Exact/VariableIntensityIV/VariableIntensity.lean"},{"n":"Causalean.PO.ID.Exact.VariableIntensityIV.VariableIntensityIVSystem.averageCausalResponse","k":"def","m":"Causalean.PO.ID.Exact.VariableIntensityIV.VariableIntensity","a":"PO/ID/Exact/VariableIntensityIV","t":1,"d":"Angrist-Imbens average causal response: Σ_j ω_j E[Δ_jY | C_j].","s":"{P : Causalean.PO.POSystem} → {𝒵 : Type u_1} → [inst : MeasurableSpace 𝒵] → [inst_1 : Fintype 𝒵] → [inst_2 : MeasurableSingletonClass 𝒵] → {J : ℕ} → Causalean.PO.ID.Exact.VariableIntensityIV.VariableIntensityIVSystem P 𝒵 J → 𝒵 → 𝒵 → ℝ","l":262,"f":"Causalean/PO/ID/Exact/VariableIntensityIV/VariableIntensity.lean"},{"n":"Causalean.PO.ID.Exact.VariableIntensityIV.VariableIntensityIVSystem.cfContrastBundle","k":"def","m":"Causalean.PO.ID.Exact.VariableIntensityIV.VariableIntensity","a":"PO/ID/Exact/VariableIntensityIV","t":1,"d":"Counterfactual bundle (D(z0), D(z1), Y(0),…,Y(J)) for the contrast (z0, z1). Index 0 is D(z0), index 1 is D(z1), index d+2 is Y(d).","s":"{P : Causalean.PO.POSystem} → {𝒵 : Type u_1} → [inst : MeasurableSpace 𝒵] → [inst_1 : Fintype 𝒵] → [inst_2 : MeasurableSingletonClass 𝒵] → {J : ℕ} → Causalean.PO.ID.Exact.VariableIntensityIV.VariableIntensityIVSystem P 𝒵 J → 𝒵 → 𝒵 → Causalean.PO.POCFBundle P","l":155,"f":"Causalean/PO/ID/Exact/VariableIntensityIV/VariableIntensity.lean"},{"n":"Causalean.PO.ID.Exact.VariableIntensityIV.VariableIntensityIVSystem.condExpDZ","k":"def","m":"Causalean.PO.ID.Exact.VariableIntensityIV.VariableIntensity","a":"PO/ID/Exact/VariableIntensityIV","t":1,"d":"E[D | Z = z] — observed first-stage conditional mean at instrument value z, defined via eventCondExp over the cell {Z = z}.","s":"{P : Causalean.PO.POSystem} → {𝒵 : Type u_1} → [inst : MeasurableSpace 𝒵] → [inst_1 : Fintype 𝒵] → [inst_2 : MeasurableSingletonClass 𝒵] → {J : ℕ} → Causalean.PO.ID.Exact.VariableIntensityIV.VariableIntensityIVSystem P 𝒵 J → 𝒵 → ℝ","l":303,"f":"Causalean/PO/ID/Exact/VariableIntensityIV/VariableIntensity.lean"},{"n":"Causalean.PO.ID.Exact.VariableIntensityIV.VariableIntensityIVSystem.condExpDZ_left_eq_integral","k":"theorem","m":"Causalean.PO.ID.Exact.VariableIntensityIV.VariableIntensity","a":"PO/ID/Exact/VariableIntensityIV","t":2,"d":"First-stage bridge (left cell z0): the observed first-stage conditional mean equals the unconditional expectation of the potential intensity D(z0).","s":"∀ {P : Causalean.PO.POSystem} {𝒵 : Type u_1} [inst : MeasurableSpace 𝒵] [inst_1 : Fintype 𝒵] [inst_2 : MeasurableSingletonClass 𝒵] {J : ℕ} (S : Causalean.PO.ID.Exact.VariableIntensityIV.VariableIntensityIVSystem P 𝒵 J) {z0 z1 : 𝒵}, S.ValidContrastAssumptions z0 z1 → 0 < (P.μ (S.zEvent z0)).toReal → S.condExpDZ z0 = ∫ (ω : P.Ω), Causalean.PO.ID.Exact.VariableIntensityIV.OrderedTreatment.intensityValue (S.DofZ z0 ω) ∂P.μ","l":388,"f":"Causalean/PO/ID/Exact/VariableIntensityIV/VariableIntensity.lean"},{"n":"Causalean.PO.ID.Exact.VariableIntensityIV.VariableIntensityIVSystem.condExpDZ_right_eq_integral","k":"theorem","m":"Causalean.PO.ID.Exact.VariableIntensityIV.VariableIntensity","a":"PO/ID/Exact/VariableIntensityIV","t":2,"d":"First-stage bridge (right cell z1).","s":"∀ {P : Causalean.PO.POSystem} {𝒵 : Type u_1} [inst : MeasurableSpace 𝒵] [inst_1 : Fintype 𝒵] [inst_2 : MeasurableSingletonClass 𝒵] {J : ℕ} (S : Causalean.PO.ID.Exact.VariableIntensityIV.VariableIntensityIVSystem P 𝒵 J) {z0 z1 : 𝒵}, S.ValidContrastAssumptions z0 z1 → 0 < (P.μ (S.zEvent z1)).toReal → S.condExpDZ z1 = ∫ (ω : P.Ω), Causalean.PO.ID.Exact.VariableIntensityIV.OrderedTreatment.intensityValue (S.DofZ z1 ω) ∂P.μ","l":431,"f":"Causalean/PO/ID/Exact/VariableIntensityIV/VariableIntensity.lean"},{"n":"Causalean.PO.ID.Exact.VariableIntensityIV.VariableIntensityIVSystem.condExpYZ","k":"def","m":"Causalean.PO.ID.Exact.VariableIntensityIV.VariableIntensity","a":"PO/ID/Exact/VariableIntensityIV","t":1,"d":"E[Y | Z = z] — observed reduced-form conditional mean at instrument value z, defined via eventCondExp over the cell {Z = z}.","s":"{P : Causalean.PO.POSystem} → {𝒵 : Type u_1} → [inst : MeasurableSpace 𝒵] → [inst_1 : Fintype 𝒵] → [inst_2 : MeasurableSingletonClass 𝒵] → {J : ℕ} → Causalean.PO.ID.Exact.VariableIntensityIV.VariableIntensityIVSystem P 𝒵 J → 𝒵 → ℝ","l":309,"f":"Causalean/PO/ID/Exact/VariableIntensityIV/VariableIntensity.lean"},{"n":"Causalean.PO.ID.Exact.VariableIntensityIV.VariableIntensityIVSystem.condExpYZ_left_eq_integral","k":"theorem","m":"Causalean.PO.ID.Exact.VariableIntensityIV.VariableIntensity","a":"PO/ID/Exact/VariableIntensityIV","t":2,"d":"Reduced-form bridge (left cell z0): the observed reduced-form conditional mean equals the unconditional expectation of Y(D(z0)).","s":"∀ {P : Causalean.PO.POSystem} {𝒵 : Type u_1} [inst : MeasurableSpace 𝒵] [inst_1 : Fintype 𝒵] [inst_2 : MeasurableSingletonClass 𝒵] {J : ℕ} (S : Causalean.PO.ID.Exact.VariableIntensityIV.VariableIntensityIVSystem P 𝒵 J) {z0 z1 : 𝒵}, S.ValidContrastAssumptions z0 z1 → 0 < (P.μ (S.zEvent z0)).toReal → S.condExpYZ z0 = ∫ (ω : P.Ω), S.YofDofZ z0 ω ∂P.μ","l":475,"f":"Causalean/PO/ID/Exact/VariableIntensityIV/VariableIntensity.lean"},{"n":"Causalean.PO.ID.Exact.VariableIntensityIV.VariableIntensityIVSystem.condExpYZ_right_eq_integral","k":"theorem","m":"Causalean.PO.ID.Exact.VariableIntensityIV.VariableIntensity","a":"PO/ID/Exact/VariableIntensityIV","t":2,"d":"Reduced-form bridge (right cell z1).","s":"∀ {P : Causalean.PO.POSystem} {𝒵 : Type u_1} [inst : MeasurableSpace 𝒵] [inst_1 : Fintype 𝒵] [inst_2 : MeasurableSingletonClass 𝒵] {J : ℕ} (S : Causalean.PO.ID.Exact.VariableIntensityIV.VariableIntensityIVSystem P 𝒵 J) {z0 z1 : 𝒵}, S.ValidContrastAssumptions z0 z1 → 0 < (P.μ (S.zEvent z1)).toReal → S.condExpYZ z1 = ∫ (ω : P.Ω), S.YofDofZ z1 ω ∂P.μ","l":554,"f":"Causalean/PO/ID/Exact/VariableIntensityIV/VariableIntensity.lean"},{"n":"Causalean.PO.ID.Exact.VariableIntensityIV.VariableIntensityIVSystem.conditionalMarginResponse","k":"def","m":"Causalean.PO.ID.Exact.VariableIntensityIV.VariableIntensity","a":"PO/ID/Exact/VariableIntensityIV","t":1,"d":"The conditional margin response is the average unit-level causal response among units whose potential treatment crosses a selected treatment margin when the instrument changes along a selected ordered contrast.","s":"{P : Causalean.PO.POSystem} → {𝒵 : Type u_1} → [inst : MeasurableSpace 𝒵] → [inst_1 : Fintype 𝒵] → [inst_2 : MeasurableSingletonClass 𝒵] → {J : ℕ} → Causalean.PO.ID.Exact.VariableIntensityIV.VariableIntensityIVSystem P 𝒵 J → 𝒵 → 𝒵 → Fin J → ℝ","l":254,"f":"Causalean/PO/ID/Exact/VariableIntensityIV/VariableIntensity.lean"},{"n":"Causalean.PO.ID.Exact.VariableIntensityIV.VariableIntensityIVSystem.crossingEvent","k":"def","m":"Causalean.PO.ID.Exact.VariableIntensityIV.VariableIntensity","a":"PO/ID/Exact/VariableIntensityIV","t":1,"d":"Crossing event C_j(z0,z1) = {D(z1) ≥ j+1 > D(z0)}.","s":"{P : Causalean.PO.POSystem} → {𝒵 : Type u_1} → [inst : MeasurableSpace 𝒵] → [inst_1 : Fintype 𝒵] → [inst_2 : MeasurableSingletonClass 𝒵] → {J : ℕ} → Causalean.PO.ID.Exact.VariableIntensityIV.VariableIntensityIVSystem P 𝒵 J → 𝒵 → 𝒵 → Fin J → Set P.Ω","l":165,"f":"Causalean/PO/ID/Exact/VariableIntensityIV/VariableIntensity.lean"},{"n":"Causalean.PO.ID.Exact.VariableIntensityIV.VariableIntensityIVSystem.crossingProb","k":"def","m":"Causalean.PO.ID.Exact.VariableIntensityIV.VariableIntensity","a":"PO/ID/Exact/VariableIntensityIV","t":1,"d":"Crossing probability for a directed instrument contrast and margin.","s":"{P : Causalean.PO.POSystem} → {𝒵 : Type u_1} → [inst : MeasurableSpace 𝒵] → [inst_1 : Fintype 𝒵] → [inst_2 : MeasurableSingletonClass 𝒵] → {J : ℕ} → Causalean.PO.ID.Exact.VariableIntensityIV.VariableIntensityIVSystem P 𝒵 J → 𝒵 → 𝒵 → Fin J → ℝ","l":169,"f":"Causalean/PO/ID/Exact/VariableIntensityIV/VariableIntensity.lean"},{"n":"Causalean.PO.ID.Exact.VariableIntensityIV.VariableIntensityIVSystem.crossingWeight","k":"def","m":"Causalean.PO.ID.Exact.VariableIntensityIV.VariableIntensity","a":"PO/ID/Exact/VariableIntensityIV","t":1,"d":"Normalized Angrist-Imbens margin-crossing weight.","s":"{P : Causalean.PO.POSystem} → {𝒵 : Type u_1} → [inst : MeasurableSpace 𝒵] → [inst_1 : Fintype 𝒵] → [inst_2 : MeasurableSingletonClass 𝒵] → {J : ℕ} → Causalean.PO.ID.Exact.VariableIntensityIV.VariableIntensityIVSystem P 𝒵 J → 𝒵 → 𝒵 → Fin J → ℝ","l":236,"f":"Causalean/PO/ID/Exact/VariableIntensityIV/VariableIntensity.lean"},{"n":"Causalean.PO.ID.Exact.VariableIntensityIV.VariableIntensityIVSystem.crossingWeight_nonneg","k":"theorem","m":"Causalean.PO.ID.Exact.VariableIntensityIV.VariableIntensity","a":"PO/ID/Exact/VariableIntensityIV","t":2,"d":"Crossing weights are nonnegative for a valid directed contrast.","s":"∀ {P : Causalean.PO.POSystem} {𝒵 : Type u_1} [inst : MeasurableSpace 𝒵] [inst_1 : Fintype 𝒵] [inst_2 : MeasurableSingletonClass 𝒵] {J : ℕ} (S : Causalean.PO.ID.Exact.VariableIntensityIV.VariableIntensityIVSystem P 𝒵 J) {z0 z1 : 𝒵}, S.ValidContrastAssumptions z0 z1 → ∀ (j : Fin J), 0 ≤ S.crossingWeight z0 z1 j","l":664,"f":"Causalean/PO/ID/Exact/VariableIntensityIV/VariableIntensity.lean"},{"n":"Causalean.PO.ID.Exact.VariableIntensityIV.VariableIntensityIVSystem.dUnderZ","k":"def","m":"Causalean.PO.ID.Exact.VariableIntensityIV.VariableIntensity","a":"PO/ID/Exact/VariableIntensityIV","t":1,"d":"Regimed variable for D(z) (treatment intensity under instrument set to z).","s":"{P : Causalean.PO.POSystem} → {𝒵 : Type u_1} → [inst : MeasurableSpace 𝒵] → [inst_1 : Fintype 𝒵] → [inst_2 : MeasurableSingletonClass 𝒵] → {J : ℕ} → Causalean.PO.ID.Exact.VariableIntensityIV.VariableIntensityIVSystem P 𝒵 J → 𝒵 → Causalean.PO.RegimedVar P (Fin (J + 1))","l":139,"f":"Causalean/PO/ID/Exact/VariableIntensityIV/VariableIntensity.lean"},{"n":"Causalean.PO.ID.Exact.VariableIntensityIV.VariableIntensityIVSystem.dVar","k":"def","m":"Causalean.PO.ID.Exact.VariableIntensityIV.VariableIntensity","a":"PO/ID/Exact/VariableIntensityIV","t":1,"d":"Treatment intensity packaged as a POVar valued in Fin (J+1).","s":"{P : Causalean.PO.POSystem} → {𝒵 : Type u_1} → [inst : MeasurableSpace 𝒵] → [inst_1 : Fintype 𝒵] → [inst_2 : MeasurableSingletonClass 𝒵] → {J : ℕ} → Causalean.PO.ID.Exact.VariableIntensityIV.VariableIntensityIVSystem P 𝒵 J → Causalean.PO.POVar P (Fin (J + 1))","l":84,"f":"Causalean/PO/ID/Exact/VariableIntensityIV/VariableIntensity.lean"},{"n":"Causalean.PO.ID.Exact.VariableIntensityIV.VariableIntensityIVSystem.factualD","k":"def","m":"Causalean.PO.ID.Exact.VariableIntensityIV.VariableIntensity","a":"PO/ID/Exact/VariableIntensityIV","t":1,"d":"Factual treatment intensity D.","s":"{P : Causalean.PO.POSystem} → {𝒵 : Type u_1} → [inst : MeasurableSpace 𝒵] → [inst_1 : Fintype 𝒵] → [inst_2 : MeasurableSingletonClass 𝒵] → {J : ℕ} → Causalean.PO.ID.Exact.VariableIntensityIV.VariableIntensityIVSystem P 𝒵 J → P.Ω → Fin (J + 1)","l":102,"f":"Causalean/PO/ID/Exact/VariableIntensityIV/VariableIntensity.lean"},{"n":"Causalean.PO.ID.Exact.VariableIntensityIV.VariableIntensityIVSystem.factualY","k":"def","m":"Causalean.PO.ID.Exact.VariableIntensityIV.VariableIntensity","a":"PO/ID/Exact/VariableIntensityIV","t":1,"d":"Factual outcome Y.","s":"{P : Causalean.PO.POSystem} → {𝒵 : Type u_1} → [inst : MeasurableSpace 𝒵] → [inst_1 : Fintype 𝒵] → [inst_2 : MeasurableSingletonClass 𝒵] → {J : ℕ} → Causalean.PO.ID.Exact.VariableIntensityIV.VariableIntensityIVSystem P 𝒵 J → P.Ω → ℝ","l":105,"f":"Causalean/PO/ID/Exact/VariableIntensityIV/VariableIntensity.lean"},{"n":"Causalean.PO.ID.Exact.VariableIntensityIV.VariableIntensityIVSystem.factualY_eq_YofD_factualD","k":"theorem","m":"Causalean.PO.ID.Exact.VariableIntensityIV.VariableIntensity","a":"PO/ID/Exact/VariableIntensityIV","t":2,"d":"Factual Y equals the counterfactual Y(factualD ω).","s":"∀ {P : Causalean.PO.POSystem} {𝒵 : Type u_1} [inst : MeasurableSpace 𝒵] [inst_1 : Fintype 𝒵] [inst_2 : MeasurableSingletonClass 𝒵] {J : ℕ} (S : Causalean.PO.ID.Exact.VariableIntensityIV.VariableIntensityIVSystem P 𝒵 J), P.Consistency → ∀ (ω : P.Ω), S.factualY ω = S.YofD (S.factualD ω) ω","l":383,"f":"Causalean/PO/ID/Exact/VariableIntensityIV/VariableIntensity.lean"},{"n":"Causalean.PO.ID.Exact.VariableIntensityIV.VariableIntensityIVSystem.factualZ","k":"def","m":"Causalean.PO.ID.Exact.VariableIntensityIV.VariableIntensity","a":"PO/ID/Exact/VariableIntensityIV","t":1,"d":"Factual instrument Z.","s":"{P : Causalean.PO.POSystem} → {𝒵 : Type u_1} → [inst : MeasurableSpace 𝒵] → [inst_1 : Fintype 𝒵] → [inst_2 : MeasurableSingletonClass 𝒵] → {J : ℕ} → Causalean.PO.ID.Exact.VariableIntensityIV.VariableIntensityIVSystem P 𝒵 J → P.Ω → 𝒵","l":99,"f":"Causalean/PO/ID/Exact/VariableIntensityIV/VariableIntensity.lean"},{"n":"Causalean.PO.ID.Exact.VariableIntensityIV.VariableIntensityIVSystem.firstStageContrast","k":"def","m":"Causalean.PO.ID.Exact.VariableIntensityIV.VariableIntensity","a":"PO/ID/Exact/VariableIntensityIV","t":1,"d":"Potential first-stage contrast E[D(z1) − D(z0)] using potential treatment intensities.","s":"{P : Causalean.PO.POSystem} → {𝒵 : Type u_1} → [inst : MeasurableSpace 𝒵] → [inst_1 : Fintype 𝒵] → [inst_2 : MeasurableSingletonClass 𝒵] → {J : ℕ} → Causalean.PO.ID.Exact.VariableIntensityIV.VariableIntensityIVSystem P 𝒵 J → 𝒵 → 𝒵 → ℝ","l":293,"f":"Causalean/PO/ID/Exact/VariableIntensityIV/VariableIntensity.lean"},{"n":"Causalean.PO.ID.Exact.VariableIntensityIV.VariableIntensityIVSystem.firstStage_eq_sum_crossingProb","k":"theorem","m":"Causalean.PO.ID.Exact.VariableIntensityIV.VariableIntensity","a":"PO/ID/Exact/VariableIntensityIV","t":2,"d":"First-stage denominator equals the sum of crossing probabilities.","s":"∀ {P : Causalean.PO.POSystem} {𝒵 : Type u_1} [inst : MeasurableSpace 𝒵] [inst_1 : Fintype 𝒵] [inst_2 : MeasurableSingletonClass 𝒵] {J : ℕ} (S : Causalean.PO.ID.Exact.VariableIntensityIV.VariableIntensityIVSystem P 𝒵 J) {z0 z1 : 𝒵}, S.ValidContrastAssumptions z0 z1 → S.firstStageContrast z0 z1 = ∑ j, S.crossingProb z0 z1 j","l":632,"f":"Causalean/PO/ID/Exact/VariableIntensityIV/VariableIntensity.lean"},{"n":"Causalean.PO.ID.Exact.VariableIntensityIV.VariableIntensityIVSystem.indicatorWeightedACR","k":"def","m":"Causalean.PO.ID.Exact.VariableIntensityIV.VariableIntensity","a":"PO/ID/Exact/VariableIntensityIV","t":1,"d":"Indicator-weighted ACR ratio, avoiding partial conditional means in the core algebra.","s":"{P : Causalean.PO.POSystem} → {𝒵 : Type u_1} → [inst : MeasurableSpace 𝒵] → [inst_1 : Fintype 𝒵] → [inst_2 : MeasurableSingletonClass 𝒵] → {J : ℕ} → Causalean.PO.ID.Exact.VariableIntensityIV.VariableIntensityIVSystem P 𝒵 J → 𝒵 → 𝒵 → ℝ","l":249,"f":"Causalean/PO/ID/Exact/VariableIntensityIV/VariableIntensity.lean"},{"n":"Causalean.PO.ID.Exact.VariableIntensityIV.VariableIntensityIVSystem.indicatorWeightedACR_eq_averageCausalResponse","k":"theorem","m":"Causalean.PO.ID.Exact.VariableIntensityIV.VariableIntensity","a":"PO/ID/Exact/VariableIntensityIV","t":1,"d":"Indicator-weighted and conditional-mean ACR forms agree. For any pair of instrument values z0 and z1, the indicator-weighted average causal response — the ratio of the summed crossing-indicator-weighted outcome contrasts to the summed crossing probabilities — equals the crossing-probability- weighted average of the conditional-mean margin responses.","s":"∀ {P : Causalean.PO.POSystem} {𝒵 : Type u_1} [inst : MeasurableSpace 𝒵] [inst_1 : Fintype 𝒵] [inst_2 : MeasurableSingletonClass 𝒵] {J : ℕ} (S : Causalean.PO.ID.Exact.VariableIntensityIV.VariableIntensityIVSystem P 𝒵 J) {z0 z1 : 𝒵}, S.indicatorWeightedACR z0 z1 = S.averageCausalResponse z0 z1","l":726,"f":"Causalean/PO/ID/Exact/VariableIntensityIV/VariableIntensity.lean"},{"n":"Causalean.PO.ID.Exact.VariableIntensityIV.VariableIntensityIVSystem.indicatorWeightedEffect","k":"def","m":"Causalean.PO.ID.Exact.VariableIntensityIV.VariableIntensity","a":"PO/ID/Exact/VariableIntensityIV","t":1,"d":"Indicator-weighted margin effect E[Δ_jY · 1_{C_j}], represented as a set integral over the crossing event.","s":"{P : Causalean.PO.POSystem} → {𝒵 : Type u_1} → [inst : MeasurableSpace 𝒵] → [inst_1 : Fintype 𝒵] → [inst_2 : MeasurableSingletonClass 𝒵] → {J : ℕ} → Causalean.PO.ID.Exact.VariableIntensityIV.VariableIntensityIVSystem P 𝒵 J → 𝒵 → 𝒵 → Fin J → ℝ","l":240,"f":"Causalean/PO/ID/Exact/VariableIntensityIV/VariableIntensity.lean"},{"n":"Causalean.PO.ID.Exact.VariableIntensityIV.VariableIntensityIVSystem.integrable_marginResponse","k":"theorem","m":"Causalean.PO.ID.Exact.VariableIntensityIV.VariableIntensity","a":"PO/ID/Exact/VariableIntensityIV","t":2,"d":"Integrability of a margin response, derived from H7.","s":"∀ {P : Causalean.PO.POSystem} {𝒵 : Type u_1} [inst : MeasurableSpace 𝒵] [inst_1 : Fintype 𝒵] [inst_2 : MeasurableSingletonClass 𝒵] {J : ℕ} (S : Causalean.PO.ID.Exact.VariableIntensityIV.VariableIntensityIVSystem P 𝒵 J) {z0 z1 : 𝒵}, S.ValidContrastAssumptions z0 z1 → ∀ (j : Fin J), MeasureTheory.Integrable (S.marginResponse j) P.μ","l":343,"f":"Causalean/PO/ID/Exact/VariableIntensityIV/VariableIntensity.lean"},{"n":"Causalean.PO.ID.Exact.VariableIntensityIV.VariableIntensityIVSystem.marginResponse","k":"def","m":"Causalean.PO.ID.Exact.VariableIntensityIV.VariableIntensity","a":"PO/ID/Exact/VariableIntensityIV","t":1,"d":"Unit causal response on margin j → j+1: Δ_jY = Y(j+1) - Y(j).","s":"{P : Causalean.PO.POSystem} → {𝒵 : Type u_1} → [inst : MeasurableSpace 𝒵] → [inst_1 : Fintype 𝒵] → [inst_2 : MeasurableSingletonClass 𝒵] → {J : ℕ} → Causalean.PO.ID.Exact.VariableIntensityIV.VariableIntensityIVSystem P 𝒵 J → Fin J → P.Ω → ℝ","l":160,"f":"Causalean/PO/ID/Exact/VariableIntensityIV/VariableIntensity.lean"},{"n":"Causalean.PO.ID.Exact.VariableIntensityIV.VariableIntensityIVSystem.measurableSet_zEvent","k":"theorem","m":"Causalean.PO.ID.Exact.VariableIntensityIV.VariableIntensity","a":"PO/ID/Exact/VariableIntensityIV","t":2,"d":"The factual instrument cell is measurable.","s":"∀ {P : Causalean.PO.POSystem} {𝒵 : Type u_1} [inst : MeasurableSpace 𝒵] [inst_1 : Fintype 𝒵] [inst_2 : MeasurableSingletonClass 𝒵] {J : ℕ} (S : Causalean.PO.ID.Exact.VariableIntensityIV.VariableIntensityIVSystem P 𝒵 J) (z : 𝒵), MeasurableSet (S.zEvent z)","l":128,"f":"Causalean/PO/ID/Exact/VariableIntensityIV/VariableIntensity.lean"},{"n":"Causalean.PO.ID.Exact.VariableIntensityIV.VariableIntensityIVSystem.measurable_DofZ","k":"theorem","m":"Causalean.PO.ID.Exact.VariableIntensityIV.VariableIntensity","a":"PO/ID/Exact/VariableIntensityIV","t":2,"d":"The potential treatment under a fixed instrument value is measurable.","s":"∀ {P : Causalean.PO.POSystem} {𝒵 : Type u_1} [inst : MeasurableSpace 𝒵] [inst_1 : Fintype 𝒵] [inst_2 : MeasurableSingletonClass 𝒵] {J : ℕ} (S : Causalean.PO.ID.Exact.VariableIntensityIV.VariableIntensityIVSystem P 𝒵 J) (z : 𝒵), Measurable (S.DofZ z)","l":111,"f":"Causalean/PO/ID/Exact/VariableIntensityIV/VariableIntensity.lean"},{"n":"Causalean.PO.ID.Exact.VariableIntensityIV.VariableIntensityIVSystem.measurable_YofD","k":"theorem","m":"Causalean.PO.ID.Exact.VariableIntensityIV.VariableIntensity","a":"PO/ID/Exact/VariableIntensityIV","t":2,"d":"The potential outcome under a fixed treatment value is measurable.","s":"∀ {P : Causalean.PO.POSystem} {𝒵 : Type u_1} [inst : MeasurableSpace 𝒵] [inst_1 : Fintype 𝒵] [inst_2 : MeasurableSingletonClass 𝒵] {J : ℕ} (S : Causalean.PO.ID.Exact.VariableIntensityIV.VariableIntensityIVSystem P 𝒵 J) (d : Fin (J + 1)), Measurable (S.YofD d)","l":115,"f":"Causalean/PO/ID/Exact/VariableIntensityIV/VariableIntensity.lean"},{"n":"Causalean.PO.ID.Exact.VariableIntensityIV.VariableIntensityIVSystem.measurable_factualD","k":"theorem","m":"Causalean.PO.ID.Exact.VariableIntensityIV.VariableIntensity","a":"PO/ID/Exact/VariableIntensityIV","t":2,"d":"The factual treatment is measurable.","s":"∀ {P : Causalean.PO.POSystem} {𝒵 : Type u_1} [inst : MeasurableSpace 𝒵] [inst_1 : Fintype 𝒵] [inst_2 : MeasurableSingletonClass 𝒵] {J : ℕ} (S : Causalean.PO.ID.Exact.VariableIntensityIV.VariableIntensityIVSystem P 𝒵 J), Measurable S.factualD","l":122,"f":"Causalean/PO/ID/Exact/VariableIntensityIV/VariableIntensity.lean"},{"n":"Causalean.PO.ID.Exact.VariableIntensityIV.VariableIntensityIVSystem.measurable_factualY","k":"theorem","m":"Causalean.PO.ID.Exact.VariableIntensityIV.VariableIntensity","a":"PO/ID/Exact/VariableIntensityIV","t":2,"d":"The factual outcome is measurable.","s":"∀ {P : Causalean.PO.POSystem} {𝒵 : Type u_1} [inst : MeasurableSpace 𝒵] [inst_1 : Fintype 𝒵] [inst_2 : MeasurableSingletonClass 𝒵] {J : ℕ} (S : Causalean.PO.ID.Exact.VariableIntensityIV.VariableIntensityIVSystem P 𝒵 J), Measurable S.factualY","l":125,"f":"Causalean/PO/ID/Exact/VariableIntensityIV/VariableIntensity.lean"},{"n":"Causalean.PO.ID.Exact.VariableIntensityIV.VariableIntensityIVSystem.measurable_factualZ","k":"theorem","m":"Causalean.PO.ID.Exact.VariableIntensityIV.VariableIntensity","a":"PO/ID/Exact/VariableIntensityIV","t":2,"d":"The factual instrument is measurable.","s":"∀ {P : Causalean.PO.POSystem} {𝒵 : Type u_1} [inst : MeasurableSpace 𝒵] [inst_1 : Fintype 𝒵] [inst_2 : MeasurableSingletonClass 𝒵] {J : ℕ} (S : Causalean.PO.ID.Exact.VariableIntensityIV.VariableIntensityIVSystem P 𝒵 J), Measurable S.factualZ","l":119,"f":"Causalean/PO/ID/Exact/VariableIntensityIV/VariableIntensity.lean"},{"n":"Causalean.PO.ID.Exact.VariableIntensityIV.VariableIntensityIVSystem.outcomeBundle","k":"def","m":"Causalean.PO.ID.Exact.VariableIntensityIV.VariableIntensity","a":"PO/ID/Exact/VariableIntensityIV","t":1,"d":"Homogeneous bundle of the J+1 treatment-indexed potential outcomes Y(0),…,Y(J).","s":"{P : Causalean.PO.POSystem} → {𝒵 : Type u_1} → [inst : MeasurableSpace 𝒵] → [inst_1 : Fintype 𝒵] → [inst_2 : MeasurableSingletonClass 𝒵] → {J : ℕ} → Causalean.PO.ID.Exact.VariableIntensityIV.VariableIntensityIVSystem P 𝒵 J → Causalean.PO.POCFBundle P","l":147,"f":"Causalean/PO/ID/Exact/VariableIntensityIV/VariableIntensity.lean"},{"n":"Causalean.PO.ID.Exact.VariableIntensityIV.VariableIntensityIVSystem.reducedFormContrast","k":"def","m":"Causalean.PO.ID.Exact.VariableIntensityIV.VariableIntensity","a":"PO/ID/Exact/VariableIntensityIV","t":1,"d":"Potential reduced-form contrast E[Y(D(z1)) − Y(D(z0))].","s":"{P : Causalean.PO.POSystem} → {𝒵 : Type u_1} → [inst : MeasurableSpace 𝒵] → [inst_1 : Fintype 𝒵] → [inst_2 : MeasurableSingletonClass 𝒵] → {J : ℕ} → Causalean.PO.ID.Exact.VariableIntensityIV.VariableIntensityIVSystem P 𝒵 J → 𝒵 → 𝒵 → ℝ","l":299,"f":"Causalean/PO/ID/Exact/VariableIntensityIV/VariableIntensity.lean"},{"n":"Causalean.PO.ID.Exact.VariableIntensityIV.VariableIntensityIVSystem.reducedForm_eq_sum_crossingEffects","k":"theorem","m":"Causalean.PO.ID.Exact.VariableIntensityIV.VariableIntensity","a":"PO/ID/Exact/VariableIntensityIV","t":1,"d":"Reduced-form decomposition across crossed margins. Fix a directed instrument contrast (z0, z1). Under the variable-intensity IV validity assumptions — SUTVA consistency of treatment and outcome, instrument independence from the potential treatments and treatment-indexed potential outcomes, almost-sure directed monotonicity of the potential treatment intensity in the instrument, a positive first stage, and integrability of every treatment-indexed potential outcome, the potential reduced-form contrast E[Y(D(z1)) − Y(D(z0))] equals the sum, over treatment-intensity margins, of the expected unit causal response on each margin restricted to the event that the instrument move from z0 to z1 crosses that margin.","s":"∀ {P : Causalean.PO.POSystem} {𝒵 : Type u_1} [inst : MeasurableSpace 𝒵] [inst_1 : Fintype 𝒵] [inst_2 : MeasurableSingletonClass 𝒵] {J : ℕ} (S : Causalean.PO.ID.Exact.VariableIntensityIV.VariableIntensityIVSystem P 𝒵 J) {z0 z1 : 𝒵}, S.ValidContrastAssumptions z0 z1 → S.reducedFormContrast z0 z1 = ∑ j, S.indicatorWeightedEffect z0 z1 j","l":685,"f":"Causalean/PO/ID/Exact/VariableIntensityIV/VariableIntensity.lean"},{"n":"Causalean.PO.ID.Exact.VariableIntensityIV.VariableIntensityIVSystem.sum_crossingWeight_eq_one","k":"theorem","m":"Causalean.PO.ID.Exact.VariableIntensityIV.VariableIntensity","a":"PO/ID/Exact/VariableIntensityIV","t":2,"d":"Crossing weights sum to one for a valid directed contrast.","s":"∀ {P : Causalean.PO.POSystem} {𝒵 : Type u_1} [inst : MeasurableSpace 𝒵] [inst_1 : Fintype 𝒵] [inst_2 : MeasurableSingletonClass 𝒵] {J : ℕ} (S : Causalean.PO.ID.Exact.VariableIntensityIV.VariableIntensityIVSystem P 𝒵 J) {z0 z1 : 𝒵}, S.ValidContrastAssumptions z0 z1 → ∑ j, S.crossingWeight z0 z1 j = 1","l":676,"f":"Causalean/PO/ID/Exact/VariableIntensityIV/VariableIntensity.lean"},{"n":"Causalean.PO.ID.Exact.VariableIntensityIV.VariableIntensityIVSystem.totalCrossingProb","k":"def","m":"Causalean.PO.ID.Exact.VariableIntensityIV.VariableIntensity","a":"PO/ID/Exact/VariableIntensityIV","t":1,"d":"Total crossing probability, equivalently the first-stage denominator under directed monotonicity.","s":"{P : Causalean.PO.POSystem} → {𝒵 : Type u_1} → [inst : MeasurableSpace 𝒵] → [inst_1 : Fintype 𝒵] → [inst_2 : MeasurableSingletonClass 𝒵] → {J : ℕ} → Causalean.PO.ID.Exact.VariableIntensityIV.VariableIntensityIVSystem P 𝒵 J → 𝒵 → 𝒵 → ℝ","l":231,"f":"Causalean/PO/ID/Exact/VariableIntensityIV/VariableIntensity.lean"},{"n":"Causalean.PO.ID.Exact.VariableIntensityIV.VariableIntensityIVSystem.unnormalizedACRContrast","k":"def","m":"Causalean.PO.ID.Exact.VariableIntensityIV.VariableIntensity","a":"PO/ID/Exact/VariableIntensityIV","t":1,"d":"Unnormalized sum of indicator-weighted crossing effects.","s":"{P : Causalean.PO.POSystem} → {𝒵 : Type u_1} → [inst : MeasurableSpace 𝒵] → [inst_1 : Fintype 𝒵] → [inst_2 : MeasurableSingletonClass 𝒵] → {J : ℕ} → Causalean.PO.ID.Exact.VariableIntensityIV.VariableIntensityIVSystem P 𝒵 J → 𝒵 → 𝒵 → ℝ","l":245,"f":"Causalean/PO/ID/Exact/VariableIntensityIV/VariableIntensity.lean"},{"n":"Causalean.PO.ID.Exact.VariableIntensityIV.VariableIntensityIVSystem.wald","k":"def","m":"Causalean.PO.ID.Exact.VariableIntensityIV.VariableIntensity","a":"PO/ID/Exact/VariableIntensityIV","t":1,"d":"Directed Wald estimand for two instrument cells.","s":"{P : Causalean.PO.POSystem} → {𝒵 : Type u_1} → [inst : MeasurableSpace 𝒵] → [inst_1 : Fintype 𝒵] → [inst_2 : MeasurableSingletonClass 𝒵] → {J : ℕ} → Causalean.PO.ID.Exact.VariableIntensityIV.VariableIntensityIVSystem P 𝒵 J → 𝒵 → 𝒵 → ℝ","l":314,"f":"Causalean/PO/ID/Exact/VariableIntensityIV/VariableIntensity.lean"},{"n":"Causalean.PO.ID.Exact.VariableIntensityIV.VariableIntensityIVSystem.wald_eq_averageCausalResponse","k":"theorem","m":"Causalean.PO.ID.Exact.VariableIntensityIV.VariableIntensity","a":"PO/ID/Exact/VariableIntensityIV","t":1,"d":"Angrist-Imbens average-causal-response characterization. Under the variable-intensity IV validity assumptions, with the instrument cell Z = z0 having positive probability and the instrument cell Z = z1 having positive probability, the directed Wald estimand — the ratio of the reduced-form to first-stage conditional-mean contrasts across the two instrument cells — equals the Angrist-Imbens average causal response: the crossing-probability-weighted average, over treatment-intensity margins, of the conditional mean causal response given that the instrument move from z0 to z1 crosses that margin.","s":"∀ {P : Causalean.PO.POSystem} {𝒵 : Type u_1} [inst : MeasurableSpace 𝒵] [inst_1 : Fintype 𝒵] [inst_2 : MeasurableSingletonClass 𝒵] {J : ℕ} (S : Causalean.PO.ID.Exact.VariableIntensityIV.VariableIntensityIVSystem P 𝒵 J) {z0 z1 : 𝒵}, S.ValidContrastAssumptions z0 z1 → 0 < (P.μ (S.zEvent z0)).toReal → 0 < (P.μ (S.zEvent z1)).toReal → S.wald z0 z1 = S.averageCausalResponse z0 z1","l":748,"f":"Causalean/PO/ID/Exact/VariableIntensityIV/VariableIntensity.lean"},{"n":"Causalean.PO.ID.Exact.VariableIntensityIV.VariableIntensityIVSystem.yUnderD","k":"def","m":"Causalean.PO.ID.Exact.VariableIntensityIV.VariableIntensity","a":"PO/ID/Exact/VariableIntensityIV","t":1,"d":"Regimed variable for Y(d) (outcome under treatment intensity set to d).","s":"{P : Causalean.PO.POSystem} → {𝒵 : Type u_1} → [inst : MeasurableSpace 𝒵] → [inst_1 : Fintype 𝒵] → [inst_2 : MeasurableSingletonClass 𝒵] → {J : ℕ} → Causalean.PO.ID.Exact.VariableIntensityIV.VariableIntensityIVSystem P 𝒵 J → Fin (J + 1) → Causalean.PO.RegimedVar P ℝ","l":143,"f":"Causalean/PO/ID/Exact/VariableIntensityIV/VariableIntensity.lean"},{"n":"Causalean.PO.ID.Exact.VariableIntensityIV.VariableIntensityIVSystem.yVar","k":"def","m":"Causalean.PO.ID.Exact.VariableIntensityIV.VariableIntensity","a":"PO/ID/Exact/VariableIntensityIV","t":1,"d":"Outcome packaged as a POVar valued in ℝ.","s":"{P : Causalean.PO.POSystem} → {𝒵 : Type u_1} → [inst : MeasurableSpace 𝒵] → [inst_1 : Fintype 𝒵] → [inst_2 : MeasurableSingletonClass 𝒵] → {J : ℕ} → Causalean.PO.ID.Exact.VariableIntensityIV.VariableIntensityIVSystem P 𝒵 J → Causalean.PO.POVar P ℝ","l":87,"f":"Causalean/PO/ID/Exact/VariableIntensityIV/VariableIntensity.lean"},{"n":"Causalean.PO.ID.Exact.VariableIntensityIV.VariableIntensityIVSystem.zEvent","k":"def","m":"Causalean.PO.ID.Exact.VariableIntensityIV.VariableIntensity","a":"PO/ID/Exact/VariableIntensityIV","t":1,"d":"Factual instrument cell {Z = z}.","s":"{P : Causalean.PO.POSystem} → {𝒵 : Type u_1} → [inst : MeasurableSpace 𝒵] → [inst_1 : Fintype 𝒵] → [inst_2 : MeasurableSingletonClass 𝒵] → {J : ℕ} → Causalean.PO.ID.Exact.VariableIntensityIV.VariableIntensityIVSystem P 𝒵 J → 𝒵 → Set P.Ω","l":108,"f":"Causalean/PO/ID/Exact/VariableIntensityIV/VariableIntensity.lean"},{"n":"Causalean.PO.ID.Exact.VariableIntensityIV.VariableIntensityIVSystem.zVar","k":"def","m":"Causalean.PO.ID.Exact.VariableIntensityIV.VariableIntensity","a":"PO/ID/Exact/VariableIntensityIV","t":1,"d":"Instrument packaged as a POVar valued in 𝒵.","s":"{P : Causalean.PO.POSystem} → {𝒵 : Type u_1} → [inst : MeasurableSpace 𝒵] → [inst_1 : Fintype 𝒵] → [inst_2 : MeasurableSingletonClass 𝒵] → {J : ℕ} → Causalean.PO.ID.Exact.VariableIntensityIV.VariableIntensityIVSystem P 𝒵 J → Causalean.PO.POVar P 𝒵","l":81,"f":"Causalean/PO/ID/Exact/VariableIntensityIV/VariableIntensity.lean"},{"n":"Causalean.PO.IsRegressionFunction","k":"structure","m":"Causalean.PO.Analysis.Regression","a":"PO/Analysis","t":1,"d":"A regression-function representative for the conditional mean of a response g given a conditioning variable X: a candidate function f on the real line that is measurable, for which the response g is integrable and the composite f ∘ X is integrable, and such that the integral of g over every measurable event determined by X equals the integral of f ∘ X over that same event.","s":"{Ω : Type u_1} → [inst : MeasurableSpace Ω] → MeasureTheory.Measure Ω → (Ω → ℝ) → (Ω → ℝ) → (ℝ → ℝ) → Prop","l":40,"f":"Causalean/PO/Analysis/Regression.lean"},{"n":"Causalean.PO.IsRegressionFunction.aeEq_of_aeEq_response","k":"theorem","m":"Causalean.PO.Analysis.Regression","a":"PO/Analysis","t":1,"d":"Fix a conditioning variable X that is μ-almost-everywhere measurable. If two response variables g₁ and g₂ are μ-almost- everywhere equal, and f₁ is a regression-function representative of the conditional mean of g₁ given X while f₂ is a regression-function representative of the conditional mean of g₂ given X, then f₁ and f₂ agree (μ.map X)-almost everywhere.","s":"∀ {Ω : Type u_1} [inst : MeasurableSpace Ω] {μ : MeasureTheory.Measure Ω} {X g₁ g₂ : Ω → ℝ} {f₁ f₂ : ℝ → ℝ}, AEMeasurable X μ → g₁ =ᵐ[μ] g₂ → Causalean.PO.IsRegressionFunction μ X g₁ f₁ → Causalean.PO.IsRegressionFunction μ X g₂ f₂ → f₁ =ᵐ[MeasureTheory.Measure.map X μ] f₂","l":98,"f":"Causalean/PO/Analysis/Regression.lean"},{"n":"Causalean.PO.IsRegressionFunction.integrable_pushforward","k":"theorem","m":"Causalean.PO.Analysis.Regression","a":"PO/Analysis","t":2,"d":"Pushforward integrability: f is integrable under μ.map X whenever X is AEMeasurable and f represents E[g|X=·].","s":"∀ {Ω : Type u_1} [inst : MeasurableSpace Ω] {μ : MeasureTheory.Measure Ω} {X g : Ω → ℝ} {f : ℝ → ℝ}, Causalean.PO.IsRegressionFunction μ X g f → AEMeasurable X μ → MeasureTheory.Integrable f (MeasureTheory.Measure.map X μ)","l":64,"f":"Causalean/PO/Analysis/Regression.lean"},{"n":"Causalean.PO.IsRegressionFunction.sub","k":"theorem","m":"Causalean.PO.Analysis.Regression","a":"PO/Analysis","t":2,"d":"Linearity (subtraction) of IsRegressionFunction: difference of representatives is the representative of the difference.","s":"∀ {Ω : Type u_1} [inst : MeasurableSpace Ω] {μ : MeasureTheory.Measure Ω} {X g₁ g₂ : Ω → ℝ} {f₁ f₂ : ℝ → ℝ}, Causalean.PO.IsRegressionFunction μ X g₁ f₁ → Causalean.PO.IsRegressionFunction μ X g₂ f₂ → Causalean.PO.IsRegressionFunction μ X (fun ω => g₁ ω - g₂ ω) fun x => f₁ x - f₂ x","l":74,"f":"Causalean/PO/Analysis/Regression.lean"},{"n":"Causalean.PO.OR","k":"def","m":"Causalean.PO.ID.Partial.Sensitivity.MSM.Setup","a":"PO/ID/Partial/Sensitivity/MSM","t":1,"d":"The odds ratio of two probabilities p₁, p₂ ∈ (0,1): OR(p₁, p₂) = (p₁/(1-p₁)) / (p₂/(1-p₂)). The MSM measures the discrepancy between the complete propensity and the observed propensity by this quantity.","s":"ℝ → ℝ → ℝ","l":37,"f":"Causalean/PO/ID/Partial/Sensitivity/MSM/Setup.lean"},{"n":"Causalean.PO.ObsIdx","k":"def","m":"Causalean.PO.Bridge.FromSCM","a":"PO/Bridge","t":1,"d":"The observed-node index set of the induced potential-outcome system consists of all random observed nodes of the structural model.","s":"{N : Type uN} → [inst : DecidableEq N] → [inst_1 : Fintype N] → {Ω : N → Type uΩ} → [inst_2 : (n : N) → MeasurableSpace (Ω n)] → Causalean.SCM N Ω → Type (max 0 uN)","l":65,"f":"Causalean/PO/Bridge/FromSCM.lean"},{"n":"Causalean.PO.POBackdoorSystem","k":"structure","m":"Causalean.PO.ID.Exact.ATE","a":"PO/ID/Exact","t":1,"d":"A backdoor system consists of a binary treatment variable, a real-valued outcome variable, and an adjustment covariate with an arbitrary measurable value space, where the three variables are pairwise distinct.","s":"Causalean.PO.POSystem → (γ : Type u_1) → [MeasurableSpace γ] → Type (max (max u_1 u_2) u_3)","l":52,"f":"Causalean/PO/ID/Exact/ATE.lean"},{"n":"Causalean.PO.POBackdoorSystem.ATE","k":"def","m":"Causalean.PO.ID.Exact.ATE","a":"PO/ID/Exact","t":1,"d":"The ATE is the population mean difference between treated and untreated potential outcomes.","s":"{P : Causalean.PO.POSystem} → {γ : Type u_1} → [inst : MeasurableSpace γ] → Causalean.PO.POBackdoorSystem P γ → ℝ","l":127,"f":"Causalean/PO/ID/Exact/ATE.lean"},{"n":"Causalean.PO.POBackdoorSystem.ATT","k":"def","m":"Causalean.PO.ID.Exact.ATT","a":"PO/ID/Exact","t":1,"d":"ATT (Average Treatment Effect on the Treated): ATT = E[A · (Y(1) − Y(0))] / π_T.","s":"{P : Causalean.PO.POSystem} → {γ : Type u_1} → [inst : MeasurableSpace γ] → Causalean.PO.POBackdoorSystem P γ → ℝ","l":47,"f":"Causalean/PO/ID/Exact/ATT.lean"},{"n":"Causalean.PO.POBackdoorSystem.ATTAssumptions","k":"structure","m":"Causalean.PO.ID.Exact.ATT","a":"PO/ID/Exact","t":1,"d":"Backdoor assumptions for ATT identification. These are the standard ATT conditions, with overlap required on only one side — every covariate stratum keeps a positive chance of the control arm (e(X) < 1) — which is strictly weaker than the two-sided overlap the ATE needs, because on the treated the outcome is observed directly.","s":"{P : Causalean.PO.POSystem} → {γ : Type u_1} → [inst : MeasurableSpace γ] → Causalean.PO.POBackdoorSystem P γ → [StandardBorelSpace P.Ω] → [MeasureTheory.IsFiniteMeasure P.μ] → Prop","l":64,"f":"Causalean/PO/ID/Exact/ATT.lean"},{"n":"Causalean.PO.POBackdoorSystem.ATTAssumptions.propScore_false_ne","k":"theorem","m":"Causalean.PO.ID.Exact.ATT","a":"PO/ID/Exact","t":2,"d":"Under one-sided overlap (e(X) < 1), the control propensity score P[D=0 | σ(X)] = 1 - e(X) is a.s. nonzero.","s":"∀ {P : Causalean.PO.POSystem} {γ : Type u_1} [inst : MeasurableSpace γ] (S : Causalean.PO.POBackdoorSystem P γ) [inst_1 : StandardBorelSpace P.Ω] [inst_2 : MeasureTheory.IsFiniteMeasure P.μ], S.ATTAssumptions → ∀ᵐ (ω : P.Ω) ∂P.μ, S.propScore false ω ≠ 0","l":128,"f":"Causalean/PO/ID/Exact/ATT.lean"},{"n":"Causalean.PO.POBackdoorSystem.ATT_eq_adjustedATT","k":"theorem","m":"Causalean.PO.ID.Exact.ATT","a":"PO/ID/Exact","t":1,"d":"ATT identification (one-sided overlap). Under consistency, unconfoundedness, one-sided overlap (e(X) < 1, the control arm), and positivity of the marginal treatment probability, the potential-outcome-level average treatment effect on the treated equals the observable adjusted-ATT functional:","s":"∀ {P : Causalean.PO.POSystem} {γ : Type u_1} [inst : MeasurableSpace γ] (S : Causalean.PO.POBackdoorSystem P γ) [inst_1 : StandardBorelSpace P.Ω] [inst_2 : MeasureTheory.IsFiniteMeasure P.μ], S.ATTAssumptions → S.ATT = S.adjustedATT","l":393,"f":"Causalean/PO/ID/Exact/ATT.lean"},{"n":"Causalean.PO.POBackdoorSystem.Assumptions","k":"structure","m":"Causalean.PO.ID.Exact.ATE","a":"PO/ID/Exact","t":1,"d":"The backdoor (Rosenbaum–Rubin / Firpo) identifying assumptions for the ATE, at the potential-outcome level, for binary treatment D and real outcome Y adjusted for covariates X: the observed outcome equals the potential outcome of the realized treatment arm, treatment is conditionally independent of the treated and untreated potential outcomes given the covariates, every covariate stratum has a positive chance of either treatment arm, and the treated and untreated potential outcomes are integrable.","s":"{P : Causalean.PO.POSystem} → {γ : Type u_1} → [inst : MeasurableSpace γ] → Causalean.PO.POBackdoorSystem P γ → [StandardBorelSpace P.Ω] → [MeasureTheory.IsFiniteMeasure P.μ] → Prop","l":140,"f":"Causalean/PO/ID/Exact/ATE.lean"},{"n":"Causalean.PO.POBackdoorSystem.Assumptions.integrable_factualY","k":"theorem","m":"Causalean.PO.ID.Exact.ATE","a":"PO/ID/Exact","t":2,"d":"The observed outcome is integrable under the backdoor assumption bundle.","s":"∀ {P : Causalean.PO.POSystem} {γ : Type u_1} [inst : MeasurableSpace γ] (S : Causalean.PO.POBackdoorSystem P γ) [inst_1 : StandardBorelSpace P.Ω] [inst_2 : MeasureTheory.IsFiniteMeasure P.μ], S.Assumptions → MeasureTheory.Integrable S.factualY P.μ","l":219,"f":"Causalean/PO/ID/Exact/ATE.lean"},{"n":"Causalean.PO.POBackdoorSystem.Assumptions.toDistributional","k":"theorem","m":"Causalean.PO.ID.Exact.QTE.DistributionalBackdoor","a":"PO/ID/Exact/QTE","t":2,"d":"The ATE backdoor bundle projects to the weaker distributional bundle.","s":"∀ {P : Causalean.PO.POSystem} {γ : Type u_1} [inst : MeasurableSpace γ] (S : Causalean.PO.POBackdoorSystem P γ) [inst_1 : StandardBorelSpace P.Ω] [inst_2 : MeasureTheory.IsFiniteMeasure P.μ], S.Assumptions → S.DistributionalAssumptions","l":92,"f":"Causalean/PO/ID/Exact/QTE/DistributionalBackdoor.lean"},{"n":"Causalean.PO.POBackdoorSystem.CATE","k":"def","m":"Causalean.PO.ID.Exact.ATE","a":"PO/ID/Exact","t":1,"d":"The conditional mean potential outcome is the mean of a treatment-arm potential outcome given the covariate.","s":"{P : Causalean.PO.POSystem} → {γ : Type u_1} → [inst : MeasurableSpace γ] → Causalean.PO.POBackdoorSystem P γ → Bool → P.Ω → ℝ","l":122,"f":"Causalean/PO/ID/Exact/ATE.lean"},{"n":"Causalean.PO.POBackdoorSystem.Calibrated","k":"def","m":"Causalean.PO.ID.Partial.Sensitivity.MSM.Sharp","a":"PO/ID/Partial/Sensitivity/MSM","t":1,"d":"Calibration (data-compatibility). A candidate complete propensity ẽ is *calibrated* if the inverse-propensity weighting of the treatment indicator averages to one within every covariate stratum: E[ Z / ẽ | σ(X) ] = 1 a.e. This is the only restriction on ẽ beyond the odds-ratio box implied by the observed-data distribution.","s":"{P : Causalean.PO.POSystem} → {γ : Type u_1} → [inst : MeasurableSpace γ] → Causalean.PO.POBackdoorSystem P γ → (P.Ω → ℝ) → Prop","l":52,"f":"Causalean/PO/ID/Partial/Sensitivity/MSM/Sharp.lean"},{"n":"Causalean.PO.POBackdoorSystem.Calibrated0","k":"def","m":"Causalean.PO.ID.Partial.Sensitivity.MSM.ControlSharp","a":"PO/ID/Partial/Sensitivity/MSM","t":1,"d":"Calibration (control arm). A candidate complete control propensity ẽ is *calibrated* if E[ (1−Z) / ẽ | σ(X) ] = 1 a.e., where 1−Z = 1_{D=0}.","s":"{P : Causalean.PO.POSystem} → {γ : Type u_1} → [inst : MeasurableSpace γ] → Causalean.PO.POBackdoorSystem P γ → (P.Ω → ℝ) → Prop","l":39,"f":"Causalean/PO/ID/Partial/Sensitivity/MSM/ControlSharp.lean"},{"n":"Causalean.PO.POBackdoorSystem.DistributionalAssumptions","k":"structure","m":"Causalean.PO.ID.Exact.QTE.DistributionalBackdoor","a":"PO/ID/Exact/QTE","t":1,"d":"Distributional backdoor assumptions. Firpo's distributional identification of the potential-outcome law under arm d uses consistency, conditional ignorability of treatment given covariates, and common support. It does not require the outcome-integrability assumptions bundled in the ATE backdoor theorem, because laws and quantiles are defined without first moments.","s":"{P : Causalean.PO.POSystem} → {γ : Type u_1} → [inst : MeasurableSpace γ] → Causalean.PO.POBackdoorSystem P γ → [StandardBorelSpace P.Ω] → [MeasureTheory.IsFiniteMeasure P.μ] → Prop","l":72,"f":"Causalean/PO/ID/Exact/QTE/DistributionalBackdoor.lean"},{"n":"Causalean.PO.POBackdoorSystem.GaussianOutcomeModel","k":"structure","m":"Causalean.PO.ID.Partial.Sensitivity.MSM.GaussianHalfWidth","a":"PO/ID/Partial/Sensitivity/MSM","t":1,"d":"The conditional-Gaussian outcome model. Strengthens GaussianTreatedModel (conditional CDF Φ((·−m)/σ)) with the two conditional moments it implies: the treated conditional mean E[A·Y|σX] = e(X)·m(X) and the truncated mean above any σ(X)-measurable cutoff. This is the faithful \"treated outcomes are conditionally Gaussian N(m(X), σ(X)²)\" premise of the Dorn–Guo corollary.","s":"{P : Causalean.PO.POSystem} → {γ : Type u_1} → [inst : MeasurableSpace γ] → Causalean.PO.POBackdoorSystem P γ → (γ → ℝ) → (γ → ℝ) → Prop","l":172,"f":"Causalean/PO/ID/Partial/Sensitivity/MSM/GaussianHalfWidth.lean"},{"n":"Causalean.PO.POBackdoorSystem.GaussianTreatedModel","k":"structure","m":"Causalean.PO.ID.Partial.Sensitivity.MSM.Gaussian","a":"PO/ID/Partial/Sensitivity/MSM","t":1,"d":"The conditional-Gaussian treated-outcome model. The treated conditional law of Y given X is N(m(X), σ(X)²): a measurable mean m and positive standard deviation σ with condCDF treatedXYLaw a t = Φ((t − m a)/σ a) for all a, t.","s":"{P : Causalean.PO.POSystem} → {γ : Type u_1} → [inst : MeasurableSpace γ] → Causalean.PO.POBackdoorSystem P γ → (γ → ℝ) → (γ → ℝ) → Prop","l":89,"f":"Causalean/PO/ID/Partial/Sensitivity/MSM/Gaussian.lean"},{"n":"Causalean.PO.POBackdoorSystem.MSMSet","k":"def","m":"Causalean.PO.ID.Partial.Sensitivity.MSM.Setup","a":"PO/ID/Partial/Sensitivity/MSM","t":1,"d":"The MSM ambiguity set at sensitivity level Λ: the set of candidate complete propensities ẽ that are a.e. in (0,1) and whose odds ratio against the observed propensity e(X) = P[D=1 | σ(X)] is a.e. bounded within [1/Λ, Λ]. Larger Λ admits more candidates, widening the resulting interval.","s":"{P : Causalean.PO.POSystem} → {γ : Type u_1} → [inst : MeasurableSpace γ] → Causalean.PO.POBackdoorSystem P γ → ℝ → Set (P.Ω → ℝ)","l":77,"f":"Causalean/PO/ID/Partial/Sensitivity/MSM/Setup.lean"},{"n":"Causalean.PO.POBackdoorSystem.MSMSet0","k":"def","m":"Causalean.PO.ID.Partial.Sensitivity.MSM.ControlSetup","a":"PO/ID/Partial/Sensitivity/MSM","t":1,"d":"The control MSM ambiguity set at sensitivity level Λ: candidate complete control propensities ẽ that are a.e. in (0,1) and whose odds ratio against the observed control propensity P[D=0 | σ(X)] = propScore false is a.e. within [1/Λ, Λ].","s":"{P : Causalean.PO.POSystem} → {γ : Type u_1} → [inst : MeasurableSpace γ] → Causalean.PO.POBackdoorSystem P γ → ℝ → Set (P.Ω → ℝ)","l":63,"f":"Causalean/PO/ID/Partial/Sensitivity/MSM/ControlSetup.lean"},{"n":"Causalean.PO.POBackdoorSystem.MSMSet0_mono","k":"theorem","m":"Causalean.PO.ID.Partial.Sensitivity.MSM.ControlSetup","a":"PO/ID/Partial/Sensitivity/MSM","t":2,"d":"Monotonicity of the control ambiguity set in the sensitivity parameter.","s":"∀ {P : Causalean.PO.POSystem} {γ : Type u_1} [inst : MeasurableSpace γ] (S : Causalean.PO.POBackdoorSystem P γ) {Λ Λ' : ℝ}, 1 ≤ Λ → Λ ≤ Λ' → S.MSMSet0 Λ ⊆ S.MSMSet0 Λ'","l":185,"f":"Causalean/PO/ID/Partial/Sensitivity/MSM/ControlSetup.lean"},{"n":"Causalean.PO.POBackdoorSystem.MSMSet0_one_eq","k":"theorem","m":"Causalean.PO.ID.Partial.Sensitivity.MSM.ControlSetup","a":"PO/ID/Partial/Sensitivity/MSM","t":1,"d":"Point identification at Λ = 1 (control arm, no-unmeasured-confounding collapse). Assume the observed control propensity score P[D=0∣σ(X)] is almost surely strictly between 0 and 1. Then for any candidate complete control propensity ẽ belonging to the (uncalibrated) control ambiguity set at sensitivity level Λ = 1, ẽ agrees almost everywhere with the observed control propensity score: at Λ = 1 the odds-ratio ambiguity set collapses to the single point given by no unmeasured confounding.","s":"∀ {P : Causalean.PO.POSystem} {γ : Type u_1} [inst : MeasurableSpace γ] (S : Causalean.PO.POBackdoorSystem P γ) (etilde : P.Ω → ℝ), (∀ᵐ (ω : P.Ω) ∂P.μ, 0 < S.propScore false ω ∧ S.propScore false ω < 1) → etilde ∈ S.MSMSet0 1 → ∀ᵐ (ω : P.Ω) ∂P.μ, etilde ω = S.propScore false ω","l":219,"f":"Causalean/PO/ID/Partial/Sensitivity/MSM/ControlSetup.lean"},{"n":"Causalean.PO.POBackdoorSystem.MSMSetCalib","k":"def","m":"Causalean.PO.ID.Partial.Sensitivity.MSM.Sharp","a":"PO/ID/Partial/Sensitivity/MSM","t":1,"d":"The calibrated (sharp) MSM ambiguity set: odds-ratio-box members that also satisfy calibration.","s":"{P : Causalean.PO.POSystem} → {γ : Type u_1} → [inst : MeasurableSpace γ] → Causalean.PO.POBackdoorSystem P γ → ℝ → Set (P.Ω → ℝ)","l":59,"f":"Causalean/PO/ID/Partial/Sensitivity/MSM/Sharp.lean"},{"n":"Causalean.PO.POBackdoorSystem.MSMSetCalib0","k":"def","m":"Causalean.PO.ID.Partial.Sensitivity.MSM.ControlSharp","a":"PO/ID/Partial/Sensitivity/MSM","t":1,"d":"The calibrated (sharp) control MSM ambiguity set.","s":"{P : Causalean.PO.POSystem} → {γ : Type u_1} → [inst : MeasurableSpace γ] → Causalean.PO.POBackdoorSystem P γ → ℝ → Set (P.Ω → ℝ)","l":44,"f":"Causalean/PO/ID/Partial/Sensitivity/MSM/ControlSharp.lean"},{"n":"Causalean.PO.POBackdoorSystem.MSMSetCalib0_subset","k":"theorem","m":"Causalean.PO.ID.Partial.Sensitivity.MSM.ControlSharp","a":"PO/ID/Partial/Sensitivity/MSM","t":2,"d":"The calibrated control set is a subset of the odds-ratio box.","s":"∀ {P : Causalean.PO.POSystem} {γ : Type u_1} [inst : MeasurableSpace γ] (S : Causalean.PO.POBackdoorSystem P γ) (Λ : ℝ), S.MSMSetCalib0 Λ ⊆ S.MSMSet0 Λ","l":136,"f":"Causalean/PO/ID/Partial/Sensitivity/MSM/ControlSharp.lean"},{"n":"Causalean.PO.POBackdoorSystem.MSMSetCalib_subset","k":"theorem","m":"Causalean.PO.ID.Partial.Sensitivity.MSM.Sharp","a":"PO/ID/Partial/Sensitivity/MSM","t":2,"d":"The calibrated set is a subset of the odds-ratio box.","s":"∀ {P : Causalean.PO.POSystem} {γ : Type u_1} [inst : MeasurableSpace γ] (S : Causalean.PO.POBackdoorSystem P γ) (Λ : ℝ), S.MSMSetCalib Λ ⊆ S.MSMSet Λ","l":157,"f":"Causalean/PO/ID/Partial/Sensitivity/MSM/Sharp.lean"},{"n":"Causalean.PO.POBackdoorSystem.MSMSet_mono","k":"theorem","m":"Causalean.PO.ID.Partial.Sensitivity.MSM.Setup","a":"PO/ID/Partial/Sensitivity/MSM","t":2,"d":"Monotonicity of the ambiguity set in the sensitivity parameter. A larger budget Λ ≤ Λ' (with 1 ≤ Λ) admits every candidate that the smaller budget admits: MSMSet Λ ⊆ MSMSet Λ'. The odds-ratio window [1/Λ, Λ] widens with Λ.","s":"∀ {P : Causalean.PO.POSystem} {γ : Type u_1} [inst : MeasurableSpace γ] (S : Causalean.PO.POBackdoorSystem P γ) {Λ Λ' : ℝ}, 1 ≤ Λ → Λ ≤ Λ' → S.MSMSet Λ ⊆ S.MSMSet Λ'","l":225,"f":"Causalean/PO/ID/Partial/Sensitivity/MSM/Setup.lean"},{"n":"Causalean.PO.POBackdoorSystem.MSMSet_one_eq","k":"theorem","m":"Causalean.PO.ID.Partial.Sensitivity.MSM.Setup","a":"PO/ID/Partial/Sensitivity/MSM","t":1,"d":"Point identification at Λ = 1 (no-unmeasured-confounding collapse). Assume the propensity score for treatment given the covariates lies strictly between 0 and 1 almost surely (overlap). If a candidate complete propensity ẽ belongs to the marginal-sensitivity ambiguity set at Λ = 1, then it equals the observed propensity score almost everywhere.","s":"∀ {P : Causalean.PO.POSystem} {γ : Type u_1} [inst : MeasurableSpace γ] (S : Causalean.PO.POBackdoorSystem P γ) (etilde : P.Ω → ℝ), (∀ᵐ (ω : P.Ω) ∂P.μ, 0 < S.propScore true ω ∧ S.propScore true ω < 1) → etilde ∈ S.MSMSet 1 → ∀ᵐ (ω : P.Ω) ∂P.μ, etilde ω = S.propScore true ω","l":265,"f":"Causalean/PO/ID/Partial/Sensitivity/MSM/Setup.lean"},{"n":"Causalean.PO.POBackdoorSystem.Y0mean","k":"def","m":"Causalean.PO.ID.Partial.Sensitivity.MSM.ControlSetup","a":"PO/ID/Partial/Sensitivity/MSM","t":1,"d":"The estimand μ₀ = E[Y(0)], the mean of the control potential outcome.","s":"{P : Causalean.PO.POSystem} → {γ : Type u_1} → [inst : MeasurableSpace γ] → Causalean.PO.POBackdoorSystem P γ → ℝ","l":60,"f":"Causalean/PO/ID/Partial/Sensitivity/MSM/ControlSetup.lean"},{"n":"Causalean.PO.POBackdoorSystem.Y0mean_mem_Icc","k":"theorem","m":"Causalean.PO.ID.Partial.Sensitivity.MSM.ControlSetup","a":"PO/ID/Partial/Sensitivity/MSM","t":1,"d":"The control MSM bound is valid. For a sensitivity parameter Λ at which the true complete control propensity P[D=0∣σ(X,Y(0))] belongs to the control ambiguity set, assuming the candidate IPW mean evaluated at the true complete control propensity recovers E[Y(0)] and that the candidate means over the ambiguity set are bounded below and bounded above, the mean control potential outcome E[Y(0)] lies in the closed interval between the control MSM lower and upper bounds.","s":"∀ {P : Causalean.PO.POSystem} {γ : Type u_1} [inst : MeasurableSpace γ] (S : Causalean.PO.POBackdoorSystem P γ) (Λ : ℝ), S.completeProp0 ∈ S.MSMSet0 Λ → S.candMean0 S.completeProp0 = S.Y0mean → BddBelow (S.candMean0 '' S.MSMSet0 Λ) → BddAbove (S.candMean0 '' S.MSMSet0 Λ) → S.Y0mean ∈ Set.Icc (S.msmLower0 Λ) (S.msmUpper0 Λ)","l":166,"f":"Causalean/PO/ID/Partial/Sensitivity/MSM/ControlSetup.lean"},{"n":"Causalean.PO.POBackdoorSystem.Y0mean_mem_Icc_calib","k":"theorem","m":"Causalean.PO.ID.Partial.Sensitivity.MSM.ControlSharp","a":"PO/ID/Partial/Sensitivity/MSM","t":1,"d":"The sharp control bound is valid. For a sensitivity parameter Λ at which the true complete control propensity P[D=0∣σ(X,Y(0))] belongs to the calibrated control ambiguity set, assuming the candidate IPW mean evaluated at the true complete control propensity recovers E[Y(0)] and that the candidate means over the calibrated ambiguity set are bounded below and bounded above, the mean control potential outcome E[Y(0)] lies in the closed interval between the sharp control lower and upper bounds.","s":"∀ {P : Causalean.PO.POSystem} {γ : Type u_1} [inst : MeasurableSpace γ] (S : Causalean.PO.POBackdoorSystem P γ) (Λ : ℝ), S.completeProp0 ∈ S.MSMSetCalib0 Λ → S.candMean0 S.completeProp0 = S.Y0mean → BddBelow (S.candMean0 '' S.MSMSetCalib0 Λ) → BddAbove (S.candMean0 '' S.MSMSetCalib0 Λ) → S.Y0mean ∈ Set.Icc (S.msmLowerCalib0 Λ) (S.msmUpperCalib0 Λ)","l":116,"f":"Causalean/PO/ID/Partial/Sensitivity/MSM/ControlSharp.lean"},{"n":"Causalean.PO.POBackdoorSystem.Y1mean","k":"def","m":"Causalean.PO.ID.Partial.Sensitivity.MSM.Setup","a":"PO/ID/Partial/Sensitivity/MSM","t":1,"d":"The estimand μ = E[Y(1)], the mean of the treated potential outcome.","s":"{P : Causalean.PO.POSystem} → {γ : Type u_1} → [inst : MeasurableSpace γ] → Causalean.PO.POBackdoorSystem P γ → ℝ","l":74,"f":"Causalean/PO/ID/Partial/Sensitivity/MSM/Setup.lean"},{"n":"Causalean.PO.POBackdoorSystem.Y1mean_mem_Icc","k":"theorem","m":"Causalean.PO.ID.Partial.Sensitivity.MSM.Setup","a":"PO/ID/Partial/Sensitivity/MSM","t":1,"d":"The MSM bound: E[Y(1)] lies in the MSM interval. Assuming the true complete propensity belongs to the marginal-sensitivity ambiguity set, the IPW/tower bridge identity candMean e₀ = E[Y(1)] holds, and the candidate mean is bounded below and above over the ambiguity set, the estimand E[Y(1)] is sandwiched between the infimum and supremum of the candidate mean over the ambiguity set:","s":"∀ {P : Causalean.PO.POSystem} {γ : Type u_1} [inst : MeasurableSpace γ] (S : Causalean.PO.POBackdoorSystem P γ) (Λ : ℝ), S.completeProp ∈ S.MSMSet Λ → S.candMean S.completeProp = S.Y1mean → BddBelow (S.candMean '' S.MSMSet Λ) → BddAbove (S.candMean '' S.MSMSet Λ) → S.Y1mean ∈ Set.Icc (S.msmLower Λ) (S.msmUpper Λ)","l":203,"f":"Causalean/PO/ID/Partial/Sensitivity/MSM/Setup.lean"},{"n":"Causalean.PO.POBackdoorSystem.Y1mean_mem_Icc_calib","k":"theorem","m":"Causalean.PO.ID.Partial.Sensitivity.MSM.Sharp","a":"PO/ID/Partial/Sensitivity/MSM","t":1,"d":"The sharp bound is valid. Assuming the true complete propensity belongs to the calibrated (sharp) marginal-sensitivity ambiguity set, the IPW/tower bridge identity candMean e₀ = E[Y(1)] holds, and the candidate mean is bounded below and above over the calibrated ambiguity set, then the estimand E[Y(1)] lies in the calibrated (sharp) interval [msmLowerCalib Λ, msmUpperCalib Λ].","s":"∀ {P : Causalean.PO.POSystem} {γ : Type u_1} [inst : MeasurableSpace γ] (S : Causalean.PO.POBackdoorSystem P γ) (Λ : ℝ), S.completeProp ∈ S.MSMSetCalib Λ → S.candMean S.completeProp = S.Y1mean → BddBelow (S.candMean '' S.MSMSetCalib Λ) → BddAbove (S.candMean '' S.MSMSetCalib Λ) → S.Y1mean ∈ Set.Icc (S.msmLowerCalib Λ) (S.msmUpperCalib Λ)","l":139,"f":"Causalean/PO/ID/Partial/Sensitivity/MSM/Sharp.lean"},{"n":"Causalean.PO.POBackdoorSystem.YofD","k":"def","m":"Causalean.PO.ID.Exact.ATE","a":"PO/ID/Exact","t":1,"d":"The potential outcome under a fixed treatment arm is a real-valued function on units.","s":"{P : Causalean.PO.POSystem} → {γ : Type u_1} → [inst : MeasurableSpace γ] → Causalean.PO.POBackdoorSystem P γ → Bool → P.Ω → ℝ","l":80,"f":"Causalean/PO/ID/Exact/ATE.lean"},{"n":"Causalean.PO.POBackdoorSystem.adjustedATE","k":"def","m":"Causalean.PO.ID.Exact.ATE","a":"PO/ID/Exact","t":1,"d":"The adjusted ATE integrates the difference of the two adjusted conditional outcome functionals.","s":"{P : Causalean.PO.POSystem} → {γ : Type u_1} → [inst : MeasurableSpace γ] → Causalean.PO.POBackdoorSystem P γ → ℝ","l":135,"f":"Causalean/PO/ID/Exact/ATE.lean"},{"n":"Causalean.PO.POBackdoorSystem.adjustedATT","k":"def","m":"Causalean.PO.ID.Exact.ATT","a":"PO/ID/Exact","t":1,"d":"Adjusted ATT (observable, control-regression form): E[A · (Y − μ₀(X))] / π_T. Only the CONTROL regression μ₀(X) = adjustedCE false appears — the treated potential outcome is observed directly on {D = 1} via consistency (A · Y = A · Y(1)), so no treated regression μ₁(X) and hence no 0 < e(X) is needed. This is the standard ATT estimand and requires only one-sided overlap e(X) < 1. The AIPW form is recovered as a corollary (adjustedATT_eq_aipwForm).","s":"{P : Causalean.PO.POSystem} → {γ : Type u_1} → [inst : MeasurableSpace γ] → Causalean.PO.POBackdoorSystem P γ → ℝ","l":53,"f":"Causalean/PO/ID/Exact/ATT.lean"},{"n":"Causalean.PO.POBackdoorSystem.adjustedATT_eq_aipwForm","k":"theorem","m":"Causalean.PO.ID.Exact.ATT","a":"PO/ID/Exact","t":1,"d":"AIPW corollary. Under the ATT identification assumptions — consistency, unconfoundedness, one-sided control-arm overlap, and a positive marginal treatment probability, provided the observed inverse-propensity-weighted correction term is integrable, the adjustedATT functional equals its augmented inverse-propensity-weighted (AIPW) form:","s":"∀ {P : Causalean.PO.POSystem} {γ : Type u_1} [inst : MeasurableSpace γ] (S : Causalean.PO.POBackdoorSystem P γ) [inst_1 : StandardBorelSpace P.Ω] [inst_2 : MeasureTheory.IsFiniteMeasure P.μ], S.ATTAssumptions → MeasureTheory.Integrable (fun ω => (1 - S.dVar.indicator true ω) * (S.propScore true ω / (1 - S.propScore true ω)) * (S.factualY ω - S.adjustedCE false ω)) P.μ → S.adjustedATT = (∫ (ω : P.Ω), S.dVar.indicator true ω * (S.factualY ω - S.adjustedCE false ω) - (1 - S.dVar.indicator true ω) * (S.propScore true ω / (1 - S.propScore true ω)) * (S.factualY ω - S.adjustedCE false ω) ∂P.μ) / S.propTreated","l":466,"f":"Causalean/PO/ID/Exact/ATT.lean"},{"n":"Causalean.PO.POBackdoorSystem.adjustedCE","k":"def","m":"Causalean.PO.ID.Exact.ATE","a":"PO/ID/Exact","t":1,"d":"The adjusted conditional functional is the observable outcome-indicator regression divided by the propensity score.","s":"{P : Causalean.PO.POSystem} → {γ : Type u_1} → [inst : MeasurableSpace γ] → Causalean.PO.POBackdoorSystem P γ → Bool → P.Ω → ℝ","l":130,"f":"Causalean/PO/ID/Exact/ATE.lean"},{"n":"Causalean.PO.POBackdoorSystem.ate","k":"def","m":"Causalean.PO.ID.Partial.Sensitivity.MSM.ATE","a":"PO/ID/Partial/Sensitivity/MSM","t":1,"d":"The average treatment effect τ = E[Y(1)] − E[Y(0)].","s":"{P : Causalean.PO.POSystem} → {γ : Type u_1} → [inst : MeasurableSpace γ] → Causalean.PO.POBackdoorSystem P γ → ℝ","l":48,"f":"Causalean/PO/ID/Partial/Sensitivity/MSM/ATE.lean"},{"n":"Causalean.PO.POBackdoorSystem.ateCalib_subset","k":"theorem","m":"Causalean.PO.ID.Partial.Sensitivity.MSM.ATE","a":"PO/ID/Partial/Sensitivity/MSM","t":1,"d":"The sharp ATE interval is contained in the ZSB ATE interval. If the treated arm's calibrated sharp upper bound does not exceed its uncalibrated ZSB upper bound, the treated arm's uncalibrated ZSB lower bound does not exceed its calibrated sharp lower bound, the control arm's calibrated sharp upper bound does not exceed its uncalibrated ZSB upper bound, and the control arm's uncalibrated ZSB lower bound does not exceed its calibrated sharp lower bound, then the sharp ATE interval [ateLowerCalib Λ, ateUpperCalib Λ] is contained in the ZSB ATE interval [ateLower Λ, ateUpper Λ].","s":"∀ {P : Causalean.PO.POSystem} {γ : Type u_1} [inst : MeasurableSpace γ] (S : Causalean.PO.POBackdoorSystem P γ) (Λ : ℝ), S.msmUpperCalib Λ ≤ S.msmUpper Λ → S.msmLower Λ ≤ S.msmLowerCalib Λ → S.msmUpperCalib0 Λ ≤ S.msmUpper0 Λ → S.msmLower0 Λ ≤ S.msmLowerCalib0 Λ → Set.Icc (S.ateLowerCalib Λ) (S.ateUpperCalib Λ) ⊆ Set.Icc (S.ateLower Λ) (S.ateUpper Λ)","l":98,"f":"Causalean/PO/ID/Partial/Sensitivity/MSM/ATE.lean"},{"n":"Causalean.PO.POBackdoorSystem.ateLower","k":"def","m":"Causalean.PO.ID.Partial.Sensitivity.MSM.ATE","a":"PO/ID/Partial/Sensitivity/MSM","t":1,"d":"The ZSB ATE lower bound ψ_T⁻ − ψ_C⁺ formed from the uncalibrated arm bounds.","s":"{P : Causalean.PO.POSystem} → {γ : Type u_1} → [inst : MeasurableSpace γ] → Causalean.PO.POBackdoorSystem P γ → ℝ → ℝ","l":62,"f":"Causalean/PO/ID/Partial/Sensitivity/MSM/ATE.lean"},{"n":"Causalean.PO.POBackdoorSystem.ateLowerCalib","k":"def","m":"Causalean.PO.ID.Partial.Sensitivity.MSM.ATE","a":"PO/ID/Partial/Sensitivity/MSM","t":1,"d":"The sharp ATE lower bound τ⁻(Λ) = ψ_T⁻(Λ) − ψ_C⁺(Λ): the treated sharp lower bound minus the control sharp *upper* bound.","s":"{P : Causalean.PO.POSystem} → {γ : Type u_1} → [inst : MeasurableSpace γ] → Causalean.PO.POBackdoorSystem P γ → ℝ → ℝ","l":55,"f":"Causalean/PO/ID/Partial/Sensitivity/MSM/ATE.lean"},{"n":"Causalean.PO.POBackdoorSystem.ateUpper","k":"def","m":"Causalean.PO.ID.Partial.Sensitivity.MSM.ATE","a":"PO/ID/Partial/Sensitivity/MSM","t":1,"d":"The ZSB ATE upper bound ψ_T⁺ − ψ_C⁻ formed from the uncalibrated arm bounds.","s":"{P : Causalean.PO.POSystem} → {γ : Type u_1} → [inst : MeasurableSpace γ] → Causalean.PO.POBackdoorSystem P γ → ℝ → ℝ","l":59,"f":"Causalean/PO/ID/Partial/Sensitivity/MSM/ATE.lean"},{"n":"Causalean.PO.POBackdoorSystem.ateUpperCalib","k":"def","m":"Causalean.PO.ID.Partial.Sensitivity.MSM.ATE","a":"PO/ID/Partial/Sensitivity/MSM","t":1,"d":"The sharp ATE upper bound τ⁺(Λ) = ψ_T⁺(Λ) − ψ_C⁻(Λ): the treated sharp upper bound minus the control sharp *lower* bound.","s":"{P : Causalean.PO.POSystem} → {γ : Type u_1} → [inst : MeasurableSpace γ] → Causalean.PO.POBackdoorSystem P γ → ℝ → ℝ","l":51,"f":"Causalean/PO/ID/Partial/Sensitivity/MSM/ATE.lean"},{"n":"Causalean.PO.POBackdoorSystem.ate_backdoor","k":"theorem","m":"Causalean.PO.ID.Exact.ATE","a":"PO/ID/Exact","t":1,"d":"Integrated backdoor ATE identification. Under the backdoor identification assumptions — consistency, unconfoundedness, two-sided overlap, and integrable potential outcomes, the average treatment effect equals the adjusted-outcome functional obtained by averaging the CATE-level backdoor identity over the covariate distribution.","s":"∀ {P : Causalean.PO.POSystem} {γ : Type u_1} [inst : MeasurableSpace γ] (S : Causalean.PO.POBackdoorSystem P γ) [inst_1 : StandardBorelSpace P.Ω] [inst_2 : MeasureTheory.IsFiniteMeasure P.μ], S.Assumptions → S.ATE = S.adjustedATE","l":436,"f":"Causalean/PO/ID/Exact/ATE.lean"},{"n":"Causalean.PO.POBackdoorSystem.ate_endpoints_eq_cutoff","k":"theorem","m":"Causalean.PO.ID.Partial.Sensitivity.MSM.ATEClosedForm","a":"PO/ID/Partial/Sensitivity/MSM","t":1,"d":"The sharp ATE interval endpoints in closed form. Fix a sensitivity parameter Λ greater than 1. Given, for the treated arm, two-sided propensity overlap, an atomless conditional outcome distribution, that the upper and lower calibration levels each lie strictly between 0 and 1 almost everywhere, that the calibrated candidate mean is bounded above over its ambiguity set, that every candidate propensity in that ambiguity set is measurable up to null sets, and integrability regularity, for every σ(X)-measurable cutoff candidate, feeding both the upper and lower calibration constructions — together with the symmetric conditions for the control arm (overlap, atomlessness, calibration-level regularity, boundedness, measurability, and cutoff integrability regularity) — then there exist σ(X)-measurable conditional-quantile cutoffs cTU, cTL, cCU, cCL such that the sharp calibrated ATE upper endpoint equals the treated upper-cutoff candidate mean minus the control lower-cutoff candidate mean, and the sharp calibrated ATE lower endpoint equals the treated lower-cutoff candidate mean minus the control upper-cutoff candidate mean.","s":"∀ {P : Causalean.PO.POSystem} {γ : Type u_1} [inst : MeasurableSpace γ] (S : Causalean.PO.POBackdoorSystem P γ) (Λ : ℝ), 1 < Λ → (∀ᵐ (ω : P.Ω) ∂P.μ, 0 < S.propScore true ω ∧ S.propScore true ω < 1) → (∀ (a : γ), Continuous ↑(ProbabilityTheory.condCDF S.treatedXYLaw a)) → (∀ᵐ (ω : P.Ω) ∂P.μ, 0 < S.calibLevel Λ ω ∧ S.calibLevel Λ ω < 1) → (∀ᵐ (ω : P.Ω) ∂P.μ, 0 < S.calibLevelLower Λ ω ∧ S.calibLevelLower Λ ω < 1) → BddAbove (S.candMean '' S.MSMSetCalib Λ) → (∀ etilde ∈ S.MSMSetCalib Λ, AEMeasurable etilde P.μ) → (∀ (c : P.Ω → ℝ), Measurable c → MeasureTheory.Integrable c P.μ ∧ MeasureTheory.Integrable (fun ω => S.dVar.indicator true ω / S.cutoffProp Λ c ω) P.μ ∧ MeasureTheory.Integrable (fun ω => S.dVar.indicator true ω * if c ω < S.factualY ω then 1 else 0) P.μ ∧ MeasureTheory.Integrable (fun ω => S.dVar.indicator true ω * S.wMin Λ ω) P.μ ∧ MeasureTheory.Integrable (fun ω => (S.wMax Λ ω - S.wMin Λ ω) * (S.dVar.indicator true ω * if c ω < S.factualY ω then 1 else 0)) P.μ ∧ MeasureTheory.Integrable (fun ω => S.dVar.indicator true ω * |S.factualY ω| * S.wMax Λ ω) P.μ ∧ MeasureTheory.Integrable (fun ω => S.dVar.indicator true ω * S.wMax Λ ω) P.μ ∧ MeasureTheory.Integrable (fun ω => |c ω| * S.dVar.indicator true ω * S.wMax Λ ω) P.μ) → (∀ (c : P.Ω → ℝ), Measurable c → MeasureTheory.Integrable c P.μ ∧ MeasureTheory.Integrable (fun ω => S.dVar.indicator true ω / S.lowerCutoffProp Λ c ω) P.μ ∧ MeasureTheory.Integrable (fun ω => S.dVar.indicator true ω * if c ω < S.factualY ω then 1 else 0) P.μ ∧ MeasureTheory.Integrable (fun ω => S.dVar.indicator true ω * S.wMax Λ ω) P.μ ∧ MeasureTheory.Integrable (fun ω => (S.wMax Λ ω - S.wMin Λ ω) * (S.dVar.indicator true ω * if c ω < S.factualY ω then 1 else 0)) P.μ ∧ MeasureTheory.Integrable (fun ω => S.dVar.indicator true ω * |S.factualY ω| * S.wMax Λ ω) P.μ ∧ MeasureTheory.Integrable (fun ω => |c ω| * S.dVar.indicator true ω * S.wMax Λ ω) P.μ) → (∀ᵐ (ω : P.Ω) ∂P.μ, 0 < S.propScore false ω ∧ S.propScore false ω < 1) → (∀ (a : γ), Continuous ↑(ProbabilityTheory.condCDF S.controlXYLaw a)) → (∀ᵐ (ω : P.Ω) ∂P.μ, 0 < S.calibLevel0 Λ ω ∧ S.calibLevel0 Λ ω < 1) → (∀ᵐ (ω : P.Ω) ∂P.μ, 0 < S.calibLevelLower0 Λ ω ∧ S.calibLevelLower0 Λ ω < 1) → BddAbove (S.candMean0 '' S.MSMSetCalib0 Λ) → (∀ etilde ∈ S.MSMSetCalib0 Λ, AEMeasurable etilde P.μ) → (∀ (c : P.Ω → ℝ), Measurable c → MeasureTheory.Integrable c P.μ ∧ MeasureTheory.Integrable (fun ω => S.dVar.indicator false ω / S.cutoffProp0 Λ c ω) P.μ ∧ MeasureTheory.Integrable (fun ω => S.dVar.indicator false ω * if c ω < S.factualY ω then 1 else 0) P.μ ∧ MeasureTheory.Integrable (fun ω => S.dVar.indicator false ω * S.wMin0 Λ ω) P.μ ∧ MeasureTheory.Integrable (fun ω => (S.wMax0 Λ ω - S.wMin0 Λ ω) * (S.dVar.indicator false ω * if c ω < S.factualY ω then 1 else 0)) P.μ ∧ MeasureTheory.Integrable (fun ω => S.dVar.indicator false ω * |S.factualY ω| * S.wMax0 Λ ω) P.μ ∧ MeasureTheory.Integrable (fun ω => S.dVar.indicator false ω * S.wMax0 Λ ω) P.μ ∧ MeasureTheory.Integrable (fun ω => |c ω| * S.dVar.indicator false ω * S.wMax0 Λ ω) P.μ) → (∀ (c : P.Ω → ℝ), Measurable c → MeasureTheory.Integrable c P.μ ∧ MeasureTheory.Integrable (fun ω => S.dVar.indicator false ω / S.lowerCutoffProp0 Λ c ω) P.μ ∧ MeasureTheory.Integrable (fun ω => S.dVar.indicator false ω * if c ω < S.factualY ω then 1 else 0) P.μ ∧ MeasureTheory.Integrable (fun ω => S.dVar.indicator false ω * S.wMax0 Λ ω) P.μ ∧ MeasureTheory.Integrable (fun ω => (S.wMax0 Λ ω - S.wMin0 Λ ω) * (S.dVar.indicator false ω * if c ω < S.factualY ω then 1 else 0)) P.μ ∧ MeasureTheory.Integrable (fun ω => S.dVar.indicator false ω * |S.factualY ω| * S.wMax0 Λ ω) P.μ ∧ MeasureTheory.Integrable (fun ω => |c ω| * S.dVar.indicator false ω * S.wMax0 Λ ω) P.μ) → ∃ cTU cTL cCU cCL, (Measurable cTU ∧ Measurable cTL ∧ Measurable cCU ∧ Measurable cCL) ∧ S.ateUpperCalib Λ = S.candMean (S.cutoffProp Λ cTU) - S.candMean0 (S.lowerCutoffProp0 Λ cCL) ∧ S.ateLowerCalib Λ = S.candMean (S.lowerCutoffProp Λ cTL) - S.candMean0 (S.cutoffProp0 Λ cCU)","l":47,"f":"Causalean/PO/ID/Partial/Sensitivity/MSM/ATEClosedForm.lean"},{"n":"Causalean.PO.POBackdoorSystem.ate_mem_Icc","k":"theorem","m":"Causalean.PO.ID.Partial.Sensitivity.MSM.ATE","a":"PO/ID/Partial/Sensitivity/MSM","t":1,"d":"The ZSB ATE interval is valid. If the treated arm's mean potential outcome E[Y(1)] lies in the uncalibrated ZSB interval [msmLower Λ, msmUpper Λ] and the control arm's mean potential outcome E[Y(0)] lies in the uncalibrated ZSB interval [msmLower0 Λ, msmUpper0 Λ], then the true average treatment effect τ = E[Y(1)] − E[Y(0)] lies in the ZSB interval [ateLower Λ, ateUpper Λ].","s":"∀ {P : Causalean.PO.POSystem} {γ : Type u_1} [inst : MeasurableSpace γ] (S : Causalean.PO.POBackdoorSystem P γ) (Λ : ℝ), S.Y1mean ∈ Set.Icc (S.msmLower Λ) (S.msmUpper Λ) → S.Y0mean ∈ Set.Icc (S.msmLower0 Λ) (S.msmUpper0 Λ) → S.ate ∈ Set.Icc (S.ateLower Λ) (S.ateUpper Λ)","l":86,"f":"Causalean/PO/ID/Partial/Sensitivity/MSM/ATE.lean"},{"n":"Causalean.PO.POBackdoorSystem.ate_mem_Icc_calib","k":"theorem","m":"Causalean.PO.ID.Partial.Sensitivity.MSM.ATE","a":"PO/ID/Partial/Sensitivity/MSM","t":1,"d":"The sharp ATE interval is valid. If the treated arm's mean potential outcome E[Y(1)] lies in the calibrated sharp interval [msmLowerCalib Λ, msmUpperCalib Λ] and the control arm's mean potential outcome E[Y(0)] lies in the calibrated sharp interval [msmLowerCalib0 Λ, msmUpperCalib0 Λ], then the true average treatment effect τ = E[Y(1)] − E[Y(0)] lies in the sharp interval [ateLowerCalib Λ, ateUpperCalib Λ].","s":"∀ {P : Causalean.PO.POSystem} {γ : Type u_1} [inst : MeasurableSpace γ] (S : Causalean.PO.POBackdoorSystem P γ) (Λ : ℝ), S.Y1mean ∈ Set.Icc (S.msmLowerCalib Λ) (S.msmUpperCalib Λ) → S.Y0mean ∈ Set.Icc (S.msmLowerCalib0 Λ) (S.msmUpperCalib0 Λ) → S.ate ∈ Set.Icc (S.ateLowerCalib Λ) (S.ateUpperCalib Λ)","l":74,"f":"Causalean/PO/ID/Partial/Sensitivity/MSM/ATE.lean"},{"n":"Causalean.PO.POBackdoorSystem.ate_mem_Icc_cutoff","k":"theorem","m":"Causalean.PO.ID.Partial.Sensitivity.MSM.ATEClosedForm","a":"PO/ID/Partial/Sensitivity/MSM","t":1,"d":"The true ATE lies in the sharp closed-form interval. Under the same treated-arm and control-arm regularity conditions as ate_endpoints_eq_cutoff — a sensitivity parameter Λ greater than 1; for the treated arm, propensity overlap, an atomless conditional outcome distribution, calibration-level regularity, boundedness of the calibrated candidate mean, measurability of every candidate propensity, and cutoff integrability regularity; and symmetrically for the control arm (overlap, atomlessness, calibration-level regularity, boundedness, measurability, and cutoff integrability regularity) — together with validity of the treated arm's calibrated sharp interval for E[Y(1)] and validity of the control arm's calibrated sharp interval for E[Y(0)], there exist σ(X)-measurable conditional-quantile cutoffs cTU, cTL, cCU, cCL such that the true average treatment effect τ = E[Y(1)] − E[Y(0)] lies between the closed-form lower endpoint (treated lower-cutoff candidate mean minus control upper-cutoff candidate mean) and the closed-form upper endpoint (treated upper-cutoff candidate mean minus control lower-cutoff candidate mean).","s":"∀ {P : Causalean.PO.POSystem} {γ : Type u_1} [inst : MeasurableSpace γ] (S : Causalean.PO.POBackdoorSystem P γ) (Λ : ℝ), 1 < Λ → (∀ᵐ (ω : P.Ω) ∂P.μ, 0 < S.propScore true ω ∧ S.propScore true ω < 1) → (∀ (a : γ), Continuous ↑(ProbabilityTheory.condCDF S.treatedXYLaw a)) → (∀ᵐ (ω : P.Ω) ∂P.μ, 0 < S.calibLevel Λ ω ∧ S.calibLevel Λ ω < 1) → (∀ᵐ (ω : P.Ω) ∂P.μ, 0 < S.calibLevelLower Λ ω ∧ S.calibLevelLower Λ ω < 1) → BddAbove (S.candMean '' S.MSMSetCalib Λ) → (∀ etilde ∈ S.MSMSetCalib Λ, AEMeasurable etilde P.μ) → (∀ (c : P.Ω → ℝ), Measurable c → MeasureTheory.Integrable c P.μ ∧ MeasureTheory.Integrable (fun ω => S.dVar.indicator true ω / S.cutoffProp Λ c ω) P.μ ∧ MeasureTheory.Integrable (fun ω => S.dVar.indicator true ω * if c ω < S.factualY ω then 1 else 0) P.μ ∧ MeasureTheory.Integrable (fun ω => S.dVar.indicator true ω * S.wMin Λ ω) P.μ ∧ MeasureTheory.Integrable (fun ω => (S.wMax Λ ω - S.wMin Λ ω) * (S.dVar.indicator true ω * if c ω < S.factualY ω then 1 else 0)) P.μ ∧ MeasureTheory.Integrable (fun ω => S.dVar.indicator true ω * |S.factualY ω| * S.wMax Λ ω) P.μ ∧ MeasureTheory.Integrable (fun ω => S.dVar.indicator true ω * S.wMax Λ ω) P.μ ∧ MeasureTheory.Integrable (fun ω => |c ω| * S.dVar.indicator true ω * S.wMax Λ ω) P.μ) → (∀ (c : P.Ω → ℝ), Measurable c → MeasureTheory.Integrable c P.μ ∧ MeasureTheory.Integrable (fun ω => S.dVar.indicator true ω / S.lowerCutoffProp Λ c ω) P.μ ∧ MeasureTheory.Integrable (fun ω => S.dVar.indicator true ω * if c ω < S.factualY ω then 1 else 0) P.μ ∧ MeasureTheory.Integrable (fun ω => S.dVar.indicator true ω * S.wMax Λ ω) P.μ ∧ MeasureTheory.Integrable (fun ω => (S.wMax Λ ω - S.wMin Λ ω) * (S.dVar.indicator true ω * if c ω < S.factualY ω then 1 else 0)) P.μ ∧ MeasureTheory.Integrable (fun ω => S.dVar.indicator true ω * |S.factualY ω| * S.wMax Λ ω) P.μ ∧ MeasureTheory.Integrable (fun ω => |c ω| * S.dVar.indicator true ω * S.wMax Λ ω) P.μ) → (∀ᵐ (ω : P.Ω) ∂P.μ, 0 < S.propScore false ω ∧ S.propScore false ω < 1) → (∀ (a : γ), Continuous ↑(ProbabilityTheory.condCDF S.controlXYLaw a)) → (∀ᵐ (ω : P.Ω) ∂P.μ, 0 < S.calibLevel0 Λ ω ∧ S.calibLevel0 Λ ω < 1) → (∀ᵐ (ω : P.Ω) ∂P.μ, 0 < S.calibLevelLower0 Λ ω ∧ S.calibLevelLower0 Λ ω < 1) → BddAbove (S.candMean0 '' S.MSMSetCalib0 Λ) → (∀ etilde ∈ S.MSMSetCalib0 Λ, AEMeasurable etilde P.μ) → (∀ (c : P.Ω → ℝ), Measurable c → MeasureTheory.Integrable c P.μ ∧ MeasureTheory.Integrable (fun ω => S.dVar.indicator false ω / S.cutoffProp0 Λ c ω) P.μ ∧ MeasureTheory.Integrable (fun ω => S.dVar.indicator false ω * if c ω < S.factualY ω then 1 else 0) P.μ ∧ MeasureTheory.Integrable (fun ω => S.dVar.indicator false ω * S.wMin0 Λ ω) P.μ ∧ MeasureTheory.Integrable (fun ω => (S.wMax0 Λ ω - S.wMin0 Λ ω) * (S.dVar.indicator false ω * if c ω < S.factualY ω then 1 else 0)) P.μ ∧ MeasureTheory.Integrable (fun ω => S.dVar.indicator false ω * |S.factualY ω| * S.wMax0 Λ ω) P.μ ∧ MeasureTheory.Integrable (fun ω => S.dVar.indicator false ω * S.wMax0 Λ ω) P.μ ∧ MeasureTheory.Integrable (fun ω => |c ω| * S.dVar.indicator false ω * S.wMax0 Λ ω) P.μ) → (∀ (c : P.Ω → ℝ), Measurable c → MeasureTheory.Integrable c P.μ ∧ MeasureTheory.Integrable (fun ω => S.dVar.indicator false ω / S.lowerCutoffProp0 Λ c ω) P.μ ∧ MeasureTheory.Integrable (fun ω => S.dVar.indicator false ω * if c ω < S.factualY ω then 1 else 0) P.μ ∧ MeasureTheory.Integrable (fun ω => S.dVar.indicator false ω * S.wMax0 Λ ω) P.μ ∧ MeasureTheory.Integrable (fun ω => (S.wMax0 Λ ω - S.wMin0 Λ ω) * (S.dVar.indicator false ω * if c ω < S.factualY ω then 1 else 0)) P.μ ∧ MeasureTheory.Integrable (fun ω => S.dVar.indicator false ω * |S.factualY ω| * S.wMax0 Λ ω) P.μ ∧ MeasureTheory.Integrable (fun ω => |c ω| * S.dVar.indicator false ω * S.wMax0 Λ ω) P.μ) → S.Y1mean ∈ Set.Icc (S.msmLowerCalib Λ) (S.msmUpperCalib Λ) → S.Y0mean ∈ Set.Icc (S.msmLowerCalib0 Λ) (S.msmUpperCalib0 Λ) → ∃ cTU cTL cCU cCL, (Measurable cTU ∧ Measurable cTL ∧ Measurable cCU ∧ Measurable cCL) ∧ S.ate ∈ Set.Icc (S.candMean (S.lowerCutoffProp Λ cTL) - S.candMean0 (S.cutoffProp0 Λ cCU)) (S.candMean (S.cutoffProp Λ cTU) - S.candMean0 (S.lowerCutoffProp0 Λ cCL))","l":140,"f":"Causalean/PO/ID/Partial/Sensitivity/MSM/ATEClosedForm.lean"},{"n":"Causalean.PO.POBackdoorSystem.calibLevel","k":"def","m":"Causalean.PO.ID.Partial.Sensitivity.MSM.CutoffConstruct","a":"PO/ID/Partial/Sensitivity/MSM","t":1,"d":"The calibration quantile level τ(ω) = 1 − survTarget(ω)/e(ω): the treated-conditional-CDF level whose quantile is the calibrating cutoff.","s":"{P : Causalean.PO.POSystem} → {γ : Type u_1} → [inst : MeasurableSpace γ] → Causalean.PO.POBackdoorSystem P γ → ℝ → P.Ω → ℝ","l":297,"f":"Causalean/PO/ID/Partial/Sensitivity/MSM/CutoffConstruct.lean"},{"n":"Causalean.PO.POBackdoorSystem.calibLevel0","k":"def","m":"Causalean.PO.ID.Partial.Sensitivity.MSM.ControlCutoffConstruct","a":"PO/ID/Partial/Sensitivity/MSM","t":1,"d":"The calibration quantile level τ(ω) = 1 − survTarget0(ω)/e(ω): the control-conditional-CDF level whose quantile is the calibrating cutoff.","s":"{P : Causalean.PO.POSystem} → {γ : Type u_1} → [inst : MeasurableSpace γ] → Causalean.PO.POBackdoorSystem P γ → ℝ → P.Ω → ℝ","l":275,"f":"Causalean/PO/ID/Partial/Sensitivity/MSM/ControlCutoffConstruct.lean"},{"n":"Causalean.PO.POBackdoorSystem.calibLevelLower","k":"def","m":"Causalean.PO.ID.Partial.Sensitivity.MSM.LowerBound","a":"PO/ID/Partial/Sensitivity/MSM","t":1,"d":"The lower calibration quantile level 1 − survTargetLower/e (= survTarget/e).","s":"{P : Causalean.PO.POSystem} → {γ : Type u_1} → [inst : MeasurableSpace γ] → Causalean.PO.POBackdoorSystem P γ → ℝ → P.Ω → ℝ","l":56,"f":"Causalean/PO/ID/Partial/Sensitivity/MSM/LowerBound.lean"},{"n":"Causalean.PO.POBackdoorSystem.calibLevelLower0","k":"def","m":"Causalean.PO.ID.Partial.Sensitivity.MSM.ControlLowerBound","a":"PO/ID/Partial/Sensitivity/MSM","t":1,"d":"The lower calibration quantile level 1 − survTargetLower0/e (= survTarget/e).","s":"{P : Causalean.PO.POSystem} → {γ : Type u_1} → [inst : MeasurableSpace γ] → Causalean.PO.POBackdoorSystem P γ → ℝ → P.Ω → ℝ","l":45,"f":"Causalean/PO/ID/Partial/Sensitivity/MSM/ControlLowerBound.lean"},{"n":"Causalean.PO.POBackdoorSystem.calibLevel_eq","k":"theorem","m":"Causalean.PO.ID.Partial.Sensitivity.MSM.Gaussian","a":"PO/ID/Partial/Sensitivity/MSM","t":2,"d":"The calibration level is exactly Λ/(Λ+1). Under overlap and 1 < Λ, the sharp-upper-bound quantile level calibLevel = 1 − survTarget/e equals Λ/(Λ+1) a.e., independent of the propensity e(X) — because survTarget = e/(Λ+1). Pure algebra from wMin, wMax.","s":"∀ {P : Causalean.PO.POSystem} {γ : Type u_1} [inst : MeasurableSpace γ] (S : Causalean.PO.POBackdoorSystem P γ) (Λ : ℝ), 1 < Λ → (∀ᵐ (ω : P.Ω) ∂P.μ, 0 < S.propScore true ω ∧ S.propScore true ω < 1) → ∀ᵐ (ω : P.Ω) ∂P.μ, S.calibLevel Λ ω = Λ / (Λ + 1)","l":75,"f":"Causalean/PO/ID/Partial/Sensitivity/MSM/Gaussian.lean"},{"n":"Causalean.PO.POBackdoorSystem.candMean","k":"def","m":"Causalean.PO.ID.Partial.Sensitivity.MSM.Setup","a":"PO/ID/Partial/Sensitivity/MSM","t":1,"d":"The candidate IPW mean E[A · Y / ẽ] for a candidate complete propensity ẽ, where A = 1_{D=1} is the treatment indicator and Y the factual outcome. Evaluated at the true complete propensity this returns E[Y(1)]; traced over the ambiguity set it sweeps out the MSM interval.","s":"{P : Causalean.PO.POSystem} → {γ : Type u_1} → [inst : MeasurableSpace γ] → Causalean.PO.POBackdoorSystem P γ → (P.Ω → ℝ) → ℝ","l":67,"f":"Causalean/PO/ID/Partial/Sensitivity/MSM/Setup.lean"},{"n":"Causalean.PO.POBackdoorSystem.candMean0","k":"def","m":"Causalean.PO.ID.Partial.Sensitivity.MSM.ControlSetup","a":"PO/ID/Partial/Sensitivity/MSM","t":1,"d":"The candidate IPW mean E[(1−A) · Y / ẽ] for a candidate complete control propensity ẽ, where 1−A = 1_{D=0} is the control indicator and Y the factual outcome. Evaluated at the true complete control propensity this returns E[Y(0)].","s":"{P : Causalean.PO.POSystem} → {γ : Type u_1} → [inst : MeasurableSpace γ] → Causalean.PO.POBackdoorSystem P γ → (P.Ω → ℝ) → ℝ","l":54,"f":"Causalean/PO/ID/Partial/Sensitivity/MSM/ControlSetup.lean"},{"n":"Causalean.PO.POBackdoorSystem.candMean0_completeProp0_eq_Y0mean","k":"theorem","m":"Causalean.PO.ID.Partial.Sensitivity.MSM.ControlSetup","a":"PO/ID/Partial/Sensitivity/MSM","t":2,"d":"IPW / tower bridge (control arm). Evaluated at the true complete control propensity e₀ = P[D=0 | σ(X, Y(0))], the candidate IPW mean recovers E[Y(0)]. The D=0 reflection of candMean_completeProp_eq_Y1mean.","s":"∀ {P : Causalean.PO.POSystem} {γ : Type u_1} [inst : MeasurableSpace γ] (S : Causalean.PO.POBackdoorSystem P γ) [StandardBorelSpace P.Ω] [MeasureTheory.IsFiniteMeasure P.μ], P.Consistency → (∀ᵐ (ω : P.Ω) ∂P.μ, 0 < S.completeProp0 ω) → MeasureTheory.Integrable (S.YofD false) P.μ → MeasureTheory.Integrable (fun ω => S.dVar.indicator false ω * S.factualY ω / S.completeProp0 ω) P.μ → S.candMean0 S.completeProp0 = S.Y0mean","l":81,"f":"Causalean/PO/ID/Partial/Sensitivity/MSM/ControlSetup.lean"},{"n":"Causalean.PO.POBackdoorSystem.candMean_completeProp_eq_Y1mean","k":"theorem","m":"Causalean.PO.ID.Partial.Sensitivity.MSM.Setup","a":"PO/ID/Partial/Sensitivity/MSM","t":1,"d":"IPW / tower bridge. Assuming potential and observed outcomes agree under the realized treatment (consistency), the true complete propensity e₀ = P[D=1 | σ(X, Y(1))] is positive almost surely, the treated potential outcome Y(1) is integrable, and the inverse-propensity-weighted candidate A·Y/e₀ is integrable, then evaluated at the true complete propensity, the candidate IPW mean recovers the estimand: candMean e₀ = E[A · Y / e₀] = E[Y(1)].","s":"∀ {P : Causalean.PO.POSystem} {γ : Type u_1} [inst : MeasurableSpace γ] (S : Causalean.PO.POBackdoorSystem P γ) [StandardBorelSpace P.Ω] [MeasureTheory.IsFiniteMeasure P.μ], P.Consistency → (∀ᵐ (ω : P.Ω) ∂P.μ, 0 < S.completeProp ω) → MeasureTheory.Integrable (S.YofD true) P.μ → MeasureTheory.Integrable (fun ω => S.dVar.indicator true ω * S.factualY ω / S.completeProp ω) P.μ → S.candMean S.completeProp = S.Y1mean","l":96,"f":"Causalean/PO/ID/Partial/Sensitivity/MSM/Setup.lean"},{"n":"Causalean.PO.POBackdoorSystem.cate_backdoor","k":"theorem","m":"Causalean.PO.ID.Exact.ATE","a":"PO/ID/Exact","t":1,"d":"CATE-level backdoor identification. Under the backdoor identification assumptions — consistency, unconfoundedness, and two-sided overlap of the propensity score, together with integrability of the potential outcomes, for a treatment arm d, the conditional average treatment effect given the covariates equals the adjusted-outcome functional:","s":"∀ {P : Causalean.PO.POSystem} {γ : Type u_1} [inst : MeasurableSpace γ] (S : Causalean.PO.POBackdoorSystem P γ) [inst_1 : StandardBorelSpace P.Ω] [inst_2 : MeasureTheory.IsFiniteMeasure P.μ], S.Assumptions → ∀ (d : Bool), S.CATE d =ᵐ[P.μ] S.adjustedCE d","l":418,"f":"Causalean/PO/ID/Exact/ATE.lean"},{"n":"Causalean.PO.POBackdoorSystem.cate_backdoor_of_propScore_ne","k":"theorem","m":"Causalean.PO.ID.Exact.ATE","a":"PO/ID/Exact","t":2,"d":"CATE-level backdoor identification (per-arm overlap form). The conditional ATE equals the adjusted-outcome functional under consistency, unconfoundedness, arm-wise outcome integrability, and the single one-sided requirement that the d-arm propensity score is a.s. nonzero (P[D=d | σ(X)] ≠ 0):","s":"∀ {P : Causalean.PO.POSystem} {γ : Type u_1} [inst : MeasurableSpace γ] (S : Causalean.PO.POBackdoorSystem P γ) [inst_1 : StandardBorelSpace P.Ω] [inst_2 : MeasureTheory.IsFiniteMeasure P.μ], P.Consistency → P.CondIndepCF (Causalean.PO.RegimedVar.ofFactual S.dVar) S.cfBundle (Causalean.PO.RegimedVar.ofFactual S.xVar) P.μ → MeasureTheory.Integrable (S.YofD true) P.μ → MeasureTheory.Integrable (S.YofD false) P.μ → ∀ (d : Bool), (∀ᵐ (ω : P.Ω) ∂P.μ, S.propScore d ω ≠ 0) → S.CATE d =ᵐ[P.μ] S.adjustedCE d","l":266,"f":"Causalean/PO/ID/Exact/ATE.lean"},{"n":"Causalean.PO.POBackdoorSystem.cfBundle","k":"def","m":"Causalean.PO.ID.Exact.ATE","a":"PO/ID/Exact","t":1,"d":"The counterfactual bundle collects the treated and untreated potential outcomes for unconfoundedness.","s":"{P : Causalean.PO.POSystem} → {γ : Type u_1} → [inst : MeasurableSpace γ] → Causalean.PO.POBackdoorSystem P γ → Causalean.PO.POCFBundle P","l":111,"f":"Causalean/PO/ID/Exact/ATE.lean"},{"n":"Causalean.PO.POBackdoorSystem.cfUnderLaw_eq_ipwLaw","k":"theorem","m":"Causalean.PO.ID.Exact.QTE.DistributionalBackdoor","a":"PO/ID/Exact/QTE","t":1,"d":"Distributional backdoor identification. Under the distributional backdoor assumption bundle — consistency, unconfoundedness, and common support, for each treatment arm d, the law of the potential outcome Y(d) equals the observable inverse-probability-weighted outcome law ipwLaw d.","s":"∀ {P : Causalean.PO.POSystem} {γ : Type u_1} [inst : MeasurableSpace γ] (S : Causalean.PO.POBackdoorSystem P γ) [inst_1 : StandardBorelSpace P.Ω] [inst_2 : MeasureTheory.IsFiniteMeasure P.μ], S.DistributionalAssumptions → ∀ (d : Bool), S.yVar.cfUnderLaw S.dVar d P.μ = S.ipwLaw d P.μ","l":436,"f":"Causalean/PO/ID/Exact/QTE/DistributionalBackdoor.lean"},{"n":"Causalean.PO.POBackdoorSystem.completeProp","k":"def","m":"Causalean.PO.ID.Partial.Sensitivity.MSM.Setup","a":"PO/ID/Partial/Sensitivity/MSM","t":1,"d":"The complete propensity e₀(X, Y(1)) = P[D=1 | σ(X, Y(1))]: the conditional probability of treatment given covariates *and* the treated potential outcome. Under unconfoundedness it collapses (a.e.) to the observed propensity e(X); in general it is the unobservable object the MSM bounds.","s":"{P : Causalean.PO.POSystem} → {γ : Type u_1} → [inst : MeasurableSpace γ] → Causalean.PO.POBackdoorSystem P γ → P.Ω → ℝ","l":60,"f":"Causalean/PO/ID/Partial/Sensitivity/MSM/Setup.lean"},{"n":"Causalean.PO.POBackdoorSystem.completeProp0","k":"def","m":"Causalean.PO.ID.Partial.Sensitivity.MSM.ControlSetup","a":"PO/ID/Partial/Sensitivity/MSM","t":1,"d":"The complete control propensity P[D=0 | σ(X, Y(0))]: the conditional probability of *being untreated* given covariates and the control potential outcome. Under unconfoundedness it collapses (a.e.) to P[D=0 | σ(X)] = propScore false.","s":"{P : Causalean.PO.POSystem} → {γ : Type u_1} → [inst : MeasurableSpace γ] → Causalean.PO.POBackdoorSystem P γ → P.Ω → ℝ","l":48,"f":"Causalean/PO/ID/Partial/Sensitivity/MSM/ControlSetup.lean"},{"n":"Causalean.PO.POBackdoorSystem.completeProp0_calibrated","k":"theorem","m":"Causalean.PO.ID.Partial.Sensitivity.MSM.ControlSharp","a":"PO/ID/Partial/Sensitivity/MSM","t":2,"d":"The true complete control propensity is calibrated. E[(1−Z) / e₀ | σ(X)] = 1 a.e., where e₀ = P[D=0 | σ(X, Y(0))]. The D=0 reflection of completeProp_calibrated.","s":"∀ {P : Causalean.PO.POSystem} {γ : Type u_1} [inst : MeasurableSpace γ] (S : Causalean.PO.POBackdoorSystem P γ) [StandardBorelSpace P.Ω] [MeasureTheory.IsFiniteMeasure P.μ], (∀ᵐ (ω : P.Ω) ∂P.μ, 0 < S.completeProp0 ω) → MeasureTheory.Integrable (fun ω => S.dVar.indicator false ω / S.completeProp0 ω) P.μ → S.Calibrated0 S.completeProp0","l":56,"f":"Causalean/PO/ID/Partial/Sensitivity/MSM/ControlSharp.lean"},{"n":"Causalean.PO.POBackdoorSystem.completeProp0_mem_MSMSet0","k":"theorem","m":"Causalean.PO.ID.Partial.Sensitivity.MSM.ControlSetup","a":"PO/ID/Partial/Sensitivity/MSM","t":2,"d":"The control MSM assumption: the truth lies in the ambiguity set.","s":"∀ {P : Causalean.PO.POSystem} {γ : Type u_1} [inst : MeasurableSpace γ] (S : Causalean.PO.POBackdoorSystem P γ) (Λ : ℝ), ((∀ᵐ (ω : P.Ω) ∂P.μ, 0 < S.completeProp0 ω ∧ S.completeProp0 ω < 1) ∧ ∀ᵐ (ω : P.Ω) ∂P.μ, 1 / Λ ≤ Causalean.PO.OR (S.completeProp0 ω) (S.propScore false ω) ∧ Causalean.PO.OR (S.completeProp0 ω) (S.propScore false ω) ≤ Λ) → S.completeProp0 ∈ S.MSMSet0 Λ","l":155,"f":"Causalean/PO/ID/Partial/Sensitivity/MSM/ControlSetup.lean"},{"n":"Causalean.PO.POBackdoorSystem.completeProp0_mem_MSMSetCalib0","k":"theorem","m":"Causalean.PO.ID.Partial.Sensitivity.MSM.ControlSharp","a":"PO/ID/Partial/Sensitivity/MSM","t":2,"d":"The true complete control propensity lies in the calibrated set.","s":"∀ {P : Causalean.PO.POSystem} {γ : Type u_1} [inst : MeasurableSpace γ] (S : Causalean.PO.POBackdoorSystem P γ) (Λ : ℝ), S.completeProp0 ∈ S.MSMSet0 Λ → S.Calibrated0 S.completeProp0 → S.completeProp0 ∈ S.MSMSetCalib0 Λ","l":110,"f":"Causalean/PO/ID/Partial/Sensitivity/MSM/ControlSharp.lean"},{"n":"Causalean.PO.POBackdoorSystem.completeProp_calibrated","k":"theorem","m":"Causalean.PO.ID.Partial.Sensitivity.MSM.Sharp","a":"PO/ID/Partial/Sensitivity/MSM","t":2,"d":"The true complete propensity is calibrated. E[Z / e₀ | σ(X)] = 1 a.e., where e₀ = P[D=1 | σ(X, Y(1))]. By the tower property (σ(X) ⊆ σ(X, Y(1))): E[Z/e₀ | σX] = E[ E[Z/e₀ | σ(X,Y(1))] | σX] = E[ (1/e₀)·E[Z|σ(X,Y(1))] | σX] = E[ e₀/e₀ | σX] = 1. Uses the same condExp pull-out + cancellation as the IPW bridge in Setup.lean.","s":"∀ {P : Causalean.PO.POSystem} {γ : Type u_1} [inst : MeasurableSpace γ] (S : Causalean.PO.POBackdoorSystem P γ) [StandardBorelSpace P.Ω] [MeasureTheory.IsFiniteMeasure P.μ], (∀ᵐ (ω : P.Ω) ∂P.μ, 0 < S.completeProp ω) → MeasureTheory.Integrable (fun ω => S.dVar.indicator true ω / S.completeProp ω) P.μ → S.Calibrated S.completeProp","l":70,"f":"Causalean/PO/ID/Partial/Sensitivity/MSM/Sharp.lean"},{"n":"Causalean.PO.POBackdoorSystem.completeProp_mem_MSMSet","k":"theorem","m":"Causalean.PO.ID.Partial.Sensitivity.MSM.Setup","a":"PO/ID/Partial/Sensitivity/MSM","t":2,"d":"The MSM assumption: the truth lies in the ambiguity set. The true complete propensity e₀ belongs to MSMSet Λ, i.e. it is a.e. interior and its odds ratio against the observed propensity e(X) is a.e. within [1/Λ, Λ]. This is precisely the marginal sensitivity assumption; the membership facts are supplied as hMSM.","s":"∀ {P : Causalean.PO.POSystem} {γ : Type u_1} [inst : MeasurableSpace γ] (S : Causalean.PO.POBackdoorSystem P γ) (Λ : ℝ), ((∀ᵐ (ω : P.Ω) ∂P.μ, 0 < S.completeProp ω ∧ S.completeProp ω < 1) ∧ ∀ᵐ (ω : P.Ω) ∂P.μ, 1 / Λ ≤ Causalean.PO.OR (S.completeProp ω) (S.propScore true ω) ∧ Causalean.PO.OR (S.completeProp ω) (S.propScore true ω) ≤ Λ) → S.completeProp ∈ S.MSMSet Λ","l":189,"f":"Causalean/PO/ID/Partial/Sensitivity/MSM/Setup.lean"},{"n":"Causalean.PO.POBackdoorSystem.completeProp_mem_MSMSetCalib","k":"theorem","m":"Causalean.PO.ID.Partial.Sensitivity.MSM.Sharp","a":"PO/ID/Partial/Sensitivity/MSM","t":2,"d":"The true complete propensity lies in the calibrated set when it satisfies MSM membership and calibration.","s":"∀ {P : Causalean.PO.POSystem} {γ : Type u_1} [inst : MeasurableSpace γ] (S : Causalean.PO.POBackdoorSystem P γ) (Λ : ℝ), S.completeProp ∈ S.MSMSet Λ → S.Calibrated S.completeProp → S.completeProp ∈ S.MSMSetCalib Λ","l":132,"f":"Causalean/PO/ID/Partial/Sensitivity/MSM/Sharp.lean"},{"n":"Causalean.PO.POBackdoorSystem.condExp_control_wMax0_eq","k":"theorem","m":"Causalean.PO.ID.Partial.Sensitivity.MSM.ControlCutoff","a":"PO/ID/Partial/Sensitivity/MSM","t":1,"d":"All-upper-weight control calibration value. For a sensitivity parameter Λ, if the product of the control indicator 1{D=0} and the upper calibration weight wMax0 Λ is integrable, then the conditional expectation of that product given the covariate σ-algebra equals, almost everywhere, the upper weight times the control propensity P[D=0∣X].","s":"∀ {P : Causalean.PO.POSystem} {γ : Type u_1} [inst : MeasurableSpace γ] (S : Causalean.PO.POBackdoorSystem P γ) (Λ : ℝ), MeasureTheory.Integrable (fun ω => S.dVar.indicator false ω * S.wMax0 Λ ω) P.μ → P.μ[fun ω => S.dVar.indicator false ω * S.wMax0 Λ ω | S.sigmaX] =ᵐ[P.μ] fun ω => S.wMax0 Λ ω * S.propScore false ω","l":140,"f":"Causalean/PO/ID/Partial/Sensitivity/MSM/ControlCutoff.lean"},{"n":"Causalean.PO.POBackdoorSystem.condExp_control_wMin0_eq","k":"theorem","m":"Causalean.PO.ID.Partial.Sensitivity.MSM.ControlCutoff","a":"PO/ID/Partial/Sensitivity/MSM","t":1,"d":"All-lower-weight control calibration value. For a sensitivity parameter Λ, if the product of the control indicator 1{D=0} and the lower calibration weight wMin0 Λ is integrable, then the conditional expectation of that product given the covariate σ-algebra equals, almost everywhere, the lower weight times the control propensity P[D=0∣X].","s":"∀ {P : Causalean.PO.POSystem} {γ : Type u_1} [inst : MeasurableSpace γ] (S : Causalean.PO.POBackdoorSystem P γ) (Λ : ℝ), MeasureTheory.Integrable (fun ω => S.dVar.indicator false ω * S.wMin0 Λ ω) P.μ → P.μ[fun ω => S.dVar.indicator false ω * S.wMin0 Λ ω | S.sigmaX] =ᵐ[P.μ] fun ω => S.wMin0 Λ ω * S.propScore false ω","l":105,"f":"Causalean/PO/ID/Partial/Sensitivity/MSM/ControlCutoff.lean"},{"n":"Causalean.PO.POBackdoorSystem.condExp_treat_wMax_eq","k":"theorem","m":"Causalean.PO.ID.Partial.Sensitivity.MSM.CutoffExists","a":"PO/ID/Partial/Sensitivity/MSM","t":1,"d":"The all-wMax conditional calibration value is wMax·e. Assuming the product of the treatment indicator and the upper marginal-sensitivity-model weight wMax is integrable, the conditional expectation, given the σ-algebra generated by the covariates, of the treatment indicator times wMax equals wMax times the propensity score, almost everywhere.","s":"∀ {P : Causalean.PO.POSystem} {γ : Type u_1} [inst : MeasurableSpace γ] (S : Causalean.PO.POBackdoorSystem P γ) (Λ : ℝ), MeasureTheory.Integrable (fun ω => S.dVar.indicator true ω * S.wMax Λ ω) P.μ → P.μ[fun ω => S.dVar.indicator true ω * S.wMax Λ ω | S.sigmaX] =ᵐ[P.μ] fun ω => S.wMax Λ ω * S.propScore true ω","l":133,"f":"Causalean/PO/ID/Partial/Sensitivity/MSM/CutoffExists.lean"},{"n":"Causalean.PO.POBackdoorSystem.condExp_treat_wMin_eq","k":"theorem","m":"Causalean.PO.ID.Partial.Sensitivity.MSM.CutoffExists","a":"PO/ID/Partial/Sensitivity/MSM","t":1,"d":"The all-wMin conditional calibration value is wMin·e. Assuming the product of the treatment indicator and the lower marginal-sensitivity-model weight wMin is integrable, the conditional expectation, given the σ-algebra generated by the covariates, of the treatment indicator times wMin equals wMin times the propensity score, almost everywhere.","s":"∀ {P : Causalean.PO.POSystem} {γ : Type u_1} [inst : MeasurableSpace γ] (S : Causalean.PO.POBackdoorSystem P γ) (Λ : ℝ), MeasureTheory.Integrable (fun ω => S.dVar.indicator true ω * S.wMin Λ ω) P.μ → P.μ[fun ω => S.dVar.indicator true ω * S.wMin Λ ω | S.sigmaX] =ᵐ[P.μ] fun ω => S.wMin Λ ω * S.propScore true ω","l":95,"f":"Causalean/PO/ID/Partial/Sensitivity/MSM/CutoffExists.lean"},{"n":"Causalean.PO.POBackdoorSystem.controlCondCDF","k":"def","m":"Causalean.PO.ID.Partial.Sensitivity.MSM.ControlCutoffConstruct","a":"PO/ID/Partial/Sensitivity/MSM","t":1,"d":"The control conditional CDF of Y given X, evaluated along the factual covariate: F(t | X(ω)) = P(Y ≤ t | X = X(ω), D = 0).","s":"{P : Causalean.PO.POSystem} → {γ : Type u_1} → [inst : MeasurableSpace γ] → Causalean.PO.POBackdoorSystem P γ → P.Ω → ℝ → ℝ","l":48,"f":"Causalean/PO/ID/Partial/Sensitivity/MSM/ControlCutoffConstruct.lean"},{"n":"Causalean.PO.POBackdoorSystem.controlCutoffProp_calibrated_of_survival","k":"theorem","m":"Causalean.PO.ID.Partial.Sensitivity.MSM.ControlCutoff","a":"PO/ID/Partial/Sensitivity/MSM","t":1,"d":"A cutoff solving the target control-survival equation is calibrated. Fix a sensitivity parameter Λ strictly greater than 1. If the control propensity P[D=0∣X] lies strictly between 0 and 1 almost everywhere (overlap), the cutoff function c is measurable with respect to the covariate σ-algebra, the ratio of the control indicator 1{D=0} to the cutoff-calibration weight cutoffProp0 Λ c is integrable, the product of the control indicator and the survival-cutoff indicator 1{c<Y} is integrable, the product of the control indicator and the lower weight wMin0 Λ is integrable, the product of the weight spread with the control-indicator survival term is integrable, and the conditional control-survival functional at c equals, almost everywhere, the target survival value survTarget0 Λ, then the cutoff-calibration propensity cutoffProp0 Λ c is calibrated: the conditional expectation of the control indicator divided by it, given the covariates, equals 1 almost everywhere.","s":"∀ {P : Causalean.PO.POSystem} {γ : Type u_1} [inst : MeasurableSpace γ] (S : Causalean.PO.POBackdoorSystem P γ) (Λ : ℝ), 1 < Λ → (∀ᵐ (ω : P.Ω) ∂P.μ, 0 < S.propScore false ω ∧ S.propScore false ω < 1) → ∀ (c : P.Ω → ℝ), Measurable c → MeasureTheory.Integrable (fun ω => S.dVar.indicator false ω / S.cutoffProp0 Λ c ω) P.μ → MeasureTheory.Integrable (fun ω => S.dVar.indicator false ω * if c ω < S.factualY ω then 1 else 0) P.μ → MeasureTheory.Integrable (fun ω => S.dVar.indicator false ω * S.wMin0 Λ ω) P.μ → MeasureTheory.Integrable (fun ω => (S.wMax0 Λ ω - S.wMin0 Λ ω) * (S.dVar.indicator false ω * if c ω < S.factualY ω then 1 else 0)) P.μ → S.controlSurv c =ᵐ[P.μ] S.survTarget0 Λ → S.Calibrated0 (S.cutoffProp0 Λ c)","l":268,"f":"Causalean/PO/ID/Partial/Sensitivity/MSM/ControlCutoff.lean"},{"n":"Causalean.PO.POBackdoorSystem.controlSet","k":"def","m":"Causalean.PO.ID.Partial.Sensitivity.MSM.ControlCutoffConstruct","a":"PO/ID/Partial/Sensitivity/MSM","t":1,"d":"The control set {ω | D(ω) = 0}.","s":"{P : Causalean.PO.POSystem} → {γ : Type u_1} → [inst : MeasurableSpace γ] → Causalean.PO.POBackdoorSystem P γ → Set P.Ω","l":40,"f":"Causalean/PO/ID/Partial/Sensitivity/MSM/ControlCutoffConstruct.lean"},{"n":"Causalean.PO.POBackdoorSystem.controlSurv","k":"def","m":"Causalean.PO.ID.Partial.Sensitivity.MSM.ControlCutoff","a":"PO/ID/Partial/Sensitivity/MSM","t":1,"d":"The conditional control-survival at a cutoff is the conditional mean of untreated units whose outcome lies above the cutoff.","s":"{P : Causalean.PO.POSystem} → {γ : Type u_1} → [inst : MeasurableSpace γ] → Causalean.PO.POBackdoorSystem P γ → (P.Ω → ℝ) → P.Ω → ℝ","l":49,"f":"Causalean/PO/ID/Partial/Sensitivity/MSM/ControlCutoff.lean"},{"n":"Causalean.PO.POBackdoorSystem.controlSurv_const_eq","k":"theorem","m":"Causalean.PO.ID.Partial.Sensitivity.MSM.ControlCutoffConstruct","a":"PO/ID/Partial/Sensitivity/MSM","t":2,"d":"The survival bridge (constant cutoff). The weighted conditional survival equals the control conditional survival scaled by the propensity: E[(1-Z)·1{Y>t} | σ(X)] = e₀(X)·(1 − F₀(t | X)) a.e. The genuine measure-theoretic content relates a conditional expectation under μ to the conditional CDF of the control push-forward law.","s":"∀ {P : Causalean.PO.POSystem} {γ : Type u_1} [inst : MeasurableSpace γ] (S : Causalean.PO.POBackdoorSystem P γ) (t : ℝ), P.μ[fun ω => S.dVar.indicator false ω * if t < S.factualY ω then 1 else 0 | S.sigmaX] =ᵐ[P.μ] fun ω => S.propScore false ω * (1 - S.controlCondCDF ω t)","l":280,"f":"Causalean/PO/ID/Partial/Sensitivity/MSM/ControlCutoffConstruct.lean"},{"n":"Causalean.PO.POBackdoorSystem.controlSurv_eq","k":"theorem","m":"Causalean.PO.ID.Partial.Sensitivity.MSM.ControlCutoffConstruct","a":"PO/ID/Partial/Sensitivity/MSM","t":2,"d":"The survival bridge (functional cutoff). The version of controlSurv_const_eq evaluated at a σ(X)-measurable cutoff c: E[(1-Z)·1{Y>c(X)} | σ(X)] = e₀(X)·(1 − F₀(c(X) | X)) a.e. Since c is σ(X)-measurable it is frozen inside the conditional expectation, reducing to the constant-cutoff bridge fibrewise. This is the form consumed by exists_calibrating_cutoff0.","s":"∀ {P : Causalean.PO.POSystem} {γ : Type u_1} [inst : MeasurableSpace γ] (S : Causalean.PO.POBackdoorSystem P γ) (c : P.Ω → ℝ), Measurable c → S.controlSurv c =ᵐ[P.μ] fun ω => S.propScore false ω * (1 - S.controlCondCDF ω (c ω))","l":579,"f":"Causalean/PO/ID/Partial/Sensitivity/MSM/ControlCutoffConstruct.lean"},{"n":"Causalean.PO.POBackdoorSystem.controlXYLaw","k":"def","m":"Causalean.PO.ID.Partial.Sensitivity.MSM.ControlCutoffConstruct","a":"PO/ID/Partial/Sensitivity/MSM","t":1,"d":"The control push-forward law of (X, Y): the law of the factual covariate and outcome on the sub-population of control units, (μ restricted to {D=0}) ∘ (X, Y)⁻¹.","s":"{P : Causalean.PO.POSystem} → {γ : Type u_1} → [inst : MeasurableSpace γ] → Causalean.PO.POBackdoorSystem P γ → MeasureTheory.Measure (γ × ℝ)","l":43,"f":"Causalean/PO/ID/Partial/Sensitivity/MSM/ControlCutoffConstruct.lean"},{"n":"Causalean.PO.POBackdoorSystem.control_calibValue_eq","k":"theorem","m":"Causalean.PO.ID.Partial.Sensitivity.MSM.ControlCutoff","a":"PO/ID/Partial/Sensitivity/MSM","t":1,"d":"Cutoff control calibration value decomposes into a lower bracket plus a survival term. Fix a sensitivity parameter Λ and a cutoff function c measurable with respect to the covariate σ-algebra. If the ratio of the control indicator 1{D=0} to the cutoff-calibration weight cutoffProp0 Λ c is integrable, the product of the control indicator and the survival-cutoff indicator 1{c<Y} is integrable, the product of the control indicator and the lower weight wMin0 Λ is integrable, and the product of the weight spread wMax0 Λ - wMin0 Λ with the control-indicator survival term is integrable, then the conditional expectation, given the covariate σ-algebra, of the control indicator divided by the cutoff-calibration weight equals, almost everywhere, the lower weight times the control propensity plus the weight spread times the conditional control-survival functional controlSurv c.","s":"∀ {P : Causalean.PO.POSystem} {γ : Type u_1} [inst : MeasurableSpace γ] (S : Causalean.PO.POBackdoorSystem P γ) (Λ : ℝ) (c : P.Ω → ℝ), Measurable c → MeasureTheory.Integrable (fun ω => S.dVar.indicator false ω / S.cutoffProp0 Λ c ω) P.μ → MeasureTheory.Integrable (fun ω => S.dVar.indicator false ω * if c ω < S.factualY ω then 1 else 0) P.μ → MeasureTheory.Integrable (fun ω => S.dVar.indicator false ω * S.wMin0 Λ ω) P.μ → MeasureTheory.Integrable (fun ω => (S.wMax0 Λ ω - S.wMin0 Λ ω) * (S.dVar.indicator false ω * if c ω < S.factualY ω then 1 else 0)) P.μ → P.μ[fun ω => S.dVar.indicator false ω / S.cutoffProp0 Λ c ω | S.sigmaX] =ᵐ[P.μ] fun ω => S.wMin0 Λ ω * S.propScore false ω + (S.wMax0 Λ ω - S.wMin0 Λ ω) * S.controlSurv c ω","l":175,"f":"Causalean/PO/ID/Partial/Sensitivity/MSM/ControlCutoff.lean"},{"n":"Causalean.PO.POBackdoorSystem.cutoffProp","k":"def","m":"Causalean.PO.ID.Partial.Sensitivity.MSM.QuantileBalance","a":"PO/ID/Partial/Sensitivity/MSM","t":1,"d":"The quantile-cutoff complete propensity at cutoff function c: the candidate whose inverse weight is wMax above the cutoff and wMin at or below it, 1 / (wMax if Y > c(X) else wMin).","s":"{P : Causalean.PO.POSystem} → {γ : Type u_1} → [inst : MeasurableSpace γ] → Causalean.PO.POBackdoorSystem P γ → ℝ → (P.Ω → ℝ) → P.Ω → ℝ","l":53,"f":"Causalean/PO/ID/Partial/Sensitivity/MSM/QuantileBalance.lean"},{"n":"Causalean.PO.POBackdoorSystem.cutoffProp0","k":"def","m":"Causalean.PO.ID.Partial.Sensitivity.MSM.ControlCutoff","a":"PO/ID/Partial/Sensitivity/MSM","t":1,"d":"The control quantile-cutoff complete propensity uses the upper weight above the cutoff and the lower weight at or below it.","s":"{P : Causalean.PO.POSystem} → {γ : Type u_1} → [inst : MeasurableSpace γ] → Causalean.PO.POBackdoorSystem P γ → ℝ → (P.Ω → ℝ) → P.Ω → ℝ","l":44,"f":"Causalean/PO/ID/Partial/Sensitivity/MSM/ControlCutoff.lean"},{"n":"Causalean.PO.POBackdoorSystem.cutoffProp0_mem_MSMSet0","k":"theorem","m":"Causalean.PO.ID.Partial.Sensitivity.MSM.ControlCutoff","a":"PO/ID/Partial/Sensitivity/MSM","t":1,"d":"The control cutoff propensity always lies inside the control odds-ratio box. Fix a sensitivity parameter Λ at least 1 and a cutoff function c. If the control propensity P[D=0∣X] lies strictly between 0 and 1 almost everywhere (overlap), then the cutoff-calibration candidate propensity cutoffProp0 Λ c always belongs to the control marginal-sensitivity-model ambiguity set MSMSet0 Λ.","s":"∀ {P : Causalean.PO.POSystem} {γ : Type u_1} [inst : MeasurableSpace γ] (S : Causalean.PO.POBackdoorSystem P γ) (Λ : ℝ), 1 ≤ Λ → (∀ᵐ (ω : P.Ω) ∂P.μ, 0 < S.propScore false ω ∧ S.propScore false ω < 1) → ∀ (c : P.Ω → ℝ), S.cutoffProp0 Λ c ∈ S.MSMSet0 Λ","l":310,"f":"Causalean/PO/ID/Partial/Sensitivity/MSM/ControlCutoff.lean"},{"n":"Causalean.PO.POBackdoorSystem.cutoffProp0_mem_MSMSetCalib0_of_survival","k":"theorem","m":"Causalean.PO.ID.Partial.Sensitivity.MSM.ControlCutoff","a":"PO/ID/Partial/Sensitivity/MSM","t":1,"d":"The cutoff belongs to the calibrated control MSM set whenever it solves the target control-survival equation. Fix a sensitivity parameter Λ strictly greater than 1. If the control propensity P[D=0∣X] lies strictly between 0 and 1 almost everywhere (overlap), the cutoff function c is measurable with respect to the covariate σ-algebra, the ratio of the control indicator 1{D=0} to the cutoff-calibration weight cutoffProp0 Λ c is integrable, the product of the control indicator and the survival-cutoff indicator 1{c<Y} is integrable, the product of the control indicator and the lower weight wMin0 Λ is integrable, the product of the weight spread with the control-indicator survival term is integrable, and the conditional control-survival functional at c equals, almost everywhere, the target survival value survTarget0 Λ, then the cutoff-calibration propensity cutoffProp0 Λ c belongs to the calibrated control MSM set MSMSetCalib0 Λ.","s":"∀ {P : Causalean.PO.POSystem} {γ : Type u_1} [inst : MeasurableSpace γ] (S : Causalean.PO.POBackdoorSystem P γ) (Λ : ℝ), 1 < Λ → (∀ᵐ (ω : P.Ω) ∂P.μ, 0 < S.propScore false ω ∧ S.propScore false ω < 1) → ∀ (c : P.Ω → ℝ), Measurable c → MeasureTheory.Integrable (fun ω => S.dVar.indicator false ω / S.cutoffProp0 Λ c ω) P.μ → MeasureTheory.Integrable (fun ω => S.dVar.indicator false ω * if c ω < S.factualY ω then 1 else 0) P.μ → MeasureTheory.Integrable (fun ω => S.dVar.indicator false ω * S.wMin0 Λ ω) P.μ → MeasureTheory.Integrable (fun ω => (S.wMax0 Λ ω - S.wMin0 Λ ω) * (S.dVar.indicator false ω * if c ω < S.factualY ω then 1 else 0)) P.μ → S.controlSurv c =ᵐ[P.μ] S.survTarget0 Λ → S.cutoffProp0 Λ c ∈ S.MSMSetCalib0 Λ","l":400,"f":"Causalean/PO/ID/Partial/Sensitivity/MSM/ControlCutoff.lean"},{"n":"Causalean.PO.POBackdoorSystem.cutoffProp_calibrated_of_survival","k":"theorem","m":"Causalean.PO.ID.Partial.Sensitivity.MSM.CutoffSelection","a":"PO/ID/Partial/Sensitivity/MSM","t":2,"d":"Calibration from the survival equation. If the conditional treated-survival of the cutoff equals the target, the cutoff is calibrated.","s":"∀ {P : Causalean.PO.POSystem} {γ : Type u_1} [inst : MeasurableSpace γ] (S : Causalean.PO.POBackdoorSystem P γ) (Λ : ℝ), 1 < Λ → (∀ᵐ (ω : P.Ω) ∂P.μ, 0 < S.propScore true ω ∧ S.propScore true ω < 1) → ∀ (c : P.Ω → ℝ), Measurable c → MeasureTheory.Integrable (fun ω => S.dVar.indicator true ω / S.cutoffProp Λ c ω) P.μ → MeasureTheory.Integrable (fun ω => S.dVar.indicator true ω * if c ω < S.factualY ω then 1 else 0) P.μ → MeasureTheory.Integrable (fun ω => S.dVar.indicator true ω * S.wMin Λ ω) P.μ → MeasureTheory.Integrable (fun ω => (S.wMax Λ ω - S.wMin Λ ω) * (S.dVar.indicator true ω * if c ω < S.factualY ω then 1 else 0)) P.μ → S.treatedSurv c =ᵐ[P.μ] S.survTarget Λ → S.Calibrated (S.cutoffProp Λ c)","l":153,"f":"Causalean/PO/ID/Partial/Sensitivity/MSM/CutoffSelection.lean"},{"n":"Causalean.PO.POBackdoorSystem.cutoffProp_mem_MSMSet","k":"theorem","m":"Causalean.PO.ID.Partial.Sensitivity.MSM.CutoffSelection","a":"PO/ID/Partial/Sensitivity/MSM","t":1,"d":"The cutoff propensity is always in the odds-ratio box. Fix a sensitivity parameter Λ at least 1 and assume the propensity score for treatment given the covariates lies strictly between 0 and 1 almost surely (overlap). Then for any measurable cutoff function c, the induced cutoff propensity always lies in the marginal-sensitivity-model odds-ratio ambiguity set, since at each point it equals either the lower or the upper marginal-sensitivity-model weight.","s":"∀ {P : Causalean.PO.POSystem} {γ : Type u_1} [inst : MeasurableSpace γ] (S : Causalean.PO.POBackdoorSystem P γ) (Λ : ℝ), 1 ≤ Λ → (∀ᵐ (ω : P.Ω) ∂P.μ, 0 < S.propScore true ω ∧ S.propScore true ω < 1) → ∀ (c : P.Ω → ℝ), S.cutoffProp Λ c ∈ S.MSMSet Λ","l":185,"f":"Causalean/PO/ID/Partial/Sensitivity/MSM/CutoffSelection.lean"},{"n":"Causalean.PO.POBackdoorSystem.cutoffProp_mem_MSMSetCalib_of_survival","k":"theorem","m":"Causalean.PO.ID.Partial.Sensitivity.MSM.CutoffSelection","a":"PO/ID/Partial/Sensitivity/MSM","t":2,"d":"Calibrated feasibility reduces to the survival equation. If the cutoff c solves the conditional treated-survival equation G(c) = survTarget, then cutoffProp Λ c ∈ MSMSetCalib Λ, discharging the hcut_mem hypothesis of msmUpperCalib_eq_cutoff.","s":"∀ {P : Causalean.PO.POSystem} {γ : Type u_1} [inst : MeasurableSpace γ] (S : Causalean.PO.POBackdoorSystem P γ) (Λ : ℝ), 1 < Λ → (∀ᵐ (ω : P.Ω) ∂P.μ, 0 < S.propScore true ω ∧ S.propScore true ω < 1) → ∀ (c : P.Ω → ℝ), Measurable c → MeasureTheory.Integrable (fun ω => S.dVar.indicator true ω / S.cutoffProp Λ c ω) P.μ → MeasureTheory.Integrable (fun ω => S.dVar.indicator true ω * if c ω < S.factualY ω then 1 else 0) P.μ → MeasureTheory.Integrable (fun ω => S.dVar.indicator true ω * S.wMin Λ ω) P.μ → MeasureTheory.Integrable (fun ω => (S.wMax Λ ω - S.wMin Λ ω) * (S.dVar.indicator true ω * if c ω < S.factualY ω then 1 else 0)) P.μ → S.treatedSurv c =ᵐ[P.μ] S.survTarget Λ → S.cutoffProp Λ c ∈ S.MSMSetCalib Λ","l":276,"f":"Causalean/PO/ID/Partial/Sensitivity/MSM/CutoffSelection.lean"},{"n":"Causalean.PO.POBackdoorSystem.cutoff_calibValue_eq","k":"theorem","m":"Causalean.PO.ID.Partial.Sensitivity.MSM.CutoffSelection","a":"PO/ID/Partial/Sensitivity/MSM","t":1,"d":"Decomposition of the cutoff calibration value. For a σ(X)-measurable cutoff function c, assume the treatment indicator divided by the cutoff propensity is integrable, the treated-survival indicator, weighted by the treatment indicator, is integrable, the product of the treatment indicator and the lower marginal-sensitivity-model weight wMin is integrable, and the gap between the upper and lower weights, weighted by the treatment-weighted treated-survival indicator, is integrable. Then pulling the σ(X)-measurable weights wMin, wMax out of the conditional expectation decomposes the cutoff calibration value: E[Z/cutoffProp Λ c | σ(X)] = wMin·e + (wMax − wMin)·G(c), where G(c) is the conditional treated-survival at cutoff c.","s":"∀ {P : Causalean.PO.POSystem} {γ : Type u_1} [inst : MeasurableSpace γ] (S : Causalean.PO.POBackdoorSystem P γ) (Λ : ℝ) (c : P.Ω → ℝ), Measurable c → MeasureTheory.Integrable (fun ω => S.dVar.indicator true ω / S.cutoffProp Λ c ω) P.μ → MeasureTheory.Integrable (fun ω => S.dVar.indicator true ω * if c ω < S.factualY ω then 1 else 0) P.μ → MeasureTheory.Integrable (fun ω => S.dVar.indicator true ω * S.wMin Λ ω) P.μ → MeasureTheory.Integrable (fun ω => (S.wMax Λ ω - S.wMin Λ ω) * (S.dVar.indicator true ω * if c ω < S.factualY ω then 1 else 0)) P.μ → P.μ[fun ω => S.dVar.indicator true ω / S.cutoffProp Λ c ω | S.sigmaX] =ᵐ[P.μ] fun ω => S.wMin Λ ω * S.propScore true ω + (S.wMax Λ ω - S.wMin Λ ω) * S.treatedSurv c ω","l":61,"f":"Causalean/PO/ID/Partial/Sensitivity/MSM/CutoffSelection.lean"},{"n":"Causalean.PO.POBackdoorSystem.cutoff_optimal","k":"theorem","m":"Causalean.PO.ID.Partial.Sensitivity.MSM.QuantileBalance","a":"PO/ID/Partial/Sensitivity/MSM","t":1,"d":"Optimality of the quantile-cutoff weight (Neyman–Pearson exchange). Fix a sensitivity parameter Λ at least 1 and assume the propensity score for treatment given the covariates lies strictly between 0 and 1 almost surely (overlap). For a σ(X)-measurable, integrable cutoff function c whose induced cutoff candidate is feasible and calibrated, and under envelope-integrability conditions bounding the treated outcome, the treatment-weighted mass, and the cutoff-weighted mass by the upper marginal-sensitivity-model weight, then for any other calibrated, box-feasible candidate complete propensity ẽ that is almost-everywhere measurable, ẽ's candidate mean is at most the cutoff candidate mean — the cutoff weight maximizes the candidate mean among calibrated candidates.","s":"∀ {P : Causalean.PO.POSystem} {γ : Type u_1} [inst : MeasurableSpace γ] (S : Causalean.PO.POBackdoorSystem P γ) (Λ : ℝ), 1 ≤ Λ → (∀ᵐ (ω : P.Ω) ∂P.μ, 0 < S.propScore true ω ∧ S.propScore true ω < 1) → ∀ (c : P.Ω → ℝ), Measurable c → MeasureTheory.Integrable c P.μ → S.cutoffProp Λ c ∈ S.MSMSetCalib Λ → MeasureTheory.Integrable (fun ω => S.dVar.indicator true ω * |S.factualY ω| * S.wMax Λ ω) P.μ → MeasureTheory.Integrable (fun ω => S.dVar.indicator true ω * S.wMax Λ ω) P.μ → MeasureTheory.Integrable (fun ω => |c ω| * S.dVar.indicator true ω * S.wMax Λ ω) P.μ → ∀ {etilde : P.Ω → ℝ}, etilde ∈ S.MSMSetCalib Λ → AEMeasurable etilde P.μ → S.candMean etilde ≤ S.candMean (S.cutoffProp Λ c)","l":59,"f":"Causalean/PO/ID/Partial/Sensitivity/MSM/QuantileBalance.lean"},{"n":"Causalean.PO.POBackdoorSystem.cutoff_optimal0","k":"theorem","m":"Causalean.PO.ID.Partial.Sensitivity.MSM.ControlQuantileBalance","a":"PO/ID/Partial/Sensitivity/MSM","t":1,"d":"Optimality of the control quantile-cutoff weight. Fix a sensitivity parameter Λ at least one. Assume the control propensity score is almost surely strictly between 0 and 1 (two-sided overlap), and let c be a σ(X)-measurable, integrable cutoff function such that the quantile-cutoff propensity it induces lies in the calibrated control ambiguity set. If the envelope, weighted-indicator, and cutoff-weighted envelope integrability conditions bounding the IPW integrands hold, then for any almost-everywhere measurable candidate complete control propensity ẽ in the calibrated ambiguity set, the candidate mean at ẽ is no greater than the candidate mean at the quantile-cutoff weight: the control cutoff attains the maximum over all calibrated candidates.","s":"∀ {P : Causalean.PO.POSystem} {γ : Type u_1} [inst : MeasurableSpace γ] (S : Causalean.PO.POBackdoorSystem P γ) (Λ : ℝ), 1 ≤ Λ → (∀ᵐ (ω : P.Ω) ∂P.μ, 0 < S.propScore false ω ∧ S.propScore false ω < 1) → ∀ (c : P.Ω → ℝ), Measurable c → MeasureTheory.Integrable c P.μ → S.cutoffProp0 Λ c ∈ S.MSMSetCalib0 Λ → MeasureTheory.Integrable (fun ω => S.dVar.indicator false ω * |S.factualY ω| * S.wMax0 Λ ω) P.μ → MeasureTheory.Integrable (fun ω => S.dVar.indicator false ω * S.wMax0 Λ ω) P.μ → MeasureTheory.Integrable (fun ω => |c ω| * S.dVar.indicator false ω * S.wMax0 Λ ω) P.μ → ∀ {etilde : P.Ω → ℝ}, etilde ∈ S.MSMSetCalib0 Λ → AEMeasurable etilde P.μ → S.candMean0 etilde ≤ S.candMean0 (S.cutoffProp0 Λ c)","l":30,"f":"Causalean/PO/ID/Partial/Sensitivity/MSM/ControlQuantileBalance.lean"},{"n":"Causalean.PO.POBackdoorSystem.cutoff_optimal0_lower","k":"theorem","m":"Causalean.PO.ID.Partial.Sensitivity.MSM.ControlLowerBound","a":"PO/ID/Partial/Sensitivity/MSM","t":1,"d":"Optimality of the lower quantile-cutoff weight. Fix a sensitivity parameter Λ at least one. Assume the control propensity score is almost surely strictly between 0 and 1 (two-sided overlap), and let c be a σ(X)-measurable, integrable cutoff function such that the lower quantile-cutoff propensity it induces lies in the calibrated control ambiguity set. If the envelope 1_{D=0}·|Y|·wMax0(Λ) is integrable, the weighted control indicator 1_{D=0}·wMax0(Λ) is integrable, and the cutoff-weighted envelope |c|·1_{D=0}·wMax0(Λ) is integrable, then for any almost-everywhere measurable candidate complete control propensity ẽ in the calibrated ambiguity set, the candidate mean at the lower quantile-cutoff weight is no greater than the candidate mean at ẽ — the lower cutoff attains the minimum over all calibrated candidates.","s":"∀ {P : Causalean.PO.POSystem} {γ : Type u_1} [inst : MeasurableSpace γ] (S : Causalean.PO.POBackdoorSystem P γ) (Λ : ℝ), 1 ≤ Λ → (∀ᵐ (ω : P.Ω) ∂P.μ, 0 < S.propScore false ω ∧ S.propScore false ω < 1) → ∀ (c : P.Ω → ℝ), Measurable c → MeasureTheory.Integrable c P.μ → S.lowerCutoffProp0 Λ c ∈ S.MSMSetCalib0 Λ → MeasureTheory.Integrable (fun ω => S.dVar.indicator false ω * |S.factualY ω| * S.wMax0 Λ ω) P.μ → MeasureTheory.Integrable (fun ω => S.dVar.indicator false ω * S.wMax0 Λ ω) P.μ → MeasureTheory.Integrable (fun ω => |c ω| * S.dVar.indicator false ω * S.wMax0 Λ ω) P.μ → ∀ {etilde : P.Ω → ℝ}, etilde ∈ S.MSMSetCalib0 Λ → AEMeasurable etilde P.μ → S.candMean0 (S.lowerCutoffProp0 Λ c) ≤ S.candMean0 etilde","l":49,"f":"Causalean/PO/ID/Partial/Sensitivity/MSM/ControlLowerBound.lean"},{"n":"Causalean.PO.POBackdoorSystem.cutoff_optimal_lower","k":"theorem","m":"Causalean.PO.ID.Partial.Sensitivity.MSM.LowerBound","a":"PO/ID/Partial/Sensitivity/MSM","t":1,"d":"Optimality of the lower quantile-cutoff weight. Fix a sensitivity parameter Λ at least 1 and assume the propensity score for treatment given the covariates lies strictly between 0 and 1 almost surely (overlap). For a σ(X)-measurable, integrable cutoff function c whose induced lower-cutoff candidate is feasible and calibrated, and under envelope-integrability conditions bounding the treated outcome, the treatment-weighted mass, and the cutoff-weighted mass by the upper marginal-sensitivity-model weight, then for any other calibrated, box-feasible candidate complete propensity ẽ that is almost-everywhere measurable, the lower-cutoff candidate mean is at most ẽ's candidate mean — the lower cutoff minimizes the candidate mean among calibrated candidates. The ≥-mirror of cutoff_optimal.","s":"∀ {P : Causalean.PO.POSystem} {γ : Type u_1} [inst : MeasurableSpace γ] (S : Causalean.PO.POBackdoorSystem P γ) (Λ : ℝ), 1 ≤ Λ → (∀ᵐ (ω : P.Ω) ∂P.μ, 0 < S.propScore true ω ∧ S.propScore true ω < 1) → ∀ (c : P.Ω → ℝ), Measurable c → MeasureTheory.Integrable c P.μ → S.lowerCutoffProp Λ c ∈ S.MSMSetCalib Λ → MeasureTheory.Integrable (fun ω => S.dVar.indicator true ω * |S.factualY ω| * S.wMax Λ ω) P.μ → MeasureTheory.Integrable (fun ω => S.dVar.indicator true ω * S.wMax Λ ω) P.μ → MeasureTheory.Integrable (fun ω => |c ω| * S.dVar.indicator true ω * S.wMax Λ ω) P.μ → ∀ {etilde : P.Ω → ℝ}, etilde ∈ S.MSMSetCalib Λ → AEMeasurable etilde P.μ → S.candMean (S.lowerCutoffProp Λ c) ≤ S.candMean etilde","l":60,"f":"Causalean/PO/ID/Partial/Sensitivity/MSM/LowerBound.lean"},{"n":"Causalean.PO.POBackdoorSystem.dVar","k":"def","m":"Causalean.PO.ID.Exact.ATE","a":"PO/ID/Exact","t":1,"d":"The treatment node is packaged as a binary potential-outcome variable.","s":"{P : Causalean.PO.POSystem} → {γ : Type u_1} → [inst : MeasurableSpace γ] → Causalean.PO.POBackdoorSystem P γ → Causalean.PO.POVar P Bool","l":73,"f":"Causalean/PO/ID/Exact/ATE.lean"},{"n":"Causalean.PO.POBackdoorSystem.eLift","k":"def","m":"Causalean.PO.ID.Exact.ATE","a":"PO/ID/Exact","t":1,"d":"Value-space propensity eLift x (a measurable representative of E[D | X=x]).","s":"{P : Causalean.PO.POSystem} → {γ : Type u_1} → [inst : MeasurableSpace γ] → Causalean.PO.POBackdoorSystem P γ → γ → ℝ","l":538,"f":"Causalean/PO/ID/Exact/ATE.lean"},{"n":"Causalean.PO.POBackdoorSystem.exists_calibrating_cutoff","k":"theorem","m":"Causalean.PO.ID.Partial.Sensitivity.MSM.CutoffConstruct","a":"PO/ID/Partial/Sensitivity/MSM","t":1,"d":"Existence of a calibrating cutoff (treated arm). Fix a sensitivity parameter Λ strictly greater than one. Assume the treated propensity score is almost surely strictly between 0 and 1 (two-sided overlap), that the treated-arm conditional law of the outcome given covariates is atomless, i.e. its conditional CDF is continuous, and that the calibration quantile level lies strictly between 0 and 1 almost everywhere. Then there exists a σ(X)-measurable cutoff function c such that the treatment-weighted conditional survival function at c agrees almost everywhere with the target survival function survTarget Λ; the cutoff is realized as the conditional quantile of the treated outcome law at the calibration level.","s":"∀ {P : Causalean.PO.POSystem} {γ : Type u_1} [inst : MeasurableSpace γ] (S : Causalean.PO.POBackdoorSystem P γ) (Λ : ℝ), 1 < Λ → (∀ᵐ (ω : P.Ω) ∂P.μ, 0 < S.propScore true ω ∧ S.propScore true ω < 1) → (∀ (a : γ), Continuous ↑(ProbabilityTheory.condCDF S.treatedXYLaw a)) → (∀ᵐ (ω : P.Ω) ∂P.μ, 0 < S.calibLevel Λ ω ∧ S.calibLevel Λ ω < 1) → ∃ c, Measurable c ∧ S.treatedSurv c =ᵐ[P.μ] S.survTarget Λ","l":738,"f":"Causalean/PO/ID/Partial/Sensitivity/MSM/CutoffConstruct.lean"},{"n":"Causalean.PO.POBackdoorSystem.exists_calibrating_cutoff0","k":"theorem","m":"Causalean.PO.ID.Partial.Sensitivity.MSM.ControlCutoffConstruct","a":"PO/ID/Partial/Sensitivity/MSM","t":2,"d":"Existence of a calibrating cutoff. Under overlap, 1 < Λ, an atomless control conditional outcome law (condCDF of the control push-forward continuous), and a strictly-interior calibration level, there is a σ(X)-measurable cutoff c solving the survival equation controlSurv c =ᵐ survTarget0 Λ. The cutoff is the conditional quantile Q_{calibLevel0}(X).","s":"∀ {P : Causalean.PO.POSystem} {γ : Type u_1} [inst : MeasurableSpace γ] (S : Causalean.PO.POBackdoorSystem P γ) (Λ : ℝ), 1 < Λ → (∀ᵐ (ω : P.Ω) ∂P.μ, 0 < S.propScore false ω ∧ S.propScore false ω < 1) → (∀ (a : γ), Continuous ↑(ProbabilityTheory.condCDF S.controlXYLaw a)) → (∀ᵐ (ω : P.Ω) ∂P.μ, 0 < S.calibLevel0 Λ ω ∧ S.calibLevel0 Λ ω < 1) → ∃ c, Measurable c ∧ S.controlSurv c =ᵐ[P.μ] S.survTarget0 Λ","l":620,"f":"Causalean/PO/ID/Partial/Sensitivity/MSM/ControlCutoffConstruct.lean"},{"n":"Causalean.PO.POBackdoorSystem.exists_calibrating_cutoff0_lower","k":"theorem","m":"Causalean.PO.ID.Partial.Sensitivity.MSM.ControlLowerBound","a":"PO/ID/Partial/Sensitivity/MSM","t":2,"d":"Existence of a calibrating lower cutoff. The survTargetLower0 analogue of exists_calibrating_cutoff: a σ(X)-measurable c with controlSurv c =ᵐ survTargetLower0 Λ.","s":"∀ {P : Causalean.PO.POSystem} {γ : Type u_1} [inst : MeasurableSpace γ] (S : Causalean.PO.POBackdoorSystem P γ) (Λ : ℝ), 1 < Λ → (∀ᵐ (ω : P.Ω) ∂P.μ, 0 < S.propScore false ω ∧ S.propScore false ω < 1) → (∀ (a : γ), Continuous ↑(ProbabilityTheory.condCDF S.controlXYLaw a)) → (∀ᵐ (ω : P.Ω) ∂P.μ, 0 < S.calibLevelLower0 Λ ω ∧ S.calibLevelLower0 Λ ω < 1) → ∃ c, Measurable c ∧ S.controlSurv c =ᵐ[P.μ] S.survTargetLower0 Λ","l":618,"f":"Causalean/PO/ID/Partial/Sensitivity/MSM/ControlLowerBound.lean"},{"n":"Causalean.PO.POBackdoorSystem.exists_calibrating_cutoff_lower","k":"theorem","m":"Causalean.PO.ID.Partial.Sensitivity.MSM.LowerBound","a":"PO/ID/Partial/Sensitivity/MSM","t":1,"d":"Existence of a calibrating lower cutoff. Fix a sensitivity parameter Λ strictly greater than 1 and assume the propensity score for treatment given the covariates lies strictly between 0 and 1 almost surely (overlap). If the treated outcome's conditional distribution given each covariate value has a continuous cumulative distribution function and the lower calibration quantile level lies strictly between 0 and 1 almost surely, then there exists a σ(X)-measurable cutoff function whose conditional treated-survival equals the lower target survival almost everywhere. The survTargetLower analogue of exists_calibrating_cutoff.","s":"∀ {P : Causalean.PO.POSystem} {γ : Type u_1} [inst : MeasurableSpace γ] (S : Causalean.PO.POBackdoorSystem P γ) (Λ : ℝ), 1 < Λ → (∀ᵐ (ω : P.Ω) ∂P.μ, 0 < S.propScore true ω ∧ S.propScore true ω < 1) → (∀ (a : γ), Continuous ↑(ProbabilityTheory.condCDF S.treatedXYLaw a)) → (∀ᵐ (ω : P.Ω) ∂P.μ, 0 < S.calibLevelLower Λ ω ∧ S.calibLevelLower Λ ω < 1) → ∃ c, Measurable c ∧ S.treatedSurv c =ᵐ[P.μ] S.survTargetLower Λ","l":619,"f":"Causalean/PO/ID/Partial/Sensitivity/MSM/LowerBound.lean"},{"n":"Causalean.PO.POBackdoorSystem.exists_factor_through_factualX","k":"theorem","m":"Causalean.PO.ID.Partial.Sensitivity.MSM.CutoffConstruct","a":"PO/ID/Partial/Sensitivity/MSM","t":2,"d":"σ(X)-measurable functions factor through X. If f is σ(X)-measurable then f = g ∘ X for a measurable g : γ → ℝ. This is standard comap factorization: the conditioning is on the value of X.","s":"∀ {P : Causalean.PO.POSystem} {γ : Type u_1} [inst : MeasurableSpace γ] (S : Causalean.PO.POBackdoorSystem P γ) {f : P.Ω → ℝ}, Measurable f → ∃ g, Measurable g ∧ f = fun ω => g (S.factualX ω)","l":729,"f":"Causalean/PO/ID/Partial/Sensitivity/MSM/CutoffConstruct.lean"},{"n":"Causalean.PO.POBackdoorSystem.exists_propScoreLift","k":"theorem","m":"Causalean.PO.ID.Exact.ATE","a":"PO/ID/Exact","t":2,"d":"The σ(X)-measurable propensity propScore true factors through factualX: a measurable value-space propensity e : γ → ℝ with propScore true = e ∘ factualX (Doob–Dynkin). This is the value-space propensity an estimator learns.","s":"∀ {P : Causalean.PO.POSystem} {γ : Type u_1} [inst : MeasurableSpace γ] (S : Causalean.PO.POBackdoorSystem P γ), ∃ e, Measurable e ∧ S.propScore true = fun ω => e (S.factualX ω)","l":527,"f":"Causalean/PO/ID/Exact/ATE.lean"},{"n":"Causalean.PO.POBackdoorSystem.exists_regFn","k":"theorem","m":"Causalean.PO.ID.Exact.ATE","a":"PO/ID/Exact","t":2,"d":"The σ(D, X)-measurable outcomeReg factors through (D, X): there is a measurable value-space regression function r : Bool × γ → ℝ with outcomeReg = r ∘ factualDX. This is Doob–Dynkin (exists_eq_measurable_comp); r is the function an ML regressor of Y on (D, X) targets.","s":"∀ {P : Causalean.PO.POSystem} {γ : Type u_1} [inst : MeasurableSpace γ] (S : Causalean.PO.POBackdoorSystem P γ), ∃ r, Measurable r ∧ S.outcomeReg = fun ω => r (S.factualDX ω)","l":500,"f":"Causalean/PO/ID/Exact/ATE.lean"},{"n":"Causalean.PO.POBackdoorSystem.factualD","k":"def","m":"Causalean.PO.ID.Exact.ATE","a":"PO/ID/Exact","t":1,"d":"The factual treatment is the observed treatment assignment for each unit.","s":"{P : Causalean.PO.POSystem} → {γ : Type u_1} → [inst : MeasurableSpace γ] → Causalean.PO.POBackdoorSystem P γ → P.Ω → Bool","l":82,"f":"Causalean/PO/ID/Exact/ATE.lean"},{"n":"Causalean.PO.POBackdoorSystem.factualDX","k":"def","m":"Causalean.PO.ID.Exact.ATE","a":"PO/ID/Exact","t":1,"d":"Factual (D, X) pair ω ↦ (factualD ω, factualX ω) : Ω → Bool × γ. The covariate–treatment cell each unit lands in.","s":"{P : Causalean.PO.POSystem} → {γ : Type u_1} → [inst : MeasurableSpace γ] → Causalean.PO.POBackdoorSystem P γ → P.Ω → Bool × γ","l":462,"f":"Causalean/PO/ID/Exact/ATE.lean"},{"n":"Causalean.PO.POBackdoorSystem.factualX","k":"def","m":"Causalean.PO.ID.Exact.ATE","a":"PO/ID/Exact","t":1,"d":"The factual covariate is the observed adjustment covariate for each unit.","s":"{P : Causalean.PO.POSystem} → {γ : Type u_1} → [inst : MeasurableSpace γ] → Causalean.PO.POBackdoorSystem P γ → P.Ω → γ","l":86,"f":"Causalean/PO/ID/Exact/ATE.lean"},{"n":"Causalean.PO.POBackdoorSystem.factualY","k":"def","m":"Causalean.PO.ID.Exact.ATE","a":"PO/ID/Exact","t":1,"d":"The factual outcome is the observed outcome for each unit.","s":"{P : Causalean.PO.POSystem} → {γ : Type u_1} → [inst : MeasurableSpace γ] → Causalean.PO.POBackdoorSystem P γ → P.Ω → ℝ","l":84,"f":"Causalean/PO/ID/Exact/ATE.lean"},{"n":"Causalean.PO.POBackdoorSystem.gaussianCutoff","k":"def","m":"Causalean.PO.ID.Partial.Sensitivity.MSM.Gaussian","a":"PO/ID/Partial/Sensitivity/MSM","t":1,"d":"The explicit Gaussian cutoff c(X) = m(X) + σ(X)·Φ⁻¹(Λ/(Λ+1)), the Λ/(Λ+1) conditional quantile of the treated Gaussian outcome law.","s":"{P : Causalean.PO.POSystem} → {γ : Type u_1} → [inst : MeasurableSpace γ] → Causalean.PO.POBackdoorSystem P γ → (γ → ℝ) → (γ → ℝ) → ℝ → P.Ω → ℝ","l":99,"f":"Causalean/PO/ID/Partial/Sensitivity/MSM/Gaussian.lean"},{"n":"Causalean.PO.POBackdoorSystem.gaussianCutoff_calibrates","k":"theorem","m":"Causalean.PO.ID.Partial.Sensitivity.MSM.Gaussian","a":"PO/ID/Partial/Sensitivity/MSM","t":2,"d":"The explicit Gaussian cutoff calibrates. Under the conditional-Gaussian model, the cutoff m(X) + σ(X)·Φ⁻¹(Λ/(Λ+1)) solves the survival equation treatedSurv c =ᵐ survTarget Λ.","s":"∀ {P : Causalean.PO.POSystem} {γ : Type u_1} [inst : MeasurableSpace γ] (S : Causalean.PO.POBackdoorSystem P γ) (Λ : ℝ), 1 < Λ → (∀ᵐ (ω : P.Ω) ∂P.μ, 0 < S.propScore true ω ∧ S.propScore true ω < 1) → ∀ {m σ : γ → ℝ}, S.GaussianTreatedModel m σ → S.treatedSurv (S.gaussianCutoff m σ Λ) =ᵐ[P.μ] S.survTarget Λ","l":114,"f":"Causalean/PO/ID/Partial/Sensitivity/MSM/Gaussian.lean"},{"n":"Causalean.PO.POBackdoorSystem.gaussianMSMHalfWidthFactor","k":"def","m":"Causalean.PO.ID.Partial.Sensitivity.MSM.GaussianHalfWidth","a":"PO/ID/Partial/Sensitivity/MSM","t":1,"d":"Dorn–Guo Gaussian half-width factor. For τ = Λ/(Λ+1) with Λ ≥ 1, the per-stratum MSM sharp half-width factor is (Λ²−1)/Λ · φ(Φ⁻¹(τ)). This packages the scalar appearing in the Gaussian ATE interval [ψ ± (Λ²−1)/Λ · φ(Φ⁻¹(Λ/(Λ+1))) · E[σ(X)]].","s":"ℝ → ℝ","l":46,"f":"Causalean/PO/ID/Partial/Sensitivity/MSM/GaussianHalfWidth.lean"},{"n":"Causalean.PO.POBackdoorSystem.integrable_condCDF_variable","k":"theorem","m":"Causalean.PO.ID.Partial.Sensitivity.MSM.CutoffConstruct","a":"PO/ID/Partial/Sensitivity/MSM","t":2,"d":"The conditional CDF evaluated at a measurable cutoff is integrable under the first marginal.","s":"∀ {γ : Type u_1} [inst : MeasurableSpace γ] (ρ : MeasureTheory.Measure (γ × ℝ)) [MeasureTheory.IsFiniteMeasure ρ] {q : γ → ℝ}, Measurable q → MeasureTheory.Integrable (fun a => ↑(ProbabilityTheory.condCDF ρ a) (q a)) ρ.fst","l":376,"f":"Causalean/PO/ID/Partial/Sensitivity/MSM/CutoffConstruct.lean"},{"n":"Causalean.PO.POBackdoorSystem.integrable_factualY_of_consistency","k":"theorem","m":"Causalean.PO.ID.Exact.ATE","a":"PO/ID/Exact","t":2,"d":"Factual Y is integrable whenever both treatment-arm potential outcomes are integrable and consistency links the observed outcome to the realized arm.","s":"∀ {P : Causalean.PO.POSystem} {γ : Type u_1} [inst : MeasurableSpace γ] (S : Causalean.PO.POBackdoorSystem P γ), P.Consistency → MeasureTheory.Integrable (S.YofD true) P.μ → MeasureTheory.Integrable (S.YofD false) P.μ → MeasureTheory.Integrable S.factualY P.μ","l":174,"f":"Causalean/PO/ID/Exact/ATE.lean"},{"n":"Causalean.PO.POBackdoorSystem.integral_comp_YofD_eq","k":"theorem","m":"Causalean.PO.ID.Exact.QTE.DistributionalBackdoor","a":"PO/ID/Exact/QTE","t":1,"d":"Core distributional backdoor identity. Under the distributional backdoor assumption bundle, for every treatment arm d and every measurable real function g that is bounded by a constant C, the mean of g applied to the potential outcome Y(d) equals the mean of g applied to the factual outcome, weighted by the inverse-probability-weighting density ipwDensity d.","s":"∀ {P : Causalean.PO.POSystem} {γ : Type u_1} [inst : MeasurableSpace γ] (S : Causalean.PO.POBackdoorSystem P γ) [inst_1 : StandardBorelSpace P.Ω] [inst_2 : MeasureTheory.IsFiniteMeasure P.μ], S.DistributionalAssumptions → ∀ (d : Bool) {g : ℝ → ℝ}, Measurable g → ∀ {C : ℝ}, (∀ (y : ℝ), |g y| ≤ C) → ∫ (ω : P.Ω), g (S.YofD d ω) ∂P.μ = ∫ (ω : P.Ω), g (S.factualY ω) * S.ipwDensity d ω ∂P.μ","l":298,"f":"Causalean/PO/ID/Exact/QTE/DistributionalBackdoor.lean"},{"n":"Causalean.PO.POBackdoorSystem.integral_mul_indicator_eq_integral_mul_propScore","k":"theorem","m":"Causalean.PO.ID.Exact.QTE.DistributionalBackdoor","a":"PO/ID/Exact/QTE","t":2,"d":"Conditional-expectation pull-out. For a σ(X)-strongly-measurable h, ∫ h·1_{D=d} dμ = ∫ h·e_d dμ.","s":"∀ {P : Causalean.PO.POSystem} {γ : Type u_1} [inst : MeasurableSpace γ] (S : Causalean.PO.POBackdoorSystem P γ) [StandardBorelSpace P.Ω] [MeasureTheory.IsFiniteMeasure P.μ] (d : Bool) {h : P.Ω → ℝ}, MeasureTheory.StronglyMeasurable h → MeasureTheory.Integrable (fun ω => h ω * S.dVar.indicator d ω) P.μ → ∫ (ω : P.Ω), h ω * S.dVar.indicator d ω ∂P.μ = ∫ (ω : P.Ω), h ω * S.propScore d ω ∂P.μ","l":139,"f":"Causalean/PO/ID/Exact/QTE/DistributionalBackdoor.lean"},{"n":"Causalean.PO.POBackdoorSystem.ipwDensity","k":"def","m":"Causalean.PO.ID.Exact.QTE.DistributionalBackdoor","a":"PO/ID/Exact/QTE","t":1,"d":"Observable inverse-probability-weighting density for arm d: 1_{T=d} / e_d(X), where e_d = P[T=d | σ(X)] is the propensity score.","s":"{P : Causalean.PO.POSystem} → {γ : Type u_1} → [inst : MeasurableSpace γ] → Causalean.PO.POBackdoorSystem P γ → Bool → P.Ω → ℝ","l":61,"f":"Causalean/PO/ID/Exact/QTE/DistributionalBackdoor.lean"},{"n":"Causalean.PO.POBackdoorSystem.ipwDensity_integrable","k":"theorem","m":"Causalean.PO.ID.Exact.QTE.DistributionalBackdoor","a":"PO/ID/Exact/QTE","t":2,"d":"Integrability of the IPW weight under bare common support.","s":"∀ {P : Causalean.PO.POSystem} {γ : Type u_1} [inst : MeasurableSpace γ] (S : Causalean.PO.POBackdoorSystem P γ) [inst_1 : StandardBorelSpace P.Ω] [inst_2 : MeasureTheory.IsFiniteMeasure P.μ], S.DistributionalAssumptions → ∀ (d : Bool), MeasureTheory.Integrable (S.ipwDensity d) P.μ","l":166,"f":"Causalean/PO/ID/Exact/QTE/DistributionalBackdoor.lean"},{"n":"Causalean.PO.POBackdoorSystem.ipwLaw","k":"def","m":"Causalean.PO.ID.Exact.QTE.DistributionalBackdoor","a":"PO/ID/Exact/QTE","t":1,"d":"Observable IPW law of arm d: the factual outcome Y pushed forward through the reweighted measure (1_{T=d}/e_d) · μ. Under the distributional backdoor assumption bundle used below, this is the law of Y(d).","s":"{P : Causalean.PO.POSystem} → {γ : Type u_1} → [inst : MeasurableSpace γ] → Causalean.PO.POBackdoorSystem P γ → Bool → MeasureTheory.Measure P.Ω → MeasureTheory.Measure ℝ","l":66,"f":"Causalean/PO/ID/Exact/QTE/DistributionalBackdoor.lean"},{"n":"Causalean.PO.POBackdoorSystem.lowerControl_calibValue_eq","k":"theorem","m":"Causalean.PO.ID.Partial.Sensitivity.MSM.ControlLowerBound","a":"PO/ID/Partial/Sensitivity/MSM","t":2,"d":"Decomposition of the lower-cutoff calibration value. E[(1-Z)/lowerCutoffProp0 Λ c | σ(X)] = wMax0·e₀ − (wMax0 − wMin0)·G(c), with G(c) = controlSurv c.","s":"∀ {P : Causalean.PO.POSystem} {γ : Type u_1} [inst : MeasurableSpace γ] (S : Causalean.PO.POBackdoorSystem P γ) (Λ : ℝ) (c : P.Ω → ℝ), Measurable c → MeasureTheory.Integrable (fun ω => S.dVar.indicator false ω / S.lowerCutoffProp0 Λ c ω) P.μ → MeasureTheory.Integrable (fun ω => S.dVar.indicator false ω * if c ω < S.factualY ω then 1 else 0) P.μ → MeasureTheory.Integrable (fun ω => S.dVar.indicator false ω * S.wMax0 Λ ω) P.μ → MeasureTheory.Integrable (fun ω => (S.wMax0 Λ ω - S.wMin0 Λ ω) * (S.dVar.indicator false ω * if c ω < S.factualY ω then 1 else 0)) P.μ → P.μ[fun ω => S.dVar.indicator false ω / S.lowerCutoffProp0 Λ c ω | S.sigmaX] =ᵐ[P.μ] fun ω => S.wMax0 Λ ω * S.propScore false ω - (S.wMax0 Λ ω - S.wMin0 Λ ω) * S.controlSurv c ω","l":397,"f":"Causalean/PO/ID/Partial/Sensitivity/MSM/ControlLowerBound.lean"},{"n":"Causalean.PO.POBackdoorSystem.lowerCutoffProp","k":"def","m":"Causalean.PO.ID.Partial.Sensitivity.MSM.LowerBound","a":"PO/ID/Partial/Sensitivity/MSM","t":1,"d":"The lower quantile-cutoff complete propensity: the candidate whose inverse weight is wMin above the cutoff and wMax at or below it, 1 / (wMin if Y > c(X) else wMax) — the minimizing worst case (opposite of cutoffProp).","s":"{P : Causalean.PO.POSystem} → {γ : Type u_1} → [inst : MeasurableSpace γ] → Causalean.PO.POBackdoorSystem P γ → ℝ → (P.Ω → ℝ) → P.Ω → ℝ","l":45,"f":"Causalean/PO/ID/Partial/Sensitivity/MSM/LowerBound.lean"},{"n":"Causalean.PO.POBackdoorSystem.lowerCutoffProp0","k":"def","m":"Causalean.PO.ID.Partial.Sensitivity.MSM.ControlLowerBound","a":"PO/ID/Partial/Sensitivity/MSM","t":1,"d":"The lower quantile-cutoff complete propensity: the candidate whose inverse weight is wMin0 above the cutoff and wMax0 at or below it, 1 / (wMin0 if Y > c(X) else wMax0) — the minimizing worst case (opposite of cutoffProp).","s":"{P : Causalean.PO.POSystem} → {γ : Type u_1} → [inst : MeasurableSpace γ] → Causalean.PO.POBackdoorSystem P γ → ℝ → (P.Ω → ℝ) → P.Ω → ℝ","l":34,"f":"Causalean/PO/ID/Partial/Sensitivity/MSM/ControlLowerBound.lean"},{"n":"Causalean.PO.POBackdoorSystem.lowerCutoffProp0_calibrated_of_survival","k":"theorem","m":"Causalean.PO.ID.Partial.Sensitivity.MSM.ControlLowerBound","a":"PO/ID/Partial/Sensitivity/MSM","t":2,"d":"Lower-cutoff calibration from the survival equation.","s":"∀ {P : Causalean.PO.POSystem} {γ : Type u_1} [inst : MeasurableSpace γ] (S : Causalean.PO.POBackdoorSystem P γ) (Λ : ℝ), 1 < Λ → (∀ᵐ (ω : P.Ω) ∂P.μ, 0 < S.propScore false ω ∧ S.propScore false ω < 1) → ∀ (c : P.Ω → ℝ), Measurable c → MeasureTheory.Integrable (fun ω => S.dVar.indicator false ω / S.lowerCutoffProp0 Λ c ω) P.μ → MeasureTheory.Integrable (fun ω => S.dVar.indicator false ω * if c ω < S.factualY ω then 1 else 0) P.μ → MeasureTheory.Integrable (fun ω => S.dVar.indicator false ω * S.wMax0 Λ ω) P.μ → MeasureTheory.Integrable (fun ω => (S.wMax0 Λ ω - S.wMin0 Λ ω) * (S.dVar.indicator false ω * if c ω < S.factualY ω then 1 else 0)) P.μ → S.controlSurv c =ᵐ[P.μ] S.survTargetLower0 Λ → S.Calibrated0 (S.lowerCutoffProp0 Λ c)","l":484,"f":"Causalean/PO/ID/Partial/Sensitivity/MSM/ControlLowerBound.lean"},{"n":"Causalean.PO.POBackdoorSystem.lowerCutoffProp0_mem_MSMSet0","k":"theorem","m":"Causalean.PO.ID.Partial.Sensitivity.MSM.ControlLowerBound","a":"PO/ID/Partial/Sensitivity/MSM","t":2,"d":"The lower cutoff propensity is always in the odds-ratio box.","s":"∀ {P : Causalean.PO.POSystem} {γ : Type u_1} [inst : MeasurableSpace γ] (S : Causalean.PO.POBackdoorSystem P γ) (Λ : ℝ), 1 ≤ Λ → (∀ᵐ (ω : P.Ω) ∂P.μ, 0 < S.propScore false ω ∧ S.propScore false ω < 1) → ∀ (c : P.Ω → ℝ), S.lowerCutoffProp0 Λ c ∈ S.MSMSet0 Λ","l":515,"f":"Causalean/PO/ID/Partial/Sensitivity/MSM/ControlLowerBound.lean"},{"n":"Causalean.PO.POBackdoorSystem.lowerCutoffProp0_mem_MSMSetCalib0_of_survival","k":"theorem","m":"Causalean.PO.ID.Partial.Sensitivity.MSM.ControlLowerBound","a":"PO/ID/Partial/Sensitivity/MSM","t":2,"d":"The lower cutoff is calibrated-feasible given the survival equation.","s":"∀ {P : Causalean.PO.POSystem} {γ : Type u_1} [inst : MeasurableSpace γ] (S : Causalean.PO.POBackdoorSystem P γ) (Λ : ℝ), 1 < Λ → (∀ᵐ (ω : P.Ω) ∂P.μ, 0 < S.propScore false ω ∧ S.propScore false ω < 1) → ∀ (c : P.Ω → ℝ), Measurable c → MeasureTheory.Integrable (fun ω => S.dVar.indicator false ω / S.lowerCutoffProp0 Λ c ω) P.μ → MeasureTheory.Integrable (fun ω => S.dVar.indicator false ω * if c ω < S.factualY ω then 1 else 0) P.μ → MeasureTheory.Integrable (fun ω => S.dVar.indicator false ω * S.wMax0 Λ ω) P.μ → MeasureTheory.Integrable (fun ω => (S.wMax0 Λ ω - S.wMin0 Λ ω) * (S.dVar.indicator false ω * if c ω < S.factualY ω then 1 else 0)) P.μ → S.controlSurv c =ᵐ[P.μ] S.survTargetLower0 Λ → S.lowerCutoffProp0 Λ c ∈ S.MSMSetCalib0 Λ","l":601,"f":"Causalean/PO/ID/Partial/Sensitivity/MSM/ControlLowerBound.lean"},{"n":"Causalean.PO.POBackdoorSystem.lowerCutoffProp_calibrated_of_survival","k":"theorem","m":"Causalean.PO.ID.Partial.Sensitivity.MSM.LowerBound","a":"PO/ID/Partial/Sensitivity/MSM","t":2,"d":"Lower-cutoff calibration from the survival equation.","s":"∀ {P : Causalean.PO.POSystem} {γ : Type u_1} [inst : MeasurableSpace γ] (S : Causalean.PO.POBackdoorSystem P γ) (Λ : ℝ), 1 < Λ → (∀ᵐ (ω : P.Ω) ∂P.μ, 0 < S.propScore true ω ∧ S.propScore true ω < 1) → ∀ (c : P.Ω → ℝ), Measurable c → MeasureTheory.Integrable (fun ω => S.dVar.indicator true ω / S.lowerCutoffProp Λ c ω) P.μ → MeasureTheory.Integrable (fun ω => S.dVar.indicator true ω * if c ω < S.factualY ω then 1 else 0) P.μ → MeasureTheory.Integrable (fun ω => S.dVar.indicator true ω * S.wMax Λ ω) P.μ → MeasureTheory.Integrable (fun ω => (S.wMax Λ ω - S.wMin Λ ω) * (S.dVar.indicator true ω * if c ω < S.factualY ω then 1 else 0)) P.μ → S.treatedSurv c =ᵐ[P.μ] S.survTargetLower Λ → S.Calibrated (S.lowerCutoffProp Λ c)","l":485,"f":"Causalean/PO/ID/Partial/Sensitivity/MSM/LowerBound.lean"},{"n":"Causalean.PO.POBackdoorSystem.lowerCutoffProp_mem_MSMSet","k":"theorem","m":"Causalean.PO.ID.Partial.Sensitivity.MSM.LowerBound","a":"PO/ID/Partial/Sensitivity/MSM","t":2,"d":"The lower cutoff propensity is always in the odds-ratio box.","s":"∀ {P : Causalean.PO.POSystem} {γ : Type u_1} [inst : MeasurableSpace γ] (S : Causalean.PO.POBackdoorSystem P γ) (Λ : ℝ), 1 ≤ Λ → (∀ᵐ (ω : P.Ω) ∂P.μ, 0 < S.propScore true ω ∧ S.propScore true ω < 1) → ∀ (c : P.Ω → ℝ), S.lowerCutoffProp Λ c ∈ S.MSMSet Λ","l":516,"f":"Causalean/PO/ID/Partial/Sensitivity/MSM/LowerBound.lean"},{"n":"Causalean.PO.POBackdoorSystem.lowerCutoffProp_mem_MSMSetCalib_of_survival","k":"theorem","m":"Causalean.PO.ID.Partial.Sensitivity.MSM.LowerBound","a":"PO/ID/Partial/Sensitivity/MSM","t":2,"d":"The lower cutoff is calibrated-feasible given the survival equation.","s":"∀ {P : Causalean.PO.POSystem} {γ : Type u_1} [inst : MeasurableSpace γ] (S : Causalean.PO.POBackdoorSystem P γ) (Λ : ℝ), 1 < Λ → (∀ᵐ (ω : P.Ω) ∂P.μ, 0 < S.propScore true ω ∧ S.propScore true ω < 1) → ∀ (c : P.Ω → ℝ), Measurable c → MeasureTheory.Integrable (fun ω => S.dVar.indicator true ω / S.lowerCutoffProp Λ c ω) P.μ → MeasureTheory.Integrable (fun ω => S.dVar.indicator true ω * if c ω < S.factualY ω then 1 else 0) P.μ → MeasureTheory.Integrable (fun ω => S.dVar.indicator true ω * S.wMax Λ ω) P.μ → MeasureTheory.Integrable (fun ω => (S.wMax Λ ω - S.wMin Λ ω) * (S.dVar.indicator true ω * if c ω < S.factualY ω then 1 else 0)) P.μ → S.treatedSurv c =ᵐ[P.μ] S.survTargetLower Λ → S.lowerCutoffProp Λ c ∈ S.MSMSetCalib Λ","l":602,"f":"Causalean/PO/ID/Partial/Sensitivity/MSM/LowerBound.lean"},{"n":"Causalean.PO.POBackdoorSystem.lowerCutoff_calibValue_eq","k":"theorem","m":"Causalean.PO.ID.Partial.Sensitivity.MSM.LowerBound","a":"PO/ID/Partial/Sensitivity/MSM","t":2,"d":"Decomposition of the lower-cutoff calibration value. E[Z/lowerCutoffProp Λ c | σ(X)] = wMax·e − (wMax − wMin)·G(c) (with G(c) = treatedSurv c).","s":"∀ {P : Causalean.PO.POSystem} {γ : Type u_1} [inst : MeasurableSpace γ] (S : Causalean.PO.POBackdoorSystem P γ) (Λ : ℝ) (c : P.Ω → ℝ), Measurable c → MeasureTheory.Integrable (fun ω => S.dVar.indicator true ω / S.lowerCutoffProp Λ c ω) P.μ → MeasureTheory.Integrable (fun ω => S.dVar.indicator true ω * if c ω < S.factualY ω then 1 else 0) P.μ → MeasureTheory.Integrable (fun ω => S.dVar.indicator true ω * S.wMax Λ ω) P.μ → MeasureTheory.Integrable (fun ω => (S.wMax Λ ω - S.wMin Λ ω) * (S.dVar.indicator true ω * if c ω < S.factualY ω then 1 else 0)) P.μ → P.μ[fun ω => S.dVar.indicator true ω / S.lowerCutoffProp Λ c ω | S.sigmaX] =ᵐ[P.μ] fun ω => S.wMax Λ ω * S.propScore true ω - (S.wMax Λ ω - S.wMin Λ ω) * S.treatedSurv c ω","l":399,"f":"Causalean/PO/ID/Partial/Sensitivity/MSM/LowerBound.lean"},{"n":"Causalean.PO.POBackdoorSystem.measurable_YofD","k":"theorem","m":"Causalean.PO.ID.Exact.ATE","a":"PO/ID/Exact","t":2,"d":"Potential outcomes under fixed treatment arms are measurable.","s":"∀ {P : Causalean.PO.POSystem} {γ : Type u_1} [inst : MeasurableSpace γ] (S : Causalean.PO.POBackdoorSystem P γ) (d : Bool), Measurable (S.YofD d)","l":89,"f":"Causalean/PO/ID/Exact/ATE.lean"},{"n":"Causalean.PO.POBackdoorSystem.measurable_condCDF_variable","k":"theorem","m":"Causalean.PO.ID.Partial.Sensitivity.MSM.CutoffConstruct","a":"PO/ID/Partial/Sensitivity/MSM","t":2,"d":"The conditional CDF remains measurable when evaluated at a measurable cutoff.","s":"∀ {γ : Type u_1} [inst : MeasurableSpace γ] (ρ : MeasureTheory.Measure (γ × ℝ)) [MeasureTheory.IsFiniteMeasure ρ] {q : γ → ℝ}, Measurable q → Measurable fun a => ↑(ProbabilityTheory.condCDF ρ a) (q a)","l":350,"f":"Causalean/PO/ID/Partial/Sensitivity/MSM/CutoffConstruct.lean"},{"n":"Causalean.PO.POBackdoorSystem.measurable_eLift","k":"theorem","m":"Causalean.PO.ID.Exact.ATE","a":"PO/ID/Exact","t":2,"d":"The value-space propensity representative is measurable.","s":"∀ {P : Causalean.PO.POSystem} {γ : Type u_1} [inst : MeasurableSpace γ] (S : Causalean.PO.POBackdoorSystem P γ), Measurable S.eLift","l":541,"f":"Causalean/PO/ID/Exact/ATE.lean"},{"n":"Causalean.PO.POBackdoorSystem.measurable_factualD","k":"theorem","m":"Causalean.PO.ID.Exact.ATE","a":"PO/ID/Exact","t":2,"d":"The observed treatment is measurable.","s":"∀ {P : Causalean.PO.POSystem} {γ : Type u_1} [inst : MeasurableSpace γ] (S : Causalean.PO.POBackdoorSystem P γ), Measurable S.factualD","l":92,"f":"Causalean/PO/ID/Exact/ATE.lean"},{"n":"Causalean.PO.POBackdoorSystem.measurable_factualDX","k":"theorem","m":"Causalean.PO.ID.Exact.ATE","a":"PO/ID/Exact","t":2,"d":"The observed treatment-covariate pair is measurable.","s":"∀ {P : Causalean.PO.POSystem} {γ : Type u_1} [inst : MeasurableSpace γ] (S : Causalean.PO.POBackdoorSystem P γ), Measurable S.factualDX","l":467,"f":"Causalean/PO/ID/Exact/ATE.lean"},{"n":"Causalean.PO.POBackdoorSystem.measurable_factualX","k":"theorem","m":"Causalean.PO.ID.Exact.ATE","a":"PO/ID/Exact","t":2,"d":"The observed covariate is measurable.","s":"∀ {P : Causalean.PO.POSystem} {γ : Type u_1} [inst : MeasurableSpace γ] (S : Causalean.PO.POBackdoorSystem P γ), Measurable S.factualX","l":96,"f":"Causalean/PO/ID/Exact/ATE.lean"},{"n":"Causalean.PO.POBackdoorSystem.measurable_factualY","k":"theorem","m":"Causalean.PO.ID.Exact.ATE","a":"PO/ID/Exact","t":2,"d":"The observed outcome is measurable.","s":"∀ {P : Causalean.PO.POSystem} {γ : Type u_1} [inst : MeasurableSpace γ] (S : Causalean.PO.POBackdoorSystem P γ), Measurable S.factualY","l":94,"f":"Causalean/PO/ID/Exact/ATE.lean"},{"n":"Causalean.PO.POBackdoorSystem.measurable_gaussianCutoff","k":"theorem","m":"Causalean.PO.ID.Partial.Sensitivity.MSM.Gaussian","a":"PO/ID/Partial/Sensitivity/MSM","t":2,"d":"The explicit Gaussian cutoff is σ(X)-measurable.","s":"∀ {P : Causalean.PO.POSystem} {γ : Type u_1} [inst : MeasurableSpace γ] (S : Causalean.PO.POBackdoorSystem P γ) {m σ : γ → ℝ}, Measurable m → Measurable σ → ∀ (Λ : ℝ), Measurable (S.gaussianCutoff m σ Λ)","l":104,"f":"Causalean/PO/ID/Partial/Sensitivity/MSM/Gaussian.lean"},{"n":"Causalean.PO.POBackdoorSystem.measurable_regFn","k":"theorem","m":"Causalean.PO.ID.Exact.ATE","a":"PO/ID/Exact","t":2,"d":"The value-space outcome regression is measurable.","s":"∀ {P : Causalean.PO.POSystem} {γ : Type u_1} [inst : MeasurableSpace γ] (S : Causalean.PO.POBackdoorSystem P γ), Measurable S.regFn","l":518,"f":"Causalean/PO/ID/Exact/ATE.lean"},{"n":"Causalean.PO.POBackdoorSystem.msmLower","k":"def","m":"Causalean.PO.ID.Partial.Sensitivity.MSM.Setup","a":"PO/ID/Partial/Sensitivity/MSM","t":1,"d":"The MSM lower bound: the infimum of the candidate mean over the ambiguity set. The smallest value of E[Y(1)] consistent with the sensitivity budget Λ.","s":"{P : Causalean.PO.POSystem} → {γ : Type u_1} → [inst : MeasurableSpace γ] → Causalean.PO.POBackdoorSystem P γ → ℝ → ℝ","l":92,"f":"Causalean/PO/ID/Partial/Sensitivity/MSM/Setup.lean"},{"n":"Causalean.PO.POBackdoorSystem.msmLower0","k":"def","m":"Causalean.PO.ID.Partial.Sensitivity.MSM.ControlSetup","a":"PO/ID/Partial/Sensitivity/MSM","t":1,"d":"The control MSM lower bound: the infimum of the candidate mean over the ambiguity set.","s":"{P : Causalean.PO.POSystem} → {γ : Type u_1} → [inst : MeasurableSpace γ] → Causalean.PO.POBackdoorSystem P γ → ℝ → ℝ","l":77,"f":"Causalean/PO/ID/Partial/Sensitivity/MSM/ControlSetup.lean"},{"n":"Causalean.PO.POBackdoorSystem.msmLower0_anti","k":"theorem","m":"Causalean.PO.ID.Partial.Sensitivity.MSM.ControlSetup","a":"PO/ID/Partial/Sensitivity/MSM","t":2,"d":"The control MSM interval widens with Λ (lower endpoint).","s":"∀ {P : Causalean.PO.POSystem} {γ : Type u_1} [inst : MeasurableSpace γ] (S : Causalean.PO.POBackdoorSystem P γ) {Λ Λ' : ℝ}, 1 ≤ Λ → Λ ≤ Λ' → (S.candMean0 '' S.MSMSet0 Λ).Nonempty → BddBelow (S.candMean0 '' S.MSMSet0 Λ') → S.msmLower0 Λ' ≤ S.msmLower0 Λ","l":209,"f":"Causalean/PO/ID/Partial/Sensitivity/MSM/ControlSetup.lean"},{"n":"Causalean.PO.POBackdoorSystem.msmLower0_le_msmLowerCalib0","k":"theorem","m":"Causalean.PO.ID.Partial.Sensitivity.MSM.ControlSharp","a":"PO/ID/Partial/Sensitivity/MSM","t":2,"d":"The sharp control lower bound is tighter than the ZSB bound.","s":"∀ {P : Causalean.PO.POSystem} {γ : Type u_1} [inst : MeasurableSpace γ] (S : Causalean.PO.POBackdoorSystem P γ) (Λ : ℝ), (S.candMean0 '' S.MSMSetCalib0 Λ).Nonempty → BddBelow (S.candMean0 '' S.MSMSet0 Λ) → S.msmLower0 Λ ≤ S.msmLowerCalib0 Λ","l":150,"f":"Causalean/PO/ID/Partial/Sensitivity/MSM/ControlSharp.lean"},{"n":"Causalean.PO.POBackdoorSystem.msmLowerCalib","k":"def","m":"Causalean.PO.ID.Partial.Sensitivity.MSM.Sharp","a":"PO/ID/Partial/Sensitivity/MSM","t":1,"d":"The sharp lower bound: the infimum of the candidate mean over the calibrated set.","s":"{P : Causalean.PO.POSystem} → {γ : Type u_1} → [inst : MeasurableSpace γ] → Causalean.PO.POBackdoorSystem P γ → ℝ → ℝ","l":67,"f":"Causalean/PO/ID/Partial/Sensitivity/MSM/Sharp.lean"},{"n":"Causalean.PO.POBackdoorSystem.msmLowerCalib0","k":"def","m":"Causalean.PO.ID.Partial.Sensitivity.MSM.ControlSharp","a":"PO/ID/Partial/Sensitivity/MSM","t":1,"d":"The sharp control lower bound: the infimum of the candidate mean over the calibrated set.","s":"{P : Causalean.PO.POSystem} → {γ : Type u_1} → [inst : MeasurableSpace γ] → Causalean.PO.POBackdoorSystem P γ → ℝ → ℝ","l":52,"f":"Causalean/PO/ID/Partial/Sensitivity/MSM/ControlSharp.lean"},{"n":"Causalean.PO.POBackdoorSystem.msmLowerCalib0_eq_cutoff","k":"theorem","m":"Causalean.PO.ID.Partial.Sensitivity.MSM.ControlLowerBound","a":"PO/ID/Partial/Sensitivity/MSM","t":1,"d":"The sharp lower bound has the quantile-balancing closed form. Fix a sensitivity parameter Λ at least one. Under two-sided overlap of the control propensity score, given a σ(X)-measurable, integrable cutoff c whose induced lower quantile-cutoff propensity is itself calibrated-feasible, and assuming the envelope, weighted-indicator, and cutoff-weighted envelope integrability conditions bounding the IPW integrands together with almost-everywhere measurability of every calibrated candidate propensity, the sharp (infimum) lower bound for E[Y(0)] over the calibrated ambiguity set equals the candidate mean evaluated at the lower quantile-cutoff propensity.","s":"∀ {P : Causalean.PO.POSystem} {γ : Type u_1} [inst : MeasurableSpace γ] (S : Causalean.PO.POBackdoorSystem P γ) (Λ : ℝ), 1 ≤ Λ → (∀ᵐ (ω : P.Ω) ∂P.μ, 0 < S.propScore false ω ∧ S.propScore false ω < 1) → ∀ (c : P.Ω → ℝ), Measurable c → MeasureTheory.Integrable c P.μ → S.lowerCutoffProp0 Λ c ∈ S.MSMSetCalib0 Λ → MeasureTheory.Integrable (fun ω => S.dVar.indicator false ω * |S.factualY ω| * S.wMax0 Λ ω) P.μ → MeasureTheory.Integrable (fun ω => S.dVar.indicator false ω * S.wMax0 Λ ω) P.μ → MeasureTheory.Integrable (fun ω => |c ω| * S.dVar.indicator false ω * S.wMax0 Λ ω) P.μ → (∀ etilde ∈ S.MSMSetCalib0 Λ, AEMeasurable etilde P.μ) → S.msmLowerCalib0 Λ = S.candMean0 (S.lowerCutoffProp0 Λ c)","l":362,"f":"Causalean/PO/ID/Partial/Sensitivity/MSM/ControlLowerBound.lean"},{"n":"Causalean.PO.POBackdoorSystem.msmLowerCalib0_eq_cutoff_unconditional","k":"theorem","m":"Causalean.PO.ID.Partial.Sensitivity.MSM.ControlLowerBound","a":"PO/ID/Partial/Sensitivity/MSM","t":1,"d":"The sharp lower bound, unconditionally. Fix a sensitivity parameter Λ strictly greater than one. Assume two-sided overlap of the control propensity score, that the control-arm conditional law of the outcome given covariates is atomless, i.e. its conditional CDF is continuous, and that the lower calibration quantile level lies strictly between 0 and 1 almost everywhere. If every calibrated candidate propensity is almost-everywhere measurable and every σ(X)-measurable cutoff satisfies the integrability conditions needed for the calibration and optimality arguments, then there exists a σ(X)-measurable cutoff function whose induced lower quantile-cutoff propensity is calibrated-feasible, at which the sharp (infimum) lower bound for E[Y(0)] equals the candidate mean.","s":"∀ {P : Causalean.PO.POSystem} {γ : Type u_1} [inst : MeasurableSpace γ] (S : Causalean.PO.POBackdoorSystem P γ) (Λ : ℝ), 1 < Λ → (∀ᵐ (ω : P.Ω) ∂P.μ, 0 < S.propScore false ω ∧ S.propScore false ω < 1) → (∀ (a : γ), Continuous ↑(ProbabilityTheory.condCDF S.controlXYLaw a)) → (∀ᵐ (ω : P.Ω) ∂P.μ, 0 < S.calibLevelLower0 Λ ω ∧ S.calibLevelLower0 Λ ω < 1) → (∀ etilde ∈ S.MSMSetCalib0 Λ, AEMeasurable etilde P.μ) → (∀ (c : P.Ω → ℝ), Measurable c → MeasureTheory.Integrable c P.μ ∧ MeasureTheory.Integrable (fun ω => S.dVar.indicator false ω / S.lowerCutoffProp0 Λ c ω) P.μ ∧ MeasureTheory.Integrable (fun ω => S.dVar.indicator false ω * if c ω < S.factualY ω then 1 else 0) P.μ ∧ MeasureTheory.Integrable (fun ω => S.dVar.indicator false ω * S.wMax0 Λ ω) P.μ ∧ MeasureTheory.Integrable (fun ω => (S.wMax0 Λ ω - S.wMin0 Λ ω) * (S.dVar.indicator false ω * if c ω < S.factualY ω then 1 else 0)) P.μ ∧ MeasureTheory.Integrable (fun ω => S.dVar.indicator false ω * |S.factualY ω| * S.wMax0 Λ ω) P.μ ∧ MeasureTheory.Integrable (fun ω => |c ω| * S.dVar.indicator false ω * S.wMax0 Λ ω) P.μ) → ∃ c, Measurable c ∧ S.lowerCutoffProp0 Λ c ∈ S.MSMSetCalib0 Λ ∧ S.msmLowerCalib0 Λ = S.candMean0 (S.lowerCutoffProp0 Λ c)","l":696,"f":"Causalean/PO/ID/Partial/Sensitivity/MSM/ControlLowerBound.lean"},{"n":"Causalean.PO.POBackdoorSystem.msmLowerCalib_eq_cutoff","k":"theorem","m":"Causalean.PO.ID.Partial.Sensitivity.MSM.LowerBound","a":"PO/ID/Partial/Sensitivity/MSM","t":2,"d":"The sharp lower bound has the quantile-balancing closed form. Given a calibrating lower cutoff, msmLowerCalib Λ = candMean (lowerCutoffProp Λ c).","s":"∀ {P : Causalean.PO.POSystem} {γ : Type u_1} [inst : MeasurableSpace γ] (S : Causalean.PO.POBackdoorSystem P γ) (Λ : ℝ), 1 ≤ Λ → (∀ᵐ (ω : P.Ω) ∂P.μ, 0 < S.propScore true ω ∧ S.propScore true ω < 1) → ∀ (c : P.Ω → ℝ), Measurable c → MeasureTheory.Integrable c P.μ → S.lowerCutoffProp Λ c ∈ S.MSMSetCalib Λ → MeasureTheory.Integrable (fun ω => S.dVar.indicator true ω * |S.factualY ω| * S.wMax Λ ω) P.μ → MeasureTheory.Integrable (fun ω => S.dVar.indicator true ω * S.wMax Λ ω) P.μ → MeasureTheory.Integrable (fun ω => |c ω| * S.dVar.indicator true ω * S.wMax Λ ω) P.μ → (∀ etilde ∈ S.MSMSetCalib Λ, AEMeasurable etilde P.μ) → S.msmLowerCalib Λ = S.candMean (S.lowerCutoffProp Λ c)","l":371,"f":"Causalean/PO/ID/Partial/Sensitivity/MSM/LowerBound.lean"},{"n":"Causalean.PO.POBackdoorSystem.msmLowerCalib_eq_cutoff_unconditional","k":"theorem","m":"Causalean.PO.ID.Partial.Sensitivity.MSM.LowerBound","a":"PO/ID/Partial/Sensitivity/MSM","t":1,"d":"The sharp lower bound, unconditionally. Fix a sensitivity parameter Λ strictly greater than 1 and assume the propensity score for treatment given the covariates lies strictly between 0 and 1 almost surely (overlap). If the treated outcome's conditional distribution given each covariate value has a continuous cumulative distribution function, the lower calibration quantile level lies strictly between 0 and 1 almost surely, every candidate complete propensity in the calibrated ambiguity set is almost-everywhere measurable, and the regularity conditions needed for the lower-cutoff candidate mean and calibration to be well defined hold for every σ(X)-measurable cutoff, then there exists a σ(X)-measurable, calibrated, box-feasible cutoff function at which the sharp lower bound on E[Y(1)] equals the candidate mean of the induced lower-cutoff propensity. The sInf-mirror of msmUpperCalib_eq_cutoff_unconditional.","s":"∀ {P : Causalean.PO.POSystem} {γ : Type u_1} [inst : MeasurableSpace γ] (S : Causalean.PO.POBackdoorSystem P γ) (Λ : ℝ), 1 < Λ → (∀ᵐ (ω : P.Ω) ∂P.μ, 0 < S.propScore true ω ∧ S.propScore true ω < 1) → (∀ (a : γ), Continuous ↑(ProbabilityTheory.condCDF S.treatedXYLaw a)) → (∀ᵐ (ω : P.Ω) ∂P.μ, 0 < S.calibLevelLower Λ ω ∧ S.calibLevelLower Λ ω < 1) → (∀ etilde ∈ S.MSMSetCalib Λ, AEMeasurable etilde P.μ) → (∀ (c : P.Ω → ℝ), Measurable c → MeasureTheory.Integrable c P.μ ∧ MeasureTheory.Integrable (fun ω => S.dVar.indicator true ω / S.lowerCutoffProp Λ c ω) P.μ ∧ MeasureTheory.Integrable (fun ω => S.dVar.indicator true ω * if c ω < S.factualY ω then 1 else 0) P.μ ∧ MeasureTheory.Integrable (fun ω => S.dVar.indicator true ω * S.wMax Λ ω) P.μ ∧ MeasureTheory.Integrable (fun ω => (S.wMax Λ ω - S.wMin Λ ω) * (S.dVar.indicator true ω * if c ω < S.factualY ω then 1 else 0)) P.μ ∧ MeasureTheory.Integrable (fun ω => S.dVar.indicator true ω * |S.factualY ω| * S.wMax Λ ω) P.μ ∧ MeasureTheory.Integrable (fun ω => |c ω| * S.dVar.indicator true ω * S.wMax Λ ω) P.μ) → ∃ c, Measurable c ∧ S.lowerCutoffProp Λ c ∈ S.MSMSetCalib Λ ∧ S.msmLowerCalib Λ = S.candMean (S.lowerCutoffProp Λ c)","l":703,"f":"Causalean/PO/ID/Partial/Sensitivity/MSM/LowerBound.lean"},{"n":"Causalean.PO.POBackdoorSystem.msmLowerForm","k":"def","m":"Causalean.PO.ID.Partial.Sensitivity.MSM.Bounds","a":"PO/ID/Partial/Sensitivity/MSM","t":1,"d":"The closed-form lower bound: wMin where Y ≥ 0, wMax where Y < 0.","s":"{P : Causalean.PO.POSystem} → {γ : Type u_1} → [inst : MeasurableSpace γ] → Causalean.PO.POBackdoorSystem P γ → ℝ → ℝ","l":72,"f":"Causalean/PO/ID/Partial/Sensitivity/MSM/Bounds.lean"},{"n":"Causalean.PO.POBackdoorSystem.msmLower_anti","k":"theorem","m":"Causalean.PO.ID.Partial.Sensitivity.MSM.Setup","a":"PO/ID/Partial/Sensitivity/MSM","t":2,"d":"The MSM interval widens with Λ (lower endpoint). Since MSMSet Λ ⊆ MSMSet Λ', the infimum of the candidate mean can only decrease. Boundedness-below of the larger image is required for the sInf comparison.","s":"∀ {P : Causalean.PO.POSystem} {γ : Type u_1} [inst : MeasurableSpace γ] (S : Causalean.PO.POBackdoorSystem P γ) {Λ Λ' : ℝ}, 1 ≤ Λ → Λ ≤ Λ' → (S.candMean '' S.MSMSet Λ).Nonempty → BddBelow (S.candMean '' S.MSMSet Λ') → S.msmLower Λ' ≤ S.msmLower Λ","l":253,"f":"Causalean/PO/ID/Partial/Sensitivity/MSM/Setup.lean"},{"n":"Causalean.PO.POBackdoorSystem.msmLower_eq","k":"theorem","m":"Causalean.PO.ID.Partial.Sensitivity.MSM.Bounds","a":"PO/ID/Partial/Sensitivity/MSM","t":1,"d":"Closed form of the MSM lower bound. Fix a sensitivity parameter Λ at least 1. If the propensity score lies strictly between 0 and 1 almost everywhere (two-sided overlap), every candidate propensity in the odds-ratio ambiguity set is measurable up to null sets, and the envelope A·|Y|·wMax(Λ) — which dominates every candidate IPW integrand — is integrable, then the infimum of the candidate IPW mean over the ambiguity set is attained pointwise: the MSM lower bound equals E[A·Y·(wMin if Y≥0 else wMax)].","s":"∀ {P : Causalean.PO.POSystem} {γ : Type u_1} [inst : MeasurableSpace γ] (S : Causalean.PO.POBackdoorSystem P γ) (Λ : ℝ), 1 ≤ Λ → (∀ᵐ (ω : P.Ω) ∂P.μ, 0 < S.propScore true ω ∧ S.propScore true ω < 1) → (∀ etilde ∈ S.MSMSet Λ, AEMeasurable etilde P.μ) → MeasureTheory.Integrable (fun ω => S.dVar.indicator true ω * |S.factualY ω| * S.wMax Λ ω) P.μ → S.msmLower Λ = S.msmLowerForm Λ","l":304,"f":"Causalean/PO/ID/Partial/Sensitivity/MSM/Bounds.lean"},{"n":"Causalean.PO.POBackdoorSystem.msmLower_le_msmLowerCalib","k":"theorem","m":"Causalean.PO.ID.Partial.Sensitivity.MSM.Sharp","a":"PO/ID/Partial/Sensitivity/MSM","t":2,"d":"The sharp lower bound is tighter than the ZSB bound: msmLower Λ ≤ msmLowerCalib Λ.","s":"∀ {P : Causalean.PO.POSystem} {γ : Type u_1} [inst : MeasurableSpace γ] (S : Causalean.PO.POBackdoorSystem P γ) (Λ : ℝ), (S.candMean '' S.MSMSetCalib Λ).Nonempty → BddBelow (S.candMean '' S.MSMSet Λ) → S.msmLower Λ ≤ S.msmLowerCalib Λ","l":175,"f":"Causalean/PO/ID/Partial/Sensitivity/MSM/Sharp.lean"},{"n":"Causalean.PO.POBackdoorSystem.msmUpper","k":"def","m":"Causalean.PO.ID.Partial.Sensitivity.MSM.Setup","a":"PO/ID/Partial/Sensitivity/MSM","t":1,"d":"The MSM upper bound: the supremum of the candidate mean over the ambiguity set. The largest value of E[Y(1)] consistent with the sensitivity budget Λ.","s":"{P : Causalean.PO.POSystem} → {γ : Type u_1} → [inst : MeasurableSpace γ] → Causalean.PO.POBackdoorSystem P γ → ℝ → ℝ","l":88,"f":"Causalean/PO/ID/Partial/Sensitivity/MSM/Setup.lean"},{"n":"Causalean.PO.POBackdoorSystem.msmUpper0","k":"def","m":"Causalean.PO.ID.Partial.Sensitivity.MSM.ControlSetup","a":"PO/ID/Partial/Sensitivity/MSM","t":1,"d":"The control MSM upper bound: the supremum of the candidate mean over the ambiguity set.","s":"{P : Causalean.PO.POSystem} → {γ : Type u_1} → [inst : MeasurableSpace γ] → Causalean.PO.POBackdoorSystem P γ → ℝ → ℝ","l":73,"f":"Causalean/PO/ID/Partial/Sensitivity/MSM/ControlSetup.lean"},{"n":"Causalean.PO.POBackdoorSystem.msmUpper0_mono","k":"theorem","m":"Causalean.PO.ID.Partial.Sensitivity.MSM.ControlSetup","a":"PO/ID/Partial/Sensitivity/MSM","t":2,"d":"The control MSM interval widens with Λ (upper endpoint).","s":"∀ {P : Causalean.PO.POSystem} {γ : Type u_1} [inst : MeasurableSpace γ] (S : Causalean.PO.POBackdoorSystem P γ) {Λ Λ' : ℝ}, 1 ≤ Λ → Λ ≤ Λ' → (S.candMean0 '' S.MSMSet0 Λ).Nonempty → BddAbove (S.candMean0 '' S.MSMSet0 Λ') → S.msmUpper0 Λ ≤ S.msmUpper0 Λ'","l":199,"f":"Causalean/PO/ID/Partial/Sensitivity/MSM/ControlSetup.lean"},{"n":"Causalean.PO.POBackdoorSystem.msmUpperCalib","k":"def","m":"Causalean.PO.ID.Partial.Sensitivity.MSM.Sharp","a":"PO/ID/Partial/Sensitivity/MSM","t":1,"d":"The sharp upper bound: the supremum of the candidate mean over the calibrated set.","s":"{P : Causalean.PO.POSystem} → {γ : Type u_1} → [inst : MeasurableSpace γ] → Causalean.PO.POBackdoorSystem P γ → ℝ → ℝ","l":64,"f":"Causalean/PO/ID/Partial/Sensitivity/MSM/Sharp.lean"},{"n":"Causalean.PO.POBackdoorSystem.msmUpperCalib0","k":"def","m":"Causalean.PO.ID.Partial.Sensitivity.MSM.ControlSharp","a":"PO/ID/Partial/Sensitivity/MSM","t":1,"d":"The sharp control upper bound: the supremum of the candidate mean over the calibrated set.","s":"{P : Causalean.PO.POSystem} → {γ : Type u_1} → [inst : MeasurableSpace γ] → Causalean.PO.POBackdoorSystem P γ → ℝ → ℝ","l":48,"f":"Causalean/PO/ID/Partial/Sensitivity/MSM/ControlSharp.lean"},{"n":"Causalean.PO.POBackdoorSystem.msmUpperCalib0_eq_cutoff","k":"theorem","m":"Causalean.PO.ID.Partial.Sensitivity.MSM.ControlQuantileBalance","a":"PO/ID/Partial/Sensitivity/MSM","t":1,"d":"The sharp calibrated control upper bound has the quantile-balancing closed form. Fix a sensitivity parameter Λ at least one. Under two-sided overlap of the control propensity score, given a σ(X)-measurable, integrable cutoff c whose induced quantile-cutoff propensity is itself calibrated-feasible, and assuming the envelope, weighted-indicator, and cutoff-weighted envelope integrability conditions bounding the IPW integrands together with almost-everywhere measurability of every calibrated candidate propensity, the sharp (supremum) upper bound for E[Y(0)] over the calibrated ambiguity set equals the candidate mean evaluated at the quantile-cutoff propensity.","s":"∀ {P : Causalean.PO.POSystem} {γ : Type u_1} [inst : MeasurableSpace γ] (S : Causalean.PO.POBackdoorSystem P γ) (Λ : ℝ), 1 ≤ Λ → (∀ᵐ (ω : P.Ω) ∂P.μ, 0 < S.propScore false ω ∧ S.propScore false ω < 1) → ∀ (c : P.Ω → ℝ), Measurable c → MeasureTheory.Integrable c P.μ → S.cutoffProp0 Λ c ∈ S.MSMSetCalib0 Λ → MeasureTheory.Integrable (fun ω => S.dVar.indicator false ω * |S.factualY ω| * S.wMax0 Λ ω) P.μ → MeasureTheory.Integrable (fun ω => S.dVar.indicator false ω * S.wMax0 Λ ω) P.μ → MeasureTheory.Integrable (fun ω => |c ω| * S.dVar.indicator false ω * S.wMax0 Λ ω) P.μ → (∀ etilde ∈ S.MSMSetCalib0 Λ, AEMeasurable etilde P.μ) → S.msmUpperCalib0 Λ = S.candMean0 (S.cutoffProp0 Λ c)","l":340,"f":"Causalean/PO/ID/Partial/Sensitivity/MSM/ControlQuantileBalance.lean"},{"n":"Causalean.PO.POBackdoorSystem.msmUpperCalib0_eq_cutoff_unconditional","k":"theorem","m":"Causalean.PO.ID.Partial.Sensitivity.MSM.ControlCutoffConstruct","a":"PO/ID/Partial/Sensitivity/MSM","t":1,"d":"The sharp control upper bound has a quantile-balancing closed form, unconditionally. Fix a sensitivity parameter Λ strictly greater than 1. If the control propensity P[D=0∣X] lies strictly between 0 and 1 almost everywhere (overlap), the control outcome's conditional law given the covariates is atomless (its conditional CDF is continuous), the calibration level lies strictly between 0 and 1 almost everywhere, every candidate propensity in the calibrated control ambiguity set is almost-everywhere measurable, and every covariate-measurable cutoff function satisfies the integrability conditions needed to evaluate the calibration and candidate-mean functionals at it, then there exists a covariate-measurable cutoff c such that the cutoff-calibration propensity cutoffProp0 Λ c lies in the calibrated control MSM set and the sharp control upper bound equals the candidate mean at that cutoff, msmUpperCalib0 Λ = candMean0 (cutoffProp0 Λ c).","s":"∀ {P : Causalean.PO.POSystem} {γ : Type u_1} [inst : MeasurableSpace γ] (S : Causalean.PO.POBackdoorSystem P γ) (Λ : ℝ), 1 < Λ → (∀ᵐ (ω : P.Ω) ∂P.μ, 0 < S.propScore false ω ∧ S.propScore false ω < 1) → (∀ (a : γ), Continuous ↑(ProbabilityTheory.condCDF S.controlXYLaw a)) → (∀ᵐ (ω : P.Ω) ∂P.μ, 0 < S.calibLevel0 Λ ω ∧ S.calibLevel0 Λ ω < 1) → BddAbove (S.candMean0 '' S.MSMSetCalib0 Λ) → (∀ etilde ∈ S.MSMSetCalib0 Λ, AEMeasurable etilde P.μ) → (∀ (c : P.Ω → ℝ), Measurable c → MeasureTheory.Integrable c P.μ ∧ MeasureTheory.Integrable (fun ω => S.dVar.indicator false ω / S.cutoffProp0 Λ c ω) P.μ ∧ MeasureTheory.Integrable (fun ω => S.dVar.indicator false ω * if c ω < S.factualY ω then 1 else 0) P.μ ∧ MeasureTheory.Integrable (fun ω => S.dVar.indicator false ω * S.wMin0 Λ ω) P.μ ∧ MeasureTheory.Integrable (fun ω => (S.wMax0 Λ ω - S.wMin0 Λ ω) * (S.dVar.indicator false ω * if c ω < S.factualY ω then 1 else 0)) P.μ ∧ MeasureTheory.Integrable (fun ω => S.dVar.indicator false ω * |S.factualY ω| * S.wMax0 Λ ω) P.μ ∧ MeasureTheory.Integrable (fun ω => S.dVar.indicator false ω * S.wMax0 Λ ω) P.μ ∧ MeasureTheory.Integrable (fun ω => |c ω| * S.dVar.indicator false ω * S.wMax0 Λ ω) P.μ) → ∃ c, Measurable c ∧ S.cutoffProp0 Λ c ∈ S.MSMSetCalib0 Λ ∧ S.msmUpperCalib0 Λ = S.candMean0 (S.cutoffProp0 Λ c)","l":699,"f":"Causalean/PO/ID/Partial/Sensitivity/MSM/ControlCutoffConstruct.lean"},{"n":"Causalean.PO.POBackdoorSystem.msmUpperCalib0_le_msmUpper0","k":"theorem","m":"Causalean.PO.ID.Partial.Sensitivity.MSM.ControlSharp","a":"PO/ID/Partial/Sensitivity/MSM","t":2,"d":"The sharp control upper bound is tighter than the ZSB bound.","s":"∀ {P : Causalean.PO.POSystem} {γ : Type u_1} [inst : MeasurableSpace γ] (S : Causalean.PO.POBackdoorSystem P γ) (Λ : ℝ), (S.candMean0 '' S.MSMSetCalib0 Λ).Nonempty → BddAbove (S.candMean0 '' S.MSMSet0 Λ) → S.msmUpperCalib0 Λ ≤ S.msmUpper0 Λ","l":140,"f":"Causalean/PO/ID/Partial/Sensitivity/MSM/ControlSharp.lean"},{"n":"Causalean.PO.POBackdoorSystem.msmUpperCalib_eq_cutoff","k":"theorem","m":"Causalean.PO.ID.Partial.Sensitivity.MSM.QuantileBalance","a":"PO/ID/Partial/Sensitivity/MSM","t":1,"d":"The sharp upper bound has the quantile-balancing closed form. Fix a sensitivity parameter Λ at least 1 and assume the propensity score for treatment given the covariates lies strictly between 0 and 1 almost surely (overlap). For a σ(X)-measurable, integrable cutoff function c whose induced cutoff candidate is feasible and calibrated, under envelope-integrability conditions bounding the treated outcome, the treatment-weighted mass, and the cutoff-weighted mass by the upper marginal-sensitivity-model weight, and assuming every candidate complete propensity in the calibrated ambiguity set is almost-everywhere measurable, then the sharp upper bound on E[Y(1)] equals the candidate mean of the cutoff propensity built from c.","s":"∀ {P : Causalean.PO.POSystem} {γ : Type u_1} [inst : MeasurableSpace γ] (S : Causalean.PO.POBackdoorSystem P γ) (Λ : ℝ), 1 ≤ Λ → (∀ᵐ (ω : P.Ω) ∂P.μ, 0 < S.propScore true ω ∧ S.propScore true ω < 1) → ∀ (c : P.Ω → ℝ), Measurable c → MeasureTheory.Integrable c P.μ → S.cutoffProp Λ c ∈ S.MSMSetCalib Λ → MeasureTheory.Integrable (fun ω => S.dVar.indicator true ω * |S.factualY ω| * S.wMax Λ ω) P.μ → MeasureTheory.Integrable (fun ω => S.dVar.indicator true ω * S.wMax Λ ω) P.μ → MeasureTheory.Integrable (fun ω => |c ω| * S.dVar.indicator true ω * S.wMax Λ ω) P.μ → (∀ etilde ∈ S.MSMSetCalib Λ, AEMeasurable etilde P.μ) → S.msmUpperCalib Λ = S.candMean (S.cutoffProp Λ c)","l":372,"f":"Causalean/PO/ID/Partial/Sensitivity/MSM/QuantileBalance.lean"},{"n":"Causalean.PO.POBackdoorSystem.msmUpperCalib_eq_cutoff_unconditional","k":"theorem","m":"Causalean.PO.ID.Partial.Sensitivity.MSM.CutoffConstruct","a":"PO/ID/Partial/Sensitivity/MSM","t":1,"d":"The sharp treated upper bound, unconditionally. Fix a sensitivity parameter Λ strictly greater than one. Assume two-sided overlap of the treated propensity score, that the treated-arm conditional outcome law given covariates is atomless, i.e. its conditional CDF is continuous, that the calibration quantile level lies strictly between 0 and 1 almost everywhere, and that the candidate means over the calibrated ambiguity set are bounded above. If every calibrated candidate propensity is almost-everywhere measurable and every σ(X)-measurable cutoff satisfies the integrability conditions needed for the calibration and optimality arguments, then there exists a σ(X)-measurable cutoff function whose induced quantile-cutoff propensity is calibrated-feasible, at which the sharp (supremum) upper bound for E[Y(1)] equals the candidate mean.","s":"∀ {P : Causalean.PO.POSystem} {γ : Type u_1} [inst : MeasurableSpace γ] (S : Causalean.PO.POBackdoorSystem P γ) (Λ : ℝ), 1 < Λ → (∀ᵐ (ω : P.Ω) ∂P.μ, 0 < S.propScore true ω ∧ S.propScore true ω < 1) → (∀ (a : γ), Continuous ↑(ProbabilityTheory.condCDF S.treatedXYLaw a)) → (∀ᵐ (ω : P.Ω) ∂P.μ, 0 < S.calibLevel Λ ω ∧ S.calibLevel Λ ω < 1) → BddAbove (S.candMean '' S.MSMSetCalib Λ) → (∀ etilde ∈ S.MSMSetCalib Λ, AEMeasurable etilde P.μ) → (∀ (c : P.Ω → ℝ), Measurable c → MeasureTheory.Integrable c P.μ ∧ MeasureTheory.Integrable (fun ω => S.dVar.indicator true ω / S.cutoffProp Λ c ω) P.μ ∧ MeasureTheory.Integrable (fun ω => S.dVar.indicator true ω * if c ω < S.factualY ω then 1 else 0) P.μ ∧ MeasureTheory.Integrable (fun ω => S.dVar.indicator true ω * S.wMin Λ ω) P.μ ∧ MeasureTheory.Integrable (fun ω => (S.wMax Λ ω - S.wMin Λ ω) * (S.dVar.indicator true ω * if c ω < S.factualY ω then 1 else 0)) P.μ ∧ MeasureTheory.Integrable (fun ω => S.dVar.indicator true ω * |S.factualY ω| * S.wMax Λ ω) P.μ ∧ MeasureTheory.Integrable (fun ω => S.dVar.indicator true ω * S.wMax Λ ω) P.μ ∧ MeasureTheory.Integrable (fun ω => |c ω| * S.dVar.indicator true ω * S.wMax Λ ω) P.μ) → ∃ c, Measurable c ∧ S.cutoffProp Λ c ∈ S.MSMSetCalib Λ ∧ S.msmUpperCalib Λ = S.candMean (S.cutoffProp Λ c)","l":822,"f":"Causalean/PO/ID/Partial/Sensitivity/MSM/CutoffConstruct.lean"},{"n":"Causalean.PO.POBackdoorSystem.msmUpperCalib_gaussian","k":"theorem","m":"Causalean.PO.ID.Partial.Sensitivity.MSM.Gaussian","a":"PO/ID/Partial/Sensitivity/MSM","t":1,"d":"The sharp upper bound at the explicit Gaussian cutoff. Fix a sensitivity parameter Λ strictly greater than 1 and assume the propensity score for treatment given the covariates lies strictly between 0 and 1 almost surely (overlap). Under the conditional-Gaussian treated-outcome model, i.e. the treated conditional law of the outcome given the covariates is Gaussian with mean m and standard deviation σ, assuming every candidate complete propensity in the calibrated ambiguity set is almost-everywhere measurable and the integrability conditions needed to make the candidate means, the survival decomposition, and the cutoff propensity well defined, the sharp (Dorn–Guo) upper bound on E[Y(1)] equals the candidate IPW mean evaluated at the cutoff propensity built from the explicit Gaussian quantile cutoff m(X) + σ(X)·Φ⁻¹(Λ/(Λ+1)).","s":"∀ {P : Causalean.PO.POSystem} {γ : Type u_1} [inst : MeasurableSpace γ] (S : Causalean.PO.POBackdoorSystem P γ) (Λ : ℝ), 1 < Λ → (∀ᵐ (ω : P.Ω) ∂P.μ, 0 < S.propScore true ω ∧ S.propScore true ω < 1) → ∀ {m σ : γ → ℝ}, S.GaussianTreatedModel m σ → (∀ etilde ∈ S.MSMSetCalib Λ, AEMeasurable etilde P.μ) → MeasureTheory.Integrable (S.gaussianCutoff m σ Λ) P.μ ∧ MeasureTheory.Integrable (fun ω => S.dVar.indicator true ω / S.cutoffProp Λ (S.gaussianCutoff m σ Λ) ω) P.μ ∧ MeasureTheory.Integrable (fun ω => S.dVar.indicator true ω * if S.gaussianCutoff m σ Λ ω < S.factualY ω then 1 else 0) P.μ ∧ MeasureTheory.Integrable (fun ω => S.dVar.indicator true ω * S.wMin Λ ω) P.μ ∧ MeasureTheory.Integrable (fun ω => (S.wMax Λ ω - S.wMin Λ ω) * (S.dVar.indicator true ω * if S.gaussianCutoff m σ Λ ω < S.factualY ω then 1 else 0)) P.μ ∧ MeasureTheory.Integrable (fun ω => S.dVar.indicator true ω * |S.factualY ω| * S.wMax Λ ω) P.μ ∧ MeasureTheory.Integrable (fun ω => S.dVar.indicator true ω * S.wMax Λ ω) P.μ ∧ MeasureTheory.Integrable (fun ω => |S.gaussianCutoff m σ Λ ω| * S.dVar.indicator true ω * S.wMax Λ ω) P.μ → S.msmUpperCalib Λ = S.candMean (S.cutoffProp Λ (S.gaussianCutoff m σ Λ))","l":148,"f":"Causalean/PO/ID/Partial/Sensitivity/MSM/Gaussian.lean"},{"n":"Causalean.PO.POBackdoorSystem.msmUpperCalib_gaussian_halfWidth","k":"theorem","m":"Causalean.PO.ID.Partial.Sensitivity.MSM.GaussianHalfWidth","a":"PO/ID/Partial/Sensitivity/MSM","t":1,"d":"The Dorn–Guo Gaussian sharp upper bound, closed form. Fix a sensitivity parameter Λ strictly greater than 1 and assume the propensity score for treatment given the covariates lies strictly between 0 and 1 almost surely (overlap). Under the conditional-Gaussian treated-outcome model with conditional mean m and conditional standard deviation σ, strengthened with the implied treated conditional mean and truncated-mean identities, assuming every candidate complete propensity in the calibrated ambiguity set is almost-everywhere measurable, the conditional mean m(X) is integrable, (1 − e(X))·σ(X) is integrable, the regularity conditions making the Gaussian-cutoff candidate mean and survival decomposition well defined, and the candidate-mean integrability at the lower weight and at the truncated difference, then the sharp upper bound on E[Y(1)] equals the point-identified mean ∫ m(X) plus the half-width (Λ²−1)/Λ · φ(Φ⁻¹(Λ/(Λ+1))) · ∫ (1 − e(X))·σ(X).","s":"∀ {P : Causalean.PO.POSystem} {γ : Type u_1} [inst : MeasurableSpace γ] (S : Causalean.PO.POBackdoorSystem P γ) (Λ : ℝ), 1 < Λ → (∀ᵐ (ω : P.Ω) ∂P.μ, 0 < S.propScore true ω ∧ S.propScore true ω < 1) → ∀ {m σ : γ → ℝ}, S.GaussianOutcomeModel m σ → (∀ etilde ∈ S.MSMSetCalib Λ, AEMeasurable etilde P.μ) → MeasureTheory.Integrable (fun ω => m (S.factualX ω)) P.μ → MeasureTheory.Integrable (fun ω => (1 - S.propScore true ω) * σ (S.factualX ω)) P.μ → MeasureTheory.Integrable (S.gaussianCutoff m σ Λ) P.μ ∧ MeasureTheory.Integrable (fun ω => S.dVar.indicator true ω / S.cutoffProp Λ (S.gaussianCutoff m σ Λ) ω) P.μ ∧ MeasureTheory.Integrable (fun ω => S.dVar.indicator true ω * if S.gaussianCutoff m σ Λ ω < S.factualY ω then 1 else 0) P.μ ∧ MeasureTheory.Integrable (fun ω => S.dVar.indicator true ω * S.wMin Λ ω) P.μ ∧ MeasureTheory.Integrable (fun ω => (S.wMax Λ ω - S.wMin Λ ω) * (S.dVar.indicator true ω * if S.gaussianCutoff m σ Λ ω < S.factualY ω then 1 else 0)) P.μ ∧ MeasureTheory.Integrable (fun ω => S.dVar.indicator true ω * |S.factualY ω| * S.wMax Λ ω) P.μ ∧ MeasureTheory.Integrable (fun ω => S.dVar.indicator true ω * S.wMax Λ ω) P.μ ∧ MeasureTheory.Integrable (fun ω => |S.gaussianCutoff m σ Λ ω| * S.dVar.indicator true ω * S.wMax Λ ω) P.μ → MeasureTheory.Integrable (fun ω => S.wMin Λ ω * (S.dVar.indicator true ω * S.factualY ω)) P.μ → MeasureTheory.Integrable (fun ω => (S.wMax Λ ω - S.wMin Λ ω) * (S.dVar.indicator true ω * S.factualY ω * if S.gaussianCutoff m σ Λ ω < S.factualY ω then 1 else 0)) P.μ → S.msmUpperCalib Λ = ∫ (ω : P.Ω), m (S.factualX ω) ∂P.μ + Causalean.PO.POBackdoorSystem.gaussianMSMHalfWidthFactor Λ * ∫ (ω : P.Ω), (1 - S.propScore true ω) * σ (S.factualX ω) ∂P.μ","l":190,"f":"Causalean/PO/ID/Partial/Sensitivity/MSM/GaussianHalfWidth.lean"},{"n":"Causalean.PO.POBackdoorSystem.msmUpperCalib_le_msmUpper","k":"theorem","m":"Causalean.PO.ID.Partial.Sensitivity.MSM.Sharp","a":"PO/ID/Partial/Sensitivity/MSM","t":1,"d":"The sharp upper bound is tighter than the ZSB bound. Assuming the calibrated candidate-mean image is nonempty and the candidate mean is bounded above over the uncalibrated odds-ratio-box ambiguity set, the sharp (calibrated) upper bound is at most the ZSB (uncalibrated) upper bound: msmUpperCalib Λ ≤ msmUpper Λ. The calibrated set is smaller, so its candidate-mean supremum can only decrease.","s":"∀ {P : Causalean.PO.POSystem} {γ : Type u_1} [inst : MeasurableSpace γ] (S : Causalean.PO.POBackdoorSystem P γ) (Λ : ℝ), (S.candMean '' S.MSMSetCalib Λ).Nonempty → BddAbove (S.candMean '' S.MSMSet Λ) → S.msmUpperCalib Λ ≤ S.msmUpper Λ","l":161,"f":"Causalean/PO/ID/Partial/Sensitivity/MSM/Sharp.lean"},{"n":"Causalean.PO.POBackdoorSystem.msmUpperForm","k":"def","m":"Causalean.PO.ID.Partial.Sensitivity.MSM.Bounds","a":"PO/ID/Partial/Sensitivity/MSM","t":1,"d":"The closed-form upper integrand: wMax where Y ≥ 0, wMin where Y < 0.","s":"{P : Causalean.PO.POSystem} → {γ : Type u_1} → [inst : MeasurableSpace γ] → Causalean.PO.POBackdoorSystem P γ → ℝ → ℝ","l":67,"f":"Causalean/PO/ID/Partial/Sensitivity/MSM/Bounds.lean"},{"n":"Causalean.PO.POBackdoorSystem.msmUpper_eq","k":"theorem","m":"Causalean.PO.ID.Partial.Sensitivity.MSM.Bounds","a":"PO/ID/Partial/Sensitivity/MSM","t":1,"d":"Closed form of the MSM upper bound. Fix a sensitivity parameter Λ at least 1. If the propensity score lies strictly between 0 and 1 almost everywhere (two-sided overlap), every candidate propensity in the odds-ratio ambiguity set is measurable up to null sets, and the envelope A·|Y|·wMax(Λ) — which dominates every candidate IPW integrand — is integrable, then the supremum of the candidate IPW mean over the ambiguity set is attained pointwise: the MSM upper bound equals E[A·Y·(wMax if Y≥0 else wMin)].","s":"∀ {P : Causalean.PO.POSystem} {γ : Type u_1} [inst : MeasurableSpace γ] (S : Causalean.PO.POBackdoorSystem P γ) (Λ : ℝ), 1 ≤ Λ → (∀ᵐ (ω : P.Ω) ∂P.μ, 0 < S.propScore true ω ∧ S.propScore true ω < 1) → (∀ etilde ∈ S.MSMSet Λ, AEMeasurable etilde P.μ) → MeasureTheory.Integrable (fun ω => S.dVar.indicator true ω * |S.factualY ω| * S.wMax Λ ω) P.μ → S.msmUpper Λ = S.msmUpperForm Λ","l":132,"f":"Causalean/PO/ID/Partial/Sensitivity/MSM/Bounds.lean"},{"n":"Causalean.PO.POBackdoorSystem.msmUpper_mono","k":"theorem","m":"Causalean.PO.ID.Partial.Sensitivity.MSM.Setup","a":"PO/ID/Partial/Sensitivity/MSM","t":2,"d":"The MSM interval widens with Λ (upper endpoint). Since MSMSet Λ ⊆ MSMSet Λ', the supremum of the candidate mean can only increase. Boundedness of the larger image is required for the sSup comparison.","s":"∀ {P : Causalean.PO.POSystem} {γ : Type u_1} [inst : MeasurableSpace γ] (S : Causalean.PO.POBackdoorSystem P γ) {Λ Λ' : ℝ}, 1 ≤ Λ → Λ ≤ Λ' → (S.candMean '' S.MSMSet Λ).Nonempty → BddAbove (S.candMean '' S.MSMSet Λ') → S.msmUpper Λ ≤ S.msmUpper Λ'","l":241,"f":"Causalean/PO/ID/Partial/Sensitivity/MSM/Setup.lean"},{"n":"Causalean.PO.POBackdoorSystem.one_le_wMax0_mul_propScore","k":"theorem","m":"Causalean.PO.ID.Partial.Sensitivity.MSM.ControlCutoff","a":"PO/ID/Partial/Sensitivity/MSM","t":1,"d":"Upper control calibration bracket is at least one. Fix a sensitivity parameter Λ at least 1. If the control propensity P[D=0∣X] lies strictly between 0 and 1 almost everywhere (overlap), then almost everywhere the upper calibration weight wMax0 Λ times the control propensity is at least 1.","s":"∀ {P : Causalean.PO.POSystem} {γ : Type u_1} [inst : MeasurableSpace γ] (S : Causalean.PO.POBackdoorSystem P γ) (Λ : ℝ), 1 ≤ Λ → (∀ᵐ (ω : P.Ω) ∂P.μ, 0 < S.propScore false ω ∧ S.propScore false ω < 1) → ∀ᵐ (ω : P.Ω) ∂P.μ, 1 ≤ S.wMax0 Λ ω * S.propScore false ω","l":83,"f":"Causalean/PO/ID/Partial/Sensitivity/MSM/ControlCutoff.lean"},{"n":"Causalean.PO.POBackdoorSystem.one_le_wMax_mul_propScore","k":"theorem","m":"Causalean.PO.ID.Partial.Sensitivity.MSM.CutoffExists","a":"PO/ID/Partial/Sensitivity/MSM","t":1,"d":"Upper calibration bracket (algebraic). Fix a sensitivity parameter Λ at least 1. Assuming the propensity score for treatment given the covariates lies strictly between 0 and 1 almost surely (overlap), almost surely the upper marginal-sensitivity-model weight satisfies 1 ≤ wMax(X)·e(X) = e(X) + Λ·(1 − e(X)).","s":"∀ {P : Causalean.PO.POSystem} {γ : Type u_1} [inst : MeasurableSpace γ] (S : Causalean.PO.POBackdoorSystem P γ) (Λ : ℝ), 1 ≤ Λ → (∀ᵐ (ω : P.Ω) ∂P.μ, 0 < S.propScore true ω ∧ S.propScore true ω < 1) → ∀ᵐ (ω : P.Ω) ∂P.μ, 1 ≤ S.wMax Λ ω * S.propScore true ω","l":73,"f":"Causalean/PO/ID/Partial/Sensitivity/MSM/CutoffExists.lean"},{"n":"Causalean.PO.POBackdoorSystem.outcomeReg","k":"def","m":"Causalean.PO.ID.Exact.ATE","a":"PO/ID/Exact","t":1,"d":"Outcome regression E[Y | D, X] := μ[factualY | σ(D, X)], the conditional mean of the *observed* outcome given the *observed* treatment and covariate. A functional of the observed data law, learnable by regressing Y on (D, X).","s":"{P : Causalean.PO.POSystem} → {γ : Type u_1} → [inst : MeasurableSpace γ] → Causalean.PO.POBackdoorSystem P γ → P.Ω → ℝ","l":488,"f":"Causalean/PO/ID/Exact/ATE.lean"},{"n":"Causalean.PO.POBackdoorSystem.outcomeReg_eq_regFn","k":"theorem","m":"Causalean.PO.ID.Exact.ATE","a":"PO/ID/Exact","t":2,"d":"The observed outcome regression equals the value-space regression evaluated at each unit's observed treatment and covariate.","s":"∀ {P : Causalean.PO.POSystem} {γ : Type u_1} [inst : MeasurableSpace γ] (S : Causalean.PO.POBackdoorSystem P γ), S.outcomeReg = fun ω => S.regFn (S.factualDX ω)","l":521,"f":"Causalean/PO/ID/Exact/ATE.lean"},{"n":"Causalean.PO.POBackdoorSystem.propScore","k":"def","m":"Causalean.PO.ID.Exact.ATE","a":"PO/ID/Exact","t":1,"d":"The propensity score is the conditional probability of a treatment arm given the covariate.","s":"{P : Causalean.PO.POSystem} → {γ : Type u_1} → [inst : MeasurableSpace γ] → Causalean.PO.POBackdoorSystem P γ → Bool → P.Ω → ℝ","l":118,"f":"Causalean/PO/ID/Exact/ATE.lean"},{"n":"Causalean.PO.POBackdoorSystem.propScore_ne_of_overlap","k":"theorem","m":"Causalean.PO.ID.Exact.ATE","a":"PO/ID/Exact","t":2,"d":"From two-sided overlap, the per-arm propensity score P[D=d | σ(X)] is a.s. nonzero: for d = true directly from 0 < e(X), for d = false from e(X) < 1 via propScore false = 1 - e(X).","s":"∀ {P : Causalean.PO.POSystem} {γ : Type u_1} [inst : MeasurableSpace γ] (S : Causalean.PO.POBackdoorSystem P γ) [StandardBorelSpace P.Ω] [MeasureTheory.IsFiniteMeasure P.μ], (∀ᵐ (ω : P.Ω) ∂P.μ, 0 < S.propScore true ω ∧ S.propScore true ω < 1) → ∀ (d : Bool), ∀ᵐ (ω : P.Ω) ∂P.μ, S.propScore d ω ≠ 0","l":226,"f":"Causalean/PO/ID/Exact/ATE.lean"},{"n":"Causalean.PO.POBackdoorSystem.propScore_nonneg","k":"theorem","m":"Causalean.PO.ID.Exact.QTE.DistributionalBackdoor","a":"PO/ID/Exact/QTE","t":2,"d":"propScore d ≥ 0 a.e. (conditional expectation of a nonnegative function).","s":"∀ {P : Causalean.PO.POSystem} {γ : Type u_1} [inst : MeasurableSpace γ] (S : Causalean.PO.POBackdoorSystem P γ) [StandardBorelSpace P.Ω] [MeasureTheory.IsFiniteMeasure P.μ] (d : Bool), 0 ≤ᵐ[P.μ] S.propScore d","l":105,"f":"Causalean/PO/ID/Exact/QTE/DistributionalBackdoor.lean"},{"n":"Causalean.PO.POBackdoorSystem.propScore_pos","k":"theorem","m":"Causalean.PO.ID.Exact.QTE.DistributionalBackdoor","a":"PO/ID/Exact/QTE","t":2,"d":"Under bare overlap, both arms have a.e.-positive propensity score.","s":"∀ {P : Causalean.PO.POSystem} {γ : Type u_1} [inst : MeasurableSpace γ] (S : Causalean.PO.POBackdoorSystem P γ) [inst_1 : StandardBorelSpace P.Ω] [inst_2 : MeasureTheory.IsFiniteMeasure P.μ], S.DistributionalAssumptions → ∀ (d : Bool), ∀ᵐ (ω : P.Ω) ∂P.μ, 0 < S.propScore d ω","l":111,"f":"Causalean/PO/ID/Exact/QTE/DistributionalBackdoor.lean"},{"n":"Causalean.PO.POBackdoorSystem.propScore_true_eq_eLift","k":"theorem","m":"Causalean.PO.ID.Exact.ATE","a":"PO/ID/Exact","t":2,"d":"The treated-arm propensity score equals the value-space propensity evaluated at each unit's observed covariate.","s":"∀ {P : Causalean.PO.POSystem} {γ : Type u_1} [inst : MeasurableSpace γ] (S : Causalean.PO.POBackdoorSystem P γ), S.propScore true = fun ω => S.eLift (S.factualX ω)","l":544,"f":"Causalean/PO/ID/Exact/ATE.lean"},{"n":"Causalean.PO.POBackdoorSystem.propTreated","k":"def","m":"Causalean.PO.ID.Exact.ATT","a":"PO/ID/Exact","t":1,"d":"Marginal probability of being treated, π_T = E[1_{D=1}] = P[D=1].","s":"{P : Causalean.PO.POSystem} → {γ : Type u_1} → [inst : MeasurableSpace γ] → Causalean.PO.POBackdoorSystem P γ → ℝ","l":43,"f":"Causalean/PO/ID/Exact/ATT.lean"},{"n":"Causalean.PO.POBackdoorSystem.qtdQuantile","k":"def","m":"Causalean.PO.ID.Exact.QTE.QuantileEffect","a":"PO/ID/Exact/QTE","t":1,"d":"The τ-quantile of the potential outcome Y(d).","s":"{P : Causalean.PO.POSystem} → {γ : Type u_1} → [inst : MeasurableSpace γ] → Causalean.PO.POBackdoorSystem P γ → Bool → MeasureTheory.Measure P.Ω → ℝ → ℝ","l":72,"f":"Causalean/PO/ID/Exact/QTE/QuantileEffect.lean"},{"n":"Causalean.PO.POBackdoorSystem.qtdQuantile_eq_quantile_cfUnderLaw","k":"theorem","m":"Causalean.PO.ID.Exact.QTE.QuantileEffect","a":"PO/ID/Exact/QTE","t":2,"d":"The quantile of Y(d) is the quantile of the law of Y(d).","s":"∀ {P : Causalean.PO.POSystem} {γ : Type u_1} [inst : MeasurableSpace γ] (S : Causalean.PO.POBackdoorSystem P γ) (d : Bool) (μ : MeasureTheory.Measure P.Ω) (τ : ℝ), S.qtdQuantile d μ τ = Causalean.Stat.quantile (S.yVar.cfUnderLaw S.dVar d μ) τ","l":82,"f":"Causalean/PO/ID/Exact/QTE/QuantileEffect.lean"},{"n":"Causalean.PO.POBackdoorSystem.qte","k":"def","m":"Causalean.PO.ID.Exact.QTE.QuantileEffect","a":"PO/ID/Exact/QTE","t":1,"d":"Totalized quantile treatment effect at level τ: the difference between the two generalized quantile functionals, defined for every real τ. Literature-facing quantile effects normally restrict τ to the unit interval.","s":"{P : Causalean.PO.POSystem} → {γ : Type u_1} → [inst : MeasurableSpace γ] → Causalean.PO.POBackdoorSystem P γ → MeasureTheory.Measure P.Ω → ℝ → ℝ","l":76,"f":"Causalean/PO/ID/Exact/QTE/QuantileEffect.lean"},{"n":"Causalean.PO.POBackdoorSystem.qte_backdoor","k":"theorem","m":"Causalean.PO.ID.Exact.QTE.QuantileEffect","a":"PO/ID/Exact/QTE","t":1,"d":"Backdoor QTE identification. Under the ATE backdoor identifying assumption bundle, the totalized generalized-quantile treatment effect at any real level τ equals the difference of the quantiles of the two observable inverse-probability-weighted outcome laws:","s":"∀ {P : Causalean.PO.POSystem} {γ : Type u_1} [inst : MeasurableSpace γ] (S : Causalean.PO.POBackdoorSystem P γ) [inst_1 : StandardBorelSpace P.Ω] [inst_2 : MeasureTheory.IsFiniteMeasure P.μ], S.Assumptions → ∀ (τ : ℝ), S.qte P.μ τ = Causalean.Stat.quantile (S.ipwLaw true P.μ) τ - Causalean.Stat.quantile (S.ipwLaw false P.μ) τ","l":110,"f":"Causalean/PO/ID/Exact/QTE/QuantileEffect.lean"},{"n":"Causalean.PO.POBackdoorSystem.qte_eq_of_law_eq","k":"theorem","m":"Causalean.PO.ID.Exact.QTE.QuantileEffect","a":"PO/ID/Exact/QTE","t":1,"d":"QTE as a functional of the identified potential-outcome laws. If the law of the potential outcome Y(1) equals a given observable measure ν₁ and the law of the potential outcome Y(0) equals a given observable measure ν₀, then for every real quantile level τ, the totalized quantile treatment effect equals the difference of the τ-quantiles of ν₁ and ν₀. This is the payoff of the quantile layer: identification of the QTE reduces to distributional identification of the two potential-outcome laws.","s":"∀ {P : Causalean.PO.POSystem} {γ : Type u_1} [inst : MeasurableSpace γ] (S : Causalean.PO.POBackdoorSystem P γ) (μ : MeasureTheory.Measure P.Ω) (τ : ℝ) {ν₁ ν₀ : MeasureTheory.Measure ℝ}, S.yVar.cfUnderLaw S.dVar true μ = ν₁ → S.yVar.cfUnderLaw S.dVar false μ = ν₀ → S.qte μ τ = Causalean.Stat.quantile ν₁ τ - Causalean.Stat.quantile ν₀ τ","l":87,"f":"Causalean/PO/ID/Exact/QTE/QuantileEffect.lean"},{"n":"Causalean.PO.POBackdoorSystem.regFn","k":"def","m":"Causalean.PO.ID.Exact.ATE","a":"PO/ID/Exact","t":1,"d":"Value-space outcome regression regFn (d, x) = E[Y | D=d, X=x].","s":"{P : Causalean.PO.POSystem} → {γ : Type u_1} → [inst : MeasurableSpace γ] → Causalean.PO.POBackdoorSystem P γ → Bool × γ → ℝ","l":515,"f":"Causalean/PO/ID/Exact/ATE.lean"},{"n":"Causalean.PO.POBackdoorSystem.regression_adjustment","k":"theorem","m":"Causalean.PO.ID.Exact.ATE","a":"PO/ID/Exact","t":1,"d":"Regression-adjustment identification (observable). For a treatment arm d, if the observed outcome is integrable and the covariate- conditional probability of receiving arm d is almost surely nonzero, then the backdoor adjustment functional for arm d equals the value-space outcome regression evaluated at the arm d:","s":"∀ {P : Causalean.PO.POSystem} {γ : Type u_1} [inst : MeasurableSpace γ] (S : Causalean.PO.POBackdoorSystem P γ) [StandardBorelSpace P.Ω] [MeasureTheory.IsFiniteMeasure P.μ] (d : Bool), MeasureTheory.Integrable S.factualY P.μ → (∀ᵐ (ω : P.Ω) ∂P.μ, S.propScore d ω ≠ 0) → S.adjustedCE d =ᵐ[P.μ] fun ω => S.regFn (d, S.factualX ω)","l":550,"f":"Causalean/PO/ID/Exact/ATE.lean"},{"n":"Causalean.PO.POBackdoorSystem.regression_adjustment_of_assumptions","k":"theorem","m":"Causalean.PO.ID.Exact.ATE","a":"PO/ID/Exact","t":2,"d":"Regression-adjustment, packaged for identified systems. The same observable identity adjustedCE d =ᵐ E[Y|D=d,X], with the two regularity hypotheses of regression_adjustment discharged from the back-door Assumptions: integrability of the observed outcome from Assumptions.integrable_factualY, and overlap from propScore_ne_of_overlap. Lets callers in an identified context apply it with just hA.","s":"∀ {P : Causalean.PO.POSystem} {γ : Type u_1} [inst : MeasurableSpace γ] (S : Causalean.PO.POBackdoorSystem P γ) [inst_1 : StandardBorelSpace P.Ω] [inst_2 : MeasureTheory.IsFiniteMeasure P.μ], S.Assumptions → ∀ (d : Bool), S.adjustedCE d =ᵐ[P.μ] fun ω => S.regFn (d, S.factualX ω)","l":670,"f":"Causalean/PO/ID/Exact/ATE.lean"},{"n":"Causalean.PO.POBackdoorSystem.setLIntegral_condCDF_variable","k":"theorem","m":"Causalean.PO.ID.Partial.Sensitivity.MSM.CutoffConstruct","a":"PO/ID/Partial/Sensitivity/MSM","t":2,"d":"Set-lintegral form of the conditional CDF identity at a measurable variable cutoff.","s":"∀ {γ : Type u_1} [inst : MeasurableSpace γ] (ρ : MeasureTheory.Measure (γ × ℝ)) [MeasureTheory.IsFiniteMeasure ρ] {B : Set γ}, MeasurableSet B → ∀ {q : γ → ℝ}, Measurable q → ∫⁻ (a : γ) in B, ENNReal.ofReal (↑(ProbabilityTheory.condCDF ρ a) (q a)) ∂ρ.fst = ρ {p | p.1 ∈ B ∧ p.2 ≤ q p.1}","l":387,"f":"Causalean/PO/ID/Partial/Sensitivity/MSM/CutoffConstruct.lean"},{"n":"Causalean.PO.POBackdoorSystem.sigmaDX","k":"def","m":"Causalean.PO.ID.Exact.ATE","a":"PO/ID/Exact","t":1,"d":"σ-algebra generated by the factual (D, X). Contains σ(X) (snd projection) and is the information a regression of Y on (D, X) may use.","s":"{P : Causalean.PO.POSystem} → {γ : Type u_1} → [inst : MeasurableSpace γ] → Causalean.PO.POBackdoorSystem P γ → MeasurableSpace P.Ω","l":471,"f":"Causalean/PO/ID/Exact/ATE.lean"},{"n":"Causalean.PO.POBackdoorSystem.sigmaDX_le","k":"theorem","m":"Causalean.PO.ID.Exact.ATE","a":"PO/ID/Exact","t":2,"d":"The joint treatment-covariate sigma-algebra is a sub-sigma-algebra of the ambient one.","s":"∀ {P : Causalean.PO.POSystem} {γ : Type u_1} [inst : MeasurableSpace γ] (S : Causalean.PO.POBackdoorSystem P γ), S.sigmaDX ≤ inferInstance","l":476,"f":"Causalean/PO/ID/Exact/ATE.lean"},{"n":"Causalean.PO.POBackdoorSystem.sigmaX","k":"def","m":"Causalean.PO.ID.Exact.ATE","a":"PO/ID/Exact","t":1,"d":"The covariate sigma-algebra is generated by the observed covariate.","s":"{P : Causalean.PO.POSystem} → {γ : Type u_1} → [inst : MeasurableSpace γ] → Causalean.PO.POBackdoorSystem P γ → MeasurableSpace P.Ω","l":99,"f":"Causalean/PO/ID/Exact/ATE.lean"},{"n":"Causalean.PO.POBackdoorSystem.sigmaXY0","k":"def","m":"Causalean.PO.ID.Partial.Sensitivity.MSM.ControlSetup","a":"PO/ID/Partial/Sensitivity/MSM","t":1,"d":"σ-algebra generated by BOTH the factual covariate X and the *control* potential outcome Y(0). Conditioning on this richer information set defines the complete control propensity; it always refines σ(X).","s":"{P : Causalean.PO.POSystem} → {γ : Type u_1} → [inst : MeasurableSpace γ] → Causalean.PO.POBackdoorSystem P γ → MeasurableSpace P.Ω","l":35,"f":"Causalean/PO/ID/Partial/Sensitivity/MSM/ControlSetup.lean"},{"n":"Causalean.PO.POBackdoorSystem.sigmaXY0_le","k":"theorem","m":"Causalean.PO.ID.Partial.Sensitivity.MSM.ControlSetup","a":"PO/ID/Partial/Sensitivity/MSM","t":2,"d":"The joint σ-algebra σ(X, Y(0)) is coarser than the ambient σ-algebra on Ω.","s":"∀ {P : Causalean.PO.POSystem} {γ : Type u_1} [inst : MeasurableSpace γ] (S : Causalean.PO.POBackdoorSystem P γ), S.sigmaXY0 ≤ inferInstance","l":42,"f":"Causalean/PO/ID/Partial/Sensitivity/MSM/ControlSetup.lean"},{"n":"Causalean.PO.POBackdoorSystem.sigmaXY1","k":"def","m":"Causalean.PO.ID.Partial.Sensitivity.MSM.Setup","a":"PO/ID/Partial/Sensitivity/MSM","t":1,"d":"σ-algebra generated by BOTH the factual covariate X and the treated potential outcome Y(1). Conditioning on this richer information set defines the *complete* propensity; it always refines σ(X).","s":"{P : Causalean.PO.POSystem} → {γ : Type u_1} → [inst : MeasurableSpace γ] → Causalean.PO.POBackdoorSystem P γ → MeasurableSpace P.Ω","l":47,"f":"Causalean/PO/ID/Partial/Sensitivity/MSM/Setup.lean"},{"n":"Causalean.PO.POBackdoorSystem.sigmaXY1_le","k":"theorem","m":"Causalean.PO.ID.Partial.Sensitivity.MSM.Setup","a":"PO/ID/Partial/Sensitivity/MSM","t":2,"d":"The joint σ-algebra σ(X, Y(1)) is coarser than the ambient σ-algebra on Ω.","s":"∀ {P : Causalean.PO.POSystem} {γ : Type u_1} [inst : MeasurableSpace γ] (S : Causalean.PO.POBackdoorSystem P γ), S.sigmaXY1 ≤ inferInstance","l":54,"f":"Causalean/PO/ID/Partial/Sensitivity/MSM/Setup.lean"},{"n":"Causalean.PO.POBackdoorSystem.sigmaX_le","k":"theorem","m":"Causalean.PO.ID.Exact.ATE","a":"PO/ID/Exact","t":2,"d":"The covariate sigma-algebra is a sub-sigma-algebra of the ambient one.","s":"∀ {P : Causalean.PO.POSystem} {γ : Type u_1} [inst : MeasurableSpace γ] (S : Causalean.PO.POBackdoorSystem P γ), S.sigmaX ≤ inferInstance","l":103,"f":"Causalean/PO/ID/Exact/ATE.lean"},{"n":"Causalean.PO.POBackdoorSystem.sigmaX_le_sigmaDX","k":"theorem","m":"Causalean.PO.ID.Exact.ATE","a":"PO/ID/Exact","t":2,"d":"σ(X) ≤ σ(D, X): the covariate σ-algebra is coarser than the joint one.","s":"∀ {P : Causalean.PO.POSystem} {γ : Type u_1} [inst : MeasurableSpace γ] (S : Causalean.PO.POBackdoorSystem P γ), S.sigmaX ≤ S.sigmaDX","l":480,"f":"Causalean/PO/ID/Exact/ATE.lean"},{"n":"Causalean.PO.POBackdoorSystem.stronglyMeasurable_outcomeReg","k":"theorem","m":"Causalean.PO.ID.Exact.ATE","a":"PO/ID/Exact","t":2,"d":"The observed outcome regression is strongly measurable with respect to the joint treatment-covariate sigma-algebra.","s":"∀ {P : Causalean.PO.POSystem} {γ : Type u_1} [inst : MeasurableSpace γ] (S : Causalean.PO.POBackdoorSystem P γ), MeasureTheory.StronglyMeasurable S.outcomeReg","l":494,"f":"Causalean/PO/ID/Exact/ATE.lean"},{"n":"Causalean.PO.POBackdoorSystem.stronglyMeasurable_propScore","k":"theorem","m":"Causalean.PO.ID.Exact.QTE.DistributionalBackdoor","a":"PO/ID/Exact/QTE","t":2,"d":"propScore d is σ(X)-strongly-measurable (it is a conditional expectation).","s":"∀ {P : Causalean.PO.POSystem} {γ : Type u_1} [inst : MeasurableSpace γ] (S : Causalean.PO.POBackdoorSystem P γ) [StandardBorelSpace P.Ω] [MeasureTheory.IsFiniteMeasure P.μ] (d : Bool), MeasureTheory.StronglyMeasurable (S.propScore d)","l":99,"f":"Causalean/PO/ID/Exact/QTE/DistributionalBackdoor.lean"},{"n":"Causalean.PO.POBackdoorSystem.sub_mem_Icc_of_mem_Icc","k":"theorem","m":"Causalean.PO.ID.Partial.Sensitivity.MSM.ATE","a":"PO/ID/Partial/Sensitivity/MSM","t":2,"d":"Interval subtraction. If a ∈ [aₗ, aᵤ] and b ∈ [bₗ, bᵤ], then a − b ∈ [aₗ − bᵤ, aᵤ − bₗ]. The arithmetic core of the ATE-interval theorems.","s":"∀ {a aₗ aᵤ b bₗ bᵤ : ℝ}, a ∈ Set.Icc aₗ aᵤ → b ∈ Set.Icc bₗ bᵤ → a - b ∈ Set.Icc (aₗ - bᵤ) (aᵤ - bₗ)","l":65,"f":"Causalean/PO/ID/Partial/Sensitivity/MSM/ATE.lean"},{"n":"Causalean.PO.POBackdoorSystem.survTarget","k":"def","m":"Causalean.PO.ID.Partial.Sensitivity.MSM.CutoffSelection","a":"PO/ID/Partial/Sensitivity/MSM","t":1,"d":"The target survival that calibrates the cutoff: (1 − wMin·e)/(wMax − wMin).","s":"{P : Causalean.PO.POSystem} → {γ : Type u_1} → [inst : MeasurableSpace γ] → Causalean.PO.POBackdoorSystem P γ → ℝ → P.Ω → ℝ","l":57,"f":"Causalean/PO/ID/Partial/Sensitivity/MSM/CutoffSelection.lean"},{"n":"Causalean.PO.POBackdoorSystem.survTarget0","k":"def","m":"Causalean.PO.ID.Partial.Sensitivity.MSM.ControlCutoff","a":"PO/ID/Partial/Sensitivity/MSM","t":1,"d":"The target control survival is the conditional survival value that makes the cutoff calibrated.","s":"{P : Causalean.PO.POSystem} → {γ : Type u_1} → [inst : MeasurableSpace γ] → Causalean.PO.POBackdoorSystem P γ → ℝ → P.Ω → ℝ","l":54,"f":"Causalean/PO/ID/Partial/Sensitivity/MSM/ControlCutoff.lean"},{"n":"Causalean.PO.POBackdoorSystem.survTargetLower","k":"def","m":"Causalean.PO.ID.Partial.Sensitivity.MSM.LowerBound","a":"PO/ID/Partial/Sensitivity/MSM","t":1,"d":"The lower target survival that calibrates the lower cutoff: survTargetLower = (wMax·e − 1)/(wMax − wMin) (equal to e − survTarget).","s":"{P : Causalean.PO.POSystem} → {γ : Type u_1} → [inst : MeasurableSpace γ] → Causalean.PO.POBackdoorSystem P γ → ℝ → P.Ω → ℝ","l":51,"f":"Causalean/PO/ID/Partial/Sensitivity/MSM/LowerBound.lean"},{"n":"Causalean.PO.POBackdoorSystem.survTargetLower0","k":"def","m":"Causalean.PO.ID.Partial.Sensitivity.MSM.ControlLowerBound","a":"PO/ID/Partial/Sensitivity/MSM","t":1,"d":"The lower target survival that calibrates the lower cutoff: survTargetLower0 = (wMax0·e − 1)/(wMax0 − wMin0) (equal to e − survTarget).","s":"{P : Causalean.PO.POSystem} → {γ : Type u_1} → [inst : MeasurableSpace γ] → Causalean.PO.POBackdoorSystem P γ → ℝ → P.Ω → ℝ","l":40,"f":"Causalean/PO/ID/Partial/Sensitivity/MSM/ControlLowerBound.lean"},{"n":"Causalean.PO.POBackdoorSystem.toBackdoorEstimationSystem","k":"def","m":"Causalean.Estimation.ATE.Setup","a":"Estimation/ATE","t":1,"d":"The compatibility/positivity fields are free. From a POBackdoorSystem with a.e. two-sided overlap and an integrable observed outcome — and *no* unconfoundedness — one constructs a BackdoorEstimationSystem: μ_val is the value-space outcome regression regFn and e_val is the propensity lift eLift clamped into (0,1) off-support. Every added field is discharged (μ_reg_compat from regression_adjustment; e_pos/e_lt_one from the clamp; e_compat from overlap). So BackdoorEstimationSystem carries no assumption beyond POBackdoorSystem + overlap + integrability — in particular the value-space lifts (μ_compat, e_compat) were never genuine assumptions.","s":"{P : Causalean.PO.POSystem} → {γ : Type u_1} → [inst : MeasurableSpace γ] → (S : Causalean.PO.POBackdoorSystem P γ) → [inst_1 : StandardBorelSpace P.Ω] → [inst_2 : MeasureTheory.IsFiniteMeasure P.μ] → (∀ᵐ (ω : P.Ω) ∂P.μ, 0 < S.propScore true ω ∧ S.propScore true ω < 1) → MeasureTheory.Integrable S.factualY P.μ → Causalean.Estimation.ATE.BackdoorEstimationSystem P γ","l":198,"f":"Causalean/Estimation/ATE/Setup.lean"},{"n":"Causalean.PO.POBackdoorSystem.toTreatedEstimationSystem","k":"def","m":"Causalean.Estimation.ATT.Setup","a":"Estimation/ATT","t":1,"d":"The compatibility/positivity fields are free. From a POBackdoorSystem with two-sided overlap and an integrable observed outcome — and *no* unconfoundedness — one constructs a TreatedEstimationSystem: μ₀_val is the control-arm regression regFn false and e_val is the propensity lift eLift clamped below 1. Every added field is discharged (μ₀_reg_compat from regression_adjustment false; e_lt_one from the clamp; e_compat from overlap). So the control regression and propensity lifts were never genuine assumptions.","s":"{P : Causalean.PO.POSystem} → {γ : Type u_1} → [inst : MeasurableSpace γ] → (S : Causalean.PO.POBackdoorSystem P γ) → [inst_1 : StandardBorelSpace P.Ω] → [inst_2 : MeasureTheory.IsFiniteMeasure P.μ] → (∀ᵐ (ω : P.Ω) ∂P.μ, 0 < S.propScore true ω ∧ S.propScore true ω < 1) → MeasureTheory.Integrable S.factualY P.μ → Causalean.Estimation.ATT.TreatedEstimationSystem P γ","l":215,"f":"Causalean/Estimation/ATT/Setup.lean"},{"n":"Causalean.PO.POBackdoorSystem.treatedCondCDF","k":"def","m":"Causalean.PO.ID.Partial.Sensitivity.MSM.CutoffConstruct","a":"PO/ID/Partial/Sensitivity/MSM","t":1,"d":"The treated conditional CDF of Y given X, evaluated along the factual covariate: F(t | X(ω)) = P(Y ≤ t | X = X(ω), D = 1).","s":"{P : Causalean.PO.POSystem} → {γ : Type u_1} → [inst : MeasurableSpace γ] → Causalean.PO.POBackdoorSystem P γ → P.Ω → ℝ → ℝ","l":70,"f":"Causalean/PO/ID/Partial/Sensitivity/MSM/CutoffConstruct.lean"},{"n":"Causalean.PO.POBackdoorSystem.treatedSet","k":"def","m":"Causalean.PO.ID.Partial.Sensitivity.MSM.CutoffConstruct","a":"PO/ID/Partial/Sensitivity/MSM","t":1,"d":"The treated set {ω | D(ω) = 1}.","s":"{P : Causalean.PO.POSystem} → {γ : Type u_1} → [inst : MeasurableSpace γ] → Causalean.PO.POBackdoorSystem P γ → Set P.Ω","l":62,"f":"Causalean/PO/ID/Partial/Sensitivity/MSM/CutoffConstruct.lean"},{"n":"Causalean.PO.POBackdoorSystem.treatedSurv","k":"def","m":"Causalean.PO.ID.Partial.Sensitivity.MSM.CutoffSelection","a":"PO/ID/Partial/Sensitivity/MSM","t":1,"d":"The conditional treated-survival at cutoff c: G(c) = E[Z·1{Y > c(X)} | σ(X)].","s":"{P : Causalean.PO.POSystem} → {γ : Type u_1} → [inst : MeasurableSpace γ] → Causalean.PO.POBackdoorSystem P γ → (P.Ω → ℝ) → P.Ω → ℝ","l":53,"f":"Causalean/PO/ID/Partial/Sensitivity/MSM/CutoffSelection.lean"},{"n":"Causalean.PO.POBackdoorSystem.treatedSurv_const_eq","k":"theorem","m":"Causalean.PO.ID.Partial.Sensitivity.MSM.CutoffConstruct","a":"PO/ID/Partial/Sensitivity/MSM","t":2,"d":"The survival bridge (constant cutoff). The weighted conditional survival equals the treated conditional survival scaled by the propensity: E[Z·1{Y>t} | σ(X)] = e(X)·(1 − F(t | X)) a.e. The genuine measure-theoretic content relates a conditional expectation under μ (weighted by the treatment indicator) to the conditional CDF of the treated push-forward law.","s":"∀ {P : Causalean.PO.POSystem} {γ : Type u_1} [inst : MeasurableSpace γ] (S : Causalean.PO.POBackdoorSystem P γ) (t : ℝ), P.μ[fun ω => S.dVar.indicator true ω * if t < S.factualY ω then 1 else 0 | S.sigmaX] =ᵐ[P.μ] fun ω => S.propScore true ω * (1 - S.treatedCondCDF ω t)","l":302,"f":"Causalean/PO/ID/Partial/Sensitivity/MSM/CutoffConstruct.lean"},{"n":"Causalean.PO.POBackdoorSystem.treatedSurv_eq","k":"theorem","m":"Causalean.PO.ID.Partial.Sensitivity.MSM.CutoffConstruct","a":"PO/ID/Partial/Sensitivity/MSM","t":2,"d":"The survival bridge (functional cutoff). The version of treatedSurv_const_eq evaluated at a σ(X)-measurable cutoff c: E[Z·1{Y>c(X)} | σ(X)] = e(X)·(1 − F(c(X) | X)) a.e. Since c is σ(X)-measurable, it is \"frozen\" inside the conditional expectation, reducing to the constant-cutoff bridge fibrewise. This is the form consumed by exists_calibrating_cutoff.","s":"∀ {P : Causalean.PO.POSystem} {γ : Type u_1} [inst : MeasurableSpace γ] (S : Causalean.PO.POBackdoorSystem P γ) (c : P.Ω → ℝ), Measurable c → S.treatedSurv c =ᵐ[P.μ] fun ω => S.propScore true ω * (1 - S.treatedCondCDF ω (c ω))","l":688,"f":"Causalean/PO/ID/Partial/Sensitivity/MSM/CutoffConstruct.lean"},{"n":"Causalean.PO.POBackdoorSystem.treatedXYLaw","k":"def","m":"Causalean.PO.ID.Partial.Sensitivity.MSM.CutoffConstruct","a":"PO/ID/Partial/Sensitivity/MSM","t":1,"d":"The treated push-forward law of (X, Y): the law of the factual covariate and outcome on the sub-population of treated units, (μ restricted to {D=1}) ∘ (X, Y)⁻¹.","s":"{P : Causalean.PO.POSystem} → {γ : Type u_1} → [inst : MeasurableSpace γ] → Causalean.PO.POBackdoorSystem P γ → MeasureTheory.Measure (γ × ℝ)","l":65,"f":"Causalean/PO/ID/Partial/Sensitivity/MSM/CutoffConstruct.lean"},{"n":"Causalean.PO.POBackdoorSystem.wMax","k":"def","m":"Causalean.PO.ID.Partial.Sensitivity.MSM.Bounds","a":"PO/ID/Partial/Sensitivity/MSM","t":1,"d":"The largest admissible inverse-propensity weight at sensitivity level Λ: wMax = 1 + Λ · (1 − e(X)) / e(X), the OR = Λ endpoint of the odds-ratio box.","s":"{P : Causalean.PO.POSystem} → {γ : Type u_1} → [inst : MeasurableSpace γ] → Causalean.PO.POBackdoorSystem P γ → ℝ → P.Ω → ℝ","l":62,"f":"Causalean/PO/ID/Partial/Sensitivity/MSM/Bounds.lean"},{"n":"Causalean.PO.POBackdoorSystem.wMax0","k":"def","m":"Causalean.PO.ID.Partial.Sensitivity.MSM.ControlCutoff","a":"PO/ID/Partial/Sensitivity/MSM","t":1,"d":"The largest admissible inverse-propensity weight for untreated units is the upper endpoint of the control odds-ratio box.","s":"{P : Causalean.PO.POSystem} → {γ : Type u_1} → [inst : MeasurableSpace γ] → Causalean.PO.POBackdoorSystem P γ → ℝ → P.Ω → ℝ","l":39,"f":"Causalean/PO/ID/Partial/Sensitivity/MSM/ControlCutoff.lean"},{"n":"Causalean.PO.POBackdoorSystem.wMin","k":"def","m":"Causalean.PO.ID.Partial.Sensitivity.MSM.Bounds","a":"PO/ID/Partial/Sensitivity/MSM","t":1,"d":"The smallest admissible inverse-propensity weight at sensitivity level Λ: wMin = 1 + (1 − e(X)) / (Λ · e(X)), the OR = 1/Λ endpoint of the odds-ratio box.","s":"{P : Causalean.PO.POSystem} → {γ : Type u_1} → [inst : MeasurableSpace γ] → Causalean.PO.POBackdoorSystem P γ → ℝ → P.Ω → ℝ","l":57,"f":"Causalean/PO/ID/Partial/Sensitivity/MSM/Bounds.lean"},{"n":"Causalean.PO.POBackdoorSystem.wMin0","k":"def","m":"Causalean.PO.ID.Partial.Sensitivity.MSM.ControlCutoff","a":"PO/ID/Partial/Sensitivity/MSM","t":1,"d":"The smallest admissible inverse-propensity weight for untreated units is the lower endpoint of the control odds-ratio box.","s":"{P : Causalean.PO.POSystem} → {γ : Type u_1} → [inst : MeasurableSpace γ] → Causalean.PO.POBackdoorSystem P γ → ℝ → P.Ω → ℝ","l":34,"f":"Causalean/PO/ID/Partial/Sensitivity/MSM/ControlCutoff.lean"},{"n":"Causalean.PO.POBackdoorSystem.wMin0_mul_propScore_le_one","k":"theorem","m":"Causalean.PO.ID.Partial.Sensitivity.MSM.ControlCutoff","a":"PO/ID/Partial/Sensitivity/MSM","t":1,"d":"Lower control calibration bracket is at most one. Fix a sensitivity parameter Λ at least 1. If the control propensity P[D=0∣X] lies strictly between 0 and 1 almost everywhere (overlap), then almost everywhere the lower calibration weight wMin0 Λ times the control propensity is at most 1.","s":"∀ {P : Causalean.PO.POSystem} {γ : Type u_1} [inst : MeasurableSpace γ] (S : Causalean.PO.POBackdoorSystem P γ) (Λ : ℝ), 1 ≤ Λ → (∀ᵐ (ω : P.Ω) ∂P.μ, 0 < S.propScore false ω ∧ S.propScore false ω < 1) → ∀ᵐ (ω : P.Ω) ∂P.μ, S.wMin0 Λ ω * S.propScore false ω ≤ 1","l":59,"f":"Causalean/PO/ID/Partial/Sensitivity/MSM/ControlCutoff.lean"},{"n":"Causalean.PO.POBackdoorSystem.wMin_mul_propScore_le_one","k":"theorem","m":"Causalean.PO.ID.Partial.Sensitivity.MSM.CutoffExists","a":"PO/ID/Partial/Sensitivity/MSM","t":1,"d":"Lower calibration bracket (algebraic). Fix a sensitivity parameter Λ at least 1. Assuming the propensity score for treatment given the covariates lies strictly between 0 and 1 almost surely (overlap), almost surely the lower marginal-sensitivity-model weight satisfies wMin(X)·e(X) = e(X) + (1 − e(X))/Λ ≤ 1, where e(X) denotes the propensity score.","s":"∀ {P : Causalean.PO.POSystem} {γ : Type u_1} [inst : MeasurableSpace γ] (S : Causalean.PO.POBackdoorSystem P γ) (Λ : ℝ), 1 ≤ Λ → (∀ᵐ (ω : P.Ω) ∂P.μ, 0 < S.propScore true ω ∧ S.propScore true ω < 1) → ∀ᵐ (ω : P.Ω) ∂P.μ, S.wMin Λ ω * S.propScore true ω ≤ 1","l":48,"f":"Causalean/PO/ID/Partial/Sensitivity/MSM/CutoffExists.lean"},{"n":"Causalean.PO.POBackdoorSystem.xVar","k":"def","m":"Causalean.PO.ID.Exact.ATE","a":"PO/ID/Exact","t":1,"d":"The adjustment covariate is the specified potential-outcome variable.","s":"{P : Causalean.PO.POSystem} → {γ : Type u_1} → [inst : MeasurableSpace γ] → Causalean.PO.POBackdoorSystem P γ → Causalean.PO.POVar P γ","l":77,"f":"Causalean/PO/ID/Exact/ATE.lean"},{"n":"Causalean.PO.POBackdoorSystem.yUnderD","k":"def","m":"Causalean.PO.ID.Exact.ATE","a":"PO/ID/Exact","t":1,"d":"The regimed form of the potential outcome fixes treatment to the chosen arm.","s":"{P : Causalean.PO.POSystem} → {γ : Type u_1} → [inst : MeasurableSpace γ] → Causalean.PO.POBackdoorSystem P γ → Bool → Causalean.PO.RegimedVar P ℝ","l":107,"f":"Causalean/PO/ID/Exact/ATE.lean"},{"n":"Causalean.PO.POBackdoorSystem.yVar","k":"def","m":"Causalean.PO.ID.Exact.ATE","a":"PO/ID/Exact","t":1,"d":"The outcome node is packaged as a real-valued potential-outcome variable.","s":"{P : Causalean.PO.POSystem} → {γ : Type u_1} → [inst : MeasurableSpace γ] → Causalean.PO.POBackdoorSystem P γ → Causalean.PO.POVar P ℝ","l":75,"f":"Causalean/PO/ID/Exact/ATE.lean"},{"n":"Causalean.PO.POBalkePearlSharp.SOmega","k":"def","m":"Causalean.PO.ID.Partial.BalkePearl.Sharp","a":"PO/ID/Partial/BalkePearl","t":1,"d":"The canonical sample space: a Z-value paired with the four latent arms (D(0), D(1), Y(0), Y(1)).","s":"Type","l":49,"f":"Causalean/PO/ID/Partial/BalkePearl/Sharp.lean"},{"n":"Causalean.PO.POBalkePearlSharp.SV","k":"def","m":"Causalean.PO.ID.Partial.BalkePearl.Sharp","a":"PO/ID/Partial/BalkePearl","t":1,"d":"The canonical variable index type: 0 = Z, 1 = D, 2 = Y.","s":"Type","l":53,"f":"Causalean/PO/ID/Partial/BalkePearl/Sharp.lean"},{"n":"Causalean.PO.POBalkePearlSharp.SX","k":"def","m":"Causalean.PO.ID.Partial.BalkePearl.Sharp","a":"PO/ID/Partial/BalkePearl","t":1,"d":"Canonical X assignment — every variable is Bool.","s":"Causalean.PO.POBalkePearlSharp.SV → Type","l":56,"f":"Causalean/PO/ID/Partial/BalkePearl/Sharp.lean"},{"n":"Causalean.PO.POBalkePearlSharp.canonicalBP","k":"def","m":"Causalean.PO.ID.Partial.BalkePearl.Sharp","a":"PO/ID/Partial/BalkePearl","t":1,"d":"The canonical Balke-Pearl system on the canonical PO system.","s":"{P : Causalean.PO.POSystem} → (S : Causalean.PO.POBalkePearlSystem P) → (π : Bool → Bool → Bool → Bool → ℝ) → (hπ_nn : ∀ (d0 d1 y0 y1 : Bool), 0 ≤ π d0 d1 y0 y1) → (hπ_sum : ∑ d0, ∑ d1, ∑ y0, ∑ y1, π d0 d1 y0 y1 = 1) → Causalean.PO.POBalkePearlSystem (Causalean.PO.POBalkePearlSharp.canonicalPOSystem S π hπ_nn hπ_sum)","l":288,"f":"Causalean/PO/ID/Partial/BalkePearl/Sharp.lean"},{"n":"Causalean.PO.POBalkePearlSharp.canonicalMeasure","k":"def","m":"Causalean.PO.ID.Partial.BalkePearl.Sharp","a":"PO/ID/Partial/BalkePearl","t":1,"d":"Canonical product measure on SOmega.","s":"{P : Causalean.PO.POSystem} → Causalean.PO.POBalkePearlSystem P → (Bool → Bool → Bool → Bool → ℝ) → MeasureTheory.Measure Causalean.PO.POBalkePearlSharp.SOmega","l":173,"f":"Causalean/PO/ID/Partial/BalkePearl/Sharp.lean"},{"n":"Causalean.PO.POBalkePearlSharp.canonicalPOSystem","k":"def","m":"Causalean.PO.ID.Partial.BalkePearl.Sharp","a":"PO/ID/Partial/BalkePearl","t":1,"d":"The canonical PO system parametrised by S and a feasible π.","s":"{P : Causalean.PO.POSystem} → Causalean.PO.POBalkePearlSystem P → (π : Bool → Bool → Bool → Bool → ℝ) → (∀ (d0 d1 y0 y1 : Bool), 0 ≤ π d0 d1 y0 y1) → ∑ d0, ∑ d1, ∑ y0, ∑ y1, π d0 d1 y0 y1 = 1 → Causalean.PO.POSystem","l":271,"f":"Causalean/PO/ID/Partial/BalkePearl/Sharp.lean"},{"n":"Causalean.PO.POBalkePearlSharp.canonical_ATE_eq","k":"theorem","m":"Causalean.PO.ID.Partial.BalkePearl.Sharp","a":"PO/ID/Partial/BalkePearl","t":2,"d":"S'.ATE = BPObjective π.","s":"∀ {P : Causalean.PO.POSystem} (S : Causalean.PO.POBalkePearlSystem P) (π : Bool → Bool → Bool → Bool → ℝ) (hπ_nn : ∀ (d0 d1 y0 y1 : Bool), 0 ≤ π d0 d1 y0 y1) (hπ_sum : ∑ d0, ∑ d1, ∑ y0, ∑ y1, π d0 d1 y0 y1 = 1), S.BaseAssumptions → (Causalean.PO.POBalkePearlSharp.S'✝ S π hπ_nn hπ_sum).ATE = Causalean.PO.POBalkePearlSystem.BPObjective π","l":823,"f":"Causalean/PO/ID/Partial/BalkePearl/Sharp.lean"},{"n":"Causalean.PO.POBalkePearlSharp.canonical_DofZ","k":"theorem","m":"Causalean.PO.ID.Partial.BalkePearl.Sharp","a":"PO/ID/Partial/BalkePearl","t":2,"d":"S'.DofZ z ω = (POBalkePearlSystem.dArm) z ω.2.1 ω.2.2.1.","s":"∀ {P : Causalean.PO.POSystem} (S : Causalean.PO.POBalkePearlSystem P) (π : Bool → Bool → Bool → Bool → ℝ) (hπ_nn : ∀ (d0 d1 y0 y1 : Bool), 0 ≤ π d0 d1 y0 y1) (hπ_sum : ∑ d0, ∑ d1, ∑ y0, ∑ y1, π d0 d1 y0 y1 = 1) (z : Bool) (ω : Causalean.PO.POBalkePearlSharp.SOmega), (Causalean.PO.POBalkePearlSharp.S'✝ S π hπ_nn hπ_sum).DofZ z ω = Causalean.PO.POBalkePearlSystem.dArm z ω.2.1 ω.2.2.1","l":490,"f":"Causalean/PO/ID/Partial/BalkePearl/Sharp.lean"},{"n":"Causalean.PO.POBalkePearlSharp.canonical_YofD","k":"theorem","m":"Causalean.PO.ID.Partial.BalkePearl.Sharp","a":"PO/ID/Partial/BalkePearl","t":2,"d":"S'.YofD d ω = (POBalkePearlSystem.yArm) d ω.2.2.2.1 ω.2.2.2.2.","s":"∀ {P : Causalean.PO.POSystem} (S : Causalean.PO.POBalkePearlSystem P) (π : Bool → Bool → Bool → Bool → ℝ) (hπ_nn : ∀ (d0 d1 y0 y1 : Bool), 0 ≤ π d0 d1 y0 y1) (hπ_sum : ∑ d0, ∑ d1, ∑ y0, ∑ y1, π d0 d1 y0 y1 = 1) (d : Bool) (ω : Causalean.PO.POBalkePearlSharp.SOmega), (Causalean.PO.POBalkePearlSharp.S'✝ S π hπ_nn hπ_sum).YofD d ω = Causalean.PO.POBalkePearlSystem.yArm d ω.2.2.2.1 ω.2.2.2.2","l":507,"f":"Causalean/PO/ID/Partial/BalkePearl/Sharp.lean"},{"n":"Causalean.PO.POBalkePearlSharp.canonical_YofZD","k":"theorem","m":"Causalean.PO.ID.Partial.BalkePearl.Sharp","a":"PO/ID/Partial/BalkePearl","t":2,"d":"S'.YofZD z d ω = (POBalkePearlSystem.yArm) d ω.2.2.2.1 ω.2.2.2.2.","s":"∀ {P : Causalean.PO.POSystem} (S : Causalean.PO.POBalkePearlSystem P) (π : Bool → Bool → Bool → Bool → ℝ) (hπ_nn : ∀ (d0 d1 y0 y1 : Bool), 0 ≤ π d0 d1 y0 y1) (hπ_sum : ∑ d0, ∑ d1, ∑ y0, ∑ y1, π d0 d1 y0 y1 = 1) (z d : Bool) (ω : Causalean.PO.POBalkePearlSharp.SOmega), (Causalean.PO.POBalkePearlSharp.S'✝ S π hπ_nn hπ_sum).YofZD z d ω = Causalean.PO.POBalkePearlSystem.yArm d ω.2.2.2.1 ω.2.2.2.2","l":524,"f":"Causalean/PO/ID/Partial/BalkePearl/Sharp.lean"},{"n":"Causalean.PO.POBalkePearlSharp.canonical_baseAssumptions","k":"theorem","m":"Causalean.PO.ID.Partial.BalkePearl.Sharp","a":"PO/ID/Partial/BalkePearl","t":2,"d":"Bundle the canonical model's BaseAssumptions.","s":"∀ {P : Causalean.PO.POSystem} (S : Causalean.PO.POBalkePearlSystem P) (π : Bool → Bool → Bool → Bool → ℝ) (hπ_nn : ∀ (d0 d1 y0 y1 : Bool), 0 ≤ π d0 d1 y0 y1) (hπ_sum : ∑ d0, ∑ d1, ∑ y0, ∑ y1, π d0 d1 y0 y1 = 1), S.BaseAssumptions → (Causalean.PO.POBalkePearlSharp.S'✝ S π hπ_nn hπ_sum).BaseAssumptions","l":689,"f":"Causalean/PO/ID/Partial/BalkePearl/Sharp.lean"},{"n":"Causalean.PO.POBalkePearlSharp.canonical_cellProb_eq","k":"theorem","m":"Causalean.PO.ID.Partial.BalkePearl.Sharp","a":"PO/ID/Partial/BalkePearl","t":2,"d":"S'.cellProb y d z = S.cellProb y d z for any feasible π.","s":"∀ {P : Causalean.PO.POSystem} (S : Causalean.PO.POBalkePearlSystem P) (π : Bool → Bool → Bool → Bool → ℝ) (hπ_nn : ∀ (d0 d1 y0 y1 : Bool), 0 ≤ π d0 d1 y0 y1) (hπ_sum : ∑ d0, ∑ d1, ∑ y0, ∑ y1, π d0 d1 y0 y1 = 1) (hA : S.BaseAssumptions), S.BPFeasible hA π → ∀ (y d z : Bool), (Causalean.PO.POBalkePearlSharp.S'✝ S π hπ_nn hπ_sum).cellProb y d z = S.cellProb y d z","l":811,"f":"Causalean/PO/ID/Partial/BalkePearl/Sharp.lean"},{"n":"Causalean.PO.POBalkePearlSharp.canonical_cfBundle_factors_through_snd","k":"theorem","m":"Causalean.PO.ID.Partial.BalkePearl.Sharp","a":"PO/ID/Partial/BalkePearl","t":2,"d":"The cfBundle's jointValue factors through Prod.snd: it depends only on the latent factor.","s":"∀ {P : Causalean.PO.POSystem} (S : Causalean.PO.POBalkePearlSystem P) (π : Bool → Bool → Bool → Bool → ℝ) (hπ_nn : ∀ (d0 d1 y0 y1 : Bool), 0 ≤ π d0 d1 y0 y1) (hπ_sum : ∑ d0, ∑ d1, ∑ y0, ∑ y1, π d0 d1 y0 y1 = 1), ∃ g, Measurable g ∧ (Causalean.PO.POBalkePearlSharp.S'✝ S π hπ_nn hπ_sum).cfBundle.jointValue = fun ω => g ω.2","l":618,"f":"Causalean/PO/ID/Partial/BalkePearl/Sharp.lean"},{"n":"Causalean.PO.POBalkePearlSharp.canonical_consistency","k":"theorem","m":"Causalean.PO.ID.Partial.BalkePearl.Sharp","a":"PO/ID/Partial/BalkePearl","t":2,"d":"Consistency for the canonical PO system.","s":"∀ {P : Causalean.PO.POSystem} (S : Causalean.PO.POBalkePearlSystem P) (π : Bool → Bool → Bool → Bool → ℝ) (hπ_nn : ∀ (d0 d1 y0 y1 : Bool), 0 ≤ π d0 d1 y0 y1) (hπ_sum : ∑ d0, ∑ d1, ∑ y0, ∑ y1, π d0 d1 y0 y1 = 1), (Causalean.PO.POBalkePearlSharp.P'✝ S π hπ_nn hπ_sum).Consistency","l":333,"f":"Causalean/PO/ID/Partial/BalkePearl/Sharp.lean"},{"n":"Causalean.PO.POBalkePearlSharp.canonical_exclusion","k":"theorem","m":"Causalean.PO.ID.Partial.BalkePearl.Sharp","a":"PO/ID/Partial/BalkePearl","t":2,"d":"Exclusion: Y(z,d) = Y(d) pointwise (and so a.e.).","s":"∀ {P : Causalean.PO.POSystem} (S : Causalean.PO.POBalkePearlSystem P) (π : Bool → Bool → Bool → Bool → ℝ) (hπ_nn : ∀ (d0 d1 y0 y1 : Bool), 0 ≤ π d0 d1 y0 y1) (hπ_sum : ∑ d0, ∑ d1, ∑ y0, ∑ y1, π d0 d1 y0 y1 = 1) (z d : Bool), (Causalean.PO.POBalkePearlSharp.S'✝ S π hπ_nn hπ_sum).YofZD z d = (Causalean.PO.POBalkePearlSharp.S'✝ S π hπ_nn hπ_sum).YofD d","l":563,"f":"Causalean/PO/ID/Partial/BalkePearl/Sharp.lean"},{"n":"Causalean.PO.POBalkePearlSharp.canonical_exogeneity","k":"theorem","m":"Causalean.PO.ID.Partial.BalkePearl.Sharp","a":"PO/ID/Partial/BalkePearl","t":2,"d":"Exogeneity: Z ⊥ cfBundle under the canonical product measure.","s":"∀ {P : Causalean.PO.POSystem} (S : Causalean.PO.POBalkePearlSystem P) (π : Bool → Bool → Bool → Bool → ℝ) (hπ_nn : ∀ (d0 d1 y0 y1 : Bool), 0 ≤ π d0 d1 y0 y1) (hπ_sum : ∑ d0, ∑ d1, ∑ y0, ∑ y1, π d0 d1 y0 y1 = 1), (Causalean.PO.POBalkePearlSharp.P'✝ S π hπ_nn hπ_sum).IndepCF (Causalean.PO.RegimedVar.ofFactual (Causalean.PO.POBalkePearlSharp.S'✝ S π hπ_nn hπ_sum).zVar) (Causalean.PO.POBalkePearlSharp.S'✝ S π hπ_nn hπ_sum).cfBundle","l":665,"f":"Causalean/PO/ID/Partial/BalkePearl/Sharp.lean"},{"n":"Causalean.PO.POBalkePearlSharp.canonical_factualD","k":"theorem","m":"Causalean.PO.ID.Partial.BalkePearl.Sharp","a":"PO/ID/Partial/BalkePearl","t":2,"d":"S'.factualD ω = D(ω.1).","s":"∀ {P : Causalean.PO.POSystem} (S : Causalean.PO.POBalkePearlSystem P) (π : Bool → Bool → Bool → Bool → ℝ) (hπ_nn : ∀ (d0 d1 y0 y1 : Bool), 0 ≤ π d0 d1 y0 y1) (hπ_sum : ∑ d0, ∑ d1, ∑ y0, ∑ y1, π d0 d1 y0 y1 = 1) (ω : Causalean.PO.POBalkePearlSharp.SOmega), (Causalean.PO.POBalkePearlSharp.S'✝ S π hπ_nn hπ_sum).factualD ω = Causalean.PO.POBalkePearlSharp.dArmω ω.1 ω","l":464,"f":"Causalean/PO/ID/Partial/BalkePearl/Sharp.lean"},{"n":"Causalean.PO.POBalkePearlSharp.canonical_factualY","k":"theorem","m":"Causalean.PO.ID.Partial.BalkePearl.Sharp","a":"PO/ID/Partial/BalkePearl","t":2,"d":"S'.factualY ω = Y(D(ω.1)).","s":"∀ {P : Causalean.PO.POSystem} (S : Causalean.PO.POBalkePearlSystem P) (π : Bool → Bool → Bool → Bool → ℝ) (hπ_nn : ∀ (d0 d1 y0 y1 : Bool), 0 ≤ π d0 d1 y0 y1) (hπ_sum : ∑ d0, ∑ d1, ∑ y0, ∑ y1, π d0 d1 y0 y1 = 1) (ω : Causalean.PO.POBalkePearlSharp.SOmega), (Causalean.PO.POBalkePearlSharp.S'✝ S π hπ_nn hπ_sum).factualY ω = Causalean.PO.POBalkePearlSharp.yArmω (Causalean.PO.POBalkePearlSharp.dArmω ω.1 ω) ω","l":476,"f":"Causalean/PO/ID/Partial/BalkePearl/Sharp.lean"},{"n":"Causalean.PO.POBalkePearlSharp.canonical_factualZ","k":"theorem","m":"Causalean.PO.ID.Partial.BalkePearl.Sharp","a":"PO/ID/Partial/BalkePearl","t":2,"d":"S'.factualZ ω = ω.1.","s":"∀ {P : Causalean.PO.POSystem} (S : Causalean.PO.POBalkePearlSystem P) (π : Bool → Bool → Bool → Bool → ℝ) (hπ_nn : ∀ (d0 d1 y0 y1 : Bool), 0 ≤ π d0 d1 y0 y1) (hπ_sum : ∑ d0, ∑ d1, ∑ y0, ∑ y1, π d0 d1 y0 y1 = 1) (ω : Causalean.PO.POBalkePearlSharp.SOmega), (Causalean.PO.POBalkePearlSharp.S'✝ S π hπ_nn hπ_sum).factualZ ω = ω.1","l":455,"f":"Causalean/PO/ID/Partial/BalkePearl/Sharp.lean"},{"n":"Causalean.PO.POBalkePearlSharp.canonical_factualZ_eq_fst","k":"theorem","m":"Causalean.PO.ID.Partial.BalkePearl.Sharp","a":"PO/ID/Partial/BalkePearl","t":2,"d":"The factualZ on the canonical model factors through Prod.fst.","s":"∀ {P : Causalean.PO.POSystem} (S : Causalean.PO.POBalkePearlSystem P) (π : Bool → Bool → Bool → Bool → ℝ) (hπ_nn : ∀ (d0 d1 y0 y1 : Bool), 0 ≤ π d0 d1 y0 y1) (hπ_sum : ∑ d0, ∑ d1, ∑ y0, ∑ y1, π d0 d1 y0 y1 = 1), (Causalean.PO.POBalkePearlSharp.S'✝ S π hπ_nn hπ_sum).factualZ = fun ω => ω.1","l":613,"f":"Causalean/PO/ID/Partial/BalkePearl/Sharp.lean"},{"n":"Causalean.PO.POBalkePearlSharp.canonical_latentProb_eq","k":"theorem","m":"Causalean.PO.ID.Partial.BalkePearl.Sharp","a":"PO/ID/Partial/BalkePearl","t":2,"d":"S'.latentProb = π.","s":"∀ {P : Causalean.PO.POSystem} (S : Causalean.PO.POBalkePearlSystem P) (π : Bool → Bool → Bool → Bool → ℝ) (hπ_nn : ∀ (d0 d1 y0 y1 : Bool), 0 ≤ π d0 d1 y0 y1) (hπ_sum : ∑ d0, ∑ d1, ∑ y0, ∑ y1, π d0 d1 y0 y1 = 1) (d0 d1 y0 y1 : Bool), (Causalean.PO.POBalkePearlSharp.S'✝ S π hπ_nn hπ_sum).latentProb d0 d1 y0 y1 = π d0 d1 y0 y1","l":804,"f":"Causalean/PO/ID/Partial/BalkePearl/Sharp.lean"},{"n":"Causalean.PO.POBalkePearlSharp.canonical_latentSet","k":"theorem","m":"Causalean.PO.ID.Partial.BalkePearl.Sharp","a":"PO/ID/Partial/BalkePearl","t":2,"d":"The canonical latent set is Set.univ ×ˢ {(d0, d1, y0, y1)}.","s":"∀ {P : Causalean.PO.POSystem} (S : Causalean.PO.POBalkePearlSystem P) (π : Bool → Bool → Bool → Bool → ℝ) (hπ_nn : ∀ (d0 d1 y0 y1 : Bool), 0 ≤ π d0 d1 y0 y1) (hπ_sum : ∑ d0, ∑ d1, ∑ y0, ∑ y1, π d0 d1 y0 y1 = 1) (d0 d1 y0 y1 : Bool), (Causalean.PO.POBalkePearlSharp.S'✝ S π hπ_nn hπ_sum).latentSet d0 d1 y0 y1 = Set.univ ×ˢ {(d0, d1, y0, y1)}","l":764,"f":"Causalean/PO/ID/Partial/BalkePearl/Sharp.lean"},{"n":"Causalean.PO.POBalkePearlSharp.canonical_latentSet_measure","k":"theorem","m":"Causalean.PO.ID.Partial.BalkePearl.Sharp","a":"PO/ID/Partial/BalkePearl","t":2,"d":"μ'(latentSet) = ENNReal.ofReal (π d0 d1 y0 y1).","s":"∀ {P : Causalean.PO.POSystem} (S : Causalean.PO.POBalkePearlSystem P) (π : Bool → Bool → Bool → Bool → ℝ) (hπ_nn : ∀ (d0 d1 y0 y1 : Bool), 0 ≤ π d0 d1 y0 y1) (hπ_sum : ∑ d0, ∑ d1, ∑ y0, ∑ y1, π d0 d1 y0 y1 = 1) (d0 d1 y0 y1 : Bool), (Causalean.PO.POBalkePearlSharp.P'✝ S π hπ_nn hπ_sum).μ ((Causalean.PO.POBalkePearlSharp.S'✝ S π hπ_nn hπ_sum).latentSet d0 d1 y0 y1) = ENNReal.ofReal (π d0 d1 y0 y1)","l":793,"f":"Causalean/PO/ID/Partial/BalkePearl/Sharp.lean"},{"n":"Causalean.PO.POBalkePearlSharp.canonical_posZ","k":"theorem","m":"Causalean.PO.ID.Partial.BalkePearl.Sharp","a":"PO/ID/Partial/BalkePearl","t":2,"d":"Positive Z probability.","s":"∀ {P : Causalean.PO.POSystem} (S : Causalean.PO.POBalkePearlSystem P) (π : Bool → Bool → Bool → Bool → ℝ) (hπ_nn : ∀ (d0 d1 y0 y1 : Bool), 0 ≤ π d0 d1 y0 y1) (hπ_sum : ∑ d0, ∑ d1, ∑ y0, ∑ y1, π d0 d1 y0 y1 = 1), S.BaseAssumptions → ∀ (z : Bool), 0 < (Causalean.PO.POBalkePearlSharp.P'✝ S π hπ_nn hπ_sum).μ ((Causalean.PO.POBalkePearlSharp.S'✝ S π hπ_nn hπ_sum).zVar.event z)","l":604,"f":"Causalean/PO/ID/Partial/BalkePearl/Sharp.lean"},{"n":"Causalean.PO.POBalkePearlSharp.canonical_zEvent","k":"theorem","m":"Causalean.PO.ID.Partial.BalkePearl.Sharp","a":"PO/ID/Partial/BalkePearl","t":2,"d":"The Z-event in the canonical model is {ω | ω.1 = z}.","s":"∀ {P : Causalean.PO.POSystem} (S : Causalean.PO.POBalkePearlSystem P) (π : Bool → Bool → Bool → Bool → ℝ) (hπ_nn : ∀ (d0 d1 y0 y1 : Bool), 0 ≤ π d0 d1 y0 y1) (hπ_sum : ∑ d0, ∑ d1, ∑ y0, ∑ y1, π d0 d1 y0 y1 = 1) (z : Bool), (Causalean.PO.POBalkePearlSharp.S'✝ S π hπ_nn hπ_sum).zEvent z = {ω | ω.1 = z}","l":571,"f":"Causalean/PO/ID/Partial/BalkePearl/Sharp.lean"},{"n":"Causalean.PO.POBalkePearlSharp.canonical_zEvent_measure","k":"theorem","m":"Causalean.PO.ID.Partial.BalkePearl.Sharp","a":"PO/ID/Partial/BalkePearl","t":2,"d":"μ' (S'.zEvent z) = P.μ (S.zEvent z).","s":"∀ {P : Causalean.PO.POSystem} (S : Causalean.PO.POBalkePearlSystem P) (π : Bool → Bool → Bool → Bool → ℝ) (hπ_nn : ∀ (d0 d1 y0 y1 : Bool), 0 ≤ π d0 d1 y0 y1) (hπ_sum : ∑ d0, ∑ d1, ∑ y0, ∑ y1, π d0 d1 y0 y1 = 1) (z : Bool), (Causalean.PO.POBalkePearlSharp.P'✝ S π hπ_nn hπ_sum).μ ((Causalean.PO.POBalkePearlSharp.S'✝ S π hπ_nn hπ_sum).zEvent z) = P.μ (S.zEvent z)","l":579,"f":"Causalean/PO/ID/Partial/BalkePearl/Sharp.lean"},{"n":"Causalean.PO.POBalkePearlSharp.dArmω","k":"def","m":"Causalean.PO.ID.Partial.BalkePearl.Sharp","a":"PO/ID/Partial/BalkePearl","t":1,"d":"The latent arm under intervention Z = z.","s":"Bool → Causalean.PO.POBalkePearlSharp.SOmega → Bool","l":61,"f":"Causalean/PO/ID/Partial/BalkePearl/Sharp.lean"},{"n":"Causalean.PO.POBalkePearlSharp.eval","k":"def","m":"Causalean.PO.ID.Partial.BalkePearl.Sharp","a":"PO/ID/Partial/BalkePearl","t":1,"d":"Canonical eval: cascade through Z → D → Y.","s":"Causalean.PO.Regime Causalean.PO.POBalkePearlSharp.SV Causalean.PO.POBalkePearlSharp.SX → Causalean.PO.POBalkePearlSharp.SOmega → (v : Causalean.PO.POBalkePearlSharp.SV) → Causalean.PO.POBalkePearlSharp.SX v","l":69,"f":"Causalean/PO/ID/Partial/BalkePearl/Sharp.lean"},{"n":"Causalean.PO.POBalkePearlSharp.eval_one","k":"theorem","m":"Causalean.PO.ID.Partial.BalkePearl.Sharp","a":"PO/ID/Partial/BalkePearl","t":2,"d":"The D-coordinate folded via eval_zero.","s":"∀ (r : Causalean.PO.Regime Causalean.PO.POBalkePearlSharp.SV Causalean.PO.POBalkePearlSharp.SX) (ω : Causalean.PO.POBalkePearlSharp.SOmega), Causalean.PO.POBalkePearlSharp.eval r ω ⟨1, of_decide_eq_true (id (Eq.refl true))⟩ = if h : ⟨1, of_decide_eq_true (id (Eq.refl true))⟩ ∈ r.target then r.assign ⟨1, of_decide_eq_true (id (Eq.refl true))⟩ h else Causalean.PO.POBalkePearlSharp.dArmω (Causalean.PO.POBalkePearlSharp.eval r ω ⟨0, of_decide_eq_true (id (Eq.refl true))⟩) ω","l":118,"f":"Causalean/PO/ID/Partial/BalkePearl/Sharp.lean"},{"n":"Causalean.PO.POBalkePearlSharp.eval_one_raw","k":"theorem","m":"Causalean.PO.ID.Partial.BalkePearl.Sharp","a":"PO/ID/Partial/BalkePearl","t":2,"d":"The D-coordinate (raw).","s":"∀ (r : Causalean.PO.Regime Causalean.PO.POBalkePearlSharp.SV Causalean.PO.POBalkePearlSharp.SX) (ω : Causalean.PO.POBalkePearlSharp.SOmega), Causalean.PO.POBalkePearlSharp.eval r ω ⟨1, of_decide_eq_true (id (Eq.refl true))⟩ = if h : ⟨1, of_decide_eq_true (id (Eq.refl true))⟩ ∈ r.target then r.assign ⟨1, of_decide_eq_true (id (Eq.refl true))⟩ h else Causalean.PO.POBalkePearlSharp.dArmω (if h0 : ⟨0, of_decide_eq_true (id (Eq.refl true))⟩ ∈ r.target then r.assign ⟨0, of_decide_eq_true (id (Eq.refl true))⟩ h0 else ω.1) ω","l":101,"f":"Causalean/PO/ID/Partial/BalkePearl/Sharp.lean"},{"n":"Causalean.PO.POBalkePearlSharp.eval_two","k":"theorem","m":"Causalean.PO.ID.Partial.BalkePearl.Sharp","a":"PO/ID/Partial/BalkePearl","t":2,"d":"The Y-coordinate folded via eval_one.","s":"∀ (r : Causalean.PO.Regime Causalean.PO.POBalkePearlSharp.SV Causalean.PO.POBalkePearlSharp.SX) (ω : Causalean.PO.POBalkePearlSharp.SOmega), Causalean.PO.POBalkePearlSharp.eval r ω ⟨2, of_decide_eq_true (id (Eq.refl true))⟩ = if h : ⟨2, of_decide_eq_true (id (Eq.refl true))⟩ ∈ r.target then r.assign ⟨2, of_decide_eq_true (id (Eq.refl true))⟩ h else Causalean.PO.POBalkePearlSharp.yArmω (Causalean.PO.POBalkePearlSharp.eval r ω ⟨1, of_decide_eq_true (id (Eq.refl true))⟩) ω","l":125,"f":"Causalean/PO/ID/Partial/BalkePearl/Sharp.lean"},{"n":"Causalean.PO.POBalkePearlSharp.eval_two_raw","k":"theorem","m":"Causalean.PO.ID.Partial.BalkePearl.Sharp","a":"PO/ID/Partial/BalkePearl","t":2,"d":"The Y-coordinate (raw).","s":"∀ (r : Causalean.PO.Regime Causalean.PO.POBalkePearlSharp.SV Causalean.PO.POBalkePearlSharp.SX) (ω : Causalean.PO.POBalkePearlSharp.SOmega), Causalean.PO.POBalkePearlSharp.eval r ω ⟨2, of_decide_eq_true (id (Eq.refl true))⟩ = if h : ⟨2, of_decide_eq_true (id (Eq.refl true))⟩ ∈ r.target then r.assign ⟨2, of_decide_eq_true (id (Eq.refl true))⟩ h else Causalean.PO.POBalkePearlSharp.yArmω (if h1 : ⟨1, of_decide_eq_true (id (Eq.refl true))⟩ ∈ r.target then r.assign ⟨1, of_decide_eq_true (id (Eq.refl true))⟩ h1 else Causalean.PO.POBalkePearlSharp.dArmω (if h0 : ⟨0, of_decide_eq_true (id (Eq.refl true))⟩ ∈ r.target then r.assign ⟨0, of_decide_eq_true (id (Eq.refl true))⟩ h0 else ω.1) ω) ω","l":108,"f":"Causalean/PO/ID/Partial/BalkePearl/Sharp.lean"},{"n":"Causalean.PO.POBalkePearlSharp.eval_zero","k":"theorem","m":"Causalean.PO.ID.Partial.BalkePearl.Sharp","a":"PO/ID/Partial/BalkePearl","t":2,"d":"The Z-coordinate eval r ω 0: assignment if intervened on, else ω.1.","s":"∀ (r : Causalean.PO.Regime Causalean.PO.POBalkePearlSharp.SV Causalean.PO.POBalkePearlSharp.SX) (ω : Causalean.PO.POBalkePearlSharp.SOmega), Causalean.PO.POBalkePearlSharp.eval r ω ⟨0, of_decide_eq_true (id (Eq.refl true))⟩ = if h : ⟨0, of_decide_eq_true (id (Eq.refl true))⟩ ∈ r.target then r.assign ⟨0, of_decide_eq_true (id (Eq.refl true))⟩ h else ω.1","l":96,"f":"Causalean/PO/ID/Partial/BalkePearl/Sharp.lean"},{"n":"Causalean.PO.POBalkePearlSharp.instIsProbCanonicalMeasure","k":"instance","m":"Causalean.PO.ID.Partial.BalkePearl.Sharp","a":"PO/ID/Partial/BalkePearl","t":2,"d":"The product of the instrument marginal and latent-table measure is a probability measure.","s":"∀ {P : Causalean.PO.POSystem} (S : Causalean.PO.POBalkePearlSystem P) {π : Bool → Bool → Bool → Bool → ℝ} [MeasureTheory.IsProbabilityMeasure (Causalean.PO.POBalkePearlSharp.piMeasure π)], MeasureTheory.IsProbabilityMeasure (Causalean.PO.POBalkePearlSharp.canonicalMeasure S π)","l":264,"f":"Causalean/PO/ID/Partial/BalkePearl/Sharp.lean"},{"n":"Causalean.PO.POBalkePearlSharp.instIsProbPiMeasure","k":"theorem","m":"Causalean.PO.ID.Partial.BalkePearl.Sharp","a":"PO/ID/Partial/BalkePearl","t":2,"d":"A nonnegative latent table that sums to one induces a probability measure.","s":"∀ {π : Bool → Bool → Bool → Bool → ℝ}, (∀ (d0 d1 y0 y1 : Bool), 0 ≤ π d0 d1 y0 y1) → ∑ d0, ∑ d1, ∑ y0, ∑ y1, π d0 d1 y0 y1 = 1 → MeasureTheory.IsProbabilityMeasure (Causalean.PO.POBalkePearlSharp.piMeasure π)","l":254,"f":"Causalean/PO/ID/Partial/BalkePearl/Sharp.lean"},{"n":"Causalean.PO.POBalkePearlSharp.instIsProbZMeasure","k":"instance","m":"Causalean.PO.ID.Partial.BalkePearl.Sharp","a":"PO/ID/Partial/BalkePearl","t":2,"d":"The original instrument marginal is a probability measure.","s":"∀ {P : Causalean.PO.POSystem} (S : Causalean.PO.POBalkePearlSystem P), MeasureTheory.IsProbabilityMeasure (Causalean.PO.POBalkePearlSharp.zMeasure S)","l":203,"f":"Causalean/PO/ID/Partial/BalkePearl/Sharp.lean"},{"n":"Causalean.PO.POBalkePearlSharp.measurable_eval","k":"theorem","m":"Causalean.PO.ID.Partial.BalkePearl.Sharp","a":"PO/ID/Partial/BalkePearl","t":2,"d":"The canonical evaluator is measurable under every intervention regime.","s":"∀ (r : Causalean.PO.Regime Causalean.PO.POBalkePearlSharp.SV Causalean.PO.POBalkePearlSharp.SX), Measurable (Causalean.PO.POBalkePearlSharp.eval r)","l":92,"f":"Causalean/PO/ID/Partial/BalkePearl/Sharp.lean"},{"n":"Causalean.PO.POBalkePearlSharp.piMeasure","k":"def","m":"Causalean.PO.ID.Partial.BalkePearl.Sharp","a":"PO/ID/Partial/BalkePearl","t":1,"d":"The discrete π-measure on Bool⁴ with weights π.","s":"(Bool → Bool → Bool → Bool → ℝ) → MeasureTheory.Measure (Bool × Bool × Bool × Bool)","l":167,"f":"Causalean/PO/ID/Partial/BalkePearl/Sharp.lean"},{"n":"Causalean.PO.POBalkePearlSharp.piMeasure_singleton","k":"theorem","m":"Causalean.PO.ID.Partial.BalkePearl.Sharp","a":"PO/ID/Partial/BalkePearl","t":2,"d":"piMeasure π {(d0, d1, y0, y1)} = ENNReal.ofReal (π d0 d1 y0 y1).","s":"∀ (π : Bool → Bool → Bool → Bool → ℝ) (d0 d1 y0 y1 : Bool), (Causalean.PO.POBalkePearlSharp.piMeasure π) {(d0, d1, y0, y1)} = ENNReal.ofReal (π d0 d1 y0 y1)","l":720,"f":"Causalean/PO/ID/Partial/BalkePearl/Sharp.lean"},{"n":"Causalean.PO.POBalkePearlSharp.piMeasure_univ_of_feasible","k":"theorem","m":"Causalean.PO.ID.Partial.BalkePearl.Sharp","a":"PO/ID/Partial/BalkePearl","t":2,"d":"piMeasure π has total mass 1 when π has nonneg entries summing to 1.","s":"∀ {π : Bool → Bool → Bool → Bool → ℝ}, (∀ (d0 d1 y0 y1 : Bool), 0 ≤ π d0 d1 y0 y1) → ∑ d0, ∑ d1, ∑ y0, ∑ y1, π d0 d1 y0 y1 = 1 → (Causalean.PO.POBalkePearlSharp.piMeasure π) Set.univ = 1","l":207,"f":"Causalean/PO/ID/Partial/BalkePearl/Sharp.lean"},{"n":"Causalean.PO.POBalkePearlSharp.yArmω","k":"def","m":"Causalean.PO.ID.Partial.BalkePearl.Sharp","a":"PO/ID/Partial/BalkePearl","t":1,"d":"The latent arm under intervention D = d.","s":"Bool → Causalean.PO.POBalkePearlSharp.SOmega → Bool","l":65,"f":"Causalean/PO/ID/Partial/BalkePearl/Sharp.lean"},{"n":"Causalean.PO.POBalkePearlSharp.zMeasure","k":"def","m":"Causalean.PO.ID.Partial.BalkePearl.Sharp","a":"PO/ID/Partial/BalkePearl","t":1,"d":"The Z-marginal of S, viewed as a measure on Bool.","s":"{P : Causalean.PO.POSystem} → Causalean.PO.POBalkePearlSystem P → MeasureTheory.Measure Bool","l":163,"f":"Causalean/PO/ID/Partial/BalkePearl/Sharp.lean"},{"n":"Causalean.PO.POBalkePearlSharp.zMeasure_univ","k":"theorem","m":"Causalean.PO.ID.Partial.BalkePearl.Sharp","a":"PO/ID/Partial/BalkePearl","t":2,"d":"zMeasure S has total mass 1.","s":"∀ {P : Causalean.PO.POSystem} (S : Causalean.PO.POBalkePearlSystem P), (Causalean.PO.POBalkePearlSharp.zMeasure S) Set.univ = 1","l":180,"f":"Causalean/PO/ID/Partial/BalkePearl/Sharp.lean"},{"n":"Causalean.PO.POBalkePearlSystem","k":"structure","m":"Causalean.PO.ID.Partial.BalkePearl.Setup","a":"PO/ID/Partial/BalkePearl","t":1,"d":"Binary-IV system for Balke–Pearl ATE bounds (def:po-iv-balke-pearl-system). Inside a potential-outcome system, this bundles a binary instrument node Z, a binary treatment node D, and a binary outcome node Y, subject to the instrument, treatment, and outcome being pairwise distinct system variables.","s":"Causalean.PO.POSystem → Type (max u_1 u_2)","l":36,"f":"Causalean/PO/ID/Partial/BalkePearl/Setup.lean"},{"n":"Causalean.PO.POBalkePearlSystem.ATE","k":"def","m":"Causalean.PO.ID.Partial.BalkePearl.Setup","a":"PO/ID/Partial/BalkePearl","t":1,"d":"Average treatment effect E[Y(1) - Y(0)] (as a difference of Bool-in-ℝ integrals).","s":"{P : Causalean.PO.POSystem} → Causalean.PO.POBalkePearlSystem P → ℝ","l":165,"f":"Causalean/PO/ID/Partial/BalkePearl/Setup.lean"},{"n":"Causalean.PO.POBalkePearlSystem.ATE_eq_BPObjective","k":"theorem","m":"Causalean.PO.ID.Partial.BalkePearl.Main","a":"PO/ID/Partial/BalkePearl","t":2,"d":"ATE = BPObjective applied to the realized latent table.","s":"∀ {P : Causalean.PO.POSystem} (S : Causalean.PO.POBalkePearlSystem P), S.BaseAssumptions → S.ATE = Causalean.PO.POBalkePearlSystem.BPObjective S.latentProb","l":90,"f":"Causalean/PO/ID/Partial/BalkePearl/Main.lean"},{"n":"Causalean.PO.POBalkePearlSystem.ATE_eq_sum_latent","k":"theorem","m":"Causalean.PO.ID.Partial.BalkePearl.LatentTable","a":"PO/ID/Partial/BalkePearl","t":1,"d":"The average treatment effect equals the probability-weighted sum, over the sixteen latent response types, of each type's treatment effect.","s":"∀ {P : Causalean.PO.POSystem} (S : Causalean.PO.POBalkePearlSystem P), S.ATE = ∑ d0, ∑ d1, ∑ y0, ∑ y1, (Causalean.PO.POBalkePearlSystem.boolToReal y1 - Causalean.PO.POBalkePearlSystem.boolToReal y0) * S.latentProb d0 d1 y0 y1","l":144,"f":"Causalean/PO/ID/Partial/BalkePearl/LatentTable.lean"},{"n":"Causalean.PO.POBalkePearlSystem.ATE_mem_BPIdentifiedInterval","k":"theorem","m":"Causalean.PO.ID.Partial.BalkePearl.Main","a":"PO/ID/Partial/BalkePearl","t":1,"d":"Necessity. Under the Balke-Pearl IV base assumptions, the true average treatment effect lies in the Balke-Pearl identified interval.","s":"∀ {P : Causalean.PO.POSystem} (S : Causalean.PO.POBalkePearlSystem P) (hA : S.BaseAssumptions), S.ATE ∈ S.BPIdentifiedInterval hA","l":98,"f":"Causalean/PO/ID/Partial/BalkePearl/Main.lean"},{"n":"Causalean.PO.POBalkePearlSystem.ATE_mem_Icc_bpLower_bpUpper","k":"theorem","m":"Causalean.PO.ID.Partial.BalkePearl.ClosedForm","a":"PO/ID/Partial/BalkePearl","t":1,"d":"The Balke-Pearl bound in closed form. Under the Balke-Pearl IV base assumptions, the true average treatment effect lies between the largest of eight closed-form lower expressions and the smallest of eight closed-form upper expressions, both computable directly from the observed cell probabilities.","s":"∀ {P : Causalean.PO.POSystem} (S : Causalean.PO.POBalkePearlSystem P), S.BaseAssumptions → S.ATE ∈ Set.Icc S.bpLower S.bpUpper","l":199,"f":"Causalean/PO/ID/Partial/BalkePearl/ClosedForm.lean"},{"n":"Causalean.PO.POBalkePearlSystem.ATE_mem_Icc_csInf_csSup","k":"theorem","m":"Causalean.PO.ID.Partial.BalkePearl.IntervalForm","a":"PO/ID/Partial/BalkePearl","t":1,"d":"Necessity, closed-interval form. Under the Balke-Pearl IV base assumptions, the true average treatment effect lies between the infimum and supremum of the Balke-Pearl identified interval — the LP minimum and maximum. This is ATE_mem_BPIdentifiedInterval rephrased through the engine bridge Causalean.PartialID.mem_Icc_csInf_csSup.","s":"∀ {P : Causalean.PO.POSystem} (S : Causalean.PO.POBalkePearlSystem P) (hA : S.BaseAssumptions), S.ATE ∈ Set.Icc (sInf (S.BPIdentifiedInterval hA)) (sSup (S.BPIdentifiedInterval hA))","l":109,"f":"Causalean/PO/ID/Partial/BalkePearl/IntervalForm.lean"},{"n":"Causalean.PO.POBalkePearlSystem.BPFeasible","k":"structure","m":"Causalean.PO.ID.Partial.BalkePearl.Main","a":"PO/ID/Partial/BalkePearl","t":1,"d":"Balke–Pearl latent-table feasibility. A latent response-type table π — the joint distribution over the instrument's and treatment's potential values together with the outcome's potential values — is feasible for a Balke–Pearl IV system under a given assumption bundle when every table entry is nonnegative, the entries sum to one, and aggregating the table over the response types compatible with each observed instrument-treatment-outcome cell reproduces the observed conditional cell probability.","s":"{P : Causalean.PO.POSystem} → (S : Causalean.PO.POBalkePearlSystem P) → S.BaseAssumptions → (Bool → Bool → Bool → Bool → ℝ) → Prop","l":47,"f":"Causalean/PO/ID/Partial/BalkePearl/Main.lean"},{"n":"Causalean.PO.POBalkePearlSystem.BPIdentifiedInterval","k":"def","m":"Causalean.PO.ID.Partial.BalkePearl.Main","a":"PO/ID/Partial/BalkePearl","t":1,"d":"The Balke-Pearl sharp identified interval for ATE.","s":"{P : Causalean.PO.POSystem} → (S : Causalean.PO.POBalkePearlSystem P) → S.BaseAssumptions → Set ℝ","l":74,"f":"Causalean/PO/ID/Partial/BalkePearl/Main.lean"},{"n":"Causalean.PO.POBalkePearlSystem.BPIdentifiedInterval_subset_Icc","k":"theorem","m":"Causalean.PO.ID.Partial.BalkePearl.ClosedForm","a":"PO/ID/Partial/BalkePearl","t":2,"d":"The identified interval is contained in the closed-form interval.","s":"∀ {P : Causalean.PO.POSystem} (S : Causalean.PO.POBalkePearlSystem P) (hA : S.BaseAssumptions), S.BPIdentifiedInterval hA ⊆ Set.Icc S.bpLower S.bpUpper","l":192,"f":"Causalean/PO/ID/Partial/BalkePearl/ClosedForm.lean"},{"n":"Causalean.PO.POBalkePearlSystem.BPObjective","k":"def","m":"Causalean.PO.ID.Partial.BalkePearl.Main","a":"PO/ID/Partial/BalkePearl","t":1,"d":"The LP objective: ∑ (y1-y0) * π = ATE in the latent model.","s":"(Bool → Bool → Bool → Bool → ℝ) → ℝ","l":67,"f":"Causalean/PO/ID/Partial/BalkePearl/Main.lean"},{"n":"Causalean.PO.POBalkePearlSystem.BPObjective_le_bpUpper","k":"theorem","m":"Causalean.PO.ID.Partial.BalkePearl.ClosedForm","a":"PO/ID/Partial/BalkePearl","t":2,"d":"The closed-form upper endpoint bounds the objective on feasible tables.","s":"∀ {P : Causalean.PO.POSystem} (S : Causalean.PO.POBalkePearlSystem P) (hA : S.BaseAssumptions) {π : Bool → Bool → Bool → Bool → ℝ}, S.BPFeasible hA π → Causalean.PO.POBalkePearlSystem.BPObjective π ≤ S.bpUpper","l":186,"f":"Causalean/PO/ID/Partial/BalkePearl/ClosedForm.lean"},{"n":"Causalean.PO.POBalkePearlSystem.BPObjective_le_bpUpperTerm","k":"theorem","m":"Causalean.PO.ID.Partial.BalkePearl.ClosedForm","a":"PO/ID/Partial/BalkePearl","t":2,"d":"Every upper expression bounds the objective from above on feasible tables.","s":"∀ {P : Causalean.PO.POSystem} (S : Causalean.PO.POBalkePearlSystem P) (hA : S.BaseAssumptions) {π : Bool → Bool → Bool → Bool → ℝ}, S.BPFeasible hA π → ∀ (i : Fin 8), Causalean.PO.POBalkePearlSystem.BPObjective π ≤ S.bpUpperTerm i","l":149,"f":"Causalean/PO/ID/Partial/BalkePearl/ClosedForm.lean"},{"n":"Causalean.PO.POBalkePearlSystem.BPObjective_le_one","k":"theorem","m":"Causalean.PO.ID.Partial.BalkePearl.IntervalForm","a":"PO/ID/Partial/BalkePearl","t":2,"d":"Upper bound on the objective. For a feasible (nonneg, sum-one) table π, the LP objective ∑ (y1 - y0) · π is at most 1, since every contrast is ≤ 1 and π is a probability vector.","s":"∀ {π : Bool → Bool → Bool → Bool → ℝ}, (∀ (d0 d1 y0 y1 : Bool), 0 ≤ π d0 d1 y0 y1) → ∑ d0, ∑ d1, ∑ y0, ∑ y1, π d0 d1 y0 y1 = 1 → Causalean.PO.POBalkePearlSystem.BPObjective π ≤ 1","l":60,"f":"Causalean/PO/ID/Partial/BalkePearl/IntervalForm.lean"},{"n":"Causalean.PO.POBalkePearlSystem.BaseAssumptions","k":"structure","m":"Causalean.PO.ID.Partial.BalkePearl.Assumptions","a":"PO/ID/Partial/BalkePearl","t":1,"d":"Balke-Pearl IV assumptions — def:po-iv-balke-pearl-assumptions.","s":"{P : Causalean.PO.POSystem} → Causalean.PO.POBalkePearlSystem P → Prop","l":55,"f":"Causalean/PO/ID/Partial/BalkePearl/Assumptions.lean"},{"n":"Causalean.PO.POBalkePearlSystem.BaseAssumptions.posZ_ne_top","k":"theorem","m":"Causalean.PO.ID.Partial.BalkePearl.Assumptions","a":"PO/ID/Partial/BalkePearl","t":2,"d":"Instrument-cell probabilities are finite.","s":"∀ {P : Causalean.PO.POSystem} {S : Causalean.PO.POBalkePearlSystem P}, S.BaseAssumptions → ∀ (z : Bool), P.μ (S.zVar.event z) ≠ ⊤","l":81,"f":"Causalean/PO/ID/Partial/BalkePearl/Assumptions.lean"},{"n":"Causalean.PO.POBalkePearlSystem.BaseAssumptions.posZ_ne_zero","k":"theorem","m":"Causalean.PO.ID.Partial.BalkePearl.Assumptions","a":"PO/ID/Partial/BalkePearl","t":2,"d":"Positive instrument probabilities are nonzero.","s":"∀ {P : Causalean.PO.POSystem} {S : Causalean.PO.POBalkePearlSystem P}, S.BaseAssumptions → ∀ (z : Bool), P.μ (S.zVar.event z) ≠ 0","l":77,"f":"Causalean/PO/ID/Partial/BalkePearl/Assumptions.lean"},{"n":"Causalean.PO.POBalkePearlSystem.BaseAssumptions.posZ_toReal_pos","k":"theorem","m":"Causalean.PO.ID.Partial.BalkePearl.Assumptions","a":"PO/ID/Partial/BalkePearl","t":1,"d":"Under the Balke-Pearl IV base assumptions — IV-specific consistency of treatment and outcome on their respective cells, the exclusion restriction, instrument exogeneity, and positive instrument probability, the real-valued probability of each instrument cell z is strictly positive.","s":"∀ {P : Causalean.PO.POSystem} {S : Causalean.PO.POBalkePearlSystem P}, S.BaseAssumptions → ∀ (z : Bool), 0 < (P.μ (S.zVar.event z)).toReal","l":86,"f":"Causalean/PO/ID/Partial/BalkePearl/Assumptions.lean"},{"n":"Causalean.PO.POBalkePearlSystem.DofZ","k":"def","m":"Causalean.PO.ID.Partial.BalkePearl.Setup","a":"PO/ID/Partial/BalkePearl","t":1,"d":"The treatment value that would be observed for a unit if the instrument were set to z.","s":"{P : Causalean.PO.POSystem} → Causalean.PO.POBalkePearlSystem P → Bool → P.Ω → Bool","l":69,"f":"Causalean/PO/ID/Partial/BalkePearl/Setup.lean"},{"n":"Causalean.PO.POBalkePearlSystem.YofD","k":"def","m":"Causalean.PO.ID.Partial.BalkePearl.Setup","a":"PO/ID/Partial/BalkePearl","t":1,"d":"Y(d) : P.Ω → Bool.","s":"{P : Causalean.PO.POSystem} → Causalean.PO.POBalkePearlSystem P → Bool → P.Ω → Bool","l":72,"f":"Causalean/PO/ID/Partial/BalkePearl/Setup.lean"},{"n":"Causalean.PO.POBalkePearlSystem.YofD_real","k":"def","m":"Causalean.PO.ID.Partial.BalkePearl.Setup","a":"PO/ID/Partial/BalkePearl","t":1,"d":"Y(d) lifted to ℝ for integration.","s":"{P : Causalean.PO.POSystem} → Causalean.PO.POBalkePearlSystem P → Bool → P.Ω → ℝ","l":107,"f":"Causalean/PO/ID/Partial/BalkePearl/Setup.lean"},{"n":"Causalean.PO.POBalkePearlSystem.YofZD","k":"def","m":"Causalean.PO.ID.Partial.BalkePearl.Setup","a":"PO/ID/Partial/BalkePearl","t":1,"d":"Two-variable counterfactual Y(z,d) := yVar.cf (r_{z,d}).","s":"{P : Causalean.PO.POSystem} → Causalean.PO.POBalkePearlSystem P → Bool → Bool → P.Ω → Bool","l":86,"f":"Causalean/PO/ID/Partial/BalkePearl/Setup.lean"},{"n":"Causalean.PO.POBalkePearlSystem.balkePearl_sharp","k":"theorem","m":"Causalean.PO.ID.Partial.BalkePearl.Sharp","a":"PO/ID/Partial/BalkePearl","t":1,"d":"Sharpness. Under the Balke-Pearl IV base assumptions, for a latent treatment-response table π that is feasible for the linear program — nonnegative, summing to one, and reproducing the observed cell probabilities as its marginals, there exists another potential-outcome system, satisfying the same base assumptions, whose observed cell probabilities agree with the original system's and whose average treatment effect equals the LP objective value of π: every feasible latent table is realised by some potential-outcome model.","s":"∀ {P : Causalean.PO.POSystem} (S : Causalean.PO.POBalkePearlSystem P) (hA : S.BaseAssumptions) (π : Bool → Bool → Bool → Bool → ℝ), S.BPFeasible hA π → ∃ P' S', ∃ (_ : S'.BaseAssumptions), (∀ (y d z : Bool), S'.cellProb y d z = S.cellProb y d z) ∧ S'.ATE = Causalean.PO.POBalkePearlSystem.BPObjective π","l":845,"f":"Causalean/PO/ID/Partial/BalkePearl/Sharp.lean"},{"n":"Causalean.PO.POBalkePearlSystem.balkePearl_sharp_of_mem","k":"theorem","m":"Causalean.PO.ID.Partial.BalkePearl.Sharp","a":"PO/ID/Partial/BalkePearl","t":1,"d":"Corollary of sharpness. Under the Balke-Pearl IV base assumptions, for a real number τ lying in the Balke-Pearl identified interval, there exists another potential-outcome system, satisfying the same base assumptions, whose observed cell probabilities agree with the original system's and whose average treatment effect equals τ: every value in the identified interval is the ATE of some BP-feasible model with matching observed cell probabilities.","s":"∀ {P : Causalean.PO.POSystem} (S : Causalean.PO.POBalkePearlSystem P) (hA : S.BaseAssumptions), ∀ τ ∈ S.BPIdentifiedInterval hA, ∃ P' S', ∃ (_ : S'.BaseAssumptions), (∀ (y d z : Bool), S'.cellProb y d z = S.cellProb y d z) ∧ S'.ATE = τ","l":870,"f":"Causalean/PO/ID/Partial/BalkePearl/Sharp.lean"},{"n":"Causalean.PO.POBalkePearlSystem.bddAbove_BPIdentifiedInterval","k":"theorem","m":"Causalean.PO.ID.Partial.BalkePearl.IntervalForm","a":"PO/ID/Partial/BalkePearl","t":2,"d":"The Balke-Pearl identified interval is bounded above (by 1).","s":"∀ {P : Causalean.PO.POSystem} (S : Causalean.PO.POBalkePearlSystem P) (hA : S.BaseAssumptions), BddAbove (S.BPIdentifiedInterval hA)","l":100,"f":"Causalean/PO/ID/Partial/BalkePearl/IntervalForm.lean"},{"n":"Causalean.PO.POBalkePearlSystem.bddBelow_BPIdentifiedInterval","k":"theorem","m":"Causalean.PO.ID.Partial.BalkePearl.IntervalForm","a":"PO/ID/Partial/BalkePearl","t":2,"d":"The Balke-Pearl identified interval is bounded below (by -1).","s":"∀ {P : Causalean.PO.POSystem} (S : Causalean.PO.POBalkePearlSystem P) (hA : S.BaseAssumptions), BddBelow (S.BPIdentifiedInterval hA)","l":93,"f":"Causalean/PO/ID/Partial/BalkePearl/IntervalForm.lean"},{"n":"Causalean.PO.POBalkePearlSystem.boolToReal","k":"def","m":"Causalean.PO.ID.Partial.BalkePearl.Setup","a":"PO/ID/Partial/BalkePearl","t":1,"d":"Canonical embedding of Bool into ℝ: true ↦ 1, false ↦ 0.","s":"Bool → ℝ","l":102,"f":"Causalean/PO/ID/Partial/BalkePearl/Setup.lean"},{"n":"Causalean.PO.POBalkePearlSystem.bpAux0t","k":"def","m":"Causalean.PO.ID.Partial.BalkePearl.Attainment.Lower","a":"PO/ID/Partial/BalkePearl/Attainment","t":1,"d":"Free-variable choice t in the witness attaining bpLowerTerm 0.","s":"{P : Causalean.PO.POSystem} → Causalean.PO.POBalkePearlSystem P → ℝ","l":45,"f":"Causalean/PO/ID/Partial/BalkePearl/Attainment/Lower.lean"},{"n":"Causalean.PO.POBalkePearlSystem.bpAux1v","k":"def","m":"Causalean.PO.ID.Partial.BalkePearl.Attainment.Lower","a":"PO/ID/Partial/BalkePearl/Attainment","t":1,"d":"Free-variable choice v in the witness attaining bpLowerTerm 1.","s":"{P : Causalean.PO.POSystem} → Causalean.PO.POBalkePearlSystem P → ℝ","l":136,"f":"Causalean/PO/ID/Partial/BalkePearl/Attainment/Lower.lean"},{"n":"Causalean.PO.POBalkePearlSystem.bpAux2u","k":"def","m":"Causalean.PO.ID.Partial.BalkePearl.Attainment.Lower","a":"PO/ID/Partial/BalkePearl/Attainment","t":1,"d":"Free-variable choice u in the witness attaining bpLowerTerm 2.","s":"{P : Causalean.PO.POSystem} → Causalean.PO.POBalkePearlSystem P → ℝ","l":227,"f":"Causalean/PO/ID/Partial/BalkePearl/Attainment/Lower.lean"},{"n":"Causalean.PO.POBalkePearlSystem.bpAux2v","k":"def","m":"Causalean.PO.ID.Partial.BalkePearl.Attainment.Lower","a":"PO/ID/Partial/BalkePearl/Attainment","t":1,"d":"Free-variable choice v in the witness attaining bpLowerTerm 2.","s":"{P : Causalean.PO.POSystem} → Causalean.PO.POBalkePearlSystem P → ℝ","l":232,"f":"Causalean/PO/ID/Partial/BalkePearl/Attainment/Lower.lean"},{"n":"Causalean.PO.POBalkePearlSystem.bpAux3u","k":"def","m":"Causalean.PO.ID.Partial.BalkePearl.Attainment.Lower","a":"PO/ID/Partial/BalkePearl/Attainment","t":1,"d":"Free-variable choice u in the witness attaining bpLowerTerm 3.","s":"{P : Causalean.PO.POSystem} → Causalean.PO.POBalkePearlSystem P → ℝ","l":327,"f":"Causalean/PO/ID/Partial/BalkePearl/Attainment/Lower.lean"},{"n":"Causalean.PO.POBalkePearlSystem.bpAux3v","k":"def","m":"Causalean.PO.ID.Partial.BalkePearl.Attainment.Lower","a":"PO/ID/Partial/BalkePearl/Attainment","t":1,"d":"Free-variable choice v in the witness attaining bpLowerTerm 3.","s":"{P : Causalean.PO.POSystem} → Causalean.PO.POBalkePearlSystem P → ℝ","l":332,"f":"Causalean/PO/ID/Partial/BalkePearl/Attainment/Lower.lean"},{"n":"Causalean.PO.POBalkePearlSystem.bpLower","k":"def","m":"Causalean.PO.ID.Partial.BalkePearl.ClosedForm","a":"PO/ID/Partial/BalkePearl","t":1,"d":"The Balke-Pearl closed-form lower endpoint: the largest of the eight lower expressions. This is the smallest average treatment effect compatible with the observed distribution under a valid binary instrument.","s":"{P : Causalean.PO.POSystem} → Causalean.PO.POBalkePearlSystem P → ℝ","l":102,"f":"Causalean/PO/ID/Partial/BalkePearl/ClosedForm.lean"},{"n":"Causalean.PO.POBalkePearlSystem.bpLowerTerm","k":"def","m":"Causalean.PO.ID.Partial.BalkePearl.ClosedForm","a":"PO/ID/Partial/BalkePearl","t":1,"d":"The i-th of the eight affine lower expressions in the Balke-Pearl closed form for the average treatment effect. Each is a valid lower bound on the effect, and their maximum is exactly the smallest effect compatible with the data.","s":"{P : Causalean.PO.POSystem} → Causalean.PO.POBalkePearlSystem P → Fin 8 → ℝ","l":64,"f":"Causalean/PO/ID/Partial/BalkePearl/ClosedForm.lean"},{"n":"Causalean.PO.POBalkePearlSystem.bpLowerTerm_le_BPObjective","k":"theorem","m":"Causalean.PO.ID.Partial.BalkePearl.ClosedForm","a":"PO/ID/Partial/BalkePearl","t":2,"d":"Every lower expression bounds the objective from below on feasible tables.","s":"∀ {P : Causalean.PO.POSystem} (S : Causalean.PO.POBalkePearlSystem P) (hA : S.BaseAssumptions) {π : Bool → Bool → Bool → Bool → ℝ}, S.BPFeasible hA π → ∀ (i : Fin 8), S.bpLowerTerm i ≤ Causalean.PO.POBalkePearlSystem.BPObjective π","l":120,"f":"Causalean/PO/ID/Partial/BalkePearl/ClosedForm.lean"},{"n":"Causalean.PO.POBalkePearlSystem.bpLowerWitness0","k":"def","m":"Causalean.PO.ID.Partial.BalkePearl.Attainment.Lower","a":"PO/ID/Partial/BalkePearl/Attainment","t":1,"d":"Latent table attaining bpLowerTerm 0.","s":"{P : Causalean.PO.POSystem} → Causalean.PO.POBalkePearlSystem P → Bool → Bool → Bool → Bool → ℝ","l":50,"f":"Causalean/PO/ID/Partial/BalkePearl/Attainment/Lower.lean"},{"n":"Causalean.PO.POBalkePearlSystem.bpLowerWitness0_feasible","k":"theorem","m":"Causalean.PO.ID.Partial.BalkePearl.Attainment.Lower","a":"PO/ID/Partial/BalkePearl/Attainment","t":2,"d":"The witness for bpLowerTerm 0 is feasible on its optimality region.","s":"∀ {P : Causalean.PO.POSystem} (S : Causalean.PO.POBalkePearlSystem P) (hA : S.BaseAssumptions) {π₀ : Bool → Bool → Bool → Bool → ℝ}, S.BPFeasible hA π₀ → (∀ (j : Fin 8), S.bpLowerTerm j ≤ S.bpLowerTerm 0) → S.BPFeasible hA S.bpLowerWitness0","l":69,"f":"Causalean/PO/ID/Partial/BalkePearl/Attainment/Lower.lean"},{"n":"Causalean.PO.POBalkePearlSystem.bpLowerWitness0_objective","k":"theorem","m":"Causalean.PO.ID.Partial.BalkePearl.Attainment.Lower","a":"PO/ID/Partial/BalkePearl/Attainment","t":2,"d":"The witness for bpLowerTerm 0 attains it.","s":"∀ {P : Causalean.PO.POSystem} (S : Causalean.PO.POBalkePearlSystem P), S.BaseAssumptions → Causalean.PO.POBalkePearlSystem.BPObjective S.bpLowerWitness0 = S.bpLowerTerm 0","l":127,"f":"Causalean/PO/ID/Partial/BalkePearl/Attainment/Lower.lean"},{"n":"Causalean.PO.POBalkePearlSystem.bpLowerWitness1","k":"def","m":"Causalean.PO.ID.Partial.BalkePearl.Attainment.Lower","a":"PO/ID/Partial/BalkePearl/Attainment","t":1,"d":"Latent table attaining bpLowerTerm 1.","s":"{P : Causalean.PO.POSystem} → Causalean.PO.POBalkePearlSystem P → Bool → Bool → Bool → Bool → ℝ","l":141,"f":"Causalean/PO/ID/Partial/BalkePearl/Attainment/Lower.lean"},{"n":"Causalean.PO.POBalkePearlSystem.bpLowerWitness1_feasible","k":"theorem","m":"Causalean.PO.ID.Partial.BalkePearl.Attainment.Lower","a":"PO/ID/Partial/BalkePearl/Attainment","t":2,"d":"The witness for bpLowerTerm 1 is feasible on its optimality region.","s":"∀ {P : Causalean.PO.POSystem} (S : Causalean.PO.POBalkePearlSystem P) (hA : S.BaseAssumptions) {π₀ : Bool → Bool → Bool → Bool → ℝ}, S.BPFeasible hA π₀ → (∀ (j : Fin 8), S.bpLowerTerm j ≤ S.bpLowerTerm 1) → S.BPFeasible hA S.bpLowerWitness1","l":160,"f":"Causalean/PO/ID/Partial/BalkePearl/Attainment/Lower.lean"},{"n":"Causalean.PO.POBalkePearlSystem.bpLowerWitness1_objective","k":"theorem","m":"Causalean.PO.ID.Partial.BalkePearl.Attainment.Lower","a":"PO/ID/Partial/BalkePearl/Attainment","t":2,"d":"The witness for bpLowerTerm 1 attains it.","s":"∀ {P : Causalean.PO.POSystem} (S : Causalean.PO.POBalkePearlSystem P), S.BaseAssumptions → Causalean.PO.POBalkePearlSystem.BPObjective S.bpLowerWitness1 = S.bpLowerTerm 1","l":218,"f":"Causalean/PO/ID/Partial/BalkePearl/Attainment/Lower.lean"},{"n":"Causalean.PO.POBalkePearlSystem.bpLowerWitness2","k":"def","m":"Causalean.PO.ID.Partial.BalkePearl.Attainment.Lower","a":"PO/ID/Partial/BalkePearl/Attainment","t":1,"d":"Latent table attaining bpLowerTerm 2.","s":"{P : Causalean.PO.POSystem} → Causalean.PO.POBalkePearlSystem P → Bool → Bool → Bool → Bool → ℝ","l":236,"f":"Causalean/PO/ID/Partial/BalkePearl/Attainment/Lower.lean"},{"n":"Causalean.PO.POBalkePearlSystem.bpLowerWitness2_feasible","k":"theorem","m":"Causalean.PO.ID.Partial.BalkePearl.Attainment.Lower","a":"PO/ID/Partial/BalkePearl/Attainment","t":2,"d":"The witness for bpLowerTerm 2 is feasible on its optimality region.","s":"∀ {P : Causalean.PO.POSystem} (S : Causalean.PO.POBalkePearlSystem P) (hA : S.BaseAssumptions) {π₀ : Bool → Bool → Bool → Bool → ℝ}, S.BPFeasible hA π₀ → (∀ (j : Fin 8), S.bpLowerTerm j ≤ S.bpLowerTerm 2) → S.BPFeasible hA S.bpLowerWitness2","l":257,"f":"Causalean/PO/ID/Partial/BalkePearl/Attainment/Lower.lean"},{"n":"Causalean.PO.POBalkePearlSystem.bpLowerWitness2_objective","k":"theorem","m":"Causalean.PO.ID.Partial.BalkePearl.Attainment.Lower","a":"PO/ID/Partial/BalkePearl/Attainment","t":2,"d":"The witness for bpLowerTerm 2 attains it.","s":"∀ {P : Causalean.PO.POSystem} (S : Causalean.PO.POBalkePearlSystem P), S.BaseAssumptions → Causalean.PO.POBalkePearlSystem.BPObjective S.bpLowerWitness2 = S.bpLowerTerm 2","l":317,"f":"Causalean/PO/ID/Partial/BalkePearl/Attainment/Lower.lean"},{"n":"Causalean.PO.POBalkePearlSystem.bpLowerWitness3","k":"def","m":"Causalean.PO.ID.Partial.BalkePearl.Attainment.Lower","a":"PO/ID/Partial/BalkePearl/Attainment","t":1,"d":"Latent table attaining bpLowerTerm 3.","s":"{P : Causalean.PO.POSystem} → Causalean.PO.POBalkePearlSystem P → Bool → Bool → Bool → Bool → ℝ","l":336,"f":"Causalean/PO/ID/Partial/BalkePearl/Attainment/Lower.lean"},{"n":"Causalean.PO.POBalkePearlSystem.bpLowerWitness3_feasible","k":"theorem","m":"Causalean.PO.ID.Partial.BalkePearl.Attainment.Lower","a":"PO/ID/Partial/BalkePearl/Attainment","t":2,"d":"The witness for bpLowerTerm 3 is feasible on its optimality region.","s":"∀ {P : Causalean.PO.POSystem} (S : Causalean.PO.POBalkePearlSystem P) (hA : S.BaseAssumptions) {π₀ : Bool → Bool → Bool → Bool → ℝ}, S.BPFeasible hA π₀ → (∀ (j : Fin 8), S.bpLowerTerm j ≤ S.bpLowerTerm 3) → S.BPFeasible hA S.bpLowerWitness3","l":357,"f":"Causalean/PO/ID/Partial/BalkePearl/Attainment/Lower.lean"},{"n":"Causalean.PO.POBalkePearlSystem.bpLowerWitness3_objective","k":"theorem","m":"Causalean.PO.ID.Partial.BalkePearl.Attainment.Lower","a":"PO/ID/Partial/BalkePearl/Attainment","t":2,"d":"The witness for bpLowerTerm 3 attains it.","s":"∀ {P : Causalean.PO.POSystem} (S : Causalean.PO.POBalkePearlSystem P), S.BaseAssumptions → Causalean.PO.POBalkePearlSystem.BPObjective S.bpLowerWitness3 = S.bpLowerTerm 3","l":417,"f":"Causalean/PO/ID/Partial/BalkePearl/Attainment/Lower.lean"},{"n":"Causalean.PO.POBalkePearlSystem.bpLowerWitness4","k":"def","m":"Causalean.PO.ID.Partial.BalkePearl.Attainment.Lower","a":"PO/ID/Partial/BalkePearl/Attainment","t":1,"d":"Latent table attaining bpLowerTerm 4.","s":"{P : Causalean.PO.POSystem} → Causalean.PO.POBalkePearlSystem P → Bool → Bool → Bool → Bool → ℝ","l":427,"f":"Causalean/PO/ID/Partial/BalkePearl/Attainment/Lower.lean"},{"n":"Causalean.PO.POBalkePearlSystem.bpLowerWitness4_feasible","k":"theorem","m":"Causalean.PO.ID.Partial.BalkePearl.Attainment.Lower","a":"PO/ID/Partial/BalkePearl/Attainment","t":2,"d":"The witness for bpLowerTerm 4 is feasible on its optimality region.","s":"∀ {P : Causalean.PO.POSystem} (S : Causalean.PO.POBalkePearlSystem P) (hA : S.BaseAssumptions) {π₀ : Bool → Bool → Bool → Bool → ℝ}, S.BPFeasible hA π₀ → (∀ (j : Fin 8), S.bpLowerTerm j ≤ S.bpLowerTerm 4) → S.BPFeasible hA S.bpLowerWitness4","l":444,"f":"Causalean/PO/ID/Partial/BalkePearl/Attainment/Lower.lean"},{"n":"Causalean.PO.POBalkePearlSystem.bpLowerWitness4_objective","k":"theorem","m":"Causalean.PO.ID.Partial.BalkePearl.Attainment.Lower","a":"PO/ID/Partial/BalkePearl/Attainment","t":2,"d":"The witness for bpLowerTerm 4 attains it.","s":"∀ {P : Causalean.PO.POSystem} (S : Causalean.PO.POBalkePearlSystem P), S.BaseAssumptions → Causalean.PO.POBalkePearlSystem.BPObjective S.bpLowerWitness4 = S.bpLowerTerm 4","l":498,"f":"Causalean/PO/ID/Partial/BalkePearl/Attainment/Lower.lean"},{"n":"Causalean.PO.POBalkePearlSystem.bpLowerWitness5","k":"def","m":"Causalean.PO.ID.Partial.BalkePearl.Attainment.Lower","a":"PO/ID/Partial/BalkePearl/Attainment","t":1,"d":"Latent table attaining bpLowerTerm 5.","s":"{P : Causalean.PO.POSystem} → Causalean.PO.POBalkePearlSystem P → Bool → Bool → Bool → Bool → ℝ","l":507,"f":"Causalean/PO/ID/Partial/BalkePearl/Attainment/Lower.lean"},{"n":"Causalean.PO.POBalkePearlSystem.bpLowerWitness5_feasible","k":"theorem","m":"Causalean.PO.ID.Partial.BalkePearl.Attainment.Lower","a":"PO/ID/Partial/BalkePearl/Attainment","t":2,"d":"The witness for bpLowerTerm 5 is feasible on its optimality region.","s":"∀ {P : Causalean.PO.POSystem} (S : Causalean.PO.POBalkePearlSystem P) (hA : S.BaseAssumptions) {π₀ : Bool → Bool → Bool → Bool → ℝ}, S.BPFeasible hA π₀ → (∀ (j : Fin 8), S.bpLowerTerm j ≤ S.bpLowerTerm 5) → S.BPFeasible hA S.bpLowerWitness5","l":522,"f":"Causalean/PO/ID/Partial/BalkePearl/Attainment/Lower.lean"},{"n":"Causalean.PO.POBalkePearlSystem.bpLowerWitness5_objective","k":"theorem","m":"Causalean.PO.ID.Partial.BalkePearl.Attainment.Lower","a":"PO/ID/Partial/BalkePearl/Attainment","t":2,"d":"The witness for bpLowerTerm 5 attains it.","s":"∀ {P : Causalean.PO.POSystem} (S : Causalean.PO.POBalkePearlSystem P), S.BaseAssumptions → Causalean.PO.POBalkePearlSystem.BPObjective S.bpLowerWitness5 = S.bpLowerTerm 5","l":576,"f":"Causalean/PO/ID/Partial/BalkePearl/Attainment/Lower.lean"},{"n":"Causalean.PO.POBalkePearlSystem.bpLowerWitness6","k":"def","m":"Causalean.PO.ID.Partial.BalkePearl.Attainment.Lower","a":"PO/ID/Partial/BalkePearl/Attainment","t":1,"d":"Latent table attaining bpLowerTerm 6.","s":"{P : Causalean.PO.POSystem} → Causalean.PO.POBalkePearlSystem P → Bool → Bool → Bool → Bool → ℝ","l":585,"f":"Causalean/PO/ID/Partial/BalkePearl/Attainment/Lower.lean"},{"n":"Causalean.PO.POBalkePearlSystem.bpLowerWitness6_feasible","k":"theorem","m":"Causalean.PO.ID.Partial.BalkePearl.Attainment.Lower","a":"PO/ID/Partial/BalkePearl/Attainment","t":2,"d":"The witness for bpLowerTerm 6 is feasible on its optimality region.","s":"∀ {P : Causalean.PO.POSystem} (S : Causalean.PO.POBalkePearlSystem P) (hA : S.BaseAssumptions) {π₀ : Bool → Bool → Bool → Bool → ℝ}, S.BPFeasible hA π₀ → (∀ (j : Fin 8), S.bpLowerTerm j ≤ S.bpLowerTerm 6) → S.BPFeasible hA S.bpLowerWitness6","l":600,"f":"Causalean/PO/ID/Partial/BalkePearl/Attainment/Lower.lean"},{"n":"Causalean.PO.POBalkePearlSystem.bpLowerWitness6_objective","k":"theorem","m":"Causalean.PO.ID.Partial.BalkePearl.Attainment.Lower","a":"PO/ID/Partial/BalkePearl/Attainment","t":2,"d":"The witness for bpLowerTerm 6 attains it.","s":"∀ {P : Causalean.PO.POSystem} (S : Causalean.PO.POBalkePearlSystem P), S.BaseAssumptions → Causalean.PO.POBalkePearlSystem.BPObjective S.bpLowerWitness6 = S.bpLowerTerm 6","l":654,"f":"Causalean/PO/ID/Partial/BalkePearl/Attainment/Lower.lean"},{"n":"Causalean.PO.POBalkePearlSystem.bpLowerWitness7","k":"def","m":"Causalean.PO.ID.Partial.BalkePearl.Attainment.Lower","a":"PO/ID/Partial/BalkePearl/Attainment","t":1,"d":"Latent table attaining bpLowerTerm 7.","s":"{P : Causalean.PO.POSystem} → Causalean.PO.POBalkePearlSystem P → Bool → Bool → Bool → Bool → ℝ","l":663,"f":"Causalean/PO/ID/Partial/BalkePearl/Attainment/Lower.lean"},{"n":"Causalean.PO.POBalkePearlSystem.bpLowerWitness7_feasible","k":"theorem","m":"Causalean.PO.ID.Partial.BalkePearl.Attainment.Lower","a":"PO/ID/Partial/BalkePearl/Attainment","t":2,"d":"The witness for bpLowerTerm 7 is feasible on its optimality region.","s":"∀ {P : Causalean.PO.POSystem} (S : Causalean.PO.POBalkePearlSystem P) (hA : S.BaseAssumptions) {π₀ : Bool → Bool → Bool → Bool → ℝ}, S.BPFeasible hA π₀ → (∀ (j : Fin 8), S.bpLowerTerm j ≤ S.bpLowerTerm 7) → S.BPFeasible hA S.bpLowerWitness7","l":680,"f":"Causalean/PO/ID/Partial/BalkePearl/Attainment/Lower.lean"},{"n":"Causalean.PO.POBalkePearlSystem.bpLowerWitness7_objective","k":"theorem","m":"Causalean.PO.ID.Partial.BalkePearl.Attainment.Lower","a":"PO/ID/Partial/BalkePearl/Attainment","t":2,"d":"The witness for bpLowerTerm 7 attains it.","s":"∀ {P : Causalean.PO.POSystem} (S : Causalean.PO.POBalkePearlSystem P), S.BaseAssumptions → Causalean.PO.POBalkePearlSystem.BPObjective S.bpLowerWitness7 = S.bpLowerTerm 7","l":734,"f":"Causalean/PO/ID/Partial/BalkePearl/Attainment/Lower.lean"},{"n":"Causalean.PO.POBalkePearlSystem.bpLower_bpUpper_eq_csInf_csSup","k":"theorem","m":"Causalean.PO.ID.Partial.BalkePearl.ClosedFormAttainment","a":"PO/ID/Partial/BalkePearl","t":1,"d":"The Balke-Pearl closed form is sharp. Under the Balke-Pearl IV base assumptions, the interval [bpLower, bpUpper], computed from the observed cell probabilities alone, has both endpoints attained by observationally-equivalent latent tables — equivalently, bpLower is the infimum and bpUpper is the supremum of the Balke-Pearl identified interval. So no smaller interval contains every average treatment effect compatible with the data: the bound cannot be improved without further assumptions.","s":"∀ {P : Causalean.PO.POSystem} (S : Causalean.PO.POBalkePearlSystem P) (hA : S.BaseAssumptions), S.bpLower = sInf (S.BPIdentifiedInterval hA) ∧ S.bpUpper = sSup (S.BPIdentifiedInterval hA)","l":69,"f":"Causalean/PO/ID/Partial/BalkePearl/ClosedFormAttainment.lean"},{"n":"Causalean.PO.POBalkePearlSystem.bpLower_le_BPObjective","k":"theorem","m":"Causalean.PO.ID.Partial.BalkePearl.ClosedForm","a":"PO/ID/Partial/BalkePearl","t":2,"d":"The closed-form lower endpoint bounds the objective on feasible tables.","s":"∀ {P : Causalean.PO.POSystem} (S : Causalean.PO.POBalkePearlSystem P) (hA : S.BaseAssumptions) {π : Bool → Bool → Bool → Bool → ℝ}, S.BPFeasible hA π → S.bpLower ≤ Causalean.PO.POBalkePearlSystem.BPObjective π","l":180,"f":"Causalean/PO/ID/Partial/BalkePearl/ClosedForm.lean"},{"n":"Causalean.PO.POBalkePearlSystem.bpLower_mem_BPIdentifiedInterval","k":"theorem","m":"Causalean.PO.ID.Partial.BalkePearl.Attainment.Lower","a":"PO/ID/Partial/BalkePearl/Attainment","t":1,"d":"Under the Balke-Pearl IV base assumptions, the closed-form Balke-Pearl lower bound, computed from the observed cell probabilities, is itself attained as the average treatment effect of some feasible latent treatment-response table — it lies in the Balke-Pearl identified interval.","s":"∀ {P : Causalean.PO.POSystem} (S : Causalean.PO.POBalkePearlSystem P) (hA : S.BaseAssumptions), S.bpLower ∈ S.BPIdentifiedInterval hA","l":745,"f":"Causalean/PO/ID/Partial/BalkePearl/Attainment/Lower.lean"},{"n":"Causalean.PO.POBalkePearlSystem.bpUAux0u","k":"def","m":"Causalean.PO.ID.Partial.BalkePearl.Attainment.Upper","a":"PO/ID/Partial/BalkePearl/Attainment","t":1,"d":"Free-variable choice in the witness attaining bpUpperTerm 0.","s":"{P : Causalean.PO.POSystem} → Causalean.PO.POBalkePearlSystem P → ℝ","l":48,"f":"Causalean/PO/ID/Partial/BalkePearl/Attainment/Upper.lean"},{"n":"Causalean.PO.POBalkePearlSystem.bpUAux1u","k":"def","m":"Causalean.PO.ID.Partial.BalkePearl.Attainment.Upper","a":"PO/ID/Partial/BalkePearl/Attainment","t":1,"d":"Free-variable choice in the witness attaining bpUpperTerm 1.","s":"{P : Causalean.PO.POSystem} → Causalean.PO.POBalkePearlSystem P → ℝ","l":140,"f":"Causalean/PO/ID/Partial/BalkePearl/Attainment/Upper.lean"},{"n":"Causalean.PO.POBalkePearlSystem.bpUAux2u","k":"def","m":"Causalean.PO.ID.Partial.BalkePearl.Attainment.Upper","a":"PO/ID/Partial/BalkePearl/Attainment","t":1,"d":"Free-variable choice in the witness attaining bpUpperTerm 2.","s":"{P : Causalean.PO.POSystem} → Causalean.PO.POBalkePearlSystem P → ℝ","l":232,"f":"Causalean/PO/ID/Partial/BalkePearl/Attainment/Upper.lean"},{"n":"Causalean.PO.POBalkePearlSystem.bpUAux2v","k":"def","m":"Causalean.PO.ID.Partial.BalkePearl.Attainment.Upper","a":"PO/ID/Partial/BalkePearl/Attainment","t":1,"d":"Free-variable choice in the witness attaining bpUpperTerm 2.","s":"{P : Causalean.PO.POSystem} → Causalean.PO.POBalkePearlSystem P → ℝ","l":237,"f":"Causalean/PO/ID/Partial/BalkePearl/Attainment/Upper.lean"},{"n":"Causalean.PO.POBalkePearlSystem.bpUAux3u","k":"def","m":"Causalean.PO.ID.Partial.BalkePearl.Attainment.Upper","a":"PO/ID/Partial/BalkePearl/Attainment","t":1,"d":"Free-variable choice in the witness attaining bpUpperTerm 3.","s":"{P : Causalean.PO.POSystem} → Causalean.PO.POBalkePearlSystem P → ℝ","l":337,"f":"Causalean/PO/ID/Partial/BalkePearl/Attainment/Upper.lean"},{"n":"Causalean.PO.POBalkePearlSystem.bpUAux3v","k":"def","m":"Causalean.PO.ID.Partial.BalkePearl.Attainment.Upper","a":"PO/ID/Partial/BalkePearl/Attainment","t":1,"d":"Free-variable choice in the witness attaining bpUpperTerm 3.","s":"{P : Causalean.PO.POSystem} → Causalean.PO.POBalkePearlSystem P → ℝ","l":342,"f":"Causalean/PO/ID/Partial/BalkePearl/Attainment/Upper.lean"},{"n":"Causalean.PO.POBalkePearlSystem.bpUpper","k":"def","m":"Causalean.PO.ID.Partial.BalkePearl.ClosedForm","a":"PO/ID/Partial/BalkePearl","t":1,"d":"The Balke-Pearl closed-form upper endpoint: the smallest of the eight upper expressions. This is the largest average treatment effect compatible with the observed distribution under a valid binary instrument.","s":"{P : Causalean.PO.POSystem} → Causalean.PO.POBalkePearlSystem P → ℝ","l":108,"f":"Causalean/PO/ID/Partial/BalkePearl/ClosedForm.lean"},{"n":"Causalean.PO.POBalkePearlSystem.bpUpperTerm","k":"def","m":"Causalean.PO.ID.Partial.BalkePearl.ClosedForm","a":"PO/ID/Partial/BalkePearl","t":1,"d":"The i-th of the eight affine upper expressions in the Balke-Pearl closed form for the average treatment effect. Each is a valid upper bound on the effect, and their minimum is exactly the largest effect compatible with the data.","s":"{P : Causalean.PO.POSystem} → Causalean.PO.POBalkePearlSystem P → Fin 8 → ℝ","l":83,"f":"Causalean/PO/ID/Partial/BalkePearl/ClosedForm.lean"},{"n":"Causalean.PO.POBalkePearlSystem.bpUpperWitness0","k":"def","m":"Causalean.PO.ID.Partial.BalkePearl.Attainment.Upper","a":"PO/ID/Partial/BalkePearl/Attainment","t":1,"d":"Latent table attaining bpUpperTerm 0.","s":"{P : Causalean.PO.POSystem} → Causalean.PO.POBalkePearlSystem P → Bool → Bool → Bool → Bool → ℝ","l":53,"f":"Causalean/PO/ID/Partial/BalkePearl/Attainment/Upper.lean"},{"n":"Causalean.PO.POBalkePearlSystem.bpUpperWitness0_feasible","k":"theorem","m":"Causalean.PO.ID.Partial.BalkePearl.Attainment.Upper","a":"PO/ID/Partial/BalkePearl/Attainment","t":2,"d":"The witness for bpUpperTerm 0 is feasible on its optimality region.","s":"∀ {P : Causalean.PO.POSystem} (S : Causalean.PO.POBalkePearlSystem P) (hA : S.BaseAssumptions) {π₀ : Bool → Bool → Bool → Bool → ℝ}, S.BPFeasible hA π₀ → (∀ (j : Fin 8), S.bpUpperTerm 0 ≤ S.bpUpperTerm j) → S.BPFeasible hA S.bpUpperWitness0","l":72,"f":"Causalean/PO/ID/Partial/BalkePearl/Attainment/Upper.lean"},{"n":"Causalean.PO.POBalkePearlSystem.bpUpperWitness0_objective","k":"theorem","m":"Causalean.PO.ID.Partial.BalkePearl.Attainment.Upper","a":"PO/ID/Partial/BalkePearl/Attainment","t":2,"d":"The witness for bpUpperTerm 0 attains it.","s":"∀ {P : Causalean.PO.POSystem} (S : Causalean.PO.POBalkePearlSystem P), S.BaseAssumptions → Causalean.PO.POBalkePearlSystem.BPObjective S.bpUpperWitness0 = S.bpUpperTerm 0","l":130,"f":"Causalean/PO/ID/Partial/BalkePearl/Attainment/Upper.lean"},{"n":"Causalean.PO.POBalkePearlSystem.bpUpperWitness1","k":"def","m":"Causalean.PO.ID.Partial.BalkePearl.Attainment.Upper","a":"PO/ID/Partial/BalkePearl/Attainment","t":1,"d":"Latent table attaining bpUpperTerm 1.","s":"{P : Causalean.PO.POSystem} → Causalean.PO.POBalkePearlSystem P → Bool → Bool → Bool → Bool → ℝ","l":145,"f":"Causalean/PO/ID/Partial/BalkePearl/Attainment/Upper.lean"},{"n":"Causalean.PO.POBalkePearlSystem.bpUpperWitness1_feasible","k":"theorem","m":"Causalean.PO.ID.Partial.BalkePearl.Attainment.Upper","a":"PO/ID/Partial/BalkePearl/Attainment","t":2,"d":"The witness for bpUpperTerm 1 is feasible on its optimality region.","s":"∀ {P : Causalean.PO.POSystem} (S : Causalean.PO.POBalkePearlSystem P) (hA : S.BaseAssumptions) {π₀ : Bool → Bool → Bool → Bool → ℝ}, S.BPFeasible hA π₀ → (∀ (j : Fin 8), S.bpUpperTerm 1 ≤ S.bpUpperTerm j) → S.BPFeasible hA S.bpUpperWitness1","l":164,"f":"Causalean/PO/ID/Partial/BalkePearl/Attainment/Upper.lean"},{"n":"Causalean.PO.POBalkePearlSystem.bpUpperWitness1_objective","k":"theorem","m":"Causalean.PO.ID.Partial.BalkePearl.Attainment.Upper","a":"PO/ID/Partial/BalkePearl/Attainment","t":2,"d":"The witness for bpUpperTerm 1 attains it.","s":"∀ {P : Causalean.PO.POSystem} (S : Causalean.PO.POBalkePearlSystem P), S.BaseAssumptions → Causalean.PO.POBalkePearlSystem.BPObjective S.bpUpperWitness1 = S.bpUpperTerm 1","l":222,"f":"Causalean/PO/ID/Partial/BalkePearl/Attainment/Upper.lean"},{"n":"Causalean.PO.POBalkePearlSystem.bpUpperWitness2","k":"def","m":"Causalean.PO.ID.Partial.BalkePearl.Attainment.Upper","a":"PO/ID/Partial/BalkePearl/Attainment","t":1,"d":"Latent table attaining bpUpperTerm 2.","s":"{P : Causalean.PO.POSystem} → Causalean.PO.POBalkePearlSystem P → Bool → Bool → Bool → Bool → ℝ","l":242,"f":"Causalean/PO/ID/Partial/BalkePearl/Attainment/Upper.lean"},{"n":"Causalean.PO.POBalkePearlSystem.bpUpperWitness2_feasible","k":"theorem","m":"Causalean.PO.ID.Partial.BalkePearl.Attainment.Upper","a":"PO/ID/Partial/BalkePearl/Attainment","t":2,"d":"The witness for bpUpperTerm 2 is feasible on its optimality region.","s":"∀ {P : Causalean.PO.POSystem} (S : Causalean.PO.POBalkePearlSystem P) (hA : S.BaseAssumptions) {π₀ : Bool → Bool → Bool → Bool → ℝ}, S.BPFeasible hA π₀ → (∀ (j : Fin 8), S.bpUpperTerm 2 ≤ S.bpUpperTerm j) → S.BPFeasible hA S.bpUpperWitness2","l":265,"f":"Causalean/PO/ID/Partial/BalkePearl/Attainment/Upper.lean"},{"n":"Causalean.PO.POBalkePearlSystem.bpUpperWitness2_objective","k":"theorem","m":"Causalean.PO.ID.Partial.BalkePearl.Attainment.Upper","a":"PO/ID/Partial/BalkePearl/Attainment","t":2,"d":"The witness for bpUpperTerm 2 attains it.","s":"∀ {P : Causalean.PO.POSystem} (S : Causalean.PO.POBalkePearlSystem P), S.BaseAssumptions → Causalean.PO.POBalkePearlSystem.BPObjective S.bpUpperWitness2 = S.bpUpperTerm 2","l":327,"f":"Causalean/PO/ID/Partial/BalkePearl/Attainment/Upper.lean"},{"n":"Causalean.PO.POBalkePearlSystem.bpUpperWitness3","k":"def","m":"Causalean.PO.ID.Partial.BalkePearl.Attainment.Upper","a":"PO/ID/Partial/BalkePearl/Attainment","t":1,"d":"Latent table attaining bpUpperTerm 3.","s":"{P : Causalean.PO.POSystem} → Causalean.PO.POBalkePearlSystem P → Bool → Bool → Bool → Bool → ℝ","l":347,"f":"Causalean/PO/ID/Partial/BalkePearl/Attainment/Upper.lean"},{"n":"Causalean.PO.POBalkePearlSystem.bpUpperWitness3_feasible","k":"theorem","m":"Causalean.PO.ID.Partial.BalkePearl.Attainment.Upper","a":"PO/ID/Partial/BalkePearl/Attainment","t":2,"d":"The witness for bpUpperTerm 3 is feasible on its optimality region.","s":"∀ {P : Causalean.PO.POSystem} (S : Causalean.PO.POBalkePearlSystem P) (hA : S.BaseAssumptions) {π₀ : Bool → Bool → Bool → Bool → ℝ}, S.BPFeasible hA π₀ → (∀ (j : Fin 8), S.bpUpperTerm 3 ≤ S.bpUpperTerm j) → S.BPFeasible hA S.bpUpperWitness3","l":370,"f":"Causalean/PO/ID/Partial/BalkePearl/Attainment/Upper.lean"},{"n":"Causalean.PO.POBalkePearlSystem.bpUpperWitness3_objective","k":"theorem","m":"Causalean.PO.ID.Partial.BalkePearl.Attainment.Upper","a":"PO/ID/Partial/BalkePearl/Attainment","t":2,"d":"The witness for bpUpperTerm 3 attains it.","s":"∀ {P : Causalean.PO.POSystem} (S : Causalean.PO.POBalkePearlSystem P), S.BaseAssumptions → Causalean.PO.POBalkePearlSystem.BPObjective S.bpUpperWitness3 = S.bpUpperTerm 3","l":432,"f":"Causalean/PO/ID/Partial/BalkePearl/Attainment/Upper.lean"},{"n":"Causalean.PO.POBalkePearlSystem.bpUpperWitness4","k":"def","m":"Causalean.PO.ID.Partial.BalkePearl.Attainment.Upper","a":"PO/ID/Partial/BalkePearl/Attainment","t":1,"d":"Latent table attaining bpUpperTerm 4.","s":"{P : Causalean.PO.POSystem} → Causalean.PO.POBalkePearlSystem P → Bool → Bool → Bool → Bool → ℝ","l":442,"f":"Causalean/PO/ID/Partial/BalkePearl/Attainment/Upper.lean"},{"n":"Causalean.PO.POBalkePearlSystem.bpUpperWitness4_feasible","k":"theorem","m":"Causalean.PO.ID.Partial.BalkePearl.Attainment.Upper","a":"PO/ID/Partial/BalkePearl/Attainment","t":2,"d":"The witness for bpUpperTerm 4 is feasible on its optimality region.","s":"∀ {P : Causalean.PO.POSystem} (S : Causalean.PO.POBalkePearlSystem P) (hA : S.BaseAssumptions) {π₀ : Bool → Bool → Bool → Bool → ℝ}, S.BPFeasible hA π₀ → (∀ (j : Fin 8), S.bpUpperTerm 4 ≤ S.bpUpperTerm j) → S.BPFeasible hA S.bpUpperWitness4","l":461,"f":"Causalean/PO/ID/Partial/BalkePearl/Attainment/Upper.lean"},{"n":"Causalean.PO.POBalkePearlSystem.bpUpperWitness4_objective","k":"theorem","m":"Causalean.PO.ID.Partial.BalkePearl.Attainment.Upper","a":"PO/ID/Partial/BalkePearl/Attainment","t":2,"d":"The witness for bpUpperTerm 4 attains it.","s":"∀ {P : Causalean.PO.POSystem} (S : Causalean.PO.POBalkePearlSystem P), S.BaseAssumptions → Causalean.PO.POBalkePearlSystem.BPObjective S.bpUpperWitness4 = S.bpUpperTerm 4","l":515,"f":"Causalean/PO/ID/Partial/BalkePearl/Attainment/Upper.lean"},{"n":"Causalean.PO.POBalkePearlSystem.bpUpperWitness5","k":"def","m":"Causalean.PO.ID.Partial.BalkePearl.Attainment.Upper","a":"PO/ID/Partial/BalkePearl/Attainment","t":1,"d":"Latent table attaining bpUpperTerm 5.","s":"{P : Causalean.PO.POSystem} → Causalean.PO.POBalkePearlSystem P → Bool → Bool → Bool → Bool → ℝ","l":524,"f":"Causalean/PO/ID/Partial/BalkePearl/Attainment/Upper.lean"},{"n":"Causalean.PO.POBalkePearlSystem.bpUpperWitness5_feasible","k":"theorem","m":"Causalean.PO.ID.Partial.BalkePearl.Attainment.Upper","a":"PO/ID/Partial/BalkePearl/Attainment","t":2,"d":"The witness for bpUpperTerm 5 is feasible on its optimality region.","s":"∀ {P : Causalean.PO.POSystem} (S : Causalean.PO.POBalkePearlSystem P) (hA : S.BaseAssumptions) {π₀ : Bool → Bool → Bool → Bool → ℝ}, S.BPFeasible hA π₀ → (∀ (j : Fin 8), S.bpUpperTerm 5 ≤ S.bpUpperTerm j) → S.BPFeasible hA S.bpUpperWitness5","l":542,"f":"Causalean/PO/ID/Partial/BalkePearl/Attainment/Upper.lean"},{"n":"Causalean.PO.POBalkePearlSystem.bpUpperWitness5_objective","k":"theorem","m":"Causalean.PO.ID.Partial.BalkePearl.Attainment.Upper","a":"PO/ID/Partial/BalkePearl/Attainment","t":2,"d":"The witness for bpUpperTerm 5 attains it.","s":"∀ {P : Causalean.PO.POSystem} (S : Causalean.PO.POBalkePearlSystem P), S.BaseAssumptions → Causalean.PO.POBalkePearlSystem.BPObjective S.bpUpperWitness5 = S.bpUpperTerm 5","l":596,"f":"Causalean/PO/ID/Partial/BalkePearl/Attainment/Upper.lean"},{"n":"Causalean.PO.POBalkePearlSystem.bpUpperWitness6","k":"def","m":"Causalean.PO.ID.Partial.BalkePearl.Attainment.Upper","a":"PO/ID/Partial/BalkePearl/Attainment","t":1,"d":"Latent table attaining bpUpperTerm 6.","s":"{P : Causalean.PO.POSystem} → Causalean.PO.POBalkePearlSystem P → Bool → Bool → Bool → Bool → ℝ","l":605,"f":"Causalean/PO/ID/Partial/BalkePearl/Attainment/Upper.lean"},{"n":"Causalean.PO.POBalkePearlSystem.bpUpperWitness6_feasible","k":"theorem","m":"Causalean.PO.ID.Partial.BalkePearl.Attainment.Upper","a":"PO/ID/Partial/BalkePearl/Attainment","t":2,"d":"The witness for bpUpperTerm 6 is feasible on its optimality region.","s":"∀ {P : Causalean.PO.POSystem} (S : Causalean.PO.POBalkePearlSystem P) (hA : S.BaseAssumptions) {π₀ : Bool → Bool → Bool → Bool → ℝ}, S.BPFeasible hA π₀ → (∀ (j : Fin 8), S.bpUpperTerm 6 ≤ S.bpUpperTerm j) → S.BPFeasible hA S.bpUpperWitness6","l":623,"f":"Causalean/PO/ID/Partial/BalkePearl/Attainment/Upper.lean"},{"n":"Causalean.PO.POBalkePearlSystem.bpUpperWitness6_objective","k":"theorem","m":"Causalean.PO.ID.Partial.BalkePearl.Attainment.Upper","a":"PO/ID/Partial/BalkePearl/Attainment","t":2,"d":"The witness for bpUpperTerm 6 attains it.","s":"∀ {P : Causalean.PO.POSystem} (S : Causalean.PO.POBalkePearlSystem P), S.BaseAssumptions → Causalean.PO.POBalkePearlSystem.BPObjective S.bpUpperWitness6 = S.bpUpperTerm 6","l":677,"f":"Causalean/PO/ID/Partial/BalkePearl/Attainment/Upper.lean"},{"n":"Causalean.PO.POBalkePearlSystem.bpUpperWitness7","k":"def","m":"Causalean.PO.ID.Partial.BalkePearl.Attainment.Upper","a":"PO/ID/Partial/BalkePearl/Attainment","t":1,"d":"Latent table attaining bpUpperTerm 7.","s":"{P : Causalean.PO.POSystem} → Causalean.PO.POBalkePearlSystem P → Bool → Bool → Bool → Bool → ℝ","l":686,"f":"Causalean/PO/ID/Partial/BalkePearl/Attainment/Upper.lean"},{"n":"Causalean.PO.POBalkePearlSystem.bpUpperWitness7_feasible","k":"theorem","m":"Causalean.PO.ID.Partial.BalkePearl.Attainment.Upper","a":"PO/ID/Partial/BalkePearl/Attainment","t":2,"d":"The witness for bpUpperTerm 7 is feasible on its optimality region.","s":"∀ {P : Causalean.PO.POSystem} (S : Causalean.PO.POBalkePearlSystem P) (hA : S.BaseAssumptions) {π₀ : Bool → Bool → Bool → Bool → ℝ}, S.BPFeasible hA π₀ → (∀ (j : Fin 8), S.bpUpperTerm 7 ≤ S.bpUpperTerm j) → S.BPFeasible hA S.bpUpperWitness7","l":705,"f":"Causalean/PO/ID/Partial/BalkePearl/Attainment/Upper.lean"},{"n":"Causalean.PO.POBalkePearlSystem.bpUpperWitness7_objective","k":"theorem","m":"Causalean.PO.ID.Partial.BalkePearl.Attainment.Upper","a":"PO/ID/Partial/BalkePearl/Attainment","t":2,"d":"The witness for bpUpperTerm 7 attains it.","s":"∀ {P : Causalean.PO.POSystem} (S : Causalean.PO.POBalkePearlSystem P), S.BaseAssumptions → Causalean.PO.POBalkePearlSystem.BPObjective S.bpUpperWitness7 = S.bpUpperTerm 7","l":759,"f":"Causalean/PO/ID/Partial/BalkePearl/Attainment/Upper.lean"},{"n":"Causalean.PO.POBalkePearlSystem.bpUpper_mem_BPIdentifiedInterval","k":"theorem","m":"Causalean.PO.ID.Partial.BalkePearl.Attainment.Upper","a":"PO/ID/Partial/BalkePearl/Attainment","t":1,"d":"Under the Balke-Pearl IV base assumptions, the closed-form Balke-Pearl upper bound, computed from the observed cell probabilities, is itself attained as the average treatment effect of some feasible latent treatment-response table — it lies in the Balke-Pearl identified interval.","s":"∀ {P : Causalean.PO.POSystem} (S : Causalean.PO.POBalkePearlSystem P) (hA : S.BaseAssumptions), S.bpUpper ∈ S.BPIdentifiedInterval hA","l":770,"f":"Causalean/PO/ID/Partial/BalkePearl/Attainment/Upper.lean"},{"n":"Causalean.PO.POBalkePearlSystem.cellProb","k":"def","m":"Causalean.PO.ID.Partial.BalkePearl.Setup","a":"PO/ID/Partial/BalkePearl","t":1,"d":"Conditional cell probability P(Y = y, D = d | Z = z).","s":"{P : Causalean.PO.POSystem} → Causalean.PO.POBalkePearlSystem P → Bool → Bool → Bool → ℝ","l":169,"f":"Causalean/PO/ID/Partial/BalkePearl/Setup.lean"},{"n":"Causalean.PO.POBalkePearlSystem.cellProb_eq_sum_latent","k":"theorem","m":"Causalean.PO.ID.Partial.BalkePearl.LatentTable","a":"PO/ID/Partial/BalkePearl","t":2,"d":"Each observed conditional cell probability equals the sum of compatible latent type probabilities.","s":"∀ {P : Causalean.PO.POSystem} (S : Causalean.PO.POBalkePearlSystem P), S.BaseAssumptions → ∀ (y d z : Bool), S.cellProb y d z = ∑ d0, ∑ d1, ∑ y0, ∑ y1, (if Causalean.PO.POBalkePearlSystem.dArm z d0 d1 = d ∧ Causalean.PO.POBalkePearlSystem.yArm d y0 y1 = y then 1 else 0) * S.latentProb d0 d1 y0 y1","l":273,"f":"Causalean/PO/ID/Partial/BalkePearl/LatentTable.lean"},{"n":"Causalean.PO.POBalkePearlSystem.cellProb_nonneg","k":"theorem","m":"Causalean.PO.ID.Partial.BalkePearl.Attainment.Basic","a":"PO/ID/Partial/BalkePearl/Attainment","t":2,"d":"Observed cell probabilities are nonnegative.","s":"∀ {P : Causalean.PO.POSystem} (S : Causalean.PO.POBalkePearlSystem P) (y d z : Bool), 0 ≤ S.cellProb y d z","l":28,"f":"Causalean/PO/ID/Partial/BalkePearl/Attainment/Basic.lean"},{"n":"Causalean.PO.POBalkePearlSystem.cfBundle","k":"def","m":"Causalean.PO.ID.Partial.BalkePearl.Assumptions","a":"PO/ID/Partial/BalkePearl","t":1,"d":"Bundle (D(0), D(1), Y(0), Y(1)) for the exogeneity assumption Z ⊥ (D(0), D(1), Y(0), Y(1)).","s":"{P : Causalean.PO.POSystem} → Causalean.PO.POBalkePearlSystem P → Causalean.PO.POCFBundle P","l":44,"f":"Causalean/PO/ID/Partial/BalkePearl/Assumptions.lean"},{"n":"Causalean.PO.POBalkePearlSystem.csInf_BPIdentifiedInterval_eq_bpLower","k":"theorem","m":"Causalean.PO.ID.Partial.BalkePearl.ClosedFormAttainment","a":"PO/ID/Partial/BalkePearl","t":1,"d":"The closed-form lower endpoint is exactly the LP minimum. Under the Balke-Pearl IV base assumptions, the infimum of the Balke-Pearl identified interval equals the closed-form lower bound bpLower.","s":"∀ {P : Causalean.PO.POSystem} (S : Causalean.PO.POBalkePearlSystem P) (hA : S.BaseAssumptions), sInf (S.BPIdentifiedInterval hA) = S.bpLower","l":43,"f":"Causalean/PO/ID/Partial/BalkePearl/ClosedFormAttainment.lean"},{"n":"Causalean.PO.POBalkePearlSystem.csSup_BPIdentifiedInterval_eq_bpUpper","k":"theorem","m":"Causalean.PO.ID.Partial.BalkePearl.ClosedFormAttainment","a":"PO/ID/Partial/BalkePearl","t":1,"d":"The closed-form upper endpoint is exactly the LP maximum. Under the Balke-Pearl IV base assumptions, the supremum of the Balke-Pearl identified interval equals the closed-form upper bound bpUpper.","s":"∀ {P : Causalean.PO.POSystem} (S : Causalean.PO.POBalkePearlSystem P) (hA : S.BaseAssumptions), sSup (S.BPIdentifiedInterval hA) = S.bpUpper","l":56,"f":"Causalean/PO/ID/Partial/BalkePearl/ClosedFormAttainment.lean"},{"n":"Causalean.PO.POBalkePearlSystem.dArm","k":"def","m":"Causalean.PO.ID.Partial.BalkePearl.LatentTable","a":"PO/ID/Partial/BalkePearl","t":1,"d":"The treatment arm under instrument value z: d₀ if z=false, d₁ if z=true.","s":"Bool → Bool → Bool → Bool","l":38,"f":"Causalean/PO/ID/Partial/BalkePearl/LatentTable.lean"},{"n":"Causalean.PO.POBalkePearlSystem.dEvent","k":"def","m":"Causalean.PO.ID.Partial.BalkePearl.Setup","a":"PO/ID/Partial/BalkePearl","t":1,"d":"The event {D = d}.","s":"{P : Causalean.PO.POSystem} → Causalean.PO.POBalkePearlSystem P → Bool → Set P.Ω","l":115,"f":"Causalean/PO/ID/Partial/BalkePearl/Setup.lean"},{"n":"Causalean.PO.POBalkePearlSystem.dUnderZ","k":"def","m":"Causalean.PO.ID.Partial.BalkePearl.Assumptions","a":"PO/ID/Partial/BalkePearl","t":1,"d":"D(z) as a RegimedVar.","s":"{P : Causalean.PO.POSystem} → Causalean.PO.POBalkePearlSystem P → Bool → Causalean.PO.RegimedVar P Bool","l":34,"f":"Causalean/PO/ID/Partial/BalkePearl/Assumptions.lean"},{"n":"Causalean.PO.POBalkePearlSystem.dVar","k":"def","m":"Causalean.PO.ID.Partial.BalkePearl.Setup","a":"PO/ID/Partial/BalkePearl","t":1,"d":"Treatment packaged as a POVar valued in Bool.","s":"{P : Causalean.PO.POSystem} → Causalean.PO.POBalkePearlSystem P → Causalean.PO.POVar P Bool","l":61,"f":"Causalean/PO/ID/Partial/BalkePearl/Setup.lean"},{"n":"Causalean.PO.POBalkePearlSystem.factualD","k":"def","m":"Causalean.PO.ID.Partial.BalkePearl.Setup","a":"PO/ID/Partial/BalkePearl","t":1,"d":"Factual treatment.","s":"{P : Causalean.PO.POSystem} → Causalean.PO.POBalkePearlSystem P → P.Ω → Bool","l":94,"f":"Causalean/PO/ID/Partial/BalkePearl/Setup.lean"},{"n":"Causalean.PO.POBalkePearlSystem.factualY","k":"def","m":"Causalean.PO.ID.Partial.BalkePearl.Setup","a":"PO/ID/Partial/BalkePearl","t":1,"d":"Factual outcome.","s":"{P : Causalean.PO.POSystem} → Causalean.PO.POBalkePearlSystem P → P.Ω → Bool","l":97,"f":"Causalean/PO/ID/Partial/BalkePearl/Setup.lean"},{"n":"Causalean.PO.POBalkePearlSystem.factualZ","k":"def","m":"Causalean.PO.ID.Partial.BalkePearl.Setup","a":"PO/ID/Partial/BalkePearl","t":1,"d":"Factual instrument.","s":"{P : Causalean.PO.POSystem} → Causalean.PO.POBalkePearlSystem P → P.Ω → Bool","l":91,"f":"Causalean/PO/ID/Partial/BalkePearl/Setup.lean"},{"n":"Causalean.PO.POBalkePearlSystem.latentProb","k":"def","m":"Causalean.PO.ID.Partial.BalkePearl.LatentTable","a":"PO/ID/Partial/BalkePearl","t":1,"d":"Probability of the latent type (d0,d1,y0,y1).","s":"{P : Causalean.PO.POSystem} → Causalean.PO.POBalkePearlSystem P → Bool → Bool → Bool → Bool → ℝ","l":51,"f":"Causalean/PO/ID/Partial/BalkePearl/LatentTable.lean"},{"n":"Causalean.PO.POBalkePearlSystem.latentProb_feasible","k":"theorem","m":"Causalean.PO.ID.Partial.BalkePearl.Main","a":"PO/ID/Partial/BalkePearl","t":2,"d":"The actual latent type distribution latentProb is feasible.","s":"∀ {P : Causalean.PO.POSystem} (S : Causalean.PO.POBalkePearlSystem P) (hA : S.BaseAssumptions), S.BPFeasible hA S.latentProb","l":81,"f":"Causalean/PO/ID/Partial/BalkePearl/Main.lean"},{"n":"Causalean.PO.POBalkePearlSystem.latentProb_nonneg","k":"theorem","m":"Causalean.PO.ID.Partial.BalkePearl.LatentTable","a":"PO/ID/Partial/BalkePearl","t":2,"d":"Latent type probabilities are nonnegative.","s":"∀ {P : Causalean.PO.POSystem} (S : Causalean.PO.POBalkePearlSystem P) (d0 d1 y0 y1 : Bool), 0 ≤ S.latentProb d0 d1 y0 y1","l":70,"f":"Causalean/PO/ID/Partial/BalkePearl/LatentTable.lean"},{"n":"Causalean.PO.POBalkePearlSystem.latentProb_sum_eq_one","k":"theorem","m":"Causalean.PO.ID.Partial.BalkePearl.LatentTable","a":"PO/ID/Partial/BalkePearl","t":2,"d":"The 16 latent sets partition Ω, so their probabilities sum to 1.","s":"∀ {P : Causalean.PO.POSystem} (S : Causalean.PO.POBalkePearlSystem P), ∑ d0, ∑ d1, ∑ y0, ∑ y1, S.latentProb d0 d1 y0 y1 = 1","l":95,"f":"Causalean/PO/ID/Partial/BalkePearl/LatentTable.lean"},{"n":"Causalean.PO.POBalkePearlSystem.latentSet","k":"def","m":"Causalean.PO.ID.Partial.BalkePearl.LatentTable","a":"PO/ID/Partial/BalkePearl","t":1,"d":"The event that unit ω has latent type (d0,d1,y0,y1).","s":"{P : Causalean.PO.POSystem} → Causalean.PO.POBalkePearlSystem P → Bool → Bool → Bool → Bool → Set P.Ω","l":46,"f":"Causalean/PO/ID/Partial/BalkePearl/LatentTable.lean"},{"n":"Causalean.PO.POBalkePearlSystem.measurableSet_dEvent","k":"theorem","m":"Causalean.PO.ID.Partial.BalkePearl.Setup","a":"PO/ID/Partial/BalkePearl","t":2,"d":"The factual treatment event is measurable.","s":"∀ {P : Causalean.PO.POSystem} (S : Causalean.PO.POBalkePearlSystem P) (d : Bool), MeasurableSet (S.dEvent d)","l":147,"f":"Causalean/PO/ID/Partial/BalkePearl/Setup.lean"},{"n":"Causalean.PO.POBalkePearlSystem.measurableSet_latentSet","k":"theorem","m":"Causalean.PO.ID.Partial.BalkePearl.LatentTable","a":"PO/ID/Partial/BalkePearl","t":2,"d":"Each latent type event is measurable.","s":"∀ {P : Causalean.PO.POSystem} (S : Causalean.PO.POBalkePearlSystem P) (d0 d1 y0 y1 : Bool), MeasurableSet (S.latentSet d0 d1 y0 y1)","l":57,"f":"Causalean/PO/ID/Partial/BalkePearl/LatentTable.lean"},{"n":"Causalean.PO.POBalkePearlSystem.measurableSet_yEvent","k":"theorem","m":"Causalean.PO.ID.Partial.BalkePearl.Setup","a":"PO/ID/Partial/BalkePearl","t":2,"d":"The factual outcome event is measurable.","s":"∀ {P : Causalean.PO.POSystem} (S : Causalean.PO.POBalkePearlSystem P) (y : Bool), MeasurableSet (S.yEvent y)","l":151,"f":"Causalean/PO/ID/Partial/BalkePearl/Setup.lean"},{"n":"Causalean.PO.POBalkePearlSystem.measurableSet_zEvent","k":"theorem","m":"Causalean.PO.ID.Partial.BalkePearl.Setup","a":"PO/ID/Partial/BalkePearl","t":2,"d":"The factual instrument event is measurable.","s":"∀ {P : Causalean.PO.POSystem} (S : Causalean.PO.POBalkePearlSystem P) (z : Bool), MeasurableSet (S.zEvent z)","l":143,"f":"Causalean/PO/ID/Partial/BalkePearl/Setup.lean"},{"n":"Causalean.PO.POBalkePearlSystem.measurable_DofZ","k":"theorem","m":"Causalean.PO.ID.Partial.BalkePearl.Setup","a":"PO/ID/Partial/BalkePearl","t":1,"d":"For a fixed instrument value z, the potential treatment D(z) is measurable.","s":"∀ {P : Causalean.PO.POSystem} (S : Causalean.PO.POBalkePearlSystem P) (z : Bool), Measurable (S.DofZ z)","l":123,"f":"Causalean/PO/ID/Partial/BalkePearl/Setup.lean"},{"n":"Causalean.PO.POBalkePearlSystem.measurable_YofD","k":"theorem","m":"Causalean.PO.ID.Partial.BalkePearl.Setup","a":"PO/ID/Partial/BalkePearl","t":2,"d":"The outcome under a fixed treatment value is measurable.","s":"∀ {P : Causalean.PO.POSystem} (S : Causalean.PO.POBalkePearlSystem P) (d : Bool), Measurable (S.YofD d)","l":128,"f":"Causalean/PO/ID/Partial/BalkePearl/Setup.lean"},{"n":"Causalean.PO.POBalkePearlSystem.measurable_YofD_real","k":"theorem","m":"Causalean.PO.ID.Partial.BalkePearl.Setup","a":"PO/ID/Partial/BalkePearl","t":2,"d":"The real-valued potential outcome under a fixed treatment is measurable.","s":"∀ {P : Causalean.PO.POSystem} (S : Causalean.PO.POBalkePearlSystem P) (d : Bool), Measurable (S.YofD_real d)","l":159,"f":"Causalean/PO/ID/Partial/BalkePearl/Setup.lean"},{"n":"Causalean.PO.POBalkePearlSystem.measurable_YofZD","k":"theorem","m":"Causalean.PO.ID.Partial.BalkePearl.Setup","a":"PO/ID/Partial/BalkePearl","t":2,"d":"The outcome under fixed instrument and treatment values is measurable.","s":"∀ {P : Causalean.PO.POSystem} (S : Causalean.PO.POBalkePearlSystem P) (z d : Bool), Measurable (S.YofZD z d)","l":132,"f":"Causalean/PO/ID/Partial/BalkePearl/Setup.lean"},{"n":"Causalean.PO.POBalkePearlSystem.measurable_boolToReal","k":"theorem","m":"Causalean.PO.ID.Partial.BalkePearl.Setup","a":"PO/ID/Partial/BalkePearl","t":2,"d":"The Boolean-to-real embedding is measurable.","s":"Measurable Causalean.PO.POBalkePearlSystem.boolToReal","l":155,"f":"Causalean/PO/ID/Partial/BalkePearl/Setup.lean"},{"n":"Causalean.PO.POBalkePearlSystem.measurable_factualD","k":"theorem","m":"Causalean.PO.ID.Partial.BalkePearl.Setup","a":"PO/ID/Partial/BalkePearl","t":2,"d":"The factual treatment is measurable.","s":"∀ {P : Causalean.PO.POSystem} (S : Causalean.PO.POBalkePearlSystem P), Measurable S.factualD","l":138,"f":"Causalean/PO/ID/Partial/BalkePearl/Setup.lean"},{"n":"Causalean.PO.POBalkePearlSystem.measurable_factualY","k":"theorem","m":"Causalean.PO.ID.Partial.BalkePearl.Setup","a":"PO/ID/Partial/BalkePearl","t":2,"d":"The factual outcome is measurable.","s":"∀ {P : Causalean.PO.POSystem} (S : Causalean.PO.POBalkePearlSystem P), Measurable S.factualY","l":140,"f":"Causalean/PO/ID/Partial/BalkePearl/Setup.lean"},{"n":"Causalean.PO.POBalkePearlSystem.measurable_factualZ","k":"theorem","m":"Causalean.PO.ID.Partial.BalkePearl.Setup","a":"PO/ID/Partial/BalkePearl","t":2,"d":"The factual instrument is measurable.","s":"∀ {P : Causalean.PO.POSystem} (S : Causalean.PO.POBalkePearlSystem P), Measurable S.factualZ","l":136,"f":"Causalean/PO/ID/Partial/BalkePearl/Setup.lean"},{"n":"Causalean.PO.POBalkePearlSystem.neg_one_le_BPObjective","k":"theorem","m":"Causalean.PO.ID.Partial.BalkePearl.IntervalForm","a":"PO/ID/Partial/BalkePearl","t":2,"d":"Lower bound on the objective. Symmetrically, -1 ≤ BPObjective π.","s":"∀ {π : Bool → Bool → Bool → Bool → ℝ}, (∀ (d0 d1 y0 y1 : Bool), 0 ≤ π d0 d1 y0 y1) → ∑ d0, ∑ d1, ∑ y0, ∑ y1, π d0 d1 y0 y1 = 1 → -1 ≤ Causalean.PO.POBalkePearlSystem.BPObjective π","l":74,"f":"Causalean/PO/ID/Partial/BalkePearl/IntervalForm.lean"},{"n":"Causalean.PO.POBalkePearlSystem.regimeZD","k":"def","m":"Causalean.PO.ID.Partial.BalkePearl.Setup","a":"PO/ID/Partial/BalkePearl","t":1,"d":"Two-variable intervention regime r_{z,d} = ({Z,D}, (z,d)).","s":"{P : Causalean.PO.POSystem} → Causalean.PO.POBalkePearlSystem P → Bool → Bool → Causalean.PO.Regime P.V P.X","l":77,"f":"Causalean/PO/ID/Partial/BalkePearl/Setup.lean"},{"n":"Causalean.PO.POBalkePearlSystem.sum_cellProb_eq_one","k":"theorem","m":"Causalean.PO.ID.Partial.BalkePearl.Attainment.Basic","a":"PO/ID/Partial/BalkePearl/Attainment","t":1,"d":"Under the Balke-Pearl IV base assumptions, for every instrument value z, the four observed outcome-treatment cell probabilities sum to one.","s":"∀ {P : Causalean.PO.POSystem} (S : Causalean.PO.POBalkePearlSystem P), S.BaseAssumptions → ∀ (z : Bool), ∑ y, ∑ d, S.cellProb y d z = 1","l":32,"f":"Causalean/PO/ID/Partial/BalkePearl/Attainment/Basic.lean"},{"n":"Causalean.PO.POBalkePearlSystem.yArm","k":"def","m":"Causalean.PO.ID.Partial.BalkePearl.LatentTable","a":"PO/ID/Partial/BalkePearl","t":1,"d":"The outcome arm under treatment d: y₀ if d=false, y₁ if d=true.","s":"Bool → Bool → Bool → Bool","l":41,"f":"Causalean/PO/ID/Partial/BalkePearl/LatentTable.lean"},{"n":"Causalean.PO.POBalkePearlSystem.yEvent","k":"def","m":"Causalean.PO.ID.Partial.BalkePearl.Setup","a":"PO/ID/Partial/BalkePearl","t":1,"d":"The event {Y = y}.","s":"{P : Causalean.PO.POSystem} → Causalean.PO.POBalkePearlSystem P → Bool → Set P.Ω","l":118,"f":"Causalean/PO/ID/Partial/BalkePearl/Setup.lean"},{"n":"Causalean.PO.POBalkePearlSystem.yUnderD","k":"def","m":"Causalean.PO.ID.Partial.BalkePearl.Assumptions","a":"PO/ID/Partial/BalkePearl","t":1,"d":"Y(d) as a RegimedVar.","s":"{P : Causalean.PO.POSystem} → Causalean.PO.POBalkePearlSystem P → Bool → Causalean.PO.RegimedVar P Bool","l":38,"f":"Causalean/PO/ID/Partial/BalkePearl/Assumptions.lean"},{"n":"Causalean.PO.POBalkePearlSystem.yVar","k":"def","m":"Causalean.PO.ID.Partial.BalkePearl.Setup","a":"PO/ID/Partial/BalkePearl","t":1,"d":"Outcome packaged as a POVar valued in Bool.","s":"{P : Causalean.PO.POSystem} → Causalean.PO.POBalkePearlSystem P → Causalean.PO.POVar P Bool","l":64,"f":"Causalean/PO/ID/Partial/BalkePearl/Setup.lean"},{"n":"Causalean.PO.POBalkePearlSystem.zEvent","k":"def","m":"Causalean.PO.ID.Partial.BalkePearl.Setup","a":"PO/ID/Partial/BalkePearl","t":1,"d":"The event {Z = z}.","s":"{P : Causalean.PO.POSystem} → Causalean.PO.POBalkePearlSystem P → Bool → Set P.Ω","l":112,"f":"Causalean/PO/ID/Partial/BalkePearl/Setup.lean"},{"n":"Causalean.PO.POBalkePearlSystem.zVar","k":"def","m":"Causalean.PO.ID.Partial.BalkePearl.Setup","a":"PO/ID/Partial/BalkePearl","t":1,"d":"Instrument packaged as a POVar valued in Bool.","s":"{P : Causalean.PO.POSystem} → Causalean.PO.POBalkePearlSystem P → Causalean.PO.POVar P Bool","l":58,"f":"Causalean/PO/ID/Partial/BalkePearl/Setup.lean"},{"n":"Causalean.PO.POCFBundle","k":"structure","m":"Causalean.PO.Assumptions.IndepCF","a":"PO/Assumptions","t":1,"d":"A finite heterogeneous bundle of n regimed potential-outcome variables indexed by Fin n: each coordinate carries its own value type equipped with a measurable-space structure, and each coordinate is itself a potential-outcome variable paired with the intervention regime under which it is evaluated.","s":"Causalean.PO.POSystem → Type (max (max (u_1 + 1) u_2) u_3)","l":39,"f":"Causalean/PO/Assumptions/IndepCF.lean"},{"n":"Causalean.PO.POCFBundle.condExpGiven","k":"def","m":"Causalean.PO.Conditioning.Bundle","a":"PO/Conditioning","t":1,"d":"The bundle conditional expectation averages a real-valued sample-space quantity given the sigma-algebra generated by the bundle's joint counterfactual-value map.","s":"{P : Causalean.PO.POSystem} → Causalean.PO.POCFBundle P → (P.Ω → ℝ) → optParam (MeasureTheory.Measure P.Ω) P.μ → P.Ω → ℝ","l":58,"f":"Causalean/PO/Conditioning/Bundle.lean"},{"n":"Causalean.PO.POCFBundle.condExpGiven_congr_ae","k":"theorem","m":"Causalean.PO.Conditioning.Bundle","a":"PO/Conditioning","t":2,"d":"a.e. congruence for condExpGiven: if f =ᵐ g, then their bundle conditional expectations agree a.e.","s":"∀ {P : Causalean.PO.POSystem} (B : Causalean.PO.POCFBundle P) {f g : P.Ω → ℝ} {μ : MeasureTheory.Measure P.Ω}, f =ᵐ[μ] g → B.condExpGiven f μ =ᵐ[μ] B.condExpGiven g μ","l":138,"f":"Causalean/PO/Conditioning/Bundle.lean"},{"n":"Causalean.PO.POCFBundle.condExpGiven_indicator_mul","k":"theorem","m":"Causalean.PO.Conditioning.Bundle","a":"PO/Conditioning","t":2,"d":"Indicator pull-out for condExpGiven: if s is B.sigma-measurable, the indicator 1_s factors out of the bundle conditional expectation.","s":"∀ {P : Causalean.PO.POSystem} (B : Causalean.PO.POCFBundle P) {s : Set P.Ω} {g : P.Ω → ℝ} {μ : MeasureTheory.Measure P.Ω}, MeasurableSet s → MeasureTheory.Integrable g μ → B.condExpGiven ((s.indicator fun x => 1) * g) μ =ᵐ[μ] (s.indicator fun x => 1) * B.condExpGiven g μ","l":119,"f":"Causalean/PO/Conditioning/Bundle.lean"},{"n":"Causalean.PO.POCFBundle.condExpGiven_mul_of_consistency_CondIndepCFBundle","k":"theorem","m":"Causalean.PO.Conditioning.EventCondExpBundle","a":"PO/Conditioning","t":1,"d":"Bundle product-form workhorse (analogue of POSystem.eventCondExp_of_consistency_IndepCF). Suppose a factual variable a is conditionally independent, given the σ-algebra of a bundle C, of a counterfactual bundle B, where h is a measurable function whose composite with B's joint value is integrable, and {x} is a measurable singleton in the range of a. If the factual integrand factualF times the indicator of {a = x} agrees almost everywhere with h composed with B's joint value, times the same indicator, then the bundle conditional expectation, given C, of factualF times the indicator of {a = x} factorises almost everywhere as the bundle conditional expectation of h composed with B's joint value, times the bundle conditional expectation of the indicator of {a = x}:","s":"∀ {P : Causalean.PO.POSystem} (B : Causalean.PO.POCFBundle P) (C : Causalean.PO.POCFBundle P) [inst : StandardBorelSpace P.Ω] {α : Type u_1} [inst_1 : MeasurableSpace α] {a : Causalean.PO.POVar P α}, P.CondIndepCFBundle (Causalean.PO.RegimedVar.ofFactual a) B C P.μ → ∀ {factualF : P.Ω → ℝ} {h : ((i : Fin B.n) → B.type i) → ℝ}, Measurable h → MeasureTheory.Integrable (fun ω => h (B.jointValue ω)) P.μ → ∀ {x : α}, MeasurableSet {x} → ((fun ω => factualF ω * a.indicator x ω) =ᵐ[P.μ] fun ω => h (B.jointValue ω) * a.indicator x ω) → (C.condExpGiven fun ω => factualF ω * a.indicator x ω) =ᵐ[P.μ] fun ω => C.condExpGiven (fun ω' => h (B.jointValue ω')) P.μ ω * C.condExpGiven (a.indicator x) P.μ ω","l":70,"f":"Causalean/PO/Conditioning/EventCondExpBundle.lean"},{"n":"Causalean.PO.POCFBundle.condExpGiven_mul_of_stronglyMeasurable_left","k":"theorem","m":"Causalean.PO.Conditioning.Bundle","a":"PO/Conditioning","t":2,"d":"Pull-out-left for condExpGiven: if f is strongly measurable w.r.t. B.sigma, then μ[f·g | B.sigma] =ᵐ f · μ[g | B.sigma].","s":"∀ {P : Causalean.PO.POSystem} (B : Causalean.PO.POCFBundle P) {f g : P.Ω → ℝ} {μ : MeasureTheory.Measure P.Ω}, MeasureTheory.StronglyMeasurable f → MeasureTheory.Integrable (f * g) μ → MeasureTheory.Integrable g μ → B.condExpGiven (f * g) μ =ᵐ[μ] f * B.condExpGiven g μ","l":98,"f":"Causalean/PO/Conditioning/Bundle.lean"},{"n":"Causalean.PO.POCFBundle.condExpGiven_mul_of_stronglyMeasurable_right","k":"theorem","m":"Causalean.PO.Conditioning.Bundle","a":"PO/Conditioning","t":2,"d":"Pull-out-right for condExpGiven.","s":"∀ {P : Causalean.PO.POSystem} (B : Causalean.PO.POCFBundle P) {f g : P.Ω → ℝ} {μ : MeasureTheory.Measure P.Ω}, MeasureTheory.StronglyMeasurable g → MeasureTheory.Integrable (f * g) μ → MeasureTheory.Integrable f μ → B.condExpGiven (f * g) μ =ᵐ[μ] B.condExpGiven f μ * g","l":109,"f":"Causalean/PO/Conditioning/Bundle.lean"},{"n":"Causalean.PO.POCFBundle.condExpGiven_tower_of_le","k":"theorem","m":"Causalean.PO.Conditioning.Bundle","a":"PO/Conditioning","t":2,"d":"Tower property over an arbitrary smaller sub-σ-algebra m ≤ B.sigma. Requires the usual SigmaFinite hypothesis on the trim of μ to B.sigma.","s":"∀ {P : Causalean.PO.POSystem} (B : Causalean.PO.POCFBundle P) {g : P.Ω → ℝ} {μ : MeasureTheory.Measure P.Ω} {m : MeasurableSpace P.Ω}, m ≤ B.sigma → ∀ [MeasureTheory.SigmaFinite (μ.trim (Causalean.PO.POCFBundle.sigma_le B))], μ[B.condExpGiven g μ | m] =ᵐ[μ] μ[g | m]","l":88,"f":"Causalean/PO/Conditioning/Bundle.lean"},{"n":"Causalean.PO.POCFBundle.condExpRatio","k":"def","m":"Causalean.PO.Conditioning.Bundle","a":"PO/Conditioning","t":1,"d":"The bundle conditional-expectation ratio compares two real-valued sample-space quantities after conditioning on the sigma-algebra generated by the bundle's joint counterfactual-value map.","s":"{P : Causalean.PO.POSystem} → Causalean.PO.POCFBundle P → (P.Ω → ℝ) → (P.Ω → ℝ) → optParam (MeasureTheory.Measure P.Ω) P.μ → P.Ω → ℝ","l":147,"f":"Causalean/PO/Conditioning/Bundle.lean"},{"n":"Causalean.PO.POCFBundle.condExpRatio_eq_of_mul","k":"theorem","m":"Causalean.PO.Conditioning.Bundle","a":"PO/Conditioning","t":2,"d":"Characterisation of bundle condExpRatio via a product identity.","s":"∀ {P : Causalean.PO.POSystem} (B : Causalean.PO.POCFBundle P) {g h target : P.Ω → ℝ} {μ : MeasureTheory.Measure P.Ω}, B.condExpGiven g μ =ᵐ[μ] B.condExpGiven h μ * target → (∀ᵐ (ω : P.Ω) ∂μ, B.condExpGiven h μ ω ≠ 0) → B.condExpRatio g h μ =ᵐ[μ] target","l":167,"f":"Causalean/PO/Conditioning/Bundle.lean"},{"n":"Causalean.PO.POCFBundle.condExpRatio_of_consistency_CondIndepCFBundle","k":"theorem","m":"Causalean.PO.Conditioning.EventCondExpBundle","a":"PO/Conditioning","t":1,"d":"Bundle ratio-form workhorse: ratio version of condExpGiven_mul_of_consistency_CondIndepCFBundle. Under the same hypotheses as that theorem — bundle-conditional independence of a from B given C, a measurable and integrable composite h ∘ B.jointValue, a measurable singleton {x}, and the consistency-on-event product identity — plus an almost-surely nonzero bundle-conditional probability of {a = x}, the conditional ratio condExpRatio (factualF · 1_{a=x}) (1_{a=x}) collapses almost everywhere to the bundle conditional mean of h ∘ B.jointValue.","s":"∀ {P : Causalean.PO.POSystem} (B : Causalean.PO.POCFBundle P) (C : Causalean.PO.POCFBundle P) [inst : StandardBorelSpace P.Ω] {α : Type u_1} [inst_1 : MeasurableSpace α] {a : Causalean.PO.POVar P α}, P.CondIndepCFBundle (Causalean.PO.RegimedVar.ofFactual a) B C P.μ → ∀ {factualF : P.Ω → ℝ} {h : ((i : Fin B.n) → B.type i) → ℝ}, Measurable h → MeasureTheory.Integrable (fun ω => h (B.jointValue ω)) P.μ → ∀ {x : α}, MeasurableSet {x} → ((fun ω => factualF ω * a.indicator x ω) =ᵐ[P.μ] fun ω => h (B.jointValue ω) * a.indicator x ω) → (∀ᵐ (ω : P.Ω) ∂P.μ, C.condExpGiven (a.indicator x) P.μ ω ≠ 0) → C.condExpRatio (fun ω => factualF ω * a.indicator x ω) (a.indicator x) =ᵐ[P.μ] C.condExpGiven fun ω' => h (B.jointValue ω')","l":160,"f":"Causalean/PO/Conditioning/EventCondExpBundle.lean"},{"n":"Causalean.PO.POCFBundle.cons","k":"def","m":"Causalean.PO.Assumptions.IndepCF","a":"PO/Assumptions","t":1,"d":"Adding a regimed potential-outcome variable to the front of a bundle produces the corresponding extended counterfactual bundle.","s":"{P : Causalean.PO.POSystem} → {α : Type u_1} → [inst : MeasurableSpace α] → Causalean.PO.RegimedVar P α → Causalean.PO.POCFBundle P → Causalean.PO.POCFBundle P","l":80,"f":"Causalean/PO/Assumptions/IndepCF.lean"},{"n":"Causalean.PO.POCFBundle.instMeasurableSpaceType","k":"instance","m":"Causalean.PO.Assumptions.IndepCF","a":"PO/Assumptions","t":2,"d":"Each coordinate type in a counterfactual bundle carries its declared measurable-space structure.","s":"{P : Causalean.PO.POSystem} → (B : Causalean.PO.POCFBundle P) → (i : Fin B.n) → MeasurableSpace (B.type i)","l":59,"f":"Causalean/PO/Assumptions/IndepCF.lean"},{"n":"Causalean.PO.POCFBundle.integrable_condExpGiven","k":"theorem","m":"Causalean.PO.Conditioning.Bundle","a":"PO/Conditioning","t":2,"d":"Mathlib's totalized conditional expectation given the bundle's σ-algebra is integrable, even when the original sample-space function is not assumed integrable.","s":"∀ {P : Causalean.PO.POSystem} (B : Causalean.PO.POCFBundle P) (g : P.Ω → ℝ) {μ : MeasureTheory.Measure P.Ω}, MeasureTheory.Integrable (B.condExpGiven g μ) μ","l":78,"f":"Causalean/PO/Conditioning/Bundle.lean"},{"n":"Causalean.PO.POCFBundle.jointValue","k":"def","m":"Causalean.PO.Assumptions.IndepCF","a":"PO/Assumptions","t":1,"d":"The joint counterfactual-value map sends each sample point to the tuple of all regimed counterfactual values in the bundle.","s":"{P : Causalean.PO.POSystem} → (B : Causalean.PO.POCFBundle P) → P.Ω → (i : Fin B.n) → B.type i","l":64,"f":"Causalean/PO/Assumptions/IndepCF.lean"},{"n":"Causalean.PO.POCFBundle.measurable_condExpRatio","k":"theorem","m":"Causalean.PO.Conditioning.Bundle","a":"PO/Conditioning","t":2,"d":"The conditional-ratio function of two real integrands given a counterfactual bundle is measurable.","s":"∀ {P : Causalean.PO.POSystem} (B : Causalean.PO.POCFBundle P) (g h : P.Ω → ℝ) {μ : MeasureTheory.Measure P.Ω}, Measurable (B.condExpRatio g h μ)","l":153,"f":"Causalean/PO/Conditioning/Bundle.lean"},{"n":"Causalean.PO.POCFBundle.measurable_jointValue","k":"theorem","m":"Causalean.PO.Assumptions.IndepCF","a":"PO/Assumptions","t":2,"d":"The joint counterfactual-value map of a bundle is measurable.","s":"∀ {P : Causalean.PO.POSystem} (B : Causalean.PO.POCFBundle P), Measurable B.jointValue","l":69,"f":"Causalean/PO/Assumptions/IndepCF.lean"},{"n":"Causalean.PO.POCFBundle.nil","k":"def","m":"Causalean.PO.Assumptions.IndepCF","a":"PO/Assumptions","t":1,"d":"The empty counterfactual bundle has no coordinates.","s":"(P : Causalean.PO.POSystem) → Causalean.PO.POCFBundle P","l":73,"f":"Causalean/PO/Assumptions/IndepCF.lean"},{"n":"Causalean.PO.POCFBundle.sigma","k":"def","m":"Causalean.PO.Conditioning.Bundle","a":"PO/Conditioning","t":1,"d":"The conditioning σ-algebra of a counterfactual bundle B (a finite collection of counterfactual variables over a potential-outcome system): the σ-algebra on the sample space generated by jointly observing all of B's counterfactual values — i.e. the information content of the whole bundle. It is the bundle analogue of the single-variable conditioning σ-algebra σ(c.factual) (the information in one variable c), generalised from one variable to the bundle's joint value map.","s":"{P : Causalean.PO.POSystem} → Causalean.PO.POCFBundle P → MeasurableSpace P.Ω","l":44,"f":"Causalean/PO/Conditioning/Bundle.lean"},{"n":"Causalean.PO.POCFBundle.sigma_le","k":"theorem","m":"Causalean.PO.Conditioning.Bundle","a":"PO/Conditioning","t":2,"d":"The conditioning σ-algebra B.sigma is a sub-σ-algebra of the ambient σ-algebra on P.Ω.","s":"∀ {P : Causalean.PO.POSystem} (B : Causalean.PO.POCFBundle P), B.sigma ≤ inferInstance","l":53,"f":"Causalean/PO/Conditioning/Bundle.lean"},{"n":"Causalean.PO.POCFBundle.stronglyMeasurable_condExpGiven","k":"theorem","m":"Causalean.PO.Conditioning.Bundle","a":"PO/Conditioning","t":2,"d":"condExpGiven yields a strongly measurable function w.r.t. the ambient σ-algebra.","s":"∀ {P : Causalean.PO.POSystem} (B : Causalean.PO.POCFBundle P) (g : P.Ω → ℝ) {μ : MeasureTheory.Measure P.Ω}, MeasureTheory.StronglyMeasurable (B.condExpGiven g μ)","l":72,"f":"Causalean/PO/Conditioning/Bundle.lean"},{"n":"Causalean.PO.POCFBundle.stronglyMeasurable_condExpGiven_comap","k":"theorem","m":"Causalean.PO.Conditioning.Bundle","a":"PO/Conditioning","t":2,"d":"condExpGiven yields a function strongly measurable w.r.t. B.sigma.","s":"∀ {P : Causalean.PO.POSystem} (B : Causalean.PO.POCFBundle P) (g : P.Ω → ℝ) {μ : MeasureTheory.Measure P.Ω}, MeasureTheory.StronglyMeasurable (B.condExpGiven g μ)","l":66,"f":"Causalean/PO/Conditioning/Bundle.lean"},{"n":"Causalean.PO.POCFBundle.stronglyMeasurable_condExpRatio","k":"theorem","m":"Causalean.PO.Conditioning.Bundle","a":"PO/Conditioning","t":2,"d":"The conditional-ratio function of two real integrands given a counterfactual bundle is strongly measurable.","s":"∀ {P : Causalean.PO.POSystem} (B : Causalean.PO.POCFBundle P) (g h : P.Ω → ℝ) {μ : MeasureTheory.Measure P.Ω}, MeasureTheory.StronglyMeasurable (B.condExpRatio g h μ)","l":161,"f":"Causalean/PO/Conditioning/Bundle.lean"},{"n":"Causalean.PO.POCSDIDSystem","k":"structure","m":"Causalean.PO.ID.Exact.CSDID","a":"PO/ID/Exact","t":1,"d":"A staggered-adoption DID system has a binary treatment variable at each of T periods, where at least two periods are observed, together with a real outcome variable at each period, such that the treatment nodes are pairwise distinct across periods and likewise for the outcome nodes, and no treatment node coincides with any outcome node.","s":"Causalean.PO.POSystem → Type (max u_1 u_2)","l":47,"f":"Causalean/PO/ID/Exact/CSDID.lean"},{"n":"Causalean.PO.POCSDIDSystem.ATT","k":"def","m":"Causalean.PO.ID.Exact.CSDID","a":"PO/ID/Exact","t":1,"d":"The group-time ATT is the cohort mean contrast between cohort and never-treated potential outcomes.","s":"{P : Causalean.PO.POSystem} → (S : Causalean.PO.POCSDIDSystem P) → (g : Fin S.T) → Fin S.T → 1 ≤ ↑g → ℝ","l":248,"f":"Causalean/PO/ID/Exact/CSDID.lean"},{"n":"Causalean.PO.POCSDIDSystem.Assumptions","k":"structure","m":"Causalean.PO.ID.Exact.CSDID","a":"PO/ID/Exact","t":1,"d":"The Callaway-Sant'Anna assumptions combine consistency of the underlying potential-outcome system: no unit is treated at period zero and once treated, a unit remains treated in every later period (irreversible adoption); pre-treatment outcomes do not anticipate future treatment; each cohort's mean untreated trend matches the never-treated group's mean untreated trend; each cohort and the never-treated group occur with positive, finite probability; and the cohort and never-treated potential outcomes are integrable.","s":"{P : Causalean.PO.POSystem} → Causalean.PO.POCSDIDSystem P → Prop","l":256,"f":"Causalean/PO/ID/Exact/CSDID.lean"},{"n":"Causalean.PO.POCSDIDSystem.Y_notin_dTargetUpTo","k":"theorem","m":"Causalean.PO.ID.Exact.CSDID","a":"PO/ID/Exact","t":2,"d":"An outcome node is not among the treatment targets before any cutoff.","s":"∀ {P : Causalean.PO.POSystem} (S : Causalean.PO.POCSDIDSystem P) (t : Fin S.T), ∀ k ≤ S.T, S.Y t ∉ S.dTargetUpTo k","l":186,"f":"Causalean/PO/ID/Exact/CSDID.lean"},{"n":"Causalean.PO.POCSDIDSystem.Y_notin_regimeBy","k":"theorem","m":"Causalean.PO.ID.Exact.CSDID","a":"PO/ID/Exact","t":2,"d":"An outcome node is not targeted by any full-horizon treatment-path regime.","s":"∀ {P : Causalean.PO.POSystem} (S : Causalean.PO.POCSDIDSystem P) (t : Fin S.T) (b : Fin S.T → Bool), S.Y t ∉ (S.regimeBy b).target","l":193,"f":"Causalean/PO/ID/Exact/CSDID.lean"},{"n":"Causalean.PO.POCSDIDSystem.YofCohort","k":"def","m":"Causalean.PO.ID.Exact.CSDID","a":"PO/ID/Exact","t":1,"d":"The cohort potential outcome is the period outcome under the treatment path for that cohort.","s":"{P : Causalean.PO.POSystem} → (S : Causalean.PO.POCSDIDSystem P) → Fin S.T → Fin S.T → P.Ω → ℝ","l":211,"f":"Causalean/PO/ID/Exact/CSDID.lean"},{"n":"Causalean.PO.POCSDIDSystem.YofNT","k":"def","m":"Causalean.PO.ID.Exact.CSDID","a":"PO/ID/Exact","t":1,"d":"The never-treated potential outcome is the period outcome under the never-treated path.","s":"{P : Causalean.PO.POSystem} → (S : Causalean.PO.POCSDIDSystem P) → Fin S.T → P.Ω → ℝ","l":215,"f":"Causalean/PO/ID/Exact/CSDID.lean"},{"n":"Causalean.PO.POCSDIDSystem.att_csdid","k":"theorem","m":"Causalean.PO.ID.Exact.CSDID","a":"PO/ID/Exact","t":1,"d":"Callaway--Sant'Anna group-time DID identification of ATT(g, t). Under the group-time assumptions — consistency, no-anticipation, and group-time parallel trends, for a treatment cohort g that starts treatment no earlier than period 1 and a calendar period t no earlier than g, the group-time average treatment effect on the treated equals the difference between the cohort-g mean outcome change from the period before g to period t and the corresponding mean outcome change for the never-treated group:","s":"∀ {P : Causalean.PO.POSystem} (S : Causalean.PO.POCSDIDSystem P), S.Assumptions → ∀ (g t : Fin S.T) (hg : 1 ≤ ↑g), ↑g ≤ ↑t → S.ATT g t hg = (Causalean.PO.eventCondExp P.μ (S.cohortEvent g hg) fun ω => S.factualY t ω - S.factualY (S.predFin g hg) ω) - Causalean.PO.eventCondExp P.μ S.neverTreatedEvent fun ω => S.factualY t ω - S.factualY (S.predFin g hg) ω","l":433,"f":"Causalean/PO/ID/Exact/CSDID.lean"},{"n":"Causalean.PO.POCSDIDSystem.cohortEvent","k":"def","m":"Causalean.PO.ID.Exact.CSDID","a":"PO/ID/Exact","t":1,"d":"The cohort event contains units untreated just before the cohort period and treated at the cohort period.","s":"{P : Causalean.PO.POSystem} → (S : Causalean.PO.POCSDIDSystem P) → (g : Fin S.T) → 1 ≤ ↑g → Set P.Ω","l":231,"f":"Causalean/PO/ID/Exact/CSDID.lean"},{"n":"Causalean.PO.POCSDIDSystem.dEvent","k":"def","m":"Causalean.PO.ID.Exact.CSDID","a":"PO/ID/Exact","t":1,"d":"The period-specific treatment event contains units with the chosen observed treatment value.","s":"{P : Causalean.PO.POSystem} → (S : Causalean.PO.POCSDIDSystem P) → Fin S.T → Bool → Set P.Ω","l":83,"f":"Causalean/PO/ID/Exact/CSDID.lean"},{"n":"Causalean.PO.POCSDIDSystem.dTargetUpTo","k":"def","m":"Causalean.PO.ID.Exact.CSDID","a":"PO/ID/Exact","t":1,"d":"The standalone target set contains the treatment nodes from periods before the cutoff.","s":"{P : Causalean.PO.POSystem} → Causalean.PO.POCSDIDSystem P → ℕ → Finset P.V","l":106,"f":"Causalean/PO/ID/Exact/CSDID.lean"},{"n":"Causalean.PO.POCSDIDSystem.dTargetUpTo_mem_iff","k":"theorem","m":"Causalean.PO.ID.Exact.CSDID","a":"PO/ID/Exact","t":2,"d":"Membership in the target set is exactly being a treatment node before the cutoff.","s":"∀ {P : Causalean.PO.POSystem} (S : Causalean.PO.POCSDIDSystem P), ∀ k ≤ S.T, ∀ (v : P.V), v ∈ S.dTargetUpTo k ↔ ∃ i, ↑i < k ∧ v = S.D i","l":113,"f":"Causalean/PO/ID/Exact/CSDID.lean"},{"n":"Causalean.PO.POCSDIDSystem.dVar","k":"def","m":"Causalean.PO.ID.Exact.CSDID","a":"PO/ID/Exact","t":1,"d":"The treatment at a period is packaged as a binary potential-outcome variable.","s":"{P : Causalean.PO.POSystem} → (S : Causalean.PO.POCSDIDSystem P) → Fin S.T → Causalean.PO.POVar P Bool","l":71,"f":"Causalean/PO/ID/Exact/CSDID.lean"},{"n":"Causalean.PO.POCSDIDSystem.factualD","k":"def","m":"Causalean.PO.ID.Exact.CSDID","a":"PO/ID/Exact","t":1,"d":"The factual treatment at a period is the observed treatment value for that period.","s":"{P : Causalean.PO.POSystem} → (S : Causalean.PO.POCSDIDSystem P) → Fin S.T → P.Ω → Bool","l":77,"f":"Causalean/PO/ID/Exact/CSDID.lean"},{"n":"Causalean.PO.POCSDIDSystem.factualY","k":"def","m":"Causalean.PO.ID.Exact.CSDID","a":"PO/ID/Exact","t":1,"d":"The factual outcome at a period is the observed outcome value for that period.","s":"{P : Causalean.PO.POSystem} → (S : Causalean.PO.POCSDIDSystem P) → Fin S.T → P.Ω → ℝ","l":80,"f":"Causalean/PO/ID/Exact/CSDID.lean"},{"n":"Causalean.PO.POCSDIDSystem.measurableSet_cohortEvent","k":"theorem","m":"Causalean.PO.ID.Exact.CSDID","a":"PO/ID/Exact","t":2,"d":"Each cohort event is measurable.","s":"∀ {P : Causalean.PO.POSystem} (S : Causalean.PO.POCSDIDSystem P) (g : Fin S.T) (hg : 1 ≤ ↑g), MeasurableSet (S.cohortEvent g hg)","l":236,"f":"Causalean/PO/ID/Exact/CSDID.lean"},{"n":"Causalean.PO.POCSDIDSystem.measurableSet_dEvent","k":"theorem","m":"Causalean.PO.ID.Exact.CSDID","a":"PO/ID/Exact","t":2,"d":"Each period-specific treatment event is measurable.","s":"∀ {P : Causalean.PO.POSystem} (S : Causalean.PO.POCSDIDSystem P) (s : Fin S.T) (b : Bool), MeasurableSet (S.dEvent s b)","l":94,"f":"Causalean/PO/ID/Exact/CSDID.lean"},{"n":"Causalean.PO.POCSDIDSystem.measurableSet_neverTreatedEvent","k":"theorem","m":"Causalean.PO.ID.Exact.CSDID","a":"PO/ID/Exact","t":2,"d":"The never-treated event is measurable.","s":"∀ {P : Causalean.PO.POSystem} (S : Causalean.PO.POCSDIDSystem P), MeasurableSet S.neverTreatedEvent","l":244,"f":"Causalean/PO/ID/Exact/CSDID.lean"},{"n":"Causalean.PO.POCSDIDSystem.measurable_YofCohort","k":"theorem","m":"Causalean.PO.ID.Exact.CSDID","a":"PO/ID/Exact","t":2,"d":"Cohort potential outcomes are measurable.","s":"∀ {P : Causalean.PO.POSystem} (S : Causalean.PO.POCSDIDSystem P) (t g : Fin S.T), Measurable (S.YofCohort t g)","l":219,"f":"Causalean/PO/ID/Exact/CSDID.lean"},{"n":"Causalean.PO.POCSDIDSystem.measurable_YofNT","k":"theorem","m":"Causalean.PO.ID.Exact.CSDID","a":"PO/ID/Exact","t":2,"d":"Never-treated potential outcomes are measurable.","s":"∀ {P : Causalean.PO.POSystem} (S : Causalean.PO.POCSDIDSystem P) (t : Fin S.T), Measurable (S.YofNT t)","l":223,"f":"Causalean/PO/ID/Exact/CSDID.lean"},{"n":"Causalean.PO.POCSDIDSystem.measurable_factualD","k":"theorem","m":"Causalean.PO.ID.Exact.CSDID","a":"PO/ID/Exact","t":2,"d":"The observed treatment at each period is measurable.","s":"∀ {P : Causalean.PO.POSystem} (S : Causalean.PO.POCSDIDSystem P) (s : Fin S.T), Measurable (S.factualD s)","l":86,"f":"Causalean/PO/ID/Exact/CSDID.lean"},{"n":"Causalean.PO.POCSDIDSystem.measurable_factualY","k":"theorem","m":"Causalean.PO.ID.Exact.CSDID","a":"PO/ID/Exact","t":2,"d":"The observed outcome at each period is measurable.","s":"∀ {P : Causalean.PO.POSystem} (S : Causalean.PO.POCSDIDSystem P) (s : Fin S.T), Measurable (S.factualY s)","l":90,"f":"Causalean/PO/ID/Exact/CSDID.lean"},{"n":"Causalean.PO.POCSDIDSystem.neverTreatedEvent","k":"def","m":"Causalean.PO.ID.Exact.CSDID","a":"PO/ID/Exact","t":1,"d":"The never-treated event contains units untreated in every period.","s":"{P : Causalean.PO.POSystem} → Causalean.PO.POCSDIDSystem P → Set P.Ω","l":241,"f":"Causalean/PO/ID/Exact/CSDID.lean"},{"n":"Causalean.PO.POCSDIDSystem.predFin","k":"def","m":"Causalean.PO.ID.Exact.CSDID","a":"PO/ID/Exact","t":1,"d":"The predecessor of an admissible cohort is the immediately preceding period.","s":"{P : Causalean.PO.POSystem} → (S : Causalean.PO.POCSDIDSystem P) → (g : Fin S.T) → 1 ≤ ↑g → Fin S.T","l":227,"f":"Causalean/PO/ID/Exact/CSDID.lean"},{"n":"Causalean.PO.POCSDIDSystem.regNT","k":"def","m":"Causalean.PO.ID.Exact.CSDID","a":"PO/ID/Exact","t":1,"d":"The never-treated regime fixes every period's treatment to untreated.","s":"{P : Causalean.PO.POSystem} → Causalean.PO.POCSDIDSystem P → Causalean.PO.Regime P.V P.X","l":180,"f":"Causalean/PO/ID/Exact/CSDID.lean"},{"n":"Causalean.PO.POCSDIDSystem.regOf","k":"def","m":"Causalean.PO.ID.Exact.CSDID","a":"PO/ID/Exact","t":1,"d":"The cohort regime leaves periods before the cohort untreated and treats all later periods.","s":"{P : Causalean.PO.POSystem} → (S : Causalean.PO.POCSDIDSystem P) → Fin S.T → Causalean.PO.Regime P.V P.X","l":176,"f":"Causalean/PO/ID/Exact/CSDID.lean"},{"n":"Causalean.PO.POCSDIDSystem.regUpToAux","k":"def","m":"Causalean.PO.ID.Exact.CSDID","a":"PO/ID/Exact","t":1,"d":"The recursive auxiliary builds a treatment-path regime over the first periods and records its target set.","s":"{P : Causalean.PO.POSystem} → (S : Causalean.PO.POCSDIDSystem P) → (Fin S.T → Bool) → (k : ℕ) → k ≤ S.T → { r // r.target = S.dTargetUpTo k }","l":135,"f":"Causalean/PO/ID/Exact/CSDID.lean"},{"n":"Causalean.PO.POCSDIDSystem.regimeBy","k":"def","m":"Causalean.PO.ID.Exact.CSDID","a":"PO/ID/Exact","t":1,"d":"A full-horizon regime fixes every period's treatment according to a chosen treatment path.","s":"{P : Causalean.PO.POSystem} → (S : Causalean.PO.POCSDIDSystem P) → (Fin S.T → Bool) → Causalean.PO.Regime P.V P.X","l":166,"f":"Causalean/PO/ID/Exact/CSDID.lean"},{"n":"Causalean.PO.POCSDIDSystem.regimeBy_target_eq","k":"theorem","m":"Causalean.PO.ID.Exact.CSDID","a":"PO/ID/Exact","t":2,"d":"The target of a full-horizon treatment-path regime is the full set of treatment nodes.","s":"∀ {P : Causalean.PO.POSystem} (S : Causalean.PO.POCSDIDSystem P) (b : Fin S.T → Bool), (S.regimeBy b).target = S.dTargetUpTo S.T","l":171,"f":"Causalean/PO/ID/Exact/CSDID.lean"},{"n":"Causalean.PO.POCSDIDSystem.yVar","k":"def","m":"Causalean.PO.ID.Exact.CSDID","a":"PO/ID/Exact","t":1,"d":"The outcome at a period is packaged as a real-valued potential-outcome variable.","s":"{P : Causalean.PO.POSystem} → (S : Causalean.PO.POCSDIDSystem P) → Fin S.T → Causalean.PO.POVar P ℝ","l":74,"f":"Causalean/PO/ID/Exact/CSDID.lean"},{"n":"Causalean.PO.POCSDIDSystem.yVar_v_notin_regNT_target","k":"theorem","m":"Causalean.PO.ID.Exact.CSDID","a":"PO/ID/Exact","t":2,"d":"The outcome variable at a period is not targeted by the never-treated regime.","s":"∀ {P : Causalean.PO.POSystem} (S : Causalean.PO.POCSDIDSystem P) (t : Fin S.T), (S.yVar t).v ∉ S.regNT.target","l":204,"f":"Causalean/PO/ID/Exact/CSDID.lean"},{"n":"Causalean.PO.POCSDIDSystem.yVar_v_notin_regOf_target","k":"theorem","m":"Causalean.PO.ID.Exact.CSDID","a":"PO/ID/Exact","t":2,"d":"The outcome variable at a period is not targeted by the cohort regime.","s":"∀ {P : Causalean.PO.POSystem} (S : Causalean.PO.POCSDIDSystem P) (t g : Fin S.T), (S.yVar t).v ∉ (S.regOf g).target","l":199,"f":"Causalean/PO/ID/Exact/CSDID.lean"},{"n":"Causalean.PO.PODIDSystem","k":"structure","m":"Causalean.PO.ID.Exact.DID","a":"PO/ID/Exact","t":1,"d":"A two-period DID system packages a treatment node whose value space is identified with the booleans, together with a pre-period outcome node and a post-period outcome node each of whose value spaces is identified with the real line; the treatment node is required to be distinct from the pre-period outcome node and distinct from the post-period outcome node.","s":"Causalean.PO.POSystem → Type (max u_1 u_2)","l":34,"f":"Causalean/PO/ID/Exact/DID.lean"},{"n":"Causalean.PO.PODIDSystem.ATT","k":"def","m":"Causalean.PO.ID.Exact.DID","a":"PO/ID/Exact","t":1,"d":"The ATT is the treated-group mean difference between treated and untreated post-period potential outcomes.","s":"{P : Causalean.PO.POSystem} → Causalean.PO.PODIDSystem P → ℝ","l":102,"f":"Causalean/PO/ID/Exact/DID.lean"},{"n":"Causalean.PO.PODIDSystem.Assumptions","k":"structure","m":"Causalean.PO.ID.Exact.DID","a":"PO/ID/Exact","t":1,"d":"Assumptions for two-period difference-in-differences identification of the ATT (def:po-did-assumptions). In words: the observed outcomes coincide with the realized-arm potential outcomes; in the pre-period the treated and control groups have the same potential outcome regardless of treatment; and absent treatment the two groups would have changed in parallel between the two periods. The treated group and the control group each occur with positive probability, and the control pre-period outcome, the control post-period outcome, and the treated post-period outcome are integrable, so the group-conditional means are well-defined and finite.","s":"{P : Causalean.PO.POSystem} → Causalean.PO.PODIDSystem P → Prop","l":107,"f":"Causalean/PO/ID/Exact/DID.lean"},{"n":"Causalean.PO.PODIDSystem.Y0ofD","k":"def","m":"Causalean.PO.ID.Exact.DID","a":"PO/ID/Exact","t":1,"d":"The pre-period potential outcome is evaluated under the intervention that fixes treatment.","s":"{P : Causalean.PO.POSystem} → Causalean.PO.PODIDSystem P → Bool → P.Ω → ℝ","l":63,"f":"Causalean/PO/ID/Exact/DID.lean"},{"n":"Causalean.PO.PODIDSystem.Y1ofD","k":"def","m":"Causalean.PO.ID.Exact.DID","a":"PO/ID/Exact","t":1,"d":"The post-period potential outcome is evaluated under the intervention that fixes treatment.","s":"{P : Causalean.PO.POSystem} → Causalean.PO.PODIDSystem P → Bool → P.Ω → ℝ","l":66,"f":"Causalean/PO/ID/Exact/DID.lean"},{"n":"Causalean.PO.PODIDSystem.att_did","k":"theorem","m":"Causalean.PO.ID.Exact.DID","a":"PO/ID/Exact","t":1,"d":"Under the two-period DID assumptions — consistency, no-anticipation, parallel trends, and positive-probability, integrable treatment and control groups, the average treatment effect on the treated equals the difference between the treated group's mean pre-to-post outcome change and the control group's mean pre-to-post outcome change.","s":"∀ {P : Causalean.PO.POSystem} (S : Causalean.PO.PODIDSystem P), S.Assumptions → S.ATT = (Causalean.PO.eventCondExp P.μ (S.dEvent true) fun ω => S.factualY₁ ω - S.factualY₀ ω) - Causalean.PO.eventCondExp P.μ (S.dEvent false) fun ω => S.factualY₁ ω - S.factualY₀ ω","l":156,"f":"Causalean/PO/ID/Exact/DID.lean"},{"n":"Causalean.PO.PODIDSystem.dEvent","k":"def","m":"Causalean.PO.ID.Exact.DID","a":"PO/ID/Exact","t":1,"d":"The treatment event contains the units whose observed treatment equals the chosen arm.","s":"{P : Causalean.PO.POSystem} → Causalean.PO.PODIDSystem P → Bool → Set P.Ω","l":78,"f":"Causalean/PO/ID/Exact/DID.lean"},{"n":"Causalean.PO.PODIDSystem.dVar","k":"def","m":"Causalean.PO.ID.Exact.DID","a":"PO/ID/Exact","t":1,"d":"The treatment node is packaged as a binary potential-outcome variable.","s":"{P : Causalean.PO.POSystem} → Causalean.PO.PODIDSystem P → Causalean.PO.POVar P Bool","l":54,"f":"Causalean/PO/ID/Exact/DID.lean"},{"n":"Causalean.PO.PODIDSystem.factualD","k":"def","m":"Causalean.PO.ID.Exact.DID","a":"PO/ID/Exact","t":1,"d":"The factual treatment is the observed binary treatment value.","s":"{P : Causalean.PO.POSystem} → Causalean.PO.PODIDSystem P → P.Ω → Bool","l":69,"f":"Causalean/PO/ID/Exact/DID.lean"},{"n":"Causalean.PO.PODIDSystem.factualY₀","k":"def","m":"Causalean.PO.ID.Exact.DID","a":"PO/ID/Exact","t":1,"d":"The factual pre-period outcome is the observed pre-period outcome value.","s":"{P : Causalean.PO.POSystem} → Causalean.PO.PODIDSystem P → P.Ω → ℝ","l":72,"f":"Causalean/PO/ID/Exact/DID.lean"},{"n":"Causalean.PO.PODIDSystem.factualY₁","k":"def","m":"Causalean.PO.ID.Exact.DID","a":"PO/ID/Exact","t":1,"d":"The factual post-period outcome is the observed post-period outcome value.","s":"{P : Causalean.PO.POSystem} → Causalean.PO.PODIDSystem P → P.Ω → ℝ","l":75,"f":"Causalean/PO/ID/Exact/DID.lean"},{"n":"Causalean.PO.PODIDSystem.measurableSet_dEvent","k":"theorem","m":"Causalean.PO.ID.Exact.DID","a":"PO/ID/Exact","t":2,"d":"Each observed treatment-arm event is measurable.","s":"∀ {P : Causalean.PO.POSystem} (S : Causalean.PO.PODIDSystem P) (d : Bool), MeasurableSet (S.dEvent d)","l":98,"f":"Causalean/PO/ID/Exact/DID.lean"},{"n":"Causalean.PO.PODIDSystem.measurable_Y0ofD","k":"theorem","m":"Causalean.PO.ID.Exact.DID","a":"PO/ID/Exact","t":2,"d":"The pre-period potential outcome under a fixed treatment arm is measurable.","s":"∀ {P : Causalean.PO.POSystem} (S : Causalean.PO.PODIDSystem P) (d : Bool), Measurable (S.Y0ofD d)","l":81,"f":"Causalean/PO/ID/Exact/DID.lean"},{"n":"Causalean.PO.PODIDSystem.measurable_Y1ofD","k":"theorem","m":"Causalean.PO.ID.Exact.DID","a":"PO/ID/Exact","t":2,"d":"The post-period potential outcome under a fixed treatment arm is measurable.","s":"∀ {P : Causalean.PO.POSystem} (S : Causalean.PO.PODIDSystem P) (d : Bool), Measurable (S.Y1ofD d)","l":85,"f":"Causalean/PO/ID/Exact/DID.lean"},{"n":"Causalean.PO.PODIDSystem.measurable_factualD","k":"theorem","m":"Causalean.PO.ID.Exact.DID","a":"PO/ID/Exact","t":2,"d":"The observed treatment is measurable.","s":"∀ {P : Causalean.PO.POSystem} (S : Causalean.PO.PODIDSystem P), Measurable S.factualD","l":89,"f":"Causalean/PO/ID/Exact/DID.lean"},{"n":"Causalean.PO.PODIDSystem.measurable_factualY₀","k":"theorem","m":"Causalean.PO.ID.Exact.DID","a":"PO/ID/Exact","t":2,"d":"The observed pre-period outcome is measurable.","s":"∀ {P : Causalean.PO.POSystem} (S : Causalean.PO.PODIDSystem P), Measurable S.factualY₀","l":92,"f":"Causalean/PO/ID/Exact/DID.lean"},{"n":"Causalean.PO.PODIDSystem.measurable_factualY₁","k":"theorem","m":"Causalean.PO.ID.Exact.DID","a":"PO/ID/Exact","t":2,"d":"The observed post-period outcome is measurable.","s":"∀ {P : Causalean.PO.POSystem} (S : Causalean.PO.PODIDSystem P), Measurable S.factualY₁","l":95,"f":"Causalean/PO/ID/Exact/DID.lean"},{"n":"Causalean.PO.PODIDSystem.y0Var","k":"def","m":"Causalean.PO.ID.Exact.DID","a":"PO/ID/Exact","t":1,"d":"The pre-period outcome node is packaged as a real-valued potential-outcome variable.","s":"{P : Causalean.PO.POSystem} → Causalean.PO.PODIDSystem P → Causalean.PO.POVar P ℝ","l":57,"f":"Causalean/PO/ID/Exact/DID.lean"},{"n":"Causalean.PO.PODIDSystem.y1Var","k":"def","m":"Causalean.PO.ID.Exact.DID","a":"PO/ID/Exact","t":1,"d":"The post-period outcome node is packaged as a real-valued potential-outcome variable.","s":"{P : Causalean.PO.POSystem} → Causalean.PO.PODIDSystem P → Causalean.PO.POVar P ℝ","l":60,"f":"Causalean/PO/ID/Exact/DID.lean"},{"n":"Causalean.PO.PODTRSystem","k":"structure","m":"Causalean.PO.ID.Exact.DTR.Setup","a":"PO/ID/Exact/DTR","t":1,"d":"A finite-horizon dynamic treatment-regime system packages the variables for sequential potential-outcome identification: a stage-indexed state history observed before each treatment, the treatment chosen at each stage whose value space is identified with a common treatment alphabet across stages, and a terminal outcome whose value space is identified with the real line, subject to the state nodes being pairwise distinct across stages, the treatment nodes being pairwise distinct across stages, and no state, treatment, or outcome node coinciding with another.","s":"Causalean.PO.POSystem → (n : ℕ) → (δ : Type) → (γ : Fin n → Type) → [inst : MeasurableSpace δ] → [MeasurableSingletonClass δ] → [(k : Fin n) → MeasurableSpace (γ k)] → Type (max u_1 u_2)","l":53,"f":"Causalean/PO/ID/Exact/DTR/Setup.lean"},{"n":"Causalean.PO.PODTRSystem.Assumptions","k":"structure","m":"Causalean.PO.ID.Exact.DTR.Setup","a":"PO/ID/Exact/DTR","t":1,"d":"Sequential backdoor assumptions for dynamic-treatment-regime identification at a general horizon n: potential-outcome consistency for the ambient system; per-sequence sequential exchangeability, i.e. at each stage the treatment is conditionally independent of the counterfactual terminal outcome under the treatment sequence given the history observed up to that stage; pointwise positivity of the stagewise propensity given the same history, almost surely; and integrability of the counterfactual terminal outcome under every treatment sequence together with integrability of the factual terminal outcome.","s":"{P : Causalean.PO.POSystem} → {n : ℕ} → {δ : Type} → {γ : Fin n → Type} → [inst : MeasurableSpace δ] → [inst_1 : MeasurableSingletonClass δ] → [inst_2 : (k : Fin n) → MeasurableSpace (γ k)] → Causalean.PO.PODTRSystem P n δ γ → [StandardBorelSpace P.Ω] → [MeasureTheory.IsFiniteMeasure P.μ] → Prop","l":309,"f":"Causalean/PO/ID/Exact/DTR/Setup.lean"},{"n":"Causalean.PO.PODTRSystem.S_of","k":"def","m":"Causalean.PO.ID.Exact.DTR.Setup","a":"PO/ID/Exact/DTR","t":1,"d":"The stage counterfactual state evaluates the state under the earlier treatment interventions.","s":"{P : Causalean.PO.POSystem} → {n : ℕ} → {δ : Type} → {γ : Fin n → Type} → [inst : MeasurableSpace δ] → [inst_1 : MeasurableSingletonClass δ] → [inst_2 : (k : Fin n) → MeasurableSpace (γ k)] → Causalean.PO.PODTRSystem P n δ γ → (Fin n → δ) → (k : Fin n) → P.Ω → γ k","l":214,"f":"Causalean/PO/ID/Exact/DTR/Setup.lean"},{"n":"Causalean.PO.PODTRSystem.Y_of","k":"def","m":"Causalean.PO.ID.Exact.DTR.Setup","a":"PO/ID/Exact/DTR","t":1,"d":"The terminal counterfactual outcome evaluates the outcome under the full treatment regime.","s":"{P : Causalean.PO.POSystem} → {n : ℕ} → {δ : Type} → {γ : Fin n → Type} → [inst : MeasurableSpace δ] → [inst_1 : MeasurableSingletonClass δ] → [inst_2 : (k : Fin n) → MeasurableSpace (γ k)] → Causalean.PO.PODTRSystem P n δ γ → (Fin n → δ) → P.Ω → ℝ","l":210,"f":"Causalean/PO/ID/Exact/DTR/Setup.lean"},{"n":"Causalean.PO.PODTRSystem.adjustedDtr","k":"def","m":"Causalean.PO.ID.Exact.DTR.Setup","a":"PO/ID/Exact/DTR","t":1,"d":"Adjusted DTR functional (observable side). Integrates the outermost ratio innerReg dbar (n - 1) — which conditions on historyBundle 0 = (S 0,) — against P.μ. For n = 0 this is 0.","s":"{P : Causalean.PO.POSystem} → {n : ℕ} → {δ : Type} → {γ : Fin n → Type} → [inst : MeasurableSpace δ] → [inst_1 : MeasurableSingletonClass δ] → [inst_2 : (k : Fin n) → MeasurableSpace (γ k)] → Causalean.PO.PODTRSystem P n δ γ → (Fin n → δ) → ℝ","l":393,"f":"Causalean/PO/ID/Exact/DTR/Setup.lean"},{"n":"Causalean.PO.PODTRSystem.cdtr_backdoor","k":"theorem","m":"Causalean.PO.ID.Exact.DTR.Main","a":"PO/ID/Exact/DTR","t":1,"d":"General-n CDTR backdoor identification. Under the dynamic-treatment-regime identification assumptions, for a positive horizon n, the σ(historyBundle 0)-conditional mean of Y(dbar) equals the outermost iterated-CE functional innerReg dbar (n - 1).","s":"∀ {P : Causalean.PO.POSystem} {n : ℕ} {δ : Type} {γ : Fin n → Type} [inst : MeasurableSpace δ] [inst_1 : MeasurableSingletonClass δ] [inst_2 : (k : Fin n) → MeasurableSpace (γ k)] [inst_3 : StandardBorelSpace P.Ω] [inst_4 : MeasureTheory.IsFiniteMeasure P.μ] (S : Causalean.PO.PODTRSystem P n δ γ), S.Assumptions → ∀ (dbar : Fin n → δ) (hn : 0 < n), (S.historyBundle 0 hn).condExpGiven (S.Y_of dbar) =ᵐ[P.μ] S.innerReg dbar (n - 1)","l":96,"f":"Causalean/PO/ID/Exact/DTR/Main.lean"},{"n":"Causalean.PO.PODTRSystem.cdtr_base","k":"theorem","m":"Causalean.PO.ID.Exact.DTR.Induction","a":"PO/ID/Exact/DTR","t":1,"d":"Base case of the backward induction for a dynamic treatment regime. Under the dynamic-treatment-regime identification assumptions — consistency and stage-wise sequential exchangeability/overlap, provided the horizon n is positive, the depth-zero adjusted-regression functional, multiplied by the indicator that the observed treatment matches the target regime dbar through stage n-1, agrees almost everywhere with that same indicator multiplied by the conditional mean of the regime outcome given the treatment-and-covariate history through stage n-1.","s":"∀ {P : Causalean.PO.POSystem} {n : ℕ} {δ : Type} {γ : Fin n → Type} [inst : MeasurableSpace δ] [inst_1 : MeasurableSingletonClass δ] [inst_2 : (k : Fin n) → MeasurableSpace (γ k)] [inst_3 : StandardBorelSpace P.Ω] [inst_4 : MeasureTheory.IsFiniteMeasure P.μ] (S : Causalean.PO.PODTRSystem P n δ γ), S.Assumptions → ∀ (dbar : Fin n → δ) (hn : 0 < n), (fun ω => S.innerReg dbar 0 ω * S.indD dbar (n - 1) ω) =ᵐ[P.μ] fun ω => S.indD dbar (n - 1) ω * (S.historyBundle (n - 1) (Nat.sub_lt hn Nat.one_pos)).condExpGiven (S.Y_of dbar) P.μ ω","l":64,"f":"Causalean/PO/ID/Exact/DTR/Induction.lean"},{"n":"Causalean.PO.PODTRSystem.cdtr_iter","k":"theorem","m":"Causalean.PO.ID.Exact.DTR.Main","a":"PO/ID/Exact/DTR","t":2,"d":"Backward-induction iterate combining cdtr_base (j = 0) with cdtr_step (j → j + 1). At depth j, asserts that innerReg dbar j multiplied by the partial indicator product indD dbar (n - 1 - j) agrees a.e. with the conditional expectation (historyBundle (n - 1 - j)).condExpGiven (Y_of dbar) multiplied by the same partial indicator.","s":"∀ {P : Causalean.PO.POSystem} {n : ℕ} {δ : Type} {γ : Fin n → Type} [inst : MeasurableSpace δ] [inst_1 : MeasurableSingletonClass δ] [inst_2 : (k : Fin n) → MeasurableSpace (γ k)] [inst_3 : StandardBorelSpace P.Ω] [inst_4 : MeasureTheory.IsFiniteMeasure P.μ] (S : Causalean.PO.PODTRSystem P n δ γ), S.Assumptions → ∀ (dbar : Fin n → δ) (hn : 0 < n), ∀ j < n, (fun ω => S.innerReg dbar j ω * S.indD dbar (n - 1 - j) ω) =ᵐ[P.μ] fun ω => S.indD dbar (n - 1 - j) ω * (S.historyBundle (n - 1 - j) (Decidable.byContradiction fun a => Causalean.PO.PODTRSystem.cdtr_iter._proof_1 hn j a)).condExpGiven (S.Y_of dbar) P.μ ω","l":45,"f":"Causalean/PO/ID/Exact/DTR/Main.lean"},{"n":"Causalean.PO.PODTRSystem.cdtr_step","k":"theorem","m":"Causalean.PO.ID.Exact.DTR.Induction","a":"PO/ID/Exact/DTR","t":2,"d":"The induction step moves the dynamic-regime cancellation identity one stage outward.","s":"∀ {P : Causalean.PO.POSystem} {n : ℕ} {δ : Type} {γ : Fin n → Type} [inst : MeasurableSpace δ] [inst_1 : MeasurableSingletonClass δ] [inst_2 : (k : Fin n) → MeasurableSpace (γ k)] [inst_3 : StandardBorelSpace P.Ω] [inst_4 : MeasureTheory.IsFiniteMeasure P.μ] (S : Causalean.PO.PODTRSystem P n δ γ), S.Assumptions → ∀ (dbar : Fin n → δ) (j : ℕ) (hj : j + 1 < n) (hk : n - j - 2 < n), MeasureTheory.Integrable (S.innerReg dbar j) P.μ → ((fun ω => S.innerReg dbar j ω * S.indD dbar (n - j - 1) ω) =ᵐ[P.μ] fun ω => S.indD dbar (n - j - 1) ω * (S.historyBundle (n - j - 1) (Decidable.byContradiction fun a => Causalean.PO.PODTRSystem.cdtr_step._proof_1 j hj a)).condExpGiven (S.Y_of dbar) P.μ ω) → (fun ω => S.innerReg dbar (j + 1) ω * S.indD dbar (n - j - 2) ω) =ᵐ[P.μ] fun ω => S.indD dbar (n - j - 2) ω * (S.historyBundle (n - j - 2) hk).condExpGiven (S.Y_of dbar) P.μ ω","l":404,"f":"Causalean/PO/ID/Exact/DTR/Induction.lean"},{"n":"Causalean.PO.PODTRSystem.cdtr_strong","k":"theorem","m":"Causalean.PO.ID.Exact.DTR.StrongCancellation","a":"PO/ID/Exact/DTR","t":1,"d":"Joint inductive invariant for the strengthened cancellation. Consider a dynamic-treatment-regime system for which the identifying assumption bundle holds, fix a positive number of decision stages, and fix a treatment history dbar. Then for every stage index j below the horizon, the partial regression term innerReg dbar j agrees almost surely with the product of the treatment-regime indicator at the mirrored stage n - 1 - j and the conditional expectation of the outcome given the history up to that stage, and this term is integrable.","s":"∀ {P : Causalean.PO.POSystem} {n : ℕ} {δ : Type} {γ : Fin n → Type} [inst : MeasurableSpace δ] [inst_1 : MeasurableSingletonClass δ] [inst_2 : (k : Fin n) → MeasurableSpace (γ k)] [inst_3 : StandardBorelSpace P.Ω] [inst_4 : MeasureTheory.IsFiniteMeasure P.μ] (S : Causalean.PO.PODTRSystem P n δ γ), S.Assumptions → ∀ (dbar : Fin n → δ) (hn : 0 < n), ∀ j < n, (S.innerReg dbar j =ᵐ[P.μ] fun ω => S.indD dbar (n - 1 - j) ω * (S.historyBundle (n - 1 - j) (Decidable.byContradiction fun a => Causalean.PO.PODTRSystem.cdtr_strong._proof_1 hn j a)).condExpGiven (S.Y_of dbar) P.μ ω) ∧ MeasureTheory.Integrable (S.innerReg dbar j) P.μ","l":90,"f":"Causalean/PO/ID/Exact/DTR/StrongCancellation.lean"},{"n":"Causalean.PO.PODTRSystem.cfYBundle","k":"def","m":"Causalean.PO.ID.Exact.DTR.Setup","a":"PO/ID/Exact/DTR","t":1,"d":"Singleton CF bundle containing just Y(dbar) — used for the per-sequence exchangeability statement.","s":"{P : Causalean.PO.POSystem} → {n : ℕ} → {δ : Type} → {γ : Fin n → Type} → [inst : MeasurableSpace δ] → [inst_1 : MeasurableSingletonClass δ] → [inst_2 : (k : Fin n) → MeasurableSpace (γ k)] → Causalean.PO.PODTRSystem P n δ γ → (Fin n → δ) → Causalean.PO.POCFBundle P","l":301,"f":"Causalean/PO/ID/Exact/DTR/Setup.lean"},{"n":"Causalean.PO.PODTRSystem.dVar","k":"def","m":"Causalean.PO.ID.Exact.DTR.Setup","a":"PO/ID/Exact/DTR","t":1,"d":"The treatment at a stage is packaged as a potential-outcome variable with the common treatment value space.","s":"{P : Causalean.PO.POSystem} → {n : ℕ} → {δ : Type} → {γ : Fin n → Type} → [inst : MeasurableSpace δ] → [inst_1 : MeasurableSingletonClass δ] → [inst_2 : (k : Fin n) → MeasurableSpace (γ k)] → Causalean.PO.PODTRSystem P n δ γ → Fin n → Causalean.PO.POVar P δ","l":109,"f":"Causalean/PO/ID/Exact/DTR/Setup.lean"},{"n":"Causalean.PO.PODTRSystem.dtrEffect","k":"def","m":"Causalean.PO.ID.Exact.DTR.Setup","a":"PO/ID/Exact/DTR","t":1,"d":"Mean potential outcome E[Y(dbar)]. Single counterfactual mean.","s":"{P : Causalean.PO.POSystem} → {n : ℕ} → {δ : Type} → {γ : Fin n → Type} → [inst : MeasurableSpace δ] → [inst_1 : MeasurableSingletonClass δ] → [inst_2 : (k : Fin n) → MeasurableSpace (γ k)] → Causalean.PO.PODTRSystem P n δ γ → (Fin n → δ) → ℝ","l":389,"f":"Causalean/PO/ID/Exact/DTR/Setup.lean"},{"n":"Causalean.PO.PODTRSystem.dtr_backdoor","k":"theorem","m":"Causalean.PO.ID.Exact.DTR.Main","a":"PO/ID/Exact/DTR","t":1,"d":"General-n integrated DTR backdoor identification. Under the dynamic-treatment-regime identification assumptions, for a positive horizon n, the mean potential outcome E[Y(dbar)] equals the integral of the outermost adjusted functional innerReg dbar (n - 1).","s":"∀ {P : Causalean.PO.POSystem} {n : ℕ} {δ : Type} {γ : Fin n → Type} [inst : MeasurableSpace δ] [inst_1 : MeasurableSingletonClass δ] [inst_2 : (k : Fin n) → MeasurableSpace (γ k)] [inst_3 : StandardBorelSpace P.Ω] [inst_4 : MeasureTheory.IsFiniteMeasure P.μ] (S : Causalean.PO.PODTRSystem P n δ γ), S.Assumptions → ∀ (dbar : Fin n → δ), 0 < n → S.dtrEffect dbar = S.adjustedDtr dbar","l":120,"f":"Causalean/PO/ID/Exact/DTR/Main.lean"},{"n":"Causalean.PO.PODTRSystem.factualAgrees_regime","k":"theorem","m":"Causalean.PO.ID.Exact.DTR.Helpers","a":"PO/ID/Exact/DTR","t":2,"d":"General multi-target consistency: every ω in the full agreement event factually agrees with S.regime dbar.","s":"∀ {P : Causalean.PO.POSystem} {n : ℕ} {δ : Type} {γ : Fin n → Type} [inst : MeasurableSpace δ] [inst_1 : MeasurableSingletonClass δ] [inst_2 : (k : Fin n) → MeasurableSpace (γ k)] (S : Causalean.PO.PODTRSystem P n δ γ) (dbar : Fin n → δ), ∀ ω ∈ {ω | ∀ (i : Fin n), S.factualD i ω = dbar i}, P.FactualAgrees (S.regime dbar) ω","l":336,"f":"Causalean/PO/ID/Exact/DTR/Helpers.lean"},{"n":"Causalean.PO.PODTRSystem.factualAgrees_regimeUpTo","k":"theorem","m":"Causalean.PO.ID.Exact.DTR.Helpers","a":"PO/ID/Exact/DTR","t":2,"d":"Helper: for each k ≤ n, FactualAgrees for regimeUpTo dbar k holds on the event \"factualD i = dbar i for all i.val < k\".","s":"∀ {P : Causalean.PO.POSystem} {n : ℕ} {δ : Type} {γ : Fin n → Type} [inst : MeasurableSpace δ] [inst_1 : MeasurableSingletonClass δ] [inst_2 : (k : Fin n) → MeasurableSpace (γ k)] (S : Causalean.PO.PODTRSystem P n δ γ) (dbar : Fin n → δ) (k : ℕ) (h : k ≤ n) (ω : P.Ω), (∀ (i : Fin n), ↑i < k → S.factualD i ω = dbar i) → P.FactualAgrees (S.regimeUpTo dbar k h) ω","l":310,"f":"Causalean/PO/ID/Exact/DTR/Helpers.lean"},{"n":"Causalean.PO.PODTRSystem.factualD","k":"def","m":"Causalean.PO.ID.Exact.DTR.Setup","a":"PO/ID/Exact/DTR","t":1,"d":"The factual treatment at a stage is the observed treatment value at that stage.","s":"{P : Causalean.PO.POSystem} → {n : ℕ} → {δ : Type} → {γ : Fin n → Type} → [inst : MeasurableSpace δ] → [inst_1 : MeasurableSingletonClass δ] → [inst_2 : (k : Fin n) → MeasurableSpace (γ k)] → Causalean.PO.PODTRSystem P n δ γ → Fin n → P.Ω → δ","l":219,"f":"Causalean/PO/ID/Exact/DTR/Setup.lean"},{"n":"Causalean.PO.PODTRSystem.factualS","k":"def","m":"Causalean.PO.ID.Exact.DTR.Setup","a":"PO/ID/Exact/DTR","t":1,"d":"The factual state at a stage is the observed state value at that stage.","s":"{P : Causalean.PO.POSystem} → {n : ℕ} → {δ : Type} → {γ : Fin n → Type} → [inst : MeasurableSpace δ] → [inst_1 : MeasurableSingletonClass δ] → [inst_2 : (k : Fin n) → MeasurableSpace (γ k)] → Causalean.PO.PODTRSystem P n δ γ → (k : Fin n) → P.Ω → γ k","l":227,"f":"Causalean/PO/ID/Exact/DTR/Setup.lean"},{"n":"Causalean.PO.PODTRSystem.factualY","k":"def","m":"Causalean.PO.ID.Exact.DTR.Setup","a":"PO/ID/Exact/DTR","t":1,"d":"The factual outcome is the observed terminal outcome.","s":"{P : Causalean.PO.POSystem} → {n : ℕ} → {δ : Type} → {γ : Fin n → Type} → [inst : MeasurableSpace δ] → [inst_1 : MeasurableSingletonClass δ] → [inst_2 : (k : Fin n) → MeasurableSpace (γ k)] → Causalean.PO.PODTRSystem P n δ γ → P.Ω → ℝ","l":223,"f":"Causalean/PO/ID/Exact/DTR/Setup.lean"},{"n":"Causalean.PO.PODTRSystem.hb_jointValue_step_eq","k":"theorem","m":"Causalean.PO.ID.Exact.DTR.Helpers","a":"PO/ID/Exact/DTR","t":2,"d":"Key algebraic identity: the stage-k joint value factors through the stage-(k+1) joint value via hb_step_proj.","s":"∀ {P : Causalean.PO.POSystem} {n : ℕ} {δ : Type} {γ : Fin n → Type} [inst : MeasurableSpace δ] [inst_1 : MeasurableSingletonClass δ] [inst_2 : (k : Fin n) → MeasurableSpace (γ k)] (S : Causalean.PO.PODTRSystem P n δ γ) (k : ℕ) (h : k + 1 < n), (S.historyBundle k (Nat.lt_of_succ_lt h)).jointValue = S.hb_step_proj k h ∘ (S.historyBundle (k + 1) h).jointValue","l":64,"f":"Causalean/PO/ID/Exact/DTR/Helpers.lean"},{"n":"Causalean.PO.PODTRSystem.hb_step_proj","k":"def","m":"Causalean.PO.ID.Exact.DTR.Helpers","a":"PO/ID/Exact/DTR","t":1,"d":"Projection from the stage-(k+1) joint value down to the stage-k joint value. Drops the fresh (S (k+1), D k) pair and keeps the history prefix.","s":"{P : Causalean.PO.POSystem} → {n : ℕ} → {δ : Type} → {γ : Fin n → Type} → [inst : MeasurableSpace δ] → [inst_1 : MeasurableSingletonClass δ] → [inst_2 : (k : Fin n) → MeasurableSpace (γ k)] → (S : Causalean.PO.PODTRSystem P n δ γ) → (k : ℕ) → (h : k + 1 < n) → ((i : Fin (S.historyBundle (k + 1) h).n) → (S.historyBundle (k + 1) h).type i) → (j : Fin (S.historyBundle k (Nat.lt_of_succ_lt h)).n) → (S.historyBundle k (Nat.lt_of_succ_lt h)).type j","l":44,"f":"Causalean/PO/ID/Exact/DTR/Helpers.lean"},{"n":"Causalean.PO.PODTRSystem.historyBundle","k":"def","m":"Causalean.PO.ID.Exact.DTR.Setup","a":"PO/ID/Exact/DTR","t":1,"d":"A history bundle collects the observed state and treatment history available just before a selected stage, ending with the current state.","s":"{P : Causalean.PO.POSystem} → {n : ℕ} → {δ : Type} → {γ : Fin n → Type} → [inst : MeasurableSpace δ] → [inst_1 : MeasurableSingletonClass δ] → [inst_2 : (k : Fin n) → MeasurableSpace (γ k)] → Causalean.PO.PODTRSystem P n δ γ → (k : ℕ) → k < n → Causalean.PO.POCFBundle P","l":280,"f":"Causalean/PO/ID/Exact/DTR/Setup.lean"},{"n":"Causalean.PO.PODTRSystem.historyBundle_sigma_mono","k":"theorem","m":"Causalean.PO.ID.Exact.DTR.Helpers","a":"PO/ID/Exact/DTR","t":2,"d":"Iterated σ-algebra monotonicity: σ(history j) ≤ σ(history k) for j ≤ k.","s":"∀ {P : Causalean.PO.POSystem} {n : ℕ} {δ : Type} {γ : Fin n → Type} [inst : MeasurableSpace δ] [inst_1 : MeasurableSingletonClass δ] [inst_2 : (k : Fin n) → MeasurableSpace (γ k)] (S : Causalean.PO.PODTRSystem P n δ γ) (j k : ℕ) (hjk : j ≤ k) (hk : k < n), (S.historyBundle j (lt_of_le_of_lt hjk hk)).sigma ≤ (S.historyBundle k hk).sigma","l":83,"f":"Causalean/PO/ID/Exact/DTR/Helpers.lean"},{"n":"Causalean.PO.PODTRSystem.historyBundle_sigma_mono_step","k":"theorem","m":"Causalean.PO.ID.Exact.DTR.Helpers","a":"PO/ID/Exact/DTR","t":2,"d":"One-step σ-algebra inclusion: σ(history k) ≤ σ(history (k+1)).","s":"∀ {P : Causalean.PO.POSystem} {n : ℕ} {δ : Type} {γ : Fin n → Type} [inst : MeasurableSpace δ] [inst_1 : MeasurableSingletonClass δ] [inst_2 : (k : Fin n) → MeasurableSpace (γ k)] (S : Causalean.PO.PODTRSystem P n δ γ) (k : ℕ) (h : k + 1 < n), (S.historyBundle k (Nat.lt_of_succ_lt h)).sigma ≤ (S.historyBundle (k + 1) h).sigma","l":72,"f":"Causalean/PO/ID/Exact/DTR/Helpers.lean"},{"n":"Causalean.PO.PODTRSystem.indD","k":"def","m":"Causalean.PO.ID.Exact.DTR.Setup","a":"PO/ID/Exact/DTR","t":1,"d":"Joint indicator 1_{D 0 = dbar 0 ∧ … ∧ D (k-1) = dbar (k-1)}.","s":"{P : Causalean.PO.POSystem} → {n : ℕ} → {δ : Type} → {γ : Fin n → Type} → [inst : MeasurableSpace δ] → [inst_1 : MeasurableSingletonClass δ] → [inst_2 : (k : Fin n) → MeasurableSpace (γ k)] → Causalean.PO.PODTRSystem P n δ γ → (Fin n → δ) → ℕ → P.Ω → ℝ","l":252,"f":"Causalean/PO/ID/Exact/DTR/Setup.lean"},{"n":"Causalean.PO.PODTRSystem.indD_eq_indicator_event","k":"theorem","m":"Causalean.PO.ID.Exact.DTR.Helpers","a":"PO/ID/Exact/DTR","t":2,"d":"Event expression for indD: as a set-indicator of the agreement event.","s":"∀ {P : Causalean.PO.POSystem} {n : ℕ} {δ : Type} {γ : Fin n → Type} [inst : MeasurableSpace δ] [inst_1 : MeasurableSingletonClass δ] [inst_2 : (k : Fin n) → MeasurableSpace (γ k)] (S : Causalean.PO.PODTRSystem P n δ γ) (dbar : Fin n → δ), ∀ k ≤ n, S.indD dbar k = {ω | ∀ (i : Fin n), ↑i < k → S.factualD i ω = dbar i}.indicator fun x => 1","l":128,"f":"Causalean/PO/ID/Exact/DTR/Helpers.lean"},{"n":"Causalean.PO.PODTRSystem.indD_eq_zero_or_one","k":"theorem","m":"Causalean.PO.ID.Exact.DTR.Helpers","a":"PO/ID/Exact/DTR","t":2,"d":"indD dbar k ω ∈ {0, 1} for every ω and every k.","s":"∀ {P : Causalean.PO.POSystem} {n : ℕ} {δ : Type} {γ : Fin n → Type} [inst : MeasurableSpace δ] [inst_1 : MeasurableSingletonClass δ] [inst_2 : (k : Fin n) → MeasurableSpace (γ k)] (S : Causalean.PO.PODTRSystem P n δ γ) (dbar : Fin n → δ) (k : ℕ) (ω : P.Ω), S.indD dbar k ω = 0 ∨ S.indD dbar k ω = 1","l":103,"f":"Causalean/PO/ID/Exact/DTR/Helpers.lean"},{"n":"Causalean.PO.PODTRSystem.indD_factor_split","k":"theorem","m":"Causalean.PO.ID.Exact.DTR.Helpers","a":"PO/ID/Exact/DTR","t":1,"d":"For any stage k within the horizon n, the indicator that a unit's observed treatment path matches the regime dbar through stage k+1 factors as the indicator through stage k times the indicator that the observed treatment at stage k equals dbar's value at stage k: indD dbar (k+1) = indD dbar k · 1_{D k = dbar k}.","s":"∀ {P : Causalean.PO.POSystem} {n : ℕ} {δ : Type} {γ : Fin n → Type} [inst : MeasurableSpace δ] [inst_1 : MeasurableSingletonClass δ] [inst_2 : (k : Fin n) → MeasurableSpace (γ k)] (S : Causalean.PO.PODTRSystem P n δ γ) (dbar : Fin n → δ) (k : ℕ) (hk : k < n), S.indD dbar (k + 1) = fun ω => S.indD dbar k ω * (S.dVar ⟨k, hk⟩).indicator (dbar ⟨k, hk⟩) ω","l":185,"f":"Causalean/PO/ID/Exact/DTR/Helpers.lean"},{"n":"Causalean.PO.PODTRSystem.indD_integrable","k":"theorem","m":"Causalean.PO.ID.Exact.DTR.Helpers","a":"PO/ID/Exact/DTR","t":2,"d":"indD dbar k is bounded by 1, hence integrable for finite μ.","s":"∀ {P : Causalean.PO.POSystem} {n : ℕ} {δ : Type} {γ : Fin n → Type} [inst : MeasurableSpace δ] [inst_1 : MeasurableSingletonClass δ] [inst_2 : (k : Fin n) → MeasurableSpace (γ k)] (S : Causalean.PO.PODTRSystem P n δ γ) (dbar : Fin n → δ) (k : ℕ) [MeasureTheory.IsFiniteMeasure P.μ], MeasureTheory.Integrable (S.indD dbar k) P.μ","l":119,"f":"Causalean/PO/ID/Exact/DTR/Helpers.lean"},{"n":"Causalean.PO.PODTRSystem.indD_mul_Y_integrable","k":"theorem","m":"Causalean.PO.ID.Exact.DTR.Helpers","a":"PO/ID/Exact/DTR","t":2,"d":"indD dbar k · Y(dbar) is integrable, bounded by |Y(dbar)|.","s":"∀ {P : Causalean.PO.POSystem} {n : ℕ} {δ : Type} {γ : Fin n → Type} [inst : MeasurableSpace δ] [inst_1 : MeasurableSingletonClass δ] [inst_2 : (k : Fin n) → MeasurableSpace (γ k)] (S : Causalean.PO.PODTRSystem P n δ γ) (dbar : Fin n → δ) (k : ℕ), MeasureTheory.Integrable (S.Y_of dbar) P.μ → MeasureTheory.Integrable (fun ω => S.indD dbar k ω * S.Y_of dbar ω) P.μ","l":346,"f":"Causalean/PO/ID/Exact/DTR/Helpers.lean"},{"n":"Causalean.PO.PODTRSystem.innerReg","k":"def","m":"Causalean.PO.ID.Exact.DTR.Setup","a":"PO/ID/Exact/DTR","t":1,"d":"The adjusted dynamic-treatment-regime regression is computed by backward recursion through the observed treatment and state history, with the zero-horizon case defined to be zero.","s":"{P : Causalean.PO.POSystem} → {n : ℕ} → {δ : Type} → {γ : Fin n → Type} → [inst : MeasurableSpace δ] → [inst_1 : MeasurableSingletonClass δ] → [inst_2 : (k : Fin n) → MeasurableSpace (γ k)] → Causalean.PO.PODTRSystem P n δ γ → (Fin n → δ) → ℕ → P.Ω → ℝ","l":344,"f":"Causalean/PO/ID/Exact/DTR/Setup.lean"},{"n":"Causalean.PO.PODTRSystem.innerReg_integrable","k":"theorem","m":"Causalean.PO.ID.Exact.DTR.StrongCancellation","a":"PO/ID/Exact/DTR","t":2,"d":"innerReg dbar j is integrable for every j < n.","s":"∀ {P : Causalean.PO.POSystem} {n : ℕ} {δ : Type} {γ : Fin n → Type} [inst : MeasurableSpace δ] [inst_1 : MeasurableSingletonClass δ] [inst_2 : (k : Fin n) → MeasurableSpace (γ k)] [inst_3 : StandardBorelSpace P.Ω] [inst_4 : MeasureTheory.IsFiniteMeasure P.μ] (S : Causalean.PO.PODTRSystem P n δ γ), S.Assumptions → ∀ (dbar : Fin n → δ), ∀ j < n, MeasureTheory.Integrable (S.innerReg dbar j) P.μ","l":495,"f":"Causalean/PO/ID/Exact/DTR/StrongCancellation.lean"},{"n":"Causalean.PO.PODTRSystem.measurable_S_of","k":"theorem","m":"Causalean.PO.ID.Exact.DTR.Setup","a":"PO/ID/Exact/DTR","t":2,"d":"Each stage counterfactual state under the earlier treatment interventions is measurable.","s":"∀ {P : Causalean.PO.POSystem} {n : ℕ} {δ : Type} {γ : Fin n → Type} [inst : MeasurableSpace δ] [inst_1 : MeasurableSingletonClass δ] [inst_2 : (k : Fin n) → MeasurableSpace (γ k)] (S : Causalean.PO.PODTRSystem P n δ γ) (dbar : Fin n → δ) (k : Fin n), Measurable (S.S_of dbar k)","l":235,"f":"Causalean/PO/ID/Exact/DTR/Setup.lean"},{"n":"Causalean.PO.PODTRSystem.measurable_Y_of","k":"theorem","m":"Causalean.PO.ID.Exact.DTR.Setup","a":"PO/ID/Exact/DTR","t":2,"d":"The terminal counterfactual outcome under a treatment sequence is measurable.","s":"∀ {P : Causalean.PO.POSystem} {n : ℕ} {δ : Type} {γ : Fin n → Type} [inst : MeasurableSpace δ] [inst_1 : MeasurableSingletonClass δ] [inst_2 : (k : Fin n) → MeasurableSpace (γ k)] (S : Causalean.PO.PODTRSystem P n δ γ) (dbar : Fin n → δ), Measurable (S.Y_of dbar)","l":231,"f":"Causalean/PO/ID/Exact/DTR/Setup.lean"},{"n":"Causalean.PO.PODTRSystem.measurable_factualD","k":"theorem","m":"Causalean.PO.ID.Exact.DTR.Setup","a":"PO/ID/Exact/DTR","t":2,"d":"Each observed treatment process is measurable.","s":"∀ {P : Causalean.PO.POSystem} {n : ℕ} {δ : Type} {γ : Fin n → Type} [inst : MeasurableSpace δ] [inst_1 : MeasurableSingletonClass δ] [inst_2 : (k : Fin n) → MeasurableSpace (γ k)] (S : Causalean.PO.PODTRSystem P n δ γ) (k : Fin n), Measurable (S.factualD k)","l":240,"f":"Causalean/PO/ID/Exact/DTR/Setup.lean"},{"n":"Causalean.PO.PODTRSystem.measurable_factualD_sigma_history","k":"theorem","m":"Causalean.PO.ID.Exact.DTR.Helpers","a":"PO/ID/Exact/DTR","t":2,"d":"factualD k' is (historyBundle m).sigma-measurable for k'.val < m.","s":"∀ {P : Causalean.PO.POSystem} {n : ℕ} {δ : Type} {γ : Fin n → Type} [inst : MeasurableSpace δ] [inst_1 : MeasurableSingletonClass δ] [inst_2 : (k : Fin n) → MeasurableSpace (γ k)] (S : Causalean.PO.PODTRSystem P n δ γ) (m : ℕ) (hm : m < n) (k' : Fin n), ↑k' < m → Measurable (S.factualD k')","l":243,"f":"Causalean/PO/ID/Exact/DTR/Helpers.lean"},{"n":"Causalean.PO.PODTRSystem.measurable_factualS","k":"theorem","m":"Causalean.PO.ID.Exact.DTR.Setup","a":"PO/ID/Exact/DTR","t":2,"d":"Each observed state process is measurable.","s":"∀ {P : Causalean.PO.POSystem} {n : ℕ} {δ : Type} {γ : Fin n → Type} [inst : MeasurableSpace δ] [inst_1 : MeasurableSingletonClass δ] [inst_2 : (k : Fin n) → MeasurableSpace (γ k)] (S : Causalean.PO.PODTRSystem P n δ γ) (k : Fin n), Measurable (S.factualS k)","l":246,"f":"Causalean/PO/ID/Exact/DTR/Setup.lean"},{"n":"Causalean.PO.PODTRSystem.measurable_factualS_sigma_history","k":"theorem","m":"Causalean.PO.ID.Exact.DTR.Helpers","a":"PO/ID/Exact/DTR","t":2,"d":"factualS k' is (historyBundle m).sigma-measurable for k'.val ≤ m.","s":"∀ {P : Causalean.PO.POSystem} {n : ℕ} {δ : Type} {γ : Fin n → Type} [inst : MeasurableSpace δ] [inst_1 : MeasurableSingletonClass δ] [inst_2 : (k : Fin n) → MeasurableSpace (γ k)] (S : Causalean.PO.PODTRSystem P n δ γ) (m : ℕ) (hm : m < n) (k' : Fin n), ↑k' ≤ m → Measurable (S.factualS k')","l":212,"f":"Causalean/PO/ID/Exact/DTR/Helpers.lean"},{"n":"Causalean.PO.PODTRSystem.measurable_factualY","k":"theorem","m":"Causalean.PO.ID.Exact.DTR.Setup","a":"PO/ID/Exact/DTR","t":2,"d":"The observed terminal outcome is measurable.","s":"∀ {P : Causalean.PO.POSystem} {n : ℕ} {δ : Type} {γ : Fin n → Type} [inst : MeasurableSpace δ] [inst_1 : MeasurableSingletonClass δ] [inst_2 : (k : Fin n) → MeasurableSpace (γ k)] (S : Causalean.PO.PODTRSystem P n δ γ), Measurable S.factualY","l":243,"f":"Causalean/PO/ID/Exact/DTR/Setup.lean"},{"n":"Causalean.PO.PODTRSystem.measurable_hb_step_proj","k":"theorem","m":"Causalean.PO.ID.Exact.DTR.Helpers","a":"PO/ID/Exact/DTR","t":2,"d":"The projection from a stage's extended history to the previous history is measurable.","s":"∀ {P : Causalean.PO.POSystem} {n : ℕ} {δ : Type} {γ : Fin n → Type} [inst : MeasurableSpace δ] [inst_1 : MeasurableSingletonClass δ] [inst_2 : (k : Fin n) → MeasurableSpace (γ k)] (S : Causalean.PO.PODTRSystem P n δ γ) (k : ℕ) (h : k + 1 < n), Measurable (S.hb_step_proj k h)","l":51,"f":"Causalean/PO/ID/Exact/DTR/Helpers.lean"},{"n":"Causalean.PO.PODTRSystem.measurable_indD","k":"theorem","m":"Causalean.PO.ID.Exact.DTR.Setup","a":"PO/ID/Exact/DTR","t":2,"d":"The joint treatment-agreement indicator up to any cutoff is measurable.","s":"∀ {P : Causalean.PO.POSystem} {n : ℕ} {δ : Type} {γ : Fin n → Type} [inst : MeasurableSpace δ] [inst_1 : MeasurableSingletonClass δ] [inst_2 : (k : Fin n) → MeasurableSpace (γ k)] (S : Causalean.PO.PODTRSystem P n δ γ) (dbar : Fin n → δ) (k : ℕ), Measurable (S.indD dbar k)","l":265,"f":"Causalean/PO/ID/Exact/DTR/Setup.lean"},{"n":"Causalean.PO.PODTRSystem.measurable_innerReg","k":"theorem","m":"Causalean.PO.ID.Exact.DTR.Helpers","a":"PO/ID/Exact/DTR","t":2,"d":"innerReg dbar j is measurable for every j.","s":"∀ {P : Causalean.PO.POSystem} {n : ℕ} {δ : Type} {γ : Fin n → Type} [inst : MeasurableSpace δ] [inst_1 : MeasurableSingletonClass δ] [inst_2 : (k : Fin n) → MeasurableSpace (γ k)] (S : Causalean.PO.PODTRSystem P n δ γ) (dbar : Fin n → δ) (j : ℕ), Measurable (S.innerReg dbar j)","l":357,"f":"Causalean/PO/ID/Exact/DTR/Helpers.lean"},{"n":"Causalean.PO.PODTRSystem.regime","k":"def","m":"Causalean.PO.ID.Exact.DTR.Setup","a":"PO/ID/Exact/DTR","t":1,"d":"The full treatment regime fixes every stage's treatment to the chosen treatment sequence.","s":"{P : Causalean.PO.POSystem} → {n : ℕ} → {δ : Type} → {γ : Fin n → Type} → [inst : MeasurableSpace δ] → [inst_1 : MeasurableSingletonClass δ] → [inst_2 : (k : Fin n) → MeasurableSpace (γ k)] → Causalean.PO.PODTRSystem P n δ γ → (Fin n → δ) → Causalean.PO.Regime P.V P.X","l":203,"f":"Causalean/PO/ID/Exact/DTR/Setup.lean"},{"n":"Causalean.PO.PODTRSystem.regimeTarget","k":"def","m":"Causalean.PO.ID.Exact.DTR.Setup","a":"PO/ID/Exact/DTR","t":1,"d":"The regime target set contains the treatment nodes before the stage cutoff.","s":"{P : Causalean.PO.POSystem} → {n : ℕ} → {δ : Type} → {γ : Fin n → Type} → [inst : MeasurableSpace δ] → [inst_1 : MeasurableSingletonClass δ] → [inst_2 : (k : Fin n) → MeasurableSpace (γ k)] → Causalean.PO.PODTRSystem P n δ γ → ℕ → Finset P.V","l":119,"f":"Causalean/PO/ID/Exact/DTR/Setup.lean"},{"n":"Causalean.PO.PODTRSystem.regimeTarget_mem_iff","k":"theorem","m":"Causalean.PO.ID.Exact.DTR.Setup","a":"PO/ID/Exact/DTR","t":1,"d":"For a dynamic-treatment-regime system S, a variable belongs to the regime target built up to stage k if and only if it is the treatment node of some earlier stage i < k.","s":"∀ {P : Causalean.PO.POSystem} {n : ℕ} {δ : Type} {γ : Fin n → Type} [inst : MeasurableSpace δ] [inst_1 : MeasurableSingletonClass δ] [inst_2 : (k : Fin n) → MeasurableSpace (γ k)] (S : Causalean.PO.PODTRSystem P n δ γ), ∀ k ≤ n, ∀ (v : P.V), v ∈ S.regimeTarget k ↔ ∃ i, ↑i < k ∧ v = S.D i","l":129,"f":"Causalean/PO/ID/Exact/DTR/Setup.lean"},{"n":"Causalean.PO.PODTRSystem.regimeUpTo","k":"def","m":"Causalean.PO.ID.Exact.DTR.Setup","a":"PO/ID/Exact/DTR","t":1,"d":"The partial regime fixes treatments before the stage cutoff to a chosen treatment sequence.","s":"{P : Causalean.PO.POSystem} → {n : ℕ} → {δ : Type} → {γ : Fin n → Type} → [inst : MeasurableSpace δ] → [inst_1 : MeasurableSingletonClass δ] → [inst_2 : (k : Fin n) → MeasurableSpace (γ k)] → Causalean.PO.PODTRSystem P n δ γ → (Fin n → δ) → (k : ℕ) → k ≤ n → Causalean.PO.Regime P.V P.X","l":192,"f":"Causalean/PO/ID/Exact/DTR/Setup.lean"},{"n":"Causalean.PO.PODTRSystem.regimeUpToAux","k":"def","m":"Causalean.PO.ID.Exact.DTR.Setup","a":"PO/ID/Exact/DTR","t":1,"d":"Mutual-recursive definition of the regime together with a proof that its target equals regimeTarget.","s":"{P : Causalean.PO.POSystem} → {n : ℕ} → {δ : Type} → {γ : Fin n → Type} → [inst : MeasurableSpace δ] → [inst_1 : MeasurableSingletonClass δ] → [inst_2 : (k : Fin n) → MeasurableSpace (γ k)] → (S : Causalean.PO.PODTRSystem P n δ γ) → (Fin n → δ) → (k : ℕ) → k ≤ n → { r // r.target = S.regimeTarget k }","l":155,"f":"Causalean/PO/ID/Exact/DTR/Setup.lean"},{"n":"Causalean.PO.PODTRSystem.regimeUpTo_target_eq","k":"theorem","m":"Causalean.PO.ID.Exact.DTR.Setup","a":"PO/ID/Exact/DTR","t":2,"d":"The target of the partial treatment regime is the standalone target set for the cutoff.","s":"∀ {P : Causalean.PO.POSystem} {n : ℕ} {δ : Type} {γ : Fin n → Type} [inst : MeasurableSpace δ] [inst_1 : MeasurableSingletonClass δ] [inst_2 : (k : Fin n) → MeasurableSpace (γ k)] (S : Causalean.PO.PODTRSystem P n δ γ) (dbar : Fin n → δ) (k : ℕ) (h : k ≤ n), (S.regimeUpTo dbar k h).target = S.regimeTarget k","l":197,"f":"Causalean/PO/ID/Exact/DTR/Setup.lean"},{"n":"Causalean.PO.PODTRSystem.stronglyMeasurable_indD_sigma_history","k":"theorem","m":"Causalean.PO.ID.Exact.DTR.Helpers","a":"PO/ID/Exact/DTR","t":2,"d":"The joint-agreement indicator indD dbar m' is (historyBundle m).sigma-strongly-measurable for m' ≤ m.","s":"∀ {P : Causalean.PO.POSystem} {n : ℕ} {δ : Type} {γ : Fin n → Type} [inst : MeasurableSpace δ] [inst_1 : MeasurableSingletonClass δ] [inst_2 : (k : Fin n) → MeasurableSpace (γ k)] (S : Causalean.PO.PODTRSystem P n δ γ) (m : ℕ) (hm : m < n) (dbar : Fin n → δ), ∀ m' ≤ m, MeasureTheory.StronglyMeasurable (S.indD dbar m')","l":284,"f":"Causalean/PO/ID/Exact/DTR/Helpers.lean"},{"n":"Causalean.PO.PODTRSystem.stronglyMeasurable_indicator_dVar_sigma_history","k":"theorem","m":"Causalean.PO.ID.Exact.DTR.Helpers","a":"PO/ID/Exact/DTR","t":2,"d":"The real-valued indicator dVar ⟨k', hk⟩.indicator (dbar ⟨k', hk⟩) is (historyBundle m).sigma-strongly-measurable for k' < m.","s":"∀ {P : Causalean.PO.POSystem} {n : ℕ} {δ : Type} {γ : Fin n → Type} [inst : MeasurableSpace δ] [inst_1 : MeasurableSingletonClass δ] [inst_2 : (k : Fin n) → MeasurableSpace (γ k)] (S : Causalean.PO.PODTRSystem P n δ γ) (m : ℕ) (hm : m < n) (k' : Fin n), ↑k' < m → ∀ (x : δ), MeasureTheory.StronglyMeasurable ((S.dVar k').indicator x)","l":268,"f":"Causalean/PO/ID/Exact/DTR/Helpers.lean"},{"n":"Causalean.PO.PODTRSystem.toDTREstimationSystem","k":"def","m":"Causalean.Estimation.DTR.Constructor","a":"Estimation/DTR","t":1,"d":"The DTR estimation-system fields are free given two-stage overlap. From a bare PODTRSystem P 2 δ γ, a fixed target regime dbar, and a.e. overlap at stages 0 and 1, construct a DTREstimationSystem. The regression fields are Doob--Dynkin lifts of the observable nested regressions, and the propensity fields are the corresponding lifts clamped into (0, 1) off support.","s":"{P : Causalean.PO.POSystem} → {δ : Type} → {γ : Fin 2 → Type} → [inst : MeasurableSpace δ] → [inst_1 : MeasurableSingletonClass δ] → [inst_2 : (k : Fin 2) → MeasurableSpace (γ k)] → [inst_3 : StandardBorelSpace P.Ω] → [inst_4 : MeasureTheory.IsFiniteMeasure P.μ] → (S : Causalean.PO.PODTRSystem P 2 δ γ) → (dbar : Fin 2 → δ) → (∀ᵐ (ω : P.Ω) ∂P.μ, 0 < (S.historyBundle 0 (of_decide_eq_true (id (Eq.refl true)))).condExpGiven ((S.dVar ⟨0, of_decide_eq_true (id (Eq.refl true))⟩).indicator (dbar ⟨0, of_decide_eq_true (id (Eq.refl true))⟩)) P.μ ω ∧ (S.historyBundle 0 (of_decide_eq_true (id (Eq.refl true)))).condExpGiven ((S.dVar ⟨0, of_decide_eq_true (id (Eq.refl true))⟩).indicator (dbar ⟨0, of_decide_eq_true (id (Eq.refl true))⟩)) P.μ ω < 1) → (∀ᵐ (ω : P.Ω) ∂P.μ, 0 < (S.historyBundle 1 (of_decide_eq_true (id (Eq.refl true)))).condExpGiven ((S.dVar ⟨1, of_decide_eq_true (id (Eq.refl true))⟩).indicator (dbar ⟨1, of_decide_eq_true (id (Eq.refl true))⟩)) P.μ ω ∧ (S.historyBundle 1 (of_decide_eq_true (id (Eq.refl true)))).condExpGiven ((S.dVar ⟨1, of_decide_eq_true (id (Eq.refl true))⟩).indicator (dbar ⟨1, of_decide_eq_true (id (Eq.refl true))⟩)) P.μ ω < 1) → Causalean.Estimation.DTR.DTREstimationSystem P δ γ","l":123,"f":"Causalean/Estimation/DTR/Constructor.lean"},{"n":"Causalean.PO.PODTRSystem.yVar","k":"def","m":"Causalean.PO.ID.Exact.DTR.Setup","a":"PO/ID/Exact/DTR","t":1,"d":"The terminal outcome is packaged as a real-valued potential-outcome variable.","s":"{P : Causalean.PO.POSystem} → {n : ℕ} → {δ : Type} → {γ : Fin n → Type} → [inst : MeasurableSpace δ] → [inst_1 : MeasurableSingletonClass δ] → [inst_2 : (k : Fin n) → MeasurableSpace (γ k)] → Causalean.PO.PODTRSystem P n δ γ → Causalean.PO.POVar P ℝ","l":114,"f":"Causalean/PO/ID/Exact/DTR/Setup.lean"},{"n":"Causalean.PO.PODTRSystem.yVar_notMem_regime","k":"theorem","m":"Causalean.PO.ID.Exact.DTR.Helpers","a":"PO/ID/Exact/DTR","t":2,"d":"The outcome node Y is not a target of S.regime dbar.","s":"∀ {P : Causalean.PO.POSystem} {n : ℕ} {δ : Type} {γ : Fin n → Type} [inst : MeasurableSpace δ] [inst_1 : MeasurableSingletonClass δ] [inst_2 : (k : Fin n) → MeasurableSpace (γ k)] (S : Causalean.PO.PODTRSystem P n δ γ) (dbar : Fin n → δ), S.yVar.v ∉ (S.regime dbar).target","l":399,"f":"Causalean/PO/ID/Exact/DTR/Helpers.lean"},{"n":"Causalean.PO.PODynLATESystem","k":"structure","m":"Causalean.PO.ID.Exact.DynamicLATE.Setup","a":"PO/ID/Exact/DynamicLATE","t":1,"d":"A two-period dynamic instrumental-variable (LATE) model in the potential-outcome framework. A unit is observed over two periods: baseline covariates S₀, then in period 1 a binary encouragement / instrument Z₁ whose value space is identified with the booleans and the treatment D₁ it shifts, likewise identified with the booleans; an intermediate state S₁; then in period 2 a second encouragement Z₂ and treatment D₂, each identified with the booleans; and finally a real-valued outcome Y, identified with the real line. Sequential instrument variation identifies dynamic complier treatment effects (def:po-dynamic-late-system), and all seven nodes are required to be pairwise distinct.","s":"Causalean.PO.POSystem → (γ₀ γ₁ : Type) → [MeasurableSpace γ₀] → [MeasurableSpace γ₁] → Type (max u_1 u_2)","l":47,"f":"Causalean/PO/ID/Exact/DynamicLATE/Setup.lean"},{"n":"Causalean.PO.PODynLATESystem.Assumptions","k":"structure","m":"Causalean.PO.ID.Exact.DynamicLATE.Setup","a":"PO/ID/Exact/DynamicLATE","t":1,"d":"The dynamic instrumental-variable / LATE assumptions for the two-period system (def:po-dynamic-late-assumptions): potential-outcome consistency for the ambient system; conditional independence of the stage-1 encouragement from the counterfactual outcome and treatment path given the baseline state, and conditional independence of the stage-2 encouragement from its counterfactual outcome and treatment given the full stage-2 history; positive stage-1 and positive stage-2 propensities almost surely; a positive stage-1 and a positive stage-2 first-stage effect of encouragement on treatment, almost surely; one-sided noncompliance, whereby each stage's counterfactual treatment never exceeds its encouragement; an exclusion restriction under which the outcome depends on the encouragements only through the resulting treatments, the first-stage treatment does not depend on the second-period encouragement, and the first-period encouragement does not depend on the second-period encouragement; and integrability of the counterfactual outcome under every fixed treatment vector, under every fixed encouragement vector, and under every fixed second-period encouragement.","s":"{P : Causalean.PO.POSystem} → {γ₀ γ₁ : Type} → [inst : MeasurableSpace γ₀] → [inst_1 : MeasurableSpace γ₁] → Causalean.PO.PODynLATESystem P γ₀ γ₁ → [StandardBorelSpace P.Ω] → [MeasureTheory.IsFiniteMeasure P.μ] → Prop","l":509,"f":"Causalean/PO/ID/Exact/DynamicLATE/Setup.lean"},{"n":"Causalean.PO.PODynLATESystem.Assumptions.integrable_factualY","k":"theorem","m":"Causalean.PO.ID.Exact.DynamicLATE.Setup","a":"PO/ID/Exact/DynamicLATE","t":2,"d":"Compatibility projection for older call sites: factual outcome integrability is derived from consistency plus integrability of the two YofZ2 cells.","s":"∀ {P : Causalean.PO.POSystem} {γ₀ γ₁ : Type} [inst : MeasurableSpace γ₀] [inst_1 : MeasurableSpace γ₁] (S : Causalean.PO.PODynLATESystem P γ₀ γ₁) [inst_2 : StandardBorelSpace P.Ω], S.Assumptions → MeasureTheory.Integrable S.factualY P.μ","l":599,"f":"Causalean/PO/ID/Exact/DynamicLATE/Setup.lean"},{"n":"Causalean.PO.PODynLATESystem.D1_ne_D2","k":"theorem","m":"Causalean.PO.ID.Exact.DynamicLATE.Setup","a":"PO/ID/Exact/DynamicLATE","t":2,"d":"The two treatment nodes are distinct.","s":"∀ {P : Causalean.PO.POSystem} {γ₀ γ₁ : Type} [inst : MeasurableSpace γ₀] [inst_1 : MeasurableSpace γ₁] (S : Causalean.PO.PODynLATESystem P γ₀ γ₁), S.D1 ≠ S.D2","l":133,"f":"Causalean/PO/ID/Exact/DynamicLATE/Setup.lean"},{"n":"Causalean.PO.PODynLATESystem.D1_ne_Y","k":"theorem","m":"Causalean.PO.ID.Exact.DynamicLATE.Setup","a":"PO/ID/Exact/DynamicLATE","t":2,"d":"Y is distinct from each treatment.","s":"∀ {P : Causalean.PO.POSystem} {γ₀ γ₁ : Type} [inst : MeasurableSpace γ₀] [inst_1 : MeasurableSpace γ₁] (S : Causalean.PO.PODynLATESystem P γ₀ γ₁), S.D1 ≠ S.Y","l":138,"f":"Causalean/PO/ID/Exact/DynamicLATE/Setup.lean"},{"n":"Causalean.PO.PODynLATESystem.D1ofZ","k":"def","m":"Causalean.PO.ID.Exact.DynamicLATE.Setup","a":"PO/ID/Exact/DynamicLATE","t":1,"d":"D₁(z) := D₁ evaluated under the encouragement regime fixing both Z's. By exclusion (D₁ has no Z₂ parent in the primitive process), this equals D₁(z 0); we keep the two-coordinate form to avoid splitting cases.","s":"{P : Causalean.PO.POSystem} → {γ₀ γ₁ : Type} → [inst : MeasurableSpace γ₀] → [inst_1 : MeasurableSpace γ₁] → Causalean.PO.PODynLATESystem P γ₀ γ₁ → (Fin 2 → Bool) → P.Ω → Bool","l":235,"f":"Causalean/PO/ID/Exact/DynamicLATE/Setup.lean"},{"n":"Causalean.PO.PODynLATESystem.D1ofZ_eq_factualD1_on_z1Event","k":"theorem","m":"Causalean.PO.ID.Exact.DynamicLATE.Consistency","a":"PO/ID/Exact/DynamicLATE","t":2,"d":"On {Z₁ = z 0}, D₁ under encouragementRegime z agrees with the factual D₁.","s":"∀ {P : Causalean.PO.POSystem} {γ₀ γ₁ : Type} [inst : MeasurableSpace γ₀] [inst_1 : MeasurableSpace γ₁] [inst_2 : StandardBorelSpace P.Ω] [inst_3 : MeasureTheory.IsFiniteMeasure P.μ] {S : Causalean.PO.PODynLATESystem P γ₀ γ₁}, S.Assumptions → ∀ (z : Fin 2 → Bool) {ω : P.Ω}, S.factualZ1 ω = z 0 → S.D1ofZ z ω = S.factualD1 ω","l":202,"f":"Causalean/PO/ID/Exact/DynamicLATE/Consistency.lean"},{"n":"Causalean.PO.PODynLATESystem.D2_ne_Y","k":"theorem","m":"Causalean.PO.ID.Exact.DynamicLATE.Setup","a":"PO/ID/Exact/DynamicLATE","t":2,"d":"The second treatment node is distinct from the outcome node.","s":"∀ {P : Causalean.PO.POSystem} {γ₀ γ₁ : Type} [inst : MeasurableSpace γ₀] [inst_1 : MeasurableSpace γ₁] (S : Causalean.PO.PODynLATESystem P γ₀ γ₁), S.D2 ≠ S.Y","l":143,"f":"Causalean/PO/ID/Exact/DynamicLATE/Setup.lean"},{"n":"Causalean.PO.PODynLATESystem.D2ofZ","k":"def","m":"Causalean.PO.ID.Exact.DynamicLATE.Setup","a":"PO/ID/Exact/DynamicLATE","t":1,"d":"D₂(z) under the encouragement regime fixing both Z's.","s":"{P : Causalean.PO.POSystem} → {γ₀ γ₁ : Type} → [inst : MeasurableSpace γ₀] → [inst_1 : MeasurableSpace γ₁] → Causalean.PO.PODynLATESystem P γ₀ γ₁ → (Fin 2 → Bool) → P.Ω → Bool","l":241,"f":"Causalean/PO/ID/Exact/DynamicLATE/Setup.lean"},{"n":"Causalean.PO.PODynLATESystem.D2ofZ2","k":"def","m":"Causalean.PO.ID.Exact.DynamicLATE.Setup","a":"PO/ID/Exact/DynamicLATE","t":1,"d":"D₂(Z₁, z₂): stage-2 treatment when only Z₂ is fixed (and Z₁ remains factual). Used in the stage-2 ignorability bundle.","s":"{P : Causalean.PO.POSystem} → {γ₀ γ₁ : Type} → [inst : MeasurableSpace γ₀] → [inst_1 : MeasurableSpace γ₁] → Causalean.PO.PODynLATESystem P γ₀ γ₁ → Bool → P.Ω → Bool","l":249,"f":"Causalean/PO/ID/Exact/DynamicLATE/Setup.lean"},{"n":"Causalean.PO.PODynLATESystem.D2ofZ2_eq_factualD2_on_z2Event","k":"theorem","m":"Causalean.PO.ID.Exact.DynamicLATE.Consistency","a":"PO/ID/Exact/DynamicLATE","t":2,"d":"On {Z₂ = z₂}, D₂(Z₁, z₂) = factualD2 pointwise.","s":"∀ {P : Causalean.PO.POSystem} {γ₀ γ₁ : Type} [inst : MeasurableSpace γ₀] [inst_1 : MeasurableSpace γ₁] [inst_2 : StandardBorelSpace P.Ω] [inst_3 : MeasureTheory.IsFiniteMeasure P.μ] {S : Causalean.PO.PODynLATESystem P γ₀ γ₁}, S.Assumptions → ∀ (z₂ : Bool) {ω : P.Ω}, S.factualZ2 ω = z₂ → S.D2ofZ2 z₂ ω = S.factualD2 ω","l":67,"f":"Causalean/PO/ID/Exact/DynamicLATE/Consistency.lean"},{"n":"Causalean.PO.PODynLATESystem.D2ofZ_eq_D2ofZ2_on_z1Event","k":"theorem","m":"Causalean.PO.ID.Exact.DynamicLATE.Consistency","a":"PO/ID/Exact/DynamicLATE","t":1,"d":"Second-stage treatment does not depend on first-stage regime detail once the actual first-period instrument matches the counterfactual one. At any outcome at which the factual first-period instrument equals the first coordinate of the instrument path z, the potential second-period treatment under the encouragement regime z coincides pointwise with the potential second-period treatment under the regime that fixes only the second-period instrument to z 1.","s":"∀ {P : Causalean.PO.POSystem} {γ₀ γ₁ : Type} [inst : MeasurableSpace γ₀] [inst_1 : MeasurableSpace γ₁] [inst_2 : StandardBorelSpace P.Ω] [inst_3 : MeasureTheory.IsFiniteMeasure P.μ] {S : Causalean.PO.PODynLATESystem P γ₀ γ₁}, S.Assumptions → ∀ (z : Fin 2 → Bool) {ω : P.Ω}, S.factualZ1 ω = z 0 → S.D2ofZ z ω = S.D2ofZ2 (z 1) ω","l":188,"f":"Causalean/PO/ID/Exact/DynamicLATE/Consistency.lean"},{"n":"Causalean.PO.PODynLATESystem.DofZ","k":"def","m":"Causalean.PO.ID.Exact.DynamicLATE.Setup","a":"PO/ID/Exact/DynamicLATE","t":1,"d":"Joint counterfactual treatment vector D(z) = (D₁(z), D₂(z)).","s":"{P : Causalean.PO.POSystem} → {γ₀ γ₁ : Type} → [inst : MeasurableSpace γ₀] → [inst_1 : MeasurableSpace γ₁] → Causalean.PO.PODynLATESystem P γ₀ γ₁ → (Fin 2 → Bool) → P.Ω → Fin 2 → Bool","l":245,"f":"Causalean/PO/ID/Exact/DynamicLATE/Setup.lean"},{"n":"Causalean.PO.PODynLATESystem.DofZEq","k":"def","m":"Causalean.PO.ID.Exact.DynamicLATE.Setup","a":"PO/ID/Exact/DynamicLATE","t":1,"d":"Indicator of the dynamic complier event {ω | D(z)(ω) = d}.","s":"{P : Causalean.PO.POSystem} → {γ₀ γ₁ : Type} → [inst : MeasurableSpace γ₀] → [inst_1 : MeasurableSpace γ₁] → Causalean.PO.PODynLATESystem P γ₀ γ₁ → (Fin 2 → Bool) → (Fin 2 → Bool) → Set P.Ω","l":460,"f":"Causalean/PO/ID/Exact/DynamicLATE/Setup.lean"},{"n":"Causalean.PO.PODynLATESystem.DofZ_in_two_values","k":"theorem","m":"Causalean.PO.ID.Exact.DynamicLATE.WhenToTreat","a":"PO/ID/Exact/DynamicLATE","t":2,"d":"For d ≼ z and one-sided noncompliance, D(z) ∈ {d, 0} a.s. when d is one of the two single-period treatment vectors (1,0) or (0,1), because the only d' ≼ z with d' ≠ d and d' ≠ 0 would have to flip a coordinate that one-sided rules out. Used to decompose Y(D(z)) on the encouragement event into the two-piece sum Y(d)·1_{D(z)=d} + Y(0)·1_{D(z)=0}.","s":"∀ {P : Causalean.PO.POSystem} {γ₀ γ₁ : Type} [inst : MeasurableSpace γ₀] [inst_1 : MeasurableSpace γ₁] [inst_2 : StandardBorelSpace P.Ω] [inst_3 : MeasureTheory.IsFiniteMeasure P.μ] {S : Causalean.PO.PODynLATESystem P γ₀ γ₁}, S.Assumptions → ∀ (d : Fin 2 → Bool), d = ![true, false] ∨ d = ![false, true] → ∀ᵐ (ω : P.Ω) ∂P.μ, S.DofZ d ω = d ∨ S.DofZ d ω = ![false, false]","l":181,"f":"Causalean/PO/ID/Exact/DynamicLATE/WhenToTreat.lean"},{"n":"Causalean.PO.PODynLATESystem.DofZ_zero_eq_zero","k":"theorem","m":"Causalean.PO.ID.Exact.DynamicLATE.WhenToTreat","a":"PO/ID/Exact/DynamicLATE","t":2,"d":"Under one-sided noncompliance, the zero-encouragement counterfactual treatment is identically zero a.s.: D(0) = 0 a.s.","s":"∀ {P : Causalean.PO.POSystem} {γ₀ γ₁ : Type} [inst : MeasurableSpace γ₀] [inst_1 : MeasurableSpace γ₁] [inst_2 : StandardBorelSpace P.Ω] [inst_3 : MeasureTheory.IsFiniteMeasure P.μ] {S : Causalean.PO.PODynLATESystem P γ₀ γ₁}, S.Assumptions → ∀ᵐ (ω : P.Ω) ∂P.μ, S.DofZ ![false, false] ω = ![false, false]","l":152,"f":"Causalean/PO/ID/Exact/DynamicLATE/WhenToTreat.lean"},{"n":"Causalean.PO.PODynLATESystem.LATE","k":"def","m":"Causalean.PO.ID.Exact.DynamicLATE.Setup","a":"PO/ID/Exact/DynamicLATE","t":1,"d":"Dynamic LATE θ(z, d) := E[Y(d) - Y(0) | D(z) = d], totalised as (∫_{D(z)=d} (Y(d) - Y(0)) dμ) / μ({D(z) = d}).toReal.","s":"{P : Causalean.PO.POSystem} → {γ₀ γ₁ : Type} → [inst : MeasurableSpace γ₀] → [inst_1 : MeasurableSpace γ₁] → Causalean.PO.PODynLATESystem P γ₀ γ₁ → (Fin 2 → Bool) → (Fin 2 → Bool) → ℝ","l":468,"f":"Causalean/PO/ID/Exact/DynamicLATE/Setup.lean"},{"n":"Causalean.PO.PODynLATESystem.Preceq","k":"def","m":"Causalean.PO.ID.Exact.DynamicLATE.Setup","a":"PO/ID/Exact/DynamicLATE","t":1,"d":"Coordinate-wise order d ≼ z: d 0 ≤ z 0 ∧ d 1 ≤ z 1.","s":"(Fin 2 → Bool) → (Fin 2 → Bool) → Prop","l":504,"f":"Causalean/PO/ID/Exact/DynamicLATE/Setup.lean"},{"n":"Causalean.PO.PODynLATESystem.YofD","k":"def","m":"Causalean.PO.ID.Exact.DynamicLATE.Setup","a":"PO/ID/Exact/DynamicLATE","t":1,"d":"Y(d) under the treatment regime fixing both D's.","s":"{P : Causalean.PO.POSystem} → {γ₀ γ₁ : Type} → [inst : MeasurableSpace γ₀] → [inst_1 : MeasurableSpace γ₁] → Causalean.PO.PODynLATESystem P γ₀ γ₁ → (Fin 2 → Bool) → P.Ω → ℝ","l":254,"f":"Causalean/PO/ID/Exact/DynamicLATE/Setup.lean"},{"n":"Causalean.PO.PODynLATESystem.YofDofZ","k":"def","m":"Causalean.PO.ID.Exact.DynamicLATE.Setup","a":"PO/ID/Exact/DynamicLATE","t":1,"d":"Y(D(z)) defined directly via the encouragement regime: under the two-target encouragement intervention, D₁ and D₂ are computed from z via the structural recursion, and Y is then computed from the resulting treatment vector. This is the natural \"outcome under encouragement z\" map; composition consistency identifies it with the explicit composition.","s":"{P : Causalean.PO.POSystem} → {γ₀ γ₁ : Type} → [inst : MeasurableSpace γ₀] → [inst_1 : MeasurableSpace γ₁] → Causalean.PO.PODynLATESystem P γ₀ γ₁ → (Fin 2 → Bool) → P.Ω → ℝ","l":258,"f":"Causalean/PO/ID/Exact/DynamicLATE/Setup.lean"},{"n":"Causalean.PO.PODynLATESystem.YofDofZ_decomposition","k":"theorem","m":"Causalean.PO.ID.Exact.DynamicLATE.WhenToTreat","a":"PO/ID/Exact/DynamicLATE","t":2,"d":"Under one-sided noncompliance, the encouragement-regime outcome under encouragement d ∈ {(1,0), (0,1)} decomposes pointwise as Y(d)·1_{D(d)=d} + Y(0)·1_{D(d)=0} a.s.","s":"∀ {P : Causalean.PO.POSystem} {γ₀ γ₁ : Type} [inst : MeasurableSpace γ₀] [inst_1 : MeasurableSpace γ₁] [MeasurableSingletonClass γ₀] [MeasurableSingletonClass γ₁] [inst_4 : StandardBorelSpace P.Ω] [inst_5 : MeasureTheory.IsFiniteMeasure P.μ] {S : Causalean.PO.PODynLATESystem P γ₀ γ₁}, S.Assumptions → ∀ (d : Fin 2 → Bool), d = ![true, false] ∨ d = ![false, true] → S.YofDofZ d =ᵐ[P.μ] fun ω => S.YofD d ω * (S.DofZEq d d).indicator (fun x => 1) ω + S.YofD ![false, false] ω * (S.DofZEq d ![false, false]).indicator (fun x => 1) ω","l":239,"f":"Causalean/PO/ID/Exact/DynamicLATE/WhenToTreat.lean"},{"n":"Causalean.PO.PODynLATESystem.YofDofZ_eq_YofD_on_DofZEq","k":"theorem","m":"Causalean.PO.ID.Exact.DynamicLATE.WhenToTreat","a":"PO/ID/Exact/DynamicLATE","t":2,"d":"On the event {D(z) = d}, the encouragement-regime outcome Y(D(z)) agrees pointwise with the treatment-regime outcome Y(d).","s":"∀ {P : Causalean.PO.POSystem} {γ₀ γ₁ : Type} [inst : MeasurableSpace γ₀] [inst_1 : MeasurableSpace γ₁] [MeasurableSingletonClass γ₀] [MeasurableSingletonClass γ₁] [inst_4 : StandardBorelSpace P.Ω] [inst_5 : MeasureTheory.IsFiniteMeasure P.μ] {S : Causalean.PO.PODynLATESystem P γ₀ γ₁}, S.Assumptions → ∀ (z d : Fin 2 → Bool), ∀ᵐ (ω : P.Ω) ∂P.μ, S.DofZ z ω = d → S.YofDofZ z ω = S.YofD d ω","l":80,"f":"Causalean/PO/ID/Exact/DynamicLATE/WhenToTreat.lean"},{"n":"Causalean.PO.PODynLATESystem.YofDofZ_eq_YofZ2_on_z1Event","k":"theorem","m":"Causalean.PO.ID.Exact.DynamicLATE.Consistency","a":"PO/ID/Exact/DynamicLATE","t":2,"d":"On {Z₁ = z 0}, Y under encouragementRegime z agrees with Y under encZ2Regime (z 1).","s":"∀ {P : Causalean.PO.POSystem} {γ₀ γ₁ : Type} [inst : MeasurableSpace γ₀] [inst_1 : MeasurableSpace γ₁] [inst_2 : StandardBorelSpace P.Ω] [inst_3 : MeasureTheory.IsFiniteMeasure P.μ] {S : Causalean.PO.PODynLATESystem P γ₀ γ₁}, S.Assumptions → ∀ (z : Fin 2 → Bool) {ω : P.Ω}, S.factualZ1 ω = z 0 → S.YofDofZ z ω = S.YofZ2 (z 1) ω","l":179,"f":"Causalean/PO/ID/Exact/DynamicLATE/Consistency.lean"},{"n":"Causalean.PO.PODynLATESystem.YofDofZ_zero_ae_eq_YofD_zero","k":"theorem","m":"Causalean.PO.ID.Exact.DynamicLATE.WhenToTreat","a":"PO/ID/Exact/DynamicLATE","t":2,"d":"A.s. Y(D(0)) = Y(0), the zero-encouragement composition collapse.","s":"∀ {P : Causalean.PO.POSystem} {γ₀ γ₁ : Type} [inst : MeasurableSpace γ₀] [inst_1 : MeasurableSpace γ₁] [MeasurableSingletonClass γ₀] [MeasurableSingletonClass γ₁] [inst_4 : StandardBorelSpace P.Ω] [inst_5 : MeasureTheory.IsFiniteMeasure P.μ] {S : Causalean.PO.PODynLATESystem P γ₀ γ₁}, S.Assumptions → S.YofDofZ ![false, false] =ᵐ[P.μ] S.YofD ![false, false]","l":173,"f":"Causalean/PO/ID/Exact/DynamicLATE/WhenToTreat.lean"},{"n":"Causalean.PO.PODynLATESystem.YofZ2","k":"def","m":"Causalean.PO.ID.Exact.DynamicLATE.Setup","a":"PO/ID/Exact/DynamicLATE","t":1,"d":"Y(D₁, D₂(Z₁, z₂)) realised as Y under the regime fixing only Z₂ = z₂.","s":"{P : Causalean.PO.POSystem} → {γ₀ γ₁ : Type} → [inst : MeasurableSpace γ₀] → [inst_1 : MeasurableSpace γ₁] → Causalean.PO.PODynLATESystem P γ₀ γ₁ → Bool → P.Ω → ℝ","l":266,"f":"Causalean/PO/ID/Exact/DynamicLATE/Setup.lean"},{"n":"Causalean.PO.PODynLATESystem.YofZ2_eq_factualY_on_z2Event","k":"theorem","m":"Causalean.PO.ID.Exact.DynamicLATE.Consistency","a":"PO/ID/Exact/DynamicLATE","t":2,"d":"On {Z₂ = z₂}, Y(D₁, D₂(Z₁, z₂)) = factualY pointwise.","s":"∀ {P : Causalean.PO.POSystem} {γ₀ γ₁ : Type} [inst : MeasurableSpace γ₀] [inst_1 : MeasurableSpace γ₁] [inst_2 : StandardBorelSpace P.Ω] [inst_3 : MeasureTheory.IsFiniteMeasure P.μ] {S : Causalean.PO.PODynLATESystem P γ₀ γ₁}, S.Assumptions → ∀ (z₂ : Bool) {ω : P.Ω}, S.factualZ2 ω = z₂ → S.YofZ2 z₂ ω = S.factualY ω","l":58,"f":"Causalean/PO/ID/Exact/DynamicLATE/Consistency.lean"},{"n":"Causalean.PO.PODynLATESystem.Z1_ne_D1","k":"theorem","m":"Causalean.PO.ID.Exact.DynamicLATE.Setup","a":"PO/ID/Exact/DynamicLATE","t":2,"d":"Encouragements are distinct from treatments.","s":"∀ {P : Causalean.PO.POSystem} {γ₀ γ₁ : Type} [inst : MeasurableSpace γ₀] [inst_1 : MeasurableSpace γ₁] (S : Causalean.PO.PODynLATESystem P γ₀ γ₁), S.Z1 ≠ S.D1","l":158,"f":"Causalean/PO/ID/Exact/DynamicLATE/Setup.lean"},{"n":"Causalean.PO.PODynLATESystem.Z1_ne_D2","k":"theorem","m":"Causalean.PO.ID.Exact.DynamicLATE.Setup","a":"PO/ID/Exact/DynamicLATE","t":2,"d":"The first encouragement node is distinct from the second treatment node.","s":"∀ {P : Causalean.PO.POSystem} {γ₀ γ₁ : Type} [inst : MeasurableSpace γ₀] [inst_1 : MeasurableSpace γ₁] (S : Causalean.PO.PODynLATESystem P γ₀ γ₁), S.Z1 ≠ S.D2","l":163,"f":"Causalean/PO/ID/Exact/DynamicLATE/Setup.lean"},{"n":"Causalean.PO.PODynLATESystem.Z1_ne_Y","k":"theorem","m":"Causalean.PO.ID.Exact.DynamicLATE.Setup","a":"PO/ID/Exact/DynamicLATE","t":2,"d":"Y is distinct from each encouragement.","s":"∀ {P : Causalean.PO.POSystem} {γ₀ γ₁ : Type} [inst : MeasurableSpace γ₀] [inst_1 : MeasurableSpace γ₁] (S : Causalean.PO.PODynLATESystem P γ₀ γ₁), S.Z1 ≠ S.Y","l":148,"f":"Causalean/PO/ID/Exact/DynamicLATE/Setup.lean"},{"n":"Causalean.PO.PODynLATESystem.Z1_ne_Z2","k":"theorem","m":"Causalean.PO.ID.Exact.DynamicLATE.Setup","a":"PO/ID/Exact/DynamicLATE","t":2,"d":"The most-used corollary: the two encouragement nodes are distinct.","s":"∀ {P : Causalean.PO.POSystem} {γ₀ γ₁ : Type} [inst : MeasurableSpace γ₀] [inst_1 : MeasurableSpace γ₁] (S : Causalean.PO.PODynLATESystem P γ₀ γ₁), S.Z1 ≠ S.Z2","l":128,"f":"Causalean/PO/ID/Exact/DynamicLATE/Setup.lean"},{"n":"Causalean.PO.PODynLATESystem.Z2_ne_D1","k":"theorem","m":"Causalean.PO.ID.Exact.DynamicLATE.Setup","a":"PO/ID/Exact/DynamicLATE","t":2,"d":"The second encouragement node is distinct from the first treatment node.","s":"∀ {P : Causalean.PO.POSystem} {γ₀ γ₁ : Type} [inst : MeasurableSpace γ₀] [inst_1 : MeasurableSpace γ₁] (S : Causalean.PO.PODynLATESystem P γ₀ γ₁), S.Z2 ≠ S.D1","l":168,"f":"Causalean/PO/ID/Exact/DynamicLATE/Setup.lean"},{"n":"Causalean.PO.PODynLATESystem.Z2_ne_D2","k":"theorem","m":"Causalean.PO.ID.Exact.DynamicLATE.Setup","a":"PO/ID/Exact/DynamicLATE","t":2,"d":"The second encouragement node is distinct from the second treatment node.","s":"∀ {P : Causalean.PO.POSystem} {γ₀ γ₁ : Type} [inst : MeasurableSpace γ₀] [inst_1 : MeasurableSpace γ₁] (S : Causalean.PO.PODynLATESystem P γ₀ γ₁), S.Z2 ≠ S.D2","l":173,"f":"Causalean/PO/ID/Exact/DynamicLATE/Setup.lean"},{"n":"Causalean.PO.PODynLATESystem.Z2_ne_Y","k":"theorem","m":"Causalean.PO.ID.Exact.DynamicLATE.Setup","a":"PO/ID/Exact/DynamicLATE","t":2,"d":"The second encouragement node is distinct from the outcome node.","s":"∀ {P : Causalean.PO.POSystem} {γ₀ γ₁ : Type} [inst : MeasurableSpace γ₀] [inst_1 : MeasurableSpace γ₁] (S : Causalean.PO.PODynLATESystem P γ₀ γ₁), S.Z2 ≠ S.Y","l":153,"f":"Causalean/PO/ID/Exact/DynamicLATE/Setup.lean"},{"n":"Causalean.PO.PODynLATESystem.cCompliance_bridge","k":"theorem","m":"Causalean.PO.ID.Exact.DynamicLATE.Bridges","a":"PO/ID/Exact/DynamicLATE","t":1,"d":"Compliance bridge (S₀-conditional). Under the dynamic LATE identifying assumptions, for an encouragement vector z and a treatment path d with d weakly dominated coordinatewise by z, the baseline-conditional probability of the dynamic complier event D(z) = d agrees almost surely with the inner-outer observable regression cObsProb z d.","s":"∀ {P : Causalean.PO.POSystem} {γ₀ γ₁ : Type} [inst : MeasurableSpace γ₀] [inst_1 : MeasurableSpace γ₁] [MeasurableSingletonClass γ₀] [MeasurableSingletonClass γ₁] [inst_4 : StandardBorelSpace P.Ω] [inst_5 : MeasureTheory.IsFiniteMeasure P.μ] {S : Causalean.PO.PODynLATESystem P γ₀ γ₁}, S.Assumptions → ∀ (z d : Fin 2 → Bool), Causalean.PO.PODynLATESystem.Preceq d z → S.historyBundle1.condExpGiven ((S.DofZEq z d).indicator fun x => 1) =ᵐ[P.μ] S.cObsProb z d","l":547,"f":"Causalean/PO/ID/Exact/DynamicLATE/Bridges.lean"},{"n":"Causalean.PO.PODynLATESystem.cLATE","k":"def","m":"Causalean.PO.ID.Exact.DynamicLATE.Setup","a":"PO/ID/Exact/DynamicLATE","t":1,"d":"Heterogeneous dynamic LATE θ(z, d, S₀): the bundle conditional version of LATE z d, realised as historyBundle1.condExpRatio of the indicator-weighted contrast.","s":"{P : Causalean.PO.POSystem} → {γ₀ γ₁ : Type} → [inst : MeasurableSpace γ₀] → [inst_1 : MeasurableSpace γ₁] → Causalean.PO.PODynLATESystem P γ₀ γ₁ → (Fin 2 → Bool) → (Fin 2 → Bool) → P.Ω → ℝ","l":474,"f":"Causalean/PO/ID/Exact/DynamicLATE/Setup.lean"},{"n":"Causalean.PO.PODynLATESystem.cMixtureLATE","k":"def","m":"Causalean.PO.ID.Exact.DynamicLATE.Setup","a":"PO/ID/Exact/DynamicLATE","t":1,"d":"Heterogeneous mixture LATE β_z(S₀).","s":"{P : Causalean.PO.POSystem} → {γ₀ γ₁ : Type} → [inst : MeasurableSpace γ₀] → [inst_1 : MeasurableSpace γ₁] → Causalean.PO.PODynLATESystem P γ₀ γ₁ → (Fin 2 → Bool) → P.Ω → ℝ","l":495,"f":"Causalean/PO/ID/Exact/DynamicLATE/Setup.lean"},{"n":"Causalean.PO.PODynLATESystem.cMixtureLATE_wald","k":"theorem","m":"Causalean.PO.ID.Exact.DynamicLATE.WhenToTreat","a":"PO/ID/Exact/DynamicLATE","t":1,"d":"Mixture dynamic LATE Wald identity (heterogeneous in S₀). Under the dynamic LATE identifying assumptions, for an encouragement vector z, the baseline-conditional mixture dynamic LATE cMixtureLATE z agrees almost surely with the ratio of the baseline-conditional mean contrast cObsMean z − cObsMean(0,0) to the baseline-conditional noncompliance probability 1 − cObsProb(z, 0,0).","s":"∀ {P : Causalean.PO.POSystem} {γ₀ γ₁ : Type} [inst : MeasurableSpace γ₀] [inst_1 : MeasurableSpace γ₁] [MeasurableSingletonClass γ₀] [MeasurableSingletonClass γ₁] [inst_4 : StandardBorelSpace P.Ω] [inst_5 : MeasureTheory.IsFiniteMeasure P.μ] {S : Causalean.PO.PODynLATESystem P γ₀ γ₁}, S.Assumptions → ∀ (z : Fin 2 → Bool), S.cMixtureLATE z =ᵐ[P.μ] fun ω => (S.cObsMean z ω - S.cObsMean ![false, false] ω) / (1 - S.cObsProb z ![false, false] ω)","l":524,"f":"Causalean/PO/ID/Exact/DynamicLATE/WhenToTreat.lean"},{"n":"Causalean.PO.PODynLATESystem.cObsMean","k":"def","m":"Causalean.PO.ID.Exact.DynamicLATE.Setup","a":"PO/ID/Exact/DynamicLATE","t":1,"d":"Outer regression of innerCondY z over (S₀, Z₁ = z₁), as a function of S₀. This is cObsMean(z; S₀).","s":"{P : Causalean.PO.POSystem} → {γ₀ γ₁ : Type} → [inst : MeasurableSpace γ₀] → [inst_1 : MeasurableSpace γ₁] → Causalean.PO.PODynLATESystem P γ₀ γ₁ → (Fin 2 → Bool) → P.Ω → ℝ","l":439,"f":"Causalean/PO/ID/Exact/DynamicLATE/Setup.lean"},{"n":"Causalean.PO.PODynLATESystem.cObsProb","k":"def","m":"Causalean.PO.ID.Exact.DynamicLATE.Setup","a":"PO/ID/Exact/DynamicLATE","t":1,"d":"Outer regression of innerCondD z d over (S₀, Z₁ = z₁).","s":"{P : Causalean.PO.POSystem} → {γ₀ γ₁ : Type} → [inst : MeasurableSpace γ₀] → [inst_1 : MeasurableSpace γ₁] → Causalean.PO.PODynLATESystem P γ₀ γ₁ → (Fin 2 → Bool) → (Fin 2 → Bool) → P.Ω → ℝ","l":446,"f":"Causalean/PO/ID/Exact/DynamicLATE/Setup.lean"},{"n":"Causalean.PO.PODynLATESystem.cOutcome_bridge","k":"theorem","m":"Causalean.PO.ID.Exact.DynamicLATE.Bridges","a":"PO/ID/Exact/DynamicLATE","t":1,"d":"Outcome bridge (S₀-conditional). Under the dynamic LATE identifying assumptions, the baseline-conditional expectation of the counterfactual outcome Y(D(z)) for an encouragement vector z agrees almost surely with the inner-outer observable regression cObsMean z.","s":"∀ {P : Causalean.PO.POSystem} {γ₀ γ₁ : Type} [inst : MeasurableSpace γ₀] [inst_1 : MeasurableSpace γ₁] [MeasurableSingletonClass γ₀] [MeasurableSingletonClass γ₁] [inst_4 : StandardBorelSpace P.Ω] [inst_5 : MeasureTheory.IsFiniteMeasure P.μ] {S : Causalean.PO.PODynLATESystem P γ₀ γ₁}, S.Assumptions → ∀ (z d : Fin 2 → Bool), Causalean.PO.PODynLATESystem.Preceq d z → S.historyBundle1.condExpGiven (S.YofDofZ z) =ᵐ[P.μ] S.cObsMean z","l":410,"f":"Causalean/PO/ID/Exact/DynamicLATE/Bridges.lean"},{"n":"Causalean.PO.PODynLATESystem.cWhenToTreatLATE","k":"def","m":"Causalean.PO.ID.Exact.DynamicLATE.Setup","a":"PO/ID/Exact/DynamicLATE","t":1,"d":"Heterogeneous when-to-treat LATE τ_d(S₀).","s":"{P : Causalean.PO.POSystem} → {γ₀ γ₁ : Type} → [inst : MeasurableSpace γ₀] → [inst_1 : MeasurableSpace γ₁] → Causalean.PO.PODynLATESystem P γ₀ γ₁ → (Fin 2 → Bool) → P.Ω → ℝ","l":486,"f":"Causalean/PO/ID/Exact/DynamicLATE/Setup.lean"},{"n":"Causalean.PO.PODynLATESystem.cWhenToTreat_wald","k":"theorem","m":"Causalean.PO.ID.Exact.DynamicLATE.WhenToTreat","a":"PO/ID/Exact/DynamicLATE","t":1,"d":"When-to-treat dynamic LATE Wald identity (heterogeneous in S₀). For a treatment path d equal to (1,0) or (0,1), the baseline-covariate-conditional when-to-treat effect τ_d(S₀) agrees almost surely with the ratio of the conditional observable-mean contrast cObsMean(d;S₀) − cObsMean(0;S₀) to the conditional observable compliance probability cObsProb(d,d;S₀).","s":"∀ {P : Causalean.PO.POSystem} {γ₀ γ₁ : Type} [inst : MeasurableSpace γ₀] [inst_1 : MeasurableSpace γ₁] [MeasurableSingletonClass γ₀] [MeasurableSingletonClass γ₁] [inst_4 : StandardBorelSpace P.Ω] [inst_5 : MeasureTheory.IsFiniteMeasure P.μ] {S : Causalean.PO.PODynLATESystem P γ₀ γ₁}, S.Assumptions → ∀ (d : Fin 2 → Bool), d = ![true, false] ∨ d = ![false, true] → S.cWhenToTreatLATE d =ᵐ[P.μ] fun ω => (S.cObsMean d ω - S.cObsMean ![false, false] ω) / S.cObsProb d d ω","l":442,"f":"Causalean/PO/ID/Exact/DynamicLATE/WhenToTreat.lean"},{"n":"Causalean.PO.PODynLATESystem.cfBundle1","k":"def","m":"Causalean.PO.ID.Exact.DynamicLATE.Setup","a":"PO/ID/Exact/DynamicLATE","t":1,"d":"Stage-1 ignorability bundle (Y(D(z)), D₁(z), D₂(z)), the counterfactual target of Z₁ ⟂ · | S₀.","s":"{P : Causalean.PO.POSystem} → {γ₀ γ₁ : Type} → [inst : MeasurableSpace γ₀] → [inst_1 : MeasurableSpace γ₁] → Causalean.PO.PODynLATESystem P γ₀ γ₁ → (Fin 2 → Bool) → Causalean.PO.POCFBundle P","l":389,"f":"Causalean/PO/ID/Exact/DynamicLATE/Setup.lean"},{"n":"Causalean.PO.PODynLATESystem.cfBundle2","k":"def","m":"Causalean.PO.ID.Exact.DynamicLATE.Setup","a":"PO/ID/Exact/DynamicLATE","t":1,"d":"Stage-2 ignorability bundle (Y(D₁, D₂(Z₁, z₂)), D₂(Z₁, z₂)), the counterfactual target of Z₂ ⟂ · | S, D₁, Z₁.","s":"{P : Causalean.PO.POSystem} → {γ₀ γ₁ : Type} → [inst : MeasurableSpace γ₀] → [inst_1 : MeasurableSpace γ₁] → Causalean.PO.PODynLATESystem P γ₀ γ₁ → Bool → Causalean.PO.POCFBundle P","l":397,"f":"Causalean/PO/ID/Exact/DynamicLATE/Setup.lean"},{"n":"Causalean.PO.PODynLATESystem.compliance_bridge","k":"theorem","m":"Causalean.PO.ID.Exact.DynamicLATE.Bridges","a":"PO/ID/Exact/DynamicLATE","t":1,"d":"Compliance bridge (unconditional). Given a treatment path d that is coordinatewise no greater than the instrument path z, the probability of the counterfactual dynamic complier event that the treatment path realized under instrument assignment z equals d coincides with the observable nested regression obsProb z d.","s":"∀ {P : Causalean.PO.POSystem} {γ₀ γ₁ : Type} [inst : MeasurableSpace γ₀] [inst_1 : MeasurableSpace γ₁] [MeasurableSingletonClass γ₀] [MeasurableSingletonClass γ₁] [inst_4 : StandardBorelSpace P.Ω] [inst_5 : MeasureTheory.IsFiniteMeasure P.μ] {S : Causalean.PO.PODynLATESystem P γ₀ γ₁}, S.Assumptions → ∀ (z d : Fin 2 → Bool), Causalean.PO.PODynLATESystem.Preceq d z → (P.μ (S.DofZEq z d)).toReal = S.obsProb z d","l":814,"f":"Causalean/PO/ID/Exact/DynamicLATE/Bridges.lean"},{"n":"Causalean.PO.PODynLATESystem.d1UnderZ","k":"def","m":"Causalean.PO.ID.Exact.DynamicLATE.Setup","a":"PO/ID/Exact/DynamicLATE","t":1,"d":"D₁ under the two-target encouragement regime, as a RegimedVar.","s":"{P : Causalean.PO.POSystem} → {γ₀ γ₁ : Type} → [inst : MeasurableSpace γ₀] → [inst_1 : MeasurableSpace γ₁] → Causalean.PO.PODynLATESystem P γ₀ γ₁ → (Fin 2 → Bool) → Causalean.PO.RegimedVar P Bool","l":353,"f":"Causalean/PO/ID/Exact/DynamicLATE/Setup.lean"},{"n":"Causalean.PO.PODynLATESystem.d1Var","k":"def","m":"Causalean.PO.ID.Exact.DynamicLATE.Setup","a":"PO/ID/Exact/DynamicLATE","t":1,"d":"The first treatment is packaged as a binary potential-outcome variable.","s":"{P : Causalean.PO.POSystem} → {γ₀ γ₁ : Type} → [inst : MeasurableSpace γ₀] → [inst_1 : MeasurableSpace γ₁] → Causalean.PO.PODynLATESystem P γ₀ γ₁ → Causalean.PO.POVar P Bool","l":93,"f":"Causalean/PO/ID/Exact/DynamicLATE/Setup.lean"},{"n":"Causalean.PO.PODynLATESystem.d2UnderZ","k":"def","m":"Causalean.PO.ID.Exact.DynamicLATE.Setup","a":"PO/ID/Exact/DynamicLATE","t":1,"d":"D₂ under the two-target encouragement regime, as a RegimedVar.","s":"{P : Causalean.PO.POSystem} → {γ₀ γ₁ : Type} → [inst : MeasurableSpace γ₀] → [inst_1 : MeasurableSpace γ₁] → Causalean.PO.PODynLATESystem P γ₀ γ₁ → (Fin 2 → Bool) → Causalean.PO.RegimedVar P Bool","l":357,"f":"Causalean/PO/ID/Exact/DynamicLATE/Setup.lean"},{"n":"Causalean.PO.PODynLATESystem.d2UnderZ2","k":"def","m":"Causalean.PO.ID.Exact.DynamicLATE.Setup","a":"PO/ID/Exact/DynamicLATE","t":1,"d":"D₂ under the stage-2-only encouragement regime.","s":"{P : Causalean.PO.POSystem} → {γ₀ γ₁ : Type} → [inst : MeasurableSpace γ₀] → [inst_1 : MeasurableSpace γ₁] → Causalean.PO.PODynLATESystem P γ₀ γ₁ → Bool → Causalean.PO.RegimedVar P Bool","l":365,"f":"Causalean/PO/ID/Exact/DynamicLATE/Setup.lean"},{"n":"Causalean.PO.PODynLATESystem.d2Var","k":"def","m":"Causalean.PO.ID.Exact.DynamicLATE.Setup","a":"PO/ID/Exact/DynamicLATE","t":1,"d":"The second treatment is packaged as a binary potential-outcome variable.","s":"{P : Causalean.PO.POSystem} → {γ₀ γ₁ : Type} → [inst : MeasurableSpace γ₀] → [inst_1 : MeasurableSpace γ₁] → Causalean.PO.PODynLATESystem P γ₀ γ₁ → Causalean.PO.POVar P Bool","l":95,"f":"Causalean/PO/ID/Exact/DynamicLATE/Setup.lean"},{"n":"Causalean.PO.PODynLATESystem.encTreatRegime","k":"def","m":"Causalean.PO.ID.Exact.DynamicLATE.Setup","a":"PO/ID/Exact/DynamicLATE","t":1,"d":"Joint encouragement-and-treatment regime r_z ⊔ r_d fixing Z₁,Z₂,D₁,D₂ simultaneously.","s":"{P : Causalean.PO.POSystem} → {γ₀ γ₁ : Type} → [inst : MeasurableSpace γ₀] → [inst_1 : MeasurableSpace γ₁] → Causalean.PO.PODynLATESystem P γ₀ γ₁ → (Fin 2 → Bool) → (Fin 2 → Bool) → Causalean.PO.Regime P.V P.X","l":227,"f":"Causalean/PO/ID/Exact/DynamicLATE/Setup.lean"},{"n":"Causalean.PO.PODynLATESystem.encZ2Regime","k":"def","m":"Causalean.PO.ID.Exact.DynamicLATE.Setup","a":"PO/ID/Exact/DynamicLATE","t":1,"d":"Regime fixing only the stage-2 encouragement: Z₂ ↦ z₂. Used in the stage-2 ignorability condition where Z₁ remains factual.","s":"{P : Causalean.PO.POSystem} → {γ₀ γ₁ : Type} → [inst : MeasurableSpace γ₀] → [inst_1 : MeasurableSpace γ₁] → Causalean.PO.PODynLATESystem P γ₀ γ₁ → Bool → Causalean.PO.Regime P.V P.X","l":198,"f":"Causalean/PO/ID/Exact/DynamicLATE/Setup.lean"},{"n":"Causalean.PO.PODynLATESystem.encouragementRegime","k":"def","m":"Causalean.PO.ID.Exact.DynamicLATE.Setup","a":"PO/ID/Exact/DynamicLATE","t":1,"d":"Regime fixing both encouragements: Z₁ ↦ z 0, Z₂ ↦ z 1.","s":"{P : Causalean.PO.POSystem} → {γ₀ γ₁ : Type} → [inst : MeasurableSpace γ₀] → [inst_1 : MeasurableSpace γ₁] → Causalean.PO.PODynLATESystem P γ₀ γ₁ → (Fin 2 → Bool) → Causalean.PO.Regime P.V P.X","l":180,"f":"Causalean/PO/ID/Exact/DynamicLATE/Setup.lean"},{"n":"Causalean.PO.PODynLATESystem.encouragementRegime_disjoint_treatmentRegime","k":"theorem","m":"Causalean.PO.ID.Exact.DynamicLATE.Setup","a":"PO/ID/Exact/DynamicLATE","t":1,"d":"For an encouragement vector z and a treatment path d, the encouragement regime fixing Z₁, Z₂ to z and the treatment regime fixing D₁, D₂ to d target disjoint sets of variables.","s":"∀ {P : Causalean.PO.POSystem} {γ₀ γ₁ : Type} [inst : MeasurableSpace γ₀] [inst_1 : MeasurableSpace γ₁] (S : Causalean.PO.PODynLATESystem P γ₀ γ₁) (z d : Fin 2 → Bool), (S.encouragementRegime z).Disjoint (S.treatmentRegime d)","l":203,"f":"Causalean/PO/ID/Exact/DynamicLATE/Setup.lean"},{"n":"Causalean.PO.PODynLATESystem.factualD1","k":"def","m":"Causalean.PO.ID.Exact.DynamicLATE.Setup","a":"PO/ID/Exact/DynamicLATE","t":1,"d":"The factual first treatment is the observed first-stage treatment value.","s":"{P : Causalean.PO.POSystem} → {γ₀ γ₁ : Type} → [inst : MeasurableSpace γ₀] → [inst_1 : MeasurableSpace γ₁] → Causalean.PO.PODynLATESystem P γ₀ γ₁ → P.Ω → Bool","l":108,"f":"Causalean/PO/ID/Exact/DynamicLATE/Setup.lean"},{"n":"Causalean.PO.PODynLATESystem.factualD2","k":"def","m":"Causalean.PO.ID.Exact.DynamicLATE.Setup","a":"PO/ID/Exact/DynamicLATE","t":1,"d":"The factual second treatment is the observed second-stage treatment value.","s":"{P : Causalean.PO.POSystem} → {γ₀ γ₁ : Type} → [inst : MeasurableSpace γ₀] → [inst_1 : MeasurableSpace γ₁] → Causalean.PO.PODynLATESystem P γ₀ γ₁ → P.Ω → Bool","l":110,"f":"Causalean/PO/ID/Exact/DynamicLATE/Setup.lean"},{"n":"Causalean.PO.PODynLATESystem.factualS0","k":"def","m":"Causalean.PO.ID.Exact.DynamicLATE.Setup","a":"PO/ID/Exact/DynamicLATE","t":1,"d":"The factual baseline state is the observed baseline covariate value.","s":"{P : Causalean.PO.POSystem} → {γ₀ γ₁ : Type} → [inst : MeasurableSpace γ₀] → [inst_1 : MeasurableSpace γ₁] → Causalean.PO.PODynLATESystem P γ₀ γ₁ → P.Ω → γ₀","l":100,"f":"Causalean/PO/ID/Exact/DynamicLATE/Setup.lean"},{"n":"Causalean.PO.PODynLATESystem.factualS1","k":"def","m":"Causalean.PO.ID.Exact.DynamicLATE.Setup","a":"PO/ID/Exact/DynamicLATE","t":1,"d":"The factual intermediate state is the observed intermediate covariate value.","s":"{P : Causalean.PO.POSystem} → {γ₀ γ₁ : Type} → [inst : MeasurableSpace γ₀] → [inst_1 : MeasurableSpace γ₁] → Causalean.PO.PODynLATESystem P γ₀ γ₁ → P.Ω → γ₁","l":102,"f":"Causalean/PO/ID/Exact/DynamicLATE/Setup.lean"},{"n":"Causalean.PO.PODynLATESystem.factualY","k":"def","m":"Causalean.PO.ID.Exact.DynamicLATE.Setup","a":"PO/ID/Exact/DynamicLATE","t":1,"d":"The factual outcome is the observed outcome value.","s":"{P : Causalean.PO.POSystem} → {γ₀ γ₁ : Type} → [inst : MeasurableSpace γ₀] → [inst_1 : MeasurableSpace γ₁] → Causalean.PO.PODynLATESystem P γ₀ γ₁ → P.Ω → ℝ","l":112,"f":"Causalean/PO/ID/Exact/DynamicLATE/Setup.lean"},{"n":"Causalean.PO.PODynLATESystem.factualZ1","k":"def","m":"Causalean.PO.ID.Exact.DynamicLATE.Setup","a":"PO/ID/Exact/DynamicLATE","t":1,"d":"The factual first encouragement is the observed first-stage instrument value.","s":"{P : Causalean.PO.POSystem} → {γ₀ γ₁ : Type} → [inst : MeasurableSpace γ₀] → [inst_1 : MeasurableSpace γ₁] → Causalean.PO.PODynLATESystem P γ₀ γ₁ → P.Ω → Bool","l":104,"f":"Causalean/PO/ID/Exact/DynamicLATE/Setup.lean"},{"n":"Causalean.PO.PODynLATESystem.factualZ2","k":"def","m":"Causalean.PO.ID.Exact.DynamicLATE.Setup","a":"PO/ID/Exact/DynamicLATE","t":1,"d":"The factual second encouragement is the observed second-stage instrument value.","s":"{P : Causalean.PO.POSystem} → {γ₀ γ₁ : Type} → [inst : MeasurableSpace γ₀] → [inst_1 : MeasurableSpace γ₁] → Causalean.PO.PODynLATESystem P γ₀ γ₁ → P.Ω → Bool","l":106,"f":"Causalean/PO/ID/Exact/DynamicLATE/Setup.lean"},{"n":"Causalean.PO.PODynLATESystem.historyBundle1","k":"def","m":"Causalean.PO.ID.Exact.DynamicLATE.Setup","a":"PO/ID/Exact/DynamicLATE","t":1,"d":"Stage-1 history bundle: the singleton (S₀,). Conditioning on this σ-algebra realises · | S₀ in the outer regression.","s":"{P : Causalean.PO.POSystem} → {γ₀ γ₁ : Type} → [inst : MeasurableSpace γ₀] → [inst_1 : MeasurableSpace γ₁] → Causalean.PO.PODynLATESystem P γ₀ γ₁ → Causalean.PO.POCFBundle P","l":371,"f":"Causalean/PO/ID/Exact/DynamicLATE/Setup.lean"},{"n":"Causalean.PO.PODynLATESystem.historyBundle2","k":"def","m":"Causalean.PO.ID.Exact.DynamicLATE.Setup","a":"PO/ID/Exact/DynamicLATE","t":1,"d":"Stage-2 history bundle: (S₀, S₁, Z₁, D₁). Conditioning on this σ-algebra realises · | S, D₁, Z₁ in the inner regression (the stage-2 encouragement Z₂ is *not* in the conditioning set; it is restricted via an indicator).","s":"{P : Causalean.PO.POSystem} → {γ₀ γ₁ : Type} → [inst : MeasurableSpace γ₀] → [inst_1 : MeasurableSpace γ₁] → Causalean.PO.PODynLATESystem P γ₀ γ₁ → Causalean.PO.POCFBundle P","l":376,"f":"Causalean/PO/ID/Exact/DynamicLATE/Setup.lean"},{"n":"Causalean.PO.PODynLATESystem.indD","k":"def","m":"Causalean.PO.ID.Exact.DynamicLATE.Setup","a":"PO/ID/Exact/DynamicLATE","t":1,"d":"Real-valued indicator of {D = d}, i.e. 1_{D₁=d 0} · 1_{D₂=d 1}.","s":"{P : Causalean.PO.POSystem} → {γ₀ γ₁ : Type} → [inst : MeasurableSpace γ₀] → [inst_1 : MeasurableSpace γ₁] → Causalean.PO.PODynLATESystem P γ₀ γ₁ → (Fin 2 → Bool) → P.Ω → ℝ","l":406,"f":"Causalean/PO/ID/Exact/DynamicLATE/Setup.lean"},{"n":"Causalean.PO.PODynLATESystem.indZ","k":"def","m":"Causalean.PO.ID.Exact.DynamicLATE.Setup","a":"PO/ID/Exact/DynamicLATE","t":1,"d":"Real-valued indicator of {Z = z}.","s":"{P : Causalean.PO.POSystem} → {γ₀ γ₁ : Type} → [inst : MeasurableSpace γ₀] → [inst_1 : MeasurableSpace γ₁] → Causalean.PO.PODynLATESystem P γ₀ γ₁ → (Fin 2 → Bool) → P.Ω → ℝ","l":410,"f":"Causalean/PO/ID/Exact/DynamicLATE/Setup.lean"},{"n":"Causalean.PO.PODynLATESystem.innerCondD","k":"def","m":"Causalean.PO.ID.Exact.DynamicLATE.Setup","a":"PO/ID/Exact/DynamicLATE","t":1,"d":"Inner regression P(D = d | S, D₁, Z = z) realised as the bundle ratio E[1_{D=d} · 1_{Z=z} | history2] / E[1_{Z=z} | history2].","s":"{P : Causalean.PO.POSystem} → {γ₀ γ₁ : Type} → [inst : MeasurableSpace γ₀] → [inst_1 : MeasurableSpace γ₁] → Causalean.PO.PODynLATESystem P γ₀ γ₁ → (Fin 2 → Bool) → (Fin 2 → Bool) → P.Ω → ℝ","l":433,"f":"Causalean/PO/ID/Exact/DynamicLATE/Setup.lean"},{"n":"Causalean.PO.PODynLATESystem.innerCondD_mul_z1_indicator","k":"theorem","m":"Causalean.PO.ID.Exact.DynamicLATE.Bridges","a":"PO/ID/Exact/DynamicLATE","t":2,"d":"Stage-2 compliance ratio bridge (multiplied by 1_{Z₁=z₁}). On the event {Z₁ = z₁},","s":"∀ {P : Causalean.PO.POSystem} {γ₀ γ₁ : Type} [inst : MeasurableSpace γ₀] [inst_1 : MeasurableSpace γ₁] [MeasurableSingletonClass γ₀] [MeasurableSingletonClass γ₁] [inst_4 : StandardBorelSpace P.Ω] [inst_5 : MeasureTheory.IsFiniteMeasure P.μ] {S : Causalean.PO.PODynLATESystem P γ₀ γ₁}, S.Assumptions → ∀ (z d : Fin 2 → Bool), (fun ω => S.innerCondD z d ω * S.z1Var.indicator (z 0) ω) =ᵐ[P.μ] fun ω => S.z1Var.indicator (z 0) ω * S.d1Var.indicator (d 0) ω * S.historyBundle2.condExpGiven (Causalean.PO.PODynLATESystem.d2ofZ2EqIndicator✝ S (z 1) (d 1)) P.μ ω","l":226,"f":"Causalean/PO/ID/Exact/DynamicLATE/Bridges.lean"},{"n":"Causalean.PO.PODynLATESystem.innerCondY","k":"def","m":"Causalean.PO.ID.Exact.DynamicLATE.Setup","a":"PO/ID/Exact/DynamicLATE","t":1,"d":"Inner regression E[Y | S, D₁, Z = z] realised as the bundle ratio E[Y · 1_{Z=z} | history2] / E[1_{Z=z} | history2].","s":"{P : Causalean.PO.POSystem} → {γ₀ γ₁ : Type} → [inst : MeasurableSpace γ₀] → [inst_1 : MeasurableSpace γ₁] → Causalean.PO.PODynLATESystem P γ₀ γ₁ → (Fin 2 → Bool) → P.Ω → ℝ","l":427,"f":"Causalean/PO/ID/Exact/DynamicLATE/Setup.lean"},{"n":"Causalean.PO.PODynLATESystem.innerCondY_mul_z1_indicator","k":"theorem","m":"Causalean.PO.ID.Exact.DynamicLATE.Bridges","a":"PO/ID/Exact/DynamicLATE","t":2,"d":"The stage-2 outcome ratio bridge identifies the inner outcome regression on a first-encouragement cell.","s":"∀ {P : Causalean.PO.POSystem} {γ₀ γ₁ : Type} [inst : MeasurableSpace γ₀] [inst_1 : MeasurableSpace γ₁] [MeasurableSingletonClass γ₀] [MeasurableSingletonClass γ₁] [inst_4 : StandardBorelSpace P.Ω] [inst_5 : MeasureTheory.IsFiniteMeasure P.μ] {S : Causalean.PO.PODynLATESystem P γ₀ γ₁}, S.Assumptions → ∀ (z : Fin 2 → Bool), (fun ω => S.innerCondY z ω * S.z1Var.indicator (z 0) ω) =ᵐ[P.μ] fun ω => S.z1Var.indicator (z 0) ω * S.historyBundle2.condExpGiven (S.YofZ2 (z 1)) P.μ ω","l":95,"f":"Causalean/PO/ID/Exact/DynamicLATE/Bridges.lean"},{"n":"Causalean.PO.PODynLATESystem.int_outcome_difference_identity","k":"theorem","m":"Causalean.PO.ID.Exact.DynamicLATE.WhenToTreat","a":"PO/ID/Exact/DynamicLATE","t":2,"d":"The numerator identity: the difference of the encouragement-regime outcome integrals collapses to the conditional contrast on {D(d) = d}.","s":"∀ {P : Causalean.PO.POSystem} {γ₀ γ₁ : Type} [inst : MeasurableSpace γ₀] [inst_1 : MeasurableSpace γ₁] [MeasurableSingletonClass γ₀] [MeasurableSingletonClass γ₁] [inst_4 : StandardBorelSpace P.Ω] [inst_5 : MeasureTheory.IsFiniteMeasure P.μ] {S : Causalean.PO.PODynLATESystem P γ₀ γ₁}, S.Assumptions → ∀ (d : Fin 2 → Bool), d = ![true, false] ∨ d = ![false, true] → ∫ (ω : P.Ω), S.YofDofZ d ω ∂P.μ - ∫ (ω : P.Ω), S.YofDofZ ![false, false] ω ∂P.μ = ∫ (ω : P.Ω) in S.DofZEq d d, S.YofD d ω - S.YofD ![false, false] ω ∂P.μ","l":276,"f":"Causalean/PO/ID/Exact/DynamicLATE/WhenToTreat.lean"},{"n":"Causalean.PO.PODynLATESystem.integrable_factualY_of_consistency_integrable_YofZ2","k":"theorem","m":"Causalean.PO.ID.Exact.DynamicLATE.Setup","a":"PO/ID/Exact/DynamicLATE","t":2,"d":"Factual Y is integrable once the two stage-2 counterfactual outcomes are integrable. The proof partitions on the factual Boolean Z₂ cell and uses PO consistency to identify Y(Z₂ = z₂) with factual Y on that cell.","s":"∀ {P : Causalean.PO.POSystem} {γ₀ γ₁ : Type} [inst : MeasurableSpace γ₀] [inst_1 : MeasurableSpace γ₁] (S : Causalean.PO.PODynLATESystem P γ₀ γ₁) [MeasureTheory.IsFiniteMeasure P.μ], P.Consistency → (∀ (z₂ : Bool), MeasureTheory.Integrable (S.YofZ2 z₂) P.μ) → MeasureTheory.Integrable S.factualY P.μ","l":310,"f":"Causalean/PO/ID/Exact/DynamicLATE/Setup.lean"},{"n":"Causalean.PO.PODynLATESystem.measurableSet_DofZEq","k":"theorem","m":"Causalean.PO.ID.Exact.DynamicLATE.Setup","a":"PO/ID/Exact/DynamicLATE","t":2,"d":"The dynamic complier event is measurable.","s":"∀ {P : Causalean.PO.POSystem} {γ₀ γ₁ : Type} [inst : MeasurableSpace γ₀] [inst_1 : MeasurableSpace γ₁] (S : Causalean.PO.PODynLATESystem P γ₀ γ₁) (z d : Fin 2 → Bool), MeasurableSet (S.DofZEq z d)","l":463,"f":"Causalean/PO/ID/Exact/DynamicLATE/Setup.lean"},{"n":"Causalean.PO.PODynLATESystem.measurable_D1ofZ","k":"theorem","m":"Causalean.PO.ID.Exact.DynamicLATE.Setup","a":"PO/ID/Exact/DynamicLATE","t":2,"d":"The first treatment under a two-target encouragement regime is measurable.","s":"∀ {P : Causalean.PO.POSystem} {γ₀ γ₁ : Type} [inst : MeasurableSpace γ₀] [inst_1 : MeasurableSpace γ₁] (S : Causalean.PO.PODynLATESystem P γ₀ γ₁) (z : Fin 2 → Bool), Measurable (S.D1ofZ z)","l":270,"f":"Causalean/PO/ID/Exact/DynamicLATE/Setup.lean"},{"n":"Causalean.PO.PODynLATESystem.measurable_D2ofZ","k":"theorem","m":"Causalean.PO.ID.Exact.DynamicLATE.Setup","a":"PO/ID/Exact/DynamicLATE","t":2,"d":"The second treatment under a two-target encouragement regime is measurable.","s":"∀ {P : Causalean.PO.POSystem} {γ₀ γ₁ : Type} [inst : MeasurableSpace γ₀] [inst_1 : MeasurableSpace γ₁] (S : Causalean.PO.PODynLATESystem P γ₀ γ₁) (z : Fin 2 → Bool), Measurable (S.D2ofZ z)","l":273,"f":"Causalean/PO/ID/Exact/DynamicLATE/Setup.lean"},{"n":"Causalean.PO.PODynLATESystem.measurable_D2ofZ2","k":"theorem","m":"Causalean.PO.ID.Exact.DynamicLATE.Setup","a":"PO/ID/Exact/DynamicLATE","t":2,"d":"The second treatment under a stage-2-only encouragement regime is measurable.","s":"∀ {P : Causalean.PO.POSystem} {γ₀ γ₁ : Type} [inst : MeasurableSpace γ₀] [inst_1 : MeasurableSpace γ₁] (S : Causalean.PO.PODynLATESystem P γ₀ γ₁) (z₂ : Bool), Measurable (S.D2ofZ2 z₂)","l":283,"f":"Causalean/PO/ID/Exact/DynamicLATE/Setup.lean"},{"n":"Causalean.PO.PODynLATESystem.measurable_DofZ","k":"theorem","m":"Causalean.PO.ID.Exact.DynamicLATE.Setup","a":"PO/ID/Exact/DynamicLATE","t":2,"d":"The joint counterfactual treatment vector under encouragement is measurable.","s":"∀ {P : Causalean.PO.POSystem} {γ₀ γ₁ : Type} [inst : MeasurableSpace γ₀] [inst_1 : MeasurableSpace γ₁] (S : Causalean.PO.PODynLATESystem P γ₀ γ₁) (z : Fin 2 → Bool), Measurable (S.DofZ z)","l":276,"f":"Causalean/PO/ID/Exact/DynamicLATE/Setup.lean"},{"n":"Causalean.PO.PODynLATESystem.measurable_YofD","k":"theorem","m":"Causalean.PO.ID.Exact.DynamicLATE.Setup","a":"PO/ID/Exact/DynamicLATE","t":2,"d":"The outcome under a fixed treatment vector is measurable.","s":"∀ {P : Causalean.PO.POSystem} {γ₀ γ₁ : Type} [inst : MeasurableSpace γ₀] [inst_1 : MeasurableSpace γ₁] (S : Causalean.PO.PODynLATESystem P γ₀ γ₁) (d : Fin 2 → Bool), Measurable (S.YofD d)","l":286,"f":"Causalean/PO/ID/Exact/DynamicLATE/Setup.lean"},{"n":"Causalean.PO.PODynLATESystem.measurable_YofDofZ","k":"theorem","m":"Causalean.PO.ID.Exact.DynamicLATE.Setup","a":"PO/ID/Exact/DynamicLATE","t":2,"d":"The outcome under a fixed encouragement vector is measurable.","s":"∀ {P : Causalean.PO.POSystem} {γ₀ γ₁ : Type} [inst : MeasurableSpace γ₀] [inst_1 : MeasurableSpace γ₁] (S : Causalean.PO.PODynLATESystem P γ₀ γ₁) (z : Fin 2 → Bool), Measurable (S.YofDofZ z)","l":289,"f":"Causalean/PO/ID/Exact/DynamicLATE/Setup.lean"},{"n":"Causalean.PO.PODynLATESystem.measurable_YofZ2","k":"theorem","m":"Causalean.PO.ID.Exact.DynamicLATE.Setup","a":"PO/ID/Exact/DynamicLATE","t":2,"d":"The outcome under a stage-2-only encouragement regime is measurable.","s":"∀ {P : Causalean.PO.POSystem} {γ₀ γ₁ : Type} [inst : MeasurableSpace γ₀] [inst_1 : MeasurableSpace γ₁] (S : Causalean.PO.PODynLATESystem P γ₀ γ₁) (z₂ : Bool), Measurable (S.YofZ2 z₂)","l":292,"f":"Causalean/PO/ID/Exact/DynamicLATE/Setup.lean"},{"n":"Causalean.PO.PODynLATESystem.measurable_factualD1","k":"theorem","m":"Causalean.PO.ID.Exact.DynamicLATE.Setup","a":"PO/ID/Exact/DynamicLATE","t":2,"d":"The observed first treatment is measurable.","s":"∀ {P : Causalean.PO.POSystem} {γ₀ γ₁ : Type} [inst : MeasurableSpace γ₀] [inst_1 : MeasurableSpace γ₁] (S : Causalean.PO.PODynLATESystem P γ₀ γ₁), Measurable S.factualD1","l":303,"f":"Causalean/PO/ID/Exact/DynamicLATE/Setup.lean"},{"n":"Causalean.PO.PODynLATESystem.measurable_factualD2","k":"theorem","m":"Causalean.PO.ID.Exact.DynamicLATE.Setup","a":"PO/ID/Exact/DynamicLATE","t":2,"d":"The observed second treatment is measurable.","s":"∀ {P : Causalean.PO.POSystem} {γ₀ γ₁ : Type} [inst : MeasurableSpace γ₀] [inst_1 : MeasurableSpace γ₁] (S : Causalean.PO.PODynLATESystem P γ₀ γ₁), Measurable S.factualD2","l":305,"f":"Causalean/PO/ID/Exact/DynamicLATE/Setup.lean"},{"n":"Causalean.PO.PODynLATESystem.measurable_factualS0","k":"theorem","m":"Causalean.PO.ID.Exact.DynamicLATE.Setup","a":"PO/ID/Exact/DynamicLATE","t":2,"d":"The observed baseline state is measurable.","s":"∀ {P : Causalean.PO.POSystem} {γ₀ γ₁ : Type} [inst : MeasurableSpace γ₀] [inst_1 : MeasurableSpace γ₁] (S : Causalean.PO.PODynLATESystem P γ₀ γ₁), Measurable S.factualS0","l":295,"f":"Causalean/PO/ID/Exact/DynamicLATE/Setup.lean"},{"n":"Causalean.PO.PODynLATESystem.measurable_factualS1","k":"theorem","m":"Causalean.PO.ID.Exact.DynamicLATE.Setup","a":"PO/ID/Exact/DynamicLATE","t":2,"d":"The observed intermediate state is measurable.","s":"∀ {P : Causalean.PO.POSystem} {γ₀ γ₁ : Type} [inst : MeasurableSpace γ₀] [inst_1 : MeasurableSpace γ₁] (S : Causalean.PO.PODynLATESystem P γ₀ γ₁), Measurable S.factualS1","l":297,"f":"Causalean/PO/ID/Exact/DynamicLATE/Setup.lean"},{"n":"Causalean.PO.PODynLATESystem.measurable_factualY","k":"theorem","m":"Causalean.PO.ID.Exact.DynamicLATE.Setup","a":"PO/ID/Exact/DynamicLATE","t":2,"d":"The observed outcome is measurable.","s":"∀ {P : Causalean.PO.POSystem} {γ₀ γ₁ : Type} [inst : MeasurableSpace γ₀] [inst_1 : MeasurableSpace γ₁] (S : Causalean.PO.PODynLATESystem P γ₀ γ₁), Measurable S.factualY","l":307,"f":"Causalean/PO/ID/Exact/DynamicLATE/Setup.lean"},{"n":"Causalean.PO.PODynLATESystem.measurable_factualZ1","k":"theorem","m":"Causalean.PO.ID.Exact.DynamicLATE.Setup","a":"PO/ID/Exact/DynamicLATE","t":2,"d":"The observed first encouragement is measurable.","s":"∀ {P : Causalean.PO.POSystem} {γ₀ γ₁ : Type} [inst : MeasurableSpace γ₀] [inst_1 : MeasurableSpace γ₁] (S : Causalean.PO.PODynLATESystem P γ₀ γ₁), Measurable S.factualZ1","l":299,"f":"Causalean/PO/ID/Exact/DynamicLATE/Setup.lean"},{"n":"Causalean.PO.PODynLATESystem.measurable_factualZ2","k":"theorem","m":"Causalean.PO.ID.Exact.DynamicLATE.Setup","a":"PO/ID/Exact/DynamicLATE","t":2,"d":"The observed second encouragement is measurable.","s":"∀ {P : Causalean.PO.POSystem} {γ₀ γ₁ : Type} [inst : MeasurableSpace γ₀] [inst_1 : MeasurableSpace γ₁] (S : Causalean.PO.PODynLATESystem P γ₀ γ₁), Measurable S.factualZ2","l":301,"f":"Causalean/PO/ID/Exact/DynamicLATE/Setup.lean"},{"n":"Causalean.PO.PODynLATESystem.measurable_indD","k":"theorem","m":"Causalean.PO.ID.Exact.DynamicLATE.Setup","a":"PO/ID/Exact/DynamicLATE","t":2,"d":"The joint treatment indicator is measurable.","s":"∀ {P : Causalean.PO.POSystem} {γ₀ γ₁ : Type} [inst : MeasurableSpace γ₀] [inst_1 : MeasurableSpace γ₁] (S : Causalean.PO.PODynLATESystem P γ₀ γ₁) (d : Fin 2 → Bool), Measurable (S.indD d)","l":414,"f":"Causalean/PO/ID/Exact/DynamicLATE/Setup.lean"},{"n":"Causalean.PO.PODynLATESystem.measurable_indZ","k":"theorem","m":"Causalean.PO.ID.Exact.DynamicLATE.Setup","a":"PO/ID/Exact/DynamicLATE","t":2,"d":"The joint encouragement indicator is measurable.","s":"∀ {P : Causalean.PO.POSystem} {γ₀ γ₁ : Type} [inst : MeasurableSpace γ₀] [inst_1 : MeasurableSpace γ₁] (S : Causalean.PO.PODynLATESystem P γ₀ γ₁) (z : Fin 2 → Bool), Measurable (S.indZ z)","l":419,"f":"Causalean/PO/ID/Exact/DynamicLATE/Setup.lean"},{"n":"Causalean.PO.PODynLATESystem.mixtureLATE","k":"def","m":"Causalean.PO.ID.Exact.DynamicLATE.Setup","a":"PO/ID/Exact/DynamicLATE","t":1,"d":"Mixture LATE β_z := E[Y(D(z)) - Y(0) | D(z) ≠ 0].","s":"{P : Causalean.PO.POSystem} → {γ₀ γ₁ : Type} → [inst : MeasurableSpace γ₀] → [inst_1 : MeasurableSpace γ₁] → Causalean.PO.PODynLATESystem P γ₀ γ₁ → (Fin 2 → Bool) → ℝ","l":489,"f":"Causalean/PO/ID/Exact/DynamicLATE/Setup.lean"},{"n":"Causalean.PO.PODynLATESystem.mixtureLATE_wald","k":"theorem","m":"Causalean.PO.ID.Exact.DynamicLATE.WhenToTreat","a":"PO/ID/Exact/DynamicLATE","t":1,"d":"Mixture dynamic LATE Wald identity (unconditional). Under the dynamic LATE identifying assumptions, for an encouragement vector z, the mixture dynamic local average treatment effect mixtureLATE z equals the Wald ratio of the observable mean contrast obsMean z − obsMean(0,0) to the observable noncompliance probability 1 − obsProb(z, 0,0).","s":"∀ {P : Causalean.PO.POSystem} {γ₀ γ₁ : Type} [inst : MeasurableSpace γ₀] [inst_1 : MeasurableSpace γ₁] [MeasurableSingletonClass γ₀] [MeasurableSingletonClass γ₁] [inst_4 : StandardBorelSpace P.Ω] [inst_5 : MeasureTheory.IsFiniteMeasure P.μ] {S : Causalean.PO.PODynLATESystem P γ₀ γ₁}, S.Assumptions → ∀ (z : Fin 2 → Bool), S.mixtureLATE z = (S.obsMean z - S.obsMean ![false, false]) / (1 - S.obsProb z ![false, false])","l":365,"f":"Causalean/PO/ID/Exact/DynamicLATE/WhenToTreat.lean"},{"n":"Causalean.PO.PODynLATESystem.obsMean","k":"def","m":"Causalean.PO.ID.Exact.DynamicLATE.Setup","a":"PO/ID/Exact/DynamicLATE","t":1,"d":"Unconditional version obsMean(z) := E[cObsMean(z; S₀)].","s":"{P : Causalean.PO.POSystem} → {γ₀ γ₁ : Type} → [inst : MeasurableSpace γ₀] → [inst_1 : MeasurableSpace γ₁] → Causalean.PO.PODynLATESystem P γ₀ γ₁ → (Fin 2 → Bool) → ℝ","l":452,"f":"Causalean/PO/ID/Exact/DynamicLATE/Setup.lean"},{"n":"Causalean.PO.PODynLATESystem.obsProb","k":"def","m":"Causalean.PO.ID.Exact.DynamicLATE.Setup","a":"PO/ID/Exact/DynamicLATE","t":1,"d":"Unconditional version obsProb(z, d) := E[cObsProb(z, d; S₀)].","s":"{P : Causalean.PO.POSystem} → {γ₀ γ₁ : Type} → [inst : MeasurableSpace γ₀] → [inst_1 : MeasurableSpace γ₁] → Causalean.PO.PODynLATESystem P γ₀ γ₁ → (Fin 2 → Bool) → (Fin 2 → Bool) → ℝ","l":455,"f":"Causalean/PO/ID/Exact/DynamicLATE/Setup.lean"},{"n":"Causalean.PO.PODynLATESystem.outcome_bridge","k":"theorem","m":"Causalean.PO.ID.Exact.DynamicLATE.Bridges","a":"PO/ID/Exact/DynamicLATE","t":1,"d":"Outcome bridge (unconditional). Under the dynamic LATE identifying assumptions, the counterfactual mean outcome E[Y(D(z))] for an encouragement vector z equals the observable nested regression obsMean z.","s":"∀ {P : Causalean.PO.POSystem} {γ₀ γ₁ : Type} [inst : MeasurableSpace γ₀] [inst_1 : MeasurableSpace γ₁] [MeasurableSingletonClass γ₀] [MeasurableSingletonClass γ₁] [inst_4 : StandardBorelSpace P.Ω] [inst_5 : MeasureTheory.IsFiniteMeasure P.μ] {S : Causalean.PO.PODynLATESystem P γ₀ γ₁}, S.Assumptions → ∀ (z : Fin 2 → Bool), ∫ (ω : P.Ω), S.YofDofZ z ω ∂P.μ = S.obsMean z","l":793,"f":"Causalean/PO/ID/Exact/DynamicLATE/Bridges.lean"},{"n":"Causalean.PO.PODynLATESystem.probDofZ_eq_zero_of_not_preceq","k":"theorem","m":"Causalean.PO.ID.Exact.DynamicLATE.Bridges","a":"PO/ID/Exact/DynamicLATE","t":2,"d":"Under one-sided noncompliance, a treatment path not below the encouragement path has zero probability.","s":"∀ {P : Causalean.PO.POSystem} {γ₀ γ₁ : Type} [inst : MeasurableSpace γ₀] [inst_1 : MeasurableSpace γ₁] [MeasurableSingletonClass γ₀] [MeasurableSingletonClass γ₁] [inst_4 : StandardBorelSpace P.Ω] [inst_5 : MeasureTheory.IsFiniteMeasure P.μ] {S : Causalean.PO.PODynLATESystem P γ₀ γ₁}, S.Assumptions → ∀ (z d : Fin 2 → Bool), ¬Causalean.PO.PODynLATESystem.Preceq d z → P.μ (S.DofZEq z d) = 0","l":66,"f":"Causalean/PO/ID/Exact/DynamicLATE/Bridges.lean"},{"n":"Causalean.PO.PODynLATESystem.treatmentRegime","k":"def","m":"Causalean.PO.ID.Exact.DynamicLATE.Setup","a":"PO/ID/Exact/DynamicLATE","t":1,"d":"Regime fixing both treatments: D₁ ↦ d 0, D₂ ↦ d 1.","s":"{P : Causalean.PO.POSystem} → {γ₀ γ₁ : Type} → [inst : MeasurableSpace γ₀] → [inst_1 : MeasurableSpace γ₁] → Causalean.PO.PODynLATESystem P γ₀ γ₁ → (Fin 2 → Bool) → Causalean.PO.Regime P.V P.X","l":189,"f":"Causalean/PO/ID/Exact/DynamicLATE/Setup.lean"},{"n":"Causalean.PO.PODynLATESystem.whenToTreatLATE","k":"def","m":"Causalean.PO.ID.Exact.DynamicLATE.Setup","a":"PO/ID/Exact/DynamicLATE","t":1,"d":"When-to-treat LATE τ_d := θ(d, d).","s":"{P : Causalean.PO.POSystem} → {γ₀ γ₁ : Type} → [inst : MeasurableSpace γ₀] → [inst_1 : MeasurableSpace γ₁] → Causalean.PO.PODynLATESystem P γ₀ γ₁ → (Fin 2 → Bool) → ℝ","l":483,"f":"Causalean/PO/ID/Exact/DynamicLATE/Setup.lean"},{"n":"Causalean.PO.PODynLATESystem.whenToTreat_wald","k":"theorem","m":"Causalean.PO.ID.Exact.DynamicLATE.WhenToTreat","a":"PO/ID/Exact/DynamicLATE","t":1,"d":"When-to-treat dynamic LATE Wald identity (unconditional). For a treatment path d equal to (1,0) or (0,1) — treatment started in period 1 and stopped, or delayed to period 2, the when-to-treat local average treatment effect τ_d equals the Wald ratio of the observable mean contrast obsMean(d) − obsMean(0,0) to the observable compliance probability obsProb(d,d).","s":"∀ {P : Causalean.PO.POSystem} {γ₀ γ₁ : Type} [inst : MeasurableSpace γ₀] [inst_1 : MeasurableSpace γ₁] [MeasurableSingletonClass γ₀] [MeasurableSingletonClass γ₁] [inst_4 : StandardBorelSpace P.Ω] [inst_5 : MeasureTheory.IsFiniteMeasure P.μ] {S : Causalean.PO.PODynLATESystem P γ₀ γ₁}, S.Assumptions → ∀ (d : Fin 2 → Bool), d = ![true, false] ∨ d = ![false, true] → S.whenToTreatLATE d = (S.obsMean d - S.obsMean ![false, false]) / S.obsProb d d","l":330,"f":"Causalean/PO/ID/Exact/DynamicLATE/WhenToTreat.lean"},{"n":"Causalean.PO.PODynLATESystem.yUnderZ","k":"def","m":"Causalean.PO.ID.Exact.DynamicLATE.Setup","a":"PO/ID/Exact/DynamicLATE","t":1,"d":"Y under the two-target encouragement regime, as a RegimedVar.","s":"{P : Causalean.PO.POSystem} → {γ₀ γ₁ : Type} → [inst : MeasurableSpace γ₀] → [inst_1 : MeasurableSpace γ₁] → Causalean.PO.PODynLATESystem P γ₀ γ₁ → (Fin 2 → Bool) → Causalean.PO.RegimedVar P ℝ","l":349,"f":"Causalean/PO/ID/Exact/DynamicLATE/Setup.lean"},{"n":"Causalean.PO.PODynLATESystem.yUnderZ2","k":"def","m":"Causalean.PO.ID.Exact.DynamicLATE.Setup","a":"PO/ID/Exact/DynamicLATE","t":1,"d":"Y under the stage-2-only encouragement regime.","s":"{P : Causalean.PO.POSystem} → {γ₀ γ₁ : Type} → [inst : MeasurableSpace γ₀] → [inst_1 : MeasurableSpace γ₁] → Causalean.PO.PODynLATESystem P γ₀ γ₁ → Bool → Causalean.PO.RegimedVar P ℝ","l":361,"f":"Causalean/PO/ID/Exact/DynamicLATE/Setup.lean"},{"n":"Causalean.PO.PODynLATESystem.yVar","k":"def","m":"Causalean.PO.ID.Exact.DynamicLATE.Setup","a":"PO/ID/Exact/DynamicLATE","t":1,"d":"The outcome is packaged as a real-valued potential-outcome variable.","s":"{P : Causalean.PO.POSystem} → {γ₀ γ₁ : Type} → [inst : MeasurableSpace γ₀] → [inst_1 : MeasurableSpace γ₁] → Causalean.PO.PODynLATESystem P γ₀ γ₁ → Causalean.PO.POVar P ℝ","l":97,"f":"Causalean/PO/ID/Exact/DynamicLATE/Setup.lean"},{"n":"Causalean.PO.PODynLATESystem.z1Var","k":"def","m":"Causalean.PO.ID.Exact.DynamicLATE.Setup","a":"PO/ID/Exact/DynamicLATE","t":1,"d":"The first encouragement is packaged as a binary potential-outcome variable.","s":"{P : Causalean.PO.POSystem} → {γ₀ γ₁ : Type} → [inst : MeasurableSpace γ₀] → [inst_1 : MeasurableSpace γ₁] → Causalean.PO.PODynLATESystem P γ₀ γ₁ → Causalean.PO.POVar P Bool","l":89,"f":"Causalean/PO/ID/Exact/DynamicLATE/Setup.lean"},{"n":"Causalean.PO.PODynLATESystem.z2Var","k":"def","m":"Causalean.PO.ID.Exact.DynamicLATE.Setup","a":"PO/ID/Exact/DynamicLATE","t":1,"d":"The second encouragement is packaged as a binary potential-outcome variable.","s":"{P : Causalean.PO.POSystem} → {γ₀ γ₁ : Type} → [inst : MeasurableSpace γ₀] → [inst_1 : MeasurableSpace γ₁] → Causalean.PO.PODynLATESystem P γ₀ γ₁ → Causalean.PO.POVar P Bool","l":91,"f":"Causalean/PO/ID/Exact/DynamicLATE/Setup.lean"},{"n":"Causalean.PO.POFrontdoorSystem","k":"structure","m":"Causalean.PO.ID.Exact.Frontdoor","a":"PO/ID/Exact","t":1,"d":"A frontdoor system packages, within an ambient potential-outcome system, a binary treatment A, a finite discrete mediator M valued in a finite type, and a real-valued outcome Y, where the treatment, mediator, and outcome are pairwise distinct variables.","s":"Causalean.PO.POSystem → (β : Type u_1) → [inst : MeasurableSpace β] → [MeasurableSingletonClass β] → [Fintype β] → [DecidableEq β] → Type (max (max u_1 u_2) u_3)","l":49,"f":"Causalean/PO/ID/Exact/Frontdoor.lean"},{"n":"Causalean.PO.POFrontdoorSystem.ATE","k":"def","m":"Causalean.PO.ID.Exact.Frontdoor","a":"PO/ID/Exact","t":1,"d":"Target parameter E[Y(1) - Y(0)].","s":"{P : Causalean.PO.POSystem} → {β : Type u_1} → [inst : MeasurableSpace β] → [inst_1 : MeasurableSingletonClass β] → [inst_2 : Fintype β] → [inst_3 : DecidableEq β] → Causalean.PO.POFrontdoorSystem P β → ℝ","l":179,"f":"Causalean/PO/ID/Exact/Frontdoor.lean"},{"n":"Causalean.PO.POFrontdoorSystem.Assumptions","k":"structure","m":"Causalean.PO.ID.Exact.Frontdoor","a":"PO/ID/Exact","t":1,"d":"Frontdoor identifying assumptions. Bundles, for a frontdoor system, consistency of the underlying potential-outcome system, the full-mediation exclusion restriction that the two-treatment-and-mediator outcome does not depend on the treatment arm, treatment–mediator exchangeability, mediator–outcome exchangeability within treatment arms, positivity of each treatment arm, positivity of the mediator within the support of its counterfactual under a treatment arm, cross-world independence of the counterfactual mediator from the full-mediation outcome, and integrability of the treatment-arm and joint treatment–mediator potential outcomes.","s":"{P : Causalean.PO.POSystem} → {β : Type u_1} → [inst : MeasurableSpace β] → [inst_1 : MeasurableSingletonClass β] → [inst_2 : Fintype β] → [inst_3 : DecidableEq β] → Causalean.PO.POFrontdoorSystem P β → Prop","l":187,"f":"Causalean/PO/ID/Exact/Frontdoor.lean"},{"n":"Causalean.PO.POFrontdoorSystem.Assumptions.integrable_factualY","k":"theorem","m":"Causalean.PO.ID.Exact.Frontdoor","a":"PO/ID/Exact","t":2,"d":"Compatibility projection for older call sites: factual outcome integrability is derived from consistency plus integrability of the finite Y(a,m) cells.","s":"∀ {P : Causalean.PO.POSystem} {β : Type u_1} [inst : MeasurableSpace β] [inst_1 : MeasurableSingletonClass β] [inst_2 : Fintype β] [inst_3 : DecidableEq β] {S : Causalean.PO.POFrontdoorSystem P β}, S.Assumptions → MeasureTheory.Integrable S.factualY P.μ","l":340,"f":"Causalean/PO/ID/Exact/Frontdoor.lean"},{"n":"Causalean.PO.POFrontdoorSystem.EYgivenAM","k":"def","m":"Causalean.PO.ID.Exact.Frontdoor","a":"PO/ID/Exact","t":1,"d":"E[Y | A = a, M = m].","s":"{P : Causalean.PO.POSystem} → {β : Type u_1} → [inst : MeasurableSpace β] → [inst_1 : MeasurableSingletonClass β] → [inst_2 : Fintype β] → [inst_3 : DecidableEq β] → Causalean.PO.POFrontdoorSystem P β → Bool → β → ℝ","l":167,"f":"Causalean/PO/ID/Exact/Frontdoor.lean"},{"n":"Causalean.PO.POFrontdoorSystem.EofY_eq_frontdoorTerm","k":"theorem","m":"Causalean.PO.ID.Exact.Frontdoor","a":"PO/ID/Exact","t":1,"d":"Frontdoor identification (individual regime). Under the frontdoor identifying assumption bundle, for each treatment arm a, the mean potential outcome under arm a equals the frontdoor functional frontdoorTerm(a), built from the mediator distribution given a and the outcome regression on mediator and treatment.","s":"∀ {P : Causalean.PO.POSystem} {β : Type u_1} [inst : MeasurableSpace β] [inst_1 : MeasurableSingletonClass β] [inst_2 : Fintype β] [inst_3 : DecidableEq β] (S : Causalean.PO.POFrontdoorSystem P β), S.Assumptions → ∀ (a : Bool), ∫ (ω : P.Ω), S.YofA a ω ∂P.μ = S.frontdoorTerm a","l":481,"f":"Causalean/PO/ID/Exact/Frontdoor.lean"},{"n":"Causalean.PO.POFrontdoorSystem.MofA","k":"def","m":"Causalean.PO.ID.Exact.Frontdoor","a":"PO/ID/Exact","t":1,"d":"The treatment-arm potential mediator fixes treatment to the chosen arm.","s":"{P : Causalean.PO.POSystem} → {β : Type u_1} → [inst : MeasurableSpace β] → [inst_1 : MeasurableSingletonClass β] → [inst_2 : Fintype β] → [inst_3 : DecidableEq β] → Causalean.PO.POFrontdoorSystem P β → Bool → P.Ω → β","l":86,"f":"Causalean/PO/ID/Exact/Frontdoor.lean"},{"n":"Causalean.PO.POFrontdoorSystem.MofA_eq_factualM_on_aEvent","k":"theorem","m":"Causalean.PO.ID.Exact.Frontdoor","a":"PO/ID/Exact","t":2,"d":"On the event {A = a}, M(a)(ω) = factualM ω. Single-target consistency.","s":"∀ {P : Causalean.PO.POSystem} {β : Type u_1} [inst : MeasurableSpace β] [inst_1 : MeasurableSingletonClass β] [inst_2 : Fintype β] [inst_3 : DecidableEq β] {S : Causalean.PO.POFrontdoorSystem P β}, P.Consistency → ∀ (a : Bool) {ω : P.Ω}, ω ∈ S.aEvent a → S.MofA a ω = S.factualM ω","l":255,"f":"Causalean/PO/ID/Exact/Frontdoor.lean"},{"n":"Causalean.PO.POFrontdoorSystem.YofA","k":"def","m":"Causalean.PO.ID.Exact.Frontdoor","a":"PO/ID/Exact","t":1,"d":"The treatment-arm potential outcome fixes treatment to the chosen arm.","s":"{P : Causalean.PO.POSystem} → {β : Type u_1} → [inst : MeasurableSpace β] → [inst_1 : MeasurableSingletonClass β] → [inst_2 : Fintype β] → [inst_3 : DecidableEq β] → Causalean.PO.POFrontdoorSystem P β → Bool → P.Ω → ℝ","l":83,"f":"Causalean/PO/ID/Exact/Frontdoor.lean"},{"n":"Causalean.PO.POFrontdoorSystem.YofAM","k":"def","m":"Causalean.PO.ID.Exact.Frontdoor","a":"PO/ID/Exact","t":1,"d":"The two-variable potential outcome fixes both treatment and mediator.","s":"{P : Causalean.PO.POSystem} → {β : Type u_1} → [inst : MeasurableSpace β] → [inst_1 : MeasurableSingletonClass β] → [inst_2 : Fintype β] → [inst_3 : DecidableEq β] → Causalean.PO.POFrontdoorSystem P β → Bool → β → P.Ω → ℝ","l":98,"f":"Causalean/PO/ID/Exact/Frontdoor.lean"},{"n":"Causalean.PO.POFrontdoorSystem.YofAM_eq_YofA_on_MofA_event","k":"theorem","m":"Causalean.PO.ID.Exact.Frontdoor","a":"PO/ID/Exact","t":2,"d":"Composition lemma (def:po-consistency, composition clause): on the event {M(a) = m}, Y(a, m)(ω) = Y(a)(ω).","s":"∀ {P : Causalean.PO.POSystem} {β : Type u_1} [inst : MeasurableSpace β] [inst_1 : MeasurableSingletonClass β] [inst_2 : Fintype β] [inst_3 : DecidableEq β] {S : Causalean.PO.POFrontdoorSystem P β}, P.Consistency → ∀ (a : Bool) (m : β) {ω : P.Ω}, S.MofA a ω = m → S.YofAM a m ω = S.YofA a ω","l":349,"f":"Causalean/PO/ID/Exact/Frontdoor.lean"},{"n":"Causalean.PO.POFrontdoorSystem.YofAM_eq_factualY_on_aMEvent","k":"theorem","m":"Causalean.PO.ID.Exact.Frontdoor","a":"PO/ID/Exact","t":2,"d":"On the event {A = a} ∩ {M = m}, Y(a,m)(ω) = factualY ω. Multi-target consistency via cf_eq_factual_of_factualAgrees.","s":"∀ {P : Causalean.PO.POSystem} {β : Type u_1} [inst : MeasurableSpace β] [inst_1 : MeasurableSingletonClass β] [inst_2 : Fintype β] [inst_3 : DecidableEq β] {S : Causalean.PO.POFrontdoorSystem P β}, P.Consistency → ∀ (a : Bool) (m : β) {ω : P.Ω}, ω ∈ S.aEvent a ∩ S.mEvent m → S.YofAM a m ω = S.factualY ω","l":261,"f":"Causalean/PO/ID/Exact/Frontdoor.lean"},{"n":"Causalean.PO.POFrontdoorSystem.aEvent","k":"def","m":"Causalean.PO.ID.Exact.Frontdoor","a":"PO/ID/Exact","t":1,"d":"The event {A = a}.","s":"{P : Causalean.PO.POSystem} → {β : Type u_1} → [inst : MeasurableSpace β] → [inst_1 : MeasurableSingletonClass β] → [inst_2 : Fintype β] → [inst_3 : DecidableEq β] → Causalean.PO.POFrontdoorSystem P β → Bool → Set P.Ω","l":131,"f":"Causalean/PO/ID/Exact/Frontdoor.lean"},{"n":"Causalean.PO.POFrontdoorSystem.aVar","k":"def","m":"Causalean.PO.ID.Exact.Frontdoor","a":"PO/ID/Exact","t":1,"d":"The treatment node is packaged as a binary potential-outcome variable.","s":"{P : Causalean.PO.POSystem} → {β : Type u_1} → [inst : MeasurableSpace β] → [inst_1 : MeasurableSingletonClass β] → [inst_2 : Fintype β] → [inst_3 : DecidableEq β] → Causalean.PO.POFrontdoorSystem P β → Causalean.PO.POVar P Bool","l":74,"f":"Causalean/PO/ID/Exact/Frontdoor.lean"},{"n":"Causalean.PO.POFrontdoorSystem.ate_frontdoor","k":"theorem","m":"Causalean.PO.ID.Exact.Frontdoor","a":"PO/ID/Exact","t":1,"d":"Frontdoor identification of the ATE. Under the frontdoor identifying assumption bundle, the average treatment effect equals the frontdoor estimand frontdoorATE, the difference of the frontdoor functional evaluated at the two treatment arms.","s":"∀ {P : Causalean.PO.POSystem} {β : Type u_1} [inst : MeasurableSpace β] [inst_1 : MeasurableSingletonClass β] [inst_2 : Fintype β] [inst_3 : DecidableEq β] (S : Causalean.PO.POFrontdoorSystem P β), S.Assumptions → S.ATE = S.frontdoorATE","l":693,"f":"Causalean/PO/ID/Exact/Frontdoor.lean"},{"n":"Causalean.PO.POFrontdoorSystem.factualA","k":"def","m":"Causalean.PO.ID.Exact.Frontdoor","a":"PO/ID/Exact","t":1,"d":"The factual treatment is the observed treatment assignment for each unit.","s":"{P : Causalean.PO.POSystem} → {β : Type u_1} → [inst : MeasurableSpace β] → [inst_1 : MeasurableSingletonClass β] → [inst_2 : Fintype β] → [inst_3 : DecidableEq β] → Causalean.PO.POFrontdoorSystem P β → P.Ω → Bool","l":104,"f":"Causalean/PO/ID/Exact/Frontdoor.lean"},{"n":"Causalean.PO.POFrontdoorSystem.factualM","k":"def","m":"Causalean.PO.ID.Exact.Frontdoor","a":"PO/ID/Exact","t":1,"d":"The factual mediator is the observed mediator value for each unit.","s":"{P : Causalean.PO.POSystem} → {β : Type u_1} → [inst : MeasurableSpace β] → [inst_1 : MeasurableSingletonClass β] → [inst_2 : Fintype β] → [inst_3 : DecidableEq β] → Causalean.PO.POFrontdoorSystem P β → P.Ω → β","l":106,"f":"Causalean/PO/ID/Exact/Frontdoor.lean"},{"n":"Causalean.PO.POFrontdoorSystem.factualY","k":"def","m":"Causalean.PO.ID.Exact.Frontdoor","a":"PO/ID/Exact","t":1,"d":"The factual outcome is the observed outcome for each unit.","s":"{P : Causalean.PO.POSystem} → {β : Type u_1} → [inst : MeasurableSpace β] → [inst_1 : MeasurableSingletonClass β] → [inst_2 : Fintype β] → [inst_3 : DecidableEq β] → Causalean.PO.POFrontdoorSystem P β → P.Ω → ℝ","l":108,"f":"Causalean/PO/ID/Exact/Frontdoor.lean"},{"n":"Causalean.PO.POFrontdoorSystem.frontdoorATE","k":"def","m":"Causalean.PO.ID.Exact.Frontdoor","a":"PO/ID/Exact","t":1,"d":"Observable (frontdoor-adjusted) ATE.","s":"{P : Causalean.PO.POSystem} → {β : Type u_1} → [inst : MeasurableSpace β] → [inst_1 : MeasurableSingletonClass β] → [inst_2 : Fintype β] → [inst_3 : DecidableEq β] → Causalean.PO.POFrontdoorSystem P β → ℝ","l":182,"f":"Causalean/PO/ID/Exact/Frontdoor.lean"},{"n":"Causalean.PO.POFrontdoorSystem.frontdoorTerm","k":"def","m":"Causalean.PO.ID.Exact.Frontdoor","a":"PO/ID/Exact","t":1,"d":"The frontdoor-adjusted functional φ(a) at treatment a:","s":"{P : Causalean.PO.POSystem} → {β : Type u_1} → [inst : MeasurableSpace β] → [inst_1 : MeasurableSingletonClass β] → [inst_2 : Fintype β] → [inst_3 : DecidableEq β] → Causalean.PO.POFrontdoorSystem P β → Bool → ℝ","l":171,"f":"Causalean/PO/ID/Exact/Frontdoor.lean"},{"n":"Causalean.PO.POFrontdoorSystem.integrable_factualY_of_consistency_integrable_YofAM","k":"theorem","m":"Causalean.PO.ID.Exact.Frontdoor","a":"PO/ID/Exact","t":2,"d":"The observed outcome is integrable when all joint treatment-mediator potential outcomes are integrable and consistency holds.","s":"∀ {P : Causalean.PO.POSystem} {β : Type u_1} [inst : MeasurableSpace β] [inst_1 : MeasurableSingletonClass β] [inst_2 : Fintype β] [inst_3 : DecidableEq β] {S : Causalean.PO.POFrontdoorSystem P β}, P.Consistency → (∀ (a : Bool) (m : β), MeasureTheory.Integrable (S.YofAM a m) P.μ) → MeasureTheory.Integrable S.factualY P.μ","l":280,"f":"Causalean/PO/ID/Exact/Frontdoor.lean"},{"n":"Causalean.PO.POFrontdoorSystem.integral_eq_sum_eventCondExp_aEvent","k":"theorem","m":"Causalean.PO.ID.Exact.Frontdoor","a":"PO/ID/Exact","t":2,"d":"Bool-partition tower identity for eventCondExp.","s":"∀ {P : Causalean.PO.POSystem} {β : Type u_1} [inst : MeasurableSpace β] [inst_1 : MeasurableSingletonClass β] [inst_2 : Fintype β] [inst_3 : DecidableEq β] {S : Causalean.PO.POFrontdoorSystem P β} (g : P.Ω → ℝ), MeasureTheory.Integrable g P.μ → ∫ (ω : P.Ω), g ω ∂P.μ = ∑ a', Causalean.PO.eventCondExp P.μ (S.aEvent a') g * (P.μ (S.aEvent a')).toReal","l":427,"f":"Causalean/PO/ID/Exact/Frontdoor.lean"},{"n":"Causalean.PO.POFrontdoorSystem.integral_eq_sum_integral_MofA","k":"theorem","m":"Causalean.PO.ID.Exact.Frontdoor","a":"PO/ID/Exact","t":2,"d":"Partition of ∫ f along fibers of MofA a (finite codomain β).","s":"∀ {P : Causalean.PO.POSystem} {β : Type u_1} [inst : MeasurableSpace β] [inst_1 : MeasurableSingletonClass β] [inst_2 : Fintype β] [inst_3 : DecidableEq β] {S : Causalean.PO.POFrontdoorSystem P β} (f : P.Ω → ℝ), MeasureTheory.Integrable f P.μ → ∀ (a : Bool), ∫ (ω : P.Ω), f ω ∂P.μ = ∑ m, ∫ (ω : P.Ω) in {ω' | S.MofA a ω' = m}, f ω ∂P.μ","l":402,"f":"Causalean/PO/ID/Exact/Frontdoor.lean"},{"n":"Causalean.PO.POFrontdoorSystem.mBundle","k":"def","m":"Causalean.PO.ID.Exact.Frontdoor","a":"PO/ID/Exact","t":1,"d":"Bundle [M(true), M(false)], used to state A ⊥ (M(1), M(0)).","s":"{P : Causalean.PO.POSystem} → {β : Type u_1} → [inst : MeasurableSpace β] → [inst_1 : MeasurableSingletonClass β] → [inst_2 : Fintype β] → [inst_3 : DecidableEq β] → Causalean.PO.POFrontdoorSystem P β → Causalean.PO.POCFBundle P","l":152,"f":"Causalean/PO/ID/Exact/Frontdoor.lean"},{"n":"Causalean.PO.POFrontdoorSystem.mEvent","k":"def","m":"Causalean.PO.ID.Exact.Frontdoor","a":"PO/ID/Exact","t":1,"d":"The event {M = m}.","s":"{P : Causalean.PO.POSystem} → {β : Type u_1} → [inst : MeasurableSpace β] → [inst_1 : MeasurableSingletonClass β] → [inst_2 : Fintype β] → [inst_3 : DecidableEq β] → Causalean.PO.POFrontdoorSystem P β → β → Set P.Ω","l":133,"f":"Causalean/PO/ID/Exact/Frontdoor.lean"},{"n":"Causalean.PO.POFrontdoorSystem.mUnderA","k":"def","m":"Causalean.PO.ID.Exact.Frontdoor","a":"PO/ID/Exact","t":1,"d":"M(a) as a RegimedVar.","s":"{P : Causalean.PO.POSystem} → {β : Type u_1} → [inst : MeasurableSpace β] → [inst_1 : MeasurableSingletonClass β] → [inst_2 : Fintype β] → [inst_3 : DecidableEq β] → Causalean.PO.POFrontdoorSystem P β → Bool → Causalean.PO.RegimedVar P β","l":148,"f":"Causalean/PO/ID/Exact/Frontdoor.lean"},{"n":"Causalean.PO.POFrontdoorSystem.mVar","k":"def","m":"Causalean.PO.ID.Exact.Frontdoor","a":"PO/ID/Exact","t":1,"d":"The mediator node is packaged as a potential-outcome variable with finite mediator values.","s":"{P : Causalean.PO.POSystem} → {β : Type u_1} → [inst : MeasurableSpace β] → [inst_1 : MeasurableSingletonClass β] → [inst_2 : Fintype β] → [inst_3 : DecidableEq β] → Causalean.PO.POFrontdoorSystem P β → Causalean.PO.POVar P β","l":76,"f":"Causalean/PO/ID/Exact/Frontdoor.lean"},{"n":"Causalean.PO.POFrontdoorSystem.measurableSet_aEvent","k":"theorem","m":"Causalean.PO.ID.Exact.Frontdoor","a":"PO/ID/Exact","t":2,"d":"The factual treatment event for a treatment arm is measurable.","s":"∀ {P : Causalean.PO.POSystem} {β : Type u_1} [inst : MeasurableSpace β] [inst_1 : MeasurableSingletonClass β] [inst_2 : Fintype β] [inst_3 : DecidableEq β] (S : Causalean.PO.POFrontdoorSystem P β) (a : Bool), MeasurableSet (S.aEvent a)","l":136,"f":"Causalean/PO/ID/Exact/Frontdoor.lean"},{"n":"Causalean.PO.POFrontdoorSystem.measurableSet_mEvent","k":"theorem","m":"Causalean.PO.ID.Exact.Frontdoor","a":"PO/ID/Exact","t":2,"d":"The factual mediator event for a mediator value is measurable.","s":"∀ {P : Causalean.PO.POSystem} {β : Type u_1} [inst : MeasurableSpace β] [inst_1 : MeasurableSingletonClass β] [inst_2 : Fintype β] [inst_3 : DecidableEq β] (S : Causalean.PO.POFrontdoorSystem P β) (m : β), MeasurableSet (S.mEvent m)","l":139,"f":"Causalean/PO/ID/Exact/Frontdoor.lean"},{"n":"Causalean.PO.POFrontdoorSystem.measurable_MofA","k":"theorem","m":"Causalean.PO.ID.Exact.Frontdoor","a":"PO/ID/Exact","t":2,"d":"Treatment-arm potential mediators are measurable.","s":"∀ {P : Causalean.PO.POSystem} {β : Type u_1} [inst : MeasurableSpace β] [inst_1 : MeasurableSingletonClass β] [inst_2 : Fintype β] [inst_3 : DecidableEq β] (S : Causalean.PO.POFrontdoorSystem P β) (a : Bool), Measurable (S.MofA a)","l":116,"f":"Causalean/PO/ID/Exact/Frontdoor.lean"},{"n":"Causalean.PO.POFrontdoorSystem.measurable_YofA","k":"theorem","m":"Causalean.PO.ID.Exact.Frontdoor","a":"PO/ID/Exact","t":2,"d":"Treatment-arm potential outcomes are measurable.","s":"∀ {P : Causalean.PO.POSystem} {β : Type u_1} [inst : MeasurableSpace β] [inst_1 : MeasurableSingletonClass β] [inst_2 : Fintype β] [inst_3 : DecidableEq β] (S : Causalean.PO.POFrontdoorSystem P β) (a : Bool), Measurable (S.YofA a)","l":113,"f":"Causalean/PO/ID/Exact/Frontdoor.lean"},{"n":"Causalean.PO.POFrontdoorSystem.measurable_YofAM","k":"theorem","m":"Causalean.PO.ID.Exact.Frontdoor","a":"PO/ID/Exact","t":2,"d":"Joint treatment-mediator potential outcomes are measurable.","s":"∀ {P : Causalean.PO.POSystem} {β : Type u_1} [inst : MeasurableSpace β] [inst_1 : MeasurableSingletonClass β] [inst_2 : Fintype β] [inst_3 : DecidableEq β] (S : Causalean.PO.POFrontdoorSystem P β) (a : Bool) (m : β), Measurable (S.YofAM a m)","l":119,"f":"Causalean/PO/ID/Exact/Frontdoor.lean"},{"n":"Causalean.PO.POFrontdoorSystem.measurable_factualA","k":"theorem","m":"Causalean.PO.ID.Exact.Frontdoor","a":"PO/ID/Exact","t":2,"d":"The observed treatment is measurable.","s":"∀ {P : Causalean.PO.POSystem} {β : Type u_1} [inst : MeasurableSpace β] [inst_1 : MeasurableSingletonClass β] [inst_2 : Fintype β] [inst_3 : DecidableEq β] (S : Causalean.PO.POFrontdoorSystem P β), Measurable S.factualA","l":122,"f":"Causalean/PO/ID/Exact/Frontdoor.lean"},{"n":"Causalean.PO.POFrontdoorSystem.measurable_factualM","k":"theorem","m":"Causalean.PO.ID.Exact.Frontdoor","a":"PO/ID/Exact","t":2,"d":"The observed mediator is measurable.","s":"∀ {P : Causalean.PO.POSystem} {β : Type u_1} [inst : MeasurableSpace β] [inst_1 : MeasurableSingletonClass β] [inst_2 : Fintype β] [inst_3 : DecidableEq β] (S : Causalean.PO.POFrontdoorSystem P β), Measurable S.factualM","l":124,"f":"Causalean/PO/ID/Exact/Frontdoor.lean"},{"n":"Causalean.PO.POFrontdoorSystem.measurable_factualY","k":"theorem","m":"Causalean.PO.ID.Exact.Frontdoor","a":"PO/ID/Exact","t":2,"d":"The observed outcome is measurable.","s":"∀ {P : Causalean.PO.POSystem} {β : Type u_1} [inst : MeasurableSpace β] [inst_1 : MeasurableSingletonClass β] [inst_2 : Fintype β] [inst_3 : DecidableEq β] (S : Causalean.PO.POFrontdoorSystem P β), Measurable S.factualY","l":126,"f":"Causalean/PO/ID/Exact/Frontdoor.lean"},{"n":"Causalean.PO.POFrontdoorSystem.pA","k":"def","m":"Causalean.PO.ID.Exact.Frontdoor","a":"PO/ID/Exact","t":1,"d":"P(A = a).","s":"{P : Causalean.PO.POSystem} → {β : Type u_1} → [inst : MeasurableSpace β] → [inst_1 : MeasurableSingletonClass β] → [inst_2 : Fintype β] → [inst_3 : DecidableEq β] → Causalean.PO.POFrontdoorSystem P β → Bool → ℝ","l":160,"f":"Causalean/PO/ID/Exact/Frontdoor.lean"},{"n":"Causalean.PO.POFrontdoorSystem.pMgivenA","k":"def","m":"Causalean.PO.ID.Exact.Frontdoor","a":"PO/ID/Exact","t":1,"d":"P(M = m | A = a), event-conditional probability.","s":"{P : Causalean.PO.POSystem} → {β : Type u_1} → [inst : MeasurableSpace β] → [inst_1 : MeasurableSingletonClass β] → [inst_2 : Fintype β] → [inst_3 : DecidableEq β] → Causalean.PO.POFrontdoorSystem P β → β → Bool → ℝ","l":163,"f":"Causalean/PO/ID/Exact/Frontdoor.lean"},{"n":"Causalean.PO.POFrontdoorSystem.regimeAM","k":"def","m":"Causalean.PO.ID.Exact.Frontdoor","a":"PO/ID/Exact","t":1,"d":"The joint treatment-mediator regime fixes treatment and mediator simultaneously.","s":"{P : Causalean.PO.POSystem} → {β : Type u_1} → [inst : MeasurableSpace β] → [inst_1 : MeasurableSingletonClass β] → [inst_2 : Fintype β] → [inst_3 : DecidableEq β] → Causalean.PO.POFrontdoorSystem P β → Bool → β → Causalean.PO.Regime P.V P.X","l":89,"f":"Causalean/PO/ID/Exact/Frontdoor.lean"},{"n":"Causalean.PO.POFrontdoorSystem.yVar","k":"def","m":"Causalean.PO.ID.Exact.Frontdoor","a":"PO/ID/Exact","t":1,"d":"The outcome node is packaged as a real-valued potential-outcome variable.","s":"{P : Causalean.PO.POSystem} → {β : Type u_1} → [inst : MeasurableSpace β] → [inst_1 : MeasurableSingletonClass β] → [inst_2 : Fintype β] → [inst_3 : DecidableEq β] → Causalean.PO.POFrontdoorSystem P β → Causalean.PO.POVar P ℝ","l":78,"f":"Causalean/PO/ID/Exact/Frontdoor.lean"},{"n":"Causalean.PO.POFuzzyRDDSystem","k":"structure","m":"Causalean.PO.ID.Exact.RDD.FuzzyRDD","a":"PO/ID/Exact/RDD","t":1,"d":"Fuzzy regression-discontinuity model in the potential-outcome framework. A unit has a continuous running variable X, an above-cutoff indicator Z recording whether X exceeds the cutoff c, a binary treatment D whose take-up probability jumps — but does not jump all the way from 0 to 1 — at the cutoff, and a real outcome Y, with the outcome, cutoff indicator, and treatment nodes pairwise distinct. The cutoff effect on the outcome divided by the cutoff jump in treatment identifies the cutoff-local LATE (def:po-fuzzy-rdd-system).","s":"Causalean.PO.POSystem → Type (max u_1 u_2)","l":69,"f":"Causalean/PO/ID/Exact/RDD/FuzzyRDD.lean"},{"n":"Causalean.PO.POFuzzyRDDSystem.Assumptions","k":"structure","m":"Causalean.PO.ID.Exact.RDD.FuzzyRDD","a":"PO/ID/Exact/RDD","t":1,"d":"Fuzzy RDD assumption bundle. For a fuzzy regression-discontinuity system, this packages consistency (SUTVA), deterministic cutoff eligibility: the above-cutoff indicator agrees almost surely with whether the running variable has crossed the cutoff, and cutoff-neighborhood monotonicity: near the cutoff, a unit who would take treatment under the untreated instrument value would also take it under the treated instrument value. It supplies latent treatment and outcome regression representatives muD/muY and observable regression representatives nuD/nuY, each certified as a genuine regression function of the corresponding response on the running variable, with the latent representatives continuous at the cutoff; it also assumes the running variable has positive local probability mass on both sides of the cutoff, and that the observable treatment regression and outcome regression have well-defined one-sided limits at the cutoff. Finally, a complier outcome-difference representative mu_Ydiff_complier is likewise a regression function that is continuous at the cutoff, and the first-stage treatment-take-up jump at the cutoff is nonzero.","s":"{P : Causalean.PO.POSystem} → Causalean.PO.POFuzzyRDDSystem P → Type","l":165,"f":"Causalean/PO/ID/Exact/RDD/FuzzyRDD.lean"},{"n":"Causalean.PO.POFuzzyRDDSystem.DofZ","k":"def","m":"Causalean.PO.ID.Exact.RDD.FuzzyRDD","a":"PO/ID/Exact/RDD","t":1,"d":"Instrument-specific potential treatment D(z).","s":"{P : Causalean.PO.POSystem} → Causalean.PO.POFuzzyRDDSystem P → Bool → P.Ω → Bool","l":107,"f":"Causalean/PO/ID/Exact/RDD/FuzzyRDD.lean"},{"n":"Causalean.PO.POFuzzyRDDSystem.YdiffComplier","k":"def","m":"Causalean.PO.ID.Exact.RDD.FuzzyRDD","a":"PO/ID/Exact/RDD","t":1,"d":"Y-difference weighted by the complier indicator 1_{D(1)=1, D(0)=0}. Equals Y(D(1)) − Y(D(0)) a.e. under monotonicity; used in the LATE bridge.","s":"{P : Causalean.PO.POSystem} → Causalean.PO.POFuzzyRDDSystem P → P.Ω → ℝ","l":119,"f":"Causalean/PO/ID/Exact/RDD/FuzzyRDD.lean"},{"n":"Causalean.PO.POFuzzyRDDSystem.YofD","k":"def","m":"Causalean.PO.ID.Exact.RDD.FuzzyRDD","a":"PO/ID/Exact/RDD","t":1,"d":"Treatment-specific potential outcome Y(d).","s":"{P : Causalean.PO.POSystem} → Causalean.PO.POFuzzyRDDSystem P → Bool → P.Ω → ℝ","l":111,"f":"Causalean/PO/ID/Exact/RDD/FuzzyRDD.lean"},{"n":"Causalean.PO.POFuzzyRDDSystem.YofDofZ","k":"def","m":"Causalean.PO.ID.Exact.RDD.FuzzyRDD","a":"PO/ID/Exact/RDD","t":1,"d":"Outcome under the treatment induced by instrument value z, Y(D(z)).","s":"{P : Causalean.PO.POSystem} → Causalean.PO.POFuzzyRDDSystem P → Bool → P.Ω → ℝ","l":115,"f":"Causalean/PO/ID/Exact/RDD/FuzzyRDD.lean"},{"n":"Causalean.PO.POFuzzyRDDSystem.factualD","k":"def","m":"Causalean.PO.ID.Exact.RDD.FuzzyRDD","a":"PO/ID/Exact/RDD","t":1,"d":"Factual treatment D.","s":"{P : Causalean.PO.POSystem} → Causalean.PO.POFuzzyRDDSystem P → P.Ω → Bool","l":101,"f":"Causalean/PO/ID/Exact/RDD/FuzzyRDD.lean"},{"n":"Causalean.PO.POFuzzyRDDSystem.factualX","k":"def","m":"Causalean.PO.ID.Exact.RDD.FuzzyRDD","a":"PO/ID/Exact/RDD","t":1,"d":"Factual running variable X.","s":"{P : Causalean.PO.POSystem} → Causalean.PO.POFuzzyRDDSystem P → P.Ω → ℝ","l":95,"f":"Causalean/PO/ID/Exact/RDD/FuzzyRDD.lean"},{"n":"Causalean.PO.POFuzzyRDDSystem.factualY","k":"def","m":"Causalean.PO.ID.Exact.RDD.FuzzyRDD","a":"PO/ID/Exact/RDD","t":1,"d":"Factual outcome Y.","s":"{P : Causalean.PO.POSystem} → Causalean.PO.POFuzzyRDDSystem P → P.Ω → ℝ","l":104,"f":"Causalean/PO/ID/Exact/RDD/FuzzyRDD.lean"},{"n":"Causalean.PO.POFuzzyRDDSystem.factualZ","k":"def","m":"Causalean.PO.ID.Exact.RDD.FuzzyRDD","a":"PO/ID/Exact/RDD","t":1,"d":"Factual cutoff-eligibility instrument Z.","s":"{P : Causalean.PO.POSystem} → Causalean.PO.POFuzzyRDDSystem P → P.Ω → Bool","l":98,"f":"Causalean/PO/ID/Exact/RDD/FuzzyRDD.lean"},{"n":"Causalean.PO.POFuzzyRDDSystem.frd_identification","k":"theorem","m":"Causalean.PO.ID.Exact.RDD.FuzzyRDD","a":"PO/ID/Exact/RDD","t":1,"d":"Fuzzy RDD identification at the cutoff, in regression-representative form. Under the fuzzy-RDD identifying assumption bundle, the Wald ratio of the latent right- and left-hand representative jumps at the cutoff equals the Wald ratio of the one-sided observable outcome- and treatment-regression limits at the cutoff:","s":"∀ {P : Causalean.PO.POSystem} (S : Causalean.PO.POFuzzyRDDSystem P) (hA : S.Assumptions), S.tau_FRD hA = (S.nuY_right_limit hA - S.nuY_left_limit hA) / (S.nuD_right_limit hA - S.nuD_left_limit hA)","l":590,"f":"Causalean/PO/ID/Exact/RDD/FuzzyRDD.lean"},{"n":"Causalean.PO.POFuzzyRDDSystem.measurableSet_zEvent","k":"theorem","m":"Causalean.PO.ID.Exact.RDD.FuzzyRDD","a":"PO/ID/Exact/RDD","t":2,"d":"The factual eligibility event is measurable.","s":"∀ {P : Causalean.PO.POSystem} (S : Causalean.PO.POFuzzyRDDSystem P) (z : Bool), MeasurableSet (S.zEvent z)","l":161,"f":"Causalean/PO/ID/Exact/RDD/FuzzyRDD.lean"},{"n":"Causalean.PO.POFuzzyRDDSystem.measurable_DofZ","k":"theorem","m":"Causalean.PO.ID.Exact.RDD.FuzzyRDD","a":"PO/ID/Exact/RDD","t":2,"d":"The instrument-specific potential treatment is measurable.","s":"∀ {P : Causalean.PO.POSystem} (S : Causalean.PO.POFuzzyRDDSystem P) (z : Bool), Measurable (S.DofZ z)","l":134,"f":"Causalean/PO/ID/Exact/RDD/FuzzyRDD.lean"},{"n":"Causalean.PO.POFuzzyRDDSystem.measurable_YdiffComplier","k":"theorem","m":"Causalean.PO.ID.Exact.RDD.FuzzyRDD","a":"PO/ID/Exact/RDD","t":2,"d":"The complier-weighted outcome difference is measurable.","s":"∀ {P : Causalean.PO.POSystem} (S : Causalean.PO.POFuzzyRDDSystem P), Measurable S.YdiffComplier","l":148,"f":"Causalean/PO/ID/Exact/RDD/FuzzyRDD.lean"},{"n":"Causalean.PO.POFuzzyRDDSystem.measurable_YofD","k":"theorem","m":"Causalean.PO.ID.Exact.RDD.FuzzyRDD","a":"PO/ID/Exact/RDD","t":2,"d":"The treatment-specific potential outcome is measurable.","s":"∀ {P : Causalean.PO.POSystem} (S : Causalean.PO.POFuzzyRDDSystem P) (d : Bool), Measurable (S.YofD d)","l":138,"f":"Causalean/PO/ID/Exact/RDD/FuzzyRDD.lean"},{"n":"Causalean.PO.POFuzzyRDDSystem.measurable_YofDofZ","k":"theorem","m":"Causalean.PO.ID.Exact.RDD.FuzzyRDD","a":"PO/ID/Exact/RDD","t":2,"d":"The outcome under the instrument-induced treatment is measurable.","s":"∀ {P : Causalean.PO.POSystem} (S : Causalean.PO.POFuzzyRDDSystem P) (z : Bool), Measurable (S.YofDofZ z)","l":142,"f":"Causalean/PO/ID/Exact/RDD/FuzzyRDD.lean"},{"n":"Causalean.PO.POFuzzyRDDSystem.measurable_factualD","k":"theorem","m":"Causalean.PO.ID.Exact.RDD.FuzzyRDD","a":"PO/ID/Exact/RDD","t":2,"d":"The factual treatment is measurable.","s":"∀ {P : Causalean.PO.POSystem} (S : Causalean.PO.POFuzzyRDDSystem P), Measurable S.factualD","l":129,"f":"Causalean/PO/ID/Exact/RDD/FuzzyRDD.lean"},{"n":"Causalean.PO.POFuzzyRDDSystem.measurable_factualX","k":"theorem","m":"Causalean.PO.ID.Exact.RDD.FuzzyRDD","a":"PO/ID/Exact/RDD","t":2,"d":"The factual running variable is measurable.","s":"∀ {P : Causalean.PO.POSystem} (S : Causalean.PO.POFuzzyRDDSystem P), Measurable S.factualX","l":125,"f":"Causalean/PO/ID/Exact/RDD/FuzzyRDD.lean"},{"n":"Causalean.PO.POFuzzyRDDSystem.measurable_factualY","k":"theorem","m":"Causalean.PO.ID.Exact.RDD.FuzzyRDD","a":"PO/ID/Exact/RDD","t":2,"d":"The factual outcome is measurable.","s":"∀ {P : Causalean.PO.POSystem} (S : Causalean.PO.POFuzzyRDDSystem P), Measurable S.factualY","l":131,"f":"Causalean/PO/ID/Exact/RDD/FuzzyRDD.lean"},{"n":"Causalean.PO.POFuzzyRDDSystem.measurable_factualZ","k":"theorem","m":"Causalean.PO.ID.Exact.RDD.FuzzyRDD","a":"PO/ID/Exact/RDD","t":2,"d":"The factual cutoff-eligibility instrument is measurable.","s":"∀ {P : Causalean.PO.POSystem} (S : Causalean.PO.POFuzzyRDDSystem P), Measurable S.factualZ","l":127,"f":"Causalean/PO/ID/Exact/RDD/FuzzyRDD.lean"},{"n":"Causalean.PO.POFuzzyRDDSystem.nuD_left_limit","k":"def","m":"Causalean.PO.ID.Exact.RDD.FuzzyRDD","a":"PO/ID/Exact/RDD","t":1,"d":"The left-hand treatment limit is the selected limit at the cutoff of the observable treatment regression as the running variable approaches from below.","s":"{P : Causalean.PO.POSystem} → (S : Causalean.PO.POFuzzyRDDSystem P) → S.Assumptions → ℝ","l":242,"f":"Causalean/PO/ID/Exact/RDD/FuzzyRDD.lean"},{"n":"Causalean.PO.POFuzzyRDDSystem.nuD_left_limit_eq","k":"theorem","m":"Causalean.PO.ID.Exact.RDD.FuzzyRDD","a":"PO/ID/Exact/RDD","t":2,"d":"Any left-hand observable treatment-regression limit at the cutoff equals the untreated latent treatment regression there.","s":"∀ {P : Causalean.PO.POSystem} (S : Causalean.PO.POFuzzyRDDSystem P) (hA : S.Assumptions) {L : ℝ}, Filter.Tendsto hA.nuD (nhdsWithin S.c (Set.Iio S.c)) (nhds L) → L = hA.muD false S.c","l":555,"f":"Causalean/PO/ID/Exact/RDD/FuzzyRDD.lean"},{"n":"Causalean.PO.POFuzzyRDDSystem.nuD_right_limit","k":"def","m":"Causalean.PO.ID.Exact.RDD.FuzzyRDD","a":"PO/ID/Exact/RDD","t":1,"d":"The right-hand treatment limit is the selected limit at the cutoff of the observable treatment regression as the running variable approaches from above.","s":"{P : Causalean.PO.POSystem} → (S : Causalean.PO.POFuzzyRDDSystem P) → S.Assumptions → ℝ","l":237,"f":"Causalean/PO/ID/Exact/RDD/FuzzyRDD.lean"},{"n":"Causalean.PO.POFuzzyRDDSystem.nuD_right_limit_eq","k":"theorem","m":"Causalean.PO.ID.Exact.RDD.FuzzyRDD","a":"PO/ID/Exact/RDD","t":2,"d":"Any right-hand observable treatment-regression limit at the cutoff equals the treated latent treatment regression there.","s":"∀ {P : Causalean.PO.POSystem} (S : Causalean.PO.POFuzzyRDDSystem P) (hA : S.Assumptions) {L : ℝ}, Filter.Tendsto hA.nuD (nhdsWithin S.c (Set.Ioi S.c)) (nhds L) → L = hA.muD true S.c","l":544,"f":"Causalean/PO/ID/Exact/RDD/FuzzyRDD.lean"},{"n":"Causalean.PO.POFuzzyRDDSystem.nuY_left_limit","k":"def","m":"Causalean.PO.ID.Exact.RDD.FuzzyRDD","a":"PO/ID/Exact/RDD","t":1,"d":"The left-hand outcome limit is the selected limit at the cutoff of the observable outcome regression as the running variable approaches from below.","s":"{P : Causalean.PO.POSystem} → (S : Causalean.PO.POFuzzyRDDSystem P) → S.Assumptions → ℝ","l":252,"f":"Causalean/PO/ID/Exact/RDD/FuzzyRDD.lean"},{"n":"Causalean.PO.POFuzzyRDDSystem.nuY_left_limit_eq","k":"theorem","m":"Causalean.PO.ID.Exact.RDD.FuzzyRDD","a":"PO/ID/Exact/RDD","t":2,"d":"Any left-hand observable outcome-regression limit at the cutoff equals the untreated latent outcome regression there.","s":"∀ {P : Causalean.PO.POSystem} (S : Causalean.PO.POFuzzyRDDSystem P) (hA : S.Assumptions) {L : ℝ}, Filter.Tendsto hA.nuY (nhdsWithin S.c (Set.Iio S.c)) (nhds L) → L = hA.muY false S.c","l":577,"f":"Causalean/PO/ID/Exact/RDD/FuzzyRDD.lean"},{"n":"Causalean.PO.POFuzzyRDDSystem.nuY_right_limit","k":"def","m":"Causalean.PO.ID.Exact.RDD.FuzzyRDD","a":"PO/ID/Exact/RDD","t":1,"d":"The right-hand outcome limit is the selected limit at the cutoff of the observable outcome regression as the running variable approaches from above.","s":"{P : Causalean.PO.POSystem} → (S : Causalean.PO.POFuzzyRDDSystem P) → S.Assumptions → ℝ","l":247,"f":"Causalean/PO/ID/Exact/RDD/FuzzyRDD.lean"},{"n":"Causalean.PO.POFuzzyRDDSystem.nuY_right_limit_eq","k":"theorem","m":"Causalean.PO.ID.Exact.RDD.FuzzyRDD","a":"PO/ID/Exact/RDD","t":2,"d":"Any right-hand observable outcome-regression limit at the cutoff equals the treated latent outcome regression there.","s":"∀ {P : Causalean.PO.POSystem} (S : Causalean.PO.POFuzzyRDDSystem P) (hA : S.Assumptions) {L : ℝ}, Filter.Tendsto hA.nuY (nhdsWithin S.c (Set.Ioi S.c)) (nhds L) → L = hA.muY true S.c","l":566,"f":"Causalean/PO/ID/Exact/RDD/FuzzyRDD.lean"},{"n":"Causalean.PO.POFuzzyRDDSystem.tau_FRD","k":"def","m":"Causalean.PO.ID.Exact.RDD.FuzzyRDD","a":"PO/ID/Exact/RDD","t":1,"d":"Cutoff-local fuzzy RDD Wald estimand in regression-representative form.","s":"{P : Causalean.PO.POSystem} → (S : Causalean.PO.POFuzzyRDDSystem P) → S.Assumptions → ℝ","l":225,"f":"Causalean/PO/ID/Exact/RDD/FuzzyRDD.lean"},{"n":"Causalean.PO.POFuzzyRDDSystem.tau_LATE","k":"def","m":"Causalean.PO.ID.Exact.RDD.FuzzyRDD","a":"PO/ID/Exact/RDD","t":1,"d":"Cutoff-local complier-effect ratio in regression-representative form. It divides the complier outcome-difference representative at the cutoff by the first-stage jump; tau_late_identification connects this ratio to tau_FRD under the fuzzy-RDD assumptions.","s":"{P : Causalean.PO.POSystem} → (S : Causalean.PO.POFuzzyRDDSystem P) → S.Assumptions → ℝ","l":230,"f":"Causalean/PO/ID/Exact/RDD/FuzzyRDD.lean"},{"n":"Causalean.PO.POFuzzyRDDSystem.tau_late_identification","k":"theorem","m":"Causalean.PO.ID.Exact.RDD.FuzzyRDD","a":"PO/ID/Exact/RDD","t":1,"d":"Fuzzy RDD identifies a cutoff representative complier ratio under an extra global monotonicity bridge. Under the local fuzzy-RDD identifying assumption bundle — consistency, deterministic cutoff eligibility, and local exclusion through potential outcomes, and additionally assuming treatment is almost-surely monotone in the instrument, i.e. whenever the potential treatment under the untreated instrument value is realized as true, the potential treatment under the treated instrument value is true as well (no defiers), the complier-weighted local average treatment effect at the cutoff tau_LATE equals the fuzzy-RDD Wald-ratio functional tau_FRD, i.e. the observable Wald ratio at the cutoff equals the complier-weighted outcome-difference representative divided by the first-stage jump.","s":"∀ {P : Causalean.PO.POSystem} (S : Causalean.PO.POFuzzyRDDSystem P) (hA : S.Assumptions), (∀ᵐ (ω : P.Ω) ∂P.μ, S.DofZ false ω = true → S.DofZ true ω = true) → S.tau_LATE hA = S.tau_FRD hA","l":646,"f":"Causalean/PO/ID/Exact/RDD/FuzzyRDD.lean"},{"n":"Causalean.PO.POFuzzyRDDSystem.tendsto_nuD_left_limit","k":"theorem","m":"Causalean.PO.ID.Exact.RDD.FuzzyRDD","a":"PO/ID/Exact/RDD","t":2,"d":"The chosen left-hand treatment-regression limit is a genuine left-hand limit at the cutoff.","s":"∀ {P : Causalean.PO.POSystem} (S : Causalean.PO.POFuzzyRDDSystem P) (hA : S.Assumptions), Filter.Tendsto hA.nuD (nhdsWithin S.c (Set.Iio S.c)) (nhds (S.nuD_left_limit hA))","l":263,"f":"Causalean/PO/ID/Exact/RDD/FuzzyRDD.lean"},{"n":"Causalean.PO.POFuzzyRDDSystem.tendsto_nuD_right_limit","k":"theorem","m":"Causalean.PO.ID.Exact.RDD.FuzzyRDD","a":"PO/ID/Exact/RDD","t":2,"d":"The chosen right-hand treatment-regression limit is a genuine right-hand limit at the cutoff.","s":"∀ {P : Causalean.PO.POSystem} (S : Causalean.PO.POFuzzyRDDSystem P) (hA : S.Assumptions), Filter.Tendsto hA.nuD (nhdsWithin S.c (Set.Ioi S.c)) (nhds (S.nuD_right_limit hA))","l":257,"f":"Causalean/PO/ID/Exact/RDD/FuzzyRDD.lean"},{"n":"Causalean.PO.POFuzzyRDDSystem.tendsto_nuY_left_limit","k":"theorem","m":"Causalean.PO.ID.Exact.RDD.FuzzyRDD","a":"PO/ID/Exact/RDD","t":2,"d":"The chosen left-hand outcome-regression limit is a genuine left-hand limit at the cutoff.","s":"∀ {P : Causalean.PO.POSystem} (S : Causalean.PO.POFuzzyRDDSystem P) (hA : S.Assumptions), Filter.Tendsto hA.nuY (nhdsWithin S.c (Set.Iio S.c)) (nhds (S.nuY_left_limit hA))","l":275,"f":"Causalean/PO/ID/Exact/RDD/FuzzyRDD.lean"},{"n":"Causalean.PO.POFuzzyRDDSystem.tendsto_nuY_right_limit","k":"theorem","m":"Causalean.PO.ID.Exact.RDD.FuzzyRDD","a":"PO/ID/Exact/RDD","t":2,"d":"The chosen right-hand outcome-regression limit is a genuine right-hand limit at the cutoff.","s":"∀ {P : Causalean.PO.POSystem} (S : Causalean.PO.POFuzzyRDDSystem P) (hA : S.Assumptions), Filter.Tendsto hA.nuY (nhdsWithin S.c (Set.Ioi S.c)) (nhds (S.nuY_right_limit hA))","l":269,"f":"Causalean/PO/ID/Exact/RDD/FuzzyRDD.lean"},{"n":"Causalean.PO.POFuzzyRDDSystem.zEvent","k":"def","m":"Causalean.PO.ID.Exact.RDD.FuzzyRDD","a":"PO/ID/Exact/RDD","t":1,"d":"Factual eligibility event {Z = z}.","s":"{P : Causalean.PO.POSystem} → Causalean.PO.POFuzzyRDDSystem P → Bool → Set P.Ω","l":158,"f":"Causalean/PO/ID/Exact/RDD/FuzzyRDD.lean"},{"n":"Causalean.PO.POHeckmanRoySystem","k":"structure","m":"Causalean.PO.ID.Exact.HeckmanRoy.Setup","a":"PO/ID/Exact/HeckmanRoy","t":1,"d":"The Heckman–Vytlacil / generalized Roy instrumental-variable model in the potential-outcome framework, packaging an instrument, a binary treatment taken when a latent selection rank U falls below a propensity threshold p(Z) valued in [0,1], and a real outcome with potential outcomes Y(0) and Y(1), where the four nodes are pairwise distinct. This is the setup behind pairwise-Wald / LATE-type identification of treatment effects from instrument-induced variation in participation (def:po-iv-heckman-roy-system).","s":"Causalean.PO.POSystem → (α : Type u_1) → [inst : MeasurableSpace α] → [MeasurableSingletonClass α] → Type (max (max u_1 u_2) u_3)","l":42,"f":"Causalean/PO/ID/Exact/HeckmanRoy/Setup.lean"},{"n":"Causalean.PO.POHeckmanRoySystem.Assumptions","k":"structure","m":"Causalean.PO.ID.Exact.HeckmanRoy.Setup","a":"PO/ID/Exact/HeckmanRoy","t":1,"d":"Heckman–Roy IV identifying assumptions (def:po-iv-heckman-roy-assumptions). Bundles consistency of the underlying potential-outcome system, exogeneity of the instrument, independent of the latent selection rank and the two potential outcomes, threshold-crossing selection: the potential treatment under instrument value z equals true exactly when the latent rank falls at or below the propensity p(z), and the latent rank being uniformly distributed on [0,1].","s":"{P : Causalean.PO.POSystem} → {α : Type u_1} → [inst : MeasurableSpace α] → [inst_1 : MeasurableSingletonClass α] → Causalean.PO.POHeckmanRoySystem P α → Prop","l":202,"f":"Causalean/PO/ID/Exact/HeckmanRoy/Setup.lean"},{"n":"Causalean.PO.POHeckmanRoySystem.DofZ","k":"def","m":"Causalean.PO.ID.Exact.HeckmanRoy.Setup","a":"PO/ID/Exact/HeckmanRoy","t":1,"d":"D(z) : P.Ω → Bool.","s":"{P : Causalean.PO.POSystem} → {α : Type u_1} → [inst : MeasurableSpace α] → [inst_1 : MeasurableSingletonClass α] → Causalean.PO.POHeckmanRoySystem P α → α → P.Ω → Bool","l":104,"f":"Causalean/PO/ID/Exact/HeckmanRoy/Setup.lean"},{"n":"Causalean.PO.POHeckmanRoySystem.DofZ_eq_factualD_on_zEvent","k":"theorem","m":"Causalean.PO.ID.Exact.HeckmanRoy.Wald","a":"PO/ID/Exact/HeckmanRoy","t":2,"d":"Pointwise D-consistency on {Z = z}: D(z)(ω) = D(ω). One-line specialisation of POVar.cf_eq_factual_on_event. Analogue of POIVSystem.DofZ_eq_factualD_on_zEvent in LATE.lean.","s":"∀ {P : Causalean.PO.POSystem} {α : Type u_1} [inst : MeasurableSpace α] [inst_1 : MeasurableSingletonClass α] (S : Causalean.PO.POHeckmanRoySystem P α), S.Assumptions → ∀ (z : α) {ω : P.Ω}, ω ∈ S.zEvent z → S.DofZ z ω = S.factualD ω","l":45,"f":"Causalean/PO/ID/Exact/HeckmanRoy/Wald.lean"},{"n":"Causalean.PO.POHeckmanRoySystem.LATE","k":"def","m":"Causalean.PO.ID.Exact.HeckmanRoy.Setup","a":"PO/ID/Exact/HeckmanRoy","t":1,"d":"Latent interval average treatment effect at (z₀, z₁) -- def:po-iv-heckman-roy-late.","s":"{P : Causalean.PO.POSystem} → {α : Type u_1} → [inst : MeasurableSpace α] → [inst_1 : MeasurableSingletonClass α] → Causalean.PO.POHeckmanRoySystem P α → α → α → ℝ","l":230,"f":"Causalean/PO/ID/Exact/HeckmanRoy/Setup.lean"},{"n":"Causalean.PO.POHeckmanRoySystem.YofD","k":"def","m":"Causalean.PO.ID.Exact.HeckmanRoy.Setup","a":"PO/ID/Exact/HeckmanRoy","t":1,"d":"Y(d) : P.Ω → ℝ.","s":"{P : Causalean.PO.POSystem} → {α : Type u_1} → [inst : MeasurableSpace α] → [inst_1 : MeasurableSingletonClass α] → Causalean.PO.POHeckmanRoySystem P α → Bool → P.Ω → ℝ","l":107,"f":"Causalean/PO/ID/Exact/HeckmanRoy/Setup.lean"},{"n":"Causalean.PO.POHeckmanRoySystem.YofDofZ","k":"def","m":"Causalean.PO.ID.Exact.HeckmanRoy.Setup","a":"PO/ID/Exact/HeckmanRoy","t":1,"d":"Y composed with D(z): 1_{D(z)=1} Y(1) + 1_{D(z)=0} Y(0).","s":"{P : Causalean.PO.POSystem} → {α : Type u_1} → [inst : MeasurableSpace α] → [inst_1 : MeasurableSingletonClass α] → Causalean.PO.POHeckmanRoySystem P α → α → P.Ω → ℝ","l":162,"f":"Causalean/PO/ID/Exact/HeckmanRoy/Setup.lean"},{"n":"Causalean.PO.POHeckmanRoySystem.cfBundle","k":"def","m":"Causalean.PO.ID.Exact.HeckmanRoy.Setup","a":"PO/ID/Exact/HeckmanRoy","t":1,"d":"Counterfactual bundle (U, Y(1), Y(0)) -- target of the instrument- independence assumption.","s":"{P : Causalean.PO.POSystem} → {α : Type u_1} → [inst : MeasurableSpace α] → [inst_1 : MeasurableSingletonClass α] → Causalean.PO.POHeckmanRoySystem P α → Causalean.PO.POCFBundle P","l":188,"f":"Causalean/PO/ID/Exact/HeckmanRoy/Setup.lean"},{"n":"Causalean.PO.POHeckmanRoySystem.complier_measure","k":"theorem","m":"Causalean.PO.ID.Exact.HeckmanRoy.Wald","a":"PO/ID/Exact/HeckmanRoy","t":2,"d":"Real-valued measure of the interval complier event: (μ {p z₀ < U ≤ p z₁}).toReal = p z₁ - p z₀, provided 0 ≤ p z₀ ≤ p z₁ ≤ 1 (which holds because p lands in [0,1] by S.hp_mem, plus the user-supplied ordering p z₀ < p z₁).","s":"∀ {P : Causalean.PO.POSystem} {α : Type u_1} [inst : MeasurableSpace α] [inst_1 : MeasurableSingletonClass α] (S : Causalean.PO.POHeckmanRoySystem P α), S.Assumptions → ∀ (z₀ z₁ : α), S.p z₀ < S.p z₁ → (P.μ (S.intervalComplierEvent z₀ z₁)).toReal = S.p z₁ - S.p z₀","l":95,"f":"Causalean/PO/ID/Exact/HeckmanRoy/Wald.lean"},{"n":"Causalean.PO.POHeckmanRoySystem.condExpDZ","k":"def","m":"Causalean.PO.ID.Exact.HeckmanRoy.Setup","a":"PO/ID/Exact/HeckmanRoy","t":1,"d":"E[D | Z = z], the treated share among units with instrument value z, as the PO event-conditional expectation eventCondExp over the event {Z = z}.","s":"{P : Causalean.PO.POSystem} → {α : Type u_1} → [inst : MeasurableSpace α] → [inst_1 : MeasurableSingletonClass α] → Causalean.PO.POHeckmanRoySystem P α → α → ℝ","l":172,"f":"Causalean/PO/ID/Exact/HeckmanRoy/Setup.lean"},{"n":"Causalean.PO.POHeckmanRoySystem.condExpYZ","k":"def","m":"Causalean.PO.ID.Exact.HeckmanRoy.Setup","a":"PO/ID/Exact/HeckmanRoy","t":1,"d":"E[Y | Z = z], the mean outcome among units with instrument value z, as the PO event-conditional expectation eventCondExp over the event {Z = z}.","s":"{P : Causalean.PO.POSystem} → {α : Type u_1} → [inst : MeasurableSpace α] → [inst_1 : MeasurableSingletonClass α] → Causalean.PO.POHeckmanRoySystem P α → α → ℝ","l":177,"f":"Causalean/PO/ID/Exact/HeckmanRoy/Setup.lean"},{"n":"Causalean.PO.POHeckmanRoySystem.dVar","k":"def","m":"Causalean.PO.ID.Exact.HeckmanRoy.Setup","a":"PO/ID/Exact/HeckmanRoy","t":1,"d":"Treatment packaged as a POVar valued in Bool.","s":"{P : Causalean.PO.POSystem} → {α : Type u_1} → [inst : MeasurableSpace α] → [inst_1 : MeasurableSingletonClass α] → Causalean.PO.POHeckmanRoySystem P α → Causalean.PO.POVar P Bool","l":85,"f":"Causalean/PO/ID/Exact/HeckmanRoy/Setup.lean"},{"n":"Causalean.PO.POHeckmanRoySystem.event_conditioning_identity","k":"theorem","m":"Causalean.PO.ID.Exact.HeckmanRoy.Wald","a":"PO/ID/Exact/HeckmanRoy","t":2,"d":"Event-conditioning identity: ∫ (Y(1) - Y(0)) · 1_{C(z₀,z₁)} ∂μ = μ(C(z₀,z₁)).toReal · LATE z₀ z₁.","s":"∀ {P : Causalean.PO.POSystem} {α : Type u_1} [inst : MeasurableSpace α] [inst_1 : MeasurableSingletonClass α] (S : Causalean.PO.POHeckmanRoySystem P α) (z₀ z₁ : α), ∫ (ω : P.Ω), (S.YofD true ω - S.YofD false ω) * (S.intervalComplierEvent z₀ z₁).indicator (fun x => 1) ω ∂P.μ = (P.μ (S.intervalComplierEvent z₀ z₁)).toReal * S.LATE z₀ z₁","l":433,"f":"Causalean/PO/ID/Exact/HeckmanRoy/Wald.lean"},{"n":"Causalean.PO.POHeckmanRoySystem.factualD","k":"def","m":"Causalean.PO.ID.Exact.HeckmanRoy.Setup","a":"PO/ID/Exact/HeckmanRoy","t":1,"d":"Factual treatment.","s":"{P : Causalean.PO.POSystem} → {α : Type u_1} → [inst : MeasurableSpace α] → [inst_1 : MeasurableSingletonClass α] → Causalean.PO.POHeckmanRoySystem P α → P.Ω → Bool","l":113,"f":"Causalean/PO/ID/Exact/HeckmanRoy/Setup.lean"},{"n":"Causalean.PO.POHeckmanRoySystem.factualU","k":"def","m":"Causalean.PO.ID.Exact.HeckmanRoy.Setup","a":"PO/ID/Exact/HeckmanRoy","t":1,"d":"Factual latent rank U.","s":"{P : Causalean.PO.POSystem} → {α : Type u_1} → [inst : MeasurableSpace α] → [inst_1 : MeasurableSingletonClass α] → Causalean.PO.POHeckmanRoySystem P α → P.Ω → ℝ","l":119,"f":"Causalean/PO/ID/Exact/HeckmanRoy/Setup.lean"},{"n":"Causalean.PO.POHeckmanRoySystem.factualY","k":"def","m":"Causalean.PO.ID.Exact.HeckmanRoy.Setup","a":"PO/ID/Exact/HeckmanRoy","t":1,"d":"Factual outcome.","s":"{P : Causalean.PO.POSystem} → {α : Type u_1} → [inst : MeasurableSpace α] → [inst_1 : MeasurableSingletonClass α] → Causalean.PO.POHeckmanRoySystem P α → P.Ω → ℝ","l":116,"f":"Causalean/PO/ID/Exact/HeckmanRoy/Setup.lean"},{"n":"Causalean.PO.POHeckmanRoySystem.factualY_eq_YofD_factualD","k":"theorem","m":"Causalean.PO.ID.Exact.HeckmanRoy.Wald","a":"PO/ID/Exact/HeckmanRoy","t":2,"d":"Pointwise Y-consistency: Y(ω) = Y(D(ω))(ω) for every ω. One-line specialisation of POVar.factual_eq_cfUnder_self_selected.","s":"∀ {P : Causalean.PO.POSystem} {α : Type u_1} [inst : MeasurableSpace α] [inst_1 : MeasurableSingletonClass α] (S : Causalean.PO.POHeckmanRoySystem P α), S.Assumptions → ∀ (ω : P.Ω), S.factualY ω = S.YofD (S.factualD ω) ω","l":53,"f":"Causalean/PO/ID/Exact/HeckmanRoy/Wald.lean"},{"n":"Causalean.PO.POHeckmanRoySystem.factualZ","k":"def","m":"Causalean.PO.ID.Exact.HeckmanRoy.Setup","a":"PO/ID/Exact/HeckmanRoy","t":1,"d":"Factual instrument.","s":"{P : Causalean.PO.POSystem} → {α : Type u_1} → [inst : MeasurableSpace α] → [inst_1 : MeasurableSingletonClass α] → Causalean.PO.POHeckmanRoySystem P α → P.Ω → α","l":110,"f":"Causalean/PO/ID/Exact/HeckmanRoy/Setup.lean"},{"n":"Causalean.PO.POHeckmanRoySystem.first_stage_identity","k":"theorem","m":"Causalean.PO.ID.Exact.HeckmanRoy.Wald","a":"PO/ID/Exact/HeckmanRoy","t":2,"d":"First-stage identity: E[D | Z=z₁] - E[D | Z=z₀] = p z₁ - p z₀.","s":"∀ {P : Causalean.PO.POSystem} {α : Type u_1} [inst : MeasurableSpace α] [inst_1 : MeasurableSingletonClass α] (S : Causalean.PO.POHeckmanRoySystem P α), S.Assumptions → ∀ (z₀ z₁ : α), 0 < (P.μ (S.zEvent z₀)).toReal → 0 < (P.μ (S.zEvent z₁)).toReal → S.p z₀ < S.p z₁ → S.condExpDZ z₁ - S.condExpDZ z₀ = S.p z₁ - S.p z₀","l":143,"f":"Causalean/PO/ID/Exact/HeckmanRoy/Wald.lean"},{"n":"Causalean.PO.POHeckmanRoySystem.instrumentRegime","k":"def","m":"Causalean.PO.ID.Exact.HeckmanRoy.Setup","a":"PO/ID/Exact/HeckmanRoy","t":1,"d":"Regime fixing instrument to z.","s":"{P : Causalean.PO.POSystem} → {α : Type u_1} → [inst : MeasurableSpace α] → [inst_1 : MeasurableSingletonClass α] → Causalean.PO.POHeckmanRoySystem P α → α → Causalean.PO.Regime P.V P.X","l":96,"f":"Causalean/PO/ID/Exact/HeckmanRoy/Setup.lean"},{"n":"Causalean.PO.POHeckmanRoySystem.intervalComplierEvent","k":"def","m":"Causalean.PO.ID.Exact.HeckmanRoy.Setup","a":"PO/ID/Exact/HeckmanRoy","t":1,"d":"Interval complier event {ω | p(z₀) < U(ω) ≤ p(z₁)} -- def:po-iv-heckman-roy-late.","s":"{P : Causalean.PO.POSystem} → {α : Type u_1} → [inst : MeasurableSpace α] → [inst_1 : MeasurableSingletonClass α] → Causalean.PO.POHeckmanRoySystem P α → α → α → Set P.Ω","l":125,"f":"Causalean/PO/ID/Exact/HeckmanRoy/Setup.lean"},{"n":"Causalean.PO.POHeckmanRoySystem.interval_indicator_sub","k":"theorem","m":"Causalean.PO.ID.Exact.HeckmanRoy.Wald","a":"PO/ID/Exact/HeckmanRoy","t":2,"d":"Pointwise: for q₀ ≤ q₁, 1_{U ω ≤ q₁} - 1_{U ω ≤ q₀} = 1_{q₀ < U ω ≤ q₁}.","s":"∀ {P : Causalean.PO.POSystem} {α : Type u_1} [inst : MeasurableSpace α] [inst_1 : MeasurableSingletonClass α] (S : Causalean.PO.POHeckmanRoySystem P α) (z₀ z₁ : α), S.p z₀ ≤ S.p z₁ → ∀ (ω : P.Ω), ((if S.factualU ω ≤ S.p z₁ then 1 else 0) - if S.factualU ω ≤ S.p z₀ then 1 else 0) = (S.intervalComplierEvent z₀ z₁).indicator (fun x => 1) ω","l":61,"f":"Causalean/PO/ID/Exact/HeckmanRoy/Wald.lean"},{"n":"Causalean.PO.POHeckmanRoySystem.measurableSet_intervalComplierEvent","k":"theorem","m":"Causalean.PO.ID.Exact.HeckmanRoy.Setup","a":"PO/ID/Exact/HeckmanRoy","t":2,"d":"The latent interval complier event is measurable.","s":"∀ {P : Causalean.PO.POSystem} {α : Type u_1} [inst : MeasurableSpace α] [inst_1 : MeasurableSingletonClass α] (S : Causalean.PO.POHeckmanRoySystem P α) (z₀ z₁ : α), MeasurableSet (S.intervalComplierEvent z₀ z₁)","l":153,"f":"Causalean/PO/ID/Exact/HeckmanRoy/Setup.lean"},{"n":"Causalean.PO.POHeckmanRoySystem.measurableSet_zEvent","k":"theorem","m":"Causalean.PO.ID.Exact.HeckmanRoy.Setup","a":"PO/ID/Exact/HeckmanRoy","t":2,"d":"The factual instrument event is measurable.","s":"∀ {P : Causalean.PO.POSystem} {α : Type u_1} [inst : MeasurableSpace α] [inst_1 : MeasurableSingletonClass α] (S : Causalean.PO.POHeckmanRoySystem P α) (z : α), MeasurableSet (S.zEvent z)","l":149,"f":"Causalean/PO/ID/Exact/HeckmanRoy/Setup.lean"},{"n":"Causalean.PO.POHeckmanRoySystem.measurable_DofZ","k":"theorem","m":"Causalean.PO.ID.Exact.HeckmanRoy.Setup","a":"PO/ID/Exact/HeckmanRoy","t":1,"d":"For a fixed instrument value z, the potential treatment D(z) is measurable.","s":"∀ {P : Causalean.PO.POSystem} {α : Type u_1} [inst : MeasurableSpace α] [inst_1 : MeasurableSingletonClass α] (S : Causalean.PO.POHeckmanRoySystem P α) (z : α), Measurable (S.DofZ z)","l":131,"f":"Causalean/PO/ID/Exact/HeckmanRoy/Setup.lean"},{"n":"Causalean.PO.POHeckmanRoySystem.measurable_YofD","k":"theorem","m":"Causalean.PO.ID.Exact.HeckmanRoy.Setup","a":"PO/ID/Exact/HeckmanRoy","t":2,"d":"The potential outcome under a fixed treatment value is measurable.","s":"∀ {P : Causalean.PO.POSystem} {α : Type u_1} [inst : MeasurableSpace α] [inst_1 : MeasurableSingletonClass α] (S : Causalean.PO.POHeckmanRoySystem P α) (d : Bool), Measurable (S.YofD d)","l":136,"f":"Causalean/PO/ID/Exact/HeckmanRoy/Setup.lean"},{"n":"Causalean.PO.POHeckmanRoySystem.measurable_YofDofZ","k":"theorem","m":"Causalean.PO.ID.Exact.HeckmanRoy.Setup","a":"PO/ID/Exact/HeckmanRoy","t":2,"d":"The outcome composed with the instrument-induced treatment is measurable.","s":"∀ {P : Causalean.PO.POSystem} {α : Type u_1} [inst : MeasurableSpace α] [inst_1 : MeasurableSingletonClass α] (S : Causalean.PO.POHeckmanRoySystem P α) (z : α), Measurable (S.YofDofZ z)","l":166,"f":"Causalean/PO/ID/Exact/HeckmanRoy/Setup.lean"},{"n":"Causalean.PO.POHeckmanRoySystem.measurable_factualD","k":"theorem","m":"Causalean.PO.ID.Exact.HeckmanRoy.Setup","a":"PO/ID/Exact/HeckmanRoy","t":2,"d":"The factual treatment is measurable.","s":"∀ {P : Causalean.PO.POSystem} {α : Type u_1} [inst : MeasurableSpace α] [inst_1 : MeasurableSingletonClass α] (S : Causalean.PO.POHeckmanRoySystem P α), Measurable S.factualD","l":142,"f":"Causalean/PO/ID/Exact/HeckmanRoy/Setup.lean"},{"n":"Causalean.PO.POHeckmanRoySystem.measurable_factualU","k":"theorem","m":"Causalean.PO.ID.Exact.HeckmanRoy.Setup","a":"PO/ID/Exact/HeckmanRoy","t":2,"d":"The factual latent rank is measurable.","s":"∀ {P : Causalean.PO.POSystem} {α : Type u_1} [inst : MeasurableSpace α] [inst_1 : MeasurableSingletonClass α] (S : Causalean.PO.POHeckmanRoySystem P α), Measurable S.factualU","l":146,"f":"Causalean/PO/ID/Exact/HeckmanRoy/Setup.lean"},{"n":"Causalean.PO.POHeckmanRoySystem.measurable_factualY","k":"theorem","m":"Causalean.PO.ID.Exact.HeckmanRoy.Setup","a":"PO/ID/Exact/HeckmanRoy","t":2,"d":"The factual outcome is measurable.","s":"∀ {P : Causalean.PO.POSystem} {α : Type u_1} [inst : MeasurableSpace α] [inst_1 : MeasurableSingletonClass α] (S : Causalean.PO.POHeckmanRoySystem P α), Measurable S.factualY","l":144,"f":"Causalean/PO/ID/Exact/HeckmanRoy/Setup.lean"},{"n":"Causalean.PO.POHeckmanRoySystem.measurable_factualZ","k":"theorem","m":"Causalean.PO.ID.Exact.HeckmanRoy.Setup","a":"PO/ID/Exact/HeckmanRoy","t":2,"d":"The factual instrument is measurable.","s":"∀ {P : Causalean.PO.POSystem} {α : Type u_1} [inst : MeasurableSpace α] [inst_1 : MeasurableSingletonClass α] (S : Causalean.PO.POHeckmanRoySystem P α), Measurable S.factualZ","l":140,"f":"Causalean/PO/ID/Exact/HeckmanRoy/Setup.lean"},{"n":"Causalean.PO.POHeckmanRoySystem.pointwise_threshold_identity","k":"theorem","m":"Causalean.PO.ID.Exact.HeckmanRoy.Wald","a":"PO/ID/Exact/HeckmanRoy","t":2,"d":"Pointwise threshold identity: Y(D(z₁)) - Y(D(z₀)) = (Y(1) - Y(0)) · 1_{p z₀ < U ≤ p z₁} a.s.","s":"∀ {P : Causalean.PO.POSystem} {α : Type u_1} [inst : MeasurableSpace α] [inst_1 : MeasurableSingletonClass α] (S : Causalean.PO.POHeckmanRoySystem P α), S.Assumptions → ∀ (z₀ z₁ : α), S.p z₀ ≤ S.p z₁ → ∀ᵐ (ω : P.Ω) ∂P.μ, S.YofDofZ z₁ ω - S.YofDofZ z₀ ω = (S.YofD true ω - S.YofD false ω) * (S.intervalComplierEvent z₀ z₁).indicator (fun x => 1) ω","l":381,"f":"Causalean/PO/ID/Exact/HeckmanRoy/Wald.lean"},{"n":"Causalean.PO.POHeckmanRoySystem.reduced_form_identity","k":"theorem","m":"Causalean.PO.ID.Exact.HeckmanRoy.Wald","a":"PO/ID/Exact/HeckmanRoy","t":2,"d":"Reduced-form identity: E[Y | Z=z₁] - E[Y | Z=z₀] = ∫ (Y(D(z₁)) - Y(D(z₀))) ∂μ.","s":"∀ {P : Causalean.PO.POSystem} {α : Type u_1} [inst : MeasurableSpace α] [inst_1 : MeasurableSingletonClass α] (S : Causalean.PO.POHeckmanRoySystem P α), S.Assumptions → ∀ (z₀ z₁ : α), 0 < (P.μ (S.zEvent z₀)).toReal → 0 < (P.μ (S.zEvent z₁)).toReal → MeasureTheory.Integrable (S.YofD true) P.μ → MeasureTheory.Integrable (S.YofD false) P.μ → S.condExpYZ z₁ - S.condExpYZ z₀ = ∫ (ω : P.Ω), S.YofDofZ z₁ ω - S.YofDofZ z₀ ω ∂P.μ","l":251,"f":"Causalean/PO/ID/Exact/HeckmanRoy/Wald.lean"},{"n":"Causalean.PO.POHeckmanRoySystem.treatmentRegime","k":"def","m":"Causalean.PO.ID.Exact.HeckmanRoy.Setup","a":"PO/ID/Exact/HeckmanRoy","t":1,"d":"Regime fixing treatment to d.","s":"{P : Causalean.PO.POSystem} → {α : Type u_1} → [inst : MeasurableSpace α] → [inst_1 : MeasurableSingletonClass α] → Causalean.PO.POHeckmanRoySystem P α → Bool → Causalean.PO.Regime P.V P.X","l":100,"f":"Causalean/PO/ID/Exact/HeckmanRoy/Setup.lean"},{"n":"Causalean.PO.POHeckmanRoySystem.uVar","k":"def","m":"Causalean.PO.ID.Exact.HeckmanRoy.Setup","a":"PO/ID/Exact/HeckmanRoy","t":1,"d":"Latent rank packaged as a POVar valued in ℝ.","s":"{P : Causalean.PO.POSystem} → {α : Type u_1} → [inst : MeasurableSpace α] → [inst_1 : MeasurableSingletonClass α] → Causalean.PO.POHeckmanRoySystem P α → Causalean.PO.POVar P ℝ","l":91,"f":"Causalean/PO/ID/Exact/HeckmanRoy/Setup.lean"},{"n":"Causalean.PO.POHeckmanRoySystem.wald_pairwise","k":"theorem","m":"Causalean.PO.ID.Exact.HeckmanRoy.Wald","a":"PO/ID/Exact/HeckmanRoy","t":1,"d":"Pairwise Wald identification of LATE (prop:po-iv-heckman-roy-wald). Under the Heckman–Roy identifying assumption bundle, for instrument values z₀, z₁ at which the event {Z = z₀} has positive probability, the event {Z = z₁} has positive probability, and the latent selection threshold at z₀ is strictly below the threshold at z₁, provided the potential outcome under treatment and the potential outcome under control are integrable, the Wald ratio of the conditional-mean outcome and treatment contrasts between Z = z₁ and Z = z₀ equals the pairwise local average treatment effect LATE z₀ z₁.","s":"∀ {P : Causalean.PO.POSystem} {α : Type u_1} [inst : MeasurableSpace α] [inst_1 : MeasurableSingletonClass α] (S : Causalean.PO.POHeckmanRoySystem P α), S.Assumptions → ∀ (z₀ z₁ : α), 0 < (P.μ (S.zEvent z₀)).toReal → 0 < (P.μ (S.zEvent z₁)).toReal → S.p z₀ < S.p z₁ → MeasureTheory.Integrable (S.YofD true) P.μ → MeasureTheory.Integrable (S.YofD false) P.μ → (S.condExpYZ z₁ - S.condExpYZ z₀) / (S.condExpDZ z₁ - S.condExpDZ z₀) = S.LATE z₀ z₁","l":468,"f":"Causalean/PO/ID/Exact/HeckmanRoy/Wald.lean"},{"n":"Causalean.PO.POHeckmanRoySystem.yUnderD","k":"def","m":"Causalean.PO.ID.Exact.HeckmanRoy.Setup","a":"PO/ID/Exact/HeckmanRoy","t":1,"d":"Regimed variable for Y(d) (outcome under treatment set to d).","s":"{P : Causalean.PO.POSystem} → {α : Type u_1} → [inst : MeasurableSpace α] → [inst_1 : MeasurableSingletonClass α] → Causalean.PO.POHeckmanRoySystem P α → Bool → Causalean.PO.RegimedVar P ℝ","l":184,"f":"Causalean/PO/ID/Exact/HeckmanRoy/Setup.lean"},{"n":"Causalean.PO.POHeckmanRoySystem.yVar","k":"def","m":"Causalean.PO.ID.Exact.HeckmanRoy.Setup","a":"PO/ID/Exact/HeckmanRoy","t":1,"d":"Outcome packaged as a POVar valued in ℝ.","s":"{P : Causalean.PO.POSystem} → {α : Type u_1} → [inst : MeasurableSpace α] → [inst_1 : MeasurableSingletonClass α] → Causalean.PO.POHeckmanRoySystem P α → Causalean.PO.POVar P ℝ","l":88,"f":"Causalean/PO/ID/Exact/HeckmanRoy/Setup.lean"},{"n":"Causalean.PO.POHeckmanRoySystem.zEvent","k":"def","m":"Causalean.PO.ID.Exact.HeckmanRoy.Setup","a":"PO/ID/Exact/HeckmanRoy","t":1,"d":"The event {Z = z}.","s":"{P : Causalean.PO.POSystem} → {α : Type u_1} → [inst : MeasurableSpace α] → [inst_1 : MeasurableSingletonClass α] → Causalean.PO.POHeckmanRoySystem P α → α → Set P.Ω","l":122,"f":"Causalean/PO/ID/Exact/HeckmanRoy/Setup.lean"},{"n":"Causalean.PO.POHeckmanRoySystem.zVar","k":"def","m":"Causalean.PO.ID.Exact.HeckmanRoy.Setup","a":"PO/ID/Exact/HeckmanRoy","t":1,"d":"Instrument packaged as a POVar valued in α.","s":"{P : Causalean.PO.POSystem} → {α : Type u_1} → [inst : MeasurableSpace α] → [inst_1 : MeasurableSingletonClass α] → Causalean.PO.POHeckmanRoySystem P α → Causalean.PO.POVar P α","l":82,"f":"Causalean/PO/ID/Exact/HeckmanRoy/Setup.lean"},{"n":"Causalean.PO.POIVSystem","k":"structure","m":"Causalean.PO.ID.Exact.LATE","a":"PO/ID/Exact","t":1,"d":"A binary instrumental-variables subsystem (def:po-iv-system) records, within an ambient potential-outcome system, an instrument, a treatment, and an outcome, where the three nodes are required to be pairwise distinct.","s":"Causalean.PO.POSystem → Type (max u_1 u_2)","l":37,"f":"Causalean/PO/ID/Exact/LATE.lean"},{"n":"Causalean.PO.POIVSystem.Assumptions","k":"structure","m":"Causalean.PO.ID.Exact.LATE","a":"PO/ID/Exact","t":1,"d":"Classical binary-instrument IV assumptions (def:po-iv-assumptions). Bundles consistency (SUTVA): the observed treatment and outcome equal the realized potential treatment and outcome, instrument exogeneity: the instrument is independent of the full counterfactual bundle of potential treatments and outcomes, monotonicity (no defiers): turning the instrument on never moves a unit out of treatment, and relevance: the complier event has positive probability.","s":"{P : Causalean.PO.POSystem} → Causalean.PO.POIVSystem P → Prop","l":159,"f":"Causalean/PO/ID/Exact/LATE.lean"},{"n":"Causalean.PO.POIVSystem.DofZ","k":"def","m":"Causalean.PO.ID.Exact.LATE","a":"PO/ID/Exact","t":1,"d":"The potential treatment under an instrument value is the treatment that would be observed if the instrument were fixed to that value.","s":"{P : Causalean.PO.POSystem} → Causalean.PO.POIVSystem P → Bool → P.Ω → Bool","l":72,"f":"Causalean/PO/ID/Exact/LATE.lean"},{"n":"Causalean.PO.POIVSystem.DofZ_eq_factualD_on_zEvent","k":"theorem","m":"Causalean.PO.ID.Exact.LATE","a":"PO/ID/Exact","t":2,"d":"On zEvent z, the counterfactual treatment D(z) equals the factual D. Pointwise specialization of Consistency.factual with r = instrumentRegime z, Y = {D}.","s":"∀ {P : Causalean.PO.POSystem} (S : Causalean.PO.POIVSystem P), S.Assumptions → ∀ (z : Bool) {ω : P.Ω}, ω ∈ S.zEvent z → S.DofZ z ω = S.factualD ω","l":202,"f":"Causalean/PO/ID/Exact/LATE.lean"},{"n":"Causalean.PO.POIVSystem.LATE","k":"def","m":"Causalean.PO.ID.Exact.LATE","a":"PO/ID/Exact","t":1,"d":"Local Average Treatment Effect -- def:po-late.","s":"{P : Causalean.PO.POSystem} → Causalean.PO.POIVSystem P → ℝ","l":193,"f":"Causalean/PO/ID/Exact/LATE.lean"},{"n":"Causalean.PO.POIVSystem.YofD","k":"def","m":"Causalean.PO.ID.Exact.LATE","a":"PO/ID/Exact","t":1,"d":"The potential outcome under a treatment value is the outcome that would be observed if treatment were fixed to that value.","s":"{P : Causalean.PO.POSystem} → Causalean.PO.POIVSystem P → Bool → P.Ω → ℝ","l":79,"f":"Causalean/PO/ID/Exact/LATE.lean"},{"n":"Causalean.PO.POIVSystem.YofDofZ","k":"def","m":"Causalean.PO.ID.Exact.LATE","a":"PO/ID/Exact","t":1,"d":"Y composed with D(z): 1_{D(z)=1} Y(1) + 1_{D(z)=0} Y(0).","s":"{P : Causalean.PO.POSystem} → Causalean.PO.POIVSystem P → Bool → P.Ω → ℝ","l":128,"f":"Causalean/PO/ID/Exact/LATE.lean"},{"n":"Causalean.PO.POIVSystem.cfBundle","k":"def","m":"Causalean.PO.ID.Exact.LATE","a":"PO/ID/Exact","t":1,"d":"Counterfactual bundle (D(1), D(0), Y(1), Y(0)).","s":"{P : Causalean.PO.POSystem} → Causalean.PO.POIVSystem P → Causalean.PO.POCFBundle P","l":151,"f":"Causalean/PO/ID/Exact/LATE.lean"},{"n":"Causalean.PO.POIVSystem.complierEvent","k":"def","m":"Causalean.PO.ID.Exact.LATE","a":"PO/ID/Exact","t":1,"d":"Complier event {ω | D(1)(ω) = 1 ∧ D(0)(ω) = 0} -- def:po-late.","s":"{P : Causalean.PO.POSystem} → Causalean.PO.POIVSystem P → Set P.Ω","l":95,"f":"Causalean/PO/ID/Exact/LATE.lean"},{"n":"Causalean.PO.POIVSystem.condExpDZ","k":"def","m":"Causalean.PO.ID.Exact.LATE","a":"PO/ID/Exact","t":1,"d":"E[D | Z = z], the event-level conditional expectation of the (0/1-coded) factual treatment on {Z = z}. Uses the shared PO conditioning tool eventCondExp (definitionally (∫_A g)/μ(A)).","s":"{P : Causalean.PO.POSystem} → Causalean.PO.POIVSystem P → Bool → ℝ","l":132,"f":"Causalean/PO/ID/Exact/LATE.lean"},{"n":"Causalean.PO.POIVSystem.condExpYZ","k":"def","m":"Causalean.PO.ID.Exact.LATE","a":"PO/ID/Exact","t":1,"d":"E[Y | Z = z], the event-level conditional expectation of the factual outcome on {Z = z}, via the shared PO conditioning tool eventCondExp.","s":"{P : Causalean.PO.POSystem} → Causalean.PO.POIVSystem P → Bool → ℝ","l":138,"f":"Causalean/PO/ID/Exact/LATE.lean"},{"n":"Causalean.PO.POIVSystem.dUnderZ","k":"def","m":"Causalean.PO.ID.Exact.LATE","a":"PO/ID/Exact","t":1,"d":"Regimed variable for D(z) (treatment under instrument set to z).","s":"{P : Causalean.PO.POSystem} → Causalean.PO.POIVSystem P → Bool → Causalean.PO.RegimedVar P Bool","l":143,"f":"Causalean/PO/ID/Exact/LATE.lean"},{"n":"Causalean.PO.POIVSystem.dVar","k":"def","m":"Causalean.PO.ID.Exact.LATE","a":"PO/ID/Exact","t":1,"d":"Treatment packaged as a POVar valued in Bool.","s":"{P : Causalean.PO.POSystem} → Causalean.PO.POIVSystem P → Causalean.PO.POVar P Bool","l":58,"f":"Causalean/PO/ID/Exact/LATE.lean"},{"n":"Causalean.PO.POIVSystem.event_conditioning_identity","k":"theorem","m":"Causalean.PO.ID.Exact.LATE","a":"PO/ID/Exact","t":2,"d":"Step 4 of rem:po-late: event-conditioning identity. E[(Y(1) - Y(0)) · 1_C] = P(C) · LATE.","s":"∀ {P : Causalean.PO.POSystem} (S : Causalean.PO.POIVSystem P), ∫ (ω : P.Ω), (S.YofD true ω - S.YofD false ω) * S.complierEvent.indicator (fun x => 1) ω ∂P.μ = (P.μ S.complierEvent).toReal * S.LATE","l":401,"f":"Causalean/PO/ID/Exact/LATE.lean"},{"n":"Causalean.PO.POIVSystem.factualD","k":"def","m":"Causalean.PO.ID.Exact.LATE","a":"PO/ID/Exact","t":1,"d":"Factual treatment D.","s":"{P : Causalean.PO.POSystem} → Causalean.PO.POIVSystem P → P.Ω → Bool","l":89,"f":"Causalean/PO/ID/Exact/LATE.lean"},{"n":"Causalean.PO.POIVSystem.factualY","k":"def","m":"Causalean.PO.ID.Exact.LATE","a":"PO/ID/Exact","t":1,"d":"Factual outcome Y.","s":"{P : Causalean.PO.POSystem} → Causalean.PO.POIVSystem P → P.Ω → ℝ","l":92,"f":"Causalean/PO/ID/Exact/LATE.lean"},{"n":"Causalean.PO.POIVSystem.factualY_eq_YofD_factualD","k":"theorem","m":"Causalean.PO.ID.Exact.LATE","a":"PO/ID/Exact","t":2,"d":"Factual Y equals the counterfactual Y(factualD ω). Pointwise specialization of Consistency.factual with r = treatmentRegime (factualD ω), Y = {Y}.","s":"∀ {P : Causalean.PO.POSystem} (S : Causalean.PO.POIVSystem P), S.Assumptions → ∀ (ω : P.Ω), S.factualY ω = S.YofD (S.factualD ω) ω","l":210,"f":"Causalean/PO/ID/Exact/LATE.lean"},{"n":"Causalean.PO.POIVSystem.factualZ","k":"def","m":"Causalean.PO.ID.Exact.LATE","a":"PO/ID/Exact","t":1,"d":"Factual instrument Z.","s":"{P : Causalean.PO.POSystem} → Causalean.PO.POIVSystem P → P.Ω → Bool","l":86,"f":"Causalean/PO/ID/Exact/LATE.lean"},{"n":"Causalean.PO.POIVSystem.first_stage_identity","k":"theorem","m":"Causalean.PO.ID.Exact.LATE","a":"PO/ID/Exact","t":2,"d":"Step 1 of rem:po-late: first-stage identity. E[D | Z=1] - E[D | Z=0] = P(C).","s":"∀ {P : Causalean.PO.POSystem} (S : Causalean.PO.POIVSystem P), S.Assumptions → 0 < (P.μ (S.zEvent true)).toReal → 0 < (P.μ (S.zEvent false)).toReal → S.condExpDZ true - S.condExpDZ false = (P.μ S.complierEvent).toReal","l":217,"f":"Causalean/PO/ID/Exact/LATE.lean"},{"n":"Causalean.PO.POIVSystem.instrumentRegime","k":"def","m":"Causalean.PO.ID.Exact.LATE","a":"PO/ID/Exact","t":1,"d":"Regime fixing instrument to z.","s":"{P : Causalean.PO.POSystem} → Causalean.PO.POIVSystem P → Bool → Causalean.PO.Regime P.V P.X","l":64,"f":"Causalean/PO/ID/Exact/LATE.lean"},{"n":"Causalean.PO.POIVSystem.late_wald","k":"theorem","m":"Causalean.PO.ID.Exact.LATE","a":"PO/ID/Exact","t":1,"d":"Wald identification of LATE (prop:po-late). Under the binary- instrument LATE identifying assumption bundle, when the event {Z=1} has positive probability, the event {Z=0} has positive probability, and the potential outcomes under treatment and under control are integrable, the Wald ratio (E[Y|Z=1] − E[Y|Z=0]) / (E[D|Z=1] − E[D|Z=0]) equals the local average treatment effect LATE.","s":"∀ {P : Causalean.PO.POSystem} (S : Causalean.PO.POIVSystem P), S.Assumptions → 0 < (P.μ (S.zEvent true)).toReal → 0 < (P.μ (S.zEvent false)).toReal → MeasureTheory.Integrable (S.YofD true) P.μ → MeasureTheory.Integrable (S.YofD false) P.μ → (S.condExpYZ true - S.condExpYZ false) / (S.condExpDZ true - S.condExpDZ false) = S.LATE","l":428,"f":"Causalean/PO/ID/Exact/LATE.lean"},{"n":"Causalean.PO.POIVSystem.measurableSet_complierEvent","k":"theorem","m":"Causalean.PO.ID.Exact.LATE","a":"PO/ID/Exact","t":2,"d":"The complier event is measurable.","s":"∀ {P : Causalean.PO.POSystem} (S : Causalean.PO.POIVSystem P), MeasurableSet S.complierEvent","l":119,"f":"Causalean/PO/ID/Exact/LATE.lean"},{"n":"Causalean.PO.POIVSystem.measurableSet_zEvent","k":"theorem","m":"Causalean.PO.ID.Exact.LATE","a":"PO/ID/Exact","t":2,"d":"The factual instrument event is measurable.","s":"∀ {P : Causalean.PO.POSystem} (S : Causalean.PO.POIVSystem P) (z : Bool), MeasurableSet (S.zEvent z)","l":124,"f":"Causalean/PO/ID/Exact/LATE.lean"},{"n":"Causalean.PO.POIVSystem.measurable_DofZ","k":"theorem","m":"Causalean.PO.ID.Exact.LATE","a":"PO/ID/Exact","t":2,"d":"The potential treatment under a fixed instrument value is measurable.","s":"∀ {P : Causalean.PO.POSystem} (S : Causalean.PO.POIVSystem P) (z : Bool), Measurable (S.DofZ z)","l":102,"f":"Causalean/PO/ID/Exact/LATE.lean"},{"n":"Causalean.PO.POIVSystem.measurable_YofD","k":"theorem","m":"Causalean.PO.ID.Exact.LATE","a":"PO/ID/Exact","t":2,"d":"The potential outcome under a fixed treatment value is measurable.","s":"∀ {P : Causalean.PO.POSystem} (S : Causalean.PO.POIVSystem P) (d : Bool), Measurable (S.YofD d)","l":115,"f":"Causalean/PO/ID/Exact/LATE.lean"},{"n":"Causalean.PO.POIVSystem.measurable_factualD","k":"theorem","m":"Causalean.PO.ID.Exact.LATE","a":"PO/ID/Exact","t":2,"d":"The factual treatment is measurable.","s":"∀ {P : Causalean.PO.POSystem} (S : Causalean.PO.POIVSystem P), Measurable S.factualD","l":109,"f":"Causalean/PO/ID/Exact/LATE.lean"},{"n":"Causalean.PO.POIVSystem.measurable_factualY","k":"theorem","m":"Causalean.PO.ID.Exact.LATE","a":"PO/ID/Exact","t":2,"d":"The factual outcome is measurable.","s":"∀ {P : Causalean.PO.POSystem} (S : Causalean.PO.POIVSystem P), Measurable S.factualY","l":112,"f":"Causalean/PO/ID/Exact/LATE.lean"},{"n":"Causalean.PO.POIVSystem.measurable_factualZ","k":"theorem","m":"Causalean.PO.ID.Exact.LATE","a":"PO/ID/Exact","t":2,"d":"The factual instrument is measurable.","s":"∀ {P : Causalean.PO.POSystem} (S : Causalean.PO.POIVSystem P), Measurable S.factualZ","l":106,"f":"Causalean/PO/ID/Exact/LATE.lean"},{"n":"Causalean.PO.POIVSystem.pointwise_monotonicity","k":"theorem","m":"Causalean.PO.ID.Exact.LATE","a":"PO/ID/Exact","t":2,"d":"Step 3 of rem:po-late: pointwise monotonicity identity. Y(D(1)) - Y(D(0)) = (Y(1) - Y(0)) · 1_C almost surely.","s":"∀ {P : Causalean.PO.POSystem} (S : Causalean.PO.POIVSystem P), S.Assumptions → ∀ᵐ (ω : P.Ω) ∂P.μ, S.YofDofZ true ω - S.YofDofZ false ω = (S.YofD true ω - S.YofD false ω) * S.complierEvent.indicator (fun x => 1) ω","l":389,"f":"Causalean/PO/ID/Exact/LATE.lean"},{"n":"Causalean.PO.POIVSystem.reduced_form_identity","k":"theorem","m":"Causalean.PO.ID.Exact.LATE","a":"PO/ID/Exact","t":2,"d":"Step 2 of rem:po-late: reduced-form identity. E[Y | Z=1] - E[Y | Z=0] = E[Y(D(1)) - Y(D(0))].","s":"∀ {P : Causalean.PO.POSystem} (S : Causalean.PO.POIVSystem P), S.Assumptions → 0 < (P.μ (S.zEvent true)).toReal → 0 < (P.μ (S.zEvent false)).toReal → MeasureTheory.Integrable (S.YofD true) P.μ → MeasureTheory.Integrable (S.YofD false) P.μ → S.condExpYZ true - S.condExpYZ false = ∫ (ω : P.Ω), S.YofDofZ true ω - S.YofDofZ false ω ∂P.μ","l":302,"f":"Causalean/PO/ID/Exact/LATE.lean"},{"n":"Causalean.PO.POIVSystem.treatmentRegime","k":"def","m":"Causalean.PO.ID.Exact.LATE","a":"PO/ID/Exact","t":1,"d":"Regime fixing treatment to d.","s":"{P : Causalean.PO.POSystem} → Causalean.PO.POIVSystem P → Bool → Causalean.PO.Regime P.V P.X","l":68,"f":"Causalean/PO/ID/Exact/LATE.lean"},{"n":"Causalean.PO.POIVSystem.yUnderD","k":"def","m":"Causalean.PO.ID.Exact.LATE","a":"PO/ID/Exact","t":1,"d":"Regimed variable for Y(d) (outcome under treatment set to d).","s":"{P : Causalean.PO.POSystem} → Causalean.PO.POIVSystem P → Bool → Causalean.PO.RegimedVar P ℝ","l":147,"f":"Causalean/PO/ID/Exact/LATE.lean"},{"n":"Causalean.PO.POIVSystem.yVar","k":"def","m":"Causalean.PO.ID.Exact.LATE","a":"PO/ID/Exact","t":1,"d":"Outcome packaged as a POVar valued in ℝ.","s":"{P : Causalean.PO.POSystem} → Causalean.PO.POIVSystem P → Causalean.PO.POVar P ℝ","l":61,"f":"Causalean/PO/ID/Exact/LATE.lean"},{"n":"Causalean.PO.POIVSystem.zEvent","k":"def","m":"Causalean.PO.ID.Exact.LATE","a":"PO/ID/Exact","t":1,"d":"Set {ω | Z(ω) = z}.","s":"{P : Causalean.PO.POSystem} → Causalean.PO.POIVSystem P → Bool → Set P.Ω","l":99,"f":"Causalean/PO/ID/Exact/LATE.lean"},{"n":"Causalean.PO.POIVSystem.zVar","k":"def","m":"Causalean.PO.ID.Exact.LATE","a":"PO/ID/Exact","t":1,"d":"Instrument packaged as a POVar valued in Bool.","s":"{P : Causalean.PO.POSystem} → Causalean.PO.POIVSystem P → Causalean.PO.POVar P Bool","l":55,"f":"Causalean/PO/ID/Exact/LATE.lean"},{"n":"Causalean.PO.POLeeSystem","k":"structure","m":"Causalean.PO.ID.Partial.Lee.Setup","a":"PO/ID/Partial/Lee","t":1,"d":"The data layer for Lee (2009) bounds: a treatment-selection model in which a binary treatment A affects whether an outcome Y is observed at all, through a binary sample-selection indicator Sel. The outcome Y is only meaningful when Sel = true (e.g. a wage observed only for the employed), so the treatment effect on Y among the always-selected subpopulation is only partially identified — the object the Lee bounds bracket. Formally this packages, inside an ambient PO system P, the three nodes A, Sel, Y, the measurable identifications of their value spaces with Bool/Bool/ℝ, and the fact that the three nodes are distinct (def:po-lee-system).","s":"Causalean.PO.POSystem → Type (max u_1 u_2)","l":33,"f":"Causalean/PO/ID/Partial/Lee/Setup.lean"},{"n":"Causalean.PO.POLeeSystem.BaseAssumptions","k":"structure","m":"Causalean.PO.ID.Partial.Lee.Assumptions","a":"PO/ID/Partial/Lee","t":1,"d":"Lee sample-selection baseline assumptions -- def:po-lee-assumptions.","s":"{P : Causalean.PO.POSystem} → Causalean.PO.POLeeSystem P → Prop","l":53,"f":"Causalean/PO/ID/Partial/Lee/Assumptions.lean"},{"n":"Causalean.PO.POLeeSystem.BaseAssumptions.integrableY","k":"theorem","m":"Causalean.PO.ID.Partial.Lee.Assumptions","a":"PO/ID/Partial/Lee","t":1,"d":"Under the baseline Lee sample-selection assumptions — consistency, pair-level random assignment, positive and finite selected cells, and integrability of both potential outcomes Y(0), Y(1), the factual outcome Y is integrable, obtained from the arm integrability of Y(0), Y(1) via consistency (factualY = Σ_a Y(a)·1{A=a} a.e.), so it need not be assumed separately.","s":"∀ {P : Causalean.PO.POSystem} {S : Causalean.PO.POLeeSystem P}, S.BaseAssumptions → MeasureTheory.Integrable S.factualY P.μ","l":92,"f":"Causalean/PO/ID/Partial/Lee/Assumptions.lean"},{"n":"Causalean.PO.POLeeSystem.BaseAssumptions.integrable_YofA","k":"theorem","m":"Causalean.PO.ID.Partial.Lee.Assumptions","a":"PO/ID/Partial/Lee","t":2,"d":"Binary-folded form of integrableY1 / integrableY0.","s":"∀ {P : Causalean.PO.POSystem} {S : Causalean.PO.POLeeSystem P}, S.BaseAssumptions → ∀ (a : Bool), MeasureTheory.Integrable (S.YofA a) P.μ","l":85,"f":"Causalean/PO/ID/Partial/Lee/Assumptions.lean"},{"n":"Causalean.PO.POLeeSystem.LeeTrimWeight","k":"structure","m":"Causalean.PO.ID.Partial.Lee.Trim","a":"PO/ID/Partial/Lee","t":1,"d":"A Lee trim weight on a finite outcome support 𝒴 -- def:po-lee-trim.","s":"{P : Causalean.PO.POSystem} → Causalean.PO.POLeeSystem P → Finset ℝ → Type","l":66,"f":"Causalean/PO/ID/Partial/Lee/Trim.lean"},{"n":"Causalean.PO.POLeeSystem.MonotoneSelection","k":"structure","m":"Causalean.PO.ID.Partial.Lee.Assumptions","a":"PO/ID/Partial/Lee","t":1,"d":"Monotone sample selection -- def:po-lee-assumptions, item 3. Sel(0) ≤ Sel(1) almost surely (with ≤ interpreted on Bool via the canonical false ≤ true order).","s":"{P : Causalean.PO.POSystem} → Causalean.PO.POLeeSystem P → Prop","l":130,"f":"Causalean/PO/ID/Partial/Lee/Assumptions.lean"},{"n":"Causalean.PO.POLeeSystem.Mw","k":"def","m":"Causalean.PO.ID.Partial.Lee.Trim","a":"PO/ID/Partial/Lee","t":1,"d":"The trimmed mean associated with a Lee trim weight: M(w) := ρ⁻¹ · ∑_{y ∈ 𝒴} y · w(y) · f₁(y).","s":"{P : Causalean.PO.POSystem} → (S : Causalean.PO.POLeeSystem P) → {𝒴 : Finset ℝ} → S.LeeTrimWeight 𝒴 → ℝ","l":78,"f":"Causalean/PO/ID/Partial/Lee/Trim.lean"},{"n":"Causalean.PO.POLeeSystem.Mw_alwaysSelectedTrimWeight_eq_condExp_Y1_AS","k":"theorem","m":"Causalean.PO.ID.Partial.Lee.TrimMean","a":"PO/ID/Partial/Lee","t":1,"d":"The constructed always-selected trim weight has the target latent mean. Given the baseline Lee assumptions, monotone sample selection, and almost-sure finite support 𝒴 for the factual outcome on the selected-treated cell, evaluating the trim-weight mean functional Mw at the constructed always-selected trim weight recovers exactly the conditional mean E[Y(1) | alwaysSelected] of the treated potential outcome among always-selected units.","s":"∀ {P : Causalean.PO.POSystem} (S : Causalean.PO.POLeeSystem P) (hA : S.BaseAssumptions) (hMono : S.MonotoneSelection) (𝒴 : Finset ℝ) (hSupp : ∀ᵐ (ω : P.Ω) ∂P.μ.restrict S.selectedTreated, S.factualY ω ∈ 𝒴), S.Mw (S.alwaysSelectedTrimWeight hA hMono 𝒴 hSupp) = Causalean.PO.eventCondExp P.μ S.alwaysSelected (S.YofA true)","l":33,"f":"Causalean/PO/ID/Partial/Lee/TrimMean.lean"},{"n":"Causalean.PO.POLeeSystem.SelOfA","k":"def","m":"Causalean.PO.ID.Partial.Lee.Setup","a":"PO/ID/Partial/Lee","t":1,"d":"Counterfactual selection under treatment arm a.","s":"{P : Causalean.PO.POSystem} → Causalean.PO.POLeeSystem P → Bool → P.Ω → Bool","l":82,"f":"Causalean/PO/ID/Partial/Lee/Setup.lean"},{"n":"Causalean.PO.POLeeSystem.YofA","k":"def","m":"Causalean.PO.ID.Partial.Lee.Setup","a":"PO/ID/Partial/Lee","t":1,"d":"Counterfactual outcome under treatment arm a.","s":"{P : Causalean.PO.POSystem} → Causalean.PO.POLeeSystem P → Bool → P.Ω → ℝ","l":87,"f":"Causalean/PO/ID/Partial/Lee/Setup.lean"},{"n":"Causalean.PO.POLeeSystem.YofA_true_in_finset_ae_alwaysSelected","k":"theorem","m":"Causalean.PO.ID.Partial.Lee.LatentSupport","a":"PO/ID/Partial/Lee","t":1,"d":"Support transfer. Under the baseline Lee assumptions, if the factual outcome lies a.e. in a finite support set 𝒴, when restricted to the observable selected-treated cell, then the latent treated potential outcome Y(1) lies a.e. in the same support 𝒴, both on the always-selected stratum and on the helped-selected stratum. This feeds the sum_eq field of alwaysSelectedTrimWeight, which needs ∑ y ∈ 𝒴, f1AS y = 1.","s":"∀ {P : Causalean.PO.POSystem} (S : Causalean.PO.POLeeSystem P), S.BaseAssumptions → ∀ (𝒴 : Finset ℝ), (∀ᵐ (ω : P.Ω) ∂P.μ.restrict S.selectedTreated, S.factualY ω ∈ 𝒴) → (∀ᵐ (ω : P.Ω) ∂P.μ.restrict S.alwaysSelected, S.YofA true ω ∈ 𝒴) ∧ ∀ᵐ (ω : P.Ω) ∂P.μ.restrict S.helpedSelected, S.YofA true ω ∈ 𝒴","l":41,"f":"Causalean/PO/ID/Partial/Lee/LatentSupport.lean"},{"n":"Causalean.PO.POLeeSystem.aEvent","k":"def","m":"Causalean.PO.ID.Partial.Lee.Setup","a":"PO/ID/Partial/Lee","t":1,"d":"The event {A = a}.","s":"{P : Causalean.PO.POSystem} → Causalean.PO.POLeeSystem P → Bool → Set P.Ω","l":101,"f":"Causalean/PO/ID/Partial/Lee/Setup.lean"},{"n":"Causalean.PO.POLeeSystem.aVar","k":"def","m":"Causalean.PO.ID.Partial.Lee.Setup","a":"PO/ID/Partial/Lee","t":1,"d":"Treatment packaged as a POVar valued in Bool.","s":"{P : Causalean.PO.POSystem} → Causalean.PO.POLeeSystem P → Causalean.PO.POVar P Bool","l":73,"f":"Causalean/PO/ID/Partial/Lee/Setup.lean"},{"n":"Causalean.PO.POLeeSystem.alwaysSelected","k":"def","m":"Causalean.PO.ID.Partial.Lee.PrincipalStrata","a":"PO/ID/Partial/Lee","t":1,"d":"The always-selected stratum contains units who would be selected under both control and treatment.","s":"{P : Causalean.PO.POSystem} → Causalean.PO.POLeeSystem P → Set P.Ω","l":37,"f":"Causalean/PO/ID/Partial/Lee/PrincipalStrata.lean"},{"n":"Causalean.PO.POLeeSystem.alwaysSelectedTrimWeight","k":"def","m":"Causalean.PO.ID.Partial.Lee.TrimWeight","a":"PO/ID/Partial/Lee","t":1,"d":"The conditional sub-distribution of mass ρ of Y(1) | AS, viewed as a Lee trim weight on 𝒴. The construction yields a LeeTrimWeight whose mean Mw equals E[Y(1) | alwaysSelected].","s":"{P : Causalean.PO.POSystem} → (S : Causalean.PO.POLeeSystem P) → S.BaseAssumptions → S.MonotoneSelection → (𝒴 : Finset ℝ) → (∀ᵐ (ω : P.Ω) ∂P.μ.restrict S.selectedTreated, S.factualY ω ∈ 𝒴) → S.LeeTrimWeight 𝒴","l":36,"f":"Causalean/PO/ID/Partial/Lee/TrimWeight.lean"},{"n":"Causalean.PO.POLeeSystem.disjoint_alwaysSelected_helpedSelected","k":"theorem","m":"Causalean.PO.ID.Partial.Lee.PrincipalStrata","a":"PO/ID/Partial/Lee","t":2,"d":"alwaysSelected and helpedSelected are disjoint.","s":"∀ {P : Causalean.PO.POSystem} (S : Causalean.PO.POLeeSystem P), Disjoint S.alwaysSelected S.helpedSelected","l":113,"f":"Causalean/PO/ID/Partial/Lee/PrincipalStrata.lean"},{"n":"Causalean.PO.POLeeSystem.f1","k":"def","m":"Causalean.PO.ID.Partial.Lee.Trim","a":"PO/ID/Partial/Lee","t":1,"d":"Observable conditional density of the outcome at y among selected treated units: f₁(y) := P(Y = y | A = true, Sel = true).","s":"{P : Causalean.PO.POSystem} → Causalean.PO.POLeeSystem P → ℝ → ℝ","l":60,"f":"Causalean/PO/ID/Partial/Lee/Trim.lean"},{"n":"Causalean.PO.POLeeSystem.f1AS","k":"def","m":"Causalean.PO.ID.Partial.Lee.TrimWeight","a":"PO/ID/Partial/Lee","t":1,"d":"Conditional density of Y(1) on alwaysSelected evaluated at y, expressed as an eventCondExp of an indicator of Y(1) = y.","s":"{P : Causalean.PO.POSystem} → Causalean.PO.POLeeSystem P → ℝ → ℝ","l":30,"f":"Causalean/PO/ID/Partial/Lee/TrimWeight.lean"},{"n":"Causalean.PO.POLeeSystem.factualA","k":"def","m":"Causalean.PO.ID.Partial.Lee.Setup","a":"PO/ID/Partial/Lee","t":1,"d":"Factual treatment.","s":"{P : Causalean.PO.POSystem} → Causalean.PO.POLeeSystem P → P.Ω → Bool","l":92,"f":"Causalean/PO/ID/Partial/Lee/Setup.lean"},{"n":"Causalean.PO.POLeeSystem.factualSel","k":"def","m":"Causalean.PO.ID.Partial.Lee.Setup","a":"PO/ID/Partial/Lee","t":1,"d":"Factual selection indicator.","s":"{P : Causalean.PO.POSystem} → Causalean.PO.POLeeSystem P → P.Ω → Bool","l":95,"f":"Causalean/PO/ID/Partial/Lee/Setup.lean"},{"n":"Causalean.PO.POLeeSystem.factualY","k":"def","m":"Causalean.PO.ID.Partial.Lee.Setup","a":"PO/ID/Partial/Lee","t":1,"d":"Factual outcome.","s":"{P : Causalean.PO.POSystem} → Causalean.PO.POLeeSystem P → P.Ω → ℝ","l":98,"f":"Causalean/PO/ID/Partial/Lee/Setup.lean"},{"n":"Causalean.PO.POLeeSystem.harmedSelected","k":"def","m":"Causalean.PO.ID.Partial.Lee.PrincipalStrata","a":"PO/ID/Partial/Lee","t":1,"d":"The harmed-selected stratum contains units selected under control but not under treatment.","s":"{P : Causalean.PO.POSystem} → Causalean.PO.POLeeSystem P → Set P.Ω","l":51,"f":"Causalean/PO/ID/Partial/Lee/PrincipalStrata.lean"},{"n":"Causalean.PO.POLeeSystem.harmedSelected_ae_empty","k":"theorem","m":"Causalean.PO.ID.Partial.Lee.PrincipalStrata","a":"PO/ID/Partial/Lee","t":1,"d":"If sample selection is monotone, i.e. selection under control implies selection under treatment almost surely (Sel(0) ≤ Sel(1)), then the harmed-selected stratum — units who would be selected under control but not under treatment — has probability zero.","s":"∀ {P : Causalean.PO.POSystem} (S : Causalean.PO.POLeeSystem P), (∀ᵐ (ω : P.Ω) ∂P.μ, S.SelOfA false ω ≤ S.SelOfA true ω) → P.μ S.harmedSelected = 0","l":133,"f":"Causalean/PO/ID/Partial/Lee/PrincipalStrata.lean"},{"n":"Causalean.PO.POLeeSystem.helpedSelected","k":"def","m":"Causalean.PO.ID.Partial.Lee.PrincipalStrata","a":"PO/ID/Partial/Lee","t":1,"d":"The helped-selected stratum contains units selected under treatment but not under control.","s":"{P : Causalean.PO.POSystem} → Causalean.PO.POLeeSystem P → Set P.Ω","l":44,"f":"Causalean/PO/ID/Partial/Lee/PrincipalStrata.lean"},{"n":"Causalean.PO.POLeeSystem.lee_bounds_ATT_AS","k":"theorem","m":"Causalean.PO.ID.Partial.Lee.Main","a":"PO/ID/Partial/Lee","t":1,"d":"Finite-support Lee bounds -- prop:po-lee-bounds. Under the baseline Lee sample-selection assumptions, monotone sample selection, and almost-sure finite support 𝒴 for the factual outcome on the selected-treated cell, the average treatment effect among always-selected units — those who would be selected for observation whether treated or not — is sandwiched between the lower Lee trimmed mean minus the observable selected-control mean m₀, and the upper Lee trimmed mean minus m₀.","s":"∀ {P : Causalean.PO.POSystem} (S : Causalean.PO.POLeeSystem P) [MeasureTheory.IsFiniteMeasure P.μ], S.BaseAssumptions → S.MonotoneSelection → ∀ (𝒴 : Finset ℝ), (∀ᵐ (ω : P.Ω) ∂P.μ.restrict S.selectedTreated, S.factualY ω ∈ 𝒴) → (S.lowerTrimMean 𝒴 - S.m0 ≤ Causalean.PO.eventCondExp P.μ S.alwaysSelected fun ω => S.YofA true ω - S.YofA false ω) ∧ (Causalean.PO.eventCondExp P.μ S.alwaysSelected fun ω => S.YofA true ω - S.YofA false ω) ≤ S.upperTrimMean 𝒴 - S.m0","l":33,"f":"Causalean/PO/ID/Partial/Lee/Main.lean"},{"n":"Causalean.PO.POLeeSystem.lowerTrimMean","k":"def","m":"Causalean.PO.ID.Partial.Lee.Trim","a":"PO/ID/Partial/Lee","t":1,"d":"Lower trimmed mean underline_m₁ := inf_{w} M(w) ranging over Lee trim weights on 𝒴.","s":"{P : Causalean.PO.POSystem} → Causalean.PO.POLeeSystem P → Finset ℝ → ℝ","l":83,"f":"Causalean/PO/ID/Partial/Lee/Trim.lean"},{"n":"Causalean.PO.POLeeSystem.m0","k":"def","m":"Causalean.PO.ID.Partial.Lee.Trim","a":"PO/ID/Partial/Lee","t":1,"d":"Observable selected-control outcome mean m₀ := E[Y | A = false, Sel = true].","s":"{P : Causalean.PO.POSystem} → Causalean.PO.POLeeSystem P → ℝ","l":93,"f":"Causalean/PO/ID/Partial/Lee/Trim.lean"},{"n":"Causalean.PO.POLeeSystem.m0_eq_eventCondExp_Y0_alwaysSelected","k":"theorem","m":"Causalean.PO.ID.Partial.Lee.ControlMean","a":"PO/ID/Partial/Lee","t":1,"d":"Under the baseline Lee assumptions (consistency and pair-level random assignment of the factual treatment to each (Y(a), Sel(a))) together with monotone sample selection, Sel(0) ≤ Sel(1) almost surely, the observable selected-control mean m₀ = E[Y | A = false, Sel = true] equals the latent conditional mean E[Y(0) | alwaysSelected] of the control potential outcome among units who would be selected under either treatment arm.","s":"∀ {P : Causalean.PO.POSystem} (S : Causalean.PO.POLeeSystem P), S.BaseAssumptions → S.MonotoneSelection → S.m0 = Causalean.PO.eventCondExp P.μ S.alwaysSelected (S.YofA false)","l":50,"f":"Causalean/PO/ID/Partial/Lee/ControlMean.lean"},{"n":"Causalean.PO.POLeeSystem.measurableSet_aEvent","k":"theorem","m":"Causalean.PO.ID.Partial.Lee.Setup","a":"PO/ID/Partial/Lee","t":2,"d":"Each factual treatment arm event is measurable.","s":"∀ {P : Causalean.PO.POSystem} (S : Causalean.PO.POLeeSystem P) (a : Bool), MeasurableSet (S.aEvent a)","l":131,"f":"Causalean/PO/ID/Partial/Lee/Setup.lean"},{"n":"Causalean.PO.POLeeSystem.measurableSet_alwaysSelected","k":"theorem","m":"Causalean.PO.ID.Partial.Lee.PrincipalStrata","a":"PO/ID/Partial/Lee","t":2,"d":"The always-selected stratum is measurable.","s":"∀ {P : Causalean.PO.POSystem} (S : Causalean.PO.POLeeSystem P), MeasurableSet S.alwaysSelected","l":60,"f":"Causalean/PO/ID/Partial/Lee/PrincipalStrata.lean"},{"n":"Causalean.PO.POLeeSystem.measurableSet_harmedSelected","k":"theorem","m":"Causalean.PO.ID.Partial.Lee.PrincipalStrata","a":"PO/ID/Partial/Lee","t":2,"d":"The harmed-selected stratum is measurable.","s":"∀ {P : Causalean.PO.POSystem} (S : Causalean.PO.POLeeSystem P), MeasurableSet S.harmedSelected","l":72,"f":"Causalean/PO/ID/Partial/Lee/PrincipalStrata.lean"},{"n":"Causalean.PO.POLeeSystem.measurableSet_helpedSelected","k":"theorem","m":"Causalean.PO.ID.Partial.Lee.PrincipalStrata","a":"PO/ID/Partial/Lee","t":2,"d":"The helped-selected stratum is measurable.","s":"∀ {P : Causalean.PO.POSystem} (S : Causalean.PO.POLeeSystem P), MeasurableSet S.helpedSelected","l":66,"f":"Causalean/PO/ID/Partial/Lee/PrincipalStrata.lean"},{"n":"Causalean.PO.POLeeSystem.measurableSet_selEvent","k":"theorem","m":"Causalean.PO.ID.Partial.Lee.Setup","a":"PO/ID/Partial/Lee","t":2,"d":"Each factual selection event is measurable.","s":"∀ {P : Causalean.PO.POSystem} (S : Causalean.PO.POLeeSystem P) (s : Bool), MeasurableSet (S.selEvent s)","l":135,"f":"Causalean/PO/ID/Partial/Lee/Setup.lean"},{"n":"Causalean.PO.POLeeSystem.measurableSet_selOfAFalseSet","k":"theorem","m":"Causalean.PO.ID.Partial.Lee.PrincipalStrata","a":"PO/ID/Partial/Lee","t":2,"d":"The selected-under-control latent set is measurable.","s":"∀ {P : Causalean.PO.POSystem} (S : Causalean.PO.POLeeSystem P), MeasurableSet S.selOfAFalseSet","l":94,"f":"Causalean/PO/ID/Partial/Lee/PrincipalStrata.lean"},{"n":"Causalean.PO.POLeeSystem.measurableSet_selOfATrueSet","k":"theorem","m":"Causalean.PO.ID.Partial.Lee.PrincipalStrata","a":"PO/ID/Partial/Lee","t":2,"d":"The selected-under-treatment latent set is measurable.","s":"∀ {P : Causalean.PO.POSystem} (S : Causalean.PO.POLeeSystem P), MeasurableSet S.selOfATrueSet","l":84,"f":"Causalean/PO/ID/Partial/Lee/PrincipalStrata.lean"},{"n":"Causalean.PO.POLeeSystem.measurableSet_selectedControl","k":"theorem","m":"Causalean.PO.ID.Partial.Lee.Setup","a":"PO/ID/Partial/Lee","t":2,"d":"The selected-control observed cell is measurable.","s":"∀ {P : Causalean.PO.POSystem} (S : Causalean.PO.POLeeSystem P), MeasurableSet S.selectedControl","l":143,"f":"Causalean/PO/ID/Partial/Lee/Setup.lean"},{"n":"Causalean.PO.POLeeSystem.measurableSet_selectedTreated","k":"theorem","m":"Causalean.PO.ID.Partial.Lee.Setup","a":"PO/ID/Partial/Lee","t":2,"d":"The selected-treated observed cell is measurable.","s":"∀ {P : Causalean.PO.POSystem} (S : Causalean.PO.POLeeSystem P), MeasurableSet S.selectedTreated","l":139,"f":"Causalean/PO/ID/Partial/Lee/Setup.lean"},{"n":"Causalean.PO.POLeeSystem.measurable_SelOfA","k":"theorem","m":"Causalean.PO.ID.Partial.Lee.Setup","a":"PO/ID/Partial/Lee","t":2,"d":"Counterfactual selection under any fixed arm is measurable.","s":"∀ {P : Causalean.PO.POSystem} (S : Causalean.PO.POLeeSystem P) (a : Bool), Measurable (S.SelOfA a)","l":115,"f":"Causalean/PO/ID/Partial/Lee/Setup.lean"},{"n":"Causalean.PO.POLeeSystem.measurable_YofA","k":"theorem","m":"Causalean.PO.ID.Partial.Lee.Setup","a":"PO/ID/Partial/Lee","t":1,"d":"For a fixed treatment arm a, the counterfactual outcome Y(a) is measurable.","s":"∀ {P : Causalean.PO.POSystem} (S : Causalean.PO.POLeeSystem P) (a : Bool), Measurable (S.YofA a)","l":119,"f":"Causalean/PO/ID/Partial/Lee/Setup.lean"},{"n":"Causalean.PO.POLeeSystem.measurable_factualA","k":"theorem","m":"Causalean.PO.ID.Partial.Lee.Setup","a":"PO/ID/Partial/Lee","t":2,"d":"Factual treatment is measurable.","s":"∀ {P : Causalean.PO.POSystem} (S : Causalean.PO.POLeeSystem P), Measurable S.factualA","l":124,"f":"Causalean/PO/ID/Partial/Lee/Setup.lean"},{"n":"Causalean.PO.POLeeSystem.measurable_factualSel","k":"theorem","m":"Causalean.PO.ID.Partial.Lee.Setup","a":"PO/ID/Partial/Lee","t":2,"d":"Factual selection is measurable.","s":"∀ {P : Causalean.PO.POSystem} (S : Causalean.PO.POLeeSystem P), Measurable S.factualSel","l":126,"f":"Causalean/PO/ID/Partial/Lee/Setup.lean"},{"n":"Causalean.PO.POLeeSystem.measurable_factualY","k":"theorem","m":"Causalean.PO.ID.Partial.Lee.Setup","a":"PO/ID/Partial/Lee","t":2,"d":"Factual outcome is measurable.","s":"∀ {P : Causalean.PO.POSystem} (S : Causalean.PO.POLeeSystem P), Measurable S.factualY","l":128,"f":"Causalean/PO/ID/Partial/Lee/Setup.lean"},{"n":"Causalean.PO.POLeeSystem.p0","k":"def","m":"Causalean.PO.ID.Partial.Lee.Trim","a":"PO/ID/Partial/Lee","t":1,"d":"p₀ = P(Sel = true | A = false).","s":"{P : Causalean.PO.POSystem} → Causalean.PO.POLeeSystem P → ℝ","l":51,"f":"Causalean/PO/ID/Partial/Lee/Trim.lean"},{"n":"Causalean.PO.POLeeSystem.p1","k":"def","m":"Causalean.PO.ID.Partial.Lee.Trim","a":"PO/ID/Partial/Lee","t":1,"d":"p₁ = P(Sel = true | A = true).","s":"{P : Causalean.PO.POSystem} → Causalean.PO.POLeeSystem P → ℝ","l":54,"f":"Causalean/PO/ID/Partial/Lee/Trim.lean"},{"n":"Causalean.PO.POLeeSystem.pSelGivenA","k":"def","m":"Causalean.PO.ID.Partial.Lee.Trim","a":"PO/ID/Partial/Lee","t":1,"d":"Conditional selection probability p_a := P(Sel = true | A = a) expressed via the event-conditional expectation of the selection indicator.","s":"{P : Causalean.PO.POSystem} → Causalean.PO.POLeeSystem P → Bool → ℝ","l":45,"f":"Causalean/PO/ID/Partial/Lee/Trim.lean"},{"n":"Causalean.PO.POLeeSystem.rho","k":"def","m":"Causalean.PO.ID.Partial.Lee.Trim","a":"PO/ID/Partial/Lee","t":1,"d":"The trimming ratio ρ := p₀ / p₁.","s":"{P : Causalean.PO.POSystem} → Causalean.PO.POLeeSystem P → ℝ","l":57,"f":"Causalean/PO/ID/Partial/Lee/Trim.lean"},{"n":"Causalean.PO.POLeeSystem.selEvent","k":"def","m":"Causalean.PO.ID.Partial.Lee.Setup","a":"PO/ID/Partial/Lee","t":1,"d":"The event {Sel = s}.","s":"{P : Causalean.PO.POSystem} → Causalean.PO.POLeeSystem P → Bool → Set P.Ω","l":104,"f":"Causalean/PO/ID/Partial/Lee/Setup.lean"},{"n":"Causalean.PO.POLeeSystem.selOfAFalseSet","k":"def","m":"Causalean.PO.ID.Partial.Lee.PrincipalStrata","a":"PO/ID/Partial/Lee","t":1,"d":"The latent selected-under-control set contains units who would be selected if untreated.","s":"{P : Causalean.PO.POSystem} → Causalean.PO.POLeeSystem P → Set P.Ω","l":88,"f":"Causalean/PO/ID/Partial/Lee/PrincipalStrata.lean"},{"n":"Causalean.PO.POLeeSystem.selOfAFalseSet_ae_eq_alwaysSelected","k":"theorem","m":"Causalean.PO.ID.Partial.Lee.PrincipalStrata","a":"PO/ID/Partial/Lee","t":2,"d":"Under monotone sample selection, {Sel(0)=true} =ᵐ alwaysSelected.","s":"∀ {P : Causalean.PO.POSystem} (S : Causalean.PO.POLeeSystem P), (∀ᵐ (ω : P.Ω) ∂P.μ, S.SelOfA false ω ≤ S.SelOfA true ω) → S.selOfAFalseSet =ᵐ[P.μ] S.alwaysSelected","l":154,"f":"Causalean/PO/ID/Partial/Lee/PrincipalStrata.lean"},{"n":"Causalean.PO.POLeeSystem.selOfATrueSet","k":"def","m":"Causalean.PO.ID.Partial.Lee.PrincipalStrata","a":"PO/ID/Partial/Lee","t":1,"d":"The latent selected-under-treatment set contains units who would be selected if treated.","s":"{P : Causalean.PO.POSystem} → Causalean.PO.POLeeSystem P → Set P.Ω","l":78,"f":"Causalean/PO/ID/Partial/Lee/PrincipalStrata.lean"},{"n":"Causalean.PO.POLeeSystem.selOfATrueSet_eq_alwaysSelected_union_helpedSelected","k":"theorem","m":"Causalean.PO.ID.Partial.Lee.PrincipalStrata","a":"PO/ID/Partial/Lee","t":2,"d":"selOfATrueSet = alwaysSelected ∪ helpedSelected as a pure set equality (no a.s. needed -- the two RHS sets are disjoint and cover the LHS by case analysis on SelOfA false ω : Bool).","s":"∀ {P : Causalean.PO.POSystem} (S : Causalean.PO.POLeeSystem P), S.selOfATrueSet = S.alwaysSelected ∪ S.helpedSelected","l":98,"f":"Causalean/PO/ID/Partial/Lee/PrincipalStrata.lean"},{"n":"Causalean.PO.POLeeSystem.selVar","k":"def","m":"Causalean.PO.ID.Partial.Lee.Setup","a":"PO/ID/Partial/Lee","t":1,"d":"Selection indicator packaged as a POVar valued in Bool.","s":"{P : Causalean.PO.POSystem} → Causalean.PO.POLeeSystem P → Causalean.PO.POVar P Bool","l":76,"f":"Causalean/PO/ID/Partial/Lee/Setup.lean"},{"n":"Causalean.PO.POLeeSystem.selectedControl","k":"def","m":"Causalean.PO.ID.Partial.Lee.Setup","a":"PO/ID/Partial/Lee","t":1,"d":"The selected-control cell {A = false, Sel = true}.","s":"{P : Causalean.PO.POSystem} → Causalean.PO.POLeeSystem P → Set P.Ω","l":110,"f":"Causalean/PO/ID/Partial/Lee/Setup.lean"},{"n":"Causalean.PO.POLeeSystem.selectedTreated","k":"def","m":"Causalean.PO.ID.Partial.Lee.Setup","a":"PO/ID/Partial/Lee","t":1,"d":"The selected-treated cell {A = true, Sel = true}.","s":"{P : Causalean.PO.POSystem} → Causalean.PO.POLeeSystem P → Set P.Ω","l":107,"f":"Causalean/PO/ID/Partial/Lee/Setup.lean"},{"n":"Causalean.PO.POLeeSystem.selectedTreated_integral_split","k":"theorem","m":"Causalean.PO.ID.Partial.Lee.MixtureIdentity","a":"PO/ID/Partial/Lee","t":2,"d":"Selected-treated integral split — the f = factualY analogue of selectedTreated_measure_split (Step B'). Consistency on {A=true, Sel=true} rewrites factualY to YofA true; pair random assignment A ⫫ (Y(1), Sel(1)) then drops the {A=true} conditioning and introduces the scalar factor (P.μ (S.aEvent true)).toReal; the latent {Sel(1)=true} set then splits as the disjoint union of alwaysSelected and helpedSelected.","s":"∀ {P : Causalean.PO.POSystem} (S : Causalean.PO.POLeeSystem P), S.BaseAssumptions → ∫ (ω : P.Ω) in S.selectedTreated, S.factualY ω ∂P.μ = (P.μ (S.aEvent true)).toReal * (∫ (ω : P.Ω) in S.alwaysSelected, S.YofA true ω ∂P.μ + ∫ (ω : P.Ω) in S.helpedSelected, S.YofA true ω ∂P.μ)","l":60,"f":"Causalean/PO/ID/Partial/Lee/MixtureIdentity.lean"},{"n":"Causalean.PO.POLeeSystem.selectedTreated_integral_split_indicator","k":"theorem","m":"Causalean.PO.ID.Partial.Lee.MixtureIdentity","a":"PO/ID/Partial/Lee","t":2,"d":"Step B for the indicator integrand 1_{factualY = y} — same shape as selectedTreated_integral_split but with the indicator integrand. Used by the le_one and sum_eq fields of alwaysSelectedTrimWeight. Proof mirrors Step B; the only change is the integrand.","s":"∀ {P : Causalean.PO.POSystem} (S : Causalean.PO.POLeeSystem P), S.BaseAssumptions → ∀ (y : ℝ), ∫ (ω : P.Ω) in S.selectedTreated, if S.factualY ω = y then 1 else 0 ∂P.μ = (P.μ (S.aEvent true)).toReal * (∫ (ω : P.Ω) in S.alwaysSelected, if S.YofA true ω = y then 1 else 0 ∂P.μ + ∫ (ω : P.Ω) in S.helpedSelected, if S.YofA true ω = y then 1 else 0 ∂P.μ)","l":216,"f":"Causalean/PO/ID/Partial/Lee/MixtureIdentity.lean"},{"n":"Causalean.PO.POLeeSystem.selectedTreated_measure_split","k":"theorem","m":"Causalean.PO.ID.Partial.Lee.MixtureIdentity","a":"PO/ID/Partial/Lee","t":1,"d":"Under the baseline Lee sample-selection assumptions, the probability mass of the observable selected-treated cell equals the probability of being treated times the sum of the probability masses of the two latent strata always-selected and helped-selected. This is the analog of selectedTreated_integral_split for the constant function 1, and it is what gives ρ = μ(AS) / μ({Sel(1)=true}).","s":"∀ {P : Causalean.PO.POSystem} (S : Causalean.PO.POLeeSystem P), S.BaseAssumptions → (P.μ S.selectedTreated).toReal = (P.μ (S.aEvent true)).toReal * ((P.μ S.alwaysSelected).toReal + (P.μ S.helpedSelected).toReal)","l":157,"f":"Causalean/PO/ID/Partial/Lee/MixtureIdentity.lean"},{"n":"Causalean.PO.POLeeSystem.trimmed_bounds_condExp_Y1_AS","k":"theorem","m":"Causalean.PO.ID.Partial.Lee.TrimBound","a":"PO/ID/Partial/Lee","t":1,"d":"The Lee trimmed-mean sandwich for the always-selected treated mean. Given the baseline Lee assumptions, monotone sample selection, and almost-sure finite support 𝒴 for the factual outcome on the selected-treated cell, the conditional mean E[Y(1) | alwaysSelected] of the treated potential outcome among always-selected units lies between the lower and upper Lee trimmed means computed over all feasible Lee trim weights on that support.","s":"∀ {P : Causalean.PO.POSystem} (S : Causalean.PO.POLeeSystem P), S.BaseAssumptions → S.MonotoneSelection → ∀ (𝒴 : Finset ℝ), (∀ᵐ (ω : P.Ω) ∂P.μ.restrict S.selectedTreated, S.factualY ω ∈ 𝒴) → S.lowerTrimMean 𝒴 ≤ Causalean.PO.eventCondExp P.μ S.alwaysSelected (S.YofA true) ∧ Causalean.PO.eventCondExp P.μ S.alwaysSelected (S.YofA true) ≤ S.upperTrimMean 𝒴","l":31,"f":"Causalean/PO/ID/Partial/Lee/TrimBound.lean"},{"n":"Causalean.PO.POLeeSystem.upperTrimMean","k":"def","m":"Causalean.PO.ID.Partial.Lee.Trim","a":"PO/ID/Partial/Lee","t":1,"d":"Upper trimmed mean overline_m₁ := sup_{w} M(w) ranging over Lee trim weights on 𝒴.","s":"{P : Causalean.PO.POSystem} → Causalean.PO.POLeeSystem P → Finset ℝ → ℝ","l":88,"f":"Causalean/PO/ID/Partial/Lee/Trim.lean"},{"n":"Causalean.PO.POLeeSystem.yVar","k":"def","m":"Causalean.PO.ID.Partial.Lee.Setup","a":"PO/ID/Partial/Lee","t":1,"d":"Outcome packaged as a POVar valued in ℝ.","s":"{P : Causalean.PO.POSystem} → Causalean.PO.POLeeSystem P → Causalean.PO.POVar P ℝ","l":79,"f":"Causalean/PO/ID/Partial/Lee/Setup.lean"},{"n":"Causalean.PO.POManskiIVSystem","k":"structure","m":"Causalean.PO.ID.Partial.Manski.Setup","a":"PO/ID/Partial/Manski","t":1,"d":"The Manski IV data layer records a discrete instrument, a binary treatment, and a real outcome inside a potential-outcome system.","s":"Causalean.PO.POSystem → (α : Type u_1) → [inst : MeasurableSpace α] → [MeasurableSingletonClass α] → Type (max (max u_1 u_2) u_3)","l":26,"f":"Causalean/PO/ID/Partial/Manski/Setup.lean"},{"n":"Causalean.PO.POManskiIVSystem.ATE","k":"def","m":"Causalean.PO.ID.Partial.Manski.Setup","a":"PO/ID/Partial/Manski","t":1,"d":"Average treatment effect E[Y(1) - Y(0)].","s":"{P : Causalean.PO.POSystem} → {α : Type u_1} → [inst : MeasurableSpace α] → [inst_1 : MeasurableSingletonClass α] → Causalean.PO.POManskiIVSystem P α → ℝ","l":105,"f":"Causalean/PO/ID/Partial/Manski/Setup.lean"},{"n":"Causalean.PO.POManskiIVSystem.BaseAssumptions","k":"structure","m":"Causalean.PO.ID.Partial.Manski.Assumptions","a":"PO/ID/Partial/Manski","t":1,"d":"Discrete-IV ATE-bound baseline assumptions -- common part of def:po-iv-manski-assumptions.","s":"{P : Causalean.PO.POSystem} → {α : Type u_1} → [inst : MeasurableSpace α] → [inst_1 : MeasurableSingletonClass α] → Causalean.PO.POManskiIVSystem P α → Type","l":31,"f":"Causalean/PO/ID/Partial/Manski/Assumptions.lean"},{"n":"Causalean.PO.POManskiIVSystem.BaseAssumptions.bounded","k":"theorem","m":"Causalean.PO.ID.Partial.Manski.Assumptions","a":"PO/ID/Partial/Manski","t":1,"d":"Given the baseline Manski assumptions bundle, which fixes bounds lo ≤ hi and asserts that both potential outcomes Y(1) and Y(0) lie a.s. in [lo, hi], the potential outcome Y(d) lies almost surely between lo and hi, uniformly for either treatment arm d — the binary-folded form of the two separate range assumptions.","s":"∀ {P : Causalean.PO.POSystem} {α : Type u_1} [inst : MeasurableSpace α] [inst_1 : MeasurableSingletonClass α] {S : Causalean.PO.POManskiIVSystem P α} (hA : S.BaseAssumptions) (d : Bool), ∀ᵐ (ω : P.Ω) ∂P.μ, hA.lo ≤ S.YofD d ω ∧ S.YofD d ω ≤ hA.hi","l":50,"f":"Causalean/PO/ID/Partial/Manski/Assumptions.lean"},{"n":"Causalean.PO.POManskiIVSystem.BaseAssumptions.integrable_YofD","k":"theorem","m":"Causalean.PO.ID.Partial.Manski.Assumptions","a":"PO/ID/Partial/Manski","t":2,"d":"Binary-folded form of integrable_Y1 / integrable_Y0.","s":"∀ {P : Causalean.PO.POSystem} {α : Type u_1} [inst : MeasurableSpace α] [inst_1 : MeasurableSingletonClass α] {S : Causalean.PO.POManskiIVSystem P α} (hA : S.BaseAssumptions) (d : Bool), MeasureTheory.Integrable (S.YofD d) P.μ","l":60,"f":"Causalean/PO/ID/Partial/Manski/Assumptions.lean"},{"n":"Causalean.PO.POManskiIVSystem.BaseAssumptions.integrable_factualY","k":"theorem","m":"Causalean.PO.ID.Partial.Manski.Assumptions","a":"PO/ID/Partial/Manski","t":2,"d":"The factual outcome Y is integrable. Derived from arm integrability via consistency (factualY = Σ_d Y(d)·1{D=d} a.e.), so it need not be assumed separately.","s":"∀ {P : Causalean.PO.POSystem} {α : Type u_1} [inst : MeasurableSpace α] [inst_1 : MeasurableSingletonClass α] {S : Causalean.PO.POManskiIVSystem P α} (hA : S.BaseAssumptions), MeasureTheory.Integrable S.factualY P.μ","l":67,"f":"Causalean/PO/ID/Partial/Manski/Assumptions.lean"},{"n":"Causalean.PO.POManskiIVSystem.MIV","k":"structure","m":"Causalean.PO.ID.Partial.Manski.Assumptions","a":"PO/ID/Partial/Manski","t":1,"d":"Monotone instrumental variable -- prop:po-iv-miv.","s":"{P : Causalean.PO.POSystem} → {α : Type u_1} → [inst : MeasurableSpace α] → [inst_1 : MeasurableSingletonClass α] → Causalean.PO.POManskiIVSystem P α → Type u_1","l":125,"f":"Causalean/PO/ID/Partial/Manski/Assumptions.lean"},{"n":"Causalean.PO.POManskiIVSystem.MTR","k":"structure","m":"Causalean.PO.ID.Partial.Manski.Assumptions","a":"PO/ID/Partial/Manski","t":1,"d":"Monotone treatment response -- prop:po-iv-mtr, item 1. Y(0) ≤ Y(1) almost surely.","s":"{P : Causalean.PO.POSystem} → {α : Type u_1} → [inst : MeasurableSpace α] → [inst_1 : MeasurableSingletonClass α] → Causalean.PO.POManskiIVSystem P α → Prop","l":112,"f":"Causalean/PO/ID/Partial/Manski/Assumptions.lean"},{"n":"Causalean.PO.POManskiIVSystem.MTS","k":"structure","m":"Causalean.PO.ID.Partial.Manski.Assumptions","a":"PO/ID/Partial/Manski","t":1,"d":"Monotone treatment selection -- prop:po-iv-mts.","s":"{P : Causalean.PO.POSystem} → {α : Type u_1} → [inst : MeasurableSpace α] → [inst_1 : MeasurableSingletonClass α] → Causalean.PO.POManskiIVSystem P α → Prop","l":117,"f":"Causalean/PO/ID/Partial/Manski/Assumptions.lean"},{"n":"Causalean.PO.POManskiIVSystem.MeanIndep","k":"structure","m":"Causalean.PO.ID.Partial.Manski.Assumptions","a":"PO/ID/Partial/Manski","t":1,"d":"Mean independence of the potential outcomes from the instrument. Stated directly on eventCondExp — matches def:po-iv-manski-assumptions letter-for-letter and avoids the stronger joint independence used in LATE.","s":"{P : Causalean.PO.POSystem} → {α : Type u_1} → [inst : MeasurableSpace α] → [inst_1 : MeasurableSingletonClass α] → Causalean.PO.POManskiIVSystem P α → Prop","l":103,"f":"Causalean/PO/ID/Partial/Manski/Assumptions.lean"},{"n":"Causalean.PO.POManskiIVSystem.YofD","k":"def","m":"Causalean.PO.ID.Partial.Manski.Setup","a":"PO/ID/Partial/Manski","t":1,"d":"Counterfactual outcome under treatment arm d.","s":"{P : Causalean.PO.POSystem} → {α : Type u_1} → [inst : MeasurableSpace α] → [inst_1 : MeasurableSingletonClass α] → Causalean.PO.POManskiIVSystem P α → Bool → P.Ω → ℝ","l":61,"f":"Causalean/PO/ID/Partial/Manski/Setup.lean"},{"n":"Causalean.PO.POManskiIVSystem.YofD_eq_factualY_on_dEvent","k":"theorem","m":"Causalean.PO.ID.Partial.Manski.MTS","a":"PO/ID/Partial/Manski","t":2,"d":"Pointwise consistency on {D = d}: YofD d = factualY on this event.","s":"∀ {P : Causalean.PO.POSystem} {α : Type u_1} [inst : MeasurableSpace α] [inst_1 : MeasurableSingletonClass α] (S : Causalean.PO.POManskiIVSystem P α) (hA : S.BaseAssumptions) (d : Bool) {ω : P.Ω}, ω ∈ S.dEvent d → S.YofD d ω = S.factualY ω","l":119,"f":"Causalean/PO/ID/Partial/Manski/MTS.lean"},{"n":"Causalean.PO.POManskiIVSystem.YofD_indD_plus_lo_le_YofD","k":"theorem","m":"Causalean.PO.ID.Partial.Manski.Helpers","a":"PO/ID/Partial/Manski","t":2,"d":"Pointwise a.e. bound: Y(d) · 1_{D=d} + lo · 1_{D=!d} ≤ Y(d) under lo ≤ Y(d).","s":"∀ {P : Causalean.PO.POSystem} {α : Type u_1} [inst : MeasurableSpace α] [inst_1 : MeasurableSingletonClass α] (S : Causalean.PO.POManskiIVSystem P α) (d : Bool) (lo : ℝ), (∀ᵐ (ω : P.Ω) ∂P.μ, lo ≤ S.YofD d ω) → (fun ω => S.YofD d ω * S.dVar.indicator d ω + lo * S.dVar.indicator (!d) ω) ≤ᵐ[P.μ] S.YofD d","l":89,"f":"Causalean/PO/ID/Partial/Manski/Helpers.lean"},{"n":"Causalean.PO.POManskiIVSystem.YofD_le_YofD_indD_plus_hi","k":"theorem","m":"Causalean.PO.ID.Partial.Manski.Helpers","a":"PO/ID/Partial/Manski","t":2,"d":"Pointwise a.e. bound: Y(d) ≤ Y(d) · 1_{D=d} + hi · 1_{D=!d} under Y(d) ≤ hi.","s":"∀ {P : Causalean.PO.POSystem} {α : Type u_1} [inst : MeasurableSpace α] [inst_1 : MeasurableSingletonClass α] (S : Causalean.PO.POManskiIVSystem P α) (d : Bool) (hi : ℝ), (∀ᵐ (ω : P.Ω) ∂P.μ, S.YofD d ω ≤ hi) → S.YofD d ≤ᵐ[P.μ] fun ω => S.YofD d ω * S.dVar.indicator d ω + hi * S.dVar.indicator (!d) ω","l":99,"f":"Causalean/PO/ID/Partial/Manski/Helpers.lean"},{"n":"Causalean.PO.POManskiIVSystem.YofD_mul_indD_integrable","k":"theorem","m":"Causalean.PO.ID.Partial.Manski.Helpers","a":"PO/ID/Partial/Manski","t":2,"d":"YofD d · indicator d is integrable.","s":"∀ {P : Causalean.PO.POSystem} {α : Type u_1} [inst : MeasurableSpace α] [inst_1 : MeasurableSingletonClass α] (S : Causalean.PO.POManskiIVSystem P α) (hA : S.BaseAssumptions) (d : Bool), MeasureTheory.Integrable (fun ω => S.YofD d ω * S.dVar.indicator d ω) P.μ","l":53,"f":"Causalean/PO/ID/Partial/Manski/Helpers.lean"},{"n":"Causalean.PO.POManskiIVSystem.boundArm","k":"def","m":"Causalean.PO.ID.Partial.Manski.Setup","a":"PO/ID/Partial/Manski","t":1,"d":"The unified arm-bound functional averages the observed outcome on arm d and the supplied outcome floor or ceiling on the opposite arm within instrument stratum z.","s":"{P : Causalean.PO.POSystem} → {α : Type u_1} → [inst : MeasurableSpace α] → [inst_1 : MeasurableSingletonClass α] → Causalean.PO.POManskiIVSystem P α → Bool → ℝ → α → ℝ","l":111,"f":"Causalean/PO/ID/Partial/Manski/Setup.lean"},{"n":"Causalean.PO.POManskiIVSystem.boundArm_lo_le_cond_YofD","k":"theorem","m":"Causalean.PO.ID.Partial.Manski.Helpers","a":"PO/ID/Partial/Manski","t":2,"d":"Stratum-level lower bound on eventCondExp μ (Z=z) (Y(d)) via the unified arm functional boundArm d lo.","s":"∀ {P : Causalean.PO.POSystem} {α : Type u_1} [inst : MeasurableSpace α] [inst_1 : MeasurableSingletonClass α] (S : Causalean.PO.POManskiIVSystem P α) [MeasureTheory.IsFiniteMeasure P.μ] (hA : S.BaseAssumptions) (d : Bool) {z : α}, z ∈ S.support → S.boundArm d hA.lo z ≤ Causalean.PO.eventCondExp P.μ (S.zEvent z) (S.YofD d)","l":117,"f":"Causalean/PO/ID/Partial/Manski/Helpers.lean"},{"n":"Causalean.PO.POManskiIVSystem.ciSup_lowerBound0_le_integral_Y0","k":"theorem","m":"Causalean.PO.ID.Partial.Manski.NonAsp","a":"PO/ID/Partial/Manski","t":2,"d":"Sup-over-support form of lowerBound0_le_integral_Y0.","s":"∀ {P : Causalean.PO.POSystem} {α : Type u_1} [inst : MeasurableSpace α] [inst_1 : MeasurableSingletonClass α] (S : Causalean.PO.POManskiIVSystem P α) [MeasureTheory.IsFiniteMeasure P.μ] (hA : S.BaseAssumptions), S.MeanIndep → S.support.Nonempty → ⨆ z, S.lowerBound0 hA.lo ↑z ≤ ∫ (ω : P.Ω), S.YofD false ω ∂P.μ","l":115,"f":"Causalean/PO/ID/Partial/Manski/NonAsp.lean"},{"n":"Causalean.PO.POManskiIVSystem.ciSup_lowerBound1_le_integral_Y1","k":"theorem","m":"Causalean.PO.ID.Partial.Manski.NonAsp","a":"PO/ID/Partial/Manski","t":2,"d":"Sup-over-support form of lowerBound1_le_integral_Y1.","s":"∀ {P : Causalean.PO.POSystem} {α : Type u_1} [inst : MeasurableSpace α] [inst_1 : MeasurableSingletonClass α] (S : Causalean.PO.POManskiIVSystem P α) [MeasureTheory.IsFiniteMeasure P.μ] (hA : S.BaseAssumptions), S.MeanIndep → S.support.Nonempty → ⨆ z, S.lowerBound1 hA.lo ↑z ≤ ∫ (ω : P.Ω), S.YofD true ω ∂P.μ","l":101,"f":"Causalean/PO/ID/Partial/Manski/NonAsp.lean"},{"n":"Causalean.PO.POManskiIVSystem.compl_dEvent_false","k":"theorem","m":"Causalean.PO.ID.Partial.Manski.MTS","a":"PO/ID/Partial/Manski","t":2,"d":"Complement of {D = false} is {D = true} (Bool case-split).","s":"∀ {P : Causalean.PO.POSystem} {α : Type u_1} [inst : MeasurableSpace α] [inst_1 : MeasurableSingletonClass α] (S : Causalean.PO.POManskiIVSystem P α), (S.dEvent false)ᶜ = S.dEvent true","l":58,"f":"Causalean/PO/ID/Partial/Manski/MTS.lean"},{"n":"Causalean.PO.POManskiIVSystem.compl_dEvent_true","k":"theorem","m":"Causalean.PO.ID.Partial.Manski.MTS","a":"PO/ID/Partial/Manski","t":2,"d":"Complement of {D = true} is {D = false} (Bool case-split).","s":"∀ {P : Causalean.PO.POSystem} {α : Type u_1} [inst : MeasurableSpace α] [inst_1 : MeasurableSingletonClass α] (S : Causalean.PO.POManskiIVSystem P α), (S.dEvent true)ᶜ = S.dEvent false","l":51,"f":"Causalean/PO/ID/Partial/Manski/MTS.lean"},{"n":"Causalean.PO.POManskiIVSystem.cond_Y0_le_upperBound0","k":"theorem","m":"Causalean.PO.ID.Partial.Manski.Helpers","a":"PO/ID/Partial/Manski","t":2,"d":"Stratum-level upper bound on the *conditional expectation* of Y(0).","s":"∀ {P : Causalean.PO.POSystem} {α : Type u_1} [inst : MeasurableSpace α] [inst_1 : MeasurableSingletonClass α] (S : Causalean.PO.POManskiIVSystem P α) [MeasureTheory.IsFiniteMeasure P.μ] (hA : S.BaseAssumptions) {z : α}, z ∈ S.support → Causalean.PO.eventCondExp P.μ (S.zEvent z) (S.YofD false) ≤ S.upperBound0 hA.hi z","l":222,"f":"Causalean/PO/ID/Partial/Manski/Helpers.lean"},{"n":"Causalean.PO.POManskiIVSystem.cond_Y1_le_upperBound1","k":"theorem","m":"Causalean.PO.ID.Partial.Manski.Helpers","a":"PO/ID/Partial/Manski","t":2,"d":"Stratum-level upper bound on the *conditional expectation* of Y(1).","s":"∀ {P : Causalean.PO.POSystem} {α : Type u_1} [inst : MeasurableSpace α] [inst_1 : MeasurableSingletonClass α] (S : Causalean.PO.POManskiIVSystem P α) [MeasureTheory.IsFiniteMeasure P.μ] (hA : S.BaseAssumptions) {z : α}, z ∈ S.support → Causalean.PO.eventCondExp P.μ (S.zEvent z) (S.YofD true) ≤ S.upperBound1 hA.hi z","l":210,"f":"Causalean/PO/ID/Partial/Manski/Helpers.lean"},{"n":"Causalean.PO.POManskiIVSystem.cond_YofD_le_boundArm_hi","k":"theorem","m":"Causalean.PO.ID.Partial.Manski.Helpers","a":"PO/ID/Partial/Manski","t":1,"d":"Under the baseline Manski assumptions, for any treatment arm d and any instrument value z in the support of the instrument, the conditional mean of the potential outcome Y(d) given Z = z is bounded above by the unified upper-envelope functional boundArm d hi evaluated at z.","s":"∀ {P : Causalean.PO.POSystem} {α : Type u_1} [inst : MeasurableSpace α] [inst_1 : MeasurableSingletonClass α] (S : Causalean.PO.POManskiIVSystem P α) [MeasureTheory.IsFiniteMeasure P.μ] (hA : S.BaseAssumptions) (d : Bool) {z : α}, z ∈ S.support → Causalean.PO.eventCondExp P.μ (S.zEvent z) (S.YofD d) ≤ S.boundArm d hA.hi z","l":158,"f":"Causalean/PO/ID/Partial/Manski/Helpers.lean"},{"n":"Causalean.PO.POManskiIVSystem.const_mul_indD_integrable","k":"theorem","m":"Causalean.PO.ID.Partial.Manski.Helpers","a":"PO/ID/Partial/Manski","t":2,"d":"Constant times indicator d is integrable.","s":"∀ {P : Causalean.PO.POSystem} {α : Type u_1} [inst : MeasurableSpace α] [inst_1 : MeasurableSingletonClass α] (S : Causalean.PO.POManskiIVSystem P α) [MeasureTheory.IsFiniteMeasure P.μ] (c : ℝ) (d : Bool), MeasureTheory.Integrable (fun ω => c * S.dVar.indicator d ω) P.μ","l":58,"f":"Causalean/PO/ID/Partial/Manski/Helpers.lean"},{"n":"Causalean.PO.POManskiIVSystem.dEvent","k":"def","m":"Causalean.PO.ID.Partial.Manski.Setup","a":"PO/ID/Partial/Manski","t":1,"d":"The event {D = d}.","s":"{P : Causalean.PO.POSystem} → {α : Type u_1} → [inst : MeasurableSpace α] → [inst_1 : MeasurableSingletonClass α] → Causalean.PO.POManskiIVSystem P α → Bool → Set P.Ω","l":78,"f":"Causalean/PO/ID/Partial/Manski/Setup.lean"},{"n":"Causalean.PO.POManskiIVSystem.dVar","k":"def","m":"Causalean.PO.ID.Partial.Manski.Setup","a":"PO/ID/Partial/Manski","t":1,"d":"Treatment packaged as a POVar valued in Bool.","s":"{P : Causalean.PO.POSystem} → {α : Type u_1} → [inst : MeasurableSpace α] → [inst_1 : MeasurableSingletonClass α] → Causalean.PO.POManskiIVSystem P α → Causalean.PO.POVar P Bool","l":55,"f":"Causalean/PO/ID/Partial/Manski/Setup.lean"},{"n":"Causalean.PO.POManskiIVSystem.eventCondExp_YofD_eq_factualY","k":"theorem","m":"Causalean.PO.ID.Partial.Manski.MTS","a":"PO/ID/Partial/Manski","t":2,"d":"E[Y(d) | D = d] = E[Y | D = d] via consistency on the event.","s":"∀ {P : Causalean.PO.POSystem} {α : Type u_1} [inst : MeasurableSpace α] [inst_1 : MeasurableSingletonClass α] (S : Causalean.PO.POManskiIVSystem P α) (hA : S.BaseAssumptions) (d : Bool), Causalean.PO.eventCondExp P.μ (S.dEvent d) (S.YofD d) = Causalean.PO.eventCondExp P.μ (S.dEvent d) S.factualY","l":125,"f":"Causalean/PO/ID/Partial/Manski/MTS.lean"},{"n":"Causalean.PO.POManskiIVSystem.factualD","k":"def","m":"Causalean.PO.ID.Partial.Manski.Setup","a":"PO/ID/Partial/Manski","t":1,"d":"Factual treatment.","s":"{P : Causalean.PO.POSystem} → {α : Type u_1} → [inst : MeasurableSpace α] → [inst_1 : MeasurableSingletonClass α] → Causalean.PO.POManskiIVSystem P α → P.Ω → Bool","l":69,"f":"Causalean/PO/ID/Partial/Manski/Setup.lean"},{"n":"Causalean.PO.POManskiIVSystem.factualY","k":"def","m":"Causalean.PO.ID.Partial.Manski.Setup","a":"PO/ID/Partial/Manski","t":1,"d":"Factual outcome.","s":"{P : Causalean.PO.POSystem} → {α : Type u_1} → [inst : MeasurableSpace α] → [inst_1 : MeasurableSingletonClass α] → Causalean.PO.POManskiIVSystem P α → P.Ω → ℝ","l":72,"f":"Causalean/PO/ID/Partial/Manski/Setup.lean"},{"n":"Causalean.PO.POManskiIVSystem.factualY_mul_indD_integrable","k":"theorem","m":"Causalean.PO.ID.Partial.Manski.Helpers","a":"PO/ID/Partial/Manski","t":2,"d":"factualY · indicator d is integrable.","s":"∀ {P : Causalean.PO.POSystem} {α : Type u_1} [inst : MeasurableSpace α] [inst_1 : MeasurableSingletonClass α] (S : Causalean.PO.POManskiIVSystem P α) (hA : S.BaseAssumptions) (d : Bool), MeasureTheory.Integrable (fun ω => S.factualY ω * S.dVar.indicator d ω) P.μ","l":48,"f":"Causalean/PO/ID/Partial/Manski/Helpers.lean"},{"n":"Causalean.PO.POManskiIVSystem.factualZ","k":"def","m":"Causalean.PO.ID.Partial.Manski.Setup","a":"PO/ID/Partial/Manski","t":1,"d":"Factual instrument.","s":"{P : Causalean.PO.POSystem} → {α : Type u_1} → [inst : MeasurableSpace α] → [inst_1 : MeasurableSingletonClass α] → Causalean.PO.POManskiIVSystem P α → P.Ω → α","l":66,"f":"Causalean/PO/ID/Partial/Manski/Setup.lean"},{"n":"Causalean.PO.POManskiIVSystem.integral_Y0_le_ciInf_upperBound0","k":"theorem","m":"Causalean.PO.ID.Partial.Manski.NonAsp","a":"PO/ID/Partial/Manski","t":2,"d":"Inf-over-support form of integral_Y0_le_upperBound0.","s":"∀ {P : Causalean.PO.POSystem} {α : Type u_1} [inst : MeasurableSpace α] [inst_1 : MeasurableSingletonClass α] (S : Causalean.PO.POManskiIVSystem P α) [MeasureTheory.IsFiniteMeasure P.μ] (hA : S.BaseAssumptions), S.MeanIndep → S.support.Nonempty → ∫ (ω : P.Ω), S.YofD false ω ∂P.μ ≤ ⨅ z, S.upperBound0 hA.hi ↑z","l":122,"f":"Causalean/PO/ID/Partial/Manski/NonAsp.lean"},{"n":"Causalean.PO.POManskiIVSystem.integral_Y0_le_upperBound0","k":"theorem","m":"Causalean.PO.ID.Partial.Manski.NonAsp","a":"PO/ID/Partial/Manski","t":2,"d":"Stratum-level upper bound on E[Y(0)].","s":"∀ {P : Causalean.PO.POSystem} {α : Type u_1} [inst : MeasurableSpace α] [inst_1 : MeasurableSingletonClass α] (S : Causalean.PO.POManskiIVSystem P α) [MeasureTheory.IsFiniteMeasure P.μ] (hA : S.BaseAssumptions), S.MeanIndep → ∀ {z : α}, z ∈ S.support → ∫ (ω : P.Ω), S.YofD false ω ∂P.μ ≤ S.upperBound0 hA.hi z","l":61,"f":"Causalean/PO/ID/Partial/Manski/NonAsp.lean"},{"n":"Causalean.PO.POManskiIVSystem.integral_Y1_le_ciInf_upperBound1","k":"theorem","m":"Causalean.PO.ID.Partial.Manski.NonAsp","a":"PO/ID/Partial/Manski","t":2,"d":"Inf-over-support form of integral_Y1_le_upperBound1.","s":"∀ {P : Causalean.PO.POSystem} {α : Type u_1} [inst : MeasurableSpace α] [inst_1 : MeasurableSingletonClass α] (S : Causalean.PO.POManskiIVSystem P α) [MeasureTheory.IsFiniteMeasure P.μ] (hA : S.BaseAssumptions), S.MeanIndep → S.support.Nonempty → ∫ (ω : P.Ω), S.YofD true ω ∂P.μ ≤ ⨅ z, S.upperBound1 hA.hi ↑z","l":108,"f":"Causalean/PO/ID/Partial/Manski/NonAsp.lean"},{"n":"Causalean.PO.POManskiIVSystem.integral_Y1_le_upperBound1","k":"theorem","m":"Causalean.PO.ID.Partial.Manski.NonAsp","a":"PO/ID/Partial/Manski","t":2,"d":"Stratum-level upper bound on E[Y(1)].","s":"∀ {P : Causalean.PO.POSystem} {α : Type u_1} [inst : MeasurableSpace α] [inst_1 : MeasurableSingletonClass α] (S : Causalean.PO.POManskiIVSystem P α) [MeasureTheory.IsFiniteMeasure P.μ] (hA : S.BaseAssumptions), S.MeanIndep → ∀ {z : α}, z ∈ S.support → ∫ (ω : P.Ω), S.YofD true ω ∂P.μ ≤ S.upperBound1 hA.hi z","l":49,"f":"Causalean/PO/ID/Partial/Manski/NonAsp.lean"},{"n":"Causalean.PO.POManskiIVSystem.integral_YofD_eq_total_law","k":"theorem","m":"Causalean.PO.ID.Partial.Manski.MTS","a":"PO/ID/Partial/Manski","t":2,"d":"Total-law decomposition for YofD d across the binary partition {D=true} ⊔ {D=false}, via the generic Fintype total law.","s":"∀ {P : Causalean.PO.POSystem} {α : Type u_1} [inst : MeasurableSpace α] [inst_1 : MeasurableSingletonClass α] (S : Causalean.PO.POManskiIVSystem P α) (hA : S.BaseAssumptions) (d : Bool), ∫ (ω : P.Ω), S.YofD d ω ∂P.μ = (P.μ (S.dEvent true)).toReal * Causalean.PO.eventCondExp P.μ (S.dEvent true) (S.YofD d) + (P.μ (S.dEvent false)).toReal * Causalean.PO.eventCondExp P.μ (S.dEvent false) (S.YofD d)","l":100,"f":"Causalean/PO/ID/Partial/Manski/MTS.lean"},{"n":"Causalean.PO.POManskiIVSystem.lowerBound0","k":"def","m":"Causalean.PO.ID.Partial.Manski.Setup","a":"PO/ID/Partial/Manski","t":1,"d":"The lower observable bound for the control potential-outcome mean in instrument stratum z uses the outcome floor on treated units.","s":"{P : Causalean.PO.POSystem} → {α : Type u_1} → [inst : MeasurableSpace α] → [inst_1 : MeasurableSingletonClass α] → Causalean.PO.POManskiIVSystem P α → ℝ → α → ℝ","l":131,"f":"Causalean/PO/ID/Partial/Manski/Setup.lean"},{"n":"Causalean.PO.POManskiIVSystem.lowerBound0_le_cond_Y0","k":"theorem","m":"Causalean.PO.ID.Partial.Manski.Helpers","a":"PO/ID/Partial/Manski","t":2,"d":"Stratum-level lower bound on the *conditional expectation* of Y(0).","s":"∀ {P : Causalean.PO.POSystem} {α : Type u_1} [inst : MeasurableSpace α] [inst_1 : MeasurableSingletonClass α] (S : Causalean.PO.POManskiIVSystem P α) [MeasureTheory.IsFiniteMeasure P.μ] (hA : S.BaseAssumptions) {z : α}, z ∈ S.support → S.lowerBound0 hA.lo z ≤ Causalean.PO.eventCondExp P.μ (S.zEvent z) (S.YofD false)","l":216,"f":"Causalean/PO/ID/Partial/Manski/Helpers.lean"},{"n":"Causalean.PO.POManskiIVSystem.lowerBound0_le_integral_Y0","k":"theorem","m":"Causalean.PO.ID.Partial.Manski.NonAsp","a":"PO/ID/Partial/Manski","t":2,"d":"Stratum-level lower bound on E[Y(0)].","s":"∀ {P : Causalean.PO.POSystem} {α : Type u_1} [inst : MeasurableSpace α] [inst_1 : MeasurableSingletonClass α] (S : Causalean.PO.POManskiIVSystem P α) [MeasureTheory.IsFiniteMeasure P.μ] (hA : S.BaseAssumptions), S.MeanIndep → ∀ {z : α}, z ∈ S.support → S.lowerBound0 hA.lo z ≤ ∫ (ω : P.Ω), S.YofD false ω ∂P.μ","l":55,"f":"Causalean/PO/ID/Partial/Manski/NonAsp.lean"},{"n":"Causalean.PO.POManskiIVSystem.lowerBound1","k":"def","m":"Causalean.PO.ID.Partial.Manski.Setup","a":"PO/ID/Partial/Manski","t":1,"d":"The lower observable bound for the treated potential-outcome mean in instrument stratum z uses the outcome floor on untreated units.","s":"{P : Causalean.PO.POSystem} → {α : Type u_1} → [inst : MeasurableSpace α] → [inst_1 : MeasurableSingletonClass α] → Causalean.PO.POManskiIVSystem P α → ℝ → α → ℝ","l":123,"f":"Causalean/PO/ID/Partial/Manski/Setup.lean"},{"n":"Causalean.PO.POManskiIVSystem.lowerBound1_le_cond_Y1","k":"theorem","m":"Causalean.PO.ID.Partial.Manski.Helpers","a":"PO/ID/Partial/Manski","t":2,"d":"Stratum-level lower bound on the *conditional expectation* of Y(1).","s":"∀ {P : Causalean.PO.POSystem} {α : Type u_1} [inst : MeasurableSpace α] [inst_1 : MeasurableSingletonClass α] (S : Causalean.PO.POManskiIVSystem P α) [MeasureTheory.IsFiniteMeasure P.μ] (hA : S.BaseAssumptions) {z : α}, z ∈ S.support → S.lowerBound1 hA.lo z ≤ Causalean.PO.eventCondExp P.μ (S.zEvent z) (S.YofD true)","l":204,"f":"Causalean/PO/ID/Partial/Manski/Helpers.lean"},{"n":"Causalean.PO.POManskiIVSystem.lowerBound1_le_integral_Y1","k":"theorem","m":"Causalean.PO.ID.Partial.Manski.NonAsp","a":"PO/ID/Partial/Manski","t":2,"d":"Stratum-level lower bound on E[Y(1)].","s":"∀ {P : Causalean.PO.POSystem} {α : Type u_1} [inst : MeasurableSpace α] [inst_1 : MeasurableSingletonClass α] (S : Causalean.PO.POManskiIVSystem P α) [MeasureTheory.IsFiniteMeasure P.μ] (hA : S.BaseAssumptions), S.MeanIndep → ∀ {z : α}, z ∈ S.support → S.lowerBound1 hA.lo z ≤ ∫ (ω : P.Ω), S.YofD true ω ∂P.μ","l":43,"f":"Causalean/PO/ID/Partial/Manski/NonAsp.lean"},{"n":"Causalean.PO.POManskiIVSystem.mLower0","k":"def","m":"Causalean.PO.ID.Partial.Manski.MIV","a":"PO/ID/Partial/Manski","t":1,"d":"Lower envelope for arm d = 0: sSup { L_{0,u} | u ∈ supp, u ≤ z }.","s":"{P : Causalean.PO.POSystem} → {α : Type u_1} → [inst : MeasurableSpace α] → [inst_1 : MeasurableSingletonClass α] → (S : Causalean.PO.POManskiIVSystem P α) → S.BaseAssumptions → [LinearOrder α] → α → ℝ","l":68,"f":"Causalean/PO/ID/Partial/Manski/MIV.lean"},{"n":"Causalean.PO.POManskiIVSystem.mLower1","k":"def","m":"Causalean.PO.ID.Partial.Manski.MIV","a":"PO/ID/Partial/Manski","t":1,"d":"Lower envelope for arm d = 1: sSup { L_{1,u} | u ∈ supp, u ≤ z }.","s":"{P : Causalean.PO.POSystem} → {α : Type u_1} → [inst : MeasurableSpace α] → [inst_1 : MeasurableSingletonClass α] → (S : Causalean.PO.POManskiIVSystem P α) → S.BaseAssumptions → [LinearOrder α] → α → ℝ","l":58,"f":"Causalean/PO/ID/Partial/Manski/MIV.lean"},{"n":"Causalean.PO.POManskiIVSystem.mUpper0","k":"def","m":"Causalean.PO.ID.Partial.Manski.MIV","a":"PO/ID/Partial/Manski","t":1,"d":"Upper envelope for arm d = 0: sInf { U_{0,u} | u ∈ supp, z ≤ u }.","s":"{P : Causalean.PO.POSystem} → {α : Type u_1} → [inst : MeasurableSpace α] → [inst_1 : MeasurableSingletonClass α] → (S : Causalean.PO.POManskiIVSystem P α) → S.BaseAssumptions → [LinearOrder α] → α → ℝ","l":73,"f":"Causalean/PO/ID/Partial/Manski/MIV.lean"},{"n":"Causalean.PO.POManskiIVSystem.mUpper1","k":"def","m":"Causalean.PO.ID.Partial.Manski.MIV","a":"PO/ID/Partial/Manski","t":1,"d":"Upper envelope for arm d = 1: sInf { U_{1,u} | u ∈ supp, z ≤ u }.","s":"{P : Causalean.PO.POSystem} → {α : Type u_1} → [inst : MeasurableSpace α] → [inst_1 : MeasurableSingletonClass α] → (S : Causalean.PO.POManskiIVSystem P α) → S.BaseAssumptions → [LinearOrder α] → α → ℝ","l":63,"f":"Causalean/PO/ID/Partial/Manski/MIV.lean"},{"n":"Causalean.PO.POManskiIVSystem.manski_ATE_mem_Icc","k":"theorem","m":"Causalean.PO.ID.Partial.Manski.IntervalForm","a":"PO/ID/Partial/Manski","t":1,"d":"Set.Icc form of manski_bounds_ATE. Under the baseline Manski assumptions and mean independence of both potential outcomes from the instrument, for any two instrument values z₁, z₀ in the support of the instrument, the average treatment effect lies in the closed interval from the z₁-lower/z₀-upper worst-case bound to the z₁-upper/z₀-lower worst-case bound — the per-stratum-pair Manski no-assumption sandwich restated as interval membership.","s":"∀ {P : Causalean.PO.POSystem} {α : Type u_1} [inst : MeasurableSpace α] [inst_1 : MeasurableSingletonClass α] (S : Causalean.PO.POManskiIVSystem P α) [MeasureTheory.IsFiniteMeasure P.μ] (hA : S.BaseAssumptions), S.MeanIndep → ∀ {z₁ z₀ : α}, z₁ ∈ S.support → z₀ ∈ S.support → S.ATE ∈ Set.Icc (S.lowerBound1 hA.lo z₁ - S.upperBound0 hA.hi z₀) (S.upperBound1 hA.hi z₁ - S.lowerBound0 hA.lo z₀)","l":35,"f":"Causalean/PO/ID/Partial/Manski/IntervalForm.lean"},{"n":"Causalean.PO.POManskiIVSystem.manski_ATE_mem_Icc_ciSup","k":"theorem","m":"Causalean.PO.ID.Partial.Manski.IntervalForm","a":"PO/ID/Partial/Manski","t":1,"d":"Set.Icc form of manski_bounds_ATE_ciSup. Under the baseline Manski assumptions, mean independence of both potential outcomes from the instrument, and a nonempty instrument support, the average treatment effect lies in the closed interval from the supremum-of-lowers-minus-infimum-of-uppers bound to the infimum-of-uppers-minus-supremum-of-lowers bound, aggregated over every instrument stratum — the sup/inf-aggregated Manski no-assumption sandwich restated as interval membership.","s":"∀ {P : Causalean.PO.POSystem} {α : Type u_1} [inst : MeasurableSpace α] [inst_1 : MeasurableSingletonClass α] (S : Causalean.PO.POManskiIVSystem P α) [MeasureTheory.IsFiniteMeasure P.μ] (hA : S.BaseAssumptions), S.MeanIndep → S.support.Nonempty → S.ATE ∈ Set.Icc ((⨆ z, S.lowerBound1 hA.lo ↑z) - ⨅ z, S.upperBound0 hA.hi ↑z) ((⨅ z, S.upperBound1 hA.hi ↑z) - ⨆ z, S.lowerBound0 hA.lo ↑z)","l":49,"f":"Causalean/PO/ID/Partial/Manski/IntervalForm.lean"},{"n":"Causalean.PO.POManskiIVSystem.manski_bounds_ATE","k":"theorem","m":"Causalean.PO.ID.Partial.Manski.NonAsp","a":"PO/ID/Partial/Manski","t":1,"d":"Manski bounds for the ATE under an imperfect instrument. For any two instrument strata z₁ and z₀ in the support of the instrument, the average treatment effect is sandwiched between the worst-case lower bound formed from the z₁-stratum lower envelope and z₀-stratum upper envelope, and the corresponding upper bound with the roles reversed — using only bounded outcomes and mean independence of the instrument, with no selection assumptions.","s":"∀ {P : Causalean.PO.POSystem} {α : Type u_1} [inst : MeasurableSpace α] [inst_1 : MeasurableSingletonClass α] (S : Causalean.PO.POManskiIVSystem P α) [MeasureTheory.IsFiniteMeasure P.μ] (hA : S.BaseAssumptions), S.MeanIndep → ∀ {z₁ z₀ : α}, z₁ ∈ S.support → z₀ ∈ S.support → S.lowerBound1 hA.lo z₁ - S.upperBound0 hA.hi z₀ ≤ S.ATE ∧ S.ATE ≤ S.upperBound1 hA.hi z₁ - S.lowerBound0 hA.lo z₀","l":72,"f":"Causalean/PO/ID/Partial/Manski/NonAsp.lean"},{"n":"Causalean.PO.POManskiIVSystem.manski_bounds_ATE_ciSup","k":"theorem","m":"Causalean.PO.ID.Partial.Manski.NonAsp","a":"PO/ID/Partial/Manski","t":1,"d":"Manski bounds for the ATE in sup/inf form. Under the baseline Manski assumptions, mean independence of the instrument, and a nonempty instrument support, the average treatment effect is sandwiched between the supremum of the lower-envelope bounds minus the infimum of the upper-envelope bounds on one side, and the infimum of the upper-envelope bounds minus the supremum of the lower-envelope bounds on the other, aggregated over the whole instrument support.","s":"∀ {P : Causalean.PO.POSystem} {α : Type u_1} [inst : MeasurableSpace α] [inst_1 : MeasurableSingletonClass α] (S : Causalean.PO.POManskiIVSystem P α) [MeasureTheory.IsFiniteMeasure P.μ] (hA : S.BaseAssumptions), S.MeanIndep → S.support.Nonempty → (⨆ z, S.lowerBound1 hA.lo ↑z) - ⨅ z, S.upperBound0 hA.hi ↑z ≤ S.ATE ∧ S.ATE ≤ (⨅ z, S.upperBound1 hA.hi ↑z) - ⨆ z, S.lowerBound0 hA.lo ↑z","l":129,"f":"Causalean/PO/ID/Partial/Manski/NonAsp.lean"},{"n":"Causalean.PO.POManskiIVSystem.measurableSet_dEvent","k":"theorem","m":"Causalean.PO.ID.Partial.Manski.Setup","a":"PO/ID/Partial/Manski","t":2,"d":"Each treatment arm event is measurable.","s":"∀ {P : Causalean.PO.POSystem} {α : Type u_1} [inst : MeasurableSpace α] [inst_1 : MeasurableSingletonClass α] (S : Causalean.PO.POManskiIVSystem P α) (d : Bool), MeasurableSet (S.dEvent d)","l":99,"f":"Causalean/PO/ID/Partial/Manski/Setup.lean"},{"n":"Causalean.PO.POManskiIVSystem.measurableSet_zEvent","k":"theorem","m":"Causalean.PO.ID.Partial.Manski.Setup","a":"PO/ID/Partial/Manski","t":2,"d":"Each instrument stratum event is measurable.","s":"∀ {P : Causalean.PO.POSystem} {α : Type u_1} [inst : MeasurableSpace α] [inst_1 : MeasurableSingletonClass α] (S : Causalean.PO.POManskiIVSystem P α) (z : α), MeasurableSet (S.zEvent z)","l":95,"f":"Causalean/PO/ID/Partial/Manski/Setup.lean"},{"n":"Causalean.PO.POManskiIVSystem.measurable_YofD","k":"theorem","m":"Causalean.PO.ID.Partial.Manski.Setup","a":"PO/ID/Partial/Manski","t":1,"d":"For a fixed treatment arm d, the counterfactual outcome Y(d) is measurable.","s":"∀ {P : Causalean.PO.POSystem} {α : Type u_1} [inst : MeasurableSpace α] [inst_1 : MeasurableSingletonClass α] (S : Causalean.PO.POManskiIVSystem P α) (d : Bool), Measurable (S.YofD d)","l":83,"f":"Causalean/PO/ID/Partial/Manski/Setup.lean"},{"n":"Causalean.PO.POManskiIVSystem.measurable_factualD","k":"theorem","m":"Causalean.PO.ID.Partial.Manski.Setup","a":"PO/ID/Partial/Manski","t":2,"d":"Factual treatment is measurable.","s":"∀ {P : Causalean.PO.POSystem} {α : Type u_1} [inst : MeasurableSpace α] [inst_1 : MeasurableSingletonClass α] (S : Causalean.PO.POManskiIVSystem P α), Measurable S.factualD","l":90,"f":"Causalean/PO/ID/Partial/Manski/Setup.lean"},{"n":"Causalean.PO.POManskiIVSystem.measurable_factualY","k":"theorem","m":"Causalean.PO.ID.Partial.Manski.Setup","a":"PO/ID/Partial/Manski","t":2,"d":"Factual outcome is measurable.","s":"∀ {P : Causalean.PO.POSystem} {α : Type u_1} [inst : MeasurableSpace α] [inst_1 : MeasurableSingletonClass α] (S : Causalean.PO.POManskiIVSystem P α), Measurable S.factualY","l":92,"f":"Causalean/PO/ID/Partial/Manski/Setup.lean"},{"n":"Causalean.PO.POManskiIVSystem.measurable_factualZ","k":"theorem","m":"Causalean.PO.ID.Partial.Manski.Setup","a":"PO/ID/Partial/Manski","t":2,"d":"Factual instrument is measurable.","s":"∀ {P : Causalean.PO.POSystem} {α : Type u_1} [inst : MeasurableSpace α] [inst_1 : MeasurableSingletonClass α] (S : Causalean.PO.POManskiIVSystem P α), Measurable S.factualZ","l":88,"f":"Causalean/PO/ID/Partial/Manski/Setup.lean"},{"n":"Causalean.PO.POManskiIVSystem.measure_mul_eventCondExp_const","k":"theorem","m":"Causalean.PO.ID.Partial.Manski.MTS","a":"PO/ID/Partial/Manski","t":2,"d":"(μ A).toReal * eventCondExp μ A (fun _ => c) = (μ A).toReal * c. Follows from eventCondExp_mul_measure_toReal with the constant integrand.","s":"∀ {P : Causalean.PO.POSystem} (A : Set P.Ω) (c : ℝ), ((P.μ A).toReal * Causalean.PO.eventCondExp P.μ A fun x => c) = (P.μ A).toReal * c","l":132,"f":"Causalean/PO/ID/Partial/Manski/MTS.lean"},{"n":"Causalean.PO.POManskiIVSystem.miv_bounds_ATE","k":"theorem","m":"Causalean.PO.ID.Partial.Manski.MIV","a":"PO/ID/Partial/Manski","t":1,"d":"MIV ATE envelope bounds (prop:po-iv-miv, integrated form). Under the baseline Manski assumptions and a monotone instrumental variable — the conditional mean of each potential outcome is nondecreasing in the instrument value across its support, the average treatment effect is sandwiched between the integrated lower-envelope contrast ∫ (mLower1(Z) − mUpper0(Z)) and the integrated upper-envelope contrast ∫ (mUpper1(Z) − mLower0(Z)).","s":"∀ {P : Causalean.PO.POSystem} {α : Type u_1} [inst : MeasurableSpace α] [inst_1 : MeasurableSingletonClass α] (S : Causalean.PO.POManskiIVSystem P α) [MeasureTheory.IsFiniteMeasure P.μ] [Fintype α] (hA : S.BaseAssumptions) (hMIV : S.MIV), ∫ (ω : P.Ω), S.mLower1 hA (S.factualZ ω) - S.mUpper0 hA (S.factualZ ω) ∂P.μ ≤ S.ATE ∧ S.ATE ≤ ∫ (ω : P.Ω), S.mUpper1 hA (S.factualZ ω) - S.mLower0 hA (S.factualZ ω) ∂P.μ","l":328,"f":"Causalean/PO/ID/Partial/Manski/MIV.lean"},{"n":"Causalean.PO.POManskiIVSystem.miv_cond_Y0_le_mUpper0","k":"theorem","m":"Causalean.PO.ID.Partial.Manski.MIV","a":"PO/ID/Partial/Manski","t":2,"d":"E[Y(0) | Z = z] ≤ mUpper0 z for every z ∈ supp Z.","s":"∀ {P : Causalean.PO.POSystem} {α : Type u_1} [inst : MeasurableSpace α] [inst_1 : MeasurableSingletonClass α] (S : Causalean.PO.POManskiIVSystem P α) [MeasureTheory.IsFiniteMeasure P.μ] (hA : S.BaseAssumptions) (hMIV : S.MIV) {z : α}, z ∈ S.support → Causalean.PO.eventCondExp P.μ (S.zEvent z) (S.YofD false) ≤ S.mUpper0 hA z","l":151,"f":"Causalean/PO/ID/Partial/Manski/MIV.lean"},{"n":"Causalean.PO.POManskiIVSystem.miv_cond_Y1_le_mUpper1","k":"theorem","m":"Causalean.PO.ID.Partial.Manski.MIV","a":"PO/ID/Partial/Manski","t":2,"d":"E[Y(1) | Z = z] ≤ mUpper1 z for every z ∈ supp Z.","s":"∀ {P : Causalean.PO.POSystem} {α : Type u_1} [inst : MeasurableSpace α] [inst_1 : MeasurableSingletonClass α] (S : Causalean.PO.POManskiIVSystem P α) [MeasureTheory.IsFiniteMeasure P.μ] (hA : S.BaseAssumptions) (hMIV : S.MIV) {z : α}, z ∈ S.support → Causalean.PO.eventCondExp P.μ (S.zEvent z) (S.YofD true) ≤ S.mUpper1 hA z","l":109,"f":"Causalean/PO/ID/Partial/Manski/MIV.lean"},{"n":"Causalean.PO.POManskiIVSystem.miv_integral_Y0_le_integral_mUpper0","k":"theorem","m":"Causalean.PO.ID.Partial.Manski.MIV","a":"PO/ID/Partial/Manski","t":2,"d":"Integrated envelope bound for arm d = 0 (upper): ∫ Y(0) ≤ ∫ mUpper0(Z).","s":"∀ {P : Causalean.PO.POSystem} {α : Type u_1} [inst : MeasurableSpace α] [inst_1 : MeasurableSingletonClass α] (S : Causalean.PO.POManskiIVSystem P α) [MeasureTheory.IsFiniteMeasure P.μ] [Fintype α] (hA : S.BaseAssumptions) (hMIV : S.MIV), ∫ (ω : P.Ω), S.YofD false ω ∂P.μ ≤ ∫ (ω : P.Ω), S.mUpper0 hA (S.factualZ ω) ∂P.μ","l":313,"f":"Causalean/PO/ID/Partial/Manski/MIV.lean"},{"n":"Causalean.PO.POManskiIVSystem.miv_integral_Y1_le_integral_mUpper1","k":"theorem","m":"Causalean.PO.ID.Partial.Manski.MIV","a":"PO/ID/Partial/Manski","t":2,"d":"Integrated envelope bound for arm d = 1 (upper): ∫ Y(1) ≤ ∫ mUpper1(Z).","s":"∀ {P : Causalean.PO.POSystem} {α : Type u_1} [inst : MeasurableSpace α] [inst_1 : MeasurableSingletonClass α] (S : Causalean.PO.POManskiIVSystem P α) [MeasureTheory.IsFiniteMeasure P.μ] [Fintype α] (hA : S.BaseAssumptions) (hMIV : S.MIV), ∫ (ω : P.Ω), S.YofD true ω ∂P.μ ≤ ∫ (ω : P.Ω), S.mUpper1 hA (S.factualZ ω) ∂P.μ","l":283,"f":"Causalean/PO/ID/Partial/Manski/MIV.lean"},{"n":"Causalean.PO.POManskiIVSystem.miv_integral_mLower0_le_integral_Y0","k":"theorem","m":"Causalean.PO.ID.Partial.Manski.MIV","a":"PO/ID/Partial/Manski","t":2,"d":"Integrated envelope bound for arm d = 0 (lower): ∫ mLower0(Z) ≤ ∫ Y(0).","s":"∀ {P : Causalean.PO.POSystem} {α : Type u_1} [inst : MeasurableSpace α] [inst_1 : MeasurableSingletonClass α] (S : Causalean.PO.POManskiIVSystem P α) [MeasureTheory.IsFiniteMeasure P.μ] [Fintype α] (hA : S.BaseAssumptions) (hMIV : S.MIV), ∫ (ω : P.Ω), S.mLower0 hA (S.factualZ ω) ∂P.μ ≤ ∫ (ω : P.Ω), S.YofD false ω ∂P.μ","l":298,"f":"Causalean/PO/ID/Partial/Manski/MIV.lean"},{"n":"Causalean.PO.POManskiIVSystem.miv_integral_mLower1_le_integral_Y1","k":"theorem","m":"Causalean.PO.ID.Partial.Manski.MIV","a":"PO/ID/Partial/Manski","t":2,"d":"Integrated envelope bound for arm d = 1 (lower): ∫ mLower1(Z) ≤ ∫ Y(1).","s":"∀ {P : Causalean.PO.POSystem} {α : Type u_1} [inst : MeasurableSpace α] [inst_1 : MeasurableSingletonClass α] (S : Causalean.PO.POManskiIVSystem P α) [MeasureTheory.IsFiniteMeasure P.μ] [Fintype α] (hA : S.BaseAssumptions) (hMIV : S.MIV), ∫ (ω : P.Ω), S.mLower1 hA (S.factualZ ω) ∂P.μ ≤ ∫ (ω : P.Ω), S.YofD true ω ∂P.μ","l":268,"f":"Causalean/PO/ID/Partial/Manski/MIV.lean"},{"n":"Causalean.PO.POManskiIVSystem.miv_mLower0_le_cond_Y0","k":"theorem","m":"Causalean.PO.ID.Partial.Manski.MIV","a":"PO/ID/Partial/Manski","t":2,"d":"mLower0 z ≤ E[Y(0) | Z = z] for every z ∈ supp Z.","s":"∀ {P : Causalean.PO.POSystem} {α : Type u_1} [inst : MeasurableSpace α] [inst_1 : MeasurableSingletonClass α] (S : Causalean.PO.POManskiIVSystem P α) [MeasureTheory.IsFiniteMeasure P.μ] (hA : S.BaseAssumptions) (hMIV : S.MIV) {z : α}, z ∈ S.support → S.mLower0 hA z ≤ Causalean.PO.eventCondExp P.μ (S.zEvent z) (S.YofD false)","l":130,"f":"Causalean/PO/ID/Partial/Manski/MIV.lean"},{"n":"Causalean.PO.POManskiIVSystem.miv_mLower1_le_cond_Y1","k":"theorem","m":"Causalean.PO.ID.Partial.Manski.MIV","a":"PO/ID/Partial/Manski","t":2,"d":"mLower1 z ≤ E[Y(1) | Z = z] for every z ∈ supp Z.","s":"∀ {P : Causalean.PO.POSystem} {α : Type u_1} [inst : MeasurableSpace α] [inst_1 : MeasurableSingletonClass α] (S : Causalean.PO.POManskiIVSystem P α) [MeasureTheory.IsFiniteMeasure P.μ] (hA : S.BaseAssumptions) (hMIV : S.MIV) {z : α}, z ∈ S.support → S.mLower1 hA z ≤ Causalean.PO.eventCondExp P.μ (S.zEvent z) (S.YofD true)","l":80,"f":"Causalean/PO/ID/Partial/Manski/MIV.lean"},{"n":"Causalean.PO.POManskiIVSystem.mtr_E_Y0_le_E_Y","k":"theorem","m":"Causalean.PO.ID.Partial.Manski.MTR","a":"PO/ID/Partial/Manski","t":2,"d":"Step 4 of prop:po-iv-mtr: E[Y(0)] ≤ E[Y].","s":"∀ {P : Causalean.PO.POSystem} {α : Type u_1} [inst : MeasurableSpace α] [inst_1 : MeasurableSingletonClass α] (S : Causalean.PO.POManskiIVSystem P α) (hA : S.BaseAssumptions), S.MTR → ∫ (ω : P.Ω), S.YofD false ω ∂P.μ ≤ ∫ (ω : P.Ω), S.factualY ω ∂P.μ","l":146,"f":"Causalean/PO/ID/Partial/Manski/MTR.lean"},{"n":"Causalean.PO.POManskiIVSystem.mtr_E_Y1_le_upper","k":"theorem","m":"Causalean.PO.ID.Partial.Manski.MTR","a":"PO/ID/Partial/Manski","t":2,"d":"Step 2 of prop:po-iv-mtr: E[Y(1)] ≤ p·E[Y|D=1] + (1-p)·b.","s":"∀ {P : Causalean.PO.POSystem} {α : Type u_1} [inst : MeasurableSpace α] [inst_1 : MeasurableSingletonClass α] (S : Causalean.PO.POManskiIVSystem P α) (hA : S.BaseAssumptions), ∫ (ω : P.Ω), S.YofD true ω ∂P.μ ≤ (P.μ (S.dEvent true)).toReal * Causalean.PO.eventCondExp P.μ (S.dEvent true) S.factualY + (P.μ (S.dEvent false)).toReal * hA.hi","l":152,"f":"Causalean/PO/ID/Partial/Manski/MTR.lean"},{"n":"Causalean.PO.POManskiIVSystem.mtr_E_Y_le_E_Y1","k":"theorem","m":"Causalean.PO.ID.Partial.Manski.MTR","a":"PO/ID/Partial/Manski","t":2,"d":"Step 1 of prop:po-iv-mtr: E[Y] ≤ E[Y(1)].","s":"∀ {P : Causalean.PO.POSystem} {α : Type u_1} [inst : MeasurableSpace α] [inst_1 : MeasurableSingletonClass α] (S : Causalean.PO.POManskiIVSystem P α) (hA : S.BaseAssumptions), S.MTR → ∫ (ω : P.Ω), S.factualY ω ∂P.μ ≤ ∫ (ω : P.Ω), S.YofD true ω ∂P.μ","l":140,"f":"Causalean/PO/ID/Partial/Manski/MTR.lean"},{"n":"Causalean.PO.POManskiIVSystem.mtr_bounds_ATE","k":"theorem","m":"Causalean.PO.ID.Partial.Manski.MTR","a":"PO/ID/Partial/Manski","t":1,"d":"The two-sided ATE sandwich from prop:po-iv-mtr. Under the baseline Manski assumptions and monotone treatment response, Y(0) ≤ Y(1) almost surely, the average treatment effect is nonnegative and is upper-bounded by the probability-weighted mix of the observed treated/control means and the range endpoints lo, hi — imputing hi for Y(1) on the control arm and lo for Y(0) on the treated arm.","s":"∀ {P : Causalean.PO.POSystem} {α : Type u_1} [inst : MeasurableSpace α] [inst_1 : MeasurableSingletonClass α] (S : Causalean.PO.POManskiIVSystem P α) (hA : S.BaseAssumptions), S.MTR → 0 ≤ S.ATE ∧ S.ATE ≤ (P.μ (S.dEvent true)).toReal * Causalean.PO.eventCondExp P.μ (S.dEvent true) S.factualY + (P.μ (S.dEvent false)).toReal * hA.hi - ((P.μ (S.dEvent false)).toReal * Causalean.PO.eventCondExp P.μ (S.dEvent false) S.factualY + (P.μ (S.dEvent true)).toReal * hA.lo)","l":251,"f":"Causalean/PO/ID/Partial/Manski/MTR.lean"},{"n":"Causalean.PO.POManskiIVSystem.mtr_lower_le_E_Y0","k":"theorem","m":"Causalean.PO.ID.Partial.Manski.MTR","a":"PO/ID/Partial/Manski","t":2,"d":"Step 3 of prop:po-iv-mtr: (1-p)·E[Y|D=0] + p·a ≤ E[Y(0)].","s":"∀ {P : Causalean.PO.POSystem} {α : Type u_1} [inst : MeasurableSpace α] [inst_1 : MeasurableSingletonClass α] (S : Causalean.PO.POManskiIVSystem P α) (hA : S.BaseAssumptions), (P.μ (S.dEvent false)).toReal * Causalean.PO.eventCondExp P.μ (S.dEvent false) S.factualY + (P.μ (S.dEvent true)).toReal * hA.lo ≤ ∫ (ω : P.Ω), S.YofD false ω ∂P.μ","l":198,"f":"Causalean/PO/ID/Partial/Manski/MTR.lean"},{"n":"Causalean.PO.POManskiIVSystem.mtr_miv_ATE_mem_Icc","k":"theorem","m":"Causalean.PO.ID.Partial.Manski.IntervalForm","a":"PO/ID/Partial/Manski","t":1,"d":"Set.Icc form of mtr_miv_bounds_ATE. Under the baseline Manski assumptions, monotone treatment response, and a monotone instrumental variable, the average treatment effect lies in the closed interval from 0 to the integrated monotone-instrument envelope contrast ∫ (mUpper1(Z) − mLower0(Z)) — the MTR + MIV sandwich restated as interval membership.","s":"∀ {P : Causalean.PO.POSystem} {α : Type u_1} [inst : MeasurableSpace α] [inst_1 : MeasurableSingletonClass α] (S : Causalean.PO.POManskiIVSystem P α) [MeasureTheory.IsFiniteMeasure P.μ] [Fintype α] (hA : S.BaseAssumptions), S.MTR → ∀ (hMIV : S.MIV), S.ATE ∈ Set.Icc 0 (∫ (ω : P.Ω), S.mUpper1 hA (S.factualZ ω) - S.mLower0 hA (S.factualZ ω) ∂P.μ)","l":79,"f":"Causalean/PO/ID/Partial/Manski/IntervalForm.lean"},{"n":"Causalean.PO.POManskiIVSystem.mtr_miv_bounds_ATE","k":"theorem","m":"Causalean.PO.ID.Partial.Manski.Combined","a":"PO/ID/Partial/Manski","t":1,"d":"cor:po-iv-mtr-miv. Under the baseline Manski assumptions, monotone treatment response, Y(0) ≤ Y(1) almost surely, and a monotone instrumental variable — the conditional mean of each potential outcome is nondecreasing in the instrument value across its support, the average treatment effect is nonnegative and is upper-bounded by the integrated monotone-instrument envelope contrast ∫ (mUpper1(Z) − mLower0(Z)).","s":"∀ {P : Causalean.PO.POSystem} {α : Type u_1} [inst : MeasurableSpace α] [inst_1 : MeasurableSingletonClass α] (S : Causalean.PO.POManskiIVSystem P α) [MeasureTheory.IsFiniteMeasure P.μ] [Fintype α] (hA : S.BaseAssumptions), S.MTR → ∀ (hMIV : S.MIV), 0 ≤ S.ATE ∧ S.ATE ≤ ∫ (ω : P.Ω), S.mUpper1 hA (S.factualZ ω) - S.mLower0 hA (S.factualZ ω) ∂P.μ","l":56,"f":"Causalean/PO/ID/Partial/Manski/Combined.lean"},{"n":"Causalean.PO.POManskiIVSystem.mtr_mts_ATE_mem_Icc","k":"theorem","m":"Causalean.PO.ID.Partial.Manski.IntervalForm","a":"PO/ID/Partial/Manski","t":1,"d":"Set.Icc form of mtr_mts_bounds_ATE. Under the baseline Manski assumptions, monotone treatment response, and monotone treatment selection, the average treatment effect lies in the closed interval from 0 to the naive observed contrast E[Y | D=1] − E[Y | D=0] — the MTR + MTS sandwich restated as interval membership.","s":"∀ {P : Causalean.PO.POSystem} {α : Type u_1} [inst : MeasurableSpace α] [inst_1 : MeasurableSingletonClass α] (S : Causalean.PO.POManskiIVSystem P α) (hA : S.BaseAssumptions), S.MTR → S.MTS → S.ATE ∈ Set.Icc 0 (Causalean.PO.eventCondExp P.μ (S.dEvent true) S.factualY - Causalean.PO.eventCondExp P.μ (S.dEvent false) S.factualY)","l":66,"f":"Causalean/PO/ID/Partial/Manski/IntervalForm.lean"},{"n":"Causalean.PO.POManskiIVSystem.mtr_mts_bounds_ATE","k":"theorem","m":"Causalean.PO.ID.Partial.Manski.Combined","a":"PO/ID/Partial/Manski","t":1,"d":"cor:po-iv-mtr-mts. Under the baseline Manski assumptions, monotone treatment response, Y(0) ≤ Y(1) almost surely, and monotone treatment selection — each potential outcome's mean given control is no larger than its mean given treatment, the average treatment effect is nonnegative and is upper-bounded by the naive observed contrast E[Y | D=1] − E[Y | D=0].","s":"∀ {P : Causalean.PO.POSystem} {α : Type u_1} [inst : MeasurableSpace α] [inst_1 : MeasurableSingletonClass α] (S : Causalean.PO.POManskiIVSystem P α) (hA : S.BaseAssumptions), S.MTR → S.MTS → 0 ≤ S.ATE ∧ S.ATE ≤ Causalean.PO.eventCondExp P.μ (S.dEvent true) S.factualY - Causalean.PO.eventCondExp P.μ (S.dEvent false) S.factualY","l":36,"f":"Causalean/PO/ID/Partial/Manski/Combined.lean"},{"n":"Causalean.PO.POManskiIVSystem.mtr_nonneg_ATE","k":"theorem","m":"Causalean.PO.ID.Partial.Manski.MTR","a":"PO/ID/Partial/Manski","t":2,"d":"Step 5 (ATE nonnegativity) of prop:po-iv-mtr: 0 ≤ ATE.","s":"∀ {P : Causalean.PO.POSystem} {α : Type u_1} [inst : MeasurableSpace α] [inst_1 : MeasurableSingletonClass α] (S : Causalean.PO.POManskiIVSystem P α) (hA : S.BaseAssumptions), S.MTR → 0 ≤ S.ATE","l":241,"f":"Causalean/PO/ID/Partial/Manski/MTR.lean"},{"n":"Causalean.PO.POManskiIVSystem.mts_E_Y0_le_upper","k":"theorem","m":"Causalean.PO.ID.Partial.Manski.MTS","a":"PO/ID/Partial/Manski","t":2,"d":"MTS upper bound on E[Y(0)]: E[Y(0)] ≤ p · b + (1-p) · E[Y|D=0].","s":"∀ {P : Causalean.PO.POSystem} {α : Type u_1} [inst : MeasurableSpace α] [inst_1 : MeasurableSingletonClass α] (S : Causalean.PO.POManskiIVSystem P α) (hA : S.BaseAssumptions), S.MTS → ∫ (ω : P.Ω), S.YofD false ω ∂P.μ ≤ (P.μ (S.dEvent true)).toReal * hA.hi + (1 - (P.μ (S.dEvent true)).toReal) * Causalean.PO.eventCondExp P.μ (S.dEvent false) S.factualY","l":263,"f":"Causalean/PO/ID/Partial/Manski/MTS.lean"},{"n":"Causalean.PO.POManskiIVSystem.mts_E_Y1_le_condY1","k":"theorem","m":"Causalean.PO.ID.Partial.Manski.MTS","a":"PO/ID/Partial/Manski","t":2,"d":"MTS upper bound on E[Y(1)]: E[Y(1)] ≤ E[Y|D=1].","s":"∀ {P : Causalean.PO.POSystem} {α : Type u_1} [inst : MeasurableSpace α] [inst_1 : MeasurableSingletonClass α] (S : Causalean.PO.POManskiIVSystem P α) (hA : S.BaseAssumptions), S.MTS → ∫ (ω : P.Ω), S.YofD true ω ∂P.μ ≤ Causalean.PO.eventCondExp P.μ (S.dEvent true) S.factualY","l":205,"f":"Causalean/PO/ID/Partial/Manski/MTS.lean"},{"n":"Causalean.PO.POManskiIVSystem.mts_bounds_ATE","k":"theorem","m":"Causalean.PO.ID.Partial.Manski.MTS","a":"PO/ID/Partial/Manski","t":1,"d":"MTS bounds for the ATE. Under the baseline Manski assumptions and monotone treatment selection — each potential outcome's mean given control is no larger than its mean given treatment, the average treatment effect is sandwiched between a lower bound built from the probability-weighted observed treated mean plus the range floor lo, minus the probability-weighted mix of the range ceiling hi and the observed control mean, and an upper bound equal to the naive observed contrast E[Y | D=1] − E[Y | D=0] — obtained by subtracting the Y(0) upper/lower bounds from the Y(1) lower/upper bounds.","s":"∀ {P : Causalean.PO.POSystem} {α : Type u_1} [inst : MeasurableSpace α] [inst_1 : MeasurableSingletonClass α] (S : Causalean.PO.POManskiIVSystem P α) (hA : S.BaseAssumptions), S.MTS → (P.μ (S.dEvent true)).toReal * Causalean.PO.eventCondExp P.μ (S.dEvent true) S.factualY + (1 - (P.μ (S.dEvent true)).toReal) * hA.lo - ((P.μ (S.dEvent true)).toReal * hA.hi + (1 - (P.μ (S.dEvent true)).toReal) * Causalean.PO.eventCondExp P.μ (S.dEvent false) S.factualY) ≤ S.ATE ∧ S.ATE ≤ Causalean.PO.eventCondExp P.μ (S.dEvent true) S.factualY - Causalean.PO.eventCondExp P.μ (S.dEvent false) S.factualY","l":311,"f":"Causalean/PO/ID/Partial/Manski/MTS.lean"},{"n":"Causalean.PO.POManskiIVSystem.mts_condY0_le_E_Y0","k":"theorem","m":"Causalean.PO.ID.Partial.Manski.MTS","a":"PO/ID/Partial/Manski","t":2,"d":"MTS lower bound on E[Y(0)]: E[Y|D=0] ≤ E[Y(0)].","s":"∀ {P : Causalean.PO.POSystem} {α : Type u_1} [inst : MeasurableSpace α] [inst_1 : MeasurableSingletonClass α] (S : Causalean.PO.POManskiIVSystem P α) (hA : S.BaseAssumptions), S.MTS → Causalean.PO.eventCondExp P.μ (S.dEvent false) S.factualY ≤ ∫ (ω : P.Ω), S.YofD false ω ∂P.μ","l":236,"f":"Causalean/PO/ID/Partial/Manski/MTS.lean"},{"n":"Causalean.PO.POManskiIVSystem.mts_lower_le_E_Y1","k":"theorem","m":"Causalean.PO.ID.Partial.Manski.MTS","a":"PO/ID/Partial/Manski","t":2,"d":"MTS lower bound on E[Y(1)]: p · E[Y|D=1] + (1-p) · a ≤ E[Y(1)].","s":"∀ {P : Causalean.PO.POSystem} {α : Type u_1} [inst : MeasurableSpace α] [inst_1 : MeasurableSingletonClass α] (S : Causalean.PO.POManskiIVSystem P α) (hA : S.BaseAssumptions), S.MTS → (P.μ (S.dEvent true)).toReal * Causalean.PO.eventCondExp P.μ (S.dEvent true) S.factualY + (1 - (P.μ (S.dEvent true)).toReal) * hA.lo ≤ ∫ (ω : P.Ω), S.YofD true ω ∂P.μ","l":143,"f":"Causalean/PO/ID/Partial/Manski/MTS.lean"},{"n":"Causalean.PO.POManskiIVSystem.prob_dEvent_true_add_false","k":"theorem","m":"Causalean.PO.ID.Partial.Manski.MTS","a":"PO/ID/Partial/Manski","t":2,"d":"Probability split: P(D=true).toReal + P(D=false).toReal = 1.","s":"∀ {P : Causalean.PO.POSystem} {α : Type u_1} [inst : MeasurableSpace α] [inst_1 : MeasurableSingletonClass α] (S : Causalean.PO.POManskiIVSystem P α), (P.μ (S.dEvent true)).toReal + (P.μ (S.dEvent false)).toReal = 1","l":65,"f":"Causalean/PO/ID/Partial/Manski/MTS.lean"},{"n":"Causalean.PO.POManskiIVSystem.support","k":"def","m":"Causalean.PO.ID.Partial.Manski.Setup","a":"PO/ID/Partial/Manski","t":1,"d":"Support of the instrument: {z | μ(Z = z) ≠ 0}.","s":"{P : Causalean.PO.POSystem} → {α : Type u_1} → [inst : MeasurableSpace α] → [inst_1 : MeasurableSingletonClass α] → Causalean.PO.POManskiIVSystem P α → Set α","l":108,"f":"Causalean/PO/ID/Partial/Manski/Setup.lean"},{"n":"Causalean.PO.POManskiIVSystem.upperBound0","k":"def","m":"Causalean.PO.ID.Partial.Manski.Setup","a":"PO/ID/Partial/Manski","t":1,"d":"The upper observable bound for the control potential-outcome mean in instrument stratum z uses the outcome ceiling on treated units.","s":"{P : Causalean.PO.POSystem} → {α : Type u_1} → [inst : MeasurableSpace α] → [inst_1 : MeasurableSingletonClass α] → Causalean.PO.POManskiIVSystem P α → ℝ → α → ℝ","l":135,"f":"Causalean/PO/ID/Partial/Manski/Setup.lean"},{"n":"Causalean.PO.POManskiIVSystem.upperBound1","k":"def","m":"Causalean.PO.ID.Partial.Manski.Setup","a":"PO/ID/Partial/Manski","t":1,"d":"The upper observable bound for the treated potential-outcome mean in instrument stratum z uses the outcome ceiling on untreated units.","s":"{P : Causalean.PO.POSystem} → {α : Type u_1} → [inst : MeasurableSpace α] → [inst_1 : MeasurableSingletonClass α] → Causalean.PO.POManskiIVSystem P α → ℝ → α → ℝ","l":127,"f":"Causalean/PO/ID/Partial/Manski/Setup.lean"},{"n":"Causalean.PO.POManskiIVSystem.yVar","k":"def","m":"Causalean.PO.ID.Partial.Manski.Setup","a":"PO/ID/Partial/Manski","t":1,"d":"Outcome packaged as a POVar valued in ℝ.","s":"{P : Causalean.PO.POSystem} → {α : Type u_1} → [inst : MeasurableSpace α] → [inst_1 : MeasurableSingletonClass α] → Causalean.PO.POManskiIVSystem P α → Causalean.PO.POVar P ℝ","l":58,"f":"Causalean/PO/ID/Partial/Manski/Setup.lean"},{"n":"Causalean.PO.POManskiIVSystem.zEvent","k":"def","m":"Causalean.PO.ID.Partial.Manski.Setup","a":"PO/ID/Partial/Manski","t":1,"d":"The event {Z = z}.","s":"{P : Causalean.PO.POSystem} → {α : Type u_1} → [inst : MeasurableSpace α] → [inst_1 : MeasurableSingletonClass α] → Causalean.PO.POManskiIVSystem P α → α → Set P.Ω","l":75,"f":"Causalean/PO/ID/Partial/Manski/Setup.lean"},{"n":"Causalean.PO.POManskiIVSystem.zVar","k":"def","m":"Causalean.PO.ID.Partial.Manski.Setup","a":"PO/ID/Partial/Manski","t":1,"d":"Instrument packaged as a POVar valued in α.","s":"{P : Causalean.PO.POSystem} → {α : Type u_1} → [inst : MeasurableSpace α] → [inst_1 : MeasurableSingletonClass α] → Causalean.PO.POManskiIVSystem P α → Causalean.PO.POVar P α","l":52,"f":"Causalean/PO/ID/Partial/Manski/Setup.lean"},{"n":"Causalean.PO.POPartialLinearModel","k":"structure","m":"Causalean.PO.ID.Exact.PartialLinear.Setup","a":"PO/ID/Exact/PartialLinear","t":1,"d":"Partially linear model under the backdoor PO framework. On top of the PO substrate, this bundles a measurable covariate function b giving the nonparametric baseline, a homogeneous per-unit treatment effect θ, a measurable structural error term U, the structural restriction that every unit's dose-response is the straight line Y(d) = b(X) + θ·d + U with the same slope for everyone, the standard backdoor (unconfoundedness) assumption that the structural error has zero mean conditional on the observed covariate and treatment, and consistency: the observed outcome is the potential outcome at the realized treatment.","s":"Causalean.PO.POSystem → (γ : Type u_1) → [MeasurableSpace γ] → Type (max (max (max u_1 u_2) u_3) u_4)","l":139,"f":"Causalean/PO/ID/Exact/PartialLinear/Setup.lean"},{"n":"Causalean.PO.POPartialLinearModel.causal_homogeneity","k":"theorem","m":"Causalean.PO.ID.Exact.PartialLinear.Setup","a":"PO/ID/Exact/PartialLinear","t":1,"d":"Causal reading of θ. Almost surely, for every pair of dose levels, the difference of the corresponding potential outcomes equals the slope θ times the difference of the doses, so θ is the constant per-unit causal effect of the treatment.","s":"∀ {P : Causalean.PO.POSystem} {γ : Type u_1} [inst : MeasurableSpace γ] (M : Causalean.PO.POPartialLinearModel P γ), ∀ᵐ (ω : P.Ω) ∂P.μ, ∀ (d d' : ℝ), M.YofD d ω - M.YofD d' ω = M.θ * (d - d')","l":198,"f":"Causalean/PO/ID/Exact/PartialLinear/Setup.lean"},{"n":"Causalean.PO.POPartialLinearModel.condExp_U_sigmaX","k":"theorem","m":"Causalean.PO.ID.Exact.PartialLinear.Identification","a":"PO/ID/Exact/PartialLinear","t":2,"d":"The structural error has zero conditional mean given the covariate. Follows from the backdoor assumption E[U | σ(X,D)] = 0 by the tower property, since σ(X) ⊆ σ(X,D).","s":"∀ {P : Causalean.PO.POSystem} {γ : Type u_1} [inst : MeasurableSpace γ] [MeasureTheory.IsFiniteMeasure P.μ] (M : Causalean.PO.POPartialLinearModel P γ), P.μ[M.U | M.sigmaX] =ᵐ[P.μ] 0","l":69,"f":"Causalean/PO/ID/Exact/PartialLinear/Identification.lean"},{"n":"Causalean.PO.POPartialLinearModel.condExp_resid_sigmaX","k":"theorem","m":"Causalean.PO.ID.Exact.PartialLinear.Identification","a":"PO/ID/Exact/PartialLinear","t":2,"d":"The treatment residual is conditionally mean-zero given the covariate: E[D − m₀(X) | σ(X)] = 0. Immediate from condExp_sub and idempotence of the conditional expectation.","s":"∀ {P : Causalean.PO.POSystem} {γ : Type u_1} [inst : MeasurableSpace γ] [MeasureTheory.IsFiniteMeasure P.μ] (M : Causalean.PO.POPartialLinearModel P γ), MeasureTheory.Integrable M.factualD P.μ → P.μ[M.resid | M.sigmaX] =ᵐ[P.μ] 0","l":82,"f":"Causalean/PO/ID/Exact/PartialLinear/Identification.lean"},{"n":"Causalean.PO.POPartialLinearModel.factualY_eq","k":"theorem","m":"Causalean.PO.ID.Exact.PartialLinear.Setup","a":"PO/ID/Exact/PartialLinear","t":2,"d":"The observed-data form of the structural model: almost surely Y = b(X) + θ·D + U. Obtained from the homogeneous dose-response evaluated at the realized treatment, using consistency Y = Y(D).","s":"∀ {P : Causalean.PO.POSystem} {γ : Type u_1} [inst : MeasurableSpace γ] (M : Causalean.PO.POPartialLinearModel P γ), M.factualY =ᵐ[P.μ] fun ω => M.b (M.factualX ω) + M.θ * M.factualD ω + M.U ω","l":182,"f":"Causalean/PO/ID/Exact/PartialLinear/Setup.lean"},{"n":"Causalean.PO.POPartialLinearModel.factualY_sub_lReg","k":"theorem","m":"Causalean.PO.ID.Exact.PartialLinear.Identification","a":"PO/ID/Exact/PartialLinear","t":2,"d":"Partialling-out identity (observed data): Y − ℓ₀(X) = θ·(D − m₀(X)) + U. Algebraic consequence of factualY_eq and lReg_eq.","s":"∀ {P : Causalean.PO.POSystem} {γ : Type u_1} [inst : MeasurableSpace γ] [MeasureTheory.IsFiniteMeasure P.μ] (M : Causalean.PO.POPartialLinearModel P γ), MeasureTheory.Integrable M.factualD P.μ → MeasureTheory.Integrable (fun ω => M.b (M.factualX ω)) P.μ → MeasureTheory.Integrable M.U P.μ → (fun ω => M.factualY ω - M.lReg ω) =ᵐ[P.μ] fun ω => M.θ * M.resid ω + M.U ω","l":196,"f":"Causalean/PO/ID/Exact/PartialLinear/Identification.lean"},{"n":"Causalean.PO.POPartialLinearModel.integral_U_resid","k":"theorem","m":"Causalean.PO.ID.Exact.PartialLinear.Identification","a":"PO/ID/Exact/PartialLinear","t":2,"d":"Orthogonality of the structural error to the covariate-treatment residual: E[U·(D − m₀(X))] = 0. Since the residual is σ(X,D)-measurable and E[U | σ(X,D)] = 0, the product integrates to zero (pull the residual out of the conditional expectation, then integrate).","s":"∀ {P : Causalean.PO.POSystem} {γ : Type u_1} [inst : MeasurableSpace γ] [MeasureTheory.IsFiniteMeasure P.μ] (M : Causalean.PO.POPartialLinearModel P γ), MeasureTheory.Integrable M.U P.μ → MeasureTheory.Integrable (fun ω => M.U ω * M.resid ω) P.μ → ∫ (ω : P.Ω), M.U ω * M.resid ω ∂P.μ = 0","l":103,"f":"Causalean/PO/ID/Exact/PartialLinear/Identification.lean"},{"n":"Causalean.PO.POPartialLinearModel.integral_partialled","k":"theorem","m":"Causalean.PO.ID.Exact.PartialLinear.Identification","a":"PO/ID/Exact/PartialLinear","t":2,"d":"The Robinson numerator equals θ times the residual second moment: E[(Y − ℓ₀(X))·(D − m₀(X))] = θ·E[(D − m₀(X))²]. Expand Y − ℓ₀ = θ·V + U and use E[U·V] = 0.","s":"∀ {P : Causalean.PO.POSystem} {γ : Type u_1} [inst : MeasurableSpace γ] [MeasureTheory.IsFiniteMeasure P.μ] (M : Causalean.PO.POPartialLinearModel P γ), MeasureTheory.Integrable M.factualD P.μ → MeasureTheory.Integrable (fun ω => M.b (M.factualX ω)) P.μ → MeasureTheory.Integrable M.U P.μ → MeasureTheory.Integrable (fun ω => M.resid ω ^ 2) P.μ → MeasureTheory.Integrable (fun ω => M.U ω * M.resid ω) P.μ → ∫ (ω : P.Ω), (M.factualY ω - M.lReg ω) * M.resid ω ∂P.μ = M.θ * ∫ (ω : P.Ω), M.resid ω ^ 2 ∂P.μ","l":209,"f":"Causalean/PO/ID/Exact/PartialLinear/Identification.lean"},{"n":"Causalean.PO.POPartialLinearModel.lReg","k":"def","m":"Causalean.PO.ID.Exact.PartialLinear.Identification","a":"PO/ID/Exact/PartialLinear","t":1,"d":"The outcome regression ℓ₀(X) = E[Y | σ(X)] (the conditional mean of the outcome given the covariate).","s":"{P : Causalean.PO.POSystem} → {γ : Type u_1} → [inst : MeasurableSpace γ] → Causalean.PO.POPartialLinearModel P γ → P.Ω → ℝ","l":61,"f":"Causalean/PO/ID/Exact/PartialLinear/Identification.lean"},{"n":"Causalean.PO.POPartialLinearModel.lReg_eq","k":"theorem","m":"Causalean.PO.ID.Exact.PartialLinear.Identification","a":"PO/ID/Exact/PartialLinear","t":2,"d":"The outcome regression decomposes as ℓ₀(X) = b(X) + θ·m₀(X). Apply the conditional expectation given σ(X) to the observed-data form Y = b(X) + θ·D + U: the covariate term is σ(X)-measurable, the treatment term contributes θ·m₀(X), and the error term vanishes by condExp_U_sigmaX.","s":"∀ {P : Causalean.PO.POSystem} {γ : Type u_1} [inst : MeasurableSpace γ] [MeasureTheory.IsFiniteMeasure P.μ] (M : Causalean.PO.POPartialLinearModel P γ), MeasureTheory.Integrable M.factualD P.μ → MeasureTheory.Integrable (fun ω => M.b (M.factualX ω)) P.μ → MeasureTheory.Integrable M.U P.μ → M.lReg =ᵐ[P.μ] fun ω => M.b (M.factualX ω) + M.θ * M.mReg ω","l":142,"f":"Causalean/PO/ID/Exact/PartialLinear/Identification.lean"},{"n":"Causalean.PO.POPartialLinearModel.mReg","k":"def","m":"Causalean.PO.ID.Exact.PartialLinear.Identification","a":"PO/ID/Exact/PartialLinear","t":1,"d":"The treatment regression m₀(X) = E[D | σ(X)] (the conditional mean of the treatment given the covariate).","s":"{P : Causalean.PO.POSystem} → {γ : Type u_1} → [inst : MeasurableSpace γ] → Causalean.PO.POPartialLinearModel P γ → P.Ω → ℝ","l":57,"f":"Causalean/PO/ID/Exact/PartialLinear/Identification.lean"},{"n":"Causalean.PO.POPartialLinearModel.resid","k":"def","m":"Causalean.PO.ID.Exact.PartialLinear.Identification","a":"PO/ID/Exact/PartialLinear","t":1,"d":"The treatment residual V = D − m₀(X) (treatment with its covariate prediction partialled out).","s":"{P : Causalean.PO.POSystem} → {γ : Type u_1} → [inst : MeasurableSpace γ] → Causalean.PO.POPartialLinearModel P γ → P.Ω → ℝ","l":65,"f":"Causalean/PO/ID/Exact/PartialLinear/Identification.lean"},{"n":"Causalean.PO.POPartialLinearModel.robinson_estimand_eq_theta","k":"theorem","m":"Causalean.PO.ID.Exact.PartialLinear.Identification","a":"PO/ID/Exact/PartialLinear","t":1,"d":"The causal bridge. Given that the factual treatment is integrable, the outcome regression on covariates is integrable, the structural error U is integrable, the squared treatment residual is integrable, and the product of the structural error and the treatment residual is integrable, provided also that the treatment has residual variation, i.e. E[(D − m₀(X))²] ≠ 0, the Robinson partialling-out estimand equals the structural treatment effect θ. This is what makes the partially linear DML estimator target the causal parameter.","s":"∀ {P : Causalean.PO.POSystem} {γ : Type u_1} [inst : MeasurableSpace γ] [MeasureTheory.IsFiniteMeasure P.μ] (M : Causalean.PO.POPartialLinearModel P γ), MeasureTheory.Integrable M.factualD P.μ → MeasureTheory.Integrable (fun ω => M.b (M.factualX ω)) P.μ → MeasureTheory.Integrable M.U P.μ → MeasureTheory.Integrable (fun ω => M.resid ω ^ 2) P.μ → MeasureTheory.Integrable (fun ω => M.U ω * M.resid ω) P.μ → ∫ (ω : P.Ω), M.resid ω ^ 2 ∂P.μ ≠ 0 → (∫ (ω : P.Ω), (M.factualY ω - M.lReg ω) * M.resid ω ∂P.μ) / ∫ (ω : P.Ω), M.resid ω ^ 2 ∂P.μ = M.θ","l":227,"f":"Causalean/PO/ID/Exact/PartialLinear/Identification.lean"},{"n":"Causalean.PO.POPartialLinearSystem","k":"structure","m":"Causalean.PO.ID.Exact.PartialLinear.Setup","a":"PO/ID/Exact/PartialLinear","t":1,"d":"Continuous-treatment backdoor subsystem. Inside a potential-outcome system, this bundles a real-valued treatment node D, a real-valued outcome node Y, and a covariate variable X taking values in an arbitrary measurable space, subject to the treatment, outcome, and covariate being pairwise distinct. This mirrors the binary backdoor subsystem but the treatment now ranges over the real line, as required by the partially linear model Y = g(X) + θ·D + noise.","s":"Causalean.PO.POSystem → (γ : Type u_1) → [MeasurableSpace γ] → Type (max (max u_1 u_2) u_3)","l":48,"f":"Causalean/PO/ID/Exact/PartialLinear/Setup.lean"},{"n":"Causalean.PO.POPartialLinearSystem.YofD","k":"def","m":"Causalean.PO.ID.Exact.PartialLinear.Setup","a":"PO/ID/Exact/PartialLinear","t":1,"d":"The potential outcome under dose d: the value the outcome would take were the treatment set to the real number d.","s":"{P : Causalean.PO.POSystem} → {γ : Type u_1} → [inst : MeasurableSpace γ] → Causalean.PO.POPartialLinearSystem P γ → ℝ → P.Ω → ℝ","l":84,"f":"Causalean/PO/ID/Exact/PartialLinear/Setup.lean"},{"n":"Causalean.PO.POPartialLinearSystem.dVar","k":"def","m":"Causalean.PO.ID.Exact.PartialLinear.Setup","a":"PO/ID/Exact/PartialLinear","t":1,"d":"The treatment regarded as a real-valued potential-outcome variable.","s":"{P : Causalean.PO.POSystem} → {γ : Type u_1} → [inst : MeasurableSpace γ] → Causalean.PO.POPartialLinearSystem P γ → Causalean.PO.POVar P ℝ","l":77,"f":"Causalean/PO/ID/Exact/PartialLinear/Setup.lean"},{"n":"Causalean.PO.POPartialLinearSystem.factualD","k":"def","m":"Causalean.PO.ID.Exact.PartialLinear.Setup","a":"PO/ID/Exact/PartialLinear","t":1,"d":"The factual (observed) treatment level of each unit.","s":"{P : Causalean.PO.POSystem} → {γ : Type u_1} → [inst : MeasurableSpace γ] → Causalean.PO.POPartialLinearSystem P γ → P.Ω → ℝ","l":87,"f":"Causalean/PO/ID/Exact/PartialLinear/Setup.lean"},{"n":"Causalean.PO.POPartialLinearSystem.factualX","k":"def","m":"Causalean.PO.ID.Exact.PartialLinear.Setup","a":"PO/ID/Exact/PartialLinear","t":1,"d":"The factual (observed) covariate of each unit.","s":"{P : Causalean.PO.POSystem} → {γ : Type u_1} → [inst : MeasurableSpace γ] → Causalean.PO.POPartialLinearSystem P γ → P.Ω → γ","l":91,"f":"Causalean/PO/ID/Exact/PartialLinear/Setup.lean"},{"n":"Causalean.PO.POPartialLinearSystem.factualXD","k":"def","m":"Causalean.PO.ID.Exact.PartialLinear.Setup","a":"PO/ID/Exact/PartialLinear","t":1,"d":"The joint observed covariate-and-treatment map ω ↦ (X(ω), D(ω)).","s":"{P : Causalean.PO.POSystem} → {γ : Type u_1} → [inst : MeasurableSpace γ] → Causalean.PO.POPartialLinearSystem P γ → P.Ω → γ × ℝ","l":112,"f":"Causalean/PO/ID/Exact/PartialLinear/Setup.lean"},{"n":"Causalean.PO.POPartialLinearSystem.factualY","k":"def","m":"Causalean.PO.ID.Exact.PartialLinear.Setup","a":"PO/ID/Exact/PartialLinear","t":1,"d":"The factual (observed) outcome of each unit.","s":"{P : Causalean.PO.POSystem} → {γ : Type u_1} → [inst : MeasurableSpace γ] → Causalean.PO.POPartialLinearSystem P γ → P.Ω → ℝ","l":89,"f":"Causalean/PO/ID/Exact/PartialLinear/Setup.lean"},{"n":"Causalean.PO.POPartialLinearSystem.measurable_YofD","k":"theorem","m":"Causalean.PO.ID.Exact.PartialLinear.Setup","a":"PO/ID/Exact/PartialLinear","t":2,"d":"The potential outcome under a fixed dose is measurable.","s":"∀ {P : Causalean.PO.POSystem} {γ : Type u_1} [inst : MeasurableSpace γ] (S : Causalean.PO.POPartialLinearSystem P γ) (d : ℝ), Measurable (S.YofD d)","l":94,"f":"Causalean/PO/ID/Exact/PartialLinear/Setup.lean"},{"n":"Causalean.PO.POPartialLinearSystem.measurable_factualD","k":"theorem","m":"Causalean.PO.ID.Exact.PartialLinear.Setup","a":"PO/ID/Exact/PartialLinear","t":2,"d":"The factual treatment level is measurable.","s":"∀ {P : Causalean.PO.POSystem} {γ : Type u_1} [inst : MeasurableSpace γ] (S : Causalean.PO.POPartialLinearSystem P γ), Measurable S.factualD","l":97,"f":"Causalean/PO/ID/Exact/PartialLinear/Setup.lean"},{"n":"Causalean.PO.POPartialLinearSystem.measurable_factualX","k":"theorem","m":"Causalean.PO.ID.Exact.PartialLinear.Setup","a":"PO/ID/Exact/PartialLinear","t":2,"d":"The factual covariate is measurable.","s":"∀ {P : Causalean.PO.POSystem} {γ : Type u_1} [inst : MeasurableSpace γ] (S : Causalean.PO.POPartialLinearSystem P γ), Measurable S.factualX","l":101,"f":"Causalean/PO/ID/Exact/PartialLinear/Setup.lean"},{"n":"Causalean.PO.POPartialLinearSystem.measurable_factualXD","k":"theorem","m":"Causalean.PO.ID.Exact.PartialLinear.Setup","a":"PO/ID/Exact/PartialLinear","t":2,"d":"The joint observed covariate-and-treatment map is measurable.","s":"∀ {P : Causalean.PO.POSystem} {γ : Type u_1} [inst : MeasurableSpace γ] (S : Causalean.PO.POPartialLinearSystem P γ), Measurable S.factualXD","l":116,"f":"Causalean/PO/ID/Exact/PartialLinear/Setup.lean"},{"n":"Causalean.PO.POPartialLinearSystem.measurable_factualY","k":"theorem","m":"Causalean.PO.ID.Exact.PartialLinear.Setup","a":"PO/ID/Exact/PartialLinear","t":2,"d":"The factual outcome is measurable.","s":"∀ {P : Causalean.PO.POSystem} {γ : Type u_1} [inst : MeasurableSpace γ] (S : Causalean.PO.POPartialLinearSystem P γ), Measurable S.factualY","l":99,"f":"Causalean/PO/ID/Exact/PartialLinear/Setup.lean"},{"n":"Causalean.PO.POPartialLinearSystem.sigmaX","k":"def","m":"Causalean.PO.ID.Exact.PartialLinear.Setup","a":"PO/ID/Exact/PartialLinear","t":1,"d":"The σ-algebra generated by the observed covariate.","s":"{P : Causalean.PO.POSystem} → {γ : Type u_1} → [inst : MeasurableSpace γ] → Causalean.PO.POPartialLinearSystem P γ → MeasurableSpace P.Ω","l":104,"f":"Causalean/PO/ID/Exact/PartialLinear/Setup.lean"},{"n":"Causalean.PO.POPartialLinearSystem.sigmaXD","k":"def","m":"Causalean.PO.ID.Exact.PartialLinear.Setup","a":"PO/ID/Exact/PartialLinear","t":1,"d":"The σ-algebra generated by the observed covariate and treatment jointly.","s":"{P : Causalean.PO.POSystem} → {γ : Type u_1} → [inst : MeasurableSpace γ] → Causalean.PO.POPartialLinearSystem P γ → MeasurableSpace P.Ω","l":120,"f":"Causalean/PO/ID/Exact/PartialLinear/Setup.lean"},{"n":"Causalean.PO.POPartialLinearSystem.sigmaXD_le","k":"theorem","m":"Causalean.PO.ID.Exact.PartialLinear.Setup","a":"PO/ID/Exact/PartialLinear","t":2,"d":"The joint covariate-treatment sigma-algebra is a sub-sigma-algebra of the ambient space.","s":"∀ {P : Causalean.PO.POSystem} {γ : Type u_1} [inst : MeasurableSpace γ] (S : Causalean.PO.POPartialLinearSystem P γ), S.sigmaXD ≤ inferInstance","l":124,"f":"Causalean/PO/ID/Exact/PartialLinear/Setup.lean"},{"n":"Causalean.PO.POPartialLinearSystem.sigmaX_le","k":"theorem","m":"Causalean.PO.ID.Exact.PartialLinear.Setup","a":"PO/ID/Exact/PartialLinear","t":2,"d":"The covariate-generated sigma-algebra is a sub-sigma-algebra of the ambient space.","s":"∀ {P : Causalean.PO.POSystem} {γ : Type u_1} [inst : MeasurableSpace γ] (S : Causalean.PO.POPartialLinearSystem P γ), S.sigmaX ≤ inferInstance","l":108,"f":"Causalean/PO/ID/Exact/PartialLinear/Setup.lean"},{"n":"Causalean.PO.POPartialLinearSystem.sigmaX_le_sigmaXD","k":"theorem","m":"Causalean.PO.ID.Exact.PartialLinear.Setup","a":"PO/ID/Exact/PartialLinear","t":2,"d":"The covariate σ-algebra is contained in the joint covariate-treatment σ-algebra.","s":"∀ {P : Causalean.PO.POSystem} {γ : Type u_1} [inst : MeasurableSpace γ] (S : Causalean.PO.POPartialLinearSystem P γ), S.sigmaX ≤ S.sigmaXD","l":129,"f":"Causalean/PO/ID/Exact/PartialLinear/Setup.lean"},{"n":"Causalean.PO.POPartialLinearSystem.xVar","k":"def","m":"Causalean.PO.ID.Exact.PartialLinear.Setup","a":"PO/ID/Exact/PartialLinear","t":1,"d":"The covariate regarded as a potential-outcome variable.","s":"{P : Causalean.PO.POSystem} → {γ : Type u_1} → [inst : MeasurableSpace γ] → Causalean.PO.POPartialLinearSystem P γ → Causalean.PO.POVar P γ","l":81,"f":"Causalean/PO/ID/Exact/PartialLinear/Setup.lean"},{"n":"Causalean.PO.POPartialLinearSystem.yVar","k":"def","m":"Causalean.PO.ID.Exact.PartialLinear.Setup","a":"PO/ID/Exact/PartialLinear","t":1,"d":"The outcome regarded as a real-valued potential-outcome variable.","s":"{P : Causalean.PO.POSystem} → {γ : Type u_1} → [inst : MeasurableSpace γ] → Causalean.PO.POPartialLinearSystem P γ → Causalean.PO.POVar P ℝ","l":79,"f":"Causalean/PO/ID/Exact/PartialLinear/Setup.lean"},{"n":"Causalean.PO.POProximalSystem","k":"structure","m":"Causalean.PO.ID.Exact.Proximal.Setup","a":"PO/ID/Exact/Proximal","t":1,"d":"Proximal ATE system. Bundles six distinguished potential-outcome variables inside a potential-outcome system: an observed covariate, a binary treatment, a treatment-side proxy, an outcome-side proxy, a real-valued outcome, and a latent confounder (def:po-proximal-system).","s":"Causalean.PO.POSystem → (γ_X : Type u_1) → (γ_Z : Type u_2) → (γ_W : Type u_3) → (γ_U : Type u_4) → [MeasurableSpace γ_X] → [MeasurableSpace γ_Z] → [MeasurableSpace γ_W] → [MeasurableSpace γ_U] → Type (max (max (max (max (max u_1 u_2) u_3) u_4) u_5) u_6)","l":29,"f":"Causalean/PO/ID/Exact/Proximal/Setup.lean"},{"n":"Causalean.PO.POProximalSystem.A","k":"def","m":"Causalean.PO.ID.Exact.Proximal.Setup","a":"PO/ID/Exact/Proximal","t":1,"d":"Factual treatment A = Avar(r∅).","s":"{P : Causalean.PO.POSystem} → {γ_X : Type u_1} → {γ_Z : Type u_2} → {γ_W : Type u_3} → {γ_U : Type u_4} → [inst : MeasurableSpace γ_X] → [inst_1 : MeasurableSpace γ_Z] → [inst_2 : MeasurableSpace γ_W] → [inst_3 : MeasurableSpace γ_U] → Causalean.PO.POProximalSystem P γ_X γ_Z γ_W γ_U → P.Ω → Bool","l":63,"f":"Causalean/PO/ID/Exact/Proximal/Setup.lean"},{"n":"Causalean.PO.POProximalSystem.ATE","k":"def","m":"Causalean.PO.ID.Partial.Proxy.Setup","a":"PO/ID/Partial/Proxy","t":1,"d":"ATE in this proximal system: E[Y(1)] - E[Y(0)].","s":"{P : Causalean.PO.POSystem} → {γ_X : Type u_1} → {γ_Z : Type u_2} → {γ_W : Type u_3} → {γ_U : Type u_4} → [inst : MeasurableSpace γ_X] → [inst_1 : MeasurableSpace γ_Z] → [inst_2 : MeasurableSpace γ_W] → [inst_3 : MeasurableSpace γ_U] → Causalean.PO.POProximalSystem P γ_X γ_Z γ_W γ_U → MeasureTheory.Measure P.Ω → ℝ","l":142,"f":"Causalean/PO/ID/Partial/Proxy/Setup.lean"},{"n":"Causalean.PO.POProximalSystem.AUX","k":"def","m":"Causalean.PO.ID.Exact.Proximal.Setup","a":"PO/ID/Exact/Proximal","t":1,"d":"(A, U, X) : P.Ω → Bool × γ_U × γ_X.","s":"{P : Causalean.PO.POSystem} → {γ_X : Type u_1} → {γ_Z : Type u_2} → {γ_W : Type u_3} → {γ_U : Type u_4} → [inst : MeasurableSpace γ_X] → [inst_1 : MeasurableSpace γ_Z] → [inst_2 : MeasurableSpace γ_W] → [inst_3 : MeasurableSpace γ_U] → Causalean.PO.POProximalSystem P γ_X γ_Z γ_W γ_U → P.Ω → Bool × γ_U × γ_X","l":108,"f":"Causalean/PO/ID/Exact/Proximal/Setup.lean"},{"n":"Causalean.PO.POProximalSystem.AX","k":"def","m":"Causalean.PO.ID.Partial.Proxy.Setup","a":"PO/ID/Partial/Proxy","t":1,"d":"(A, X) : P.Ω → Bool × γ_X.","s":"{P : Causalean.PO.POSystem} → {γ_X : Type u_1} → {γ_Z : Type u_2} → {γ_W : Type u_3} → {γ_U : Type u_4} → [inst : MeasurableSpace γ_X] → [inst_1 : MeasurableSpace γ_Z] → [inst_2 : MeasurableSpace γ_W] → [inst_3 : MeasurableSpace γ_U] → Causalean.PO.POProximalSystem P γ_X γ_Z γ_W γ_U → P.Ω → Bool × γ_X","l":46,"f":"Causalean/PO/ID/Partial/Proxy/Setup.lean"},{"n":"Causalean.PO.POProximalSystem.AZUX","k":"def","m":"Causalean.PO.ID.Exact.Proximal.Setup","a":"PO/ID/Exact/Proximal","t":1,"d":"(A, Z, U, X) : P.Ω → Bool × γ_Z × γ_U × γ_X.","s":"{P : Causalean.PO.POSystem} → {γ_X : Type u_1} → {γ_Z : Type u_2} → {γ_W : Type u_3} → {γ_U : Type u_4} → [inst : MeasurableSpace γ_X] → [inst_1 : MeasurableSpace γ_Z] → [inst_2 : MeasurableSpace γ_W] → [inst_3 : MeasurableSpace γ_U] → Causalean.PO.POProximalSystem P γ_X γ_Z γ_W γ_U → P.Ω → Bool × γ_Z × γ_U × γ_X","l":116,"f":"Causalean/PO/ID/Exact/Proximal/Setup.lean"},{"n":"Causalean.PO.POProximalSystem.AZX","k":"def","m":"Causalean.PO.ID.Exact.Proximal.Setup","a":"PO/ID/Exact/Proximal","t":1,"d":"(A, Z, X) : P.Ω → Bool × γ_Z × γ_X.","s":"{P : Causalean.PO.POSystem} → {γ_X : Type u_1} → {γ_Z : Type u_2} → {γ_W : Type u_3} → {γ_U : Type u_4} → [inst : MeasurableSpace γ_X] → [inst_1 : MeasurableSpace γ_Z] → [inst_2 : MeasurableSpace γ_W] → [inst_3 : MeasurableSpace γ_U] → Causalean.PO.POProximalSystem P γ_X γ_Z γ_W γ_U → P.Ω → Bool × γ_Z × γ_X","l":104,"f":"Causalean/PO/ID/Exact/Proximal/Setup.lean"},{"n":"Causalean.PO.POProximalSystem.Assumptions","k":"structure","m":"Causalean.PO.ID.Exact.Proximal.Assumptions","a":"PO/ID/Exact/Proximal","t":1,"d":"Proximal ATE assumption bundle (Miao–Geng–Tchetgen Tchetgen proximal identification, def:po-proximal-assumptions). For a proximal system with covariate, binary treatment, treatment-side and outcome-side proxies, outcome, and latent confounder, this packages consistency (SUTVA), latent exchangeability: each potential outcome is independent of treatment given the latent confounder and covariate, the two proxy restrictions that the outcome-side proxy carries no information about the outcome beyond treatment, latent confounder, and covariate and the treatment-side proxy is independent of treatment and the outcome-side proxy given the latent confounder and covariate, a measurable outcome bridge function h satisfying the bridge equation that the outcome minus h evaluated at treatment, outcome-side proxy, and covariate has zero mean conditional on treatment, treatment-side proxy, and covariate, a positivity condition that every latent-confounder-and-covariate- measurable event of positive probability meets each treatment arm with positive probability, a completeness condition that within each treatment arm, functions of the latent confounder and covariate with zero bridge-conditional mean vanish almost surely, and integrability of the two potential outcomes, the composite h(A,W,X), and the bridge function evaluated at each fixed treatment arm.","s":"{P : Causalean.PO.POSystem} → {γ_X : Type u_1} → {γ_Z : Type u_2} → {γ_W : Type u_3} → {γ_U : Type u_4} → [inst : MeasurableSpace γ_X] → [inst_1 : MeasurableSpace γ_Z] → [inst_2 : MeasurableSpace γ_W] → [inst_3 : MeasurableSpace γ_U] → Causalean.PO.POProximalSystem P γ_X γ_Z γ_W γ_U → (μ : optParam (MeasureTheory.Measure P.Ω) P.μ) → [MeasureTheory.IsFiniteMeasure μ] → [StandardBorelSpace P.Ω] → Type (max u_1 u_3)","l":35,"f":"Causalean/PO/ID/Exact/Proximal/Assumptions.lean"},{"n":"Causalean.PO.POProximalSystem.Assumptions.Eofyofa_eq_Eh","k":"theorem","m":"Causalean.PO.ID.Exact.Proximal.Main","a":"PO/ID/Exact/Proximal","t":1,"d":"Proximal ATE identification (prop:po-proximal-ate step 1). Under the proximal identifying assumption bundle, including completeness within treatment level and integrability of the potential outcomes and bridge-function values, provided the treatment and outcome are distinct nodes, for each treatment level a the counterfactual mean outcome equals the mean of the proximal bridge function evaluated at that level: E[Y(a)] = E[h(a,W,X)].","s":"∀ {P : Causalean.PO.POSystem} {γ_X : Type u_1} {γ_Z : Type u_2} {γ_W : Type u_3} {γ_U : Type u_4} [inst : MeasurableSpace γ_X] [inst_1 : MeasurableSpace γ_Z] [inst_2 : MeasurableSpace γ_W] [inst_3 : MeasurableSpace γ_U] {S : Causalean.PO.POProximalSystem P γ_X γ_Z γ_W γ_U} {μ : MeasureTheory.Measure P.Ω} [inst_4 : MeasureTheory.IsFiniteMeasure μ] [inst_5 : StandardBorelSpace P.Ω] (HA : S.Assumptions μ) (a : Bool), S.Avar.v ≠ S.Yvar.v → ∫ (ω : P.Ω), S.YofA a ω ∂μ = ∫ (ω : P.Ω), HA.h (a, S.W ω, S.X ω) ∂μ","l":52,"f":"Causalean/PO/ID/Exact/Proximal/Main.lean"},{"n":"Causalean.PO.POProximalSystem.Assumptions.ate_proximal","k":"theorem","m":"Causalean.PO.ID.Exact.Proximal.Main","a":"PO/ID/Exact/Proximal","t":1,"d":"Proximal ATE identification. Under the proximal identifying assumption bundle and given the treatment and outcome are distinct nodes, the average treatment effect E[Y(1)] − E[Y(0)] equals the bridge-function contrast E[h(1,W,X)] − E[h(0,W,X)].","s":"∀ {P : Causalean.PO.POSystem} {γ_X : Type u_1} {γ_Z : Type u_2} {γ_W : Type u_3} {γ_U : Type u_4} [inst : MeasurableSpace γ_X] [inst_1 : MeasurableSpace γ_Z] [inst_2 : MeasurableSpace γ_W] [inst_3 : MeasurableSpace γ_U] {S : Causalean.PO.POProximalSystem P γ_X γ_Z γ_W γ_U} {μ : MeasureTheory.Measure P.Ω} [inst_4 : MeasureTheory.IsFiniteMeasure μ] [inst_5 : StandardBorelSpace P.Ω] (HA : S.Assumptions μ), S.Avar.v ≠ S.Yvar.v → ∫ (ω : P.Ω), S.YofA true ω ∂μ - ∫ (ω : P.Ω), S.YofA false ω ∂μ = ∫ (ω : P.Ω), HA.h (true, S.W ω, S.X ω) ∂μ - ∫ (ω : P.Ω), HA.h (false, S.W ω, S.X ω) ∂μ","l":514,"f":"Causalean/PO/ID/Exact/Proximal/Main.lean"},{"n":"Causalean.PO.POProximalSystem.Assumptions.integrable_Y","k":"theorem","m":"Causalean.PO.ID.Exact.Proximal.Assumptions","a":"PO/ID/Exact/Proximal","t":1,"d":"Compatibility projection. Under the proximal identifying assumption bundle, and given the treatment and outcome are distinct nodes, the factual outcome Y is integrable, as a consequence of the consistency assumption together with the integrability of the two potential-outcome cells Y(0) and Y(1).","s":"∀ {P : Causalean.PO.POSystem} {γ_X : Type u_1} {γ_Z : Type u_2} {γ_W : Type u_3} {γ_U : Type u_4} [inst : MeasurableSpace γ_X] [inst_1 : MeasurableSpace γ_Z] [inst_2 : MeasurableSpace γ_W] [inst_3 : MeasurableSpace γ_U] {S : Causalean.PO.POProximalSystem P γ_X γ_Z γ_W γ_U} {μ : MeasureTheory.Measure P.Ω} [inst_4 : MeasureTheory.IsFiniteMeasure μ] [inst_5 : StandardBorelSpace P.Ω] (HA : S.Assumptions μ), S.Avar.v ≠ S.Yvar.v → MeasureTheory.Integrable S.Y μ","l":132,"f":"Causalean/PO/ID/Exact/Proximal/Assumptions.lean"},{"n":"Causalean.PO.POProximalSystem.IsLowerEnvW","k":"def","m":"Causalean.PO.ID.Partial.Proxy.Setup","a":"PO/ID/Partial/Proxy","t":1,"d":"Lower envelope for the W-proxy density ratio. Mirror of IsUpperEnvW with the inequality reversed, including integrability of the weighted left-hand side.","s":"{P : Causalean.PO.POSystem} → {γ_X : Type u_1} → {γ_Z : Type u_2} → {γ_W : Type u_3} → {γ_U : Type u_4} → [inst : MeasurableSpace γ_X] → [inst_1 : MeasurableSpace γ_Z] → [inst_2 : MeasurableSpace γ_W] → [inst_3 : MeasurableSpace γ_U] → Causalean.PO.POProximalSystem P γ_X γ_Z γ_W γ_U → MeasureTheory.Measure P.Ω → Bool → (Bool × γ_X → ℝ) → Prop","l":179,"f":"Causalean/PO/ID/Partial/Proxy/Setup.lean"},{"n":"Causalean.PO.POProximalSystem.IsLowerEnvWZ","k":"def","m":"Causalean.PO.ID.Partial.Proxy.Setup","a":"PO/ID/Partial/Proxy","t":1,"d":"Lower envelope for the same-arm joint-vs-product W-Z density ratio. Mirror of IsUpperEnvWZ with the inequality reversed: the canonical sharp choice is Lenv(a, x) = inf_{w, z} ρ(w, z, x).","s":"{P : Causalean.PO.POSystem} → {γ_X : Type u_1} → {γ_Z : Type u_2} → {γ_W : Type u_3} → {γ_U : Type u_4} → [inst : MeasurableSpace γ_X] → [inst_1 : MeasurableSpace γ_Z] → [inst_2 : MeasurableSpace γ_W] → [inst_3 : MeasurableSpace γ_U] → Causalean.PO.POProximalSystem P γ_X γ_Z γ_W γ_U → MeasureTheory.Measure P.Ω → Bool → (Bool × γ_X → ℝ) → Prop","l":238,"f":"Causalean/PO/ID/Partial/Proxy/Setup.lean"},{"n":"Causalean.PO.POProximalSystem.IsLowerEnvZ","k":"def","m":"Causalean.PO.ID.Partial.Proxy.Setup","a":"PO/ID/Partial/Proxy","t":1,"d":"Lower envelope for E[Y | Z, X, A = a]. Mirror of IsUpperEnvZ.","s":"{P : Causalean.PO.POSystem} → {γ_X : Type u_1} → {γ_Z : Type u_2} → {γ_W : Type u_3} → {γ_U : Type u_4} → [inst : MeasurableSpace γ_X] → [inst_1 : MeasurableSpace γ_Z] → [inst_2 : MeasurableSpace γ_W] → [inst_3 : MeasurableSpace γ_U] → Causalean.PO.POProximalSystem P γ_X γ_Z γ_W γ_U → MeasureTheory.Measure P.Ω → Bool → (Bool × γ_X → ℝ) → Prop","l":203,"f":"Causalean/PO/ID/Partial/Proxy/Setup.lean"},{"n":"Causalean.PO.POProximalSystem.IsUpperEnvW","k":"def","m":"Causalean.PO.ID.Partial.Proxy.Setup","a":"PO/ID/Partial/Proxy","t":1,"d":"Upper envelope for the W-proxy density ratio p(W | A = ¬a, X) / p(W | A = a, X) under treatment level a.","s":"{P : Causalean.PO.POSystem} → {γ_X : Type u_1} → {γ_Z : Type u_2} → {γ_W : Type u_3} → {γ_U : Type u_4} → [inst : MeasurableSpace γ_X] → [inst_1 : MeasurableSpace γ_Z] → [inst_2 : MeasurableSpace γ_W] → [inst_3 : MeasurableSpace γ_U] → Causalean.PO.POProximalSystem P γ_X γ_Z γ_W γ_U → MeasureTheory.Measure P.Ω → Bool → (Bool × γ_X → ℝ) → Prop","l":158,"f":"Causalean/PO/ID/Partial/Proxy/Setup.lean"},{"n":"Causalean.PO.POProximalSystem.IsUpperEnvWZ","k":"def","m":"Causalean.PO.ID.Partial.Proxy.Setup","a":"PO/ID/Partial/Proxy","t":1,"d":"Upper envelope for the same-arm joint-vs-product W-Z density ratio under treatment level a: ρ(w, z, x) := p(w, z | A = a, x) / (p(w | A = a, x) · p(z | A = a, x)).","s":"{P : Causalean.PO.POSystem} → {γ_X : Type u_1} → {γ_Z : Type u_2} → {γ_W : Type u_3} → {γ_U : Type u_4} → [inst : MeasurableSpace γ_X] → [inst_1 : MeasurableSpace γ_Z] → [inst_2 : MeasurableSpace γ_W] → [inst_3 : MeasurableSpace γ_U] → Causalean.PO.POProximalSystem P γ_X γ_Z γ_W γ_U → MeasureTheory.Measure P.Ω → Bool → (Bool × γ_X → ℝ) → Prop","l":211,"f":"Causalean/PO/ID/Partial/Proxy/Setup.lean"},{"n":"Causalean.PO.POProximalSystem.IsUpperEnvZ","k":"def","m":"Causalean.PO.ID.Partial.Proxy.Setup","a":"PO/ID/Partial/Proxy","t":1,"d":"Upper envelope for E[Y | Z, X, A = a] under treatment level a. A σ_X-measurable function Uenv(a, X) such that, μ-a.e. on {A = a}, (μ[Y | σ_AZX]) ≤ Uenv(a, X). The paper's max_z E[Y | z, X, A = a] is the canonical (sharp) instance.","s":"{P : Causalean.PO.POSystem} → {γ_X : Type u_1} → {γ_Z : Type u_2} → {γ_W : Type u_3} → {γ_U : Type u_4} → [inst : MeasurableSpace γ_X] → [inst_1 : MeasurableSpace γ_Z] → [inst_2 : MeasurableSpace γ_W] → [inst_3 : MeasurableSpace γ_U] → Causalean.PO.POProximalSystem P γ_X γ_Z γ_W γ_U → MeasureTheory.Measure P.Ω → Bool → (Bool × γ_X → ℝ) → Prop","l":194,"f":"Causalean/PO/ID/Partial/Proxy/Setup.lean"},{"n":"Causalean.PO.POProximalSystem.TwoProxyAssumptions","k":"structure","m":"Causalean.PO.ID.Partial.Proxy.Assumptions","a":"PO/ID/Partial/Proxy","t":1,"d":"Assumption bundle for the two-proxy partial-identification theorem (Thm 3). Both proxies W and Z are present, but the strong exclusion restrictions W ⟂ A | (U, X) and Y ⟂ Z | (A, U, X) may FAIL — that is the \"invalid proxy\" angle. The only proxy-side conditional independence required is proxy_WZ_indep : W ⟂ Z | (A, X, U) (paper Assumption 6); in particular, no W ⟂ A | (U, X) requirement is bundled. Both bridge functions h and q are kept as fields, with their latent-side defining equations.","s":"{P : Causalean.PO.POSystem} → {γ_X : Type u_1} → {γ_Z : Type u_2} → {γ_W : Type u_3} → {γ_U : Type u_4} → [inst : MeasurableSpace γ_X] → [inst_1 : MeasurableSpace γ_Z] → [inst_2 : MeasurableSpace γ_W] → [inst_3 : MeasurableSpace γ_U] → Causalean.PO.POProximalSystem P γ_X γ_Z γ_W γ_U → (μ : optParam (MeasureTheory.Measure P.Ω) P.μ) → [MeasureTheory.IsFiniteMeasure μ] → [StandardBorelSpace P.Ω] → Type (max (max (max u_1 u_2) u_3) u_7)","l":260,"f":"Causalean/PO/ID/Partial/Proxy/Assumptions.lean"},{"n":"Causalean.PO.POProximalSystem.U","k":"def","m":"Causalean.PO.ID.Exact.Proximal.Setup","a":"PO/ID/Exact/Proximal","t":1,"d":"Latent confounder U = Uvar(r∅).","s":"{P : Causalean.PO.POSystem} → {γ_X : Type u_1} → {γ_Z : Type u_2} → {γ_W : Type u_3} → {γ_U : Type u_4} → [inst : MeasurableSpace γ_X] → [inst_1 : MeasurableSpace γ_Z] → [inst_2 : MeasurableSpace γ_W] → [inst_3 : MeasurableSpace γ_U] → Causalean.PO.POProximalSystem P γ_X γ_Z γ_W γ_U → P.Ω → γ_U","l":75,"f":"Causalean/PO/ID/Exact/Proximal/Setup.lean"},{"n":"Causalean.PO.POProximalSystem.UX","k":"def","m":"Causalean.PO.ID.Exact.Proximal.Setup","a":"PO/ID/Exact/Proximal","t":1,"d":"(U, X) : P.Ω → γ_U × γ_X.","s":"{P : Causalean.PO.POSystem} → {γ_X : Type u_1} → {γ_Z : Type u_2} → {γ_W : Type u_3} → {γ_U : Type u_4} → [inst : MeasurableSpace γ_X] → [inst_1 : MeasurableSpace γ_Z] → [inst_2 : MeasurableSpace γ_W] → [inst_3 : MeasurableSpace γ_U] → Causalean.PO.POProximalSystem P γ_X γ_Z γ_W γ_U → P.Ω → γ_U × γ_X","l":112,"f":"Causalean/PO/ID/Exact/Proximal/Setup.lean"},{"n":"Causalean.PO.POProximalSystem.W","k":"def","m":"Causalean.PO.ID.Exact.Proximal.Setup","a":"PO/ID/Exact/Proximal","t":1,"d":"Factual outcome-side proxy W = Wvar(r∅).","s":"{P : Causalean.PO.POSystem} → {γ_X : Type u_1} → {γ_Z : Type u_2} → {γ_W : Type u_3} → {γ_U : Type u_4} → [inst : MeasurableSpace γ_X] → [inst_1 : MeasurableSpace γ_Z] → [inst_2 : MeasurableSpace γ_W] → [inst_3 : MeasurableSpace γ_U] → Causalean.PO.POProximalSystem P γ_X γ_Z γ_W γ_U → P.Ω → γ_W","l":69,"f":"Causalean/PO/ID/Exact/Proximal/Setup.lean"},{"n":"Causalean.PO.POProximalSystem.WBasedAssumptions","k":"structure","m":"Causalean.PO.ID.Partial.Proxy.Assumptions","a":"PO/ID/Partial/Proxy","t":1,"d":"Assumption bundle for the W-proxy partial-identification theorem (Ghassami-Shpitser-Tchetgen Tchetgen 2024, Thm 1).","s":"{P : Causalean.PO.POSystem} → {γ_X : Type u_1} → {γ_Z : Type u_2} → {γ_W : Type u_3} → {γ_U : Type u_4} → [inst : MeasurableSpace γ_X] → [inst_1 : MeasurableSpace γ_Z] → [inst_2 : MeasurableSpace γ_W] → [inst_3 : MeasurableSpace γ_U] → Causalean.PO.POProximalSystem P γ_X γ_Z γ_W γ_U → (μ : optParam (MeasureTheory.Measure P.Ω) P.μ) → [MeasureTheory.IsFiniteMeasure μ] → [StandardBorelSpace P.Ω] → Type (max u_1 u_3)","l":49,"f":"Causalean/PO/ID/Partial/Proxy/Assumptions.lean"},{"n":"Causalean.PO.POProximalSystem.X","k":"def","m":"Causalean.PO.ID.Exact.Proximal.Setup","a":"PO/ID/Exact/Proximal","t":1,"d":"Factual covariate X = Xvar(r∅).","s":"{P : Causalean.PO.POSystem} → {γ_X : Type u_1} → {γ_Z : Type u_2} → {γ_W : Type u_3} → {γ_U : Type u_4} → [inst : MeasurableSpace γ_X] → [inst_1 : MeasurableSpace γ_Z] → [inst_2 : MeasurableSpace γ_W] → [inst_3 : MeasurableSpace γ_U] → Causalean.PO.POProximalSystem P γ_X γ_Z γ_W γ_U → P.Ω → γ_X","l":60,"f":"Causalean/PO/ID/Exact/Proximal/Setup.lean"},{"n":"Causalean.PO.POProximalSystem.Y","k":"def","m":"Causalean.PO.ID.Exact.Proximal.Setup","a":"PO/ID/Exact/Proximal","t":1,"d":"Factual outcome Y = Yvar(r∅).","s":"{P : Causalean.PO.POSystem} → {γ_X : Type u_1} → {γ_Z : Type u_2} → {γ_W : Type u_3} → {γ_U : Type u_4} → [inst : MeasurableSpace γ_X] → [inst_1 : MeasurableSpace γ_Z] → [inst_2 : MeasurableSpace γ_W] → [inst_3 : MeasurableSpace γ_U] → Causalean.PO.POProximalSystem P γ_X γ_Z γ_W γ_U → P.Ω → ℝ","l":72,"f":"Causalean/PO/ID/Exact/Proximal/Setup.lean"},{"n":"Causalean.PO.POProximalSystem.YofA","k":"def","m":"Causalean.PO.ID.Exact.Proximal.Setup","a":"PO/ID/Exact/Proximal","t":1,"d":"Treatment-specific potential outcome Y(a) := Yvar.cfUnder Avar a.","s":"{P : Causalean.PO.POSystem} → {γ_X : Type u_1} → {γ_Z : Type u_2} → {γ_W : Type u_3} → {γ_U : Type u_4} → [inst : MeasurableSpace γ_X] → [inst_1 : MeasurableSpace γ_Z] → [inst_2 : MeasurableSpace γ_W] → [inst_3 : MeasurableSpace γ_U] → Causalean.PO.POProximalSystem P γ_X γ_Z γ_W γ_U → Bool → P.Ω → ℝ","l":95,"f":"Causalean/PO/ID/Exact/Proximal/Setup.lean"},{"n":"Causalean.PO.POProximalSystem.YofA_essbound_above","k":"theorem","m":"Causalean.PO.ID.Partial.Proxy.Helpers.Common","a":"PO/ID/Partial/Proxy/Helpers","t":2,"d":"Y(a) inherits Y's a.e. upper bound under latent exchangeability and consistency. Requires overlap_strong: every σ_UX-measurable null-on-{A=a} set is globally null, so the single-arm bound lifts globally.","s":"∀ {P : Causalean.PO.POSystem} {γ_X : Type u_1} {γ_Z : Type u_2} {γ_W : Type u_3} {γ_U : Type u_4} [inst : MeasurableSpace γ_X] [inst_1 : MeasurableSpace γ_Z] [inst_2 : MeasurableSpace γ_W] [inst_3 : MeasurableSpace γ_U] {S : Causalean.PO.POProximalSystem P γ_X γ_Z γ_W γ_U} {μ : MeasureTheory.Measure P.Ω} [inst_4 : MeasureTheory.IsFiniteMeasure μ] [inst_5 : StandardBorelSpace P.Ω] {a : Bool}, P.Consistency → ProbabilityTheory.CondIndepFun S.σ_UX (Causalean.PO.POProximalSystem.σ_UX_le S) (S.YofA a) S.A μ → S.Avar.v ≠ S.Yvar.v → (∀ (s : Set P.Ω), MeasurableSet s → μ (s ∩ {ω | S.A ω = a}) = 0 → μ s = 0) → ∀ {M : ℝ}, (∀ᵐ (ω : P.Ω) ∂μ, S.Y ω ≤ M) → ∀ᵐ (ω : P.Ω) ∂μ, S.YofA a ω ≤ M","l":86,"f":"Causalean/PO/ID/Partial/Proxy/Helpers/Common.lean"},{"n":"Causalean.PO.POProximalSystem.YofA_essbound_below","k":"theorem","m":"Causalean.PO.ID.Partial.Proxy.Helpers.Common","a":"PO/ID/Partial/Proxy/Helpers","t":2,"d":"Y(a) inherits Y's a.e. lower bound. Mirror of YofA_essbound_above.","s":"∀ {P : Causalean.PO.POSystem} {γ_X : Type u_1} {γ_Z : Type u_2} {γ_W : Type u_3} {γ_U : Type u_4} [inst : MeasurableSpace γ_X] [inst_1 : MeasurableSpace γ_Z] [inst_2 : MeasurableSpace γ_W] [inst_3 : MeasurableSpace γ_U] {S : Causalean.PO.POProximalSystem P γ_X γ_Z γ_W γ_U} {μ : MeasureTheory.Measure P.Ω} [inst_4 : MeasureTheory.IsFiniteMeasure μ] [inst_5 : StandardBorelSpace P.Ω] {a : Bool}, P.Consistency → ProbabilityTheory.CondIndepFun S.σ_UX (Causalean.PO.POProximalSystem.σ_UX_le S) (S.YofA a) S.A μ → S.Avar.v ≠ S.Yvar.v → (∀ (s : Set P.Ω), MeasurableSet s → μ (s ∩ {ω | S.A ω = a}) = 0 → μ s = 0) → ∀ {M : ℝ}, (∀ᵐ (ω : P.Ω) ∂μ, M ≤ S.Y ω) → ∀ᵐ (ω : P.Ω) ∂μ, M ≤ S.YofA a ω","l":104,"f":"Causalean/PO/ID/Partial/Proxy/Helpers/Common.lean"},{"n":"Causalean.PO.POProximalSystem.Z","k":"def","m":"Causalean.PO.ID.Exact.Proximal.Setup","a":"PO/ID/Exact/Proximal","t":1,"d":"Factual treatment-side proxy Z = Zvar(r∅).","s":"{P : Causalean.PO.POSystem} → {γ_X : Type u_1} → {γ_Z : Type u_2} → {γ_W : Type u_3} → {γ_U : Type u_4} → [inst : MeasurableSpace γ_X] → [inst_1 : MeasurableSpace γ_Z] → [inst_2 : MeasurableSpace γ_W] → [inst_3 : MeasurableSpace γ_U] → Causalean.PO.POProximalSystem P γ_X γ_Z γ_W γ_U → P.Ω → γ_Z","l":66,"f":"Causalean/PO/ID/Exact/Proximal/Setup.lean"},{"n":"Causalean.PO.POProximalSystem.ZBasedAssumptions","k":"structure","m":"Causalean.PO.ID.Partial.Proxy.Assumptions","a":"PO/ID/Partial/Proxy","t":1,"d":"Assumption bundle for the Z-proxy partial-identification theorem (Thm 2). Drops proxy_WAZ, completeness, and W-related fields.","s":"{P : Causalean.PO.POSystem} → {γ_X : Type u_1} → {γ_Z : Type u_2} → {γ_W : Type u_3} → {γ_U : Type u_4} → [inst : MeasurableSpace γ_X] → [inst_1 : MeasurableSpace γ_Z] → [inst_2 : MeasurableSpace γ_W] → [inst_3 : MeasurableSpace γ_U] → Causalean.PO.POProximalSystem P γ_X γ_Z γ_W γ_U → (μ : optParam (MeasureTheory.Measure P.Ω) P.μ) → [MeasureTheory.IsFiniteMeasure μ] → [StandardBorelSpace P.Ω] → Type (max (max u_1 u_2) u_7)","l":132,"f":"Causalean/PO/ID/Partial/Proxy/Assumptions.lean"},{"n":"Causalean.PO.POProximalSystem.condExp_Y_eq_condExp_h_arm_AX","k":"theorem","m":"Causalean.PO.ID.Partial.Proxy.Helpers.Common","a":"PO/ID/Partial/Proxy/Helpers","t":2,"d":"In the W-proxy bundle, the observed conditional mean of the outcome equals the conditional mean of the bridge function on the matching treatment arm.","s":"∀ {P : Causalean.PO.POSystem} {γ_X : Type u_1} {γ_Z : Type u_2} {γ_W : Type u_3} {γ_U : Type u_4} [inst : MeasurableSpace γ_X] [inst_1 : MeasurableSpace γ_Z] [inst_2 : MeasurableSpace γ_W] [inst_3 : MeasurableSpace γ_U] {S : Causalean.PO.POProximalSystem P γ_X γ_Z γ_W γ_U} {μ : MeasureTheory.Measure P.Ω} [inst_4 : MeasureTheory.IsFiniteMeasure μ] [inst_5 : StandardBorelSpace P.Ω] (HA : S.WBasedAssumptions μ) (a : Bool), S.Avar.v ≠ S.Yvar.v → μ[S.Y | S.σ_AX] =ᵐ[μ.restrict {ω | S.A ω = a}] μ[fun ω => HA.h (a, S.W ω, S.X ω) | S.σ_AX]","l":287,"f":"Causalean/PO/ID/Partial/Proxy/Helpers/Common.lean"},{"n":"Causalean.PO.POProximalSystem.condExp_Y_eq_condExp_h_arm_AX_core","k":"theorem","m":"Causalean.PO.ID.Partial.Proxy.Helpers.Common","a":"PO/ID/Partial/Proxy/Helpers","t":2,"d":"Core observed-data collapse identity (no assumption-bundle dependency).","s":"∀ {P : Causalean.PO.POSystem} {γ_X : Type u_1} {γ_Z : Type u_2} {γ_W : Type u_3} {γ_U : Type u_4} [inst : MeasurableSpace γ_X] [inst_1 : MeasurableSpace γ_Z] [inst_2 : MeasurableSpace γ_W] [inst_3 : MeasurableSpace γ_U] {S : Causalean.PO.POProximalSystem P γ_X γ_Z γ_W γ_U} {μ : MeasureTheory.Measure P.Ω} [MeasureTheory.IsFiniteMeasure μ] [StandardBorelSpace P.Ω] {h : Bool × γ_W × γ_X → ℝ} (a : Bool), MeasureTheory.Integrable S.Y μ → MeasureTheory.Integrable (fun ω => h (S.A ω, S.W ω, S.X ω)) μ → MeasureTheory.Integrable (fun ω => h (a, S.W ω, S.X ω)) μ → μ[fun ω => S.Y ω - h (S.A ω, S.W ω, S.X ω) | S.σ_AUX] =ᵐ[μ] 0 → μ[S.Y | S.σ_AX] =ᵐ[μ.restrict {ω | S.A ω = a}] μ[fun ω => h (a, S.W ω, S.X ω) | S.σ_AX]","l":122,"f":"Causalean/PO/ID/Partial/Proxy/Helpers/Common.lean"},{"n":"Causalean.PO.POProximalSystem.condExp_Y_eq_condExp_h_arm_AX_twoProxy","k":"theorem","m":"Causalean.PO.ID.Partial.Proxy.Helpers.Common","a":"PO/ID/Partial/Proxy/Helpers","t":1,"d":"Fix a treatment arm a and assume the two-proxy bridge assumption bundle, with the treatment and outcome variables distinct. Then, restricted to the stratum {A = a}, the conditional expectation of the observed outcome Y given the σ-algebra generated by treatment and covariates equals the conditional expectation of the outcome bridge function evaluated at (a, W, X) given that same σ-algebra.","s":"∀ {P : Causalean.PO.POSystem} {γ_X : Type u_1} {γ_Z : Type u_2} {γ_W : Type u_3} {γ_U : Type u_4} [inst : MeasurableSpace γ_X] [inst_1 : MeasurableSpace γ_Z] [inst_2 : MeasurableSpace γ_W] [inst_3 : MeasurableSpace γ_U] {S : Causalean.PO.POProximalSystem P γ_X γ_Z γ_W γ_U} {μ : MeasureTheory.Measure P.Ω} [inst_4 : MeasureTheory.IsFiniteMeasure μ] [inst_5 : StandardBorelSpace P.Ω] (HA : S.TwoProxyAssumptions μ) (a : Bool), S.Avar.v ≠ S.Yvar.v → μ[S.Y | S.σ_AX] =ᵐ[μ.restrict {ω | S.A ω = a}] μ[fun ω => HA.h (a, S.W ω, S.X ω) | S.σ_AX]","l":302,"f":"Causalean/PO/ID/Partial/Proxy/Helpers/Common.lean"},{"n":"Causalean.PO.POProximalSystem.condExp_drop_Z","k":"theorem","m":"Causalean.PO.ID.Exact.Proximal.Helpers","a":"PO/ID/Exact/Proximal","t":2,"d":"From proxy_YZ : Y ⟂ Z | (A,U,X), conclude E[Y | σ(A,Z,U,X)] =ᵐ[μ] E[Y | σ(A,U,X)].","s":"∀ {P : Causalean.PO.POSystem} {γ_X : Type u_1} {γ_Z : Type u_2} {γ_W : Type u_3} {γ_U : Type u_4} [inst : MeasurableSpace γ_X] [inst_1 : MeasurableSpace γ_Z] [inst_2 : MeasurableSpace γ_W] [inst_3 : MeasurableSpace γ_U] {S : Causalean.PO.POProximalSystem P γ_X γ_Z γ_W γ_U} {μ : MeasureTheory.Measure P.Ω} [inst_4 : MeasureTheory.IsFiniteMeasure μ] [inst_5 : StandardBorelSpace P.Ω] (HA : S.Assumptions μ), S.Avar.v ≠ S.Yvar.v → μ[S.Y | S.σ_AZUX] =ᵐ[μ] μ[S.Y | S.σ_AUX]","l":80,"f":"Causalean/PO/ID/Exact/Proximal/Helpers.lean"},{"n":"Causalean.PO.POProximalSystem.condExp_drop_Z'","k":"theorem","m":"Causalean.PO.ID.Exact.Proximal.Helpers","a":"PO/ID/Exact/Proximal","t":2,"d":"Field-level form: from proxy_YZ : Y ⟂ Z | (A,U,X) and integrability of Y, E[Y | σ(A,Z,U,X)] =ᵐ[μ] E[Y | σ(A,U,X)].","s":"∀ {P : Causalean.PO.POSystem} {γ_X : Type u_1} {γ_Z : Type u_2} {γ_W : Type u_3} {γ_U : Type u_4} [inst : MeasurableSpace γ_X] [inst_1 : MeasurableSpace γ_Z] [inst_2 : MeasurableSpace γ_W] [inst_3 : MeasurableSpace γ_U] {S : Causalean.PO.POProximalSystem P γ_X γ_Z γ_W γ_U} {μ : MeasureTheory.Measure P.Ω} [inst_4 : MeasureTheory.IsFiniteMeasure μ] [inst_5 : StandardBorelSpace P.Ω], ProbabilityTheory.CondIndepFun S.σ_AUX (Causalean.PO.POProximalSystem.σ_AUX_le S) S.Y S.Z μ → MeasureTheory.Integrable S.Y μ → μ[S.Y | S.σ_AZUX] =ᵐ[μ] μ[S.Y | S.σ_AUX]","l":52,"f":"Causalean/PO/ID/Exact/Proximal/Helpers.lean"},{"n":"Causalean.PO.POProximalSystem.condExp_h_drop_A","k":"theorem","m":"Causalean.PO.ID.Exact.Proximal.Helpers","a":"PO/ID/Exact/Proximal","t":2,"d":"From proxy_WAZ : W ⟂ (A,Z) | (U,X), projecting to W ⟂ A | (U,X), E[h(a,W,X) | σ(A,U,X)] =ᵐ[μ] E[h(a,W,X) | σ(U,X)].","s":"∀ {P : Causalean.PO.POSystem} {γ_X : Type u_1} {γ_Z : Type u_2} {γ_W : Type u_3} {γ_U : Type u_4} [inst : MeasurableSpace γ_X] [inst_1 : MeasurableSpace γ_Z] [inst_2 : MeasurableSpace γ_W] [inst_3 : MeasurableSpace γ_U] {S : Causalean.PO.POProximalSystem P γ_X γ_Z γ_W γ_U} {μ : MeasureTheory.Measure P.Ω} [inst_4 : MeasureTheory.IsFiniteMeasure μ] [inst_5 : StandardBorelSpace P.Ω] (HA : S.Assumptions μ) (a : Bool), μ[fun ω => HA.h (a, S.W ω, S.X ω) | S.σ_AUX] =ᵐ[μ] μ[fun ω => HA.h (a, S.W ω, S.X ω) | S.σ_UX]","l":205,"f":"Causalean/PO/ID/Exact/Proximal/Helpers.lean"},{"n":"Causalean.PO.POProximalSystem.condExp_h_drop_A'","k":"theorem","m":"Causalean.PO.ID.Exact.Proximal.Helpers","a":"PO/ID/Exact/Proximal","t":2,"d":"Field-level form: from proxy_WA : W ⟂ A | (U,X), measurability and arm-integrability of the bridge h, conclude E[h(a,W,X) | σ(A,U,X)] =ᵐ[μ] E[h(a,W,X) | σ(U,X)].","s":"∀ {P : Causalean.PO.POSystem} {γ_X : Type u_1} {γ_Z : Type u_2} {γ_W : Type u_3} {γ_U : Type u_4} [inst : MeasurableSpace γ_X] [inst_1 : MeasurableSpace γ_Z] [inst_2 : MeasurableSpace γ_W] [inst_3 : MeasurableSpace γ_U] {S : Causalean.PO.POProximalSystem P γ_X γ_Z γ_W γ_U} {μ : MeasureTheory.Measure P.Ω} [inst_4 : MeasureTheory.IsFiniteMeasure μ] [inst_5 : StandardBorelSpace P.Ω] {h_fun : Bool × γ_W × γ_X → ℝ}, ProbabilityTheory.CondIndepFun S.σ_UX (Causalean.PO.POProximalSystem.σ_UX_le S) S.W S.A μ → Measurable h_fun → ∀ (a : Bool), MeasureTheory.Integrable (fun ω => h_fun (a, S.W ω, S.X ω)) μ → μ[fun ω => h_fun (a, S.W ω, S.X ω) | S.σ_AUX] =ᵐ[μ] μ[fun ω => h_fun (a, S.W ω, S.X ω) | S.σ_UX]","l":171,"f":"Causalean/PO/ID/Exact/Proximal/Helpers.lean"},{"n":"Causalean.PO.POProximalSystem.condExp_h_drop_Z","k":"theorem","m":"Causalean.PO.ID.Exact.Proximal.Helpers","a":"PO/ID/Exact/Proximal","t":2,"d":"From proxy_WAZ : W ⟂ (A,Z) | (U,X), for any a : Bool, E[h(a,W,X) | σ(A,Z,U,X)] =ᵐ[μ] E[h(a,W,X) | σ(A,U,X)].","s":"∀ {P : Causalean.PO.POSystem} {γ_X : Type u_1} {γ_Z : Type u_2} {γ_W : Type u_3} {γ_U : Type u_4} [inst : MeasurableSpace γ_X] [inst_1 : MeasurableSpace γ_Z] [inst_2 : MeasurableSpace γ_W] [inst_3 : MeasurableSpace γ_U] {S : Causalean.PO.POProximalSystem P γ_X γ_Z γ_W γ_U} {μ : MeasureTheory.Measure P.Ω} [inst_4 : MeasureTheory.IsFiniteMeasure μ] [inst_5 : StandardBorelSpace P.Ω] (HA : S.Assumptions μ) (a : Bool), μ[fun ω => HA.h (a, S.W ω, S.X ω) | S.σ_AZUX] =ᵐ[μ] μ[fun ω => HA.h (a, S.W ω, S.X ω) | S.σ_AUX]","l":158,"f":"Causalean/PO/ID/Exact/Proximal/Helpers.lean"},{"n":"Causalean.PO.POProximalSystem.condExp_h_drop_Z'","k":"theorem","m":"Causalean.PO.ID.Exact.Proximal.Helpers","a":"PO/ID/Exact/Proximal","t":2,"d":"Field-level form: from proxy_WAZ : W ⟂ (A,Z) | (U,X), measurability and arm-integrability of the bridge h, conclude E[h(a,W,X) | σ(A,Z,U,X)] =ᵐ[μ] E[h(a,W,X) | σ(A,U,X)].","s":"∀ {P : Causalean.PO.POSystem} {γ_X : Type u_1} {γ_Z : Type u_2} {γ_W : Type u_3} {γ_U : Type u_4} [inst : MeasurableSpace γ_X] [inst_1 : MeasurableSpace γ_Z] [inst_2 : MeasurableSpace γ_W] [inst_3 : MeasurableSpace γ_U] {S : Causalean.PO.POProximalSystem P γ_X γ_Z γ_W γ_U} {μ : MeasureTheory.Measure P.Ω} [inst_4 : MeasureTheory.IsFiniteMeasure μ] [inst_5 : StandardBorelSpace P.Ω] {h_fun : Bool × γ_W × γ_X → ℝ}, ProbabilityTheory.CondIndepFun S.σ_UX (Causalean.PO.POProximalSystem.σ_UX_le S) S.W (fun ω => (S.A ω, S.Z ω)) μ → Measurable h_fun → ∀ (a : Bool), MeasureTheory.Integrable (fun ω => h_fun (a, S.W ω, S.X ω)) μ → μ[fun ω => h_fun (a, S.W ω, S.X ω) | S.σ_AZUX] =ᵐ[μ] μ[fun ω => h_fun (a, S.W ω, S.X ω) | S.σ_AUX]","l":91,"f":"Causalean/PO/ID/Exact/Proximal/Helpers.lean"},{"n":"Causalean.PO.POProximalSystem.condExp_q_eq_stratumOddsRatio_arm_AX","k":"theorem","m":"Causalean.PO.ID.Partial.Proxy.Helpers.CondExpQ","a":"PO/ID/Partial/Proxy/Helpers","t":1,"d":"q-collapse on the on-arm σ(A,X) (TwoProxy). Fix a treatment arm a and assume the two-proxy bridge assumption bundle, with the treatment and outcome variables distinct. Under the probability-ratio convention for the treatment-proxy bridge q encoded by likelihoodRatio_swapA_spec, restricted to the stratum {A = a}, the conditional expectation of q(Z, a, X) given the σ-algebra generated by treatment and covariates collapses to the σ(X)-measurable stratum odds ratio p(A≠a∣X)/p(A=a∣X):","s":"∀ {P : Causalean.PO.POSystem} {γ_X : Type u_1} {γ_Z : Type u_2} {γ_W : Type u_3} {γ_U : Type u_4} [inst : MeasurableSpace γ_X] [inst_1 : MeasurableSpace γ_Z] [inst_2 : MeasurableSpace γ_W] [inst_3 : MeasurableSpace γ_U] {S : Causalean.PO.POProximalSystem P γ_X γ_Z γ_W γ_U} {μ : MeasureTheory.Measure P.Ω} [inst_4 : MeasureTheory.IsFiniteMeasure μ] [inst_5 : StandardBorelSpace P.Ω] (HA : S.TwoProxyAssumptions μ) (a : Bool), S.Avar.v ≠ S.Yvar.v → μ[fun ω => HA.q (S.Z ω, a, S.X ω) | S.σ_AX] =ᵐ[μ.restrict {ω | S.A ω = a}] S.stratumOddsRatio μ a","l":139,"f":"Causalean/PO/ID/Partial/Proxy/Helpers/CondExpQ.lean"},{"n":"Causalean.PO.POProximalSystem.condIntYofA_eq_h_arm","k":"theorem","m":"Causalean.PO.ID.Partial.Proxy.Helpers.BridgeW","a":"PO/ID/Partial/Proxy/Helpers","t":1,"d":"Off-arm bridge-substitution identity (W-proxy). Fix a treatment arm a and assume the W-only proximal bridge assumption bundle — latent exchangeability of the potential outcome Y(a) given treatment conditional on the latent confounder and covariates, independence of the proxy W from treatment given that same conditioning, the outcome-bridge equation, and consistency — together with the treatment and outcome variables being distinct. Then, restricted to the off-arm stratum {A ≠ a}, the average potential outcome Y(a) equals the average of the bridge function h evaluated at (a, W, X): ∫_{A≠a} Y(a) dμ = ∫_{A≠a} h(a, W, X) dμ.","s":"∀ {P : Causalean.PO.POSystem} {γ_X : Type u_1} {γ_Z : Type u_2} {γ_W : Type u_3} {γ_U : Type u_4} [inst : MeasurableSpace γ_X] [inst_1 : MeasurableSpace γ_Z] [inst_2 : MeasurableSpace γ_W] [inst_3 : MeasurableSpace γ_U] {S : Causalean.PO.POProximalSystem P γ_X γ_Z γ_W γ_U} {μ : MeasureTheory.Measure P.Ω} [inst_4 : MeasureTheory.IsFiniteMeasure μ] [inst_5 : StandardBorelSpace P.Ω] (HA : S.WBasedAssumptions μ) (a : Bool), S.Avar.v ≠ S.Yvar.v → ∫ (ω : P.Ω) in {ω | S.A ω ≠ a}, S.YofA a ω ∂μ = ∫ (ω : P.Ω) in {ω | S.A ω ≠ a}, HA.h (a, S.W ω, S.X ω) ∂μ","l":39,"f":"Causalean/PO/ID/Partial/Proxy/Helpers/BridgeW.lean"},{"n":"Causalean.PO.POProximalSystem.condIntYofA_eq_hq_armSwap_twoProxy","k":"theorem","m":"Causalean.PO.ID.Partial.Proxy.Helpers.BridgeWZ","a":"PO/ID/Partial/Proxy/Helpers","t":1,"d":"Two-proxy bridge-substitution identity, same-arm form (Ghassami-Shpitser-Tchetgen Tchetgen 2024, Theorem 3, equation (★), in the codebase's probability-ratio q convention). Fix a treatment arm a and assume the two-proxy bridge assumption bundle — consistency, latent exchangeability of Y(a), the likelihood-ratio arm-swap relation linking the off-arm and on-arm measures, the outcome bridge h, the treatment-proxy bridge q, and conditional independence of the two proxies W and Z given treatment and covariates — together with the treatment and outcome variables being distinct. Then the average potential outcome Y(a) over the off-arm stratum {A ≠ a} equals the on-arm average of the product of the outcome bridge h(a, W, X) and the treatment-proxy bridge q(Z, a, X): ∫_{A≠a} Y(a) dμ = ∫_{A=a} h(a, W, X) · q(Z, a, X) dμ.","s":"∀ {P : Causalean.PO.POSystem} {γ_X : Type u_1} {γ_Z : Type u_2} {γ_W : Type u_3} {γ_U : Type u_4} [inst : MeasurableSpace γ_X] [inst_1 : MeasurableSpace γ_Z] [inst_2 : MeasurableSpace γ_W] [inst_3 : MeasurableSpace γ_U] {S : Causalean.PO.POProximalSystem P γ_X γ_Z γ_W γ_U} {μ : MeasureTheory.Measure P.Ω} [inst_4 : MeasureTheory.IsFiniteMeasure μ] [inst_5 : StandardBorelSpace P.Ω] (HA : S.TwoProxyAssumptions μ) (a : Bool), S.Avar.v ≠ S.Yvar.v → ∫ (ω : P.Ω) in {ω | S.A ω ≠ a}, S.YofA a ω ∂μ = ∫ (ω : P.Ω) in {ω | S.A ω = a}, HA.h (a, S.W ω, S.X ω) * HA.q (S.Z ω, a, S.X ω) ∂μ","l":39,"f":"Causalean/PO/ID/Partial/Proxy/Helpers/BridgeWZ.lean"},{"n":"Causalean.PO.POProximalSystem.condIntYofA_le_envelope_arm","k":"theorem","m":"Causalean.PO.ID.Partial.Proxy.ZBased.ArmChain","a":"PO/ID/Partial/Proxy/ZBased","t":1,"d":"Off-arm bridge-substitution identity, upper envelope side. Fix a treatment arm a and assume the Z-based assumption bundle — consistency, latent exchangeability, the outcome-proxy independence condition, and the treatment-side bridge q — together with the treatment and outcome variables being distinct; let Uenv be an upper envelope function bounding the σ(A,Z,X)-conditional mean of the outcome on the on-arm stratum, assumed integrable, with the envelope weighted by the treatment-proxy bridge and the envelope weighted by the likelihood-ratio arm-swap factor both integrable. Then the average potential outcome Y(a) over the off-arm stratum {A ≠ a} is at most the average of the envelope Uenv(a, X) over that same stratum: ∫_{A≠a} Y(a) dμ ≤ ∫_{A≠a} Uenv(a, X) dμ.","s":"∀ {P : Causalean.PO.POSystem} {γ_X : Type u_1} {γ_Z : Type u_2} {γ_W : Type u_3} {γ_U : Type u_4} [inst : MeasurableSpace γ_X] [inst_1 : MeasurableSpace γ_Z] [inst_2 : MeasurableSpace γ_W] [inst_3 : MeasurableSpace γ_U] {S : Causalean.PO.POProximalSystem P γ_X γ_Z γ_W γ_U} {μ : MeasureTheory.Measure P.Ω} [inst_4 : MeasureTheory.IsFiniteMeasure μ] [inst_5 : StandardBorelSpace P.Ω] (HA : S.ZBasedAssumptions μ) (a : Bool), S.Avar.v ≠ S.Yvar.v → ∀ {Uenv : Bool × γ_X → ℝ}, S.IsUpperEnvZ μ a Uenv → MeasureTheory.Integrable (fun ω => Uenv (a, S.X ω)) μ → MeasureTheory.Integrable (fun ω => Uenv (a, S.X ω) * HA.q (S.Z ω, a, S.X ω)) μ → MeasureTheory.Integrable (fun ω => Uenv (a, S.X ω) * HA.likelihoodRatio_swapA a ω) μ → ∫ (ω : P.Ω) in {ω | S.A ω ≠ a}, S.YofA a ω ∂μ ≤ ∫ (ω : P.Ω) in {ω | S.A ω ≠ a}, Uenv (a, S.X ω) ∂μ","l":185,"f":"Causalean/PO/ID/Partial/Proxy/ZBased/ArmChain.lean"},{"n":"Causalean.PO.POProximalSystem.condMeanYofA","k":"def","m":"Causalean.PO.ID.Partial.Proxy.Setup","a":"PO/ID/Partial/Proxy","t":1,"d":"Conditional potential outcome mean E[Y(a) | A = ¬a]. Defined as (μ {A ≠ a})⁻¹ · ∫_{A ≠ a} Y(a) dμ.","s":"{P : Causalean.PO.POSystem} → {γ_X : Type u_1} → {γ_Z : Type u_2} → {γ_W : Type u_3} → {γ_U : Type u_4} → [inst : MeasurableSpace γ_X] → [inst_1 : MeasurableSpace γ_Z] → [inst_2 : MeasurableSpace γ_W] → [inst_3 : MeasurableSpace γ_U] → Causalean.PO.POProximalSystem P γ_X γ_Z γ_W γ_U → MeasureTheory.Measure P.Ω → Bool → ℝ","l":137,"f":"Causalean/PO/ID/Partial/Proxy/Setup.lean"},{"n":"Causalean.PO.POProximalSystem.condMeanYofA_WZ_bounds","k":"theorem","m":"Causalean.PO.ID.Partial.Proxy.TwoProxy","a":"PO/ID/Partial/Proxy","t":1,"d":"Theorem 3 (Ghassami-Shpitser-Tchetgen Tchetgen 2024, two-proxy partial identification). Fix a treatment arm a and assume the two-proxy assumption bundle — consistency, latent exchangeability, conditional independence of the proxies W and Z given treatment and covariates, the outcome and treatment-proxy bridges, and essential bounds on Y — together with the treatment and outcome variables being distinct; let Lenv, Uenv be lower and upper envelope functions bounding the same-arm joint-versus-product density ratio of the two proxies, with the off-arm stratum of positive mass and the envelope-weighted product of the outcome- and treatment-proxy conditional bridge means integrable. Then the conditional target E[Y(a) ∣ A ≠ a] is sandwiched between the essential Y-bounds and the integrated envelope bound, expressed entirely through observable quantities — the stratum odds ratio, the envelope functions, and the observed conditional mean of Y.","s":"∀ {P : Causalean.PO.POSystem} {γ_X : Type u_1} {γ_Z : Type u_2} {γ_W : Type u_3} {γ_U : Type u_4} [inst : MeasurableSpace γ_X] [inst_1 : MeasurableSpace γ_Z] [inst_2 : MeasurableSpace γ_W] [inst_3 : MeasurableSpace γ_U] {S : Causalean.PO.POProximalSystem P γ_X γ_Z γ_W γ_U} {μ : MeasureTheory.Measure P.Ω} [inst_4 : MeasureTheory.IsFiniteMeasure μ] [inst_5 : StandardBorelSpace P.Ω] (HA : S.TwoProxyAssumptions μ) (a : Bool), S.Avar.v ≠ S.Yvar.v → ∀ (Lenv Uenv : Bool × γ_X → ℝ), S.IsLowerEnvWZ μ a Lenv → S.IsUpperEnvWZ μ a Uenv → 0 < (μ {ω | S.A ω ≠ a}).toReal → MeasureTheory.Integrable (fun ω => Uenv (a, S.X ω) * μ[fun ω' => HA.h (a, S.W ω', S.X ω') | S.σ_AX] ω * μ[fun ω' => HA.q (S.Z ω', a, S.X ω') | S.σ_AX] ω) μ → MeasureTheory.Integrable (fun ω => Lenv (a, S.X ω) * μ[fun ω' => HA.h (a, S.W ω', S.X ω') | S.σ_AX] ω * μ[fun ω' => HA.q (S.Z ω', a, S.X ω') | S.σ_AX] ω) μ → max (Classical.choose HA.Y_bdd_below) ((μ {ω | S.A ω ≠ a}).toReal⁻¹ * ∫ (ω : P.Ω) in {ω | S.A ω = a}, S.stratumOddsRatio μ a ω * Lenv (a, S.X ω) * μ[S.Y | S.σ_AX] ω ∂μ) ≤ S.condMeanYofA μ a ∧ S.condMeanYofA μ a ≤ min (Classical.choose HA.Y_bdd_above) ((μ {ω | S.A ω ≠ a}).toReal⁻¹ * ∫ (ω : P.Ω) in {ω | S.A ω = a}, S.stratumOddsRatio μ a ω * Uenv (a, S.X ω) * μ[S.Y | S.σ_AX] ω ∂μ)","l":74,"f":"Causalean/PO/ID/Partial/Proxy/TwoProxy.lean"},{"n":"Causalean.PO.POProximalSystem.condMeanYofA_WZ_mem_Icc","k":"theorem","m":"Causalean.PO.ID.Partial.Proxy.IntervalForm","a":"PO/ID/Partial/Proxy","t":1,"d":"Set.Icc form of Theorem 3 (condMeanYofA_WZ_bounds). Fix a treatment arm a and assume the two-proxy bridge assumption bundle, with the treatment and outcome variables distinct; let Lenv, Uenv be lower and upper envelope functions bounding the same-arm joint-versus-product W-Z density ratio, with the off-arm stratum of positive mass and the envelope-weighted product of the two conditional bridge means integrable. Then the conditional target E[Y(a) ∣ A ≠ a] lies in the closed interval spanned by the essential Y-bounds and the joint-WZ envelope clamps, the Set.Icc membership restatement of the scalar sandwich bound condMeanYofA_WZ_bounds.","s":"∀ {P : Causalean.PO.POSystem} {γ_X : Type u_1} {γ_Z : Type u_2} {γ_W : Type u_3} {γ_U : Type u_4} [inst : MeasurableSpace γ_X] [inst_1 : MeasurableSpace γ_Z] [inst_2 : MeasurableSpace γ_W] [inst_3 : MeasurableSpace γ_U] {S : Causalean.PO.POProximalSystem P γ_X γ_Z γ_W γ_U} {μ : MeasureTheory.Measure P.Ω} [inst_4 : MeasureTheory.IsFiniteMeasure μ] [inst_5 : StandardBorelSpace P.Ω] (HA : S.TwoProxyAssumptions μ) (a : Bool), S.Avar.v ≠ S.Yvar.v → ∀ (Lenv Uenv : Bool × γ_X → ℝ), S.IsLowerEnvWZ μ a Lenv → S.IsUpperEnvWZ μ a Uenv → 0 < (μ {ω | S.A ω ≠ a}).toReal → MeasureTheory.Integrable (fun ω => Uenv (a, S.X ω) * μ[fun ω' => HA.h (a, S.W ω', S.X ω') | S.σ_AX] ω * μ[fun ω' => HA.q (S.Z ω', a, S.X ω') | S.σ_AX] ω) μ → MeasureTheory.Integrable (fun ω => Lenv (a, S.X ω) * μ[fun ω' => HA.h (a, S.W ω', S.X ω') | S.σ_AX] ω * μ[fun ω' => HA.q (S.Z ω', a, S.X ω') | S.σ_AX] ω) μ → S.condMeanYofA μ a ∈ Set.Icc (max (Classical.choose HA.Y_bdd_below) ((μ {ω | S.A ω ≠ a}).toReal⁻¹ * ∫ (ω : P.Ω) in {ω | S.A ω = a}, S.stratumOddsRatio μ a ω * Lenv (a, S.X ω) * μ[S.Y | S.σ_AX] ω ∂μ)) (min (Classical.choose HA.Y_bdd_above) ((μ {ω | S.A ω ≠ a}).toReal⁻¹ * ∫ (ω : P.Ω) in {ω | S.A ω = a}, S.stratumOddsRatio μ a ω * Uenv (a, S.X ω) * μ[S.Y | S.σ_AX] ω ∂μ))","l":198,"f":"Causalean/PO/ID/Partial/Proxy/IntervalForm.lean"},{"n":"Causalean.PO.POProximalSystem.condMeanYofA_W_bounds","k":"theorem","m":"Causalean.PO.ID.Partial.Proxy.WBased","a":"PO/ID/Partial/Proxy","t":1,"d":"Theorem 1 (Ghassami-Shpitser-Tchetgen Tchetgen 2024, W-only partial identification). Fix a treatment arm a and assume the W-only assumption bundle — consistency, latent exchangeability, independence of the proxy W from treatment given the latent confounder and covariates, the outcome bridge, and essential bounds on Y — together with the treatment and outcome variables being distinct; let Lenv, Uenv be lower and upper envelope functions bounding the W-proxy density ratio, with the off-arm stratum of positive mass and the envelope-weighted bridge moments integrable. Then the conditional target E[Y(a) ∣ A ≠ a] lies between the trivial essential-Y bound and the integrated envelope bound on each side.","s":"∀ {P : Causalean.PO.POSystem} {γ_X : Type u_1} {γ_Z : Type u_2} {γ_W : Type u_3} {γ_U : Type u_4} [inst : MeasurableSpace γ_X] [inst_1 : MeasurableSpace γ_Z] [inst_2 : MeasurableSpace γ_W] [inst_3 : MeasurableSpace γ_U] {S : Causalean.PO.POProximalSystem P γ_X γ_Z γ_W γ_U} {μ : MeasureTheory.Measure P.Ω} [inst_4 : MeasureTheory.IsFiniteMeasure μ] [inst_5 : StandardBorelSpace P.Ω] (HA : S.WBasedAssumptions μ) (a : Bool), S.Avar.v ≠ S.Yvar.v → ∀ (Lenv Uenv : Bool × γ_X → ℝ), S.IsLowerEnvW μ a Lenv → S.IsUpperEnvW μ a Uenv → 0 < (μ {ω | S.A ω ≠ a}).toReal → MeasureTheory.Integrable (fun ω => S.stratumOddsRatio μ a ω * Uenv (a, S.X ω) * HA.h (a, S.W ω, S.X ω)) μ → MeasureTheory.Integrable (fun ω => S.stratumOddsRatio μ a ω * Lenv (a, S.X ω) * HA.h (a, S.W ω, S.X ω)) μ → max (Classical.choose HA.Y_bdd_below) ((μ {ω | S.A ω ≠ a}).toReal⁻¹ * ∫ (ω : P.Ω) in {ω | S.A ω = a}, S.stratumOddsRatio μ a ω * Lenv (a, S.X ω) * μ[S.Y | S.σ_AX] ω ∂μ) ≤ S.condMeanYofA μ a ∧ S.condMeanYofA μ a ≤ min (Classical.choose HA.Y_bdd_above) ((μ {ω | S.A ω ≠ a}).toReal⁻¹ * ∫ (ω : P.Ω) in {ω | S.A ω = a}, S.stratumOddsRatio μ a ω * Uenv (a, S.X ω) * μ[S.Y | S.σ_AX] ω ∂μ)","l":64,"f":"Causalean/PO/ID/Partial/Proxy/WBased.lean"},{"n":"Causalean.PO.POProximalSystem.condMeanYofA_W_mem_Icc","k":"theorem","m":"Causalean.PO.ID.Partial.Proxy.IntervalForm","a":"PO/ID/Partial/Proxy","t":1,"d":"Set.Icc form of Theorem 1 (condMeanYofA_W_bounds). Fix a treatment arm a and assume the W-only proximal bridge assumption bundle, with the treatment and outcome variables distinct; let Lenv, Uenv be lower and upper envelope functions bounding the W-proxy density ratio, with the off-arm stratum of positive mass and the envelope-weighted bridge moments integrable. Then the conditional target E[Y(a) ∣ A ≠ a] lies in the closed interval spanned by the essential Y-bounds and the W-proxy envelope clamps, the Set.Icc membership restatement of the scalar sandwich bound condMeanYofA_W_bounds.","s":"∀ {P : Causalean.PO.POSystem} {γ_X : Type u_1} {γ_Z : Type u_2} {γ_W : Type u_3} {γ_U : Type u_4} [inst : MeasurableSpace γ_X] [inst_1 : MeasurableSpace γ_Z] [inst_2 : MeasurableSpace γ_W] [inst_3 : MeasurableSpace γ_U] {S : Causalean.PO.POProximalSystem P γ_X γ_Z γ_W γ_U} {μ : MeasureTheory.Measure P.Ω} [inst_4 : MeasureTheory.IsFiniteMeasure μ] [inst_5 : StandardBorelSpace P.Ω] (HA : S.WBasedAssumptions μ) (a : Bool), S.Avar.v ≠ S.Yvar.v → ∀ (Lenv Uenv : Bool × γ_X → ℝ), S.IsLowerEnvW μ a Lenv → S.IsUpperEnvW μ a Uenv → 0 < (μ {ω | S.A ω ≠ a}).toReal → MeasureTheory.Integrable (fun ω => S.stratumOddsRatio μ a ω * Uenv (a, S.X ω) * HA.h (a, S.W ω, S.X ω)) μ → MeasureTheory.Integrable (fun ω => S.stratumOddsRatio μ a ω * Lenv (a, S.X ω) * HA.h (a, S.W ω, S.X ω)) μ → S.condMeanYofA μ a ∈ Set.Icc (max (Classical.choose HA.Y_bdd_below) ((μ {ω | S.A ω ≠ a}).toReal⁻¹ * ∫ (ω : P.Ω) in {ω | S.A ω = a}, S.stratumOddsRatio μ a ω * Lenv (a, S.X ω) * μ[S.Y | S.σ_AX] ω ∂μ)) (min (Classical.choose HA.Y_bdd_above) ((μ {ω | S.A ω ≠ a}).toReal⁻¹ * ∫ (ω : P.Ω) in {ω | S.A ω = a}, S.stratumOddsRatio μ a ω * Uenv (a, S.X ω) * μ[S.Y | S.σ_AX] ω ∂μ))","l":54,"f":"Causalean/PO/ID/Partial/Proxy/IntervalForm.lean"},{"n":"Causalean.PO.POProximalSystem.condMeanYofA_Z_bounds","k":"theorem","m":"Causalean.PO.ID.Partial.Proxy.ZBased","a":"PO/ID/Partial/Proxy/ZBased","t":1,"d":"Theorem 2 (Ghassami-Shpitser-Tchetgen Tchetgen, arXiv 2304.04374). Fix a treatment arm a and assume the Z-based assumption bundle — consistency, latent exchangeability, the outcome-proxy independence condition Y ⟂ Z ∣ (A, U, X), and the treatment-side bridge q — together with the treatment and outcome variables being distinct; let Lenv, Uenv be lower and upper envelope functions bounding the σ(A,Z,X)-conditional mean of the outcome on the on-arm stratum, assumed integrable, with the envelope weighted by the treatment-proxy bridge and the envelope weighted by the likelihood-ratio arm-swap factor both integrable, and the off-arm stratum of positive mass. Then the conditional target E[Y(a) ∣ A ≠ a] is sandwiched between the normalised {A ≠ a}-integrals of Lenv(a,X) and Uenv(a,X).","s":"∀ {P : Causalean.PO.POSystem} {γ_X : Type u_1} {γ_Z : Type u_2} {γ_W : Type u_3} {γ_U : Type u_4} [inst : MeasurableSpace γ_X] [inst_1 : MeasurableSpace γ_Z] [inst_2 : MeasurableSpace γ_W] [inst_3 : MeasurableSpace γ_U] {S : Causalean.PO.POProximalSystem P γ_X γ_Z γ_W γ_U} {μ : MeasureTheory.Measure P.Ω} [inst_4 : MeasureTheory.IsFiniteMeasure μ] [inst_5 : StandardBorelSpace P.Ω] (HA : S.ZBasedAssumptions μ) (a : Bool), S.Avar.v ≠ S.Yvar.v → ∀ (Lenv Uenv : Bool × γ_X → ℝ), S.IsLowerEnvZ μ a Lenv → S.IsUpperEnvZ μ a Uenv → MeasureTheory.Integrable (fun ω => Lenv (a, S.X ω)) μ → MeasureTheory.Integrable (fun ω => Uenv (a, S.X ω)) μ → MeasureTheory.Integrable (fun ω => Lenv (a, S.X ω) * HA.q (S.Z ω, a, S.X ω)) μ → MeasureTheory.Integrable (fun ω => Uenv (a, S.X ω) * HA.q (S.Z ω, a, S.X ω)) μ → MeasureTheory.Integrable (fun ω => Lenv (a, S.X ω) * HA.likelihoodRatio_swapA a ω) μ → MeasureTheory.Integrable (fun ω => Uenv (a, S.X ω) * HA.likelihoodRatio_swapA a ω) μ → 0 < (μ {ω | S.A ω ≠ a}).toReal → (μ {ω | S.A ω ≠ a}).toReal⁻¹ * ∫ (ω : P.Ω) in {ω | S.A ω ≠ a}, Lenv (a, S.X ω) ∂μ ≤ S.condMeanYofA μ a ∧ S.condMeanYofA μ a ≤ (μ {ω | S.A ω ≠ a}).toReal⁻¹ * ∫ (ω : P.Ω) in {ω | S.A ω ≠ a}, Uenv (a, S.X ω) ∂μ","l":69,"f":"Causalean/PO/ID/Partial/Proxy/ZBased.lean"},{"n":"Causalean.PO.POProximalSystem.condMeanYofA_Z_mem_Icc","k":"theorem","m":"Causalean.PO.ID.Partial.Proxy.IntervalForm","a":"PO/ID/Partial/Proxy","t":1,"d":"Set.Icc form of Theorem 2 (condMeanYofA_Z_bounds). Fix a treatment arm a and assume the Z-only proximal bridge assumption bundle, with the treatment and outcome variables distinct; let Lenv, Uenv be lower and upper envelope functions bounding the σ(A,Z,X)-conditional mean of the outcome on the on-arm stratum, assumed integrable, with the envelope weighted by the treatment-proxy bridge and the envelope weighted by the likelihood-ratio arm-swap factor both integrable, and the off-arm stratum of positive mass. Then the conditional target E[Y(a) ∣ A ≠ a] lies in the closed interval spanned by the normalised on-arm integrals of Lenv and Uenv, the Set.Icc membership restatement of the scalar sandwich bound condMeanYofA_Z_bounds.","s":"∀ {P : Causalean.PO.POSystem} {γ_X : Type u_1} {γ_Z : Type u_2} {γ_W : Type u_3} {γ_U : Type u_4} [inst : MeasurableSpace γ_X] [inst_1 : MeasurableSpace γ_Z] [inst_2 : MeasurableSpace γ_W] [inst_3 : MeasurableSpace γ_U] {S : Causalean.PO.POProximalSystem P γ_X γ_Z γ_W γ_U} {μ : MeasureTheory.Measure P.Ω} [inst_4 : MeasureTheory.IsFiniteMeasure μ] [inst_5 : StandardBorelSpace P.Ω] (HA : S.ZBasedAssumptions μ) (a : Bool), S.Avar.v ≠ S.Yvar.v → ∀ (Lenv Uenv : Bool × γ_X → ℝ), S.IsLowerEnvZ μ a Lenv → S.IsUpperEnvZ μ a Uenv → MeasureTheory.Integrable (fun ω => Lenv (a, S.X ω)) μ → MeasureTheory.Integrable (fun ω => Uenv (a, S.X ω)) μ → MeasureTheory.Integrable (fun ω => Lenv (a, S.X ω) * HA.q (S.Z ω, a, S.X ω)) μ → MeasureTheory.Integrable (fun ω => Uenv (a, S.X ω) * HA.q (S.Z ω, a, S.X ω)) μ → MeasureTheory.Integrable (fun ω => Lenv (a, S.X ω) * HA.likelihoodRatio_swapA a ω) μ → MeasureTheory.Integrable (fun ω => Uenv (a, S.X ω) * HA.likelihoodRatio_swapA a ω) μ → 0 < (μ {ω | S.A ω ≠ a}).toReal → S.condMeanYofA μ a ∈ Set.Icc ((μ {ω | S.A ω ≠ a}).toReal⁻¹ * ∫ (ω : P.Ω) in {ω | S.A ω ≠ a}, Lenv (a, S.X ω) ∂μ) ((μ {ω | S.A ω ≠ a}).toReal⁻¹ * ∫ (ω : P.Ω) in {ω | S.A ω ≠ a}, Uenv (a, S.X ω) ∂μ)","l":128,"f":"Causalean/PO/ID/Partial/Proxy/IntervalForm.lean"},{"n":"Causalean.PO.POProximalSystem.consistency_event","k":"theorem","m":"Causalean.PO.ID.Exact.Proximal.Helpers","a":"PO/ID/Exact/Proximal","t":1,"d":"Factual-outcome bridge on the treatment-arm event. From the consistency assumption in the proximal identifying assumption bundle, and given the treatment and outcome are distinct nodes, on the event where the treatment equals arm a, the conditional expectation of the factual outcome given σ(A,U,X) agrees almost surely with the conditional expectation of the potential outcome Y(a) given the same σ-algebra, since Y = Y(a) pointwise there.","s":"∀ {P : Causalean.PO.POSystem} {γ_X : Type u_1} {γ_Z : Type u_2} {γ_W : Type u_3} {γ_U : Type u_4} [inst : MeasurableSpace γ_X] [inst_1 : MeasurableSpace γ_Z] [inst_2 : MeasurableSpace γ_W] [inst_3 : MeasurableSpace γ_U] {S : Causalean.PO.POProximalSystem P γ_X γ_Z γ_W γ_U} {μ : MeasureTheory.Measure P.Ω} [inst_4 : MeasureTheory.IsFiniteMeasure μ] [inst_5 : StandardBorelSpace P.Ω] (HA : S.Assumptions μ) (a : Bool), S.Avar.v ≠ S.Yvar.v → μ[S.Y | S.σ_AUX] =ᵐ[μ.restrict {ω | S.A ω = a}] μ[S.YofA a | S.σ_AUX]","l":298,"f":"Causalean/PO/ID/Exact/Proximal/Helpers.lean"},{"n":"Causalean.PO.POProximalSystem.consistency_event'","k":"theorem","m":"Causalean.PO.ID.Exact.Proximal.Helpers","a":"PO/ID/Exact/Proximal","t":2,"d":"Field-level form: from consistency, integrability of Y and Y(a), on {A=a} we have E[Y | σ(A,U,X)] =ᵐ[μ.restrict {A=a}] E[Y(a) | σ(A,U,X)].","s":"∀ {P : Causalean.PO.POSystem} {γ_X : Type u_1} {γ_Z : Type u_2} {γ_W : Type u_3} {γ_U : Type u_4} [inst : MeasurableSpace γ_X] [inst_1 : MeasurableSpace γ_Z] [inst_2 : MeasurableSpace γ_W] [inst_3 : MeasurableSpace γ_U] {S : Causalean.PO.POProximalSystem P γ_X γ_Z γ_W γ_U} {μ : MeasureTheory.Measure P.Ω} [MeasureTheory.IsFiniteMeasure μ] [StandardBorelSpace P.Ω], P.Consistency → ∀ (a : Bool), S.Avar.v ≠ S.Yvar.v → MeasureTheory.Integrable S.Y μ → MeasureTheory.Integrable (S.YofA a) μ → μ[S.Y | S.σ_AUX] =ᵐ[μ.restrict {ω | S.A ω = a}] μ[S.YofA a | S.σ_AUX]","l":252,"f":"Causalean/PO/ID/Exact/Proximal/Helpers.lean"},{"n":"Causalean.PO.POProximalSystem.envelope_le_condIntYofA_arm","k":"theorem","m":"Causalean.PO.ID.Partial.Proxy.ZBased.ArmChain","a":"PO/ID/Partial/Proxy/ZBased","t":2,"d":"Off-arm bridge-substitution identity, lower envelope side. Mirror of condIntYofA_le_envelope_arm.","s":"∀ {P : Causalean.PO.POSystem} {γ_X : Type u_1} {γ_Z : Type u_2} {γ_W : Type u_3} {γ_U : Type u_4} [inst : MeasurableSpace γ_X] [inst_1 : MeasurableSpace γ_Z] [inst_2 : MeasurableSpace γ_W] [inst_3 : MeasurableSpace γ_U] {S : Causalean.PO.POProximalSystem P γ_X γ_Z γ_W γ_U} {μ : MeasureTheory.Measure P.Ω} [inst_4 : MeasureTheory.IsFiniteMeasure μ] [inst_5 : StandardBorelSpace P.Ω] (HA : S.ZBasedAssumptions μ) (a : Bool), S.Avar.v ≠ S.Yvar.v → ∀ {Lenv : Bool × γ_X → ℝ}, S.IsLowerEnvZ μ a Lenv → MeasureTheory.Integrable (fun ω => Lenv (a, S.X ω)) μ → MeasureTheory.Integrable (fun ω => Lenv (a, S.X ω) * HA.q (S.Z ω, a, S.X ω)) μ → MeasureTheory.Integrable (fun ω => Lenv (a, S.X ω) * HA.likelihoodRatio_swapA a ω) μ → ∫ (ω : P.Ω) in {ω | S.A ω ≠ a}, Lenv (a, S.X ω) ∂μ ≤ ∫ (ω : P.Ω) in {ω | S.A ω ≠ a}, S.YofA a ω ∂μ","l":501,"f":"Causalean/PO/ID/Partial/Proxy/ZBased/ArmChain.lean"},{"n":"Causalean.PO.POProximalSystem.eq_zero_globally_of_eq_zero_on_arm","k":"theorem","m":"Causalean.PO.ID.Exact.Proximal.Helpers","a":"PO/ID/Exact/Proximal","t":2,"d":"From positivity_arm a and a σ_UX-measurable function that is μ-a.e. zero on the arm {A=a}, conclude it is μ-a.e. zero globally.","s":"∀ {P : Causalean.PO.POSystem} {γ_X : Type u_1} {γ_Z : Type u_2} {γ_W : Type u_3} {γ_U : Type u_4} [inst : MeasurableSpace γ_X] [inst_1 : MeasurableSpace γ_Z] [inst_2 : MeasurableSpace γ_W] [inst_3 : MeasurableSpace γ_U] {S : Causalean.PO.POProximalSystem P γ_X γ_Z γ_W γ_U} {μ : MeasureTheory.Measure P.Ω} [inst_4 : MeasureTheory.IsFiniteMeasure μ] [inst_5 : StandardBorelSpace P.Ω] (HA : S.Assumptions μ) (a : Bool) {f : P.Ω → ℝ}, Measurable f → f =ᵐ[μ.restrict {ω | S.A ω = a}] 0 → f =ᵐ[μ] 0","l":315,"f":"Causalean/PO/ID/Exact/Proximal/Helpers.lean"},{"n":"Causalean.PO.POProximalSystem.latent_exch_to_condExp","k":"theorem","m":"Causalean.PO.ID.Exact.Proximal.Helpers","a":"PO/ID/Exact/Proximal","t":2,"d":"From latent_exch a : Y(a) ⟂ A | (U,X): E[Y(a) | σ(A,U,X)] =ᵐ[μ] E[Y(a) | σ(U,X)].","s":"∀ {P : Causalean.PO.POSystem} {γ_X : Type u_1} {γ_Z : Type u_2} {γ_W : Type u_3} {γ_U : Type u_4} [inst : MeasurableSpace γ_X] [inst_1 : MeasurableSpace γ_Z] [inst_2 : MeasurableSpace γ_W] [inst_3 : MeasurableSpace γ_U] {S : Causalean.PO.POProximalSystem P γ_X γ_Z γ_W γ_U} {μ : MeasureTheory.Measure P.Ω} [inst_4 : MeasureTheory.IsFiniteMeasure μ] [inst_5 : StandardBorelSpace P.Ω] (HA : S.Assumptions μ) (a : Bool), μ[S.YofA a | S.σ_AUX] =ᵐ[μ] μ[S.YofA a | S.σ_UX]","l":240,"f":"Causalean/PO/ID/Exact/Proximal/Helpers.lean"},{"n":"Causalean.PO.POProximalSystem.latent_exch_to_condExp'","k":"theorem","m":"Causalean.PO.ID.Exact.Proximal.Helpers","a":"PO/ID/Exact/Proximal","t":2,"d":"Field-level form: from latent_exch a : Y(a) ⟂ A | (U,X) and integrability of Y(a), conclude E[Y(a) | σ(A,U,X)] =ᵐ[μ] E[Y(a) | σ(U,X)].","s":"∀ {P : Causalean.PO.POSystem} {γ_X : Type u_1} {γ_Z : Type u_2} {γ_W : Type u_3} {γ_U : Type u_4} [inst : MeasurableSpace γ_X] [inst_1 : MeasurableSpace γ_Z] [inst_2 : MeasurableSpace γ_W] [inst_3 : MeasurableSpace γ_U] {S : Causalean.PO.POProximalSystem P γ_X γ_Z γ_W γ_U} {μ : MeasureTheory.Measure P.Ω} [inst_4 : MeasureTheory.IsFiniteMeasure μ] [inst_5 : StandardBorelSpace P.Ω] (a : Bool), ProbabilityTheory.CondIndepFun S.σ_UX (Causalean.PO.POProximalSystem.σ_UX_le S) (S.YofA a) S.A μ → MeasureTheory.Integrable (S.YofA a) μ → μ[S.YofA a | S.σ_AUX] =ᵐ[μ] μ[S.YofA a | S.σ_UX]","l":222,"f":"Causalean/PO/ID/Exact/Proximal/Helpers.lean"},{"n":"Causalean.PO.POProximalSystem.meanYofA","k":"def","m":"Causalean.PO.ID.Partial.Proxy.Setup","a":"PO/ID/Partial/Proxy","t":1,"d":"Marginal potential outcome mean E[Y(a)] = ∫ Y(a) dμ.","s":"{P : Causalean.PO.POSystem} → {γ_X : Type u_1} → {γ_Z : Type u_2} → {γ_W : Type u_3} → {γ_U : Type u_4} → [inst : MeasurableSpace γ_X] → [inst_1 : MeasurableSpace γ_Z] → [inst_2 : MeasurableSpace γ_W] → [inst_3 : MeasurableSpace γ_U] → Causalean.PO.POProximalSystem P γ_X γ_Z γ_W γ_U → MeasureTheory.Measure P.Ω → Bool → ℝ","l":133,"f":"Causalean/PO/ID/Partial/Proxy/Setup.lean"},{"n":"Causalean.PO.POProximalSystem.meanYofA_W_bounds","k":"theorem","m":"Causalean.PO.ID.Partial.Proxy.WBased","a":"PO/ID/Partial/Proxy","t":1,"d":"Corollary 1 (Ghassami-Shpitser-Tchetgen Tchetgen 2024, W-only marginal version). Fix a treatment arm a and assume the W-only assumption bundle, with the treatment and outcome variables distinct; let Lenv, Uenv be lower and upper envelope functions bounding the W-proxy density ratio, and assume the envelope-weighted bridge moments and the envelope-weighted observed conditional means are integrable. Then the marginal target E[Y(a)] lies between a trivial clamp and an envelope clamp on each side, plus the point-identified on-arm contribution.","s":"∀ {P : Causalean.PO.POSystem} {γ_X : Type u_1} {γ_Z : Type u_2} {γ_W : Type u_3} {γ_U : Type u_4} [inst : MeasurableSpace γ_X] [inst_1 : MeasurableSpace γ_Z] [inst_2 : MeasurableSpace γ_W] [inst_3 : MeasurableSpace γ_U] {S : Causalean.PO.POProximalSystem P γ_X γ_Z γ_W γ_U} {μ : MeasureTheory.Measure P.Ω} [inst_4 : MeasureTheory.IsFiniteMeasure μ] [inst_5 : StandardBorelSpace P.Ω] (HA : S.WBasedAssumptions μ) (a : Bool), S.Avar.v ≠ S.Yvar.v → ∀ (Lenv Uenv : Bool × γ_X → ℝ), S.IsLowerEnvW μ a Lenv → S.IsUpperEnvW μ a Uenv → MeasureTheory.Integrable (fun ω => S.stratumOddsRatio μ a ω * Uenv (a, S.X ω) * HA.h (a, S.W ω, S.X ω)) μ → MeasureTheory.Integrable (fun ω => S.stratumOddsRatio μ a ω * Lenv (a, S.X ω) * HA.h (a, S.W ω, S.X ω)) μ → MeasureTheory.Integrable (fun ω => S.stratumOddsRatio μ a ω * Uenv (a, S.X ω) * μ[S.Y | S.σ_AX] ω) μ → MeasureTheory.Integrable (fun ω => S.stratumOddsRatio μ a ω * Lenv (a, S.X ω) * μ[S.Y | S.σ_AX] ω) μ → max (Classical.choose HA.Y_bdd_below * (μ {ω | S.A ω ≠ a}).toReal) (∫ (ω : P.Ω) in {ω | S.A ω = a}, S.stratumOddsRatio μ a ω * Lenv (a, S.X ω) * μ[S.Y | S.σ_AX] ω ∂μ) + ∫ (ω : P.Ω) in {ω | S.A ω = a}, S.Y ω ∂μ ≤ S.meanYofA μ a ∧ S.meanYofA μ a ≤ min (Classical.choose HA.Y_bdd_above * (μ {ω | S.A ω ≠ a}).toReal) (∫ (ω : P.Ω) in {ω | S.A ω = a}, S.stratumOddsRatio μ a ω * Uenv (a, S.X ω) * μ[S.Y | S.σ_AX] ω ∂μ) + ∫ (ω : P.Ω) in {ω | S.A ω = a}, S.Y ω ∂μ","l":324,"f":"Causalean/PO/ID/Partial/Proxy/WBased.lean"},{"n":"Causalean.PO.POProximalSystem.meanYofA_W_mem_Icc","k":"theorem","m":"Causalean.PO.ID.Partial.Proxy.IntervalForm","a":"PO/ID/Partial/Proxy","t":1,"d":"Set.Icc form of Corollary 1 (meanYofA_W_bounds). Fix a treatment arm a and assume the W-only proximal bridge assumption bundle, with the treatment and outcome variables distinct; let Lenv, Uenv be lower and upper envelope functions bounding the W-proxy density ratio, and assume the envelope-weighted bridge moments and the envelope-weighted observed conditional means are integrable. Then the marginal target E[Y(a)] lies in the closed interval spanned by the W-proxy marginal lower and upper clamps, the Set.Icc membership restatement of the scalar sandwich bound meanYofA_W_bounds.","s":"∀ {P : Causalean.PO.POSystem} {γ_X : Type u_1} {γ_Z : Type u_2} {γ_W : Type u_3} {γ_U : Type u_4} [inst : MeasurableSpace γ_X] [inst_1 : MeasurableSpace γ_Z] [inst_2 : MeasurableSpace γ_W] [inst_3 : MeasurableSpace γ_U] {S : Causalean.PO.POProximalSystem P γ_X γ_Z γ_W γ_U} {μ : MeasureTheory.Measure P.Ω} [inst_4 : MeasureTheory.IsFiniteMeasure μ] [inst_5 : StandardBorelSpace P.Ω] (HA : S.WBasedAssumptions μ) (a : Bool), S.Avar.v ≠ S.Yvar.v → ∀ (Lenv Uenv : Bool × γ_X → ℝ), S.IsLowerEnvW μ a Lenv → S.IsUpperEnvW μ a Uenv → MeasureTheory.Integrable (fun ω => S.stratumOddsRatio μ a ω * Uenv (a, S.X ω) * HA.h (a, S.W ω, S.X ω)) μ → MeasureTheory.Integrable (fun ω => S.stratumOddsRatio μ a ω * Lenv (a, S.X ω) * HA.h (a, S.W ω, S.X ω)) μ → MeasureTheory.Integrable (fun ω => S.stratumOddsRatio μ a ω * Uenv (a, S.X ω) * μ[S.Y | S.σ_AX] ω) μ → MeasureTheory.Integrable (fun ω => S.stratumOddsRatio μ a ω * Lenv (a, S.X ω) * μ[S.Y | S.σ_AX] ω) μ → S.meanYofA μ a ∈ Set.Icc (max (Classical.choose HA.Y_bdd_below * (μ {ω | S.A ω ≠ a}).toReal) (∫ (ω : P.Ω) in {ω | S.A ω = a}, S.stratumOddsRatio μ a ω * Lenv (a, S.X ω) * μ[S.Y | S.σ_AX] ω ∂μ) + ∫ (ω : P.Ω) in {ω | S.A ω = a}, S.Y ω ∂μ) (min (Classical.choose HA.Y_bdd_above * (μ {ω | S.A ω ≠ a}).toReal) (∫ (ω : P.Ω) in {ω | S.A ω = a}, S.stratumOddsRatio μ a ω * Uenv (a, S.X ω) * μ[S.Y | S.σ_AX] ω ∂μ) + ∫ (ω : P.Ω) in {ω | S.A ω = a}, S.Y ω ∂μ)","l":87,"f":"Causalean/PO/ID/Partial/Proxy/IntervalForm.lean"},{"n":"Causalean.PO.POProximalSystem.meanYofA_Z_bounds","k":"theorem","m":"Causalean.PO.ID.Partial.Proxy.ZBased","a":"PO/ID/Partial/Proxy/ZBased","t":1,"d":"Corollary 2 (Ghassami-Shpitser-Tchetgen Tchetgen, arXiv 2304.04374). Fix a treatment arm a and assume the Z-based assumption bundle, with the treatment and outcome variables distinct; let Lenv, Uenv be lower and upper envelope functions bounding the σ(A,Z,X)-conditional mean of the outcome on the on-arm stratum, assumed integrable, with the envelope weighted by the treatment-proxy bridge and the envelope weighted by the likelihood-ratio arm-swap factor both integrable. Then the marginal target E[Y(a)] is sandwiched between the {A ≠ a}-integral of Lenv(a,X) and of Uenv(a,X), each plus the point-identified {A = a}-integral of Y.","s":"∀ {P : Causalean.PO.POSystem} {γ_X : Type u_1} {γ_Z : Type u_2} {γ_W : Type u_3} {γ_U : Type u_4} [inst : MeasurableSpace γ_X] [inst_1 : MeasurableSpace γ_Z] [inst_2 : MeasurableSpace γ_W] [inst_3 : MeasurableSpace γ_U] {S : Causalean.PO.POProximalSystem P γ_X γ_Z γ_W γ_U} {μ : MeasureTheory.Measure P.Ω} [inst_4 : MeasureTheory.IsFiniteMeasure μ] [inst_5 : StandardBorelSpace P.Ω] (HA : S.ZBasedAssumptions μ) (a : Bool), S.Avar.v ≠ S.Yvar.v → ∀ (Lenv Uenv : Bool × γ_X → ℝ), S.IsLowerEnvZ μ a Lenv → S.IsUpperEnvZ μ a Uenv → MeasureTheory.Integrable (fun ω => Lenv (a, S.X ω)) μ → MeasureTheory.Integrable (fun ω => Uenv (a, S.X ω)) μ → MeasureTheory.Integrable (fun ω => Lenv (a, S.X ω) * HA.q (S.Z ω, a, S.X ω)) μ → MeasureTheory.Integrable (fun ω => Uenv (a, S.X ω) * HA.q (S.Z ω, a, S.X ω)) μ → MeasureTheory.Integrable (fun ω => Lenv (a, S.X ω) * HA.likelihoodRatio_swapA a ω) μ → MeasureTheory.Integrable (fun ω => Uenv (a, S.X ω) * HA.likelihoodRatio_swapA a ω) μ → ∫ (ω : P.Ω) in {ω | S.A ω ≠ a}, Lenv (a, S.X ω) ∂μ + ∫ (ω : P.Ω) in {ω | S.A ω = a}, S.Y ω ∂μ ≤ S.meanYofA μ a ∧ S.meanYofA μ a ≤ ∫ (ω : P.Ω) in {ω | S.A ω ≠ a}, Uenv (a, S.X ω) ∂μ + ∫ (ω : P.Ω) in {ω | S.A ω = a}, S.Y ω ∂μ","l":118,"f":"Causalean/PO/ID/Partial/Proxy/ZBased.lean"},{"n":"Causalean.PO.POProximalSystem.meanYofA_Z_mem_Icc","k":"theorem","m":"Causalean.PO.ID.Partial.Proxy.IntervalForm","a":"PO/ID/Partial/Proxy","t":1,"d":"Set.Icc form of Corollary 2 (meanYofA_Z_bounds). Fix a treatment arm a and assume the Z-only proximal bridge assumption bundle, with the treatment and outcome variables distinct; let Lenv, Uenv be lower and upper envelope functions bounding the σ(A,Z,X)-conditional mean of the outcome on the on-arm stratum, assumed integrable, with the envelope weighted by the treatment-proxy bridge and the envelope weighted by the likelihood-ratio arm-swap factor both integrable. Then the marginal target E[Y(a)] lies in the closed interval spanned by the Z-proxy marginal envelope integrals, the Set.Icc membership restatement of the scalar sandwich bound meanYofA_Z_bounds.","s":"∀ {P : Causalean.PO.POSystem} {γ_X : Type u_1} {γ_Z : Type u_2} {γ_W : Type u_3} {γ_U : Type u_4} [inst : MeasurableSpace γ_X] [inst_1 : MeasurableSpace γ_Z] [inst_2 : MeasurableSpace γ_W] [inst_3 : MeasurableSpace γ_U] {S : Causalean.PO.POProximalSystem P γ_X γ_Z γ_W γ_U} {μ : MeasureTheory.Measure P.Ω} [inst_4 : MeasureTheory.IsFiniteMeasure μ] [inst_5 : StandardBorelSpace P.Ω] (HA : S.ZBasedAssumptions μ) (a : Bool), S.Avar.v ≠ S.Yvar.v → ∀ (Lenv Uenv : Bool × γ_X → ℝ), S.IsLowerEnvZ μ a Lenv → S.IsUpperEnvZ μ a Uenv → MeasureTheory.Integrable (fun ω => Lenv (a, S.X ω)) μ → MeasureTheory.Integrable (fun ω => Uenv (a, S.X ω)) μ → MeasureTheory.Integrable (fun ω => Lenv (a, S.X ω) * HA.q (S.Z ω, a, S.X ω)) μ → MeasureTheory.Integrable (fun ω => Uenv (a, S.X ω) * HA.q (S.Z ω, a, S.X ω)) μ → MeasureTheory.Integrable (fun ω => Lenv (a, S.X ω) * HA.likelihoodRatio_swapA a ω) μ → MeasureTheory.Integrable (fun ω => Uenv (a, S.X ω) * HA.likelihoodRatio_swapA a ω) μ → S.meanYofA μ a ∈ Set.Icc (∫ (ω : P.Ω) in {ω | S.A ω ≠ a}, Lenv (a, S.X ω) ∂μ + ∫ (ω : P.Ω) in {ω | S.A ω = a}, S.Y ω ∂μ) (∫ (ω : P.Ω) in {ω | S.A ω ≠ a}, Uenv (a, S.X ω) ∂μ + ∫ (ω : P.Ω) in {ω | S.A ω = a}, S.Y ω ∂μ)","l":162,"f":"Causalean/PO/ID/Partial/Proxy/IntervalForm.lean"},{"n":"Causalean.PO.POProximalSystem.meanYofA_eq_strata","k":"theorem","m":"Causalean.PO.ID.Partial.Proxy.Helpers.Common","a":"PO/ID/Partial/Proxy/Helpers","t":2,"d":"E[Y(a)] = ∫_{A = ¬a} Y(a) dμ + ∫_{A = a} Y dμ. Combines the stratum decomposition μ = μ.restrict {A=a} + μ.restrict {A=¬a} with consistency on the {A = a} arm.","s":"∀ {P : Causalean.PO.POSystem} {γ_X : Type u_1} {γ_Z : Type u_2} {γ_W : Type u_3} {γ_U : Type u_4} [inst : MeasurableSpace γ_X] [inst_1 : MeasurableSpace γ_Z] [inst_2 : MeasurableSpace γ_W] [inst_3 : MeasurableSpace γ_U] {S : Causalean.PO.POProximalSystem P γ_X γ_Z γ_W γ_U} {μ : MeasureTheory.Measure P.Ω}, P.Consistency → ∀ (a : Bool), S.Avar.v ≠ S.Yvar.v → MeasureTheory.Integrable (S.YofA a) μ → S.meanYofA μ a = ∫ (ω : P.Ω) in {ω | S.A ω ≠ a}, S.YofA a ω ∂μ + ∫ (ω : P.Ω) in {ω | S.A ω = a}, S.Y ω ∂μ","l":51,"f":"Causalean/PO/ID/Partial/Proxy/Helpers/Common.lean"},{"n":"Causalean.PO.POProximalSystem.measurable_A","k":"theorem","m":"Causalean.PO.ID.Exact.Proximal.Setup","a":"PO/ID/Exact/Proximal","t":2,"d":"The factual treatment is measurable.","s":"∀ {P : Causalean.PO.POSystem} {γ_X : Type u_1} {γ_Z : Type u_2} {γ_W : Type u_3} {γ_U : Type u_4} [inst : MeasurableSpace γ_X] [inst_1 : MeasurableSpace γ_Z] [inst_2 : MeasurableSpace γ_W] [inst_3 : MeasurableSpace γ_U] (S : Causalean.PO.POProximalSystem P γ_X γ_Z γ_W γ_U), Measurable S.A","l":82,"f":"Causalean/PO/ID/Exact/Proximal/Setup.lean"},{"n":"Causalean.PO.POProximalSystem.measurable_AUX","k":"theorem","m":"Causalean.PO.ID.Exact.Proximal.Setup","a":"PO/ID/Exact/Proximal","t":2,"d":"The treatment, latent confounder, and covariate tuple is measurable.","s":"∀ {P : Causalean.PO.POSystem} {γ_X : Type u_1} {γ_Z : Type u_2} {γ_W : Type u_3} {γ_U : Type u_4} [inst : MeasurableSpace γ_X] [inst_1 : MeasurableSpace γ_Z] [inst_2 : MeasurableSpace γ_W] [inst_3 : MeasurableSpace γ_U] (S : Causalean.PO.POProximalSystem P γ_X γ_Z γ_W γ_U), Measurable S.AUX","l":126,"f":"Causalean/PO/ID/Exact/Proximal/Setup.lean"},{"n":"Causalean.PO.POProximalSystem.measurable_AX","k":"theorem","m":"Causalean.PO.ID.Partial.Proxy.Setup","a":"PO/ID/Partial/Proxy","t":2,"d":"The treatment-covariate tuple map is measurable.","s":"∀ {P : Causalean.PO.POSystem} {γ_X : Type u_1} {γ_Z : Type u_2} {γ_W : Type u_3} {γ_U : Type u_4} [inst : MeasurableSpace γ_X] [inst_1 : MeasurableSpace γ_Z] [inst_2 : MeasurableSpace γ_W] [inst_3 : MeasurableSpace γ_U] (S : Causalean.PO.POProximalSystem P γ_X γ_Z γ_W γ_U), Measurable S.AX","l":49,"f":"Causalean/PO/ID/Partial/Proxy/Setup.lean"},{"n":"Causalean.PO.POProximalSystem.measurable_AZUX","k":"theorem","m":"Causalean.PO.ID.Exact.Proximal.Setup","a":"PO/ID/Exact/Proximal","t":2,"d":"The treatment, treatment-side proxy, latent confounder, and covariate tuple is measurable.","s":"∀ {P : Causalean.PO.POSystem} {γ_X : Type u_1} {γ_Z : Type u_2} {γ_W : Type u_3} {γ_U : Type u_4} [inst : MeasurableSpace γ_X] [inst_1 : MeasurableSpace γ_Z] [inst_2 : MeasurableSpace γ_W] [inst_3 : MeasurableSpace γ_U] (S : Causalean.PO.POProximalSystem P γ_X γ_Z γ_W γ_U), Measurable S.AZUX","l":134,"f":"Causalean/PO/ID/Exact/Proximal/Setup.lean"},{"n":"Causalean.PO.POProximalSystem.measurable_AZX","k":"theorem","m":"Causalean.PO.ID.Exact.Proximal.Setup","a":"PO/ID/Exact/Proximal","t":2,"d":"The treatment, treatment-side proxy, and covariate tuple is measurable.","s":"∀ {P : Causalean.PO.POSystem} {γ_X : Type u_1} {γ_Z : Type u_2} {γ_W : Type u_3} {γ_U : Type u_4} [inst : MeasurableSpace γ_X] [inst_1 : MeasurableSpace γ_Z] [inst_2 : MeasurableSpace γ_W] [inst_3 : MeasurableSpace γ_U] (S : Causalean.PO.POProximalSystem P γ_X γ_Z γ_W γ_U), Measurable S.AZX","l":122,"f":"Causalean/PO/ID/Exact/Proximal/Setup.lean"},{"n":"Causalean.PO.POProximalSystem.measurable_U","k":"theorem","m":"Causalean.PO.ID.Exact.Proximal.Setup","a":"PO/ID/Exact/Proximal","t":2,"d":"The factual latent confounder is measurable.","s":"∀ {P : Causalean.PO.POSystem} {γ_X : Type u_1} {γ_Z : Type u_2} {γ_W : Type u_3} {γ_U : Type u_4} [inst : MeasurableSpace γ_X] [inst_1 : MeasurableSpace γ_Z] [inst_2 : MeasurableSpace γ_W] [inst_3 : MeasurableSpace γ_U] (S : Causalean.PO.POProximalSystem P γ_X γ_Z γ_W γ_U), Measurable S.U","l":90,"f":"Causalean/PO/ID/Exact/Proximal/Setup.lean"},{"n":"Causalean.PO.POProximalSystem.measurable_UX","k":"theorem","m":"Causalean.PO.ID.Exact.Proximal.Setup","a":"PO/ID/Exact/Proximal","t":2,"d":"The latent confounder and covariate tuple is measurable.","s":"∀ {P : Causalean.PO.POSystem} {γ_X : Type u_1} {γ_Z : Type u_2} {γ_W : Type u_3} {γ_U : Type u_4} [inst : MeasurableSpace γ_X] [inst_1 : MeasurableSpace γ_Z] [inst_2 : MeasurableSpace γ_W] [inst_3 : MeasurableSpace γ_U] (S : Causalean.PO.POProximalSystem P γ_X γ_Z γ_W γ_U), Measurable S.UX","l":130,"f":"Causalean/PO/ID/Exact/Proximal/Setup.lean"},{"n":"Causalean.PO.POProximalSystem.measurable_W","k":"theorem","m":"Causalean.PO.ID.Exact.Proximal.Setup","a":"PO/ID/Exact/Proximal","t":2,"d":"The factual outcome-side proxy is measurable.","s":"∀ {P : Causalean.PO.POSystem} {γ_X : Type u_1} {γ_Z : Type u_2} {γ_W : Type u_3} {γ_U : Type u_4} [inst : MeasurableSpace γ_X] [inst_1 : MeasurableSpace γ_Z] [inst_2 : MeasurableSpace γ_W] [inst_3 : MeasurableSpace γ_U] (S : Causalean.PO.POProximalSystem P γ_X γ_Z γ_W γ_U), Measurable S.W","l":86,"f":"Causalean/PO/ID/Exact/Proximal/Setup.lean"},{"n":"Causalean.PO.POProximalSystem.measurable_X","k":"theorem","m":"Causalean.PO.ID.Exact.Proximal.Setup","a":"PO/ID/Exact/Proximal","t":2,"d":"The factual covariate is measurable.","s":"∀ {P : Causalean.PO.POSystem} {γ_X : Type u_1} {γ_Z : Type u_2} {γ_W : Type u_3} {γ_U : Type u_4} [inst : MeasurableSpace γ_X] [inst_1 : MeasurableSpace γ_Z] [inst_2 : MeasurableSpace γ_W] [inst_3 : MeasurableSpace γ_U] (S : Causalean.PO.POProximalSystem P γ_X γ_Z γ_W γ_U), Measurable S.X","l":80,"f":"Causalean/PO/ID/Exact/Proximal/Setup.lean"},{"n":"Causalean.PO.POProximalSystem.measurable_Y","k":"theorem","m":"Causalean.PO.ID.Exact.Proximal.Setup","a":"PO/ID/Exact/Proximal","t":2,"d":"The factual outcome is measurable.","s":"∀ {P : Causalean.PO.POSystem} {γ_X : Type u_1} {γ_Z : Type u_2} {γ_W : Type u_3} {γ_U : Type u_4} [inst : MeasurableSpace γ_X] [inst_1 : MeasurableSpace γ_Z] [inst_2 : MeasurableSpace γ_W] [inst_3 : MeasurableSpace γ_U] (S : Causalean.PO.POProximalSystem P γ_X γ_Z γ_W γ_U), Measurable S.Y","l":88,"f":"Causalean/PO/ID/Exact/Proximal/Setup.lean"},{"n":"Causalean.PO.POProximalSystem.measurable_YofA","k":"theorem","m":"Causalean.PO.ID.Exact.Proximal.Setup","a":"PO/ID/Exact/Proximal","t":2,"d":"The treatment-specific potential outcome is measurable.","s":"∀ {P : Causalean.PO.POSystem} {γ_X : Type u_1} {γ_Z : Type u_2} {γ_W : Type u_3} {γ_U : Type u_4} [inst : MeasurableSpace γ_X] [inst_1 : MeasurableSpace γ_Z] [inst_2 : MeasurableSpace γ_W] [inst_3 : MeasurableSpace γ_U] (S : Causalean.PO.POProximalSystem P γ_X γ_Z γ_W γ_U) (a : Bool), Measurable (S.YofA a)","l":98,"f":"Causalean/PO/ID/Exact/Proximal/Setup.lean"},{"n":"Causalean.PO.POProximalSystem.measurable_Z","k":"theorem","m":"Causalean.PO.ID.Exact.Proximal.Setup","a":"PO/ID/Exact/Proximal","t":2,"d":"The factual treatment-side proxy is measurable.","s":"∀ {P : Causalean.PO.POSystem} {γ_X : Type u_1} {γ_Z : Type u_2} {γ_W : Type u_3} {γ_U : Type u_4} [inst : MeasurableSpace γ_X] [inst_1 : MeasurableSpace γ_Z] [inst_2 : MeasurableSpace γ_W] [inst_3 : MeasurableSpace γ_U] (S : Causalean.PO.POProximalSystem P γ_X γ_Z γ_W γ_U), Measurable S.Z","l":84,"f":"Causalean/PO/ID/Exact/Proximal/Setup.lean"},{"n":"Causalean.PO.POProximalSystem.stratumOddsRatio","k":"def","m":"Causalean.PO.ID.Partial.Proxy.Setup","a":"PO/ID/Partial/Proxy","t":1,"d":"The σ_X-measurable function ω ↦ E[𝟙{A ≠ a} | X](ω) / E[𝟙{A = a} | X](ω), which equals p(A = ¬a | X) / p(A = a | X) in the discrete case. Used to re-normalise integrals against μ.restrict {A = a} to the {A ≠ a} arm.","s":"{P : Causalean.PO.POSystem} → {γ_X : Type u_1} → {γ_Z : Type u_2} → {γ_W : Type u_3} → {γ_U : Type u_4} → [inst : MeasurableSpace γ_X] → [inst_1 : MeasurableSpace γ_Z] → [inst_2 : MeasurableSpace γ_W] → [inst_3 : MeasurableSpace γ_U] → Causalean.PO.POProximalSystem P γ_X γ_Z γ_W γ_U → MeasureTheory.Measure P.Ω → Bool → P.Ω → ℝ","l":148,"f":"Causalean/PO/ID/Partial/Proxy/Setup.lean"},{"n":"Causalean.PO.POProximalSystem.σ_AUX","k":"def","m":"Causalean.PO.ID.Exact.Proximal.Setup","a":"PO/ID/Exact/Proximal","t":1,"d":"σ-algebra generated by (A, U, X).","s":"{P : Causalean.PO.POSystem} → {γ_X : Type u_1} → {γ_Z : Type u_2} → {γ_W : Type u_3} → {γ_U : Type u_4} → [inst : MeasurableSpace γ_X] → [inst_1 : MeasurableSpace γ_Z] → [inst_2 : MeasurableSpace γ_W] → [inst_3 : MeasurableSpace γ_U] → Causalean.PO.POProximalSystem P γ_X γ_Z γ_W γ_U → MeasurableSpace P.Ω","l":146,"f":"Causalean/PO/ID/Exact/Proximal/Setup.lean"},{"n":"Causalean.PO.POProximalSystem.σ_AUX_le","k":"theorem","m":"Causalean.PO.ID.Exact.Proximal.Setup","a":"PO/ID/Exact/Proximal","t":2,"d":"The sigma-algebra generated by treatment, latent confounder, and covariate is a sub-sigma-algebra of the ambient space.","s":"∀ {P : Causalean.PO.POSystem} {γ_X : Type u_1} {γ_Z : Type u_2} {γ_W : Type u_3} {γ_U : Type u_4} [inst : MeasurableSpace γ_X] [inst_1 : MeasurableSpace γ_Z] [inst_2 : MeasurableSpace γ_W] [inst_3 : MeasurableSpace γ_U] (S : Causalean.PO.POProximalSystem P γ_X γ_Z γ_W γ_U), S.σ_AUX ≤ inferInstance","l":165,"f":"Causalean/PO/ID/Exact/Proximal/Setup.lean"},{"n":"Causalean.PO.POProximalSystem.σ_AUX_le_σ_AZUX","k":"theorem","m":"Causalean.PO.ID.Partial.Proxy.Setup","a":"PO/ID/Partial/Proxy","t":2,"d":"The sigma-algebra on treatment, unobserved confounder, and covariates is coarser than the sigma-algebra that also includes the Z proxy.","s":"∀ {P : Causalean.PO.POSystem} {γ_X : Type u_1} {γ_Z : Type u_2} {γ_W : Type u_3} {γ_U : Type u_4} [inst : MeasurableSpace γ_X] [inst_1 : MeasurableSpace γ_Z] [inst_2 : MeasurableSpace γ_W] [inst_3 : MeasurableSpace γ_U] (S : Causalean.PO.POProximalSystem P γ_X γ_Z γ_W γ_U), S.σ_AUX ≤ S.σ_AZUX","l":97,"f":"Causalean/PO/ID/Partial/Proxy/Setup.lean"},{"n":"Causalean.PO.POProximalSystem.σ_AX","k":"def","m":"Causalean.PO.ID.Partial.Proxy.Setup","a":"PO/ID/Partial/Proxy","t":1,"d":"σ-algebra generated by (A, X).","s":"{P : Causalean.PO.POSystem} → {γ_X : Type u_1} → {γ_Z : Type u_2} → {γ_W : Type u_3} → {γ_U : Type u_4} → [inst : MeasurableSpace γ_X] → [inst_1 : MeasurableSpace γ_Z] → [inst_2 : MeasurableSpace γ_W] → [inst_3 : MeasurableSpace γ_U] → Causalean.PO.POProximalSystem P γ_X γ_Z γ_W γ_U → MeasurableSpace P.Ω","l":53,"f":"Causalean/PO/ID/Partial/Proxy/Setup.lean"},{"n":"Causalean.PO.POProximalSystem.σ_AX_le","k":"theorem","m":"Causalean.PO.ID.Partial.Proxy.Setup","a":"PO/ID/Partial/Proxy","t":2,"d":"The sigma-algebra generated by treatment and covariates is a sub-sigma-algebra of the ambient measurable space.","s":"∀ {P : Causalean.PO.POSystem} {γ_X : Type u_1} {γ_Z : Type u_2} {γ_W : Type u_3} {γ_U : Type u_4} [inst : MeasurableSpace γ_X] [inst_1 : MeasurableSpace γ_Z] [inst_2 : MeasurableSpace γ_W] [inst_3 : MeasurableSpace γ_U] (S : Causalean.PO.POProximalSystem P γ_X γ_Z γ_W γ_U), S.σ_AX ≤ inferInstance","l":57,"f":"Causalean/PO/ID/Partial/Proxy/Setup.lean"},{"n":"Causalean.PO.POProximalSystem.σ_AX_le_σ_AUX","k":"theorem","m":"Causalean.PO.ID.Partial.Proxy.Setup","a":"PO/ID/Partial/Proxy","t":2,"d":"σ_AX is coarser than σ_AUX, since (A, X) factors through (A, U, X) via the projection (a, u, x) ↦ (a, x).","s":"∀ {P : Causalean.PO.POSystem} {γ_X : Type u_1} {γ_Z : Type u_2} {γ_W : Type u_3} {γ_U : Type u_4} [inst : MeasurableSpace γ_X] [inst_1 : MeasurableSpace γ_Z] [inst_2 : MeasurableSpace γ_W] [inst_3 : MeasurableSpace γ_U] (S : Causalean.PO.POProximalSystem P γ_X γ_Z γ_W γ_U), S.σ_AX ≤ S.σ_AUX","l":110,"f":"Causalean/PO/ID/Partial/Proxy/Setup.lean"},{"n":"Causalean.PO.POProximalSystem.σ_AZUX","k":"def","m":"Causalean.PO.ID.Exact.Proximal.Setup","a":"PO/ID/Exact/Proximal","t":1,"d":"σ-algebra generated by (A, Z, U, X).","s":"{P : Causalean.PO.POSystem} → {γ_X : Type u_1} → {γ_Z : Type u_2} → {γ_W : Type u_3} → {γ_U : Type u_4} → [inst : MeasurableSpace γ_X] → [inst_1 : MeasurableSpace γ_Z] → [inst_2 : MeasurableSpace γ_W] → [inst_3 : MeasurableSpace γ_U] → Causalean.PO.POProximalSystem P γ_X γ_Z γ_W γ_U → MeasurableSpace P.Ω","l":154,"f":"Causalean/PO/ID/Exact/Proximal/Setup.lean"},{"n":"Causalean.PO.POProximalSystem.σ_AZUX_le","k":"theorem","m":"Causalean.PO.ID.Exact.Proximal.Setup","a":"PO/ID/Exact/Proximal","t":2,"d":"The sigma-algebra generated by treatment, treatment-side proxy, latent confounder, and covariate is a sub-sigma-algebra of the ambient space.","s":"∀ {P : Causalean.PO.POSystem} {γ_X : Type u_1} {γ_Z : Type u_2} {γ_W : Type u_3} {γ_U : Type u_4} [inst : MeasurableSpace γ_X] [inst_1 : MeasurableSpace γ_Z] [inst_2 : MeasurableSpace γ_W] [inst_3 : MeasurableSpace γ_U] (S : Causalean.PO.POProximalSystem P γ_X γ_Z γ_W γ_U), S.σ_AZUX ≤ inferInstance","l":175,"f":"Causalean/PO/ID/Exact/Proximal/Setup.lean"},{"n":"Causalean.PO.POProximalSystem.σ_AZX","k":"def","m":"Causalean.PO.ID.Exact.Proximal.Setup","a":"PO/ID/Exact/Proximal","t":1,"d":"σ-algebra generated by (A, Z, X).","s":"{P : Causalean.PO.POSystem} → {γ_X : Type u_1} → {γ_Z : Type u_2} → {γ_W : Type u_3} → {γ_U : Type u_4} → [inst : MeasurableSpace γ_X] → [inst_1 : MeasurableSpace γ_Z] → [inst_2 : MeasurableSpace γ_W] → [inst_3 : MeasurableSpace γ_U] → Causalean.PO.POProximalSystem P γ_X γ_Z γ_W γ_U → MeasurableSpace P.Ω","l":142,"f":"Causalean/PO/ID/Exact/Proximal/Setup.lean"},{"n":"Causalean.PO.POProximalSystem.σ_AZX_le","k":"theorem","m":"Causalean.PO.ID.Exact.Proximal.Setup","a":"PO/ID/Exact/Proximal","t":2,"d":"The sigma-algebra generated by treatment, treatment-side proxy, and covariate is a sub-sigma-algebra of the ambient space.","s":"∀ {P : Causalean.PO.POSystem} {γ_X : Type u_1} {γ_Z : Type u_2} {γ_W : Type u_3} {γ_U : Type u_4} [inst : MeasurableSpace γ_X] [inst_1 : MeasurableSpace γ_Z] [inst_2 : MeasurableSpace γ_W] [inst_3 : MeasurableSpace γ_U] (S : Causalean.PO.POProximalSystem P γ_X γ_Z γ_W γ_U), S.σ_AZX ≤ inferInstance","l":160,"f":"Causalean/PO/ID/Exact/Proximal/Setup.lean"},{"n":"Causalean.PO.POProximalSystem.σ_AZX_le_σ_AZUX","k":"theorem","m":"Causalean.PO.ID.Partial.Proxy.Setup","a":"PO/ID/Partial/Proxy","t":2,"d":"σ_AZX ≤ σ_AZUX, since (A, Z, X) factors through (A, Z, U, X).","s":"∀ {P : Causalean.PO.POSystem} {γ_X : Type u_1} {γ_Z : Type u_2} {γ_W : Type u_3} {γ_U : Type u_4} [inst : MeasurableSpace γ_X] [inst_1 : MeasurableSpace γ_Z] [inst_2 : MeasurableSpace γ_W] [inst_3 : MeasurableSpace γ_U] (S : Causalean.PO.POProximalSystem P γ_X γ_Z γ_W γ_U), S.σ_AZX ≤ S.σ_AZUX","l":87,"f":"Causalean/PO/ID/Partial/Proxy/Setup.lean"},{"n":"Causalean.PO.POProximalSystem.σ_UX","k":"def","m":"Causalean.PO.ID.Exact.Proximal.Setup","a":"PO/ID/Exact/Proximal","t":1,"d":"σ-algebra generated by (U, X).","s":"{P : Causalean.PO.POSystem} → {γ_X : Type u_1} → {γ_Z : Type u_2} → {γ_W : Type u_3} → {γ_U : Type u_4} → [inst : MeasurableSpace γ_X] → [inst_1 : MeasurableSpace γ_Z] → [inst_2 : MeasurableSpace γ_W] → [inst_3 : MeasurableSpace γ_U] → Causalean.PO.POProximalSystem P γ_X γ_Z γ_W γ_U → MeasurableSpace P.Ω","l":150,"f":"Causalean/PO/ID/Exact/Proximal/Setup.lean"},{"n":"Causalean.PO.POProximalSystem.σ_UX_le","k":"theorem","m":"Causalean.PO.ID.Exact.Proximal.Setup","a":"PO/ID/Exact/Proximal","t":1,"d":"The σ-algebra generated by the latent confounder and covariate is a sub-σ-algebra of the ambient measurable space.","s":"∀ {P : Causalean.PO.POSystem} {γ_X : Type u_1} {γ_Z : Type u_2} {γ_W : Type u_3} {γ_U : Type u_4} [inst : MeasurableSpace γ_X] [inst_1 : MeasurableSpace γ_Z] [inst_2 : MeasurableSpace γ_W] [inst_3 : MeasurableSpace γ_U] (S : Causalean.PO.POProximalSystem P γ_X γ_Z γ_W γ_U), S.σ_UX ≤ inferInstance","l":170,"f":"Causalean/PO/ID/Exact/Proximal/Setup.lean"},{"n":"Causalean.PO.POProximalSystem.σ_X","k":"def","m":"Causalean.PO.ID.Partial.Proxy.Setup","a":"PO/ID/Partial/Proxy","t":1,"d":"σ-algebra generated by X alone.","s":"{P : Causalean.PO.POSystem} → {γ_X : Type u_1} → {γ_Z : Type u_2} → {γ_W : Type u_3} → {γ_U : Type u_4} → [inst : MeasurableSpace γ_X] → [inst_1 : MeasurableSpace γ_Z] → [inst_2 : MeasurableSpace γ_W] → [inst_3 : MeasurableSpace γ_U] → Causalean.PO.POProximalSystem P γ_X γ_Z γ_W γ_U → MeasurableSpace P.Ω","l":62,"f":"Causalean/PO/ID/Partial/Proxy/Setup.lean"},{"n":"Causalean.PO.POProximalSystem.σ_X_le","k":"theorem","m":"Causalean.PO.ID.Partial.Proxy.Setup","a":"PO/ID/Partial/Proxy","t":1,"d":"The σ-algebra generated by the covariates is a sub-σ-algebra of the ambient measurable space.","s":"∀ {P : Causalean.PO.POSystem} {γ_X : Type u_1} {γ_Z : Type u_2} {γ_W : Type u_3} {γ_U : Type u_4} [inst : MeasurableSpace γ_X] [inst_1 : MeasurableSpace γ_Z] [inst_2 : MeasurableSpace γ_W] [inst_3 : MeasurableSpace γ_U] (S : Causalean.PO.POProximalSystem P γ_X γ_Z γ_W γ_U), S.σ_X ≤ inferInstance","l":66,"f":"Causalean/PO/ID/Partial/Proxy/Setup.lean"},{"n":"Causalean.PO.POProximalSystem.σ_X_le_σ_AUX","k":"theorem","m":"Causalean.PO.ID.Partial.Proxy.Setup","a":"PO/ID/Partial/Proxy","t":2,"d":"σ_X ≤ σ_AUX, by composition σ_X ≤ σ_AX ≤ σ_AUX.","s":"∀ {P : Causalean.PO.POSystem} {γ_X : Type u_1} {γ_Z : Type u_2} {γ_W : Type u_3} {γ_U : Type u_4} [inst : MeasurableSpace γ_X] [inst_1 : MeasurableSpace γ_Z] [inst_2 : MeasurableSpace γ_W] [inst_3 : MeasurableSpace γ_U] (S : Causalean.PO.POProximalSystem P γ_X γ_Z γ_W γ_U), S.σ_X ≤ S.σ_AUX","l":127,"f":"Causalean/PO/ID/Partial/Proxy/Setup.lean"},{"n":"Causalean.PO.POProximalSystem.σ_X_le_σ_AX","k":"theorem","m":"Causalean.PO.ID.Partial.Proxy.Setup","a":"PO/ID/Partial/Proxy","t":2,"d":"σ_X is coarser than σ_AX: X factors through (A, X) via the second projection.","s":"∀ {P : Causalean.PO.POSystem} {γ_X : Type u_1} {γ_Z : Type u_2} {γ_W : Type u_3} {γ_U : Type u_4} [inst : MeasurableSpace γ_X] [inst_1 : MeasurableSpace γ_Z] [inst_2 : MeasurableSpace γ_W] [inst_3 : MeasurableSpace γ_U] (S : Causalean.PO.POProximalSystem P γ_X γ_Z γ_W γ_U), S.σ_X ≤ S.σ_AX","l":71,"f":"Causalean/PO/ID/Partial/Proxy/Setup.lean"},{"n":"Causalean.PO.POProximalSystem.σ_X_le_σ_UX","k":"theorem","m":"Causalean.PO.ID.Partial.Proxy.Setup","a":"PO/ID/Partial/Proxy","t":2,"d":"σ_X ≤ σ_UX, since X factors through (U, X) via the second projection.","s":"∀ {P : Causalean.PO.POSystem} {γ_X : Type u_1} {γ_Z : Type u_2} {γ_W : Type u_3} {γ_U : Type u_4} [inst : MeasurableSpace γ_X] [inst_1 : MeasurableSpace γ_Z] [inst_2 : MeasurableSpace γ_W] [inst_3 : MeasurableSpace γ_U] (S : Causalean.PO.POProximalSystem P γ_X γ_Z γ_W γ_U), S.σ_X ≤ S.σ_UX","l":80,"f":"Causalean/PO/ID/Partial/Proxy/Setup.lean"},{"n":"Causalean.PO.POSharpRDDSystem","k":"structure","m":"Causalean.PO.ID.Exact.RDD.SharpRDD","a":"PO/ID/Exact/RDD","t":1,"d":"Sharp regression-discontinuity model in the potential-outcome framework. A unit has a continuous running variable X, a deterministic treatment D that switches on exactly when X crosses the cutoff c, jumping from 0 to 1 at the cutoff, and a real outcome Y, with the outcome and treatment nodes distinct. The jump in the outcome regression at the cutoff identifies the cutoff-local average treatment effect (def:po-sharp-rdd-system).","s":"Causalean.PO.POSystem → Type (max u_1 u_2)","l":29,"f":"Causalean/PO/ID/Exact/RDD/SharpRDD.lean"},{"n":"Causalean.PO.POSharpRDDSystem.Assumptions","k":"structure","m":"Causalean.PO.ID.Exact.RDD.SharpRDD","a":"PO/ID/Exact/RDD","t":1,"d":"Sharp RDD assumption bundle (def:po-sharp-rdd-assumptions). For a sharp regression-discontinuity system, this packages consistency (SUTVA) and the deterministic cutoff rule: the treatment indicator agrees almost surely with whether the running variable has reached the cutoff. It supplies treatment-specific latent regression representatives mu and an observable outcome regression representative nu, each certified as a genuine regression function of the corresponding response on the running variable, with the latent representatives continuous at the cutoff; it also assumes the running variable has positive local probability mass on both sides of the cutoff and the observable outcome regression has well-defined one-sided limits at the cutoff.","s":"{P : Causalean.PO.POSystem} → Causalean.PO.POSharpRDDSystem P → Type","l":79,"f":"Causalean/PO/ID/Exact/RDD/SharpRDD.lean"},{"n":"Causalean.PO.POSharpRDDSystem.YofD","k":"def","m":"Causalean.PO.ID.Exact.RDD.SharpRDD","a":"PO/ID/Exact/RDD","t":1,"d":"Treatment-specific potential outcome Y(d).","s":"{P : Causalean.PO.POSystem} → Causalean.PO.POSharpRDDSystem P → Bool → P.Ω → ℝ","l":59,"f":"Causalean/PO/ID/Exact/RDD/SharpRDD.lean"},{"n":"Causalean.PO.POSharpRDDSystem.dEvent","k":"def","m":"Causalean.PO.ID.Exact.RDD.SharpRDD","a":"PO/ID/Exact/RDD","t":1,"d":"Factual treatment event {D = d}.","s":"{P : Causalean.PO.POSystem} → Causalean.PO.POSharpRDDSystem P → Bool → Set P.Ω","l":72,"f":"Causalean/PO/ID/Exact/RDD/SharpRDD.lean"},{"n":"Causalean.PO.POSharpRDDSystem.factualD","k":"def","m":"Causalean.PO.ID.Exact.RDD.SharpRDD","a":"PO/ID/Exact/RDD","t":1,"d":"Factual treatment D.","s":"{P : Causalean.PO.POSystem} → Causalean.PO.POSharpRDDSystem P → P.Ω → Bool","l":53,"f":"Causalean/PO/ID/Exact/RDD/SharpRDD.lean"},{"n":"Causalean.PO.POSharpRDDSystem.factualX","k":"def","m":"Causalean.PO.ID.Exact.RDD.SharpRDD","a":"PO/ID/Exact/RDD","t":1,"d":"Factual running variable X.","s":"{P : Causalean.PO.POSystem} → Causalean.PO.POSharpRDDSystem P → P.Ω → ℝ","l":50,"f":"Causalean/PO/ID/Exact/RDD/SharpRDD.lean"},{"n":"Causalean.PO.POSharpRDDSystem.factualY","k":"def","m":"Causalean.PO.ID.Exact.RDD.SharpRDD","a":"PO/ID/Exact/RDD","t":1,"d":"Factual outcome Y.","s":"{P : Causalean.PO.POSystem} → Causalean.PO.POSharpRDDSystem P → P.Ω → ℝ","l":56,"f":"Causalean/PO/ID/Exact/RDD/SharpRDD.lean"},{"n":"Causalean.PO.POSharpRDDSystem.measurableSet_dEvent","k":"theorem","m":"Causalean.PO.ID.Exact.RDD.SharpRDD","a":"PO/ID/Exact/RDD","t":2,"d":"The factual treatment event is measurable.","s":"∀ {P : Causalean.PO.POSystem} (S : Causalean.PO.POSharpRDDSystem P) (d : Bool), MeasurableSet (S.dEvent d)","l":75,"f":"Causalean/PO/ID/Exact/RDD/SharpRDD.lean"},{"n":"Causalean.PO.POSharpRDDSystem.measurable_YofD","k":"theorem","m":"Causalean.PO.ID.Exact.RDD.SharpRDD","a":"PO/ID/Exact/RDD","t":2,"d":"The treatment-specific potential outcome is measurable.","s":"∀ {P : Causalean.PO.POSystem} (S : Causalean.PO.POSharpRDDSystem P) (d : Bool), Measurable (S.YofD d)","l":68,"f":"Causalean/PO/ID/Exact/RDD/SharpRDD.lean"},{"n":"Causalean.PO.POSharpRDDSystem.measurable_factualD","k":"theorem","m":"Causalean.PO.ID.Exact.RDD.SharpRDD","a":"PO/ID/Exact/RDD","t":2,"d":"The factual treatment is measurable.","s":"∀ {P : Causalean.PO.POSystem} (S : Causalean.PO.POSharpRDDSystem P), Measurable S.factualD","l":64,"f":"Causalean/PO/ID/Exact/RDD/SharpRDD.lean"},{"n":"Causalean.PO.POSharpRDDSystem.measurable_factualX","k":"theorem","m":"Causalean.PO.ID.Exact.RDD.SharpRDD","a":"PO/ID/Exact/RDD","t":2,"d":"The factual running variable is measurable.","s":"∀ {P : Causalean.PO.POSystem} (S : Causalean.PO.POSharpRDDSystem P), Measurable S.factualX","l":62,"f":"Causalean/PO/ID/Exact/RDD/SharpRDD.lean"},{"n":"Causalean.PO.POSharpRDDSystem.measurable_factualY","k":"theorem","m":"Causalean.PO.ID.Exact.RDD.SharpRDD","a":"PO/ID/Exact/RDD","t":2,"d":"The factual outcome is measurable.","s":"∀ {P : Causalean.PO.POSystem} (S : Causalean.PO.POSharpRDDSystem P), Measurable S.factualY","l":66,"f":"Causalean/PO/ID/Exact/RDD/SharpRDD.lean"},{"n":"Causalean.PO.POSharpRDDSystem.nu_left_limit","k":"def","m":"Causalean.PO.ID.Exact.RDD.SharpRDD","a":"PO/ID/Exact/RDD","t":1,"d":"The left-hand limit lim_{x ↑ c} ν(x) chosen from the existence witness of Assumptions.","s":"{P : Causalean.PO.POSystem} → (S : Causalean.PO.POSharpRDDSystem P) → S.Assumptions → ℝ","l":122,"f":"Causalean/PO/ID/Exact/RDD/SharpRDD.lean"},{"n":"Causalean.PO.POSharpRDDSystem.nu_left_limit_eq","k":"theorem","m":"Causalean.PO.ID.Exact.RDD.SharpRDD","a":"PO/ID/Exact/RDD","t":2,"d":"Any left-limit of ν at c coincides with μ_0 c.","s":"∀ {P : Causalean.PO.POSystem} (S : Causalean.PO.POSharpRDDSystem P) (hA : S.Assumptions) {L : ℝ}, Filter.Tendsto hA.nu (nhdsWithin S.c (Set.Iio S.c)) (nhds L) → L = hA.mu false S.c","l":293,"f":"Causalean/PO/ID/Exact/RDD/SharpRDD.lean"},{"n":"Causalean.PO.POSharpRDDSystem.nu_right_limit","k":"def","m":"Causalean.PO.ID.Exact.RDD.SharpRDD","a":"PO/ID/Exact/RDD","t":1,"d":"The right-hand limit lim_{x ↓ c} ν(x) chosen from the existence witness of Assumptions.","s":"{P : Causalean.PO.POSystem} → (S : Causalean.PO.POSharpRDDSystem P) → S.Assumptions → ℝ","l":117,"f":"Causalean/PO/ID/Exact/RDD/SharpRDD.lean"},{"n":"Causalean.PO.POSharpRDDSystem.nu_right_limit_eq","k":"theorem","m":"Causalean.PO.ID.Exact.RDD.SharpRDD","a":"PO/ID/Exact/RDD","t":2,"d":"Any right-limit of ν at c coincides with μ_1 c.","s":"∀ {P : Causalean.PO.POSystem} (S : Causalean.PO.POSharpRDDSystem P) (hA : S.Assumptions) {L : ℝ}, Filter.Tendsto hA.nu (nhdsWithin S.c (Set.Ioi S.c)) (nhds L) → L = hA.mu true S.c","l":283,"f":"Causalean/PO/ID/Exact/RDD/SharpRDD.lean"},{"n":"Causalean.PO.POSharpRDDSystem.rdd_identification","k":"theorem","m":"Causalean.PO.ID.Exact.RDD.SharpRDD","a":"PO/ID/Exact/RDD","t":1,"d":"Sharp RDD identification at the cutoff (textbook form). Under the sharp-RDD assumption bundle — SUTVA consistency, deterministic treatment assignment D = 1{X ≥ c} almost surely, regression-function representatives for the potential and observed outcomes, continuity of the potential-outcome regression functions at the cutoff, positive local mass of the running variable on each side of c, and existence of both one-sided limits of the observed-outcome regression function at c, the cutoff-local treatment effect equals the difference of the right- and left-hand limits of the observed-outcome regression function at the cutoff:","s":"∀ {P : Causalean.PO.POSystem} (S : Causalean.PO.POSharpRDDSystem P) (hA : S.Assumptions), S.tau_RDD hA = S.nu_right_limit hA - S.nu_left_limit hA","l":305,"f":"Causalean/PO/ID/Exact/RDD/SharpRDD.lean"},{"n":"Causalean.PO.POSharpRDDSystem.tau_RDD","k":"def","m":"Causalean.PO.ID.Exact.RDD.SharpRDD","a":"PO/ID/Exact/RDD","t":1,"d":"Cutoff-local RDD estimand — by definition the difference of the treatment-specific regression representatives at the cutoff. In the standard reading of μ_d c = E[Y(d) | X = c], this is E[Y(1) - Y(0) | X = c].","s":"{P : Causalean.PO.POSystem} → (S : Causalean.PO.POSharpRDDSystem P) → S.Assumptions → ℝ","l":111,"f":"Causalean/PO/ID/Exact/RDD/SharpRDD.lean"},{"n":"Causalean.PO.POSharpRDDSystem.tendsto_nu_left_limit","k":"theorem","m":"Causalean.PO.ID.Exact.RDD.SharpRDD","a":"PO/ID/Exact/RDD","t":2,"d":"The chosen left-hand observable regression limit is a genuine left-hand limit at the cutoff.","s":"∀ {P : Causalean.PO.POSystem} (S : Causalean.PO.POSharpRDDSystem P) (hA : S.Assumptions), Filter.Tendsto hA.nu (nhdsWithin S.c (Set.Iio S.c)) (nhds (S.nu_left_limit hA))","l":133,"f":"Causalean/PO/ID/Exact/RDD/SharpRDD.lean"},{"n":"Causalean.PO.POSharpRDDSystem.tendsto_nu_right_limit","k":"theorem","m":"Causalean.PO.ID.Exact.RDD.SharpRDD","a":"PO/ID/Exact/RDD","t":2,"d":"The chosen right-hand observable regression limit is a genuine right-hand limit at the cutoff.","s":"∀ {P : Causalean.PO.POSystem} (S : Causalean.PO.POSharpRDDSystem P) (hA : S.Assumptions), Filter.Tendsto hA.nu (nhdsWithin S.c (Set.Ioi S.c)) (nhds (S.nu_right_limit hA))","l":127,"f":"Causalean/PO/ID/Exact/RDD/SharpRDD.lean"},{"n":"Causalean.PO.POSystem","k":"structure","m":"Causalean.PO.Core.System","a":"PO/Core","t":1,"d":"A potential-outcome system consists of a finite set of variables, a measurable value space for each variable, a measurable sample space carrying a probability measure, and, for every intervention regime and sample point, a jointly measurable assignment of potential-outcome values to all variables.","s":"Type (max (max (u_1 + 1) (u_2 + 1)) (u_3 + 1))","l":27,"f":"Causalean/PO/Core/System.lean"},{"n":"Causalean.PO.POSystem.CondIndepCF","k":"def","m":"Causalean.PO.Assumptions.IndepCF","a":"PO/Assumptions","t":1,"d":"A regimed potential-outcome variable is conditionally independent of a counterfactual bundle when its value is conditionally independent of the bundle's joint counterfactual value given the sigma-algebra generated by the conditioning variable.","s":"{α : Type u_1} → {γ : Type u_2} → [inst : MeasurableSpace α] → [inst_1 : MeasurableSpace γ] → (P : Causalean.PO.POSystem) → [StandardBorelSpace P.Ω] → Causalean.PO.RegimedVar P α → Causalean.PO.POCFBundle P → Causalean.PO.RegimedVar P γ → (μ : optParam (MeasureTheory.Measure P.Ω) P.μ) → [MeasureTheory.IsFiniteMeasure μ] → Prop","l":98,"f":"Causalean/PO/Assumptions/IndepCF.lean"},{"n":"Causalean.PO.POSystem.CondIndepCF.component","k":"theorem","m":"Causalean.PO.Assumptions.IndepCF","a":"PO/Assumptions","t":2,"d":"Component projection: conditional independence of a.value from a single coordinate of the bundle follows from CondIndepCF.","s":"∀ {P : Causalean.PO.POSystem} {α : Type u_1} {γ : Type u_2} [inst : MeasurableSpace α] [inst_1 : MeasurableSpace γ] [inst_2 : StandardBorelSpace P.Ω] {a : Causalean.PO.RegimedVar P α} {B : Causalean.PO.POCFBundle P} {c : Causalean.PO.RegimedVar P γ} {μ : MeasureTheory.Measure P.Ω} [inst_3 : MeasureTheory.IsFiniteMeasure μ], P.CondIndepCF a B c μ → ∀ (i : Fin B.n), ProbabilityTheory.CondIndepFun (MeasurableSpace.comap c.value inferInstance) (Measurable.comap_le (Causalean.PO.RegimedVar.measurable_value c)) a.value (fun ω => B.jointValue ω i) μ","l":186,"f":"Causalean/PO/Assumptions/IndepCF.lean"},{"n":"Causalean.PO.POSystem.CondIndepCF.project","k":"theorem","m":"Causalean.PO.Assumptions.IndepCF","a":"PO/Assumptions","t":1,"d":"Fix a potential-outcome system in which a regimed variable a is conditionally independent of a counterfactual bundle B given a regimed variable c. Then for any measurable function ψ of the bundle's joint value, the value of a remains conditionally independent, given c, of ψ composed with the bundle's joint value.","s":"∀ {P : Causalean.PO.POSystem} {α : Type u_1} {β : Type u_2} {γ : Type u_3} [inst : MeasurableSpace α] [inst_1 : MeasurableSpace β] [inst_2 : MeasurableSpace γ] [inst_3 : StandardBorelSpace P.Ω] {a : Causalean.PO.RegimedVar P α} {B : Causalean.PO.POCFBundle P} {c : Causalean.PO.RegimedVar P γ} {μ : MeasureTheory.Measure P.Ω} [inst_4 : MeasureTheory.IsFiniteMeasure μ] {ψ : ((i : Fin B.n) → B.type i) → β}, P.CondIndepCF a B c μ → Measurable ψ → ProbabilityTheory.CondIndepFun (MeasurableSpace.comap c.value inferInstance) (Measurable.comap_le (Causalean.PO.RegimedVar.measurable_value c)) a.value (ψ ∘ B.jointValue) μ","l":169,"f":"Causalean/PO/Assumptions/IndepCF.lean"},{"n":"Causalean.PO.POSystem.CondIndepCF.toCondIndepFun","k":"theorem","m":"Causalean.PO.Assumptions.IndepCF","a":"PO/Assumptions","t":2,"d":"Bridge: CondIndepCF unfolds to CondIndepFun with the comap σ-algebra generated by c.value.","s":"∀ {P : Causalean.PO.POSystem} {α : Type u_1} {γ : Type u_2} [inst : MeasurableSpace α] [inst_1 : MeasurableSpace γ] [inst_2 : StandardBorelSpace P.Ω] {a : Causalean.PO.RegimedVar P α} {B : Causalean.PO.POCFBundle P} {c : Causalean.PO.RegimedVar P γ} {μ : MeasureTheory.Measure P.Ω} [inst_3 : MeasureTheory.IsFiniteMeasure μ], P.CondIndepCF a B c μ → ProbabilityTheory.CondIndepFun (MeasurableSpace.comap c.value inferInstance) (Measurable.comap_le (Causalean.PO.RegimedVar.measurable_value c)) a.value B.jointValue μ","l":127,"f":"Causalean/PO/Assumptions/IndepCF.lean"},{"n":"Causalean.PO.POSystem.CondIndepCFBundle","k":"def","m":"Causalean.PO.Conditioning.Bundle","a":"PO/Conditioning","t":1,"d":"A regimed potential-outcome variable and one counterfactual bundle are conditionally independent given the sigma-algebra generated by another counterfactual bundle's joint counterfactual-value map.","s":"(P : Causalean.PO.POSystem) → [StandardBorelSpace P.Ω] → {α : Type u_1} → [inst : MeasurableSpace α] → Causalean.PO.RegimedVar P α → Causalean.PO.POCFBundle P → Causalean.PO.POCFBundle P → (μ : optParam (MeasureTheory.Measure P.Ω) P.μ) → [MeasureTheory.IsFiniteMeasure μ] → Prop","l":180,"f":"Causalean/PO/Conditioning/Bundle.lean"},{"n":"Causalean.PO.POSystem.CondIndepCFBundle.project","k":"theorem","m":"Causalean.PO.Conditioning.Bundle","a":"PO/Conditioning","t":1,"d":"Bundle-conditional projection. Given a regimed variable a that is conditionally independent, given the σ-algebra generated by a counterfactual bundle C, of another counterfactual bundle B, then for any measurable function ψ of the joint value of B, the value of a remains conditionally independent, given C's σ-algebra, of ψ composed with the joint value of B.","s":"∀ {P : Causalean.PO.POSystem} [inst : StandardBorelSpace P.Ω] {α : Type u_1} {β : Type u_2} [inst_1 : MeasurableSpace α] [inst_2 : MeasurableSpace β] {a : Causalean.PO.RegimedVar P α} {B : Causalean.PO.POCFBundle P} {C : Causalean.PO.POCFBundle P} {μ : MeasureTheory.Measure P.Ω} [inst_3 : MeasureTheory.IsFiniteMeasure μ] {ψ : ((i : Fin B.n) → B.type i) → β}, P.CondIndepCFBundle a B C μ → Measurable ψ → ProbabilityTheory.CondIndepFun C.sigma (Causalean.PO.POCFBundle.sigma_le C) a.value (ψ ∘ B.jointValue) μ","l":205,"f":"Causalean/PO/Conditioning/Bundle.lean"},{"n":"Causalean.PO.POSystem.CondIndepCFBundle.toCondIndepFun","k":"theorem","m":"Causalean.PO.Conditioning.Bundle","a":"PO/Conditioning","t":2,"d":"Bridge: CondIndepCFBundle unfolds to CondIndepFun w.r.t. C.sigma.","s":"∀ {P : Causalean.PO.POSystem} [inst : StandardBorelSpace P.Ω] {α : Type u_1} [inst_1 : MeasurableSpace α] {a : Causalean.PO.RegimedVar P α} {B : Causalean.PO.POCFBundle P} {C : Causalean.PO.POCFBundle P} {μ : MeasureTheory.Measure P.Ω} [inst_2 : MeasureTheory.IsFiniteMeasure μ], P.CondIndepCFBundle a B C μ → ProbabilityTheory.CondIndepFun C.sigma (Causalean.PO.POCFBundle.sigma_le C) a.value B.jointValue μ","l":196,"f":"Causalean/PO/Conditioning/Bundle.lean"},{"n":"Causalean.PO.POSystem.Consistency","k":"structure","m":"Causalean.PO.Assumptions.Consistency","a":"PO/Assumptions","t":1,"d":"A potential-outcome system is consistent when two conditions hold. First, for every intervention regime and every finite set of variables disjoint from the regime's target, a unit's potential outcomes for that set under the regime equal its factual potential outcomes whenever it factually agrees with the regime. Second, for every pair of disjoint regimes and every finite set of variables disjoint from the union of their targets, a unit's potential outcomes for that set under the composed regime equal its potential outcomes under the first regime alone, whenever it agrees with the second regime after the first has been applied.","s":"Causalean.PO.POSystem → Prop","l":43,"f":"Causalean/PO/Assumptions/Consistency.lean"},{"n":"Causalean.PO.POSystem.FactualAgrees","k":"def","m":"Causalean.PO.Assumptions.Consistency","a":"PO/Assumptions","t":1,"d":"For a potential-outcome system, an intervention regime, and a unit in the sample space, factual agreement means that for every variable targeted by the regime, the factual value of that variable for the unit equals the value assigned by the regime.","s":"(P : Causalean.PO.POSystem) → Causalean.PO.Regime P.V P.X → P.Ω → Prop","l":25,"f":"Causalean/PO/Assumptions/Consistency.lean"},{"n":"Causalean.PO.POSystem.IndepCF","k":"def","m":"Causalean.PO.Assumptions.IndepCF","a":"PO/Assumptions","t":1,"d":"A regimed potential-outcome variable is independent of a counterfactual bundle when its value is independent of the bundle's joint counterfactual value under the chosen measure.","s":"{α : Type u_1} → [inst : MeasurableSpace α] → (P : Causalean.PO.POSystem) → Causalean.PO.RegimedVar P α → Causalean.PO.POCFBundle P → optParam (MeasureTheory.Measure P.Ω) P.μ → Prop","l":91,"f":"Causalean/PO/Assumptions/IndepCF.lean"},{"n":"Causalean.PO.POSystem.IndepCF.component","k":"theorem","m":"Causalean.PO.Assumptions.IndepCF","a":"PO/Assumptions","t":2,"d":"Component projection: independence of a.value from a single coordinate of the bundle follows from IndepCF.","s":"∀ {P : Causalean.PO.POSystem} {α : Type u_1} [inst : MeasurableSpace α] {a : Causalean.PO.RegimedVar P α} {B : Causalean.PO.POCFBundle P} {μ : MeasureTheory.Measure P.Ω}, P.IndepCF a B μ → ∀ (i : Fin B.n), ProbabilityTheory.IndepFun a.value (fun ω => B.jointValue ω i) μ","l":161,"f":"Causalean/PO/Assumptions/IndepCF.lean"},{"n":"Causalean.PO.POSystem.IndepCF.ofIndepFun","k":"theorem","m":"Causalean.PO.Assumptions.IndepCF","a":"PO/Assumptions","t":2,"d":"Bridge: IndepFun ⇒ IndepCF (reverse direction, also trivial).","s":"∀ {P : Causalean.PO.POSystem} {α : Type u_1} [inst : MeasurableSpace α] {a : Causalean.PO.RegimedVar P α} {B : Causalean.PO.POCFBundle P} {μ : MeasureTheory.Measure P.Ω}, ProbabilityTheory.IndepFun a.value B.jointValue μ → P.IndepCF a B μ","l":122,"f":"Causalean/PO/Assumptions/IndepCF.lean"},{"n":"Causalean.PO.POSystem.IndepCF.project","k":"theorem","m":"Causalean.PO.Assumptions.IndepCF","a":"PO/Assumptions","t":2,"d":"Projection: independence of a.value from any measurable function of the bundle's joint value follows from IndepCF.","s":"∀ {P : Causalean.PO.POSystem} {α : Type u_1} {β : Type u_2} [inst : MeasurableSpace α] [inst_1 : MeasurableSpace β] {a : Causalean.PO.RegimedVar P α} {B : Causalean.PO.POCFBundle P} {μ : MeasureTheory.Measure P.Ω} {ψ : ((i : Fin B.n) → B.type i) → β}, P.IndepCF a B μ → Measurable ψ → ProbabilityTheory.IndepFun a.value (ψ ∘ B.jointValue) μ","l":153,"f":"Causalean/PO/Assumptions/IndepCF.lean"},{"n":"Causalean.PO.POSystem.IndepCF.toIndepFun","k":"theorem","m":"Causalean.PO.Assumptions.IndepCF","a":"PO/Assumptions","t":2,"d":"Bridge: IndepCF is by definition IndepFun a.value B.jointValue μ.","s":"∀ {P : Causalean.PO.POSystem} {α : Type u_1} [inst : MeasurableSpace α] {a : Causalean.PO.RegimedVar P α} {B : Causalean.PO.POCFBundle P} {μ : MeasureTheory.Measure P.Ω}, P.IndepCF a B μ → ProbabilityTheory.IndepFun a.value B.jointValue μ","l":117,"f":"Causalean/PO/Assumptions/IndepCF.lean"},{"n":"Causalean.PO.POSystem.IntermediateAgrees","k":"def","m":"Causalean.PO.Assumptions.Consistency","a":"PO/Assumptions","t":1,"d":"For a potential-outcome system, two intervention regimes, and a unit in the sample space, intermediate agreement means that after applying the first regime, every variable targeted by the second regime has the value assigned to it by the second regime.","s":"(P : Causalean.PO.POSystem) → Causalean.PO.Regime P.V P.X → Causalean.PO.Regime P.V P.X → P.Ω → Prop","l":34,"f":"Causalean/PO/Assumptions/Consistency.lean"},{"n":"Causalean.PO.POSystem.component","k":"def","m":"Causalean.PO.Core.System","a":"PO/Core","t":1,"d":"A coordinate potential outcome maps each unit to the value a selected variable would take under a selected intervention regime.","s":"(P : Causalean.PO.POSystem) → Causalean.PO.Regime P.V P.X → (v : P.V) → P.Ω → P.X v","l":55,"f":"Causalean/PO/Core/System.lean"},{"n":"Causalean.PO.POSystem.condIndepCF_congr_cond","k":"theorem","m":"Causalean.PO.Assumptions.IndepCF","a":"PO/Assumptions","t":2,"d":"Transport CondIndepCF across equality of the conditioning variables' generated σ-algebras.","s":"∀ {P : Causalean.PO.POSystem} {α : Type u_1} {γ : Type u_2} {γ' : Type u_3} [inst : MeasurableSpace α] [inst_1 : MeasurableSpace γ] [inst_2 : MeasurableSpace γ'] [inst_3 : StandardBorelSpace P.Ω] {a : Causalean.PO.RegimedVar P α} {B : Causalean.PO.POCFBundle P} {c : Causalean.PO.RegimedVar P γ} {c' : Causalean.PO.RegimedVar P γ'} {μ : MeasureTheory.Measure P.Ω} [inst_4 : MeasureTheory.IsFiniteMeasure μ], MeasurableSpace.comap c.value inferInstance = MeasurableSpace.comap c'.value inferInstance → P.CondIndepCF a B c μ → P.CondIndepCF a B c' μ","l":139,"f":"Causalean/PO/Assumptions/IndepCF.lean"},{"n":"Causalean.PO.POSystem.counterfactualDist","k":"def","m":"Causalean.PO.Core.Counterfactual","a":"PO/Core","t":1,"d":"For a potential-outcome system and a finite list of counterfactual queries, the counterfactual distribution is the probability measure obtained by pushing the system's probability measure on the sample space through the cross-world evaluation map for those queries.","s":"(P : Causalean.PO.POSystem) → (qs : List (Causalean.PO.Regime P.V P.X × Finset P.V)) → MeasureTheory.Measure ((i : Fin qs.length) → Causalean.ValuesOn qs[i].2 P.X)","l":44,"f":"Causalean/PO/Core/Counterfactual.lean"},{"n":"Causalean.PO.POSystem.counterfactualDist_marginal","k":"theorem","m":"Causalean.PO.Core.Counterfactual","a":"PO/Core","t":1,"d":"For a finite list of counterfactual queries qs and an index i into that list, the i-th coordinate marginal of the joint counterfactual distribution over all queries equals the potential-outcome law for query i alone.","s":"∀ (P : Causalean.PO.POSystem) (qs : List (Causalean.PO.Regime P.V P.X × Finset P.V)) (i : Fin qs.length), MeasureTheory.Measure.map (fun f => f i) (P.counterfactualDist qs) = P.poOperator qs[i].1 qs[i].2","l":62,"f":"Causalean/PO/Core/Counterfactual.lean"},{"n":"Causalean.PO.POSystem.crossWorldEval","k":"def","m":"Causalean.PO.Core.Counterfactual","a":"PO/Core","t":1,"d":"For a potential-outcome system, a finite list of queries, and a unit in the sample space, this function returns the tuple whose $i$th component is the potential outcome, for the finite variable set in the $i$th query, under the intervention regime in the $i$th query.","s":"(P : Causalean.PO.POSystem) → (qs : List (Causalean.PO.Regime P.V P.X × Finset P.V)) → P.Ω → (i : Fin qs.length) → Causalean.ValuesOn qs[i].2 P.X","l":24,"f":"Causalean/PO/Core/Counterfactual.lean"},{"n":"Causalean.PO.POSystem.eventCondExp_eq_integral_of_IndepCF","k":"theorem","m":"Causalean.PO.Conditioning.EventCondExp","a":"PO/Conditioning","t":1,"d":"Drop-of-conditioning (quotient form). Suppose a regimed variable rv is independent of a counterfactual bundle B, where h is a measurable function of the bundle's joint value and {x} is a measurable singleton in the range of rv. If the event {rv = x} has positive and finite measure, then the event-level conditional expectation of h composed with the bundle's joint value, given {rv = x}, equals the unconditional integral of the same composite.","s":"∀ {P : Causalean.PO.POSystem} {α : Type u_2} [inst : MeasurableSpace α] {rv : Causalean.PO.RegimedVar P α} {B : Causalean.PO.POCFBundle P} {μ : MeasureTheory.Measure P.Ω}, P.IndepCF rv B μ → ∀ {h : ((i : Fin B.n) → B.type i) → ℝ}, Measurable h → ∀ {x : α}, MeasurableSet {x} → μ (rv.value ⁻¹' {x}) ≠ 0 → μ (rv.value ⁻¹' {x}) ≠ ⊤ → (Causalean.PO.eventCondExp μ (rv.value ⁻¹' {x}) fun ω => h (B.jointValue ω)) = ∫ (ω : P.Ω), h (B.jointValue ω) ∂μ","l":290,"f":"Causalean/PO/Conditioning/EventCondExp.lean"},{"n":"Causalean.PO.POSystem.eventCondExp_event_eq_integral_of_IndepCF","k":"theorem","m":"Causalean.PO.Conditioning.EventCondExp","a":"PO/Conditioning","t":2,"d":"Drop-of-conditioning on factual events (quotient form). Specialisation of the above to the factual event a.event x.","s":"∀ {P : Causalean.PO.POSystem} {α : Type u_2} [inst : MeasurableSpace α] {a : Causalean.PO.POVar P α} {B : Causalean.PO.POCFBundle P} {μ : MeasureTheory.Measure P.Ω}, P.IndepCF (Causalean.PO.RegimedVar.ofFactual a) B μ → ∀ {h : ((i : Fin B.n) → B.type i) → ℝ}, Measurable h → ∀ {x : α}, MeasurableSet {x} → μ (a.event x) ≠ 0 → μ (a.event x) ≠ ⊤ → (Causalean.PO.eventCondExp μ (a.event x) fun ω => h (B.jointValue ω)) = ∫ (ω : P.Ω), h (B.jointValue ω) ∂μ","l":317,"f":"Causalean/PO/Conditioning/EventCondExp.lean"},{"n":"Causalean.PO.POSystem.eventCondExp_of_consistency_IndepCF","k":"theorem","m":"Causalean.PO.Conditioning.EventCondExp","a":"PO/Conditioning","t":1,"d":"Drop-of-conditioning on an event, given a bundle relabeling of the integrand. Suppose a factual variable a is independent of a counterfactual bundle B, where h is a measurable function of the bundle's joint value and {x} is a measurable singleton in the range of a. If the factual integrand factualF agrees, almost everywhere on the event {a = x}, with h composed with the bundle's joint value, and that event has positive and finite measure, then the event-level conditional expectation of factualF given {a = x} equals the unconditional integral of h composed with the bundle's joint value.","s":"∀ {P : Causalean.PO.POSystem} {α : Type u_2} [inst : MeasurableSpace α] {a : Causalean.PO.POVar P α} {B : Causalean.PO.POCFBundle P} {μ : MeasureTheory.Measure P.Ω}, P.IndepCF (Causalean.PO.RegimedVar.ofFactual a) B μ → ∀ {factualF : P.Ω → ℝ} {h : ((i : Fin B.n) → B.type i) → ℝ}, Measurable h → ∀ {x : α}, MeasurableSet {x} → (factualF =ᵐ[μ.restrict (a.event x)] fun ω => h (B.jointValue ω)) → μ (a.event x) ≠ 0 → μ (a.event x) ≠ ⊤ → Causalean.PO.eventCondExp μ (a.event x) factualF = ∫ (ω : P.Ω), h (B.jointValue ω) ∂μ","l":357,"f":"Causalean/PO/Conditioning/EventCondExp.lean"},{"n":"Causalean.PO.POSystem.factualAgrees_empty","k":"theorem","m":"Causalean.PO.Assumptions.ConsistencyLemmas","a":"PO/Assumptions","t":2,"d":"FactualAgrees holds vacuously for the empty regime.","s":"∀ {P : Causalean.PO.POSystem} (ω : P.Ω), P.FactualAgrees Causalean.PO.Regime.empty ω","l":46,"f":"Causalean/PO/Assumptions/ConsistencyLemmas.lean"},{"n":"Causalean.PO.POSystem.factualAgrees_sqcup","k":"theorem","m":"Causalean.PO.Assumptions.ConsistencyLemmas","a":"PO/Assumptions","t":2,"d":"Combinator: FactualAgrees for a disjoint union reduces to FactualAgrees for each component.","s":"∀ {P : Causalean.PO.POSystem} {r₁ r₂ : Causalean.PO.Regime P.V P.X} (h : r₁.Disjoint r₂) {ω : P.Ω}, P.FactualAgrees r₁ ω → P.FactualAgrees r₂ ω → P.FactualAgrees (r₁.sqcup r₂ h) ω","l":51,"f":"Causalean/PO/Assumptions/ConsistencyLemmas.lean"},{"n":"Causalean.PO.POSystem.instIsProbabilityMeasureForallValuesOnVSndRegimeXFinsetGetElemListProdFinLengthLtNatValCounterfactualDist","k":"instance","m":"Causalean.PO.Core.Counterfactual","a":"PO/Core","t":2,"d":"The finite cross-world counterfactual distribution is a probability measure.","s":"∀ (P : Causalean.PO.POSystem) (qs : List (Causalean.PO.Regime P.V P.X × Finset P.V)), MeasureTheory.IsProbabilityMeasure (P.counterfactualDist qs)","l":55,"f":"Causalean/PO/Core/Counterfactual.lean"},{"n":"Causalean.PO.POSystem.instIsProbabilityMeasureValuesOnVXPoOperator","k":"instance","m":"Causalean.PO.Core.System","a":"PO/Core","t":2,"d":"The potential-outcome law of a finite set of variables under a regime is a probability measure.","s":"∀ (P : Causalean.PO.POSystem) (r : Causalean.PO.Regime P.V P.X) (Y : Finset P.V), MeasureTheory.IsProbabilityMeasure (P.poOperator r Y)","l":93,"f":"Causalean/PO/Core/System.lean"},{"n":"Causalean.PO.POSystem.integral_event_eq_mul_of_IndepCF","k":"theorem","m":"Causalean.PO.Conditioning.EventCondExp","a":"PO/Conditioning","t":2,"d":"Drop-of-conditioning for the factual event. Specialisation of POSystem.integral_restrict_value_eq_mul_of_IndepCF to a factual POVar, using POVar.event directly.","s":"∀ {P : Causalean.PO.POSystem} {α : Type u_2} [inst : MeasurableSpace α] {a : Causalean.PO.POVar P α} {B : Causalean.PO.POCFBundle P} {μ : MeasureTheory.Measure P.Ω}, P.IndepCF (Causalean.PO.RegimedVar.ofFactual a) B μ → ∀ {h : ((i : Fin B.n) → B.type i) → ℝ}, Measurable h → ∀ (x : α), MeasurableSet {x} → ∫ (ω : P.Ω) in a.event x, h (B.jointValue ω) ∂μ = (μ (a.event x)).toReal * ∫ (ω : P.Ω), h (B.jointValue ω) ∂μ","l":269,"f":"Causalean/PO/Conditioning/EventCondExp.lean"},{"n":"Causalean.PO.POSystem.integral_restrict_value_eq_mul_of_IndepCF","k":"theorem","m":"Causalean.PO.Conditioning.EventCondExp","a":"PO/Conditioning","t":2,"d":"Drop-of-conditioning (multiplied form). If a regimed variable rv is independent of the counterfactual bundle B, then the integral of h ∘ B.jointValue over the preimage rv.value ⁻¹' {x} factors as (μ (rv.value ⁻¹' {x})).toReal * ∫ h ∘ B.jointValue ∂μ.","s":"∀ {P : Causalean.PO.POSystem} {α : Type u_2} [inst : MeasurableSpace α] {rv : Causalean.PO.RegimedVar P α} {B : Causalean.PO.POCFBundle P} {μ : MeasureTheory.Measure P.Ω}, P.IndepCF rv B μ → ∀ {h : ((i : Fin B.n) → B.type i) → ℝ}, Measurable h → ∀ (x : α), MeasurableSet {x} → ∫ (ω : P.Ω) in rv.value ⁻¹' {x}, h (B.jointValue ω) ∂μ = (μ (rv.value ⁻¹' {x})).toReal * ∫ (ω : P.Ω), h (B.jointValue ω) ∂μ","l":247,"f":"Causalean/PO/Conditioning/EventCondExp.lean"},{"n":"Causalean.PO.POSystem.liftRegime","k":"def","m":"Causalean.PO.Bridge.Induce","a":"PO/Bridge","t":1,"d":"For a potential-outcome system, a finite set of variables, and an intervention regime whose variables all lie in that finite set, this construction returns the corresponding intervention regime on the original system by viewing each restricted variable as a variable of the original system.","s":"(P : Causalean.PO.POSystem) → (R : Finset P.V) → (Causalean.PO.Regime ↥R fun v => P.X ↑v) → Causalean.PO.Regime P.V P.X","l":28,"f":"Causalean/PO/Bridge/Induce.lean"},{"n":"Causalean.PO.POSystem.liftRegime_assign","k":"theorem","m":"Causalean.PO.Bridge.Induce","a":"PO/Bridge","t":2,"d":"Reading the lifted assignment at a known sub-regime member returns the original sub-assignment. Membership proofs are propositional, so the ambient hv is irrelevant.","s":"∀ (P : Causalean.PO.POSystem) (R : Finset P.V) (r' : Causalean.PO.Regime ↥R fun v => P.X ↑v) (w : ↥R) (hw : w ∈ r'.target) (hv : ↑w ∈ (P.liftRegime R r').target), (P.liftRegime R r').assign (↑w) hv = r'.assign w hw","l":60,"f":"Causalean/PO/Bridge/Induce.lean"},{"n":"Causalean.PO.POSystem.liftRegime_disjoint","k":"theorem","m":"Causalean.PO.Bridge.Induce","a":"PO/Bridge","t":2,"d":"Disjoint restricted regimes remain disjoint after lifting them to the original system.","s":"∀ (P : Causalean.PO.POSystem) (R : Finset P.V) {r₁' r₂' : Causalean.PO.Regime ↥R fun v => P.X ↑v}, r₁'.Disjoint r₂' → (P.liftRegime R r₁').Disjoint (P.liftRegime R r₂')","l":128,"f":"Causalean/PO/Bridge/Induce.lean"},{"n":"Causalean.PO.POSystem.liftRegime_empty","k":"theorem","m":"Causalean.PO.Bridge.Induce","a":"PO/Bridge","t":2,"d":"Lifting the empty restricted regime gives the empty regime on the original system.","s":"∀ (P : Causalean.PO.POSystem) (R : Finset P.V), P.liftRegime R Causalean.PO.Regime.empty = Causalean.PO.Regime.empty","l":71,"f":"Causalean/PO/Bridge/Induce.lean"},{"n":"Causalean.PO.POSystem.liftRegime_sqcup","k":"theorem","m":"Causalean.PO.Bridge.Induce","a":"PO/Bridge","t":2,"d":"The lift commutes with Regime.sqcup (full equality).","s":"∀ (P : Causalean.PO.POSystem) (R : Finset P.V) {r₁' r₂' : Causalean.PO.Regime ↥R fun v => P.X ↑v} (h : r₁'.Disjoint r₂'), P.liftRegime R (r₁'.sqcup r₂' h) = (P.liftRegime R r₁').sqcup (P.liftRegime R r₂') (Causalean.PO.POSystem.liftRegime_disjoint P R h)","l":150,"f":"Causalean/PO/Bridge/Induce.lean"},{"n":"Causalean.PO.POSystem.liftRegime_sqcup_target","k":"theorem","m":"Causalean.PO.Bridge.Induce","a":"PO/Bridge","t":2,"d":"The lift commutes with Regime.sqcup (target equality).","s":"∀ (P : Causalean.PO.POSystem) (R : Finset P.V) {r₁' r₂' : Causalean.PO.Regime ↥R fun v => P.X ↑v} (h : r₁'.Disjoint r₂'), (P.liftRegime R (r₁'.sqcup r₂' h)).target = (P.liftRegime R r₁').target ∪ (P.liftRegime R r₂').target","l":142,"f":"Causalean/PO/Bridge/Induce.lean"},{"n":"Causalean.PO.POSystem.liftRegime_target","k":"theorem","m":"Causalean.PO.Bridge.Induce","a":"PO/Bridge","t":2,"d":"The target of a lifted restricted regime is the image of the restricted target in the original variable set.","s":"∀ (P : Causalean.PO.POSystem) (R : Finset P.V) (r' : Causalean.PO.Regime ↥R fun v => P.X ↑v), (P.liftRegime R r').target = Finset.map { toFun := Subtype.val, inj' := Subtype.val_injective } r'.target","l":53,"f":"Causalean/PO/Bridge/Induce.lean"},{"n":"Causalean.PO.POSystem.measurable_component","k":"theorem","m":"Causalean.PO.Core.System","a":"PO/Core","t":1,"d":"For an intervention regime r and a variable v, the coordinate potential outcome of v under r is a measurable function of the unit.","s":"∀ (P : Causalean.PO.POSystem) (r : Causalean.PO.Regime P.V P.X) (v : P.V), Measurable (P.component r v)","l":62,"f":"Causalean/PO/Core/System.lean"},{"n":"Causalean.PO.POSystem.measurable_crossWorldEval","k":"theorem","m":"Causalean.PO.Core.Counterfactual","a":"PO/Core","t":2,"d":"The cross-world evaluation map for a finite list of counterfactual queries is measurable.","s":"∀ (P : Causalean.PO.POSystem) (qs : List (Causalean.PO.Regime P.V P.X × Finset P.V)), Measurable (P.crossWorldEval qs)","l":36,"f":"Causalean/PO/Core/Counterfactual.lean"},{"n":"Causalean.PO.POSystem.measurable_poVariable","k":"theorem","m":"Causalean.PO.Core.System","a":"PO/Core","t":2,"d":"The joint potential outcome for any finite set of variables under any intervention regime is measurable.","s":"∀ (P : Causalean.PO.POSystem) (r : Causalean.PO.Regime P.V P.X) (Y : Finset P.V), Measurable (P.poVariable r Y)","l":77,"f":"Causalean/PO/Core/System.lean"},{"n":"Causalean.PO.POSystem.ofSCM","k":"def","m":"Causalean.PO.Bridge.FromSCM","a":"PO/Bridge","t":1,"d":"A structural model and a background assignment induce a potential-outcome system whose variables are all observed random nodes of the model.","s":"{N : Type uN} → [inst : DecidableEq N] → [inst_1 : Fintype N] → {Ω : N → Type uΩ} → [inst_2 : (n : N) → MeasurableSpace (Ω n)] → (M : Causalean.SCM N Ω) → M.FixedValues → Causalean.PO.POSystem","l":246,"f":"Causalean/PO/Bridge/FromSCM.lean"},{"n":"Causalean.PO.POSystem.ofSCM_condIndepCF_of_dSep","k":"theorem","m":"Causalean.PO.Bridge.FromSCMCondIndep","a":"PO/Bridge","t":1,"d":"SCM-to-PO conditional-independence bridge under d-separation. Fix a structural causal model M with a fixed-value assignment s, and SWIG node sets that are each contained in the model's random variables and pairwise disjoint from one another. Suppose the first node set is d-separated from the second by the third in the model's DAG. Suppose further that there is a measurable value-space map aMap for the first node set and a measurable value-space map BMap for the second node set such that, under the latent draw, the potential-outcome value of a regimed variable a equals aMap applied to the projection of the evaluated model state onto the first node set, the joint value of a counterfactual bundle B equals BMap applied to the projection onto the second node set, and the conditioning value of a regimed variable c equals the projection onto the third node set. Then, in the potential-outcome system induced by M at s, a and B are conditionally independent given c.","s":"∀ {N : Type uN} [inst : DecidableEq N] [inst_1 : Fintype N] {Ω : N → Type uΩ} [inst_2 : (n : N) → MeasurableSpace (Ω n)] (M : Causalean.SCM N Ω) (s : M.FixedValues) [StandardBorelSpace M.RandomValues] [StandardBorelSpace M.LatentValues] [inst_5 : StandardBorelSpace (Causalean.PO.POSystem.ofSCM M s).Ω] [∀ (n : Causalean.SWIGNode N), StandardBorelSpace (Causalean.swigΩ Ω n)] [∀ (n : Causalean.SWIGNode N), Nonempty (Causalean.swigΩ Ω n)] [∀ (s' : M.FixedValues), MeasureTheory.IsFiniteMeasure (M.jointKernel s')] {X Y Z : Finset (Causalean.SWIGNode N)} (hX : X ⊆ M.randomVars) (hY : Y ⊆ M.randomVars) (hZ : Z ⊆ M.randomVars), Disjoint X Y → Disjoint X Z → Disjoint Y Z → M.dag.dSep X Y Z → ∀ {α : Type u_1} [inst_9 : MeasurableSpace α] [StandardBorelSpace α] [Nonempty α] (a : Causalean.PO.RegimedVar (Causalean.PO.POSystem.ofSCM M s) α) (B : Causalean.PO.POCFBundle (Causalean.PO.POSystem.ofSCM M s)) [StandardBorelSpace ((i : Fin B.n) → B.type i)] [Nonempty ((i : Fin B.n) → B.type i)] (c : Causalean.PO.RegimedVar (Causalean.PO.POSystem.ofSCM M s) (Causalean.ValuesOn Z (Causalean.swigΩ Ω))) (aMap : Causalean.ValuesOn X (Causalean.swigΩ Ω) → α) (BMap : Causalean.ValuesOn Y (Causalean.swigΩ Ω) → (i : Fin B.n) → B.type i), Measurable aMap → Measurable BMap → (a.value = aMap ∘ Causalean.valuesProjection hX ∘ fun ℓ => M.evalMap s ℓ) → (B.jointValue = BMap ∘ Causalean.valuesProjection hY ∘ fun ℓ => M.evalMap s ℓ) → (c.value = Causalean.valuesProjection hZ ∘ fun ℓ => M.evalMap s ℓ) → (Causalean.PO.POSystem.ofSCM M s).CondIndepCF a B c (Causalean.PO.POSystem.ofSCM M s).μ","l":91,"f":"Causalean/PO/Bridge/FromSCMCondIndep.lean"},{"n":"Causalean.PO.POSystem.ofSCM_consistency","k":"theorem","m":"Causalean.PO.Bridge.FromSCM","a":"PO/Bridge","t":1,"d":"For a structural causal model M and an assignment of values to its fixed background variables s, the potential-outcome system induced by M and s satisfies the consistency assumption.","s":"∀ {N : Type uN} [inst : DecidableEq N] [inst_1 : Fintype N] {Ω : N → Type uΩ} [inst_2 : (n : N) → MeasurableSpace (Ω n)] (M : Causalean.SCM N Ω) (s : M.FixedValues), (Causalean.PO.POSystem.ofSCM M s).Consistency","l":398,"f":"Causalean/PO/Bridge/FromSCM.lean"},{"n":"Causalean.PO.POSystem.poOperator","k":"def","m":"Causalean.PO.Core.System","a":"PO/Core","t":1,"d":"A potential-outcome law is the distribution of a selected finite set of variables under a selected intervention regime.","s":"(P : Causalean.PO.POSystem) → Causalean.PO.Regime P.V P.X → (Y : Finset P.V) → MeasureTheory.Measure (Causalean.ValuesOn Y P.X)","l":85,"f":"Causalean/PO/Core/System.lean"},{"n":"Causalean.PO.POSystem.poVariable","k":"def","m":"Causalean.PO.Core.System","a":"PO/Core","t":1,"d":"A joint potential outcome maps each unit to the vector of values a selected finite set of variables would take under a selected intervention regime.","s":"(P : Causalean.PO.POSystem) → Causalean.PO.Regime P.V P.X → (Y : Finset P.V) → P.Ω → Causalean.ValuesOn Y P.X","l":69,"f":"Causalean/PO/Core/System.lean"},{"n":"Causalean.PO.POSystem.restrict","k":"def","m":"Causalean.PO.Bridge.Induce","a":"PO/Bridge","t":1,"d":"For a potential-outcome system and a finite set of variables, the restricted potential-outcome system has that finite set as its variable collection, keeps the same sample space and probability measure, and evaluates restricted intervention regimes by first lifting them back to intervention regimes of the original system.","s":"(P : Causalean.PO.POSystem) → Finset P.V → Causalean.PO.POSystem","l":81,"f":"Causalean/PO/Bridge/Induce.lean"},{"n":"Causalean.PO.POSystem.restrict_V","k":"theorem","m":"Causalean.PO.Bridge.Induce","a":"PO/Bridge","t":2,"d":"The variable type of the restricted potential-outcome system is the chosen finite set of variables.","s":"∀ (P : Causalean.PO.POSystem) (R : Finset P.V), (P.restrict R).V = ↥R","l":100,"f":"Causalean/PO/Bridge/Induce.lean"},{"n":"Causalean.PO.POSystem.restrict_X","k":"theorem","m":"Causalean.PO.Bridge.Induce","a":"PO/Bridge","t":2,"d":"The value space in the restricted system is the original value space at the underlying variable.","s":"∀ (P : Causalean.PO.POSystem) (R : Finset P.V) (v : ↥R), (P.restrict R).X v = P.X ↑v","l":103,"f":"Causalean/PO/Bridge/Induce.lean"},{"n":"Causalean.PO.POSystem.restrict_component","k":"theorem","m":"Causalean.PO.Bridge.Induce","a":"PO/Bridge","t":2,"d":"A coordinate potential outcome in the restricted system agrees with the original coordinate after lifting the regime.","s":"∀ (P : Causalean.PO.POSystem) (R : Finset P.V) (r' : Causalean.PO.Regime ↥R fun v => P.X ↑v) (v : ↥R), (P.restrict R).component r' v = P.component (P.liftRegime R r') ↑v","l":119,"f":"Causalean/PO/Bridge/Induce.lean"},{"n":"Causalean.PO.POSystem.restrict_consistency","k":"theorem","m":"Causalean.PO.Bridge.Induce","a":"PO/Bridge","t":1,"d":"Restriction preserves consistency. If the ambient potential-outcome system P is consistent (SUTVA holds), then the system P restricted to R is consistent as well.","s":"∀ (P : Causalean.PO.POSystem) (R : Finset P.V), P.Consistency → (P.restrict R).Consistency","l":203,"f":"Causalean/PO/Bridge/Induce.lean"},{"n":"Causalean.PO.POSystem.restrict_eval","k":"theorem","m":"Causalean.PO.Bridge.Induce","a":"PO/Bridge","t":2,"d":"Evaluation in the restricted system is evaluation in the original system after lifting the restricted regime.","s":"∀ (P : Causalean.PO.POSystem) (R : Finset P.V) (r' : Causalean.PO.Regime ↥R fun v => P.X ↑v) (ω : P.Ω) (v : ↥R), (P.restrict R).eval r' ω v = P.eval (P.liftRegime R r') ω ↑v","l":112,"f":"Causalean/PO/Bridge/Induce.lean"},{"n":"Causalean.PO.POSystem.restrict_Ω","k":"theorem","m":"Causalean.PO.Bridge.Induce","a":"PO/Bridge","t":2,"d":"The restricted potential-outcome system uses the same sample space as the original system.","s":"∀ (P : Causalean.PO.POSystem) (R : Finset P.V), (P.restrict R).Ω = P.Ω","l":106,"f":"Causalean/PO/Bridge/Induce.lean"},{"n":"Causalean.PO.POSystem.restrict_μ","k":"theorem","m":"Causalean.PO.Bridge.Induce","a":"PO/Bridge","t":2,"d":"The restricted potential-outcome system uses the same probability measure as the original system.","s":"∀ (P : Causalean.PO.POSystem) (R : Finset P.V), (P.restrict R).μ = P.μ","l":108,"f":"Causalean/PO/Bridge/Induce.lean"},{"n":"Causalean.PO.POVar","k":"structure","m":"Causalean.PO.Core.Variable","a":"PO/Core","t":1,"d":"A potential-outcome variable records a system variable together with a measurable relabeling of its native value space into a chosen analysis scale α.","s":"Causalean.PO.POSystem → (α : Type u_1) → [MeasurableSpace α] → Type (max (max u_1 u_2) u_3)","l":28,"f":"Causalean/PO/Core/Variable.lean"},{"n":"Causalean.PO.POVar.cf","k":"def","m":"Causalean.PO.Core.Variable","a":"PO/Core","t":1,"d":"A counterfactual value function assigns each unit the value that a selected variable would have under a selected intervention regime, reported on its chosen analysis scale.","s":"{P : Causalean.PO.POSystem} → {α : Type u_1} → [inst : MeasurableSpace α] → Causalean.PO.POVar P α → Causalean.PO.Regime P.V P.X → P.Ω → α","l":44,"f":"Causalean/PO/Core/Variable.lean"},{"n":"Causalean.PO.POVar.cfCDF","k":"def","m":"Causalean.PO.Analysis.Quantile","a":"PO/Analysis","t":1,"d":"For a real-valued potential-outcome variable, an intervention regime, and a measure on the sample space, this is the cumulative distribution function of the variable's potential outcome under that regime with respect to that measure.","s":"{P : Causalean.PO.POSystem} → Causalean.PO.POVar P ℝ → Causalean.PO.Regime P.V P.X → MeasureTheory.Measure P.Ω → StieltjesFunction ℝ","l":45,"f":"Causalean/PO/Analysis/Quantile.lean"},{"n":"Causalean.PO.POVar.cfCDF_eq_measureReal","k":"theorem","m":"Causalean.PO.Analysis.Quantile","a":"PO/Analysis","t":1,"d":"For an intervention regime r and a measure μ on the sample space, the cdf of the potential outcome a(r) evaluated at y equals the μ-probability of the event a(r) ≤ y.","s":"∀ {P : Causalean.PO.POSystem} (a : Causalean.PO.POVar P ℝ) (r : Causalean.PO.Regime P.V P.X) (μ : MeasureTheory.Measure P.Ω) [MeasureTheory.IsProbabilityMeasure μ] (y : ℝ), ↑(a.cfCDF r μ) y = (a.cfLaw r μ).real (Set.Iic y)","l":63,"f":"Causalean/PO/Analysis/Quantile.lean"},{"n":"Causalean.PO.POVar.cfLaw","k":"def","m":"Causalean.PO.Analysis.Quantile","a":"PO/Analysis","t":1,"d":"For a real-valued potential-outcome variable, an intervention regime, and a measure on the sample space, this is the distribution of the variable's potential outcome under the regime, obtained by pushing that measure forward through the counterfactual-value function.","s":"{P : Causalean.PO.POSystem} → Causalean.PO.POVar P ℝ → Causalean.PO.Regime P.V P.X → MeasureTheory.Measure P.Ω → MeasureTheory.Measure ℝ","l":29,"f":"Causalean/PO/Analysis/Quantile.lean"},{"n":"Causalean.PO.POVar.cfQuantile","k":"def","m":"Causalean.PO.Analysis.Quantile","a":"PO/Analysis","t":1,"d":"For a real-valued potential-outcome variable, an intervention regime, a measure on the sample space, and a real index $\\tau$, this is the $\\tau$-quantile of the variable's potential outcome under that regime with respect to that measure.","s":"{P : Causalean.PO.POSystem} → Causalean.PO.POVar P ℝ → Causalean.PO.Regime P.V P.X → MeasureTheory.Measure P.Ω → ℝ → ℝ","l":54,"f":"Causalean/PO/Analysis/Quantile.lean"},{"n":"Causalean.PO.POVar.cfUnder","k":"def","m":"Causalean.PO.Core.Variable","a":"PO/Core","t":1,"d":"The potential outcome of variable y when the intervention variable w is set to the value d: the single-variable counterfactual y(w := d). (Binders follow the usual econometric convention — y is the outcome variable and d is the treatment/intervention value; w is the variable being intervened on.)","s":"{P : Causalean.PO.POSystem} → {α : Type u_1} → [inst : MeasurableSpace α] → {β : Type u_2} → [inst_1 : MeasurableSpace β] → Causalean.PO.POVar P α → Causalean.PO.POVar P β → β → P.Ω → α","l":80,"f":"Causalean/PO/Core/Variable.lean"},{"n":"Causalean.PO.POVar.cfUnderLaw","k":"def","m":"Causalean.PO.Analysis.Quantile","a":"PO/Analysis","t":1,"d":"For a real-valued potential-outcome variable, another potential-outcome variable, a value of the second variable, and a measure on the sample space, this is the distribution of the first variable's potential outcome under the single-variable intervention that sets the second variable to that value.","s":"{P : Causalean.PO.POSystem} → Causalean.PO.POVar P ℝ → {β : Type u_1} → [inst : MeasurableSpace β] → Causalean.PO.POVar P β → β → MeasureTheory.Measure P.Ω → MeasureTheory.Measure ℝ","l":75,"f":"Causalean/PO/Analysis/Quantile.lean"},{"n":"Causalean.PO.POVar.cfUnderQuantile","k":"def","m":"Causalean.PO.Analysis.Quantile","a":"PO/Analysis","t":1,"d":"For a real-valued potential-outcome variable, another potential-outcome variable, a value of the second variable, a measure on the sample space, and a real index $\\tau$, this is the $\\tau$-quantile of the first variable's potential outcome under the single-variable intervention that sets the second variable to that value.","s":"{P : Causalean.PO.POSystem} → Causalean.PO.POVar P ℝ → {β : Type u_1} → [inst : MeasurableSpace β] → Causalean.PO.POVar P β → β → MeasureTheory.Measure P.Ω → ℝ → ℝ","l":84,"f":"Causalean/PO/Analysis/Quantile.lean"},{"n":"Causalean.PO.POVar.cfUnderQuantile_eq","k":"theorem","m":"Causalean.PO.Analysis.Quantile","a":"PO/Analysis","t":2,"d":"The single-intervention quantile is exactly the quantile computed from the corresponding single-intervention law.","s":"∀ {P : Causalean.PO.POSystem} (a : Causalean.PO.POVar P ℝ) {β : Type u_1} [inst : MeasurableSpace β] (w : Causalean.PO.POVar P β) (y : β) (μ : MeasureTheory.Measure P.Ω) (τ : ℝ), a.cfUnderQuantile w y μ τ = Causalean.Stat.quantile (a.cfUnderLaw w y μ) τ","l":95,"f":"Causalean/PO/Analysis/Quantile.lean"},{"n":"Causalean.PO.POVar.cf_eq_factual_of_factualAgrees","k":"theorem","m":"Causalean.PO.Assumptions.ConsistencyLemmas","a":"PO/Assumptions","t":1,"d":"Multi-target consistency. Under the consistency (SUTVA) assumption on the potential-outcome system, for a regimed variable a whose index does not lie in the target of a regime r, if the outcome ω factually agrees with the regime r, then the counterfactual value of a under r at ω equals its factual value at ω.","s":"∀ {P : Causalean.PO.POSystem} {α : Type u_1} [inst : MeasurableSpace α], P.Consistency → ∀ (a : Causalean.PO.POVar P α) (r : Causalean.PO.Regime P.V P.X), a.v ∉ r.target → ∀ (ω : P.Ω), P.FactualAgrees r ω → a.cf r ω = a.factual ω","l":201,"f":"Causalean/PO/Assumptions/ConsistencyLemmas.lean"},{"n":"Causalean.PO.POVar.cf_eq_factual_on_event","k":"theorem","m":"Causalean.PO.Assumptions.ConsistencyLemmas","a":"PO/Assumptions","t":2,"d":"Under consistency, changing a distinct variable to a value it already has does not change the counterfactual value of the target variable on that event.","s":"∀ {P : Causalean.PO.POSystem} {α : Type u_1} {β : Type u_2} [inst : MeasurableSpace α] [inst_1 : MeasurableSpace β], P.Consistency → ∀ (a : Causalean.PO.POVar P α) (w : Causalean.PO.POVar P β) (y : β), a.v ≠ w.v → ∀ {ω : P.Ω}, ω ∈ w.event y → a.cfUnder w y ω = a.factual ω","l":99,"f":"Causalean/PO/Assumptions/ConsistencyLemmas.lean"},{"n":"Causalean.PO.POVar.comap_factual_le","k":"theorem","m":"Causalean.PO.Conditioning.CondExpTooling","a":"PO/Conditioning","t":2,"d":"The conditioning σ-algebra σ(c.factual) is a sub-σ-algebra of the ambient σ-algebra on P.Ω.","s":"∀ {P : Causalean.PO.POSystem} {γ : Type u_1} [inst : MeasurableSpace γ] (c : Causalean.PO.POVar P γ), MeasurableSpace.comap c.factual inferInstance ≤ inferInstance","l":42,"f":"Causalean/PO/Conditioning/CondExpTooling.lean"},{"n":"Causalean.PO.POVar.condExpGiven","k":"def","m":"Causalean.PO.Conditioning.CondExpTooling","a":"PO/Conditioning","t":1,"d":"For a conditioning potential-outcome variable, a real-valued function on the sample space, and a measure on the sample space, this is the conditional expectation of that function given the σ-algebra generated by the factual value of the conditioning variable.","s":"{P : Causalean.PO.POSystem} → {γ : Type u_1} → [inst : MeasurableSpace γ] → Causalean.PO.POVar P γ → (P.Ω → ℝ) → optParam (MeasureTheory.Measure P.Ω) P.μ → P.Ω → ℝ","l":30,"f":"Causalean/PO/Conditioning/CondExpTooling.lean"},{"n":"Causalean.PO.POVar.condExpGiven_add","k":"theorem","m":"Causalean.PO.Conditioning.CondExpTooling","a":"PO/Conditioning","t":2,"d":"Linearity of condExpGiven: additivity.","s":"∀ {P : Causalean.PO.POSystem} {γ : Type u_1} [inst : MeasurableSpace γ] (c : Causalean.PO.POVar P γ) {f g : P.Ω → ℝ} {μ : MeasureTheory.Measure P.Ω}, MeasureTheory.Integrable f μ → MeasureTheory.Integrable g μ → c.condExpGiven (f + g) μ =ᵐ[μ] c.condExpGiven f μ + c.condExpGiven g μ","l":65,"f":"Causalean/PO/Conditioning/CondExpTooling.lean"},{"n":"Causalean.PO.POVar.condExpGiven_indicator_mul","k":"theorem","m":"Causalean.PO.Conditioning.CondExpTooling","a":"PO/Conditioning","t":2,"d":"Pull-out version specialised to indicator functions of sets measurable w.r.t. σ(c.factual). If the event is measurable with respect to the conditioning variable's factual σ-algebra, its indicator factors out of the conditional expectation just like any other strongly measurable multiplier.","s":"∀ {P : Causalean.PO.POSystem} {γ : Type u_1} [inst : MeasurableSpace γ] (c : Causalean.PO.POVar P γ) {s : Set P.Ω} {g : P.Ω → ℝ} {μ : MeasureTheory.Measure P.Ω}, MeasurableSet s → MeasureTheory.Integrable g μ → c.condExpGiven ((s.indicator fun x => 1) * g) μ =ᵐ[μ] (s.indicator fun x => 1) * c.condExpGiven g μ","l":130,"f":"Causalean/PO/Conditioning/CondExpTooling.lean"},{"n":"Causalean.PO.POVar.condExpGiven_mul_of_stronglyMeasurable_left","k":"theorem","m":"Causalean.PO.Conditioning.CondExpTooling","a":"PO/Conditioning","t":2,"d":"Pull-out-left for condExpGiven: if f is strongly measurable w.r.t. σ(c.factual), then μ[f·g | σ(c.factual)] =ᵐ f · μ[g | σ(c.factual)].","s":"∀ {P : Causalean.PO.POSystem} {γ : Type u_1} [inst : MeasurableSpace γ] (c : Causalean.PO.POVar P γ) {f g : P.Ω → ℝ} {μ : MeasureTheory.Measure P.Ω}, MeasureTheory.StronglyMeasurable f → MeasureTheory.Integrable (f * g) μ → MeasureTheory.Integrable g μ → c.condExpGiven (f * g) μ =ᵐ[μ] f * c.condExpGiven g μ","l":109,"f":"Causalean/PO/Conditioning/CondExpTooling.lean"},{"n":"Causalean.PO.POVar.condExpGiven_mul_of_stronglyMeasurable_right","k":"theorem","m":"Causalean.PO.Conditioning.CondExpTooling","a":"PO/Conditioning","t":2,"d":"Pull-out-right for condExpGiven.","s":"∀ {P : Causalean.PO.POSystem} {γ : Type u_1} [inst : MeasurableSpace γ] (c : Causalean.PO.POVar P γ) {f g : P.Ω → ℝ} {μ : MeasureTheory.Measure P.Ω}, MeasureTheory.StronglyMeasurable g → MeasureTheory.Integrable (f * g) μ → MeasureTheory.Integrable f μ → c.condExpGiven (f * g) μ =ᵐ[μ] c.condExpGiven f μ * g","l":120,"f":"Causalean/PO/Conditioning/CondExpTooling.lean"},{"n":"Causalean.PO.POVar.condExpGiven_smul","k":"theorem","m":"Causalean.PO.Conditioning.CondExpTooling","a":"PO/Conditioning","t":2,"d":"Scalar homogeneity of condExpGiven.","s":"∀ {P : Causalean.PO.POSystem} {γ : Type u_1} [inst : MeasurableSpace γ] (c : Causalean.PO.POVar P γ) (k : ℝ) (g : P.Ω → ℝ) {μ : MeasureTheory.Measure P.Ω}, c.condExpGiven (k • g) μ =ᵐ[μ] k • c.condExpGiven g μ","l":79,"f":"Causalean/PO/Conditioning/CondExpTooling.lean"},{"n":"Causalean.PO.POVar.condExpGiven_sub","k":"theorem","m":"Causalean.PO.Conditioning.CondExpTooling","a":"PO/Conditioning","t":2,"d":"Linearity of condExpGiven: subtraction.","s":"∀ {P : Causalean.PO.POSystem} {γ : Type u_1} [inst : MeasurableSpace γ] (c : Causalean.PO.POVar P γ) {f g : P.Ω → ℝ} {μ : MeasureTheory.Measure P.Ω}, MeasureTheory.Integrable f μ → MeasureTheory.Integrable g μ → c.condExpGiven (f - g) μ =ᵐ[μ] c.condExpGiven f μ - c.condExpGiven g μ","l":72,"f":"Causalean/PO/Conditioning/CondExpTooling.lean"},{"n":"Causalean.PO.POVar.condExpGiven_tower","k":"theorem","m":"Causalean.PO.Conditioning.CondExpTooling","a":"PO/Conditioning","t":2,"d":"Tower property between two POVars: if σ(c₁.factual) ≤ σ(c₂.factual), then μ[μ[g|σ(c₂)] | σ(c₁)] =ᵐ μ[g|σ(c₁)].","s":"∀ {P : Causalean.PO.POSystem} {γ : Type u_1} [inst : MeasurableSpace γ] {δ : Type u_2} [inst_1 : MeasurableSpace δ] (c₁ : Causalean.PO.POVar P γ) (c₂ : Causalean.PO.POVar P δ) {g : P.Ω → ℝ} {μ : MeasureTheory.Measure P.Ω}, MeasurableSpace.comap c₁.factual inferInstance ≤ MeasurableSpace.comap c₂.factual inferInstance → ∀ [MeasureTheory.SigmaFinite (μ.trim (Causalean.PO.POVar.comap_factual_le c₂))], c₁.condExpGiven (c₂.condExpGiven g μ) μ =ᵐ[μ] c₁.condExpGiven g μ","l":163,"f":"Causalean/PO/Conditioning/CondExpTooling.lean"},{"n":"Causalean.PO.POVar.condExpGiven_tower_of_le","k":"theorem","m":"Causalean.PO.Conditioning.CondExpTooling","a":"PO/Conditioning","t":2,"d":"Tower property over an arbitrary smaller sub-σ-algebra m ≤ σ(c.factual). Requires the usual SigmaFinite hypothesis on the trim of μ to σ(c.factual).","s":"∀ {P : Causalean.PO.POSystem} {γ : Type u_1} [inst : MeasurableSpace γ] (c : Causalean.PO.POVar P γ) {g : P.Ω → ℝ} {μ : MeasureTheory.Measure P.Ω} {m : MeasurableSpace P.Ω}, m ≤ MeasurableSpace.comap c.factual inferInstance → ∀ [MeasureTheory.SigmaFinite (μ.trim (Causalean.PO.POVar.comap_factual_le c))], μ[c.condExpGiven g μ | m] =ᵐ[μ] μ[g | m]","l":153,"f":"Causalean/PO/Conditioning/CondExpTooling.lean"},{"n":"Causalean.PO.POVar.condExpRatio","k":"def","m":"Causalean.PO.Conditioning.CondExpTooling","a":"PO/Conditioning","t":1,"d":"For a conditioning potential-outcome variable, two real-valued functions on the sample space, and a measure on the sample space, this is the pointwise ratio of the conditional expectation of the first function to the conditional expectation of the second function, both given the σ-algebra generated by the factual value of the conditioning variable.","s":"{P : Causalean.PO.POSystem} → {γ : Type u_1} → [inst : MeasurableSpace γ] → Causalean.PO.POVar P γ → (P.Ω → ℝ) → (P.Ω → ℝ) → optParam (MeasureTheory.Measure P.Ω) P.μ → P.Ω → ℝ","l":176,"f":"Causalean/PO/Conditioning/CondExpTooling.lean"},{"n":"Causalean.PO.POVar.condExpRatio_eq_of_mul","k":"theorem","m":"Causalean.PO.Conditioning.CondExpTooling","a":"PO/Conditioning","t":2,"d":"Characterisation of condExpRatio via a product identity. If μ[g|σ(c)] = μ[h|σ(c)] · target a.s. and μ[h|σ(c)] ≠ 0 a.s., then condExpRatio c g h =ᵐ target. This is the reusable algebraic step for turning a conditional-expectation product identity into a ratio identity.","s":"∀ {P : Causalean.PO.POSystem} {γ : Type u_1} [inst : MeasurableSpace γ] (c : Causalean.PO.POVar P γ) {g h target : P.Ω → ℝ} {μ : MeasureTheory.Measure P.Ω}, c.condExpGiven g μ =ᵐ[μ] c.condExpGiven h μ * target → (∀ᵐ (ω : P.Ω) ∂μ, c.condExpGiven h μ ω ≠ 0) → c.condExpRatio g h μ =ᵐ[μ] target","l":202,"f":"Causalean/PO/Conditioning/CondExpTooling.lean"},{"n":"Causalean.PO.POVar.event","k":"def","m":"Causalean.PO.Core.Variable","a":"PO/Core","t":1,"d":"A factual-value event is the set of units whose observed value of a selected variable equals a selected analysis-scale value.","s":"{P : Causalean.PO.POSystem} → {α : Type u_1} → [inst : MeasurableSpace α] → Causalean.PO.POVar P α → α → Set P.Ω","l":68,"f":"Causalean/PO/Core/Variable.lean"},{"n":"Causalean.PO.POVar.eventCondExp_cfUnder_eq_factual_on_event","k":"theorem","m":"Causalean.PO.Conditioning.EventCondExp","a":"PO/Conditioning","t":2,"d":"Consistency-on-event for eventCondExp. Generic consumer of POVar.cf_eq_factual_on_event: on the event {a = a₀}, the counterfactual y.cfUnder a a₀ agrees with y.factual pointwise, so their conditional means on that event coincide. This is the shared rewrite underlying Manski MTR/MTS and (via the finite-partition total law) the integrated MIV bounds.","s":"∀ {P : Causalean.PO.POSystem} {β : Type u_2} [inst : MeasurableSpace β], P.Consistency → ∀ (y : Causalean.PO.POVar P ℝ) (a : Causalean.PO.POVar P β) (a₀ : β), MeasurableSet (a.event a₀) → y.v ≠ a.v → ∀ (μ : MeasureTheory.Measure P.Ω), Causalean.PO.eventCondExp μ (a.event a₀) (y.cfUnder a a₀) = Causalean.PO.eventCondExp μ (a.event a₀) y.factual","l":336,"f":"Causalean/PO/Conditioning/EventCondExp.lean"},{"n":"Causalean.PO.POVar.factual","k":"def","m":"Causalean.PO.Core.Variable","a":"PO/Core","t":1,"d":"A factual value function assigns each unit the observed no-intervention value of a selected variable, reported on its chosen analysis scale.","s":"{P : Causalean.PO.POSystem} → {α : Type u_1} → [inst : MeasurableSpace α] → Causalean.PO.POVar P α → P.Ω → α","l":52,"f":"Causalean/PO/Core/Variable.lean"},{"n":"Causalean.PO.POVar.factualAgrees_single","k":"theorem","m":"Causalean.PO.Assumptions.ConsistencyLemmas","a":"PO/Assumptions","t":2,"d":"Combinator: from a factual equality a.factual ω = x, build FactualAgrees for the singleton regime {a.v ← a.equiv.symm x}.","s":"∀ {P : Causalean.PO.POSystem} {α : Type u_1} [inst : MeasurableSpace α] (a : Causalean.PO.POVar P α) (x : α) {ω : P.Ω}, a.factual ω = x → P.FactualAgrees (Causalean.PO.Regime.single a.v (a.equiv.symm x)) ω","l":83,"f":"Causalean/PO/Assumptions/ConsistencyLemmas.lean"},{"n":"Causalean.PO.POVar.factual_eq_cfUnder_self_selected","k":"theorem","m":"Causalean.PO.Assumptions.ConsistencyLemmas","a":"PO/Assumptions","t":2,"d":"Under consistency, setting a distinct variable to its realized factual value leaves the target variable at its factual value.","s":"∀ {P : Causalean.PO.POSystem} {α : Type u_1} {β : Type u_2} [inst : MeasurableSpace α] [inst_1 : MeasurableSpace β], P.Consistency → ∀ (a : Causalean.PO.POVar P α) (w : Causalean.PO.POVar P β), a.v ≠ w.v → ∀ (ω : P.Ω), a.factual ω = a.cfUnder w (w.factual ω) ω","l":135,"f":"Causalean/PO/Assumptions/ConsistencyLemmas.lean"},{"n":"Causalean.PO.POVar.factual_mul_indicator_eq_cfUnder_mul_indicator","k":"theorem","m":"Causalean.PO.Assumptions.ConsistencyLemmas","a":"PO/Assumptions","t":2,"d":"Integrated form of consistency: Y · 1_{W=y} = Y(w=y) · 1_{W=y} pointwise, where a plays the role of Y and w the role of the treatment. Used in backdoor-style identification proofs where the factual outcome is replaced by the counterfactual on the event {W = y}.","s":"∀ {P : Causalean.PO.POSystem} {β : Type u_1} [inst : MeasurableSpace β], P.Consistency → ∀ (a : Causalean.PO.POVar P ℝ) (w : Causalean.PO.POVar P β) (y : β), a.v ≠ w.v → (fun ω => a.factual ω * (w.event y).indicator (fun x => 1) ω) = fun ω => a.cfUnder w y ω * (w.event y).indicator (fun x => 1) ω","l":163,"f":"Causalean/PO/Assumptions/ConsistencyLemmas.lean"},{"n":"Causalean.PO.POVar.factual_mul_indicator_eq_cfUnder_mul_indicator_fn","k":"theorem","m":"Causalean.PO.Assumptions.ConsistencyLemmas","a":"PO/Assumptions","t":2,"d":"Pointwise-function variant of POVar.factual_mul_indicator_eq_cfUnder_mul_indicator, phrased directly in terms of POVar.indicator (rather than Set.indicator on w.event y).","s":"∀ {P : Causalean.PO.POSystem} {β : Type u_1} [inst : MeasurableSpace β] [MeasurableSingletonClass β], P.Consistency → ∀ (y : Causalean.PO.POVar P ℝ) (w : Causalean.PO.POVar P β) (x : β), y.v ≠ w.v → (fun ω => y.factual ω * w.indicator x ω) = fun ω => y.cfUnder w x ω * w.indicator x ω","l":185,"f":"Causalean/PO/Assumptions/ConsistencyLemmas.lean"},{"n":"Causalean.PO.POVar.factual_mul_indicator_eq_cf_mul_indicator","k":"theorem","m":"Causalean.PO.Assumptions.ConsistencyLemmas","a":"PO/Assumptions","t":2,"d":"Multi-target integrated consistency: Y · 1_E = Y(r) · 1_E pointwise, whenever every ω ∈ E factually agrees with r and a.v ∉ r.target.","s":"∀ {P : Causalean.PO.POSystem}, P.Consistency → ∀ (a : Causalean.PO.POVar P ℝ) (r : Causalean.PO.Regime P.V P.X), a.v ∉ r.target → ∀ (E : Set P.Ω), (∀ ω ∈ E, P.FactualAgrees r ω) → (fun ω => a.factual ω * E.indicator (fun x => 1) ω) = fun ω => a.cf r ω * E.indicator (fun x => 1) ω","l":225,"f":"Causalean/PO/Assumptions/ConsistencyLemmas.lean"},{"n":"Causalean.PO.POVar.indicator","k":"def","m":"Causalean.PO.Core.Variable","a":"PO/Core","t":1,"d":"A factual-value indicator is the zero-one function that marks units whose observed value of a selected variable equals a selected analysis-scale value.","s":"{P : Causalean.PO.POSystem} → {α : Type u_1} → [inst : MeasurableSpace α] → Causalean.PO.POVar P α → α → P.Ω → ℝ","l":103,"f":"Causalean/PO/Core/Variable.lean"},{"n":"Causalean.PO.POVar.indicator_add_indicator_not","k":"theorem","m":"Causalean.PO.Core.Variable","a":"PO/Core","t":2,"d":"Binary case: a.indicator true ω + a.indicator false ω = 1.","s":"∀ {P : Causalean.PO.POSystem} (a : Causalean.PO.POVar P Bool) (ω : P.Ω), a.indicator true ω + a.indicator false ω = 1","l":174,"f":"Causalean/PO/Core/Variable.lean"},{"n":"Causalean.PO.POVar.indicator_apply_eq_one","k":"theorem","m":"Causalean.PO.Core.Variable","a":"PO/Core","t":2,"d":"Pointwise: a.indicator x ω = 1 on {a = x}.","s":"∀ {P : Causalean.PO.POSystem} {α : Type u_1} [inst : MeasurableSpace α] (a : Causalean.PO.POVar P α) {x : α} {ω : P.Ω}, a.factual ω = x → a.indicator x ω = 1","l":118,"f":"Causalean/PO/Core/Variable.lean"},{"n":"Causalean.PO.POVar.indicator_apply_eq_zero","k":"theorem","m":"Causalean.PO.Core.Variable","a":"PO/Core","t":2,"d":"Pointwise: a.indicator x ω = 0 off {a = x}.","s":"∀ {P : Causalean.PO.POSystem} {α : Type u_1} [inst : MeasurableSpace α] (a : Causalean.PO.POVar P α) {x : α} {ω : P.Ω}, a.factual ω ≠ x → a.indicator x ω = 0","l":124,"f":"Causalean/PO/Core/Variable.lean"},{"n":"Causalean.PO.POVar.indicator_eq_event_indicator","k":"theorem","m":"Causalean.PO.Core.Variable","a":"PO/Core","t":1,"d":"For a potential-outcome variable a and a value x in its range, the real-valued factual indicator a.indicator x equals the set-indicator of the factual event {a = x}.","s":"∀ {P : Causalean.PO.POSystem} {α : Type u_1} [inst : MeasurableSpace α] (a : Causalean.PO.POVar P α) (x : α), a.indicator x = (a.event x).indicator fun x => 1","l":112,"f":"Causalean/PO/Core/Variable.lean"},{"n":"Causalean.PO.POVar.indicator_eq_one_or_zero","k":"theorem","m":"Causalean.PO.Core.Variable","a":"PO/Core","t":2,"d":"a.indicator x ω is always 0 or 1.","s":"∀ {P : Causalean.PO.POSystem} {α : Type u_1} [inst : MeasurableSpace α] (a : Causalean.PO.POVar P α) (x : α) (ω : P.Ω), a.indicator x ω = 1 ∨ a.indicator x ω = 0","l":166,"f":"Causalean/PO/Core/Variable.lean"},{"n":"Causalean.PO.POVar.instIsProbabilityMeasureCfLaw","k":"instance","m":"Causalean.PO.Analysis.Quantile","a":"PO/Analysis","t":2,"d":"The law of a real-valued potential outcome is a probability measure whenever the original sample-space measure is a probability measure.","s":"∀ {P : Causalean.PO.POSystem} (a : Causalean.PO.POVar P ℝ) (r : Causalean.PO.Regime P.V P.X) (μ : MeasureTheory.Measure P.Ω) [MeasureTheory.IsProbabilityMeasure μ], MeasureTheory.IsProbabilityMeasure (a.cfLaw r μ)","l":39,"f":"Causalean/PO/Analysis/Quantile.lean"},{"n":"Causalean.PO.POVar.integrable_condExpGiven","k":"theorem","m":"Causalean.PO.Conditioning.CondExpTooling","a":"PO/Conditioning","t":2,"d":"Mathlib's totalized conditional expectation given the variable's σ-algebra is integrable, even when the original sample-space function is not assumed integrable.","s":"∀ {P : Causalean.PO.POSystem} {γ : Type u_1} [inst : MeasurableSpace γ] (c : Causalean.PO.POVar P γ) (g : P.Ω → ℝ) {μ : MeasureTheory.Measure P.Ω}, MeasureTheory.Integrable (c.condExpGiven g μ) μ","l":98,"f":"Causalean/PO/Conditioning/CondExpTooling.lean"},{"n":"Causalean.PO.POVar.integrable_indicator","k":"theorem","m":"Causalean.PO.Core.Variable","a":"PO/Core","t":2,"d":"a.indicator x is integrable under any finite measure (bounded by 1).","s":"∀ {P : Causalean.PO.POSystem} {α : Type u_1} [inst : MeasurableSpace α] {μ : MeasureTheory.Measure P.Ω} [MeasureTheory.IsFiniteMeasure μ] (a : Causalean.PO.POVar P α) (x : α), MeasurableSet {x} → MeasureTheory.Integrable (a.indicator x) μ","l":153,"f":"Causalean/PO/Core/Variable.lean"},{"n":"Causalean.PO.POVar.integrable_mul_indicator","k":"theorem","m":"Causalean.PO.Conditioning.CondExpTooling","a":"PO/Conditioning","t":2,"d":"Multiplying an integrable function by a POVar factual indicator preserves integrability.","s":"∀ {P : Causalean.PO.POSystem} {α : Type u_2} [inst : MeasurableSpace α] (a : Causalean.PO.POVar P α) (x : α), MeasurableSet {x} → ∀ {f : P.Ω → ℝ} {μ : MeasureTheory.Measure P.Ω}, MeasureTheory.Integrable f μ → MeasureTheory.Integrable (fun ω => f ω * a.indicator x ω) μ","l":50,"f":"Causalean/PO/Conditioning/CondExpTooling.lean"},{"n":"Causalean.PO.POVar.integral_sub_eq_integral_sub_of_condExpGiven_ae_eq","k":"theorem","m":"Causalean.PO.Conditioning.CondExpTooling","a":"PO/Conditioning","t":1,"d":"CATE-to-ATE integrator. Fix a conditioning variable c and integrable functions f and g on the same probability space. If f' is an almost-everywhere representative of the conditional expectation of f given c and g' is an almost-everywhere representative of the conditional expectation of g given c, then the integral of f - g equals the integral of f' - g'.","s":"∀ {P : Causalean.PO.POSystem} {γ : Type u_1} [inst : MeasurableSpace γ] (c : Causalean.PO.POVar P γ) {f g f' g' : P.Ω → ℝ} {μ : MeasureTheory.Measure P.Ω} [MeasureTheory.SigmaFinite (μ.trim (Causalean.PO.POVar.comap_factual_le c))], MeasureTheory.Integrable f μ → MeasureTheory.Integrable g μ → c.condExpGiven f μ =ᵐ[μ] f' → c.condExpGiven g μ =ᵐ[μ] g' → ∫ (ω : P.Ω), f ω - g ω ∂μ = ∫ (ω : P.Ω), f' ω - g' ω ∂μ","l":217,"f":"Causalean/PO/Conditioning/CondExpTooling.lean"},{"n":"Causalean.PO.POVar.measurableSet_event","k":"theorem","m":"Causalean.PO.Core.Variable","a":"PO/Core","t":2,"d":"The event that a potential-outcome variable's factual value equals a given singleton-measurable value is measurable.","s":"∀ {P : Causalean.PO.POSystem} {α : Type u_1} [inst : MeasurableSpace α] (a : Causalean.PO.POVar P α) (x : α), MeasurableSet {x} → MeasurableSet (a.event x)","l":74,"f":"Causalean/PO/Core/Variable.lean"},{"n":"Causalean.PO.POVar.measurable_cf","k":"theorem","m":"Causalean.PO.Core.Variable","a":"PO/Core","t":2,"d":"The counterfactual-value function of a potential-outcome variable under any intervention regime is measurable.","s":"∀ {P : Causalean.PO.POSystem} {α : Type u_1} [inst : MeasurableSpace α] (a : Causalean.PO.POVar P α) (r : Causalean.PO.Regime P.V P.X), Measurable (a.cf r)","l":58,"f":"Causalean/PO/Core/Variable.lean"},{"n":"Causalean.PO.POVar.measurable_cfUnder","k":"theorem","m":"Causalean.PO.Core.Variable","a":"PO/Core","t":2,"d":"The single-intervention counterfactual-value function is measurable.","s":"∀ {P : Causalean.PO.POSystem} {α : Type u_1} [inst : MeasurableSpace α] {β : Type u_2} [inst_1 : MeasurableSpace β] (y : Causalean.PO.POVar P α) (w : Causalean.PO.POVar P β) (d : β), Measurable (y.cfUnder w d)","l":92,"f":"Causalean/PO/Core/Variable.lean"},{"n":"Causalean.PO.POVar.measurable_condExpRatio","k":"theorem","m":"Causalean.PO.Conditioning.CondExpTooling","a":"PO/Conditioning","t":2,"d":"condExpRatio is measurable (pointwise division of two strongly measurable functions is measurable).","s":"∀ {P : Causalean.PO.POSystem} {γ : Type u_1} [inst : MeasurableSpace γ] (c : Causalean.PO.POVar P γ) (g h : P.Ω → ℝ) {μ : MeasureTheory.Measure P.Ω}, Measurable (c.condExpRatio g h μ)","l":189,"f":"Causalean/PO/Conditioning/CondExpTooling.lean"},{"n":"Causalean.PO.POVar.measurable_factual","k":"theorem","m":"Causalean.PO.Core.Variable","a":"PO/Core","t":2,"d":"The factual-value function of a potential-outcome variable is measurable.","s":"∀ {P : Causalean.PO.POSystem} {α : Type u_1} [inst : MeasurableSpace α] (a : Causalean.PO.POVar P α), Measurable a.factual","l":64,"f":"Causalean/PO/Core/Variable.lean"},{"n":"Causalean.PO.POVar.measurable_indicator","k":"theorem","m":"Causalean.PO.Core.Variable","a":"PO/Core","t":2,"d":"a.indicator x is measurable.","s":"∀ {P : Causalean.PO.POSystem} {α : Type u_1} [inst : MeasurableSpace α] (a : Causalean.PO.POVar P α) (x : α), MeasurableSet {x} → Measurable (a.indicator x)","l":130,"f":"Causalean/PO/Core/Variable.lean"},{"n":"Causalean.PO.POVar.stronglyMeasurable_condExpGiven","k":"theorem","m":"Causalean.PO.Conditioning.CondExpTooling","a":"PO/Conditioning","t":2,"d":"condExpGiven yields a strongly measurable function w.r.t. the ambient σ-algebra.","s":"∀ {P : Causalean.PO.POSystem} {γ : Type u_1} [inst : MeasurableSpace γ] (c : Causalean.PO.POVar P γ) (g : P.Ω → ℝ) {μ : MeasureTheory.Measure P.Ω}, MeasureTheory.StronglyMeasurable (c.condExpGiven g μ)","l":93,"f":"Causalean/PO/Conditioning/CondExpTooling.lean"},{"n":"Causalean.PO.POVar.stronglyMeasurable_condExpGiven_comap","k":"theorem","m":"Causalean.PO.Conditioning.CondExpTooling","a":"PO/Conditioning","t":2,"d":"condExpGiven yields a function strongly measurable w.r.t. σ(c.factual).","s":"∀ {P : Causalean.PO.POSystem} {γ : Type u_1} [inst : MeasurableSpace γ] (c : Causalean.PO.POVar P γ) (g : P.Ω → ℝ) {μ : MeasureTheory.Measure P.Ω}, MeasureTheory.StronglyMeasurable (c.condExpGiven g μ)","l":86,"f":"Causalean/PO/Conditioning/CondExpTooling.lean"},{"n":"Causalean.PO.POVar.stronglyMeasurable_condExpRatio","k":"theorem","m":"Causalean.PO.Conditioning.CondExpTooling","a":"PO/Conditioning","t":2,"d":"condExpRatio is strongly measurable.","s":"∀ {P : Causalean.PO.POSystem} {γ : Type u_1} [inst : MeasurableSpace γ] (c : Causalean.PO.POVar P γ) (g h : P.Ω → ℝ) {μ : MeasureTheory.Measure P.Ω}, MeasureTheory.StronglyMeasurable (c.condExpRatio g h μ)","l":197,"f":"Causalean/PO/Conditioning/CondExpTooling.lean"},{"n":"Causalean.PO.POVar.stronglyMeasurable_indicator_comap","k":"theorem","m":"Causalean.PO.Core.Variable","a":"PO/Core","t":2,"d":"a.indicator x is strongly measurable w.r.t. the σ-algebra generated by a.factual.","s":"∀ {P : Causalean.PO.POSystem} {α : Type u_1} [inst : MeasurableSpace α] (a : Causalean.PO.POVar P α) (x : α), MeasurableSet {x} → MeasureTheory.StronglyMeasurable (a.indicator x)","l":137,"f":"Causalean/PO/Core/Variable.lean"},{"n":"Causalean.PO.RDDLimits.oneSidedLimit_eq_left","k":"theorem","m":"Causalean.PO.ID.Exact.RDD.RDDLimits","a":"PO/ID/Exact/RDD","t":1,"d":"Left-side limit identification. Symmetric form of oneSidedLimit_eq_right. If a real-valued function f agrees with a reference function g almost everywhere, with respect to a measure π, on the half-line to the left of the cutoff c, g is continuous at c, and every open interval immediately to the left of c has positive π-measure, then any left-hand limit L of f at c must equal g evaluated at c.","s":"∀ {π : MeasureTheory.Measure ℝ} {f g : ℝ → ℝ} {c : ℝ}, f =ᵐ[π.restrict (Set.Iio c)] g → ContinuousAt g c → (∀ ε > 0, π (Set.Ioo (c - ε) c) ≠ 0) → ∀ {L : ℝ}, Filter.Tendsto f (nhdsWithin c (Set.Iio c)) (nhds L) → L = g c","l":122,"f":"Causalean/PO/ID/Exact/RDD/RDDLimits.lean"},{"n":"Causalean.PO.RDDLimits.oneSidedLimit_eq_right","k":"theorem","m":"Causalean.PO.ID.Exact.RDD.RDDLimits","a":"PO/ID/Exact/RDD","t":2,"d":"Right-side limit identification. If f =ᵐ[π.restrict (Ici c)] g, g is continuous at c, every right-open neighborhood of c has positive π-mass, and f has any right-side limit L at c, then L = g c.","s":"∀ {π : MeasureTheory.Measure ℝ} {f g : ℝ → ℝ} {c : ℝ}, f =ᵐ[π.restrict (Set.Ici c)] g → ContinuousAt g c → (∀ ε > 0, π (Set.Ioo c (c + ε)) ≠ 0) → ∀ {L : ℝ}, Filter.Tendsto f (nhdsWithin c (Set.Ioi c)) (nhds L) → L = g c","l":89,"f":"Causalean/PO/ID/Exact/RDD/RDDLimits.lean"},{"n":"Causalean.PO.RDDLimits.value_eq_of_aeEq_left","k":"theorem","m":"Causalean.PO.ID.Exact.RDD.RDDLimits","a":"PO/ID/Exact/RDD","t":2,"d":"Symmetric pointwise version of value_eq_of_aeEq_right.","s":"∀ {π : MeasureTheory.Measure ℝ} {f g : ℝ → ℝ} {c : ℝ}, f =ᵐ[π.restrict (Set.Iio c)] g → ContinuousAt f c → ContinuousAt g c → (∀ ε > 0, π (Set.Ioo (c - ε) c) ≠ 0) → f c = g c","l":170,"f":"Causalean/PO/ID/Exact/RDD/RDDLimits.lean"},{"n":"Causalean.PO.RDDLimits.value_eq_of_aeEq_right","k":"theorem","m":"Causalean.PO.ID.Exact.RDD.RDDLimits","a":"PO/ID/Exact/RDD","t":2,"d":"Pointwise equality from a.e. agreement plus continuity. If f and g agree π.restrict (Ici c)-a.e., both are continuous at c, and every right-open neighborhood of c has positive π-mass, then f c = g c.","s":"∀ {π : MeasureTheory.Measure ℝ} {f g : ℝ → ℝ} {c : ℝ}, f =ᵐ[π.restrict (Set.Ici c)] g → ContinuousAt f c → ContinuousAt g c → (∀ ε > 0, π (Set.Ioo c (c + ε)) ≠ 0) → f c = g c","l":158,"f":"Causalean/PO/ID/Exact/RDD/RDDLimits.lean"},{"n":"Causalean.PO.Regime","k":"structure","m":"Causalean.PO.Core.Regime","a":"PO/Core","t":1,"d":"An intervention regime specifies a finite set of targeted variables together with an assigned value in the corresponding value space for each targeted variable.","s":"(V : Type u_3) → [DecidableEq V] → (V → Type u_4) → Type (max u_3 u_4)","l":28,"f":"Causalean/PO/Core/Regime.lean"},{"n":"Causalean.PO.Regime.Disjoint","k":"def","m":"Causalean.PO.Core.Regime","a":"PO/Core","t":1,"d":"Two intervention regimes are disjoint exactly when their target sets have no variable in common.","s":"{V : Type u_1} → [inst : DecidableEq V] → {X : V → Type u_2} → Causalean.PO.Regime V X → Causalean.PO.Regime V X → Prop","l":49,"f":"Causalean/PO/Core/Regime.lean"},{"n":"Causalean.PO.Regime.disjoint_single_of_not_mem","k":"theorem","m":"Causalean.PO.Core.Regime","a":"PO/Core","t":2,"d":"Any regime that does not target a variable is disjoint from the singleton intervention on that variable.","s":"∀ {V : Type u_1} [inst : DecidableEq V] {X : V → Type u_2} {v : V} (x : X v) (r : Causalean.PO.Regime V X), v ∉ r.target → r.Disjoint (Causalean.PO.Regime.single v x)","l":142,"f":"Causalean/PO/Core/Regime.lean"},{"n":"Causalean.PO.Regime.empty","k":"def","m":"Causalean.PO.Core.Regime","a":"PO/Core","t":1,"d":"The empty intervention regime fixes no variables and therefore has no assignments.","s":"{V : Type u_1} → [inst : DecidableEq V] → {X : V → Type u_2} → Causalean.PO.Regime V X","l":40,"f":"Causalean/PO/Core/Regime.lean"},{"n":"Causalean.PO.Regime.empty_disjoint_left","k":"theorem","m":"Causalean.PO.Core.Regime","a":"PO/Core","t":2,"d":"Every regime is disjoint from the empty intervention regime on its right.","s":"∀ {V : Type u_1} [inst : DecidableEq V] {X : V → Type u_2} (r : Causalean.PO.Regime V X), r.Disjoint Causalean.PO.Regime.empty","l":82,"f":"Causalean/PO/Core/Regime.lean"},{"n":"Causalean.PO.Regime.empty_disjoint_right","k":"theorem","m":"Causalean.PO.Core.Regime","a":"PO/Core","t":2,"d":"The empty intervention regime is disjoint from every regime on its right.","s":"∀ {V : Type u_1} [inst : DecidableEq V] {X : V → Type u_2} (r : Causalean.PO.Regime V X), Causalean.PO.Regime.empty.Disjoint r","l":78,"f":"Causalean/PO/Core/Regime.lean"},{"n":"Causalean.PO.Regime.empty_target","k":"theorem","m":"Causalean.PO.Core.Regime","a":"PO/Core","t":2,"d":"The empty intervention regime has no target variables.","s":"∀ {V : Type u_1} [inst : DecidableEq V] {X : V → Type u_2}, Causalean.PO.Regime.empty.target = ∅","l":75,"f":"Causalean/PO/Core/Regime.lean"},{"n":"Causalean.PO.Regime.ext","k":"theorem","m":"Causalean.PO.Core.Regime","a":"PO/Core","t":2,"d":"Extensionality for Regime: equal targets and pointwise-equal assignments.","s":"∀ {V : Type u_1} [inst : DecidableEq V] {X : V → Type u_2} {r₁ r₂ : Causalean.PO.Regime V X}, r₁.target = r₂.target → (∀ (v : V) (h₁ : v ∈ r₁.target) (h₂ : v ∈ r₂.target), r₁.assign v h₁ = r₂.assign v h₂) → r₁ = r₂","l":104,"f":"Causalean/PO/Core/Regime.lean"},{"n":"Causalean.PO.Regime.leftBiasedUnion","k":"def","m":"Causalean.PO.Core.Regime","a":"PO/Core","t":1,"d":"The left-biased union of two regimes targets their union and uses the first regime's assignment wherever both regimes target the same variable.","s":"{V : Type u_1} → [inst : DecidableEq V] → {X : V → Type u_2} → Causalean.PO.Regime V X → Causalean.PO.Regime V X → Causalean.PO.Regime V X","l":53,"f":"Causalean/PO/Core/Regime.lean"},{"n":"Causalean.PO.Regime.listLookup","k":"def","m":"Causalean.PO.Core.Regime","a":"PO/Core","t":1,"d":"Lookup in a list of variable-value assignments returns the listed value for a variable that appears among the listed labels.","s":"{V : Type u_1} → [DecidableEq V] → {X : V → Type u_2} → (l : List ((v : V) × X v)) → (v : V) → v ∈ List.map Sigma.fst l → X v","l":150,"f":"Causalean/PO/Core/Regime.lean"},{"n":"Causalean.PO.Regime.listLookup_cons_of_ne","k":"theorem","m":"Causalean.PO.Core.Regime","a":"PO/Core","t":2,"d":"Looking up a different variable skips the head of a dependent assignment list and continues in the tail.","s":"∀ {V : Type u_1} [inst : DecidableEq V] {X : V → Type u_2} {v w : V} {x : X w} {rest : List ((v : V) × X v)}, v ≠ w → ∀ (hv' : v ∈ List.map Sigma.fst rest), Causalean.PO.Regime.listLookup (⟨w, x⟩ :: rest) v (of_eq_true (Eq.trans (Eq.trans (congrFun' (congrArg Membership.mem List.map_cons) v) List.mem_cons._simp_1) (Eq.trans (congrArg (Or (v = w)) (eq_true hv')) (or_true (v = w))))) = Causalean.PO.Regime.listLookup rest v hv'","l":208,"f":"Causalean/PO/Core/Regime.lean"},{"n":"Causalean.PO.Regime.listLookup_cons_self","k":"theorem","m":"Causalean.PO.Core.Regime","a":"PO/Core","t":2,"d":"Looking up the head variable of a dependent assignment list returns the head value.","s":"∀ {V : Type u_1} [inst : DecidableEq V] {X : V → Type u_2} {v : V} {x : X v} {rest : List ((v : V) × X v)}, Causalean.PO.Regime.listLookup (⟨v, x⟩ :: rest) v (of_eq_true (Eq.trans (Eq.trans (congrFun' (congrArg Membership.mem List.map_cons) v) List.mem_cons._simp_1) (Eq.trans (congr (congrArg Or (eq_self v)) (Eq.trans List.mem_map._simp_1 (Eq.trans Sigma.exists._simp_1 (Eq.trans (congrArg Exists (funext fun a => exists_and_right._simp_1)) exists_eq_right._simp_1)))) (true_or (∃ x, ⟨v, x⟩ ∈ rest))))) = x","l":202,"f":"Causalean/PO/Core/Regime.lean"},{"n":"Causalean.PO.Regime.ofList","k":"def","m":"Causalean.PO.Core.Regime","a":"PO/Core","t":1,"d":"A duplicate-free list of variable-value assignments determines the intervention regime that targets exactly the listed variables and assigns each target its listed value.","s":"{V : Type u_1} → [inst : DecidableEq V] → {X : V → Type u_2} → (l : List ((v : V) × X v)) → (List.map Sigma.fst l).Nodup → Causalean.PO.Regime V X","l":174,"f":"Causalean/PO/Core/Regime.lean"},{"n":"Causalean.PO.Regime.ofListLeftBiased","k":"def","m":"Causalean.PO.Core.Regime","a":"PO/Core","t":1,"d":"A list of variable-value assignments determines a regime by using the first listed assignment for each targeted variable.","s":"{V : Type u_1} → [inst : DecidableEq V] → {X : V → Type u_2} → List ((v : V) × X v) → Causalean.PO.Regime V X","l":167,"f":"Causalean/PO/Core/Regime.lean"},{"n":"Causalean.PO.Regime.ofList_cons_target","k":"theorem","m":"Causalean.PO.Core.Regime","a":"PO/Core","t":2,"d":"The target of a regime built from a nonempty list inserts the head variable into the target from the tail.","s":"∀ {V : Type u_1} [inst : DecidableEq V] {X : V → Type u_2} {v : V} {x : X v} {rest : List ((v : V) × X v)} (h : (List.map Sigma.fst (⟨v, x⟩ :: rest)).Nodup), (Causalean.PO.Regime.ofList (⟨v, x⟩ :: rest) h).target = insert v (List.map Sigma.fst rest).toFinset","l":193,"f":"Causalean/PO/Core/Regime.lean"},{"n":"Causalean.PO.Regime.ofList_nil","k":"theorem","m":"Causalean.PO.Core.Regime","a":"PO/Core","t":2,"d":"Building a regime from the empty list gives the empty intervention regime.","s":"∀ {V : Type u_1} [inst : DecidableEq V] {X : V → Type u_2}, Causalean.PO.Regime.ofList [] (of_eq_true (Eq.trans (congrArg List.Nodup List.map_nil) List.nodup_nil._simp_1)) = Causalean.PO.Regime.empty","l":188,"f":"Causalean/PO/Core/Regime.lean"},{"n":"Causalean.PO.Regime.ofList_target","k":"theorem","m":"Causalean.PO.Core.Regime","a":"PO/Core","t":1,"d":"For a duplicate-free list of variable-value assignments l, the target of the regime it determines is exactly the finite set of variables listed in l.","s":"∀ {V : Type u_1} [inst : DecidableEq V] {X : V → Type u_2} (l : List ((v : V) × X v)) (h : (List.map Sigma.fst l).Nodup), (Causalean.PO.Regime.ofList l h).target = (List.map Sigma.fst l).toFinset","l":181,"f":"Causalean/PO/Core/Regime.lean"},{"n":"Causalean.PO.Regime.single","k":"def","m":"Causalean.PO.Core.Regime","a":"PO/Core","t":1,"d":"A singleton intervention regime fixes exactly one variable to the supplied value.","s":"{V : Type u_1} → [inst : DecidableEq V] → {X : V → Type u_2} → (v : V) → X v → Causalean.PO.Regime V X","l":117,"f":"Causalean/PO/Core/Regime.lean"},{"n":"Causalean.PO.Regime.single_assign_self","k":"theorem","m":"Causalean.PO.Core.Regime","a":"PO/Core","t":2,"d":"Evaluating the singleton intervention assignment at its target returns the supplied value.","s":"∀ {V : Type u_1} [inst : DecidableEq V] {X : V → Type u_2} (v : V) (x : X v), (Causalean.PO.Regime.single v x).assign v (Finset.mem_singleton_self v) = x","l":127,"f":"Causalean/PO/Core/Regime.lean"},{"n":"Causalean.PO.Regime.single_disjoint_of_not_mem","k":"theorem","m":"Causalean.PO.Core.Regime","a":"PO/Core","t":2,"d":"A singleton intervention regime is disjoint from any regime that does not target its variable.","s":"∀ {V : Type u_1} [inst : DecidableEq V] {X : V → Type u_2} {v : V} (x : X v) (r : Causalean.PO.Regime V X), v ∉ r.target → (Causalean.PO.Regime.single v x).Disjoint r","l":136,"f":"Causalean/PO/Core/Regime.lean"},{"n":"Causalean.PO.Regime.single_disjoint_single","k":"theorem","m":"Causalean.PO.Core.Regime","a":"PO/Core","t":2,"d":"Singleton intervention regimes on two distinct variables are disjoint.","s":"∀ {V : Type u_1} [inst : DecidableEq V] {X : V → Type u_2} {v w : V}, v ≠ w → ∀ (x : X v) (y : X w), (Causalean.PO.Regime.single v x).Disjoint (Causalean.PO.Regime.single w y)","l":131,"f":"Causalean/PO/Core/Regime.lean"},{"n":"Causalean.PO.Regime.single_target","k":"theorem","m":"Causalean.PO.Core.Regime","a":"PO/Core","t":2,"d":"The singleton intervention regime targets exactly the one variable it fixes.","s":"∀ {V : Type u_1} [inst : DecidableEq V] {X : V → Type u_2} (v : V) (x : X v), (Causalean.PO.Regime.single v x).target = {v}","l":123,"f":"Causalean/PO/Core/Regime.lean"},{"n":"Causalean.PO.Regime.sqcup","k":"def","m":"Causalean.PO.Core.Regime","a":"PO/Core","t":1,"d":"The disjoint union of two compatible intervention regimes targets the union of their target sets and uses the assignment from the unique component regime that targets each variable.","s":"{V : Type u_1} → [inst : DecidableEq V] → {X : V → Type u_2} → (r₁ r₂ : Causalean.PO.Regime V X) → r₁.Disjoint r₂ → Causalean.PO.Regime V X","l":65,"f":"Causalean/PO/Core/Regime.lean"},{"n":"Causalean.PO.Regime.sqcup_assign_neg","k":"theorem","m":"Causalean.PO.Core.Regime","a":"PO/Core","t":2,"d":"sqcup agrees with r₂ whenever v ∉ r₁.target (and hence v ∈ r₂.target).","s":"∀ {V : Type u_1} [inst : DecidableEq V] {X : V → Type u_2} (r₁ r₂ : Causalean.PO.Regime V X) (h : r₁.Disjoint r₂), ∀ v ∉ r₁.target, ∀ (h2 : v ∈ r₂.target), (r₁.sqcup r₂ h).assign v (of_eq_true (Eq.trans Finset.mem_union._simp_1 (Eq.trans (congrArg (Or (v ∈ r₁.target)) (eq_true h2)) (or_true (v ∈ r₁.target))))) = r₂.assign v h2","l":97,"f":"Causalean/PO/Core/Regime.lean"},{"n":"Causalean.PO.Regime.sqcup_assign_pos","k":"theorem","m":"Causalean.PO.Core.Regime","a":"PO/Core","t":2,"d":"sqcup agrees with r₁ whenever v ∈ r₁.target.","s":"∀ {V : Type u_1} [inst : DecidableEq V] {X : V → Type u_2} (r₁ r₂ : Causalean.PO.Regime V X) (h : r₁.Disjoint r₂) (v : V) (h1 : v ∈ r₁.target), (r₁.sqcup r₂ h).assign v (of_eq_true (Eq.trans Finset.mem_union._simp_1 (Eq.trans (congrFun' (congrArg Or (eq_true h1)) (v ∈ r₂.target)) (true_or (v ∈ r₂.target))))) = r₁.assign v h1","l":90,"f":"Causalean/PO/Core/Regime.lean"},{"n":"Causalean.PO.Regime.sqcup_target","k":"theorem","m":"Causalean.PO.Core.Regime","a":"PO/Core","t":2,"d":"The target of the disjoint union of two regimes is the union of their target sets.","s":"∀ {V : Type u_1} [inst : DecidableEq V] {X : V → Type u_2} (r₁ r₂ : Causalean.PO.Regime V X) (h : r₁.Disjoint r₂), (r₁.sqcup r₂ h).target = r₁.target ∪ r₂.target","l":86,"f":"Causalean/PO/Core/Regime.lean"},{"n":"Causalean.PO.RegimedVar","k":"structure","m":"Causalean.PO.Core.Variable","a":"PO/Core","t":1,"d":"A regimed variable pairs a potential-outcome variable with the intervention regime under which it should be evaluated.","s":"Causalean.PO.POSystem → (α : Type u_1) → [MeasurableSpace α] → Type (max (max u_1 u_2) u_3)","l":191,"f":"Causalean/PO/Core/Variable.lean"},{"n":"Causalean.PO.RegimedVar.measurable_value","k":"theorem","m":"Causalean.PO.Core.Variable","a":"PO/Core","t":2,"d":"The value function of a regimed variable is measurable.","s":"∀ {P : Causalean.PO.POSystem} {α : Type u_1} [inst : MeasurableSpace α] (rv : Causalean.PO.RegimedVar P α), Measurable rv.value","l":211,"f":"Causalean/PO/Core/Variable.lean"},{"n":"Causalean.PO.RegimedVar.ofFactual","k":"def","m":"Causalean.PO.Core.Variable","a":"PO/Core","t":1,"d":"Factual bundling views a potential-outcome variable as evaluated under the no-intervention regime.","s":"{P : Causalean.PO.POSystem} → {α : Type u_1} → [inst : MeasurableSpace α] → Causalean.PO.POVar P α → Causalean.PO.RegimedVar P α","l":215,"f":"Causalean/PO/Core/Variable.lean"},{"n":"Causalean.PO.RegimedVar.ofSingle","k":"def","m":"Causalean.PO.Core.Variable","a":"PO/Core","t":1,"d":"Single-intervention bundling views a potential-outcome variable as evaluated after fixing one system variable to a chosen native value.","s":"{P : Causalean.PO.POSystem} → {α : Type u_1} → [inst : MeasurableSpace α] → Causalean.PO.POVar P α → (w : P.V) → P.X w → Causalean.PO.RegimedVar P α","l":221,"f":"Causalean/PO/Core/Variable.lean"},{"n":"Causalean.PO.RegimedVar.value","k":"def","m":"Causalean.PO.Core.Variable","a":"PO/Core","t":1,"d":"A regimed variable's value function assigns each unit the counterfactual value implied by the variable-regime pair.","s":"{P : Causalean.PO.POSystem} → {α : Type u_1} → [inst : MeasurableSpace α] → Causalean.PO.RegimedVar P α → P.Ω → α","l":205,"f":"Causalean/PO/Core/Variable.lean"},{"n":"Causalean.PO.combinedFixed","k":"def","m":"Causalean.PO.Bridge.FromSCM","a":"PO/Bridge","t":1,"d":"The combined fixed-variable assignment feeds the original background values and the regime's intervention values into the intervened structural model.","s":"{N : Type uN} → [inst : DecidableEq N] → [inst_1 : Fintype N] → {Ω : N → Type uΩ} → [inst_2 : (n : N) → MeasurableSpace (Ω n)] → (M : Causalean.SCM N Ω) → M.FixedValues → (r : Causalean.PO.Regime (Causalean.PO.ObsIdx M) (Causalean.PO.obsValue M)) → (M.fixSet (Causalean.PO.regimeTargetN M r) (Causalean.PO.regimeTargetN_obs M r) (Causalean.PO.regimeTargetN_notFixed M r)).FixedValues","l":175,"f":"Causalean/PO/Bridge/FromSCM.lean"},{"n":"Causalean.PO.combinedFixed_new","k":"theorem","m":"Causalean.PO.Bridge.FromSCM","a":"PO/Bridge","t":2,"d":"On new intervention coordinates, combinedFixed at ⟨.fixed D, _⟩ equals r.assign v' hv'tgt, both at type Ω D. The internal Classical.choose witness is identified with v' by uniqueness (both have .val = .random D); the cast chain collapses via cast_heq + proof_irrel_heq.","s":"∀ {N : Type uN} [inst : DecidableEq N] [inst_1 : Fintype N] {Ω : N → Type uΩ} [inst_2 : (n : N) → MeasurableSpace (Ω n)] (M : Causalean.SCM N Ω) (s : M.FixedValues) (r : Causalean.PO.Regime (Causalean.PO.ObsIdx M) (Causalean.PO.obsValue M)) (v' : Causalean.PO.ObsIdx M) (hv'tgt : v' ∈ r.target) (D : N) (hD : D ∈ Causalean.PO.regimeTargetN M r) (hDval : ↑v' = Causalean.SWIGNode.random D), Causalean.PO.combinedFixed M s r ⟨Causalean.SWIGNode.fixed D, Finset.mem_union_right M.fixed (Finset.mem_image.mpr (Exists.intro D ⟨hD, rfl⟩))⟩ = cast (congrArg (Causalean.swigΩ Ω) hDval) (r.assign v' hv'tgt)","l":315,"f":"Causalean/PO/Bridge/FromSCM.lean"},{"n":"Causalean.PO.combinedFixed_old","k":"theorem","m":"Causalean.PO.Bridge.FromSCM","a":"PO/Bridge","t":2,"d":"On original fixed coordinates, combinedFixed agrees with s.","s":"∀ {N : Type uN} [inst : DecidableEq N] [inst_1 : Fintype N] {Ω : N → Type uΩ} [inst_2 : (n : N) → MeasurableSpace (Ω n)] (M : Causalean.SCM N Ω) (s : M.FixedValues) (r : Causalean.PO.Regime (Causalean.PO.ObsIdx M) (Causalean.PO.obsValue M)) (v : Causalean.SWIGNode N) (hv : v ∈ M.fixed), Causalean.PO.combinedFixed M s r ⟨v, Finset.mem_union_left (Finset.image Causalean.SWIGNode.fixed (Causalean.PO.regimeTargetN M r)) hv⟩ = s ⟨v, hv⟩","l":267,"f":"Causalean/PO/Bridge/FromSCM.lean"},{"n":"Causalean.PO.eventCondExp","k":"def","m":"Causalean.PO.Conditioning.EventCondExp","a":"PO/Conditioning","t":1,"d":"Event-level conditional expectation is the average of a real-valued sample-space quantity over an event, computed as its integral over the event divided by the event's real mass.","s":"{Ω : Type u_1} → [inst : MeasurableSpace Ω] → MeasureTheory.Measure Ω → Set Ω → (Ω → ℝ) → ℝ","l":51,"f":"Causalean/PO/Conditioning/EventCondExp.lean"},{"n":"Causalean.PO.eventCondExp_add","k":"theorem","m":"Causalean.PO.Conditioning.EventCondExp","a":"PO/Conditioning","t":2,"d":"Event-level conditional expectation is additive for integrable summands on the event.","s":"∀ {Ω : Type u_1} [inst : MeasurableSpace Ω] (μ : MeasureTheory.Measure Ω) (A : Set Ω) {g₁ g₂ : Ω → ℝ}, MeasureTheory.IntegrableOn g₁ A μ → MeasureTheory.IntegrableOn g₂ A μ → Causalean.PO.eventCondExp μ A (g₁ + g₂) = Causalean.PO.eventCondExp μ A g₁ + Causalean.PO.eventCondExp μ A g₂","l":193,"f":"Causalean/PO/Conditioning/EventCondExp.lean"},{"n":"Causalean.PO.eventCondExp_congr_ae","k":"theorem","m":"Causalean.PO.Conditioning.EventCondExp","a":"PO/Conditioning","t":2,"d":"a.e.-equal integrands have equal event-level conditional expectations.","s":"∀ {Ω : Type u_1} [inst : MeasurableSpace Ω] (μ : MeasureTheory.Measure Ω) (A : Set Ω) {f g : Ω → ℝ}, f =ᵐ[μ.restrict A] g → Causalean.PO.eventCondExp μ A f = Causalean.PO.eventCondExp μ A g","l":162,"f":"Causalean/PO/Conditioning/EventCondExp.lean"},{"n":"Causalean.PO.eventCondExp_congr_on","k":"theorem","m":"Causalean.PO.Conditioning.EventCondExp","a":"PO/Conditioning","t":2,"d":"Equal-on-A integrands have equal event-level conditional expectations. This specialises eventCondExp_congr_ae to a pointwise identity on a measurable set A.","s":"∀ {Ω : Type u_1} [inst : MeasurableSpace Ω] (μ : MeasureTheory.Measure Ω) {A : Set Ω}, MeasurableSet A → ∀ {f g : Ω → ℝ}, (∀ ω ∈ A, f ω = g ω) → Causalean.PO.eventCondExp μ A f = Causalean.PO.eventCondExp μ A g","l":169,"f":"Causalean/PO/Conditioning/EventCondExp.lean"},{"n":"Causalean.PO.eventCondExp_eq_sum_condProb_mul_eventCondExp","k":"theorem","m":"Causalean.PO.Conditioning.EventCondExp","a":"PO/Conditioning","t":1,"d":"Totalized finite-partition identity for event-level averages. Fix a measurable event A and a finite family of pairwise disjoint measurable sets C i covering the whole space, such that each intersection A ∩ C i has finite measure, and let f be an integrable function. Then the event-level average of f on A equals the sum, over i, of the event-level average of f on A ∩ C i weighted by the ratio of the measure of A ∩ C i to the measure of A:","s":"∀ {Ω : Type u_1} [inst : MeasurableSpace Ω] {ι : Type u_2} [inst_1 : Fintype ι] (μ : MeasureTheory.Measure Ω) (A : Set Ω) (C : ι → Set Ω), MeasurableSet A → (∀ (i : ι), MeasurableSet (C i)) → Pairwise (Function.onFun Disjoint C) → ⋃ i, C i = Set.univ → (∀ (i : ι), μ (A ∩ C i) ≠ ⊤) → ∀ (f : Ω → ℝ), MeasureTheory.Integrable f μ → Causalean.PO.eventCondExp μ A f = ∑ i, (μ (A ∩ C i)).toReal / (μ A).toReal * Causalean.PO.eventCondExp μ (A ∩ C i) f","l":107,"f":"Causalean/PO/Conditioning/EventCondExp.lean"},{"n":"Causalean.PO.eventCondExp_mono_ae","k":"theorem","m":"Causalean.PO.Conditioning.EventCondExp","a":"PO/Conditioning","t":2,"d":"Monotonicity of eventCondExp under an a.e. inequality of integrable functions.","s":"∀ {Ω : Type u_1} [inst : MeasurableSpace Ω] (μ : MeasureTheory.Measure Ω) {A : Set Ω} {f g : Ω → ℝ}, MeasureTheory.IntegrableOn f A μ → MeasureTheory.IntegrableOn g A μ → f ≤ᵐ[μ.restrict A] g → Causalean.PO.eventCondExp μ A f ≤ Causalean.PO.eventCondExp μ A g","l":179,"f":"Causalean/PO/Conditioning/EventCondExp.lean"},{"n":"Causalean.PO.eventCondExp_mul_measure_toReal","k":"theorem","m":"Causalean.PO.Conditioning.EventCondExp","a":"PO/Conditioning","t":2,"d":"eventCondExp μ A f · (μ A).toReal = ∫_A f, including the zero-measure case where both sides collapse to 0.","s":"∀ {Ω : Type u_1} [inst : MeasurableSpace Ω] (μ : MeasureTheory.Measure Ω) (A : Set Ω), μ A ≠ ⊤ → ∀ (f : Ω → ℝ), Causalean.PO.eventCondExp μ A f * (μ A).toReal = ∫ (ω : Ω) in A, f ω ∂μ","l":67,"f":"Causalean/PO/Conditioning/EventCondExp.lean"},{"n":"Causalean.PO.eventCondExp_of_ae_eq_IndepFun","k":"theorem","m":"Causalean.PO.Conditioning.EventCondExp","a":"PO/Conditioning","t":2,"d":"Consistency + drop-of-conditioning for plain IndepFun. If z is independent of a counterfactual bundle B, and a factual integrand agrees with a measurable projection h ∘ B a.e. on the cell {z = x}, then its event-level conditional expectation on that cell equals the unconditional integral of the projection.","s":"∀ {Ω : Type u_1} [inst : MeasurableSpace Ω] {α : Type u_2} {β : Type u_3} [inst_1 : MeasurableSpace α] [inst_2 : MeasurableSpace β] {μ : MeasureTheory.Measure Ω} {z : Ω → α} {B : Ω → β}, ProbabilityTheory.IndepFun z B μ → Measurable z → Measurable B → ∀ {factualF : Ω → ℝ} {h : β → ℝ}, Measurable h → ∀ {x : α}, MeasurableSet {x} → (factualF =ᵐ[μ.restrict (z ⁻¹' {x})] fun ω => h (B ω)) → (μ (z ⁻¹' {x})).toReal ≠ 0 → Causalean.PO.eventCondExp μ (z ⁻¹' {x}) factualF = ∫ (ω : Ω), h (B ω) ∂μ","l":216,"f":"Causalean/PO/Conditioning/EventCondExp.lean"},{"n":"Causalean.PO.eventCondExp_smul","k":"theorem","m":"Causalean.PO.Conditioning.EventCondExp","a":"PO/Conditioning","t":2,"d":"Event-level conditional expectation is homogeneous with respect to real scalar multiplication.","s":"∀ {Ω : Type u_1} [inst : MeasurableSpace Ω] (μ : MeasureTheory.Measure Ω) (A : Set Ω) (c : ℝ) (g : Ω → ℝ), (Causalean.PO.eventCondExp μ A fun ω => c * g ω) = c * Causalean.PO.eventCondExp μ A g","l":208,"f":"Causalean/PO/Conditioning/EventCondExp.lean"},{"n":"Causalean.PO.eventCondExp_sub","k":"theorem","m":"Causalean.PO.Conditioning.EventCondExp","a":"PO/Conditioning","t":2,"d":"Event-level conditional expectation is additive over subtraction for integrable functions on the event.","s":"∀ {Ω : Type u_1} [inst : MeasurableSpace Ω] (μ : MeasureTheory.Measure Ω) (A : Set Ω) {g₁ g₂ : Ω → ℝ}, MeasureTheory.IntegrableOn g₁ A μ → MeasureTheory.IntegrableOn g₂ A μ → Causalean.PO.eventCondExp μ A (g₁ - g₂) = Causalean.PO.eventCondExp μ A g₁ - Causalean.PO.eventCondExp μ A g₂","l":200,"f":"Causalean/PO/Conditioning/EventCondExp.lean"},{"n":"Causalean.PO.inducedEval","k":"def","m":"Causalean.PO.Bridge.FromSCM","a":"PO/Bridge","t":1,"d":"The induced joint evaluation map assigns values to all observed-node indices by evaluating the intervened structural model at a background assignment, a potential-outcome regime, and a latent draw.","s":"{N : Type uN} → [inst : DecidableEq N] → [inst_1 : Fintype N] → {Ω : N → Type uΩ} → [inst_2 : (n : N) → MeasurableSpace (Ω n)] → (M : Causalean.SCM N Ω) → M.FixedValues → Causalean.PO.Regime (Causalean.PO.ObsIdx M) (Causalean.PO.obsValue M) → M.LatentValues → (v : Causalean.PO.ObsIdx M) → Causalean.PO.obsValue M v","l":204,"f":"Causalean/PO/Bridge/FromSCM.lean"},{"n":"Causalean.PO.inducedEval_empty_eq_evalMap","k":"theorem","m":"Causalean.PO.Bridge.FromSCM","a":"PO/Bridge","t":2,"d":"inducedEval at Regime.empty equals M.evalMap s at the same latent.","s":"∀ {N : Type uN} [inst : DecidableEq N] [inst_1 : Fintype N] {Ω : N → Type uΩ} [inst_2 : (n : N) → MeasurableSpace (Ω n)] (M : Causalean.SCM N Ω) (s : M.FixedValues) (ℓ : M.LatentValues) (v : Causalean.PO.ObsIdx M), Causalean.PO.inducedEval M s Causalean.PO.Regime.empty ℓ v = M.evalMap s ℓ ⟨↑v, Finset.mem_union_left M.unobserved v.property⟩","l":348,"f":"Causalean/PO/Bridge/FromSCM.lean"},{"n":"Causalean.PO.inducedEval_measurable","k":"theorem","m":"Causalean.PO.Bridge.FromSCM","a":"PO/Bridge","t":2,"d":"The induced joint evaluation map of a structural model is measurable in the latent draw.","s":"∀ {N : Type uN} [inst : DecidableEq N] [inst_1 : Fintype N] {Ω : N → Type uΩ} [inst_2 : (n : N) → MeasurableSpace (Ω n)] (M : Causalean.SCM N Ω) (s : M.FixedValues) (r : Causalean.PO.Regime (Causalean.PO.ObsIdx M) (Causalean.PO.obsValue M)), Measurable (Causalean.PO.inducedEval M s r)","l":225,"f":"Causalean/PO/Bridge/FromSCM.lean"},{"n":"Causalean.PO.instDecidableEqObsIdx","k":"instance","m":"Causalean.PO.Bridge.FromSCM","a":"PO/Bridge","t":2,"d":"Observed-node indices of an induced potential-outcome system have decidable equality.","s":"{N : Type uN} → [inst : DecidableEq N] → [inst_1 : Fintype N] → {Ω : N → Type uΩ} → [inst_2 : (n : N) → MeasurableSpace (Ω n)] → (M : Causalean.SCM N Ω) → DecidableEq (Causalean.PO.ObsIdx M)","l":90,"f":"Causalean/PO/Bridge/FromSCM.lean"},{"n":"Causalean.PO.instFintypeObsIdx","k":"instance","m":"Causalean.PO.Bridge.FromSCM","a":"PO/Bridge","t":2,"d":"The observed-node indices of an induced potential-outcome system form a finite type.","s":"{N : Type uN} → [inst : DecidableEq N] → [inst_1 : Fintype N] → {Ω : N → Type uΩ} → [inst_2 : (n : N) → MeasurableSpace (Ω n)] → (M : Causalean.SCM N Ω) → Fintype (Causalean.PO.ObsIdx M)","l":86,"f":"Causalean/PO/Bridge/FromSCM.lean"},{"n":"Causalean.PO.instMeasurableObsValue","k":"instance","m":"Causalean.PO.Bridge.FromSCM","a":"PO/Bridge","t":2,"d":"The value space attached to each observed-node index has the measurable structure inherited from the structural model.","s":"{N : Type uN} → [inst : DecidableEq N] → [inst_1 : Fintype N] → {Ω : N → Type uΩ} → [inst_2 : (n : N) → MeasurableSpace (Ω n)] → (M : Causalean.SCM N Ω) → (v : Causalean.PO.ObsIdx M) → MeasurableSpace (Causalean.PO.obsValue M v)","l":94,"f":"Causalean/PO/Bridge/FromSCM.lean"},{"n":"Causalean.PO.integral_eq_sum_measure_mul_eventCondExp","k":"theorem","m":"Causalean.PO.Conditioning.EventCondExp","a":"PO/Conditioning","t":2,"d":"Finite-partition total law. For any disjoint covering of univ by a Fintype-indexed family of measurable sets, ∫ f = ∑ i, (μ (A i)).toReal · eventCondExp μ (A i) f.","s":"∀ {Ω : Type u_1} [inst : MeasurableSpace Ω] {ι : Type u_2} [inst_1 : Fintype ι] (μ : MeasureTheory.Measure Ω) [MeasureTheory.IsFiniteMeasure μ] (A : ι → Set Ω), (∀ (i : ι), MeasurableSet (A i)) → Pairwise (Function.onFun Disjoint A) → ⋃ i, A i = Set.univ → ∀ (f : Ω → ℝ), MeasureTheory.Integrable f μ → ∫ (ω : Ω), f ω ∂μ = ∑ i, (μ (A i)).toReal * Causalean.PO.eventCondExp μ (A i) f","l":82,"f":"Causalean/PO/Conditioning/EventCondExp.lean"},{"n":"Causalean.PO.obsIdx_mk_random","k":"def","m":"Causalean.PO.Bridge.FromSCM","a":"PO/Bridge","t":1,"d":"Every observed random node is an index of the potential-outcome system induced by an SCM.","s":"{N : Type uN} → [inst : DecidableEq N] → [inst_1 : Fintype N] → {Ω : N → Type uΩ} → [inst_2 : (n : N) → MeasurableSpace (Ω n)] → (M : Causalean.SCM N Ω) → (n : N) → Causalean.SWIGNode.random n ∈ M.observed → Causalean.PO.ObsIdx M","l":74,"f":"Causalean/PO/Bridge/FromSCM.lean"},{"n":"Causalean.PO.obsIdx_val_injective","k":"theorem","m":"Causalean.PO.Bridge.FromSCM","a":"PO/Bridge","t":2,"d":"Two elements of ObsIdx M with the same .val are equal (injectivity of the coercion).","s":"∀ {N : Type uN} [inst : DecidableEq N] [inst_1 : Fintype N] {Ω : N → Type uΩ} [inst_2 : (n : N) → MeasurableSpace (Ω n)] {M : Causalean.SCM N Ω} {v w : Causalean.PO.ObsIdx M}, ↑v = ↑w → v = w","l":290,"f":"Causalean/PO/Bridge/FromSCM.lean"},{"n":"Causalean.PO.obsName_injective","k":"theorem","m":"Causalean.PO.Bridge.FromSCM","a":"PO/Bridge","t":2,"d":"obsName M is injective: obsName v = obsName w → v = w.","s":"∀ {N : Type uN} [inst : DecidableEq N] [inst_1 : Fintype N] {Ω : N → Type uΩ} [inst_2 : (n : N) → MeasurableSpace (Ω n)] (M : Causalean.SCM N Ω), Function.Injective (Causalean.PO.obsName✝ M)","l":294,"f":"Causalean/PO/Bridge/FromSCM.lean"},{"n":"Causalean.PO.obsValue","k":"def","m":"Causalean.PO.Bridge.FromSCM","a":"PO/Bridge","t":1,"d":"The value space of an observed-node index in the induced potential-outcome system is the structural-model value space attached to that observed node.","s":"{N : Type uN} → [inst : DecidableEq N] → [inst_1 : Fintype N] → {Ω : N → Type uΩ} → [inst_2 : (n : N) → MeasurableSpace (Ω n)] → (M : Causalean.SCM N Ω) → Causalean.PO.ObsIdx M → Type uΩ","l":79,"f":"Causalean/PO/Bridge/FromSCM.lean"},{"n":"Causalean.PO.regimeTargetN","k":"def","m":"Causalean.PO.Bridge.FromSCM","a":"PO/Bridge","t":1,"d":"The structural variable names targeted by a potential-outcome regime are the underlying names of the regime targets whose fixed counterparts are not already fixed in the structural model.","s":"{N : Type uN} → [inst : DecidableEq N] → [inst_1 : Fintype N] → {Ω : N → Type uΩ} → [inst_2 : (n : N) → MeasurableSpace (Ω n)] → (M : Causalean.SCM N Ω) → Causalean.PO.Regime (Causalean.PO.ObsIdx M) (Causalean.PO.obsValue M) → Finset N","l":104,"f":"Causalean/PO/Bridge/FromSCM.lean"},{"n":"Causalean.PO.regimeTargetN_empty","k":"theorem","m":"Causalean.PO.Bridge.FromSCM","a":"PO/Bridge","t":2,"d":"regimeTargetN M Regime.empty = ∅.","s":"∀ {N : Type uN} [inst : DecidableEq N] [inst_1 : Fintype N] {Ω : N → Type uΩ} [inst_2 : (n : N) → MeasurableSpace (Ω n)] (M : Causalean.SCM N Ω), Causalean.PO.regimeTargetN M Causalean.PO.Regime.empty = ∅","l":343,"f":"Causalean/PO/Bridge/FromSCM.lean"},{"n":"Causalean.PO.regimeTargetN_eq_image_obsName","k":"theorem","m":"Causalean.PO.Bridge.FromSCM","a":"PO/Bridge","t":2,"d":"regimeTargetN M r is the image of the structurally eligible regime targets under obsName M.","s":"∀ {N : Type uN} [inst : DecidableEq N] [inst_1 : Fintype N] {Ω : N → Type uΩ} [inst_2 : (n : N) → MeasurableSpace (Ω n)] (M : Causalean.SCM N Ω) (r : Causalean.PO.Regime (Causalean.PO.ObsIdx M) (Causalean.PO.obsValue M)), Causalean.PO.regimeTargetN M r = Finset.image (Causalean.PO.obsName✝ M) ({v ∈ r.target | Causalean.SWIGNode.fixed (Causalean.PO.obsName✝ M v) ∉ M.fixed})","l":274,"f":"Causalean/PO/Bridge/FromSCM.lean"},{"n":"Causalean.PO.regimeTargetN_mem_val","k":"theorem","m":"Causalean.PO.Bridge.FromSCM","a":"PO/Bridge","t":2,"d":"Every structural variable name targeted by a translated regime comes from an observed-node index whose observed node is the corresponding random node.","s":"∀ {N : Type uN} [inst : DecidableEq N] [inst_1 : Fintype N] {Ω : N → Type uΩ} [inst_2 : (n : N) → MeasurableSpace (Ω n)] (M : Causalean.SCM N Ω) (r : Causalean.PO.Regime (Causalean.PO.ObsIdx M) (Causalean.PO.obsValue M)), ∀ D ∈ Causalean.PO.regimeTargetN M r, ∃ v' ∈ r.target, ↑v' = Causalean.SWIGNode.random D","l":281,"f":"Causalean/PO/Bridge/FromSCM.lean"},{"n":"Causalean.PO.regimeTargetN_notFixed","k":"theorem","m":"Causalean.PO.Bridge.FromSCM","a":"PO/Bridge","t":2,"d":"fixSet-obligation: .fixed D ∉ M.fixed for every structurally eligible regime target D. The target-name translation filters out observed variables whose fixed counterpart is already in M.fixed.","s":"∀ {N : Type uN} [inst : DecidableEq N] [inst_1 : Fintype N] {Ω : N → Type uΩ} [inst_2 : (n : N) → MeasurableSpace (Ω n)] (M : Causalean.SCM N Ω) (r : Causalean.PO.Regime (Causalean.PO.ObsIdx M) (Causalean.PO.obsValue M)), ∀ D ∈ Causalean.PO.regimeTargetN M r, Causalean.SWIGNode.fixed D ∉ M.fixed","l":129,"f":"Causalean/PO/Bridge/FromSCM.lean"},{"n":"Causalean.PO.regimeTargetN_obs","k":"theorem","m":"Causalean.PO.Bridge.FromSCM","a":"PO/Bridge","t":2,"d":"fixSet-obligation for regimeTargetN: every target D has .random D ∈ M.observed.","s":"∀ {N : Type uN} [inst : DecidableEq N] [inst_1 : Fintype N] {Ω : N → Type uΩ} [inst_2 : (n : N) → MeasurableSpace (Ω n)] (M : Causalean.SCM N Ω) (r : Causalean.PO.Regime (Causalean.PO.ObsIdx M) (Causalean.PO.obsValue M)), ∀ D ∈ Causalean.PO.regimeTargetN M r, Causalean.SWIGNode.random D ∈ M.observed","l":118,"f":"Causalean/PO/Bridge/FromSCM.lean"},{"n":"Causalean.PO.setIntegral_mul_condExp_of_stronglyMeasurableLeft","k":"theorem","m":"Causalean.PO.ID.Partial.Proxy.Helpers.Common","a":"PO/ID/Partial/Proxy/Helpers","t":2,"d":"A sigma-measurable left factor can be pulled through conditional expectation inside a set integral over a measurable event.","s":"∀ {Ω : Type u_1} {m mΩ : MeasurableSpace Ω}, m ≤ mΩ → ∀ {μ : MeasureTheory.Measure Ω} [MeasureTheory.IsFiniteMeasure μ] {f g : Ω → ℝ}, MeasureTheory.StronglyMeasurable f → MeasureTheory.Integrable g μ → MeasureTheory.Integrable (f * g) μ → ∀ {s : Set Ω}, MeasurableSet s → ∫ (ω : Ω) in s, f ω * g ω ∂μ = ∫ (ω : Ω) in s, f ω * μ[g | m] ω ∂μ","l":329,"f":"Causalean/PO/ID/Partial/Proxy/Helpers/Common.lean"},{"n":"Causalean.PO.sqcup_assign_left","k":"theorem","m":"Causalean.PO.Bridge.FromSCM","a":"PO/Bridge","t":2,"d":"Regime.sqcup agrees with r₁ on r₁.target.","s":"∀ {V : Type u_1} [inst : DecidableEq V] {X : V → Type u_2} [(v : V) → MeasurableSpace (X v)] (r₁ r₂ : Causalean.PO.Regime V X) (h : r₁.Disjoint r₂) (v : V) (hv : v ∈ r₁.target), (r₁.sqcup r₂ h).assign v (Finset.mem_union_left r₂.target hv) = r₁.assign v hv","l":298,"f":"Causalean/PO/Bridge/FromSCM.lean"},{"n":"Causalean.PO.sqcup_assign_right","k":"theorem","m":"Causalean.PO.Bridge.FromSCM","a":"PO/Bridge","t":2,"d":"Regime.sqcup agrees with r₂ on r₂.target.","s":"∀ {V : Type u_1} [inst : DecidableEq V] {X : V → Type u_2} [(v : V) → MeasurableSpace (X v)] (r₁ r₂ : Causalean.PO.Regime V X) (h : r₁.Disjoint r₂) (v : V) (hv : v ∈ r₂.target), (r₁.sqcup r₂ h).assign v (Finset.mem_union_right r₁.target hv) = r₂.assign v hv","l":306,"f":"Causalean/PO/Bridge/FromSCM.lean"},{"n":"Causalean.Panel.AdoptionPath.absorbingTreatment","k":"def","m":"Causalean.Panel.AdoptionPath","a":"Panel","t":1,"d":"Absorbing treatment path 1{h < ∞ and h ≤ t}. Since ⊤ ≤ t is false, the never-treated path is untreated in every finite period.","s":"{T : ℕ} → WithTop (Fin T) → Fin T → ℝ","l":97,"f":"Causalean/Panel/AdoptionPath.lean"},{"n":"Causalean.Panel.AdoptionPath.absorbingTreatment_eq","k":"theorem","m":"Causalean.Panel.AdoptionPath","a":"Panel","t":1,"d":"For an adoption date h and period t within a horizon of T periods, the absorbing treatment indicator equals one exactly when adoption has occurred by that period, and zero otherwise.","s":"∀ {T : ℕ} (h : WithTop (Fin T)) (t : Fin T), Causalean.Panel.AdoptionPath.absorbingTreatment h t = if h ≤ ↑t then 1 else 0","l":102,"f":"Causalean/Panel/AdoptionPath.lean"},{"n":"Causalean.Panel.AdoptionPath.finite","k":"def","m":"Causalean.Panel.AdoptionPath","a":"Panel","t":1,"d":"A finite adoption path, embedded in WithTop (Fin T).","s":"{T : ℕ} → Fin T → WithTop (Fin T)","l":35,"f":"Causalean/Panel/AdoptionPath.lean"},{"n":"Causalean.Panel.AdoptionPath.finite_eq","k":"theorem","m":"Causalean.Panel.AdoptionPath","a":"Panel","t":2,"d":"Embedding a finite adoption period into the shared adoption-path type is definitionally the ordinary finite-period inclusion.","s":"∀ {T : ℕ} (g : Fin T), Causalean.Panel.AdoptionPath.finite g = ↑g","l":39,"f":"Causalean/Panel/AdoptionPath.lean"},{"n":"Causalean.Panel.AdoptionPath.isEventuallyTreated","k":"def","m":"Causalean.Panel.AdoptionPath","a":"Panel","t":1,"d":"h < ∞, the path is eventually treated.","s":"{T : ℕ} → WithTop (Fin T) → Prop","l":87,"f":"Causalean/Panel/AdoptionPath.lean"},{"n":"Causalean.Panel.AdoptionPath.isEventuallyTreated_eq","k":"theorem","m":"Causalean.Panel.AdoptionPath","a":"Panel","t":2,"d":"The Sun-Abraham-compatible eventually-treated name is the finite adoption-date predicate.","s":"∀ {T : ℕ} (h : WithTop (Fin T)), Causalean.Panel.AdoptionPath.isEventuallyTreated h = (h ≠ ⊤)","l":91,"f":"Causalean/Panel/AdoptionPath.lean"},{"n":"Causalean.Panel.AdoptionPath.isFinite","k":"def","m":"Causalean.Panel.AdoptionPath","a":"Panel","t":1,"d":"The path is finite, i.e. eventually treated.","s":"{T : ℕ} → WithTop (Fin T) → Prop","l":62,"f":"Causalean/Panel/AdoptionPath.lean"},{"n":"Causalean.Panel.AdoptionPath.isFinite_eq","k":"theorem","m":"Causalean.Panel.AdoptionPath","a":"Panel","t":2,"d":"A path is eventually treated exactly when its adoption date is not infinite.","s":"∀ {T : ℕ} (a : WithTop (Fin T)), Causalean.Panel.AdoptionPath.isFinite a = (a ≠ ⊤)","l":66,"f":"Causalean/Panel/AdoptionPath.lean"},{"n":"Causalean.Panel.AdoptionPath.isInfinite","k":"def","m":"Causalean.Panel.AdoptionPath","a":"Panel","t":1,"d":"The path is infinite, i.e. never treated.","s":"{T : ℕ} → WithTop (Fin T) → Prop","l":70,"f":"Causalean/Panel/AdoptionPath.lean"},{"n":"Causalean.Panel.AdoptionPath.isInfinite_eq","k":"theorem","m":"Causalean.Panel.AdoptionPath","a":"Panel","t":2,"d":"A path is never treated exactly when its adoption date is infinite.","s":"∀ {T : ℕ} (a : WithTop (Fin T)), Causalean.Panel.AdoptionPath.isInfinite a = (a = ⊤)","l":74,"f":"Causalean/Panel/AdoptionPath.lean"},{"n":"Causalean.Panel.AdoptionPath.isNeverTreated","k":"def","m":"Causalean.Panel.AdoptionPath","a":"Panel","t":1,"d":"A path is never treated when its adoption date is infinite.","s":"{T : ℕ} → WithTop (Fin T) → Prop","l":78,"f":"Causalean/Panel/AdoptionPath.lean"},{"n":"Causalean.Panel.AdoptionPath.isNeverTreated_eq","k":"theorem","m":"Causalean.Panel.AdoptionPath","a":"Panel","t":2,"d":"The Sun-Abraham-compatible never-treated name is the infinite adoption-date predicate.","s":"∀ {T : ℕ} (h : WithTop (Fin T)), Causalean.Panel.AdoptionPath.isNeverTreated h = (h = ⊤)","l":82,"f":"Causalean/Panel/AdoptionPath.lean"},{"n":"Causalean.Panel.AdoptionPath.le","k":"def","m":"Causalean.Panel.AdoptionPath","a":"Panel","t":1,"d":"Adoption-date predicate A ≤ t, i.e. adoption has occurred by period t.","s":"{T : ℕ} → WithTop (Fin T) → Fin T → Prop","l":44,"f":"Causalean/Panel/AdoptionPath.lean"},{"n":"Causalean.Panel.AdoptionPath.le_eq","k":"theorem","m":"Causalean.Panel.AdoptionPath","a":"Panel","t":2,"d":"The treated-by-period predicate is exactly the order comparison with the finite period viewed as an adoption date.","s":"∀ {T : ℕ} (a : WithTop (Fin T)) (t : Fin T), Causalean.Panel.AdoptionPath.le a t = (a ≤ ↑t)","l":48,"f":"Causalean/Panel/AdoptionPath.lean"},{"n":"Causalean.Panel.AdoptionPath.lt","k":"def","m":"Causalean.Panel.AdoptionPath","a":"Panel","t":1,"d":"Adoption-date predicate t < A, i.e. the path is untreated at period t.","s":"{T : ℕ} → WithTop (Fin T) → Fin T → Prop","l":53,"f":"Causalean/Panel/AdoptionPath.lean"},{"n":"Causalean.Panel.AdoptionPath.lt_eq","k":"theorem","m":"Causalean.Panel.AdoptionPath","a":"Panel","t":2,"d":"The untreated-before-adoption predicate is exactly the strict order comparison with the finite period viewed as an adoption date.","s":"∀ {T : ℕ} (a : WithTop (Fin T)) (t : Fin T), Causalean.Panel.AdoptionPath.lt a t = (↑t < a)","l":57,"f":"Causalean/Panel/AdoptionPath.lean"},{"n":"Causalean.Panel.AdoptionPath.lt_of_isInfinite","k":"theorem","m":"Causalean.Panel.AdoptionPath","a":"Panel","t":2,"d":"Never-treated paths are untreated in every finite period.","s":"∀ {T : ℕ} {a : WithTop (Fin T)} {t : Fin T}, Causalean.Panel.AdoptionPath.isInfinite a → Causalean.Panel.AdoptionPath.lt a t","l":115,"f":"Causalean/Panel/AdoptionPath.lean"},{"n":"Causalean.Panel.AdoptionPath.not_le_of_lt","k":"theorem","m":"Causalean.Panel.AdoptionPath","a":"Panel","t":2,"d":"If t < A, then adoption has not occurred by t.","s":"∀ {T : ℕ} {a : WithTop (Fin T)} {t : Fin T}, Causalean.Panel.AdoptionPath.lt a t → ¬Causalean.Panel.AdoptionPath.le a t","l":110,"f":"Causalean/Panel/AdoptionPath.lean"},{"n":"Causalean.Panel.BaselineExposure","k":"def","m":"Causalean.Panel.PO.Exposure","a":"Panel/PO","t":1,"d":"A baseline exposure: a distinguished element e_r^0 ∈ E r for every cell r. In the binary finite-memory case this is (0, …, 0).","s":"{I : Type u_1} → {T : Type u_2} → (I × T → Type u_3) → Type (max (max u_1 u_2) u_3)","l":36,"f":"Causalean/Panel/PO/Exposure.lean"},{"n":"Causalean.Panel.CellBridge.cellIndicator","k":"def","m":"Causalean.Panel.CellBridge","a":"Panel","t":1,"d":"Real-valued singleton indicator for the cell {X = i}.","s":"{Ω : Type u_1} → {ι : Type u_2} → [MeasurableSpace Ω] → (Ω → ι) → ι → Ω → ℝ","l":70,"f":"Causalean/Panel/CellBridge.lean"},{"n":"Causalean.Panel.CellBridge.cellMass","k":"def","m":"Causalean.Panel.CellBridge","a":"Panel","t":1,"d":"Cell mass (μ {G = g}).toReal.","s":"{Ω : Type u_1} → {𝒢 : Type u_2} → [inst : MeasurableSpace Ω] → MeasureTheory.Measure Ω → (Ω → 𝒢) → 𝒢 → ℝ","l":253,"f":"Causalean/Panel/CellBridge.lean"},{"n":"Causalean.Panel.CellBridge.cellMean","k":"def","m":"Causalean.Panel.CellBridge","a":"Panel","t":1,"d":"Cell mean of a real-valued function on a finite cell. Zero-mass cells use Lean's totalized division convention.","s":"{Ω : Type u_1} → {𝒢 : Type u_2} → [inst : MeasurableSpace Ω] → MeasureTheory.Measure Ω → (Ω → ℝ) → (Ω → 𝒢) → 𝒢 → ℝ","l":263,"f":"Causalean/Panel/CellBridge.lean"},{"n":"Causalean.Panel.CellBridge.cellMean_mul_cellMass","k":"theorem","m":"Causalean.Panel.CellBridge","a":"Panel","t":2,"d":"Cell mean multiplied by cell mass recovers the indicator-weighted raw integral.","s":"∀ {Ω : Type u_1} {𝒢 : Type u_2} [inst : MeasurableSpace Ω] (μ : MeasureTheory.Measure Ω) (F : Ω → ℝ) (G : Ω → 𝒢) (g : 𝒢), μ {ω | G ω = g} ≠ ⊤ → Causalean.Panel.CellBridge.cellMean μ F G g * Causalean.Panel.CellBridge.cellMass μ G g = ∫ (ω : Ω), F ω * {ω' | G ω' = g}.indicator (fun x => 1) ω ∂μ","l":331,"f":"Causalean/Panel/CellBridge.lean"},{"n":"Causalean.Panel.CellBridge.cell_integral_div_mul_cellMass","k":"theorem","m":"Causalean.Panel.CellBridge","a":"Panel","t":2,"d":"Dividing an indicator-weighted cell integral by a nonzero cell mass and multiplying back recovers the numerator; on zero-mass cells the numerator is zero because the indicator is a.e. zero.","s":"∀ {Ω : Type u_1} {𝒢 : Type u_2} [inst : MeasurableSpace Ω] (μ : MeasureTheory.Measure Ω) (F : Ω → ℝ) (G : Ω → 𝒢) (g : 𝒢), μ {ω | G ω = g} ≠ ⊤ → (∫ (ω : Ω), F ω * {ω' | G ω' = g}.indicator (fun x => 1) ω ∂μ) / Causalean.Panel.CellBridge.cellMass μ G g * Causalean.Panel.CellBridge.cellMass μ G g = ∫ (ω : Ω), F ω * {ω' | G ω' = g}.indicator (fun x => 1) ω ∂μ","l":294,"f":"Causalean/Panel/CellBridge.lean"},{"n":"Causalean.Panel.CellBridge.finiteCellIndicator","k":"def","m":"Causalean.Panel.CellBridge","a":"Panel","t":1,"d":"Finite-weighted-support cell indicator from Causalean.Panel.Weighted.IndicatorSpan.","s":"{R : Type u_1} → {𝒢 : Type u_2} → [DecidableEq 𝒢] → (R → 𝒢) → 𝒢 → R → ℝ","l":49,"f":"Causalean/Panel/CellBridge.lean"},{"n":"Causalean.Panel.CellBridge.finiteIndicatorSpan","k":"def","m":"Causalean.Panel.CellBridge","a":"Panel","t":1,"d":"Finite-weighted one-axis indicator span from Causalean.Panel.Weighted.IndicatorSpan.","s":"{R : Type u_1} → {𝒢 : Type u_2} → [Fintype 𝒢] → [DecidableEq 𝒢] → (R → 𝒢) → Submodule ℝ (R → ℝ)","l":54,"f":"Causalean/Panel/CellBridge.lean"},{"n":"Causalean.Panel.CellBridge.finiteTwoAxisIndicatorSpan","k":"def","m":"Causalean.Panel.CellBridge","a":"Panel","t":1,"d":"Finite-weighted two-axis indicator span from Causalean.Panel.Weighted.IndicatorSpan.","s":"{R : Type u_1} → {𝒢 : Type u_2} → {𝒯 : Type u_3} → [Fintype 𝒢] → [DecidableEq 𝒢] → [Fintype 𝒯] → [DecidableEq 𝒯] → (R → 𝒢) → (R → 𝒯) → Submodule ℝ (R → ℝ)","l":60,"f":"Causalean/Panel/CellBridge.lean"},{"n":"Causalean.Panel.CellBridge.indicatorSpan","k":"def","m":"Causalean.Panel.CellBridge","a":"Panel","t":1,"d":"One-axis finite indicator span generated by a measurable cell map G.","s":"{Ω : Type u_1} → {𝒢 : Type u_2} → [inst : MeasurableSpace Ω] → [Fintype 𝒢] → [DecidableEq 𝒢] → [inst_3 : MeasurableSpace 𝒢] → [MeasurableSingletonClass 𝒢] → (μ : MeasureTheory.Measure Ω) → [MeasureTheory.IsFiniteMeasure μ] → (G : Ω → 𝒢) → Measurable G → Causalean.Panel.LinearL2Class μ","l":75,"f":"Causalean/Panel/CellBridge.lean"},{"n":"Causalean.Panel.CellBridge.indicator_cell_memLp","k":"theorem","m":"Causalean.Panel.CellBridge","a":"Panel","t":2,"d":"Cell indicators are in L² under a finite measure.","s":"∀ {Ω : Type u_1} {𝒢 : Type u_2} [inst : MeasurableSpace Ω] [inst_1 : MeasurableSpace 𝒢] [MeasurableSingletonClass 𝒢] (μ : MeasureTheory.Measure Ω) [MeasureTheory.IsFiniteMeasure μ] (G : Ω → 𝒢), Measurable G → ∀ (g : 𝒢), MeasureTheory.MemLp (fun ω => {ω' | G ω' = g}.indicator (fun x => 1) ω) 2 μ","l":271,"f":"Causalean/Panel/CellBridge.lean"},{"n":"Causalean.Panel.CellBridge.integral_cell_indicator_one_eq_cellMass","k":"theorem","m":"Causalean.Panel.CellBridge","a":"Panel","t":2,"d":"Cell indicators integrate to their real cell mass.","s":"∀ {Ω : Type u_1} {𝒢 : Type u_2} [inst : MeasurableSpace Ω] [inst_1 : MeasurableSpace 𝒢] [MeasurableSingletonClass 𝒢] (μ : MeasureTheory.Measure Ω) (G : Ω → 𝒢), Measurable G → ∀ (g : 𝒢), ∫ (ω : Ω), {ω' | G ω' = g}.indicator (fun x => 1) ω ∂μ = Causalean.Panel.CellBridge.cellMass μ G g","l":282,"f":"Causalean/Panel/CellBridge.lean"},{"n":"Causalean.Panel.CellBridge.integral_eq_sum_cell","k":"theorem","m":"Causalean.Panel.CellBridge","a":"Panel","t":2,"d":"Integrate an integrable function by summing over a finite measurable partition generated by G.","s":"∀ {Ω : Type u_1} {𝒢 : Type u_2} [inst : MeasurableSpace Ω] [inst_1 : Fintype 𝒢] [inst_2 : MeasurableSpace 𝒢] [MeasurableSingletonClass 𝒢] (μ : MeasureTheory.Measure Ω) (F : Ω → ℝ) (G : Ω → 𝒢), Measurable G → MeasureTheory.Integrable F μ → ∫ (ω : Ω), F ω ∂μ = ∑ g, ∫ (ω : Ω), F ω * {ω' | G ω' = g}.indicator (fun x => 1) ω ∂μ","l":479,"f":"Causalean/Panel/CellBridge.lean"},{"n":"Causalean.Panel.CellBridge.integral_mul_indicatorSpan_eq_zero_of_cell","k":"theorem","m":"Causalean.Panel.CellBridge","a":"Panel","t":2,"d":"If a square-integrable residual is orthogonal to every cell indicator, it is orthogonal to every finite indicator-span member.","s":"∀ {Ω : Type u_1} {𝒢 : Type u_2} [inst : MeasurableSpace Ω] [inst_1 : Fintype 𝒢] [inst_2 : MeasurableSpace 𝒢] [MeasurableSingletonClass 𝒢] (μ : MeasureTheory.Measure Ω) [MeasureTheory.IsFiniteMeasure μ] (V : Ω → ℝ) (G : Ω → 𝒢), Measurable G → MeasureTheory.MemLp V 2 μ → ∀ (c : 𝒢 → ℝ), (∀ (g : 𝒢), ∫ (ω : Ω), V ω * {ω' | G ω' = g}.indicator (fun x => 1) ω ∂μ = 0) → ∫ (ω : Ω), V ω * ∑ g, c g * {ω' | G ω' = g}.indicator (fun x => 1) ω ∂μ = 0","l":342,"f":"Causalean/Panel/CellBridge.lean"},{"n":"Causalean.Panel.CellBridge.integral_mul_twoAxisIndicatorSpan_eq_zero_of_axes","k":"theorem","m":"Causalean.Panel.CellBridge","a":"Panel","t":1,"d":"If a square-integrable residual V is orthogonal in expectation to the indicator of every level g of one axis G and orthogonal in expectation to the indicator of every level t of the other axis T, then it is orthogonal in expectation to every finite linear combination of those axis indicators, with coefficients cG and cT.","s":"∀ {Ω : Type u_1} {𝒢 : Type u_2} {𝒯 : Type u_3} [inst : MeasurableSpace Ω] [inst_1 : Fintype 𝒢] [inst_2 : MeasurableSpace 𝒢] [MeasurableSingletonClass 𝒢] [inst_4 : Fintype 𝒯] [inst_5 : MeasurableSpace 𝒯] [MeasurableSingletonClass 𝒯] (μ : MeasureTheory.Measure Ω) [MeasureTheory.IsFiniteMeasure μ] (V : Ω → ℝ) (G : Ω → 𝒢) (T : Ω → 𝒯), Measurable G → Measurable T → MeasureTheory.MemLp V 2 μ → ∀ (cG : 𝒢 → ℝ) (cT : 𝒯 → ℝ), (∀ (g : 𝒢), ∫ (ω : Ω), V ω * {ω' | G ω' = g}.indicator (fun x => 1) ω ∂μ = 0) → (∀ (t : 𝒯), ∫ (ω : Ω), V ω * {ω' | T ω' = t}.indicator (fun x => 1) ω ∂μ = 0) → ∫ (ω : Ω), V ω * (∑ g, cG g * {ω' | G ω' = g}.indicator (fun x => 1) ω + ∑ t, cT t * {ω' | T ω' = t}.indicator (fun x => 1) ω) ∂μ = 0","l":403,"f":"Causalean/Panel/CellBridge.lean"},{"n":"Causalean.Panel.CellBridge.jointCellMass","k":"def","m":"Causalean.Panel.CellBridge","a":"Panel","t":1,"d":"Joint cell mass (μ {G = g ∧ T = t}).toReal.","s":"{Ω : Type u_1} → {𝒢 : Type u_2} → {𝒯 : Type u_3} → [inst : MeasurableSpace Ω] → MeasureTheory.Measure Ω → (Ω → 𝒢) → (Ω → 𝒯) → 𝒢 → 𝒯 → ℝ","l":258,"f":"Causalean/Panel/CellBridge.lean"},{"n":"Causalean.Panel.CellBridge.twoAxisIndicatorSpan","k":"def","m":"Causalean.Panel.CellBridge","a":"Panel","t":1,"d":"Two-axis finite indicator span generated by measurable maps G and T.","s":"{Ω : Type u_1} → {𝒢 : Type u_2} → {𝒯 : Type u_3} → [inst : MeasurableSpace Ω] → [Fintype 𝒢] → [DecidableEq 𝒢] → [inst_3 : MeasurableSpace 𝒢] → [MeasurableSingletonClass 𝒢] → [Fintype 𝒯] → [DecidableEq 𝒯] → [inst_7 : MeasurableSpace 𝒯] → [MeasurableSingletonClass 𝒯] → (μ : MeasureTheory.Measure Ω) → [MeasureTheory.IsFiniteMeasure μ] → (G : Ω → 𝒢) → (T : Ω → 𝒯) → Measurable G → Measurable T → Causalean.Panel.LinearL2Class μ","l":157,"f":"Causalean/Panel/CellBridge.lean"},{"n":"Causalean.Panel.Cells","k":"def","m":"Causalean.Panel.Cells","a":"Panel","t":1,"d":"Observed cells of a panel together with positive normalized weights.","s":"(I : Type u_1) → (T : Type u_2) → [Fintype I] → [Fintype T] → [DecidableEq I] → [DecidableEq T] → Type (max u_2 u_1)","l":56,"f":"Causalean/Panel/Cells.lean"},{"n":"Causalean.Panel.Cells.CohortLaw","k":"structure","m":"Causalean.Panel.FixedEffect.IndicatorClosedForms","a":"Panel/FixedEffect","t":1,"d":"Cohort law on a finite cohort index Fin C: probabilities π(g) ≥ 0 summing to one.","s":"ℕ → Type","l":457,"f":"Causalean/Panel/FixedEffect/IndicatorClosedForms.lean"},{"n":"Causalean.Panel.Cells.H_twfe_eq","k":"theorem","m":"Causalean.Panel.FixedEffect","a":"Panel/FixedEffect","t":1,"d":"The two-way fixed-effect subspace H_twfe equals the generic two-axis additive span over units and periods.","s":"∀ {I : Type u_1} {T : Type u_2}, Causalean.Panel.Cells.H_twfe = Causalean.Panel.Weighted.twoAxisAdditiveSpan I T","l":60,"f":"Causalean/Panel/FixedEffect.lean"},{"n":"Causalean.Panel.Cells.H_twfe_orthogonal_iff","k":"theorem","m":"Causalean.Panel.FixedEffect.IndicatorClosedForms","a":"Panel/FixedEffect","t":1,"d":"For a panel weighting scheme and any panel array g, g is orthogonal, under the weighted inner product, to every member of the two-way fixed-effect subspace H_twfe exactly when it is orthogonal to every unit indicator and every period indicator.","s":"∀ {I : Type u_1} {T : Type u_2} [inst : Fintype I] [inst_1 : Fintype T] [inst_2 : DecidableEq I] [inst_3 : DecidableEq T] (c : Causalean.Panel.Cells I T) (g : Causalean.Panel.Cells.V I T), (∀ h ∈ Causalean.Panel.Cells.H_twfe, c.ip g h = 0) ↔ (∀ (i₀ : I), (c.ip g fun r => if r.1 = i₀ then 1 else 0) = 0) ∧ ∀ (t₀ : T), (c.ip g fun r => if r.2 = t₀ then 1 else 0) = 0","l":128,"f":"Causalean/Panel/FixedEffect/IndicatorClosedForms.lean"},{"n":"Causalean.Panel.Cells.balanced_observed","k":"theorem","m":"Causalean.Panel.Cells","a":"Panel","t":2,"d":"The balanced panel observes every unit-period cell.","s":"∀ {I : Type u_1} {T : Type u_2} [inst : Fintype I] [inst_1 : Fintype T] [inst_2 : DecidableEq I] [inst_3 : DecidableEq T] [inst_4 : Nonempty I] [inst_5 : Nonempty T], Causalean.Panel.Cells.balanced.observed = Finset.univ","l":112,"f":"Causalean/Panel/Cells.lean"},{"n":"Causalean.Panel.Cells.balanced_weight","k":"theorem","m":"Causalean.Panel.Cells","a":"Panel","t":1,"d":"For any unit-period cell r, the balanced panel design assigns it weight equal to one divided by the total number of unit-period cells.","s":"∀ {I : Type u_1} {T : Type u_2} [inst : Fintype I] [inst_1 : Fintype T] [inst_2 : DecidableEq I] [inst_3 : DecidableEq T] [inst_4 : Nonempty I] [inst_5 : Nonempty T] (r : I × T), Causalean.Panel.Cells.balanced.weight r = 1 / (↑(Fintype.card I) * ↑(Fintype.card T))","l":116,"f":"Causalean/Panel/Cells.lean"},{"n":"Causalean.Panel.Cells.cohortPeriod_ip_period_eq","k":"theorem","m":"Causalean.Panel.FixedEffect.IndicatorClosedForms","a":"Panel/FixedEffect","t":2,"d":"In a cohort-by-period panel with cohort-period weights, the weighted inner product of any outcome array with a period indicator equals the cohort-weighted sum of outcomes in that period.","s":"∀ {C S : ℕ} [inst : Nonempty (Fin C)] [inst_1 : Nonempty (Fin S)] (law : Causalean.Panel.Cells.CohortLaw C) (hpi : ∀ (g : Fin C), 0 < ↑(law.pi g)) (Y : Causalean.Panel.Cells.V (Fin C) (Fin S)) (t₁ : Fin S), ((Causalean.Panel.Cells.cohortPeriodCells law hpi).ip Y fun r => if r.2 = t₁ then 1 else 0) = 1 / ↑(Fintype.card (Fin S)) * ∑ g, ↑(law.pi g) * Y (g, t₁)","l":705,"f":"Causalean/Panel/FixedEffect/IndicatorClosedForms.lean"},{"n":"Causalean.Panel.Cells.cohortPeriod_ip_unit_eq","k":"theorem","m":"Causalean.Panel.FixedEffect.IndicatorClosedForms","a":"Panel/FixedEffect","t":2,"d":"In a cohort-by-period panel with cohort-period weights, the weighted inner product of any outcome array with a cohort's unit indicator equals that cohort's weighted sum of outcomes across periods.","s":"∀ {C S : ℕ} [inst : Nonempty (Fin C)] [inst_1 : Nonempty (Fin S)] (law : Causalean.Panel.Cells.CohortLaw C) (hpi : ∀ (g : Fin C), 0 < ↑(law.pi g)) (Y : Causalean.Panel.Cells.V (Fin C) (Fin S)) (g₁ : Fin C), ((Causalean.Panel.Cells.cohortPeriodCells law hpi).ip Y fun r => if r.1 = g₁ then 1 else 0) = ↑(law.pi g₁) / ↑(Fintype.card (Fin S)) * ∑ t, Y (g₁, t)","l":683,"f":"Causalean/Panel/FixedEffect/IndicatorClosedForms.lean"},{"n":"Causalean.Panel.Cells.cohort_sum_pi_centered_eq_zero","k":"theorem","m":"Causalean.Panel.FixedEffect.IndicatorClosedForms","a":"Panel/FixedEffect","t":2,"d":"A singleton indicator centered by the selected category's weight has a zero weighted average whenever the weights across categories sum to one.","s":"∀ {α : Type u_3} [inst : Fintype α] [inst_1 : DecidableEq α] (w : α → ℝ), ∑ g, w g = 1 → ∀ (g₀ : α), ∑ g, w g * ((if g = g₀ then 1 else 0) - w g₀) = 0","l":504,"f":"Causalean/Panel/FixedEffect/IndicatorClosedForms.lean"},{"n":"Causalean.Panel.Cells.const_mem_H_twfe","k":"theorem","m":"Causalean.Panel.FixedEffect","a":"Panel/FixedEffect","t":1,"d":"For any constant c₀, the constant array c₀ belongs to the two-way fixed-effect subspace H_twfe.","s":"∀ {I : Type u_1} {T : Type u_2} (c₀ : ℝ), (fun x => c₀) ∈ Causalean.Panel.Cells.H_twfe","l":65,"f":"Causalean/Panel/FixedEffect.lean"},{"n":"Causalean.Panel.Cells.ipMat_add_left","k":"theorem","m":"Causalean.Panel.InnerProduct","a":"Panel","t":2,"d":"The matrix-valued panel inner product is additive in its left tuple of arrays.","s":"∀ {I : Type u_1} {T : Type u_2} [inst : Fintype I] [inst_1 : Fintype T] [inst_2 : DecidableEq I] [inst_3 : DecidableEq T] {K : ℕ} (c : Causalean.Panel.Cells I T) (A A' B : Fin K → I × T → ℝ), c.ipMat (A + A') B = c.ipMat A B + c.ipMat A' B","l":144,"f":"Causalean/Panel/InnerProduct.lean"},{"n":"Causalean.Panel.Cells.ipMat_add_right","k":"theorem","m":"Causalean.Panel.InnerProduct","a":"Panel","t":2,"d":"The matrix-valued panel inner product is additive in its right tuple of arrays.","s":"∀ {I : Type u_1} {T : Type u_2} [inst : Fintype I] [inst_1 : Fintype T] [inst_2 : DecidableEq I] [inst_3 : DecidableEq T] {K : ℕ} (c : Causalean.Panel.Cells I T) (A B B' : Fin K → I × T → ℝ), c.ipMat A (B + B') = c.ipMat A B + c.ipMat A B'","l":150,"f":"Causalean/Panel/InnerProduct.lean"},{"n":"Causalean.Panel.Cells.ipMat_apply","k":"theorem","m":"Causalean.Panel.InnerProduct","a":"Panel","t":2,"d":"The matrix-valued panel inner product has entries equal to scalar inner products of the corresponding array columns.","s":"∀ {I : Type u_1} {T : Type u_2} [inst : Fintype I] [inst_1 : Fintype T] [inst_2 : DecidableEq I] [inst_3 : DecidableEq T] {K : ℕ} (c : Causalean.Panel.Cells I T) (A B : Fin K → I × T → ℝ) (j k : Fin K), c.ipMat A B j k = c.ip (A j) (B k)","l":132,"f":"Causalean/Panel/InnerProduct.lean"},{"n":"Causalean.Panel.Cells.ipMat_eq_weighted","k":"theorem","m":"Causalean.Panel.InnerProduct","a":"Panel","t":2,"d":"Cells.ipMat is definitionally equal to WeightedSupport.ipMat.","s":"∀ {I : Type u_1} {T : Type u_2} [inst : Fintype I] [inst_1 : Fintype T] [inst_2 : DecidableEq I] [inst_3 : DecidableEq T] {K : ℕ} (c : Causalean.Panel.Cells I T) (A B : Fin K → I × T → ℝ), c.ipMat A B = Causalean.Panel.Weighted.WeightedSupport.ipMat c A B","l":80,"f":"Causalean/Panel/InnerProduct.lean"},{"n":"Causalean.Panel.Cells.ipMat_smul_left","k":"theorem","m":"Causalean.Panel.InnerProduct","a":"Panel","t":2,"d":"The matrix-valued panel inner product is homogeneous in its left tuple of arrays.","s":"∀ {I : Type u_1} {T : Type u_2} [inst : Fintype I] [inst_1 : Fintype T] [inst_2 : DecidableEq I] [inst_3 : DecidableEq T] {K : ℕ} (c : Causalean.Panel.Cells I T) (s : ℝ) (A B : Fin K → I × T → ℝ), c.ipMat (s • A) B = s • c.ipMat A B","l":156,"f":"Causalean/Panel/InnerProduct.lean"},{"n":"Causalean.Panel.Cells.ipMat_smul_right","k":"theorem","m":"Causalean.Panel.InnerProduct","a":"Panel","t":2,"d":"The matrix-valued panel inner product is homogeneous in its right tuple of arrays.","s":"∀ {I : Type u_1} {T : Type u_2} [inst : Fintype I] [inst_1 : Fintype T] [inst_2 : DecidableEq I] [inst_3 : DecidableEq T] {K : ℕ} (c : Causalean.Panel.Cells I T) (s : ℝ) (A B : Fin K → I × T → ℝ), c.ipMat A (s • B) = s • c.ipMat A B","l":162,"f":"Causalean/Panel/InnerProduct.lean"},{"n":"Causalean.Panel.Cells.ipMat_transpose","k":"theorem","m":"Causalean.Panel.InnerProduct","a":"Panel","t":2,"d":"Swapping the two tuples of arrays transposes the matrix-valued panel inner product.","s":"∀ {I : Type u_1} {T : Type u_2} [inst : Fintype I] [inst_1 : Fintype T] [inst_2 : DecidableEq I] [inst_3 : DecidableEq T] {K : ℕ} (c : Causalean.Panel.Cells I T) (A B : Fin K → I × T → ℝ), (c.ipMat A B).transpose = c.ipMat B A","l":138,"f":"Causalean/Panel/InnerProduct.lean"},{"n":"Causalean.Panel.Cells.ip_add_left","k":"theorem","m":"Causalean.Panel.InnerProduct","a":"Panel","t":2,"d":"The panel weighted inner product is additive in its left array.","s":"∀ {I : Type u_1} {T : Type u_2} [inst : Fintype I] [inst_1 : Fintype T] [inst_2 : DecidableEq I] [inst_3 : DecidableEq T] (c : Causalean.Panel.Cells I T) (A A' B : I × T → ℝ), c.ip (A + A') B = c.ip A B + c.ip A' B","l":101,"f":"Causalean/Panel/InnerProduct.lean"},{"n":"Causalean.Panel.Cells.ip_add_right","k":"theorem","m":"Causalean.Panel.InnerProduct","a":"Panel","t":2,"d":"The panel weighted inner product is additive in its right array.","s":"∀ {I : Type u_1} {T : Type u_2} [inst : Fintype I] [inst_1 : Fintype T] [inst_2 : DecidableEq I] [inst_3 : DecidableEq T] (c : Causalean.Panel.Cells I T) (A B B' : I × T → ℝ), c.ip A (B + B') = c.ip A B + c.ip A B'","l":106,"f":"Causalean/Panel/InnerProduct.lean"},{"n":"Causalean.Panel.Cells.ip_def","k":"theorem","m":"Causalean.Panel.InnerProduct","a":"Panel","t":2,"d":"The panel weighted inner product unfolds to the weighted sum over observed unit-period cells.","s":"∀ {I : Type u_1} {T : Type u_2} [inst : Fintype I] [inst_1 : Fintype T] [inst_2 : DecidableEq I] [inst_3 : DecidableEq T] (c : Causalean.Panel.Cells I T) (A B : I × T → ℝ), c.ip A B = ∑ r ∈ c.observed, c.weight r * A r * B r","l":91,"f":"Causalean/Panel/InnerProduct.lean"},{"n":"Causalean.Panel.Cells.ip_eq_weighted","k":"theorem","m":"Causalean.Panel.InnerProduct","a":"Panel","t":2,"d":"Cells.ip is definitionally equal to WeightedSupport.ip.","s":"∀ {I : Type u_1} {T : Type u_2} [inst : Fintype I] [inst_1 : Fintype T] [inst_2 : DecidableEq I] [inst_3 : DecidableEq T] (c : Causalean.Panel.Cells I T) (A B : I × T → ℝ), c.ip A B = Causalean.Panel.Weighted.WeightedSupport.ip c A B","l":67,"f":"Causalean/Panel/InnerProduct.lean"},{"n":"Causalean.Panel.Cells.ip_self_eq_zero_iff","k":"theorem","m":"Causalean.Panel.InnerProduct","a":"Panel","t":1,"d":"For a panel cell structure and a panel array A, the self inner product c.ip A A is zero exactly when A vanishes on every observed cell of c.","s":"∀ {I : Type u_1} {T : Type u_2} [inst : Fintype I] [inst_1 : Fintype T] [inst_2 : DecidableEq I] [inst_3 : DecidableEq T] (c : Causalean.Panel.Cells I T) (A : I × T → ℝ), c.ip A A = 0 ↔ ∀ r ∈ c.observed, A r = 0","l":126,"f":"Causalean/Panel/InnerProduct.lean"},{"n":"Causalean.Panel.Cells.ip_self_nonneg","k":"theorem","m":"Causalean.Panel.InnerProduct","a":"Panel","t":2,"d":"The self inner product of any panel array is nonnegative.","s":"∀ {I : Type u_1} {T : Type u_2} [inst : Fintype I] [inst_1 : Fintype T] [inst_2 : DecidableEq I] [inst_3 : DecidableEq T] (c : Causalean.Panel.Cells I T) (A : I × T → ℝ), 0 ≤ c.ip A A","l":121,"f":"Causalean/Panel/InnerProduct.lean"},{"n":"Causalean.Panel.Cells.ip_smul_left","k":"theorem","m":"Causalean.Panel.InnerProduct","a":"Panel","t":2,"d":"The panel weighted inner product is homogeneous in its left array.","s":"∀ {I : Type u_1} {T : Type u_2} [inst : Fintype I] [inst_1 : Fintype T] [inst_2 : DecidableEq I] [inst_3 : DecidableEq T] (c : Causalean.Panel.Cells I T) (s : ℝ) (A B : I × T → ℝ), c.ip (s • A) B = s * c.ip A B","l":111,"f":"Causalean/Panel/InnerProduct.lean"},{"n":"Causalean.Panel.Cells.ip_smul_right","k":"theorem","m":"Causalean.Panel.InnerProduct","a":"Panel","t":2,"d":"The panel weighted inner product is homogeneous in its right array.","s":"∀ {I : Type u_1} {T : Type u_2} [inst : Fintype I] [inst_1 : Fintype T] [inst_2 : DecidableEq I] [inst_3 : DecidableEq T] (c : Causalean.Panel.Cells I T) (s : ℝ) (A B : I × T → ℝ), c.ip A (s • B) = s * c.ip A B","l":116,"f":"Causalean/Panel/InnerProduct.lean"},{"n":"Causalean.Panel.Cells.ip_symm","k":"theorem","m":"Causalean.Panel.InnerProduct","a":"Panel","t":2,"d":"The panel weighted inner product is symmetric in its two arrays.","s":"∀ {I : Type u_1} {T : Type u_2} [inst : Fintype I] [inst_1 : Fintype T] [inst_2 : DecidableEq I] [inst_3 : DecidableEq T] (c : Causalean.Panel.Cells I T) (A B : I × T → ℝ), c.ip A B = c.ip B A","l":96,"f":"Causalean/Panel/InnerProduct.lean"},{"n":"Causalean.Panel.Cells.proj_apply_eq_of_mem_orthogonal","k":"theorem","m":"Causalean.Panel.Subspace","a":"Panel","t":2,"d":"Any nuisance-space candidate with the projection orthogonality condition matches the chosen projection on observed cells.","s":"∀ {I : Type u_1} {T : Type u_2} [inst : Fintype I] [inst_1 : Fintype T] [inst_2 : DecidableEq I] [inst_3 : DecidableEq T] (c : Causalean.Panel.Cells I T) (H : Submodule ℝ (Causalean.Panel.Cells.V I T)) (X : Causalean.Panel.Cells.V I T) {Y : Causalean.Panel.Cells.V I T}, Y ∈ H → (∀ h ∈ H, c.ip (X - Y) h = 0) → ∀ r ∈ c.observed, (c.proj H) X r = Y r","l":163,"f":"Causalean/Panel/Subspace.lean"},{"n":"Causalean.Panel.Cells.proj_apply_of_mem","k":"theorem","m":"Causalean.Panel.Subspace","a":"Panel","t":2,"d":"Projecting a nuisance-space array returns the same values on observed cells.","s":"∀ {I : Type u_1} {T : Type u_2} [inst : Fintype I] [inst_1 : Fintype T] [inst_2 : DecidableEq I] [inst_3 : DecidableEq T] (c : Causalean.Panel.Cells I T) (H : Submodule ℝ (Causalean.Panel.Cells.V I T)) {Y : Causalean.Panel.Cells.V I T}, Y ∈ H → ∀ r ∈ c.observed, (c.proj H) Y r = Y r","l":156,"f":"Causalean/Panel/Subspace.lean"},{"n":"Causalean.Panel.Cells.proj_eq_weighted","k":"theorem","m":"Causalean.Panel.Subspace","a":"Panel","t":2,"d":"The panel projection alias is definitionally equal to the generic weighted support projection.","s":"∀ {I : Type u_1} {T : Type u_2} [inst : Fintype I] [inst_1 : Fintype T] [inst_2 : DecidableEq I] [inst_3 : DecidableEq T] (c : Causalean.Panel.Cells I T) (H : Submodule ℝ (Causalean.Panel.Cells.V I T)), c.proj H = Causalean.Panel.Weighted.WeightedSupport.proj c H","l":52,"f":"Causalean/Panel/Subspace.lean"},{"n":"Causalean.Panel.Cells.proj_idem_apply","k":"theorem","m":"Causalean.Panel.Subspace","a":"Panel","t":2,"d":"Applying the chosen panel projection twice agrees with applying it once on observed cells.","s":"∀ {I : Type u_1} {T : Type u_2} [inst : Fintype I] [inst_1 : Fintype T] [inst_2 : DecidableEq I] [inst_3 : DecidableEq T] (c : Causalean.Panel.Cells I T) (H : Submodule ℝ (Causalean.Panel.Cells.V I T)) (X : Causalean.Panel.Cells.V I T), ∀ r ∈ c.observed, (c.proj H) ((c.proj H) X) r = (c.proj H) X r","l":173,"f":"Causalean/Panel/Subspace.lean"},{"n":"Causalean.Panel.Cells.proj_mem","k":"theorem","m":"Causalean.Panel.Subspace","a":"Panel","t":2,"d":"The chosen panel projection of an array lies in the nuisance subspace.","s":"∀ {I : Type u_1} {T : Type u_2} [inst : Fintype I] [inst_1 : Fintype T] [inst_2 : DecidableEq I] [inst_3 : DecidableEq T] (c : Causalean.Panel.Cells I T) (H : Submodule ℝ (Causalean.Panel.Cells.V I T)) (X : Causalean.Panel.Cells.V I T), (c.proj H) X ∈ H","l":144,"f":"Causalean/Panel/Subspace.lean"},{"n":"Causalean.Panel.Cells.proj_orthogonal","k":"theorem","m":"Causalean.Panel.Subspace","a":"Panel","t":2,"d":"The projection residual is orthogonal to every nuisance-space array under the panel weighted inner product.","s":"∀ {I : Type u_1} {T : Type u_2} [inst : Fintype I] [inst_1 : Fintype T] [inst_2 : DecidableEq I] [inst_3 : DecidableEq T] (c : Causalean.Panel.Cells I T) (H : Submodule ℝ (Causalean.Panel.Cells.V I T)) (X : Causalean.Panel.Cells.V I T) {h : Causalean.Panel.Cells.V I T}, h ∈ H → c.ip (X - (c.proj H) X) h = 0","l":149,"f":"Causalean/Panel/Subspace.lean"},{"n":"Causalean.Panel.Cells.residualize_apply","k":"theorem","m":"Causalean.Panel.Subspace","a":"Panel","t":2,"d":"Applying the panel residual maker subtracts the nuisance-space projection.","s":"∀ {I : Type u_1} {T : Type u_2} [inst : Fintype I] [inst_1 : Fintype T] [inst_2 : DecidableEq I] [inst_3 : DecidableEq T] (c : Causalean.Panel.Cells I T) (H : Submodule ℝ (Causalean.Panel.Cells.V I T)) (X : Causalean.Panel.Cells.V I T), (c.residualize H) X = X - (c.proj H) X","l":110,"f":"Causalean/Panel/Subspace.lean"},{"n":"Causalean.Panel.Cells.residualize_eq_weighted","k":"theorem","m":"Causalean.Panel.Subspace","a":"Panel","t":2,"d":"The panel residual-maker alias is definitionally equal to the generic weighted support residual maker.","s":"∀ {I : Type u_1} {T : Type u_2} [inst : Fintype I] [inst_1 : Fintype T] [inst_2 : DecidableEq I] [inst_3 : DecidableEq T] (c : Causalean.Panel.Cells I T) (H : Submodule ℝ (Causalean.Panel.Cells.V I T)), c.residualize H = Causalean.Panel.Weighted.WeightedSupport.residualize c H","l":63,"f":"Causalean/Panel/Subspace.lean"},{"n":"Causalean.Panel.Cells.residualize_idem_apply","k":"theorem","m":"Causalean.Panel.Subspace","a":"Panel","t":2,"d":"Applying the panel residual maker twice agrees with applying it once on observed cells.","s":"∀ {I : Type u_1} {T : Type u_2} [inst : Fintype I] [inst_1 : Fintype T] [inst_2 : DecidableEq I] [inst_3 : DecidableEq T] (c : Causalean.Panel.Cells I T) (H : Submodule ℝ (Causalean.Panel.Cells.V I T)) (X : Causalean.Panel.Cells.V I T), ∀ r ∈ c.observed, (c.residualize H) ((c.residualize H) X) r = (c.residualize H) X r","l":136,"f":"Causalean/Panel/Subspace.lean"},{"n":"Causalean.Panel.Cells.residualize_in_orthogonal","k":"theorem","m":"Causalean.Panel.Subspace","a":"Panel","t":1,"d":"For an array h lying in the nuisance subspace H, the array X residualized against H is orthogonal to h under the panel weighted inner product c.ip.","s":"∀ {I : Type u_1} {T : Type u_2} [inst : Fintype I] [inst_1 : Fintype T] [inst_2 : DecidableEq I] [inst_3 : DecidableEq T] (c : Causalean.Panel.Cells I T) (H : Submodule ℝ (Causalean.Panel.Cells.V I T)) (X : Causalean.Panel.Cells.V I T) {h : Causalean.Panel.Cells.V I T}, h ∈ H → c.ip (c.tildeX H X) h = 0","l":122,"f":"Causalean/Panel/Subspace.lean"},{"n":"Causalean.Panel.Cells.residualize_self_of_mem","k":"theorem","m":"Causalean.Panel.Subspace","a":"Panel","t":2,"d":"A nuisance-space array residualizes to zero on observed cells.","s":"∀ {I : Type u_1} {T : Type u_2} [inst : Fintype I] [inst_1 : Fintype T] [inst_2 : DecidableEq I] [inst_3 : DecidableEq T] (c : Causalean.Panel.Cells I T) (H : Submodule ℝ (Causalean.Panel.Cells.V I T)) {X : Causalean.Panel.Cells.V I T}, X ∈ H → ∀ r ∈ c.observed, c.tildeX H X r = 0","l":129,"f":"Causalean/Panel/Subspace.lean"},{"n":"Causalean.Panel.Cells.sum_centered_indicator_mem_real","k":"theorem","m":"Causalean.Panel.FixedEffect.IndicatorClosedForms","a":"Panel/FixedEffect","t":2,"d":"Centering the indicator of a finite subset by its uniform average makes its sum across the index set equal to zero.","s":"∀ {α : Type u_3} [inst : Fintype α] [inst_1 : DecidableEq α] (S : Finset α), ∑ x, ((if x ∈ S then 1 else 0) - ↑S.card / ↑(Fintype.card α)) = 0","l":87,"f":"Causalean/Panel/FixedEffect/IndicatorClosedForms.lean"},{"n":"Causalean.Panel.Cells.tildeXVec_apply","k":"theorem","m":"Causalean.Panel.Subspace","a":"Panel","t":2,"d":"Column-wise residualization residualizes each regressor column separately.","s":"∀ {I : Type u_1} {T : Type u_2} [inst : Fintype I] [inst_1 : Fintype T] [inst_2 : DecidableEq I] [inst_3 : DecidableEq T] {K : ℕ} (c : Causalean.Panel.Cells I T) (H : Submodule ℝ (Causalean.Panel.Cells.V I T)) (X : Fin K → Causalean.Panel.Cells.V I T) (k : Fin K), c.tildeXVec H X k = c.tildeX H (X k)","l":116,"f":"Causalean/Panel/Subspace.lean"},{"n":"Causalean.Panel.Cells.tildeXVec_eq_weighted","k":"theorem","m":"Causalean.Panel.Subspace","a":"Panel","t":2,"d":"The panel column-wise residualization alias is definitionally equal to the generic weighted support column-wise residualization.","s":"∀ {I : Type u_1} {T : Type u_2} [inst : Fintype I] [inst_1 : Fintype T] [inst_2 : DecidableEq I] [inst_3 : DecidableEq T] {K : ℕ} (c : Causalean.Panel.Cells I T) (H : Submodule ℝ (Causalean.Panel.Cells.V I T)) (X : Fin K → Causalean.Panel.Cells.V I T), c.tildeXVec H X = Causalean.Panel.Weighted.WeightedSupport.tildeXVec c H X","l":90,"f":"Causalean/Panel/Subspace.lean"},{"n":"Causalean.Panel.Cells.tildeX_cell_indicator_cohortPeriod","k":"theorem","m":"Causalean.Panel.FixedEffect.IndicatorClosedForms","a":"Panel/FixedEffect","t":1,"d":"For a target cohort g₀, a target period t₀, and an evaluation cell r, the cohort-period-weighted two-way-fixed-effect residual of the cell indicator 1{g = g₀, t = t₀}, evaluated at r, equals the product of centered cohort and period indicators minus the corresponding cross terms fixed by the gauge normalization ∑_t β(t) = π(g₀).","s":"∀ {C S : ℕ} [inst : Nonempty (Fin C)] [inst_1 : Nonempty (Fin S)] (law : Causalean.Panel.Cells.CohortLaw C) (hpi : ∀ (g : Fin C), 0 < ↑(law.pi g)) (g₀ : Fin C) (t₀ : Fin S) (r : Fin C × Fin S), (Causalean.Panel.Cells.cohortPeriodCells law hpi).tildeX Causalean.Panel.Cells.H_twfe (fun s => if s.1 = g₀ ∧ s.2 = t₀ then 1 else 0) r = ((((if r.1 = g₀ then 1 else 0) * if r.2 = t₀ then 1 else 0) - ↑(law.pi g₀) * if r.2 = t₀ then 1 else 0) - 1 / ↑(Fintype.card (Fin S)) * if r.1 = g₀ then 1 else 0) + ↑(law.pi g₀) / ↑(Fintype.card (Fin S))","l":614,"f":"Causalean/Panel/FixedEffect/IndicatorClosedForms.lean"},{"n":"Causalean.Panel.Cells.tildeX_const","k":"theorem","m":"Causalean.Panel.FixedEffect.IndicatorClosedForms","a":"Panel/FixedEffect","t":2,"d":"The TWFE residual of a constant array vanishes on every observed cell.","s":"∀ {I : Type u_1} {T : Type u_2} [inst : Fintype I] [inst_1 : Fintype T] [inst_2 : DecidableEq I] [inst_3 : DecidableEq T] (c : Causalean.Panel.Cells I T) (k : ℝ), ∀ r ∈ c.observed, c.tildeX Causalean.Panel.Cells.H_twfe (fun x => k) r = 0","l":260,"f":"Causalean/Panel/FixedEffect/IndicatorClosedForms.lean"},{"n":"Causalean.Panel.Cells.tildeX_diagonal_indicator_cohortPeriod","k":"theorem","m":"Causalean.Panel.FixedEffect.IndicatorClosedForms","a":"Panel/FixedEffect","t":1,"d":"For an evaluation cell r, the cohort-period-weighted two-way-fixed-effect residual of the diagonal (switch-on) indicator 1{g.val = t.val}, evaluated at r, equals the indicator minus the cohort-conditional mean, minus the period-conditional mean net of the grand mean.","s":"∀ {C S : ℕ} [inst : Nonempty (Fin C)] [inst_1 : Nonempty (Fin S)] (law : Causalean.Panel.Cells.CohortLaw C) (hpi : ∀ (g : Fin C), 0 < ↑(law.pi g)) (r : Fin C × Fin S), (Causalean.Panel.Cells.cohortPeriodCells law hpi).tildeX Causalean.Panel.Cells.H_twfe (fun s => if ↑s.1 = ↑s.2 then 1 else 0) r = (((if ↑r.1 = ↑r.2 then 1 else 0) - if ↑r.1 < S then 1 / ↑(Fintype.card (Fin S)) else 0) - if h : ↑r.2 < C then ↑(law.pi ⟨↑r.2, h⟩) else 0) + 1 / ↑(Fintype.card (Fin S)) * ∑ g, if ↑g < S then ↑(law.pi g) else 0","l":1076,"f":"Causalean/Panel/FixedEffect/IndicatorClosedForms.lean"},{"n":"Causalean.Panel.Cells.tildeX_eq","k":"theorem","m":"Causalean.Panel.Subspace","a":"Panel","t":2,"d":"Residualizing a panel array subtracts its nuisance-space projection.","s":"∀ {I : Type u_1} {T : Type u_2} [inst : Fintype I] [inst_1 : Fintype T] [inst_2 : DecidableEq I] [inst_3 : DecidableEq T] (c : Causalean.Panel.Cells I T) (H : Submodule ℝ (Causalean.Panel.Cells.V I T)) (X : Causalean.Panel.Cells.V I T), c.tildeX H X = X - (c.proj H) X","l":104,"f":"Causalean/Panel/Subspace.lean"},{"n":"Causalean.Panel.Cells.tildeX_eq_weighted","k":"theorem","m":"Causalean.Panel.Subspace","a":"Panel","t":2,"d":"The panel residualized-array alias is definitionally equal to the generic weighted support residualized array.","s":"∀ {I : Type u_1} {T : Type u_2} [inst : Fintype I] [inst_1 : Fintype T] [inst_2 : DecidableEq I] [inst_3 : DecidableEq T] (c : Causalean.Panel.Cells I T) (H : Submodule ℝ (Causalean.Panel.Cells.V I T)) (X : Causalean.Panel.Cells.V I T), c.tildeX H X = Causalean.Panel.Weighted.WeightedSupport.tildeX c H X","l":75,"f":"Causalean/Panel/Subspace.lean"},{"n":"Causalean.Panel.Cells.tildeX_period_indicator","k":"theorem","m":"Causalean.Panel.FixedEffect.IndicatorClosedForms","a":"Panel/FixedEffect","t":2,"d":"Period indicator is absorbed by H_twfe: ind(t = t₀) lies in H_twfe (via a := 0, b(t) := ind(t = t₀)), so its residual vanishes on every observed cell. Independent of the weights.","s":"∀ {I : Type u_1} {T : Type u_2} [inst : Fintype I] [inst_1 : Fintype T] [inst_2 : DecidableEq I] [inst_3 : DecidableEq T] (c : Causalean.Panel.Cells I T) (t₀ : T), ∀ r ∈ c.observed, c.tildeX Causalean.Panel.Cells.H_twfe (fun s => if s.2 = t₀ then 1 else 0) r = 0","l":280,"f":"Causalean/Panel/FixedEffect/IndicatorClosedForms.lean"},{"n":"Causalean.Panel.Cells.tildeX_product_indicator_balanced","k":"theorem","m":"Causalean.Panel.FixedEffect.IndicatorClosedForms","a":"Panel/FixedEffect","t":1,"d":"For a subset of units S_I, a subset of periods S_T, and an evaluation cell r, under balanced (equal) weights, the two-way-fixed-effect residual of the product indicator 1{i ∈ S_I}·1{t ∈ S_T}, evaluated at r, factorizes into the product of the centered unit indicator and the centered period indicator.","s":"∀ {I : Type u_1} {T : Type u_2} [inst : Fintype I] [inst_1 : Fintype T] [inst_2 : DecidableEq I] [inst_3 : DecidableEq T] [inst_4 : Nonempty I] [inst_5 : Nonempty T] (S_I : Finset I) (S_T : Finset T) (r : I × T), Causalean.Panel.Cells.balanced.tildeX Causalean.Panel.Cells.H_twfe (fun s => (if s.1 ∈ S_I then 1 else 0) * if s.2 ∈ S_T then 1 else 0) r = ((if r.1 ∈ S_I then 1 else 0) - ↑S_I.card / ↑(Fintype.card I)) * ((if r.2 ∈ S_T then 1 else 0) - ↑S_T.card / ↑(Fintype.card T))","l":387,"f":"Causalean/Panel/FixedEffect/IndicatorClosedForms.lean"},{"n":"Causalean.Panel.Cells.tildeX_triangular_indicator_cohortPeriod","k":"theorem","m":"Causalean.Panel.FixedEffect.IndicatorClosedForms","a":"Panel/FixedEffect","t":1,"d":"For an evaluation cell r, the cohort-period-weighted two-way-fixed-effect residual of the triangular (continued-treatment) indicator 1{g.val < t.val}, evaluated at r, equals the indicator minus the cohort-conditional mean, minus the period-conditional mean net of the grand mean.","s":"∀ {C S : ℕ} [inst : Nonempty (Fin C)] [inst_1 : Nonempty (Fin S)] (law : Causalean.Panel.Cells.CohortLaw C) (hpi : ∀ (g : Fin C), 0 < ↑(law.pi g)) (r : Fin C × Fin S), (Causalean.Panel.Cells.cohortPeriodCells law hpi).tildeX Causalean.Panel.Cells.H_twfe (fun s => if ↑s.1 < ↑s.2 then 1 else 0) r = (if ↑r.1 < ↑r.2 then 1 else 0) - (∑ s, if ↑r.1 < ↑s then 1 else 0) / ↑(Fintype.card (Fin S)) - ∑ g, (if ↑g < ↑r.2 then 1 else 0) * ↑(law.pi g) + 1 / ↑(Fintype.card (Fin S)) * ∑ g, ∑ s, if ↑g < ↑s then ↑(law.pi g) else 0","l":1137,"f":"Causalean/Panel/FixedEffect/IndicatorClosedForms.lean"},{"n":"Causalean.Panel.Cells.tildeX_unit_indicator","k":"theorem","m":"Causalean.Panel.FixedEffect.IndicatorClosedForms","a":"Panel/FixedEffect","t":2,"d":"Unit indicator is absorbed by H_twfe: ind(i = i₀) lies in H_twfe (via a(i) := ind(i = i₀), b(t) := 0), so its residual vanishes on every observed cell. Independent of the weights.","s":"∀ {I : Type u_1} {T : Type u_2} [inst : Fintype I] [inst_1 : Fintype T] [inst_2 : DecidableEq I] [inst_3 : DecidableEq T] (c : Causalean.Panel.Cells I T) (i₀ : I), ∀ r ∈ c.observed, c.tildeX Causalean.Panel.Cells.H_twfe (fun s => if s.1 = i₀ then 1 else 0) r = 0","l":269,"f":"Causalean/Panel/FixedEffect/IndicatorClosedForms.lean"},{"n":"Causalean.Panel.EstimandCharacterization.EventStudyContamination.EventStudyPopulation","k":"structure","m":"Causalean.Panel.EstimandCharacterization.EventStudyContamination.PopulationBridge","a":"Panel/EstimandCharacterization/EventStudyContamination","t":1,"d":"A staggered-adoption event-study population: a probability space carrying a realized adoption cohort G — with every cohort cell measurable — a calendar-time map time that is strictly increasing in the period index, a finite set cohorts of adoption cohorts in the event-study support, and adoption-path–indexed potential outcomes Ypath t h ω = Y_{ωt}(h) satisfying structural no-anticipation: in any period where a path is untreated, its outcome equals the never-treated outcome, for every unit.","s":"ℕ → Type (u_1 + 1)","l":50,"f":"Causalean/Panel/EstimandCharacterization/EventStudyContamination/PopulationBridge.lean"},{"n":"Causalean.Panel.EstimandCharacterization.EventStudyContamination.EventStudyPopulation.IW_convex_characterization_population","k":"theorem","m":"Causalean.Panel.EstimandCharacterization.EventStudyContamination.PopulationBridge","a":"Panel/EstimandCharacterization/EventStudyContamination","t":1,"d":"Population interaction-weighted characterization (headline). For a staggered-adoption event-study population E and an interaction-weighted design I on the system it induces, if the induced system satisfies comparison-group parallel trends, the eligibility, baseline, target, and comparison-group support conditions of IWSupport hold, the aggregation weights rho are nonnegative and sum to one over the eligible cohorts, and every eligible cohort's population CATT at the fixed event time lies between bounds lo and hi, then the interaction-weighted estimand nuIW is the rho-weighted convex average of the genuine population effects CATT g ℓ, and in particular lies between lo and hi — with no contamination from other event times.","s":"∀ {T : ℕ} (E : Causalean.Panel.EstimandCharacterization.EventStudyContamination.EventStudyPopulation T) (I : E.toSystem.IWDesign), E.toSystem.IWComparisonParallelTrends I → E.toSystem.IWSupport I → (∀ g ∈ I.cohortsIW, 0 ≤ I.rho g) → ∑ g ∈ I.cohortsIW, I.rho g = 1 → ∀ {lo hi : ℝ}, (∀ g ∈ I.cohortsIW, lo ≤ E.toSystem.CATT g I.eventTime) → (∀ g ∈ I.cohortsIW, E.toSystem.CATT g I.eventTime ≤ hi) → (∀ g ∈ I.cohortsIW, E.toSystem.Delta I g = E.toSystem.CATT g I.eventTime) ∧ E.toSystem.nuIW I = ∑ g ∈ I.cohortsIW, I.rho g * E.toSystem.CATT g I.eventTime ∧ lo ≤ E.toSystem.nuIW I ∧ E.toSystem.nuIW I ≤ hi","l":295,"f":"Causalean/Panel/EstimandCharacterization/EventStudyContamination/PopulationBridge.lean"},{"n":"Causalean.Panel.EstimandCharacterization.EventStudyContamination.EventStudyPopulation.OutcomesIntegrable","k":"def","m":"Causalean.Panel.EstimandCharacterization.EventStudyContamination.PopulationBridge","a":"Panel/EstimandCharacterization/EventStudyContamination","t":1,"d":"Outcome integrability (Sun-Abraham assumption H5). Every potential-outcome slice Ypath t h is μ-integrable. This is the population content of the source theorem's integrability hypothesis: it is exactly the condition under which each cohort-cell mean E.cellMean (Ypath t h) · = E[Y_{·t}(h) ∣ G = ·] is a genuine finite expectation rather than only a totalized value. Not every population satisfies it; for example, heavy-tailed potential outcomes can fail this condition.","s":"{T : ℕ} → Causalean.Panel.EstimandCharacterization.EventStudyContamination.EventStudyPopulation T → Prop","l":212,"f":"Causalean/Panel/EstimandCharacterization/EventStudyContamination/PopulationBridge.lean"},{"n":"Causalean.Panel.EstimandCharacterization.EventStudyContamination.EventStudyPopulation.cell","k":"def","m":"Causalean.Panel.EstimandCharacterization.EventStudyContamination.PopulationBridge","a":"Panel/EstimandCharacterization/EventStudyContamination","t":1,"d":"The event {ω | G ω = h} for an adoption path.","s":"{T : ℕ} → (E : Causalean.Panel.EstimandCharacterization.EventStudyContamination.EventStudyPopulation T) → WithTop (Fin T) → Set E.Ω","l":102,"f":"Causalean/Panel/EstimandCharacterization/EventStudyContamination/PopulationBridge.lean"},{"n":"Causalean.Panel.EstimandCharacterization.EventStudyContamination.EventStudyPopulation.cellMass","k":"def","m":"Causalean.Panel.EstimandCharacterization.EventStudyContamination.PopulationBridge","a":"Panel/EstimandCharacterization/EventStudyContamination","t":1,"d":"The population mass of an adoption-path event, as a real number.","s":"{T : ℕ} → Causalean.Panel.EstimandCharacterization.EventStudyContamination.EventStudyPopulation T → WithTop (Fin T) → ℝ","l":106,"f":"Causalean/Panel/EstimandCharacterization/EventStudyContamination/PopulationBridge.lean"},{"n":"Causalean.Panel.EstimandCharacterization.EventStudyContamination.EventStudyPopulation.cellMean","k":"def","m":"Causalean.Panel.EstimandCharacterization.EventStudyContamination.PopulationBridge","a":"Panel/EstimandCharacterization/EventStudyContamination","t":1,"d":"Event-level conditional mean on an adoption-path cell. This is totalized by eventCondExp, so zero-mass paths are allowed at the population-bridge layer.","s":"{T : ℕ} → (E : Causalean.Panel.EstimandCharacterization.EventStudyContamination.EventStudyPopulation T) → (E.Ω → ℝ) → WithTop (Fin T) → ℝ","l":110,"f":"Causalean/Panel/EstimandCharacterization/EventStudyContamination/PopulationBridge.lean"},{"n":"Causalean.Panel.EstimandCharacterization.EventStudyContamination.EventStudyPopulation.cellMean_congr_on","k":"theorem","m":"Causalean.Panel.EstimandCharacterization.EventStudyContamination.PopulationBridge","a":"Panel/EstimandCharacterization/EventStudyContamination","t":2,"d":"Event-level means agree when the integrands agree pointwise on the adoption-path cell.","s":"∀ {T : ℕ} (E : Causalean.Panel.EstimandCharacterization.EventStudyContamination.EventStudyPopulation T) {f g : E.Ω → ℝ} (h : WithTop (Fin T)), (∀ ω ∈ E.cell h, f ω = g ω) → E.cellMean f h = E.cellMean g h","l":116,"f":"Causalean/Panel/EstimandCharacterization/EventStudyContamination/PopulationBridge.lean"},{"n":"Causalean.Panel.EstimandCharacterization.EventStudyContamination.EventStudyPopulation.cellMean_sub","k":"theorem","m":"Causalean.Panel.EstimandCharacterization.EventStudyContamination.PopulationBridge","a":"Panel/EstimandCharacterization/EventStudyContamination","t":2,"d":"Event-level means are additive over subtraction of integrable integrands.","s":"∀ {T : ℕ} (E : Causalean.Panel.EstimandCharacterization.EventStudyContamination.EventStudyPopulation T) {f g : E.Ω → ℝ} (h : WithTop (Fin T)), MeasureTheory.IntegrableOn f (E.cell h) E.μ → MeasureTheory.IntegrableOn g (E.cell h) E.μ → E.cellMean (f - g) h = E.cellMean f h - E.cellMean g h","l":123,"f":"Causalean/Panel/EstimandCharacterization/EventStudyContamination/PopulationBridge.lean"},{"n":"Causalean.Panel.EstimandCharacterization.EventStudyContamination.EventStudyPopulation.contamination_representation_population","k":"theorem","m":"Causalean.Panel.EstimandCharacterization.EventStudyContamination.PopulationBridge","a":"Panel/EstimandCharacterization/EventStudyContamination","t":1,"d":"Population contamination representation (headline). For a staggered-adoption event-study population E and a conventional design D on the system it induces, if the never-treated potential outcome follows the additive parallel-trends restriction, the included, displayed, and admissible event times all lie within the declared finite support, and the cell-grid weighted-projection residualization input is supplied, then the conventional TWFE event-study coefficient D.mu equals the contamination-weighted sum of genuine population cohort-relative-time effects CATT g e = E[Y_{·t}(g) − Y_{·t}(∞) ∣ G = g]. Consistency and no-anticipation are derived from the potential-outcome structure rather than assumed; this is the Sun-Abraham contamination theorem stated over a genuinely potential-outcome-anchored system.","s":"∀ {T : ℕ} (E : Causalean.Panel.EstimandCharacterization.EventStudyContamination.EventStudyPopulation T) (D : E.toSystem.ConventionalDesign), E.toSystem.MeanParallelUntreated → E.toSystem.ConventionalFiniteSupport D → E.toSystem.CellGridResidualization D → D.mu = ∑ ge ∈ E.toSystem.admissibleCells D.eventSupport, E.toSystem.omega D ge.1 ge.2 * E.toSystem.CATT ge.1 ge.2","l":269,"f":"Causalean/Panel/EstimandCharacterization/EventStudyContamination/PopulationBridge.lean"},{"n":"Causalean.Panel.EstimandCharacterization.EventStudyContamination.EventStudyPopulation.observed","k":"def","m":"Causalean.Panel.EstimandCharacterization.EventStudyContamination.PopulationBridge","a":"Panel/EstimandCharacterization/EventStudyContamination","t":1,"d":"The observed outcome at period t: the potential outcome under the unit's realized adoption path. Consistency is thus definitional.","s":"{T : ℕ} → (E : Causalean.Panel.EstimandCharacterization.EventStudyContamination.EventStudyPopulation T) → Fin T → E.Ω → ℝ","l":130,"f":"Causalean/Panel/EstimandCharacterization/EventStudyContamination/PopulationBridge.lean"},{"n":"Causalean.Panel.EstimandCharacterization.EventStudyContamination.EventStudyPopulation.observed_eqOn_cell","k":"theorem","m":"Causalean.Panel.EstimandCharacterization.EventStudyContamination.PopulationBridge","a":"Panel/EstimandCharacterization/EventStudyContamination","t":2,"d":"On the cohort cell {G = finitePath g}, the observed outcome equals the own-path potential outcome — the pointwise content of consistency.","s":"∀ {T : ℕ} (E : Causalean.Panel.EstimandCharacterization.EventStudyContamination.EventStudyPopulation T) (g t : Fin T), ∀ ω ∈ E.cell (Causalean.Panel.EstimandCharacterization.EventStudyContamination.EventStudySystem.finitePath g), E.observed t ω = E.Ypath t (Causalean.Panel.EstimandCharacterization.EventStudyContamination.EventStudySystem.finitePath g) ω","l":161,"f":"Causalean/Panel/EstimandCharacterization/EventStudyContamination/PopulationBridge.lean"},{"n":"Causalean.Panel.EstimandCharacterization.EventStudyContamination.EventStudyPopulation.toSystem","k":"def","m":"Causalean.Panel.EstimandCharacterization.EventStudyContamination.PopulationBridge","a":"Panel/EstimandCharacterization/EventStudyContamination","t":1,"d":"The event-study system induced by a population: every mean field is the cohort-cell conditional mean of the appropriate potential-outcome slice.","s":"{T : ℕ} → Causalean.Panel.EstimandCharacterization.EventStudyContamination.EventStudyPopulation T → Causalean.Panel.EstimandCharacterization.EventStudyContamination.EventStudySystem T","l":135,"f":"Causalean/Panel/EstimandCharacterization/EventStudyContamination/PopulationBridge.lean"},{"n":"Causalean.Panel.EstimandCharacterization.EventStudyContamination.EventStudyPopulation.toSystem_CATT_eq_meanDiff","k":"theorem","m":"Causalean.Panel.EstimandCharacterization.EventStudyContamination.PopulationBridge","a":"Panel/EstimandCharacterization/EventStudyContamination","t":2,"d":"Integrability makes CATT a genuine expected contrast. Under outcome integrability (assumption H5), the two cohort-cell means combine into a single cell mean of the potential-outcome difference: each summand is E[Y_{·t}(g) − Y_{·t}(∞) ∣ G = g], a genuine expectation of the individual treatment-effect random variable. This is where the integrability hypothesis does real work — cellMean_sub requires each slice to be integrable on the cell, so without H5 the two means could not be merged.","s":"∀ {T : ℕ} (E : Causalean.Panel.EstimandCharacterization.EventStudyContamination.EventStudyPopulation T), E.OutcomesIntegrable → ∀ (g : Fin T) (e : ℤ), E.toSystem.CATT g e = (↑(E.toSystem.targetPeriods g e).card)⁻¹ * ∑ t ∈ E.toSystem.targetPeriods g e, E.cellMean (E.Ypath t (Causalean.Panel.EstimandCharacterization.EventStudyContamination.EventStudySystem.finitePath g) - E.Ypath t ⊤) (Causalean.Panel.EstimandCharacterization.EventStudyContamination.EventStudySystem.finitePath g)","l":248,"f":"Causalean/Panel/EstimandCharacterization/EventStudyContamination/PopulationBridge.lean"},{"n":"Causalean.Panel.EstimandCharacterization.EventStudyContamination.EventStudyPopulation.toSystem_CATT_eq_po_contrast","k":"theorem","m":"Causalean.Panel.EstimandCharacterization.EventStudyContamination.PopulationBridge","a":"Panel/EstimandCharacterization/EventStudyContamination","t":1,"d":"Causal-meaning certificate. For a population satisfying the event-study setup, in the system it induces, the treatment-effect estimand at cohort g and relative time e equals the cohort-cell average of the population potential-outcome contrast E[Y_{·t}(g) ∣ G = g] − E[Y_{·t}(∞) ∣ G = g] over the relevant periods, so the estimand carries genuine causal content rather than a free-standing definition on reals.","s":"∀ {T : ℕ} (E : Causalean.Panel.EstimandCharacterization.EventStudyContamination.EventStudyPopulation T) (g : Fin T) (e : ℤ), E.toSystem.CATT g e = (↑(E.toSystem.targetPeriods g e).card)⁻¹ * ∑ t ∈ E.toSystem.targetPeriods g e, (E.cellMean (E.Ypath t (Causalean.Panel.EstimandCharacterization.EventStudyContamination.EventStudySystem.finitePath g)) (Causalean.Panel.EstimandCharacterization.EventStudyContamination.EventStudySystem.finitePath g) - E.cellMean (E.Ypath t ⊤) (Causalean.Panel.EstimandCharacterization.EventStudyContamination.EventStudySystem.finitePath g))","l":232,"f":"Causalean/Panel/EstimandCharacterization/EventStudyContamination/PopulationBridge.lean"},{"n":"Causalean.Panel.EstimandCharacterization.EventStudyContamination.EventStudyPopulation.toSystem_causalRestrictions","k":"theorem","m":"Causalean.Panel.EstimandCharacterization.EventStudyContamination.PopulationBridge","a":"Panel/EstimandCharacterization/EventStudyContamination","t":2,"d":"Bundle: the induced system satisfies the Sun-Abraham causal restrictions, given the additive parallel-trends hypothesis. Consistency and no-anticipation are derived; only parallel trends is assumed.","s":"∀ {T : ℕ} (E : Causalean.Panel.EstimandCharacterization.EventStudyContamination.EventStudyPopulation T), E.toSystem.MeanParallelUntreated → E.toSystem.EventStudyCausalRestrictions","l":222,"f":"Causalean/Panel/EstimandCharacterization/EventStudyContamination/PopulationBridge.lean"},{"n":"Causalean.Panel.EstimandCharacterization.EventStudyContamination.EventStudyPopulation.toSystem_consistency","k":"theorem","m":"Causalean.Panel.EstimandCharacterization.EventStudyContamination.PopulationBridge","a":"Panel/EstimandCharacterization/EventStudyContamination","t":2,"d":"Consistency is derived. The observed cohort mean equals the own-path potential-outcome mean, because the observed outcome is the potential outcome under the realized path.","s":"∀ {T : ℕ} (E : Causalean.Panel.EstimandCharacterization.EventStudyContamination.EventStudyPopulation T), E.toSystem.Consistency","l":171,"f":"Causalean/Panel/EstimandCharacterization/EventStudyContamination/PopulationBridge.lean"},{"n":"Causalean.Panel.EstimandCharacterization.EventStudyContamination.EventStudyPopulation.toSystem_noAnticipation","k":"theorem","m":"Causalean.Panel.EstimandCharacterization.EventStudyContamination.PopulationBridge","a":"Panel/EstimandCharacterization/EventStudyContamination","t":2,"d":"No-anticipation is derived from structural no-anticipation: on a pre-adoption period the own-path and never-treated potential-outcome means coincide, so their cohort means do.","s":"∀ {T : ℕ} (E : Causalean.Panel.EstimandCharacterization.EventStudyContamination.EventStudyPopulation T), E.toSystem.NoAnticipation","l":181,"f":"Causalean/Panel/EstimandCharacterization/EventStudyContamination/PopulationBridge.lean"},{"n":"Causalean.Panel.EstimandCharacterization.EventStudyContamination.EventStudyPopulation.toSystem_pathConsistency","k":"theorem","m":"Causalean.Panel.EstimandCharacterization.EventStudyContamination.PopulationBridge","a":"Panel/EstimandCharacterization/EventStudyContamination","t":2,"d":"Path-consistency is derived: on a period where comparison path h is untreated, the observed path mean equals the never-treated path mean.","s":"∀ {T : ℕ} (E : Causalean.Panel.EstimandCharacterization.EventStudyContamination.EventStudyPopulation T), E.toSystem.PathConsistency","l":199,"f":"Causalean/Panel/EstimandCharacterization/EventStudyContamination/PopulationBridge.lean"},{"n":"Causalean.Panel.EstimandCharacterization.EventStudyContamination.EventStudySystem","k":"structure","m":"Causalean.Panel.EstimandCharacterization.EventStudyContamination.Setup","a":"Panel/EstimandCharacterization/EventStudyContamination","t":1,"d":"A finite-cell record of a staggered-adoption event-study design over T periods, where adoption paths are finite periods or the never-treated path. It bundles a numeric encoding of each period used to form relative event time, the finite set of adoption cohorts in the event-study support, each adoption path's population share, and the cohort-period cell mass, together with, by cohort or by comparison adoption path, the factual observed outcome mean, the mean potential outcome under the cohort's own treatment path, and the mean never-treated potential outcome.","s":"ℕ → Type","l":37,"f":"Causalean/Panel/EstimandCharacterization/EventStudyContamination/Setup.lean"},{"n":"Causalean.Panel.EstimandCharacterization.EventStudyContamination.EventStudySystem.AdmissibleCell","k":"def","m":"Causalean.Panel.EstimandCharacterization.EventStudyContamination.Setup","a":"Panel/EstimandCharacterization/EventStudyContamination","t":1,"d":"A finite cohort-relative-time cell is admissible when the cohort is in support and at least one finite period realizes that relative time.","s":"{T : ℕ} → Causalean.Panel.EstimandCharacterization.EventStudyContamination.EventStudySystem T → Fin T → ℤ → Prop","l":100,"f":"Causalean/Panel/EstimandCharacterization/EventStudyContamination/Setup.lean"},{"n":"Causalean.Panel.EstimandCharacterization.EventStudyContamination.EventStudySystem.CATT","k":"def","m":"Causalean.Panel.EstimandCharacterization.EventStudyContamination.Setup","a":"Panel/EstimandCharacterization/EventStudyContamination","t":1,"d":"Cohort average treatment effect on the treated for the finite cohort-relative-time cell (g,e).","s":"{T : ℕ} → Causalean.Panel.EstimandCharacterization.EventStudyContamination.EventStudySystem T → Fin T → ℤ → ℝ","l":136,"f":"Causalean/Panel/EstimandCharacterization/EventStudyContamination/Setup.lean"},{"n":"Causalean.Panel.EstimandCharacterization.EventStudyContamination.EventStudySystem.CATT_eq_sourceCATT_of_injective","k":"theorem","m":"Causalean.Panel.EstimandCharacterization.EventStudyContamination.Setup","a":"Panel/EstimandCharacterization/EventStudyContamination","t":2,"d":"G1 faithfulness corollary. When the calendar-time map time is injective (the usual one-period-per-relative-time encoding), the cell-averaged CATT g e collapses to the source's *point* CATT_{g,e} at the unique period t realizing relative time e, i.e. treatedMean g t - untreatedMean g t.","s":"∀ {T : ℕ} (P : Causalean.Panel.EstimandCharacterization.EventStudyContamination.EventStudySystem T), Function.Injective P.time → ∀ {g : Fin T} {e : ℤ} {t : Fin T}, t ∈ P.targetPeriods g e → P.CATT g e = P.treatedMean g t - P.untreatedMean g t","l":283,"f":"Causalean/Panel/EstimandCharacterization/EventStudyContamination/Setup.lean"},{"n":"Causalean.Panel.EstimandCharacterization.EventStudyContamination.EventStudySystem.CATT_eq_zero_of_noAnticipation","k":"theorem","m":"Causalean.Panel.EstimandCharacterization.EventStudyContamination.Setup","a":"Panel/EstimandCharacterization/EventStudyContamination","t":1,"d":"No anticipation implies zero pre-treatment CATT. If the mean treated and never-treated potential outcomes for every adopting cohort coincide in every period strictly preceding that cohort's own adoption period (no anticipation), then for a cohort g in the finite adoption-cohort support and a relative event time e strictly before adoption ($e < 0$), the cohort-average treatment effect on the treated at cell (g, e) is zero.","s":"∀ {T : ℕ} (P : Causalean.Panel.EstimandCharacterization.EventStudyContamination.EventStudySystem T), P.NoAnticipation → ∀ {g : Fin T} {e : ℤ}, g ∈ P.cohorts → e < 0 → P.CATT g e = 0","l":202,"f":"Causalean/Panel/EstimandCharacterization/EventStudyContamination/Setup.lean"},{"n":"Causalean.Panel.EstimandCharacterization.EventStudyContamination.EventStudySystem.CellGridResidualization","k":"structure","m":"Causalean.Panel.EstimandCharacterization.EventStudyContamination.CellGrid","a":"Panel/EstimandCharacterization/EventStudyContamination","t":1,"d":"Cell-grid residualization input.","s":"{T : ℕ} → (P : Causalean.Panel.EstimandCharacterization.EventStudyContamination.EventStudySystem T) → P.ConventionalDesign → Prop","l":249,"f":"Causalean/Panel/EstimandCharacterization/EventStudyContamination/CellGrid.lean"},{"n":"Causalean.Panel.EstimandCharacterization.EventStudyContamination.EventStudySystem.CellIndex","k":"def","m":"Causalean.Panel.EstimandCharacterization.EventStudyContamination.CellGrid","a":"Panel/EstimandCharacterization/EventStudyContamination","t":1,"d":"The finite cell index type for the conventional design D: the admissible cohort-relative-time cells.","s":"{T : ℕ} → (P : Causalean.Panel.EstimandCharacterization.EventStudyContamination.EventStudySystem T) → P.ConventionalDesign → Type","l":50,"f":"Causalean/Panel/EstimandCharacterization/EventStudyContamination/CellGrid.lean"},{"n":"Causalean.Panel.EstimandCharacterization.EventStudyContamination.EventStudySystem.Consistency","k":"def","m":"Causalean.Panel.EstimandCharacterization.EventStudyContamination.Setup","a":"Panel/EstimandCharacterization/EventStudyContamination","t":1,"d":"Consistency for observed finite-cohort outcome means.","s":"{T : ℕ} → Causalean.Panel.EstimandCharacterization.EventStudyContamination.EventStudySystem T → Prop","l":160,"f":"Causalean/Panel/EstimandCharacterization/EventStudyContamination/Setup.lean"},{"n":"Causalean.Panel.EstimandCharacterization.EventStudyContamination.EventStudySystem.ConventionalDesign","k":"structure","m":"Causalean.Panel.EstimandCharacterization.EventStudyContamination.Conventional","a":"Panel/EstimandCharacterization/EventStudyContamination","t":1,"d":"Conventional event-study finite design for the coefficient on displayedEvent.","s":"{T : ℕ} → Causalean.Panel.EstimandCharacterization.EventStudyContamination.EventStudySystem T → Type","l":38,"f":"Causalean/Panel/EstimandCharacterization/EventStudyContamination/Conventional.lean"},{"n":"Causalean.Panel.EstimandCharacterization.EventStudyContamination.EventStudySystem.ConventionalFiniteSupport","k":"structure","m":"Causalean.Panel.EstimandCharacterization.EventStudyContamination.Conventional","a":"Panel/EstimandCharacterization/EventStudyContamination","t":1,"d":"Transparent finite support bookkeeping for the conventional cell expansion. It records that the explicit event-time support is the finite universe over which the displayed theorem is expanded, without asserting the headline contamination formula itself.","s":"{T : ℕ} → (P : Causalean.Panel.EstimandCharacterization.EventStudyContamination.EventStudySystem T) → P.ConventionalDesign → Prop","l":118,"f":"Causalean/Panel/EstimandCharacterization/EventStudyContamination/Conventional.lean"},{"n":"Causalean.Panel.EstimandCharacterization.EventStudyContamination.EventStudySystem.ConventionalResidualization","k":"structure","m":"Causalean.Panel.EstimandCharacterization.EventStudyContamination.Conventional","a":"Panel/EstimandCharacterization/EventStudyContamination","t":1,"d":"Finite-cell residualization record for the conventional coefficient.","s":"{T : ℕ} → (P : Causalean.Panel.EstimandCharacterization.EventStudyContamination.EventStudySystem T) → P.ConventionalDesign → Prop","l":72,"f":"Causalean/Panel/EstimandCharacterization/EventStudyContamination/Conventional.lean"},{"n":"Causalean.Panel.EstimandCharacterization.EventStudyContamination.EventStudySystem.DIDContrast","k":"def","m":"Causalean.Panel.EstimandCharacterization.EventStudyContamination.InteractionWeighted","a":"Panel/EstimandCharacterization/EventStudyContamination","t":1,"d":"Cohort-specific DID contrast using the treated cohort and its comparison group.","s":"{T : ℕ} → (P : Causalean.Panel.EstimandCharacterization.EventStudyContamination.EventStudySystem T) → P.IWDesign → Fin T → ℝ","l":89,"f":"Causalean/Panel/EstimandCharacterization/EventStudyContamination/InteractionWeighted.lean"},{"n":"Causalean.Panel.EstimandCharacterization.EventStudyContamination.EventStudySystem.Delta","k":"def","m":"Causalean.Panel.EstimandCharacterization.EventStudyContamination.InteractionWeighted","a":"Panel/EstimandCharacterization/EventStudyContamination","t":1,"d":"Cohort-specific IW DID contrast Delta(g,l). It is definitional rather than stored separately, so the IW theorem identifies the actual DID contrast.","s":"{T : ℕ} → (P : Causalean.Panel.EstimandCharacterization.EventStudyContamination.EventStudySystem T) → P.IWDesign → Fin T → ℝ","l":96,"f":"Causalean/Panel/EstimandCharacterization/EventStudyContamination/InteractionWeighted.lean"},{"n":"Causalean.Panel.EstimandCharacterization.EventStudyContamination.EventStudySystem.EventStudyCausalRestrictions","k":"structure","m":"Causalean.Panel.EstimandCharacterization.EventStudyContamination.Setup","a":"Panel/EstimandCharacterization/EventStudyContamination","t":1,"d":"Sun-Abraham event-study causal restrictions. Field names mirror the NL artifact's assumption names.","s":"{T : ℕ} → Causalean.Panel.EstimandCharacterization.EventStudyContamination.EventStudySystem T → Prop","l":195,"f":"Causalean/Panel/EstimandCharacterization/EventStudyContamination/Setup.lean"},{"n":"Causalean.Panel.EstimandCharacterization.EventStudyContamination.EventStudySystem.IWComparisonParallelTrends","k":"structure","m":"Causalean.Panel.EstimandCharacterization.EventStudyContamination.InteractionWeighted","a":"Panel/EstimandCharacterization/EventStudyContamination","t":1,"d":"Comparison-group parallel trends for the IW DID contrast.","s":"{T : ℕ} → (P : Causalean.Panel.EstimandCharacterization.EventStudyContamination.EventStudySystem T) → P.IWDesign → Prop","l":131,"f":"Causalean/Panel/EstimandCharacterization/EventStudyContamination/InteractionWeighted.lean"},{"n":"Causalean.Panel.EstimandCharacterization.EventStudyContamination.EventStudySystem.IWDesign","k":"structure","m":"Causalean.Panel.EstimandCharacterization.EventStudyContamination.InteractionWeighted","a":"Panel/EstimandCharacterization/EventStudyContamination","t":1,"d":"Interaction-weighted finite DID design for a fixed event time.","s":"{T : ℕ} → Causalean.Panel.EstimandCharacterization.EventStudyContamination.EventStudySystem T → Type","l":52,"f":"Causalean/Panel/EstimandCharacterization/EventStudyContamination/InteractionWeighted.lean"},{"n":"Causalean.Panel.EstimandCharacterization.EventStudyContamination.EventStudySystem.IWSupport","k":"structure","m":"Causalean.Panel.EstimandCharacterization.EventStudyContamination.InteractionWeighted","a":"Panel/EstimandCharacterization/EventStudyContamination","t":1,"d":"IW support restrictions for eligible cohorts and their comparison groups.","s":"{T : ℕ} → (P : Causalean.Panel.EstimandCharacterization.EventStudyContamination.EventStudySystem T) → P.IWDesign → Prop","l":106,"f":"Causalean/Panel/EstimandCharacterization/EventStudyContamination/InteractionWeighted.lean"},{"n":"Causalean.Panel.EstimandCharacterization.EventStudyContamination.EventStudySystem.IW_Delta_eq_CATT","k":"theorem","m":"Causalean.Panel.EstimandCharacterization.EventStudyContamination.InteractionWeighted","a":"Panel/EstimandCharacterization/EventStudyContamination","t":1,"d":"For an event-study system P and interaction-weighted design I, if consistency, no-anticipation, and path-consistency hold, every eligible cohort's comparison-group mean change from baseline to target period matches its own untreated-mean change, every eligible cohort is a genuine cohort of the system, and every comparison unit is untreated throughout the baseline and target periods, then for any eligible cohort g, its interaction-weighted DID contrast Delta I g equals the cohort-relative-time average treatment effect CATT g I.eventTime.","s":"∀ {T : ℕ} (P : Causalean.Panel.EstimandCharacterization.EventStudyContamination.EventStudySystem T) (I : P.IWDesign), P.Consistency → P.NoAnticipation → P.PathConsistency → (∀ g ∈ I.cohortsIW, (↑(P.targetPeriods g I.eventTime).card)⁻¹ * ∑ t ∈ P.targetPeriods g I.eventTime, P.untreatedMean g t - (↑(P.baselinePeriods g).card)⁻¹ * ∑ t ∈ P.baselinePeriods g, P.untreatedMean g t = (P.comparisonMass I g)⁻¹ * ∑ h ∈ I.comparisonGroup g, P.cohortShare h * ((↑(P.targetPeriods g I.eventTime).card)⁻¹ * ∑ t ∈ P.targetPeriods g I.eventTime, P.untreatedPathMean h t - (↑(P.baselinePeriods g).card)⁻¹ * ∑ t ∈ P.baselinePeriods g, P.untreatedPathMean h t)) → (∀ g ∈ I.cohortsIW, g ∈ P.cohorts) → (∀ g ∈ I.cohortsIW, ∀ h ∈ I.comparisonGroup g, ∀ t ∈ P.baselinePeriods g, Causalean.Panel.EstimandCharacterization.EventStudyContamination.EventStudySystem.absorbingTreatment h t = 0) → (∀ g ∈ I.cohortsIW, ∀ h ∈ I.comparisonGroup g, ∀ t ∈ P.targetPeriods g I.eventTime, Causalean.Panel.EstimandCharacterization.EventStudyContamination.EventStudySystem.absorbingTreatment h t = 0) → ∀ {g : Fin T}, g ∈ I.cohortsIW → P.Delta I g = P.CATT g I.eventTime","l":156,"f":"Causalean/Panel/EstimandCharacterization/EventStudyContamination/InteractionWeighted.lean"},{"n":"Causalean.Panel.EstimandCharacterization.EventStudyContamination.EventStudySystem.IW_convex_characterization","k":"theorem","m":"Causalean.Panel.EstimandCharacterization.EventStudyContamination.InteractionWeighted","a":"Panel/EstimandCharacterization/EventStudyContamination","t":1,"d":"For an event-study system P and interaction-weighted design I, if consistency, no-anticipation, and path-consistency hold, every eligible cohort's comparison-group mean change from baseline to target period matches its own untreated-mean change, the eligibility, baseline, target, and comparison-group support conditions of IWSupport hold, the aggregation weights rho are nonnegative and sum to one over the eligible cohorts, and every eligible cohort's CATT at the fixed event time lies between bounds lo and hi, then each cohort's DID contrast equals its CATT, the interaction-weighted estimand nuIW I equals the rho-weighted average of those CATTs, and nuIW I itself lies between lo and hi — a genuine convex average with no contamination from other relative times.","s":"∀ {T : ℕ} (P : Causalean.Panel.EstimandCharacterization.EventStudyContamination.EventStudySystem T) (I : P.IWDesign), P.Consistency → P.NoAnticipation → P.PathConsistency → (∀ g ∈ I.cohortsIW, (↑(P.targetPeriods g I.eventTime).card)⁻¹ * ∑ t ∈ P.targetPeriods g I.eventTime, P.untreatedMean g t - (↑(P.baselinePeriods g).card)⁻¹ * ∑ t ∈ P.baselinePeriods g, P.untreatedMean g t = (P.comparisonMass I g)⁻¹ * ∑ h ∈ I.comparisonGroup g, P.cohortShare h * ((↑(P.targetPeriods g I.eventTime).card)⁻¹ * ∑ t ∈ P.targetPeriods g I.eventTime, P.untreatedPathMean h t - (↑(P.baselinePeriods g).card)⁻¹ * ∑ t ∈ P.baselinePeriods g, P.untreatedPathMean h t)) → P.IWSupport I → (∀ g ∈ I.cohortsIW, 0 ≤ I.rho g) → ∑ g ∈ I.cohortsIW, I.rho g = 1 → ∀ {lo hi : ℝ}, (∀ g ∈ I.cohortsIW, lo ≤ P.CATT g I.eventTime) → (∀ g ∈ I.cohortsIW, P.CATT g I.eventTime ≤ hi) → (∀ g ∈ I.cohortsIW, P.Delta I g = P.CATT g I.eventTime) ∧ P.nuIW I = ∑ g ∈ I.cohortsIW, I.rho g * P.CATT g I.eventTime ∧ lo ≤ P.nuIW I ∧ P.nuIW I ≤ hi","l":283,"f":"Causalean/Panel/EstimandCharacterization/EventStudyContamination/InteractionWeighted.lean"},{"n":"Causalean.Panel.EstimandCharacterization.EventStudyContamination.EventStudySystem.IsEventStudyNuisance","k":"def","m":"Causalean.Panel.EstimandCharacterization.EventStudyContamination.Conventional","a":"Panel/EstimandCharacterization/EventStudyContamination","t":1,"d":"Finite-span nuisance class for the coefficient on R^l: cohort effects, period effects, and other included relative-time indicators.","s":"{T : ℕ} → (P : Causalean.Panel.EstimandCharacterization.EventStudyContamination.EventStudySystem T) → P.ConventionalDesign → (Fin T → Fin T → ℝ) → Prop","l":54,"f":"Causalean/Panel/EstimandCharacterization/EventStudyContamination/Conventional.lean"},{"n":"Causalean.Panel.EstimandCharacterization.EventStudyContamination.EventStudySystem.MeanParallelUntreated","k":"def","m":"Causalean.Panel.EstimandCharacterization.EventStudyContamination.Setup","a":"Panel/EstimandCharacterization/EventStudyContamination","t":1,"d":"Additive mean parallel untreated paths.","s":"{T : ℕ} → Causalean.Panel.EstimandCharacterization.EventStudyContamination.EventStudySystem T → Prop","l":190,"f":"Causalean/Panel/EstimandCharacterization/EventStudyContamination/Setup.lean"},{"n":"Causalean.Panel.EstimandCharacterization.EventStudyContamination.EventStudySystem.NoAnticipation","k":"def","m":"Causalean.Panel.EstimandCharacterization.EventStudyContamination.Setup","a":"Panel/EstimandCharacterization/EventStudyContamination","t":1,"d":"No anticipation in mean potential outcomes.","s":"{T : ℕ} → Causalean.Panel.EstimandCharacterization.EventStudyContamination.EventStudySystem T → Prop","l":185,"f":"Causalean/Panel/EstimandCharacterization/EventStudyContamination/Setup.lean"},{"n":"Causalean.Panel.EstimandCharacterization.EventStudyContamination.EventStudySystem.PathConsistency","k":"def","m":"Causalean.Panel.EstimandCharacterization.EventStudyContamination.Setup","a":"Panel/EstimandCharacterization/EventStudyContamination","t":1,"d":"Path-level consistency for comparison-group adoption paths.","s":"{T : ℕ} → Causalean.Panel.EstimandCharacterization.EventStudyContamination.EventStudySystem T → Prop","l":164,"f":"Causalean/Panel/EstimandCharacterization/EventStudyContamination/Setup.lean"},{"n":"Causalean.Panel.EstimandCharacterization.EventStudyContamination.EventStudySystem.absorbingTreatment","k":"def","m":"Causalean.Panel.EstimandCharacterization.EventStudyContamination.Setup","a":"Panel/EstimandCharacterization/EventStudyContamination","t":1,"d":"Absorbing treatment path 1{h < ∞ and h ≤ t}. Since ⊤ ≤ t is false, the never-treated path is untreated in every finite period.","s":"{T : ℕ} → WithTop (Fin T) → Fin T → ℝ","l":83,"f":"Causalean/Panel/EstimandCharacterization/EventStudyContamination/Setup.lean"},{"n":"Causalean.Panel.EstimandCharacterization.EventStudyContamination.EventStudySystem.admissibleCells","k":"def","m":"Causalean.Panel.EstimandCharacterization.EventStudyContamination.Setup","a":"Panel/EstimandCharacterization/EventStudyContamination","t":1,"d":"Finite support of admissible cells, filtered through an explicit finite relative-time support E.","s":"{T : ℕ} → Causalean.Panel.EstimandCharacterization.EventStudyContamination.EventStudySystem T → Finset ℤ → Finset (Fin T × ℤ)","l":106,"f":"Causalean/Panel/EstimandCharacterization/EventStudyContamination/Setup.lean"},{"n":"Causalean.Panel.EstimandCharacterization.EventStudyContamination.EventStudySystem.apparent_pretrends_from_post_treatment","k":"theorem","m":"Causalean.Panel.EstimandCharacterization.EventStudyContamination.Conventional","a":"Panel/EstimandCharacterization/EventStudyContamination","t":1,"d":"For an event-study system P and conventional design D, if the consistency, mean-parallel-trends, and no-anticipation causal restrictions hold, the residualized displayed-event indicator Rdot is orthogonal in expectation to every function in the event-study nuisance class, the residualized denominator is strictly positive, the coefficient D.mu equals its FWL residualized-ratio form, and the included, displayed, and admissible event times all lie within the declared finite support, then D.mu equals the contamination-weighted sum of cohort-relative-time CATTs restricted to nonnegative relative times, i.e. once every negative-relative-time CATT vanishes under no-anticipation, the displayed lead's coefficient is a weighted sum of post-treatment effects.","s":"∀ {T : ℕ} (P : Causalean.Panel.EstimandCharacterization.EventStudyContamination.EventStudySystem T) (D : P.ConventionalDesign), P.EventStudyCausalRestrictions → (∀ (h : Fin T → Fin T → ℝ), P.IsEventStudyNuisance D h → ∑ ge ∈ P.admissibleCells D.eventSupport, P.cellMassAtEvent ge.1 ge.2 * D.Rdot ge.1 ge.2 * P.cellAverage h ge.1 ge.2 = 0) → 0 < P.residualDenom D → D.mu = P.conventionalMuRatio D → P.ConventionalFiniteSupport D → D.mu = ∑ ge ∈ P.admissibleCells D.eventSupport with 0 ≤ ge.2, P.omega D ge.1 ge.2 * P.CATT ge.1 ge.2","l":377,"f":"Causalean/Panel/EstimandCharacterization/EventStudyContamination/Conventional.lean"},{"n":"Causalean.Panel.EstimandCharacterization.EventStudyContamination.EventStudySystem.apparent_pretrends_from_post_treatment_of_cellGrid","k":"theorem","m":"Causalean.Panel.EstimandCharacterization.EventStudyContamination.Contamination","a":"Panel/EstimandCharacterization/EventStudyContamination","t":1,"d":"Apparent pretrends (genuine). For the conventional event-study coefficient D.mu of a design D on system P, if the consistency, mean-parallel-trends, and no-anticipation causal restrictions hold, the included, displayed, and admissible event times all lie within the declared finite event-time support, and the cell-grid weighted-projection residualization input is supplied, then D.mu equals the contamination-weighted sum of cohort-relative-time CATTs restricted to nonnegative relative times — since no-anticipation forces every negative-relative-time CATT to vanish, this exhibits D.mu as a weighted average of post-treatment effects.","s":"∀ {T : ℕ} {P : Causalean.Panel.EstimandCharacterization.EventStudyContamination.EventStudySystem T} {D : P.ConventionalDesign}, P.EventStudyCausalRestrictions → P.ConventionalFiniteSupport D → P.CellGridResidualization D → D.mu = ∑ ge ∈ P.admissibleCells D.eventSupport with 0 ≤ ge.2, P.omega D ge.1 ge.2 * P.CATT ge.1 ge.2","l":75,"f":"Causalean/Panel/EstimandCharacterization/EventStudyContamination/Contamination.lean"},{"n":"Causalean.Panel.EstimandCharacterization.EventStudyContamination.EventStudySystem.baselinePeriods","k":"def","m":"Causalean.Panel.EstimandCharacterization.EventStudyContamination.Setup","a":"Panel/EstimandCharacterization/EventStudyContamination","t":1,"d":"Baseline periods with relative time -1 for finite cohort g.","s":"{T : ℕ} → Causalean.Panel.EstimandCharacterization.EventStudyContamination.EventStudySystem T → Fin T → Finset (Fin T)","l":95,"f":"Causalean/Panel/EstimandCharacterization/EventStudyContamination/Setup.lean"},{"n":"Causalean.Panel.EstimandCharacterization.EventStudyContamination.EventStudySystem.cellAverage","k":"def","m":"Causalean.Panel.EstimandCharacterization.EventStudyContamination.Conventional","a":"Panel/EstimandCharacterization/EventStudyContamination","t":1,"d":"Average of a cohort-period nuisance function over the finite periods that realize cell (g,e).","s":"{T : ℕ} → Causalean.Panel.EstimandCharacterization.EventStudyContamination.EventStudySystem T → (Fin T → Fin T → ℝ) → Fin T → ℤ → ℝ","l":65,"f":"Causalean/Panel/EstimandCharacterization/EventStudyContamination/Conventional.lean"},{"n":"Causalean.Panel.EstimandCharacterization.EventStudyContamination.EventStudySystem.cellAverage_eventIndicator","k":"theorem","m":"Causalean.Panel.EstimandCharacterization.EventStudyContamination.CellGrid","a":"Panel/EstimandCharacterization/EventStudyContamination","t":2,"d":"The cell average of the relative-time indicator 1{relTime = e}, evaluated on a nonempty cell (g, e'), is the cell-level indicator 1{e' = e}.","s":"∀ {T : ℕ} (P : Causalean.Panel.EstimandCharacterization.EventStudyContamination.EventStudySystem T) {e : ℤ} {g : Fin T} {e' : ℤ}, (P.targetPeriods g e').Nonempty → P.cellAverage (fun g t => Causalean.Panel.EstimandCharacterization.EventStudyContamination.EventStudySystem.eventIndicator e (P.relTime g t)) g e' = Causalean.Panel.EstimandCharacterization.EventStudyContamination.EventStudySystem.eventIndicator e e'","l":146,"f":"Causalean/Panel/EstimandCharacterization/EventStudyContamination/CellGrid.lean"},{"n":"Causalean.Panel.EstimandCharacterization.EventStudyContamination.EventStudySystem.cellAverage_mem_cellNuisance","k":"theorem","m":"Causalean.Panel.EstimandCharacterization.EventStudyContamination.CellGrid","a":"Panel/EstimandCharacterization/EventStudyContamination","t":2,"d":"Every cell-evaluated nuisance function lies in the cell-nuisance subspace.","s":"∀ {T : ℕ} (P : Causalean.Panel.EstimandCharacterization.EventStudyContamination.EventStudySystem T) (D : P.ConventionalDesign) {hCell : Fin T → Fin T → ℝ}, P.IsEventStudyNuisance D hCell → (fun cell => P.cellAverage hCell (↑cell).1 (↑cell).2) ∈ P.cellNuisance D","l":139,"f":"Causalean/Panel/EstimandCharacterization/EventStudyContamination/CellGrid.lean"},{"n":"Causalean.Panel.EstimandCharacterization.EventStudyContamination.EventStudySystem.cellGrid_mu_eq_conventionalMuRatio","k":"theorem","m":"Causalean.Panel.EstimandCharacterization.EventStudyContamination.CellGrid","a":"Panel/EstimandCharacterization/EventStudyContamination","t":1,"d":"Cell-grid FWL bridge. Under cell-grid residualization of the event-study design, the conventional event-study coefficient equals the Frisch–Waugh–Lovell residualized ratio computed directly on the cohort × relative-time cell grid.","s":"∀ {T : ℕ} {P : Causalean.Panel.EstimandCharacterization.EventStudyContamination.EventStudySystem T} {D : P.ConventionalDesign}, P.CellGridResidualization D → D.mu = P.conventionalMuRatio D","l":293,"f":"Causalean/Panel/EstimandCharacterization/EventStudyContamination/CellGrid.lean"},{"n":"Causalean.Panel.EstimandCharacterization.EventStudyContamination.EventStudySystem.cellGrid_provides_residualization","k":"theorem","m":"Causalean.Panel.EstimandCharacterization.EventStudyContamination.CellGrid","a":"Panel/EstimandCharacterization/EventStudyContamination","t":1,"d":"For a conventional event-study design D over an event-study system P, if every admissible cohort-relative-time cell has strictly positive population mass, the collection of admissible cells is nonempty, and the design's residualized displayed-event indicator Rdot agrees, cell by cell, with the weighted projection residual on the cell grid, then the three finite-cell orthogonality conditions packaged as ConventionalResidualization — derived here from a genuine weighted projection rather than assumed — hold for D.","s":"∀ {T : ℕ} {P : Causalean.Panel.EstimandCharacterization.EventStudyContamination.EventStudySystem T} {D : P.ConventionalDesign} (hCellMassPos : ∀ ge ∈ P.admissibleCells D.eventSupport, 0 < P.cellMassAtEvent ge.1 ge.2) (hCellNonempty : (P.admissibleCells D.eventSupport).Nonempty), (∀ (cell : P.CellIndex D), D.Rdot (↑cell).1 (↑cell).2 = (P.cellSupport D hCellMassPos hCellNonempty).tildeX (P.cellNuisance D) (P.cellRegressor D) cell) → P.ConventionalResidualization D","l":340,"f":"Causalean/Panel/EstimandCharacterization/EventStudyContamination/CellGrid.lean"},{"n":"Causalean.Panel.EstimandCharacterization.EventStudyContamination.EventStudySystem.cellMassAtEvent","k":"def","m":"Causalean.Panel.EstimandCharacterization.EventStudyContamination.Setup","a":"Panel/EstimandCharacterization/EventStudyContamination","t":1,"d":"Balanced finite-cell mass for cohort-relative-time cell (g,e).","s":"{T : ℕ} → Causalean.Panel.EstimandCharacterization.EventStudyContamination.EventStudySystem T → Fin T → ℤ → ℝ","l":119,"f":"Causalean/Panel/EstimandCharacterization/EventStudyContamination/Setup.lean"},{"n":"Causalean.Panel.EstimandCharacterization.EventStudyContamination.EventStudySystem.cellNuisance","k":"def","m":"Causalean.Panel.EstimandCharacterization.EventStudyContamination.CellGrid","a":"Panel/EstimandCharacterization/EventStudyContamination","t":1,"d":"The cell-nuisance subspace H on the cell grid: the span of cell-evaluated event-study nuisance functions.","s":"{T : ℕ} → (P : Causalean.Panel.EstimandCharacterization.EventStudyContamination.EventStudySystem T) → (D : P.ConventionalDesign) → Submodule ℝ (P.CellIndex D → ℝ)","l":133,"f":"Causalean/Panel/EstimandCharacterization/EventStudyContamination/CellGrid.lean"},{"n":"Causalean.Panel.EstimandCharacterization.EventStudyContamination.EventStudySystem.cellNuisanceGen","k":"def","m":"Causalean.Panel.EstimandCharacterization.EventStudyContamination.CellGrid","a":"Panel/EstimandCharacterization/EventStudyContamination","t":1,"d":"Generators of the cell-nuisance subspace: cell-evaluated event-study nuisance functions.","s":"{T : ℕ} → (P : Causalean.Panel.EstimandCharacterization.EventStudyContamination.EventStudySystem T) → (D : P.ConventionalDesign) → Set (P.CellIndex D → ℝ)","l":126,"f":"Causalean/Panel/EstimandCharacterization/EventStudyContamination/CellGrid.lean"},{"n":"Causalean.Panel.EstimandCharacterization.EventStudyContamination.EventStudySystem.cellOutcome","k":"def","m":"Causalean.Panel.EstimandCharacterization.EventStudyContamination.CellGrid","a":"Panel/EstimandCharacterization/EventStudyContamination","t":1,"d":"Cell-grid outcome: the observed cell mean as a cell function.","s":"{T : ℕ} → (P : Causalean.Panel.EstimandCharacterization.EventStudyContamination.EventStudySystem T) → (D : P.ConventionalDesign) → P.CellIndex D → ℝ","l":244,"f":"Causalean/Panel/EstimandCharacterization/EventStudyContamination/CellGrid.lean"},{"n":"Causalean.Panel.EstimandCharacterization.EventStudyContamination.EventStudySystem.cellRegressor","k":"def","m":"Causalean.Panel.EstimandCharacterization.EventStudyContamination.CellGrid","a":"Panel/EstimandCharacterization/EventStudyContamination","t":1,"d":"Cell-grid regressor: the displayed-event relative-time indicator as a cell function.","s":"{T : ℕ} → (P : Causalean.Panel.EstimandCharacterization.EventStudyContamination.EventStudySystem T) → (D : P.ConventionalDesign) → P.CellIndex D → ℝ","l":238,"f":"Causalean/Panel/EstimandCharacterization/EventStudyContamination/CellGrid.lean"},{"n":"Causalean.Panel.EstimandCharacterization.EventStudyContamination.EventStudySystem.cellSupport","k":"def","m":"Causalean.Panel.EstimandCharacterization.EventStudyContamination.CellGrid","a":"Panel/EstimandCharacterization/EventStudyContamination","t":1,"d":"The cell-grid weighted support: the empirical population over admissible cells with weight cellMassAtEvent / Z.","s":"{T : ℕ} → (P : Causalean.Panel.EstimandCharacterization.EventStudyContamination.EventStudySystem T) → (D : P.ConventionalDesign) → (∀ ge ∈ P.admissibleCells D.eventSupport, 0 < P.cellMassAtEvent ge.1 ge.2) → (P.admissibleCells D.eventSupport).Nonempty → Causalean.Panel.Weighted.WeightedSupport (P.CellIndex D)","l":59,"f":"Causalean/Panel/EstimandCharacterization/EventStudyContamination/CellGrid.lean"},{"n":"Causalean.Panel.EstimandCharacterization.EventStudyContamination.EventStudySystem.cellTotalMass","k":"def","m":"Causalean.Panel.EstimandCharacterization.EventStudyContamination.CellGrid","a":"Panel/EstimandCharacterization/EventStudyContamination","t":1,"d":"Total admissible cell mass Z = Σ_{(g,e)} cellMassAtEvent g e.","s":"{T : ℕ} → (P : Causalean.Panel.EstimandCharacterization.EventStudyContamination.EventStudySystem T) → P.ConventionalDesign → ℝ","l":55,"f":"Causalean/Panel/EstimandCharacterization/EventStudyContamination/CellGrid.lean"},{"n":"Causalean.Panel.EstimandCharacterization.EventStudyContamination.EventStudySystem.cohortsAtEvent","k":"def","m":"Causalean.Panel.EstimandCharacterization.EventStudyContamination.Setup","a":"Panel/EstimandCharacterization/EventStudyContamination","t":1,"d":"Cohorts observed at relative time e, with membership checked against an explicit finite relative-time support.","s":"{T : ℕ} → Causalean.Panel.EstimandCharacterization.EventStudyContamination.EventStudySystem T → Finset ℤ → ℤ → Finset (Fin T)","l":113,"f":"Causalean/Panel/EstimandCharacterization/EventStudyContamination/Setup.lean"},{"n":"Causalean.Panel.EstimandCharacterization.EventStudyContamination.EventStudySystem.comparisonMass","k":"def","m":"Causalean.Panel.EstimandCharacterization.EventStudyContamination.InteractionWeighted","a":"Panel/EstimandCharacterization/EventStudyContamination","t":1,"d":"Total comparison-group population mass for cohort g's IW contrast.","s":"{T : ℕ} → (P : Causalean.Panel.EstimandCharacterization.EventStudyContamination.EventStudySystem T) → P.IWDesign → Fin T → ℝ","l":76,"f":"Causalean/Panel/EstimandCharacterization/EventStudyContamination/InteractionWeighted.lean"},{"n":"Causalean.Panel.EstimandCharacterization.EventStudyContamination.EventStudySystem.comparisonMeanChange","k":"def","m":"Causalean.Panel.EstimandCharacterization.EventStudyContamination.InteractionWeighted","a":"Panel/EstimandCharacterization/EventStudyContamination","t":1,"d":"Population-share weighted comparison-group mean change from baseline g-1 to target g+l.","s":"{T : ℕ} → (P : Causalean.Panel.EstimandCharacterization.EventStudyContamination.EventStudySystem T) → P.IWDesign → Fin T → ℝ","l":81,"f":"Causalean/Panel/EstimandCharacterization/EventStudyContamination/InteractionWeighted.lean"},{"n":"Causalean.Panel.EstimandCharacterization.EventStudyContamination.EventStudySystem.contamination_representation","k":"theorem","m":"Causalean.Panel.EstimandCharacterization.EventStudyContamination.Conventional","a":"Panel/EstimandCharacterization/EventStudyContamination","t":1,"d":"Conventional Sun-Abraham contamination representation. For an event-study system P and conventional design D, if observed outcomes equal the potential outcome under the realized treatment path (consistency), the never-treated potential outcome follows an additive parallel-trends restriction, the residualized displayed-event indicator Rdot is orthogonal in expectation to every function in the event-study nuisance class, the residualized denominator is strictly positive, the coefficient D.mu equals its FWL residualized-ratio form, and the included, displayed, and admissible event times all lie within the declared finite support, then D.mu equals the Sun-Abraham contamination-weighted sum of cohort-relative-time CATTs over every admissible cell.","s":"∀ {T : ℕ} (P : Causalean.Panel.EstimandCharacterization.EventStudyContamination.EventStudySystem T) (D : P.ConventionalDesign), P.Consistency → P.MeanParallelUntreated → (∀ (h : Fin T → Fin T → ℝ), P.IsEventStudyNuisance D h → ∑ ge ∈ P.admissibleCells D.eventSupport, P.cellMassAtEvent ge.1 ge.2 * D.Rdot ge.1 ge.2 * P.cellAverage h ge.1 ge.2 = 0) → 0 < P.residualDenom D → D.mu = P.conventionalMuRatio D → P.ConventionalFiniteSupport D → D.mu = ∑ ge ∈ P.admissibleCells D.eventSupport, P.omega D ge.1 ge.2 * P.CATT ge.1 ge.2","l":178,"f":"Causalean/Panel/EstimandCharacterization/EventStudyContamination/Conventional.lean"},{"n":"Causalean.Panel.EstimandCharacterization.EventStudyContamination.EventStudySystem.contamination_representation_of_cellGrid","k":"theorem","m":"Causalean.Panel.EstimandCharacterization.EventStudyContamination.Contamination","a":"Panel/EstimandCharacterization/EventStudyContamination","t":1,"d":"Contamination representation (genuine). For the conventional event-study coefficient D.mu of a design D on system P, if observed outcomes equal the potential outcome under the realized treatment path (consistency), the never-treated potential outcome follows an additive parallel-trends restriction, the included, displayed, and admissible event times all lie within the declared finite support, and the cell-grid weighted-projection residualization input is supplied, then D.mu equals the Sun-Abraham contamination-weighted sum of cohort-relative-time CATTs over every admissible cell.","s":"∀ {T : ℕ} {P : Causalean.Panel.EstimandCharacterization.EventStudyContamination.EventStudySystem T} {D : P.ConventionalDesign}, P.Consistency → P.MeanParallelUntreated → P.ConventionalFiniteSupport D → P.CellGridResidualization D → D.mu = ∑ ge ∈ P.admissibleCells D.eventSupport, P.omega D ge.1 ge.2 * P.CATT ge.1 ge.2","l":35,"f":"Causalean/Panel/EstimandCharacterization/EventStudyContamination/Contamination.lean"},{"n":"Causalean.Panel.EstimandCharacterization.EventStudyContamination.EventStudySystem.contamination_representation_split","k":"theorem","m":"Causalean.Panel.EstimandCharacterization.EventStudyContamination.Conventional","a":"Panel/EstimandCharacterization/EventStudyContamination","t":1,"d":"For an event-study system P and conventional design D, if observed outcomes equal the potential outcome under the realized treatment path (consistency), the never-treated outcome satisfies additive parallel trends, the design satisfies the finite-cell orthogonality conditions ConventionalResidualization, the residualized denominator is strictly positive, the coefficient D.mu equals its FWL residualized-ratio form, and the included, displayed, and admissible event times all lie within the declared finite support, then D.mu splits as the displayed-event-time contamination term plus the contamination-weighted sum over every other admissible cohort-relative-time cell.","s":"∀ {T : ℕ} (P : Causalean.Panel.EstimandCharacterization.EventStudyContamination.EventStudySystem T) (D : P.ConventionalDesign), P.Consistency → P.MeanParallelUntreated → P.ConventionalResidualization D → 0 < P.residualDenom D → D.mu = P.conventionalMuRatio D → P.ConventionalFiniteSupport D → D.mu = ∑ g ∈ P.cohortsAtEvent D.eventSupport D.displayedEvent, P.omega D g D.displayedEvent * P.CATT g D.displayedEvent + ∑ ge ∈ P.admissibleCells D.eventSupport with ge.2 ≠ D.displayedEvent, P.omega D ge.1 ge.2 * P.CATT ge.1 ge.2","l":304,"f":"Causalean/Panel/EstimandCharacterization/EventStudyContamination/Conventional.lean"},{"n":"Causalean.Panel.EstimandCharacterization.EventStudyContamination.EventStudySystem.contamination_representation_split_of_cellGrid","k":"theorem","m":"Causalean.Panel.EstimandCharacterization.EventStudyContamination.Contamination","a":"Panel/EstimandCharacterization/EventStudyContamination","t":1,"d":"Contamination split (genuine). For the conventional event-study coefficient D.mu of a design D on system P, if the consistency, mean-parallel-trends, and no-anticipation causal restrictions hold, the included, displayed, and admissible event times all lie within the declared finite support, and the cell-grid weighted-projection residualization input is supplied, then D.mu splits as the displayed-event-time contamination term plus the contamination-weighted sum over every other admissible cohort-relative-time cell.","s":"∀ {T : ℕ} {P : Causalean.Panel.EstimandCharacterization.EventStudyContamination.EventStudySystem T} {D : P.ConventionalDesign}, P.EventStudyCausalRestrictions → P.ConventionalFiniteSupport D → P.CellGridResidualization D → D.mu = ∑ g ∈ P.cohortsAtEvent D.eventSupport D.displayedEvent, P.omega D g D.displayedEvent * P.CATT g D.displayedEvent + ∑ ge ∈ P.admissibleCells D.eventSupport with ge.2 ≠ D.displayedEvent, P.omega D ge.1 ge.2 * P.CATT ge.1 ge.2","l":54,"f":"Causalean/Panel/EstimandCharacterization/EventStudyContamination/Contamination.lean"},{"n":"Causalean.Panel.EstimandCharacterization.EventStudyContamination.EventStudySystem.conventionalMuRatio","k":"def","m":"Causalean.Panel.EstimandCharacterization.EventStudyContamination.Conventional","a":"Panel/EstimandCharacterization/EventStudyContamination","t":1,"d":"Residualized-ratio form of the conventional event-study coefficient.","s":"{T : ℕ} → (P : Causalean.Panel.EstimandCharacterization.EventStudyContamination.EventStudySystem T) → P.ConventionalDesign → ℝ","l":113,"f":"Causalean/Panel/EstimandCharacterization/EventStudyContamination/Conventional.lean"},{"n":"Causalean.Panel.EstimandCharacterization.EventStudyContamination.EventStudySystem.desired_event_weights_sum_one","k":"theorem","m":"Causalean.Panel.EstimandCharacterization.EventStudyContamination.Conventional","a":"Panel/EstimandCharacterization/EventStudyContamination","t":2,"d":"Desired-event-time weights sum to one.","s":"∀ {T : ℕ} (P : Causalean.Panel.EstimandCharacterization.EventStudyContamination.EventStudySystem T) (D : P.ConventionalDesign), ∑ ge ∈ P.admissibleCells D.eventSupport, P.cellMassAtEvent ge.1 ge.2 * D.Rdot ge.1 ge.2 * Causalean.Panel.EstimandCharacterization.EventStudyContamination.EventStudySystem.eventIndicator D.displayedEvent ge.2 = ∑ g ∈ P.cohortsAtEvent D.eventSupport D.displayedEvent, P.cellMassAtEvent g D.displayedEvent * D.Rdot g D.displayedEvent → 0 < P.residualDenom D → ∑ g ∈ P.cohortsAtEvent D.eventSupport D.displayedEvent, P.omega D g D.displayedEvent = 1","l":135,"f":"Causalean/Panel/EstimandCharacterization/EventStudyContamination/Conventional.lean"},{"n":"Causalean.Panel.EstimandCharacterization.EventStudyContamination.EventStudySystem.eventIndicator","k":"def","m":"Causalean.Panel.EstimandCharacterization.EventStudyContamination.Conventional","a":"Panel/EstimandCharacterization/EventStudyContamination","t":1,"d":"Included-event-time indicator on a cohort-relative-time cell.","s":"ℤ → ℤ → ℝ","l":34,"f":"Causalean/Panel/EstimandCharacterization/EventStudyContamination/Conventional.lean"},{"n":"Causalean.Panel.EstimandCharacterization.EventStudyContamination.EventStudySystem.eventIndicator_mem_cellNuisance","k":"theorem","m":"Causalean.Panel.EstimandCharacterization.EventStudyContamination.CellGrid","a":"Panel/EstimandCharacterization/EventStudyContamination","t":2,"d":"A non-displayed included relative-time indicator lies in the cell-nuisance subspace (it is the cell average of the corresponding event-study nuisance).","s":"∀ {T : ℕ} (P : Causalean.Panel.EstimandCharacterization.EventStudyContamination.EventStudySystem T) (D : P.ConventionalDesign) {e : ℤ}, e ∈ D.includedEvents → e ≠ D.displayedEvent → (fun cell => Causalean.Panel.EstimandCharacterization.EventStudyContamination.EventStudySystem.eventIndicator e (↑cell).2) ∈ P.cellNuisance D","l":165,"f":"Causalean/Panel/EstimandCharacterization/EventStudyContamination/CellGrid.lean"},{"n":"Causalean.Panel.EstimandCharacterization.EventStudyContamination.EventStudySystem.finitePath","k":"def","m":"Causalean.Panel.EstimandCharacterization.EventStudyContamination.Setup","a":"Panel/EstimandCharacterization/EventStudyContamination","t":1,"d":"A finite adoption path, embedded in WithTop (Fin T).","s":"{T : ℕ} → Fin T → WithTop (Fin T)","l":69,"f":"Causalean/Panel/EstimandCharacterization/EventStudyContamination/Setup.lean"},{"n":"Causalean.Panel.EstimandCharacterization.EventStudyContamination.EventStudySystem.ip_cellSupport","k":"theorem","m":"Causalean.Panel.EstimandCharacterization.EventStudyContamination.CellGrid","a":"Panel/EstimandCharacterization/EventStudyContamination","t":2,"d":"ip → cell-sum. The cell-grid weighted inner product of two cell functions reproduces the finite-cell sum (divided by the total mass Z). This is the bridge that turns weighted-projection facts into the finite-cell orthogonality conditions.","s":"∀ {T : ℕ} (P : Causalean.Panel.EstimandCharacterization.EventStudyContamination.EventStudySystem T) (D : P.ConventionalDesign) (hpos : ∀ ge ∈ P.admissibleCells D.eventSupport, 0 < P.cellMassAtEvent ge.1 ge.2) (hne : (P.admissibleCells D.eventSupport).Nonempty) (A B : Fin T × ℤ → ℝ), ((P.cellSupport D hpos hne).ip (fun cell => A ↑cell) fun cell => B ↑cell) = (∑ ge ∈ P.admissibleCells D.eventSupport, P.cellMassAtEvent ge.1 ge.2 * A ge * B ge) / P.cellTotalMass D","l":96,"f":"Causalean/Panel/EstimandCharacterization/EventStudyContamination/CellGrid.lean"},{"n":"Causalean.Panel.EstimandCharacterization.EventStudyContamination.EventStudySystem.isEventuallyTreated","k":"def","m":"Causalean.Panel.EstimandCharacterization.EventStudyContamination.Setup","a":"Panel/EstimandCharacterization/EventStudyContamination","t":1,"d":"h < ∞, the path is eventually treated.","s":"{T : ℕ} → WithTop (Fin T) → Prop","l":75,"f":"Causalean/Panel/EstimandCharacterization/EventStudyContamination/Setup.lean"},{"n":"Causalean.Panel.EstimandCharacterization.EventStudyContamination.EventStudySystem.isNeverTreated","k":"def","m":"Causalean.Panel.EstimandCharacterization.EventStudyContamination.Setup","a":"Panel/EstimandCharacterization/EventStudyContamination","t":1,"d":"h = ∞, the never-treated adoption path.","s":"{T : ℕ} → WithTop (Fin T) → Prop","l":72,"f":"Causalean/Panel/EstimandCharacterization/EventStudyContamination/Setup.lean"},{"n":"Causalean.Panel.EstimandCharacterization.EventStudyContamination.EventStudySystem.meanCellContrast","k":"def","m":"Causalean.Panel.EstimandCharacterization.EventStudyContamination.Setup","a":"Panel/EstimandCharacterization/EventStudyContamination","t":1,"d":"Average treated-minus-never potential-outcome contrast over (g,e).","s":"{T : ℕ} → Causalean.Panel.EstimandCharacterization.EventStudyContamination.EventStudySystem T → Fin T → ℤ → ℝ","l":130,"f":"Causalean/Panel/EstimandCharacterization/EventStudyContamination/Setup.lean"},{"n":"Causalean.Panel.EstimandCharacterization.EventStudyContamination.EventStudySystem.nuIW","k":"def","m":"Causalean.Panel.EstimandCharacterization.EventStudyContamination.InteractionWeighted","a":"Panel/EstimandCharacterization/EventStudyContamination","t":1,"d":"Interaction-weighted event-study estimand.","s":"{T : ℕ} → (P : Causalean.Panel.EstimandCharacterization.EventStudyContamination.EventStudySystem T) → P.IWDesign → ℝ","l":102,"f":"Causalean/Panel/EstimandCharacterization/EventStudyContamination/InteractionWeighted.lean"},{"n":"Causalean.Panel.EstimandCharacterization.EventStudyContamination.EventStudySystem.observedBaselineMean","k":"def","m":"Causalean.Panel.EstimandCharacterization.EventStudyContamination.InteractionWeighted","a":"Panel/EstimandCharacterization/EventStudyContamination","t":1,"d":"Observed treated-cohort baseline mean from the finite-cohort factual means, using relative time -1.","s":"{T : ℕ} → Causalean.Panel.EstimandCharacterization.EventStudyContamination.EventStudySystem T → Fin T → ℝ","l":69,"f":"Causalean/Panel/EstimandCharacterization/EventStudyContamination/InteractionWeighted.lean"},{"n":"Causalean.Panel.EstimandCharacterization.EventStudyContamination.EventStudySystem.observedCellMean","k":"def","m":"Causalean.Panel.EstimandCharacterization.EventStudyContamination.Setup","a":"Panel/EstimandCharacterization/EventStudyContamination","t":1,"d":"Average factual observed outcome over the periods realizing (g,e).","s":"{T : ℕ} → Causalean.Panel.EstimandCharacterization.EventStudyContamination.EventStudySystem T → Fin T → ℤ → ℝ","l":124,"f":"Causalean/Panel/EstimandCharacterization/EventStudyContamination/Setup.lean"},{"n":"Causalean.Panel.EstimandCharacterization.EventStudyContamination.EventStudySystem.observedCellMean_eq_fixedEffects_add_CATT","k":"theorem","m":"Causalean.Panel.EstimandCharacterization.EventStudyContamination.Setup","a":"Panel/EstimandCharacterization/EventStudyContamination","t":1,"d":"Cell-mean decomposition into additive untreated fixed effects and CATT. If the factual observed outcome mean on a cohort's own periods equals its mean treated potential outcome (consistency) and the mean never-treated potential outcome admits an additive cohort/period fixed-effects representation (mean-parallel untreated paths), then for any cohort g in the finite adoption-cohort support, the average observed outcome mean over the target periods of cell (g, e) decomposes as the average, over those periods, of the additive fixed effects alpha g + lambda t plus the cohort-average treatment effect on the treated CATT g e.","s":"∀ {T : ℕ} (P : Causalean.Panel.EstimandCharacterization.EventStudyContamination.EventStudySystem T), P.Consistency → P.MeanParallelUntreated → ∀ {g : Fin T} {e : ℤ}, g ∈ P.cohorts → ∃ alpha lambda, P.observedCellMean g e = (↑(P.targetPeriods g e).card)⁻¹ * ∑ t ∈ P.targetPeriods g e, (alpha g + lambda t) + P.CATT g e","l":225,"f":"Causalean/Panel/EstimandCharacterization/EventStudyContamination/Setup.lean"},{"n":"Causalean.Panel.EstimandCharacterization.EventStudyContamination.EventStudySystem.observedTargetMean","k":"def","m":"Causalean.Panel.EstimandCharacterization.EventStudyContamination.InteractionWeighted","a":"Panel/EstimandCharacterization/EventStudyContamination","t":1,"d":"Observed treated-cohort target mean from the finite-cohort factual means.","s":"{T : ℕ} → (P : Causalean.Panel.EstimandCharacterization.EventStudyContamination.EventStudySystem T) → P.IWDesign → Fin T → ℝ","l":63,"f":"Causalean/Panel/EstimandCharacterization/EventStudyContamination/InteractionWeighted.lean"},{"n":"Causalean.Panel.EstimandCharacterization.EventStudyContamination.EventStudySystem.omega","k":"def","m":"Causalean.Panel.EstimandCharacterization.EventStudyContamination.Conventional","a":"Panel/EstimandCharacterization/EventStudyContamination","t":1,"d":"Sun-Abraham contamination weight for the conventional event-study coefficient.","s":"{T : ℕ} → (P : Causalean.Panel.EstimandCharacterization.EventStudyContamination.EventStudySystem T) → P.ConventionalDesign → Fin T → ℤ → ℝ","l":129,"f":"Causalean/Panel/EstimandCharacterization/EventStudyContamination/Conventional.lean"},{"n":"Causalean.Panel.EstimandCharacterization.EventStudyContamination.EventStudySystem.other_included_event_weights_sum_zero","k":"theorem","m":"Causalean.Panel.EstimandCharacterization.EventStudyContamination.Conventional","a":"Panel/EstimandCharacterization/EventStudyContamination","t":2,"d":"Other included-event-time weights sum to zero.","s":"∀ {T : ℕ} (P : Causalean.Panel.EstimandCharacterization.EventStudyContamination.EventStudySystem T) (D : P.ConventionalDesign) {e : ℤ}, ∑ g ∈ P.cohortsAtEvent D.eventSupport e, P.cellMassAtEvent g e * D.Rdot g e = 0 → 0 < P.residualDenom D → ∑ g ∈ P.cohortsAtEvent D.eventSupport e, P.omega D g e = 0","l":160,"f":"Causalean/Panel/EstimandCharacterization/EventStudyContamination/Conventional.lean"},{"n":"Causalean.Panel.EstimandCharacterization.EventStudyContamination.EventStudySystem.pathBaselineMean","k":"def","m":"Causalean.Panel.EstimandCharacterization.EventStudyContamination.Setup","a":"Panel/EstimandCharacterization/EventStudyContamination","t":1,"d":"Baseline observed mean for an arbitrary adoption path, using relative time -1 for treated cohort g.","s":"{T : ℕ} → Causalean.Panel.EstimandCharacterization.EventStudyContamination.EventStudySystem T → WithTop (Fin T) → Fin T → ℝ","l":153,"f":"Causalean/Panel/EstimandCharacterization/EventStudyContamination/Setup.lean"},{"n":"Causalean.Panel.EstimandCharacterization.EventStudyContamination.EventStudySystem.pathConsistency_observed_eq_untreated","k":"theorem","m":"Causalean.Panel.EstimandCharacterization.EventStudyContamination.Setup","a":"Panel/EstimandCharacterization/EventStudyContamination","t":2,"d":"Path-consistency, applied to an untreated comparison period, yields the observed-equals-untreated path-mean bridge used by the IW comparison-group argument.","s":"∀ {T : ℕ} (P : Causalean.Panel.EstimandCharacterization.EventStudyContamination.EventStudySystem T), P.PathConsistency → ∀ {h : WithTop (Fin T)} {t : Fin T}, Causalean.Panel.EstimandCharacterization.EventStudyContamination.EventStudySystem.absorbingTreatment h t = 0 → P.observedPathMean h t = P.untreatedPathMean h t","l":176,"f":"Causalean/Panel/EstimandCharacterization/EventStudyContamination/Setup.lean"},{"n":"Causalean.Panel.EstimandCharacterization.EventStudyContamination.EventStudySystem.pathTargetMean","k":"def","m":"Causalean.Panel.EstimandCharacterization.EventStudyContamination.Setup","a":"Panel/EstimandCharacterization/EventStudyContamination","t":1,"d":"Target-period observed mean for an arbitrary adoption path, used by the IW comparison-group DID contrast.","s":"{T : ℕ} → Causalean.Panel.EstimandCharacterization.EventStudyContamination.EventStudySystem T → WithTop (Fin T) → Fin T → ℤ → ℝ","l":146,"f":"Causalean/Panel/EstimandCharacterization/EventStudyContamination/Setup.lean"},{"n":"Causalean.Panel.EstimandCharacterization.EventStudyContamination.EventStudySystem.relTime","k":"def","m":"Causalean.Panel.EstimandCharacterization.EventStudyContamination.Setup","a":"Panel/EstimandCharacterization/EventStudyContamination","t":1,"d":"Relative event time t - g under the system's integer-valued period map.","s":"{T : ℕ} → Causalean.Panel.EstimandCharacterization.EventStudyContamination.EventStudySystem T → Fin T → Fin T → ℤ","l":78,"f":"Causalean/Panel/EstimandCharacterization/EventStudyContamination/Setup.lean"},{"n":"Causalean.Panel.EstimandCharacterization.EventStudyContamination.EventStudySystem.residualDenom","k":"def","m":"Causalean.Panel.EstimandCharacterization.EventStudyContamination.Conventional","a":"Panel/EstimandCharacterization/EventStudyContamination","t":1,"d":"Finite residualized denominator E[Rdot^l R^l].","s":"{T : ℕ} → (P : Causalean.Panel.EstimandCharacterization.EventStudyContamination.EventStudySystem T) → P.ConventionalDesign → ℝ","l":99,"f":"Causalean/Panel/EstimandCharacterization/EventStudyContamination/Conventional.lean"},{"n":"Causalean.Panel.EstimandCharacterization.EventStudyContamination.EventStudySystem.residualNumerator","k":"def","m":"Causalean.Panel.EstimandCharacterization.EventStudyContamination.Conventional","a":"Panel/EstimandCharacterization/EventStudyContamination","t":1,"d":"Finite residualized numerator E[Rdot^l Y].","s":"{T : ℕ} → (P : Causalean.Panel.EstimandCharacterization.EventStudyContamination.EventStudySystem T) → P.ConventionalDesign → ℝ","l":106,"f":"Causalean/Panel/EstimandCharacterization/EventStudyContamination/Conventional.lean"},{"n":"Causalean.Panel.EstimandCharacterization.EventStudyContamination.EventStudySystem.sum_admissible_mul_eventIndicator","k":"theorem","m":"Causalean.Panel.EstimandCharacterization.EventStudyContamination.CellGrid","a":"Panel/EstimandCharacterization/EventStudyContamination","t":2,"d":"Cell-grid filtering identity. Weighting an admissible-cell sum by the relative-time indicator 1{e = e'} collapses it to a sum over the cohorts observed at relative time e'. Pure finite algebra; no residualization needed. This is the engine behind hDisplayedExpansion and hOtherIncludedOrthogonal.","s":"∀ {T : ℕ} (P : Causalean.Panel.EstimandCharacterization.EventStudyContamination.EventStudySystem T) (D : P.ConventionalDesign) (e' : ℤ) (f : Fin T → ℤ → ℝ), ∑ ge ∈ P.admissibleCells D.eventSupport, f ge.1 ge.2 * Causalean.Panel.EstimandCharacterization.EventStudyContamination.EventStudySystem.eventIndicator e' ge.2 = ∑ g ∈ P.cohortsAtEvent D.eventSupport e', f g e'","l":194,"f":"Causalean/Panel/EstimandCharacterization/EventStudyContamination/CellGrid.lean"},{"n":"Causalean.Panel.EstimandCharacterization.EventStudyContamination.EventStudySystem.targetPeriods","k":"def","m":"Causalean.Panel.EstimandCharacterization.EventStudyContamination.Setup","a":"Panel/EstimandCharacterization/EventStudyContamination","t":1,"d":"The target periods for finite cohort g at relative time e.","s":"{T : ℕ} → Causalean.Panel.EstimandCharacterization.EventStudyContamination.EventStudySystem T → Fin T → ℤ → Finset (Fin T)","l":89,"f":"Causalean/Panel/EstimandCharacterization/EventStudyContamination/Setup.lean"},{"n":"Causalean.Panel.EstimandCharacterization.EventStudyContamination.EventStudySystem.targetPeriods_subsingleton_of_injective","k":"theorem","m":"Causalean.Panel.EstimandCharacterization.EventStudyContamination.Setup","a":"Panel/EstimandCharacterization/EventStudyContamination","t":2,"d":"Under an injective calendar-time encoding, at most one finite period can realize a given relative time, so the target-period set of any cell is a subsingleton.","s":"∀ {T : ℕ} (P : Causalean.Panel.EstimandCharacterization.EventStudyContamination.EventStudySystem T), Function.Injective P.time → ∀ (g : Fin T) (e : ℤ), (↑(P.targetPeriods g e)).Subsingleton","l":268,"f":"Causalean/Panel/EstimandCharacterization/EventStudyContamination/Setup.lean"},{"n":"Causalean.Panel.EstimandCharacterization.EventStudyContamination.EventStudySystem.tildeX_eq_Rdot","k":"theorem","m":"Causalean.Panel.EstimandCharacterization.EventStudyContamination.CellGrid","a":"Panel/EstimandCharacterization/EventStudyContamination","t":2,"d":"Under cell-grid residualization, the residualized displayed-event regressor equals the design residual at every admissible cohort--relative-time cell.","s":"∀ {T : ℕ} {P : Causalean.Panel.EstimandCharacterization.EventStudyContamination.EventStudySystem T} {D : P.ConventionalDesign} (h : P.CellGridResidualization D), (P.cellSupport D h.hCellMassPos h.hCellNonempty).tildeX (P.cellNuisance D) (P.cellRegressor D) = fun cell => D.Rdot (↑cell).1 (↑cell).2","l":285,"f":"Causalean/Panel/EstimandCharacterization/EventStudyContamination/CellGrid.lean"},{"n":"Causalean.Panel.EstimandCharacterization.EventStudyContamination.sum_convex_mem_Icc","k":"theorem","m":"Causalean.Panel.EstimandCharacterization.EventStudyContamination.InteractionWeighted","a":"Panel/EstimandCharacterization/EventStudyContamination","t":2,"d":"Finite convex-combination bound. If the weights w are nonnegative and sum to one, then any nonnegative-weighted average of values lying in [lo, hi] again lies in [lo, hi]. This is the algebraic content of \"convex combination\" used to certify that the IW estimand is a genuine convex average of the target CATTs (no contamination).","s":"∀ {ι : Type u_1} (s : Finset ι) (w f : ι → ℝ) {lo hi : ℝ}, (∀ i ∈ s, 0 ≤ w i) → ∑ i ∈ s, w i = 1 → (∀ i ∈ s, lo ≤ f i) → (∀ i ∈ s, f i ≤ hi) → lo ≤ ∑ i ∈ s, w i * f i ∧ ∑ i ∈ s, w i * f i ≤ hi","l":25,"f":"Causalean/Panel/EstimandCharacterization/EventStudyContamination/InteractionWeighted.lean"},{"n":"Causalean.Panel.EstimandCharacterization.FlexibleDIDMundlak.AggregateWeights","k":"structure","m":"Causalean.Panel.EstimandCharacterization.FlexibleDIDMundlak.DID","a":"Panel/EstimandCharacterization/FlexibleDIDMundlak","t":1,"d":"Nonnegative aggregate weights summing to one on the target treated cells.","s":"{Cohort : Type u_1} → {Time : Type u_2} → {Covar : Type u_3} → [inst : Fintype Cohort] → [inst_1 : Fintype Time] → [inst_2 : Fintype Covar] → Causalean.Panel.EstimandCharacterization.FlexibleDIDMundlak.StaggeredATTCells Cohort Time Covar → Type (max u_1 u_2)","l":97,"f":"Causalean/Panel/EstimandCharacterization/FlexibleDIDMundlak/DID.lean"},{"n":"Causalean.Panel.EstimandCharacterization.FlexibleDIDMundlak.ConditionalParallelTrendsAdditive","k":"def","m":"Causalean.Panel.EstimandCharacterization.FlexibleDIDMundlak.DID","a":"Panel/EstimandCharacterization/FlexibleDIDMundlak","t":1,"d":"Conditional parallel trends, represented by the equivalent additive untreated mean form m0(g,t,c) = α(g,c) + λ(t,c).","s":"{Cohort : Type u_1} → {Time : Type u_2} → {Covar : Type u_3} → [inst : Fintype Cohort] → [inst_1 : Fintype Time] → [inst_2 : Fintype Covar] → Causalean.Panel.EstimandCharacterization.FlexibleDIDMundlak.StaggeredATTCells Cohort Time Covar → Prop","l":111,"f":"Causalean/Panel/EstimandCharacterization/FlexibleDIDMundlak/DID.lean"},{"n":"Causalean.Panel.EstimandCharacterization.FlexibleDIDMundlak.FlexibleDIDEstimands","k":"structure","m":"Causalean.Panel.EstimandCharacterization.FlexibleDIDMundlak.DID","a":"Panel/EstimandCharacterization/FlexibleDIDMundlak","t":1,"d":"On top of a staggered-cell design P and its saturated untreated-outcome regression S, this structure packages three families of treated-cell coefficients — an imputation coefficient, a pooled-least-squares (POLS) coefficient, and an extended two-way-fixed-effects (ETWFE) coefficient — together with the conditions pinning them down: on every treated cell the imputation coefficient equals the covariate-weighted imputation residual mean, and the POLS and ETWFE coefficients each solve the finite-cell covariate-weighted residual normal equation.","s":"{Cohort : Type u_1} → {Time : Type u_2} → {Covar : Type u_3} → [inst : Fintype Cohort] → [inst_1 : Fintype Time] → [inst_2 : Fintype Covar] → (P : Causalean.Panel.EstimandCharacterization.FlexibleDIDMundlak.StaggeredATTCells Cohort Time Covar) → Causalean.Panel.EstimandCharacterization.FlexibleDIDMundlak.SaturatedUntreatedRegression P → Type (max u_1 u_2)","l":443,"f":"Causalean/Panel/EstimandCharacterization/FlexibleDIDMundlak/DID.lean"},{"n":"Causalean.Panel.EstimandCharacterization.FlexibleDIDMundlak.FlexibleDIDEstimands.etwfe_cell_eq_pols","k":"theorem","m":"Causalean.Panel.EstimandCharacterization.FlexibleDIDMundlak.DID","a":"Panel/EstimandCharacterization/FlexibleDIDMundlak","t":2,"d":"Compatibility alias: ETWFE/POLS equality is now derived by solving both finite-cell normal equations, rather than stored as a field.","s":"∀ {Cohort : Type u_1} {Time : Type u_2} {Covar : Type u_3} [inst : Fintype Cohort] [inst_1 : Fintype Time] [inst_2 : Fintype Covar] (P : Causalean.Panel.EstimandCharacterization.FlexibleDIDMundlak.StaggeredATTCells Cohort Time Covar) (S : Causalean.Panel.EstimandCharacterization.FlexibleDIDMundlak.SaturatedUntreatedRegression P) (E : Causalean.Panel.EstimandCharacterization.FlexibleDIDMundlak.FlexibleDIDEstimands P S) {g : Cohort} {t : Time}, P.treatedCell g t → E.thetaETWFE g t = E.thetaPOLS g t","l":518,"f":"Causalean/Panel/EstimandCharacterization/FlexibleDIDMundlak/DID.lean"},{"n":"Causalean.Panel.EstimandCharacterization.FlexibleDIDMundlak.FlexibleDIDEstimands.pols_cell_eq_imputation","k":"theorem","m":"Causalean.Panel.EstimandCharacterization.FlexibleDIDMundlak.DID","a":"Panel/EstimandCharacterization/FlexibleDIDMundlak","t":2,"d":"Compatibility alias: the POLS/imputation equality is now derived from the POLS normal equation, rather than stored as a field.","s":"∀ {Cohort : Type u_1} {Time : Type u_2} {Covar : Type u_3} [inst : Fintype Cohort] [inst_1 : Fintype Time] [inst_2 : Fintype Covar] (P : Causalean.Panel.EstimandCharacterization.FlexibleDIDMundlak.StaggeredATTCells Cohort Time Covar) (S : Causalean.Panel.EstimandCharacterization.FlexibleDIDMundlak.SaturatedUntreatedRegression P) (E : Causalean.Panel.EstimandCharacterization.FlexibleDIDMundlak.FlexibleDIDEstimands P S) {g : Cohort} {t : Time}, P.treatedCell g t → E.thetaPOLS g t = E.thetaImp g t","l":508,"f":"Causalean/Panel/EstimandCharacterization/FlexibleDIDMundlak/DID.lean"},{"n":"Causalean.Panel.EstimandCharacterization.FlexibleDIDMundlak.FlexibleDIDEstimands.thetaETWFE_eq_imputationTheta","k":"theorem","m":"Causalean.Panel.EstimandCharacterization.FlexibleDIDMundlak.DID","a":"Panel/EstimandCharacterization/FlexibleDIDMundlak","t":1,"d":"ETWFE cell coefficients equal imputation. On any treated cohort-time cell (g,t), the extended two-way-fixed-effects (ETWFE) cell coefficient equals the imputation residual mean at that cell.","s":"∀ {Cohort : Type u_1} {Time : Type u_2} {Covar : Type u_3} [inst : Fintype Cohort] [inst_1 : Fintype Time] [inst_2 : Fintype Covar] (P : Causalean.Panel.EstimandCharacterization.FlexibleDIDMundlak.StaggeredATTCells Cohort Time Covar) (S : Causalean.Panel.EstimandCharacterization.FlexibleDIDMundlak.SaturatedUntreatedRegression P) (E : Causalean.Panel.EstimandCharacterization.FlexibleDIDMundlak.FlexibleDIDEstimands P S) {g : Cohort} {t : Time}, P.treatedCell g t → E.thetaETWFE g t = Causalean.Panel.EstimandCharacterization.FlexibleDIDMundlak.imputationTheta P S g t","l":492,"f":"Causalean/Panel/EstimandCharacterization/FlexibleDIDMundlak/DID.lean"},{"n":"Causalean.Panel.EstimandCharacterization.FlexibleDIDMundlak.FlexibleDIDEstimands.thetaPOLS_eq_imputationTheta","k":"theorem","m":"Causalean.Panel.EstimandCharacterization.FlexibleDIDMundlak.DID","a":"Panel/EstimandCharacterization/FlexibleDIDMundlak","t":1,"d":"POLS cell coefficients equal imputation. On any treated cohort-time cell (g,t), the flexible POLS cell coefficient equals the imputation residual mean at that cell.","s":"∀ {Cohort : Type u_1} {Time : Type u_2} {Covar : Type u_3} [inst : Fintype Cohort] [inst_1 : Fintype Time] [inst_2 : Fintype Covar] (P : Causalean.Panel.EstimandCharacterization.FlexibleDIDMundlak.StaggeredATTCells Cohort Time Covar) (S : Causalean.Panel.EstimandCharacterization.FlexibleDIDMundlak.SaturatedUntreatedRegression P) (E : Causalean.Panel.EstimandCharacterization.FlexibleDIDMundlak.FlexibleDIDEstimands P S) {g : Cohort} {t : Time}, P.treatedCell g t → E.thetaPOLS g t = Causalean.Panel.EstimandCharacterization.FlexibleDIDMundlak.imputationTheta P S g t","l":477,"f":"Causalean/Panel/EstimandCharacterization/FlexibleDIDMundlak/DID.lean"},{"n":"Causalean.Panel.EstimandCharacterization.FlexibleDIDMundlak.IsCellAdditive","k":"def","m":"Causalean.Panel.EstimandCharacterization.FlexibleDIDMundlak.DID","a":"Panel/EstimandCharacterization/FlexibleDIDMundlak","t":1,"d":"An additive function d(g,t,c) = γ(g,c) + δ(t,c) of the cohort/time/covariate cell, the difference class used to compare two additive untreated-mean representations.","s":"{Cohort : Type u_1} → {Time : Type u_2} → {Covar : Type u_3} → (Cohort → Time → Covar → ℝ) → Prop","l":118,"f":"Causalean/Panel/EstimandCharacterization/FlexibleDIDMundlak/DID.lean"},{"n":"Causalean.Panel.EstimandCharacterization.FlexibleDIDMundlak.IsTwoWayMundlakNuisance","k":"def","m":"Causalean.Panel.EstimandCharacterization.FlexibleDIDMundlak.TWFE","a":"Panel/EstimandCharacterization/FlexibleDIDMundlak","t":1,"d":"Two-way Mundlak nuisance span for a scalar regressor: constants, unit means of X, time means of X, optional time-constant controls Z_i, and optional time-only controls M_t.","s":"{Unit : Type u_1} → {Time : Type u_2} → [Fintype Unit] → [Fintype Time] → {Z : Type u_3} → {M : Type u_4} → [Fintype Z] → [Fintype M] → (Unit → Time → ℝ) → (Z → Unit → ℝ) → (M → Time → ℝ) → (Unit → Time → ℝ) → Prop","l":137,"f":"Causalean/Panel/EstimandCharacterization/FlexibleDIDMundlak/TWFE.lean"},{"n":"Causalean.Panel.EstimandCharacterization.FlexibleDIDMundlak.IsVectorTwoWayMundlakNuisance","k":"def","m":"Causalean.Panel.EstimandCharacterization.FlexibleDIDMundlak.VectorMundlak","a":"Panel/EstimandCharacterization/FlexibleDIDMundlak","t":1,"d":"Two-way Mundlak nuisance span for a K-vector regressor: constants, the unit means and time means of every coordinate of X, optional time-constant controls Z_i, and optional time-only controls M_t.","s":"{Unit : Type u_1} → {Time : Type u_2} → [Fintype Unit] → [Fintype Time] → {K : Type u_3} → [Fintype K] → {Z : Type u_4} → {M : Type u_5} → [Fintype Z] → [Fintype M] → (Unit → Time → K → ℝ) → (Z → Unit → ℝ) → (M → Time → ℝ) → (Unit → Time → ℝ) → Prop","l":176,"f":"Causalean/Panel/EstimandCharacterization/FlexibleDIDMundlak/VectorMundlak.lean"},{"n":"Causalean.Panel.EstimandCharacterization.FlexibleDIDMundlak.NoAnticipation","k":"def","m":"Causalean.Panel.EstimandCharacterization.FlexibleDIDMundlak.DID","a":"Panel/EstimandCharacterization/FlexibleDIDMundlak","t":1,"d":"No anticipation: before adoption, the cohort-g potential outcome equals the untreated potential outcome. Here untreatedCell marks the relevant pre-treatment / not-yet-treated observations.","s":"{Cohort : Type u_1} → {Time : Type u_2} → {Covar : Type u_3} → [inst : Fintype Cohort] → [inst_1 : Fintype Time] → [inst_2 : Fintype Covar] → Causalean.Panel.EstimandCharacterization.FlexibleDIDMundlak.StaggeredATTCells Cohort Time Covar → Prop","l":104,"f":"Causalean/Panel/EstimandCharacterization/FlexibleDIDMundlak/DID.lean"},{"n":"Causalean.Panel.EstimandCharacterization.FlexibleDIDMundlak.SaturatedUntreatedRegression","k":"structure","m":"Causalean.Panel.EstimandCharacterization.FlexibleDIDMundlak.DID","a":"Panel/EstimandCharacterization/FlexibleDIDMundlak","t":1,"d":"A finite-cell weighted least-squares fit of the untreated-outcome mean for a staggered cell design P, restricted to the untreated observations. It bundles a fitted untreated-outcome mean that is additive in cohort and time given the covariate cell, projection weights that are strictly positive on the untreated design, the requirement that the fit solves the covariate/cell-weighted normal equations against every additive test function, summed over the untreated design, full-rank identification of the additive class from vanishing on the untreated design alone, and a positive cohort share on every treated cell.","s":"{Cohort : Type u_1} → {Time : Type u_2} → {Covar : Type u_3} → [inst : Fintype Cohort] → [inst_1 : Fintype Time] → [inst_2 : Fintype Covar] → Causalean.Panel.EstimandCharacterization.FlexibleDIDMundlak.StaggeredATTCells Cohort Time Covar → Type (max (max u_1 u_2) u_3)","l":137,"f":"Causalean/Panel/EstimandCharacterization/FlexibleDIDMundlak/DID.lean"},{"n":"Causalean.Panel.EstimandCharacterization.FlexibleDIDMundlak.SaturatedUntreatedRegression.recovers_target_Y0","k":"theorem","m":"Causalean.Panel.EstimandCharacterization.FlexibleDIDMundlak.DID","a":"Panel/EstimandCharacterization/FlexibleDIDMundlak","t":1,"d":"Saturated untreated regression recovers the untreated potential outcome on treated cells. If no anticipation holds: the treated and untreated potential-outcome means agree on every cell in the untreated-outcome regression's design and conditional parallel trends holds — the mean untreated potential outcome admits an additive cohort/time fixed-effects representation given covariates, then on any treated cohort-time cell (g,t) covered by the saturated untreated regression S, the fitted value S.m0 g t c equals the mean untreated potential outcome Y0Mean g t c, for every covariate cell c.","s":"∀ {Cohort : Type u_1} {Time : Type u_2} {Covar : Type u_3} [inst : Fintype Cohort] [inst_1 : Fintype Time] [inst_2 : Fintype Covar] {P : Causalean.Panel.EstimandCharacterization.FlexibleDIDMundlak.StaggeredATTCells Cohort Time Covar} (S : Causalean.Panel.EstimandCharacterization.FlexibleDIDMundlak.SaturatedUntreatedRegression P), Causalean.Panel.EstimandCharacterization.FlexibleDIDMundlak.NoAnticipation P → Causalean.Panel.EstimandCharacterization.FlexibleDIDMundlak.ConditionalParallelTrendsAdditive P → ∀ ⦃g : Cohort⦄ ⦃t : Time⦄, P.treatedCell g t → ∀ (c : Covar), S.m0 g t c = P.Y0Mean g t c","l":294,"f":"Causalean/Panel/EstimandCharacterization/FlexibleDIDMundlak/DID.lean"},{"n":"Causalean.Panel.EstimandCharacterization.FlexibleDIDMundlak.SaturatedUntreatedRegression.toUntreatedFitWitness","k":"def","m":"Causalean.Panel.EstimandCharacterization.FlexibleDIDMundlak.DID","a":"Panel/EstimandCharacterization/FlexibleDIDMundlak","t":1,"d":"Forget the target-support and design-identification fields that are not needed for exact fit on untreated cells.","s":"{Cohort : Type u_1} → {Time : Type u_2} → {Covar : Type u_3} → [inst : Fintype Cohort] → [inst_1 : Fintype Time] → [inst_2 : Fintype Covar] → {P : Causalean.Panel.EstimandCharacterization.FlexibleDIDMundlak.StaggeredATTCells Cohort Time Covar} → Causalean.Panel.EstimandCharacterization.FlexibleDIDMundlak.SaturatedUntreatedRegression P → Causalean.Panel.EstimandCharacterization.FlexibleDIDMundlak.UntreatedFitWitness P","l":203,"f":"Causalean/Panel/EstimandCharacterization/FlexibleDIDMundlak/DID.lean"},{"n":"Causalean.Panel.EstimandCharacterization.FlexibleDIDMundlak.SaturatedUntreatedRegression.untreatedFit","k":"theorem","m":"Causalean.Panel.EstimandCharacterization.FlexibleDIDMundlak.DID","a":"Panel/EstimandCharacterization/FlexibleDIDMundlak","t":2,"d":"The weighted projection m0 reproduces the factual cohort-g outcome mean on every untreated cell.","s":"∀ {Cohort : Type u_1} {Time : Type u_2} {Covar : Type u_3} [inst : Fintype Cohort] [inst_1 : Fintype Time] [inst_2 : Fintype Covar] {P : Causalean.Panel.EstimandCharacterization.FlexibleDIDMundlak.StaggeredATTCells Cohort Time Covar} (S : Causalean.Panel.EstimandCharacterization.FlexibleDIDMundlak.UntreatedFitWitness P), Causalean.Panel.EstimandCharacterization.FlexibleDIDMundlak.NoAnticipation P → Causalean.Panel.EstimandCharacterization.FlexibleDIDMundlak.ConditionalParallelTrendsAdditive P → ∀ ⦃g : Cohort⦄ ⦃t : Time⦄, P.untreatedCell g t → ∀ (c : Covar), S.m0 g t c = P.YgMean g t c","l":216,"f":"Causalean/Panel/EstimandCharacterization/FlexibleDIDMundlak/DID.lean"},{"n":"Causalean.Panel.EstimandCharacterization.FlexibleDIDMundlak.ScalarTWFEProblem","k":"structure","m":"Causalean.Panel.EstimandCharacterization.FlexibleDIDMundlak.TWFE","a":"Panel/EstimandCharacterization/FlexibleDIDMundlak","t":1,"d":"A scalar two-way-fixed-effects regression problem on a finite balanced panel of units and time periods, given a scalar outcome and a scalar regressor, where the sum of squared double-demeaned regressor values is strictly positive — the scalar full-rank condition ensuring the two-way within estimator is well defined.","s":"(Unit : Type u_3) → (Time : Type u_4) → [Fintype Unit] → [Fintype Time] → Type (max u_3 u_4)","l":33,"f":"Causalean/Panel/EstimandCharacterization/FlexibleDIDMundlak/TWFE.lean"},{"n":"Causalean.Panel.EstimandCharacterization.FlexibleDIDMundlak.ScalarTWFEProblem.betaTWFE","k":"def","m":"Causalean.Panel.EstimandCharacterization.FlexibleDIDMundlak.TWFE","a":"Panel/EstimandCharacterization/FlexibleDIDMundlak","t":1,"d":"Population scalar TWFE coefficient from the double-demeaned normal equation.","s":"{Unit : Type u_1} → {Time : Type u_2} → [inst : Fintype Unit] → [inst_1 : Fintype Time] → Causalean.Panel.EstimandCharacterization.FlexibleDIDMundlak.ScalarTWFEProblem Unit Time → ℝ","l":53,"f":"Causalean/Panel/EstimandCharacterization/FlexibleDIDMundlak/TWFE.lean"},{"n":"Causalean.Panel.EstimandCharacterization.FlexibleDIDMundlak.ScalarTWFEProblem.betaTWFE_normalEq","k":"theorem","m":"Causalean.Panel.EstimandCharacterization.FlexibleDIDMundlak.TWFE","a":"Panel/EstimandCharacterization/FlexibleDIDMundlak","t":2,"d":"The closed-form coefficient satisfies the scalar TWFE normal equation by dividing through the positive residualized sum of squares.","s":"∀ {Unit : Type u_1} {Time : Type u_2} [inst : Fintype Unit] [inst_1 : Fintype Time] (P : Causalean.Panel.EstimandCharacterization.FlexibleDIDMundlak.ScalarTWFEProblem Unit Time), P.twfeNormalEq P.betaTWFE","l":62,"f":"Causalean/Panel/EstimandCharacterization/FlexibleDIDMundlak/TWFE.lean"},{"n":"Causalean.Panel.EstimandCharacterization.FlexibleDIDMundlak.ScalarTWFEProblem.betaTWFE_unique","k":"theorem","m":"Causalean.Panel.EstimandCharacterization.FlexibleDIDMundlak.TWFE","a":"Panel/EstimandCharacterization/FlexibleDIDMundlak","t":2,"d":"Scalar full-rank uniqueness of the TWFE normal-equation solution.","s":"∀ {Unit : Type u_1} {Time : Type u_2} [inst : Fintype Unit] [inst_1 : Fintype Time] (P : Causalean.Panel.EstimandCharacterization.FlexibleDIDMundlak.ScalarTWFEProblem Unit Time) {β : ℝ}, P.twfeNormalEq β → β = P.betaTWFE","l":94,"f":"Causalean/Panel/EstimandCharacterization/FlexibleDIDMundlak/TWFE.lean"},{"n":"Causalean.Panel.EstimandCharacterization.FlexibleDIDMundlak.ScalarTWFEProblem.toVector","k":"def","m":"Causalean.Panel.EstimandCharacterization.FlexibleDIDMundlak.VectorTWFE","a":"Panel/EstimandCharacterization/FlexibleDIDMundlak","t":1,"d":"The scalar TWFE problem embeds as the singleton-K = Fin 1 vector problem: the regressor is the same scalar in the single coordinate and the matrix full-rank condition reduces to the scalar ddotX_ss_pos.","s":"{Unit : Type u_1} → {Time : Type u_2} → [inst : Fintype Unit] → [inst_1 : Fintype Time] → Causalean.Panel.EstimandCharacterization.FlexibleDIDMundlak.ScalarTWFEProblem Unit Time → Causalean.Panel.EstimandCharacterization.FlexibleDIDMundlak.VectorTWFEProblem Unit Time (Fin 1)","l":165,"f":"Causalean/Panel/EstimandCharacterization/FlexibleDIDMundlak/VectorTWFE.lean"},{"n":"Causalean.Panel.EstimandCharacterization.FlexibleDIDMundlak.ScalarTWFEProblem.toVector_betaTWFE","k":"theorem","m":"Causalean.Panel.EstimandCharacterization.FlexibleDIDMundlak.VectorTWFE","a":"Panel/EstimandCharacterization/FlexibleDIDMundlak","t":2,"d":"The singleton-coordinate vector TWFE coefficient recovers the scalar TWFE coefficient, so the scalar theorem is the K = Fin 1 case of the vector one.","s":"∀ {Unit : Type u_1} {Time : Type u_2} [inst : Fintype Unit] [inst_1 : Fintype Time] (P : Causalean.Panel.EstimandCharacterization.FlexibleDIDMundlak.ScalarTWFEProblem Unit Time), P.toVector.betaTWFE 0 = P.betaTWFE","l":183,"f":"Causalean/Panel/EstimandCharacterization/FlexibleDIDMundlak/VectorTWFE.lean"},{"n":"Causalean.Panel.EstimandCharacterization.FlexibleDIDMundlak.ScalarTWFEProblem.twfeDenominator","k":"def","m":"Causalean.Panel.EstimandCharacterization.FlexibleDIDMundlak.TWFE","a":"Panel/EstimandCharacterization/FlexibleDIDMundlak","t":1,"d":"Residualized-design denominator for scalar TWFE.","s":"{Unit : Type u_1} → {Time : Type u_2} → [inst : Fintype Unit] → [inst_1 : Fintype Time] → Causalean.Panel.EstimandCharacterization.FlexibleDIDMundlak.ScalarTWFEProblem Unit Time → ℝ","l":45,"f":"Causalean/Panel/EstimandCharacterization/FlexibleDIDMundlak/TWFE.lean"},{"n":"Causalean.Panel.EstimandCharacterization.FlexibleDIDMundlak.ScalarTWFEProblem.twfeNormalEq","k":"def","m":"Causalean.Panel.EstimandCharacterization.FlexibleDIDMundlak.TWFE","a":"Panel/EstimandCharacterization/FlexibleDIDMundlak","t":1,"d":"Scalar TWFE normal equation after double demeaning.","s":"{Unit : Type u_1} → {Time : Type u_2} → [inst : Fintype Unit] → [inst_1 : Fintype Time] → Causalean.Panel.EstimandCharacterization.FlexibleDIDMundlak.ScalarTWFEProblem Unit Time → ℝ → Prop","l":58,"f":"Causalean/Panel/EstimandCharacterization/FlexibleDIDMundlak/TWFE.lean"},{"n":"Causalean.Panel.EstimandCharacterization.FlexibleDIDMundlak.ScalarTWFEProblem.twfeNumerator","k":"def","m":"Causalean.Panel.EstimandCharacterization.FlexibleDIDMundlak.TWFE","a":"Panel/EstimandCharacterization/FlexibleDIDMundlak","t":1,"d":"Residualized numerator using double-demeaned outcome and regressor.","s":"{Unit : Type u_1} → {Time : Type u_2} → [inst : Fintype Unit] → [inst_1 : Fintype Time] → Causalean.Panel.EstimandCharacterization.FlexibleDIDMundlak.ScalarTWFEProblem Unit Time → ℝ","l":49,"f":"Causalean/Panel/EstimandCharacterization/FlexibleDIDMundlak/TWFE.lean"},{"n":"Causalean.Panel.EstimandCharacterization.FlexibleDIDMundlak.ScalarTWMFit","k":"structure","m":"Causalean.Panel.EstimandCharacterization.FlexibleDIDMundlak.TWFE","a":"Panel/EstimandCharacterization/FlexibleDIDMundlak","t":1,"d":"A scalar two-way Mundlak fit of the TWFE problem P against optional time-constant controls Zvar and time-only controls Mvar, stated by normal equations rather than by a particular coding of the nuisance regressors. It bundles a scalar coefficient on the regressor and a nuisance function lying in the two-way Mundlak span, subject to the pooled normal equation against the regressor and the pooled normal equation against every nuisance function in that span.","s":"{Unit : Type u_1} → {Time : Type u_2} → [inst : Fintype Unit] → [inst_1 : Fintype Time] → {Z : Type u_3} → {M : Type u_4} → [Fintype Z] → [Fintype M] → Causalean.Panel.EstimandCharacterization.FlexibleDIDMundlak.ScalarTWFEProblem Unit Time → (Z → Unit → ℝ) → (M → Time → ℝ) → Type (max u_1 u_2)","l":163,"f":"Causalean/Panel/EstimandCharacterization/FlexibleDIDMundlak/TWFE.lean"},{"n":"Causalean.Panel.EstimandCharacterization.FlexibleDIDMundlak.StaggeredATTCells","k":"structure","m":"Causalean.Panel.EstimandCharacterization.FlexibleDIDMundlak.DID","a":"Panel/EstimandCharacterization/FlexibleDIDMundlak","t":1,"d":"Finite staggered-adoption cell system: cohort shares and within-cohort covariate weights over treated and untreated cohort-time cells, with cell-level means of the untreated and cohort-specific treated potential outcomes. It requires that every treated cell's cohort has positive share, that the covariate weights are nonnegative and sum to one within each cohort, and that the observed cell mean coincides with the treated mean on treated cells and with the untreated mean on untreated cells.","s":"(Cohort : Type u_4) → (Time : Type u_5) → (Covar : Type u_6) → [Fintype Cohort] → [Fintype Time] → [Fintype Covar] → Type (max (max u_4 u_5) u_6)","l":37,"f":"Causalean/Panel/EstimandCharacterization/FlexibleDIDMundlak/DID.lean"},{"n":"Causalean.Panel.EstimandCharacterization.FlexibleDIDMundlak.StaggeredATTCells.ofMeasure","k":"def","m":"Causalean.Panel.EstimandCharacterization.FlexibleDIDMundlak.PopulationOrigin","a":"Panel/EstimandCharacterization/FlexibleDIDMundlak","t":1,"d":"Builds a finite staggered-DID cell system from a population model.","s":"{Cohort : Type u_1} → {Time : Type u_2} → {Covar : Type u_3} → [inst : Fintype Cohort] → [inst_1 : Fintype Time] → [inst_2 : Fintype Covar] → {Ω : Type u_4} → [inst_3 : MeasurableSpace Ω] → (μ : MeasureTheory.Measure Ω) → (cellEvent : Cohort → Time → Covar → Set Ω) → (Y0pop Ygpop Yobspop : Ω → ℝ) → (treatedCell untreatedCell : Cohort → Time → Prop) → (cohortShare : Cohort → ℝ) → (covarWeight : Cohort → Covar → ℝ) → (∀ (g : Cohort) (t : Time) (c : Covar), MeasurableSet (cellEvent g t c)) → (∀ (g : Cohort) (t : Time) (c : Covar), 0 < (μ (cellEvent g t c)).toReal) → (∀ (g : Cohort) (t : Time) (c : Covar), MeasureTheory.IntegrableOn Y0pop (cellEvent g t c) μ) → (∀ (g : Cohort) (t : Time) (c : Covar), MeasureTheory.IntegrableOn Ygpop (cellEvent g t c) μ) → (∀ (g : Cohort) (t : Time) (c : Covar), MeasureTheory.IntegrableOn Yobspop (cellEvent g t c) μ) → (∀ ⦃g : Cohort⦄ ⦃t : Time⦄, treatedCell g t → 0 < cohortShare g) → (∀ (g : Cohort) (c : Covar), 0 ≤ covarWeight g c) → (∀ (g : Cohort), ∑ c, covarWeight g c = 1) → (∀ ⦃g : Cohort⦄ ⦃t : Time⦄, treatedCell g t → ∀ (c : Covar), ∀ ω ∈ cellEvent g t c, Yobspop ω = Ygpop ω) → (∀ ⦃g : Cohort⦄ ⦃t : Time⦄, untreatedCell g t → ∀ (c : Covar), ∀ ω ∈ cellEvent g t c, Yobspop ω = Y0pop ω) → Causalean.Panel.EstimandCharacterization.FlexibleDIDMundlak.StaggeredATTCells Cohort Time Covar","l":58,"f":"Causalean/Panel/EstimandCharacterization/FlexibleDIDMundlak/PopulationOrigin.lean"},{"n":"Causalean.Panel.EstimandCharacterization.FlexibleDIDMundlak.StaggeredATTCells.ofPopulation","k":"def","m":"Causalean.Panel.EstimandCharacterization.FlexibleDIDMundlak.PopulationOrigin","a":"Panel/EstimandCharacterization/FlexibleDIDMundlak","t":1,"d":"Probability-measure specialization of StaggeredATTCells.ofMeasure.","s":"{Cohort : Type u_1} → {Time : Type u_2} → {Covar : Type u_3} → [inst : Fintype Cohort] → [inst_1 : Fintype Time] → [inst_2 : Fintype Covar] → {Ω : Type u_4} → [inst_3 : MeasurableSpace Ω] → (μ : MeasureTheory.Measure Ω) → [MeasureTheory.IsProbabilityMeasure μ] → (cellEvent : Cohort → Time → Covar → Set Ω) → (Y0pop Ygpop Yobspop : Ω → ℝ) → (treatedCell untreatedCell : Cohort → Time → Prop) → (cohortShare : Cohort → ℝ) → (covarWeight : Cohort → Covar → ℝ) → (∀ (g : Cohort) (t : Time) (c : Covar), MeasurableSet (cellEvent g t c)) → (∀ (g : Cohort) (t : Time) (c : Covar), 0 < (μ (cellEvent g t c)).toReal) → (∀ (g : Cohort) (t : Time) (c : Covar), MeasureTheory.IntegrableOn Y0pop (cellEvent g t c) μ) → (∀ (g : Cohort) (t : Time) (c : Covar), MeasureTheory.IntegrableOn Ygpop (cellEvent g t c) μ) → (∀ (g : Cohort) (t : Time) (c : Covar), MeasureTheory.IntegrableOn Yobspop (cellEvent g t c) μ) → (∀ ⦃g : Cohort⦄ ⦃t : Time⦄, treatedCell g t → 0 < cohortShare g) → (∀ (g : Cohort) (c : Covar), 0 ≤ covarWeight g c) → (∀ (g : Cohort), ∑ c, covarWeight g c = 1) → (∀ ⦃g : Cohort⦄ ⦃t : Time⦄, treatedCell g t → ∀ (c : Covar), ∀ ω ∈ cellEvent g t c, Yobspop ω = Ygpop ω) → (∀ ⦃g : Cohort⦄ ⦃t : Time⦄, untreatedCell g t → ∀ (c : Covar), ∀ ω ∈ cellEvent g t c, Yobspop ω = Y0pop ω) → Causalean.Panel.EstimandCharacterization.FlexibleDIDMundlak.StaggeredATTCells Cohort Time Covar","l":119,"f":"Causalean/Panel/EstimandCharacterization/FlexibleDIDMundlak/PopulationOrigin.lean"},{"n":"Causalean.Panel.EstimandCharacterization.FlexibleDIDMundlak.StaggeredATTCells.tauAgg","k":"def","m":"Causalean.Panel.EstimandCharacterization.FlexibleDIDMundlak.DID","a":"Panel/EstimandCharacterization/FlexibleDIDMundlak","t":1,"d":"Aggregate ATT for requested finite treated-cell weights.","s":"{Cohort : Type u_1} → {Time : Type u_2} → {Covar : Type u_3} → [inst : Fintype Cohort] → [inst_1 : Fintype Time] → [inst_2 : Fintype Covar] → Causalean.Panel.EstimandCharacterization.FlexibleDIDMundlak.StaggeredATTCells Cohort Time Covar → (Cohort → Time → ℝ) → ℝ","l":90,"f":"Causalean/Panel/EstimandCharacterization/FlexibleDIDMundlak/DID.lean"},{"n":"Causalean.Panel.EstimandCharacterization.FlexibleDIDMundlak.StaggeredATTCells.tauCell","k":"def","m":"Causalean.Panel.EstimandCharacterization.FlexibleDIDMundlak.DID","a":"Panel/EstimandCharacterization/FlexibleDIDMundlak","t":1,"d":"ATT cell τ_gt, averaged over baseline covariate cells within cohort.","s":"{Cohort : Type u_1} → {Time : Type u_2} → {Covar : Type u_3} → [inst : Fintype Cohort] → [inst_1 : Fintype Time] → [inst_2 : Fintype Covar] → Causalean.Panel.EstimandCharacterization.FlexibleDIDMundlak.StaggeredATTCells Cohort Time Covar → Cohort → Time → ℝ","l":85,"f":"Causalean/Panel/EstimandCharacterization/FlexibleDIDMundlak/DID.lean"},{"n":"Causalean.Panel.EstimandCharacterization.FlexibleDIDMundlak.StaggeredATTCells.treatedCells","k":"def","m":"Causalean.Panel.EstimandCharacterization.FlexibleDIDMundlak.DID","a":"Panel/EstimandCharacterization/FlexibleDIDMundlak","t":1,"d":"Treated cohort-time support set C_tr.","s":"{Cohort : Type u_1} → {Time : Type u_2} → {Covar : Type u_3} → [inst : Fintype Cohort] → [inst_1 : Fintype Time] → [inst_2 : Fintype Covar] → Causalean.Panel.EstimandCharacterization.FlexibleDIDMundlak.StaggeredATTCells Cohort Time Covar → Finset (Cohort × Time)","l":73,"f":"Causalean/Panel/EstimandCharacterization/FlexibleDIDMundlak/DID.lean"},{"n":"Causalean.Panel.EstimandCharacterization.FlexibleDIDMundlak.StaggeredATTCells.untreatedCells","k":"def","m":"Causalean.Panel.EstimandCharacterization.FlexibleDIDMundlak.DID","a":"Panel/EstimandCharacterization/FlexibleDIDMundlak","t":1,"d":"Untreated cohort-time design set used to fit the untreated-outcome regression (the support of the weighted projection that produces m0).","s":"{Cohort : Type u_1} → {Time : Type u_2} → {Covar : Type u_3} → [inst : Fintype Cohort] → [inst_1 : Fintype Time] → [inst_2 : Fintype Covar] → Causalean.Panel.EstimandCharacterization.FlexibleDIDMundlak.StaggeredATTCells Cohort Time Covar → Finset (Cohort × Time)","l":79,"f":"Causalean/Panel/EstimandCharacterization/FlexibleDIDMundlak/DID.lean"},{"n":"Causalean.Panel.EstimandCharacterization.FlexibleDIDMundlak.UntreatedDesignIdentifies","k":"def","m":"Causalean.Panel.EstimandCharacterization.FlexibleDIDMundlak.DID","a":"Panel/EstimandCharacterization/FlexibleDIDMundlak","t":1,"d":"Connected untreated design and full-rank identification condition.","s":"{Cohort : Type u_1} → {Time : Type u_2} → {Covar : Type u_3} → [inst : Fintype Cohort] → [inst_1 : Fintype Time] → [inst_2 : Fintype Covar] → Causalean.Panel.EstimandCharacterization.FlexibleDIDMundlak.StaggeredATTCells Cohort Time Covar → Prop","l":125,"f":"Causalean/Panel/EstimandCharacterization/FlexibleDIDMundlak/DID.lean"},{"n":"Causalean.Panel.EstimandCharacterization.FlexibleDIDMundlak.UntreatedFitWitness","k":"structure","m":"Causalean.Panel.EstimandCharacterization.FlexibleDIDMundlak.DID","a":"Panel/EstimandCharacterization/FlexibleDIDMundlak","t":1,"d":"The part of an untreated-regression witness needed to prove exact fit on the untreated design.","s":"{Cohort : Type u_1} → {Time : Type u_2} → {Covar : Type u_3} → [inst : Fintype Cohort] → [inst_1 : Fintype Time] → [inst_2 : Fintype Covar] → Causalean.Panel.EstimandCharacterization.FlexibleDIDMundlak.StaggeredATTCells Cohort Time Covar → Type (max (max u_1 u_2) u_3)","l":187,"f":"Causalean/Panel/EstimandCharacterization/FlexibleDIDMundlak/DID.lean"},{"n":"Causalean.Panel.EstimandCharacterization.FlexibleDIDMundlak.VectorTWFEProblem","k":"structure","m":"Causalean.Panel.EstimandCharacterization.FlexibleDIDMundlak.VectorTWFE","a":"Panel/EstimandCharacterization/FlexibleDIDMundlak","t":1,"d":"A K-vector two-way-fixed-effects regression problem on a finite balanced panel of units and time periods, given a scalar outcome and a K-vector of regressors, where the residualized Gram matrix of the double-demeaned regressors is nonsingular — the vector full-rank condition ensuring the two-way within estimator is well defined.","s":"(Unit : Type u_4) → (Time : Type u_5) → [Fintype Unit] → [Fintype Time] → (K : Type u_6) → [Fintype K] → [DecidableEq K] → Type (max (max u_4 u_5) u_6)","l":56,"f":"Causalean/Panel/EstimandCharacterization/FlexibleDIDMundlak/VectorTWFE.lean"},{"n":"Causalean.Panel.EstimandCharacterization.FlexibleDIDMundlak.VectorTWFEProblem.betaTWFE","k":"def","m":"Causalean.Panel.EstimandCharacterization.FlexibleDIDMundlak.VectorTWFE","a":"Panel/EstimandCharacterization/FlexibleDIDMundlak","t":1,"d":"Closed-form vector TWFE coefficient Q_{\\ddot X}⁻¹ (Σ_it ddot X ddot Y).","s":"{Unit : Type u_1} → {Time : Type u_2} → [inst : Fintype Unit] → [inst_1 : Fintype Time] → {K : Type u_3} → [inst_2 : Fintype K] → [inst_3 : DecidableEq K] → Causalean.Panel.EstimandCharacterization.FlexibleDIDMundlak.VectorTWFEProblem Unit Time K → K → ℝ","l":70,"f":"Causalean/Panel/EstimandCharacterization/FlexibleDIDMundlak/VectorTWFE.lean"},{"n":"Causalean.Panel.EstimandCharacterization.FlexibleDIDMundlak.VectorTWFEProblem.betaTWFE_normalEq","k":"theorem","m":"Causalean.Panel.EstimandCharacterization.FlexibleDIDMundlak.VectorTWFE","a":"Panel/EstimandCharacterization/FlexibleDIDMundlak","t":1,"d":"Existence of a TWFE solution. For a vector two-way-fixed-effects problem, whose residualized Gram matrix is nonsingular by assumption, the closed-form coefficient P.betaTWFE solves the matrix normal equation defining the TWFE coefficient.","s":"∀ {Unit : Type u_1} {Time : Type u_2} [inst : Fintype Unit] [inst_1 : Fintype Time] {K : Type u_3} [inst_2 : Fintype K] [inst_3 : DecidableEq K] (P : Causalean.Panel.EstimandCharacterization.FlexibleDIDMundlak.VectorTWFEProblem Unit Time K), P.vecTwfeNormalEq P.betaTWFE","l":140,"f":"Causalean/Panel/EstimandCharacterization/FlexibleDIDMundlak/VectorTWFE.lean"},{"n":"Causalean.Panel.EstimandCharacterization.FlexibleDIDMundlak.VectorTWFEProblem.betaTWFE_unique","k":"theorem","m":"Causalean.Panel.EstimandCharacterization.FlexibleDIDMundlak.VectorTWFE","a":"Panel/EstimandCharacterization/FlexibleDIDMundlak","t":1,"d":"Full-rank uniqueness of the vector TWFE coefficient. For a vector TWFE problem P with nonsingular residualized Gram matrix, if a coefficient vector β satisfies the coordinate-wise TWFE normal equation — in every coordinate the double-demeaned regressor is orthogonal to the double-demeaned residual, then β equals the closed-form vector TWFE coefficient P.betaTWFE.","s":"∀ {Unit : Type u_1} {Time : Type u_2} [inst : Fintype Unit] [inst_1 : Fintype Time] {K : Type u_3} [inst_2 : Fintype K] [inst_3 : DecidableEq K] (P : Causalean.Panel.EstimandCharacterization.FlexibleDIDMundlak.VectorTWFEProblem Unit Time K) {β : K → ℝ}, P.vecTwfeNormalEq β → β = P.betaTWFE","l":149,"f":"Causalean/Panel/EstimandCharacterization/FlexibleDIDMundlak/VectorTWFE.lean"},{"n":"Causalean.Panel.EstimandCharacterization.FlexibleDIDMundlak.VectorTWFEProblem.vecTwfeNormalEq","k":"def","m":"Causalean.Panel.EstimandCharacterization.FlexibleDIDMundlak.VectorTWFE","a":"Panel/EstimandCharacterization/FlexibleDIDMundlak","t":1,"d":"Vector TWFE normal equation after double demeaning: in every coordinate the residualized regressor is orthogonal to the residual.","s":"{Unit : Type u_1} → {Time : Type u_2} → [inst : Fintype Unit] → [inst_1 : Fintype Time] → {K : Type u_3} → [inst_2 : Fintype K] → [inst_3 : DecidableEq K] → Causalean.Panel.EstimandCharacterization.FlexibleDIDMundlak.VectorTWFEProblem Unit Time K → (K → ℝ) → Prop","l":74,"f":"Causalean/Panel/EstimandCharacterization/FlexibleDIDMundlak/VectorTWFE.lean"},{"n":"Causalean.Panel.EstimandCharacterization.FlexibleDIDMundlak.VectorTWMFit","k":"structure","m":"Causalean.Panel.EstimandCharacterization.FlexibleDIDMundlak.VectorMundlak","a":"Panel/EstimandCharacterization/FlexibleDIDMundlak","t":1,"d":"A coding-free K-vector two-way Mundlak fit of the vector TWFE problem P against optional time-constant controls Zvar and time-only controls Mvar, stated by normal equations rather than by a particular coding of the nuisance regressors. It bundles a coefficient vector and a nuisance function lying in the vector two-way Mundlak span, subject to the pooled normal equation against every regressor coordinate and the pooled normal equation against every nuisance function in that span.","s":"{Unit : Type u_1} → {Time : Type u_2} → [inst : Fintype Unit] → [inst_1 : Fintype Time] → {K : Type u_3} → [inst_2 : Fintype K] → [inst_3 : DecidableEq K] → {Z : Type u_4} → {M : Type u_5} → [Fintype Z] → [Fintype M] → Causalean.Panel.EstimandCharacterization.FlexibleDIDMundlak.VectorTWFEProblem Unit Time K → (Z → Unit → ℝ) → (M → Time → ℝ) → Type (max (max u_1 u_2) u_3)","l":212,"f":"Causalean/Panel/EstimandCharacterization/FlexibleDIDMundlak/VectorMundlak.lean"},{"n":"Causalean.Panel.EstimandCharacterization.FlexibleDIDMundlak.cellIndicator","k":"def","m":"Causalean.Panel.EstimandCharacterization.FlexibleDIDMundlak.DID","a":"Panel/EstimandCharacterization/FlexibleDIDMundlak","t":1,"d":"Saturated treated-cell indicator 1{(g',t') = (g,t)} (the POLS/ETWFE treated regressor for cell (g,t)).","s":"{Cohort : Type u_1} → {Time : Type u_2} → [DecidableEq Cohort] → [DecidableEq Time] → Cohort → Time → Cohort → Time → ℝ","l":390,"f":"Causalean/Panel/EstimandCharacterization/FlexibleDIDMundlak/DID.lean"},{"n":"Causalean.Panel.EstimandCharacterization.FlexibleDIDMundlak.cellIndicator_normalEq_eq_cellResidual","k":"theorem","m":"Causalean.Panel.EstimandCharacterization.FlexibleDIDMundlak.DID","a":"Panel/EstimandCharacterization/FlexibleDIDMundlak","t":2,"d":"Saturated block-diagonalization for treated-cell indicators.","s":"∀ {Cohort : Type u_1} {Time : Type u_2} {Covar : Type u_3} [inst : Fintype Cohort] [inst_1 : Fintype Time] [inst_2 : Fintype Covar] [inst_3 : DecidableEq Cohort] [inst_4 : DecidableEq Time] (outcome fitted : Cohort → Time → Covar → ℝ) (covarWeight : Cohort → Covar → ℝ) (theta : ℝ) (g : Cohort) (t : Time), ∑ g', ∑ t', ∑ c, Causalean.Panel.EstimandCharacterization.FlexibleDIDMundlak.cellIndicator g t g' t' * covarWeight g' c * (outcome g' t' c - fitted g' t' c - Causalean.Panel.EstimandCharacterization.FlexibleDIDMundlak.cellIndicator g t g' t' * theta) = 0 ↔ ∑ c, covarWeight g c * (outcome g t c - fitted g t c - theta) = 0","l":396,"f":"Causalean/Panel/EstimandCharacterization/FlexibleDIDMundlak/DID.lean"},{"n":"Causalean.Panel.EstimandCharacterization.FlexibleDIDMundlak.cellResidualNormalEq","k":"def","m":"Causalean.Panel.EstimandCharacterization.FlexibleDIDMundlak.DID","a":"Panel/EstimandCharacterization/FlexibleDIDMundlak","t":1,"d":"Finite-cell residual normal equation for a cell coefficient. With baseline-covariate weights summing to one inside cohort g, this pins down the unique coefficient as the imputation residual mean.","s":"{Cohort : Type u_1} → {Time : Type u_2} → {Covar : Type u_3} → [inst : Fintype Cohort] → [inst_1 : Fintype Time] → [inst_2 : Fintype Covar] → (P : Causalean.Panel.EstimandCharacterization.FlexibleDIDMundlak.StaggeredATTCells Cohort Time Covar) → Causalean.Panel.EstimandCharacterization.FlexibleDIDMundlak.SaturatedUntreatedRegression P → ℝ → Cohort → Time → Prop","l":351,"f":"Causalean/Panel/EstimandCharacterization/FlexibleDIDMundlak/DID.lean"},{"n":"Causalean.Panel.EstimandCharacterization.FlexibleDIDMundlak.cellResidualNormalEq_eq_imputationTheta","k":"theorem","m":"Causalean.Panel.EstimandCharacterization.FlexibleDIDMundlak.DID","a":"Panel/EstimandCharacterization/FlexibleDIDMundlak","t":2,"d":"A finite-cell residual normal equation identifies the coefficient with the imputation residual mean.","s":"∀ {Cohort : Type u_1} {Time : Type u_2} {Covar : Type u_3} [inst : Fintype Covar] (outcome fitted : Cohort → Time → Covar → ℝ) (covarWeight : Cohort → Covar → ℝ), (∀ (g : Cohort), ∑ c, covarWeight g c = 1) → ∀ {theta : ℝ} {g : Cohort} {t : Time}, ∑ c, covarWeight g c * (outcome g t c - fitted g t c - theta) = 0 → theta = ∑ c, covarWeight g c * (outcome g t c - fitted g t c)","l":360,"f":"Causalean/Panel/EstimandCharacterization/FlexibleDIDMundlak/DID.lean"},{"n":"Causalean.Panel.EstimandCharacterization.FlexibleDIDMundlak.ddotVec","k":"def","m":"Causalean.Panel.EstimandCharacterization.FlexibleDIDMundlak.VectorTWFE","a":"Panel/EstimandCharacterization/FlexibleDIDMundlak","t":1,"d":"Component-wise double-demeaned vector regressor: the k-th coordinate is the scalar double demean of the k-th component field.","s":"{Unit : Type u_1} → {Time : Type u_2} → [Fintype Unit] → [Fintype Time] → {K : Type u_3} → (Unit → Time → K → ℝ) → Unit → Time → K → ℝ","l":43,"f":"Causalean/Panel/EstimandCharacterization/FlexibleDIDMundlak/VectorTWFE.lean"},{"n":"Causalean.Panel.EstimandCharacterization.FlexibleDIDMundlak.flexible_did_aggregate_characterization","k":"theorem","m":"Causalean.Panel.EstimandCharacterization.FlexibleDIDMundlak.DID","a":"Panel/EstimandCharacterization/FlexibleDIDMundlak","t":1,"d":"Aggregate characterization: every weighting of imputation, POLS, and ETWFE equals the weighted ATT aggregate. If no anticipation holds and conditional parallel trends holds — the mean untreated potential outcome admits an additive cohort/time fixed-effects representation given covariates, then for any treated-cell weighting function a, the a-weighted aggregates of the imputation, POLS, and ETWFE cell estimands all equal the a-weighted ATT aggregate.","s":"∀ {Cohort : Type u_1} {Time : Type u_2} {Covar : Type u_3} [inst : Fintype Cohort] [inst_1 : Fintype Time] [inst_2 : Fintype Covar] (P : Causalean.Panel.EstimandCharacterization.FlexibleDIDMundlak.StaggeredATTCells Cohort Time Covar) (S : Causalean.Panel.EstimandCharacterization.FlexibleDIDMundlak.SaturatedUntreatedRegression P) (E : Causalean.Panel.EstimandCharacterization.FlexibleDIDMundlak.FlexibleDIDEstimands P S), Causalean.Panel.EstimandCharacterization.FlexibleDIDMundlak.NoAnticipation P → Causalean.Panel.EstimandCharacterization.FlexibleDIDMundlak.ConditionalParallelTrendsAdditive P → ∀ (a : Cohort → Time → ℝ), Causalean.Panel.EstimandCharacterization.FlexibleDIDMundlak.psiImp P E a = P.tauAgg a ∧ Causalean.Panel.EstimandCharacterization.FlexibleDIDMundlak.psiPOLS P E a = P.tauAgg a ∧ Causalean.Panel.EstimandCharacterization.FlexibleDIDMundlak.psiETWFE P E a = P.tauAgg a","l":588,"f":"Causalean/Panel/EstimandCharacterization/FlexibleDIDMundlak/DID.lean"},{"n":"Causalean.Panel.EstimandCharacterization.FlexibleDIDMundlak.flexible_did_cell_characterization","k":"theorem","m":"Causalean.Panel.EstimandCharacterization.FlexibleDIDMundlak.DID","a":"Panel/EstimandCharacterization/FlexibleDIDMundlak","t":1,"d":"Cell-level characterization: imputation, POLS, and ETWFE agree with the ATT cell. If no anticipation holds, conditional parallel trends holds — the mean untreated potential outcome admits an additive cohort/time fixed-effects representation given covariates, and (g,t) is a treated cohort-time cell covered by the untreated-regression witness S and the imputation/POLS/ETWFE estimands E, then the imputation, POLS, and ETWFE cell coefficients at (g,t) all equal the ATT cell τ_gt.","s":"∀ {Cohort : Type u_1} {Time : Type u_2} {Covar : Type u_3} [inst : Fintype Cohort] [inst_1 : Fintype Time] [inst_2 : Fintype Covar] (P : Causalean.Panel.EstimandCharacterization.FlexibleDIDMundlak.StaggeredATTCells Cohort Time Covar) (S : Causalean.Panel.EstimandCharacterization.FlexibleDIDMundlak.SaturatedUntreatedRegression P) (E : Causalean.Panel.EstimandCharacterization.FlexibleDIDMundlak.FlexibleDIDEstimands P S), Causalean.Panel.EstimandCharacterization.FlexibleDIDMundlak.NoAnticipation P → Causalean.Panel.EstimandCharacterization.FlexibleDIDMundlak.ConditionalParallelTrendsAdditive P → ∀ {g : Cohort} {t : Time}, P.treatedCell g t → E.thetaImp g t = P.tauCell g t ∧ E.thetaPOLS g t = P.tauCell g t ∧ E.thetaETWFE g t = P.tauCell g t","l":564,"f":"Causalean/Panel/EstimandCharacterization/FlexibleDIDMundlak/DID.lean"},{"n":"Causalean.Panel.EstimandCharacterization.FlexibleDIDMundlak.flexible_did_scaffold_characterization","k":"theorem","m":"Causalean.Panel.EstimandCharacterization.FlexibleDIDMundlak.DID","a":"Panel/EstimandCharacterization/FlexibleDIDMundlak","t":1,"d":"Headline finite-cell characterization (Wooldridge, Theorem B). If no anticipation holds: the treated and untreated potential-outcome means agree on every cell in the untreated-outcome regression's design and conditional parallel trends holds — the mean untreated potential outcome admits an additive cohort/time fixed-effects representation given covariates, then, given the saturated untreated regression S and the POLS/ETWFE finite-cell residual normal equations carried by E, on every treated cohort-time cell the flexible imputation, POLS, and ETWFE estimands all equal the ATT cell, and consequently every treated-cell weighted aggregate of the three estimands equals the correspondingly weighted ATT aggregate.","s":"∀ {Cohort : Type u_1} {Time : Type u_2} {Covar : Type u_3} [inst : Fintype Cohort] [inst_1 : Fintype Time] [inst_2 : Fintype Covar] (P : Causalean.Panel.EstimandCharacterization.FlexibleDIDMundlak.StaggeredATTCells Cohort Time Covar) (S : Causalean.Panel.EstimandCharacterization.FlexibleDIDMundlak.SaturatedUntreatedRegression P) (E : Causalean.Panel.EstimandCharacterization.FlexibleDIDMundlak.FlexibleDIDEstimands P S), Causalean.Panel.EstimandCharacterization.FlexibleDIDMundlak.NoAnticipation P → Causalean.Panel.EstimandCharacterization.FlexibleDIDMundlak.ConditionalParallelTrendsAdditive P → (∀ ⦃g : Cohort⦄ ⦃t : Time⦄, P.treatedCell g t → E.thetaImp g t = P.tauCell g t ∧ E.thetaPOLS g t = P.tauCell g t ∧ E.thetaETWFE g t = P.tauCell g t) ∧ ∀ (a : Cohort → Time → ℝ), Causalean.Panel.EstimandCharacterization.FlexibleDIDMundlak.psiImp P E a = P.tauAgg a ∧ Causalean.Panel.EstimandCharacterization.FlexibleDIDMundlak.psiPOLS P E a = P.tauAgg a ∧ Causalean.Panel.EstimandCharacterization.FlexibleDIDMundlak.psiETWFE P E a = P.tauAgg a","l":629,"f":"Causalean/Panel/EstimandCharacterization/FlexibleDIDMundlak/DID.lean"},{"n":"Causalean.Panel.EstimandCharacterization.FlexibleDIDMundlak.gram","k":"def","m":"Causalean.Panel.EstimandCharacterization.FlexibleDIDMundlak.VectorTWFE","a":"Panel/EstimandCharacterization/FlexibleDIDMundlak","t":1,"d":"Residualized Gram matrix Q_{\\ddot X} = Σ_it ddot(X_it) ddot(X_it)ᵀ.","s":"{Unit : Type u_1} → {Time : Type u_2} → [Fintype Unit] → [Fintype Time] → {K : Type u_3} → (Unit → Time → K → ℝ) → Matrix K K ℝ","l":48,"f":"Causalean/Panel/EstimandCharacterization/FlexibleDIDMundlak/VectorTWFE.lean"},{"n":"Causalean.Panel.EstimandCharacterization.FlexibleDIDMundlak.gramOf","k":"def","m":"Causalean.Panel.EstimandCharacterization.FlexibleDIDMundlak.VectorMundlak","a":"Panel/EstimandCharacterization/FlexibleDIDMundlak","t":1,"d":"Generic residualized Gram matrix of a supplied residualized regressor.","s":"{Unit : Type u_1} → {Time : Type u_2} → [Fintype Unit] → [Fintype Time] → {K : Type u_3} → (Unit → Time → K → ℝ) → Matrix K K ℝ","l":37,"f":"Causalean/Panel/EstimandCharacterization/FlexibleDIDMundlak/VectorMundlak.lean"},{"n":"Causalean.Panel.EstimandCharacterization.FlexibleDIDMundlak.gram_eq_gramOf","k":"theorem","m":"Causalean.Panel.EstimandCharacterization.FlexibleDIDMundlak.VectorMundlak","a":"Panel/EstimandCharacterization/FlexibleDIDMundlak","t":2,"d":"gram is the residualized instance of the generic version.","s":"∀ {Unit : Type u_1} {Time : Type u_2} [inst : Fintype Unit] [inst_1 : Fintype Time] {K : Type u_3} (X : Unit → Time → K → ℝ), Causalean.Panel.EstimandCharacterization.FlexibleDIDMundlak.gram X = Causalean.Panel.EstimandCharacterization.FlexibleDIDMundlak.gramOf (Causalean.Panel.EstimandCharacterization.FlexibleDIDMundlak.ddotVec X)","l":46,"f":"Causalean/Panel/EstimandCharacterization/FlexibleDIDMundlak/VectorMundlak.lean"},{"n":"Causalean.Panel.EstimandCharacterization.FlexibleDIDMundlak.imputationTheta","k":"def","m":"Causalean.Panel.EstimandCharacterization.FlexibleDIDMundlak.DID","a":"Panel/EstimandCharacterization/FlexibleDIDMundlak","t":1,"d":"Imputation residual mean for a treated cohort-time cell.","s":"{Cohort : Type u_1} → {Time : Type u_2} → {Covar : Type u_3} → [inst : Fintype Cohort] → [inst_1 : Fintype Time] → [inst_2 : Fintype Covar] → (P : Causalean.Panel.EstimandCharacterization.FlexibleDIDMundlak.StaggeredATTCells Cohort Time Covar) → Causalean.Panel.EstimandCharacterization.FlexibleDIDMundlak.SaturatedUntreatedRegression P → Cohort → Time → ℝ","l":341,"f":"Causalean/Panel/EstimandCharacterization/FlexibleDIDMundlak/DID.lean"},{"n":"Causalean.Panel.EstimandCharacterization.FlexibleDIDMundlak.imputationTheta_eq_eventCondExp","k":"theorem","m":"Causalean.Panel.EstimandCharacterization.FlexibleDIDMundlak.PopulationBridge","a":"Panel/EstimandCharacterization/FlexibleDIDMundlak","t":2,"d":"The finite imputation residual mean equals a population conditional expectation.","s":"∀ {Cohort : Type u_1} {Time : Type u_2} {Covar : Type u_3} [inst : Fintype Cohort] [inst_1 : Fintype Time] [inst_2 : Fintype Covar] {Ω : Type u_4} [inst_3 : MeasurableSpace Ω] (μ : MeasureTheory.Measure Ω) [MeasureTheory.IsFiniteMeasure μ] (P : Causalean.Panel.EstimandCharacterization.FlexibleDIDMundlak.StaggeredATTCells Cohort Time Covar) (S : Causalean.Panel.EstimandCharacterization.FlexibleDIDMundlak.SaturatedUntreatedRegression P) (g : Cohort) (t : Time) (cohortEvent : Set Ω) (covarCell : Covar → Set Ω) (Δ : Ω → ℝ), MeasurableSet cohortEvent → (∀ (c : Covar), MeasurableSet (covarCell c)) → Pairwise (Function.onFun Disjoint covarCell) → ⋃ c, covarCell c = Set.univ → MeasureTheory.Integrable Δ μ → (∀ (c : Covar), P.covarWeight g c = (μ (cohortEvent ∩ covarCell c)).toReal / (μ cohortEvent).toReal) → (∀ (c : Covar), P.observedMean g t c - S.m0 g t c = Causalean.PO.eventCondExp μ (cohortEvent ∩ covarCell c) Δ) → Causalean.Panel.EstimandCharacterization.FlexibleDIDMundlak.imputationTheta P S g t = Causalean.PO.eventCondExp μ cohortEvent Δ","l":46,"f":"Causalean/Panel/EstimandCharacterization/FlexibleDIDMundlak/PopulationBridge.lean"},{"n":"Causalean.Panel.EstimandCharacterization.FlexibleDIDMundlak.imputationTheta_eq_tauCell","k":"theorem","m":"Causalean.Panel.EstimandCharacterization.FlexibleDIDMundlak.DID","a":"Panel/EstimandCharacterization/FlexibleDIDMundlak","t":2,"d":"Imputation recovers the ATT cell once the saturated untreated prediction equals the untreated potential-outcome mean in target cells.","s":"∀ {Cohort : Type u_1} {Time : Type u_2} {Covar : Type u_3} [inst : Fintype Cohort] [inst_1 : Fintype Time] [inst_2 : Fintype Covar] (P : Causalean.Panel.EstimandCharacterization.FlexibleDIDMundlak.StaggeredATTCells Cohort Time Covar) (S : Causalean.Panel.EstimandCharacterization.FlexibleDIDMundlak.SaturatedUntreatedRegression P) (E : Causalean.Panel.EstimandCharacterization.FlexibleDIDMundlak.FlexibleDIDEstimands P S), Causalean.Panel.EstimandCharacterization.FlexibleDIDMundlak.NoAnticipation P → Causalean.Panel.EstimandCharacterization.FlexibleDIDMundlak.ConditionalParallelTrendsAdditive P → ∀ {g : Cohort} {t : Time}, P.treatedCell g t → E.thetaImp g t = P.tauCell g t","l":549,"f":"Causalean/Panel/EstimandCharacterization/FlexibleDIDMundlak/DID.lean"},{"n":"Causalean.Panel.EstimandCharacterization.FlexibleDIDMundlak.m0_eq_eventCondExp_treated","k":"theorem","m":"Causalean.Panel.EstimandCharacterization.FlexibleDIDMundlak.PopulationBridge","a":"Panel/EstimandCharacterization/FlexibleDIDMundlak","t":2,"d":"On a treated cell, the fitted untreated mean equals the population conditional mean of the untreated potential outcome.","s":"∀ {Cohort : Type u_1} {Time : Type u_2} {Covar : Type u_3} [inst : Fintype Cohort] [inst_1 : Fintype Time] [inst_2 : Fintype Covar] {Ω : Type u_4} [inst_3 : MeasurableSpace Ω] (μ : MeasureTheory.Measure Ω) {P : Causalean.Panel.EstimandCharacterization.FlexibleDIDMundlak.StaggeredATTCells Cohort Time Covar} (S : Causalean.Panel.EstimandCharacterization.FlexibleDIDMundlak.SaturatedUntreatedRegression P), Causalean.Panel.EstimandCharacterization.FlexibleDIDMundlak.NoAnticipation P → Causalean.Panel.EstimandCharacterization.FlexibleDIDMundlak.ConditionalParallelTrendsAdditive P → ∀ {g : Cohort} {t : Time}, P.treatedCell g t → ∀ (c : Covar) (cellEvent : Cohort → Time → Covar → Set Ω) (Y0pop : Ω → ℝ), P.Y0Mean g t c = Causalean.PO.eventCondExp μ (cellEvent g t c) Y0pop → S.m0 g t c = Causalean.PO.eventCondExp μ (cellEvent g t c) Y0pop","l":130,"f":"Causalean/Panel/EstimandCharacterization/FlexibleDIDMundlak/PopulationBridge.lean"},{"n":"Causalean.Panel.EstimandCharacterization.FlexibleDIDMundlak.m0_eq_eventCondExp_treated_ofPopulation","k":"theorem","m":"Causalean.Panel.EstimandCharacterization.FlexibleDIDMundlak.PopulationOrigin","a":"Panel/EstimandCharacterization/FlexibleDIDMundlak","t":1,"d":"On a treated cell in a population-built system, the fitted untreated mean equals the population conditional mean of the untreated potential outcome. Fix a population model on a sample space Ω, with population outcomes Y0pop, Ygpop, Yobspop; suppose every cell event cellEvent g t c is measurable, every cell has strictly positive probability mass, and the three population outcomes are each integrable on every cell. Suppose also the cohort share is strictly positive on every treated cell, the covariate weights are nonnegative and sum to one within each cohort, and the observed outcome agrees pointwise with the cohort-g outcome on treated cells and with the untreated outcome on untreated cells (pointwise consistency). If the finite cell system P is exactly the one built from this population data by StaggeredATTCells.ofPopulation and, for a saturated untreated regression S on P, no anticipation holds and conditional parallel trends holds, then on any treated cell (g,t), the saturated regression's fitted value S.m0 g t c equals the population conditional mean E[Y0pop | cellEvent g t c], for every covariate cell c.","s":"∀ {Cohort : Type u_1} {Time : Type u_2} {Covar : Type u_3} [inst : Fintype Cohort] [inst_1 : Fintype Time] [inst_2 : Fintype Covar] {Ω : Type u_4} [inst_3 : MeasurableSpace Ω] (μ : MeasureTheory.Measure Ω) [inst_4 : MeasureTheory.IsProbabilityMeasure μ] (cellEvent : Cohort → Time → Covar → Set Ω) (Y0pop Ygpop Yobspop : Ω → ℝ) (treatedCell untreatedCell : Cohort → Time → Prop) (cohortShare : Cohort → ℝ) (covarWeight : Cohort → Covar → ℝ) (hmeas : ∀ (g : Cohort) (t : Time) (c : Covar), MeasurableSet (cellEvent g t c)) (hcell_pos : ∀ (g : Cohort) (t : Time) (c : Covar), 0 < (μ (cellEvent g t c)).toReal) (hY0_int : ∀ (g : Cohort) (t : Time) (c : Covar), MeasureTheory.IntegrableOn Y0pop (cellEvent g t c) μ) (hYg_int : ∀ (g : Cohort) (t : Time) (c : Covar), MeasureTheory.IntegrableOn Ygpop (cellEvent g t c) μ) (hYobs_int : ∀ (g : Cohort) (t : Time) (c : Covar), MeasureTheory.IntegrableOn Yobspop (cellEvent g t c) μ) (cohortShare_pos_on_treated : ∀ ⦃g : Cohort⦄ ⦃t : Time⦄, treatedCell g t → 0 < cohortShare g) (covarWeight_nonneg : ∀ (g : Cohort) (c : Covar), 0 ≤ covarWeight g c) (covarWeight_sum_one : ∀ (g : Cohort), ∑ c, covarWeight g c = 1) (hcons_tr : ∀ ⦃g : Cohort⦄ ⦃t : Time⦄, treatedCell g t → ∀ (c : Covar), ∀ ω ∈ cellEvent g t c, Yobspop ω = Ygpop ω) (hcons_ut : ∀ ⦃g : Cohort⦄ ⦃t : Time⦄, untreatedCell g t → ∀ (c : Covar), ∀ ω ∈ cellEvent g t c, Yobspop ω = Y0pop ω) {P : Causalean.Panel.EstimandCharacterization.FlexibleDIDMundlak.StaggeredATTCells Cohort Time Covar}, P = Causalean.Panel.EstimandCharacterization.FlexibleDIDMundlak.StaggeredATTCells.ofPopulation μ cellEvent Y0pop Ygpop Yobspop treatedCell untreatedCell cohortShare covarWeight hmeas hcell_pos hY0_int hYg_int hYobs_int cohortShare_pos_on_treated covarWeight_nonneg covarWeight_sum_one hcons_tr hcons_ut → ∀ (S : Causalean.Panel.EstimandCharacterization.FlexibleDIDMundlak.SaturatedUntreatedRegression P), Causalean.Panel.EstimandCharacterization.FlexibleDIDMundlak.NoAnticipation P → Causalean.Panel.EstimandCharacterization.FlexibleDIDMundlak.ConditionalParallelTrendsAdditive P → ∀ {g : Cohort} {t : Time}, P.treatedCell g t → ∀ (c : Covar), S.m0 g t c = Causalean.PO.eventCondExp μ (cellEvent g t c) Y0pop","l":144,"f":"Causalean/Panel/EstimandCharacterization/FlexibleDIDMundlak/PopulationOrigin.lean"},{"n":"Causalean.Panel.EstimandCharacterization.FlexibleDIDMundlak.m0_eq_eventCondExp_untreated","k":"theorem","m":"Causalean.Panel.EstimandCharacterization.FlexibleDIDMundlak.PopulationBridge","a":"Panel/EstimandCharacterization/FlexibleDIDMundlak","t":2,"d":"On an untreated cell, the fitted untreated mean equals the population conditional mean of the untreated potential outcome.","s":"∀ {Cohort : Type u_1} {Time : Type u_2} {Covar : Type u_3} [inst : Fintype Cohort] [inst_1 : Fintype Time] [inst_2 : Fintype Covar] {Ω : Type u_4} [inst_3 : MeasurableSpace Ω] (μ : MeasureTheory.Measure Ω) {P : Causalean.Panel.EstimandCharacterization.FlexibleDIDMundlak.StaggeredATTCells Cohort Time Covar} (S : Causalean.Panel.EstimandCharacterization.FlexibleDIDMundlak.SaturatedUntreatedRegression P), Causalean.Panel.EstimandCharacterization.FlexibleDIDMundlak.NoAnticipation P → Causalean.Panel.EstimandCharacterization.FlexibleDIDMundlak.ConditionalParallelTrendsAdditive P → ∀ {g : Cohort} {t : Time}, P.untreatedCell g t → ∀ (c : Covar) (cellEvent : Cohort → Time → Covar → Set Ω) (Y0pop : Ω → ℝ), P.Y0Mean g t c = Causalean.PO.eventCondExp μ (cellEvent g t c) Y0pop → S.m0 g t c = Causalean.PO.eventCondExp μ (cellEvent g t c) Y0pop","l":149,"f":"Causalean/Panel/EstimandCharacterization/FlexibleDIDMundlak/PopulationBridge.lean"},{"n":"Causalean.Panel.EstimandCharacterization.FlexibleDIDMundlak.m0_eq_eventCondExp_untreated_ofPopulation","k":"theorem","m":"Causalean.Panel.EstimandCharacterization.FlexibleDIDMundlak.PopulationOrigin","a":"Panel/EstimandCharacterization/FlexibleDIDMundlak","t":1,"d":"On an untreated cell in a population-built system, the fitted untreated mean equals the population conditional mean of the untreated potential outcome. Fix a population model on a sample space Ω, with population outcomes Y0pop, Ygpop, Yobspop; suppose every cell event cellEvent g t c is measurable, every cell has strictly positive probability mass, and the three population outcomes are each integrable on every cell. Suppose also the cohort share is strictly positive on every treated cell, the covariate weights are nonnegative and sum to one within each cohort, and the observed outcome agrees pointwise with the cohort-g outcome on treated cells and with the untreated outcome on untreated cells (pointwise consistency). If the finite cell system P is exactly the one built from this population data by StaggeredATTCells.ofPopulation and, for a saturated untreated regression S on P, no anticipation holds and conditional parallel trends holds, then on any untreated cell (g,t), the saturated regression's fitted value S.m0 g t c equals the population conditional mean E[Y0pop | cellEvent g t c], for every covariate cell c.","s":"∀ {Cohort : Type u_1} {Time : Type u_2} {Covar : Type u_3} [inst : Fintype Cohort] [inst_1 : Fintype Time] [inst_2 : Fintype Covar] {Ω : Type u_4} [inst_3 : MeasurableSpace Ω] (μ : MeasureTheory.Measure Ω) [inst_4 : MeasureTheory.IsProbabilityMeasure μ] (cellEvent : Cohort → Time → Covar → Set Ω) (Y0pop Ygpop Yobspop : Ω → ℝ) (treatedCell untreatedCell : Cohort → Time → Prop) (cohortShare : Cohort → ℝ) (covarWeight : Cohort → Covar → ℝ) (hmeas : ∀ (g : Cohort) (t : Time) (c : Covar), MeasurableSet (cellEvent g t c)) (hcell_pos : ∀ (g : Cohort) (t : Time) (c : Covar), 0 < (μ (cellEvent g t c)).toReal) (hY0_int : ∀ (g : Cohort) (t : Time) (c : Covar), MeasureTheory.IntegrableOn Y0pop (cellEvent g t c) μ) (hYg_int : ∀ (g : Cohort) (t : Time) (c : Covar), MeasureTheory.IntegrableOn Ygpop (cellEvent g t c) μ) (hYobs_int : ∀ (g : Cohort) (t : Time) (c : Covar), MeasureTheory.IntegrableOn Yobspop (cellEvent g t c) μ) (cohortShare_pos_on_treated : ∀ ⦃g : Cohort⦄ ⦃t : Time⦄, treatedCell g t → 0 < cohortShare g) (covarWeight_nonneg : ∀ (g : Cohort) (c : Covar), 0 ≤ covarWeight g c) (covarWeight_sum_one : ∀ (g : Cohort), ∑ c, covarWeight g c = 1) (hcons_tr : ∀ ⦃g : Cohort⦄ ⦃t : Time⦄, treatedCell g t → ∀ (c : Covar), ∀ ω ∈ cellEvent g t c, Yobspop ω = Ygpop ω) (hcons_ut : ∀ ⦃g : Cohort⦄ ⦃t : Time⦄, untreatedCell g t → ∀ (c : Covar), ∀ ω ∈ cellEvent g t c, Yobspop ω = Y0pop ω) {P : Causalean.Panel.EstimandCharacterization.FlexibleDIDMundlak.StaggeredATTCells Cohort Time Covar}, P = Causalean.Panel.EstimandCharacterization.FlexibleDIDMundlak.StaggeredATTCells.ofPopulation μ cellEvent Y0pop Ygpop Yobspop treatedCell untreatedCell cohortShare covarWeight hmeas hcell_pos hY0_int hYg_int hYobs_int cohortShare_pos_on_treated covarWeight_nonneg covarWeight_sum_one hcons_tr hcons_ut → ∀ (S : Causalean.Panel.EstimandCharacterization.FlexibleDIDMundlak.SaturatedUntreatedRegression P), Causalean.Panel.EstimandCharacterization.FlexibleDIDMundlak.NoAnticipation P → Causalean.Panel.EstimandCharacterization.FlexibleDIDMundlak.ConditionalParallelTrendsAdditive P → ∀ {g : Cohort} {t : Time}, P.untreatedCell g t → ∀ (c : Covar), S.m0 g t c = Causalean.PO.eventCondExp μ (cellEvent g t c) Y0pop","l":204,"f":"Causalean/Panel/EstimandCharacterization/FlexibleDIDMundlak/PopulationOrigin.lean"},{"n":"Causalean.Panel.EstimandCharacterization.FlexibleDIDMundlak.matrix_fwl_eq_of_normalEqs","k":"theorem","m":"Causalean.Panel.EstimandCharacterization.FlexibleDIDMundlak.VectorMundlak","a":"Panel/EstimandCharacterization/FlexibleDIDMundlak","t":2,"d":"Matrix Frisch-Waugh-Lovell handoff. If a coefficient vector β and nuisance term Hβ satisfy the finite normal equations against the raw vector regressor D and a nuisance class H, while each coordinate of the residualized regressor Dtilde is orthogonal to H and the residualized Gram matrix is nonsingular, then β is the residualized matrix coefficient.","s":"∀ {Unit : Type u_1} {Time : Type u_2} [inst : Fintype Unit] [inst_1 : Fintype Time] {K : Type u_3} [inst_2 : Fintype K] [inst_3 : DecidableEq K] (H : (Unit → Time → ℝ) → Prop) {Y Yproj Ytilde : Unit → Time → ℝ} {D Dproj Dtilde : Unit → Time → K → ℝ} {Hβ : Unit → Time → ℝ} {β : K → ℝ}, (∀ (i : Unit) (t : Time), Y i t = Yproj i t + Ytilde i t) → (∀ (i : Unit) (t : Time) (k : K), D i t k = Dproj i t k + Dtilde i t k) → (∀ (k : K), H fun i t => Dproj i t k) → H Hβ → (∀ (k : K) (h : Unit → Time → ℝ), H h → ∑ i, ∑ t, Dtilde i t k * h i t = 0) → (∀ (k : K), ∑ i, ∑ t, Dtilde i t k * Yproj i t = 0) → IsUnit (Causalean.Panel.EstimandCharacterization.FlexibleDIDMundlak.gramOf Dtilde).det → (∀ (k : K), ∑ i, ∑ t, D i t k * (Y i t - ∑ j, D i t j * β j - Hβ i t) = 0) → (∀ (h : Unit → Time → ℝ), H h → ∑ i, ∑ t, h i t * (Y i t - ∑ j, D i t j * β j - Hβ i t) = 0) → β = (Causalean.Panel.EstimandCharacterization.FlexibleDIDMundlak.gramOf Dtilde)⁻¹.mulVec (Causalean.Panel.EstimandCharacterization.FlexibleDIDMundlak.numerOf Dtilde Ytilde)","l":77,"f":"Causalean/Panel/EstimandCharacterization/FlexibleDIDMundlak/VectorMundlak.lean"},{"n":"Causalean.Panel.EstimandCharacterization.FlexibleDIDMundlak.mundlak_nuisance_unit_time","k":"theorem","m":"Causalean.Panel.EstimandCharacterization.FlexibleDIDMundlak.TWFE","a":"Panel/EstimandCharacterization/FlexibleDIDMundlak","t":2,"d":"Mundlak nuisance functions are unit/time additive, so optional time-constant and time-only controls lie inside the same orthogonality class.","s":"∀ {Unit : Type u_1} {Time : Type u_2} [inst : Fintype Unit] [inst_1 : Fintype Time] {Z : Type u_3} {M : Type u_4} [inst_2 : Fintype Z] [inst_3 : Fintype M] (X : Unit → Time → ℝ) (Zvar : Z → Unit → ℝ) (Mvar : M → Time → ℝ) {h : Unit → Time → ℝ}, Causalean.Panel.EstimandCharacterization.FlexibleDIDMundlak.IsTwoWayMundlakNuisance X Zvar Mvar h → Causalean.Panel.UniformTwoWayPanel.IsUnitTimeAdditive h","l":149,"f":"Causalean/Panel/EstimandCharacterization/FlexibleDIDMundlak/TWFE.lean"},{"n":"Causalean.Panel.EstimandCharacterization.FlexibleDIDMundlak.numer","k":"def","m":"Causalean.Panel.EstimandCharacterization.FlexibleDIDMundlak.VectorTWFE","a":"Panel/EstimandCharacterization/FlexibleDIDMundlak","t":1,"d":"Residualized numerator vector Σ_it ddot(X_it) ddot(Y_it).","s":"{Unit : Type u_1} → {Time : Type u_2} → [Fintype Unit] → [Fintype Time] → {K : Type u_3} → (Unit → Time → K → ℝ) → (Unit → Time → ℝ) → K → ℝ","l":52,"f":"Causalean/Panel/EstimandCharacterization/FlexibleDIDMundlak/VectorTWFE.lean"},{"n":"Causalean.Panel.EstimandCharacterization.FlexibleDIDMundlak.numerOf","k":"def","m":"Causalean.Panel.EstimandCharacterization.FlexibleDIDMundlak.VectorMundlak","a":"Panel/EstimandCharacterization/FlexibleDIDMundlak","t":1,"d":"Generic residualized numerator vector.","s":"{Unit : Type u_1} → {Time : Type u_2} → [Fintype Unit] → [Fintype Time] → {K : Type u_3} → (Unit → Time → K → ℝ) → (Unit → Time → ℝ) → K → ℝ","l":41,"f":"Causalean/Panel/EstimandCharacterization/FlexibleDIDMundlak/VectorMundlak.lean"},{"n":"Causalean.Panel.EstimandCharacterization.FlexibleDIDMundlak.numer_eq_numerOf","k":"theorem","m":"Causalean.Panel.EstimandCharacterization.FlexibleDIDMundlak.VectorMundlak","a":"Panel/EstimandCharacterization/FlexibleDIDMundlak","t":2,"d":"numer is the residualized instance of the generic version.","s":"∀ {Unit : Type u_1} {Time : Type u_2} [inst : Fintype Unit] [inst_1 : Fintype Time] {K : Type u_3} (X : Unit → Time → K → ℝ) (Y : Unit → Time → ℝ), Causalean.Panel.EstimandCharacterization.FlexibleDIDMundlak.numer X Y = Causalean.Panel.EstimandCharacterization.FlexibleDIDMundlak.numerOf (Causalean.Panel.EstimandCharacterization.FlexibleDIDMundlak.ddotVec X) (Causalean.Panel.UniformTwoWayPanel.ddot Y)","l":50,"f":"Causalean/Panel/EstimandCharacterization/FlexibleDIDMundlak/VectorMundlak.lean"},{"n":"Causalean.Panel.EstimandCharacterization.FlexibleDIDMundlak.psiETWFE","k":"def","m":"Causalean.Panel.EstimandCharacterization.FlexibleDIDMundlak.DID","a":"Panel/EstimandCharacterization/FlexibleDIDMundlak","t":1,"d":"Aggregate estimand for the flexible ETWFE coefficients.","s":"{Cohort : Type u_1} → {Time : Type u_2} → {Covar : Type u_3} → [inst : Fintype Cohort] → [inst_1 : Fintype Time] → [inst_2 : Fintype Covar] → (P : Causalean.Panel.EstimandCharacterization.FlexibleDIDMundlak.StaggeredATTCells Cohort Time Covar) → {S : Causalean.Panel.EstimandCharacterization.FlexibleDIDMundlak.SaturatedUntreatedRegression P} → Causalean.Panel.EstimandCharacterization.FlexibleDIDMundlak.FlexibleDIDEstimands P S → (Cohort → Time → ℝ) → ℝ","l":543,"f":"Causalean/Panel/EstimandCharacterization/FlexibleDIDMundlak/DID.lean"},{"n":"Causalean.Panel.EstimandCharacterization.FlexibleDIDMundlak.psiImp","k":"def","m":"Causalean.Panel.EstimandCharacterization.FlexibleDIDMundlak.DID","a":"Panel/EstimandCharacterization/FlexibleDIDMundlak","t":1,"d":"Aggregate estimand for the imputation coefficients.","s":"{Cohort : Type u_1} → {Time : Type u_2} → {Covar : Type u_3} → [inst : Fintype Cohort] → [inst_1 : Fintype Time] → [inst_2 : Fintype Covar] → (P : Causalean.Panel.EstimandCharacterization.FlexibleDIDMundlak.StaggeredATTCells Cohort Time Covar) → {S : Causalean.Panel.EstimandCharacterization.FlexibleDIDMundlak.SaturatedUntreatedRegression P} → Causalean.Panel.EstimandCharacterization.FlexibleDIDMundlak.FlexibleDIDEstimands P S → (Cohort → Time → ℝ) → ℝ","l":531,"f":"Causalean/Panel/EstimandCharacterization/FlexibleDIDMundlak/DID.lean"},{"n":"Causalean.Panel.EstimandCharacterization.FlexibleDIDMundlak.psiPOLS","k":"def","m":"Causalean.Panel.EstimandCharacterization.FlexibleDIDMundlak.DID","a":"Panel/EstimandCharacterization/FlexibleDIDMundlak","t":1,"d":"Aggregate estimand for the flexible POLS coefficients.","s":"{Cohort : Type u_1} → {Time : Type u_2} → {Covar : Type u_3} → [inst : Fintype Cohort] → [inst_1 : Fintype Time] → [inst_2 : Fintype Covar] → (P : Causalean.Panel.EstimandCharacterization.FlexibleDIDMundlak.StaggeredATTCells Cohort Time Covar) → {S : Causalean.Panel.EstimandCharacterization.FlexibleDIDMundlak.SaturatedUntreatedRegression P} → Causalean.Panel.EstimandCharacterization.FlexibleDIDMundlak.FlexibleDIDEstimands P S → (Cohort → Time → ℝ) → ℝ","l":537,"f":"Causalean/Panel/EstimandCharacterization/FlexibleDIDMundlak/DID.lean"},{"n":"Causalean.Panel.EstimandCharacterization.FlexibleDIDMundlak.sum_dotRegressor","k":"theorem","m":"Causalean.Panel.EstimandCharacterization.FlexibleDIDMundlak.VectorMundlak","a":"Panel/EstimandCharacterization/FlexibleDIDMundlak","t":2,"d":"Reshuffle: a residualized regressor against a β-combination of regressors factors through the cross-Gram.","s":"∀ {Unit : Type u_1} {Time : Type u_2} [inst : Fintype Unit] [inst_1 : Fintype Time] {K : Type u_3} [inst_2 : Fintype K] (Dt D : Unit → Time → K → ℝ) (β : K → ℝ) (k : K), ∑ i, ∑ t, Dt i t k * ∑ j, D i t j * β j = ∑ j, (∑ i, ∑ t, Dt i t k * D i t j) * β j","l":55,"f":"Causalean/Panel/EstimandCharacterization/FlexibleDIDMundlak/VectorMundlak.lean"},{"n":"Causalean.Panel.EstimandCharacterization.FlexibleDIDMundlak.sum_ite_one_mul","k":"theorem","m":"Causalean.Panel.EstimandCharacterization.FlexibleDIDMundlak.VectorMundlak","a":"Panel/EstimandCharacterization/FlexibleDIDMundlak","t":2,"d":"Selecting a single coordinate via a 0/1 indicator collapses the coordinate sum to that coordinate's value.","s":"∀ {K : Type u_3} [inst : Fintype K] [inst_1 : DecidableEq K] (k : K) (f : K → ℝ), ∑ k', (if k' = k then 1 else 0) * f k' = f k","l":203,"f":"Causalean/Panel/EstimandCharacterization/FlexibleDIDMundlak/VectorMundlak.lean"},{"n":"Causalean.Panel.EstimandCharacterization.FlexibleDIDMundlak.thetaImp_eq_eventCondExp","k":"theorem","m":"Causalean.Panel.EstimandCharacterization.FlexibleDIDMundlak.PopulationBridge","a":"Panel/EstimandCharacterization/FlexibleDIDMundlak","t":1,"d":"The imputation estimand equals a population conditional expectation. On a treated cohort-time cell (g,t), suppose the cohort event is measurable, each covariate cell is measurable, the covariate cells are pairwise disjoint, the covariate cells cover the whole sample space, the treatment-effect integrand Δ is integrable, each finite covariate weight equals the conditional probability of that covariate cell given the cohort event, and each finite cell residual — the observed mean minus the fitted untreated mean — equals the within-cell conditional mean of Δ given the cohort event and that covariate cell. Then the imputation cell estimand thetaImp g t equals the population conditional expectation E[Δ | cohortEvent].","s":"∀ {Cohort : Type u_1} {Time : Type u_2} {Covar : Type u_3} [inst : Fintype Cohort] [inst_1 : Fintype Time] [inst_2 : Fintype Covar] {Ω : Type u_4} [inst_3 : MeasurableSpace Ω] (μ : MeasureTheory.Measure Ω) [MeasureTheory.IsFiniteMeasure μ] (P : Causalean.Panel.EstimandCharacterization.FlexibleDIDMundlak.StaggeredATTCells Cohort Time Covar) (S : Causalean.Panel.EstimandCharacterization.FlexibleDIDMundlak.SaturatedUntreatedRegression P) (E : Causalean.Panel.EstimandCharacterization.FlexibleDIDMundlak.FlexibleDIDEstimands P S) {g : Cohort} {t : Time}, P.treatedCell g t → ∀ (cohortEvent : Set Ω) (covarCell : Covar → Set Ω) (Δ : Ω → ℝ), MeasurableSet cohortEvent → (∀ (c : Covar), MeasurableSet (covarCell c)) → Pairwise (Function.onFun Disjoint covarCell) → ⋃ c, covarCell c = Set.univ → MeasureTheory.Integrable Δ μ → (∀ (c : Covar), P.covarWeight g c = (μ (cohortEvent ∩ covarCell c)).toReal / (μ cohortEvent).toReal) → (∀ (c : Covar), P.observedMean g t c - S.m0 g t c = Causalean.PO.eventCondExp μ (cohortEvent ∩ covarCell c) Δ) → E.thetaImp g t = Causalean.PO.eventCondExp μ cohortEvent Δ","l":82,"f":"Causalean/Panel/EstimandCharacterization/FlexibleDIDMundlak/PopulationBridge.lean"},{"n":"Causalean.Panel.EstimandCharacterization.FlexibleDIDMundlak.twfe_twm_equivalence","k":"theorem","m":"Causalean.Panel.EstimandCharacterization.FlexibleDIDMundlak.TWFE","a":"Panel/EstimandCharacterization/FlexibleDIDMundlak","t":1,"d":"Wooldridge finite-panel scalar TWFE-two-way-Mundlak equivalence. For a scalar two-way-fixed-effects panel regression problem P with optional time-constant controls Zvar and time-only controls Mvar, given any pooled two-way Mundlak regression fit stated by its normal equations, that fit's coefficient on the regressor equals the two-way-fixed-effects coefficient.","s":"∀ {Unit : Type u_1} {Time : Type u_2} [inst : Fintype Unit] [inst_1 : Fintype Time] {Z : Type u_3} {M : Type u_4} [inst_2 : Fintype Z] [inst_3 : Fintype M] (P : Causalean.Panel.EstimandCharacterization.FlexibleDIDMundlak.ScalarTWFEProblem Unit Time) (Zvar : Z → Unit → ℝ) (Mvar : M → Time → ℝ) (fit : Causalean.Panel.EstimandCharacterization.FlexibleDIDMundlak.ScalarTWMFit P Zvar Mvar), fit.beta = P.betaTWFE","l":198,"f":"Causalean/Panel/EstimandCharacterization/FlexibleDIDMundlak/TWFE.lean"},{"n":"Causalean.Panel.EstimandCharacterization.FlexibleDIDMundlak.twfe_twm_optional_controls_invariant","k":"theorem","m":"Causalean.Panel.EstimandCharacterization.FlexibleDIDMundlak.TWFE","a":"Panel/EstimandCharacterization/FlexibleDIDMundlak","t":2,"d":"Adding or removing optional time-constant or time-only controls does not change the scalar coefficient, because both fits equal the TWFE coefficient.","s":"∀ {Unit : Type u_1} {Time : Type u_2} [inst : Fintype Unit] [inst_1 : Fintype Time] {Z₁ : Type u_5} {M₁ : Type u_6} {Z₂ : Type u_7} {M₂ : Type u_8} [inst_2 : Fintype Z₁] [inst_3 : Fintype M₁] [inst_4 : Fintype Z₂] [inst_5 : Fintype M₂] (P : Causalean.Panel.EstimandCharacterization.FlexibleDIDMundlak.ScalarTWFEProblem Unit Time) (Zvar₁ : Z₁ → Unit → ℝ) (Mvar₁ : M₁ → Time → ℝ) (Zvar₂ : Z₂ → Unit → ℝ) (Mvar₂ : M₂ → Time → ℝ) (fit₁ : Causalean.Panel.EstimandCharacterization.FlexibleDIDMundlak.ScalarTWMFit P Zvar₁ Mvar₁) (fit₂ : Causalean.Panel.EstimandCharacterization.FlexibleDIDMundlak.ScalarTWMFit P Zvar₂ Mvar₂), fit₁.beta = fit₂.beta","l":255,"f":"Causalean/Panel/EstimandCharacterization/FlexibleDIDMundlak/TWFE.lean"},{"n":"Causalean.Panel.EstimandCharacterization.FlexibleDIDMundlak.twfe_twm_residual_common","k":"theorem","m":"Causalean.Panel.EstimandCharacterization.FlexibleDIDMundlak.TWFE","a":"Panel/EstimandCharacterization/FlexibleDIDMundlak","t":2,"d":"Residualizing the scalar regressor against the two-way Mundlak nuisance span leaves the same residual as double demeaning.","s":"∀ {Unit : Type u_1} {Time : Type u_2} [inst : Fintype Unit] [inst_1 : Fintype Time] {Z : Type u_3} {M : Type u_4} [inst_2 : Fintype Z] [inst_3 : Fintype M] (P : Causalean.Panel.EstimandCharacterization.FlexibleDIDMundlak.ScalarTWFEProblem Unit Time) (Zvar : Z → Unit → ℝ) (Mvar : M → Time → ℝ), (Causalean.Panel.UniformTwoWayPanel.IsUnitTimeAdditive fun i t => P.X i t - Causalean.Panel.UniformTwoWayPanel.ddot P.X i t) ∧ ∀ (h : Unit → Time → ℝ), Causalean.Panel.EstimandCharacterization.FlexibleDIDMundlak.IsTwoWayMundlakNuisance P.X Zvar Mvar h → Causalean.Panel.UniformTwoWayPanel.inner (Causalean.Panel.UniformTwoWayPanel.ddot P.X) h = 0","l":180,"f":"Causalean/Panel/EstimandCharacterization/FlexibleDIDMundlak/TWFE.lean"},{"n":"Causalean.Panel.EstimandCharacterization.FlexibleDIDMundlak.vecNormalEq_iff_mulVec","k":"theorem","m":"Causalean.Panel.EstimandCharacterization.FlexibleDIDMundlak.VectorTWFE","a":"Panel/EstimandCharacterization/FlexibleDIDMundlak","t":2,"d":"The coordinate-wise normal equation is equivalent to the matrix normal equation Q_{\\ddot X} β = Σ_it ddot X ddot Y.","s":"∀ {Unit : Type u_1} {Time : Type u_2} [inst : Fintype Unit] [inst_1 : Fintype Time] {K : Type u_3} [inst_2 : Fintype K] (X : Unit → Time → K → ℝ) (Y : Unit → Time → ℝ) (β : K → ℝ), (∀ (k : K), ∑ i, ∑ t, Causalean.Panel.EstimandCharacterization.FlexibleDIDMundlak.ddotVec X i t k * (Causalean.Panel.UniformTwoWayPanel.ddot Y i t - ∑ j, Causalean.Panel.EstimandCharacterization.FlexibleDIDMundlak.ddotVec X i t j * β j) = 0) ↔ (Causalean.Panel.EstimandCharacterization.FlexibleDIDMundlak.gram X).mulVec β = Causalean.Panel.EstimandCharacterization.FlexibleDIDMundlak.numer X Y","l":83,"f":"Causalean/Panel/EstimandCharacterization/FlexibleDIDMundlak/VectorTWFE.lean"},{"n":"Causalean.Panel.EstimandCharacterization.FlexibleDIDMundlak.vec_twfe_twm_equivalence","k":"theorem","m":"Causalean.Panel.EstimandCharacterization.FlexibleDIDMundlak.VectorMundlak","a":"Panel/EstimandCharacterization/FlexibleDIDMundlak","t":1,"d":"Wooldridge finite-panel K-vector TWFE-two-way-Mundlak equivalence (Theorem A). For a K-vector two-way-fixed-effects panel regression problem P with optional time-constant controls Zvar and time-only controls Mvar, given any pooled two-way Mundlak regression fit stated by its normal equations, that fit's coefficient vector on the regressors equals the K-vector two-way-fixed-effects coefficient vector.","s":"∀ {Unit : Type u_1} {Time : Type u_2} [inst : Fintype Unit] [inst_1 : Fintype Time] {K : Type u_3} [inst_2 : Fintype K] [inst_3 : DecidableEq K] {Z : Type u_4} {M : Type u_5} [inst_4 : Fintype Z] [inst_5 : Fintype M] (P : Causalean.Panel.EstimandCharacterization.FlexibleDIDMundlak.VectorTWFEProblem Unit Time K) (Zvar : Z → Unit → ℝ) (Mvar : M → Time → ℝ) (fit : Causalean.Panel.EstimandCharacterization.FlexibleDIDMundlak.VectorTWMFit P Zvar Mvar), fit.beta = P.betaTWFE","l":232,"f":"Causalean/Panel/EstimandCharacterization/FlexibleDIDMundlak/VectorMundlak.lean"},{"n":"Causalean.Panel.EstimandCharacterization.FlexibleDIDMundlak.vec_twfe_twm_optional_controls_invariant","k":"theorem","m":"Causalean.Panel.EstimandCharacterization.FlexibleDIDMundlak.VectorMundlak","a":"Panel/EstimandCharacterization/FlexibleDIDMundlak","t":2,"d":"Adding or removing optional time-constant or time-only controls does not change the K-vector Mundlak coefficient, since both fits equal the vector TWFE coefficient.","s":"∀ {Unit : Type u_1} {Time : Type u_2} [inst : Fintype Unit] [inst_1 : Fintype Time] {K : Type u_3} [inst_2 : Fintype K] [inst_3 : DecidableEq K] {Z₁ : Type u_6} {M₁ : Type u_7} {Z₂ : Type u_8} {M₂ : Type u_9} [inst_4 : Fintype Z₁] [inst_5 : Fintype M₁] [inst_6 : Fintype Z₂] [inst_7 : Fintype M₂] (P : Causalean.Panel.EstimandCharacterization.FlexibleDIDMundlak.VectorTWFEProblem Unit Time K) (Zvar₁ : Z₁ → Unit → ℝ) (Mvar₁ : M₁ → Time → ℝ) (Zvar₂ : Z₂ → Unit → ℝ) (Mvar₂ : M₂ → Time → ℝ) (fit₁ : Causalean.Panel.EstimandCharacterization.FlexibleDIDMundlak.VectorTWMFit P Zvar₁ Mvar₁) (fit₂ : Causalean.Panel.EstimandCharacterization.FlexibleDIDMundlak.VectorTWMFit P Zvar₂ Mvar₂), fit₁.beta = fit₂.beta","l":301,"f":"Causalean/Panel/EstimandCharacterization/FlexibleDIDMundlak/VectorMundlak.lean"},{"n":"Causalean.Panel.EstimandCharacterization.FlexibleDIDMundlak.vector_mundlak_nuisance_unit_time","k":"theorem","m":"Causalean.Panel.EstimandCharacterization.FlexibleDIDMundlak.VectorMundlak","a":"Panel/EstimandCharacterization/FlexibleDIDMundlak","t":2,"d":"Vector two-way Mundlak nuisance terms are unit/time additive, so the optional controls lie inside the same orthogonality class as for the scalar case.","s":"∀ {Unit : Type u_1} {Time : Type u_2} [inst : Fintype Unit] [inst_1 : Fintype Time] {K : Type u_3} [inst_2 : Fintype K] {Z : Type u_4} {M : Type u_5} [inst_3 : Fintype Z] [inst_4 : Fintype M] (X : Unit → Time → K → ℝ) (Zvar : Z → Unit → ℝ) (Mvar : M → Time → ℝ) {h : Unit → Time → ℝ}, Causalean.Panel.EstimandCharacterization.FlexibleDIDMundlak.IsVectorTwoWayMundlakNuisance X Zvar Mvar h → Causalean.Panel.UniformTwoWayPanel.IsUnitTimeAdditive h","l":188,"f":"Causalean/Panel/EstimandCharacterization/FlexibleDIDMundlak/VectorMundlak.lean"},{"n":"Causalean.Panel.EstimandCharacterization.HeterogeneousTWFE.DCDHPanel","k":"structure","m":"Causalean.Panel.EstimandCharacterization.HeterogeneousTWFE.FinitePanel","a":"Panel/EstimandCharacterization/HeterogeneousTWFE","t":1,"d":"A finite de Chaisemartin-D'Haultfoeuille group-time panel: it bundles group-time cell weights, a binary treatment indicator, the observed outcome, the untreated potential outcome, cell-level treatment effects, and a residualized-treatment witness, subject to strict positivity and unit sum of the weights, the treatment indicator taking only the values zero and one, potential-outcome consistency — the observed outcome equals the untreated outcome plus the treatment indicator times the treatment effect, the residualized witness differing from the treatment indicator by a group-plus-time additive function, its orthogonality, in the weighted inner product, to every group-plus-time additive function, and a strictly positive weighted sum of its squares.","s":"(G : Type u_1) → (T : Type u_2) → [Fintype G] → [Fintype T] → Type (max u_1 u_2)","l":91,"f":"Causalean/Panel/EstimandCharacterization/HeterogeneousTWFE/FinitePanel.lean"},{"n":"Causalean.Panel.EstimandCharacterization.HeterogeneousTWFE.DCDHPanel.DWeightedTau","k":"def","m":"Causalean.Panel.EstimandCharacterization.HeterogeneousTWFE.FinitePanel","a":"Panel/EstimandCharacterization/HeterogeneousTWFE","t":1,"d":"All-cell treatment-effect component using the binary treatment indicator.","s":"{G : Type u_1} → {T : Type u_2} → [inst : Fintype G] → [inst_1 : Fintype T] → Causalean.Panel.EstimandCharacterization.HeterogeneousTWFE.DCDHPanel G T → ℝ","l":143,"f":"Causalean/Panel/EstimandCharacterization/HeterogeneousTWFE/FinitePanel.lean"},{"n":"Causalean.Panel.EstimandCharacterization.HeterogeneousTWFE.DCDHPanel.DWeightedTau_eq_treatedWeightedTau","k":"theorem","m":"Causalean.Panel.EstimandCharacterization.HeterogeneousTWFE.FinitePanel","a":"Panel/EstimandCharacterization/HeterogeneousTWFE","t":2,"d":"The all-cell D-weighted component is the same as the treated-cell sum.","s":"∀ {G : Type u_1} {T : Type u_2} [inst : Fintype G] [inst_1 : Fintype T] (P : Causalean.Panel.EstimandCharacterization.HeterogeneousTWFE.DCDHPanel G T), P.DWeightedTau = P.treatedWeightedTau","l":220,"f":"Causalean/Panel/EstimandCharacterization/HeterogeneousTWFE/FinitePanel.lean"},{"n":"Causalean.Panel.EstimandCharacterization.HeterogeneousTWFE.DCDHPanel.SD","k":"def","m":"Causalean.Panel.EstimandCharacterization.HeterogeneousTWFE.FinitePanel","a":"Panel/EstimandCharacterization/HeterogeneousTWFE","t":1,"d":"Residualized-treatment denominator S_D.","s":"{G : Type u_1} → {T : Type u_2} → [inst : Fintype G] → [inst_1 : Fintype T] → Causalean.Panel.EstimandCharacterization.HeterogeneousTWFE.DCDHPanel G T → ℝ","l":122,"f":"Causalean/Panel/EstimandCharacterization/HeterogeneousTWFE/FinitePanel.lean"},{"n":"Causalean.Panel.EstimandCharacterization.HeterogeneousTWFE.DCDHPanel.betaTWFE","k":"def","m":"Causalean.Panel.EstimandCharacterization.HeterogeneousTWFE.FinitePanel","a":"Panel/EstimandCharacterization/HeterogeneousTWFE","t":1,"d":"Finite FWL/TWFE coefficient.","s":"{G : Type u_1} → {T : Type u_2} → [inst : Fintype G] → [inst_1 : Fintype T] → Causalean.Panel.EstimandCharacterization.HeterogeneousTWFE.DCDHPanel G T → ℝ","l":126,"f":"Causalean/Panel/EstimandCharacterization/HeterogeneousTWFE/FinitePanel.lean"},{"n":"Causalean.Panel.EstimandCharacterization.HeterogeneousTWFE.DCDHPanel.inner_Dtilde_D_eq_SD","k":"theorem","m":"Causalean.Panel.EstimandCharacterization.HeterogeneousTWFE.FinitePanel","a":"Panel/EstimandCharacterization/HeterogeneousTWFE","t":2,"d":"Orthogonality of the residualized treatment against D - Dtilde gives the DCDH denominator identity.","s":"∀ {G : Type u_1} {T : Type u_2} [inst : Fintype G] [inst_1 : Fintype T] (P : Causalean.Panel.EstimandCharacterization.HeterogeneousTWFE.DCDHPanel G T), ∑ g, ∑ t, P.pi g t * P.Dtilde g t * P.D g t = P.SD","l":175,"f":"Causalean/Panel/EstimandCharacterization/HeterogeneousTWFE/FinitePanel.lean"},{"n":"Causalean.Panel.EstimandCharacterization.HeterogeneousTWFE.DCDHPanel.inner_Dtilde_D_eq_SD_core","k":"theorem","m":"Causalean.Panel.EstimandCharacterization.HeterogeneousTWFE.FinitePanel","a":"Panel/EstimandCharacterization/HeterogeneousTWFE","t":2,"d":"Orthogonality of the residualized treatment against D - Dtilde gives the DCDH denominator identity.","s":"∀ {G : Type u_1} {T : Type u_2} [inst : Fintype G] [inst_1 : Fintype T] (pi D Dtilde : G → T → ℝ), (Causalean.Panel.EstimandCharacterization.HeterogeneousTWFE.IsGTFE fun g t => D g t - Dtilde g t) → (∀ (h : G → T → ℝ), Causalean.Panel.EstimandCharacterization.HeterogeneousTWFE.IsGTFE h → ∑ g, ∑ t, pi g t * Dtilde g t * h g t = 0) → ∑ g, ∑ t, pi g t * Dtilde g t * D g t = ∑ g, ∑ t, pi g t * Dtilde g t ^ 2","l":155,"f":"Causalean/Panel/EstimandCharacterization/HeterogeneousTWFE/FinitePanel.lean"},{"n":"Causalean.Panel.EstimandCharacterization.HeterogeneousTWFE.DCDHPanel.normalized_weight_neg_iff_residual_neg","k":"theorem","m":"Causalean.Panel.EstimandCharacterization.HeterogeneousTWFE.FinitePanel","a":"Panel/EstimandCharacterization/HeterogeneousTWFE","t":2,"d":"Dividing a positive cell mass times a residual by a positive normalizer preserves whether the residual is negative.","s":"∀ {pi Dtilde SD : ℝ}, 0 < pi → 0 < SD → (pi * Dtilde / SD < 0 ↔ Dtilde < 0)","l":280,"f":"Causalean/Panel/EstimandCharacterization/HeterogeneousTWFE/FinitePanel.lean"},{"n":"Causalean.Panel.EstimandCharacterization.HeterogeneousTWFE.DCDHPanel.normalized_weight_pos_iff_residual_pos","k":"theorem","m":"Causalean.Panel.EstimandCharacterization.HeterogeneousTWFE.FinitePanel","a":"Panel/EstimandCharacterization/HeterogeneousTWFE","t":2,"d":"Dividing a positive cell mass times a residual by a positive normalizer preserves whether the residual is positive.","s":"∀ {pi Dtilde SD : ℝ}, 0 < pi → 0 < SD → (0 < pi * Dtilde / SD ↔ 0 < Dtilde)","l":306,"f":"Causalean/Panel/EstimandCharacterization/HeterogeneousTWFE/FinitePanel.lean"},{"n":"Causalean.Panel.EstimandCharacterization.HeterogeneousTWFE.DCDHPanel.normalized_weight_zero_iff_residual_zero","k":"theorem","m":"Causalean.Panel.EstimandCharacterization.HeterogeneousTWFE.FinitePanel","a":"Panel/EstimandCharacterization/HeterogeneousTWFE","t":2,"d":"A positive cell mass and positive normalizer make the normalized weight zero exactly when its residual is zero.","s":"∀ {pi Dtilde SD : ℝ}, 0 < pi → 0 < SD → (pi * Dtilde / SD = 0 ↔ Dtilde = 0)","l":334,"f":"Causalean/Panel/EstimandCharacterization/HeterogeneousTWFE/FinitePanel.lean"},{"n":"Causalean.Panel.EstimandCharacterization.HeterogeneousTWFE.DCDHPanel.ofPopulation","k":"def","m":"Causalean.Panel.EstimandCharacterization.HeterogeneousTWFE.PopulationBridge","a":"Panel/EstimandCharacterization/HeterogeneousTWFE","t":1,"d":"Builds a finite DCDH panel from a population model.","s":"{G : Type u_1} → {T : Type u_2} → [inst : Fintype G] → [inst_1 : Fintype T] → {Ω : Type u_3} → [inst_2 : MeasurableSpace Ω] → (μ : MeasureTheory.Measure Ω) → [MeasureTheory.IsProbabilityMeasure μ] → (cellEvent : G → T → Set Ω) → (Yfac Y0pop Y1pop : Ω → ℝ) → (d Dtilde : G → T → ℝ) → (∀ (g : G) (t : T), MeasurableSet (cellEvent g t)) → Pairwise (Function.onFun Disjoint fun p => cellEvent p.1 p.2) → ⋃ p, cellEvent p.1 p.2 = Set.univ → (∀ (g : G) (t : T), 0 < (μ (cellEvent g t)).toReal) → (∀ (g : G) (t : T), d g t = 0 ∨ d g t = 1) → (∀ (g : G) (t : T), ∀ ω ∈ cellEvent g t, Yfac ω = Y0pop ω + d g t * (Y1pop ω - Y0pop ω)) → (Causalean.Panel.EstimandCharacterization.HeterogeneousTWFE.IsGTFE fun g t => d g t - Dtilde g t) → (∀ (h : G → T → ℝ), Causalean.Panel.EstimandCharacterization.HeterogeneousTWFE.IsGTFE h → ∑ g, ∑ t, (μ (cellEvent g t)).toReal * Dtilde g t * h g t = 0) → 0 < ∑ g, ∑ t, (μ (cellEvent g t)).toReal * Dtilde g t ^ 2 → Causalean.Panel.EstimandCharacterization.HeterogeneousTWFE.DCDHPanel G T","l":45,"f":"Causalean/Panel/EstimandCharacterization/HeterogeneousTWFE/PopulationBridge.lean"},{"n":"Causalean.Panel.EstimandCharacterization.HeterogeneousTWFE.DCDHPanel.omega","k":"def","m":"Causalean.Panel.EstimandCharacterization.HeterogeneousTWFE.FinitePanel","a":"Panel/EstimandCharacterization/HeterogeneousTWFE","t":1,"d":"Normalized DCDH cell weight, interpreted on treated cells.","s":"{G : Type u_1} → {T : Type u_2} → [inst : Fintype G] → [inst_1 : Fintype T] → Causalean.Panel.EstimandCharacterization.HeterogeneousTWFE.DCDHPanel G T → G → T → ℝ","l":135,"f":"Causalean/Panel/EstimandCharacterization/HeterogeneousTWFE/FinitePanel.lean"},{"n":"Causalean.Panel.EstimandCharacterization.HeterogeneousTWFE.DCDHPanel.treatedCells","k":"def","m":"Causalean.Panel.EstimandCharacterization.HeterogeneousTWFE.FinitePanel","a":"Panel/EstimandCharacterization/HeterogeneousTWFE","t":1,"d":"Treated cells {(g,t) | D_gt = 1}.","s":"{G : Type u_1} → {T : Type u_2} → [inst : Fintype G] → [inst_1 : Fintype T] → Causalean.Panel.EstimandCharacterization.HeterogeneousTWFE.DCDHPanel G T → Finset (G × T)","l":139,"f":"Causalean/Panel/EstimandCharacterization/HeterogeneousTWFE/FinitePanel.lean"},{"n":"Causalean.Panel.EstimandCharacterization.HeterogeneousTWFE.DCDHPanel.treatedWeightedTau","k":"def","m":"Causalean.Panel.EstimandCharacterization.HeterogeneousTWFE.FinitePanel","a":"Panel/EstimandCharacterization/HeterogeneousTWFE","t":1,"d":"Treated-cell weighted treatment-effect component.","s":"{G : Type u_1} → {T : Type u_2} → [inst : Fintype G] → [inst_1 : Fintype T] → Causalean.Panel.EstimandCharacterization.HeterogeneousTWFE.DCDHPanel G T → ℝ","l":147,"f":"Causalean/Panel/EstimandCharacterization/HeterogeneousTWFE/FinitePanel.lean"},{"n":"Causalean.Panel.EstimandCharacterization.HeterogeneousTWFE.DCDHPanel.treated_omega_neg_iff_Dtilde_neg","k":"theorem","m":"Causalean.Panel.EstimandCharacterization.HeterogeneousTWFE.FinitePanel","a":"Panel/EstimandCharacterization/HeterogeneousTWFE","t":1,"d":"Sign equivalence for the DCDH weight (negative direction). For a DCDH panel and any cohort-period cell, the normalized weight ω_gt is negative exactly when the residualized treatment D̃_gt is negative.","s":"∀ {G : Type u_1} {T : Type u_2} [inst : Fintype G] [inst_1 : Fintype T] (P : Causalean.Panel.EstimandCharacterization.HeterogeneousTWFE.DCDHPanel G T) {g : G} {t : T}, P.omega g t < 0 ↔ P.Dtilde g t < 0","l":294,"f":"Causalean/Panel/EstimandCharacterization/HeterogeneousTWFE/FinitePanel.lean"},{"n":"Causalean.Panel.EstimandCharacterization.HeterogeneousTWFE.DCDHPanel.treated_omega_pos_iff_Dtilde_pos","k":"theorem","m":"Causalean.Panel.EstimandCharacterization.HeterogeneousTWFE.FinitePanel","a":"Panel/EstimandCharacterization/HeterogeneousTWFE","t":1,"d":"Positive-weight direction of the DCDH sign characterization. For a DCDH panel and any cohort-period cell, the normalized weight ω_gt is positive exactly when the residualized treatment D̃_gt is positive.","s":"∀ {G : Type u_1} {T : Type u_2} [inst : Fintype G] [inst_1 : Fintype T] (P : Causalean.Panel.EstimandCharacterization.HeterogeneousTWFE.DCDHPanel G T) {g : G} {t : T}, 0 < P.omega g t ↔ 0 < P.Dtilde g t","l":321,"f":"Causalean/Panel/EstimandCharacterization/HeterogeneousTWFE/FinitePanel.lean"},{"n":"Causalean.Panel.EstimandCharacterization.HeterogeneousTWFE.DCDHPanel.treated_omega_sum_eq_one","k":"theorem","m":"Causalean.Panel.EstimandCharacterization.HeterogeneousTWFE.FinitePanel","a":"Panel/EstimandCharacterization/HeterogeneousTWFE","t":1,"d":"DCDH weights sum to one. For a DCDH panel, the normalized DCDH weights sum to one over all treated cells.","s":"∀ {G : Type u_1} {T : Type u_2} [inst : Fintype G] [inst_1 : Fintype T] (P : Causalean.Panel.EstimandCharacterization.HeterogeneousTWFE.DCDHPanel G T), ∑ gt ∈ P.treatedCells, P.omega gt.1 gt.2 = 1","l":253,"f":"Causalean/Panel/EstimandCharacterization/HeterogeneousTWFE/FinitePanel.lean"},{"n":"Causalean.Panel.EstimandCharacterization.HeterogeneousTWFE.DCDHPanel.treated_omega_zero_iff_Dtilde_zero","k":"theorem","m":"Causalean.Panel.EstimandCharacterization.HeterogeneousTWFE.FinitePanel","a":"Panel/EstimandCharacterization/HeterogeneousTWFE","t":1,"d":"Zero-weight direction of the DCDH sign characterization. For a DCDH panel and any cohort-period cell, the normalized weight ω_gt is zero exactly when the residualized treatment D̃_gt is zero.","s":"∀ {G : Type u_1} {T : Type u_2} [inst : Fintype G] [inst_1 : Fintype T] (P : Causalean.Panel.EstimandCharacterization.HeterogeneousTWFE.DCDHPanel G T) {g : G} {t : T}, P.omega g t = 0 ↔ P.Dtilde g t = 0","l":351,"f":"Causalean/Panel/EstimandCharacterization/HeterogeneousTWFE/FinitePanel.lean"},{"n":"Causalean.Panel.EstimandCharacterization.HeterogeneousTWFE.DCDHPanel.twfe_eq_treated_weighted_tau_of_zeroUntreatedContrast","k":"theorem","m":"Causalean.Panel.EstimandCharacterization.HeterogeneousTWFE.FinitePanel","a":"Panel/EstimandCharacterization/HeterogeneousTWFE","t":1,"d":"Zero untreated bias implies TWFE equals the treated-cell weighted effect. If the untreated-outcome residual contrast vanishes — the residualized-treatment-weighted average of the untreated potential outcome Y0 over the whole panel is zero, then the finite two-way fixed-effects (TWFE) coefficient equals the treated-cell weighted sum of treatment effects.","s":"∀ {G : Type u_1} {T : Type u_2} [inst : Fintype G] [inst_1 : Fintype T] (P : Causalean.Panel.EstimandCharacterization.HeterogeneousTWFE.DCDHPanel G T), P.zeroUntreatedResidualContrast → P.betaTWFE = P.treatedWeightedTau","l":240,"f":"Causalean/Panel/EstimandCharacterization/HeterogeneousTWFE/FinitePanel.lean"},{"n":"Causalean.Panel.EstimandCharacterization.HeterogeneousTWFE.DCDHPanel.twfe_eq_untreatedBias_add_DWeightedTau","k":"theorem","m":"Causalean.Panel.EstimandCharacterization.HeterogeneousTWFE.FinitePanel","a":"Panel/EstimandCharacterization/HeterogeneousTWFE","t":1,"d":"DCDH finite TWFE decomposition (all-cell weighting). For a DCDH panel, the finite two-way fixed-effects (TWFE) coefficient decomposes as the sum of the untreated bias and the all-cell D-weighted treatment-effect component.","s":"∀ {G : Type u_1} {T : Type u_2} [inst : Fintype G] [inst_1 : Fintype T] (P : Causalean.Panel.EstimandCharacterization.HeterogeneousTWFE.DCDHPanel G T), P.betaTWFE = P.untreatedBias + P.DWeightedTau","l":201,"f":"Causalean/Panel/EstimandCharacterization/HeterogeneousTWFE/FinitePanel.lean"},{"n":"Causalean.Panel.EstimandCharacterization.HeterogeneousTWFE.DCDHPanel.twfe_eq_untreatedBias_add_treated_weighted_tau","k":"theorem","m":"Causalean.Panel.EstimandCharacterization.HeterogeneousTWFE.FinitePanel","a":"Panel/EstimandCharacterization/HeterogeneousTWFE","t":1,"d":"DCDH finite TWFE decomposition (treated-cell weighting). For a DCDH panel, the finite TWFE coefficient decomposes as the sum of the untreated bias and the normalized treated-cell weighted sum of treatment effects.","s":"∀ {G : Type u_1} {T : Type u_2} [inst : Fintype G] [inst_1 : Fintype T] (P : Causalean.Panel.EstimandCharacterization.HeterogeneousTWFE.DCDHPanel G T), P.betaTWFE = P.untreatedBias + P.treatedWeightedTau","l":233,"f":"Causalean/Panel/EstimandCharacterization/HeterogeneousTWFE/FinitePanel.lean"},{"n":"Causalean.Panel.EstimandCharacterization.HeterogeneousTWFE.DCDHPanel.untreatedBias","k":"def","m":"Causalean.Panel.EstimandCharacterization.HeterogeneousTWFE.FinitePanel","a":"Panel/EstimandCharacterization/HeterogeneousTWFE","t":1,"d":"Untreated residual contrast divided by the residualized-treatment denominator.","s":"{G : Type u_1} → {T : Type u_2} → [inst : Fintype G] → [inst_1 : Fintype T] → Causalean.Panel.EstimandCharacterization.HeterogeneousTWFE.DCDHPanel G T → ℝ","l":130,"f":"Causalean/Panel/EstimandCharacterization/HeterogeneousTWFE/FinitePanel.lean"},{"n":"Causalean.Panel.EstimandCharacterization.HeterogeneousTWFE.DCDHPanel.weighted_residual_contrast_eq_zero_of_isGTFE","k":"theorem","m":"Causalean.Panel.EstimandCharacterization.HeterogeneousTWFE.FinitePanel","a":"Panel/EstimandCharacterization/HeterogeneousTWFE","t":2,"d":"A weighted residual is orthogonal to every additive array, so its contrast with an additive untreated-outcome array vanishes.","s":"∀ {G : Type u_1} {T : Type u_2} [inst : Fintype G] [inst_1 : Fintype T] (pi Dtilde Y0 : G → T → ℝ), (∀ (h : G → T → ℝ), Causalean.Panel.EstimandCharacterization.HeterogeneousTWFE.IsGTFE h → ∑ g, ∑ t, pi g t * Dtilde g t * h g t = 0) → Causalean.Panel.EstimandCharacterization.HeterogeneousTWFE.IsGTFE Y0 → ∑ g, ∑ t, pi g t * Dtilde g t * Y0 g t = 0","l":182,"f":"Causalean/Panel/EstimandCharacterization/HeterogeneousTWFE/FinitePanel.lean"},{"n":"Causalean.Panel.EstimandCharacterization.HeterogeneousTWFE.DCDHPanel.zeroUntreatedResidualContrast","k":"def","m":"Causalean.Panel.EstimandCharacterization.HeterogeneousTWFE.FinitePanel","a":"Panel/EstimandCharacterization/HeterogeneousTWFE","t":1,"d":"Zero untreated residual contrast, the bias-free DCDH condition.","s":"{G : Type u_1} → {T : Type u_2} → [inst : Fintype G] → [inst_1 : Fintype T] → Causalean.Panel.EstimandCharacterization.HeterogeneousTWFE.DCDHPanel G T → Prop","l":151,"f":"Causalean/Panel/EstimandCharacterization/HeterogeneousTWFE/FinitePanel.lean"},{"n":"Causalean.Panel.EstimandCharacterization.HeterogeneousTWFE.DCDHPanel.zeroUntreatedResidualContrast_of_Y0_mem_gtfe","k":"theorem","m":"Causalean.Panel.EstimandCharacterization.HeterogeneousTWFE.FinitePanel","a":"Panel/EstimandCharacterization/HeterogeneousTWFE","t":2,"d":"Group-plus-period untreated means imply the zero untreated residual contrast.","s":"∀ {G : Type u_1} {T : Type u_2} [inst : Fintype G] [inst_1 : Fintype T] (P : Causalean.Panel.EstimandCharacterization.HeterogeneousTWFE.DCDHPanel G T), Causalean.Panel.EstimandCharacterization.HeterogeneousTWFE.IsGTFE P.Y0 → P.zeroUntreatedResidualContrast","l":192,"f":"Causalean/Panel/EstimandCharacterization/HeterogeneousTWFE/FinitePanel.lean"},{"n":"Causalean.Panel.EstimandCharacterization.HeterogeneousTWFE.IsGTFE","k":"def","m":"Causalean.Panel.EstimandCharacterization.HeterogeneousTWFE.FinitePanel","a":"Panel/EstimandCharacterization/HeterogeneousTWFE","t":1,"d":"Group-time fixed-effect span, represented as additive group and period components.","s":"{G : Type u_1} → {T : Type u_2} → (G → T → ℝ) → Prop","l":84,"f":"Causalean/Panel/EstimandCharacterization/HeterogeneousTWFE/FinitePanel.lean"},{"n":"Causalean.Panel.EstimandCharacterization.HeterogeneousTWFE.exists_panel_with_positive_treated_effects_twfe_negative_of_negative_component","k":"theorem","m":"Causalean.Panel.EstimandCharacterization.HeterogeneousTWFE.FinitePanel","a":"Panel/EstimandCharacterization/HeterogeneousTWFE","t":1,"d":"Panel-level DCDH sign reversal. If the untreated-outcome residual contrast vanishes and the treated cells with negative normalized DCDH weight carry a strictly negative total weight, then there exists another finite DCDH panel — sharing the same cell weights, treatment indicator, untreated potential outcomes, and residualized treatment, but with strictly positive treatment effects on every cell and zero untreated residual contrast — whose two-way fixed-effects (TWFE) coefficient is strictly negative.","s":"∀ {G : Type u_1} {T : Type u_2} [inst : Fintype G] [inst_1 : Fintype T] (P : Causalean.Panel.EstimandCharacterization.HeterogeneousTWFE.DCDHPanel G T), P.zeroUntreatedResidualContrast → ∑ gt ∈ P.treatedCells with P.omega gt.1 gt.2 < 0, P.omega gt.1 gt.2 < 0 → ∃ P', (∀ (g : G) (t : T), P'.pi g t = P.pi g t) ∧ (∀ (g : G) (t : T), P'.D g t = P.D g t) ∧ (∀ (g : G) (t : T), P'.Y0 g t = P.Y0 g t) ∧ (∀ (g : G) (t : T), P'.Dtilde g t = P.Dtilde g t) ∧ (∀ (g : G) (t : T), 0 < P'.tau g t) ∧ P'.zeroUntreatedResidualContrast ∧ P'.betaTWFE < 0","l":455,"f":"Causalean/Panel/EstimandCharacterization/HeterogeneousTWFE/FinitePanel.lean"},{"n":"Causalean.Panel.EstimandCharacterization.HeterogeneousTWFE.exists_positive_effects_negative_weighted_sum_of_negative_component","k":"theorem","m":"Causalean.Panel.EstimandCharacterization.HeterogeneousTWFE.FinitePanel","a":"Panel/EstimandCharacterization/HeterogeneousTWFE","t":2,"d":"Finite signed-average construction: if normalized weights have a strictly negative component, some strictly positive effects have a negative weighted sum.","s":"∀ {ι : Type u_1} [inst : Fintype ι] (w : ι → ℝ), ∑ i with w i < 0, w i < 0 → ∃ tau, (∀ (i : ι), 0 < tau i) ∧ ∑ i, w i * tau i < 0","l":384,"f":"Causalean/Panel/EstimandCharacterization/HeterogeneousTWFE/FinitePanel.lean"},{"n":"Causalean.Panel.EstimandCharacterization.HeterogeneousTWFE.neg_component_of_exists_Dtilde_neg","k":"theorem","m":"Causalean.Panel.EstimandCharacterization.HeterogeneousTWFE.FinitePanel","a":"Panel/EstimandCharacterization/HeterogeneousTWFE","t":2,"d":"If some treated cell has negative residualized treatment, then the total weight over negatively weighted treated cells is strictly negative.","s":"∀ {G : Type u_1} {T : Type u_2} [inst : Fintype G] [inst_1 : Fintype T] (P : Causalean.Panel.EstimandCharacterization.HeterogeneousTWFE.DCDHPanel G T), (∃ g t, P.D g t = 1 ∧ P.Dtilde g t < 0) → ∑ gt ∈ P.treatedCells with P.omega gt.1 gt.2 < 0, P.omega gt.1 gt.2 < 0","l":426,"f":"Causalean/Panel/EstimandCharacterization/HeterogeneousTWFE/FinitePanel.lean"},{"n":"Causalean.Panel.EstimandCharacterization.HeterogeneousTWFE.ofTwoWayPanel","k":"def","m":"Causalean.Panel.EstimandCharacterization.HeterogeneousTWFE.FWLBridge","a":"Panel/EstimandCharacterization/HeterogeneousTWFE","t":1,"d":"Builds the uniform balanced-panel DCDH structure whose residualized treatment is the Frisch-Waugh-Lovell double-demeaned treatment.","s":"{G : Type u_1} → {T : Type u_2} → [inst : Fintype G] → [inst_1 : Fintype T] → Causalean.Panel.UniformTwoWayPanel.BalancedPanel G T → (D Y Y0 tau : G → T → ℝ) → (∀ (g : G) (t : T), D g t = 0 ∨ D g t = 1) → (∀ (g : G) (t : T), Y g t = Y0 g t + D g t * tau g t) → 0 < ∑ g, ∑ t, Causalean.Panel.EstimandCharacterization.HeterogeneousTWFE.uniformWeight G T * Causalean.Panel.UniformTwoWayPanel.ddot D g t ^ 2 → Causalean.Panel.EstimandCharacterization.HeterogeneousTWFE.DCDHPanel G T","l":55,"f":"Causalean/Panel/EstimandCharacterization/HeterogeneousTWFE/FWLBridge.lean"},{"n":"Causalean.Panel.EstimandCharacterization.HeterogeneousTWFE.ofTwoWayPanel_Dtilde","k":"theorem","m":"Causalean.Panel.EstimandCharacterization.HeterogeneousTWFE.FWLBridge","a":"Panel/EstimandCharacterization/HeterogeneousTWFE","t":2,"d":"The residualized treatment of ofTwoWayPanel is exactly the double-demeaned treatment D̈.","s":"∀ {G : Type u_1} {T : Type u_2} [inst : Fintype G] [inst_1 : Fintype T] (balanced : Causalean.Panel.UniformTwoWayPanel.BalancedPanel G T) (D Y Y0 tau : G → T → ℝ) (hD_binary : ∀ (g : G) (t : T), D g t = 0 ∨ D g t = 1) (hconsistency : ∀ (g : G) (t : T), Y g t = Y0 g t + D g t * tau g t) (hSD : 0 < ∑ g, ∑ t, Causalean.Panel.EstimandCharacterization.HeterogeneousTWFE.uniformWeight G T * Causalean.Panel.UniformTwoWayPanel.ddot D g t ^ 2), (Causalean.Panel.EstimandCharacterization.HeterogeneousTWFE.ofTwoWayPanel balanced D Y Y0 tau hD_binary hconsistency hSD).Dtilde = Causalean.Panel.UniformTwoWayPanel.ddot D","l":119,"f":"Causalean/Panel/EstimandCharacterization/HeterogeneousTWFE/FWLBridge.lean"},{"n":"Causalean.Panel.EstimandCharacterization.HeterogeneousTWFE.ofTwoWayPanel_betaTWFE","k":"theorem","m":"Causalean.Panel.EstimandCharacterization.HeterogeneousTWFE.FWLBridge","a":"Panel/EstimandCharacterization/HeterogeneousTWFE","t":1,"d":"The DCDH TWFE coefficient of the FWL-derived panel equals the uniform-panel Frisch–Waugh–Lovell residualized coefficient of Y on the double-demeaned treatment. For a uniform balanced panel and outcome, treatment, untreated-outcome, and effect arrays Y, D, Y0, tau, suppose the treatment is binary, the observed outcome equals the untreated outcome plus the treatment times the effect (consistency), and the double-demeaned treatment has strictly positive uniformly-weighted variation. Then the two-way fixed-effects (TWFE) coefficient of the DCDH panel built from this data by ofTwoWayPanel equals the Frisch–Waugh–Lovell residualized coefficient of Y regressed on the double-demeaned treatment ddot D.","s":"∀ {G : Type u_1} {T : Type u_2} [inst : Fintype G] [inst_1 : Fintype T] (balanced : Causalean.Panel.UniformTwoWayPanel.BalancedPanel G T) (D Y Y0 tau : G → T → ℝ) (hD_binary : ∀ (g : G) (t : T), D g t = 0 ∨ D g t = 1) (hconsistency : ∀ (g : G) (t : T), Y g t = Y0 g t + D g t * tau g t) (hSD : 0 < ∑ g, ∑ t, Causalean.Panel.EstimandCharacterization.HeterogeneousTWFE.uniformWeight G T * Causalean.Panel.UniformTwoWayPanel.ddot D g t ^ 2), (Causalean.Panel.EstimandCharacterization.HeterogeneousTWFE.ofTwoWayPanel balanced D Y Y0 tau hD_binary hconsistency hSD).betaTWFE = Causalean.Panel.UniformTwoWayPanel.finiteResidualizedCoefficient (Causalean.Panel.UniformTwoWayPanel.ddot D) Y","l":130,"f":"Causalean/Panel/EstimandCharacterization/HeterogeneousTWFE/FWLBridge.lean"},{"n":"Causalean.Panel.EstimandCharacterization.HeterogeneousTWFE.two_cell_signed_weights_positive_effects_negative_sum","k":"theorem","m":"Causalean.Panel.EstimandCharacterization.HeterogeneousTWFE.FinitePanel","a":"Panel/EstimandCharacterization/HeterogeneousTWFE","t":2,"d":"Two-cell signed-weight calculation from the DCDH sign-reversal example.","s":"∀ {c ε M : ℝ}, 0 < c → 0 < ε → (1 + c) * ε / c < M → 0 < ε ∧ 0 < M ∧ (1 + c) * ε + -c * M < 0","l":365,"f":"Causalean/Panel/EstimandCharacterization/HeterogeneousTWFE/FinitePanel.lean"},{"n":"Causalean.Panel.EstimandCharacterization.HeterogeneousTWFE.uniformWeight","k":"def","m":"Causalean.Panel.EstimandCharacterization.HeterogeneousTWFE.FWLBridge","a":"Panel/EstimandCharacterization/HeterogeneousTWFE","t":1,"d":"Uniform unit-period cell weight 1/(|G|·|T|).","s":"(G : Type u_3) → (T : Type u_4) → [Fintype G] → [Fintype T] → ℝ","l":41,"f":"Causalean/Panel/EstimandCharacterization/HeterogeneousTWFE/FWLBridge.lean"},{"n":"Causalean.Panel.EstimandCharacterization.HeterogeneousTWFE.uniformWeight_pos","k":"theorem","m":"Causalean.Panel.EstimandCharacterization.HeterogeneousTWFE.FWLBridge","a":"Panel/EstimandCharacterization/HeterogeneousTWFE","t":2,"d":"For finite nonempty group and time types, the uniform unit-period cell weight is strictly positive.","s":"∀ {G : Type u_1} {T : Type u_2} [inst : Fintype G] [inst_1 : Fintype T] [Nonempty G] [Nonempty T], 0 < Causalean.Panel.EstimandCharacterization.HeterogeneousTWFE.uniformWeight G T","l":45,"f":"Causalean/Panel/EstimandCharacterization/HeterogeneousTWFE/FWLBridge.lean"},{"n":"Causalean.Panel.EstimandCharacterization.ImputationEventStudy.BJSPanel","k":"structure","m":"Causalean.Panel.EstimandCharacterization.ImputationEventStudy.Imputation","a":"Panel/EstimandCharacterization/ImputationEventStudy","t":1,"d":"A finite treated/untreated cell design for the BJS imputation decomposition, storing treated cells' regressor rows and untreated cells' regressor rows, an arbitrary — possibly signed and non-normalized — target weight on treated cells, the observed-law means on treated and untreated cells, the untreated-potential-outcome means on treated and untreated cells, a nuisance vector for the untreated-outcome model, and treated-cell causal effects.","s":"(Treated : Type u_4) → (Untreated : Type u_5) → (Regressor : Type u_6) → [Fintype Treated] → [Fintype Untreated] → [Fintype Regressor] → Type (max (max u_4 u_5) u_6)","l":46,"f":"Causalean/Panel/EstimandCharacterization/ImputationEventStudy/Imputation.lean"},{"n":"Causalean.Panel.EstimandCharacterization.ImputationEventStudy.BJSPanel.ImputationWeights","k":"structure","m":"Causalean.Panel.EstimandCharacterization.ImputationEventStudy.Imputation","a":"Panel/EstimandCharacterization/ImputationEventStudy","t":1,"d":"Witness-form prediction span / imputation weights.","s":"{Treated : Type u_1} → {Untreated : Type u_2} → {Regressor : Type u_3} → [inst : Fintype Treated] → [inst_1 : Fintype Untreated] → [inst_2 : Fintype Regressor] → Causalean.Panel.EstimandCharacterization.ImputationEventStudy.BJSPanel Treated Untreated Regressor → Type (max u_1 u_2)","l":104,"f":"Causalean/Panel/EstimandCharacterization/ImputationEventStudy/Imputation.lean"},{"n":"Causalean.Panel.EstimandCharacterization.ImputationEventStudy.BJSPanel.LinearEstimator","k":"structure","m":"Causalean.Panel.EstimandCharacterization.ImputationEventStudy.Imputation","a":"Panel/EstimandCharacterization/ImputationEventStudy","t":1,"d":"A linear functional of treated and untreated observed cell outcomes.","s":"{Treated : Type u_1} → {Untreated : Type u_2} → {Regressor : Type u_3} → [inst : Fintype Treated] → [inst_1 : Fintype Untreated] → [inst_2 : Fintype Regressor] → Causalean.Panel.EstimandCharacterization.ImputationEventStudy.BJSPanel Treated Untreated Regressor → Type (max u_1 u_2)","l":199,"f":"Causalean/Panel/EstimandCharacterization/ImputationEventStudy/Imputation.lean"},{"n":"Causalean.Panel.EstimandCharacterization.ImputationEventStudy.BJSPanel.LinearEstimator.HasImputationRepresentation","k":"structure","m":"Causalean.Panel.EstimandCharacterization.ImputationEventStudy.Imputation","a":"Panel/EstimandCharacterization/ImputationEventStudy","t":1,"d":"Explicit imputation representation witness for the linear-unbiased representation helper.","s":"{Treated : Type u_1} → {Untreated : Type u_2} → {Regressor : Type u_3} → [inst : Fintype Treated] → [inst_1 : Fintype Untreated] → [inst_2 : Fintype Regressor] → {P : Causalean.Panel.EstimandCharacterization.ImputationEventStudy.BJSPanel Treated Untreated Regressor} → P.LinearEstimator → Type (max u_1 u_2)","l":253,"f":"Causalean/Panel/EstimandCharacterization/ImputationEventStudy/Imputation.lean"},{"n":"Causalean.Panel.EstimandCharacterization.ImputationEventStudy.BJSPanel.LinearEstimator.modelValue","k":"def","m":"Causalean.Panel.EstimandCharacterization.ImputationEventStudy.Imputation","a":"Panel/EstimandCharacterization/ImputationEventStudy","t":1,"d":"Model-implied value for a nuisance vector and unrestricted treated-effect vector.","s":"{Treated : Type u_1} → {Untreated : Type u_2} → {Regressor : Type u_3} → [inst : Fintype Treated] → [inst_1 : Fintype Untreated] → [inst_2 : Fintype Regressor] → {P : Causalean.Panel.EstimandCharacterization.ImputationEventStudy.BJSPanel Treated Untreated Regressor} → P.LinearEstimator → (Regressor → ℝ) → (Treated → ℝ) → ℝ","l":219,"f":"Causalean/Panel/EstimandCharacterization/ImputationEventStudy/Imputation.lean"},{"n":"Causalean.Panel.EstimandCharacterization.ImputationEventStudy.BJSPanel.LinearEstimator.nuisance_coord","k":"theorem","m":"Causalean.Panel.EstimandCharacterization.ImputationEventStudy.Imputation","a":"Panel/EstimandCharacterization/ImputationEventStudy","t":2,"d":"Nuisance unbiasedness for every beta gives the BJS left-null-space coordinate constraint aᵀ Q_T + vUᵀ Q_U = 0 (row by row). Hoisted for the efficiency layer.","s":"∀ {Treated : Type u_1} {Untreated : Type u_2} {Regressor : Type u_3} [inst : Fintype Treated] [inst_1 : Fintype Untreated] [inst_2 : Fintype Regressor] {P : Causalean.Panel.EstimandCharacterization.ImputationEventStudy.BJSPanel Treated Untreated Regressor} (L : P.LinearEstimator), L.unbiasedForAllTau → ∀ (r : Regressor), ∑ c, P.a c * P.qT c r + ∑ u, L.vU u * P.qU u r = 0","l":243,"f":"Causalean/Panel/EstimandCharacterization/ImputationEventStudy/Imputation.lean"},{"n":"Causalean.Panel.EstimandCharacterization.ImputationEventStudy.BJSPanel.LinearEstimator.observedValue","k":"def","m":"Causalean.Panel.EstimandCharacterization.ImputationEventStudy.Imputation","a":"Panel/EstimandCharacterization/ImputationEventStudy","t":1,"d":"Observed-law value of the linear estimator on the panel means.","s":"{Treated : Type u_1} → {Untreated : Type u_2} → {Regressor : Type u_3} → [inst : Fintype Treated] → [inst_1 : Fintype Untreated] → [inst_2 : Fintype Regressor] → {P : Causalean.Panel.EstimandCharacterization.ImputationEventStudy.BJSPanel Treated Untreated Regressor} → P.LinearEstimator → ℝ","l":215,"f":"Causalean/Panel/EstimandCharacterization/ImputationEventStudy/Imputation.lean"},{"n":"Causalean.Panel.EstimandCharacterization.ImputationEventStudy.BJSPanel.LinearEstimator.unbiasedForAllTau","k":"def","m":"Causalean.Panel.EstimandCharacterization.ImputationEventStudy.Imputation","a":"Panel/EstimandCharacterization/ImputationEventStudy","t":1,"d":"Linear unbiasedness over unrestricted treated effects and nuisance vectors in the finite-cell model.","s":"{Treated : Type u_1} → {Untreated : Type u_2} → {Regressor : Type u_3} → [inst : Fintype Treated] → [inst_1 : Fintype Untreated] → [inst_2 : Fintype Regressor] → {P : Causalean.Panel.EstimandCharacterization.ImputationEventStudy.BJSPanel Treated Untreated Regressor} → P.LinearEstimator → Prop","l":227,"f":"Causalean/Panel/EstimandCharacterization/ImputationEventStudy/Imputation.lean"},{"n":"Causalean.Panel.EstimandCharacterization.ImputationEventStudy.BJSPanel.LinearEstimator.vT_eq_a","k":"theorem","m":"Causalean.Panel.EstimandCharacterization.ImputationEventStudy.Imputation","a":"Panel/EstimandCharacterization/ImputationEventStudy","t":2,"d":"Unrestricted treatment-effect heterogeneity forces the treated-cell weights to equal the target weights (vT = a). Proved by an indicator-tau test. Hoisted here so the efficiency layer can reuse it without re-running the argument.","s":"∀ {Treated : Type u_1} {Untreated : Type u_2} {Regressor : Type u_3} [inst : Fintype Treated] [inst_1 : Fintype Untreated] [inst_2 : Fintype Regressor] {P : Causalean.Panel.EstimandCharacterization.ImputationEventStudy.BJSPanel Treated Untreated Regressor} (L : P.LinearEstimator), L.unbiasedForAllTau → ∀ (c : Treated), L.vT c = P.a c","l":233,"f":"Causalean/Panel/EstimandCharacterization/ImputationEventStudy/Imputation.lean"},{"n":"Causalean.Panel.EstimandCharacterization.ImputationEventStudy.BJSPanel.LinearEstimator.value","k":"def","m":"Causalean.Panel.EstimandCharacterization.ImputationEventStudy.Imputation","a":"Panel/EstimandCharacterization/ImputationEventStudy","t":1,"d":"Value of a treated-plus-untreated linear estimator at arbitrary cell outcome arrays.","s":"{Treated : Type u_1} → {Untreated : Type u_2} → {Regressor : Type u_3} → [inst : Fintype Treated] → [inst_1 : Fintype Untreated] → [inst_2 : Fintype Regressor] → {P : Causalean.Panel.EstimandCharacterization.ImputationEventStudy.BJSPanel Treated Untreated Regressor} → P.LinearEstimator → (Treated → ℝ) → (Untreated → ℝ) → ℝ","l":210,"f":"Causalean/Panel/EstimandCharacterization/ImputationEventStudy/Imputation.lean"},{"n":"Causalean.Panel.EstimandCharacterization.ImputationEventStudy.BJSPanel.PredictionIdentified","k":"def","m":"Causalean.Panel.EstimandCharacterization.ImputationEventStudy.Imputation","a":"Panel/EstimandCharacterization/ImputationEventStudy","t":1,"d":"Existence form of the target-relevant prediction-span condition.","s":"{Treated : Type u_1} → {Untreated : Type u_2} → {Regressor : Type u_3} → [inst : Fintype Treated] → [inst_1 : Fintype Untreated] → [inst_2 : Fintype Regressor] → Causalean.Panel.EstimandCharacterization.ImputationEventStudy.BJSPanel Treated Untreated Regressor → Prop","l":117,"f":"Causalean/Panel/EstimandCharacterization/ImputationEventStudy/Imputation.lean"},{"n":"Causalean.Panel.EstimandCharacterization.ImputationEventStudy.BJSPanel.TreatmentEffectFixed","k":"def","m":"Causalean.Panel.EstimandCharacterization.ImputationEventStudy.Imputation","a":"Panel/EstimandCharacterization/ImputationEventStudy","t":1,"d":"Treated-cell effects are fixed at the observed-law mean level: treated observed means equal untreated means plus tau.","s":"{Treated : Type u_1} → {Untreated : Type u_2} → {Regressor : Type u_3} → [inst : Fintype Treated] → [inst_1 : Fintype Untreated] → [inst_2 : Fintype Regressor] → Causalean.Panel.EstimandCharacterization.ImputationEventStudy.BJSPanel Treated Untreated Regressor → Prop","l":99,"f":"Causalean/Panel/EstimandCharacterization/ImputationEventStudy/Imputation.lean"},{"n":"Causalean.Panel.EstimandCharacterization.ImputationEventStudy.BJSPanel.UntreatedOutcomeModel","k":"def","m":"Causalean.Panel.EstimandCharacterization.ImputationEventStudy.Imputation","a":"Panel/EstimandCharacterization/ImputationEventStudy","t":1,"d":"Deterministic untreated-outcome mean model and untreated-cell no-anticipation / observation equation.","s":"{Treated : Type u_1} → {Untreated : Type u_2} → {Regressor : Type u_3} → [inst : Fintype Treated] → [inst_1 : Fintype Untreated] → [inst_2 : Fintype Regressor] → Causalean.Panel.EstimandCharacterization.ImputationEventStudy.BJSPanel Treated Untreated Regressor → Prop","l":92,"f":"Causalean/Panel/EstimandCharacterization/ImputationEventStudy/Imputation.lean"},{"n":"Causalean.Panel.EstimandCharacterization.ImputationEventStudy.BJSPanel.bjs_imputation_identification","k":"theorem","m":"Causalean.Panel.EstimandCharacterization.ImputationEventStudy.Imputation","a":"Panel/EstimandCharacterization/ImputationEventStudy","t":1,"d":"Population BJS imputation identification. Suppose the untreated outcome mean follows a deterministic linear model in the regressors, and each untreated cell's observed mean equals that untreated model mean (untreated outcome model), there exist imputation weights whose row identity expresses every target-relevant treated cell's regressor row as a weighted combination of untreated-cell rows (target-relevant prediction span), and each treated cell's observed mean equals its untreated potential-outcome mean plus its treatment effect (treatment effect fixed). Then there is an imputation-weight witness for which the observed-law imputation functional psiImp equals the target weighted sum of treatment effects theta.","s":"∀ {Treated : Type u_1} {Untreated : Type u_2} {Regressor : Type u_3} [inst : Fintype Treated] [inst_1 : Fintype Untreated] [inst_2 : Fintype Regressor] (P : Causalean.Panel.EstimandCharacterization.ImputationEventStudy.BJSPanel Treated Untreated Regressor), P.UntreatedOutcomeModel → P.PredictionIdentified → P.TreatmentEffectFixed → ∃ h, P.psiImp h.weight = P.theta","l":128,"f":"Causalean/Panel/EstimandCharacterization/ImputationEventStudy/Imputation.lean"},{"n":"Causalean.Panel.EstimandCharacterization.ImputationEventStudy.BJSPanel.bjs_linear_unbiased_iff_imputation_form","k":"theorem","m":"Causalean.Panel.EstimandCharacterization.ImputationEventStudy.PanelBridge","a":"Panel/EstimandCharacterization/ImputationEventStudy","t":1,"d":"Combined iff characterization of the BJS linear-unbiased imputation class. Given a target-relevant prediction-span witness and at least one treated cell with nonzero target weight, a linear estimator L is unbiased for every value of the treatment-effect vector if and only if it admits a BJS imputation-weight representation: its treated coefficients match the target weights, its untreated coefficients equal the negative weighted imputation sum, its value equals the imputation contrast for every pair of treated/untreated observed outcomes, and its observed value equals the population imputation functional psiImp.","s":"∀ {Treated : Type u_1} {Untreated : Type u_2} {Regressor : Type u_3} [inst : Fintype Treated] [inst_1 : Fintype Untreated] [inst_2 : Fintype Regressor] (P : Causalean.Panel.EstimandCharacterization.ImputationEventStudy.BJSPanel Treated Untreated Regressor) (L : P.LinearEstimator), P.PredictionIdentified → (∃ c, P.a c ≠ 0) → (L.unbiasedForAllTau ↔ ∃ H, (∀ (c : Treated), L.vT c = P.a c) ∧ (∀ (u : Untreated), L.vU u = -∑ c, P.a c * H.weight c u) ∧ (∀ (YT : Treated → ℝ) (YU : Untreated → ℝ), L.value YT YU = ∑ c, P.a c * (YT c - ∑ u, H.weight c u * YU u)) ∧ L.observedValue = P.psiImp H.weight)","l":370,"f":"Causalean/Panel/EstimandCharacterization/ImputationEventStudy/PanelBridge.lean"},{"n":"Causalean.Panel.EstimandCharacterization.ImputationEventStudy.BJSPanel.bjs_ols_imputation_min_variance_spherical","k":"theorem","m":"Causalean.Panel.EstimandCharacterization.ImputationEventStudy.Efficiency","a":"Panel/EstimandCharacterization/ImputationEventStudy","t":1,"d":"BJS efficiency: OLS imputation is BLUE under spherical errors. For a BJS event-study panel P and a family of cell outcomes Y on a probability space, suppose each cell outcome is square-integrable, the cell outcomes form a spherical family with common variance σ² — equal variances and zero cross-covariances, and the full event-study design matrix designFull P has full column rank. Then for any linear estimator L unbiased for every value of the treatment-effect vector, the variance of the OLS imputation estimator built from designFull P and the target functional cFull P is no larger than the variance of L.","s":"∀ {Treated : Type u_1} {Untreated : Type u_2} {Regressor : Type u_3} [inst : Fintype Treated] [inst_1 : Fintype Untreated] [inst_2 : Fintype Regressor] [inst_3 : DecidableEq Treated] [inst_4 : DecidableEq Regressor] {P : Causalean.Panel.EstimandCharacterization.ImputationEventStudy.BJSPanel Treated Untreated Regressor} {Ω : Type u_4} {mΩ : MeasurableSpace Ω} {μ : MeasureTheory.Measure Ω} [MeasureTheory.IsProbabilityMeasure μ] (Y : Treated ⊕ Untreated → Ω → ℝ), (∀ (i : Treated ⊕ Untreated), MeasureTheory.MemLp (Y i) 2 μ) → ∀ {σ : ℝ}, Causalean.GaussMarkov.SphericalFamily Y μ σ → IsUnit (P.designFull.transpose * P.designFull).det → ∀ (L : P.LinearEstimator), L.unbiasedForAllTau → ProbabilityTheory.variance (fun ω => ∑ i, Causalean.GaussMarkov.olsWeight P.designFull P.cFull i * Y i ω) μ ≤ ProbabilityTheory.variance (fun ω => ∑ i, Causalean.Panel.EstimandCharacterization.ImputationEventStudy.BJSPanel.weightOf L i * Y i ω) μ","l":86,"f":"Causalean/Panel/EstimandCharacterization/ImputationEventStudy/Efficiency.lean"},{"n":"Causalean.Panel.EstimandCharacterization.ImputationEventStudy.BJSPanel.cFull","k":"def","m":"Causalean.Panel.EstimandCharacterization.ImputationEventStudy.Efficiency","a":"Panel/EstimandCharacterization/ImputationEventStudy","t":1,"d":"Target functional in design coordinates: target weights a on the τ block, zero on the covariate block.","s":"{Treated : Type u_1} → {Untreated : Type u_2} → {Regressor : Type u_3} → [inst : Fintype Treated] → [inst_1 : Fintype Untreated] → [inst_2 : Fintype Regressor] → Causalean.Panel.EstimandCharacterization.ImputationEventStudy.BJSPanel Treated Untreated Regressor → Treated ⊕ Regressor → ℝ","l":48,"f":"Causalean/Panel/EstimandCharacterization/ImputationEventStudy/Efficiency.lean"},{"n":"Causalean.Panel.EstimandCharacterization.ImputationEventStudy.BJSPanel.columnSpan","k":"def","m":"Causalean.Panel.EstimandCharacterization.ImputationEventStudy.PanelBridge","a":"Panel/EstimandCharacterization/ImputationEventStudy","t":1,"d":"The regressor column span inside Untreated → ℝ: the span of all columns u ↦ q_{ur}. Membership of a treated row's representation in (the analogous span over Regressor) is the BJS row-span condition; orthogonality to *this* span is the left-null-space condition.","s":"{Treated : Type u_1} → {Untreated : Type u_2} → {Regressor : Type u_3} → [inst : Fintype Treated] → [inst_1 : Fintype Untreated] → [inst_2 : Fintype Regressor] → Causalean.Panel.EstimandCharacterization.ImputationEventStudy.BJSPanel Treated Untreated Regressor → Submodule ℝ (Untreated → ℝ)","l":79,"f":"Causalean/Panel/EstimandCharacterization/ImputationEventStudy/PanelBridge.lean"},{"n":"Causalean.Panel.EstimandCharacterization.ImputationEventStudy.BJSPanel.columnSpan_ip_orthogonal_iff","k":"theorem","m":"Causalean.Panel.EstimandCharacterization.ImputationEventStudy.PanelBridge","a":"Panel/EstimandCharacterization/ImputationEventStudy","t":2,"d":"Keystone bridge. A vector over the untreated cells is ip-orthogonal to the whole regressor column span iff it satisfies the BJS left-null-space condition ∀ r, ∑_u v_u q_{ur} = 0. This is the panel-substrate restatement of \"v lies in the left null space of Q_U\".","s":"∀ {Treated : Type u_1} {Untreated : Type u_2} {Regressor : Type u_3} [inst : Fintype Treated] [inst_1 : Fintype Untreated] [inst_2 : Fintype Regressor] [inst_3 : DecidableEq Untreated] (P : Causalean.Panel.EstimandCharacterization.ImputationEventStudy.BJSPanel Treated Untreated Regressor) [inst_4 : Nonempty Untreated] (v : Untreated → ℝ), (∀ h ∈ P.columnSpan, (Causalean.Panel.EstimandCharacterization.ImputationEventStudy.untreatedSupport Untreated).ip v h = 0) ↔ ∀ (r : Regressor), ∑ u, v u * P.qU u r = 0","l":114,"f":"Causalean/Panel/EstimandCharacterization/ImputationEventStudy/PanelBridge.lean"},{"n":"Causalean.Panel.EstimandCharacterization.ImputationEventStudy.BJSPanel.designFull","k":"def","m":"Causalean.Panel.EstimandCharacterization.ImputationEventStudy.Efficiency","a":"Panel/EstimandCharacterization/ImputationEventStudy","t":1,"d":"The event-study design matrix: rows are observed cells (Treated ⊕ Untreated), columns are treated-cell fixed effects (Treated, the τ part) stacked with covariates (Regressor, the β part). Treated rows carry the cell-indicator in the τ block and qT in the covariate block; untreated rows carry zeros in the τ block and qU in the covariate block.","s":"{Treated : Type u_1} → {Untreated : Type u_2} → {Regressor : Type u_3} → [inst : Fintype Treated] → [inst_1 : Fintype Untreated] → [inst_2 : Fintype Regressor] → [DecidableEq Treated] → Causalean.Panel.EstimandCharacterization.ImputationEventStudy.BJSPanel Treated Untreated Regressor → Matrix (Treated ⊕ Untreated) (Treated ⊕ Regressor) ℝ","l":37,"f":"Causalean/Panel/EstimandCharacterization/ImputationEventStudy/Efficiency.lean"},{"n":"Causalean.Panel.EstimandCharacterization.ImputationEventStudy.BJSPanel.designFull_col_eq_cellIndicator","k":"theorem","m":"Causalean.Panel.EstimandCharacterization.ImputationEventStudy.Efficiency","a":"Panel/EstimandCharacterization/ImputationEventStudy","t":2,"d":"Each treated-cell fixed-effect column of designFull is the panel cell indicator of treatedClassifier.","s":"∀ {Treated : Type u_1} {Untreated : Type u_2} {Regressor : Type u_3} [inst : Fintype Treated] [inst_1 : Fintype Untreated] [inst_2 : Fintype Regressor] [inst_3 : DecidableEq Treated] (P : Causalean.Panel.EstimandCharacterization.ImputationEventStudy.BJSPanel Treated Untreated Regressor) (d : Treated), (fun i => P.designFull i (Sum.inl d)) = Causalean.Panel.Weighted.cellIndicator Causalean.Panel.EstimandCharacterization.ImputationEventStudy.BJSPanel.treatedClassifier (some d)","l":128,"f":"Causalean/Panel/EstimandCharacterization/ImputationEventStudy/Efficiency.lean"},{"n":"Causalean.Panel.EstimandCharacterization.ImputationEventStudy.BJSPanel.designFull_col_mem_indicatorSpan","k":"theorem","m":"Causalean.Panel.EstimandCharacterization.ImputationEventStudy.Efficiency","a":"Panel/EstimandCharacterization/ImputationEventStudy","t":2,"d":"The treated-cell fixed-effect block of designFull lies in the panel IndicatorSpan of the treated-cell classifier: the BJS event-study FE design is the panel indicator-span substrate.","s":"∀ {Treated : Type u_1} {Untreated : Type u_2} {Regressor : Type u_3} [inst : Fintype Treated] [inst_1 : Fintype Untreated] [inst_2 : Fintype Regressor] [inst_3 : DecidableEq Treated] (P : Causalean.Panel.EstimandCharacterization.ImputationEventStudy.BJSPanel Treated Untreated Regressor) (d : Treated), (fun i => P.designFull i (Sum.inl d)) ∈ Causalean.Panel.Weighted.indicatorSpan Causalean.Panel.EstimandCharacterization.ImputationEventStudy.BJSPanel.treatedClassifier","l":143,"f":"Causalean/Panel/EstimandCharacterization/ImputationEventStudy/Efficiency.lean"},{"n":"Causalean.Panel.EstimandCharacterization.ImputationEventStudy.BJSPanel.exists_imputationWeights_of_gap_orthogonal","k":"theorem","m":"Causalean.Panel.EstimandCharacterization.ImputationEventStudy.PanelBridge","a":"Panel/EstimandCharacterization/ImputationEventStudy","t":2,"d":"Left-null-space row adjustment (audit M3), phrased through the panel substrate. Given base imputation weights H0, a nonzero target weight at c0, and a target untreated-coefficient vector vU whose gap gap u = vU u + ∑_c a_c · H0.weight c u is ip-orthogonal to the regressor column span (equivalently: gap lies in the left null space of Q_U), one nonzero target row of H0 can absorb the gap without disturbing any target-relevant row identity. The resulting imputation weights H represent the untreated coefficients: ∑_c a_c · H.weight c u = - vU u for every u.","s":"∀ {Treated : Type u_1} {Untreated : Type u_2} {Regressor : Type u_3} [inst : Fintype Treated] [inst_1 : Fintype Untreated] [inst_2 : Fintype Regressor] [inst_3 : DecidableEq Untreated] (P : Causalean.Panel.EstimandCharacterization.ImputationEventStudy.BJSPanel Treated Untreated Regressor) [inst_4 : Nonempty Untreated] (vU : Untreated → ℝ) {c0 : Treated}, P.a c0 ≠ 0 → ∀ (H0 : P.ImputationWeights), (∀ h ∈ P.columnSpan, (Causalean.Panel.EstimandCharacterization.ImputationEventStudy.untreatedSupport Untreated).ip (fun u => vU u + ∑ c, P.a c * H0.weight c u) h = 0) → ∃ H, ∀ (u : Untreated), ∑ c, P.a c * H.weight c u = -vU u","l":140,"f":"Causalean/Panel/EstimandCharacterization/ImputationEventStudy/PanelBridge.lean"},{"n":"Causalean.Panel.EstimandCharacterization.ImputationEventStudy.BJSPanel.ip_regressorColumn_eq_zero_iff","k":"theorem","m":"Causalean.Panel.EstimandCharacterization.ImputationEventStudy.PanelBridge","a":"Panel/EstimandCharacterization/ImputationEventStudy","t":2,"d":"ip-orthogonality to a single regressor column ⟺ that column's BJS left-null-space coordinate vanishes.","s":"∀ {Treated : Type u_1} {Untreated : Type u_2} {Regressor : Type u_3} [inst : Fintype Treated] [inst_1 : Fintype Untreated] [inst_2 : Fintype Regressor] [inst_3 : DecidableEq Untreated] (P : Causalean.Panel.EstimandCharacterization.ImputationEventStudy.BJSPanel Treated Untreated Regressor) [inst_4 : Nonempty Untreated] (v : Untreated → ℝ) (r : Regressor), (Causalean.Panel.EstimandCharacterization.ImputationEventStudy.untreatedSupport Untreated).ip v (P.regressorColumn r) = 0 ↔ ∑ u, v u * P.qU u r = 0","l":99,"f":"Causalean/Panel/EstimandCharacterization/ImputationEventStudy/PanelBridge.lean"},{"n":"Causalean.Panel.EstimandCharacterization.ImputationEventStudy.BJSPanel.ip_untreatedSupport_regressorColumn","k":"theorem","m":"Causalean.Panel.EstimandCharacterization.ImputationEventStudy.PanelBridge","a":"Panel/EstimandCharacterization/ImputationEventStudy","t":2,"d":"The panel inner product of an untreated-cell vector v with a regressor column is the BJS left-null-space sum, rescaled by the uniform weight.","s":"∀ {Treated : Type u_1} {Untreated : Type u_2} {Regressor : Type u_3} [inst : Fintype Treated] [inst_1 : Fintype Untreated] [inst_2 : Fintype Regressor] [inst_3 : DecidableEq Untreated] (P : Causalean.Panel.EstimandCharacterization.ImputationEventStudy.BJSPanel Treated Untreated Regressor) [inst_4 : Nonempty Untreated] (v : Untreated → ℝ) (r : Regressor), (Causalean.Panel.EstimandCharacterization.ImputationEventStudy.untreatedSupport Untreated).ip v (P.regressorColumn r) = (↑(Fintype.card Untreated))⁻¹ * ∑ u, v u * P.qU u r","l":88,"f":"Causalean/Panel/EstimandCharacterization/ImputationEventStudy/PanelBridge.lean"},{"n":"Causalean.Panel.EstimandCharacterization.ImputationEventStudy.BJSPanel.linear_unbiased_of_imputation_representation","k":"theorem","m":"Causalean.Panel.EstimandCharacterization.ImputationEventStudy.Imputation","a":"Panel/EstimandCharacterization/ImputationEventStudy","t":2,"d":"Universal unbiasedness supplies the treated-coefficient identity required by linear_unbiased_of_imputation_representation_of_vT_eq_a.","s":"∀ {Treated : Type u_1} {Untreated : Type u_2} {Regressor : Type u_3} [inst : Fintype Treated] [inst_1 : Fintype Untreated] [inst_2 : Fintype Regressor] (P : Causalean.Panel.EstimandCharacterization.ImputationEventStudy.BJSPanel Treated Untreated Regressor) (L : P.LinearEstimator), L.unbiasedForAllTau → ∀ (hImputationWitness : L.HasImputationRepresentation), ∃ H, (∀ (c : Treated), L.vT c = P.a c) ∧ (∀ (u : Untreated), L.vU u = -∑ c, P.a c * H.weight c u) ∧ (∀ (YT : Treated → ℝ) (YU : Untreated → ℝ), L.value YT YU = ∑ c, P.a c * (YT c - ∑ u, H.weight c u * YU u)) ∧ L.observedValue = P.psiImp H.weight","l":377,"f":"Causalean/Panel/EstimandCharacterization/ImputationEventStudy/Imputation.lean"},{"n":"Causalean.Panel.EstimandCharacterization.ImputationEventStudy.BJSPanel.linear_unbiased_of_imputation_representation_of_vT_eq_a","k":"theorem","m":"Causalean.Panel.EstimandCharacterization.ImputationEventStudy.Imputation","a":"Panel/EstimandCharacterization/ImputationEventStudy","t":2,"d":"Witness-based BJS linear-estimator representation from its treated coefficients and an imputation witness, exposed as an estimator identity for arbitrary outcome arrays.","s":"∀ {Treated : Type u_1} {Untreated : Type u_2} {Regressor : Type u_3} [inst : Fintype Treated] [inst_1 : Fintype Untreated] [inst_2 : Fintype Regressor] (P : Causalean.Panel.EstimandCharacterization.ImputationEventStudy.BJSPanel Treated Untreated Regressor) (L : P.LinearEstimator), (∀ (c : Treated), L.vT c = P.a c) → ∀ (hImputationWitness : L.HasImputationRepresentation), ∃ H, (∀ (c : Treated), L.vT c = P.a c) ∧ (∀ (u : Untreated), L.vU u = -∑ c, P.a c * H.weight c u) ∧ (∀ (YT : Treated → ℝ) (YU : Untreated → ℝ), L.value YT YU = ∑ c, P.a c * (YT c - ∑ u, H.weight c u * YU u)) ∧ L.observedValue = P.psiImp H.weight","l":271,"f":"Causalean/Panel/EstimandCharacterization/ImputationEventStudy/Imputation.lean"},{"n":"Causalean.Panel.EstimandCharacterization.ImputationEventStudy.BJSPanel.linear_unbiased_of_prediction_identified","k":"theorem","m":"Causalean.Panel.EstimandCharacterization.ImputationEventStudy.PanelBridge","a":"Panel/EstimandCharacterization/ImputationEventStudy","t":2,"d":"BJS linear-unbiased representation from the primitive prediction-span condition.","s":"∀ {Treated : Type u_1} {Untreated : Type u_2} {Regressor : Type u_3} [inst : Fintype Treated] [inst_1 : Fintype Untreated] [inst_2 : Fintype Regressor] (P : Causalean.Panel.EstimandCharacterization.ImputationEventStudy.BJSPanel Treated Untreated Regressor) (L : P.LinearEstimator), L.unbiasedForAllTau → P.PredictionIdentified → (∃ c, P.a c ≠ 0) → ∃ H, (∀ (c : Treated), L.vT c = P.a c) ∧ (∀ (u : Untreated), L.vU u = -∑ c, P.a c * H.weight c u) ∧ (∀ (YT : Treated → ℝ) (YU : Untreated → ℝ), L.value YT YU = ∑ c, P.a c * (YT c - ∑ u, H.weight c u * YU u)) ∧ L.observedValue = P.psiImp H.weight","l":257,"f":"Causalean/Panel/EstimandCharacterization/ImputationEventStudy/PanelBridge.lean"},{"n":"Causalean.Panel.EstimandCharacterization.ImputationEventStudy.BJSPanel.psiImp","k":"def","m":"Causalean.Panel.EstimandCharacterization.ImputationEventStudy.Imputation","a":"Panel/EstimandCharacterization/ImputationEventStudy","t":1,"d":"Observed-law population imputation functional for arbitrary imputation weights. The row identity is a theorem hypothesis, not part of this functional's definition.","s":"{Treated : Type u_1} → {Untreated : Type u_2} → {Regressor : Type u_3} → [inst : Fintype Treated] → [inst_1 : Fintype Untreated] → [inst_2 : Fintype Regressor] → Causalean.Panel.EstimandCharacterization.ImputationEventStudy.BJSPanel Treated Untreated Regressor → (Treated → Untreated → ℝ) → ℝ","l":121,"f":"Causalean/Panel/EstimandCharacterization/ImputationEventStudy/Imputation.lean"},{"n":"Causalean.Panel.EstimandCharacterization.ImputationEventStudy.BJSPanel.regressorColumn","k":"def","m":"Causalean.Panel.EstimandCharacterization.ImputationEventStudy.PanelBridge","a":"Panel/EstimandCharacterization/ImputationEventStudy","t":1,"d":"The r-th regressor column as an array over the untreated cells: u ↦ q_{ur}.","s":"{Treated : Type u_1} → {Untreated : Type u_2} → {Regressor : Type u_3} → [inst : Fintype Treated] → [inst_1 : Fintype Untreated] → [inst_2 : Fintype Regressor] → Causalean.Panel.EstimandCharacterization.ImputationEventStudy.BJSPanel Treated Untreated Regressor → Regressor → Untreated → ℝ","l":75,"f":"Causalean/Panel/EstimandCharacterization/ImputationEventStudy/PanelBridge.lean"},{"n":"Causalean.Panel.EstimandCharacterization.ImputationEventStudy.BJSPanel.targetForTau","k":"def","m":"Causalean.Panel.EstimandCharacterization.ImputationEventStudy.Imputation","a":"Panel/EstimandCharacterization/ImputationEventStudy","t":1,"d":"Target value for an arbitrary unrestricted treated-effect vector.","s":"{Treated : Type u_1} → {Untreated : Type u_2} → {Regressor : Type u_3} → [inst : Fintype Treated] → [inst_1 : Fintype Untreated] → [inst_2 : Fintype Regressor] → Causalean.Panel.EstimandCharacterization.ImputationEventStudy.BJSPanel Treated Untreated Regressor → (Treated → ℝ) → ℝ","l":88,"f":"Causalean/Panel/EstimandCharacterization/ImputationEventStudy/Imputation.lean"},{"n":"Causalean.Panel.EstimandCharacterization.ImputationEventStudy.BJSPanel.theta","k":"def","m":"Causalean.Panel.EstimandCharacterization.ImputationEventStudy.Imputation","a":"Panel/EstimandCharacterization/ImputationEventStudy","t":1,"d":"Target weighted sum over treated-cell effects. No positivity or normalization of a is imposed.","s":"{Treated : Type u_1} → {Untreated : Type u_2} → {Regressor : Type u_3} → [inst : Fintype Treated] → [inst_1 : Fintype Untreated] → [inst_2 : Fintype Regressor] → Causalean.Panel.EstimandCharacterization.ImputationEventStudy.BJSPanel Treated Untreated Regressor → ℝ","l":83,"f":"Causalean/Panel/EstimandCharacterization/ImputationEventStudy/Imputation.lean"},{"n":"Causalean.Panel.EstimandCharacterization.ImputationEventStudy.BJSPanel.treatedClassifier","k":"def","m":"Causalean.Panel.EstimandCharacterization.ImputationEventStudy.Efficiency","a":"Panel/EstimandCharacterization/ImputationEventStudy","t":1,"d":"Classifier on observed cells sending each treated cell to its own label and every untreated cell to none. Its Some-indicators are the τ-columns of designFull.","s":"{Treated : Type u_1} → {Untreated : Type u_2} → Treated ⊕ Untreated → Option Treated","l":120,"f":"Causalean/Panel/EstimandCharacterization/ImputationEventStudy/Efficiency.lean"},{"n":"Causalean.Panel.EstimandCharacterization.ImputationEventStudy.BJSPanel.weightOf","k":"def","m":"Causalean.Panel.EstimandCharacterization.ImputationEventStudy.Efficiency","a":"Panel/EstimandCharacterization/ImputationEventStudy","t":1,"d":"A linear estimator's weight vector over all observed cells.","s":"{Treated : Type u_1} → {Untreated : Type u_2} → {Regressor : Type u_3} → [inst : Fintype Treated] → [inst_1 : Fintype Untreated] → [inst_2 : Fintype Regressor] → {P : Causalean.Panel.EstimandCharacterization.ImputationEventStudy.BJSPanel Treated Untreated Regressor} → P.LinearEstimator → Treated ⊕ Untreated → ℝ","l":53,"f":"Causalean/Panel/EstimandCharacterization/ImputationEventStudy/Efficiency.lean"},{"n":"Causalean.Panel.EstimandCharacterization.ImputationEventStudy.BJSPanel.weightOf_vecMul_designFull","k":"theorem","m":"Causalean.Panel.EstimandCharacterization.ImputationEventStudy.Efficiency","a":"Panel/EstimandCharacterization/ImputationEventStudy","t":2,"d":"Unbiasedness bridge. A linear estimator is unbiased for every tau iff its weight vector satisfies the design constraint w ᵥ* designFull = cFull. (Only the forward direction is needed below, but it captures both vT = a and the nuisance left-null constraint.)","s":"∀ {Treated : Type u_1} {Untreated : Type u_2} {Regressor : Type u_3} [inst : Fintype Treated] [inst_1 : Fintype Untreated] [inst_2 : Fintype Regressor] [inst_3 : DecidableEq Treated] {P : Causalean.Panel.EstimandCharacterization.ImputationEventStudy.BJSPanel Treated Untreated Regressor} (L : P.LinearEstimator), L.unbiasedForAllTau → Matrix.vecMul (Causalean.Panel.EstimandCharacterization.ImputationEventStudy.BJSPanel.weightOf L) P.designFull = P.cFull","l":59,"f":"Causalean/Panel/EstimandCharacterization/ImputationEventStudy/Efficiency.lean"},{"n":"Causalean.Panel.EstimandCharacterization.ImputationEventStudy.BJSPopulation","k":"structure","m":"Causalean.Panel.EstimandCharacterization.ImputationEventStudy.PopulationBridge","a":"Panel/EstimandCharacterization/ImputationEventStudy","t":1,"d":"A population for the BJS imputation design: a probability space with a treated/untreated cell classifier and potential outcomes Y0 (untreated), Y1 (treated), and observed Yobs, related by cell-level consistency. The design rows qT, qU, target weights a, and nuisance vector beta0 are carried through to the induced panel unchanged.","s":"(Treated : Type u_4) → (Untreated : Type u_5) → (Regressor : Type u_6) → [Fintype Treated] → [Fintype Untreated] → [Fintype Regressor] → Type (max (max (max u_4 u_5) u_6) (u_7 + 1))","l":48,"f":"Causalean/Panel/EstimandCharacterization/ImputationEventStudy/PopulationBridge.lean"},{"n":"Causalean.Panel.EstimandCharacterization.ImputationEventStudy.BJSPopulation.bjs_imputation_identification_population","k":"theorem","m":"Causalean.Panel.EstimandCharacterization.ImputationEventStudy.PopulationBridge","a":"Panel/EstimandCharacterization/ImputationEventStudy","t":1,"d":"Population BJS imputation identification (headline). For a population BJS design E, suppose the conditional mean of the untreated potential outcome on each treated cell equals a linear function q_T · β₀ of the treated-cell regressors and likewise, on each untreated cell, the conditional mean of the untreated potential outcome equals q_U · β₀ — jointly the additive untreated-outcome (parallel-trends) model — together with a target-relevant prediction-span witness for the induced panel. Then the observed-law imputation functional identifies the target ∑ a_c · (E[Y(1) ∣ cell c] − E[Y(0) ∣ cell c]): there is an imputation-weight witness for which the population imputation functional psiImp equals the target theta.","s":"∀ {Treated : Type u_1} {Untreated : Type u_2} {Regressor : Type u_3} [inst : Fintype Treated] [inst_1 : Fintype Untreated] [inst_2 : Fintype Regressor] (E : Causalean.Panel.EstimandCharacterization.ImputationEventStudy.BJSPopulation Treated Untreated Regressor), (∀ (c : Treated), E.cells.mean E.Y0 (Sum.inl c) = Causalean.Panel.EstimandCharacterization.ImputationEventStudy.dot (E.qT c) E.beta0) → (∀ (u : Untreated), E.cells.mean E.Y0 (Sum.inr u) = Causalean.Panel.EstimandCharacterization.ImputationEventStudy.dot (E.qU u) E.beta0) → E.toPanel.PredictionIdentified → ∃ h, E.toPanel.psiImp h.weight = E.toPanel.theta","l":167,"f":"Causalean/Panel/EstimandCharacterization/ImputationEventStudy/PopulationBridge.lean"},{"n":"Causalean.Panel.EstimandCharacterization.ImputationEventStudy.BJSPopulation.cells","k":"def","m":"Causalean.Panel.EstimandCharacterization.ImputationEventStudy.PopulationBridge","a":"Panel/EstimandCharacterization/ImputationEventStudy","t":1,"d":"The treated/untreated cell partition induced by the classifier.","s":"{Treated : Type u_1} → {Untreated : Type u_2} → {Regressor : Type u_3} → [inst : Fintype Treated] → [inst_1 : Fintype Untreated] → [inst_2 : Fintype Regressor] → (E : Causalean.Panel.EstimandCharacterization.ImputationEventStudy.BJSPopulation Treated Untreated Regressor) → Causalean.Panel.PO.CellPartition E.μ (Treated ⊕ Untreated)","l":98,"f":"Causalean/Panel/EstimandCharacterization/ImputationEventStudy/PopulationBridge.lean"},{"n":"Causalean.Panel.EstimandCharacterization.ImputationEventStudy.BJSPopulation.toPanel","k":"def","m":"Causalean.Panel.EstimandCharacterization.ImputationEventStudy.PopulationBridge","a":"Panel/EstimandCharacterization/ImputationEventStudy","t":1,"d":"The BJS panel induced by a population: every mean field is a treated/untreated cell conditional mean of the appropriate potential-outcome slice, and the treated effect tau c is the genuine population contrast E[Y(1) ∣ cell c] − E[Y(0) ∣ cell c].","s":"{Treated : Type u_1} → {Untreated : Type u_2} → {Regressor : Type u_3} → [inst : Fintype Treated] → [inst_1 : Fintype Untreated] → [inst_2 : Fintype Regressor] → Causalean.Panel.EstimandCharacterization.ImputationEventStudy.BJSPopulation Treated Untreated Regressor → Causalean.Panel.EstimandCharacterization.ImputationEventStudy.BJSPanel Treated Untreated Regressor","l":102,"f":"Causalean/Panel/EstimandCharacterization/ImputationEventStudy/PopulationBridge.lean"},{"n":"Causalean.Panel.EstimandCharacterization.ImputationEventStudy.BJSPopulation.toPanel_EY_T_eq_mean_Y1","k":"theorem","m":"Causalean.Panel.EstimandCharacterization.ImputationEventStudy.PopulationBridge","a":"Panel/EstimandCharacterization/ImputationEventStudy","t":2,"d":"On a treated cell the observed mean equals the treated potential-outcome mean.","s":"∀ {Treated : Type u_1} {Untreated : Type u_2} {Regressor : Type u_3} [inst : Fintype Treated] [inst_1 : Fintype Untreated] [inst_2 : Fintype Regressor] (E : Causalean.Panel.EstimandCharacterization.ImputationEventStudy.BJSPopulation Treated Untreated Regressor) (c : Treated), E.toPanel.EY_T c = E.cells.mean E.Y1 (Sum.inl c)","l":117,"f":"Causalean/Panel/EstimandCharacterization/ImputationEventStudy/PopulationBridge.lean"},{"n":"Causalean.Panel.EstimandCharacterization.ImputationEventStudy.BJSPopulation.toPanel_EY_U_eq_mean_Y0","k":"theorem","m":"Causalean.Panel.EstimandCharacterization.ImputationEventStudy.PopulationBridge","a":"Panel/EstimandCharacterization/ImputationEventStudy","t":2,"d":"On an untreated cell the observed mean equals the untreated potential-outcome mean.","s":"∀ {Treated : Type u_1} {Untreated : Type u_2} {Regressor : Type u_3} [inst : Fintype Treated] [inst_1 : Fintype Untreated] [inst_2 : Fintype Regressor] (E : Causalean.Panel.EstimandCharacterization.ImputationEventStudy.BJSPopulation Treated Untreated Regressor) (u : Untreated), E.toPanel.EY_U u = E.cells.mean E.Y0 (Sum.inr u)","l":126,"f":"Causalean/Panel/EstimandCharacterization/ImputationEventStudy/PopulationBridge.lean"},{"n":"Causalean.Panel.EstimandCharacterization.ImputationEventStudy.BJSPopulation.toPanel_tau_eq_po_contrast","k":"theorem","m":"Causalean.Panel.EstimandCharacterization.ImputationEventStudy.PopulationBridge","a":"Panel/EstimandCharacterization/ImputationEventStudy","t":2,"d":"Causal-meaning certificate. In the induced panel, tau c is literally the population treatment-effect contrast E[Y(1) ∣ cell c] − E[Y(0) ∣ cell c], so the estimand carries genuine causal content.","s":"∀ {Treated : Type u_1} {Untreated : Type u_2} {Regressor : Type u_3} [inst : Fintype Treated] [inst_1 : Fintype Untreated] [inst_2 : Fintype Regressor] (E : Causalean.Panel.EstimandCharacterization.ImputationEventStudy.BJSPopulation Treated Untreated Regressor) (c : Treated), E.toPanel.tau c = E.cells.mean E.Y1 (Sum.inl c) - E.cells.mean E.Y0 (Sum.inl c)","l":135,"f":"Causalean/Panel/EstimandCharacterization/ImputationEventStudy/PopulationBridge.lean"},{"n":"Causalean.Panel.EstimandCharacterization.ImputationEventStudy.BJSPopulation.toPanel_treatmentEffectFixed","k":"theorem","m":"Causalean.Panel.EstimandCharacterization.ImputationEventStudy.PopulationBridge","a":"Panel/EstimandCharacterization/ImputationEventStudy","t":2,"d":"Treatment-effect-fixed is derived. EY_T = EY0_T + tau holds because on treated cells the observed mean is the treated potential-outcome mean and tau is the treated-minus-untreated contrast.","s":"∀ {Treated : Type u_1} {Untreated : Type u_2} {Regressor : Type u_3} [inst : Fintype Treated] [inst_1 : Fintype Untreated] [inst_2 : Fintype Regressor] (E : Causalean.Panel.EstimandCharacterization.ImputationEventStudy.BJSPopulation Treated Untreated Regressor), E.toPanel.TreatmentEffectFixed","l":142,"f":"Causalean/Panel/EstimandCharacterization/ImputationEventStudy/PopulationBridge.lean"},{"n":"Causalean.Panel.EstimandCharacterization.ImputationEventStudy.BJSPopulation.toPanel_untreatedModel","k":"theorem","m":"Causalean.Panel.EstimandCharacterization.ImputationEventStudy.PopulationBridge","a":"Panel/EstimandCharacterization/ImputationEventStudy","t":2,"d":"Untreated-outcome model is derived from the linear untreated-mean hypotheses plus consistency. The two linear-model conjuncts E[Y(0) ∣ cell] = q · β₀ are the genuine modeling hypotheses hLinT/hLinU (additive fixed-effect / parallel-trends form); the observed-equals-untreated conjunct is derived from untreated-cell consistency.","s":"∀ {Treated : Type u_1} {Untreated : Type u_2} {Regressor : Type u_3} [inst : Fintype Treated] [inst_1 : Fintype Untreated] [inst_2 : Fintype Regressor] (E : Causalean.Panel.EstimandCharacterization.ImputationEventStudy.BJSPopulation Treated Untreated Regressor), (∀ (c : Treated), E.cells.mean E.Y0 (Sum.inl c) = Causalean.Panel.EstimandCharacterization.ImputationEventStudy.dot (E.qT c) E.beta0) → (∀ (u : Untreated), E.cells.mean E.Y0 (Sum.inr u) = Causalean.Panel.EstimandCharacterization.ImputationEventStudy.dot (E.qU u) E.beta0) → E.toPanel.UntreatedOutcomeModel","l":151,"f":"Causalean/Panel/EstimandCharacterization/ImputationEventStudy/PopulationBridge.lean"},{"n":"Causalean.Panel.EstimandCharacterization.ImputationEventStudy.TreatedCell","k":"def","m":"Causalean.Panel.EstimandCharacterization.ImputationEventStudy.PanelGrounding","a":"Panel/EstimandCharacterization/ImputationEventStudy","t":1,"d":"Cells whose treatment has switched on by period t (g i ≤ t).","s":"{I : Type u_1} → {T : ℕ} → (I → WithTop (Fin T)) → Type u_1","l":48,"f":"Causalean/Panel/EstimandCharacterization/ImputationEventStudy/PanelGrounding.lean"},{"n":"Causalean.Panel.EstimandCharacterization.ImputationEventStudy.UntreatedCell","k":"def","m":"Causalean.Panel.EstimandCharacterization.ImputationEventStudy.PanelGrounding","a":"Panel/EstimandCharacterization/ImputationEventStudy","t":1,"d":"Cells still untreated at period t (t < g i). Includes every cell of a never-treated unit and every pre-adoption cell of a treated unit.","s":"{I : Type u_1} → {T : ℕ} → (I → WithTop (Fin T)) → Type u_1","l":52,"f":"Causalean/Panel/EstimandCharacterization/ImputationEventStudy/PanelGrounding.lean"},{"n":"Causalean.Panel.EstimandCharacterization.ImputationEventStudy.dot","k":"def","m":"Causalean.Panel.EstimandCharacterization.ImputationEventStudy.Imputation","a":"Panel/EstimandCharacterization/ImputationEventStudy","t":1,"d":"Finite dot product for regressor rows and nuisance vectors.","s":"{Regressor : Type u_3} → [Fintype Regressor] → (Regressor → ℝ) → (Regressor → ℝ) → ℝ","l":42,"f":"Causalean/Panel/EstimandCharacterization/ImputationEventStudy/Imputation.lean"},{"n":"Causalean.Panel.EstimandCharacterization.ImputationEventStudy.dot_feRow","k":"theorem","m":"Causalean.Panel.EstimandCharacterization.ImputationEventStudy.PanelGrounding","a":"Panel/EstimandCharacterization/ImputationEventStudy","t":2,"d":"The two-way FE row evaluates the additive fixed-effect model: q_{(i,t)} · (α, λ) = α_i + λ_t.","s":"∀ {I : Type u_1} [inst : Fintype I] [inst_1 : DecidableEq I] {T : ℕ} (c : I × Fin T) (α : I → ℝ) (lam : Fin T → ℝ), Causalean.Panel.EstimandCharacterization.ImputationEventStudy.dot (Causalean.Panel.EstimandCharacterization.ImputationEventStudy.feRow c) (Sum.elim α lam) = α c.1 + lam c.2","l":93,"f":"Causalean/Panel/EstimandCharacterization/ImputationEventStudy/PanelGrounding.lean"},{"n":"Causalean.Panel.EstimandCharacterization.ImputationEventStudy.feRow","k":"def","m":"Causalean.Panel.EstimandCharacterization.ImputationEventStudy.PanelGrounding","a":"Panel/EstimandCharacterization/ImputationEventStudy","t":1,"d":"Two-way fixed-effect design row for cell (i,t): the unit-i indicator on the I block stacked with the time-t indicator on the Fin T block.","s":"{I : Type u_1} → [DecidableEq I] → {T : ℕ} → I × Fin T → I ⊕ Fin T → ℝ","l":87,"f":"Causalean/Panel/EstimandCharacterization/ImputationEventStudy/PanelGrounding.lean"},{"n":"Causalean.Panel.EstimandCharacterization.ImputationEventStudy.instDecidablePredProdFinLeFstSnd","k":"instance","m":"Causalean.Panel.EstimandCharacterization.ImputationEventStudy.PanelGrounding","a":"Panel/EstimandCharacterization/ImputationEventStudy","t":2,"d":"The treated-cell predicate is decidable for each unit-period cell.","s":"{I : Type u_1} → {T : ℕ} → (g : I → WithTop (Fin T)) → DecidablePred fun c => Causalean.Panel.AdoptionPath.le (g c.1) c.2","l":57,"f":"Causalean/Panel/EstimandCharacterization/ImputationEventStudy/PanelGrounding.lean"},{"n":"Causalean.Panel.EstimandCharacterization.ImputationEventStudy.instDecidablePredProdFinLtFstSnd","k":"instance","m":"Causalean.Panel.EstimandCharacterization.ImputationEventStudy.PanelGrounding","a":"Panel/EstimandCharacterization/ImputationEventStudy","t":2,"d":"The untreated-cell predicate is decidable for each unit-period cell.","s":"{I : Type u_1} → {T : ℕ} → (g : I → WithTop (Fin T)) → DecidablePred fun c => Causalean.Panel.AdoptionPath.lt (g c.1) c.2","l":61,"f":"Causalean/Panel/EstimandCharacterization/ImputationEventStudy/PanelGrounding.lean"},{"n":"Causalean.Panel.EstimandCharacterization.ImputationEventStudy.instFintypeTreatedCell","k":"instance","m":"Causalean.Panel.EstimandCharacterization.ImputationEventStudy.PanelGrounding","a":"Panel/EstimandCharacterization/ImputationEventStudy","t":2,"d":"The treated cells form a finite type whenever units and periods are finite.","s":"{I : Type u_1} → [Fintype I] → {T : ℕ} → (g : I → WithTop (Fin T)) → Fintype (Causalean.Panel.EstimandCharacterization.ImputationEventStudy.TreatedCell g)","l":65,"f":"Causalean/Panel/EstimandCharacterization/ImputationEventStudy/PanelGrounding.lean"},{"n":"Causalean.Panel.EstimandCharacterization.ImputationEventStudy.instFintypeUntreatedCell","k":"instance","m":"Causalean.Panel.EstimandCharacterization.ImputationEventStudy.PanelGrounding","a":"Panel/EstimandCharacterization/ImputationEventStudy","t":2,"d":"The untreated cells form a finite type whenever units and periods are finite.","s":"{I : Type u_1} → [Fintype I] → {T : ℕ} → (g : I → WithTop (Fin T)) → Fintype (Causalean.Panel.EstimandCharacterization.ImputationEventStudy.UntreatedCell g)","l":67,"f":"Causalean/Panel/EstimandCharacterization/ImputationEventStudy/PanelGrounding.lean"},{"n":"Causalean.Panel.EstimandCharacterization.ImputationEventStudy.ofStaggeredTWFE","k":"def","m":"Causalean.Panel.EstimandCharacterization.ImputationEventStudy.PanelGrounding","a":"Panel/EstimandCharacterization/ImputationEventStudy","t":1,"d":"Grounding constructor. From an adoption path g, unit effects α, time effects λ, target weights a, and treated-cell effects τ, build the BJSPanel whose treated/untreated cells are the staggered-adoption partition of I × Fin T and whose regressors are the two-way fixed effects. The untreated-outcome model E[Y(0)] = α_i + λ_t, no anticipation, and the fixed-effect equation all hold definitionally.","s":"{I : Type u_1} → [inst : Fintype I] → [DecidableEq I] → {T : ℕ} → (g : I → WithTop (Fin T)) → (I → ℝ) → (Fin T → ℝ) → (Causalean.Panel.EstimandCharacterization.ImputationEventStudy.TreatedCell g → ℝ) → (Causalean.Panel.EstimandCharacterization.ImputationEventStudy.TreatedCell g → ℝ) → Causalean.Panel.EstimandCharacterization.ImputationEventStudy.BJSPanel (Causalean.Panel.EstimandCharacterization.ImputationEventStudy.TreatedCell g) (Causalean.Panel.EstimandCharacterization.ImputationEventStudy.UntreatedCell g) (I ⊕ Fin T)","l":102,"f":"Causalean/Panel/EstimandCharacterization/ImputationEventStudy/PanelGrounding.lean"},{"n":"Causalean.Panel.EstimandCharacterization.ImputationEventStudy.ofStaggeredTWFE_treatmentFixed","k":"theorem","m":"Causalean.Panel.EstimandCharacterization.ImputationEventStudy.PanelGrounding","a":"Panel/EstimandCharacterization/ImputationEventStudy","t":2,"d":"The grounded panel satisfies the BJS fixed-effect equation E[Y_T] = E[Y_T(0)] + τ.","s":"∀ {I : Type u_1} [inst : Fintype I] [inst_1 : DecidableEq I] {T : ℕ} (g : I → WithTop (Fin T)) (α : I → ℝ) (lam : Fin T → ℝ) (a tau : Causalean.Panel.EstimandCharacterization.ImputationEventStudy.TreatedCell g → ℝ), (Causalean.Panel.EstimandCharacterization.ImputationEventStudy.ofStaggeredTWFE g α lam a tau).TreatmentEffectFixed","l":135,"f":"Causalean/Panel/EstimandCharacterization/ImputationEventStudy/PanelGrounding.lean"},{"n":"Causalean.Panel.EstimandCharacterization.ImputationEventStudy.ofStaggeredTWFE_untreatedModel","k":"theorem","m":"Causalean.Panel.EstimandCharacterization.ImputationEventStudy.PanelGrounding","a":"Panel/EstimandCharacterization/ImputationEventStudy","t":1,"d":"Untreated-outcome model holds by construction. For unit fixed effects α, period fixed effects lam, and target weights and treated-cell effects a, tau, the staggered-adoption grounded panel ofStaggeredTWFE g α lam a tau satisfies the BJS untreated-outcome model: treated and untreated cell means both equal the two-way fixed-effect model α_i + λ_t, and untreated cells exhibit no anticipation.","s":"∀ {I : Type u_1} [inst : Fintype I] [inst_1 : DecidableEq I] {T : ℕ} (g : I → WithTop (Fin T)) (α : I → ℝ) (lam : Fin T → ℝ) (a tau : Causalean.Panel.EstimandCharacterization.ImputationEventStudy.TreatedCell g → ℝ), (Causalean.Panel.EstimandCharacterization.ImputationEventStudy.ofStaggeredTWFE g α lam a tau).UntreatedOutcomeModel","l":121,"f":"Causalean/Panel/EstimandCharacterization/ImputationEventStudy/PanelGrounding.lean"},{"n":"Causalean.Panel.EstimandCharacterization.ImputationEventStudy.treated_not_untreated","k":"theorem","m":"Causalean.Panel.EstimandCharacterization.ImputationEventStudy.PanelGrounding","a":"Panel/EstimandCharacterization/ImputationEventStudy","t":2,"d":"The cell partition is exclusive: no cell is both adopted-by-t and untreated-at-t.","s":"∀ {I : Type u_1} {T : ℕ} (g : I → WithTop (Fin T)) {c : I × Fin T}, Causalean.Panel.AdoptionPath.le (g c.1) c.2 → Causalean.Panel.AdoptionPath.lt (g c.1) c.2 → False","l":71,"f":"Causalean/Panel/EstimandCharacterization/ImputationEventStudy/PanelGrounding.lean"},{"n":"Causalean.Panel.EstimandCharacterization.ImputationEventStudy.treated_or_untreated","k":"theorem","m":"Causalean.Panel.EstimandCharacterization.ImputationEventStudy.PanelGrounding","a":"Panel/EstimandCharacterization/ImputationEventStudy","t":2,"d":"The cell partition is exhaustive: every cell is treated or untreated.","s":"∀ {I : Type u_1} {T : ℕ} (g : I → WithTop (Fin T)) (c : I × Fin T), Causalean.Panel.AdoptionPath.le (g c.1) c.2 ∨ Causalean.Panel.AdoptionPath.lt (g c.1) c.2","l":78,"f":"Causalean/Panel/EstimandCharacterization/ImputationEventStudy/PanelGrounding.lean"},{"n":"Causalean.Panel.EstimandCharacterization.ImputationEventStudy.untreatedSupport","k":"def","m":"Causalean.Panel.EstimandCharacterization.ImputationEventStudy.PanelBridge","a":"Panel/EstimandCharacterization/ImputationEventStudy","t":1,"d":"The untreated cells equipped with the uniform weighted support ω_u ≡ 1/|U|. The OLS imputation estimator is the uniform-weight instance of the panel WLS substrate, so this is the WeightedSupport through which BJS consumes the ip / Subspace API.","s":"(Untreated : Type u_4) → [inst : Fintype Untreated] → [inst_1 : DecidableEq Untreated] → [Nonempty Untreated] → Causalean.Panel.Weighted.WeightedSupport Untreated","l":52,"f":"Causalean/Panel/EstimandCharacterization/ImputationEventStudy/PanelBridge.lean"},{"n":"Causalean.Panel.EstimandCharacterization.OLSWeightDecomposition.FinitePartition","k":"structure","m":"Causalean.Panel.EstimandCharacterization.OLSWeightDecomposition.FinitePartition","a":"Panel/EstimandCharacterization/OLSWeightDecomposition","t":1,"d":"A finite covariate partition for Słoczyński's saturated-OLS weight decomposition, carrying cell probabilities, within-cell treated shares, and within-cell conditional treatment effects, together with the side conditions used by the saturated-OLS algebra: the cell probabilities are nonnegative and sum to one, the treated shares lie in the unit interval, and the overlap denominator Σ π_g p_g (1−p_g) is strictly positive.","s":"(𝒢 : Type u_1) → [Fintype 𝒢] → Type u_1","l":65,"f":"Causalean/Panel/EstimandCharacterization/OLSWeightDecomposition/FinitePartition.lean"},{"n":"Causalean.Panel.EstimandCharacterization.OLSWeightDecomposition.FinitePartition.cellOverlap","k":"def","m":"Causalean.Panel.EstimandCharacterization.OLSWeightDecomposition.FinitePartition","a":"Panel/EstimandCharacterization/OLSWeightDecomposition","t":1,"d":"Within-cell treatment variance p_g (1 − p_g).","s":"{𝒢 : Type u_1} → [inst : Fintype 𝒢] → Causalean.Panel.EstimandCharacterization.OLSWeightDecomposition.FinitePartition 𝒢 → 𝒢 → ℝ","l":91,"f":"Causalean/Panel/EstimandCharacterization/OLSWeightDecomposition/FinitePartition.lean"},{"n":"Causalean.Panel.EstimandCharacterization.OLSWeightDecomposition.FinitePartition.cellOverlap_eq_perWeights_mul","k":"theorem","m":"Causalean.Panel.EstimandCharacterization.OLSWeightDecomposition.FinitePartition","a":"Panel/EstimandCharacterization/OLSWeightDecomposition","t":2,"d":"The cell overlap variance is the product of the two per-observation leverage factors: p_g (1−p_g) = perUntreatedWeight_g · perTreatedWeight_g. This links the finite-cell denominator directly to the paper's per-unit weight interpretation.","s":"∀ {𝒢 : Type u_2} [inst : Fintype 𝒢] (P : Causalean.Panel.EstimandCharacterization.OLSWeightDecomposition.FinitePartition 𝒢) (g : 𝒢), P.cellOverlap g = P.perUntreatedWeight g * P.perTreatedWeight g","l":203,"f":"Causalean/Panel/EstimandCharacterization/OLSWeightDecomposition/FinitePartition.lean"},{"n":"Causalean.Panel.EstimandCharacterization.OLSWeightDecomposition.FinitePartition.equal_groups_collapses","k":"theorem","m":"Causalean.Panel.EstimandCharacterization.OLSWeightDecomposition.FinitePartition","a":"Panel/EstimandCharacterization/OLSWeightDecomposition","t":1,"d":"Equal-cell-share collapse to unweighted average (finite half of prop:po-estimand-sloczynski-ols-equal-groups). If every cell has an equal treated share p_g = 1/2, then the saturated-OLS estimand β_sat equals the probability-weighted average of the cell treatment effects Σ π_g τ_g.","s":"∀ {𝒢 : Type u_1} [inst : Fintype 𝒢] (P : Causalean.Panel.EstimandCharacterization.OLSWeightDecomposition.FinitePartition 𝒢), (∀ (g : 𝒢), P.p g = 1 / 2) → P.overlapWeightedATE = ∑ g, P.π g * P.τ g","l":155,"f":"Causalean/Panel/EstimandCharacterization/OLSWeightDecomposition/FinitePartition.lean"},{"n":"Causalean.Panel.EstimandCharacterization.OLSWeightDecomposition.FinitePartition.equal_groups_weight_eq_pi","k":"theorem","m":"Causalean.Panel.EstimandCharacterization.OLSWeightDecomposition.FinitePartition","a":"Panel/EstimandCharacterization/OLSWeightDecomposition","t":2,"d":"Equal-cell-share weight collapse: if p_g = 1/2 for every cell, the normalized overlap weight reduces to the cell probability π_g.","s":"∀ {𝒢 : Type u_1} [inst : Fintype 𝒢] (P : Causalean.Panel.EstimandCharacterization.OLSWeightDecomposition.FinitePartition 𝒢), (∀ (g : 𝒢), P.p g = 1 / 2) → ∀ (g : 𝒢), P.overlapWeight g = P.π g","l":139,"f":"Causalean/Panel/EstimandCharacterization/OLSWeightDecomposition/FinitePartition.lean"},{"n":"Causalean.Panel.EstimandCharacterization.OLSWeightDecomposition.FinitePartition.finite_weights_eq_sum","k":"theorem","m":"Causalean.Panel.EstimandCharacterization.OLSWeightDecomposition.FinitePartition","a":"Panel/EstimandCharacterization/OLSWeightDecomposition","t":1,"d":"Finite-partition OLS weights (prop:po-estimand-sloczynski-ols-finite-weights). The saturated-OLS estimand equals the sum of normalized overlap weights times cell effects, Σ_g ω_g τ_g.","s":"∀ {𝒢 : Type u_1} [inst : Fintype 𝒢] (P : Causalean.Panel.EstimandCharacterization.OLSWeightDecomposition.FinitePartition 𝒢), P.overlapWeightedATE = ∑ g, P.overlapWeight g * P.τ g","l":130,"f":"Causalean/Panel/EstimandCharacterization/OLSWeightDecomposition/FinitePartition.lean"},{"n":"Causalean.Panel.EstimandCharacterization.OLSWeightDecomposition.FinitePartition.homogeneous_collapses","k":"theorem","m":"Causalean.Panel.EstimandCharacterization.OLSWeightDecomposition.FinitePartition","a":"Panel/EstimandCharacterization/OLSWeightDecomposition","t":1,"d":"Homogeneous-effect collapse (saturated half of prop:po-estimand-sloczynski-ols-homogeneous). If the cell-level treatment effect equals a common constant τ₀ in every cell, then the saturated-OLS estimand β_sat equals τ₀.","s":"∀ {𝒢 : Type u_1} [inst : Fintype 𝒢] (P : Causalean.Panel.EstimandCharacterization.OLSWeightDecomposition.FinitePartition 𝒢) {τ₀ : ℝ}, (∀ (g : 𝒢), P.τ g = τ₀) → P.overlapWeightedATE = τ₀","l":167,"f":"Causalean/Panel/EstimandCharacterization/OLSWeightDecomposition/FinitePartition.lean"},{"n":"Causalean.Panel.EstimandCharacterization.OLSWeightDecomposition.FinitePartition.omega_sum_eq_one","k":"theorem","m":"Causalean.Panel.EstimandCharacterization.OLSWeightDecomposition.FinitePartition","a":"Panel/EstimandCharacterization/OLSWeightDecomposition","t":2,"d":"The overlap weights sum to one. Direct from positivity of the denominator and pulling the common denominator out of the sum.","s":"∀ {𝒢 : Type u_1} [inst : Fintype 𝒢] (P : Causalean.Panel.EstimandCharacterization.OLSWeightDecomposition.FinitePartition 𝒢), ∑ g, P.overlapWeight g = 1","l":122,"f":"Causalean/Panel/EstimandCharacterization/OLSWeightDecomposition/FinitePartition.lean"},{"n":"Causalean.Panel.EstimandCharacterization.OLSWeightDecomposition.FinitePartition.overlapDenominator","k":"def","m":"Causalean.Panel.EstimandCharacterization.OLSWeightDecomposition.FinitePartition","a":"Panel/EstimandCharacterization/OLSWeightDecomposition","t":1,"d":"Denominator Σ_g π_g · p_g (1−p_g) of the saturated-OLS estimand in finite-cell form. Positive by overlap_pos.","s":"{𝒢 : Type u_1} → [inst : Fintype 𝒢] → Causalean.Panel.EstimandCharacterization.OLSWeightDecomposition.FinitePartition 𝒢 → ℝ","l":100,"f":"Causalean/Panel/EstimandCharacterization/OLSWeightDecomposition/FinitePartition.lean"},{"n":"Causalean.Panel.EstimandCharacterization.OLSWeightDecomposition.FinitePartition.overlapDenominator_pos","k":"theorem","m":"Causalean.Panel.EstimandCharacterization.OLSWeightDecomposition.FinitePartition","a":"Panel/EstimandCharacterization/OLSWeightDecomposition","t":2,"d":"The denominator is positive. Restated from overlap_pos.","s":"∀ {𝒢 : Type u_1} [inst : Fintype 𝒢] (P : Causalean.Panel.EstimandCharacterization.OLSWeightDecomposition.FinitePartition 𝒢), 0 < P.overlapDenominator","l":118,"f":"Causalean/Panel/EstimandCharacterization/OLSWeightDecomposition/FinitePartition.lean"},{"n":"Causalean.Panel.EstimandCharacterization.OLSWeightDecomposition.FinitePartition.overlapNumerator","k":"def","m":"Causalean.Panel.EstimandCharacterization.OLSWeightDecomposition.FinitePartition","a":"Panel/EstimandCharacterization/OLSWeightDecomposition","t":1,"d":"Numerator Σ_g π_g · p_g (1−p_g) · τ_g of the saturated-OLS estimand in finite-cell form.","s":"{𝒢 : Type u_1} → [inst : Fintype 𝒢] → Causalean.Panel.EstimandCharacterization.OLSWeightDecomposition.FinitePartition 𝒢 → ℝ","l":95,"f":"Causalean/Panel/EstimandCharacterization/OLSWeightDecomposition/FinitePartition.lean"},{"n":"Causalean.Panel.EstimandCharacterization.OLSWeightDecomposition.FinitePartition.overlapWeight","k":"def","m":"Causalean.Panel.EstimandCharacterization.OLSWeightDecomposition.FinitePartition","a":"Panel/EstimandCharacterization/OLSWeightDecomposition","t":1,"d":"Normalized weight ω_g = π_g p_g (1−p_g) / Σ_h π_h p_h (1−p_h).","s":"{𝒢 : Type u_1} → [inst : Fintype 𝒢] → Causalean.Panel.EstimandCharacterization.OLSWeightDecomposition.FinitePartition 𝒢 → 𝒢 → ℝ","l":105,"f":"Causalean/Panel/EstimandCharacterization/OLSWeightDecomposition/FinitePartition.lean"},{"n":"Causalean.Panel.EstimandCharacterization.OLSWeightDecomposition.FinitePartition.overlapWeightedATE","k":"def","m":"Causalean.Panel.EstimandCharacterization.OLSWeightDecomposition.FinitePartition","a":"Panel/EstimandCharacterization/OLSWeightDecomposition","t":1,"d":"Saturated-OLS estimand in finite-cell form, β_sat = (Σ π·p(1−p)·τ) / (Σ π·p(1−p)).","s":"{𝒢 : Type u_1} → [inst : Fintype 𝒢] → Causalean.Panel.EstimandCharacterization.OLSWeightDecomposition.FinitePartition 𝒢 → ℝ","l":110,"f":"Causalean/Panel/EstimandCharacterization/OLSWeightDecomposition/FinitePartition.lean"},{"n":"Causalean.Panel.EstimandCharacterization.OLSWeightDecomposition.FinitePartition.perTreatedWeight","k":"def","m":"Causalean.Panel.EstimandCharacterization.OLSWeightDecomposition.FinitePartition","a":"Panel/EstimandCharacterization/OLSWeightDecomposition","t":1,"d":"Per-treated-observation leverage factor in cell g.","s":"{𝒢 : Type u_2} → [inst : Fintype 𝒢] → Causalean.Panel.EstimandCharacterization.OLSWeightDecomposition.FinitePartition 𝒢 → 𝒢 → ℝ","l":181,"f":"Causalean/Panel/EstimandCharacterization/OLSWeightDecomposition/FinitePartition.lean"},{"n":"Causalean.Panel.EstimandCharacterization.OLSWeightDecomposition.FinitePartition.perTreatedWeight_antitone","k":"theorem","m":"Causalean.Panel.EstimandCharacterization.OLSWeightDecomposition.FinitePartition","a":"Panel/EstimandCharacterization/OLSWeightDecomposition","t":1,"d":"Headline monotonicity — smaller treated group gets larger per-treated weight (rem:po-estimand-sloczynski-ols-group-size). The per-treated-observation leverage factor 1 − p_g is antitone in the treated share: if cell g has a treated share no larger than cell h's, then g's per-treated-observation leverage factor is at least h's: perTreatedWeight P h ≤ perTreatedWeight P g.","s":"∀ {𝒢 : Type u_2} [inst : Fintype 𝒢] (P : Causalean.Panel.EstimandCharacterization.OLSWeightDecomposition.FinitePartition 𝒢) {g h : 𝒢}, P.p g ≤ P.p h → P.perTreatedWeight h ≤ P.perTreatedWeight g","l":213,"f":"Causalean/Panel/EstimandCharacterization/OLSWeightDecomposition/FinitePartition.lean"},{"n":"Causalean.Panel.EstimandCharacterization.OLSWeightDecomposition.FinitePartition.perUntreatedWeight","k":"def","m":"Causalean.Panel.EstimandCharacterization.OLSWeightDecomposition.FinitePartition","a":"Panel/EstimandCharacterization/OLSWeightDecomposition","t":1,"d":"Per-untreated-observation leverage factor in cell g.","s":"{𝒢 : Type u_2} → [inst : Fintype 𝒢] → Causalean.Panel.EstimandCharacterization.OLSWeightDecomposition.FinitePartition 𝒢 → 𝒢 → ℝ","l":193,"f":"Causalean/Panel/EstimandCharacterization/OLSWeightDecomposition/FinitePartition.lean"},{"n":"Causalean.Panel.EstimandCharacterization.OLSWeightDecomposition.FinitePartition.perUntreatedWeight_monotone","k":"theorem","m":"Causalean.Panel.EstimandCharacterization.OLSWeightDecomposition.FinitePartition","a":"Panel/EstimandCharacterization/OLSWeightDecomposition","t":1,"d":"Monotonicity — larger treated group gives larger per-untreated weight (rem:po-estimand-sloczynski-ols-group-size, untreated side). The per-untreated-observation leverage factor p_g is monotone in the treated share: if cell g has a treated share no larger than cell h's, then h's per-untreated-observation leverage factor is at least g's: perUntreatedWeight P g ≤ perUntreatedWeight P h.","s":"∀ {𝒢 : Type u_2} [inst : Fintype 𝒢] (P : Causalean.Panel.EstimandCharacterization.OLSWeightDecomposition.FinitePartition 𝒢) {g h : 𝒢}, P.p g ≤ P.p h → P.perUntreatedWeight g ≤ P.perUntreatedWeight h","l":235,"f":"Causalean/Panel/EstimandCharacterization/OLSWeightDecomposition/FinitePartition.lean"},{"n":"Causalean.Panel.EstimandCharacterization.OLSWeightDecomposition.OppositeGroupRepr","k":"structure","m":"Causalean.Panel.EstimandCharacterization.OLSWeightDecomposition.FinitePartition","a":"Panel/EstimandCharacterization/OLSWeightDecomposition","t":1,"d":"Słoczyński's two-component ATT/ATU representation (ass:po-estimand-sloczynski-ols-opposite-group).","s":"Type","l":253,"f":"Causalean/Panel/EstimandCharacterization/OLSWeightDecomposition/FinitePartition.lean"},{"n":"Causalean.Panel.EstimandCharacterization.OLSWeightDecomposition.OppositeGroupRepr.equal_groups_eq_ATE","k":"theorem","m":"Causalean.Panel.EstimandCharacterization.OLSWeightDecomposition.FinitePartition","a":"Panel/EstimandCharacterization/OLSWeightDecomposition","t":1,"d":"Equal-group-size collapse to the ATE (top half of prop:po-estimand-sloczynski-ols-equal-groups). If the treated share ρ equals one half, then the opposite-group OLS coefficient β_ols equals the ATE τ_ATE = ρ·τ_ATT + (1−ρ)·τ_ATU.","s":"∀ (R : Causalean.Panel.EstimandCharacterization.OLSWeightDecomposition.OppositeGroupRepr), R.ρ = 1 / 2 → R.β_ols = R.tau_ATE","l":324,"f":"Causalean/Panel/EstimandCharacterization/OLSWeightDecomposition/FinitePartition.lean"},{"n":"Causalean.Panel.EstimandCharacterization.OLSWeightDecomposition.OppositeGroupRepr.homogeneous_eq_constant","k":"theorem","m":"Causalean.Panel.EstimandCharacterization.OLSWeightDecomposition.FinitePartition","a":"Panel/EstimandCharacterization/OLSWeightDecomposition","t":1,"d":"Homogeneous-effect collapse for the opposite-group representation (ATT/ATU half of prop:po-estimand-sloczynski-ols-homogeneous). If the ATT equals a common constant τ₀ and the ATU equals that same constant τ₀, then the opposite-group OLS coefficient β_ols equals τ₀, regardless of the treated share ρ.","s":"∀ (R : Causalean.Panel.EstimandCharacterization.OLSWeightDecomposition.OppositeGroupRepr) {τ₀ : ℝ}, R.τ_ATT = τ₀ → R.τ_ATU = τ₀ → R.β_ols = τ₀","l":332,"f":"Causalean/Panel/EstimandCharacterization/OLSWeightDecomposition/FinitePartition.lean"},{"n":"Causalean.Panel.EstimandCharacterization.OLSWeightDecomposition.OppositeGroupRepr.represents","k":"theorem","m":"Causalean.Panel.EstimandCharacterization.OLSWeightDecomposition.FinitePartition","a":"Panel/EstimandCharacterization/OLSWeightDecomposition","t":1,"d":"Słoczyński opposite-group identity (prop:po-estimand-sloczynski-ols-opposite-group) — derived rather than assumed. Substituting the equal-dispersion weights into the two-component representation shows that the OLS coefficient equals (1−ρ)·τ_ATT + ρ·τ_ATU: the treated-group effect receives the untreated share as its weight, and vice versa.","s":"∀ (R : Causalean.Panel.EstimandCharacterization.OLSWeightDecomposition.OppositeGroupRepr), R.β_ols = (1 - R.ρ) * R.τ_ATT + R.ρ * R.τ_ATU","l":317,"f":"Causalean/Panel/EstimandCharacterization/OLSWeightDecomposition/FinitePartition.lean"},{"n":"Causalean.Panel.EstimandCharacterization.OLSWeightDecomposition.OppositeGroupRepr.tau_ATE","k":"def","m":"Causalean.Panel.EstimandCharacterization.OLSWeightDecomposition.FinitePartition","a":"Panel/EstimandCharacterization/OLSWeightDecomposition","t":1,"d":"ATE expressed as ρ τ_ATT + (1−ρ) τ_ATU (def:po-estimand-sloczynski-ols-att-atu).","s":"Causalean.Panel.EstimandCharacterization.OLSWeightDecomposition.OppositeGroupRepr → ℝ","l":305,"f":"Causalean/Panel/EstimandCharacterization/OLSWeightDecomposition/FinitePartition.lean"},{"n":"Causalean.Panel.EstimandCharacterization.OLSWeightDecomposition.OppositeGroupRepr.w0_eq_rho","k":"theorem","m":"Causalean.Panel.EstimandCharacterization.OLSWeightDecomposition.FinitePartition","a":"Panel/EstimandCharacterization/OLSWeightDecomposition","t":2,"d":"The untreated weight in the equal-dispersion case is the treated share ρ, forced by w₁ + w₀ = 1 and w₁ = 1 − ρ.","s":"∀ (R : Causalean.Panel.EstimandCharacterization.OLSWeightDecomposition.OppositeGroupRepr), R.w₀ = R.ρ","l":311,"f":"Causalean/Panel/EstimandCharacterization/OLSWeightDecomposition/FinitePartition.lean"},{"n":"Causalean.Panel.EstimandCharacterization.OLSWeightDecomposition.Y_memLp_of_consistency","k":"theorem","m":"Causalean.Panel.EstimandCharacterization.OLSWeightDecomposition.Support.Orthogonality","a":"Panel/EstimandCharacterization/OLSWeightDecomposition/Support","t":2,"d":"The observed outcome Y = D · Y1 + (1 − D) · Y0 is in MemLp 2 μ under consistency, binary treatment, and MemLp 2 μ assumptions for both potential outcomes. The pointwise bound |Y| ≤ |Y0| + |Y1| (a.e.) plus closure of MemLp 2 under sums gives the result. Stated separately because the bridge theorem takes Y_memLp as a hypothesis but downstream consumers (e.g. tests) may need to derive it.","s":"∀ {Ω : Type u_1} [inst : MeasurableSpace Ω] (μ : MeasureTheory.Measure Ω) {D Y Y0 Y1 : Ω → ℝ}, Measurable D → (∀ᵐ (ω : Ω) ∂μ, D ω = 0 ∨ D ω = 1) → MeasureTheory.MemLp Y0 2 μ → MeasureTheory.MemLp Y1 2 μ → (Y =ᵐ[μ] fun ω => D ω * Y1 ω + (1 - D ω) * Y0 ω) → MeasureTheory.MemLp Y 2 μ","l":139,"f":"Causalean/Panel/EstimandCharacterization/OLSWeightDecomposition/Support/Orthogonality.lean"},{"n":"Causalean.Panel.EstimandCharacterization.OLSWeightDecomposition.bridge_finite_residualized_eq_overlap","k":"theorem","m":"Causalean.Panel.EstimandCharacterization.OLSWeightDecomposition.OverlapWeightedATE","a":"Panel/EstimandCharacterization/OLSWeightDecomposition","t":1,"d":"Finite-cell saturated-OLS bridge to the overlap-weighted average treatment effect. For a probability space (Ω, μ) with a measurable binary treatment D, an outcome Y, square-integrable potential outcomes Y0, Y1, and a measurable finite covariate G, suppose treatment is binary almost everywhere, the observed outcome is consistent — Y equals D·Y1 + (1−D)·Y0 almost everywhere, the integrated-indicator finite-cell bridge condition holds, a finite-cell operational substitute for full conditional-mean independence of the potential outcomes given treatment and the covariate, and the covariate cells have nondegenerate treatment overlap. Then the saturated-OLS residualized coefficient of Y on D given G equals the overlap-weighted average treatment effect Σ_g ω_g · E[Y(1)−Y(0) ∣ G=g] built from the potential outcomes Y0, Y1:","s":"∀ {Ω : Type u_1} {𝒢 : Type u_2} [inst : MeasurableSpace Ω] [inst_1 : Fintype 𝒢] [inst_2 : DecidableEq 𝒢] [inst_3 : MeasurableSpace 𝒢] [inst_4 : MeasurableSingletonClass 𝒢] (μ : MeasureTheory.Measure Ω) [inst_5 : MeasureTheory.IsProbabilityMeasure μ] (D Y Y0 Y1 : Ω → ℝ) (G : Ω → 𝒢) (B_meas_G : Measurable G) (B_meas_D : Measurable D) (B_binary : ∀ᵐ (ω : Ω) ∂μ, D ω = 0 ∨ D ω = 1) (B_Y0_L2 : MeasureTheory.MemLp Y0 2 μ) (B_Y1_L2 : MeasureTheory.MemLp Y1 2 μ) (B_consis : Y =ᵐ[μ] fun ω => D ω * Y1 ω + (1 - D ω) * Y0 ω), (∀ (d : ℝ) (g : 𝒢), d = 0 ∨ d = 1 → ∫ (ω : Ω), (if d = 1 then Y1 ω else Y0 ω) * {ω' | D ω' = d}.indicator (fun x => 1) ω * {ω' | G ω' = g}.indicator (fun x => 1) ω ∂μ = (∫ (ω : Ω), (if d = 1 then Y1 ω else Y0 ω) * {ω' | G ω' = g}.indicator (fun x => 1) ω ∂μ) * if d = 1 then Causalean.Panel.EstimandCharacterization.OLSWeightDecomposition.cellShare μ D G g else 1 - Causalean.Panel.EstimandCharacterization.OLSWeightDecomposition.cellShare μ D G g) → ∀ (B_overlap : 0 < ∑ g, Causalean.Panel.EstimandCharacterization.OLSWeightDecomposition.cellMass μ G g * (Causalean.Panel.EstimandCharacterization.OLSWeightDecomposition.cellShare μ D G g * (1 - Causalean.Panel.EstimandCharacterization.OLSWeightDecomposition.cellShare μ D G g))), Causalean.Panel.residualizedCoefficient μ (Causalean.Panel.EstimandCharacterization.OLSWeightDecomposition.saturatedClass μ G B_meas_G) (Causalean.Panel.EstimandCharacterization.OLSWeightDecomposition.residWitnessY μ Y G B_meas_G (Causalean.Panel.EstimandCharacterization.OLSWeightDecomposition.Y_memLp_of_consistency μ B_meas_D B_binary B_Y0_L2 B_Y1_L2 B_consis)) (Causalean.Panel.EstimandCharacterization.OLSWeightDecomposition.residWitnessD μ D G B_meas_G B_meas_D B_binary) = (Causalean.Panel.EstimandCharacterization.OLSWeightDecomposition.partitionOf μ D Y0 Y1 G B_meas_G B_binary B_overlap).overlapWeightedATE","l":37,"f":"Causalean/Panel/EstimandCharacterization/OLSWeightDecomposition/OverlapWeightedATE.lean"},{"n":"Causalean.Panel.EstimandCharacterization.OLSWeightDecomposition.cellMass","k":"def","m":"Causalean.Panel.EstimandCharacterization.OLSWeightDecomposition.Support.Basic","a":"Panel/EstimandCharacterization/OLSWeightDecomposition/Support","t":1,"d":"Cell mass (μ {G = g}).toReal, the probability weight of covariate cell g.","s":"{Ω : Type u_1} → {𝒢 : Type u_2} → [inst : MeasurableSpace Ω] → MeasureTheory.Measure Ω → (Ω → 𝒢) → 𝒢 → ℝ","l":52,"f":"Causalean/Panel/EstimandCharacterization/OLSWeightDecomposition/Support/Basic.lean"},{"n":"Causalean.Panel.EstimandCharacterization.OLSWeightDecomposition.cellMass_nonneg","k":"theorem","m":"Causalean.Panel.EstimandCharacterization.OLSWeightDecomposition.Support.Basic","a":"Panel/EstimandCharacterization/OLSWeightDecomposition/Support","t":2,"d":"Cell mass is nonnegative — (μ S).toReal ≥ 0.","s":"∀ {Ω : Type u_1} {𝒢 : Type u_2} [inst : MeasurableSpace Ω] (μ : MeasureTheory.Measure Ω) (G : Ω → 𝒢) (g : 𝒢), 0 ≤ Causalean.Panel.EstimandCharacterization.OLSWeightDecomposition.cellMass μ G g","l":140,"f":"Causalean/Panel/EstimandCharacterization/OLSWeightDecomposition/Support/Basic.lean"},{"n":"Causalean.Panel.EstimandCharacterization.OLSWeightDecomposition.cellMass_sum_eq_one","k":"theorem","m":"Causalean.Panel.EstimandCharacterization.OLSWeightDecomposition.Support.Basic","a":"Panel/EstimandCharacterization/OLSWeightDecomposition/Support","t":2,"d":"Cell masses sum to 1 under IsProbabilityMeasure μ and a measurable G valued in a finite type with MeasurableSingletonClass. The {G = g} family partitions Ω (up to μ-null sets) and the μ-mass of each is finite. This supplies the π_sum_one field for the finite-partition algebra.","s":"∀ {Ω : Type u_1} {𝒢 : Type u_2} [inst : MeasurableSpace Ω] [inst_1 : Fintype 𝒢] [inst_2 : MeasurableSpace 𝒢] [MeasurableSingletonClass 𝒢] (μ : MeasureTheory.Measure Ω) [MeasureTheory.IsProbabilityMeasure μ] (G : Ω → 𝒢), Measurable G → ∑ g, Causalean.Panel.EstimandCharacterization.OLSWeightDecomposition.cellMass μ G g = 1","l":146,"f":"Causalean/Panel/EstimandCharacterization/OLSWeightDecomposition/Support/Basic.lean"},{"n":"Causalean.Panel.EstimandCharacterization.OLSWeightDecomposition.cellMean_eq_eventCondExp","k":"theorem","m":"Causalean.Panel.EstimandCharacterization.OLSWeightDecomposition.Support.Basic","a":"Panel/EstimandCharacterization/OLSWeightDecomposition/Support","t":1,"d":"For an ambient probability space and covariate space, a probability measure μ on Ω, an integrand F, a measurable covariate map G, and a covariate cell g, the CellBridge indicator-weighted cell mean of F on {G = g} equals the shared event-level conditional expectation of F given {G = g}.","s":"∀ {Ω : Type u_1} {𝒢 : Type u_2} [inst : MeasurableSpace Ω] [inst_1 : MeasurableSpace 𝒢] [MeasurableSingletonClass 𝒢] (μ : MeasureTheory.Measure Ω) (F : Ω → ℝ) (G : Ω → 𝒢), Measurable G → ∀ (g : 𝒢), Causalean.Panel.CellBridge.cellMean μ F G g = Causalean.PO.eventCondExp μ {ω | G ω = g} F","l":83,"f":"Causalean/Panel/EstimandCharacterization/OLSWeightDecomposition/Support/Basic.lean"},{"n":"Causalean.Panel.EstimandCharacterization.OLSWeightDecomposition.cellShare","k":"def","m":"Causalean.Panel.EstimandCharacterization.OLSWeightDecomposition.Support.Basic","a":"Panel/EstimandCharacterization/OLSWeightDecomposition/Support","t":1,"d":"Cell-wise treated share, defined as the indicator-weighted integral of D on covariate cell g, divided by cellMass μ G g. On zero-mass cells, the value is 0 by Mathlib's 0/0 = 0 convention.","s":"{Ω : Type u_1} → {𝒢 : Type u_2} → [inst : MeasurableSpace Ω] → MeasureTheory.Measure Ω → (Ω → ℝ) → (Ω → 𝒢) → 𝒢 → ℝ","l":57,"f":"Causalean/Panel/EstimandCharacterization/OLSWeightDecomposition/Support/Basic.lean"},{"n":"Causalean.Panel.EstimandCharacterization.OLSWeightDecomposition.cellShare_eq_eventCondExp","k":"theorem","m":"Causalean.Panel.EstimandCharacterization.OLSWeightDecomposition.Support.Basic","a":"Panel/EstimandCharacterization/OLSWeightDecomposition/Support","t":2,"d":"OLS treated share is a shared population cell mean. cellShare μ D G g = E[D | G = g] in the shared eventCondExp operator.","s":"∀ {Ω : Type u_1} {𝒢 : Type u_2} [inst : MeasurableSpace Ω] [inst_1 : MeasurableSpace 𝒢] [MeasurableSingletonClass 𝒢] (μ : MeasureTheory.Measure Ω) (D : Ω → ℝ) (G : Ω → 𝒢), Measurable G → ∀ (g : 𝒢), Causalean.Panel.EstimandCharacterization.OLSWeightDecomposition.cellShare μ D G g = Causalean.PO.eventCondExp μ {ω | G ω = g} D","l":101,"f":"Causalean/Panel/EstimandCharacterization/OLSWeightDecomposition/Support/Basic.lean"},{"n":"Causalean.Panel.EstimandCharacterization.OLSWeightDecomposition.cellShare_le_one","k":"theorem","m":"Causalean.Panel.EstimandCharacterization.OLSWeightDecomposition.Support.Basic","a":"Panel/EstimandCharacterization/OLSWeightDecomposition/Support","t":2,"d":"Cell-wise treated share is at most 1 when D lies a.e. between zero and one. The integrand D · 𝟙{G = g} ≤ 𝟙{G = g} a.e., so the numerator is at most cellMass μ G g.","s":"∀ {Ω : Type u_1} {𝒢 : Type u_2} [inst : MeasurableSpace Ω] [inst_1 : MeasurableSpace 𝒢] [MeasurableSingletonClass 𝒢] (μ : MeasureTheory.Measure Ω) [MeasureTheory.IsFiniteMeasure μ] (D : Ω → ℝ) (G : Ω → 𝒢), Measurable G → (∀ᵐ (ω : Ω) ∂μ, 0 ≤ D ω ∧ D ω ≤ 1) → ∀ (g : 𝒢), Causalean.Panel.EstimandCharacterization.OLSWeightDecomposition.cellShare μ D G g ≤ 1","l":184,"f":"Causalean/Panel/EstimandCharacterization/OLSWeightDecomposition/Support/Basic.lean"},{"n":"Causalean.Panel.EstimandCharacterization.OLSWeightDecomposition.cellShare_nonneg","k":"theorem","m":"Causalean.Panel.EstimandCharacterization.OLSWeightDecomposition.Support.Basic","a":"Panel/EstimandCharacterization/OLSWeightDecomposition/Support","t":2,"d":"Cell-wise treated share is nonnegative when D is a.e. nonnegative.","s":"∀ {Ω : Type u_1} {𝒢 : Type u_2} [inst : MeasurableSpace Ω] (μ : MeasureTheory.Measure Ω) (D : Ω → ℝ) (G : Ω → 𝒢), (∀ᵐ (ω : Ω) ∂μ, 0 ≤ D ω) → ∀ (g : 𝒢), 0 ≤ Causalean.Panel.EstimandCharacterization.OLSWeightDecomposition.cellShare μ D G g","l":171,"f":"Causalean/Panel/EstimandCharacterization/OLSWeightDecomposition/Support/Basic.lean"},{"n":"Causalean.Panel.EstimandCharacterization.OLSWeightDecomposition.cellTau","k":"def","m":"Causalean.Panel.EstimandCharacterization.OLSWeightDecomposition.Support.Basic","a":"Panel/EstimandCharacterization/OLSWeightDecomposition/Support","t":1,"d":"Cell-wise treatment effect E[Y(1) − Y(0) | G = g] in the shared indicator-weighted cell-mean convention. Like cellShare, zero-mass cells collapse to 0.","s":"{Ω : Type u_1} → {𝒢 : Type u_2} → [inst : MeasurableSpace Ω] → MeasureTheory.Measure Ω → (Ω → ℝ) → (Ω → ℝ) → (Ω → 𝒢) → 𝒢 → ℝ","l":64,"f":"Causalean/Panel/EstimandCharacterization/OLSWeightDecomposition/Support/Basic.lean"},{"n":"Causalean.Panel.EstimandCharacterization.OLSWeightDecomposition.cellTau_eq_eventCondExp","k":"theorem","m":"Causalean.Panel.EstimandCharacterization.OLSWeightDecomposition.Support.Basic","a":"Panel/EstimandCharacterization/OLSWeightDecomposition/Support","t":2,"d":"OLS cell treatment effect is a shared population cell mean. cellTau μ Y0 Y1 G g = E[Y(1) − Y(0) | G = g] in the shared eventCondExp operator — a genuine potential-outcome contrast.","s":"∀ {Ω : Type u_1} {𝒢 : Type u_2} [inst : MeasurableSpace Ω] [inst_1 : MeasurableSpace 𝒢] [MeasurableSingletonClass 𝒢] (μ : MeasureTheory.Measure Ω) (Y0 Y1 : Ω → ℝ) (G : Ω → 𝒢), Measurable G → ∀ (g : 𝒢), Causalean.Panel.EstimandCharacterization.OLSWeightDecomposition.cellTau μ Y0 Y1 G g = Causalean.PO.eventCondExp μ {ω | G ω = g} fun ω => Y1 ω - Y0 ω","l":109,"f":"Causalean/Panel/EstimandCharacterization/OLSWeightDecomposition/Support/Basic.lean"},{"n":"Causalean.Panel.EstimandCharacterization.OLSWeightDecomposition.cellTau_mul_cellMass","k":"theorem","m":"Causalean.Panel.EstimandCharacterization.OLSWeightDecomposition.Support.Integrals","a":"Panel/EstimandCharacterization/OLSWeightDecomposition/Support","t":2,"d":"Cell-effect numerator divided by cell mass and multiplied back recovers the raw effect numerator.","s":"∀ {Ω : Type u_1} {𝒢 : Type u_2} [inst : MeasurableSpace Ω] (μ : MeasureTheory.Measure Ω) [MeasureTheory.IsFiniteMeasure μ] (Y0 Y1 : Ω → ℝ) (G : Ω → 𝒢) (g : 𝒢), Causalean.Panel.EstimandCharacterization.OLSWeightDecomposition.cellTau μ Y0 Y1 G g * Causalean.Panel.EstimandCharacterization.OLSWeightDecomposition.cellMass μ G g = ∫ (ω : Ω), (Y1 ω - Y0 ω) * {ω' | G ω' = g}.indicator (fun x => 1) ω ∂μ","l":65,"f":"Causalean/Panel/EstimandCharacterization/OLSWeightDecomposition/Support/Integrals.lean"},{"n":"Causalean.Panel.EstimandCharacterization.OLSWeightDecomposition.cell_integral_div_mul_cellMass","k":"theorem","m":"Causalean.Panel.EstimandCharacterization.OLSWeightDecomposition.Support.Integrals","a":"Panel/EstimandCharacterization/OLSWeightDecomposition/Support","t":2,"d":"Dividing an indicator-weighted cell integral by a nonzero cell mass and multiplying back recovers the numerator; on zero-mass cells the numerator is zero because the indicator is a.e. zero.","s":"∀ {Ω : Type u_1} {𝒢 : Type u_2} [inst : MeasurableSpace Ω] (μ : MeasureTheory.Measure Ω) [MeasureTheory.IsFiniteMeasure μ] (F : Ω → ℝ) (G : Ω → 𝒢) (g : 𝒢), (∫ (ω : Ω), F ω * {ω' | G ω' = g}.indicator (fun x => 1) ω ∂μ) / Causalean.Panel.EstimandCharacterization.OLSWeightDecomposition.cellMass μ G g * Causalean.Panel.EstimandCharacterization.OLSWeightDecomposition.cellMass μ G g = ∫ (ω : Ω), F ω * {ω' | G ω' = g}.indicator (fun x => 1) ω ∂μ","l":52,"f":"Causalean/Panel/EstimandCharacterization/OLSWeightDecomposition/Support/Integrals.lean"},{"n":"Causalean.Panel.EstimandCharacterization.OLSWeightDecomposition.condExp_D_bounds","k":"theorem","m":"Causalean.Panel.EstimandCharacterization.OLSWeightDecomposition.GeneralCondExp","a":"Panel/EstimandCharacterization/OLSWeightDecomposition","t":2,"d":"A real-valued variable that is zero or one almost surely has conditional expectation, given a sub-σ-algebra, between zero and one almost surely.","s":"∀ {Ω : Type u_1} {m mΩ : MeasurableSpace Ω} (μ : MeasureTheory.Measure Ω) [MeasureTheory.IsFiniteMeasure μ], m ≤ mΩ → ∀ {D : Ω → ℝ}, MeasureTheory.Integrable D μ → (∀ᵐ (ω : Ω) ∂μ, D ω = 0 ∨ D ω = 1) → (∀ᵐ (ω : Ω) ∂μ, 0 ≤ μ[D | m] ω) ∧ ∀ᵐ (ω : Ω) ∂μ, μ[D | m] ω ≤ 1","l":84,"f":"Causalean/Panel/EstimandCharacterization/OLSWeightDecomposition/GeneralCondExp.lean"},{"n":"Causalean.Panel.EstimandCharacterization.OLSWeightDecomposition.condExp_D_norm_le","k":"theorem","m":"Causalean.Panel.EstimandCharacterization.OLSWeightDecomposition.GeneralCondExp","a":"Panel/EstimandCharacterization/OLSWeightDecomposition","t":2,"d":"The conditional expectation of a real-valued variable that is zero or one almost surely has absolute value at most one almost surely.","s":"∀ {Ω : Type u_1} {m mΩ : MeasurableSpace Ω} (μ : MeasureTheory.Measure Ω) [MeasureTheory.IsFiniteMeasure μ], m ≤ mΩ → ∀ {D : Ω → ℝ}, MeasureTheory.Integrable D μ → (∀ᵐ (ω : Ω) ∂μ, D ω = 0 ∨ D ω = 1) → ∀ᵐ (ω : Ω) ∂μ, ‖μ[D | m] ω‖ ≤ 1","l":99,"f":"Causalean/Panel/EstimandCharacterization/OLSWeightDecomposition/GeneralCondExp.lean"},{"n":"Causalean.Panel.EstimandCharacterization.OLSWeightDecomposition.condExp_one_sub","k":"theorem","m":"Causalean.Panel.EstimandCharacterization.OLSWeightDecomposition.GeneralCondExp","a":"Panel/EstimandCharacterization/OLSWeightDecomposition","t":2,"d":"The conditional expectation of one minus an integrable real-valued variable, given a sub-σ-algebra, equals one minus that variable's conditional expectation almost surely.","s":"∀ {Ω : Type u_1} {m mΩ : MeasurableSpace Ω} (μ : MeasureTheory.Measure Ω) [MeasureTheory.IsFiniteMeasure μ], m ≤ mΩ → ∀ {D : Ω → ℝ}, MeasureTheory.Integrable D μ → μ[fun ω => 1 - D ω | m] =ᵐ[μ] fun ω => 1 - μ[D | m] ω","l":71,"f":"Causalean/Panel/EstimandCharacterization/OLSWeightDecomposition/GeneralCondExp.lean"},{"n":"Causalean.Panel.EstimandCharacterization.OLSWeightDecomposition.denom_identity","k":"theorem","m":"Causalean.Panel.EstimandCharacterization.OLSWeightDecomposition.GeneralCondExp","a":"Panel/EstimandCharacterization/OLSWeightDecomposition","t":2,"d":"Denominator identity (A). For binary D, ∫ (D − μ[D | m])² ∂μ = ∫ μ[D | m]·(1 − μ[D | m]) ∂μ.","s":"∀ {Ω : Type u_1} {m mΩ : MeasurableSpace Ω} (μ : MeasureTheory.Measure Ω) [MeasureTheory.IsFiniteMeasure μ], m ≤ mΩ → ∀ (D : Ω → ℝ), (∀ᵐ (ω : Ω) ∂μ, D ω = 0 ∨ D ω = 1) → MeasureTheory.Integrable D μ → ∫ (ω : Ω), (D ω - μ[D | m] ω) ^ 2 ∂μ = ∫ (ω : Ω), μ[D | m] ω * (1 - μ[D | m] ω) ∂μ","l":108,"f":"Causalean/Panel/EstimandCharacterization/OLSWeightDecomposition/GeneralCondExp.lean"},{"n":"Causalean.Panel.EstimandCharacterization.OLSWeightDecomposition.denom_per_cell","k":"theorem","m":"Causalean.Panel.EstimandCharacterization.OLSWeightDecomposition.Support.PerCell","a":"Panel/EstimandCharacterization/OLSWeightDecomposition/Support","t":2,"d":"Per-cell denominator identity. On the cell {G = g}, (D − propensity)(ω) = D ω − cellShare μ D G g; squaring and using D ∈ {0, 1} plus the defining identity of cellShare gives","s":"∀ {Ω : Type u_1} {𝒢 : Type u_2} [inst : MeasurableSpace Ω] [inst_1 : Fintype 𝒢] [DecidableEq 𝒢] [inst_3 : MeasurableSpace 𝒢] [MeasurableSingletonClass 𝒢] (μ : MeasureTheory.Measure Ω) [MeasureTheory.IsFiniteMeasure μ] (D : Ω → ℝ) (G : Ω → 𝒢), Measurable G → Measurable D → (∀ᵐ (ω : Ω) ∂μ, D ω = 0 ∨ D ω = 1) → ∀ (g : 𝒢), ∫ (ω : Ω), (D ω - Causalean.Panel.EstimandCharacterization.OLSWeightDecomposition.propensity μ D G ω) * (D ω - Causalean.Panel.EstimandCharacterization.OLSWeightDecomposition.propensity μ D G ω) * {ω' | G ω' = g}.indicator (fun x => 1) ω ∂μ = Causalean.Panel.EstimandCharacterization.OLSWeightDecomposition.cellMass μ G g * (Causalean.Panel.EstimandCharacterization.OLSWeightDecomposition.cellShare μ D G g * (1 - Causalean.Panel.EstimandCharacterization.OLSWeightDecomposition.cellShare μ D G g))","l":31,"f":"Causalean/Panel/EstimandCharacterization/OLSWeightDecomposition/Support/PerCell.lean"},{"n":"Causalean.Panel.EstimandCharacterization.OLSWeightDecomposition.indicator_cell_memLp","k":"theorem","m":"Causalean.Panel.EstimandCharacterization.OLSWeightDecomposition.Support.Integrals","a":"Panel/EstimandCharacterization/OLSWeightDecomposition/Support","t":2,"d":"Indicator helper: Set.indicator {G = g} 1 is in MemLp 2 μ for a finite measure (it is bounded by 1 and finite measure ⇒ MemLp p for every p). Stated as a separate lemma because it appears repeatedly in orthogonality and per-cell calculations.","s":"∀ {Ω : Type u_1} {𝒢 : Type u_2} [inst : MeasurableSpace Ω] [inst_1 : MeasurableSpace 𝒢] [MeasurableSingletonClass 𝒢] (μ : MeasureTheory.Measure Ω) [MeasureTheory.IsFiniteMeasure μ] (G : Ω → 𝒢), Measurable G → ∀ (g : 𝒢), MeasureTheory.MemLp (fun ω => {ω' | G ω' = g}.indicator (fun x => 1) ω) 2 μ","l":32,"f":"Causalean/Panel/EstimandCharacterization/OLSWeightDecomposition/Support/Integrals.lean"},{"n":"Causalean.Panel.EstimandCharacterization.OLSWeightDecomposition.integrable_mul_indicator_D_G","k":"theorem","m":"Causalean.Panel.EstimandCharacterization.OLSWeightDecomposition.Support.Integrals","a":"Panel/EstimandCharacterization/OLSWeightDecomposition/Support","t":2,"d":"Product of two singleton indicators against an L² function is integrable under a finite measure.","s":"∀ {Ω : Type u_1} {𝒢 : Type u_2} [inst : MeasurableSpace Ω] [inst_1 : MeasurableSpace 𝒢] [MeasurableSingletonClass 𝒢] (μ : MeasureTheory.Measure Ω) [MeasureTheory.IsFiniteMeasure μ] (F D : Ω → ℝ) (G : Ω → 𝒢), Measurable D → Measurable G → MeasureTheory.MemLp F 2 μ → ∀ (d : ℝ) (g : 𝒢), MeasureTheory.Integrable (fun ω => F ω * {ω' | D ω' = d}.indicator (fun x => 1) ω * {ω' | G ω' = g}.indicator (fun x => 1) ω) μ","l":76,"f":"Causalean/Panel/EstimandCharacterization/OLSWeightDecomposition/Support/Integrals.lean"},{"n":"Causalean.Panel.EstimandCharacterization.OLSWeightDecomposition.integral_cell_indicator_one_eq_cellMass","k":"theorem","m":"Causalean.Panel.EstimandCharacterization.OLSWeightDecomposition.Support.Integrals","a":"Panel/EstimandCharacterization/OLSWeightDecomposition/Support","t":2,"d":"Cell indicators integrate to the corresponding real cell mass.","s":"∀ {Ω : Type u_1} {𝒢 : Type u_2} [inst : MeasurableSpace Ω] [inst_1 : MeasurableSpace 𝒢] [MeasurableSingletonClass 𝒢] (μ : MeasureTheory.Measure Ω) (G : Ω → 𝒢), Measurable G → ∀ (g : 𝒢), ∫ (ω : Ω), {ω' | G ω' = g}.indicator (fun x => 1) ω ∂μ = Causalean.Panel.EstimandCharacterization.OLSWeightDecomposition.cellMass μ G g","l":43,"f":"Causalean/Panel/EstimandCharacterization/OLSWeightDecomposition/Support/Integrals.lean"},{"n":"Causalean.Panel.EstimandCharacterization.OLSWeightDecomposition.integral_eq_sum_cell","k":"theorem","m":"Causalean.Panel.EstimandCharacterization.OLSWeightDecomposition.Support.Integrals","a":"Panel/EstimandCharacterization/OLSWeightDecomposition/Support","t":1,"d":"For an ambient probability space and finite covariate space, a measure μ, a measurable covariate map G, and an integrable function F, the integral of F against μ equals the sum, over covariate cells g, of the integral of F restricted to the cell {G = g}.","s":"∀ {Ω : Type u_1} {𝒢 : Type u_2} [inst : MeasurableSpace Ω] [inst_1 : Fintype 𝒢] [inst_2 : MeasurableSpace 𝒢] [MeasurableSingletonClass 𝒢] (μ : MeasureTheory.Measure Ω) (F : Ω → ℝ) (G : Ω → 𝒢), Measurable G → MeasureTheory.Integrable F μ → ∫ (ω : Ω), F ω ∂μ = ∑ g, ∫ (ω : Ω), F ω * {ω' | G ω' = g}.indicator (fun x => 1) ω ∂μ","l":162,"f":"Causalean/Panel/EstimandCharacterization/OLSWeightDecomposition/Support/Integrals.lean"},{"n":"Causalean.Panel.EstimandCharacterization.OLSWeightDecomposition.integral_mul_saturated_eq_zero_of_cell","k":"theorem","m":"Causalean.Panel.EstimandCharacterization.OLSWeightDecomposition.Support.Integrals","a":"Panel/EstimandCharacterization/OLSWeightDecomposition/Support","t":2,"d":"If a square-integrable residual is orthogonal to every cell indicator, it is orthogonal to every saturated finite linear combination.","s":"∀ {Ω : Type u_1} {𝒢 : Type u_2} [inst : MeasurableSpace Ω] [inst_1 : Fintype 𝒢] [inst_2 : MeasurableSpace 𝒢] [MeasurableSingletonClass 𝒢] (μ : MeasureTheory.Measure Ω) [MeasureTheory.IsFiniteMeasure μ] (V : Ω → ℝ) (G : Ω → 𝒢), Measurable G → MeasureTheory.MemLp V 2 μ → ∀ (c : 𝒢 → ℝ), (∀ (g : 𝒢), ∫ (ω : Ω), V ω * {ω' | G ω' = g}.indicator (fun x => 1) ω ∂μ = 0) → ∫ (ω : Ω), V ω * ∑ g, c g * {ω' | G ω' = g}.indicator (fun x => 1) ω ∂μ = 0","l":144,"f":"Causalean/Panel/EstimandCharacterization/OLSWeightDecomposition/Support/Integrals.lean"},{"n":"Causalean.Panel.EstimandCharacterization.OLSWeightDecomposition.meanReg","k":"def","m":"Causalean.Panel.EstimandCharacterization.OLSWeightDecomposition.Support.Basic","a":"Panel/EstimandCharacterization/OLSWeightDecomposition/Support","t":1,"d":"Saturated mean regression meanReg μ Y G ω is the cell-wise average of Y, evaluated at G ω. Plays the role of m(G(·)) for the outcome Y. Lies in saturatedClass μ G G_meas.","s":"{Ω : Type u_1} → {𝒢 : Type u_2} → [inst : MeasurableSpace Ω] → [Fintype 𝒢] → MeasureTheory.Measure Ω → (Ω → ℝ) → (Ω → 𝒢) → Ω → ℝ","l":127,"f":"Causalean/Panel/EstimandCharacterization/OLSWeightDecomposition/Support/Basic.lean"},{"n":"Causalean.Panel.EstimandCharacterization.OLSWeightDecomposition.meanReg_eq_cellMean_of_mem","k":"theorem","m":"Causalean.Panel.EstimandCharacterization.OLSWeightDecomposition.Support.Integrals","a":"Panel/EstimandCharacterization/OLSWeightDecomposition/Support","t":2,"d":"On its own cell, the saturated mean-regression representative equals the corresponding cell mean.","s":"∀ {Ω : Type u_1} {𝒢 : Type u_2} [inst : MeasurableSpace Ω] [inst_1 : Fintype 𝒢] (μ : MeasureTheory.Measure Ω) (Y : Ω → ℝ) (G : Ω → 𝒢) {g : 𝒢} {ω : Ω}, G ω = g → Causalean.Panel.EstimandCharacterization.OLSWeightDecomposition.meanReg μ Y G ω = (∫ (ω' : Ω), Y ω' * {ω' | G ω' = g}.indicator (fun x => 1) ω' ∂μ) / Causalean.Panel.EstimandCharacterization.OLSWeightDecomposition.cellMass μ G g","l":125,"f":"Causalean/Panel/EstimandCharacterization/OLSWeightDecomposition/Support/Integrals.lean"},{"n":"Causalean.Panel.EstimandCharacterization.OLSWeightDecomposition.meanReg_mem_saturatedClass","k":"theorem","m":"Causalean.Panel.EstimandCharacterization.OLSWeightDecomposition.Support.Basic","a":"Panel/EstimandCharacterization/OLSWeightDecomposition/Support","t":2,"d":"The mean-regression meanReg μ Y G lies in saturatedClass μ G G_meas. Take the coefficient map c g := (∫ Y · 𝟙{G = g} dμ) / cellMass μ G g.","s":"∀ {Ω : Type u_1} {𝒢 : Type u_2} [inst : MeasurableSpace Ω] [inst_1 : Fintype 𝒢] [inst_2 : DecidableEq 𝒢] [inst_3 : MeasurableSpace 𝒢] [inst_4 : MeasurableSingletonClass 𝒢] (μ : MeasureTheory.Measure Ω) [inst_5 : MeasureTheory.IsFiniteMeasure μ] (Y : Ω → ℝ) (G : Ω → 𝒢) (G_meas : Measurable G), (Causalean.Panel.EstimandCharacterization.OLSWeightDecomposition.saturatedClass μ G G_meas).mem (Causalean.Panel.EstimandCharacterization.OLSWeightDecomposition.meanReg μ Y G)","l":236,"f":"Causalean/Panel/EstimandCharacterization/OLSWeightDecomposition/Support/Basic.lean"},{"n":"Causalean.Panel.EstimandCharacterization.OLSWeightDecomposition.num_per_cell","k":"theorem","m":"Causalean.Panel.EstimandCharacterization.OLSWeightDecomposition.Support.PerCell","a":"Panel/EstimandCharacterization/OLSWeightDecomposition/Support","t":1,"d":"Per-cell numerator identity. For a probability space (Ω, μ) with measurable treatment D, an outcome Y, square-integrable potential outcomes Y0, Y1, and a measurable finite covariate G, suppose treatment is binary almost everywhere, the observed outcome is consistent — Y equals D·Y1 + (1−D)·Y0 almost everywhere, and the finite-cell conditional-mean-independence bridge condition holds, the integrated finite-cell substitute for E[Y(d) ∣ D, G] = E[Y(d) ∣ G]. Then, for any cell g, the propensity-residual-weighted, cell-indicator-integrated outcome ∫ (D − propensity) · Y · 𝟙{G = g} dμ equals cellMass g · cellShare g · (1 − cellShare g) · cellTau g.","s":"∀ {Ω : Type u_1} {𝒢 : Type u_2} [inst : MeasurableSpace Ω] [inst_1 : Fintype 𝒢] [DecidableEq 𝒢] [inst_3 : MeasurableSpace 𝒢] [MeasurableSingletonClass 𝒢] (μ : MeasureTheory.Measure Ω) [MeasureTheory.IsProbabilityMeasure μ] (D Y Y0 Y1 : Ω → ℝ) (G : Ω → 𝒢), Measurable G → Measurable D → (∀ᵐ (ω : Ω) ∂μ, D ω = 0 ∨ D ω = 1) → MeasureTheory.MemLp Y0 2 μ → MeasureTheory.MemLp Y1 2 μ → (Y =ᵐ[μ] fun ω => D ω * Y1 ω + (1 - D ω) * Y0 ω) → (∀ (d : ℝ) (g : 𝒢), d = 0 ∨ d = 1 → ∫ (ω : Ω), (if d = 1 then Y1 ω else Y0 ω) * {ω' | D ω' = d}.indicator (fun x => 1) ω * {ω' | G ω' = g}.indicator (fun x => 1) ω ∂μ = (∫ (ω : Ω), (if d = 1 then Y1 ω else Y0 ω) * {ω' | G ω' = g}.indicator (fun x => 1) ω ∂μ) * if d = 1 then Causalean.Panel.EstimandCharacterization.OLSWeightDecomposition.cellShare μ D G g else 1 - Causalean.Panel.EstimandCharacterization.OLSWeightDecomposition.cellShare μ D G g) → ∀ (g : 𝒢), ∫ (ω : Ω), (D ω - Causalean.Panel.EstimandCharacterization.OLSWeightDecomposition.propensity μ D G ω) * Y ω * {ω' | G ω' = g}.indicator (fun x => 1) ω ∂μ = Causalean.Panel.EstimandCharacterization.OLSWeightDecomposition.cellMass μ G g * (Causalean.Panel.EstimandCharacterization.OLSWeightDecomposition.cellShare μ D G g * (1 - Causalean.Panel.EstimandCharacterization.OLSWeightDecomposition.cellShare μ D G g)) * Causalean.Panel.EstimandCharacterization.OLSWeightDecomposition.cellTau μ Y0 Y1 G g","l":110,"f":"Causalean/Panel/EstimandCharacterization/OLSWeightDecomposition/Support/PerCell.lean"},{"n":"Causalean.Panel.EstimandCharacterization.OLSWeightDecomposition.numer_identity","k":"theorem","m":"Causalean.Panel.EstimandCharacterization.OLSWeightDecomposition.GeneralCondExp","a":"Panel/EstimandCharacterization/OLSWeightDecomposition","t":2,"d":"Numerator identity (B). Under consistency and the factored conditional-mean-independence hypotheses, ∫ (D − μ[D | m])·Y ∂μ = ∫ μ[D | m]·(1 − μ[D | m])·(μ[Y1 | m] − μ[Y0 | m]) ∂μ.","s":"∀ {Ω : Type u_1} {m mΩ : MeasurableSpace Ω} (μ : MeasureTheory.Measure Ω) [MeasureTheory.IsFiniteMeasure μ], m ≤ mΩ → ∀ (D Y Y0 Y1 : Ω → ℝ), (∀ᵐ (ω : Ω) ∂μ, D ω = 0 ∨ D ω = 1) → MeasureTheory.Integrable D μ → MeasureTheory.Integrable (fun ω => D ω * Y1 ω) μ → MeasureTheory.Integrable (fun ω => (1 - D ω) * Y0 ω) μ → (Y =ᵐ[μ] fun ω => D ω * Y1 ω + (1 - D ω) * Y0 ω) → (μ[fun ω => D ω * Y1 ω | m] =ᵐ[μ] fun ω => μ[D | m] ω * μ[Y1 | m] ω) → (μ[fun ω => (1 - D ω) * Y0 ω | m] =ᵐ[μ] fun ω => μ[fun ω => 1 - D ω | m] ω * μ[Y0 | m] ω) → ∫ (ω : Ω), (D ω - μ[D | m] ω) * Y ω ∂μ = ∫ (ω : Ω), μ[D | m] ω * (1 - μ[D | m] ω) * (μ[Y1 | m] ω - μ[Y0 | m] ω) ∂μ","l":165,"f":"Causalean/Panel/EstimandCharacterization/OLSWeightDecomposition/GeneralCondExp.lean"},{"n":"Causalean.Panel.EstimandCharacterization.OLSWeightDecomposition.partitionOf","k":"def","m":"Causalean.Panel.EstimandCharacterization.OLSWeightDecomposition.Support.Partition","a":"Panel/EstimandCharacterization/OLSWeightDecomposition/Support","t":1,"d":"Measurable binary treatment data with a finite cell classifier and positive saturated-overlap denominator determine the finite Słoczyński partition of cell masses, treated shares, and within-cell treatment effects.","s":"{Ω : Type u_1} → {𝒢 : Type u_2} → [inst : MeasurableSpace Ω] → [inst_1 : Fintype 𝒢] → [DecidableEq 𝒢] → [inst_3 : MeasurableSpace 𝒢] → [MeasurableSingletonClass 𝒢] → (μ : MeasureTheory.Measure Ω) → [MeasureTheory.IsProbabilityMeasure μ] → (D : Ω → ℝ) → (Ω → ℝ) → (Ω → ℝ) → (G : Ω → 𝒢) → Measurable G → (∀ᵐ (ω : Ω) ∂μ, D ω = 0 ∨ D ω = 1) → 0 < ∑ g, Causalean.Panel.EstimandCharacterization.OLSWeightDecomposition.cellMass μ G g * (Causalean.Panel.EstimandCharacterization.OLSWeightDecomposition.cellShare μ D G g * (1 - Causalean.Panel.EstimandCharacterization.OLSWeightDecomposition.cellShare μ D G g)) → Causalean.Panel.EstimandCharacterization.OLSWeightDecomposition.FinitePartition 𝒢","l":32,"f":"Causalean/Panel/EstimandCharacterization/OLSWeightDecomposition/Support/Partition.lean"},{"n":"Causalean.Panel.EstimandCharacterization.OLSWeightDecomposition.partitionOf_p_eq_eventCondExp","k":"theorem","m":"Causalean.Panel.EstimandCharacterization.OLSWeightDecomposition.Support.Partition","a":"Panel/EstimandCharacterization/OLSWeightDecomposition/Support","t":1,"d":"Population-cell certificate for partitionOf. For a probability space (Ω, μ) with a measurable finite covariate G and treatment/potential-outcome data D, Y0, Y1, suppose treatment is binary almost everywhere and the covariate cells have nondegenerate treatment overlap. Then, for any cell g, the treated share p_g of the Słoczyński partition partitionOf equals the shared population cell mean E[D ∣ G = g] (via eventCondExp).","s":"∀ {Ω : Type u_1} {𝒢 : Type u_2} [inst : MeasurableSpace Ω] [inst_1 : Fintype 𝒢] [inst_2 : DecidableEq 𝒢] [inst_3 : MeasurableSpace 𝒢] [inst_4 : MeasurableSingletonClass 𝒢] (μ : MeasureTheory.Measure Ω) [inst_5 : MeasureTheory.IsProbabilityMeasure μ] (D Y0 Y1 : Ω → ℝ) (G : Ω → 𝒢) (G_meas : Measurable G) (D_binary : ∀ᵐ (ω : Ω) ∂μ, D ω = 0 ∨ D ω = 1) (overlap : 0 < ∑ g, Causalean.Panel.EstimandCharacterization.OLSWeightDecomposition.cellMass μ G g * (Causalean.Panel.EstimandCharacterization.OLSWeightDecomposition.cellShare μ D G g * (1 - Causalean.Panel.EstimandCharacterization.OLSWeightDecomposition.cellShare μ D G g))) (g : 𝒢), (Causalean.Panel.EstimandCharacterization.OLSWeightDecomposition.partitionOf μ D Y0 Y1 G G_meas D_binary overlap).p g = Causalean.PO.eventCondExp μ {ω | G ω = g} D","l":61,"f":"Causalean/Panel/EstimandCharacterization/OLSWeightDecomposition/Support/Partition.lean"},{"n":"Causalean.Panel.EstimandCharacterization.OLSWeightDecomposition.partitionOf_tau_eq_eventCondExp","k":"theorem","m":"Causalean.Panel.EstimandCharacterization.OLSWeightDecomposition.Support.Partition","a":"Panel/EstimandCharacterization/OLSWeightDecomposition/Support","t":2,"d":"Population-cell certificate for partitionOf. The cell treatment effect τ_g of the Słoczyński partition is the shared population potential-outcome contrast E[Y(1) − Y(0) | G = g] (via eventCondExp), so the overlap-weighted estimand is built from genuine potential-outcome cell means.","s":"∀ {Ω : Type u_1} {𝒢 : Type u_2} [inst : MeasurableSpace Ω] [inst_1 : Fintype 𝒢] [inst_2 : DecidableEq 𝒢] [inst_3 : MeasurableSpace 𝒢] [inst_4 : MeasurableSingletonClass 𝒢] (μ : MeasureTheory.Measure Ω) [inst_5 : MeasureTheory.IsProbabilityMeasure μ] (D Y0 Y1 : Ω → ℝ) (G : Ω → 𝒢) (G_meas : Measurable G) (D_binary : ∀ᵐ (ω : Ω) ∂μ, D ω = 0 ∨ D ω = 1) (overlap : 0 < ∑ g, Causalean.Panel.EstimandCharacterization.OLSWeightDecomposition.cellMass μ G g * (Causalean.Panel.EstimandCharacterization.OLSWeightDecomposition.cellShare μ D G g * (1 - Causalean.Panel.EstimandCharacterization.OLSWeightDecomposition.cellShare μ D G g))) (g : 𝒢), (Causalean.Panel.EstimandCharacterization.OLSWeightDecomposition.partitionOf μ D Y0 Y1 G G_meas D_binary overlap).τ g = Causalean.PO.eventCondExp μ {ω | G ω = g} fun ω => Y1 ω - Y0 ω","l":79,"f":"Causalean/Panel/EstimandCharacterization/OLSWeightDecomposition/Support/Partition.lean"},{"n":"Causalean.Panel.EstimandCharacterization.OLSWeightDecomposition.propensity","k":"def","m":"Causalean.Panel.EstimandCharacterization.OLSWeightDecomposition.Support.Basic","a":"Panel/EstimandCharacterization/OLSWeightDecomposition/Support","t":1,"d":"Saturated propensity propensity μ D G ω = cellShare μ D G (G ω) (pointwise, by disjointness of the {G = g} family). Plays the role of p(G(·)) and lies in saturatedClass μ G G_meas.","s":"{Ω : Type u_1} → {𝒢 : Type u_2} → [inst : MeasurableSpace Ω] → [Fintype 𝒢] → MeasureTheory.Measure Ω → (Ω → ℝ) → (Ω → 𝒢) → Ω → ℝ","l":119,"f":"Causalean/Panel/EstimandCharacterization/OLSWeightDecomposition/Support/Basic.lean"},{"n":"Causalean.Panel.EstimandCharacterization.OLSWeightDecomposition.propensity_eq_cellShare_of_mem","k":"theorem","m":"Causalean.Panel.EstimandCharacterization.OLSWeightDecomposition.Support.Integrals","a":"Panel/EstimandCharacterization/OLSWeightDecomposition/Support","t":2,"d":"On its own cell, the saturated propensity representative equals the corresponding cell share.","s":"∀ {Ω : Type u_1} {𝒢 : Type u_2} [inst : MeasurableSpace Ω] [inst_1 : Fintype 𝒢] (μ : MeasureTheory.Measure Ω) (D : Ω → ℝ) (G : Ω → 𝒢) {g : 𝒢} {ω : Ω}, G ω = g → Causalean.Panel.EstimandCharacterization.OLSWeightDecomposition.propensity μ D G ω = Causalean.Panel.EstimandCharacterization.OLSWeightDecomposition.cellShare μ D G g","l":109,"f":"Causalean/Panel/EstimandCharacterization/OLSWeightDecomposition/Support/Integrals.lean"},{"n":"Causalean.Panel.EstimandCharacterization.OLSWeightDecomposition.propensity_mem_saturatedClass","k":"theorem","m":"Causalean.Panel.EstimandCharacterization.OLSWeightDecomposition.Support.Basic","a":"Panel/EstimandCharacterization/OLSWeightDecomposition/Support","t":2,"d":"The pointwise representative propensity μ D G lies in saturatedClass μ G G_meas. Take the coefficient map c g := cellShare μ D G g; equality holds pointwise (and so a.e.).","s":"∀ {Ω : Type u_1} {𝒢 : Type u_2} [inst : MeasurableSpace Ω] [inst_1 : Fintype 𝒢] [inst_2 : DecidableEq 𝒢] [inst_3 : MeasurableSpace 𝒢] [inst_4 : MeasurableSingletonClass 𝒢] (μ : MeasureTheory.Measure Ω) [inst_5 : MeasureTheory.IsFiniteMeasure μ] (D : Ω → ℝ) (G : Ω → 𝒢) (G_meas : Measurable G), (Causalean.Panel.EstimandCharacterization.OLSWeightDecomposition.saturatedClass μ G G_meas).mem (Causalean.Panel.EstimandCharacterization.OLSWeightDecomposition.propensity μ D G)","l":224,"f":"Causalean/Panel/EstimandCharacterization/OLSWeightDecomposition/Support/Basic.lean"},{"n":"Causalean.Panel.EstimandCharacterization.OLSWeightDecomposition.residD_cell_orthogonal","k":"theorem","m":"Causalean.Panel.EstimandCharacterization.OLSWeightDecomposition.Support.Orthogonality","a":"Panel/EstimandCharacterization/OLSWeightDecomposition/Support","t":2,"d":"The treatment residual is orthogonal to each saturated cell indicator.","s":"∀ {Ω : Type u_1} {𝒢 : Type u_2} [inst : MeasurableSpace Ω] [inst_1 : Fintype 𝒢] [inst_2 : MeasurableSpace 𝒢] [MeasurableSingletonClass 𝒢] (μ : MeasureTheory.Measure Ω) [MeasureTheory.IsFiniteMeasure μ] (D : Ω → ℝ) (G : Ω → 𝒢), Measurable G → Measurable D → (∀ᵐ (ω : Ω) ∂μ, D ω = 0 ∨ D ω = 1) → ∀ (g : 𝒢), ∫ (ω : Ω), (D ω - Causalean.Panel.EstimandCharacterization.OLSWeightDecomposition.propensity μ D G ω) * {ω' | G ω' = g}.indicator (fun x => 1) ω ∂μ = 0","l":29,"f":"Causalean/Panel/EstimandCharacterization/OLSWeightDecomposition/Support/Orthogonality.lean"},{"n":"Causalean.Panel.EstimandCharacterization.OLSWeightDecomposition.residWitnessD","k":"def","m":"Causalean.Panel.EstimandCharacterization.OLSWeightDecomposition.Support.Partition","a":"Panel/EstimandCharacterization/OLSWeightDecomposition/Support","t":1,"d":"Residualization witness for the treatment variable D.","s":"{Ω : Type u_1} → {𝒢 : Type u_2} → [inst : MeasurableSpace Ω] → [inst_1 : Fintype 𝒢] → [inst_2 : DecidableEq 𝒢] → [inst_3 : MeasurableSpace 𝒢] → [inst_4 : MeasurableSingletonClass 𝒢] → (μ : MeasureTheory.Measure Ω) → [inst_5 : MeasureTheory.IsProbabilityMeasure μ] → (D : Ω → ℝ) → (G : Ω → 𝒢) → (G_meas : Measurable G) → Measurable D → (∀ᵐ (ω : Ω) ∂μ, D ω = 0 ∨ D ω = 1) → Causalean.Panel.ResidualizationWitness μ (Causalean.Panel.EstimandCharacterization.OLSWeightDecomposition.saturatedClass μ G G_meas) D","l":94,"f":"Causalean/Panel/EstimandCharacterization/OLSWeightDecomposition/Support/Partition.lean"},{"n":"Causalean.Panel.EstimandCharacterization.OLSWeightDecomposition.residWitnessY","k":"def","m":"Causalean.Panel.EstimandCharacterization.OLSWeightDecomposition.Support.Partition","a":"Panel/EstimandCharacterization/OLSWeightDecomposition/Support","t":1,"d":"Residualization witness for the outcome variable Y.","s":"{Ω : Type u_1} → {𝒢 : Type u_2} → [inst : MeasurableSpace Ω] → [inst_1 : Fintype 𝒢] → [inst_2 : DecidableEq 𝒢] → [inst_3 : MeasurableSpace 𝒢] → [inst_4 : MeasurableSingletonClass 𝒢] → (μ : MeasureTheory.Measure Ω) → [inst_5 : MeasureTheory.IsProbabilityMeasure μ] → (Y : Ω → ℝ) → (G : Ω → 𝒢) → (G_meas : Measurable G) → MeasureTheory.MemLp Y 2 μ → Causalean.Panel.ResidualizationWitness μ (Causalean.Panel.EstimandCharacterization.OLSWeightDecomposition.saturatedClass μ G G_meas) Y","l":164,"f":"Causalean/Panel/EstimandCharacterization/OLSWeightDecomposition/Support/Partition.lean"},{"n":"Causalean.Panel.EstimandCharacterization.OLSWeightDecomposition.residY_cell_orthogonal","k":"theorem","m":"Causalean.Panel.EstimandCharacterization.OLSWeightDecomposition.Support.Orthogonality","a":"Panel/EstimandCharacterization/OLSWeightDecomposition/Support","t":1,"d":"For an ambient probability space and finite covariate space, a square-integrable outcome Y, a measurable covariate map G, and a covariate cell g, the regression residual of Y on the saturated cell-mean model is orthogonal to the indicator of cell {G = g}.","s":"∀ {Ω : Type u_1} {𝒢 : Type u_2} [inst : MeasurableSpace Ω] [inst_1 : Fintype 𝒢] [inst_2 : MeasurableSpace 𝒢] [MeasurableSingletonClass 𝒢] (μ : MeasureTheory.Measure Ω) [MeasureTheory.IsFiniteMeasure μ] (Y : Ω → ℝ) (G : Ω → 𝒢), Measurable G → MeasureTheory.MemLp Y 2 μ → ∀ (g : 𝒢), ∫ (ω : Ω), (Y ω - Causalean.Panel.EstimandCharacterization.OLSWeightDecomposition.meanReg μ Y G ω) * {ω' | G ω' = g}.indicator (fun x => 1) ω ∂μ = 0","l":87,"f":"Causalean/Panel/EstimandCharacterization/OLSWeightDecomposition/Support/Orthogonality.lean"},{"n":"Causalean.Panel.EstimandCharacterization.OLSWeightDecomposition.saturatedClass","k":"def","m":"Causalean.Panel.EstimandCharacterization.OLSWeightDecomposition.Support.Basic","a":"Panel/EstimandCharacterization/OLSWeightDecomposition/Support","t":1,"d":"Linear L² control class spanned, up to almost-everywhere equality, by the finite family of cell indicators 𝟙{G = g} for g : 𝒢.","s":"{Ω : Type u_1} → {𝒢 : Type u_2} → [inst : MeasurableSpace Ω] → [Fintype 𝒢] → [DecidableEq 𝒢] → [inst_3 : MeasurableSpace 𝒢] → [MeasurableSingletonClass 𝒢] → (μ : MeasureTheory.Measure Ω) → [MeasureTheory.IsFiniteMeasure μ] → (G : Ω → 𝒢) → Measurable G → Causalean.Panel.LinearL2Class μ","l":34,"f":"Causalean/Panel/EstimandCharacterization/OLSWeightDecomposition/Support/Basic.lean"},{"n":"Causalean.Panel.EstimandCharacterization.OLSWeightDecomposition.saturated_ols_overlap_weighted_ate","k":"theorem","m":"Causalean.Panel.EstimandCharacterization.OLSWeightDecomposition.GeneralCondExp","a":"Panel/EstimandCharacterization/OLSWeightDecomposition","t":1,"d":"Saturated-control OLS overlap-weighted ATE identity. Fix a covariate σ-algebra m at most as fine as the ambient one, and outcome, treatment, and potential-outcome functions D, Y, Y0, Y1. Suppose treatment is binary almost everywhere, D is integrable, D·Y1 is integrable, and (1−D)·Y0 is integrable. Suppose also consistency: the observed outcome equals D·Y1 + (1−D)·Y0 almost everywhere, and the factored conditional-mean-independence conditions μ[D·Y1 ∣ m] equals μ[D∣m]·μ[Y1∣m] almost everywhere and μ[(1−D)·Y0 ∣ m] equals μ[1−D∣m]·μ[Y0∣m] almost everywhere. Then the saturated-control OLS residualized treatment-outcome ratio E[(D−μ[D∣m])·Y] / E[(D−μ[D∣m])²] equals the overlap-weighted average of conditional treatment effects E[μ[D∣m]·(1−μ[D∣m])·(μ[Y1∣m]−μ[Y0∣m])] / E[μ[D∣m]·(1−μ[D∣m])], using Lean's totalized division convention so the identity also covers the zero-denominator case. This is the paper-facing statement corresponding to LaTeX thm:po-estimand-sloczynski-ols-overlap-weighted-ate.","s":"∀ {Ω : Type u_1} {m mΩ : MeasurableSpace Ω} (μ : MeasureTheory.Measure Ω) [MeasureTheory.IsFiniteMeasure μ], m ≤ mΩ → ∀ (D Y Y0 Y1 : Ω → ℝ), (∀ᵐ (ω : Ω) ∂μ, D ω = 0 ∨ D ω = 1) → MeasureTheory.Integrable D μ → MeasureTheory.Integrable (fun ω => D ω * Y1 ω) μ → MeasureTheory.Integrable (fun ω => (1 - D ω) * Y0 ω) μ → (Y =ᵐ[μ] fun ω => D ω * Y1 ω + (1 - D ω) * Y0 ω) → (μ[fun ω => D ω * Y1 ω | m] =ᵐ[μ] fun ω => μ[D | m] ω * μ[Y1 | m] ω) → (μ[fun ω => (1 - D ω) * Y0 ω | m] =ᵐ[μ] fun ω => μ[fun ω => 1 - D ω | m] ω * μ[Y0 | m] ω) → (∫ (ω : Ω), (D ω - μ[D | m] ω) * Y ω ∂μ) / ∫ (ω : Ω), (D ω - μ[D | m] ω) ^ 2 ∂μ = (∫ (ω : Ω), μ[D | m] ω * (1 - μ[D | m] ω) * (μ[Y1 | m] ω - μ[Y0 | m] ω) ∂μ) / ∫ (ω : Ω), μ[D | m] ω * (1 - μ[D | m] ω) ∂μ","l":284,"f":"Causalean/Panel/EstimandCharacterization/OLSWeightDecomposition/GeneralCondExp.lean"},{"n":"Causalean.Panel.EstimandCharacterization.OLSWeightDecomposition.saturated_ols_overlap_weighted_ate_core","k":"theorem","m":"Causalean.Panel.EstimandCharacterization.OLSWeightDecomposition.GeneralCondExp","a":"Panel/EstimandCharacterization/OLSWeightDecomposition","t":2,"d":"Positivity-free core of the saturated-control OLS overlap-weighted ratio identity. Both ratios are totalized, so the algebraic equality also holds when the common denominator is zero.","s":"∀ {Ω : Type u_1} {m mΩ : MeasurableSpace Ω} (μ : MeasureTheory.Measure Ω) [MeasureTheory.IsFiniteMeasure μ], m ≤ mΩ → ∀ (D Y Y0 Y1 : Ω → ℝ), (∀ᵐ (ω : Ω) ∂μ, D ω = 0 ∨ D ω = 1) → MeasureTheory.Integrable D μ → MeasureTheory.Integrable (fun ω => D ω * Y1 ω) μ → MeasureTheory.Integrable (fun ω => (1 - D ω) * Y0 ω) μ → (Y =ᵐ[μ] fun ω => D ω * Y1 ω + (1 - D ω) * Y0 ω) → (μ[fun ω => D ω * Y1 ω | m] =ᵐ[μ] fun ω => μ[D | m] ω * μ[Y1 | m] ω) → (μ[fun ω => (1 - D ω) * Y0 ω | m] =ᵐ[μ] fun ω => μ[fun ω => 1 - D ω | m] ω * μ[Y0 | m] ω) → (∫ (ω : Ω), (D ω - μ[D | m] ω) * Y ω ∂μ) / ∫ (ω : Ω), (D ω - μ[D | m] ω) ^ 2 ∂μ = (∫ (ω : Ω), μ[D | m] ω * (1 - μ[D | m] ω) * (μ[Y1 | m] ω - μ[Y0 | m] ω) ∂μ) / ∫ (ω : Ω), μ[D | m] ω * (1 - μ[D | m] ω) ∂μ","l":263,"f":"Causalean/Panel/EstimandCharacterization/OLSWeightDecomposition/GeneralCondExp.lean"},{"n":"Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.ATT_window","k":"def","m":"Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.Causal","a":"Panel/EstimandCharacterization/StaggeredTWFEDecomposition","t":1,"d":"Window-specific ATT in the two-state Goodman-Bacon causal layer: ATT_{g,S} := |S|⁻¹ ∑_{t ∈ S} (Y_{gt}(A_g) − Y_{gt}(∞)), parameterized by potential-outcome maps Y0 = Y(∞) and Y1 = Y(A_g).","s":"{𝒢 : Type u_1} → {T : ℕ} → (𝒢 → Fin T → ℝ) → (𝒢 → Fin T → ℝ) → 𝒢 → Finset (Fin T) → ℝ","l":102,"f":"Causalean/Panel/EstimandCharacterization/StaggeredTWFEDecomposition/Causal.lean"},{"n":"Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.AdoptionDate.isFin","k":"def","m":"Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.FinitePanel","a":"Panel/EstimandCharacterization/StaggeredTWFEDecomposition","t":1,"d":"A_g is finite, i.e. cohort g is eventually treated.","s":"{T : ℕ} → WithTop (Fin T) → Prop","l":96,"f":"Causalean/Panel/EstimandCharacterization/StaggeredTWFEDecomposition/FinitePanel.lean"},{"n":"Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.AdoptionDate.isInf","k":"def","m":"Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.FinitePanel","a":"Panel/EstimandCharacterization/StaggeredTWFEDecomposition","t":1,"d":"A_g = ∞, i.e. cohort g is never treated.","s":"{T : ℕ} → WithTop (Fin T) → Prop","l":99,"f":"Causalean/Panel/EstimandCharacterization/StaggeredTWFEDecomposition/FinitePanel.lean"},{"n":"Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.AdoptionDate.le","k":"def","m":"Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.FinitePanel","a":"Panel/EstimandCharacterization/StaggeredTWFEDecomposition","t":1,"d":"Adoption-date predicate A_g ≤ t, i.e. cohort g has adopted by period t. The lift (t : WithTop (Fin T)) carries ⊤ = ∞ so that ∞ ≤ t is false, matching the LaTeX convention 1_{∞ ≤ t} = 0.","s":"{T : ℕ} → WithTop (Fin T) → Fin T → Prop","l":87,"f":"Causalean/Panel/EstimandCharacterization/StaggeredTWFEDecomposition/FinitePanel.lean"},{"n":"Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.AdoptionDate.lt","k":"def","m":"Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.FinitePanel","a":"Panel/EstimandCharacterization/StaggeredTWFEDecomposition","t":1,"d":"Adoption-date predicate t < A_g, i.e. cohort g is untreated at period t.","s":"{T : ℕ} → WithTop (Fin T) → Fin T → Prop","l":92,"f":"Causalean/Panel/EstimandCharacterization/StaggeredTWFEDecomposition/FinitePanel.lean"},{"n":"Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.CausalAssumptions","k":"structure","m":"Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.Causal","a":"Panel/EstimandCharacterization/StaggeredTWFEDecomposition","t":1,"d":"Causal-side assumptions for the Goodman-Bacon decomposition (LaTeX ass:po-estimand-goodman-bacon-causal), given a cohort panel P and potential-outcome maps Y0 (the never-treated path) and Y1 (each cohort's own adoption-date path). It packages consistency on treated cells — the factual outcome equals the post-adoption potential outcome once the cohort has adopted, consistency on untreated cells — the factual outcome equals the never-treated potential outcome before adoption, no anticipation — the pre-adoption potential outcomes under the two paths coincide, and pairwise untreated parallel trends for the treated-versus-never comparison, the early-versus-late-before-late comparison, and the late-versus-early-after-early comparison.","s":"{𝒢 : Type u_1} → [inst : Fintype 𝒢] → {T : ℕ} → Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.CohortPanel 𝒢 T → (𝒢 → Fin T → ℝ) → (𝒢 → Fin T → ℝ) → Prop","l":109,"f":"Causalean/Panel/EstimandCharacterization/StaggeredTWFEDecomposition/Causal.lean"},{"n":"Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.CausalAssumptions.AdoptionDate.lt_of_isInf","k":"theorem","m":"Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.Causal","a":"Panel/EstimandCharacterization/StaggeredTWFEDecomposition","t":2,"d":"Helper: every period is strictly less than ⊤ in WithTop (Fin T).","s":"∀ {T : ℕ} {a : WithTop (Fin T)} {t : Fin T}, Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.AdoptionDate.isInf a → Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.AdoptionDate.lt a t","l":171,"f":"Causalean/Panel/EstimandCharacterization/StaggeredTWFEDecomposition/Causal.lean"},{"n":"Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.CausalAssumptions.AdoptionDate.not_le_of_lt","k":"theorem","m":"Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.Causal","a":"Panel/EstimandCharacterization/StaggeredTWFEDecomposition","t":2,"d":"Helper: AdoptionDate.lt (P.A g) t rules out AdoptionDate.le (P.A g) t.","s":"∀ {T : ℕ} {a : WithTop (Fin T)} {t : Fin T}, Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.AdoptionDate.lt a t → ¬Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.AdoptionDate.le a t","l":165,"f":"Causalean/Panel/EstimandCharacterization/StaggeredTWFEDecomposition/Causal.lean"},{"n":"Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.CausalAssumptions.Ybar_eq_Ybar0_add_ATT_of_treated","k":"theorem","m":"Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.Causal","a":"Panel/EstimandCharacterization/StaggeredTWFEDecomposition","t":2,"d":"On a window where every cell is treated for cohort g, the factual Ybar decomposes as the never-treated mean plus the window ATT.","s":"∀ {𝒢 : Type u_1} [inst : Fintype 𝒢] {T : ℕ} {P : Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.CohortPanel 𝒢 T} {Y0 Y1 : 𝒢 → Fin T → ℝ}, (∀ (g : 𝒢) (t : Fin T), Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.AdoptionDate.le (P.A g) t → P.Y g t = Y1 g t) → ∀ (g : 𝒢) (S : Finset (Fin T)), (∀ t ∈ S, Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.AdoptionDate.le (P.A g) t) → Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.Ybar P g S = Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.Ybar0 Y0 g S + Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.ATT_window Y0 Y1 g S","l":210,"f":"Causalean/Panel/EstimandCharacterization/StaggeredTWFEDecomposition/Causal.lean"},{"n":"Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.CausalAssumptions.Ybar_eq_Ybar0_late_on_S0_EL","k":"theorem","m":"Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.Causal","a":"Panel/EstimandCharacterization/StaggeredTWFEDecomposition","t":2,"d":"The late-cohort factual Ybar on S0_EL P e = {t : t < A_e} equals the never-treated Ybar0 when A_e < A_ℓ.","s":"∀ {𝒢 : Type u_1} [inst : Fintype 𝒢] {T : ℕ} {P : Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.CohortPanel 𝒢 T} {Y0 : 𝒢 → Fin T → ℝ}, (∀ (g : 𝒢) (t : Fin T), Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.AdoptionDate.lt (P.A g) t → P.Y g t = Y0 g t) → ∀ (e ℓ : 𝒢), P.A e < P.A ℓ → Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.Ybar P ℓ (Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.S0_EL P e) = Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.Ybar0 Y0 ℓ (Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.S0_EL P e)","l":264,"f":"Causalean/Panel/EstimandCharacterization/StaggeredTWFEDecomposition/Causal.lean"},{"n":"Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.CausalAssumptions.Ybar_eq_Ybar0_late_on_S0_LE","k":"theorem","m":"Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.Causal","a":"Panel/EstimandCharacterization/StaggeredTWFEDecomposition","t":2,"d":"The late-cohort factual Ybar on S0_LE P e ℓ = {A_e ≤ t < A_ℓ} equals the never-treated Ybar0.","s":"∀ {𝒢 : Type u_1} [inst : Fintype 𝒢] {T : ℕ} {P : Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.CohortPanel 𝒢 T} {Y0 : 𝒢 → Fin T → ℝ}, (∀ (g : 𝒢) (t : Fin T), Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.AdoptionDate.lt (P.A g) t → P.Y g t = Y0 g t) → ∀ (e ℓ : 𝒢), Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.Ybar P ℓ (Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.S0_LE P e ℓ) = Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.Ybar0 Y0 ℓ (Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.S0_LE P e ℓ)","l":283,"f":"Causalean/Panel/EstimandCharacterization/StaggeredTWFEDecomposition/Causal.lean"},{"n":"Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.CausalAssumptions.Ybar_eq_Ybar0_late_on_S1_EL","k":"theorem","m":"Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.Causal","a":"Panel/EstimandCharacterization/StaggeredTWFEDecomposition","t":2,"d":"The late-cohort factual Ybar on S1_EL P e ℓ = {A_e ≤ t < A_ℓ} equals the never-treated Ybar0: each cell satisfies t < A_ℓ.","s":"∀ {𝒢 : Type u_1} [inst : Fintype 𝒢] {T : ℕ} {P : Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.CohortPanel 𝒢 T} {Y0 : 𝒢 → Fin T → ℝ}, (∀ (g : 𝒢) (t : Fin T), Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.AdoptionDate.lt (P.A g) t → P.Y g t = Y0 g t) → ∀ (e ℓ : 𝒢), Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.Ybar P ℓ (Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.S1_EL P e ℓ) = Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.Ybar0 Y0 ℓ (Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.S1_EL P e ℓ)","l":249,"f":"Causalean/Panel/EstimandCharacterization/StaggeredTWFEDecomposition/Causal.lean"},{"n":"Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.CausalAssumptions.Ybar_eq_Ybar0_of_inf","k":"theorem","m":"Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.Causal","a":"Panel/EstimandCharacterization/StaggeredTWFEDecomposition","t":2,"d":"On any window S where every period is untreated for cohort u, the factual Ybar equals the never-treated Ybar0.","s":"∀ {𝒢 : Type u_1} [inst : Fintype 𝒢] {T : ℕ} {P : Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.CohortPanel 𝒢 T} {Y0 : 𝒢 → Fin T → ℝ}, (∀ (g : 𝒢) (t : Fin T), Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.AdoptionDate.lt (P.A g) t → P.Y g t = Y0 g t) → ∀ {u : 𝒢} (S : Finset (Fin T)), (∀ t ∈ S, Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.AdoptionDate.lt (P.A u) t) → Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.Ybar P u S = Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.Ybar0 Y0 u S","l":195,"f":"Causalean/Panel/EstimandCharacterization/StaggeredTWFEDecomposition/Causal.lean"},{"n":"Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.CausalAssumptions.Ybar_eq_Ybar0_on_S0_EL","k":"theorem","m":"Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.Causal","a":"Panel/EstimandCharacterization/StaggeredTWFEDecomposition","t":2,"d":"On the early-cohort untreated window S0_EL P e = {t : t < A_e}, the factual Ybar of cohort e equals the never-treated Ybar0.","s":"∀ {𝒢 : Type u_1} [inst : Fintype 𝒢] {T : ℕ} {P : Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.CohortPanel 𝒢 T} {Y0 : 𝒢 → Fin T → ℝ}, (∀ (g : 𝒢) (t : Fin T), Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.AdoptionDate.lt (P.A g) t → P.Y g t = Y0 g t) → ∀ (e : 𝒢), Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.Ybar P e (Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.S0_EL P e) = Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.Ybar0 Y0 e (Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.S0_EL P e)","l":234,"f":"Causalean/Panel/EstimandCharacterization/StaggeredTWFEDecomposition/Causal.lean"},{"n":"Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.CausalAssumptions.Ybar_eq_Ybar0_on_S0_TN","k":"theorem","m":"Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.Causal","a":"Panel/EstimandCharacterization/StaggeredTWFEDecomposition","t":2,"d":"On the untreated window S0_TN P g = {t : t < A_g}, the factual Ybar of cohort g equals the never-treated Ybar0: by consistencyUntreated on each cell.","s":"∀ {𝒢 : Type u_1} [inst : Fintype 𝒢] {T : ℕ} {P : Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.CohortPanel 𝒢 T} {Y0 : 𝒢 → Fin T → ℝ}, (∀ (g : 𝒢) (t : Fin T), Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.AdoptionDate.lt (P.A g) t → P.Y g t = Y0 g t) → ∀ (g : 𝒢), Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.Ybar P g (Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.S0_TN P g) = Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.Ybar0 Y0 g (Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.S0_TN P g)","l":179,"f":"Causalean/Panel/EstimandCharacterization/StaggeredTWFEDecomposition/Causal.lean"},{"n":"Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.CohortPanel","k":"structure","m":"Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.FinitePanel","a":"Panel/EstimandCharacterization/StaggeredTWFEDecomposition","t":1,"d":"A staggered-adoption cohort panel: a cell-statistics record carrying, per cohort, a population share, an adoption date — a finite period, or ⊤ for the never-treated case, and the cohort-period factual outcome mean, subject to a positive number of periods, strictly positive cohort shares, and cohort shares summing to one.","s":"(𝒢 : Type u_1) → ℕ → [Fintype 𝒢] → Type u_1","l":66,"f":"Causalean/Panel/EstimandCharacterization/StaggeredTWFEDecomposition/FinitePanel.lean"},{"n":"Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.CompTag","k":"inductive","m":"Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.FinitePanel","a":"Panel/EstimandCharacterization/StaggeredTWFEDecomposition","t":1,"d":"Three-valued tag for the type of an admissible 2x2 comparison: treated-versus-never, early-versus-late before late, and late-versus-early after early.","s":"Type","l":57,"f":"Causalean/Panel/EstimandCharacterization/StaggeredTWFEDecomposition/FinitePanel.lean"},{"n":"Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.D","k":"def","m":"Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.FinitePanel","a":"Panel/EstimandCharacterization/StaggeredTWFEDecomposition","t":1,"d":"Treatment indicator D_{gt} = 1_{A_g ≤ t}, binary and absorbing. Marked noncomputable because the WithTop-order predicate is taken via classical decidability.","s":"{𝒢 : Type u_1} → [inst : Fintype 𝒢] → {T : ℕ} → Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.CohortPanel 𝒢 T → 𝒢 → Fin T → ℝ","l":107,"f":"Causalean/Panel/EstimandCharacterization/StaggeredTWFEDecomposition/FinitePanel.lean"},{"n":"Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.D_eq_of_A_eq","k":"theorem","m":"Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.AlgebraicDecomposition","a":"Panel/EstimandCharacterization/StaggeredTWFEDecomposition","t":2,"d":"Cohorts that share the same adoption date have identical treatment status in every time period.","s":"∀ {𝒢 : Type u_1} [inst : Fintype 𝒢] {T : ℕ} (P : Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.CohortPanel 𝒢 T) {g u : 𝒢}, P.A g = P.A u → ∀ (t : Fin T), Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.D P g t = Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.D P u t","l":255,"f":"Causalean/Panel/EstimandCharacterization/StaggeredTWFEDecomposition/AlgebraicDecomposition.lean"},{"n":"Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.D_eq_zero_of_isInf","k":"theorem","m":"Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.Pairwise","a":"Panel/EstimandCharacterization/StaggeredTWFEDecomposition","t":2,"d":"In a cohort panel, a cohort that is never treated has a zero treatment indicator in every period.","s":"∀ {𝒢 : Type u_1} [inst : Fintype 𝒢] {T : ℕ} (P : Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.CohortPanel 𝒢 T) {u : 𝒢}, Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.AdoptionDate.isInf (P.A u) → ∀ (t : Fin T), Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.D P u t = 0","l":141,"f":"Causalean/Panel/EstimandCharacterization/StaggeredTWFEDecomposition/Pairwise.lean"},{"n":"Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.D_sq_eq_D","k":"theorem","m":"Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.Pairwise","a":"Panel/EstimandCharacterization/StaggeredTWFEDecomposition","t":2,"d":"In a cohort panel, the binary treatment indicator for any cohort and time period equals its own square.","s":"∀ {𝒢 : Type u_1} [inst : Fintype 𝒢] {T : ℕ} (P : Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.CohortPanel 𝒢 T) (g : 𝒢) (t : Fin T), Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.D P g t ^ 2 = Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.D P g t","l":159,"f":"Causalean/Panel/EstimandCharacterization/StaggeredTWFEDecomposition/Pairwise.lean"},{"n":"Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.Dtilde","k":"def","m":"Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.FinitePanel","a":"Panel/EstimandCharacterization/StaggeredTWFEDecomposition","t":1,"d":"Residualized treatment via the explicit double-demeaning formula \\widetilde{D}_{gt} := D_{gt} − \\overline{D}_g − E[D|T=t] + E[D].","s":"{𝒢 : Type u_1} → [inst : Fintype 𝒢] → {T : ℕ} → Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.CohortPanel 𝒢 T → 𝒢 → Fin T → ℝ","l":138,"f":"Causalean/Panel/EstimandCharacterization/StaggeredTWFEDecomposition/FinitePanel.lean"},{"n":"Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.Dtilde_eq","k":"theorem","m":"Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.FinitePanel","a":"Panel/EstimandCharacterization/StaggeredTWFEDecomposition","t":1,"d":"For a cohort panel and a cohort-period cell, the double-demeaned residualized treatment Dtilde P g t equals the original Goodman-Bacon closed form: the raw treatment minus the cohort mean, minus the period cross-cohort mean, plus the grand mean.","s":"∀ {𝒢 : Type u_1} [inst : Fintype 𝒢] {T : ℕ} (P : Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.CohortPanel 𝒢 T) (g : 𝒢) (t : Fin T), Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.Dtilde P g t = Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.D P g t - Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.barD P g - ∑ g', P.p g' * Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.D P g' t + Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.pCohort P","l":144,"f":"Causalean/Panel/EstimandCharacterization/StaggeredTWFEDecomposition/FinitePanel.lean"},{"n":"Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.Dtilde_eq_centeredD_sub_weighted_mean","k":"theorem","m":"Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.Pairwise","a":"Panel/EstimandCharacterization/StaggeredTWFEDecomposition","t":2,"d":"The explicit double-demeaning formula for Dtilde is the weighted centering, across cohorts, of the cohort-demeaned treatment path.","s":"∀ {𝒢 : Type u_1} [inst : Fintype 𝒢] [DecidableEq 𝒢] {T : ℕ} (P : Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.CohortPanel 𝒢 T) (g : 𝒢) (t : Fin T), Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.Dtilde P g t = Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.centeredD P g t - ∑ h, P.p h * Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.centeredD P h t","l":373,"f":"Causalean/Panel/EstimandCharacterization/StaggeredTWFEDecomposition/Pairwise.lean"},{"n":"Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.IsBalancedPanelLaw","k":"def","m":"Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.Support.Orthogonality","a":"Panel/EstimandCharacterization/StaggeredTWFEDecomposition/Support","t":1,"d":"The \"balanced cell-mass product\" hypothesis. Says the joint law of (G, T_rv) factors as a product on each cohort × period cell:","s":"{Ω : Type u_1} → {𝒢 : Type u_2} → [inst : MeasurableSpace Ω] → {T : ℕ} → MeasureTheory.Measure Ω → (Ω → 𝒢) → (Ω → Fin T) → Prop","l":38,"f":"Causalean/Panel/EstimandCharacterization/StaggeredTWFEDecomposition/Support/Orthogonality.lean"},{"n":"Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.Lambda","k":"def","m":"Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.FinitePanel","a":"Panel/EstimandCharacterization/StaggeredTWFEDecomposition","t":1,"d":"Aggregate raw-weight denominator Λ := ∑_{TN admissible} λ^TN + ∑_{e<ℓ<∞} (λ^EL + λ^LE).","s":"{𝒢 : Type u_1} → [inst : Fintype 𝒢] → {T : ℕ} → Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.CohortPanel 𝒢 T → ℝ","l":243,"f":"Causalean/Panel/EstimandCharacterization/StaggeredTWFEDecomposition/FinitePanel.lean"},{"n":"Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.Lambda_eq_gap_sums","k":"theorem","m":"Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.AlgebraicDecomposition","a":"Panel/EstimandCharacterization/StaggeredTWFEDecomposition","t":2,"d":"The raw denominator of the staggered-adoption two-way fixed-effects decomposition equals the sum of treated-versus-never comparison terms and ordered early-versus-late comparison terms.","s":"∀ {𝒢 : Type u_1} [inst : Fintype 𝒢] {T : ℕ} (P : Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.CohortPanel 𝒢 T), Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.Lambda P = (∑ g, ∑ u, if Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.AdoptionDate.isFin (P.A g) ∧ Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.AdoptionDate.isInf (P.A u) then P.p g * P.p u * Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.q P g u * (1 - Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.q P g u) else 0) + ∑ e, ∑ ℓ, if P.A e < P.A ℓ ∧ Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.AdoptionDate.isFin (P.A ℓ) then P.p e * P.p ℓ * Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.q P e ℓ * (1 - Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.q P e ℓ) else 0","l":229,"f":"Causalean/Panel/EstimandCharacterization/StaggeredTWFEDecomposition/AlgebraicDecomposition.lean"},{"n":"Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.S0_EL","k":"def","m":"Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.FinitePanel","a":"Panel/EstimandCharacterization/StaggeredTWFEDecomposition","t":1,"d":"Early-vs-late untreated window \\mathcal{T}_{e\\ell}^0 = {t : t < A_e}.","s":"{𝒢 : Type u_1} → [inst : Fintype 𝒢] → {T : ℕ} → Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.CohortPanel 𝒢 T → 𝒢 → Finset (Fin T)","l":184,"f":"Causalean/Panel/EstimandCharacterization/StaggeredTWFEDecomposition/FinitePanel.lean"},{"n":"Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.S0_LE","k":"def","m":"Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.FinitePanel","a":"Panel/EstimandCharacterization/StaggeredTWFEDecomposition","t":1,"d":"Late-vs-early early-treated window \\mathcal{T}_{\\ell e}^0 = {t : A_e ≤ t < A_ℓ}.","s":"{𝒢 : Type u_1} → [inst : Fintype 𝒢] → {T : ℕ} → Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.CohortPanel 𝒢 T → 𝒢 → 𝒢 → Finset (Fin T)","l":194,"f":"Causalean/Panel/EstimandCharacterization/StaggeredTWFEDecomposition/FinitePanel.lean"},{"n":"Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.S0_TN","k":"def","m":"Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.FinitePanel","a":"Panel/EstimandCharacterization/StaggeredTWFEDecomposition","t":1,"d":"Treated-versus-never untreated window \\mathcal{T}_g^0 = {t : t < A_g}.","s":"{𝒢 : Type u_1} → [inst : Fintype 𝒢] → {T : ℕ} → Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.CohortPanel 𝒢 T → 𝒢 → Finset (Fin T)","l":174,"f":"Causalean/Panel/EstimandCharacterization/StaggeredTWFEDecomposition/FinitePanel.lean"},{"n":"Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.S1_EL","k":"def","m":"Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.FinitePanel","a":"Panel/EstimandCharacterization/StaggeredTWFEDecomposition","t":1,"d":"Early-vs-late treated window \\mathcal{T}_{e\\ell}^1 = {t : A_e ≤ t < A_ℓ}.","s":"{𝒢 : Type u_1} → [inst : Fintype 𝒢] → {T : ℕ} → Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.CohortPanel 𝒢 T → 𝒢 → 𝒢 → Finset (Fin T)","l":189,"f":"Causalean/Panel/EstimandCharacterization/StaggeredTWFEDecomposition/FinitePanel.lean"},{"n":"Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.S1_LE","k":"def","m":"Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.FinitePanel","a":"Panel/EstimandCharacterization/StaggeredTWFEDecomposition","t":1,"d":"Late-vs-early both-treated window \\mathcal{T}_{\\ell e}^1 = {t : A_ℓ ≤ t}.","s":"{𝒢 : Type u_1} → [inst : Fintype 𝒢] → {T : ℕ} → Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.CohortPanel 𝒢 T → 𝒢 → Finset (Fin T)","l":199,"f":"Causalean/Panel/EstimandCharacterization/StaggeredTWFEDecomposition/FinitePanel.lean"},{"n":"Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.S1_TN","k":"def","m":"Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.FinitePanel","a":"Panel/EstimandCharacterization/StaggeredTWFEDecomposition","t":1,"d":"Treated-versus-never treated window \\mathcal{T}_g^1 = {t : A_g ≤ t}.","s":"{𝒢 : Type u_1} → [inst : Fintype 𝒢] → {T : ℕ} → Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.CohortPanel 𝒢 T → 𝒢 → Finset (Fin T)","l":179,"f":"Causalean/Panel/EstimandCharacterization/StaggeredTWFEDecomposition/FinitePanel.lean"},{"n":"Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.TN_pair_contribution_eq_lambda_delta","k":"theorem","m":"Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.Pairwise","a":"Panel/EstimandCharacterization/StaggeredTWFEDecomposition","t":1,"d":"TN numerator pair. For a treated cohort g with finite adoption date and a never-treated cohort u with infinite adoption date, the sum of the two ordered pairwise-covariance contributions between g and u in the finite cohort panel P equals the product of the TN comparison weight and the treated-versus-never contrast, λ_TN P g u · Δ_TN P g u.","s":"∀ {𝒢 : Type u_1} [inst : Fintype 𝒢] [DecidableEq 𝒢] {T : ℕ} (P : Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.CohortPanel 𝒢 T) {g u : 𝒢}, Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.AdoptionDate.isFin (P.A g) → Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.AdoptionDate.isInf (P.A u) → Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.numPairContribution P g u + Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.numPairContribution P u g = Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.lambdaTN P g u * Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.Δ_TN P g u","l":1002,"f":"Causalean/Panel/EstimandCharacterization/StaggeredTWFEDecomposition/Pairwise.lean"},{"n":"Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.TN_pair_vd_contribution_eq_gap","k":"theorem","m":"Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.Pairwise","a":"Panel/EstimandCharacterization/StaggeredTWFEDecomposition","t":2,"d":"TN denominator pair: combining the two ordered pairwise-variance contributions gives the treated-vs-never raw denominator factor.","s":"∀ {𝒢 : Type u_1} [inst : Fintype 𝒢] [DecidableEq 𝒢] {T : ℕ} (P : Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.CohortPanel 𝒢 T) {g u : 𝒢}, Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.AdoptionDate.isInf (P.A u) → Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.vdPairContribution P g u + Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.vdPairContribution P u g = P.p g * P.p u * Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.q P g u * (1 - Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.q P g u)","l":960,"f":"Causalean/Panel/EstimandCharacterization/StaggeredTWFEDecomposition/Pairwise.lean"},{"n":"Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.TT_pair_contribution_eq_lambda_delta_sum","k":"theorem","m":"Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.Pairwise","a":"Panel/EstimandCharacterization/StaggeredTWFEDecomposition","t":2,"d":"Treated-treated numerator pair: combining the two ordered pairwise-covariance contributions splits into the EL and LE comparison windows.","s":"∀ {𝒢 : Type u_1} [inst : Fintype 𝒢] [DecidableEq 𝒢] {T : ℕ} (P : Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.CohortPanel 𝒢 T) {e ℓ : 𝒢}, P.A e < P.A ℓ → Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.AdoptionDate.isFin (P.A ℓ) → Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.numPairContribution P e ℓ + Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.numPairContribution P ℓ e = Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.lambdaEL P e ℓ * Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.Δ_EL P e ℓ + Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.lambdaLE P e ℓ * Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.Δ_LE P e ℓ","l":1028,"f":"Causalean/Panel/EstimandCharacterization/StaggeredTWFEDecomposition/Pairwise.lean"},{"n":"Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.TT_pair_vd_contribution_eq_gap","k":"theorem","m":"Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.Pairwise","a":"Panel/EstimandCharacterization/StaggeredTWFEDecomposition","t":2,"d":"Treated-treated denominator pair: combining the two ordered pairwise-variance contributions gives the timing-pair raw denominator factor.","s":"∀ {𝒢 : Type u_1} [inst : Fintype 𝒢] [DecidableEq 𝒢] {T : ℕ} (P : Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.CohortPanel 𝒢 T) {e ℓ : 𝒢}, P.A e < P.A ℓ → Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.vdPairContribution P e ℓ + Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.vdPairContribution P ℓ e = P.p e * P.p ℓ * Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.q P e ℓ * (1 - Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.q P e ℓ)","l":981,"f":"Causalean/Panel/EstimandCharacterization/StaggeredTWFEDecomposition/Pairwise.lean"},{"n":"Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.VD","k":"def","m":"Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.FinitePanel","a":"Panel/EstimandCharacterization/StaggeredTWFEDecomposition","t":1,"d":"Residualized treatment variance V_D := ∑_{g,t} (p_g/T) \\widetilde{D}_{gt}^2.","s":"{𝒢 : Type u_1} → [inst : Fintype 𝒢] → {T : ℕ} → Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.CohortPanel 𝒢 T → ℝ","l":155,"f":"Causalean/Panel/EstimandCharacterization/StaggeredTWFEDecomposition/FinitePanel.lean"},{"n":"Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.VD_eq_pairwise_centeredD","k":"theorem","m":"Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.Pairwise","a":"Panel/EstimandCharacterization/StaggeredTWFEDecomposition","t":1,"d":"For a cohort panel, the residualized-treatment variance VD P equals the sum, over all ordered pairs of cohorts, of their pairwise centered-treatment contribution vdPairContribution P g u.","s":"∀ {𝒢 : Type u_1} [inst : Fintype 𝒢] [DecidableEq 𝒢] {T : ℕ} (P : Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.CohortPanel 𝒢 T), Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.VD P = ∑ g, ∑ u, Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.vdPairContribution P g u","l":383,"f":"Causalean/Panel/EstimandCharacterization/StaggeredTWFEDecomposition/Pairwise.lean"},{"n":"Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.Ybar","k":"def","m":"Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.FinitePanel","a":"Panel/EstimandCharacterization/StaggeredTWFEDecomposition","t":1,"d":"Window mean \\overline{Y}_{g,S} := |S|⁻¹ ∑_{t ∈ S} Y_{gt}. Defined unconditionally; nonemptiness S.Nonempty is supplied as a hypothesis when used.","s":"{𝒢 : Type u_1} → [inst : Fintype 𝒢] → {T : ℕ} → Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.CohortPanel 𝒢 T → 𝒢 → Finset (Fin T) → ℝ","l":165,"f":"Causalean/Panel/EstimandCharacterization/StaggeredTWFEDecomposition/FinitePanel.lean"},{"n":"Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.Ybar0","k":"def","m":"Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.Causal","a":"Panel/EstimandCharacterization/StaggeredTWFEDecomposition","t":1,"d":"Window mean of a potential-outcome map Y* : 𝒢 → Fin T → ℝ, Ȳ_{g(*),S} := |S|⁻¹ ∑_{t ∈ S} Y*_{gt}. Defined unconditionally; nonemptiness of S is supplied at theorem-use time.","s":"{𝒢 : Type u_1} → {T : ℕ} → (𝒢 → Fin T → ℝ) → 𝒢 → Finset (Fin T) → ℝ","l":96,"f":"Causalean/Panel/EstimandCharacterization/StaggeredTWFEDecomposition/Causal.lean"},{"n":"Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.admissible","k":"def","m":"Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.FinitePanel","a":"Panel/EstimandCharacterization/StaggeredTWFEDecomposition","t":1,"d":"Admissibility predicate for a tagged cohort pair (tag, g₁, g₂). For TN: g₁ eventually treated and g₂ never treated, both with positive shares. For EL/LE: A_{g₁} < A_{g₂} < ∞, both with positive shares.","s":"{𝒢 : Type u_1} → [inst : Fintype 𝒢] → {T : ℕ} → Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.CohortPanel 𝒢 T → Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.CompTag × 𝒢 × 𝒢 → Prop","l":265,"f":"Causalean/Panel/EstimandCharacterization/StaggeredTWFEDecomposition/FinitePanel.lean"},{"n":"Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.adoption_pair_sum_decomp","k":"theorem","m":"Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.AlgebraicDecomposition","a":"Panel/EstimandCharacterization/StaggeredTWFEDecomposition","t":2,"d":"When a pairwise cohort contribution is zero for cohorts sharing an adoption date, its total over all cohort pairs decomposes into the four possible ordered timing comparisons.","s":"∀ {𝒢 : Type u_1} [inst : Fintype 𝒢] {T : ℕ} (P : Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.CohortPanel 𝒢 T) (f : 𝒢 → 𝒢 → ℝ), (∀ (g u : 𝒢), P.A g = P.A u → f g u = 0) → ∑ g, ∑ u, f g u = (((∑ g, ∑ u, if Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.AdoptionDate.isFin (P.A g) ∧ Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.AdoptionDate.isInf (P.A u) then f g u else 0) + ∑ g, ∑ u, if Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.AdoptionDate.isFin (P.A u) ∧ Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.AdoptionDate.isInf (P.A g) then f g u else 0) + ∑ g, ∑ u, if P.A g < P.A u ∧ Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.AdoptionDate.isFin (P.A u) then f g u else 0) + ∑ g, ∑ u, if P.A u < P.A g ∧ Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.AdoptionDate.isFin (P.A g) then f g u else 0","l":372,"f":"Causalean/Panel/EstimandCharacterization/StaggeredTWFEDecomposition/AlgebraicDecomposition.lean"},{"n":"Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.barD","k":"def","m":"Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.FinitePanel","a":"Panel/EstimandCharacterization/StaggeredTWFEDecomposition","t":1,"d":"Cohort treatment share \\overline{D}_g := T⁻¹ ∑_t D_{gt}.","s":"{𝒢 : Type u_1} → [inst : Fintype 𝒢] → {T : ℕ} → Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.CohortPanel 𝒢 T → 𝒢 → ℝ","l":113,"f":"Causalean/Panel/EstimandCharacterization/StaggeredTWFEDecomposition/FinitePanel.lean"},{"n":"Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.barD_eq_unitMean","k":"theorem","m":"Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.FinitePanel","a":"Panel/EstimandCharacterization/StaggeredTWFEDecomposition","t":2,"d":"Goodman-Bacon's cohort treatment share is the shared unit mean.","s":"∀ {𝒢 : Type u_1} [inst : Fintype 𝒢] {T : ℕ} (P : Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.CohortPanel 𝒢 T) (g : 𝒢), Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.barD P g = Causalean.Panel.WeightedTwoWayPanel.unitMean (Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.D P) g","l":127,"f":"Causalean/Panel/EstimandCharacterization/StaggeredTWFEDecomposition/FinitePanel.lean"},{"n":"Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.barD_eq_zero_of_isInf","k":"theorem","m":"Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.AlgebraicDecomposition","a":"Panel/EstimandCharacterization/StaggeredTWFEDecomposition","t":2,"d":"A never-treated cohort has zero average treatment over the panel.","s":"∀ {𝒢 : Type u_1} [inst : Fintype 𝒢] {T : ℕ} (P : Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.CohortPanel 𝒢 T) {g : 𝒢}, Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.AdoptionDate.isInf (P.A g) → Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.barD P g = 0","l":202,"f":"Causalean/Panel/EstimandCharacterization/StaggeredTWFEDecomposition/AlgebraicDecomposition.lean"},{"n":"Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.betaTWFE","k":"def","m":"Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.FinitePanel","a":"Panel/EstimandCharacterization/StaggeredTWFEDecomposition","t":1,"d":"Population TWFE coefficient (finite-cell form) from def:po-estimand-goodman-bacon-twfe. Defined unconditionally; positivity of the denominator is supplied at theorem-use time via hVD_pos.","s":"{𝒢 : Type u_1} → [inst : Fintype 𝒢] → {T : ℕ} → Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.CohortPanel 𝒢 T → ℝ","l":159,"f":"Causalean/Panel/EstimandCharacterization/StaggeredTWFEDecomposition/FinitePanel.lean"},{"n":"Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.binary_time_cov_filter_mean","k":"theorem","m":"Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.Pairwise","a":"Panel/EstimandCharacterization/StaggeredTWFEDecomposition","t":2,"d":"For an indicator of a nonempty set of periods, the time-average product of its centered value and another quantity equals its variance times the difference between the selected-period and unselected-period averages of that quantity.","s":"∀ {T : ℕ}, 0 < T → ∀ (p : Fin T → Prop) [inst : DecidablePred p] (z : Fin T → ℝ), ↑(Finset.filter p Finset.univ).card ≠ 0 → (↑T)⁻¹ * ∑ t, ((if p t then 1 else 0) - (↑T)⁻¹ * ∑ t, if p t then 1 else 0) * z t = ((↑T)⁻¹ * ∑ t, if p t then 1 else 0) * (1 - (↑T)⁻¹ * ∑ t, if p t then 1 else 0) * ((↑(Finset.filter p Finset.univ).card)⁻¹ * ∑ t with p t, z t - (↑{t | ¬p t}.card)⁻¹ * ∑ t with ¬p t, z t)","l":205,"f":"Causalean/Panel/EstimandCharacterization/StaggeredTWFEDecomposition/Pairwise.lean"},{"n":"Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.binary_time_variance","k":"theorem","m":"Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.Pairwise","a":"Panel/EstimandCharacterization/StaggeredTWFEDecomposition","t":2,"d":"For a binary quantity observed over the panel's time periods, its average squared deviation from its time mean equals that mean times one minus that mean.","s":"∀ {T : ℕ}, 0 < T → ∀ (x : Fin T → ℝ), (∀ (t : Fin T), x t ^ 2 = x t) → (↑T)⁻¹ * ∑ t, (x t - (↑T)⁻¹ * ∑ t, x t) ^ 2 = ((↑T)⁻¹ * ∑ t, x t) * (1 - (↑T)⁻¹ * ∑ t, x t)","l":167,"f":"Causalean/Panel/EstimandCharacterization/StaggeredTWFEDecomposition/Pairwise.lean"},{"n":"Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.bridge_Dtilde_sq_eq_VD","k":"theorem","m":"Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.PopulationBridge","a":"Panel/EstimandCharacterization/StaggeredTWFEDecomposition","t":1,"d":"On a probability space carrying cohort label G, period label T_rv, treatment D, and outcome Y, assume D is binary almost everywhere, there is at least one time period, every cohort has strictly positive population mass, the cohort masses sum to one, each cohort's mass is split evenly across periods (a balanced panel law), on each cohort-period cell D is almost-everywhere equal to its own cell mean (cell-measurability of D), and the population per-cell residualized treatment panelDtilde agrees with the finite-panel residualized treatment Dtilde. Then the population second moment of the residualized-treatment witness — the FWL denominator ∫ Vtilde² dμ — equals the finite-cell Goodman-Bacon denominator VD of the induced finite panel.","s":"∀ {Ω : Type u_1} {𝒢 : Type u_2} [inst : MeasurableSpace Ω] [inst_1 : Fintype 𝒢] [inst_2 : DecidableEq 𝒢] [inst_3 : MeasurableSpace 𝒢] [inst_4 : MeasurableSingletonClass 𝒢] {T : ℕ} (μ : MeasureTheory.Measure Ω) [inst_5 : MeasureTheory.IsProbabilityMeasure μ] (D Y : Ω → ℝ) (G : Ω → 𝒢) (T_rv : Ω → Fin T) (A : 𝒢 → WithTop (Fin T)) (G_meas : Measurable G) (T_meas : Measurable T_rv) (D_meas : Measurable D) (D_binary : ∀ᵐ (ω : Ω) ∂μ, D ω = 0 ∨ D ω = 1) (B_balanced : Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.IsBalancedPanelLaw μ G T_rv) (hT_pos : 0 < T) (hp_pos : ∀ (g : 𝒢), 0 < Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.cohortMass μ G g) (hp_sum : ∑ g, Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.cohortMass μ G g = 1), (∀ (g : 𝒢) (t : Fin T), Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.cellMass μ G T_rv g t = Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.cohortMass μ G g / ↑T) → (∀ (g : 𝒢) (t : Fin T), ∀ᵐ (ω : Ω) ∂μ.restrict {ω' | G ω' = g ∧ T_rv ω' = t}, D ω = (∫ (ω' : Ω), D ω' * {ω' | G ω' = g ∧ T_rv ω' = t}.indicator (fun x => 1) ω' ∂μ) / Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.cellMass μ G T_rv g t) → (∀ (g : 𝒢) (t : Fin T), Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.panelDtilde μ D G T_rv g t = Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.Dtilde (Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.panelOf μ Y G T_rv A hT_pos hp_pos hp_sum) g t) → ∫ (ω : Ω), (Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.residWitnessD_panel μ D G T_rv G_meas T_meas D_meas D_binary B_balanced).Vtilde ω * (Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.residWitnessD_panel μ D G T_rv G_meas T_meas D_meas D_binary B_balanced).Vtilde ω ∂μ = Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.VD (Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.panelOf μ Y G T_rv A hT_pos hp_pos hp_sum)","l":59,"f":"Causalean/Panel/EstimandCharacterization/StaggeredTWFEDecomposition/PopulationBridge.lean"},{"n":"Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.bridge_VD_pos_iff_Dtilde_sq_pos","k":"theorem","m":"Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.PopulationBridge","a":"Panel/EstimandCharacterization/StaggeredTWFEDecomposition","t":1,"d":"On a probability space carrying cohort label G, period label T_rv, treatment D, and outcome Y, assume D is binary almost everywhere, there is at least one time period, every cohort has strictly positive population mass, the cohort masses sum to one, each cohort's mass is split evenly across periods (a balanced panel law), on each cohort-period cell D is almost-everywhere equal to its own cell mean (cell-measurability of D), and the population per-cell residualized treatment panelDtilde agrees with the finite-panel residualized treatment Dtilde. Then the finite-cell Goodman-Bacon denominator VD is strictly positive if and only if the population FWL denominator ∫ Vtilde² dμ is strictly positive.","s":"∀ {Ω : Type u_1} {𝒢 : Type u_2} [inst : MeasurableSpace Ω] [inst_1 : Fintype 𝒢] [inst_2 : DecidableEq 𝒢] [inst_3 : MeasurableSpace 𝒢] [inst_4 : MeasurableSingletonClass 𝒢] {T : ℕ} (μ : MeasureTheory.Measure Ω) [inst_5 : MeasureTheory.IsProbabilityMeasure μ] (D Y : Ω → ℝ) (G : Ω → 𝒢) (T_rv : Ω → Fin T) (A : 𝒢 → WithTop (Fin T)) (G_meas : Measurable G) (T_meas : Measurable T_rv) (D_meas : Measurable D) (D_binary : ∀ᵐ (ω : Ω) ∂μ, D ω = 0 ∨ D ω = 1) (B_balanced : Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.IsBalancedPanelLaw μ G T_rv) (hT_pos : 0 < T) (hp_pos : ∀ (g : 𝒢), 0 < Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.cohortMass μ G g) (hp_sum : ∑ g, Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.cohortMass μ G g = 1), (∀ (g : 𝒢) (t : Fin T), Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.cellMass μ G T_rv g t = Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.cohortMass μ G g / ↑T) → (∀ (g : 𝒢) (t : Fin T), ∀ᵐ (ω : Ω) ∂μ.restrict {ω' | G ω' = g ∧ T_rv ω' = t}, D ω = (∫ (ω' : Ω), D ω' * {ω' | G ω' = g ∧ T_rv ω' = t}.indicator (fun x => 1) ω' ∂μ) / Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.cellMass μ G T_rv g t) → (∀ (g : 𝒢) (t : Fin T), Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.panelDtilde μ D G T_rv g t = Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.Dtilde (Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.panelOf μ Y G T_rv A hT_pos hp_pos hp_sum) g t) → (0 < Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.VD (Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.panelOf μ Y G T_rv A hT_pos hp_pos hp_sum) ↔ 0 < ∫ (ω : Ω), (Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.residWitnessD_panel μ D G T_rv G_meas T_meas D_meas D_binary B_balanced).Vtilde ω * (Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.residWitnessD_panel μ D G T_rv G_meas T_meas D_meas D_binary B_balanced).Vtilde ω ∂μ)","l":126,"f":"Causalean/Panel/EstimandCharacterization/StaggeredTWFEDecomposition/PopulationBridge.lean"},{"n":"Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.bridge_finite_residualized_eq_twfe","k":"theorem","m":"Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.PopulationBridge","a":"Panel/EstimandCharacterization/StaggeredTWFEDecomposition","t":1,"d":"On a probability space carrying cohort label G, period label T_rv, treatment D, and outcome Y, assume D is binary almost everywhere, there is at least one time period, every cohort has strictly positive population mass, the cohort masses sum to one, each cohort's mass is split evenly across periods (a balanced panel law), on each cohort-period cell D is almost-everywhere equal to its own cell mean (cell-measurability of D), and the population per-cell residualized treatment panelDtilde agrees with the finite-panel residualized treatment Dtilde. Then the population residualized regression coefficient of D on Y, controlling for the cohort/period additive class panelClass, equals the finite-cell Goodman-Bacon two-way-fixed-effects coefficient betaTWFE of the induced finite panel.","s":"∀ {Ω : Type u_1} {𝒢 : Type u_2} [inst : MeasurableSpace Ω] [inst_1 : Fintype 𝒢] [inst_2 : DecidableEq 𝒢] [inst_3 : MeasurableSpace 𝒢] [inst_4 : MeasurableSingletonClass 𝒢] {T : ℕ} (μ : MeasureTheory.Measure Ω) [inst_5 : MeasureTheory.IsProbabilityMeasure μ] (D Y : Ω → ℝ) (G : Ω → 𝒢) (T_rv : Ω → Fin T) (A : 𝒢 → WithTop (Fin T)) (G_meas : Measurable G) (T_meas : Measurable T_rv) (D_meas : Measurable D) (D_binary : ∀ᵐ (ω : Ω) ∂μ, D ω = 0 ∨ D ω = 1) (Y_memLp : MeasureTheory.MemLp Y 2 μ) (B_balanced : Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.IsBalancedPanelLaw μ G T_rv) (hT_pos : 0 < T) (hp_pos : ∀ (g : 𝒢), 0 < Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.cohortMass μ G g) (hp_sum : ∑ g, Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.cohortMass μ G g = 1), (∀ (g : 𝒢) (t : Fin T), Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.cellMass μ G T_rv g t = Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.cohortMass μ G g / ↑T) → (∀ (g : 𝒢) (t : Fin T), ∀ᵐ (ω : Ω) ∂μ.restrict {ω' | G ω' = g ∧ T_rv ω' = t}, D ω = (∫ (ω' : Ω), D ω' * {ω' | G ω' = g ∧ T_rv ω' = t}.indicator (fun x => 1) ω' ∂μ) / Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.cellMass μ G T_rv g t) → (∀ (g : 𝒢) (t : Fin T), Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.panelDtilde μ D G T_rv g t = Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.Dtilde (Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.panelOf μ Y G T_rv A hT_pos hp_pos hp_sum) g t) → Causalean.Panel.residualizedCoefficient μ (Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.panelClass μ G T_rv G_meas T_meas) (Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.residWitnessY_panel μ Y G T_rv G_meas T_meas Y_memLp B_balanced) (Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.residWitnessD_panel μ D G T_rv G_meas T_meas D_meas D_binary B_balanced) = Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.betaTWFE (Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.panelOf μ Y G T_rv A hT_pos hp_pos hp_sum)","l":173,"f":"Causalean/Panel/EstimandCharacterization/StaggeredTWFEDecomposition/PopulationBridge.lean"},{"n":"Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.cellMass","k":"def","m":"Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.Support.Basic","a":"Panel/EstimandCharacterization/StaggeredTWFEDecomposition/Support","t":1,"d":"B2. Cell mass. cellMass μ G T_rv g t = (μ {G = g ∧ T_rv = t}).toReal. Plays the role of ℙ(G = g, T = t) = p_g / |𝒯| in def:po-estimand-goodman-bacon-panel.","s":"{Ω : Type u_1} → {𝒢 : Type u_2} → [inst : MeasurableSpace Ω] → {T : ℕ} → MeasureTheory.Measure Ω → (Ω → 𝒢) → (Ω → Fin T) → 𝒢 → Fin T → ℝ","l":65,"f":"Causalean/Panel/EstimandCharacterization/StaggeredTWFEDecomposition/Support/Basic.lean"},{"n":"Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.cellMass_nonneg","k":"theorem","m":"Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.Support.Basic","a":"Panel/EstimandCharacterization/StaggeredTWFEDecomposition/Support","t":2,"d":"Cell mass is nonnegative.","s":"∀ {Ω : Type u_1} {𝒢 : Type u_2} [inst : MeasurableSpace Ω] {T : ℕ} (μ : MeasureTheory.Measure Ω) (G : Ω → 𝒢) (T_rv : Ω → Fin T) (g : 𝒢) (t : Fin T), 0 ≤ Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.cellMass μ G T_rv g t","l":218,"f":"Causalean/Panel/EstimandCharacterization/StaggeredTWFEDecomposition/Support/Basic.lean"},{"n":"Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.cellMean","k":"def","m":"Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.Support.Basic","a":"Panel/EstimandCharacterization/StaggeredTWFEDecomposition/Support","t":1,"d":"B2. Cell mean. cellMean μ Y G T_rv g t plays the role of Y_{gt} = E[Y | G = g, T = t] from the LaTeX panel definition. On zero-mass cells the value is 0 by Mathlib's 0/0 = 0 convention.","s":"{Ω : Type u_1} → {𝒢 : Type u_2} → [inst : MeasurableSpace Ω] → {T : ℕ} → MeasureTheory.Measure Ω → (Ω → ℝ) → (Ω → 𝒢) → (Ω → Fin T) → 𝒢 → Fin T → ℝ","l":83,"f":"Causalean/Panel/EstimandCharacterization/StaggeredTWFEDecomposition/Support/Basic.lean"},{"n":"Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.cellMean_eq_cellBridge","k":"theorem","m":"Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.Support.Basic","a":"Panel/EstimandCharacterization/StaggeredTWFEDecomposition/Support","t":2,"d":"The panel cell mean for a cohort and period equals the generic cell-mean operator applied to the joint cohort-period cell.","s":"∀ {Ω : Type u_1} {𝒢 : Type u_2} [inst : MeasurableSpace Ω] {T : ℕ} (μ : MeasureTheory.Measure Ω) (Y : Ω → ℝ) (G : Ω → 𝒢) (T_rv : Ω → Fin T) (g : 𝒢) (t : Fin T), Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.cellMean μ Y G T_rv g t = Causalean.Panel.CellBridge.cellMean μ Y (fun ω => (G ω, T_rv ω)) (g, t)","l":93,"f":"Causalean/Panel/EstimandCharacterization/StaggeredTWFEDecomposition/Support/Basic.lean"},{"n":"Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.centeredD","k":"def","m":"Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.Pairwise","a":"Panel/EstimandCharacterization/StaggeredTWFEDecomposition","t":1,"d":"Cohort-demeaned treatment path D_{gt} − \\overline D_g.","s":"{𝒢 : Type u_1} → [inst : Fintype 𝒢] → {T : ℕ} → Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.CohortPanel 𝒢 T → 𝒢 → Fin T → ℝ","l":119,"f":"Causalean/Panel/EstimandCharacterization/StaggeredTWFEDecomposition/Pairwise.lean"},{"n":"Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.cohortBarD","k":"def","m":"Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.Support.Basic","a":"Panel/EstimandCharacterization/StaggeredTWFEDecomposition/Support","t":1,"d":"B2. Cohort mean treatment share. cohortBarD μ D G g plays the role of \\overline{D}_g = E[D | G = g].","s":"{Ω : Type u_1} → {𝒢 : Type u_2} → [inst : MeasurableSpace Ω] → MeasureTheory.Measure Ω → (Ω → ℝ) → (Ω → 𝒢) → 𝒢 → ℝ","l":108,"f":"Causalean/Panel/EstimandCharacterization/StaggeredTWFEDecomposition/Support/Basic.lean"},{"n":"Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.cohortBarD_mul_cohortMass","k":"theorem","m":"Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.Support.Integrals","a":"Panel/EstimandCharacterization/StaggeredTWFEDecomposition/Support","t":2,"d":"cohortBarD g · cohortMass g = ∫ D · 𝟙{G=g} dμ.","s":"∀ {Ω : Type u_1} {𝒢 : Type u_2} [inst : MeasurableSpace Ω] (μ : MeasureTheory.Measure Ω) [MeasureTheory.IsFiniteMeasure μ] (D : Ω → ℝ) (G : Ω → 𝒢) (g : 𝒢), Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.cohortBarD μ D G g * Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.cohortMass μ G g = ∫ (ω : Ω), D ω * {ω' | G ω' = g}.indicator (fun x => 1) ω ∂μ","l":188,"f":"Causalean/Panel/EstimandCharacterization/StaggeredTWFEDecomposition/Support/Integrals.lean"},{"n":"Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.cohortMass","k":"def","m":"Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.Support.Basic","a":"Panel/EstimandCharacterization/StaggeredTWFEDecomposition/Support","t":1,"d":"B2. Cohort mass. cohortMass μ G g = (μ {G = g}).toReal. Plays the role of p_g. Equals ∑ t, cellMass μ G T_rv g t under the balanced-law hypothesis.","s":"{Ω : Type u_1} → {𝒢 : Type u_2} → [inst : MeasurableSpace Ω] → MeasureTheory.Measure Ω → (Ω → 𝒢) → 𝒢 → ℝ","l":72,"f":"Causalean/Panel/EstimandCharacterization/StaggeredTWFEDecomposition/Support/Basic.lean"},{"n":"Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.cohortMass_nonneg","k":"theorem","m":"Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.Support.Basic","a":"Panel/EstimandCharacterization/StaggeredTWFEDecomposition/Support","t":2,"d":"Cohort mass is nonnegative.","s":"∀ {Ω : Type u_1} {𝒢 : Type u_2} [inst : MeasurableSpace Ω] (μ : MeasureTheory.Measure Ω) (G : Ω → 𝒢) (g : 𝒢), 0 ≤ Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.cohortMass μ G g","l":223,"f":"Causalean/Panel/EstimandCharacterization/StaggeredTWFEDecomposition/Support/Basic.lean"},{"n":"Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.cohortWeights","k":"def","m":"Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.FinitePanel","a":"Panel/EstimandCharacterization/StaggeredTWFEDecomposition","t":1,"d":"Cohort shares as unit weights for the shared weighted two-way panel module.","s":"{𝒢 : Type u_1} → [inst : Fintype 𝒢] → {T : ℕ} → Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.CohortPanel 𝒢 T → Causalean.Panel.WeightedTwoWayPanel.UnitWeights 𝒢","l":121,"f":"Causalean/Panel/EstimandCharacterization/StaggeredTWFEDecomposition/FinitePanel.lean"},{"n":"Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.cohort_integral_div_mul_cohortMass","k":"theorem","m":"Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.Support.Integrals","a":"Panel/EstimandCharacterization/StaggeredTWFEDecomposition/Support","t":2,"d":"Defining identity for cohortBarD: dividing by cohort mass and multiplying back recovers the cohort-indicator-weighted integral. On zero-mass cohorts both sides are zero (a.e.-vanishing indicator).","s":"∀ {Ω : Type u_1} {𝒢 : Type u_2} [inst : MeasurableSpace Ω] (μ : MeasureTheory.Measure Ω) [MeasureTheory.IsFiniteMeasure μ] (F : Ω → ℝ) (G : Ω → 𝒢) (g : 𝒢), (∫ (ω : Ω), F ω * {ω' | G ω' = g}.indicator (fun x => 1) ω ∂μ) / Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.cohortMass μ G g * Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.cohortMass μ G g = ∫ (ω : Ω), F ω * {ω' | G ω' = g}.indicator (fun x => 1) ω ∂μ","l":116,"f":"Causalean/Panel/EstimandCharacterization/StaggeredTWFEDecomposition/Support/Integrals.lean"},{"n":"Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.contrast","k":"def","m":"Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.FinitePanel","a":"Panel/EstimandCharacterization/StaggeredTWFEDecomposition","t":1,"d":"Unified 2x2 contrast on the full index CompTag × 𝒢 × 𝒢: returns the matching Δ_TN/Δ_EL/Δ_LE on admissible triples and 0 otherwise.","s":"{𝒢 : Type u_1} → [inst : Fintype 𝒢] → {T : ℕ} → Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.CohortPanel 𝒢 T → Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.CompTag × 𝒢 × 𝒢 → ℝ","l":298,"f":"Causalean/Panel/EstimandCharacterization/StaggeredTWFEDecomposition/FinitePanel.lean"},{"n":"Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.contrastCausal","k":"def","m":"Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.CausalDecomposition","a":"Panel/EstimandCharacterization/StaggeredTWFEDecomposition","t":1,"d":"The causal 2x2 contrast on the full index CompTag × 𝒢 × 𝒢: on an admissible comparison it returns the potential-outcome window contrast identified by the Layer C corollaries — ATT_window for TN/EL and the bad-comparison adjustment for LE — and 0 otherwise. This is the causal counterpart of the algebraic contrast.","s":"{𝒢 : Type u_1} → [inst : Fintype 𝒢] → {T : ℕ} → Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.CohortPanel 𝒢 T → (𝒢 → Fin T → ℝ) → (𝒢 → Fin T → ℝ) → Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.CompTag × 𝒢 × 𝒢 → ℝ","l":47,"f":"Causalean/Panel/EstimandCharacterization/StaggeredTWFEDecomposition/CausalDecomposition.lean"},{"n":"Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.contrast_eq_contrastCausal","k":"theorem","m":"Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.CausalDecomposition","a":"Panel/EstimandCharacterization/StaggeredTWFEDecomposition","t":2,"d":"On an admissible comparison, the algebraic contrast equals the causal potential-outcome contrast, by the Layer C corollaries.","s":"∀ {𝒢 : Type u_1} [inst : Fintype 𝒢] {T : ℕ} (P : Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.CohortPanel 𝒢 T) (Y0 Y1 : 𝒢 → Fin T → ℝ), Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.CausalAssumptions P Y0 Y1 → ∀ {k : Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.CompTag × 𝒢 × 𝒢}, Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.admissible P k → Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.contrast P k = Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.contrastCausal P Y0 Y1 k","l":65,"f":"Causalean/Panel/EstimandCharacterization/StaggeredTWFEDecomposition/CausalDecomposition.lean"},{"n":"Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.denom_per_cell_panel","k":"theorem","m":"Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.Support.PerCell","a":"Panel/EstimandCharacterization/StaggeredTWFEDecomposition/Support","t":2,"d":"Per-cell denominator identity. On the cell {G = g ∧ T_rv = t}, (D - panelPropensity)(ω) = panelDtilde μ D G T_rv g t; squaring and integrating over the cell gives","s":"∀ {Ω : Type u_1} {𝒢 : Type u_2} [inst : MeasurableSpace Ω] [inst_1 : Fintype 𝒢] [inst_2 : MeasurableSpace 𝒢] [MeasurableSingletonClass 𝒢] {T : ℕ} (μ : MeasureTheory.Measure Ω) [MeasureTheory.IsFiniteMeasure μ] (D : Ω → ℝ) (G : Ω → 𝒢) (T_rv : Ω → Fin T), Measurable G → Measurable T_rv → ∀ (g : 𝒢) (t : Fin T), (∀ᵐ (ω : Ω) ∂μ.restrict {ω' | G ω' = g ∧ T_rv ω' = t}, D ω = (∫ (ω' : Ω), D ω' * {ω' | G ω' = g ∧ T_rv ω' = t}.indicator (fun x => 1) ω' ∂μ) / Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.cellMass μ G T_rv g t) → ∫ (ω : Ω), (D ω - Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.panelPropensity μ D G T_rv ω) * (D ω - Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.panelPropensity μ D G T_rv ω) * {ω' | G ω' = g ∧ T_rv ω' = t}.indicator (fun x => 1) ω ∂μ = Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.cellMass μ G T_rv g t * Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.panelDtilde μ D G T_rv g t ^ 2","l":48,"f":"Causalean/Panel/EstimandCharacterization/StaggeredTWFEDecomposition/Support/PerCell.lean"},{"n":"Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.disjoint_union_mean_eq_card_weighted_mean","k":"theorem","m":"Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.Pairwise","a":"Panel/EstimandCharacterization/StaggeredTWFEDecomposition","t":2,"d":"When two finite sets are disjoint and the second is nonempty, the mean of a quantity over their union is the cardinality-weighted combination of its means over the two sets.","s":"∀ {α : Type u_2} [inst : DecidableEq α] (A B : Finset α) (z : α → ℝ), Disjoint A B → ↑B.card ≠ 0 → (↑(A ∪ B).card)⁻¹ * ∑ t ∈ A ∪ B, z t = ↑A.card / ↑(A ∪ B).card * ((↑A.card)⁻¹ * ∑ t ∈ A, z t) + (1 - ↑A.card / ↑(A ∪ B).card) * ((↑B.card)⁻¹ * ∑ t ∈ B, z t)","l":599,"f":"Causalean/Panel/EstimandCharacterization/StaggeredTWFEDecomposition/Pairwise.lean"},{"n":"Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.gap_sums_eq_VD","k":"theorem","m":"Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.AlgebraicDecomposition","a":"Panel/EstimandCharacterization/StaggeredTWFEDecomposition","t":2,"d":"The total of cohort-share products times each pair's treatment-rate gap and one minus that gap, over treated-versus-never and ordered early-versus-later pairs, equals the residualized treatment variance.","s":"∀ {𝒢 : Type u_1} [inst : Fintype 𝒢] [DecidableEq 𝒢] {T : ℕ} (P : Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.CohortPanel 𝒢 T), ((∑ g, ∑ u, if Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.AdoptionDate.isFin (P.A g) ∧ Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.AdoptionDate.isInf (P.A u) then P.p g * P.p u * Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.q P g u * (1 - Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.q P g u) else 0) + ∑ e, ∑ ℓ, if P.A e < P.A ℓ ∧ Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.AdoptionDate.isFin (P.A ℓ) then P.p e * P.p ℓ * Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.q P e ℓ * (1 - Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.q P e ℓ) else 0) = Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.VD P","l":483,"f":"Causalean/Panel/EstimandCharacterization/StaggeredTWFEDecomposition/AlgebraicDecomposition.lean"},{"n":"Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.indicator_cohort_memLp","k":"theorem","m":"Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.Support.Integrals","a":"Panel/EstimandCharacterization/StaggeredTWFEDecomposition/Support","t":2,"d":"Cohort indicator 𝟙{G = g} is in MemLp 2 μ (bounded + finite measure).","s":"∀ {Ω : Type u_1} {𝒢 : Type u_2} [inst : MeasurableSpace Ω] [inst_1 : MeasurableSpace 𝒢] [MeasurableSingletonClass 𝒢] (μ : MeasureTheory.Measure Ω) [MeasureTheory.IsFiniteMeasure μ] (G : Ω → 𝒢), Measurable G → ∀ (g : 𝒢), MeasureTheory.MemLp (fun ω => {ω' | G ω' = g}.indicator (fun x => 1) ω) 2 μ","l":41,"f":"Causalean/Panel/EstimandCharacterization/StaggeredTWFEDecomposition/Support/Integrals.lean"},{"n":"Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.indicator_panel_cell_memLp","k":"theorem","m":"Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.Support.Integrals","a":"Panel/EstimandCharacterization/StaggeredTWFEDecomposition/Support","t":2,"d":"Joint cell indicator 𝟙{G=g ∧ T_rv=t} is in MemLp 2 μ.","s":"∀ {Ω : Type u_1} {𝒢 : Type u_2} [inst : MeasurableSpace Ω] [inst_1 : MeasurableSpace 𝒢] [MeasurableSingletonClass 𝒢] {T : ℕ} (μ : MeasureTheory.Measure Ω) [MeasureTheory.IsFiniteMeasure μ] (G : Ω → 𝒢) (T_rv : Ω → Fin T), Measurable G → Measurable T_rv → ∀ (g : 𝒢) (t : Fin T), MeasureTheory.MemLp (fun ω => {ω' | G ω' = g ∧ T_rv ω' = t}.indicator (fun x => 1) ω) 2 μ","l":56,"f":"Causalean/Panel/EstimandCharacterization/StaggeredTWFEDecomposition/Support/Integrals.lean"},{"n":"Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.indicator_period_memLp","k":"theorem","m":"Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.Support.Integrals","a":"Panel/EstimandCharacterization/StaggeredTWFEDecomposition/Support","t":2,"d":"Period indicator 𝟙{T_rv = t} is in MemLp 2 μ.","s":"∀ {Ω : Type u_1} [inst : MeasurableSpace Ω] {T : ℕ} (μ : MeasureTheory.Measure Ω) [MeasureTheory.IsFiniteMeasure μ] (T_rv : Ω → Fin T), Measurable T_rv → ∀ (t : Fin T), MeasureTheory.MemLp (fun ω => {ω' | T_rv ω' = t}.indicator (fun x => 1) ω) 2 μ","l":48,"f":"Causalean/Panel/EstimandCharacterization/StaggeredTWFEDecomposition/Support/Integrals.lean"},{"n":"Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.instDecidableEqCompTag","k":"instance","m":"Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.FinitePanel","a":"Panel/EstimandCharacterization/StaggeredTWFEDecomposition","t":2,"d":"","s":"DecidableEq Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.CompTag","l":60,"f":"Causalean/Panel/EstimandCharacterization/StaggeredTWFEDecomposition/FinitePanel.lean"},{"n":"Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.instFintypeCompTag","k":"instance","m":"Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.FinitePanel","a":"Panel/EstimandCharacterization/StaggeredTWFEDecomposition","t":2,"d":"The three Goodman-Bacon comparison tags form a finite type.","s":"Fintype Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.CompTag","l":62,"f":"Causalean/Panel/EstimandCharacterization/StaggeredTWFEDecomposition/FinitePanel.lean"},{"n":"Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.integral_cohort_indicator_one_eq_cohortMass","k":"theorem","m":"Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.Support.Integrals","a":"Panel/EstimandCharacterization/StaggeredTWFEDecomposition/Support","t":2,"d":"∫ 𝟙{G = g} dμ = cohortMass μ G g.","s":"∀ {Ω : Type u_1} {𝒢 : Type u_2} [inst : MeasurableSpace Ω] [inst_1 : MeasurableSpace 𝒢] [MeasurableSingletonClass 𝒢] (μ : MeasureTheory.Measure Ω) (G : Ω → 𝒢), Measurable G → ∀ (g : 𝒢), ∫ (ω : Ω), {ω' | G ω' = g}.indicator (fun x => 1) ω ∂μ = Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.cohortMass μ G g","l":77,"f":"Causalean/Panel/EstimandCharacterization/StaggeredTWFEDecomposition/Support/Integrals.lean"},{"n":"Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.integral_eq_sum_cohort","k":"theorem","m":"Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.Support.Integrals","a":"Panel/EstimandCharacterization/StaggeredTWFEDecomposition/Support","t":2,"d":"Integrate by summing over cohort cells: ∫ F dμ = ∑_g ∫ F · 𝟙{G = g} dμ.","s":"∀ {Ω : Type u_1} {𝒢 : Type u_2} [inst : MeasurableSpace Ω] [inst_1 : Fintype 𝒢] [inst_2 : MeasurableSpace 𝒢] [MeasurableSingletonClass 𝒢] (μ : MeasureTheory.Measure Ω) (F : Ω → ℝ) (G : Ω → 𝒢), Measurable G → MeasureTheory.Integrable F μ → ∫ (ω : Ω), F ω ∂μ = ∑ g, ∫ (ω : Ω), F ω * {ω' | G ω' = g}.indicator (fun x => 1) ω ∂μ","l":300,"f":"Causalean/Panel/EstimandCharacterization/StaggeredTWFEDecomposition/Support/Integrals.lean"},{"n":"Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.integral_eq_sum_panel_cell","k":"theorem","m":"Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.Support.Integrals","a":"Panel/EstimandCharacterization/StaggeredTWFEDecomposition/Support","t":2,"d":"Integrate by summing over cohort × period cells.","s":"∀ {Ω : Type u_1} {𝒢 : Type u_2} [inst : MeasurableSpace Ω] [inst_1 : Fintype 𝒢] [inst_2 : MeasurableSpace 𝒢] [MeasurableSingletonClass 𝒢] {T : ℕ} (μ : MeasureTheory.Measure Ω) (F : Ω → ℝ) (G : Ω → 𝒢) (T_rv : Ω → Fin T), Measurable G → Measurable T_rv → MeasureTheory.Integrable F μ → ∫ (ω : Ω), F ω ∂μ = ∑ g, ∑ t, ∫ (ω : Ω), F ω * {ω' | G ω' = g ∧ T_rv ω' = t}.indicator (fun x => 1) ω ∂μ","l":321,"f":"Causalean/Panel/EstimandCharacterization/StaggeredTWFEDecomposition/Support/Integrals.lean"},{"n":"Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.integral_eq_sum_period","k":"theorem","m":"Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.Support.Integrals","a":"Panel/EstimandCharacterization/StaggeredTWFEDecomposition/Support","t":2,"d":"Integrate by summing over period cells.","s":"∀ {Ω : Type u_1} [inst : MeasurableSpace Ω] {T : ℕ} (μ : MeasureTheory.Measure Ω) (F : Ω → ℝ) (T_rv : Ω → Fin T), Measurable T_rv → MeasureTheory.Integrable F μ → ∫ (ω : Ω), F ω ∂μ = ∑ t, ∫ (ω : Ω), F ω * {ω' | T_rv ω' = t}.indicator (fun x => 1) ω ∂μ","l":311,"f":"Causalean/Panel/EstimandCharacterization/StaggeredTWFEDecomposition/Support/Integrals.lean"},{"n":"Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.integral_mul_panelClass_eq_zero_of_axes","k":"theorem","m":"Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.Support.Integrals","a":"Panel/EstimandCharacterization/StaggeredTWFEDecomposition/Support","t":1,"d":"If a square-integrable random variable V has zero mean product against every cohort indicator 𝟙{G = g} and zero mean product against every period indicator 𝟙{T_rv = t}, then V is orthogonal, in the L²(μ) sense, to every unit/period-additive combination ∑_g cG(g)·𝟙{G = g} + ∑_t cT(t)·𝟙{T_rv = t}, i.e. to every member of the additive nuisance class panelClass.","s":"∀ {Ω : Type u_1} {𝒢 : Type u_2} [inst : MeasurableSpace Ω] [inst_1 : Fintype 𝒢] [inst_2 : MeasurableSpace 𝒢] [MeasurableSingletonClass 𝒢] {T : ℕ} (μ : MeasureTheory.Measure Ω) [MeasureTheory.IsFiniteMeasure μ] (V : Ω → ℝ) (G : Ω → 𝒢) (T_rv : Ω → Fin T), Measurable G → Measurable T_rv → MeasureTheory.MemLp V 2 μ → ∀ (cG : 𝒢 → ℝ) (cT : Fin T → ℝ), (∀ (g : 𝒢), ∫ (ω : Ω), V ω * {ω' | G ω' = g}.indicator (fun x => 1) ω ∂μ = 0) → (∀ (t : Fin T), ∫ (ω : Ω), V ω * {ω' | T_rv ω' = t}.indicator (fun x => 1) ω ∂μ = 0) → ∫ (ω : Ω), V ω * (∑ g, cG g * {ω' | G ω' = g}.indicator (fun x => 1) ω + ∑ t, cT t * {ω' | T_rv ω' = t}.indicator (fun x => 1) ω) ∂μ = 0","l":272,"f":"Causalean/Panel/EstimandCharacterization/StaggeredTWFEDecomposition/Support/Integrals.lean"},{"n":"Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.integral_panel_cell_indicator_one_eq_cellMass","k":"theorem","m":"Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.Support.Integrals","a":"Panel/EstimandCharacterization/StaggeredTWFEDecomposition/Support","t":2,"d":"∫ 𝟙{G = g ∧ T_rv = t} dμ = cellMass μ G T_rv g t.","s":"∀ {Ω : Type u_1} {𝒢 : Type u_2} [inst : MeasurableSpace Ω] [inst_1 : MeasurableSpace 𝒢] [MeasurableSingletonClass 𝒢] {T : ℕ} (μ : MeasureTheory.Measure Ω) (G : Ω → 𝒢) (T_rv : Ω → Fin T), Measurable G → Measurable T_rv → ∀ (g : 𝒢) (t : Fin T), ∫ (ω : Ω), {ω' | G ω' = g ∧ T_rv ω' = t}.indicator (fun x => 1) ω ∂μ = Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.cellMass μ G T_rv g t","l":94,"f":"Causalean/Panel/EstimandCharacterization/StaggeredTWFEDecomposition/Support/Integrals.lean"},{"n":"Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.integral_period_indicator_one_eq_periodMass","k":"theorem","m":"Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.Support.Integrals","a":"Panel/EstimandCharacterization/StaggeredTWFEDecomposition/Support","t":2,"d":"∫ 𝟙{T_rv = t} dμ = periodMass μ T_rv t.","s":"∀ {Ω : Type u_1} [inst : MeasurableSpace Ω] {T : ℕ} (μ : MeasureTheory.Measure Ω) (T_rv : Ω → Fin T), Measurable T_rv → ∀ (t : Fin T), ∫ (ω : Ω), {ω' | T_rv ω' = t}.indicator (fun x => 1) ω ∂μ = Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.periodMass μ T_rv t","l":85,"f":"Causalean/Panel/EstimandCharacterization/StaggeredTWFEDecomposition/Support/Integrals.lean"},{"n":"Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.lambdaEL","k":"def","m":"Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.FinitePanel","a":"Panel/EstimandCharacterization/StaggeredTWFEDecomposition","t":1,"d":"Early-vs-late raw weight λ^EL_{e,ℓ} := p_e p_ℓ q (1−q) μ.","s":"{𝒢 : Type u_1} → [inst : Fintype 𝒢] → {T : ℕ} → Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.CohortPanel 𝒢 T → 𝒢 → 𝒢 → ℝ","l":234,"f":"Causalean/Panel/EstimandCharacterization/StaggeredTWFEDecomposition/FinitePanel.lean"},{"n":"Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.lambdaEL_add_lambdaLE_eq_gap","k":"theorem","m":"Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.AlgebraicDecomposition","a":"Panel/EstimandCharacterization/StaggeredTWFEDecomposition","t":2,"d":"The two ordered raw comparison weights between two cohorts sum to the product of their cohort shares, the gap in their average treatment rates, and one minus that gap.","s":"∀ {𝒢 : Type u_1} [inst : Fintype 𝒢] {T : ℕ} (P : Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.CohortPanel 𝒢 T) (e ℓ : 𝒢), Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.lambdaEL P e ℓ + Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.lambdaLE P e ℓ = P.p e * P.p ℓ * Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.q P e ℓ * (1 - Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.q P e ℓ)","l":219,"f":"Causalean/Panel/EstimandCharacterization/StaggeredTWFEDecomposition/AlgebraicDecomposition.lean"},{"n":"Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.lambdaLE","k":"def","m":"Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.FinitePanel","a":"Panel/EstimandCharacterization/StaggeredTWFEDecomposition","t":1,"d":"Late-vs-early raw weight λ^LE_{ℓ,e} := p_e p_ℓ q (1−q) (1−μ).","s":"{𝒢 : Type u_1} → [inst : Fintype 𝒢] → {T : ℕ} → Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.CohortPanel 𝒢 T → 𝒢 → 𝒢 → ℝ","l":238,"f":"Causalean/Panel/EstimandCharacterization/StaggeredTWFEDecomposition/FinitePanel.lean"},{"n":"Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.lambdaTN","k":"def","m":"Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.FinitePanel","a":"Panel/EstimandCharacterization/StaggeredTWFEDecomposition","t":1,"d":"Treated-vs-never raw weight λ^TN_{g,u} := p_g p_u \\overline{D}_g (1−\\overline{D}_g).","s":"{𝒢 : Type u_1} → [inst : Fintype 𝒢] → {T : ℕ} → Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.CohortPanel 𝒢 T → 𝒢 → 𝒢 → ℝ","l":222,"f":"Causalean/Panel/EstimandCharacterization/StaggeredTWFEDecomposition/FinitePanel.lean"},{"n":"Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.lambdaTN_eq_gap_of_isInf","k":"theorem","m":"Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.AlgebraicDecomposition","a":"Panel/EstimandCharacterization/StaggeredTWFEDecomposition","t":2,"d":"When one cohort is never treated, its raw comparison weight with another cohort equals the product of their cohort shares, the gap in their average treatment rates, and one minus that gap.","s":"∀ {𝒢 : Type u_1} [inst : Fintype 𝒢] {T : ℕ} (P : Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.CohortPanel 𝒢 T) {g u : 𝒢}, Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.AdoptionDate.isInf (P.A u) → Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.lambdaTN P g u = P.p g * P.p u * Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.q P g u * (1 - Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.q P g u)","l":209,"f":"Causalean/Panel/EstimandCharacterization/StaggeredTWFEDecomposition/AlgebraicDecomposition.lean"},{"n":"Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.lambdaWeight","k":"def","m":"Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.FinitePanel","a":"Panel/EstimandCharacterization/StaggeredTWFEDecomposition","t":1,"d":"Unified raw weight on the full index CompTag × 𝒢 × 𝒢: returns the matching λ_TN/λ_EL/λ_LE on admissible triples and 0 otherwise.","s":"{𝒢 : Type u_1} → [inst : Fintype 𝒢] → {T : ℕ} → Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.CohortPanel 𝒢 T → Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.CompTag × 𝒢 × 𝒢 → ℝ","l":309,"f":"Causalean/Panel/EstimandCharacterization/StaggeredTWFEDecomposition/FinitePanel.lean"},{"n":"Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.memLp_two_of_binary","k":"theorem","m":"Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.Support.Orthogonality","a":"Panel/EstimandCharacterization/StaggeredTWFEDecomposition/Support","t":2,"d":"Under a probability measure, a measurable real-valued variable that equals either zero or one almost surely has a finite second moment.","s":"∀ {Ω : Type u_1} [inst : MeasurableSpace Ω] (μ : MeasureTheory.Measure Ω) [MeasureTheory.IsProbabilityMeasure μ] (D : Ω → ℝ), Measurable D → (∀ᵐ (ω : Ω) ∂μ, D ω = 0 ∨ D ω = 1) → MeasureTheory.MemLp D 2 μ","l":49,"f":"Causalean/Panel/EstimandCharacterization/StaggeredTWFEDecomposition/Support/Orthogonality.lean"},{"n":"Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.mu","k":"def","m":"Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.FinitePanel","a":"Panel/EstimandCharacterization/StaggeredTWFEDecomposition","t":1,"d":"Splitting fraction μ_{eℓ} := (1−\\overline{D}_e)/(1−q_{eℓ}).","s":"{𝒢 : Type u_1} → [inst : Fintype 𝒢] → {T : ℕ} → Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.CohortPanel 𝒢 T → 𝒢 → 𝒢 → ℝ","l":230,"f":"Causalean/Panel/EstimandCharacterization/StaggeredTWFEDecomposition/FinitePanel.lean"},{"n":"Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.numPairContribution","k":"def","m":"Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.Pairwise","a":"Panel/EstimandCharacterization/StaggeredTWFEDecomposition","t":1,"d":"Ordered-pair contribution in the pairwise representation of the TWFE numerator. As for vdPairContribution, the two orders of each cohort pair are combined by the Goodman-Bacon-specific window lemmas below.","s":"{𝒢 : Type u_1} → [inst : Fintype 𝒢] → {T : ℕ} → Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.CohortPanel 𝒢 T → 𝒢 → 𝒢 → ℝ","l":133,"f":"Causalean/Panel/EstimandCharacterization/StaggeredTWFEDecomposition/Pairwise.lean"},{"n":"Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.num_per_cell_panel","k":"theorem","m":"Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.Support.PerCell","a":"Panel/EstimandCharacterization/StaggeredTWFEDecomposition/Support","t":1,"d":"Per-cell numerator identity. On the cohort-period cell (g, t), if the treatment D is almost-everywhere equal, on that cell, to its own cell mean (cell-measurability of D), then the integral of the propensity-residual D - panelPropensity times the outcome Y over the cell equals the cell mass times the cell's residualized treatment panelDtilde times the outcome's cell mean:","s":"∀ {Ω : Type u_1} {𝒢 : Type u_2} [inst : MeasurableSpace Ω] [inst_1 : Fintype 𝒢] [inst_2 : MeasurableSpace 𝒢] [MeasurableSingletonClass 𝒢] {T : ℕ} (μ : MeasureTheory.Measure Ω) [MeasureTheory.IsFiniteMeasure μ] (D Y : Ω → ℝ) (G : Ω → 𝒢) (T_rv : Ω → Fin T), Measurable G → Measurable T_rv → ∀ (g : 𝒢) (t : Fin T), (∀ᵐ (ω : Ω) ∂μ.restrict {ω' | G ω' = g ∧ T_rv ω' = t}, D ω = (∫ (ω' : Ω), D ω' * {ω' | G ω' = g ∧ T_rv ω' = t}.indicator (fun x => 1) ω' ∂μ) / Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.cellMass μ G T_rv g t) → ∫ (ω : Ω), (D ω - Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.panelPropensity μ D G T_rv ω) * Y ω * {ω' | G ω' = g ∧ T_rv ω' = t}.indicator (fun x => 1) ω ∂μ = Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.cellMass μ G T_rv g t * Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.panelDtilde μ D G T_rv g t * Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.cellMean μ Y G T_rv g t","l":120,"f":"Causalean/Panel/EstimandCharacterization/StaggeredTWFEDecomposition/Support/PerCell.lean"},{"n":"Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.pCohort","k":"def","m":"Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.FinitePanel","a":"Panel/EstimandCharacterization/StaggeredTWFEDecomposition","t":1,"d":"Overall treatment share E[D] = ∑_g p_g \\overline{D}_g.","s":"{𝒢 : Type u_1} → [inst : Fintype 𝒢] → {T : ℕ} → Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.CohortPanel 𝒢 T → ℝ","l":117,"f":"Causalean/Panel/EstimandCharacterization/StaggeredTWFEDecomposition/FinitePanel.lean"},{"n":"Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.pCohort_eq_grandMean","k":"theorem","m":"Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.FinitePanel","a":"Panel/EstimandCharacterization/StaggeredTWFEDecomposition","t":2,"d":"Goodman-Bacon's overall treatment share is the shared weighted grand mean.","s":"∀ {𝒢 : Type u_1} [inst : Fintype 𝒢] {T : ℕ} (P : Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.CohortPanel 𝒢 T), Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.pCohort P = Causalean.Panel.WeightedTwoWayPanel.grandMean (Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.cohortWeights P) (Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.D P)","l":133,"f":"Causalean/Panel/EstimandCharacterization/StaggeredTWFEDecomposition/FinitePanel.lean"},{"n":"Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.pairwise_sum_normalize","k":"theorem","m":"Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.Pairwise","a":"Panel/EstimandCharacterization/StaggeredTWFEDecomposition","t":2,"d":"In a finite cohort panel, averaging half the ordered-pair weighted total at each period equals the ordered-pair weighted total of the time sums with the same normalization.","s":"∀ {𝒢 : Type u_1} [inst : Fintype 𝒢] {T : ℕ} (P : Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.CohortPanel 𝒢 T) (f : 𝒢 → 𝒢 → Fin T → ℝ), (↑T)⁻¹ * ∑ t, 1 / 2 * ∑ g, ∑ u, P.p g * P.p u * f g u t = ∑ g, ∑ u, P.p g * P.p u / (2 * ↑T) * ∑ t, f g u t","l":99,"f":"Causalean/Panel/EstimandCharacterization/StaggeredTWFEDecomposition/Pairwise.lean"},{"n":"Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.panelClass","k":"def","m":"Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.Support.Basic","a":"Panel/EstimandCharacterization/StaggeredTWFEDecomposition/Support","t":1,"d":"B1. Saturated cohort + period class. Linear L² class spanned (a.e.) by the family of cohort indicators 𝟙{G = g} (for g : 𝒢) and period indicators 𝟙{T_rv = t} (for t : Fin T).","s":"{Ω : Type u_1} → {𝒢 : Type u_2} → [inst : MeasurableSpace Ω] → [Fintype 𝒢] → [DecidableEq 𝒢] → [inst_3 : MeasurableSpace 𝒢] → [MeasurableSingletonClass 𝒢] → {T : ℕ} → (μ : MeasureTheory.Measure Ω) → [MeasureTheory.IsFiniteMeasure μ] → (G : Ω → 𝒢) → (T_rv : Ω → Fin T) → Measurable G → Measurable T_rv → Causalean.Panel.LinearL2Class μ","l":48,"f":"Causalean/Panel/EstimandCharacterization/StaggeredTWFEDecomposition/Support/Basic.lean"},{"n":"Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.panelDtilde","k":"def","m":"Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.Support.PerCell","a":"Panel/EstimandCharacterization/StaggeredTWFEDecomposition/Support","t":1,"d":"The \"panel-residualized treatment hat\" Dtilde_{gt} value of the in-class residual on cell (g, t): equals D_{gt} - barD_g - (E[D | T=t] - E[D]), the LaTeX double-demeaning formula.","s":"{Ω : Type u_1} → {𝒢 : Type u_2} → [inst : MeasurableSpace Ω] → {T : ℕ} → MeasureTheory.Measure Ω → (Ω → ℝ) → (Ω → 𝒢) → (Ω → Fin T) → 𝒢 → Fin T → ℝ","l":31,"f":"Causalean/Panel/EstimandCharacterization/StaggeredTWFEDecomposition/Support/PerCell.lean"},{"n":"Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.panelMeanReg","k":"def","m":"Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.Support.Basic","a":"Panel/EstimandCharacterization/StaggeredTWFEDecomposition/Support","t":1,"d":"B3. Saturated cohort+period mean regression for Y. Analogous to panelPropensity, with Y in place of D.","s":"{Ω : Type u_1} → {𝒢 : Type u_2} → [inst : MeasurableSpace Ω] → [Fintype 𝒢] → {T : ℕ} → MeasureTheory.Measure Ω → (Ω → ℝ) → (Ω → 𝒢) → (Ω → Fin T) → Ω → ℝ","l":157,"f":"Causalean/Panel/EstimandCharacterization/StaggeredTWFEDecomposition/Support/Basic.lean"},{"n":"Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.panelMeanRegHat","k":"def","m":"Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.Support.Integrals","a":"Panel/EstimandCharacterization/StaggeredTWFEDecomposition/Support","t":1,"d":"\"panel-meanReg hat\" analogue for Y.","s":"{Ω : Type u_1} → {𝒢 : Type u_2} → [inst : MeasurableSpace Ω] → {T : ℕ} → MeasureTheory.Measure Ω → (Ω → ℝ) → (Ω → 𝒢) → (Ω → Fin T) → 𝒢 → Fin T → ℝ","l":215,"f":"Causalean/Panel/EstimandCharacterization/StaggeredTWFEDecomposition/Support/Integrals.lean"},{"n":"Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.panelMeanReg_cohort_axis_orthogonal","k":"theorem","m":"Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.Support.Orthogonality","a":"Panel/EstimandCharacterization/StaggeredTWFEDecomposition/Support","t":2,"d":"Under a balanced panel law, the residual from the panel mean regression has zero average product with the indicator of any fixed cohort.","s":"∀ {Ω : Type u_1} {𝒢 : Type u_2} [inst : MeasurableSpace Ω] [inst_1 : Fintype 𝒢] [inst_2 : MeasurableSpace 𝒢] [MeasurableSingletonClass 𝒢] {T : ℕ} (μ : MeasureTheory.Measure Ω) [MeasureTheory.IsProbabilityMeasure μ] (F : Ω → ℝ) (G : Ω → 𝒢) (T_rv : Ω → Fin T), Measurable G → Measurable T_rv → MeasureTheory.MemLp F 2 μ → Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.IsBalancedPanelLaw μ G T_rv → ∀ (g : 𝒢), ∫ (ω : Ω), (F ω - Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.panelMeanReg μ F G T_rv ω) * {ω' | G ω' = g}.indicator (fun x => 1) ω ∂μ = 0","l":372,"f":"Causalean/Panel/EstimandCharacterization/StaggeredTWFEDecomposition/Support/Orthogonality.lean"},{"n":"Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.panelMeanReg_eq_hat_of_mem","k":"theorem","m":"Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.Support.Integrals","a":"Panel/EstimandCharacterization/StaggeredTWFEDecomposition/Support","t":2,"d":"On the cell {G = g ∧ T_rv = t}, panelMeanReg evaluates to panelMeanRegHat g t.","s":"∀ {Ω : Type u_1} {𝒢 : Type u_2} [inst : MeasurableSpace Ω] [inst_1 : Fintype 𝒢] {T : ℕ} (μ : MeasureTheory.Measure Ω) (Y : Ω → ℝ) (G : Ω → 𝒢) (T_rv : Ω → Fin T) {g : 𝒢} {t : Fin T} {ω : Ω}, G ω = g → T_rv ω = t → Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.panelMeanReg μ Y G T_rv ω = Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.panelMeanRegHat μ Y G T_rv g t","l":249,"f":"Causalean/Panel/EstimandCharacterization/StaggeredTWFEDecomposition/Support/Integrals.lean"},{"n":"Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.panelMeanReg_mem_panelClass","k":"theorem","m":"Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.Support.Basic","a":"Panel/EstimandCharacterization/StaggeredTWFEDecomposition/Support","t":1,"d":"For a finite measure μ, an outcome variable Y, a measurable cohort map G, and a measurable period map T_rv, the pointwise saturated cohort-and-period mean regression of Y belongs to the two-axis additive (cohort-plus-period) function class.","s":"∀ {Ω : Type u_1} {𝒢 : Type u_2} [inst : MeasurableSpace Ω] [inst_1 : Fintype 𝒢] [inst_2 : DecidableEq 𝒢] [inst_3 : MeasurableSpace 𝒢] [inst_4 : MeasurableSingletonClass 𝒢] {T : ℕ} (μ : MeasureTheory.Measure Ω) [inst_5 : MeasureTheory.IsFiniteMeasure μ] (Y : Ω → ℝ) (G : Ω → 𝒢) (T_rv : Ω → Fin T) (G_meas : Measurable G) (T_meas : Measurable T_rv), (Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.panelClass μ G T_rv G_meas T_meas).mem (Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.panelMeanReg μ Y G T_rv)","l":195,"f":"Causalean/Panel/EstimandCharacterization/StaggeredTWFEDecomposition/Support/Basic.lean"},{"n":"Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.panelMeanReg_period_axis_orthogonal","k":"theorem","m":"Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.Support.Orthogonality","a":"Panel/EstimandCharacterization/StaggeredTWFEDecomposition/Support","t":2,"d":"Under a balanced panel law, the residual from the panel mean regression has zero average product with the indicator of any fixed period.","s":"∀ {Ω : Type u_1} {𝒢 : Type u_2} [inst : MeasurableSpace Ω] [inst_1 : Fintype 𝒢] [inst_2 : MeasurableSpace 𝒢] [MeasurableSingletonClass 𝒢] {T : ℕ} (μ : MeasureTheory.Measure Ω) [MeasureTheory.IsProbabilityMeasure μ] (F : Ω → ℝ) (G : Ω → 𝒢) (T_rv : Ω → Fin T), Measurable G → Measurable T_rv → MeasureTheory.MemLp F 2 μ → Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.IsBalancedPanelLaw μ G T_rv → ∀ (t : Fin T), ∫ (ω : Ω), (F ω - Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.panelMeanReg μ F G T_rv ω) * {ω' | T_rv ω' = t}.indicator (fun x => 1) ω ∂μ = 0","l":484,"f":"Causalean/Panel/EstimandCharacterization/StaggeredTWFEDecomposition/Support/Orthogonality.lean"},{"n":"Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.panelOf","k":"def","m":"Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.Support.Basic","a":"Panel/EstimandCharacterization/StaggeredTWFEDecomposition/Support","t":1,"d":"B2. Bridge to Layer A. Constructs a CohortPanel 𝒢 T from a measure carrying the cell-level data plus an explicit adoption date A : 𝒢 → WithTop (Fin T).","s":"{Ω : Type u_1} → {𝒢 : Type u_2} → [inst : MeasurableSpace Ω] → [inst_1 : Fintype 𝒢] → {T : ℕ} → (μ : MeasureTheory.Measure Ω) → (Ω → ℝ) → (G : Ω → 𝒢) → (Ω → Fin T) → (𝒢 → WithTop (Fin T)) → 0 < T → (∀ (g : 𝒢), 0 < Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.cohortMass μ G g) → ∑ g, Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.cohortMass μ G g = 1 → Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.CohortPanel 𝒢 T","l":117,"f":"Causalean/Panel/EstimandCharacterization/StaggeredTWFEDecomposition/Support/Basic.lean"},{"n":"Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.panelPropensity","k":"def","m":"Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.Support.Basic","a":"Panel/EstimandCharacterization/StaggeredTWFEDecomposition/Support","t":1,"d":"B3. Saturated cohort+period propensity for D. The pointwise representative","s":"{Ω : Type u_1} → {𝒢 : Type u_2} → [inst : MeasurableSpace Ω] → [Fintype 𝒢] → {T : ℕ} → MeasureTheory.Measure Ω → (Ω → ℝ) → (Ω → 𝒢) → (Ω → Fin T) → Ω → ℝ","l":137,"f":"Causalean/Panel/EstimandCharacterization/StaggeredTWFEDecomposition/Support/Basic.lean"},{"n":"Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.panelPropensityHat","k":"def","m":"Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.Support.Integrals","a":"Panel/EstimandCharacterization/StaggeredTWFEDecomposition/Support","t":1,"d":"The \"panel-propensity hat\" pHat g t = cohortBarD g + cT t where cT t := (∫ D · 𝟙{T_rv=t} dμ) / periodMass t - ∫ D dμ.","s":"{Ω : Type u_1} → {𝒢 : Type u_2} → [inst : MeasurableSpace Ω] → {T : ℕ} → MeasureTheory.Measure Ω → (Ω → ℝ) → (Ω → 𝒢) → (Ω → Fin T) → 𝒢 → Fin T → ℝ","l":204,"f":"Causalean/Panel/EstimandCharacterization/StaggeredTWFEDecomposition/Support/Integrals.lean"},{"n":"Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.panelPropensity_eq_hat_of_mem","k":"theorem","m":"Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.Support.Integrals","a":"Panel/EstimandCharacterization/StaggeredTWFEDecomposition/Support","t":2,"d":"On the cell {G = g ∧ T_rv = t}, panelPropensity evaluates to panelPropensityHat g t. Pointwise (no a.e. needed) by single-cell membership selecting one term in each finite sum.","s":"∀ {Ω : Type u_1} {𝒢 : Type u_2} [inst : MeasurableSpace Ω] [inst_1 : Fintype 𝒢] {T : ℕ} (μ : MeasureTheory.Measure Ω) (D : Ω → ℝ) (G : Ω → 𝒢) (T_rv : Ω → Fin T) {g : 𝒢} {t : Fin T} {ω : Ω}, G ω = g → T_rv ω = t → Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.panelPropensity μ D G T_rv ω = Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.panelPropensityHat μ D G T_rv g t","l":227,"f":"Causalean/Panel/EstimandCharacterization/StaggeredTWFEDecomposition/Support/Integrals.lean"},{"n":"Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.panelPropensity_mem_panelClass","k":"theorem","m":"Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.Support.Basic","a":"Panel/EstimandCharacterization/StaggeredTWFEDecomposition/Support","t":1,"d":"For a finite measure μ, a treatment variable D, a measurable cohort map G, and a measurable period map T_rv, the pointwise saturated cohort-and-period propensity regression of D belongs to the two-axis additive (cohort-plus-period) function class.","s":"∀ {Ω : Type u_1} {𝒢 : Type u_2} [inst : MeasurableSpace Ω] [inst_1 : Fintype 𝒢] [inst_2 : DecidableEq 𝒢] [inst_3 : MeasurableSpace 𝒢] [inst_4 : MeasurableSingletonClass 𝒢] {T : ℕ} (μ : MeasureTheory.Measure Ω) [inst_5 : MeasureTheory.IsFiniteMeasure μ] (D : Ω → ℝ) (G : Ω → 𝒢) (T_rv : Ω → Fin T) (G_meas : Measurable G) (T_meas : Measurable T_rv), (Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.panelClass μ G T_rv G_meas T_meas).mem (Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.panelPropensity μ D G T_rv)","l":174,"f":"Causalean/Panel/EstimandCharacterization/StaggeredTWFEDecomposition/Support/Basic.lean"},{"n":"Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.panel_cell_integral_div_mul_cellMass","k":"theorem","m":"Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.Support.Integrals","a":"Panel/EstimandCharacterization/StaggeredTWFEDecomposition/Support","t":2,"d":"Cell analogue (joint cohort × period cell).","s":"∀ {Ω : Type u_1} {𝒢 : Type u_2} [inst : MeasurableSpace Ω] [inst_1 : MeasurableSpace 𝒢] [MeasurableSingletonClass 𝒢] {T : ℕ} (μ : MeasureTheory.Measure Ω) [MeasureTheory.IsFiniteMeasure μ] (F : Ω → ℝ) (G : Ω → 𝒢) (T_rv : Ω → Fin T), Measurable G → Measurable T_rv → ∀ (g : 𝒢) (t : Fin T), (∫ (ω : Ω), F ω * {ω' | G ω' = g ∧ T_rv ω' = t}.indicator (fun x => 1) ω ∂μ) / Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.cellMass μ G T_rv g t * Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.cellMass μ G T_rv g t = ∫ (ω : Ω), F ω * {ω' | G ω' = g ∧ T_rv ω' = t}.indicator (fun x => 1) ω ∂μ","l":139,"f":"Causalean/Panel/EstimandCharacterization/StaggeredTWFEDecomposition/Support/Integrals.lean"},{"n":"Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.periodMass","k":"def","m":"Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.Support.Basic","a":"Panel/EstimandCharacterization/StaggeredTWFEDecomposition/Support","t":1,"d":"B2. Period mass. periodMass μ T_rv t = (μ {T_rv = t}).toReal. Plays the role of 1/|𝒯| under the balanced-law hypothesis.","s":"{Ω : Type u_1} → [inst : MeasurableSpace Ω] → {T : ℕ} → MeasureTheory.Measure Ω → (Ω → Fin T) → Fin T → ℝ","l":78,"f":"Causalean/Panel/EstimandCharacterization/StaggeredTWFEDecomposition/Support/Basic.lean"},{"n":"Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.periodMass_nonneg","k":"theorem","m":"Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.Support.Basic","a":"Panel/EstimandCharacterization/StaggeredTWFEDecomposition/Support","t":2,"d":"Period mass is nonnegative.","s":"∀ {Ω : Type u_1} [inst : MeasurableSpace Ω] {T : ℕ} (μ : MeasureTheory.Measure Ω) (T_rv : Ω → Fin T) (t : Fin T), 0 ≤ Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.periodMass μ T_rv t","l":227,"f":"Causalean/Panel/EstimandCharacterization/StaggeredTWFEDecomposition/Support/Basic.lean"},{"n":"Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.period_centered_sum_eq_zero","k":"theorem","m":"Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.Support.Orthogonality","a":"Panel/EstimandCharacterization/StaggeredTWFEDecomposition/Support","t":2,"d":"Under a probability distribution, period-specific means of an integrable outcome, centered by the overall mean and weighted by their period probabilities, sum to zero.","s":"∀ {Ω : Type u_1} [inst : MeasurableSpace Ω] {T : ℕ} (μ : MeasureTheory.Measure Ω) [MeasureTheory.IsProbabilityMeasure μ] (F : Ω → ℝ) (T_rv : Ω → Fin T), Measurable T_rv → MeasureTheory.Integrable F μ → ∑ t, ((∫ (ω : Ω), F ω * {ω' | T_rv ω' = t}.indicator (fun x => 1) ω ∂μ) / Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.periodMass μ T_rv t - ∫ (ω : Ω), F ω ∂μ) * Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.periodMass μ T_rv t = 0","l":338,"f":"Causalean/Panel/EstimandCharacterization/StaggeredTWFEDecomposition/Support/Orthogonality.lean"},{"n":"Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.period_integral_div_mul_periodMass","k":"theorem","m":"Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.Support.Integrals","a":"Panel/EstimandCharacterization/StaggeredTWFEDecomposition/Support","t":2,"d":"Period analogue.","s":"∀ {Ω : Type u_1} [inst : MeasurableSpace Ω] {T : ℕ} (μ : MeasureTheory.Measure Ω) [MeasureTheory.IsFiniteMeasure μ] (F : Ω → ℝ) (T_rv : Ω → Fin T) (t : Fin T), (∫ (ω : Ω), F ω * {ω' | T_rv ω' = t}.indicator (fun x => 1) ω ∂μ) / Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.periodMass μ T_rv t * Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.periodMass μ T_rv t = ∫ (ω : Ω), F ω * {ω' | T_rv ω' = t}.indicator (fun x => 1) ω ∂μ","l":128,"f":"Causalean/Panel/EstimandCharacterization/StaggeredTWFEDecomposition/Support/Integrals.lean"},{"n":"Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.q","k":"def","m":"Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.FinitePanel","a":"Panel/EstimandCharacterization/StaggeredTWFEDecomposition","t":1,"d":"Treated-treated timing-pair gap q_{eℓ} := \\overline{D}_e − \\overline{D}_ℓ.","s":"{𝒢 : Type u_1} → [inst : Fintype 𝒢] → {T : ℕ} → Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.CohortPanel 𝒢 T → 𝒢 → 𝒢 → ℝ","l":226,"f":"Causalean/Panel/EstimandCharacterization/StaggeredTWFEDecomposition/FinitePanel.lean"},{"n":"Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.raw_weight_sum_eq_VD","k":"theorem","m":"Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.AlgebraicDecomposition","a":"Panel/EstimandCharacterization/StaggeredTWFEDecomposition","t":2,"d":"Prop A5.2 (raw_weight_sum_eq_VD). The aggregate raw-weight denominator equals the residualized treatment variance: Λ P = V_D P. Key denominator identity in thm:po-estimand-goodman-bacon-decomposition.","s":"∀ {𝒢 : Type u_1} [inst : Fintype 𝒢] [DecidableEq 𝒢] {T : ℕ} (P : Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.CohortPanel 𝒢 T), Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.Lambda P = Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.VD P","l":514,"f":"Causalean/Panel/EstimandCharacterization/StaggeredTWFEDecomposition/AlgebraicDecomposition.lean"},{"n":"Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.residD_cohort_orthogonal","k":"theorem","m":"Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.Support.Orthogonality","a":"Panel/EstimandCharacterization/StaggeredTWFEDecomposition/Support","t":1,"d":"For a treatment indicator D that is binary almost everywhere, the residual between D and its cohort-period propensity score panelPropensity is orthogonal, in the L²(μ) sense, to every cohort indicator 𝟙{G = g}.","s":"∀ {Ω : Type u_1} {𝒢 : Type u_2} [inst : MeasurableSpace Ω] [inst_1 : Fintype 𝒢] [inst_2 : MeasurableSpace 𝒢] [MeasurableSingletonClass 𝒢] {T : ℕ} (μ : MeasureTheory.Measure Ω) [MeasureTheory.IsProbabilityMeasure μ] (D : Ω → ℝ) (G : Ω → 𝒢) (T_rv : Ω → Fin T), Measurable G → Measurable T_rv → Measurable D → (∀ᵐ (ω : Ω) ∂μ, D ω = 0 ∨ D ω = 1) → Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.IsBalancedPanelLaw μ G T_rv → ∀ (g : 𝒢), ∫ (ω : Ω), (D ω - Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.panelPropensity μ D G T_rv ω) * {ω' | G ω' = g}.indicator (fun x => 1) ω ∂μ = 0","l":621,"f":"Causalean/Panel/EstimandCharacterization/StaggeredTWFEDecomposition/Support/Orthogonality.lean"},{"n":"Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.residD_period_orthogonal","k":"theorem","m":"Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.Support.Orthogonality","a":"Panel/EstimandCharacterization/StaggeredTWFEDecomposition/Support","t":2,"d":"The treatment residual is orthogonal to every period indicator 𝟙{T_rv = t}. Requires B_balanced to make the cross-cohort sum cancel.","s":"∀ {Ω : Type u_1} {𝒢 : Type u_2} [inst : MeasurableSpace Ω] [inst_1 : Fintype 𝒢] [inst_2 : MeasurableSpace 𝒢] [MeasurableSingletonClass 𝒢] {T : ℕ} (μ : MeasureTheory.Measure Ω) [MeasureTheory.IsProbabilityMeasure μ] (D : Ω → ℝ) (G : Ω → 𝒢) (T_rv : Ω → Fin T), Measurable G → Measurable T_rv → Measurable D → (∀ᵐ (ω : Ω) ∂μ, D ω = 0 ∨ D ω = 1) → Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.IsBalancedPanelLaw μ G T_rv → ∀ (t : Fin T), ∫ (ω : Ω), (D ω - Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.panelPropensity μ D G T_rv ω) * {ω' | T_rv ω' = t}.indicator (fun x => 1) ω ∂μ = 0","l":662,"f":"Causalean/Panel/EstimandCharacterization/StaggeredTWFEDecomposition/Support/Orthogonality.lean"},{"n":"Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.residWitnessD_panel","k":"def","m":"Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.Support.Partition","a":"Panel/EstimandCharacterization/StaggeredTWFEDecomposition/Support","t":1,"d":"B3. Residualization witness for D against panelClass. With VH := panelPropensity μ D G T_rv and Vtilde ω := D ω - panelPropensity ω, this packages the four witness obligations:","s":"{Ω : Type u_1} → {𝒢 : Type u_2} → [inst : MeasurableSpace Ω] → [inst_1 : Fintype 𝒢] → [inst_2 : DecidableEq 𝒢] → [inst_3 : MeasurableSpace 𝒢] → [inst_4 : MeasurableSingletonClass 𝒢] → {T : ℕ} → (μ : MeasureTheory.Measure Ω) → [inst_5 : MeasureTheory.IsProbabilityMeasure μ] → (D : Ω → ℝ) → (G : Ω → 𝒢) → (T_rv : Ω → Fin T) → (G_meas : Measurable G) → (T_meas : Measurable T_rv) → Measurable D → (∀ᵐ (ω : Ω) ∂μ, D ω = 0 ∨ D ω = 1) → Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.IsBalancedPanelLaw μ G T_rv → Causalean.Panel.ResidualizationWitness μ (Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.panelClass μ G T_rv G_meas T_meas) D","l":31,"f":"Causalean/Panel/EstimandCharacterization/StaggeredTWFEDecomposition/Support/Partition.lean"},{"n":"Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.residWitnessY_panel","k":"def","m":"Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.Support.Partition","a":"Panel/EstimandCharacterization/StaggeredTWFEDecomposition/Support","t":1,"d":"B3. Residualization witness for Y against panelClass.","s":"{Ω : Type u_1} → {𝒢 : Type u_2} → [inst : MeasurableSpace Ω] → [inst_1 : Fintype 𝒢] → [inst_2 : DecidableEq 𝒢] → [inst_3 : MeasurableSpace 𝒢] → [inst_4 : MeasurableSingletonClass 𝒢] → {T : ℕ} → (μ : MeasureTheory.Measure Ω) → [inst_5 : MeasureTheory.IsProbabilityMeasure μ] → (Y : Ω → ℝ) → (G : Ω → 𝒢) → (T_rv : Ω → Fin T) → (G_meas : Measurable G) → (T_meas : Measurable T_rv) → MeasureTheory.MemLp Y 2 μ → Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.IsBalancedPanelLaw μ G T_rv → Causalean.Panel.ResidualizationWitness μ (Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.panelClass μ G T_rv G_meas T_meas) Y","l":101,"f":"Causalean/Panel/EstimandCharacterization/StaggeredTWFEDecomposition/Support/Partition.lean"},{"n":"Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.residY_cohort_orthogonal","k":"theorem","m":"Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.Support.Orthogonality","a":"Panel/EstimandCharacterization/StaggeredTWFEDecomposition/Support","t":2,"d":"The outcome residual is orthogonal to every cohort indicator.","s":"∀ {Ω : Type u_1} {𝒢 : Type u_2} [inst : MeasurableSpace Ω] [inst_1 : Fintype 𝒢] [inst_2 : MeasurableSpace 𝒢] [MeasurableSingletonClass 𝒢] {T : ℕ} (μ : MeasureTheory.Measure Ω) [MeasureTheory.IsProbabilityMeasure μ] (Y : Ω → ℝ) (G : Ω → 𝒢) (T_rv : Ω → Fin T), Measurable G → Measurable T_rv → MeasureTheory.MemLp Y 2 μ → Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.IsBalancedPanelLaw μ G T_rv → ∀ (g : 𝒢), ∫ (ω : Ω), (Y ω - Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.panelMeanReg μ Y G T_rv ω) * {ω' | G ω' = g}.indicator (fun x => 1) ω ∂μ = 0","l":646,"f":"Causalean/Panel/EstimandCharacterization/StaggeredTWFEDecomposition/Support/Orthogonality.lean"},{"n":"Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.residY_period_orthogonal","k":"theorem","m":"Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.Support.Orthogonality","a":"Panel/EstimandCharacterization/StaggeredTWFEDecomposition/Support","t":2,"d":"The outcome residual is orthogonal to every period indicator.","s":"∀ {Ω : Type u_1} {𝒢 : Type u_2} [inst : MeasurableSpace Ω] [inst_1 : Fintype 𝒢] [inst_2 : MeasurableSpace 𝒢] [MeasurableSingletonClass 𝒢] {T : ℕ} (μ : MeasureTheory.Measure Ω) [MeasureTheory.IsProbabilityMeasure μ] (Y : Ω → ℝ) (G : Ω → 𝒢) (T_rv : Ω → Fin T), Measurable G → Measurable T_rv → MeasureTheory.MemLp Y 2 μ → Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.IsBalancedPanelLaw μ G T_rv → ∀ (t : Fin T), ∫ (ω : Ω), (Y ω - Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.panelMeanReg μ Y G T_rv ω) * {ω' | T_rv ω' = t}.indicator (fun x => 1) ω ∂μ = 0","l":684,"f":"Causalean/Panel/EstimandCharacterization/StaggeredTWFEDecomposition/Support/Orthogonality.lean"},{"n":"Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.sum_lambdaWeight_eq_Lambda","k":"theorem","m":"Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.AlgebraicDecomposition","a":"Panel/EstimandCharacterization/StaggeredTWFEDecomposition","t":2,"d":"Across all admissible comparison types and cohort pairs, the raw comparison weights sum to the aggregate normalizing denominator.","s":"∀ {𝒢 : Type u_1} [inst : Fintype 𝒢] {T : ℕ} (P : Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.CohortPanel 𝒢 T), ∑ k ∈ Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.𝒦 P, Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.lambdaWeight P k = Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.Lambda P","l":570,"f":"Causalean/Panel/EstimandCharacterization/StaggeredTWFEDecomposition/AlgebraicDecomposition.lean"},{"n":"Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.sum_periodMass_eq_one","k":"theorem","m":"Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.Support.Orthogonality","a":"Panel/EstimandCharacterization/StaggeredTWFEDecomposition/Support","t":2,"d":"Under a probability distribution, the probabilities assigned to every period by a measurable finite-valued period variable sum to one.","s":"∀ {Ω : Type u_1} [inst : MeasurableSpace Ω] {T : ℕ} (μ : MeasureTheory.Measure Ω) [MeasureTheory.IsProbabilityMeasure μ] (T_rv : Ω → Fin T), Measurable T_rv → ∑ t, Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.periodMass μ T_rv t = 1","l":318,"f":"Causalean/Panel/EstimandCharacterization/StaggeredTWFEDecomposition/Support/Orthogonality.lean"},{"n":"Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.sum_weight_over_T_commute","k":"theorem","m":"Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.Pairwise","a":"Panel/EstimandCharacterization/StaggeredTWFEDecomposition","t":2,"d":"In a finite cohort panel, a cohort-weighted sum over cohorts and periods with equal period weight equals the average over periods of the corresponding cohort-weighted sums.","s":"∀ {𝒢 : Type u_1} [inst : Fintype 𝒢] {T : ℕ} (P : Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.CohortPanel 𝒢 T) (f : 𝒢 → Fin T → ℝ), ∑ g, ∑ t, P.p g / ↑T * f g t = (↑T)⁻¹ * ∑ t, ∑ g, P.p g * f g t","l":88,"f":"Causalean/Panel/EstimandCharacterization/StaggeredTWFEDecomposition/Pairwise.lean"},{"n":"Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.twfe_eq_weighted_avg","k":"theorem","m":"Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.AlgebraicDecomposition","a":"Panel/EstimandCharacterization/StaggeredTWFEDecomposition","t":1,"d":"Theorem A5.5 (twfe_eq_weighted_avg, thm:po-estimand-goodman-bacon-decomposition). For a cohort panel, the two-way fixed-effects (TWFE) coefficient, under the totalized zero-variance convention, equals the weighted sum of admissible two-by-two DID contrasts across comparison groups.","s":"∀ {𝒢 : Type u_1} [inst : Fintype 𝒢] [DecidableEq 𝒢] {T : ℕ} (P : Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.CohortPanel 𝒢 T), Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.betaTWFE P = ∑ k ∈ Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.𝒦 P, Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.weight P k * Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.contrast P k","l":701,"f":"Causalean/Panel/EstimandCharacterization/StaggeredTWFEDecomposition/AlgebraicDecomposition.lean"},{"n":"Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.twfe_eq_weighted_avg_core","k":"theorem","m":"Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.AlgebraicDecomposition","a":"Panel/EstimandCharacterization/StaggeredTWFEDecomposition","t":2,"d":"The positivity-free algebraic core of the Goodman-Bacon decomposition: the totalized TWFE ratio equals the totalized weighted sum of admissible two-by-two DID contrasts, including in zero-variance cases.","s":"∀ {𝒢 : Type u_1} [inst : Fintype 𝒢] [DecidableEq 𝒢] {T : ℕ} (P : Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.CohortPanel 𝒢 T), Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.betaTWFE P = ∑ k ∈ Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.𝒦 P, Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.weight P k * Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.contrast P k","l":664,"f":"Causalean/Panel/EstimandCharacterization/StaggeredTWFEDecomposition/AlgebraicDecomposition.lean"},{"n":"Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.twfe_numerator_eq_lambda_delta_sum","k":"theorem","m":"Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.AlgebraicDecomposition","a":"Panel/EstimandCharacterization/StaggeredTWFEDecomposition","t":2,"d":"Prop A5.3 (twfe_numerator_eq_lambda_delta_sum). The TWFE numerator decomposes by unordered cohort pairs into raw-weight times 2x2 DID contrast contributions.","s":"∀ {𝒢 : Type u_1} [inst : Fintype 𝒢] [DecidableEq 𝒢] {T : ℕ} (P : Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.CohortPanel 𝒢 T), ∑ g, ∑ t, P.p g / ↑T * Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.Dtilde P g t * P.Y g t = (∑ g, ∑ u, if Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.AdoptionDate.isFin (P.A g) ∧ Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.AdoptionDate.isInf (P.A u) then Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.lambdaTN P g u * Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.Δ_TN P g u else 0) + ∑ e, ∑ ℓ, if P.A e < P.A ℓ ∧ Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.AdoptionDate.isFin (P.A ℓ) then Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.lambdaEL P e ℓ * Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.Δ_EL P e ℓ + Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.lambdaLE P e ℓ * Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.Δ_LE P e ℓ else 0","l":524,"f":"Causalean/Panel/EstimandCharacterization/StaggeredTWFEDecomposition/AlgebraicDecomposition.lean"},{"n":"Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.twfe_numerator_eq_pairwise_centeredD_Y","k":"theorem","m":"Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.Pairwise","a":"Panel/EstimandCharacterization/StaggeredTWFEDecomposition","t":2,"d":"Pairwise representation of the TWFE numerator.","s":"∀ {𝒢 : Type u_1} [inst : Fintype 𝒢] [DecidableEq 𝒢] {T : ℕ} (P : Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.CohortPanel 𝒢 T), ∑ g, ∑ t, P.p g / ↑T * Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.Dtilde P g t * P.Y g t = ∑ g, ∑ u, Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.numPairContribution P g u","l":418,"f":"Causalean/Panel/EstimandCharacterization/StaggeredTWFEDecomposition/Pairwise.lean"},{"n":"Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.twfe_po_decomposition","k":"theorem","m":"Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.CausalDecomposition","a":"Panel/EstimandCharacterization/StaggeredTWFEDecomposition","t":1,"d":"Fused causal Goodman-Bacon decomposition. Fix a cohort panel P and potential-outcome maps Y0 (never-treated path) and Y1 (own-adoption-date path). Assume consistency on treated and untreated cells, no anticipation, and pairwise untreated parallel trends across the treated-versus-never, early-versus-late, and late-versus-early comparison types; then the two-way fixed-effects coefficient betaTWFE P equals the sum, over admissible pairwise comparisons, of each comparison's Goodman-Bacon weight times its potential-outcome window contrast — the treated-versus-never and early-versus-late window-specific ATTs, or the late-versus-early bad-comparison adjustment.","s":"∀ {𝒢 : Type u_1} [inst : Fintype 𝒢] [DecidableEq 𝒢] {T : ℕ} (P : Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.CohortPanel 𝒢 T) (Y0 Y1 : 𝒢 → Fin T → ℝ), Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.CausalAssumptions P Y0 Y1 → Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.betaTWFE P = ∑ k ∈ Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.𝒦 P, Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.weight P k * Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.contrastCausal P Y0 Y1 k","l":82,"f":"Causalean/Panel/EstimandCharacterization/StaggeredTWFEDecomposition/CausalDecomposition.lean"},{"n":"Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.vdPairContribution","k":"def","m":"Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.Pairwise","a":"Panel/EstimandCharacterization/StaggeredTWFEDecomposition","t":1,"d":"Ordered-pair contribution in the pairwise representation of V_D.","s":"{𝒢 : Type u_1} → [inst : Fintype 𝒢] → {T : ℕ} → Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.CohortPanel 𝒢 T → 𝒢 → 𝒢 → ℝ","l":123,"f":"Causalean/Panel/EstimandCharacterization/StaggeredTWFEDecomposition/Pairwise.lean"},{"n":"Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.w_EL","k":"def","m":"Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.FinitePanel","a":"Panel/EstimandCharacterization/StaggeredTWFEDecomposition","t":1,"d":"Normalized early-vs-late weight w^EL_{e,ℓ} := λ^EL_{e,ℓ} / Λ.","s":"{𝒢 : Type u_1} → [inst : Fintype 𝒢] → {T : ℕ} → Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.CohortPanel 𝒢 T → 𝒢 → 𝒢 → ℝ","l":255,"f":"Causalean/Panel/EstimandCharacterization/StaggeredTWFEDecomposition/FinitePanel.lean"},{"n":"Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.w_LE","k":"def","m":"Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.FinitePanel","a":"Panel/EstimandCharacterization/StaggeredTWFEDecomposition","t":1,"d":"Normalized late-vs-early weight w^LE_{ℓ,e} := λ^LE_{ℓ,e} / Λ.","s":"{𝒢 : Type u_1} → [inst : Fintype 𝒢] → {T : ℕ} → Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.CohortPanel 𝒢 T → 𝒢 → 𝒢 → ℝ","l":259,"f":"Causalean/Panel/EstimandCharacterization/StaggeredTWFEDecomposition/FinitePanel.lean"},{"n":"Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.w_TN","k":"def","m":"Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.FinitePanel","a":"Panel/EstimandCharacterization/StaggeredTWFEDecomposition","t":1,"d":"Normalized treated-vs-never weight w^TN_{g,u} := λ^TN_{g,u} / Λ.","s":"{𝒢 : Type u_1} → [inst : Fintype 𝒢] → {T : ℕ} → Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.CohortPanel 𝒢 T → 𝒢 → 𝒢 → ℝ","l":251,"f":"Causalean/Panel/EstimandCharacterization/StaggeredTWFEDecomposition/FinitePanel.lean"},{"n":"Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.weight","k":"def","m":"Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.FinitePanel","a":"Panel/EstimandCharacterization/StaggeredTWFEDecomposition","t":1,"d":"Unified normalized weight on the full index CompTag × 𝒢 × 𝒢: returns the matching w_TN/w_EL/w_LE on admissible triples and 0 otherwise.","s":"{𝒢 : Type u_1} → [inst : Fintype 𝒢] → {T : ℕ} → Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.CohortPanel 𝒢 T → Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.CompTag × 𝒢 × 𝒢 → ℝ","l":287,"f":"Causalean/Panel/EstimandCharacterization/StaggeredTWFEDecomposition/FinitePanel.lean"},{"n":"Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.weighted_center_cov_uncentered_right","k":"theorem","m":"Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.Pairwise","a":"Panel/EstimandCharacterization/StaggeredTWFEDecomposition","t":2,"d":"For a finite collection with weights summing to one, the weighted sum of deviations of one quantity from its weighted mean times another quantity equals one half of the weighted sum of pairwise differences in the two quantities.","s":"∀ {ι : Type u_2} [inst : Fintype ι] (p x y : ι → ℝ), ∑ i, p i = 1 → ∑ i, p i * (x i - ∑ j, p j * x j) * y i = 1 / 2 * ∑ i, ∑ j, p i * p j * (x i - x j) * (y i - y j)","l":40,"f":"Causalean/Panel/EstimandCharacterization/StaggeredTWFEDecomposition/Pairwise.lean"},{"n":"Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.weights_nonneg","k":"theorem","m":"Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.AlgebraicDecomposition","a":"Panel/EstimandCharacterization/StaggeredTWFEDecomposition","t":1,"d":"Prop A5.1 (weights_nonneg). For a cohort panel and any comparison-type index, the raw Goodman-Bacon comparison weight is nonnegative.","s":"∀ {𝒢 : Type u_1} [inst : Fintype 𝒢] {T : ℕ} (P : Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.CohortPanel 𝒢 T) (k : Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.CompTag × 𝒢 × 𝒢), 0 ≤ Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.lambdaWeight P k","l":57,"f":"Causalean/Panel/EstimandCharacterization/StaggeredTWFEDecomposition/AlgebraicDecomposition.lean"},{"n":"Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.weights_sum_one","k":"theorem","m":"Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.AlgebraicDecomposition","a":"Panel/EstimandCharacterization/StaggeredTWFEDecomposition","t":1,"d":"Prop A5.4 (weights_sum_one). For a finite Goodman–Bacon cohort panel P, if the residualized-treatment variance VD P is strictly positive, then the normalized comparison weights sum to one over all admissible 2×2 comparisons: Σ_{k ∈ 𝒦 P} weight P k = 1.","s":"∀ {𝒢 : Type u_1} [inst : Fintype 𝒢] [DecidableEq 𝒢] {T : ℕ} (P : Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.CohortPanel 𝒢 T), 0 < Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.VD P → ∑ k ∈ Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.𝒦 P, Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.weight P k = 1","l":638,"f":"Causalean/Panel/EstimandCharacterization/StaggeredTWFEDecomposition/AlgebraicDecomposition.lean"},{"n":"Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.Δ_EL","k":"def","m":"Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.FinitePanel","a":"Panel/EstimandCharacterization/StaggeredTWFEDecomposition","t":1,"d":"Early-vs-late before late 2x2 DID contrast Δ^EL_{e,ℓ}.","s":"{𝒢 : Type u_1} → [inst : Fintype 𝒢] → {T : ℕ} → Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.CohortPanel 𝒢 T → 𝒢 → 𝒢 → ℝ","l":210,"f":"Causalean/Panel/EstimandCharacterization/StaggeredTWFEDecomposition/FinitePanel.lean"},{"n":"Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.Δ_EL_eq_ATT","k":"theorem","m":"Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.Causal","a":"Panel/EstimandCharacterization/StaggeredTWFEDecomposition","t":1,"d":"Layer C corollary 2 — EL identifies the early cohort's ATT. Fix a cohort panel P and potential-outcome maps Y0, Y1, and assume consistency on treated and untreated cells, no anticipation, and pairwise untreated parallel trends across the three comparison types. For an early cohort e whose adoption date strictly precedes that of a late cohort ℓ, with ℓ's adoption date finite, the early-versus-late contrast Δ_EL equals e's window-specific average treatment effect on the treated over the window running from e's own adoption date up to ℓ's adoption date:","s":"∀ {𝒢 : Type u_1} [inst : Fintype 𝒢] {T : ℕ} (P : Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.CohortPanel 𝒢 T) (Y0 Y1 : 𝒢 → Fin T → ℝ), Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.CausalAssumptions P Y0 Y1 → ∀ (e ℓ : 𝒢), P.A e < P.A ℓ → Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.AdoptionDate.isFin (P.A ℓ) → Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.Δ_EL P e ℓ = Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.ATT_window Y0 Y1 e (Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.S1_EL P e ℓ)","l":344,"f":"Causalean/Panel/EstimandCharacterization/StaggeredTWFEDecomposition/Causal.lean"},{"n":"Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.Δ_LE","k":"def","m":"Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.FinitePanel","a":"Panel/EstimandCharacterization/StaggeredTWFEDecomposition","t":1,"d":"Late-vs-early after early 2x2 DID contrast Δ^LE_{ℓ,e}.","s":"{𝒢 : Type u_1} → [inst : Fintype 𝒢] → {T : ℕ} → Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.CohortPanel 𝒢 T → 𝒢 → 𝒢 → ℝ","l":215,"f":"Causalean/Panel/EstimandCharacterization/StaggeredTWFEDecomposition/FinitePanel.lean"},{"n":"Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.Δ_LE_eq_bad_comparison","k":"theorem","m":"Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.Causal","a":"Panel/EstimandCharacterization/StaggeredTWFEDecomposition","t":1,"d":"Layer C corollary 3 — LE has a bad-comparison term. Fix a cohort panel P and potential-outcome maps Y0, Y1, and assume consistency on treated and untreated cells, no anticipation, and pairwise untreated parallel trends across the three comparison types. For an early cohort e whose adoption date strictly precedes that of a late cohort ℓ, with ℓ's adoption date finite, the late-versus-early contrast Δ_LE equals ℓ's window-specific average treatment effect on the treated over its post-adoption window, minus the \"bad-comparison\" gap between e's own treatment effects on the two comparison windows:","s":"∀ {𝒢 : Type u_1} [inst : Fintype 𝒢] {T : ℕ} (P : Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.CohortPanel 𝒢 T) (Y0 Y1 : 𝒢 → Fin T → ℝ), Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.CausalAssumptions P Y0 Y1 → ∀ (e ℓ : 𝒢), P.A e < P.A ℓ → Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.AdoptionDate.isFin (P.A ℓ) → Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.Δ_LE P e ℓ = Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.ATT_window Y0 Y1 ℓ (Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.S1_LE P ℓ) - (Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.ATT_window Y0 Y1 e (Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.S1_LE P ℓ) - Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.ATT_window Y0 Y1 e (Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.S0_LE P e ℓ))","l":387,"f":"Causalean/Panel/EstimandCharacterization/StaggeredTWFEDecomposition/Causal.lean"},{"n":"Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.Δ_TN","k":"def","m":"Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.FinitePanel","a":"Panel/EstimandCharacterization/StaggeredTWFEDecomposition","t":1,"d":"Treated-vs-never 2x2 DID contrast Δ^TN_{g,u}.","s":"{𝒢 : Type u_1} → [inst : Fintype 𝒢] → {T : ℕ} → Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.CohortPanel 𝒢 T → 𝒢 → 𝒢 → ℝ","l":205,"f":"Causalean/Panel/EstimandCharacterization/StaggeredTWFEDecomposition/FinitePanel.lean"},{"n":"Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.Δ_TN_eq_ATT","k":"theorem","m":"Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.Causal","a":"Panel/EstimandCharacterization/StaggeredTWFEDecomposition","t":1,"d":"Layer C corollary 1 — TN identifies ATT. Fix a cohort panel P and potential-outcome maps Y0 (never-treated path) and Y1 (own-adoption-date path), and assume consistency on treated and untreated cells, no anticipation, and pairwise untreated parallel trends across the three comparison types. For a treated cohort g whose adoption date is finite compared against a never-treated cohort u whose adoption date is infinite, the treated-versus-never 2x2 difference-in-differences contrast Δ_TN equals g's window-specific average treatment effect on the treated over its post-adoption window:","s":"∀ {𝒢 : Type u_1} [inst : Fintype 𝒢] {T : ℕ} (P : Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.CohortPanel 𝒢 T) (Y0 Y1 : 𝒢 → Fin T → ℝ), Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.CausalAssumptions P Y0 Y1 → ∀ (g u : 𝒢), Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.AdoptionDate.isFin (P.A g) → Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.AdoptionDate.isInf (P.A u) → Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.Δ_TN P g u = Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.ATT_window Y0 Y1 g (Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.S1_TN P g)","l":300,"f":"Causalean/Panel/EstimandCharacterization/StaggeredTWFEDecomposition/Causal.lean"},{"n":"Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.𝒦","k":"def","m":"Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.FinitePanel","a":"Panel/EstimandCharacterization/StaggeredTWFEDecomposition","t":1,"d":"Comparison index set 𝒦 P : Finset (CompTag × 𝒢 × 𝒢), the set of all admissible 2x2 comparisons.","s":"{𝒢 : Type u_1} → [inst : Fintype 𝒢] → {T : ℕ} → Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.CohortPanel 𝒢 T → Finset (Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.CompTag × 𝒢 × 𝒢)","l":281,"f":"Causalean/Panel/EstimandCharacterization/StaggeredTWFEDecomposition/FinitePanel.lean"},{"n":"Causalean.Panel.Exposure","k":"def","m":"Causalean.Panel.PO.Exposure","a":"Panel/PO","t":1,"d":"An exposure mapping: a known function from realized treatment paths to a cell-dependent exposure set E r.","s":"(I : Type u_1) → (T : Type u_2) → Type u_3 → (I × T → Type u_4) → Type (max (max (max u_1 u_2) u_3) u_4)","l":31,"f":"Causalean/Panel/PO/Exposure.lean"},{"n":"Causalean.Panel.FWLInstanceL2.fwlCoef_eq_of_original_minimizer","k":"theorem","m":"Causalean.Panel.FWLInstanceL2","a":"Panel","t":1,"d":"FWL uniqueness on L²(μ). Fix square-integrable regressors X and outcome Y, and a finite-dimensional nuisance subspace H of L²(μ), and assume the residualized regressor Gram matrix is invertible. If the nuisance term h lies in H and the pair (β, h) minimizes the original least-squares objective jointly over all coefficient vectors and nuisance terms in H, then β equals the Frisch–Waugh–Lovell coefficient computed by residualizing against H. Specialization of Causalean.Mathlib.FWL.fwlCoef_eq_of_original_minimizer.","s":"∀ {α : Type u_1} [inst : MeasurableSpace α] {μ : MeasureTheory.Measure α} {K : ℕ} (H : Submodule ℝ ↥(MeasureTheory.Lp ℝ 2 μ)) [inst_1 : H.HasOrthogonalProjection] (X : Fin K → ↥(MeasureTheory.Lp ℝ 2 μ)) (Y : ↥(MeasureTheory.Lp ℝ 2 μ)), IsUnit (Causalean.Mathlib.FWL.gramResid H X).det → ∀ (β : Fin K → ℝ) {h : ↥(MeasureTheory.Lp ℝ 2 μ)}, h ∈ H → (∀ (γ : Fin K → ℝ) {g : ↥(MeasureTheory.Lp ℝ 2 μ)}, g ∈ H → Causalean.Mathlib.FWL.originalObjective X Y β h ≤ Causalean.Mathlib.FWL.originalObjective X Y γ g) → β = Causalean.Mathlib.FWL.fwlCoef H X Y","l":107,"f":"Causalean/Panel/FWLInstanceL2.lean"},{"n":"Causalean.Panel.FWLInstanceL2.fwlCoef_original_minimizes","k":"theorem","m":"Causalean.Panel.FWLInstanceL2","a":"Panel","t":2,"d":"FWL least-squares optimality on L²(μ). The FWL coefficient (paired with the nuisance projection of its raw residual) minimizes the original least-squares objective over coefficients and nuisance terms in H. Specialization of Causalean.Mathlib.FWL.fwlCoef_original_minimizes.","s":"∀ {α : Type u_1} [inst : MeasurableSpace α] {μ : MeasureTheory.Measure α} {K : ℕ} (H : Submodule ℝ ↥(MeasureTheory.Lp ℝ 2 μ)) [inst_1 : H.HasOrthogonalProjection] (X : Fin K → ↥(MeasureTheory.Lp ℝ 2 μ)) (Y : ↥(MeasureTheory.Lp ℝ 2 μ)), IsUnit (Causalean.Mathlib.FWL.gramResid H X).det → ∀ (β : Fin K → ℝ) {h : ↥(MeasureTheory.Lp ℝ 2 μ)}, h ∈ H → Causalean.Mathlib.FWL.originalObjective X Y (Causalean.Mathlib.FWL.fwlCoef H X Y) (Submodule.orthogonalProjectionFn (Y - Causalean.Mathlib.FWL.fittedValue X (Causalean.Mathlib.FWL.fwlCoef H X Y))) ≤ Causalean.Mathlib.FWL.originalObjective X Y β h","l":94,"f":"Causalean/Panel/FWLInstanceL2.lean"},{"n":"Causalean.Panel.FWLInstanceL2.gramResid_mulVec_fwlCoef","k":"theorem","m":"Causalean.Panel.FWLInstanceL2","a":"Panel","t":2,"d":"Residualized normal equations on L²(μ). The FWL coefficient solves gramResid H X *ᵥ fwlCoef H X Y = residInnerVec H X Y whenever the residualized Gram matrix is invertible. Specialization of Causalean.Mathlib.FWL.gramResid_mulVec_fwlCoef.","s":"∀ {α : Type u_1} [inst : MeasurableSpace α] {μ : MeasureTheory.Measure α} {K : ℕ} (H : Submodule ℝ ↥(MeasureTheory.Lp ℝ 2 μ)) [inst_1 : H.HasOrthogonalProjection] (X : Fin K → ↥(MeasureTheory.Lp ℝ 2 μ)) (Y : ↥(MeasureTheory.Lp ℝ 2 μ)), IsUnit (Causalean.Mathlib.FWL.gramResid H X).det → (Causalean.Mathlib.FWL.gramResid H X).mulVec (Causalean.Mathlib.FWL.fwlCoef H X Y) = Causalean.Mathlib.FWL.residInnerVec H X Y","l":84,"f":"Causalean/Panel/FWLInstanceL2.lean"},{"n":"Causalean.Panel.FWLInstanceL2.hasOrthogonalProjection_of_finiteDimensional","k":"instance","m":"Causalean.Panel.FWLInstanceL2","a":"Panel","t":2,"d":"A finite-dimensional subspace of Lp ℝ 2 μ admits an orthogonal projection: it is complete (finite-dimensional over the complete field ℝ), so Mathlib's completeness instance applies.","s":"∀ {α : Type u_1} [inst : MeasurableSpace α] {μ : MeasureTheory.Measure α} (H : Submodule ℝ ↥(MeasureTheory.Lp ℝ 2 μ)) [FiniteDimensional ℝ ↥H], H.HasOrthogonalProjection","l":59,"f":"Causalean/Panel/FWLInstanceL2.lean"},{"n":"Causalean.Panel.FWLInstanceL2.inner_eq_integral","k":"theorem","m":"Causalean.Panel.FWLInstanceL2","a":"Panel","t":2,"d":"L² inner product = integral pairing. For two square-integrable real random variables, the Hilbert-space inner product equals the integral of their product, independent of the chosen representatives. This is the bridge between abstract FWL inner products and the population second moments used in estimand papers.","s":"∀ {α : Type u_1} [inst : MeasurableSpace α] {μ : MeasureTheory.Measure α} (f g : ↥(MeasureTheory.Lp ℝ 2 μ)), inner ℝ f g = ∫ (a : α), ↑↑f a * ↑↑g a ∂μ","l":67,"f":"Causalean/Panel/FWLInstanceL2.lean"},{"n":"Causalean.Panel.LinearL2Class","k":"structure","m":"Causalean.Panel.Analysis.Residualization","a":"Panel/Analysis","t":1,"d":"A linear square-integrable class is a nuisance space of real-valued functions, cut out by a membership predicate mem, that is closed under the zero function, closed under addition, and closed under real scalar multiplication, with every member square-integrable.","s":"{Ω : Type u_1} → [inst : MeasurableSpace Ω] → MeasureTheory.Measure Ω → Type u_1","l":33,"f":"Causalean/Panel/Analysis/Residualization.lean"},{"n":"Causalean.Panel.PO.CellPartition","k":"structure","m":"Causalean.Panel.PO.PopulationCells","a":"Panel/PO","t":1,"d":"A finite measurable partition of a probability space (Ω, μ) into positive-mass cells indexed by a finite type ι: an assignment of a cell to each index such that every cell is measurable, distinct cells are pairwise disjoint, the cells cover the whole space, and every cell has strictly positive probability mass.","s":"{Ω : Type u_1} → [inst : MeasurableSpace Ω] → MeasureTheory.Measure Ω → (ι : Type u_2) → [Fintype ι] → Type (max u_1 u_2)","l":46,"f":"Causalean/Panel/PO/PopulationCells.lean"},{"n":"Causalean.Panel.PO.CellPartition.cellPartitionOfClassifier_cell","k":"theorem","m":"Causalean.Panel.PO.PopulationCells","a":"Panel/PO","t":2,"d":"","s":"∀ {Ω : Type u_1} [inst : MeasurableSpace Ω] {ι : Type u_3} [inst_1 : Fintype ι] (μ : MeasureTheory.Measure Ω) (g : Ω → ι) (hmeas : ∀ (i : ι), MeasurableSet (g ⁻¹' {i})) (hpos : ∀ (i : ι), 0 < (μ (g ⁻¹' {i})).toReal) (i : ι), (Causalean.Panel.PO.cellPartitionOfClassifier μ g hmeas hpos).cell i = g ⁻¹' {i}","l":83,"f":"Causalean/Panel/PO/PopulationCells.lean"},{"n":"Causalean.Panel.PO.CellPartition.integral_eq_sum_mass_mul_mean","k":"theorem","m":"Causalean.Panel.PO.PopulationCells","a":"Panel/PO","t":2,"d":"Finite-partition total law. The integral of f is the mass-weighted sum of its cell means: ∫ f = ∑ i, π_i · E[f ∣ cell i].","s":"∀ {Ω : Type u_1} [inst : MeasurableSpace Ω] {μ : MeasureTheory.Measure Ω} {ι : Type u_2} [inst_1 : Fintype ι] (P : Causalean.Panel.PO.CellPartition μ ι) [MeasureTheory.IsFiniteMeasure μ] (f : Ω → ℝ), MeasureTheory.Integrable f μ → ∫ (ω : Ω), f ω ∂μ = ∑ i, P.mass i * P.mean f i","l":118,"f":"Causalean/Panel/PO/PopulationCells.lean"},{"n":"Causalean.Panel.PO.CellPartition.mass","k":"def","m":"Causalean.Panel.PO.PopulationCells","a":"Panel/PO","t":1,"d":"Cell probability π_i = ℙ(cell i), as a real number.","s":"{Ω : Type u_1} → [inst : MeasurableSpace Ω] → {μ : MeasureTheory.Measure Ω} → {ι : Type u_2} → [inst_1 : Fintype ι] → Causalean.Panel.PO.CellPartition μ ι → ι → ℝ","l":89,"f":"Causalean/Panel/PO/PopulationCells.lean"},{"n":"Causalean.Panel.PO.CellPartition.mass_ne_zero","k":"theorem","m":"Causalean.Panel.PO.PopulationCells","a":"Panel/PO","t":2,"d":"Every cell has nonzero mass.","s":"∀ {Ω : Type u_1} [inst : MeasurableSpace Ω] {μ : MeasureTheory.Measure Ω} {ι : Type u_2} [inst_1 : Fintype ι] (P : Causalean.Panel.PO.CellPartition μ ι) (i : ι), P.mass i ≠ 0","l":100,"f":"Causalean/Panel/PO/PopulationCells.lean"},{"n":"Causalean.Panel.PO.CellPartition.mass_pos","k":"theorem","m":"Causalean.Panel.PO.PopulationCells","a":"Panel/PO","t":2,"d":"Every cell has strictly positive mass.","s":"∀ {Ω : Type u_1} [inst : MeasurableSpace Ω] {μ : MeasureTheory.Measure Ω} {ι : Type u_2} [inst_1 : Fintype ι] (P : Causalean.Panel.PO.CellPartition μ ι) (i : ι), 0 < P.mass i","l":97,"f":"Causalean/Panel/PO/PopulationCells.lean"},{"n":"Causalean.Panel.PO.CellPartition.mass_sum_one","k":"theorem","m":"Causalean.Panel.PO.PopulationCells","a":"Panel/PO","t":2,"d":"Masses sum to one. The cell probabilities of a partition of a probability space add to 1.","s":"∀ {Ω : Type u_1} [inst : MeasurableSpace Ω] {μ : MeasureTheory.Measure Ω} {ι : Type u_2} [inst_1 : Fintype ι] (P : Causalean.Panel.PO.CellPartition μ ι) [MeasureTheory.IsProbabilityMeasure μ], ∑ i, P.mass i = 1","l":104,"f":"Causalean/Panel/PO/PopulationCells.lean"},{"n":"Causalean.Panel.PO.CellPartition.mean","k":"def","m":"Causalean.Panel.PO.PopulationCells","a":"Panel/PO","t":1,"d":"Cell-conditional mean E[f ∣ cell i], i.e. the average of f over the cell computed as its integral over the cell divided by the cell's real mass.","s":"{Ω : Type u_1} → [inst : MeasurableSpace Ω] → {μ : MeasureTheory.Measure Ω} → {ι : Type u_2} → [inst_1 : Fintype ι] → Causalean.Panel.PO.CellPartition μ ι → (Ω → ℝ) → ι → ℝ","l":92,"f":"Causalean/Panel/PO/PopulationCells.lean"},{"n":"Causalean.Panel.PO.CellPartition.mean_add","k":"theorem","m":"Causalean.Panel.PO.PopulationCells","a":"Panel/PO","t":2,"d":"Cell means are additive over sums of integrable integrands.","s":"∀ {Ω : Type u_1} [inst : MeasurableSpace Ω] {μ : MeasureTheory.Measure Ω} {ι : Type u_2} [inst_1 : Fintype ι] (P : Causalean.Panel.PO.CellPartition μ ι) {f g : Ω → ℝ} (i : ι), MeasureTheory.IntegrableOn f (P.cell i) μ → MeasureTheory.IntegrableOn g (P.cell i) μ → P.mean (f + g) i = P.mean f i + P.mean g i","l":146,"f":"Causalean/Panel/PO/PopulationCells.lean"},{"n":"Causalean.Panel.PO.CellPartition.mean_congr_ae","k":"theorem","m":"Causalean.Panel.PO.PopulationCells","a":"Panel/PO","t":2,"d":"Cell means agree for a.e.-equal integrands.","s":"∀ {Ω : Type u_1} [inst : MeasurableSpace Ω] {μ : MeasureTheory.Measure Ω} {ι : Type u_2} [inst_1 : Fintype ι] (P : Causalean.Panel.PO.CellPartition μ ι) {f g : Ω → ℝ} (i : ι), f =ᵐ[μ] g → P.mean f i = P.mean g i","l":134,"f":"Causalean/Panel/PO/PopulationCells.lean"},{"n":"Causalean.Panel.PO.CellPartition.mean_congr_on","k":"theorem","m":"Causalean.Panel.PO.PopulationCells","a":"Panel/PO","t":2,"d":"Cell means agree when the integrands agree pointwise on the cell. This is the workhorse \"consistency descent\" step: on a cell where an observed quantity equals a potential-outcome slice, their cell means coincide.","s":"∀ {Ω : Type u_1} [inst : MeasurableSpace Ω] {μ : MeasureTheory.Measure Ω} {ι : Type u_2} [inst_1 : Fintype ι] (P : Causalean.Panel.PO.CellPartition μ ι) {f g : Ω → ℝ} (i : ι), (∀ ω ∈ P.cell i, f ω = g ω) → P.mean f i = P.mean g i","l":126,"f":"Causalean/Panel/PO/PopulationCells.lean"},{"n":"Causalean.Panel.PO.CellPartition.mean_consistency","k":"theorem","m":"Causalean.Panel.PO.PopulationCells","a":"Panel/PO","t":1,"d":"Cell-mean consistency descent. On a cell where the treatment value d is binary, d ∈ {0,1}, and pointwise potential-outcome consistency Y = Y0 + d·(Y1 − Y0) holds throughout the cell, the cell-conditional means satisfy the same identity: E[Y ∣ cell] = E[Y0 ∣ cell] + d·(E[Y1 ∣ cell] − E[Y0 ∣ cell]).","s":"∀ {Ω : Type u_1} [inst : MeasurableSpace Ω] {μ : MeasureTheory.Measure Ω} {ι : Type u_2} [inst_1 : Fintype ι] (P : Causalean.Panel.PO.CellPartition μ ι) (i : ι) (Y Y0 Y1 : Ω → ℝ) {d : ℝ}, d = 0 ∨ d = 1 → (∀ ω ∈ P.cell i, Y ω = Y0 ω + d * (Y1 ω - Y0 ω)) → P.mean Y i = P.mean Y0 i + d * (P.mean Y1 i - P.mean Y0 i)","l":157,"f":"Causalean/Panel/PO/PopulationCells.lean"},{"n":"Causalean.Panel.PO.CellPartition.mean_smul","k":"theorem","m":"Causalean.Panel.PO.PopulationCells","a":"Panel/PO","t":2,"d":"Cell means are homogeneous under real scalar multiplication.","s":"∀ {Ω : Type u_1} [inst : MeasurableSpace Ω] {μ : MeasureTheory.Measure Ω} {ι : Type u_2} [inst_1 : Fintype ι] (P : Causalean.Panel.PO.CellPartition μ ι) (c : ℝ) (f : Ω → ℝ) (i : ι), P.mean (fun ω => c * f ω) i = c * P.mean f i","l":152,"f":"Causalean/Panel/PO/PopulationCells.lean"},{"n":"Causalean.Panel.PO.CellPartition.mean_sub","k":"theorem","m":"Causalean.Panel.PO.PopulationCells","a":"Panel/PO","t":2,"d":"Cell means are additive over subtraction of integrable integrands.","s":"∀ {Ω : Type u_1} [inst : MeasurableSpace Ω] {μ : MeasureTheory.Measure Ω} {ι : Type u_2} [inst_1 : Fintype ι] (P : Causalean.Panel.PO.CellPartition μ ι) {f g : Ω → ℝ} (i : ι), MeasureTheory.IntegrableOn f (P.cell i) μ → MeasureTheory.IntegrableOn g (P.cell i) μ → P.mean (f - g) i = P.mean f i - P.mean g i","l":140,"f":"Causalean/Panel/PO/PopulationCells.lean"},{"n":"Causalean.Panel.PO.Mobius.coeff_sum","k":"theorem","m":"Causalean.Panel.PO.Mobius","a":"Panel/PO","t":2,"d":"Coefficient sum: for A ⊆ B, ∑ S, A ⊆ S ⊆ B, (-1)^(|S|-|A|) = if A = B then 1 else 0.","s":"∀ {ι : Type u_1} [inst : DecidableEq ι] (A B : Finset ι), A ⊆ B → ∑ S ∈ B.powerset with A ⊆ S, (-1) ^ (S.card - A.card) = if A = B then 1 else 0","l":93,"f":"Causalean/Panel/PO/Mobius.lean"},{"n":"Causalean.Panel.PO.Mobius.delta","k":"def","m":"Causalean.Panel.PO.Mobius","a":"Panel/PO","t":1,"d":"The interaction effect at a subset S. The main lemma uses this only for non-empty S, but the definition is total.","s":"{ι : Type u_1} → [DecidableEq ι] → ((ι → Fin 2) → ℝ) → Finset ι → ℝ","l":59,"f":"Causalean/Panel/PO/Mobius.lean"},{"n":"Causalean.Panel.PO.Mobius.indicator","k":"def","m":"Causalean.Panel.PO.Mobius","a":"Panel/PO","t":1,"d":"Indicator history 𝟙_A : ι → Fin 2, equal to 1 on A and 0 elsewhere.","s":"{ι : Type u_1} → [DecidableEq ι] → Finset ι → ι → Fin 2","l":48,"f":"Causalean/Panel/PO/Mobius.lean"},{"n":"Causalean.Panel.PO.Mobius.indicator_empty","k":"theorem","m":"Causalean.Panel.PO.Mobius","a":"Panel/PO","t":2,"d":"The indicator history of the empty subset is the all-zero treatment history.","s":"∀ {ι : Type u_1} [inst : DecidableEq ι], Causalean.Panel.PO.Mobius.indicator ∅ = fun x => 0","l":53,"f":"Causalean/Panel/PO/Mobius.lean"},{"n":"Causalean.Panel.PO.Mobius.mobius_expansion","k":"theorem","m":"Causalean.Panel.PO.Mobius","a":"Panel/PO","t":1,"d":"Any binary finite-memory response τ normalized to zero at the all-zero history can be recovered as the sum of its nonempty inclusion-exclusion interaction coefficients, with each interaction contributing only when every one of its lags is active in the history.","s":"∀ {ι : Type u_1} [inst : DecidableEq ι] [inst_1 : Fintype ι] (τ : (ι → Fin 2) → ℝ), (τ fun x => 0) = 0 → ∀ (h : ι → Fin 2), τ h = ∑ S ∈ Finset.univ.powerset with S.Nonempty, Causalean.Panel.PO.Mobius.delta τ S * ∏ k ∈ S, ↑↑(h k)","l":173,"f":"Causalean/Panel/PO/Mobius.lean"},{"n":"Causalean.Panel.PO.Mobius.prod_indicator_eq","k":"theorem","m":"Causalean.Panel.PO.Mobius","a":"Panel/PO","t":2,"d":"Indicator product: ∏ k ∈ S, ((h k).val : ℝ) equals 1 if S ⊆ B and 0 otherwise, where B = {k : h k = 1}.","s":"∀ {ι : Type u_1} [inst : DecidableEq ι] (h : ι → Fin 2) (B : Finset ι), (∀ (k : ι), k ∈ B ↔ h k = 1) → ∀ (S : Finset ι), ∏ k ∈ S, ↑↑(h k) = if S ⊆ B then 1 else 0","l":65,"f":"Causalean/Panel/PO/Mobius.lean"},{"n":"Causalean.Panel.PO.cellPartitionOfClassifier","k":"def","m":"Causalean.Panel.PO.PopulationCells","a":"Panel/PO","t":1,"d":"Build a cell partition from a finite classifier g : Ω → ι: the cells are the level sets {ω | g ω = i}. Disjointness and covering are automatic; the caller supplies measurability and positive mass of each level set. This is the common way panel bridges obtain their partition (cohort map, treated/untreated classifier, cohort-period map, …).","s":"{Ω : Type u_1} → [inst : MeasurableSpace Ω] → {ι : Type u_2} → [inst_1 : Fintype ι] → (μ : MeasureTheory.Measure Ω) → (g : Ω → ι) → (∀ (i : ι), MeasurableSet (g ⁻¹' {i})) → (∀ (i : ι), 0 < (μ (g ⁻¹' {i})).toReal) → Causalean.Panel.PO.CellPartition μ ι","l":63,"f":"Causalean/Panel/PO/PopulationCells.lean"},{"n":"Causalean.Panel.PanelConsistency","k":"def","m":"Causalean.Panel.PO.Consistency","a":"Panel/PO","t":1,"d":"Panel consistency says that on every observed unit-period cell, the factual outcome equals the potential outcome evaluated at the realized exposure.","s":"Causalean.Panel.PanelPOSystem → Prop","l":39,"f":"Causalean/Panel/PO/Consistency.lean"},{"n":"Causalean.Panel.PanelPOSystem","k":"structure","m":"Causalean.Panel.PO.CellPO","a":"Panel/PO","t":1,"d":"A panel potential-outcomes system pairs finite unit and time index sets with a finite collection of observed unit-period cells carrying positive normalized weights, a treatment-path-valued random variable D that is measurable in every unit-period coordinate, a cell-specific exposure mapping e together with a baseline exposure e0, a family of potential outcomes Y at each cell and exposure that is measurable and integrable, and a family of factual observed outcomes Yobs stored independently of the potential-outcome schedule that is likewise measurable and integrable.","s":"Type (max (max (max (max (u_1 + 1) (u_2 + 1)) (u_3 + 1)) (u_4 + 1)) (u_5 + 1))","l":52,"f":"Causalean/Panel/PO/CellPO.lean"},{"n":"Causalean.Panel.PanelPOSystem.DSigmaAlg","k":"def","m":"Causalean.Panel.PO.CellPO","a":"Panel/PO","t":1,"d":"The σ-algebra generated by the treatment path D. This is the \"information of the assignment vector\" — the σ-algebra in \\bbE[· | D] of Definition 2.4.","s":"(P : Causalean.Panel.PanelPOSystem) → MeasurableSpace P.Ω","l":127,"f":"Causalean/Panel/PO/CellPO.lean"},{"n":"Causalean.Panel.PanelPOSystem.DSigmaAlg_le","k":"theorem","m":"Causalean.Panel.PO.CellPO","a":"Panel/PO","t":1,"d":"The σ-algebra generated by the treatment path D, σ(D), is a sub-σ-algebra of the ambient σ-algebra on the outcome space Ω.","s":"∀ (P : Causalean.Panel.PanelPOSystem), P.DSigmaAlg ≤ inferInstance","l":133,"f":"Causalean/Panel/PO/CellPO.lean"},{"n":"Causalean.Panel.PanelPOSystem.m0","k":"def","m":"Causalean.Panel.PO.CellPO","a":"Panel/PO","t":1,"d":"Baseline conditional mean m_r^0(D) := E[Y_r(e_r^0) | D].","s":"(P : Causalean.Panel.PanelPOSystem) → (r : P.I × P.T) → r ∈ P.cells.observed → P.Ω → ℝ","l":140,"f":"Causalean/Panel/PO/CellPO.lean"},{"n":"Causalean.Panel.PanelPOSystem.observedExposure","k":"def","m":"Causalean.Panel.PO.CellPO","a":"Panel/PO","t":1,"d":"The realized exposure at an observed cell r: E_r = e_r(D).","s":"(P : Causalean.Panel.PanelPOSystem) → (r : P.I × P.T) → (hr : r ∈ P.cells.observed) → P.Ω → P.E r hr","l":116,"f":"Causalean/Panel/PO/CellPO.lean"},{"n":"Causalean.Panel.PanelPOSystem.observedY","k":"def","m":"Causalean.Panel.PO.CellPO","a":"Panel/PO","t":1,"d":"The factual observed outcome at an observed cell. It is stored independently from the potential-outcome schedule; the consistency assumption relates it to the potential outcome at the realized exposure.","s":"(P : Causalean.Panel.PanelPOSystem) → (r : P.I × P.T) → r ∈ P.cells.observed → P.Ω → ℝ","l":121,"f":"Causalean/Panel/PO/CellPO.lean"},{"n":"Causalean.Panel.PanelPOSystem.observedY_eq_potentialOutcome","k":"def","m":"Causalean.Panel.PO.Consistency","a":"Panel/PO","t":1,"d":"Pointwise consistency at a given observed cell is the equality between the factual observed outcome and the potential outcome indexed by the realized exposure.","s":"(P : Causalean.Panel.PanelPOSystem) → (r : P.I × P.T) → r ∈ P.cells.observed → P.Ω → Prop","l":31,"f":"Causalean/Panel/PO/Consistency.lean"},{"n":"Causalean.Panel.PanelPOSystem.realizedTau","k":"def","m":"Causalean.Panel.PO.CellPO","a":"Panel/PO","t":1,"d":"The realized exposure response τ_r(E_r | D), obtained by evaluating tau at the exposure induced by the realized treatment path.","s":"(P : Causalean.Panel.PanelPOSystem) → (r : P.I × P.T) → r ∈ P.cells.observed → P.Ω → ℝ","l":151,"f":"Causalean/Panel/PO/CellPO.lean"},{"n":"Causalean.Panel.PanelPOSystem.tau","k":"def","m":"Causalean.Panel.PO.CellPO","a":"Panel/PO","t":1,"d":"Cell-level exposure response τ_r(e | D) := E[Y_r(e) − Y_r(e_r^0) | D].","s":"(P : Causalean.Panel.PanelPOSystem) → (r : P.I × P.T) → (hr : r ∈ P.cells.observed) → P.E r hr → P.Ω → ℝ","l":144,"f":"Causalean/Panel/PO/CellPO.lean"},{"n":"Causalean.Panel.ResidualizationWitness","k":"structure","m":"Causalean.Panel.Analysis.Residualization","a":"Panel/Analysis","t":1,"d":"A residualization witness decomposes a real-valued function V into an in-class projection VH and a square-integrable residual Vtilde such that V equals their sum almost everywhere and the residual is orthogonal in expectation to every member of the nuisance class.","s":"{Ω : Type u_1} → [inst : MeasurableSpace Ω] → (μ : MeasureTheory.Measure Ω) → Causalean.Panel.LinearL2Class μ → (Ω → ℝ) → Type u_1","l":52,"f":"Causalean/Panel/Analysis/Residualization.lean"},{"n":"Causalean.Panel.TreatmentPath","k":"def","m":"Causalean.Panel.PO.TreatmentPath","a":"Panel/PO","t":1,"d":"A treatment path: each unit i : I and time index t : T is assigned an action in A, representing the realized assignment D_{it}.","s":"Type u_1 → Type u_2 → Type u_3 → Type (max (max u_1 u_2) u_3)","l":22,"f":"Causalean/Panel/PO/TreatmentPath.lean"},{"n":"Causalean.Panel.TreatmentPath.BinaryHistory_zero","k":"theorem","m":"Causalean.Panel.PO.TreatmentPath","a":"Panel/PO","t":2,"d":"Lag 0 of the binary zero-baseline history is the contemporaneous treatment.","s":"∀ {I : Type u_1} {T₀ : ℕ} (p : ℕ) (D : Causalean.Panel.TreatmentPath I (Fin T₀) (Fin 2)) (i : I) (t : Fin T₀), Causalean.Panel.TreatmentPath.BinaryHistory p D i t 0 = D i t","l":74,"f":"Causalean/Panel/PO/TreatmentPath.lean"},{"n":"Causalean.Panel.TreatmentPath.HistoryDefault_zero","k":"theorem","m":"Causalean.Panel.PO.TreatmentPath","a":"Panel/PO","t":2,"d":"Lag 0 of the default-boundary history is the contemporaneous treatment.","s":"∀ {I : Type u_1} {A : Type u_2} {T₀ : ℕ} [inst : Inhabited A] (p : ℕ) (D : Causalean.Panel.TreatmentPath I (Fin T₀) A) (i : I) (t : Fin T₀), Causalean.Panel.TreatmentPath.HistoryDefault p D i t 0 = D i t","l":68,"f":"Causalean/Panel/PO/TreatmentPath.lean"},{"n":"Causalean.Panel.TreatmentPath.History_zero","k":"theorem","m":"Causalean.Panel.PO.TreatmentPath","a":"Panel/PO","t":1,"d":"Lag 0 of the finite-memory treatment history equals the unit's contemporaneous treatment value.","s":"∀ {I : Type u_1} {A : Type u_2} {T₀ : ℕ} (a0 : A) (p : ℕ) (D : Causalean.Panel.TreatmentPath I (Fin T₀) A) (i : I) (t : Fin T₀), Causalean.Panel.TreatmentPath.History a0 p D i t 0 = D i t","l":60,"f":"Causalean/Panel/PO/TreatmentPath.lean"},{"n":"Causalean.Panel.UniformTwoWayPanel.BalancedPanel","k":"structure","m":"Causalean.Panel.UniformTwoWayPanel","a":"Panel","t":1,"d":"Balanced panel. The side conditions from the source definition that a panel counts as balanced: the unit index type has at least two elements and the time index type has at least two elements.","s":"(Unit : Type u_3) → (Time : Type u_4) → [Fintype Unit] → [Fintype Time] → Prop","l":47,"f":"Causalean/Panel/UniformTwoWayPanel.lean"},{"n":"Causalean.Panel.UniformTwoWayPanel.IsUnitTimeAdditive","k":"def","m":"Causalean.Panel.UniformTwoWayPanel","a":"Panel","t":1,"d":"Unit-time additive nuisance class h_it = a_i + b_t.","s":"{Unit : Type u_1} → {Time : Type u_2} → (Unit → Time → ℝ) → Prop","l":283,"f":"Causalean/Panel/UniformTwoWayPanel.lean"},{"n":"Causalean.Panel.UniformTwoWayPanel.ddot","k":"def","m":"Causalean.Panel.UniformTwoWayPanel","a":"Panel","t":1,"d":"Two-way residual / double-demeaned array.","s":"{Unit : Type u_1} → {Time : Type u_2} → [Fintype Unit] → [Fintype Time] → (Unit → Time → ℝ) → Unit → Time → ℝ","l":88,"f":"Causalean/Panel/UniformTwoWayPanel.lean"},{"n":"Causalean.Panel.UniformTwoWayPanel.ddot_eq_weighted","k":"theorem","m":"Causalean.Panel.UniformTwoWayPanel","a":"Panel","t":2,"d":"Uniform double-demeaning is weighted double-demeaning with uniform unit weights.","s":"∀ {Unit : Type u_1} {Time : Type u_2} [inst : Fintype Unit] [inst_1 : Fintype Time] (V : Unit → Time → ℝ) (i : Unit) (t : Time), Causalean.Panel.UniformTwoWayPanel.ddot V i t = Causalean.Panel.WeightedTwoWayPanel.ddot (Causalean.Panel.UniformTwoWayPanel.uniformWeights (Fintype.card_pos_iff.mpr (Nonempty.intro i))) V i t","l":128,"f":"Causalean/Panel/UniformTwoWayPanel.lean"},{"n":"Causalean.Panel.UniformTwoWayPanel.ddot_orthogonal_time","k":"theorem","m":"Causalean.Panel.UniformTwoWayPanel","a":"Panel","t":2,"d":"Double-demeaned arrays are orthogonal to arbitrary time-only functions.","s":"∀ {Unit : Type u_1} {Time : Type u_2} [inst : Fintype Unit] [inst_1 : Fintype Time], 0 < Fintype.card Unit → ∀ (V : Unit → Time → ℝ) (b : Time → ℝ), ∑ i, ∑ t, Causalean.Panel.UniformTwoWayPanel.ddot V i t * b t = 0","l":343,"f":"Causalean/Panel/UniformTwoWayPanel.lean"},{"n":"Causalean.Panel.UniformTwoWayPanel.ddot_orthogonal_unit","k":"theorem","m":"Causalean.Panel.UniformTwoWayPanel","a":"Panel","t":2,"d":"Double-demeaned arrays are orthogonal to arbitrary unit-only functions.","s":"∀ {Unit : Type u_1} {Time : Type u_2} [inst : Fintype Unit] [inst_1 : Fintype Time], 0 < Fintype.card Unit → 0 < Fintype.card Time → ∀ (V : Unit → Time → ℝ) (a : Unit → ℝ), ∑ i, ∑ t, Causalean.Panel.UniformTwoWayPanel.ddot V i t * a i = 0","l":315,"f":"Causalean/Panel/UniformTwoWayPanel.lean"},{"n":"Causalean.Panel.UniformTwoWayPanel.ddot_orthogonal_unit_time","k":"theorem","m":"Causalean.Panel.UniformTwoWayPanel","a":"Panel","t":1,"d":"Assume at least one unit and at least one period. Then for any array h of unit/time additive form h_it = a_i + b_t, the double-demeaned array ddot V is orthogonal to h under the unnormalized uniform panel inner product.","s":"∀ {Unit : Type u_1} {Time : Type u_2} [inst : Fintype Unit] [inst_1 : Fintype Time], 0 < Fintype.card Unit → 0 < Fintype.card Time → ∀ (V h : Unit → Time → ℝ), Causalean.Panel.UniformTwoWayPanel.IsUnitTimeAdditive h → Causalean.Panel.UniformTwoWayPanel.inner (Causalean.Panel.UniformTwoWayPanel.ddot V) h = 0","l":370,"f":"Causalean/Panel/UniformTwoWayPanel.lean"},{"n":"Causalean.Panel.UniformTwoWayPanel.ddot_reconstruct","k":"theorem","m":"Causalean.Panel.UniformTwoWayPanel","a":"Panel","t":2,"d":"Reconstruction identity V_it = ddot V_it + unitMean V_i + timeMean V_t - grandMean V.","s":"∀ {Unit : Type u_1} {Time : Type u_2} [inst : Fintype Unit] [inst_1 : Fintype Time] (V : Unit → Time → ℝ) (i : Unit) (t : Time), Causalean.Panel.UniformTwoWayPanel.ddot V i t + Causalean.Panel.UniformTwoWayPanel.unitMean V i + Causalean.Panel.UniformTwoWayPanel.timeMean V t - Causalean.Panel.UniformTwoWayPanel.grandMean V = V i t","l":293,"f":"Causalean/Panel/UniformTwoWayPanel.lean"},{"n":"Causalean.Panel.UniformTwoWayPanel.finiteResidualizedCoefficient","k":"def","m":"Causalean.Panel.UniformTwoWayPanel","a":"Panel","t":1,"d":"Finite scalar residualized coefficient for a supplied residualized regressor Dtilde and residualized outcome Ytilde.","s":"{Unit : Type u_1} → {Time : Type u_2} → [Fintype Unit] → [Fintype Time] → (Unit → Time → ℝ) → (Unit → Time → ℝ) → ℝ","l":181,"f":"Causalean/Panel/UniformTwoWayPanel.lean"},{"n":"Causalean.Panel.UniformTwoWayPanel.finite_residualized_coefficient_eq_of_normalEqs","k":"theorem","m":"Causalean.Panel.UniformTwoWayPanel","a":"Panel","t":1,"d":"Finite scalar Frisch–Waugh–Lovell handoff. Suppose the outcome and regressor decompose as Y = Yproj + Ytilde and D = Dproj + Dtilde, where Dproj and the fitted nuisance term Hβ both satisfy the nuisance predicate H, Dtilde is orthogonal to every array satisfying H, Dtilde is orthogonal to Yproj, and Dtilde has strictly positive self-inner-product (a nondegenerate residualized regressor). If the coefficient β and nuisance fit Hβ satisfy the normal equation against the raw regressor D and the normal equation against its nuisance component Dproj, then β equals the finite residualized coefficient inner Dtilde Ytilde / inner Dtilde Dtilde.","s":"∀ {Unit : Type u_1} {Time : Type u_2} [inst : Fintype Unit] [inst_1 : Fintype Time] (H : (Unit → Time → ℝ) → Prop) {Y D Yproj Ytilde Dproj Dtilde Hβ : Unit → Time → ℝ} {β : ℝ}, (∀ (i : Unit) (t : Time), Y i t = Yproj i t + Ytilde i t) → (∀ (i : Unit) (t : Time), D i t = Dproj i t + Dtilde i t) → H Dproj → H Hβ → (∀ (h : Unit → Time → ℝ), H h → Causalean.Panel.UniformTwoWayPanel.inner Dtilde h = 0) → Causalean.Panel.UniformTwoWayPanel.inner Dtilde Yproj = 0 → 0 < Causalean.Panel.UniformTwoWayPanel.inner Dtilde Dtilde → (Causalean.Panel.UniformTwoWayPanel.inner D fun i t => Y i t - D i t * β - Hβ i t) = 0 → (Causalean.Panel.UniformTwoWayPanel.inner Dproj fun i t => Y i t - D i t * β - Hβ i t) = 0 → β = Causalean.Panel.UniformTwoWayPanel.finiteResidualizedCoefficient Dtilde Ytilde","l":187,"f":"Causalean/Panel/UniformTwoWayPanel.lean"},{"n":"Causalean.Panel.UniformTwoWayPanel.grandMean","k":"def","m":"Causalean.Panel.UniformTwoWayPanel","a":"Panel","t":1,"d":"Grand mean \\bar V under the uniform unit-period measure.","s":"{Unit : Type u_1} → {Time : Type u_2} → [Fintype Unit] → [Fintype Time] → (Unit → Time → ℝ) → ℝ","l":83,"f":"Causalean/Panel/UniformTwoWayPanel.lean"},{"n":"Causalean.Panel.UniformTwoWayPanel.grandMean_eq_weighted","k":"theorem","m":"Causalean.Panel.UniformTwoWayPanel","a":"Panel","t":2,"d":"In a finite balanced panel with uniform unit weights, the usual grand mean equals the grand mean computed under those weights.","s":"∀ {Unit : Type u_1} {Time : Type u_2} [inst : Fintype Unit] [inst_1 : Fintype Time] (hU : 0 < Fintype.card Unit) (V : Unit → Time → ℝ), Causalean.Panel.UniformTwoWayPanel.grandMean V = Causalean.Panel.WeightedTwoWayPanel.grandMean (Causalean.Panel.UniformTwoWayPanel.uniformWeights hU) V","l":106,"f":"Causalean/Panel/UniformTwoWayPanel.lean"},{"n":"Causalean.Panel.UniformTwoWayPanel.inner","k":"def","m":"Causalean.Panel.UniformTwoWayPanel","a":"Panel","t":1,"d":"Uniform finite-panel inner product, without the harmless normalizing constant.","s":"{Unit : Type u_1} → {Time : Type u_2} → [Fintype Unit] → [Fintype Time] → (Unit → Time → ℝ) → (Unit → Time → ℝ) → ℝ","l":92,"f":"Causalean/Panel/UniformTwoWayPanel.lean"},{"n":"Causalean.Panel.UniformTwoWayPanel.inner_eq_card_smul_weighted","k":"theorem","m":"Causalean.Panel.UniformTwoWayPanel","a":"Panel","t":2,"d":"The unnormalized uniform inner product is the unit count times the weighted inner product under uniform unit weights.","s":"∀ {Unit : Type u_1} {Time : Type u_2} [inst : Fintype Unit] [inst_1 : Fintype Time] (hU : 0 < Fintype.card Unit) (V W : Unit → Time → ℝ), Causalean.Panel.UniformTwoWayPanel.inner V W = ↑(Fintype.card Unit) * Causalean.Panel.WeightedTwoWayPanel.inner (Causalean.Panel.UniformTwoWayPanel.uniformWeights hU) V W","l":170,"f":"Causalean/Panel/UniformTwoWayPanel.lean"},{"n":"Causalean.Panel.UniformTwoWayPanel.sub_ddot_eq_unitTimeProjection","k":"theorem","m":"Causalean.Panel.UniformTwoWayPanel","a":"Panel","t":2,"d":"Pointwise residual decomposition V - ddot V into the unit/time projection.","s":"∀ {Unit : Type u_1} {Time : Type u_2} [inst : Fintype Unit] [inst_1 : Fintype Time] (V : Unit → Time → ℝ) (i : Unit) (t : Time), V i t - Causalean.Panel.UniformTwoWayPanel.ddot V i t = Causalean.Panel.UniformTwoWayPanel.unitTimeProjection V i t","l":308,"f":"Causalean/Panel/UniformTwoWayPanel.lean"},{"n":"Causalean.Panel.UniformTwoWayPanel.sum_eq_card_mul_uniform_weighted","k":"theorem","m":"Causalean.Panel.UniformTwoWayPanel","a":"Panel","t":2,"d":"In a finite balanced panel, the unweighted sum across all unit-period cells equals the number of units times the corresponding sum under uniform unit weights.","s":"∀ {Unit : Type u_1} {Time : Type u_2} [inst : Fintype Unit] [inst_1 : Fintype Time] (hU : 0 < Fintype.card Unit) (F : Unit → Time → ℝ), ∑ i, ∑ t, F i t = ↑(Fintype.card Unit) * ∑ i, ∑ t, (Causalean.Panel.UniformTwoWayPanel.uniformWeights hU).p i * F i t","l":139,"f":"Causalean/Panel/UniformTwoWayPanel.lean"},{"n":"Causalean.Panel.UniformTwoWayPanel.timeMean","k":"def","m":"Causalean.Panel.UniformTwoWayPanel","a":"Panel","t":1,"d":"Time mean \\bar V_{·t} under the uniform unit measure.","s":"{Unit : Type u_1} → {Time : Type u_2} → [Fintype Unit] → (Unit → Time → ℝ) → Time → ℝ","l":79,"f":"Causalean/Panel/UniformTwoWayPanel.lean"},{"n":"Causalean.Panel.UniformTwoWayPanel.timeMean_eq_weighted","k":"theorem","m":"Causalean.Panel.UniformTwoWayPanel","a":"Panel","t":2,"d":"In a finite balanced panel with uniformly weighted units, the usual time mean in any period equals the time mean computed under the uniform unit weights.","s":"∀ {Unit : Type u_1} {Time : Type u_2} [inst : Fintype Unit] (hU : 0 < Fintype.card Unit) (V : Unit → Time → ℝ) (t : Time), Causalean.Panel.UniformTwoWayPanel.timeMean V t = Causalean.Panel.WeightedTwoWayPanel.timeMean (Causalean.Panel.UniformTwoWayPanel.uniformWeights hU) V t","l":98,"f":"Causalean/Panel/UniformTwoWayPanel.lean"},{"n":"Causalean.Panel.UniformTwoWayPanel.uniformWeights","k":"def","m":"Causalean.Panel.UniformTwoWayPanel","a":"Panel","t":1,"d":"Uniform unit weights as a probability vector, used to view this module as the uniform specialization of WeightedTwoWayPanel.","s":"{Unit : Type u_1} → [inst : Fintype Unit] → 0 < Fintype.card Unit → Causalean.Panel.WeightedTwoWayPanel.UnitWeights Unit","l":57,"f":"Causalean/Panel/UniformTwoWayPanel.lean"},{"n":"Causalean.Panel.UniformTwoWayPanel.unitMean","k":"def","m":"Causalean.Panel.UniformTwoWayPanel","a":"Panel","t":1,"d":"Unit mean \\bar V_{i·} under the uniform period measure.","s":"{Unit : Type u_1} → {Time : Type u_2} → [Fintype Time] → (Unit → Time → ℝ) → Unit → ℝ","l":75,"f":"Causalean/Panel/UniformTwoWayPanel.lean"},{"n":"Causalean.Panel.UniformTwoWayPanel.unitTimeProjection","k":"def","m":"Causalean.Panel.UniformTwoWayPanel","a":"Panel","t":1,"d":"The unit/time component removed by double demeaning.","s":"{Unit : Type u_1} → {Time : Type u_2} → [Fintype Unit] → [Fintype Time] → (Unit → Time → ℝ) → Unit → Time → ℝ","l":289,"f":"Causalean/Panel/UniformTwoWayPanel.lean"},{"n":"Causalean.Panel.UniformTwoWayPanel.unitTimeProjection_additive","k":"theorem","m":"Causalean.Panel.UniformTwoWayPanel","a":"Panel","t":2,"d":"The removed component is itself unit/time additive.","s":"∀ {Unit : Type u_1} {Time : Type u_2} [inst : Fintype Unit] [inst_1 : Fintype Time] (V : Unit → Time → ℝ), Causalean.Panel.UniformTwoWayPanel.IsUnitTimeAdditive (Causalean.Panel.UniformTwoWayPanel.unitTimeProjection V)","l":300,"f":"Causalean/Panel/UniformTwoWayPanel.lean"},{"n":"Causalean.Panel.Weighted.AdditiveSpan","k":"def","m":"Causalean.Panel.Weighted.AdditiveSpan","a":"Panel/Weighted","t":1,"d":"The additive span of two projections f₁ : R → A and f₂ : R → B: the subspace of R → ℝ consisting of arrays h r = a (f₁ r) + b (f₂ r) for some a : A → ℝ and b : B → ℝ.","s":"{R : Type u_1} → {A : Type u_2} → {B : Type u_3} → (R → A) → (R → B) → Submodule ℝ (R → ℝ)","l":58,"f":"Causalean/Panel/Weighted/AdditiveSpan.lean"},{"n":"Causalean.Panel.Weighted.AdditiveSpan.const_mem","k":"theorem","m":"Causalean.Panel.Weighted.AdditiveSpan","a":"Panel/Weighted","t":1,"d":"The constant function equal to c₀ belongs to the additive span generated by two feature maps f₁ and f₂; it is witnessed by taking the f₁-component function constantly c₀ and the f₂-component function constantly 0.","s":"∀ {R : Type u_1} {A : Type u_2} {B : Type u_3} (f₁ : R → A) (f₂ : R → B) (c₀ : ℝ), (fun x => c₀) ∈ Causalean.Panel.Weighted.AdditiveSpan f₁ f₂","l":86,"f":"Causalean/Panel/Weighted/AdditiveSpan.lean"},{"n":"Causalean.Panel.Weighted.AdditiveSpan.mem_iff","k":"theorem","m":"Causalean.Panel.Weighted.AdditiveSpan","a":"Panel/Weighted","t":2,"d":"Membership unfolding for AdditiveSpan.","s":"∀ {R : Type u_1} {A : Type u_2} {B : Type u_3} {f₁ : R → A} {f₂ : R → B} {h : R → ℝ}, h ∈ Causalean.Panel.Weighted.AdditiveSpan f₁ f₂ ↔ ∃ a b, ∀ (r : R), h r = a (f₁ r) + b (f₂ r)","l":80,"f":"Causalean/Panel/Weighted/AdditiveSpan.lean"},{"n":"Causalean.Panel.Weighted.IsUnitTimeAdditive","k":"def","m":"Causalean.Panel.Weighted.AdditiveSpan","a":"Panel/Weighted","t":1,"d":"Predicate for the unit/time additive class h i t = a i + b t.","s":"{Unit : Type u_1} → {Time : Type u_2} → (Unit → Time → ℝ) → Prop","l":110,"f":"Causalean/Panel/Weighted/AdditiveSpan.lean"},{"n":"Causalean.Panel.Weighted.NormalizedWeights.normalizedWeight","k":"def","m":"Causalean.Panel.Weighted.NormalizedWeights","a":"Panel/Weighted","t":1,"d":"Generic normalized finite weight a_i / Σ_k a_k.","s":"{ι : Type u_3} → [Fintype ι] → {K : Type u_4} → [Field K] → (ι → K) → ι → K","l":57,"f":"Causalean/Panel/Weighted/NormalizedWeights.lean"},{"n":"Causalean.Panel.Weighted.NormalizedWeights.normalizedWeight_nonneg","k":"theorem","m":"Causalean.Panel.Weighted.NormalizedWeights","a":"Panel/Weighted","t":2,"d":"Nonnegativity of normalized weights from nonnegative raw weights and a positive normalizing sum.","s":"∀ {ι : Type u_3} [inst : Fintype ι] {K : Type u_4} [inst_1 : Field K] [inst_2 : LinearOrder K] [IsStrictOrderedRing K] (a : ι → K), (∀ (i : ι), 0 ≤ a i) → 0 < ∑ i, a i → ∀ (i : ι), 0 ≤ Causalean.Panel.Weighted.NormalizedWeights.normalizedWeight a i","l":61,"f":"Causalean/Panel/Weighted/NormalizedWeights.lean"},{"n":"Causalean.Panel.Weighted.NormalizedWeights.pairwise_cov_right","k":"theorem","m":"Causalean.Panel.Weighted.NormalizedWeights","a":"Panel/Weighted","t":2,"d":"When finite weights sum to one, the weighted sum of pairwise products of differences equals twice the weighted cross-moment minus twice the product of the weighted means.","s":"∀ {ι : Type u_3} [inst : Fintype ι] {K : Type u_4} [inst_1 : CommRing K] (p x y : ι → K), ∑ i, p i = 1 → ∑ i, ∑ j, p i * p j * (x i - x j) * (y i - y j) = 2 * (∑ i, p i * x i * y i - (∑ j, p j * x j) * ∑ j, p j * y j)","l":123,"f":"Causalean/Panel/Weighted/NormalizedWeights.lean"},{"n":"Causalean.Panel.Weighted.NormalizedWeights.sum_normalizedWeight_eq_one","k":"theorem","m":"Causalean.Panel.Weighted.NormalizedWeights","a":"Panel/Weighted","t":2,"d":"Normalized finite weights sum to one when the normalizing sum is nonzero.","s":"∀ {ι : Type u_3} [inst : Fintype ι] {K : Type u_4} [inst_1 : Field K] (a : ι → K), ∑ i, a i ≠ 0 → ∑ i, Causalean.Panel.Weighted.NormalizedWeights.normalizedWeight a i = 1","l":70,"f":"Causalean/Panel/Weighted/NormalizedWeights.lean"},{"n":"Causalean.Panel.Weighted.NormalizedWeights.weighted_center_cov","k":"theorem","m":"Causalean.Panel.Weighted.NormalizedWeights","a":"Panel/Weighted","t":1,"d":"For weights p summing to one over a field where 2 is nonzero, the weighted centered covariance of x and y equals half the average, weighted by p ⊗ p, of the pairwise cross-products of their differences: Σᵢ pᵢ (xᵢ − x̄)(yᵢ − ȳ) = 1/2 Σᵢ Σⱼ pᵢpⱼ (xᵢ − xⱼ)(yᵢ − yⱼ).","s":"∀ {ι : Type u_3} [inst : Fintype ι] {K : Type u_4} [inst_1 : Field K] (p x y : ι → K), ∑ i, p i = 1 → 2 ≠ 0 → ∑ i, p i * (x i - ∑ j, p j * x j) * (y i - ∑ j, p j * y j) = 1 / 2 * ∑ i, ∑ j, p i * p j * (x i - x j) * (y i - y j)","l":156,"f":"Causalean/Panel/Weighted/NormalizedWeights.lean"},{"n":"Causalean.Panel.Weighted.NormalizedWeights.weighted_center_cov_left","k":"theorem","m":"Causalean.Panel.Weighted.NormalizedWeights","a":"Panel/Weighted","t":2,"d":"When finite weights sum to one, the weighted covariance of two centered variables equals their weighted cross-moment minus the product of their weighted means.","s":"∀ {ι : Type u_3} [inst : Fintype ι] {K : Type u_4} [inst_1 : CommRing K] (p x y : ι → K), ∑ i, p i = 1 → ∑ i, p i * (x i - ∑ j, p j * x j) * (y i - ∑ j, p j * y j) = ∑ i, p i * x i * y i - (∑ j, p j * x j) * ∑ j, p j * y j","l":78,"f":"Causalean/Panel/Weighted/NormalizedWeights.lean"},{"n":"Causalean.Panel.Weighted.NormalizedWeights.weighted_center_var","k":"theorem","m":"Causalean.Panel.Weighted.NormalizedWeights","a":"Panel/Weighted","t":2,"d":"Weighted centered variance as half the average pairwise squared gap.","s":"∀ {ι : Type u_3} [inst : Fintype ι] {K : Type u_4} [inst_1 : Field K] (p x : ι → K), ∑ i, p i = 1 → 2 ≠ 0 → ∑ i, p i * (x i - ∑ j, p j * x j) ^ 2 = 1 / 2 * ∑ i, ∑ j, p i * p j * (x i - x j) ^ 2","l":168,"f":"Causalean/Panel/Weighted/NormalizedWeights.lean"},{"n":"Causalean.Panel.Weighted.WeightedSupport","k":"structure","m":"Causalean.Panel.Weighted.Support","a":"Panel/Weighted","t":1,"d":"A finite weighted index together with positive normalized weights: a nonempty observed subset of indices together with a weight function that is strictly positive on every observed index, vanishes off the observed set, and sums to one over the observed indices.","s":"(R : Type u_1) → [Fintype R] → [DecidableEq R] → Type u_1","l":53,"f":"Causalean/Panel/Weighted/Support.lean"},{"n":"Causalean.Panel.Weighted.WeightedSupport.Q_XX","k":"def","m":"Causalean.Panel.Weighted.FWL","a":"Panel/Weighted","t":1,"d":"Residualized regressor Gram matrix Q_XX = ⟨M_H X, M_H X⟩_ω.","s":"{R : Type u_1} → [inst : Fintype R] → [inst_1 : DecidableEq R] → {K : ℕ} → Causalean.Panel.Weighted.WeightedSupport R → (Fin K → R → ℝ) → Submodule ℝ (R → ℝ) → Matrix (Fin K) (Fin K) ℝ","l":66,"f":"Causalean/Panel/Weighted/FWL.lean"},{"n":"Causalean.Panel.Weighted.WeightedSupport.Q_XX_apply","k":"theorem","m":"Causalean.Panel.Weighted.FWL","a":"Panel/Weighted","t":2,"d":"Each entry of the residualized Gram matrix is the weighted inner product of the corresponding residualized regressors.","s":"∀ {R : Type u_1} [inst : Fintype R] [inst_1 : DecidableEq R] {K : ℕ} (c : Causalean.Panel.Weighted.WeightedSupport R) (X : Fin K → R → ℝ) (H : Submodule ℝ (R → ℝ)) (j k : Fin K), c.Q_XX X H j k = c.ip (c.tildeX H (X j)) (c.tildeX H (X k))","l":71,"f":"Causalean/Panel/Weighted/FWL.lean"},{"n":"Causalean.Panel.Weighted.WeightedSupport.Q_XX_mulVec_thetaHat","k":"theorem","m":"Causalean.Panel.Weighted.FWL","a":"Panel/Weighted","t":2,"d":"Under nonsingularity, thetaHat solves the residualized normal equations.","s":"∀ {R : Type u_1} [inst : Fintype R] [inst_1 : DecidableEq R] {K : ℕ} (c : Causalean.Panel.Weighted.WeightedSupport R) (X : Fin K → R → ℝ) (H : Submodule ℝ (R → ℝ)) (Y : R → ℝ), IsUnit (c.Q_XX X H).det → (c.Q_XX X H).mulVec (c.thetaHat X H Y) = c.rhsVec X H Y","l":100,"f":"Causalean/Panel/Weighted/FWL.lean"},{"n":"Causalean.Panel.Weighted.WeightedSupport.RankCondition","k":"def","m":"Causalean.Panel.Weighted.FWL","a":"Panel/Weighted","t":1,"d":"Rank condition for the residualized regressors: Q_XX is invertible.","s":"{R : Type u_1} → [inst : Fintype R] → [inst_1 : DecidableEq R] → {K : ℕ} → Causalean.Panel.Weighted.WeightedSupport R → Submodule ℝ (R → ℝ) → (Fin K → R → ℝ) → Prop","l":95,"f":"Causalean/Panel/Weighted/FWL.lean"},{"n":"Causalean.Panel.Weighted.WeightedSupport.cellMass","k":"def","m":"Causalean.Panel.Weighted.IndicatorSpan","a":"Panel/Weighted","t":1,"d":"The cell mass is the total weight of observed records in one classifier cell.","s":"{R : Type u_1} → [inst : Fintype R] → [inst_1 : DecidableEq R] → {𝒢 : Type u_2} → [DecidableEq 𝒢] → Causalean.Panel.Weighted.WeightedSupport R → (R → 𝒢) → 𝒢 → ℝ","l":162,"f":"Causalean/Panel/Weighted/IndicatorSpan.lean"},{"n":"Causalean.Panel.Weighted.WeightedSupport.cellMass_def","k":"theorem","m":"Causalean.Panel.Weighted.IndicatorSpan","a":"Panel/Weighted","t":2,"d":"The cell mass unfolds to the weighted sum of the corresponding cell indicator over observed records.","s":"∀ {R : Type u_1} [inst : Fintype R] [inst_1 : DecidableEq R] {𝒢 : Type u_2} [inst_2 : DecidableEq 𝒢] (c : Causalean.Panel.Weighted.WeightedSupport R) (G : R → 𝒢) (g : 𝒢), c.cellMass G g = ∑ r ∈ c.observed, c.weight r * Causalean.Panel.Weighted.cellIndicator G g r","l":168,"f":"Causalean/Panel/Weighted/IndicatorSpan.lean"},{"n":"Causalean.Panel.Weighted.WeightedSupport.cellMass_nonneg","k":"theorem","m":"Causalean.Panel.Weighted.IndicatorSpan","a":"Panel/Weighted","t":2,"d":"Every classifier cell has nonnegative total weight.","s":"∀ {R : Type u_1} [inst : Fintype R] [inst_1 : DecidableEq R] {𝒢 : Type u_2} [inst_2 : DecidableEq 𝒢] (c : Causalean.Panel.Weighted.WeightedSupport R) (G : R → 𝒢) (g : 𝒢), 0 ≤ c.cellMass G g","l":174,"f":"Causalean/Panel/Weighted/IndicatorSpan.lean"},{"n":"Causalean.Panel.Weighted.WeightedSupport.cellMean","k":"def","m":"Causalean.Panel.Weighted.IndicatorSpan","a":"Panel/Weighted","t":1,"d":"On a classifier cell with positive observed mass, the cell mean is the ordinary weighted average of the outcome over observed records in that cell.","s":"{R : Type u_1} → [inst : Fintype R] → [inst_1 : DecidableEq R] → {𝒢 : Type u_2} → [inst_2 : DecidableEq 𝒢] → (c : Causalean.Panel.Weighted.WeightedSupport R) → (R → ℝ) → (G : R → 𝒢) → (g : 𝒢) → 0 < c.cellMass G g → ℝ","l":185,"f":"Causalean/Panel/Weighted/IndicatorSpan.lean"},{"n":"Causalean.Panel.Weighted.WeightedSupport.cellMean_mul_cellMass","k":"theorem","m":"Causalean.Panel.Weighted.IndicatorSpan","a":"Panel/Weighted","t":1,"d":"On a classifier cell g with strictly positive total weight, multiplying the cell mean of F by the cell's mass recovers the weighted sum of F over observed records in that cell.","s":"∀ {R : Type u_1} [inst : Fintype R] [inst_1 : DecidableEq R] {𝒢 : Type u_2} [inst_2 : DecidableEq 𝒢] (c : Causalean.Panel.Weighted.WeightedSupport R) (F : R → ℝ) (G : R → 𝒢) (g : 𝒢) (hpos : 0 < c.cellMass G g), c.cellMean F G g hpos * c.cellMass G g = ∑ r ∈ c.observed, c.weight r * F r * Causalean.Panel.Weighted.cellIndicator G g r","l":195,"f":"Causalean/Panel/Weighted/IndicatorSpan.lean"},{"n":"Causalean.Panel.Weighted.WeightedSupport.fwl_identity","k":"theorem","m":"Causalean.Panel.Weighted.FWL","a":"Panel/Weighted","t":1,"d":"Frisch–Waugh–Lovell at the WeightedSupport level. Assume the residualized-regressor Gram matrix Q_XX is invertible (the rank condition). Then whenever a coefficient vector β together with a nuisance term α ∈ H jointly minimizes the weighted least-squares objective c.ip (Y − ∑ₖ βₖ·Xₖ − α) (Y − ∑ₖ βₖ·Xₖ − α) over all coefficient/nuisance pairs, β must equal the short-regression residualized coefficient thetaHat c X H Y.","s":"∀ {R : Type u_1} [inst : Fintype R] [inst_1 : DecidableEq R] {K : ℕ} (c : Causalean.Panel.Weighted.WeightedSupport R) (H : Submodule ℝ (R → ℝ)) (X : Fin K → R → ℝ) (Y : R → ℝ), c.RankCondition H X → ∀ (β : Fin K → ℝ), ∀ α ∈ H, (∀ (β' : Fin K → ℝ), ∀ α' ∈ H, c.ip (Y - ∑ k, β k • X k - α) (Y - ∑ k, β k • X k - α) ≤ c.ip (Y - ∑ k, β' k • X k - α') (Y - ∑ k, β' k • X k - α')) → β = c.thetaHat X H Y","l":154,"f":"Causalean/Panel/Weighted/FWL.lean"},{"n":"Causalean.Panel.Weighted.WeightedSupport.ip","k":"def","m":"Causalean.Panel.Weighted.InnerProduct","a":"Panel/Weighted","t":1,"d":"The weighted inner product multiplies two arrays record by record and sums the products with the support weights over observed records.","s":"{R : Type u_1} → [inst : Fintype R] → [inst_1 : DecidableEq R] → Causalean.Panel.Weighted.WeightedSupport R → (R → ℝ) → (R → ℝ) → ℝ","l":65,"f":"Causalean/Panel/Weighted/InnerProduct.lean"},{"n":"Causalean.Panel.Weighted.WeightedSupport.ipMat","k":"def","m":"Causalean.Panel.Weighted.InnerProduct","a":"Panel/Weighted","t":1,"d":"The matrix-valued inner product takes the scalar weighted inner product between each pair of columns.","s":"{R : Type u_1} → [inst : Fintype R] → [inst_1 : DecidableEq R] → {J : Type u_2} → Causalean.Panel.Weighted.WeightedSupport R → (J → R → ℝ) → (J → R → ℝ) → Matrix J J ℝ","l":157,"f":"Causalean/Panel/Weighted/InnerProduct.lean"},{"n":"Causalean.Panel.Weighted.WeightedSupport.ipMat_add_left","k":"theorem","m":"Causalean.Panel.Weighted.InnerProduct","a":"Panel/Weighted","t":2,"d":"Additivity in the left tuple argument (entrywise).","s":"∀ {R : Type u_1} [inst : Fintype R] [inst_1 : DecidableEq R] {J : Type u_2} (c : Causalean.Panel.Weighted.WeightedSupport R) (A A' B : J → R → ℝ), c.ipMat (A + A') B = c.ipMat A B + c.ipMat A' B","l":175,"f":"Causalean/Panel/Weighted/InnerProduct.lean"},{"n":"Causalean.Panel.Weighted.WeightedSupport.ipMat_add_right","k":"theorem","m":"Causalean.Panel.Weighted.InnerProduct","a":"Panel/Weighted","t":2,"d":"Additivity in the right tuple argument (entrywise).","s":"∀ {R : Type u_1} [inst : Fintype R] [inst_1 : DecidableEq R] {J : Type u_2} (c : Causalean.Panel.Weighted.WeightedSupport R) (A B B' : J → R → ℝ), c.ipMat A (B + B') = c.ipMat A B + c.ipMat A B'","l":183,"f":"Causalean/Panel/Weighted/InnerProduct.lean"},{"n":"Causalean.Panel.Weighted.WeightedSupport.ipMat_apply","k":"theorem","m":"Causalean.Panel.Weighted.InnerProduct","a":"Panel/Weighted","t":2,"d":"Each entry of the matrix-valued inner product is the scalar weighted inner product of the corresponding two columns.","s":"∀ {R : Type u_1} [inst : Fintype R] [inst_1 : DecidableEq R] {J : Type u_2} (c : Causalean.Panel.Weighted.WeightedSupport R) (A B : J → R → ℝ) (j k : J), c.ipMat A B j k = c.ip (A j) (B k)","l":163,"f":"Causalean/Panel/Weighted/InnerProduct.lean"},{"n":"Causalean.Panel.Weighted.WeightedSupport.ipMat_smul_left","k":"theorem","m":"Causalean.Panel.Weighted.InnerProduct","a":"Panel/Weighted","t":2,"d":"Scalar homogeneity in the left tuple argument (entrywise).","s":"∀ {R : Type u_1} [inst : Fintype R] [inst_1 : DecidableEq R] {J : Type u_2} (c : Causalean.Panel.Weighted.WeightedSupport R) (s : ℝ) (A B : J → R → ℝ), c.ipMat (s • A) B = s • c.ipMat A B","l":191,"f":"Causalean/Panel/Weighted/InnerProduct.lean"},{"n":"Causalean.Panel.Weighted.WeightedSupport.ipMat_smul_right","k":"theorem","m":"Causalean.Panel.Weighted.InnerProduct","a":"Panel/Weighted","t":2,"d":"Scalar homogeneity in the right tuple argument (entrywise).","s":"∀ {R : Type u_1} [inst : Fintype R] [inst_1 : DecidableEq R] {J : Type u_2} (c : Causalean.Panel.Weighted.WeightedSupport R) (s : ℝ) (A B : J → R → ℝ), c.ipMat A (s • B) = s • c.ipMat A B","l":199,"f":"Causalean/Panel/Weighted/InnerProduct.lean"},{"n":"Causalean.Panel.Weighted.WeightedSupport.ipMat_transpose","k":"theorem","m":"Causalean.Panel.Weighted.InnerProduct","a":"Panel/Weighted","t":2,"d":"The transpose of ⟨A, B⟩_ω is ⟨B, A⟩_ω.","s":"∀ {R : Type u_1} [inst : Fintype R] [inst_1 : DecidableEq R] {J : Type u_2} (c : Causalean.Panel.Weighted.WeightedSupport R) (A B : J → R → ℝ), (c.ipMat A B).transpose = c.ipMat B A","l":169,"f":"Causalean/Panel/Weighted/InnerProduct.lean"},{"n":"Causalean.Panel.Weighted.WeightedSupport.ip_add_left","k":"theorem","m":"Causalean.Panel.Weighted.InnerProduct","a":"Panel/Weighted","t":2,"d":"Additivity in the left argument.","s":"∀ {R : Type u_1} [inst : Fintype R] [inst_1 : DecidableEq R] (c : Causalean.Panel.Weighted.WeightedSupport R) (A A' B : R → ℝ), c.ip (A + A') B = c.ip A B + c.ip A' B","l":83,"f":"Causalean/Panel/Weighted/InnerProduct.lean"},{"n":"Causalean.Panel.Weighted.WeightedSupport.ip_add_right","k":"theorem","m":"Causalean.Panel.Weighted.InnerProduct","a":"Panel/Weighted","t":2,"d":"Additivity in the right argument.","s":"∀ {R : Type u_1} [inst : Fintype R] [inst_1 : DecidableEq R] (c : Causalean.Panel.Weighted.WeightedSupport R) (A B B' : R → ℝ), c.ip A (B + B') = c.ip A B + c.ip A B'","l":92,"f":"Causalean/Panel/Weighted/InnerProduct.lean"},{"n":"Causalean.Panel.Weighted.WeightedSupport.ip_def","k":"theorem","m":"Causalean.Panel.Weighted.InnerProduct","a":"Panel/Weighted","t":2,"d":"The weighted inner product unfolds to its finite weighted sum over observed records.","s":"∀ {R : Type u_1} [inst : Fintype R] [inst_1 : DecidableEq R] (c : Causalean.Panel.Weighted.WeightedSupport R) (A B : R → ℝ), c.ip A B = ∑ r ∈ c.observed, c.weight r * A r * B r","l":70,"f":"Causalean/Panel/Weighted/InnerProduct.lean"},{"n":"Causalean.Panel.Weighted.WeightedSupport.ip_eq_integral","k":"theorem","m":"Causalean.Panel.Weighted.OfProbabilityMeasure","a":"Panel/Weighted","t":1,"d":"For a finite probability measure μ, the weighted inner product induced by μ on any two functions A and B equals the L²(μ) integral pairing ∫ A·B dμ.","s":"∀ {R : Type u_1} [inst : Fintype R] [inst_1 : DecidableEq R] [inst_2 : MeasurableSpace R] [inst_3 : MeasurableSingletonClass R] (μ : MeasureTheory.Measure R) [inst_4 : MeasureTheory.IsProbabilityMeasure μ] (A B : R → ℝ), (Causalean.Panel.Weighted.WeightedSupport.ofProbabilityMeasure μ).ip A B = ∫ (r : R), A r * B r ∂μ","l":111,"f":"Causalean/Panel/Weighted/OfProbabilityMeasure.lean"},{"n":"Causalean.Panel.Weighted.WeightedSupport.ip_eq_zero_of_zero_on_observed","k":"theorem","m":"Causalean.Panel.Weighted.WLS","a":"Panel/Weighted","t":2,"d":"If h vanishes on every observed index, then c.ip A h = 0 for any A.","s":"∀ {R : Type u_1} [inst : Fintype R] [inst_1 : DecidableEq R] (c : Causalean.Panel.Weighted.WeightedSupport R) (A h : R → ℝ), (∀ r ∈ c.observed, h r = 0) → c.ip A h = 0","l":117,"f":"Causalean/Panel/Weighted/WLS.lean"},{"n":"Causalean.Panel.Weighted.WeightedSupport.ip_self_eq_zero_iff","k":"theorem","m":"Causalean.Panel.Weighted.InnerProduct","a":"Panel/Weighted","t":1,"d":"For a weighted-support system c, the weighted self inner product ⟨A, A⟩_ω of an array A vanishes if and only if A is zero at every index in the observed set. Arrays that differ only off c.observed are thus identified by this inner product.","s":"∀ {R : Type u_1} [inst : Fintype R] [inst_1 : DecidableEq R] (c : Causalean.Panel.Weighted.WeightedSupport R) (A : R → ℝ), c.ip A A = 0 ↔ ∀ r ∈ c.observed, A r = 0","l":126,"f":"Causalean/Panel/Weighted/InnerProduct.lean"},{"n":"Causalean.Panel.Weighted.WeightedSupport.ip_self_nonneg","k":"theorem","m":"Causalean.Panel.Weighted.InnerProduct","a":"Panel/Weighted","t":2,"d":"Positivity: ⟨A, A⟩_ω ≥ 0.","s":"∀ {R : Type u_1} [inst : Fintype R] [inst_1 : DecidableEq R] (c : Causalean.Panel.Weighted.WeightedSupport R) (A : R → ℝ), 0 ≤ c.ip A A","l":120,"f":"Causalean/Panel/Weighted/InnerProduct.lean"},{"n":"Causalean.Panel.Weighted.WeightedSupport.ip_self_sub_le_of_orth","k":"theorem","m":"Causalean.Panel.Weighted.WLS","a":"Panel/Weighted","t":2,"d":"If p ∈ H and the residual X - p is c.ip-orthogonal to all of H, then p minimizes the WLS objective over H.","s":"∀ {R : Type u_1} [inst : Fintype R] [inst_1 : DecidableEq R] (c : Causalean.Panel.Weighted.WeightedSupport R) (H : Submodule ℝ (R → ℝ)) (X p : R → ℝ), p ∈ H → (∀ h ∈ H, c.ip (X - p) h = 0) → ∀ {h : R → ℝ}, h ∈ H → c.ip (X - p) (X - p) ≤ c.ip (X - h) (X - h)","l":65,"f":"Causalean/Panel/Weighted/WLS.lean"},{"n":"Causalean.Panel.Weighted.WeightedSupport.ip_self_summand_nonneg","k":"theorem","m":"Causalean.Panel.Weighted.InnerProduct","a":"Panel/Weighted","t":2,"d":"Each summand of ⟨A, A⟩_ω is nonnegative.","s":"∀ {R : Type u_1} [inst : Fintype R] [inst_1 : DecidableEq R] (c : Causalean.Panel.Weighted.WeightedSupport R) (A : R → ℝ), ∀ r ∈ c.observed, 0 ≤ c.weight r * A r * A r","l":111,"f":"Causalean/Panel/Weighted/InnerProduct.lean"},{"n":"Causalean.Panel.Weighted.WeightedSupport.ip_smul_left","k":"theorem","m":"Causalean.Panel.Weighted.InnerProduct","a":"Panel/Weighted","t":2,"d":"Scalar homogeneity in the left argument.","s":"∀ {R : Type u_1} [inst : Fintype R] [inst_1 : DecidableEq R] (c : Causalean.Panel.Weighted.WeightedSupport R) (s : ℝ) (A B : R → ℝ), c.ip (s • A) B = s * c.ip A B","l":97,"f":"Causalean/Panel/Weighted/InnerProduct.lean"},{"n":"Causalean.Panel.Weighted.WeightedSupport.ip_smul_right","k":"theorem","m":"Causalean.Panel.Weighted.InnerProduct","a":"Panel/Weighted","t":2,"d":"Scalar homogeneity in the right argument.","s":"∀ {R : Type u_1} [inst : Fintype R] [inst_1 : DecidableEq R] (c : Causalean.Panel.Weighted.WeightedSupport R) (s : ℝ) (A B : R → ℝ), c.ip A (s • B) = s * c.ip A B","l":106,"f":"Causalean/Panel/Weighted/InnerProduct.lean"},{"n":"Causalean.Panel.Weighted.WeightedSupport.ip_sub_smul_expand","k":"theorem","m":"Causalean.Panel.Weighted.WLS","a":"Panel/Weighted","t":2,"d":"Quadratic-in-t expansion used in the perturbation argument: ⟨X - p - t • h, X - p - t • h⟩_ω = ⟨X-p, X-p⟩_ω - 2 t · ⟨X-p, h⟩_ω + t² · ⟨h, h⟩_ω.","s":"∀ {R : Type u_1} [inst : Fintype R] [inst_1 : DecidableEq R] (c : Causalean.Panel.Weighted.WeightedSupport R) (X p h : R → ℝ) (t : ℝ), c.ip (X - p - t • h) (X - p - t • h) = c.ip (X - p) (X - p) - 2 * t * c.ip (X - p) h + t ^ 2 * c.ip h h","l":126,"f":"Causalean/Panel/Weighted/WLS.lean"},{"n":"Causalean.Panel.Weighted.WeightedSupport.ip_sum_left_finset","k":"theorem","m":"Causalean.Panel.Weighted.FWL","a":"Panel/Weighted","t":2,"d":"For a weighted inner product, the inner product of a finite sum of functions with another function equals the corresponding finite sum of inner products.","s":"∀ {R : Type u_1} [inst : Fintype R] [inst_1 : DecidableEq R] (c : Causalean.Panel.Weighted.WeightedSupport R) {ι : Type u_2} (s : Finset ι) (f : ι → R → ℝ) (B : R → ℝ), c.ip (∑ i ∈ s, f i) B = ∑ i ∈ s, c.ip (f i) B","l":114,"f":"Causalean/Panel/Weighted/FWL.lean"},{"n":"Causalean.Panel.Weighted.WeightedSupport.ip_sum_right","k":"theorem","m":"Causalean.Panel.Weighted.FWL","a":"Panel/Weighted","t":2,"d":"The weighted inner product of a function with a finite sum of functions equals the sum of its weighted inner products with the summands.","s":"∀ {R : Type u_1} [inst : Fintype R] [inst_1 : DecidableEq R] {K : ℕ} (c : Causalean.Panel.Weighted.WeightedSupport R) (A : R → ℝ) (f : Fin K → R → ℝ), c.ip A (∑ k, f k) = ∑ k, c.ip A (f k)","l":130,"f":"Causalean/Panel/Weighted/FWL.lean"},{"n":"Causalean.Panel.Weighted.WeightedSupport.ip_symm","k":"theorem","m":"Causalean.Panel.Weighted.InnerProduct","a":"Panel/Weighted","t":2,"d":"Symmetry of the weighted inner product.","s":"∀ {R : Type u_1} [inst : Fintype R] [inst_1 : DecidableEq R] (c : Causalean.Panel.Weighted.WeightedSupport R) (A B : R → ℝ), c.ip A B = c.ip B A","l":75,"f":"Causalean/Panel/Weighted/InnerProduct.lean"},{"n":"Causalean.Panel.Weighted.WeightedSupport.ip_tildeX_eq_ip_tildeX_residual","k":"theorem","m":"Causalean.Panel.Weighted.FWL","a":"Panel/Weighted","t":2,"d":"A regressor residualized against a control space has the same weighted inner product with any function as with that function after residualizing it against the same control space.","s":"∀ {R : Type u_1} [inst : Fintype R] [inst_1 : DecidableEq R] {K : ℕ} (c : Causalean.Panel.Weighted.WeightedSupport R) (H : Submodule ℝ (R → ℝ)) (X : Fin K → R → ℝ) (A : R → ℝ) (k : Fin K), c.ip (c.tildeX H (X k)) A = c.ip (c.tildeX H (X k)) (c.tildeX H A)","l":139,"f":"Causalean/Panel/Weighted/FWL.lean"},{"n":"Causalean.Panel.Weighted.WeightedSupport.ip_tildeX_self","k":"theorem","m":"Causalean.Panel.Weighted.ScalarFWL","a":"Panel/Weighted","t":2,"d":"⟨X̃, X⟩_ω = ⟨X̃, X̃⟩_ω: the residualized regressor sees X and its residual identically, because the projection part lies in H and is orthogonal to the residual.","s":"∀ {R : Type u_1} [inst : Fintype R] [inst_1 : DecidableEq R] (c : Causalean.Panel.Weighted.WeightedSupport R) (H : Submodule ℝ (R → ℝ)) (X : R → ℝ), c.ip (c.tildeX H X) X = c.ip (c.tildeX H X) (c.tildeX H X)","l":53,"f":"Causalean/Panel/Weighted/ScalarFWL.lean"},{"n":"Causalean.Panel.Weighted.WeightedSupport.jointCellMass","k":"def","m":"Causalean.Panel.Weighted.IndicatorSpan","a":"Panel/Weighted","t":1,"d":"The joint cell mass is the total weight of observed records lying in a given pair of classifier cells.","s":"{R : Type u_1} → [inst : Fintype R] → [inst_1 : DecidableEq R] → {𝒢 : Type u_2} → {𝒯 : Type u_3} → [DecidableEq 𝒢] → [DecidableEq 𝒯] → Causalean.Panel.Weighted.WeightedSupport R → (R → 𝒢) → (R → 𝒯) → 𝒢 → 𝒯 → ℝ","l":206,"f":"Causalean/Panel/Weighted/IndicatorSpan.lean"},{"n":"Causalean.Panel.Weighted.WeightedSupport.jointCellMass_def","k":"theorem","m":"Causalean.Panel.Weighted.IndicatorSpan","a":"Panel/Weighted","t":2,"d":"The joint cell mass unfolds to the weighted sum of the two corresponding cell indicators over observed records.","s":"∀ {R : Type u_1} [inst : Fintype R] [inst_1 : DecidableEq R] {𝒢 : Type u_2} {𝒯 : Type u_3} [inst_2 : DecidableEq 𝒢] [inst_3 : DecidableEq 𝒯] (c : Causalean.Panel.Weighted.WeightedSupport R) (G : R → 𝒢) (T : R → 𝒯) (g : 𝒢) (t : 𝒯), c.jointCellMass G T g t = ∑ r ∈ c.observed, c.weight r * Causalean.Panel.Weighted.cellIndicator G g r * Causalean.Panel.Weighted.cellIndicator T t r","l":213,"f":"Causalean/Panel/Weighted/IndicatorSpan.lean"},{"n":"Causalean.Panel.Weighted.WeightedSupport.ofProbabilityMeasure","k":"def","m":"Causalean.Panel.Weighted.OfProbabilityMeasure","a":"Panel/Weighted","t":1,"d":"A probability measure on a finite measurable space induces a weighted support whose weights are point masses and whose observed records have positive mass.","s":"{R : Type u_1} → [inst : Fintype R] → [inst_1 : DecidableEq R] → [inst_2 : MeasurableSpace R] → [MeasurableSingletonClass R] → (μ : MeasureTheory.Measure R) → [MeasureTheory.IsProbabilityMeasure μ] → Causalean.Panel.Weighted.WeightedSupport R","l":49,"f":"Causalean/Panel/Weighted/OfProbabilityMeasure.lean"},{"n":"Causalean.Panel.Weighted.WeightedSupport.ofProbabilityMeasure_weight","k":"theorem","m":"Causalean.Panel.Weighted.OfProbabilityMeasure","a":"Panel/Weighted","t":2,"d":"The induced weighted support assigns each record the point mass of that record under the probability measure.","s":"∀ {R : Type u_1} [inst : Fintype R] [inst_1 : DecidableEq R] [inst_2 : MeasurableSpace R] [inst_3 : MeasurableSingletonClass R] (μ : MeasureTheory.Measure R) [inst_4 : MeasureTheory.IsProbabilityMeasure μ] (r : R), (Causalean.Panel.Weighted.WeightedSupport.ofProbabilityMeasure μ).weight r = (μ {r}).toReal","l":106,"f":"Causalean/Panel/Weighted/OfProbabilityMeasure.lean"},{"n":"Causalean.Panel.Weighted.WeightedSupport.proj","k":"def","m":"Causalean.Panel.Weighted.Subspace","a":"Panel/Weighted","t":1,"d":"The c.ip-orthogonal projection onto H, chosen via classical choice from weighted_orthogonal_projection_exists.","s":"{R : Type u_1} → [inst : Fintype R] → [inst_1 : DecidableEq R] → Causalean.Panel.Weighted.WeightedSupport R → Submodule ℝ (R → ℝ) → (R → ℝ) →ₗ[ℝ] R → ℝ","l":100,"f":"Causalean/Panel/Weighted/Subspace.lean"},{"n":"Causalean.Panel.Weighted.WeightedSupport.proj_apply_eq_of_mem_orthogonal","k":"theorem","m":"Causalean.Panel.Weighted.Subspace","a":"Panel/Weighted","t":2,"d":"Projection uniqueness on observed indices.","s":"∀ {R : Type u_1} [inst : Fintype R] [inst_1 : DecidableEq R] (c : Causalean.Panel.Weighted.WeightedSupport R) (H : Submodule ℝ (R → ℝ)) (X : R → ℝ) {Y : R → ℝ}, Y ∈ H → (∀ h ∈ H, c.ip (X - Y) h = 0) → ∀ r ∈ c.observed, (c.proj H) X r = Y r","l":133,"f":"Causalean/Panel/Weighted/Subspace.lean"},{"n":"Causalean.Panel.Weighted.WeightedSupport.proj_apply_of_mem","k":"theorem","m":"Causalean.Panel.Weighted.Subspace","a":"Panel/Weighted","t":2,"d":"If Y ∈ H then c.proj H Y agrees with Y on every observed index.","s":"∀ {R : Type u_1} [inst : Fintype R] [inst_1 : DecidableEq R] (c : Causalean.Panel.Weighted.WeightedSupport R) (H : Submodule ℝ (R → ℝ)) {Y : R → ℝ}, Y ∈ H → ∀ r ∈ c.observed, (c.proj H) Y r = Y r","l":120,"f":"Causalean/Panel/Weighted/Subspace.lean"},{"n":"Causalean.Panel.Weighted.WeightedSupport.proj_eq_argmin","k":"theorem","m":"Causalean.Panel.Weighted.WLS","a":"Panel/Weighted","t":1,"d":"The weighted orthogonal projection minimizes the WLS objective over H. Given a weighted support on a finite index set and a submodule H of candidate real-valued functions, for any competitor h lying in H, the weighted sum of squared residuals of the projection of a target function onto H is at most the weighted sum of squared residuals of h.","s":"∀ {R : Type u_1} [inst : Fintype R] [inst_1 : DecidableEq R] (c : Causalean.Panel.Weighted.WeightedSupport R) (H : Submodule ℝ (R → ℝ)) (X h : R → ℝ), h ∈ H → c.ip (X - (c.proj H) X) (X - (c.proj H) X) ≤ c.ip (X - h) (X - h)","l":91,"f":"Causalean/Panel/Weighted/WLS.lean"},{"n":"Causalean.Panel.Weighted.WeightedSupport.proj_idem_apply","k":"theorem","m":"Causalean.Panel.Weighted.Subspace","a":"Panel/Weighted","t":2,"d":"Idempotence of the projection on the observed indices.","s":"∀ {R : Type u_1} [inst : Fintype R] [inst_1 : DecidableEq R] (c : Causalean.Panel.Weighted.WeightedSupport R) (H : Submodule ℝ (R → ℝ)) (X : R → ℝ), ∀ r ∈ c.observed, (c.proj H) ((c.proj H) X) r = (c.proj H) X r","l":164,"f":"Causalean/Panel/Weighted/Subspace.lean"},{"n":"Causalean.Panel.Weighted.WeightedSupport.proj_mem","k":"theorem","m":"Causalean.Panel.Weighted.Subspace","a":"Panel/Weighted","t":2,"d":"The chosen weighted projection always lands in the nuisance subspace.","s":"∀ {R : Type u_1} [inst : Fintype R] [inst_1 : DecidableEq R] (c : Causalean.Panel.Weighted.WeightedSupport R) (H : Submodule ℝ (R → ℝ)) (X : R → ℝ), (c.proj H) X ∈ H","l":106,"f":"Causalean/Panel/Weighted/Subspace.lean"},{"n":"Causalean.Panel.Weighted.WeightedSupport.proj_orthogonal","k":"theorem","m":"Causalean.Panel.Weighted.Subspace","a":"Panel/Weighted","t":2,"d":"The residual from the chosen weighted projection is orthogonal to every element of the nuisance subspace.","s":"∀ {R : Type u_1} [inst : Fintype R] [inst_1 : DecidableEq R] (c : Causalean.Panel.Weighted.WeightedSupport R) (H : Submodule ℝ (R → ℝ)) (X : R → ℝ) {h : R → ℝ}, h ∈ H → c.ip (X - (c.proj H) X) h = 0","l":111,"f":"Causalean/Panel/Weighted/Subspace.lean"},{"n":"Causalean.Panel.Weighted.WeightedSupport.residualize","k":"def","m":"Causalean.Panel.Weighted.Subspace","a":"Panel/Weighted","t":1,"d":"The residual maker M_H = id - P_H.","s":"{R : Type u_1} → [inst : Fintype R] → [inst_1 : DecidableEq R] → Causalean.Panel.Weighted.WeightedSupport R → Submodule ℝ (R → ℝ) → (R → ℝ) →ₗ[ℝ] R → ℝ","l":172,"f":"Causalean/Panel/Weighted/Subspace.lean"},{"n":"Causalean.Panel.Weighted.WeightedSupport.residualize_apply","k":"theorem","m":"Causalean.Panel.Weighted.Subspace","a":"Panel/Weighted","t":2,"d":"Applying the residual maker subtracts the weighted projection from the original array.","s":"∀ {R : Type u_1} [inst : Fintype R] [inst_1 : DecidableEq R] (c : Causalean.Panel.Weighted.WeightedSupport R) (H : Submodule ℝ (R → ℝ)) (X : R → ℝ), (c.residualize H) X = X - (c.proj H) X","l":177,"f":"Causalean/Panel/Weighted/Subspace.lean"},{"n":"Causalean.Panel.Weighted.WeightedSupport.residualize_idem_apply","k":"theorem","m":"Causalean.Panel.Weighted.Subspace","a":"Panel/Weighted","t":2,"d":"Idempotence of M_H on the observed indices: M_H (M_H X) = M_H X on c.observed.","s":"∀ {R : Type u_1} [inst : Fintype R] [inst_1 : DecidableEq R] (c : Causalean.Panel.Weighted.WeightedSupport R) (H : Submodule ℝ (R → ℝ)) (X : R → ℝ), ∀ r ∈ c.observed, (c.residualize H) ((c.residualize H) X) r = (c.residualize H) X r","l":214,"f":"Causalean/Panel/Weighted/Subspace.lean"},{"n":"Causalean.Panel.Weighted.WeightedSupport.residualize_in_orthogonal","k":"theorem","m":"Causalean.Panel.Weighted.Subspace","a":"Panel/Weighted","t":1,"d":"Key orthogonality. For any array h lying in the nuisance subspace H, the residualized array X̃ = M_H X is orthogonal to h under the weighted inner product c.ip.","s":"∀ {R : Type u_1} [inst : Fintype R] [inst_1 : DecidableEq R] (c : Causalean.Panel.Weighted.WeightedSupport R) (H : Submodule ℝ (R → ℝ)) (X : R → ℝ) {h : R → ℝ}, h ∈ H → c.ip (c.tildeX H X) h = 0","l":196,"f":"Causalean/Panel/Weighted/Subspace.lean"},{"n":"Causalean.Panel.Weighted.WeightedSupport.residualize_orth_iff_argmin","k":"theorem","m":"Causalean.Panel.Weighted.WLS","a":"Panel/Weighted","t":1,"d":"First-order WLS optimality. For a weighted support and a submodule H of candidate functions, fix a candidate function p belonging to H; then the residual X - p being weighted-orthogonal to every element of H is equivalent to p attaining the minimal weighted sum of squared residuals over H.","s":"∀ {R : Type u_1} [inst : Fintype R] [inst_1 : DecidableEq R] (c : Causalean.Panel.Weighted.WeightedSupport R) (H : Submodule ℝ (R → ℝ)) (X p : R → ℝ), p ∈ H → ((∀ h ∈ H, c.ip (X - p) h = 0) ↔ ∀ h ∈ H, c.ip (X - p) (X - p) ≤ c.ip (X - h) (X - h))","l":146,"f":"Causalean/Panel/Weighted/WLS.lean"},{"n":"Causalean.Panel.Weighted.WeightedSupport.residualize_self_of_mem","k":"theorem","m":"Causalean.Panel.Weighted.Subspace","a":"Panel/Weighted","t":2,"d":"If X ∈ H then X̃ = M_H X vanishes on c.observed.","s":"∀ {R : Type u_1} [inst : Fintype R] [inst_1 : DecidableEq R] (c : Causalean.Panel.Weighted.WeightedSupport R) (H : Submodule ℝ (R → ℝ)) {X : R → ℝ}, X ∈ H → ∀ r ∈ c.observed, c.tildeX H X r = 0","l":205,"f":"Causalean/Panel/Weighted/Subspace.lean"},{"n":"Causalean.Panel.Weighted.WeightedSupport.rhsVec","k":"def","m":"Causalean.Panel.Weighted.FWL","a":"Panel/Weighted","t":1,"d":"Residualized FWL right-hand side ⟨M_H X, Y⟩_ω.","s":"{R : Type u_1} → [inst : Fintype R] → [inst_1 : DecidableEq R] → {K : ℕ} → Causalean.Panel.Weighted.WeightedSupport R → (Fin K → R → ℝ) → Submodule ℝ (R → ℝ) → (R → ℝ) → Fin K → ℝ","l":78,"f":"Causalean/Panel/Weighted/FWL.lean"},{"n":"Causalean.Panel.Weighted.WeightedSupport.rhsVec_apply","k":"theorem","m":"Causalean.Panel.Weighted.FWL","a":"Panel/Weighted","t":2,"d":"Each entry of the residualized right-hand side is the weighted inner product of a residualized regressor with the outcome.","s":"∀ {R : Type u_1} [inst : Fintype R] [inst_1 : DecidableEq R] {K : ℕ} (c : Causalean.Panel.Weighted.WeightedSupport R) (X : Fin K → R → ℝ) (H : Submodule ℝ (R → ℝ)) (Y : R → ℝ) (j : Fin K), c.rhsVec X H Y j = c.ip (c.tildeX H (X j)) Y","l":83,"f":"Causalean/Panel/Weighted/FWL.lean"},{"n":"Causalean.Panel.Weighted.WeightedSupport.scalar_fwl_of_normalEqs","k":"theorem","m":"Causalean.Panel.Weighted.ScalarFWL","a":"Panel/Weighted","t":1,"d":"Scalar FWL from the normal equations. Suppose the nuisance term α lies in H, the H-residualized regressor X̃ has nonzero weighted self-inner-product, and the coefficient β with nuisance term α satisfies the weighted normal equation against the raw regressor X and the weighted normal equation against every element of the nuisance space H. Then β equals the residualized coefficient ratio ⟨X̃, Y⟩_ω / ⟨X̃, X̃⟩_ω.","s":"∀ {R : Type u_1} [inst : Fintype R] [inst_1 : DecidableEq R] (c : Causalean.Panel.Weighted.WeightedSupport R) (H : Submodule ℝ (R → ℝ)) (X Y : R → ℝ) (β : ℝ), ∀ α ∈ H, c.ip (c.tildeX H X) (c.tildeX H X) ≠ 0 → c.ip (Y - β • X - α) X = 0 → (∀ h ∈ H, c.ip (Y - β • X - α) h = 0) → β = c.ip (c.tildeX H X) Y / c.ip (c.tildeX H X) (c.tildeX H X)","l":69,"f":"Causalean/Panel/Weighted/ScalarFWL.lean"},{"n":"Causalean.Panel.Weighted.WeightedSupport.sum_weight","k":"theorem","m":"Causalean.Panel.Weighted.Support","a":"Panel/Weighted","t":1,"d":"For a weighted-support system c, the weights sum to one over the observed index set (restated from the structure's normalization field for convenient reuse).","s":"∀ {R : Type u_1} [inst : Fintype R] [inst_1 : DecidableEq R] (c : Causalean.Panel.Weighted.WeightedSupport R), ∑ r ∈ c.observed, c.weight r = 1","l":102,"f":"Causalean/Panel/Weighted/Support.lean"},{"n":"Causalean.Panel.Weighted.WeightedSupport.sum_weight_univ","k":"theorem","m":"Causalean.Panel.Weighted.Support","a":"Panel/Weighted","t":2,"d":"Summing ω over the full type R equals summing over observed, because ω vanishes outside observed.","s":"∀ {R : Type u_1} [inst : Fintype R] [inst_1 : DecidableEq R] (c : Causalean.Panel.Weighted.WeightedSupport R), ∑ r, c.weight r = ∑ r ∈ c.observed, c.weight r","l":93,"f":"Causalean/Panel/Weighted/Support.lean"},{"n":"Causalean.Panel.Weighted.WeightedSupport.sum_weight_univ_eq_one","k":"theorem","m":"Causalean.Panel.Weighted.Support","a":"Panel/Weighted","t":2,"d":"Summing the weights over Finset.univ gives 1.","s":"∀ {R : Type u_1} [inst : Fintype R] [inst_1 : DecidableEq R] (c : Causalean.Panel.Weighted.WeightedSupport R), ∑ r, c.weight r = 1","l":108,"f":"Causalean/Panel/Weighted/Support.lean"},{"n":"Causalean.Panel.Weighted.WeightedSupport.thetaHat","k":"def","m":"Causalean.Panel.Weighted.FWL","a":"Panel/Weighted","t":1,"d":"Residualized weighted least-squares coefficient θ̂ = Q_XX⁻¹ ⟨M_H X, Y⟩_ω.","s":"{R : Type u_1} → [inst : Fintype R] → [inst_1 : DecidableEq R] → {K : ℕ} → Causalean.Panel.Weighted.WeightedSupport R → (Fin K → R → ℝ) → Submodule ℝ (R → ℝ) → (R → ℝ) → Fin K → ℝ","l":89,"f":"Causalean/Panel/Weighted/FWL.lean"},{"n":"Causalean.Panel.Weighted.WeightedSupport.tildeX","k":"def","m":"Causalean.Panel.Weighted.Subspace","a":"Panel/Weighted","t":1,"d":"The residualized scalar array X̃ := M_H X.","s":"{R : Type u_1} → [inst : Fintype R] → [inst_1 : DecidableEq R] → Causalean.Panel.Weighted.WeightedSupport R → Submodule ℝ (R → ℝ) → (R → ℝ) → R → ℝ","l":184,"f":"Causalean/Panel/Weighted/Subspace.lean"},{"n":"Causalean.Panel.Weighted.WeightedSupport.tildeXVec","k":"def","m":"Causalean.Panel.Weighted.Subspace","a":"Panel/Weighted","t":1,"d":"Column-by-column residualization for vector arrays X : J → (R → ℝ).","s":"{R : Type u_1} → [inst : Fintype R] → [inst_1 : DecidableEq R] → {J : Type u_2} → Causalean.Panel.Weighted.WeightedSupport R → Submodule ℝ (R → ℝ) → (J → R → ℝ) → J → R → ℝ","l":248,"f":"Causalean/Panel/Weighted/Subspace.lean"},{"n":"Causalean.Panel.Weighted.WeightedSupport.tildeXVec_apply","k":"theorem","m":"Causalean.Panel.Weighted.Subspace","a":"Panel/Weighted","t":2,"d":"The residualized vector array applies scalar residualization to the chosen column.","s":"∀ {R : Type u_1} [inst : Fintype R] [inst_1 : DecidableEq R] {J : Type u_2} (c : Causalean.Panel.Weighted.WeightedSupport R) (H : Submodule ℝ (R → ℝ)) (X : J → R → ℝ) (k : J), c.tildeXVec H X k = c.tildeX H (X k)","l":253,"f":"Causalean/Panel/Weighted/Subspace.lean"},{"n":"Causalean.Panel.Weighted.WeightedSupport.tildeX_eq","k":"theorem","m":"Causalean.Panel.Weighted.Subspace","a":"Panel/Weighted","t":2,"d":"The residualized scalar array is the original array minus its weighted projection onto the nuisance subspace.","s":"∀ {R : Type u_1} [inst : Fintype R] [inst_1 : DecidableEq R] (c : Causalean.Panel.Weighted.WeightedSupport R) (H : Submodule ℝ (R → ℝ)) (X : R → ℝ), c.tildeX H X = X - (c.proj H) X","l":189,"f":"Causalean/Panel/Weighted/Subspace.lean"},{"n":"Causalean.Panel.Weighted.WeightedSupport.weight_nonneg","k":"theorem","m":"Causalean.Panel.Weighted.Support","a":"Panel/Weighted","t":2,"d":"The weight function ω is nonnegative everywhere on R.","s":"∀ {R : Type u_1} [inst : Fintype R] [inst_1 : DecidableEq R] (c : Causalean.Panel.Weighted.WeightedSupport R) (r : R), 0 ≤ c.weight r","l":87,"f":"Causalean/Panel/Weighted/Support.lean"},{"n":"Causalean.Panel.Weighted.WeightedSupport.weighted_orthogonal_projection_exists","k":"theorem","m":"Causalean.Panel.Weighted.Subspace","a":"Panel/Weighted","t":2,"d":"Existence of a c.ip-orthogonal projection onto a subspace H.","s":"∀ {R : Type u_1} [inst : Fintype R] [inst_1 : DecidableEq R] (c : Causalean.Panel.Weighted.WeightedSupport R) (H : Submodule ℝ (R → ℝ)), ∃ P, (∀ (X : R → ℝ), P X ∈ H) ∧ ∀ (X h : R → ℝ), h ∈ H → c.ip (X - P X) h = 0","l":59,"f":"Causalean/Panel/Weighted/Subspace.lean"},{"n":"Causalean.Panel.Weighted.cellIndicator","k":"def","m":"Causalean.Panel.Weighted.IndicatorSpan","a":"Panel/Weighted","t":1,"d":"The cell indicator is one on records classified into a given cell and zero elsewhere.","s":"{R : Type u_1} → {𝒢 : Type u_2} → [DecidableEq 𝒢] → (R → 𝒢) → 𝒢 → R → ℝ","l":52,"f":"Causalean/Panel/Weighted/IndicatorSpan.lean"},{"n":"Causalean.Panel.Weighted.cellIndicator_apply","k":"theorem","m":"Causalean.Panel.Weighted.IndicatorSpan","a":"Panel/Weighted","t":2,"d":"The cell indicator evaluates by checking whether the record belongs to the chosen classifier cell.","s":"∀ {R : Type u_1} {𝒢 : Type u_2} [inst : DecidableEq 𝒢] (G : R → 𝒢) (g : 𝒢) (r : R), Causalean.Panel.Weighted.cellIndicator G g r = if G r = g then 1 else 0","l":58,"f":"Causalean/Panel/Weighted/IndicatorSpan.lean"},{"n":"Causalean.Panel.Weighted.cellIndicator_mem_indicatorSpan","k":"theorem","m":"Causalean.Panel.Weighted.IndicatorSpan","a":"Panel/Weighted","t":2,"d":"Each singleton cell indicator belongs to the span generated by all cells of the same classifier.","s":"∀ {R : Type u_1} {𝒢 : Type u_2} [inst : Fintype 𝒢] [inst_1 : DecidableEq 𝒢] (G : R → 𝒢) (g : 𝒢), Causalean.Panel.Weighted.cellIndicator G g ∈ Causalean.Panel.Weighted.indicatorSpan G","l":111,"f":"Causalean/Panel/Weighted/IndicatorSpan.lean"},{"n":"Causalean.Panel.Weighted.indicatorSpan","k":"def","m":"Causalean.Panel.Weighted.IndicatorSpan","a":"Panel/Weighted","t":1,"d":"The one-axis indicator span contains exactly the finite linear combinations of singleton indicators for one classifier.","s":"{R : Type u_1} → {𝒢 : Type u_2} → [Fintype 𝒢] → [DecidableEq 𝒢] → (R → 𝒢) → Submodule ℝ (R → ℝ)","l":66,"f":"Causalean/Panel/Weighted/IndicatorSpan.lean"},{"n":"Causalean.Panel.Weighted.indicatorSpan_le_twoAxisIndicatorSpan","k":"theorem","m":"Causalean.Panel.Weighted.IndicatorSpan","a":"Panel/Weighted","t":2,"d":"The first classifier's indicator span is contained in the two-axis indicator span.","s":"∀ {R : Type u_1} {𝒢 : Type u_2} {𝒯 : Type u_3} [inst : Fintype 𝒢] [inst_1 : DecidableEq 𝒢] [inst_2 : Fintype 𝒯] [inst_3 : DecidableEq 𝒯] (G : R → 𝒢) (T : R → 𝒯), Causalean.Panel.Weighted.indicatorSpan G ≤ Causalean.Panel.Weighted.twoAxisIndicatorSpan G T","l":142,"f":"Causalean/Panel/Weighted/IndicatorSpan.lean"},{"n":"Causalean.Panel.Weighted.indicatorSpan_le_twoAxisIndicatorSpan_right","k":"theorem","m":"Causalean.Panel.Weighted.IndicatorSpan","a":"Panel/Weighted","t":2,"d":"The second classifier's indicator span is contained in the two-axis indicator span.","s":"∀ {R : Type u_1} {𝒢 : Type u_2} {𝒯 : Type u_3} [inst : Fintype 𝒢] [inst_1 : DecidableEq 𝒢] [inst_2 : Fintype 𝒯] [inst_3 : DecidableEq 𝒯] (G : R → 𝒢) (T : R → 𝒯), Causalean.Panel.Weighted.indicatorSpan T ≤ Causalean.Panel.Weighted.twoAxisIndicatorSpan G T","l":149,"f":"Causalean/Panel/Weighted/IndicatorSpan.lean"},{"n":"Causalean.Panel.Weighted.mem_indicatorSpan_iff","k":"theorem","m":"Causalean.Panel.Weighted.IndicatorSpan","a":"Panel/Weighted","t":2,"d":"Membership in the one-axis indicator span is equivalent to having coefficients on the classifier cells whose weighted sum reproduces the array.","s":"∀ {R : Type u_1} {𝒢 : Type u_2} [inst : Fintype 𝒢] [inst_1 : DecidableEq 𝒢] (G : R → 𝒢) (f : R → ℝ), f ∈ Causalean.Panel.Weighted.indicatorSpan G ↔ ∃ c, ∀ (r : R), f r = ∑ g, c g * Causalean.Panel.Weighted.cellIndicator G g r","l":103,"f":"Causalean/Panel/Weighted/IndicatorSpan.lean"},{"n":"Causalean.Panel.Weighted.twoAxisAdditiveSpan","k":"def","m":"Causalean.Panel.Weighted.AdditiveSpan","a":"Panel/Weighted","t":1,"d":"The two-axis additive span for the product index R = I × T: specializes AdditiveSpan to f₁ = Prod.fst, f₂ = Prod.snd.","s":"(I : Type u_1) → (T : Type u_2) → Submodule ℝ (I × T → ℝ)","l":105,"f":"Causalean/Panel/Weighted/AdditiveSpan.lean"},{"n":"Causalean.Panel.Weighted.twoAxisIndicatorSpan","k":"def","m":"Causalean.Panel.Weighted.IndicatorSpan","a":"Panel/Weighted","t":1,"d":"The two-axis indicator span contains sums of one-axis indicator functions from two classifiers.","s":"{R : Type u_1} → {𝒢 : Type u_2} → {𝒯 : Type u_3} → [Fintype 𝒢] → [DecidableEq 𝒢] → [Fintype 𝒯] → [DecidableEq 𝒯] → (R → 𝒢) → (R → 𝒯) → Submodule ℝ (R → ℝ)","l":135,"f":"Causalean/Panel/Weighted/IndicatorSpan.lean"},{"n":"Causalean.Panel.WeightedTwoWayPanel.IsUnitTimeAdditive","k":"def","m":"Causalean.Panel.WeightedTwoWayPanel","a":"Panel","t":1,"d":"Unit-time additive nuisance class h_it = a_i + b_t (shared predicate).","s":"{Unit : Type u_1} → {Time : Type u_2} → (Unit → Time → ℝ) → Prop","l":83,"f":"Causalean/Panel/WeightedTwoWayPanel.lean"},{"n":"Causalean.Panel.WeightedTwoWayPanel.UnitWeights","k":"structure","m":"Causalean.Panel.WeightedTwoWayPanel","a":"Panel","t":1,"d":"Unit weights. A weight function p assigning each unit a share that forms a probability vector: every unit's weight is strictly positive and the weights sum to one across units.","s":"(Unit : Type u_3) → [Fintype Unit] → Type u_3","l":52,"f":"Causalean/Panel/WeightedTwoWayPanel.lean"},{"n":"Causalean.Panel.WeightedTwoWayPanel.cellSupport","k":"def","m":"Causalean.Panel.WeightedTwoWayPanel","a":"Panel","t":1,"d":"The panel viewed as a cell-indexed weighted support on R = Unit × Time with the factorized weight ω_{(i,t)} = p_i / |Time| (every cell observed).","s":"{Unit : Type u_1} → {Time : Type u_2} → [inst : Fintype Unit] → [inst_1 : Fintype Time] → [inst_2 : DecidableEq Unit] → [inst_3 : DecidableEq Time] → [Nonempty Unit] → [Nonempty Time] → Causalean.Panel.WeightedTwoWayPanel.UnitWeights Unit → Causalean.Panel.Weighted.WeightedSupport (Unit × Time)","l":273,"f":"Causalean/Panel/WeightedTwoWayPanel.lean"},{"n":"Causalean.Panel.WeightedTwoWayPanel.cellSupport_observed","k":"theorem","m":"Causalean.Panel.WeightedTwoWayPanel","a":"Panel","t":2,"d":"Every unit-period cell is observed in the cell-support bridge.","s":"∀ {Unit : Type u_1} {Time : Type u_2} [inst : Fintype Unit] [inst_1 : Fintype Time] [inst_2 : DecidableEq Unit] [inst_3 : DecidableEq Time] [inst_4 : Nonempty Unit] [inst_5 : Nonempty Time] (w : Causalean.Panel.WeightedTwoWayPanel.UnitWeights Unit), (Causalean.Panel.WeightedTwoWayPanel.cellSupport w).observed = Finset.univ","l":307,"f":"Causalean/Panel/WeightedTwoWayPanel.lean"},{"n":"Causalean.Panel.WeightedTwoWayPanel.cellSupport_weight","k":"theorem","m":"Causalean.Panel.WeightedTwoWayPanel","a":"Panel","t":2,"d":"The cell-support bridge assigns each unit-period cell its unit weight divided equally across periods.","s":"∀ {Unit : Type u_1} {Time : Type u_2} [inst : Fintype Unit] [inst_1 : Fintype Time] [inst_2 : DecidableEq Unit] [inst_3 : DecidableEq Time] [inst_4 : Nonempty Unit] [inst_5 : Nonempty Time] (w : Causalean.Panel.WeightedTwoWayPanel.UnitWeights Unit) (r : Unit × Time), (Causalean.Panel.WeightedTwoWayPanel.cellSupport w).weight r = w.p r.1 / ↑(Fintype.card Time)","l":302,"f":"Causalean/Panel/WeightedTwoWayPanel.lean"},{"n":"Causalean.Panel.WeightedTwoWayPanel.ddot","k":"def","m":"Causalean.Panel.WeightedTwoWayPanel","a":"Panel","t":1,"d":"Two-way residual / double-demeaned array under the p-weighted means.","s":"{Unit : Type u_1} → {Time : Type u_2} → [inst : Fintype Unit] → [Fintype Time] → Causalean.Panel.WeightedTwoWayPanel.UnitWeights Unit → (Unit → Time → ℝ) → Unit → Time → ℝ","l":74,"f":"Causalean/Panel/WeightedTwoWayPanel.lean"},{"n":"Causalean.Panel.WeightedTwoWayPanel.ddot_eq_residualize","k":"theorem","m":"Causalean.Panel.WeightedTwoWayPanel","a":"Panel","t":1,"d":"Concrete ↔ abstract bridge. For unit weights w and an outcome array V, the closed-form two-way double-demeaned residual ddot w V i t equals the generic weighted-support residual against the two-axis additive span, evaluated at cell (i, t). Every cell is observed, so the identity holds pointwise.","s":"∀ {Unit : Type u_1} {Time : Type u_2} [inst : Fintype Unit] [inst_1 : Fintype Time] [inst_2 : DecidableEq Unit] [inst_3 : DecidableEq Time] [inst_4 : Nonempty Unit] [inst_5 : Nonempty Time] (w : Causalean.Panel.WeightedTwoWayPanel.UnitWeights Unit) (V : Unit → Time → ℝ) (i : Unit) (t : Time), ((Causalean.Panel.WeightedTwoWayPanel.cellSupport w).residualize (Causalean.Panel.Weighted.twoAxisAdditiveSpan Unit Time)) (fun r => V r.1 r.2) (i, t) = Causalean.Panel.WeightedTwoWayPanel.ddot w V i t","l":349,"f":"Causalean/Panel/WeightedTwoWayPanel.lean"},{"n":"Causalean.Panel.WeightedTwoWayPanel.ddot_orthogonal_time","k":"theorem","m":"Causalean.Panel.WeightedTwoWayPanel","a":"Panel","t":2,"d":"Double-demeaned arrays are orthogonal (in the p-weighted inner product) to arbitrary time-only functions: ∑_i ∑_t p_i · ddot V_{it} · b_t = 0. Per-period the p-weighted unit sum of ddot vanishes (∑ p = 1 cancels the time mean against the grand mean).","s":"∀ {Unit : Type u_1} {Time : Type u_2} [inst : Fintype Unit] [inst_1 : Fintype Time] (w : Causalean.Panel.WeightedTwoWayPanel.UnitWeights Unit) (V : Unit → Time → ℝ) (b : Time → ℝ), ∑ i, ∑ t, w.p i * (Causalean.Panel.WeightedTwoWayPanel.ddot w V i t * b t) = 0","l":190,"f":"Causalean/Panel/WeightedTwoWayPanel.lean"},{"n":"Causalean.Panel.WeightedTwoWayPanel.ddot_orthogonal_unit","k":"theorem","m":"Causalean.Panel.WeightedTwoWayPanel","a":"Panel","t":2,"d":"Double-demeaned arrays are orthogonal (in the p-weighted inner product) to arbitrary unit-only functions.","s":"∀ {Unit : Type u_1} {Time : Type u_2} [inst : Fintype Unit] [inst_1 : Fintype Time] (w : Causalean.Panel.WeightedTwoWayPanel.UnitWeights Unit) (V : Unit → Time → ℝ) (a : Unit → ℝ), ∑ i, ∑ t, w.p i * (Causalean.Panel.WeightedTwoWayPanel.ddot w V i t * a i) = 0","l":176,"f":"Causalean/Panel/WeightedTwoWayPanel.lean"},{"n":"Causalean.Panel.WeightedTwoWayPanel.ddot_orthogonal_unit_time","k":"theorem","m":"Causalean.Panel.WeightedTwoWayPanel","a":"Panel","t":1,"d":"Double-demeaned arrays are orthogonal to every unit/time additive nuisance function. For a probability-weighted panel of units and periods, any array that decomposes as the sum of a unit-specific term and a time-specific term is orthogonal, under the p-weighted inner product, to the double-demeaned residual of any panel array V: their weighted inner product is zero.","s":"∀ {Unit : Type u_1} {Time : Type u_2} [inst : Fintype Unit] [inst_1 : Fintype Time] (w : Causalean.Panel.WeightedTwoWayPanel.UnitWeights Unit) (V h : Unit → Time → ℝ), Causalean.Panel.WeightedTwoWayPanel.IsUnitTimeAdditive h → Causalean.Panel.WeightedTwoWayPanel.inner w (Causalean.Panel.WeightedTwoWayPanel.ddot w V) h = 0","l":233,"f":"Causalean/Panel/WeightedTwoWayPanel.lean"},{"n":"Causalean.Panel.WeightedTwoWayPanel.ddot_reconstruct","k":"theorem","m":"Causalean.Panel.WeightedTwoWayPanel","a":"Panel","t":2,"d":"Reconstruction identity V_it = ddot V_it + unitMean + timeMean - grandMean.","s":"∀ {Unit : Type u_1} {Time : Type u_2} [inst : Fintype Unit] [inst_1 : Fintype Time] (w : Causalean.Panel.WeightedTwoWayPanel.UnitWeights Unit) (V : Unit → Time → ℝ) (i : Unit) (t : Time), Causalean.Panel.WeightedTwoWayPanel.ddot w V i t + Causalean.Panel.WeightedTwoWayPanel.unitMean V i + Causalean.Panel.WeightedTwoWayPanel.timeMean w V t - Causalean.Panel.WeightedTwoWayPanel.grandMean w V = V i t","l":92,"f":"Causalean/Panel/WeightedTwoWayPanel.lean"},{"n":"Causalean.Panel.WeightedTwoWayPanel.grandMean","k":"def","m":"Causalean.Panel.WeightedTwoWayPanel","a":"Panel","t":1,"d":"p-weighted grand mean \\bar V = ∑_i p_i \\bar V_{i·}.","s":"{Unit : Type u_1} → {Time : Type u_2} → [inst : Fintype Unit] → [Fintype Time] → Causalean.Panel.WeightedTwoWayPanel.UnitWeights Unit → (Unit → Time → ℝ) → ℝ","l":70,"f":"Causalean/Panel/WeightedTwoWayPanel.lean"},{"n":"Causalean.Panel.WeightedTwoWayPanel.inner","k":"def","m":"Causalean.Panel.WeightedTwoWayPanel","a":"Panel","t":1,"d":"p-weighted finite-panel inner product (uniform-time normalizer dropped as harmless).","s":"{Unit : Type u_1} → {Time : Type u_2} → [inst : Fintype Unit] → [Fintype Time] → Causalean.Panel.WeightedTwoWayPanel.UnitWeights Unit → (Unit → Time → ℝ) → (Unit → Time → ℝ) → ℝ","l":78,"f":"Causalean/Panel/WeightedTwoWayPanel.lean"},{"n":"Causalean.Panel.WeightedTwoWayPanel.inner_eq_card_smul_ip","k":"theorem","m":"Causalean.Panel.WeightedTwoWayPanel","a":"Panel","t":2,"d":"The p-weighted panel inner product is |Time| times the generic weighted-support inner product on the cell support.","s":"∀ {Unit : Type u_1} {Time : Type u_2} [inst : Fintype Unit] [inst_1 : Fintype Time] [inst_2 : DecidableEq Unit] [inst_3 : DecidableEq Time] [inst_4 : Nonempty Unit] [inst_5 : Nonempty Time] (w : Causalean.Panel.WeightedTwoWayPanel.UnitWeights Unit) (V W : Unit → Time → ℝ), Causalean.Panel.WeightedTwoWayPanel.inner w V W = ↑(Fintype.card Time) * (Causalean.Panel.WeightedTwoWayPanel.cellSupport w).ip (fun r => V r.1 r.2) fun r => W r.1 r.2","l":327,"f":"Causalean/Panel/WeightedTwoWayPanel.lean"},{"n":"Causalean.Panel.WeightedTwoWayPanel.mem_twoAxisAdditiveSpan_iff","k":"theorem","m":"Causalean.Panel.WeightedTwoWayPanel","a":"Panel","t":2,"d":"Membership in the two-axis additive span is exactly the panel's IsUnitTimeAdditive predicate after uncurrying.","s":"∀ {Unit : Type u_1} {Time : Type u_2} {h : Unit × Time → ℝ}, h ∈ Causalean.Panel.Weighted.twoAxisAdditiveSpan Unit Time ↔ Causalean.Panel.WeightedTwoWayPanel.IsUnitTimeAdditive fun i t => h (i, t)","l":313,"f":"Causalean/Panel/WeightedTwoWayPanel.lean"},{"n":"Causalean.Panel.WeightedTwoWayPanel.sub_ddot_eq_unitTimeProjection","k":"theorem","m":"Causalean.Panel.WeightedTwoWayPanel","a":"Panel","t":2,"d":"Pointwise residual decomposition V - ddot V into the unit/time projection.","s":"∀ {Unit : Type u_1} {Time : Type u_2} [inst : Fintype Unit] [inst_1 : Fintype Time] (w : Causalean.Panel.WeightedTwoWayPanel.UnitWeights Unit) (V : Unit → Time → ℝ) (i : Unit) (t : Time), V i t - Causalean.Panel.WeightedTwoWayPanel.ddot w V i t = Causalean.Panel.WeightedTwoWayPanel.unitTimeProjection w V i t","l":106,"f":"Causalean/Panel/WeightedTwoWayPanel.lean"},{"n":"Causalean.Panel.WeightedTwoWayPanel.timeMean","k":"def","m":"Causalean.Panel.WeightedTwoWayPanel","a":"Panel","t":1,"d":"p-weighted time mean \\bar V_{·t} = ∑_i p_i V_{it}.","s":"{Unit : Type u_1} → {Time : Type u_2} → [inst : Fintype Unit] → Causalean.Panel.WeightedTwoWayPanel.UnitWeights Unit → (Unit → Time → ℝ) → Time → ℝ","l":66,"f":"Causalean/Panel/WeightedTwoWayPanel.lean"},{"n":"Causalean.Panel.WeightedTwoWayPanel.unitMean","k":"def","m":"Causalean.Panel.WeightedTwoWayPanel","a":"Panel","t":1,"d":"Unit mean \\bar V_{i·} under the uniform period measure (weight-free in time).","s":"{Unit : Type u_1} → {Time : Type u_2} → [Fintype Time] → (Unit → Time → ℝ) → Unit → ℝ","l":62,"f":"Causalean/Panel/WeightedTwoWayPanel.lean"},{"n":"Causalean.Panel.WeightedTwoWayPanel.unitTimeProjection","k":"def","m":"Causalean.Panel.WeightedTwoWayPanel","a":"Panel","t":1,"d":"The unit/time component removed by double demeaning.","s":"{Unit : Type u_1} → {Time : Type u_2} → [inst : Fintype Unit] → [Fintype Time] → Causalean.Panel.WeightedTwoWayPanel.UnitWeights Unit → (Unit → Time → ℝ) → Unit → Time → ℝ","l":87,"f":"Causalean/Panel/WeightedTwoWayPanel.lean"},{"n":"Causalean.Panel.WeightedTwoWayPanel.unitTimeProjection_additive","k":"theorem","m":"Causalean.Panel.WeightedTwoWayPanel","a":"Panel","t":2,"d":"The removed component is itself unit/time additive.","s":"∀ {Unit : Type u_1} {Time : Type u_2} [inst : Fintype Unit] [inst_1 : Fintype Time] (w : Causalean.Panel.WeightedTwoWayPanel.UnitWeights Unit) (V : Unit → Time → ℝ), Causalean.Panel.WeightedTwoWayPanel.IsUnitTimeAdditive (Causalean.Panel.WeightedTwoWayPanel.unitTimeProjection w V)","l":98,"f":"Causalean/Panel/WeightedTwoWayPanel.lean"},{"n":"Causalean.Panel.panelConsistency_holds","k":"theorem","m":"Causalean.Panel.PO.Consistency","a":"Panel/PO","t":1,"d":"If panel consistency holds for the panel potential-outcome system P, then the factual observed outcome equals the potential outcome evaluated at the realized exposure, for every observed unit-period cell and every sample point.","s":"∀ (P : Causalean.Panel.PanelPOSystem), Causalean.Panel.PanelConsistency P → ∀ (r : P.I × P.T) (hr : r ∈ P.cells.observed) (ω : P.Ω), P.observedY r hr ω = P.Y r hr (P.observedExposure r hr ω) ω","l":45,"f":"Causalean/Panel/PO/Consistency.lean"},{"n":"Causalean.Panel.residualizedCoefficient","k":"def","m":"Causalean.Panel.Analysis.Residualization","a":"Panel/Analysis","t":1,"d":"The residualized population coefficient is the ratio of the covariance-like integral of treatment and outcome residuals to the second moment of the treatment residual.","s":"{Ω : Type u_1} → [inst : MeasurableSpace Ω] → (μ : MeasureTheory.Measure Ω) → (H : Causalean.Panel.LinearL2Class μ) → {Y D : Ω → ℝ} → Causalean.Panel.ResidualizationWitness μ H Y → Causalean.Panel.ResidualizationWitness μ H D → ℝ","l":78,"f":"Causalean/Panel/Analysis/Residualization.lean"},{"n":"Causalean.Panel.residualizedCoefficient_eq_of_normalEqs","k":"theorem","m":"Causalean.Panel.Analysis.Residualization","a":"Panel/Analysis","t":1,"d":"Population Frisch-Waugh-Lovell theorem (prop:po-estimand-fwl). Let wY and wD be residualization witnesses decomposing outcome Y and treatment D each into a component of a common linear square-integrable nuisance class H plus an orthogonal residual. If the second moment of the treatment residual is strictly positive, and a nuisance-class element Hβ together with a coefficient β are such that the combined residual Y − β·D − Hβ is square-integrable, orthogonal in expectation to D, and orthogonal in expectation to every member of H, then β equals the ratio of the covariance-like integral of the outcome and treatment residuals Ỹ and D̃ to the second moment of D̃, i.e. the residualized coefficient residualizedCoefficient μ H wY wD.","s":"∀ {Ω : Type u_1} [inst : MeasurableSpace Ω] (μ : MeasureTheory.Measure Ω) (H : Causalean.Panel.LinearL2Class μ) {Y D : Ω → ℝ} (wY : Causalean.Panel.ResidualizationWitness μ H Y) (wD : Causalean.Panel.ResidualizationWitness μ H D), 0 < ∫ (ω : Ω), wD.Vtilde ω * wD.Vtilde ω ∂μ → ∀ (β : ℝ) (Hβ : Ω → ℝ), H.mem Hβ → MeasureTheory.MemLp (fun ω => Y ω - β * D ω - Hβ ω) 2 μ → ∫ (ω : Ω), (Y ω - β * D ω - Hβ ω) * D ω ∂μ = 0 → (∀ ⦃h : Ω → ℝ⦄, H.mem h → ∫ (ω : Ω), (Y ω - β * D ω - Hβ ω) * h ω ∂μ = 0) → β = Causalean.Panel.residualizedCoefficient μ H wY wD","l":89,"f":"Causalean/Panel/Analysis/Residualization.lean"},{"n":"Causalean.PartialID.ConicProgram","k":"structure","m":"Causalean.PO.ID.Partial.LP.ConicDuality","a":"PO/ID/Partial/LP","t":1,"d":"Primal data of a conic linear program over real Hilbert spaces: minimize ⟪c, x⟫ subject to A x = b and x ∈ K.","s":"(E : Type u_3) → (F : Type u_4) → [inst : NormedAddCommGroup E] → [InnerProductSpace ℝ E] → [CompleteSpace E] → [inst : NormedAddCommGroup F] → [InnerProductSpace ℝ F] → [CompleteSpace F] → Type (max u_3 u_4)","l":44,"f":"Causalean/PO/ID/Partial/LP/ConicDuality.lean"},{"n":"Causalean.PartialID.ConicProgram.DualFeasible","k":"def","m":"Causalean.PO.ID.Partial.LP.ConicDuality","a":"PO/ID/Partial/LP","t":1,"d":"A dual multiplier is dual feasible when the reduced cost c - Aᵀ y lies in the dual cone K⋆ = innerDual K.","s":"{E : Type u_1} → {F : Type u_2} → [inst : NormedAddCommGroup E] → [inst_1 : InnerProductSpace ℝ E] → [inst_2 : CompleteSpace E] → [inst_3 : NormedAddCommGroup F] → [inst_4 : InnerProductSpace ℝ F] → [inst_5 : CompleteSpace F] → Causalean.PartialID.ConicProgram E F → F → Prop","l":66,"f":"Causalean/PO/ID/Partial/LP/ConicDuality.lean"},{"n":"Causalean.PartialID.ConicProgram.PrimalFeasible","k":"def","m":"Causalean.PO.ID.Partial.LP.ConicDuality","a":"PO/ID/Partial/LP","t":1,"d":"A point is primal feasible when it satisfies the equality constraint and lies in the cone.","s":"{E : Type u_1} → {F : Type u_2} → [inst : NormedAddCommGroup E] → [inst_1 : InnerProductSpace ℝ E] → [inst_2 : CompleteSpace E] → [inst_3 : NormedAddCommGroup F] → [inst_4 : InnerProductSpace ℝ F] → [inst_5 : CompleteSpace F] → Causalean.PartialID.ConicProgram E F → E → Prop","l":62,"f":"Causalean/PO/ID/Partial/LP/ConicDuality.lean"},{"n":"Causalean.PartialID.ConicProgram.augmentedImage","k":"def","m":"Causalean.PO.ID.Partial.LP.ConicDuality","a":"PO/ID/Partial/LP","t":1,"d":"The augmented image cone { (A x, ⟪c, x⟫) : x ∈ K } in F × ℝ. Its closedness is the constraint qualification for primal attainment, and the geometry (a boundary point (b, primalValue)) is where the dual certificate is read off.","s":"{E : Type u_1} → {F : Type u_2} → [inst : NormedAddCommGroup E] → [inst_1 : InnerProductSpace ℝ E] → [inst_2 : CompleteSpace E] → [inst_3 : NormedAddCommGroup F] → [inst_4 : InnerProductSpace ℝ F] → [inst_5 : CompleteSpace F] → Causalean.PartialID.ConicProgram E F → Set (F × ℝ)","l":122,"f":"Causalean/PO/ID/Partial/LP/ConicDuality.lean"},{"n":"Causalean.PartialID.ConicProgram.dualValue","k":"def","m":"Causalean.PO.ID.Partial.LP.ConicDuality","a":"PO/ID/Partial/LP","t":1,"d":"The dual optimal value sup { ⟪b, y⟫ : y dual feasible }.","s":"{E : Type u_1} → {F : Type u_2} → [inst : NormedAddCommGroup E] → [inst_1 : InnerProductSpace ℝ E] → [inst_2 : CompleteSpace E] → [inst_3 : NormedAddCommGroup F] → [inst_4 : InnerProductSpace ℝ F] → [inst_5 : CompleteSpace F] → Causalean.PartialID.ConicProgram E F → ℝ","l":75,"f":"Causalean/PO/ID/Partial/LP/ConicDuality.lean"},{"n":"Causalean.PartialID.ConicProgram.dualValue_le_primalValue","k":"theorem","m":"Causalean.PO.ID.Partial.LP.ConicDuality","a":"PO/ID/Partial/LP","t":2,"d":"Weak duality (value form). When both programs are feasible, dualValue ≤ primalValue. (Boundedness is not needed: the pointwise bound exhibits primalValue as an explicit upper bound for the dual values.)","s":"∀ {E : Type u_1} {F : Type u_2} [inst : NormedAddCommGroup E] [inst_1 : InnerProductSpace ℝ E] [inst_2 : CompleteSpace E] [inst_3 : NormedAddCommGroup F] [inst_4 : InnerProductSpace ℝ F] [inst_5 : CompleteSpace F] (P : Causalean.PartialID.ConicProgram E F), {x | P.PrimalFeasible x}.Nonempty → {y | P.DualFeasible y}.Nonempty → P.dualValue ≤ P.primalValue","l":94,"f":"Causalean/PO/ID/Partial/LP/ConicDuality.lean"},{"n":"Causalean.PartialID.ConicProgram.farkas","k":"theorem","m":"Causalean.PO.ID.Partial.LP.ConicDuality","a":"PO/ID/Partial/LP","t":1,"d":"Farkas alternative / feasibility engine. For a target point b, b lies in the closed image cone A(K) if and only if every dual direction y whose pullback Aᵀ y lies in the dual cone of K pairs nonnegatively with b.","s":"∀ {E : Type u_1} {F : Type u_2} [inst : NormedAddCommGroup E] [inst_1 : InnerProductSpace ℝ E] [inst_2 : CompleteSpace E] [inst_3 : NormedAddCommGroup F] [inst_4 : InnerProductSpace ℝ F] [inst_5 : CompleteSpace F] (P : Causalean.PartialID.ConicProgram E F) {b : F}, b ∈ ProperCone.map P.A P.K ↔ ∀ (y : F), (ContinuousLinearMap.adjoint P.A) y ∈ ProperCone.innerDual ↑P.K → 0 ≤ inner ℝ b y","l":107,"f":"Causalean/PO/ID/Partial/LP/ConicDuality.lean"},{"n":"Causalean.PartialID.ConicProgram.primalValue","k":"def","m":"Causalean.PO.ID.Partial.LP.ConicDuality","a":"PO/ID/Partial/LP","t":1,"d":"The primal optimal value inf { ⟪c, x⟫ : x primal feasible }.","s":"{E : Type u_1} → {F : Type u_2} → [inst : NormedAddCommGroup E] → [inst_1 : InnerProductSpace ℝ E] → [inst_2 : CompleteSpace E] → [inst_3 : NormedAddCommGroup F] → [inst_4 : InnerProductSpace ℝ F] → [inst_5 : CompleteSpace F] → Causalean.PartialID.ConicProgram E F → ℝ","l":71,"f":"Causalean/PO/ID/Partial/LP/ConicDuality.lean"},{"n":"Causalean.PartialID.ConicProgram.strong_duality_primal_attained","k":"theorem","m":"Causalean.PO.ID.Partial.LP.ConicDuality","a":"PO/ID/Partial/LP","t":1,"d":"Strong duality I — primal attainment (closedness CQ). For a conic program that is primal feasible and whose feasible objective values are bounded below, if the augmented image cone {(Ax, ⟪c,x⟫) : x ∈ K} is closed — the constraint qualification separating attained optima from mere infima — then the primal optimum is attained: some primal-feasible point x achieves the objective value ⟪c,x⟫ = primalValue exactly. This is the \"there is an extremal data-generating distribution\" half of sharpness.","s":"∀ {E : Type u_1} {F : Type u_2} [inst : NormedAddCommGroup E] [inst_1 : InnerProductSpace ℝ E] [inst_2 : CompleteSpace E] [inst_3 : NormedAddCommGroup F] [inst_4 : InnerProductSpace ℝ F] [inst_5 : CompleteSpace F] (P : Causalean.PartialID.ConicProgram E F), {x | P.PrimalFeasible x}.Nonempty → BddBelow ((fun x => inner ℝ P.c x) '' {x | P.PrimalFeasible x}) → IsClosed P.augmentedImage → ∃ x, P.PrimalFeasible x ∧ inner ℝ P.c x = P.primalValue","l":128,"f":"Causalean/PO/ID/Partial/LP/ConicDuality.lean"},{"n":"Causalean.PartialID.ConicProgram.strong_duality_zero_gap","k":"theorem","m":"Causalean.PO.ID.Partial.LP.ConicDuality","a":"PO/ID/Partial/LP","t":1,"d":"Strong duality II — zero gap (closedness CQ). For a conic program that is primal feasible and whose feasible objective values are bounded below, if the augmented image cone is closed — the same constraint qualification as primal attainment — then there is no duality gap: the primal optimal value equals the dual optimal value, primalValue = dualValue.","s":"∀ {E : Type u_1} {F : Type u_2} [inst : NormedAddCommGroup E] [inst_1 : InnerProductSpace ℝ E] [inst_2 : CompleteSpace E] [inst_3 : NormedAddCommGroup F] [inst_4 : InnerProductSpace ℝ F] [inst_5 : CompleteSpace F] (P : Causalean.PartialID.ConicProgram E F), {x | P.PrimalFeasible x}.Nonempty → BddBelow ((fun x => inner ℝ P.c x) '' {x | P.PrimalFeasible x}) → IsClosed P.augmentedImage → P.primalValue = P.dualValue","l":158,"f":"Causalean/PO/ID/Partial/LP/ConicDuality.lean"},{"n":"Causalean.PartialID.ConicProgram.weak_duality","k":"theorem","m":"Causalean.PO.ID.Partial.LP.ConicDuality","a":"PO/ID/Partial/LP","t":2,"d":"Weak duality (pointwise). Any dual-feasible objective value lower-bounds any primal-feasible objective value.","s":"∀ {E : Type u_1} {F : Type u_2} [inst : NormedAddCommGroup E] [inst_1 : InnerProductSpace ℝ E] [inst_2 : CompleteSpace E] [inst_3 : NormedAddCommGroup F] [inst_4 : InnerProductSpace ℝ F] [inst_5 : CompleteSpace F] (P : Causalean.PartialID.ConicProgram E F) {x : E} {y : F}, P.PrimalFeasible x → P.DualFeasible y → inner ℝ P.b y ≤ inner ℝ P.c x","l":79,"f":"Causalean/PO/ID/Partial/LP/ConicDuality.lean"},{"n":"Causalean.PartialID.CriterionSet.LinearMinorant","k":"def","m":"Causalean.PO.ID.Partial.CriterionSet.Consistency","a":"PO/ID/Partial/CriterionSet","t":1,"d":"A criterion Q satisfies a linear minorant with modulus δ relative to its identified set: δ · d(θ, Θ_I) ≤ Q θ for all θ. This is the Chernozhukov–Hong–Tamer (2007) polynomial-minorant identifiability condition of degree γ = 1 — it forces Q to grow at least linearly away from Θ_I, so a small criterion value pins θ near the identified set.","s":"{Θ : Type u_1} → [PseudoMetricSpace Θ] → (Θ → ℝ) → ℝ → Prop","l":99,"f":"Causalean/PO/ID/Partial/CriterionSet/Consistency.lean"},{"n":"Causalean.PartialID.CriterionSet.directedHausdorff_eq_zero_of_subset","k":"theorem","m":"Causalean.PO.ID.Partial.CriterionSet.Consistency","a":"PO/ID/Partial/CriterionSet","t":2,"d":"If A ⊆ B then the directed Hausdorff distance from A to B is 0.","s":"∀ {Θ : Type u_1} [inst : PseudoMetricSpace Θ] {A B : Set Θ}, A ⊆ B → Causalean.PartialID.RandomSet.directedHausdorff A B = 0","l":89,"f":"Causalean/PO/ID/Partial/CriterionSet/Consistency.lean"},{"n":"Causalean.PartialID.CriterionSet.directedHausdorff_le_of_forall","k":"theorem","m":"Causalean.PO.ID.Partial.CriterionSet.Consistency","a":"PO/ID/Partial/CriterionSet","t":2,"d":"If every point of A is within M of B (and M ≥ 0), the directed Hausdorff distance from A to B is at most M.","s":"∀ {Θ : Type u_1} [inst : PseudoMetricSpace Θ] {A B : Set Θ} {M : ℝ}, 0 ≤ M → (∀ a ∈ A, Metric.infDist a B ≤ M) → Causalean.PartialID.RandomSet.directedHausdorff A B ≤ M","l":83,"f":"Causalean/PO/ID/Partial/CriterionSet/Consistency.lean"},{"n":"Causalean.PartialID.CriterionSet.directedHausdorff_nonneg","k":"theorem","m":"Causalean.PO.ID.Partial.CriterionSet.Consistency","a":"PO/ID/Partial/CriterionSet","t":2,"d":"The directed Hausdorff distance is nonnegative (a sup of nonnegative point-to-set distances).","s":"∀ {Θ : Type u_1} [inst : PseudoMetricSpace Θ] (A B : Set Θ), 0 ≤ Causalean.PartialID.RandomSet.directedHausdorff A B","l":74,"f":"Causalean/PO/ID/Partial/CriterionSet/Consistency.lean"},{"n":"Causalean.PartialID.CriterionSet.hausdorffDist_levelSet_le","k":"theorem","m":"Causalean.PO.ID.Partial.CriterionSet.Consistency","a":"PO/ID/Partial/CriterionSet","t":1,"d":"Deterministic CHT rate bound. Fix a positive linear-minorant modulus δ, a nonnegative sup-norm error bound ε, and a cutoff c at least ε. If the population criterion Q satisfies a linear minorant of modulus δ relative to its identified set — δ times the distance to the identified set never exceeds Q, and the sample criterion Qn is within ε of Q in sup norm at every point, then the level-set estimator {Qn ≤ c} is within Hausdorff distance (c + ε) / δ of the population identified set {Q = 0}:","s":"∀ {Θ : Type u_1} [inst : PseudoMetricSpace Θ] {Q Qn : Θ → ℝ} {δ c ε : ℝ}, 0 < δ → 0 ≤ ε → ε ≤ c → Causalean.PartialID.CriterionSet.LinearMinorant Q δ → (∀ (θ : Θ), |Qn θ - Q θ| ≤ ε) → Causalean.PartialID.RandomSet.hausdorffDist (Causalean.PartialID.CriterionSet.levelSet Qn c) (Causalean.PartialID.CriterionSet.identifiedSet Q) ≤ (c + ε) / δ","l":107,"f":"Causalean/PO/ID/Partial/CriterionSet/Consistency.lean"},{"n":"Causalean.PartialID.CriterionSet.hausdorffDist_nonneg","k":"theorem","m":"Causalean.PO.ID.Partial.CriterionSet.Consistency","a":"PO/ID/Partial/CriterionSet","t":2,"d":"The symmetric Hausdorff distance is nonnegative.","s":"∀ {Θ : Type u_1} [inst : PseudoMetricSpace Θ] (A B : Set Θ), 0 ≤ Causalean.PartialID.RandomSet.hausdorffDist A B","l":79,"f":"Causalean/PO/ID/Partial/CriterionSet/Consistency.lean"},{"n":"Causalean.PartialID.CriterionSet.identifiedSet","k":"def","m":"Causalean.PO.ID.Partial.CriterionSet.Basic","a":"PO/ID/Partial/CriterionSet","t":1,"d":"A criterion function's identified set is the set of parameters where the population criterion reaches zero.","s":"{Θ : Type u_1} → (Θ → ℝ) → Set Θ","l":23,"f":"Causalean/PO/ID/Partial/CriterionSet/Basic.lean"},{"n":"Causalean.PartialID.CriterionSet.identifiedSet_eq_levelSet_zero_of_nonneg","k":"theorem","m":"Causalean.PO.ID.Partial.CriterionSet.Basic","a":"PO/ID/Partial/CriterionSet","t":2,"d":"For a nonnegative criterion, the identified set is the level-0 set of the same criterion (since Q θ = 0 ↔ Q θ ≤ 0 under nonnegativity).","s":"∀ {Θ : Type u_1} {Q : Θ → ℝ}, (∀ (θ : Θ), 0 ≤ Q θ) → Causalean.PartialID.CriterionSet.identifiedSet Q = Causalean.PartialID.CriterionSet.levelSet Q 0","l":51,"f":"Causalean/PO/ID/Partial/CriterionSet/Basic.lean"},{"n":"Causalean.PartialID.CriterionSet.identifiedSet_subset_levelSet_self","k":"theorem","m":"Causalean.PO.ID.Partial.CriterionSet.Basic","a":"PO/ID/Partial/CriterionSet","t":2,"d":"The identified set sits inside any nonnegative-cutoff level set of the same criterion.","s":"∀ {Θ : Type u_1} {Q : Θ → ℝ} {c : ℝ}, 0 ≤ c → Causalean.PartialID.CriterionSet.identifiedSet Q ⊆ Causalean.PartialID.CriterionSet.levelSet Q c","l":59,"f":"Causalean/PO/ID/Partial/CriterionSet/Basic.lean"},{"n":"Causalean.PartialID.CriterionSet.levelSet","k":"def","m":"Causalean.PO.ID.Partial.CriterionSet.Basic","a":"PO/ID/Partial/CriterionSet","t":1,"d":"The level-set estimator keeps the parameters whose sample criterion is no larger than the cutoff.","s":"{Θ : Type u_1} → (Θ → ℝ) → ℝ → Set Θ","l":30,"f":"Causalean/PO/ID/Partial/CriterionSet/Basic.lean"},{"n":"Causalean.PartialID.CriterionSet.levelSet_mono","k":"theorem","m":"Causalean.PO.ID.Partial.CriterionSet.Basic","a":"PO/ID/Partial/CriterionSet","t":2,"d":"The level set is monotone in the cutoff c.","s":"∀ {Θ : Type u_1} {Qn : Θ → ℝ} {c₁ c₂ : ℝ}, c₁ ≤ c₂ → Causalean.PartialID.CriterionSet.levelSet Qn c₁ ⊆ Causalean.PartialID.CriterionSet.levelSet Qn c₂","l":47,"f":"Causalean/PO/ID/Partial/CriterionSet/Basic.lean"},{"n":"Causalean.PartialID.CriterionSet.mem_identifiedSet","k":"theorem","m":"Causalean.PO.ID.Partial.CriterionSet.Basic","a":"PO/ID/Partial/CriterionSet","t":1,"d":"For a criterion function Q and a candidate parameter θ, θ belongs to the identified set of Q if and only if Q vanishes at θ.","s":"∀ {Θ : Type u_1} {Q : Θ → ℝ} {θ : Θ}, θ ∈ Causalean.PartialID.CriterionSet.identifiedSet Q ↔ Q θ = 0","l":37,"f":"Causalean/PO/ID/Partial/CriterionSet/Basic.lean"},{"n":"Causalean.PartialID.CriterionSet.mem_levelSet","k":"theorem","m":"Causalean.PO.ID.Partial.CriterionSet.Basic","a":"PO/ID/Partial/CriterionSet","t":2,"d":"Membership in a level set is the same as having criterion value below the cutoff.","s":"∀ {Θ : Type u_1} {Qn : Θ → ℝ} {c : ℝ} {θ : Θ}, θ ∈ Causalean.PartialID.CriterionSet.levelSet Qn c ↔ Qn θ ≤ c","l":43,"f":"Causalean/PO/ID/Partial/CriterionSet/Basic.lean"},{"n":"Causalean.PartialID.CriterionSet.tendsto_hausdorffDist_levelSet","k":"theorem","m":"Causalean.PO.ID.Partial.CriterionSet.Consistency","a":"PO/ID/Partial/CriterionSet","t":1,"d":"Hausdorff consistency of the criterion-set estimator. Fix a positive linear-minorant modulus δ such that the population criterion Q satisfies a linear minorant of modulus δ. Suppose each sup-norm error tolerance is nonnegative, each cutoff is at least the corresponding error tolerance, the sample criterion Qn n is within tolerance ε n of Q in sup norm at every sample size n, the cutoffs tend to zero, and the error tolerances tend to zero. Then the Hausdorff distance between the level-set estimator {Qn n ≤ c n} and the population identified set {Q = 0} tends to zero as the sample size grows: H(levelSet Qnₙ cₙ, identifiedSet Q) → 0.","s":"∀ {Θ : Type u_1} [inst : PseudoMetricSpace Θ] {Q : Θ → ℝ} {Qn : ℕ → Θ → ℝ} {δ : ℝ} {c ε : ℕ → ℝ}, 0 < δ → Causalean.PartialID.CriterionSet.LinearMinorant Q δ → (∀ (n : ℕ), 0 ≤ ε n) → (∀ (n : ℕ), ε n ≤ c n) → (∀ (n : ℕ) (θ : Θ), |Qn n θ - Q θ| ≤ ε n) → Filter.Tendsto c Filter.atTop (nhds 0) → Filter.Tendsto ε Filter.atTop (nhds 0) → Filter.Tendsto (fun n => Causalean.PartialID.RandomSet.hausdorffDist (Causalean.PartialID.CriterionSet.levelSet (Qn n) (c n)) (Causalean.PartialID.CriterionSet.identifiedSet Q)) Filter.atTop (nhds 0)","l":143,"f":"Causalean/PO/ID/Partial/CriterionSet/Consistency.lean"},{"n":"Causalean.PartialID.IdentifiedInterval","k":"def","m":"Causalean.PO.ID.Partial.Basic","a":"PO/ID/Partial","t":1,"d":"Sharp identified interval: the set of all objective values attainable by a feasible parameter.","s":"{α : Type u_1} → (α → ℝ) → (α → Prop) → Set ℝ","l":25,"f":"Causalean/PO/ID/Partial/Basic.lean"},{"n":"Causalean.PartialID.Inference.RandomCoversIcc","k":"def","m":"Causalean.PO.ID.Partial.Inference.Basic","a":"PO/ID/Partial/Inference","t":1,"d":"Coverage of the identified set (Horowitz–Manski notion). The event that the random interval [A ω, B ω] contains the whole population interval [L, U]. Equivalently A ω ≤ L and U ≤ B ω.","s":"{Ω : Type u_1} → (Ω → ℝ) → (Ω → ℝ) → ℝ → ℝ → Set Ω","l":89,"f":"Causalean/PO/ID/Partial/Inference/Basic.lean"},{"n":"Causalean.PartialID.Inference.RandomCoversPoint","k":"def","m":"Causalean.PO.ID.Partial.Inference.Basic","a":"PO/ID/Partial/Inference","t":1,"d":"Coverage of the parameter (Imbens–Manski notion). The event that the random interval [A ω, B ω] contains the fixed real value θ.","s":"{Ω : Type u_1} → (Ω → ℝ) → (Ω → ℝ) → ℝ → Set Ω","l":78,"f":"Causalean/PO/ID/Partial/Inference/Basic.lean"},{"n":"Causalean.PartialID.Inference.bernstein_honest_ci_point_cover","k":"theorem","m":"Causalean.PO.ID.Partial.Inference.IntervalCI","a":"PO/ID/Partial/Inference","t":1,"d":"Finite-sample honest CI for the parameter (Bernstein). Variance-adaptive version: with measurable and integrable bounding statistics f_L, f_U, deviation bounds |f_L − ∫f_L| ≤ cL and |f_U − ∫f_U| ≤ cU holding a.s. for nonnegative constants cL, cU, and variance proxies σL, σU > 0 dominating the respective second central moments, for a positive sample size n and failure probabilities δ_L, δ_U in (0, 1], if the true parameter θ₀ lies in the population sandwich [∫ f_L dP, ∫ f_U dP], then widening the sample means by the corresponding Bernstein half-widths yields a random interval covering θ₀ with probability at least 1 − δ_L − δ_U.","s":"∀ {Ω : Type u_1} {X' : Type u_2} [inst : MeasurableSpace Ω] [inst_1 : MeasurableSpace X'] {μ : MeasureTheory.Measure Ω} {P : MeasureTheory.Measure X'} (S : Causalean.Stat.IIDSample Ω X' μ P) {fL fU : X' → ℝ}, Measurable fL → Measurable fU → MeasureTheory.Integrable fL P → MeasureTheory.Integrable fU P → ∀ {cL σL cU σU : ℝ}, 0 ≤ cL → 0 < σL → 0 ≤ cU → 0 < σU → (∀ᵐ (x : X') ∂P, |fL x - ∫ (y : X'), fL y ∂P| ≤ cL) → (∀ᵐ (x : X') ∂P, |fU x - ∫ (y : X'), fU y ∂P| ≤ cU) → ∫ (x : X'), (fL x - ∫ (y : X'), fL y ∂P) ^ 2 ∂P ≤ σL ^ 2 → ∫ (x : X'), (fU x - ∫ (y : X'), fU y ∂P) ^ 2 ∂P ≤ σU ^ 2 → ∀ (n : ℕ), 0 < n → ∀ {δL δU : ℝ}, 0 < δL → δL ≤ 1 → 0 < δU → δU ≤ 1 → ∀ {θ₀ : ℝ}, θ₀ ∈ Set.Icc (∫ (x : X'), fL x ∂P) (∫ (x : X'), fU x ∂P) → 1 - δL - δU ≤ μ.real (Causalean.PartialID.Inference.RandomCoversPoint (fun ω => S.sampleMean fL n ω - Causalean.Stat.Concentration.bernsteinCIHalfWidth cL σL n δL) (fun ω => S.sampleMean fU n ω + Causalean.Stat.Concentration.bernsteinCIHalfWidth cU σU n δU) θ₀)","l":136,"f":"Causalean/PO/ID/Partial/Inference/IntervalCI.lean"},{"n":"Causalean.PartialID.Inference.farEnd_vanishes_of_tendsto_atBot","k":"theorem","m":"Causalean.PO.ID.Partial.Inference.ImbensManski","a":"PO/ID/Partial/Inference","t":2,"d":"Far-endpoint failure vanishes (the Δ > 0 regime). If the normalized upper-endpoint statistic Sᵤ ⇒ N(0,1) and the (signed) threshold drifts to −∞, then the probability that Sᵤ,ₙ falls below it tends to 0. Concretely tₙ = −(c + bₙ) with bₙ → ∞ (the positive-width regime); the proof bounds the moving tail by a fixed Gaussian tail N(0,1)((-∞,-R]) made arbitrarily small via finite_measure_halfline_tails_small, then transported by portmanteau.","s":"∀ {Ω : Type u_1} [inst : MeasurableSpace Ω] {μ : MeasureTheory.Measure Ω} [inst_1 : MeasureTheory.IsProbabilityMeasure μ] {Su : ℕ → Ω → ℝ} (hSu : ∀ (n : ℕ), Measurable (Su n)), (Causalean.Stat.Tendsto_dist Su (Causalean.Stat.gaussianMeasure 0 1) μ fun n => Measurable.aemeasurable (hSu n)) → ∀ {t : ℕ → ℝ}, Filter.Tendsto t Filter.atTop Filter.atBot → Filter.Tendsto (fun n => (μ {ω | Su n ω < t n}).toReal) Filter.atTop (nhds 0)","l":106,"f":"Causalean/PO/ID/Partial/Inference/ImbensManski.lean"},{"n":"Causalean.PartialID.Inference.gaussianMeasure_zero_one_frontier_Iic","k":"theorem","m":"Causalean.PO.ID.Partial.Inference.ImbensManski","a":"PO/ID/Partial/Inference","t":2,"d":"The standard normal gives zero mass to the boundary {c} of the closed half-line (-∞, c]; the continuity-point fact needed to apply portmanteau to a one-sided threshold.","s":"∀ (c : ℝ), (Causalean.Stat.gaussianMeasure 0 1) (frontier (Set.Iic c)) = 0","l":98,"f":"Causalean/PO/ID/Partial/Inference/ImbensManski.lean"},{"n":"Causalean.PartialID.Inference.hoeffding_honest_ci_point_cover","k":"theorem","m":"Causalean.PO.ID.Partial.Inference.IntervalCI","a":"PO/ID/Partial/Inference","t":1,"d":"Finite-sample honest CI for the parameter (Hoeffding). The parameter-coverage corollary of hoeffding_honest_ci_set_cover: under the same setup — measurable bounding statistics f_L, f_U a.s. confined to [aL, bL] and [aU, bU] respectively, a positive sample size n, and failure probabilities δ_L, δ_U in (0, 1] — if the true scalar parameter θ₀ lies in the population sandwich [∫ f_L dP, ∫ f_U dP], then the same widened random interval covers θ₀ with probability at least 1 − δ_L − δ_U.","s":"∀ {Ω : Type u_1} {X' : Type u_2} [inst : MeasurableSpace Ω] [inst_1 : MeasurableSpace X'] {μ : MeasureTheory.Measure Ω} {P : MeasureTheory.Measure X'} (S : Causalean.Stat.IIDSample Ω X' μ P) {fL fU : X' → ℝ}, Measurable fL → Measurable fU → ∀ {aL bL aU bU : ℝ}, aL < bL → aU < bU → (∀ᵐ (x : X') ∂P, fL x ∈ Set.Icc aL bL) → (∀ᵐ (x : X') ∂P, fU x ∈ Set.Icc aU bU) → ∀ (n : ℕ), 0 < n → ∀ {δL δU : ℝ}, 0 < δL → δL ≤ 1 → 0 < δU → δU ≤ 1 → ∀ {θ₀ : ℝ}, θ₀ ∈ Set.Icc (∫ (x : X'), fL x ∂P) (∫ (x : X'), fU x ∂P) → 1 - δL - δU ≤ μ.real (Causalean.PartialID.Inference.RandomCoversPoint (fun ω => S.sampleMean fL n ω - Causalean.Stat.Concentration.hoeffdingCIHalfWidth aL bL n δL) (fun ω => S.sampleMean fU n ω + Causalean.Stat.Concentration.hoeffdingCIHalfWidth aU bU n δU) θ₀)","l":104,"f":"Causalean/PO/ID/Partial/Inference/IntervalCI.lean"},{"n":"Causalean.PartialID.Inference.hoeffding_honest_ci_set_cover","k":"theorem","m":"Causalean.PO.ID.Partial.Inference.IntervalCI","a":"PO/ID/Partial/Inference","t":1,"d":"Finite-sample honest CI for the identified set (Hoeffding). Given lower and upper bounding statistics f_L, f_U that are measurable, each confined a.s. to a known bounded range — f_L ∈ [aL, bL] with aL < bL, f_U ∈ [aU, bU] with aU < bU — at any positive sample size n and any pair of failure probabilities δ_L, δ_U in (0, 1], widening the sample means X̄ₙ(f_L) and X̄ₙ(f_U) outward by the matching Hoeffding half-widths produces a random interval that covers the whole identified interval [∫ f_L dP, ∫ f_U dP] with probability at least 1 − δ_L − δ_U.","s":"∀ {Ω : Type u_1} {X' : Type u_2} [inst : MeasurableSpace Ω] [inst_1 : MeasurableSpace X'] {μ : MeasureTheory.Measure Ω} {P : MeasureTheory.Measure X'} (S : Causalean.Stat.IIDSample Ω X' μ P) {fL fU : X' → ℝ}, Measurable fL → Measurable fU → ∀ {aL bL aU bU : ℝ}, aL < bL → aU < bU → (∀ᵐ (x : X') ∂P, fL x ∈ Set.Icc aL bL) → (∀ᵐ (x : X') ∂P, fU x ∈ Set.Icc aU bU) → ∀ (n : ℕ), 0 < n → ∀ {δL δU : ℝ}, 0 < δL → δL ≤ 1 → 0 < δU → δU ≤ 1 → 1 - δL - δU ≤ μ.real (Causalean.PartialID.Inference.RandomCoversIcc (fun ω => S.sampleMean fL n ω - Causalean.Stat.Concentration.hoeffdingCIHalfWidth aL bL n δL) (fun ω => S.sampleMean fU n ω + Causalean.Stat.Concentration.hoeffdingCIHalfWidth aU bU n δU) (∫ (x : X'), fL x ∂P) (∫ (x : X'), fU x ∂P))","l":71,"f":"Causalean/PO/ID/Partial/Inference/IntervalCI.lean"},{"n":"Causalean.PartialID.Inference.honest_ci_point_cover","k":"theorem","m":"Causalean.PO.ID.Partial.Inference.Basic","a":"PO/ID/Partial/Inference","t":1,"d":"Abstract honest confidence interval (parameter coverage). Let lo, hi be random lower/upper endpoint estimators of a population interval [L, U], with lo measurable and hi measurable. Suppose the true value θ lies in [L, U], the probability that the widened lower endpoint lo − wL overshoots L is at most δL, and the probability that the widened upper endpoint hi + wU undershoots U is at most δU. Then the widened random interval [lo − wL, hi + wU] covers θ with probability at least 1 − δL − δU.","s":"∀ {Ω : Type u_1} [inst : MeasurableSpace Ω] {μ : MeasureTheory.Measure Ω} [MeasureTheory.IsProbabilityMeasure μ] {lo hi : Ω → ℝ}, Measurable lo → Measurable hi → ∀ {L U wL wU δL δU θ : ℝ}, θ ∈ Set.Icc L U → μ.real {ω | L < lo ω - wL} ≤ δL → μ.real {ω | hi ω + wU < U} ≤ δU → 1 - δL - δU ≤ μ.real (Causalean.PartialID.Inference.RandomCoversPoint (fun ω => lo ω - wL) (fun ω => hi ω + wU) θ)","l":189,"f":"Causalean/PO/ID/Partial/Inference/Basic.lean"},{"n":"Causalean.PartialID.Inference.honest_ci_set_cover","k":"theorem","m":"Causalean.PO.ID.Partial.Inference.Basic","a":"PO/ID/Partial/Inference","t":1,"d":"Abstract honest confidence interval (set coverage). Let lo, hi be the (random) lower/upper endpoint estimators of the population bounds L, U, with lo measurable and hi measurable. Suppose the probability that the widened lower endpoint lo − wL overshoots L is at most δL, and the probability that the widened upper endpoint hi + wU undershoots U is at most δU. Then the widened random interval [lo − wL, hi + wU] covers the entire identified set [L, U] with probability at least 1 − δL − δU.","s":"∀ {Ω : Type u_1} [inst : MeasurableSpace Ω] {μ : MeasureTheory.Measure Ω} [MeasureTheory.IsProbabilityMeasure μ] {lo hi : Ω → ℝ}, Measurable lo → Measurable hi → ∀ {L U wL wU δL δU : ℝ}, μ.real {ω | L < lo ω - wL} ≤ δL → μ.real {ω | hi ω + wU < U} ≤ δU → 1 - δL - δU ≤ μ.real (Causalean.PartialID.Inference.RandomCoversIcc (fun ω => lo ω - wL) (fun ω => hi ω + wU) L U)","l":146,"f":"Causalean/PO/ID/Partial/Inference/Basic.lean"},{"n":"Causalean.PartialID.Inference.imbensManski_pointwise_coverage","k":"theorem","m":"Causalean.PO.ID.Partial.Inference.ImbensManski","a":"PO/ID/Partial/Inference","t":1,"d":"Lower-protected Imbens-Manski pointwise coverage. Let Sl, Su be the normalized lower- and upper-endpoint statistics for an interval-identified scalar parameter, with Sl n measurable for every sample size n, Su n measurable for every sample size n, and Sl converging in distribution to the standard normal law. Fix a one-sided critical value c. Suppose the normalized lower offset aL n, of the target point inside the identified interval, is nonnegative for every n, and the far-endpoint failure probability — that Su n falls below -(c + bU n) — tends to zero as n → ∞. Then for any positive tolerance ε, eventually, as n → ∞, the probability that the Imbens-Manski interval covers the target point is at least Φ(c) − ε, where Φ is the standard-normal distribution function.","s":"∀ {Ω : Type u_1} [inst : MeasurableSpace Ω] {μ : MeasureTheory.Measure Ω} [inst_1 : MeasureTheory.IsProbabilityMeasure μ] {Sl Su : ℕ → Ω → ℝ} (hSl : ∀ (n : ℕ), Measurable (Sl n)), (∀ (n : ℕ), Measurable (Su n)) → (Causalean.Stat.Tendsto_dist Sl (Causalean.Stat.gaussianMeasure 0 1) μ fun n => Measurable.aemeasurable (hSl n)) → ∀ (c : ℝ) {aL bU : ℕ → ℝ}, (∀ (n : ℕ), 0 ≤ aL n) → Filter.Tendsto (fun n => (μ {ω | Su n ω < -(c + bU n)}).toReal) Filter.atTop (nhds 0) → ∀ {ε : ℝ}, 0 < ε → ∀ᶠ (n : ℕ) in Filter.atTop, ((Causalean.Stat.gaussianMeasure 0 1) (Set.Iic c)).toReal - ε ≤ (μ {ω | Sl n ω ≤ c + aL n ∧ -(c + bU n) ≤ Su n ω}).toReal","l":171,"f":"Causalean/PO/ID/Partial/Inference/ImbensManski.lean"},{"n":"Causalean.PartialID.Inference.lowerOvershoot_subset_absMiss","k":"theorem","m":"Causalean.PO.ID.Partial.Inference.Basic","a":"PO/ID/Partial/Inference","t":2,"d":"A lower endpoint overshoot after widening implies the corresponding two-sided absolute-deviation miss.","s":"∀ {Ω : Type u_1} {lo : Ω → ℝ} {L wL : ℝ}, {ω | L < lo ω - wL} ⊆ {ω | wL ≤ |lo ω - L|}","l":125,"f":"Causalean/PO/ID/Partial/Inference/Basic.lean"},{"n":"Causalean.PartialID.Inference.randomCoversIcc_compl_subset","k":"theorem","m":"Causalean.PO.ID.Partial.Inference.Basic","a":"PO/ID/Partial/Inference","t":2,"d":"The complement of the set-coverage event is contained in the union of the two one-sided endpoint-failure events: lower overshoot after widening or upper undershoot after widening.","s":"∀ {Ω : Type u_1} {lo hi : Ω → ℝ} {L U wL wU : ℝ}, (Causalean.PartialID.Inference.RandomCoversIcc (fun ω => lo ω - wL) (fun ω => hi ω + wU) L U)ᶜ ⊆ {ω | L < lo ω - wL} ∪ {ω | hi ω + wU < U}","l":110,"f":"Causalean/PO/ID/Partial/Inference/Basic.lean"},{"n":"Causalean.PartialID.Inference.randomCoversIcc_subset_randomCoversPoint","k":"theorem","m":"Causalean.PO.ID.Partial.Inference.Basic","a":"PO/ID/Partial/Inference","t":2,"d":"Set coverage ⟹ point coverage (the easy half of Imbens–Manski 2004, Lemma 1). If the true value θ lies in the identified set [L, U], then every sample realization whose random interval covers all of [L, U] also covers θ. Consequently a confidence interval that is honest for the *set* is automatically honest for the *parameter* — which is why parameter-coverage intervals can be no larger, and generically strictly smaller.","s":"∀ {Ω : Type u_1} (A B : Ω → ℝ) {L U θ : ℝ}, θ ∈ Set.Icc L U → Causalean.PartialID.Inference.RandomCoversIcc A B L U ⊆ Causalean.PartialID.Inference.RandomCoversPoint A B θ","l":95,"f":"Causalean/PO/ID/Partial/Inference/Basic.lean"},{"n":"Causalean.PartialID.Inference.upperUndershoot_subset_absMiss","k":"theorem","m":"Causalean.PO.ID.Partial.Inference.Basic","a":"PO/ID/Partial/Inference","t":2,"d":"An upper endpoint undershoot after widening implies the corresponding two-sided absolute-deviation miss.","s":"∀ {Ω : Type u_1} {hi : Ω → ℝ} {U wU : ℝ}, {ω | hi ω + wU < U} ⊆ {ω | wU ≤ |hi ω - U|}","l":135,"f":"Causalean/PO/ID/Partial/Inference/Basic.lean"},{"n":"Causalean.PartialID.RandomSet.IsBody","k":"structure","m":"Causalean.PO.ID.Partial.RandomSet.SetValued","a":"PO/ID/Partial/RandomSet","t":1,"d":"A body: a nonempty, compact, convex subset of E — the value type 𝒦ₖ(E) of a Beresteanu–Molinari set-valued random variable. Compactness makes supportFn C d well-defined (the linear functional attains its sup); convexity is what lets the support function characterise the set.","s":"{E : Type u_1} → [inst : NormedAddCommGroup E] → [InnerProductSpace ℝ E] → Set E → Prop","l":63,"f":"Causalean/PO/ID/Partial/RandomSet/SetValued.lean"},{"n":"Causalean.PartialID.RandomSet.IsBody.bddAbove","k":"theorem","m":"Causalean.PO.ID.Partial.RandomSet.SetValued","a":"PO/ID/Partial/RandomSet","t":2,"d":"The support function of a body is bounded above, hence meaningful.","s":"∀ {E : Type u_1} [inst : NormedAddCommGroup E] [inst_1 : InnerProductSpace ℝ E] {C : Set E}, Causalean.PartialID.RandomSet.IsBody C → ∀ (d : E), BddAbove ((fun x => inner ℝ d x) '' C)","l":82,"f":"Causalean/PO/ID/Partial/RandomSet/SetValued.lean"},{"n":"Causalean.PartialID.RandomSet.IsSelection","k":"def","m":"Causalean.PO.ID.Partial.RandomSet.Interval","a":"PO/ID/Partial/RandomSet","t":1,"d":"f is a measurable everywhere-selection of the interval random set [L, U]: it is measurable and L ω ≤ f ω ≤ U ω for every ω.","s":"{Ω : Type u_1} → [MeasurableSpace Ω] → (Ω → ℝ) → (Ω → ℝ) → (Ω → ℝ) → Prop","l":67,"f":"Causalean/PO/ID/Partial/RandomSet/Interval.lean"},{"n":"Causalean.PartialID.RandomSet.artstein_supportFn_neg_one","k":"theorem","m":"Causalean.PO.ID.Partial.RandomSet.Aumann","a":"PO/ID/Partial/RandomSet","t":1,"d":"Artstein identity, d = 1, direction −1. For measurable lower and upper endpoint functions L, U of an interval-valued random set that are integrable and satisfy L pointwise at most U, the support function of the Aumann (selection) expectation of the random interval [L, U] in the direction −1, namely the negated lower endpoint, equals the expectation of the support function of the random interval itself in that direction: s(−1, E[F]) = E[s(−1, F)].","s":"∀ {Ω : Type u_1} [inst : MeasurableSpace Ω] {μ : MeasureTheory.Measure Ω} {L U : Ω → ℝ}, Measurable L → Measurable U → MeasureTheory.Integrable L μ → MeasureTheory.Integrable U μ → (∀ (ω : Ω), L ω ≤ U ω) → Causalean.PartialID.supportFn (Causalean.PartialID.RandomSet.selectionExpectation L U μ) (-1) = ∫ (ω : Ω), Causalean.PartialID.supportFn (Causalean.PartialID.RandomSet.randomInterval L U ω) (-1) ∂μ","l":113,"f":"Causalean/PO/ID/Partial/RandomSet/Aumann.lean"},{"n":"Causalean.PartialID.RandomSet.artstein_supportFn_one","k":"theorem","m":"Causalean.PO.ID.Partial.RandomSet.Aumann","a":"PO/ID/Partial/RandomSet","t":1,"d":"Artstein identity, d = 1, direction +1. For measurable lower and upper endpoint functions L, U of an interval-valued random set that are integrable and satisfy L pointwise at most U, the support function of the Aumann (selection) expectation of the random interval [L, U] in the direction +1, namely the upper endpoint, equals the expectation of the support function of the random interval itself in that direction: s(+1, E[F]) = E[s(+1, F)].","s":"∀ {Ω : Type u_1} [inst : MeasurableSpace Ω] {μ : MeasureTheory.Measure Ω} {L U : Ω → ℝ}, Measurable L → Measurable U → MeasureTheory.Integrable L μ → MeasureTheory.Integrable U μ → (∀ (ω : Ω), L ω ≤ U ω) → Causalean.PartialID.supportFn (Causalean.PartialID.RandomSet.selectionExpectation L U μ) 1 = ∫ (ω : Ω), Causalean.PartialID.supportFn (Causalean.PartialID.RandomSet.randomInterval L U ω) 1 ∂μ","l":97,"f":"Causalean/PO/ID/Partial/RandomSet/Aumann.lean"},{"n":"Causalean.PartialID.RandomSet.bddAbove_inner_image","k":"theorem","m":"Causalean.PO.ID.Partial.RandomSet.SetValued","a":"PO/ID/Partial/RandomSet","t":2,"d":"On a compact set the linear functional ⟪d, ·⟫ is bounded above.","s":"∀ {E : Type u_1} [inst : NormedAddCommGroup E] [inst_1 : InnerProductSpace ℝ E] {C : Set E}, IsCompact C → ∀ (d : E), BddAbove ((fun x => inner ℝ d x) '' C)","l":72,"f":"Causalean/PO/ID/Partial/RandomSet/SetValued.lean"},{"n":"Causalean.PartialID.RandomSet.continuous_dirStat","k":"theorem","m":"Causalean.PO.ID.Partial.RandomSet.IntervalInference","a":"PO/ID/Partial/RandomSet","t":2,"d":"The directed-Hausdorff endpoint functional is continuous.","s":"Continuous Causalean.PartialID.RandomSet.dirStat","l":184,"f":"Causalean/PO/ID/Partial/RandomSet/IntervalInference.lean"},{"n":"Causalean.PartialID.RandomSet.continuous_maxAbs","k":"theorem","m":"Causalean.PO.ID.Partial.RandomSet.IntervalCLT","a":"PO/ID/Partial/RandomSet","t":2,"d":"The max-absolute-value functional on endpoint deviations is continuous.","s":"Continuous Causalean.PartialID.RandomSet.maxAbs","l":76,"f":"Causalean/PO/ID/Partial/RandomSet/IntervalCLT.lean"},{"n":"Causalean.PartialID.RandomSet.continuous_maxAbsK","k":"theorem","m":"Causalean.PO.ID.Partial.RandomSet.SupportProcess","a":"PO/ID/Partial/RandomSet","t":2,"d":"The grid supremum statistic is continuous.","s":"∀ {k : ℕ} [inst : NeZero k], Continuous Causalean.PartialID.RandomSet.maxAbsK","l":71,"f":"Causalean/PO/ID/Partial/RandomSet/SupportProcess.lean"},{"n":"Causalean.PartialID.RandomSet.coverage_event_eq","k":"theorem","m":"Causalean.PO.ID.Partial.RandomSet.IntervalInference","a":"PO/ID/Partial/RandomSet","t":1,"d":"Coverage event identity (Beresteanu–Molinari coverage corollary). Fix the population identified interval E[Y] = [μL, μU] with μL ≤ μU, the sample-mean interval Ȳₙ(ω) = [yl(ω), yu(ω)] with yl(ω) ≤ yu(ω) for every ω, a sample size of at least one, and a nonnegative critical value c. With the BM bandwidth r = c/√n, the (one-sided) coverage event {E[Y] ⊆ Uₙ} equals the event {√n · dᴴ(E[Y], Ȳₙ) ≤ c} on the directed Hausdorff statistic. Hence the coverage probability is μ {ω | √n · dᴴ(E[Y], Ȳₙ(ω)) ≤ c}.","s":"∀ {Ω : Type u_1} (μL μU : ℝ) (yl yu : Ω → ℝ) (n : ℕ), 1 ≤ n → ∀ (c : ℝ), 0 ≤ c → μL ≤ μU → (∀ (ω : Ω), yl ω ≤ yu ω) → {ω | Set.Icc μL μU ⊆ Causalean.PartialID.RandomSet.dilate (Set.Icc (yl ω) (yu ω)) (c / √↑n)} = {ω | √↑n * Causalean.PartialID.RandomSet.directedHausdorff (Set.Icc μL μU) (Set.Icc (yl ω) (yu ω)) ≤ c}","l":139,"f":"Causalean/PO/ID/Partial/RandomSet/IntervalInference.lean"},{"n":"Causalean.PartialID.RandomSet.dilate","k":"def","m":"Causalean.PO.ID.Partial.RandomSet.IntervalInference","a":"PO/ID/Partial/RandomSet","t":1,"d":"Minkowski dilation of a real set by a radius r: A ⊕ [−r, r] = {a + t | a ∈ A, |t| ≤ r}. In Beresteanu–Molinari this is the confidence region Uₙ = Ȳₙ ⊕ B(0, r) built around the estimated interval.","s":"Set ℝ → ℝ → Set ℝ","l":66,"f":"Causalean/PO/ID/Partial/RandomSet/IntervalInference.lean"},{"n":"Causalean.PartialID.RandomSet.dilate_Icc","k":"theorem","m":"Causalean.PO.ID.Partial.RandomSet.IntervalInference","a":"PO/ID/Partial/RandomSet","t":2,"d":"Dilation of an interval (Beresteanu–Molinari Uₙ). For r ≥ 0, [a,b] ⊕ [−r,r] = [a−r, b+r]: every endpoint is pushed out by the radius.","s":"∀ {a b r : ℝ}, a ≤ b → 0 ≤ r → Causalean.PartialID.RandomSet.dilate (Set.Icc a b) r = Set.Icc (a - r) (b + r)","l":72,"f":"Causalean/PO/ID/Partial/RandomSet/IntervalInference.lean"},{"n":"Causalean.PartialID.RandomSet.dirStat","k":"def","m":"Causalean.PO.ID.Partial.RandomSet.IntervalInference","a":"PO/ID/Partial/RandomSet","t":1,"d":"The directed-Hausdorff functional dirStat w = max 0 (max w₀ (−w₁)) on ℝ². On the centered endpoint normalised sum it is √n · dᴴ(E[Y], Ȳₙ).","s":"EuclideanSpace ℝ (Fin 2) → ℝ","l":180,"f":"Causalean/PO/ID/Partial/RandomSet/IntervalInference.lean"},{"n":"Causalean.PartialID.RandomSet.dirStat_normalizedSum_eq","k":"theorem","m":"Causalean.PO.ID.Partial.RandomSet.IntervalInference","a":"PO/ID/Partial/RandomSet","t":2,"d":"Directed Hausdorff bridge. dirStat of the centered endpoint normalised sum equals √n · dᴴ(E[Y], Ȳₙ).","s":"∀ {Ω : Type u_1} {X : Type u_2} [inst : MeasurableSpace Ω] [inst_1 : MeasurableSpace X] {μ : MeasureTheory.Measure Ω} {P : MeasureTheory.Measure X} (S : Causalean.Stat.IIDSample Ω X μ P) (yL yU : X → ℝ), (∀ (z : X), yL z ≤ yU z) → MeasureTheory.Integrable yL P → MeasureTheory.Integrable yU P → ∀ (n : ℕ) (ω : Ω), Causalean.PartialID.RandomSet.dirStat (Causalean.Stat.IsAsymLinearVec.normalizedSum S (Causalean.PartialID.RandomSet.intervalIFVec yL yU P) (fun m => Finset.range m) n ω) = √↑n * Causalean.PartialID.RandomSet.directedHausdorff (Set.Icc (∫ (x : X), yL x ∂P) (∫ (x : X), yU x ∂P)) (Set.Icc (Causalean.PartialID.RandomSet.sampleMean S yL n ω) (Causalean.PartialID.RandomSet.sampleMean S yU n ω))","l":248,"f":"Causalean/PO/ID/Partial/RandomSet/IntervalInference.lean"},{"n":"Causalean.PartialID.RandomSet.directedHausdorff","k":"def","m":"Causalean.PO.ID.Partial.RandomSet.Hausdorff","a":"PO/ID/Partial/RandomSet","t":1,"d":"Directed (one-sided) Hausdorff distance from A to B: the supremum over points of A of their distance to B. Real-valued (via Metric.infDist); on the empty image sSup returns 0.","s":"{α : Type u_1} → [PseudoMetricSpace α] → Set α → Set α → ℝ","l":59,"f":"Causalean/PO/ID/Partial/RandomSet/Hausdorff.lean"},{"n":"Causalean.PartialID.RandomSet.directedHausdorff_Icc","k":"theorem","m":"Causalean.PO.ID.Partial.RandomSet.Hausdorff","a":"PO/ID/Partial/RandomSet","t":2,"d":"Directed Hausdorff distance between intervals. dᴴ([a,b], [c,d]) = max 0 (max (c − a) (b − d)) (for a ≤ b, c ≤ d): the worst over-reach of [a,b] beyond [c,d], achieved at the endpoints.","s":"∀ {a b c d : ℝ}, a ≤ b → c ≤ d → Causalean.PartialID.RandomSet.directedHausdorff (Set.Icc a b) (Set.Icc c d) = max 0 (max (c - a) (b - d))","l":133,"f":"Causalean/PO/ID/Partial/RandomSet/Hausdorff.lean"},{"n":"Causalean.PartialID.RandomSet.directedRegion_coverage","k":"theorem","m":"Causalean.PO.ID.Partial.RandomSet.IntervalInference","a":"PO/ID/Partial/RandomSet","t":1,"d":"Asymptotic coverage of the directed confidence region. For an i.i.d. sample with interval endpoints yL, yU satisfying the lower endpoint pointwise at most the upper endpoint and both integrable, assume the centered endpoint influence function is measurable with finite second moment and mean zero, its normalized partial sums are almost-everywhere measurable at every sample size, and fix a nonnegative bandwidth constant c that is a continuity point of the directed-Hausdorff Gaussian limit law. With population identified interval E[Y] = [E y_L, E y_U], sample-mean interval Ȳₙ, and the BM bandwidth c/√n, the coverage probability of the whole identified set converges to the limit-law mass of (-∞, c]:","s":"∀ {Ω : Type u_1} {X : Type u_2} [inst : MeasurableSpace Ω] [inst_1 : MeasurableSpace X] {μ : MeasureTheory.Measure Ω} {P : MeasureTheory.Measure X} [MeasureTheory.IsProbabilityMeasure μ] (S : Causalean.Stat.IIDSample Ω X μ P) (yL yU : X → ℝ), (∀ (z : X), yL z ≤ yU z) → MeasureTheory.Integrable yL P → MeasureTheory.Integrable yU P → ∀ (hψ : Measurable (Causalean.PartialID.RandomSet.intervalIFVec yL yU P)) (hvar : MeasureTheory.Integrable (fun x => ‖Causalean.PartialID.RandomSet.intervalIFVec yL yU P x‖ ^ 2) P), ∫ (x : X), Causalean.PartialID.RandomSet.intervalIFVec yL yU P x ∂P = 0 → (∀ (n : ℕ), AEMeasurable (Causalean.Stat.IsAsymLinearVec.normalizedSum S (Causalean.PartialID.RandomSet.intervalIFVec yL yU P) (fun m => Finset.range m) n) μ) → ∀ {c : ℝ}, 0 ≤ c → (MeasureTheory.Measure.map Causalean.PartialID.RandomSet.dirStat (Causalean.Stat.gaussianLimit hψ hvar)) {c} = 0 → Filter.Tendsto (fun n => μ {ω | Set.Icc (∫ (x : X), yL x ∂P) (∫ (x : X), yU x ∂P) ⊆ Causalean.PartialID.RandomSet.dilate (Set.Icc (Causalean.PartialID.RandomSet.sampleMean S yL n ω) (Causalean.PartialID.RandomSet.sampleMean S yU n ω)) (c / √↑n)}) Filter.atTop (nhds ((MeasureTheory.Measure.map Causalean.PartialID.RandomSet.dirStat (Causalean.Stat.gaussianLimit hψ hvar)) (Set.Iic c)))","l":270,"f":"Causalean/PO/ID/Partial/RandomSet/IntervalInference.lean"},{"n":"Causalean.PartialID.RandomSet.eucl₂","k":"def","m":"Causalean.PO.ID.Partial.RandomSet.IntervalCLT","a":"PO/ID/Partial/RandomSet","t":1,"d":"A pair of real endpoint deviations is viewed as a two-dimensional Euclidean vector.","s":"(Fin 2 → ℝ) → EuclideanSpace ℝ (Fin 2)","l":66,"f":"Causalean/PO/ID/Partial/RandomSet/IntervalCLT.lean"},{"n":"Causalean.PartialID.RandomSet.eucl₂_apply","k":"theorem","m":"Causalean.PO.ID.Partial.RandomSet.IntervalCLT","a":"PO/ID/Partial/RandomSet","t":2,"d":"Coordinate access for eucl₂: (eucl₂ v) j = v j.","s":"∀ (v : Fin 2 → ℝ) (j : Fin 2), (Causalean.PartialID.RandomSet.eucl₂ v).ofLp j = v j","l":207,"f":"Causalean/PO/ID/Partial/RandomSet/IntervalCLT.lean"},{"n":"Causalean.PartialID.RandomSet.gridTestReject","k":"def","m":"Causalean.PO.ID.Partial.RandomSet.GridTest","a":"PO/ID/Partial/RandomSet","t":1,"d":"The rejection region of the finite-grid test at critical value c: reject H₀ : E[F] = EF when the statistic Tₙ exceeds c.","s":"{Ω : Type u_1} → {X : Type u_2} → [inst : MeasurableSpace Ω] → [inst_1 : MeasurableSpace X] → {μ : MeasureTheory.Measure Ω} → {P : MeasureTheory.Measure X} → {k : ℕ} → [NeZero k] → {V : Type u_3} → [inst_3 : NormedAddCommGroup V] → [InnerProductSpace ℝ V] → Causalean.Stat.IIDSample Ω X μ P → (X → Set V) → Set V → (Fin k → V) → ℕ → ℝ → Set Ω","l":91,"f":"Causalean/PO/ID/Partial/RandomSet/GridTest.lean"},{"n":"Causalean.PartialID.RandomSet.gridTestStat","k":"def","m":"Causalean.PO.ID.Partial.RandomSet.GridTest","a":"PO/ID/Partial/RandomSet","t":1,"d":"The finite-grid test statistic is the grid supremum of the normalized centered support-process sum.","s":"{Ω : Type u_1} → {X : Type u_2} → [inst : MeasurableSpace Ω] → [inst_1 : MeasurableSpace X] → {μ : MeasureTheory.Measure Ω} → {P : MeasureTheory.Measure X} → {k : ℕ} → [NeZero k] → {V : Type u_3} → [inst_3 : NormedAddCommGroup V] → [InnerProductSpace ℝ V] → Causalean.Stat.IIDSample Ω X μ P → (X → Set V) → Set V → (Fin k → V) → ℕ → Ω → ℝ","l":75,"f":"Causalean/PO/ID/Partial/RandomSet/GridTest.lean"},{"n":"Causalean.PartialID.RandomSet.gridTestStat_clt","k":"theorem","m":"Causalean.PO.ID.Partial.RandomSet.GridTest","a":"PO/ID/Partial/RandomSet","t":1,"d":"The normalized finite-grid support-process statistic converges in distribution to the grid supremum of its Gaussian limit.","s":"∀ {Ω : Type u_1} {X : Type u_2} [inst : MeasurableSpace Ω] [inst_1 : MeasurableSpace X] {μ : MeasureTheory.Measure Ω} {P : MeasureTheory.Measure X} [inst_2 : MeasureTheory.IsProbabilityMeasure μ] {k : ℕ} [inst_3 : NeZero k] {V : Type u_3} [inst_4 : NormedAddCommGroup V] [inst_5 : InnerProductSpace ℝ V] (S : Causalean.Stat.IIDSample Ω X μ P) (F : X → Set V) (EF : Set V) (p : Fin k → V) (hψ : Measurable (Causalean.PartialID.RandomSet.supportProcess F EF p)) (hvar : MeasureTheory.Integrable (fun x => ‖Causalean.PartialID.RandomSet.supportProcess F EF p x‖ ^ 2) P), ∫ (x : X), Causalean.PartialID.RandomSet.supportProcess F EF p x ∂P = 0 → ∀ (hSum_meas : ∀ (n : ℕ), AEMeasurable (Causalean.Stat.IsAsymLinearVec.normalizedSum S (Causalean.PartialID.RandomSet.supportProcess F EF p) (fun m => Finset.range m) n) μ), Causalean.Stat.Tendsto_dist_vec (Causalean.PartialID.RandomSet.gridTestStat S F EF p) (MeasureTheory.Measure.map Causalean.PartialID.RandomSet.maxAbsK (Causalean.Stat.gaussianLimit hψ hvar)) μ fun n => Measurable.comp_aemeasurable Causalean.PartialID.RandomSet.measurable_maxAbsK (hSum_meas n)","l":110,"f":"Causalean/PO/ID/Partial/RandomSet/GridTest.lean"},{"n":"Causalean.PartialID.RandomSet.gridTest_asymptotic_level","k":"theorem","m":"Causalean.PO.ID.Partial.RandomSet.GridTest","a":"PO/ID/Partial/RandomSet","t":1,"d":"Asymptotic level of the finite-grid tail test. At any continuity point c of the Gaussian limit law of the grid test statistic — i.e. the limit law assigns zero mass to {c} — the tail (rejection) probability of the normalized finite-grid support-process statistic converges to the corresponding tail mass of the Gaussian limit law: μ (gridTestReject … c) → L(c, ∞).","s":"∀ {Ω : Type u_1} {X : Type u_2} [inst : MeasurableSpace Ω] [inst_1 : MeasurableSpace X] {μ : MeasureTheory.Measure Ω} {P : MeasureTheory.Measure X} [MeasureTheory.IsProbabilityMeasure μ] {k : ℕ} [inst_3 : NeZero k] {V : Type u_3} [inst_4 : NormedAddCommGroup V] [inst_5 : InnerProductSpace ℝ V] (S : Causalean.Stat.IIDSample Ω X μ P) (F : X → Set V) (EF : Set V) (p : Fin k → V) (hψ : Measurable (Causalean.PartialID.RandomSet.supportProcess F EF p)) (hvar : MeasureTheory.Integrable (fun x => ‖Causalean.PartialID.RandomSet.supportProcess F EF p x‖ ^ 2) P), ∫ (x : X), Causalean.PartialID.RandomSet.supportProcess F EF p x ∂P = 0 → (∀ (n : ℕ), AEMeasurable (Causalean.Stat.IsAsymLinearVec.normalizedSum S (Causalean.PartialID.RandomSet.supportProcess F EF p) (fun m => Finset.range m) n) μ) → ∀ {c : ℝ}, (MeasureTheory.Measure.map Causalean.PartialID.RandomSet.maxAbsK (Causalean.Stat.gaussianLimit hψ hvar)) {c} = 0 → Filter.Tendsto (fun n => μ (Causalean.PartialID.RandomSet.gridTestReject S F EF p n c)) Filter.atTop (nhds ((MeasureTheory.Measure.map Causalean.PartialID.RandomSet.maxAbsK (Causalean.Stat.gaussianLimit hψ hvar)) (Set.Ioi c)))","l":126,"f":"Causalean/PO/ID/Partial/RandomSet/GridTest.lean"},{"n":"Causalean.PartialID.RandomSet.hausdorffDist","k":"def","m":"Causalean.PO.ID.Partial.RandomSet.Hausdorff","a":"PO/ID/Partial/RandomSet","t":1,"d":"Symmetric Hausdorff distance H(A,B) = max(dᴴ(A,B), dᴴ(B,A)).","s":"{α : Type u_1} → [PseudoMetricSpace α] → Set α → Set α → ℝ","l":66,"f":"Causalean/PO/ID/Partial/RandomSet/Hausdorff.lean"},{"n":"Causalean.PartialID.RandomSet.hausdorffDist_Icc","k":"theorem","m":"Causalean.PO.ID.Partial.RandomSet.Hausdorff","a":"PO/ID/Partial/RandomSet","t":1,"d":"The geometric keystone (Beresteanu–Molinari eq. (A.1), d = 1). For real numbers a ≤ b and c ≤ d forming two well-ordered closed intervals, the symmetric Hausdorff distance between [a,b] and [c,d] equals the larger of the two endpoint gaps: H([a,b], [c,d]) = max(|a − c|, |b − d|).","s":"∀ {a b c d : ℝ}, a ≤ b → c ≤ d → Causalean.PartialID.RandomSet.hausdorffDist (Set.Icc a b) (Set.Icc c d) = max |a - c| |b - d|","l":171,"f":"Causalean/PO/ID/Partial/RandomSet/Hausdorff.lean"},{"n":"Causalean.PartialID.RandomSet.hausdorffDist_Icc_eq_supportFn","k":"theorem","m":"Causalean.PO.ID.Partial.RandomSet.Aumann","a":"PO/ID/Partial/RandomSet","t":1,"d":"The d = 1 Hörmander identity (Beresteanu–Molinari eq. (A.1)). For real numbers a ≤ b and c ≤ d forming two well-ordered closed intervals, the Hausdorff distance between the intervals [a,b] and [c,d] equals the largest, over the two unit directions +1 and −1, of the absolute difference between their support functions in that direction.","s":"∀ {a b c d : ℝ}, a ≤ b → c ≤ d → Causalean.PartialID.RandomSet.hausdorffDist (Set.Icc a b) (Set.Icc c d) = max |Causalean.PartialID.supportFn (Set.Icc a b) (-1) - Causalean.PartialID.supportFn (Set.Icc c d) (-1)| |Causalean.PartialID.supportFn (Set.Icc a b) 1 - Causalean.PartialID.supportFn (Set.Icc c d) 1|","l":84,"f":"Causalean/PO/ID/Partial/RandomSet/Aumann.lean"},{"n":"Causalean.PartialID.RandomSet.infDist_Icc","k":"theorem","m":"Causalean.PO.ID.Partial.RandomSet.Hausdorff","a":"PO/ID/Partial/RandomSet","t":2,"d":"Distance from a real point to a closed interval. For c ≤ d, infDist x [c,d] = max 0 (max (c − x) (x − d)) — zero inside the interval, and the signed gap to the nearer endpoint outside it.","s":"∀ {c d : ℝ}, c ≤ d → ∀ (x : ℝ), Metric.infDist x (Set.Icc c d) = max 0 (max (c - x) (x - d))","l":71,"f":"Causalean/PO/ID/Partial/RandomSet/Hausdorff.lean"},{"n":"Causalean.PartialID.RandomSet.instIsProbabilityMeasureRealMapEuclideanSpaceFinMaxAbsKGaussianLimit","k":"instance","m":"Causalean.PO.ID.Partial.RandomSet.SupportProcess","a":"PO/ID/Partial/RandomSet","t":2,"d":"The finite-grid support statistic has a probability measure as its Gaussian pushforward limit law.","s":"∀ {X : Type u_2} [inst : MeasurableSpace X] {P : MeasureTheory.Measure X} {k : ℕ} [inst_1 : NeZero k] {ψ : X → EuclideanSpace ℝ (Fin k)} (hψ : Measurable ψ) (hvar : MeasureTheory.Integrable (fun x => ‖ψ x‖ ^ 2) P), MeasureTheory.IsProbabilityMeasure (MeasureTheory.Measure.map Causalean.PartialID.RandomSet.maxAbsK (Causalean.Stat.gaussianLimit hψ hvar))","l":85,"f":"Causalean/PO/ID/Partial/RandomSet/SupportProcess.lean"},{"n":"Causalean.PartialID.RandomSet.instIsProbabilityMeasureRealMapEuclideanSpaceFinOfNatNatDirStatGaussianLimit","k":"instance","m":"Causalean.PO.ID.Partial.RandomSet.IntervalInference","a":"PO/ID/Partial/RandomSet","t":2,"d":"The directed limit law is a probability measure.","s":"∀ {X : Type u_2} [inst : MeasurableSpace X] {P : MeasureTheory.Measure X} {ψ : X → EuclideanSpace ℝ (Fin 2)} (hψ : Measurable ψ) (hvar : MeasureTheory.Integrable (fun x => ‖ψ x‖ ^ 2) P), MeasureTheory.IsProbabilityMeasure (MeasureTheory.Measure.map Causalean.PartialID.RandomSet.dirStat (Causalean.Stat.gaussianLimit hψ hvar))","l":195,"f":"Causalean/PO/ID/Partial/RandomSet/IntervalInference.lean"},{"n":"Causalean.PartialID.RandomSet.instIsProbabilityMeasureRealMapEuclideanSpaceFinOfNatNatMaxAbsGaussianLimit","k":"instance","m":"Causalean.PO.ID.Partial.RandomSet.IntervalCLT","a":"PO/ID/Partial/RandomSet","t":2,"d":"The limit law of the Hausdorff statistic is a probability measure (pushforward of the Gaussian limit by the continuous maxAbs).","s":"∀ {X : Type u_2} [inst : MeasurableSpace X] {P : MeasureTheory.Measure X} {ψ : X → EuclideanSpace ℝ (Fin 2)} (hψ : Measurable ψ) (hvar : MeasureTheory.Integrable (fun x => ‖ψ x‖ ^ 2) P), MeasureTheory.IsProbabilityMeasure (MeasureTheory.Measure.map Causalean.PartialID.RandomSet.maxAbs (Causalean.Stat.gaussianLimit hψ hvar))","l":87,"f":"Causalean/PO/ID/Partial/RandomSet/IntervalCLT.lean"},{"n":"Causalean.PartialID.RandomSet.integral_le_integral_of_le","k":"theorem","m":"Causalean.PO.ID.Partial.RandomSet.Interval","a":"PO/ID/Partial/RandomSet","t":2,"d":"Monotonicity of the endpoint integrals (used to order the reported bounds).","s":"∀ {Ω : Type u_1} [inst : MeasurableSpace Ω] {μ : MeasureTheory.Measure Ω} {L U : Ω → ℝ}, MeasureTheory.Integrable L μ → MeasureTheory.Integrable U μ → (∀ (ω : Ω), L ω ≤ U ω) → ∫ (ω : Ω), L ω ∂μ ≤ ∫ (ω : Ω), U ω ∂μ","l":126,"f":"Causalean/PO/ID/Partial/RandomSet/Interval.lean"},{"n":"Causalean.PartialID.RandomSet.intervalIFVec","k":"def","m":"Causalean.PO.ID.Partial.RandomSet.IntervalCLT","a":"PO/ID/Partial/RandomSet","t":1,"d":"The centered interval-endpoint influence function ψ(z) = (y_L(z) − E y_L, y_U(z) − E y_U), valued in ℝ². Its vector normalised sum is the centered-and-scaled endpoint pair √n((ȳ_{nL}, ȳ_{nU}) − (E y_L, E y_U)).","s":"{X : Type u_2} → [inst : MeasurableSpace X] → (X → ℝ) → (X → ℝ) → MeasureTheory.Measure X → X → EuclideanSpace ℝ (Fin 2)","l":134,"f":"Causalean/PO/ID/Partial/RandomSet/IntervalCLT.lean"},{"n":"Causalean.PartialID.RandomSet.intervalIFVec_hHmeas","k":"theorem","m":"Causalean.PO.ID.Partial.RandomSet.IntervalCLT","a":"PO/ID/Partial/RandomSet","t":2,"d":"The scaled Hausdorff statistic is AEMeasurable for each n. Equals maxAbs ∘ (normalised sum) everywhere by maxAbs_normalizedSum_eq.","s":"∀ {Ω : Type u_1} {X : Type u_2} [inst : MeasurableSpace Ω] [inst_1 : MeasurableSpace X] {μ : MeasureTheory.Measure Ω} {P : MeasureTheory.Measure X} (S : Causalean.Stat.IIDSample Ω X μ P) (yL yU : X → ℝ), (∀ (z : X), yL z ≤ yU z) → Measurable yL → Measurable yU → MeasureTheory.Integrable yL P → MeasureTheory.Integrable yU P → ∀ (n : ℕ), AEMeasurable (fun ω => √↑n * Causalean.PartialID.RandomSet.hausdorffDist (Set.Icc (Causalean.PartialID.RandomSet.sampleMean S yL n ω) (Causalean.PartialID.RandomSet.sampleMean S yU n ω)) (Set.Icc (∫ (x : X), yL x ∂P) (∫ (x : X), yU x ∂P))) μ","l":315,"f":"Causalean/PO/ID/Partial/RandomSet/IntervalCLT.lean"},{"n":"Causalean.PartialID.RandomSet.intervalIFVec_integrable","k":"theorem","m":"Causalean.PO.ID.Partial.RandomSet.IntervalCLT","a":"PO/ID/Partial/RandomSet","t":2,"d":"intervalIFVec is Bochner-integrable, from MemLp 2 ⇒ Integrable of the endpoints on a probability measure.","s":"∀ {X : Type u_2} [inst : MeasurableSpace X] {P : MeasureTheory.Measure X} [MeasureTheory.IsProbabilityMeasure P] (yL yU : X → ℝ), MeasureTheory.MemLp yL 2 P → MeasureTheory.MemLp yU 2 P → MeasureTheory.Integrable (Causalean.PartialID.RandomSet.intervalIFVec yL yU P) P","l":252,"f":"Causalean/PO/ID/Partial/RandomSet/IntervalCLT.lean"},{"n":"Causalean.PartialID.RandomSet.intervalIFVec_mean_zero","k":"theorem","m":"Causalean.PO.ID.Partial.RandomSet.IntervalCLT","a":"PO/ID/Partial/RandomSet","t":2,"d":"The endpoint influence function is centered: ∫ ψ dP = 0.","s":"∀ {X : Type u_2} [inst : MeasurableSpace X] {P : MeasureTheory.Measure X} [MeasureTheory.IsProbabilityMeasure P] (yL yU : X → ℝ), MeasureTheory.MemLp yL 2 P → MeasureTheory.MemLp yU 2 P → ∫ (x : X), Causalean.PartialID.RandomSet.intervalIFVec yL yU P x ∂P = 0","l":266,"f":"Causalean/PO/ID/Partial/RandomSet/IntervalCLT.lean"},{"n":"Causalean.PartialID.RandomSet.intervalIFVec_sum_aemeasurable","k":"theorem","m":"Causalean.PO.ID.Partial.RandomSet.IntervalCLT","a":"PO/ID/Partial/RandomSet","t":2,"d":"The vector normalised sum of intervalIFVec is AEMeasurable for each n, from measurability of the endpoints and of the sample coordinates S.Z i.","s":"∀ {Ω : Type u_1} {X : Type u_2} [inst : MeasurableSpace Ω] [inst_1 : MeasurableSpace X] {μ : MeasureTheory.Measure Ω} {P : MeasureTheory.Measure X} (S : Causalean.Stat.IIDSample Ω X μ P) (yL yU : X → ℝ), Measurable yL → Measurable yU → ∀ (n : ℕ), AEMeasurable (Causalean.Stat.IsAsymLinearVec.normalizedSum S (Causalean.PartialID.RandomSet.intervalIFVec yL yU P) (fun m => Finset.range m) n) μ","l":301,"f":"Causalean/PO/ID/Partial/RandomSet/IntervalCLT.lean"},{"n":"Causalean.PartialID.RandomSet.intervalIFVec_var_integrable","k":"theorem","m":"Causalean.PO.ID.Partial.RandomSet.IntervalCLT","a":"PO/ID/Partial/RandomSet","t":2,"d":"The variance term ‖ψ‖² ∈ L¹(P), from MemLp 2 of the two endpoints.","s":"∀ {X : Type u_2} [inst : MeasurableSpace X] {P : MeasureTheory.Measure X} [MeasureTheory.IsProbabilityMeasure P] (yL yU : X → ℝ), MeasureTheory.MemLp yL 2 P → MeasureTheory.MemLp yU 2 P → MeasureTheory.Integrable (fun x => ‖Causalean.PartialID.RandomSet.intervalIFVec yL yU P x‖ ^ 2) P","l":234,"f":"Causalean/PO/ID/Partial/RandomSet/IntervalCLT.lean"},{"n":"Causalean.PartialID.RandomSet.interval_data_clt","k":"theorem","m":"Causalean.PO.ID.Partial.RandomSet.IntervalCLT","a":"PO/ID/Partial/RandomSet","t":1,"d":"Beresteanu–Molinari Theorem 3.2 (scalar interval data). For an i.i.d. sample of interval data Yᵢ = [y_{iL}, y_{iU}] with the lower endpoint pointwise at most the upper endpoint and both endpoints integrable, assume the centered endpoint influence function is measurable, has finite second moment and is integrable, is centered, and its normalized partial sums and the resulting scaled Hausdorff statistic are almost-everywhere measurable at every sample size. Then the scaled Hausdorff distance between the sample-mean interval Ȳₙ and the population identified interval E[Y] = [E y_L, E y_U] converges in distribution to the max-abs of the bivariate Gaussian limit of the influence function:","s":"∀ {Ω : Type u_1} {X : Type u_2} [inst : MeasurableSpace Ω] [inst_1 : MeasurableSpace X] {μ : MeasureTheory.Measure Ω} {P : MeasureTheory.Measure X} [inst_2 : MeasureTheory.IsProbabilityMeasure μ] (S : Causalean.Stat.IIDSample Ω X μ P) (yL yU : X → ℝ), (∀ (z : X), yL z ≤ yU z) → MeasureTheory.Integrable yL P → MeasureTheory.Integrable yU P → ∀ (hψ : Measurable (Causalean.PartialID.RandomSet.intervalIFVec yL yU P)) (hvar : MeasureTheory.Integrable (fun x => ‖Causalean.PartialID.RandomSet.intervalIFVec yL yU P x‖ ^ 2) P), MeasureTheory.Integrable (Causalean.PartialID.RandomSet.intervalIFVec yL yU P) P → ∫ (x : X), Causalean.PartialID.RandomSet.intervalIFVec yL yU P x ∂P = 0 → (∀ (n : ℕ), AEMeasurable (Causalean.Stat.IsAsymLinearVec.normalizedSum S (Causalean.PartialID.RandomSet.intervalIFVec yL yU P) (fun m => Finset.range m) n) μ) → ∀ (hHmeas : ∀ (n : ℕ), AEMeasurable (fun ω => √↑n * Causalean.PartialID.RandomSet.hausdorffDist (Set.Icc (Causalean.PartialID.RandomSet.sampleMean S yL n ω) (Causalean.PartialID.RandomSet.sampleMean S yU n ω)) (Set.Icc (∫ (x : X), yL x ∂P) (∫ (x : X), yU x ∂P))) μ), Causalean.Stat.Tendsto_dist_vec (fun n ω => √↑n * Causalean.PartialID.RandomSet.hausdorffDist (Set.Icc (Causalean.PartialID.RandomSet.sampleMean S yL n ω) (Causalean.PartialID.RandomSet.sampleMean S yU n ω)) (Set.Icc (∫ (x : X), yL x ∂P) (∫ (x : X), yU x ∂P))) (MeasureTheory.Measure.map Causalean.PartialID.RandomSet.maxAbs (Causalean.Stat.gaussianLimit hψ hvar)) μ hHmeas","l":331,"f":"Causalean/PO/ID/Partial/RandomSet/IntervalCLT.lean"},{"n":"Causalean.PartialID.RandomSet.interval_data_clt_of_memLp","k":"theorem","m":"Causalean.PO.ID.Partial.RandomSet.IntervalCLT","a":"PO/ID/Partial/RandomSet","t":1,"d":"Beresteanu–Molinari Theorem 3.2, self-contained MemLp 2 form. For an i.i.d. sample of interval data Yᵢ = [y_{iL}, y_{iU}] with the lower endpoint pointwise at most the upper endpoint, measurable endpoints, and finite second moments (MemLp 2) for both endpoints, the scaled Hausdorff distance between the sample-mean interval and the population identified interval E[Y] = [E y_L, E y_U] converges in distribution to the max-abs of the bivariate Gaussian limit of the endpoint influence function. All four multivariate-CLT hypotheses of interval_data_clt are discharged from these clean moment conditions on yL, yU.","s":"∀ {Ω : Type u_1} {X : Type u_2} [inst : MeasurableSpace Ω] [inst_1 : MeasurableSpace X] {μ : MeasureTheory.Measure Ω} {P : MeasureTheory.Measure X} [inst_2 : MeasureTheory.IsProbabilityMeasure μ] [inst_3 : MeasureTheory.IsProbabilityMeasure P] (S : Causalean.Stat.IIDSample Ω X μ P) (yL yU : X → ℝ) (hLU : ∀ (z : X), yL z ≤ yU z) (hLmeas : Measurable yL) (hUmeas : Measurable yU) (hLsq : MeasureTheory.MemLp yL 2 P) (hUsq : MeasureTheory.MemLp yU 2 P), Causalean.Stat.Tendsto_dist_vec (fun n ω => √↑n * Causalean.PartialID.RandomSet.hausdorffDist (Set.Icc (Causalean.PartialID.RandomSet.sampleMean S yL n ω) (Causalean.PartialID.RandomSet.sampleMean S yU n ω)) (Set.Icc (∫ (x : X), yL x ∂P) (∫ (x : X), yU x ∂P))) (MeasureTheory.Measure.map Causalean.PartialID.RandomSet.maxAbs (Causalean.Stat.gaussianLimit (Causalean.PartialID.RandomSet.measurable_intervalIFVec yL yU hLmeas hUmeas) (Causalean.PartialID.RandomSet.intervalIFVec_var_integrable yL yU hLsq hUsq))) μ (Causalean.PartialID.RandomSet.intervalIFVec_hHmeas S yL yU hLU hLmeas hUmeas (MeasureTheory.MemLp.integrable (Mathlib.Meta.NormNum.isNat_le_true (Mathlib.Meta.NormNum.isNat_ofNat ENNReal Nat.cast_one) (Mathlib.Meta.NormNum.isNat_ofNat ENNReal (Eq.refl 2)) (Eq.refl true)) hLsq) (MeasureTheory.MemLp.integrable (Mathlib.Meta.NormNum.isNat_le_true (Mathlib.Meta.NormNum.isNat_ofNat ENNReal Nat.cast_one) (Mathlib.Meta.NormNum.isNat_ofNat ENNReal (Eq.refl 2)) (Eq.refl true)) hUsq))","l":371,"f":"Causalean/PO/ID/Partial/RandomSet/IntervalCLT.lean"},{"n":"Causalean.PartialID.RandomSet.isBody_finsetSum","k":"theorem","m":"Causalean.PO.ID.Partial.RandomSet.SetValued","a":"PO/ID/Partial/RandomSet","t":2,"d":"A finite Minkowski sum of bodies is a body.","s":"∀ {E : Type u_1} [inst : NormedAddCommGroup E] [inst_1 : InnerProductSpace ℝ E] {ι : Type u_2} (s : Finset ι) (F : ι → Set E), (∀ i ∈ s, Causalean.PartialID.RandomSet.IsBody (F i)) → Causalean.PartialID.RandomSet.IsBody (∑ i ∈ s, F i)","l":87,"f":"Causalean/PO/ID/Partial/RandomSet/SetValued.lean"},{"n":"Causalean.PartialID.RandomSet.isSelection_iff_exists_param","k":"theorem","m":"Causalean.PO.ID.Partial.RandomSet.Interval","a":"PO/ID/Partial/RandomSet","t":2,"d":"Measurable selection of an interval random set. A function f is a selection of [L, U] iff f = L + t·(U − L) for some measurable t : Ω → [0,1]. Elementary — no Kuratowski–Ryll-Nardzewski.","s":"∀ {Ω : Type u_1} [inst : MeasurableSpace Ω] {L U : Ω → ℝ}, Measurable L → Measurable U → (∀ (ω : Ω), L ω ≤ U ω) → ∀ (f : Ω → ℝ), Causalean.PartialID.RandomSet.IsSelection L U f ↔ ∃ t, Measurable t ∧ (∀ (ω : Ω), t ω ∈ Set.Icc 0 1) ∧ ∀ (ω : Ω), f ω = L ω + t ω * (U ω - L ω)","l":78,"f":"Causalean/PO/ID/Partial/RandomSet/Interval.lean"},{"n":"Causalean.PartialID.RandomSet.isSelection_left","k":"theorem","m":"Causalean.PO.ID.Partial.RandomSet.Interval","a":"PO/ID/Partial/RandomSet","t":2,"d":"The lower endpoint is always a selection, so the random set has a measurable selection.","s":"∀ {Ω : Type u_1} [inst : MeasurableSpace Ω] {L U : Ω → ℝ}, Measurable L → (∀ (ω : Ω), L ω ≤ U ω) → Causalean.PartialID.RandomSet.IsSelection L U L","l":72,"f":"Causalean/PO/ID/Partial/RandomSet/Interval.lean"},{"n":"Causalean.PartialID.RandomSet.maxAbs","k":"def","m":"Causalean.PO.ID.Partial.RandomSet.IntervalCLT","a":"PO/ID/Partial/RandomSet","t":1,"d":"The max-abs functional w ↦ max(|w₀|, |w₁|) on ℝ². In the d = 1 random-set picture this is the Hausdorff distance between the intervals whose endpoint gaps are w₀ and w₁ (cf. hausdorffDist_Icc).","s":"EuclideanSpace ℝ (Fin 2) → ℝ","l":71,"f":"Causalean/PO/ID/Partial/RandomSet/IntervalCLT.lean"},{"n":"Causalean.PartialID.RandomSet.maxAbsK","k":"def","m":"Causalean.PO.ID.Partial.RandomSet.SupportProcess","a":"PO/ID/Partial/RandomSet","t":1,"d":"The grid supremum statistic returns the largest absolute support-process deviation across the chosen directions.","s":"{k : ℕ} → [NeZero k] → EuclideanSpace ℝ (Fin k) → ℝ","l":60,"f":"Causalean/PO/ID/Partial/RandomSet/SupportProcess.lean"},{"n":"Causalean.PartialID.RandomSet.maxAbs_normalizedSum_eq","k":"theorem","m":"Causalean.PO.ID.Partial.RandomSet.IntervalCLT","a":"PO/ID/Partial/RandomSet","t":1,"d":"The Hausdorff bridge (Beresteanu–Molinari Theorem 3.2, statistic form). For an i.i.d. sample with interval endpoints yL, yU satisfying the lower endpoint pointwise at most the upper endpoint and both integrable, the max-abs functional applied to the centered endpoint normalised sum equals the scaled Hausdorff distance √n · H(Ȳₙ, E[Y]) between the sample-mean interval Ȳₙ = [ȳ_{nL}, ȳ_{nU}] and the population identified interval E[Y] = [E y_L, E y_U].","s":"∀ {Ω : Type u_1} {X : Type u_2} [inst : MeasurableSpace Ω] [inst_1 : MeasurableSpace X] {μ : MeasureTheory.Measure Ω} {P : MeasureTheory.Measure X} (S : Causalean.Stat.IIDSample Ω X μ P) (yL yU : X → ℝ), (∀ (z : X), yL z ≤ yU z) → MeasureTheory.Integrable yL P → MeasureTheory.Integrable yU P → ∀ (n : ℕ) (ω : Ω), Causalean.PartialID.RandomSet.maxAbs (Causalean.Stat.IsAsymLinearVec.normalizedSum S (Causalean.PartialID.RandomSet.intervalIFVec yL yU P) (fun m => Finset.range m) n ω) = √↑n * Causalean.PartialID.RandomSet.hausdorffDist (Set.Icc (Causalean.PartialID.RandomSet.sampleMean S yL n ω) (Causalean.PartialID.RandomSet.sampleMean S yU n ω)) (Set.Icc (∫ (x : X), yL x ∂P) (∫ (x : X), yU x ∂P))","l":183,"f":"Causalean/PO/ID/Partial/RandomSet/IntervalCLT.lean"},{"n":"Causalean.PartialID.RandomSet.measurable_dirStat","k":"theorem","m":"Causalean.PO.ID.Partial.RandomSet.IntervalInference","a":"PO/ID/Partial/RandomSet","t":2,"d":"The directed-Hausdorff endpoint functional is measurable.","s":"Measurable Causalean.PartialID.RandomSet.dirStat","l":187,"f":"Causalean/PO/ID/Partial/RandomSet/IntervalInference.lean"},{"n":"Causalean.PartialID.RandomSet.measurable_intervalIFVec","k":"theorem","m":"Causalean.PO.ID.Partial.RandomSet.IntervalCLT","a":"PO/ID/Partial/RandomSet","t":2,"d":"intervalIFVec is measurable from measurability of the two endpoint maps.","s":"∀ {X : Type u_2} [inst : MeasurableSpace X] {P : MeasureTheory.Measure X} (yL yU : X → ℝ), Measurable yL → Measurable yU → Measurable (Causalean.PartialID.RandomSet.intervalIFVec yL yU P)","l":223,"f":"Causalean/PO/ID/Partial/RandomSet/IntervalCLT.lean"},{"n":"Causalean.PartialID.RandomSet.measurable_maxAbs","k":"theorem","m":"Causalean.PO.ID.Partial.RandomSet.IntervalCLT","a":"PO/ID/Partial/RandomSet","t":2,"d":"The max-absolute-value functional on endpoint deviations is measurable.","s":"Measurable Causalean.PartialID.RandomSet.maxAbs","l":79,"f":"Causalean/PO/ID/Partial/RandomSet/IntervalCLT.lean"},{"n":"Causalean.PartialID.RandomSet.measurable_maxAbsK","k":"theorem","m":"Causalean.PO.ID.Partial.RandomSet.SupportProcess","a":"PO/ID/Partial/RandomSet","t":2,"d":"The grid supremum statistic is measurable.","s":"∀ {k : ℕ} [inst : NeZero k], Measurable Causalean.PartialID.RandomSet.maxAbsK","l":76,"f":"Causalean/PO/ID/Partial/RandomSet/SupportProcess.lean"},{"n":"Causalean.PartialID.RandomSet.minkowskiMean","k":"def","m":"Causalean.PO.ID.Partial.RandomSet.SetValued","a":"PO/ID/Partial/RandomSet","t":1,"d":"The empirical Minkowski average (1/|s|) • ∑_{i∈s} Fᵢ.","s":"{E : Type u_1} → [inst : NormedAddCommGroup E] → [InnerProductSpace ℝ E] → {ι : Type u_2} → Finset ι → (ι → Set E) → Set E","l":120,"f":"Causalean/PO/ID/Partial/RandomSet/SetValued.lean"},{"n":"Causalean.PartialID.RandomSet.norm_sq_intervalIFVec","k":"theorem","m":"Causalean.PO.ID.Partial.RandomSet.IntervalCLT","a":"PO/ID/Partial/RandomSet","t":2,"d":"Pointwise squared norm of the endpoint influence function as the sum of the two squared centered endpoints.","s":"∀ {X : Type u_2} [inst : MeasurableSpace X] {P : MeasureTheory.Measure X} (yL yU : X → ℝ) (x : X), ‖Causalean.PartialID.RandomSet.intervalIFVec yL yU P x‖ ^ 2 = (yL x - ∫ (z : X), yL z ∂P) ^ 2 + (yU x - ∫ (z : X), yU z ∂P) ^ 2","l":211,"f":"Causalean/PO/ID/Partial/RandomSet/IntervalCLT.lean"},{"n":"Causalean.PartialID.RandomSet.normalizedSum_dirStat_clt","k":"theorem","m":"Causalean.PO.ID.Partial.RandomSet.IntervalInference","a":"PO/ID/Partial/RandomSet","t":2,"d":"Directed continuous-mapping CLT. dirStat of the vector normalised sum converges in distribution to (gaussianLimit ψ).map dirStat.","s":"∀ {Ω : Type u_1} {X : Type u_2} [inst : MeasurableSpace Ω] [inst_1 : MeasurableSpace X] {μ : MeasureTheory.Measure Ω} {P : MeasureTheory.Measure X} [inst_2 : MeasureTheory.IsProbabilityMeasure μ] {ψ : X → EuclideanSpace ℝ (Fin 2)} (hψ : Measurable ψ) (hvar : MeasureTheory.Integrable (fun x => ‖ψ x‖ ^ 2) P) (S : Causalean.Stat.IIDSample Ω X μ P), ∫ (x : X), ψ x ∂P = 0 → ∀ (hSum_meas : ∀ (n : ℕ), AEMeasurable (Causalean.Stat.IsAsymLinearVec.normalizedSum S ψ (fun m => Finset.range m) n) μ), Causalean.Stat.Tendsto_dist_vec (fun n ω => Causalean.PartialID.RandomSet.dirStat (Causalean.Stat.IsAsymLinearVec.normalizedSum S ψ (fun m => Finset.range m) n ω)) (MeasureTheory.Measure.map Causalean.PartialID.RandomSet.dirStat (Causalean.Stat.gaussianLimit hψ hvar)) μ fun n => Measurable.comp_aemeasurable Causalean.PartialID.RandomSet.measurable_dirStat (hSum_meas n)","l":200,"f":"Causalean/PO/ID/Partial/RandomSet/IntervalInference.lean"},{"n":"Causalean.PartialID.RandomSet.normalizedSum_maxAbs_clt","k":"theorem","m":"Causalean.PO.ID.Partial.RandomSet.IntervalCLT","a":"PO/ID/Partial/RandomSet","t":2,"d":"Abstract continuous-mapping CLT. maxAbs of the vector normalised sum converges in distribution to the pushforward (gaussianLimit ψ).map maxAbs — the law of max(|z_L|, |z_U|) for the bivariate Gaussian limit. Immediate from the multivariate CLT (clt_normalizedSum_vec) and the continuous-mapping theorem (Tendsto_dist_vec.map_continuous).","s":"∀ {Ω : Type u_1} {X : Type u_2} [inst : MeasurableSpace Ω] [inst_1 : MeasurableSpace X] {μ : MeasureTheory.Measure Ω} {P : MeasureTheory.Measure X} [inst_2 : MeasureTheory.IsProbabilityMeasure μ] {ψ : X → EuclideanSpace ℝ (Fin 2)} (hψ : Measurable ψ) (hvar : MeasureTheory.Integrable (fun x => ‖ψ x‖ ^ 2) P) (S : Causalean.Stat.IIDSample Ω X μ P), MeasureTheory.Integrable ψ P → ∫ (x : X), ψ x ∂P = 0 → ∀ (hSum_meas : ∀ (n : ℕ), AEMeasurable (Causalean.Stat.IsAsymLinearVec.normalizedSum S ψ (fun m => Finset.range m) n) μ), Causalean.Stat.Tendsto_dist_vec (fun n ω => Causalean.PartialID.RandomSet.maxAbs (Causalean.Stat.IsAsymLinearVec.normalizedSum S ψ (fun m => Finset.range m) n ω)) (MeasureTheory.Measure.map Causalean.PartialID.RandomSet.maxAbs (Causalean.Stat.gaussianLimit hψ hvar)) μ fun n => Measurable.comp_aemeasurable Causalean.PartialID.RandomSet.measurable_maxAbs (hSum_meas n)","l":93,"f":"Causalean/PO/ID/Partial/RandomSet/IntervalCLT.lean"},{"n":"Causalean.PartialID.RandomSet.randomInterval","k":"def","m":"Causalean.PO.ID.Partial.RandomSet.Interval","a":"PO/ID/Partial/RandomSet","t":1,"d":"The interval-valued random closed set ω ↦ [L ω, U ω].","s":"{Ω : Type u_1} → (Ω → ℝ) → (Ω → ℝ) → Ω → Set ℝ","l":64,"f":"Causalean/PO/ID/Partial/RandomSet/Interval.lean"},{"n":"Causalean.PartialID.RandomSet.sInf_selectionExpectation","k":"theorem","m":"Causalean.PO.ID.Partial.RandomSet.Interval","a":"PO/ID/Partial/RandomSet","t":1,"d":"For measurable lower and upper endpoint functions L, U that are integrable and satisfy L pointwise at most U, the sharp lower endpoint of the identified set — the infimum of the selection (Aumann) expectation over all measurable selections of the interval-valued random set [L, U] — equals the expectation of the lower endpoint L: sInf (selectionExpectation L U μ) = ∫ L dμ.","s":"∀ {Ω : Type u_1} [inst : MeasurableSpace Ω] {μ : MeasureTheory.Measure Ω} {L U : Ω → ℝ}, Measurable L → Measurable U → MeasureTheory.Integrable L μ → MeasureTheory.Integrable U μ → (∀ (ω : Ω), L ω ≤ U ω) → sInf (Causalean.PartialID.RandomSet.selectionExpectation L U μ) = ∫ (ω : Ω), L ω ∂μ","l":175,"f":"Causalean/PO/ID/Partial/RandomSet/Interval.lean"},{"n":"Causalean.PartialID.RandomSet.sSup_selectionExpectation","k":"theorem","m":"Causalean.PO.ID.Partial.RandomSet.Interval","a":"PO/ID/Partial/RandomSet","t":1,"d":"For measurable lower and upper endpoint functions L, U that are integrable and satisfy L pointwise at most U, the sharp upper endpoint of the identified set — the supremum of the selection (Aumann) expectation over all measurable selections of the interval-valued random set [L, U] — equals the expectation of the upper endpoint U: sSup (selectionExpectation L U μ) = ∫ U dμ.","s":"∀ {Ω : Type u_1} [inst : MeasurableSpace Ω] {μ : MeasureTheory.Measure Ω} {L U : Ω → ℝ}, Measurable L → Measurable U → MeasureTheory.Integrable L μ → MeasureTheory.Integrable U μ → (∀ (ω : Ω), L ω ≤ U ω) → sSup (Causalean.PartialID.RandomSet.selectionExpectation L U μ) = ∫ (ω : Ω), U ω ∂μ","l":187,"f":"Causalean/PO/ID/Partial/RandomSet/Interval.lean"},{"n":"Causalean.PartialID.RandomSet.sampleMean","k":"def","m":"Causalean.PO.ID.Partial.RandomSet.IntervalCLT","a":"PO/ID/Partial/RandomSet","t":1,"d":"Sample mean of y over the first n draws of the i.i.d. sample.","s":"{Ω : Type u_1} → {X : Type u_2} → [inst : MeasurableSpace Ω] → [inst_1 : MeasurableSpace X] → {μ : MeasureTheory.Measure Ω} → {P : MeasureTheory.Measure X} → Causalean.Stat.IIDSample Ω X μ P → (X → ℝ) → ℕ → Ω → ℝ","l":130,"f":"Causalean/PO/ID/Partial/RandomSet/IntervalCLT.lean"},{"n":"Causalean.PartialID.RandomSet.sampleMean_le","k":"theorem","m":"Causalean.PO.ID.Partial.RandomSet.IntervalCLT","a":"PO/ID/Partial/RandomSet","t":2,"d":"The sample-mean interval is well-ordered (lower ≤ upper) when y_L ≤ y_U.","s":"∀ {Ω : Type u_1} {X : Type u_2} [inst : MeasurableSpace Ω] [inst_1 : MeasurableSpace X] {μ : MeasureTheory.Measure Ω} {P : MeasureTheory.Measure X} (S : Causalean.Stat.IIDSample Ω X μ P) (yL yU : X → ℝ), (∀ (z : X), yL z ≤ yU z) → ∀ (n : ℕ) (ω : Ω), Causalean.PartialID.RandomSet.sampleMean S yL n ω ≤ Causalean.PartialID.RandomSet.sampleMean S yU n ω","l":174,"f":"Causalean/PO/ID/Partial/RandomSet/IntervalCLT.lean"},{"n":"Causalean.PartialID.RandomSet.selectionExpectation","k":"def","m":"Causalean.PO.ID.Partial.RandomSet.Interval","a":"PO/ID/Partial/RandomSet","t":1,"d":"The selection (Aumann) expectation of the interval random set [L, U]: the set of integrals of integrable measurable selections.","s":"{Ω : Type u_1} → [inst : MeasurableSpace Ω] → (Ω → ℝ) → (Ω → ℝ) → MeasureTheory.Measure Ω → Set ℝ","l":121,"f":"Causalean/PO/ID/Partial/RandomSet/Interval.lean"},{"n":"Causalean.PartialID.RandomSet.selectionExpectation_eq_Icc","k":"theorem","m":"Causalean.PO.ID.Partial.RandomSet.Interval","a":"PO/ID/Partial/RandomSet","t":1,"d":"Selection expectation equals [∫L, ∫U]. For measurable lower and upper endpoint functions L, U that are integrable and satisfy L pointwise at most U, the selection (Aumann) expectation of the interval-valued random set [L, U] — the set of integrals of its integrable measurable selections — equals the closed interval [∫L dμ, ∫U dμ]. The forward inclusion is integral monotonicity; the reverse inclusion realises every intermediate value with a constant mixing weight t ∈ [0,1], so no atomlessness is needed.","s":"∀ {Ω : Type u_1} [inst : MeasurableSpace Ω] {μ : MeasureTheory.Measure Ω} {L U : Ω → ℝ}, Measurable L → Measurable U → MeasureTheory.Integrable L μ → MeasureTheory.Integrable U μ → (∀ (ω : Ω), L ω ≤ U ω) → Causalean.PartialID.RandomSet.selectionExpectation L U μ = Set.Icc (∫ (ω : Ω), L ω ∂μ) (∫ (ω : Ω), U ω ∂μ)","l":131,"f":"Causalean/PO/ID/Partial/RandomSet/Interval.lean"},{"n":"Causalean.PartialID.RandomSet.setValued_supportProcess_clt","k":"theorem","m":"Causalean.PO.ID.Partial.RandomSet.SupportProcess","a":"PO/ID/Partial/RandomSet","t":1,"d":"For a set-valued random variable F with center EF, observed via an IID sample on a finite grid of directions p, if the centered support process is measurable, it has finite second moment, it is mean zero, E[ψ] = 0, and the normalized partial sums of the process are almost-everywhere measurable at every sample size, then the grid supremum statistic applied to those normalized sums converges in distribution to the corresponding supremum functional of the process's Gaussian limit.","s":"∀ {Ω : Type u_1} {X : Type u_2} [inst : MeasurableSpace Ω] [inst_1 : MeasurableSpace X] {μ : MeasureTheory.Measure Ω} {P : MeasureTheory.Measure X} [inst_2 : MeasureTheory.IsProbabilityMeasure μ] {k : ℕ} [inst_3 : NeZero k] {V : Type u_3} [inst_4 : NormedAddCommGroup V] [inst_5 : InnerProductSpace ℝ V] (S : Causalean.Stat.IIDSample Ω X μ P) (F : X → Set V) (EF : Set V) (p : Fin k → V) (hψ : Measurable (Causalean.PartialID.RandomSet.supportProcess F EF p)) (hvar : MeasureTheory.Integrable (fun x => ‖Causalean.PartialID.RandomSet.supportProcess F EF p x‖ ^ 2) P), ∫ (x : X), Causalean.PartialID.RandomSet.supportProcess F EF p x ∂P = 0 → ∀ (hSum_meas : ∀ (n : ℕ), AEMeasurable (Causalean.Stat.IsAsymLinearVec.normalizedSum S (Causalean.PartialID.RandomSet.supportProcess F EF p) (fun m => Finset.range m) n) μ), Causalean.Stat.Tendsto_dist_vec (fun n ω => Causalean.PartialID.RandomSet.maxAbsK (Causalean.Stat.IsAsymLinearVec.normalizedSum S (Causalean.PartialID.RandomSet.supportProcess F EF p) (fun m => Finset.range m) n ω)) (MeasureTheory.Measure.map Causalean.PartialID.RandomSet.maxAbsK (Causalean.Stat.gaussianLimit hψ hvar)) μ fun n => Measurable.comp_aemeasurable Causalean.PartialID.RandomSet.measurable_maxAbsK (hSum_meas n)","l":187,"f":"Causalean/PO/ID/Partial/RandomSet/SupportProcess.lean"},{"n":"Causalean.PartialID.RandomSet.sqrt_inv_centered","k":"theorem","m":"Causalean.PO.ID.Partial.RandomSet.IntervalCLT","a":"PO/ID/Partial/RandomSet","t":2,"d":"The centered-sum / √n identity (√n)⁻¹·(s − n·c) = √n·(s/n − c), the algebra turning a normalised centered sum into √n × (sample mean − population mean).","s":"∀ (n : ℕ) (s c : ℝ), (√↑n)⁻¹ * (s - ↑n * c) = √↑n * (s / ↑n - c)","l":116,"f":"Causalean/PO/ID/Partial/RandomSet/IntervalCLT.lean"},{"n":"Causalean.PartialID.RandomSet.subset_dilate_iff_directedHausdorff_le","k":"theorem","m":"Causalean.PO.ID.Partial.RandomSet.IntervalInference","a":"PO/ID/Partial/RandomSet","t":1,"d":"Coverage characterization (Beresteanu–Molinari Proposition 2.7 core). For real numbers with μL ≤ μU and yl ≤ yu forming two closed intervals, and a nonnegative dilation radius r, the population identified interval [μL, μU] lies inside the dilated estimate [yl, yu] ⊕ [−r, r] if and only if the directed Hausdorff distance from [μL,μU] to [yl,yu] is at most r. This is the one-sided coverage event of the Beresteanu–Molinari confidence region.","s":"∀ {μL μU yl yu r : ℝ}, μL ≤ μU → yl ≤ yu → 0 ≤ r → (Set.Icc μL μU ⊆ Causalean.PartialID.RandomSet.dilate (Set.Icc yl yu) r ↔ Causalean.PartialID.RandomSet.directedHausdorff (Set.Icc μL μU) (Set.Icc yl yu) ≤ r)","l":94,"f":"Causalean/PO/ID/Partial/RandomSet/IntervalInference.lean"},{"n":"Causalean.PartialID.RandomSet.subset_dilate_iff_hausdorff_le","k":"theorem","m":"Causalean.PO.ID.Partial.RandomSet.IntervalInference","a":"PO/ID/Partial/RandomSet","t":1,"d":"Symmetric (two-sided) coverage characterization (Beresteanu–Molinari Uₙ / Theorem 2.4). For real numbers with μL ≤ μU and yl ≤ yu forming two closed intervals, and a nonnegative dilation radius r, each interval lies inside the other's dilation by r if and only if the symmetric Hausdorff distance between [μL,μU] and [yl,yu] is at most r. The mutual containment [μL,μU] ⊆ Uᵧ ∧ [yl,yu] ⊆ Uᵤ is exactly the two-sided event H ≤ r.","s":"∀ {μL μU yl yu r : ℝ}, μL ≤ μU → yl ≤ yu → 0 ≤ r → (Set.Icc μL μU ⊆ Causalean.PartialID.RandomSet.dilate (Set.Icc yl yu) r ∧ Set.Icc yl yu ⊆ Causalean.PartialID.RandomSet.dilate (Set.Icc μL μU) r ↔ Causalean.PartialID.RandomSet.hausdorffDist (Set.Icc μL μU) (Set.Icc yl yu) ≤ r)","l":112,"f":"Causalean/PO/ID/Partial/RandomSet/IntervalInference.lean"},{"n":"Causalean.PartialID.RandomSet.supportFn_Icc_neg_one","k":"theorem","m":"Causalean.PO.ID.Partial.RandomSet.Aumann","a":"PO/ID/Partial/RandomSet","t":2,"d":"Support function of a real interval at −1: the negated lower endpoint.","s":"∀ {a b : ℝ}, a ≤ b → Causalean.PartialID.supportFn (Set.Icc a b) (-1) = -a","l":76,"f":"Causalean/PO/ID/Partial/RandomSet/Aumann.lean"},{"n":"Causalean.PartialID.RandomSet.supportFn_Icc_one","k":"theorem","m":"Causalean.PO.ID.Partial.RandomSet.Aumann","a":"PO/ID/Partial/RandomSet","t":2,"d":"Support function of a real interval at +1: the upper endpoint.","s":"∀ {a b : ℝ}, a ≤ b → Causalean.PartialID.supportFn (Set.Icc a b) 1 = b","l":69,"f":"Causalean/PO/ID/Partial/RandomSet/Aumann.lean"},{"n":"Causalean.PartialID.RandomSet.supportFn_finsetSum","k":"theorem","m":"Causalean.PO.ID.Partial.RandomSet.SetValued","a":"PO/ID/Partial/RandomSet","t":2,"d":"Support function commutes with finite Minkowski sums: s(∑ᵢ Fᵢ, d) = ∑ᵢ s(Fᵢ, d).","s":"∀ {E : Type u_1} [inst : NormedAddCommGroup E] [inst_1 : InnerProductSpace ℝ E] {ι : Type u_2} (s : Finset ι) (F : ι → Set E) (d : E), (∀ i ∈ s, Causalean.PartialID.RandomSet.IsBody (F i)) → Causalean.PartialID.supportFn (∑ i ∈ s, F i) d = ∑ i ∈ s, Causalean.PartialID.supportFn (F i) d","l":102,"f":"Causalean/PO/ID/Partial/RandomSet/SetValued.lean"},{"n":"Causalean.PartialID.RandomSet.supportFn_minkowskiMean","k":"theorem","m":"Causalean.PO.ID.Partial.RandomSet.SetValued","a":"PO/ID/Partial/RandomSet","t":1,"d":"Keystone — Minkowski-mean support bridge. Given a finite index set s all of whose values F i are bodies — nonempty, compact, convex subsets, the support function of the empirical Minkowski average (1/|s|) · ∑ᵢ Fᵢ in a direction d equals the arithmetic average of the individual support functions: s(d, F̄ₙ) = (1/|s|) · ∑ᵢ s(d, Fᵢ). This is the identity that turns the support process into an ordinary sample mean, so the multivariate CLT applies.","s":"∀ {E : Type u_1} [inst : NormedAddCommGroup E] [inst_1 : InnerProductSpace ℝ E] {ι : Type u_2} (s : Finset ι) (F : ι → Set E) (d : E), (∀ i ∈ s, Causalean.PartialID.RandomSet.IsBody (F i)) → Causalean.PartialID.supportFn (Causalean.PartialID.RandomSet.minkowskiMean s F) d = (↑s.card)⁻¹ * ∑ i ∈ s, Causalean.PartialID.supportFn (F i) d","l":124,"f":"Causalean/PO/ID/Partial/RandomSet/SetValued.lean"},{"n":"Causalean.PartialID.RandomSet.supportProcess","k":"def","m":"Causalean.PO.ID.Partial.RandomSet.SupportProcess","a":"PO/ID/Partial/RandomSet","t":1,"d":"The centered support process records each random set's support deviation from its center on a finite grid of directions.","s":"{X : Type u_2} → {k : ℕ} → {V : Type u_3} → [inst : NormedAddCommGroup V] → [InnerProductSpace ℝ V] → (X → Set V) → Set V → (Fin k → V) → X → EuclideanSpace ℝ (Fin k)","l":135,"f":"Causalean/PO/ID/Partial/RandomSet/SupportProcess.lean"},{"n":"Causalean.PartialID.RandomSet.supportProcess_normalizedSum_apply","k":"theorem","m":"Causalean.PO.ID.Partial.RandomSet.SupportProcess","a":"PO/ID/Partial/RandomSet","t":2,"d":"Each coordinate of the normalized support-process sum is the rescaled support deviation of the empirical Minkowski average.","s":"∀ {Ω : Type u_1} {X : Type u_2} [inst : MeasurableSpace Ω] [inst_1 : MeasurableSpace X] {μ : MeasureTheory.Measure Ω} {P : MeasureTheory.Measure X} {k : ℕ} {V : Type u_3} [inst_2 : NormedAddCommGroup V] [inst_3 : InnerProductSpace ℝ V] (S : Causalean.Stat.IIDSample Ω X μ P) (F : X → Set V) (EF : Set V) (p : Fin k → V), (∀ (x : X), Causalean.PartialID.RandomSet.IsBody (F x)) → ∀ (n : ℕ) (ω : Ω) (j : Fin k), (Causalean.Stat.IsAsymLinearVec.normalizedSum S (Causalean.PartialID.RandomSet.supportProcess F EF p) (fun m => Finset.range m) n ω).ofLp j = √↑n * (Causalean.PartialID.supportFn (Causalean.PartialID.RandomSet.minkowskiMean (Finset.range n) fun i => F (S.Z i ω)) (p j) - Causalean.PartialID.supportFn EF (p j))","l":157,"f":"Causalean/PO/ID/Partial/RandomSet/SupportProcess.lean"},{"n":"Causalean.PartialID.RandomSet.supportProcess_ofLp","k":"theorem","m":"Causalean.PO.ID.Partial.RandomSet.SupportProcess","a":"PO/ID/Partial/RandomSet","t":2,"d":"Each coordinate of the centered support process is the corresponding support deviation.","s":"∀ {X : Type u_2} {k : ℕ} {V : Type u_3} [inst : NormedAddCommGroup V] [inst_1 : InnerProductSpace ℝ V] (F : X → Set V) (EF : Set V) (p : Fin k → V) (x : X) (j : Fin k), (Causalean.PartialID.RandomSet.supportProcess F EF p x).ofLp j = Causalean.PartialID.supportFn (F x) (p j) - Causalean.PartialID.supportFn EF (p j)","l":150,"f":"Causalean/PO/ID/Partial/RandomSet/SupportProcess.lean"},{"n":"Causalean.PartialID.RandomSet.supportProcess_sup_clt","k":"theorem","m":"Causalean.PO.ID.Partial.RandomSet.SupportProcess","a":"PO/ID/Partial/RandomSet","t":1,"d":"For a centered support process ψ on k fixed directions observed via an IID sample S, if ψ is mean zero, E[ψ] = 0, and the normalized partial sums built from S are almost-everywhere measurable at every sample size, then the grid supremum statistic maxAbsK applied to those normalized sums converges in distribution to maxAbsK applied to the Gaussian limit of ψ.","s":"∀ {Ω : Type u_1} {X : Type u_2} [inst : MeasurableSpace Ω] [inst_1 : MeasurableSpace X] {μ : MeasureTheory.Measure Ω} {P : MeasureTheory.Measure X} [inst_2 : MeasureTheory.IsProbabilityMeasure μ] {k : ℕ} [inst_3 : NeZero k] {ψ : X → EuclideanSpace ℝ (Fin k)} (hψ : Measurable ψ) (hvar : MeasureTheory.Integrable (fun x => ‖ψ x‖ ^ 2) P) (S : Causalean.Stat.IIDSample Ω X μ P), ∫ (x : X), ψ x ∂P = 0 → ∀ (hSum_meas : ∀ (n : ℕ), AEMeasurable (Causalean.Stat.IsAsymLinearVec.normalizedSum S ψ (fun m => Finset.range m) n) μ), Causalean.Stat.Tendsto_dist_vec (fun n ω => Causalean.PartialID.RandomSet.maxAbsK (Causalean.Stat.IsAsymLinearVec.normalizedSum S ψ (fun m => Finset.range m) n ω)) (MeasureTheory.Measure.map Causalean.PartialID.RandomSet.maxAbsK (Causalean.Stat.gaussianLimit hψ hvar)) μ fun n => Measurable.comp_aemeasurable Causalean.PartialID.RandomSet.measurable_maxAbsK (hSum_meas n)","l":94,"f":"Causalean/PO/ID/Partial/RandomSet/SupportProcess.lean"},{"n":"Causalean.PartialID.affineBall","k":"def","m":"Causalean.PO.ID.Partial.SupportFunction.AffineBall","a":"PO/ID/Partial/SupportFunction","t":1,"d":"The affine-ball fiber contains the solutions to a linear equation that also satisfy a norm bound.","s":"{H : Type u_1} → {F : Type u_2} → [inst : NormedAddCommGroup H] → [inst_1 : InnerProductSpace ℝ H] → [inst_2 : NormedAddCommGroup F] → [inst_3 : InnerProductSpace ℝ F] → (H →L[ℝ] F) → F → ℝ → Set H","l":67,"f":"Causalean/PO/ID/Partial/SupportFunction/AffineBall.lean"},{"n":"Causalean.PartialID.affineBall_point_identified_iff","k":"theorem","m":"Causalean.PO.ID.Partial.SupportFunction.AffineBall","a":"PO/ID/Partial/SupportFunction","t":1,"d":"Point identification. Let A be a continuous linear map between real inner-product spaces with b in its range, B a radius, and c a target direction. Suppose h₀ solves the linear constraint A h₀ = b, h₀ is orthogonal to the kernel of A (the minimum-norm solution), and h₀'s norm is strictly less than B (strict slack in the norm bound). Then the identified set of ⟪c, ·⟫ over the fiber {h : A h = b, ‖h‖ ≤ B} collapses to a point if and only if the orthogonal projection of c onto the kernel of A vanishes, i.e. c is orthogonal to that kernel.","s":"∀ {H : Type u_1} {F : Type u_2} [inst : NormedAddCommGroup H] [inst_1 : InnerProductSpace ℝ H] [inst_2 : CompleteSpace H] [inst_3 : NormedAddCommGroup F] [inst_4 : InnerProductSpace ℝ F] (A : H →L[ℝ] F) {b : F} {B : ℝ} {c h₀ : H}, A h₀ = b → h₀ ∈ (Causalean.PartialID.opKer A)ᗮ → ‖h₀‖ < B → (Causalean.PartialID.width (Causalean.PartialID.affineBall A b B) c = 0 ↔ (Causalean.PartialID.opKer A).orthogonalProjection c = 0)","l":216,"f":"Causalean/PO/ID/Partial/SupportFunction/AffineBall.lean"},{"n":"Causalean.PartialID.frechet_lower","k":"theorem","m":"Causalean.PO.ID.Partial.Frechet","a":"PO/ID/Partial","t":1,"d":"Fréchet–Hoeffding lower bound. For a fixed joint law of two real random variables X, Y under a probability measure, if Y is measurable, then the joint probability P(X ≤ u, Y ≤ v) is at least the larger of zero and the sum of the two marginal CDF values at u and v minus one: max (P(X ≤ u) + P(Y ≤ v) - 1) 0 ≤ P(X ≤ u, Y ≤ v).","s":"∀ {Ω : Type u_1} [inst : MeasurableSpace Ω] (P : MeasureTheory.Measure Ω) [MeasureTheory.IsProbabilityMeasure P] {X Y : Ω → ℝ}, Measurable Y → ∀ (u v : ℝ), max ((P {ω | X ω ≤ u}).toReal + (P {ω | Y ω ≤ v}).toReal - 1) 0 ≤ (P {ω | X ω ≤ u ∧ Y ω ≤ v}).toReal","l":87,"f":"Causalean/PO/ID/Partial/Frechet.lean"},{"n":"Causalean.PartialID.frechet_upper","k":"theorem","m":"Causalean.PO.ID.Partial.Frechet","a":"PO/ID/Partial","t":1,"d":"Fréchet–Hoeffding upper bound. For any threshold u for X and any threshold v for Y, the joint probability P(X ≤ u, Y ≤ v) is at most the smaller of the two marginal probabilities P(X ≤ u) and P(Y ≤ v).","s":"∀ {Ω : Type u_1} [inst : MeasurableSpace Ω] (P : MeasureTheory.Measure Ω) [MeasureTheory.IsProbabilityMeasure P] {X Y : Ω → ℝ} (u v : ℝ), (P {ω | X ω ≤ u ∧ Y ω ≤ v}).toReal ≤ min (P {ω | X ω ≤ u}).toReal (P {ω | Y ω ≤ v}).toReal","l":71,"f":"Causalean/PO/ID/Partial/Frechet.lean"},{"n":"Causalean.PartialID.identifiedInterval_eq_Icc","k":"theorem","m":"Causalean.PO.ID.Partial.Basic","a":"PO/ID/Partial","t":1,"d":"Sharp interval (order-connected form). For an abstract objective function over a feasible parameter set, suppose the objective is bounded below by L on every feasible parameter, bounded above by U on every feasible parameter, the value L itself is attained by some feasible parameter, the value U itself is attained by some feasible parameter, and the set of attainable objective values is order-connected — it contains every real number between any two of its members. Then the identified interval — the set of all objective values attainable over the feasible parameter set — equals the closed interval [L, U] exactly. Order-connectedness is the abstract substitute for \"no gaps\", supplied concretely by identifiedInterval_param_Icc through continuity + connectedness of a parameterization.","s":"∀ {α : Type u_1} {obj : α → ℝ} {feasible : α → Prop} {L U : ℝ}, (∀ (x : α), feasible x → L ≤ obj x) → (∀ (x : α), feasible x → obj x ≤ U) → L ∈ Causalean.PartialID.IdentifiedInterval obj feasible → U ∈ Causalean.PartialID.IdentifiedInterval obj feasible → (Causalean.PartialID.IdentifiedInterval obj feasible).OrdConnected → Causalean.PartialID.IdentifiedInterval obj feasible = Set.Icc L U","l":63,"f":"Causalean/PO/ID/Partial/Basic.lean"},{"n":"Causalean.PartialID.identifiedInterval_eq_image","k":"theorem","m":"Causalean.PO.ID.Partial.SupportFunction.Interval","a":"PO/ID/Partial/SupportFunction","t":1,"d":"Bridge to the scalar engine. For an objective functional obj and a feasible set C, the scalar identified interval of obj relative to membership in C equals exactly the image of obj on C. Specialised to obj = ⟪d, ·⟫, this identifies the scalar engine's output with the linear image studied here.","s":"∀ {α : Type u_2} (obj : α → ℝ) (C : Set α), (Causalean.PartialID.IdentifiedInterval obj fun x => x ∈ C) = obj '' C","l":168,"f":"Causalean/PO/ID/Partial/SupportFunction/Interval.lean"},{"n":"Causalean.PartialID.identifiedInterval_param_Icc","k":"theorem","m":"Causalean.PO.ID.Partial.Basic","a":"PO/ID/Partial","t":1,"d":"Mixing-pattern constructor. Suppose the feasible parameter set is exactly the image of the unit interval [0, 1] under a path γ, the objective composed with γ is continuous on [0, 1], the objective value at the path's start equals L, the objective value at the path's end equals U, and the objective stays between L and U at every point along the path. Then the sharp identified interval is exactly [L, U]. This is the canonical partial-identification \"mixing\" shape: an unidentified nuisance ranging over a connected parameter set sweeps the objective continuously across the whole interval between its extreme values.","s":"∀ {α : Type u_1} {obj : α → ℝ} {feasible : α → Prop} {γ : ℝ → α} {L U : ℝ}, (∀ (x : α), feasible x ↔ ∃ t ∈ Set.Icc 0 1, γ t = x) → ContinuousOn (fun t => obj (γ t)) (Set.Icc 0 1) → obj (γ 0) = L → obj (γ 1) = U → (∀ t ∈ Set.Icc 0 1, L ≤ obj (γ t) ∧ obj (γ t) ≤ U) → Causalean.PartialID.IdentifiedInterval obj feasible = Set.Icc L U","l":82,"f":"Causalean/PO/ID/Partial/Basic.lean"},{"n":"Causalean.PartialID.identifiedInterval_subset_Icc","k":"theorem","m":"Causalean.PO.ID.Partial.Basic","a":"PO/ID/Partial","t":2,"d":"Outer bound. If the objective is uniformly bounded below by L and above by U over the feasible set, the sharp identified interval is contained in [L, U].","s":"∀ {α : Type u_1} {obj : α → ℝ} {feasible : α → Prop} {L U : ℝ}, (∀ (x : α), feasible x → L ≤ obj x) → (∀ (x : α), feasible x → obj x ≤ U) → Causalean.PartialID.IdentifiedInterval obj feasible ⊆ Set.Icc L U","l":54,"f":"Causalean/PO/ID/Partial/Basic.lean"},{"n":"Causalean.PartialID.kerHasOrthogonalProjection","k":"instance","m":"Causalean.PO.ID.Partial.SupportFunction.AffineBall","a":"PO/ID/Partial/SupportFunction","t":2,"d":"The free-direction subspace of a continuous linear map admits an orthogonal projection because the kernel is closed in a complete Hilbert space.","s":"∀ {H : Type u_1} {F : Type u_2} [inst : NormedAddCommGroup H] [inst_1 : InnerProductSpace ℝ H] [CompleteSpace H] [inst_3 : NormedAddCommGroup F] [inst_4 : InnerProductSpace ℝ F] (A : H →L[ℝ] F), (Causalean.PartialID.opKer A).HasOrthogonalProjection","l":60,"f":"Causalean/PO/ID/Partial/SupportFunction/AffineBall.lean"},{"n":"Causalean.PartialID.l2Ball","k":"def","m":"Causalean.PO.ID.Partial.SupportFunction.Sensitivity","a":"PO/ID/Partial/SupportFunction","t":1,"d":"The χ²/L² ambiguity set: weights normalized against the unit base direction e and within an L²-ball of radius ρ around it.","s":"{H : Type u_2} → [inst : NormedAddCommGroup H] → [InnerProductSpace ℝ H] → H → ℝ → Set H","l":173,"f":"Causalean/PO/ID/Partial/SupportFunction/Sensitivity.lean"},{"n":"Causalean.PartialID.l2Ball_eq_translate","k":"theorem","m":"Causalean.PO.ID.Partial.SupportFunction.Sensitivity","a":"PO/ID/Partial/SupportFunction","t":1,"d":"Reduction to the affine-ball engine. When the base direction has unit inner product with itself, ⟪e, e⟫ = 1, the χ²/L² ambiguity set l2Ball e ρ equals the translate by e of the affine ball of radius ρ for the linear functional ⟪e, ·⟫ centered at the origin — the substitution v = w - e turns the normalization constraint ⟪e, w⟫ = 1 into the linear constraint ⟪e, v⟫ = 0.","s":"∀ {H : Type u_2} [inst : NormedAddCommGroup H] [inst_1 : InnerProductSpace ℝ H] {e : H}, inner ℝ e e = 1 → ∀ (ρ : ℝ), Causalean.PartialID.l2Ball e ρ = (fun v => e + v) '' Causalean.PartialID.affineBall ((innerSL ℝ) e) 0 ρ","l":192,"f":"Causalean/PO/ID/Partial/SupportFunction/Sensitivity.lean"},{"n":"Causalean.PartialID.l2Ball_point_identified_iff","k":"theorem","m":"Causalean.PO.ID.Partial.SupportFunction.Sensitivity","a":"PO/ID/Partial/SupportFunction","t":1,"d":"Point identification. For a unit-norm base direction e and a strictly positive radius ρ, the χ²/L² robust interval for the target c collapses to a single point exactly when ‖c‖² equals ⟪e, c⟫², i.e. when c is collinear with e — equivalently, equality holds in the Cauchy–Schwarz inequality, meaning the represented outcome is almost-surely constant.","s":"∀ {H : Type u_2} [inst : NormedAddCommGroup H] [inst_1 : InnerProductSpace ℝ H] [CompleteSpace H] {e : H}, ‖e‖ = 1 → ∀ (c : H) {ρ : ℝ}, 0 < ρ → (Causalean.PartialID.width (Causalean.PartialID.l2Ball e ρ) c = 0 ↔ ‖c‖ ^ 2 = inner ℝ e c ^ 2)","l":271,"f":"Causalean/PO/ID/Partial/SupportFunction/Sensitivity.lean"},{"n":"Causalean.PartialID.le_robustUpper","k":"theorem","m":"Causalean.PO.ID.Partial.SupportFunction.Sensitivity","a":"PO/ID/Partial/SupportFunction","t":2,"d":"Any admissible weight has a reweighted mean no larger than the upper robust bound.","s":"∀ {E : Type u_1} [inst : NormedAddCommGroup E] [inst_1 : InnerProductSpace ℝ E] {W : Set E} {c w : E}, w ∈ W → BddAbove ((fun x => inner ℝ c x) '' W) → inner ℝ c w ≤ Causalean.PartialID.robustUpper W c","l":114,"f":"Causalean/PO/ID/Partial/SupportFunction/Sensitivity.lean"},{"n":"Causalean.PartialID.le_supportFn","k":"theorem","m":"Causalean.PO.ID.Partial.SupportFunction.Basic","a":"PO/ID/Partial/SupportFunction","t":1,"d":"Lower sandwich. For a feasible set C in a real inner-product space and a direction d, if a point x belongs to C and the linear functional ⟪d, ·⟫ is bounded above on C, then x's functional value is at most the support value: ⟪d, x⟫ ≤ supportFn C d.","s":"∀ {E : Type u_1} [inst : NormedAddCommGroup E] [inst_1 : InnerProductSpace ℝ E] {C : Set E} {d x : E}, x ∈ C → BddAbove ((fun x => inner ℝ d x) '' C) → inner ℝ d x ≤ Causalean.PartialID.supportFn C d","l":69,"f":"Causalean/PO/ID/Partial/SupportFunction/Basic.lean"},{"n":"Causalean.PartialID.linearImage_eq_Icc","k":"theorem","m":"Causalean.PO.ID.Partial.SupportFunction.Interval","a":"PO/ID/Partial/SupportFunction","t":1,"d":"Sharp identified interval (attainment form). For a convex feasible set C in a real inner-product space and a linear target direction d, assuming the target functional is bounded above on C in the direction d and in the opposite direction -d, and that the upper support value is attained by some feasible point and the lower support value -supportFn C (-d) is likewise attained, then the identified set of the target functional over C is exactly the closed interval [-supportFn C (-d), supportFn C d].","s":"∀ {E : Type u_1} [inst : NormedAddCommGroup E] [inst_1 : InnerProductSpace ℝ E] {C : Set E} {d : E}, Convex ℝ C → BddAbove ((fun x => inner ℝ d x) '' C) → BddAbove ((fun x => inner ℝ (-d) x) '' C) → Causalean.PartialID.supportFn C d ∈ (fun x => inner ℝ d x) '' C → -Causalean.PartialID.supportFn C (-d) ∈ (fun x => inner ℝ d x) '' C → (fun x => inner ℝ d x) '' C = Set.Icc (-Causalean.PartialID.supportFn C (-d)) (Causalean.PartialID.supportFn C d)","l":86,"f":"Causalean/PO/ID/Partial/SupportFunction/Interval.lean"},{"n":"Causalean.PartialID.linearImage_eq_Icc_of_isCompact","k":"theorem","m":"Causalean.PO.ID.Partial.SupportFunction.Interval","a":"PO/ID/Partial/SupportFunction","t":1,"d":"Sharp identified interval (compact convex form). For a linear target direction d and a feasible set that is compact, convex, and nonempty, the identified set of the target functional ⟪d, ·⟫ over C is exactly the closed interval [-supportFn C (-d), supportFn C d]: a continuous functional attains its supremum and infimum on a compact set and is automatically bounded there.","s":"∀ {E : Type u_1} [inst : NormedAddCommGroup E] [inst_1 : InnerProductSpace ℝ E] {C : Set E} {d : E}, IsCompact C → Convex ℝ C → C.Nonempty → (fun x => inner ℝ d x) '' C = Set.Icc (-Causalean.PartialID.supportFn C (-d)) (Causalean.PartialID.supportFn C d)","l":102,"f":"Causalean/PO/ID/Partial/SupportFunction/Interval.lean"},{"n":"Causalean.PartialID.linearImage_ordConnected","k":"theorem","m":"Causalean.PO.ID.Partial.SupportFunction.Interval","a":"PO/ID/Partial/SupportFunction","t":2,"d":"Order-connectedness. The linear image of a convex set is convex in ℝ, hence order-connected: the identified set has \"no gaps\".","s":"∀ {E : Type u_1} [inst : NormedAddCommGroup E] [inst_1 : InnerProductSpace ℝ E] {C : Set E} {d : E}, Convex ℝ C → ((fun x => inner ℝ d x) '' C).OrdConnected","l":77,"f":"Causalean/PO/ID/Partial/SupportFunction/Interval.lean"},{"n":"Causalean.PartialID.linearImage_subset_Icc","k":"theorem","m":"Causalean.PO.ID.Partial.SupportFunction.Interval","a":"PO/ID/Partial/SupportFunction","t":2,"d":"Outer bound. The identified set of the linear target lies inside the support interval [-supportFn C (-d), supportFn C d].","s":"∀ {E : Type u_1} [inst : NormedAddCommGroup E] [inst_1 : InnerProductSpace ℝ E] {C : Set E} {d : E}, BddAbove ((fun x => inner ℝ d x) '' C) → BddAbove ((fun x => inner ℝ (-d) x) '' C) → (fun x => inner ℝ d x) '' C ⊆ Set.Icc (-Causalean.PartialID.supportFn C (-d)) (Causalean.PartialID.supportFn C d)","l":68,"f":"Causalean/PO/ID/Partial/SupportFunction/Interval.lean"},{"n":"Causalean.PartialID.makarov_lower_iSup","k":"theorem","m":"Causalean.PO.ID.Partial.Frechet","a":"PO/ID/Partial","t":1,"d":"Makarov lower bound (easy direction), sup-convolution envelope form. For a fixed joint law of two real random variables X, Y, if Y is measurable, then taking the supremum, over every reference point, of the per-threshold Makarov lower bound still lower-bounds the CDF of the difference X - Y at the given threshold.","s":"∀ {Ω : Type u_1} [inst : MeasurableSpace Ω] (P : MeasureTheory.Measure Ω) [MeasureTheory.IsProbabilityMeasure P] {X Y : Ω → ℝ}, Measurable Y → ∀ (s : ℝ), ⨆ a, max ((P {ω | X ω ≤ a}).toReal - (P {ω | Y ω < a - s}).toReal) 0 ≤ (P {ω | X ω - Y ω ≤ s}).toReal","l":136,"f":"Causalean/PO/ID/Partial/Frechet.lean"},{"n":"Causalean.PartialID.makarov_lower_param","k":"theorem","m":"Causalean.PO.ID.Partial.Frechet","a":"PO/ID/Partial","t":1,"d":"Makarov lower bound (easy direction), per-threshold form. For a fixed joint law of two real random variables X, Y, if Y is measurable, then for every reference point a and threshold s, the CDF of the difference X - Y at s is at least the larger of zero and the gap between the CDF of X at a and the CDF of Y just below a - s: max (P(X ≤ a) - P(Y < a - s)) 0 ≤ P(X - Y ≤ s).","s":"∀ {Ω : Type u_1} [inst : MeasurableSpace Ω] (P : MeasureTheory.Measure Ω) [MeasureTheory.IsProbabilityMeasure P] {X Y : Ω → ℝ}, Measurable Y → ∀ (s a : ℝ), max ((P {ω | X ω ≤ a}).toReal - (P {ω | Y ω < a - s}).toReal) 0 ≤ (P {ω | X ω - Y ω ≤ s}).toReal","l":102,"f":"Causalean/PO/ID/Partial/Frechet.lean"},{"n":"Causalean.PartialID.mem_Icc_csInf_csSup","k":"theorem","m":"Causalean.PO.ID.Partial.Basic","a":"PO/ID/Partial","t":2,"d":"Worst/best case over a nuisance. If a set s ⊆ ℝ is bounded, every one of its members lies between sInf s and sSup s. Applied with s = range obj this is the engine form of \"the truth is bracketed by the extreme feasible values\".","s":"∀ {s : Set ℝ} {y : ℝ}, BddBelow s → BddAbove s → y ∈ s → y ∈ Set.Icc (sInf s) (sSup s)","l":43,"f":"Causalean/PO/ID/Partial/Basic.lean"},{"n":"Causalean.PartialID.mem_Icc_of_sandwich","k":"theorem","m":"Causalean.PO.ID.Partial.Basic","a":"PO/ID/Partial","t":2,"d":"Sandwich → membership. The literal content of a two-sided bound L ≤ θ ≤ U: the target functional θ lies in the reported interval [L, U]. Names the step that turns the inequality pair every concrete bound produces into the Set.Icc vocabulary.","s":"∀ {θ L U : ℝ}, L ≤ θ → θ ≤ U → θ ∈ Set.Icc L U","l":35,"f":"Causalean/PO/ID/Partial/Basic.lean"},{"n":"Causalean.PartialID.mem_affineBall","k":"theorem","m":"Causalean.PO.ID.Partial.SupportFunction.AffineBall","a":"PO/ID/Partial/SupportFunction","t":2,"d":"Membership in the affine-ball fiber is exactly satisfying the equation and the norm bound.","s":"∀ {H : Type u_1} {F : Type u_2} [inst : NormedAddCommGroup H] [inst_1 : InnerProductSpace ℝ H] [inst_2 : NormedAddCommGroup F] [inst_3 : InnerProductSpace ℝ F] {A : H →L[ℝ] F} {b : F} {B : ℝ} {h : H}, h ∈ Causalean.PartialID.affineBall A b B ↔ A h = b ∧ ‖h‖ ≤ B","l":72,"f":"Causalean/PO/ID/Partial/SupportFunction/AffineBall.lean"},{"n":"Causalean.PartialID.mem_identifiedInterval","k":"theorem","m":"Causalean.PO.ID.Partial.Basic","a":"PO/ID/Partial","t":2,"d":"A feasible parameter's objective value belongs to the identified interval.","s":"∀ {α : Type u_1} {obj : α → ℝ} {feasible : α → Prop} {x : α}, feasible x → obj x ∈ Causalean.PartialID.IdentifiedInterval obj feasible","l":30,"f":"Causalean/PO/ID/Partial/Basic.lean"},{"n":"Causalean.PartialID.mem_l2Ball","k":"theorem","m":"Causalean.PO.ID.Partial.SupportFunction.Sensitivity","a":"PO/ID/Partial/SupportFunction","t":2,"d":"Membership in the chi-square or L2 ambiguity set is exactly normalization and the radius bound.","s":"∀ {H : Type u_2} [inst : NormedAddCommGroup H] [inst_1 : InnerProductSpace ℝ H] {e : H} {ρ : ℝ} {w : H}, w ∈ Causalean.PartialID.l2Ball e ρ ↔ inner ℝ e w = 1 ∧ ‖w - e‖ ≤ ρ","l":178,"f":"Causalean/PO/ID/Partial/SupportFunction/Sensitivity.lean"},{"n":"Causalean.PartialID.neg_supportFn_neg_le","k":"theorem","m":"Causalean.PO.ID.Partial.SupportFunction.Interval","a":"PO/ID/Partial/SupportFunction","t":2,"d":"Lower endpoint bound. If ⟪-d, ·⟫ is bounded above on C, then the lower endpoint -supportFn C (-d) is a lower bound for the functional ⟪d, ·⟫.","s":"∀ {E : Type u_1} [inst : NormedAddCommGroup E] [inst_1 : InnerProductSpace ℝ E] {C : Set E} {d x : E}, x ∈ C → BddAbove ((fun x => inner ℝ (-d) x) '' C) → -Causalean.PartialID.supportFn C (-d) ≤ inner ℝ d x","l":59,"f":"Causalean/PO/ID/Partial/SupportFunction/Interval.lean"},{"n":"Causalean.PartialID.norm_orthogonalProjection_opKer_innerSL","k":"theorem","m":"Causalean.PO.ID.Partial.SupportFunction.Sensitivity","a":"PO/ID/Partial/SupportFunction","t":2,"d":"The norm of the free-direction projection equals the standard deviation √(‖c‖² − ⟪e,c⟫²) under ‖e‖ = 1.","s":"∀ {H : Type u_2} [inst : NormedAddCommGroup H] [inst_1 : InnerProductSpace ℝ H] [inst_2 : CompleteSpace H] {e : H}, ‖e‖ = 1 → ∀ (c : H), ‖↑((Causalean.PartialID.opKer ((innerSL ℝ) e)).orthogonalProjection c)‖ = √(‖c‖ ^ 2 - inner ℝ e c ^ 2)","l":212,"f":"Causalean/PO/ID/Partial/SupportFunction/Sensitivity.lean"},{"n":"Causalean.PartialID.opKer","k":"def","m":"Causalean.PO.ID.Partial.SupportFunction.AffineBall","a":"PO/ID/Partial/SupportFunction","t":1,"d":"The free-direction subspace of a continuous operator: perturbations in ker A do not change the linear constraint A h = b.","s":"{H : Type u_1} → {F : Type u_2} → [inst : NormedAddCommGroup H] → [inst_1 : InnerProductSpace ℝ H] → [inst_2 : NormedAddCommGroup F] → [inst_3 : InnerProductSpace ℝ F] → (H →L[ℝ] F) → Submodule ℝ H","l":56,"f":"Causalean/PO/ID/Partial/SupportFunction/AffineBall.lean"},{"n":"Causalean.PartialID.opKer_innerSL_eq","k":"theorem","m":"Causalean.PO.ID.Partial.SupportFunction.Sensitivity","a":"PO/ID/Partial/SupportFunction","t":2,"d":"The free-direction kernel for normalization against the base direction is its orthogonal complement: ker (innerSL ℝ e) = (ℝ ∙ e)ᗮ.","s":"∀ {H : Type u_2} [inst : NormedAddCommGroup H] [inst_1 : InnerProductSpace ℝ H] (e : H), Causalean.PartialID.opKer ((innerSL ℝ) e) = (ℝ ∙ e)ᗮ","l":184,"f":"Causalean/PO/ID/Partial/SupportFunction/Sensitivity.lean"},{"n":"Causalean.PartialID.robustInterval_eq_image","k":"theorem","m":"Causalean.PO.ID.Partial.SupportFunction.Sensitivity","a":"PO/ID/Partial/SupportFunction","t":1,"d":"The robust interval is the identified set. When the ambiguity set W is compact, convex, and nonempty, the set of reweighted means ⟪c, w⟫ attained as w ranges over W is exactly the closed interval [robustLower W c, robustUpper W c] — the robust bounds are sharp.","s":"∀ {E : Type u_1} [inst : NormedAddCommGroup E] [inst_1 : InnerProductSpace ℝ E] {W : Set E} {c : E}, IsCompact W → Convex ℝ W → W.Nonempty → (fun x => inner ℝ c x) '' W = Set.Icc (Causalean.PartialID.robustLower W c) (Causalean.PartialID.robustUpper W c)","l":156,"f":"Causalean/PO/ID/Partial/SupportFunction/Sensitivity.lean"},{"n":"Causalean.PartialID.robustLower","k":"def","m":"Causalean.PO.ID.Partial.SupportFunction.Sensitivity","a":"PO/ID/Partial/SupportFunction","t":1,"d":"The lower robust bound of the reweighted mean ⟪c, ·⟫ over an ambiguity set W: the worst case from below, -supportFn W (-c).","s":"{E : Type u_1} → [inst : NormedAddCommGroup E] → [InnerProductSpace ℝ E] → Set E → E → ℝ","l":99,"f":"Causalean/PO/ID/Partial/SupportFunction/Sensitivity.lean"},{"n":"Causalean.PartialID.robustLower_antitone","k":"theorem","m":"Causalean.PO.ID.Partial.SupportFunction.Sensitivity","a":"PO/ID/Partial/SupportFunction","t":2,"d":"Monotonicity in the budget (lower). A larger ambiguity set lowers the lower robust bound, so the robust interval widens.","s":"∀ {E : Type u_1} [inst : NormedAddCommGroup E] [inst_1 : InnerProductSpace ℝ E] {W W' : Set E} {c : E}, W ⊆ W' → W.Nonempty → BddAbove ((fun x => inner ℝ (-c) x) '' W') → Causalean.PartialID.robustLower W' c ≤ Causalean.PartialID.robustLower W c","l":133,"f":"Causalean/PO/ID/Partial/SupportFunction/Sensitivity.lean"},{"n":"Causalean.PartialID.robustLower_le","k":"theorem","m":"Causalean.PO.ID.Partial.SupportFunction.Sensitivity","a":"PO/ID/Partial/SupportFunction","t":2,"d":"Any admissible weight has a reweighted mean no smaller than the lower robust bound.","s":"∀ {E : Type u_1} [inst : NormedAddCommGroup E] [inst_1 : InnerProductSpace ℝ E] {W : Set E} {c w : E}, w ∈ W → BddAbove ((fun x => inner ℝ (-c) x) '' W) → Causalean.PartialID.robustLower W c ≤ inner ℝ c w","l":120,"f":"Causalean/PO/ID/Partial/SupportFunction/Sensitivity.lean"},{"n":"Causalean.PartialID.robustLower_le_robustUpper","k":"theorem","m":"Causalean.PO.ID.Partial.SupportFunction.Sensitivity","a":"PO/ID/Partial/SupportFunction","t":2,"d":"The robust interval [robustLower, robustUpper] is well-ordered under the usual boundedness conditions: its width is width W c ≥ 0.","s":"∀ {E : Type u_1} [inst : NormedAddCommGroup E] [inst_1 : InnerProductSpace ℝ E] {W : Set E} {c : E}, W.Nonempty → BddAbove ((fun x => inner ℝ c x) '' W) → BddAbove ((fun x => inner ℝ (-c) x) '' W) → Causalean.PartialID.robustLower W c ≤ Causalean.PartialID.robustUpper W c","l":103,"f":"Causalean/PO/ID/Partial/SupportFunction/Sensitivity.lean"},{"n":"Causalean.PartialID.robustLower_singleton","k":"theorem","m":"Causalean.PO.ID.Partial.SupportFunction.Sensitivity","a":"PO/ID/Partial/SupportFunction","t":1,"d":"No ambiguity implies point identification (lower). When the ambiguity set collapses to the singleton {w₀}, the lower robust bound for the linear functional c equals the identified value ⟪c, w₀⟫.","s":"∀ {E : Type u_1} [inst : NormedAddCommGroup E] [inst_1 : InnerProductSpace ℝ E] (c w₀ : E), Causalean.PartialID.robustLower {w₀} c = inner ℝ c w₀","l":149,"f":"Causalean/PO/ID/Partial/SupportFunction/Sensitivity.lean"},{"n":"Causalean.PartialID.robustUpper","k":"def","m":"Causalean.PO.ID.Partial.SupportFunction.Sensitivity","a":"PO/ID/Partial/SupportFunction","t":1,"d":"The upper robust bound of the reweighted mean ⟪c, ·⟫ over an ambiguity set W: the worst case from above, supportFn W c.","s":"{E : Type u_1} → [inst : NormedAddCommGroup E] → [InnerProductSpace ℝ E] → Set E → E → ℝ","l":95,"f":"Causalean/PO/ID/Partial/SupportFunction/Sensitivity.lean"},{"n":"Causalean.PartialID.robustUpper_mono","k":"theorem","m":"Causalean.PO.ID.Partial.SupportFunction.Sensitivity","a":"PO/ID/Partial/SupportFunction","t":2,"d":"Monotonicity in the budget (upper). A larger ambiguity set raises the upper robust bound.","s":"∀ {E : Type u_1} [inst : NormedAddCommGroup E] [inst_1 : InnerProductSpace ℝ E] {W W' : Set E} {c : E}, W ⊆ W' → W.Nonempty → BddAbove ((fun x => inner ℝ c x) '' W') → Causalean.PartialID.robustUpper W c ≤ Causalean.PartialID.robustUpper W' c","l":126,"f":"Causalean/PO/ID/Partial/SupportFunction/Sensitivity.lean"},{"n":"Causalean.PartialID.robustUpper_singleton","k":"theorem","m":"Causalean.PO.ID.Partial.SupportFunction.Sensitivity","a":"PO/ID/Partial/SupportFunction","t":1,"d":"No ambiguity implies point identification (upper). When the ambiguity set collapses to the singleton {w₀}, the upper robust bound for the linear functional c equals the identified value ⟪c, w₀⟫.","s":"∀ {E : Type u_1} [inst : NormedAddCommGroup E] [inst_1 : InnerProductSpace ℝ E] (c w₀ : E), Causalean.PartialID.robustUpper {w₀} c = inner ℝ c w₀","l":142,"f":"Causalean/PO/ID/Partial/SupportFunction/Sensitivity.lean"},{"n":"Causalean.PartialID.supportFn","k":"def","m":"Causalean.PO.ID.Partial.SupportFunction.Basic","a":"PO/ID/Partial/SupportFunction","t":1,"d":"The support function of a set C in direction d: supportFn C d = sup { ⟪d, x⟫ : x ∈ C }. Outside the nonempty and bounded-above regime, the raw sSup value is a convention and should not be used mathematically; meaningful statements carry Nonempty/BddAbove hypotheses (see le_supportFn, supportFn_le).","s":"{E : Type u_1} → [inst : NormedAddCommGroup E] → [InnerProductSpace ℝ E] → Set E → E → ℝ","l":55,"f":"Causalean/PO/ID/Partial/SupportFunction/Basic.lean"},{"n":"Causalean.PartialID.supportFn_add_dir_le","k":"theorem","m":"Causalean.PO.ID.Partial.SupportFunction.Calculus","a":"PO/ID/Partial/SupportFunction","t":1,"d":"Subadditivity in the direction (sublinearity of the support function). For a feasible set C in a real inner-product space and directions d₁, d₂, assuming C is nonempty and the linear functionals ⟪d₁, ·⟫ and ⟪d₂, ·⟫ are each bounded above on C, then the support value in the combined direction is at most the sum of the support values in each direction: supportFn C (d₁ + d₂) ≤ supportFn C d₁ + supportFn C d₂.","s":"∀ {E : Type u_1} [inst : NormedAddCommGroup E] [inst_1 : InnerProductSpace ℝ E] {C : Set E} {d₁ d₂ : E}, C.Nonempty → BddAbove ((fun x => inner ℝ d₁ x) '' C) → BddAbove ((fun x => inner ℝ d₂ x) '' C) → Causalean.PartialID.supportFn C (d₁ + d₂) ≤ Causalean.PartialID.supportFn C d₁ + Causalean.PartialID.supportFn C d₂","l":102,"f":"Causalean/PO/ID/Partial/SupportFunction/Calculus.lean"},{"n":"Causalean.PartialID.supportFn_affineBall_eq","k":"theorem","m":"Causalean.PO.ID.Partial.SupportFunction.AffineBall","a":"PO/ID/Partial/SupportFunction","t":1,"d":"Closed-form support value over the affine-ball fiber. Let A be a continuous linear map between real inner-product spaces, b a target value, B a radius, and c a direction. Suppose h₀ solves the linear constraint A h₀ = b, h₀ is orthogonal to the kernel of A (the minimum-norm solution), and h₀'s norm is at most B (h₀ fits the ball). Then the support value of ⟪c, ·⟫ over the fiber {h : A h = b, ‖h‖ ≤ B} equals ⟪c, h₀⟫ + √(B² − ‖h₀‖²)·‖P c‖, where P is the orthogonal projection onto the kernel of A.","s":"∀ {H : Type u_1} {F : Type u_2} [inst : NormedAddCommGroup H] [inst_1 : InnerProductSpace ℝ H] [inst_2 : CompleteSpace H] [inst_3 : NormedAddCommGroup F] [inst_4 : InnerProductSpace ℝ F] (A : H →L[ℝ] F) {b : F} {B : ℝ} {c h₀ : H}, A h₀ = b → h₀ ∈ (Causalean.PartialID.opKer A)ᗮ → ‖h₀‖ ≤ B → Causalean.PartialID.supportFn (Causalean.PartialID.affineBall A b B) c = inner ℝ c h₀ + √(B ^ 2 - ‖h₀‖ ^ 2) * ‖↑((Causalean.PartialID.opKer A).orthogonalProjection c)‖","l":76,"f":"Causalean/PO/ID/Partial/SupportFunction/AffineBall.lean"},{"n":"Causalean.PartialID.supportFn_eq_iSup_image","k":"theorem","m":"Causalean.PO.ID.Partial.SupportFunction.Basic","a":"PO/ID/Partial/SupportFunction","t":2,"d":"The support function is definitionally the supremum of the linear target over the set.","s":"∀ {E : Type u_1} [inst : NormedAddCommGroup E] [inst_1 : InnerProductSpace ℝ E] (C : Set E) (d : E), Causalean.PartialID.supportFn C d = sSup ((fun x => inner ℝ d x) '' C)","l":63,"f":"Causalean/PO/ID/Partial/SupportFunction/Basic.lean"},{"n":"Causalean.PartialID.supportFn_inter_le","k":"theorem","m":"Causalean.PO.ID.Partial.SupportFunction.Calculus","a":"PO/ID/Partial/SupportFunction","t":2,"d":"Intersection rule (one-sided). The support function of an intersection is at most the minimum of the parts.","s":"∀ {E : Type u_1} [inst : NormedAddCommGroup E] [inst_1 : InnerProductSpace ℝ E] {C D : Set E} {d : E}, (C ∩ D).Nonempty → BddAbove ((fun x => inner ℝ d x) '' C) → BddAbove ((fun x => inner ℝ d x) '' D) → Causalean.PartialID.supportFn (C ∩ D) d ≤ min (Causalean.PartialID.supportFn C d) (Causalean.PartialID.supportFn D d)","l":145,"f":"Causalean/PO/ID/Partial/SupportFunction/Calculus.lean"},{"n":"Causalean.PartialID.supportFn_l2Ball_eq","k":"theorem","m":"Causalean.PO.ID.Partial.SupportFunction.Sensitivity","a":"PO/ID/Partial/SupportFunction","t":1,"d":"Closed-form worst case (mean plus ρ·SD). For a unit-norm base direction e and a nonnegative radius ρ, the largest reweighted value ⟪c, w⟫ attains over the χ²/L² ambiguity set l2Ball e ρ equals ⟪e, c⟫ + ρ · √(‖c‖² − ⟪e, c⟫²) — the base-direction mean plus ρ standard deviations of c.","s":"∀ {H : Type u_2} [inst : NormedAddCommGroup H] [inst_1 : InnerProductSpace ℝ H] [CompleteSpace H] {e : H}, ‖e‖ = 1 → ∀ (c : H) {ρ : ℝ}, 0 ≤ ρ → Causalean.PartialID.supportFn (Causalean.PartialID.l2Ball e ρ) c = inner ℝ e c + ρ * √(‖c‖ ^ 2 - inner ℝ e c ^ 2)","l":238,"f":"Causalean/PO/ID/Partial/SupportFunction/Sensitivity.lean"},{"n":"Causalean.PartialID.supportFn_le","k":"theorem","m":"Causalean.PO.ID.Partial.SupportFunction.Basic","a":"PO/ID/Partial/SupportFunction","t":2,"d":"Upper sandwich. A uniform upper bound b on the functional over a nonempty C bounds the support value: supportFn C d ≤ b.","s":"∀ {E : Type u_1} [inst : NormedAddCommGroup E] [inst_1 : InnerProductSpace ℝ E] {C : Set E} {d : E} {b : ℝ}, C.Nonempty → (∀ x ∈ C, inner ℝ d x ≤ b) → Causalean.PartialID.supportFn C d ≤ b","l":78,"f":"Causalean/PO/ID/Partial/SupportFunction/Basic.lean"},{"n":"Causalean.PartialID.supportFn_minkowski","k":"theorem","m":"Causalean.PO.ID.Partial.SupportFunction.Calculus","a":"PO/ID/Partial/SupportFunction","t":2,"d":"Minkowski additivity. The support function is additive over Minkowski sums: supportFn (C + D) d = supportFn C d + supportFn D d.","s":"∀ {E : Type u_1} [inst : NormedAddCommGroup E] [inst_1 : InnerProductSpace ℝ E] {C D : Set E} {d : E}, C.Nonempty → D.Nonempty → BddAbove ((fun x => inner ℝ d x) '' C) → BddAbove ((fun x => inner ℝ d x) '' D) → Causalean.PartialID.supportFn (C + D) d = Causalean.PartialID.supportFn C d + Causalean.PartialID.supportFn D d","l":229,"f":"Causalean/PO/ID/Partial/SupportFunction/Calculus.lean"},{"n":"Causalean.PartialID.supportFn_mono","k":"theorem","m":"Causalean.PO.ID.Partial.SupportFunction.Calculus","a":"PO/ID/Partial/SupportFunction","t":2,"d":"Monotone in the set. If C ⊆ D (with C nonempty and the functional bounded above on D), then supportFn C d ≤ supportFn D d.","s":"∀ {E : Type u_1} [inst : NormedAddCommGroup E] [inst_1 : InnerProductSpace ℝ E] {C D : Set E} {d : E}, C ⊆ D → C.Nonempty → BddAbove ((fun x => inner ℝ d x) '' D) → Causalean.PartialID.supportFn C d ≤ Causalean.PartialID.supportFn D d","l":121,"f":"Causalean/PO/ID/Partial/SupportFunction/Calculus.lean"},{"n":"Causalean.PartialID.supportFn_nonneg_of_zero_mem","k":"theorem","m":"Causalean.PO.ID.Partial.SupportFunction.Calculus","a":"PO/ID/Partial/SupportFunction","t":2,"d":"If 0 ∈ C then the support value is nonnegative (the functional value 0 at the origin is a lower bound for the sup).","s":"∀ {E : Type u_1} [inst : NormedAddCommGroup E] [inst_1 : InnerProductSpace ℝ E] {C : Set E} {d : E}, 0 ∈ C → BddAbove ((fun x => inner ℝ d x) '' C) → 0 ≤ Causalean.PartialID.supportFn C d","l":57,"f":"Causalean/PO/ID/Partial/SupportFunction/Calculus.lean"},{"n":"Causalean.PartialID.supportFn_singleton","k":"theorem","m":"Causalean.PO.ID.Partial.SupportFunction.Calculus","a":"PO/ID/Partial/SupportFunction","t":2,"d":"Support function of a singleton is the functional value at the point.","s":"∀ {E : Type u_1} [inst : NormedAddCommGroup E] [inst_1 : InnerProductSpace ℝ E] (d x₀ : E), Causalean.PartialID.supportFn {x₀} d = inner ℝ d x₀","l":52,"f":"Causalean/PO/ID/Partial/SupportFunction/Calculus.lean"},{"n":"Causalean.PartialID.supportFn_smul_dir","k":"theorem","m":"Causalean.PO.ID.Partial.SupportFunction.Calculus","a":"PO/ID/Partial/SupportFunction","t":2,"d":"Positive homogeneity in the direction. For t ≥ 0, supportFn C (t • d) = t * supportFn C d.","s":"∀ {E : Type u_1} [inst : NormedAddCommGroup E] [inst_1 : InnerProductSpace ℝ E] {C : Set E} {d : E} {t : ℝ}, 0 ≤ t → C.Nonempty → BddAbove ((fun x => inner ℝ d x) '' C) → Causalean.PartialID.supportFn C (t • d) = t * Causalean.PartialID.supportFn C d","l":65,"f":"Causalean/PO/ID/Partial/SupportFunction/Calculus.lean"},{"n":"Causalean.PartialID.supportFn_smul_set","k":"theorem","m":"Causalean.PO.ID.Partial.SupportFunction.Calculus","a":"PO/ID/Partial/SupportFunction","t":2,"d":"Positive scaling rule. Scaling the set by a ≥ 0 scales the support value by a.","s":"∀ {E : Type u_1} [inst : NormedAddCommGroup E] [inst_1 : InnerProductSpace ℝ E] {C : Set E} {d : E} {a : ℝ}, 0 ≤ a → C.Nonempty → BddAbove ((fun x => inner ℝ d x) '' C) → Causalean.PartialID.supportFn (a • C) d = a * Causalean.PartialID.supportFn C d","l":182,"f":"Causalean/PO/ID/Partial/SupportFunction/Calculus.lean"},{"n":"Causalean.PartialID.supportFn_translate","k":"theorem","m":"Causalean.PO.ID.Partial.SupportFunction.Calculus","a":"PO/ID/Partial/SupportFunction","t":2,"d":"Translation rule. Translating the set by x₀ shifts the support value by ⟪d, x₀⟫.","s":"∀ {E : Type u_1} [inst : NormedAddCommGroup E] [inst_1 : InnerProductSpace ℝ E] {C : Set E} {d : E} (x₀ : E), C.Nonempty → BddAbove ((fun x => inner ℝ d x) '' C) → Causalean.PartialID.supportFn ((fun x => x₀ + x) '' C) d = inner ℝ d x₀ + Causalean.PartialID.supportFn C d","l":155,"f":"Causalean/PO/ID/Partial/SupportFunction/Calculus.lean"},{"n":"Causalean.PartialID.supportFn_union","k":"theorem","m":"Causalean.PO.ID.Partial.SupportFunction.Calculus","a":"PO/ID/Partial/SupportFunction","t":2,"d":"Union rule. The support function of a union is the maximum of the parts.","s":"∀ {E : Type u_1} [inst : NormedAddCommGroup E] [inst_1 : InnerProductSpace ℝ E] {C D : Set E} {d : E}, C.Nonempty → D.Nonempty → BddAbove ((fun x => inner ℝ d x) '' C) → BddAbove ((fun x => inner ℝ d x) '' D) → Causalean.PartialID.supportFn (C ∪ D) d = max (Causalean.PartialID.supportFn C d) (Causalean.PartialID.supportFn D d)","l":129,"f":"Causalean/PO/ID/Partial/SupportFunction/Calculus.lean"},{"n":"Causalean.PartialID.width","k":"def","m":"Causalean.PO.ID.Partial.SupportFunction.Interval","a":"PO/ID/Partial/SupportFunction","t":1,"d":"The width of the identified set: supportFn C d + supportFn C (-d), i.e. the upper endpoint minus the lower endpoint.","s":"{E : Type u_1} → [inst : NormedAddCommGroup E] → [InnerProductSpace ℝ E] → Set E → E → ℝ","l":125,"f":"Causalean/PO/ID/Partial/SupportFunction/Interval.lean"},{"n":"Causalean.PartialID.width_affineBall_eq","k":"theorem","m":"Causalean.PO.ID.Partial.SupportFunction.AffineBall","a":"PO/ID/Partial/SupportFunction","t":1,"d":"Closed-form width of the affine-ball identified set. Under the same hypotheses as supportFn_affineBall_eq — h₀ solves A h₀ = b, h₀ is orthogonal to the kernel of A, and h₀'s norm is at most B — the width of the identified set of ⟪c, ·⟫ over the fiber {h : A h = b, ‖h‖ ≤ B} equals 2·√(B² − ‖h₀‖²)·‖P c‖, twice the residual radius times the norm of c's projection onto the kernel of A.","s":"∀ {H : Type u_1} {F : Type u_2} [inst : NormedAddCommGroup H] [inst_1 : InnerProductSpace ℝ H] [inst_2 : CompleteSpace H] [inst_3 : NormedAddCommGroup F] [inst_4 : InnerProductSpace ℝ F] (A : H →L[ℝ] F) {b : F} {B : ℝ} {c h₀ : H}, A h₀ = b → h₀ ∈ (Causalean.PartialID.opKer A)ᗮ → ‖h₀‖ ≤ B → Causalean.PartialID.width (Causalean.PartialID.affineBall A b B) c = 2 * √(B ^ 2 - ‖h₀‖ ^ 2) * ‖↑((Causalean.PartialID.opKer A).orthogonalProjection c)‖","l":196,"f":"Causalean/PO/ID/Partial/SupportFunction/AffineBall.lean"},{"n":"Causalean.PartialID.width_eq_zero_iff","k":"theorem","m":"Causalean.PO.ID.Partial.SupportFunction.Interval","a":"PO/ID/Partial/SupportFunction","t":1,"d":"Point identification. For a feasible set C and direction d in a real inner-product space, assuming C is nonempty and the target functional ⟪d, ·⟫ is bounded above on C in the direction d and in the opposite direction -d, then the identified set collapses to a point (zero width) if and only if the target functional is constant on C, equal to its support value.","s":"∀ {E : Type u_1} [inst : NormedAddCommGroup E] [inst_1 : InnerProductSpace ℝ E] {C : Set E} {d : E}, C.Nonempty → BddAbove ((fun x => inner ℝ d x) '' C) → BddAbove ((fun x => inner ℝ (-d) x) '' C) → (Causalean.PartialID.width C d = 0 ↔ ∀ x ∈ C, inner ℝ d x = Causalean.PartialID.supportFn C d)","l":140,"f":"Causalean/PO/ID/Partial/SupportFunction/Interval.lean"},{"n":"Causalean.PartialID.width_l2Ball_eq","k":"theorem","m":"Causalean.PO.ID.Partial.SupportFunction.Sensitivity","a":"PO/ID/Partial/SupportFunction","t":1,"d":"Closed-form width. For a unit-norm base direction e and a nonnegative radius ρ, the width of the χ²/L² robust interval for the target c equals 2ρ · √(‖c‖² − ⟪e, c⟫²), twice the radius times the standard deviation of c.","s":"∀ {H : Type u_2} [inst : NormedAddCommGroup H] [inst_1 : InnerProductSpace ℝ H] [CompleteSpace H] {e : H}, ‖e‖ = 1 → ∀ (c : H) {ρ : ℝ}, 0 ≤ ρ → Causalean.PartialID.width (Causalean.PartialID.l2Ball e ρ) c = 2 * ρ * √(‖c‖ ^ 2 - inner ℝ e c ^ 2)","l":260,"f":"Causalean/PO/ID/Partial/SupportFunction/Sensitivity.lean"},{"n":"Causalean.PartialID.width_nonneg","k":"theorem","m":"Causalean.PO.ID.Partial.SupportFunction.Interval","a":"PO/ID/Partial/SupportFunction","t":2,"d":"The identified width is nonnegative.","s":"∀ {E : Type u_1} [inst : NormedAddCommGroup E] [inst_1 : InnerProductSpace ℝ E] {C : Set E} {d : E}, C.Nonempty → BddAbove ((fun x => inner ℝ d x) '' C) → BddAbove ((fun x => inner ℝ (-d) x) '' C) → 0 ≤ Causalean.PartialID.width C d","l":129,"f":"Causalean/PO/ID/Partial/SupportFunction/Interval.lean"},{"n":"Causalean.SCM","k":"structure","m":"Causalean.SCM.Model.SCM","a":"SCM/Model","t":1,"d":"A generalized structural causal model bundles a SWIG graph together with an edge-type assignment recording functional restrictions on each edge, the requirement that each fixed parameter and its corresponding random node share the same value space, a deterministic structural function for each observed node mapping its parents' values to its own, and an independent probability law on each latent root node.","s":"(N : Type u_2) → [DecidableEq N] → [Fintype N] → (Ω : N → Type u_3) → [(n : N) → MeasurableSpace (Ω n)] → Type (max u_2 u_3)","l":41,"f":"Causalean/SCM/Model/SCM.lean"},{"n":"Causalean.SCM.Assumptions.BoolChainNode","k":"inductive","m":"Causalean.SCM.ID.Assumptions.Monotonicity","a":"SCM/ID/Assumptions","t":1,"d":"Two variables for the Boolean monotonicity sanity check.","s":"Type","l":56,"f":"Causalean/SCM/ID/Assumptions/Monotonicity.lean"},{"n":"Causalean.SCM.Assumptions.BoolChainNode.dag","k":"def","m":"Causalean.SCM.ID.Assumptions.Monotonicity","a":"SCM/ID/Assumptions","t":1,"d":"The two-node DAG used by the Boolean monotonicity witnesses.","s":"Causalean.DAG Causalean.SCM.Assumptions.BoolChainNode","l":100,"f":"Causalean/SCM/ID/Assumptions/Monotonicity.lean"},{"n":"Causalean.SCM.Assumptions.BoolChainNode.edge","k":"def","m":"Causalean.SCM.ID.Assumptions.Monotonicity","a":"SCM/ID/Assumptions","t":1,"d":"The Boolean witness has one directed edge, from treatment d to outcome y.","s":"Causalean.SCM.Assumptions.BoolChainNode → Causalean.SCM.Assumptions.BoolChainNode → Prop","l":74,"f":"Causalean/SCM/ID/Assumptions/Monotonicity.lean"},{"n":"Causalean.SCM.Assumptions.BoolChainNode.edgeDecidable","k":"instance","m":"Causalean.SCM.ID.Assumptions.Monotonicity","a":"SCM/ID/Assumptions","t":2,"d":"","s":"DecidableRel Causalean.SCM.Assumptions.BoolChainNode.edge","l":79,"f":"Causalean/SCM/ID/Assumptions/Monotonicity.lean"},{"n":"Causalean.SCM.Assumptions.BoolChainNode.instFintype","k":"instance","m":"Causalean.SCM.ID.Assumptions.Monotonicity","a":"SCM/ID/Assumptions","t":2,"d":"The Boolean witness type has exactly the two elements d and y.","s":"Fintype Causalean.SCM.Assumptions.BoolChainNode","l":64,"f":"Causalean/SCM/ID/Assumptions/Monotonicity.lean"},{"n":"Causalean.SCM.Assumptions.BoolChainNode.topo","k":"def","m":"Causalean.SCM.ID.Assumptions.Monotonicity","a":"SCM/ID/Assumptions","t":1,"d":"The topological order places the parent before the child.","s":"Causalean.SCM.Assumptions.BoolChainNode → ℕ","l":87,"f":"Causalean/SCM/ID/Assumptions/Monotonicity.lean"},{"n":"Causalean.SCM.Assumptions.MonotoneMechanism","k":"def","m":"Causalean.SCM.ID.Assumptions.Monotonicity","a":"SCM/ID/Assumptions","t":1,"d":"Raising one parent coordinate cannot decrease the child equation.","s":"{N : Type u_1} → [inst : DecidableEq N] → [inst_1 : Fintype N] → {Ω : N → Type u_2} → [inst_2 : (n : N) → MeasurableSpace (Ω n)] → [(n : N) → Preorder (Ω n)] → Causalean.SWIGNode N → Causalean.SWIGNode N → Causalean.SCM N Ω → Prop","l":38,"f":"Causalean/SCM/ID/Assumptions/Monotonicity.lean"},{"n":"Causalean.SCM.Assumptions.antitoneBoolSCM","k":"def","m":"Causalean.SCM.ID.Assumptions.Monotonicity","a":"SCM/ID/Assumptions","t":1,"d":"A Boolean SCM whose outcome equation reverses the parent value.","s":"Causalean.SCM Causalean.SCM.Assumptions.BoolChainNode Causalean.SCM.Assumptions.boolChainΩ","l":224,"f":"Causalean/SCM/ID/Assumptions/Monotonicity.lean"},{"n":"Causalean.SCM.Assumptions.antitoneBoolSCM_violates","k":"theorem","m":"Causalean.SCM.ID.Assumptions.Monotonicity","a":"SCM/ID/Assumptions","t":1,"d":"The reversing Boolean structural causal model violates monotonicity of the outcome mechanism in the designated parent coordinate.","s":"¬Causalean.SCM.Assumptions.MonotoneMechanism (Causalean.SWIGNode.random Causalean.SCM.Assumptions.BoolChainNode.y) (Causalean.SWIGNode.random Causalean.SCM.Assumptions.BoolChainNode.d) Causalean.SCM.Assumptions.antitoneBoolSCM","l":276,"f":"Causalean/SCM/ID/Assumptions/Monotonicity.lean"},{"n":"Causalean.SCM.Assumptions.boolChainDParent","k":"def","m":"Causalean.SCM.ID.Assumptions.Monotonicity","a":"SCM/ID/Assumptions","t":1,"d":"The designated parent coordinate for the Boolean witness outcome.","s":"↥(Causalean.SCM.Assumptions.boolChainSWIG.dag.parents (Causalean.SWIGNode.random Causalean.SCM.Assumptions.BoolChainNode.y))","l":146,"f":"Causalean/SCM/ID/Assumptions/Monotonicity.lean"},{"n":"Causalean.SCM.Assumptions.boolChainMeasurableSpace","k":"instance","m":"Causalean.SCM.ID.Assumptions.Monotonicity","a":"SCM/ID/Assumptions","t":2,"d":"","s":"(n : Causalean.SCM.Assumptions.BoolChainNode) → MeasurableSpace (Causalean.SCM.Assumptions.boolChainΩ n)","l":138,"f":"Causalean/SCM/ID/Assumptions/Monotonicity.lean"},{"n":"Causalean.SCM.Assumptions.boolChainPreorder","k":"instance","m":"Causalean.SCM.ID.Assumptions.Monotonicity","a":"SCM/ID/Assumptions","t":2,"d":"","s":"(n : Causalean.SCM.Assumptions.BoolChainNode) → Preorder (Causalean.SCM.Assumptions.boolChainΩ n)","l":142,"f":"Causalean/SCM/ID/Assumptions/Monotonicity.lean"},{"n":"Causalean.SCM.Assumptions.boolChainSWIG","k":"def","m":"Causalean.SCM.ID.Assumptions.Monotonicity","a":"SCM/ID/Assumptions","t":1,"d":"The standard SWIG graph for the two-node Boolean witness.","s":"Causalean.SWIGGraph Causalean.SCM.Assumptions.BoolChainNode","l":110,"f":"Causalean/SCM/ID/Assumptions/Monotonicity.lean"},{"n":"Causalean.SCM.Assumptions.boolChainΩ","k":"def","m":"Causalean.SCM.ID.Assumptions.Monotonicity","a":"SCM/ID/Assumptions","t":1,"d":"Every variable in the Boolean witness has Boolean values.","s":"Causalean.SCM.Assumptions.BoolChainNode → Type","l":135,"f":"Causalean/SCM/ID/Assumptions/Monotonicity.lean"},{"n":"Causalean.SCM.Assumptions.boolParentAssignment","k":"def","m":"Causalean.SCM.ID.Assumptions.Monotonicity","a":"SCM/ID/Assumptions","t":1,"d":"The Boolean parent assignment that sets the designated parent to b.","s":"Bool → (w : ↥(Causalean.SCM.Assumptions.boolChainSWIG.dag.parents (Causalean.SWIGNode.random Causalean.SCM.Assumptions.BoolChainNode.y))) → Causalean.swigΩ Causalean.SCM.Assumptions.boolChainΩ ↑w","l":241,"f":"Causalean/SCM/ID/Assumptions/Monotonicity.lean"},{"n":"Causalean.SCM.Assumptions.boolParentAssignment_boolChainDParent","k":"theorem","m":"Causalean.SCM.ID.Assumptions.Monotonicity","a":"SCM/ID/Assumptions","t":2,"d":"Evaluating the Boolean parent assignment at the designated parent returns the assigned value.","s":"∀ (b : Bool), Causalean.SCM.Assumptions.boolParentAssignment b Causalean.SCM.Assumptions.boolChainDParent = b","l":258,"f":"Causalean/SCM/ID/Assumptions/Monotonicity.lean"},{"n":"Causalean.SCM.Assumptions.copyStructFun","k":"def","m":"Causalean.SCM.ID.Assumptions.Monotonicity","a":"SCM/ID/Assumptions","t":1,"d":"The Boolean copying structural equation for the witness SCM.","s":"(v : ↥Causalean.SCM.Assumptions.boolChainSWIG.observed) → ((w : ↥(Causalean.SCM.Assumptions.boolChainSWIG.dag.parents ↑v)) → Causalean.swigΩ Causalean.SCM.Assumptions.boolChainΩ ↑w) → Causalean.swigΩ Causalean.SCM.Assumptions.boolChainΩ ↑v","l":151,"f":"Causalean/SCM/ID/Assumptions/Monotonicity.lean"},{"n":"Causalean.SCM.Assumptions.flipStructFun","k":"def","m":"Causalean.SCM.ID.Assumptions.Monotonicity","a":"SCM/ID/Assumptions","t":1,"d":"The Boolean reversing structural equation for the witness SCM.","s":"(v : ↥Causalean.SCM.Assumptions.boolChainSWIG.observed) → ((w : ↥(Causalean.SCM.Assumptions.boolChainSWIG.dag.parents ↑v)) → Causalean.swigΩ Causalean.SCM.Assumptions.boolChainΩ ↑w) → Causalean.swigΩ Causalean.SCM.Assumptions.boolChainΩ ↑v","l":167,"f":"Causalean/SCM/ID/Assumptions/Monotonicity.lean"},{"n":"Causalean.SCM.Assumptions.instDecidableEqBoolChainNode","k":"instance","m":"Causalean.SCM.ID.Assumptions.Monotonicity","a":"SCM/ID/Assumptions","t":2,"d":"","s":"DecidableEq Causalean.SCM.Assumptions.BoolChainNode","l":60,"f":"Causalean/SCM/ID/Assumptions/Monotonicity.lean"},{"n":"Causalean.SCM.Assumptions.instPreorderSwigΩ","k":"instance","m":"Causalean.SCM.ID.Assumptions.Monotonicity","a":"SCM/ID/Assumptions","t":2,"d":"The SWIG value spaces inherit the base-variable order.","s":"{N : Type u_1} → {Ω : N → Type u_2} → [(n : N) → Preorder (Ω n)] → (s : Causalean.SWIGNode N) → Preorder (Causalean.swigΩ Ω s)","l":29,"f":"Causalean/SCM/ID/Assumptions/Monotonicity.lean"},{"n":"Causalean.SCM.Assumptions.instReprBoolChainNode","k":"instance","m":"Causalean.SCM.ID.Assumptions.Monotonicity","a":"SCM/ID/Assumptions","t":2,"d":"","s":"Repr Causalean.SCM.Assumptions.BoolChainNode","l":60,"f":"Causalean/SCM/ID/Assumptions/Monotonicity.lean"},{"n":"Causalean.SCM.Assumptions.monotoneBoolSCM","k":"def","m":"Causalean.SCM.ID.Assumptions.Monotonicity","a":"SCM/ID/Assumptions","t":1,"d":"A Boolean SCM whose outcome equation copies the parent value.","s":"Causalean.SCM Causalean.SCM.Assumptions.BoolChainNode Causalean.SCM.Assumptions.boolChainΩ","l":207,"f":"Causalean/SCM/ID/Assumptions/Monotonicity.lean"},{"n":"Causalean.SCM.Assumptions.monotoneBoolSCM_satisfies","k":"theorem","m":"Causalean.SCM.ID.Assumptions.Monotonicity","a":"SCM/ID/Assumptions","t":1,"d":"The copying Boolean structural causal model satisfies monotonicity of the outcome mechanism in the designated parent coordinate.","s":"Causalean.SCM.Assumptions.MonotoneMechanism (Causalean.SWIGNode.random Causalean.SCM.Assumptions.BoolChainNode.y) (Causalean.SWIGNode.random Causalean.SCM.Assumptions.BoolChainNode.d) Causalean.SCM.Assumptions.monotoneBoolSCM","l":264,"f":"Causalean/SCM/ID/Assumptions/Monotonicity.lean"},{"n":"Causalean.SCM.BackdoorPositivityAE","k":"def","m":"Causalean.SCM.ID.Backdoor","a":"SCM/ID","t":1,"d":"Joint (product) positivity / overlap at base s₀.","s":"{N : Type u_1} → [inst : DecidableEq N] → [inst_1 : Fintype N] → {Ω : N → Type u_2} → [inst_2 : (n : N) → MeasurableSpace (Ω n)] → (M : Causalean.SCM N Ω) → (X : Finset N) → (Z : Finset (Causalean.SWIGNode N)) → Z ⊆ M.observed → Finset.image Causalean.SWIGNode.random X ∪ Z ⊆ M.observed → M.FixedValues → Prop","l":66,"f":"Causalean/SCM/ID/Backdoor.lean"},{"n":"Causalean.SCM.DistMarkovEquiv","k":"def","m":"Causalean.Graph.MarkovEquiv.Distributional","a":"Graph/MarkovEquiv","t":1,"d":"Two DAGs are distributionally Markov equivalent (over value spaces Ω) when they are global I-maps of exactly the same distributions over every structural causal model on the same node set. The value-space family Ω is an explicit parameter since it is not determined by the graphs.","s":"{N : Type uN} → [inst : DecidableEq N] → [inst_1 : Fintype N] → (Ω : N → Type uΩ) → [(n : N) → MeasurableSpace (Ω n)] → Causalean.DAG (Causalean.SWIGNode N) → Causalean.DAG (Causalean.SWIGNode N) → Prop","l":66,"f":"Causalean/Graph/MarkovEquiv/Distributional.lean"},{"n":"Causalean.SCM.DominatedObs","k":"def","m":"Causalean.SCM.ID.Density.ReferenceMeasure","a":"SCM/ID/Density","t":1,"d":"A structural causal model is dominated when each observational law is absolutely continuous with respect to the joint reference measure.","s":"{N : Type u_1} → [inst : DecidableEq N] → [inst_1 : Fintype N] → {Ω : N → Type u_2} → [inst_2 : (n : N) → MeasurableSpace (Ω n)] → Causalean.SCM N Ω → Causalean.SCM.ReferenceMeasures Ω → Prop","l":65,"f":"Causalean/SCM/ID/Density/ReferenceMeasure.lean"},{"n":"Causalean.SCM.Equiv","k":"def","m":"Causalean.SCM.Model.SCM","a":"SCM/Model","t":1,"d":"Two structural causal models are structurally equivalent when their graph, edge labels, structural functions, and latent laws agree.","s":"{N : Type uN} → [inst : DecidableEq N] → [inst_1 : Fintype N] → {Ω : N → Type uΩ} → [inst_2 : (n : N) → MeasurableSpace (Ω n)] → Causalean.SCM N Ω → Causalean.SCM N Ω → Prop","l":253,"f":"Causalean/SCM/Model/SCM.lean"},{"n":"Causalean.SCM.Equiv.heq_jointKernel","k":"theorem","m":"Causalean.SCM.Model.EquivKernel","a":"SCM/Model","t":2,"d":"HEq transport for jointKernel.","s":"∀ {N : Type u_1} [inst : DecidableEq N] [inst_1 : Fintype N] {Ω : N → Type u_2} [inst_2 : (n : N) → MeasurableSpace (Ω n)] {M₁ M₂ : Causalean.SCM N Ω}, M₁.Equivalent M₂.toSWIGGraph → M₁.structFun ≍ M₂.structFun → M₁.latentDist ≍ M₂.latentDist → M₁.jointKernel ≍ M₂.jointKernel","l":245,"f":"Causalean/SCM/Model/EquivKernel.lean"},{"n":"Causalean.SCM.Equiv.heq_latentProduct","k":"theorem","m":"Causalean.SCM.Model.EquivKernel","a":"SCM/Model","t":2,"d":"HEq transport for latentProduct.","s":"∀ {N : Type u_1} [inst : DecidableEq N] [inst_1 : Fintype N] {Ω : N → Type u_2} [inst_2 : (n : N) → MeasurableSpace (Ω n)] {M₁ M₂ : Causalean.SCM N Ω}, M₁.unobserved = M₂.unobserved → M₁.latentDist ≍ M₂.latentDist → M₁.latentProduct ≍ M₂.latentProduct","l":216,"f":"Causalean/SCM/Model/EquivKernel.lean"},{"n":"Causalean.SCM.Equiv.heq_obsCondKernel","k":"theorem","m":"Causalean.SCM.Model.EquivKernel","a":"SCM/Model","t":2,"d":"HEq transport for obsCondKernel.","s":"∀ {N : Type u_1} [inst : DecidableEq N] [inst_1 : Fintype N] {Ω : N → Type u_2} [inst_2 : (n : N) → MeasurableSpace (Ω n)] {M₁ M₂ : Causalean.SCM N Ω} (hGraph : M₁.Equivalent M₂.toSWIGGraph), M₁.structFun ≍ M₂.structFun → M₁.latentDist ≍ M₂.latentDist → ∀ (Y CC : Finset (Causalean.SWIGNode N)) (hY₁ : Y ⊆ M₁.observed) (hCC₁ : CC ⊆ M₁.observed) [inst_3 : StandardBorelSpace (Causalean.ValuesOn Y (Causalean.swigΩ Ω))] [inst_4 : Nonempty (Causalean.ValuesOn Y (Causalean.swigΩ Ω))] [inst_5 : MeasurableSpace.CountableOrCountablyGenerated M₁.FixedValues (Causalean.ValuesOn CC (Causalean.swigΩ Ω))] [inst_6 : MeasurableSpace.CountableOrCountablyGenerated M₂.FixedValues (Causalean.ValuesOn CC (Causalean.swigΩ Ω))], have hY₂ := fun ⦃v⦄ hv => Eq.mpr (id (congrArg (fun _a => v ∈ _a) (Eq.symm hGraph.right.right.left))) (hY₁ hv); have hCC₂ := fun ⦃v⦄ hv => Eq.mpr (id (congrArg (fun _a => v ∈ _a) (Eq.symm hGraph.right.right.left))) (hCC₁ hv); M₁.obsCondKernel Y CC hY₁ hCC₁ ≍ M₂.obsCondKernel Y CC hY₂ hCC₂","l":378,"f":"Causalean/SCM/Model/EquivKernel.lean"},{"n":"Causalean.SCM.Equiv.heq_obsKernel","k":"theorem","m":"Causalean.SCM.Model.EquivKernel","a":"SCM/Model","t":1,"d":"HEq transport for obsKernel. For two structural causal models M₁, M₂ that are structurally equivalent, have heterogeneously equal structural functions, and have heterogeneously equal latent-root distributions, then their observational kernels are heterogeneously equal.","s":"∀ {N : Type u_1} [inst : DecidableEq N] [inst_1 : Fintype N] {Ω : N → Type u_2} [inst_2 : (n : N) → MeasurableSpace (Ω n)] {M₁ M₂ : Causalean.SCM N Ω}, M₁.Equivalent M₂.toSWIGGraph → M₁.structFun ≍ M₂.structFun → M₁.latentDist ≍ M₂.latentDist → M₁.obsKernel ≍ M₂.obsKernel","l":296,"f":"Causalean/SCM/Model/EquivKernel.lean"},{"n":"Causalean.SCM.Equiv.refl","k":"theorem","m":"Causalean.SCM.Model.SCM","a":"SCM/Model","t":2,"d":"Structural equivalence is reflexive.","s":"∀ {N : Type uN} [inst : DecidableEq N] [inst_1 : Fintype N] {Ω : N → Type uΩ} [inst_2 : (n : N) → MeasurableSpace (Ω n)] (M : Causalean.SCM N Ω), M.Equiv M","l":267,"f":"Causalean/SCM/Model/SCM.lean"},{"n":"Causalean.SCM.Equiv.symm","k":"theorem","m":"Causalean.SCM.Model.SCM","a":"SCM/Model","t":2,"d":"Structural equivalence is symmetric.","s":"∀ {N : Type uN} [inst : DecidableEq N] [inst_1 : Fintype N] {Ω : N → Type uΩ} [inst_2 : (n : N) → MeasurableSpace (Ω n)] {M₁ M₂ : Causalean.SCM N Ω}, M₁.Equiv M₂ → M₂.Equiv M₁","l":275,"f":"Causalean/SCM/Model/SCM.lean"},{"n":"Causalean.SCM.Equiv.trans","k":"theorem","m":"Causalean.SCM.Model.SCM","a":"SCM/Model","t":1,"d":"Structural equivalence is transitive: if M₁ and M₂ are structurally equivalent and M₂ and M₃ are structurally equivalent, then M₁ and M₃ are structurally equivalent.","s":"∀ {N : Type uN} [inst : DecidableEq N] [inst_1 : Fintype N] {Ω : N → Type uΩ} [inst_2 : (n : N) → MeasurableSpace (Ω n)] {M₁ M₂ M₃ : Causalean.SCM N Ω}, M₁.Equiv M₂ → M₂.Equiv M₃ → M₁.Equiv M₃","l":285,"f":"Causalean/SCM/Model/SCM.lean"},{"n":"Causalean.SCM.Examples.BackDoor.BDNode","k":"inductive","m":"Causalean.SCM.Examples.BackDoor","a":"SCM/Examples","t":1,"d":"The backdoor example has treatment, outcome, observed-confounder, and three latent-root vertices.","s":"Type","l":70,"f":"Causalean/SCM/Examples/BackDoor.lean"},{"n":"Causalean.SCM.Examples.BackDoor.bdDAG","k":"def","m":"Causalean.SCM.Examples.BackDoor","a":"SCM/Examples","t":1,"d":"This directed acyclic graph formalizes the canonical backdoor-adjustment example.","s":"Causalean.DAG Causalean.SCM.Examples.BackDoor.BDNode","l":151,"f":"Causalean/SCM/Examples/BackDoor.lean"},{"n":"Causalean.SCM.Examples.BackDoor.bdEdge","k":"def","m":"Causalean.SCM.Examples.BackDoor","a":"SCM/Examples","t":1,"d":"The backdoor graph has confounding paths through the observed confounder, a direct treatment-outcome edge, and one latent root for each observed node.","s":"Causalean.SCM.Examples.BackDoor.BDNode → Causalean.SCM.Examples.BackDoor.BDNode → Prop","l":114,"f":"Causalean/SCM/Examples/BackDoor.lean"},{"n":"Causalean.SCM.Examples.BackDoor.bdEdgeTypes","k":"def","m":"Causalean.SCM.Examples.BackDoor","a":"SCM/Examples","t":1,"d":"The backdoor example treats every graph edge as nonparametric.","s":"Causalean.EdgeTypeAssignment Causalean.SCM.Examples.BackDoor.bdDAG","l":373,"f":"Causalean/SCM/Examples/BackDoor.lean"},{"n":"Causalean.SCM.Examples.BackDoor.bdSWIG","k":"def","m":"Causalean.SCM.Examples.BackDoor","a":"SCM/Examples","t":1,"d":"This SWIG graph represents the backdoor example before any intervention.","s":"Causalean.SWIGGraph Causalean.SCM.Examples.BackDoor.BDNode","l":208,"f":"Causalean/SCM/Examples/BackDoor.lean"},{"n":"Causalean.SCM.Examples.BackDoor.bdTopo","k":"def","m":"Causalean.SCM.Examples.BackDoor","a":"SCM/Examples","t":1,"d":"The backdoor graph orders latent roots first, then confounder, treatment, and outcome.","s":"Causalean.SCM.Examples.BackDoor.BDNode → ℕ","l":132,"f":"Causalean/SCM/Examples/BackDoor.lean"},{"n":"Causalean.SCM.Examples.BackDoor.bdTopo_lt","k":"theorem","m":"Causalean.SCM.Examples.BackDoor","a":"SCM/Examples","t":1,"d":"Every edge of the backdoor-example graph connects a node with a strictly smaller assigned order label to one with a strictly larger label — the chosen ordering is a valid topological order for the graph.","s":"∀ (u v : Causalean.SCM.Examples.BackDoor.BDNode), Causalean.SCM.Examples.BackDoor.bdEdge u v → Causalean.SCM.Examples.BackDoor.bdTopo u < Causalean.SCM.Examples.BackDoor.bdTopo v","l":141,"f":"Causalean/SCM/Examples/BackDoor.lean"},{"n":"Causalean.SCM.Examples.BackDoor.instDecidableEqBDNode","k":"instance","m":"Causalean.SCM.Examples.BackDoor","a":"SCM/Examples","t":2,"d":"","s":"DecidableEq Causalean.SCM.Examples.BackDoor.BDNode","l":78,"f":"Causalean/SCM/Examples/BackDoor.lean"},{"n":"Causalean.SCM.Examples.BackDoor.instDecidableRelBDNodeBdEdge","k":"instance","m":"Causalean.SCM.Examples.BackDoor","a":"SCM/Examples","t":2,"d":"Whether a proposed backdoor-example edge is present is decidable by case analysis on the endpoints.","s":"DecidableRel Causalean.SCM.Examples.BackDoor.bdEdge","l":124,"f":"Causalean/SCM/Examples/BackDoor.lean"},{"n":"Causalean.SCM.Examples.BackDoor.instFintypeBDNode","k":"instance","m":"Causalean.SCM.Examples.BackDoor","a":"SCM/Examples","t":2,"d":"The backdoor-example vertex set is finite, with six named vertices.","s":"Fintype Causalean.SCM.Examples.BackDoor.BDNode","l":105,"f":"Causalean/SCM/Examples/BackDoor.lean"},{"n":"Causalean.SCM.Examples.BackDoor.instReprBDNode","k":"instance","m":"Causalean.SCM.Examples.BackDoor","a":"SCM/Examples","t":2,"d":"Backdoor-example vertices can be rendered as their fully qualified constructor names.","s":"Repr Causalean.SCM.Examples.BackDoor.BDNode","l":101,"f":"Causalean/SCM/Examples/BackDoor.lean"},{"n":"Causalean.SCM.Examples.ContinuousBackdoor.CBNode","k":"def","m":"Causalean.SCM.Examples.ContinuousBackdoor","a":"SCM/Examples","t":1,"d":"The continuous-backdoor example has three nodes: confounder, treatment, and outcome.","s":"Type","l":78,"f":"Causalean/SCM/Examples/ContinuousBackdoor.lean"},{"n":"Causalean.SCM.Examples.ContinuousBackdoor.CBΩ","k":"def","m":"Causalean.SCM.Examples.ContinuousBackdoor","a":"SCM/Examples","t":1,"d":"Every node in the continuous-backdoor example takes real values.","s":"Causalean.SCM.Examples.ContinuousBackdoor.CBNode → Type","l":81,"f":"Causalean/SCM/Examples/ContinuousBackdoor.lean"},{"n":"Causalean.SCM.Examples.ContinuousBackdoor.Xidx","k":"def","m":"Causalean.SCM.Examples.ContinuousBackdoor","a":"SCM/Examples","t":1,"d":"This node index denotes the treatment in the continuous-backdoor example.","s":"Causalean.SCM.Examples.ContinuousBackdoor.CBNode","l":86,"f":"Causalean/SCM/Examples/ContinuousBackdoor.lean"},{"n":"Causalean.SCM.Examples.ContinuousBackdoor.Yidx","k":"def","m":"Causalean.SCM.Examples.ContinuousBackdoor","a":"SCM/Examples","t":1,"d":"This node index denotes the outcome in the continuous-backdoor example.","s":"Causalean.SCM.Examples.ContinuousBackdoor.CBNode","l":88,"f":"Causalean/SCM/Examples/ContinuousBackdoor.lean"},{"n":"Causalean.SCM.Examples.ContinuousBackdoor.Zidx","k":"def","m":"Causalean.SCM.Examples.ContinuousBackdoor","a":"SCM/Examples","t":1,"d":"This node index denotes the confounder in the continuous-backdoor example.","s":"Causalean.SCM.Examples.ContinuousBackdoor.CBNode","l":84,"f":"Causalean/SCM/Examples/ContinuousBackdoor.lean"},{"n":"Causalean.SCM.Examples.ContinuousBackdoor.cbDAG","k":"def","m":"Causalean.SCM.Examples.ContinuousBackdoor","a":"SCM/Examples","t":1,"d":"This directed acyclic graph formalizes the three-node continuous backdoor example.","s":"Causalean.DAG Causalean.SCM.Examples.ContinuousBackdoor.CBNode","l":117,"f":"Causalean/SCM/Examples/ContinuousBackdoor.lean"},{"n":"Causalean.SCM.Examples.ContinuousBackdoor.cbEdge","k":"def","m":"Causalean.SCM.Examples.ContinuousBackdoor","a":"SCM/Examples","t":1,"d":"The edge relation says exactly that the confounder points to treatment and outcome, and treatment points to outcome.","s":"Causalean.SCM.Examples.ContinuousBackdoor.CBNode → Causalean.SCM.Examples.ContinuousBackdoor.CBNode → Prop","l":102,"f":"Causalean/SCM/Examples/ContinuousBackdoor.lean"},{"n":"Causalean.SCM.Examples.ContinuousBackdoor.cbEdgeBool","k":"def","m":"Causalean.SCM.Examples.ContinuousBackdoor","a":"SCM/Examples","t":1,"d":"The edge indicator records confounder-to-treatment, confounder-to-outcome, and treatment-to-outcome arrows.","s":"Causalean.SCM.Examples.ContinuousBackdoor.CBNode → Causalean.SCM.Examples.ContinuousBackdoor.CBNode → Bool","l":95,"f":"Causalean/SCM/Examples/ContinuousBackdoor.lean"},{"n":"Causalean.SCM.Examples.ContinuousBackdoor.cbSWIGGraph","k":"def","m":"Causalean.SCM.Examples.ContinuousBackdoor","a":"SCM/Examples","t":1,"d":"This computable SWIG graph represents the continuous-backdoor example before any intervention.","s":"Causalean.SWIGGraph Causalean.SCM.Examples.ContinuousBackdoor.CBNode","l":127,"f":"Causalean/SCM/Examples/ContinuousBackdoor.lean"},{"n":"Causalean.SCM.Examples.ContinuousBackdoor.cbTopo","k":"def","m":"Causalean.SCM.Examples.ContinuousBackdoor","a":"SCM/Examples","t":1,"d":"The continuous-backdoor graph orders confounder before treatment before outcome.","s":"Causalean.SCM.Examples.ContinuousBackdoor.CBNode → ℕ","l":109,"f":"Causalean/SCM/Examples/ContinuousBackdoor.lean"},{"n":"Causalean.SCM.Examples.ContinuousBackdoor.cbTopo_lt","k":"theorem","m":"Causalean.SCM.Examples.ContinuousBackdoor","a":"SCM/Examples","t":2,"d":"Every edge in the continuous-backdoor graph points from an earlier to a later node in the chosen topological order.","s":"∀ (u v : Causalean.SCM.Examples.ContinuousBackdoor.CBNode), Causalean.SCM.Examples.ContinuousBackdoor.cbEdge u v → Causalean.SCM.Examples.ContinuousBackdoor.cbTopo u < Causalean.SCM.Examples.ContinuousBackdoor.cbTopo v","l":112,"f":"Causalean/SCM/Examples/ContinuousBackdoor.lean"},{"n":"Causalean.SCM.Examples.ContinuousBackdoor.cb_Xfixed","k":"theorem","m":"Causalean.SCM.Examples.ContinuousBackdoor","a":"SCM/Examples","t":2,"d":"The treatment's fixed node is not already fixed in the computable continuous-backdoor SWIG graph.","s":"∀ D ∈ {Causalean.SCM.Examples.ContinuousBackdoor.Xidx}, Causalean.SWIGNode.fixed D ∉ Causalean.SCM.Examples.ContinuousBackdoor.cbSWIGGraph.fixed","l":218,"f":"Causalean/SCM/Examples/ContinuousBackdoor.lean"},{"n":"Causalean.SCM.Examples.ContinuousBackdoor.cb_XrZ_obs","k":"theorem","m":"Causalean.SCM.Examples.ContinuousBackdoor","a":"SCM/Examples","t":2,"d":"Treatment random-image together with the adjustment set is observed.","s":"Finset.image Causalean.SWIGNode.random {Causalean.SCM.Examples.ContinuousBackdoor.Xidx} ∪ {Causalean.SWIGNode.random Causalean.SCM.Examples.ContinuousBackdoor.Zidx} ⊆ Causalean.SCM.Examples.ContinuousBackdoor.continuousBackdoorSCM.observed","l":287,"f":"Causalean/SCM/Examples/ContinuousBackdoor.lean"},{"n":"Causalean.SCM.Examples.ContinuousBackdoor.cb_Xr_obs","k":"theorem","m":"Causalean.SCM.Examples.ContinuousBackdoor","a":"SCM/Examples","t":2,"d":"The treatment random-image is observed.","s":"Finset.image Causalean.SWIGNode.random {Causalean.SCM.Examples.ContinuousBackdoor.Xidx} ⊆ Causalean.SCM.Examples.ContinuousBackdoor.continuousBackdoorSCM.observed","l":279,"f":"Causalean/SCM/Examples/ContinuousBackdoor.lean"},{"n":"Causalean.SCM.Examples.ContinuousBackdoor.cb_Xrand_obs","k":"theorem","m":"Causalean.SCM.Examples.ContinuousBackdoor","a":"SCM/Examples","t":2,"d":"The treatment's observed random node belongs to the computable continuous-backdoor SWIG graph.","s":"∀ D ∈ {Causalean.SCM.Examples.ContinuousBackdoor.Xidx}, Causalean.SWIGNode.random D ∈ Causalean.SCM.Examples.ContinuousBackdoor.cbSWIGGraph.observed","l":205,"f":"Causalean/SCM/Examples/ContinuousBackdoor.lean"},{"n":"Causalean.SCM.Examples.ContinuousBackdoor.cb_Yobs","k":"theorem","m":"Causalean.SCM.Examples.ContinuousBackdoor","a":"SCM/Examples","t":2,"d":"{random Y} is an observed node set.","s":"{Causalean.SWIGNode.random Causalean.SCM.Examples.ContinuousBackdoor.Yidx} ⊆ Causalean.SCM.Examples.ContinuousBackdoor.continuousBackdoorSCM.observed","l":265,"f":"Causalean/SCM/Examples/ContinuousBackdoor.lean"},{"n":"Causalean.SCM.Examples.ContinuousBackdoor.cb_Zobs","k":"theorem","m":"Causalean.SCM.Examples.ContinuousBackdoor","a":"SCM/Examples","t":2,"d":"{random Z} (the adjustment set) is an observed node set.","s":"{Causalean.SWIGNode.random Causalean.SCM.Examples.ContinuousBackdoor.Zidx} ⊆ Causalean.SCM.Examples.ContinuousBackdoor.continuousBackdoorSCM.observed","l":272,"f":"Causalean/SCM/Examples/ContinuousBackdoor.lean"},{"n":"Causalean.SCM.Examples.ContinuousBackdoor.cb_backdoor_criterion","k":"theorem","m":"Causalean.SCM.Examples.ContinuousBackdoor","a":"SCM/Examples","t":1,"d":"The observed confounder satisfies the graphical backdoor criterion relative to the treatment and outcome nodes in the continuous-backdoor example graph.","s":"Causalean.SCM.Examples.ContinuousBackdoor.cbSWIGGraph.backdoorCriterion {Causalean.SCM.Examples.ContinuousBackdoor.Xidx} Causalean.SCM.Examples.ContinuousBackdoor.cb_Xrand_obs Causalean.SCM.Examples.ContinuousBackdoor.cb_Xfixed {Causalean.SWIGNode.random Causalean.SCM.Examples.ContinuousBackdoor.Yidx} {Causalean.SWIGNode.random Causalean.SCM.Examples.ContinuousBackdoor.Zidx}","l":224,"f":"Causalean/SCM/Examples/ContinuousBackdoor.lean"},{"n":"Causalean.SCM.Examples.ContinuousBackdoor.cb_backdoor_identified","k":"theorem","m":"Causalean.SCM.Examples.ContinuousBackdoor","a":"SCM/Examples","t":1,"d":"Backdoor adjustment identity on the continuous-backdoor example. Fix an assignment s0 of values to the model's fixed background variables. If for every post-intervention background assignment, the do(X)-intervened marginal law of Z is absolutely continuous with respect to its purely observational marginal law — the Rule-2 joint overlap condition, and the product of the observational marginal laws of the treatment's random image and of Z is absolutely continuous with respect to their joint observational law — the joint positivity condition, then almost everywhere under that product measure, the conditional law of the outcome Y given Z under the intervention that fixes the treatment equals the purely observational conditional law of Y given both the treatment and Z — the backdoor Rule-2 adjustment identity.","s":"∀ (s0 : Causalean.SCM.Examples.ContinuousBackdoor.continuousBackdoorSCM.FixedValues), (∀ (s : (Causalean.SCM.Examples.ContinuousBackdoor.continuousBackdoorSCM.fixSet {Causalean.SCM.Examples.ContinuousBackdoor.Xidx} Causalean.SCM.Examples.ContinuousBackdoor.cb_Xrand_obs Causalean.SCM.Examples.ContinuousBackdoor.cb_Xfixed).FixedValues), Causalean.SCM.ID.Rule2JointOverlap Causalean.SCM.Examples.ContinuousBackdoor.continuousBackdoorSCM {Causalean.SCM.Examples.ContinuousBackdoor.Xidx} Causalean.SCM.Examples.ContinuousBackdoor.cb_Xrand_obs Causalean.SCM.Examples.ContinuousBackdoor.cb_Xfixed {Causalean.SWIGNode.random Causalean.SCM.Examples.ContinuousBackdoor.Zidx} Causalean.SCM.Examples.ContinuousBackdoor.cb_XrZ_obs s) → (MeasureTheory.Measure.map (fun p => Causalean.valuesUnionMk p.1 p.2) ((MeasureTheory.Measure.map (Causalean.valuesProjection Causalean.SCM.Examples.ContinuousBackdoor.cb_Xr_obs) (Causalean.SCM.Examples.ContinuousBackdoor.continuousBackdoorSCM.obsKernel s0)).compProd (ProbabilityTheory.Kernel.const (Causalean.ValuesOn (Finset.image Causalean.SWIGNode.random {Causalean.SCM.Examples.ContinuousBackdoor.Xidx}) (Causalean.swigΩ Causalean.SCM.Examples.ContinuousBackdoor.CBΩ)) (MeasureTheory.Measure.map (Causalean.valuesProjection Causalean.SCM.Examples.ContinuousBackdoor.cb_Zobs) (Causalean.SCM.Examples.ContinuousBackdoor.continuousBackdoorSCM.obsKernel s0))))).AbsolutelyContinuous (MeasureTheory.Measure.map (Causalean.valuesProjection Causalean.SCM.Examples.ContinuousBackdoor.cb_XrZ_obs) (Causalean.SCM.Examples.ContinuousBackdoor.continuousBackdoorSCM.obsKernel s0)) → ∀ᵐ (p : Causalean.ValuesOn (Finset.image Causalean.SWIGNode.random {Causalean.SCM.Examples.ContinuousBackdoor.Xidx}) (Causalean.swigΩ Causalean.SCM.Examples.ContinuousBackdoor.CBΩ) × Causalean.ValuesOn {Causalean.SWIGNode.random Causalean.SCM.Examples.ContinuousBackdoor.Zidx} (Causalean.swigΩ Causalean.SCM.Examples.ContinuousBackdoor.CBΩ)) ∂(MeasureTheory.Measure.map (Causalean.valuesProjection Causalean.SCM.Examples.ContinuousBackdoor.cb_Xr_obs) (Causalean.SCM.Examples.ContinuousBackdoor.continuousBackdoorSCM.obsKernel s0)).compProd (ProbabilityTheory.Kernel.const (Causalean.ValuesOn (Finset.image Causalean.SWIGNode.random {Causalean.SCM.Examples.ContinuousBackdoor.Xidx}) (Causalean.swigΩ Causalean.SCM.Examples.ContinuousBackdoor.CBΩ)) (MeasureTheory.Measure.map (Causalean.valuesProjection Causalean.SCM.Examples.ContinuousBackdoor.cb_Zobs) (Causalean.SCM.Examples.ContinuousBackdoor.continuousBackdoorSCM.obsKernel s0))), ((Causalean.SCM.Examples.ContinuousBackdoor.continuousBackdoorSCM.fixSet {Causalean.SCM.Examples.ContinuousBackdoor.Xidx} Causalean.SCM.Examples.ContinuousBackdoor.cb_Xrand_obs Causalean.SCM.Examples.ContinuousBackdoor.cb_Xfixed).obsCondKernel {Causalean.SWIGNode.random Causalean.SCM.Examples.ContinuousBackdoor.Yidx} {Causalean.SWIGNode.random Causalean.SCM.Examples.ContinuousBackdoor.Zidx} (Eq.symm (Causalean.SCM.fixSet_observed Causalean.SCM.Examples.ContinuousBackdoor.continuousBackdoorSCM {Causalean.SCM.Examples.ContinuousBackdoor.Xidx} Causalean.SCM.Examples.ContinuousBackdoor.cb_Xrand_obs Causalean.SCM.Examples.ContinuousBackdoor.cb_Xfixed) ▸ Causalean.SCM.Examples.ContinuousBackdoor.cb_Yobs) (Eq.symm (Causalean.SCM.fixSet_observed Causalean.SCM.Examples.ContinuousBackdoor.continuousBackdoorSCM {Causalean.SCM.Examples.ContinuousBackdoor.Xidx} Causalean.SCM.Examples.ContinuousBackdoor.cb_Xrand_obs Causalean.SCM.Examples.ContinuousBackdoor.cb_Xfixed) ▸ Causalean.SCM.Examples.ContinuousBackdoor.cb_Zobs)) (Causalean.SCM.Examples.ContinuousBackdoor.continuousBackdoorSCM.fixSetExtend {Causalean.SCM.Examples.ContinuousBackdoor.Xidx} Causalean.SCM.Examples.ContinuousBackdoor.cb_Xrand_obs Causalean.SCM.Examples.ContinuousBackdoor.cb_Xfixed s0 p.1, p.2) = (Causalean.SCM.Examples.ContinuousBackdoor.continuousBackdoorSCM.obsCondKernel {Causalean.SWIGNode.random Causalean.SCM.Examples.ContinuousBackdoor.Yidx} (Finset.image Causalean.SWIGNode.random {Causalean.SCM.Examples.ContinuousBackdoor.Xidx} ∪ {Causalean.SWIGNode.random Causalean.SCM.Examples.ContinuousBackdoor.Zidx}) Causalean.SCM.Examples.ContinuousBackdoor.cb_Yobs Causalean.SCM.Examples.ContinuousBackdoor.cb_XrZ_obs) (s0, Causalean.valuesUnionMk p.1 p.2)","l":302,"f":"Causalean/SCM/Examples/ContinuousBackdoor.lean"},{"n":"Causalean.SCM.Examples.ContinuousBackdoor.cb_disj_XrZ","k":"theorem","m":"Causalean.SCM.Examples.ContinuousBackdoor","a":"SCM/Examples","t":2,"d":"The treatment random-image is disjoint from the adjustment set.","s":"Disjoint (Finset.image Causalean.SWIGNode.random {Causalean.SCM.Examples.ContinuousBackdoor.Xidx}) {Causalean.SWIGNode.random Causalean.SCM.Examples.ContinuousBackdoor.Zidx}","l":297,"f":"Causalean/SCM/Examples/ContinuousBackdoor.lean"},{"n":"Causalean.SCM.Examples.ContinuousBackdoor.cb_disj_YXr","k":"theorem","m":"Causalean.SCM.Examples.ContinuousBackdoor","a":"SCM/Examples","t":2,"d":"The outcome is disjoint from the treatment random-image.","s":"Disjoint {Causalean.SWIGNode.random Causalean.SCM.Examples.ContinuousBackdoor.Yidx} (Finset.image Causalean.SWIGNode.random {Causalean.SCM.Examples.ContinuousBackdoor.Xidx})","l":292,"f":"Causalean/SCM/Examples/ContinuousBackdoor.lean"},{"n":"Causalean.SCM.Examples.ContinuousBackdoor.continuousBackdoorSCM","k":"def","m":"Causalean.SCM.Examples.ContinuousBackdoor","a":"SCM/Examples","t":1,"d":"This real-valued backdoor structural model has observed nodes for a confounder, treatment, and outcome, with no latent variables.","s":"Causalean.SCM Causalean.SCM.Examples.ContinuousBackdoor.CBNode Causalean.SCM.Examples.ContinuousBackdoor.CBΩ","l":158,"f":"Causalean/SCM/Examples/ContinuousBackdoor.lean"},{"n":"Causalean.SCM.Examples.ContinuousBackdoor.instDecidableRelCBNodeCbEdge","k":"instance","m":"Causalean.SCM.Examples.ContinuousBackdoor","a":"SCM/Examples","t":2,"d":"Whether a proposed continuous-backdoor edge is present is decidable by evaluating the edge indicator.","s":"DecidableRel Causalean.SCM.Examples.ContinuousBackdoor.cbEdge","l":105,"f":"Causalean/SCM/Examples/ContinuousBackdoor.lean"},{"n":"Causalean.SCM.Examples.Frontdoor.FDNode","k":"inductive","m":"Causalean.SCM.Examples.Frontdoor","a":"SCM/Examples","t":1,"d":"The frontdoor example has latent confounder, treatment, mediator, and outcome vertices.","s":"Type","l":34,"f":"Causalean/SCM/Examples/Frontdoor.lean"},{"n":"Causalean.SCM.Examples.Frontdoor.fdDAG","k":"def","m":"Causalean.SCM.Examples.Frontdoor","a":"SCM/Examples","t":1,"d":"This directed acyclic graph formalizes the canonical frontdoor example.","s":"Causalean.DAG Causalean.SCM.Examples.Frontdoor.FDNode","l":85,"f":"Causalean/SCM/Examples/Frontdoor.lean"},{"n":"Causalean.SCM.Examples.Frontdoor.fdEdge","k":"def","m":"Causalean.SCM.Examples.Frontdoor","a":"SCM/Examples","t":1,"d":"The frontdoor graph has latent confounding of X and Y, and the directed path X → M → Y.","s":"Causalean.SCM.Examples.Frontdoor.FDNode → Causalean.SCM.Examples.Frontdoor.FDNode → Prop","l":53,"f":"Causalean/SCM/Examples/Frontdoor.lean"},{"n":"Causalean.SCM.Examples.Frontdoor.fdSWIG","k":"def","m":"Causalean.SCM.Examples.Frontdoor","a":"SCM/Examples","t":1,"d":"This SWIG graph represents the frontdoor example before any intervention.","s":"Causalean.SWIGGraph Causalean.SCM.Examples.Frontdoor.FDNode","l":95,"f":"Causalean/SCM/Examples/Frontdoor.lean"},{"n":"Causalean.SCM.Examples.Frontdoor.fdTopo","k":"def","m":"Causalean.SCM.Examples.Frontdoor","a":"SCM/Examples","t":1,"d":"The frontdoor graph orders latent confounder, treatment, mediator, then outcome.","s":"Causalean.SCM.Examples.Frontdoor.FDNode → ℕ","l":69,"f":"Causalean/SCM/Examples/Frontdoor.lean"},{"n":"Causalean.SCM.Examples.Frontdoor.fdTopo_lt","k":"theorem","m":"Causalean.SCM.Examples.Frontdoor","a":"SCM/Examples","t":1,"d":"Every edge of the frontdoor-example graph connects a node with a smaller assigned order label to one with a larger label, so the chosen ordering is a valid topological order.","s":"∀ (u v : Causalean.SCM.Examples.Frontdoor.FDNode), Causalean.SCM.Examples.Frontdoor.fdEdge u v → Causalean.SCM.Examples.Frontdoor.fdTopo u < Causalean.SCM.Examples.Frontdoor.fdTopo v","l":76,"f":"Causalean/SCM/Examples/Frontdoor.lean"},{"n":"Causalean.SCM.Examples.Frontdoor.instDecidableEqFDNode","k":"instance","m":"Causalean.SCM.Examples.Frontdoor","a":"SCM/Examples","t":2,"d":"","s":"DecidableEq Causalean.SCM.Examples.Frontdoor.FDNode","l":40,"f":"Causalean/SCM/Examples/Frontdoor.lean"},{"n":"Causalean.SCM.Examples.Frontdoor.instDecidableRelFDNodeFdEdge","k":"instance","m":"Causalean.SCM.Examples.Frontdoor","a":"SCM/Examples","t":2,"d":"Whether a proposed frontdoor-example edge is present is decidable by endpoint cases.","s":"DecidableRel Causalean.SCM.Examples.Frontdoor.fdEdge","l":61,"f":"Causalean/SCM/Examples/Frontdoor.lean"},{"n":"Causalean.SCM.Examples.Frontdoor.instFintypeFDNode","k":"instance","m":"Causalean.SCM.Examples.Frontdoor","a":"SCM/Examples","t":2,"d":"The frontdoor-example vertex set is finite, with four named vertices.","s":"Fintype Causalean.SCM.Examples.Frontdoor.FDNode","l":44,"f":"Causalean/SCM/Examples/Frontdoor.lean"},{"n":"Causalean.SCM.Examples.Frontdoor.instReprFDNode","k":"instance","m":"Causalean.SCM.Examples.Frontdoor","a":"SCM/Examples","t":2,"d":"","s":"Repr Causalean.SCM.Examples.Frontdoor.FDNode","l":40,"f":"Causalean/SCM/Examples/Frontdoor.lean"},{"n":"Causalean.SCM.Examples.IV.IVNode","k":"inductive","m":"Causalean.SCM.Examples.IV","a":"SCM/Examples","t":1,"d":"The instrumental-variable example has instrument, treatment, outcome, and unobserved-confounder vertices.","s":"Type","l":47,"f":"Causalean/SCM/Examples/IV.lean"},{"n":"Causalean.SCM.Examples.IV.instDecidableEqIVNode","k":"instance","m":"Causalean.SCM.Examples.IV","a":"SCM/Examples","t":2,"d":"","s":"DecidableEq Causalean.SCM.Examples.IV.IVNode","l":54,"f":"Causalean/SCM/Examples/IV.lean"},{"n":"Causalean.SCM.Examples.IV.instDecidableRelIVNodeIvEdge","k":"instance","m":"Causalean.SCM.Examples.IV","a":"SCM/Examples","t":2,"d":"A proposed instrumental-variable edge is decidable by case analysis on its endpoints.","s":"DecidableRel Causalean.SCM.Examples.IV.ivEdge","l":97,"f":"Causalean/SCM/Examples/IV.lean"},{"n":"Causalean.SCM.Examples.IV.instFintypeIVNode","k":"instance","m":"Causalean.SCM.Examples.IV","a":"SCM/Examples","t":2,"d":"The instrumental-variable vertex set is finite, with four named vertices.","s":"Fintype Causalean.SCM.Examples.IV.IVNode","l":79,"f":"Causalean/SCM/Examples/IV.lean"},{"n":"Causalean.SCM.Examples.IV.instReprIVNode","k":"instance","m":"Causalean.SCM.Examples.IV","a":"SCM/Examples","t":2,"d":"Instrumental-variable vertices can be rendered as their fully qualified constructor names.","s":"Repr Causalean.SCM.Examples.IV.IVNode","l":75,"f":"Causalean/SCM/Examples/IV.lean"},{"n":"Causalean.SCM.Examples.IV.ivDAG","k":"def","m":"Causalean.SCM.Examples.IV","a":"SCM/Examples","t":1,"d":"This directed acyclic graph formalizes the standard instrumental-variable example.","s":"Causalean.DAG Causalean.SCM.Examples.IV.IVNode","l":126,"f":"Causalean/SCM/Examples/IV.lean"},{"n":"Causalean.SCM.Examples.IV.ivDoDGraph","k":"def","m":"Causalean.SCM.Examples.IV","a":"SCM/Examples","t":1,"d":"This graph is the SWIG obtained by intervening on the treatment in the instrumental-variable DAG.","s":"Causalean.DAG (Causalean.SWIGNode Causalean.SCM.Examples.IV.IVNode)","l":395,"f":"Causalean/SCM/Examples/IV.lean"},{"n":"Causalean.SCM.Examples.IV.ivEdge","k":"def","m":"Causalean.SCM.Examples.IV","a":"SCM/Examples","t":1,"d":"The instrumental-variable graph has instrument-to-treatment, treatment-to-outcome, and latent-confounder-to-treatment/outcome edges.","s":"Causalean.SCM.Examples.IV.IVNode → Causalean.SCM.Examples.IV.IVNode → Prop","l":88,"f":"Causalean/SCM/Examples/IV.lean"},{"n":"Causalean.SCM.Examples.IV.ivEdgeTypes","k":"def","m":"Causalean.SCM.Examples.IV","a":"SCM/Examples","t":1,"d":"The instrumental-variable edge assignment makes the instrument's effect on treatment strictly increasing and leaves all other edges nonparametric.","s":"Causalean.EdgeTypeAssignment Causalean.SCM.Examples.IV.ivDAG","l":285,"f":"Causalean/SCM/Examples/IV.lean"},{"n":"Causalean.SCM.Examples.IV.ivSCM","k":"def","m":"Causalean.SCM.Examples.IV","a":"SCM/Examples","t":1,"d":"This concrete instrumental-variable structural model realizes the example with one-point value spaces.","s":"Causalean.SCM Causalean.SCM.Examples.IV.IVNode Causalean.SCM.Examples.IV.ivΩ","l":333,"f":"Causalean/SCM/Examples/IV.lean"},{"n":"Causalean.SCM.Examples.IV.ivSWIGGraph","k":"def","m":"Causalean.SCM.Examples.IV","a":"SCM/Examples","t":1,"d":"This SWIG graph represents the instrumental-variable example before any intervention.","s":"Causalean.SWIGGraph Causalean.SCM.Examples.IV.IVNode","l":232,"f":"Causalean/SCM/Examples/IV.lean"},{"n":"Causalean.SCM.Examples.IV.ivTopo","k":"def","m":"Causalean.SCM.Examples.IV","a":"SCM/Examples","t":1,"d":"The instrumental-variable graph orders the unobserved confounder and instrument before treatment and outcome.","s":"Causalean.SCM.Examples.IV.IVNode → ℕ","l":108,"f":"Causalean/SCM/Examples/IV.lean"},{"n":"Causalean.SCM.Examples.IV.ivTopo_lt","k":"theorem","m":"Causalean.SCM.Examples.IV","a":"SCM/Examples","t":1,"d":"Every edge of the instrumental-variable example graph connects a node with a smaller assigned order label to one with a larger label, so the chosen ordering is a valid topological order.","s":"∀ (u v : Causalean.SCM.Examples.IV.IVNode), Causalean.SCM.Examples.IV.ivEdge u v → Causalean.SCM.Examples.IV.ivTopo u < Causalean.SCM.Examples.IV.ivTopo v","l":116,"f":"Causalean/SCM/Examples/IV.lean"},{"n":"Causalean.SCM.Examples.IV.ivΩ","k":"def","m":"Causalean.SCM.Examples.IV","a":"SCM/Examples","t":1,"d":"This toy instrumental-variable model uses a one-point value space for every node.","s":"Causalean.SCM.Examples.IV.IVNode → Type","l":324,"f":"Causalean/SCM/Examples/IV.lean"},{"n":"Causalean.SCM.Examples.IV.ivΩ_measurable","k":"instance","m":"Causalean.SCM.Examples.IV","a":"SCM/Examples","t":2,"d":"Every one-point value space in the toy instrumental-variable model carries the trivial measurable structure.","s":"(n : Causalean.SCM.Examples.IV.IVNode) → MeasurableSpace (Causalean.SCM.Examples.IV.ivΩ n)","l":330,"f":"Causalean/SCM/Examples/IV.lean"},{"n":"Causalean.SCM.Examples.MonotoneCounterfactualBound.boolParentAssignmentIn","k":"def","m":"Causalean.SCM.Examples.MonotoneCounterfactualBound","a":"SCM/Examples","t":1,"d":"A parent assignment inside an arbitrary Boolean-chain SCM.","s":"(M : Causalean.SCM Causalean.SCM.Assumptions.BoolChainNode Causalean.SCM.Assumptions.boolChainΩ) → Bool → (w : ↥(M.dag.parents (Causalean.SWIGNode.random Causalean.SCM.Assumptions.BoolChainNode.y))) → Causalean.swigΩ Causalean.SCM.Assumptions.boolChainΩ ↑w","l":38,"f":"Causalean/SCM/Examples/MonotoneCounterfactualBound.lean"},{"n":"Causalean.SCM.Examples.MonotoneCounterfactualBound.boolParentAssignmentIn_parent","k":"theorem","m":"Causalean.SCM.Examples.MonotoneCounterfactualBound","a":"SCM/Examples","t":2,"d":"The designated parent coordinate receives the assigned Boolean value.","s":"∀ (M : Causalean.SCM Causalean.SCM.Assumptions.BoolChainNode Causalean.SCM.Assumptions.boolChainΩ) (hparent : Causalean.SWIGNode.random Causalean.SCM.Assumptions.BoolChainNode.d ∈ M.dag.parents (Causalean.SWIGNode.random Causalean.SCM.Assumptions.BoolChainNode.y)) (b : Bool), Causalean.SCM.Examples.MonotoneCounterfactualBound.boolParentAssignmentIn M b ⟨Causalean.SWIGNode.random Causalean.SCM.Assumptions.BoolChainNode.d, hparent⟩ = b","l":55,"f":"Causalean/SCM/Examples/MonotoneCounterfactualBound.lean"},{"n":"Causalean.SCM.Examples.MonotoneCounterfactualBound.boolScore","k":"def","m":"Causalean.SCM.Examples.MonotoneCounterfactualBound","a":"SCM/Examples","t":1,"d":"A Boolean value is scored as one for true and zero for false.","s":"Bool → ℝ","l":31,"f":"Causalean/SCM/Examples/MonotoneCounterfactualBound.lean"},{"n":"Causalean.SCM.Examples.MonotoneCounterfactualBound.monotoneCounterfactualBound","k":"theorem","m":"Causalean.SCM.Examples.MonotoneCounterfactualBound","a":"SCM/Examples","t":1,"d":"Structural monotonicity of the outcome mechanism in the treatment parent forces every response contrast in the compatible class to lie in the valid [0, 1] partial-identification interval.","s":"Causalean.SCM.PartialID.compatibleInterval Causalean.SCM.Assumptions.boolChainSWIG (Causalean.SCM.Assumptions.MonotoneMechanism (Causalean.SWIGNode.random Causalean.SCM.Assumptions.BoolChainNode.y) (Causalean.SWIGNode.random Causalean.SCM.Assumptions.BoolChainNode.d)) Causalean.SCM.Assumptions.monotoneBoolSCM Causalean.SCM.Examples.MonotoneCounterfactualBound.monotoneResponseContrast ⊆ Set.Icc 0 1","l":108,"f":"Causalean/SCM/Examples/MonotoneCounterfactualBound.lean"},{"n":"Causalean.SCM.Examples.MonotoneCounterfactualBound.monotoneCounterfactualBound_assumption_satisfiable","k":"theorem","m":"Causalean.SCM.Examples.MonotoneCounterfactualBound","a":"SCM/Examples","t":1,"d":"The class of structural causal models compatible with the Boolean-chain graph and monotone in the outcome's treatment parent, evaluated against the copying model as reference, is nonempty.","s":"Causalean.SCM.PartialID.CompatibleSCM Causalean.SCM.Assumptions.boolChainSWIG (Causalean.SCM.Assumptions.MonotoneMechanism (Causalean.SWIGNode.random Causalean.SCM.Assumptions.BoolChainNode.y) (Causalean.SWIGNode.random Causalean.SCM.Assumptions.BoolChainNode.d)) Causalean.SCM.Assumptions.monotoneBoolSCM Causalean.SCM.Assumptions.monotoneBoolSCM","l":153,"f":"Causalean/SCM/Examples/MonotoneCounterfactualBound.lean"},{"n":"Causalean.SCM.Examples.MonotoneCounterfactualBound.monotoneCounterfactualBound_fails_without_monotonicity","k":"theorem","m":"Causalean.SCM.Examples.MonotoneCounterfactualBound","a":"SCM/Examples","t":1,"d":"Without the monotonicity restriction, the unconstrained compatible class's response contrast need not lie in [0, 1].","s":"¬Causalean.SCM.PartialID.compatibleInterval Causalean.SCM.Assumptions.boolChainSWIG (fun x => True) Causalean.SCM.Assumptions.antitoneBoolSCM Causalean.SCM.Examples.MonotoneCounterfactualBound.monotoneResponseContrast ⊆ Set.Icc 0 1","l":189,"f":"Causalean/SCM/Examples/MonotoneCounterfactualBound.lean"},{"n":"Causalean.SCM.Examples.MonotoneCounterfactualBound.monotoneResponseContrast","k":"def","m":"Causalean.SCM.Examples.MonotoneCounterfactualBound","a":"SCM/Examples","t":1,"d":"The real-valued response contrast of the Boolean outcome equation.","s":"Causalean.SCM Causalean.SCM.Assumptions.BoolChainNode Causalean.SCM.Assumptions.boolChainΩ → ℝ","l":70,"f":"Causalean/SCM/Examples/MonotoneCounterfactualBound.lean"},{"n":"Causalean.SCM.Examples.MonotoneCounterfactualBound.monotoneResponseContrast_antitoneBoolSCM","k":"theorem","m":"Causalean.SCM.Examples.MonotoneCounterfactualBound","a":"SCM/Examples","t":2,"d":"The reversing Boolean SCM has response contrast minus one.","s":"Causalean.SCM.Examples.MonotoneCounterfactualBound.monotoneResponseContrast Causalean.SCM.Assumptions.antitoneBoolSCM = -1","l":177,"f":"Causalean/SCM/Examples/MonotoneCounterfactualBound.lean"},{"n":"Causalean.SCM.Examples.MonotoneCounterfactualBound.monotoneResponseContrast_monotoneBoolSCM","k":"theorem","m":"Causalean.SCM.Examples.MonotoneCounterfactualBound","a":"SCM/Examples","t":2,"d":"The copying Boolean SCM has response contrast one.","s":"Causalean.SCM.Examples.MonotoneCounterfactualBound.monotoneResponseContrast Causalean.SCM.Assumptions.monotoneBoolSCM = 1","l":165,"f":"Causalean/SCM/Examples/MonotoneCounterfactualBound.lean"},{"n":"Causalean.SCM.FixedValues","k":"def","m":"Causalean.SCM.Model.SCM","a":"SCM/Model","t":1,"d":"Fixed values assign intervention or fixed-node values in a structural causal model.","s":"{N : Type uN} → [inst : DecidableEq N] → [inst_1 : Fintype N] → {Ω : N → Type uΩ} → [inst_2 : (n : N) → MeasurableSpace (Ω n)] → Causalean.SCM N Ω → Type (max uΩ uN)","l":110,"f":"Causalean/SCM/Model/SCM.lean"},{"n":"Causalean.SCM.FullCondIndep","k":"def","m":"Causalean.SCM.Do.FullCondIndep","a":"SCM/Do","t":1,"d":"Full conditional independence on RandomValues M (= V ∪ L).","s":"{N : Type uN} → [inst : DecidableEq N] → [inst_1 : Fintype N] → {Ω : N → Type uΩ} → [inst_2 : (n : N) → MeasurableSpace (Ω n)] → (M : Causalean.SCM N Ω) → [StandardBorelSpace M.RandomValues] → (X Y Z : Finset (Causalean.SWIGNode N)) → X ⊆ M.randomVars → Y ⊆ M.randomVars → Z ⊆ M.randomVars → (μ : MeasureTheory.Measure M.RandomValues) → [MeasureTheory.IsFiniteMeasure μ] → Prop","l":32,"f":"Causalean/SCM/Do/FullCondIndep.lean"},{"n":"Causalean.SCM.ID.CFactorReachableRec","k":"inductive","m":"Causalean.SCM.ID.GraphicalThms.DoGFormulaRec","a":"SCM/ID/GraphicalThms","t":1,"d":"Recursive IDENTIFY reachability (full Tian–Shpitser success certificate). CFactorReachableRec G T C holds when identify(C, T, Q[T]) succeeds, i.e. the c-factor Q[C] is recoverable from Q[T].","s":"{N : Type u_1} → [inst : DecidableEq N] → [inst_1 : Fintype N] → Causalean.SWIGGraph N → Finset (Causalean.SWIGNode N) → Finset (Causalean.SWIGNode N) → Prop","l":75,"f":"Causalean/SCM/ID/GraphicalThms/DoGFormulaRec.lean"},{"n":"Causalean.SCM.ID.CFactorReachableRec.target_subset","k":"theorem","m":"Causalean.SCM.ID.Density.QFactor","a":"SCM/ID/Density","t":2,"d":"The target set of a recursive c-factor reachability certificate is contained in the source set.","s":"∀ {N : Type u_1} [inst : DecidableEq N] [inst_1 : Fintype N] {G : Causalean.SWIGGraph N} {T C : Finset (Causalean.SWIGNode N)}, Causalean.SCM.ID.CFactorReachableRec G T C → C ⊆ T","l":1047,"f":"Causalean/SCM/ID/Density/QFactor.lean"},{"n":"Causalean.SCM.ID.CausalQuery","k":"def","m":"Causalean.SCM.ID.Identifiable","a":"SCM/ID","t":1,"d":"A causal query is a functional of a causal model, producing a value in some type α.","s":"(N : Type u_2) → [DecidableEq N] → [Fintype N] → (Ω : N → Type u_3) → [(n : N) → MeasurableSpace (Ω n)] → Type u_4 → Type (max (max u_4 u_3) u_2)","l":62,"f":"Causalean/SCM/ID/Identifiable.lean"},{"n":"Causalean.SCM.ID.DiscreteID.DiscretePositive","k":"def","m":"Causalean.SCM.ID.DiscreteID.Positive","a":"SCM/ID/DiscreteID","t":1,"d":"A discrete SCM has full observational support on every observed assignment, at every fixed-value slice. This is strong but non-vacuous, and it is the safe default assumption for first-pass discrete ID soundness.","s":"{N : Type u_1} → [inst : DecidableEq N] → [inst_1 : Fintype N] → {Ω : N → Type u_2} → [inst_2 : (n : N) → MeasurableSpace (Ω n)] → Causalean.SCM N Ω → Prop","l":45,"f":"Causalean/SCM/ID/DiscreteID/Positive.lean"},{"n":"Causalean.SCM.ID.DiscreteID.PositiveMass","k":"def","m":"Causalean.SCM.ID.DiscreteID.Positive","a":"SCM/ID/DiscreteID","t":1,"d":"A measure has full point-mass support when every singleton has nonzero mass.","s":"{α : Type u_3} → [inst : MeasurableSpace α] → MeasureTheory.Measure α → Prop","l":33,"f":"Causalean/SCM/ID/DiscreteID/Positive.lean"},{"n":"Causalean.SCM.ID.DiscreteID.PositiveMass.eq_of_ae_eq","k":"theorem","m":"Causalean.SCM.ID.DiscreteID.Positive","a":"SCM/ID/DiscreteID","t":2,"d":"Under positive point mass at every point, almost-everywhere equality is pointwise equality.","s":"∀ {α : Type u_3} {β : Type u_4} [inst : MeasurableSpace α] {μ : MeasureTheory.Measure α}, Causalean.SCM.ID.DiscreteID.PositiveMass μ → ∀ {f g : α → β}, f =ᵐ[μ] g → ∀ (x : α), f x = g x","l":37,"f":"Causalean/SCM/ID/DiscreteID/Positive.lean"},{"n":"Causalean.SCM.ID.DiscreteID.PositiveMass.map_of_surjective","k":"theorem","m":"Causalean.SCM.ID.DiscreteID.Positive","a":"SCM/ID/DiscreteID","t":2,"d":"Positive point mass is preserved by a measurable surjection.","s":"∀ {α : Type u_3} {β : Type u_4} [inst : MeasurableSpace α] [inst_1 : MeasurableSpace β] [MeasurableSingletonClass β] {μ : MeasureTheory.Measure α} {f : α → β}, Measurable f → Causalean.SCM.ID.DiscreteID.PositiveMass μ → Function.Surjective f → Causalean.SCM.ID.DiscreteID.PositiveMass (MeasureTheory.Measure.map f μ)","l":51,"f":"Causalean/SCM/ID/DiscreteID/Positive.lean"},{"n":"Causalean.SCM.ID.DiscreteID.PositiveMass.map_valuesProjection","k":"theorem","m":"Causalean.SCM.ID.DiscreteID.Positive","a":"SCM/ID/DiscreteID","t":1,"d":"If a measure assigns nonzero point mass to every value assignment on the full index set, then restricting to the values on a subset of coordinates reached by projection preserves this: the pushed-forward measure still assigns nonzero point mass to every value assignment on that subset.","s":"∀ {M : Type u_3} {I J : Finset M} {Ω' : M → Type u_4} [inst : (m : M) → MeasurableSpace (Ω' m)] [∀ (m : M), Nonempty (Ω' m)] [MeasurableSingletonClass (Causalean.ValuesOn J Ω')] {μ : MeasureTheory.Measure (Causalean.ValuesOn I Ω')}, Causalean.SCM.ID.DiscreteID.PositiveMass μ → ∀ (hJI : J ⊆ I), Causalean.SCM.ID.DiscreteID.PositiveMass (MeasureTheory.Measure.map (Causalean.valuesProjection hJI) μ)","l":73,"f":"Causalean/SCM/ID/DiscreteID/Positive.lean"},{"n":"Causalean.SCM.ID.DiscreteID.PositiveMass.of_map_injective","k":"theorem","m":"Causalean.SCM.ID.DiscreteID.Positive","a":"SCM/ID/DiscreteID","t":2,"d":"Positive point mass can be pulled back across an injective measurable map when the pushed-forward measure is positive at every image value.","s":"∀ {α : Type u_3} {β : Type u_4} [inst : MeasurableSpace α] [inst_1 : MeasurableSpace β] [MeasurableSingletonClass β] {μ : MeasureTheory.Measure α} {f : α → β}, Measurable f → Function.Injective f → Causalean.SCM.ID.DiscreteID.PositiveMass (MeasureTheory.Measure.map f μ) → Causalean.SCM.ID.DiscreteID.PositiveMass μ","l":90,"f":"Causalean/SCM/ID/DiscreteID/Positive.lean"},{"n":"Causalean.SCM.ID.DiscreteID.StandardDiscretePositive","k":"def","m":"Causalean.SCM.ID.DiscreteID.Positive","a":"SCM/ID/DiscreteID","t":1,"d":"The structural assumptions for the first discrete ID soundness target.","s":"{N : Type u_1} → [inst : DecidableEq N] → [inst_1 : Fintype N] → {Ω : N → Type u_2} → [inst_2 : (n : N) → MeasurableSpace (Ω n)] → Causalean.SCM N Ω → Prop","l":123,"f":"Causalean/SCM/ID/DiscreteID/Positive.lean"},{"n":"Causalean.SCM.ID.DiscreteID.conditionalDenominator","k":"def","m":"Causalean.SCM.ID.DiscreteID.Mass","a":"SCM/ID/DiscreteID","t":1,"d":"The denominator used by conditionalMass.","s":"{α : Type u_1} → {β : Type u_2} → [inst : MeasurableSpace α] → [inst_1 : MeasurableSpace β] → MeasureTheory.Measure (α × β) → β → ENNReal","l":54,"f":"Causalean/SCM/ID/DiscreteID/Mass.lean"},{"n":"Causalean.SCM.ID.DiscreteID.conditionalDenominator_apply","k":"theorem","m":"Causalean.SCM.ID.DiscreteID.Mass","a":"SCM/ID/DiscreteID","t":2,"d":"","s":"∀ {α : Type u_1} {β : Type u_2} [inst : MeasurableSpace α] [inst_1 : MeasurableSpace β] (μ : MeasureTheory.Measure (α × β)) (b : β), Causalean.SCM.ID.DiscreteID.conditionalDenominator μ b = Causalean.SCM.ID.DiscreteID.singletonMass (MeasureTheory.Measure.map Prod.snd μ) b","l":278,"f":"Causalean/SCM/ID/DiscreteID/Mass.lean"},{"n":"Causalean.SCM.ID.DiscreteID.conditionalDenominator_ne_zero_of_positive_marginal","k":"theorem","m":"Causalean.SCM.ID.DiscreteID.Positive","a":"SCM/ID/DiscreteID","t":2,"d":"A positive marginal point mass is exactly the nonzero denominator needed by the finite conditional-mass ratio.","s":"∀ {α : Type u_3} {β : Type u_4} [inst : MeasurableSpace α] [inst_1 : MeasurableSpace β] {μ : MeasureTheory.Measure (α × β)}, Causalean.SCM.ID.DiscreteID.PositiveMass (MeasureTheory.Measure.map Prod.snd μ) → ∀ (b : β), Causalean.SCM.ID.DiscreteID.conditionalDenominator μ b ≠ 0","l":114,"f":"Causalean/SCM/ID/DiscreteID/Positive.lean"},{"n":"Causalean.SCM.ID.DiscreteID.conditionalMass","k":"def","m":"Causalean.SCM.ID.DiscreteID.Mass","a":"SCM/ID/DiscreteID","t":1,"d":"A discrete conditional mass, written as a ratio of a joint point mass to a marginal point mass. A sound ID formula must separately carry positivity for the denominator when it uses this value.","s":"{α : Type u_1} → {β : Type u_2} → [inst : MeasurableSpace α] → [inst_1 : MeasurableSpace β] → MeasureTheory.Measure (α × β) → α → β → ENNReal","l":46,"f":"Causalean/SCM/ID/DiscreteID/Mass.lean"},{"n":"Causalean.SCM.ID.DiscreteID.conditionalMass_mul_denominator","k":"theorem","m":"Causalean.SCM.ID.DiscreteID.Mass","a":"SCM/ID/DiscreteID","t":2,"d":"Multiplying a discrete conditional mass by its actual denominator recovers the joint point mass, provided that denominator is nonzero and finite.","s":"∀ {α : Type u_1} {β : Type u_2} [inst : MeasurableSpace α] [inst_1 : MeasurableSpace β] (μ : MeasureTheory.Measure (α × β)) (a : α) (b : β), Causalean.SCM.ID.DiscreteID.conditionalDenominator μ b ≠ 0 → Causalean.SCM.ID.DiscreteID.conditionalDenominator μ b ≠ ⊤ → Causalean.SCM.ID.DiscreteID.conditionalMass μ a b * Causalean.SCM.ID.DiscreteID.conditionalDenominator μ b = Causalean.SCM.ID.DiscreteID.singletonMass μ (a, b)","l":284,"f":"Causalean/SCM/ID/DiscreteID/Mass.lean"},{"n":"Causalean.SCM.ID.DiscreteID.eq_of_ae_eq_of_forall_singletonMass_ne_zero","k":"theorem","m":"Causalean.SCM.ID.DiscreteID.Mass","a":"SCM/ID/DiscreteID","t":2,"d":"An almost-everywhere equality is pointwise when every singleton has nonzero mass.","s":"∀ {α : Type u_1} {β : Type u_2} [inst : MeasurableSpace α] {μ : MeasureTheory.Measure α} {f g : α → β}, (∀ (x : α), Causalean.SCM.ID.DiscreteID.singletonMass μ x ≠ 0) → f =ᵐ[μ] g → ∀ (x : α), f x = g x","l":72,"f":"Causalean/SCM/ID/DiscreteID/Mass.lean"},{"n":"Causalean.SCM.ID.DiscreteID.marginalMass","k":"def","m":"Causalean.SCM.ID.DiscreteID.Mass","a":"SCM/ID/DiscreteID","t":1,"d":"The marginal mass of a selected coordinate value.","s":"{α : Type u_1} → {β : Type u_2} → [inst : MeasurableSpace α] → [inst_1 : MeasurableSpace β] → MeasureTheory.Measure α → (f : α → β) → Measurable f → β → ENNReal","l":41,"f":"Causalean/SCM/ID/DiscreteID/Mass.lean"},{"n":"Causalean.SCM.ID.DiscreteID.measure_eq_of_singletonMass_eq","k":"theorem","m":"Causalean.SCM.ID.DiscreteID.Mass","a":"SCM/ID/DiscreteID","t":2,"d":"Measures on a countable space are equal when all singleton masses agree.","s":"∀ {α : Type u_1} [inst : MeasurableSpace α] [Countable α] {μ ν : MeasureTheory.Measure α}, (∀ (x : α), Causalean.SCM.ID.DiscreteID.singletonMass μ x = Causalean.SCM.ID.DiscreteID.singletonMass ν x) → μ = ν","l":65,"f":"Causalean/SCM/ID/DiscreteID/Mass.lean"},{"n":"Causalean.SCM.ID.DiscreteID.singletonMass","k":"def","m":"Causalean.SCM.ID.DiscreteID.Mass","a":"SCM/ID/DiscreteID","t":1,"d":"The mass that a measure assigns to a single point.","s":"{α : Type u_1} → [inst : MeasurableSpace α] → MeasureTheory.Measure α → α → ENNReal","l":36,"f":"Causalean/SCM/ID/DiscreteID/Mass.lean"},{"n":"Causalean.SCM.ID.DiscreteID.singletonMass_apply","k":"theorem","m":"Causalean.SCM.ID.DiscreteID.Mass","a":"SCM/ID/DiscreteID","t":2,"d":"","s":"∀ {α : Type u_1} [inst : MeasurableSpace α] (μ : MeasureTheory.Measure α) (x : α), Causalean.SCM.ID.DiscreteID.singletonMass μ x = μ {x}","l":60,"f":"Causalean/SCM/ID/DiscreteID/Mass.lean"},{"n":"Causalean.SCM.ID.DiscreteID.singletonMass_comp_eq_of_const","k":"theorem","m":"Causalean.SCM.ID.DiscreteID.Mass","a":"SCM/ID/DiscreteID","t":2,"d":"A finite kernel mixture has the common singleton mass of its slices when the finite singleton masses of the mixing measure sum to one.","s":"∀ {α : Type u_1} {β : Type u_2} [inst : MeasurableSpace α] [inst_1 : MeasurableSpace β] [inst_2 : Fintype α] [MeasurableSingletonClass α] [MeasurableSingletonClass β] (μ : MeasureTheory.Measure α) (κ : ProbabilityTheory.Kernel α β) (y : β) (c : ENNReal), ∑ x, Causalean.SCM.ID.DiscreteID.singletonMass μ x = 1 → (∀ (x : α), Causalean.SCM.ID.DiscreteID.singletonMass (κ x) y = c) → Causalean.SCM.ID.DiscreteID.singletonMass (μ.bind ⇑κ) y = c","l":205,"f":"Causalean/SCM/ID/DiscreteID/Mass.lean"},{"n":"Causalean.SCM.ID.DiscreteID.singletonMass_comp_eq_of_pointMass","k":"theorem","m":"Causalean.SCM.ID.DiscreteID.Mass","a":"SCM/ID/DiscreteID","t":2,"d":"A finite kernel mixture has the singleton mass of one slice when the mixing measure is a point mass at that slice.","s":"∀ {α : Type u_1} {β : Type u_2} [inst : MeasurableSpace α] [inst_1 : MeasurableSpace β] [Finite α] [MeasurableSingletonClass α] [MeasurableSingletonClass β] (μ : MeasureTheory.Measure α) (κ : ProbabilityTheory.Kernel α β) (x0 : α) (y : β), Causalean.SCM.ID.DiscreteID.singletonMass μ x0 = 1 → (∀ (x : α), x ≠ x0 → Causalean.SCM.ID.DiscreteID.singletonMass μ x = 0) → Causalean.SCM.ID.DiscreteID.singletonMass (μ.bind ⇑κ) y = Causalean.SCM.ID.DiscreteID.singletonMass (κ x0) y","l":227,"f":"Causalean/SCM/ID/DiscreteID/Mass.lean"},{"n":"Causalean.SCM.ID.DiscreteID.singletonMass_comp_eq_sum","k":"theorem","m":"Causalean.SCM.ID.DiscreteID.Mass","a":"SCM/ID/DiscreteID","t":2,"d":"On a finite source space, the singleton mass of a kernel mixture is the finite weighted sum of the singleton masses of the kernel slices.","s":"∀ {α : Type u_1} {β : Type u_2} [inst : MeasurableSpace α] [inst_1 : MeasurableSpace β] [inst_2 : Fintype α] [MeasurableSingletonClass α] [MeasurableSingletonClass β] (μ : MeasureTheory.Measure α) (κ : ProbabilityTheory.Kernel α β) (y : β), Causalean.SCM.ID.DiscreteID.singletonMass (μ.bind ⇑κ) y = ∑ x, Causalean.SCM.ID.DiscreteID.singletonMass μ x * Causalean.SCM.ID.DiscreteID.singletonMass (κ x) y","l":192,"f":"Causalean/SCM/ID/DiscreteID/Mass.lean"},{"n":"Causalean.SCM.ID.DiscreteID.singletonMass_map","k":"theorem","m":"Causalean.SCM.ID.DiscreteID.Mass","a":"SCM/ID/DiscreteID","t":2,"d":"The singleton mass of a mapped measure is the mass of the corresponding fibre.","s":"∀ {α : Type u_1} {β : Type u_2} [inst : MeasurableSpace α] [inst_1 : MeasurableSpace β] [MeasurableSingletonClass β] (μ : MeasureTheory.Measure α) {f : α → β}, Measurable f → ∀ (y : β), Causalean.SCM.ID.DiscreteID.singletonMass (MeasureTheory.Measure.map f μ) y = μ (f ⁻¹' {y})","l":93,"f":"Causalean/SCM/ID/DiscreteID/Mass.lean"},{"n":"Causalean.SCM.ID.DiscreteID.singletonMass_map_const_eq_one","k":"theorem","m":"Causalean.SCM.ID.DiscreteID.Mass","a":"SCM/ID/DiscreteID","t":2,"d":"A pointwise constant measurable map from a probability measure gives unit singleton mass at the constant value.","s":"∀ {α : Type u_1} {β : Type u_2} [inst : MeasurableSpace α] [inst_1 : MeasurableSpace β] [MeasurableSingletonClass β] (μ : MeasureTheory.Measure α) {f : α → β}, Measurable f → ∀ (y0 : β), (∀ (x : α), f x = y0) → μ Set.univ = 1 → Causalean.SCM.ID.DiscreteID.singletonMass (MeasureTheory.Measure.map f μ) y0 = 1","l":129,"f":"Causalean/SCM/ID/DiscreteID/Mass.lean"},{"n":"Causalean.SCM.ID.DiscreteID.singletonMass_map_const_eq_univ","k":"theorem","m":"Causalean.SCM.ID.DiscreteID.Mass","a":"SCM/ID/DiscreteID","t":2,"d":"If a measurable map is pointwise constant, the pushed-forward measure has all mass at that constant value.","s":"∀ {α : Type u_1} {β : Type u_2} [inst : MeasurableSpace α] [inst_1 : MeasurableSpace β] [MeasurableSingletonClass β] (μ : MeasureTheory.Measure α) {f : α → β}, Measurable f → ∀ (y0 : β), (∀ (x : α), f x = y0) → Causalean.SCM.ID.DiscreteID.singletonMass (MeasureTheory.Measure.map f μ) y0 = μ Set.univ","l":101,"f":"Causalean/SCM/ID/DiscreteID/Mass.lean"},{"n":"Causalean.SCM.ID.DiscreteID.singletonMass_map_const_eq_zero_of_ne","k":"theorem","m":"Causalean.SCM.ID.DiscreteID.Mass","a":"SCM/ID/DiscreteID","t":2,"d":"If a measurable map is pointwise constant at y0, the pushed-forward measure has zero singleton mass at every different value.","s":"∀ {α : Type u_1} {β : Type u_2} [inst : MeasurableSpace α] [inst_1 : MeasurableSpace β] [MeasurableSingletonClass β] (μ : MeasureTheory.Measure α) {f : α → β}, Measurable f → ∀ (y0 y : β), (∀ (x : α), f x = y0) → y ≠ y0 → Causalean.SCM.ID.DiscreteID.singletonMass (MeasureTheory.Measure.map f μ) y = 0","l":114,"f":"Causalean/SCM/ID/DiscreteID/Mass.lean"},{"n":"Causalean.SCM.ID.DiscreteID.singletonMass_map_eq_sum_fiber","k":"theorem","m":"Causalean.SCM.ID.DiscreteID.Mass","a":"SCM/ID/DiscreteID","t":2,"d":"On a finite measurable-singleton space, the singleton mass of a pushed-forward measure is the finite sum of singleton masses over the fiber.","s":"∀ {α : Type u_1} {β : Type u_2} [inst : MeasurableSpace α] [inst_1 : MeasurableSpace β] [inst_2 : Fintype α] [MeasurableSingletonClass α] [MeasurableSingletonClass β] [inst_5 : DecidableEq β] (μ : MeasureTheory.Measure α) {f : α → β}, Measurable f → ∀ (y : β), Causalean.SCM.ID.DiscreteID.singletonMass (MeasureTheory.Measure.map f μ) y = ∑ x, if f x = y then Causalean.SCM.ID.DiscreteID.singletonMass μ x else 0","l":140,"f":"Causalean/SCM/ID/DiscreteID/Mass.lean"},{"n":"Causalean.SCM.ID.DiscreteID.sum_singletonMass_eq_one","k":"theorem","m":"Causalean.SCM.ID.DiscreteID.Mass","a":"SCM/ID/DiscreteID","t":2,"d":"On a finite measurable-singleton probability space, singleton masses sum to one.","s":"∀ {α : Type u_1} [inst : MeasurableSpace α] [inst_1 : Fintype α] [MeasurableSingletonClass α] (μ : MeasureTheory.Measure α) [MeasureTheory.IsProbabilityMeasure μ], ∑ x, Causalean.SCM.ID.DiscreteID.singletonMass μ x = 1","l":183,"f":"Causalean/SCM/ID/DiscreteID/Mass.lean"},{"n":"Causalean.SCM.ID.DiscreteID.sum_singletonMass_eq_univ","k":"theorem","m":"Causalean.SCM.ID.DiscreteID.Mass","a":"SCM/ID/DiscreteID","t":2,"d":"On a finite measurable-singleton space, the sum of all singleton masses is the total mass of the measure.","s":"∀ {α : Type u_1} [inst : MeasurableSpace α] [inst_1 : Fintype α] [MeasurableSingletonClass α] (μ : MeasureTheory.Measure α), ∑ x, Causalean.SCM.ID.DiscreteID.singletonMass μ x = μ Set.univ","l":167,"f":"Causalean/SCM/ID/DiscreteID/Mass.lean"},{"n":"Causalean.SCM.ID.DiscreteID.valuesOn_measure_eq_of_singletonMass_eq","k":"theorem","m":"Causalean.SCM.ID.DiscreteID.Mass","a":"SCM/ID/DiscreteID","t":1,"d":"For two measures on a finite-coordinate product over countable value spaces, if every value assignment carries the same singleton point mass under both measures, then the two measures are equal.","s":"∀ {M : Type u_1} {Ω' : M → Type u_2} [inst : (m : M) → MeasurableSpace (Ω' m)] [∀ (m : M), Countable (Ω' m)] (I : Finset M) {μ ν : MeasureTheory.Measure (Causalean.ValuesOn I Ω')}, (∀ (x : Causalean.ValuesOn I Ω'), Causalean.SCM.ID.DiscreteID.singletonMass μ x = Causalean.SCM.ID.DiscreteID.singletonMass ν x) → μ = ν","l":266,"f":"Causalean/SCM/ID/DiscreteID/Mass.lean"},{"n":"Causalean.SCM.ID.DiscreteID.valuesProjection_surjective","k":"theorem","m":"Causalean.SCM.ID.DiscreteID.Mass","a":"SCM/ID/DiscreteID","t":2,"d":"Coordinate restriction is surjective when every omitted coordinate has at least one default value.","s":"∀ {M : Type u_1} {I J : Finset M} {Ω' : M → Type u_2} [inst : (m : M) → MeasurableSpace (Ω' m)] [∀ (m : ↥I), Nonempty (Ω' ↑m)] (hJI : J ⊆ I), Function.Surjective (Causalean.valuesProjection hJI)","l":247,"f":"Causalean/SCM/ID/DiscreteID/Mass.lean"},{"n":"Causalean.SCM.ID.DistrictIdConclusion","k":"def","m":"Causalean.SCM.ID.GraphicalThms.QFactorIdentity","a":"SCM/ID/GraphicalThms","t":1,"d":"The conclusion of district identification: for T ∈ C(G), the c-factor Q[T] of the full SCM equals a.e. the matching do-side conditional after fixing Wn. Special case R = M.observed of QFactorIdentityConclusion.","s":"{N : Type u_1} → [inst : DecidableEq N] → [inst_1 : Fintype N] → {Ω : N → Type u_2} → [inst_2 : (n : N) → MeasurableSpace (Ω n)] → Causalean.SCM N Ω → Finset (Causalean.SWIGNode N) → Finset N → Prop","l":622,"f":"Causalean/SCM/ID/GraphicalThms/QFactorIdentity.lean"},{"n":"Causalean.SCM.ID.ENNReal.div_mul_common","k":"theorem","m":"Causalean.SCM.ID.Density.MechCFactor","a":"SCM/ID/Density","t":2,"d":"Cancelling a common nonzero finite factor preserves a quotient of extended nonnegative reals.","s":"∀ {a b r : ENNReal}, r ≠ 0 → r ≠ ⊤ → a * r / (b * r) = a / b","l":360,"f":"Causalean/SCM/ID/Density/MechCFactor.lean"},{"n":"Causalean.SCM.ID.ENNReal.prod_div_prod","k":"theorem","m":"Causalean.SCM.ID.Density.MechCFactor","a":"SCM/ID/Density","t":2,"d":"A finite product of quotients is the quotient of the finite products when denominators are nonzero and finite.","s":"∀ {ι : Type u_3} [DecidableEq ι] (t : Finset ι) (f g : ι → ENNReal), (∀ i ∈ t, g i ≠ 0) → (∀ i ∈ t, g i ≠ ⊤) → ∏ i ∈ t, f i / g i = (∏ i ∈ t, f i) / ∏ i ∈ t, g i","l":675,"f":"Causalean/SCM/ID/Density/MechCFactor.lean"},{"n":"Causalean.SCM.ID.Finset.prod_ne_top_of_ne_top","k":"theorem","m":"Causalean.SCM.ID.Density.MechCFactor","a":"SCM/ID/Density","t":2,"d":"A finite product of finite extended nonnegative reals is finite.","s":"∀ {ι : Type u_3} [DecidableEq ι] (s : Finset ι) (f : ι → ENNReal), (∀ i ∈ s, f i ≠ ⊤) → ∏ i ∈ s, f i ≠ ⊤","l":372,"f":"Causalean/SCM/ID/Density/MechCFactor.lean"},{"n":"Causalean.SCM.ID.Identifiable","k":"def","m":"Causalean.SCM.ID.Identifiable","a":"SCM/ID","t":1,"d":"A causal query Φ is identifiable from the observational distribution if any two causal models sharing the same SWIG graph and observational kernel must agree on Φ.","s":"{N : Type u_1} → [inst : DecidableEq N] → [inst_1 : Fintype N] → {Ω : N → Type u_2} → [inst_2 : (n : N) → MeasurableSpace (Ω n)] → {α : Type u_3} → Causalean.SWIGGraph N → Causalean.SCM.ID.CausalQuery N Ω α → Prop","l":95,"f":"Causalean/SCM/ID/Identifiable.lean"},{"n":"Causalean.SCM.ID.IdentifiableUnder","k":"def","m":"Causalean.SCM.ID.Identifiable","a":"SCM/ID","t":1,"d":"Identifiability given both functional and structural assumptions.","s":"{N : Type u_1} → [inst : DecidableEq N] → [inst_1 : Fintype N] → {Ω : N → Type u_2} → [inst_2 : (n : N) → MeasurableSpace (Ω n)] → {α : Type u_3} → Causalean.SWIGGraph N → (Causalean.SCM N Ω → Prop) → (Causalean.SCM N Ω → Prop) → Causalean.SCM.ID.CausalQuery N Ω α → Prop","l":146,"f":"Causalean/SCM/ID/Identifiable.lean"},{"n":"Causalean.SCM.ID.InterventionTargetSimpConclusion","k":"def","m":"Causalean.SCM.ID.GraphicalThms.QFactorIdentity","a":"SCM/ID/GraphicalThms","t":1,"d":"The conclusion of fact4 (intervention target simplification), stated as a structural equivalence of gSCMs.","s":"{N : Type u_1} → [inst : DecidableEq N] → [inst_1 : Fintype N] → {Ω : N → Type u_2} → [inst_2 : (n : N) → MeasurableSpace (Ω n)] → Causalean.SCM N Ω → Finset N → Finset N → Prop","l":435,"f":"Causalean/SCM/ID/GraphicalThms/QFactorIdentity.lean"},{"n":"Causalean.SCM.ID.KernelGlobalMarkovOn","k":"def","m":"Causalean.SCM.ID.Density.QFactor","a":"SCM/ID/Density","t":1,"d":"This predicate says that every graphical separation in a graph gives the corresponding coordinate conditional independence under a finite-coordinate measure.","s":"{N : Type u_1} → [inst : DecidableEq N] → [inst_1 : Fintype N] → {Ω : N → Type u_2} → [inst_2 : (n : N) → MeasurableSpace (Ω n)] → Causalean.SWIGGraph N → (D : Finset (Causalean.SWIGNode N)) → (μ : MeasureTheory.Measure (Causalean.ValuesOn D (Causalean.swigΩ Ω))) → [StandardBorelSpace (Causalean.ValuesOn D (Causalean.swigΩ Ω))] → [MeasureTheory.IsFiniteMeasure μ] → Prop","l":122,"f":"Causalean/SCM/ID/Density/QFactor.lean"},{"n":"Causalean.SCM.ID.KernelObsCondIndepOn","k":"def","m":"Causalean.SCM.ID.Density.QFactor","a":"SCM/ID/Density","t":1,"d":"This predicate says that two coordinate blocks are conditionally independent given a third block under a finite-coordinate measure.","s":"{N : Type u_1} → {Ω : N → Type u_2} → [inst : (n : N) → MeasurableSpace (Ω n)] → (D X Y Z : Finset (Causalean.SWIGNode N)) → X ⊆ D → Y ⊆ D → Z ⊆ D → (μ : MeasureTheory.Measure (Causalean.ValuesOn D (Causalean.swigΩ Ω))) → [StandardBorelSpace (Causalean.ValuesOn D (Causalean.swigΩ Ω))] → [MeasureTheory.IsFiniteMeasure μ] → Prop","l":104,"f":"Causalean/SCM/ID/Density/QFactor.lean"},{"n":"Causalean.SCM.ID.NonIdentifiable","k":"def","m":"Causalean.SCM.ID.Identifiable","a":"SCM/ID","t":1,"d":"A causal query is non-identifiable if there exist two models with the same SWIG graph and observational kernel but different query values.","s":"{N : Type u_1} → [inst : DecidableEq N] → [inst_1 : Fintype N] → {Ω : N → Type u_2} → [inst_2 : (n : N) → MeasurableSpace (Ω n)] → {α : Type u_3} → Causalean.SWIGGraph N → Causalean.SCM.ID.CausalQuery N Ω α → Prop","l":115,"f":"Causalean/SCM/ID/Identifiable.lean"},{"n":"Causalean.SCM.ID.QFactorIdentityConclusion","k":"def","m":"Causalean.SCM.ID.GraphicalThms.QFactorIdentity","a":"SCM/ID/GraphicalThms","t":1,"d":"The conditional conclusion of Tian's fixing lemma (Prop 2.19).","s":"{N : Type u_1} → [inst : DecidableEq N] → [inst_1 : Fintype N] → {Ω : N → Type u_2} → [inst_2 : (n : N) → MeasurableSpace (Ω n)] → Causalean.SCM N Ω → Finset (Causalean.SWIGNode N) → Finset (Causalean.SWIGNode N) → Finset N → Prop","l":106,"f":"Causalean/SCM/ID/GraphicalThms/QFactorIdentity.lean"},{"n":"Causalean.SCM.ID.QFactorMarginalFixingConclusion","k":"def","m":"Causalean.SCM.ID.GraphicalThms.QFactorIdentity","a":"SCM/ID/GraphicalThms","t":1,"d":"The proven marginal fixing helper behind Tian's fixing lemma.","s":"{N : Type u_1} → [inst : DecidableEq N] → [inst_1 : Fintype N] → {Ω : N → Type u_2} → [inst_2 : (n : N) → MeasurableSpace (Ω n)] → Causalean.SCM N Ω → Finset (Causalean.SWIGNode N) → Finset (Causalean.SWIGNode N) → Finset N → Prop","l":69,"f":"Causalean/SCM/ID/GraphicalThms/QFactorIdentity.lean"},{"n":"Causalean.SCM.ID.Rule2JointOverlap","k":"def","m":"Causalean.SCM.ID.Overlap","a":"SCM/ID","t":1,"d":"Rule 2 joint overlap (canonical, continuous-friendly). Absolute continuity of the (Z.image .random ∪ W)-marginal of (M'.fixSet Z).obsKernel s' w.r.t. the same marginal of M'.obsKernel (fixSetProj s').","s":"{N : Type u_1} → [inst : DecidableEq N] → [inst_1 : Fintype N] → {Ω : N → Type u_2} → [inst_2 : (n : N) → MeasurableSpace (Ω n)] → (M' : Causalean.SCM N Ω) → (Z : Finset N) → (hZ_obs : ∀ D ∈ Z, Causalean.SWIGNode.random D ∈ M'.observed) → (hZ_fixed : ∀ D ∈ Z, Causalean.SWIGNode.fixed D ∉ M'.fixed) → (W : Finset (Causalean.SWIGNode N)) → Finset.image Causalean.SWIGNode.random Z ∪ W ⊆ M'.observed → (M'.fixSet Z hZ_obs hZ_fixed).FixedValues → Prop","l":48,"f":"Causalean/SCM/ID/Overlap.lean"},{"n":"Causalean.SCM.ID.cComponentDensityFactor_eq_mechCFactor","k":"theorem","m":"Causalean.SCM.ID.Density.MechCFactor","a":"SCM/ID/Density","t":1,"d":"(A) Tian Lemma 1 (Eq. 37 = Eq. 36). For a standard causal model M, i.e. one with no fixed nodes, a reference family that is faithful — every coordinate value has nonzero reference mass, and an observational kernel at a fixed-value slice s with full point-mass support (every observed assignment has nonzero probability), fix a node set S contained in the observed coordinates that is a full c-component of the model's SWIG graph. Then the observational full-prefix c-component density cComponentDensityFactor on S equals the mechanism c-factor Q[S] (mechCFactor) — the do-complement marginal density of S.","s":"∀ {N : Type u_1} [inst : DecidableEq N] [inst_1 : Fintype N] {Ω : N → Type u_2} [inst_2 : (n : N) → MeasurableSpace (Ω n)] [(n : N) → Fintype (Ω n)] [∀ (n : N), MeasurableSingletonClass (Ω n)] (M : Causalean.SCM N Ω) (ref : Causalean.SCM.ReferenceMeasures Ω) (s : M.FixedValues), M.isStandard → ∀ (S : Finset (Causalean.SWIGNode N)) (hS : S ⊆ M.observed), S ∈ M.cComponentSet → Causalean.SCM.ReferenceFaithful ref → Causalean.SCM.ID.DiscreteID.PositiveMass (M.obsKernel s) → ∀ [inst_5 : ∀ (s' : M.FixedValues), MeasureTheory.IsFiniteMeasure (M.obsKernel s')] [inst_6 : ∀ (k : ℕ) (hk : k < M.observed.card), StandardBorelSpace (Causalean.ValuesOn {↑(M.observedAt ⟨k, hk⟩)} (Causalean.swigΩ Ω))] [inst_7 : ∀ (k : ℕ) (hk : k < M.observed.card), Nonempty (Causalean.ValuesOn {↑(M.observedAt ⟨k, hk⟩)} (Causalean.swigΩ Ω))] [inst_8 : ∀ (k : ℕ), MeasurableSpace.CountableOrCountablyGenerated M.FixedValues (Causalean.ValuesOn (M.prefixNodes k) (Causalean.swigΩ Ω))] (x : Causalean.ValuesOn M.observed (Causalean.swigΩ Ω)), M.cComponentDensityFactor ref s S x = M.mechCFactor ref S hS s x","l":919,"f":"Causalean/SCM/ID/Density/MechCFactor.lean"},{"n":"Causalean.SCM.ID.cComponentSet_nonempty","k":"theorem","m":"Causalean.SCM.ID.DiscreteID.Checker","a":"SCM/ID/DiscreteID","t":2,"d":"Every c-component in a graph's c-component set contains at least one node.","s":"∀ {N : Type u_1} [inst : DecidableEq N] [inst_1 : Fintype N] (G : Causalean.SWIGGraph N) {C : Finset (Causalean.SWIGNode N)}, C ∈ G.cComponentSet → C.Nonempty","l":65,"f":"Causalean/SCM/ID/DiscreteID/Checker.lean"},{"n":"Causalean.SCM.ID.cFactorReachable","k":"def","m":"Causalean.SCM.ID.GraphicalThms.DoGFormula","a":"SCM/ID/GraphicalThms","t":1,"d":"Structural c-factor reachability, no-fixing approximation.","s":"{N : Type u_1} → [inst : DecidableEq N] → [inst_1 : Fintype N] → Causalean.SWIGGraph N → Finset (Causalean.SWIGNode N) → Finset (Causalean.SWIGNode N) → Prop","l":57,"f":"Causalean/SCM/ID/GraphicalThms/DoGFormula.lean"},{"n":"Causalean.SCM.ID.cFactorReachableRecB","k":"def","m":"Causalean.SCM.ID.DiscreteID.Checker","a":"SCM/ID/DiscreteID","t":1,"d":"Computable fuel-bounded IDENTIFY reachability. Returns true when the c-factor Q[C] can be recovered from Q[T] within fuel IDENTIFY steps. This is the executable mirror of the inductive CFactorReachableRec: it is choice-free (it searches (G.induce A).cComponentSet with any instead of naming the component via the noncomputable containingCComponent).","s":"{N : Type u_1} → [inst : DecidableEq N] → [inst_1 : Fintype N] → Causalean.SWIGGraph N → ℕ → Finset (Causalean.SWIGNode N) → Finset (Causalean.SWIGNode N) → Bool","l":36,"f":"Causalean/SCM/ID/DiscreteID/Checker.lean"},{"n":"Causalean.SCM.ID.cFactorReachableRecB_sound","k":"theorem","m":"Causalean.SCM.ID.DiscreteID.Checker","a":"SCM/ID/DiscreteID","t":2,"d":"Soundness of the computable reachability check. If the fuel-bounded checker accepts, the inductive reachability certificate holds.","s":"∀ {N : Type u_1} [inst : DecidableEq N] [inst_1 : Fintype N] (G : Causalean.SWIGGraph N) (fuel : ℕ) (T C : Finset (Causalean.SWIGNode N)), Causalean.SCM.ID.cFactorReachableRecB G fuel T C = true → Causalean.SCM.ID.CFactorReachableRec G T C","l":75,"f":"Causalean/SCM/ID/DiscreteID/Checker.lean"},{"n":"Causalean.SCM.ID.cFactorReachable_base_toRec","k":"theorem","m":"Causalean.SCM.ID.GraphicalThms.DoGFormulaRec","a":"SCM/ID/GraphicalThms","t":1,"d":"The no-fixing certificate is the base case of the recursive one. For a SWIG graph G and a district S, if S is already reachable from its containing district under the plain no-fixing certificate, then S is recursively reachable from its containing district — which, since S is a full c-component, is S itself.","s":"∀ {N : Type u_1} [inst : DecidableEq N] [inst_1 : Fintype N] (G : Causalean.SWIGGraph N) (S : Finset (Causalean.SWIGNode N)), Causalean.SCM.ID.cFactorReachable G (Causalean.SCM.ID.containingCComponent G S) S → Causalean.SCM.ID.CFactorReachableRec G (Causalean.SCM.ID.containingCComponent G S) S","l":141,"f":"Causalean/SCM/ID/GraphicalThms/DoGFormulaRec.lean"},{"n":"Causalean.SCM.ID.component_ref_atom_product_eq_jointRef","k":"theorem","m":"Causalean.SCM.ID.Density.MechCFactor","a":"SCM/ID/Density","t":2,"d":"Multiplying the singleton reference masses for all observed variables in one c-component equals the singleton mass of their joint reference measure.","s":"∀ {N : Type u_1} [inst : DecidableEq N] [inst_1 : Fintype N] {Ω : N → Type u_2} [inst_2 : (n : N) → MeasurableSpace (Ω n)] [∀ (n : N), MeasurableSingletonClass (Ω n)] (M : Causalean.SCM N Ω) (ref : Causalean.SCM.ReferenceMeasures Ω) (S : Finset (Causalean.SWIGNode N)) (hS : S ⊆ M.observed), S ∈ M.cComponentSet → ∀ (x : Causalean.ValuesOn M.observed (Causalean.swigΩ Ω)), ∏ i with M.cComponentOf ↑(M.observedAt i) = S, (ref.μ ↑(M.observedAt i)) {x (M.observedAt i)} = (Causalean.SCM.jointRef ref S) {Causalean.valuesProjection hS x}","l":856,"f":"Causalean/SCM/ID/Density/MechCFactor.lean"},{"n":"Causalean.SCM.ID.component_ref_atom_product_eq_jointRef_prefixIn","k":"theorem","m":"Causalean.SCM.ID.Density.QFactor","a":"SCM/ID/Density","t":2,"d":"For a subset of an ordered finite graph set, the product of coordinate reference-measure masses at selected values equals the reference measure's mass at their joint singleton outcome.","s":"∀ {N : Type u_1} [inst : DecidableEq N] [inst_1 : Fintype N] {Ω : N → Type u_2} [inst_2 : (n : N) → MeasurableSpace (Ω n)] [∀ (n : N), MeasurableSingletonClass (Ω n)] (H : Causalean.SWIGGraph N) (D : Finset (Causalean.SWIGNode N)) (ref : Causalean.SCM.ReferenceMeasures Ω) (S : Finset (Causalean.SWIGNode N)) (hSD : S ⊆ D) (xD : Causalean.ValuesOn D (Causalean.swigΩ Ω)), ∏ i with ↑(H.nodesAt D i) ∈ S, (ref.μ ↑(H.nodesAt D i)) {xD (H.nodesAt D i)} = (Causalean.SCM.jointRef ref S) {Causalean.valuesProjection hSD xD}","l":1170,"f":"Causalean/SCM/ID/Density/QFactor.lean"},{"n":"Causalean.SCM.ID.containingCComponent","k":"def","m":"Causalean.SCM.ID.GraphicalThms.DoGFormula","a":"SCM/ID/GraphicalThms","t":1,"d":"The c-component of G containing the finite set S, seeded by an arbitrary member of S; empty S has no containing component and returns ∅.","s":"{N : Type u_1} → [inst : DecidableEq N] → [inst_1 : Fintype N] → Causalean.SWIGGraph N → Finset (Causalean.SWIGNode N) → Finset (Causalean.SWIGNode N)","l":50,"f":"Causalean/SCM/ID/GraphicalThms/DoGFormula.lean"},{"n":"Causalean.SCM.ID.containingCComponent_eq_of_mem_of_subset","k":"theorem","m":"Causalean.SCM.ID.DiscreteID.Checker","a":"SCM/ID/DiscreteID","t":2,"d":"The unique c-component containing a nonempty set S is containingCComponent. Bridges the checker's cComponentSet.any search to the reachability predicate's choice-based containingCComponent.","s":"∀ {N : Type u_1} [inst : DecidableEq N] [inst_1 : Fintype N] (G : Causalean.SWIGGraph N) {S C' : Finset (Causalean.SWIGNode N)}, S.Nonempty → C' ∈ G.cComponentSet → S ⊆ C' → Causalean.SCM.ID.containingCComponent G S = C'","l":52,"f":"Causalean/SCM/ID/DiscreteID/Checker.lean"},{"n":"Causalean.SCM.ID.containingCComponent_induce_subset","k":"theorem","m":"Causalean.SCM.ID.Density.IdentifyMass","a":"SCM/ID/Density","t":2,"d":"A containing c-component in an induced graph lies inside the inducing set.","s":"∀ {N : Type u_1} [inst : DecidableEq N] [inst_1 : Fintype N] (G : Causalean.SWIGGraph N) (A C : Finset (Causalean.SWIGNode N)), Causalean.SCM.ID.containingCComponent (G.induce A) C ⊆ A","l":99,"f":"Causalean/SCM/ID/Density/IdentifyMass.lean"},{"n":"Causalean.SCM.ID.containingCComponent_of_mem_cComponentSet","k":"theorem","m":"Causalean.SCM.ID.GraphicalThms.DoGFormulaRec","a":"SCM/ID/GraphicalThms","t":2,"d":"When S is a full c-component of G, its containing district is S itself.","s":"∀ {N : Type u_1} [inst : DecidableEq N] [inst_1 : Fintype N] (G : Causalean.SWIGGraph N), ∀ S ∈ G.cComponentSet, Causalean.SCM.ID.containingCComponent G S = S","l":113,"f":"Causalean/SCM/ID/GraphicalThms/DoGFormulaRec.lean"},{"n":"Causalean.SCM.ID.containingCComponent_subset_observed","k":"theorem","m":"Causalean.SCM.ID.Density.IdentifyMass","a":"SCM/ID/Density","t":2,"d":"The containing c-component is always a set of observed nodes.","s":"∀ {N : Type u_1} [inst : DecidableEq N] [inst_1 : Fintype N] (G : Causalean.SWIGGraph N) (S : Finset (Causalean.SWIGNode N)), Causalean.SCM.ID.containingCComponent G S ⊆ G.observed","l":90,"f":"Causalean/SCM/ID/Density/IdentifyMass.lean"},{"n":"Causalean.SCM.ID.defaultInterventionalKernel","k":"def","m":"Causalean.SCM.ID.Query","a":"SCM/ID","t":1,"d":"This fixed fallback kernel is used only outside the standard identification query domain.","s":"{N : Type u_1} → [inst : DecidableEq N] → {Ω : N → Type u_2} → [inst_1 : (n : N) → MeasurableSpace (Ω n)] → [∀ (n : N), Nonempty (Ω n)] → (X : Finset N) → (Y : Finset (Causalean.SWIGNode N)) → ProbabilityTheory.Kernel (Causalean.ValuesOn (Finset.image Causalean.SWIGNode.random X) (Causalean.swigΩ Ω)) (Causalean.ValuesOn Y (Causalean.swigΩ Ω))","l":38,"f":"Causalean/SCM/ID/Query.lean"},{"n":"Causalean.SCM.ID.district_id","k":"theorem","m":"Causalean.SCM.ID.GraphicalThms.QFactorIdentity","a":"SCM/ID/GraphicalThms","t":1,"d":"Corollary (District identification). For a structural causal model M, a c-component T of M's full graph, and an intervention set Wn, the c-factor of T equals, almost everywhere, the observational conditional obtained by intervening on Wn.","s":"∀ {N : Type u_1} [inst : DecidableEq N] [inst_1 : Fintype N] {Ω : N → Type u_2} [inst_2 : (n : N) → MeasurableSpace (Ω n)] (M : Causalean.SCM N Ω) (T : Finset (Causalean.SWIGNode N)) (Wn : Finset N), Causalean.SCM.ID.DistrictIdConclusion M T Wn","l":631,"f":"Causalean/SCM/ID/GraphicalThms/QFactorIdentity.lean"},{"n":"Causalean.SCM.ID.doAncestralDistrictDensity_recovered_from_obs","k":"theorem","m":"Causalean.SCM.ID.GraphicalThms.DoGFormulaTian","a":"SCM/ID/GraphicalThms","t":1,"d":"T2, abstract density recovery statement. Fix a standard structural causal model M and an intervention target set X for which every targeted node is currently a random observed node and none of its fixed copies is already fixed, an output set Y, and a reference-measure family faithful to the graph. For a district S of the truncated c-component set of the post-intervention ancestral graph and a c-component C of the base graph that is factor-reachable from S, assume every fixed-value assignment gives an observational kernel with everywhere-positive point masses, no intervention target's random form lies in Y, and that an extension map from ancestral assignments to full observed assignments restricts back to the identity and agrees with the intervention values sDo on the targeted coordinates. Then the district factor of S computed from the density of the do-law's ancestral marginal equals, almost everywhere, the full-graph c-component density factor of C evaluated at the extension of the ancestral assignment.","s":"∀ {N : Type u_1} [inst : DecidableEq N] [inst_1 : Fintype N] {Ω : N → Type u_2} [inst_2 : (n : N) → MeasurableSpace (Ω n)] [∀ (n : N), Nonempty (Ω n)] [(n : N) → Fintype (Ω n)] [∀ (n : N), MeasurableSingletonClass (Ω n)] (M : Causalean.SCM N Ω) (X : Finset N), M.isStandard → ∀ (hObs : ∀ D ∈ X, Causalean.SWIGNode.random D ∈ M.observed) (hFix : ∀ D ∈ X, Causalean.SWIGNode.fixed D ∉ M.fixed) (Y : Finset (Causalean.SWIGNode N)) (ref : Causalean.SCM.ReferenceMeasures Ω), Causalean.SCM.ReferenceFaithful ref → ∀ (sDo : (M.fixSet X hObs hFix).FixedValues) (S C : Finset (Causalean.SWIGNode N)), S ∈ Causalean.SCM.ID.fixTruncCComponentSet M X hObs hFix Y → Causalean.SCM.ID.cFactorReachable M.toSWIGGraph C S → C ∈ M.cComponentSet → ∀ [inst_6 : MeasureTheory.IsFiniteMeasure ((Causalean.SCM.ID.doObsKernelAncestralMarginal M X hObs hFix Y) sDo)] [inst_7 : ∀ (s' : M.FixedValues), MeasureTheory.IsFiniteMeasure (M.obsKernel s')] [inst_8 : ∀ (k : ℕ) (hk : k < (Causalean.SCM.ID.fixObservedAncestralSet M X hObs hFix Y).card), StandardBorelSpace (Causalean.ValuesOn {↑(((M.fixSet X hObs hFix).induce (Causalean.SCM.ID.fixAncestralSet M X hObs hFix Y)).nodesAt (Causalean.SCM.ID.fixObservedAncestralSet M X hObs hFix Y) ⟨k, hk⟩)} (Causalean.swigΩ Ω))] [inst_9 : ∀ (k : ℕ) (hk : k < (Causalean.SCM.ID.fixObservedAncestralSet M X hObs hFix Y).card), Nonempty (Causalean.ValuesOn {↑(((M.fixSet X hObs hFix).induce (Causalean.SCM.ID.fixAncestralSet M X hObs hFix Y)).nodesAt (Causalean.SCM.ID.fixObservedAncestralSet M X hObs hFix Y) ⟨k, hk⟩)} (Causalean.swigΩ Ω))] [inst_10 : ∀ (k : ℕ) (hk : k < M.observed.card), StandardBorelSpace (Causalean.ValuesOn {↑(M.observedAt ⟨k, hk⟩)} (Causalean.swigΩ Ω))] [inst_11 : ∀ (k : ℕ) (hk : k < M.observed.card), Nonempty (Causalean.ValuesOn {↑(M.observedAt ⟨k, hk⟩)} (Causalean.swigΩ Ω))] [inst_12 : ∀ (k : ℕ), MeasurableSpace.CountableOrCountablyGenerated M.FixedValues (Causalean.ValuesOn (M.prefixNodes k) (Causalean.swigΩ Ω))], (∀ (s' : M.FixedValues), Causalean.SCM.ID.DiscreteID.PositiveMass (M.obsKernel s')) → (∀ D ∈ X, Causalean.SWIGNode.random D ∉ Y) → ∀ (extend : Causalean.ValuesOn (Causalean.SCM.ID.fixObservedAncestralSet M X hObs hFix Y) (Causalean.swigΩ Ω) → Causalean.ValuesOn M.observed (Causalean.swigΩ Ω)), (∀ (xD : Causalean.ValuesOn (Causalean.SCM.ID.fixObservedAncestralSet M X hObs hFix Y) (Causalean.swigΩ Ω)), Causalean.valuesProjection (have this := Finset.inter_subset_right; this) (extend xD) = xD) → (∀ (xD : Causalean.ValuesOn (Causalean.SCM.ID.fixObservedAncestralSet M X hObs hFix Y) (Causalean.swigΩ Ω)) (D : N) (hD : D ∈ X), extend xD ⟨Causalean.SWIGNode.random D, hObs D hD⟩ = sDo ⟨Causalean.SWIGNode.fixed D, Finset.mem_union_right M.fixed (Finset.mem_image.mpr (Exists.intro D ⟨hD, rfl⟩))⟩) → let D := Causalean.SCM.ID.fixObservedAncestralSet M X hObs hFix Y; let H := (M.fixSet X hObs hFix).induce (Causalean.SCM.ID.fixAncestralSet M X hObs hFix Y); Causalean.SCM.ID.tianDistrictDensity H D ((Causalean.SCM.ID.doObsKernelAncestralMarginal M X hObs hFix Y) sDo) ref S =ᵐ[Causalean.SCM.jointRef ref D] fun xD => M.cComponentDensityFactor ref (M.fixSetProj X hObs hFix sDo) C (extend xD)","l":1137,"f":"Causalean/SCM/ID/GraphicalThms/DoGFormulaTian.lean"},{"n":"Causalean.SCM.ID.doAncestralDistrictDensity_recovered_from_obs_core","k":"theorem","m":"Causalean.SCM.ID.GraphicalThms.DoGFormulaTian","a":"SCM/ID/GraphicalThms","t":2,"d":"T2 density-recovery core.","s":"∀ {N : Type u_1} [inst : DecidableEq N] [inst_1 : Fintype N] {Ω : N → Type u_2} [inst_2 : (n : N) → MeasurableSpace (Ω n)] [∀ (n : N), Nonempty (Ω n)] [(n : N) → Fintype (Ω n)] [∀ (n : N), MeasurableSingletonClass (Ω n)] (M : Causalean.SCM N Ω) (X : Finset N), M.isStandard → ∀ (hObs : ∀ D ∈ X, Causalean.SWIGNode.random D ∈ M.observed) (hFix : ∀ D ∈ X, Causalean.SWIGNode.fixed D ∉ M.fixed) (Y : Finset (Causalean.SWIGNode N)) (ref : Causalean.SCM.ReferenceMeasures Ω), Causalean.SCM.ReferenceFaithful ref → ∀ (sDo : (M.fixSet X hObs hFix).FixedValues) (S C : Finset (Causalean.SWIGNode N)), S ∈ Causalean.SCM.ID.fixTruncCComponentSet M X hObs hFix Y → Causalean.SCM.ID.cFactorReachable M.toSWIGGraph C S → C ∈ M.cComponentSet → ∀ [inst_6 : MeasureTheory.IsFiniteMeasure ((Causalean.SCM.ID.doObsKernelAncestralMarginal M X hObs hFix Y) sDo)] [inst_7 : ∀ (s' : M.FixedValues), MeasureTheory.IsFiniteMeasure (M.obsKernel s')] [inst_8 : ∀ (k : ℕ) (hk : k < (Causalean.SCM.ID.fixObservedAncestralSet M X hObs hFix Y).card), StandardBorelSpace (Causalean.ValuesOn {↑(((M.fixSet X hObs hFix).induce (Causalean.SCM.ID.fixAncestralSet M X hObs hFix Y)).nodesAt (Causalean.SCM.ID.fixObservedAncestralSet M X hObs hFix Y) ⟨k, hk⟩)} (Causalean.swigΩ Ω))] [inst_9 : ∀ (k : ℕ) (hk : k < (Causalean.SCM.ID.fixObservedAncestralSet M X hObs hFix Y).card), Nonempty (Causalean.ValuesOn {↑(((M.fixSet X hObs hFix).induce (Causalean.SCM.ID.fixAncestralSet M X hObs hFix Y)).nodesAt (Causalean.SCM.ID.fixObservedAncestralSet M X hObs hFix Y) ⟨k, hk⟩)} (Causalean.swigΩ Ω))] [inst_10 : ∀ (k : ℕ) (hk : k < M.observed.card), StandardBorelSpace (Causalean.ValuesOn {↑(M.observedAt ⟨k, hk⟩)} (Causalean.swigΩ Ω))] [inst_11 : ∀ (k : ℕ) (hk : k < M.observed.card), Nonempty (Causalean.ValuesOn {↑(M.observedAt ⟨k, hk⟩)} (Causalean.swigΩ Ω))] [inst_12 : ∀ (k : ℕ), MeasurableSpace.CountableOrCountablyGenerated M.FixedValues (Causalean.ValuesOn (M.prefixNodes k) (Causalean.swigΩ Ω))], (∀ (s' : M.FixedValues), Causalean.SCM.ID.DiscreteID.PositiveMass (M.obsKernel s')) → (∀ D ∈ X, Causalean.SWIGNode.random D ∉ Y) → ∀ (extend : Causalean.ValuesOn (Causalean.SCM.ID.fixObservedAncestralSet M X hObs hFix Y) (Causalean.swigΩ Ω) → Causalean.ValuesOn M.observed (Causalean.swigΩ Ω)), (∀ (xD : Causalean.ValuesOn (Causalean.SCM.ID.fixObservedAncestralSet M X hObs hFix Y) (Causalean.swigΩ Ω)), Causalean.valuesProjection (have this := Finset.inter_subset_right; this) (extend xD) = xD) → (∀ (xD : Causalean.ValuesOn (Causalean.SCM.ID.fixObservedAncestralSet M X hObs hFix Y) (Causalean.swigΩ Ω)) (D : N) (hD : D ∈ X), extend xD ⟨Causalean.SWIGNode.random D, hObs D hD⟩ = sDo ⟨Causalean.SWIGNode.fixed D, Finset.mem_union_right M.fixed (Finset.mem_image.mpr (Exists.intro D ⟨hD, rfl⟩))⟩) → let D := Causalean.SCM.ID.fixObservedAncestralSet M X hObs hFix Y; let H := (M.fixSet X hObs hFix).induce (Causalean.SCM.ID.fixAncestralSet M X hObs hFix Y); Causalean.SCM.ID.tianDistrictDensity H D ((Causalean.SCM.ID.doObsKernelAncestralMarginal M X hObs hFix Y) sDo) ref S =ᵐ[Causalean.SCM.jointRef ref D] fun xD => M.cComponentDensityFactor ref (M.fixSetProj X hObs hFix sDo) C (extend xD)","l":1052,"f":"Causalean/SCM/ID/GraphicalThms/DoGFormulaTian.lean"},{"n":"Causalean.SCM.ID.doAncestralDistrictDensity_recovered_from_obs_core_self","k":"theorem","m":"Causalean.SCM.ID.GraphicalThms.DoGFormulaTian","a":"SCM/ID/GraphicalThms","t":2,"d":"Finite same-district recovery: the do-law's S-district density factor equals S's observational c-component density factor (after extension), for a district S that is already a full observational c-component.","s":"∀ {N : Type u_1} [inst : DecidableEq N] [inst_1 : Fintype N] {Ω : N → Type u_2} [inst_2 : (n : N) → MeasurableSpace (Ω n)] [∀ (n : N), Nonempty (Ω n)] [(n : N) → Fintype (Ω n)] [∀ (n : N), MeasurableSingletonClass (Ω n)] (M : Causalean.SCM N Ω) (X : Finset N), M.isStandard → ∀ (hObs : ∀ D ∈ X, Causalean.SWIGNode.random D ∈ M.observed) (hFix : ∀ D ∈ X, Causalean.SWIGNode.fixed D ∉ M.fixed) (Y : Finset (Causalean.SWIGNode N)) (ref : Causalean.SCM.ReferenceMeasures Ω), Causalean.SCM.ReferenceFaithful ref → ∀ (sDo : (M.fixSet X hObs hFix).FixedValues), ∀ S ∈ Causalean.SCM.ID.fixTruncCComponentSet M X hObs hFix Y, S ∈ M.cComponentSet → ∀ [inst_6 : MeasureTheory.IsFiniteMeasure ((Causalean.SCM.ID.doObsKernelAncestralMarginal M X hObs hFix Y) sDo)] [inst_7 : ∀ (s' : M.FixedValues), MeasureTheory.IsFiniteMeasure (M.obsKernel s')] [inst_8 : ∀ (k : ℕ) (hk : k < (Causalean.SCM.ID.fixObservedAncestralSet M X hObs hFix Y).card), StandardBorelSpace (Causalean.ValuesOn {↑(((M.fixSet X hObs hFix).induce (Causalean.SCM.ID.fixAncestralSet M X hObs hFix Y)).nodesAt (Causalean.SCM.ID.fixObservedAncestralSet M X hObs hFix Y) ⟨k, hk⟩)} (Causalean.swigΩ Ω))] [inst_9 : ∀ (k : ℕ) (hk : k < (Causalean.SCM.ID.fixObservedAncestralSet M X hObs hFix Y).card), Nonempty (Causalean.ValuesOn {↑(((M.fixSet X hObs hFix).induce (Causalean.SCM.ID.fixAncestralSet M X hObs hFix Y)).nodesAt (Causalean.SCM.ID.fixObservedAncestralSet M X hObs hFix Y) ⟨k, hk⟩)} (Causalean.swigΩ Ω))] [inst_10 : ∀ (k : ℕ) (hk : k < M.observed.card), StandardBorelSpace (Causalean.ValuesOn {↑(M.observedAt ⟨k, hk⟩)} (Causalean.swigΩ Ω))] [inst_11 : ∀ (k : ℕ) (hk : k < M.observed.card), Nonempty (Causalean.ValuesOn {↑(M.observedAt ⟨k, hk⟩)} (Causalean.swigΩ Ω))] [inst_12 : ∀ (k : ℕ), MeasurableSpace.CountableOrCountablyGenerated M.FixedValues (Causalean.ValuesOn (M.prefixNodes k) (Causalean.swigΩ Ω))], (∀ (s' : M.FixedValues), Causalean.SCM.ID.DiscreteID.PositiveMass (M.obsKernel s')) → (∀ D ∈ X, Causalean.SWIGNode.random D ∉ Y) → ∀ (extend : Causalean.ValuesOn (Causalean.SCM.ID.fixObservedAncestralSet M X hObs hFix Y) (Causalean.swigΩ Ω) → Causalean.ValuesOn M.observed (Causalean.swigΩ Ω)), (∀ (xD : Causalean.ValuesOn (Causalean.SCM.ID.fixObservedAncestralSet M X hObs hFix Y) (Causalean.swigΩ Ω)), Causalean.valuesProjection (have this := Finset.inter_subset_right; this) (extend xD) = xD) → (∀ (xD : Causalean.ValuesOn (Causalean.SCM.ID.fixObservedAncestralSet M X hObs hFix Y) (Causalean.swigΩ Ω)) (D : N) (hD : D ∈ X), extend xD ⟨Causalean.SWIGNode.random D, hObs D hD⟩ = sDo ⟨Causalean.SWIGNode.fixed D, Finset.mem_union_right M.fixed (Finset.mem_image.mpr (Exists.intro D ⟨hD, rfl⟩))⟩) → let D := Causalean.SCM.ID.fixObservedAncestralSet M X hObs hFix Y; let H := (M.fixSet X hObs hFix).induce (Causalean.SCM.ID.fixAncestralSet M X hObs hFix Y); Causalean.SCM.ID.tianDistrictDensity H D ((Causalean.SCM.ID.doObsKernelAncestralMarginal M X hObs hFix Y) sDo) ref S =ᵐ[Causalean.SCM.jointRef ref D] fun xD => M.cComponentDensityFactor ref (M.fixSetProj X hObs hFix sDo) S (extend xD)","l":977,"f":"Causalean/SCM/ID/GraphicalThms/DoGFormulaTian.lean"},{"n":"Causalean.SCM.ID.doAncestralDistrictDensity_recovered_from_obs_rec","k":"theorem","m":"Causalean.SCM.ID.GraphicalThms.IDAlgorithmRec","a":"SCM/ID/GraphicalThms","t":1,"d":"Recursive district-density recovery from the observational kernel. Fix a standard structural causal model M and an intervention target set X for which every targeted node is currently a random observed node and none of its fixed copies is already fixed, an output set Y, and a reference-measure family faithful to the graph. For a district S of the truncated c-component set of the post-intervention ancestral graph and a c-component C of the base graph that is recursively factor-reachable from S, assume every fixed-value assignment gives an observational kernel with everywhere-positive point masses, no intervention target's random form lies in Y, and that an extension map from ancestral assignments to full observed assignments restricts back to the identity and agrees with the intervention values sDo on the targeted coordinates. Then the district factor of S computed from the density of the do-law's ancestral marginal equals, almost everywhere, the full-graph c-component density factor of C evaluated at the extension of the ancestral assignment.","s":"∀ {N : Type u_1} [inst : DecidableEq N] [inst_1 : Fintype N] {Ω : N → Type u_2} [inst_2 : (n : N) → MeasurableSpace (Ω n)] [∀ (n : N), Nonempty (Ω n)] [inst_4 : (n : N) → Fintype (Ω n)] [∀ (n : N), MeasurableSingletonClass (Ω n)] (M : Causalean.SCM N Ω) (X : Finset N), M.isStandard → ∀ (hObs : ∀ D ∈ X, Causalean.SWIGNode.random D ∈ M.observed) (hFix : ∀ D ∈ X, Causalean.SWIGNode.fixed D ∉ M.fixed) (Y : Finset (Causalean.SWIGNode N)) (ref : Causalean.SCM.ReferenceMeasures Ω), Causalean.SCM.ReferenceFaithful ref → ∀ (sDo : (M.fixSet X hObs hFix).FixedValues) (S C : Finset (Causalean.SWIGNode N)), S ∈ Causalean.SCM.ID.fixTruncCComponentSet M X hObs hFix Y → Causalean.SCM.ID.CFactorReachableRec M.toSWIGGraph C S → C ∈ M.cComponentSet → ∀ [inst_6 : MeasureTheory.IsFiniteMeasure ((Causalean.SCM.ID.doObsKernelAncestralMarginal M X hObs hFix Y) sDo)] [inst_7 : ∀ (s' : M.FixedValues), MeasureTheory.IsFiniteMeasure (M.obsKernel s')] [inst_8 : ∀ (k : ℕ) (hk : k < (Causalean.SCM.ID.fixObservedAncestralSet M X hObs hFix Y).card), StandardBorelSpace (Causalean.ValuesOn {↑(((M.fixSet X hObs hFix).induce (Causalean.SCM.ID.fixAncestralSet M X hObs hFix Y)).nodesAt (Causalean.SCM.ID.fixObservedAncestralSet M X hObs hFix Y) ⟨k, hk⟩)} (Causalean.swigΩ Ω))] [inst_9 : ∀ (k : ℕ) (hk : k < (Causalean.SCM.ID.fixObservedAncestralSet M X hObs hFix Y).card), Nonempty (Causalean.ValuesOn {↑(((M.fixSet X hObs hFix).induce (Causalean.SCM.ID.fixAncestralSet M X hObs hFix Y)).nodesAt (Causalean.SCM.ID.fixObservedAncestralSet M X hObs hFix Y) ⟨k, hk⟩)} (Causalean.swigΩ Ω))] [inst_10 : ∀ (k : ℕ) (hk : k < M.observed.card), StandardBorelSpace (Causalean.ValuesOn {↑(M.observedAt ⟨k, hk⟩)} (Causalean.swigΩ Ω))] [inst_11 : ∀ (k : ℕ) (hk : k < M.observed.card), Nonempty (Causalean.ValuesOn {↑(M.observedAt ⟨k, hk⟩)} (Causalean.swigΩ Ω))] [inst_12 : ∀ (k : ℕ), MeasurableSpace.CountableOrCountablyGenerated M.FixedValues (Causalean.ValuesOn (M.prefixNodes k) (Causalean.swigΩ Ω))], (∀ (s' : M.FixedValues), Causalean.SCM.ID.DiscreteID.PositiveMass (M.obsKernel s')) → (∀ D ∈ X, Causalean.SWIGNode.random D ∉ Y) → ∀ (extend : Causalean.ValuesOn (Causalean.SCM.ID.fixObservedAncestralSet M X hObs hFix Y) (Causalean.swigΩ Ω) → Causalean.ValuesOn M.observed (Causalean.swigΩ Ω)), (∀ (xD : Causalean.ValuesOn (Causalean.SCM.ID.fixObservedAncestralSet M X hObs hFix Y) (Causalean.swigΩ Ω)), Causalean.valuesProjection (have this := Finset.inter_subset_right; this) (extend xD) = xD) → (∀ (xD : Causalean.ValuesOn (Causalean.SCM.ID.fixObservedAncestralSet M X hObs hFix Y) (Causalean.swigΩ Ω)) (D : N) (hD : D ∈ X), extend xD ⟨Causalean.SWIGNode.random D, hObs D hD⟩ = sDo ⟨Causalean.SWIGNode.fixed D, Finset.mem_union_right M.fixed (Finset.mem_image.mpr (Exists.intro D ⟨hD, rfl⟩))⟩) → let D := Causalean.SCM.ID.fixObservedAncestralSet M X hObs hFix Y; let H := (M.fixSet X hObs hFix).induce (Causalean.SCM.ID.fixAncestralSet M X hObs hFix Y); Causalean.SCM.ID.tianDistrictDensity H D ((Causalean.SCM.ID.doObsKernelAncestralMarginal M X hObs hFix Y) sDo) ref S =ᵐ[Causalean.SCM.jointRef ref D] fun xD => Causalean.SCM.ID.recoveredFactorRec M ref (M.fixSetProj X hObs hFix sDo) C S (extend xD)","l":322,"f":"Causalean/SCM/ID/GraphicalThms/IDAlgorithmRec.lean"},{"n":"Causalean.SCM.ID.doKernelY_eq_cfactor_decomposition","k":"theorem","m":"Causalean.SCM.ID.GraphicalThms.IDAlgorithm","a":"SCM/ID/GraphicalThms","t":1,"d":"Valid-branch kernel equality under a successful no-fixing ID certificate. For two finite structural causal models M₁, M₂ that share the same SWIG graph G, are each dominated by a reference-measure family ref that is faithful to the graph, satisfy discrete positivity of their observational kernels, and have heterogeneously equal observational kernels, if the total interventional query on outcome set Y under intervention X is well formed in both models and X, Y admit a successful no-fixing ID certificate on G, then the two models' post-intervention outcome kernels for Y are heterogeneously equal.","s":"∀ {N : Type u_1} [inst : DecidableEq N] [inst_1 : Fintype N] {Ω : N → Type u_2} [inst_2 : (n : N) → MeasurableSpace (Ω n)] [∀ (n : N), StandardBorelSpace (Ω n)] [∀ (n : N), Nonempty (Ω n)] [(n : N) → Fintype (Ω n)] [∀ (n : N), MeasurableSingletonClass (Ω n)] (X : Finset N) (Y : Finset (Causalean.SWIGNode N)) (G : Causalean.SWIGGraph N) (ref : Causalean.SCM.ReferenceMeasures Ω), Causalean.SCM.ReferenceFaithful ref → Causalean.SCM.ID.idSucceeds X Y G → ∀ (M₁ M₂ : Causalean.SCM N Ω), M₁.toSWIGGraph = G → M₂.toSWIGGraph = G → M₁.DominatedObs ref → M₂.DominatedObs ref → Causalean.SCM.ID.DiscreteID.DiscretePositive M₁ → Causalean.SCM.ID.DiscreteID.DiscretePositive M₂ → M₁.obsKernel ≍ M₂.obsKernel → ∀ (hvalid₁ : Causalean.SCM.ID.interventionalQueryValid X Y M₁) (hvalid₂ : Causalean.SCM.ID.interventionalQueryValid X Y M₂), M₁.doKernelY X hvalid₁.left hvalid₁.right.left Y hvalid₁.right.right.left (Causalean.SCM.ID.standardFixedValues M₁ hvalid₁.right.right.right) = M₂.doKernelY X hvalid₂.left hvalid₂.right.left Y hvalid₂.right.right.left (Causalean.SCM.ID.standardFixedValues M₂ hvalid₂.right.right.right)","l":504,"f":"Causalean/SCM/ID/GraphicalThms/IDAlgorithm.lean"},{"n":"Causalean.SCM.ID.doKernelY_eq_cfactor_decomposition_rec","k":"theorem","m":"Causalean.SCM.ID.GraphicalThms.IDAlgorithmRec","a":"SCM/ID/GraphicalThms","t":1,"d":"Recursive valid-branch kernel equality. For two finite structural causal models M₁, M₂ that share the same SWIG graph G, are each dominated by a reference-measure family ref that is faithful to the graph, satisfy discrete positivity of their observational kernels, and have heterogeneously equal observational kernels, if the total interventional query on outcome set Y under intervention X is well formed in both models and X, Y admit a successful full recursive ID certificate on G, then the two models' post-intervention outcome kernels for Y are heterogeneously equal. This is the same transport as doKernelY_eq_cfactor_decomposition, with the recursive Y-marginal wrapper in place of the no-fixing one.","s":"∀ {N : Type u_1} [inst : DecidableEq N] [inst_1 : Fintype N] {Ω : N → Type u_2} [inst_2 : (n : N) → MeasurableSpace (Ω n)] [∀ (n : N), StandardBorelSpace (Ω n)] [∀ (n : N), Nonempty (Ω n)] [(n : N) → Fintype (Ω n)] [∀ (n : N), MeasurableSingletonClass (Ω n)] (X : Finset N) (Y : Finset (Causalean.SWIGNode N)) (G : Causalean.SWIGGraph N) (ref : Causalean.SCM.ReferenceMeasures Ω), Causalean.SCM.ReferenceFaithful ref → Causalean.SCM.ID.idSucceedsRec X Y G → ∀ (M₁ M₂ : Causalean.SCM N Ω), M₁.toSWIGGraph = G → M₂.toSWIGGraph = G → M₁.DominatedObs ref → M₂.DominatedObs ref → Causalean.SCM.ID.DiscreteID.DiscretePositive M₁ → Causalean.SCM.ID.DiscreteID.DiscretePositive M₂ → M₁.obsKernel ≍ M₂.obsKernel → ∀ (hvalid₁ : Causalean.SCM.ID.interventionalQueryValid X Y M₁) (hvalid₂ : Causalean.SCM.ID.interventionalQueryValid X Y M₂), M₁.doKernelY X hvalid₁.left hvalid₁.right.left Y hvalid₁.right.right.left (Causalean.SCM.ID.standardFixedValues M₁ hvalid₁.right.right.right) = M₂.doKernelY X hvalid₂.left hvalid₂.right.left Y hvalid₂.right.right.left (Causalean.SCM.ID.standardFixedValues M₂ hvalid₂.right.right.right)","l":821,"f":"Causalean/SCM/ID/GraphicalThms/IDAlgorithmRec.lean"},{"n":"Causalean.SCM.ID.doKernelY_eq_of_doObsKernel_heq","k":"theorem","m":"Causalean.SCM.ID.DoLawTransport","a":"SCM/ID","t":1,"d":"Structural transport. Fix two causal models that share the same underlying SWIG graph, together with a treatment set X that is a valid intervention in the first model — every node of X is observed and not already fixed there and likewise valid in the second model, and an outcome set Y that is observed in the first model and observed in the second. If the two models' base fixed-value slices coincide and their Y-marginals of the do-observational law agree, up to the type-level identification the shared graph provides, then the resulting post-intervention Y-marginal kernels doKernelY agree at that shared base slice.","s":"∀ {N : Type u_1} [inst : DecidableEq N] [inst_1 : Fintype N] {Ω : N → Type u_2} [inst_2 : (n : N) → MeasurableSpace (Ω n)] (X : Finset N) (Y : Finset (Causalean.SWIGNode N)) (M₁ M₂ : Causalean.SCM N Ω), M₁.toSWIGGraph = M₂.toSWIGGraph → ∀ (hObs₁ : ∀ D ∈ X, Causalean.SWIGNode.random D ∈ M₁.observed) (hFix₁ : ∀ D ∈ X, Causalean.SWIGNode.fixed D ∉ M₁.fixed) (hObs₂ : ∀ D ∈ X, Causalean.SWIGNode.random D ∈ M₂.observed) (hFix₂ : ∀ D ∈ X, Causalean.SWIGNode.fixed D ∉ M₂.fixed) (hY₁ : Y ⊆ M₁.observed) (hY₂ : Y ⊆ M₂.observed) (s0₁ : M₁.FixedValues) (s0₂ : M₂.FixedValues), s0₁ ≍ s0₂ → Causalean.SCM.ID.doObsKernelYMarginal M₁ X hObs₁ hFix₁ Y hY₁ ≍ Causalean.SCM.ID.doObsKernelYMarginal M₂ X hObs₂ hFix₂ Y hY₂ → M₁.doKernelY X hObs₁ hFix₁ Y hY₁ s0₁ = M₂.doKernelY X hObs₂ hFix₂ Y hY₂ s0₂","l":55,"f":"Causalean/SCM/ID/DoLawTransport.lean"},{"n":"Causalean.SCM.ID.doObsKernelAncestralMarginal","k":"def","m":"Causalean.SCM.ID.GraphicalThms.DoGFormula","a":"SCM/ID/GraphicalThms","t":1,"d":"The post-intervention marginal on the observed ancestors of the query: the do-observational law pushed forward to the observed part of Ystar = An_{G_X}(Y). This is the object the truncated g-formula identifies first; the requested Y-marginal is a further projection of it along Y ⊆ Ystar.","s":"{N : Type u_1} → [inst : DecidableEq N] → [inst_1 : Fintype N] → {Ω : N → Type u_2} → [inst_2 : (n : N) → MeasurableSpace (Ω n)] → (M : Causalean.SCM N Ω) → (X : Finset N) → (hObs : ∀ D ∈ X, Causalean.SWIGNode.random D ∈ M.observed) → (hFix : ∀ D ∈ X, Causalean.SWIGNode.fixed D ∉ M.fixed) → (Y : Finset (Causalean.SWIGNode N)) → ProbabilityTheory.Kernel (M.fixSet X hObs hFix).FixedValues (Causalean.ValuesOn (Causalean.SCM.ID.fixObservedAncestralSet M X hObs hFix Y) (Causalean.swigΩ Ω))","l":393,"f":"Causalean/SCM/ID/GraphicalThms/DoGFormula.lean"},{"n":"Causalean.SCM.ID.doObsKernelAncestralMarginal_dominated","k":"theorem","m":"Causalean.SCM.ID.GraphicalThms.IDAlgorithm","a":"SCM/ID/GraphicalThms","t":2,"d":"A faithful finite product reference dominates the post-intervention observed-ancestral marginal.","s":"∀ {N : Type u_1} [inst : DecidableEq N] [inst_1 : Fintype N] {Ω : N → Type u_2} [inst_2 : (n : N) → MeasurableSpace (Ω n)] [∀ (n : N), MeasurableSingletonClass (Ω n)] (M : Causalean.SCM N Ω) (X : Finset N) (hObs : ∀ D ∈ X, Causalean.SWIGNode.random D ∈ M.observed) (hFix : ∀ D ∈ X, Causalean.SWIGNode.fixed D ∉ M.fixed) (Y : Finset (Causalean.SWIGNode N)) (ref : Causalean.SCM.ReferenceMeasures Ω), Causalean.SCM.ReferenceFaithful ref → ∀ (s : (M.fixSet X hObs hFix).FixedValues), ((Causalean.SCM.ID.doObsKernelAncestralMarginal M X hObs hFix Y) s).AbsolutelyContinuous (Causalean.SCM.jointRef ref (Causalean.SCM.ID.fixObservedAncestralSet M X hObs hFix Y))","l":133,"f":"Causalean/SCM/ID/GraphicalThms/IDAlgorithm.lean"},{"n":"Causalean.SCM.ID.doObsKernelAncestralMarginal_globalMarkovOn","k":"theorem","m":"Causalean.SCM.ID.GraphicalThms.DoGFormulaTian","a":"SCM/ID/GraphicalThms","t":2,"d":"The do-law ancestral marginal is globally Markov with respect to the pure ancestral graph G_X[D]. This is the SCM-to-measure bridge for T1; it does not assert that D is an ancestrally closed SCM support.","s":"∀ {N : Type u_1} [inst : DecidableEq N] [inst_1 : Fintype N] {Ω : N → Type u_2} [inst_2 : (n : N) → MeasurableSpace (Ω n)] [inst_3 : ∀ (n : Causalean.SWIGNode N), StandardBorelSpace (Causalean.swigΩ Ω n)] [∀ (n : Causalean.SWIGNode N), Nonempty (Causalean.swigΩ Ω n)] (M : Causalean.SCM N Ω) (X : Finset N) (hObs : ∀ D ∈ X, Causalean.SWIGNode.random D ∈ M.observed) (hFix : ∀ D ∈ X, Causalean.SWIGNode.fixed D ∉ M.fixed) (Y : Finset (Causalean.SWIGNode N)) [StandardBorelSpace (M.fixSet X hObs hFix).RandomValues] [StandardBorelSpace (M.fixSet X hObs hFix).ObservedValues] [∀ (s : (M.fixSet X hObs hFix).FixedValues), MeasureTheory.IsFiniteMeasure ((M.fixSet X hObs hFix).jointKernel s)] [∀ (s : (M.fixSet X hObs hFix).FixedValues), MeasureTheory.IsFiniteMeasure ((M.fixSet X hObs hFix).obsKernel s)] (s : (M.fixSet X hObs hFix).FixedValues) [inst_9 : MeasureTheory.IsFiniteMeasure ((Causalean.SCM.ID.doObsKernelAncestralMarginal M X hObs hFix Y) s)], have A := Causalean.SCM.ID.fixAncestralSet M X hObs hFix Y; let D := Causalean.SCM.ID.fixObservedAncestralSet M X hObs hFix Y; have H := (M.fixSet X hObs hFix).induce A; Causalean.SCM.ID.KernelGlobalMarkovOn H D ((Causalean.SCM.ID.doObsKernelAncestralMarginal M X hObs hFix Y) s)","l":809,"f":"Causalean/SCM/ID/GraphicalThms/DoGFormulaTian.lean"},{"n":"Causalean.SCM.ID.doObsKernelAncestralMarginal_heq_of_obsDensity_heq","k":"theorem","m":"Causalean.SCM.ID.GraphicalThms.IDAlgorithm","a":"SCM/ID/GraphicalThms","t":2,"d":"Two dominated models with the same SWIG graph and observational density have the same post-intervention law on the observed ancestors selected by a successful ID certificate.","s":"∀ {N : Type u_1} [inst : DecidableEq N] [inst_1 : Fintype N] {Ω : N → Type u_2} [inst_2 : (n : N) → MeasurableSpace (Ω n)] [∀ (n : N), StandardBorelSpace (Ω n)] [∀ (n : N), Nonempty (Ω n)] [(n : N) → Fintype (Ω n)] [∀ (n : N), MeasurableSingletonClass (Ω n)] (X : Finset N) (Y : Finset (Causalean.SWIGNode N)) (G : Causalean.SWIGGraph N) (ref : Causalean.SCM.ReferenceMeasures Ω), Causalean.SCM.ReferenceFaithful ref → Causalean.SCM.ID.idSucceeds X Y G → ∀ (M₁ M₂ : Causalean.SCM N Ω), M₁.toSWIGGraph = G → M₂.toSWIGGraph = G → M₁.DominatedObs ref → M₂.DominatedObs ref → Causalean.SCM.ID.DiscreteID.DiscretePositive M₁ → Causalean.SCM.ID.DiscreteID.DiscretePositive M₂ → M₁.obsDensity ref ≍ M₂.obsDensity ref → ∀ (hvalid₁ : Causalean.SCM.ID.interventionalQueryValid X Y M₁) (hvalid₂ : Causalean.SCM.ID.interventionalQueryValid X Y M₂), Causalean.SCM.ID.doObsKernelAncestralMarginal M₁ X hvalid₁.left hvalid₁.right.left Y ≍ Causalean.SCM.ID.doObsKernelAncestralMarginal M₂ X hvalid₂.left hvalid₂.right.left Y","l":154,"f":"Causalean/SCM/ID/GraphicalThms/IDAlgorithm.lean"},{"n":"Causalean.SCM.ID.doObsKernelAncestralMarginal_heq_of_obsDensity_heq_rec","k":"theorem","m":"Causalean.SCM.ID.GraphicalThms.IDAlgorithmRec","a":"SCM/ID/GraphicalThms","t":2,"d":"Recursive Tian–Shpitser density core for the observed-ancestral do-law.","s":"∀ {N : Type u_1} [inst : DecidableEq N] [inst_1 : Fintype N] {Ω : N → Type u_2} [inst_2 : (n : N) → MeasurableSpace (Ω n)] [∀ (n : N), StandardBorelSpace (Ω n)] [∀ (n : N), Nonempty (Ω n)] [(n : N) → Fintype (Ω n)] [∀ (n : N), MeasurableSingletonClass (Ω n)] (X : Finset N) (Y : Finset (Causalean.SWIGNode N)) (G : Causalean.SWIGGraph N) (ref : Causalean.SCM.ReferenceMeasures Ω), Causalean.SCM.ReferenceFaithful ref → Causalean.SCM.ID.idSucceedsRec X Y G → ∀ (M₁ M₂ : Causalean.SCM N Ω), M₁.toSWIGGraph = G → M₂.toSWIGGraph = G → M₁.DominatedObs ref → M₂.DominatedObs ref → Causalean.SCM.ID.DiscreteID.DiscretePositive M₁ → Causalean.SCM.ID.DiscreteID.DiscretePositive M₂ → M₁.obsDensity ref ≍ M₂.obsDensity ref → ∀ (hvalid₁ : Causalean.SCM.ID.interventionalQueryValid X Y M₁) (hvalid₂ : Causalean.SCM.ID.interventionalQueryValid X Y M₂), Causalean.SCM.ID.doObsKernelAncestralMarginal M₁ X hvalid₁.left hvalid₁.right.left Y ≍ Causalean.SCM.ID.doObsKernelAncestralMarginal M₂ X hvalid₂.left hvalid₂.right.left Y","l":535,"f":"Causalean/SCM/ID/GraphicalThms/IDAlgorithmRec.lean"},{"n":"Causalean.SCM.ID.doObsKernelAncestralMarginal_positiveMass","k":"theorem","m":"Causalean.SCM.ID.Density.QMass","a":"SCM/ID/Density","t":1,"d":"For an intervention set X whose random copies are observed and whose fixed copies are not already frozen, a positive observational kernel at every fixed-value assignment, and an outcome set Y disjoint from the random copies of X, the do(X)-law ancestral marginal kernel used in the identification density assembly also has everywhere-positive point mass.","s":"∀ {N : Type u_1} [inst : DecidableEq N] [inst_1 : Fintype N] {Ω : N → Type u_2} [inst_2 : (n : N) → MeasurableSpace (Ω n)] [∀ (n : N), Nonempty (Ω n)] [(n : N) → Fintype (Ω n)] [∀ (n : N), MeasurableSingletonClass (Ω n)] (M : Causalean.SCM N Ω) (X : Finset N) (hObs : ∀ D ∈ X, Causalean.SWIGNode.random D ∈ M.observed) (hFix : ∀ D ∈ X, Causalean.SWIGNode.fixed D ∉ M.fixed) (Y : Finset (Causalean.SWIGNode N)), (∀ (s' : M.FixedValues), Causalean.SCM.ID.DiscreteID.PositiveMass (M.obsKernel s')) → (∀ d ∈ X, Causalean.SWIGNode.random d ∉ Y) → ∀ (sDo : (M.fixSet X hObs hFix).FixedValues), Causalean.SCM.ID.DiscreteID.PositiveMass ((Causalean.SCM.ID.doObsKernelAncestralMarginal M X hObs hFix Y) sDo)","l":1664,"f":"Causalean/SCM/ID/Density/QMass.lean"},{"n":"Causalean.SCM.ID.doObsKernelAncestralMarginal_tian_cfactorization_density","k":"theorem","m":"Causalean.SCM.ID.GraphicalThms.DoGFormulaTian","a":"SCM/ID/GraphicalThms","t":1,"d":"ID-specific T1 wrapper. For an intervention target set X where every targeted node is currently a random observed node with no fixed copy already fixed, if the ancestral marginal of the do-law ν_M = (M.fixSet X).obsKernel.map π_D is absolutely continuous with respect to the product reference measure on the ancestral observed set, then its Radon–Nikodym density equals, almost everywhere, the product over the c-components of the induced post-intervention ancestral graph G_X[D] of their Tian district-density factors.","s":"∀ {N : Type u_1} [inst : DecidableEq N] [inst_1 : Fintype N] {Ω : N → Type u_2} [inst_2 : (n : N) → MeasurableSpace (Ω n)] [∀ (n : Causalean.SWIGNode N), StandardBorelSpace (Causalean.swigΩ Ω n)] [∀ (n : Causalean.SWIGNode N), Nonempty (Causalean.swigΩ Ω n)] [(n : N) → Fintype (Ω n)] [∀ (n : N), MeasurableSingletonClass (Ω n)] (M : Causalean.SCM N Ω) (X : Finset N) (hObs : ∀ D ∈ X, Causalean.SWIGNode.random D ∈ M.observed) (hFix : ∀ D ∈ X, Causalean.SWIGNode.fixed D ∉ M.fixed) (Y : Finset (Causalean.SWIGNode N)) (ref : Causalean.SCM.ReferenceMeasures Ω) (s : (M.fixSet X hObs hFix).FixedValues) [StandardBorelSpace (Causalean.ValuesOn (Causalean.SCM.ID.fixObservedAncestralSet M X hObs hFix Y) (Causalean.swigΩ Ω))] [StandardBorelSpace (M.fixSet X hObs hFix).RandomValues] [StandardBorelSpace (M.fixSet X hObs hFix).ObservedValues] [∀ (s' : (M.fixSet X hObs hFix).FixedValues), MeasureTheory.IsFiniteMeasure ((M.fixSet X hObs hFix).jointKernel s')] [∀ (s' : (M.fixSet X hObs hFix).FixedValues), MeasureTheory.IsFiniteMeasure ((M.fixSet X hObs hFix).obsKernel s')] [inst_12 : MeasureTheory.IsFiniteMeasure ((Causalean.SCM.ID.doObsKernelAncestralMarginal M X hObs hFix Y) s)] [inst_13 : ∀ (k : ℕ) (hk : k < (Causalean.SCM.ID.fixObservedAncestralSet M X hObs hFix Y).card), StandardBorelSpace (Causalean.ValuesOn {↑(((M.fixSet X hObs hFix).induce (Causalean.SCM.ID.fixAncestralSet M X hObs hFix Y)).nodesAt (Causalean.SCM.ID.fixObservedAncestralSet M X hObs hFix Y) ⟨k, hk⟩)} (Causalean.swigΩ Ω))] [inst_14 : ∀ (k : ℕ) (hk : k < (Causalean.SCM.ID.fixObservedAncestralSet M X hObs hFix Y).card), Nonempty (Causalean.ValuesOn {↑(((M.fixSet X hObs hFix).induce (Causalean.SCM.ID.fixAncestralSet M X hObs hFix Y)).nodesAt (Causalean.SCM.ID.fixObservedAncestralSet M X hObs hFix Y) ⟨k, hk⟩)} (Causalean.swigΩ Ω))], ((Causalean.SCM.ID.doObsKernelAncestralMarginal M X hObs hFix Y) s).AbsolutelyContinuous (Causalean.SCM.jointRef ref (Causalean.SCM.ID.fixObservedAncestralSet M X hObs hFix Y)) → let D := Causalean.SCM.ID.fixObservedAncestralSet M X hObs hFix Y; let H := (M.fixSet X hObs hFix).induce (Causalean.SCM.ID.fixAncestralSet M X hObs hFix Y); ((Causalean.SCM.ID.doObsKernelAncestralMarginal M X hObs hFix Y) s).rnDeriv (Causalean.SCM.jointRef ref D) =ᵐ[Causalean.SCM.jointRef ref D] fun x => ∏ S ∈ H.cComponentSet, Causalean.SCM.ID.tianDistrictDensity H D ((Causalean.SCM.ID.doObsKernelAncestralMarginal M X hObs hFix Y) s) ref S x","l":1223,"f":"Causalean/SCM/ID/GraphicalThms/DoGFormulaTian.lean"},{"n":"Causalean.SCM.ID.doObsKernelYMarginal","k":"def","m":"Causalean.SCM.ID.DoLawTransport","a":"SCM/ID","t":1,"d":"The Y-marginal of the do-observational law: push (M.fixSet X).obsKernel forward along the projection to the outcome coordinates Y. This is the only part of the do-law that the post-intervention Y-marginal kernel doKernelY depends on — doKernelY is this marginal, reindexed in the treatment value by the comap extension. Crucially this is the *identifiable* object: the full do-law over all observed nodes is not a functional of the observational law, but its Y-marginal (more precisely its An_{G_X}(Y)-marginal, of which this is a further projection) is.","s":"{N : Type u_1} → [inst : DecidableEq N] → [inst_1 : Fintype N] → {Ω : N → Type u_2} → [inst_2 : (n : N) → MeasurableSpace (Ω n)] → (M : Causalean.SCM N Ω) → (X : Finset N) → (hObs : ∀ D ∈ X, Causalean.SWIGNode.random D ∈ M.observed) → (hFix : ∀ D ∈ X, Causalean.SWIGNode.fixed D ∉ M.fixed) → (Y : Finset (Causalean.SWIGNode N)) → Y ⊆ M.observed → ProbabilityTheory.Kernel (M.fixSet X hObs hFix).FixedValues (Causalean.ValuesOn Y (Causalean.swigΩ Ω))","l":37,"f":"Causalean/SCM/ID/DoLawTransport.lean"},{"n":"Causalean.SCM.ID.doObsKernelYMarginal_eq_ancestralMarginal_map","k":"theorem","m":"Causalean.SCM.ID.GraphicalThms.IDAlgorithm","a":"SCM/ID/GraphicalThms","t":2,"d":"The outcome marginal of the do-law is the projection of the observed-ancestral do-law marginal to the outcome coordinates.","s":"∀ {N : Type u_1} [inst : DecidableEq N] [inst_1 : Fintype N] {Ω : N → Type u_2} [inst_2 : (n : N) → MeasurableSpace (Ω n)] (M : Causalean.SCM N Ω) (X : Finset N) (hObs : ∀ D ∈ X, Causalean.SWIGNode.random D ∈ M.observed) (hFix : ∀ D ∈ X, Causalean.SWIGNode.fixed D ∉ M.fixed) (Y : Finset (Causalean.SWIGNode N)) (hY : Y ⊆ M.observed), Causalean.SCM.ID.doObsKernelYMarginal M X hObs hFix Y hY = (Causalean.SCM.ID.doObsKernelAncestralMarginal M X hObs hFix Y).map (Causalean.valuesProjection (Causalean.SCM.ID.subset_fixObservedAncestralSet M X hObs hFix Y hY))","l":63,"f":"Causalean/SCM/ID/GraphicalThms/IDAlgorithm.lean"},{"n":"Causalean.SCM.ID.doObsKernelYMarginal_heq_of_ancestralMarginal_heq","k":"theorem","m":"Causalean.SCM.ID.GraphicalThms.IDAlgorithm","a":"SCM/ID/GraphicalThms","t":2,"d":"If two models with the same SWIG graph have equal observed-ancestral do-law marginals, then their outcome marginals agree.","s":"∀ {N : Type u_1} [inst : DecidableEq N] [inst_1 : Fintype N] {Ω : N → Type u_2} [inst_2 : (n : N) → MeasurableSpace (Ω n)] (X : Finset N) (Y : Finset (Causalean.SWIGNode N)) (M₁ M₂ : Causalean.SCM N Ω), M₁.toSWIGGraph = M₂.toSWIGGraph → ∀ (hObs₁ : ∀ D ∈ X, Causalean.SWIGNode.random D ∈ M₁.observed) (hFix₁ : ∀ D ∈ X, Causalean.SWIGNode.fixed D ∉ M₁.fixed) (hObs₂ : ∀ D ∈ X, Causalean.SWIGNode.random D ∈ M₂.observed) (hFix₂ : ∀ D ∈ X, Causalean.SWIGNode.fixed D ∉ M₂.fixed) (hY₁ : Y ⊆ M₁.observed) (hY₂ : Y ⊆ M₂.observed), Causalean.SCM.ID.doObsKernelAncestralMarginal M₁ X hObs₁ hFix₁ Y ≍ Causalean.SCM.ID.doObsKernelAncestralMarginal M₂ X hObs₂ hFix₂ Y → Causalean.SCM.ID.doObsKernelYMarginal M₁ X hObs₁ hFix₁ Y hY₁ ≍ Causalean.SCM.ID.doObsKernelYMarginal M₂ X hObs₂ hFix₂ Y hY₂","l":88,"f":"Causalean/SCM/ID/GraphicalThms/IDAlgorithm.lean"},{"n":"Causalean.SCM.ID.doObsKernelYMarginal_heq_of_obsDensity_heq","k":"theorem","m":"Causalean.SCM.ID.GraphicalThms.IDAlgorithm","a":"SCM/ID/GraphicalThms","t":2,"d":"Two dominated models with the same SWIG graph and observational density have the same post-intervention outcome marginal under a successful ID certificate.","s":"∀ {N : Type u_1} [inst : DecidableEq N] [inst_1 : Fintype N] {Ω : N → Type u_2} [inst_2 : (n : N) → MeasurableSpace (Ω n)] [∀ (n : N), StandardBorelSpace (Ω n)] [∀ (n : N), Nonempty (Ω n)] [(n : N) → Fintype (Ω n)] [∀ (n : N), MeasurableSingletonClass (Ω n)] (X : Finset N) (Y : Finset (Causalean.SWIGNode N)) (G : Causalean.SWIGGraph N) (ref : Causalean.SCM.ReferenceMeasures Ω), Causalean.SCM.ReferenceFaithful ref → Causalean.SCM.ID.idSucceeds X Y G → ∀ (M₁ M₂ : Causalean.SCM N Ω), M₁.toSWIGGraph = G → M₂.toSWIGGraph = G → M₁.DominatedObs ref → M₂.DominatedObs ref → Causalean.SCM.ID.DiscreteID.DiscretePositive M₁ → Causalean.SCM.ID.DiscreteID.DiscretePositive M₂ → M₁.obsDensity ref ≍ M₂.obsDensity ref → ∀ (hvalid₁ : Causalean.SCM.ID.interventionalQueryValid X Y M₁) (hvalid₂ : Causalean.SCM.ID.interventionalQueryValid X Y M₂), Causalean.SCM.ID.doObsKernelYMarginal M₁ X hvalid₁.left hvalid₁.right.left Y hvalid₁.right.right.left ≍ Causalean.SCM.ID.doObsKernelYMarginal M₂ X hvalid₂.left hvalid₂.right.left Y hvalid₂.right.right.left","l":426,"f":"Causalean/SCM/ID/GraphicalThms/IDAlgorithm.lean"},{"n":"Causalean.SCM.ID.doObsKernelYMarginal_heq_of_obsDensity_heq_rec","k":"theorem","m":"Causalean.SCM.ID.GraphicalThms.IDAlgorithmRec","a":"SCM/ID/GraphicalThms","t":2,"d":"Recursive density-to-Y-marginal wrapper. The projection from the observed-ancestral marginal to the query coordinates is identical to the no-fixing proof; only the ancestral density core changes.","s":"∀ {N : Type u_1} [inst : DecidableEq N] [inst_1 : Fintype N] {Ω : N → Type u_2} [inst_2 : (n : N) → MeasurableSpace (Ω n)] [∀ (n : N), StandardBorelSpace (Ω n)] [∀ (n : N), Nonempty (Ω n)] [(n : N) → Fintype (Ω n)] [∀ (n : N), MeasurableSingletonClass (Ω n)] (X : Finset N) (Y : Finset (Causalean.SWIGNode N)) (G : Causalean.SWIGGraph N) (ref : Causalean.SCM.ReferenceMeasures Ω), Causalean.SCM.ReferenceFaithful ref → Causalean.SCM.ID.idSucceedsRec X Y G → ∀ (M₁ M₂ : Causalean.SCM N Ω), M₁.toSWIGGraph = G → M₂.toSWIGGraph = G → M₁.DominatedObs ref → M₂.DominatedObs ref → Causalean.SCM.ID.DiscreteID.DiscretePositive M₁ → Causalean.SCM.ID.DiscreteID.DiscretePositive M₂ → M₁.obsDensity ref ≍ M₂.obsDensity ref → ∀ (hvalid₁ : Causalean.SCM.ID.interventionalQueryValid X Y M₁) (hvalid₂ : Causalean.SCM.ID.interventionalQueryValid X Y M₂), Causalean.SCM.ID.doObsKernelYMarginal M₁ X hvalid₁.left hvalid₁.right.left Y hvalid₁.right.right.left ≍ Causalean.SCM.ID.doObsKernelYMarginal M₂ X hvalid₂.left hvalid₂.right.left Y hvalid₂.right.right.left","l":773,"f":"Causalean/SCM/ID/GraphicalThms/IDAlgorithmRec.lean"},{"n":"Causalean.SCM.ID.doObsKernelYMarginal_heq_of_obsKernel_heq","k":"theorem","m":"Causalean.SCM.ID.GraphicalThms.IDAlgorithm","a":"SCM/ID/GraphicalThms","t":2,"d":"Two dominated models with the same SWIG graph and observational law have the same post-intervention outcome marginal under a successful ID certificate.","s":"∀ {N : Type u_1} [inst : DecidableEq N] [inst_1 : Fintype N] {Ω : N → Type u_2} [inst_2 : (n : N) → MeasurableSpace (Ω n)] [∀ (n : N), StandardBorelSpace (Ω n)] [∀ (n : N), Nonempty (Ω n)] [(n : N) → Fintype (Ω n)] [∀ (n : N), MeasurableSingletonClass (Ω n)] (X : Finset N) (Y : Finset (Causalean.SWIGNode N)) (G : Causalean.SWIGGraph N) (ref : Causalean.SCM.ReferenceMeasures Ω), Causalean.SCM.ReferenceFaithful ref → Causalean.SCM.ID.idSucceeds X Y G → ∀ (M₁ M₂ : Causalean.SCM N Ω), M₁.toSWIGGraph = G → M₂.toSWIGGraph = G → M₁.DominatedObs ref → M₂.DominatedObs ref → Causalean.SCM.ID.DiscreteID.DiscretePositive M₁ → Causalean.SCM.ID.DiscreteID.DiscretePositive M₂ → M₁.obsKernel ≍ M₂.obsKernel → ∀ (hvalid₁ : Causalean.SCM.ID.interventionalQueryValid X Y M₁) (hvalid₂ : Causalean.SCM.ID.interventionalQueryValid X Y M₂), Causalean.SCM.ID.doObsKernelYMarginal M₁ X hvalid₁.left hvalid₁.right.left Y hvalid₁.right.right.left ≍ Causalean.SCM.ID.doObsKernelYMarginal M₂ X hvalid₂.left hvalid₂.right.left Y hvalid₂.right.right.left","l":475,"f":"Causalean/SCM/ID/GraphicalThms/IDAlgorithm.lean"},{"n":"Causalean.SCM.ID.doObsKernelYMarginal_heq_of_obsKernel_heq_rec","k":"theorem","m":"Causalean.SCM.ID.GraphicalThms.IDAlgorithmRec","a":"SCM/ID/GraphicalThms","t":2,"d":"Recursive observational-kernel wrapper. Equal observational kernels give equal observational densities, which feed the recursive density core.","s":"∀ {N : Type u_1} [inst : DecidableEq N] [inst_1 : Fintype N] {Ω : N → Type u_2} [inst_2 : (n : N) → MeasurableSpace (Ω n)] [∀ (n : N), StandardBorelSpace (Ω n)] [∀ (n : N), Nonempty (Ω n)] [(n : N) → Fintype (Ω n)] [∀ (n : N), MeasurableSingletonClass (Ω n)] (X : Finset N) (Y : Finset (Causalean.SWIGNode N)) (G : Causalean.SWIGGraph N) (ref : Causalean.SCM.ReferenceMeasures Ω), Causalean.SCM.ReferenceFaithful ref → Causalean.SCM.ID.idSucceedsRec X Y G → ∀ (M₁ M₂ : Causalean.SCM N Ω), M₁.toSWIGGraph = G → M₂.toSWIGGraph = G → M₁.DominatedObs ref → M₂.DominatedObs ref → Causalean.SCM.ID.DiscreteID.DiscretePositive M₁ → Causalean.SCM.ID.DiscreteID.DiscretePositive M₂ → M₁.obsKernel ≍ M₂.obsKernel → ∀ (hvalid₁ : Causalean.SCM.ID.interventionalQueryValid X Y M₁) (hvalid₂ : Causalean.SCM.ID.interventionalQueryValid X Y M₂), Causalean.SCM.ID.doObsKernelYMarginal M₁ X hvalid₁.left hvalid₁.right.left Y hvalid₁.right.right.left ≍ Causalean.SCM.ID.doObsKernelYMarginal M₂ X hvalid₂.left hvalid₂.right.left Y hvalid₂.right.right.left","l":798,"f":"Causalean/SCM/ID/GraphicalThms/IDAlgorithmRec.lean"},{"n":"Causalean.SCM.ID.extendTianPrefix","k":"def","m":"Causalean.SCM.ID.GraphicalThms.DoGFormulaTian","a":"SCM/ID/GraphicalThms","t":1,"d":"Extend a D-prefix assignment by the next singleton coordinate.","s":"{N : Type u_1} → [inst : DecidableEq N] → [inst_1 : Fintype N] → {Ω : N → Type u_2} → [(n : N) → MeasurableSpace (Ω n)] → (H : Causalean.SWIGGraph N) → (D : Finset (Causalean.SWIGNode N)) → {n : ℕ} → (hn : n < D.card) → Causalean.ValuesOn (H.prefixIn D n) (Causalean.swigΩ Ω) × Causalean.ValuesOn {↑(H.nodesAt D ⟨n, hn⟩)} (Causalean.swigΩ Ω) → Causalean.ValuesOn (H.prefixIn D (n + 1)) (Causalean.swigΩ Ω)","l":169,"f":"Causalean/SCM/ID/GraphicalThms/DoGFormulaTian.lean"},{"n":"Causalean.SCM.ID.extractDistrictObserved","k":"def","m":"Causalean.SCM.ID.GraphicalThms.IDAlgorithmRec","a":"SCM/ID/GraphicalThms","t":1,"d":"Observed-set form of SCM.extractDistrict.","s":"{N : Type u_1} → [inst : DecidableEq N] → [inst_1 : Fintype N] → {Ω : N → Type u_2} → [(n : N) → Fintype (Ω n)] → (O : Finset (Causalean.SWIGNode N)) → Causalean.SWIGGraph N → (A : Finset (Causalean.SWIGNode N)) → Finset (Causalean.SWIGNode N) → A ⊆ O → (Causalean.ValuesOn O (Causalean.swigΩ Ω) → ENNReal) → Causalean.ValuesOn O (Causalean.swigΩ Ω) → ENNReal","l":71,"f":"Causalean/SCM/ID/GraphicalThms/IDAlgorithmRec.lean"},{"n":"Causalean.SCM.ID.extractDistrictObserved_eq_extractDistrict","k":"theorem","m":"Causalean.SCM.ID.GraphicalThms.IDAlgorithmRec","a":"SCM/ID/GraphicalThms","t":2,"d":"","s":"∀ {N : Type u_1} [inst : DecidableEq N] [inst_1 : Fintype N] {Ω : N → Type u_2} [inst_2 : (n : N) → MeasurableSpace (Ω n)] [inst_3 : (n : N) → Fintype (Ω n)] (M : Causalean.SCM N Ω) (G' : Causalean.SWIGGraph N) (A C' : Finset (Causalean.SWIGNode N)) (hA : A ⊆ M.observed) (q : Causalean.ValuesOn M.observed (Causalean.swigΩ Ω) → ENNReal), Causalean.SCM.ID.extractDistrictObserved M.observed G' A C' hA q = Causalean.SCM.extractDistrict M.observed G' A C' hA q","l":123,"f":"Causalean/SCM/ID/GraphicalThms/IDAlgorithmRec.lean"},{"n":"Causalean.SCM.ID.extractDistrict_qLocalMass","k":"theorem","m":"Causalean.SCM.ID.Density.QFactor","a":"SCM/ID/Density","t":2,"d":"Extracting an induced district from the local mass on an ancestral set recovers that district's local q-mass.","s":"∀ {N : Type u_1} [inst : DecidableEq N] [inst_1 : Fintype N] {Ω : N → Type u_2} [inst_2 : (n : N) → MeasurableSpace (Ω n)] [inst_3 : (n : N) → Fintype (Ω n)] [∀ (n : N), MeasurableSingletonClass (Ω n)] (M : Causalean.SCM N Ω) (s : M.FixedValues) (A C' : Finset (Causalean.SWIGNode N)) (hA : A ⊆ M.observed) (hC' : C' ∈ (M.induce A).cComponentSet), Causalean.SCM.ID.DiscreteID.PositiveMass (M.obsKernel s) → ∀ (x : Causalean.ValuesOn M.observed (Causalean.swigΩ Ω)), Causalean.SCM.extractDistrict M.observed (M.induce A) A C' hA (M.qLocalMass s A hA) x = M.qLocalMass s C' (fun x hv => hA (have hHobs := of_eq_true (Eq.trans (congrFun' (congrArg Eq (congrArg Causalean.SWIGGraph.observed (Causalean.SWIGGraph.mk.congr_simp (M.inducedDag ({s ∈ M.fixed | Causalean.iotaMap s ∈ A ∩ M.observed} ∪ A ∩ M.observed ∪ {u ∈ M.unobserved | ∃ v ∈ A ∩ M.observed, M.dag.edge u v})) (M.inducedDag ({x ∈ M.fixed | Causalean.iotaMap x ∈ A} ∪ (A ∪ {x ∈ M.unobserved | ∃ v ∈ A, M.dag.edge x v}))) (congrArg M.inducedDag (Eq.trans (congr (congrArg Union.union (congr (congrArg Union.union (Finset.filter_congr fun x a => Iff.of_eq (congrFun' (congrArg Membership.mem (Finset.inter_eq_left.mpr hA)) (Causalean.iotaMap x)))) (Finset.inter_eq_left.mpr hA))) (Finset.filter_congr fun x a => Iff.of_eq (congrArg Exists (funext fun v => congrFun' (congrArg And (congrFun' (congrArg Membership.mem (Finset.inter_eq_left.mpr hA)) v)) (M.dag.edge x v))))) (Finset.union_assoc ({x ∈ M.fixed | Causalean.iotaMap x ∈ A}) A ({x ∈ M.unobserved | ∃ v ∈ A, M.dag.edge x v})))) ({s ∈ M.fixed | Causalean.iotaMap s ∈ A ∩ M.observed}) ({x ∈ M.fixed | Causalean.iotaMap x ∈ A}) (Finset.filter_congr fun x a => Iff.of_eq (congrFun' (congrArg Membership.mem (Finset.inter_eq_left.mpr hA)) (Causalean.iotaMap x))) (A ∩ M.observed) A (Finset.inter_eq_left.mpr hA) ({u ∈ M.unobserved | ∃ v ∈ A ∩ M.observed, M.dag.edge u v}) ({x ∈ M.unobserved | ∃ v ∈ A, M.dag.edge x v}) (Finset.filter_congr fun x a => Iff.of_eq (congrArg Exists (funext fun v => congrFun' (congrArg And (congrFun' (congrArg Membership.mem (Finset.inter_eq_left.mpr hA)) v)) (M.dag.edge x v)))) (Causalean.SWIGGraph.induce._proof_1 M.toSWIGGraph A) (Causalean.SWIGGraph.induce._proof_2 M.toSWIGGraph A) (Causalean.SWIGGraph.induce._proof_3 M.toSWIGGraph A) (Causalean.SWIGGraph.induce._proof_4 M.toSWIGGraph A) (Causalean.SWIGGraph.induce._proof_5 M.toSWIGGraph A) (Causalean.SWIGGraph.induce._proof_6 M.toSWIGGraph A) (Causalean.SWIGGraph.induce._proof_7 M.toSWIGGraph A) (Causalean.SWIGGraph.induce._proof_8 M.toSWIGGraph A) (Causalean.SWIGGraph.induce._proof_9 M.toSWIGGraph A) (Causalean.SWIGGraph.induce._proof_10 M.toSWIGGraph A)))) A) (eq_self A)); hHobs ▸ Causalean.SWIGGraph.cComponentSet_subset_observed (M.induce A) C' hC' hv)) x","l":938,"f":"Causalean/SCM/ID/Density/QFactor.lean"},{"n":"Causalean.SCM.ID.fixAncestralSet","k":"def","m":"Causalean.SCM.ID.GraphicalThms.DoGFormula","a":"SCM/ID/GraphicalThms","t":1,"d":"The post-intervention ancestral set of the query. In the graph obtained by intervening on the treatment set X, this is the set consisting of the query nodes Y together with all of their ancestors. It is the support of the part of the do-law that Tian's algorithm identifies: nodes outside it do not influence P(Y ∣ do(X)).","s":"{N : Type u_1} → [inst : DecidableEq N] → [inst_1 : Fintype N] → {Ω : N → Type u_2} → [inst_2 : (n : N) → MeasurableSpace (Ω n)] → (M : Causalean.SCM N Ω) → (X : Finset N) → (∀ D ∈ X, Causalean.SWIGNode.random D ∈ M.observed) → (∀ D ∈ X, Causalean.SWIGNode.fixed D ∉ M.fixed) → Finset (Causalean.SWIGNode N) → Finset (Causalean.SWIGNode N)","l":92,"f":"Causalean/SCM/ID/GraphicalThms/DoGFormula.lean"},{"n":"Causalean.SCM.ID.fixObservedAncestralSet","k":"def","m":"Causalean.SCM.ID.GraphicalThms.DoGFormula","a":"SCM/ID/GraphicalThms","t":1,"d":"The observed part of the post-intervention ancestral support of the query. The post-intervention ancestral set An_{G_X}(Y) is a SWIG-node set and may include fixed intervention nodes, but the observational law is carried only on observed coordinates, so the measure-theoretic support is its intersection with the observed nodes.","s":"{N : Type u_1} → [inst : DecidableEq N] → [inst_1 : Fintype N] → {Ω : N → Type u_2} → [inst_2 : (n : N) → MeasurableSpace (Ω n)] → (M : Causalean.SCM N Ω) → (X : Finset N) → (∀ D ∈ X, Causalean.SWIGNode.random D ∈ M.observed) → (∀ D ∈ X, Causalean.SWIGNode.fixed D ∉ M.fixed) → Finset (Causalean.SWIGNode N) → Finset (Causalean.SWIGNode N)","l":125,"f":"Causalean/SCM/ID/GraphicalThms/DoGFormula.lean"},{"n":"Causalean.SCM.ID.fixObservedAncestralSet_obsParent_closed","k":"theorem","m":"Causalean.SCM.ID.GraphicalThms.DoGFormula","a":"SCM/ID/GraphicalThms","t":2,"d":"The observed post-intervention ancestral support is closed under observed parents in the do-model.","s":"∀ {N : Type u_1} [inst : DecidableEq N] [inst_1 : Fintype N] {Ω : N → Type u_2} [inst_2 : (n : N) → MeasurableSpace (Ω n)] (M : Causalean.SCM N Ω) (X : Finset N) (hObs : ∀ D ∈ X, Causalean.SWIGNode.random D ∈ M.observed) (hFix : ∀ D ∈ X, Causalean.SWIGNode.fixed D ∉ M.fixed) (Y : Finset (Causalean.SWIGNode N)), (M.fixSet X hObs hFix).ObsParentClosed (Causalean.SCM.ID.fixObservedAncestralSet M X hObs hFix Y)","l":374,"f":"Causalean/SCM/ID/GraphicalThms/DoGFormula.lean"},{"n":"Causalean.SCM.ID.fixSet_cComponentSet_eq","k":"theorem","m":"Causalean.SCM.ID.GraphicalThms.DoGFormula","a":"SCM/ID/GraphicalThms","t":2,"d":"Intervening does not change the full c-component partition.","s":"∀ {N : Type u_1} [inst : DecidableEq N] [inst_1 : Fintype N] {Ω : N → Type u_2} [inst_2 : (n : N) → MeasurableSpace (Ω n)] (M : Causalean.SCM N Ω) (X : Finset N) (hObs : ∀ D ∈ X, Causalean.SWIGNode.random D ∈ M.observed) (hFix : ∀ D ∈ X, Causalean.SWIGNode.fixed D ∉ M.fixed), (M.fixSet X hObs hFix).cComponentSet = M.cComponentSet","l":354,"f":"Causalean/SCM/ID/GraphicalThms/DoGFormula.lean"},{"n":"Causalean.SCM.ID.fixSet_cComponentSet_mem","k":"theorem","m":"Causalean.SCM.ID.GraphicalThms.DoGFormula","a":"SCM/ID/GraphicalThms","t":2,"d":"Membership in the full c-component partition transports across fixSet.","s":"∀ {N : Type u_1} [inst : DecidableEq N] [inst_1 : Fintype N] {Ω : N → Type u_2} [inst_2 : (n : N) → MeasurableSpace (Ω n)] (M : Causalean.SCM N Ω) (X : Finset N) (hObs : ∀ D ∈ X, Causalean.SWIGNode.random D ∈ M.observed) (hFix : ∀ D ∈ X, Causalean.SWIGNode.fixed D ∉ M.fixed) (S : Finset (Causalean.SWIGNode N)), S ∈ (M.fixSet X hObs hFix).cComponentSet ↔ S ∈ M.cComponentSet","l":364,"f":"Causalean/SCM/ID/GraphicalThms/DoGFormula.lean"},{"n":"Causalean.SCM.ID.fixTruncCComponentSet","k":"def","m":"Causalean.SCM.ID.GraphicalThms.DoGFormula","a":"SCM/ID/GraphicalThms","t":1,"d":"The truncated c-component index set. The c-components of the mutilated graph G_X after inducing on the post-intervention ancestral set of the query. This is the index set of the truncated product in the do-law g-formula: the do-law Y-marginal factorizes into one c-factor per element of this set.","s":"{N : Type u_1} → [inst : DecidableEq N] → [inst_1 : Fintype N] → {Ω : N → Type u_2} → [inst_2 : (n : N) → MeasurableSpace (Ω n)] → (M : Causalean.SCM N Ω) → (X : Finset N) → (∀ D ∈ X, Causalean.SWIGNode.random D ∈ M.observed) → (∀ D ∈ X, Causalean.SWIGNode.fixed D ∉ M.fixed) → Finset (Causalean.SWIGNode N) → Finset (Finset (Causalean.SWIGNode N))","l":104,"f":"Causalean/SCM/ID/GraphicalThms/DoGFormula.lean"},{"n":"Causalean.SCM.ID.idAlgorithm","k":"def","m":"Causalean.SCM.ID.DiscreteID.Checker","a":"SCM/ID/DiscreteID","t":1,"d":"The executable ID checker. Runs the recursive Tian–Shpitser algorithm on (G, X, Y) with fuel reduction steps: it requires a valid intervention split, an observed query disjoint from X, and that every c-component of the post-intervention ancestral graph is recursively reachable from its containing district. Computable — usable with #eval / decide on concrete graphs.","s":"{N : Type u_1} → [inst : DecidableEq N] → [inst_1 : Fintype N] → ℕ → Causalean.SWIGGraph N → Finset N → Finset (Causalean.SWIGNode N) → Bool","l":119,"f":"Causalean/SCM/ID/DiscreteID/Checker.lean"},{"n":"Causalean.SCM.ID.idAlgorithm_sound","k":"theorem","m":"Causalean.SCM.ID.DiscreteID.Checker","a":"SCM/ID/DiscreteID","t":1,"d":"Soundness of the executable ID algorithm. When the runnable checker idAlgorithm returns true on the graph G, intervention set X, query Y, and the given fuel bound, the interventional query P(Y ∣ do(X)) is identified within the standard discrete positive model class.","s":"∀ {N : Type u_1} [inst : DecidableEq N] [inst_1 : Fintype N] {Ω : N → Type u_2} [inst_2 : (n : N) → MeasurableSpace (Ω n)] [∀ (n : N), StandardBorelSpace (Ω n)] [inst_4 : ∀ (n : N), Nonempty (Ω n)] [(n : N) → Fintype (Ω n)] [∀ (n : N), MeasurableSingletonClass (Ω n)] (fuel : ℕ) (G : Causalean.SWIGGraph N) (X : Finset N) (Y : Finset (Causalean.SWIGNode N)), Causalean.SCM.ID.idAlgorithm fuel G X Y = true → Causalean.SCM.ID.IdentifiableUnder G (fun x => True) Causalean.SCM.ID.DiscreteID.StandardDiscretePositive (Causalean.SCM.ID.interventionalQuery X Y)","l":157,"f":"Causalean/SCM/ID/DiscreteID/Checker.lean"},{"n":"Causalean.SCM.ID.idAlgorithm_success_toRec","k":"theorem","m":"Causalean.SCM.ID.DiscreteID.Checker","a":"SCM/ID/DiscreteID","t":2,"d":"A successful run yields the graphical certificate. idAlgorithm … = true implies the declarative recursive success certificate idSucceedsRec.","s":"∀ {N : Type u_1} [inst : DecidableEq N] [inst_1 : Fintype N] (fuel : ℕ) (G : Causalean.SWIGGraph N) (X : Finset N) (Y : Finset (Causalean.SWIGNode N)), Causalean.SCM.ID.idAlgorithm fuel G X Y = true → Causalean.SCM.ID.idSucceedsRec X Y G","l":136,"f":"Causalean/SCM/ID/DiscreteID/Checker.lean"},{"n":"Causalean.SCM.ID.idSucceeds","k":"def","m":"Causalean.SCM.ID.GraphicalThms.DoGFormula","a":"SCM/ID/GraphicalThms","t":1,"d":"Structural success certificate for the no-additional-fixing (full-district) fragment of the Tian–Shpitser ID algorithm.","s":"{N : Type u_1} → [inst : DecidableEq N] → [inst_1 : Fintype N] → Finset N → Finset (Causalean.SWIGNode N) → Causalean.SWIGGraph N → Prop","l":68,"f":"Causalean/SCM/ID/GraphicalThms/DoGFormula.lean"},{"n":"Causalean.SCM.ID.idSucceedsRec","k":"def","m":"Causalean.SCM.ID.GraphicalThms.DoGFormulaRec","a":"SCM/ID/GraphicalThms","t":1,"d":"Full recursive success certificate for the ID algorithm. As idSucceeds, but each c-component S of the post-intervention ancestral graph need only be *recursively reachable* from its containing district (CFactorReachableRec), not already a full c-component. This is the honest Tian–Shpitser ID success condition (for the soundness direction).","s":"{N : Type u_1} → [inst : DecidableEq N] → [inst_1 : Fintype N] → Finset N → Finset (Causalean.SWIGNode N) → Causalean.SWIGGraph N → Prop","l":98,"f":"Causalean/SCM/ID/GraphicalThms/DoGFormulaRec.lean"},{"n":"Causalean.SCM.ID.idSucceeds_toRec","k":"theorem","m":"Causalean.SCM.ID.GraphicalThms.DoGFormulaRec","a":"SCM/ID/GraphicalThms","t":1,"d":"idSucceedsRec generalizes idSucceeds. For an intervention target set X, an outcome node set Y, and a SWIG graph G, if the plain no-fixing ID certificate succeeds for X, Y on G, then the full recursive ID certificate also succeeds for X, Y on G, so soundness proved for idSucceedsRec subsumes the no-fixing headline.","s":"∀ {N : Type u_1} [inst : DecidableEq N] [inst_1 : Fintype N] (X : Finset N) (Y : Finset (Causalean.SWIGNode N)) (G : Causalean.SWIGGraph N), Causalean.SCM.ID.idSucceeds X Y G → Causalean.SCM.ID.idSucceedsRec X Y G","l":154,"f":"Causalean/SCM/ID/GraphicalThms/DoGFormulaRec.lean"},{"n":"Causalean.SCM.ID.id_sound","k":"theorem","m":"Causalean.SCM.ID.GraphicalThms.IDAlgorithm","a":"SCM/ID/GraphicalThms","t":1,"d":"Soundness of the no-fixing ID algorithm. Fix an intervention target set X, an outcome node set Y, a SWIG graph G, and a reference-measure family ref that is faithful to the graph. Then whenever the no-fixing ID certificate succeeds for X, Y on G, the interventional query mapping X to Y is identifiable within the class of models dominated by ref with discretely positive observational kernels: any two such models that share graph G and observational kernel agree on the query.","s":"∀ {N : Type u_1} [inst : DecidableEq N] [inst_1 : Fintype N] {Ω : N → Type u_2} [inst_2 : (n : N) → MeasurableSpace (Ω n)] [∀ (n : N), StandardBorelSpace (Ω n)] [inst_4 : ∀ (n : N), Nonempty (Ω n)] [(n : N) → Fintype (Ω n)] [∀ (n : N), MeasurableSingletonClass (Ω n)] (X : Finset N) (Y : Finset (Causalean.SWIGNode N)) (G : Causalean.SWIGGraph N) (ref : Causalean.SCM.ReferenceMeasures Ω), Causalean.SCM.ReferenceFaithful ref → Causalean.SCM.ID.idSucceeds X Y G → Causalean.SCM.ID.IdentifiableUnder G (fun x => True) (fun M => M.DominatedObs ref ∧ Causalean.SCM.ID.DiscreteID.DiscretePositive M) (Causalean.SCM.ID.interventionalQuery X Y)","l":554,"f":"Causalean/SCM/ID/GraphicalThms/IDAlgorithm.lean"},{"n":"Causalean.SCM.ID.id_sound_discrete","k":"theorem","m":"Causalean.SCM.ID.GraphicalThms.IDSoundDiscrete","a":"SCM/ID/GraphicalThms","t":1,"d":"Discrete soundness of the ID assembly for the no-additional-fixing (full-district) fragment (on-contract). For treatment nodes X, outcome nodes Y, and a SWIG graph G, a successful idSucceeds certificate implies that the interventional query is identified from the observational distribution within the standard discrete positive model class.","s":"∀ {N : Type u_1} [inst : DecidableEq N] [inst_1 : Fintype N] {Ω : N → Type u_2} [inst_2 : (n : N) → MeasurableSpace (Ω n)] [∀ (n : N), StandardBorelSpace (Ω n)] [inst_4 : ∀ (n : N), Nonempty (Ω n)] [(n : N) → Fintype (Ω n)] [∀ (n : N), MeasurableSingletonClass (Ω n)] (X : Finset N) (Y : Finset (Causalean.SWIGNode N)) (G : Causalean.SWIGGraph N), Causalean.SCM.ID.idSucceeds X Y G → Causalean.SCM.ID.IdentifiableUnder G (fun x => True) Causalean.SCM.ID.DiscreteID.StandardDiscretePositive (Causalean.SCM.ID.interventionalQuery X Y)","l":30,"f":"Causalean/SCM/ID/GraphicalThms/IDSoundDiscrete.lean"},{"n":"Causalean.SCM.ID.id_sound_rec","k":"theorem","m":"Causalean.SCM.ID.GraphicalThms.IDAlgorithmRec","a":"SCM/ID/GraphicalThms","t":1,"d":"Soundness of the full recursive ID algorithm for finite discrete-positive models. Fix an intervention target set X, an outcome node set Y, a SWIG graph G, and a reference-measure family ref that is faithful to the graph. Then whenever the full recursive Tian–Shpitser IDENTIFY certificate succeeds for X, Y on G, the interventional query mapping X to Y is identifiable within the class of models dominated by ref with discretely positive observational kernels: any two such models that share graph G and observational kernel agree on the query. This generalizes id_sound from the no-fixing fragment to the full Tian-Shpitser IDENTIFY recursion; see the module docstring for the proof architecture.","s":"∀ {N : Type u_1} [inst : DecidableEq N] [inst_1 : Fintype N] {Ω : N → Type u_2} [inst_2 : (n : N) → MeasurableSpace (Ω n)] [∀ (n : N), StandardBorelSpace (Ω n)] [inst_4 : ∀ (n : N), Nonempty (Ω n)] [(n : N) → Fintype (Ω n)] [∀ (n : N), MeasurableSingletonClass (Ω n)] (X : Finset N) (Y : Finset (Causalean.SWIGNode N)) (G : Causalean.SWIGGraph N) (ref : Causalean.SCM.ReferenceMeasures Ω), Causalean.SCM.ReferenceFaithful ref → Causalean.SCM.ID.idSucceedsRec X Y G → Causalean.SCM.ID.IdentifiableUnder G (fun x => True) (fun M => M.DominatedObs ref ∧ Causalean.SCM.ID.DiscreteID.DiscretePositive M) (Causalean.SCM.ID.interventionalQuery X Y)","l":858,"f":"Causalean/SCM/ID/GraphicalThms/IDAlgorithmRec.lean"},{"n":"Causalean.SCM.ID.id_sound_rec_discrete","k":"theorem","m":"Causalean.SCM.ID.GraphicalThms.IDAlgorithmRec","a":"SCM/ID/GraphicalThms","t":1,"d":"Discrete soundness of the full recursive ID algorithm (on-contract). For an intervention target set X, an outcome node set Y, and a SWIG graph G, if the full recursive Tian–Shpitser IDENTIFY certificate succeeds for X, Y on G, then the interventional query mapping X to Y is identifiable within the standard discrete positive model class. Obtained from id_sound_rec at the counting reference by collapsing DominatedObs · countingRef to StandardDiscretePositive, exactly as id_sound_discrete is obtained from id_sound. This subsumes id_sound_discrete (via idSucceeds_toRec).","s":"∀ {N : Type u_1} [inst : DecidableEq N] [inst_1 : Fintype N] {Ω : N → Type u_2} [inst_2 : (n : N) → MeasurableSpace (Ω n)] [∀ (n : N), StandardBorelSpace (Ω n)] [inst_4 : ∀ (n : N), Nonempty (Ω n)] [(n : N) → Fintype (Ω n)] [∀ (n : N), MeasurableSingletonClass (Ω n)] (X : Finset N) (Y : Finset (Causalean.SWIGNode N)) (G : Causalean.SWIGGraph N), Causalean.SCM.ID.idSucceedsRec X Y G → Causalean.SCM.ID.IdentifiableUnder G (fun x => True) Causalean.SCM.ID.DiscreteID.StandardDiscretePositive (Causalean.SCM.ID.interventionalQuery X Y)","l":895,"f":"Causalean/SCM/ID/GraphicalThms/IDAlgorithmRec.lean"},{"n":"Causalean.SCM.ID.identifiableUnder_mono","k":"theorem","m":"Causalean.SCM.ID.Identifiable","a":"SCM/ID","t":1,"d":"Monotonicity of identifiability under assumptions. Fix functional-assumption predicates Af₁, Af₂ and structural-assumption predicates As₁, As₂ on causal models sharing a graph G, together with a causal query Φ. If every model satisfying Af₂ also satisfies Af₁, every model satisfying As₂ also satisfies As₁, and Φ is identifiable under the assumption pair (Af₁, As₁), then Φ is identifiable under (Af₂, As₂): passing to the more restrictive assumption predicates Af₂, As₂ cannot destroy identifiability.","s":"∀ {N : Type u_1} [inst : DecidableEq N] [inst_1 : Fintype N] {Ω : N → Type u_2} [inst_2 : (n : N) → MeasurableSpace (Ω n)] {α : Type u_3} (G : Causalean.SWIGGraph N) (Af₁ Af₂ As₁ As₂ : Causalean.SCM N Ω → Prop) (Φ : Causalean.SCM.ID.CausalQuery N Ω α), (∀ (M : Causalean.SCM N Ω), Af₂ M → Af₁ M) → (∀ (M : Causalean.SCM N Ω), As₂ M → As₁ M) → Causalean.SCM.ID.IdentifiableUnder G Af₁ As₁ Φ → Causalean.SCM.ID.IdentifiableUnder G Af₂ As₂ Φ","l":174,"f":"Causalean/SCM/ID/Identifiable.lean"},{"n":"Causalean.SCM.ID.identifiable_eq_identifiableUnder_true","k":"theorem","m":"Causalean.SCM.ID.Identifiable","a":"SCM/ID","t":2,"d":"Identifiability without additional assumptions is the special case where both Af and As are trivially satisfied.","s":"∀ {N : Type u_1} [inst : DecidableEq N] [inst_1 : Fintype N] {Ω : N → Type u_2} [inst_2 : (n : N) → MeasurableSpace (Ω n)] {α : Type u_3} (G : Causalean.SWIGGraph N) (Φ : Causalean.SCM.ID.CausalQuery N Ω α), Causalean.SCM.ID.Identifiable G Φ ↔ Causalean.SCM.ID.IdentifiableUnder G (fun x => True) (fun x => True) Φ","l":166,"f":"Causalean/SCM/ID/Identifiable.lean"},{"n":"Causalean.SCM.ID.identifyMassRecObserved","k":"def","m":"Causalean.SCM.ID.GraphicalThms.IDAlgorithmRec","a":"SCM/ID/GraphicalThms","t":1,"d":"Observed-set form of SCM.identifyMassRec.","s":"{N : Type u_1} → [inst : DecidableEq N] → [inst_1 : Fintype N] → {Ω : N → Type u_2} → [(n : N) → Fintype (Ω n)] → (O : Finset (Causalean.SWIGNode N)) → Causalean.SWIGGraph N → (T : Finset (Causalean.SWIGNode N)) → Finset (Causalean.SWIGNode N) → T ⊆ O → (Causalean.ValuesOn O (Causalean.swigΩ Ω) → ENNReal) → Causalean.ValuesOn O (Causalean.swigΩ Ω) → ENNReal","l":84,"f":"Causalean/SCM/ID/GraphicalThms/IDAlgorithmRec.lean"},{"n":"Causalean.SCM.ID.identifyMassRecObserved_eq_identifyMassRec","k":"theorem","m":"Causalean.SCM.ID.GraphicalThms.IDAlgorithmRec","a":"SCM/ID/GraphicalThms","t":2,"d":"","s":"∀ {N : Type u_1} [inst : DecidableEq N] [inst_1 : Fintype N] {Ω : N → Type u_2} [inst_2 : (n : N) → MeasurableSpace (Ω n)] [inst_3 : (n : N) → Fintype (Ω n)] (M : Causalean.SCM N Ω) (G : Causalean.SWIGGraph N) (T C : Finset (Causalean.SWIGNode N)) (hT : T ⊆ M.observed) (q : Causalean.ValuesOn M.observed (Causalean.swigΩ Ω) → ENNReal), Causalean.SCM.ID.identifyMassRecObserved M.observed G T C hT q = Causalean.SCM.identifyMassRec M.observed G T C hT q","l":133,"f":"Causalean/SCM/ID/GraphicalThms/IDAlgorithmRec.lean"},{"n":"Causalean.SCM.ID.identifyMassRec_qLocalMass","k":"theorem","m":"Causalean.SCM.ID.Density.QFactor","a":"SCM/ID/Density","t":2,"d":"The obs-side IDENTIFY recursion recovers the local q-mass of the target district from the local q-mass of any recursively reachable source district.","s":"∀ {N : Type u_1} [inst : DecidableEq N] [inst_1 : Fintype N] {Ω : N → Type u_2} [inst_2 : (n : N) → MeasurableSpace (Ω n)] [inst_3 : (n : N) → Fintype (Ω n)] [∀ (n : N), MeasurableSingletonClass (Ω n)] (M : Causalean.SCM N Ω) (s : M.FixedValues), Causalean.SCM.ID.DiscreteID.PositiveMass (M.obsKernel s) → ∀ (T C : Finset (Causalean.SWIGNode N)) (hT : T ⊆ M.observed) (hReach : Causalean.SCM.ID.CFactorReachableRec M.toSWIGGraph T C) (x : Causalean.ValuesOn M.observed (Causalean.swigΩ Ω)), Causalean.SCM.identifyMassRec M.observed M.toSWIGGraph T C hT (M.qLocalMass s T hT) x = M.qLocalMass s C (fun x hv => hT (Causalean.SCM.ID.CFactorReachableRec.target_subset hReach hv)) x","l":1066,"f":"Causalean/SCM/ID/Density/QFactor.lean"},{"n":"Causalean.SCM.ID.inducedAncestral","k":"def","m":"Causalean.SCM.ID.GraphicalThms.DoGFormulaRec","a":"SCM/ID/GraphicalThms","t":1,"d":"Observed ancestors of C within the subgraph induced on T. Tian's An(C)_{G_T}: restrict G to the node set T, take the ancestors of C in that restricted graph, and keep the observed nodes. This is the set the IDENTIFY subroutine compares against C (project) and T (hedge / fail).","s":"{N : Type u_1} → [inst : DecidableEq N] → [inst_1 : Fintype N] → Causalean.SWIGGraph N → Finset (Causalean.SWIGNode N) → Finset (Causalean.SWIGNode N) → Finset (Causalean.SWIGNode N)","l":35,"f":"Causalean/SCM/ID/GraphicalThms/DoGFormulaRec.lean"},{"n":"Causalean.SCM.ID.inducedAncestral_parent_closed","k":"theorem","m":"Causalean.SCM.ID.GraphicalThms.DoGFormulaRec","a":"SCM/ID/GraphicalThms","t":2,"d":"The induced ancestral set is closed under observed parents inside the ambient district T.","s":"∀ {N : Type u_1} [inst : DecidableEq N] [inst_1 : Fintype N] (G : Causalean.SWIGGraph N) {T C : Finset (Causalean.SWIGNode N)}, T ⊆ G.observed → ∀ v ∈ T, ∀ w ∈ Causalean.SCM.ID.inducedAncestral G T C, G.dag.edge v w → v ∈ Causalean.SCM.ID.inducedAncestral G T C","l":43,"f":"Causalean/SCM/ID/GraphicalThms/DoGFormulaRec.lean"},{"n":"Causalean.SCM.ID.inducedAncestral_self_of_mem_cComponentSet","k":"theorem","m":"Causalean.SCM.ID.GraphicalThms.DoGFormulaRec","a":"SCM/ID/GraphicalThms","t":2,"d":"The observed ancestors of a full c-component S within G_S are S itself.","s":"∀ {N : Type u_1} [inst : DecidableEq N] [inst_1 : Fintype N] (G : Causalean.SWIGGraph N), ∀ S ∈ G.cComponentSet, Causalean.SCM.ID.inducedAncestral G S S = S","l":126,"f":"Causalean/SCM/ID/GraphicalThms/DoGFormulaRec.lean"},{"n":"Causalean.SCM.ID.inducedAncestral_subset_left","k":"theorem","m":"Causalean.SCM.ID.Density.IdentifyMass","a":"SCM/ID/Density","t":2,"d":"The induced ancestral set lies inside the ambient observed node set.","s":"∀ {N : Type u_1} [inst : DecidableEq N] [inst_1 : Fintype N] (G : Causalean.SWIGGraph N) (T C : Finset (Causalean.SWIGNode N)), Causalean.SCM.ID.inducedAncestral G T C ⊆ T","l":69,"f":"Causalean/SCM/ID/Density/IdentifyMass.lean"},{"n":"Causalean.SCM.ID.instDecidableInterventionValid","k":"instance","m":"Causalean.SCM.ID.DiscreteID.Checker","a":"SCM/ID/DiscreteID","t":2,"d":"Validity of an intervention split is decidable — enables the executable idAlgorithm to branch on it.","s":"{N : Type u_1} → [inst : DecidableEq N] → [inst_1 : Fintype N] → (X : Finset N) → (G : Causalean.SWIGGraph N) → Decidable (Causalean.SCM.ID.interventionValid X G)","l":112,"f":"Causalean/SCM/ID/DiscreteID/Checker.lean"},{"n":"Causalean.SCM.ID.instIsFiniteMeasure_doObsKernelAncestralMarginal","k":"instance","m":"Causalean.SCM.ID.GraphicalThms.IDAlgorithm","a":"SCM/ID/GraphicalThms","t":2,"d":"The observed-ancestral marginal of a post-intervention observational law is a finite measure.","s":"∀ {N : Type u_1} [inst : DecidableEq N] [inst_1 : Fintype N] {Ω : N → Type u_2} [inst_2 : (n : N) → MeasurableSpace (Ω n)] (M : Causalean.SCM N Ω) (X : Finset N) (hObs : ∀ D ∈ X, Causalean.SWIGNode.random D ∈ M.observed) (hFix : ∀ D ∈ X, Causalean.SWIGNode.fixed D ∉ M.fixed) (Y : Finset (Causalean.SWIGNode N)) (s : (M.fixSet X hObs hFix).FixedValues), MeasureTheory.IsFiniteMeasure ((Causalean.SCM.ID.doObsKernelAncestralMarginal M X hObs hFix Y) s)","l":118,"f":"Causalean/SCM/ID/GraphicalThms/IDAlgorithm.lean"},{"n":"Causalean.SCM.ID.instIsMarkovKernel_doObsKernelAncestralMarginal","k":"instance","m":"Causalean.SCM.ID.GraphicalThms.DoGFormula","a":"SCM/ID/GraphicalThms","t":2,"d":"The post-intervention ancestral marginal is a Markov kernel: it is the coordinate pushforward of the (Markov) do-observational kernel, so each slice is a probability measure.","s":"∀ {N : Type u_1} [inst : DecidableEq N] [inst_1 : Fintype N] {Ω : N → Type u_2} [inst_2 : (n : N) → MeasurableSpace (Ω n)] (M : Causalean.SCM N Ω) (X : Finset N) (hObs : ∀ D ∈ X, Causalean.SWIGNode.random D ∈ M.observed) (hFix : ∀ D ∈ X, Causalean.SWIGNode.fixed D ∉ M.fixed) (Y : Finset (Causalean.SWIGNode N)), ProbabilityTheory.IsMarkovKernel (Causalean.SCM.ID.doObsKernelAncestralMarginal M X hObs hFix Y)","l":406,"f":"Causalean/SCM/ID/GraphicalThms/DoGFormula.lean"},{"n":"Causalean.SCM.ID.interventionValid","k":"def","m":"Causalean.SCM.ID.GraphicalThms.DoGFormula","a":"SCM/ID/GraphicalThms","t":1,"d":"The graph-level precondition that the SWIG split by X is valid.","s":"{N : Type u_1} → [inst : DecidableEq N] → [inst_1 : Fintype N] → Finset N → Causalean.SWIGGraph N → Prop","l":45,"f":"Causalean/SCM/ID/GraphicalThms/DoGFormula.lean"},{"n":"Causalean.SCM.ID.intervention_target_simp","k":"theorem","m":"Causalean.SCM.ID.GraphicalThms.QFactorIdentity","a":"SCM/ID/GraphicalThms","t":2,"d":"Proposition (fact4 — intervention target simplification).","s":"∀ {N : Type u_1} [inst : DecidableEq N] [inst_1 : Fintype N] {Ω : N → Type u_2} [inst_2 : (n : N) → MeasurableSpace (Ω n)] (M : Causalean.SCM N Ω) (Dn Yn : Finset N), Causalean.SCM.ID.InterventionTargetSimpConclusion M Dn Yn","l":462,"f":"Causalean/SCM/ID/GraphicalThms/QFactorIdentity.lean"},{"n":"Causalean.SCM.ID.interventionalQuery","k":"def","m":"Causalean.SCM.ID.Query","a":"SCM/ID","t":1,"d":"The interventional query returns the post-intervention outcome law as a kernel indexed by treatment values.","s":"{N : Type u_1} → [inst : DecidableEq N] → [inst_1 : Fintype N] → {Ω : N → Type u_2} → [inst_2 : (n : N) → MeasurableSpace (Ω n)] → [∀ (n : N), Nonempty (Ω n)] → (X : Finset N) → (Y : Finset (Causalean.SWIGNode N)) → Causalean.SCM.ID.CausalQuery N Ω (ProbabilityTheory.Kernel (Causalean.ValuesOn (Finset.image Causalean.SWIGNode.random X) (Causalean.swigΩ Ω)) (Causalean.ValuesOn Y (Causalean.swigΩ Ω)))","l":72,"f":"Causalean/SCM/ID/Query.lean"},{"n":"Causalean.SCM.ID.interventionalQueryValid","k":"def","m":"Causalean.SCM.ID.Query","a":"SCM/ID","t":1,"d":"This predicate states when the interventional query is in its meaningful standard-model branch.","s":"{N : Type u_1} → [inst : DecidableEq N] → [inst_1 : Fintype N] → {Ω : N → Type u_2} → [inst_2 : (n : N) → MeasurableSpace (Ω n)] → Finset N → Finset (Causalean.SWIGNode N) → Causalean.SCM N Ω → Prop","l":63,"f":"Causalean/SCM/ID/Query.lean"},{"n":"Causalean.SCM.ID.interventionalQueryValid_iff_of_toSWIGGraph_eq","k":"theorem","m":"Causalean.SCM.ID.Query","a":"SCM/ID","t":1,"d":"Well-formedness invariance under matching SWIG graphs. For an intervention target set X and outcome set Y, if two structural causal models M₁, M₂ have the same SWIG graph, then they agree on whether the total interventional query for X, Y is well formed.","s":"∀ {N : Type u_1} [inst : DecidableEq N] [inst_1 : Fintype N] {Ω : N → Type u_2} [inst_2 : (n : N) → MeasurableSpace (Ω n)] (X : Finset N) (Y : Finset (Causalean.SWIGNode N)) (M₁ M₂ : Causalean.SCM N Ω), M₁.toSWIGGraph = M₂.toSWIGGraph → (Causalean.SCM.ID.interventionalQueryValid X Y M₁ ↔ Causalean.SCM.ID.interventionalQueryValid X Y M₂)","l":120,"f":"Causalean/SCM/ID/Query.lean"},{"n":"Causalean.SCM.ID.interventionalQuery_eq_default_of_not_valid","k":"theorem","m":"Causalean.SCM.ID.Query","a":"SCM/ID","t":2,"d":"Outside the well-formed branch, the total interventional query is the fixed fallback kernel.","s":"∀ {N : Type u_1} [inst : DecidableEq N] [inst_1 : Fintype N] {Ω : N → Type u_2} [inst_2 : (n : N) → MeasurableSpace (Ω n)] [inst_3 : ∀ (n : N), Nonempty (Ω n)] (X : Finset N) (Y : Finset (Causalean.SWIGNode N)) (M : Causalean.SCM N Ω), ¬Causalean.SCM.ID.interventionalQueryValid X Y M → Causalean.SCM.ID.interventionalQuery X Y M = Causalean.SCM.ID.defaultInterventionalKernel X Y","l":109,"f":"Causalean/SCM/ID/Query.lean"},{"n":"Causalean.SCM.ID.interventionalQuery_eq_doKernelY_of_valid","k":"theorem","m":"Causalean.SCM.ID.Query","a":"SCM/ID","t":1,"d":"For treatment nodes X and outcome nodes Y satisfying the well-formedness conditions for a valid interventional query in a model M, the total interventional query evaluated at M equals the post-intervention outcome-marginal kernel doKernelY.","s":"∀ {N : Type u_1} [inst : DecidableEq N] [inst_1 : Fintype N] {Ω : N → Type u_2} [inst_2 : (n : N) → MeasurableSpace (Ω n)] [inst_3 : ∀ (n : N), Nonempty (Ω n)] (X : Finset N) (Y : Finset (Causalean.SWIGNode N)) (M : Causalean.SCM N Ω) (h : Causalean.SCM.ID.interventionalQueryValid X Y M), Causalean.SCM.ID.interventionalQuery X Y M = M.doKernelY X h.left h.right.left Y h.right.right.left (Causalean.SCM.ID.standardFixedValues M h.right.right.right)","l":95,"f":"Causalean/SCM/ID/Query.lean"},{"n":"Causalean.SCM.ID.jointRef_extendTianPrefix","k":"theorem","m":"Causalean.SCM.ID.GraphicalThms.DoGFormulaTian","a":"SCM/ID/GraphicalThms","t":2,"d":"The successor prefix extension carries the product of the old-prefix reference and the next singleton reference to the successor-prefix reference.","s":"∀ {N : Type u_1} [inst : DecidableEq N] [inst_1 : Fintype N] {Ω : N → Type u_2} [inst_2 : (n : N) → MeasurableSpace (Ω n)] (H : Causalean.SWIGGraph N) (D : Finset (Causalean.SWIGNode N)) (ref : Causalean.SCM.ReferenceMeasures Ω) {k : ℕ} (hk : k < D.card), MeasureTheory.Measure.map (Causalean.SCM.ID.extendTianPrefix H D hk) ((Causalean.SCM.jointRef ref (H.prefixIn D k)).prod (Causalean.SCM.jointRef ref {↑(H.nodesAt D ⟨k, hk⟩)})) = Causalean.SCM.jointRef ref (H.prefixIn D (k + 1))","l":201,"f":"Causalean/SCM/ID/GraphicalThms/DoGFormulaTian.lean"},{"n":"Causalean.SCM.ID.jointRef_map_prefixIn_absolutelyContinuous","k":"theorem","m":"Causalean.SCM.ID.GraphicalThms.DoGFormulaTian","a":"SCM/ID/GraphicalThms","t":2,"d":"The reference marginal of any D-prefix is absolutely continuous with respect to the corresponding product reference.","s":"∀ {N : Type u_1} [inst : DecidableEq N] [inst_1 : Fintype N] {Ω : N → Type u_2} [inst_2 : (n : N) → MeasurableSpace (Ω n)] (H : Causalean.SWIGGraph N) (D : Finset (Causalean.SWIGNode N)) (ref : Causalean.SCM.ReferenceMeasures Ω) (k : ℕ), (MeasureTheory.Measure.map (Causalean.valuesProjection (Causalean.SWIGGraph.prefixIn_subset H D k)) (Causalean.SCM.jointRef ref D)).AbsolutelyContinuous (Causalean.SCM.jointRef ref (H.prefixIn D k))","l":245,"f":"Causalean/SCM/ID/GraphicalThms/DoGFormulaTian.lean"},{"n":"Causalean.SCM.ID.marginalizeOnObserved","k":"def","m":"Causalean.SCM.ID.GraphicalThms.IDAlgorithmRec","a":"SCM/ID/GraphicalThms","t":1,"d":"Observed-set form of SCM.marginalizeOn, used only to expose that the recursive mass functional is independent of the rest of the SCM record.","s":"{N : Type u_1} → [DecidableEq N] → {Ω : N → Type u_2} → [(n : N) → Fintype (Ω n)] → (O W : Finset (Causalean.SWIGNode N)) → W ⊆ O → (Causalean.ValuesOn O (Causalean.swigΩ Ω) → ENNReal) → Causalean.ValuesOn O (Causalean.swigΩ Ω) → ENNReal","l":63,"f":"Causalean/SCM/ID/GraphicalThms/IDAlgorithmRec.lean"},{"n":"Causalean.SCM.ID.marginalizeOnObserved_eq_marginalizeOn","k":"theorem","m":"Causalean.SCM.ID.GraphicalThms.IDAlgorithmRec","a":"SCM/ID/GraphicalThms","t":2,"d":"","s":"∀ {N : Type u_1} [inst : DecidableEq N] [inst_1 : Fintype N] {Ω : N → Type u_2} [inst_2 : (n : N) → MeasurableSpace (Ω n)] [inst_3 : (n : N) → Fintype (Ω n)] (M : Causalean.SCM N Ω) (W : Finset (Causalean.SWIGNode N)) (hW : W ⊆ M.observed) (q : Causalean.ValuesOn M.observed (Causalean.swigΩ Ω) → ENNReal), Causalean.SCM.ID.marginalizeOnObserved M.observed W hW q = Causalean.SCM.marginalizeOn M.observed W hW q","l":117,"f":"Causalean/SCM/ID/GraphicalThms/IDAlgorithmRec.lean"},{"n":"Causalean.SCM.ID.markov_tian_cfactorization_density","k":"theorem","m":"Causalean.SCM.ID.GraphicalThms.DoGFormulaTian","a":"SCM/ID/GraphicalThms","t":1,"d":"Density form of the Markov-to-c-factorization theorem. Let H be a pure SWIG graph, D a finite set of SWIG nodes, μ a finite (probability) measure on the assignments to D, and ref a family of reference measures. If D is exactly the observed-node set of H and μ is absolutely continuous with respect to the product reference measure on D, then the Radon–Nikodym density of μ against that product reference equals, almost everywhere, the product over the c-components of H of their Tian district-density factors.","s":"∀ {N : Type u_1} [inst : DecidableEq N] [inst_1 : Fintype N] {Ω : N → Type u_2} [inst_2 : (n : N) → MeasurableSpace (Ω n)] (H : Causalean.SWIGGraph N) (D : Finset (Causalean.SWIGNode N)), H.observed = D → ∀ (μ : MeasureTheory.Measure (Causalean.ValuesOn D (Causalean.swigΩ Ω))) (ref : Causalean.SCM.ReferenceMeasures Ω) [(n : N) → Fintype (Ω n)] [∀ (n : N), MeasurableSingletonClass (Ω n)] [StandardBorelSpace (Causalean.ValuesOn D (Causalean.swigΩ Ω))] [inst_6 : MeasureTheory.IsProbabilityMeasure μ] [inst_7 : ∀ (k : ℕ) (hk : k < D.card), StandardBorelSpace (Causalean.ValuesOn {↑(H.nodesAt D ⟨k, hk⟩)} (Causalean.swigΩ Ω))] [inst_8 : ∀ (k : ℕ) (hk : k < D.card), Nonempty (Causalean.ValuesOn {↑(H.nodesAt D ⟨k, hk⟩)} (Causalean.swigΩ Ω))], μ.AbsolutelyContinuous (Causalean.SCM.jointRef ref D) → μ.rnDeriv (Causalean.SCM.jointRef ref D) =ᵐ[Causalean.SCM.jointRef ref D] fun x => ∏ S ∈ H.cComponentSet, Causalean.SCM.ID.tianDistrictDensity H D μ ref S x","l":777,"f":"Causalean/SCM/ID/GraphicalThms/DoGFormulaTian.lean"},{"n":"Causalean.SCM.ID.measurable_extendTianPrefix","k":"theorem","m":"Causalean.SCM.ID.GraphicalThms.DoGFormulaTian","a":"SCM/ID/GraphicalThms","t":2,"d":"Prefix extension is measurable.","s":"∀ {N : Type u_1} [inst : DecidableEq N] [inst_1 : Fintype N] {Ω : N → Type u_2} [inst_2 : (n : N) → MeasurableSpace (Ω n)] (H : Causalean.SWIGGraph N) (D : Finset (Causalean.SWIGNode N)) {n : ℕ} (hn : n < D.card), Measurable (Causalean.SCM.ID.extendTianPrefix H D hn)","l":179,"f":"Causalean/SCM/ID/GraphicalThms/DoGFormulaTian.lean"},{"n":"Causalean.SCM.ID.measure_map_prefixIn_absolutelyContinuous_jointRef","k":"theorem","m":"Causalean.SCM.ID.GraphicalThms.DoGFormulaTian","a":"SCM/ID/GraphicalThms","t":2,"d":"Domination of a dominated law's prefix marginal by the prefix reference.","s":"∀ {N : Type u_1} [inst : DecidableEq N] [inst_1 : Fintype N] {Ω : N → Type u_2} [inst_2 : (n : N) → MeasurableSpace (Ω n)] (H : Causalean.SWIGGraph N) (D : Finset (Causalean.SWIGNode N)) (μ : MeasureTheory.Measure (Causalean.ValuesOn D (Causalean.swigΩ Ω))) (ref : Causalean.SCM.ReferenceMeasures Ω), μ.AbsolutelyContinuous (Causalean.SCM.jointRef ref D) → ∀ (k : ℕ), (MeasureTheory.Measure.map (Causalean.valuesProjection (Causalean.SWIGGraph.prefixIn_subset H D k)) μ).AbsolutelyContinuous (Causalean.SCM.jointRef ref (H.prefixIn D k))","l":293,"f":"Causalean/SCM/ID/GraphicalThms/DoGFormulaTian.lean"},{"n":"Causalean.SCM.ID.measure_prefixIn_rnDeriv_eq_tianPrefixDensityProductInPrefix","k":"theorem","m":"Causalean.SCM.ID.GraphicalThms.DoGFormulaTian","a":"SCM/ID/GraphicalThms","t":2,"d":"Prefix-level Radon--Nikodym chain rule for Tian's arbitrary-measure conditional density product.","s":"∀ {N : Type u_1} [inst : DecidableEq N] [inst_1 : Fintype N] {Ω : N → Type u_2} [inst_2 : (n : N) → MeasurableSpace (Ω n)] (H : Causalean.SWIGGraph N) (D : Finset (Causalean.SWIGNode N)) (μ : MeasureTheory.Measure (Causalean.ValuesOn D (Causalean.swigΩ Ω))) (ref : Causalean.SCM.ReferenceMeasures Ω) [(n : N) → Fintype (Ω n)] [∀ (n : N), MeasurableSingletonClass (Ω n)] [StandardBorelSpace (Causalean.ValuesOn D (Causalean.swigΩ Ω))] [inst_6 : MeasureTheory.IsProbabilityMeasure μ] [inst_7 : ∀ (k : ℕ) (hk : k < D.card), StandardBorelSpace (Causalean.ValuesOn {↑(H.nodesAt D ⟨k, hk⟩)} (Causalean.swigΩ Ω))] [inst_8 : ∀ (k : ℕ) (hk : k < D.card), Nonempty (Causalean.ValuesOn {↑(H.nodesAt D ⟨k, hk⟩)} (Causalean.swigΩ Ω))], μ.AbsolutelyContinuous (Causalean.SCM.jointRef ref D) → ∀ k ≤ D.card, (MeasureTheory.Measure.map (Causalean.valuesProjection (Causalean.SWIGGraph.prefixIn_subset H D k)) μ).rnDeriv (Causalean.SCM.jointRef ref (H.prefixIn D k)) =ᵐ[Causalean.SCM.jointRef ref (H.prefixIn D k)] Causalean.SCM.ID.tianPrefixDensityProductInPrefix H D μ ref k","l":497,"f":"Causalean/SCM/ID/GraphicalThms/DoGFormulaTian.lean"},{"n":"Causalean.SCM.ID.mechCFactor_eq_qLocalMass_div_jointRef","k":"theorem","m":"Causalean.SCM.ID.Density.MechCFactor","a":"SCM/ID/Density","t":2,"d":"","s":"∀ {N : Type u_1} [inst : DecidableEq N] [inst_1 : Fintype N] {Ω : N → Type u_2} [inst_2 : (n : N) → MeasurableSpace (Ω n)] [(n : N) → Fintype (Ω n)] [∀ (n : N), MeasurableSingletonClass (Ω n)] (M : Causalean.SCM N Ω) (ref : Causalean.SCM.ReferenceMeasures Ω) (s : M.FixedValues) (S : Finset (Causalean.SWIGNode N)) (hS : S ⊆ M.observed), Causalean.SCM.ReferenceFaithful ref → (∀ (n : N), Causalean.SWIGNode.fixed n ∈ M.fixed → ∀ (v : Causalean.SWIGNode N), ¬M.dag.edge (Causalean.SWIGNode.random n) v) → ∀ (x : Causalean.ValuesOn M.observed (Causalean.swigΩ Ω)), M.mechCFactor ref S hS s x = M.qLocalMass s S hS x / (Causalean.SCM.jointRef ref S) {Causalean.valuesProjection hS x}","l":710,"f":"Causalean/SCM/ID/Density/MechCFactor.lean"},{"n":"Causalean.SCM.ID.mechCFactor_fixSet_invariant","k":"theorem","m":"Causalean.SCM.ID.Density.MechCFactor","a":"SCM/ID/Density","t":1,"d":"Do(X)-invariance of the c-factor Q[S] (Tian Lemma 4). Let an intervention set X have random copies that are all observed and fixed copies that are not already fixed in the base model, giving the intervened model M.fixSet X. For a standard base model M, a reference family that is faithful, and a node set S contained in the observed coordinates whose random copies avoid every intervened node in X, if the base assignment x records, at each intervened node, the same value that the intervened model's fixed values sDo assign to the corresponding fixed coordinate, then the S-c-factor Q[S] of the intervened model M.fixSet X at sDo, x equals the S-c-factor of the base model M at the projected fixed values M.fixSetProj X sDo, x.","s":"∀ {N : Type u_1} [inst : DecidableEq N] [inst_1 : Fintype N] {Ω : N → Type u_2} [inst_2 : (n : N) → MeasurableSpace (Ω n)] [(n : N) → Fintype (Ω n)] [∀ (n : N), MeasurableSingletonClass (Ω n)] (M : Causalean.SCM N Ω) (ref : Causalean.SCM.ReferenceMeasures Ω) (X : Finset N), Causalean.SCM.ReferenceFaithful ref → M.isStandard → ∀ (hObs : ∀ D ∈ X, Causalean.SWIGNode.random D ∈ M.observed) (hFix : ∀ D ∈ X, Causalean.SWIGNode.fixed D ∉ M.fixed) (S : Finset (Causalean.SWIGNode N)) (hS : S ⊆ M.observed), (∀ n ∈ X, Causalean.SWIGNode.random n ∉ S) → ∀ (sDo : (M.fixSet X hObs hFix).FixedValues) (x : Causalean.ValuesOn M.observed (Causalean.swigΩ Ω)), (∀ (D : N) (hD : D ∈ X), x ⟨Causalean.SWIGNode.random D, hObs D hD⟩ = sDo ⟨Causalean.SWIGNode.fixed D, Finset.mem_union_right M.fixed (Finset.mem_image.mpr (Exists.intro D ⟨hD, rfl⟩))⟩) → (M.fixSet X hObs hFix).mechCFactor ref S (id hS) sDo x = M.mechCFactor ref S hS (M.fixSetProj X hObs hFix sDo) x","l":1069,"f":"Causalean/SCM/ID/Density/MechCFactor.lean"},{"n":"Causalean.SCM.ID.mem_cComponent_iff_cComponentOf_eq","k":"theorem","m":"Causalean.SCM.ID.Density.QFactor","a":"SCM/ID/Density","t":2,"d":"For an observed node and a c-component listed in cComponentSet, membership in that component is the same as saying that the node's computed c-component is that listed component. This converts Tian's district factor, which filters by membership in a district, into the fiber form needed for finite-product regrouping.","s":"∀ {N : Type u_1} [inst : DecidableEq N] [inst_1 : Fintype N] (G : Causalean.SWIGGraph N) {v : Causalean.SWIGNode N} {S : Finset (Causalean.SWIGNode N)}, v ∈ G.observed → S ∈ G.cComponentSet → (v ∈ S ↔ G.cComponentOf v = S)","l":211,"f":"Causalean/SCM/ID/Density/QFactor.lean"},{"n":"Causalean.SCM.ID.mem_prefixIn_iff","k":"theorem","m":"Causalean.SCM.ID.GraphicalThms.DoGFormulaTian","a":"SCM/ID/GraphicalThms","t":2,"d":"Membership in a graph-ordered prefix of D is exactly index membership below the prefix length.","s":"∀ {N : Type u_1} [inst : DecidableEq N] [inst_1 : Fintype N] (H : Causalean.SWIGGraph N) (D : Finset (Causalean.SWIGNode N)) (n : ℕ) (v : Causalean.SWIGNode N), v ∈ H.prefixIn D n ↔ ∃ (h : v ∈ D), ↑(H.nodeIndex D ⟨v, h⟩) < n","l":63,"f":"Causalean/SCM/ID/GraphicalThms/DoGFormulaTian.lean"},{"n":"Causalean.SCM.ID.nodesAt_mem_prefixIn_iff","k":"theorem","m":"Causalean.SCM.ID.GraphicalThms.DoGFormulaTian","a":"SCM/ID/GraphicalThms","t":2,"d":"The node at index i belongs to the first n D-nodes iff i < n.","s":"∀ {N : Type u_1} [inst : DecidableEq N] [inst_1 : Fintype N] (H : Causalean.SWIGGraph N) (D : Finset (Causalean.SWIGNode N)) (n : ℕ) (i : Fin D.card), ↑(H.nodesAt D i) ∈ H.prefixIn D n ↔ ↑i < n","l":87,"f":"Causalean/SCM/ID/GraphicalThms/DoGFormulaTian.lean"},{"n":"Causalean.SCM.ID.nodesAt_not_mem_prefixIn","k":"theorem","m":"Causalean.SCM.ID.GraphicalThms.DoGFormulaTian","a":"SCM/ID/GraphicalThms","t":2,"d":"The next D-node is not in the previous D-prefix.","s":"∀ {N : Type u_1} [inst : DecidableEq N] [inst_1 : Fintype N] (H : Causalean.SWIGGraph N) (D : Finset (Causalean.SWIGNode N)) {n : ℕ} (hn : n < D.card), ↑(H.nodesAt D ⟨n, hn⟩) ∉ H.prefixIn D n","l":126,"f":"Causalean/SCM/ID/GraphicalThms/DoGFormulaTian.lean"},{"n":"Causalean.SCM.ID.nodesAt_toSWIGGraph_observed_eq_observedAt","k":"theorem","m":"Causalean.SCM.ID.GraphicalThms.DoGFormulaTian","a":"SCM/ID/GraphicalThms","t":2,"d":"The graph-level topological enumeration agrees definitionally with the SCM-level observed-node enumeration.","s":"∀ {N : Type u_1} [inst : DecidableEq N] [inst_1 : Fintype N] {Ω : N → Type u_2} [inst_2 : (n : N) → MeasurableSpace (Ω n)] (M : Causalean.SCM N Ω) (i : Fin M.observed.card), M.nodesAt M.observed i = M.observedAt i","l":888,"f":"Causalean/SCM/ID/GraphicalThms/DoGFormulaTian.lean"},{"n":"Causalean.SCM.ID.nonIdentifiable_iff","k":"theorem","m":"Causalean.SCM.ID.Identifiable","a":"SCM/ID","t":1,"d":"For a SWIG graph G and a causal query Φ, the causal query is non-identifiable from the observational distribution if and only if there exist two causal models with SWIG graph G and the same observational law that disagree on the value of Φ.","s":"∀ {N : Type u_1} [inst : DecidableEq N] [inst_1 : Fintype N] {Ω : N → Type u_2} [inst_2 : (n : N) → MeasurableSpace (Ω n)] {α : Type u_3} (G : Causalean.SWIGGraph N) (Φ : Causalean.SCM.ID.CausalQuery N Ω α), Causalean.SCM.ID.NonIdentifiable G Φ ↔ ∃ M₁ M₂, M₁.toSWIGGraph = G ∧ M₂.toSWIGGraph = G ∧ Causalean.SCM.ID.obsEquiv M₁ M₂ ∧ Φ M₁ ≠ Φ M₂","l":124,"f":"Causalean/SCM/ID/Identifiable.lean"},{"n":"Causalean.SCM.ID.obsCondKernel_slice_ae_eq_of_pairMeasure_eq","k":"theorem","m":"Causalean.SCM.ID.GraphicalThms.QFactorIdentity","a":"SCM/ID/GraphicalThms","t":2,"d":"Equal pair laws give equal conditional-kernel slices a.e.","s":"∀ {N : Type u_1} [inst : DecidableEq N] [inst_1 : Fintype N] {Ω : N → Type u_2} [inst_2 : (n : N) → MeasurableSpace (Ω n)] (M₁ M₂ : Causalean.SCM N Ω) (Y CC : Finset (Causalean.SWIGNode N)) (hY₁ : Y ⊆ M₁.observed) (hCC₁ : CC ⊆ M₁.observed) (hY₂ : Y ⊆ M₂.observed) (hCC₂ : CC ⊆ M₂.observed) [inst_3 : StandardBorelSpace (Causalean.ValuesOn Y (Causalean.swigΩ Ω))] [inst_4 : Nonempty (Causalean.ValuesOn Y (Causalean.swigΩ Ω))] [∀ (s : M₁.FixedValues), MeasureTheory.IsFiniteMeasure (M₁.obsKernel s)] [∀ (s : M₂.FixedValues), MeasureTheory.IsFiniteMeasure (M₂.obsKernel s)] [inst_7 : MeasurableSpace.CountableOrCountablyGenerated M₁.FixedValues (Causalean.ValuesOn CC (Causalean.swigΩ Ω))] [inst_8 : MeasurableSpace.CountableOrCountablyGenerated M₂.FixedValues (Causalean.ValuesOn CC (Causalean.swigΩ Ω))] (s₁ : M₁.FixedValues) (s₂ : M₂.FixedValues), (M₁.obsCondPairKernel Y CC hY₁ hCC₁) s₁ = (M₂.obsCondPairKernel Y CC hY₂ hCC₂) s₂ → (fun c => ((M₁.obsCondKernel Y CC hY₁ hCC₁).sectR s₁) c) =ᵐ[MeasureTheory.Measure.map (Causalean.valuesProjection hCC₁) (M₁.obsKernel s₁)] fun c => ((M₂.obsCondKernel Y CC hY₂ hCC₂).sectR s₂) c","l":206,"f":"Causalean/SCM/ID/GraphicalThms/QFactorIdentity.lean"},{"n":"Causalean.SCM.ID.obsEquiv","k":"def","m":"Causalean.SCM.ID.Identifiable","a":"SCM/ID","t":1,"d":"Two SCMs are observationally equivalent if their derived observational kernels agree. Because obsKernel has dependent domain/codomain (FixedValues M and ObservedValues M), we use HEq to accommodate two models whose fixed/observed sets may only be propositionally equal.","s":"{N : Type u_1} → [inst : DecidableEq N] → [inst_1 : Fintype N] → {Ω : N → Type u_2} → [inst_2 : (n : N) → MeasurableSpace (Ω n)] → Causalean.SCM N Ω → Causalean.SCM N Ω → Prop","l":83,"f":"Causalean/SCM/ID/Identifiable.lean"},{"n":"Causalean.SCM.ID.pinnedExtend","k":"def","m":"Causalean.SCM.ID.GraphicalThms.DoGFormula","a":"SCM/ID/GraphicalThms","t":1,"d":"Extend an assignment on the post-intervention observed ancestral support to the original observed coordinates, pinning intervened random coordinates to the fixed values of the do-slice.","s":"{N : Type u_1} → [inst : DecidableEq N] → [inst_1 : Fintype N] → {Ω : N → Type u_2} → [inst_2 : (n : N) → MeasurableSpace (Ω n)] → [∀ (n : N), Nonempty (Ω n)] → (M : Causalean.SCM N Ω) → (X : Finset N) → (hObs : ∀ D ∈ X, Causalean.SWIGNode.random D ∈ M.observed) → (hFix : ∀ D ∈ X, Causalean.SWIGNode.fixed D ∉ M.fixed) → (Y : Finset (Causalean.SWIGNode N)) → (M.fixSet X hObs hFix).FixedValues → Causalean.ValuesOn (Causalean.SCM.ID.fixObservedAncestralSet M X hObs hFix Y) (Causalean.swigΩ Ω) → Causalean.ValuesOn M.observed (Causalean.swigΩ Ω)","l":137,"f":"Causalean/SCM/ID/GraphicalThms/DoGFormula.lean"},{"n":"Causalean.SCM.ID.pinnedExtend_pin_eq","k":"theorem","m":"Causalean.SCM.ID.GraphicalThms.DoGFormula","a":"SCM/ID/GraphicalThms","t":2,"d":"The pinned extension reads intervened random coordinates from the matching fixed coordinate of the do-slice.","s":"∀ {N : Type u_1} [inst : DecidableEq N] [inst_1 : Fintype N] {Ω : N → Type u_2} [inst_2 : (n : N) → MeasurableSpace (Ω n)] [inst_3 : ∀ (n : N), Nonempty (Ω n)] (M : Causalean.SCM N Ω) (X : Finset N) (hObs : ∀ D ∈ X, Causalean.SWIGNode.random D ∈ M.observed) (hFix : ∀ D ∈ X, Causalean.SWIGNode.fixed D ∉ M.fixed) (Y : Finset (Causalean.SWIGNode N)) (sDo : (M.fixSet X hObs hFix).FixedValues) (xD : Causalean.ValuesOn (Causalean.SCM.ID.fixObservedAncestralSet M X hObs hFix Y) (Causalean.swigΩ Ω)) (D : N) (hD : D ∈ X), Causalean.SCM.ID.pinnedExtend M X hObs hFix Y sDo xD ⟨Causalean.SWIGNode.random D, hObs D hD⟩ = sDo ⟨Causalean.SWIGNode.fixed D, Finset.mem_union_right M.fixed (Finset.mem_image.mpr (Exists.intro D ⟨hD, rfl⟩))⟩","l":336,"f":"Causalean/SCM/ID/GraphicalThms/DoGFormula.lean"},{"n":"Causalean.SCM.ID.pinnedExtend_projection_eq","k":"theorem","m":"Causalean.SCM.ID.GraphicalThms.DoGFormula","a":"SCM/ID/GraphicalThms","t":2,"d":"The pinned extension projects back to the ancestral assignment when natural intervened coordinates are excluded from the query.","s":"∀ {N : Type u_1} [inst : DecidableEq N] [inst_1 : Fintype N] {Ω : N → Type u_2} [inst_2 : (n : N) → MeasurableSpace (Ω n)] [inst_3 : ∀ (n : N), Nonempty (Ω n)] (M : Causalean.SCM N Ω) (X : Finset N) (hObs : ∀ D ∈ X, Causalean.SWIGNode.random D ∈ M.observed) (hFix : ∀ D ∈ X, Causalean.SWIGNode.fixed D ∉ M.fixed) (Y : Finset (Causalean.SWIGNode N)) (sDo : (M.fixSet X hObs hFix).FixedValues), (∀ D ∈ X, Causalean.SWIGNode.random D ∉ Y) → ∀ (xD : Causalean.ValuesOn (Causalean.SCM.ID.fixObservedAncestralSet M X hObs hFix Y) (Causalean.swigΩ Ω)), Causalean.valuesProjection (have this := Finset.inter_subset_right; this) (Causalean.SCM.ID.pinnedExtend M X hObs hFix Y sDo xD) = xD","l":303,"f":"Causalean/SCM/ID/GraphicalThms/DoGFormula.lean"},{"n":"Causalean.SCM.ID.prefixIn_card","k":"theorem","m":"Causalean.SCM.ID.GraphicalThms.DoGFormulaTian","a":"SCM/ID/GraphicalThms","t":2,"d":"The full D-prefix is D.","s":"∀ {N : Type u_1} [inst : DecidableEq N] [inst_1 : Fintype N] (H : Causalean.SWIGGraph N) (D : Finset (Causalean.SWIGNode N)), H.prefixIn D D.card = D","l":108,"f":"Causalean/SCM/ID/GraphicalThms/DoGFormulaTian.lean"},{"n":"Causalean.SCM.ID.prefixIn_disjoint_singleton_next","k":"theorem","m":"Causalean.SCM.ID.GraphicalThms.DoGFormulaTian","a":"SCM/ID/GraphicalThms","t":2,"d":"The previous D-prefix is disjoint from the singleton next node.","s":"∀ {N : Type u_1} [inst : DecidableEq N] [inst_1 : Fintype N] (H : Causalean.SWIGGraph N) (D : Finset (Causalean.SWIGNode N)) {n : ℕ} (hn : n < D.card), Disjoint (H.prefixIn D n) {↑(H.nodesAt D ⟨n, hn⟩)}","l":161,"f":"Causalean/SCM/ID/GraphicalThms/DoGFormulaTian.lean"},{"n":"Causalean.SCM.ID.prefixIn_mono","k":"theorem","m":"Causalean.SCM.ID.GraphicalThms.DoGFormulaTian","a":"SCM/ID/GraphicalThms","t":2,"d":"Prefix sets are monotone in the prefix length.","s":"∀ {N : Type u_1} [inst : DecidableEq N] [inst_1 : Fintype N] (H : Causalean.SWIGGraph N) (D : Finset (Causalean.SWIGNode N)) {m k : ℕ}, m ≤ k → H.prefixIn D m ⊆ H.prefixIn D k","l":118,"f":"Causalean/SCM/ID/GraphicalThms/DoGFormulaTian.lean"},{"n":"Causalean.SCM.ID.prefixIn_parent_closed_induce_observed","k":"theorem","m":"Causalean.SCM.ID.Density.QFactor","a":"SCM/ID/Density","t":2,"d":"In an induced set of observed variables, every topological prefix contains all parents within that set of each variable it contains.","s":"∀ {N : Type u_1} [inst : DecidableEq N] [inst_1 : Fintype N] {Ω : N → Type u_2} [inst_2 : (n : N) → MeasurableSpace (Ω n)] (M : Causalean.SCM N Ω), ∀ A ⊆ M.observed, ∀ (k : ℕ), ∀ v ∈ A, ∀ w ∈ (M.induce A).prefixIn A k, M.dag.edge v w → v ∈ (M.induce A).prefixIn A k","l":379,"f":"Causalean/SCM/ID/Density/QFactor.lean"},{"n":"Causalean.SCM.ID.prefixIn_qProduct_ratio_eq_component_ratio_of_family","k":"theorem","m":"Causalean.SCM.ID.Density.QFactor","a":"SCM/ID/Density","t":2,"d":"In a product over an abstract pairwise-disjoint family, the prefix-ratio step at a node of S cancels every factor except the S factor.","s":"∀ {N : Type u_1} [inst : DecidableEq N] [inst_1 : Fintype N] {Ω : N → Type u_2} [inst_2 : (n : N) → MeasurableSpace (Ω n)] [(n : N) → Fintype (Ω n)] [∀ (n : N), MeasurableSingletonClass (Ω n)] (M : Causalean.SCM N Ω) (H : Causalean.SWIGGraph N) (D : Finset (Causalean.SWIGNode N)) (s : M.FixedValues) (𝒞 : Finset (Finset (Causalean.SWIGNode N))), ∀ S ∈ 𝒞, (∀ C ∈ 𝒞, C ≠ S → Disjoint C S) → ∀ (i : Fin D.card) (hDobs : D ⊆ M.observed), ↑(H.nodesAt D i) ∈ S → Causalean.SCM.ID.DiscreteID.PositiveMass (M.obsKernel s) → ∀ (x : Causalean.ValuesOn M.observed (Causalean.swigΩ Ω)), (∏ C ∈ 𝒞, M.qLocalMass s (C ∩ H.prefixIn D (↑i + 1)) (fun x hv => hDobs (Causalean.SWIGGraph.prefixIn_subset H D (↑i + 1) (Finset.mem_of_mem_inter_right hv))) x) / ∏ C ∈ 𝒞, M.qLocalMass s (C ∩ H.prefixIn D ↑i) (fun x hv => hDobs (Causalean.SWIGGraph.prefixIn_subset H D (↑i) (Finset.mem_of_mem_inter_right hv))) x = M.qLocalMass s (S ∩ H.prefixIn D (↑i + 1)) (fun x hv => hDobs (Causalean.SWIGGraph.prefixIn_subset H D (↑i + 1) (Finset.mem_of_mem_inter_right hv))) x / M.qLocalMass s (S ∩ H.prefixIn D ↑i) (fun x hv => hDobs (Causalean.SWIGGraph.prefixIn_subset H D (↑i) (Finset.mem_of_mem_inter_right hv))) x","l":545,"f":"Causalean/SCM/ID/Density/QFactor.lean"},{"n":"Causalean.SCM.ID.prefixIn_qProduct_ratio_eq_component_ratio_of_family_of_ne_zero","k":"theorem","m":"Causalean.SCM.ID.Density.QFactor","a":"SCM/ID/Density","t":2,"d":"Nonzero-denominator variant of prefixIn_qProduct_ratio_eq_component_ratio_of_family.","s":"∀ {N : Type u_1} [inst : DecidableEq N] [inst_1 : Fintype N] {Ω : N → Type u_2} [inst_2 : (n : N) → MeasurableSpace (Ω n)] [(n : N) → Fintype (Ω n)] [∀ (n : N), MeasurableSingletonClass (Ω n)] (M : Causalean.SCM N Ω) (H : Causalean.SWIGGraph N) (D : Finset (Causalean.SWIGNode N)) (s : M.FixedValues) (𝒞 : Finset (Finset (Causalean.SWIGNode N))), ∀ S ∈ 𝒞, (∀ C ∈ 𝒞, C ≠ S → Disjoint C S) → ∀ (i : Fin D.card) (hDobs : D ⊆ M.observed), ↑(H.nodesAt D i) ∈ S → ∀ (x : Causalean.ValuesOn M.observed (Causalean.swigΩ Ω)), ∏ C ∈ 𝒞 \\ {S}, M.qLocalMass s (C ∩ H.prefixIn D ↑i) (fun x hv => hDobs (Causalean.SWIGGraph.prefixIn_subset H D (↑i) (Finset.mem_of_mem_inter_right hv))) x ≠ 0 → (∏ C ∈ 𝒞, M.qLocalMass s (C ∩ H.prefixIn D (↑i + 1)) (fun x hv => hDobs (Causalean.SWIGGraph.prefixIn_subset H D (↑i + 1) (Finset.mem_of_mem_inter_right hv))) x) / ∏ C ∈ 𝒞, M.qLocalMass s (C ∩ H.prefixIn D ↑i) (fun x hv => hDobs (Causalean.SWIGGraph.prefixIn_subset H D (↑i) (Finset.mem_of_mem_inter_right hv))) x = M.qLocalMass s (S ∩ H.prefixIn D (↑i + 1)) (fun x hv => hDobs (Causalean.SWIGGraph.prefixIn_subset H D (↑i + 1) (Finset.mem_of_mem_inter_right hv))) x / M.qLocalMass s (S ∩ H.prefixIn D ↑i) (fun x hv => hDobs (Causalean.SWIGGraph.prefixIn_subset H D (↑i) (Finset.mem_of_mem_inter_right hv))) x","l":684,"f":"Causalean/SCM/ID/Density/QFactor.lean"},{"n":"Causalean.SCM.ID.prefixIn_succ","k":"theorem","m":"Causalean.SCM.ID.GraphicalThms.DoGFormulaTian","a":"SCM/ID/GraphicalThms","t":2,"d":"The successor D-prefix is obtained by adjoining the next D-node.","s":"∀ {N : Type u_1} [inst : DecidableEq N] [inst_1 : Fintype N] (H : Causalean.SWIGGraph N) (D : Finset (Causalean.SWIGNode N)) {n : ℕ} (hn : n < D.card), H.prefixIn D (n + 1) = H.prefixIn D n ∪ {↑(H.nodesAt D ⟨n, hn⟩)}","l":133,"f":"Causalean/SCM/ID/GraphicalThms/DoGFormulaTian.lean"},{"n":"Causalean.SCM.ID.prefixIn_zero","k":"theorem","m":"Causalean.SCM.ID.GraphicalThms.DoGFormulaTian","a":"SCM/ID/GraphicalThms","t":2,"d":"The D-prefix of length zero is empty.","s":"∀ {N : Type u_1} [inst : DecidableEq N] [inst_1 : Fintype N] (H : Causalean.SWIGGraph N) (D : Finset (Causalean.SWIGNode N)), H.prefixIn D 0 = ∅","l":77,"f":"Causalean/SCM/ID/GraphicalThms/DoGFormulaTian.lean"},{"n":"Causalean.SCM.ID.prefixNodes_obsParentClosed","k":"theorem","m":"Causalean.SCM.ID.Density.MechCFactor","a":"SCM/ID/Density","t":2,"d":"The first k observed nodes in a structural causal model's topological order form a set closed under observed parents.","s":"∀ {N : Type u_1} [inst : DecidableEq N] [inst_1 : Fintype N] {Ω : N → Type u_2} [inst_2 : (n : N) → MeasurableSpace (Ω n)] (M : Causalean.SCM N Ω) (k : ℕ), M.ObsParentClosed (M.prefixNodes k)","l":299,"f":"Causalean/SCM/ID/Density/MechCFactor.lean"},{"n":"Causalean.SCM.ID.prefix_pair_singleton_mass_eq_succ_prefix_mass","k":"theorem","m":"Causalean.SCM.ID.Density.QFactor","a":"SCM/ID/Density","t":2,"d":"The mass of a realized prefix together with its next coordinate equals the mass of the same realization of the successor prefix under any measure on the ordered graph values.","s":"∀ {N : Type u_1} [inst : DecidableEq N] [inst_1 : Fintype N] {Ω : N → Type u_2} [inst_2 : (n : N) → MeasurableSpace (Ω n)] [∀ (n : N), MeasurableSingletonClass (Ω n)] (H : Causalean.SWIGGraph N) (D : Finset (Causalean.SWIGNode N)) (μ : MeasureTheory.Measure (Causalean.ValuesOn D (Causalean.swigΩ Ω))) (i : Fin D.card) (x : Causalean.ValuesOn D (Causalean.swigΩ Ω)), (MeasureTheory.Measure.map (fun ω => (Causalean.valuesProjection (Causalean.SWIGGraph.prefixIn_subset H D ↑i) ω, Causalean.valuesProjection (have this := fun ⦃v⦄ hv => Eq.symm (Eq.mp (congrArg (fun _a => _a) (propext Finset.mem_singleton)) hv) ▸ (H.nodesAt D i).property; this) ω)) μ) {(Causalean.valuesProjection (Causalean.SWIGGraph.prefixIn_subset H D ↑i) x, Causalean.valuesProjection (have this := fun ⦃v⦄ hv => Eq.symm (Eq.mp (congrArg (fun _a => _a) (propext Finset.mem_singleton)) hv) ▸ (H.nodesAt D i).property; this) x)} = (MeasureTheory.Measure.map (Causalean.valuesProjection (Causalean.SWIGGraph.prefixIn_subset H D (↑i + 1))) μ) {Causalean.valuesProjection (Causalean.SWIGGraph.prefixIn_subset H D (↑i + 1)) x}","l":1224,"f":"Causalean/SCM/ID/Density/QFactor.lean"},{"n":"Causalean.SCM.ID.prod_filter_div_telescope","k":"theorem","m":"Causalean.SCM.ID.Density.QMass","a":"SCM/ID/Density","t":2,"d":"Pure ENNReal telescope for products of selected adjacent ratios.","s":"∀ (a : ℕ → ENNReal) (m : ℕ), ∀ T ⊆ Finset.range m, (∀ i ≤ m, a i ≠ 0) → (∀ i ≤ m, a i ≠ ⊤) → (∀ i < m, i ∉ T → a (i + 1) = a i) → ∏ i ∈ T, a (i + 1) / a i = a m / a 0","l":1776,"f":"Causalean/SCM/ID/Density/QMass.lean"},{"n":"Causalean.SCM.ID.prod_tianDistrictDensity_eq_tianDensityProduct","k":"theorem","m":"Causalean.SCM.ID.Density.QFactor","a":"SCM/ID/Density","t":2,"d":"Tian's full prefix-chain density regroups exactly as the product of the Tian district factors over the graph c-components. This is pure finite-product algebra: each prefix index maps to the c-component of its node, and the district factor is precisely the product over the corresponding fiber.","s":"∀ {N : Type u_1} [inst : DecidableEq N] [inst_1 : Fintype N] {Ω : N → Type u_2} [inst_2 : (n : N) → MeasurableSpace (Ω n)] (H : Causalean.SWIGGraph N) (D : Finset (Causalean.SWIGNode N)), H.observed = D → ∀ (μ : MeasureTheory.Measure (Causalean.ValuesOn D (Causalean.swigΩ Ω))) (ref : Causalean.SCM.ReferenceMeasures Ω) [inst_3 : MeasureTheory.IsFiniteMeasure μ] [inst_4 : ∀ (k : ℕ) (hk : k < D.card), StandardBorelSpace (Causalean.ValuesOn {↑(H.nodesAt D ⟨k, hk⟩)} (Causalean.swigΩ Ω))] [inst_5 : ∀ (k : ℕ) (hk : k < D.card), Nonempty (Causalean.ValuesOn {↑(H.nodesAt D ⟨k, hk⟩)} (Causalean.swigΩ Ω))], (fun x => ∏ S ∈ H.cComponentSet, Causalean.SCM.ID.tianDistrictDensity H D μ ref S x) = fun x => Causalean.SCM.ID.tianDensityProduct H D μ ref x","l":230,"f":"Causalean/SCM/ID/Density/QFactor.lean"},{"n":"Causalean.SCM.ID.qLocalMass_fixSet_invariant","k":"theorem","m":"Causalean.SCM.ID.Density.MechCFactor","a":"SCM/ID/Density","t":2,"d":"Local q-mass is invariant under a fixSet intervention when no coordinate in S is one of the intervened random nodes and the full assignment pins each intervened random node to the corresponding fixed value.","s":"∀ {N : Type u_1} [inst : DecidableEq N] [inst_1 : Fintype N] {Ω : N → Type u_2} [inst_2 : (n : N) → MeasurableSpace (Ω n)] (M : Causalean.SCM N Ω) (X : Finset N) (hObs : ∀ D ∈ X, Causalean.SWIGNode.random D ∈ M.observed) (hFix : ∀ D ∈ X, Causalean.SWIGNode.fixed D ∉ M.fixed) (sDo : (M.fixSet X hObs hFix).FixedValues) (S : Finset (Causalean.SWIGNode N)) (hS : S ⊆ M.observed), (∀ n ∈ X, Causalean.SWIGNode.random n ∉ S) → ∀ (x : Causalean.ValuesOn M.observed (Causalean.swigΩ Ω)), (∀ (D : N) (hD : D ∈ X), x ⟨Causalean.SWIGNode.random D, hObs D hD⟩ = sDo ⟨Causalean.SWIGNode.fixed D, Finset.mem_union_right M.fixed (Finset.mem_image.mpr (Exists.intro D ⟨hD, rfl⟩))⟩) → (M.fixSet X hObs hFix).qLocalMass sDo S (id hS) x = M.qLocalMass (M.fixSetProj X hObs hFix sDo) S hS x","l":1020,"f":"Causalean/SCM/ID/Density/MechCFactor.lean"},{"n":"Causalean.SCM.ID.qLocalMass_ne_top","k":"theorem","m":"Causalean.SCM.ID.Density.MechCFactor","a":"SCM/ID/Density","t":2,"d":"Every local q-mass associated with fixed values, an observed node set, and an observed assignment is finite.","s":"∀ {N : Type u_1} [inst : DecidableEq N] [inst_1 : Fintype N] {Ω : N → Type u_2} [inst_2 : (n : N) → MeasurableSpace (Ω n)] (M : Causalean.SCM N Ω) (s : M.FixedValues) (T : Finset (Causalean.SWIGNode N)) (hT : T ⊆ M.observed) (x : Causalean.ValuesOn M.observed (Causalean.swigΩ Ω)), M.qLocalMass s T hT x ≠ ⊤","l":351,"f":"Causalean/SCM/ID/Density/MechCFactor.lean"},{"n":"Causalean.SCM.ID.qLocalMass_obsProof_irrel","k":"theorem","m":"Causalean.SCM.ID.Density.QFactor","a":"SCM/ID/Density","t":2,"d":"The local q-mass of an observed variable set does not depend on which proof establishes that the set is observed.","s":"∀ {N : Type u_1} [inst : DecidableEq N] [inst_1 : Fintype N] {Ω : N → Type u_2} [inst_2 : (n : N) → MeasurableSpace (Ω n)] (M : Causalean.SCM N Ω) (s : M.FixedValues) (T : Finset (Causalean.SWIGNode N)) (hT hT' : T ⊆ M.observed) (x : Causalean.ValuesOn M.observed (Causalean.swigΩ Ω)), M.qLocalMass s T hT x = M.qLocalMass s T hT' x","l":1056,"f":"Causalean/SCM/ID/Density/QFactor.lean"},{"n":"Causalean.SCM.ID.q_factor_identity","k":"theorem","m":"Causalean.SCM.ID.GraphicalThms.QFactorIdentity","a":"SCM/ID/GraphicalThms","t":1,"d":"Proposition 2.19 (Q-factor identity / Tian's lemma). For an ancestrally-closed node set R and a c-component T of the induced subgraph on R in a structural causal model M, after intervening on a node set Wn, the structurally-defined c-factor on T equals, almost everywhere, the conditional of T given its q-factor parents extracted from that intervention. See QFactorIdentityConclusion for the precise hypothesis frame.","s":"∀ {N : Type u_1} [inst : DecidableEq N] [inst_1 : Fintype N] {Ω : N → Type u_2} [inst_2 : (n : N) → MeasurableSpace (Ω n)] (M : Causalean.SCM N Ω) (R T : Finset (Causalean.SWIGNode N)) (Wn : Finset N), Causalean.SCM.ID.QFactorIdentityConclusion M R T Wn","l":274,"f":"Causalean/SCM/ID/GraphicalThms/QFactorIdentity.lean"},{"n":"Causalean.SCM.ID.q_factor_marginal_fixing","k":"theorem","m":"Causalean.SCM.ID.GraphicalThms.QFactorIdentity","a":"SCM/ID/GraphicalThms","t":2,"d":"Marginal fixing form of Tian's Q-factor identity.","s":"∀ {N : Type u_1} [inst : DecidableEq N] [inst_1 : Fintype N] {Ω : N → Type u_2} [inst_2 : (n : N) → MeasurableSpace (Ω n)] (M : Causalean.SCM N Ω) (R T : Finset (Causalean.SWIGNode N)) (Wn : Finset N), Causalean.SCM.ID.QFactorMarginalFixingConclusion M R T Wn","l":162,"f":"Causalean/SCM/ID/GraphicalThms/QFactorIdentity.lean"},{"n":"Causalean.SCM.ID.random_intervened_mem_fixAncestralSet_iff_mem_Y","k":"theorem","m":"Causalean.SCM.ID.GraphicalThms.DoGFormula","a":"SCM/ID/GraphicalThms","t":2,"d":"For an intervened variable, its random copy is a post-intervention ancestor of the query exactly when it is explicitly queried.","s":"∀ {N : Type u_1} [inst : DecidableEq N] [inst_1 : Fintype N] {Ω : N → Type u_2} [inst_2 : (n : N) → MeasurableSpace (Ω n)] (M : Causalean.SCM N Ω) (X : Finset N) (hObs : ∀ D ∈ X, Causalean.SWIGNode.random D ∈ M.observed) (hFix : ∀ D ∈ X, Causalean.SWIGNode.fixed D ∉ M.fixed) (Y : Finset (Causalean.SWIGNode N)) {d : N}, d ∈ X → (Causalean.SWIGNode.random d ∈ Causalean.SCM.ID.fixAncestralSet M X hObs hFix Y ↔ Causalean.SWIGNode.random d ∈ Y)","l":277,"f":"Causalean/SCM/ID/GraphicalThms/DoGFormula.lean"},{"n":"Causalean.SCM.ID.recoveredFactorRec","k":"def","m":"Causalean.SCM.ID.GraphicalThms.IDAlgorithmRec","a":"SCM/ID/GraphicalThms","t":1,"d":"Obs-side recursive recovered factor for a target district.","s":"{N : Type u_1} → [inst : DecidableEq N] → [inst_1 : Fintype N] → {Ω : N → Type u_2} → [inst_2 : (n : N) → MeasurableSpace (Ω n)] → (M : Causalean.SCM N Ω) → Causalean.SCM.ReferenceMeasures Ω → M.FixedValues → Finset (Causalean.SWIGNode N) → Finset (Causalean.SWIGNode N) → [hfin : (n : N) → Fintype (Ω n)] → [∀ (s' : M.FixedValues), MeasureTheory.IsFiniteMeasure (M.obsKernel s')] → [∀ (k : ℕ) (hk : k < M.observed.card), StandardBorelSpace (Causalean.ValuesOn {↑(M.observedAt ⟨k, hk⟩)} (Causalean.swigΩ Ω))] → [∀ (k : ℕ) (hk : k < M.observed.card), Nonempty (Causalean.ValuesOn {↑(M.observedAt ⟨k, hk⟩)} (Causalean.swigΩ Ω))] → [∀ (k : ℕ), MeasurableSpace.CountableOrCountablyGenerated M.FixedValues (Causalean.ValuesOn (M.prefixNodes k) (Causalean.swigΩ Ω))] → Causalean.ValuesOn M.observed (Causalean.swigΩ Ω) → ENNReal","l":170,"f":"Causalean/SCM/ID/GraphicalThms/IDAlgorithmRec.lean"},{"n":"Causalean.SCM.ID.recoveredFactorRec_heq_of_obsKernel_heq","k":"theorem","m":"Causalean.SCM.ID.GraphicalThms.IDAlgorithmRec","a":"SCM/ID/GraphicalThms","t":2,"d":"Equal observational kernels transport the recursive recovered factor.","s":"∀ {N : Type u_1} [inst : DecidableEq N] [inst_1 : Fintype N] {Ω : N → Type u_2} [inst_2 : (n : N) → MeasurableSpace (Ω n)] (M₁ M₂ : Causalean.SCM N Ω) (ref : Causalean.SCM.ReferenceMeasures Ω) (C S : Finset (Causalean.SWIGNode N)), M₁.toSWIGGraph = M₂.toSWIGGraph → M₁.obsKernel ≍ M₂.obsKernel → ∀ [hfin : (n : N) → Fintype (Ω n)] [inst_3 : ∀ (s' : M₁.FixedValues), MeasureTheory.IsFiniteMeasure (M₁.obsKernel s')] [inst_4 : ∀ (k : ℕ) (hk : k < M₁.observed.card), StandardBorelSpace (Causalean.ValuesOn {↑(M₁.observedAt ⟨k, hk⟩)} (Causalean.swigΩ Ω))] [inst_5 : ∀ (k : ℕ) (hk : k < M₁.observed.card), Nonempty (Causalean.ValuesOn {↑(M₁.observedAt ⟨k, hk⟩)} (Causalean.swigΩ Ω))] [inst_6 : ∀ (k : ℕ), MeasurableSpace.CountableOrCountablyGenerated M₁.FixedValues (Causalean.ValuesOn (M₁.prefixNodes k) (Causalean.swigΩ Ω))] [inst_7 : ∀ (s' : M₂.FixedValues), MeasureTheory.IsFiniteMeasure (M₂.obsKernel s')] [inst_8 : ∀ (k : ℕ) (hk : k < M₂.observed.card), StandardBorelSpace (Causalean.ValuesOn {↑(M₂.observedAt ⟨k, hk⟩)} (Causalean.swigΩ Ω))] [inst_9 : ∀ (k : ℕ) (hk : k < M₂.observed.card), Nonempty (Causalean.ValuesOn {↑(M₂.observedAt ⟨k, hk⟩)} (Causalean.swigΩ Ω))] [inst_10 : ∀ (k : ℕ), MeasurableSpace.CountableOrCountablyGenerated M₂.FixedValues (Causalean.ValuesOn (M₂.prefixNodes k) (Causalean.swigΩ Ω))], (fun s => Causalean.SCM.ID.recoveredFactorRec M₁ ref s C S) ≍ fun s => Causalean.SCM.ID.recoveredFactorRec M₂ ref s C S","l":207,"f":"Causalean/SCM/ID/GraphicalThms/IDAlgorithmRec.lean"},{"n":"Causalean.SCM.ID.rnDeriv_eq_tianDensityProduct","k":"theorem","m":"Causalean.SCM.ID.GraphicalThms.DoGFormulaTian","a":"SCM/ID/GraphicalThms","t":2,"d":"The density of any dominated finite law on a finite coordinate set factors as the product of its one-coordinate conditional densities along the chosen topological order. The conditional density at each coordinate is computed from Mathlib's regular conditional distribution given the preceding prefix.","s":"∀ {N : Type u_1} [inst : DecidableEq N] [inst_1 : Fintype N] {Ω : N → Type u_2} [inst_2 : (n : N) → MeasurableSpace (Ω n)] (H : Causalean.SWIGGraph N) (D : Finset (Causalean.SWIGNode N)) (μ : MeasureTheory.Measure (Causalean.ValuesOn D (Causalean.swigΩ Ω))) (ref : Causalean.SCM.ReferenceMeasures Ω) [(n : N) → Fintype (Ω n)] [∀ (n : N), MeasurableSingletonClass (Ω n)] [StandardBorelSpace (Causalean.ValuesOn D (Causalean.swigΩ Ω))] [inst_6 : MeasureTheory.IsProbabilityMeasure μ] [inst_7 : ∀ (k : ℕ) (hk : k < D.card), StandardBorelSpace (Causalean.ValuesOn {↑(H.nodesAt D ⟨k, hk⟩)} (Causalean.swigΩ Ω))] [inst_8 : ∀ (k : ℕ) (hk : k < D.card), Nonempty (Causalean.ValuesOn {↑(H.nodesAt D ⟨k, hk⟩)} (Causalean.swigΩ Ω))], μ.AbsolutelyContinuous (Causalean.SCM.jointRef ref D) → μ.rnDeriv (Causalean.SCM.jointRef ref D) =ᵐ[Causalean.SCM.jointRef ref D] Causalean.SCM.ID.tianDensityProduct H D μ ref","l":701,"f":"Causalean/SCM/ID/GraphicalThms/DoGFormulaTian.lean"},{"n":"Causalean.SCM.ID.splitMono_bidirectedReachable_iff","k":"theorem","m":"Causalean.SCM.ID.GraphicalThms.DoGFormula","a":"SCM/ID/GraphicalThms","t":2,"d":"A valid monolithic intervention split leaves bidirected reachability between any two SWIG nodes unchanged.","s":"∀ {N : Type u_1} [inst : DecidableEq N] [inst_1 : Fintype N] (G : Causalean.SWIGGraph N) (X : Finset N) (hObs : ∀ D ∈ X, Causalean.SWIGNode.random D ∈ G.observed) (hFix : ∀ D ∈ X, Causalean.SWIGNode.fixed D ∉ G.fixed) (v w : Causalean.SWIGNode N), (G.splitMono X hObs hFix).bidirectedReachable v w ↔ G.bidirectedReachable v w","l":200,"f":"Causalean/SCM/ID/GraphicalThms/DoGFormula.lean"},{"n":"Causalean.SCM.ID.splitMono_cComponentOf_eq","k":"theorem","m":"Causalean.SCM.ID.GraphicalThms.DoGFormula","a":"SCM/ID/GraphicalThms","t":2,"d":"Under a valid monolithic intervention split, the bidirected component containing any SWIG node is the same as it was before intervention.","s":"∀ {N : Type u_1} [inst : DecidableEq N] [inst_1 : Fintype N] (G : Causalean.SWIGGraph N) (X : Finset N) (hObs : ∀ D ∈ X, Causalean.SWIGNode.random D ∈ G.observed) (hFix : ∀ D ∈ X, Causalean.SWIGNode.fixed D ∉ G.fixed) (v : Causalean.SWIGNode N), (G.splitMono X hObs hFix).cComponentOf v = G.cComponentOf v","l":223,"f":"Causalean/SCM/ID/GraphicalThms/DoGFormula.lean"},{"n":"Causalean.SCM.ID.splitMono_cComponentSet_eq","k":"theorem","m":"Causalean.SCM.ID.GraphicalThms.DoGFormula","a":"SCM/ID/GraphicalThms","t":2,"d":"Splitting a graph under a valid monolithic intervention leaves its partition into bidirected connected components unchanged.","s":"∀ {N : Type u_1} [inst : DecidableEq N] [inst_1 : Fintype N] (G : Causalean.SWIGGraph N) (X : Finset N) (hObs : ∀ D ∈ X, Causalean.SWIGNode.random D ∈ G.observed) (hFix : ∀ D ∈ X, Causalean.SWIGNode.fixed D ∉ G.fixed), (G.splitMono X hObs hFix).cComponentSet = G.cComponentSet","l":243,"f":"Causalean/SCM/ID/GraphicalThms/DoGFormula.lean"},{"n":"Causalean.SCM.ID.splitMono_directlyConfounded_iff","k":"theorem","m":"Causalean.SCM.ID.GraphicalThms.DoGFormula","a":"SCM/ID/GraphicalThms","t":2,"d":"Under a valid monolithic intervention split, two nodes are directly confounded exactly when they were directly confounded in the original graph.","s":"∀ {N : Type u_1} [inst : DecidableEq N] [inst_1 : Fintype N] (G : Causalean.SWIGGraph N) (X : Finset N) (hObs : ∀ D ∈ X, Causalean.SWIGNode.random D ∈ G.observed) (hFix : ∀ D ∈ X, Causalean.SWIGNode.fixed D ∉ G.fixed) (v₁ v₂ : Causalean.SWIGNode N), (G.splitMono X hObs hFix).directlyConfounded v₁ v₂ ↔ G.directlyConfounded v₁ v₂","l":181,"f":"Causalean/SCM/ID/GraphicalThms/DoGFormula.lean"},{"n":"Causalean.SCM.ID.splitMono_edge_from_unobserved_iff","k":"theorem","m":"Causalean.SCM.ID.GraphicalThms.DoGFormula","a":"SCM/ID/GraphicalThms","t":2,"d":"Under a valid monolithic intervention split, a directed edge from an unobserved node exists exactly when that edge existed in the original graph.","s":"∀ {N : Type u_1} [inst : DecidableEq N] [inst_1 : Fintype N] (G : Causalean.SWIGGraph N) (X : Finset N) (hObs : ∀ D ∈ X, Causalean.SWIGNode.random D ∈ G.observed) (hFix : ∀ D ∈ X, Causalean.SWIGNode.fixed D ∉ G.fixed) {u v : Causalean.SWIGNode N}, u ∈ G.unobserved → ((G.splitMono X hObs hFix).dag.edge u v ↔ G.dag.edge u v)","l":166,"f":"Causalean/SCM/ID/GraphicalThms/DoGFormula.lean"},{"n":"Causalean.SCM.ID.splitMono_no_edge_from_intervened_random","k":"theorem","m":"Causalean.SCM.ID.GraphicalThms.DoGFormula","a":"SCM/ID/GraphicalThms","t":2,"d":"Under a valid monolithic intervention split, the random copy of an intervened variable has no outgoing directed edge.","s":"∀ {N : Type u_1} [inst : DecidableEq N] [inst_1 : Fintype N] (G : Causalean.SWIGGraph N) (X : Finset N) (hObs : ∀ D ∈ X, Causalean.SWIGNode.random D ∈ G.observed) (hFix : ∀ D ∈ X, Causalean.SWIGNode.fixed D ∉ G.fixed) {d : N}, d ∈ X → ∀ (v : Causalean.SWIGNode N), ¬(G.splitMono X hObs hFix).dag.edge (Causalean.SWIGNode.random d) v","l":253,"f":"Causalean/SCM/ID/GraphicalThms/DoGFormula.lean"},{"n":"Causalean.SCM.ID.splitMono_not_isAncestor_from_intervened_random","k":"theorem","m":"Causalean.SCM.ID.GraphicalThms.DoGFormula","a":"SCM/ID/GraphicalThms","t":2,"d":"After a valid monolithic intervention split, the random copy of an intervened variable is not an ancestor of any node.","s":"∀ {N : Type u_1} [inst : DecidableEq N] [inst_1 : Fintype N] (G : Causalean.SWIGGraph N) (X : Finset N) (hObs : ∀ D ∈ X, Causalean.SWIGNode.random D ∈ G.observed) (hFix : ∀ D ∈ X, Causalean.SWIGNode.fixed D ∉ G.fixed) {d : N}, d ∈ X → ∀ (v : Causalean.SWIGNode N), ¬(G.splitMono X hObs hFix).dag.isAncestor (Causalean.SWIGNode.random d) v","l":264,"f":"Causalean/SCM/ID/GraphicalThms/DoGFormula.lean"},{"n":"Causalean.SCM.ID.standardFixedValues","k":"def","m":"Causalean.SCM.ID.Query","a":"SCM/ID","t":1,"d":"A standard structural causal model has a canonical fixed-value assignment.","s":"{N : Type u_1} → [inst : DecidableEq N] → [inst_1 : Fintype N] → {Ω : N → Type u_2} → [inst_2 : (n : N) → MeasurableSpace (Ω n)] → (M : Causalean.SCM N Ω) → M.isStandard → M.FixedValues","l":51,"f":"Causalean/SCM/ID/Query.lean"},{"n":"Causalean.SCM.ID.standardFixedValues_heq","k":"theorem","m":"Causalean.SCM.ID.GraphicalThms.IDAlgorithm","a":"SCM/ID/GraphicalThms","t":2,"d":"Two standard models with the same SWIG graph have the same canonical fixed-value assignment after type transport.","s":"∀ {N : Type u_1} [inst : DecidableEq N] [inst_1 : Fintype N] {Ω : N → Type u_2} [inst_2 : (n : N) → MeasurableSpace (Ω n)] (M₁ M₂ : Causalean.SCM N Ω) (hf : M₁.fixed = M₂.fixed) (h₁ : M₁.isStandard), Causalean.SCM.ID.standardFixedValues M₁ h₁ ≍ Causalean.SCM.ID.standardFixedValues M₂ (id (Eq.mpr (id (congrArg (fun _a => _a = ∅) (Eq.symm hf))) h₁))","l":46,"f":"Causalean/SCM/ID/GraphicalThms/IDAlgorithm.lean"},{"n":"Causalean.SCM.ID.subset_fixAncestralSet","k":"theorem","m":"Causalean.SCM.ID.GraphicalThms.DoGFormula","a":"SCM/ID/GraphicalThms","t":2,"d":"The post-intervention ancestral set contains the query.","s":"∀ {N : Type u_1} [inst : DecidableEq N] [inst_1 : Fintype N] {Ω : N → Type u_2} [inst_2 : (n : N) → MeasurableSpace (Ω n)] (M : Causalean.SCM N Ω) (X : Finset N) (hObs : ∀ D ∈ X, Causalean.SWIGNode.random D ∈ M.observed) (hFix : ∀ D ∈ X, Causalean.SWIGNode.fixed D ∉ M.fixed) (Y : Finset (Causalean.SWIGNode N)), Y ⊆ Causalean.SCM.ID.fixAncestralSet M X hObs hFix Y","l":116,"f":"Causalean/SCM/ID/GraphicalThms/DoGFormula.lean"},{"n":"Causalean.SCM.ID.subset_fixObservedAncestralSet","k":"theorem","m":"Causalean.SCM.ID.GraphicalThms.DoGFormula","a":"SCM/ID/GraphicalThms","t":1,"d":"For a valid intervention set X (observed, not already fixed) and an outcome set Y contained in the observed nodes, Y is contained in its own post-intervention observed-ancestral closure: it lies in the post-intervention ancestral set (subset_fixAncestralSet) and in the observed nodes.","s":"∀ {N : Type u_1} [inst : DecidableEq N] [inst_1 : Fintype N] {Ω : N → Type u_2} [inst_2 : (n : N) → MeasurableSpace (Ω n)] (M : Causalean.SCM N Ω) (X : Finset N) (hObs : ∀ D ∈ X, Causalean.SWIGNode.random D ∈ M.observed) (hFix : ∀ D ∈ X, Causalean.SWIGNode.fixed D ∉ M.fixed), ∀ Y ⊆ M.observed, Y ⊆ Causalean.SCM.ID.fixObservedAncestralSet M X hObs hFix Y","l":419,"f":"Causalean/SCM/ID/GraphicalThms/DoGFormula.lean"},{"n":"Causalean.SCM.ID.subset_inducedAncestral","k":"theorem","m":"Causalean.SCM.ID.Density.IdentifyMass","a":"SCM/ID/Density","t":2,"d":"If C is observed and contained in T, then it is contained in its induced ancestral set inside T.","s":"∀ {N : Type u_1} [inst : DecidableEq N] [inst_1 : Fintype N] (G : Causalean.SWIGGraph N) {T C : Finset (Causalean.SWIGNode N)}, C ⊆ T → C ⊆ G.observed → C ⊆ Causalean.SCM.ID.inducedAncestral G T C","l":78,"f":"Causalean/SCM/ID/Density/IdentifyMass.lean"},{"n":"Causalean.SCM.ID.swigValueNonempty","k":"def","m":"Causalean.SCM.ID.Query","a":"SCM/ID","t":1,"d":"Every random or fixed SWIG-node value space is nonempty when every base-node value space is nonempty.","s":"∀ {N : Type u_1} {Ω : N → Type u_2} [∀ (n : N), Nonempty (Ω n)] (w : Causalean.SWIGNode N), Nonempty (Causalean.swigΩ Ω w)","l":25,"f":"Causalean/SCM/ID/Query.lean"},{"n":"Causalean.SCM.ID.tianDensityProduct","k":"def","m":"Causalean.SCM.ID.Density.QFactor","a":"SCM/ID/Density","t":1,"d":"Tian's full prefix-chain density multiplies all one-node prefix conditional factors for a finite node set.","s":"{N : Type u_1} → [inst : DecidableEq N] → [inst_1 : Fintype N] → {Ω : N → Type u_2} → [inst_2 : (n : N) → MeasurableSpace (Ω n)] → (G : Causalean.SWIGGraph N) → (D : Finset (Causalean.SWIGNode N)) → (μ : MeasureTheory.Measure (Causalean.ValuesOn D (Causalean.swigΩ Ω))) → Causalean.SCM.ReferenceMeasures Ω → [MeasureTheory.IsFiniteMeasure μ] → [∀ (k : ℕ) (hk : k < D.card), StandardBorelSpace (Causalean.ValuesOn {↑(G.nodesAt D ⟨k, hk⟩)} (Causalean.swigΩ Ω))] → [∀ (k : ℕ) (hk : k < D.card), Nonempty (Causalean.ValuesOn {↑(G.nodesAt D ⟨k, hk⟩)} (Causalean.swigΩ Ω))] → Causalean.ValuesOn D (Causalean.swigΩ Ω) → ENNReal","l":195,"f":"Causalean/SCM/ID/Density/QFactor.lean"},{"n":"Causalean.SCM.ID.tianDistrictDensity","k":"def","m":"Causalean.SCM.ID.Density.QFactor","a":"SCM/ID/Density","t":1,"d":"Tian's district density multiplies the prefix conditional factors whose nodes lie in one district.","s":"{N : Type u_1} → [inst : DecidableEq N] → [inst_1 : Fintype N] → {Ω : N → Type u_2} → [inst_2 : (n : N) → MeasurableSpace (Ω n)] → (G : Causalean.SWIGGraph N) → (D : Finset (Causalean.SWIGNode N)) → (μ : MeasureTheory.Measure (Causalean.ValuesOn D (Causalean.swigΩ Ω))) → Causalean.SCM.ReferenceMeasures Ω → Finset (Causalean.SWIGNode N) → [MeasureTheory.IsFiniteMeasure μ] → [∀ (k : ℕ) (hk : k < D.card), StandardBorelSpace (Causalean.ValuesOn {↑(G.nodesAt D ⟨k, hk⟩)} (Causalean.swigΩ Ω))] → [∀ (k : ℕ) (hk : k < D.card), Nonempty (Causalean.ValuesOn {↑(G.nodesAt D ⟨k, hk⟩)} (Causalean.swigΩ Ω))] → Causalean.ValuesOn D (Causalean.swigΩ Ω) → ENNReal","l":172,"f":"Causalean/SCM/ID/Density/QFactor.lean"},{"n":"Causalean.SCM.ID.tianDistrictDensity_eq_mechCFactor_doModel","k":"theorem","m":"Causalean.SCM.ID.Density.QFactor","a":"SCM/ID/Density","t":1,"d":"(B) Tian Lemma 1 for the do-model ancestral marginal. Consider a structural causal model M with an intervention set X whose random copies are observed and whose fixed copies are not already frozen, an outcome set Y disjoint from the random copies of X, under a faithful reference-measure family, a positive observational kernel at every fixed-value assignment, and the standing assumption that M is a standard model. For a set S that is simultaneously a district of the post-intervention ancestral graph and a full c-component of M, and an extension map that inverts the projection onto the ancestral observed coordinates, the Tian district density read off the do(X)-law ancestral marginal at S agrees with the mechanism c-factor of the post-intervention model at the extended point.","s":"∀ {N : Type u_1} [inst : DecidableEq N] [inst_1 : Fintype N] {Ω : N → Type u_2} [inst_2 : (n : N) → MeasurableSpace (Ω n)] [∀ (n : N), Nonempty (Ω n)] [(n : N) → Fintype (Ω n)] [∀ (n : N), MeasurableSingletonClass (Ω n)] (M : Causalean.SCM N Ω) (X : Finset N) (hObs : ∀ D ∈ X, Causalean.SWIGNode.random D ∈ M.observed) (hFix : ∀ D ∈ X, Causalean.SWIGNode.fixed D ∉ M.fixed) (Y : Finset (Causalean.SWIGNode N)) (ref : Causalean.SCM.ReferenceMeasures Ω), Causalean.SCM.ReferenceFaithful ref → ∀ (sDo : (M.fixSet X hObs hFix).FixedValues), (∀ (s' : M.FixedValues), Causalean.SCM.ID.DiscreteID.PositiveMass (M.obsKernel s')) → (∀ d ∈ X, Causalean.SWIGNode.random d ∉ Y) → M.isStandard → ∀ S ∈ Causalean.SCM.ID.fixTruncCComponentSet M X hObs hFix Y, ∀ (hSfull : S ∈ M.cComponentSet) [inst_6 : MeasureTheory.IsFiniteMeasure ((Causalean.SCM.ID.doObsKernelAncestralMarginal M X hObs hFix Y) sDo)] [inst_7 : ∀ (k : ℕ) (hk : k < (Causalean.SCM.ID.fixObservedAncestralSet M X hObs hFix Y).card), StandardBorelSpace (Causalean.ValuesOn {↑(((M.fixSet X hObs hFix).induce (Causalean.SCM.ID.fixAncestralSet M X hObs hFix Y)).nodesAt (Causalean.SCM.ID.fixObservedAncestralSet M X hObs hFix Y) ⟨k, hk⟩)} (Causalean.swigΩ Ω))] [inst_8 : ∀ (k : ℕ) (hk : k < (Causalean.SCM.ID.fixObservedAncestralSet M X hObs hFix Y).card), Nonempty (Causalean.ValuesOn {↑(((M.fixSet X hObs hFix).induce (Causalean.SCM.ID.fixAncestralSet M X hObs hFix Y)).nodesAt (Causalean.SCM.ID.fixObservedAncestralSet M X hObs hFix Y) ⟨k, hk⟩)} (Causalean.swigΩ Ω))] (extend : Causalean.ValuesOn (Causalean.SCM.ID.fixObservedAncestralSet M X hObs hFix Y) (Causalean.swigΩ Ω) → Causalean.ValuesOn M.observed (Causalean.swigΩ Ω)), (∀ (xD : Causalean.ValuesOn (Causalean.SCM.ID.fixObservedAncestralSet M X hObs hFix Y) (Causalean.swigΩ Ω)), Causalean.valuesProjection (have this := Finset.inter_subset_right; this) (extend xD) = xD) → ∀ (xD : Causalean.ValuesOn (Causalean.SCM.ID.fixObservedAncestralSet M X hObs hFix Y) (Causalean.swigΩ Ω)), Causalean.SCM.ID.tianDistrictDensity ((M.fixSet X hObs hFix).induce (Causalean.SCM.ID.fixAncestralSet M X hObs hFix Y)) (Causalean.SCM.ID.fixObservedAncestralSet M X hObs hFix Y) ((Causalean.SCM.ID.doObsKernelAncestralMarginal M X hObs hFix Y) sDo) ref S xD = (M.fixSet X hObs hFix).mechCFactor ref S (have hSobs := Causalean.SWIGGraph.cComponentSet_subset_observed M.toSWIGGraph S hSfull; Eq.mpr (id ge_iff_le._simp_1) hSobs) sDo (extend xD)","l":1678,"f":"Causalean/SCM/ID/Density/QFactor.lean"},{"n":"Causalean.SCM.ID.tianDistrictDensity_eq_qLocalMass_div_jointRef_district","k":"theorem","m":"Causalean.SCM.ID.Density.QFactor","a":"SCM/ID/Density","t":1,"d":"Consider an intervention set X whose random copies are observed and whose fixed copies are not already frozen together with an outcome set Y disjoint from the random copies of X, under a faithful reference-measure family and a positive observational kernel at every fixed-value assignment. For any district S of the post-intervention ancestral graph and an extension map inverting the projection onto the ancestral observed coordinates, the Tian district density read off the do(X)-law ancestral marginal at S equals the do-model local q-mass on S divided by the reference atom mass of S.","s":"∀ {N : Type u_1} [inst : DecidableEq N] [inst_1 : Fintype N] {Ω : N → Type u_2} [inst_2 : (n : N) → MeasurableSpace (Ω n)] [∀ (n : N), Nonempty (Ω n)] [(n : N) → Fintype (Ω n)] [∀ (n : N), MeasurableSingletonClass (Ω n)] (M : Causalean.SCM N Ω) (X : Finset N) (hObs : ∀ D ∈ X, Causalean.SWIGNode.random D ∈ M.observed) (hFix : ∀ D ∈ X, Causalean.SWIGNode.fixed D ∉ M.fixed) (Y : Finset (Causalean.SWIGNode N)) (ref : Causalean.SCM.ReferenceMeasures Ω), Causalean.SCM.ReferenceFaithful ref → ∀ (sDo : (M.fixSet X hObs hFix).FixedValues), (∀ (s' : M.FixedValues), Causalean.SCM.ID.DiscreteID.PositiveMass (M.obsKernel s')) → (∀ d ∈ X, Causalean.SWIGNode.random d ∉ Y) → ∀ (S : Finset (Causalean.SWIGNode N)) (hS : S ∈ Causalean.SCM.ID.fixTruncCComponentSet M X hObs hFix Y) [inst_6 : MeasureTheory.IsFiniteMeasure ((Causalean.SCM.ID.doObsKernelAncestralMarginal M X hObs hFix Y) sDo)] [inst_7 : ∀ (k : ℕ) (hk : k < (Causalean.SCM.ID.fixObservedAncestralSet M X hObs hFix Y).card), StandardBorelSpace (Causalean.ValuesOn {↑(((M.fixSet X hObs hFix).induce (Causalean.SCM.ID.fixAncestralSet M X hObs hFix Y)).nodesAt (Causalean.SCM.ID.fixObservedAncestralSet M X hObs hFix Y) ⟨k, hk⟩)} (Causalean.swigΩ Ω))] [inst_8 : ∀ (k : ℕ) (hk : k < (Causalean.SCM.ID.fixObservedAncestralSet M X hObs hFix Y).card), Nonempty (Causalean.ValuesOn {↑(((M.fixSet X hObs hFix).induce (Causalean.SCM.ID.fixAncestralSet M X hObs hFix Y)).nodesAt (Causalean.SCM.ID.fixObservedAncestralSet M X hObs hFix Y) ⟨k, hk⟩)} (Causalean.swigΩ Ω))] (extend : Causalean.ValuesOn (Causalean.SCM.ID.fixObservedAncestralSet M X hObs hFix Y) (Causalean.swigΩ Ω) → Causalean.ValuesOn M.observed (Causalean.swigΩ Ω)), (∀ (xD : Causalean.ValuesOn (Causalean.SCM.ID.fixObservedAncestralSet M X hObs hFix Y) (Causalean.swigΩ Ω)), Causalean.valuesProjection (have this := Finset.inter_subset_right; this) (extend xD) = xD) → ∀ (xD : Causalean.ValuesOn (Causalean.SCM.ID.fixObservedAncestralSet M X hObs hFix Y) (Causalean.swigΩ Ω)), let MX := M.fixSet X hObs hFix; let D := Causalean.SCM.ID.fixObservedAncestralSet M X hObs hFix Y; let H := MX.induce (Causalean.SCM.ID.fixAncestralSet M X hObs hFix Y); Causalean.SCM.ID.tianDistrictDensity H D ((Causalean.SCM.ID.doObsKernelAncestralMarginal M X hObs hFix Y) sDo) ref S xD = MX.qLocalMass sDo S (have this := fun ⦃v⦄ hv => have hSHobs := Causalean.SWIGGraph.cComponentSet_subset_observed H S (id hS); have hvD := id (hSHobs hv); (Finset.mem_inter.mp hvD).right; this) (extend xD) / (Causalean.SCM.jointRef ref S) {Causalean.valuesProjection (have this := fun ⦃v⦄ hv => have hSHobs := Causalean.SWIGGraph.cComponentSet_subset_observed H S (id hS); id (hSHobs hv); this) xD}","l":1949,"f":"Causalean/SCM/ID/Density/QFactor.lean"},{"n":"Causalean.SCM.ID.tianPrefixDensityProductInPrefix","k":"def","m":"Causalean.SCM.ID.GraphicalThms.DoGFormulaTian","a":"SCM/ID/GraphicalThms","t":1,"d":"Recursive Tian prefix density product on an intermediate prefix.","s":"{N : Type u_1} → [inst : DecidableEq N] → [inst_1 : Fintype N] → {Ω : N → Type u_2} → [inst_2 : (n : N) → MeasurableSpace (Ω n)] → (H : Causalean.SWIGGraph N) → (D : Finset (Causalean.SWIGNode N)) → (μ : MeasureTheory.Measure (Causalean.ValuesOn D (Causalean.swigΩ Ω))) → Causalean.SCM.ReferenceMeasures Ω → [MeasureTheory.IsFiniteMeasure μ] → [∀ (k : ℕ) (hk : k < D.card), StandardBorelSpace (Causalean.ValuesOn {↑(H.nodesAt D ⟨k, hk⟩)} (Causalean.swigΩ Ω))] → [∀ (k : ℕ) (hk : k < D.card), Nonempty (Causalean.ValuesOn {↑(H.nodesAt D ⟨k, hk⟩)} (Causalean.swigΩ Ω))] → (k : ℕ) → Causalean.ValuesOn (H.prefixIn D k) (Causalean.swigΩ Ω) → ENNReal","l":305,"f":"Causalean/SCM/ID/GraphicalThms/DoGFormulaTian.lean"},{"n":"Causalean.SCM.ID.tianPrefixDensityProductInPrefix_card_eq_tianDensityProduct","k":"theorem","m":"Causalean.SCM.ID.GraphicalThms.DoGFormulaTian","a":"SCM/ID/GraphicalThms","t":2,"d":"At the full D prefix, the recursive prefix density product is Tian's finite product over all D indices.","s":"∀ {N : Type u_1} [inst : DecidableEq N] [inst_1 : Fintype N] {Ω : N → Type u_2} [inst_2 : (n : N) → MeasurableSpace (Ω n)] (H : Causalean.SWIGGraph N) (D : Finset (Causalean.SWIGNode N)) (μ : MeasureTheory.Measure (Causalean.ValuesOn D (Causalean.swigΩ Ω))) (ref : Causalean.SCM.ReferenceMeasures Ω) [inst_3 : MeasureTheory.IsFiniteMeasure μ] [inst_4 : ∀ (k : ℕ) (hk : k < D.card), StandardBorelSpace (Causalean.ValuesOn {↑(H.nodesAt D ⟨k, hk⟩)} (Causalean.swigΩ Ω))] [inst_5 : ∀ (k : ℕ) (hk : k < D.card), Nonempty (Causalean.ValuesOn {↑(H.nodesAt D ⟨k, hk⟩)} (Causalean.swigΩ Ω))] (y : Causalean.ValuesOn (H.prefixIn D D.card) (Causalean.swigΩ Ω)), Causalean.SCM.ID.tianPrefixDensityProductInPrefix H D μ ref D.card y = Causalean.SCM.ID.tianDensityProduct H D μ ref ((Causalean.valuesEquivOfEq (Causalean.SCM.ID.prefixIn_card H D)) y)","l":447,"f":"Causalean/SCM/ID/GraphicalThms/DoGFormulaTian.lean"},{"n":"Causalean.SCM.ID.tianPrefixDensityProductInPrefix_eq_range_product","k":"theorem","m":"Causalean.SCM.ID.GraphicalThms.DoGFormulaTian","a":"SCM/ID/GraphicalThms","t":2,"d":"The recursive prefix density product is the range product of its one-step factors.","s":"∀ {N : Type u_1} [inst : DecidableEq N] [inst_1 : Fintype N] {Ω : N → Type u_2} [inst_2 : (n : N) → MeasurableSpace (Ω n)] (H : Causalean.SWIGGraph N) (D : Finset (Causalean.SWIGNode N)) (μ : MeasureTheory.Measure (Causalean.ValuesOn D (Causalean.swigΩ Ω))) (ref : Causalean.SCM.ReferenceMeasures Ω) [inst_3 : MeasureTheory.IsFiniteMeasure μ] [inst_4 : ∀ (k : ℕ) (hk : k < D.card), StandardBorelSpace (Causalean.ValuesOn {↑(H.nodesAt D ⟨k, hk⟩)} (Causalean.swigΩ Ω))] [inst_5 : ∀ (k : ℕ) (hk : k < D.card), Nonempty (Causalean.ValuesOn {↑(H.nodesAt D ⟨k, hk⟩)} (Causalean.swigΩ Ω))], ∀ k ≤ D.card, ∀ (z : Causalean.ValuesOn (H.prefixIn D k) (Causalean.swigΩ Ω)), Causalean.SCM.ID.tianPrefixDensityProductInPrefix H D μ ref k z = ∏ i ∈ Finset.range k, Causalean.SCM.ID.tianPrefixStepDensityInPrefix H D μ ref k z i","l":384,"f":"Causalean/SCM/ID/GraphicalThms/DoGFormulaTian.lean"},{"n":"Causalean.SCM.ID.tianPrefixStepDensity","k":"def","m":"Causalean.SCM.ID.Density.QFactor","a":"SCM/ID/Density","t":1,"d":"This density is Tian's one-node conditional factor along the topological prefix of a finite node set.","s":"{N : Type u_1} → [inst : DecidableEq N] → [inst_1 : Fintype N] → {Ω : N → Type u_2} → [inst_2 : (n : N) → MeasurableSpace (Ω n)] → (G : Causalean.SWIGGraph N) → (D : Finset (Causalean.SWIGNode N)) → (μ : MeasureTheory.Measure (Causalean.ValuesOn D (Causalean.swigΩ Ω))) → Causalean.SCM.ReferenceMeasures Ω → (i : Fin D.card) → [MeasureTheory.IsFiniteMeasure μ] → [StandardBorelSpace (Causalean.ValuesOn {↑(G.nodesAt D i)} (Causalean.swigΩ Ω))] → [Nonempty (Causalean.ValuesOn {↑(G.nodesAt D i)} (Causalean.swigΩ Ω))] → Causalean.ValuesOn D (Causalean.swigΩ Ω) → ENNReal","l":139,"f":"Causalean/SCM/ID/Density/QFactor.lean"},{"n":"Causalean.SCM.ID.tianPrefixStepDensityInPrefix","k":"def","m":"Causalean.SCM.ID.GraphicalThms.DoGFormulaTian","a":"SCM/ID/GraphicalThms","t":1,"d":"One-step Tian density read from a k-prefix assignment.","s":"{N : Type u_1} → [inst : DecidableEq N] → [inst_1 : Fintype N] → {Ω : N → Type u_2} → [inst_2 : (n : N) → MeasurableSpace (Ω n)] → (H : Causalean.SWIGGraph N) → (D : Finset (Causalean.SWIGNode N)) → (μ : MeasureTheory.Measure (Causalean.ValuesOn D (Causalean.swigΩ Ω))) → Causalean.SCM.ReferenceMeasures Ω → [MeasureTheory.IsFiniteMeasure μ] → [∀ (i : ℕ) (hi : i < D.card), StandardBorelSpace (Causalean.ValuesOn {↑(H.nodesAt D ⟨i, hi⟩)} (Causalean.swigΩ Ω))] → [∀ (i : ℕ) (hi : i < D.card), Nonempty (Causalean.ValuesOn {↑(H.nodesAt D ⟨i, hi⟩)} (Causalean.swigΩ Ω))] → (k : ℕ) → Causalean.ValuesOn (H.prefixIn D k) (Causalean.swigΩ Ω) → ℕ → ENNReal","l":345,"f":"Causalean/SCM/ID/GraphicalThms/DoGFormulaTian.lean"},{"n":"Causalean.SCM.ID.tianPrefixStepDensityInPrefix_eq_mass_ratio","k":"theorem","m":"Causalean.SCM.ID.Density.TianMassBridge","a":"SCM/ID/Density","t":1,"d":"Fix a step index i earlier than the prefix length k that also lies within the node domain D. Provided the conditional distribution of the i-th coordinate given the shorter prefix is absolutely continuous with respect to the reference measure on that coordinate, and the reference-measure mass of the singleton value at that coordinate is nonzero and finite, then the Tian prefix-step density at index i equals the prefix conditional singleton mass divided by the corresponding singleton mass of the product reference measure.","s":"∀ {N : Type u_1} [inst : DecidableEq N] [inst_1 : Fintype N] {Ω : N → Type u_2} [inst_2 : (n : N) → MeasurableSpace (Ω n)] (H : Causalean.SWIGGraph N) (D : Finset (Causalean.SWIGNode N)) (μ : MeasureTheory.Measure (Causalean.ValuesOn D (Causalean.swigΩ Ω))) (ref : Causalean.SCM.ReferenceMeasures Ω) [inst_3 : MeasureTheory.IsFiniteMeasure μ] [inst_4 : ∀ (j : ℕ) (hj : j < D.card), StandardBorelSpace (Causalean.ValuesOn {↑(H.nodesAt D ⟨j, hj⟩)} (Causalean.swigΩ Ω))] [inst_5 : ∀ (j : ℕ) (hj : j < D.card), Nonempty (Causalean.ValuesOn {↑(H.nodesAt D ⟨j, hj⟩)} (Causalean.swigΩ Ω))] (k i : ℕ) (hi : i < k) (hcard : i < D.card) [MeasurableSingletonClass (Causalean.ValuesOn {↑(H.nodesAt D ⟨i, hcard⟩)} (Causalean.swigΩ Ω))] (z : Causalean.ValuesOn (H.prefixIn D k) (Causalean.swigΩ Ω)), ((ProbabilityTheory.condDistrib (Causalean.valuesProjection (have this := fun ⦃v⦄ hv => Eq.symm (Eq.mp (congrArg (fun _a => _a) (propext Finset.mem_singleton)) hv) ▸ (H.nodesAt D ⟨i, hcard⟩).property; this)) (Causalean.valuesProjection (Causalean.SWIGGraph.prefixIn_subset H D i)) μ) (Causalean.valuesProjection (Causalean.SCM.ID.prefixIn_mono H D (Nat.le_of_lt hi)) z)).AbsolutelyContinuous (Causalean.SCM.jointRef ref {↑(H.nodesAt D ⟨i, hcard⟩)}) → (Causalean.SCM.jointRef ref {↑(H.nodesAt D ⟨i, hcard⟩)}) {Causalean.valuesProjection (have this := fun ⦃v⦄ hv => Eq.symm (Eq.mp (congrArg (fun _a => _a) (propext Finset.mem_singleton)) hv) ▸ Eq.mpr (id (congrArg (fun _a => _a) (propext (Causalean.SCM.ID.nodesAt_mem_prefixIn_iff H D k ⟨i, hcard⟩)))) hi; this) z} ≠ 0 → (Causalean.SCM.jointRef ref {↑(H.nodesAt D ⟨i, hcard⟩)}) {Causalean.valuesProjection (have this := fun ⦃v⦄ hv => Eq.symm (Eq.mp (congrArg (fun _a => _a) (propext Finset.mem_singleton)) hv) ▸ Eq.mpr (id (congrArg (fun _a => _a) (propext (Causalean.SCM.ID.nodesAt_mem_prefixIn_iff H D k ⟨i, hcard⟩)))) hi; this) z} ≠ ⊤ → Causalean.SCM.ID.tianPrefixStepDensityInPrefix H D μ ref k z i = ((ProbabilityTheory.condDistrib (Causalean.valuesProjection (have this := fun ⦃v⦄ hv => Eq.symm (Eq.mp (congrArg (fun _a => _a) (propext Finset.mem_singleton)) hv) ▸ (H.nodesAt D ⟨i, hcard⟩).property; this)) (Causalean.valuesProjection (Causalean.SWIGGraph.prefixIn_subset H D i)) μ) (Causalean.valuesProjection (Causalean.SCM.ID.prefixIn_mono H D (Nat.le_of_lt hi)) z)) {Causalean.valuesProjection (have this := fun ⦃v⦄ hv => Eq.symm (Eq.mp (congrArg (fun _a => _a) (propext Finset.mem_singleton)) hv) ▸ Eq.mpr (id (congrArg (fun _a => _a) (propext (Causalean.SCM.ID.nodesAt_mem_prefixIn_iff H D k ⟨i, hcard⟩)))) hi; this) z} / (Causalean.SCM.jointRef ref {↑(H.nodesAt D ⟨i, hcard⟩)}) {Causalean.valuesProjection (have this := fun ⦃v⦄ hv => Eq.symm (Eq.mp (congrArg (fun _a => _a) (propext Finset.mem_singleton)) hv) ▸ Eq.mpr (id (congrArg (fun _a => _a) (propext (Causalean.SCM.ID.nodesAt_mem_prefixIn_iff H D k ⟨i, hcard⟩)))) hi; this) z}","l":35,"f":"Causalean/SCM/ID/Density/TianMassBridge.lean"},{"n":"Causalean.SCM.ID.tianPrefixStepDensity_eq_prefix_mass_ratio","k":"theorem","m":"Causalean.SCM.ID.Density.QFactor","a":"SCM/ID/Density","t":2,"d":"When the preceding prefix has nonzero singleton mass, the one-step Tian density is the ratio of the successive prefix singleton masses, divided by the singleton reference mass of the added variable.","s":"∀ {N : Type u_1} [inst : DecidableEq N] [inst_1 : Fintype N] {Ω : N → Type u_2} [inst_2 : (n : N) → MeasurableSpace (Ω n)] [(n : N) → Fintype (Ω n)] [∀ (n : N), MeasurableSingletonClass (Ω n)] (H : Causalean.SWIGGraph N) (D : Finset (Causalean.SWIGNode N)) (μ : MeasureTheory.Measure (Causalean.ValuesOn D (Causalean.swigΩ Ω))) (ref : Causalean.SCM.ReferenceMeasures Ω), Causalean.SCM.ReferenceFaithful ref → ∀ [inst_5 : MeasureTheory.IsFiniteMeasure μ] [inst_6 : ∀ (k : ℕ) (hk : k < D.card), StandardBorelSpace (Causalean.ValuesOn {↑(H.nodesAt D ⟨k, hk⟩)} (Causalean.swigΩ Ω))] [inst_7 : ∀ (k : ℕ) (hk : k < D.card), Nonempty (Causalean.ValuesOn {↑(H.nodesAt D ⟨k, hk⟩)} (Causalean.swigΩ Ω))] (i : Fin D.card) (x : Causalean.ValuesOn D (Causalean.swigΩ Ω)), (MeasureTheory.Measure.map (Causalean.valuesProjection (Causalean.SWIGGraph.prefixIn_subset H D ↑i)) μ) {Causalean.valuesProjection (Causalean.SWIGGraph.prefixIn_subset H D ↑i) x} ≠ 0 → Causalean.SCM.ID.tianPrefixStepDensity H D μ ref i x = (MeasureTheory.Measure.map (Causalean.valuesProjection (Causalean.SWIGGraph.prefixIn_subset H D (↑i + 1))) μ) {Causalean.valuesProjection (Causalean.SWIGGraph.prefixIn_subset H D (↑i + 1)) x} / (MeasureTheory.Measure.map (Causalean.valuesProjection (Causalean.SWIGGraph.prefixIn_subset H D ↑i)) μ) {Causalean.valuesProjection (Causalean.SWIGGraph.prefixIn_subset H D ↑i) x} / (Causalean.SCM.jointRef ref {↑(H.nodesAt D i)}) {Causalean.valuesProjection (have this := fun ⦃v⦄ hv => Eq.symm (Eq.mp (congrArg (fun _a => _a) (propext Finset.mem_singleton)) hv) ▸ (H.nodesAt D i).property; this) x}","l":1324,"f":"Causalean/SCM/ID/Density/QFactor.lean"},{"n":"Causalean.SCM.ID.tian_full_cComponent_density_recovery_core","k":"theorem","m":"Causalean.SCM.ID.GraphicalThms.DoGFormulaTian","a":"SCM/ID/GraphicalThms","t":2,"d":"Core Tian c-factor recovery for a full observational c-component.","s":"∀ {N : Type u_1} [inst : DecidableEq N] [inst_1 : Fintype N] {Ω : N → Type u_2} [inst_2 : (n : N) → MeasurableSpace (Ω n)] [∀ (n : N), Nonempty (Ω n)] [(n : N) → Fintype (Ω n)] [∀ (n : N), MeasurableSingletonClass (Ω n)] (M : Causalean.SCM N Ω) (X : Finset N), M.isStandard → ∀ (hObs : ∀ D ∈ X, Causalean.SWIGNode.random D ∈ M.observed) (hFix : ∀ D ∈ X, Causalean.SWIGNode.fixed D ∉ M.fixed) (Y : Finset (Causalean.SWIGNode N)) (ref : Causalean.SCM.ReferenceMeasures Ω), Causalean.SCM.ReferenceFaithful ref → ∀ (sDo : (M.fixSet X hObs hFix).FixedValues), ∀ S ∈ Causalean.SCM.ID.fixTruncCComponentSet M X hObs hFix Y, S ∈ M.cComponentSet → ∀ [inst_6 : MeasureTheory.IsFiniteMeasure ((Causalean.SCM.ID.doObsKernelAncestralMarginal M X hObs hFix Y) sDo)] [inst_7 : ∀ (s' : M.FixedValues), MeasureTheory.IsFiniteMeasure (M.obsKernel s')] [inst_8 : ∀ (k : ℕ) (hk : k < (Causalean.SCM.ID.fixObservedAncestralSet M X hObs hFix Y).card), StandardBorelSpace (Causalean.ValuesOn {↑(((M.fixSet X hObs hFix).induce (Causalean.SCM.ID.fixAncestralSet M X hObs hFix Y)).nodesAt (Causalean.SCM.ID.fixObservedAncestralSet M X hObs hFix Y) ⟨k, hk⟩)} (Causalean.swigΩ Ω))] [inst_9 : ∀ (k : ℕ) (hk : k < (Causalean.SCM.ID.fixObservedAncestralSet M X hObs hFix Y).card), Nonempty (Causalean.ValuesOn {↑(((M.fixSet X hObs hFix).induce (Causalean.SCM.ID.fixAncestralSet M X hObs hFix Y)).nodesAt (Causalean.SCM.ID.fixObservedAncestralSet M X hObs hFix Y) ⟨k, hk⟩)} (Causalean.swigΩ Ω))] [inst_10 : ∀ (k : ℕ) (hk : k < M.observed.card), StandardBorelSpace (Causalean.ValuesOn {↑(M.observedAt ⟨k, hk⟩)} (Causalean.swigΩ Ω))] [inst_11 : ∀ (k : ℕ) (hk : k < M.observed.card), Nonempty (Causalean.ValuesOn {↑(M.observedAt ⟨k, hk⟩)} (Causalean.swigΩ Ω))] [inst_12 : ∀ (k : ℕ), MeasurableSpace.CountableOrCountablyGenerated M.FixedValues (Causalean.ValuesOn (M.prefixNodes k) (Causalean.swigΩ Ω))], (∀ (s' : M.FixedValues), Causalean.SCM.ID.DiscreteID.PositiveMass (M.obsKernel s')) → (∀ D ∈ X, Causalean.SWIGNode.random D ∉ Y) → ∀ (extend : Causalean.ValuesOn (Causalean.SCM.ID.fixObservedAncestralSet M X hObs hFix Y) (Causalean.swigΩ Ω) → Causalean.ValuesOn M.observed (Causalean.swigΩ Ω)), (∀ (xD : Causalean.ValuesOn (Causalean.SCM.ID.fixObservedAncestralSet M X hObs hFix Y) (Causalean.swigΩ Ω)), Causalean.valuesProjection (have this := Finset.inter_subset_right; this) (extend xD) = xD) → (∀ (xD : Causalean.ValuesOn (Causalean.SCM.ID.fixObservedAncestralSet M X hObs hFix Y) (Causalean.swigΩ Ω)) (D : N) (hD : D ∈ X), extend xD ⟨Causalean.SWIGNode.random D, hObs D hD⟩ = sDo ⟨Causalean.SWIGNode.fixed D, Finset.mem_union_right M.fixed (Finset.mem_image.mpr (Exists.intro D ⟨hD, rfl⟩))⟩) → let D := Causalean.SCM.ID.fixObservedAncestralSet M X hObs hFix Y; let H := (M.fixSet X hObs hFix).induce (Causalean.SCM.ID.fixAncestralSet M X hObs hFix Y); Causalean.SCM.ID.tianDistrictDensity H D ((Causalean.SCM.ID.doObsKernelAncestralMarginal M X hObs hFix Y) sDo) ref S =ᵐ[Causalean.SCM.jointRef ref D] fun xD => M.cComponentDensityFactor ref (M.fixSetProj X hObs hFix sDo) S (extend xD)","l":895,"f":"Causalean/SCM/ID/GraphicalThms/DoGFormulaTian.lean"},{"n":"Causalean.SCM.ID.tian_full_cComponent_density_recovery_core_direct","k":"theorem","m":"Causalean.SCM.ID.Density.QFactor","a":"SCM/ID/Density","t":1,"d":"District recovery (Tian Lemma 4 projection consistency). Consider a standard structural causal model M with an intervention set X whose random copies are observed and whose fixed copies are not already frozen, under a faithful reference-measure family and a positive observational kernel at every fixed-value assignment, for an outcome set Y disjoint from the random copies of X. For a set S that is simultaneously a district of the post-intervention ancestral graph and a full c-component of M, and an extension map that inverts the projection onto the ancestral observed coordinates and reproduces the intervention values sDo on the intervened coordinates, the Tian district density read off the do(X)-law ancestral marginal at S agrees, almost everywhere with respect to the product reference measure on the ancestral observed coordinates, with the observational c-component density factor at S pulled back through the extension.","s":"∀ {N : Type u_1} [inst : DecidableEq N] [inst_1 : Fintype N] {Ω : N → Type u_2} [inst_2 : (n : N) → MeasurableSpace (Ω n)] [∀ (n : N), Nonempty (Ω n)] [(n : N) → Fintype (Ω n)] [∀ (n : N), MeasurableSingletonClass (Ω n)] (M : Causalean.SCM N Ω) (X : Finset N), M.isStandard → ∀ (hObs : ∀ D ∈ X, Causalean.SWIGNode.random D ∈ M.observed) (hFix : ∀ D ∈ X, Causalean.SWIGNode.fixed D ∉ M.fixed) (Y : Finset (Causalean.SWIGNode N)) (ref : Causalean.SCM.ReferenceMeasures Ω), Causalean.SCM.ReferenceFaithful ref → ∀ (sDo : (M.fixSet X hObs hFix).FixedValues), ∀ S ∈ Causalean.SCM.ID.fixTruncCComponentSet M X hObs hFix Y, S ∈ M.cComponentSet → ∀ [inst_6 : MeasureTheory.IsFiniteMeasure ((Causalean.SCM.ID.doObsKernelAncestralMarginal M X hObs hFix Y) sDo)] [inst_7 : ∀ (s' : M.FixedValues), MeasureTheory.IsFiniteMeasure (M.obsKernel s')] [inst_8 : ∀ (k : ℕ) (hk : k < (Causalean.SCM.ID.fixObservedAncestralSet M X hObs hFix Y).card), StandardBorelSpace (Causalean.ValuesOn {↑(((M.fixSet X hObs hFix).induce (Causalean.SCM.ID.fixAncestralSet M X hObs hFix Y)).nodesAt (Causalean.SCM.ID.fixObservedAncestralSet M X hObs hFix Y) ⟨k, hk⟩)} (Causalean.swigΩ Ω))] [inst_9 : ∀ (k : ℕ) (hk : k < (Causalean.SCM.ID.fixObservedAncestralSet M X hObs hFix Y).card), Nonempty (Causalean.ValuesOn {↑(((M.fixSet X hObs hFix).induce (Causalean.SCM.ID.fixAncestralSet M X hObs hFix Y)).nodesAt (Causalean.SCM.ID.fixObservedAncestralSet M X hObs hFix Y) ⟨k, hk⟩)} (Causalean.swigΩ Ω))] [inst_10 : ∀ (k : ℕ) (hk : k < M.observed.card), StandardBorelSpace (Causalean.ValuesOn {↑(M.observedAt ⟨k, hk⟩)} (Causalean.swigΩ Ω))] [inst_11 : ∀ (k : ℕ) (hk : k < M.observed.card), Nonempty (Causalean.ValuesOn {↑(M.observedAt ⟨k, hk⟩)} (Causalean.swigΩ Ω))] [inst_12 : ∀ (k : ℕ), MeasurableSpace.CountableOrCountablyGenerated M.FixedValues (Causalean.ValuesOn (M.prefixNodes k) (Causalean.swigΩ Ω))], (∀ (s' : M.FixedValues), Causalean.SCM.ID.DiscreteID.PositiveMass (M.obsKernel s')) → (∀ D ∈ X, Causalean.SWIGNode.random D ∉ Y) → ∀ (extend : Causalean.ValuesOn (Causalean.SCM.ID.fixObservedAncestralSet M X hObs hFix Y) (Causalean.swigΩ Ω) → Causalean.ValuesOn M.observed (Causalean.swigΩ Ω)), (∀ (xD : Causalean.ValuesOn (Causalean.SCM.ID.fixObservedAncestralSet M X hObs hFix Y) (Causalean.swigΩ Ω)), Causalean.valuesProjection (have this := Finset.inter_subset_right; this) (extend xD) = xD) → (∀ (xD : Causalean.ValuesOn (Causalean.SCM.ID.fixObservedAncestralSet M X hObs hFix Y) (Causalean.swigΩ Ω)) (D : N) (hD : D ∈ X), extend xD ⟨Causalean.SWIGNode.random D, hObs D hD⟩ = sDo ⟨Causalean.SWIGNode.fixed D, Finset.mem_union_right M.fixed (Finset.mem_image.mpr (Exists.intro D ⟨hD, rfl⟩))⟩) → let D := Causalean.SCM.ID.fixObservedAncestralSet M X hObs hFix Y; let H := (M.fixSet X hObs hFix).induce (Causalean.SCM.ID.fixAncestralSet M X hObs hFix Y); Causalean.SCM.ID.tianDistrictDensity H D ((Causalean.SCM.ID.doObsKernelAncestralMarginal M X hObs hFix Y) sDo) ref S =ᵐ[Causalean.SCM.jointRef ref D] fun xD => M.cComponentDensityFactor ref (M.fixSetProj X hObs hFix sDo) S (extend xD)","l":2222,"f":"Causalean/SCM/ID/Density/QFactor.lean"},{"n":"Causalean.SCM.ID.valuesOnNonempty","k":"def","m":"Causalean.SCM.ID.Query","a":"SCM/ID","t":1,"d":"A finite coordinate product of SWIG-node value spaces is nonempty when all base-node value spaces are nonempty.","s":"∀ {N : Type u_1} {Ω : N → Type u_2} [∀ (n : N), Nonempty (Ω n)] (Y : Finset (Causalean.SWIGNode N)), Nonempty (Causalean.ValuesOn Y (Causalean.swigΩ Ω))","l":31,"f":"Causalean/SCM/ID/Query.lean"},{"n":"Causalean.SCM.ID.valuesProjection_extend_eq_of_subset","k":"theorem","m":"Causalean.SCM.ID.Density.QFactor","a":"SCM/ID/Density","t":2,"d":"If an extension preserves all values on a larger observed set, then restricting the extension to any subset gives the same values as restricting the original assignment directly.","s":"∀ {N : Type u_1} [inst : DecidableEq N] [inst_1 : Fintype N] {Ω : N → Type u_2} [inst_2 : (n : N) → MeasurableSpace (Ω n)] (M : Causalean.SCM N Ω) {D S : Finset (Causalean.SWIGNode N)} (hDobs : D ⊆ M.observed) (hSD : S ⊆ D) (hSobs : S ⊆ M.observed) (extend : Causalean.ValuesOn D (Causalean.swigΩ Ω) → Causalean.ValuesOn M.observed (Causalean.swigΩ Ω)), (∀ (xD : Causalean.ValuesOn D (Causalean.swigΩ Ω)), Causalean.valuesProjection hDobs (extend xD) = xD) → ∀ (xD : Causalean.ValuesOn D (Causalean.swigΩ Ω)), Causalean.valuesProjection hSobs (extend xD) = Causalean.valuesProjection hSD xD","l":1426,"f":"Causalean/SCM/ID/Density/QFactor.lean"},{"n":"Causalean.SCM.ID.valuesUnionEquiv_extendTianPrefix","k":"theorem","m":"Causalean.SCM.ID.GraphicalThms.DoGFormulaTian","a":"SCM/ID/GraphicalThms","t":2,"d":"The successor-prefix extension is inverse to the union-equivalence view of the successor prefix.","s":"∀ {N : Type u_1} [inst : DecidableEq N] [inst_1 : Fintype N] {Ω : N → Type u_2} [inst_2 : (n : N) → MeasurableSpace (Ω n)] (H : Causalean.SWIGGraph N) (D : Finset (Causalean.SWIGNode N)) {n : ℕ} (hn : n < D.card) (p : Causalean.ValuesOn (H.prefixIn D n) (Causalean.swigΩ Ω) × Causalean.ValuesOn {↑(H.nodesAt D ⟨n, hn⟩)} (Causalean.swigΩ Ω)), (Causalean.SCM.valuesUnionEquiv (Causalean.SCM.ID.prefixIn_disjoint_singleton_next H D hn)) ((Causalean.valuesEquivOfEq (Causalean.SCM.ID.prefixIn_succ H D hn)) (Causalean.SCM.ID.extendTianPrefix H D hn p)) = p","l":187,"f":"Causalean/SCM/ID/GraphicalThms/DoGFormulaTian.lean"},{"n":"Causalean.SCM.IsFaithful","k":"def","m":"Causalean.Graph.MarkovEquiv.Distributional","a":"Graph/MarkovEquiv","t":1,"d":"A measure μ is faithful to a DAG G when every conditional independence of μ reflects a genuine d-separation of G (the converse of being an I-map): for pairwise-disjoint X, Y, Z, X ⟂ Y | Z under μ forces G to d-separate X and Y given Z. A measure that is both an I-map of and faithful to G has conditional independences exactly matching G's d-separations.","s":"{N : Type uN} → [inst : DecidableEq N] → [inst_1 : Fintype N] → {Ω : N → Type uΩ} → [inst_2 : (n : N) → MeasurableSpace (Ω n)] → Causalean.DAG (Causalean.SWIGNode N) → (M : Causalean.SCM N Ω) → [StandardBorelSpace M.RandomValues] → (μ : MeasureTheory.Measure M.RandomValues) → [MeasureTheory.IsFiniteMeasure μ] → Prop","l":53,"f":"Causalean/Graph/MarkovEquiv/Distributional.lean"},{"n":"Causalean.SCM.IsGlobalIMap","k":"def","m":"Causalean.Graph.MarkovEquiv.Distributional","a":"Graph/MarkovEquiv","t":1,"d":"A measure μ on the random values of M is a global I-map of a DAG G (on the same node set) when every d-separation in G is a conditional independence under μ. The required pairwise disjointness is already part of d-separation.","s":"{N : Type uN} → [inst : DecidableEq N] → [inst_1 : Fintype N] → {Ω : N → Type uΩ} → [inst_2 : (n : N) → MeasurableSpace (Ω n)] → Causalean.DAG (Causalean.SWIGNode N) → (M : Causalean.SCM N Ω) → [StandardBorelSpace M.RandomValues] → (μ : MeasureTheory.Measure M.RandomValues) → [MeasureTheory.IsFiniteMeasure μ] → Prop","l":43,"f":"Causalean/Graph/MarkovEquiv/Distributional.lean"},{"n":"Causalean.SCM.LatentValues","k":"def","m":"Causalean.SCM.Model.SCM","a":"SCM/Model","t":1,"d":"Latent values assign values to the unobserved root nodes of a structural causal model.","s":"{N : Type uN} → [inst : DecidableEq N] → [inst_1 : Fintype N] → {Ω : N → Type uΩ} → [inst_2 : (n : N) → MeasurableSpace (Ω n)] → Causalean.SCM N Ω → Type (max uΩ uN)","l":118,"f":"Causalean/SCM/Model/SCM.lean"},{"n":"Causalean.SCM.ObsCondIndep","k":"def","m":"Causalean.SCM.Do.SemiGraphoid","a":"SCM/Do","t":1,"d":"Observational conditional independence.","s":"{N : Type uN} → [inst : DecidableEq N] → [inst_1 : Fintype N] → {Ω : N → Type uΩ} → [inst_2 : (n : N) → MeasurableSpace (Ω n)] → (M : Causalean.SCM N Ω) → [StandardBorelSpace M.ObservedValues] → (X Y Z : Finset (Causalean.SWIGNode N)) → X ⊆ M.observed → Y ⊆ M.observed → Z ⊆ M.observed → (μ : MeasureTheory.Measure M.ObservedValues) → [MeasureTheory.IsFiniteMeasure μ] → Prop","l":40,"f":"Causalean/SCM/Do/SemiGraphoid.lean"},{"n":"Causalean.SCM.ObsParentClosed","k":"def","m":"Causalean.SCM.ID.Density.LatentBlocks","a":"SCM/ID/Density","t":1,"d":"A finite set of observed nodes is closed under observed parents.","s":"{N : Type u_1} → [inst : DecidableEq N] → [inst_1 : Fintype N] → {Ω : N → Type u_2} → [inst_2 : (n : N) → MeasurableSpace (Ω n)] → Causalean.SCM N Ω → Finset (Causalean.SWIGNode N) → Prop","l":676,"f":"Causalean/SCM/ID/Density/LatentBlocks.lean"},{"n":"Causalean.SCM.ObsStepFiberRN","k":"def","m":"Causalean.SCM.ID.Density.ChainRuleDensity","a":"SCM/ID/Density","t":1,"d":"Per-step analytic hypotheses needed to expose the fibre Radon--Nikodym derivative against a σ-finite one-node reference.","s":"{N : Type u_1} → [inst : DecidableEq N] → [inst_1 : Fintype N] → {Ω : N → Type u_2} → [inst_2 : (n : N) → MeasurableSpace (Ω n)] → (M : Causalean.SCM N Ω) → Causalean.SCM.ReferenceMeasures Ω → M.FixedValues → [∀ (s' : M.FixedValues), MeasureTheory.IsFiniteMeasure (M.obsKernel s')] → [∀ (k : ℕ) (hk : k < M.observed.card), StandardBorelSpace (Causalean.ValuesOn {↑(M.observedAt ⟨k, hk⟩)} (Causalean.swigΩ Ω))] → [∀ (k : ℕ) (hk : k < M.observed.card), Nonempty (Causalean.ValuesOn {↑(M.observedAt ⟨k, hk⟩)} (Causalean.swigΩ Ω))] → [∀ (k : ℕ), MeasurableSpace.CountableOrCountablyGenerated M.FixedValues (Causalean.ValuesOn (M.prefixNodes k) (Causalean.swigΩ Ω))] → Prop","l":121,"f":"Causalean/SCM/ID/Density/ChainRuleDensity.lean"},{"n":"Causalean.SCM.ObservedPrefixValues","k":"def","m":"Causalean.SCM.Factored.PrefixState","a":"SCM/Factored","t":1,"d":"Values for the first n observed nodes in canonical topological order.","s":"{N : Type uN} → [inst : DecidableEq N] → [inst_1 : Fintype N] → {Ω : N → Type uΩ} → [inst_2 : (n : N) → MeasurableSpace (Ω n)] → (M : Causalean.SCM N Ω) → (n : ℕ) → n ≤ M.observed.card → Type uΩ","l":32,"f":"Causalean/SCM/Factored/PrefixState.lean"},{"n":"Causalean.SCM.ObservedValues","k":"def","m":"Causalean.SCM.Model.SCM","a":"SCM/Model","t":1,"d":"Observed values assign values to the observed nodes of a structural causal model.","s":"{N : Type uN} → [inst : DecidableEq N] → [inst_1 : Fintype N] → {Ω : N → Type uΩ} → [inst_2 : (n : N) → MeasurableSpace (Ω n)] → Causalean.SCM N Ω → Type (max uΩ uN)","l":114,"f":"Causalean/SCM/Model/SCM.lean"},{"n":"Causalean.SCM.OrderedLatentPrefixValues","k":"def","m":"Causalean.SCM.Factored.PrefixState","a":"SCM/Factored","t":1,"d":"Random values consisting of the latent tuple paired with an observed prefix. This is the state-space of the kernel at step n in the factored construction.","s":"{N : Type uN} → [inst : DecidableEq N] → [inst_1 : Fintype N] → {Ω : N → Type uΩ} → [inst_2 : (n : N) → MeasurableSpace (Ω n)] → (M : Causalean.SCM N Ω) → (n : ℕ) → n ≤ M.observed.card → Type (max (max uΩ uN) uΩ)","l":53,"f":"Causalean/SCM/Factored/PrefixState.lean"},{"n":"Causalean.SCM.PartialID.CompatibleSCM","k":"def","m":"Causalean.SCM.PartialID.CanonicalModel","a":"SCM/PartialID","t":1,"d":"The compatible class for a graphical partial-identification problem: the structural causal models M that share the SWIG graph G, satisfy the structural assumptions As, and are observationally equivalent to the reference model M₀ (their derived observational kernels agree). This is the class over which a bound must hold to be *sound*.","s":"{N : Type u_1} → [inst : DecidableEq N] → [inst_1 : Fintype N] → {Ω : N → Type u_2} → [inst_2 : (n : N) → MeasurableSpace (Ω n)] → Causalean.SWIGGraph N → (Causalean.SCM N Ω → Prop) → Causalean.SCM N Ω → Causalean.SCM N Ω → Prop","l":43,"f":"Causalean/SCM/PartialID/CanonicalModel.lean"},{"n":"Causalean.SCM.PartialID.IsSharp","k":"def","m":"Causalean.SCM.PartialID.SharpnessCertificate","a":"SCM/PartialID","t":1,"d":"A bound [L, U] is sharp for the real-valued query obj over the compatible class of (G, As, M₀) when the identified set (the range of obj over the compatible class) is exactly the closed interval [L, U].","s":"{N : Type u_1} → [inst : DecidableEq N] → [inst_1 : Fintype N] → {Ω : N → Type u_2} → [inst_2 : (n : N) → MeasurableSpace (Ω n)] → Causalean.SWIGGraph N → (Causalean.SCM N Ω → Prop) → Causalean.SCM N Ω → (Causalean.SCM N Ω → ℝ) → ℝ → ℝ → Prop","l":28,"f":"Causalean/SCM/PartialID/SharpnessCertificate.lean"},{"n":"Causalean.SCM.PartialID.compatibleInterval","k":"def","m":"Causalean.SCM.PartialID.CanonicalModel","a":"SCM/PartialID","t":1,"d":"The compatible interval of a real-valued causal query: its range over the compatible class. This is the graphical identified set; a partial-identification bound [L, U] is *sound* exactly when this set is contained in Set.Icc L U, and *sharp* when they are equal. Built on the abstract IdentifiedInterval, so the order-convexity / sharpness lemmas of the partial-identification layer apply.","s":"{N : Type u_1} → [inst : DecidableEq N] → [inst_1 : Fintype N] → {Ω : N → Type u_2} → [inst_2 : (n : N) → MeasurableSpace (Ω n)] → Causalean.SWIGGraph N → (Causalean.SCM N Ω → Prop) → Causalean.SCM N Ω → (Causalean.SCM N Ω → ℝ) → Set ℝ","l":68,"f":"Causalean/SCM/PartialID/CanonicalModel.lean"},{"n":"Causalean.SCM.PartialID.compatibleSCM_mono","k":"theorem","m":"Causalean.SCM.PartialID.CanonicalModel","a":"SCM/PartialID","t":1,"d":"If the structural-assumption predicate As' is stronger than As, i.e. every model satisfying As' also satisfies As, then every model compatible with the reference model M₀ under the stricter assumptions As' is also compatible under the weaker assumptions As — strengthening the structural assumptions can only shrink the compatible class.","s":"∀ {N : Type u_1} [inst : DecidableEq N] [inst_1 : Fintype N] {Ω : N → Type u_2} [inst_2 : (n : N) → MeasurableSpace (Ω n)] {G : Causalean.SWIGGraph N} {As As' : Causalean.SCM N Ω → Prop} {M₀ : Causalean.SCM N Ω}, (∀ (M : Causalean.SCM N Ω), As' M → As M) → ∀ (M : Causalean.SCM N Ω), Causalean.SCM.PartialID.CompatibleSCM G As' M₀ M → Causalean.SCM.PartialID.CompatibleSCM G As M₀ M","l":59,"f":"Causalean/SCM/PartialID/CanonicalModel.lean"},{"n":"Causalean.SCM.PartialID.compatibleSCM_self","k":"theorem","m":"Causalean.SCM.PartialID.CanonicalModel","a":"SCM/PartialID","t":1,"d":"For any graph G, structural-assumption predicate As, and reference model M₀, if M₀'s own SWIG graph is G and M₀ satisfies the structural assumptions As, then M₀ belongs to its own compatible class CompatibleSCM G As M₀.","s":"∀ {N : Type u_1} [inst : DecidableEq N] [inst_1 : Fintype N] {Ω : N → Type u_2} [inst_2 : (n : N) → MeasurableSpace (Ω n)] (G : Causalean.SWIGGraph N) (As : Causalean.SCM N Ω → Prop) (M₀ : Causalean.SCM N Ω), M₀.toSWIGGraph = G → As M₀ → Causalean.SCM.PartialID.CompatibleSCM G As M₀ M₀","l":51,"f":"Causalean/SCM/PartialID/CanonicalModel.lean"},{"n":"Causalean.SCM.PartialID.isSharp_of_attaining","k":"theorem","m":"Causalean.SCM.PartialID.SharpnessCertificate","a":"SCM/PartialID","t":1,"d":"Sharpness certificate. For a real-valued query obj over the compatible class of (G, As, M₀), if the identified set is contained in the interval [L, U] (soundness), the identified set is order-connected, some compatible model attains the value L, and some compatible model attains the value U, then the identified set equals [L, U], i.e. the bound is sharp.","s":"∀ {N : Type u_1} [inst : DecidableEq N] [inst_1 : Fintype N] {Ω : N → Type u_2} [inst_2 : (n : N) → MeasurableSpace (Ω n)] (G : Causalean.SWIGGraph N) (As : Causalean.SCM N Ω → Prop) (M₀ : Causalean.SCM N Ω) (obj : Causalean.SCM N Ω → ℝ) (L U : ℝ), Causalean.SCM.PartialID.compatibleInterval G As M₀ obj ⊆ Set.Icc L U → (Causalean.SCM.PartialID.compatibleInterval G As M₀ obj).OrdConnected → (∃ M, Causalean.SCM.PartialID.CompatibleSCM G As M₀ M ∧ obj M = L) → (∃ M, Causalean.SCM.PartialID.CompatibleSCM G As M₀ M ∧ obj M = U) → Causalean.SCM.PartialID.IsSharp G As M₀ obj L U","l":35,"f":"Causalean/SCM/PartialID/SharpnessCertificate.lean"},{"n":"Causalean.SCM.QmechMeasure","k":"def","m":"Causalean.SCM.ID.Density.MechCFactor","a":"SCM/ID/Density","t":1,"d":"Tian's Q[S] measure (Eq. 36 / Eq. 55): the do(observed∖S) marginal on S. Intervene on every not-already-fixed observed node outside S, then project the intervened observational law to the coordinates in S.","s":"{N : Type u_1} → [inst : DecidableEq N] → [inst_1 : Fintype N] → {Ω : N → Type u_2} → [inst_2 : (n : N) → MeasurableSpace (Ω n)] → (M : Causalean.SCM N Ω) → (S : Finset (Causalean.SWIGNode N)) → S ⊆ M.observed → (M.fixSet (M.mechComplementNames S) (Causalean.SCM.mechComplementNames_random_mem_observed M S) (Causalean.SCM.mechComplementNames_fixed_not_mem M S)).FixedValues → MeasureTheory.Measure (Causalean.ValuesOn S (Causalean.swigΩ Ω))","l":139,"f":"Causalean/SCM/ID/Density/MechCFactor.lean"},{"n":"Causalean.SCM.QmechMeasure_singleton_eq_qLocalMass","k":"theorem","m":"Causalean.SCM.ID.Density.MechCFactor","a":"SCM/ID/Density","t":2,"d":"The do-complement Q[S] atom equals the local q-mass on S.","s":"∀ {N : Type u_1} [inst : DecidableEq N] [inst_1 : Fintype N] {Ω : N → Type u_2} [inst_2 : (n : N) → MeasurableSpace (Ω n)] [(n : N) → Fintype (Ω n)] [∀ (n : N), MeasurableSingletonClass (Ω n)] (M : Causalean.SCM N Ω) (S : Finset (Causalean.SWIGNode N)) (hS : S ⊆ M.observed), (∀ (n : N), Causalean.SWIGNode.fixed n ∈ M.fixed → ∀ (v : Causalean.SWIGNode N), ¬M.dag.edge (Causalean.SWIGNode.random n) v) → ∀ (s : M.FixedValues) (x : Causalean.ValuesOn M.observed (Causalean.swigΩ Ω)), (M.QmechMeasure S hS (M.mechDoValues S s x)) {Causalean.valuesProjection hS x} = M.qLocalMass s S hS x","l":170,"f":"Causalean/SCM/ID/Density/MechCFactor.lean"},{"n":"Causalean.SCM.RandomValues","k":"def","m":"Causalean.SCM.Model.SCM","a":"SCM/Model","t":1,"d":"Random values assign values to every observed or latent node in a structural causal model.","s":"{N : Type uN} → [inst : DecidableEq N] → [inst_1 : Fintype N] → {Ω : N → Type uΩ} → [inst_2 : (n : N) → MeasurableSpace (Ω n)] → Causalean.SCM N Ω → Type (max uΩ uN)","l":135,"f":"Causalean/SCM/Model/SCM.lean"},{"n":"Causalean.SCM.ReferenceFaithful","k":"def","m":"Causalean.SCM.ID.Density.FiniteReference","a":"SCM/ID/Density","t":1,"d":"A reference family is faithful when every single coordinate value has nonzero reference mass.","s":"{N : Type u_1} → {Ω : N → Type u_2} → [inst : (n : N) → MeasurableSpace (Ω n)] → Causalean.SCM.ReferenceMeasures Ω → Prop","l":72,"f":"Causalean/SCM/ID/Density/FiniteReference.lean"},{"n":"Causalean.SCM.ReferenceMeasures","k":"structure","m":"Causalean.SCM.ID.Density.ReferenceMeasure","a":"SCM/ID/Density","t":1,"d":"A reference measure family assigns a measure on every random or fixed node's value space, required to be σ-finite.","s":"{N : Type u_1} → (Ω : N → Type u_3) → [(n : N) → MeasurableSpace (Ω n)] → Type (max u_1 u_3)","l":35,"f":"Causalean/SCM/ID/Density/ReferenceMeasure.lean"},{"n":"Causalean.SCM.UnobservedValues","k":"def","m":"Causalean.SCM.Model.SCM","a":"SCM/Model","t":1,"d":"Unobserved values are the same object as latent values and are kept as a compatibility alias.","s":"{N : Type uN} → [inst : DecidableEq N] → [inst_1 : Fintype N] → {Ω : N → Type uΩ} → [inst_2 : (n : N) → MeasurableSpace (Ω n)] → Causalean.SCM N Ω → Type (max uΩ uN)","l":122,"f":"Causalean/SCM/Model/SCM.lean"},{"n":"Causalean.SCM.absolutelyContinuous_jointRef_of_faithful","k":"theorem","m":"Causalean.SCM.ID.Density.FiniteReference","a":"SCM/ID/Density","t":1,"d":"Faithful references dominate. On a finite coordinate product indexed by a node set I, if the reference family ref is faithful — every single coordinate value carries nonzero reference mass, then every measure μ on that product is absolutely continuous with respect to the joint reference measure.","s":"∀ {N : Type u_1} {Ω : N → Type u_2} [inst : (n : N) → MeasurableSpace (Ω n)] [∀ (n : N), MeasurableSingletonClass (Ω n)] (ref : Causalean.SCM.ReferenceMeasures Ω), Causalean.SCM.ReferenceFaithful ref → ∀ (I : Finset (Causalean.SWIGNode N)) (μ : MeasureTheory.Measure (Causalean.ValuesOn I (Causalean.swigΩ Ω))), μ.AbsolutelyContinuous (Causalean.SCM.jointRef ref I)","l":118,"f":"Causalean/SCM/ID/Density/FiniteReference.lean"},{"n":"Causalean.SCM.absolutelyContinuous_of_map_measurableEmbedding","k":"theorem","m":"Causalean.SCM.ID.Density.ChainRuleDensity","a":"SCM/ID/Density","t":2,"d":"Pull absolute continuity back through a measurable embedding.","s":"∀ {α : Type u_3} {β : Type u_4} [inst : MeasurableSpace α] [inst_1 : MeasurableSpace β] {f : α → β}, MeasurableEmbedding f → ∀ {μ ν : MeasureTheory.Measure α}, (MeasureTheory.Measure.map f μ).AbsolutelyContinuous (MeasureTheory.Measure.map f ν) → μ.AbsolutelyContinuous ν","l":418,"f":"Causalean/SCM/ID/Density/ChainRuleDensity.lean"},{"n":"Causalean.SCM.absolutelyContinuous_of_singleton_ne_zero","k":"theorem","m":"Causalean.SCM.ID.Density.FiniteReference","a":"SCM/ID/Density","t":2,"d":"Any measure is absolutely continuous with respect to a measure that gives every singleton nonzero mass.","s":"∀ {α : Type u_3} [inst : MeasurableSpace α] (μ ν : MeasureTheory.Measure α), (∀ (x : α), ν {x} ≠ 0) → μ.AbsolutelyContinuous ν","l":78,"f":"Causalean/SCM/ID/Density/FiniteReference.lean"},{"n":"Causalean.SCM.adjustmentKernelY","k":"def","m":"Causalean.SCM.ID.Backdoor","a":"SCM/ID","t":1,"d":"Backdoor-adjustment Y-marginal as a kernel in the treatment value t, at base s₀.","s":"{N : Type u_1} → [inst : DecidableEq N] → [inst_1 : Fintype N] → {Ω : N → Type u_2} → [inst_2 : (n : N) → MeasurableSpace (Ω n)] → [∀ (n : Causalean.SWIGNode N), StandardBorelSpace (Causalean.swigΩ Ω n)] → [∀ (n : Causalean.SWIGNode N), Nonempty (Causalean.swigΩ Ω n)] → (M : Causalean.SCM N Ω) → (X : Finset N) → (∀ D ∈ X, Causalean.SWIGNode.random D ∈ M.observed) → (∀ D ∈ X, Causalean.SWIGNode.fixed D ∉ M.fixed) → (Y Z : Finset (Causalean.SWIGNode N)) → Y ⊆ M.observed → Z ⊆ M.observed → M.FixedValues → ProbabilityTheory.Kernel (Causalean.ValuesOn (Finset.image Causalean.SWIGNode.random X) (Causalean.swigΩ Ω)) (Causalean.ValuesOn Y (Causalean.swigΩ Ω))","l":99,"f":"Causalean/SCM/ID/Backdoor.lean"},{"n":"Causalean.SCM.adjustmentKernelY_empty_eq","k":"theorem","m":"Causalean.SCM.ID.Frontdoor","a":"SCM/ID","t":2,"d":"With an empty adjustment set, the outcome adjustment kernel equals the observed conditional kernel of the outcome variables given the treated variables at the same fixed values.","s":"∀ {N : Type u_1} [inst : DecidableEq N] [inst_1 : Fintype N] {Ω : N → Type u_2} [inst_2 : (n : N) → MeasurableSpace (Ω n)] [inst_3 : ∀ (n : Causalean.SWIGNode N), StandardBorelSpace (Causalean.swigΩ Ω n)] [inst_4 : ∀ (n : Causalean.SWIGNode N), Nonempty (Causalean.swigΩ Ω n)] (M : Causalean.SCM N Ω) (X : Finset N) (hObs : ∀ D ∈ X, Causalean.SWIGNode.random D ∈ M.observed) (hFix : ∀ D ∈ X, Causalean.SWIGNode.fixed D ∉ M.fixed) (Wr : Finset (Causalean.SWIGNode N)) (hWr : Wr ⊆ M.observed) (hXr : Finset.image Causalean.SWIGNode.random X ⊆ M.observed) (s0 : M.FixedValues) (t : Causalean.ValuesOn (Finset.image Causalean.SWIGNode.random X) (Causalean.swigΩ Ω)), (M.adjustmentKernelY X hObs hFix Wr ∅ hWr (Finset.empty_subset M.observed) s0) t = (M.obsCondKernel Wr (Finset.image Causalean.SWIGNode.random X) hWr hXr) (s0, t)","l":279,"f":"Causalean/SCM/ID/Frontdoor.lean"},{"n":"Causalean.SCM.aemeasurable_fiber_rnDeriv_of_finite","k":"theorem","m":"Causalean.SCM.ID.Density.FiniteReference","a":"SCM/ID/Density","t":1,"d":"For finite index types α and β, a joint measure μ on α × β, a measure ρ on β, and a Markov kernel κ from α to β, the fibre Radon-Nikodym derivative selector (κ p.1).rnDeriv ρ p.2 is almost-everywhere measurable with respect to μ.","s":"∀ {α : Type u_3} {β : Type u_4} [inst : MeasurableSpace α] [inst_1 : MeasurableSpace β] [Finite α] [Finite β] [MeasurableSingletonClass α] [MeasurableSingletonClass β] (μ : MeasureTheory.Measure (α × β)) (ρ : MeasureTheory.Measure β) (κ : ProbabilityTheory.Kernel α β), AEMeasurable (fun p => (κ p.1).rnDeriv ρ p.2) μ","l":129,"f":"Causalean/SCM/ID/Density/FiniteReference.lean"},{"n":"Causalean.SCM.ancestralFactorization","k":"theorem","m":"Causalean.SCM.Model.Evaluation","a":"SCM/Model","t":1,"d":"Ancestral factorization (Lemma lem:scm-ancestral-factor). Fix a structural causal model M, a target node set T, two fixed-value assignments s, s', two latent assignments ℓ, ℓ', and a node v with v belonging to T and v an observed node. If s and s' agree on every fixed node that equals or is an ancestor of some node of T and ℓ and ℓ' agree on every unobserved node that equals or is an ancestor of some node of T, then the evaluation of M at v with (s, ℓ) equals the evaluation with (s', ℓ').","s":"∀ {N : Type u_1} [inst : DecidableEq N] [inst_1 : Fintype N] {Ω : N → Type u_2} [inst_2 : (n : N) → MeasurableSpace (Ω n)] (M : Causalean.SCM N Ω) (T : Finset (Causalean.SWIGNode N)) {s s' : M.FixedValues} {ℓ ℓ' : M.LatentValues}, (∀ (d : Causalean.SWIGNode N) (hd : d ∈ M.fixed), (∃ v ∈ T, d = v ∨ M.dag.isAncestor d v) → s ⟨d, hd⟩ = s' ⟨d, hd⟩) → (∀ (u : Causalean.SWIGNode N) (hu : u ∈ M.unobserved), (∃ v ∈ T, u = v ∨ M.dag.isAncestor u v) → ℓ ⟨u, hu⟩ = ℓ' ⟨u, hu⟩) → ∀ {v : Causalean.SWIGNode N}, v ∈ T → ∀ (hv_obs : v ∈ M.observed), M.evalMap s ℓ ⟨v, Finset.mem_union_left M.unobserved hv_obs⟩ = M.evalMap s' ℓ' ⟨v, Finset.mem_union_left M.unobserved hv_obs⟩","l":521,"f":"Causalean/SCM/Model/Evaluation.lean"},{"n":"Causalean.SCM.backdoorAdjustment","k":"def","m":"Causalean.SCM.ID.Adjustment","a":"SCM/ID","t":1,"d":"Backdoor adjustment functional.","s":"{N : Type u_1} → [inst : DecidableEq N] → [inst_1 : Fintype N] → {Ω : N → Type u_2} → [inst_2 : (n : N) → MeasurableSpace (Ω n)] → [∀ (n : Causalean.SWIGNode N), StandardBorelSpace (Causalean.swigΩ Ω n)] → [∀ (n : Causalean.SWIGNode N), Nonempty (Causalean.swigΩ Ω n)] → (M : Causalean.SCM N Ω) → (X : Finset N) → (hX_obs : ∀ D ∈ X, Causalean.SWIGNode.random D ∈ M.observed) → (hX_fixed : ∀ D ∈ X, Causalean.SWIGNode.fixed D ∉ M.fixed) → (Y Z : Finset (Causalean.SWIGNode N)) → Y ⊆ M.observed → Z ⊆ M.observed → ProbabilityTheory.Kernel (M.fixSet X hX_obs hX_fixed).FixedValues (Causalean.ValuesOn Y (Causalean.swigΩ Ω))","l":68,"f":"Causalean/SCM/ID/Adjustment.lean"},{"n":"Causalean.SCM.backdoorAdjustment_invariant","k":"theorem","m":"Causalean.SCM.ID.Adjustment","a":"SCM/ID","t":1,"d":"Cross-SCM invariance of backdoorAdjustment. Fix two causal models M₁ and M₂ that share the same SWIG graph and the same observational kernel, and a treatment set X, an outcome set Y, and an adjustment set Z that are valid intervention data and observed variables in both models. Then the backdoor-adjustment kernel for Y given Z after intervening on X, computed within M₁, is the same as the one computed within M₂.","s":"∀ {N : Type u_1} [inst : DecidableEq N] [inst_1 : Fintype N] {Ω : N → Type u_2} [inst_2 : (n : N) → MeasurableSpace (Ω n)] [inst_3 : ∀ (n : Causalean.SWIGNode N), StandardBorelSpace (Causalean.swigΩ Ω n)] [inst_4 : ∀ (n : Causalean.SWIGNode N), Nonempty (Causalean.swigΩ Ω n)] (M₁ M₂ : Causalean.SCM N Ω), M₁.toSWIGGraph = M₂.toSWIGGraph → M₁.obsKernel ≍ M₂.obsKernel → ∀ (X : Finset N) (hX_obs₁ : ∀ D ∈ X, Causalean.SWIGNode.random D ∈ M₁.observed) (hX_fixed₁ : ∀ D ∈ X, Causalean.SWIGNode.fixed D ∉ M₁.fixed) (hX_obs₂ : ∀ D ∈ X, Causalean.SWIGNode.random D ∈ M₂.observed) (hX_fixed₂ : ∀ D ∈ X, Causalean.SWIGNode.fixed D ∉ M₂.fixed) (Y Z : Finset (Causalean.SWIGNode N)) (hY₁ : Y ⊆ M₁.observed) (hZ₁ : Z ⊆ M₁.observed) (hY₂ : Y ⊆ M₂.observed) (hZ₂ : Z ⊆ M₂.observed), M₁.backdoorAdjustment X hX_obs₁ hX_fixed₁ Y Z hY₁ hZ₁ ≍ M₂.backdoorAdjustment X hX_obs₂ hX_fixed₂ Y Z hY₂ hZ₂","l":258,"f":"Causalean/SCM/ID/Adjustment.lean"},{"n":"Causalean.SCM.backdoorCriterion_W_nonDesc","k":"theorem","m":"Causalean.SCM.ID.Toolkit.Derivation","a":"SCM/ID/Toolkit","t":2,"d":"Backdoor criterion (i), post-intervention non-descendance form.","s":"∀ {N : Type u_1} [inst : DecidableEq N] [inst_1 : Fintype N] {Ω : N → Type u_2} [inst_2 : (n : N) → MeasurableSpace (Ω n)] [∀ (n : Causalean.SWIGNode N), StandardBorelSpace (Causalean.swigΩ Ω n)] [∀ (n : Causalean.SWIGNode N), Nonempty (Causalean.swigΩ Ω n)] (M : Causalean.SCM N Ω) (X : Finset N) (hObs : ∀ D ∈ X, Causalean.SWIGNode.random D ∈ M.observed) (hFix : ∀ D ∈ X, Causalean.SWIGNode.fixed D ∉ M.fixed) (Y Z : Finset (Causalean.SWIGNode N)), M.backdoorCriterion X hObs hFix Y Z → ∀ x ∈ X, ∀ v ∈ Z, ¬(M.fixSet X hObs hFix).dag.isAncestor (Causalean.SWIGNode.fixed x) v","l":107,"f":"Causalean/SCM/ID/Toolkit/Derivation.lean"},{"n":"Causalean.SCM.backdoorCriterion_W_nonDescM1","k":"theorem","m":"Causalean.SCM.ID.Toolkit.Derivation","a":"SCM/ID/Toolkit","t":2,"d":"Backdoor criterion (i), base-graph non-descendance form.","s":"∀ {N : Type u_1} [inst : DecidableEq N] [inst_1 : Fintype N] {Ω : N → Type u_2} [inst_2 : (n : N) → MeasurableSpace (Ω n)] [∀ (n : Causalean.SWIGNode N), StandardBorelSpace (Causalean.swigΩ Ω n)] [∀ (n : Causalean.SWIGNode N), Nonempty (Causalean.swigΩ Ω n)] (M : Causalean.SCM N Ω) (X : Finset N) (hObs : ∀ D ∈ X, Causalean.SWIGNode.random D ∈ M.observed) (hFix : ∀ D ∈ X, Causalean.SWIGNode.fixed D ∉ M.fixed) (Y Z : Finset (Causalean.SWIGNode N)), M.backdoorCriterion X hObs hFix Y Z → ∀ D ∈ X, ∀ w ∈ Z, ¬M.dag.isAncestor (Causalean.SWIGNode.random D) w","l":124,"f":"Causalean/SCM/ID/Toolkit/Derivation.lean"},{"n":"Causalean.SCM.backdoorCriterion_dSep_fixSet","k":"theorem","m":"Causalean.SCM.ID.Toolkit.Derivation","a":"SCM/ID/Toolkit","t":1,"d":"Backdoor criterion (ii), extended to the post-intervention fixed nodes. Fix a structural causal model M and an intervention target set X such that every targeted node is currently a random observed node and none of its fixed copies is already fixed, an outcome set Y and a conditioning set Z with Y contained in the observed nodes and the randomized image of X contained in the observed nodes. If M's DAG satisfies the backdoor criterion for X, Y given Z, then in the post-intervention graph obtained by fixing X, Y is d-separated from the randomized image of X given Z together with the post-intervention fixed node set.","s":"∀ {N : Type u_1} [inst : DecidableEq N] [inst_1 : Fintype N] {Ω : N → Type u_2} [inst_2 : (n : N) → MeasurableSpace (Ω n)] [∀ (n : Causalean.SWIGNode N), StandardBorelSpace (Causalean.swigΩ Ω n)] [∀ (n : Causalean.SWIGNode N), Nonempty (Causalean.swigΩ Ω n)] (M : Causalean.SCM N Ω) (X : Finset N) (hObs : ∀ D ∈ X, Causalean.SWIGNode.random D ∈ M.observed) (hFix : ∀ D ∈ X, Causalean.SWIGNode.fixed D ∉ M.fixed) (Y Z : Finset (Causalean.SWIGNode N)), Y ⊆ M.observed → Finset.image Causalean.SWIGNode.random X ⊆ M.observed → M.backdoorCriterion X hObs hFix Y Z → (M.fixSet X hObs hFix).dag.dSep Y (Finset.image Causalean.SWIGNode.random X) (Z ∪ (M.fixSet X hObs hFix).fixed)","l":55,"f":"Causalean/SCM/ID/Toolkit/Derivation.lean"},{"n":"Causalean.SCM.backdoor_completeness_ae","k":"theorem","m":"Causalean.SCM.ID.Backdoor","a":"SCM/ID","t":1,"d":"Backdoor completeness, a.e. in the treatment value. Fix a causal model M and a treatment set X whose random copy is observed and whose fixed copy is not already held fixed in M, and let Y, Z be observed outcome and adjustment sets, neither of which overlaps the treatment's random nodes, satisfying the backdoor criterion relative to (X,Y). Given a base configuration s0 at which overlap holds along every post-intervention slice and the product of the observational treatment and adjustment-set marginals is absolutely continuous with respect to their joint observational law, then for almost every treatment value t under the observational treatment marginal, the post-intervention outcome distribution at t equals the backdoor-adjustment functional evaluated at t.","s":"∀ {N : Type u_1} [inst : DecidableEq N] [inst_1 : Fintype N] {Ω : N → Type u_2} [inst_2 : (n : N) → MeasurableSpace (Ω n)] [inst_3 : ∀ (n : Causalean.SWIGNode N), StandardBorelSpace (Causalean.swigΩ Ω n)] [inst_4 : ∀ (n : Causalean.SWIGNode N), Nonempty (Causalean.swigΩ Ω n)] (M : Causalean.SCM N Ω) (X : Finset N) (hObs : ∀ D ∈ X, Causalean.SWIGNode.random D ∈ M.observed) (hFix : ∀ D ∈ X, Causalean.SWIGNode.fixed D ∉ M.fixed) (Y Z : Finset (Causalean.SWIGNode N)) (hY : Y ⊆ M.observed) (hZ : Z ⊆ M.observed), M.backdoorCriterion X hObs hFix Y Z → Disjoint Y (Finset.image Causalean.SWIGNode.random X) → Disjoint (Finset.image Causalean.SWIGNode.random X) Z → ∀ (s0 : M.FixedValues), (∀ (s : (M.fixSet X hObs hFix).FixedValues), Causalean.SCM.ID.Rule2JointOverlap M X hObs hFix Z (Finset.union_subset (Finset.image_subset_iff.mpr hObs) hZ) s) → M.BackdoorPositivityAE X Z hZ (Finset.union_subset (Finset.image_subset_iff.mpr hObs) hZ) s0 → ∀ᵐ (t : Causalean.ValuesOn (Finset.image Causalean.SWIGNode.random X) (Causalean.swigΩ Ω)) ∂M.treatmentMarginal X (LE.le.trans Finset.subset_union_left (Finset.union_subset (Finset.image_subset_iff.mpr hObs) hZ)) s0, (M.doKernelY X hObs hFix Y hY s0) t = (M.adjustmentKernelY X hObs hFix Y Z hY hZ s0) t","l":354,"f":"Causalean/SCM/ID/Backdoor.lean"},{"n":"Causalean.SCM.backdoor_completeness_ae_compProd","k":"theorem","m":"Causalean.SCM.ID.Backdoor","a":"SCM/ID","t":2,"d":"Joint (compProd) completeness — primary form.","s":"∀ {N : Type u_1} [inst : DecidableEq N] [inst_1 : Fintype N] {Ω : N → Type u_2} [inst_2 : (n : N) → MeasurableSpace (Ω n)] [inst_3 : ∀ (n : Causalean.SWIGNode N), StandardBorelSpace (Causalean.swigΩ Ω n)] [inst_4 : ∀ (n : Causalean.SWIGNode N), Nonempty (Causalean.swigΩ Ω n)] (M : Causalean.SCM N Ω) (X : Finset N) (hObs : ∀ D ∈ X, Causalean.SWIGNode.random D ∈ M.observed) (hFix : ∀ D ∈ X, Causalean.SWIGNode.fixed D ∉ M.fixed) (Y Z : Finset (Causalean.SWIGNode N)) (hY : Y ⊆ M.observed) (hZ : Z ⊆ M.observed), M.backdoorCriterion X hObs hFix Y Z → Disjoint Y (Finset.image Causalean.SWIGNode.random X) → Disjoint (Finset.image Causalean.SWIGNode.random X) Z → ∀ (s0 : M.FixedValues), (∀ (s : (M.fixSet X hObs hFix).FixedValues), Causalean.SCM.ID.Rule2JointOverlap M X hObs hFix Z (Finset.union_subset (Finset.image_subset_iff.mpr hObs) hZ) s) → M.BackdoorPositivityAE X Z hZ (Finset.union_subset (Finset.image_subset_iff.mpr hObs) hZ) s0 → (M.treatmentMarginal X (LE.le.trans Finset.subset_union_left (Finset.union_subset (Finset.image_subset_iff.mpr hObs) hZ)) s0).compProd (M.doKernelY X hObs hFix Y hY s0) = (M.treatmentMarginal X (LE.le.trans Finset.subset_union_left (Finset.union_subset (Finset.image_subset_iff.mpr hObs) hZ)) s0).compProd (M.adjustmentKernelY X hObs hFix Y Z hY hZ s0)","l":190,"f":"Causalean/SCM/ID/Backdoor.lean"},{"n":"Causalean.SCM.backdoor_identifiable_ae","k":"theorem","m":"Causalean.SCM.ID.Backdoor","a":"SCM/ID","t":1,"d":"Backdoor identifiability, a.e. in the treatment value (cross-SCM corollary). Fix two causal models M₁, M₂ that share the same SWIG graph and the same observational kernel, with base configurations that agree under this shared identification. Fix a treatment set X valid in both models and disjoint outcome/adjustment sets Y, Z that are observed in both models and disjoint from the treatment's random nodes, such that Z satisfies the backdoor criterion for (X,Y) in each model separately, with overlap holding along every post-intervention slice in each model and product positivity of the observational treatment and adjustment-set marginals holding in each model. Then for almost every treatment value t under the shared observational treatment marginal, the post-intervention outcome distribution computed in M₁ at t equals the one computed in M₂ at t, since each model individually satisfies backdoor completeness and the two share the same graph and observational kernel.","s":"∀ {N : Type u_1} [inst : DecidableEq N] [inst_1 : Fintype N] {Ω : N → Type u_2} [inst_2 : (n : N) → MeasurableSpace (Ω n)] [∀ (n : Causalean.SWIGNode N), StandardBorelSpace (Causalean.swigΩ Ω n)] [∀ (n : Causalean.SWIGNode N), Nonempty (Causalean.swigΩ Ω n)] (M₁ M₂ : Causalean.SCM N Ω), M₁.toSWIGGraph = M₂.toSWIGGraph → ∀ (X : Finset N) (Y Z : Finset (Causalean.SWIGNode N)) (hObs₁ : ∀ D ∈ X, Causalean.SWIGNode.random D ∈ M₁.observed) (hFix₁ : ∀ D ∈ X, Causalean.SWIGNode.fixed D ∉ M₁.fixed) (hObs₂ : ∀ D ∈ X, Causalean.SWIGNode.random D ∈ M₂.observed) (hFix₂ : ∀ D ∈ X, Causalean.SWIGNode.fixed D ∉ M₂.fixed) (hY₁ : Y ⊆ M₁.observed) (hZ₁ : Z ⊆ M₁.observed) (hY₂ : Y ⊆ M₂.observed) (hZ₂ : Z ⊆ M₂.observed), M₁.backdoorCriterion X hObs₁ hFix₁ Y Z → M₂.backdoorCriterion X hObs₂ hFix₂ Y Z → Disjoint Y (Finset.image Causalean.SWIGNode.random X) → Disjoint (Finset.image Causalean.SWIGNode.random X) Z → ∀ (s0₁ : M₁.FixedValues) (s0₂ : M₂.FixedValues), (∀ (s : (M₁.fixSet X hObs₁ hFix₁).FixedValues), Causalean.SCM.ID.Rule2JointOverlap M₁ X hObs₁ hFix₁ Z (Finset.union_subset (Finset.image_subset_iff.mpr hObs₁) hZ₁) s) → (∀ (s : (M₂.fixSet X hObs₂ hFix₂).FixedValues), Causalean.SCM.ID.Rule2JointOverlap M₂ X hObs₂ hFix₂ Z (Finset.union_subset (Finset.image_subset_iff.mpr hObs₂) hZ₂) s) → M₁.BackdoorPositivityAE X Z hZ₁ (Finset.union_subset (Finset.image_subset_iff.mpr hObs₁) hZ₁) s0₁ → M₂.BackdoorPositivityAE X Z hZ₂ (Finset.union_subset (Finset.image_subset_iff.mpr hObs₂) hZ₂) s0₂ → M₁.obsKernel ≍ M₂.obsKernel → s0₁ ≍ s0₂ → ⇑(M₁.doKernelY X hObs₁ hFix₁ Y hY₁ s0₁) =ᵐ[M₁.treatmentMarginal X (LE.le.trans Finset.subset_union_left (Finset.union_subset (Finset.image_subset_iff.mpr hObs₁) hZ₁)) s0₁] ⇑(M₂.doKernelY X hObs₂ hFix₂ Y hY₂ s0₂)","l":398,"f":"Causalean/SCM/ID/Backdoor.lean"},{"n":"Causalean.SCM.backdoor_rule2_ae","k":"theorem","m":"Causalean.SCM.ID.Toolkit.Derivation","a":"SCM/ID/Toolkit","t":2,"d":"Rule-2 applicator (backdoor form).","s":"∀ {N : Type u_1} [inst : DecidableEq N] [inst_1 : Fintype N] {Ω : N → Type u_2} [inst_2 : (n : N) → MeasurableSpace (Ω n)] [inst_3 : ∀ (n : Causalean.SWIGNode N), StandardBorelSpace (Causalean.swigΩ Ω n)] [inst_4 : ∀ (n : Causalean.SWIGNode N), Nonempty (Causalean.swigΩ Ω n)] (M : Causalean.SCM N Ω) (X : Finset N) (hObs : ∀ D ∈ X, Causalean.SWIGNode.random D ∈ M.observed) (hFix : ∀ D ∈ X, Causalean.SWIGNode.fixed D ∉ M.fixed) (Y Z : Finset (Causalean.SWIGNode N)) (hY : Y ⊆ M.observed) (hZ : Z ⊆ M.observed) (hXr : Finset.image Causalean.SWIGNode.random X ⊆ M.observed) (hXrZ : Finset.image Causalean.SWIGNode.random X ∪ Z ⊆ M.observed), Disjoint Y (Finset.image Causalean.SWIGNode.random X) → Disjoint (Finset.image Causalean.SWIGNode.random X) Z → M.backdoorCriterion X hObs hFix Y Z → ∀ (s0 : M.FixedValues), (∀ (s : (M.fixSet X hObs hFix).FixedValues), Causalean.SCM.ID.Rule2JointOverlap M X hObs hFix Z hXrZ s) → (MeasureTheory.Measure.map (fun p => Causalean.valuesUnionMk p.1 p.2) ((MeasureTheory.Measure.map (Causalean.valuesProjection hXr) (M.obsKernel s0)).compProd (ProbabilityTheory.Kernel.const (Causalean.ValuesOn (Finset.image Causalean.SWIGNode.random X) (Causalean.swigΩ Ω)) (MeasureTheory.Measure.map (Causalean.valuesProjection hZ) (M.obsKernel s0))))).AbsolutelyContinuous (MeasureTheory.Measure.map (Causalean.valuesProjection hXrZ) (M.obsKernel s0)) → ∀ᵐ (p : Causalean.ValuesOn (Finset.image Causalean.SWIGNode.random X) (Causalean.swigΩ Ω) × Causalean.ValuesOn Z (Causalean.swigΩ Ω)) ∂(MeasureTheory.Measure.map (Causalean.valuesProjection hXr) (M.obsKernel s0)).compProd (ProbabilityTheory.Kernel.const (Causalean.ValuesOn (Finset.image Causalean.SWIGNode.random X) (Causalean.swigΩ Ω)) (MeasureTheory.Measure.map (Causalean.valuesProjection hZ) (M.obsKernel s0))), ((M.fixSet X hObs hFix).obsCondKernel Y Z (Eq.symm (Causalean.SCM.fixSet_observed M X hObs hFix) ▸ hY) (Eq.symm (Causalean.SCM.fixSet_observed M X hObs hFix) ▸ hZ)) (M.fixSetExtend X hObs hFix s0 p.1, p.2) = (M.obsCondKernel Y (Finset.image Causalean.SWIGNode.random X ∪ Z) hY hXrZ) (s0, Causalean.valuesUnionMk p.1 p.2)","l":138,"f":"Causalean/SCM/ID/Toolkit/Derivation.lean"},{"n":"Causalean.SCM.backdoor_rule3_Z_marginal","k":"theorem","m":"Causalean.SCM.ID.BackdoorCriterion","a":"SCM/ID","t":1,"d":"Backdoor Rule-3 leg. Fix a causal model M and a treatment set X whose random copy is observed and whose fixed copy is not already held fixed in M, and a set Z that is observed in M such that no node of Z is a descendant of any treatment random node — the non-descendant clause of the backdoor criterion. Then, at any post-intervention configuration s_post, the Z-marginal of the post-intervention observational law at s_post equals the Z-marginal of the original observational law at the pre-intervention configuration underlying s_post.","s":"∀ {N : Type u_1} [inst : DecidableEq N] [inst_1 : Fintype N] {Ω : N → Type u_2} [inst_2 : (n : N) → MeasurableSpace (Ω n)] (M : Causalean.SCM N Ω) (X : Finset N) (hX_obs : ∀ D ∈ X, Causalean.SWIGNode.random D ∈ M.observed) (hX_fixed : ∀ D ∈ X, Causalean.SWIGNode.fixed D ∉ M.fixed) (Z : Finset (Causalean.SWIGNode N)) (hZ : Z ⊆ M.observed), (∀ z ∈ Z, ∀ D ∈ X, ¬M.dag.isAncestor (Causalean.SWIGNode.random D) z) → ∀ (s_post : (M.fixSet X hX_obs hX_fixed).FixedValues), MeasureTheory.Measure.map (Causalean.valuesProjection (Eq.symm (Causalean.SCM.fixSet_observed M X hX_obs hX_fixed) ▸ hZ)) ((M.fixSet X hX_obs hX_fixed).obsKernel s_post) = MeasureTheory.Measure.map (Causalean.valuesProjection hZ) (M.obsKernel (M.fixSetProj X hX_obs hX_fixed s_post))","l":112,"f":"Causalean/SCM/ID/BackdoorCriterion.lean"},{"n":"Causalean.SCM.bbZAncestors_fixSet_transport","k":"theorem","m":"Causalean.SCM.Do.CutsetDSep","a":"SCM/Do","t":2,"d":"A node in the Bayes-ball ancestor set of a target set and the model's fixed nodes remains in the corresponding ancestor set after intervention, once the fixed treatment copies are added to the target set, provided no treatment's random copy is an ancestor of a target node.","s":"∀ {N : Type u_1} [inst : DecidableEq N] [inst_1 : Fintype N] {Ω : N → Type u_2} [inst_2 : (n : N) → MeasurableSpace (Ω n)] (M : Causalean.SCM N Ω) (Z : Finset N) (hZ_obs : ∀ D ∈ Z, Causalean.SWIGNode.random D ∈ M.observed) (hZ_fixed : ∀ D ∈ Z, Causalean.SWIGNode.fixed D ∉ M.fixed) (W : Finset (Causalean.SWIGNode N)), (∀ D ∈ Z, ∀ w ∈ W, ¬M.dag.isAncestor (Causalean.SWIGNode.random D) w) → ∀ {m : Causalean.SWIGNode N}, m ∈ M.dag.bbZAncestors (W ∪ M.fixed) → m ∈ (M.fixSet Z hZ_obs hZ_fixed).dag.bbZAncestors (W ∪ M.fixed ∪ Finset.image Causalean.SWIGNode.fixed Z)","l":256,"f":"Causalean/SCM/Do/CutsetDSep.lean"},{"n":"Causalean.SCM.cComponentDensityFactor","k":"def","m":"Causalean.SCM.ID.Density.CComponentDensity","a":"SCM/ID/Density","t":1,"d":"The per-c-component density factor: the product of the one-node conditional density factors over exactly the observed nodes lying in the c-component C. This is the density-side analogue of Tian's c-factor Q[C].","s":"{N : Type u_1} → [inst : DecidableEq N] → [inst_1 : Fintype N] → {Ω : N → Type u_2} → [inst_2 : (n : N) → MeasurableSpace (Ω n)] → (M : Causalean.SCM N Ω) → Causalean.SCM.ReferenceMeasures Ω → M.FixedValues → Finset (Causalean.SWIGNode N) → [∀ (s' : M.FixedValues), MeasureTheory.IsFiniteMeasure (M.obsKernel s')] → [∀ (k : ℕ) (hk : k < M.observed.card), StandardBorelSpace (Causalean.ValuesOn {↑(M.observedAt ⟨k, hk⟩)} (Causalean.swigΩ Ω))] → [∀ (k : ℕ) (hk : k < M.observed.card), Nonempty (Causalean.ValuesOn {↑(M.observedAt ⟨k, hk⟩)} (Causalean.swigΩ Ω))] → [∀ (k : ℕ), MeasurableSpace.CountableOrCountablyGenerated M.FixedValues (Causalean.ValuesOn (M.prefixNodes k) (Causalean.swigΩ Ω))] → Causalean.ValuesOn M.observed (Causalean.swigΩ Ω) → ENNReal","l":31,"f":"Causalean/SCM/ID/Density/CComponentDensity.lean"},{"n":"Causalean.SCM.cComponentDensityFactor_heq_of_obsKernel_heq","k":"theorem","m":"Causalean.SCM.ID.Density.CComponentDensity","a":"SCM/ID/Density","t":2,"d":"Cross-model c-component density-factor transport. If two structural causal models share the same SWIG graph and have heterogeneously-equal observational kernels, then every c-component density factor recovered from the observational chain-rule density is heterogeneously equal across the two models.","s":"∀ {N : Type u_1} [inst : DecidableEq N] [inst_1 : Fintype N] {Ω : N → Type u_2} [inst_2 : (n : N) → MeasurableSpace (Ω n)] (M₁ M₂ : Causalean.SCM N Ω) (ref : Causalean.SCM.ReferenceMeasures Ω) (C : Finset (Causalean.SWIGNode N)), M₁.toSWIGGraph = M₂.toSWIGGraph → M₁.obsKernel ≍ M₂.obsKernel → ∀ [inst_3 : ∀ (s' : M₁.FixedValues), MeasureTheory.IsFiniteMeasure (M₁.obsKernel s')] [inst_4 : ∀ (k : ℕ) (hk : k < M₁.observed.card), StandardBorelSpace (Causalean.ValuesOn {↑(M₁.observedAt ⟨k, hk⟩)} (Causalean.swigΩ Ω))] [inst_5 : ∀ (k : ℕ) (hk : k < M₁.observed.card), Nonempty (Causalean.ValuesOn {↑(M₁.observedAt ⟨k, hk⟩)} (Causalean.swigΩ Ω))] [inst_6 : ∀ (k : ℕ), MeasurableSpace.CountableOrCountablyGenerated M₁.FixedValues (Causalean.ValuesOn (M₁.prefixNodes k) (Causalean.swigΩ Ω))] [inst_7 : ∀ (s' : M₂.FixedValues), MeasureTheory.IsFiniteMeasure (M₂.obsKernel s')] [inst_8 : ∀ (k : ℕ) (hk : k < M₂.observed.card), StandardBorelSpace (Causalean.ValuesOn {↑(M₂.observedAt ⟨k, hk⟩)} (Causalean.swigΩ Ω))] [inst_9 : ∀ (k : ℕ) (hk : k < M₂.observed.card), Nonempty (Causalean.ValuesOn {↑(M₂.observedAt ⟨k, hk⟩)} (Causalean.swigΩ Ω))] [inst_10 : ∀ (k : ℕ), MeasurableSpace.CountableOrCountablyGenerated M₂.FixedValues (Causalean.ValuesOn (M₂.prefixNodes k) (Causalean.swigΩ Ω))], (fun s => M₁.cComponentDensityFactor ref s C) ≍ fun s => M₂.cComponentDensityFactor ref s C","l":86,"f":"Causalean/SCM/ID/Density/CComponentDensity.lean"},{"n":"Causalean.SCM.c_component_factorization","k":"theorem","m":"Causalean.SCM.ID.GraphicalThms.CComponentFactor","a":"SCM/ID/GraphicalThms","t":1,"d":"Theorem (Tian 2002, per-node chain-rule factorization). For a structural causal model M, at a fixed assignment s, its observational kernel equals the full chain-rule product of one-node conditional kernels along the topological order of observed nodes. Tian's Pa⁺(V) is interpreted as the full observed history before V, so this statement is the ordinary iterated-disintegration factorization and does not use graphical Markov or do-calculus reasoning.","s":"∀ {N : Type u_1} [inst : DecidableEq N] [inst_1 : Fintype N] {Ω : N → Type u_2} [inst_2 : (n : N) → MeasurableSpace (Ω n)] (M : Causalean.SCM N Ω) (s : M.FixedValues) [inst_3 : ∀ (k : ℕ) (hk : k < M.observed.card), StandardBorelSpace (Causalean.ValuesOn {↑(M.observedAt ⟨k, hk⟩)} (Causalean.swigΩ Ω))] [inst_4 : ∀ (k : ℕ) (hk : k < M.observed.card), Nonempty (Causalean.ValuesOn {↑(M.observedAt ⟨k, hk⟩)} (Causalean.swigΩ Ω))] [inst_5 : ∀ (k : ℕ), MeasurableSpace.CountableOrCountablyGenerated M.FixedValues (Causalean.ValuesOn (M.prefixNodes k) (Causalean.swigΩ Ω))], M.obsKernel s = M.qFactorProduct s","l":130,"f":"Causalean/SCM/ID/GraphicalThms/CComponentFactor.lean"},{"n":"Causalean.SCM.compProd_deterministic_apply","k":"theorem","m":"Causalean.SCM.Factored.EvalMapCorrespond","a":"SCM/Factored","t":2,"d":"Composing a kernel with a deterministic second kernel gives the distribution obtained by drawing from the first kernel and appending the deterministic output to that draw. This map-valued identity is useful when constructing factored kernels recursively.","s":"∀ {α : Type u_1} {β : Type u_2} {γ : Type u_3} [inst : MeasurableSpace α] [inst_1 : MeasurableSpace β] [inst_2 : MeasurableSpace γ] (κ : ProbabilityTheory.Kernel α β) [ProbabilityTheory.IsSFiniteKernel κ] {f : α × β → γ} (hf : Measurable f) (a : α), (κ.compProd (ProbabilityTheory.Kernel.deterministic f hf)) a = MeasureTheory.Measure.map (fun b => (b, f (a, b))) (κ a)","l":140,"f":"Causalean/SCM/Factored/EvalMapCorrespond.lean"},{"n":"Causalean.SCM.compProd_singleton_mass","k":"theorem","m":"Causalean.SCM.ID.Density.MassBridge","a":"SCM/ID/Density","t":2,"d":"The singleton mass of a measure-kernel composition product factors as the conditioning singleton mass times the fibre singleton mass.","s":"∀ {α : Type u_3} {β : Type u_4} [inst : MeasurableSpace α] [inst_1 : MeasurableSpace β] [MeasurableSingletonClass α] [MeasurableSingletonClass β] (μ : MeasureTheory.Measure α) [MeasureTheory.SFinite μ] (κ : ProbabilityTheory.Kernel α β) [ProbabilityTheory.IsSFiniteKernel κ] (b : α) (y : β), (μ.compProd κ) {(b, y)} = μ {b} * (κ b) {y}","l":101,"f":"Causalean/SCM/ID/Density/MassBridge.lean"},{"n":"Causalean.SCM.condDistrib_eq_of_map_prod_eq","k":"theorem","m":"Causalean.SCM.Do.Rule3Conditional","a":"SCM/Do","t":2,"d":"Two conditional distributions with the same joint law are equal.","s":"∀ {α : Type u_3} {α' : Type u_4} {β : Type u_5} {γ : Type u_6} [inst : MeasurableSpace α] [inst_1 : MeasurableSpace α'] [inst_2 : MeasurableSpace β] [inst_3 : MeasurableSpace γ] [inst_4 : StandardBorelSpace β] [inst_5 : Nonempty β] {X : α → γ} {Y : α → β} {X' : α' → γ} {Y' : α' → β} {μ : MeasureTheory.Measure α} {ν : MeasureTheory.Measure α'} [inst_6 : MeasureTheory.IsFiniteMeasure μ] [inst_7 : MeasureTheory.IsFiniteMeasure ν], MeasureTheory.Measure.map (fun a => (X a, Y a)) μ = MeasureTheory.Measure.map (fun a => (X' a, Y' a)) ν → ProbabilityTheory.condDistrib Y X μ = ProbabilityTheory.condDistrib Y' X' ν","l":52,"f":"Causalean/SCM/Do/Rule3Conditional.lean"},{"n":"Causalean.SCM.condDistrib_fixSet_cross_SCM_bridge","k":"theorem","m":"Causalean.SCM.Do.Rule2AE","a":"SCM/Do","t":2,"d":"Joint-μ_C-a.e. cross-SCM condDistrib bridge (isolated hard lemma).","s":"∀ {N : Type u_1} [inst : DecidableEq N] [inst_1 : Fintype N] {Ω : N → Type u_2} [inst_2 : (n : N) → MeasurableSpace (Ω n)] (M' : Causalean.SCM N Ω) (Z : Finset N) (hZ_obs : ∀ D ∈ Z, Causalean.SWIGNode.random D ∈ M'.observed) (hZ_fixed : ∀ D ∈ Z, Causalean.SWIGNode.fixed D ∉ M'.fixed) (Y W : Finset (Causalean.SWIGNode N)) (hY : Y ⊆ M'.observed) (hW : W ⊆ M'.observed) (hZr : Finset.image Causalean.SWIGNode.random Z ⊆ M'.observed) (hZrW : Finset.image Causalean.SWIGNode.random Z ∪ W ⊆ M'.observed), (M'.fixSet Z hZ_obs hZ_fixed).dag.dSep Y (Finset.image Causalean.SWIGNode.random Z) (W ∪ (M'.fixSet Z hZ_obs hZ_fixed).fixed) → (∀ z ∈ Z, ∀ v ∈ W, ¬(M'.fixSet Z hZ_obs hZ_fixed).dag.isAncestor (Causalean.SWIGNode.fixed z) v) → (∀ D ∈ Z, ∀ w ∈ W, ¬M'.dag.isAncestor (Causalean.SWIGNode.random D) w) → ∀ [inst_3 : ∀ (n : Causalean.SWIGNode N), StandardBorelSpace (Causalean.swigΩ Ω n)] [inst_4 : ∀ (n : Causalean.SWIGNode N), Nonempty (Causalean.swigΩ Ω n)] [MeasurableSpace.CountableOrCountablyGenerated M'.FixedValues (Causalean.ValuesOn (Finset.image Causalean.SWIGNode.random Z ∪ W) (Causalean.swigΩ Ω))] [MeasurableSpace.CountableOrCountablyGenerated (M'.fixSet Z hZ_obs hZ_fixed).FixedValues (Causalean.ValuesOn (Finset.image Causalean.SWIGNode.random Z ∪ W) (Causalean.swigΩ Ω))] [inst_7 : MeasurableSpace.CountableOrCountablyGenerated (M'.fixSet Z hZ_obs hZ_fixed).FixedValues (Causalean.ValuesOn W (Causalean.swigΩ Ω))] [MeasurableSingletonClass (Causalean.ValuesOn (Finset.image Causalean.SWIGNode.random Z ∪ W) (Causalean.swigΩ Ω))] (s0 : M'.FixedValues), (MeasureTheory.Measure.map (fun p => Causalean.valuesUnionMk p.1 p.2) ((MeasureTheory.Measure.map (Causalean.valuesProjection hZr) (M'.obsKernel s0)).compProd (ProbabilityTheory.Kernel.const (Causalean.ValuesOn (Finset.image Causalean.SWIGNode.random Z) (Causalean.swigΩ Ω)) (MeasureTheory.Measure.map (Causalean.valuesProjection hW) (M'.obsKernel s0))))).AbsolutelyContinuous (MeasureTheory.Measure.map (Causalean.valuesProjection hZrW) (M'.obsKernel s0)) → ∀ᵐ (p : Causalean.ValuesOn (Finset.image Causalean.SWIGNode.random Z) (Causalean.swigΩ Ω) × Causalean.ValuesOn W (Causalean.swigΩ Ω)) ∂(MeasureTheory.Measure.map (Causalean.valuesProjection hZr) (M'.obsKernel s0)).compProd (ProbabilityTheory.Kernel.const (Causalean.ValuesOn (Finset.image Causalean.SWIGNode.random Z) (Causalean.swigΩ Ω)) (MeasureTheory.Measure.map (Causalean.valuesProjection hW) (M'.obsKernel s0))), ((M'.fixSet Z hZ_obs hZ_fixed).obsCondKernel Y W (Eq.symm (Causalean.SCM.fixSet_observed M' Z hZ_obs hZ_fixed) ▸ hY) (Eq.symm (Causalean.SCM.fixSet_observed M' Z hZ_obs hZ_fixed) ▸ hW)) (M'.fixSetExtend Z hZ_obs hZ_fixed s0 p.1, p.2) = (ProbabilityTheory.condDistrib (Causalean.valuesProjection hY) (Causalean.valuesProjection hZrW) (M'.obsKernel s0)) (Causalean.valuesUnionMk p.1 p.2)","l":41,"f":"Causalean/SCM/Do/Rule2AE.lean"},{"n":"Causalean.SCM.condDistrib_intervention_ancestral_eq","k":"theorem","m":"Causalean.SCM.Do.Rule3","a":"SCM/Do","t":1,"d":"Rule 3 core — intervention on non-ancestors of T is irrelevant (Option B in the design; covers the simplified Z_Y = Z case of the tex's Rule 3, sufficient for backdoor / frontdoor demos). Fix a do-set Z of nodes whose random copies are observed in the base model and whose fixed nodes have not already been intervened on and a target block T of observed variables. If none of the fixed copies of Z's nodes is an ancestor, in the intervention SWIG graph, of any node in T, then the T-marginal law of the intervened model equals the T-marginal law of the base model evaluated at the corresponding fixed values.","s":"∀ {N : Type u_1} [inst : DecidableEq N] [inst_1 : Fintype N] {Ω : N → Type u_2} [inst_2 : (n : N) → MeasurableSpace (Ω n)] (M' : Causalean.SCM N Ω) (Z : Finset N) (hZ_obs : ∀ D ∈ Z, Causalean.SWIGNode.random D ∈ M'.observed) (hZ_fixed : ∀ D ∈ Z, Causalean.SWIGNode.fixed D ∉ M'.fixed) (T : Finset (Causalean.SWIGNode N)) (hT : T ⊆ M'.observed), (∀ z ∈ Z, ∀ v ∈ T, ¬(M'.fixSet Z hZ_obs hZ_fixed).dag.isAncestor (Causalean.SWIGNode.fixed z) v) → ∀ (s' : (M'.fixSet Z hZ_obs hZ_fixed).FixedValues), MeasureTheory.Measure.map (Causalean.valuesProjection (Eq.symm (Causalean.SCM.fixSet_observed M' Z hZ_obs hZ_fixed) ▸ hT)) ((M'.fixSet Z hZ_obs hZ_fixed).obsKernel s') = MeasureTheory.Measure.map (Causalean.valuesProjection hT) (M'.obsKernel (M'.fixSetProj Z hZ_obs hZ_fixed s'))","l":338,"f":"Causalean/SCM/Do/Rule3.lean"},{"n":"Causalean.SCM.condDistrib_singleton_mass_eq_conditionalMass","k":"theorem","m":"Causalean.SCM.ID.Density.MassBridge","a":"SCM/ID/Density","t":1,"d":"For a measurable map Y and a measurable map Z out of a finite measure space, at a conditioning value z with nonzero pushforward mass under Z, the singleton mass that Mathlib's conditional distribution condDistrib Y Z μ assigns to a value y at z equals the discrete conditional-mass ratio conditionalMass, computed from the joint pushforward law of (Y, Z), evaluated at (y, z).","s":"∀ {α : Type u_3} {β : Type u_4} {γ : Type u_5} [inst : MeasurableSpace α] [inst_1 : MeasurableSpace β] [inst_2 : MeasurableSpace γ] [inst_3 : StandardBorelSpace γ] [inst_4 : Nonempty γ] [MeasurableSingletonClass β] [MeasurableSingletonClass γ] {μ : MeasureTheory.Measure α} [inst_7 : MeasureTheory.IsFiniteMeasure μ] {Y : α → γ} {Z : α → β}, Measurable Y → Measurable Z → ∀ (z : β) (y : γ), (MeasureTheory.Measure.map Z μ) {z} ≠ 0 → ((ProbabilityTheory.condDistrib Y Z μ) z) {y} = Causalean.SCM.ID.DiscreteID.conditionalMass (MeasureTheory.Measure.map (fun ω => (Y ω, Z ω)) μ) y z","l":160,"f":"Causalean/SCM/ID/Density/MassBridge.lean"},{"n":"Causalean.SCM.condDistrib_singleton_mass_of_ne_zero","k":"theorem","m":"Causalean.SCM.ID.Density.MassBridge","a":"SCM/ID/Density","t":2,"d":"Mathlib's measure-level conditional distribution has singleton mass equal to the corresponding joint singleton mass divided by the conditioning mass.","s":"∀ {α : Type u_3} {β : Type u_4} {γ : Type u_5} [inst : MeasurableSpace α] [inst_1 : MeasurableSpace β] [inst_2 : MeasurableSpace γ] [inst_3 : StandardBorelSpace γ] [inst_4 : Nonempty γ] [MeasurableSingletonClass β] [MeasurableSingletonClass γ] {μ : MeasureTheory.Measure α} [inst_7 : MeasureTheory.IsFiniteMeasure μ] {Y : α → γ} {Z : α → β}, Measurable Y → ∀ (z : β) (y : γ), (MeasureTheory.Measure.map Z μ) {z} ≠ 0 → ((ProbabilityTheory.condDistrib Y Z μ) z) {y} = (MeasureTheory.Measure.map (fun ω => (Z ω, Y ω)) μ) {(z, y)} / (MeasureTheory.Measure.map Z μ) {z}","l":141,"f":"Causalean/SCM/ID/Density/MassBridge.lean"},{"n":"Causalean.SCM.condIndepFun_of_map","k":"theorem","m":"Causalean.SCM.Do.LocalMarkov","a":"SCM/Do","t":2,"d":"Pushforward bridge for CondIndepFun.","s":"∀ {α : Type u_1} [inst : MeasurableSpace α] [inst_1 : StandardBorelSpace α] {β : Type u_2} [inst_2 : MeasurableSpace β] [inst_3 : StandardBorelSpace β] {γ : Type u_3} [inst_4 : MeasurableSpace γ] [StandardBorelSpace γ] [Nonempty γ] {δ : Type u_4} [inst_7 : MeasurableSpace δ] [StandardBorelSpace δ] [Nonempty δ] {ε : Type u_5} [inst_10 : MeasurableSpace ε] {φ : α → β} (hφ : Measurable φ) {X : β → γ}, Measurable X → ∀ {Y : β → δ}, Measurable Y → ∀ {Z : β → ε} (hZ : Measurable Z) {ν : MeasureTheory.Measure α} [inst_11 : MeasureTheory.IsFiniteMeasure ν] [inst_12 : MeasureTheory.IsFiniteMeasure (MeasureTheory.Measure.map φ ν)], ProbabilityTheory.CondIndepFun (MeasurableSpace.comap (Z ∘ φ) inferInstance) (Measurable.comap_le (Measurable.comp hZ hφ)) (X ∘ φ) (Y ∘ φ) ν → ProbabilityTheory.CondIndepFun (MeasurableSpace.comap Z inferInstance) (Measurable.comap_le hZ) X Y (MeasureTheory.Measure.map φ ν)","l":54,"f":"Causalean/SCM/Do/LocalMarkov.lean"},{"n":"Causalean.SCM.condIndepFun_of_shared_base_valuesProjection_pi","k":"theorem","m":"Causalean.SCM.Do.GlobalMarkov","a":"SCM/Do","t":2,"d":"Under an SCM's latent product measure, two outcomes that depend on a shared latent block and otherwise on disjoint latent blocks are conditionally independent after conditioning on any measurable summary of the shared block.","s":"∀ {N : Type uN} [inst : DecidableEq N] [inst_1 : Fintype N] {Ω : N → Type uΩ} [inst_2 : (n : N) → MeasurableSpace (Ω n)] (M : Causalean.SCM N Ω) {U Rx Ry : Finset (Causalean.SWIGNode N)} [inst_3 : StandardBorelSpace M.LatentValues] {β : Type u_1} {γ : Type u_2} {δ : Type u_3} [inst_4 : MeasurableSpace β] [inst_5 : MeasurableSpace γ] [inst_6 : MeasurableSpace δ] (hU : U ⊆ M.unobserved) (hRx : Rx ⊆ M.unobserved) (hRy : Ry ⊆ M.unobserved) {baseMap : Causalean.ValuesOn U (Causalean.swigΩ Ω) → β} (hbaseMap : Measurable baseMap) {leftMap : β × Causalean.ValuesOn Rx (Causalean.swigΩ Ω) → γ}, Measurable leftMap → ∀ {rightMap : β × Causalean.ValuesOn Ry (Causalean.swigΩ Ω) → δ}, Measurable rightMap → Disjoint U Rx → Disjoint U Ry → Disjoint Rx Ry → ProbabilityTheory.CondIndepFun (MeasurableSpace.comap (baseMap ∘ Causalean.valuesProjection hU) inferInstance) (Measurable.comap_le (Measurable.comp hbaseMap (Causalean.measurable_valuesProjection hU))) (fun ω => leftMap (baseMap (Causalean.valuesProjection hU ω), Causalean.valuesProjection hRx ω)) (fun ω => rightMap (baseMap (Causalean.valuesProjection hU ω), Causalean.valuesProjection hRy ω)) M.latentProduct","l":370,"f":"Causalean/SCM/Do/GlobalMarkov.lean"},{"n":"Causalean.SCM.condIndep_valuesProjection_contraction_axiom","k":"theorem","m":"Causalean.SCM.Do.SemiGraphoid","a":"SCM/Do","t":2,"d":"Contraction for CondIndepFun when coordinates are given by valuesProjection.","s":"∀ {N : Type uN} [inst : DecidableEq N] {Ω : N → Type uΩ} [inst_1 : (n : N) → MeasurableSpace (Ω n)] {I X Y W Z : Finset (Causalean.SWIGNode N)} (hX : X ⊆ I) (hY : Y ⊆ I) (hW : W ⊆ I) (hZ : Z ⊆ I) [inst_2 : StandardBorelSpace (Causalean.ValuesOn I (Causalean.swigΩ Ω))] {μ : MeasureTheory.Measure (Causalean.ValuesOn I (Causalean.swigΩ Ω))} [inst_3 : MeasureTheory.IsFiniteMeasure μ], ProbabilityTheory.CondIndepFun (MeasurableSpace.comap (Causalean.valuesProjection (Finset.union_subset hZ hW)) inferInstance) (Causalean.comap_valuesProjection_le (Finset.union_subset hZ hW)) (Causalean.valuesProjection hX) (Causalean.valuesProjection hY) μ → ProbabilityTheory.CondIndepFun (MeasurableSpace.comap (Causalean.valuesProjection hZ) inferInstance) (Causalean.comap_valuesProjection_le hZ) (Causalean.valuesProjection hX) (Causalean.valuesProjection hW) μ → ProbabilityTheory.CondIndepFun (MeasurableSpace.comap (Causalean.valuesProjection hZ) inferInstance) (Causalean.comap_valuesProjection_le hZ) (Causalean.valuesProjection hX) (Causalean.valuesProjection (Finset.union_subset hY hW)) μ","l":175,"f":"Causalean/SCM/Do/SemiGraphoid.lean"},{"n":"Causalean.SCM.condIndep_valuesProjection_decomposition","k":"theorem","m":"Causalean.SCM.Do.SemiGraphoid","a":"SCM/Do","t":2,"d":"Decomposition for CondIndepFun when coordinates are given by valuesProjection.","s":"∀ {N : Type uN} [inst : DecidableEq N] {Ω : N → Type uΩ} [inst_1 : (n : N) → MeasurableSpace (Ω n)] {I X Y W Z : Finset (Causalean.SWIGNode N)} (hX : X ⊆ I) (hYW : Y ∪ W ⊆ I) (hZ : Z ⊆ I) [inst_2 : StandardBorelSpace (Causalean.ValuesOn I (Causalean.swigΩ Ω))] {μ : MeasureTheory.Measure (Causalean.ValuesOn I (Causalean.swigΩ Ω))} [inst_3 : MeasureTheory.IsFiniteMeasure μ], ProbabilityTheory.CondIndepFun (MeasurableSpace.comap (Causalean.valuesProjection hZ) inferInstance) (Causalean.comap_valuesProjection_le hZ) (Causalean.valuesProjection hX) (Causalean.valuesProjection hYW) μ → ProbabilityTheory.CondIndepFun (MeasurableSpace.comap (Causalean.valuesProjection hZ) inferInstance) (Causalean.comap_valuesProjection_le hZ) (Causalean.valuesProjection hX) (Causalean.valuesProjection (LE.le.trans Finset.subset_union_left hYW)) μ","l":126,"f":"Causalean/SCM/Do/SemiGraphoid.lean"},{"n":"Causalean.SCM.condIndep_valuesProjection_subset_right","k":"theorem","m":"Causalean.SCM.Do.SemiGraphoid","a":"SCM/Do","t":2,"d":"Subset-right for CondIndepFun when coordinates are given by valuesProjection.","s":"∀ {N : Type uN} {Ω : N → Type uΩ} [inst : (n : N) → MeasurableSpace (Ω n)] {I X Y Y' Z : Finset (Causalean.SWIGNode N)} (hX : X ⊆ I) (hY : Y ⊆ I) (hY' : Y' ⊆ I) (hZ : Z ⊆ I), Y' ⊆ Y → ∀ [inst_1 : StandardBorelSpace (Causalean.ValuesOn I (Causalean.swigΩ Ω))] {μ : MeasureTheory.Measure (Causalean.ValuesOn I (Causalean.swigΩ Ω))} [inst_2 : MeasureTheory.IsFiniteMeasure μ], ProbabilityTheory.CondIndepFun (MeasurableSpace.comap (Causalean.valuesProjection hZ) inferInstance) (Causalean.comap_valuesProjection_le hZ) (Causalean.valuesProjection hX) (Causalean.valuesProjection hY) μ → ProbabilityTheory.CondIndepFun (MeasurableSpace.comap (Causalean.valuesProjection hZ) inferInstance) (Causalean.comap_valuesProjection_le hZ) (Causalean.valuesProjection hX) (Causalean.valuesProjection hY') μ","l":102,"f":"Causalean/SCM/Do/SemiGraphoid.lean"},{"n":"Causalean.SCM.condIndep_valuesProjection_symm","k":"theorem","m":"Causalean.SCM.Do.SemiGraphoid","a":"SCM/Do","t":2,"d":"Symmetry for CondIndepFun when coordinates are given by valuesProjection.","s":"∀ {N : Type uN} {Ω : N → Type uΩ} [inst : (n : N) → MeasurableSpace (Ω n)] {I X Y Z : Finset (Causalean.SWIGNode N)} (hX : X ⊆ I) (hY : Y ⊆ I) (hZ : Z ⊆ I) [inst_1 : StandardBorelSpace (Causalean.ValuesOn I (Causalean.swigΩ Ω))] {μ : MeasureTheory.Measure (Causalean.ValuesOn I (Causalean.swigΩ Ω))} [inst_2 : MeasureTheory.IsFiniteMeasure μ], ProbabilityTheory.CondIndepFun (MeasurableSpace.comap (Causalean.valuesProjection hZ) inferInstance) (Causalean.comap_valuesProjection_le hZ) (Causalean.valuesProjection hX) (Causalean.valuesProjection hY) μ → ProbabilityTheory.CondIndepFun (MeasurableSpace.comap (Causalean.valuesProjection hZ) inferInstance) (Causalean.comap_valuesProjection_le hZ) (Causalean.valuesProjection hY) (Causalean.valuesProjection hX) μ","l":80,"f":"Causalean/SCM/Do/SemiGraphoid.lean"},{"n":"Causalean.SCM.condIndep_valuesProjection_weak_union_axiom","k":"theorem","m":"Causalean.SCM.Do.SemiGraphoid","a":"SCM/Do","t":2,"d":"Weak union for CondIndepFun when coordinates are given by valuesProjection.","s":"∀ {N : Type uN} [inst : DecidableEq N] {Ω : N → Type uΩ} [inst_1 : (n : N) → MeasurableSpace (Ω n)] {I X Y W Z : Finset (Causalean.SWIGNode N)} (hX : X ⊆ I) (hYW : Y ∪ W ⊆ I) (hZ : Z ⊆ I) [inst_2 : StandardBorelSpace (Causalean.ValuesOn I (Causalean.swigΩ Ω))] {μ : MeasureTheory.Measure (Causalean.ValuesOn I (Causalean.swigΩ Ω))} [inst_3 : MeasureTheory.IsFiniteMeasure μ], ProbabilityTheory.CondIndepFun (MeasurableSpace.comap (Causalean.valuesProjection hZ) inferInstance) (Causalean.comap_valuesProjection_le hZ) (Causalean.valuesProjection hX) (Causalean.valuesProjection hYW) μ → ProbabilityTheory.CondIndepFun (MeasurableSpace.comap (Causalean.valuesProjection (Finset.union_subset hZ (LE.le.trans Finset.subset_union_right hYW))) inferInstance) (Causalean.comap_valuesProjection_le (Finset.union_subset hZ (LE.le.trans Finset.subset_union_right hYW))) (Causalean.valuesProjection hX) (Causalean.valuesProjection (LE.le.trans Finset.subset_union_left hYW)) μ","l":149,"f":"Causalean/SCM/Do/SemiGraphoid.lean"},{"n":"Causalean.SCM.condKernel_singleton_mass_of_ne_zero","k":"theorem","m":"Causalean.SCM.ID.Density.MassBridge","a":"SCM/ID/Density","t":2,"d":"A disintegration conditional kernel at a positive conditioning atom is the joint singleton mass divided by the conditioning singleton mass.","s":"∀ {α : Type u_3} {β : Type u_4} [inst : MeasurableSpace α] [inst_1 : MeasurableSpace β] [inst_2 : StandardBorelSpace β] [inst_3 : Nonempty β] [MeasurableSingletonClass α] [MeasurableSingletonClass β] (ρ : MeasureTheory.Measure (α × β)) [inst_6 : MeasureTheory.IsFiniteMeasure ρ] (b : α) (y : β), ρ.fst {b} ≠ 0 → (ρ.condKernel b) {y} = ρ {(b, y)} / ρ.fst {b}","l":125,"f":"Causalean/SCM/ID/Density/MassBridge.lean"},{"n":"Causalean.SCM.countingRef","k":"def","m":"Causalean.SCM.ID.Density.CountingReference","a":"SCM/ID/Density","t":1,"d":"The counting reference family: each SWIG-node coordinate carries the counting measure. On countable value spaces this is a σ-finite measure.","s":"{N : Type u_1} → {Ω : N → Type u_2} → [inst : (n : N) → MeasurableSpace (Ω n)] → [∀ (sn : Causalean.SWIGNode N), Countable (Causalean.swigΩ Ω sn)] → [∀ (n : N), MeasurableSingletonClass (Ω n)] → Causalean.SCM.ReferenceMeasures Ω","l":26,"f":"Causalean/SCM/ID/Density/CountingReference.lean"},{"n":"Causalean.SCM.cutsetLatent","k":"def","m":"Causalean.SCM.Model.CutsetLatent","a":"SCM/Model","t":1,"d":"The latent cutset contains the unobserved roots that can drive a target set along paths avoiding an override block.","s":"{N : Type u_2} → [inst : DecidableEq N] → [inst_1 : Fintype N] → {Ω : N → Type u_3} → [inst_2 : (n : N) → MeasurableSpace (Ω n)] → Causalean.SCM N Ω → Finset (Causalean.SWIGNode N) → Finset (Causalean.SWIGNode N) → Finset (Causalean.SWIGNode N)","l":173,"f":"Causalean/SCM/Model/CutsetLatent.lean"},{"n":"Causalean.SCM.cutsetLatent_dSep_of_dSep","k":"theorem","m":"Causalean.SCM.Do.CutsetDSep","a":"SCM/Do","t":1,"d":"Concatenation d-separation for the latent cutset. Suppose W consists of observed nodes and F consists of fixed nodes, and that the target set Y is d-separated from the randomised do-block Zr given W together with F. Then the latent cutset cutsetLatent Y (Zr ∪ W) — the latent roots reaching Y along a directed path whose interior avoids Zr ∪ W — is likewise d-separated from Zr given the same W ∪ F.","s":"∀ {N : Type u_1} [inst : DecidableEq N] [inst_1 : Fintype N] {Ω : N → Type u_2} [inst_2 : (n : N) → MeasurableSpace (Ω n)] (M : Causalean.SCM N Ω) (Y Zr W F : Finset (Causalean.SWIGNode N)), W ⊆ M.observed → F ⊆ M.fixed → M.dag.dSep Y Zr (W ∪ F) → M.dag.dSep (M.cutsetLatent Y (Zr ∪ W)) Zr (W ∪ F)","l":44,"f":"Causalean/SCM/Do/CutsetDSep.lean"},{"n":"Causalean.SCM.cutsetLatent_dSep_of_fixSet_dSep","k":"theorem","m":"Causalean.SCM.Do.CutsetDSep","a":"SCM/Do","t":1,"d":"Cross-model concatenation d-separation for the latent cutset. Suppose that, in the intervened model M.fixSet Z _ _, the target set Y is d-separated from the randomised do-block Zr = Z.image .random given the adjustment set W together with the post-intervention fixed block, where W consists of observed nodes and no w ∈ W is, in the base graph M.dag, a descendant of any treatment's random copy .random D (D ∈ Z) — backdoor criterion (i). Then, in the base model M, the latent cutset cutsetLatent Y (Zr ∪ W) is d-separated from Zr given W ∪ M.fixed.","s":"∀ {N : Type u_1} [inst : DecidableEq N] [inst_1 : Fintype N] {Ω : N → Type u_2} [inst_2 : (n : N) → MeasurableSpace (Ω n)] (M : Causalean.SCM N Ω) (Z : Finset N) (hZ_obs : ∀ D ∈ Z, Causalean.SWIGNode.random D ∈ M.observed) (hZ_fixed : ∀ D ∈ Z, Causalean.SWIGNode.fixed D ∉ M.fixed) (Y W : Finset (Causalean.SWIGNode N)), W ⊆ M.observed → (∀ D ∈ Z, ∀ w ∈ W, ¬M.dag.isAncestor (Causalean.SWIGNode.random D) w) → (M.fixSet Z hZ_obs hZ_fixed).dag.dSep Y (Finset.image Causalean.SWIGNode.random Z) (W ∪ (M.fixSet Z hZ_obs hZ_fixed).fixed) → M.dag.dSep (M.cutsetLatent Y (Finset.image Causalean.SWIGNode.random Z ∪ W)) (Finset.image Causalean.SWIGNode.random Z) (W ∪ M.fixed)","l":768,"f":"Causalean/SCM/Do/CutsetDSep.lean"},{"n":"Causalean.SCM.cutsetLatent_subset","k":"theorem","m":"Causalean.SCM.Model.CutsetLatent","a":"SCM/Model","t":2,"d":"The latent cutset is contained in the model's unobserved node set.","s":"∀ {N : Type u_2} [inst : DecidableEq N] [inst_1 : Fintype N] {Ω : N → Type u_3} [inst_2 : (n : N) → MeasurableSpace (Ω n)] (M : Causalean.SCM N Ω) (Y C : Finset (Causalean.SWIGNode N)), M.cutsetLatent Y C ⊆ M.unobserved","l":202,"f":"Causalean/SCM/Model/CutsetLatent.lean"},{"n":"Causalean.SCM.cutset_condIndep_condDistrib","k":"theorem","m":"Causalean.SCM.Do.Rule2Kernel.WitnessBridge","a":"SCM/Do/Rule2Kernel","t":2,"d":"Cut-set conditional independence, in condDistrib-pair form.","s":"∀ {N : Type u_1} [inst : DecidableEq N] [inst_1 : Fintype N] {Ω : N → Type u_2} [inst_2 : (n : N) → MeasurableSpace (Ω n)] (M : Causalean.SCM N Ω) [StandardBorelSpace M.RandomValues] [∀ (n : Causalean.SWIGNode N), StandardBorelSpace (Causalean.swigΩ Ω n)] [∀ (n : Causalean.SWIGNode N), Nonempty (Causalean.swigΩ Ω n)] [∀ (s : M.FixedValues), MeasureTheory.IsFiniteMeasure (M.jointKernel s)] (Y W : Finset (Causalean.SWIGNode N)) (Z : Finset N) [inst_7 : StandardBorelSpace (Causalean.ValuesOn (M.cutsetLatent Y (Finset.image Causalean.SWIGNode.random Z ∪ W)) (Causalean.swigΩ Ω))] [inst_8 : Nonempty (Causalean.ValuesOn (M.cutsetLatent Y (Finset.image Causalean.SWIGNode.random Z ∪ W)) (Causalean.swigΩ Ω))] (hZr : Finset.image Causalean.SWIGNode.random Z ⊆ M.observed) (hW : W ⊆ M.observed), M.dag.dSep (M.cutsetLatent Y (Finset.image Causalean.SWIGNode.random Z ∪ W)) (Finset.image Causalean.SWIGNode.random Z) (W ∪ M.fixed) → ∀ (s : M.FixedValues), (fun p => (ProbabilityTheory.condDistrib (Causalean.valuesProjection (Causalean.SCM.cutsetLatent_subset M Y (Finset.image Causalean.SWIGNode.random Z ∪ W))) (fun ℓ => (Causalean.valuesProjection hZr (M.randomToObserved (M.evalMap s ℓ)), Causalean.valuesProjection hW (M.randomToObserved (M.evalMap s ℓ)))) M.latentProduct) p) =ᵐ[MeasureTheory.Measure.map (fun ℓ => (Causalean.valuesProjection hZr (M.randomToObserved (M.evalMap s ℓ)), Causalean.valuesProjection hW (M.randomToObserved (M.evalMap s ℓ)))) M.latentProduct] fun p => (ProbabilityTheory.condDistrib (Causalean.valuesProjection (Causalean.SCM.cutsetLatent_subset M Y (Finset.image Causalean.SWIGNode.random Z ∪ W))) (fun ℓ => Causalean.valuesProjection hW (M.randomToObserved (M.evalMap s ℓ))) M.latentProduct) p.2","l":372,"f":"Causalean/SCM/Do/Rule2Kernel/WitnessBridge.lean"},{"n":"Causalean.SCM.cutset_factor_pointwise","k":"theorem","m":"Causalean.SCM.Do.Rule2Kernel.WitnessBridge","a":"SCM/Do/Rule2Kernel","t":2,"d":"Cut-set factorization, pointwise, at the realized override.","s":"∀ {N : Type u_1} [inst : DecidableEq N] [inst_1 : Fintype N] {Ω : N → Type u_2} [inst_2 : (n : N) → MeasurableSpace (Ω n)] (M : Causalean.SCM N Ω) [∀ (n : N), Nonempty (Ω n)] (Y W : Finset (Causalean.SWIGNode N)) (Z : Finset N) (hY : Y ⊆ M.observed) (hZrW : Finset.image Causalean.SWIGNode.random Z ∪ W ⊆ M.observed) (s : M.FixedValues), ∃ h, (Measurable fun p => h p.1.1 p.1.2 p.2) ∧ (∀ (ℓ : M.LatentValues), Causalean.valuesProjection hY (M.randomToObserved (M.evalMap s ℓ)) = h (Causalean.valuesProjection (fun x hz => hZrW (Finset.mem_union_left W hz)) (M.randomToObserved (M.evalMap s ℓ))) (Causalean.valuesProjection (fun x hw => hZrW (Finset.mem_union_right (Finset.image Causalean.SWIGNode.random Z) hw)) (M.randomToObserved (M.evalMap s ℓ))) (Causalean.valuesProjection (Causalean.SCM.cutsetLatent_subset M Y (Finset.image Causalean.SWIGNode.random Z ∪ W)) ℓ)) ∧ ∀ (zr : Causalean.ValuesOn (Finset.image Causalean.SWIGNode.random Z) (Causalean.swigΩ Ω)) (w' : Causalean.ValuesOn W (Causalean.swigΩ Ω)) (ℓ : M.LatentValues), h zr w' (Causalean.valuesProjection (Causalean.SCM.cutsetLatent_subset M Y (Finset.image Causalean.SWIGNode.random Z ∪ W)) ℓ) = M.evalMap_overrideC hY hZrW s (Causalean.valuesUnionMk zr w') ℓ","l":215,"f":"Causalean/SCM/Do/Rule2Kernel/WitnessBridge.lean"},{"n":"Causalean.SCM.disjoint_base_of_disjoint_random_image","k":"theorem","m":"Causalean.SCM.ID.Toolkit.FrontdoorGraph","a":"SCM/ID/Toolkit","t":2,"d":"Disjoint random-node images have disjoint underlying base-variable sets.","s":"∀ {N : Type u_1} [inst : DecidableEq N] [Fintype N] (X W : Finset N), Disjoint (Finset.image Causalean.SWIGNode.random W) (Finset.image Causalean.SWIGNode.random X) → Disjoint W X","l":41,"f":"Causalean/SCM/ID/Toolkit/FrontdoorGraph.lean"},{"n":"Causalean.SCM.disjoint_fixed_observed","k":"theorem","m":"Causalean.SCM.ID.Toolkit.Derivation","a":"SCM/ID/Toolkit","t":2,"d":"Fixed and observed node sets of an SCM are disjoint.","s":"∀ {N : Type u_1} [inst : DecidableEq N] [inst_1 : Fintype N] {Ω : N → Type u_2} [inst_2 : (n : N) → MeasurableSpace (Ω n)] [∀ (n : Causalean.SWIGNode N), StandardBorelSpace (Causalean.swigΩ Ω n)] [∀ (n : Causalean.SWIGNode N), Nonempty (Causalean.swigΩ Ω n)] (M : Causalean.SCM N Ω), Disjoint M.fixed M.observed","l":44,"f":"Causalean/SCM/ID/Toolkit/Derivation.lean"},{"n":"Causalean.SCM.distMarkovEquiv_of_markovEquiv","k":"theorem","m":"Causalean.Graph.MarkovEquiv.Distributional","a":"Graph/MarkovEquiv","t":1,"d":"Easy half. For two DAGs G₁, G₂ on the same node set, if they are Markov equivalent — they declare exactly the same d-separations, then they are distributionally Markov equivalent: a distribution is a global I-map of one exactly when it is a global I-map of the other.","s":"∀ {N : Type uN} [inst : DecidableEq N] [inst_1 : Fintype N] {Ω : N → Type uΩ} [inst_2 : (n : N) → MeasurableSpace (Ω n)] {G₁ G₂ : Causalean.DAG (Causalean.SWIGNode N)}, Causalean.MarkovEquiv G₁ G₂ → Causalean.SCM.DistMarkovEquiv Ω G₁ G₂","l":89,"f":"Causalean/Graph/MarkovEquiv/Distributional.lean"},{"n":"Causalean.SCM.doKernelY","k":"def","m":"Causalean.SCM.ID.Backdoor","a":"SCM/ID","t":1,"d":"Post-intervention Y-marginal as a kernel in the treatment value t, at base s₀: t ↦ ((M.fixSet X).obsKernel (s_post s₀ t)).map π_Y.","s":"{N : Type u_1} → [inst : DecidableEq N] → [inst_1 : Fintype N] → {Ω : N → Type u_2} → [inst_2 : (n : N) → MeasurableSpace (Ω n)] → (M : Causalean.SCM N Ω) → (X : Finset N) → (∀ D ∈ X, Causalean.SWIGNode.random D ∈ M.observed) → (∀ D ∈ X, Causalean.SWIGNode.fixed D ∉ M.fixed) → (Y : Finset (Causalean.SWIGNode N)) → Y ⊆ M.observed → M.FixedValues → ProbabilityTheory.Kernel (Causalean.ValuesOn (Finset.image Causalean.SWIGNode.random X) (Causalean.swigΩ Ω)) (Causalean.ValuesOn Y (Causalean.swigΩ Ω))","l":87,"f":"Causalean/SCM/ID/Backdoor.lean"},{"n":"Causalean.SCM.doKernelY_disintegrate","k":"theorem","m":"Causalean.SCM.ID.Backdoor","a":"SCM/ID","t":2,"d":"Treatment-indexed chain rule. At treatment value t, the post-do(X) Y-marginal kernel disintegrates through Z: it is the conditional Y ∣ Z (under do(X)) composed with the post-do(X) Z-marginal kernel. The doKernelY-level form of obsKernel_map_eq_obsCondKernel_comp.","s":"∀ {N : Type u_1} [inst : DecidableEq N] [inst_1 : Fintype N] {Ω : N → Type u_2} [inst_2 : (n : N) → MeasurableSpace (Ω n)] [inst_3 : ∀ (n : Causalean.SWIGNode N), StandardBorelSpace (Causalean.swigΩ Ω n)] [inst_4 : ∀ (n : Causalean.SWIGNode N), Nonempty (Causalean.swigΩ Ω n)] (M : Causalean.SCM N Ω) (X : Finset N) (hObs : ∀ D ∈ X, Causalean.SWIGNode.random D ∈ M.observed) (hFix : ∀ D ∈ X, Causalean.SWIGNode.fixed D ∉ M.fixed) (Y Z : Finset (Causalean.SWIGNode N)) (hY : Y ⊆ M.observed) (hZ : Z ⊆ M.observed) (s0 : M.FixedValues) (t : Causalean.ValuesOn (Finset.image Causalean.SWIGNode.random X) (Causalean.swigΩ Ω)), (M.doKernelY X hObs hFix Y hY s0) t = ((M.doKernelY X hObs hFix Z hZ s0) t).bind ⇑(((M.fixSet X hObs hFix).obsCondKernel Y Z (Eq.symm (Causalean.SCM.fixSet_observed M X hObs hFix) ▸ hY) (Eq.symm (Causalean.SCM.fixSet_observed M X hObs hFix) ▸ hZ)).sectR (M.fixSetExtend X hObs hFix s0 t))","l":135,"f":"Causalean/SCM/ID/Backdoor.lean"},{"n":"Causalean.SCM.doKernelY_marginal_const","k":"theorem","m":"Causalean.SCM.ID.Backdoor","a":"SCM/ID","t":2,"d":"Treatment-indexed Rule 3 (marginal invariance). If no X-copy is an ancestor of any Z-node, the post-do(X) Z-marginal kernel is constant in the treatment value and equals the observational Z-marginal at base s₀. The doKernelY-level form of backdoor_rule3_Z_marginal.","s":"∀ {N : Type u_1} [inst : DecidableEq N] [inst_1 : Fintype N] {Ω : N → Type u_2} [inst_2 : (n : N) → MeasurableSpace (Ω n)] [∀ (n : Causalean.SWIGNode N), StandardBorelSpace (Causalean.swigΩ Ω n)] [∀ (n : Causalean.SWIGNode N), Nonempty (Causalean.swigΩ Ω n)] (M : Causalean.SCM N Ω) (X : Finset N) (hObs : ∀ D ∈ X, Causalean.SWIGNode.random D ∈ M.observed) (hFix : ∀ D ∈ X, Causalean.SWIGNode.fixed D ∉ M.fixed) (Z : Finset (Causalean.SWIGNode N)) (hZ : Z ⊆ M.observed), (∀ z ∈ Z, ∀ D ∈ X, ¬M.dag.isAncestor (Causalean.SWIGNode.random D) z) → ∀ (s0 : M.FixedValues) (t : Causalean.ValuesOn (Finset.image Causalean.SWIGNode.random X) (Causalean.swigΩ Ω)), (M.doKernelY X hObs hFix Z hZ s0) t = MeasureTheory.Measure.map (Causalean.valuesProjection hZ) (M.obsKernel s0)","l":163,"f":"Causalean/SCM/ID/Backdoor.lean"},{"n":"Causalean.SCM.doSide_M2_condDistrib_eq_M1_witness","k":"theorem","m":"Causalean.SCM.Do.Rule2Kernel.WitnessBridge","a":"SCM/Do/Rule2Kernel","t":1,"d":"Do-side analytic core: the M2 W-conditional equals the M1 witness kernel. Let M' be a structural causal model and Z a set of treatment names with each treatment's pre-intervention node observed and each treatment's post-intervention node not already fixed in M', so that intervening on Z is well-formed; let Y, W be node sets with Y, W, the treatments' pre-intervention nodes, and their union with W all observed and Y disjoint from the treatments' pre-intervention nodes. Suppose no node of W is a descendant, in the intervened model's graph, of any treatment's post-intervention node, and let h be a jointly measurable map that, at a baseline assignment s0 to the original model's fixed nodes, factors the realized outcome as h applied to the realized treatment value, the realized W value, and the latent cut-set's value and additionally, for every candidate treatment/W pair, agrees there with the outcome obtained by instead overriding the baseline assignment s0 to that pair. Then, at the intervened model's slice fixing the treatments to a value t and the remaining fixed nodes to s0, the measure-level conditional law of Y given W equals, for almost every w under the W-marginal of the original model's observational kernel at s0, the pushforward under h t w of the original model's posterior conditional law of the latent cut-set given W.","s":"∀ {N : Type u_1} [inst : DecidableEq N] [inst_1 : Fintype N] {Ω : N → Type u_2} [inst_2 : (n : N) → MeasurableSpace (Ω n)] (M' : Causalean.SCM N Ω) (Z : Finset N) (hZ_obs : ∀ D ∈ Z, Causalean.SWIGNode.random D ∈ M'.observed) (hZ_fixed : ∀ D ∈ Z, Causalean.SWIGNode.fixed D ∉ M'.fixed) (Y W : Finset (Causalean.SWIGNode N)) (hY : Y ⊆ M'.observed) (hW : W ⊆ M'.observed) (hZr : Finset.image Causalean.SWIGNode.random Z ⊆ M'.observed) (hZrW : Finset.image Causalean.SWIGNode.random Z ∪ W ⊆ M'.observed), Disjoint Y (Finset.image Causalean.SWIGNode.random Z) → (∀ z ∈ Z, ∀ v ∈ W, ¬(M'.fixSet Z hZ_obs hZ_fixed).dag.isAncestor (Causalean.SWIGNode.fixed z) v) → ∀ [StandardBorelSpace M'.RandomValues] [inst_4 : StandardBorelSpace (Causalean.ValuesOn Y (Causalean.swigΩ Ω))] [inst_5 : Nonempty (Causalean.ValuesOn Y (Causalean.swigΩ Ω))] [inst_6 : StandardBorelSpace (Causalean.ValuesOn (M'.cutsetLatent Y (Finset.image Causalean.SWIGNode.random Z ∪ W)) (Causalean.swigΩ Ω))] [inst_7 : Nonempty (Causalean.ValuesOn (M'.cutsetLatent Y (Finset.image Causalean.SWIGNode.random Z ∪ W)) (Causalean.swigΩ Ω))] [StandardBorelSpace (M'.fixSet Z hZ_obs hZ_fixed).RandomValues] [StandardBorelSpace (M'.fixSet Z hZ_obs hZ_fixed).ObservedValues] [StandardBorelSpace (Causalean.ValuesOn (Finset.image Causalean.SWIGNode.random Z) (Causalean.swigΩ Ω))] [Nonempty (Causalean.ValuesOn (Finset.image Causalean.SWIGNode.random Z) (Causalean.swigΩ Ω))] [MeasurableSpace.CountableOrCountablyGenerated (M'.fixSet Z hZ_obs hZ_fixed).FixedValues (Causalean.ValuesOn W (Causalean.swigΩ Ω))] [MeasurableSingletonClass (Causalean.ValuesOn (Finset.image Causalean.SWIGNode.random Z ∪ W) (Causalean.swigΩ Ω))] (s0 : M'.FixedValues) (t : Causalean.ValuesOn (Finset.image Causalean.SWIGNode.random Z) (Causalean.swigΩ Ω)) (h : Causalean.ValuesOn (Finset.image Causalean.SWIGNode.random Z) (Causalean.swigΩ Ω) → Causalean.ValuesOn W (Causalean.swigΩ Ω) → Causalean.ValuesOn (M'.cutsetLatent Y (Finset.image Causalean.SWIGNode.random Z ∪ W)) (Causalean.swigΩ Ω) → Causalean.ValuesOn Y (Causalean.swigΩ Ω)), (Measurable fun p => h p.1.1 p.1.2 p.2) → (∀ (ℓ : M'.LatentValues), Causalean.valuesProjection hY (M'.randomToObserved (M'.evalMap s0 ℓ)) = h (Causalean.valuesProjection hZr (M'.randomToObserved (M'.evalMap s0 ℓ))) (Causalean.valuesProjection hW (M'.randomToObserved (M'.evalMap s0 ℓ))) (Causalean.valuesProjection (Causalean.SCM.cutsetLatent_subset M' Y (Finset.image Causalean.SWIGNode.random Z ∪ W)) ℓ)) → (∀ (zr : Causalean.ValuesOn (Finset.image Causalean.SWIGNode.random Z) (Causalean.swigΩ Ω)) (w' : Causalean.ValuesOn W (Causalean.swigΩ Ω)) (ℓ : M'.LatentValues), h zr w' (Causalean.valuesProjection (Causalean.SCM.cutsetLatent_subset M' Y (Finset.image Causalean.SWIGNode.random Z ∪ W)) ℓ) = M'.evalMap_overrideC hY hZrW s0 (Causalean.valuesUnionMk zr w') ℓ) → (fun w => (ProbabilityTheory.condDistrib (Causalean.valuesProjection (Eq.symm (Causalean.SCM.fixSet_observed M' Z hZ_obs hZ_fixed) ▸ hY)) (Causalean.valuesProjection (Eq.symm (Causalean.SCM.fixSet_observed M' Z hZ_obs hZ_fixed) ▸ hW)) ((M'.fixSet Z hZ_obs hZ_fixed).obsKernel (M'.fixSetExtend Z hZ_obs hZ_fixed s0 t))) w) =ᵐ[MeasureTheory.Measure.map (Causalean.valuesProjection hW) (M'.obsKernel s0)] fun w => MeasureTheory.Measure.map (h t w) ((ProbabilityTheory.condDistrib (Causalean.valuesProjection (Causalean.SCM.cutsetLatent_subset M' Y (Finset.image Causalean.SWIGNode.random Z ∪ W))) (fun ℓ => Causalean.valuesProjection hW (M'.randomToObserved (M'.evalMap s0 ℓ))) M'.latentProduct) w)","l":1077,"f":"Causalean/SCM/Do/Rule2Kernel/WitnessBridge.lean"},{"n":"Causalean.SCM.doSide_M2_pullback_eq_M1_witness","k":"theorem","m":"Causalean.SCM.Do.Rule2Kernel.WitnessBridge","a":"SCM/Do/Rule2Kernel","t":2,"d":"Do-side cross-SCM core: the M2-pullback W-conditional on M2.latentProduct equals the M1 posterior witness kernel.","s":"∀ {N : Type u_1} [inst : DecidableEq N] [inst_1 : Fintype N] {Ω : N → Type u_2} [inst_2 : (n : N) → MeasurableSpace (Ω n)] (M' : Causalean.SCM N Ω) (Z : Finset N) (hZ_obs : ∀ D ∈ Z, Causalean.SWIGNode.random D ∈ M'.observed) (hZ_fixed : ∀ D ∈ Z, Causalean.SWIGNode.fixed D ∉ M'.fixed) (Y W : Finset (Causalean.SWIGNode N)) (hY : Y ⊆ M'.observed) (hZrW : Finset.image Causalean.SWIGNode.random Z ∪ W ⊆ M'.observed), Disjoint Y (Finset.image Causalean.SWIGNode.random Z) → (∀ z ∈ Z, ∀ v ∈ W, ¬(M'.fixSet Z hZ_obs hZ_fixed).dag.isAncestor (Causalean.SWIGNode.fixed z) v) → ∀ [StandardBorelSpace M'.RandomValues] [inst_4 : StandardBorelSpace (Causalean.ValuesOn Y (Causalean.swigΩ Ω))] [inst_5 : Nonempty (Causalean.ValuesOn Y (Causalean.swigΩ Ω))] [inst_6 : StandardBorelSpace (Causalean.ValuesOn (M'.cutsetLatent Y (Finset.image Causalean.SWIGNode.random Z ∪ W)) (Causalean.swigΩ Ω))] [inst_7 : Nonempty (Causalean.ValuesOn (M'.cutsetLatent Y (Finset.image Causalean.SWIGNode.random Z ∪ W)) (Causalean.swigΩ Ω))] [StandardBorelSpace (M'.fixSet Z hZ_obs hZ_fixed).RandomValues] [StandardBorelSpace (M'.fixSet Z hZ_obs hZ_fixed).ObservedValues] [StandardBorelSpace (Causalean.ValuesOn (Finset.image Causalean.SWIGNode.random Z) (Causalean.swigΩ Ω))] [Nonempty (Causalean.ValuesOn (Finset.image Causalean.SWIGNode.random Z) (Causalean.swigΩ Ω))] [MeasurableSpace.CountableOrCountablyGenerated (M'.fixSet Z hZ_obs hZ_fixed).FixedValues (Causalean.ValuesOn W (Causalean.swigΩ Ω))] [MeasurableSingletonClass (Causalean.ValuesOn (Finset.image Causalean.SWIGNode.random Z ∪ W) (Causalean.swigΩ Ω))] (s0 : M'.FixedValues) (t : Causalean.ValuesOn (Finset.image Causalean.SWIGNode.random Z) (Causalean.swigΩ Ω)) (h : Causalean.ValuesOn (Finset.image Causalean.SWIGNode.random Z) (Causalean.swigΩ Ω) → Causalean.ValuesOn W (Causalean.swigΩ Ω) → Causalean.ValuesOn (M'.cutsetLatent Y (Finset.image Causalean.SWIGNode.random Z ∪ W)) (Causalean.swigΩ Ω) → Causalean.ValuesOn Y (Causalean.swigΩ Ω)), (Measurable fun p => h p.1.1 p.1.2 p.2) → (∀ (zr : Causalean.ValuesOn (Finset.image Causalean.SWIGNode.random Z) (Causalean.swigΩ Ω)) (w' : Causalean.ValuesOn W (Causalean.swigΩ Ω)) (ℓ : M'.LatentValues), h zr w' (Causalean.valuesProjection (Causalean.SCM.cutsetLatent_subset M' Y (Finset.image Causalean.SWIGNode.random Z ∪ W)) ℓ) = M'.evalMap_overrideC hY hZrW s0 (Causalean.valuesUnionMk zr w') ℓ) → (fun w => (ProbabilityTheory.condDistrib (Causalean.valuesProjection (Eq.symm (Causalean.SCM.fixSet_observed M' Z hZ_obs hZ_fixed) ▸ hY) ∘ fun ℓ => (M'.fixSet Z hZ_obs hZ_fixed).randomToObserved ((M'.fixSet Z hZ_obs hZ_fixed).evalMap (M'.fixSetExtend Z hZ_obs hZ_fixed s0 t) ℓ)) (Causalean.valuesProjection (Eq.symm (Causalean.SCM.fixSet_observed M' Z hZ_obs hZ_fixed) ▸ fun x hw => hZrW (Finset.mem_union_right (Finset.image Causalean.SWIGNode.random Z) hw)) ∘ fun ℓ => (M'.fixSet Z hZ_obs hZ_fixed).randomToObserved ((M'.fixSet Z hZ_obs hZ_fixed).evalMap (M'.fixSetExtend Z hZ_obs hZ_fixed s0 t) ℓ)) (M'.fixSet Z hZ_obs hZ_fixed).latentProduct) w) =ᵐ[MeasureTheory.Measure.map (Causalean.valuesProjection fun x hw => hZrW (Finset.mem_union_right (Finset.image Causalean.SWIGNode.random Z) hw)) (M'.obsKernel s0)] fun w => MeasureTheory.Measure.map (h t w) ((ProbabilityTheory.condDistrib (Causalean.valuesProjection (Causalean.SCM.cutsetLatent_subset M' Y (Finset.image Causalean.SWIGNode.random Z ∪ W))) (fun ℓ => Causalean.valuesProjection (fun x hw => hZrW (Finset.mem_union_right (Finset.image Causalean.SWIGNode.random Z) hw)) (M'.randomToObserved (M'.evalMap s0 ℓ))) M'.latentProduct) w)","l":851,"f":"Causalean/SCM/Do/Rule2Kernel/WitnessBridge.lean"},{"n":"Causalean.SCM.doSide_eq_witness","k":"theorem","m":"Causalean.SCM.Do.Rule2Kernel.WitnessBridge","a":"SCM/Do/Rule2Kernel","t":1,"d":"Do-side per-slice witness identity (M1-witness-kernel form). Let M' be a structural causal model and Z a set of treatment names with each treatment's pre-intervention node observed and each treatment's post-intervention node not already fixed in M'; let Y, W be node sets with Y, W, the treatments' pre-intervention nodes, and their union with W all observed, Y disjoint from the treatments' pre-intervention nodes, and those pre-intervention nodes disjoint from W. Suppose no node of W is a descendant, in the intervened model's graph, of any treatment's post-intervention node, and let h be a jointly measurable map that, at a baseline assignment s0 to the original model's fixed nodes, factors the realized outcome as h applied to the realized treatment value, the realized W value, and the latent cut-set's value, and additionally for every candidate treatment/W pair agrees there with the outcome obtained by instead overriding the baseline assignment s0 to that pair. Then, at the intervened model's slice fixing the treatments to a value t and the remaining fixed nodes to s0, for almost every w under the W-marginal of the original model's observational kernel at s0, the intervened model's conditional-probability kernel for Y given W at (s', w) equals the pushforward under h t w of the original model's posterior conditional law of the latent cut-set given W.","s":"∀ {N : Type u_1} [inst : DecidableEq N] [inst_1 : Fintype N] {Ω : N → Type u_2} [inst_2 : (n : N) → MeasurableSpace (Ω n)] (M' : Causalean.SCM N Ω) (Z : Finset N) (hZ_obs : ∀ D ∈ Z, Causalean.SWIGNode.random D ∈ M'.observed) (hZ_fixed : ∀ D ∈ Z, Causalean.SWIGNode.fixed D ∉ M'.fixed) (Y W : Finset (Causalean.SWIGNode N)) (hY : Y ⊆ M'.observed) (hW : W ⊆ M'.observed) (hZr : Finset.image Causalean.SWIGNode.random Z ⊆ M'.observed) (hZrW : Finset.image Causalean.SWIGNode.random Z ∪ W ⊆ M'.observed), Disjoint (Finset.image Causalean.SWIGNode.random Z) W → Disjoint Y (Finset.image Causalean.SWIGNode.random Z) → (∀ z ∈ Z, ∀ v ∈ W, ¬(M'.fixSet Z hZ_obs hZ_fixed).dag.isAncestor (Causalean.SWIGNode.fixed z) v) → ∀ [StandardBorelSpace M'.RandomValues] [inst_4 : StandardBorelSpace (Causalean.ValuesOn Y (Causalean.swigΩ Ω))] [inst_5 : Nonempty (Causalean.ValuesOn Y (Causalean.swigΩ Ω))] [inst_6 : StandardBorelSpace (Causalean.ValuesOn (M'.cutsetLatent Y (Finset.image Causalean.SWIGNode.random Z ∪ W)) (Causalean.swigΩ Ω))] [inst_7 : Nonempty (Causalean.ValuesOn (M'.cutsetLatent Y (Finset.image Causalean.SWIGNode.random Z ∪ W)) (Causalean.swigΩ Ω))] [StandardBorelSpace (M'.fixSet Z hZ_obs hZ_fixed).RandomValues] [StandardBorelSpace (M'.fixSet Z hZ_obs hZ_fixed).ObservedValues] [StandardBorelSpace (Causalean.ValuesOn (Finset.image Causalean.SWIGNode.random Z) (Causalean.swigΩ Ω))] [Nonempty (Causalean.ValuesOn (Finset.image Causalean.SWIGNode.random Z) (Causalean.swigΩ Ω))] [MeasurableSpace.CountableOrCountablyGenerated M'.FixedValues (Causalean.ValuesOn (Finset.image Causalean.SWIGNode.random Z ∪ W) (Causalean.swigΩ Ω))] [MeasurableSpace.CountableOrCountablyGenerated (M'.fixSet Z hZ_obs hZ_fixed).FixedValues (Causalean.ValuesOn (Finset.image Causalean.SWIGNode.random Z ∪ W) (Causalean.swigΩ Ω))] [inst_14 : MeasurableSpace.CountableOrCountablyGenerated (M'.fixSet Z hZ_obs hZ_fixed).FixedValues (Causalean.ValuesOn W (Causalean.swigΩ Ω))] [MeasurableSingletonClass (Causalean.ValuesOn (Finset.image Causalean.SWIGNode.random Z ∪ W) (Causalean.swigΩ Ω))] (s0 : M'.FixedValues) (t : Causalean.ValuesOn (Finset.image Causalean.SWIGNode.random Z) (Causalean.swigΩ Ω)) (h : Causalean.ValuesOn (Finset.image Causalean.SWIGNode.random Z) (Causalean.swigΩ Ω) → Causalean.ValuesOn W (Causalean.swigΩ Ω) → Causalean.ValuesOn (M'.cutsetLatent Y (Finset.image Causalean.SWIGNode.random Z ∪ W)) (Causalean.swigΩ Ω) → Causalean.ValuesOn Y (Causalean.swigΩ Ω)), (Measurable fun p => h p.1.1 p.1.2 p.2) → (∀ (ℓ : M'.LatentValues), Causalean.valuesProjection hY (M'.randomToObserved (M'.evalMap s0 ℓ)) = h (Causalean.valuesProjection hZr (M'.randomToObserved (M'.evalMap s0 ℓ))) (Causalean.valuesProjection hW (M'.randomToObserved (M'.evalMap s0 ℓ))) (Causalean.valuesProjection (Causalean.SCM.cutsetLatent_subset M' Y (Finset.image Causalean.SWIGNode.random Z ∪ W)) ℓ)) → (∀ (zr : Causalean.ValuesOn (Finset.image Causalean.SWIGNode.random Z) (Causalean.swigΩ Ω)) (w' : Causalean.ValuesOn W (Causalean.swigΩ Ω)) (ℓ : M'.LatentValues), h zr w' (Causalean.valuesProjection (Causalean.SCM.cutsetLatent_subset M' Y (Finset.image Causalean.SWIGNode.random Z ∪ W)) ℓ) = M'.evalMap_overrideC hY hZrW s0 (Causalean.valuesUnionMk zr w') ℓ) → ∀ᵐ (w : Causalean.ValuesOn W (Causalean.swigΩ Ω)) ∂MeasureTheory.Measure.map (Causalean.valuesProjection hW) (M'.obsKernel s0), ((M'.fixSet Z hZ_obs hZ_fixed).obsCondKernel Y W (Eq.symm (Causalean.SCM.fixSet_observed M' Z hZ_obs hZ_fixed) ▸ hY) (Eq.symm (Causalean.SCM.fixSet_observed M' Z hZ_obs hZ_fixed) ▸ hW)) (M'.fixSetExtend Z hZ_obs hZ_fixed s0 t, w) = MeasureTheory.Measure.map (h t w) ((ProbabilityTheory.condDistrib (Causalean.valuesProjection (Causalean.SCM.cutsetLatent_subset M' Y (Finset.image Causalean.SWIGNode.random Z ∪ W))) (fun ℓ => Causalean.valuesProjection hW (M'.randomToObserved (M'.evalMap s0 ℓ))) M'.latentProduct) w)","l":1235,"f":"Causalean/SCM/Do/Rule2Kernel/WitnessBridge.lean"},{"n":"Causalean.SCM.do_rule1","k":"theorem","m":"Causalean.SCM.Do.DoCalculus","a":"SCM/Do","t":1,"d":"Rule 1: Insertion/deletion of observations (single-SCM form). On any structural causal model M', suppose Y, Z, and W are all observed nodes of M', and that Y is d-separated from Z given W together with the fixed nodes M'.fixed, in the split graph M'.dag. Then, at any fixed-value point s, Y and Z are conditionally independent given W under the observational kernel M'.obsKernel s.","s":"∀ {N : Type u_1} [inst : DecidableEq N] [inst_1 : Fintype N] {Ω : N → Type u_2} [inst_2 : (n : N) → MeasurableSpace (Ω n)] (M' : Causalean.SCM N Ω) [inst_3 : ∀ (n : Causalean.SWIGNode N), StandardBorelSpace (Causalean.swigΩ Ω n)] [∀ (n : Causalean.SWIGNode N), Nonempty (Causalean.swigΩ Ω n)] (Y Z W : Finset (Causalean.SWIGNode N)) [StandardBorelSpace (Causalean.ValuesOn Y (Causalean.swigΩ Ω))] [Nonempty (Causalean.ValuesOn Y (Causalean.swigΩ Ω))] [StandardBorelSpace (Causalean.ValuesOn Z (Causalean.swigΩ Ω))] [Nonempty (Causalean.ValuesOn Z (Causalean.swigΩ Ω))] (hY : Y ⊆ M'.observed) (hZ : Z ⊆ M'.observed) (hW : W ⊆ M'.observed), M'.dag.dSep Y Z (W ∪ M'.fixed) → ∀ (s : M'.FixedValues), M'.ObsCondIndep Y Z W hY hZ hW (M'.obsKernel s)","l":57,"f":"Causalean/SCM/Do/DoCalculus.lean"},{"n":"Causalean.SCM.do_rule2_kernel","k":"theorem","m":"Causalean.SCM.Do.DoCalculus","a":"SCM/Do","t":1,"d":"Rule 2: Action/observation exchange (single-SCM form, kernel-native). Fix a structural causal model M' and a treatment set Z for which each member's random copy is already observed in M' and each member's fixed copy is not yet among M''s fixed nodes, with outcome and conditioning sets Y and W, and with the random copies of Z and their union with W all observed in M'. Suppose that, in the model intervened on Z, Y is d-separated from the random copies of Z given W together with the post-intervention fixed nodes, that no fixed copy of a Z-variable is a post-intervention ancestor of any node in W, and that no random copy of a Z-variable is an M'-ancestor of any node in W, so W is not downstream of the intervention in either graph. At a fixed-value point s0, assume the law obtained by independently pairing a treatment value drawn from the observational marginal of Z's random copies with a conditioning value drawn from the observational marginal of W is absolutely continuous with respect to the actual observational joint law of Z's random copies and W. Then for almost every such independently-paired pair (t, w), the Y-given-W conditional kernel of the model intervened at t, evaluated at the fixed value extended by t and at w, equals the Y-given-(Z ∪ W) conditional kernel of M' evaluated at s0 and the point filled by combining t and w.","s":"∀ {N : Type u_1} [inst : DecidableEq N] [inst_1 : Fintype N] {Ω : N → Type u_2} [inst_2 : (n : N) → MeasurableSpace (Ω n)] (M' : Causalean.SCM N Ω) (Z : Finset N) (hZ_obs : ∀ D ∈ Z, Causalean.SWIGNode.random D ∈ M'.observed) (hZ_fixed : ∀ D ∈ Z, Causalean.SWIGNode.fixed D ∉ M'.fixed) (Y W : Finset (Causalean.SWIGNode N)) (hY : Y ⊆ M'.observed) (hW : W ⊆ M'.observed) (hZr : Finset.image Causalean.SWIGNode.random Z ⊆ M'.observed) (hZrW : Finset.image Causalean.SWIGNode.random Z ∪ W ⊆ M'.observed), (M'.fixSet Z hZ_obs hZ_fixed).dag.dSep Y (Finset.image Causalean.SWIGNode.random Z) (W ∪ (M'.fixSet Z hZ_obs hZ_fixed).fixed) → (∀ z ∈ Z, ∀ v ∈ W, ¬(M'.fixSet Z hZ_obs hZ_fixed).dag.isAncestor (Causalean.SWIGNode.fixed z) v) → (∀ D ∈ Z, ∀ w ∈ W, ¬M'.dag.isAncestor (Causalean.SWIGNode.random D) w) → ∀ [inst_3 : ∀ (n : Causalean.SWIGNode N), StandardBorelSpace (Causalean.swigΩ Ω n)] [inst_4 : ∀ (n : Causalean.SWIGNode N), Nonempty (Causalean.swigΩ Ω n)] [inst_5 : MeasurableSpace.CountableOrCountablyGenerated M'.FixedValues (Causalean.ValuesOn (Finset.image Causalean.SWIGNode.random Z ∪ W) (Causalean.swigΩ Ω))] [MeasurableSpace.CountableOrCountablyGenerated (M'.fixSet Z hZ_obs hZ_fixed).FixedValues (Causalean.ValuesOn (Finset.image Causalean.SWIGNode.random Z ∪ W) (Causalean.swigΩ Ω))] [inst_7 : MeasurableSpace.CountableOrCountablyGenerated (M'.fixSet Z hZ_obs hZ_fixed).FixedValues (Causalean.ValuesOn W (Causalean.swigΩ Ω))] [MeasurableSingletonClass (Causalean.ValuesOn (Finset.image Causalean.SWIGNode.random Z ∪ W) (Causalean.swigΩ Ω))] (s0 : M'.FixedValues), (MeasureTheory.Measure.map (fun p => Causalean.valuesUnionMk p.1 p.2) ((MeasureTheory.Measure.map (Causalean.valuesProjection hZr) (M'.obsKernel s0)).compProd (ProbabilityTheory.Kernel.const (Causalean.ValuesOn (Finset.image Causalean.SWIGNode.random Z) (Causalean.swigΩ Ω)) (MeasureTheory.Measure.map (Causalean.valuesProjection hW) (M'.obsKernel s0))))).AbsolutelyContinuous (MeasureTheory.Measure.map (Causalean.valuesProjection hZrW) (M'.obsKernel s0)) → ∀ᵐ (p : Causalean.ValuesOn (Finset.image Causalean.SWIGNode.random Z) (Causalean.swigΩ Ω) × Causalean.ValuesOn W (Causalean.swigΩ Ω)) ∂(MeasureTheory.Measure.map (Causalean.valuesProjection hZr) (M'.obsKernel s0)).compProd (ProbabilityTheory.Kernel.const (Causalean.ValuesOn (Finset.image Causalean.SWIGNode.random Z) (Causalean.swigΩ Ω)) (MeasureTheory.Measure.map (Causalean.valuesProjection hW) (M'.obsKernel s0))), ((M'.fixSet Z hZ_obs hZ_fixed).obsCondKernel Y W (Eq.symm (Causalean.SCM.fixSet_observed M' Z hZ_obs hZ_fixed) ▸ hY) (Eq.symm (Causalean.SCM.fixSet_observed M' Z hZ_obs hZ_fixed) ▸ hW)) (M'.fixSetExtend Z hZ_obs hZ_fixed s0 p.1, p.2) = (M'.obsCondKernel Y (Finset.image Causalean.SWIGNode.random Z ∪ W) hY hZrW) (s0, Causalean.valuesUnionMk p.1 p.2)","l":95,"f":"Causalean/SCM/Do/DoCalculus.lean"},{"n":"Causalean.SCM.do_rule3","k":"theorem","m":"Causalean.SCM.Do.DoCalculus","a":"SCM/Do","t":1,"d":"Rule 3: insertion/deletion of actions (simplified joint-marginal form). Fix a structural causal model M' and a treatment set Z for which each member's random copy is already observed in M' and each member's fixed copy is not yet among M''s fixed nodes, together with outcome and conditioning sets Y and W, both observed in M'. Suppose no node of Y ∪ W is, in the model intervened on Z, a descendant of the fixed copy of any variable in Z. Then the joint law of (Y, W) under the intervened model, at any post-intervention fixed value, equals the joint law of (Y, W) under the base model M' at the corresponding pre-intervention fixed value.","s":"∀ {N : Type u_1} [inst : DecidableEq N] [inst_1 : Fintype N] {Ω : N → Type u_2} [inst_2 : (n : N) → MeasurableSpace (Ω n)] (M' : Causalean.SCM N Ω) (Z : Finset N) (hZ_obs : ∀ D ∈ Z, Causalean.SWIGNode.random D ∈ M'.observed) (hZ_fixed : ∀ D ∈ Z, Causalean.SWIGNode.fixed D ∉ M'.fixed) (Y W : Finset (Causalean.SWIGNode N)) (hY : Y ⊆ M'.observed) (hW : W ⊆ M'.observed), (∀ v ∈ Y ∪ W, ∀ d ∈ Z, ¬(M'.fixSet Z hZ_obs hZ_fixed).dag.isAncestor (Causalean.SWIGNode.fixed d) v) → ∀ (s' : (M'.fixSet Z hZ_obs hZ_fixed).FixedValues), MeasureTheory.Measure.map (Causalean.valuesProjection (Eq.symm (Causalean.SCM.fixSet_observed M' Z hZ_obs hZ_fixed) ▸ Finset.union_subset hY hW)) ((M'.fixSet Z hZ_obs hZ_fixed).obsKernel s') = MeasureTheory.Measure.map (Causalean.valuesProjection (Finset.union_subset hY hW)) (M'.obsKernel (M'.fixSetProj Z hZ_obs hZ_fixed s'))","l":181,"f":"Causalean/SCM/Do/DoCalculus.lean"},{"n":"Causalean.SCM.do_rule3_conditional","k":"theorem","m":"Causalean.SCM.Do.Rule3Conditional","a":"SCM/Do","t":1,"d":"Conditional Rule 3 (headline, a.e. obsCondKernel form). Pearl's Rule 3 for the deletion of actions, stated against the project's jointly-measurable conditional kernel. Fix a do-set Z of nodes whose random copies are observed in the base model and whose fixed nodes have not already been intervened on, an outcome block Y and a conditioning block W of observed variables. If none of the fixed copies of Z's nodes is an ancestor, in the intervention SWIG graph, of any node in Y ∪ W, then for almost every value w of W under the intervened model's W-marginal, the Y-given-W conditional kernel of the model intervened at do(Z) equals the Y-given-W conditional kernel of the base model, both evaluated at the corresponding fixed values.","s":"∀ {N : Type u_1} [inst : DecidableEq N] [inst_1 : Fintype N] {Ω : N → Type u_2} [inst_2 : (n : N) → MeasurableSpace (Ω n)] (M' : Causalean.SCM N Ω) (Z : Finset N) (hZ_obs : ∀ D ∈ Z, Causalean.SWIGNode.random D ∈ M'.observed) (hZ_fixed : ∀ D ∈ Z, Causalean.SWIGNode.fixed D ∉ M'.fixed) (Y W : Finset (Causalean.SWIGNode N)) (hY : Y ⊆ M'.observed) (hW : W ⊆ M'.observed) [inst_3 : StandardBorelSpace (Causalean.ValuesOn Y (Causalean.swigΩ Ω))] [inst_4 : Nonempty (Causalean.ValuesOn Y (Causalean.swigΩ Ω))] [inst_5 : MeasurableSpace.CountableOrCountablyGenerated M'.FixedValues (Causalean.ValuesOn W (Causalean.swigΩ Ω))] [inst_6 : MeasurableSpace.CountableOrCountablyGenerated (M'.fixSet Z hZ_obs hZ_fixed).FixedValues (Causalean.ValuesOn W (Causalean.swigΩ Ω))], (∀ v ∈ Y ∪ W, ∀ d ∈ Z, ¬(M'.fixSet Z hZ_obs hZ_fixed).dag.isAncestor (Causalean.SWIGNode.fixed d) v) → ∀ (s' : (M'.fixSet Z hZ_obs hZ_fixed).FixedValues), (fun w => ((M'.fixSet Z hZ_obs hZ_fixed).obsCondKernel Y W (Eq.symm (Causalean.SCM.fixSet_observed M' Z hZ_obs hZ_fixed) ▸ hY) (Eq.symm (Causalean.SCM.fixSet_observed M' Z hZ_obs hZ_fixed) ▸ hW)) (s', w)) =ᵐ[MeasureTheory.Measure.map (Causalean.valuesProjection hW) (M'.obsKernel (M'.fixSetProj Z hZ_obs hZ_fixed s'))] fun w => (M'.obsCondKernel Y W hY hW) (M'.fixSetProj Z hZ_obs hZ_fixed s', w)","l":213,"f":"Causalean/SCM/Do/Rule3Conditional.lean"},{"n":"Causalean.SCM.do_rule3_conditional_condDistrib","k":"theorem","m":"Causalean.SCM.Do.Rule3Conditional","a":"SCM/Do","t":2,"d":"Conditional Rule 3 (literal condDistrib form).","s":"∀ {N : Type u_1} [inst : DecidableEq N] [inst_1 : Fintype N] {Ω : N → Type u_2} [inst_2 : (n : N) → MeasurableSpace (Ω n)] (M' : Causalean.SCM N Ω) (Z : Finset N) (hZ_obs : ∀ D ∈ Z, Causalean.SWIGNode.random D ∈ M'.observed) (hZ_fixed : ∀ D ∈ Z, Causalean.SWIGNode.fixed D ∉ M'.fixed) (Y W : Finset (Causalean.SWIGNode N)) (hY : Y ⊆ M'.observed) (hW : W ⊆ M'.observed) [inst_3 : StandardBorelSpace (Causalean.ValuesOn Y (Causalean.swigΩ Ω))] [inst_4 : Nonempty (Causalean.ValuesOn Y (Causalean.swigΩ Ω))], (∀ v ∈ Y ∪ W, ∀ d ∈ Z, ¬(M'.fixSet Z hZ_obs hZ_fixed).dag.isAncestor (Causalean.SWIGNode.fixed d) v) → ∀ (s' : (M'.fixSet Z hZ_obs hZ_fixed).FixedValues), ProbabilityTheory.condDistrib (Causalean.valuesProjection (Eq.symm (Causalean.SCM.fixSet_observed M' Z hZ_obs hZ_fixed) ▸ hY)) (Causalean.valuesProjection (Eq.symm (Causalean.SCM.fixSet_observed M' Z hZ_obs hZ_fixed) ▸ hW)) ((M'.fixSet Z hZ_obs hZ_fixed).obsKernel s') = ProbabilityTheory.condDistrib (Causalean.valuesProjection hY) (Causalean.valuesProjection hW) (M'.obsKernel (M'.fixSetProj Z hZ_obs hZ_fixed s'))","l":180,"f":"Causalean/SCM/Do/Rule3Conditional.lean"},{"n":"Causalean.SCM.dominatedObs_countingRef","k":"theorem","m":"Causalean.SCM.ID.Density.CountingReference","a":"SCM/ID/Density","t":1,"d":"For every structural causal model M, each of its observational laws is absolutely continuous with respect to the counting reference family's product measure, because a faithful reference dominates every measure on a countable coordinate product. Note this holds for *all* M, with no positivity or graph hypothesis.","s":"∀ {N : Type u_1} [inst : DecidableEq N] [inst_1 : Fintype N] {Ω : N → Type u_2} [inst_2 : (n : N) → MeasurableSpace (Ω n)] [inst_3 : ∀ (sn : Causalean.SWIGNode N), Countable (Causalean.swigΩ Ω sn)] [inst_4 : ∀ (n : N), MeasurableSingletonClass (Ω n)] (M : Causalean.SCM N Ω), M.DominatedObs Causalean.SCM.countingRef","l":46,"f":"Causalean/SCM/ID/Density/CountingReference.lean"},{"n":"Causalean.SCM.double_edge_to_doW_of_not_fixedX","k":"theorem","m":"Causalean.SCM.ID.Toolkit.FrontdoorGraph","a":"SCM/ID/Toolkit","t":2,"d":"For two disjoint intervention blocks, an edge remaining after first fixing the X block and then the W block also remains after fixing W alone when its source is not a fixed copy from X.","s":"∀ {N : Type u_1} [inst : DecidableEq N] [inst_1 : Fintype N] {Ω : N → Type u_2} [inst_2 : (n : N) → MeasurableSpace (Ω n)] (M : Causalean.SCM N Ω) (X W : Finset N) (hX_obs : ∀ D ∈ X, Causalean.SWIGNode.random D ∈ M.observed) (hX_fixed : ∀ D ∈ X, Causalean.SWIGNode.fixed D ∉ M.fixed) (hW_obs : ∀ D ∈ W, Causalean.SWIGNode.random D ∈ M.observed) (hW_fixed : ∀ D ∈ W, Causalean.SWIGNode.fixed D ∉ M.fixed) (hDisj : Disjoint X W) {u v : Causalean.SWIGNode N}, (∀ D ∈ X, u ≠ Causalean.SWIGNode.fixed D) → ((M.fixSet X hX_obs hX_fixed).fixSet W (fun D hD => id (hW_obs D hD)) (Causalean.SCM.fixSet_fixed_not_mem_of_disjoint M W X hX_obs hX_fixed hW_fixed hDisj)).dag.edge u v → (M.fixSet W hW_obs hW_fixed).dag.edge u v","l":173,"f":"Causalean/SCM/ID/Toolkit/FrontdoorGraph.lean"},{"n":"Causalean.SCM.double_edge_to_doX_of_not_fixedW","k":"theorem","m":"Causalean.SCM.ID.Toolkit.FrontdoorGraph","a":"SCM/ID/Toolkit","t":2,"d":"For two disjoint intervention blocks, an edge remaining after first fixing the W block and then the X block also remains after fixing X alone when its source is not a fixed copy from W.","s":"∀ {N : Type u_1} [inst : DecidableEq N] [inst_1 : Fintype N] {Ω : N → Type u_2} [inst_2 : (n : N) → MeasurableSpace (Ω n)] (M : Causalean.SCM N Ω) (X W : Finset N) (hX_obs : ∀ D ∈ X, Causalean.SWIGNode.random D ∈ M.observed) (hX_fixed : ∀ D ∈ X, Causalean.SWIGNode.fixed D ∉ M.fixed) (hW_obs : ∀ D ∈ W, Causalean.SWIGNode.random D ∈ M.observed) (hW_fixed : ∀ D ∈ W, Causalean.SWIGNode.fixed D ∉ M.fixed) (hDisj : Disjoint W X) {u v : Causalean.SWIGNode N}, (∀ D ∈ W, u ≠ Causalean.SWIGNode.fixed D) → ((M.fixSet W hW_obs hW_fixed).fixSet X (fun D hD => id (hX_obs D hD)) (Causalean.SCM.fixSet_fixed_not_mem_of_disjoint M X W hW_obs hW_fixed hX_fixed hDisj)).dag.edge u v → (M.fixSet X hX_obs hX_fixed).dag.edge u v","l":60,"f":"Causalean/SCM/ID/Toolkit/FrontdoorGraph.lean"},{"n":"Causalean.SCM.edge_fixSet_of_edge","k":"theorem","m":"Causalean.SCM.Do.CutsetDSep","a":"SCM/Do","t":2,"d":"A directed edge in the original graph remains an edge after fixing treatments when its source is not the random copy of any treatment being fixed.","s":"∀ {N : Type u_1} [inst : DecidableEq N] [inst_1 : Fintype N] {Ω : N → Type u_2} [inst_2 : (n : N) → MeasurableSpace (Ω n)] (M : Causalean.SCM N Ω) (Z : Finset N) (hZ_obs : ∀ D ∈ Z, Causalean.SWIGNode.random D ∈ M.observed) (hZ_fixed : ∀ D ∈ Z, Causalean.SWIGNode.fixed D ∉ M.fixed) {u v : Causalean.SWIGNode N}, M.dag.edge u v → (∀ D ∈ Z, u ≠ Causalean.SWIGNode.random D) → (M.fixSet Z hZ_obs hZ_fixed).dag.edge u v","l":165,"f":"Causalean/SCM/Do/CutsetDSep.lean"},{"n":"Causalean.SCM.edge_of_edge_fixSet","k":"theorem","m":"Causalean.SCM.Do.CutsetDSep","a":"SCM/Do","t":2,"d":"Every directed edge after fixing treatments is either an original edge or the redirected outgoing edge from the fixed copy of a treatment being fixed.","s":"∀ {N : Type u_1} [inst : DecidableEq N] [inst_1 : Fintype N] {Ω : N → Type u_2} [inst_2 : (n : N) → MeasurableSpace (Ω n)] (M : Causalean.SCM N Ω) (Z : Finset N) (hZ_obs : ∀ D ∈ Z, Causalean.SWIGNode.random D ∈ M.observed) (hZ_fixed : ∀ D ∈ Z, Causalean.SWIGNode.fixed D ∉ M.fixed) {u v : Causalean.SWIGNode N}, (M.fixSet Z hZ_obs hZ_fixed).dag.edge u v → M.dag.edge u v ∨ ∃ D ∈ Z, u = Causalean.SWIGNode.fixed D ∧ M.dag.edge (Causalean.SWIGNode.random D) v","l":190,"f":"Causalean/SCM/Do/CutsetDSep.lean"},{"n":"Causalean.SCM.emptyPrefixValues","k":"def","m":"Causalean.SCM.Factored.ObsChainKernel","a":"SCM/Factored","t":1,"d":"The unique value on the empty prefix.","s":"{N : Type u_1} → [inst : DecidableEq N] → [inst_1 : Fintype N] → {Ω : N → Type u_2} → [inst_2 : (n : N) → MeasurableSpace (Ω n)] → (M : Causalean.SCM N Ω) → Causalean.ValuesOn (M.prefixNodes 0) (Causalean.swigΩ Ω)","l":388,"f":"Causalean/SCM/Factored/ObsChainKernel.lean"},{"n":"Causalean.SCM.evalMap","k":"def","m":"Causalean.SCM.Model.Evaluation","a":"SCM/Model","t":1,"d":"The evaluation map sends fixed values and a latent realization to the resulting values of all observed and latent random nodes.","s":"{N : Type u_1} → [inst : DecidableEq N] → [inst_1 : Fintype N] → {Ω : N → Type u_2} → [inst_2 : (n : N) → MeasurableSpace (Ω n)] → (M : Causalean.SCM N Ω) → M.FixedValues → M.LatentValues → M.RandomValues","l":170,"f":"Causalean/SCM/Model/Evaluation.lean"},{"n":"Causalean.SCM.evalMap_agree_iff_localConsistent","k":"theorem","m":"Causalean.SCM.ID.Density.LatentBlocks","a":"SCM/ID/Density","t":1,"d":"On a set P of observed nodes closed under taking observed parents (every observed parent of a member of P is itself in P), agreement of the structural evaluation evalMap with the recorded observed assignment x at every node of P is equivalent to pointwise local consistency of x against a fixed-value slice s and latent realization ℓ at every node of P.","s":"∀ {N : Type u_1} [inst : DecidableEq N] [inst_1 : Fintype N] {Ω : N → Type u_2} [inst_2 : (n : N) → MeasurableSpace (Ω n)] (M : Causalean.SCM N Ω) (s : M.FixedValues) (P : Finset (Causalean.SWIGNode N)) (hP : M.ObsParentClosed P) (x : Causalean.ValuesOn M.observed (Causalean.swigΩ Ω)) (ℓ : M.LatentValues), (∀ (v : Causalean.SWIGNode N) (hv : v ∈ P), M.evalMap s ℓ ⟨v, Finset.mem_union_left M.unobserved (hP.left hv)⟩ = x ⟨v, hP.left hv⟩) ↔ ∀ (v : Causalean.SWIGNode N) (hv : v ∈ P), M.localConsistent s x v (hP.left hv) ℓ","l":681,"f":"Causalean/SCM/ID/Density/LatentBlocks.lean"},{"n":"Causalean.SCM.evalMap_eq_iff_localConsistent_of_observed_parent_agree","k":"theorem","m":"Causalean.SCM.ID.Density.LatentBlocks","a":"SCM/ID/Density","t":2,"d":"When all observed parents agree with their assigned values, evaluation at an observed node equals its assignment exactly when the latent values are locally consistent.","s":"∀ {N : Type u_1} [inst : DecidableEq N] [inst_1 : Fintype N] {Ω : N → Type u_2} [inst_2 : (n : N) → MeasurableSpace (Ω n)] (M : Causalean.SCM N Ω) (s : M.FixedValues) (x : Causalean.ValuesOn M.observed (Causalean.swigΩ Ω)) (ℓ : M.LatentValues) (v : Causalean.SWIGNode N) (hv : v ∈ M.observed), (∀ (w : Causalean.SWIGNode N) (hw : w ∈ M.observed), M.dag.edge w v → M.evalMap s ℓ ⟨w, Finset.mem_union_left M.unobserved hw⟩ = x ⟨w, hw⟩) → (M.evalMap s ℓ ⟨v, Finset.mem_union_left M.unobserved hv⟩ = x ⟨v, hv⟩ ↔ M.localConsistent s x v hv ℓ)","l":416,"f":"Causalean/SCM/ID/Density/LatentBlocks.lean"},{"n":"Causalean.SCM.evalMap_eq_of_equiv","k":"theorem","m":"Causalean.SCM.Model.EquivKernel","a":"SCM/Model","t":1,"d":"Cross-SCM pointwise equality of evalMap under SCM.Equiv. For two structural causal models M₁, M₂ that are structurally equivalent — the same graph up to topological reordering, matching edge types and have heterogeneously equal structural functions, fixed-value assignments s₁, s₂ and latent assignments ℓ₁, ℓ₂ on the two models, and a shared node w that is a random-variable node of M₁ and a random-variable node of M₂: if s₁ and s₂ agree coordinatewise on every fixed node shared by both models and ℓ₁ and ℓ₂ agree coordinatewise on every unobserved node shared by both models, then the two models' evaluation maps agree at w: M₁'s evaluation with (s₁, ℓ₁) equals M₂'s evaluation with (s₂, ℓ₂).","s":"∀ {N : Type u_1} [inst : DecidableEq N] [inst_1 : Fintype N] {Ω : N → Type u_2} [inst_2 : (n : N) → MeasurableSpace (Ω n)] {M₁ M₂ : Causalean.SCM N Ω}, M₁.Equivalent M₂.toSWIGGraph → M₁.structFun ≍ M₂.structFun → ∀ (s₁ : M₁.FixedValues) (ℓ₁ : M₁.LatentValues) (s₂ : M₂.FixedValues) (ℓ₂ : M₂.LatentValues), (∀ {d : Causalean.SWIGNode N} (hd₁ : d ∈ M₁.fixed) (hd₂ : d ∈ M₂.fixed), s₁ ⟨d, hd₁⟩ = s₂ ⟨d, hd₂⟩) → (∀ {u : Causalean.SWIGNode N} (hu₁ : u ∈ M₁.unobserved) (hu₂ : u ∈ M₂.unobserved), ℓ₁ ⟨u, hu₁⟩ = ℓ₂ ⟨u, hu₂⟩) → ∀ {w : Causalean.SWIGNode N} (hw₁ : w ∈ M₁.randomVars) (hw₂ : w ∈ M₂.randomVars), M₁.evalMap s₁ ℓ₁ ⟨w, hw₁⟩ = M₂.evalMap s₂ ℓ₂ ⟨w, hw₂⟩","l":105,"f":"Causalean/SCM/Model/EquivKernel.lean"},{"n":"Causalean.SCM.evalMap_eq_x_of_observedAt_eq","k":"theorem","m":"Causalean.SCM.ID.Density.LatentBlocks","a":"SCM/ID/Density","t":2,"d":"If structural evaluation at an indexed observed node equals its recorded value, then the same equality holds after replacing that node by any equal observed node.","s":"∀ {N : Type u_1} [inst : DecidableEq N] [inst_1 : Fintype N] {Ω : N → Type u_2} [inst_2 : (n : N) → MeasurableSpace (Ω n)] (M : Causalean.SCM N Ω) (s : M.FixedValues) (x : Causalean.ValuesOn M.observed (Causalean.swigΩ Ω)) (ℓ : M.LatentValues) (j : Fin M.observed.card) {v : Causalean.SWIGNode N} (hv : v ∈ M.observed), ↑(M.observedAt j) = v → M.evalMap s ℓ ⟨↑(M.observedAt j), Finset.mem_union_left M.unobserved (M.observedAt j).property⟩ = x (M.observedAt j) → M.evalMap s ℓ ⟨v, Finset.mem_union_left M.unobserved hv⟩ = x ⟨v, hv⟩","l":474,"f":"Causalean/SCM/ID/Density/LatentBlocks.lean"},{"n":"Causalean.SCM.evalMap_factors_excluding_latent","k":"theorem","m":"Causalean.SCM.Model.EvalLatent","a":"SCM/Model","t":1,"d":"Latent-restricted factorization away from a chosen latent root. Fix a structural causal model M, a fixed-value assignment s, and a latent root node a. For a set T of random-variable nodes such that no node of T is a descendant of a, and none equals a, then the projection of the evaluation evalMap s to T factors through a measurable function of the latent coordinates other than a's — i.e. it does not depend on the latent value at a.","s":"∀ {N : Type u_1} [inst : DecidableEq N] [inst_1 : Fintype N] {Ω : N → Type u_2} [inst_2 : (n : N) → MeasurableSpace (Ω n)] (M : Causalean.SCM N Ω) (s : M.FixedValues) (a : Causalean.SWIGNode N) (ha : a ∈ M.unobserved) (T : Finset (Causalean.SWIGNode N)) (hT_sub : T ⊆ M.randomVars), (∀ v ∈ T, ¬M.dag.isAncestor a v ∧ v ≠ a) → ∃ g, Measurable g ∧ ∀ (ℓ : M.LatentValues), Causalean.valuesProjection hT_sub (M.evalMap s ℓ) = g fun i => ℓ ↑i","l":49,"f":"Causalean/SCM/Model/EvalLatent.lean"},{"n":"Causalean.SCM.evalMap_factors_through_ancestors","k":"theorem","m":"Causalean.SCM.Model.EvalFactorization","a":"SCM/Model","t":1,"d":"Ancestor factorization (existence form). Fix a structural causal model M and a node v with v an observed node. Then there is a measurable function g of the fixed-ancestor projection and the latent-ancestor projection of v such that, for every fixed-value assignment s and latent assignment ℓ, the evaluated value at v equals g applied to those two projections of s and ℓ.","s":"∀ {N : Type u_1} [inst : DecidableEq N] [inst_1 : Fintype N] {Ω : N → Type u_2} [inst_2 : (n : N) → MeasurableSpace (Ω n)] (M : Causalean.SCM N Ω) [∀ (n : N), Nonempty (Ω n)] (v : Causalean.SWIGNode N) (hv : v ∈ M.observed), ∃ g, Measurable (Function.uncurry g) ∧ ∀ (s : M.FixedValues) (ℓ : M.LatentValues), M.evalMap s ℓ ⟨v, Finset.mem_union_left M.unobserved hv⟩ = g (Causalean.valuesProjection (Causalean.SCM.fixedAncestorsOfNode_subset M v) s) (Causalean.valuesProjection (Causalean.SCM.latentAncestorsOfNode_subset M v) ℓ)","l":206,"f":"Causalean/SCM/Model/EvalFactorization.lean"},{"n":"Causalean.SCM.evalMap_factors_through_parents","k":"theorem","m":"Causalean.SCM.Model.EvalFactorization","a":"SCM/Model","t":1,"d":"Parent factorization of evalMap at observed nodes. Fix a structural causal model M, a fixed-value assignment s, and a node v with v an observed node. Then there is a measurable function g of the assignment restricted to v's random parent coordinates such that, for every latent assignment ℓ, the evaluated value at v equals g applied to that restriction of the joint evaluation.","s":"∀ {N : Type u_1} [inst : DecidableEq N] [inst_1 : Fintype N] {Ω : N → Type u_2} [inst_2 : (n : N) → MeasurableSpace (Ω n)] (M : Causalean.SCM N Ω) (s : M.FixedValues) (v : Causalean.SWIGNode N) (hv : v ∈ M.observed), ∃ g, Measurable g ∧ ∀ (ℓ : M.LatentValues), M.evalMap s ℓ ⟨v, Finset.mem_union_left M.unobserved hv⟩ = g (Causalean.valuesProjection Finset.inter_subset_right (M.evalMap s ℓ))","l":58,"f":"Causalean/SCM/Model/EvalFactorization.lean"},{"n":"Causalean.SCM.evalMap_fixSet_factual_eq","k":"theorem","m":"Causalean.SCM.Model.CounterfactualLemmas","a":"SCM/Model","t":1,"d":"prop:scm-cf-consistency (Basic Concepts.tex L475–480). Fix a structural causal model M, an intervention target set X such that every targeted node is currently a random observed node and none of its fixed copies is already fixed, a base fixed-value assignment s, a latent assignment ℓ, and an intervened fixed-value assignment sx for the model obtained by fixing X. If sx agrees with s on the model's original fixed coordinates and for every targeted node the base evaluation at its random form already equals sx's value at its fixed form — the factual consistency condition, then for every observed node v, evaluating the intervened model with sx and ℓ at v agrees with evaluating the base model with s and ℓ at v.","s":"∀ {N : Type uN} [inst : DecidableEq N] [inst_1 : Fintype N] {Ω : N → Type uΩ} [inst_2 : (n : N) → MeasurableSpace (Ω n)] (M : Causalean.SCM N Ω) (X : Finset N) (hObs : ∀ D ∈ X, Causalean.SWIGNode.random D ∈ M.observed) (hFix : ∀ D ∈ X, Causalean.SWIGNode.fixed D ∉ M.fixed) (s : M.FixedValues) (ℓ : M.LatentValues) (sx : (M.fixSet X hObs hFix).FixedValues), (∀ (v : Causalean.SWIGNode N) (hv : v ∈ M.fixed), sx ⟨v, Finset.mem_union_left (Finset.image Causalean.SWIGNode.fixed X) hv⟩ = s ⟨v, hv⟩) → (∀ (D : N) (hD : D ∈ X), M.evalMap s ℓ ⟨Causalean.SWIGNode.random D, Finset.mem_union_left M.unobserved (hObs D hD)⟩ = sx ⟨Causalean.SWIGNode.fixed D, Finset.mem_union_right M.fixed (Finset.mem_image.mpr (Exists.intro D ⟨hD, rfl⟩))⟩) → ∀ (v : ↥M.observed), (M.fixSet X hObs hFix).evalMap sx ℓ ⟨↑v, Finset.mem_union_left (M.fixSet X hObs hFix).unobserved v.property⟩ = M.evalMap s ℓ ⟨↑v, Finset.mem_union_left M.unobserved v.property⟩","l":110,"f":"Causalean/SCM/Model/CounterfactualLemmas.lean"},{"n":"Causalean.SCM.evalMap_fixSet_observed_apply","k":"theorem","m":"Causalean.SCM.Model.CounterfactualLemmas","a":"SCM/Model","t":2,"d":"For one recursive evaluation step after an intervention, the value of an observed variable is the original structural equation evaluated at the parent values where intervened parents are pinned to their assigned intervention values.","s":"∀ {N : Type uN} [inst : DecidableEq N] [inst_1 : Fintype N] {Ω : N → Type uΩ} [inst_2 : (n : N) → MeasurableSpace (Ω n)] (M : Causalean.SCM N Ω) (X : Finset N) (hObs : ∀ D ∈ X, Causalean.SWIGNode.random D ∈ M.observed) (hFix : ∀ D ∈ X, Causalean.SWIGNode.fixed D ∉ M.fixed) (sx : (M.fixSet X hObs hFix).FixedValues) (ℓ : M.LatentValues) (v : ↥M.observed), (M.fixSet X hObs hFix).evalMap sx ℓ ⟨↑v, Finset.mem_union_left (M.fixSet X hObs hFix).unobserved v.property⟩ = M.structFun v (Causalean.SCM.fixMonoParentMap M.toSWIGGraph X hObs hFix ↑v fun w => if huo : ↑w ∈ (M.fixSet X hObs hFix).unobserved then ℓ ⟨↑w, huo⟩ else if hfix : ↑w ∈ (M.fixSet X hObs hFix).fixed then sx ⟨↑w, hfix⟩ else have hedge := (Causalean.DAG.mem_parents (M.fixSet X hObs hFix).dag).mp w.property; have hobs := Or.casesOn (Finset.mem_union.mp ((M.fixSet X hObs hFix).dag_edges_classified (↑w) (↑v) hedge).left) (fun h1 => Or.casesOn (Finset.mem_union.mp h1) (fun hfx => absurd hfx hfix) fun hob => hob) fun h2 => absurd h2 huo; (M.fixSet X hObs hFix).evalMap sx ℓ ⟨↑w, Finset.mem_union_left (M.fixSet X hObs hFix).unobserved hobs⟩)","l":73,"f":"Causalean/SCM/Model/CounterfactualLemmas.lean"},{"n":"Causalean.SCM.evalMap_fixSet_union_eq","k":"theorem","m":"Causalean.SCM.Model.CounterfactualLemmas","a":"SCM/Model","t":1,"d":"Composition of a combined intervention with a single-stage intervention. Fix a structural causal model M, a latent assignment ℓ, and two intervention target sets X₁, X₂ such that every node of X₁ is currently a random observed node with no fixed copy already fixed, and likewise every node of X₁ ∪ X₂, with an X₁-only intervened fixed-value assignment sx₁ and a combined-intervention fixed-value assignment sxU for X₁ ∪ X₂. If sxU agrees with sx₁ on the original fixed coordinates and on the X₁ intervention coordinates and the X₁-intervened model's value at each X₂ node already equals sxU's assignment there — the intermediate condition, then for every observed node v, evaluating the combined-intervention model with sxU and ℓ at v agrees with evaluating the X₁-only intervened model with sx₁ and ℓ at v.","s":"∀ {N : Type uN} [inst : DecidableEq N] [inst_1 : Fintype N] {Ω : N → Type uΩ} [inst_2 : (n : N) → MeasurableSpace (Ω n)] (M : Causalean.SCM N Ω) (X₁ X₂ : Finset N) (hObs₁ : ∀ D ∈ X₁, Causalean.SWIGNode.random D ∈ M.observed) (hFix₁ : ∀ D ∈ X₁, Causalean.SWIGNode.fixed D ∉ M.fixed) (hObsU : ∀ D ∈ X₁ ∪ X₂, Causalean.SWIGNode.random D ∈ M.observed) (hFixU : ∀ D ∈ X₁ ∪ X₂, Causalean.SWIGNode.fixed D ∉ M.fixed) (ℓ : M.LatentValues) (sx₁ : (M.fixSet X₁ hObs₁ hFix₁).FixedValues) (sxU : (M.fixSet (X₁ ∪ X₂) hObsU hFixU).FixedValues), (∀ (v : Causalean.SWIGNode N) (hv : v ∈ M.fixed), sxU ⟨v, Finset.mem_union_left (Finset.image Causalean.SWIGNode.fixed (X₁ ∪ X₂)) hv⟩ = sx₁ ⟨v, Finset.mem_union_left (Finset.image Causalean.SWIGNode.fixed X₁) hv⟩) → (∀ (D : N) (hD : D ∈ X₁), sxU ⟨Causalean.SWIGNode.fixed D, Finset.mem_union_right M.fixed (Finset.mem_image.mpr (Exists.intro D ⟨Finset.mem_union_left X₂ hD, rfl⟩))⟩ = sx₁ ⟨Causalean.SWIGNode.fixed D, Finset.mem_union_right M.fixed (Finset.mem_image.mpr (Exists.intro D ⟨hD, rfl⟩))⟩) → (∀ (D : N) (hD : D ∈ X₂), (M.fixSet X₁ hObs₁ hFix₁).evalMap sx₁ ℓ ⟨Causalean.SWIGNode.random D, Finset.mem_union_left (M.fixSet X₁ hObs₁ hFix₁).unobserved (hObsU D (Finset.mem_union_right X₁ hD))⟩ = sxU ⟨Causalean.SWIGNode.fixed D, Finset.mem_union_right M.fixed (Finset.mem_image.mpr (Exists.intro D ⟨Finset.mem_union_right X₁ hD, rfl⟩))⟩) → ∀ (v : ↥M.observed), (M.fixSet (X₁ ∪ X₂) hObsU hFixU).evalMap sxU ℓ ⟨↑v, Finset.mem_union_left (M.fixSet (X₁ ∪ X₂) hObsU hFixU).unobserved v.property⟩ = (M.fixSet X₁ hObs₁ hFix₁).evalMap sx₁ ℓ ⟨↑v, Finset.mem_union_left (M.fixSet X₁ hObs₁ hFix₁).unobserved v.property⟩","l":293,"f":"Causalean/SCM/Model/CounterfactualLemmas.lean"},{"n":"Causalean.SCM.evalMap_measurable","k":"theorem","m":"Causalean.SCM.Model.Evaluation","a":"SCM/Model","t":2,"d":"The evaluation map is jointly measurable in the fixed-value assignment and latent realization.","s":"∀ {N : Type u_1} [inst : DecidableEq N] [inst_1 : Fintype N] {Ω : N → Type u_2} [inst_2 : (n : N) → MeasurableSpace (Ω n)] (M : Causalean.SCM N Ω), Measurable (Function.uncurry M.evalMap)","l":415,"f":"Causalean/SCM/Model/Evaluation.lean"},{"n":"Causalean.SCM.evalMap_observed","k":"theorem","m":"Causalean.SCM.Model.Evaluation","a":"SCM/Model","t":2,"d":"On an observed node, the evaluation map is the auxiliary topological-order evaluation transported to that node.","s":"∀ {N : Type u_1} [inst : DecidableEq N] [inst_1 : Fintype N] {Ω : N → Type u_2} [inst_2 : (n : N) → MeasurableSpace (Ω n)] (M : Causalean.SCM N Ω) (s : M.FixedValues) (ℓ : M.LatentValues) (w : ↥M.randomVars) (hobs : ↑w ∈ M.observed), M.evalMap s ℓ w = Causalean.SCM.observedAt_observedIndex M ⟨↑w, hobs⟩ ▸ M.evalObservedAux s ℓ (↑(M.observedIndex ⟨↑w, hobs⟩)) (M.observedIndex ⟨↑w, hobs⟩).isLt","l":198,"f":"Causalean/SCM/Model/Evaluation.lean"},{"n":"Causalean.SCM.evalMap_observed_unfold","k":"theorem","m":"Causalean.SCM.Model.Evaluation","a":"SCM/Model","t":2,"d":"At an observed node, the evaluation map unfolds to the structural function applied to fixed, latent, or recursively evaluated parent values.","s":"∀ {N : Type u_1} [inst : DecidableEq N] [inst_1 : Fintype N] {Ω : N → Type u_2} [inst_2 : (n : N) → MeasurableSpace (Ω n)] (M : Causalean.SCM N Ω) (s : M.FixedValues) (ℓ : M.LatentValues) (v : ↥M.observed), M.evalMap s ℓ ⟨↑v, Finset.mem_union_left M.unobserved v.property⟩ = M.structFun v fun w => if huo : ↑w ∈ M.unobserved then ℓ ⟨↑w, huo⟩ else if hfix : ↑w ∈ M.fixed then s ⟨↑w, hfix⟩ else have hedge := (Causalean.DAG.mem_parents M.dag).mp w.property; have hobs := Or.casesOn (Finset.mem_union.mp (M.dag_edges_classified (↑w) (↑v) hedge).left) (fun h1 => Or.casesOn (Finset.mem_union.mp h1) (fun hfx => absurd hfx hfix) fun hob => hob) fun h2 => absurd h2 huo; M.evalMap s ℓ ⟨↑w, Finset.mem_union_left M.unobserved hobs⟩","l":298,"f":"Causalean/SCM/Model/Evaluation.lean"},{"n":"Causalean.SCM.evalMap_overrideC","k":"def","m":"Causalean.SCM.Model.EvalOverrideC","a":"SCM/Model","t":1,"d":"The overridden evaluation map returns target observed values while holding the override block fixed.","s":"{N : Type u_1} → [inst : DecidableEq N] → [inst_1 : Fintype N] → {Ω : N → Type u_2} → [inst_2 : (n : N) → MeasurableSpace (Ω n)] → (M : Causalean.SCM N Ω) → {Y C : Finset (Causalean.SWIGNode N)} → Y ⊆ M.observed → C ⊆ M.observed → M.FixedValues → Causalean.ValuesOn C (Causalean.swigΩ Ω) → M.LatentValues → Causalean.ValuesOn Y (Causalean.swigΩ Ω)","l":218,"f":"Causalean/SCM/Model/EvalOverrideC.lean"},{"n":"Causalean.SCM.evalMap_overrideC_agree_anc","k":"theorem","m":"Causalean.SCM.Do.Rule2Kernel.Structural.StructPointwise","a":"SCM/Do/Rule2Kernel/Structural","t":2,"d":"Two assignments that override observed variables give the same evaluated values at every target variable when they agree on which relevant ancestors are overridden and on the values assigned there.","s":"∀ {N : Type u_1} [inst : DecidableEq N] [inst_1 : Fintype N] {Ω : N → Type u_2} [inst_2 : (n : N) → MeasurableSpace (Ω n)] (M : Causalean.SCM N Ω) {Y C₁ C₂ : Finset (Causalean.SWIGNode N)} (hY : Y ⊆ M.observed) (hC₁ : C₁ ⊆ M.observed) (hC₂ : C₂ ⊆ M.observed) (s : M.FixedValues) (ℓ : M.LatentValues) (c₁ : Causalean.ValuesOn C₁ (Causalean.swigΩ Ω)) (c₂ : Causalean.ValuesOn C₂ (Causalean.swigΩ Ω)), (∀ (x : Causalean.SWIGNode N), (∃ y ∈ Y, x = y ∨ M.dag.isAncestor x y) → (x ∈ C₁ ↔ x ∈ C₂) ∧ ∀ (h₁ : x ∈ C₁) (h₂ : x ∈ C₂), c₁ ⟨x, h₁⟩ = c₂ ⟨x, h₂⟩) → M.evalMap_overrideC hY hC₁ s c₁ ℓ = M.evalMap_overrideC hY hC₂ s c₂ ℓ","l":192,"f":"Causalean/SCM/Do/Rule2Kernel/Structural/StructPointwise.lean"},{"n":"Causalean.SCM.evalMap_overrideC_agree_cutset","k":"theorem","m":"Causalean.SCM.Model.CutsetLatent","a":"SCM/Model","t":2,"d":"The overridden evaluation on the target set depends on latent values only through the latent cutset.","s":"∀ {N : Type u_2} [inst : DecidableEq N] [inst_1 : Fintype N] {Ω : N → Type u_3} [inst_2 : (n : N) → MeasurableSpace (Ω n)] (M : Causalean.SCM N Ω) {Y C : Finset (Causalean.SWIGNode N)} (hY : Y ⊆ M.observed) (hC : C ⊆ M.observed) (s : M.FixedValues) (c : Causalean.ValuesOn C (Causalean.swigΩ Ω)) (ℓ₁ ℓ₂ : M.LatentValues), Causalean.valuesProjection (Causalean.SCM.cutsetLatent_subset M Y C) ℓ₁ = Causalean.valuesProjection (Causalean.SCM.cutsetLatent_subset M Y C) ℓ₂ → M.evalMap_overrideC hY hC s c ℓ₁ = M.evalMap_overrideC hY hC s c ℓ₂","l":296,"f":"Causalean/SCM/Model/CutsetLatent.lean"},{"n":"Causalean.SCM.evalMap_overrideC_apply_of_mem_C","k":"theorem","m":"Causalean.SCM.Model.EvalOverrideC","a":"SCM/Model","t":2,"d":"At a target node inside the override block, the overridden evaluation returns the assigned override value.","s":"∀ {N : Type u_1} [inst : DecidableEq N] [inst_1 : Fintype N] {Ω : N → Type u_2} [inst_2 : (n : N) → MeasurableSpace (Ω n)] (M : Causalean.SCM N Ω) {Y C : Finset (Causalean.SWIGNode N)} (hY : Y ⊆ M.observed) (hC : C ⊆ M.observed) (s : M.FixedValues) (c : Causalean.ValuesOn C (Causalean.swigΩ Ω)) (ℓ : M.LatentValues) (v : ↥Y) (hvC : ↑v ∈ C), M.evalMap_overrideC hY hC s c ℓ v = c ⟨↑v, hvC⟩","l":288,"f":"Causalean/SCM/Model/EvalOverrideC.lean"},{"n":"Causalean.SCM.evalMap_overrideC_apply_of_not_mem_C","k":"theorem","m":"Causalean.SCM.Model.EvalOverrideC","a":"SCM/Model","t":2,"d":"At a target node outside the override block, the overridden evaluation applies the structural function to overridden parent values.","s":"∀ {N : Type u_1} [inst : DecidableEq N] [inst_1 : Fintype N] {Ω : N → Type u_2} [inst_2 : (n : N) → MeasurableSpace (Ω n)] (M : Causalean.SCM N Ω) {Y C : Finset (Causalean.SWIGNode N)} (hY : Y ⊆ M.observed) (hC : C ⊆ M.observed) (s : M.FixedValues) (c : Causalean.ValuesOn C (Causalean.swigΩ Ω)) (ℓ : M.LatentValues) (v : ↥Y), ↑v ∉ C → M.evalMap_overrideC hY hC s c ℓ v = Causalean.SCM.observedAt_observedIndex M ⟨↑v, hY v.property⟩ ▸ M.structFun (M.observedAt (M.observedIndex ⟨↑v, hY v.property⟩)) fun w => M.parentMapOverride s c ℓ (M.observedIndex ⟨↑v, hY v.property⟩).isLt (fun m x hm_card => M.evalObservedAuxOverride hC s c ℓ m hm_card) w","l":318,"f":"Causalean/SCM/Model/EvalOverrideC.lean"},{"n":"Causalean.SCM.evalMap_overrideC_at_self","k":"theorem","m":"Causalean.SCM.Model.EvalOverrideC","a":"SCM/Model","t":1,"d":"Cornerstone: overriding by the model's own values changes nothing. Fix a structural causal model M, a target set Y and an override set C with Y contained in the observed nodes and C contained in the observed nodes, a fixed-value assignment s, and a latent assignment ℓ. Then overriding C with the values that evalMap s ℓ would itself have produced there leaves the resulting evaluation on Y equal to the plain evaluation evalMap s ℓ on Y. No d-separation hypothesis is needed.","s":"∀ {N : Type u_1} [inst : DecidableEq N] [inst_1 : Fintype N] {Ω : N → Type u_2} [inst_2 : (n : N) → MeasurableSpace (Ω n)] (M : Causalean.SCM N Ω) {Y C : Finset (Causalean.SWIGNode N)} (hY : Y ⊆ M.observed) (hC : C ⊆ M.observed) (s : M.FixedValues) (ℓ : M.LatentValues), M.evalMap_overrideC hY hC s (fun v' => Causalean.SCM.observedAt_observedIndex M ⟨↑v', hC v'.property⟩ ▸ M.evalObservedAux s ℓ (↑(M.observedIndex ⟨↑v', hC v'.property⟩)) (M.observedIndex ⟨↑v', hC v'.property⟩).isLt) ℓ = fun v => Causalean.SCM.observedAt_observedIndex M ⟨↑v, hY v.property⟩ ▸ M.evalObservedAux s ℓ (↑(M.observedIndex ⟨↑v, hY v.property⟩)) (M.observedIndex ⟨↑v, hY v.property⟩).isLt","l":449,"f":"Causalean/SCM/Model/EvalOverrideC.lean"},{"n":"Causalean.SCM.evalMap_overrideC_dropZr_on_fillZrW","k":"theorem","m":"Causalean.SCM.Do.Rule2Kernel.Structural.StructPointwise","a":"SCM/Do/Rule2Kernel/Structural","t":2,"d":"Filled intervention coordinates do not affect the post-intervention override on Y.","s":"∀ {N : Type u_1} [inst : DecidableEq N] [inst_1 : Fintype N] {Ω : N → Type u_2} [inst_2 : (n : N) → MeasurableSpace (Ω n)] (M' : Causalean.SCM N Ω) (Z : Finset N) (hZ_obs : ∀ D ∈ Z, Causalean.SWIGNode.random D ∈ M'.observed) (hZ_fixed : ∀ D ∈ Z, Causalean.SWIGNode.fixed D ∉ M'.fixed) (Y W : Finset (Causalean.SWIGNode N)) (hY_M2 : Y ⊆ (M'.fixSet Z hZ_obs hZ_fixed).observed) (hZrW : Finset.image Causalean.SWIGNode.random Z ∪ W ⊆ (M'.fixSet Z hZ_obs hZ_fixed).observed), Disjoint Y (Finset.image Causalean.SWIGNode.random Z) → ∀ (s : (M'.fixSet Z hZ_obs hZ_fixed).FixedValues) (ℓ : (M'.fixSet Z hZ_obs hZ_fixed).LatentValues) (w : Causalean.ValuesOn W (Causalean.swigΩ Ω)), (M'.fixSet Z hZ_obs hZ_fixed).evalMap_overrideC hY_M2 hZrW s (M'.fillZrW Z hZ_obs hZ_fixed W s w) ℓ = (M'.fixSet Z hZ_obs hZ_fixed).evalMap_overrideC hY_M2 (LE.le.trans Finset.subset_union_right hZrW) s w ℓ","l":217,"f":"Causalean/SCM/Do/Rule2Kernel/Structural/StructPointwise.lean"},{"n":"Causalean.SCM.evalMap_overrideC_eq","k":"theorem","m":"Causalean.SCM.Model.EvalOverrideC","a":"SCM/Model","t":2,"d":"At any target node, the overridden evaluation map is the transported override auxiliary value at that node's topological index.","s":"∀ {N : Type u_1} [inst : DecidableEq N] [inst_1 : Fintype N] {Ω : N → Type u_2} [inst_2 : (n : N) → MeasurableSpace (Ω n)] (M : Causalean.SCM N Ω) {Y C : Finset (Causalean.SWIGNode N)} (hY : Y ⊆ M.observed) (hC : C ⊆ M.observed) (s : M.FixedValues) (c : Causalean.ValuesOn C (Causalean.swigΩ Ω)) (ℓ : M.LatentValues) (v : ↥Y), M.evalMap_overrideC hY hC s c ℓ v = Causalean.SCM.observedAt_observedIndex M ⟨↑v, hY v.property⟩ ▸ M.evalObservedAuxOverride hC s c ℓ (↑(M.observedIndex ⟨↑v, hY v.property⟩)) (M.observedIndex ⟨↑v, hY v.property⟩).isLt","l":234,"f":"Causalean/SCM/Model/EvalOverrideC.lean"},{"n":"Causalean.SCM.evalMap_overrideC_fixSet_compat_on_fillZrW","k":"theorem","m":"Causalean.SCM.Do.Rule2Kernel.Structural.StructCrossSCM","a":"SCM/Do/Rule2Kernel/Structural","t":1,"d":"Original and post-intervention override evaluations agree at fillZrW. For the intervention on names Z, whose random copies are observed in the base model and whose fixed copies are not yet part of the base model's fixed coordinates, with the union of the random copies of Z and a set W contained in the intervened model's observed variables, fix an intervened fixed assignment s, a conditioning value w on W, and a latent draw ℓ. Then at any observed node v of the intervened model, the base model's override-evaluation at the projected fixed assignment and the filled point built from s and w, applied to the reindexed latent draw, equals the intervened model's override-evaluation at s, the same filled point, and ℓ, evaluated at v.","s":"∀ {N : Type u_1} [inst : DecidableEq N] [inst_1 : Fintype N] {Ω : N → Type u_2} [inst_2 : (n : N) → MeasurableSpace (Ω n)] (M' : Causalean.SCM N Ω) (Z : Finset N) (hZ_obs : ∀ D ∈ Z, Causalean.SWIGNode.random D ∈ M'.observed) (hZ_fixed : ∀ D ∈ Z, Causalean.SWIGNode.fixed D ∉ M'.fixed) (W : Finset (Causalean.SWIGNode N)) (hZrW_M2 : Finset.image Causalean.SWIGNode.random Z ∪ W ⊆ (M'.fixSet Z hZ_obs hZ_fixed).observed) (s : (M'.fixSet Z hZ_obs hZ_fixed).FixedValues) (w : Causalean.ValuesOn W (Causalean.swigΩ Ω)) (ℓ : (M'.fixSet Z hZ_obs hZ_fixed).LatentValues) {v : Causalean.SWIGNode N} (hv : v ∈ (M'.fixSet Z hZ_obs hZ_fixed).observed), M'.evalMap_overrideC (Finset.Subset.refl M'.observed) (Causalean.SCM.fixSet_observed M' Z hZ_obs hZ_fixed ▸ hZrW_M2) (M'.fixSetProj Z hZ_obs hZ_fixed s) (M'.fillZrW Z hZ_obs hZ_fixed W s w) (Causalean.valuesProjection (le_of_eq (Eq.symm (Causalean.SCM.fixSet_unobserved M' Z hZ_obs hZ_fixed))) ℓ) ⟨v, hv⟩ = (M'.fixSet Z hZ_obs hZ_fixed).evalMap_overrideC (Finset.Subset.refl (M'.fixSet Z hZ_obs hZ_fixed).observed) hZrW_M2 s (M'.fillZrW Z hZ_obs hZ_fixed W s w) ℓ ⟨v, hv⟩","l":178,"f":"Causalean/SCM/Do/Rule2Kernel/Structural/StructCrossSCM.lean"},{"n":"Causalean.SCM.evalMap_overrideC_observed_unfold","k":"theorem","m":"Causalean.SCM.Do.Rule2Kernel.Structural.StructCrossSCM","a":"SCM/Do/Rule2Kernel/Structural","t":2,"d":"Cast-free observed unfold for evalMap_overrideC (Y = M.observed case).","s":"∀ {N : Type u_1} [inst : DecidableEq N] [inst_1 : Fintype N] {Ω : N → Type u_2} [inst_2 : (n : N) → MeasurableSpace (Ω n)] (M : Causalean.SCM N Ω) {C : Finset (Causalean.SWIGNode N)} (hC : C ⊆ M.observed) (s : M.FixedValues) (c : Causalean.ValuesOn C (Causalean.swigΩ Ω)) (ℓ : M.LatentValues) (v : ↥M.observed), M.evalMap_overrideC (Finset.Subset.refl M.observed) hC s c ℓ ⟨↑v, v.property⟩ = if hvC : ↑v ∈ C then c ⟨↑v, hvC⟩ else M.structFun v (Causalean.SCM.parentDispatchOverride✝ M hC s c ℓ v)","l":129,"f":"Causalean/SCM/Do/Rule2Kernel/Structural/StructCrossSCM.lean"},{"n":"Causalean.SCM.evalMap_unobserved","k":"theorem","m":"Causalean.SCM.Model.Evaluation","a":"SCM/Model","t":2,"d":"On a latent node, the evaluation map is direct projection from the latent assignment.","s":"∀ {N : Type u_1} [inst : DecidableEq N] [inst_1 : Fintype N] {Ω : N → Type u_2} [inst_2 : (n : N) → MeasurableSpace (Ω n)] (M : Causalean.SCM N Ω) (s : M.FixedValues) (ℓ : M.LatentValues) (w : ↥M.randomVars) (huo : ↑w ∈ M.unobserved), M.evalMap s ℓ w = ℓ ⟨↑w, huo⟩","l":210,"f":"Causalean/SCM/Model/Evaluation.lean"},{"n":"Causalean.SCM.evalObservedAux","k":"def","m":"Causalean.SCM.Model.Evaluation","a":"SCM/Model","t":1,"d":"The auxiliary evaluator recursively applies structural functions in topological order.","s":"{N : Type u_1} → [inst : DecidableEq N] → [inst_1 : Fintype N] → {Ω : N → Type u_2} → [inst_2 : (n : N) → MeasurableSpace (Ω n)] → (M : Causalean.SCM N Ω) → M.FixedValues → M.LatentValues → (n : ℕ) → (hn : n < M.observed.card) → Causalean.swigΩ Ω ↑(M.observedAt ⟨n, hn⟩)","l":136,"f":"Causalean/SCM/Model/Evaluation.lean"},{"n":"Causalean.SCM.evalObservedAuxOverride","k":"def","m":"Causalean.SCM.Model.EvalOverrideC","a":"SCM/Model","t":1,"d":"The override auxiliary evaluator computes each observed node in topological order, short-circuiting nodes in the override block.","s":"{N : Type u_1} → [inst : DecidableEq N] → [inst_1 : Fintype N] → {Ω : N → Type u_2} → [inst_2 : (n : N) → MeasurableSpace (Ω n)] → (M : Causalean.SCM N Ω) → {C : Finset (Causalean.SWIGNode N)} → C ⊆ M.observed → M.FixedValues → Causalean.ValuesOn C (Causalean.swigΩ Ω) → M.LatentValues → (n : ℕ) → (hn : n < M.observed.card) → Causalean.swigΩ Ω ↑(M.observedAt ⟨n, hn⟩)","l":177,"f":"Causalean/SCM/Model/EvalOverrideC.lean"},{"n":"Causalean.SCM.evalObservedAuxOverride_agree_anc","k":"theorem","m":"Causalean.SCM.Do.Rule2Kernel.Structural.StructPointwise","a":"SCM/Do/Rule2Kernel/Structural","t":2,"d":"If two overrides use the same values on every observed ancestor of a target set, their recursive structural evaluations agree at every such ancestor and target.","s":"∀ {N : Type u_1} [inst : DecidableEq N] [inst_1 : Fintype N] {Ω : N → Type u_2} [inst_2 : (n : N) → MeasurableSpace (Ω n)] (M : Causalean.SCM N Ω) (T : Finset (Causalean.SWIGNode N)) {C₁ C₂ : Finset (Causalean.SWIGNode N)} (hC₁ : C₁ ⊆ M.observed) (hC₂ : C₂ ⊆ M.observed) (s : M.FixedValues) (ℓ : M.LatentValues) (c₁ : Causalean.ValuesOn C₁ (Causalean.swigΩ Ω)) (c₂ : Causalean.ValuesOn C₂ (Causalean.swigΩ Ω)), (∀ (x : Causalean.SWIGNode N), (∃ t ∈ T, x = t ∨ M.dag.isAncestor x t) → (x ∈ C₁ ↔ x ∈ C₂) ∧ ∀ (h₁ : x ∈ C₁) (h₂ : x ∈ C₂), c₁ ⟨x, h₁⟩ = c₂ ⟨x, h₂⟩) → ∀ (n : ℕ) (hn : n < M.observed.card), (∃ t ∈ T, ↑(M.observedAt ⟨n, hn⟩) = t ∨ M.dag.isAncestor (↑(M.observedAt ⟨n, hn⟩)) t) → M.evalObservedAuxOverride hC₁ s c₁ ℓ n hn = M.evalObservedAuxOverride hC₂ s c₂ ℓ n hn","l":109,"f":"Causalean/SCM/Do/Rule2Kernel/Structural/StructPointwise.lean"},{"n":"Causalean.SCM.evalObservedAuxOverride_agree_cutset","k":"theorem","m":"Causalean.SCM.Model.CutsetLatent","a":"SCM/Model","t":2,"d":"If two latent assignments agree at every unobserved node that is a target or reaches a target along a directed path whose interior avoids the override block, their recursive override evaluations agree at every observed node that is a target or reaches one along such a path.","s":"∀ {N : Type u_2} [inst : DecidableEq N] [inst_1 : Fintype N] {Ω : N → Type u_3} [inst_2 : (n : N) → MeasurableSpace (Ω n)] (M : Causalean.SCM N Ω) (T : Finset (Causalean.SWIGNode N)) {C : Finset (Causalean.SWIGNode N)} (hC : C ⊆ M.observed) (s : M.FixedValues) (c : Causalean.ValuesOn C (Causalean.swigΩ Ω)) (ℓ₁ ℓ₂ : M.LatentValues), (∀ (u : Causalean.SWIGNode N) (hu : u ∈ M.unobserved), (∃ t ∈ T, u = t ∨ M.dag.isAncestorAvoiding C u t) → ℓ₁ ⟨u, hu⟩ = ℓ₂ ⟨u, hu⟩) → ∀ (n : ℕ) (hn : n < M.observed.card), (∃ t ∈ T, ↑(M.observedAt ⟨n, hn⟩) = t ∨ M.dag.isAncestorAvoiding C (↑(M.observedAt ⟨n, hn⟩)) t) → M.evalObservedAuxOverride hC s c ℓ₁ n hn = M.evalObservedAuxOverride hC s c ℓ₂ n hn","l":213,"f":"Causalean/SCM/Model/CutsetLatent.lean"},{"n":"Causalean.SCM.evalObservedAuxOverride_eq","k":"theorem","m":"Causalean.SCM.Model.EvalOverrideC","a":"SCM/Model","t":2,"d":"The override auxiliary evaluator unfolds to either the override value or the structural function applied to overridden parents.","s":"∀ {N : Type u_1} [inst : DecidableEq N] [inst_1 : Fintype N] {Ω : N → Type u_2} [inst_2 : (n : N) → MeasurableSpace (Ω n)] (M : Causalean.SCM N Ω) {C : Finset (Causalean.SWIGNode N)} (hC : C ⊆ M.observed) (s : M.FixedValues) (c : Causalean.ValuesOn C (Causalean.swigΩ Ω)) (ℓ : M.LatentValues) (n : ℕ) (hn : n < M.observed.card), M.evalObservedAuxOverride hC s c ℓ n hn = if hcSelf : ↑(M.observedAt ⟨n, hn⟩) ∈ C then c ⟨↑(M.observedAt ⟨n, hn⟩), hcSelf⟩ else M.structFun (M.observedAt ⟨n, hn⟩) fun w => M.parentMapOverride s c ℓ hn (fun m x hm_card => M.evalObservedAuxOverride hC s c ℓ m hm_card) w","l":198,"f":"Causalean/SCM/Model/EvalOverrideC.lean"},{"n":"Causalean.SCM.evalObservedAuxOverride_eq_evalObservedAux_at_self","k":"theorem","m":"Causalean.SCM.Model.EvalOverrideC","a":"SCM/Model","t":2,"d":"When an observed-node override uses the model's own recursively evaluated values, the overridden recursive evaluator agrees with the original evaluator at every topological position.","s":"∀ {N : Type u_1} [inst : DecidableEq N] [inst_1 : Fintype N] {Ω : N → Type u_2} [inst_2 : (n : N) → MeasurableSpace (Ω n)] (M : Causalean.SCM N Ω) {C : Finset (Causalean.SWIGNode N)} (hC : C ⊆ M.observed) (s : M.FixedValues) (ℓ : M.LatentValues) (n : ℕ) (hn : n < M.observed.card), M.evalObservedAuxOverride hC s (fun v' => Causalean.SCM.observedAt_observedIndex M ⟨↑v', hC v'.property⟩ ▸ M.evalObservedAux s ℓ (↑(M.observedIndex ⟨↑v', hC v'.property⟩)) (M.observedIndex ⟨↑v', hC v'.property⟩).isLt) ℓ n hn = M.evalObservedAux s ℓ n hn","l":386,"f":"Causalean/SCM/Model/EvalOverrideC.lean"},{"n":"Causalean.SCM.evalObservedAuxOverride_measurable","k":"theorem","m":"Causalean.SCM.Model.EvalOverrideC","a":"SCM/Model","t":2,"d":"At every position in an SCM's topological order, its observed-variable evaluator with specified observed values overridden is jointly measurable in fixed, override, and latent inputs.","s":"∀ {N : Type u_1} [inst : DecidableEq N] [inst_1 : Fintype N] {Ω : N → Type u_2} [inst_2 : (n : N) → MeasurableSpace (Ω n)] (M : Causalean.SCM N Ω) {C : Finset (Causalean.SWIGNode N)} (hC : C ⊆ M.observed) (n : ℕ) (hn : n < M.observed.card), Measurable fun p => M.evalObservedAuxOverride hC p.1.1 p.1.2 p.2 n hn","l":489,"f":"Causalean/SCM/Model/EvalOverrideC.lean"},{"n":"Causalean.SCM.evalObservedAux_agree_anc","k":"theorem","m":"Causalean.SCM.Model.Evaluation","a":"SCM/Model","t":2,"d":"An observed variable has the same recursively evaluated value under two inputs when those inputs agree on every fixed and latent cause that can affect the target variables. This expresses the local ancestral dependence of a structural causal model.","s":"∀ {N : Type u_1} [inst : DecidableEq N] [inst_1 : Fintype N] {Ω : N → Type u_2} [inst_2 : (n : N) → MeasurableSpace (Ω n)] (M : Causalean.SCM N Ω) (T : Finset (Causalean.SWIGNode N)) {s s' : M.FixedValues} {ℓ ℓ' : M.LatentValues}, (∀ (d : Causalean.SWIGNode N) (hd : d ∈ M.fixed), (∃ v ∈ T, d = v ∨ M.dag.isAncestor d v) → s ⟨d, hd⟩ = s' ⟨d, hd⟩) → (∀ (u : Causalean.SWIGNode N) (hu : u ∈ M.unobserved), (∃ v ∈ T, u = v ∨ M.dag.isAncestor u v) → ℓ ⟨u, hu⟩ = ℓ' ⟨u, hu⟩) → ∀ (n : ℕ) (hn : n < M.observed.card), (∃ v ∈ T, ↑(M.observedAt ⟨n, hn⟩) = v ∨ M.dag.isAncestor (↑(M.observedAt ⟨n, hn⟩)) v) → M.evalObservedAux s ℓ n hn = M.evalObservedAux s' ℓ' n hn","l":454,"f":"Causalean/SCM/Model/Evaluation.lean"},{"n":"Causalean.SCM.evalObservedAux_cast_collapse_at_observedAt","k":"theorem","m":"Causalean.SCM.Model.EvalOverrideC","a":"SCM/Model","t":2,"d":"For an observed node, transporting its recursively evaluated value through the round-trip topological-index lookup leaves that value unchanged.","s":"∀ {N : Type u_1} [inst : DecidableEq N] [inst_1 : Fintype N] {Ω : N → Type u_2} [inst_2 : (n : N) → MeasurableSpace (Ω n)] (M : Causalean.SCM N Ω) (s : M.FixedValues) (ℓ : M.LatentValues) (j : Fin M.observed.card) (hvObs : ↑(M.observedAt j) ∈ M.observed), Causalean.SCM.observedAt_observedIndex M ⟨↑(M.observedAt j), hvObs⟩ ▸ M.evalObservedAux s ℓ (↑(M.observedIndex ⟨↑(M.observedAt j), hvObs⟩)) (M.observedIndex ⟨↑(M.observedAt j), hvObs⟩).isLt = M.evalObservedAux s ℓ (↑j) j.isLt","l":348,"f":"Causalean/SCM/Model/EvalOverrideC.lean"},{"n":"Causalean.SCM.evalObservedAux_eq","k":"theorem","m":"Causalean.SCM.Model.Evaluation","a":"SCM/Model","t":2,"d":"The auxiliary evaluator unfolds to its structural function applied to the parent tuple.","s":"∀ {N : Type u_1} [inst : DecidableEq N] [inst_1 : Fintype N] {Ω : N → Type u_2} [inst_2 : (n : N) → MeasurableSpace (Ω n)] (M : Causalean.SCM N Ω) (s : M.FixedValues) (ℓ : M.LatentValues) (n : ℕ) (hn : n < M.observed.card), M.evalObservedAux s ℓ n hn = M.structFun (M.observedAt ⟨n, hn⟩) fun w => M.parentMap s ℓ hn (fun m x hm_card => M.evalObservedAux s ℓ m hm_card) w","l":152,"f":"Causalean/SCM/Model/Evaluation.lean"},{"n":"Causalean.SCM.evalObservedAux_measurable","k":"theorem","m":"Causalean.SCM.Model.Evaluation","a":"SCM/Model","t":2,"d":"At every position in a causal model's topological ordering of observed variables, the recursively evaluated observed value is measurable as a function of the model's fixed and latent inputs.","s":"∀ {N : Type u_1} [inst : DecidableEq N] [inst_1 : Fintype N] {Ω : N → Type u_2} [inst_2 : (n : N) → MeasurableSpace (Ω n)] (M : Causalean.SCM N Ω) (n : ℕ) (hn : n < M.observed.card), Measurable fun p => M.evalObservedAux p.1 p.2 n hn","l":349,"f":"Causalean/SCM/Model/Evaluation.lean"},{"n":"Causalean.SCM.exists_evalMap_overrideC_factors_cutset","k":"theorem","m":"Causalean.SCM.Model.CutsetLatent","a":"SCM/Model","t":1,"d":"Cutset factorization of an override evaluation. Fix a structural causal model M, a target node set Y and an override node set C with Y contained in the observed nodes and C contained in the observed nodes, a fixed-value assignment s, and an override assignment c on C. Then there is a measurable map from assignments on the latent cutset of Y relative to C to assignments on Y such that, for every latent assignment, the overridden evaluation at Y equals this map applied to the latent assignment's projection onto the cutset.","s":"∀ {N : Type u_2} [inst : DecidableEq N] [inst_1 : Fintype N] {Ω : N → Type u_3} [inst_2 : (n : N) → MeasurableSpace (Ω n)] (M : Causalean.SCM N Ω) {Y C : Finset (Causalean.SWIGNode N)} (hY : Y ⊆ M.observed) (hC : C ⊆ M.observed) (s : M.FixedValues) (c : Causalean.ValuesOn C (Causalean.swigΩ Ω)), ∃ h, Measurable h ∧ ∀ (ℓ : M.LatentValues), M.evalMap_overrideC hY hC s c ℓ = h (Causalean.valuesProjection (Causalean.SCM.cutsetLatent_subset M Y C) ℓ)","l":334,"f":"Causalean/SCM/Model/CutsetLatent.lean"},{"n":"Causalean.SCM.extendObsPrefix","k":"def","m":"Causalean.SCM.Factored.ObsChainKernel","a":"SCM/Factored","t":1,"d":"Extend a prefix assignment by the next observed value.","s":"{N : Type u_1} → [inst : DecidableEq N] → [inst_1 : Fintype N] → {Ω : N → Type u_2} → [inst_2 : (n : N) → MeasurableSpace (Ω n)] → (M : Causalean.SCM N Ω) → {n : ℕ} → (hn : n < M.observed.card) → Causalean.ValuesOn (M.prefixNodes n) (Causalean.swigΩ Ω) × Causalean.swigΩ Ω ↑(M.observedAt ⟨n, hn⟩) → Causalean.ValuesOn (M.prefixNodes (n + 1)) (Causalean.swigΩ Ω)","l":407,"f":"Causalean/SCM/Factored/ObsChainKernel.lean"},{"n":"Causalean.SCM.extendOrderedLatentPrefix","k":"def","m":"Causalean.SCM.Factored.PrefixState","a":"SCM/Factored","t":1,"d":"Append the next observed value to an ordered-latent prefix state. This is the state-space normalization map used after one compProd step in the factored construction: ((ℓ, ξ), y) ↦ (ℓ, (ξ, y)).","s":"{N : Type uN} → [inst : DecidableEq N] → [inst_1 : Fintype N] → {Ω : N → Type uΩ} → [inst_2 : (n : N) → MeasurableSpace (Ω n)] → (M : Causalean.SCM N Ω) → {n : ℕ} → (hn : n + 1 ≤ M.observed.card) → M.OrderedLatentPrefixValues n (Nat.le_of_succ_le hn) × Causalean.swigΩ Ω ↑(M.observedAt ⟨n, hn⟩) → M.OrderedLatentPrefixValues (n + 1) hn","l":105,"f":"Causalean/SCM/Factored/PrefixState.lean"},{"n":"Causalean.SCM.extractDistrict","k":"def","m":"Causalean.SCM.ID.Density.IdentifyMass","a":"SCM/ID/Density","t":1,"d":"Extract the district factor for C' from a mass function on A by multiplying adjacent prefix marginal ratios along the topological order of G'.","s":"{N : Type u_1} → [inst : DecidableEq N] → [inst_1 : Fintype N] → {Ω : N → Type u_2} → [(n : N) → Fintype (Ω n)] → (O : Finset (Causalean.SWIGNode N)) → Causalean.SWIGGraph N → (A : Finset (Causalean.SWIGNode N)) → Finset (Causalean.SWIGNode N) → A ⊆ O → (Causalean.ValuesOn O (Causalean.swigΩ Ω) → ENNReal) → Causalean.ValuesOn O (Causalean.swigΩ Ω) → ENNReal","l":119,"f":"Causalean/SCM/ID/Density/IdentifyMass.lean"},{"n":"Causalean.SCM.fillZrW","k":"def","m":"Causalean.SCM.Do.Rule2Kernel.Helpers","a":"SCM/Do/Rule2Kernel","t":1,"d":"The Rule 2 filler inserts intervention values into a conditioning assignment.","s":"{N : Type u_1} → [inst : DecidableEq N] → [inst_1 : Fintype N] → {Ω : N → Type u_2} → [inst_2 : (n : N) → MeasurableSpace (Ω n)] → (M' : Causalean.SCM N Ω) → (Z : Finset N) → (hZ_obs : ∀ D ∈ Z, Causalean.SWIGNode.random D ∈ M'.observed) → (hZ_fixed : ∀ D ∈ Z, Causalean.SWIGNode.fixed D ∉ M'.fixed) → (W : Finset (Causalean.SWIGNode N)) → (M'.fixSet Z hZ_obs hZ_fixed).FixedValues → Causalean.ValuesOn W (Causalean.swigΩ Ω) → Causalean.ValuesOn (Finset.image Causalean.SWIGNode.random Z ∪ W) (Causalean.swigΩ Ω)","l":79,"f":"Causalean/SCM/Do/Rule2Kernel/Helpers.lean"},{"n":"Causalean.SCM.fillZrW_fixSetExtend","k":"theorem","m":"Causalean.SCM.Do.Rule2Kernel.Helpers","a":"SCM/Do/Rule2Kernel","t":1,"d":"For the intervention names X, whose random copies are observed in the base model and whose fixed copies are not yet part of the base model's fixed coordinates, filling the conditioning assignment z against the extended post-intervention fixed assignment built from the base assignment s0 and treatment value t equals directly combining t and z into one assignment on their union.","s":"∀ {N : Type u_1} [inst : DecidableEq N] [inst_1 : Fintype N] {Ω : N → Type u_2} [inst_2 : (n : N) → MeasurableSpace (Ω n)] (M : Causalean.SCM N Ω) (X : Finset N) (hObs : ∀ D ∈ X, Causalean.SWIGNode.random D ∈ M.observed) (hFix : ∀ D ∈ X, Causalean.SWIGNode.fixed D ∉ M.fixed) (Z : Finset (Causalean.SWIGNode N)) (s0 : M.FixedValues) (t : Causalean.ValuesOn (Finset.image Causalean.SWIGNode.random X) (Causalean.swigΩ Ω)) (z : Causalean.ValuesOn Z (Causalean.swigΩ Ω)), M.fillZrW X hObs hFix Z (M.fixSetExtend X hObs hFix s0 t) z = Causalean.valuesUnionMk t z","l":300,"f":"Causalean/SCM/Do/Rule2Kernel/Helpers.lean"},{"n":"Causalean.SCM.fillZrW_image_eq","k":"theorem","m":"Causalean.SCM.Do.Rule2Kernel.WMarginal","a":"SCM/Do/Rule2Kernel","t":2,"d":"The image of a W-event under the filled-assignment map fillZrW Z hZ_obs hZ_fixed W s is exactly the set of assignments on Z.image .random ∪ W whose W-projection lies in the event and whose Z.image .random projection equals the fixed intervention value zFixedAsRandom (valuesProjection (fixSet_image_fixed_subset ...) s).","s":"∀ {N : Type u_1} [inst : DecidableEq N] [inst_1 : Fintype N] {Ω : N → Type u_2} [inst_2 : (n : N) → MeasurableSpace (Ω n)] (M' : Causalean.SCM N Ω) (Z : Finset N) (hZ_obs : ∀ D ∈ Z, Causalean.SWIGNode.random D ∈ M'.observed) (hZ_fixed : ∀ D ∈ Z, Causalean.SWIGNode.fixed D ∉ M'.fixed) (W : Finset (Causalean.SWIGNode N)), Disjoint (Finset.image Causalean.SWIGNode.random Z) W → ∀ (s' : (M'.fixSet Z hZ_obs hZ_fixed).FixedValues) (A : Set (Causalean.ValuesOn W (Causalean.swigΩ Ω))), M'.fillZrW Z hZ_obs hZ_fixed W s' '' A = (fun c => Causalean.valuesProjection Finset.subset_union_right c) ⁻¹' A ∩ (fun c => Causalean.valuesProjection Finset.subset_union_left c) ⁻¹' {Causalean.SCM.zFixedAsRandom (Causalean.valuesProjection (Causalean.SCM.fixSet_image_fixed_subset M' Z hZ_obs hZ_fixed) s')}","l":47,"f":"Causalean/SCM/Do/Rule2Kernel/WMarginal.lean"},{"n":"Causalean.SCM.fillZrW_injective","k":"theorem","m":"Causalean.SCM.Do.Rule2Kernel.WMarginal","a":"SCM/Do/Rule2Kernel","t":2,"d":"fillZrW Z _ _ W s' is injective when Z.image .random and W are disjoint.","s":"∀ {N : Type u_1} [inst : DecidableEq N] [inst_1 : Fintype N] {Ω : N → Type u_2} [inst_2 : (n : N) → MeasurableSpace (Ω n)] (M' : Causalean.SCM N Ω) (Z : Finset N) (hZ_obs : ∀ D ∈ Z, Causalean.SWIGNode.random D ∈ M'.observed) (hZ_fixed : ∀ D ∈ Z, Causalean.SWIGNode.fixed D ∉ M'.fixed) (W : Finset (Causalean.SWIGNode N)), Disjoint (Finset.image Causalean.SWIGNode.random Z) W → ∀ (s' : (M'.fixSet Z hZ_obs hZ_fixed).FixedValues), Function.Injective (M'.fillZrW Z hZ_obs hZ_fixed W s')","l":112,"f":"Causalean/SCM/Do/Rule2Kernel/WMarginal.lean"},{"n":"Causalean.SCM.fillZrW_random_eq_fixed","k":"theorem","m":"Causalean.SCM.Do.Rule2Kernel.Structural.StructPointwise","a":"SCM/Do/Rule2Kernel/Structural","t":2,"d":"The filled assignment pins intervention values on the random copies of Z.","s":"∀ {N : Type u_1} [inst : DecidableEq N] [inst_1 : Fintype N] {Ω : N → Type u_2} [inst_2 : (n : N) → MeasurableSpace (Ω n)] (M' : Causalean.SCM N Ω) (Z : Finset N) (hZ_obs : ∀ D ∈ Z, Causalean.SWIGNode.random D ∈ M'.observed) (hZ_fixed : ∀ D ∈ Z, Causalean.SWIGNode.fixed D ∉ M'.fixed) (W : Finset (Causalean.SWIGNode N)) (s : (M'.fixSet Z hZ_obs hZ_fixed).FixedValues) (w : Causalean.ValuesOn W (Causalean.swigΩ Ω)) {D : N} (hD : D ∈ Z) (hRD : Causalean.SWIGNode.random D ∈ Finset.image Causalean.SWIGNode.random Z ∪ W), M'.fillZrW Z hZ_obs hZ_fixed W s w ⟨Causalean.SWIGNode.random D, hRD⟩ = s ⟨Causalean.SWIGNode.fixed D, Causalean.SCM.fixed_mem_fixSet M' Z hZ_obs hZ_fixed hD⟩","l":289,"f":"Causalean/SCM/Do/Rule2Kernel/Structural/StructPointwise.lean"},{"n":"Causalean.SCM.fixMono","k":"def","m":"Causalean.SCM.Model.InterventionMono","a":"SCM/Model","t":1,"d":"The monolithic generalized intervention applies all target splits at once while inheriting latent laws and reindexing structural parents.","s":"{N : Type u_1} → [inst : DecidableEq N] → [inst_1 : Fintype N] → {Ω : N → Type u_2} → [inst_2 : (n : N) → MeasurableSpace (Ω n)] → (M : Causalean.SCM N Ω) → (X : Finset N) → (∀ D ∈ X, Causalean.SWIGNode.random D ∈ M.observed) → (∀ D ∈ X, Causalean.SWIGNode.fixed D ∉ M.fixed) → Causalean.SCM N Ω","l":203,"f":"Causalean/SCM/Model/InterventionMono.lean"},{"n":"Causalean.SCM.fixMonoParentMap","k":"def","m":"Causalean.SCM.Model.InterventionMono","a":"SCM/Model","t":1,"d":"The monolithic parent reindexer converts split-graph parent values into the parent values expected by the original structural function.","s":"{N : Type u_1} → [inst : DecidableEq N] → [inst_1 : Fintype N] → {Ω : N → Type u_2} → (G : Causalean.SWIGGraph N) → (X : Finset N) → (hObs : ∀ D ∈ X, Causalean.SWIGNode.random D ∈ G.observed) → (hFix : ∀ D ∈ X, Causalean.SWIGNode.fixed D ∉ G.fixed) → (v : Causalean.SWIGNode N) → ((w' : ↥((G.splitMono X hObs hFix).dag.parents v)) → Causalean.swigΩ Ω ↑w') → (w : ↥(G.dag.parents v)) → Causalean.swigΩ Ω ↑w","l":44,"f":"Causalean/SCM/Model/InterventionMono.lean"},{"n":"Causalean.SCM.fixMonoParentMap_apply_fixed","k":"theorem","m":"Causalean.SCM.Model.InterventionMono","a":"SCM/Model","t":2,"d":"At a fixed-coordinate parent, the monolithic parent reindexer reads the same fixed coordinate from the split graph.","s":"∀ {N : Type u_1} [inst : DecidableEq N] [inst_1 : Fintype N] {Ω : N → Type u_2} (G : Causalean.SWIGGraph N) (X : Finset N) (hObs : ∀ D ∈ X, Causalean.SWIGNode.random D ∈ G.observed) (hFix : ∀ D ∈ X, Causalean.SWIGNode.fixed D ∉ G.fixed) (v : Causalean.SWIGNode N) (ξ : (w' : ↥((G.splitMono X hObs hFix).dag.parents v)) → Causalean.swigΩ Ω ↑w') (d : N) (hwVal : Causalean.SWIGNode.fixed d ∈ G.dag.parents v), Causalean.SCM.fixMonoParentMap G X hObs hFix v ξ ⟨Causalean.SWIGNode.fixed d, hwVal⟩ = ξ ⟨Causalean.SWIGNode.fixed d, (Causalean.SWIGGraph.splitMono_parents_char G X hObs hFix v (Causalean.SWIGNode.fixed d)).mpr (Or.inl ⟨hwVal, fun x x_1 heq => Eq.casesOn (motive := fun a t => Causalean.SWIGNode.random x = a → heq ≍ t → False) heq (fun h => False.elim (noConfusion_of_Nat Causalean.SWIGNode.ctorIdx h)) (Eq.refl (Causalean.SWIGNode.random x)) (HEq.refl heq)⟩)⟩","l":79,"f":"Causalean/SCM/Model/InterventionMono.lean"},{"n":"Causalean.SCM.fixMonoParentMap_apply_random","k":"theorem","m":"Causalean.SCM.Model.InterventionMono","a":"SCM/Model","t":2,"d":"At a treated random-coordinate parent, the monolithic parent reindexer reads the corresponding fixed coordinate from the split graph.","s":"∀ {N : Type u_1} [inst : DecidableEq N] [inst_1 : Fintype N] {Ω : N → Type u_2} (G : Causalean.SWIGGraph N) (X : Finset N) (hObs : ∀ D ∈ X, Causalean.SWIGNode.random D ∈ G.observed) (hFix : ∀ D ∈ X, Causalean.SWIGNode.fixed D ∉ G.fixed) (v : Causalean.SWIGNode N) (D : N) (hD : D ∈ X) (ξ : (w' : ↥((G.splitMono X hObs hFix).dag.parents v)) → Causalean.swigΩ Ω ↑w') (hD_parent : Causalean.SWIGNode.random D ∈ G.dag.parents v), Causalean.SCM.fixMonoParentMap G X hObs hFix v ξ ⟨Causalean.SWIGNode.random D, hD_parent⟩ = ξ ⟨Causalean.SWIGNode.fixed D, (Causalean.SWIGGraph.splitMono_parents_char G X hObs hFix v (Causalean.SWIGNode.fixed D)).mpr (Or.inr (Exists.intro D ⟨hD, ⟨rfl, hD_parent⟩⟩))⟩","l":124,"f":"Causalean/SCM/Model/InterventionMono.lean"},{"n":"Causalean.SCM.fixMonoParentMap_apply_random_notMem","k":"theorem","m":"Causalean.SCM.Model.InterventionMono","a":"SCM/Model","t":2,"d":"At an untreated random-coordinate parent, the monolithic parent reindexer reads the same random coordinate from the split graph.","s":"∀ {N : Type u_1} [inst : DecidableEq N] [inst_1 : Fintype N] {Ω : N → Type u_2} (G : Causalean.SWIGGraph N) (X : Finset N) (hObs : ∀ D ∈ X, Causalean.SWIGNode.random D ∈ G.observed) (hFix : ∀ D ∈ X, Causalean.SWIGNode.fixed D ∉ G.fixed) (v : Causalean.SWIGNode N) (ξ : (w' : ↥((G.splitMono X hObs hFix).dag.parents v)) → Causalean.swigΩ Ω ↑w') (u : N) (hu : u ∉ X) (hwVal : Causalean.SWIGNode.random u ∈ G.dag.parents v), Causalean.SCM.fixMonoParentMap G X hObs hFix v ξ ⟨Causalean.SWIGNode.random u, hwVal⟩ = ξ ⟨Causalean.SWIGNode.random u, (Causalean.SWIGGraph.splitMono_parents_char G X hObs hFix v (Causalean.SWIGNode.random u)).mpr (Or.inl ⟨hwVal, fun x hD heq => hu (Eq.symm (Causalean.SWIGNode.random.inj heq) ▸ hD)⟩)⟩","l":100,"f":"Causalean/SCM/Model/InterventionMono.lean"},{"n":"Causalean.SCM.fixMono_fixed","k":"theorem","m":"Causalean.SCM.Model.InterventionMono","a":"SCM/Model","t":1,"d":"Fixed-node set of the monolithic intervention. For a SWIG graph G and an intervention target set X such that every targeted node is currently a random observed node and none of its fixed copies is already fixed, the monolithic intervention graph's fixed node set equals G's fixed node set together with the fixed copies of the targeted nodes in X.","s":"∀ {N : Type u_1} [inst : DecidableEq N] [inst_1 : Fintype N] (G : Causalean.SWIGGraph N) (X : Finset N) (hObs : ∀ D ∈ X, Causalean.SWIGNode.random D ∈ G.observed) (hFix : ∀ D ∈ X, Causalean.SWIGNode.fixed D ∉ G.fixed), (G.splitMono X hObs hFix).fixed = G.fixed ∪ Finset.image Causalean.SWIGNode.fixed X","l":288,"f":"Causalean/SCM/Model/InterventionMono.lean"},{"n":"Causalean.SCM.fixMono_fixed_subset","k":"theorem","m":"Causalean.SCM.Model.InterventionMono","a":"SCM/Model","t":2,"d":"The original fixed node set is contained in the fixed node set after the monolithic intervention.","s":"∀ {N : Type u_1} [inst : DecidableEq N] [inst_1 : Fintype N] (G : Causalean.SWIGGraph N) (X : Finset N) (hObs : ∀ D ∈ X, Causalean.SWIGNode.random D ∈ G.observed) (hFix : ∀ D ∈ X, Causalean.SWIGNode.fixed D ∉ G.fixed), G.fixed ⊆ (G.splitMono X hObs hFix).fixed","l":310,"f":"Causalean/SCM/Model/InterventionMono.lean"},{"n":"Causalean.SCM.fixMono_image_fixed_subset","k":"theorem","m":"Causalean.SCM.Model.InterventionMono","a":"SCM/Model","t":2,"d":"The fixed copies of the intervention targets are contained in the fixed node set after the monolithic intervention.","s":"∀ {N : Type u_1} [inst : DecidableEq N] [inst_1 : Fintype N] (G : Causalean.SWIGGraph N) (X : Finset N) (hObs : ∀ D ∈ X, Causalean.SWIGNode.random D ∈ G.observed) (hFix : ∀ D ∈ X, Causalean.SWIGNode.fixed D ∉ G.fixed), Finset.image Causalean.SWIGNode.fixed X ⊆ (G.splitMono X hObs hFix).fixed","l":320,"f":"Causalean/SCM/Model/InterventionMono.lean"},{"n":"Causalean.SCM.fixMono_latentDist","k":"theorem","m":"Causalean.SCM.Model.InterventionMono","a":"SCM/Model","t":1,"d":"Latent-distribution invariance of the monolithic intervention. For a structural causal model M and an intervention target set X such that every targeted node is currently a random observed node and none of its fixed copies is already fixed, and any latent-root node u of the monolithically intervened model, the intervened model's latent distribution at u equals M's original latent distribution at u.","s":"∀ {N : Type u_1} [inst : DecidableEq N] [inst_1 : Fintype N] {Ω : N → Type u_2} [inst_2 : (n : N) → MeasurableSpace (Ω n)] (M : Causalean.SCM N Ω) (X : Finset N) (hObs : ∀ D ∈ X, Causalean.SWIGNode.random D ∈ M.observed) (hFix : ∀ D ∈ X, Causalean.SWIGNode.fixed D ∉ M.fixed) (u : ↥(M.fixMono X hObs hFix).unobserved), (M.fixMono X hObs hFix).latentDist u = M.latentDist u","l":298,"f":"Causalean/SCM/Model/InterventionMono.lean"},{"n":"Causalean.SCM.fixMono_observed","k":"theorem","m":"Causalean.SCM.Model.InterventionMono","a":"SCM/Model","t":1,"d":"Observed-node invariance of the monolithic intervention. For a SWIG graph G and an intervention target set X such that every targeted node is currently a random observed node and none of its fixed copies is already fixed, the monolithic intervention graph obtained by fixing X has the same observed node set as G.","s":"∀ {N : Type u_1} [inst : DecidableEq N] [inst_1 : Fintype N] (G : Causalean.SWIGGraph N) (X : Finset N) (hObs : ∀ D ∈ X, Causalean.SWIGNode.random D ∈ G.observed) (hFix : ∀ D ∈ X, Causalean.SWIGNode.fixed D ∉ G.fixed), (G.splitMono X hObs hFix).observed = G.observed","l":273,"f":"Causalean/SCM/Model/InterventionMono.lean"},{"n":"Causalean.SCM.fixMono_parents_eq_of_no_fixed_parent","k":"theorem","m":"Causalean.SCM.Model.InterventionMono","a":"SCM/Model","t":2,"d":"If no fixed copy of a target is a parent of a vertex after intervention, that vertex has the same parents as before.","s":"∀ {N : Type u_1} [inst : DecidableEq N] [inst_1 : Fintype N] {Ω : N → Type u_2} [inst_2 : (n : N) → MeasurableSpace (Ω n)] (M : Causalean.SCM N Ω) (X : Finset N) (hObs : ∀ D ∈ X, Causalean.SWIGNode.random D ∈ M.observed) (hFix : ∀ D ∈ X, Causalean.SWIGNode.fixed D ∉ M.fixed) {v : Causalean.SWIGNode N}, (∀ D ∈ X, Causalean.SWIGNode.fixed D ∉ (M.fixMono X hObs hFix).dag.parents v) → (M.fixMono X hObs hFix).dag.parents v = M.dag.parents v","l":330,"f":"Causalean/SCM/Model/InterventionMono.lean"},{"n":"Causalean.SCM.fixMono_structFun_apply","k":"theorem","m":"Causalean.SCM.Model.CounterfactualLemmas","a":"SCM/Model","t":2,"d":"After an intervention on a set of observed variables, each observed variable is still computed by the original structural equation, with intervened parents replaced by their assigned intervention values.","s":"∀ {N : Type uN} [inst : DecidableEq N] [inst_1 : Fintype N] {Ω : N → Type uΩ} [inst_2 : (n : N) → MeasurableSpace (Ω n)] (M : Causalean.SCM N Ω) (X : Finset N) (hObs : ∀ D ∈ X, Causalean.SWIGNode.random D ∈ M.observed) (hFix : ∀ D ∈ X, Causalean.SWIGNode.fixed D ∉ M.fixed) (v : ↥(M.fixMono X hObs hFix).observed) (ξ : (w : ↥((M.fixMono X hObs hFix).dag.parents ↑v)) → Causalean.swigΩ Ω ↑w), (M.fixMono X hObs hFix).structFun v ξ = M.structFun ⟨↑v, v.property⟩ (Causalean.SCM.fixMonoParentMap M.toSWIGGraph X hObs hFix (↑v) ξ)","l":45,"f":"Causalean/SCM/Model/CounterfactualLemmas.lean"},{"n":"Causalean.SCM.fixMono_unobserved","k":"theorem","m":"Causalean.SCM.Model.InterventionMono","a":"SCM/Model","t":2,"d":"The monolithic intervention preserves the unobserved node set.","s":"∀ {N : Type u_1} [inst : DecidableEq N] [inst_1 : Fintype N] (G : Causalean.SWIGGraph N) (X : Finset N) (hObs : ∀ D ∈ X, Causalean.SWIGNode.random D ∈ G.observed) (hFix : ∀ D ∈ X, Causalean.SWIGNode.fixed D ∉ G.fixed), (G.splitMono X hObs hFix).unobserved = G.unobserved","l":282,"f":"Causalean/SCM/Model/InterventionMono.lean"},{"n":"Causalean.SCM.fixSet","k":"def","m":"Causalean.SCM.Model.InterventionSet","a":"SCM/Model","t":1,"d":"Standard (Pearl) multi-target do — definitional alias for fixMono.","s":"{N : Type u_1} → [inst : DecidableEq N] → [inst_1 : Fintype N] → {Ω : N → Type u_2} → [inst_2 : (n : N) → MeasurableSpace (Ω n)] → (M : Causalean.SCM N Ω) → (X : Finset N) → (∀ D ∈ X, Causalean.SWIGNode.random D ∈ M.observed) → (∀ D ∈ X, Causalean.SWIGNode.fixed D ∉ M.fixed) → Causalean.SCM N Ω","l":41,"f":"Causalean/SCM/Model/InterventionSet.lean"},{"n":"Causalean.SCM.fixSetExtend","k":"def","m":"Causalean.SCM.Do.Rule2Kernel.Helpers","a":"SCM/Do/Rule2Kernel","t":1,"d":"This map extends a base fixed-value assignment with treatment values for a post-intervention model.","s":"{N : Type u_1} → [inst : DecidableEq N] → [inst_1 : Fintype N] → {Ω : N → Type u_2} → [inst_2 : (n : N) → MeasurableSpace (Ω n)] → (M : Causalean.SCM N Ω) → (X : Finset N) → (hObs : ∀ D ∈ X, Causalean.SWIGNode.random D ∈ M.observed) → (hFix : ∀ D ∈ X, Causalean.SWIGNode.fixed D ∉ M.fixed) → M.FixedValues → Causalean.ValuesOn (Finset.image Causalean.SWIGNode.random X) (Causalean.swigΩ Ω) → (M.fixSet X hObs hFix).FixedValues","l":218,"f":"Causalean/SCM/Do/Rule2Kernel/Helpers.lean"},{"n":"Causalean.SCM.fixSetExtend_apply_new_fixed","k":"theorem","m":"Causalean.SCM.Do.Rule2Kernel.Helpers","a":"SCM/Do/Rule2Kernel","t":2,"d":"A newly fixed coordinate in fixSetExtend reads the matching observed intervention coordinate.","s":"∀ {N : Type u_1} [inst : DecidableEq N] [inst_1 : Fintype N] {Ω : N → Type u_2} [inst_2 : (n : N) → MeasurableSpace (Ω n)] (M : Causalean.SCM N Ω) (X : Finset N) (hObs : ∀ D ∈ X, Causalean.SWIGNode.random D ∈ M.observed) (hFix : ∀ D ∈ X, Causalean.SWIGNode.fixed D ∉ M.fixed) (s0 : M.FixedValues) (t : Causalean.ValuesOn (Finset.image Causalean.SWIGNode.random X) (Causalean.swigΩ Ω)) {D : N} (hD : D ∈ X), M.fixSetExtend X hObs hFix s0 t ⟨Causalean.SWIGNode.fixed D, Causalean.SCM.fixed_mem_fixSet M X hObs hFix hD⟩ = t ⟨Causalean.SWIGNode.random D, Finset.mem_image.mpr (Exists.intro D ⟨hD, rfl⟩)⟩","l":233,"f":"Causalean/SCM/Do/Rule2Kernel/Helpers.lean"},{"n":"Causalean.SCM.fixSetProj","k":"def","m":"Causalean.SCM.Model.InterventionSet","a":"SCM/Model","t":1,"d":"Canonical projection of post-intervention fixed assignments onto the original fixed coordinates.","s":"{N : Type u_1} → [inst : DecidableEq N] → [inst_1 : Fintype N] → {Ω : N → Type u_2} → [inst_2 : (n : N) → MeasurableSpace (Ω n)] → (M : Causalean.SCM N Ω) → (X : Finset N) → (hObs : ∀ D ∈ X, Causalean.SWIGNode.random D ∈ M.observed) → (hFix : ∀ D ∈ X, Causalean.SWIGNode.fixed D ∉ M.fixed) → (M.fixSet X hObs hFix).FixedValues → M.FixedValues","l":725,"f":"Causalean/SCM/Model/InterventionSet.lean"},{"n":"Causalean.SCM.fixSetProj_fixSetExtend","k":"theorem","m":"Causalean.SCM.Do.Rule2Kernel.Helpers","a":"SCM/Do/Rule2Kernel","t":1,"d":"For the intervention names X, whose random copies are observed in the base model and whose fixed copies are not yet part of the base model's fixed coordinates, projecting the extended post-intervention fixed assignment — built from the base assignment s0 and treatment value t — back onto the original fixed coordinates recovers s0.","s":"∀ {N : Type u_1} [inst : DecidableEq N] [inst_1 : Fintype N] {Ω : N → Type u_2} [inst_2 : (n : N) → MeasurableSpace (Ω n)] (M : Causalean.SCM N Ω) (X : Finset N) (hObs : ∀ D ∈ X, Causalean.SWIGNode.random D ∈ M.observed) (hFix : ∀ D ∈ X, Causalean.SWIGNode.fixed D ∉ M.fixed) (s0 : M.FixedValues) (t : Causalean.ValuesOn (Finset.image Causalean.SWIGNode.random X) (Causalean.swigΩ Ω)), M.fixSetProj X hObs hFix (M.fixSetExtend X hObs hFix s0 t) = s0","l":265,"f":"Causalean/SCM/Do/Rule2Kernel/Helpers.lean"},{"n":"Causalean.SCM.fixSetZSlice","k":"def","m":"Causalean.SCM.Model.InterventionSet","a":"SCM/Model","t":1,"d":"Z-fixed-slice extractor. Reads a FixedValues assignment of the double-intervention ((M.fixSet X).fixSet Z) on the Z.image .fixed slice, producing a ValuesOn (Z.image SWIGNode.fixed) (swigΩ Ω). General-purpose extractor for the inner do(z) coordinate slice of a two-layer intervention.","s":"{N : Type u_1} → [inst : DecidableEq N] → [inst_1 : Fintype N] → {Ω : N → Type u_2} → [inst_2 : (n : N) → MeasurableSpace (Ω n)] → (M : Causalean.SCM N Ω) → (X Z : Finset N) → (hX_obs : ∀ D ∈ X, Causalean.SWIGNode.random D ∈ M.observed) → (hX_fixed : ∀ D ∈ X, Causalean.SWIGNode.fixed D ∉ M.fixed) → (hZ_obs : ∀ D ∈ Z, Causalean.SWIGNode.random D ∈ (M.fixSet X hX_obs hX_fixed).observed) → (hZ_fixed : ∀ D ∈ Z, Causalean.SWIGNode.fixed D ∉ (M.fixSet X hX_obs hX_fixed).fixed) → ((M.fixSet X hX_obs hX_fixed).fixSet Z hZ_obs hZ_fixed).FixedValues → Causalean.ValuesOn (Finset.image Causalean.SWIGNode.fixed Z) (Causalean.swigΩ Ω)","l":751,"f":"Causalean/SCM/Model/InterventionSet.lean"},{"n":"Causalean.SCM.fixSet_edgeType_fixed_mem_eq","k":"theorem","m":"Causalean.SCM.Model.InterventionSet","a":"SCM/Model","t":2,"d":"For a variable included in the intervention set, the edge label from its fixed-coordinate node equals the original label from its random-coordinate node.","s":"∀ {N : Type u_1} [inst : DecidableEq N] [inst_1 : Fintype N] {Ω : N → Type u_2} [inst_2 : (n : N) → MeasurableSpace (Ω n)] (M : Causalean.SCM N Ω) (X : Finset N) (hObs : ∀ D ∈ X, Causalean.SWIGNode.random D ∈ M.observed) (hFix : ∀ D ∈ X, Causalean.SWIGNode.fixed D ∉ M.fixed) (d : N) (v : Causalean.SWIGNode N), d ∈ X → (M.fixSet X hObs hFix).edgeTypes.edgeType (Causalean.SWIGNode.fixed d) v = M.edgeTypes.edgeType (Causalean.SWIGNode.random d) v","l":465,"f":"Causalean/SCM/Model/InterventionSet.lean"},{"n":"Causalean.SCM.fixSet_edgeType_fixed_notMem_eq","k":"theorem","m":"Causalean.SCM.Model.InterventionSet","a":"SCM/Model","t":2,"d":"For a variable outside the intervention set, the edge label from its fixed-coordinate node is unchanged by the intervention.","s":"∀ {N : Type u_1} [inst : DecidableEq N] [inst_1 : Fintype N] {Ω : N → Type u_2} [inst_2 : (n : N) → MeasurableSpace (Ω n)] (M : Causalean.SCM N Ω) (X : Finset N) (hObs : ∀ D ∈ X, Causalean.SWIGNode.random D ∈ M.observed) (hFix : ∀ D ∈ X, Causalean.SWIGNode.fixed D ∉ M.fixed) (d : N) (v : Causalean.SWIGNode N), d ∉ X → (M.fixSet X hObs hFix).edgeTypes.edgeType (Causalean.SWIGNode.fixed d) v = M.edgeTypes.edgeType (Causalean.SWIGNode.fixed d) v","l":484,"f":"Causalean/SCM/Model/InterventionSet.lean"},{"n":"Causalean.SCM.fixSet_edgeType_random_eq","k":"theorem","m":"Causalean.SCM.Model.InterventionSet","a":"SCM/Model","t":2,"d":"After intervening on a set of variables, the label of every edge leaving a random-coordinate node is unchanged from the original causal model.","s":"∀ {N : Type u_1} [inst : DecidableEq N] [inst_1 : Fintype N] {Ω : N → Type u_2} [inst_2 : (n : N) → MeasurableSpace (Ω n)] (M : Causalean.SCM N Ω) (X : Finset N) (hObs : ∀ D ∈ X, Causalean.SWIGNode.random D ∈ M.observed) (hFix : ∀ D ∈ X, Causalean.SWIGNode.fixed D ∉ M.fixed) (u : N) (v : Causalean.SWIGNode N), (M.fixSet X hObs hFix).edgeTypes.edgeType (Causalean.SWIGNode.random u) v = M.edgeTypes.edgeType (Causalean.SWIGNode.random u) v","l":448,"f":"Causalean/SCM/Model/InterventionSet.lean"},{"n":"Causalean.SCM.fixSet_edge_iff","k":"theorem","m":"Causalean.SCM.Model.InterventionAncestry","a":"SCM/Model","t":2,"d":"An edge in a causal model after intervention is exactly the corresponding edge produced by splitting the intervened variables in the original graph.","s":"∀ {N : Type u_1} [inst : DecidableEq N] [inst_1 : Fintype N] {Ω : N → Type u_2} [inst_2 : (n : N) → MeasurableSpace (Ω n)] (M : Causalean.SCM N Ω) (X : Finset N) (hX_obs : ∀ D ∈ X, Causalean.SWIGNode.random D ∈ M.observed) (hX_fixed : ∀ D ∈ X, Causalean.SWIGNode.fixed D ∉ M.fixed) (u v : Causalean.SWIGNode N), (M.fixSet X hX_obs hX_fixed).dag.edge u v ↔ Causalean.SWIGGraph.splitMonoEdgeRel M.dag.edge X u v","l":30,"f":"Causalean/SCM/Model/InterventionAncestry.lean"},{"n":"Causalean.SCM.fixSet_empty_edge","k":"theorem","m":"Causalean.SCM.Model.InterventionSet","a":"SCM/Model","t":2,"d":"Edges of fixSet ∅ coincide with the base.","s":"∀ {N : Type u_1} [inst : DecidableEq N] [inst_1 : Fintype N] (G : Causalean.SWIGGraph N) (u v : Causalean.SWIGNode N), (G.splitMono ∅ (fun x hD => absurd hD (Finset.notMem_empty x)) fun x hD => absurd hD (Finset.notMem_empty x)).dag.edge u v ↔ G.dag.edge u v","l":157,"f":"Causalean/SCM/Model/InterventionSet.lean"},{"n":"Causalean.SCM.fixSet_empty_equiv","k":"theorem","m":"Causalean.SCM.Model.InterventionSet","a":"SCM/Model","t":1,"d":"For a structural causal model M, intervening on the empty set of nodes yields a structural causal model that is equivalent to M.","s":"∀ {N : Type u_1} [inst : DecidableEq N] [inst_1 : Fintype N] {Ω : N → Type u_2} [inst_2 : (n : N) → MeasurableSpace (Ω n)] (M : Causalean.SCM N Ω), (M.fixSet ∅ (fun x hD => absurd hD (Finset.notMem_empty x)) fun x hD => absurd hD (Finset.notMem_empty x)).Equiv M","l":174,"f":"Causalean/SCM/Model/InterventionSet.lean"},{"n":"Causalean.SCM.fixSet_empty_parents","k":"theorem","m":"Causalean.SCM.Model.InterventionSet","a":"SCM/Model","t":2,"d":"Parent set of fixSet ∅ coincides with the base.","s":"∀ {N : Type u_1} [inst : DecidableEq N] [inst_1 : Fintype N] {Ω : N → Type u_2} [inst_2 : (n : N) → MeasurableSpace (Ω n)] (M : Causalean.SCM N Ω) (v : Causalean.SWIGNode N), (M.fixSet ∅ (fun x hD => absurd hD (Finset.notMem_empty x)) fun x hD => absurd hD (Finset.notMem_empty x)).dag.parents v = M.dag.parents v","l":143,"f":"Causalean/SCM/Model/InterventionSet.lean"},{"n":"Causalean.SCM.fixSet_equiv_congr","k":"theorem","m":"Causalean.SCM.Model.InterventionSet","a":"SCM/Model","t":1,"d":"fixSet preserves SCM.Equiv. Given a structural equivalence h between M₁ and M₂ and a finite set of nodes X whose members are, in M₁, observed in their random coordinate and not already held fixed, intervening on X in both models produces two structurally equivalent post-intervention models — the analogous observed/not-fixed conditions transfer automatically to M₂ via h.","s":"∀ {N : Type u_1} [inst : DecidableEq N] [inst_1 : Fintype N] {Ω : N → Type u_2} [inst_2 : (n : N) → MeasurableSpace (Ω n)] {M₁ M₂ : Causalean.SCM N Ω} (h : M₁.Equiv M₂) (X : Finset N) (hObs₁ : ∀ D ∈ X, Causalean.SWIGNode.random D ∈ M₁.observed) (hFix₁ : ∀ D ∈ X, Causalean.SWIGNode.fixed D ∉ M₁.fixed), have hObs₂ := fun D hD => Eq.mpr (id (congrArg (fun _a => Causalean.SWIGNode.random D ∈ _a) (Eq.symm h.left.right.right.left))) (hObs₁ D hD); have hFix₂ := fun D hD hDfix => hFix₁ D hD (Eq.mpr (id (congrArg (fun _a => Causalean.SWIGNode.fixed D ∈ _a) h.left.right.left)) hDfix); (M₁.fixSet X hObs₁ hFix₁).Equiv (M₂.fixSet X hObs₂ hFix₂)","l":268,"f":"Causalean/SCM/Model/InterventionSet.lean"},{"n":"Causalean.SCM.fixSet_evalMap_levelset_compat","k":"theorem","m":"Causalean.SCM.Do.Rule2Kernel.LevelsetCompat","a":"SCM/Do/Rule2Kernel","t":1,"d":"Cross-SCM evalMap level-set bridge for Rule 2. For the intervention on names Z, whose random copies are observed in the base model and whose fixed copies are not yet part of the base model's fixed coordinates, fix an intervened fixed assignment s' and a latent draw ℓ. Suppose on the \"level set\" picked out by ℓ, the base model's evaluation map at the projected fixed assignment already reproduces, at every random copy of a name in Z, the intervention value recorded in s'; then at any observed node v in the intervened model, the intervened model's evaluation map at (s', ℓ) agrees with the base model's evaluation map at the projected fixed assignment and the correspondingly reindexed latent draw.","s":"∀ {N : Type u_1} [inst : DecidableEq N] [inst_1 : Fintype N] {Ω : N → Type u_2} [inst_2 : (n : N) → MeasurableSpace (Ω n)] (M' : Causalean.SCM N Ω) (Z : Finset N) (hZ_obs : ∀ D ∈ Z, Causalean.SWIGNode.random D ∈ M'.observed) (hZ_fixed : ∀ D ∈ Z, Causalean.SWIGNode.fixed D ∉ M'.fixed) (s' : (M'.fixSet Z hZ_obs hZ_fixed).FixedValues) (ℓ : (M'.fixSet Z hZ_obs hZ_fixed).LatentValues), (∀ (D : N) (hD : D ∈ Z), M'.evalMap (M'.fixSetProj Z hZ_obs hZ_fixed s') (Causalean.valuesProjection (le_of_eq (Eq.symm (Causalean.SCM.fixSet_unobserved M' Z hZ_obs hZ_fixed))) ℓ) ⟨Causalean.SWIGNode.random D, Finset.mem_union_left M'.unobserved (hZ_obs D hD)⟩ = s' ⟨Causalean.SWIGNode.fixed D, Causalean.SCM.fixed_mem_fixSet M' Z hZ_obs hZ_fixed hD⟩) → ∀ {v : Causalean.SWIGNode N} (hv : v ∈ (M'.fixSet Z hZ_obs hZ_fixed).observed), (M'.fixSet Z hZ_obs hZ_fixed).evalMap s' ℓ ⟨v, Finset.mem_union_left (M'.fixSet Z hZ_obs hZ_fixed).unobserved hv⟩ = M'.evalMap (M'.fixSetProj Z hZ_obs hZ_fixed s') (Causalean.valuesProjection (le_of_eq (Eq.symm (Causalean.SCM.fixSet_unobserved M' Z hZ_obs hZ_fixed))) ℓ) ⟨v, Finset.mem_union_left M'.unobserved (le_of_eq (Causalean.SCM.fixSet_observed M' Z hZ_obs hZ_fixed) hv)⟩","l":33,"f":"Causalean/SCM/Do/Rule2Kernel/LevelsetCompat.lean"},{"n":"Causalean.SCM.fixSet_evalMap_levelset_compat_M2","k":"theorem","m":"Causalean.SCM.Do.Rule2Kernel.LevelsetCompat","a":"SCM/Do/Rule2Kernel","t":1,"d":"M2-direction level-set evalMap bridge for Rule 2. For the intervention on names Z, whose random copies are observed in the base model and whose fixed copies are not yet part of the base model's fixed coordinates, fix an intervened fixed assignment s' and a latent draw ℓ. Suppose the intervened model's own evaluation map at (s', ℓ) already reproduces, at every random copy of a name in Z, the intervention value recorded in s'; then at any observed node v in the intervened model, the intervened model's evaluation map at (s', ℓ) agrees with the base model's evaluation map at the projected fixed assignment and the correspondingly reindexed latent draw.","s":"∀ {N : Type u_1} [inst : DecidableEq N] [inst_1 : Fintype N] {Ω : N → Type u_2} [inst_2 : (n : N) → MeasurableSpace (Ω n)] (M' : Causalean.SCM N Ω) (Z : Finset N) (hZ_obs : ∀ D ∈ Z, Causalean.SWIGNode.random D ∈ M'.observed) (hZ_fixed : ∀ D ∈ Z, Causalean.SWIGNode.fixed D ∉ M'.fixed) (s' : (M'.fixSet Z hZ_obs hZ_fixed).FixedValues) (ℓ : (M'.fixSet Z hZ_obs hZ_fixed).LatentValues), (∀ (D : N) (hD : D ∈ Z), (M'.fixSet Z hZ_obs hZ_fixed).evalMap s' ℓ ⟨Causalean.SWIGNode.random D, Finset.mem_union_left (M'.fixSet Z hZ_obs hZ_fixed).unobserved (hZ_obs D hD)⟩ = s' ⟨Causalean.SWIGNode.fixed D, Causalean.SCM.fixed_mem_fixSet M' Z hZ_obs hZ_fixed hD⟩) → ∀ {v : Causalean.SWIGNode N} (hv : v ∈ (M'.fixSet Z hZ_obs hZ_fixed).observed), (M'.fixSet Z hZ_obs hZ_fixed).evalMap s' ℓ ⟨v, Finset.mem_union_left (M'.fixSet Z hZ_obs hZ_fixed).unobserved hv⟩ = M'.evalMap (M'.fixSetProj Z hZ_obs hZ_fixed s') (Causalean.valuesProjection (le_of_eq (Eq.symm (Causalean.SCM.fixSet_unobserved M' Z hZ_obs hZ_fixed))) ℓ) ⟨v, Finset.mem_union_left M'.unobserved (le_of_eq (Causalean.SCM.fixSet_observed M' Z hZ_obs hZ_fixed) hv)⟩","l":273,"f":"Causalean/SCM/Do/Rule2Kernel/LevelsetCompat.lean"},{"n":"Causalean.SCM.fixSet_evalMap_nonAnc_compat","k":"theorem","m":"Causalean.SCM.Do.Rule3","a":"SCM/Do","t":2,"d":"Cross-SCM evalMap bridge for Rule 3.","s":"∀ {N : Type u_1} [inst : DecidableEq N] [inst_1 : Fintype N] {Ω : N → Type u_2} [inst_2 : (n : N) → MeasurableSpace (Ω n)] (M' : Causalean.SCM N Ω) (Z : Finset N) (hZ_obs : ∀ D ∈ Z, Causalean.SWIGNode.random D ∈ M'.observed) (hZ_fixed : ∀ D ∈ Z, Causalean.SWIGNode.fixed D ∉ M'.fixed) (s' : (M'.fixSet Z hZ_obs hZ_fixed).FixedValues) (ℓ : (M'.fixSet Z hZ_obs hZ_fixed).LatentValues) {v : Causalean.SWIGNode N} (hv : v ∈ (M'.fixSet Z hZ_obs hZ_fixed).observed), (∀ z ∈ Z, ¬(M'.fixSet Z hZ_obs hZ_fixed).dag.isAncestor (Causalean.SWIGNode.fixed z) v) → (M'.fixSet Z hZ_obs hZ_fixed).evalMap s' ℓ ⟨v, Finset.mem_union_left (M'.fixSet Z hZ_obs hZ_fixed).unobserved hv⟩ = M'.evalMap (M'.fixSetProj Z hZ_obs hZ_fixed s') (Causalean.valuesProjection (le_of_eq (Eq.symm (Causalean.SCM.fixSet_unobserved M' Z hZ_obs hZ_fixed))) ℓ) ⟨v, Finset.mem_union_left M'.unobserved (le_of_eq (Causalean.SCM.fixSet_observed M' Z hZ_obs hZ_fixed) hv)⟩","l":91,"f":"Causalean/SCM/Do/Rule3.lean"},{"n":"Causalean.SCM.fixSet_fixed","k":"theorem","m":"Causalean.SCM.Model.InterventionSet","a":"SCM/Model","t":2,"d":"fixSet enlarges fixed by exactly X.image SWIGNode.fixed.","s":"∀ {N : Type u_1} [inst : DecidableEq N] [inst_1 : Fintype N] {Ω : N → Type u_2} [inst_2 : (n : N) → MeasurableSpace (Ω n)] (M : Causalean.SCM N Ω) (X : Finset N) (hObs : ∀ D ∈ X, Causalean.SWIGNode.random D ∈ M.observed) (hFix : ∀ D ∈ X, Causalean.SWIGNode.fixed D ∉ M.fixed), (M.fixSet X hObs hFix).fixed = M.fixed ∪ Finset.image Causalean.SWIGNode.fixed X","l":64,"f":"Causalean/SCM/Model/InterventionSet.lean"},{"n":"Causalean.SCM.fixSet_fixed_not_mem_of_disjoint","k":"theorem","m":"Causalean.SCM.ID.Toolkit.FrontdoorGraph","a":"SCM/ID/Toolkit","t":2,"d":"Fixing one intervention block does not add fixed copies from a disjoint block.","s":"∀ {N : Type u_1} [inst : DecidableEq N] [inst_1 : Fintype N] {Ω : N → Type u_2} [inst_2 : (n : N) → MeasurableSpace (Ω n)] (M : Causalean.SCM N Ω) (X W : Finset N) (hW_obs : ∀ D ∈ W, Causalean.SWIGNode.random D ∈ M.observed) (hW_fixed : ∀ D ∈ W, Causalean.SWIGNode.fixed D ∉ M.fixed), (∀ D ∈ X, Causalean.SWIGNode.fixed D ∉ M.fixed) → Disjoint W X → ∀ D ∈ X, Causalean.SWIGNode.fixed D ∉ (M.fixSet W hW_obs hW_fixed).fixed","l":47,"f":"Causalean/SCM/ID/Toolkit/FrontdoorGraph.lean"},{"n":"Causalean.SCM.fixSet_fixed_subset","k":"theorem","m":"Causalean.SCM.Model.InterventionSet","a":"SCM/Model","t":2,"d":"fixSet only enlarges the fixed set.","s":"∀ {N : Type u_1} [inst : DecidableEq N] [inst_1 : Fintype N] {Ω : N → Type u_2} [inst_2 : (n : N) → MeasurableSpace (Ω n)] (M : Causalean.SCM N Ω) (X : Finset N) (hObs : ∀ D ∈ X, Causalean.SWIGNode.random D ∈ M.observed) (hFix : ∀ D ∈ X, Causalean.SWIGNode.fixed D ∉ M.fixed), M.fixed ⊆ (M.fixSet X hObs hFix).fixed","l":78,"f":"Causalean/SCM/Model/InterventionSet.lean"},{"n":"Causalean.SCM.fixSet_image_fixed_subset","k":"theorem","m":"Causalean.SCM.Model.InterventionSet","a":"SCM/Model","t":2,"d":"X.image SWIGNode.fixed ⊆ (M.fixSet X _ _).fixed.","s":"∀ {N : Type u_1} [inst : DecidableEq N] [inst_1 : Fintype N] {Ω : N → Type u_2} [inst_2 : (n : N) → MeasurableSpace (Ω n)] (M : Causalean.SCM N Ω) (X : Finset N) (hObs : ∀ D ∈ X, Causalean.SWIGNode.random D ∈ M.observed) (hFix : ∀ D ∈ X, Causalean.SWIGNode.fixed D ∉ M.fixed), Finset.image Causalean.SWIGNode.fixed X ⊆ (M.fixSet X hObs hFix).fixed","l":85,"f":"Causalean/SCM/Model/InterventionSet.lean"},{"n":"Causalean.SCM.fixSet_isAncestor_fixed_forward","k":"theorem","m":"Causalean.SCM.Model.InterventionAncestry","a":"SCM/Model","t":1,"d":"Forward direction: .fixed D-ancestry in fixSet X lifts to .random D-ancestry in the base graph. Fix a structural causal model M and an intervention target set X such that every targeted node is currently a random observed node and none of its fixed copies is already fixed. For a targeted node D and a node v, if the fixed copy of D is a proper ancestor of v in the post-intervention graph obtained by fixing X, then the random copy of D is a proper ancestor of v in the original base graph.","s":"∀ {N : Type u_1} [inst : DecidableEq N] [inst_1 : Fintype N] {Ω : N → Type u_2} [inst_2 : (n : N) → MeasurableSpace (Ω n)] (M : Causalean.SCM N Ω) (X : Finset N) (hX_obs : ∀ D ∈ X, Causalean.SWIGNode.random D ∈ M.observed) (hX_fixed : ∀ D ∈ X, Causalean.SWIGNode.fixed D ∉ M.fixed) {D : N}, D ∈ X → ∀ {v : Causalean.SWIGNode N}, (M.fixSet X hX_obs hX_fixed).dag.isAncestor (Causalean.SWIGNode.fixed D) v → M.dag.isAncestor (Causalean.SWIGNode.random D) v","l":42,"f":"Causalean/SCM/Model/InterventionAncestry.lean"},{"n":"Causalean.SCM.fixSet_latentDist","k":"theorem","m":"Causalean.SCM.Model.InterventionSet","a":"SCM/Model","t":2,"d":"fixSet inherits latentDist verbatim. Both sides have the same type ((M.fixSet X _ _).unobserved = M.unobserved by rfl) and equal body.","s":"∀ {N : Type u_1} [inst : DecidableEq N] [inst_1 : Fintype N] {Ω : N → Type u_2} [inst_2 : (n : N) → MeasurableSpace (Ω n)] (M : Causalean.SCM N Ω) (X : Finset N) (hObs : ∀ D ∈ X, Causalean.SWIGNode.random D ∈ M.observed) (hFix : ∀ D ∈ X, Causalean.SWIGNode.fixed D ∉ M.fixed) (u : ↥(M.fixSet X hObs hFix).unobserved), (M.fixSet X hObs hFix).latentDist u = M.latentDist u","l":70,"f":"Causalean/SCM/Model/InterventionSet.lean"},{"n":"Causalean.SCM.fixSet_latentProduct_compat","k":"theorem","m":"Causalean.SCM.Do.Rule3","a":"SCM/Do","t":2,"d":"Cross-SCM latent-product transport for Rule 3.","s":"∀ {N : Type u_1} [inst : DecidableEq N] [inst_1 : Fintype N] {Ω : N → Type u_2} [inst_2 : (n : N) → MeasurableSpace (Ω n)] (M' : Causalean.SCM N Ω) (Z : Finset N) (hZ_obs : ∀ D ∈ Z, Causalean.SWIGNode.random D ∈ M'.observed) (hZ_fixed : ∀ D ∈ Z, Causalean.SWIGNode.fixed D ∉ M'.fixed), MeasureTheory.Measure.map (Causalean.valuesProjection (le_of_eq (Eq.symm (Causalean.SCM.fixSet_unobserved M' Z hZ_obs hZ_fixed)))) (M'.fixSet Z hZ_obs hZ_fixed).latentProduct = M'.latentProduct","l":30,"f":"Causalean/SCM/Do/Rule3.lean"},{"n":"Causalean.SCM.fixSet_observed","k":"theorem","m":"Causalean.SCM.Model.InterventionSet","a":"SCM/Model","t":2,"d":"fixSet preserves observed.","s":"∀ {N : Type u_1} [inst : DecidableEq N] [inst_1 : Fintype N] {Ω : N → Type u_2} [inst_2 : (n : N) → MeasurableSpace (Ω n)] (M : Causalean.SCM N Ω) (X : Finset N) (hObs : ∀ D ∈ X, Causalean.SWIGNode.random D ∈ M.observed) (hFix : ∀ D ∈ X, Causalean.SWIGNode.fixed D ∉ M.fixed), (M.fixSet X hObs hFix).observed = M.observed","l":52,"f":"Causalean/SCM/Model/InterventionSet.lean"},{"n":"Causalean.SCM.fixSet_parents_eq_of_no_fixed_parent","k":"theorem","m":"Causalean.SCM.Model.InterventionSet","a":"SCM/Model","t":2,"d":"SCM-level parent-set coincidence at non-.fixed-targeted vertices.","s":"∀ {N : Type u_1} [inst : DecidableEq N] [inst_1 : Fintype N] {Ω : N → Type u_2} [inst_2 : (n : N) → MeasurableSpace (Ω n)] (M : Causalean.SCM N Ω) (X : Finset N) (hObs : ∀ D ∈ X, Causalean.SWIGNode.random D ∈ M.observed) (hFix : ∀ D ∈ X, Causalean.SWIGNode.fixed D ∉ M.fixed) {v : Causalean.SWIGNode N}, (∀ D ∈ X, Causalean.SWIGNode.fixed D ∉ (M.fixSet X hObs hFix).dag.parents v) → (M.fixSet X hObs hFix).dag.parents v = M.dag.parents v","l":101,"f":"Causalean/SCM/Model/InterventionSet.lean"},{"n":"Causalean.SCM.fixSet_random_no_children","k":"theorem","m":"Causalean.SCM.Do.Rule2Kernel.Structural.StructPointwise","a":"SCM/Do/Rule2Kernel/Structural","t":1,"d":"In the intervened model, consider a name D belonging to the intervention set Z, where random copies of names in Z are observed in the base model and their fixed copies are not yet part of the base model's fixed coordinates. Then the random copy of D has no outgoing edge, in the post-intervention DAG, to any node v.","s":"∀ {N : Type u_1} [inst : DecidableEq N] [inst_1 : Fintype N] {Ω : N → Type u_2} [inst_2 : (n : N) → MeasurableSpace (Ω n)] (M : Causalean.SCM N Ω) (Z : Finset N) (hZ_obs : ∀ D ∈ Z, Causalean.SWIGNode.random D ∈ M.observed) (hZ_fixed : ∀ D ∈ Z, Causalean.SWIGNode.fixed D ∉ M.fixed) {D : N}, D ∈ Z → ∀ (v : Causalean.SWIGNode N), ¬(M.fixSet Z hZ_obs hZ_fixed).dag.edge (Causalean.SWIGNode.random D) v","l":70,"f":"Causalean/SCM/Do/Rule2Kernel/Structural/StructPointwise.lean"},{"n":"Causalean.SCM.fixSet_random_not_isAncestor","k":"theorem","m":"Causalean.SCM.Do.Rule2Kernel.Structural.StructPointwise","a":"SCM/Do/Rule2Kernel/Structural","t":2,"d":"In M.fixSet Z _ _, every .random D (D ∈ Z) is not a proper ancestor of any node. Direct consequence of fixSet_random_no_children via isAncestor_child.","s":"∀ {N : Type u_1} [inst : DecidableEq N] [inst_1 : Fintype N] {Ω : N → Type u_2} [inst_2 : (n : N) → MeasurableSpace (Ω n)] (M : Causalean.SCM N Ω) (Z : Finset N) (hZ_obs : ∀ D ∈ Z, Causalean.SWIGNode.random D ∈ M.observed) (hZ_fixed : ∀ D ∈ Z, Causalean.SWIGNode.fixed D ∉ M.fixed) {D : N}, D ∈ Z → ∀ (v : Causalean.SWIGNode N), ¬(M.fixSet Z hZ_obs hZ_fixed).dag.isAncestor (Causalean.SWIGNode.random D) v","l":91,"f":"Causalean/SCM/Do/Rule2Kernel/Structural/StructPointwise.lean"},{"n":"Causalean.SCM.fixSet_structFun_apply","k":"theorem","m":"Causalean.SCM.Model.CounterfactualLemmas","a":"SCM/Model","t":2,"d":"After an intervention on a set of observed variables, each observed variable is still computed by the original structural equation, with intervened parents replaced by their assigned intervention values.","s":"∀ {N : Type uN} [inst : DecidableEq N] [inst_1 : Fintype N] {Ω : N → Type uΩ} [inst_2 : (n : N) → MeasurableSpace (Ω n)] (M : Causalean.SCM N Ω) (X : Finset N) (hObs : ∀ D ∈ X, Causalean.SWIGNode.random D ∈ M.observed) (hFix : ∀ D ∈ X, Causalean.SWIGNode.fixed D ∉ M.fixed) (v : ↥(M.fixSet X hObs hFix).observed) (ξ : (w : ↥((M.fixSet X hObs hFix).dag.parents ↑v)) → Causalean.swigΩ Ω ↑w), (M.fixSet X hObs hFix).structFun v ξ = M.structFun ⟨↑v, v.property⟩ (Causalean.SCM.fixMonoParentMap M.toSWIGGraph X hObs hFix (↑v) ξ)","l":59,"f":"Causalean/SCM/Model/CounterfactualLemmas.lean"},{"n":"Causalean.SCM.fixSet_unobserved","k":"theorem","m":"Causalean.SCM.Model.InterventionSet","a":"SCM/Model","t":2,"d":"fixSet preserves unobserved.","s":"∀ {N : Type u_1} [inst : DecidableEq N] [inst_1 : Fintype N] {Ω : N → Type u_2} [inst_2 : (n : N) → MeasurableSpace (Ω n)] (M : Causalean.SCM N Ω) (X : Finset N) (hObs : ∀ D ∈ X, Causalean.SWIGNode.random D ∈ M.observed) (hFix : ∀ D ∈ X, Causalean.SWIGNode.fixed D ∉ M.fixed), (M.fixSet X hObs hFix).unobserved = M.unobserved","l":58,"f":"Causalean/SCM/Model/InterventionSet.lean"},{"n":"Causalean.SCM.fixedAncestorsOfNode","k":"def","m":"Causalean.SCM.Model.EvalFactorization","a":"SCM/Model","t":1,"d":"The fixed ancestors of a node are the fixed nodes equal to that node or ancestral to it.","s":"{N : Type u_1} → [inst : DecidableEq N] → [inst_1 : Fintype N] → {Ω : N → Type u_2} → [inst_2 : (n : N) → MeasurableSpace (Ω n)] → Causalean.SCM N Ω → Causalean.SWIGNode N → Finset (Causalean.SWIGNode N)","l":167,"f":"Causalean/SCM/Model/EvalFactorization.lean"},{"n":"Causalean.SCM.fixedAncestorsOfNode_subset","k":"theorem","m":"Causalean.SCM.Model.EvalFactorization","a":"SCM/Model","t":2,"d":"The fixed-ancestor set is contained in the fixed node set.","s":"∀ {N : Type u_1} [inst : DecidableEq N] [inst_1 : Fintype N] {Ω : N → Type u_2} [inst_2 : (n : N) → MeasurableSpace (Ω n)] (M : Causalean.SCM N Ω) (v : Causalean.SWIGNode N), M.fixedAncestorsOfNode v ⊆ M.fixed","l":194,"f":"Causalean/SCM/Model/EvalFactorization.lean"},{"n":"Causalean.SCM.fixed_mem_fixSet","k":"theorem","m":"Causalean.SCM.Model.InterventionSet","a":"SCM/Model","t":2,"d":"Every SWIGNode.fixed D with D ∈ X is in the fixed set of fixSet X.","s":"∀ {N : Type u_1} [inst : DecidableEq N] [inst_1 : Fintype N] {Ω : N → Type u_2} [inst_2 : (n : N) → MeasurableSpace (Ω n)] (M : Causalean.SCM N Ω) (X : Finset N) (hObs : ∀ D ∈ X, Causalean.SWIGNode.random D ∈ M.observed) (hFix : ∀ D ∈ X, Causalean.SWIGNode.fixed D ∉ M.fixed) {D : N}, D ∈ X → Causalean.SWIGNode.fixed D ∈ (M.fixSet X hObs hFix).fixed","l":92,"f":"Causalean/SCM/Model/InterventionSet.lean"},{"n":"Causalean.SCM.frontdoorAdjustment","k":"def","m":"Causalean.SCM.ID.Adjustment","a":"SCM/ID","t":1,"d":"Frontdoor adjustment functional.","s":"{N : Type u_1} → [inst : DecidableEq N] → [inst_1 : Fintype N] → {Ω : N → Type u_2} → [inst_2 : (n : N) → MeasurableSpace (Ω n)] → [∀ (n : Causalean.SWIGNode N), StandardBorelSpace (Causalean.swigΩ Ω n)] → [∀ (n : Causalean.SWIGNode N), Nonempty (Causalean.swigΩ Ω n)] → (M : Causalean.SCM N Ω) → (X : Finset N) → (hX_obs : ∀ D ∈ X, Causalean.SWIGNode.random D ∈ M.observed) → (hX_fixed : ∀ D ∈ X, Causalean.SWIGNode.fixed D ∉ M.fixed) → (Y Z : Finset (Causalean.SWIGNode N)) → Y ⊆ M.observed → Z ⊆ M.observed → ProbabilityTheory.Kernel (M.fixSet X hX_obs hX_fixed).FixedValues (Causalean.ValuesOn Y (Causalean.swigΩ Ω))","l":139,"f":"Causalean/SCM/ID/Adjustment.lean"},{"n":"Causalean.SCM.frontdoorAdjustment_invariant","k":"theorem","m":"Causalean.SCM.ID.Adjustment","a":"SCM/ID","t":1,"d":"Cross-SCM invariance of frontdoorAdjustment. Fix two causal models M₁ and M₂ that share the same SWIG graph and the same observational kernel, and a treatment set X, an outcome set Y, and a mediator set Z that are valid intervention data and observed variables in both models. Then the frontdoor-adjustment kernel for Y through mediator Z after intervening on X, computed within M₁, is the same as the one computed within M₂.","s":"∀ {N : Type u_1} [inst : DecidableEq N] [inst_1 : Fintype N] {Ω : N → Type u_2} [inst_2 : (n : N) → MeasurableSpace (Ω n)] [inst_3 : ∀ (n : Causalean.SWIGNode N), StandardBorelSpace (Causalean.swigΩ Ω n)] [inst_4 : ∀ (n : Causalean.SWIGNode N), Nonempty (Causalean.swigΩ Ω n)] (M₁ M₂ : Causalean.SCM N Ω), M₁.toSWIGGraph = M₂.toSWIGGraph → M₁.obsKernel ≍ M₂.obsKernel → ∀ (X : Finset N) (hX_obs₁ : ∀ D ∈ X, Causalean.SWIGNode.random D ∈ M₁.observed) (hX_fixed₁ : ∀ D ∈ X, Causalean.SWIGNode.fixed D ∉ M₁.fixed) (hX_obs₂ : ∀ D ∈ X, Causalean.SWIGNode.random D ∈ M₂.observed) (hX_fixed₂ : ∀ D ∈ X, Causalean.SWIGNode.fixed D ∉ M₂.fixed) (Y Z : Finset (Causalean.SWIGNode N)) (hY₁ : Y ⊆ M₁.observed) (hZ₁ : Z ⊆ M₁.observed) (hY₂ : Y ⊆ M₂.observed) (hZ₂ : Z ⊆ M₂.observed), M₁.frontdoorAdjustment X hX_obs₁ hX_fixed₁ Y Z hY₁ hZ₁ ≍ M₂.frontdoorAdjustment X hX_obs₂ hX_fixed₂ Y Z hY₂ hZ₂","l":327,"f":"Causalean/SCM/ID/Adjustment.lean"},{"n":"Causalean.SCM.frontdoorKernelY","k":"def","m":"Causalean.SCM.ID.Frontdoor","a":"SCM/ID","t":1,"d":"Frontdoor-adjustment Y-marginal as a kernel in the treatment value t, at base s₀ — the frontdoor analogue of adjustmentKernelY. Reindexes the graph-level frontdoorAdjustment (with mediator Z := W.image SWIGNode.random) along fixSetExtend s₀, so its input is the treatment value t rather than the full post-intervention slice.","s":"{N : Type u_1} → [inst : DecidableEq N] → [inst_1 : Fintype N] → {Ω : N → Type u_2} → [inst_2 : (n : N) → MeasurableSpace (Ω n)] → [∀ (n : Causalean.SWIGNode N), StandardBorelSpace (Causalean.swigΩ Ω n)] → [∀ (n : Causalean.SWIGNode N), Nonempty (Causalean.swigΩ Ω n)] → (M : Causalean.SCM N Ω) → (X : Finset N) → (∀ D ∈ X, Causalean.SWIGNode.random D ∈ M.observed) → (∀ D ∈ X, Causalean.SWIGNode.fixed D ∉ M.fixed) → (Y W : Finset (Causalean.SWIGNode N)) → Y ⊆ M.observed → W ⊆ M.observed → M.FixedValues → ProbabilityTheory.Kernel (Causalean.ValuesOn (Finset.image Causalean.SWIGNode.random X) (Causalean.swigΩ Ω)) (Causalean.ValuesOn Y (Causalean.swigΩ Ω))","l":132,"f":"Causalean/SCM/ID/Frontdoor.lean"},{"n":"Causalean.SCM.frontdoor_completeness_ae_compProd","k":"theorem","m":"Causalean.SCM.ID.Frontdoor","a":"SCM/ID","t":1,"d":"Frontdoor completeness — joint (compProd), version-safe primary form. Let X be a valid intervention set — observed and not already fixed, and Wbase likewise a valid intervention set of mediators, with the outcome set Y observed, the random copies of Wbase observed, and the random copies of X observed. Suppose the frontdoor criterion holds for (X, Wbase, Y), that Y is disjoint from the random copies of Wbase and the random copies of Wbase are disjoint from those of X, and that each of the three legs of the frontdoor decomposition — the do(X) leg, the do(Wbase) leg adjusting for X, and the nested do(X)-then-do(Wbase) leg — satisfies the matching backdoor overlap and positivity conditions. Then the joint law of the treatment marginal with the post-intervention Y-marginal equals the joint law of the treatment marginal with the frontdoor-adjustment functional.","s":"∀ {N : Type u_1} [inst : DecidableEq N] [inst_1 : Fintype N] {Ω : N → Type u_2} [inst_2 : (n : N) → MeasurableSpace (Ω n)] [inst_3 : ∀ (n : Causalean.SWIGNode N), StandardBorelSpace (Causalean.swigΩ Ω n)] [inst_4 : ∀ (n : Causalean.SWIGNode N), Nonempty (Causalean.swigΩ Ω n)] (M : Causalean.SCM N Ω) (X : Finset N) (hObs : ∀ D ∈ X, Causalean.SWIGNode.random D ∈ M.observed) (hFix : ∀ D ∈ X, Causalean.SWIGNode.fixed D ∉ M.fixed) (Wbase : Finset N) (hWobs : ∀ D ∈ Wbase, Causalean.SWIGNode.random D ∈ M.observed) (hWfix : ∀ D ∈ Wbase, Causalean.SWIGNode.fixed D ∉ M.fixed) (Y : Finset (Causalean.SWIGNode N)) (hY : Y ⊆ M.observed) (hWr : Finset.image Causalean.SWIGNode.random Wbase ⊆ M.observed) (hXr : Finset.image Causalean.SWIGNode.random X ⊆ M.observed), M.frontdoorCriterion X hObs hFix Wbase hWobs hWfix Y → Disjoint Y (Finset.image Causalean.SWIGNode.random Wbase) → ∀ (hDisj_WrXr : Disjoint (Finset.image Causalean.SWIGNode.random Wbase) (Finset.image Causalean.SWIGNode.random X)) (s0 : M.FixedValues), (∀ (s : (M.fixSet X hObs hFix).FixedValues), Causalean.SCM.ID.Rule2JointOverlap M X hObs hFix ∅ (Eq.mpr (id (congrFun' (congrArg LE.le (Finset.union_empty (Finset.image Causalean.SWIGNode.random X))) M.observed)) hXr) s) → M.BackdoorPositivityAE X ∅ (Finset.empty_subset M.observed) (Eq.mpr (id (congrFun' (congrArg LE.le (Finset.union_empty (Finset.image Causalean.SWIGNode.random X))) M.observed)) hXr) s0 → (∀ (s : (M.fixSet Wbase hWobs hWfix).FixedValues), Causalean.SCM.ID.Rule2JointOverlap M Wbase hWobs hWfix (Finset.image Causalean.SWIGNode.random X) (Finset.union_subset hWr hXr) s) → M.BackdoorPositivityAE Wbase (Finset.image Causalean.SWIGNode.random X) hXr (Finset.union_subset hWr hXr) s0 → (∀ (s : ((M.fixSet X hObs hFix).fixSet Wbase (fun D hD => id (hWobs D hD)) (Causalean.SCM.fixSet_fixed_not_mem_of_disjoint M Wbase X hObs hFix hWfix (Disjoint.symm (Causalean.SCM.disjoint_base_of_disjoint_random_image X Wbase hDisj_WrXr)))).FixedValues), Causalean.SCM.ID.Rule2JointOverlap (M.fixSet X hObs hFix) Wbase (fun D hD => id (hWobs D hD)) (Causalean.SCM.fixSet_fixed_not_mem_of_disjoint M Wbase X hObs hFix hWfix (Disjoint.symm (Causalean.SCM.disjoint_base_of_disjoint_random_image X Wbase hDisj_WrXr))) ∅ (Eq.mpr (id (congrFun' (congrArg LE.le (Finset.union_empty (Finset.image Causalean.SWIGNode.random Wbase))) M.observed)) hWr) s) → (∀ (s : (M.fixSet X hObs hFix).FixedValues), (MeasureTheory.Measure.map (fun p => Causalean.valuesUnionMk p.1 p.2) ((MeasureTheory.Measure.map (Causalean.valuesProjection (id hWr)) ((M.fixSet X hObs hFix).obsKernel s)).compProd (ProbabilityTheory.Kernel.const (Causalean.ValuesOn (Finset.image Causalean.SWIGNode.random Wbase) (Causalean.swigΩ Ω)) (MeasureTheory.Measure.map (Causalean.valuesProjection (Finset.empty_subset (M.fixSet X hObs hFix).observed)) ((M.fixSet X hObs hFix).obsKernel s))))).AbsolutelyContinuous (MeasureTheory.Measure.map (Causalean.valuesProjection (Eq.mpr (id (congrFun' (congrArg LE.le (Finset.union_empty (Finset.image Causalean.SWIGNode.random Wbase))) M.observed)) hWr)) ((M.fixSet X hObs hFix).obsKernel s))) → (M.treatmentMarginal X hXr s0).compProd (M.doKernelY X hObs hFix Y hY s0) = (M.treatmentMarginal X hXr s0).compProd (M.frontdoorKernelY X hObs hFix Y (Finset.image Causalean.SWIGNode.random Wbase) hY hWr s0)","l":1085,"f":"Causalean/SCM/ID/Frontdoor.lean"},{"n":"Causalean.SCM.frontdoor_fd1_rule3_nonDesc","k":"theorem","m":"Causalean.SCM.ID.Toolkit.FrontdoorGraph","a":"SCM/ID/Toolkit","t":2,"d":"G2: corrected-FD1 Rule-3 non-ancestry premise.","s":"∀ {N : Type u_1} [inst : DecidableEq N] [inst_1 : Fintype N] {Ω : N → Type u_2} [inst_2 : (n : N) → MeasurableSpace (Ω n)] (M : Causalean.SCM N Ω) (X Wbase : Finset N) (hX_obs : ∀ D ∈ X, Causalean.SWIGNode.random D ∈ M.observed) (hX_fixed : ∀ D ∈ X, Causalean.SWIGNode.fixed D ∉ M.fixed) (hW_obs : ∀ D ∈ Wbase, Causalean.SWIGNode.random D ∈ M.observed) (hW_fixed : ∀ D ∈ Wbase, Causalean.SWIGNode.fixed D ∉ M.fixed) (Y : Finset (Causalean.SWIGNode N)), (M.fixSet X hX_obs hX_fixed).dag.dSep Y (Finset.image Causalean.SWIGNode.fixed X) (Finset.image Causalean.SWIGNode.random Wbase) → ∀ (hDisj_WX : Disjoint (Finset.image Causalean.SWIGNode.random Wbase) (Finset.image Causalean.SWIGNode.random X)), ∀ v ∈ Y, ∀ d ∈ X, ¬((M.fixSet Wbase hW_obs hW_fixed).fixSet X (fun D hD => id (hX_obs D hD)) (Causalean.SCM.fixSet_fixed_not_mem_of_disjoint M X Wbase hW_obs hW_fixed hX_fixed (Causalean.SCM.disjoint_base_of_disjoint_random_image X Wbase hDisj_WX))).dag.isAncestor (Causalean.SWIGNode.fixed d) v","l":217,"f":"Causalean/SCM/ID/Toolkit/FrontdoorGraph.lean"},{"n":"Causalean.SCM.frontdoor_fd3_rule2_dSep","k":"theorem","m":"Causalean.SCM.ID.Toolkit.FrontdoorGraph","a":"SCM/ID/Toolkit","t":1,"d":"Rule-2 d-separation premise for the frontdoor third condition. Fix a structural causal model M and two intervention target sets X, Wbase such that every node of X is currently a random observed node with no fixed copy already fixed, and likewise every node of Wbase, and an outcome set Y with Y contained in the observed nodes, such that the base graph satisfies the backdoor criterion for Wbase, Y given the randomized image of X and the randomized images of Wbase and X are disjoint. Then in the graph obtained by first fixing X and then fixing Wbase, Y is d-separated from the randomized image of Wbase given exactly that double-intervention graph's fixed node set.","s":"∀ {N : Type u_1} [inst : DecidableEq N] [inst_1 : Fintype N] {Ω : N → Type u_2} [inst_2 : (n : N) → MeasurableSpace (Ω n)] (M : Causalean.SCM N Ω) (X Wbase : Finset N) (hX_obs : ∀ D ∈ X, Causalean.SWIGNode.random D ∈ M.observed) (hX_fixed : ∀ D ∈ X, Causalean.SWIGNode.fixed D ∉ M.fixed) (hW_obs : ∀ D ∈ Wbase, Causalean.SWIGNode.random D ∈ M.observed) (hW_fixed : ∀ D ∈ Wbase, Causalean.SWIGNode.fixed D ∉ M.fixed), ∀ Y ⊆ M.observed, M.backdoorCriterion Wbase hW_obs hW_fixed Y (Finset.image Causalean.SWIGNode.random X) → ∀ (hDisj_WX : Disjoint (Finset.image Causalean.SWIGNode.random Wbase) (Finset.image Causalean.SWIGNode.random X)), ((M.fixSet X hX_obs hX_fixed).fixSet Wbase (fun D hD => id (hW_obs D hD)) (Causalean.SCM.fixSet_fixed_not_mem_of_disjoint M Wbase X hX_obs hX_fixed hW_fixed (Disjoint.symm (Causalean.SCM.disjoint_base_of_disjoint_random_image X Wbase hDisj_WX)))).dag.dSep Y (Finset.image Causalean.SWIGNode.random Wbase) (∅ ∪ ((M.fixSet X hX_obs hX_fixed).fixSet Wbase (fun D hD => id (hW_obs D hD)) (Causalean.SCM.fixSet_fixed_not_mem_of_disjoint M Wbase X hX_obs hX_fixed hW_fixed (Disjoint.symm (Causalean.SCM.disjoint_base_of_disjoint_random_image X Wbase hDisj_WX)))).fixed)","l":316,"f":"Causalean/SCM/ID/Toolkit/FrontdoorGraph.lean"},{"n":"Causalean.SCM.frontdoor_identifiable_ae","k":"theorem","m":"Causalean.SCM.ID.Frontdoor","a":"SCM/ID","t":1,"d":"Frontdoor identification, a.e. in the treatment value. Let X be a valid intervention set — observed and not already fixed, and Wbase likewise a valid intervention set of mediators, with the outcome set Y observed, the random copies of Wbase observed, and the random copies of X observed. Suppose the frontdoor criterion holds for (X, Wbase, Y), that Y is disjoint from the random copies of Wbase and the random copies of Wbase are disjoint from those of X, and that each of the three legs of the frontdoor decomposition — the do(X) leg, the do(Wbase) leg adjusting for X, and the nested do(X)-then-do(Wbase) leg — satisfies the matching backdoor overlap and positivity conditions. Then for treatmentMarginal-almost-every treatment value t, the post-intervention Y-marginal doKernelY at t equals the frontdoor-adjustment functional frontdoorKernelY at t.","s":"∀ {N : Type u_1} [inst : DecidableEq N] [inst_1 : Fintype N] {Ω : N → Type u_2} [inst_2 : (n : N) → MeasurableSpace (Ω n)] [inst_3 : ∀ (n : Causalean.SWIGNode N), StandardBorelSpace (Causalean.swigΩ Ω n)] [inst_4 : ∀ (n : Causalean.SWIGNode N), Nonempty (Causalean.swigΩ Ω n)] (M : Causalean.SCM N Ω) (X : Finset N) (hObs : ∀ D ∈ X, Causalean.SWIGNode.random D ∈ M.observed) (hFix : ∀ D ∈ X, Causalean.SWIGNode.fixed D ∉ M.fixed) (Wbase : Finset N) (hWobs : ∀ D ∈ Wbase, Causalean.SWIGNode.random D ∈ M.observed) (hWfix : ∀ D ∈ Wbase, Causalean.SWIGNode.fixed D ∉ M.fixed) (Y : Finset (Causalean.SWIGNode N)) (hY : Y ⊆ M.observed) (hWr : Finset.image Causalean.SWIGNode.random Wbase ⊆ M.observed) (hXr : Finset.image Causalean.SWIGNode.random X ⊆ M.observed), M.frontdoorCriterion X hObs hFix Wbase hWobs hWfix Y → Disjoint Y (Finset.image Causalean.SWIGNode.random Wbase) → ∀ (hDisj_WrXr : Disjoint (Finset.image Causalean.SWIGNode.random Wbase) (Finset.image Causalean.SWIGNode.random X)) (s0 : M.FixedValues), (∀ (s : (M.fixSet X hObs hFix).FixedValues), Causalean.SCM.ID.Rule2JointOverlap M X hObs hFix ∅ (Eq.mpr (id (congrFun' (congrArg LE.le (Finset.union_empty (Finset.image Causalean.SWIGNode.random X))) M.observed)) hXr) s) → M.BackdoorPositivityAE X ∅ (Finset.empty_subset M.observed) (Eq.mpr (id (congrFun' (congrArg LE.le (Finset.union_empty (Finset.image Causalean.SWIGNode.random X))) M.observed)) hXr) s0 → (∀ (s : (M.fixSet Wbase hWobs hWfix).FixedValues), Causalean.SCM.ID.Rule2JointOverlap M Wbase hWobs hWfix (Finset.image Causalean.SWIGNode.random X) (Finset.union_subset hWr hXr) s) → M.BackdoorPositivityAE Wbase (Finset.image Causalean.SWIGNode.random X) hXr (Finset.union_subset hWr hXr) s0 → (∀ (s : ((M.fixSet X hObs hFix).fixSet Wbase (fun D hD => id (hWobs D hD)) (Causalean.SCM.fixSet_fixed_not_mem_of_disjoint M Wbase X hObs hFix hWfix (Disjoint.symm (Causalean.SCM.disjoint_base_of_disjoint_random_image X Wbase hDisj_WrXr)))).FixedValues), Causalean.SCM.ID.Rule2JointOverlap (M.fixSet X hObs hFix) Wbase (fun D hD => id (hWobs D hD)) (Causalean.SCM.fixSet_fixed_not_mem_of_disjoint M Wbase X hObs hFix hWfix (Disjoint.symm (Causalean.SCM.disjoint_base_of_disjoint_random_image X Wbase hDisj_WrXr))) ∅ (Eq.mpr (id (congrFun' (congrArg LE.le (Finset.union_empty (Finset.image Causalean.SWIGNode.random Wbase))) M.observed)) hWr) s) → (∀ (s : (M.fixSet X hObs hFix).FixedValues), (MeasureTheory.Measure.map (fun p => Causalean.valuesUnionMk p.1 p.2) ((MeasureTheory.Measure.map (Causalean.valuesProjection (id hWr)) ((M.fixSet X hObs hFix).obsKernel s)).compProd (ProbabilityTheory.Kernel.const (Causalean.ValuesOn (Finset.image Causalean.SWIGNode.random Wbase) (Causalean.swigΩ Ω)) (MeasureTheory.Measure.map (Causalean.valuesProjection (Finset.empty_subset (M.fixSet X hObs hFix).observed)) ((M.fixSet X hObs hFix).obsKernel s))))).AbsolutelyContinuous (MeasureTheory.Measure.map (Causalean.valuesProjection (Eq.mpr (id (congrFun' (congrArg LE.le (Finset.union_empty (Finset.image Causalean.SWIGNode.random Wbase))) M.observed)) hWr)) ((M.fixSet X hObs hFix).obsKernel s))) → ∀ᵐ (t : Causalean.ValuesOn (Finset.image Causalean.SWIGNode.random X) (Causalean.swigΩ Ω)) ∂M.treatmentMarginal X hXr s0, (M.doKernelY X hObs hFix Y hY s0) t = (M.frontdoorKernelY X hObs hFix Y (Finset.image Causalean.SWIGNode.random Wbase) hY hWr s0) t","l":1163,"f":"Causalean/SCM/ID/Frontdoor.lean"},{"n":"Causalean.SCM.fullCondIndep_congr_left","k":"theorem","m":"Causalean.SCM.Do.FullCondIndep","a":"SCM/Do","t":2,"d":"Transport FullCondIndep along a Finset equality in the first argument. Used in the Verma–Pearl induction to convert {a} ∪ A' to insert a A'.","s":"∀ {N : Type uN} [inst : DecidableEq N] [inst_1 : Fintype N] {Ω : N → Type uΩ} [inst_2 : (n : N) → MeasurableSpace (Ω n)] (M : Causalean.SCM N Ω) [inst_3 : StandardBorelSpace M.RandomValues] {X X' Y Z : Finset (Causalean.SWIGNode N)} (heq : X = X') {hX : X ⊆ M.randomVars} {hY : Y ⊆ M.randomVars} {hZ : Z ⊆ M.randomVars} {μ : MeasureTheory.Measure M.RandomValues} [inst_4 : MeasureTheory.IsFiniteMeasure μ], M.FullCondIndep X Y Z hX hY hZ μ → M.FullCondIndep X' Y Z (heq ▸ hX) hY hZ μ","l":139,"f":"Causalean/SCM/Do/FullCondIndep.lean"},{"n":"Causalean.SCM.fullCondIndep_congr_right","k":"theorem","m":"Causalean.SCM.Do.FullCondIndep","a":"SCM/Do","t":2,"d":"Transport FullCondIndep along a Finset equality in the conditioning set. Used to align an empty conditioning set with parents v ∩ randomVars when v is a latent root.","s":"∀ {N : Type uN} [inst : DecidableEq N] [inst_1 : Fintype N] {Ω : N → Type uΩ} [inst_2 : (n : N) → MeasurableSpace (Ω n)] (M : Causalean.SCM N Ω) [inst_3 : StandardBorelSpace M.RandomValues] {X Y Z Z' : Finset (Causalean.SWIGNode N)} (heq : Z = Z') {hX : X ⊆ M.randomVars} {hY : Y ⊆ M.randomVars} {hZ : Z ⊆ M.randomVars} {μ : MeasureTheory.Measure M.RandomValues} [inst_4 : MeasureTheory.IsFiniteMeasure μ], M.FullCondIndep X Y Z hX hY hZ μ → M.FullCondIndep X Y Z' hX hY (heq ▸ hZ) μ","l":151,"f":"Causalean/SCM/Do/FullCondIndep.lean"},{"n":"Causalean.SCM.fullCondIndep_const_left","k":"theorem","m":"Causalean.SCM.Do.FullCondIndep","a":"SCM/Do","t":2,"d":"The empty source set is conditionally independent of anything: with X = ∅, the projection valuesProjection ∅ is constant, so FullCondIndep holds trivially. Interprets the nil constructor of OrderedLocalSG.","s":"∀ {N : Type uN} [inst : DecidableEq N] [inst_1 : Fintype N] {Ω : N → Type uΩ} [inst_2 : (n : N) → MeasurableSpace (Ω n)] (M : Causalean.SCM N Ω) [inst_3 : StandardBorelSpace M.RandomValues] {Y Z : Finset (Causalean.SWIGNode N)} (hY : Y ⊆ M.randomVars) (hZ : Z ⊆ M.randomVars) {μ : MeasureTheory.Measure M.RandomValues} [inst_4 : MeasureTheory.IsFiniteMeasure μ], M.FullCondIndep ∅ Y Z (Finset.empty_subset M.randomVars) hY hZ μ","l":164,"f":"Causalean/SCM/Do/FullCondIndep.lean"},{"n":"Causalean.SCM.fullCondIndep_contraction","k":"theorem","m":"Causalean.SCM.Do.FullCondIndep","a":"SCM/Do","t":1,"d":"Contraction for full conditional independence. Let M be a structural causal model and let μ be a finite measure on the full random state. Suppose X, Y, W, and Z are all sets of M's random variables, and that, under μ, X is conditionally independent of Y given Z ∪ W and X is conditionally independent of W given Z. Then X is conditionally independent of Y ∪ W given Z.","s":"∀ {N : Type uN} [inst : DecidableEq N] [inst_1 : Fintype N] {Ω : N → Type uΩ} [inst_2 : (n : N) → MeasurableSpace (Ω n)] (M : Causalean.SCM N Ω) [inst_3 : StandardBorelSpace M.RandomValues] {X Y W Z : Finset (Causalean.SWIGNode N)} (hX : X ⊆ M.randomVars) (hY : Y ⊆ M.randomVars) (hW : W ⊆ M.randomVars) (hZ : Z ⊆ M.randomVars) {μ : MeasureTheory.Measure M.RandomValues} [inst_4 : MeasureTheory.IsFiniteMeasure μ], M.FullCondIndep X Y (Z ∪ W) hX hY (Finset.union_subset hZ hW) μ → M.FullCondIndep X W Z hX hW hZ μ → M.FullCondIndep X (Y ∪ W) Z hX (Finset.union_subset hY hW) hZ μ","l":115,"f":"Causalean/SCM/Do/FullCondIndep.lean"},{"n":"Causalean.SCM.fullCondIndep_decomposition","k":"theorem","m":"Causalean.SCM.Do.FullCondIndep","a":"SCM/Do","t":2,"d":"Decomposition for FullCondIndep.","s":"∀ {N : Type uN} [inst : DecidableEq N] [inst_1 : Fintype N] {Ω : N → Type uΩ} [inst_2 : (n : N) → MeasurableSpace (Ω n)] (M : Causalean.SCM N Ω) [inst_3 : StandardBorelSpace M.RandomValues] {X Y W Z : Finset (Causalean.SWIGNode N)} (hX : X ⊆ M.randomVars) (hYW : Y ∪ W ⊆ M.randomVars) (hZ : Z ⊆ M.randomVars) {μ : MeasureTheory.Measure M.RandomValues} [inst_4 : MeasureTheory.IsFiniteMeasure μ], M.FullCondIndep X (Y ∪ W) Z hX hYW hZ μ → M.FullCondIndep X Y Z hX (LE.le.trans Finset.subset_union_left hYW) hZ μ","l":83,"f":"Causalean/SCM/Do/FullCondIndep.lean"},{"n":"Causalean.SCM.fullCondIndep_of_orderedLocalSG","k":"theorem","m":"Causalean.SCM.Do.GlobalMarkov","a":"SCM/Do","t":2,"d":"A conditional-independence conclusion derived from the graph's ordered local Markov statements and semi-graphoid rules also holds in the structural causal model's full joint distribution.","s":"∀ {N : Type uN} [inst : DecidableEq N] [inst_1 : Fintype N] {Ω : N → Type uΩ} [inst_2 : (n : N) → MeasurableSpace (Ω n)] (M : Causalean.SCM N Ω) [inst_3 : StandardBorelSpace M.RandomValues] [StandardBorelSpace M.LatentValues] [∀ (v : Causalean.SWIGNode N), StandardBorelSpace (Causalean.ValuesOn {v} (Causalean.swigΩ Ω))] [∀ (v : Causalean.SWIGNode N), Nonempty (Causalean.ValuesOn {v} (Causalean.swigΩ Ω))] [∀ (v : Causalean.SWIGNode N), StandardBorelSpace (Causalean.ValuesOn (M.dag.nonDescendants v ∩ M.randomVars) (Causalean.swigΩ Ω))] [∀ (v : Causalean.SWIGNode N), Nonempty (Causalean.ValuesOn (M.dag.nonDescendants v ∩ M.randomVars) (Causalean.swigΩ Ω))] (s : M.FixedValues) {X Y Z : Finset (Causalean.SWIGNode N)}, M.dag.OrderedLocalSG M.randomVars X Y Z → ∀ (hX : X ⊆ M.randomVars) (hY : Y ⊆ M.randomVars) (hZ : Z ⊆ M.randomVars), M.FullCondIndep X Y Z hX hY hZ (M.jointKernel s)","l":534,"f":"Causalean/SCM/Do/GlobalMarkov.lean"},{"n":"Causalean.SCM.fullCondIndep_ordered_local","k":"theorem","m":"Causalean.SCM.Do.GlobalMarkov","a":"SCM/Do","t":2,"d":"For a random node, any set of random non-descendants that contains its random parents is conditionally independent of the remaining nodes in that set given those parents, under the model's joint kernel.","s":"∀ {N : Type uN} [inst : DecidableEq N] [inst_1 : Fintype N] {Ω : N → Type uΩ} [inst_2 : (n : N) → MeasurableSpace (Ω n)] (M : Causalean.SCM N Ω) [inst_3 : StandardBorelSpace M.RandomValues] [StandardBorelSpace M.LatentValues] [inst_5 : ∀ (s : M.FixedValues), MeasureTheory.IsFiniteMeasure (M.jointKernel s)] [∀ (v : Causalean.SWIGNode N), StandardBorelSpace (Causalean.ValuesOn {v} (Causalean.swigΩ Ω))] [∀ (v : Causalean.SWIGNode N), Nonempty (Causalean.ValuesOn {v} (Causalean.swigΩ Ω))] [∀ (v : Causalean.SWIGNode N), StandardBorelSpace (Causalean.ValuesOn (M.dag.nonDescendants v ∩ M.randomVars) (Causalean.swigΩ Ω))] [∀ (v : Causalean.SWIGNode N), Nonempty (Causalean.ValuesOn (M.dag.nonDescendants v ∩ M.randomVars) (Causalean.swigΩ Ω))] (s : M.FixedValues) (v : Causalean.SWIGNode N) (hv : v ∈ M.randomVars) (P : Finset (Causalean.SWIGNode N)) (hP : P ⊆ M.randomVars), P ⊆ M.dag.nonDescendants v → M.dag.parents v ∩ M.randomVars ⊆ P → M.FullCondIndep {v} (P \\ (M.dag.parents v ∩ M.randomVars)) (M.dag.parents v ∩ M.randomVars) (Finset.singleton_subset_iff.mpr hv) (Finset.Subset.trans Finset.sdiff_subset hP) Finset.inter_subset_right (M.jointKernel s)","l":476,"f":"Causalean/SCM/Do/GlobalMarkov.lean"},{"n":"Causalean.SCM.fullCondIndep_singleton_of_dSep_with_fixed","k":"theorem","m":"Causalean.SCM.Do.GlobalMarkov","a":"SCM/Do","t":2,"d":"D-separation of one random node from a target set implies conditional independence after conditioning on the random nodes, even when the graph's conditioning set also contains fixed intervention nodes. The fixed nodes affect the graphical separation but do not appear among the random values being conditioned on.","s":"∀ {N : Type uN} [inst : DecidableEq N] [inst_1 : Fintype N] {Ω : N → Type uΩ} [inst_2 : (n : N) → MeasurableSpace (Ω n)] (M : Causalean.SCM N Ω) [inst_3 : StandardBorelSpace M.RandomValues] [∀ (n : Causalean.SWIGNode N), StandardBorelSpace (Causalean.swigΩ Ω n)] [∀ (n : Causalean.SWIGNode N), Nonempty (Causalean.swigΩ Ω n)] (a : Causalean.SWIGNode N) (Y W_rand W_fix : Finset (Causalean.SWIGNode N)) (ha : a ∈ M.randomVars) (hY : Y ⊆ M.randomVars) (hW_rand : W_rand ⊆ M.randomVars), W_fix ⊆ M.fixed → M.dag.dSep {a} Y (W_rand ∪ W_fix) → ∀ (s : M.FixedValues), M.FullCondIndep {a} Y W_rand (Finset.singleton_subset_iff.mpr ha) hY hW_rand (M.jointKernel s)","l":605,"f":"Causalean/SCM/Do/GlobalMarkov.lean"},{"n":"Causalean.SCM.fullCondIndep_subset_right","k":"theorem","m":"Causalean.SCM.Do.FullCondIndep","a":"SCM/Do","t":2,"d":"Subset right for FullCondIndep.","s":"∀ {N : Type uN} [inst : DecidableEq N] [inst_1 : Fintype N] {Ω : N → Type uΩ} [inst_2 : (n : N) → MeasurableSpace (Ω n)] (M : Causalean.SCM N Ω) [inst_3 : StandardBorelSpace M.RandomValues] {X Y Y' Z : Finset (Causalean.SWIGNode N)} (hX : X ⊆ M.randomVars) (hY : Y ⊆ M.randomVars) (hY' : Y' ⊆ M.randomVars) (hZ : Z ⊆ M.randomVars), Y' ⊆ Y → ∀ {μ : MeasureTheory.Measure M.RandomValues} [inst_4 : MeasureTheory.IsFiniteMeasure μ], M.FullCondIndep X Y Z hX hY hZ μ → M.FullCondIndep X Y' Z hX hY' hZ μ","l":70,"f":"Causalean/SCM/Do/FullCondIndep.lean"},{"n":"Causalean.SCM.fullCondIndep_symm","k":"theorem","m":"Causalean.SCM.Do.FullCondIndep","a":"SCM/Do","t":2,"d":"Symmetry for FullCondIndep.","s":"∀ {N : Type uN} [inst : DecidableEq N] [inst_1 : Fintype N] {Ω : N → Type uΩ} [inst_2 : (n : N) → MeasurableSpace (Ω n)] (M : Causalean.SCM N Ω) [inst_3 : StandardBorelSpace M.RandomValues] {X Y Z : Finset (Causalean.SWIGNode N)} (hX : X ⊆ M.randomVars) (hY : Y ⊆ M.randomVars) (hZ : Z ⊆ M.randomVars) {μ : MeasureTheory.Measure M.RandomValues} [inst_4 : MeasureTheory.IsFiniteMeasure μ], M.FullCondIndep X Y Z hX hY hZ μ → M.FullCondIndep Y X Z hY hX hZ μ","l":59,"f":"Causalean/SCM/Do/FullCondIndep.lean"},{"n":"Causalean.SCM.fullCondIndep_weak_union","k":"theorem","m":"Causalean.SCM.Do.FullCondIndep","a":"SCM/Do","t":1,"d":"Weak union for full conditional independence. Let M be a structural causal model and let μ be a finite measure on the full random state (observed together with latent variables). Suppose X, Y ∪ W, and Z ∪ W are all sets of M's random variables, and that, under μ, X is conditionally independent of Y ∪ W given Z. Then X is conditionally independent of Y given Z ∪ W.","s":"∀ {N : Type uN} [inst : DecidableEq N] [inst_1 : Fintype N] {Ω : N → Type uΩ} [inst_2 : (n : N) → MeasurableSpace (Ω n)] (M : Causalean.SCM N Ω) [inst_3 : StandardBorelSpace M.RandomValues] {X Y W Z : Finset (Causalean.SWIGNode N)} (hX : X ⊆ M.randomVars) (hYW : Y ∪ W ⊆ M.randomVars) (hZW : Z ∪ W ⊆ M.randomVars) {μ : MeasureTheory.Measure M.RandomValues} [inst_4 : MeasureTheory.IsFiniteMeasure μ], M.FullCondIndep X (Y ∪ W) Z hX hYW (LE.le.trans Finset.subset_union_left hZW) μ → M.FullCondIndep X Y (Z ∪ W) hX (LE.le.trans Finset.subset_union_left hYW) hZW μ","l":96,"f":"Causalean/SCM/Do/FullCondIndep.lean"},{"n":"Causalean.SCM.full_globalMarkov","k":"theorem","m":"Causalean.SCM.Do.GlobalMarkov","a":"SCM/Do","t":1,"d":"Full Global Markov Property. If X, Y, and Z are sets of nodes drawn from the model's random (observed and latent) nodes and X is d-separated from Y by Z in the model's causal graph, then under the joint distribution over all random coordinates at fixed value s, the X-coordinates and the Y-coordinates are conditionally independent given the Z-coordinates.","s":"∀ {N : Type uN} [inst : DecidableEq N] [inst_1 : Fintype N] {Ω : N → Type uΩ} [inst_2 : (n : N) → MeasurableSpace (Ω n)] (M : Causalean.SCM N Ω) [inst_3 : StandardBorelSpace M.RandomValues] [∀ (n : Causalean.SWIGNode N), StandardBorelSpace (Causalean.swigΩ Ω n)] [∀ (n : Causalean.SWIGNode N), Nonempty (Causalean.swigΩ Ω n)] (X Y Z : Finset (Causalean.SWIGNode N)) (hX : X ⊆ M.randomVars) (hY : Y ⊆ M.randomVars) (hZ : Z ⊆ M.randomVars), M.dag.dSep X Y Z → ∀ (s : M.FixedValues), M.FullCondIndep X Y Z hX hY hZ (M.jointKernel s)","l":753,"f":"Causalean/SCM/Do/GlobalMarkov.lean"},{"n":"Causalean.SCM.full_globalMarkov_with_fixed","k":"theorem","m":"Causalean.SCM.Do.GlobalMarkov","a":"SCM/Do","t":1,"d":"Full Global Markov with fixed-node conditioning shadow. If X, Y, and Z_rand are sets of nodes drawn from the model's random (observed and latent) nodes and Z_fix is a set of the model's fixed (intervened) nodes, and X is d-separated from Y by the union Z_rand ∪ Z_fix in the model's causal graph, then under the joint distribution over all random coordinates at fixed value s, the X-coordinates and the Y-coordinates are conditionally independent given only the Z_rand-coordinates — the fixed nodes contribute to the graphical separation but, since their values are already pinned by s, drop out of the probabilistic conditioning set.","s":"∀ {N : Type uN} [inst : DecidableEq N] [inst_1 : Fintype N] {Ω : N → Type uΩ} [inst_2 : (n : N) → MeasurableSpace (Ω n)] (M : Causalean.SCM N Ω) [inst_3 : StandardBorelSpace M.RandomValues] [∀ (n : Causalean.SWIGNode N), StandardBorelSpace (Causalean.swigΩ Ω n)] [∀ (n : Causalean.SWIGNode N), Nonempty (Causalean.swigΩ Ω n)] (X Y Z_rand Z_fix : Finset (Causalean.SWIGNode N)) (hX : X ⊆ M.randomVars) (hY : Y ⊆ M.randomVars) (hZ_rand : Z_rand ⊆ M.randomVars), Z_fix ⊆ M.fixed → M.dag.dSep X Y (Z_rand ∪ Z_fix) → ∀ (s : M.FixedValues), M.FullCondIndep X Y Z_rand hX hY hZ_rand (M.jointKernel s)","l":640,"f":"Causalean/SCM/Do/GlobalMarkov.lean"},{"n":"Causalean.SCM.full_local_markov","k":"theorem","m":"Causalean.SCM.Do.LocalMarkov","a":"SCM/Do","t":1,"d":"Full Local Markov Property. If v is an observed node of the model, then under the joint distribution over all random (observed and latent) coordinates at fixed value s, the v-coordinate is conditionally independent of its non-descendants — restricted to random nodes — given all of its parents, including any latent parents, likewise restricted to random nodes.","s":"∀ {N : Type uN} [inst : DecidableEq N] [inst_1 : Fintype N] {Ω : N → Type uΩ} [inst_2 : (n : N) → MeasurableSpace (Ω n)] (M : Causalean.SCM N Ω) [inst_3 : StandardBorelSpace M.RandomValues] [StandardBorelSpace M.LatentValues] (v : Causalean.SWIGNode N) (hv : v ∈ M.observed) [StandardBorelSpace (Causalean.ValuesOn {v} (Causalean.swigΩ Ω))] [Nonempty (Causalean.ValuesOn {v} (Causalean.swigΩ Ω))] [StandardBorelSpace (Causalean.ValuesOn (M.dag.nonDescendants v ∩ M.randomVars) (Causalean.swigΩ Ω))] [Nonempty (Causalean.ValuesOn (M.dag.nonDescendants v ∩ M.randomVars) (Causalean.swigΩ Ω))] (s : M.FixedValues), M.FullCondIndep {v} (M.dag.nonDescendants v ∩ M.randomVars) (M.dag.parents v ∩ M.randomVars) (Finset.singleton_subset_iff.mpr (Causalean.SCM.observed_subset_randomVars M hv)) Finset.inter_subset_right Finset.inter_subset_right (M.jointKernel s)","l":108,"f":"Causalean/SCM/Do/LocalMarkov.lean"},{"n":"Causalean.SCM.full_local_markov_latent","k":"theorem","m":"Causalean.SCM.Do.LocalMarkov","a":"SCM/Do","t":1,"d":"Latent Local Markov Property. If a is a latent (unobserved) node of the model, then under the joint distribution over all random coordinates at fixed value s, the a-coordinate is unconditionally independent of its non-descendants — restricted to random nodes — i.e. conditionally independent given the empty conditioning set.","s":"∀ {N : Type uN} [inst : DecidableEq N] [inst_1 : Fintype N] {Ω : N → Type uΩ} [inst_2 : (n : N) → MeasurableSpace (Ω n)] (M : Causalean.SCM N Ω) [inst_3 : StandardBorelSpace M.RandomValues] [StandardBorelSpace M.LatentValues] (a : Causalean.SWIGNode N) (ha : a ∈ M.unobserved) [StandardBorelSpace (Causalean.ValuesOn {a} (Causalean.swigΩ Ω))] [Nonempty (Causalean.ValuesOn {a} (Causalean.swigΩ Ω))] [StandardBorelSpace (Causalean.ValuesOn (M.dag.nonDescendants a ∩ M.randomVars) (Causalean.swigΩ Ω))] [Nonempty (Causalean.ValuesOn (M.dag.nonDescendants a ∩ M.randomVars) (Causalean.swigΩ Ω))] (s : M.FixedValues), M.FullCondIndep {a} (M.dag.nonDescendants a ∩ M.randomVars) ∅ (Finset.singleton_subset_iff.mpr (Finset.mem_union_right M.observed ha)) Finset.inter_subset_right (Finset.empty_subset M.randomVars) (M.jointKernel s)","l":242,"f":"Causalean/SCM/Do/LocalMarkov.lean"},{"n":"Causalean.SCM.get_appendTail","k":"theorem","m":"Causalean.SCM.Do.CutsetDSep","a":"SCM/Do","t":2,"d":"Appending the tail of one nonempty list to another nonempty list preserves the first list's entries and then reads the remaining entries from the second list with a one-place shift.","s":"∀ {V : Type u_3} (pa q : List V), pa ≠ [] → ∀ (hq : q ≠ []), (pa ++ q.tail).length = pa.length + q.length - 1 ∧ (∀ (j : ℕ) (hj : j < pa.length), (pa ++ q.tail).get ⟨j, Eq.mpr (id (congrArg (fun _a => j < _a) List.length_append)) (have this := List.length_tail; Decidable.byContradiction fun a => Causalean.SCM.get_appendTail._proof_1 pa q j hj this a)⟩ = pa.get ⟨j, hj⟩) ∧ ∀ (j : ℕ) (hjL : pa.length ≤ j) (hj : j < (pa ++ q.tail).length), (pa ++ q.tail).get ⟨j, hj⟩ = q.get ⟨j - pa.length + 1, have this := List.length_pos_iff.mpr hq; Decidable.byContradiction fun a => Causalean.SCM.get_appendTail._proof_2 pa q j hjL (Eq.mp (congrArg (fun _a => j < pa.length + _a) List.length_tail) (Eq.mp (congrArg (fun _a => j < _a) List.length_append) hj)) this a⟩","l":446,"f":"Causalean/SCM/Do/CutsetDSep.lean"},{"n":"Causalean.SCM.globalMarkov","k":"theorem","m":"Causalean.SCM.Do.ObsMarkov","a":"SCM/Do","t":1,"d":"Global Markov Property. If X, Y, and Z are sets of observed nodes and X is d-separated from Y by Z in the model's full causal graph, which also includes any latent nodes, then under the observational distribution — the law of the observed coordinates alone, at fixed value s — the X-coordinates and Y-coordinates are conditionally independent given the Z-coordinates.","s":"∀ {N : Type uN} [inst : DecidableEq N] [inst_1 : Fintype N] {Ω : N → Type uΩ} [inst_2 : (n : N) → MeasurableSpace (Ω n)] (M : Causalean.SCM N Ω) [inst_3 : ∀ (n : Causalean.SWIGNode N), StandardBorelSpace (Causalean.swigΩ Ω n)] [∀ (n : Causalean.SWIGNode N), Nonempty (Causalean.swigΩ Ω n)] (X Y Z : Finset (Causalean.SWIGNode N)) [StandardBorelSpace (Causalean.ValuesOn X (Causalean.swigΩ Ω))] [Nonempty (Causalean.ValuesOn X (Causalean.swigΩ Ω))] [StandardBorelSpace (Causalean.ValuesOn Y (Causalean.swigΩ Ω))] [Nonempty (Causalean.ValuesOn Y (Causalean.swigΩ Ω))] (hX : X ⊆ M.observed) (hY : Y ⊆ M.observed) (hZ : Z ⊆ M.observed), M.dag.dSep X Y Z → ∀ (s : M.FixedValues), M.ObsCondIndep X Y Z hX hY hZ (M.obsKernel s)","l":96,"f":"Causalean/SCM/Do/ObsMarkov.lean"},{"n":"Causalean.SCM.globalMarkov_with_fixed","k":"theorem","m":"Causalean.SCM.Do.ObsMarkov","a":"SCM/Do","t":1,"d":"Global Markov with fixed-node conditioning. If X, Y, and Z_obs are sets of observed nodes and Z_fix is a set of the model's fixed (intervened) nodes, and X is d-separated from Y by the union Z_obs ∪ Z_fix in the model's full causal graph, then under the observational distribution at fixed value s, the X-coordinates and Y-coordinates are conditionally independent given only the Z_obs-coordinates — the fixed nodes contribute to the graphical separation but, since their values are already pinned by s, drop out of the probabilistic conditioning set.","s":"∀ {N : Type uN} [inst : DecidableEq N] [inst_1 : Fintype N] {Ω : N → Type uΩ} [inst_2 : (n : N) → MeasurableSpace (Ω n)] (M : Causalean.SCM N Ω) [inst_3 : ∀ (n : Causalean.SWIGNode N), StandardBorelSpace (Causalean.swigΩ Ω n)] [∀ (n : Causalean.SWIGNode N), Nonempty (Causalean.swigΩ Ω n)] (X Y Z_obs Z_fix : Finset (Causalean.SWIGNode N)) [StandardBorelSpace (Causalean.ValuesOn X (Causalean.swigΩ Ω))] [Nonempty (Causalean.ValuesOn X (Causalean.swigΩ Ω))] [StandardBorelSpace (Causalean.ValuesOn Y (Causalean.swigΩ Ω))] [Nonempty (Causalean.ValuesOn Y (Causalean.swigΩ Ω))] (hX : X ⊆ M.observed) (hY : Y ⊆ M.observed) (hZ_obs : Z_obs ⊆ M.observed), Z_fix ⊆ M.fixed → M.dag.dSep X Y (Z_obs ∪ Z_fix) → ∀ (s : M.FixedValues), M.ObsCondIndep X Y Z_obs hX hY hZ_obs (M.obsKernel s)","l":125,"f":"Causalean/SCM/Do/ObsMarkov.lean"},{"n":"Causalean.SCM.hNoDesc_descend_to_parent","k":"theorem","m":"Causalean.SCM.Do.Rule3","a":"SCM/Do","t":2,"d":"If none of the fixed intervention nodes is an ancestor of a node, then none is an ancestor of any parent of that node.","s":"∀ {N : Type u_1} [inst : DecidableEq N] [inst_1 : Fintype N] {Ω : N → Type u_2} [inst_2 : (n : N) → MeasurableSpace (Ω n)] {M2 : Causalean.SCM N Ω} {Z : Finset N} {v w : Causalean.SWIGNode N}, w ∈ M2.dag.parents v → (∀ z ∈ Z, ¬M2.dag.isAncestor (Causalean.SWIGNode.fixed z) v) → ∀ z ∈ Z, ¬M2.dag.isAncestor (Causalean.SWIGNode.fixed z) w","l":80,"f":"Causalean/SCM/Do/Rule3.lean"},{"n":"Causalean.SCM.hNoDesc_implies_no_fixed_parent","k":"theorem","m":"Causalean.SCM.Do.Rule3","a":"SCM/Do","t":2,"d":"If none of the fixed intervention nodes is an ancestor of a node, none is a parent of that node.","s":"∀ {N : Type u_1} [inst : DecidableEq N] [inst_1 : Fintype N] {Ω : N → Type u_2} [inst_2 : (n : N) → MeasurableSpace (Ω n)] {M2 : Causalean.SCM N Ω} {Z : Finset N} {v : Causalean.SWIGNode N}, (∀ z ∈ Z, ¬M2.dag.isAncestor (Causalean.SWIGNode.fixed z) v) → ∀ z ∈ Z, Causalean.SWIGNode.fixed z ∉ M2.dag.parents v","l":72,"f":"Causalean/SCM/Do/Rule3.lean"},{"n":"Causalean.SCM.identifyMassRec","k":"def","m":"Causalean.SCM.ID.Density.IdentifyMass","a":"SCM/ID/Density","t":1,"d":"The mass-level IDENTIFY recursion. Starting with a mass function for T, it recursively projects to the induced ancestral set of C, extracts the containing district there, and stops when the induced ancestral set is exactly C. The hedge branch A = T returns the current mass function; successful reachability proofs never use that branch.","s":"{N : Type u_1} → [inst : DecidableEq N] → [inst_1 : Fintype N] → {Ω : N → Type u_2} → [(n : N) → Fintype (Ω n)] → (O : Finset (Causalean.SWIGNode N)) → Causalean.SWIGGraph N → (T : Finset (Causalean.SWIGNode N)) → Finset (Causalean.SWIGNode N) → T ⊆ O → (Causalean.ValuesOn O (Causalean.swigΩ Ω) → ENNReal) → Causalean.ValuesOn O (Causalean.swigΩ Ω) → ENNReal","l":134,"f":"Causalean/SCM/ID/Density/IdentifyMass.lean"},{"n":"Causalean.SCM.identifyMassRec_base","k":"theorem","m":"Causalean.SCM.ID.Density.IdentifyMass","a":"SCM/ID/Density","t":1,"d":"For a target set T contained in the observed coordinates, if the graph-induced ancestral set of T relative to C already equals C, then the mass-level IDENTIFY recursion identifyMassRec on T stops immediately and returns the marginal of the input mass function q obtained by summing out the coordinates in T \\ C.","s":"∀ {N : Type u_1} [inst : DecidableEq N] [inst_1 : Fintype N] {Ω : N → Type u_2} [inst_2 : (n : N) → MeasurableSpace (Ω n)] [inst_3 : (n : N) → Fintype (Ω n)] (M : Causalean.SCM N Ω) (G : Causalean.SWIGGraph N) (T C : Finset (Causalean.SWIGNode N)) (hT : T ⊆ M.observed) (q : Causalean.ValuesOn M.observed (Causalean.swigΩ Ω) → ENNReal), Causalean.SCM.ID.inducedAncestral G T C = C → Causalean.SCM.identifyMassRec M.observed G T C hT q = Causalean.SCM.marginalizeOn M.observed (T \\ C) (fun x hv => hT (Finset.mem_sdiff.mp hv).left) q","l":171,"f":"Causalean/SCM/ID/Density/IdentifyMass.lean"},{"n":"Causalean.SCM.identifyMassRec_hedge","k":"theorem","m":"Causalean.SCM.ID.Density.IdentifyMass","a":"SCM/ID/Density","t":1,"d":"For a target set T contained in the observed coordinates, if the graph-induced ancestral set of T relative to C does not equal C but it equals T itself — the hedge case, reached only after the base case has failed — then the recursion identifyMassRec on T returns the input mass function q unchanged.","s":"∀ {N : Type u_1} [inst : DecidableEq N] [inst_1 : Fintype N] {Ω : N → Type u_2} [inst_2 : (n : N) → MeasurableSpace (Ω n)] [inst_3 : (n : N) → Fintype (Ω n)] (M : Causalean.SCM N Ω) (G : Causalean.SWIGGraph N) (T C : Finset (Causalean.SWIGNode N)) (hT : T ⊆ M.observed) (q : Causalean.ValuesOn M.observed (Causalean.swigΩ Ω) → ENNReal), Causalean.SCM.ID.inducedAncestral G T C ≠ C → Causalean.SCM.ID.inducedAncestral G T C = T → Causalean.SCM.identifyMassRec M.observed G T C hT q = q","l":187,"f":"Causalean/SCM/ID/Density/IdentifyMass.lean"},{"n":"Causalean.SCM.identifyMassRec_step","k":"theorem","m":"Causalean.SCM.ID.Density.IdentifyMass","a":"SCM/ID/Density","t":1,"d":"For a target set T contained in the observed coordinates, if the graph-induced ancestral set of T relative to C does not equal C and does not equal T either — i.e. neither the base nor the hedge case applies — then the recursion identifyMassRec on T unfolds one step: it extracts, from the mass function q marginalized onto the induced ancestral set, the district factor of the c-component of C inside that induced ancestral graph, and recurses on that district.","s":"∀ {N : Type u_1} [inst : DecidableEq N] [inst_1 : Fintype N] {Ω : N → Type u_2} [inst_2 : (n : N) → MeasurableSpace (Ω n)] [inst_3 : (n : N) → Fintype (Ω n)] (M : Causalean.SCM N Ω) (G : Causalean.SWIGGraph N) (T C : Finset (Causalean.SWIGNode N)) (hT : T ⊆ M.observed) (q : Causalean.ValuesOn M.observed (Causalean.swigΩ Ω) → ENNReal), Causalean.SCM.ID.inducedAncestral G T C ≠ C → Causalean.SCM.ID.inducedAncestral G T C ≠ T → Causalean.SCM.identifyMassRec M.observed G T C hT q = let A := Causalean.SCM.ID.inducedAncestral G T C; have hA := fun x hv => hT (Causalean.SCM.ID.inducedAncestral_subset_left G T C hv); let C₁ := Causalean.SCM.ID.containingCComponent (G.induce A) C; have hC₁ := fun x hv => hT (Causalean.SCM.ID.inducedAncestral_subset_left G T C (Causalean.SCM.ID.containingCComponent_induce_subset G A C hv)); Causalean.SCM.identifyMassRec M.observed G C₁ C hC₁ (Causalean.SCM.extractDistrict M.observed (G.induce A) A C₁ hA (Causalean.SCM.marginalizeOn M.observed (T \\ A) (fun x hv => hT (Finset.mem_sdiff.mp hv).left) q))","l":205,"f":"Causalean/SCM/ID/Density/IdentifyMass.lean"},{"n":"Causalean.SCM.indepFun_valuesProjection_latentProduct","k":"theorem","m":"Causalean.SCM.Do.GlobalMarkov","a":"SCM/Do","t":2,"d":"Under M.latentProduct = ⊗_{u ∈ 𝐋} ℙ(L_u), the coordinate-tuple projections at two disjoint latent blocks A, B ⊆ M.unobserved are independent. The latent product is a Measure.pi over the subtype of unobserved nodes, so this is indepFun_pi_of_disjoint transported along the reindexSubtypeProj relabelling.","s":"∀ {N : Type uN} [inst : DecidableEq N] [inst_1 : Fintype N] {Ω : N → Type uΩ} [inst_2 : (n : N) → MeasurableSpace (Ω n)] (M : Causalean.SCM N Ω) {A B : Finset (Causalean.SWIGNode N)} (hA : A ⊆ M.unobserved) (hB : B ⊆ M.unobserved), Disjoint A B → ProbabilityTheory.IndepFun (Causalean.valuesProjection hA) (Causalean.valuesProjection hB) M.latentProduct","l":332,"f":"Causalean/SCM/Do/GlobalMarkov.lean"},{"n":"Causalean.SCM.induce","k":"def","m":"Causalean.SCM.Model.Induced","a":"SCM/Model","t":1,"d":"The induced sub-SCM for an ancestrally closed node set keeps the observed and fixed parts selected by the induced SWIG graph and keeps exactly the original latent roots that feed retained observed nodes.","s":"{N : Type u_1} → [inst : DecidableEq N] → [inst_1 : Fintype N] → {Ω : N → Type u_2} → [inst_2 : (n : N) → MeasurableSpace (Ω n)] → (M : Causalean.SCM N Ω) → (R : Finset (Causalean.SWIGNode N)) → M.isAncestrallyClosedSCM R → Causalean.SCM N Ω","l":146,"f":"Causalean/SCM/Model/Induced.lean"},{"n":"Causalean.SCM.induce_evalMap_compat","k":"theorem","m":"Causalean.SCM.Model.Induced","a":"SCM/Model","t":1,"d":"Evaluation-map bridge for the induced sub-SCM. Fix a structural causal model M and a node set R that is ancestrally closed in the SCM sense. Then at every random node v retained by the induced model M.induce R hR, the induced evaluation map at the restriction of sTilde to R and the projected latent assignment computes the same value as the original evaluation map at sTilde.","s":"∀ {N : Type u_1} [inst : DecidableEq N] [inst_1 : Fintype N] {Ω : N → Type u_2} [inst_2 : (n : N) → MeasurableSpace (Ω n)] (M : Causalean.SCM N Ω) (R : Finset (Causalean.SWIGNode N)) (hR : M.isAncestrallyClosedSCM R) (sTilde : M.FixedValues) (ℓ : M.LatentValues) {v : Causalean.SWIGNode N} (hvI : v ∈ (M.induce R hR).randomVars), have hvM := Or.casesOn (Finset.mem_union.mp hvI) (fun hvIObs => Finset.mem_union_left M.unobserved (Finset.inter_subset_right hvIObs)) fun hvIUo => Finset.mem_union_right M.observed (Causalean.SCM.induce_unobserved_subset M R hR hvIUo); (M.induce R hR).evalMap (Causalean.valuesProjection (Finset.filter_subset (fun s => Causalean.iotaMap s ∈ R ∩ M.observed) M.fixed) sTilde) (Causalean.valuesProjection (Causalean.SCM.induce_unobserved_subset M R hR) ℓ) ⟨v, hvI⟩ = M.evalMap sTilde ℓ ⟨v, hvM⟩","l":273,"f":"Causalean/SCM/Model/Induced.lean"},{"n":"Causalean.SCM.induce_latentProduct_eq_map","k":"theorem","m":"Causalean.SCM.Model.Induced","a":"SCM/Model","t":2,"d":"The latent product of an induced sub-SCM is the marginal of the original latent product.","s":"∀ {N : Type u_1} [inst : DecidableEq N] [inst_1 : Fintype N] {Ω : N → Type u_2} [inst_2 : (n : N) → MeasurableSpace (Ω n)] (M : Causalean.SCM N Ω) (R : Finset (Causalean.SWIGNode N)) (hR : M.isAncestrallyClosedSCM R), (M.induce R hR).latentProduct = MeasureTheory.Measure.map (Causalean.valuesProjection (Causalean.SCM.induce_unobserved_subset M R hR)) M.latentProduct","l":249,"f":"Causalean/SCM/Model/Induced.lean"},{"n":"Causalean.SCM.induce_marginal_compat","k":"theorem","m":"Causalean.SCM.Model.Induced","a":"SCM/Model","t":1,"d":"Marginal compatibility of the induced sub-SCM (prop:scm-induced-marginal). Fix a structural causal model M, an ancestrally closed node set R, and a fixed-value assignment sTilde on M. Then the observational kernel of the induced sub-SCM at the restriction of sTilde to R equals the pushforward of M's observational kernel at sTilde onto the induced observed coordinates R ∩ M.observed.","s":"∀ {N : Type u_1} [inst : DecidableEq N] [inst_1 : Fintype N] {Ω : N → Type u_2} [inst_2 : (n : N) → MeasurableSpace (Ω n)] (M : Causalean.SCM N Ω) (R : Finset (Causalean.SWIGNode N)) (hR : M.isAncestrallyClosedSCM R) (sTilde : M.FixedValues), (M.induce R hR).obsKernel (Causalean.valuesProjection (Finset.filter_subset (fun s => Causalean.iotaMap s ∈ R ∩ M.observed) M.fixed) sTilde) = MeasureTheory.Measure.map (Causalean.valuesProjection (have this := Finset.inter_subset_right; this)) (M.obsKernel sTilde)","l":450,"f":"Causalean/SCM/Model/Induced.lean"},{"n":"Causalean.SCM.induce_parents_eq_of_ancClosed","k":"theorem","m":"Causalean.SCM.Model.Induced","a":"SCM/Model","t":2,"d":"An ancestrally closed retained observed set keeps every parent needed to evaluate each retained observed node, so its induced graph has exactly the same parent set at those nodes.","s":"∀ {N : Type u_1} [inst : DecidableEq N] [inst_1 : Fintype N] {Ω : N → Type u_2} [inst_2 : (n : N) → MeasurableSpace (Ω n)] (M : Causalean.SCM N Ω) (R : Finset (Causalean.SWIGNode N)), M.isAncestrallyClosedSCM R → ∀ {v : Causalean.SWIGNode N}, v ∈ (M.induce R).observed → (M.induce R).dag.parents v = M.dag.parents v","l":87,"f":"Causalean/SCM/Model/Induced.lean"},{"n":"Causalean.SCM.induce_unobserved_subset","k":"theorem","m":"Causalean.SCM.Model.Induced","a":"SCM/Model","t":2,"d":"The latent roots of an induced sub-SCM are original latent roots.","s":"∀ {N : Type u_1} [inst : DecidableEq N] [inst_1 : Fintype N] {Ω : N → Type u_2} [inst_2 : (n : N) → MeasurableSpace (Ω n)] (M : Causalean.SCM N Ω) (R : Finset (Causalean.SWIGNode N)) (hR : M.isAncestrallyClosedSCM R), (M.induce R hR).unobserved ⊆ M.unobserved","l":185,"f":"Causalean/SCM/Model/Induced.lean"},{"n":"Causalean.SCM.instFintypeSwigΩ","k":"instance","m":"Causalean.SCM.ID.Density.FiniteReference","a":"SCM/ID/Density","t":2,"d":"Every random or fixed SWIG-node value space is finite when every base-node value space is finite.","s":"{N : Type u_1} → {Ω : N → Type u_2} → [(n : N) → Fintype (Ω n)] → (sn : Causalean.SWIGNode N) → Fintype (Causalean.swigΩ Ω sn)","l":26,"f":"Causalean/SCM/ID/Density/FiniteReference.lean"},{"n":"Causalean.SCM.instFintypeValuesOnSwigΩ","k":"instance","m":"Causalean.SCM.ID.Density.PiUnion","a":"SCM/ID/Density","t":2,"d":"Value assignments on finite SWIG-node sets are finite when each base node value space is finite.","s":"{N : Type u_1} → [DecidableEq N] → {Ω : N → Type u_2} → [(n : N) → Fintype (Ω n)] → (I : Finset (Causalean.SWIGNode N)) → Fintype (Causalean.ValuesOn I (Causalean.swigΩ Ω))","l":32,"f":"Causalean/SCM/ID/Density/PiUnion.lean"},{"n":"Causalean.SCM.instIsFiniteKernelAdjustmentKernelY","k":"instance","m":"Causalean.SCM.ID.Backdoor","a":"SCM/ID","t":2,"d":"The treatment-indexed backdoor-adjustment Y-marginal kernel is finite.","s":"∀ {N : Type u_1} [inst : DecidableEq N] [inst_1 : Fintype N] {Ω : N → Type u_2} [inst_2 : (n : N) → MeasurableSpace (Ω n)] [inst_3 : ∀ (n : Causalean.SWIGNode N), StandardBorelSpace (Causalean.swigΩ Ω n)] [inst_4 : ∀ (n : Causalean.SWIGNode N), Nonempty (Causalean.swigΩ Ω n)] (M : Causalean.SCM N Ω) (X : Finset N) (hObs : ∀ D ∈ X, Causalean.SWIGNode.random D ∈ M.observed) (hFix : ∀ D ∈ X, Causalean.SWIGNode.fixed D ∉ M.fixed) (Y Z : Finset (Causalean.SWIGNode N)) (hY : Y ⊆ M.observed) (hZ : Z ⊆ M.observed) (s0 : M.FixedValues), ProbabilityTheory.IsFiniteKernel (M.adjustmentKernelY X hObs hFix Y Z hY hZ s0)","l":118,"f":"Causalean/SCM/ID/Backdoor.lean"},{"n":"Causalean.SCM.instIsFiniteKernelBackdoorAdjustment","k":"instance","m":"Causalean.SCM.ID.Adjustment","a":"SCM/ID","t":2,"d":"The backdoor-adjustment functional is a finite kernel (a compProd of finite kernels, pushed through Prod.snd).","s":"∀ {N : Type u_1} [inst : DecidableEq N] [inst_1 : Fintype N] {Ω : N → Type u_2} [inst_2 : (n : N) → MeasurableSpace (Ω n)] [inst_3 : ∀ (n : Causalean.SWIGNode N), StandardBorelSpace (Causalean.swigΩ Ω n)] [inst_4 : ∀ (n : Causalean.SWIGNode N), Nonempty (Causalean.swigΩ Ω n)] (M : Causalean.SCM N Ω) (X : Finset N) (hX_obs : ∀ D ∈ X, Causalean.SWIGNode.random D ∈ M.observed) (hX_fixed : ∀ D ∈ X, Causalean.SWIGNode.fixed D ∉ M.fixed) (Y Z : Finset (Causalean.SWIGNode N)) (hY : Y ⊆ M.observed) (hZ : Z ⊆ M.observed), ProbabilityTheory.IsFiniteKernel (M.backdoorAdjustment X hX_obs hX_fixed Y Z hY hZ)","l":130,"f":"Causalean/SCM/ID/Adjustment.lean"},{"n":"Causalean.SCM.instIsFiniteKernelDoKernelY","k":"instance","m":"Causalean.SCM.ID.Backdoor","a":"SCM/ID","t":2,"d":"The treatment-indexed post-intervention Y-marginal kernel is finite.","s":"∀ {N : Type u_1} [inst : DecidableEq N] [inst_1 : Fintype N] {Ω : N → Type u_2} [inst_2 : (n : N) → MeasurableSpace (Ω n)] (M : Causalean.SCM N Ω) (X : Finset N) (hObs : ∀ D ∈ X, Causalean.SWIGNode.random D ∈ M.observed) (hFix : ∀ D ∈ X, Causalean.SWIGNode.fixed D ∉ M.fixed) (Y : Finset (Causalean.SWIGNode N)) (hY : Y ⊆ M.observed) (s0 : M.FixedValues), ProbabilityTheory.IsFiniteKernel (M.doKernelY X hObs hFix Y hY s0)","l":110,"f":"Causalean/SCM/ID/Backdoor.lean"},{"n":"Causalean.SCM.instIsFiniteKernelFrontdoorAdjustment","k":"instance","m":"Causalean.SCM.ID.Adjustment","a":"SCM/ID","t":2,"d":"The frontdoor-adjustment functional is a finite kernel (nested compProds of finite kernels, pushed through Prod.snd).","s":"∀ {N : Type u_1} [inst : DecidableEq N] [inst_1 : Fintype N] {Ω : N → Type u_2} [inst_2 : (n : N) → MeasurableSpace (Ω n)] [inst_3 : ∀ (n : Causalean.SWIGNode N), StandardBorelSpace (Causalean.swigΩ Ω n)] [inst_4 : ∀ (n : Causalean.SWIGNode N), Nonempty (Causalean.swigΩ Ω n)] (M : Causalean.SCM N Ω) (X : Finset N) (hX_obs : ∀ D ∈ X, Causalean.SWIGNode.random D ∈ M.observed) (hX_fixed : ∀ D ∈ X, Causalean.SWIGNode.fixed D ∉ M.fixed) (Y Z : Finset (Causalean.SWIGNode N)) (hY : Y ⊆ M.observed) (hZ : Z ⊆ M.observed), ProbabilityTheory.IsFiniteKernel (M.frontdoorAdjustment X hX_obs hX_fixed Y Z hY hZ)","l":249,"f":"Causalean/SCM/ID/Adjustment.lean"},{"n":"Causalean.SCM.instIsFiniteKernelFrontdoorKernelY","k":"instance","m":"Causalean.SCM.ID.Frontdoor","a":"SCM/ID","t":2,"d":"The treatment-indexed frontdoor-adjustment Y-marginal kernel is finite.","s":"∀ {N : Type u_1} [inst : DecidableEq N] [inst_1 : Fintype N] {Ω : N → Type u_2} [inst_2 : (n : N) → MeasurableSpace (Ω n)] [inst_3 : ∀ (n : Causalean.SWIGNode N), StandardBorelSpace (Causalean.swigΩ Ω n)] [inst_4 : ∀ (n : Causalean.SWIGNode N), Nonempty (Causalean.swigΩ Ω n)] (M : Causalean.SCM N Ω) (X : Finset N) (hObs : ∀ D ∈ X, Causalean.SWIGNode.random D ∈ M.observed) (hFix : ∀ D ∈ X, Causalean.SWIGNode.fixed D ∉ M.fixed) (Y W : Finset (Causalean.SWIGNode N)) (hY : Y ⊆ M.observed) (hW : W ⊆ M.observed) (s0 : M.FixedValues), ProbabilityTheory.IsFiniteKernel (M.frontdoorKernelY X hObs hFix Y W hY hW s0)","l":147,"f":"Causalean/SCM/ID/Frontdoor.lean"},{"n":"Causalean.SCM.instIsFiniteMeasure_jointRef","k":"instance","m":"Causalean.SCM.ID.Density.FiniteReference","a":"SCM/ID/Density","t":2,"d":"The finite product reference measure is finite on finite measurable-singleton node spaces.","s":"∀ {N : Type u_1} {Ω : N → Type u_2} [inst : (n : N) → MeasurableSpace (Ω n)] [∀ (n : N), Finite (Ω n)] [∀ (n : N), MeasurableSingletonClass (Ω n)] (ref : Causalean.SCM.ReferenceMeasures Ω) (I : Finset (Causalean.SWIGNode N)), MeasureTheory.IsFiniteMeasure (Causalean.SCM.jointRef ref I)","l":64,"f":"Causalean/SCM/ID/Density/FiniteReference.lean"},{"n":"Causalean.SCM.instIsFiniteMeasure_refMu","k":"instance","m":"Causalean.SCM.ID.Density.FiniteReference","a":"SCM/ID/Density","t":2,"d":"Each coordinate reference measure is finite on finite measurable-singleton node spaces.","s":"∀ {N : Type u_1} {Ω : N → Type u_2} [inst : (n : N) → MeasurableSpace (Ω n)] [∀ (n : N), Finite (Ω n)] [∀ (n : N), MeasurableSingletonClass (Ω n)] (ref : Causalean.SCM.ReferenceMeasures Ω) (v : Causalean.SWIGNode N), MeasureTheory.IsFiniteMeasure (ref.μ v)","l":54,"f":"Causalean/SCM/ID/Density/FiniteReference.lean"},{"n":"Causalean.SCM.instIsMarkovKernelJointKernel","k":"instance","m":"Causalean.SCM.Model.Kernel","a":"SCM/Model","t":2,"d":"The joint kernel is a Markov kernel.","s":"∀ {N : Type u_1} [inst : DecidableEq N] [inst_1 : Fintype N] {Ω : N → Type u_2} [inst_2 : (n : N) → MeasurableSpace (Ω n)] (M : Causalean.SCM N Ω), ProbabilityTheory.IsMarkovKernel M.jointKernel","l":154,"f":"Causalean/SCM/Model/Kernel.lean"},{"n":"Causalean.SCM.instIsMarkovKernelObsCondKernel","k":"instance","m":"Causalean.SCM.ID.Adjustment","a":"SCM/ID","t":2,"d":"The observational conditional kernel obsCondKernel Y CC is a Markov kernel (its values are probability measures), inherited from Mathlib's condKernel.","s":"∀ {N : Type u_1} [inst : DecidableEq N] [inst_1 : Fintype N] {Ω : N → Type u_2} [inst_2 : (n : N) → MeasurableSpace (Ω n)] [inst_3 : ∀ (n : Causalean.SWIGNode N), StandardBorelSpace (Causalean.swigΩ Ω n)] [inst_4 : ∀ (n : Causalean.SWIGNode N), Nonempty (Causalean.swigΩ Ω n)] (M : Causalean.SCM N Ω) (Y CC : Finset (Causalean.SWIGNode N)) (hY : Y ⊆ M.observed) (hCC : CC ⊆ M.observed), ProbabilityTheory.IsMarkovKernel (M.obsCondKernel Y CC hY hCC)","l":61,"f":"Causalean/SCM/ID/Adjustment.lean"},{"n":"Causalean.SCM.instIsMarkovKernelObsKernel","k":"instance","m":"Causalean.SCM.Model.Kernel","a":"SCM/Model","t":2,"d":"The observational kernel is a Markov kernel.","s":"∀ {N : Type u_1} [inst : DecidableEq N] [inst_1 : Fintype N] {Ω : N → Type u_2} [inst_2 : (n : N) → MeasurableSpace (Ω n)] (M : Causalean.SCM N Ω), ProbabilityTheory.IsMarkovKernel M.obsKernel","l":160,"f":"Causalean/SCM/Model/Kernel.lean"},{"n":"Causalean.SCM.instMeasurableSingletonClassSwigΩ","k":"instance","m":"Causalean.SCM.ID.Density.FiniteReference","a":"SCM/ID/Density","t":2,"d":"Every random or fixed SWIG-node value space has measurable singletons when every base-node value space has measurable singletons.","s":"∀ {N : Type u_1} {Ω : N → Type u_2} [inst : (n : N) → MeasurableSpace (Ω n)] [∀ (n : N), MeasurableSingletonClass (Ω n)] (sn : Causalean.SWIGNode N), MeasurableSingletonClass (Causalean.swigΩ Ω sn)","l":33,"f":"Causalean/SCM/ID/Density/FiniteReference.lean"},{"n":"Causalean.SCM.instMeasurableSpaceObservedPrefixValues","k":"instance","m":"Causalean.SCM.Factored.PrefixState","a":"SCM/Factored","t":2,"d":"Measurable-space structure on ObservedPrefixValues, by the same recursion as the type itself.","s":"{N : Type uN} → [inst : DecidableEq N] → [inst_1 : Fintype N] → {Ω : N → Type uΩ} → [inst_2 : (n : N) → MeasurableSpace (Ω n)] → (M : Causalean.SCM N Ω) → {n : ℕ} → (hn : n ≤ M.observed.card) → MeasurableSpace (M.ObservedPrefixValues n hn)","l":41,"f":"Causalean/SCM/Factored/PrefixState.lean"},{"n":"Causalean.SCM.instProbabilityLatentProduct","k":"instance","m":"Causalean.SCM.Model.SCM","a":"SCM/Model","t":2,"d":"The latent product measure has total mass one.","s":"∀ {N : Type uN} [inst : DecidableEq N] [inst_1 : Fintype N] {Ω : N → Type uΩ} [inst_2 : (n : N) → MeasurableSpace (Ω n)] (M : Causalean.SCM N Ω), MeasureTheory.IsProbabilityMeasure M.latentProduct","l":178,"f":"Causalean/SCM/Model/SCM.lean"},{"n":"Causalean.SCM.instSetoidSCM","k":"instance","m":"Causalean.SCM.Model.SCM","a":"SCM/Model","t":2,"d":"Structural causal models form a setoid under structural equivalence.","s":"{N : Type uN} → [inst : DecidableEq N] → [inst_1 : Fintype N] → {Ω : N → Type uΩ} → [inst_2 : (n : N) → MeasurableSpace (Ω n)] → Setoid (Causalean.SCM N Ω)","l":299,"f":"Causalean/SCM/Model/SCM.lean"},{"n":"Causalean.SCM.instSigmaFiniteJointRef","k":"instance","m":"Causalean.SCM.ID.Density.ReferenceMeasure","a":"SCM/ID/Density","t":2,"d":"Finite products of sigma-finite coordinate reference measures are sigma-finite.","s":"∀ {N : Type u_1} {Ω : N → Type u_2} [inst : (n : N) → MeasurableSpace (Ω n)] (ref : Causalean.SCM.ReferenceMeasures Ω) (I : Finset (Causalean.SWIGNode N)), MeasureTheory.SigmaFinite (Causalean.SCM.jointRef ref I)","l":58,"f":"Causalean/SCM/ID/Density/ReferenceMeasure.lean"},{"n":"Causalean.SCM.isAncestorAvoiding_of_sinks","k":"theorem","m":"Causalean.SCM.ID.Toolkit.FrontdoorGraph","a":"SCM/ID/Toolkit","t":2,"d":"","s":"∀ {V : Type u_3} [inst : DecidableEq V] [inst_1 : Fintype V] (G : Causalean.DAG V) (C : Finset V), (∀ c ∈ C, ∀ (v : V), ¬G.edge c v) → ∀ {u v : V}, G.isAncestor u v → G.isAncestorAvoiding C u v","l":102,"f":"Causalean/SCM/ID/Toolkit/FrontdoorGraph.lean"},{"n":"Causalean.SCM.isAncestor_fixSet_of_isAncestor","k":"theorem","m":"Causalean.SCM.Do.CutsetDSep","a":"SCM/Do","t":2,"d":"A directed ancestry path in a causal graph remains after intervening on a set when every possible edge source along that path is not a random copy of an intervened variable. Thus the original ancestor remains an ancestor in the intervened graph.","s":"∀ {N : Type u_1} [inst : DecidableEq N] [inst_1 : Fintype N] {Ω : N → Type u_2} [inst_2 : (n : N) → MeasurableSpace (Ω n)] (M : Causalean.SCM N Ω) (Z : Finset N) (hZ_obs : ∀ D ∈ Z, Causalean.SWIGNode.random D ∈ M.observed) (hZ_fixed : ∀ D ∈ Z, Causalean.SWIGNode.fixed D ∉ M.fixed) {u v : Causalean.SWIGNode N}, M.dag.isAncestor u v → (∀ (s : Causalean.SWIGNode N), s = u ∨ M.dag.isAncestor u s → M.dag.isAncestor s v → ∀ D ∈ Z, s ≠ Causalean.SWIGNode.random D) → (M.fixSet Z hZ_obs hZ_fixed).dag.isAncestor u v","l":211,"f":"Causalean/SCM/Do/CutsetDSep.lean"},{"n":"Causalean.SCM.isAncestrallyClosedSCM","k":"def","m":"Causalean.SCM.Model.Induced","a":"SCM/Model","t":1,"d":"R ⊆ V ∪ S ∪ L is ancestrally closed in the SCM sense if it satisfies both clauses of def:scm-anc-closed:","s":"{N : Type u_1} → [inst : DecidableEq N] → [inst_1 : Fintype N] → {Ω : N → Type u_2} → [inst_2 : (n : N) → MeasurableSpace (Ω n)] → Causalean.SCM N Ω → Finset (Causalean.SWIGNode N) → Prop","l":67,"f":"Causalean/SCM/Model/Induced.lean"},{"n":"Causalean.SCM.isFiniteMeasure_of_finite_measurableSingleton","k":"theorem","m":"Causalean.SCM.ID.Density.FiniteReference","a":"SCM/ID/Density","t":2,"d":"A σ-finite measure on a finite measurable-singleton space is finite.","s":"∀ {α : Type u_3} [inst : MeasurableSpace α] [Finite α] [MeasurableSingletonClass α] (μ : MeasureTheory.Measure α) [MeasureTheory.SigmaFinite μ], MeasureTheory.IsFiniteMeasure μ","l":40,"f":"Causalean/SCM/ID/Density/FiniteReference.lean"},{"n":"Causalean.SCM.isGlobalIMap_dag_self","k":"theorem","m":"Causalean.Graph.MarkovEquiv.Distributional","a":"Graph/MarkovEquiv","t":1,"d":"The bridge, restated. For any structural causal model M and any point s of its fixed values, the joint distribution of M's random values under s is a global I-map of M's own DAG — this is exactly the global Markov property full_globalMarkov.","s":"∀ {N : Type uN} [inst : DecidableEq N] [inst_1 : Fintype N] {Ω : N → Type uΩ} [inst_2 : (n : N) → MeasurableSpace (Ω n)] (M : Causalean.SCM N Ω) [inst_3 : StandardBorelSpace M.RandomValues] [∀ (n : Causalean.SWIGNode N), StandardBorelSpace (Causalean.swigΩ Ω n)] [∀ (n : Causalean.SWIGNode N), Nonempty (Causalean.swigΩ Ω n)] [inst_6 : ∀ (s : M.FixedValues), MeasureTheory.IsFiniteMeasure (M.jointKernel s)] (s : M.FixedValues), Causalean.SCM.IsGlobalIMap M.dag M (M.jointKernel s)","l":76,"f":"Causalean/Graph/MarkovEquiv/Distributional.lean"},{"n":"Causalean.SCM.isMarkov_jointKernelPrefix","k":"instance","m":"Causalean.SCM.Factored.PrefixKernel","a":"SCM/Factored","t":2,"d":"The prefix kernel is Markov at every step.","s":"∀ {N : Type uN} [inst : DecidableEq N] [inst_1 : Fintype N] {Ω : N → Type uΩ} [inst_2 : (n : N) → MeasurableSpace (Ω n)] (M : Causalean.SCM N Ω) (n : ℕ) (hn : n ≤ M.observed.card), ProbabilityTheory.IsMarkovKernel (M.jointKernelPrefix n hn)","l":102,"f":"Causalean/SCM/Factored/PrefixKernel.lean"},{"n":"Causalean.SCM.isMarkov_jointKernelPrefixZero","k":"instance","m":"Causalean.SCM.Factored.PrefixKernel","a":"SCM/Factored","t":2,"d":"The base prefix kernel is Markov.","s":"∀ {N : Type uN} [inst : DecidableEq N] [inst_1 : Fintype N] {Ω : N → Type uΩ} [inst_2 : (n : N) → MeasurableSpace (Ω n)] (M : Causalean.SCM N Ω), ProbabilityTheory.IsMarkovKernel M.jointKernelPrefixZero","l":76,"f":"Causalean/SCM/Factored/PrefixKernel.lean"},{"n":"Causalean.SCM.isMarkov_latentKernelOnFixed","k":"instance","m":"Causalean.SCM.Factored.PrefixKernel","a":"SCM/Factored","t":2,"d":"The latent constant kernel is Markov.","s":"∀ {N : Type uN} [inst : DecidableEq N] [inst_1 : Fintype N] {Ω : N → Type uΩ} [inst_2 : (n : N) → MeasurableSpace (Ω n)] (M : Causalean.SCM N Ω), ProbabilityTheory.IsMarkovKernel M.latentKernelOnFixed","l":57,"f":"Causalean/SCM/Factored/PrefixKernel.lean"},{"n":"Causalean.SCM.isMarkov_obsChainKernel","k":"instance","m":"Causalean.SCM.Factored.ObsChainKernel","a":"SCM/Factored","t":2,"d":"The recursive observational chain kernel is Markov.","s":"∀ {N : Type u_1} [inst : DecidableEq N] [inst_1 : Fintype N] {Ω : N → Type u_2} [inst_2 : (n : N) → MeasurableSpace (Ω n)] (M : Causalean.SCM N Ω) [inst_3 : ∀ (k : ℕ) (hk : k < M.observed.card), StandardBorelSpace (Causalean.ValuesOn {↑(M.observedAt ⟨k, hk⟩)} (Causalean.swigΩ Ω))] [inst_4 : ∀ (k : ℕ) (hk : k < M.observed.card), Nonempty (Causalean.ValuesOn {↑(M.observedAt ⟨k, hk⟩)} (Causalean.swigΩ Ω))] [inst_5 : ∀ (k : Fin M.observed.card), MeasurableSpace.CountableOrCountablyGenerated M.FixedValues (Causalean.ValuesOn (M.prefixNodes ↑k) (Causalean.swigΩ Ω))] (n : ℕ) (hn : n ≤ M.observed.card), ProbabilityTheory.IsMarkovKernel (M.obsChainKernel n hn)","l":557,"f":"Causalean/SCM/Factored/ObsChainKernel.lean"},{"n":"Causalean.SCM.isMarkov_obsChainKernelZero","k":"instance","m":"Causalean.SCM.Factored.ObsChainKernel","a":"SCM/Factored","t":2,"d":"The base observational chain kernel is Markov.","s":"∀ {N : Type u_1} [inst : DecidableEq N] [inst_1 : Fintype N] {Ω : N → Type u_2} [inst_2 : (n : N) → MeasurableSpace (Ω n)] (M : Causalean.SCM N Ω), ProbabilityTheory.IsMarkovKernel M.obsChainKernelZero","l":401,"f":"Causalean/SCM/Factored/ObsChainKernel.lean"},{"n":"Causalean.SCM.isMarkov_obsStepCondKernel","k":"instance","m":"Causalean.SCM.Factored.ObsChainKernel","a":"SCM/Factored","t":2,"d":"The one-node observational conditional kernel is Markov.","s":"∀ {N : Type u_1} [inst : DecidableEq N] [inst_1 : Fintype N] {Ω : N → Type u_2} [inst_2 : (n : N) → MeasurableSpace (Ω n)] (M : Causalean.SCM N Ω) {n : ℕ} (hn : n < M.observed.card) [inst_3 : StandardBorelSpace (Causalean.ValuesOn {↑(M.observedAt ⟨n, hn⟩)} (Causalean.swigΩ Ω))] [inst_4 : Nonempty (Causalean.ValuesOn {↑(M.observedAt ⟨n, hn⟩)} (Causalean.swigΩ Ω))] [inst_5 : MeasurableSpace.CountableOrCountablyGenerated M.FixedValues (Causalean.ValuesOn (M.prefixNodes n) (Causalean.swigΩ Ω))], ProbabilityTheory.IsMarkovKernel (M.obsStepCondKernel hn)","l":275,"f":"Causalean/SCM/Factored/ObsChainKernel.lean"},{"n":"Causalean.SCM.isMarkov_stepKernel","k":"instance","m":"Causalean.SCM.Factored.StepKernel","a":"SCM/Factored","t":2,"d":"The step kernel is Markov (inherits from Kernel.deterministic).","s":"∀ {N : Type uN} [inst : DecidableEq N] [inst_1 : Fintype N] {Ω : N → Type uΩ} [inst_2 : (n : N) → MeasurableSpace (Ω n)] (M : Causalean.SCM N Ω) {n : ℕ} (hn : n + 1 ≤ M.observed.card), ProbabilityTheory.IsMarkovKernel (M.stepKernel hn)","l":87,"f":"Causalean/SCM/Factored/StepKernel.lean"},{"n":"Causalean.SCM.isStandard","k":"def","m":"Causalean.SCM.Model.SCM","a":"SCM/Model","t":1,"d":"A standard structural causal model has no fixed intervention variables.","s":"{N : Type uN} → [inst : DecidableEq N] → [inst_1 : Fintype N] → {Ω : N → Type uΩ} → [inst_2 : (n : N) → MeasurableSpace (Ω n)] → Causalean.SCM N Ω → Prop","l":128,"f":"Causalean/SCM/Model/SCM.lean"},{"n":"Causalean.SCM.jointKernel","k":"def","m":"Causalean.SCM.Model.Kernel","a":"SCM/Model","t":1,"d":"The joint kernel maps each fixed-value assignment to the law of the model evaluation under the latent product distribution.","s":"{N : Type u_1} → [inst : DecidableEq N] → [inst_1 : Fintype N] → {Ω : N → Type u_2} → [inst_2 : (n : N) → MeasurableSpace (Ω n)] → (M : Causalean.SCM N Ω) → ProbabilityTheory.Kernel M.FixedValues M.RandomValues","l":70,"f":"Causalean/SCM/Model/Kernel.lean"},{"n":"Causalean.SCM.jointKernelPrefix","k":"def","m":"Causalean.SCM.Factored.PrefixKernel","a":"SCM/Factored","t":1,"d":"The prefix kernel at level n: sends fixed values s to the joint law of the latent tuple together with the values of the first n observed nodes, built by sequential compProd with the deterministic stepKernels.","s":"{N : Type uN} → [inst : DecidableEq N] → [inst_1 : Fintype N] → {Ω : N → Type uΩ} → [inst_2 : (n : N) → MeasurableSpace (Ω n)] → (M : Causalean.SCM N Ω) → (n : ℕ) → (hn : n ≤ M.observed.card) → ProbabilityTheory.Kernel M.FixedValues (M.OrderedLatentPrefixValues n hn)","l":87,"f":"Causalean/SCM/Factored/PrefixKernel.lean"},{"n":"Causalean.SCM.jointKernelPrefixZero","k":"def","m":"Causalean.SCM.Factored.PrefixKernel","a":"SCM/Factored","t":1,"d":"Base case of the recursive prefix kernel: only the latent variables have been generated so far, paired with the trivial PUnit observed prefix.","s":"{N : Type uN} → [inst : DecidableEq N] → [inst_1 : Fintype N] → {Ω : N → Type uΩ} → [inst_2 : (n : N) → MeasurableSpace (Ω n)] → (M : Causalean.SCM N Ω) → ProbabilityTheory.Kernel M.FixedValues (M.OrderedLatentPrefixValues 0 (Nat.zero_le M.observed.card))","l":66,"f":"Causalean/SCM/Factored/PrefixKernel.lean"},{"n":"Causalean.SCM.jointKernelPrefix_apply_eq","k":"theorem","m":"Causalean.SCM.Factored.EvalMapCorrespond","a":"SCM/Factored","t":1,"d":"Main correspondence. For a structural causal model M, its length-n prefix kernel evaluated at a fixed assignment s equals the pushforward of the latent-value product measure through the deterministic partial evaluation map at s.","s":"∀ {N : Type uN} [inst : DecidableEq N] [inst_1 : Fintype N] {Ω : N → Type uΩ} [inst_2 : (n : N) → MeasurableSpace (Ω n)] (M : Causalean.SCM N Ω) (s : M.FixedValues) (n : ℕ) (hn : n ≤ M.observed.card), (M.jointKernelPrefix n hn) s = MeasureTheory.Measure.map (fun ℓ => M.partialEvalMap n hn s ℓ) M.latentProduct","l":316,"f":"Causalean/SCM/Factored/EvalMapCorrespond.lean"},{"n":"Causalean.SCM.jointKernel_apply_eq","k":"theorem","m":"Causalean.SCM.Model.Kernel","a":"SCM/Model","t":1,"d":"For a structural causal model M and a fixed-node assignment s, the joint kernel evaluated at s equals the pushforward of the latent-variable product measure through the evaluation map at s.","s":"∀ {N : Type u_1} [inst : DecidableEq N] [inst_1 : Fintype N] {Ω : N → Type u_2} [inst_2 : (n : N) → MeasurableSpace (Ω n)] (M : Causalean.SCM N Ω) (s : M.FixedValues), M.jointKernel s = MeasureTheory.Measure.map (fun ℓ => M.evalMap s ℓ) M.latentProduct","l":101,"f":"Causalean/SCM/Model/Kernel.lean"},{"n":"Causalean.SCM.jointKernel_eq_factored_kernel","k":"theorem","m":"Causalean.SCM.Factored.Factorization","a":"SCM/Factored","t":1,"d":"Kernel-level factorization. For a structural causal model M, its joint kernel equals the prefix kernel at full length, pushed through the reindexing map identifying the full prefix state with the random coordinates. Follows from the pointwise form via kernel extensionality.","s":"∀ {N : Type uN} [inst : DecidableEq N] [inst_1 : Fintype N] {Ω : N → Type uΩ} [inst_2 : (n : N) → MeasurableSpace (Ω n)] (M : Causalean.SCM N Ω), M.jointKernel = (M.jointKernelPrefix M.observed.card (le_refl M.observed.card)).map M.orderedLatentPrefixFullToRandom","l":209,"f":"Causalean/SCM/Factored/Factorization.lean"},{"n":"Causalean.SCM.jointKernel_factored","k":"theorem","m":"Causalean.SCM.Factored.Factorization","a":"SCM/Factored","t":1,"d":"Main factorization theorem (pointwise). For a structural causal model M, at each fixed assignment s, the joint kernel equals the pushforward of the full-length prefix kernel through the reindexing map identifying the full prefix state with the random coordinates.","s":"∀ {N : Type uN} [inst : DecidableEq N] [inst_1 : Fintype N] {Ω : N → Type uΩ} [inst_2 : (n : N) → MeasurableSpace (Ω n)] (M : Causalean.SCM N Ω) (s : M.FixedValues), M.jointKernel s = MeasureTheory.Measure.map M.orderedLatentPrefixFullToRandom ((M.jointKernelPrefix M.observed.card (le_refl M.observed.card)) s)","l":167,"f":"Causalean/SCM/Factored/Factorization.lean"},{"n":"Causalean.SCM.jointKernel_map_commute","k":"theorem","m":"Causalean.SCM.Model.Kernel","a":"SCM/Model","t":2,"d":"The observational kernel can be written as one pushforward that evaluates the model and then projects to observed coordinates.","s":"∀ {N : Type u_1} [inst : DecidableEq N] [inst_1 : Fintype N] {Ω : N → Type u_2} [inst_2 : (n : N) → MeasurableSpace (Ω n)] (M : Causalean.SCM N Ω), M.obsKernel = ((ProbabilityTheory.Kernel.const M.FixedValues M.latentProduct).compProd (ProbabilityTheory.Kernel.deterministic (Function.uncurry M.evalMap) (Causalean.SCM.evalMap_measurable M))).map (M.randomToObserved ∘ Prod.snd)","l":177,"f":"Causalean/SCM/Model/Kernel.lean"},{"n":"Causalean.SCM.jointRef","k":"def","m":"Causalean.SCM.ID.Density.ReferenceMeasure","a":"SCM/ID/Density","t":1,"d":"The joint reference measure is the finite product of the per-node reference measures over a node set.","s":"{N : Type u_1} → {Ω : N → Type u_2} → [inst : (n : N) → MeasurableSpace (Ω n)] → Causalean.SCM.ReferenceMeasures Ω → (I : Finset (Causalean.SWIGNode N)) → MeasureTheory.Measure (Causalean.ValuesOn I (Causalean.swigΩ Ω))","l":49,"f":"Causalean/SCM/ID/Density/ReferenceMeasure.lean"},{"n":"Causalean.SCM.jointRef_extendObsPrefix","k":"theorem","m":"Causalean.SCM.ID.Density.ChainRuleDensity","a":"SCM/ID/Density","t":2,"d":"The successor prefix extension carries the product of the old-prefix reference and the next-node reference to the successor-prefix reference.","s":"∀ {N : Type u_1} [inst : DecidableEq N] [inst_1 : Fintype N] {Ω : N → Type u_2} [inst_2 : (n : N) → MeasurableSpace (Ω n)] (M : Causalean.SCM N Ω) (ref : Causalean.SCM.ReferenceMeasures Ω) {k : ℕ} (hk : k < M.observed.card), MeasureTheory.Measure.map (M.extendObsPrefix hk) ((Causalean.SCM.jointRef ref (M.prefixNodes k)).prod (ref.μ ↑(M.observedAt ⟨k, hk⟩))) = Causalean.SCM.jointRef ref (M.prefixNodes (k + 1))","l":214,"f":"Causalean/SCM/ID/Density/ChainRuleDensity.lean"},{"n":"Causalean.SCM.jointRef_prefix_card_map","k":"theorem","m":"Causalean.SCM.ID.Density.ChainRuleDensity","a":"SCM/ID/Density","t":2,"d":"Transport the full-prefix reference measure to the observed-value reference.","s":"∀ {N : Type u_1} [inst : DecidableEq N] [inst_1 : Fintype N] {Ω : N → Type u_2} [inst_2 : (n : N) → MeasurableSpace (Ω n)] (M : Causalean.SCM N Ω) (ref : Causalean.SCM.ReferenceMeasures Ω), MeasureTheory.Measure.map (⇑(Causalean.valuesEquivOfEq (Causalean.SCM.prefixNodes_card M M.observed.card (le_refl M.observed.card)))) (Causalean.SCM.jointRef ref (M.prefixNodes M.observed.card)) = Causalean.SCM.jointRef ref M.observed","l":200,"f":"Causalean/SCM/ID/Density/ChainRuleDensity.lean"},{"n":"Causalean.SCM.jointRef_singleton_eq_prod","k":"theorem","m":"Causalean.SCM.ID.Density.MassBridge","a":"SCM/ID/Density","t":2,"d":"The singleton mass of a finite product reference is the product of the coordinate singleton masses.","s":"∀ {N : Type u_1} [DecidableEq N] [Fintype N] {Ω : N → Type u_2} [inst : (n : N) → MeasurableSpace (Ω n)] (ref : Causalean.SCM.ReferenceMeasures Ω) (I : Finset (Causalean.SWIGNode N)) (x : Causalean.ValuesOn I (Causalean.swigΩ Ω)), (Causalean.SCM.jointRef ref I) {x} = ∏ i, (ref.μ ↑i) {x i}","l":51,"f":"Causalean/SCM/ID/Density/MassBridge.lean"},{"n":"Causalean.SCM.jointRef_singleton_ne_zero","k":"theorem","m":"Causalean.SCM.ID.Density.FiniteReference","a":"SCM/ID/Density","t":2,"d":"A faithful reference family gives every point in a finite coordinate product nonzero joint reference mass.","s":"∀ {N : Type u_1} {Ω : N → Type u_2} [inst : (n : N) → MeasurableSpace (Ω n)] [∀ (n : N), MeasurableSingletonClass (Ω n)] (ref : Causalean.SCM.ReferenceMeasures Ω), Causalean.SCM.ReferenceFaithful ref → ∀ (I : Finset (Causalean.SWIGNode N)) (x : Causalean.ValuesOn I (Causalean.swigΩ Ω)), (Causalean.SCM.jointRef ref I) {x} ≠ 0","l":104,"f":"Causalean/SCM/ID/Density/FiniteReference.lean"},{"n":"Causalean.SCM.kernel_apply_univ","k":"theorem","m":"Causalean.SCM.Model.Kernel","a":"SCM/Model","t":2,"d":"Every slice of a Markov kernel has total mass one.","s":"∀ {α : Type u_3} {β : Type u_4} [inst : MeasurableSpace α] [inst_1 : MeasurableSpace β] (κ : ProbabilityTheory.Kernel α β) [ProbabilityTheory.IsMarkovKernel κ] (a : α), (κ a) Set.univ = 1","l":166,"f":"Causalean/SCM/Model/Kernel.lean"},{"n":"Causalean.SCM.latentAncestorsOfNode","k":"def","m":"Causalean.SCM.Model.EvalFactorization","a":"SCM/Model","t":1,"d":"The latent ancestors of a node are the unobserved nodes equal to that node or ancestral to it.","s":"{N : Type u_1} → [inst : DecidableEq N] → [inst_1 : Fintype N] → {Ω : N → Type u_2} → [inst_2 : (n : N) → MeasurableSpace (Ω n)] → Causalean.SCM N Ω → Causalean.SWIGNode N → Finset (Causalean.SWIGNode N)","l":151,"f":"Causalean/SCM/Model/EvalFactorization.lean"},{"n":"Causalean.SCM.latentAncestorsOfNode_subset","k":"theorem","m":"Causalean.SCM.Model.EvalFactorization","a":"SCM/Model","t":2,"d":"The latent-ancestor set is contained in the unobserved node set.","s":"∀ {N : Type u_1} [inst : DecidableEq N] [inst_1 : Fintype N] {Ω : N → Type u_2} [inst_2 : (n : N) → MeasurableSpace (Ω n)] (M : Causalean.SCM N Ω) (v : Causalean.SWIGNode N), M.latentAncestorsOfNode v ⊆ M.unobserved","l":200,"f":"Causalean/SCM/Model/EvalFactorization.lean"},{"n":"Causalean.SCM.latentAncestorsOfSet","k":"def","m":"Causalean.SCM.Do.GlobalMarkov","a":"SCM/Do","t":1,"d":"The latent ancestors of a node set are the unobserved nodes that either belong to the set or are ancestors of one of its nodes.","s":"{N : Type uN} → [inst : DecidableEq N] → [inst_1 : Fintype N] → {Ω : N → Type uΩ} → [inst_2 : (n : N) → MeasurableSpace (Ω n)] → Causalean.SCM N Ω → Finset (Causalean.SWIGNode N) → Finset (Causalean.SWIGNode N)","l":38,"f":"Causalean/SCM/Do/GlobalMarkov.lean"},{"n":"Causalean.SCM.latentBlock","k":"def","m":"Causalean.SCM.ID.Density.LatentBlocks","a":"SCM/ID/Density","t":1,"d":"The latent variables with an edge into a candidate observed c-component.","s":"{N : Type u_1} → [inst : DecidableEq N] → [inst_1 : Fintype N] → {Ω : N → Type u_2} → [inst_2 : (n : N) → MeasurableSpace (Ω n)] → Causalean.SCM N Ω → Finset (Causalean.SWIGNode N) → Finset (Causalean.SWIGNode N)","l":36,"f":"Causalean/SCM/ID/Density/LatentBlocks.lean"},{"n":"Causalean.SCM.latentBlock_pairwise_disjoint","k":"theorem","m":"Causalean.SCM.ID.Density.LatentBlocks","a":"SCM/ID/Density","t":1,"d":"If C is a full c-component of the causal model's SWIG graph and C' is likewise a full c-component, and C and C' are distinct, then their latent-parent blocks — the unobserved nodes with an edge into the component — are disjoint.","s":"∀ {N : Type u_1} [inst : DecidableEq N] [inst_1 : Fintype N] {Ω : N → Type u_2} [inst_2 : (n : N) → MeasurableSpace (Ω n)] (M : Causalean.SCM N Ω) {C C' : Finset (Causalean.SWIGNode N)}, C ∈ M.cComponentSet → C' ∈ M.cComponentSet → C ≠ C' → Disjoint (M.latentBlock C) (M.latentBlock C')","l":42,"f":"Causalean/SCM/ID/Density/LatentBlocks.lean"},{"n":"Causalean.SCM.latentBlock_pairwise_disjoint_fixSet_induce_components","k":"theorem","m":"Causalean.SCM.ID.Density.LatentBlocks","a":"SCM/ID/Density","t":2,"d":"Do-model specialization of induced-component latent-block disjointness.","s":"∀ {N : Type u_1} [inst : DecidableEq N] [inst_1 : Fintype N] {Ω : N → Type u_2} [inst_2 : (n : N) → MeasurableSpace (Ω n)] (M : Causalean.SCM N Ω) (X : Finset N) (hObs : ∀ D ∈ X, Causalean.SWIGNode.random D ∈ M.observed) (hFix : ∀ D ∈ X, Causalean.SWIGNode.fixed D ∉ M.fixed) (R : Finset (Causalean.SWIGNode N)) {C C' : Finset (Causalean.SWIGNode N)}, C ∈ ((M.fixSet X hObs hFix).induce R).cComponentSet → C' ∈ ((M.fixSet X hObs hFix).induce R).cComponentSet → C ≠ C' → Disjoint ((M.fixSet X hObs hFix).latentBlock C) ((M.fixSet X hObs hFix).latentBlock C')","l":98,"f":"Causalean/SCM/ID/Density/LatentBlocks.lean"},{"n":"Causalean.SCM.latentBlock_pairwise_disjoint_induce_components","k":"theorem","m":"Causalean.SCM.ID.Density.LatentBlocks","a":"SCM/ID/Density","t":2,"d":"Distinct c-components of an induced SWIG have disjoint latent-parent blocks in the ambient SCM.","s":"∀ {N : Type u_1} [inst : DecidableEq N] [inst_1 : Fintype N] {Ω : N → Type u_2} [inst_2 : (n : N) → MeasurableSpace (Ω n)] (M : Causalean.SCM N Ω) (R : Finset (Causalean.SWIGNode N)) {C C' : Finset (Causalean.SWIGNode N)}, C ∈ (M.induce R).cComponentSet → C' ∈ (M.induce R).cComponentSet → C ≠ C' → Disjoint (M.latentBlock C) (M.latentBlock C')","l":64,"f":"Causalean/SCM/ID/Density/LatentBlocks.lean"},{"n":"Causalean.SCM.latentKernelOnFixed","k":"def","m":"Causalean.SCM.Factored.PrefixKernel","a":"SCM/Factored","t":1,"d":"The latent product law as a kernel indexed by fixed values. Ignores the fixed input and always returns the product distribution latentProduct. Shorthand for readability of the recursion below.","s":"{N : Type uN} → [inst : DecidableEq N] → [inst_1 : Fintype N] → {Ω : N → Type uΩ} → [inst_2 : (n : N) → MeasurableSpace (Ω n)] → (M : Causalean.SCM N Ω) → ProbabilityTheory.Kernel M.FixedValues M.LatentValues","l":50,"f":"Causalean/SCM/Factored/PrefixKernel.lean"},{"n":"Causalean.SCM.latentProduct","k":"def","m":"Causalean.SCM.Model.SCM","a":"SCM/Model","t":1,"d":"The latent product is the joint distribution over all latent roots formed from their individual laws.","s":"{N : Type uN} → [inst : DecidableEq N] → [inst_1 : Fintype N] → {Ω : N → Type uΩ} → [inst_2 : (n : N) → MeasurableSpace (Ω n)] → (M : Causalean.SCM N Ω) → MeasureTheory.Measure M.LatentValues","l":165,"f":"Causalean/SCM/Model/SCM.lean"},{"n":"Causalean.SCM.latentProduct_singleton_eq_prod","k":"theorem","m":"Causalean.SCM.ID.Density.QMass","a":"SCM/ID/Density","t":2,"d":"The latent-product mass of a singleton latent assignment equals the product of the singleton masses assigned by the latent distributions at every unobserved node.","s":"∀ {N : Type u_1} [inst : DecidableEq N] [inst_1 : Fintype N] {Ω : N → Type u_2} [inst_2 : (n : N) → MeasurableSpace (Ω n)] [(n : N) → Fintype (Ω n)] [∀ (n : N), MeasurableSingletonClass (Ω n)] (M : Causalean.SCM N Ω) (ℓ₀ : M.LatentValues), M.latentProduct {ℓ₀} = ∏ u, (M.latentDist u) {ℓ₀ u}","l":637,"f":"Causalean/SCM/ID/Density/QMass.lean"},{"n":"Causalean.SCM.localConsistent","k":"def","m":"Causalean.SCM.ID.Density.LatentBlocks","a":"SCM/ID/Density","t":1,"d":"v's mechanism, parents read locally from x/s/ℓ, reproduces x v.","s":"{N : Type u_1} → [inst : DecidableEq N] → [inst_1 : Fintype N] → {Ω : N → Type u_2} → [inst_2 : (n : N) → MeasurableSpace (Ω n)] → (M : Causalean.SCM N Ω) → M.FixedValues → Causalean.ValuesOn M.observed (Causalean.swigΩ Ω) → (v : Causalean.SWIGNode N) → v ∈ M.observed → M.LatentValues → Prop","l":113,"f":"Causalean/SCM/ID/Density/LatentBlocks.lean"},{"n":"Causalean.SCM.localConsistent_depends_only_on_block","k":"theorem","m":"Causalean.SCM.ID.Density.LatentBlocks","a":"SCM/ID/Density","t":2,"d":"Local consistency at v depends only on the latent block of v's c-component.","s":"∀ {N : Type u_1} [inst : DecidableEq N] [inst_1 : Fintype N] {Ω : N → Type u_2} [inst_2 : (n : N) → MeasurableSpace (Ω n)] (M : Causalean.SCM N Ω) (s : M.FixedValues) (x : Causalean.ValuesOn M.observed (Causalean.swigΩ Ω)) (v : Causalean.SWIGNode N) (hv : v ∈ M.observed) (ℓ ℓ' : M.LatentValues), (∀ (u : Causalean.SWIGNode N) (hu : u ∈ M.latentBlock (M.cComponentOf v)), ℓ ⟨u, (Finset.mem_filter.mp hu).left⟩ = ℓ' ⟨u, (Finset.mem_filter.mp hu).left⟩) → (M.localConsistent s x v hv ℓ ↔ M.localConsistent s x v hv ℓ')","l":124,"f":"Causalean/SCM/ID/Density/LatentBlocks.lean"},{"n":"Causalean.SCM.localConsistent_depends_only_on_latentBlock_of_mem","k":"theorem","m":"Causalean.SCM.ID.Density.QMass","a":"SCM/ID/Density","t":2,"d":"Local consistency at an observed node in a node set is unchanged when two latent assignments agree on every unobserved parent of a node in that set.","s":"∀ {N : Type u_1} [inst : DecidableEq N] [inst_1 : Fintype N] {Ω : N → Type u_2} [inst_2 : (n : N) → MeasurableSpace (Ω n)] (M : Causalean.SCM N Ω) (s : M.FixedValues) (x : Causalean.ValuesOn M.observed (Causalean.swigΩ Ω)) {U : Finset (Causalean.SWIGNode N)} {v : Causalean.SWIGNode N}, v ∈ U → ∀ (hv : v ∈ M.observed) (ℓ ℓ' : M.LatentValues), (∀ (u : Causalean.SWIGNode N) (hu : u ∈ M.latentBlock U), ℓ ⟨u, (Finset.mem_filter.mp hu).left⟩ = ℓ' ⟨u, (Finset.mem_filter.mp hu).left⟩) → (M.localConsistent s x v hv ℓ ↔ M.localConsistent s x v hv ℓ')","l":876,"f":"Causalean/SCM/ID/Density/QMass.lean"},{"n":"Causalean.SCM.localConsistent_event_eq_component_biInter","k":"theorem","m":"Causalean.SCM.ID.Density.QMass","a":"SCM/ID/Density","t":2,"d":"For an observed-parent-closed node set, local consistency at all of its nodes is equivalent to local consistency within each of its confounded components.","s":"∀ {N : Type u_1} [inst : DecidableEq N] [inst_1 : Fintype N] {Ω : N → Type u_2} [inst_2 : (n : N) → MeasurableSpace (Ω n)] (M : Causalean.SCM N Ω) (s : M.FixedValues) (P : Finset (Causalean.SWIGNode N)) (hP : M.ObsParentClosed P) (x : Causalean.ValuesOn M.observed (Causalean.swigΩ Ω)), {ℓ | ∀ (v : Causalean.SWIGNode N) (hv : v ∈ P), M.localConsistent s x v (hP.left hv) ℓ} = ⋂ C ∈ M.cComponentSet, {ℓ | ∀ (v : Causalean.SWIGNode N) (hv : v ∈ C ∩ P), M.localConsistent s x v (hP.left (Finset.mem_of_mem_inter_right hv)) ℓ}","l":1045,"f":"Causalean/SCM/ID/Density/QMass.lean"},{"n":"Causalean.SCM.localConsistent_event_eq_family_biInter","k":"theorem","m":"Causalean.SCM.ID.Density.QMass","a":"SCM/ID/Density","t":2,"d":"For a finite family of observed node sets, local consistency over their union is equivalent to local consistency over every member of the family.","s":"∀ {N : Type u_1} [inst : DecidableEq N] [inst_1 : Fintype N] {Ω : N → Type u_2} [inst_2 : (n : N) → MeasurableSpace (Ω n)] (M : Causalean.SCM N Ω) (s : M.FixedValues) (𝒞 : Finset (Finset (Causalean.SWIGNode N))) (h𝒞obs : ∀ U ∈ 𝒞, U ⊆ M.observed) (hSup : 𝒞.sup id ⊆ M.observed) (x : Causalean.ValuesOn M.observed (Causalean.swigΩ Ω)), {ℓ | ∀ (v : Causalean.SWIGNode N) (hv : v ∈ 𝒞.sup id), M.localConsistent s x v (hSup hv) ℓ} = ⋂ U ∈ 𝒞, if hU : U ∈ 𝒞 then {ℓ | ∀ (v : Causalean.SWIGNode N) (hv : v ∈ U), M.localConsistent s x v (h𝒞obs U hU hv) ℓ} else Set.univ","l":1078,"f":"Causalean/SCM/ID/Density/QMass.lean"},{"n":"Causalean.SCM.localConsistent_fixSet_iff","k":"theorem","m":"Causalean.SCM.ID.Density.LatentBlocks","a":"SCM/ID/Density","t":1,"d":"Consider an intervention on a node set W whose random copies are all observed and whose fixed copies are not already fixed in the base model, giving the intervened model M.fixSet W. Fix a node v that is not itself one of the intervened random copies and that is observed both in the intervened model and in the base model. If the intervened model's observed assignment x' agrees with the base assignment x on every base-observed coordinate, the base assignment records, at each intervened node, the same value that the intervened model's fixed values assign to the corresponding fixed coordinate, and the intervened model's fixed values sW project, via fixSetProj, onto the base fixed values s, then local consistency of the structural evaluation at v in the intervened model, under sW, x', and a latent realization ℓ, is equivalent to local consistency at v in the base model, under s, x, and the same ℓ.","s":"∀ {N : Type u_1} [inst : DecidableEq N] [inst_1 : Fintype N] {Ω : N → Type u_2} [inst_2 : (n : N) → MeasurableSpace (Ω n)] (M : Causalean.SCM N Ω) (W : Finset N) (hObs : ∀ D ∈ W, Causalean.SWIGNode.random D ∈ M.observed) (hFix : ∀ D ∈ W, Causalean.SWIGNode.fixed D ∉ M.fixed) (sW : (M.fixSet W hObs hFix).FixedValues) (s : M.FixedValues) (x' : Causalean.ValuesOn (M.fixSet W hObs hFix).observed (Causalean.swigΩ Ω)) (x : Causalean.ValuesOn M.observed (Causalean.swigΩ Ω)) (v : Causalean.SWIGNode N) (hv' : v ∈ (M.fixSet W hObs hFix).observed) (hv : v ∈ M.observed), v ∉ Finset.image Causalean.SWIGNode.random W → (∀ (w : Causalean.SWIGNode N) (hw : w ∈ M.observed), x' ⟨w, id hw⟩ = x ⟨w, hw⟩) → (∀ (D : N) (hD : D ∈ W), x ⟨Causalean.SWIGNode.random D, hObs D hD⟩ = sW ⟨Causalean.SWIGNode.fixed D, Finset.mem_union_right M.fixed (Finset.mem_image.mpr (Exists.intro D ⟨hD, rfl⟩))⟩) → M.fixSetProj W hObs hFix sW = s → ∀ (ℓ : M.LatentValues), (M.fixSet W hObs hFix).localConsistent sW x' v hv' ℓ ↔ M.localConsistent s x v hv ℓ","l":497,"f":"Causalean/SCM/ID/Density/LatentBlocks.lean"},{"n":"Causalean.SCM.localConsistent_iff_structFun_dispatch","k":"theorem","m":"Causalean.SCM.ID.Density.LatentBlocks","a":"SCM/ID/Density","t":2,"d":"","s":"∀ {N : Type u_1} [inst : DecidableEq N] [inst_1 : Fintype N] {Ω : N → Type u_2} [inst_2 : (n : N) → MeasurableSpace (Ω n)] (M : Causalean.SCM N Ω) (s : M.FixedValues) (x : Causalean.ValuesOn M.observed (Causalean.swigΩ Ω)) (v : Causalean.SWIGNode N) (hv : v ∈ M.observed) (ℓ : M.LatentValues), M.localConsistent s x v hv ℓ ↔ (M.structFun ⟨v, hv⟩ fun w => if huo : ↑w ∈ M.unobserved then ℓ ⟨↑w, huo⟩ else if hfix : ↑w ∈ M.fixed then s ⟨↑w, hfix⟩ else have hedge := (Causalean.DAG.mem_parents M.dag).mp w.property; have hobs := Or.casesOn (Finset.mem_union.mp (M.dag_edges_classified (↑w) v hedge).left) (fun h1 => Or.casesOn (Finset.mem_union.mp h1) (fun hfx => absurd hfx hfix) fun hob => hob) fun h2 => absurd h2 huo; x ⟨↑w, hobs⟩) = x ⟨v, hv⟩","l":368,"f":"Causalean/SCM/ID/Density/LatentBlocks.lean"},{"n":"Causalean.SCM.map_pi_valuesEquivOfEq","k":"theorem","m":"Causalean.SCM.ID.Density.ChainRuleDensity","a":"SCM/ID/Density","t":2,"d":"Measure.pi reindexes along valuesEquivOfEq, with no probability-measure hypothesis (the index equality reduces the equiv to the identity). This is the σ-finite-friendly companion of measurePreserving_valuesEquivOfEq.","s":"∀ {M' : Type u_3} [DecidableEq M'] {I J : Finset M'} {Ω' : M' → Type u_4} [inst : (n : M') → MeasurableSpace (Ω' n)] (h : I = J) (μ : (i : ↥I) → MeasureTheory.Measure (Ω' ↑i)), MeasureTheory.Measure.map (⇑(Causalean.valuesEquivOfEq h)) (MeasureTheory.Measure.pi μ) = MeasureTheory.Measure.pi fun j => μ ⟨↑j, (fun h_1 => Eq.ndrec (motive := fun {J} => ∀ (j : ↥J), ↑j ∈ J → ↑j ∈ I) (fun j h => h) h j h_1) j.property⟩","l":156,"f":"Causalean/SCM/ID/Density/ChainRuleDensity.lean"},{"n":"Causalean.SCM.marginalizeOn","k":"def","m":"Causalean.SCM.ID.Density.IdentifyMass","a":"SCM/ID/Density","t":1,"d":"Marginalize a full-observed mass function by summing over the coordinates in W and overriding those coordinates in the evaluation point.","s":"{N : Type u_1} → [DecidableEq N] → {Ω : N → Type u_2} → [(n : N) → Fintype (Ω n)] → (O W : Finset (Causalean.SWIGNode N)) → W ⊆ O → (Causalean.ValuesOn O (Causalean.swigΩ Ω) → ENNReal) → Causalean.ValuesOn O (Causalean.swigΩ Ω) → ENNReal","l":111,"f":"Causalean/SCM/ID/Density/IdentifyMass.lean"},{"n":"Causalean.SCM.marginalizeOn_empty","k":"theorem","m":"Causalean.SCM.ID.Density.QMass","a":"SCM/ID/Density","t":2,"d":"Marginalizing a mass function over an empty set of observed coordinates leaves its value at every observed assignment unchanged.","s":"∀ {N : Type u_1} [inst : DecidableEq N] [inst_1 : Fintype N] {Ω : N → Type u_2} [inst_2 : (n : N) → MeasurableSpace (Ω n)] [inst_3 : (n : N) → Fintype (Ω n)] (M : Causalean.SCM N Ω) (q : Causalean.ValuesOn M.observed (Causalean.swigΩ Ω) → ENNReal) (x : Causalean.ValuesOn M.observed (Causalean.swigΩ Ω)), Causalean.SCM.marginalizeOn M.observed ∅ (of_eq_true (Finset.empty_subset._simp_1 M.observed)) q x = q x","l":161,"f":"Causalean/SCM/ID/Density/QMass.lean"},{"n":"Causalean.SCM.measurableEmbedding_fillZrW","k":"theorem","m":"Causalean.SCM.Do.Rule2Kernel.WMarginal","a":"SCM/Do/Rule2Kernel","t":2,"d":"fillZrW Z _ _ W s' is a MeasurableEmbedding when Z.image .random and W are disjoint.","s":"∀ {N : Type u_1} [inst : DecidableEq N] [inst_1 : Fintype N] {Ω : N → Type u_2} [inst_2 : (n : N) → MeasurableSpace (Ω n)] (M' : Causalean.SCM N Ω) (Z : Finset N) (hZ_obs : ∀ D ∈ Z, Causalean.SWIGNode.random D ∈ M'.observed) (hZ_fixed : ∀ D ∈ Z, Causalean.SWIGNode.fixed D ∉ M'.fixed) (W : Finset (Causalean.SWIGNode N)), Disjoint (Finset.image Causalean.SWIGNode.random Z) W → ∀ [MeasurableSingletonClass (Causalean.ValuesOn (Finset.image Causalean.SWIGNode.random Z ∪ W) (Causalean.swigΩ Ω))] (s' : (M'.fixSet Z hZ_obs hZ_fixed).FixedValues), MeasurableEmbedding (M'.fillZrW Z hZ_obs hZ_fixed W s')","l":206,"f":"Causalean/SCM/Do/Rule2Kernel/WMarginal.lean"},{"n":"Causalean.SCM.measurableSet_comap_piFinset_of_depends","k":"theorem","m":"Causalean.SCM.ID.Density.QMass","a":"SCM/ID/Density","t":2,"d":"A set of finite product outcomes whose membership depends only on a specified finite set of coordinates is measurable with respect to the σ-algebra on those coordinates.","s":"∀ {ι : Type u_3} [Fintype ι] {α : ι → Type u_4} [inst : (i : ι) → MeasurableSpace (α i)] [(i : ι) → Fintype (α i)] [∀ (i : ι), MeasurableSingletonClass (α i)] (S : Finset ι) (A : Set ((i : ι) → α i)), (∀ (ξ ξ' : (i : ι) → α i), (∀ i ∈ S, ξ i = ξ' i) → (ξ ∈ A ↔ ξ' ∈ A)) → MeasurableSet A","l":659,"f":"Causalean/SCM/ID/Density/QMass.lean"},{"n":"Causalean.SCM.measurableSet_fillZrW_image","k":"theorem","m":"Causalean.SCM.Do.Rule2Kernel.WMarginal","a":"SCM/Do/Rule2Kernel","t":2,"d":"A measurable W-event has a measurable image under fillZrW.","s":"∀ {N : Type u_1} [inst : DecidableEq N] [inst_1 : Fintype N] {Ω : N → Type u_2} [inst_2 : (n : N) → MeasurableSpace (Ω n)] (M' : Causalean.SCM N Ω) (Z : Finset N) (hZ_obs : ∀ D ∈ Z, Causalean.SWIGNode.random D ∈ M'.observed) (hZ_fixed : ∀ D ∈ Z, Causalean.SWIGNode.fixed D ∉ M'.fixed) (W : Finset (Causalean.SWIGNode N)), Disjoint (Finset.image Causalean.SWIGNode.random Z) W → ∀ [MeasurableSingletonClass (Causalean.ValuesOn (Finset.image Causalean.SWIGNode.random Z ∪ W) (Causalean.swigΩ Ω))] (s' : (M'.fixSet Z hZ_obs hZ_fixed).FixedValues) {A : Set (Causalean.ValuesOn W (Causalean.swigΩ Ω))}, MeasurableSet A → MeasurableSet (M'.fillZrW Z hZ_obs hZ_fixed W s' '' A)","l":135,"f":"Causalean/SCM/Do/Rule2Kernel/WMarginal.lean"},{"n":"Causalean.SCM.measurableSet_measure_eq","k":"theorem","m":"Causalean.SCM.Do.Rule2Kernel.WitnessBridge","a":"SCM/Do/Rule2Kernel","t":2,"d":"The agreement set of two measurable measure-valued maps is measurable.","s":"∀ {γ : Type u_3} {β : Type u_4} [inst : MeasurableSpace γ] [inst_1 : MeasurableSpace β] [MeasurableSpace.CountablyGenerated β] {f g : γ → MeasureTheory.Measure β}, Measurable f → Measurable g → (∀ (p : γ), MeasureTheory.IsFiniteMeasure (f p)) → MeasurableSet {p | f p = g p}","l":88,"f":"Causalean/SCM/Do/Rule2Kernel/WitnessBridge.lean"},{"n":"Causalean.SCM.measurable_cast_family","k":"theorem","m":"Causalean.Graph.SWIG","a":"Graph","t":2,"d":"Transporting a value along an equality of indices is measurable.","s":"∀ {I : Type u_1} {X : I → Type u_2} [inst : (i : I) → MeasurableSpace (X i)] {a b : I} (hab : a = b), Measurable (cast (congrArg X hab))","l":143,"f":"Causalean/Graph/SWIG.lean"},{"n":"Causalean.SCM.measurable_evalMap_overrideC","k":"theorem","m":"Causalean.SCM.Model.EvalOverrideC","a":"SCM/Model","t":2,"d":"The overridden evaluation map is jointly measurable in fixed values, override values, and the latent realization.","s":"∀ {N : Type u_1} [inst : DecidableEq N] [inst_1 : Fintype N] {Ω : N → Type u_2} [inst_2 : (n : N) → MeasurableSpace (Ω n)] (M : Causalean.SCM N Ω) {Y C : Finset (Causalean.SWIGNode N)} (hY : Y ⊆ M.observed) (hC : C ⊆ M.observed), Measurable fun p => M.evalMap_overrideC hY hC p.1.1 p.1.2 p.2","l":580,"f":"Causalean/SCM/Model/EvalOverrideC.lean"},{"n":"Causalean.SCM.measurable_extendObsPrefix","k":"theorem","m":"Causalean.SCM.Factored.ObsChainKernel","a":"SCM/Factored","t":2,"d":"Prefix extension is measurable.","s":"∀ {N : Type u_1} [inst : DecidableEq N] [inst_1 : Fintype N] {Ω : N → Type u_2} [inst_2 : (n : N) → MeasurableSpace (Ω n)] (M : Causalean.SCM N Ω) {n : ℕ} (hn : n < M.observed.card), Measurable (M.extendObsPrefix hn)","l":418,"f":"Causalean/SCM/Factored/ObsChainKernel.lean"},{"n":"Causalean.SCM.measurable_extendOrderedLatentPrefix","k":"theorem","m":"Causalean.SCM.Factored.PrefixState","a":"SCM/Factored","t":1,"d":"Fix a structural causal model M and a step index n such that there are at least n + 1 observed nodes, so n names the next node to be appended in the canonical topological order of observed nodes. Then the map that appends the freshly generated value of that node to a length-n prefix of previously observed values, together with the latent assignment, producing a length-(n + 1) prefix, is measurable.","s":"∀ {N : Type uN} [inst : DecidableEq N] [inst_1 : Fintype N] {Ω : N → Type uΩ} [inst_2 : (n : N) → MeasurableSpace (Ω n)] (M : Causalean.SCM N Ω) {n : ℕ} (hn : n + 1 ≤ M.observed.card), Measurable (M.extendOrderedLatentPrefix hn)","l":115,"f":"Causalean/SCM/Factored/PrefixState.lean"},{"n":"Causalean.SCM.measurable_family_cast","k":"theorem","m":"Causalean.Graph.SWIG","a":"Graph","t":2,"d":"A measurable function remains measurable after transporting its codomain index.","s":"∀ {I : Type u_1} {γ : Type u_2} {X : I → Type u_3} [inst : (i : I) → MeasurableSpace (X i)] [inst_1 : MeasurableSpace γ] {v w : I} (h : v = w) {f : γ → X v}, Measurable f → Measurable fun x => h ▸ f x","l":150,"f":"Causalean/Graph/SWIG.lean"},{"n":"Causalean.SCM.measurable_fillZrW","k":"theorem","m":"Causalean.SCM.Do.Rule2Kernel.Helpers","a":"SCM/Do/Rule2Kernel","t":2,"d":"The Rule 2 filler is measurable as a function of the free conditioning assignment.","s":"∀ {N : Type u_1} [inst : DecidableEq N] [inst_1 : Fintype N] {Ω : N → Type u_2} [inst_2 : (n : N) → MeasurableSpace (Ω n)] (M' : Causalean.SCM N Ω) (Z : Finset N) (hZ_obs : ∀ D ∈ Z, Causalean.SWIGNode.random D ∈ M'.observed) (hZ_fixed : ∀ D ∈ Z, Causalean.SWIGNode.fixed D ∉ M'.fixed) (W : Finset (Causalean.SWIGNode N)) (s' : (M'.fixSet Z hZ_obs hZ_fixed).FixedValues), Measurable (M'.fillZrW Z hZ_obs hZ_fixed W s')","l":154,"f":"Causalean/SCM/Do/Rule2Kernel/Helpers.lean"},{"n":"Causalean.SCM.measurable_fillZrW_prod","k":"theorem","m":"Causalean.SCM.Do.Rule2Kernel.Helpers","a":"SCM/Do/Rule2Kernel","t":2,"d":"The Rule 2 filler is jointly measurable in the post-intervention fixed slice and the free conditioning coordinates.","s":"∀ {N : Type u_1} [inst : DecidableEq N] [inst_1 : Fintype N] {Ω : N → Type u_2} [inst_2 : (n : N) → MeasurableSpace (Ω n)] (M' : Causalean.SCM N Ω) (Z : Finset N) (hZ_obs : ∀ D ∈ Z, Causalean.SWIGNode.random D ∈ M'.observed) (hZ_fixed : ∀ D ∈ Z, Causalean.SWIGNode.fixed D ∉ M'.fixed) (W : Finset (Causalean.SWIGNode N)), Measurable fun p => M'.fillZrW Z hZ_obs hZ_fixed W p.1 p.2","l":167,"f":"Causalean/SCM/Do/Rule2Kernel/Helpers.lean"},{"n":"Causalean.SCM.measurable_fixMonoParentMap","k":"theorem","m":"Causalean.SCM.Model.InterventionMono","a":"SCM/Model","t":2,"d":"The parent values used by a monolithic intervention depend measurably on the original parent values, so this reindexing can be used safely when constructing intervened structural equations and probability kernels.","s":"∀ {N : Type u_1} [inst : DecidableEq N] [inst_1 : Fintype N] {Ω : N → Type u_2} [inst_2 : (n : N) → MeasurableSpace (Ω n)] (G : Causalean.SWIGGraph N) (X : Finset N) (hObs : ∀ D ∈ X, Causalean.SWIGNode.random D ∈ G.observed) (hFix : ∀ D ∈ X, Causalean.SWIGNode.fixed D ∉ G.fixed) (v : Causalean.SWIGNode N), Measurable (Causalean.SCM.fixMonoParentMap G X hObs hFix v)","l":147,"f":"Causalean/SCM/Model/InterventionMono.lean"},{"n":"Causalean.SCM.measurable_fixSetExtend","k":"theorem","m":"Causalean.SCM.Do.Rule2Kernel.Helpers","a":"SCM/Do/Rule2Kernel","t":2,"d":"Extending a base fixed-value assignment is measurable in the treatment value when the base assignment is held fixed.","s":"∀ {N : Type u_1} [inst : DecidableEq N] [inst_1 : Fintype N] {Ω : N → Type u_2} [inst_2 : (n : N) → MeasurableSpace (Ω n)] (M : Causalean.SCM N Ω) (X : Finset N) (hObs : ∀ D ∈ X, Causalean.SWIGNode.random D ∈ M.observed) (hFix : ∀ D ∈ X, Causalean.SWIGNode.fixed D ∉ M.fixed) (s0 : M.FixedValues), Measurable (M.fixSetExtend X hObs hFix s0)","l":246,"f":"Causalean/SCM/Do/Rule2Kernel/Helpers.lean"},{"n":"Causalean.SCM.measurable_fixSetProj","k":"theorem","m":"Causalean.SCM.Model.InterventionSet","a":"SCM/Model","t":2,"d":"fixSetProj is measurable.","s":"∀ {N : Type u_1} [inst : DecidableEq N] [inst_1 : Fintype N] {Ω : N → Type u_2} [inst_2 : (n : N) → MeasurableSpace (Ω n)] (M : Causalean.SCM N Ω) (X : Finset N) (hObs : ∀ D ∈ X, Causalean.SWIGNode.random D ∈ M.observed) (hFix : ∀ D ∈ X, Causalean.SWIGNode.fixed D ∉ M.fixed), Measurable (M.fixSetProj X hObs hFix)","l":740,"f":"Causalean/SCM/Model/InterventionSet.lean"},{"n":"Causalean.SCM.measurable_fixSetZSlice","k":"theorem","m":"Causalean.SCM.Model.InterventionSet","a":"SCM/Model","t":2,"d":"fixSetZSlice is measurable.","s":"∀ {N : Type u_1} [inst : DecidableEq N] [inst_1 : Fintype N] {Ω : N → Type u_2} [inst_2 : (n : N) → MeasurableSpace (Ω n)] (M : Causalean.SCM N Ω) (X Z : Finset N) (hX_obs : ∀ D ∈ X, Causalean.SWIGNode.random D ∈ M.observed) (hX_fixed : ∀ D ∈ X, Causalean.SWIGNode.fixed D ∉ M.fixed) (hZ_obs : ∀ D ∈ Z, Causalean.SWIGNode.random D ∈ (M.fixSet X hX_obs hX_fixed).observed) (hZ_fixed : ∀ D ∈ Z, Causalean.SWIGNode.fixed D ∉ (M.fixSet X hX_obs hX_fixed).fixed), Measurable (M.fixSetZSlice X Z hX_obs hX_fixed hZ_obs hZ_fixed)","l":769,"f":"Causalean/SCM/Model/InterventionSet.lean"},{"n":"Causalean.SCM.measurable_observedPrefixValue","k":"theorem","m":"Causalean.SCM.Factored.PrefixState","a":"SCM/Factored","t":2,"d":"observedPrefixValue is measurable in its prefix-state argument.","s":"∀ {N : Type uN} [inst : DecidableEq N] [inst_1 : Fintype N] {Ω : N → Type uΩ} [inst_2 : (n : N) → MeasurableSpace (Ω n)] (M : Causalean.SCM N Ω) {n : ℕ} (hn : n ≤ M.observed.card) (i : Fin n), Measurable fun ξ => M.observedPrefixValue hn ξ i","l":78,"f":"Causalean/SCM/Factored/PrefixState.lean"},{"n":"Causalean.SCM.measurable_orderedLatentPrefixFullToRandom","k":"theorem","m":"Causalean.SCM.Factored.Factorization","a":"SCM/Factored","t":2,"d":"orderedLatentPrefixFullToRandom is measurable. Case-split mirrors the definition; the observed branch composes observedPrefixValue with a cast.","s":"∀ {N : Type uN} [inst : DecidableEq N] [inst_1 : Fintype N] {Ω : N → Type uΩ} [inst_2 : (n : N) → MeasurableSpace (Ω n)] (M : Causalean.SCM N Ω), Measurable M.orderedLatentPrefixFullToRandom","l":78,"f":"Causalean/SCM/Factored/Factorization.lean"},{"n":"Causalean.SCM.measurable_parentValuesFromPrefix","k":"theorem","m":"Causalean.SCM.Factored.ParentLookup","a":"SCM/Factored","t":1,"d":"Fix a structural causal model M and a step index n such that there are at least n + 1 observed nodes, so n names a valid position in the canonical topological order of observed nodes. Then the map parentValuesFromPrefix that reads off the parent values of the n-th observed node from a fixed-value assignment, a latent assignment, and the already-generated length-n prefix of observed values is jointly measurable in these three arguments.","s":"∀ {N : Type uN} [inst : DecidableEq N] [inst_1 : Fintype N] {Ω : N → Type uΩ} [inst_2 : (n : N) → MeasurableSpace (Ω n)] (M : Causalean.SCM N Ω) {n : ℕ} (hn : n + 1 ≤ M.observed.card), Measurable (M.parentValuesFromPrefix hn)","l":116,"f":"Causalean/SCM/Factored/ParentLookup.lean"},{"n":"Causalean.SCM.measurable_partialEvalMap","k":"theorem","m":"Causalean.SCM.Factored.EvalMapCorrespond","a":"SCM/Factored","t":2,"d":"partialEvalMap is jointly measurable in (s, ℓ). Proved by induction on n: the base case is a product of projections, and the step case composes the measurable stepFun, extendOrderedLatentPrefix, and the inductive hypothesis.","s":"∀ {N : Type uN} [inst : DecidableEq N] [inst_1 : Fintype N] {Ω : N → Type uΩ} [inst_2 : (n : N) → MeasurableSpace (Ω n)] (M : Causalean.SCM N Ω) (n : ℕ) (hn : n ≤ M.observed.card), Measurable fun sℓ => M.partialEvalMap n hn sℓ.1 sℓ.2","l":109,"f":"Causalean/SCM/Factored/EvalMapCorrespond.lean"},{"n":"Causalean.SCM.measurable_randomToObserved","k":"theorem","m":"Causalean.SCM.Model.Kernel","a":"SCM/Model","t":2,"d":"The observed-coordinate projection is measurable.","s":"∀ {N : Type u_1} [inst : DecidableEq N] [inst_1 : Fintype N] {Ω : N → Type u_2} [inst_2 : (n : N) → MeasurableSpace (Ω n)] (M : Causalean.SCM N Ω), Measurable M.randomToObserved","l":93,"f":"Causalean/SCM/Model/Kernel.lean"},{"n":"Causalean.SCM.measurable_singletonValue","k":"theorem","m":"Causalean.SCM.Factored.ObsChainKernel","a":"SCM/Factored","t":2,"d":"Reading a singleton value is measurable.","s":"∀ {ι : Type u_3} {α : ι → Type u_4} [inst : (i : ι) → MeasurableSpace (α i)] {v : ι}, Measurable Causalean.SCM.singletonValue","l":217,"f":"Causalean/SCM/Factored/ObsChainKernel.lean"},{"n":"Causalean.SCM.measurable_singletonValues","k":"theorem","m":"Causalean.SCM.Factored.ObsChainKernel","a":"SCM/Factored","t":2,"d":"Building a singleton tuple is measurable.","s":"∀ {ι : Type u_3} {α : ι → Type u_4} [inst : (i : ι) → MeasurableSpace (α i)] {v : ι}, Measurable Causalean.SCM.singletonValues","l":225,"f":"Causalean/SCM/Factored/ObsChainKernel.lean"},{"n":"Causalean.SCM.measurable_stepFun","k":"theorem","m":"Causalean.SCM.Factored.StepKernel","a":"SCM/Factored","t":1,"d":"Fix a structural causal model M and a step index n such that there are at least n + 1 observed nodes, so n names a valid position in the canonical topological order of observed nodes. Then the deterministic map stepFun, which produces the value of the n-th observed node by assembling its parent tuple from the fixed values, latent values, and previously generated observed prefix and applying the node's structural equation, is measurable.","s":"∀ {N : Type uN} [inst : DecidableEq N] [inst_1 : Fintype N] {Ω : N → Type uΩ} [inst_2 : (n : N) → MeasurableSpace (Ω n)] (M : Causalean.SCM N Ω) {n : ℕ} (hn : n + 1 ≤ M.observed.card), Measurable (M.stepFun hn)","l":60,"f":"Causalean/SCM/Factored/StepKernel.lean"},{"n":"Causalean.SCM.measurable_xRandomAsFixed","k":"theorem","m":"Causalean.SCM.Do.Rule2Kernel.Helpers","a":"SCM/Do/Rule2Kernel","t":2,"d":"Reading observed treatment values as intervention-coordinate values is measurable.","s":"∀ {N : Type u_1} [inst : DecidableEq N] {Ω : N → Type u_2} [inst_1 : (n : N) → MeasurableSpace (Ω n)] {X : Finset N}, Measurable Causalean.SCM.xRandomAsFixed","l":210,"f":"Causalean/SCM/Do/Rule2Kernel/Helpers.lean"},{"n":"Causalean.SCM.measurable_zFixedAsRandom","k":"theorem","m":"Causalean.SCM.Do.Rule2Kernel.Helpers","a":"SCM/Do/Rule2Kernel","t":2,"d":"Reading intervention values as observed random-variable values is a measurable operation.","s":"∀ {N : Type u_1} [inst : DecidableEq N] {Ω : N → Type u_2} [inst_1 : (n : N) → MeasurableSpace (Ω n)] {Z : Finset N}, Measurable Causalean.SCM.zFixedAsRandom","l":67,"f":"Causalean/SCM/Do/Rule2Kernel/Helpers.lean"},{"n":"Causalean.SCM.measurePreserving_valuesUnionEquiv","k":"theorem","m":"Causalean.SCM.ID.Density.PiUnion","a":"SCM/ID/Density","t":1,"d":"Measure.pi splits over a disjoint union. Given a family of σ-finite reference measures μ, one per coordinate, together with two finite index sets A and B assumed disjoint, the canonical equivalence carrying value assignments on A ∪ B to pairs of assignments on A and on B is measure-preserving from the product reference measure on A ∪ B to the product of the component reference measures on A and on B.","s":"∀ {N : Type u_1} [inst : DecidableEq N] {Ω : N → Type u_2} [inst_1 : (n : N) → MeasurableSpace (Ω n)] {A B : Finset (Causalean.SWIGNode N)} (hDisj : Disjoint A B) (μ : (v : Causalean.SWIGNode N) → MeasureTheory.Measure (Causalean.swigΩ Ω v)) [∀ (v : Causalean.SWIGNode N), MeasureTheory.SigmaFinite (μ v)], MeasureTheory.MeasurePreserving (⇑(Causalean.SCM.valuesUnionEquiv hDisj)) (MeasureTheory.Measure.pi fun i => μ ↑i) ((MeasureTheory.Measure.pi fun a => μ ↑a).prod (MeasureTheory.Measure.pi fun b => μ ↑b))","l":103,"f":"Causalean/SCM/ID/Density/PiUnion.lean"},{"n":"Causalean.SCM.measure_pi_map_valuesProjection","k":"theorem","m":"Causalean.SCM.Model.Induced","a":"SCM/Model","t":2,"d":"Product measures marginalize under coordinate restriction.","s":"∀ {N : Type u_1} [Fintype N] {Ω : N → Type u_2} [inst : (n : N) → MeasurableSpace (Ω n)] {I J : Finset (Causalean.SWIGNode N)} (hJI : J ⊆ I) (μ : (i : ↥I) → MeasureTheory.Measure (Causalean.swigΩ Ω ↑i)) [∀ (i : ↥I), MeasureTheory.IsProbabilityMeasure (μ i)], MeasureTheory.Measure.map (Causalean.valuesProjection hJI) (MeasureTheory.Measure.pi μ) = MeasureTheory.Measure.pi fun j => μ ⟨↑j, hJI j.property⟩","l":193,"f":"Causalean/SCM/Model/Induced.lean"},{"n":"Causalean.SCM.mechCFactor","k":"def","m":"Causalean.SCM.ID.Density.MechCFactor","a":"SCM/ID/Density","t":1,"d":"The mechanism c-factor density Q[S] as a function of a full observed assignment x: read the do-values for V∖S from x (keeping M's fixed slice s), form the do-complement marginal on S, and take its rnDeriv against the product reference on S, evaluated at the S-projection of x.","s":"{N : Type u_1} → [inst : DecidableEq N] → [inst_1 : Fintype N] → {Ω : N → Type u_2} → [inst_2 : (n : N) → MeasurableSpace (Ω n)] → (M : Causalean.SCM N Ω) → Causalean.SCM.ReferenceMeasures Ω → (S : Finset (Causalean.SWIGNode N)) → S ⊆ M.observed → M.FixedValues → Causalean.ValuesOn M.observed (Causalean.swigΩ Ω) → ENNReal","l":158,"f":"Causalean/SCM/ID/Density/MechCFactor.lean"},{"n":"Causalean.SCM.mechComplementNames","k":"def","m":"Causalean.SCM.ID.Density.MechCFactor","a":"SCM/ID/Density","t":1,"d":"Base-node names whose random copy lies in M.observed \\ S and whose fixed copy is not already fixed in M — the variables intervened on in Tian's do-complement object Q[S] = P_{v∖s}(s). The \"not already fixed\" clause lets this apply to non-standard models such as M.fixSet X.","s":"{N : Type u_1} → [inst : DecidableEq N] → [inst_1 : Fintype N] → {Ω : N → Type u_2} → [inst_2 : (n : N) → MeasurableSpace (Ω n)] → Causalean.SCM N Ω → Finset (Causalean.SWIGNode N) → Finset N","l":52,"f":"Causalean/SCM/ID/Density/MechCFactor.lean"},{"n":"Causalean.SCM.mechComplementNames_fixed_not_mem","k":"theorem","m":"Causalean.SCM.ID.Density.MechCFactor","a":"SCM/ID/Density","t":2,"d":"Every mechanism-complement variable has a fixed copy that is not already fixed.","s":"∀ {N : Type u_1} [inst : DecidableEq N] [inst_1 : Fintype N] {Ω : N → Type u_2} [inst_2 : (n : N) → MeasurableSpace (Ω n)] (M : Causalean.SCM N Ω) (S : Finset (Causalean.SWIGNode N)), ∀ n ∈ M.mechComplementNames S, Causalean.SWIGNode.fixed n ∉ M.fixed","l":68,"f":"Causalean/SCM/ID/Density/MechCFactor.lean"},{"n":"Causalean.SCM.mechComplementNames_image_random_subset_observed","k":"theorem","m":"Causalean.SCM.ID.Density.MechCFactor","a":"SCM/ID/Density","t":2,"d":"The random copies of the mechanism-complement variables form a subset of the observed nodes.","s":"∀ {N : Type u_1} [inst : DecidableEq N] [inst_1 : Fintype N] {Ω : N → Type u_2} [inst_2 : (n : N) → MeasurableSpace (Ω n)] (M : Causalean.SCM N Ω) (S : Finset (Causalean.SWIGNode N)), Finset.image Causalean.SWIGNode.random (M.mechComplementNames S) ⊆ M.observed","l":75,"f":"Causalean/SCM/ID/Density/MechCFactor.lean"},{"n":"Causalean.SCM.mechComplementNames_random_mem_observed","k":"theorem","m":"Causalean.SCM.ID.Density.MechCFactor","a":"SCM/ID/Density","t":2,"d":"Every mechanism-complement variable has its random copy among the observed nodes.","s":"∀ {N : Type u_1} [inst : DecidableEq N] [inst_1 : Fintype N] {Ω : N → Type u_2} [inst_2 : (n : N) → MeasurableSpace (Ω n)] (M : Causalean.SCM N Ω) (S : Finset (Causalean.SWIGNode N)), ∀ n ∈ M.mechComplementNames S, Causalean.SWIGNode.random n ∈ M.observed","l":61,"f":"Causalean/SCM/ID/Density/MechCFactor.lean"},{"n":"Causalean.SCM.mechDoValues","k":"def","m":"Causalean.SCM.ID.Density.MechCFactor","a":"SCM/ID/Density","t":1,"d":"Fixed-value slice for the do-complement SCM, read from a full observed assignment: keep M's existing fixed slice s and extend it on the do-complement coordinates by projecting x to the random copies in V∖S.","s":"{N : Type u_1} → [inst : DecidableEq N] → [inst_1 : Fintype N] → {Ω : N → Type u_2} → [inst_2 : (n : N) → MeasurableSpace (Ω n)] → (M : Causalean.SCM N Ω) → (S : Finset (Causalean.SWIGNode N)) → M.FixedValues → Causalean.ValuesOn M.observed (Causalean.swigΩ Ω) → (M.fixSet (M.mechComplementNames S) (Causalean.SCM.mechComplementNames_random_mem_observed M S) (Causalean.SCM.mechComplementNames_fixed_not_mem M S)).FixedValues","l":125,"f":"Causalean/SCM/ID/Density/MechCFactor.lean"},{"n":"Causalean.SCM.mem_cutsetLatent","k":"theorem","m":"Causalean.SCM.Model.CutsetLatent","a":"SCM/Model","t":2,"d":"Membership in the latent cutset means being unobserved and reaching the target set along an avoiding path.","s":"∀ {N : Type u_2} [inst : DecidableEq N] [inst_1 : Fintype N] {Ω : N → Type u_3} [inst_2 : (n : N) → MeasurableSpace (Ω n)] (M : Causalean.SCM N Ω) {Y C : Finset (Causalean.SWIGNode N)} {u : Causalean.SWIGNode N}, u ∈ M.cutsetLatent Y C ↔ u ∈ M.unobserved ∧ ∃ y ∈ Y, u = y ∨ M.dag.isAncestorAvoiding C u y","l":189,"f":"Causalean/SCM/Model/CutsetLatent.lean"},{"n":"Causalean.SCM.mem_fixedAncestorsOfNode","k":"theorem","m":"Causalean.SCM.Model.EvalFactorization","a":"SCM/Model","t":2,"d":"Membership in the fixed-ancestor set means being fixed and equal to or ancestral to the node.","s":"∀ {N : Type u_1} [inst : DecidableEq N] [inst_1 : Fintype N] {Ω : N → Type u_2} [inst_2 : (n : N) → MeasurableSpace (Ω n)] (M : Causalean.SCM N Ω) {v d : Causalean.SWIGNode N}, d ∈ M.fixedAncestorsOfNode v ↔ d ∈ M.fixed ∧ (d = v ∨ M.dag.isAncestor d v)","l":185,"f":"Causalean/SCM/Model/EvalFactorization.lean"},{"n":"Causalean.SCM.mem_latentAncestorsOfNode","k":"theorem","m":"Causalean.SCM.Model.EvalFactorization","a":"SCM/Model","t":2,"d":"Membership in the latent-ancestor set means being unobserved and equal to or ancestral to the node.","s":"∀ {N : Type u_1} [inst : DecidableEq N] [inst_1 : Fintype N] {Ω : N → Type u_2} [inst_2 : (n : N) → MeasurableSpace (Ω n)] (M : Causalean.SCM N Ω) {v u : Causalean.SWIGNode N}, u ∈ M.latentAncestorsOfNode v ↔ u ∈ M.unobserved ∧ (u = v ∨ M.dag.isAncestor u v)","l":175,"f":"Causalean/SCM/Model/EvalFactorization.lean"},{"n":"Causalean.SCM.mem_prefixNodes_iff","k":"theorem","m":"Causalean.SCM.Factored.ObsChainKernel","a":"SCM/Factored","t":2,"d":"Membership in prefixNodes is exactly having observed index below n.","s":"∀ {N : Type u_1} [inst : DecidableEq N] [inst_1 : Fintype N] {Ω : N → Type u_2} [inst_2 : (n : N) → MeasurableSpace (Ω n)] (M : Causalean.SCM N Ω) (n : ℕ) (v : Causalean.SWIGNode N), v ∈ M.prefixNodes n ↔ ∃ (h : v ∈ M.observed), ↑(M.observedIndex ⟨v, h⟩) < n","l":63,"f":"Causalean/SCM/Factored/ObsChainKernel.lean"},{"n":"Causalean.SCM.mu_C_comap_F_eq_nu_C_comap_F","k":"theorem","m":"Causalean.SCM.Do.Rule2Kernel.DiscreteZHelpers","a":"SCM/Do/Rule2Kernel","t":2,"d":"Cross-SCM pullback equality along the filled assignment.","s":"∀ {N : Type u_1} [inst : DecidableEq N] [inst_1 : Fintype N] {Ω : N → Type u_2} [inst_2 : (n : N) → MeasurableSpace (Ω n)] (M' : Causalean.SCM N Ω) (Z : Finset N) (hZ_obs : ∀ D ∈ Z, Causalean.SWIGNode.random D ∈ M'.observed) (hZ_fixed : ∀ D ∈ Z, Causalean.SWIGNode.fixed D ∉ M'.fixed) (W : Finset (Causalean.SWIGNode N)) (hZrW : Finset.image Causalean.SWIGNode.random Z ∪ W ⊆ M'.observed), Disjoint (Finset.image Causalean.SWIGNode.random Z) W → ∀ [MeasurableSingletonClass (Causalean.ValuesOn (Finset.image Causalean.SWIGNode.random Z ∪ W) (Causalean.swigΩ Ω))] (s : (M'.fixSet Z hZ_obs hZ_fixed).FixedValues), MeasureTheory.Measure.comap (M'.fillZrW Z hZ_obs hZ_fixed W s) (MeasureTheory.Measure.map (Causalean.valuesProjection hZrW) (M'.obsKernel (M'.fixSetProj Z hZ_obs hZ_fixed s))) = MeasureTheory.Measure.comap (M'.fillZrW Z hZ_obs hZ_fixed W s) (MeasureTheory.Measure.map (Causalean.valuesProjection (Eq.symm (Causalean.SCM.fixSet_observed M' Z hZ_obs hZ_fixed) ▸ hZrW)) ((M'.fixSet Z hZ_obs hZ_fixed).obsKernel s))","l":113,"f":"Causalean/SCM/Do/Rule2Kernel/DiscreteZHelpers.lean"},{"n":"Causalean.SCM.not_fixedTreatment_of_uadj","k":"theorem","m":"Causalean.SCM.Do.CutsetDSep","a":"SCM/Do","t":2,"d":"A node incident to a directed edge in the base graph cannot be the fixed copy of a treatment variable that is newly fixed by the intervention.","s":"∀ {N : Type u_1} [inst : DecidableEq N] [inst_1 : Fintype N] {Ω : N → Type u_2} [inst_2 : (n : N) → MeasurableSpace (Ω n)] (M : Causalean.SCM N Ω) (Z : Finset N), (∀ D ∈ Z, Causalean.SWIGNode.fixed D ∉ M.fixed) → ∀ {a v : Causalean.SWIGNode N}, M.dag.UAdj a v → ∀ {D : N}, D ∈ Z → v ≠ Causalean.SWIGNode.fixed D","l":278,"f":"Causalean/SCM/Do/CutsetDSep.lean"},{"n":"Causalean.SCM.not_fixedW_of_incoming_doX","k":"theorem","m":"Causalean.SCM.ID.Toolkit.FrontdoorGraph","a":"SCM/ID/Toolkit","t":2,"d":"If fixed copies of the W block are absent in the original model, no edge after fixing X can end at a fixed copy from W.","s":"∀ {N : Type u_1} [inst : DecidableEq N] [inst_1 : Fintype N] {Ω : N → Type u_2} [inst_2 : (n : N) → MeasurableSpace (Ω n)] (M : Causalean.SCM N Ω) (X W : Finset N) (hX_obs : ∀ D ∈ X, Causalean.SWIGNode.random D ∈ M.observed) (hX_fixed : ∀ D ∈ X, Causalean.SWIGNode.fixed D ∉ M.fixed), (∀ D ∈ W, Causalean.SWIGNode.fixed D ∉ M.fixed) → ∀ {u w : Causalean.SWIGNode N}, (M.fixSet X hX_obs hX_fixed).dag.edge u w → ∀ D ∈ W, w ≠ Causalean.SWIGNode.fixed D","l":116,"f":"Causalean/SCM/ID/Toolkit/FrontdoorGraph.lean"},{"n":"Causalean.SCM.not_fixed_of_obs","k":"theorem","m":"Causalean.SCM.Model.SCM","a":"SCM/Model","t":2,"d":"An observed node cannot also be a fixed node.","s":"∀ {N : Type uN} [inst : DecidableEq N] [inst_1 : Fintype N] (G : Causalean.SWIGGraph N) {n : Causalean.SWIGNode N}, n ∈ G.observed → n ∉ G.fixed","l":149,"f":"Causalean/SCM/Model/SCM.lean"},{"n":"Causalean.SCM.not_obs_of_unobs","k":"theorem","m":"Causalean.SCM.Model.SCM","a":"SCM/Model","t":2,"d":"An unobserved node cannot also be an observed node.","s":"∀ {N : Type uN} [inst : DecidableEq N] [inst_1 : Fintype N] (G : Causalean.SWIGGraph N) {n : Causalean.SWIGNode N}, n ∈ G.unobserved → n ∉ G.observed","l":144,"f":"Causalean/SCM/Model/SCM.lean"},{"n":"Causalean.SCM.not_unobs_of_obs","k":"theorem","m":"Causalean.SCM.Model.SCM","a":"SCM/Model","t":2,"d":"An observed node cannot also be an unobserved node.","s":"∀ {N : Type uN} [inst : DecidableEq N] [inst_1 : Fintype N] (G : Causalean.SWIGGraph N) {n : Causalean.SWIGNode N}, n ∈ G.observed → n ∉ G.unobserved","l":139,"f":"Causalean/SCM/Model/SCM.lean"},{"n":"Causalean.SCM.obsChainKernel","k":"def","m":"Causalean.SCM.Factored.ObsChainKernel","a":"SCM/Factored","t":1,"d":"The recursive observational chain-rule kernel through the first n observed nodes.","s":"{N : Type u_1} → [inst : DecidableEq N] → [inst_1 : Fintype N] → {Ω : N → Type u_2} → [inst_2 : (n : N) → MeasurableSpace (Ω n)] → (M : Causalean.SCM N Ω) → [∀ (k : ℕ) (hk : k < M.observed.card), StandardBorelSpace (Causalean.ValuesOn {↑(M.observedAt ⟨k, hk⟩)} (Causalean.swigΩ Ω))] → [∀ (k : ℕ) (hk : k < M.observed.card), Nonempty (Causalean.ValuesOn {↑(M.observedAt ⟨k, hk⟩)} (Causalean.swigΩ Ω))] → [∀ (k : Fin M.observed.card), MeasurableSpace.CountableOrCountablyGenerated M.FixedValues (Causalean.ValuesOn (M.prefixNodes ↑k) (Causalean.swigΩ Ω))] → (n : ℕ) → n ≤ M.observed.card → ProbabilityTheory.Kernel M.FixedValues (Causalean.ValuesOn (M.prefixNodes n) (Causalean.swigΩ Ω))","l":526,"f":"Causalean/SCM/Factored/ObsChainKernel.lean"},{"n":"Causalean.SCM.obsChainKernelZero","k":"def","m":"Causalean.SCM.Factored.ObsChainKernel","a":"SCM/Factored","t":1,"d":"The base observational chain kernel: a Dirac mass on the empty prefix.","s":"{N : Type u_1} → [inst : DecidableEq N] → [inst_1 : Fintype N] → {Ω : N → Type u_2} → [inst_2 : (n : N) → MeasurableSpace (Ω n)] → (M : Causalean.SCM N Ω) → ProbabilityTheory.Kernel M.FixedValues (Causalean.ValuesOn (M.prefixNodes 0) (Causalean.swigΩ Ω))","l":396,"f":"Causalean/SCM/Factored/ObsChainKernel.lean"},{"n":"Causalean.SCM.obsChainKernel_absolutelyContinuous_jointRef_prefix","k":"theorem","m":"Causalean.SCM.ID.Density.ChainRuleDensity","a":"SCM/ID/Density","t":2,"d":"Domination of the recursive observed-prefix chain by the prefix reference.","s":"∀ {N : Type u_1} [inst : DecidableEq N] [inst_1 : Fintype N] {Ω : N → Type u_2} [inst_2 : (n : N) → MeasurableSpace (Ω n)] (M : Causalean.SCM N Ω) (ref : Causalean.SCM.ReferenceMeasures Ω), M.DominatedObs ref → ∀ (s : M.FixedValues) [∀ (s' : M.FixedValues), MeasureTheory.IsFiniteMeasure (M.obsKernel s')] [inst_4 : ∀ (k : ℕ) (hk : k < M.observed.card), StandardBorelSpace (Causalean.ValuesOn {↑(M.observedAt ⟨k, hk⟩)} (Causalean.swigΩ Ω))] [inst_5 : ∀ (k : ℕ) (hk : k < M.observed.card), Nonempty (Causalean.ValuesOn {↑(M.observedAt ⟨k, hk⟩)} (Causalean.swigΩ Ω))] [inst_6 : ∀ (k : ℕ), MeasurableSpace.CountableOrCountablyGenerated M.FixedValues (Causalean.ValuesOn (M.prefixNodes k) (Causalean.swigΩ Ω))] (k : ℕ) (hk : k ≤ M.observed.card), ((M.obsChainKernel k hk) s).AbsolutelyContinuous (Causalean.SCM.jointRef ref (M.prefixNodes k))","l":489,"f":"Causalean/SCM/ID/Density/ChainRuleDensity.lean"},{"n":"Causalean.SCM.obsChainKernel_card_rnDeriv_eq_qFactorDensityProduct_prefix","k":"theorem","m":"Causalean.SCM.ID.Density.ChainRuleDensity","a":"SCM/ID/Density","t":2,"d":"Prefix-level analytic chain rule at the full observed prefix.","s":"∀ {N : Type u_1} [inst : DecidableEq N] [inst_1 : Fintype N] {Ω : N → Type u_2} [inst_2 : (n : N) → MeasurableSpace (Ω n)] (M : Causalean.SCM N Ω) (ref : Causalean.SCM.ReferenceMeasures Ω), M.DominatedObs ref → ∀ (s : M.FixedValues) [inst_3 : ∀ (s' : M.FixedValues), MeasureTheory.IsFiniteMeasure (M.obsKernel s')] [inst_4 : ∀ (k : ℕ) (hk : k < M.observed.card), StandardBorelSpace (Causalean.ValuesOn {↑(M.observedAt ⟨k, hk⟩)} (Causalean.swigΩ Ω))] [inst_5 : ∀ (k : ℕ) (hk : k < M.observed.card), Nonempty (Causalean.ValuesOn {↑(M.observedAt ⟨k, hk⟩)} (Causalean.swigΩ Ω))] [inst_6 : ∀ (k : ℕ), MeasurableSpace.CountableOrCountablyGenerated M.FixedValues (Causalean.ValuesOn (M.prefixNodes k) (Causalean.swigΩ Ω))], M.ObsStepFiberRN ref s → ((M.obsChainKernel M.observed.card (le_refl M.observed.card)) s).rnDeriv (Causalean.SCM.jointRef ref (M.prefixNodes M.observed.card)) =ᵐ[Causalean.SCM.jointRef ref (M.prefixNodes M.observed.card)] fun y => M.qFactorDensityProduct ref s ((Causalean.valuesEquivOfEq (Causalean.SCM.prefixNodes_card M M.observed.card (le_refl M.observed.card))) y)","l":811,"f":"Causalean/SCM/ID/Density/ChainRuleDensity.lean"},{"n":"Causalean.SCM.obsChainKernel_card_rnDeriv_eq_qFactorDensityProduct_prefix_induction","k":"theorem","m":"Causalean.SCM.ID.Density.ChainRuleDensity","a":"SCM/ID/Density","t":2,"d":"The analytic prefix induction for the density chain rule.","s":"∀ {N : Type u_1} [inst : DecidableEq N] [inst_1 : Fintype N] {Ω : N → Type u_2} [inst_2 : (n : N) → MeasurableSpace (Ω n)] (M : Causalean.SCM N Ω) (ref : Causalean.SCM.ReferenceMeasures Ω), M.DominatedObs ref → ∀ (s : M.FixedValues) [inst_3 : ∀ (s' : M.FixedValues), MeasureTheory.IsFiniteMeasure (M.obsKernel s')] [inst_4 : ∀ (k : ℕ) (hk : k < M.observed.card), StandardBorelSpace (Causalean.ValuesOn {↑(M.observedAt ⟨k, hk⟩)} (Causalean.swigΩ Ω))] [inst_5 : ∀ (k : ℕ) (hk : k < M.observed.card), Nonempty (Causalean.ValuesOn {↑(M.observedAt ⟨k, hk⟩)} (Causalean.swigΩ Ω))] [inst_6 : ∀ (k : ℕ), MeasurableSpace.CountableOrCountablyGenerated M.FixedValues (Causalean.ValuesOn (M.prefixNodes k) (Causalean.swigΩ Ω))], M.ObsStepFiberRN ref s → ((M.obsChainKernel M.observed.card (le_refl M.observed.card)) s).rnDeriv (Causalean.SCM.jointRef ref (M.prefixNodes M.observed.card)) =ᵐ[Causalean.SCM.jointRef ref (M.prefixNodes M.observed.card)] fun y => M.qFactorDensityProduct ref s ((Causalean.valuesEquivOfEq (Causalean.SCM.prefixNodes_card M M.observed.card (le_refl M.observed.card))) y)","l":777,"f":"Causalean/SCM/ID/Density/ChainRuleDensity.lean"},{"n":"Causalean.SCM.obsChainKernel_card_rnDeriv_pullback_eq_qFactorDensityProduct","k":"theorem","m":"Causalean.SCM.ID.Density.ChainRuleDensity","a":"SCM/ID/Density","t":2,"d":"Push the full-prefix a.e. density identity forward to observed coordinates.","s":"∀ {N : Type u_1} [inst : DecidableEq N] [inst_1 : Fintype N] {Ω : N → Type u_2} [inst_2 : (n : N) → MeasurableSpace (Ω n)] (M : Causalean.SCM N Ω) (ref : Causalean.SCM.ReferenceMeasures Ω), M.DominatedObs ref → ∀ (s : M.FixedValues) [inst_3 : ∀ (s' : M.FixedValues), MeasureTheory.IsFiniteMeasure (M.obsKernel s')] [inst_4 : ∀ (k : ℕ) (hk : k < M.observed.card), StandardBorelSpace (Causalean.ValuesOn {↑(M.observedAt ⟨k, hk⟩)} (Causalean.swigΩ Ω))] [inst_5 : ∀ (k : ℕ) (hk : k < M.observed.card), Nonempty (Causalean.ValuesOn {↑(M.observedAt ⟨k, hk⟩)} (Causalean.swigΩ Ω))] [inst_6 : ∀ (k : ℕ), MeasurableSpace.CountableOrCountablyGenerated M.FixedValues (Causalean.ValuesOn (M.prefixNodes k) (Causalean.swigΩ Ω))], M.ObsStepFiberRN ref s → (fun x => ((M.obsChainKernel M.observed.card (le_refl M.observed.card)) s).rnDeriv (Causalean.SCM.jointRef ref (M.prefixNodes M.observed.card)) ((Causalean.valuesEquivOfEq (Causalean.SCM.prefixNodes_card M M.observed.card (le_refl M.observed.card))).symm x)) =ᵐ[Causalean.SCM.jointRef ref M.observed] M.qFactorDensityProduct ref s","l":884,"f":"Causalean/SCM/ID/Density/ChainRuleDensity.lean"},{"n":"Causalean.SCM.obsChainKernel_card_rnDeriv_pullback_eq_qFactorDensityProduct_of_prefix","k":"theorem","m":"Causalean.SCM.ID.Density.ChainRuleDensity","a":"SCM/ID/Density","t":2,"d":"Push the full-prefix a.e. density identity forward, assuming the reference transport and prefix-level chain rule.","s":"∀ {N : Type u_1} [inst : DecidableEq N] [inst_1 : Fintype N] {Ω : N → Type u_2} [inst_2 : (n : N) → MeasurableSpace (Ω n)] (M : Causalean.SCM N Ω) (ref : Causalean.SCM.ReferenceMeasures Ω) (s : M.FixedValues), MeasureTheory.Measure.map (⇑(Causalean.valuesEquivOfEq (Causalean.SCM.prefixNodes_card M M.observed.card (le_refl M.observed.card)))) (Causalean.SCM.jointRef ref (M.prefixNodes M.observed.card)) = Causalean.SCM.jointRef ref M.observed → ∀ [inst_3 : ∀ (s' : M.FixedValues), MeasureTheory.IsFiniteMeasure (M.obsKernel s')] [inst_4 : ∀ (k : ℕ) (hk : k < M.observed.card), StandardBorelSpace (Causalean.ValuesOn {↑(M.observedAt ⟨k, hk⟩)} (Causalean.swigΩ Ω))] [inst_5 : ∀ (k : ℕ) (hk : k < M.observed.card), Nonempty (Causalean.ValuesOn {↑(M.observedAt ⟨k, hk⟩)} (Causalean.swigΩ Ω))] [inst_6 : ∀ (k : ℕ), MeasurableSpace.CountableOrCountablyGenerated M.FixedValues (Causalean.ValuesOn (M.prefixNodes k) (Causalean.swigΩ Ω))], (((M.obsChainKernel M.observed.card (le_refl M.observed.card)) s).rnDeriv (Causalean.SCM.jointRef ref (M.prefixNodes M.observed.card)) =ᵐ[Causalean.SCM.jointRef ref (M.prefixNodes M.observed.card)] fun y => M.qFactorDensityProduct ref s ((Causalean.valuesEquivOfEq (Causalean.SCM.prefixNodes_card M M.observed.card (le_refl M.observed.card))) y)) → (fun x => ((M.obsChainKernel M.observed.card (le_refl M.observed.card)) s).rnDeriv (Causalean.SCM.jointRef ref (M.prefixNodes M.observed.card)) ((Causalean.valuesEquivOfEq (Causalean.SCM.prefixNodes_card M M.observed.card (le_refl M.observed.card))).symm x)) =ᵐ[Causalean.SCM.jointRef ref M.observed] M.qFactorDensityProduct ref s","l":840,"f":"Causalean/SCM/ID/Density/ChainRuleDensity.lean"},{"n":"Causalean.SCM.obsChainKernel_rnDeriv_eq_prefixDensityProduct","k":"theorem","m":"Causalean.SCM.ID.Density.ChainRuleDensity","a":"SCM/ID/Density","t":2,"d":"General prefix-level RN derivative for the recursive observational chain.","s":"∀ {N : Type u_1} [inst : DecidableEq N] [inst_1 : Fintype N] {Ω : N → Type u_2} [inst_2 : (n : N) → MeasurableSpace (Ω n)] (M : Causalean.SCM N Ω) (ref : Causalean.SCM.ReferenceMeasures Ω), M.DominatedObs ref → ∀ (s : M.FixedValues) [inst_3 : ∀ (s' : M.FixedValues), MeasureTheory.IsFiniteMeasure (M.obsKernel s')] [inst_4 : ∀ (k : ℕ) (hk : k < M.observed.card), StandardBorelSpace (Causalean.ValuesOn {↑(M.observedAt ⟨k, hk⟩)} (Causalean.swigΩ Ω))] [inst_5 : ∀ (k : ℕ) (hk : k < M.observed.card), Nonempty (Causalean.ValuesOn {↑(M.observedAt ⟨k, hk⟩)} (Causalean.swigΩ Ω))] [inst_6 : ∀ (k : ℕ), MeasurableSpace.CountableOrCountablyGenerated M.FixedValues (Causalean.ValuesOn (M.prefixNodes k) (Causalean.swigΩ Ω))], M.ObsStepFiberRN ref s → ∀ (k : ℕ) (hk : k ≤ M.observed.card), ((M.obsChainKernel k hk) s).rnDeriv (Causalean.SCM.jointRef ref (M.prefixNodes k)) =ᵐ[Causalean.SCM.jointRef ref (M.prefixNodes k)] M.prefixDensityProduct ref s k","l":554,"f":"Causalean/SCM/ID/Density/ChainRuleDensity.lean"},{"n":"Causalean.SCM.obsCondIndep_contraction","k":"theorem","m":"Causalean.SCM.Do.SemiGraphoid","a":"SCM/Do","t":1,"d":"Contraction. Fix subsets X, Y, W, Z of the node set of the structural causal model M, with X, Y, W, and Z each contained in the observed nodes, and let μ be a finite measure on the observed values. If X is conditionally independent of Y given the union of Z and W and X is conditionally independent of W given Z, then X is conditionally independent of the union of Y and W given Z.","s":"∀ {N : Type uN} [inst : DecidableEq N] [inst_1 : Fintype N] {Ω : N → Type uΩ} [inst_2 : (n : N) → MeasurableSpace (Ω n)] (M : Causalean.SCM N Ω) [inst_3 : StandardBorelSpace M.ObservedValues] {X Y W Z : Finset (Causalean.SWIGNode N)} (hX : X ⊆ M.observed) (hY : Y ⊆ M.observed) (hW : W ⊆ M.observed) (hZ : Z ⊆ M.observed) {μ : MeasureTheory.Measure M.ObservedValues} [inst_4 : MeasureTheory.IsFiniteMeasure μ], M.ObsCondIndep X Y (Z ∪ W) hX hY (Finset.union_subset hZ hW) μ → M.ObsCondIndep X W Z hX hW hZ μ → M.ObsCondIndep X (Y ∪ W) Z hX (Finset.union_subset hY hW) hZ μ","l":252,"f":"Causalean/SCM/Do/SemiGraphoid.lean"},{"n":"Causalean.SCM.obsCondIndep_decomposition","k":"theorem","m":"Causalean.SCM.Do.SemiGraphoid","a":"SCM/Do","t":1,"d":"Decomposition. Let X, Y, W, Z be finite node sets of the structural causal model M, with X and the union of Y and W contained in the observed nodes, and Z contained in the observed nodes, under a finite measure μ on the observed values. If X is conditionally independent of the union of Y and W given Z, then X is conditionally independent of Y given Z.","s":"∀ {N : Type uN} [inst : DecidableEq N] [inst_1 : Fintype N] {Ω : N → Type uΩ} [inst_2 : (n : N) → MeasurableSpace (Ω n)] (M : Causalean.SCM N Ω) [inst_3 : StandardBorelSpace M.ObservedValues] {X Y W Z : Finset (Causalean.SWIGNode N)} (hX : X ⊆ M.observed) (hYW : Y ∪ W ⊆ M.observed) (hZ : Z ⊆ M.observed) {μ : MeasureTheory.Measure M.ObservedValues} [inst_4 : MeasureTheory.IsFiniteMeasure μ], M.ObsCondIndep X (Y ∪ W) Z hX hYW hZ μ → M.ObsCondIndep X Y Z hX (LE.le.trans Finset.subset_union_left hYW) hZ μ","l":218,"f":"Causalean/SCM/Do/SemiGraphoid.lean"},{"n":"Causalean.SCM.obsCondIndep_subset_right","k":"theorem","m":"Causalean.SCM.Do.SemiGraphoid","a":"SCM/Do","t":2,"d":"Subset right. If X ⊥ Y | Z and Y' ⊆ Y, then X ⊥ Y' | Z.","s":"∀ {N : Type uN} [inst : DecidableEq N] [inst_1 : Fintype N] {Ω : N → Type uΩ} [inst_2 : (n : N) → MeasurableSpace (Ω n)] (M : Causalean.SCM N Ω) [inst_3 : StandardBorelSpace M.ObservedValues] {X Y Y' Z : Finset (Causalean.SWIGNode N)} (hX : X ⊆ M.observed) (hY : Y ⊆ M.observed) (hY' : Y' ⊆ M.observed) (hZ : Z ⊆ M.observed), Y' ⊆ Y → ∀ {μ : MeasureTheory.Measure M.ObservedValues} [inst_4 : MeasureTheory.IsFiniteMeasure μ], M.ObsCondIndep X Y Z hX hY hZ μ → M.ObsCondIndep X Y' Z hX hY' hZ μ","l":205,"f":"Causalean/SCM/Do/SemiGraphoid.lean"},{"n":"Causalean.SCM.obsCondIndep_symm","k":"theorem","m":"Causalean.SCM.Do.SemiGraphoid","a":"SCM/Do","t":2,"d":"Symmetry. Conditional independence is symmetric in X and Y.","s":"∀ {N : Type uN} [inst : DecidableEq N] [inst_1 : Fintype N] {Ω : N → Type uΩ} [inst_2 : (n : N) → MeasurableSpace (Ω n)] (M : Causalean.SCM N Ω) [inst_3 : StandardBorelSpace M.ObservedValues] {X Y W : Finset (Causalean.SWIGNode N)} (hX : X ⊆ M.observed) (hY : Y ⊆ M.observed) (hW : W ⊆ M.observed) {μ : MeasureTheory.Measure M.ObservedValues} [inst_4 : MeasureTheory.IsFiniteMeasure μ], M.ObsCondIndep X Y W hX hY hW μ → M.ObsCondIndep Y X W hY hX hW μ","l":69,"f":"Causalean/SCM/Do/SemiGraphoid.lean"},{"n":"Causalean.SCM.obsCondIndep_weak_union","k":"theorem","m":"Causalean.SCM.Do.SemiGraphoid","a":"SCM/Do","t":1,"d":"Weak union. Let X, Y, W, Z be finite node sets of the structural causal model M, with X and the union of Y and W contained in the observed nodes, and Z contained in the observed nodes, under a finite measure μ on the observed values. If X is conditionally independent of the union of Y and W given Z, then X is conditionally independent of Y given the union of Z and W.","s":"∀ {N : Type uN} [inst : DecidableEq N] [inst_1 : Fintype N] {Ω : N → Type uΩ} [inst_2 : (n : N) → MeasurableSpace (Ω n)] (M : Causalean.SCM N Ω) [inst_3 : StandardBorelSpace M.ObservedValues] {X Y W Z : Finset (Causalean.SWIGNode N)} (hX : X ⊆ M.observed) (hYW : Y ∪ W ⊆ M.observed) (hZ : Z ⊆ M.observed) {μ : MeasureTheory.Measure M.ObservedValues} [inst_4 : MeasureTheory.IsFiniteMeasure μ], M.ObsCondIndep X (Y ∪ W) Z hX hYW hZ μ → M.ObsCondIndep X Y (Z ∪ W) hX (LE.le.trans Finset.subset_union_left hYW) (Finset.union_subset hZ (LE.le.trans Finset.subset_union_right hYW)) μ","l":235,"f":"Causalean/SCM/Do/SemiGraphoid.lean"},{"n":"Causalean.SCM.obsCondKernel","k":"def","m":"Causalean.SCM.Model.Kernel","a":"SCM/Model","t":1,"d":"The observational conditional kernel is a jointly measurable conditional law of target coordinates given conditioning coordinates.","s":"{N : Type u_1} → [inst : DecidableEq N] → [inst_1 : Fintype N] → {Ω : N → Type u_2} → [inst_2 : (n : N) → MeasurableSpace (Ω n)] → (M : Causalean.SCM N Ω) → (Y CC : Finset (Causalean.SWIGNode N)) → Y ⊆ M.observed → CC ⊆ M.observed → [StandardBorelSpace (Causalean.ValuesOn Y (Causalean.swigΩ Ω))] → [Nonempty (Causalean.ValuesOn Y (Causalean.swigΩ Ω))] → [MeasurableSpace.CountableOrCountablyGenerated M.FixedValues (Causalean.ValuesOn CC (Causalean.swigΩ Ω))] → ProbabilityTheory.Kernel (M.FixedValues × Causalean.ValuesOn CC (Causalean.swigΩ Ω)) (Causalean.ValuesOn Y (Causalean.swigΩ Ω))","l":209,"f":"Causalean/SCM/Model/Kernel.lean"},{"n":"Causalean.SCM.obsCondKernel_ae_eq_condDistrib","k":"theorem","m":"Causalean.SCM.Model.Kernel","a":"SCM/Model","t":1,"d":"At a fixed value s of the intervention-free (fixed) coordinates, provided the target coordinates Y are observed and the conditioning coordinates CC are observed, the jointly-measurable conditional kernel obsCondKernel built from Y and CC, evaluated at s, agrees almost everywhere — under the pushforward of the observational kernel onto CC — with the measure-theoretic conditional distribution of Y given CC under the observational kernel at s.","s":"∀ {N : Type u_1} [inst : DecidableEq N] [inst_1 : Fintype N] {Ω : N → Type u_2} [inst_2 : (n : N) → MeasurableSpace (Ω n)] (M : Causalean.SCM N Ω) (Y CC : Finset (Causalean.SWIGNode N)) (hY : Y ⊆ M.observed) (hCC : CC ⊆ M.observed) [inst_3 : StandardBorelSpace (Causalean.ValuesOn Y (Causalean.swigΩ Ω))] [inst_4 : Nonempty (Causalean.ValuesOn Y (Causalean.swigΩ Ω))] [inst_5 : MeasurableSpace.CountableOrCountablyGenerated M.FixedValues (Causalean.ValuesOn CC (Causalean.swigΩ Ω))] (s : M.FixedValues), (fun c => (M.obsCondKernel Y CC hY hCC) (s, c)) =ᵐ[MeasureTheory.Measure.map (Causalean.valuesProjection hCC) (M.obsKernel s)] ⇑(ProbabilityTheory.condDistrib (Causalean.valuesProjection hY) (Causalean.valuesProjection hCC) (M.obsKernel s))","l":234,"f":"Causalean/SCM/Model/Kernel.lean"},{"n":"Causalean.SCM.obsCondKernel_cross_SCM_ae_eq_on_fillZrW","k":"theorem","m":"Causalean.SCM.Do.Rule2Kernel.RectIdentity","a":"SCM/Do/Rule2Kernel","t":1,"d":"Cross-SCM conditional kernels agree along the fillZrW filled assignment. For the intervention on names Z, whose random copies are observed in the base model and whose fixed copies are not yet part of the base model's fixed coordinates, with outcome set Y and conditioning set W contained in the observed variables, the union of the random copies of Z and W contained in the observed variables with the random copies of Z disjoint from W, fix an intervened fixed assignment s; then for a measurable outcome set B and almost every conditioning value w, taken with respect to the pullback of the base model's law on Z.random ∪ W under the filled-assignment map, the base model's conditional distribution of Y given Z.random ∪ W, evaluated at the filled point built from w, and the intervened model's corresponding conditional distribution, evaluated at the same filled point, assign B the same probability.","s":"∀ {N : Type u_1} [inst : DecidableEq N] [inst_1 : Fintype N] {Ω : N → Type u_2} [inst_2 : (n : N) → MeasurableSpace (Ω n)] (M' : Causalean.SCM N Ω) (Z : Finset N) (hZ_obs : ∀ D ∈ Z, Causalean.SWIGNode.random D ∈ M'.observed) (hZ_fixed : ∀ D ∈ Z, Causalean.SWIGNode.fixed D ∉ M'.fixed) (Y W : Finset (Causalean.SWIGNode N)) (hY : Y ⊆ M'.observed), W ⊆ M'.observed → ∀ (hZrW : Finset.image Causalean.SWIGNode.random Z ∪ W ⊆ M'.observed), Disjoint (Finset.image Causalean.SWIGNode.random Z) W → ∀ [inst_3 : StandardBorelSpace (Causalean.ValuesOn Y (Causalean.swigΩ Ω))] [inst_4 : Nonempty (Causalean.ValuesOn Y (Causalean.swigΩ Ω))] [inst_5 : MeasurableSpace.CountableOrCountablyGenerated M'.FixedValues (Causalean.ValuesOn (Finset.image Causalean.SWIGNode.random Z ∪ W) (Causalean.swigΩ Ω))] [inst_6 : MeasurableSpace.CountableOrCountablyGenerated (M'.fixSet Z hZ_obs hZ_fixed).FixedValues (Causalean.ValuesOn (Finset.image Causalean.SWIGNode.random Z ∪ W) (Causalean.swigΩ Ω))] [MeasurableSingletonClass (Causalean.ValuesOn (Finset.image Causalean.SWIGNode.random Z ∪ W) (Causalean.swigΩ Ω))] (s : (M'.fixSet Z hZ_obs hZ_fixed).FixedValues) {B : Set (Causalean.ValuesOn Y (Causalean.swigΩ Ω))}, MeasurableSet B → ∀ᵐ (w : Causalean.ValuesOn W (Causalean.swigΩ Ω)) ∂MeasureTheory.Measure.comap (M'.fillZrW Z hZ_obs hZ_fixed W s) (MeasureTheory.Measure.map (Causalean.valuesProjection hZrW) (M'.obsKernel (M'.fixSetProj Z hZ_obs hZ_fixed s))), ((M'.obsCondKernel Y (Finset.image Causalean.SWIGNode.random Z ∪ W) hY hZrW) (M'.fixSetProj Z hZ_obs hZ_fixed s, M'.fillZrW Z hZ_obs hZ_fixed W s w)) B = (((M'.fixSet Z hZ_obs hZ_fixed).obsCondKernel Y (Finset.image Causalean.SWIGNode.random Z ∪ W) (Eq.symm (Causalean.SCM.fixSet_observed M' Z hZ_obs hZ_fixed) ▸ hY) (Eq.symm (Causalean.SCM.fixSet_observed M' Z hZ_obs hZ_fixed) ▸ hZrW)) (s, M'.fillZrW Z hZ_obs hZ_fixed W s w)) B","l":319,"f":"Causalean/SCM/Do/Rule2Kernel/RectIdentity.lean"},{"n":"Causalean.SCM.obsCondKernel_cross_eq_ae_of_discrete","k":"theorem","m":"Causalean.SCM.Do.Rule2Kernel.DiscreteZHelpers","a":"SCM/Do/Rule2Kernel","t":2,"d":"Discrete-treatment cross-SCM conditional-kernel equality.","s":"∀ {N : Type u_1} [inst : DecidableEq N] [inst_1 : Fintype N] {Ω : N → Type u_2} [inst_2 : (n : N) → MeasurableSpace (Ω n)] (M' : Causalean.SCM N Ω) (Z : Finset N) (hZ_obs : ∀ D ∈ Z, Causalean.SWIGNode.random D ∈ M'.observed) (hZ_fixed : ∀ D ∈ Z, Causalean.SWIGNode.fixed D ∉ M'.fixed) (Y W : Finset (Causalean.SWIGNode N)) (hY : Y ⊆ M'.observed) (hW : W ⊆ M'.observed) (hZrW : Finset.image Causalean.SWIGNode.random Z ∪ W ⊆ M'.observed) [∀ (n : Causalean.SWIGNode N), StandardBorelSpace (Causalean.swigΩ Ω n)] [∀ (n : Causalean.SWIGNode N), Nonempty (Causalean.swigΩ Ω n)], (M'.fixSet Z hZ_obs hZ_fixed).dag.dSep Y (Finset.image Causalean.SWIGNode.random Z) (W ∪ (M'.fixSet Z hZ_obs hZ_fixed).fixed) → (∀ z ∈ Z, ∀ v ∈ W, ¬(M'.fixSet Z hZ_obs hZ_fixed).dag.isAncestor (Causalean.SWIGNode.fixed z) v) → ∀ [StandardBorelSpace (M'.fixSet Z hZ_obs hZ_fixed).RandomValues] [StandardBorelSpace (M'.fixSet Z hZ_obs hZ_fixed).ObservedValues] [inst_7 : StandardBorelSpace (Causalean.ValuesOn Y (Causalean.swigΩ Ω))] [inst_8 : Nonempty (Causalean.ValuesOn Y (Causalean.swigΩ Ω))] [StandardBorelSpace (Causalean.ValuesOn (Finset.image Causalean.SWIGNode.random Z) (Causalean.swigΩ Ω))] [Nonempty (Causalean.ValuesOn (Finset.image Causalean.SWIGNode.random Z) (Causalean.swigΩ Ω))] [∀ (s : M'.FixedValues), MeasureTheory.IsFiniteMeasure (M'.obsKernel s)] [∀ (s : (M'.fixSet Z hZ_obs hZ_fixed).FixedValues), MeasureTheory.IsFiniteMeasure ((M'.fixSet Z hZ_obs hZ_fixed).jointKernel s)] [∀ (s : (M'.fixSet Z hZ_obs hZ_fixed).FixedValues), MeasureTheory.IsFiniteMeasure ((M'.fixSet Z hZ_obs hZ_fixed).obsKernel s)] [inst_14 : MeasurableSpace.CountableOrCountablyGenerated M'.FixedValues (Causalean.ValuesOn (Finset.image Causalean.SWIGNode.random Z ∪ W) (Causalean.swigΩ Ω))] [MeasurableSpace.CountableOrCountablyGenerated (M'.fixSet Z hZ_obs hZ_fixed).FixedValues (Causalean.ValuesOn (Finset.image Causalean.SWIGNode.random Z ∪ W) (Causalean.swigΩ Ω))] [inst_16 : MeasurableSpace.CountableOrCountablyGenerated (M'.fixSet Z hZ_obs hZ_fixed).FixedValues (Causalean.ValuesOn W (Causalean.swigΩ Ω))] [MeasurableSingletonClass (Causalean.ValuesOn (Finset.image Causalean.SWIGNode.random Z ∪ W) (Causalean.swigΩ Ω))] (s : (M'.fixSet Z hZ_obs hZ_fixed).FixedValues), (MeasureTheory.Measure.map (Causalean.valuesProjection hW) (M'.obsKernel (M'.fixSetProj Z hZ_obs hZ_fixed s))).AbsolutelyContinuous (MeasureTheory.Measure.comap (M'.fillZrW Z hZ_obs hZ_fixed W s) (MeasureTheory.Measure.map (Causalean.valuesProjection hZrW) (M'.obsKernel (M'.fixSetProj Z hZ_obs hZ_fixed s)))) → ∀ {B : Set (Causalean.ValuesOn Y (Causalean.swigΩ Ω))}, MeasurableSet B → ∀ᵐ (w : Causalean.ValuesOn W (Causalean.swigΩ Ω)) ∂MeasureTheory.Measure.map (Causalean.valuesProjection (Eq.symm (Causalean.SCM.fixSet_observed M' Z hZ_obs hZ_fixed) ▸ hW)) ((M'.fixSet Z hZ_obs hZ_fixed).obsKernel s), ((M'.obsCondKernel Y (Finset.image Causalean.SWIGNode.random Z ∪ W) hY hZrW) (M'.fixSetProj Z hZ_obs hZ_fixed s, M'.fillZrW Z hZ_obs hZ_fixed W s w)) B = (((M'.fixSet Z hZ_obs hZ_fixed).obsCondKernel Y W (Eq.symm (Causalean.SCM.fixSet_observed M' Z hZ_obs hZ_fixed) ▸ hY) (Eq.symm (Causalean.SCM.fixSet_observed M' Z hZ_obs hZ_fixed) ▸ hW)) (s, w)) B","l":176,"f":"Causalean/SCM/Do/Rule2Kernel/DiscreteZHelpers.lean"},{"n":"Causalean.SCM.obsCondKernel_dSep_collapse_ae","k":"theorem","m":"Causalean.SCM.Do.Rule2Kernel.RectIdentity","a":"SCM/Do/Rule2Kernel","t":1,"d":"D-separation collapse for the post-intervention conditional kernel. For the intervention on names Z, whose random copies are observed in the base model and whose fixed copies are not yet part of the base model's fixed coordinates, with outcome set Y and conditioning set W contained in the observed variables, suppose in the post-intervention SWIG DAG, Y is d-separated from the random copies of Z given W together with the post-intervention fixed set. Then, fixing an intervened fixed assignment s and a measurable outcome set B, for almost every conditioning value on Z.random ∪ W, the intervened model's conditional distribution of Y given Z.random ∪ W assigns B the same probability as its conditional distribution of Y given W alone, evaluated at the W-projection of that conditioning value.","s":"∀ {N : Type u_1} [inst : DecidableEq N] [inst_1 : Fintype N] {Ω : N → Type u_2} [inst_2 : (n : N) → MeasurableSpace (Ω n)] (M' : Causalean.SCM N Ω) (Z : Finset N) (hZ_obs : ∀ D ∈ Z, Causalean.SWIGNode.random D ∈ M'.observed) (hZ_fixed : ∀ D ∈ Z, Causalean.SWIGNode.fixed D ∉ M'.fixed) (Y W : Finset (Causalean.SWIGNode N)) (hY : Y ⊆ M'.observed) (hW : W ⊆ M'.observed) [inst_3 : ∀ (n : Causalean.SWIGNode N), StandardBorelSpace (Causalean.swigΩ Ω n)] [inst_4 : ∀ (n : Causalean.SWIGNode N), Nonempty (Causalean.swigΩ Ω n)], (M'.fixSet Z hZ_obs hZ_fixed).dag.dSep Y (Finset.image Causalean.SWIGNode.random Z) (W ∪ (M'.fixSet Z hZ_obs hZ_fixed).fixed) → ∀ [inst_5 : MeasurableSpace.CountableOrCountablyGenerated (M'.fixSet Z hZ_obs hZ_fixed).FixedValues (Causalean.ValuesOn (Finset.image Causalean.SWIGNode.random Z ∪ W) (Causalean.swigΩ Ω))] [inst_6 : MeasurableSpace.CountableOrCountablyGenerated (M'.fixSet Z hZ_obs hZ_fixed).FixedValues (Causalean.ValuesOn W (Causalean.swigΩ Ω))] (s : (M'.fixSet Z hZ_obs hZ_fixed).FixedValues) {B : Set (Causalean.ValuesOn Y (Causalean.swigΩ Ω))}, MeasurableSet B → have hZr := fun ⦃v⦄ hv => Exists.casesOn (Finset.mem_image.mp hv) fun D h => And.casesOn h fun hD right => Eq.ndrec (motive := fun ⦃v⦄ => v ∈ Finset.image Causalean.SWIGNode.random Z → v ∈ M'.observed) (fun hv => hZ_obs D hD) right hv; have hZrW := Finset.union_subset hZr hW; ∀ᵐ (c : Causalean.ValuesOn (Finset.image Causalean.SWIGNode.random Z ∪ W) (Causalean.swigΩ Ω)) ∂MeasureTheory.Measure.map (Causalean.valuesProjection (Eq.symm (Causalean.SCM.fixSet_observed M' Z hZ_obs hZ_fixed) ▸ hZrW)) ((M'.fixSet Z hZ_obs hZ_fixed).obsKernel s), (((M'.fixSet Z hZ_obs hZ_fixed).obsCondKernel Y (Finset.image Causalean.SWIGNode.random Z ∪ W) (Eq.symm (Causalean.SCM.fixSet_observed M' Z hZ_obs hZ_fixed) ▸ hY) (Eq.symm (Causalean.SCM.fixSet_observed M' Z hZ_obs hZ_fixed) ▸ hZrW)) (s, c)) B = (((M'.fixSet Z hZ_obs hZ_fixed).obsCondKernel Y W (Eq.symm (Causalean.SCM.fixSet_observed M' Z hZ_obs hZ_fixed) ▸ hY) (Eq.symm (Causalean.SCM.fixSet_observed M' Z hZ_obs hZ_fixed) ▸ hW)) (s, Causalean.valuesProjection Finset.subset_union_right c)) B","l":47,"f":"Causalean/SCM/Do/Rule2Kernel/RectIdentity.lean"},{"n":"Causalean.SCM.obsCondKernel_empty_eq_marginal","k":"theorem","m":"Causalean.SCM.ID.Toolkit.ObsChainRule","a":"SCM/ID/Toolkit","t":2,"d":"Empty-conditioning collapse. Conditioning obsCondKernel Y on the empty set collapses to the plain Y-marginal (M.obsKernel s).map π_Y. This is the reusable step every do-calculus factorization performs when a conditioning block becomes empty (e.g. the first node of a fixing sequence); it is graph-agnostic in M, so it applies verbatim to any stacked-intervention model M.fixSet ….","s":"∀ {N : Type u_1} [inst : DecidableEq N] [inst_1 : Fintype N] {Ω : N → Type u_2} [inst_2 : (n : N) → MeasurableSpace (Ω n)] (M : Causalean.SCM N Ω) (Y : Finset (Causalean.SWIGNode N)) (hY : Y ⊆ M.observed) [inst_3 : StandardBorelSpace (Causalean.ValuesOn Y (Causalean.swigΩ Ω))] [inst_4 : Nonempty (Causalean.ValuesOn Y (Causalean.swigΩ Ω))] (s : M.FixedValues) (c : Causalean.ValuesOn ∅ (Causalean.swigΩ Ω)), (M.obsCondKernel Y ∅ hY (Finset.empty_subset M.observed)) (s, c) = MeasureTheory.Measure.map (Causalean.valuesProjection hY) (M.obsKernel s)","l":155,"f":"Causalean/SCM/ID/Toolkit/ObsChainRule.lean"},{"n":"Causalean.SCM.obsCondKernel_fixSet_M1_eq_ae_product","k":"theorem","m":"Causalean.SCM.Do.Rule2Kernel.WitnessBridge","a":"SCM/Do/Rule2Kernel","t":1,"d":"Product-form cross-SCM bridge for the do-side conditional kernel. Let M' be a structural causal model and Z a set of treatment names with each treatment's pre-intervention node observed and each treatment's post-intervention node not already fixed in M'; let Y, W be node sets with Y, W, the treatments' pre-intervention nodes, and their union with W all observed, Y disjoint from the treatments' pre-intervention nodes, and those pre-intervention nodes disjoint from W. Suppose that, in the intervened model's graph, no node of W is a descendant of any treatment's post-intervention node, that, in the original model's graph, no node of W is a descendant of any treatment's pre-intervention node, and that in the intervened model's graph, Y is d-separated from the treatments' pre-intervention nodes given W together with the intervened model's fixed nodes. Assume also, at a baseline assignment s0 to the original model's fixed nodes, an overlap condition: the pushforward, under combining a treatment value with a W value, of the product of the treatments' pre-intervention marginal law and the W-marginal law (both taken from the original model's observational kernel at s0) is absolutely continuous with respect to the original model's marginal law on the treatments' pre-intervention nodes together with W, again at s0. Then for almost every pair (t, w) drawn from that product law, the intervened model's conditional-probability kernel for Y given W, evaluated at treatment value t and conditioning value w, equals the original model's conditional-probability kernel for Y given the union of the treatments' pre-intervention nodes and W, evaluated at the combined value (t, w).","s":"∀ {N : Type u_1} [inst : DecidableEq N] [inst_1 : Fintype N] {Ω : N → Type u_2} [inst_2 : (n : N) → MeasurableSpace (Ω n)] (M' : Causalean.SCM N Ω) (Z : Finset N) (hZ_obs : ∀ D ∈ Z, Causalean.SWIGNode.random D ∈ M'.observed) (hZ_fixed : ∀ D ∈ Z, Causalean.SWIGNode.fixed D ∉ M'.fixed) (Y W : Finset (Causalean.SWIGNode N)) (hY : Y ⊆ M'.observed) (hW : W ⊆ M'.observed) (hZr : Finset.image Causalean.SWIGNode.random Z ⊆ M'.observed) (hZrW : Finset.image Causalean.SWIGNode.random Z ∪ W ⊆ M'.observed), Disjoint (Finset.image Causalean.SWIGNode.random Z) W → Disjoint Y (Finset.image Causalean.SWIGNode.random Z) → (∀ z ∈ Z, ∀ v ∈ W, ¬(M'.fixSet Z hZ_obs hZ_fixed).dag.isAncestor (Causalean.SWIGNode.fixed z) v) → (∀ D ∈ Z, ∀ w ∈ W, ¬M'.dag.isAncestor (Causalean.SWIGNode.random D) w) → (M'.fixSet Z hZ_obs hZ_fixed).dag.dSep Y (Finset.image Causalean.SWIGNode.random Z) (W ∪ (M'.fixSet Z hZ_obs hZ_fixed).fixed) → ∀ [StandardBorelSpace M'.RandomValues] [∀ (n : Causalean.SWIGNode N), StandardBorelSpace (Causalean.swigΩ Ω n)] [∀ (n : Causalean.SWIGNode N), Nonempty (Causalean.swigΩ Ω n)] [inst_6 : StandardBorelSpace (Causalean.ValuesOn Y (Causalean.swigΩ Ω))] [inst_7 : Nonempty (Causalean.ValuesOn Y (Causalean.swigΩ Ω))] [StandardBorelSpace (Causalean.ValuesOn (M'.cutsetLatent Y (Finset.image Causalean.SWIGNode.random Z ∪ W)) (Causalean.swigΩ Ω))] [Nonempty (Causalean.ValuesOn (M'.cutsetLatent Y (Finset.image Causalean.SWIGNode.random Z ∪ W)) (Causalean.swigΩ Ω))] [StandardBorelSpace (M'.fixSet Z hZ_obs hZ_fixed).RandomValues] [StandardBorelSpace (M'.fixSet Z hZ_obs hZ_fixed).ObservedValues] [StandardBorelSpace (Causalean.ValuesOn (Finset.image Causalean.SWIGNode.random Z) (Causalean.swigΩ Ω))] [Nonempty (Causalean.ValuesOn (Finset.image Causalean.SWIGNode.random Z) (Causalean.swigΩ Ω))] [∀ (s : M'.FixedValues), MeasureTheory.IsFiniteMeasure (M'.jointKernel s)] [∀ (s : M'.FixedValues), MeasureTheory.IsFiniteMeasure (M'.obsKernel s)] [∀ (s : (M'.fixSet Z hZ_obs hZ_fixed).FixedValues), MeasureTheory.IsFiniteMeasure ((M'.fixSet Z hZ_obs hZ_fixed).jointKernel s)] [∀ (s : (M'.fixSet Z hZ_obs hZ_fixed).FixedValues), MeasureTheory.IsFiniteMeasure ((M'.fixSet Z hZ_obs hZ_fixed).obsKernel s)] [inst_18 : MeasurableSpace.CountableOrCountablyGenerated M'.FixedValues (Causalean.ValuesOn (Finset.image Causalean.SWIGNode.random Z ∪ W) (Causalean.swigΩ Ω))] [MeasurableSpace.CountableOrCountablyGenerated (M'.fixSet Z hZ_obs hZ_fixed).FixedValues (Causalean.ValuesOn (Finset.image Causalean.SWIGNode.random Z ∪ W) (Causalean.swigΩ Ω))] [inst_20 : MeasurableSpace.CountableOrCountablyGenerated (M'.fixSet Z hZ_obs hZ_fixed).FixedValues (Causalean.ValuesOn W (Causalean.swigΩ Ω))] [MeasurableSingletonClass (Causalean.ValuesOn (Finset.image Causalean.SWIGNode.random Z ∪ W) (Causalean.swigΩ Ω))] (s0 : M'.FixedValues), (MeasureTheory.Measure.map (fun p => Causalean.valuesUnionMk p.1 p.2) ((MeasureTheory.Measure.map (Causalean.valuesProjection hZr) (M'.obsKernel s0)).compProd (ProbabilityTheory.Kernel.const (Causalean.ValuesOn (Finset.image Causalean.SWIGNode.random Z) (Causalean.swigΩ Ω)) (MeasureTheory.Measure.map (Causalean.valuesProjection hW) (M'.obsKernel s0))))).AbsolutelyContinuous (MeasureTheory.Measure.map (Causalean.valuesProjection hZrW) (M'.obsKernel s0)) → ∀ᵐ (p : Causalean.ValuesOn (Finset.image Causalean.SWIGNode.random Z) (Causalean.swigΩ Ω) × Causalean.ValuesOn W (Causalean.swigΩ Ω)) ∂(MeasureTheory.Measure.map (Causalean.valuesProjection hZr) (M'.obsKernel s0)).compProd (ProbabilityTheory.Kernel.const (Causalean.ValuesOn (Finset.image Causalean.SWIGNode.random Z) (Causalean.swigΩ Ω)) (MeasureTheory.Measure.map (Causalean.valuesProjection hW) (M'.obsKernel s0))), ((M'.fixSet Z hZ_obs hZ_fixed).obsCondKernel Y W (Eq.symm (Causalean.SCM.fixSet_observed M' Z hZ_obs hZ_fixed) ▸ hY) (Eq.symm (Causalean.SCM.fixSet_observed M' Z hZ_obs hZ_fixed) ▸ hW)) (M'.fixSetExtend Z hZ_obs hZ_fixed s0 p.1, p.2) = (M'.obsCondKernel Y (Finset.image Causalean.SWIGNode.random Z ∪ W) hY hZrW) (s0, Causalean.valuesUnionMk p.1 p.2)","l":1389,"f":"Causalean/SCM/Do/Rule2Kernel/WitnessBridge.lean"},{"n":"Causalean.SCM.obsCondKernel_fixSet_eq_ae_witness","k":"theorem","m":"Causalean.SCM.Do.Rule2AE","a":"SCM/Do","t":1,"d":"Rule 2, a.e. in the treatment value (product form + positivity). Consider intervening on the finite set of names Z, where each name's random copy is already observed in the base model and its fixed copy is not yet part of the base model's fixed coordinates, with outcome set Y and conditioning set W contained in the observed variables, together with the random copies of Z and their union with W. Assume in the post-intervention SWIG DAG, Y is d-separated from the random copies of Z given W together with the post-intervention fixed set, no fixed copy of a name in Z is an ancestor of any node of W in the post-intervention DAG, no random copy of a name in Z is an ancestor of any node of W in the original DAG, and positivity: the pushforward of the product of the treatment and conditioning marginals under the fill map is absolutely continuous with respect to the base model's law on Z.random ∪ W. Then for almost every pair (t, w) under that product measure, the W-conditional kernel of the model intervened at treatment value t, evaluated together with w, restricted to Y, equals the base model's conditional distribution of Y given Z.random ∪ W evaluated at the combined point (t, w).","s":"∀ {N : Type u_1} [inst : DecidableEq N] [inst_1 : Fintype N] {Ω : N → Type u_2} [inst_2 : (n : N) → MeasurableSpace (Ω n)] (M' : Causalean.SCM N Ω) (Z : Finset N) (hZ_obs : ∀ D ∈ Z, Causalean.SWIGNode.random D ∈ M'.observed) (hZ_fixed : ∀ D ∈ Z, Causalean.SWIGNode.fixed D ∉ M'.fixed) (Y W : Finset (Causalean.SWIGNode N)) (hY : Y ⊆ M'.observed) (hW : W ⊆ M'.observed) (hZr : Finset.image Causalean.SWIGNode.random Z ⊆ M'.observed) (hZrW : Finset.image Causalean.SWIGNode.random Z ∪ W ⊆ M'.observed), (M'.fixSet Z hZ_obs hZ_fixed).dag.dSep Y (Finset.image Causalean.SWIGNode.random Z) (W ∪ (M'.fixSet Z hZ_obs hZ_fixed).fixed) → (∀ z ∈ Z, ∀ v ∈ W, ¬(M'.fixSet Z hZ_obs hZ_fixed).dag.isAncestor (Causalean.SWIGNode.fixed z) v) → (∀ D ∈ Z, ∀ w ∈ W, ¬M'.dag.isAncestor (Causalean.SWIGNode.random D) w) → ∀ [inst_3 : ∀ (n : Causalean.SWIGNode N), StandardBorelSpace (Causalean.swigΩ Ω n)] [inst_4 : ∀ (n : Causalean.SWIGNode N), Nonempty (Causalean.swigΩ Ω n)] [inst_5 : MeasurableSpace.CountableOrCountablyGenerated M'.FixedValues (Causalean.ValuesOn (Finset.image Causalean.SWIGNode.random Z ∪ W) (Causalean.swigΩ Ω))] [MeasurableSpace.CountableOrCountablyGenerated (M'.fixSet Z hZ_obs hZ_fixed).FixedValues (Causalean.ValuesOn (Finset.image Causalean.SWIGNode.random Z ∪ W) (Causalean.swigΩ Ω))] [inst_7 : MeasurableSpace.CountableOrCountablyGenerated (M'.fixSet Z hZ_obs hZ_fixed).FixedValues (Causalean.ValuesOn W (Causalean.swigΩ Ω))] [MeasurableSingletonClass (Causalean.ValuesOn (Finset.image Causalean.SWIGNode.random Z ∪ W) (Causalean.swigΩ Ω))] (s0 : M'.FixedValues), (MeasureTheory.Measure.map (fun p => Causalean.valuesUnionMk p.1 p.2) ((MeasureTheory.Measure.map (Causalean.valuesProjection hZr) (M'.obsKernel s0)).compProd (ProbabilityTheory.Kernel.const (Causalean.ValuesOn (Finset.image Causalean.SWIGNode.random Z) (Causalean.swigΩ Ω)) (MeasureTheory.Measure.map (Causalean.valuesProjection hW) (M'.obsKernel s0))))).AbsolutelyContinuous (MeasureTheory.Measure.map (Causalean.valuesProjection hZrW) (M'.obsKernel s0)) → ∀ᵐ (p : Causalean.ValuesOn (Finset.image Causalean.SWIGNode.random Z) (Causalean.swigΩ Ω) × Causalean.ValuesOn W (Causalean.swigΩ Ω)) ∂(MeasureTheory.Measure.map (Causalean.valuesProjection hZr) (M'.obsKernel s0)).compProd (ProbabilityTheory.Kernel.const (Causalean.ValuesOn (Finset.image Causalean.SWIGNode.random Z) (Causalean.swigΩ Ω)) (MeasureTheory.Measure.map (Causalean.valuesProjection hW) (M'.obsKernel s0))), ((M'.fixSet Z hZ_obs hZ_fixed).obsCondKernel Y W (Eq.symm (Causalean.SCM.fixSet_observed M' Z hZ_obs hZ_fixed) ▸ hY) (Eq.symm (Causalean.SCM.fixSet_observed M' Z hZ_obs hZ_fixed) ▸ hW)) (M'.fixSetExtend Z hZ_obs hZ_fixed s0 p.1, p.2) = (M'.obsCondKernel Y (Finset.image Causalean.SWIGNode.random Z ∪ W) hY hZrW) (s0, Causalean.valuesUnionMk p.1 p.2)","l":139,"f":"Causalean/SCM/Do/Rule2AE.lean"},{"n":"Causalean.SCM.obsCondKernel_singleton_mass_of_ne_zero","k":"theorem","m":"Causalean.SCM.ID.Density.MassBridge","a":"SCM/ID/Density","t":1,"d":"For a query coordinate set Y contained in the observed nodes and a conditioning coordinate set CC contained in the observed nodes, at a conditioning value c with nonzero pushforward mass of the observational kernel under projection onto CC, the observational conditional kernel's singleton mass at a value y equals the observational kernel's joint singleton mass at the pair (c, y) divided by its singleton mass at c.","s":"∀ {N : Type u_1} [inst : DecidableEq N] [inst_1 : Fintype N] {Ω : N → Type u_2} [inst_2 : (n : N) → MeasurableSpace (Ω n)] (M : Causalean.SCM N Ω) (Y CC : Finset (Causalean.SWIGNode N)) (hY : Y ⊆ M.observed) (hCC : CC ⊆ M.observed) [MeasurableSingletonClass (Causalean.ValuesOn CC (Causalean.swigΩ Ω))] [MeasurableSingletonClass (Causalean.ValuesOn Y (Causalean.swigΩ Ω))] [inst_5 : StandardBorelSpace (Causalean.ValuesOn Y (Causalean.swigΩ Ω))] [inst_6 : Nonempty (Causalean.ValuesOn Y (Causalean.swigΩ Ω))] [∀ (s : M.FixedValues), MeasureTheory.IsFiniteMeasure (M.obsKernel s)] [inst_8 : MeasurableSpace.CountableOrCountablyGenerated M.FixedValues (Causalean.ValuesOn CC (Causalean.swigΩ Ω))] (s : M.FixedValues) (c : Causalean.ValuesOn CC (Causalean.swigΩ Ω)) (y : Causalean.ValuesOn Y (Causalean.swigΩ Ω)), (MeasureTheory.Measure.map (Causalean.valuesProjection hCC) (M.obsKernel s)) {c} ≠ 0 → ((M.obsCondKernel Y CC hY hCC) (s, c)) {y} = (MeasureTheory.Measure.map (fun ω => (Causalean.valuesProjection hCC ω, Causalean.valuesProjection hY ω)) (M.obsKernel s)) {(c, y)} / (MeasureTheory.Measure.map (Causalean.valuesProjection hCC) (M.obsKernel s)) {c}","l":191,"f":"Causalean/SCM/ID/Density/MassBridge.lean"},{"n":"Causalean.SCM.obsCondKernel_union_eq_witness","k":"theorem","m":"Causalean.SCM.Do.Rule2Kernel.WitnessBridge","a":"SCM/Do/Rule2Kernel","t":2,"d":"The (Zr∪W)-conditional packaged as the witness kernel.","s":"∀ {N : Type u_1} [inst : DecidableEq N] [inst_1 : Fintype N] {Ω : N → Type u_2} [inst_2 : (n : N) → MeasurableSpace (Ω n)] (M : Causalean.SCM N Ω) [StandardBorelSpace M.RandomValues] [∀ (n : Causalean.SWIGNode N), StandardBorelSpace (Causalean.swigΩ Ω n)] [∀ (n : Causalean.SWIGNode N), Nonempty (Causalean.swigΩ Ω n)] [∀ (s : M.FixedValues), MeasureTheory.IsFiniteMeasure (M.jointKernel s)] (Y W : Finset (Causalean.SWIGNode N)) (Z : Finset N) [inst_7 : StandardBorelSpace (Causalean.ValuesOn (M.cutsetLatent Y (Finset.image Causalean.SWIGNode.random Z ∪ W)) (Causalean.swigΩ Ω))] [inst_8 : Nonempty (Causalean.ValuesOn (M.cutsetLatent Y (Finset.image Causalean.SWIGNode.random Z ∪ W)) (Causalean.swigΩ Ω))] [inst_9 : StandardBorelSpace (Causalean.ValuesOn Y (Causalean.swigΩ Ω))] [inst_10 : Nonempty (Causalean.ValuesOn Y (Causalean.swigΩ Ω))] [inst_11 : MeasurableSpace.CountableOrCountablyGenerated M.FixedValues (Causalean.ValuesOn (Finset.image Causalean.SWIGNode.random Z ∪ W) (Causalean.swigΩ Ω))] (hY : Y ⊆ M.observed) (hZrW : Finset.image Causalean.SWIGNode.random Z ∪ W ⊆ M.observed), Disjoint (Finset.image Causalean.SWIGNode.random Z) W → M.dag.dSep (M.cutsetLatent Y (Finset.image Causalean.SWIGNode.random Z ∪ W)) (Finset.image Causalean.SWIGNode.random Z) (W ∪ M.fixed) → ∀ (s : M.FixedValues) (h : Causalean.ValuesOn (Finset.image Causalean.SWIGNode.random Z) (Causalean.swigΩ Ω) → Causalean.ValuesOn W (Causalean.swigΩ Ω) → Causalean.ValuesOn (M.cutsetLatent Y (Finset.image Causalean.SWIGNode.random Z ∪ W)) (Causalean.swigΩ Ω) → Causalean.ValuesOn Y (Causalean.swigΩ Ω)), (Measurable fun p => h p.1.1 p.1.2 p.2) → (∀ (ℓ : M.LatentValues), Causalean.valuesProjection hY (M.randomToObserved (M.evalMap s ℓ)) = h (Causalean.valuesProjection (LE.le.trans Finset.subset_union_left hZrW) (M.randomToObserved (M.evalMap s ℓ))) (Causalean.valuesProjection (LE.le.trans Finset.subset_union_right hZrW) (M.randomToObserved (M.evalMap s ℓ))) (Causalean.valuesProjection (Causalean.SCM.cutsetLatent_subset M Y (Finset.image Causalean.SWIGNode.random Z ∪ W)) ℓ)) → (fun c => (M.obsCondKernel Y (Finset.image Causalean.SWIGNode.random Z ∪ W) hY hZrW) (s, c)) =ᵐ[MeasureTheory.Measure.map (Causalean.valuesProjection hZrW) (M.obsKernel s)] fun c => MeasureTheory.Measure.map (h (Causalean.valuesProjection Finset.subset_union_left c) (Causalean.valuesProjection Finset.subset_union_right c)) ((ProbabilityTheory.condDistrib (Causalean.valuesProjection (Causalean.SCM.cutsetLatent_subset M Y (Finset.image Causalean.SWIGNode.random Z ∪ W))) (fun ℓ => Causalean.valuesProjection (LE.le.trans Finset.subset_union_right hZrW) (M.randomToObserved (M.evalMap s ℓ))) M.latentProduct) (Causalean.valuesProjection Finset.subset_union_right c))","l":706,"f":"Causalean/SCM/Do/Rule2Kernel/WitnessBridge.lean"},{"n":"Causalean.SCM.obsCondPairKernel","k":"def","m":"Causalean.SCM.Model.Kernel","a":"SCM/Model","t":1,"d":"The conditional-pair kernel pushes the observational law to conditioning coordinates paired with target coordinates.","s":"{N : Type u_1} → [inst : DecidableEq N] → [inst_1 : Fintype N] → {Ω : N → Type u_2} → [inst_2 : (n : N) → MeasurableSpace (Ω n)] → (M : Causalean.SCM N Ω) → (Y CC : Finset (Causalean.SWIGNode N)) → Y ⊆ M.observed → CC ⊆ M.observed → ProbabilityTheory.Kernel M.FixedValues (Causalean.ValuesOn CC (Causalean.swigΩ Ω) × Causalean.ValuesOn Y (Causalean.swigΩ Ω))","l":199,"f":"Causalean/SCM/Model/Kernel.lean"},{"n":"Causalean.SCM.obsCondPairKernel_apply_eq_compProd","k":"theorem","m":"Causalean.SCM.Factored.ObsChainKernel","a":"SCM/Factored","t":2,"d":"Slice-level disintegration for the pair kernel defining obsCondKernel.","s":"∀ {N : Type u_1} [inst : DecidableEq N] [inst_1 : Fintype N] {Ω : N → Type u_2} [inst_2 : (n : N) → MeasurableSpace (Ω n)] (M : Causalean.SCM N Ω) (Y CC : Finset (Causalean.SWIGNode N)) (hY : Y ⊆ M.observed) (hCC : CC ⊆ M.observed) [inst_3 : StandardBorelSpace (Causalean.ValuesOn Y (Causalean.swigΩ Ω))] [inst_4 : Nonempty (Causalean.ValuesOn Y (Causalean.swigΩ Ω))] [inst_5 : MeasurableSpace.CountableOrCountablyGenerated M.FixedValues (Causalean.ValuesOn CC (Causalean.swigΩ Ω))] (s : M.FixedValues), (M.obsCondPairKernel Y CC hY hCC) s = (MeasureTheory.Measure.map (Causalean.valuesProjection hCC) (M.obsKernel s)).compProd ((M.obsCondKernel Y CC hY hCC).sectR s)","l":481,"f":"Causalean/SCM/Factored/ObsChainKernel.lean"},{"n":"Causalean.SCM.obsDensity","k":"def","m":"Causalean.SCM.ID.Density.ReferenceMeasure","a":"SCM/ID/Density","t":1,"d":"The joint observational density is the Radon-Nikodym derivative of the observational law with respect to the observed-node reference product.","s":"{N : Type u_1} → [inst : DecidableEq N] → [inst_1 : Fintype N] → {Ω : N → Type u_2} → [inst_2 : (n : N) → MeasurableSpace (Ω n)] → (M : Causalean.SCM N Ω) → Causalean.SCM.ReferenceMeasures Ω → M.FixedValues → Causalean.ValuesOn M.observed (Causalean.swigΩ Ω) → ENNReal","l":73,"f":"Causalean/SCM/ID/Density/ReferenceMeasure.lean"},{"n":"Causalean.SCM.obsDensity_eq_qFactorDensityProduct","k":"theorem","m":"Causalean.SCM.ID.Density.ChainRuleDensity","a":"SCM/ID/Density","t":1,"d":"Observational density chain rule. In a structural causal model whose observational law is absolutely continuous with respect to the joint reference measure on the observed nodes, if in addition the stepwise fibre Radon--Nikodym condition holds along the observed topological order, then the joint observational density agrees, almost everywhere with respect to that joint reference measure, with the product of the one-node conditional density factors taken in observed topological order.","s":"∀ {N : Type u_1} [inst : DecidableEq N] [inst_1 : Fintype N] {Ω : N → Type u_2} [inst_2 : (n : N) → MeasurableSpace (Ω n)] (M : Causalean.SCM N Ω) (ref : Causalean.SCM.ReferenceMeasures Ω), M.DominatedObs ref → ∀ (s : M.FixedValues) [inst_3 : ∀ (s' : M.FixedValues), MeasureTheory.IsFiniteMeasure (M.obsKernel s')] [inst_4 : ∀ (k : ℕ) (hk : k < M.observed.card), StandardBorelSpace (Causalean.ValuesOn {↑(M.observedAt ⟨k, hk⟩)} (Causalean.swigΩ Ω))] [inst_5 : ∀ (k : ℕ) (hk : k < M.observed.card), Nonempty (Causalean.ValuesOn {↑(M.observedAt ⟨k, hk⟩)} (Causalean.swigΩ Ω))] [inst_6 : ∀ (k : ℕ), MeasurableSpace.CountableOrCountablyGenerated M.FixedValues (Causalean.ValuesOn (M.prefixNodes k) (Causalean.swigΩ Ω))], M.ObsStepFiberRN ref s → M.obsDensity ref s =ᵐ[Causalean.SCM.jointRef ref M.observed] M.qFactorDensityProduct ref s","l":945,"f":"Causalean/SCM/ID/Density/ChainRuleDensity.lean"},{"n":"Causalean.SCM.obsDensity_heq_of_obsKernel_heq","k":"theorem","m":"Causalean.SCM.ID.Density.DoLawMarginal","a":"SCM/ID/Density","t":1,"d":"Cross-model density transport. If two structural causal models M₁ and M₂ share the same SWIG graph and have heterogeneously-equal observational kernels, then their observational densities are heterogeneously equal. After unifying the SWIG-graph data the observed-value types coincide, the observational kernels become literally equal, and the density is the Radon–Nikodym derivative of that kernel against a fixed reference measure.","s":"∀ {N : Type u_1} [inst : DecidableEq N] [inst_1 : Fintype N] {Ω : N → Type u_2} [inst_2 : (n : N) → MeasurableSpace (Ω n)] (M₁ M₂ : Causalean.SCM N Ω) (ref : Causalean.SCM.ReferenceMeasures Ω), M₁.toSWIGGraph = M₂.toSWIGGraph → M₁.obsKernel ≍ M₂.obsKernel → M₁.obsDensity ref ≍ M₂.obsDensity ref","l":28,"f":"Causalean/SCM/ID/Density/DoLawMarginal.lean"},{"n":"Causalean.SCM.obsKernel","k":"def","m":"Causalean.SCM.Model.Kernel","a":"SCM/Model","t":1,"d":"The observational kernel maps each fixed-value assignment to the induced law of the observed nodes.","s":"{N : Type u_1} → [inst : DecidableEq N] → [inst_1 : Fintype N] → {Ω : N → Type u_2} → [inst_2 : (n : N) → MeasurableSpace (Ω n)] → (M : Causalean.SCM N Ω) → ProbabilityTheory.Kernel M.FixedValues M.ObservedValues","l":143,"f":"Causalean/SCM/Model/Kernel.lean"},{"n":"Causalean.SCM.obsKernel_apply_univ","k":"theorem","m":"Causalean.SCM.Model.Kernel","a":"SCM/Model","t":2,"d":"Each observational-kernel slice has total mass one.","s":"∀ {N : Type u_1} [inst : DecidableEq N] [inst_1 : Fintype N] {Ω : N → Type u_2} [inst_2 : (n : N) → MeasurableSpace (Ω n)] (M : Causalean.SCM N Ω) (s : M.FixedValues), (M.obsKernel s) Set.univ = 1","l":172,"f":"Causalean/SCM/Model/Kernel.lean"},{"n":"Causalean.SCM.obsKernel_disintegrate_rect","k":"theorem","m":"Causalean.SCM.Do.Rule2Kernel.InterSingleton","a":"SCM/Do/Rule2Kernel","t":2,"d":"Base-model disintegration over condition coordinates.","s":"∀ {N : Type u_1} [inst : DecidableEq N] [inst_1 : Fintype N] {Ω : N → Type u_2} [inst_2 : (n : N) → MeasurableSpace (Ω n)] (M : Causalean.SCM N Ω) (Y CC : Finset (Causalean.SWIGNode N)) (hY : Y ⊆ M.observed) (hCC : CC ⊆ M.observed) [inst_3 : StandardBorelSpace (Causalean.ValuesOn Y (Causalean.swigΩ Ω))] [inst_4 : Nonempty (Causalean.ValuesOn Y (Causalean.swigΩ Ω))] [inst_5 : MeasurableSpace.CountableOrCountablyGenerated M.FixedValues (Causalean.ValuesOn CC (Causalean.swigΩ Ω))] (s : M.FixedValues) {D : Set (Causalean.ValuesOn CC (Causalean.swigΩ Ω))} {B : Set (Causalean.ValuesOn Y (Causalean.swigΩ Ω))}, MeasurableSet D → MeasurableSet B → (M.obsKernel s) (Causalean.valuesProjection hCC ⁻¹' D ∩ Causalean.valuesProjection hY ⁻¹' B) = ∫⁻ (c : Causalean.ValuesOn CC (Causalean.swigΩ Ω)) in D, ((M.obsCondKernel Y CC hY hCC) (s, c)) B ∂MeasureTheory.Measure.map (Causalean.valuesProjection hCC) (M.obsKernel s)","l":693,"f":"Causalean/SCM/Do/Rule2Kernel/InterSingleton.lean"},{"n":"Causalean.SCM.obsKernel_empty_projection_eq_dirac","k":"theorem","m":"Causalean.SCM.ID.Toolkit.ObsChainRule","a":"SCM/ID/Toolkit","t":2,"d":"Empty-projection collapse. Projecting the observational kernel onto the empty node set yields a Dirac measure: ValuesOn ∅ is a subsingleton, so the projected Markov kernel is the point mass at its unique value. The base case of any conditioning set shrinking to ∅ in a fixing/adjustment step.","s":"∀ {N : Type u_1} [inst : DecidableEq N] [inst_1 : Fintype N] {Ω : N → Type u_2} [inst_2 : (n : N) → MeasurableSpace (Ω n)] (M : Causalean.SCM N Ω) (hEmpty : ∅ ⊆ M.observed) (s : M.FixedValues) (c : Causalean.ValuesOn ∅ (Causalean.swigΩ Ω)), MeasureTheory.Measure.map (Causalean.valuesProjection hEmpty) (M.obsKernel s) = MeasureTheory.Measure.dirac c","l":124,"f":"Causalean/SCM/ID/Toolkit/ObsChainRule.lean"},{"n":"Causalean.SCM.obsKernel_eq_latentProduct_map","k":"theorem","m":"Causalean.SCM.Do.Rule2Kernel.WitnessBridge","a":"SCM/Do/Rule2Kernel","t":2,"d":"obsKernel s is the latent product pushed through randomToObserved ∘ evalMap s.","s":"∀ {N : Type u_1} [inst : DecidableEq N] [inst_1 : Fintype N] {Ω : N → Type u_2} [inst_2 : (n : N) → MeasurableSpace (Ω n)] (M : Causalean.SCM N Ω) (s : M.FixedValues), M.obsKernel s = MeasureTheory.Measure.map (fun ℓ => M.randomToObserved (M.evalMap s ℓ)) M.latentProduct","l":178,"f":"Causalean/SCM/Do/Rule2Kernel/WitnessBridge.lean"},{"n":"Causalean.SCM.obsKernel_eq_of_obsDensity_ae_eq","k":"theorem","m":"Causalean.SCM.ID.Density.ReferenceMeasure","a":"SCM/ID/Density","t":1,"d":"Within a model whose observational law is dominated by the joint reference measure, if the joint observational densities at two fixed-value slices agree almost everywhere with respect to that reference measure, then the two slices induce the same observational law.","s":"∀ {N : Type u_1} [inst : DecidableEq N] [inst_1 : Fintype N] {Ω : N → Type u_2} [inst_2 : (n : N) → MeasurableSpace (Ω n)] (M : Causalean.SCM N Ω) (ref : Causalean.SCM.ReferenceMeasures Ω), M.DominatedObs ref → ∀ (s₁ s₂ : M.FixedValues), M.obsDensity ref s₁ =ᵐ[Causalean.SCM.jointRef ref M.observed] M.obsDensity ref s₂ → M.obsKernel s₁ = M.obsKernel s₂","l":88,"f":"Causalean/SCM/ID/Density/ReferenceMeasure.lean"},{"n":"Causalean.SCM.obsKernel_eq_qFactorProduct","k":"theorem","m":"Causalean.SCM.Factored.ObsChainKernel","a":"SCM/Factored","t":1,"d":"For a structural causal model M, at a fixed assignment s, its observational kernel equals the full chain-rule product of one-node conditional kernels along the observed topological order.","s":"∀ {N : Type u_1} [inst : DecidableEq N] [inst_1 : Fintype N] {Ω : N → Type u_2} [inst_2 : (n : N) → MeasurableSpace (Ω n)] (M : Causalean.SCM N Ω) (s : M.FixedValues) [inst_3 : ∀ (k : ℕ) (hk : k < M.observed.card), StandardBorelSpace (Causalean.ValuesOn {↑(M.observedAt ⟨k, hk⟩)} (Causalean.swigΩ Ω))] [inst_4 : ∀ (k : ℕ) (hk : k < M.observed.card), Nonempty (Causalean.ValuesOn {↑(M.observedAt ⟨k, hk⟩)} (Causalean.swigΩ Ω))] [inst_5 : ∀ (k : Fin M.observed.card), MeasurableSpace.CountableOrCountablyGenerated M.FixedValues (Causalean.ValuesOn (M.prefixNodes ↑k) (Causalean.swigΩ Ω))], M.obsKernel s = M.qFactorProduct s","l":744,"f":"Causalean/SCM/Factored/ObsChainKernel.lean"},{"n":"Causalean.SCM.obsKernel_fixSet_W_marginal_eq_M1_marginal","k":"theorem","m":"Causalean.SCM.Do.Rule2Kernel.DiscreteZHelpers","a":"SCM/Do/Rule2Kernel","t":1,"d":"Rule 3 W-marginal equality for Rule 2. For the intervention on names Z, whose random copies are observed in the base model and whose fixed copies are not yet part of the base model's fixed coordinates, and a conditioning set W contained in the observed variables, if no fixed copy of a name in Z is an ancestor, in the post-intervention DAG, of any node in W, then the W-marginal of the intervened model's observational kernel at a fixed assignment s equals the W-marginal of the base model's observational kernel at the corresponding projected assignment.","s":"∀ {N : Type u_1} [inst : DecidableEq N] [inst_1 : Fintype N] {Ω : N → Type u_2} [inst_2 : (n : N) → MeasurableSpace (Ω n)] (M' : Causalean.SCM N Ω) (Z : Finset N) (hZ_obs : ∀ D ∈ Z, Causalean.SWIGNode.random D ∈ M'.observed) (hZ_fixed : ∀ D ∈ Z, Causalean.SWIGNode.fixed D ∉ M'.fixed) (W : Finset (Causalean.SWIGNode N)) (hW : W ⊆ M'.observed), (∀ z ∈ Z, ∀ v ∈ W, ¬(M'.fixSet Z hZ_obs hZ_fixed).dag.isAncestor (Causalean.SWIGNode.fixed z) v) → ∀ (s : (M'.fixSet Z hZ_obs hZ_fixed).FixedValues), MeasureTheory.Measure.map (Causalean.valuesProjection (Eq.symm (Causalean.SCM.fixSet_observed M' Z hZ_obs hZ_fixed) ▸ hW)) ((M'.fixSet Z hZ_obs hZ_fixed).obsKernel s) = MeasureTheory.Measure.map (Causalean.valuesProjection hW) (M'.obsKernel (M'.fixSetProj Z hZ_obs hZ_fixed s))","l":71,"f":"Causalean/SCM/Do/Rule2Kernel/DiscreteZHelpers.lean"},{"n":"Causalean.SCM.obsKernel_fixSet_W_marginal_pushforward_eq","k":"theorem","m":"Causalean.SCM.Do.Rule2Kernel.WMarginal","a":"SCM/Do/Rule2Kernel","t":1,"d":"W-marginal pushforward identity. Fix a do-set Z of nodes whose random copies are observed in the base model and whose fixed nodes have not already been intervened on and an observation block W such that the union of Z's random copies and W is observed and disjoint from Z's random copies. If, under the intervened model's law of the observed variables given the fixed values s', the post-intervention random copies of Z almost surely equal their assigned intervention values, then for every measurable W-event A, the intervened model's marginal probability of A on W equals the base model's probability of the pullback of A through the map that fills in the pinned Z-random-copy coordinates alongside W.","s":"∀ {N : Type u_1} [inst : DecidableEq N] [inst_1 : Fintype N] {Ω : N → Type u_2} [inst_2 : (n : N) → MeasurableSpace (Ω n)] (M' : Causalean.SCM N Ω) (Z : Finset N) (hZ_obs : ∀ D ∈ Z, Causalean.SWIGNode.random D ∈ M'.observed) (hZ_fixed : ∀ D ∈ Z, Causalean.SWIGNode.fixed D ∉ M'.fixed) (W : Finset (Causalean.SWIGNode N)) (hZrW : Finset.image Causalean.SWIGNode.random Z ∪ W ⊆ M'.observed), Disjoint (Finset.image Causalean.SWIGNode.random Z) W → ∀ [MeasurableSingletonClass (Causalean.ValuesOn (Finset.image Causalean.SWIGNode.random Z ∪ W) (Causalean.swigΩ Ω))] (s' : (M'.fixSet Z hZ_obs hZ_fixed).FixedValues), (∀ᵐ (ω : (M'.fixSet Z hZ_obs hZ_fixed).ObservedValues) ∂(M'.fixSet Z hZ_obs hZ_fixed).obsKernel s', ∀ (D : N) (hD : D ∈ Z), ω ⟨Causalean.SWIGNode.random D, hZ_obs D hD⟩ = s' ⟨Causalean.SWIGNode.fixed D, Causalean.SCM.fixed_mem_fixSet M' Z hZ_obs hZ_fixed hD⟩) → ∀ {A : Set (Causalean.ValuesOn W (Causalean.swigΩ Ω))}, MeasurableSet A → (MeasureTheory.Measure.map (Causalean.valuesProjection (Eq.symm (Causalean.SCM.fixSet_observed M' Z hZ_obs hZ_fixed) ▸ fun ⦃v⦄ hv => hZrW (Finset.mem_union_right (Finset.image Causalean.SWIGNode.random Z) hv))) ((M'.fixSet Z hZ_obs hZ_fixed).obsKernel s')) A = (MeasureTheory.Measure.map (Causalean.valuesProjection hZrW) (M'.obsKernel (M'.fixSetProj Z hZ_obs hZ_fixed s'))) (M'.fillZrW Z hZ_obs hZ_fixed W s' '' A)","l":223,"f":"Causalean/SCM/Do/Rule2Kernel/WMarginal.lean"},{"n":"Causalean.SCM.obsKernel_fixSet_W_rect_integral_eq","k":"theorem","m":"Causalean.SCM.Do.Rule2Kernel.WMarginal","a":"SCM/Do/Rule2Kernel","t":1,"d":"Cross-SCM rectangle bridge for the W-marginal. Fix a do-set Z of nodes whose random copies are observed in the base model and whose fixed nodes have not already been intervened on, together with an outcome block Y and a conditioning block W of observed variables such that the union of Z's random copies and W is observed and disjoint from Z's random copies. Assume that, under the intervened model's law given the fixed values s, the post-intervention random copies of Z almost surely equal their assigned intervention values. Then for every measurable W-event A and every measurable Y-event B, integrating, over A and with respect to the intervened model's W-marginal, the base model's conditional probability of B given the outcome of filling in Z's pinned random-copy values alongside each w reconstructs the intervened model's probability that both the W-event A and the Y-event B occur.","s":"∀ {N : Type u_1} [inst : DecidableEq N] [inst_1 : Fintype N] {Ω : N → Type u_2} [inst_2 : (n : N) → MeasurableSpace (Ω n)] (M' : Causalean.SCM N Ω) (Z : Finset N) (hZ_obs : ∀ D ∈ Z, Causalean.SWIGNode.random D ∈ M'.observed) (hZ_fixed : ∀ D ∈ Z, Causalean.SWIGNode.fixed D ∉ M'.fixed) (Y W : Finset (Causalean.SWIGNode N)) (hY : Y ⊆ M'.observed) (hW : W ⊆ M'.observed) (hZrW : Finset.image Causalean.SWIGNode.random Z ∪ W ⊆ M'.observed), Disjoint (Finset.image Causalean.SWIGNode.random Z) W → ∀ [inst_3 : StandardBorelSpace (Causalean.ValuesOn Y (Causalean.swigΩ Ω))] [inst_4 : Nonempty (Causalean.ValuesOn Y (Causalean.swigΩ Ω))] [inst_5 : MeasurableSpace.CountableOrCountablyGenerated M'.FixedValues (Causalean.ValuesOn (Finset.image Causalean.SWIGNode.random Z ∪ W) (Causalean.swigΩ Ω))] [MeasurableSingletonClass (Causalean.ValuesOn (Finset.image Causalean.SWIGNode.random Z ∪ W) (Causalean.swigΩ Ω))] (s : (M'.fixSet Z hZ_obs hZ_fixed).FixedValues), (∀ᵐ (ω : (M'.fixSet Z hZ_obs hZ_fixed).ObservedValues) ∂(M'.fixSet Z hZ_obs hZ_fixed).obsKernel s, ∀ (D : N) (hD : D ∈ Z), ω ⟨Causalean.SWIGNode.random D, hZ_obs D hD⟩ = s ⟨Causalean.SWIGNode.fixed D, Causalean.SCM.fixed_mem_fixSet M' Z hZ_obs hZ_fixed hD⟩) → ∀ {A : Set (Causalean.ValuesOn W (Causalean.swigΩ Ω))} {B : Set (Causalean.ValuesOn Y (Causalean.swigΩ Ω))}, MeasurableSet A → MeasurableSet B → ∫⁻ (w : Causalean.ValuesOn W (Causalean.swigΩ Ω)) in A, ((M'.obsCondKernel Y (Finset.image Causalean.SWIGNode.random Z ∪ W) hY hZrW) (M'.fixSetProj Z hZ_obs hZ_fixed s, M'.fillZrW Z hZ_obs hZ_fixed W s w)) B ∂MeasureTheory.Measure.map (Causalean.valuesProjection (Eq.symm (Causalean.SCM.fixSet_observed M' Z hZ_obs hZ_fixed) ▸ hW)) ((M'.fixSet Z hZ_obs hZ_fixed).obsKernel s) = ((M'.fixSet Z hZ_obs hZ_fixed).obsKernel s) (Causalean.valuesProjection (Eq.symm (Causalean.SCM.fixSet_observed M' Z hZ_obs hZ_fixed) ▸ hW) ⁻¹' A ∩ Causalean.valuesProjection (Eq.symm (Causalean.SCM.fixSet_observed M' Z hZ_obs hZ_fixed) ▸ hY) ⁻¹' B)","l":384,"f":"Causalean/SCM/Do/Rule2Kernel/WMarginal.lean"},{"n":"Causalean.SCM.obsKernel_heq_of_obsDensity_heq","k":"theorem","m":"Causalean.SCM.ID.Density.DoLawMarginal","a":"SCM/ID/Density","t":1,"d":"Cross-model law transport (converse). If two structural causal models M₁ and M₂ are each dominated by the same reference measure, share the same SWIG graph, and have heterogeneously-equal observational densities, then their observational kernels are heterogeneously equal. After unifying the SWIG-graph data the observed-value types coincide and the densities become literally equal; weighting the common joint reference by that density recovers each observational law (withDensity_obsDensity_eq), so the two laws agree. This is the converse of obsDensity_heq_of_obsKernel_heq: under dominance, equal density and equal law are interchangeable, letting the kernel-level identification tools be driven from a density hypothesis.","s":"∀ {N : Type u_1} [inst : DecidableEq N] [inst_1 : Fintype N] {Ω : N → Type u_2} [inst_2 : (n : N) → MeasurableSpace (Ω n)] (M₁ M₂ : Causalean.SCM N Ω) (ref : Causalean.SCM.ReferenceMeasures Ω), M₁.toSWIGGraph = M₂.toSWIGGraph → M₁.DominatedObs ref → M₂.DominatedObs ref → M₁.obsDensity ref ≍ M₂.obsDensity ref → M₁.obsKernel ≍ M₂.obsKernel","l":53,"f":"Causalean/SCM/ID/Density/DoLawMarginal.lean"},{"n":"Causalean.SCM.obsKernel_inter_Wset_Zrand_levelset_eq","k":"theorem","m":"Causalean.SCM.Do.Rule2Kernel.InterSingleton","a":"SCM/Do/Rule2Kernel","t":1,"d":"For the intervention on names Z, whose random copies are observed in the base model and whose fixed copies are not yet part of the base model's fixed coordinates, with the union of the random copies of Z and a conditioning set W contained in the observed variables and the random copies of Z disjoint from W, fix an intervened fixed assignment s'; for measurable subsets S of the observed-value space and measurable subsets A of the values on W, the intervened model's observational kernel at s', evaluated on S intersected with the preimage under the Z.random ∪ W-projection of the image of A under the filled-assignment map, equals the base model's observational kernel at the projected fixed assignment, evaluated on the analogous set.","s":"∀ {N : Type u_1} [inst : DecidableEq N] [inst_1 : Fintype N] {Ω : N → Type u_2} [inst_2 : (n : N) → MeasurableSpace (Ω n)] (M' : Causalean.SCM N Ω) (Z : Finset N) (hZ_obs : ∀ D ∈ Z, Causalean.SWIGNode.random D ∈ M'.observed) (hZ_fixed : ∀ D ∈ Z, Causalean.SWIGNode.fixed D ∉ M'.fixed) (W : Finset (Causalean.SWIGNode N)) (hZrW : Finset.image Causalean.SWIGNode.random Z ∪ W ⊆ M'.observed), Disjoint (Finset.image Causalean.SWIGNode.random Z) W → ∀ [MeasurableSingletonClass (Causalean.ValuesOn (Finset.image Causalean.SWIGNode.random Z ∪ W) (Causalean.swigΩ Ω))] (s' : (M'.fixSet Z hZ_obs hZ_fixed).FixedValues) {S : Set M'.ObservedValues} {A : Set (Causalean.ValuesOn W (Causalean.swigΩ Ω))}, MeasurableSet S → MeasurableSet A → ((M'.fixSet Z hZ_obs hZ_fixed).obsKernel s') (S ∩ Causalean.valuesProjection (Eq.symm (Causalean.SCM.fixSet_observed M' Z hZ_obs hZ_fixed) ▸ hZrW) ⁻¹' M'.fillZrW Z hZ_obs hZ_fixed W s' '' A) = (M'.obsKernel (M'.fixSetProj Z hZ_obs hZ_fixed s')) (S ∩ Causalean.valuesProjection hZrW ⁻¹' M'.fillZrW Z hZ_obs hZ_fixed W s' '' A)","l":324,"f":"Causalean/SCM/Do/Rule2Kernel/InterSingleton.lean"},{"n":"Causalean.SCM.obsKernel_inter_singleton_Zrand_eq","k":"theorem","m":"Causalean.SCM.Do.Rule2Kernel.InterSingleton","a":"SCM/Do/Rule2Kernel","t":2,"d":"Joint kernel agreement on the Z.random-level-set event.","s":"∀ {N : Type u_1} [inst : DecidableEq N] [inst_1 : Fintype N] {Ω : N → Type u_2} [inst_2 : (n : N) → MeasurableSpace (Ω n)] (M' : Causalean.SCM N Ω) (Z : Finset N) (hZ_obs : ∀ D ∈ Z, Causalean.SWIGNode.random D ∈ M'.observed) (hZ_fixed : ∀ D ∈ Z, Causalean.SWIGNode.fixed D ∉ M'.fixed) (W : Finset (Causalean.SWIGNode N)) (hZrW : Finset.image Causalean.SWIGNode.random Z ∪ W ⊆ M'.observed) [MeasurableSingletonClass (Causalean.ValuesOn (Finset.image Causalean.SWIGNode.random Z ∪ W) (Causalean.swigΩ Ω))] (s' : (M'.fixSet Z hZ_obs hZ_fixed).FixedValues) (w : Causalean.ValuesOn W (Causalean.swigΩ Ω)) {S : Set M'.ObservedValues}, MeasurableSet S → ((M'.fixSet Z hZ_obs hZ_fixed).obsKernel s') (S ∩ Causalean.valuesProjection (Eq.symm (Causalean.SCM.fixSet_observed M' Z hZ_obs hZ_fixed) ▸ hZrW) ⁻¹' {M'.fillZrW Z hZ_obs hZ_fixed W s' w}) = (M'.obsKernel (M'.fixSetProj Z hZ_obs hZ_fixed s')) (S ∩ Causalean.valuesProjection hZrW ⁻¹' {M'.fillZrW Z hZ_obs hZ_fixed W s' w})","l":39,"f":"Causalean/SCM/Do/Rule2Kernel/InterSingleton.lean"},{"n":"Causalean.SCM.obsKernel_map_W_eq","k":"theorem","m":"Causalean.SCM.Do.Rule3Conditional","a":"SCM/Do","t":2,"d":"Rule 3\\* on the W-marginal.","s":"∀ {N : Type u_1} [inst : DecidableEq N] [inst_1 : Fintype N] {Ω : N → Type u_2} [inst_2 : (n : N) → MeasurableSpace (Ω n)] (M' : Causalean.SCM N Ω) (Z : Finset N) (hZ_obs : ∀ D ∈ Z, Causalean.SWIGNode.random D ∈ M'.observed) (hZ_fixed : ∀ D ∈ Z, Causalean.SWIGNode.fixed D ∉ M'.fixed) (W : Finset (Causalean.SWIGNode N)) (hW : W ⊆ M'.observed), (∀ v ∈ W, ∀ d ∈ Z, ¬(M'.fixSet Z hZ_obs hZ_fixed).dag.isAncestor (Causalean.SWIGNode.fixed d) v) → ∀ (s' : (M'.fixSet Z hZ_obs hZ_fixed).FixedValues), MeasureTheory.Measure.map (Causalean.valuesProjection (Eq.symm (Causalean.SCM.fixSet_observed M' Z hZ_obs hZ_fixed) ▸ hW)) ((M'.fixSet Z hZ_obs hZ_fixed).obsKernel s') = MeasureTheory.Measure.map (Causalean.valuesProjection hW) (M'.obsKernel (M'.fixSetProj Z hZ_obs hZ_fixed s'))","l":154,"f":"Causalean/SCM/Do/Rule3Conditional.lean"},{"n":"Causalean.SCM.obsKernel_map_eq_obsCondKernel_comp","k":"theorem","m":"Causalean.SCM.ID.Toolkit.ObsChainRule","a":"SCM/ID/Toolkit","t":1,"d":"Observational chain rule. Fix a structural causal model M, a target node set Y and a conditioning node set CC with Y contained in the observed nodes and CC contained in the observed nodes. Then, at a fixed parameter slice s, the observational distribution of Y equals the observational conditional law of Y given CC composed with the observational marginal law of CC.","s":"∀ {N : Type u_1} [inst : DecidableEq N] [inst_1 : Fintype N] {Ω : N → Type u_2} [inst_2 : (n : N) → MeasurableSpace (Ω n)] (M : Causalean.SCM N Ω) (Y CC : Finset (Causalean.SWIGNode N)) (hY : Y ⊆ M.observed) (hCC : CC ⊆ M.observed) [inst_3 : StandardBorelSpace (Causalean.ValuesOn Y (Causalean.swigΩ Ω))] [inst_4 : Nonempty (Causalean.ValuesOn Y (Causalean.swigΩ Ω))] [inst_5 : MeasurableSpace.CountableOrCountablyGenerated M.FixedValues (Causalean.ValuesOn CC (Causalean.swigΩ Ω))] (s : M.FixedValues), MeasureTheory.Measure.map (Causalean.valuesProjection hY) (M.obsKernel s) = (MeasureTheory.Measure.map (Causalean.valuesProjection hCC) (M.obsKernel s)).bind ⇑((M.obsCondKernel Y CC hY hCC).sectR s)","l":40,"f":"Causalean/SCM/ID/Toolkit/ObsChainRule.lean"},{"n":"Causalean.SCM.obsKernel_map_prefixNodes","k":"theorem","m":"Causalean.SCM.Factored.ObsChainKernel","a":"SCM/Factored","t":2,"d":"Prefix form of the observational chain rule.","s":"∀ {N : Type u_1} [inst : DecidableEq N] [inst_1 : Fintype N] {Ω : N → Type u_2} [inst_2 : (n : N) → MeasurableSpace (Ω n)] (M : Causalean.SCM N Ω) (s : M.FixedValues) [inst_3 : ∀ (k : ℕ) (hk : k < M.observed.card), StandardBorelSpace (Causalean.ValuesOn {↑(M.observedAt ⟨k, hk⟩)} (Causalean.swigΩ Ω))] [inst_4 : ∀ (k : ℕ) (hk : k < M.observed.card), Nonempty (Causalean.ValuesOn {↑(M.observedAt ⟨k, hk⟩)} (Causalean.swigΩ Ω))] [inst_5 : ∀ (k : Fin M.observed.card), MeasurableSpace.CountableOrCountablyGenerated M.FixedValues (Causalean.ValuesOn (M.prefixNodes ↑k) (Causalean.swigΩ Ω))] (n : ℕ) (hn : n ≤ M.observed.card), MeasureTheory.Measure.map (Causalean.valuesProjection (Causalean.SCM.prefixNodes_subset_observed M n)) (M.obsKernel s) = (M.obsChainKernel n hn) s","l":611,"f":"Causalean/SCM/Factored/ObsChainKernel.lean"},{"n":"Causalean.SCM.obsKernel_map_prodWY_eq","k":"theorem","m":"Causalean.SCM.Do.Rule3Conditional","a":"SCM/Do","t":2,"d":"Rule 3\\* on the (W, Y) joint pushforward.","s":"∀ {N : Type u_1} [inst : DecidableEq N] [inst_1 : Fintype N] {Ω : N → Type u_2} [inst_2 : (n : N) → MeasurableSpace (Ω n)] (M' : Causalean.SCM N Ω) (Z : Finset N) (hZ_obs : ∀ D ∈ Z, Causalean.SWIGNode.random D ∈ M'.observed) (hZ_fixed : ∀ D ∈ Z, Causalean.SWIGNode.fixed D ∉ M'.fixed) (Y W : Finset (Causalean.SWIGNode N)) (hY : Y ⊆ M'.observed) (hW : W ⊆ M'.observed), (∀ v ∈ Y ∪ W, ∀ d ∈ Z, ¬(M'.fixSet Z hZ_obs hZ_fixed).dag.isAncestor (Causalean.SWIGNode.fixed d) v) → ∀ (s' : (M'.fixSet Z hZ_obs hZ_fixed).FixedValues), MeasureTheory.Measure.map (fun ω => (Causalean.valuesProjection (Eq.symm (Causalean.SCM.fixSet_observed M' Z hZ_obs hZ_fixed) ▸ hW) ω, Causalean.valuesProjection (Eq.symm (Causalean.SCM.fixSet_observed M' Z hZ_obs hZ_fixed) ▸ hY) ω)) ((M'.fixSet Z hZ_obs hZ_fixed).obsKernel s') = MeasureTheory.Measure.map (fun ω => (Causalean.valuesProjection hW ω, Causalean.valuesProjection hY ω)) (M'.obsKernel (M'.fixSetProj Z hZ_obs hZ_fixed s'))","l":76,"f":"Causalean/SCM/Do/Rule3Conditional.lean"},{"n":"Causalean.SCM.obsKernel_map_proj_eq_of_equiv","k":"theorem","m":"Causalean.SCM.ID.Frontdoor","a":"SCM/ID","t":2,"d":"Structurally equivalent structural causal models assign the same observational marginal law to a shared set of observed variables when their fixed-variable assignments correspond.","s":"∀ {N : Type u_1} [inst : DecidableEq N] [inst_1 : Fintype N] {Ω : N → Type u_2} [inst_2 : (n : N) → MeasurableSpace (Ω n)] [∀ (n : Causalean.SWIGNode N), StandardBorelSpace (Causalean.swigΩ Ω n)] [∀ (n : Causalean.SWIGNode N), Nonempty (Causalean.swigΩ Ω n)] {M₁ M₂ : Causalean.SCM N Ω}, M₁.Equiv M₂ → ∀ (Y : Finset (Causalean.SWIGNode N)) (hY₁ : Y ⊆ M₁.observed) (hY₂ : Y ⊆ M₂.observed) (s₁ : M₁.FixedValues) (s₂ : M₂.FixedValues), s₁ ≍ s₂ → MeasureTheory.Measure.map (Causalean.valuesProjection hY₁) (M₁.obsKernel s₁) = MeasureTheory.Measure.map (Causalean.valuesProjection hY₂) (M₂.obsKernel s₂)","l":230,"f":"Causalean/SCM/ID/Frontdoor.lean"},{"n":"Causalean.SCM.obsKernel_marginal_singleton_eq_latentProduct_agree","k":"theorem","m":"Causalean.SCM.ID.Density.MassBridge","a":"SCM/ID/Density","t":2,"d":"Mapping an observational kernel to a subcollection of observed coordinates turns a singleton mass into the latent-product mass of the corresponding agreement event.","s":"∀ {N : Type u_1} [inst : DecidableEq N] [inst_1 : Fintype N] {Ω : N → Type u_2} [inst_2 : (n : N) → MeasurableSpace (Ω n)] (M : Causalean.SCM N Ω) (s : M.FixedValues) {P : Finset (Causalean.SWIGNode N)} (hP : P ⊆ M.observed) [MeasurableSingletonClass (Causalean.ValuesOn P (Causalean.swigΩ Ω))] (x : M.ObservedValues), (MeasureTheory.Measure.map (Causalean.valuesProjection hP) (M.obsKernel s)) {Causalean.valuesProjection hP x} = M.latentProduct {ℓ | ∀ (v : ↥P), M.evalMap s ℓ ⟨↑v, Finset.mem_union_left M.unobserved (hP v.property)⟩ = x ⟨↑v, hP v.property⟩}","l":61,"f":"Causalean/SCM/ID/Density/MassBridge.lean"},{"n":"Causalean.SCM.obsKernel_marginal_singleton_eq_prod_qLocalMass","k":"theorem","m":"Causalean.SCM.ID.Density.QMass","a":"SCM/ID/Density","t":1,"d":"For a set of observed nodes P that is closed under observed parents, the singleton mass of the projection of the observational law onto P equals the product, over the full c-components C of the graph, of the local q-mass on C ∩ P.","s":"∀ {N : Type u_1} [inst : DecidableEq N] [inst_1 : Fintype N] {Ω : N → Type u_2} [inst_2 : (n : N) → MeasurableSpace (Ω n)] [(n : N) → Fintype (Ω n)] [∀ (n : N), MeasurableSingletonClass (Ω n)] (M : Causalean.SCM N Ω) (s : M.FixedValues) (P : Finset (Causalean.SWIGNode N)) (hP : M.ObsParentClosed P) (x : Causalean.ValuesOn M.observed (Causalean.swigΩ Ω)), (MeasureTheory.Measure.map (Causalean.valuesProjection hP.left) (M.obsKernel s)) {Causalean.valuesProjection hP.left x} = ∏ C ∈ M.cComponentSet, M.qLocalMass s (C ∩ P) (fun x hv => hP.left (Finset.mem_of_mem_inter_right hv)) x","l":1573,"f":"Causalean/SCM/ID/Density/QMass.lean"},{"n":"Causalean.SCM.obsSide_eq_witness","k":"theorem","m":"Causalean.SCM.Do.Rule2Kernel.WitnessBridge","a":"SCM/Do/Rule2Kernel","t":1,"d":"Obs-side witness-kernel identity (pair-conditioned, on the latent space). Let M be a structural causal model with node sets Y, W and a set of treatment names Z, where Y is observed, the pre-intervention nodes of Z are observed, and W is observed. Suppose the latent cut-set for Y relative to Z's pre-intervention nodes together with W is d-separated, in M's causal graph, from Z's pre-intervention nodes given W and the fixed nodes, and let h be a jointly measurable map such that, at a fixed baseline assignment s, the realized value of Y factors pointwise as h applied to the realized pre-intervention-Z value, the realized W value, and the cut-set's latent value. Then, under M's latent product measure, the conditional law of Y given the pair of realized pre-intervention-Z and W values equals, for almost every such pair, the pushforward under h applied to that pair of the conditional law of the latent cut-set given the realized W value.","s":"∀ {N : Type u_1} [inst : DecidableEq N] [inst_1 : Fintype N] {Ω : N → Type u_2} [inst_2 : (n : N) → MeasurableSpace (Ω n)] (M : Causalean.SCM N Ω) [StandardBorelSpace M.RandomValues] [∀ (n : Causalean.SWIGNode N), StandardBorelSpace (Causalean.swigΩ Ω n)] [∀ (n : Causalean.SWIGNode N), Nonempty (Causalean.swigΩ Ω n)] [∀ (s : M.FixedValues), MeasureTheory.IsFiniteMeasure (M.jointKernel s)] (Y W : Finset (Causalean.SWIGNode N)) (Z : Finset N) [inst_7 : StandardBorelSpace (Causalean.ValuesOn (M.cutsetLatent Y (Finset.image Causalean.SWIGNode.random Z ∪ W)) (Causalean.swigΩ Ω))] [inst_8 : Nonempty (Causalean.ValuesOn (M.cutsetLatent Y (Finset.image Causalean.SWIGNode.random Z ∪ W)) (Causalean.swigΩ Ω))] [inst_9 : StandardBorelSpace (Causalean.ValuesOn Y (Causalean.swigΩ Ω))] [inst_10 : Nonempty (Causalean.ValuesOn Y (Causalean.swigΩ Ω))] (hY : Y ⊆ M.observed) (hZr : Finset.image Causalean.SWIGNode.random Z ⊆ M.observed) (hW : W ⊆ M.observed), M.dag.dSep (M.cutsetLatent Y (Finset.image Causalean.SWIGNode.random Z ∪ W)) (Finset.image Causalean.SWIGNode.random Z) (W ∪ M.fixed) → ∀ (s : M.FixedValues) (h : Causalean.ValuesOn (Finset.image Causalean.SWIGNode.random Z) (Causalean.swigΩ Ω) → Causalean.ValuesOn W (Causalean.swigΩ Ω) → Causalean.ValuesOn (M.cutsetLatent Y (Finset.image Causalean.SWIGNode.random Z ∪ W)) (Causalean.swigΩ Ω) → Causalean.ValuesOn Y (Causalean.swigΩ Ω)), (Measurable fun p => h p.1.1 p.1.2 p.2) → (∀ (ℓ : M.LatentValues), Causalean.valuesProjection hY (M.randomToObserved (M.evalMap s ℓ)) = h (Causalean.valuesProjection hZr (M.randomToObserved (M.evalMap s ℓ))) (Causalean.valuesProjection hW (M.randomToObserved (M.evalMap s ℓ))) (Causalean.valuesProjection (Causalean.SCM.cutsetLatent_subset M Y (Finset.image Causalean.SWIGNode.random Z ∪ W)) ℓ)) → (fun p => (ProbabilityTheory.condDistrib (fun ℓ => Causalean.valuesProjection hY (M.randomToObserved (M.evalMap s ℓ))) (fun ℓ => (Causalean.valuesProjection hZr (M.randomToObserved (M.evalMap s ℓ)), Causalean.valuesProjection hW (M.randomToObserved (M.evalMap s ℓ)))) M.latentProduct) p) =ᵐ[MeasureTheory.Measure.map (fun ℓ => (Causalean.valuesProjection hZr (M.randomToObserved (M.evalMap s ℓ)), Causalean.valuesProjection hW (M.randomToObserved (M.evalMap s ℓ)))) M.latentProduct] fun p => MeasureTheory.Measure.map (h p.1 p.2) ((ProbabilityTheory.condDistrib (Causalean.valuesProjection (Causalean.SCM.cutsetLatent_subset M Y (Finset.image Causalean.SWIGNode.random Z ∪ W))) (fun ℓ => Causalean.valuesProjection hW (M.randomToObserved (M.evalMap s ℓ))) M.latentProduct) p.2)","l":610,"f":"Causalean/SCM/Do/Rule2Kernel/WitnessBridge.lean"},{"n":"Causalean.SCM.obsStepCondDensity","k":"def","m":"Causalean.SCM.ID.Density.ChainRuleDensity","a":"SCM/ID/Density","t":1,"d":"The one-node conditional density factor at an observed coordinate.","s":"{N : Type u_1} → [inst : DecidableEq N] → [inst_1 : Fintype N] → {Ω : N → Type u_2} → [inst_2 : (n : N) → MeasurableSpace (Ω n)] → (M : Causalean.SCM N Ω) → Causalean.SCM.ReferenceMeasures Ω → M.FixedValues → (i : Fin M.observed.card) → [∀ (s' : M.FixedValues), MeasureTheory.IsFiniteMeasure (M.obsKernel s')] → [StandardBorelSpace (Causalean.ValuesOn {↑(M.observedAt i)} (Causalean.swigΩ Ω))] → [Nonempty (Causalean.ValuesOn {↑(M.observedAt i)} (Causalean.swigΩ Ω))] → [MeasurableSpace.CountableOrCountablyGenerated M.FixedValues (Causalean.ValuesOn (M.prefixNodes ↑i) (Causalean.swigΩ Ω))] → Causalean.ValuesOn M.observed (Causalean.swigΩ Ω) → ENNReal","l":39,"f":"Causalean/SCM/ID/Density/ChainRuleDensity.lean"},{"n":"Causalean.SCM.obsStepCondDensity_eq_mass_ratio","k":"theorem","m":"Causalean.SCM.ID.Density.MassBridge","a":"SCM/ID/Density","t":1,"d":"At the i-th observed coordinate, if the one-step observational conditional kernel is absolutely continuous with respect to the reference measure on that coordinate, and the reference measure's singleton mass at the recorded value is nonzero and finite, then the one-node observational step density obsStepCondDensity equals the conditional kernel's singleton mass at the recorded value divided by the reference measure's singleton mass there.","s":"∀ {N : Type u_1} [inst : DecidableEq N] [inst_1 : Fintype N] {Ω : N → Type u_2} [inst_2 : (n : N) → MeasurableSpace (Ω n)] (M : Causalean.SCM N Ω) (ref : Causalean.SCM.ReferenceMeasures Ω) (s : M.FixedValues) (i : Fin M.observed.card) [MeasurableSingletonClass (Causalean.swigΩ Ω ↑(M.observedAt i))] [inst_4 : ∀ (s' : M.FixedValues), MeasureTheory.IsFiniteMeasure (M.obsKernel s')] [inst_5 : StandardBorelSpace (Causalean.ValuesOn {↑(M.observedAt i)} (Causalean.swigΩ Ω))] [inst_6 : Nonempty (Causalean.ValuesOn {↑(M.observedAt i)} (Causalean.swigΩ Ω))] [inst_7 : MeasurableSpace.CountableOrCountablyGenerated M.FixedValues (Causalean.ValuesOn (M.prefixNodes ↑i) (Causalean.swigΩ Ω))] (x : Causalean.ValuesOn M.observed (Causalean.swigΩ Ω)), ((M.obsStepCondKernel i.isLt) (s, Causalean.valuesProjection (Causalean.SCM.prefixNodes_subset_observed M ↑i) x)).AbsolutelyContinuous (ref.μ ↑(M.observedAt i)) → (ref.μ ↑(M.observedAt i)) {x (M.observedAt i)} ≠ 0 → (ref.μ ↑(M.observedAt i)) {x (M.observedAt i)} ≠ ⊤ → M.obsStepCondDensity ref s i x = ((M.obsStepCondKernel i.isLt) (s, Causalean.valuesProjection (Causalean.SCM.prefixNodes_subset_observed M ↑i) x)) {x (M.observedAt i)} / (ref.μ ↑(M.observedAt i)) {x (M.observedAt i)}","l":251,"f":"Causalean/SCM/ID/Density/MassBridge.lean"},{"n":"Causalean.SCM.obsStepCondKernel","k":"def","m":"Causalean.SCM.Factored.ObsChainKernel","a":"SCM/Factored","t":1,"d":"The single-node observational conditional kernel for the n-th observed node given the first n observed nodes.","s":"{N : Type u_1} → [inst : DecidableEq N] → [inst_1 : Fintype N] → {Ω : N → Type u_2} → [inst_2 : (n : N) → MeasurableSpace (Ω n)] → (M : Causalean.SCM N Ω) → {n : ℕ} → (hn : n < M.observed.card) → [StandardBorelSpace (Causalean.ValuesOn {↑(M.observedAt ⟨n, hn⟩)} (Causalean.swigΩ Ω))] → [Nonempty (Causalean.ValuesOn {↑(M.observedAt ⟨n, hn⟩)} (Causalean.swigΩ Ω))] → [MeasurableSpace.CountableOrCountablyGenerated M.FixedValues (Causalean.ValuesOn (M.prefixNodes n) (Causalean.swigΩ Ω))] → ProbabilityTheory.Kernel (M.FixedValues × Causalean.ValuesOn (M.prefixNodes n) (Causalean.swigΩ Ω)) (Causalean.swigΩ Ω ↑(M.observedAt ⟨n, hn⟩))","l":253,"f":"Causalean/SCM/Factored/ObsChainKernel.lean"},{"n":"Causalean.SCM.obsStepCondKernel_map_singletonValues","k":"theorem","m":"Causalean.SCM.Factored.ObsChainKernel","a":"SCM/Factored","t":2,"d":"Mapping the scalar step kernel back to the singleton tuple recovers the conditional kernel it was built from.","s":"∀ {N : Type u_1} [inst : DecidableEq N] [inst_1 : Fintype N] {Ω : N → Type u_2} [inst_2 : (n : N) → MeasurableSpace (Ω n)] (M : Causalean.SCM N Ω) {n : ℕ} (hn : n < M.observed.card) [inst_3 : StandardBorelSpace (Causalean.ValuesOn {↑(M.observedAt ⟨n, hn⟩)} (Causalean.swigΩ Ω))] [inst_4 : Nonempty (Causalean.ValuesOn {↑(M.observedAt ⟨n, hn⟩)} (Causalean.swigΩ Ω))] [inst_5 : MeasurableSpace.CountableOrCountablyGenerated M.FixedValues (Causalean.ValuesOn (M.prefixNodes n) (Causalean.swigΩ Ω))], (M.obsStepCondKernel hn).map Causalean.SCM.singletonValues = M.obsCondKernel {↑(M.observedAt ⟨n, hn⟩)} (M.prefixNodes n) (fun ⦃v⦄ hv => have hv_eq := Eq.mp Finset.mem_singleton._simp_1 hv; of_eq_true (Eq.trans (congrArg (Membership.mem M.observed) hv_eq) (eq_true (M.observedAt ⟨n, hn⟩).property))) (Causalean.SCM.prefixNodes_subset_observed M n)","l":307,"f":"Causalean/SCM/Factored/ObsChainKernel.lean"},{"n":"Causalean.SCM.obsStepCondKernel_sectR_map_singletonValues","k":"theorem","m":"Causalean.SCM.Factored.ObsChainKernel","a":"SCM/Factored","t":2,"d":"Slice form of obsStepCondKernel_map_singletonValues.","s":"∀ {N : Type u_1} [inst : DecidableEq N] [inst_1 : Fintype N] {Ω : N → Type u_2} [inst_2 : (n : N) → MeasurableSpace (Ω n)] (M : Causalean.SCM N Ω) {n : ℕ} (hn : n < M.observed.card) [inst_3 : StandardBorelSpace (Causalean.ValuesOn {↑(M.observedAt ⟨n, hn⟩)} (Causalean.swigΩ Ω))] [inst_4 : Nonempty (Causalean.ValuesOn {↑(M.observedAt ⟨n, hn⟩)} (Causalean.swigΩ Ω))] [inst_5 : MeasurableSpace.CountableOrCountablyGenerated M.FixedValues (Causalean.ValuesOn (M.prefixNodes n) (Causalean.swigΩ Ω))] (s : M.FixedValues), ((M.obsStepCondKernel hn).sectR s).map Causalean.SCM.singletonValues = (M.obsCondKernel {↑(M.observedAt ⟨n, hn⟩)} (M.prefixNodes n) (fun ⦃v⦄ hv => have hv_eq := Eq.mp Finset.mem_singleton._simp_1 hv; of_eq_true (Eq.trans (congrArg (Membership.mem M.observed) hv_eq) (eq_true (M.observedAt ⟨n, hn⟩).property))) (Causalean.SCM.prefixNodes_subset_observed M n)).sectR s","l":345,"f":"Causalean/SCM/Factored/ObsChainKernel.lean"},{"n":"Causalean.SCM.obs_condIndep_of_full","k":"theorem","m":"Causalean.SCM.Do.ObsMarkov","a":"SCM/Do","t":2,"d":"Projection lemma. For X, Y, Z ⊆ V (all observed), conditional independence at the full distribution level (jointKernel) implies conditional independence at the observational level (obsKernel).","s":"∀ {N : Type uN} [inst : DecidableEq N] [inst_1 : Fintype N] {Ω : N → Type uΩ} [inst_2 : (n : N) → MeasurableSpace (Ω n)] (M : Causalean.SCM N Ω) [inst_3 : StandardBorelSpace M.RandomValues] [inst_4 : StandardBorelSpace M.ObservedValues] {X Y Z : Finset (Causalean.SWIGNode N)} [StandardBorelSpace (Causalean.ValuesOn X (Causalean.swigΩ Ω))] [Nonempty (Causalean.ValuesOn X (Causalean.swigΩ Ω))] [StandardBorelSpace (Causalean.ValuesOn Y (Causalean.swigΩ Ω))] [Nonempty (Causalean.ValuesOn Y (Causalean.swigΩ Ω))] (hX : X ⊆ M.observed) (hY : Y ⊆ M.observed) (hZ : Z ⊆ M.observed) (s : M.FixedValues), M.FullCondIndep X Y Z (LE.le.trans hX (Causalean.SCM.observed_subset_randomVars M)) (LE.le.trans hY (Causalean.SCM.observed_subset_randomVars M)) (LE.le.trans hZ (Causalean.SCM.observed_subset_randomVars M)) (M.jointKernel s) → M.ObsCondIndep X Y Z hX hY hZ (M.obsKernel s)","l":45,"f":"Causalean/SCM/Do/ObsMarkov.lean"},{"n":"Causalean.SCM.observedAt","k":"def","m":"Causalean.SCM.Model.SCM","a":"SCM/Model","t":1,"d":"The observed-node enumeration returns the node at a given position in canonical topological order.","s":"{N : Type uN} → [inst : DecidableEq N] → [inst_1 : Fintype N] → {Ω : N → Type uΩ} → [inst_2 : (n : N) → MeasurableSpace (Ω n)] → (M : Causalean.SCM N Ω) → Fin M.observed.card → ↥M.observed","l":194,"f":"Causalean/SCM/Model/SCM.lean"},{"n":"Causalean.SCM.observedAt_mem_prefixNodes_iff","k":"theorem","m":"Causalean.SCM.Factored.ObsChainKernel","a":"SCM/Factored","t":2,"d":"An observed node at index i belongs to the first n nodes iff i < n.","s":"∀ {N : Type u_1} [inst : DecidableEq N] [inst_1 : Fintype N] {Ω : N → Type u_2} [inst_2 : (n : N) → MeasurableSpace (Ω n)] (M : Causalean.SCM N Ω) (n : ℕ) (i : Fin M.observed.card), ↑(M.observedAt i) ∈ M.prefixNodes n ↔ ↑i < n","l":91,"f":"Causalean/SCM/Factored/ObsChainKernel.lean"},{"n":"Causalean.SCM.observedAt_not_mem_prefixNodes","k":"theorem","m":"Causalean.SCM.Factored.ObsChainKernel","a":"SCM/Factored","t":2,"d":"The next observed node is not in the previous prefix.","s":"∀ {N : Type u_1} [inst : DecidableEq N] [inst_1 : Fintype N] {Ω : N → Type u_2} [inst_2 : (n : N) → MeasurableSpace (Ω n)] (M : Causalean.SCM N Ω) {n : ℕ} (hn : n < M.observed.card), ↑(M.observedAt ⟨n, hn⟩) ∉ M.prefixNodes n","l":120,"f":"Causalean/SCM/Factored/ObsChainKernel.lean"},{"n":"Causalean.SCM.observedAt_observedIndex","k":"theorem","m":"Causalean.SCM.Model.SCM","a":"SCM/Model","t":2,"d":"Looking up an observed node by its canonical index recovers the same node.","s":"∀ {N : Type uN} [inst : DecidableEq N] [inst_1 : Fintype N] {Ω : N → Type uΩ} [inst_2 : (n : N) → MeasurableSpace (Ω n)] (M : Causalean.SCM N Ω) (v : ↥M.observed), ↑(M.observedAt (M.observedIndex v)) = ↑v","l":209,"f":"Causalean/SCM/Model/SCM.lean"},{"n":"Causalean.SCM.observedAt_observedIndex_subtype","k":"theorem","m":"Causalean.SCM.ID.Density.LatentBlocks","a":"SCM/ID/Density","t":2,"d":"Taking an observed node to its topological index and back recovers the same observed-node subtype value.","s":"∀ {N : Type u_1} [inst : DecidableEq N] [inst_1 : Fintype N] {Ω : N → Type u_2} [inst_2 : (n : N) → MeasurableSpace (Ω n)] (M : Causalean.SCM N Ω) {v : Causalean.SWIGNode N} (hv : v ∈ M.observed), M.observedAt (M.observedIndex ⟨v, hv⟩) = ⟨v, hv⟩","l":188,"f":"Causalean/SCM/ID/Density/LatentBlocks.lean"},{"n":"Causalean.SCM.observedIndex","k":"def","m":"Causalean.SCM.Model.SCM","a":"SCM/Model","t":1,"d":"The observed-node index returns the canonical topological position of an observed node.","s":"{N : Type uN} → [inst : DecidableEq N] → [inst_1 : Fintype N] → {Ω : N → Type uΩ} → [inst_2 : (n : N) → MeasurableSpace (Ω n)] → (M : Causalean.SCM N Ω) → ↥M.observed → Fin M.observed.card","l":202,"f":"Causalean/SCM/Model/SCM.lean"},{"n":"Causalean.SCM.observedIndex_observedAt","k":"theorem","m":"Causalean.SCM.Model.SCM","a":"SCM/Model","t":2,"d":"Looking up the canonical index of the observed node at a position recovers that position.","s":"∀ {N : Type uN} [inst : DecidableEq N] [inst_1 : Fintype N] {Ω : N → Type uΩ} [inst_2 : (n : N) → MeasurableSpace (Ω n)] (M : Causalean.SCM N Ω) (k : Fin M.observed.card), M.observedIndex (M.observedAt k) = k","l":217,"f":"Causalean/SCM/Model/SCM.lean"},{"n":"Causalean.SCM.observedPredecessors_observedAt","k":"theorem","m":"Causalean.SCM.Factored.ObsChainKernel","a":"SCM/Factored","t":2,"d":"For the node at index n, Tian's full-history predecessor set is exactly the first n observed nodes.","s":"∀ {N : Type u_1} [inst : DecidableEq N] [inst_1 : Fintype N] {Ω : N → Type u_2} [inst_2 : (n : N) → MeasurableSpace (Ω n)] (M : Causalean.SCM N Ω) {n : ℕ} (hn : n < M.observed.card), M.observedPredecessors ↑(M.observedAt ⟨n, hn⟩) = M.prefixNodes n","l":161,"f":"Causalean/SCM/Factored/ObsChainKernel.lean"},{"n":"Causalean.SCM.observedPrefixValue","k":"def","m":"Causalean.SCM.Factored.PrefixState","a":"SCM/Factored","t":1,"d":"Read the value of an observed node at position i < n from a prefix state of length n.","s":"{N : Type uN} → [inst : DecidableEq N] → [inst_1 : Fintype N] → {Ω : N → Type uΩ} → [inst_2 : (n : N) → MeasurableSpace (Ω n)] → (M : Causalean.SCM N Ω) → {n : ℕ} → (hn : n ≤ M.observed.card) → M.ObservedPrefixValues n hn → (i : Fin n) → Causalean.swigΩ Ω ↑(M.observedAt ⟨↑i, Nat.lt_of_lt_of_le i.isLt hn⟩)","l":63,"f":"Causalean/SCM/Factored/PrefixState.lean"},{"n":"Causalean.SCM.observedPrefixValue_succ_castSucc","k":"theorem","m":"Causalean.SCM.Factored.EvalMapCorrespond","a":"SCM/Factored","t":2,"d":"Appending an observed coordinate to a prefix leaves every earlier coordinate of the prefix unchanged.","s":"∀ {N : Type uN} [inst : DecidableEq N] [inst_1 : Fintype N] {Ω : N → Type uΩ} [inst_2 : (n : N) → MeasurableSpace (Ω n)] (M : Causalean.SCM N Ω) {k : ℕ} (hn : k + 1 ≤ M.observed.card) (ξ : M.ObservedPrefixValues k (Nat.le_of_succ_le hn)) (y : Causalean.swigΩ Ω ↑(M.observedAt ⟨k, hn⟩)) (j : Fin k), M.observedPrefixValue hn (ξ, y) j.castSucc = M.observedPrefixValue (Nat.le_of_succ_le hn) ξ j","l":181,"f":"Causalean/SCM/Factored/EvalMapCorrespond.lean"},{"n":"Causalean.SCM.observedPrefixValue_succ_last","k":"theorem","m":"Causalean.SCM.Factored.EvalMapCorrespond","a":"SCM/Factored","t":2,"d":"Appending an observed coordinate to a prefix makes the final coordinate of the expanded prefix equal to the appended value.","s":"∀ {N : Type uN} [inst : DecidableEq N] [inst_1 : Fintype N] {Ω : N → Type uΩ} [inst_2 : (n : N) → MeasurableSpace (Ω n)] (M : Causalean.SCM N Ω) {k : ℕ} (hn : k + 1 ≤ M.observed.card) (ξ : M.ObservedPrefixValues k (Nat.le_of_succ_le hn)) (y : Causalean.swigΩ Ω ↑(M.observedAt ⟨k, hn⟩)), M.observedPrefixValue hn (ξ, y) (Fin.last k) = y","l":171,"f":"Causalean/SCM/Factored/EvalMapCorrespond.lean"},{"n":"Causalean.SCM.observed_parent_index_lt","k":"theorem","m":"Causalean.SCM.Model.SCM","a":"SCM/Model","t":1,"d":"For a structural causal model M, fix a valid position n among the observed nodes, and let p be a node such that there is an edge from p to the n-th node in the canonical observed order and p itself is observed; then the canonical index of p among the observed nodes is strictly less than n — an observed parent always precedes its child in the canonical observed topological order.","s":"∀ {N : Type uN} [inst : DecidableEq N] [inst_1 : Fintype N] {Ω : N → Type uΩ} [inst_2 : (n : N) → MeasurableSpace (Ω n)] (M : Causalean.SCM N Ω) {n : ℕ} (hn : n < M.observed.card) {p : Causalean.SWIGNode N}, M.dag.edge p ↑(M.observedAt ⟨n, hn⟩) → ∀ (hobs : p ∈ M.observed), M.observedIndex ⟨p, hobs⟩ < ⟨n, hn⟩","l":228,"f":"Causalean/SCM/Model/SCM.lean"},{"n":"Causalean.SCM.observed_subset_randomVars","k":"theorem","m":"Causalean.SCM.Do.LocalMarkov","a":"SCM/Do","t":2,"d":"Observed nodes lie in randomVars.","s":"∀ {N : Type uN} [inst : DecidableEq N] [inst_1 : Fintype N] {Ω : N → Type uΩ} [inst_2 : (n : N) → MeasurableSpace (Ω n)] (M : Causalean.SCM N Ω), M.observed ⊆ M.randomVars","l":45,"f":"Causalean/SCM/Do/LocalMarkov.lean"},{"n":"Causalean.SCM.orderedLatentPrefixFullToRandom","k":"def","m":"Causalean.SCM.Factored.Factorization","a":"SCM/Factored","t":1,"d":"Reindex the full prefix state (all observed nodes generated, plus the latent tuple) to RandomValues M = ValuesOn (observed ∪ unobserved) (swigΩ Ω).","s":"{N : Type uN} → [inst : DecidableEq N] → [inst_1 : Fintype N] → {Ω : N → Type uΩ} → [inst_2 : (n : N) → MeasurableSpace (Ω n)] → (M : Causalean.SCM N Ω) → M.OrderedLatentPrefixValues M.observed.card (le_refl M.observed.card) → M.RandomValues","l":51,"f":"Causalean/SCM/Factored/Factorization.lean"},{"n":"Causalean.SCM.overrideOn","k":"def","m":"Causalean.SCM.ID.Density.PiUnion","a":"SCM/ID/Density","t":1,"d":"Override the coordinates in W of an assignment on I, leaving the other coordinates unchanged.","s":"{N : Type u_1} → [DecidableEq N] → {Ω : N → Type u_2} → {I W : Finset (Causalean.SWIGNode N)} → Causalean.ValuesOn I (Causalean.swigΩ Ω) → Causalean.ValuesOn W (Causalean.swigΩ Ω) → Causalean.ValuesOn I (Causalean.swigΩ Ω)","l":42,"f":"Causalean/SCM/ID/Density/PiUnion.lean"},{"n":"Causalean.SCM.overrideOn_mem","k":"theorem","m":"Causalean.SCM.ID.Density.PiUnion","a":"SCM/ID/Density","t":2,"d":"On overridden coordinates, overrideOn reads from the replacement assignment.","s":"∀ {N : Type u_1} [inst : DecidableEq N] {Ω : N → Type u_2} [(n : N) → MeasurableSpace (Ω n)] {I W : Finset (Causalean.SWIGNode N)} (x : Causalean.ValuesOn I (Causalean.swigΩ Ω)) (y : Causalean.ValuesOn W (Causalean.swigΩ Ω)) (i : ↥I) (hiW : ↑i ∈ W), Causalean.SCM.overrideOn x y i = y ⟨↑i, hiW⟩","l":49,"f":"Causalean/SCM/ID/Density/PiUnion.lean"},{"n":"Causalean.SCM.overrideOn_notMem","k":"theorem","m":"Causalean.SCM.ID.Density.PiUnion","a":"SCM/ID/Density","t":2,"d":"Away from overridden coordinates, overrideOn keeps the original assignment.","s":"∀ {N : Type u_1} [inst : DecidableEq N] {Ω : N → Type u_2} [(n : N) → MeasurableSpace (Ω n)] {I W : Finset (Causalean.SWIGNode N)} (x : Causalean.ValuesOn I (Causalean.swigΩ Ω)) (y : Causalean.ValuesOn W (Causalean.swigΩ Ω)) (i : ↥I), ↑i ∉ W → Causalean.SCM.overrideOn x y i = x i","l":57,"f":"Causalean/SCM/ID/Density/PiUnion.lean"},{"n":"Causalean.SCM.overrideOn_overrideOn","k":"theorem","m":"Causalean.SCM.ID.Density.PiUnion","a":"SCM/ID/Density","t":2,"d":"Re-overriding the same coordinate set keeps the last replacement assignment.","s":"∀ {N : Type u_1} [inst : DecidableEq N] {Ω : N → Type u_2} [(n : N) → MeasurableSpace (Ω n)] {I W : Finset (Causalean.SWIGNode N)} (x : Causalean.ValuesOn I (Causalean.swigΩ Ω)) (y z : Causalean.ValuesOn W (Causalean.swigΩ Ω)), Causalean.SCM.overrideOn (Causalean.SCM.overrideOn x y) z = Causalean.SCM.overrideOn x z","l":65,"f":"Causalean/SCM/ID/Density/PiUnion.lean"},{"n":"Causalean.SCM.parentMap","k":"def","m":"Causalean.SCM.Model.Evaluation","a":"SCM/Model","t":1,"d":"A parent-value tuple is assembled from fixed, latent, and earlier observed values.","s":"{N : Type u_1} → [inst : DecidableEq N] → [inst_1 : Fintype N] → {Ω : N → Type u_2} → [inst_2 : (n : N) → MeasurableSpace (Ω n)] → (M : Causalean.SCM N Ω) → M.FixedValues → M.LatentValues → {n : ℕ} → (hn : n < M.observed.card) → ((m : ℕ) → m < n → (hm : m < M.observed.card) → Causalean.swigΩ Ω ↑(M.observedAt ⟨m, hm⟩)) → (w : ↥(M.dag.parents ↑(M.observedAt ⟨n, hn⟩))) → Causalean.swigΩ Ω ↑w","l":45,"f":"Causalean/SCM/Model/Evaluation.lean"},{"n":"Causalean.SCM.parentMapOverride","k":"def","m":"Causalean.SCM.Model.EvalOverrideC","a":"SCM/Model","t":1,"d":"The override parent-value tuple reads latent, fixed, overridden, or recursively evaluated observed parent values as appropriate.","s":"{N : Type u_1} → [inst : DecidableEq N] → [inst_1 : Fintype N] → {Ω : N → Type u_2} → [inst_2 : (n : N) → MeasurableSpace (Ω n)] → (M : Causalean.SCM N Ω) → {C : Finset (Causalean.SWIGNode N)} → M.FixedValues → Causalean.ValuesOn C (Causalean.swigΩ Ω) → M.LatentValues → {n : ℕ} → (hn : n < M.observed.card) → ((m : ℕ) → m < n → (hm : m < M.observed.card) → Causalean.swigΩ Ω ↑(M.observedAt ⟨m, hm⟩)) → (w : ↥(M.dag.parents ↑(M.observedAt ⟨n, hn⟩))) → Causalean.swigΩ Ω ↑w","l":70,"f":"Causalean/SCM/Model/EvalOverrideC.lean"},{"n":"Causalean.SCM.parentMapOverride_C","k":"theorem","m":"Causalean.SCM.Model.EvalOverrideC","a":"SCM/Model","t":2,"d":"The override parent-value tuple reads an overridden observed parent directly from the override assignment.","s":"∀ {N : Type u_1} [inst : DecidableEq N] [inst_1 : Fintype N] {Ω : N → Type u_2} [inst_2 : (n : N) → MeasurableSpace (Ω n)] (M : Causalean.SCM N Ω) {C : Finset (Causalean.SWIGNode N)}, C ⊆ M.observed → ∀ (s : M.FixedValues) (c : Causalean.ValuesOn C (Causalean.swigΩ Ω)) (ℓ : M.LatentValues) {n : ℕ} (hn : n < M.observed.card) (prev : (m : ℕ) → m < n → (hm : m < M.observed.card) → Causalean.swigΩ Ω ↑(M.observedAt ⟨m, hm⟩)) (w : ↥(M.dag.parents ↑(M.observedAt ⟨n, hn⟩))) (hc : ↑w ∈ C), M.parentMapOverride s c ℓ hn prev w = c ⟨↑w, hc⟩","l":137,"f":"Causalean/SCM/Model/EvalOverrideC.lean"},{"n":"Causalean.SCM.parentMapOverride_fixed","k":"theorem","m":"Causalean.SCM.Model.EvalOverrideC","a":"SCM/Model","t":2,"d":"The override parent-value tuple reads a fixed parent directly from the fixed-value assignment.","s":"∀ {N : Type u_1} [inst : DecidableEq N] [inst_1 : Fintype N] {Ω : N → Type u_2} [inst_2 : (n : N) → MeasurableSpace (Ω n)] (M : Causalean.SCM N Ω) {C : Finset (Causalean.SWIGNode N)}, C ⊆ M.observed → ∀ (s : M.FixedValues) (c : Causalean.ValuesOn C (Causalean.swigΩ Ω)) (ℓ : M.LatentValues) {n : ℕ} (hn : n < M.observed.card) (prev : (m : ℕ) → m < n → (hm : m < M.observed.card) → Causalean.swigΩ Ω ↑(M.observedAt ⟨m, hm⟩)) (w : ↥(M.dag.parents ↑(M.observedAt ⟨n, hn⟩))) (hfix : ↑w ∈ M.fixed), M.parentMapOverride s c ℓ hn prev w = s ⟨↑w, hfix⟩","l":117,"f":"Causalean/SCM/Model/EvalOverrideC.lean"},{"n":"Causalean.SCM.parentMapOverride_observed","k":"theorem","m":"Causalean.SCM.Model.EvalOverrideC","a":"SCM/Model","t":2,"d":"The override parent-value tuple reads a non-overridden observed parent from the previous recursive values.","s":"∀ {N : Type u_1} [inst : DecidableEq N] [inst_1 : Fintype N] {Ω : N → Type u_2} [inst_2 : (n : N) → MeasurableSpace (Ω n)] (M : Causalean.SCM N Ω) {C : Finset (Causalean.SWIGNode N)}, C ⊆ M.observed → ∀ (s : M.FixedValues) (c : Causalean.ValuesOn C (Causalean.swigΩ Ω)) (ℓ : M.LatentValues) {n : ℕ} (hn : n < M.observed.card) (prev : (m : ℕ) → m < n → (hm : m < M.observed.card) → Causalean.swigΩ Ω ↑(M.observedAt ⟨m, hm⟩)) (w : ↥(M.dag.parents ↑(M.observedAt ⟨n, hn⟩))) (hobs : ↑w ∈ M.observed), ↑w ∉ C → M.parentMapOverride s c ℓ hn prev w = Causalean.SCM.observedAt_observedIndex M ⟨↑w, hobs⟩ ▸ prev (↑(M.observedIndex ⟨↑w, hobs⟩)) (Causalean.SCM.observed_parent_index_lt M hn ((Causalean.DAG.mem_parents M.dag).mp w.property) hobs) (M.observedIndex ⟨↑w, hobs⟩).isLt","l":153,"f":"Causalean/SCM/Model/EvalOverrideC.lean"},{"n":"Causalean.SCM.parentMapOverride_unobserved","k":"theorem","m":"Causalean.SCM.Model.EvalOverrideC","a":"SCM/Model","t":2,"d":"The override parent-value tuple reads a latent parent directly from the latent assignment.","s":"∀ {N : Type u_1} [inst : DecidableEq N] [inst_1 : Fintype N] {Ω : N → Type u_2} [inst_2 : (n : N) → MeasurableSpace (Ω n)] (M : Causalean.SCM N Ω) {C : Finset (Causalean.SWIGNode N)}, C ⊆ M.observed → ∀ (s : M.FixedValues) (c : Causalean.ValuesOn C (Causalean.swigΩ Ω)) (ℓ : M.LatentValues) {n : ℕ} (hn : n < M.observed.card) (prev : (m : ℕ) → m < n → (hm : m < M.observed.card) → Causalean.swigΩ Ω ↑(M.observedAt ⟨m, hm⟩)) (w : ↥(M.dag.parents ↑(M.observedAt ⟨n, hn⟩))) (huo : ↑w ∈ M.unobserved), M.parentMapOverride s c ℓ hn prev w = ℓ ⟨↑w, huo⟩","l":104,"f":"Causalean/SCM/Model/EvalOverrideC.lean"},{"n":"Causalean.SCM.parentMap_fixed","k":"theorem","m":"Causalean.SCM.Model.Evaluation","a":"SCM/Model","t":2,"d":"The parent-value tuple reads a fixed parent directly from the fixed-value assignment.","s":"∀ {N : Type u_1} [inst : DecidableEq N] [inst_1 : Fintype N] {Ω : N → Type u_2} [inst_2 : (n : N) → MeasurableSpace (Ω n)] (M : Causalean.SCM N Ω) (s : M.FixedValues) (ℓ : M.LatentValues) {n : ℕ} (hn : n < M.observed.card) (prev : (m : ℕ) → m < n → (hm : m < M.observed.card) → Causalean.swigΩ Ω ↑(M.observedAt ⟨m, hm⟩)) (w : ↥(M.dag.parents ↑(M.observedAt ⟨n, hn⟩))) (hfix : ↑w ∈ M.fixed), M.parentMap s ℓ hn prev w = s ⟨↑w, hfix⟩","l":95,"f":"Causalean/SCM/Model/Evaluation.lean"},{"n":"Causalean.SCM.parentMap_observed","k":"theorem","m":"Causalean.SCM.Model.Evaluation","a":"SCM/Model","t":2,"d":"The parent-value tuple reads an observed parent from the previously computed observed values.","s":"∀ {N : Type u_1} [inst : DecidableEq N] [inst_1 : Fintype N] {Ω : N → Type u_2} [inst_2 : (n : N) → MeasurableSpace (Ω n)] (M : Causalean.SCM N Ω) (s : M.FixedValues) (ℓ : M.LatentValues) {n : ℕ} (hn : n < M.observed.card) (prev : (m : ℕ) → m < n → (hm : m < M.observed.card) → Causalean.swigΩ Ω ↑(M.observedAt ⟨m, hm⟩)) (w : ↥(M.dag.parents ↑(M.observedAt ⟨n, hn⟩))) (hobs : ↑w ∈ M.observed), M.parentMap s ℓ hn prev w = Causalean.SCM.observedAt_observedIndex M ⟨↑w, hobs⟩ ▸ prev (↑(M.observedIndex ⟨↑w, hobs⟩)) (Causalean.SCM.observed_parent_index_lt M hn ((Causalean.DAG.mem_parents M.dag).mp w.property) hobs) (M.observedIndex ⟨↑w, hobs⟩).isLt","l":113,"f":"Causalean/SCM/Model/Evaluation.lean"},{"n":"Causalean.SCM.parentMap_prevFromObservedValues_eq_dispatch","k":"theorem","m":"Causalean.SCM.ID.Density.LatentBlocks","a":"SCM/ID/Density","t":2,"d":"","s":"∀ {N : Type u_1} [inst : DecidableEq N] [inst_1 : Fintype N] {Ω : N → Type u_2} [inst_2 : (n : N) → MeasurableSpace (Ω n)] (M : Causalean.SCM N Ω) (s : M.FixedValues) (x : Causalean.ValuesOn M.observed (Causalean.swigΩ Ω)) (ℓ : M.LatentValues) (j : Fin M.observed.card) (w : ↥(M.dag.parents ↑(M.observedAt j))), M.parentMap s ℓ j.isLt (M.prevFromObservedValues x) w = if huo : ↑w ∈ M.unobserved then ℓ ⟨↑w, huo⟩ else if hfix : ↑w ∈ M.fixed then s ⟨↑w, hfix⟩ else have hedge := (Causalean.DAG.mem_parents M.dag).mp w.property; have hobs := Or.casesOn (Finset.mem_union.mp (M.dag_edges_classified (↑w) (↑(M.observedAt j)) hedge).left) (fun h1 => Or.casesOn (Finset.mem_union.mp h1) (fun hfx => absurd hfx hfix) fun hob => hob) fun h2 => absurd h2 huo; x ⟨↑w, hobs⟩","l":216,"f":"Causalean/SCM/ID/Density/LatentBlocks.lean"},{"n":"Causalean.SCM.parentMap_unobserved","k":"theorem","m":"Causalean.SCM.Model.Evaluation","a":"SCM/Model","t":2,"d":"The parent-value tuple reads a latent parent directly from the latent assignment.","s":"∀ {N : Type u_1} [inst : DecidableEq N] [inst_1 : Fintype N] {Ω : N → Type u_2} [inst_2 : (n : N) → MeasurableSpace (Ω n)] (M : Causalean.SCM N Ω) (s : M.FixedValues) (ℓ : M.LatentValues) {n : ℕ} (hn : n < M.observed.card) (prev : (m : ℕ) → m < n → (hm : m < M.observed.card) → Causalean.swigΩ Ω ↑(M.observedAt ⟨m, hm⟩)) (w : ↥(M.dag.parents ↑(M.observedAt ⟨n, hn⟩))) (huo : ↑w ∈ M.unobserved), M.parentMap s ℓ hn prev w = ℓ ⟨↑w, huo⟩","l":83,"f":"Causalean/SCM/Model/Evaluation.lean"},{"n":"Causalean.SCM.parentValuesFromPrefix","k":"def","m":"Causalean.SCM.Factored.ParentLookup","a":"SCM/Factored","t":1,"d":"Assemble the full parent tuple of the next observed node v_n = M.observedAt ⟨n, hn⟩ from fixed values s, latent values ℓ, and the already-generated observed prefix ξ.","s":"{N : Type uN} → [inst : DecidableEq N] → [inst_1 : Fintype N] → {Ω : N → Type uΩ} → [inst_2 : (n : N) → MeasurableSpace (Ω n)] → (M : Causalean.SCM N Ω) → {n : ℕ} → (hn : n + 1 ≤ M.observed.card) → M.FixedValues × M.OrderedLatentPrefixValues n (Nat.le_of_succ_le hn) → (w : ↥(M.dag.parents ↑(M.observedAt ⟨n, hn⟩))) → Causalean.swigΩ Ω ↑w","l":77,"f":"Causalean/SCM/Factored/ParentLookup.lean"},{"n":"Causalean.SCM.parent_unobserved_of_not_fixed_not_observed","k":"theorem","m":"Causalean.SCM.Factored.ParentLookup","a":"SCM/Factored","t":2,"d":"A parent that is neither fixed nor observed must be an unobserved node. This lets an evaluator classify a parent's value source without depending on where the target appears in an observation order.","s":"∀ {N : Type uN} [inst : DecidableEq N] [inst_1 : Fintype N] (G : Causalean.SWIGGraph N) {u v : Causalean.SWIGNode N}, G.dag.edge u v → u ∉ G.fixed → u ∉ G.observed → u ∈ G.unobserved","l":57,"f":"Causalean/SCM/Factored/ParentLookup.lean"},{"n":"Causalean.SCM.partialEvalMap","k":"def","m":"Causalean.SCM.Factored.EvalMapCorrespond","a":"SCM/Factored","t":1,"d":"Deterministic twin of jointKernelPrefix: produces the prefix state (ℓ, observed-prefix) at level n as a plain function. Its Dirac-pushforward equals jointKernelPrefix n hn (see jointKernelPrefix_apply_eq).","s":"{N : Type uN} → [inst : DecidableEq N] → [inst_1 : Fintype N] → {Ω : N → Type uΩ} → [inst_2 : (n : N) → MeasurableSpace (Ω n)] → (M : Causalean.SCM N Ω) → (n : ℕ) → (hn : n ≤ M.observed.card) → M.FixedValues → M.LatentValues → M.OrderedLatentPrefixValues n hn","l":61,"f":"Causalean/SCM/Factored/EvalMapCorrespond.lean"},{"n":"Causalean.SCM.partialEvalMap_full_eq","k":"theorem","m":"Causalean.SCM.Factored.Factorization","a":"SCM/Factored","t":2,"d":"Bridge lemma: reindexing the deterministic full-prefix value built from partialEvalMap at length observed.card yields exactly evalMap s ℓ.","s":"∀ {N : Type uN} [inst : DecidableEq N] [inst_1 : Fintype N] {Ω : N → Type uΩ} [inst_2 : (n : N) → MeasurableSpace (Ω n)] (M : Causalean.SCM N Ω) (s : M.FixedValues) (ℓ : M.LatentValues), M.orderedLatentPrefixFullToRandom (M.partialEvalMap M.observed.card (le_refl M.observed.card) s ℓ) = M.evalMap s ℓ","l":109,"f":"Causalean/SCM/Factored/Factorization.lean"},{"n":"Causalean.SCM.partialEvalMap_latent","k":"theorem","m":"Causalean.SCM.Factored.EvalMapCorrespond","a":"SCM/Factored","t":2,"d":"The first component of partialEvalMap is always the input latent tuple: the recursion only writes to the ObservedPrefixValues factor.","s":"∀ {N : Type uN} [inst : DecidableEq N] [inst_1 : Fintype N] {Ω : N → Type uΩ} [inst_2 : (n : N) → MeasurableSpace (Ω n)] (M : Causalean.SCM N Ω) (s : M.FixedValues) (ℓ : M.LatentValues) (n : ℕ) (hn : n ≤ M.observed.card), (M.partialEvalMap n hn s ℓ).1 = ℓ","l":80,"f":"Causalean/SCM/Factored/EvalMapCorrespond.lean"},{"n":"Causalean.SCM.partialEvalMap_observedPrefixValue","k":"theorem","m":"Causalean.SCM.Factored.EvalMapCorrespond","a":"SCM/Factored","t":2,"d":"Bridges partialEvalMap (kernel-side) to evalObservedAux (existing evaluator). Proven by induction on n mirroring the definitions: at each step, the newly-appended coordinate (index n) is stepFun hn (s, prev) = structFun v_n (parentValuesFromPrefix hn (s, prev)), which equals evalObservedAux M s ℓ n _ once one shows the parent lookups agree. Earlier indices are handled by the inductive hypothesis through observedPrefixValue of the extension.","s":"∀ {N : Type uN} [inst : DecidableEq N] [inst_1 : Fintype N] {Ω : N → Type uΩ} [inst_2 : (n : N) → MeasurableSpace (Ω n)] (M : Causalean.SCM N Ω) (s : M.FixedValues) (ℓ : M.LatentValues) (n : ℕ) (hn : n ≤ M.observed.card) (i : Fin n), M.observedPrefixValue hn (M.partialEvalMap n hn s ℓ).2 i = M.evalObservedAux s ℓ (↑i) (Nat.lt_of_lt_of_le i.isLt hn)","l":216,"f":"Causalean/SCM/Factored/EvalMapCorrespond.lean"},{"n":"Causalean.SCM.partialEvalMap_succ_snd_fst","k":"theorem","m":"Causalean.SCM.Factored.EvalMapCorrespond","a":"SCM/Factored","t":2,"d":"Extending a deterministic evaluation prefix by one observed variable leaves the previously computed observed prefix unchanged.","s":"∀ {N : Type uN} [inst : DecidableEq N] [inst_1 : Fintype N] {Ω : N → Type uΩ} [inst_2 : (n : N) → MeasurableSpace (Ω n)] (M : Causalean.SCM N Ω) (s : M.FixedValues) (ℓ : M.LatentValues) {k : ℕ} (hn : k + 1 ≤ M.observed.card), (M.partialEvalMap (k + 1) hn s ℓ).2.1 = (M.partialEvalMap k (Nat.le_of_succ_le hn) s ℓ).2","l":192,"f":"Causalean/SCM/Factored/EvalMapCorrespond.lean"},{"n":"Causalean.SCM.partialEvalMap_succ_snd_snd","k":"theorem","m":"Causalean.SCM.Factored.EvalMapCorrespond","a":"SCM/Factored","t":2,"d":"Extending a deterministic evaluation prefix appends the value determined for the newly added observed variable.","s":"∀ {N : Type uN} [inst : DecidableEq N] [inst_1 : Fintype N] {Ω : N → Type uΩ} [inst_2 : (n : N) → MeasurableSpace (Ω n)] (M : Causalean.SCM N Ω) (s : M.FixedValues) (ℓ : M.LatentValues) {k : ℕ} (hn : k + 1 ≤ M.observed.card), (M.partialEvalMap (k + 1) hn s ℓ).2.2 = M.stepFun hn (s, M.partialEvalMap k (Nat.le_of_succ_le hn) s ℓ)","l":202,"f":"Causalean/SCM/Factored/EvalMapCorrespond.lean"},{"n":"Causalean.SCM.path_fixSet_active","k":"theorem","m":"Causalean.SCM.Do.CutsetDSep","a":"SCM/Do","t":2,"d":"An active path in the original structural causal model remains active after fixing the treatment set, when each of its directed edges has a non-treatment source.","s":"∀ {N : Type u_1} [inst : DecidableEq N] [inst_1 : Fintype N] {Ω : N → Type u_2} [inst_2 : (n : N) → MeasurableSpace (Ω n)] (M : Causalean.SCM N Ω) (Z : Finset N) (hZ_obs : ∀ D ∈ Z, Causalean.SWIGNode.random D ∈ M.observed) (hZ_fixed : ∀ D ∈ Z, Causalean.SWIGNode.fixed D ∉ M.fixed) (W : Finset (Causalean.SWIGNode N)), (∀ D ∈ Z, ∀ w ∈ W, ¬M.dag.isAncestor (Causalean.SWIGNode.random D) w) → ∀ {P : List (Causalean.SWIGNode N)}, M.dag.IsActivePath (W ∪ M.fixed) P → (∀ (i : ℕ) (hi : i + 1 < P.length), (M.dag.edge (P.get ⟨i, Decidable.byContradiction fun a => Causalean.SCM.path_fixSet_active._proof_1 i hi a⟩) (P.get ⟨i + 1, hi⟩) → ∀ D ∈ Z, P.get ⟨i, Decidable.byContradiction fun a => Causalean.SCM.path_fixSet_active._proof_1 i hi a⟩ ≠ Causalean.SWIGNode.random D) ∧ (M.dag.edge (P.get ⟨i + 1, hi⟩) (P.get ⟨i, Decidable.byContradiction fun a => Causalean.SCM.path_fixSet_active._proof_1 i hi a⟩) → ∀ D ∈ Z, P.get ⟨i + 1, hi⟩ ≠ Causalean.SWIGNode.random D)) → (M.fixSet Z hZ_obs hZ_fixed).dag.IsActivePath (W ∪ M.fixed ∪ Finset.image Causalean.SWIGNode.fixed Z) P","l":294,"f":"Causalean/SCM/Do/CutsetDSep.lean"},{"n":"Causalean.SCM.prefixDensityProduct","k":"def","m":"Causalean.SCM.ID.Density.ChainRuleDensity","a":"SCM/ID/Density","t":1,"d":"The recursive prefix density product matching obsChainKernel.","s":"{N : Type u_1} → [inst : DecidableEq N] → [inst_1 : Fintype N] → {Ω : N → Type u_2} → [inst_2 : (n : N) → MeasurableSpace (Ω n)] → (M : Causalean.SCM N Ω) → Causalean.SCM.ReferenceMeasures Ω → M.FixedValues → [∀ (s' : M.FixedValues), MeasureTheory.IsFiniteMeasure (M.obsKernel s')] → [∀ (k : ℕ) (hk : k < M.observed.card), StandardBorelSpace (Causalean.ValuesOn {↑(M.observedAt ⟨k, hk⟩)} (Causalean.swigΩ Ω))] → [∀ (k : ℕ) (hk : k < M.observed.card), Nonempty (Causalean.ValuesOn {↑(M.observedAt ⟨k, hk⟩)} (Causalean.swigΩ Ω))] → [∀ (k : ℕ), MeasurableSpace.CountableOrCountablyGenerated M.FixedValues (Causalean.ValuesOn (M.prefixNodes k) (Causalean.swigΩ Ω))] → (k : ℕ) → Causalean.ValuesOn (M.prefixNodes k) (Causalean.swigΩ Ω) → ENNReal","l":88,"f":"Causalean/SCM/ID/Density/ChainRuleDensity.lean"},{"n":"Causalean.SCM.prefixDensityProduct_card_eq_qFactorDensityProduct","k":"theorem","m":"Causalean.SCM.ID.Density.ChainRuleDensity","a":"SCM/ID/Density","t":2,"d":"At the full observed prefix, the recursive prefix density product is the existing finite product over observed indices.","s":"∀ {N : Type u_1} [inst : DecidableEq N] [inst_1 : Fintype N] {Ω : N → Type u_2} [inst_2 : (n : N) → MeasurableSpace (Ω n)] (M : Causalean.SCM N Ω) (ref : Causalean.SCM.ReferenceMeasures Ω) (s : M.FixedValues) [inst_3 : ∀ (s' : M.FixedValues), MeasureTheory.IsFiniteMeasure (M.obsKernel s')] [inst_4 : ∀ (k : ℕ) (hk : k < M.observed.card), StandardBorelSpace (Causalean.ValuesOn {↑(M.observedAt ⟨k, hk⟩)} (Causalean.swigΩ Ω))] [inst_5 : ∀ (k : ℕ) (hk : k < M.observed.card), Nonempty (Causalean.ValuesOn {↑(M.observedAt ⟨k, hk⟩)} (Causalean.swigΩ Ω))] [inst_6 : ∀ (k : ℕ), MeasurableSpace.CountableOrCountablyGenerated M.FixedValues (Causalean.ValuesOn (M.prefixNodes k) (Causalean.swigΩ Ω))] (y : Causalean.ValuesOn (M.prefixNodes M.observed.card) (Causalean.swigΩ Ω)), M.prefixDensityProduct ref s M.observed.card y = M.qFactorDensityProduct ref s ((Causalean.valuesEquivOfEq (Causalean.SCM.prefixNodes_card M M.observed.card (le_refl M.observed.card))) y)","l":351,"f":"Causalean/SCM/ID/Density/ChainRuleDensity.lean"},{"n":"Causalean.SCM.prefixDensityProduct_eq_range_product","k":"theorem","m":"Causalean.SCM.ID.Density.ChainRuleDensity","a":"SCM/ID/Density","t":2,"d":"The recursive prefix density product is the range product of its one-step factors.","s":"∀ {N : Type u_1} [inst : DecidableEq N] [inst_1 : Fintype N] {Ω : N → Type u_2} [inst_2 : (n : N) → MeasurableSpace (Ω n)] (M : Causalean.SCM N Ω) (ref : Causalean.SCM.ReferenceMeasures Ω) (s : M.FixedValues) [inst_3 : ∀ (s' : M.FixedValues), MeasureTheory.IsFiniteMeasure (M.obsKernel s')] [inst_4 : ∀ (k : ℕ) (hk : k < M.observed.card), StandardBorelSpace (Causalean.ValuesOn {↑(M.observedAt ⟨k, hk⟩)} (Causalean.swigΩ Ω))] [inst_5 : ∀ (k : ℕ) (hk : k < M.observed.card), Nonempty (Causalean.ValuesOn {↑(M.observedAt ⟨k, hk⟩)} (Causalean.swigΩ Ω))] [inst_6 : ∀ (k : ℕ), MeasurableSpace.CountableOrCountablyGenerated M.FixedValues (Causalean.ValuesOn (M.prefixNodes k) (Causalean.swigΩ Ω))], ∀ k ≤ M.observed.card, ∀ (z : Causalean.ValuesOn (M.prefixNodes k) (Causalean.swigΩ Ω)), M.prefixDensityProduct ref s k z = ∏ i ∈ Finset.range k, M.prefixStepDensityInPrefix ref s k z i","l":307,"f":"Causalean/SCM/ID/Density/ChainRuleDensity.lean"},{"n":"Causalean.SCM.prefixNodes","k":"def","m":"Causalean.SCM.Factored.ObsChainKernel","a":"SCM/Factored","t":1,"d":"The first n observed nodes in the canonical topological order. If n ≥ M.observed.card, this is all observed nodes.","s":"{N : Type u_1} → [inst : DecidableEq N] → [inst_1 : Fintype N] → {Ω : N → Type u_2} → [inst_2 : (n : N) → MeasurableSpace (Ω n)] → Causalean.SCM N Ω → ℕ → Finset (Causalean.SWIGNode N)","l":56,"f":"Causalean/SCM/Factored/ObsChainKernel.lean"},{"n":"Causalean.SCM.prefixNodes_card","k":"theorem","m":"Causalean.SCM.Factored.ObsChainKernel","a":"SCM/Factored","t":2,"d":"Every prefix at least as long as the observed-node list is the full observed node set.","s":"∀ {N : Type u_1} [inst : DecidableEq N] [inst_1 : Fintype N] {Ω : N → Type u_2} [inst_2 : (n : N) → MeasurableSpace (Ω n)] (M : Causalean.SCM N Ω) (n : ℕ), M.observed.card ≤ n → M.prefixNodes n = M.observed","l":109,"f":"Causalean/SCM/Factored/ObsChainKernel.lean"},{"n":"Causalean.SCM.prefixNodes_disjoint_singleton_next","k":"theorem","m":"Causalean.SCM.Factored.ObsChainKernel","a":"SCM/Factored","t":2,"d":"The previous prefix is disjoint from the singleton next node.","s":"∀ {N : Type u_1} [inst : DecidableEq N] [inst_1 : Fintype N] {Ω : N → Type u_2} [inst_2 : (n : N) → MeasurableSpace (Ω n)] (M : Causalean.SCM N Ω) {n : ℕ} (hn : n < M.observed.card), Disjoint (M.prefixNodes n) {↑(M.observedAt ⟨n, hn⟩)}","l":154,"f":"Causalean/SCM/Factored/ObsChainKernel.lean"},{"n":"Causalean.SCM.prefixNodes_mono","k":"theorem","m":"Causalean.SCM.ID.Density.ChainRuleDensity","a":"SCM/ID/Density","t":2,"d":"Prefix node sets are monotone in the prefix length.","s":"∀ {N : Type u_1} [inst : DecidableEq N] [inst_1 : Fintype N] {Ω : N → Type u_2} [inst_2 : (n : N) → MeasurableSpace (Ω n)] (M : Causalean.SCM N Ω) {m k : ℕ}, m ≤ k → M.prefixNodes m ⊆ M.prefixNodes k","l":81,"f":"Causalean/SCM/ID/Density/ChainRuleDensity.lean"},{"n":"Causalean.SCM.prefixNodes_subset_observed","k":"theorem","m":"Causalean.SCM.Factored.ObsChainKernel","a":"SCM/Factored","t":2,"d":"Prefix nodes are observed nodes.","s":"∀ {N : Type u_1} [inst : DecidableEq N] [inst_1 : Fintype N] {Ω : N → Type u_2} [inst_2 : (n : N) → MeasurableSpace (Ω n)] (M : Causalean.SCM N Ω) (n : ℕ), M.prefixNodes n ⊆ M.observed","l":75,"f":"Causalean/SCM/Factored/ObsChainKernel.lean"},{"n":"Causalean.SCM.prefixNodes_succ","k":"theorem","m":"Causalean.SCM.Factored.ObsChainKernel","a":"SCM/Factored","t":2,"d":"The prefix successor is obtained by adjoining the next observed node.","s":"∀ {N : Type u_1} [inst : DecidableEq N] [inst_1 : Fintype N] {Ω : N → Type u_2} [inst_2 : (n : N) → MeasurableSpace (Ω n)] (M : Causalean.SCM N Ω) {n : ℕ} (hn : n < M.observed.card), M.prefixNodes (n + 1) = M.prefixNodes n ∪ {↑(M.observedAt ⟨n, hn⟩)}","l":127,"f":"Causalean/SCM/Factored/ObsChainKernel.lean"},{"n":"Causalean.SCM.prefixNodes_zero","k":"theorem","m":"Causalean.SCM.Factored.ObsChainKernel","a":"SCM/Factored","t":2,"d":"The empty prefix has no nodes.","s":"∀ {N : Type u_1} [inst : DecidableEq N] [inst_1 : Fintype N] {Ω : N → Type u_2} [inst_2 : (n : N) → MeasurableSpace (Ω n)] (M : Causalean.SCM N Ω), M.prefixNodes 0 = ∅","l":81,"f":"Causalean/SCM/Factored/ObsChainKernel.lean"},{"n":"Causalean.SCM.prefixStepDensityInPrefix","k":"def","m":"Causalean.SCM.ID.Density.ChainRuleDensity","a":"SCM/ID/Density","t":1,"d":"The i-th one-step density factor read from a k-prefix assignment.","s":"{N : Type u_1} → [inst : DecidableEq N] → [inst_1 : Fintype N] → {Ω : N → Type u_2} → [inst_2 : (n : N) → MeasurableSpace (Ω n)] → (M : Causalean.SCM N Ω) → Causalean.SCM.ReferenceMeasures Ω → M.FixedValues → [∀ (s' : M.FixedValues), MeasureTheory.IsFiniteMeasure (M.obsKernel s')] → [∀ (i : ℕ) (hi : i < M.observed.card), StandardBorelSpace (Causalean.ValuesOn {↑(M.observedAt ⟨i, hi⟩)} (Causalean.swigΩ Ω))] → [∀ (i : ℕ) (hi : i < M.observed.card), Nonempty (Causalean.ValuesOn {↑(M.observedAt ⟨i, hi⟩)} (Causalean.swigΩ Ω))] → [∀ (i : ℕ), MeasurableSpace.CountableOrCountablyGenerated M.FixedValues (Causalean.ValuesOn (M.prefixNodes i) (Causalean.swigΩ Ω))] → (k : ℕ) → Causalean.ValuesOn (M.prefixNodes k) (Causalean.swigΩ Ω) → ℕ → ENNReal","l":279,"f":"Causalean/SCM/ID/Density/ChainRuleDensity.lean"},{"n":"Causalean.SCM.prefixSucc_projection_pair","k":"theorem","m":"Causalean.SCM.Factored.ObsChainKernel","a":"SCM/Factored","t":2,"d":"Projecting the successor prefix through the union equivalence gives the previous-prefix block and the singleton next-node block.","s":"∀ {N : Type u_1} [inst : DecidableEq N] [inst_1 : Fintype N] {Ω : N → Type u_2} [inst_2 : (n : N) → MeasurableSpace (Ω n)] (M : Causalean.SCM N Ω) {n : ℕ} (hn : n < M.observed.card), (fun ω => (Causalean.SCM.valuesUnionEquiv (Causalean.SCM.prefixNodes_disjoint_singleton_next M hn)) ((Causalean.valuesEquivOfEq (Causalean.SCM.prefixNodes_succ M hn)) (Causalean.valuesProjection (Causalean.SCM.prefixNodes_subset_observed M (n + 1)) ω))) = fun ω => (Causalean.valuesProjection (Causalean.SCM.prefixNodes_subset_observed M n) ω, Causalean.valuesProjection (fun ⦃w⦄ hw => have hw_eq := Eq.mp Finset.mem_singleton._simp_1 hw; of_eq_true (Eq.trans (congrArg (Membership.mem M.observed) hw_eq) (eq_true (M.observedAt ⟨n, hn⟩).property))) ω)","l":428,"f":"Causalean/SCM/Factored/ObsChainKernel.lean"},{"n":"Causalean.SCM.prevFromObservedValues","k":"def","m":"Causalean.SCM.ID.Density.LatentBlocks","a":"SCM/ID/Density","t":1,"d":"The recursive evaluator can read all earlier observed coordinates from a full observed assignment.","s":"{N : Type u_1} → [inst : DecidableEq N] → [inst_1 : Fintype N] → {Ω : N → Type u_2} → [inst_2 : (n : N) → MeasurableSpace (Ω n)] → (M : Causalean.SCM N Ω) → Causalean.ValuesOn M.observed (Causalean.swigΩ Ω) → {n : ℕ} → (m : ℕ) → m < n → (hm : m < M.observed.card) → Causalean.swigΩ Ω ↑(M.observedAt ⟨m, hm⟩)","l":27,"f":"Causalean/SCM/ID/Density/LatentBlocks.lean"},{"n":"Causalean.SCM.prevFromObservedValues_apply_observed","k":"theorem","m":"Causalean.SCM.ID.Density.LatentBlocks","a":"SCM/ID/Density","t":2,"d":"","s":"∀ {N : Type u_1} [inst : DecidableEq N] [inst_1 : Fintype N] {Ω : N → Type u_2} [inst_2 : (n : N) → MeasurableSpace (Ω n)] (M : Causalean.SCM N Ω) (x : Causalean.ValuesOn M.observed (Causalean.swigΩ Ω)) {n : ℕ} {hn : n < M.observed.card} {w : ↥(M.dag.parents ↑(M.observedAt ⟨n, hn⟩))} (hobs : ↑w ∈ M.observed) (hlt : ↑(M.observedIndex ⟨↑w, hobs⟩) < n), Causalean.SCM.observedAt_observedIndex M ⟨↑w, hobs⟩ ▸ M.prevFromObservedValues x (↑(M.observedIndex ⟨↑w, hobs⟩)) hlt (M.observedIndex ⟨↑w, hobs⟩).isLt = x ⟨↑w, hobs⟩","l":195,"f":"Causalean/SCM/ID/Density/LatentBlocks.lean"},{"n":"Causalean.SCM.qFactor","k":"def","m":"Causalean.SCM.ID.GraphicalThms.CComponentFactor","a":"SCM/ID/GraphicalThms","t":1,"d":"Conditional-kernel proxy for a Tian c-component factor.","s":"{N : Type u_1} → [inst : DecidableEq N] → [inst_1 : Fintype N] → {Ω : N → Type u_2} → [inst_2 : (n : N) → MeasurableSpace (Ω n)] → (M : Causalean.SCM N Ω) → (C : Finset (Causalean.SWIGNode N)) → C ⊆ M.observed → [StandardBorelSpace (Causalean.ValuesOn C (Causalean.swigΩ Ω))] → [Nonempty (Causalean.ValuesOn C (Causalean.swigΩ Ω))] → M.FixedValues → [∀ (s' : M.FixedValues), MeasureTheory.IsFiniteMeasure (M.obsKernel s')] → [MeasurableSpace.CountableOrCountablyGenerated M.FixedValues (Causalean.ValuesOn (M.qFactorParents C) (Causalean.swigΩ Ω))] → ProbabilityTheory.Kernel (Causalean.ValuesOn (M.qFactorParents C) (Causalean.swigΩ Ω)) (Causalean.ValuesOn C (Causalean.swigΩ Ω))","l":103,"f":"Causalean/SCM/ID/GraphicalThms/CComponentFactor.lean"},{"n":"Causalean.SCM.qFactorDensityProduct","k":"def","m":"Causalean.SCM.ID.Density.ChainRuleDensity","a":"SCM/ID/Density","t":1,"d":"The chain-rule density product for the observational law.","s":"{N : Type u_1} → [inst : DecidableEq N] → [inst_1 : Fintype N] → {Ω : N → Type u_2} → [inst_2 : (n : N) → MeasurableSpace (Ω n)] → (M : Causalean.SCM N Ω) → Causalean.SCM.ReferenceMeasures Ω → M.FixedValues → [∀ (s' : M.FixedValues), MeasureTheory.IsFiniteMeasure (M.obsKernel s')] → [∀ (k : ℕ) (hk : k < M.observed.card), StandardBorelSpace (Causalean.ValuesOn {↑(M.observedAt ⟨k, hk⟩)} (Causalean.swigΩ Ω))] → [∀ (k : ℕ) (hk : k < M.observed.card), Nonempty (Causalean.ValuesOn {↑(M.observedAt ⟨k, hk⟩)} (Causalean.swigΩ Ω))] → [∀ (k : ℕ), MeasurableSpace.CountableOrCountablyGenerated M.FixedValues (Causalean.ValuesOn (M.prefixNodes k) (Causalean.swigΩ Ω))] → Causalean.ValuesOn M.observed (Causalean.swigΩ Ω) → ENNReal","l":61,"f":"Causalean/SCM/ID/Density/ChainRuleDensity.lean"},{"n":"Causalean.SCM.qFactorDensityProduct_eq_prod_cComponentFactor","k":"theorem","m":"Causalean.SCM.ID.Density.CComponentDensity","a":"SCM/ID/Density","t":1,"d":"c-component regrouping of the chain-rule density product. For a structural causal model M, a reference measure family ref, a fixed assignment s, and a value assignment x to the observed coordinates, the product of all one-node conditional density factors equals the product, over the c-components of the graph, of the per-component density factors.","s":"∀ {N : Type u_1} [inst : DecidableEq N] [inst_1 : Fintype N] {Ω : N → Type u_2} [inst_2 : (n : N) → MeasurableSpace (Ω n)] (M : Causalean.SCM N Ω) (ref : Causalean.SCM.ReferenceMeasures Ω) (s : M.FixedValues) [inst_3 : ∀ (s' : M.FixedValues), MeasureTheory.IsFiniteMeasure (M.obsKernel s')] [inst_4 : ∀ (k : ℕ) (hk : k < M.observed.card), StandardBorelSpace (Causalean.ValuesOn {↑(M.observedAt ⟨k, hk⟩)} (Causalean.swigΩ Ω))] [inst_5 : ∀ (k : ℕ) (hk : k < M.observed.card), Nonempty (Causalean.ValuesOn {↑(M.observedAt ⟨k, hk⟩)} (Causalean.swigΩ Ω))] [inst_6 : ∀ (k : ℕ), MeasurableSpace.CountableOrCountablyGenerated M.FixedValues (Causalean.ValuesOn (M.prefixNodes k) (Causalean.swigΩ Ω))] (x : Causalean.ValuesOn M.observed (Causalean.swigΩ Ω)), M.qFactorDensityProduct ref s x = ∏ C ∈ M.cComponentSet, M.cComponentDensityFactor ref s C x","l":54,"f":"Causalean/SCM/ID/Density/CComponentDensity.lean"},{"n":"Causalean.SCM.qFactorProduct","k":"def","m":"Causalean.SCM.Factored.ObsChainKernel","a":"SCM/Factored","t":1,"d":"The full observational chain-rule product as a kernel on observed values.","s":"{N : Type u_1} → [inst : DecidableEq N] → [inst_1 : Fintype N] → {Ω : N → Type u_2} → [inst_2 : (n : N) → MeasurableSpace (Ω n)] → (M : Causalean.SCM N Ω) → [∀ (k : ℕ) (hk : k < M.observed.card), StandardBorelSpace (Causalean.ValuesOn {↑(M.observedAt ⟨k, hk⟩)} (Causalean.swigΩ Ω))] → [∀ (k : ℕ) (hk : k < M.observed.card), Nonempty (Causalean.ValuesOn {↑(M.observedAt ⟨k, hk⟩)} (Causalean.swigΩ Ω))] → [∀ (k : Fin M.observed.card), MeasurableSpace.CountableOrCountablyGenerated M.FixedValues (Causalean.ValuesOn (M.prefixNodes ↑k) (Causalean.swigΩ Ω))] → ProbabilityTheory.Kernel M.FixedValues M.ObservedValues","l":595,"f":"Causalean/SCM/Factored/ObsChainKernel.lean"},{"n":"Causalean.SCM.qFactorProduct_rnDeriv_eq_obsChainKernel_card_pullback","k":"theorem","m":"Causalean.SCM.ID.Density.ChainRuleDensity","a":"SCM/ID/Density","t":2,"d":"Peel the final qFactorProduct map back to the full prefix chain kernel.","s":"∀ {N : Type u_1} [inst : DecidableEq N] [inst_1 : Fintype N] {Ω : N → Type u_2} [inst_2 : (n : N) → MeasurableSpace (Ω n)] (M : Causalean.SCM N Ω) (ref : Causalean.SCM.ReferenceMeasures Ω) (s : M.FixedValues) [∀ (s' : M.FixedValues), MeasureTheory.IsFiniteMeasure (M.obsKernel s')] [inst_4 : ∀ (k : ℕ) (hk : k < M.observed.card), StandardBorelSpace (Causalean.ValuesOn {↑(M.observedAt ⟨k, hk⟩)} (Causalean.swigΩ Ω))] [inst_5 : ∀ (k : ℕ) (hk : k < M.observed.card), Nonempty (Causalean.ValuesOn {↑(M.observedAt ⟨k, hk⟩)} (Causalean.swigΩ Ω))] [inst_6 : ∀ (k : ℕ), MeasurableSpace.CountableOrCountablyGenerated M.FixedValues (Causalean.ValuesOn (M.prefixNodes k) (Causalean.swigΩ Ω))], (M.qFactorProduct s).rnDeriv (Causalean.SCM.jointRef ref M.observed) =ᵐ[Causalean.SCM.jointRef ref M.observed] fun x => ((M.obsChainKernel M.observed.card (le_refl M.observed.card)) s).rnDeriv (Causalean.SCM.jointRef ref (M.prefixNodes M.observed.card)) ((Causalean.valuesEquivOfEq (Causalean.SCM.prefixNodes_card M M.observed.card (le_refl M.observed.card))).symm x)","l":750,"f":"Causalean/SCM/ID/Density/ChainRuleDensity.lean"},{"n":"Causalean.SCM.qFactorProduct_rnDeriv_eq_obsChainKernel_card_pullback_of_jointRef","k":"theorem","m":"Causalean.SCM.ID.Density.ChainRuleDensity","a":"SCM/ID/Density","t":2,"d":"RN-derivative transport for qFactorProduct, assuming the final reference transport has already been identified.","s":"∀ {N : Type u_1} [inst : DecidableEq N] [inst_1 : Fintype N] {Ω : N → Type u_2} [inst_2 : (n : N) → MeasurableSpace (Ω n)] (M : Causalean.SCM N Ω) (ref : Causalean.SCM.ReferenceMeasures Ω) (s : M.FixedValues), MeasureTheory.Measure.map (⇑(Causalean.valuesEquivOfEq (Causalean.SCM.prefixNodes_card M M.observed.card (le_refl M.observed.card)))) (Causalean.SCM.jointRef ref (M.prefixNodes M.observed.card)) = Causalean.SCM.jointRef ref M.observed → ∀ [∀ (s' : M.FixedValues), MeasureTheory.IsFiniteMeasure (M.obsKernel s')] [inst_4 : ∀ (k : ℕ) (hk : k < M.observed.card), StandardBorelSpace (Causalean.ValuesOn {↑(M.observedAt ⟨k, hk⟩)} (Causalean.swigΩ Ω))] [inst_5 : ∀ (k : ℕ) (hk : k < M.observed.card), Nonempty (Causalean.ValuesOn {↑(M.observedAt ⟨k, hk⟩)} (Causalean.swigΩ Ω))] [inst_6 : ∀ (k : ℕ), MeasurableSpace.CountableOrCountablyGenerated M.FixedValues (Causalean.ValuesOn (M.prefixNodes k) (Causalean.swigΩ Ω))], (M.qFactorProduct s).rnDeriv (Causalean.SCM.jointRef ref M.observed) =ᵐ[Causalean.SCM.jointRef ref M.observed] fun x => ((M.obsChainKernel M.observed.card (le_refl M.observed.card)) s).rnDeriv (Causalean.SCM.jointRef ref (M.prefixNodes M.observed.card)) ((Causalean.valuesEquivOfEq (Causalean.SCM.prefixNodes_card M M.observed.card (le_refl M.observed.card))).symm x)","l":710,"f":"Causalean/SCM/ID/Density/ChainRuleDensity.lean"},{"n":"Causalean.SCM.qFactorProduct_rnDeriv_eq_qFactorDensityProduct","k":"theorem","m":"Causalean.SCM.ID.Density.ChainRuleDensity","a":"SCM/ID/Density","t":2,"d":"Analytic chain rule for the mapped observational product kernel.","s":"∀ {N : Type u_1} [inst : DecidableEq N] [inst_1 : Fintype N] {Ω : N → Type u_2} [inst_2 : (n : N) → MeasurableSpace (Ω n)] (M : Causalean.SCM N Ω) (ref : Causalean.SCM.ReferenceMeasures Ω), M.DominatedObs ref → ∀ (s : M.FixedValues) [inst_3 : ∀ (s' : M.FixedValues), MeasureTheory.IsFiniteMeasure (M.obsKernel s')] [inst_4 : ∀ (k : ℕ) (hk : k < M.observed.card), StandardBorelSpace (Causalean.ValuesOn {↑(M.observedAt ⟨k, hk⟩)} (Causalean.swigΩ Ω))] [inst_5 : ∀ (k : ℕ) (hk : k < M.observed.card), Nonempty (Causalean.ValuesOn {↑(M.observedAt ⟨k, hk⟩)} (Causalean.swigΩ Ω))] [inst_6 : ∀ (k : ℕ), MeasurableSpace.CountableOrCountablyGenerated M.FixedValues (Causalean.ValuesOn (M.prefixNodes k) (Causalean.swigΩ Ω))], M.ObsStepFiberRN ref s → (M.qFactorProduct s).rnDeriv (Causalean.SCM.jointRef ref M.observed) =ᵐ[Causalean.SCM.jointRef ref M.observed] M.qFactorDensityProduct ref s","l":911,"f":"Causalean/SCM/ID/Density/ChainRuleDensity.lean"},{"n":"Causalean.SCM.qLocalMass","k":"def","m":"Causalean.SCM.ID.Density.QMass","a":"SCM/ID/Density","t":1,"d":"The latent-product mass of satisfying local consistency on an observed subset.","s":"{N : Type u_1} → [inst : DecidableEq N] → [inst_1 : Fintype N] → {Ω : N → Type u_2} → [inst_2 : (n : N) → MeasurableSpace (Ω n)] → (M : Causalean.SCM N Ω) → M.FixedValues → (T : Finset (Causalean.SWIGNode N)) → T ⊆ M.observed → Causalean.ValuesOn M.observed (Causalean.swigΩ Ω) → ENNReal","l":39,"f":"Causalean/SCM/ID/Density/QMass.lean"},{"n":"Causalean.SCM.qLocalMass_anti","k":"theorem","m":"Causalean.SCM.ID.Density.QMass","a":"SCM/ID/Density","t":2,"d":"Local q-mass is antitone in the constrained observed set.","s":"∀ {N : Type u_1} [inst : DecidableEq N] [inst_1 : Fintype N] {Ω : N → Type u_2} [inst_2 : (n : N) → MeasurableSpace (Ω n)] (M : Causalean.SCM N Ω) (s : M.FixedValues) {T T' : Finset (Causalean.SWIGNode N)}, T ⊆ T' → ∀ (hT : T ⊆ M.observed) (hT' : T' ⊆ M.observed) (x : Causalean.ValuesOn M.observed (Causalean.swigΩ Ω)), M.qLocalMass s T' hT' x ≤ M.qLocalMass s T hT x","l":55,"f":"Causalean/SCM/ID/Density/QMass.lean"},{"n":"Causalean.SCM.qLocalMass_empty","k":"theorem","m":"Causalean.SCM.ID.Density.QMass","a":"SCM/ID/Density","t":2,"d":"The empty local q-mass is one.","s":"∀ {N : Type u_1} [inst : DecidableEq N] [inst_1 : Fintype N] {Ω : N → Type u_2} [inst_2 : (n : N) → MeasurableSpace (Ω n)] (M : Causalean.SCM N Ω) (s : M.FixedValues) (x : Causalean.ValuesOn M.observed (Causalean.swigΩ Ω)), M.qLocalMass s ∅ (of_eq_true (Finset.empty_subset._simp_1 M.observed)) x = 1","l":48,"f":"Causalean/SCM/ID/Density/QMass.lean"},{"n":"Causalean.SCM.qLocalMass_marginalize_ancestralClosed","k":"theorem","m":"Causalean.SCM.ID.Density.QMass","a":"SCM/ID/Density","t":1,"d":"For a set of observed nodes T and a subset W of T such that every parent, within T, of a node in W also lies in W, marginalizing the local q-mass on T over the coordinates in T \\ W yields the local q-mass on W.","s":"∀ {N : Type u_1} [inst : DecidableEq N] [inst_1 : Fintype N] {Ω : N → Type u_2} [inst_2 : (n : N) → MeasurableSpace (Ω n)] [inst_3 : (n : N) → Fintype (Ω n)] [∀ (n : N), MeasurableSingletonClass (Ω n)] (M : Causalean.SCM N Ω) (s : M.FixedValues) (T W : Finset (Causalean.SWIGNode N)) (hT : T ⊆ M.observed) (hWT : W ⊆ T), (∀ v ∈ T, ∀ w ∈ W, M.dag.edge v w → v ∈ W) → ∀ (x : Causalean.ValuesOn M.observed (Causalean.swigΩ Ω)), Causalean.SCM.marginalizeOn M.observed (T \\ W) (fun x hv => hT (Finset.mem_sdiff.mp hv).left) (M.qLocalMass s T hT) x = M.qLocalMass s W (fun x hv => hT (hWT hv)) x","l":516,"f":"Causalean/SCM/ID/Density/QMass.lean"},{"n":"Causalean.SCM.qLocalMass_pos_of_positiveObs","k":"theorem","m":"Causalean.SCM.ID.Density.QMass","a":"SCM/ID/Density","t":2,"d":"Positive observational mass implies nonzero local q-mass.","s":"∀ {N : Type u_1} [inst : DecidableEq N] [inst_1 : Fintype N] {Ω : N → Type u_2} [inst_2 : (n : N) → MeasurableSpace (Ω n)] (M : Causalean.SCM N Ω) (s : M.FixedValues) [MeasurableSingletonClass (Causalean.ValuesOn M.observed (Causalean.swigΩ Ω))], Causalean.SCM.ID.DiscreteID.PositiveMass (M.obsKernel s) → ∀ (T : Finset (Causalean.SWIGNode N)) (hT : T ⊆ M.observed) (x : Causalean.ValuesOn M.observed (Causalean.swigΩ Ω)), M.qLocalMass s T hT x ≠ 0","l":1605,"f":"Causalean/SCM/ID/Density/QMass.lean"},{"n":"Causalean.SCM.qLocalMass_prod_inter_of_latentBlock_disjoint","k":"theorem","m":"Causalean.SCM.ID.Density.QMass","a":"SCM/ID/Density","t":2,"d":"Local q-mass on a covered set factors over an abstract family after intersecting each family member with the covered set.","s":"∀ {N : Type u_1} [inst : DecidableEq N] [inst_1 : Fintype N] {Ω : N → Type u_2} [inst_2 : (n : N) → MeasurableSpace (Ω n)] [(n : N) → Fintype (Ω n)] [∀ (n : N), MeasurableSingletonClass (Ω n)] (M : Causalean.SCM N Ω) (s : M.FixedValues) (P : Finset (Causalean.SWIGNode N)) (hPobs : P ⊆ M.observed) (𝒞 : Finset (Finset (Causalean.SWIGNode N))) (h𝒞obs : ∀ U ∈ 𝒞, U ⊆ M.observed), P ⊆ 𝒞.sup id → ((↑𝒞).Pairwise fun U U' => Disjoint (M.latentBlock U) (M.latentBlock U')) → ∀ (x : Causalean.ValuesOn M.observed (Causalean.swigΩ Ω)), M.qLocalMass s P hPobs x = ∏ U ∈ 𝒞, if hU : U ∈ 𝒞 then M.qLocalMass s (U ∩ P) (fun x hv => h𝒞obs U hU (Finset.mem_of_mem_inter_left hv)) x else 1","l":1338,"f":"Causalean/SCM/ID/Density/QMass.lean"},{"n":"Causalean.SCM.qLocalMass_prod_of_latentBlock_disjoint","k":"theorem","m":"Causalean.SCM.ID.Density.QMass","a":"SCM/ID/Density","t":1,"d":"For a finite family 𝒞 of observed node sets whose latent parent blocks are pairwise disjoint, the local q-mass on the union of the family equals the product, over the members U of 𝒞, of the local q-mass on U.","s":"∀ {N : Type u_1} [inst : DecidableEq N] [inst_1 : Fintype N] {Ω : N → Type u_2} [inst_2 : (n : N) → MeasurableSpace (Ω n)] [(n : N) → Fintype (Ω n)] [∀ (n : N), MeasurableSingletonClass (Ω n)] (M : Causalean.SCM N Ω) (s : M.FixedValues) (𝒞 : Finset (Finset (Causalean.SWIGNode N))) (h𝒞obs : ∀ U ∈ 𝒞, U ⊆ M.observed), ((↑𝒞).Pairwise fun U U' => Disjoint (M.latentBlock U) (M.latentBlock U')) → ∀ (x : Causalean.ValuesOn M.observed (Causalean.swigΩ Ω)), M.qLocalMass s (𝒞.sup id) (fun v hv => Exists.casesOn (Eq.mp (congrArg (fun _a => _a) (propext Finset.mem_sup)) hv) fun U h => And.casesOn h fun hU hvU => h𝒞obs U hU hvU) x = ∏ U ∈ 𝒞, if hU : U ∈ 𝒞 then M.qLocalMass s U (h𝒞obs U hU) x else 1","l":1211,"f":"Causalean/SCM/ID/Density/QMass.lean"},{"n":"Causalean.SCM.qLocalMass_sum_point_eliminate","k":"theorem","m":"Causalean.SCM.ID.Density.QMass","a":"SCM/ID/Density","t":2,"d":"Summing a local q-mass over one childless observed coordinate removes that coordinate from the constrained set.","s":"∀ {N : Type u_1} [inst : DecidableEq N] [inst_1 : Fintype N] {Ω : N → Type u_2} [inst_2 : (n : N) → MeasurableSpace (Ω n)] [inst_3 : (n : N) → Fintype (Ω n)] [∀ (n : N), MeasurableSingletonClass (Ω n)] (M : Causalean.SCM N Ω) (s : M.FixedValues) (T : Finset (Causalean.SWIGNode N)) (hT : T ⊆ M.observed) {v : Causalean.SWIGNode N}, v ∈ T → (∀ w ∈ T, ¬M.dag.edge v w) → ∀ (x : Causalean.ValuesOn M.observed (Causalean.swigΩ Ω)), ∑ ω, M.qLocalMass s T hT (Causalean.SCM.overrideOn x (Causalean.SCM.singletonValuePt✝ v ω)) = M.qLocalMass s (T.erase v) (fun x hv => hT (Finset.mem_of_mem_erase hv)) x","l":421,"f":"Causalean/SCM/ID/Density/QMass.lean"},{"n":"Causalean.SCM.randomToObserved","k":"def","m":"Causalean.SCM.Model.Kernel","a":"SCM/Model","t":1,"d":"The observed projection keeps only the observed coordinates of a full random assignment.","s":"{N : Type u_1} → [inst : DecidableEq N] → [inst_1 : Fintype N] → {Ω : N → Type u_2} → [inst_2 : (n : N) → MeasurableSpace (Ω n)] → (M : Causalean.SCM N Ω) → M.RandomValues → M.ObservedValues","l":88,"f":"Causalean/SCM/Model/Kernel.lean"},{"n":"Causalean.SCM.randomVars","k":"def","m":"Causalean.SCM.Model.SCM","a":"SCM/Model","t":1,"d":"The random variables of a structural causal model are its observed and latent nodes.","s":"{N : Type uN} → [inst : DecidableEq N] → [inst_1 : Fintype N] → {Ω : N → Type uΩ} → [inst_2 : (n : N) → MeasurableSpace (Ω n)] → Causalean.SCM N Ω → Finset (Causalean.SWIGNode N)","l":131,"f":"Causalean/SCM/Model/SCM.lean"},{"n":"Causalean.SCM.referenceFaithful_countingRef","k":"theorem","m":"Causalean.SCM.ID.Density.CountingReference","a":"SCM/ID/Density","t":1,"d":"The counting reference family is faithful: every singleton coordinate value has counting mass one, in particular nonzero.","s":"∀ {N : Type u_1} {Ω : N → Type u_2} [inst : (n : N) → MeasurableSpace (Ω n)] [inst_1 : ∀ (sn : Causalean.SWIGNode N), Countable (Causalean.swigΩ Ω sn)] [inst_2 : ∀ (n : N), MeasurableSingletonClass (Ω n)], Causalean.SCM.ReferenceFaithful Causalean.SCM.countingRef","l":36,"f":"Causalean/SCM/ID/Density/CountingReference.lean"},{"n":"Causalean.SCM.reindexSubtypeProj","k":"def","m":"Causalean.SCM.Do.GlobalMarkov","a":"SCM/Do","t":1,"d":"Measurable reindexing equivalence between the coordinate tuple on the subtype-image S.subtype (· ∈ P) of S ⊆ P and the coordinate tuple on S itself. It is the identity on values, only relabelling the index from the doubly-nested subtype to the direct subtype {j // j ∈ S}.","s":"{M' : Type u_1} → [inst : DecidableEq M'] → [Fintype M'] → {Ω' : M' → Type u_2} → [inst_2 : (n : M') → MeasurableSpace (Ω' n)] → {P : Finset M'} → (S : Finset M') → S ⊆ P → ((i : ↥(Finset.subtype (fun x => x ∈ P) S)) → Ω' ↑↑i) ≃ᵐ ((j : ↥S) → Ω' ↑j)","l":312,"f":"Causalean/SCM/Do/GlobalMarkov.lean"},{"n":"Causalean.SCM.rnDeriv_compProd_prod_sigmaFinite_of_fiber_ac","k":"theorem","m":"Causalean.SCM.ID.Density.ChainRuleDensity","a":"SCM/ID/Density","t":2,"d":"σ-finite product-reference RN derivative for a finite kernel, under explicit fibre domination and fibre-density measurability.","s":"∀ {α : Type u_3} {β : Type u_4} [inst : MeasurableSpace α] [inst_1 : MeasurableSpace β] (μ ν : MeasureTheory.Measure α) (ρ : MeasureTheory.Measure β) (κ : ProbabilityTheory.Kernel α β) (f : α → ENNReal) [MeasureTheory.IsFiniteMeasure μ] [MeasureTheory.SigmaFinite ν] [MeasureTheory.SigmaFinite ρ] [ProbabilityTheory.IsFiniteKernel κ], μ.AbsolutelyContinuous ν → (∀ᵐ (a : α) ∂μ, (κ a).AbsolutelyContinuous ρ) → AEMeasurable (fun p => (κ p.1).rnDeriv ρ p.2) (ν.prod ρ) → μ.rnDeriv ν =ᵐ[ν] f → (μ.compProd κ).rnDeriv (ν.prod ρ) =ᵐ[ν.prod ρ] fun p => f p.1 * (κ p.1).rnDeriv ρ p.2","l":466,"f":"Causalean/SCM/ID/Density/ChainRuleDensity.lean"},{"n":"Causalean.SCM.rnDeriv_compProd_same_left","k":"theorem","m":"Causalean.SCM.ID.Density.ChainRuleDensity","a":"SCM/ID/Density","t":2,"d":"Conditional factor of the composition-product Radon–Nikodym derivative.","s":"∀ {α : Type u_3} {β : Type u_4} [inst : MeasurableSpace α] [inst_1 : MeasurableSpace β] [inst_2 : MeasurableSpace.CountableOrCountablyGenerated α β] (μ : MeasureTheory.Measure α) (κ η : ProbabilityTheory.Kernel α β) [MeasureTheory.IsFiniteMeasure μ] [ProbabilityTheory.IsFiniteKernel κ] [ProbabilityTheory.IsFiniteKernel η], (∀ (a : α), (κ a).AbsolutelyContinuous (η a)) → (μ.compProd κ).rnDeriv (μ.compProd η) =ᵐ[μ.compProd η] fun p => κ.rnDeriv η p.1 p.2","l":387,"f":"Causalean/SCM/ID/Density/ChainRuleDensity.lean"},{"n":"Causalean.SCM.rnDeriv_compProd_same_left_ae","k":"theorem","m":"Causalean.SCM.ID.Density.ChainRuleDensity","a":"SCM/ID/Density","t":2,"d":"Same-left composition-product RN derivative under a.e. fibre absolute continuity. This is the a.e. variant needed after extracting fibre domination from product domination.","s":"∀ {α : Type u_3} {β : Type u_4} [inst : MeasurableSpace α] [inst_1 : MeasurableSpace β] [inst_2 : MeasurableSpace.CountableOrCountablyGenerated α β] (μ : MeasureTheory.Measure α) (κ η : ProbabilityTheory.Kernel α β) [MeasureTheory.IsFiniteMeasure μ] [ProbabilityTheory.IsFiniteKernel κ] [ProbabilityTheory.IsFiniteKernel η], (∀ᵐ (a : α) ∂μ, (κ a).AbsolutelyContinuous (η a)) → (μ.compProd κ).rnDeriv (μ.compProd η) =ᵐ[μ.compProd η] fun p => κ.rnDeriv η p.1 p.2","l":432,"f":"Causalean/SCM/ID/Density/ChainRuleDensity.lean"},{"n":"Causalean.SCM.rnDeriv_mul_measure_singleton","k":"theorem","m":"Causalean.SCM.ID.Density.MassBridge","a":"SCM/ID/Density","t":2,"d":"On a measurable singleton, the RN derivative multiplied by the base mass recovers the numerator singleton mass.","s":"∀ {α : Type u_3} [inst : MeasurableSpace α] [MeasurableSingletonClass α] (μ ν : MeasureTheory.Measure α) [MeasureTheory.SFinite ν] [μ.HaveLebesgueDecomposition ν], μ.AbsolutelyContinuous ν → ∀ (x : α), ν {x} * μ.rnDeriv ν x = μ {x}","l":27,"f":"Causalean/SCM/ID/Density/MassBridge.lean"},{"n":"Causalean.SCM.rnDeriv_singleton_eq_div","k":"theorem","m":"Causalean.SCM.ID.Density.MassBridge","a":"SCM/ID/Density","t":2,"d":"On a positive finite singleton of the reference measure, the RN derivative is the ratio of numerator mass to reference mass.","s":"∀ {α : Type u_3} [inst : MeasurableSpace α] [MeasurableSingletonClass α] (μ ν : MeasureTheory.Measure α) [MeasureTheory.SFinite ν] [μ.HaveLebesgueDecomposition ν], μ.AbsolutelyContinuous ν → ∀ (x : α), ν {x} ≠ 0 → ν {x} ≠ ⊤ → μ.rnDeriv ν x = μ {x} / ν {x}","l":41,"f":"Causalean/SCM/ID/Density/MassBridge.lean"},{"n":"Causalean.SCM.singletonValue","k":"def","m":"Causalean.SCM.Factored.ObsChainKernel","a":"SCM/Factored","t":1,"d":"Read the only coordinate of a singleton ValuesOn tuple.","s":"{ι : Type u_3} → {α : ι → Type u_4} → {v : ι} → Causalean.ValuesOn {v} α → α v","l":203,"f":"Causalean/SCM/Factored/ObsChainKernel.lean"},{"n":"Causalean.SCM.singletonValue_singletonValues","k":"theorem","m":"Causalean.SCM.Factored.ObsChainKernel","a":"SCM/Factored","t":2,"d":"Reading the tuple built from a singleton value returns that value.","s":"∀ {ι : Type u_3} {α : ι → Type u_4} {v : ι} (x : α v), Causalean.SCM.singletonValue (Causalean.SCM.singletonValues x) = x","l":236,"f":"Causalean/SCM/Factored/ObsChainKernel.lean"},{"n":"Causalean.SCM.singletonValues","k":"def","m":"Causalean.SCM.Factored.ObsChainKernel","a":"SCM/Factored","t":1,"d":"Build a singleton ValuesOn tuple from a value.","s":"{ι : Type u_3} → {α : ι → Type u_4} → {v : ι} → α v → Causalean.ValuesOn {v} α","l":209,"f":"Causalean/SCM/Factored/ObsChainKernel.lean"},{"n":"Causalean.SCM.singletonValues_map_ref_eq_jointRef","k":"theorem","m":"Causalean.SCM.ID.Density.ChainRuleDensity","a":"SCM/ID/Density","t":2,"d":"The scalar reference on one node maps to the singleton product reference.","s":"∀ {N : Type u_1} [DecidableEq N] [Fintype N] {Ω : N → Type u_2} [inst : (n : N) → MeasurableSpace (Ω n)] (ref : Causalean.SCM.ReferenceMeasures Ω) (v : Causalean.SWIGNode N), MeasureTheory.Measure.map Causalean.SCM.singletonValues (ref.μ v) = Causalean.SCM.jointRef ref {v}","l":170,"f":"Causalean/SCM/ID/Density/ChainRuleDensity.lean"},{"n":"Causalean.SCM.singletonValues_singletonValue","k":"theorem","m":"Causalean.SCM.Factored.ObsChainKernel","a":"SCM/Factored","t":2,"d":"Building a singleton tuple from its only coordinate returns the tuple.","s":"∀ {ι : Type u_3} {α : ι → Type u_4} {v : ι} (x : Causalean.ValuesOn {v} α), Causalean.SCM.singletonValues (Causalean.SCM.singletonValue x) = x","l":243,"f":"Causalean/SCM/Factored/ObsChainKernel.lean"},{"n":"Causalean.SCM.standard_fixed_random_edgeless","k":"theorem","m":"Causalean.SCM.ID.Density.MechCFactor","a":"SCM/ID/Density","t":2,"d":"","s":"∀ {N : Type u_1} [inst : DecidableEq N] [inst_1 : Fintype N] {Ω : N → Type u_2} [inst_2 : (n : N) → MeasurableSpace (Ω n)] (M : Causalean.SCM N Ω), M.isStandard → ∀ (n : N), Causalean.SWIGNode.fixed n ∈ M.fixed → ∀ (v : Causalean.SWIGNode N), ¬M.dag.edge (Causalean.SWIGNode.random n) v","l":259,"f":"Causalean/SCM/ID/Density/MechCFactor.lean"},{"n":"Causalean.SCM.stepFun","k":"def","m":"Causalean.SCM.Factored.StepKernel","a":"SCM/Factored","t":1,"d":"The deterministic assignment producing the value of the n-th observed node v_n = M.observedAt ⟨n, hn⟩ from the current prefix state (s, ℓ, ξ): assemble the parent tuple via parentValuesFromPrefix, then apply the structural function M.structFun v_n.","s":"{N : Type uN} → [inst : DecidableEq N] → [inst_1 : Fintype N] → {Ω : N → Type uΩ} → [inst_2 : (n : N) → MeasurableSpace (Ω n)] → (M : Causalean.SCM N Ω) → {n : ℕ} → (hn : n + 1 ≤ M.observed.card) → M.FixedValues × M.OrderedLatentPrefixValues n (Nat.le_of_succ_le hn) → Causalean.swigΩ Ω ↑(M.observedAt ⟨n, hn⟩)","l":50,"f":"Causalean/SCM/Factored/StepKernel.lean"},{"n":"Causalean.SCM.stepKernel","k":"def","m":"Causalean.SCM.Factored.StepKernel","a":"SCM/Factored","t":1,"d":"The step kernel for the n-th observed node. Since structFun v_n is a deterministic measurable map, the kernel is Kernel.deterministic (stepFun hn): on input (s, ℓ, ξ) it returns Dirac (structFun v_n (Pa(v_n))).","s":"{N : Type uN} → [inst : DecidableEq N] → [inst_1 : Fintype N] → {Ω : N → Type uΩ} → [inst_2 : (n : N) → MeasurableSpace (Ω n)] → (M : Causalean.SCM N Ω) → {n : ℕ} → (hn : n + 1 ≤ M.observed.card) → ProbabilityTheory.Kernel (M.FixedValues × M.OrderedLatentPrefixValues n (Nat.le_of_succ_le hn)) (Causalean.swigΩ Ω ↑(M.observedAt ⟨n, hn⟩))","l":77,"f":"Causalean/SCM/Factored/StepKernel.lean"},{"n":"Causalean.SCM.structFun_apply_eq_of_equiv","k":"theorem","m":"Causalean.SCM.Model.EquivKernel","a":"SCM/Model","t":2,"d":"Equivalent structural graphs with matching structural functions produce the same output at an observed node whenever corresponding parent values agree.","s":"∀ {N : Type u_1} [inst : DecidableEq N] [inst_1 : Fintype N] {Ω : N → Type u_2} [inst_2 : (n : N) → MeasurableSpace (Ω n)] {M₁ M₂ : Causalean.SCM N Ω}, M₁.Equivalent M₂.toSWIGGraph → M₁.structFun ≍ M₂.structFun → ∀ (v : Causalean.SWIGNode N) (hv_obs₁ : v ∈ M₁.observed) (hv_obs₂ : v ∈ M₂.observed) {ξ₁ : (w : ↥(M₁.dag.parents v)) → Causalean.swigΩ Ω ↑w} {ξ₂ : (w : ↥(M₂.dag.parents v)) → Causalean.swigΩ Ω ↑w}, (∀ (p : Causalean.SWIGNode N) (hp₁ : p ∈ M₁.dag.parents v) (hp₂ : p ∈ M₂.dag.parents v), ξ₁ ⟨p, hp₁⟩ = ξ₂ ⟨p, hp₂⟩) → M₁.structFun ⟨v, hv_obs₁⟩ ξ₁ = M₂.structFun ⟨v, hv_obs₂⟩ ξ₂","l":40,"f":"Causalean/SCM/Model/EquivKernel.lean"},{"n":"Causalean.SCM.swigInterventionSet_insert_equiv","k":"theorem","m":"Causalean.SCM.Model.InterventionSet","a":"SCM/Model","t":2,"d":"Insert form of monolithic intervention composition.","s":"∀ {N : Type u_1} [inst : DecidableEq N] [inst_1 : Fintype N] {Ω : N → Type u_2} [inst_2 : (n : N) → MeasurableSpace (Ω n)] (M : Causalean.SCM N Ω) (X : Finset N) (y : N) (hyX : y ∉ X) (hInsert_obs : ∀ D ∈ insert y X, Causalean.SWIGNode.random D ∈ M.observed) (hInsert_fixed : ∀ D ∈ insert y X, Causalean.SWIGNode.fixed D ∉ M.fixed), have hX_obs := fun D hD => hInsert_obs D (Finset.mem_insert_of_mem hD); have hX_fixed := fun D hD => hInsert_fixed D (Finset.mem_insert_of_mem hD); have hy_obs := fun D hD => have hDy := Finset.mem_singleton.mp hD; Eq.ndrec (motive := fun D => D ∈ {y} → Causalean.SWIGNode.random D ∈ (M.fixSet X hX_obs hX_fixed).observed) (fun hD => id (hInsert_obs y (Finset.mem_insert_self y X))) (Eq.symm hDy) hD; have hy_fixed := fun D hD hDfixed => have hDy := Finset.mem_singleton.mp hD; Eq.ndrec (motive := fun D => D ∈ {y} → Causalean.SWIGNode.fixed D ∈ (M.fixSet X hX_obs hX_fixed).fixed → False) (fun hD hDfixed => Or.casesOn (Finset.mem_union.mp (Eq.mp (congrArg (fun _a => Causalean.SWIGNode.fixed y ∈ _a) (Causalean.SCM.fixSet_fixed M X hX_obs hX_fixed)) hDfixed)) (fun hyfixed => hInsert_fixed y (Finset.mem_insert_self y X) hyfixed) fun hyimage => Exists.casesOn (Finset.mem_image.mp hyimage) fun z h => And.casesOn h fun hz hzy => hyX (Eq.symm (Eq.symm (Causalean.SWIGNode.fixed.inj hzy)) ▸ hz)) (Eq.symm hDy) hD hDfixed; ((M.fixSet X hX_obs hX_fixed).fixSet {y} hy_obs hy_fixed).Equiv (M.fixSet (insert y X) hInsert_obs hInsert_fixed)","l":681,"f":"Causalean/SCM/Model/InterventionSet.lean"},{"n":"Causalean.SCM.topoLinearOrder","k":"def","m":"Causalean.SCM.Model.SCM","a":"SCM/Model","t":1,"d":"The graph's topological order gives a canonical linear order on SWIG nodes.","s":"{N : Type uN} → [inst : DecidableEq N] → [inst_1 : Fintype N] → {Ω : N → Type uΩ} → [inst_2 : (n : N) → MeasurableSpace (Ω n)] → Causalean.SCM N Ω → LinearOrder (Causalean.SWIGNode N)","l":190,"f":"Causalean/SCM/Model/SCM.lean"},{"n":"Causalean.SCM.treatmentMarginal","k":"def","m":"Causalean.SCM.ID.Backdoor","a":"SCM/ID","t":1,"d":"The observational treatment marginal νX = (M.obsKernel s₀).map π_{X.random}.","s":"{N : Type u_1} → [inst : DecidableEq N] → [inst_1 : Fintype N] → {Ω : N → Type u_2} → [inst_2 : (n : N) → MeasurableSpace (Ω n)] → (M : Causalean.SCM N Ω) → (X : Finset N) → Finset.image Causalean.SWIGNode.random X ⊆ M.observed → M.FixedValues → MeasureTheory.Measure (Causalean.ValuesOn (Finset.image Causalean.SWIGNode.random X) (Causalean.swigΩ Ω))","l":60,"f":"Causalean/SCM/ID/Backdoor.lean"},{"n":"Causalean.SCM.treatment_not_isAncestor_cond","k":"theorem","m":"Causalean.SCM.Do.CutsetDSep","a":"SCM/Do","t":2,"d":"Under the backdoor non-descendancy condition, a treatment node cannot be a proper ancestor of any node in the observed-or-fixed conditioning block.","s":"∀ {N : Type u_1} [inst : DecidableEq N] [inst_1 : Fintype N] {Ω : N → Type u_2} [inst_2 : (n : N) → MeasurableSpace (Ω n)] (M : Causalean.SCM N Ω) (Z : Finset N) (W : Finset (Causalean.SWIGNode N)), (∀ D ∈ Z, ∀ w ∈ W, ¬M.dag.isAncestor (Causalean.SWIGNode.random D) w) → ∀ {D : N}, D ∈ Z → ∀ {c : Causalean.SWIGNode N}, c ∈ W ∪ M.fixed → M.dag.isAncestor (Causalean.SWIGNode.random D) c → False","l":236,"f":"Causalean/SCM/Do/CutsetDSep.lean"},{"n":"Causalean.SCM.unionSumEquiv","k":"def","m":"Causalean.SCM.ID.Density.PiUnion","a":"SCM/ID/Density","t":1,"d":"The index equivalence {a ∈ A} ⊕ {b ∈ B} ≃ {i ∈ A ∪ B} for disjoint A, B.","s":"{N : Type u_1} → [inst : DecidableEq N] → {A B : Finset (Causalean.SWIGNode N)} → Disjoint A B → ↥A ⊕ ↥B ≃ ↥(A ∪ B)","l":83,"f":"Causalean/SCM/ID/Density/PiUnion.lean"},{"n":"Causalean.SCM.valuesEquivOfEq_heq","k":"theorem","m":"Causalean.SCM.Do.ValuesReindex","a":"SCM/Do","t":2,"d":"A value block is HEq to its reindexing along a Finset equality. The bridge that lets the equiv-mediated Eq laws above discharge legacy HEq goals in one line.","s":"∀ {M : Type u_3} {Ω : M → Type u_4} [inst : (m : M) → MeasurableSpace (Ω m)] {I J : Finset M} (h : I = J) (x : Causalean.ValuesOn I Ω), (Causalean.valuesEquivOfEq h) x ≍ x","l":61,"f":"Causalean/SCM/Do/ValuesReindex.lean"},{"n":"Causalean.SCM.valuesOn_heq_of_coord","k":"theorem","m":"Causalean.SCM.Do.ValuesReindex","a":"SCM/Do","t":1,"d":"HEq of value assignments from coordinatewise agreement. Given two node sets I and J that are equal as sets and value assignments f on I and g on J, if f and g agree at every node common to both index sets, then f and g are heterogeneously equal (HEq).","s":"∀ {M : Type u_3} {Ω : M → Type u_4} [(m : M) → MeasurableSpace (Ω m)] {I J : Finset M}, I = J → ∀ (f : Causalean.ValuesOn I Ω) (g : Causalean.ValuesOn J Ω), (∀ (v : M) (hI : v ∈ I) (hJ : v ∈ J), f ⟨v, hI⟩ = g ⟨v, hJ⟩) → f ≍ g","l":93,"f":"Causalean/SCM/Do/ValuesReindex.lean"},{"n":"Causalean.SCM.valuesProjection_overrideOn","k":"theorem","m":"Causalean.SCM.ID.Density.PiUnion","a":"SCM/ID/Density","t":2,"d":"Projecting an override back to the overridden coordinates returns the replacement assignment.","s":"∀ {N : Type u_1} [inst : DecidableEq N] {Ω : N → Type u_2} [inst_1 : (n : N) → MeasurableSpace (Ω n)] {I W : Finset (Causalean.SWIGNode N)} (hWI : W ⊆ I) (x : Causalean.ValuesOn I (Causalean.swigΩ Ω)) (y : Causalean.ValuesOn W (Causalean.swigΩ Ω)), Causalean.valuesProjection hWI (Causalean.SCM.overrideOn x y) = y","l":75,"f":"Causalean/SCM/ID/Density/PiUnion.lean"},{"n":"Causalean.SCM.valuesProjection_randomToObserved_evalMap_meas","k":"theorem","m":"Causalean.SCM.Do.Rule2Kernel.WitnessBridge","a":"SCM/Do/Rule2Kernel","t":2,"d":"The pulled-back observed-coordinate projection through randomToObserved ∘ evalMap s is the latent projection of the evaluation, valuesProjection hY ∘ randomToObserved ∘ E. This is definitional unfolding, exposed so downstream condDistrib_map_comp rewrites can name the composite.","s":"∀ {N : Type u_1} [inst : DecidableEq N] [inst_1 : Fintype N] {Ω : N → Type u_2} [inst_2 : (n : N) → MeasurableSpace (Ω n)] (M : Causalean.SCM N Ω) {Y : Finset (Causalean.SWIGNode N)} (hY : Y ⊆ M.observed) (s : M.FixedValues), Measurable fun ℓ => Causalean.valuesProjection hY (M.randomToObserved (M.evalMap s ℓ))","l":196,"f":"Causalean/SCM/Do/Rule2Kernel/WitnessBridge.lean"},{"n":"Causalean.SCM.valuesUnionEquiv","k":"def","m":"Causalean.SCM.Do.Rule2Kernel.Helpers","a":"SCM/Do/Rule2Kernel","t":1,"d":"A value assignment on a disjoint union is measurably equivalent to the pair of assignments on the two parts.","s":"{N : Type u_1} → [inst : DecidableEq N] → {Ω : N → Type u_2} → [inst_1 : (n : N) → MeasurableSpace (Ω n)] → {A B : Finset (Causalean.SWIGNode N)} → Disjoint A B → Causalean.ValuesOn (A ∪ B) (Causalean.swigΩ Ω) ≃ᵐ Causalean.ValuesOn A (Causalean.swigΩ Ω) × Causalean.ValuesOn B (Causalean.swigΩ Ω)","l":99,"f":"Causalean/SCM/Do/Rule2Kernel/Helpers.lean"},{"n":"Causalean.SCM.valuesUnionEquiv_extendObsPrefix","k":"theorem","m":"Causalean.SCM.Factored.ObsChainKernel","a":"SCM/Factored","t":2,"d":"The successor-prefix extension is inverse to the union-equivalence view of the successor prefix.","s":"∀ {N : Type u_1} [inst : DecidableEq N] [inst_1 : Fintype N] {Ω : N → Type u_2} [inst_2 : (n : N) → MeasurableSpace (Ω n)] (M : Causalean.SCM N Ω) {n : ℕ} (hn : n < M.observed.card) (p : Causalean.ValuesOn (M.prefixNodes n) (Causalean.swigΩ Ω) × Causalean.swigΩ Ω ↑(M.observedAt ⟨n, hn⟩)), (Causalean.SCM.valuesUnionEquiv (Causalean.SCM.prefixNodes_disjoint_singleton_next M hn)) ((Causalean.valuesEquivOfEq (Causalean.SCM.prefixNodes_succ M hn)) (M.extendObsPrefix hn p)) = (p.1, Causalean.SCM.singletonValues p.2)","l":464,"f":"Causalean/SCM/Factored/ObsChainKernel.lean"},{"n":"Causalean.SCM.valuesUnionEquiv_valuesEquivOfEq_symm_valuesUnionMk","k":"theorem","m":"Causalean.SCM.Factored.ObsChainKernel","a":"SCM/Factored","t":2,"d":"Transporting a combined assignment to an equal index set and back, then splitting the disjoint union, recovers the original pair of assignments.","s":"∀ {N : Type u_1} [inst : DecidableEq N] {Ω : N → Type u_2} [inst_1 : (n : N) → MeasurableSpace (Ω n)] {A B C : Finset (Causalean.SWIGNode N)} (hDisj : Disjoint A B) (hUnion : C = A ∪ B) (p : Causalean.ValuesOn A (Causalean.swigΩ Ω) × Causalean.ValuesOn B (Causalean.swigΩ Ω)), (Causalean.SCM.valuesUnionEquiv hDisj) ((Causalean.valuesEquivOfEq hUnion) ((Causalean.valuesEquivOfEq hUnion).symm (Causalean.valuesUnionMk p.1 p.2))) = p","l":452,"f":"Causalean/SCM/Factored/ObsChainKernel.lean"},{"n":"Causalean.SCM.valuesUnionMk_comm","k":"theorem","m":"Causalean.SCM.Do.ValuesReindex","a":"SCM/Do","t":1,"d":"Union commutativity. When A and B are disjoint, reindexing valuesUnionMk a b (a block on A ∪ B) along A ∪ B = B ∪ A yields valuesUnionMk b a.","s":"∀ {M : Type u_3} [inst : DecidableEq M] {Ω : M → Type u_4} [inst_1 : (m : M) → MeasurableSpace (Ω m)] {A B : Finset M}, Disjoint A B → ∀ (a : Causalean.ValuesOn A Ω) (b : Causalean.ValuesOn B Ω), (Causalean.valuesEquivOfEq (Finset.union_comm A B)) (Causalean.valuesUnionMk a b) = Causalean.valuesUnionMk b a","l":33,"f":"Causalean/SCM/Do/ValuesReindex.lean"},{"n":"Causalean.SCM.valuesUnionMk_comm_heq","k":"theorem","m":"Causalean.SCM.Do.ValuesReindex","a":"SCM/Do","t":2,"d":"Union commutativity, HEq form. Supersedes the ad-hoc per-proof HEq construction: a one-line corollary of valuesUnionMk_comm.","s":"∀ {M : Type u_3} [inst : DecidableEq M] {Ω : M → Type u_4} [inst_1 : (m : M) → MeasurableSpace (Ω m)] {A B : Finset M}, Disjoint A B → ∀ (a : Causalean.ValuesOn A Ω) (b : Causalean.ValuesOn B Ω), Causalean.valuesUnionMk a b ≍ Causalean.valuesUnionMk b a","l":70,"f":"Causalean/SCM/Do/ValuesReindex.lean"},{"n":"Causalean.SCM.valuesUnionMk_empty_right","k":"theorem","m":"Causalean.SCM.Do.ValuesReindex","a":"SCM/Do","t":2,"d":"Right ∅ collapse. Reindexing valuesUnionMk a e (with e the trivial block on ∅) along A ∪ ∅ = A recovers a.","s":"∀ {M : Type u_3} [inst : DecidableEq M] {Ω : M → Type u_4} [inst_1 : (m : M) → MeasurableSpace (Ω m)] {A : Finset M} (a : Causalean.ValuesOn A Ω) (e : Causalean.ValuesOn ∅ Ω), (Causalean.valuesEquivOfEq (Finset.union_empty A)) (Causalean.valuesUnionMk a e) = a","l":50,"f":"Causalean/SCM/Do/ValuesReindex.lean"},{"n":"Causalean.SCM.valuesUnionMk_empty_right_heq","k":"theorem","m":"Causalean.SCM.Do.ValuesReindex","a":"SCM/Do","t":2,"d":"Right ∅ collapse, HEq form. valuesUnionMk a e (trivial ∅ block) is HEq to a; a one-line corollary of valuesUnionMk_empty_right.","s":"∀ {M : Type u_3} [inst : DecidableEq M] {Ω : M → Type u_4} [inst_1 : (m : M) → MeasurableSpace (Ω m)] {A : Finset M} (a : Causalean.ValuesOn A Ω) (e : Causalean.ValuesOn ∅ Ω), Causalean.valuesUnionMk a e ≍ a","l":79,"f":"Causalean/SCM/Do/ValuesReindex.lean"},{"n":"Causalean.SCM.withDensity_obsDensity_eq","k":"theorem","m":"Causalean.SCM.ID.Density.ReferenceMeasure","a":"SCM/ID/Density","t":1,"d":"In a structural causal model whose observational law is absolutely continuous with respect to the joint reference measure on the observed nodes, weighting that joint reference measure by the observational density recovers the observational law exactly.","s":"∀ {N : Type u_1} [inst : DecidableEq N] [inst_1 : Fintype N] {Ω : N → Type u_2} [inst_2 : (n : N) → MeasurableSpace (Ω n)] (M : Causalean.SCM N Ω) (ref : Causalean.SCM.ReferenceMeasures Ω), M.DominatedObs ref → ∀ (s : M.FixedValues), (Causalean.SCM.jointRef ref M.observed).withDensity (M.obsDensity ref s) = M.obsKernel s","l":79,"f":"Causalean/SCM/ID/Density/ReferenceMeasure.lean"},{"n":"Causalean.SCM.xRandomAsFixed","k":"def","m":"Causalean.SCM.Do.Rule2Kernel.Helpers","a":"SCM/Do/Rule2Kernel","t":1,"d":"This map reads observed treatment values as values for the corresponding intervention coordinates.","s":"{N : Type u_1} → [inst : DecidableEq N] → {Ω : N → Type u_2} → {X : Finset N} → Causalean.ValuesOn (Finset.image Causalean.SWIGNode.random X) (Causalean.swigΩ Ω) → Causalean.ValuesOn (Finset.image Causalean.SWIGNode.fixed X) (Causalean.swigΩ Ω)","l":193,"f":"Causalean/SCM/Do/Rule2Kernel/Helpers.lean"},{"n":"Causalean.SCM.zFixedAsRandom","k":"def","m":"Causalean.SCM.Do.Rule2Kernel.Helpers","a":"SCM/Do/Rule2Kernel","t":1,"d":"This map reads intervention values as values of the corresponding observed random variables.","s":"{N : Type u_1} → [inst : DecidableEq N] → {Ω : N → Type u_2} → {Z : Finset N} → Causalean.ValuesOn (Finset.image Causalean.SWIGNode.fixed Z) (Causalean.swigΩ Ω) → Causalean.ValuesOn (Finset.image Causalean.SWIGNode.random Z) (Causalean.swigΩ Ω)","l":43,"f":"Causalean/SCM/Do/Rule2Kernel/Helpers.lean"},{"n":"Causalean.SCM.zFixedAsRandom_proj_fixSetExtend","k":"theorem","m":"Causalean.SCM.Do.Rule2Kernel.Helpers","a":"SCM/Do/Rule2Kernel","t":2,"d":"Relabeling the treatment part of the extended fixed assignment back to observed coordinates recovers the treatment assignment.","s":"∀ {N : Type u_1} [inst : DecidableEq N] [inst_1 : Fintype N] {Ω : N → Type u_2} [inst_2 : (n : N) → MeasurableSpace (Ω n)] (M : Causalean.SCM N Ω) (X : Finset N) (hObs : ∀ D ∈ X, Causalean.SWIGNode.random D ∈ M.observed) (hFix : ∀ D ∈ X, Causalean.SWIGNode.fixed D ∉ M.fixed) (s0 : M.FixedValues) (t : Causalean.ValuesOn (Finset.image Causalean.SWIGNode.random X) (Causalean.swigΩ Ω)), Causalean.SCM.zFixedAsRandom (Causalean.valuesProjection (Causalean.SCM.fixSet_image_fixed_subset M X hObs hFix) (M.fixSetExtend X hObs hFix s0 t)) = t","l":278,"f":"Causalean/SCM/Do/Rule2Kernel/Helpers.lean"},{"n":"Causalean.SWIGGraph","k":"structure","m":"Causalean.Graph.SWIG","a":"Graph","t":1,"d":"A Single-World Intervention Graph (SWIG), G = (S, V, U, E, ι) (Definition 4 from Basic Concepts.tex): a directed acyclic graph on the SWIG nodes whose vertices are partitioned into fixed intervention nodes, observed random nodes, and unobserved random nodes, where every fixed node is genuinely of fixed form, every observed node is of random form, every unobserved node is of random form, and the observed and unobserved sets are disjoint. Every edge of the graph has both endpoints classified as fixed, observed, or unobserved; the map sending each fixed intervention node to its random counterpart lands inside the observed nodes; fixed nodes and unobserved nodes have no parents; a fixed-form node absent from the fixed set is isolated, with neither parents nor children; and every child of a classified node is observed.","s":"(N : Type u_2) → [DecidableEq N] → [Fintype N] → Type u_2","l":355,"f":"Causalean/Graph/SWIG.lean"},{"n":"Causalean.SWIGGraph.Equivalent","k":"def","m":"Causalean.Graph.SWIG","a":"Graph","t":1,"d":"Equivalence of SWIG graphs, ignoring the particular topological order.","s":"{N : Type u_2} → [inst : DecidableEq N] → [inst_1 : Fintype N] → Causalean.SWIGGraph N → Causalean.SWIGGraph N → Prop","l":454,"f":"Causalean/Graph/SWIG.lean"},{"n":"Causalean.SWIGGraph.Equivalent.parents_eq","k":"theorem","m":"Causalean.Graph.SWIG","a":"Graph","t":2,"d":"Equivalent SWIGGraphs have the same parents Finset at every node.","s":"∀ {N : Type u_2} [inst : DecidableEq N] [inst_1 : Fintype N] {G H : Causalean.SWIGGraph N}, (∀ (u v : Causalean.SWIGNode N), G.dag.edge u v ↔ H.dag.edge u v) → ∀ (v : Causalean.SWIGNode N), G.dag.parents v = H.dag.parents v","l":509,"f":"Causalean/Graph/SWIG.lean"},{"n":"Causalean.SWIGGraph.Equivalent.refl","k":"theorem","m":"Causalean.Graph.SWIG","a":"Graph","t":2,"d":"SWIG graph equivalence is reflexive.","s":"∀ {N : Type u_2} [inst : DecidableEq N] [inst_1 : Fintype N] (G : Causalean.SWIGGraph N), G.Equivalent G","l":469,"f":"Causalean/Graph/SWIG.lean"},{"n":"Causalean.SWIGGraph.Equivalent.splitMono_congr","k":"theorem","m":"Causalean.Graph.SWIGSplitMono","a":"Graph","t":2,"d":"Congruence of splitMono under SWIGGraph.Equivalent.","s":"∀ {N : Type u_1} [inst : DecidableEq N] [inst_1 : Fintype N] {G₁ G₂ : Causalean.SWIGGraph N}, G₁.Equivalent G₂ → ∀ (X : Finset N) (hObs₁ : ∀ D ∈ X, Causalean.SWIGNode.random D ∈ G₁.observed) (hFix₁ : ∀ D ∈ X, Causalean.SWIGNode.fixed D ∉ G₁.fixed) (hObs₂ : ∀ D ∈ X, Causalean.SWIGNode.random D ∈ G₂.observed) (hFix₂ : ∀ D ∈ X, Causalean.SWIGNode.fixed D ∉ G₂.fixed), (G₁.splitMono X hObs₁ hFix₁).Equivalent (G₂.splitMono X hObs₂ hFix₂)","l":611,"f":"Causalean/Graph/SWIGSplitMono.lean"},{"n":"Causalean.SWIGGraph.Equivalent.symm","k":"theorem","m":"Causalean.Graph.SWIG","a":"Graph","t":2,"d":"SWIG graph equivalence is symmetric.","s":"∀ {N : Type u_2} [inst : DecidableEq N] [inst_1 : Fintype N] {G H : Causalean.SWIGGraph N}, G.Equivalent H → H.Equivalent G","l":476,"f":"Causalean/Graph/SWIG.lean"},{"n":"Causalean.SWIGGraph.Equivalent.trans","k":"theorem","m":"Causalean.Graph.SWIG","a":"Graph","t":2,"d":"SWIG graph equivalence is transitive.","s":"∀ {N : Type u_2} [inst : DecidableEq N] [inst_1 : Fintype N] {G H K : Causalean.SWIGGraph N}, G.Equivalent H → H.Equivalent K → G.Equivalent K","l":491,"f":"Causalean/Graph/SWIG.lean"},{"n":"Causalean.SWIGGraph.activePath_node_not_fixed","k":"theorem","m":"Causalean.Graph.DSep.InduceTransport","a":"Graph/DSep","t":2,"d":"Every node on an active path conditioned on the union of a conditioning set and the fixed intervention nodes, with observed endpoints, is not fixed.","s":"∀ {N : Type u_1} [inst : DecidableEq N] [inst_1 : Fintype N] (G : Causalean.SWIGGraph N) {R X Y Z : Finset (Causalean.SWIGNode N)} {x y v : Causalean.SWIGNode N} {p : List (Causalean.SWIGNode N)}, X ⊆ R ∩ G.observed → Y ⊆ R ∩ G.observed → x ∈ X → y ∈ Y → G.dag.IsActivePath (Z ∪ G.fixed) p → p.head? = some x → p.getLast? = some y → v ∈ p → v ∉ G.fixed","l":246,"f":"Causalean/Graph/DSep/InduceTransport.lean"},{"n":"Causalean.SWIGGraph.backdoorCriterion","k":"def","m":"Causalean.SCM.ID.BackdoorCriterion","a":"SCM/ID","t":1,"d":"Backdoor criterion of Z relative to (X, Y) in the SWIG graph G.","s":"{N : Type u_1} → [inst : DecidableEq N] → [inst_1 : Fintype N] → (G : Causalean.SWIGGraph N) → (X : Finset N) → (∀ D ∈ X, Causalean.SWIGNode.random D ∈ G.observed) → (∀ D ∈ X, Causalean.SWIGNode.fixed D ∉ G.fixed) → Finset (Causalean.SWIGNode N) → Finset (Causalean.SWIGNode N) → Prop","l":69,"f":"Causalean/SCM/ID/BackdoorCriterion.lean"},{"n":"Causalean.SWIGGraph.bidirectedBFS","k":"def","m":"Causalean.Graph.CComponents","a":"Graph","t":1,"d":"Compute the bidirected-reachable set from a single vertex via BFS on the bidirected projection graph.","s":"{N : Type u_1} → [inst : DecidableEq N] → [inst_1 : Fintype N] → Causalean.SWIGGraph N → Causalean.SWIGNode N → Finset (Causalean.SWIGNode N)","l":89,"f":"Causalean/Graph/CComponents.lean"},{"n":"Causalean.SWIGGraph.bidirectedBFS_go_closed","k":"theorem","m":"Causalean.Graph.CComponents","a":"Graph","t":2,"d":"Closure at saturation. Under the BFS invariants — the visited set is observed, and every *already-expanded* visited node (one outside the frontier) has all its directly-confounded neighbors in visited — and given enough remaining fuel (card observed - card visited ≤ fuel), the result of go is closed under the directly-confounded relation: every neighbor of a node in the result is again in the result.","s":"∀ {N : Type u_1} [inst : DecidableEq N] [inst_1 : Fintype N] (G : Causalean.SWIGGraph N) (fuel : ℕ) (frontier visited : Finset (Causalean.SWIGNode N)), visited ⊆ G.observed → (∀ a ∈ visited, a ∉ frontier → ∀ (b : Causalean.SWIGNode N), G.directlyConfounded a b → b ∈ visited) → G.observed.card - visited.card ≤ fuel → ∀ a ∈ Causalean.SWIGGraph.bidirectedBFS.go G frontier visited fuel, ∀ (b : Causalean.SWIGNode N), G.directlyConfounded a b → b ∈ Causalean.SWIGGraph.bidirectedBFS.go G frontier visited fuel","l":288,"f":"Causalean/Graph/CComponents.lean"},{"n":"Causalean.SWIGGraph.bidirectedBFS_go_reachable","k":"theorem","m":"Causalean.Graph.CComponents","a":"Graph","t":2,"d":"Soundness of BFS. Every node produced by bidirectedBFS.go from a frontier and visited set all of whose elements are bidirected-reachable from start is itself bidirected-reachable from start. Fully proved by induction on the fuel.","s":"∀ {N : Type u_1} [inst : DecidableEq N] [inst_1 : Fintype N] (G : Causalean.SWIGGraph N) {start : Causalean.SWIGNode N} (fuel : ℕ) (frontier visited : Finset (Causalean.SWIGNode N)), (∀ y ∈ visited, G.bidirectedReachable start y) → (∀ y ∈ frontier, G.bidirectedReachable start y) → ∀ z ∈ Causalean.SWIGGraph.bidirectedBFS.go G frontier visited fuel, G.bidirectedReachable start z","l":254,"f":"Causalean/Graph/CComponents.lean"},{"n":"Causalean.SWIGGraph.bidirectedBFS_go_subset_observed","k":"theorem","m":"Causalean.Graph.CComponents","a":"Graph","t":2,"d":"If the visited set and frontier are within observed, so is the result of go.","s":"∀ {N : Type u_1} [inst : DecidableEq N] [inst_1 : Fintype N] (G : Causalean.SWIGGraph N) (frontier visited : Finset (Causalean.SWIGNode N)) (fuel : ℕ), visited ⊆ G.observed → Causalean.SWIGGraph.bidirectedBFS.go G frontier visited fuel ⊆ G.observed","l":217,"f":"Causalean/Graph/CComponents.lean"},{"n":"Causalean.SWIGGraph.bidirectedBFS_subset_observed","k":"theorem","m":"Causalean.Graph.CComponents","a":"Graph","t":2,"d":"The bidirected BFS from start is contained in observed.","s":"∀ {N : Type u_1} [inst : DecidableEq N] [inst_1 : Fintype N] (G : Causalean.SWIGGraph N) (start : Causalean.SWIGNode N), G.bidirectedBFS start ⊆ G.observed","l":236,"f":"Causalean/Graph/CComponents.lean"},{"n":"Causalean.SWIGGraph.bidirectedNeighbors","k":"def","m":"Causalean.Graph.CComponents","a":"Graph","t":1,"d":"The bidirected neighbors of an observed variable v: all observed variables that are directly confounded with v.","s":"{N : Type u_1} → [inst : DecidableEq N] → [inst_1 : Fintype N] → Causalean.SWIGGraph N → Causalean.SWIGNode N → Finset (Causalean.SWIGNode N)","l":76,"f":"Causalean/Graph/CComponents.lean"},{"n":"Causalean.SWIGGraph.bidirectedNeighbors_subset_observed","k":"theorem","m":"Causalean.Graph.CComponents","a":"Graph","t":2,"d":"The bidirected neighbors of a node are observed.","s":"∀ {N : Type u_1} [inst : DecidableEq N] [inst_1 : Fintype N] (G : Causalean.SWIGGraph N) (v : Causalean.SWIGNode N), G.bidirectedNeighbors v ⊆ G.observed","l":199,"f":"Causalean/Graph/CComponents.lean"},{"n":"Causalean.SWIGGraph.bidirectedReachable","k":"inductive","m":"Causalean.Graph.CComponents","a":"Graph","t":1,"d":"Bidirected reachability: the transitive closure of the directly-confounded relation, restricted to observed variables. Two observed variables are in the same C-component iff they are bidirected-reachable from each other.","s":"{N : Type u_1} → [inst : DecidableEq N] → [inst_1 : Fintype N] → Causalean.SWIGGraph N → Causalean.SWIGNode N → Causalean.SWIGNode N → Prop","l":81,"f":"Causalean/Graph/CComponents.lean"},{"n":"Causalean.SWIGGraph.bidirectedReachable_head","k":"theorem","m":"Causalean.Graph.CComponents","a":"Graph","t":2,"d":"Prepend a directly-confounded step at the head of a reachability chain: if u and v are directly confounded and v reaches w, then u reaches w. Proved by induction on the v-to-w derivation.","s":"∀ {N : Type u_1} [inst : DecidableEq N] [inst_1 : Fintype N] (G : Causalean.SWIGGraph N) {u v w : Causalean.SWIGNode N}, G.directlyConfounded u v → G.bidirectedReachable v w → G.bidirectedReachable u w","l":163,"f":"Causalean/Graph/CComponents.lean"},{"n":"Causalean.SWIGGraph.bidirectedReachable_observed_left","k":"theorem","m":"Causalean.Graph.CComponents","a":"Graph","t":2,"d":"Both endpoints of a bidirected-reachability derivation are observed (left endpoint).","s":"∀ {N : Type u_1} [inst : DecidableEq N] [inst_1 : Fintype N] (G : Causalean.SWIGGraph N) {u v : Causalean.SWIGNode N}, G.bidirectedReachable u v → u ∈ G.observed","l":143,"f":"Causalean/Graph/CComponents.lean"},{"n":"Causalean.SWIGGraph.bidirectedReachable_observed_right","k":"theorem","m":"Causalean.Graph.CComponents","a":"Graph","t":2,"d":"Both endpoints of a bidirected-reachability derivation are observed (right endpoint).","s":"∀ {N : Type u_1} [inst : DecidableEq N] [inst_1 : Fintype N] (G : Causalean.SWIGGraph N) {u v : Causalean.SWIGNode N}, G.bidirectedReachable u v → v ∈ G.observed","l":151,"f":"Causalean/Graph/CComponents.lean"},{"n":"Causalean.SWIGGraph.bidirectedReachable_symm","k":"theorem","m":"Causalean.Graph.CComponents","a":"Graph","t":2,"d":"Bidirected reachability is symmetric.","s":"∀ {N : Type u_1} [inst : DecidableEq N] [inst_1 : Fintype N] (G : Causalean.SWIGGraph N) {u v : Causalean.SWIGNode N}, G.bidirectedReachable u v → G.bidirectedReachable v u","l":181,"f":"Causalean/Graph/CComponents.lean"},{"n":"Causalean.SWIGGraph.bidirectedReachable_trans","k":"theorem","m":"Causalean.Graph.CComponents","a":"Graph","t":2,"d":"Bidirected reachability is transitive.","s":"∀ {N : Type u_1} [inst : DecidableEq N] [inst_1 : Fintype N] (G : Causalean.SWIGGraph N) {u v w : Causalean.SWIGNode N}, G.bidirectedReachable u v → G.bidirectedReachable v w → G.bidirectedReachable u w","l":189,"f":"Causalean/Graph/CComponents.lean"},{"n":"Causalean.SWIGGraph.cComponentOf","k":"def","m":"Causalean.Graph.CComponents","a":"Graph","t":1,"d":"The C-component of a given observed variable v: the maximal set of observed variables connected to v via chains of directly-confounded pairs.","s":"{N : Type u_1} → [inst : DecidableEq N] → [inst_1 : Fintype N] → Causalean.SWIGGraph N → Causalean.SWIGNode N → Finset (Causalean.SWIGNode N)","l":107,"f":"Causalean/Graph/CComponents.lean"},{"n":"Causalean.SWIGGraph.cComponentOf_eq_of_mem_cComponentSet","k":"theorem","m":"Causalean.Graph.CComponents","a":"Graph","t":2,"d":"A node in a listed c-component has that component as its computed c-component.","s":"∀ {N : Type u_1} [inst : DecidableEq N] [inst_1 : Fintype N] (G : Causalean.SWIGGraph N) {C : Finset (Causalean.SWIGNode N)}, C ∈ G.cComponentSet → ∀ {v : Causalean.SWIGNode N}, v ∈ C → G.cComponentOf v = C","l":502,"f":"Causalean/Graph/CComponents.lean"},{"n":"Causalean.SWIGGraph.cComponentOf_eq_of_reachable","k":"theorem","m":"Causalean.Graph.CComponents","a":"Graph","t":2,"d":"Two c-components seeded by reachable observed nodes are equal.","s":"∀ {N : Type u_1} [inst : DecidableEq N] [inst_1 : Fintype N] (G : Causalean.SWIGGraph N) {v w : Causalean.SWIGNode N}, G.bidirectedReachable v w → G.cComponentOf v = G.cComponentOf w","l":486,"f":"Causalean/Graph/CComponents.lean"},{"n":"Causalean.SWIGGraph.cComponentOf_subset_observed","k":"theorem","m":"Causalean.Graph.CComponents","a":"Graph","t":2,"d":"The c-component of v is contained in the observed nodes.","s":"∀ {N : Type u_1} [inst : DecidableEq N] [inst_1 : Fintype N] (G : Causalean.SWIGGraph N) (v : Causalean.SWIGNode N), G.cComponentOf v ⊆ G.observed","l":444,"f":"Causalean/Graph/CComponents.lean"},{"n":"Causalean.SWIGGraph.cComponentSet","k":"def","m":"Causalean.Graph.CComponents","a":"Graph","t":1,"d":"The set of c-components as a Finset of Finsets — the canonical, order-independent index for the c-component factorization. It is the image of the observed nodes under cComponentOf (each observed node maps to its own component; distinct components have distinct images, so duplicates collapse). Used as the index set in c_component_factorization.","s":"{N : Type u_1} → [inst : DecidableEq N] → [inst_1 : Fintype N] → Causalean.SWIGGraph N → Finset (Finset (Causalean.SWIGNode N))","l":125,"f":"Causalean/Graph/CComponents.lean"},{"n":"Causalean.SWIGGraph.cComponentSet_biUnion","k":"theorem","m":"Causalean.Graph.CComponents","a":"Graph","t":1,"d":"The c-components cover exactly the observed nodes: their union recovers the set of observed nodes exactly.","s":"∀ {N : Type u_1} [inst : DecidableEq N] [inst_1 : Fintype N] (G : Causalean.SWIGGraph N), G.cComponentSet.biUnion id = G.observed","l":463,"f":"Causalean/Graph/CComponents.lean"},{"n":"Causalean.SWIGGraph.cComponentSet_pairwise_disjoint","k":"theorem","m":"Causalean.Graph.CComponents","a":"Graph","t":1,"d":"Distinct c-components are pairwise disjoint: no observed node belongs to two different c-components.","s":"∀ {N : Type u_1} [inst : DecidableEq N] [inst_1 : Fintype N] (G : Causalean.SWIGGraph N), (↑G.cComponentSet).PairwiseDisjoint id","l":590,"f":"Causalean/Graph/CComponents.lean"},{"n":"Causalean.SWIGGraph.cComponentSet_subset_observed","k":"theorem","m":"Causalean.Graph.CComponents","a":"Graph","t":2,"d":"Every c-component (in the canonical set) is contained in the observed nodes.","s":"∀ {N : Type u_1} [inst : DecidableEq N] [inst_1 : Fintype N] (G : Causalean.SWIGGraph N), ∀ C ∈ G.cComponentSet, C ⊆ G.observed","l":454,"f":"Causalean/Graph/CComponents.lean"},{"n":"Causalean.SWIGGraph.cComponents","k":"def","m":"Causalean.Graph.CComponents","a":"Graph","t":1,"d":"Compute all C-components in the order obtained by scanning the observed variables.","s":"{N : Type u_1} → [inst : DecidableEq N] → [inst_1 : Fintype N] → Causalean.SWIGGraph N → Array (Finset (Causalean.SWIGNode N))","l":115,"f":"Causalean/Graph/CComponents.lean"},{"n":"Causalean.SWIGGraph.child_classified","k":"theorem","m":"Causalean.Graph.SWIG","a":"Graph","t":2,"d":"If w is a child of u in G, then w is classified (fixed, observed, or unobserved).","s":"∀ {N : Type u_2} [inst : DecidableEq N] [inst_1 : Fintype N] (G : Causalean.SWIGGraph N) {u w : Causalean.SWIGNode N}, w ∈ G.dag.children u → w ∈ G.fixed ∪ G.observed ∪ G.unobserved","l":527,"f":"Causalean/Graph/SWIG.lean"},{"n":"Causalean.SWIGGraph.dSep_union_fixed_of_induce_dSep","k":"theorem","m":"Causalean.Graph.DSep.InduceTransport","a":"Graph/DSep","t":1,"d":"D-separation in an ancestral induced SWIG lifts to the ambient SWIG once fixed intervention nodes are included in the conditioning set. Fix a SWIG G and a node set R that is closed under G's ancestral relation, with X, Y, and Z each contained in the observed nodes of R. If X and Y are d-separated by Z in the graph induced on R, then X and Y are d-separated by Z together with G's fixed intervention nodes, in the ambient graph.","s":"∀ {N : Type u_1} [inst : DecidableEq N] [inst_1 : Fintype N] (G : Causalean.SWIGGraph N) (R X Y Z : Finset (Causalean.SWIGNode N)), X ⊆ R ∩ G.observed → Y ⊆ R ∩ G.observed → Z ⊆ R ∩ G.observed → G.dag.ancestralSet R ⊆ R → (G.induce R).dag.dSep X Y Z → G.dag.dSep X Y (Z ∪ G.fixed)","l":300,"f":"Causalean/Graph/DSep/InduceTransport.lean"},{"n":"Causalean.SWIGGraph.decDirectlyConfounded","k":"instance","m":"Causalean.Graph.CComponents","a":"Graph","t":2,"d":"Direct confounding can be decided by finite search for a shared unobserved parent.","s":"{N : Type u_1} → [inst : DecidableEq N] → [inst_1 : Fintype N] → (G : Causalean.SWIGGraph N) → (v₁ v₂ : Causalean.SWIGNode N) → Decidable (G.directlyConfounded v₁ v₂)","l":71,"f":"Causalean/Graph/CComponents.lean"},{"n":"Causalean.SWIGGraph.decIsAncestrallyClosed","k":"instance","m":"Causalean.SCM.ID.GraphicalThms.InducedSubgraph","a":"SCM/ID/GraphicalThms","t":2,"d":"Ancestral closure of a finite SWIG graph is decidable.","s":"{N : Type u_1} → [inst : DecidableEq N] → [inst_1 : Fintype N] → (G : Causalean.SWIGGraph N) → Decidable G.isAncestrallyClosed","l":54,"f":"Causalean/SCM/ID/GraphicalThms/InducedSubgraph.lean"},{"n":"Causalean.SWIGGraph.directlyConfounded","k":"def","m":"Causalean.Graph.CComponents","a":"Graph","t":1,"d":"Two distinct variables are directly confounded if they share an unobserved parent.","s":"{N : Type u_1} → [inst : DecidableEq N] → [inst_1 : Fintype N] → Causalean.SWIGGraph N → Causalean.SWIGNode N → Causalean.SWIGNode N → Prop","l":62,"f":"Causalean/Graph/CComponents.lean"},{"n":"Causalean.SWIGGraph.directlyConfounded_symm","k":"theorem","m":"Causalean.Graph.CComponents","a":"Graph","t":2,"d":"The directly-confounded relation is symmetric: if v₁ and v₂ share an unobserved parent, then so do v₂ and v₁ (the shared parent and the observed-ness conditions are symmetric in the two arguments).","s":"∀ {N : Type u_1} [inst : DecidableEq N] [inst_1 : Fintype N] (G : Causalean.SWIGGraph N) {v₁ v₂ : Causalean.SWIGNode N}, G.directlyConfounded v₁ v₂ → G.directlyConfounded v₂ v₁","l":135,"f":"Causalean/Graph/CComponents.lean"},{"n":"Causalean.SWIGGraph.frontdoorCriterion","k":"def","m":"Causalean.SCM.ID.Frontdoor","a":"SCM/ID","t":1,"d":"frontdoorCriterion says that the base mediator set W satisfies Pearl's frontdoor criterion for treatment variables X and outcome nodes Y: the mediator random nodes intercept every directed treatment-outcome path, have no open back-door path from treatment, admit treatment adjustment for the mediator-outcome effect, and are excluded from the treatment and outcome sets.","s":"{N : Type u_1} → [inst : DecidableEq N] → [inst_1 : Fintype N] → (G : Causalean.SWIGGraph N) → (X : Finset N) → (∀ D ∈ X, Causalean.SWIGNode.random D ∈ G.observed) → (∀ D ∈ X, Causalean.SWIGNode.fixed D ∉ G.fixed) → (W : Finset N) → (∀ D ∈ W, Causalean.SWIGNode.random D ∈ G.observed) → (∀ D ∈ W, Causalean.SWIGNode.fixed D ∉ G.fixed) → Finset (Causalean.SWIGNode N) → Prop","l":65,"f":"Causalean/SCM/ID/Frontdoor.lean"},{"n":"Causalean.SWIGGraph.induce","k":"def","m":"Causalean.Graph.Induce","a":"Graph","t":1,"d":"Induce a sub-SWIG on a subset R. The new observed is R ∩ observed; the new fixed drops any fixed node whose iotaMap image was removed; the new unobserved keeps exactly the original latent roots with an edge into the retained observed set; the DAG keeps only edges with both endpoints in the new active set.","s":"{N : Type u_1} → [inst : DecidableEq N] → [inst_1 : Fintype N] → Causalean.SWIGGraph N → Finset (Causalean.SWIGNode N) → Causalean.SWIGGraph N","l":110,"f":"Causalean/Graph/Induce.lean"},{"n":"Causalean.SWIGGraph.induce_cComponentOf_eq_of_shared_unobserved_parent","k":"theorem","m":"Causalean.Graph.CComponents","a":"Graph","t":2,"d":"If a latent node has edges into two observed nodes retained by an induced graph, those observed nodes seed the same induced c-component.","s":"∀ {N : Type u_1} [inst : DecidableEq N] [inst_1 : Fintype N] (G : Causalean.SWIGGraph N) (R : Finset (Causalean.SWIGNode N)) {u v w : Causalean.SWIGNode N}, u ∈ G.unobserved → v ∈ R → w ∈ R → G.dag.edge u v → G.dag.edge u w → (G.induce R).cComponentOf v = (G.induce R).cComponentOf w","l":549,"f":"Causalean/Graph/CComponents.lean"},{"n":"Causalean.SWIGGraph.induce_isAncestor_mem_R","k":"theorem","m":"Causalean.Graph.Induce","a":"Graph","t":2,"d":"In the induced subgraph, every vertex with a proper ancestor lies in the retained observed support.","s":"∀ {N : Type u_1} [inst : DecidableEq N] [inst_1 : Fintype N] (G : Causalean.SWIGGraph N) (R : Finset (Causalean.SWIGNode N)) {u v : Causalean.SWIGNode N}, (G.induce R).dag.isAncestor u v → v ∈ R ∩ G.observed","l":248,"f":"Causalean/Graph/Induce.lean"},{"n":"Causalean.SWIGGraph.inducedDag","k":"def","m":"Causalean.Graph.Induce","a":"Graph","t":1,"d":"The DAG obtained by filtering G.dag's edges to those with both endpoints in active. Acyclicity follows from the parent graph via the parent's topological order (every restricted edge is an original edge).","s":"{N : Type u_1} → [inst : DecidableEq N] → [inst_1 : Fintype N] → Causalean.SWIGGraph N → Finset (Causalean.SWIGNode N) → Causalean.DAG (Causalean.SWIGNode N)","l":66,"f":"Causalean/Graph/Induce.lean"},{"n":"Causalean.SWIGGraph.inducedDag_children_subset","k":"theorem","m":"Causalean.Graph.Induce","a":"Graph","t":2,"d":"Every child in the restricted DAG is also a child in the original graph.","s":"∀ {N : Type u_1} [inst : DecidableEq N] [inst_1 : Fintype N] (G : Causalean.SWIGGraph N) (active : Finset (Causalean.SWIGNode N)) (u : Causalean.SWIGNode N), (G.inducedDag active).children u ⊆ G.dag.children u","l":89,"f":"Causalean/Graph/Induce.lean"},{"n":"Causalean.SWIGGraph.inducedDag_edge_iff","k":"theorem","m":"Causalean.Graph.Induce","a":"Graph","t":1,"d":"For a set of active nodes and vertices u, v, u and v are joined by an edge of the DAG restricted to the active nodes exactly when they are joined by an edge of the original DAG and both are active.","s":"∀ {N : Type u_1} [inst : DecidableEq N] [inst_1 : Fintype N] (G : Causalean.SWIGGraph N) (active : Finset (Causalean.SWIGNode N)) (u v : Causalean.SWIGNode N), (G.inducedDag active).edge u v ↔ G.dag.edge u v ∧ u ∈ active ∧ v ∈ active","l":75,"f":"Causalean/Graph/Induce.lean"},{"n":"Causalean.SWIGGraph.inducedDag_isAncestor_mem_active","k":"theorem","m":"Causalean.Graph.Induce","a":"Graph","t":2,"d":"If (G.inducedDag active).isAncestor u v, then both endpoints belong to active.","s":"∀ {N : Type u_1} [inst : DecidableEq N] [inst_1 : Fintype N] (G : Causalean.SWIGGraph N) (active : Finset (Causalean.SWIGNode N)) {u v : Causalean.SWIGNode N}, (G.inducedDag active).isAncestor u v → u ∈ active ∧ v ∈ active","l":96,"f":"Causalean/Graph/Induce.lean"},{"n":"Causalean.SWIGGraph.inducedDag_parents_subset","k":"theorem","m":"Causalean.Graph.Induce","a":"Graph","t":2,"d":"Every parent in the restricted DAG is also a parent in the original graph.","s":"∀ {N : Type u_1} [inst : DecidableEq N] [inst_1 : Fintype N] (G : Causalean.SWIGGraph N) (active : Finset (Causalean.SWIGNode N)) (v : Causalean.SWIGNode N), (G.inducedDag active).parents v ⊆ G.dag.parents v","l":82,"f":"Causalean/Graph/Induce.lean"},{"n":"Causalean.SWIGGraph.inducedEdge","k":"def","m":"Causalean.Graph.Induce","a":"Graph","t":1,"d":"The edge relation of G.dag restricted to a given active vertex set: keep an edge iff both endpoints are in active.","s":"{N : Type u_1} → [inst : DecidableEq N] → [inst_1 : Fintype N] → Causalean.SWIGGraph N → Finset (Causalean.SWIGNode N) → Causalean.SWIGNode N → Causalean.SWIGNode N → Prop","l":54,"f":"Causalean/Graph/Induce.lean"},{"n":"Causalean.SWIGGraph.inducedEdge_decidable","k":"instance","m":"Causalean.Graph.Induce","a":"Graph","t":2,"d":"Whether an edge remains after restricting to an active vertex set is decidable.","s":"{N : Type u_1} → [inst : DecidableEq N] → [inst_1 : Fintype N] → (G : Causalean.SWIGGraph N) → (active : Finset (Causalean.SWIGNode N)) → DecidableRel (G.inducedEdge active)","l":59,"f":"Causalean/Graph/Induce.lean"},{"n":"Causalean.SWIGGraph.iota","k":"def","m":"Causalean.Graph.SWIG","a":"Graph","t":1,"d":"The canonical map ι : S → V sending each fixed intervention parameter to its random counterpart in observed, via iotaMap.","s":"{N : Type u_2} → [inst : DecidableEq N] → [inst_1 : Fintype N] → (G : Causalean.SWIGGraph N) → ↥G.fixed → ↥G.observed","l":423,"f":"Causalean/Graph/SWIG.lean"},{"n":"Causalean.SWIGGraph.iotaN","k":"def","m":"Causalean.Graph.SWIG","a":"Graph","t":1,"d":"ι at the level of original nodes N, using the fact that every s ∈ fixed is of the form .fixed n.","s":"{N : Type u_2} → [inst : DecidableEq N] → [inst_1 : Fintype N] → (G : Causalean.SWIGGraph N) → { n // Causalean.SWIGNode.fixed n ∈ G.fixed } → { n // Causalean.SWIGNode.random n ∈ G.observed }","l":437,"f":"Causalean/Graph/SWIG.lean"},{"n":"Causalean.SWIGGraph.iotaNode","k":"def","m":"Causalean.Graph.SWIG","a":"Graph","t":1,"d":"Evaluate ι as a SWIGNode (forgetting membership).","s":"{N : Type u_2} → [inst : DecidableEq N] → [inst_1 : Fintype N] → (G : Causalean.SWIGGraph N) → ↥G.fixed → Causalean.SWIGNode N","l":429,"f":"Causalean/Graph/SWIG.lean"},{"n":"Causalean.SWIGGraph.iotaNode_eq_iotaMap","k":"theorem","m":"Causalean.Graph.SWIG","a":"Graph","t":2,"d":"Forgetting the membership proof in the graph-level link map gives the node-level link map.","s":"∀ {N : Type u_2} [inst : DecidableEq N] [inst_1 : Fintype N] (G : Causalean.SWIGGraph N) (s : ↥G.fixed), G.iotaNode s = Causalean.iotaMap ↑s","l":433,"f":"Causalean/Graph/SWIG.lean"},{"n":"Causalean.SWIGGraph.isAncestrallyClosed","k":"def","m":"Causalean.SCM.ID.GraphicalThms.InducedSubgraph","a":"SCM/ID/GraphicalThms","t":1,"d":"A graph is ancestrally closed when every observed parent of an observed node is also observed.","s":"{N : Type u_1} → [inst : DecidableEq N] → [inst_1 : Fintype N] → Causalean.SWIGGraph N → Prop","l":46,"f":"Causalean/SCM/ID/GraphicalThms/InducedSubgraph.lean"},{"n":"Causalean.SWIGGraph.isStandard","k":"def","m":"Causalean.Graph.SWIG","a":"Graph","t":1,"d":"A standard SWIG graph has no fixed (intervention) variables: S = ∅.","s":"{N : Type u_2} → [inst : DecidableEq N] → [inst_1 : Fintype N] → Causalean.SWIGGraph N → Prop","l":447,"f":"Causalean/Graph/SWIG.lean"},{"n":"Causalean.SWIGGraph.mem_bidirectedBFS_iff_reachable","k":"theorem","m":"Causalean.Graph.CComponents","a":"Graph","t":1,"d":"BFS computes bidirected reachability. Fix a single-world intervention graph G and a node start that is observed in G. Then a node w is found by the breadth-first search from start exactly when w is bidirected-reachable from start — connected to it by a chain of directly-confounded (shared-unobserved-parent) pairs.","s":"∀ {N : Type u_1} [inst : DecidableEq N] [inst_1 : Fintype N] (G : Causalean.SWIGGraph N) {start w : Causalean.SWIGNode N}, start ∈ G.observed → (w ∈ G.bidirectedBFS start ↔ G.bidirectedReachable start w)","l":398,"f":"Causalean/Graph/CComponents.lean"},{"n":"Causalean.SWIGGraph.mem_bidirectedBFS_self","k":"theorem","m":"Causalean.Graph.CComponents","a":"Graph","t":2,"d":"The start node belongs to its own BFS result (when observed).","s":"∀ {N : Type u_1} [inst : DecidableEq N] [inst_1 : Fintype N] (G : Causalean.SWIGGraph N) {start : Causalean.SWIGNode N}, start ∈ G.observed → start ∈ G.bidirectedBFS start","l":245,"f":"Causalean/Graph/CComponents.lean"},{"n":"Causalean.SWIGGraph.mem_cComponentOf_iff_reachable","k":"theorem","m":"Causalean.Graph.CComponents","a":"Graph","t":1,"d":"Membership in a c-component is exactly bidirected reachability from its seed. For a node v that is observed in G, a node w belongs to the c-component seeded at v exactly when w is bidirected-reachable from v.","s":"∀ {N : Type u_1} [inst : DecidableEq N] [inst_1 : Fintype N] (G : Causalean.SWIGGraph N) {v w : Causalean.SWIGNode N}, v ∈ G.observed → (w ∈ G.cComponentOf v ↔ G.bidirectedReachable v w)","l":479,"f":"Causalean/Graph/CComponents.lean"},{"n":"Causalean.SWIGGraph.mem_cComponentOf_self","k":"theorem","m":"Causalean.Graph.CComponents","a":"Graph","t":2,"d":"An observed node belongs to its own c-component.","s":"∀ {N : Type u_1} [inst : DecidableEq N] [inst_1 : Fintype N] (G : Causalean.SWIGGraph N) {v : Causalean.SWIGNode N}, v ∈ G.observed → v ∈ G.cComponentOf v","l":449,"f":"Causalean/Graph/CComponents.lean"},{"n":"Causalean.SWIGGraph.no_shared_unobserved_parent_of_mem_cComponentSet_of_not_mem","k":"theorem","m":"Causalean.Graph.CComponents","a":"Graph","t":2,"d":"No latent root can be a shared parent of a c-component node and a node outside that c-component.","s":"∀ {N : Type u_1} [inst : DecidableEq N] [inst_1 : Fintype N] (G : Causalean.SWIGGraph N) {C : Finset (Causalean.SWIGNode N)}, C ∈ G.cComponentSet → ∀ {v w u : Causalean.SWIGNode N}, v ∈ C → w ∉ C → u ∈ G.unobserved → G.dag.edge u v → G.dag.edge u w → False","l":533,"f":"Causalean/Graph/CComponents.lean"},{"n":"Causalean.SWIGGraph.nodeIndex","k":"def","m":"Causalean.SCM.ID.Density.IdentifyMass","a":"SCM/ID/Density","t":1,"d":"This selector returns a node's position inside a finite node set sorted by graph topological order.","s":"{N : Type u_1} → [inst : DecidableEq N] → [inst_1 : Fintype N] → Causalean.SWIGGraph N → (D : Finset (Causalean.SWIGNode N)) → ↥D → Fin D.card","l":46,"f":"Causalean/SCM/ID/Density/IdentifyMass.lean"},{"n":"Causalean.SWIGGraph.nodesAt","k":"def","m":"Causalean.SCM.ID.Density.IdentifyMass","a":"SCM/ID/Density","t":1,"d":"This selector returns the node at a given position in a finite node set sorted by graph topological order.","s":"{N : Type u_1} → [inst : DecidableEq N] → [inst_1 : Fintype N] → Causalean.SWIGGraph N → (D : Finset (Causalean.SWIGNode N)) → Fin D.card → ↥D","l":38,"f":"Causalean/SCM/ID/Density/IdentifyMass.lean"},{"n":"Causalean.SWIGGraph.nonDescIn","k":"def","m":"Causalean.SCM.ID.GraphicalThms.InducedSubgraph","a":"SCM/ID/GraphicalThms","t":1,"d":"This set contains selected nodes excluding the target and its proper descendants.","s":"{N : Type u_1} → [inst : DecidableEq N] → [inst_1 : Fintype N] → Causalean.SWIGGraph N → Finset (Causalean.SWIGNode N) → Causalean.SWIGNode N → Finset (Causalean.SWIGNode N)","l":79,"f":"Causalean/SCM/ID/GraphicalThms/InducedSubgraph.lean"},{"n":"Causalean.SWIGGraph.nonDescIn_subset","k":"theorem","m":"Causalean.SCM.ID.GraphicalThms.InducedSubgraph","a":"SCM/ID/GraphicalThms","t":2,"d":"Every induced non-descendant lies in the selected set.","s":"∀ {N : Type u_1} [inst : DecidableEq N] [inst_1 : Fintype N] (G : Causalean.SWIGGraph N) (R : Finset (Causalean.SWIGNode N)) (v₀ : Causalean.SWIGNode N), G.nonDescIn R v₀ ⊆ R","l":113,"f":"Causalean/SCM/ID/GraphicalThms/InducedSubgraph.lean"},{"n":"Causalean.SWIGGraph.nonDescIn_subset_erase","k":"theorem","m":"Causalean.SCM.ID.GraphicalThms.InducedSubgraph","a":"SCM/ID/GraphicalThms","t":2,"d":"Every induced non-descendant lies in the selected set with the target node removed.","s":"∀ {N : Type u_1} [inst : DecidableEq N] [inst_1 : Fintype N] (G : Causalean.SWIGGraph N) (R : Finset (Causalean.SWIGNode N)) (v₀ : Causalean.SWIGNode N), G.nonDescIn R v₀ ⊆ R.erase v₀","l":107,"f":"Causalean/SCM/ID/GraphicalThms/InducedSubgraph.lean"},{"n":"Causalean.SWIGGraph.not_directlyConfounded_of_mem_cComponentSet_of_not_mem","k":"theorem","m":"Causalean.Graph.CComponents","a":"Graph","t":2,"d":"A node outside a c-component is not directly confounded with a node inside it.","s":"∀ {N : Type u_1} [inst : DecidableEq N] [inst_1 : Fintype N] (G : Causalean.SWIGGraph N) {C : Finset (Causalean.SWIGNode N)}, C ∈ G.cComponentSet → ∀ {v w : Causalean.SWIGNode N}, v ∈ C → w ∉ C → ¬G.directlyConfounded v w","l":514,"f":"Causalean/Graph/CComponents.lean"},{"n":"Causalean.SWIGGraph.not_mem_fixed_of_mem_observed","k":"theorem","m":"Causalean.Graph.DSep.InduceTransport","a":"Graph/DSep","t":2,"d":"Observed SWIG vertices are not members of the fixed intervention set.","s":"∀ {N : Type u_1} [inst : DecidableEq N] [inst_1 : Fintype N] (G : Causalean.SWIGGraph N) {v : Causalean.SWIGNode N}, v ∈ G.observed → v ∉ G.fixed","l":59,"f":"Causalean/Graph/DSep/InduceTransport.lean"},{"n":"Causalean.SWIGGraph.observedPredecessors","k":"def","m":"Causalean.SCM.Factored.ObsChainKernel","a":"SCM/Factored","t":1,"d":"Observed predecessors Pa⁺_G(v). The observed nodes appearing strictly before v in the topological order of G.dag.","s":"{N : Type u_1} → [inst : DecidableEq N] → [inst_1 : Fintype N] → Causalean.SWIGGraph N → Causalean.SWIGNode N → Finset (Causalean.SWIGNode N)","l":31,"f":"Causalean/SCM/Factored/ObsChainKernel.lean"},{"n":"Causalean.SWIGGraph.observedPredecessors_subset_observed","k":"theorem","m":"Causalean.SCM.Factored.ObsChainKernel","a":"SCM/Factored","t":2,"d":"Observed predecessors are observed by construction.","s":"∀ {N : Type u_1} [inst : DecidableEq N] [inst_1 : Fintype N] (G : Causalean.SWIGGraph N) (v : Causalean.SWIGNode N), G.observedPredecessors v ⊆ G.observed","l":38,"f":"Causalean/SCM/Factored/ObsChainKernel.lean"},{"n":"Causalean.SWIGGraph.parent_classified","k":"theorem","m":"Causalean.Graph.SWIG","a":"Graph","t":2,"d":"If u is a parent of v in G, then u is classified (fixed, observed, or unobserved).","s":"∀ {N : Type u_2} [inst : DecidableEq N] [inst_1 : Fintype N] (G : Causalean.SWIGGraph N) {u v : Causalean.SWIGNode N}, u ∈ G.dag.parents v → u ∈ G.fixed ∪ G.observed ∪ G.unobserved","l":520,"f":"Causalean/Graph/SWIG.lean"},{"n":"Causalean.SWIGGraph.prefixIn","k":"def","m":"Causalean.SCM.ID.Density.IdentifyMass","a":"SCM/ID/Density","t":1,"d":"This finite set contains the first selected nodes in graph topological order.","s":"{N : Type u_1} → [inst : DecidableEq N] → [inst_1 : Fintype N] → Causalean.SWIGGraph N → Finset (Causalean.SWIGNode N) → ℕ → Finset (Causalean.SWIGNode N)","l":54,"f":"Causalean/SCM/ID/Density/IdentifyMass.lean"},{"n":"Causalean.SWIGGraph.prefixIn_obsParentClosed","k":"theorem","m":"Causalean.SCM.ID.Density.QFactor","a":"SCM/ID/Density","t":2,"d":"Topological prefixes of an observed-parent-closed set remain observed-parent-closed.","s":"∀ {N : Type u_1} [inst : DecidableEq N] [inst_1 : Fintype N] {Ω : N → Type u_2} [inst_2 : (n : N) → MeasurableSpace (Ω n)] (M : Causalean.SCM N Ω) (X : Finset N) (hObs : ∀ D ∈ X, Causalean.SWIGNode.random D ∈ M.observed) (hFix : ∀ D ∈ X, Causalean.SWIGNode.fixed D ∉ M.fixed) (Y : Finset (Causalean.SWIGNode N)) (k : ℕ), (M.fixSet X hObs hFix).ObsParentClosed (((M.fixSet X hObs hFix).induce (Causalean.SCM.ID.fixAncestralSet M X hObs hFix Y)).prefixIn (Causalean.SCM.ID.fixObservedAncestralSet M X hObs hFix Y) k)","l":41,"f":"Causalean/SCM/ID/Density/QFactor.lean"},{"n":"Causalean.SWIGGraph.prefixIn_subset","k":"theorem","m":"Causalean.SCM.ID.Density.IdentifyMass","a":"SCM/ID/Density","t":2,"d":"Every node in a topological prefix of a selected node set belongs to the selected node set.","s":"∀ {N : Type u_1} [inst : DecidableEq N] [inst_1 : Fintype N] (G : Causalean.SWIGGraph N) (D : Finset (Causalean.SWIGNode N)) (n : ℕ), G.prefixIn D n ⊆ D","l":59,"f":"Causalean/SCM/ID/Density/IdentifyMass.lean"},{"n":"Causalean.SWIGGraph.properDescIn","k":"def","m":"Causalean.SCM.ID.GraphicalThms.InducedSubgraph","a":"SCM/ID/GraphicalThms","t":1,"d":"This set contains the proper descendants of a node inside an induced SWIG subgraph.","s":"{N : Type u_1} → [inst : DecidableEq N] → [inst_1 : Fintype N] → Causalean.SWIGGraph N → Finset (Causalean.SWIGNode N) → Causalean.SWIGNode N → Finset (Causalean.SWIGNode N)","l":71,"f":"Causalean/SCM/ID/GraphicalThms/InducedSubgraph.lean"},{"n":"Causalean.SWIGGraph.properDescIn_disjoint_nonDescIn","k":"theorem","m":"Causalean.SCM.ID.GraphicalThms.InducedSubgraph","a":"SCM/ID/GraphicalThms","t":2,"d":"The proper-descendant set and non-descendant set are disjoint inside the selected nodes.","s":"∀ {N : Type u_1} [inst : DecidableEq N] [inst_1 : Fintype N] (G : Causalean.SWIGGraph N) (R : Finset (Causalean.SWIGNode N)) (v₀ : Causalean.SWIGNode N), Disjoint (G.properDescIn R v₀) (G.nonDescIn R v₀)","l":119,"f":"Causalean/SCM/ID/GraphicalThms/InducedSubgraph.lean"},{"n":"Causalean.SWIGGraph.properDescIn_subset_erase","k":"theorem","m":"Causalean.SCM.ID.GraphicalThms.InducedSubgraph","a":"SCM/ID/GraphicalThms","t":2,"d":"Proper descendants lie in the selected set with the target node removed.","s":"∀ {N : Type u_1} [inst : DecidableEq N] [inst_1 : Fintype N] (G : Causalean.SWIGGraph N) (R : Finset (Causalean.SWIGNode N)) (v₀ : Causalean.SWIGNode N), G.properDescIn R v₀ ⊆ R.erase v₀","l":97,"f":"Causalean/SCM/ID/GraphicalThms/InducedSubgraph.lean"},{"n":"Causalean.SWIGGraph.properDescIn_union_nonDescIn_eq_erase","k":"theorem","m":"Causalean.SCM.ID.GraphicalThms.InducedSubgraph","a":"SCM/ID/GraphicalThms","t":1,"d":"Within a selected node set R of a SWIG graph G, the proper descendants of a node v₀ together with its non-descendants exhaust the selected nodes other than v₀ itself.","s":"∀ {N : Type u_1} [inst : DecidableEq N] [inst_1 : Fintype N] (G : Causalean.SWIGGraph N) (R : Finset (Causalean.SWIGNode N)) (v₀ : Causalean.SWIGNode N), G.properDescIn R v₀ ∪ G.nonDescIn R v₀ = R.erase v₀","l":129,"f":"Causalean/SCM/ID/GraphicalThms/InducedSubgraph.lean"},{"n":"Causalean.SWIGGraph.qFactorParents","k":"def","m":"Causalean.SCM.ID.GraphicalThms.CComponentFactor","a":"SCM/ID/GraphicalThms","t":1,"d":"The conditioning set of the c-component factor Q[C]:","s":"{N : Type u_1} → [inst : DecidableEq N] → [inst_1 : Fintype N] → Causalean.SWIGGraph N → Finset (Causalean.SWIGNode N) → Finset (Causalean.SWIGNode N)","l":78,"f":"Causalean/SCM/ID/GraphicalThms/CComponentFactor.lean"},{"n":"Causalean.SWIGGraph.qFactorParents_subset_observed","k":"theorem","m":"Causalean.SCM.ID.GraphicalThms.CComponentFactor","a":"SCM/ID/GraphicalThms","t":2,"d":"qFactorParents C consists of observed nodes outside C.","s":"∀ {N : Type u_1} [inst : DecidableEq N] [inst_1 : Fintype N] (G : Causalean.SWIGGraph N) (C : Finset (Causalean.SWIGNode N)), G.qFactorParents C ⊆ G.observed","l":87,"f":"Causalean/SCM/ID/GraphicalThms/CComponentFactor.lean"},{"n":"Causalean.SWIGGraph.randomVars","k":"def","m":"Causalean.SCM.Model.SCM","a":"SCM/Model","t":1,"d":"The random variables of a SWIG graph are its observed and latent nodes.","s":"{N : Type u_1} → [inst : DecidableEq N] → [inst_1 : Fintype N] → Causalean.SWIGGraph N → Finset (Causalean.SWIGNode N)","l":93,"f":"Causalean/SCM/Model/SCM.lean"},{"n":"Causalean.SWIGGraph.splitMono","k":"def","m":"Causalean.Graph.SWIGSplitMono","a":"Graph","t":1,"d":"Monolithic multi-target split. (Definition 8, one-shot form.)","s":"{N : Type u_1} → [inst : DecidableEq N] → [inst_1 : Fintype N] → (G : Causalean.SWIGGraph N) → (X : Finset N) → (∀ D ∈ X, Causalean.SWIGNode.random D ∈ G.observed) → (∀ D ∈ X, Causalean.SWIGNode.fixed D ∉ G.fixed) → Causalean.SWIGGraph N","l":214,"f":"Causalean/Graph/SWIGSplitMono.lean"},{"n":"Causalean.SWIGGraph.splitMonoDAG","k":"def","m":"Causalean.Graph.SWIGSplitMono","a":"Graph","t":1,"d":"The monolithically-split DAG: the DAG on SWIGNode N obtained by rerouting all .random D → w edges (for D ∈ X) to .fixed D → w in a single pass.","s":"{N : Type u_1} → [inst : DecidableEq N] → [inst_1 : Fintype N] → Causalean.SWIGGraph N → Finset N → Causalean.DAG (Causalean.SWIGNode N)","l":133,"f":"Causalean/Graph/SWIGSplitMono.lean"},{"n":"Causalean.SWIGGraph.splitMonoEdgeRel","k":"def","m":"Causalean.Graph.SWIGSplitMono","a":"Graph","t":1,"d":"Edge relation after monolithically splitting every node D ∈ X.","s":"{N : Type u_1} → [DecidableEq N] → (Causalean.SWIGNode N → Causalean.SWIGNode N → Prop) → Finset N → Causalean.SWIGNode N → Causalean.SWIGNode N → Prop","l":75,"f":"Causalean/Graph/SWIGSplitMono.lean"},{"n":"Causalean.SWIGGraph.splitMonoEdgeRel_decidable","k":"instance","m":"Causalean.Graph.SWIGSplitMono","a":"Graph","t":2,"d":"The monolithic split edge relation is decidable whenever the original edge relation is decidable.","s":"{N : Type u_1} → [inst : DecidableEq N] → (dagEdge : Causalean.SWIGNode N → Causalean.SWIGNode N → Prop) → [DecidableRel dagEdge] → (X : Finset N) → DecidableRel (Causalean.SWIGGraph.splitMonoEdgeRel dagEdge X)","l":90,"f":"Causalean/Graph/SWIGSplitMono.lean"},{"n":"Causalean.SWIGGraph.splitMonoTopo","k":"def","m":"Causalean.Graph.SWIGSplitMono","a":"Graph","t":1,"d":"Topological order for the monolithically-split DAG.","s":"{N : Type u_1} → [inst : DecidableEq N] → [inst_1 : Fintype N] → Causalean.SWIGGraph N → Finset N → Causalean.SWIGNode N → ℕ","l":116,"f":"Causalean/Graph/SWIGSplitMono.lean"},{"n":"Causalean.SWIGGraph.splitMono_empty_parents","k":"theorem","m":"Causalean.SCM.Model.InterventionSet","a":"SCM/Model","t":2,"d":"Splitting no variables leaves every parent set unchanged.","s":"∀ {N : Type u_1} [inst : DecidableEq N] [inst_1 : Fintype N] (G : Causalean.SWIGGraph N) (v : Causalean.SWIGNode N), (G.splitMono ∅ (fun x hD => absurd hD (Finset.notMem_empty x)) fun x hD => absurd hD (Finset.notMem_empty x)).dag.parents v = G.dag.parents v","l":123,"f":"Causalean/SCM/Model/InterventionSet.lean"},{"n":"Causalean.SWIGGraph.splitMono_fixed","k":"theorem","m":"Causalean.Graph.SWIGSplitMono","a":"Graph","t":2,"d":"Monolithic splitting adds the fixed copies of the target variables to the fixed node set.","s":"∀ {N : Type u_1} [inst : DecidableEq N] [inst_1 : Fintype N] (G : Causalean.SWIGGraph N) (X : Finset N) (hObs : ∀ D ∈ X, Causalean.SWIGNode.random D ∈ G.observed) (hFix : ∀ D ∈ X, Causalean.SWIGNode.fixed D ∉ G.fixed), (G.splitMono X hObs hFix).fixed = G.fixed ∪ Finset.image Causalean.SWIGNode.fixed X","l":492,"f":"Causalean/Graph/SWIGSplitMono.lean"},{"n":"Causalean.SWIGGraph.splitMono_observed","k":"theorem","m":"Causalean.Graph.SWIGSplitMono","a":"Graph","t":2,"d":"Monolithic splitting preserves the observed node set.","s":"∀ {N : Type u_1} [inst : DecidableEq N] [inst_1 : Fintype N] (G : Causalean.SWIGGraph N) (X : Finset N) (hObs : ∀ D ∈ X, Causalean.SWIGNode.random D ∈ G.observed) (hFix : ∀ D ∈ X, Causalean.SWIGNode.fixed D ∉ G.fixed), (G.splitMono X hObs hFix).observed = G.observed","l":480,"f":"Causalean/Graph/SWIGSplitMono.lean"},{"n":"Causalean.SWIGGraph.splitMono_parents_char","k":"theorem","m":"Causalean.Graph.SWIGSplitMono","a":"Graph","t":1,"d":"Characterization of parents in splitMono. Fix a SWIG G and a set X of variables to split, where the random copy of every variable in X is observed in G and the fixed copy of every variable in X is not already among G's fixed nodes. Then, for any node v, a node x is a parent of v in the graph obtained by monolithically splitting X exactly when either x is a parent of v in the original graph and is not the random copy of any variable in X, or x is the fixed copy of some variable D ∈ X whose random copy is a parent of v in the original graph.","s":"∀ {N : Type u_1} [inst : DecidableEq N] [inst_1 : Fintype N] (G : Causalean.SWIGGraph N) (X : Finset N) (hObs : ∀ D ∈ X, Causalean.SWIGNode.random D ∈ G.observed) (hFix : ∀ D ∈ X, Causalean.SWIGNode.fixed D ∉ G.fixed) (v x : Causalean.SWIGNode N), x ∈ (G.splitMono X hObs hFix).dag.parents v ↔ (x ∈ G.dag.parents v ∧ ∀ D ∈ X, x ≠ Causalean.SWIGNode.random D) ∨ ∃ D ∈ X, x = Causalean.SWIGNode.fixed D ∧ Causalean.SWIGNode.random D ∈ G.dag.parents v","l":502,"f":"Causalean/Graph/SWIGSplitMono.lean"},{"n":"Causalean.SWIGGraph.splitMono_parents_eq_of_no_fixed_parent","k":"theorem","m":"Causalean.Graph.SWIGSplitMono","a":"Graph","t":2,"d":"Parent-set coincidence at non-.fixed-targeted vertices.","s":"∀ {N : Type u_1} [inst : DecidableEq N] [inst_1 : Fintype N] (G : Causalean.SWIGGraph N) (X : Finset N) (hObs : ∀ D ∈ X, Causalean.SWIGNode.random D ∈ G.observed) (hFix : ∀ D ∈ X, Causalean.SWIGNode.fixed D ∉ G.fixed) (v : Causalean.SWIGNode N), (∀ D ∈ X, Causalean.SWIGNode.fixed D ∉ (G.splitMono X hObs hFix).dag.parents v) → (G.splitMono X hObs hFix).dag.parents v = G.dag.parents v","l":576,"f":"Causalean/Graph/SWIGSplitMono.lean"},{"n":"Causalean.SWIGGraph.splitMono_unobserved","k":"theorem","m":"Causalean.Graph.SWIGSplitMono","a":"Graph","t":2,"d":"Monolithic splitting preserves the unobserved node set.","s":"∀ {N : Type u_1} [inst : DecidableEq N] [inst_1 : Fintype N] (G : Causalean.SWIGGraph N) (X : Finset N) (hObs : ∀ D ∈ X, Causalean.SWIGNode.random D ∈ G.observed) (hFix : ∀ D ∈ X, Causalean.SWIGNode.fixed D ∉ G.fixed), (G.splitMono X hObs hFix).unobserved = G.unobserved","l":486,"f":"Causalean/Graph/SWIGSplitMono.lean"},{"n":"Causalean.SWIGGraph.subset_bidirectedBFS_go","k":"theorem","m":"Causalean.Graph.CComponents","a":"Graph","t":2,"d":"The visited set only grows: it is contained in the result of go.","s":"∀ {N : Type u_1} [inst : DecidableEq N] [inst_1 : Fintype N] (G : Causalean.SWIGGraph N) (frontier visited : Finset (Causalean.SWIGNode N)) (fuel : ℕ), visited ⊆ Causalean.SWIGGraph.bidirectedBFS.go G frontier visited fuel","l":205,"f":"Causalean/Graph/CComponents.lean"},{"n":"Causalean.SWIGGraph.topoLinearOrder","k":"def","m":"Causalean.SCM.ID.Density.IdentifyMass","a":"SCM/ID/Density","t":1,"d":"This order compares graph nodes by their topological position in the SWIG graph.","s":"{N : Type u_1} → [inst : DecidableEq N] → [inst_1 : Fintype N] → Causalean.SWIGGraph N → LinearOrder (Causalean.SWIGNode N)","l":34,"f":"Causalean/SCM/ID/Density/IdentifyMass.lean"},{"n":"Causalean.SWIGGraph.v₀_not_mem_nonDescIn","k":"theorem","m":"Causalean.SCM.ID.GraphicalThms.InducedSubgraph","a":"SCM/ID/GraphicalThms","t":2,"d":"The target node is not in its induced non-descendant set because it is explicitly removed.","s":"∀ {N : Type u_1} [inst : DecidableEq N] [inst_1 : Fintype N] (G : Causalean.SWIGGraph N) (R : Finset (Causalean.SWIGNode N)) (v₀ : Causalean.SWIGNode N), v₀ ∉ G.nonDescIn R v₀","l":92,"f":"Causalean/SCM/ID/GraphicalThms/InducedSubgraph.lean"},{"n":"Causalean.SWIGGraph.v₀_not_mem_properDescIn","k":"theorem","m":"Causalean.SCM.ID.GraphicalThms.InducedSubgraph","a":"SCM/ID/GraphicalThms","t":2,"d":"A node is not a proper descendant of itself inside the induced subgraph.","s":"∀ {N : Type u_1} [inst : DecidableEq N] [inst_1 : Fintype N] (G : Causalean.SWIGGraph N) (R : Finset (Causalean.SWIGNode N)) (v₀ : Causalean.SWIGNode N), v₀ ∉ G.properDescIn R v₀","l":86,"f":"Causalean/SCM/ID/GraphicalThms/InducedSubgraph.lean"},{"n":"Causalean.SWIGNode","k":"inductive","m":"Causalean.Graph.SWIG","a":"Graph","t":1,"d":"A node of a SWIG (Single World Intervention Graph) is either the *random* version of a base variable or its *fixed* intervention version, so the SWIG node set is the disjoint union of two copies of the base variable set.","s":"Type u_1 → Type u_1","l":73,"f":"Causalean/Graph/SWIG.lean"},{"n":"Causalean.SWIGNode.equiv","k":"def","m":"Causalean.Graph.SWIG","a":"Graph","t":1,"d":"Equivalence between SWIGNode N and N ⊕ N.","s":"{N : Type u_1} → Causalean.SWIGNode N ≃ N ⊕ N","l":104,"f":"Causalean/Graph/SWIG.lean"},{"n":"Causalean.SWIGNode.fixed_injective","k":"theorem","m":"Causalean.Graph.SWIG","a":"Graph","t":2,"d":"The fixed-node constructor is injective: equal fixed SWIG nodes come from the same base variable.","s":"∀ {N : Type u_1}, Function.Injective Causalean.SWIGNode.fixed","l":99,"f":"Causalean/Graph/SWIG.lean"},{"n":"Causalean.SWIGNode.instFintype","k":"instance","m":"Causalean.Graph.SWIG","a":"Graph","t":2,"d":"If the base variables are finite, then the split SWIG node set is finite.","s":"{N : Type u_1} → [Fintype N] → Fintype (Causalean.SWIGNode N)","l":115,"f":"Causalean/Graph/SWIG.lean"},{"n":"Causalean.SWIGNode.random_injective","k":"theorem","m":"Causalean.Graph.SWIG","a":"Graph","t":2,"d":"The random-node constructor is injective: equal random SWIG nodes come from the same base variable.","s":"∀ {N : Type u_1}, Function.Injective Causalean.SWIGNode.random","l":94,"f":"Causalean/Graph/SWIG.lean"},{"n":"Causalean.SameImmoralities","k":"def","m":"Causalean.Graph.MarkovEquiv.Defs","a":"Graph/MarkovEquiv","t":1,"d":"Two DAGs have the same v-structures when their immorality relations agree.","s":"{V : Type u_1} → [inst : DecidableEq V] → [inst_1 : Fintype V] → Causalean.DAG V → Causalean.DAG V → Prop","l":57,"f":"Causalean/Graph/MarkovEquiv/Defs.lean"},{"n":"Causalean.SameImmoralities.symm","k":"theorem","m":"Causalean.Graph.MarkovEquiv.Decompose","a":"Graph/MarkovEquiv","t":2,"d":"SameImmoralities is symmetric.","s":"∀ {V : Type u_1} [inst : DecidableEq V] [inst_1 : Fintype V] {G₁ G₂ : Causalean.DAG V}, Causalean.SameImmoralities G₁ G₂ → Causalean.SameImmoralities G₂ G₁","l":31,"f":"Causalean/Graph/MarkovEquiv/Decompose.lean"},{"n":"Causalean.SameImmoralities.trans","k":"theorem","m":"Causalean.Graph.MarkovEquiv.Decompose","a":"Graph/MarkovEquiv","t":2,"d":"SameImmoralities is transitive.","s":"∀ {V : Type u_1} [inst : DecidableEq V] [inst_1 : Fintype V] {G₁ G₂ G₃ : Causalean.DAG V}, Causalean.SameImmoralities G₁ G₂ → Causalean.SameImmoralities G₂ G₃ → Causalean.SameImmoralities G₁ G₃","l":35,"f":"Causalean/Graph/MarkovEquiv/Decompose.lean"},{"n":"Causalean.SameSkeleton","k":"def","m":"Causalean.Graph.MarkovEquiv.Defs","a":"Graph/MarkovEquiv","t":1,"d":"Two DAGs have the same skeleton when their undirected adjacency relations agree: for every pair a, b, there is an edge (in either direction) between them in G₁ iff there is one in G₂.","s":"{V : Type u_1} → [inst : DecidableEq V] → [inst_1 : Fintype V] → Causalean.DAG V → Causalean.DAG V → Prop","l":52,"f":"Causalean/Graph/MarkovEquiv/Defs.lean"},{"n":"Causalean.SameSkeleton.symm","k":"theorem","m":"Causalean.Graph.MarkovEquiv.Decompose","a":"Graph/MarkovEquiv","t":2,"d":"SameSkeleton is symmetric.","s":"∀ {V : Type u_1} [inst : DecidableEq V] [inst_1 : Fintype V] {G₁ G₂ : Causalean.DAG V}, Causalean.SameSkeleton G₁ G₂ → Causalean.SameSkeleton G₂ G₁","l":23,"f":"Causalean/Graph/MarkovEquiv/Decompose.lean"},{"n":"Causalean.SameSkeleton.trans","k":"theorem","m":"Causalean.Graph.MarkovEquiv.Decompose","a":"Graph/MarkovEquiv","t":2,"d":"SameSkeleton is transitive.","s":"∀ {V : Type u_1} [inst : DecidableEq V] [inst_1 : Fintype V] {G₁ G₂ G₃ : Causalean.DAG V}, Causalean.SameSkeleton G₁ G₂ → Causalean.SameSkeleton G₂ G₃ → Causalean.SameSkeleton G₁ G₃","l":27,"f":"Causalean/Graph/MarkovEquiv/Decompose.lean"},{"n":"Causalean.Stat.AsymptoticEquicont","k":"def","m":"Causalean.Stat.EmpiricalProcess.Equicontinuity.Modulus","a":"Stat/EmpiricalProcess/Equicontinuity","t":1,"d":"Class-level asymptotic equicontinuity of the score family ψ at θ₀.","s":"{Ω : Type u_1} → {X : Type u_2} → [inst : MeasurableSpace Ω] → [inst_1 : MeasurableSpace X] → {E : Type u_3} → [inst_2 : NormedAddCommGroup E] → [InnerProductSpace ℝ E] → (E → X → E) → E → (P : MeasureTheory.Measure X) → (μ : MeasureTheory.Measure Ω) → Causalean.Stat.IIDSample Ω X μ P → Prop","l":68,"f":"Causalean/Stat/EmpiricalProcess/Equicontinuity/Modulus.lean"},{"n":"Causalean.Stat.Concentration.AntitoneOn.leftRiemann_sum_le_integral","k":"theorem","m":"Causalean.Stat.Concentration.Covering.DudleyEntropy","a":"Stat/Concentration/Covering","t":2,"d":"For an antitone grid and antitone integrand, the full left Riemann sum is bounded by the interval integral.","s":"∀ (n : ℕ) (f : ℕ → ℝ) (g : ℝ → ℝ), Antitone f → AntitoneOn g (Set.Icc (f n) (f 0)) → ∑ j, (f ↑j - f (↑j + 1)) * g (f ↑j) ≤ ∫ (x : ℝ) in f n..f 0, g x","l":1596,"f":"Causalean/Stat/Concentration/Covering/DudleyEntropy.lean"},{"n":"Causalean.Stat.Concentration.BinaryFactoredVCClass","k":"structure","m":"Causalean.Stat.Concentration.Covering.VCLocalizedRegime","a":"Stat/Concentration/Covering","t":1,"d":"A binary factorization of a real-valued function class through a Boolean labeling π, such that on every finite sample each function's values factor through the Boolean labels, and whose realized Boolean patterns have VC dimension at most d on every finite sample.","s":"{ι : Type u} → {𝒳 : Type v} → (ι → 𝒳 → ℝ) → ℕ → Type (max u v)","l":40,"f":"Causalean/Stat/Concentration/Covering/VCLocalizedRegime.lean"},{"n":"Causalean.Stat.Concentration.BinaryTraceEntropyControl","k":"def","m":"Causalean.Stat.Concentration.Covering.VCLocalizedRegime","a":"Stat/Concentration/Covering","t":1,"d":"Binary trace entropy evidence used by the shared localized star-hull Dudley residual.","s":"{ι : Type u} → {𝒳 : Type v} → (ι → 𝒳 → Bool) → ℕ → Prop","l":388,"f":"Causalean/Stat/Concentration/Covering/VCLocalizedRegime.lean"},{"n":"Causalean.Stat.Concentration.CoeffBox","k":"def","m":"Causalean.Stat.Concentration.Covering.RealValuedVCSubgraph.Parametric","a":"Stat/Concentration/Covering/RealValuedVCSubgraph","t":1,"d":"The box of coefficient vectors whose coordinates have absolute value at most B.","s":"Type w → ℝ → Type (max 0 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n → 𝒳} (q : Causalean.Stat.Concentration.EmpiricalFunctionSpace F S), F q.index = F q.index","l":147,"f":"Causalean/Stat/Concentration/Covering/EmpiricalPseudoMetric.lean"},{"n":"Causalean.Stat.Concentration.EuclideanRadialPolynomial.ClosedBallParam","k":"def","m":"Causalean.Stat.Concentration.Covering.EuclideanRadialPolynomial.Geometry","a":"Stat/Concentration/Covering/EuclideanRadialPolynomial","t":1,"d":"A center and a nonnegative radius parameterize a genuine closed Euclidean ball.","s":"ℕ → Type","l":205,"f":"Causalean/Stat/Concentration/Covering/EuclideanRadialPolynomial/Geometry.lean"},{"n":"Causalean.Stat.Concentration.EuclideanRadialPolynomial.EuclideanPoint","k":"def","m":"Causalean.Stat.Concentration.Covering.EuclideanRadialPolynomial.Geometry","a":"Stat/Concentration/Covering/EuclideanRadialPolynomial","t":1,"d":"The d-dimensional real Euclidean space used by the radial classes.","s":"ℕ → Type","l":25,"f":"Causalean/Stat/Concentration/Covering/EuclideanRadialPolynomial/Geometry.lean"},{"n":"Causalean.Stat.Concentration.EuclideanRadialPolynomial.HasPolynomialL2Cover.monoEnvelope","k":"theorem","m":"Causalean.Stat.Concentration.Covering.EuclideanRadialPolynomial.Cover","a":"Stat/Concentration/Covering/EuclideanRadialPolynomial","t":2,"d":"Enlarging the positive envelope of a polynomial L² covering certificate preserves the certificate and its polynomial constants.","s":"∀ {𝒳 : Type u} [inst : MeasurableSpace 𝒳] {ι : Type v} {F : ι → 𝒳 → ℝ} {U V : ℝ}, Causalean.Stat.Concentration.HasPolynomialL2Cover F U → U ≤ V → Causalean.Stat.Concentration.HasPolynomialL2Cover F V","l":233,"f":"Causalean/Stat/Concentration/Covering/EuclideanRadialPolynomial/Cover.lean"},{"n":"Causalean.Stat.Concentration.EuclideanRadialPolynomial.HasPolynomialL2Cover.pullback","k":"theorem","m":"Causalean.Stat.Concentration.Covering.EuclideanRadialPolynomial.Cover","a":"Stat/Concentration/Covering/EuclideanRadialPolynomial","t":2,"d":"Pulling a polynomial-cover class back along an arbitrary parameter map preserves a polynomial L² covering certificate when the new parameter type is nonempty. Cover centers are replaced by occupied-class representatives, which costs only a factor two in radius.","s":"∀ {𝒳 : Type u} [inst : MeasurableSpace 𝒳] {ι : Type v} {κ : Type w} [Nonempty κ] {F : ι → 𝒳 → ℝ} {U : ℝ}, Causalean.Stat.Concentration.HasPolynomialL2Cover F U → ∀ (e : κ → ι), Causalean.Stat.Concentration.HasPolynomialL2Cover (fun k => F (e k)) U","l":96,"f":"Causalean/Stat/Concentration/Covering/EuclideanRadialPolynomial/Cover.lean"},{"n":"Causalean.Stat.Concentration.EuclideanRadialPolynomial.HasPolynomialL2CoverWith.monoEnvelope","k":"theorem","m":"Causalean.Stat.Concentration.Covering.EuclideanRadialPolynomial.Cover","a":"Stat/Concentration/Covering/EuclideanRadialPolynomial","t":2,"d":"Enlarging an envelope preserves named entropy witnesses.","s":"∀ {𝒳 : Type u} [inst : MeasurableSpace 𝒳] {ι : Type v} {F : ι → 𝒳 → ℝ} {U V A : ℝ} {p : ℕ}, Causalean.Stat.Concentration.HasPolynomialL2CoverWith F U A p → U ≤ V → Causalean.Stat.Concentration.HasPolynomialL2CoverWith F V A p","l":251,"f":"Causalean/Stat/Concentration/Covering/EuclideanRadialPolynomial/Cover.lean"},{"n":"Causalean.Stat.Concentration.EuclideanRadialPolynomial.HasPolynomialL2CoverWith.pullback","k":"theorem","m":"Causalean.Stat.Concentration.Covering.EuclideanRadialPolynomial.Cover","a":"Stat/Concentration/Covering/EuclideanRadialPolynomial","t":2,"d":"Pullback preserves named entropy witnesses, with the explicit factor-two radius cost used by the ordinary pullback construction.","s":"∀ {𝒳 : Type u} [inst : MeasurableSpace 𝒳] {ι : Type v} {κ : Type w} [Nonempty κ] {F : ι → 𝒳 → ℝ} {U A : ℝ} {p : ℕ}, Causalean.Stat.Concentration.HasPolynomialL2CoverWith F U A p → ∀ (e : κ → ι), Causalean.Stat.Concentration.HasPolynomialL2CoverWith (fun k => F (e k)) U (2 * A) p","l":159,"f":"Causalean/Stat/Concentration/Covering/EuclideanRadialPolynomial/Cover.lean"},{"n":"Causalean.Stat.Concentration.EuclideanRadialPolynomial.HasPseudoDimAtMost.compDomain","k":"theorem","m":"Causalean.Stat.Concentration.Covering.EuclideanRadialPolynomial.Trace","a":"Stat/Concentration/Covering/EuclideanRadialPolynomial","t":2,"d":"Precomposing every function in a real-valued class with a fixed map cannot increase its pseudo-dimension.","s":"∀ {𝒳 : Type u} {ι : Type v} {𝒴 : Type w} {F : ι → 𝒳 → ℝ} {d : ℕ}, Causalean.Stat.Concentration.HasPseudoDimAtMost F d → ∀ (g : 𝒴 → 𝒳), Causalean.Stat.Concentration.HasPseudoDimAtMost (fun i y => F i (g y)) d","l":71,"f":"Causalean/Stat/Concentration/Covering/EuclideanRadialPolynomial/Trace.lean"},{"n":"Causalean.Stat.Concentration.EuclideanRadialPolynomial.HasPseudoDimAtMost.reindex","k":"theorem","m":"Causalean.Stat.Concentration.Covering.EuclideanRadialPolynomial.Trace","a":"Stat/Concentration/Covering/EuclideanRadialPolynomial","t":2,"d":"Reparameterizing a real-valued class by an arbitrary map cannot increase its pseudo-dimension.","s":"∀ {𝒳 : Type u} {ι : Type v} {κ : Type w} {F : ι → 𝒳 → ℝ} {d : ℕ}, Causalean.Stat.Concentration.HasPseudoDimAtMost F d → ∀ (e : κ → ι), Causalean.Stat.Concentration.HasPseudoDimAtMost (fun k => F (e k)) d","l":57,"f":"Causalean/Stat/Concentration/Covering/EuclideanRadialPolynomial/Trace.lean"},{"n":"Causalean.Stat.Concentration.EuclideanRadialPolynomial.HasUniformPolynomialL2CoverOver.monoEnvelope","k":"theorem","m":"Causalean.Stat.Concentration.Covering.EuclideanRadialPolynomial.Cover","a":"Stat/Concentration/Covering/EuclideanRadialPolynomial","t":2,"d":"A parameterwise envelope enlargement preserves uniform named witnesses.","s":"∀ {𝒳 : Type u} [inst : MeasurableSpace 𝒳] {S : Type u_1} {ι : S → Type v} {F : (s : S) → ι s → 𝒳 → ℝ} {U V : S → ℝ}, Causalean.Stat.Concentration.HasUniformPolynomialL2CoverOver S F U → (∀ (s : S), U s ≤ V s) → Causalean.Stat.Concentration.HasUniformPolynomialL2CoverOver S F V","l":266,"f":"Causalean/Stat/Concentration/Covering/EuclideanRadialPolynomial/Cover.lean"},{"n":"Causalean.Stat.Concentration.EuclideanRadialPolynomial.HasUniformPolynomialL2CoverOver.pullback","k":"theorem","m":"Causalean.Stat.Concentration.Covering.EuclideanRadialPolynomial.Cover","a":"Stat/Concentration/Covering/EuclideanRadialPolynomial","t":2,"d":"A parameterwise pullback preserves entropy witnesses uniformly over the outer parameter family.","s":"∀ {𝒳 : Type u} [inst : MeasurableSpace 𝒳] {S : Type u_1} {ι : S → Type v} {κ : S → Type w} {F : (s : S) → ι s → 𝒳 → ℝ} {U : S → ℝ}, Causalean.Stat.Concentration.HasUniformPolynomialL2CoverOver S F U → (∀ (s : S), Nonempty (κ s)) → ∀ (e : (s : S) → κ s → ι s), Causalean.Stat.Concentration.HasUniformPolynomialL2CoverOver S (fun s k => F s (e s k)) U","l":218,"f":"Causalean/Stat/Concentration/Covering/EuclideanRadialPolynomial/Cover.lean"},{"n":"Causalean.Stat.Concentration.EuclideanRadialPolynomial.HasVCAtMost.compDomain","k":"theorem","m":"Causalean.Stat.Concentration.Covering.EuclideanRadialPolynomial.Trace","a":"Stat/Concentration/Covering/EuclideanRadialPolynomial","t":2,"d":"Precomposing the observation argument of a Boolean class cannot increase its finite-trace VC dimension.","s":"∀ {𝒳 : Type u} {ι : Type v} {𝒴 : Type w} {π : ι → 𝒳 → Bool} {d : ℕ}, Causalean.Stat.Concentration.HasVCAtMost π d → ∀ (g : 𝒴 → 𝒳), Causalean.Stat.Concentration.HasVCAtMost (fun i y => π i (g y)) d","l":48,"f":"Causalean/Stat/Concentration/Covering/EuclideanRadialPolynomial/Trace.lean"},{"n":"Causalean.Stat.Concentration.EuclideanRadialPolynomial.HasVCAtMost.reindex","k":"theorem","m":"Causalean.Stat.Concentration.Covering.EuclideanRadialPolynomial.Trace","a":"Stat/Concentration/Covering/EuclideanRadialPolynomial","t":2,"d":"Reparameterizing a Boolean class by an arbitrary map cannot increase its finite-trace VC dimension.","s":"∀ {𝒳 : Type u} {ι : Type v} {κ : Type w} {π : ι → 𝒳 → Bool} {d : ℕ}, Causalean.Stat.Concentration.HasVCAtMost π d → ∀ (e : κ → ι), Causalean.Stat.Concentration.HasVCAtMost (fun k => π (e k)) d","l":35,"f":"Causalean/Stat/Concentration/Covering/EuclideanRadialPolynomial/Trace.lean"},{"n":"Causalean.Stat.Concentration.EuclideanRadialPolynomial.RadialMonomialParam","k":"def","m":"Causalean.Stat.Concentration.Covering.EuclideanRadialPolynomial.Geometry","a":"Stat/Concentration/Covering/EuclideanRadialPolynomial","t":1,"d":"The parameter space for a moving center and a degree between zero and p, inclusive.","s":"ℕ → ℕ → Type","l":284,"f":"Causalean/Stat/Concentration/Covering/EuclideanRadialPolynomial/Geometry.lean"},{"n":"Causalean.Stat.Concentration.EuclideanRadialPolynomial.RadialPolynomialParam","k":"def","m":"Causalean.Stat.Concentration.Covering.EuclideanRadialPolynomial.Score","a":"Stat/Concentration/Covering/EuclideanRadialPolynomial","t":1,"d":"The parameter space for a shared center and a boxed radial-polynomial coefficient vector.","s":"ℕ → ℕ → ℝ → Type","l":34,"f":"Causalean/Stat/Concentration/Covering/EuclideanRadialPolynomial/Score.lean"},{"n":"Causalean.Stat.Concentration.EuclideanRadialPolynomial.RadialResidualScoreParam","k":"def","m":"Causalean.Stat.Concentration.Covering.EuclideanRadialPolynomial.Score","a":"Stat/Concentration/Covering/EuclideanRadialPolynomial","t":1,"d":"The parameter space of a moving center, a boxed polynomial coefficient vector, and one member of a finite signed-arm family.","s":"ℕ → ℕ → ℝ → Type v → Type (max 0 v)","l":151,"f":"Causalean/Stat/Concentration/Covering/EuclideanRadialPolynomial/Score.lean"},{"n":"Causalean.Stat.Concentration.EuclideanRadialPolynomial.RadiusPoint","k":"def","m":"Causalean.Stat.Concentration.Covering.EuclideanRadialPolynomial.Geometry","a":"Stat/Concentration/Covering/EuclideanRadialPolynomial","t":1,"d":"A Euclidean point paired with a real radius threshold. Negative thresholds are allowed; they give a constant ball or exterior label.","s":"ℕ → Type","l":28,"f":"Causalean/Stat/Concentration/Covering/EuclideanRadialPolynomial/Geometry.lean"},{"n":"Causalean.Stat.Concentration.EuclideanRadialPolynomial.abs_boundedRadialPolynomial_le","k":"theorem","m":"Causalean.Stat.Concentration.Covering.EuclideanRadialPolynomial.Score","a":"Stat/Concentration/Covering/EuclideanRadialPolynomial","t":2,"d":"Every boxed shared-center radial polynomial is bounded by the number of basis functions times the coefficient bound times the monomial envelope.","s":"∀ (d p : ℕ) {q a b B : ℝ}, 0 < q → 0 ≤ a → a ≤ b → 0 ≤ B → ∀ (x : Causalean.Stat.Concentration.EuclideanRadialPolynomial.EuclideanPoint d) (β : Causalean.Stat.Concentration.CoeffBox (Fin (p + 1)) B) (z : Causalean.Stat.Concentration.EuclideanRadialPolynomial.EuclideanPoint d), |Causalean.Stat.Concentration.EuclideanRadialPolynomial.boundedRadialPolynomial d p q a b B x β z| ≤ Causalean.Stat.Concentration.EuclideanRadialPolynomial.radialPolynomialEnvelope b p B","l":51,"f":"Causalean/Stat/Concentration/Covering/EuclideanRadialPolynomial/Score.lean"},{"n":"Causalean.Stat.Concentration.EuclideanRadialPolynomial.abs_radialAnnulusMonomial_le","k":"theorem","m":"Causalean.Stat.Concentration.Covering.EuclideanRadialPolynomial.Cover","a":"Stat/Concentration/Covering/EuclideanRadialPolynomial","t":2,"d":"Every degree k ≤ p radial monomial on an ordered nonnegative annulus is bounded in absolute value by radialMonomialEnvelope b p.","s":"∀ (d p k : ℕ) {q a b : ℝ}, 0 < q → 0 ≤ a → a ≤ b → k ≤ p → ∀ (x z : Causalean.Stat.Concentration.EuclideanRadialPolynomial.EuclideanPoint d), |Causalean.Stat.Concentration.EuclideanRadialPolynomial.radialAnnulusMonomial d q a b k x z| ≤ Causalean.Stat.Concentration.EuclideanRadialPolynomial.radialMonomialEnvelope b p","l":43,"f":"Causalean/Stat/Concentration/Covering/EuclideanRadialPolynomial/Cover.lean"},{"n":"Causalean.Stat.Concentration.EuclideanRadialPolynomial.booleanCombinationVCBound","k":"def","m":"Causalean.Stat.Concentration.Covering.EuclideanRadialPolynomial.Trace","a":"Stat/Concentration/Covering/EuclideanRadialPolynomial","t":1,"d":"A deliberately coarse uniform VC bound for a Boolean formula in m independently parameterized classes of VC dimension at most d.","s":"ℕ → ℕ → ℕ","l":25,"f":"Causalean/Stat/Concentration/Covering/EuclideanRadialPolynomial/Trace.lean"},{"n":"Causalean.Stat.Concentration.EuclideanRadialPolynomial.booleanCombination_hasVCAtMost","k":"theorem","m":"Causalean.Stat.Concentration.Covering.EuclideanRadialPolynomial.Trace","a":"Stat/Concentration/Covering/EuclideanRadialPolynomial","t":1,"d":"VC bound for a point-dependent Boolean combination of classes. Given m independently parameterized Boolean classifier families, each of VC dimension at most d, applying to them any combining rule that may itself depend on the sampled point still yields a Boolean class of VC dimension at most booleanCombinationVCBound m d.","s":"∀ {𝒳 : Type u} {m d : ℕ} {ι : Fin m → Type v} (π : (j : Fin m) → ι j → 𝒳 → Bool), (∀ (j : Fin m), Causalean.Stat.Concentration.HasVCAtMost (π j) d) → ∀ (combine : 𝒳 → (Fin m → Bool) → Bool), Causalean.Stat.Concentration.HasVCAtMost (fun θ x => combine x fun j => π j (θ j) x) (Causalean.Stat.Concentration.EuclideanRadialPolynomial.booleanCombinationVCBound m d)","l":426,"f":"Causalean/Stat/Concentration/Covering/EuclideanRadialPolynomial/Trace.lean"},{"n":"Causalean.Stat.Concentration.EuclideanRadialPolynomial.boundedRadialPolynomial","k":"def","m":"Causalean.Stat.Concentration.Covering.EuclideanRadialPolynomial.Score","a":"Stat/Concentration/Covering/EuclideanRadialPolynomial","t":1,"d":"A bounded coefficient vector evaluated against all radial monomials from degree zero through p, with one center shared by every term.","s":"(d p : ℕ) → ℝ → ℝ → ℝ → (B : ℝ) → Causalean.Stat.Concentration.EuclideanRadialPolynomial.EuclideanPoint d → Causalean.Stat.Concentration.CoeffBox (Fin (p + 1)) B → Causalean.Stat.Concentration.EuclideanRadialPolynomial.EuclideanPoint d → ℝ","l":25,"f":"Causalean/Stat/Concentration/Covering/EuclideanRadialPolynomial/Score.lean"},{"n":"Causalean.Stat.Concentration.EuclideanRadialPolynomial.boundedRadialPolynomialOn","k":"def","m":"Causalean.Stat.Concentration.Covering.EuclideanRadialPolynomial.Score","a":"Stat/Concentration/Covering/EuclideanRadialPolynomial","t":1,"d":"A shared-center bounded radial polynomial evaluated after a Euclidean location map on a general observation space.","s":"{Ω : Type u} → [MeasurableSpace Ω] → (d p : ℕ) → (Ω → Causalean.Stat.Concentration.EuclideanRadialPolynomial.EuclideanPoint d) → ℝ → ℝ → ℝ → (B : ℝ) → Causalean.Stat.Concentration.EuclideanRadialPolynomial.RadialPolynomialParam d p B → Ω → ℝ","l":39,"f":"Causalean/Stat/Concentration/Covering/EuclideanRadialPolynomial/Score.lean"},{"n":"Causalean.Stat.Concentration.EuclideanRadialPolynomial.boundedRadialPolynomialOn_hasPolynomialL2Cover","k":"theorem","m":"Causalean.Stat.Concentration.Covering.EuclideanRadialPolynomial.Score","a":"Stat/Concentration/Covering/EuclideanRadialPolynomial","t":2,"d":"Boxed finite radial polynomials with one moving center shared by all degrees have a uniform polynomial L² covering certificate.","s":"∀ {Ω : Type u} [inst : MeasurableSpace Ω] (d p : ℕ) (loc : Ω → Causalean.Stat.Concentration.EuclideanRadialPolynomial.EuclideanPoint d) {q a b B : ℝ}, Measurable loc → 0 < q → 0 ≤ a → a ≤ b → 0 < B → Causalean.Stat.Concentration.HasPolynomialL2Cover (Causalean.Stat.Concentration.EuclideanRadialPolynomial.boundedRadialPolynomialOn d p loc q a b B) (Causalean.Stat.Concentration.EuclideanRadialPolynomial.radialPolynomialEnvelope b p B)","l":92,"f":"Causalean/Stat/Concentration/Covering/EuclideanRadialPolynomial/Score.lean"},{"n":"Causalean.Stat.Concentration.EuclideanRadialPolynomial.euclideanClosedBallClassifier","k":"def","m":"Causalean.Stat.Concentration.Covering.EuclideanRadialPolynomial.Geometry","a":"Stat/Concentration/Covering/EuclideanRadialPolynomial","t":1,"d":"The Boolean classifier of closed Euclidean balls in finite-dimensional real Euclidean space.","s":"(d : ℕ) → Causalean.Stat.Concentration.EuclideanRadialPolynomial.ClosedBallParam d → Causalean.Stat.Concentration.EuclideanRadialPolynomial.EuclideanPoint d → Bool","l":209,"f":"Causalean/Stat/Concentration/Covering/EuclideanRadialPolynomial/Geometry.lean"},{"n":"Causalean.Stat.Concentration.EuclideanRadialPolynomial.euclideanClosedBall_hasVCAtMost","k":"theorem","m":"Causalean.Stat.Concentration.Covering.EuclideanRadialPolynomial.Geometry","a":"Stat/Concentration/Covering/EuclideanRadialPolynomial","t":2,"d":"Closed balls in d-dimensional Euclidean space have VC dimension at most d + 2. This neutral Causalean-only statement replaces the paper-local planar ball lemma with a finite-dimensional bound.","s":"∀ (d : ℕ), Causalean.Stat.Concentration.HasVCAtMost (Causalean.Stat.Concentration.EuclideanRadialPolynomial.euclideanClosedBallClassifier d) (d + 2)","l":258,"f":"Causalean/Stat/Concentration/Covering/EuclideanRadialPolynomial/Geometry.lean"},{"n":"Causalean.Stat.Concentration.EuclideanRadialPolynomial.finiteClass_hasPolynomialL2Cover","k":"theorem","m":"Causalean.Stat.Concentration.Covering.EuclideanRadialPolynomial.Cover","a":"Stat/Concentration/Covering/EuclideanRadialPolynomial","t":2,"d":"Every finite family of measurable functions bounded by a positive common envelope has a uniform polynomial L² covering certificate.","s":"∀ {𝒳 : Type u} [inst : MeasurableSpace 𝒳] {ι : Type v} [Fintype ι] (F : ι → 𝒳 → ℝ) {U : ℝ}, 0 < U → (∀ (i : ι), Measurable (F i)) → (∀ (i : ι) (x : 𝒳), |F i x| ≤ U) → Causalean.Stat.Concentration.HasPolynomialL2Cover F U","l":279,"f":"Causalean/Stat/Concentration/Covering/EuclideanRadialPolynomial/Cover.lean"},{"n":"Causalean.Stat.Concentration.EuclideanRadialPolynomial.finiteClass_hasPolynomialL2CoverWith","k":"theorem","m":"Causalean.Stat.Concentration.Covering.EuclideanRadialPolynomial.Cover","a":"Stat/Concentration/Covering/EuclideanRadialPolynomial","t":2,"d":"A finite bounded measurable class has named entropy witnesses depending only on its cardinality.","s":"∀ {𝒳 : Type u} [inst : MeasurableSpace 𝒳] {ι : Type v} [inst_1 : Fintype ι] (F : ι → 𝒳 → ℝ) {U : ℝ}, 0 < U → (∀ (i : ι), Measurable (F i)) → (∀ (i : ι) (x : 𝒳), |F i x| ≤ U) → Causalean.Stat.Concentration.HasPolynomialL2CoverWith F U (max 1 ↑(Fintype.card ι)) 1","l":309,"f":"Causalean/Stat/Concentration/Covering/EuclideanRadialPolynomial/Cover.lean"},{"n":"Causalean.Stat.Concentration.EuclideanRadialPolynomial.finiteSignedArmRadial_hasPolynomialL2Cover","k":"theorem","m":"Causalean.Stat.Concentration.Covering.EuclideanRadialPolynomial.Cover","a":"Stat/Concentration/Covering/EuclideanRadialPolynomial","t":2,"d":"Multiplying a moving-center radial monomial by any bounded measurable finite family of signed arms preserves a uniform polynomial cover. Boolean arms and the two signs are obtained by taking a finite arm type and values in {0,1} or {-1,1}.","s":"∀ {Ω : Type u} [inst : MeasurableSpace Ω] {A : Type v} [Fintype A] (d p : ℕ) (loc : Ω → Causalean.Stat.Concentration.EuclideanRadialPolynomial.EuclideanPoint d) (arm : A → Ω → ℝ) {q a b : ℝ}, Measurable loc → (∀ (s : A), Measurable (arm s)) → (∀ (s : A) (ω : Ω), |arm s ω| ≤ 1) → 0 < q → 0 ≤ a → a ≤ b → Causalean.Stat.Concentration.HasPolynomialL2Cover (fun θ ω => Causalean.Stat.Concentration.EuclideanRadialPolynomial.radialMonomialOn d p loc q a b θ.1 ω * arm θ.2 ω) (Causalean.Stat.Concentration.EuclideanRadialPolynomial.radialMonomialEnvelope b p)","l":420,"f":"Causalean/Stat/Concentration/Covering/EuclideanRadialPolynomial/Cover.lean"},{"n":"Causalean.Stat.Concentration.EuclideanRadialPolynomial.finiteUnionVCBound","k":"def","m":"Causalean.Stat.Concentration.Covering.EuclideanRadialPolynomial.Trace","a":"Stat/Concentration/Covering/EuclideanRadialPolynomial","t":1,"d":"A deliberately coarse uniform VC bound for a union of m classes, each of VC dimension at most d.","s":"ℕ → ℕ → ℕ","l":30,"f":"Causalean/Stat/Concentration/Covering/EuclideanRadialPolynomial/Trace.lean"},{"n":"Causalean.Stat.Concentration.EuclideanRadialPolynomial.finiteUnion_hasVCAtMost","k":"theorem","m":"Causalean.Stat.Concentration.Covering.EuclideanRadialPolynomial.Trace","a":"Stat/Concentration/Covering/EuclideanRadialPolynomial","t":2,"d":"A finite union of m Boolean classes of VC dimension at most d has finite VC dimension bounded by finiteUnionVCBound m d.","s":"∀ {𝒳 : Type u} {K : Type v} [inst : Fintype K] {ι : K → Type w} {d : ℕ} (π : (k : K) → ι k → 𝒳 → Bool), (∀ (k : K), Causalean.Stat.Concentration.HasVCAtMost (π k) d) → Causalean.Stat.Concentration.HasVCAtMost (fun θ => π θ.fst θ.snd) (Causalean.Stat.Concentration.EuclideanRadialPolynomial.finiteUnionVCBound (Fintype.card K) d)","l":446,"f":"Causalean/Stat/Concentration/Covering/EuclideanRadialPolynomial/Trace.lean"},{"n":"Causalean.Stat.Concentration.EuclideanRadialPolynomial.fixedRadialPseudoDimBound","k":"def","m":"Causalean.Stat.Concentration.Covering.EuclideanRadialPolynomial.Geometry","a":"Stat/Concentration/Covering/EuclideanRadialPolynomial","t":1,"d":"An explicit trace bound for one fixed radial degree.","s":"ℕ → ℕ","l":295,"f":"Causalean/Stat/Concentration/Covering/EuclideanRadialPolynomial/Geometry.lean"},{"n":"Causalean.Stat.Concentration.EuclideanRadialPolynomial.linearSignClass","k":"def","m":"Causalean.Stat.Concentration.Covering.EuclideanRadialPolynomial.Trace","a":"Stat/Concentration/Covering/EuclideanRadialPolynomial","t":1,"d":"The homogeneous linear threshold class generated by a finite feature vector labels a point according to the sign of its parameter-feature pairing.","s":"{𝒳 : Type u} → {K : Type v} → [Fintype K] → (K → 𝒳 → ℝ) → (K → ℝ) → 𝒳 → Bool","l":80,"f":"Causalean/Stat/Concentration/Covering/EuclideanRadialPolynomial/Trace.lean"},{"n":"Causalean.Stat.Concentration.EuclideanRadialPolynomial.linearSignClass_hasVCAtMost","k":"theorem","m":"Causalean.Stat.Concentration.Covering.EuclideanRadialPolynomial.Trace","a":"Stat/Concentration/Covering/EuclideanRadialPolynomial","t":2,"d":"Homogeneous linear threshold classifiers in K real coordinates have VC dimension at most the number of coordinates.","s":"∀ {𝒳 : Type u} {K : Type v} [inst : Fintype K] (φ : K → 𝒳 → ℝ), Causalean.Stat.Concentration.HasVCAtMost (Causalean.Stat.Concentration.EuclideanRadialPolynomial.linearSignClass φ) (Fintype.card K)","l":86,"f":"Causalean/Stat/Concentration/Covering/EuclideanRadialPolynomial/Trace.lean"},{"n":"Causalean.Stat.Concentration.EuclideanRadialPolynomial.movingCenterBallExteriorClassifier","k":"def","m":"Causalean.Stat.Concentration.Covering.EuclideanRadialPolynomial.Geometry","a":"Stat/Concentration/Covering/EuclideanRadialPolynomial","t":1,"d":"A moving center labels a point-radius pair when it lies strictly outside the corresponding ball.","s":"(d : ℕ) → Causalean.Stat.Concentration.EuclideanRadialPolynomial.EuclideanPoint d → Causalean.Stat.Concentration.EuclideanRadialPolynomial.RadiusPoint d → Bool","l":38,"f":"Causalean/Stat/Concentration/Covering/EuclideanRadialPolynomial/Geometry.lean"},{"n":"Causalean.Stat.Concentration.EuclideanRadialPolynomial.movingCenterBallExterior_hasVCAtMost","k":"theorem","m":"Causalean.Stat.Concentration.Covering.EuclideanRadialPolynomial.Geometry","a":"Stat/Concentration/Covering/EuclideanRadialPolynomial","t":2,"d":"Variable-radius strict-exterior tests indexed by a moving Euclidean center have VC dimension at most d + 2.","s":"∀ (d : ℕ), Causalean.Stat.Concentration.HasVCAtMost (Causalean.Stat.Concentration.EuclideanRadialPolynomial.movingCenterBallExteriorClassifier d) (d + 2)","l":199,"f":"Causalean/Stat/Concentration/Covering/EuclideanRadialPolynomial/Geometry.lean"},{"n":"Causalean.Stat.Concentration.EuclideanRadialPolynomial.movingCenterClosedBallClassifier","k":"def","m":"Causalean.Stat.Concentration.Covering.EuclideanRadialPolynomial.Geometry","a":"Stat/Concentration/Covering/EuclideanRadialPolynomial","t":1,"d":"A moving center labels a point-radius pair when the point lies in the corresponding closed ball about that center.","s":"(d : ℕ) → Causalean.Stat.Concentration.EuclideanRadialPolynomial.EuclideanPoint d → Causalean.Stat.Concentration.EuclideanRadialPolynomial.RadiusPoint d → Bool","l":32,"f":"Causalean/Stat/Concentration/Covering/EuclideanRadialPolynomial/Geometry.lean"},{"n":"Causalean.Stat.Concentration.EuclideanRadialPolynomial.movingCenterClosedBall_hasVCAtMost","k":"theorem","m":"Causalean.Stat.Concentration.Covering.EuclideanRadialPolynomial.Geometry","a":"Stat/Concentration/Covering/EuclideanRadialPolynomial","t":2,"d":"Variable-radius closed-ball tests indexed by a moving Euclidean center have VC dimension at most d + 2.","s":"∀ (d : ℕ), Causalean.Stat.Concentration.HasVCAtMost (Causalean.Stat.Concentration.EuclideanRadialPolynomial.movingCenterClosedBallClassifier d) (d + 2)","l":186,"f":"Causalean/Stat/Concentration/Covering/EuclideanRadialPolynomial/Geometry.lean"},{"n":"Causalean.Stat.Concentration.EuclideanRadialPolynomial.radialAnnulusMonomial","k":"def","m":"Causalean.Stat.Concentration.Covering.EuclideanRadialPolynomial.Geometry","a":"Stat/Concentration/Covering/EuclideanRadialPolynomial","t":1,"d":"The compactly supported radial monomial with center x, bandwidth q, relative annulus endpoints a,b, and natural degree k.","s":"(d : ℕ) → ℝ → ℝ → ℝ → ℕ → Causalean.Stat.Concentration.EuclideanRadialPolynomial.EuclideanPoint d → Causalean.Stat.Concentration.EuclideanRadialPolynomial.EuclideanPoint d → ℝ","l":276,"f":"Causalean/Stat/Concentration/Covering/EuclideanRadialPolynomial/Geometry.lean"},{"n":"Causalean.Stat.Concentration.EuclideanRadialPolynomial.radialAnnulusMonomial_hasPolynomialL2Cover","k":"theorem","m":"Causalean.Stat.Concentration.Covering.EuclideanRadialPolynomial.Cover","a":"Stat/Concentration/Covering/EuclideanRadialPolynomial","t":2,"d":"A fixed degree k ≤ p of the moving-center radial class has the same uniform polynomial cover and envelope as the full degree vector.","s":"∀ (d p k : ℕ) {q a b : ℝ}, 0 < q → 0 ≤ a → a ≤ b → k ≤ p → Causalean.Stat.Concentration.HasPolynomialL2Cover (fun x => Causalean.Stat.Concentration.EuclideanRadialPolynomial.radialAnnulusMonomial d q a b k x) (Causalean.Stat.Concentration.EuclideanRadialPolynomial.radialMonomialEnvelope b p)","l":356,"f":"Causalean/Stat/Concentration/Covering/EuclideanRadialPolynomial/Cover.lean"},{"n":"Causalean.Stat.Concentration.EuclideanRadialPolynomial.radialAnnulusMonomial_hasPseudoDimAtMost","k":"theorem","m":"Causalean.Stat.Concentration.Covering.EuclideanRadialPolynomial.Geometry","a":"Stat/Concentration/Covering/EuclideanRadialPolynomial","t":2,"d":"For every positive bandwidth and ordered nonnegative annulus, the class of fixed-degree radial monomials with moving center has pseudo-dimension bounded solely by the Euclidean dimension.","s":"∀ (d k : ℕ) {q a b : ℝ}, 0 < q → 0 ≤ a → a ≤ b → Causalean.Stat.Concentration.HasPseudoDimAtMost (fun x => Causalean.Stat.Concentration.EuclideanRadialPolynomial.radialAnnulusMonomial d q a b k x) (Causalean.Stat.Concentration.EuclideanRadialPolynomial.fixedRadialPseudoDimBound d)","l":336,"f":"Causalean/Stat/Concentration/Covering/EuclideanRadialPolynomial/Geometry.lean"},{"n":"Causalean.Stat.Concentration.EuclideanRadialPolynomial.radialAnnulusMonomial_measurable","k":"theorem","m":"Causalean.Stat.Concentration.Covering.EuclideanRadialPolynomial.Cover","a":"Stat/Concentration/Covering/EuclideanRadialPolynomial","t":2,"d":"A compactly supported Euclidean radial monomial is Borel measurable in the observation for every fixed center and degree.","s":"∀ (d k : ℕ) (q a b : ℝ) (x : Causalean.Stat.Concentration.EuclideanRadialPolynomial.EuclideanPoint d), Measurable (Causalean.Stat.Concentration.EuclideanRadialPolynomial.radialAnnulusMonomial d q a b k x)","l":30,"f":"Causalean/Stat/Concentration/Covering/EuclideanRadialPolynomial/Cover.lean"},{"n":"Causalean.Stat.Concentration.EuclideanRadialPolynomial.radialMonomialClass","k":"def","m":"Causalean.Stat.Concentration.Covering.EuclideanRadialPolynomial.Geometry","a":"Stat/Concentration/Covering/EuclideanRadialPolynomial","t":1,"d":"The radial-monomial class in which the center and the degree up to p both vary.","s":"(d p : ℕ) → ℝ → ℝ → ℝ → Causalean.Stat.Concentration.EuclideanRadialPolynomial.RadialMonomialParam d p → Causalean.Stat.Concentration.EuclideanRadialPolynomial.EuclideanPoint d → ℝ","l":289,"f":"Causalean/Stat/Concentration/Covering/EuclideanRadialPolynomial/Geometry.lean"},{"n":"Causalean.Stat.Concentration.EuclideanRadialPolynomial.radialMonomialClass_hasPolynomialL2Cover","k":"theorem","m":"Causalean.Stat.Concentration.Covering.EuclideanRadialPolynomial.Cover","a":"Stat/Concentration/Covering/EuclideanRadialPolynomial","t":2,"d":"Moving-center radial monomials of every degree from zero through p have a polynomial L²(Q) cover with a positive constant envelope, uniformly over every probability measure Q.","s":"∀ (d p : ℕ) {q a b : ℝ}, 0 < q → 0 ≤ a → a ≤ b → Causalean.Stat.Concentration.HasPolynomialL2Cover (Causalean.Stat.Concentration.EuclideanRadialPolynomial.radialMonomialClass d p q a b) (Causalean.Stat.Concentration.EuclideanRadialPolynomial.radialMonomialEnvelope b p)","l":339,"f":"Causalean/Stat/Concentration/Covering/EuclideanRadialPolynomial/Cover.lean"},{"n":"Causalean.Stat.Concentration.EuclideanRadialPolynomial.radialMonomialClass_hasPseudoDimAtMost","k":"theorem","m":"Causalean.Stat.Concentration.Covering.EuclideanRadialPolynomial.Geometry","a":"Stat/Concentration/Covering/EuclideanRadialPolynomial","t":1,"d":"Pseudo-dimension bound for the moving-center radial-monomial class. For a positive bandwidth q, a nonnegative annulus inner radius a, and inner radius at most outer radius b, allowing both the Euclidean center and the monomial degree — ranging from zero through p — to vary gives the radial-monomial class a pseudo-dimension of at most radialPseudoDimBound d p.","s":"∀ (d p : ℕ) {q a b : ℝ}, 0 < q → 0 ≤ a → a ≤ b → Causalean.Stat.Concentration.HasPseudoDimAtMost (Causalean.Stat.Concentration.EuclideanRadialPolynomial.radialMonomialClass d p q a b) (Causalean.Stat.Concentration.EuclideanRadialPolynomial.radialPseudoDimBound d p)","l":408,"f":"Causalean/Stat/Concentration/Covering/EuclideanRadialPolynomial/Geometry.lean"},{"n":"Causalean.Stat.Concentration.EuclideanRadialPolynomial.radialMonomialEnvelope","k":"def","m":"Causalean.Stat.Concentration.Covering.EuclideanRadialPolynomial.Cover","a":"Stat/Concentration/Covering/EuclideanRadialPolynomial","t":1,"d":"A positive common envelope for all radial powers from zero through p on an annulus whose upper relative radius is b.","s":"ℝ → ℕ → ℝ","l":25,"f":"Causalean/Stat/Concentration/Covering/EuclideanRadialPolynomial/Cover.lean"},{"n":"Causalean.Stat.Concentration.EuclideanRadialPolynomial.radialMonomialOn","k":"def","m":"Causalean.Stat.Concentration.Covering.EuclideanRadialPolynomial.Cover","a":"Stat/Concentration/Covering/EuclideanRadialPolynomial","t":1,"d":"Evaluation of the radial-monomial class after a measurable Euclidean location map on a general observation space.","s":"{Ω : Type u} → [MeasurableSpace Ω] → (d p : ℕ) → (Ω → Causalean.Stat.Concentration.EuclideanRadialPolynomial.EuclideanPoint d) → ℝ → ℝ → ℝ → Causalean.Stat.Concentration.EuclideanRadialPolynomial.RadialMonomialParam d p → Ω → ℝ","l":369,"f":"Causalean/Stat/Concentration/Covering/EuclideanRadialPolynomial/Cover.lean"},{"n":"Causalean.Stat.Concentration.EuclideanRadialPolynomial.radialMonomialOn_hasPolynomialL2Cover","k":"theorem","m":"Causalean.Stat.Concentration.Covering.EuclideanRadialPolynomial.Cover","a":"Stat/Concentration/Covering/EuclideanRadialPolynomial","t":1,"d":"Covering certificate transported through a location map. For a measurable map from the underlying observation space into d-dimensional Euclidean space, given a positive bandwidth q, a nonnegative annulus inner radius a, and inner radius at most outer radius b, composing the moving-center radial-monomial class of degree at most p with the location map still carries a uniform polynomial L² covering certificate at envelope radialMonomialEnvelope b p.","s":"∀ {Ω : Type u} [inst : MeasurableSpace Ω] (d p : ℕ) (loc : Ω → Causalean.Stat.Concentration.EuclideanRadialPolynomial.EuclideanPoint d) {q a b : ℝ}, Measurable loc → 0 < q → 0 ≤ a → a ≤ b → Causalean.Stat.Concentration.HasPolynomialL2Cover (Causalean.Stat.Concentration.EuclideanRadialPolynomial.radialMonomialOn d p loc q a b) (Causalean.Stat.Concentration.EuclideanRadialPolynomial.radialMonomialEnvelope b p)","l":377,"f":"Causalean/Stat/Concentration/Covering/EuclideanRadialPolynomial/Cover.lean"},{"n":"Causalean.Stat.Concentration.EuclideanRadialPolynomial.radialMonomialOn_hasPolynomialL2CoverWith","k":"theorem","m":"Causalean.Stat.Concentration.Covering.EuclideanRadialPolynomial.Cover","a":"Stat/Concentration/Covering/EuclideanRadialPolynomial","t":2,"d":"The moving-center radial class after a measurable location map has the canonical named VC-subgraph entropy witnesses.","s":"∀ {Ω : Type u} [inst : MeasurableSpace Ω] (d p : ℕ) (loc : Ω → Causalean.Stat.Concentration.EuclideanRadialPolynomial.EuclideanPoint d) {q a b : ℝ}, Measurable loc → 0 < q → 0 ≤ a → a ≤ b → Causalean.Stat.Concentration.HasPolynomialL2CoverWith (Causalean.Stat.Concentration.EuclideanRadialPolynomial.radialMonomialOn d p loc q a b) (Causalean.Stat.Concentration.EuclideanRadialPolynomial.radialMonomialEnvelope b p) 16 (8 * (Causalean.Stat.Concentration.EuclideanRadialPolynomial.radialPseudoDimBound d p + 1))","l":400,"f":"Causalean/Stat/Concentration/Covering/EuclideanRadialPolynomial/Cover.lean"},{"n":"Causalean.Stat.Concentration.EuclideanRadialPolynomial.radialPolynomialEnvelope","k":"def","m":"Causalean.Stat.Concentration.Covering.EuclideanRadialPolynomial.Score","a":"Stat/Concentration/Covering/EuclideanRadialPolynomial","t":1,"d":"The constant envelope of a boxed degree-p radial polynomial.","s":"ℝ → ℕ → ℝ → ℝ","l":47,"f":"Causalean/Stat/Concentration/Covering/EuclideanRadialPolynomial/Score.lean"},{"n":"Causalean.Stat.Concentration.EuclideanRadialPolynomial.radialPseudoDimBound","k":"def","m":"Causalean.Stat.Concentration.Covering.EuclideanRadialPolynomial.Geometry","a":"Stat/Concentration/Covering/EuclideanRadialPolynomial","t":1,"d":"An explicit trace bound for all radial degrees from zero through p.","s":"ℕ → ℕ → ℕ","l":299,"f":"Causalean/Stat/Concentration/Covering/EuclideanRadialPolynomial/Geometry.lean"},{"n":"Causalean.Stat.Concentration.EuclideanRadialPolynomial.radialResidualScore","k":"def","m":"Causalean.Stat.Concentration.Covering.EuclideanRadialPolynomial.Score","a":"Stat/Concentration/Covering/EuclideanRadialPolynomial","t":1,"d":"A finite-arm residual score: a radial monomial multiplies a bounded response minus a shared-center boxed radial polynomial.","s":"{Ω : Type u} → [MeasurableSpace Ω] → {A : Type v} → (d p : ℕ) → (Ω → Causalean.Stat.Concentration.EuclideanRadialPolynomial.EuclideanPoint d) → (A → Ω → ℝ) → (Ω → ℝ) → ℝ → ℝ → ℝ → (B : ℝ) → Fin (p + 1) → Causalean.Stat.Concentration.EuclideanRadialPolynomial.RadialResidualScoreParam d p B A → Ω → ℝ","l":156,"f":"Causalean/Stat/Concentration/Covering/EuclideanRadialPolynomial/Score.lean"},{"n":"Causalean.Stat.Concentration.EuclideanRadialPolynomial.radialResidualScoreEnvelope","k":"def","m":"Causalean.Stat.Concentration.Covering.EuclideanRadialPolynomial.Score","a":"Stat/Concentration/Covering/EuclideanRadialPolynomial","t":1,"d":"The constant envelope for a bounded finite-arm radial residual score.","s":"ℝ → ℕ → ℝ → ℝ → ℝ","l":169,"f":"Causalean/Stat/Concentration/Covering/EuclideanRadialPolynomial/Score.lean"},{"n":"Causalean.Stat.Concentration.EuclideanRadialPolynomial.radialResidualScore_hasPolynomialL2Cover","k":"theorem","m":"Causalean.Stat.Concentration.Covering.EuclideanRadialPolynomial.Score","a":"Stat/Concentration/Covering/EuclideanRadialPolynomial","t":1,"d":"Covering certificate for a bounded finite-arm residual score. Given a measurable Euclidean location map, a finite family of arms, each measurable and bounded in absolute value by 1, a measurable response variable that is bounded in absolute value by R, together with a positive bandwidth q, a nonnegative annulus inner radius a, inner radius at most outer radius b, a positive polynomial-coefficient bound B, and a positive response bound R, then the score formed by multiplying an arm indicator, a radial monomial, and the residual of the response against a boxed radial-polynomial fit carries a positive-envelope uniform polynomial L²(Q) covering certificate — with envelope radialResidualScoreEnvelope b p B R — over every probability measure Q, including atomic ones.","s":"∀ {Ω : Type u} [inst : MeasurableSpace Ω] {A : Type v} [Fintype A] (d p : ℕ) (loc : Ω → Causalean.Stat.Concentration.EuclideanRadialPolynomial.EuclideanPoint d) (arm : A → Ω → ℝ) (response : Ω → ℝ) {q a b B R : ℝ} (j : Fin (p + 1)), Measurable loc → (∀ (s : A), Measurable (arm s)) → (∀ (s : A) (ω : Ω), |arm s ω| ≤ 1) → Measurable response → (∀ (ω : Ω), |response ω| ≤ R) → 0 < q → 0 ≤ a → a ≤ b → 0 < B → 0 < R → Causalean.Stat.Concentration.HasPolynomialL2Cover (Causalean.Stat.Concentration.EuclideanRadialPolynomial.radialResidualScore d p loc arm response q a b B j) (Causalean.Stat.Concentration.EuclideanRadialPolynomial.radialResidualScoreEnvelope b p B R)","l":186,"f":"Causalean/Stat/Concentration/Covering/EuclideanRadialPolynomial/Score.lean"},{"n":"Causalean.Stat.Concentration.EuclideanRadialPolynomial.radialResidualScore_hasUniformPolynomialL2CoverWith","k":"theorem","m":"Causalean.Stat.Concentration.Covering.EuclideanRadialPolynomial.Score","a":"Stat/Concentration/Covering/EuclideanRadialPolynomial","t":2,"d":"The radial residual-score construction admits entropy witnesses depending only on the Euclidean dimension, polynomial degree, and finite arm type. In particular, the witnesses precede all location maps, arm functions, responses, bandwidths, annuli, and envelope radii.","s":"∀ {Ω : Type u} [inst : MeasurableSpace Ω] {A : Type v} [Fintype A] [Nonempty A] (d p : ℕ), ∃ C n, ∀ (loc : Ω → Causalean.Stat.Concentration.EuclideanRadialPolynomial.EuclideanPoint d) (arm : A → Ω → ℝ) (response : Ω → ℝ) {q a b B R : ℝ}, Measurable loc → (∀ (s : A), Measurable (arm s)) → (∀ (s : A) (ω : Ω), |arm s ω| ≤ 1) → Measurable response → (∀ (ω : Ω), |response ω| ≤ R) → 0 < q → 0 ≤ a → a ≤ b → 0 < B → 0 < R → Causalean.Stat.Concentration.HasPolynomialL2CoverWith (fun θ => Causalean.Stat.Concentration.EuclideanRadialPolynomial.radialResidualScore d p loc arm response q a b B θ.2 θ.1) (Causalean.Stat.Concentration.EuclideanRadialPolynomial.radialResidualScoreEnvelope b p B R) C n","l":293,"f":"Causalean/Stat/Concentration/Covering/EuclideanRadialPolynomial/Score.lean"},{"n":"Causalean.Stat.Concentration.F_on","k":"def","m":"Causalean.Stat.Concentration.TailBounds.Massart","a":"Stat/Concentration/TailBounds","t":1,"d":"Restricting a function class to a finite index set yields the corresponding subtype-indexed class.","s":"{Z : Type v} → {ι : Type u} → (ι → Z → ℝ) → (f : Finset ι) → ↥f → Z → ℝ","l":178,"f":"Causalean/Stat/Concentration/TailBounds/Massart.lean"},{"n":"Causalean.Stat.Concentration.HasCountableEmpiricalSupReduction","k":"def","m":"Causalean.Stat.Concentration.VarianceAdaptiveVCExpectedMaximal.Separability","a":"Stat/Concentration/VarianceAdaptiveVCExpectedMaximal","t":1,"d":"A countable subfamily realizes the continuum empirical-process supremum almost surely under every finite product law.","s":"{Ω : Type u_1} → {ι : Type u_2} → [inst : MeasurableSpace Ω] → MeasureTheory.Measure Ω → (ι → Ω → ℝ) → Prop","l":31,"f":"Causalean/Stat/Concentration/VarianceAdaptiveVCExpectedMaximal/Separability.lean"},{"n":"Causalean.Stat.Concentration.HasPolynomialEmpiricalL2Cover","k":"def","m":"Causalean.Stat.Concentration.VarianceAdaptiveVCExpectedMaximal.Basic","a":"Stat/Concentration/VarianceAdaptiveVCExpectedMaximal","t":1,"d":"A class has empirical polynomial L² covering numbers with constants A and v when every positive-size finite empirical law admits a cover of relative radius ε with cardinality at most the real power (A / ε) ^ v.","s":"{𝒳 : Type u} → [MeasurableSpace 𝒳] → {ι : Type v} → (ι → 𝒳 → ℝ) → ℝ → ℝ → ℝ → Prop","l":23,"f":"Causalean/Stat/Concentration/VarianceAdaptiveVCExpectedMaximal/Basic.lean"},{"n":"Causalean.Stat.Concentration.HasPolynomialEmpiricalL2Cover.coveringNumber_le","k":"theorem","m":"Causalean.Stat.Concentration.VarianceAdaptiveVCExpectedMaximal.EmpiricalCover","a":"Stat/Concentration/VarianceAdaptiveVCExpectedMaximal","t":1,"d":"Covering-number bound from an empirical cover. Suppose F has polynomial empirical $L^2$ covering numbers with envelope U, base A, and exponent v, every member of F is measurable, the envelope U is strictly positive, the sample S has positive size n, and the relative scale ε lies in the interval $(0,1]$ ; then Dudley's empirical covering number of the class at radius ε * U, taken in the totally bounded empirical pseudometric space this cover furnishes, is at most the real power (A / ε) ^ v.","s":"∀ {𝒳 : Type u} [inst : MeasurableSpace 𝒳] {ι : Type v} {F : ι → 𝒳 → ℝ} {U A v : ℝ} (hcover : Causalean.Stat.Concentration.HasPolynomialEmpiricalL2Cover F U A v) (hmeas : ∀ (i : ι), Measurable (F i)) (hU : 0 < U) {n : ℕ} (S : Fin n → 𝒳) (hn : 0 < n) (ε : ℝ), 0 < ε → ε ≤ 1 → have htot := Causalean.Stat.Concentration.HasPolynomialEmpiricalL2Cover.totallyBounded hcover hmeas hU S hn; ↑(Causalean.Stat.Concentration.coveringNumber htot (ε * U)) ≤ (A / ε).rpow v","l":62,"f":"Causalean/Stat/Concentration/VarianceAdaptiveVCExpectedMaximal/EmpiricalCover.lean"},{"n":"Causalean.Stat.Concentration.HasPolynomialEmpiricalL2Cover.enlargeEnvelope","k":"theorem","m":"Causalean.Stat.Concentration.VarianceAdaptiveVCExpectedMaximal.Basic","a":"Stat/Concentration/VarianceAdaptiveVCExpectedMaximal","t":2,"d":"Enlarging the envelope of an empirical polynomial cover preserves its entropy witnesses.","s":"∀ {𝒳 : Type u} [inst : MeasurableSpace 𝒳] {ι : Type v} {F : ι → 𝒳 → ℝ} {U V A v : ℝ}, Causalean.Stat.Concentration.HasPolynomialEmpiricalL2Cover F U A v → U ≤ V → Causalean.Stat.Concentration.HasPolynomialEmpiricalL2Cover F V A v","l":34,"f":"Causalean/Stat/Concentration/VarianceAdaptiveVCExpectedMaximal/Basic.lean"},{"n":"Causalean.Stat.Concentration.HasPolynomialEmpiricalL2Cover.monoEnvelope","k":"theorem","m":"Causalean.Stat.Concentration.VarianceAdaptiveVCExpectedMaximal.Basic","a":"Stat/Concentration/VarianceAdaptiveVCExpectedMaximal","t":2,"d":"Enlarging the envelope in a polynomial empirical-cover certificate keeps the same centers, entropy base, and exponent.","s":"∀ {𝒳 : Type u} [inst : MeasurableSpace 𝒳] {ι : Type v} {F : ι → 𝒳 → ℝ} {U V A v : ℝ}, Causalean.Stat.Concentration.HasPolynomialEmpiricalL2Cover F U A v → U ≤ V → Causalean.Stat.Concentration.HasPolynomialEmpiricalL2Cover F V A v","l":116,"f":"Causalean/Stat/Concentration/VarianceAdaptiveVCExpectedMaximal/Basic.lean"},{"n":"Causalean.Stat.Concentration.HasPolynomialEmpiricalL2Cover.pullback","k":"theorem","m":"Causalean.Stat.Concentration.VarianceAdaptiveVCExpectedMaximal.Basic","a":"Stat/Concentration/VarianceAdaptiveVCExpectedMaximal","t":2,"d":"Restricting a measurable polynomial empirical-cover class along a nonempty parameter map preserves its exponent and costs only a factor two in the entropy base. The replacement centers are chosen inside the restricted class.","s":"∀ {𝒳 : Type u} [inst : MeasurableSpace 𝒳] {ι : Type v} {κ : Type w} [Nonempty κ] {F : ι → 𝒳 → ℝ} {U A v : ℝ}, Causalean.Stat.Concentration.HasPolynomialEmpiricalL2Cover F U A v → (∀ (i : ι), Measurable (F i)) → ∀ (e : κ → ι), Causalean.Stat.Concentration.HasPolynomialEmpiricalL2Cover (fun k => F (e k)) U (2 * A) v","l":48,"f":"Causalean/Stat/Concentration/VarianceAdaptiveVCExpectedMaximal/Basic.lean"},{"n":"Causalean.Stat.Concentration.HasPolynomialEmpiricalL2Cover.totallyBounded","k":"theorem","m":"Causalean.Stat.Concentration.VarianceAdaptiveVCExpectedMaximal.EmpiricalCover","a":"Stat/Concentration/VarianceAdaptiveVCExpectedMaximal","t":2,"d":"A positive-envelope polynomial empirical covering certificate makes the class image totally bounded in the empirical root-mean-square pseudometric on every nonempty finite sample.","s":"∀ {𝒳 : Type u} [inst : MeasurableSpace 𝒳] {ι : Type v} {F : ι → 𝒳 → ℝ} {U A v : ℝ}, Causalean.Stat.Concentration.HasPolynomialEmpiricalL2Cover F U A v → (∀ (i : ι), Measurable (F i)) → 0 < U → ∀ {n : ℕ} (S : Fin n → 𝒳), 0 < n → TotallyBounded Set.univ","l":21,"f":"Causalean/Stat/Concentration/VarianceAdaptiveVCExpectedMaximal/EmpiricalCover.lean"},{"n":"Causalean.Stat.Concentration.HasPolynomialL2Cover","k":"structure","m":"Causalean.Stat.Concentration.Covering.RealValuedVCSubgraph.Algebra","a":"Stat/Concentration/Covering/RealValuedVCSubgraph","t":1,"d":"A class of functions has uniform polynomial L² entropy with envelope U when U is positive, every function in the class is measurable and pointwise bounded in absolute value by U, and the class admits one polynomial covering-number bound, in the relative radius, holding simultaneously for every probability measure and every relative radius in (0,1] .","s":"{𝒳 : Type u} → [MeasurableSpace 𝒳] → {ι : Type v} → (ι → 𝒳 → ℝ) → ℝ → Prop","l":192,"f":"Causalean/Stat/Concentration/Covering/RealValuedVCSubgraph/Algebra.lean"},{"n":"Causalean.Stat.Concentration.HasPolynomialL2Cover.add","k":"theorem","m":"Causalean.Stat.Concentration.Covering.RealValuedVCSubgraph.Algebra","a":"Stat/Concentration/Covering/RealValuedVCSubgraph","t":2,"d":"Pointwise addition of two independently indexed polynomial-entropy classes has uniform polynomial L² entropy with summed envelope.","s":"∀ {𝒳 : Type u} [inst : MeasurableSpace 𝒳] {ι : Type v} {κ : Type w} {F : ι → 𝒳 → ℝ} {G : κ → 𝒳 → ℝ} {U V : ℝ}, Causalean.Stat.Concentration.HasPolynomialL2Cover F U → Causalean.Stat.Concentration.HasPolynomialL2Cover G V → Causalean.Stat.Concentration.HasPolynomialL2Cover (fun p x => F p.1 x + G p.2 x) (U + V)","l":270,"f":"Causalean/Stat/Concentration/Covering/RealValuedVCSubgraph/Algebra.lean"},{"n":"Causalean.Stat.Concentration.HasPolynomialL2Cover.finProd","k":"theorem","m":"Causalean.Stat.Concentration.Covering.RealValuedVCSubgraph.Algebra","a":"Stat/Concentration/Covering/RealValuedVCSubgraph","t":2,"d":"A fixed finite product of independently indexed polynomial-entropy classes again has uniform polynomial L² entropy, with the product envelope.","s":"∀ {𝒳 : Type u} [inst : MeasurableSpace 𝒳] {K : Type w} [inst_1 : Fintype K] [Nonempty K] {ι : K → Type v} {F : (k : K) → ι k → 𝒳 → ℝ} {U : K → ℝ}, (∀ (k : K), Causalean.Stat.Concentration.HasPolynomialL2Cover (F k) (U k)) → Causalean.Stat.Concentration.HasPolynomialL2Cover (fun θ x => ∏ k, F k (θ k) x) (∏ k, U k)","l":413,"f":"Causalean/Stat/Concentration/Covering/RealValuedVCSubgraph/Algebra.lean"},{"n":"Causalean.Stat.Concentration.HasPolynomialL2Cover.finSum","k":"theorem","m":"Causalean.Stat.Concentration.Covering.RealValuedVCSubgraph.Algebra","a":"Stat/Concentration/Covering/RealValuedVCSubgraph","t":2,"d":"A fixed finite sum of independently indexed polynomial-entropy classes again has uniform polynomial L² entropy, with the sum of the envelopes.","s":"∀ {𝒳 : Type u} [inst : MeasurableSpace 𝒳] {K : Type w} [inst_1 : Fintype K] [Nonempty K] {ι : K → Type v} {F : (k : K) → ι k → 𝒳 → ℝ} {U : K → ℝ}, (∀ (k : K), Causalean.Stat.Concentration.HasPolynomialL2Cover (F k) (U k)) → Causalean.Stat.Concentration.HasPolynomialL2Cover (fun θ x => ∑ k, F k (θ k) x) (∑ k, U k)","l":355,"f":"Causalean/Stat/Concentration/Covering/RealValuedVCSubgraph/Algebra.lean"},{"n":"Causalean.Stat.Concentration.HasPolynomialL2Cover.hasPolynomialEmpiricalL2Cover","k":"theorem","m":"Causalean.Stat.Concentration.VarianceAdaptiveVCExpectedMaximal.Basic","a":"Stat/Concentration/VarianceAdaptiveVCExpectedMaximal","t":1,"d":"Empirical covering from population covering. If a family F of real-valued functions with envelope U admits a uniform polynomial $L^2$ covering-number bound over every probability measure, then there exist a base A at least Euler's number and an exponent v at least one such that F has polynomial empirical $L^2$ covering numbers with envelope U, base A, and exponent v.","s":"∀ {𝒳 : Type u} [inst : MeasurableSpace 𝒳] {ι : Type v} {F : ι → 𝒳 → ℝ} {U : ℝ}, Causalean.Stat.Concentration.HasPolynomialL2Cover F U → ∃ A v, Real.exp 1 ≤ A ∧ 1 ≤ v ∧ Causalean.Stat.Concentration.HasPolynomialEmpiricalL2Cover F U A v","l":153,"f":"Causalean/Stat/Concentration/VarianceAdaptiveVCExpectedMaximal/Basic.lean"},{"n":"Causalean.Stat.Concentration.HasPolynomialL2Cover.mul","k":"theorem","m":"Causalean.Stat.Concentration.Covering.RealValuedVCSubgraph.Algebra","a":"Stat/Concentration/Covering/RealValuedVCSubgraph","t":2,"d":"Pointwise multiplication of two independently indexed bounded polynomial-entropy classes has uniform polynomial L² entropy with product envelope.","s":"∀ {𝒳 : Type u} [inst : MeasurableSpace 𝒳] {ι : Type v} {κ : Type w} {F : ι → 𝒳 → ℝ} {G : κ → 𝒳 → ℝ} {U V : ℝ}, Causalean.Stat.Concentration.HasPolynomialL2Cover F U → Causalean.Stat.Concentration.HasPolynomialL2Cover G V → Causalean.Stat.Concentration.HasPolynomialL2Cover (fun p x => F p.1 x * G p.2 x) (U * V)","l":311,"f":"Causalean/Stat/Concentration/Covering/RealValuedVCSubgraph/Algebra.lean"},{"n":"Causalean.Stat.Concentration.HasPolynomialL2Cover.mulIndicator","k":"theorem","m":"Causalean.Stat.Concentration.Covering.RealValuedVCSubgraph.Parametric","a":"Stat/Concentration/Covering/RealValuedVCSubgraph","t":2,"d":"Multiplying a polynomial-entropy real class by a measurable finite-VC family of indicators preserves uniform polynomial L² entropy.","s":"∀ {𝒳 : Type u} [inst : MeasurableSpace 𝒳] {ι : Type v} {κ : Type w} {F : ι → 𝒳 → ℝ} {π : κ → 𝒳 → Bool} {U : ℝ} {d : ℕ}, Causalean.Stat.Concentration.HasPolynomialL2Cover F U → (∀ (j : κ), MeasurableSet {x | π j x = true}) → Causalean.Stat.Concentration.HasVCAtMost π d → Causalean.Stat.Concentration.HasPolynomialL2Cover (fun p x => F p.1 x * if π p.2 x = true then 1 else 0) U","l":250,"f":"Causalean/Stat/Concentration/Covering/RealValuedVCSubgraph/Parametric.lean"},{"n":"Causalean.Stat.Concentration.HasPolynomialL2Cover.neg","k":"theorem","m":"Causalean.Stat.Concentration.Covering.RealValuedVCSubgraph.Algebra","a":"Stat/Concentration/Covering/RealValuedVCSubgraph","t":2,"d":"Negating every member of a polynomial-entropy class preserves its envelope and uniform polynomial L² entropy.","s":"∀ {𝒳 : Type u} [inst : MeasurableSpace 𝒳] {ι : Type v} {F : ι → 𝒳 → ℝ} {U : ℝ}, Causalean.Stat.Concentration.HasPolynomialL2Cover F U → Causalean.Stat.Concentration.HasPolynomialL2Cover (fun i x => -F i x) U","l":242,"f":"Causalean/Stat/Concentration/Covering/RealValuedVCSubgraph/Algebra.lean"},{"n":"Causalean.Stat.Concentration.HasPolynomialL2Cover.varianceAdaptiveExpectedMaximal_le","k":"theorem","m":"Causalean.Stat.Concentration.VarianceAdaptiveVCExpectedMaximal.ExpectedMaximal","a":"Stat/Concentration/VarianceAdaptiveVCExpectedMaximal","t":2,"d":"An existing HasPolynomialL2Cover certificate yields some explicit polynomial constants for which the countable class satisfies the variance-adaptive expected maximal inequality at every positive sample size.","s":"∀ {𝒳 : Type u} [inst : MeasurableSpace 𝒳] {ι : Type v} [Nonempty ι] [Countable ι] (P : MeasureTheory.Measure 𝒳) [MeasureTheory.IsProbabilityMeasure P] (F : ι → 𝒳 → ℝ) {U σ : ℝ}, Causalean.Stat.Concentration.HasPolynomialL2Cover F U → 0 < σ → σ < U → (∀ (i : ι), (Causalean.Stat.Concentration.measureL2Dist P (F i) fun x => 0) ≤ σ) → ∃ A v, Real.exp 1 ≤ A ∧ 1 ≤ v ∧ ∀ (n : ℕ), 0 < n → (∫ (S : Fin n → 𝒳), Causalean.Stat.Concentration.countableEmpiricalSup P F S ∂MeasureTheory.Measure.pi fun x => P) ≤ Causalean.Stat.Concentration.varianceAdaptiveVCConstant * Causalean.Stat.Concentration.vcExpectedMaximalRate U σ A v n","l":73,"f":"Causalean/Stat/Concentration/VarianceAdaptiveVCExpectedMaximal/ExpectedMaximal.lean"},{"n":"Causalean.Stat.Concentration.HasPolynomialL2CoverWith","k":"structure","m":"Causalean.Stat.Concentration.Covering.RealValuedVCSubgraph.Algebra","a":"Stat/Concentration/Covering/RealValuedVCSubgraph","t":1,"d":"A polynomial L² covering certificate with named entropy witnesses. Unlike HasPolynomialL2Cover, this form retains the particular base and integer exponent, so a family of constructions can share witnesses before its observation-specific parameters are introduced.","s":"{𝒳 : Type u} → [MeasurableSpace 𝒳] → {ι : Type v} → (ι → 𝒳 → ℝ) → ℝ → ℝ → ℕ → Prop","l":471,"f":"Causalean/Stat/Concentration/Covering/RealValuedVCSubgraph/Algebra.lean"},{"n":"Causalean.Stat.Concentration.HasPolynomialL2CoverWith.add","k":"theorem","m":"Causalean.Stat.Concentration.Covering.RealValuedVCSubgraph.Algebra","a":"Stat/Concentration/Covering/RealValuedVCSubgraph","t":2,"d":"Addition combines named polynomial-cover witnesses by the same explicit formula as the ordinary cover algebra.","s":"∀ {𝒳 : Type u} [inst : MeasurableSpace 𝒳] {ι : Type v} {κ : Type w} {F : ι → 𝒳 → ℝ} {G : κ → 𝒳 → ℝ} {U V A B : ℝ} {p q : ℕ}, Causalean.Stat.Concentration.HasPolynomialL2CoverWith F U A p → Causalean.Stat.Concentration.HasPolynomialL2CoverWith G V B q → Causalean.Stat.Concentration.HasPolynomialL2CoverWith (fun z x => F z.1 x + G z.2 x) (U + V) (8 * A * B) (p + q + 2)","l":566,"f":"Causalean/Stat/Concentration/Covering/RealValuedVCSubgraph/Algebra.lean"},{"n":"Causalean.Stat.Concentration.HasPolynomialL2CoverWith.enlargeEnvelope","k":"theorem","m":"Causalean.Stat.Concentration.Covering.RealValuedVCSubgraph.Algebra","a":"Stat/Concentration/Covering/RealValuedVCSubgraph","t":2,"d":"Enlarging a named envelope preserves its entropy witnesses.","s":"∀ {𝒳 : Type u} [inst : MeasurableSpace 𝒳] {ι : Type v} {F : ι → 𝒳 → ℝ} {U V A : ℝ} {p : ℕ}, Causalean.Stat.Concentration.HasPolynomialL2CoverWith F U A p → U ≤ V → Causalean.Stat.Concentration.HasPolynomialL2CoverWith F V A p","l":491,"f":"Causalean/Stat/Concentration/Covering/RealValuedVCSubgraph/Algebra.lean"},{"n":"Causalean.Stat.Concentration.HasPolynomialL2CoverWith.forget","k":"theorem","m":"Causalean.Stat.Concentration.Covering.RealValuedVCSubgraph.Algebra","a":"Stat/Concentration/Covering/RealValuedVCSubgraph","t":2,"d":"Forgetting the named witnesses gives the ordinary polynomial-cover certificate.","s":"∀ {𝒳 : Type u} [inst : MeasurableSpace 𝒳] {ι : Type v} {F : ι → 𝒳 → ℝ} {U A : ℝ} {p : ℕ}, Causalean.Stat.Concentration.HasPolynomialL2CoverWith F U A p → Causalean.Stat.Concentration.HasPolynomialL2Cover F U","l":483,"f":"Causalean/Stat/Concentration/Covering/RealValuedVCSubgraph/Algebra.lean"},{"n":"Causalean.Stat.Concentration.HasPolynomialL2CoverWith.hasPolynomialEmpiricalL2Cover","k":"theorem","m":"Causalean.Stat.Concentration.VarianceAdaptiveVCExpectedMaximal.Basic","a":"Stat/Concentration/VarianceAdaptiveVCExpectedMaximal","t":2,"d":"Named arbitrary-law entropy witnesses give correspondingly named empirical witnesses after the canonical Euler-base and positive-exponent normalization.","s":"∀ {𝒳 : Type u} [inst : MeasurableSpace 𝒳] {ι : Type v} {F : ι → 𝒳 → ℝ} {U A₀ : ℝ} {p : ℕ}, Causalean.Stat.Concentration.HasPolynomialL2CoverWith F U A₀ p → Causalean.Stat.Concentration.HasPolynomialEmpiricalL2Cover F U (max (Real.exp 1) (2 * A₀)) ↑(p + 1)","l":212,"f":"Causalean/Stat/Concentration/VarianceAdaptiveVCExpectedMaximal/Basic.lean"},{"n":"Causalean.Stat.Concentration.HasPolynomialL2CoverWith.mul","k":"theorem","m":"Causalean.Stat.Concentration.Covering.RealValuedVCSubgraph.Algebra","a":"Stat/Concentration/Covering/RealValuedVCSubgraph","t":2,"d":"Multiplication combines named polynomial-cover witnesses by the same explicit formula as the ordinary cover algebra.","s":"∀ {𝒳 : Type u} [inst : MeasurableSpace 𝒳] {ι : Type v} {κ : Type w} {F : ι → 𝒳 → ℝ} {G : κ → 𝒳 → ℝ} {U V A B : ℝ} {p q : ℕ}, Causalean.Stat.Concentration.HasPolynomialL2CoverWith F U A p → Causalean.Stat.Concentration.HasPolynomialL2CoverWith G V B q → Causalean.Stat.Concentration.HasPolynomialL2CoverWith (fun z x => F z.1 x * G z.2 x) (U * V) (8 * A * B) (p + q + 2)","l":609,"f":"Causalean/Stat/Concentration/Covering/RealValuedVCSubgraph/Algebra.lean"},{"n":"Causalean.Stat.Concentration.HasPolynomialL2CoverWith.neg","k":"theorem","m":"Causalean.Stat.Concentration.Covering.RealValuedVCSubgraph.Algebra","a":"Stat/Concentration/Covering/RealValuedVCSubgraph","t":2,"d":"Negation preserves named polynomial-cover witnesses.","s":"∀ {𝒳 : Type u} [inst : MeasurableSpace 𝒳] {ι : Type v} {F : ι → 𝒳 → ℝ} {U A : ℝ} {p : ℕ}, Causalean.Stat.Concentration.HasPolynomialL2CoverWith F U A p → Causalean.Stat.Concentration.HasPolynomialL2CoverWith (fun i x => -F i x) U A p","l":541,"f":"Causalean/Stat/Concentration/Covering/RealValuedVCSubgraph/Algebra.lean"},{"n":"Causalean.Stat.Concentration.HasPolynomialL2CoverWith.tightenEnvelopeBy","k":"theorem","m":"Causalean.Stat.Concentration.Covering.RealValuedVCSubgraph.Algebra","a":"Stat/Concentration/Covering/RealValuedVCSubgraph","t":2,"d":"If an assembled class has envelope c * V but a direct pointwise bound by V, shrinking the envelope by the fixed factor c costs the same factor in the named entropy base.","s":"∀ {𝒳 : Type u} [inst : MeasurableSpace 𝒳] {ι : Type v} {F : ι → 𝒳 → ℝ} {V A c : ℝ} {p : ℕ}, Causalean.Stat.Concentration.HasPolynomialL2CoverWith F (c * V) A p → 1 ≤ c → 0 < V → (∀ (i : ι) (x : 𝒳), |F i x| ≤ V) → Causalean.Stat.Concentration.HasPolynomialL2CoverWith F V (c * A) p","l":655,"f":"Causalean/Stat/Concentration/Covering/RealValuedVCSubgraph/Algebra.lean"},{"n":"Causalean.Stat.Concentration.HasPseudoDimAtMost","k":"def","m":"Causalean.Stat.Concentration.Covering.RealValuedVCSubgraph.Basic","a":"Stat/Concentration/Covering/RealValuedVCSubgraph","t":1,"d":"A real-valued class has pseudo-dimension at most d when every finite trace of its strict subgraphs has VC dimension at most d.","s":"{𝒳 : Type u} → {ι : Type v} → (ι → 𝒳 → ℝ) → ℕ → Prop","l":26,"f":"Causalean/Stat/Concentration/Covering/RealValuedVCSubgraph/Basic.lean"},{"n":"Causalean.Stat.Concentration.HasPseudoDimAtMost.hasPolynomialL2Cover","k":"theorem","m":"Causalean.Stat.Concentration.Covering.RealValuedVCSubgraph.Algebra","a":"Stat/Concentration/Covering/RealValuedVCSubgraph","t":1,"d":"Bounded finite-pseudo-dimension classes admit a polynomial L² cover. If a real-valued function class has pseudo-dimension at most d, every member is measurable, and the class is uniformly bounded by a positive envelope U, then the class carries a uniform polynomial L² covering certificate at envelope U.","s":"∀ {𝒳 : Type u} [inst : MeasurableSpace 𝒳] {ι : Type v} [Nonempty ι] {F : ι → 𝒳 → ℝ} {d : ℕ} {U : ℝ}, Causalean.Stat.Concentration.HasPseudoDimAtMost F d → (∀ (i : ι), Measurable (F i)) → 0 < U → (∀ (i : ι) (x : 𝒳), |F i x| ≤ U) → Causalean.Stat.Concentration.HasPolynomialL2Cover F U","l":226,"f":"Causalean/Stat/Concentration/Covering/RealValuedVCSubgraph/Algebra.lean"},{"n":"Causalean.Stat.Concentration.HasPseudoDimAtMost.hasPolynomialL2CoverWith","k":"theorem","m":"Causalean.Stat.Concentration.Covering.RealValuedVCSubgraph.Algebra","a":"Stat/Concentration/Covering/RealValuedVCSubgraph","t":2,"d":"A bounded measurable pseudo-dimension class has the canonical named entropy witnesses used by the VC-subgraph covering theorem.","s":"∀ {𝒳 : Type u} [inst : MeasurableSpace 𝒳] {ι : Type v} [Nonempty ι] {F : ι → 𝒳 → ℝ} {d : ℕ} {U : ℝ}, Causalean.Stat.Concentration.HasPseudoDimAtMost F d → (∀ (i : ι), Measurable (F i)) → 0 < U → (∀ (i : ι) (x : 𝒳), |F i x| ≤ U) → Causalean.Stat.Concentration.HasPolynomialL2CoverWith F U 16 (8 * (d + 1))","l":511,"f":"Causalean/Stat/Concentration/Covering/RealValuedVCSubgraph/Algebra.lean"},{"n":"Causalean.Stat.Concentration.HasSubexponentialMGF","k":"structure","m":"Causalean.Stat.Concentration.TailBounds.SubExponential","a":"Stat/Concentration/TailBounds","t":1,"d":"A random variable X has a sub-exponential moment-generating function with parameters (v, b) with respect to μ if, for every t with b · |t| < 1, exp (t * X) is μ-integrable and the moment-generating function obeys mgf X μ t ≤ exp (v t² / 2).","s":"{Ω : Type u_1} → {mΩ : MeasurableSpace Ω} → (Ω → ℝ) → NNReal → NNReal → autoParam (MeasureTheory.Measure Ω) Causalean.Stat.Concentration.HasSubexponentialMGF._auto_1 → Prop","l":46,"f":"Causalean/Stat/Concentration/TailBounds/SubExponential.lean"},{"n":"Causalean.Stat.Concentration.HasSubexponentialMGF.add_of_indepFun","k":"theorem","m":"Causalean.Stat.Concentration.TailBounds.SubExponential","a":"Stat/Concentration/TailBounds","t":2,"d":"Independent sub-exponential variables add: X + Y has parameters (vX + vY, max bX bY).","s":"∀ {Ω : Type u_1} {mΩ : MeasurableSpace Ω} {μ : MeasureTheory.Measure Ω} {X Y : Ω → ℝ} {vX bX vY bY : NNReal}, Causalean.Stat.Concentration.HasSubexponentialMGF X vX bX μ → Causalean.Stat.Concentration.HasSubexponentialMGF Y vY bY μ → ProbabilityTheory.IndepFun X Y μ → Causalean.Stat.Concentration.HasSubexponentialMGF (fun ω => X ω + Y ω) (vX + vY) (max bX bY) μ","l":114,"f":"Causalean/Stat/Concentration/TailBounds/SubExponential.lean"},{"n":"Causalean.Stat.Concentration.HasSubexponentialMGF.congr","k":"theorem","m":"Causalean.Stat.Concentration.TailBounds.SubExponential","a":"Stat/Concentration/TailBounds","t":2,"d":"Sub-exponentiality transfers along an a.e.-equality.","s":"∀ {Ω : Type u_1} {mΩ : MeasurableSpace Ω} {μ : MeasureTheory.Measure Ω} {X : Ω → ℝ} {v b : NNReal}, Causalean.Stat.Concentration.HasSubexponentialMGF X v b μ → ∀ {Y : Ω → ℝ}, X =ᵐ[μ] Y → Causalean.Stat.Concentration.HasSubexponentialMGF Y v b μ","l":84,"f":"Causalean/Stat/Concentration/TailBounds/SubExponential.lean"},{"n":"Causalean.Stat.Concentration.HasSubexponentialMGF.const_mul","k":"theorem","m":"Causalean.Stat.Concentration.TailBounds.SubExponential","a":"Stat/Concentration/TailBounds","t":2,"d":"Sub-exponentiality is preserved under scaling: r • X has parameters (r² v, |r| b).","s":"∀ {Ω : Type u_1} {mΩ : MeasurableSpace Ω} {μ : MeasureTheory.Measure Ω} {X : Ω → ℝ} {v b : NNReal}, Causalean.Stat.Concentration.HasSubexponentialMGF X v b μ → ∀ (r : ℝ), Causalean.Stat.Concentration.HasSubexponentialMGF (fun ω => r * X ω) (⟨r ^ 2, sq_nonneg r⟩ * v) (⟨|r|, abs_nonneg r⟩ * b) μ","l":94,"f":"Causalean/Stat/Concentration/TailBounds/SubExponential.lean"},{"n":"Causalean.Stat.Concentration.HasSubexponentialMGF.isFiniteMeasure","k":"theorem","m":"Causalean.Stat.Concentration.TailBounds.SubExponential","a":"Stat/Concentration/TailBounds","t":2,"d":"A sub-exponential random variable lives on a finite measure (integrability of exp (0 · X) = 1).","s":"∀ {Ω : Type u_1} {mΩ : MeasurableSpace Ω} {μ : MeasureTheory.Measure Ω} {X : Ω → ℝ} {v b : NNReal}, Causalean.Stat.Concentration.HasSubexponentialMGF X v b μ → MeasureTheory.IsFiniteMeasure μ","l":62,"f":"Causalean/Stat/Concentration/TailBounds/SubExponential.lean"},{"n":"Causalean.Stat.Concentration.HasSubexponentialMGF.measure_abs_ge_le","k":"theorem","m":"Causalean.Stat.Concentration.TailBounds.MaximalInequality","a":"Stat/Concentration/TailBounds","t":1,"d":"Two-sided Chernoff bound. If the random variable X has a sub-exponential moment-generating function with parameters (v, b) with respect to μ and ε is nonnegative, then the probability that |X| is at least ε is at most $2\\exp(-ε^2/(2(v+bε)))$.","s":"∀ {Ω : Type u_1} {mΩ : MeasurableSpace Ω} {μ : MeasureTheory.Measure Ω} {X : Ω → ℝ} {v b : NNReal}, Causalean.Stat.Concentration.HasSubexponentialMGF X v b μ → ∀ {ε : ℝ}, 0 ≤ ε → μ.real {ω | ε ≤ |X ω|} ≤ 2 * Real.exp (-ε ^ 2 / (2 * (↑v + ↑b * ε)))","l":44,"f":"Causalean/Stat/Concentration/TailBounds/MaximalInequality.lean"},{"n":"Causalean.Stat.Concentration.HasSubexponentialMGF.measure_ge_le","k":"theorem","m":"Causalean.Stat.Concentration.TailBounds.SubExponential","a":"Stat/Concentration/TailBounds","t":1,"d":"Chernoff bound for the right tail of a sub-exponential random variable. If X has a sub-exponential moment-generating function with parameters (v, b) with respect to μ and ε is nonnegative, then the probability that X is at least ε is at most $\\exp(-ε^2/(2(v+bε)))$.","s":"∀ {Ω : Type u_1} {mΩ : MeasurableSpace Ω} {μ : MeasureTheory.Measure Ω} {X : Ω → ℝ} {v b : NNReal}, Causalean.Stat.Concentration.HasSubexponentialMGF X v b μ → ∀ {ε : ℝ}, 0 ≤ ε → μ.real {ω | ε ≤ X ω} ≤ Real.exp (-ε ^ 2 / (2 * (↑v + ↑b * ε)))","l":154,"f":"Causalean/Stat/Concentration/TailBounds/SubExponential.lean"},{"n":"Causalean.Stat.Concentration.HasSubexponentialMGF.neg","k":"theorem","m":"Causalean.Stat.Concentration.TailBounds.SubExponential","a":"Stat/Concentration/TailBounds","t":2,"d":"Sub-exponentiality is preserved under negation (same parameters).","s":"∀ {Ω : Type u_1} {mΩ : MeasurableSpace Ω} {μ : MeasureTheory.Measure Ω} {X : Ω → ℝ} {v b : NNReal}, Causalean.Stat.Concentration.HasSubexponentialMGF X v b μ → Causalean.Stat.Concentration.HasSubexponentialMGF (fun ω => -X ω) v b μ","l":69,"f":"Causalean/Stat/Concentration/TailBounds/SubExponential.lean"},{"n":"Causalean.Stat.Concentration.HasSubexponentialMGF.sum_range_of_iIndepFun","k":"theorem","m":"Causalean.Stat.Concentration.TailBounds.SubExponential","a":"Stat/Concentration/TailBounds","t":2,"d":"A sum of n independent sub-exponential variables sharing common parameters (v, b) is sub-exponential with parameters (n • v, b) (the b-parameters coincide, so they do not grow).","s":"∀ {Ω : Type u_1} {mΩ : MeasurableSpace Ω} {μ : MeasureTheory.Measure Ω} {Z : ℕ → Ω → ℝ}, ProbabilityTheory.iIndepFun Z μ → (∀ (i : ℕ), AEMeasurable (Z i) μ) → ∀ {v b : NNReal} {n : ℕ}, (∀ i < n, Causalean.Stat.Concentration.HasSubexponentialMGF (Z i) v b μ) → Causalean.Stat.Concentration.HasSubexponentialMGF (fun ω => ∑ i ∈ Finset.range n, Z i ω) (n • v) b μ","l":202,"f":"Causalean/Stat/Concentration/TailBounds/SubExponential.lean"},{"n":"Causalean.Stat.Concentration.HasSubexponentialMGF.zero","k":"theorem","m":"Causalean.Stat.Concentration.TailBounds.SubExponential","a":"Stat/Concentration/TailBounds","t":2,"d":"The constant 0 is sub-exponential with parameters (0, b) (any b).","s":"∀ {Ω : Type u_1} {mΩ : MeasurableSpace Ω} {μ : MeasureTheory.Measure Ω} {b : NNReal} [MeasureTheory.IsProbabilityMeasure μ], Causalean.Stat.Concentration.HasSubexponentialMGF (fun x => 0) 0 b μ","l":148,"f":"Causalean/Stat/Concentration/TailBounds/SubExponential.lean"},{"n":"Causalean.Stat.Concentration.HasUniformPolynomialL2CoverOver","k":"def","m":"Causalean.Stat.Concentration.Covering.RealValuedVCSubgraph.Algebra","a":"Stat/Concentration/Covering/RealValuedVCSubgraph","t":1,"d":"A collection has polynomial-cover witnesses uniform over an auxiliary parameter type when the named base and exponent precede that parameter.","s":"{𝒳 : Type u} → [MeasurableSpace 𝒳] → (S : Type u_1) → {ι : S → Type v} → ((s : S) → ι s → 𝒳 → ℝ) → (S → ℝ) → Prop","l":686,"f":"Causalean/Stat/Concentration/Covering/RealValuedVCSubgraph/Algebra.lean"},{"n":"Causalean.Stat.Concentration.HasUniformPolynomialL2CoverOver.add","k":"theorem","m":"Causalean.Stat.Concentration.Covering.RealValuedVCSubgraph.Algebra","a":"Stat/Concentration/Covering/RealValuedVCSubgraph","t":2,"d":"Uniform named covers are stable under pointwise addition.","s":"∀ {𝒳 : Type u} [inst : MeasurableSpace 𝒳] {S : Type u_1} {ι : S → Type v} {κ : S → Type w} {F : (s : S) → ι s → 𝒳 → ℝ} {G : (s : S) → κ s → 𝒳 → ℝ} {U V : S → ℝ}, Causalean.Stat.Concentration.HasUniformPolynomialL2CoverOver S F U → Causalean.Stat.Concentration.HasUniformPolynomialL2CoverOver S G V → Causalean.Stat.Concentration.HasUniformPolynomialL2CoverOver S (fun s z x => F s z.1 x + G s z.2 x) fun s => U s + V s","l":703,"f":"Causalean/Stat/Concentration/Covering/RealValuedVCSubgraph/Algebra.lean"},{"n":"Causalean.Stat.Concentration.HasUniformPolynomialL2CoverOver.finSum","k":"theorem","m":"Causalean.Stat.Concentration.Covering.RealValuedVCSubgraph.Algebra","a":"Stat/Concentration/Covering/RealValuedVCSubgraph","t":2,"d":"A nonempty finite sum preserves entropy witnesses uniformly over all auxiliary parameters.","s":"∀ {𝒳 : Type u} [inst : MeasurableSpace 𝒳] {S : Type u_1} {K : Type w} [inst_1 : Fintype K] [Nonempty K] {ι : S → K → Type v} {F : (s : S) → (k : K) → ι s k → 𝒳 → ℝ} {U : S → K → ℝ}, (∀ (k : K), Causalean.Stat.Concentration.HasUniformPolynomialL2CoverOver S (fun s => F s k) fun s => U s k) → Causalean.Stat.Concentration.HasUniformPolynomialL2CoverOver S (fun s θ x => ∑ k, F s k (θ k) x) fun s => ∑ k, U s k","l":731,"f":"Causalean/Stat/Concentration/Covering/RealValuedVCSubgraph/Algebra.lean"},{"n":"Causalean.Stat.Concentration.HasUniformPolynomialL2CoverOver.mul","k":"theorem","m":"Causalean.Stat.Concentration.Covering.RealValuedVCSubgraph.Algebra","a":"Stat/Concentration/Covering/RealValuedVCSubgraph","t":2,"d":"Uniform named covers are stable under pointwise multiplication.","s":"∀ {𝒳 : Type u} [inst : MeasurableSpace 𝒳] {S : Type u_1} {ι : S → Type v} {κ : S → Type w} {F : (s : S) → ι s → 𝒳 → ℝ} {G : (s : S) → κ s → 𝒳 → ℝ} {U V : S → ℝ}, Causalean.Stat.Concentration.HasUniformPolynomialL2CoverOver S F U → Causalean.Stat.Concentration.HasUniformPolynomialL2CoverOver S G V → Causalean.Stat.Concentration.HasUniformPolynomialL2CoverOver S (fun s z x => F s z.1 x * G s z.2 x) fun s => U s * V s","l":717,"f":"Causalean/Stat/Concentration/Covering/RealValuedVCSubgraph/Algebra.lean"},{"n":"Causalean.Stat.Concentration.HasUniformPolynomialL2CoverOver.neg","k":"theorem","m":"Causalean.Stat.Concentration.Covering.RealValuedVCSubgraph.Algebra","a":"Stat/Concentration/Covering/RealValuedVCSubgraph","t":2,"d":"Uniform named covers are stable under pointwise negation.","s":"∀ {𝒳 : Type u} [inst : MeasurableSpace 𝒳] {S : Type u_1} {ι : S → Type v} {F : (s : S) → ι s → 𝒳 → ℝ} {U : S → ℝ}, Causalean.Stat.Concentration.HasUniformPolynomialL2CoverOver S F U → Causalean.Stat.Concentration.HasUniformPolynomialL2CoverOver S (fun s i x => -F s i x) U","l":693,"f":"Causalean/Stat/Concentration/Covering/RealValuedVCSubgraph/Algebra.lean"},{"n":"Causalean.Stat.Concentration.HasVCAtMost","k":"def","m":"Causalean.Stat.Concentration.Covering.RealValuedVCSubgraph.Parametric","a":"Stat/Concentration/Covering/RealValuedVCSubgraph","t":1,"d":"A Boolean family has VC dimension at most d when every finite sample growth family has VC dimension at most d.","s":"{𝒳 : Type u} → {κ : Type w} → (κ → 𝒳 → Bool) → ℕ → Prop","l":169,"f":"Causalean/Stat/Concentration/Covering/RealValuedVCSubgraph/Parametric.lean"},{"n":"Causalean.Stat.Concentration.HasVCAtMost.indicatorClass_hasPseudoDimAtMost","k":"theorem","m":"Causalean.Stat.Concentration.Covering.RealValuedVCSubgraph.Parametric","a":"Stat/Concentration/Covering/RealValuedVCSubgraph","t":2,"d":"A Boolean VC class, viewed as a real-valued zero-one indicator class, has the same pseudo-dimension bound.","s":"∀ {𝒳 : Type u} [MeasurableSpace 𝒳] {κ : Type w} (π : κ → 𝒳 → Bool) (d : ℕ), Causalean.Stat.Concentration.HasVCAtMost π d → Causalean.Stat.Concentration.HasPseudoDimAtMost (fun j x => if π j x = true then 1 else 0) d","l":174,"f":"Causalean/Stat/Concentration/Covering/RealValuedVCSubgraph/Parametric.lean"},{"n":"Causalean.Stat.Concentration.HasVCUniformEntropy","k":"def","m":"Causalean.Stat.Concentration.VarianceAdaptiveVCExpectedMaximal.EntropyChaining","a":"Stat/Concentration/VarianceAdaptiveVCExpectedMaximal","t":1,"d":"A function class has uniform VC-type entropy when it has a measurable uniform envelope, a strict population L2 radius, and polynomial empirical L2 covers along every countable enumeration.","s":"{Ω : Type u_1} → {ι : Type u_2} → [inst : MeasurableSpace Ω] → MeasureTheory.Measure Ω → (ι → Ω → ℝ) → ℝ → ℝ → ℝ → ℝ → Prop","l":23,"f":"Causalean/Stat/Concentration/VarianceAdaptiveVCExpectedMaximal/EntropyChaining.lean"},{"n":"Causalean.Stat.Concentration.IsL2Cover","k":"def","m":"Causalean.Stat.Concentration.Covering.RealValuedVCSubgraph.Basic","a":"Stat/Concentration/Covering/RealValuedVCSubgraph","t":1,"d":"A finite set of class indices is an open L²(Q) cover at radius r when every class member is within distance strictly less than r of one center.","s":"{𝒳 : Type u} → {ι : Type v} → [inst : MeasurableSpace 𝒳] → MeasureTheory.Measure 𝒳 → (ι → 𝒳 → ℝ) → ℝ → Finset ι → Prop","l":44,"f":"Causalean/Stat/Concentration/Covering/RealValuedVCSubgraph/Basic.lean"},{"n":"Causalean.Stat.Concentration.L2CoveringNumberLe","k":"def","m":"Causalean.Stat.Concentration.Covering.RealValuedVCSubgraph.Basic","a":"Stat/Concentration/Covering/RealValuedVCSubgraph","t":1,"d":"The L²(Q) covering number is at most N when an index-valued cover with at most N centers exists.","s":"{𝒳 : Type u} → {ι : Type v} → [inst : MeasurableSpace 𝒳] → MeasureTheory.Measure 𝒳 → (ι → 𝒳 → ℝ) → ℝ → ℕ → Prop","l":49,"f":"Causalean/Stat/Concentration/Covering/RealValuedVCSubgraph/Basic.lean"},{"n":"Causalean.Stat.Concentration.LipschitzAt0","k":"def","m":"Causalean.Stat.Concentration.Rademacher.Contraction","a":"Stat/Concentration/Rademacher","t":1,"d":"A real-valued function φ is \"Lipschitz at 0\" with constant L if φ 0 = 0 and |φ x - φ y| ≤ L |x - y| for all x, y. The \"at 0\" pin-down is used by the absolute-value contraction theorem; the signed theorem needs only the global Lipschitz inequality.","s":"(ℝ → ℝ) → ℝ → Prop","l":56,"f":"Causalean/Stat/Concentration/Rademacher/Contraction.lean"},{"n":"Causalean.Stat.Concentration.LocalizedRegime","k":"structure","m":"Causalean.Stat.Concentration.UniformDeviation.UniformDeviationLocalized","a":"Stat/Concentration/UniformDeviation","t":1,"d":"The localized regime packages the assumptions needed for a critical-radius uniform-deviation bound over a bounded loss class: a non-negative uniform bound b such that every loss in the class is bounded in absolute value by b on the sample, together with a sub-root function ψ that upper-bounds the localized Rademacher complexity of the class at every sample size.","s":"(Ω : Type u_1) → (ι : Type u_2) → (𝒳 : Type u_3) → [inst : MeasurableSpace Ω] → (ι → 𝒳 → ℝ) → ((𝒳 → ℝ) → ℝ) → MeasureTheory.Measure Ω → (Ω → 𝒳) → Type","l":59,"f":"Causalean/Stat/Concentration/UniformDeviation/UniformDeviationLocalized.lean"},{"n":"Causalean.Stat.Concentration.LocalizedVCDudleyHypotheses","k":"structure","m":"Causalean.Stat.Concentration.Covering.VCLocalizedRegime","a":"Stat/Concentration/Covering","t":1,"d":"The deterministic prerequisites used to run the Dudley entropy-integral step after localization: a samplewise $L^2$ radius bound on the localized star-hull-zeroed class, needed to run Dudley with $c = r$, and a total-boundedness (covering-number) precondition on that same localized class.","s":"{ι : Type u} → {𝒳 : Type v} → (ι → 𝒳 → ℝ) → ((𝒳 → ℝ) → ℝ) → Prop","l":54,"f":"Causalean/Stat/Concentration/Covering/VCLocalizedRegime.lean"},{"n":"Causalean.Stat.Concentration.MassartNotation.X","k":"def","m":"Causalean.Stat.Concentration.TailBounds.Massart","a":"Stat/Concentration/TailBounds","t":1,"d":"The aggregated Rademacher variable for one class index is the sum of its coordinate increments.","s":"{Z : Type v} → {m : ℕ} → {ι : Type u} → (ι → Z → ℝ) → (Fin m → Z) → ι → Signs m → ℝ","l":142,"f":"Causalean/Stat/Concentration/TailBounds/Massart.lean"},{"n":"Causalean.Stat.Concentration.MassartNotation.Y","k":"def","m":"Causalean.Stat.Concentration.TailBounds.Massart","a":"Stat/Concentration/TailBounds","t":1,"d":"A Rademacher increment is the signed, sample-scaled value of one function at one sample coordinate.","s":"{Z : Type v} → {m : ℕ} → {ι : Type u} → (ι → Z → ℝ) → (Fin m → Z) → Fin m → ι → Signs m → ℝ","l":136,"f":"Causalean/Stat/Concentration/TailBounds/Massart.lean"},{"n":"Causalean.Stat.Concentration.MassartNotation.r","k":"def","m":"Causalean.Stat.Concentration.TailBounds.Massart","a":"Stat/Concentration/TailBounds","t":1,"d":"The finite-class coordinate envelope is the sample-scaled supremum absolute value over a finite index set.","s":"{Z : Type v} → {m : ℕ} → {ι : Type u} → (ι → Z → ℝ) → (Fin m → Z) → (f : Finset ι) → f.Nonempty → Fin m → ℝ","l":148,"f":"Causalean/Stat/Concentration/TailBounds/Massart.lean"},{"n":"Causalean.Stat.Concentration.MassartNotation.r'","k":"def","m":"Causalean.Stat.Concentration.TailBounds.Massart","a":"Stat/Concentration/TailBounds","t":1,"d":"The pointwise coordinate radius is the sample-scaled absolute value for one index and one sample coordinate.","s":"{Z : Type v} → {m : ℕ} → {ι : Type u} → (ι → Z → ℝ) → (Fin m → Z) → Fin m → ι → ℝ","l":153,"f":"Causalean/Stat/Concentration/TailBounds/Massart.lean"},{"n":"Causalean.Stat.Concentration.MassartNotation.xy_identity","k":"theorem","m":"Causalean.Stat.Concentration.TailBounds.Massart","a":"Stat/Concentration/TailBounds","t":2,"d":"The aggregate Rademacher variable is exactly the sum of its coordinate increments.","s":"∀ {Z : Type v} {m : ℕ} {ι : Type u} (F : ι → Z → ℝ) (S : Fin m → Z) (j : ι), Causalean.Stat.Concentration.MassartNotation.X F S j = ∑ i, Causalean.Stat.Concentration.MassartNotation.Y F S i j","l":160,"f":"Causalean/Stat/Concentration/TailBounds/Massart.lean"},{"n":"Causalean.Stat.Concentration.MonotoneOn.leftRiemann_sum_le_integral_antitoneOn","k":"theorem","m":"Causalean.Stat.Concentration.Covering.DudleyEntropy","a":"Stat/Concentration/Covering","t":2,"d":"For a monotone grid and an antitone integrand, a left Riemann term is bounded by the corresponding interval integral.","s":"∀ (n : ℕ) (f : ℕ → ℝ) (g : ℝ → ℝ), Monotone f → AntitoneOn g (Set.Icc (f 0) (f n)) → ∀ (j : Fin n), (f (↑j + 1) - f ↑j) * g (f (↑j + 1)) ≤ ∫ (x : ℝ) in f ↑j..f (↑j + 1), g x","l":1536,"f":"Causalean/Stat/Concentration/Covering/DudleyEntropy.lean"},{"n":"Causalean.Stat.Concentration.ProbabilityTheory.Kernel.iIndepFun.ae_iIndepFun_real","k":"theorem","m":"Causalean.Stat.Concentration.ConditionalKernel","a":"Stat/Concentration","t":2,"d":"Kernel independence of a finite family of measurable real random variables specializes to ordinary independence on almost every probability fiber. The countable rational half-line generator is what permits one common outer null set for all measurable events.","s":"∀ {α : Type u_1} {Ω : Type u_2} {ι : Type u_3} [inst : MeasurableSpace α] [inst_1 : MeasurableSpace Ω] {κ : ProbabilityTheory.Kernel α Ω} {ν : MeasureTheory.Measure α} [Finite ι] {X : ι → Ω → ℝ}, (∀ (i : ι), Measurable (X i)) → ProbabilityTheory.Kernel.iIndepFun X κ ν → ∀ᵐ (a : α) ∂ν, ProbabilityTheory.iIndepFun X (κ a)","l":28,"f":"Causalean/Stat/Concentration/ConditionalKernel.lean"},{"n":"Causalean.Stat.Concentration.RademacherUpperBound","k":"def","m":"Causalean.Stat.Concentration.UniformDeviation.CriticalRadius","a":"Stat/Concentration/UniformDeviation","t":1,"d":"Deterministic upper envelope on the localized Rademacher complexity. For every radius r ≥ 0, the population Rademacher complexity of the zero-out family on the radius-r star-hull ball is at most ψ r.","s":"{Ω : Type u_1} → {ι : Type u_2} → {𝒳 : Type u_3} → [inst : MeasurableSpace Ω] → (ι → 𝒳 → ℝ) → ((𝒳 → ℝ) → ℝ) → MeasureTheory.Measure Ω → (Ω → 𝒳) → ℕ → (ℝ → ℝ) → Prop","l":139,"f":"Causalean/Stat/Concentration/UniformDeviation/CriticalRadius.lean"},{"n":"Causalean.Stat.Concentration.SubRoot","k":"def","m":"Causalean.Stat.Concentration.UniformDeviation.CriticalRadius","a":"Stat/Concentration/UniformDeviation","t":1,"d":"A radius envelope is sub-root when it is non-negative and non-decreasing on non-negative radii, and its value divided by the radius is non-increasing as the positive radius grows.","s":"(ℝ → ℝ) → Prop","l":185,"f":"Causalean/Stat/Concentration/UniformDeviation/CriticalRadius.lean"},{"n":"Causalean.Stat.Concentration.absInner_bddAbove","k":"theorem","m":"Causalean.Stat.Concentration.Rademacher.Contraction","a":"Stat/Concentration/Rademacher","t":2,"d":"Signed empirical averages of a uniformly bounded function class are bounded above by the common absolute bound.","s":"∀ {ι : Type u_1} {𝒳 : Type u_2} (H : ι → 𝒳 → ℝ) {M : ℝ}, 0 ≤ M → (∀ (i : ι) (x : 𝒳), |H i x| ≤ M) → ∀ (n : ℕ) (S : Fin n → 𝒳) (σ : Signs n), BddAbove (Set.range fun i => |(↑n)⁻¹ * ∑ k, ↑↑(σ k) * H i (S k)|)","l":803,"f":"Causalean/Stat/Concentration/Rademacher/Contraction.lean"},{"n":"Causalean.Stat.Concentration.absInner_le_of_bound","k":"theorem","m":"Causalean.Stat.Concentration.Rademacher.Contraction","a":"Stat/Concentration/Rademacher","t":2,"d":"A signed empirical average of a uniformly bounded function has absolute value no larger than the same uniform bound.","s":"∀ {ι : Type u_1} {𝒳 : Type u_2} (H : ι → 𝒳 → ℝ) {M : ℝ}, 0 ≤ M → (∀ (i : ι) (x : 𝒳), |H i x| ≤ M) → ∀ (n : ℕ) (S : Fin n → 𝒳) (σ : Signs n) (i : ι), |(↑n)⁻¹ * ∑ k, ↑↑(σ k) * H i (S k)| ≤ M","l":779,"f":"Causalean/Stat/Concentration/Rademacher/Contraction.lean"},{"n":"Causalean.Stat.Concentration.abs_inner_starHullZeroOut_mono","k":"theorem","m":"Causalean.Stat.Concentration.Rademacher.LocalRademacher","a":"Stat/Concentration/Rademacher","t":2,"d":"Per-(p, σ, S) zero-out inner-expression bound. For r₁ ≤ r₂, the inner expression of empiricalRademacherComplexity for the zero-out family at r₁ is dominated in absolute value by that at r₂. This is the precise ingredient the helper above consumes for localRademacherComplexity_mono_r.","s":"∀ {ι : Type u_2} {𝒳 : Type u_3} (F : ι → 𝒳 → ℝ) (norm : (𝒳 → ℝ) → ℝ) {r₁ r₂ : ℝ}, r₁ ≤ r₂ → ∀ {n : ℕ} (S : Fin n → 𝒳) (σ : Signs n) (p : Causalean.Stat.Concentration.starHullParam ι), |(↑n)⁻¹ * ∑ k, ↑↑(σ k) * Causalean.Stat.Concentration.starHullZeroOut F norm r₁ p (S k)| ≤ |(↑n)⁻¹ * ∑ k, ↑↑(σ k) * Causalean.Stat.Concentration.starHullZeroOut F norm r₂ p (S k)|","l":161,"f":"Causalean/Stat/Concentration/Rademacher/LocalRademacher.lean"},{"n":"Causalean.Stat.Concentration.abs_starHullZeroOut_le_bound","k":"theorem","m":"Causalean.Stat.Concentration.UniformDeviation.CriticalRadius","a":"Stat/Concentration/UniformDeviation","t":2,"d":"A zeroed-out star-hull value inherits a bound on its corresponding base-family value.","s":"∀ {𝒳 : Type u_4} {ι : Type u_5} (F : ι → 𝒳 → ℝ) (norm : (𝒳 → ℝ) → ℝ) {b r : ℝ}, 0 ≤ b → ∀ {p : Causalean.Stat.Concentration.starHullParam ι} {x : 𝒳}, |F p.2 x| ≤ b → |Causalean.Stat.Concentration.starHullZeroOut F norm r p x| ≤ b","l":361,"f":"Causalean/Stat/Concentration/UniformDeviation/CriticalRadius.lean"},{"n":"Causalean.Stat.Concentration.abs_starHullZeroOut_le_of_bound","k":"theorem","m":"Causalean.Stat.Concentration.Covering.VCLocalizedRegime","a":"Stat/Concentration/Covering","t":2,"d":"If every function in a class is pointwise bounded in absolute value by a common constant, then every zeroed localized star-hull function has the same bound at every point.","s":"∀ {ι : Type u} {𝒳 : Type v} {F : ι → 𝒳 → ℝ} {norm : (𝒳 → ℝ) → ℝ} {B r : ℝ}, (∀ (i : ι) (x : 𝒳), |F i x| ≤ B) → ∀ (p : Causalean.Stat.Concentration.starHullParam ι) (x : 𝒳), |Causalean.Stat.Concentration.starHullZeroOut F norm r p x| ≤ B","l":71,"f":"Causalean/Stat/Concentration/Covering/VCLocalizedRegime.lean"},{"n":"Causalean.Stat.Concentration.abs_starHullZeroOut_le_starHullEval","k":"theorem","m":"Causalean.Stat.Concentration.Rademacher.LocalRademacher","a":"Stat/Concentration/Rademacher","t":2,"d":"Pointwise dominance by the inclusion. The zero-out family is dominated coordinatewise (in absolute value) by the un-localised inclusion starHullEval F.","s":"∀ {ι : Type u_2} {𝒳 : Type u_3} (F : ι → 𝒳 → ℝ) (norm : (𝒳 → ℝ) → ℝ) (r : ℝ) (p : Causalean.Stat.Concentration.starHullParam ι) (x : 𝒳), |Causalean.Stat.Concentration.starHullZeroOut F norm r p x| ≤ |Causalean.Stat.Concentration.starHullEval F p x|","l":216,"f":"Causalean/Stat/Concentration/Rademacher/LocalRademacher.lean"},{"n":"Causalean.Stat.Concentration.abs_starHullZeroOut_mono","k":"theorem","m":"Causalean.Stat.Concentration.Rademacher.LocalRademacher","a":"Stat/Concentration/Rademacher","t":2,"d":"Pointwise monotonicity of the zero-out family in the radius. For r₁ ≤ r₂, |starHullZeroOut F norm r₁ p x| ≤ |starHullZeroOut F norm r₂ p x| coordinatewise.","s":"∀ {ι : Type u_2} {𝒳 : Type u_3} (F : ι → 𝒳 → ℝ) (norm : (𝒳 → ℝ) → ℝ) {r₁ r₂ : ℝ}, r₁ ≤ r₂ → ∀ (p : Causalean.Stat.Concentration.starHullParam ι) (x : 𝒳), |Causalean.Stat.Concentration.starHullZeroOut F norm r₁ p x| ≤ |Causalean.Stat.Concentration.starHullZeroOut F norm r₂ p x|","l":201,"f":"Causalean/Stat/Concentration/Rademacher/LocalRademacher.lean"},{"n":"Causalean.Stat.Concentration.absolute_dudley_vc_starHullZeroOut_linear_residual_shared","k":"theorem","m":"Causalean.Stat.Concentration.Covering.VCLocalizedRegime","a":"Stat/Concentration/Covering","t":2,"d":"Shared absolute-form Dudley/VC bridge for the localized star-hull class.","s":"∀ {ι : Type u} {𝒳 : Type v} (F : ι → 𝒳 → ℝ) (norm : (𝒳 → ℝ) → ℝ) (π : ι → 𝒳 → Bool), (∀ {m : ℕ} (S : Fin m → 𝒳), ∃ φ, ∀ (i : ι) (j : Fin m), F i (S j) = φ j (π i (S j))) → ∀ (K : ℝ) (d n : ℕ), 1 ≤ K → Causalean.Stat.Concentration.BinaryTraceEntropyControl π d → Causalean.Stat.Concentration.LocalizedVCDudleyHypotheses F norm → ∀ (S : Fin n → 𝒳) (r : ℝ), 0 ≤ r → empiricalRademacherComplexity n (Causalean.Stat.Concentration.starHullZeroOut F norm r) S ≤ Causalean.Stat.Concentration.vcLocalizedPsi K d n r","l":1200,"f":"Causalean/Stat/Concentration/Covering/VCLocalizedRegime.lean"},{"n":"Causalean.Stat.Concentration.ae_ae_condExpKernel_of_ae","k":"theorem","m":"Causalean.Stat.Concentration.ConditionalKernel","a":"Stat/Concentration","t":2,"d":"An ambient almost-sure proposition holds on almost every fiber of the regular conditional distribution.","s":"∀ {Ω : Type u_1} [mΩ : MeasurableSpace Ω] [inst : StandardBorelSpace Ω] {μ : MeasureTheory.Measure Ω} [inst_1 : MeasureTheory.IsFiniteMeasure μ] {m : MeasurableSpace Ω} (hm : m ≤ mΩ) {p : Ω → Prop}, (∀ᵐ (ω : Ω) ∂μ, p ω) → ∀ᵐ (ω : Ω) ∂μ.trim hm, ∀ᵐ (ω' : Ω) ∂(ProbabilityTheory.condExpKernel μ m) ω, p ω'","l":149,"f":"Causalean/Stat/Concentration/ConditionalKernel.lean"},{"n":"Causalean.Stat.Concentration.ae_condExpKernel_projection_energy_gt_le","k":"theorem","m":"Causalean.Stat.Concentration.ConditionalProjectionTail","a":"Stat/Concentration","t":1,"d":"Fix a sub-σ-algebra m of the ambient σ-algebra and a finite family eps of real random variables on Ω that is measurable and almost surely bounded by 1 in absolute value under μ; suppose further that, on almost every m-conditioning fiber, each eps i has conditional mean zero under the regular conditional kernel given m, and that the family eps is conditionally independent given m. Let Pi be a matrix-valued map that is m-measurable entrywise and, on almost every m-conditioning fiber, symmetric, idempotent (Pi·Pi = Pi), and of rank at most r, and fix a positive tolerance zeta. Then, on almost every m-conditioning fiber, the conditional probability — under the regular conditional kernel given m — that the projected noise's squared energy $\\sum_i(\\sum_j \\mathrm{Pi}_{ij}\\,\\mathrm{eps}_j)^2$ exceeds the threshold $8(r\\log 5+\\log(2/\\zeta))$ is at most zeta.","s":"∀ {Ω : Type u_1} [mΩ : MeasurableSpace Ω] [inst : StandardBorelSpace Ω] {μ : MeasureTheory.Measure Ω} [inst_1 : MeasureTheory.IsProbabilityMeasure μ] {m : MeasurableSpace Ω} (hm : m ≤ mΩ) {n r : ℕ} (eps : Fin n → Ω → ℝ) (Pi : Ω → Matrix (Fin n) (Fin n) ℝ), (∀ (i : Fin n), Measurable (eps i)) → (∀ (i : Fin n), ∀ᵐ (ω : Ω) ∂μ, |eps i ω| ≤ 1) → (∀ (i : Fin n), ∀ᵐ (ω : Ω) ∂μ.trim hm, ∫ (ω' : Ω), eps i ω' ∂(ProbabilityTheory.condExpKernel μ m) ω = 0) → ProbabilityTheory.iCondIndepFun m hm eps μ → (∀ (i j : Fin n), Measurable fun ω => Pi ω i j) → (∀ᵐ (ω : Ω) ∂μ.trim hm, (Pi ω).transpose = Pi ω) → (∀ᵐ (ω : Ω) ∂μ.trim hm, Pi ω * Pi ω = Pi ω) → (∀ᵐ (ω : Ω) ∂μ.trim hm, (Pi ω).rank ≤ r) → ∀ {zeta : ℝ}, 0 < zeta → ∀ᵐ (ω : Ω) ∂μ.trim hm, ((ProbabilityTheory.condExpKernel μ m) ω).real {ω' | 8 * (↑r * Real.log 5 + Real.log (2 / zeta)) < ∑ i, (∑ j, Pi ω' i j * eps j ω') ^ 2} ≤ zeta","l":27,"f":"Causalean/Stat/Concentration/ConditionalProjectionTail.lean"},{"n":"Causalean.Stat.Concentration.ae_eq_const_condExpKernel_of_measurable","k":"theorem","m":"Causalean.Stat.Concentration.ConditionalKernel","a":"Stat/Concentration","t":2,"d":"A random variable measurable with respect to the conditioning σ-algebra equals its observed value on almost every conditional fiber.","s":"∀ {Ω : Type u_1} [mΩ : MeasurableSpace Ω] [inst : StandardBorelSpace Ω] {μ : MeasureTheory.Measure Ω} [inst_1 : MeasureTheory.IsFiniteMeasure μ] {m : MeasurableSpace Ω} (hm : m ≤ mΩ) {β : Type u_2} [inst_2 : MeasurableSpace β] [MeasurableEq β] {f : Ω → β}, Measurable f → ∀ᵐ (ω : Ω) ∂μ.trim hm, ∀ᵐ (ω' : Ω) ∂(ProbabilityTheory.condExpKernel μ m) ω, f ω' = f ω","l":161,"f":"Causalean/Stat/Concentration/ConditionalKernel.lean"},{"n":"Causalean.Stat.Concentration.averaging_exists_zero_count","k":"theorem","m":"Causalean.Stat.Concentration.Covering.HausslerPacking","a":"Stat/Concentration/Covering","t":2,"d":"A nonnegative finite probability weighting with average count below one must assign count zero to at least one index.","s":"∀ {ι : Type u_1} [inst : Fintype ι] (μ : ι → ℝ) (cnt : ι → ℕ), ∑ i, μ i = 1 → (∀ (i : ι), 0 ≤ μ i) → ∑ i, μ i * ↑(cnt i) < 1 → ∃ i, cnt i = 0","l":149,"f":"Causalean/Stat/Concentration/Covering/HausslerPacking.lean"},{"n":"Causalean.Stat.Concentration.bernoulliCount","k":"def","m":"Causalean.Stat.Concentration.TailBounds.BinomialCount","a":"Stat/Concentration/TailBounds","t":1,"d":"The number of the first m observations on which f is one.","s":"{Ω : Type u_1} → {𝒳 : Type u_2} → [inst : MeasurableSpace Ω] → [inst_1 : MeasurableSpace 𝒳] → {μ : MeasureTheory.Measure Ω} → {P : MeasureTheory.Measure 𝒳} → Causalean.Stat.IIDSample Ω 𝒳 μ P → (𝒳 → ℝ) → ℕ → Ω → ℝ","l":43,"f":"Causalean/Stat/Concentration/TailBounds/BinomialCount.lean"},{"n":"Causalean.Stat.Concentration.bernoulliCount_le","k":"theorem","m":"Causalean.Stat.Concentration.TailBounds.BinomialCount","a":"Stat/Concentration/TailBounds","t":2,"d":"A count of m observations of a statistic bounded by one never exceeds m.","s":"∀ {Ω : Type u_1} {𝒳 : Type u_2} [inst : MeasurableSpace Ω] [inst_1 : MeasurableSpace 𝒳] {μ : MeasureTheory.Measure Ω} {P : MeasureTheory.Measure 𝒳} (S : Causalean.Stat.IIDSample Ω 𝒳 μ P) {f : 𝒳 → ℝ}, (∀ (x : 𝒳), f x ≤ 1) → ∀ (m : ℕ) (ω : Ω), Causalean.Stat.Concentration.bernoulliCount S f m ω ≤ ↑m","l":61,"f":"Causalean/Stat/Concentration/TailBounds/BinomialCount.lean"},{"n":"Causalean.Stat.Concentration.bernoulliCount_lower_tail","k":"theorem","m":"Causalean.Stat.Concentration.TailBounds.BinomialCount","a":"Stat/Concentration/TailBounds","t":1,"d":"Lower multiplicative tail for the count of an i.i.d. [0,1]-valued statistic. Let S be an i.i.d. sample and let f be a measurable statistic taking values in the unit interval. If p is nonnegative, the population mean of f is at least p, and twice the threshold a is less than m times p, then the probability that the sum of f over the first m draws is at most a is at most $\\exp(-mp/8)$.","s":"∀ {Ω : Type u_1} {𝒳 : Type u_2} [inst : MeasurableSpace Ω] [inst_1 : MeasurableSpace 𝒳] {μ : MeasureTheory.Measure Ω} {P : MeasureTheory.Measure 𝒳} (S : Causalean.Stat.IIDSample Ω 𝒳 μ P) {f : 𝒳 → ℝ}, Measurable f → (∀ (x : 𝒳), f x ∈ Set.Icc 0 1) → ∀ {p a : ℝ}, 0 ≤ p → p ≤ ∫ (x : 𝒳), f x ∂P → ∀ {m : ℕ}, 2 * a < ↑m * p → μ.real {ω | Causalean.Stat.Concentration.bernoulliCount S f m ω ≤ a} ≤ Real.exp (-(↑m * p) / 8)","l":295,"f":"Causalean/Stat/Concentration/TailBounds/BinomialCount.lean"},{"n":"Causalean.Stat.Concentration.bernoulliCount_measurable","k":"theorem","m":"Causalean.Stat.Concentration.TailBounds.BinomialCount","a":"Stat/Concentration/TailBounds","t":2,"d":"","s":"∀ {Ω : Type u_1} {𝒳 : Type u_2} [inst : MeasurableSpace Ω] [inst_1 : MeasurableSpace 𝒳] {μ : MeasureTheory.Measure Ω} {P : MeasureTheory.Measure 𝒳} (S : Causalean.Stat.IIDSample Ω 𝒳 μ P) {f : 𝒳 → ℝ}, Measurable f → ∀ (m : ℕ), Measurable (Causalean.Stat.Concentration.bernoulliCount S f m)","l":48,"f":"Causalean/Stat/Concentration/TailBounds/BinomialCount.lean"},{"n":"Causalean.Stat.Concentration.bernoulliCount_nonneg","k":"theorem","m":"Causalean.Stat.Concentration.TailBounds.BinomialCount","a":"Stat/Concentration/TailBounds","t":2,"d":"A count built from a nonnegative statistic is nonnegative.","s":"∀ {Ω : Type u_1} {𝒳 : Type u_2} [inst : MeasurableSpace Ω] [inst_1 : MeasurableSpace 𝒳] {μ : MeasureTheory.Measure Ω} {P : MeasureTheory.Measure 𝒳} (S : Causalean.Stat.IIDSample Ω 𝒳 μ P) {f : 𝒳 → ℝ}, (∀ (x : 𝒳), 0 ≤ f x) → ∀ (m : ℕ) (ω : Ω), 0 ≤ Causalean.Stat.Concentration.bernoulliCount S f m ω","l":54,"f":"Causalean/Stat/Concentration/TailBounds/BinomialCount.lean"},{"n":"Causalean.Stat.Concentration.bernoulliCount_upper_tail","k":"theorem","m":"Causalean.Stat.Concentration.TailBounds.BinomialCount","a":"Stat/Concentration/TailBounds","t":1,"d":"Upper multiplicative tail for the count of an i.i.d. [0,1]-valued statistic. Let S be an i.i.d. sample and let f be a measurable statistic taking values in the unit interval. If the population mean of f is at most p and m times p is less than half the threshold a, then the probability that the sum of f over the first m draws exceeds a is at most $\\exp(-a(\\log 2 - 1/2))$.","s":"∀ {Ω : Type u_1} {𝒳 : Type u_2} [inst : MeasurableSpace Ω] [inst_1 : MeasurableSpace 𝒳] {μ : MeasureTheory.Measure Ω} {P : MeasureTheory.Measure 𝒳} (S : Causalean.Stat.IIDSample Ω 𝒳 μ P) {f : 𝒳 → ℝ}, Measurable f → (∀ (x : 𝒳), f x ∈ Set.Icc 0 1) → ∀ {p a : ℝ}, ∫ (x : 𝒳), f x ∂P ≤ p → ∀ {m : ℕ}, ↑m * p < a / 2 → μ.real {ω | a < Causalean.Stat.Concentration.bernoulliCount S f m ω} ≤ Real.exp (-a * (Real.log 2 - 1 / 2))","l":270,"f":"Causalean/Stat/Concentration/TailBounds/BinomialCount.lean"},{"n":"Causalean.Stat.Concentration.bernsteinCIHalfWidth","k":"def","m":"Causalean.Stat.Concentration.UniformDeviation.ConfidenceInterval","a":"Stat/Concentration/UniformDeviation","t":1,"d":"The two-sided Bernstein confidence half-width at level δ for a statistic with range bound c (|f − m| ≤ c) and variance proxy σ²: 2σ·√(log(2/δ)/n) + 2c·(log(2/δ)/n). Variance-adaptive: the leading term scales with σ, not the range c.","s":"ℝ → ℝ → ℕ → ℝ → ℝ","l":64,"f":"Causalean/Stat/Concentration/UniformDeviation/ConfidenceInterval.lean"},{"n":"Causalean.Stat.Concentration.bernsteinCIHalfWidth_mono_sigma","k":"theorem","m":"Causalean.Stat.Concentration.TailBounds.EmpiricalBernstein","a":"Stat/Concentration/TailBounds","t":2,"d":"The Bernstein confidence-interval half-width does not decrease when its standard-deviation input is increased, so an upper variance bound gives a conservative interval.","s":"∀ {c : ℝ} {n : ℕ} {δ σ σ' : ℝ}, σ ≤ σ' → Causalean.Stat.Concentration.bernsteinCIHalfWidth c σ n δ ≤ Causalean.Stat.Concentration.bernsteinCIHalfWidth c σ' n δ","l":366,"f":"Causalean/Stat/Concentration/TailBounds/EmpiricalBernstein.lean"},{"n":"Causalean.Stat.Concentration.bernstein_abs_ge","k":"theorem","m":"Causalean.Stat.Concentration.TailBounds.Bernstein","a":"Stat/Concentration/TailBounds","t":1,"d":"Two-sided Bernstein inequality for the sample mean of a bounded statistic. Given an i.i.d. sample S and a statistic f that is measurable and integrable under the population law P, suppose c is nonnegative, f stays within c of its population mean m = ∫ f ∂P, P-almost everywhere, and the population variance of f is at most σ². Then for any sample size n ≥ 1 and any threshold ε ≥ 0, the probability that the sample mean of f over n draws deviates from m by at least ε in absolute value is at most $2\\exp(-n\\varepsilon^2/(2(2\\sigma^2+c\\varepsilon)))$.","s":"∀ {Ω : Type u_1} {mΩ : MeasurableSpace Ω} {μ : MeasureTheory.Measure Ω} {X' : Type u_2} [inst : MeasurableSpace X'] {P : MeasureTheory.Measure X'} (S : Causalean.Stat.IIDSample Ω X' μ P) {f : X' → ℝ}, Measurable f → MeasureTheory.Integrable f P → ∀ {c σ : ℝ}, 0 ≤ c → (∀ᵐ (x : X') ∂P, |f x - ∫ (y : X'), f y ∂P| ≤ c) → ∫ (x : X'), (f x - ∫ (y : X'), f y ∂P) ^ 2 ∂P ≤ σ ^ 2 → ∀ (n : ℕ), 0 < n → ∀ {ε : ℝ}, 0 ≤ ε → μ.real {ω | ε ≤ |S.sampleMean f n ω - ∫ (x : X'), f x ∂P|} ≤ 2 * Real.exp (-↑n * ε ^ 2 / (2 * (2 * σ ^ 2 + c * ε)))","l":224,"f":"Causalean/Stat/Concentration/TailBounds/Bernstein.lean"},{"n":"Causalean.Stat.Concentration.bernstein_ci_cover","k":"theorem","m":"Causalean.Stat.Concentration.UniformDeviation.ConfidenceInterval","a":"Stat/Concentration/UniformDeviation","t":1,"d":"Finite-sample Bernstein confidence interval (coverage form). Let S be an i.i.d. sample drawn from P, and write m = ∫ f dP for the population mean. If f is measurable and P-integrable, if f is almost-everywhere within c of m, for some nonnegative c, if the variance of f is bounded by σ², for some positive σ, if the sample size n is positive, and if the confidence level δ lies in (0, 1], then with probability at least 1 − δ the sample mean X̄ₙ of f over S's first n draws satisfies |X̄ₙ − m| < w, where w = bernsteinCIHalfWidth c σ n δ.","s":"∀ {Ω : Type u_1} {X' : Type u_2} [inst : MeasurableSpace Ω] [inst_1 : MeasurableSpace X'] {μ : MeasureTheory.Measure Ω} {P : MeasureTheory.Measure X'} (S : Causalean.Stat.IIDSample Ω X' μ P) {f : X' → ℝ}, Measurable f → MeasureTheory.Integrable f P → ∀ {c σ : ℝ}, 0 ≤ c → 0 < σ → (∀ᵐ (x : X') ∂P, |f x - ∫ (y : X'), f y ∂P| ≤ c) → ∫ (x : X'), (f x - ∫ (y : X'), f y ∂P) ^ 2 ∂P ≤ σ ^ 2 → ∀ (n : ℕ), 0 < n → ∀ {δ : ℝ}, 0 < δ → δ ≤ 1 → 1 - δ ≤ μ.real {ω | |S.sampleMean f n ω - ∫ (x : X'), f x ∂P| < Causalean.Stat.Concentration.bernsteinCIHalfWidth c σ n δ}","l":206,"f":"Causalean/Stat/Concentration/UniformDeviation/ConfidenceInterval.lean"},{"n":"Causalean.Stat.Concentration.bernstein_ci_miss","k":"theorem","m":"Causalean.Stat.Concentration.UniformDeviation.ConfidenceInterval","a":"Stat/Concentration/UniformDeviation","t":1,"d":"Finite-sample Bernstein confidence interval (miss-probability form). Let S be an i.i.d. sample drawn from P, and write m = ∫ f dP for the population mean. If f is measurable and P-integrable, if f is almost-everywhere within c of m, for some nonnegative c, if the variance of f is bounded by σ², for some positive σ, if the sample size n is positive, and if the confidence level δ lies in (0, 1], then with probability at most δ the sample mean X̄ₙ of f over S's first n draws satisfies w ≤ |X̄ₙ − m|, where w = bernsteinCIHalfWidth c σ n δ.","s":"∀ {Ω : Type u_1} {X' : Type u_2} [inst : MeasurableSpace Ω] [inst_1 : MeasurableSpace X'] {μ : MeasureTheory.Measure Ω} {P : MeasureTheory.Measure X'} (S : Causalean.Stat.IIDSample Ω X' μ P) {f : X' → ℝ}, Measurable f → MeasureTheory.Integrable f P → ∀ {c σ : ℝ}, 0 ≤ c → 0 < σ → (∀ᵐ (x : X') ∂P, |f x - ∫ (y : X'), f y ∂P| ≤ c) → ∫ (x : X'), (f x - ∫ (y : X'), f y ∂P) ^ 2 ∂P ≤ σ ^ 2 → ∀ (n : ℕ), 0 < n → ∀ {δ : ℝ}, 0 < δ → δ ≤ 1 → μ.real {ω | Causalean.Stat.Concentration.bernsteinCIHalfWidth c σ n δ ≤ |S.sampleMean f n ω - ∫ (x : X'), f x ∂P|} ≤ δ","l":117,"f":"Causalean/Stat/Concentration/UniformDeviation/ConfidenceInterval.lean"},{"n":"Causalean.Stat.Concentration.bernstein_ge","k":"theorem","m":"Causalean.Stat.Concentration.TailBounds.Bernstein","a":"Stat/Concentration/TailBounds","t":1,"d":"One-sided Bernstein inequality for the sample mean of a bounded statistic. Given an i.i.d. sample S and a statistic f that is measurable and integrable under the population law P, suppose c is nonnegative, f stays within c of its population mean m = ∫ f ∂P, P-almost everywhere, and the population variance of f is at most σ². Then for any sample size n ≥ 1 and any threshold ε ≥ 0, the probability that the sample mean of f over n draws exceeds m by at least ε is at most $\\exp(-n\\varepsilon^2/(2(2\\sigma^2+c\\varepsilon)))$.","s":"∀ {Ω : Type u_1} {mΩ : MeasurableSpace Ω} {μ : MeasureTheory.Measure Ω} {X' : Type u_2} [inst : MeasurableSpace X'] {P : MeasureTheory.Measure X'} (S : Causalean.Stat.IIDSample Ω X' μ P) {f : X' → ℝ}, Measurable f → MeasureTheory.Integrable f P → ∀ {c σ : ℝ}, 0 ≤ c → (∀ᵐ (x : X') ∂P, |f x - ∫ (y : X'), f y ∂P| ≤ c) → ∫ (x : X'), (f x - ∫ (y : X'), f y ∂P) ^ 2 ∂P ≤ σ ^ 2 → ∀ (n : ℕ), 0 < n → ∀ {ε : ℝ}, 0 ≤ ε → μ.real {ω | ε ≤ S.sampleMean f n ω - ∫ (x : X'), f x ∂P} ≤ Real.exp (-↑n * ε ^ 2 / (2 * (2 * σ ^ 2 + c * ε)))","l":155,"f":"Causalean/Stat/Concentration/TailBounds/Bernstein.lean"},{"n":"Causalean.Stat.Concentration.boundedCount_lower_tail_of_tilt","k":"theorem","m":"Causalean.Stat.Concentration.TailBounds.BinomialCount","a":"Stat/Concentration/TailBounds","t":2,"d":"Chernoff lower tail at an arbitrary nonpositive tilt. For a measurable [0,1]-valued statistic with population mean at least p, the probability that the count of the first m observations falls at or below a level a is at most exp (-s * a + m * p * (exp s - 1)), for every nonpositive s. Optimising over s recovers the usual multiplicative Chernoff bounds.","s":"∀ {Ω : Type u_1} {𝒳 : Type u_2} [inst : MeasurableSpace Ω] [inst_1 : MeasurableSpace 𝒳] {μ : MeasureTheory.Measure Ω} {P : MeasureTheory.Measure 𝒳} (S : Causalean.Stat.IIDSample Ω 𝒳 μ P) {f : 𝒳 → ℝ}, Measurable f → (∀ (x : 𝒳), f x ∈ Set.Icc 0 1) → ∀ {p a : ℝ}, p ≤ ∫ (x : 𝒳), f x ∂P → ∀ (m : ℕ), ∀ s ≤ 0, μ.real {ω | Causalean.Stat.Concentration.bernoulliCount S f m ω ≤ a} ≤ Real.exp (-s * a + ↑m * (p * (Real.exp s - 1)))","l":229,"f":"Causalean/Stat/Concentration/TailBounds/BinomialCount.lean"},{"n":"Causalean.Stat.Concentration.boundedCount_mgf_le","k":"theorem","m":"Causalean.Stat.Concentration.TailBounds.BinomialCount","a":"Stat/Concentration/TailBounds","t":2,"d":"If the population mean of a measurable [0,1]-valued statistic is at most p, then at every nonnegative tilt s the moment generating function of the count of the first m observations is at most exp (m * p * (exp s - 1)).","s":"∀ {Ω : Type u_1} {𝒳 : Type u_2} [inst : MeasurableSpace Ω] [inst_1 : MeasurableSpace 𝒳] {μ : MeasureTheory.Measure Ω} {P : MeasureTheory.Measure 𝒳} (S : Causalean.Stat.IIDSample Ω 𝒳 μ P) {f : 𝒳 → ℝ}, Measurable f → (∀ (x : 𝒳), f x ∈ Set.Icc 0 1) → ∀ {p : ℝ}, ∫ (x : 𝒳), f x ∂P ≤ p → ∀ (m : ℕ) (s : ℝ), 0 ≤ s → ProbabilityTheory.mgf (Causalean.Stat.Concentration.bernoulliCount S f m) μ s ≤ Real.exp (↑m * (p * (Real.exp s - 1)))","l":174,"f":"Causalean/Stat/Concentration/TailBounds/BinomialCount.lean"},{"n":"Causalean.Stat.Concentration.boundedCount_mgf_le_integral","k":"theorem","m":"Causalean.Stat.Concentration.TailBounds.BinomialCount","a":"Stat/Concentration/TailBounds","t":2,"d":"The moment generating function of the count of the first m observations of a measurable [0,1]-valued statistic is at most exp (m * mean * (exp s - 1)), where mean is the statistic's population mean. This is the i.i.d. tensorisation of the one-observation bound and holds at every tilt s.","s":"∀ {Ω : Type u_1} {𝒳 : Type u_2} [inst : MeasurableSpace Ω] [inst_1 : MeasurableSpace 𝒳] {μ : MeasureTheory.Measure Ω} {P : MeasureTheory.Measure 𝒳} (S : Causalean.Stat.IIDSample Ω 𝒳 μ P) {f : 𝒳 → ℝ}, Measurable f → (∀ (x : 𝒳), f x ∈ Set.Icc 0 1) → ∀ (m : ℕ) (s : ℝ), ProbabilityTheory.mgf (Causalean.Stat.Concentration.bernoulliCount S f m) μ s ≤ Real.exp (↑m * ((∫ (x : 𝒳), f x ∂P) * (Real.exp s - 1)))","l":143,"f":"Causalean/Stat/Concentration/TailBounds/BinomialCount.lean"},{"n":"Causalean.Stat.Concentration.boundedCount_upper_tail_of_tilt","k":"theorem","m":"Causalean.Stat.Concentration.TailBounds.BinomialCount","a":"Stat/Concentration/TailBounds","t":2,"d":"Chernoff upper tail at an arbitrary nonnegative tilt. For a measurable [0,1]-valued statistic with population mean at most p, the probability that the count of the first m observations exceeds a level a is at most exp (-s * a + m * p * (exp s - 1)), for every nonnegative s. Optimising over s recovers the usual multiplicative Chernoff bounds.","s":"∀ {Ω : Type u_1} {𝒳 : Type u_2} [inst : MeasurableSpace Ω] [inst_1 : MeasurableSpace 𝒳] {μ : MeasureTheory.Measure Ω} {P : MeasureTheory.Measure 𝒳} (S : Causalean.Stat.IIDSample Ω 𝒳 μ P) {f : 𝒳 → ℝ}, Measurable f → (∀ (x : 𝒳), f x ∈ Set.Icc 0 1) → ∀ {p a : ℝ}, ∫ (x : 𝒳), f x ∂P ≤ p → ∀ (m : ℕ) (s : ℝ), 0 ≤ s → μ.real {ω | a < Causalean.Stat.Concentration.bernoulliCount S f m ω} ≤ Real.exp (-s * a + ↑m * (p * (Real.exp s - 1)))","l":193,"f":"Causalean/Stat/Concentration/TailBounds/BinomialCount.lean"},{"n":"Causalean.Stat.Concentration.bounded_hasSubexponentialMGF","k":"theorem","m":"Causalean.Stat.Concentration.TailBounds.Bernstein","a":"Stat/Concentration/TailBounds","t":2,"d":"A bounded mean-zero random variable is sub-exponential with parameters (2σ², c). The MGF branch is bounded_mgf_le_exp_sq (note 2σ² · t²/2 = σ²t²), valid on c|t| < 1.","s":"∀ {Ω : Type u_1} {mΩ : MeasurableSpace Ω} {μ : MeasureTheory.Measure Ω} {X : Ω → ℝ} [MeasureTheory.IsProbabilityMeasure μ] {c σ : ℝ} (hc : 0 ≤ c), AEMeasurable X μ → ∫ (x : Ω), X x ∂μ = 0 → (∀ᵐ (ω : Ω) ∂μ, |X ω| ≤ c) → ∫ (x : Ω), (fun ω => X ω ^ 2) x ∂μ ≤ σ ^ 2 → Causalean.Stat.Concentration.HasSubexponentialMGF X ⟨2 * σ ^ 2, mul_nonneg (le_of_lt (Mathlib.Meta.Positivity.pos_of_isNat (Mathlib.Meta.NormNum.isNat_ofNat ℝ (Eq.refl 2)) (Eq.refl (Nat.ble 1 2)))) (Even.pow_nonneg (even_two_mul 1) σ)⟩ ⟨c, hc⟩ μ","l":132,"f":"Causalean/Stat/Concentration/TailBounds/Bernstein.lean"},{"n":"Causalean.Stat.Concentration.bounded_mgf_le_exp_sq","k":"theorem","m":"Causalean.Stat.Concentration.TailBounds.Bernstein","a":"Stat/Concentration/TailBounds","t":2,"d":"MGF bound for a bounded mean-zero random variable on the validity interval. For a mean-zero X with |X| ≤ c μ-a.e. and second moment E[X²] ≤ σ², the moment-generating function obeys mgf X μ t ≤ exp(σ² t²) whenever c |t| ≤ 1.","s":"∀ {Ω : Type u_1} {mΩ : MeasurableSpace Ω} {μ : MeasureTheory.Measure Ω} {X : Ω → ℝ} [MeasureTheory.IsProbabilityMeasure μ] {c σ : ℝ}, 0 ≤ c → AEMeasurable X μ → ∫ (x : Ω), X x ∂μ = 0 → (∀ᵐ (ω : Ω) ∂μ, |X ω| ≤ c) → ∫ (x : Ω), (fun ω => X ω ^ 2) x ∂μ ≤ σ ^ 2 → ∀ {t : ℝ}, c * |t| ≤ 1 → ProbabilityTheory.mgf X μ t ≤ Real.exp (σ ^ 2 * t ^ 2)","l":74,"f":"Causalean/Stat/Concentration/TailBounds/Bernstein.lean"},{"n":"Causalean.Stat.Concentration.card_growthFamily_le_sum_choose","k":"theorem","m":"Causalean.Stat.Concentration.Covering.VCCovering","a":"Stat/Concentration/Covering","t":2,"d":"A finite set family with bounded VC dimension has cardinality controlled by the Sauer-Shelah binomial sum on the sample.","s":"∀ {n d : ℕ} (𝒜 : Finset (Finset (Fin n))), 𝒜.vcDim ≤ d → 𝒜.card ≤ ∑ k ≤ d, n.choose k","l":59,"f":"Causalean/Stat/Concentration/Covering/VCCovering.lean"},{"n":"Causalean.Stat.Concentration.card_subsample_family_le_succ_pow","k":"theorem","m":"Causalean.Stat.Concentration.Covering.HausslerPacking","a":"Stat/Concentration/Covering","t":2,"d":"A sampled Boolean trace family with bounded VC dimension has polynomially many realized restricted patterns.","s":"∀ {n m d : ℕ} (J : Fin m → Fin n) (P : Finset (Fin n → Bool)), (Finset.image (Causalean.Stat.Concentration.subsamplePattern J) P).vcDim ≤ d → (Finset.image (Causalean.Stat.Concentration.subsamplePattern J) P).card ≤ (m + 1) ^ d","l":220,"f":"Causalean/Stat/Concentration/Covering/HausslerPacking.lean"},{"n":"Causalean.Stat.Concentration.centeredEmpiricalAverage","k":"def","m":"Causalean.Stat.Concentration.VarianceAdaptiveVCExpectedMaximal.Separability","a":"Stat/Concentration/VarianceAdaptiveVCExpectedMaximal","t":1,"d":"The centered empirical average is the sample average of a function minus its population integral.","s":"{Ω : Type u_1} → [inst : MeasurableSpace Ω] → MeasureTheory.Measure Ω → {n : ℕ} → (Fin n → Ω) → (Ω → ℝ) → ℝ","l":25,"f":"Causalean/Stat/Concentration/VarianceAdaptiveVCExpectedMaximal/Separability.lean"},{"n":"Causalean.Stat.Concentration.ciSup_mul_const_of_le_one","k":"theorem","m":"Causalean.Stat.Concentration.UniformDeviation.CriticalRadius","a":"Stat/Concentration/UniformDeviation","t":2,"d":"A supremum commutes with multiplication by a nonnegative constant.","s":"∀ {A : Type u_4} (c : A → ℝ) (b : ℝ), 0 ≤ b → ⨆ a, c a * b = (⨆ a, c a) * b","l":67,"f":"Causalean/Stat/Concentration/UniformDeviation/CriticalRadius.lean"},{"n":"Causalean.Stat.Concentration.ciSup_prod_eq_of_bddAbove","k":"theorem","m":"Causalean.Stat.Concentration.Covering.VCUniformDeviation","a":"Stat/Concentration/Covering","t":2,"d":"The bounded supremum of a real-valued quantity indexed by two choices is unchanged when the two choices are optimized one after the other.","s":"∀ {A : Type u_1} {B : Type u_2} [Nonempty A] [Nonempty B] (f : A → B → ℝ), BddAbove (Set.range fun p => f p.1 p.2) → ⨆ p, f p.1 p.2 = ⨆ b, ⨆ a, f a b","l":67,"f":"Causalean/Stat/Concentration/Covering/VCUniformDeviation.lean"},{"n":"Causalean.Stat.Concentration.collision_bound","k":"theorem","m":"Causalean.Stat.Concentration.Covering.HausslerPacking","a":"Stat/Concentration/Covering","t":2,"d":"When two Boolean vectors are separated under nonnegative coordinate weights, their probability of agreeing on every coordinate of a repeated weighted sample decays exponentially with the sample length.","s":"∀ {n m : ℕ} (w : Fin n → ℝ), (∀ (j : Fin n), 0 ≤ w j) → ∀ (r ε W : ℝ), 0 < r → W = ∑ j, w j → 0 < W → W ≤ r ^ 2 → ∀ (a b : Fin n → Bool), ε ^ 2 ≤ Causalean.Stat.Concentration.weightedHammingSq w a b → (∑ j with a j = b j, w j / W) ^ m ≤ Real.exp (-↑m * ε ^ 2 / r ^ 2)","l":172,"f":"Causalean/Stat/Concentration/Covering/HausslerPacking.lean"},{"n":"Causalean.Stat.Concentration.completeSpace_matrix_linftyOp","k":"theorem","m":"Causalean.Stat.Concentration.Matrix.InversePerturbation","a":"Stat/Concentration/Matrix","t":2,"d":"Completeness of finite matrices under the ℓ∞-operator norm follows from the coordinatewise function-space uniformity. Needed to invoke the Neumann/geometric-series unit API.","s":"∀ {α : Type u_1} {β : Type u_2} {R : Type u_3} [inst : Fintype α] [inst_1 : Fintype β] [inst_2 : NormedAddCommGroup R] [CompleteSpace R], CompleteSpace (Matrix α β R)","l":49,"f":"Causalean/Stat/Concentration/Matrix/InversePerturbation.lean"},{"n":"Causalean.Stat.Concentration.countableEmpiricalProcessSup","k":"def","m":"Causalean.Stat.Concentration.VarianceAdaptiveVCExpectedMaximal.EntropyChaining","a":"Stat/Concentration/VarianceAdaptiveVCExpectedMaximal","t":1,"d":"Countable empirical-process supremum associated with an enumeration.","s":"{Ω : Type u_1} → {ι : Type u_2} → [inst : MeasurableSpace Ω] → MeasureTheory.Measure Ω → (ι → Ω → ℝ) → (ℕ → ι) → {n : ℕ} → (Fin n → Ω) → ENNReal","l":36,"f":"Causalean/Stat/Concentration/VarianceAdaptiveVCExpectedMaximal/EntropyChaining.lean"},{"n":"Causalean.Stat.Concentration.countableEmpiricalSup","k":"def","m":"Causalean.Stat.Concentration.VarianceAdaptiveVCExpectedMaximal.Basic","a":"Stat/Concentration/VarianceAdaptiveVCExpectedMaximal","t":1,"d":"The empirical supremum of a countable real-valued class is the largest absolute difference between its sample average and population mean.","s":"{𝒳 : Type u} → [inst : MeasurableSpace 𝒳] → {ι : Type v} → MeasureTheory.Measure 𝒳 → (ι → 𝒳 → ℝ) → {n : ℕ} → (Fin n → 𝒳) → ℝ","l":147,"f":"Causalean/Stat/Concentration/VarianceAdaptiveVCExpectedMaximal/Basic.lean"},{"n":"Causalean.Stat.Concentration.coveringFinset","k":"def","m":"Causalean.Stat.Concentration.Covering.CoveringNumber","a":"Stat/Concentration/Covering","t":1,"d":"A chosen finite positive-radius net attains the covering number.","s":"{X : Type u_1} → [inst : PseudoMetricSpace X] → {A : Set X} → TotallyBounded A → {ε : ℝ} → ε > 0 → Finset X","l":96,"f":"Causalean/Stat/Concentration/Covering/CoveringNumber.lean"},{"n":"Causalean.Stat.Concentration.coveringFinset_card","k":"theorem","m":"Causalean.Stat.Concentration.Covering.CoveringNumber","a":"Stat/Concentration/Covering","t":2,"d":"The chosen covering finset has cardinality equal to the covering number.","s":"∀ {X : Type u_1} [inst : PseudoMetricSpace X] {A : Set X} (ha : TotallyBounded A) {ε : ℝ} (hε : ε > 0), (Causalean.Stat.Concentration.coveringFinset ha hε).card = Causalean.Stat.Concentration.coveringNumber ha ε","l":112,"f":"Causalean/Stat/Concentration/Covering/CoveringNumber.lean"},{"n":"Causalean.Stat.Concentration.coveringFinset_cover","k":"theorem","m":"Causalean.Stat.Concentration.Covering.CoveringNumber","a":"Stat/Concentration/Covering","t":2,"d":"The chosen covering finset covers the target set by balls of the requested positive radius.","s":"∀ {X : Type u_1} [inst : PseudoMetricSpace X] {A : Set X} (ha : TotallyBounded A) {ε : ℝ} (hε : ε > 0), A ⊆ ⋃ y ∈ Causalean.Stat.Concentration.coveringFinset ha hε, Metric.ball y ε","l":102,"f":"Causalean/Stat/Concentration/Covering/CoveringNumber.lean"},{"n":"Causalean.Stat.Concentration.coveringNumber","k":"def","m":"Causalean.Stat.Concentration.Covering.CoveringNumber","a":"Stat/Concentration/Covering","t":1,"d":"The covering number is the smallest size of a finite positive-radius net for a totally bounded set, and is zero at nonpositive radii.","s":"{X : Type u_1} → [inst : PseudoMetricSpace X] → {A : Set X} → TotallyBounded A → ℝ → ℕ","l":39,"f":"Causalean/Stat/Concentration/Covering/CoveringNumber.lean"},{"n":"Causalean.Stat.Concentration.coveringNumber_aemeasurable","k":"theorem","m":"Causalean.Stat.Concentration.Covering.CoveringNumber","a":"Stat/Concentration/Covering","t":2,"d":"The covering-number function of the radius is almost-everywhere measurable on the real line.","s":"∀ {X : Type u_1} [inst : PseudoMetricSpace X] {A : Set X} (ha : TotallyBounded A) (μ : MeasureTheory.Measure ℝ), AEMeasurable (Causalean.Stat.Concentration.coveringNumber ha) μ","l":81,"f":"Causalean/Stat/Concentration/Covering/CoveringNumber.lean"},{"n":"Causalean.Stat.Concentration.coveringNumber_antitone","k":"theorem","m":"Causalean.Stat.Concentration.Covering.CoveringNumber","a":"Stat/Concentration/Covering","t":2,"d":"Covering numbers weakly decrease as the positive covering radius grows.","s":"∀ {X : Type u_1} [inst : PseudoMetricSpace X] {A : Set X} (ha : TotallyBounded A), AntitoneOn (Causalean.Stat.Concentration.coveringNumber ha) (Set.Ioi 0)","l":56,"f":"Causalean/Stat/Concentration/Covering/CoveringNumber.lean"},{"n":"Causalean.Stat.Concentration.coveringNumber_eq","k":"theorem","m":"Causalean.Stat.Concentration.Covering.CoveringNumber","a":"Stat/Concentration/Covering","t":1,"d":"Value of the covering number at a positive radius. For a totally bounded subset of a pseudometric space, at any positive covering radius ε, the covering number equals the least cardinality of a finite ε-net witnessing total boundedness, as selected by coveringNumber_exists.","s":"∀ {X : Type u_1} [inst : PseudoMetricSpace X] {A : Set X} (ha : TotallyBounded A) {ε : ℝ} (hε : ε > 0), Causalean.Stat.Concentration.coveringNumber ha ε = Nat.find (Causalean.Stat.Concentration.coveringNumber_exists ha hε)","l":47,"f":"Causalean/Stat/Concentration/Covering/CoveringNumber.lean"},{"n":"Causalean.Stat.Concentration.coveringNumber_exists","k":"theorem","m":"Causalean.Stat.Concentration.Covering.CoveringNumber","a":"Stat/Concentration/Covering","t":2,"d":"A totally bounded set has a finite positive-radius net, with the net size recorded as a natural number.","s":"∀ {X : Type u_1} {A : Set X} [inst : PseudoMetricSpace X], TotallyBounded A → ∀ {ε : ℝ}, ε > 0 → ∃ n t, t.card = n ∧ A ⊆ ⋃ y ∈ t, Metric.ball y ε","l":23,"f":"Causalean/Stat/Concentration/Covering/CoveringNumber.lean"},{"n":"Causalean.Stat.Concentration.coveringNumber_nonzero","k":"theorem","m":"Causalean.Stat.Concentration.Covering.CoveringNumber","a":"Stat/Concentration/Covering","t":2,"d":"A nonempty totally bounded set has a positive covering number at every positive radius.","s":"∀ {X : Type u_1} [inst : PseudoMetricSpace X] {A : Set X}, A.Nonempty → ∀ (ha : TotallyBounded A) {ε : ℝ}, ε > 0 → 0 < Causalean.Stat.Concentration.coveringNumber ha ε","l":72,"f":"Causalean/Stat/Concentration/Covering/CoveringNumber.lean"},{"n":"Causalean.Stat.Concentration.criticalRadius","k":"def","m":"Causalean.Stat.Concentration.UniformDeviation.CriticalRadius","a":"Stat/Concentration/UniformDeviation","t":1,"d":"The critical radius of ψ: the infimum over δ > 0 with ψ δ ≤ δ². Defined via sInf; if the set is empty (e.g. ψ grows faster than δ² everywhere), the value is 0 by Mathlib convention.","s":"(ℝ → ℝ) → ℝ","l":156,"f":"Causalean/Stat/Concentration/UniformDeviation/CriticalRadius.lean"},{"n":"Causalean.Stat.Concentration.criticalRadius_fp_of_subRoot","k":"theorem","m":"Causalean.Stat.Concentration.UniformDeviation.CriticalRadius","a":"Stat/Concentration/UniformDeviation","t":1,"d":"Fixed-point property at the critical radius. If ψ is sub-root and its critical radius criticalRadius ψ is positive, then the critical radius is itself a solution of its own defining inequality: ψ (criticalRadius ψ) ≤ (criticalRadius ψ) ^ 2.","s":"∀ {ψ : ℝ → ℝ}, Causalean.Stat.Concentration.SubRoot ψ → 0 < Causalean.Stat.Concentration.criticalRadius ψ → ψ (Causalean.Stat.Concentration.criticalRadius ψ) ≤ Causalean.Stat.Concentration.criticalRadius ψ ^ 2","l":269,"f":"Causalean/Stat/Concentration/UniformDeviation/CriticalRadius.lean"},{"n":"Causalean.Stat.Concentration.criticalRadius_le","k":"theorem","m":"Causalean.Stat.Concentration.UniformDeviation.CriticalRadius","a":"Stat/Concentration/UniformDeviation","t":2,"d":"Any δ > 0 with ψ δ ≤ δ² upper-bounds the critical radius.","s":"∀ {ψ : ℝ → ℝ} {δ : ℝ}, 0 < δ → ψ δ ≤ δ ^ 2 → Causalean.Stat.Concentration.criticalRadius ψ ≤ δ","l":174,"f":"Causalean/Stat/Concentration/UniformDeviation/CriticalRadius.lean"},{"n":"Causalean.Stat.Concentration.criticalRadius_linear_eq","k":"theorem","m":"Causalean.Stat.Concentration.Covering.VCUniformDeviation","a":"Stat/Concentration/Covering","t":2,"d":"The critical radius of a positive-slope linear envelope equals its slope, giving the exact fixed-point scale for a localized empirical-process bound.","s":"∀ {C : ℝ}, 0 < C → (Causalean.Stat.Concentration.criticalRadius fun r => C * r) = C","l":38,"f":"Causalean/Stat/Concentration/Covering/VCUniformDeviation.lean"},{"n":"Causalean.Stat.Concentration.criticalRadius_linear_le","k":"theorem","m":"Causalean.Stat.Concentration.Covering.VCLocalizedRegime","a":"Stat/Concentration/Covering","t":2,"d":"The critical radius of a positive-slope linear envelope is at most its slope.","s":"∀ {C : ℝ}, 0 < C → (Causalean.Stat.Concentration.criticalRadius fun r => C * r) ≤ C","l":333,"f":"Causalean/Stat/Concentration/Covering/VCLocalizedRegime.lean"},{"n":"Causalean.Stat.Concentration.criticalRadius_linear_sq_le","k":"theorem","m":"Causalean.Stat.Concentration.Covering.VCLocalizedRegime","a":"Stat/Concentration/Covering","t":2,"d":"The squared critical radius of a positive-slope linear envelope is at most the squared slope.","s":"∀ {C : ℝ}, 0 < C → (Causalean.Stat.Concentration.criticalRadius fun r => C * r) ^ 2 ≤ C ^ 2","l":340,"f":"Causalean/Stat/Concentration/Covering/VCLocalizedRegime.lean"},{"n":"Causalean.Stat.Concentration.criticalRadius_nonneg","k":"theorem","m":"Causalean.Stat.Concentration.UniformDeviation.CriticalRadius","a":"Stat/Concentration/UniformDeviation","t":2,"d":"The critical radius is non-negative.","s":"∀ (ψ : ℝ → ℝ), 0 ≤ Causalean.Stat.Concentration.criticalRadius ψ","l":162,"f":"Causalean/Stat/Concentration/UniformDeviation/CriticalRadius.lean"},{"n":"Causalean.Stat.Concentration.criticalRadius_pos_of_subRoot","k":"theorem","m":"Causalean.Stat.Concentration.UniformDeviation.CriticalRadius","a":"Stat/Concentration/UniformDeviation","t":1,"d":"Positivity of the critical radius. If ψ is sub-root, r₀ is a positive radius with ψ r₀ ≤ r₀ ^ 2, and ψ grows strictly faster than the square near the origin, i.e. there is some ε with 0 < ε < r₀ and ε ^ 2 < ψ ε, then the critical radius of ψ is strictly positive.","s":"∀ {ψ : ℝ → ℝ}, Causalean.Stat.Concentration.SubRoot ψ → ∀ {r₀ : ℝ}, 0 < r₀ → ψ r₀ ≤ r₀ ^ 2 → (∃ ε > 0, ε < r₀ ∧ ε ^ 2 < ψ ε) → 0 < Causalean.Stat.Concentration.criticalRadius ψ","l":315,"f":"Causalean/Stat/Concentration/UniformDeviation/CriticalRadius.lean"},{"n":"Causalean.Stat.Concentration.criticalRadius_vcLocalizedPsi_le","k":"theorem","m":"Causalean.Stat.Concentration.Covering.VCLocalizedRegime","a":"Stat/Concentration/Covering","t":2,"d":"The finite-VC critical radius is bounded by the finite-VC slope.","s":"∀ {K : ℝ} {d n : ℕ}, 0 ≤ K → 0 < n → Causalean.Stat.Concentration.criticalRadius (Causalean.Stat.Concentration.vcLocalizedPsi K d n) ≤ Causalean.Stat.Concentration.vcLocalizedSlope K d n","l":350,"f":"Causalean/Stat/Concentration/Covering/VCLocalizedRegime.lean"},{"n":"Causalean.Stat.Concentration.criticalRadius_vcLocalizedPsi_sq_le","k":"theorem","m":"Causalean.Stat.Concentration.Covering.VCLocalizedRegime","a":"Stat/Concentration/Covering","t":2,"d":"The finite-VC squared critical radius is bounded by the squared slope.","s":"∀ {K : ℝ} {d n : ℕ}, 0 ≤ K → 0 < n → Causalean.Stat.Concentration.criticalRadius (Causalean.Stat.Concentration.vcLocalizedPsi K d n) ^ 2 ≤ Causalean.Stat.Concentration.vcLocalizedSlope K d n ^ 2","l":357,"f":"Causalean/Stat/Concentration/Covering/VCLocalizedRegime.lean"},{"n":"Causalean.Stat.Concentration.criticalRadius_vcLocalizedPsi_sq_le_rate","k":"theorem","m":"Causalean.Stat.Concentration.Covering.VCLocalizedRegime","a":"Stat/Concentration/Covering","t":1,"d":"Rate bound for the finite-VC critical radius. For a nonnegative localization constant K and a positive sample size n, the squared critical radius of the finite-VC localized envelope vcLocalizedPsi K d n is at most 36·(K·d·log(n+1)+1)/n — the advertised (d·log n)/n-order bound.","s":"∀ {K : ℝ} {d n : ℕ}, 0 ≤ K → 0 < n → Causalean.Stat.Concentration.criticalRadius (Causalean.Stat.Concentration.vcLocalizedPsi K d n) ^ 2 ≤ 36 * ((K * ↑d * Real.log (↑n + 1) + 1) / ↑n)","l":365,"f":"Causalean/Stat/Concentration/Covering/VCLocalizedRegime.lean"},{"n":"Causalean.Stat.Concentration.designInv00_perturb","k":"theorem","m":"Causalean.Stat.Concentration.Matrix.InversePerturbation","a":"Stat/Concentration/Matrix","t":1,"d":"The (0,0) specialization of the entrywise inverse perturbation bound. If S is invertible, every row of S⁻¹ has its absolute entries summing to at most c, every entry of M differs from the corresponding entry of S by at most η, and the product c · ((p+1) · η) is at most 1/2, then M is also invertible and its (0,0) inverse entry differs from S's by at most 2 · c² · ((p+1) · η).","s":"∀ {p : ℕ} (S M : Matrix (Fin (p + 1)) (Fin (p + 1)) ℝ), IsUnit S.det → ∀ {η c : ℝ}, (∀ (i : Fin (p + 1)), ∑ j, |S⁻¹ i j| ≤ c) → (∀ (j k : Fin (p + 1)), |M j k - S j k| ≤ η) → c * (↑(p + 1) * η) ≤ 1 / 2 → IsUnit M.det ∧ |M⁻¹ 0 0 - S⁻¹ 0 0| ≤ 2 * c ^ 2 * (↑(p + 1) * η)","l":233,"f":"Causalean/Stat/Concentration/Matrix/InversePerturbation.lean"},{"n":"Causalean.Stat.Concentration.designInv_perturb","k":"theorem","m":"Causalean.Stat.Concentration.Matrix.InversePerturbation","a":"Stat/Concentration/Matrix","t":2,"d":"Entrywise perturbation of an inverse entry. Let S be an invertible (p+1)×(p+1) real matrix whose inverse has rows with absolute sums bounded by c, and let M be a matrix all of whose entries are within η of S (|Mⱼₖ − Sⱼₖ| ≤ η). If c·(p+1)·η ≤ 1/2, then M is invertible and every selected pair of inverse entries satisfies","s":"∀ {p : ℕ} (S M : Matrix (Fin (p + 1)) (Fin (p + 1)) ℝ), IsUnit S.det → ∀ {η c : ℝ}, (∀ (i : Fin (p + 1)), ∑ j, |S⁻¹ i j| ≤ c) → (∀ (j k : Fin (p + 1)), |M j k - S j k| ≤ η) → c * (↑(p + 1) * η) ≤ 1 / 2 → ∀ {i j : Fin (p + 1)}, IsUnit M.det ∧ |M⁻¹ i j - S⁻¹ i j| ≤ 2 * c ^ 2 * (↑(p + 1) * η)","l":102,"f":"Causalean/Stat/Concentration/Matrix/InversePerturbation.lean"},{"n":"Causalean.Stat.Concentration.designMatrix_inv_concentration","k":"theorem","m":"Causalean.Stat.Concentration.Matrix.DesignInverse","a":"Stat/Concentration/Matrix","t":1,"d":"Matrix-inverse concentration of the random design moment matrix. Let g j k : Ω → ℝ be the per-entry design statistics, each square-integrable under μ, so the empirical moment matrix is M(ω) = fun j k => ∑ᵢ g j k (ωᵢ) and, given that the population matrix S equals N times its entrywise expectation, S j k = N·𝔼[g j k], suppose S is invertible with inverse row sums bounded by c ≥ 0, η is positive, and the scale satisfies c·(p+1)·η ≤ 1/2. Then the event on which M fails to be invertible *or* its leverage (M⁻¹)₀₀ is farther than 2 c² (p+1) η from (S⁻¹)₀₀ has probability at most the union-bound tail ∑_{j,k} N·Var[g j k]/η². This is the high-probability statement that the random design is non-degenerate with O(1/(Nh)) leverage on the good event.","s":"∀ {N p : ℕ} {Ω : Type u_1} [inst : MeasurableSpace Ω] (μ : MeasureTheory.Measure Ω) [MeasureTheory.IsProbabilityMeasure μ] (g : Fin (p + 1) → Fin (p + 1) → Ω → ℝ), (∀ (j k : Fin (p + 1)), MeasureTheory.MemLp (g j k) 2 μ) → ∀ (S : Matrix (Fin (p + 1)) (Fin (p + 1)) ℝ), (∀ (j k : Fin (p + 1)), S j k = ↑N * ∫ (x : Ω), g j k x ∂μ) → IsUnit S.det → ∀ {η c : ℝ}, 0 < η → (∀ (i : Fin (p + 1)), ∑ j, |S⁻¹ i j| ≤ c) → c * (↑(p + 1) * η) ≤ 1 / 2 → (MeasureTheory.Measure.pi fun x => μ) {ω | ¬(IsUnit (Matrix.of fun j k => ∑ i, g j k (ω i)).det ∧ |(Matrix.of fun j k => ∑ i, g j k (ω i))⁻¹ 0 0 - S⁻¹ 0 0| ≤ 2 * c ^ 2 * (↑(p + 1) * η))} ≤ ∑ a, ENNReal.ofReal (↑N * ProbabilityTheory.variance (g a.1 a.2) μ / η ^ 2)","l":38,"f":"Causalean/Stat/Concentration/Matrix/DesignInverse.lean"},{"n":"Causalean.Stat.Concentration.dist_def","k":"theorem","m":"Causalean.Stat.Concentration.Covering.EmpiricalPseudoMetric","a":"Stat/Concentration/Covering","t":2,"d":"","s":"∀ {𝒳 : Type v} {n : ℕ} {ι : Type u} {F : ι → 𝒳 → ℝ} {S : Fin n → 𝒳} (f g : Causalean.Stat.Concentration.EmpiricalFunctionSpace F S), dist f g = Causalean.Stat.Concentration.empiricalNorm S (F f.index - F g.index)","l":155,"f":"Causalean/Stat/Concentration/Covering/EmpiricalPseudoMetric.lean"},{"n":"Causalean.Stat.Concentration.dudley_entropy_integral_bound","k":"theorem","m":"Causalean.Stat.Concentration.Covering.DudleyEntropy","a":"Stat/Concentration/Covering","t":1,"d":"Dudley entropy-integral bound, without the outer absolute value. Fix a positive scale ε that is strictly less than half the common empirical-norm envelope c, where the sample size m is positive, every member of the class has empirical norm on the sample at most c, and the sample-restricted function class is totally bounded in the empirical pseudometric. Then the empirical Rademacher complexity computed without the outer absolute value is at most 4ε + (12/√m) ∫_ε^(c/2) √(log(coveringNumber x)) dx, the usual Dudley chaining bound in terms of the covering-number entropy integral.","s":"∀ {Z : Type v} {m : ℕ} {ι : Type u} [Nonempty ι] {F : ι → Z → ℝ} {S : Fin m → Z} {c ε : ℝ}, 0 < ε → ∀ (h' : TotallyBounded Set.univ), 0 < m → (∀ (f : ι), Causalean.Stat.Concentration.empiricalNorm S (F f) ≤ c) → ε < c / 2 → empiricalRademacherComplexity_without_abs m F S ≤ 4 * ε + 12 / √↑m * ∫ (x : ℝ) in ε..c / 2, √(Real.log ↑(Causalean.Stat.Concentration.coveringNumber h' x))","l":1840,"f":"Causalean/Stat/Concentration/Covering/DudleyEntropy.lean"},{"n":"Causalean.Stat.Concentration.empiricalBernsteinCIHalfWidth","k":"def","m":"Causalean.Stat.Concentration.TailBounds.EmpiricalBernstein","a":"Stat/Concentration/TailBounds","t":1,"d":"The data-driven Bernstein half-width: the Bernstein half-width with the oracle standard deviation σ replaced by the *observed* upper bound √(V̂ₙ(ω) + τ), where τ = empiricalVarianceSlack a b n δ. This is a function of ω (it depends on the sample through V̂ₙ).","s":"{Ω : Type u_1} → {X : Type u_2} → [inst : MeasurableSpace Ω] → [inst_1 : MeasurableSpace X] → {μ : MeasureTheory.Measure Ω} → {P : MeasureTheory.Measure X} → Causalean.Stat.IIDSample Ω X μ P → (X → ℝ) → ℝ → ℝ → ℝ → ℕ → ℝ → Ω → ℝ","l":357,"f":"Causalean/Stat/Concentration/TailBounds/EmpiricalBernstein.lean"},{"n":"Causalean.Stat.Concentration.empiricalDist","k":"def","m":"Causalean.Stat.Concentration.Covering.EmpiricalPseudoMetric","a":"Stat/Concentration/Covering","t":1,"d":"The empirical distance between two functions is the empirical norm of their difference on the fixed sample.","s":"{𝒳 : Type v} → {n : ℕ} → (Fin n → 𝒳) → (𝒳 → ℝ) → (𝒳 → ℝ) → ℝ","l":37,"f":"Causalean/Stat/Concentration/Covering/EmpiricalPseudoMetric.lean"},{"n":"Causalean.Stat.Concentration.empiricalDist_comm","k":"theorem","m":"Causalean.Stat.Concentration.Covering.EmpiricalPseudoMetric","a":"Stat/Concentration/Covering","t":2,"d":"The empirical distance is symmetric.","s":"∀ {𝒳 : Type v} {n : ℕ} (S : Fin n → 𝒳) (f g : 𝒳 → ℝ), Causalean.Stat.Concentration.empiricalDist S f g = Causalean.Stat.Concentration.empiricalDist S g f","l":98,"f":"Causalean/Stat/Concentration/Covering/EmpiricalPseudoMetric.lean"},{"n":"Causalean.Stat.Concentration.empiricalDist_def","k":"theorem","m":"Causalean.Stat.Concentration.Covering.EmpiricalPseudoMetric","a":"Stat/Concentration/Covering","t":1,"d":"For a fixed sample S and functions f and g, the empirical distance between f and g equals the empirical norm of their pointwise difference.","s":"∀ {𝒳 : Type v} {n : ℕ} (S : Fin n → 𝒳) (f g : 𝒳 → ℝ), Causalean.Stat.Concentration.empiricalDist S f g = Causalean.Stat.Concentration.empiricalNorm S (f - g)","l":42,"f":"Causalean/Stat/Concentration/Covering/EmpiricalPseudoMetric.lean"},{"n":"Causalean.Stat.Concentration.empiricalDist_proj","k":"theorem","m":"Causalean.Stat.Concentration.Covering.EmpiricalPseudoMetric","a":"Stat/Concentration/Covering","t":2,"d":"A single sample coordinate is bounded by the empirical norm up to the sample-size scaling.","s":"∀ {𝒳 : Type v} {n : ℕ} (S : Fin n → 𝒳) (f : 𝒳 → ℝ) (i : Fin n), |f (S i)| / √↑n ≤ Causalean.Stat.Concentration.empiricalNorm S f","l":104,"f":"Causalean/Stat/Concentration/Covering/EmpiricalPseudoMetric.lean"},{"n":"Causalean.Stat.Concentration.empiricalDist_sq_eq_weightedHammingSq","k":"theorem","m":"Causalean.Stat.Concentration.Covering.VCLocalizedCovering","a":"Stat/Concentration/Covering","t":1,"d":"Empirical L² distance as weighted Hamming distance. For a real-valued class that factors through a Boolean classifier at each sample coordinate, F i (S j) = φ j (π i (S j)), the squared empirical L²(Pₙ) distance between two members F i and F i' equals the weighted Hamming distance² of their induced Boolean sample-patterns, with per-coordinate weight (φ j true − φ j false)² / n.","s":"∀ {ι : Type u_1} {𝒳 : Type u_2} {n : ℕ} (F : ι → 𝒳 → ℝ) (S : Fin n → 𝒳) (π : ι → 𝒳 → Bool) (φ : Fin n → Bool → ℝ), (∀ (i : ι) (j : Fin n), F i (S j) = φ j (π i (S j))) → ∀ (i i' : ι), Causalean.Stat.Concentration.empiricalDist S (F i) (F i') ^ 2 = Causalean.Stat.Concentration.weightedHammingSq (fun j => (φ j true - φ j false) ^ 2 / ↑n) (fun j => π i (S j)) fun j => π i' (S j)","l":16,"f":"Causalean/Stat/Concentration/Covering/VCLocalizedCovering.lean"},{"n":"Causalean.Stat.Concentration.empiricalFunctionSpace_dist_le_of_forall_abs_sub_le","k":"theorem","m":"Causalean.Stat.Concentration.Covering.VCLocalizedRegime","a":"Stat/Concentration/Covering","t":2,"d":"If two functions differ by at most a nonnegative amount at every observation in a finite sample, then their empirical distance is at most that amount.","s":"∀ {ι : Type u} {𝒳 : Type v} {G : Causalean.Stat.Concentration.starHullParam ι → 𝒳 → ℝ} {n : ℕ} {S : Fin n → 𝒳} (q q' : Causalean.Stat.Concentration.EmpiricalFunctionSpace G S) {η : ℝ}, 0 ≤ η → (∀ (j : Fin n), |G q.index (S j) - G q'.index (S j)| ≤ η) → dist q q' ≤ η","l":126,"f":"Causalean/Stat/Concentration/Covering/VCLocalizedRegime.lean"},{"n":"Causalean.Stat.Concentration.empiricalNorm","k":"def","m":"Causalean.Stat.Concentration.Covering.EmpiricalPseudoMetric","a":"Stat/Concentration/Covering","t":1,"d":"The empirical norm is the root mean square value of a function on a fixed sample.","s":"{𝒳 : Type v} → {n : ℕ} → (Fin n → 𝒳) → (𝒳 → ℝ) → ℝ","l":26,"f":"Causalean/Stat/Concentration/Covering/EmpiricalPseudoMetric.lean"},{"n":"Causalean.Stat.Concentration.empiricalNorm_const_mul","k":"theorem","m":"Causalean.Stat.Concentration.Covering.VCLocalizedRegime","a":"Stat/Concentration/Covering","t":2,"d":"Scaling a function by a constant scales its empirical root-mean-square norm by the absolute value of that constant.","s":"∀ {𝒳 : Type u_2} {n : ℕ} (S : Fin n → 𝒳) (c : ℝ) (f : 𝒳 → ℝ), (Causalean.Stat.Concentration.empiricalNorm S fun x => c * f x) = |c| * Causalean.Stat.Concentration.empiricalNorm S f","l":614,"f":"Causalean/Stat/Concentration/Covering/VCLocalizedRegime.lean"},{"n":"Causalean.Stat.Concentration.empiricalNorm_def","k":"theorem","m":"Causalean.Stat.Concentration.Covering.EmpiricalPseudoMetric","a":"Stat/Concentration/Covering","t":2,"d":"The empirical norm unfolds to the square root of the average squared sample values.","s":"∀ {𝒳 : Type v} {n : ℕ} (S : Fin n → 𝒳) (f : 𝒳 → ℝ), Causalean.Stat.Concentration.empiricalNorm S f = √(1 / ↑n * ∑ i, f (S i) ^ 2)","l":31,"f":"Causalean/Stat/Concentration/Covering/EmpiricalPseudoMetric.lean"},{"n":"Causalean.Stat.Concentration.empiricalNorm_le_of_forall_abs_le","k":"theorem","m":"Causalean.Stat.Concentration.Covering.VCLocalizedRegime","a":"Stat/Concentration/Covering","t":2,"d":"A function whose absolute value is bounded at every point of a finite sample has empirical root-mean-square norm no larger than that bound.","s":"∀ {𝒳 : Type v} {n : ℕ} (S : Fin n → 𝒳) {f : 𝒳 → ℝ} {η : ℝ}, 0 ≤ η → (∀ (j : Fin n), |f (S j)| ≤ η) → Causalean.Stat.Concentration.empiricalNorm S f ≤ η","l":91,"f":"Causalean/Stat/Concentration/Covering/VCLocalizedRegime.lean"},{"n":"Causalean.Stat.Concentration.empiricalPMet","k":"def","m":"Causalean.Stat.Concentration.Covering.EmpiricalPseudoMetric","a":"Stat/Concentration/Covering","t":1,"d":"The empirical distance defines a pseudometric on functions evaluated on the fixed sample.","s":"{𝒳 : Type v} → {n : ℕ} → (Fin n → 𝒳) → PseudoMetricSpace (𝒳 → ℝ)","l":50,"f":"Causalean/Stat/Concentration/Covering/EmpiricalPseudoMetric.lean"},{"n":"Causalean.Stat.Concentration.empiricalRademacherComplexity_F_on_univ_eq","k":"theorem","m":"Causalean.Stat.Concentration.Covering.VCLocalizedRegime","a":"Stat/Concentration/Covering","t":2,"d":"Restricting a finite function family to its full index set leaves its empirical Rademacher complexity unchanged.","s":"∀ {ι' : Type u_2} {Z : Type u_3} [inst : Fintype ι'] {m : ℕ} (H : ι' → Z → ℝ) (S' : Fin m → Z), empiricalRademacherComplexity m (Causalean.Stat.Concentration.F_on H Finset.univ) S' = empiricalRademacherComplexity m H S'","l":581,"f":"Causalean/Stat/Concentration/Covering/VCLocalizedRegime.lean"},{"n":"Causalean.Stat.Concentration.empiricalRademacherComplexity_congr_sample","k":"theorem","m":"Causalean.Stat.Concentration.Rademacher.Rademacher","a":"Stat/Concentration/Rademacher","t":2,"d":"The empirical Rademacher complexity sees the function family only through its values on the observed sample: two families that agree at every sample point have equal empirical Rademacher complexity.","s":"∀ {𝒳 : Type u_1} {ι : Type u_2} (n : ℕ) (f f' : ι → 𝒳 → ℝ) (sample : Fin n → 𝒳), (∀ (i : ι) (k : Fin n), f i (sample k) = f' i (sample k)) → empiricalRademacherComplexity n f sample = empiricalRademacherComplexity n f' sample","l":51,"f":"Causalean/Stat/Concentration/Rademacher/Rademacher.lean"},{"n":"Causalean.Stat.Concentration.empiricalRademacherComplexity_contraction_abs_of_bddAbove","k":"theorem","m":"Causalean.Stat.Concentration.Rademacher.Contraction","a":"Stat/Concentration/Rademacher","t":2,"d":"Ledoux–Talagrand contraction over an arbitrary (possibly infinite) index. For an L-Lipschitz φ with φ 0 = 0 and a class F with a uniform bound M, R̂_n(φ ∘ F) ≤ 2L · R̂_n(F) for *any* nonempty index ι. The infinite-index case is reduced to the finite-index rademacher_contraction_abs by choosing, for each of the finitely many sign vectors, an ε-approximate maximizer; their finite collection is a finite subindex on which the Fintype contraction applies.","s":"∀ {ι : Type u_1} {𝒳 : Type u_2} [Nonempty ι] (φ : ℝ → ℝ) {L : ℝ}, Causalean.Stat.Concentration.LipschitzAt0 φ L → ∀ (F : ι → 𝒳 → ℝ) {M : ℝ}, 0 ≤ M → (∀ (i : ι) (x : 𝒳), |F i x| ≤ M) → ∀ (n : ℕ) (S : Fin n → 𝒳), empiricalRademacherComplexity n (fun i x => φ (F i x)) S ≤ 2 * L * empiricalRademacherComplexity n F S","l":842,"f":"Causalean/Stat/Concentration/Rademacher/Contraction.lean"},{"n":"Causalean.Stat.Concentration.empiricalRademacherComplexity_mono_of_inner","k":"theorem","m":"Causalean.Stat.Concentration.Rademacher.LocalRademacher","a":"Stat/Concentration/Rademacher","t":2,"d":"Per-σ, per-i inner-expression bound implies empirical Rademacher comparison.","s":"∀ {𝒳 : Type u_3} {ι' : Type u_4} {n : ℕ} (f g : ι' → 𝒳 → ℝ) (S : Fin n → 𝒳), (∀ (σ : Signs n), BddAbove (Set.range fun i => |(↑n)⁻¹ * ∑ k, ↑↑(σ k) * g i (S k)|)) → (∀ (σ : Signs n) (i : ι'), |(↑n)⁻¹ * ∑ k, ↑↑(σ k) * f i (S k)| ≤ |(↑n)⁻¹ * ∑ k, ↑↑(σ k) * g i (S k)|) → empiricalRademacherComplexity n f S ≤ empiricalRademacherComplexity n g S","l":133,"f":"Causalean/Stat/Concentration/Rademacher/LocalRademacher.lean"},{"n":"Causalean.Stat.Concentration.empiricalRademacherComplexity_smul_class","k":"theorem","m":"Causalean.Stat.Concentration.Rademacher.Contraction","a":"Stat/Concentration/Rademacher","t":2,"d":"Scalar-multiplication law for empirical Rademacher complexity. Scaling each element of a function class by c scales the absolute-value empirical Rademacher complexity by |c|.","s":"∀ {ι : Type u_1} {𝒳 : Type u_2} (F : ι → 𝒳 → ℝ) (c : ℝ) (n : ℕ) (S : Fin n → 𝒳), empiricalRademacherComplexity n (fun i x => c * F i x) S = |c| * empiricalRademacherComplexity n F S","l":729,"f":"Causalean/Stat/Concentration/Rademacher/Contraction.lean"},{"n":"Causalean.Stat.Concentration.empiricalRademacherComplexity_sub_le","k":"theorem","m":"Causalean.Stat.Concentration.Rademacher.Contraction","a":"Stat/Concentration/Rademacher","t":2,"d":"Empirical Rademacher complexity is sub-additive over differences of classes. For classes F, G sharing the index ι, each with a uniform bound, the complexity of fun i x => F i x - G i x is at most the sum of the individual complexities.","s":"∀ {ι : Type u_1} {𝒳 : Type u_2} [Nonempty ι] (F G : ι → 𝒳 → ℝ) {MF MG : ℝ}, 0 ≤ MF → 0 ≤ MG → (∀ (i : ι) (x : 𝒳), |F i x| ≤ MF) → (∀ (i : ι) (x : 𝒳), |G i x| ≤ MG) → ∀ (n : ℕ) (S : Fin n → 𝒳), empiricalRademacherComplexity n (fun i x => F i x - G i x) S ≤ empiricalRademacherComplexity n F S + empiricalRademacherComplexity n G S","l":814,"f":"Causalean/Stat/Concentration/Rademacher/Contraction.lean"},{"n":"Causalean.Stat.Concentration.empiricalRademacherComplexity_without_abs_smul_class","k":"theorem","m":"Causalean.Stat.Concentration.Rademacher.Contraction","a":"Stat/Concentration/Rademacher","t":2,"d":"Without-abs analogue of empiricalRademacherComplexity_smul_class: the signed scaling carries c, not |c|.","s":"∀ {ι : Type u_1} {𝒳 : Type u_2} (F : ι → 𝒳 → ℝ) (c : ℝ), 0 ≤ c → ∀ (n : ℕ) (S : Fin n → 𝒳), empiricalRademacherComplexity_without_abs n (fun i x => c * F i x) S = c * empiricalRademacherComplexity_without_abs n F S","l":438,"f":"Causalean/Stat/Concentration/Rademacher/Contraction.lean"},{"n":"Causalean.Stat.Concentration.empiricalRademacher_withAbs_finiteClass_le","k":"theorem","m":"Causalean.Stat.Concentration.Covering.VCLocalizedRegime","a":"Stat/Concentration/Covering","t":2,"d":"The absolute empirical Rademacher complexity of a finite class is bounded by a Massart logarithmic factor times a common radius.","s":"∀ {ι' : Type u_2} {Z : Type u_3} {m : ℕ}, 0 < m → ∀ (H : ι' → Z → ℝ) (S' : Fin m → Z) (f : Finset ι'), f.Nonempty → ∀ (ρ : ℝ), (∀ i ∈ f, √(∑ k, ((↑m)⁻¹ * |H i (S' k)|) ^ 2) ≤ ρ) → empiricalRademacherComplexity m (Causalean.Stat.Concentration.F_on H f) S' ≤ ρ * √(2 * Real.log (2 * ↑f.card))","l":444,"f":"Causalean/Stat/Concentration/Covering/VCLocalizedRegime.lean"},{"n":"Causalean.Stat.Concentration.empiricalVarianceSlack","k":"def","m":"Causalean.Stat.Concentration.TailBounds.EmpiricalBernstein","a":"Stat/Concentration/TailBounds","t":1,"d":"The deterministic slack added to the observed sample variance to upper-bound the population variance with probability ≥ 1 − δ. With M := max |a| |b| and d := √(log(4/δ)/(2n)) the (level δ/2) Hoeffding deviation factor, it is M²·d + 2 M (b − a)·d. The first term controls the second-moment deviation (f² on [0, M²]), the second the squared-mean deviation (|X̄ₙ² − m²| ≤ 2M·|X̄ₙ − m|).","s":"ℝ → ℝ → ℕ → ℝ → ℝ","l":152,"f":"Causalean/Stat/Concentration/TailBounds/EmpiricalBernstein.lean"},{"n":"Causalean.Stat.Concentration.empirical_abs_withZero_le_no_abs_plus_neg","k":"theorem","m":"Causalean.Stat.Concentration.Rademacher.Contraction","a":"Stat/Concentration/Rademacher","t":2,"d":"If a function class contains the zero function, its empirical Rademacher complexity with absolute values is bounded by the sum of the corresponding unsigned complexities for the class and its negation.","s":"∀ {ι : Type u_1} {𝒳 : Type u_2} [Finite ι] (F : Option ι → 𝒳 → ℝ), (∀ (x : 𝒳), F none x = 0) → ∀ (n : ℕ) (S : Fin n → 𝒳), empiricalRademacherComplexity n F S ≤ empiricalRademacherComplexity_without_abs n F S + empiricalRademacherComplexity_without_abs n (fun i x => -F i x) S","l":617,"f":"Causalean/Stat/Concentration/Rademacher/Contraction.lean"},{"n":"Causalean.Stat.Concentration.empirical_bernstein_ci_miss","k":"theorem","m":"Causalean.Stat.Concentration.TailBounds.EmpiricalBernstein","a":"Stat/Concentration/TailBounds","t":1,"d":"Data-driven (empirical) Bernstein confidence interval, miss-probability form. Let S be an i.i.d. sample and let f be a measurable statistic taking values in an interval [a,b] with a < b, with population mean m = ∫ f ∂P. Suppose c is a nonnegative bound with f deviating from m by at most c, P-almost everywhere, the population variance σ² = ∫ (f − m)² ∂P is strictly positive, the sample size n is positive, and the confidence level δ lies in (0, 1]. Then the population mean m falls outside the random, data-driven interval [X̄ₙ − ŵ(ω), X̄ₙ + ŵ(ω)] — whose half-width ŵ(ω) = empiricalBernsteinCIHalfWidth S f a b c n δ ω is computed from the observed sample variance via √(V̂ₙ(ω) + τ) — with probability at most 2δ.","s":"∀ {Ω : Type u_1} {X : Type u_2} [inst : MeasurableSpace Ω] [inst_1 : MeasurableSpace X] {μ : MeasureTheory.Measure Ω} {P : MeasureTheory.Measure X} (S : Causalean.Stat.IIDSample Ω X μ P) {f : X → ℝ}, Measurable f → ∀ {a b : ℝ}, a < b → (∀ᵐ (x : X) ∂P, f x ∈ Set.Icc a b) → ∀ {c : ℝ}, 0 ≤ c → (∀ᵐ (x : X) ∂P, |f x - ∫ (y : X), f y ∂P| ≤ c) → 0 < ∫ (x : X), (f x - ∫ (y : X), f y ∂P) ^ 2 ∂P → ∀ (n : ℕ), 0 < n → ∀ {δ : ℝ}, 0 < δ → δ ≤ 1 → μ.real {ω | Causalean.Stat.Concentration.empiricalBernsteinCIHalfWidth S f a b c n δ ω ≤ |S.sampleMean f n ω - ∫ (x : X), f x ∂P|} ≤ 2 * δ","l":374,"f":"Causalean/Stat/Concentration/TailBounds/EmpiricalBernstein.lean"},{"n":"Causalean.Stat.Concentration.empirical_dist_eq_zero_of_factor_pattern","k":"theorem","m":"Causalean.Stat.Concentration.Covering.VCCovering","a":"Stat/Concentration/Covering","t":2,"d":"If two functions induce the same Boolean pattern at every observation in a finite sample, then their empirical distance is zero whenever their values factor through those patterns.","s":"∀ {𝒳 : Type v} {ι : Type u} {n : ℕ} {F : ι → 𝒳 → ℝ} {S : Fin n → 𝒳} {π : ι → 𝒳 → Bool} {φ : Fin n → Bool → ℝ}, (∀ (i : ι) (j : Fin n), F i (S j) = φ j (π i (S j))) → ∀ {i i' : ι}, Causalean.Stat.Concentration.restrictionPattern (π i') S = Causalean.Stat.Concentration.restrictionPattern (π i) S → dist { index := i } { index := i' } = 0","l":76,"f":"Causalean/Stat/Concentration/Covering/VCCovering.lean"},{"n":"Causalean.Stat.Concentration.empirical_variance_concentration","k":"theorem","m":"Causalean.Stat.Concentration.TailBounds.EmpiricalBernstein","a":"Stat/Concentration/TailBounds","t":1,"d":"Data-driven population-variance bound. Let S be an i.i.d. sample and let f be a measurable statistic taking values in an interval [a, b] with a < b P-almost everywhere. For any confidence level δ in (0, 1] and any positive sample size n, the event that the observed sample variance plus the deterministic slack empiricalVarianceSlack a b n δ falls short of the population variance σ² = ∫ (f − ∫ f)² ∂P has probability at most δ; equivalently, with probability at least 1 − δ the population variance is bounded by the observed sample variance plus that slack.","s":"∀ {Ω : Type u_1} {X : Type u_2} [inst : MeasurableSpace Ω] [inst_1 : MeasurableSpace X] {μ : MeasureTheory.Measure Ω} {P : MeasureTheory.Measure X} (S : Causalean.Stat.IIDSample Ω X μ P) {f : X → ℝ}, Measurable f → ∀ {a b : ℝ}, a < b → (∀ᵐ (x : X) ∂P, f x ∈ Set.Icc a b) → ∀ (n : ℕ), 0 < n → ∀ {δ : ℝ}, 0 < δ → δ ≤ 1 → μ.real {ω | S.sampleVariance f n ω + Causalean.Stat.Concentration.empiricalVarianceSlack a b n δ < ∫ (x : X), (f x - ∫ (y : X), f y ∂P) ^ 2 ∂P} ≤ δ","l":173,"f":"Causalean/Stat/Concentration/TailBounds/EmpiricalBernstein.lean"},{"n":"Causalean.Stat.Concentration.erm_excess_le_two_uniformDeviation","k":"theorem","m":"Causalean.Stat.Concentration.UniformDeviation.ERMOracle","a":"Stat/Concentration/UniformDeviation","t":2,"d":"ERM basic inequality (deterministic). If ihat beats the comparator istar in empirical risk on the sample X ∘ ω (Rₙ(ihat) ≤ Rₙ(istar)), then its excess population risk is at most twice the uniform deviation of the loss class on that sample.","s":"∀ {n : ℕ} {Ω : Type u_1} [inst : MeasurableSpace Ω] {ι : Type u_2} {𝒳 : Type u_3} {μ : MeasureTheory.Measure Ω} {f : ι → 𝒳 → ℝ} [MeasureTheory.IsProbabilityMeasure μ] (X : Ω → 𝒳), (∀ (i : ι), AEMeasurable (f i ∘ X) μ) → ∀ (ω : Fin n → Ω) {b : ℝ}, 0 ≤ b → (∀ (i : ι) (x : 𝒳), |f i x| ≤ b) → ∀ (ihat istar : ι), (↑n)⁻¹ * ∑ k, f ihat (X (ω k)) ≤ (↑n)⁻¹ * ∑ k, f istar (X (ω k)) → ∫ (x : Ω), (fun ω' => f ihat (X ω')) x ∂μ - ∫ (x : Ω), (fun ω' => f istar (X ω')) x ∂μ ≤ 2 * uniformDeviation n f μ X (X ∘ ω)","l":38,"f":"Causalean/Stat/Concentration/UniformDeviation/ERMOracle.lean"},{"n":"Causalean.Stat.Concentration.erm_oracle_inequality","k":"theorem","m":"Causalean.Stat.Concentration.UniformDeviation.ERMOracle","a":"Stat/Concentration/UniformDeviation","t":1,"d":"Generic ERM oracle inequality (Rademacher). Consider a countable hypothesis class indexed by ι, evaluated through measurable loss functions f i composed with a measurable data map X, where every loss value is bounded in absolute value by a nonnegative constant b. Let ihat assign to each sample of size n an index that attains empirical risk no larger than that of a fixed comparator istar. Then, provided the McDiarmid tail parameter t satisfies t·b² ≤ 1/2 and ε is nonnegative, the probability, over the n-fold product sample, that the excess population risk of ihat over istar exceeds 4·𝔯ₙ + 2ε — where 𝔯ₙ is the Rademacher complexity of the loss class — is at most exp(−ε²·t·n).","s":"∀ {n : ℕ} {Ω : Type u_1} [inst : MeasurableSpace Ω] {ι : Type u_2} {𝒳 : Type u_3} {μ : MeasureTheory.Measure Ω} {f : ι → 𝒳 → ℝ} [inst_1 : MeasurableSpace 𝒳] [Nonempty 𝒳] [Countable ι] [MeasureTheory.IsProbabilityMeasure μ], (∀ (i : ι), Measurable (f i)) → ∀ (X : Ω → 𝒳), Measurable X → ∀ {b : ℝ}, 0 ≤ b → (∀ (i : ι) (x : 𝒳), |f i x| ≤ b) → ∀ {t : ℝ}, t * b ^ 2 ≤ 1 / 2 → ∀ {ε : ℝ}, 0 ≤ ε → ∀ (ihat : (Fin n → Ω) → ι) (istar : ι), (∀ (ω : Fin n → Ω), (↑n)⁻¹ * ∑ k, f (ihat ω) (X (ω k)) ≤ (↑n)⁻¹ * ∑ k, f istar (X (ω k))) → ((MeasureTheory.Measure.pi fun x => μ) fun ω => 4 • rademacherComplexity n f μ X + 2 * ε < ∫ (x : Ω), (fun ω' => f (ihat ω) (X ω')) x ∂μ - ∫ (x : Ω), (fun ω' => f istar (X ω')) x ∂μ).toReal ≤ Real.exp (-ε ^ 2 * t * ↑n)","l":118,"f":"Causalean/Stat/Concentration/UniformDeviation/ERMOracle.lean"},{"n":"Causalean.Stat.Concentration.erm_oracle_inequality_separable","k":"theorem","m":"Causalean.Stat.Concentration.UniformDeviation.ERMOracle","a":"Stat/Concentration/UniformDeviation","t":1,"d":"Generic ERM oracle inequality (separable class). As in erm_oracle_inequality, but the hypothesis index ι need only be a separable, first-countable topological space rather than countable — the form that covers the (uncountable but separable) L²/L¹-ball linear classes. Given measurable loss functions f i composed with a measurable data map X, with every loss value bounded in absolute value by a nonnegative constant b and each loss value f i x depending continuously on the index i, let ihat assign to each sample of size n an index that attains empirical risk no larger than that of a fixed comparator istar. Then, provided the McDiarmid tail parameter t satisfies t·b² ≤ 1/2 and ε is nonnegative, the probability, over the n-fold product sample, that the excess population risk of ihat over istar exceeds 4·𝔯ₙ + 2ε — where 𝔯ₙ is the Rademacher complexity of the loss class — is at most exp(−ε²·t·n).","s":"∀ {n : ℕ} {Ω : Type u_1} [inst : MeasurableSpace Ω] {ι : Type u_2} {𝒳 : Type u_3} {μ : MeasureTheory.Measure Ω} {f : ι → 𝒳 → ℝ} [inst_1 : MeasurableSpace 𝒳] [Nonempty 𝒳] [inst_3 : TopologicalSpace ι] [TopologicalSpace.SeparableSpace ι] [FirstCountableTopology ι] [MeasureTheory.IsProbabilityMeasure μ], (∀ (i : ι), Measurable (f i)) → ∀ (X : Ω → 𝒳), Measurable X → ∀ {b : ℝ}, 0 ≤ b → (∀ (i : ι) (x : 𝒳), |f i x| ≤ b) → (∀ (x : 𝒳), Continuous fun i => f i x) → ∀ {t : ℝ}, t * b ^ 2 ≤ 1 / 2 → ∀ {ε : ℝ}, 0 ≤ ε → ∀ (ihat : (Fin n → Ω) → ι) (istar : ι), (∀ (ω : Fin n → Ω), (↑n)⁻¹ * ∑ k, f (ihat ω) (X (ω k)) ≤ (↑n)⁻¹ * ∑ k, f istar (X (ω k))) → ((MeasureTheory.Measure.pi fun x => μ) fun ω => 4 • rademacherComplexity n f μ X + 2 * ε < ∫ (x : Ω), (fun ω' => f (ihat ω) (X ω')) x ∂μ - ∫ (x : Ω), (fun ω' => f istar (X ω')) x ∂μ).toReal ≤ Real.exp (-ε ^ 2 * t * ↑n)","l":167,"f":"Causalean/Stat/Concentration/UniformDeviation/ERMOracle.lean"},{"n":"Causalean.Stat.Concentration.exists_separating_subsample","k":"theorem","m":"Causalean.Stat.Concentration.Covering.HausslerPacking","a":"Stat/Concentration/Covering","t":2,"d":"Honest isolated core: the weighted random-coordinate extraction.","s":"∀ {n : ℕ} (d : ℕ) (w : Fin n → ℝ), (∀ (j : Fin n), 0 ≤ w j) → ∀ (r ε : ℝ), 0 < r → 0 < ε → ∑ j, w j ≤ r ^ 2 → ∀ (P : Finset (Fin n → Bool)), 2 ≤ P.card → (Finset.image (fun a => {j | a j = true}) P).vcDim ≤ d → (∀ a ∈ P, ∀ b ∈ P, a ≠ b → ε ^ 2 ≤ Causalean.Stat.Concentration.weightedHammingSq w a b) → ∃ m J, ↑m ≤ 1 + 2 * r ^ 2 / ε ^ 2 * Real.log ↑P.card ∧ (Finset.image (Causalean.Stat.Concentration.subsamplePattern J) P).vcDim ≤ d ∧ Set.InjOn (Causalean.Stat.Concentration.subsamplePattern J) ↑P","l":523,"f":"Causalean/Stat/Concentration/Covering/HausslerPacking.lean"},{"n":"Causalean.Stat.Concentration.exp_le_one_add_add_sq","k":"theorem","m":"Causalean.Stat.Concentration.TailBounds.Bernstein","a":"Stat/Concentration/TailBounds","t":2,"d":"Pointwise elementary bound exp u ≤ 1 + u + u² valid for |u| ≤ 1.","s":"∀ {u : ℝ}, |u| ≤ 1 → Real.exp u ≤ 1 + u + u ^ 2","l":63,"f":"Causalean/Stat/Concentration/TailBounds/Bernstein.lean"},{"n":"Causalean.Stat.Concentration.exp_mul_le_secant","k":"theorem","m":"Causalean.Stat.Concentration.TailBounds.BinomialCount","a":"Stat/Concentration/TailBounds","t":2,"d":"On the unit interval the exponential function stays below the chord joining its values at the two endpoints: for a number x between zero and one and any tilt s, exp (s * x) is at most 1 + x * (exp s - 1). This is the convexity step behind every Bernoulli-type Chernoff bound.","s":"∀ {x s : ℝ}, x ∈ Set.Icc 0 1 → Real.exp (s * x) ≤ 1 + x * (Real.exp s - 1)","l":72,"f":"Causalean/Stat/Concentration/TailBounds/BinomialCount.lean"},{"n":"Causalean.Stat.Concentration.exp_neg_log_two_div","k":"theorem","m":"Causalean.Stat.Concentration.UniformDeviation.ConfidenceInterval","a":"Stat/Concentration/UniformDeviation","t":2,"d":"For a positive confidence level, exponentiating the negative logarithm of twice its inverse returns half that confidence level.","s":"∀ {δ : ℝ}, 0 < δ → Real.exp (-Real.log (2 / δ)) = δ / 2","l":78,"f":"Causalean/Stat/Concentration/UniformDeviation/ConfidenceInterval.lean"},{"n":"Causalean.Stat.Concentration.finiteSampleMeasure","k":"def","m":"Causalean.Stat.Concentration.Covering.RealValuedVCSubgraph.Empirical","a":"Stat/Concentration/Covering/RealValuedVCSubgraph","t":1,"d":"The empirical probability law of a nonempty sample is the normalized sum of Dirac masses at its observations.","s":"{𝒳 : Type u} → [inst : MeasurableSpace 𝒳] → {n : ℕ} → (Fin n → 𝒳) → MeasureTheory.Measure 𝒳","l":23,"f":"Causalean/Stat/Concentration/Covering/RealValuedVCSubgraph/Empirical.lean"},{"n":"Causalean.Stat.Concentration.finiteSampleMeasure_isProbabilityMeasure","k":"theorem","m":"Causalean.Stat.Concentration.Covering.RealValuedVCSubgraph.Empirical","a":"Stat/Concentration/Covering/RealValuedVCSubgraph","t":2,"d":"The empirical law of a positive-size finite sample is a probability measure.","s":"∀ {𝒳 : Type u} [inst : MeasurableSpace 𝒳] {n : ℕ} (S : Fin n → 𝒳), 0 < n → MeasureTheory.IsProbabilityMeasure (Causalean.Stat.Concentration.finiteSampleMeasure S)","l":28,"f":"Causalean/Stat/Concentration/Covering/RealValuedVCSubgraph/Empirical.lean"},{"n":"Causalean.Stat.Concentration.finite_averaging_exists_separating_subsample","k":"theorem","m":"Causalean.Stat.Concentration.Covering.HausslerPacking","a":"Stat/Concentration/Covering","t":2,"d":"Finite averaging core for the weighted random-coordinate extraction.","s":"∀ {n : ℕ} (w : Fin n → ℝ), (∀ (j : Fin n), 0 ≤ w j) → ∀ (r ε : ℝ), 0 < r → 0 < ε → ∑ j, w j ≤ r ^ 2 → ∀ (P : Finset (Fin n → Bool)), 2 ≤ P.card → (∀ a ∈ P, ∀ b ∈ P, a ≠ b → ε ^ 2 ≤ Causalean.Stat.Concentration.weightedHammingSq w a b) → ∃ m J, ↑m ≤ 1 + 2 * r ^ 2 / ε ^ 2 * Real.log ↑P.card ∧ Set.InjOn (Causalean.Stat.Concentration.subsamplePattern J) ↑P","l":314,"f":"Causalean/Stat/Concentration/Covering/HausslerPacking.lean"},{"n":"Causalean.Stat.Concentration.finite_iSup_eq_value","k":"theorem","m":"Causalean.Stat.Concentration.Rademacher.Contraction","a":"Stat/Concentration/Rademacher","t":2,"d":"A real-valued function on a nonempty finite population attains a largest value, and its supremum is that value.","s":"∀ {α : Type u_3} [Nonempty α] [Finite α] (f : α → ℝ), ∃ a, ⨆ x, f x = f a ∧ ∀ (x : α), f x ≤ f a","l":103,"f":"Causalean/Stat/Concentration/Rademacher/Contraction.lean"},{"n":"Causalean.Stat.Concentration.growthFamily","k":"def","m":"Causalean.Stat.Concentration.Covering.VCCovering","a":"Stat/Concentration/Covering","t":1,"d":"The finite family of Boolean restriction patterns realized on a sample.","s":"{𝒳 : Type v} → {ι : Type u} → {n : ℕ} → (ι → 𝒳 → Bool) → (Fin n → 𝒳) → Finset (Finset (Fin n))","l":36,"f":"Causalean/Stat/Concentration/Covering/VCCovering.lean"},{"n":"Causalean.Stat.Concentration.growthFamilyRep","k":"def","m":"Causalean.Stat.Concentration.Covering.VCLocalizedRegime","a":"Stat/Concentration/Covering","t":1,"d":"A representative classifier index is chosen for each realized Boolean growth-family pattern.","s":"{ι : Type u_2} → {𝒳 : Type u_3} → {n : ℕ} → (π : ι → 𝒳 → Bool) → (S : Fin n → 𝒳) → ↥(Causalean.Stat.Concentration.growthFamily π S) → ι","l":643,"f":"Causalean/Stat/Concentration/Covering/VCLocalizedRegime.lean"},{"n":"Causalean.Stat.Concentration.growthFamilyRep_spec","k":"theorem","m":"Causalean.Stat.Concentration.Covering.VCLocalizedRegime","a":"Stat/Concentration/Covering","t":2,"d":"The chosen growth-family representative realizes the pattern it represents.","s":"∀ {ι : Type u_2} {𝒳 : Type u_3} {n : ℕ} (π : ι → 𝒳 → Bool) (S : Fin n → 𝒳) (A : ↥(Causalean.Stat.Concentration.growthFamily π S)), Causalean.Stat.Concentration.restrictionPattern (π (Causalean.Stat.Concentration.growthFamilyRep π S A)) S = ↑A","l":650,"f":"Causalean/Stat/Concentration/Covering/VCLocalizedRegime.lean"},{"n":"Causalean.Stat.Concentration.growthFamily_card_le_succ_pow_of_trace","k":"theorem","m":"Causalean.Stat.Concentration.Covering.VCLocalizedRegime","a":"Stat/Concentration/Covering","t":2,"d":"A Boolean class whose trace family has bounded VC dimension, or a direct trace-size bound, realizes no more patterns on a finite sample than a polynomial of degree d in one plus the sample size.","s":"∀ {ι : Type u} {𝒳 : Type v} (π : ι → 𝒳 → Bool) (d n : ℕ), Causalean.Stat.Concentration.BinaryTraceEntropyControl π d → ∀ (S : Fin n → 𝒳), (Causalean.Stat.Concentration.growthFamily π S).card ≤ (n + 1) ^ d","l":399,"f":"Causalean/Stat/Concentration/Covering/VCLocalizedRegime.lean"},{"n":"Causalean.Stat.Concentration.hasCountableEmpiricalSupReduction_of_pointwise_dense","k":"theorem","m":"Causalean.Stat.Concentration.VarianceAdaptiveVCExpectedMaximal.Separability","a":"Stat/Concentration/VarianceAdaptiveVCExpectedMaximal","t":1,"d":"Countable supremum reduction from pointwise density. Let μ be a σ-finite measure on Ω, g : ι → Ω → ℝ a family of functions, and g0 : ℕ → ι a countable subfamily. Suppose S is a μ-conull subset of Ω, on S, every g i is the pointwise limit, along some subsequence, of the countable subfamily g ∘ g0, each g i is measurable, and there is a single μ-integrable envelope G dominating |g i| uniformly in i. Then the countable subfamily indexed by g0 realizes the full continuum empirical-process supremum of g almost surely under every finite product law of μ.","s":"∀ {Ω : Type u_1} {ι : Type u_2} [inst : MeasurableSpace Ω] (μ : MeasureTheory.Measure Ω) [MeasureTheory.SigmaFinite μ] (g : ι → Ω → ℝ) (g0 : ℕ → ι) (S : Set Ω), (∀ᵐ (z : Ω) ∂μ, z ∈ S) → (∀ (i : ι), ∃ kseq, ∀ z ∈ S, Filter.Tendsto (fun m => g (g0 (kseq m)) z) Filter.atTop (nhds (g i z))) → (∀ (i : ι), Measurable (g i)) → (∃ G, MeasureTheory.Integrable G μ ∧ ∀ (i : ι) (z : Ω), |g i z| ≤ G z) → Causalean.Stat.Concentration.HasCountableEmpiricalSupReduction μ g","l":41,"f":"Causalean/Stat/Concentration/VarianceAdaptiveVCExpectedMaximal/Separability.lean"},{"n":"Causalean.Stat.Concentration.hasPseudoDimAtMost_iff_growthFamily","k":"theorem","m":"Causalean.Stat.Concentration.Covering.RealValuedVCSubgraph.Basic","a":"Stat/Concentration/Covering/RealValuedVCSubgraph","t":2,"d":"The pseudo-dimension certificate unfolds to the existing finite Boolean growth-family VC certificate on every thresholded sample.","s":"∀ {𝒳 : Type u} {ι : Type v} (F : ι → 𝒳 → ℝ) (d : ℕ), Causalean.Stat.Concentration.HasPseudoDimAtMost F d ↔ ∀ (n : ℕ) (T : Fin n → 𝒳 × ℝ), (Causalean.Stat.Concentration.growthFamily (Causalean.Stat.Concentration.subgraphClassifier F) T).vcDim ≤ d","l":31,"f":"Causalean/Stat/Concentration/Covering/RealValuedVCSubgraph/Basic.lean"},{"n":"Causalean.Stat.Concentration.hasSubgaussianMGF_linearCombination_of_iIndep","k":"theorem","m":"Causalean.Stat.Concentration.ConditionalKernel","a":"Stat/Concentration","t":1,"d":"For a finite family of real random variables eps and real coefficients v, if each eps i is a.e. measurable, each eps i is a.s. bounded by 1 in absolute value, each eps i has mean zero, and the family eps is independent, then the linear combination $\\sum_i v_i \\cdot \\mathrm{eps}_i$ is sub-Gaussian with variance proxy $\\sum_i v_i^2$.","s":"∀ {Ω : Type u_1} {ι : Type u_2} [inst : MeasurableSpace Ω] [inst_1 : Fintype ι] {P : MeasureTheory.Measure Ω} [MeasureTheory.IsProbabilityMeasure P] (eps : ι → Ω → ℝ) (v : ι → ℝ), (∀ (i : ι), AEMeasurable (eps i) P) → (∀ (i : ι), ∀ᵐ (ω : Ω) ∂P, |eps i ω| ≤ 1) → (∀ (i : ι), ∫ (ω : Ω), eps i ω ∂P = 0) → ProbabilityTheory.iIndepFun eps P → ProbabilityTheory.HasSubgaussianMGF (fun ω => ∑ i, v i * eps i ω) ⟨∑ i, v i ^ 2, Finset.sum_nonneg fun i x => Even.pow_nonneg (even_two_mul 1) (v i)⟩ P","l":97,"f":"Causalean/Stat/Concentration/ConditionalKernel.lean"},{"n":"Causalean.Stat.Concentration.hoeffdingCIHalfWidth","k":"def","m":"Causalean.Stat.Concentration.UniformDeviation.ConfidenceInterval","a":"Stat/Concentration/UniformDeviation","t":1,"d":"The two-sided Hoeffding confidence half-width at level δ for an [a, b]-valued statistic and sample size n: (b − a)·√(log(2/δ) / (2n)).","s":"ℝ → ℝ → ℕ → ℝ → ℝ","l":58,"f":"Causalean/Stat/Concentration/UniformDeviation/ConfidenceInterval.lean"},{"n":"Causalean.Stat.Concentration.hoeffdingCIHalfWidth_half","k":"theorem","m":"Causalean.Stat.Concentration.TailBounds.EmpiricalBernstein","a":"Stat/Concentration/TailBounds","t":2,"d":"Halving the confidence level in a Hoeffding half-width replaces its logarithmic factor with log(4/δ), so the resulting half-width can be used in split-confidence and union-bound calculations.","s":"∀ (a b : ℝ) (n : ℕ) {δ : ℝ}, Causalean.Stat.Concentration.hoeffdingCIHalfWidth a b n (δ / 2) = (b - a) * √(Real.log (4 / δ) / (2 * ↑n))","l":162,"f":"Causalean/Stat/Concentration/TailBounds/EmpiricalBernstein.lean"},{"n":"Causalean.Stat.Concentration.hoeffding_abs_ge","k":"theorem","m":"Causalean.Stat.Concentration.TailBounds.Hoeffding","a":"Stat/Concentration/TailBounds","t":1,"d":"Two-sided Hoeffding inequality for the sample mean of a bounded statistic. Let S be an i.i.d. sample and let f be a measurable statistic taking values in an interval [a, b] with a < b P-almost everywhere. Then for any sample size n ≥ 1 and any threshold ε ≥ 0, the probability that the sample mean of f over n draws deviates from its population mean $E[f]$ by at least ε in absolute value is at most $2\\exp(-2n\\varepsilon^2/(b-a)^2)$.","s":"∀ {Ω : Type u_1} {X : Type u_2} [inst : MeasurableSpace Ω] [inst_1 : MeasurableSpace X] {μ : MeasureTheory.Measure Ω} {P : MeasureTheory.Measure X} (S : Causalean.Stat.IIDSample Ω X μ P) {f : X → ℝ}, Measurable f → ∀ {a b : ℝ}, a < b → (∀ᵐ (x : X) ∂P, f x ∈ Set.Icc a b) → ∀ (n : ℕ), 0 < n → ∀ {ε : ℝ}, 0 ≤ ε → μ.real {ω | ε ≤ |S.sampleMean f n ω - ∫ (x : X), f x ∂P|} ≤ 2 * Real.exp (-2 * ↑n * ε ^ 2 / (b - a) ^ 2)","l":161,"f":"Causalean/Stat/Concentration/TailBounds/Hoeffding.lean"},{"n":"Causalean.Stat.Concentration.hoeffding_ci_cover","k":"theorem","m":"Causalean.Stat.Concentration.UniformDeviation.ConfidenceInterval","a":"Stat/Concentration/UniformDeviation","t":1,"d":"Finite-sample Hoeffding confidence interval (coverage form). Let S be an i.i.d. sample drawn from P, and write m = ∫ f dP for the population mean. If f is measurable, if f is almost-everywhere valued in the interval [a, b] with a < b, if the sample size n is positive, and if the confidence level δ lies in (0, 1], then with probability at least 1 − δ the sample mean X̄ₙ of f over S's first n draws satisfies |X̄ₙ − m| < w, where w = hoeffdingCIHalfWidth a b n δ.","s":"∀ {Ω : Type u_1} {X' : Type u_2} [inst : MeasurableSpace Ω] [inst_1 : MeasurableSpace X'] {μ : MeasureTheory.Measure Ω} {P : MeasureTheory.Measure X'} (S : Causalean.Stat.IIDSample Ω X' μ P) {f : X' → ℝ}, Measurable f → ∀ {a b : ℝ}, a < b → (∀ᵐ (x : X') ∂P, f x ∈ Set.Icc a b) → ∀ (n : ℕ), 0 < n → ∀ {δ : ℝ}, 0 < δ → δ ≤ 1 → 1 - δ ≤ μ.real {ω | |S.sampleMean f n ω - ∫ (x : X'), f x ∂P| < Causalean.Stat.Concentration.hoeffdingCIHalfWidth a b n δ}","l":178,"f":"Causalean/Stat/Concentration/UniformDeviation/ConfidenceInterval.lean"},{"n":"Causalean.Stat.Concentration.hoeffding_ci_miss","k":"theorem","m":"Causalean.Stat.Concentration.UniformDeviation.ConfidenceInterval","a":"Stat/Concentration/UniformDeviation","t":1,"d":"Finite-sample Hoeffding confidence interval (miss-probability form). Let S be an i.i.d. sample drawn from P, and write m = ∫ f dP for the population mean. If f is measurable, if f is almost-everywhere valued in the interval [a, b] with a < b, if the sample size n is positive, and if the confidence level δ lies in (0, 1], then with probability at most δ the sample mean X̄ₙ of f over S's first n draws satisfies w ≤ |X̄ₙ − m|, where w = hoeffdingCIHalfWidth a b n δ.","s":"∀ {Ω : Type u_1} {X' : Type u_2} [inst : MeasurableSpace Ω] [inst_1 : MeasurableSpace X'] {μ : MeasureTheory.Measure Ω} {P : MeasureTheory.Measure X'} (S : Causalean.Stat.IIDSample Ω X' μ P) {f : X' → ℝ}, Measurable f → ∀ {a b : ℝ}, a < b → (∀ᵐ (x : X') ∂P, f x ∈ Set.Icc a b) → ∀ (n : ℕ), 0 < n → ∀ {δ : ℝ}, 0 < δ → δ ≤ 1 → μ.real {ω | Causalean.Stat.Concentration.hoeffdingCIHalfWidth a b n δ ≤ |S.sampleMean f n ω - ∫ (x : X'), f x ∂P|} ≤ δ","l":85,"f":"Causalean/Stat/Concentration/UniformDeviation/ConfidenceInterval.lean"},{"n":"Causalean.Stat.Concentration.hoeffding_ge","k":"theorem","m":"Causalean.Stat.Concentration.TailBounds.Hoeffding","a":"Stat/Concentration/TailBounds","t":1,"d":"One-sided Hoeffding inequality for the sample mean of a bounded statistic. Let S be an i.i.d. sample and let f be a measurable statistic taking values in an interval [a, b] with a < b P-almost everywhere. Then for any sample size n ≥ 1 and any threshold ε ≥ 0, the probability that the sample mean of f over n draws exceeds its population mean $E[f]$ by at least ε is at most $\\exp(-2n\\varepsilon^2/(b-a)^2)$.","s":"∀ {Ω : Type u_1} {X : Type u_2} [inst : MeasurableSpace Ω] [inst_1 : MeasurableSpace X] {μ : MeasureTheory.Measure Ω} {P : MeasureTheory.Measure X} (S : Causalean.Stat.IIDSample Ω X μ P) {f : X → ℝ}, Measurable f → ∀ {a b : ℝ}, a < b → (∀ᵐ (x : X) ∂P, f x ∈ Set.Icc a b) → ∀ (n : ℕ), 0 < n → ∀ {ε : ℝ}, 0 ≤ ε → μ.real {ω | ε ≤ S.sampleMean f n ω - ∫ (x : X), f x ∂P} ≤ Real.exp (-2 * ↑n * ε ^ 2 / (b - a) ^ 2)","l":89,"f":"Causalean/Stat/Concentration/TailBounds/Hoeffding.lean"},{"n":"Causalean.Stat.Concentration.iSup_abs_le_iSup_add_iSup_neg_of_exists_zero","k":"theorem","m":"Causalean.Stat.Concentration.Rademacher.Contraction","a":"Stat/Concentration/Rademacher","t":2,"d":"If a finite collection of real numbers includes zero, its largest absolute value is at most the sum of its largest value and the largest value after negation.","s":"∀ {α : Type u_3} [Finite α] (x : α → ℝ), (∃ a, x a = 0) → ⨆ a, |x a| ≤ (⨆ a, x a) + ⨆ a, -x a","l":531,"f":"Causalean/Stat/Concentration/Rademacher/Contraction.lean"},{"n":"Causalean.Stat.Concentration.iid_sum_chebyshev","k":"theorem","m":"Causalean.Stat.Concentration.Matrix.IidSums","a":"Stat/Concentration/Matrix","t":1,"d":"Chebyshev concentration of an iid sum. Combining the iid expectation and variance laws with Chebyshev's inequality: for a fixed square-integrable statistic g and any positive threshold ε, the sum ∑ᵢ g(Xᵢ) over an iid sample of size N deviates from its mean N·𝔼[g] by at least ε with probability at most N·Var[g]/ε². For the design weight g = K((·−t)/h) this is the concentration of the total kernel weight M₀₀ around N·𝔼[K((A−t)/h)] = Θ(Nh).","s":"∀ {N : ℕ} {Ω : Type u_1} [inst : MeasurableSpace Ω] (μ : MeasureTheory.Measure Ω) [MeasureTheory.IsProbabilityMeasure μ] (g : Ω → ℝ), MeasureTheory.MemLp g 2 μ → ∀ {ε : ℝ}, 0 < ε → (MeasureTheory.Measure.pi fun x => μ) {ω | ε ≤ |∑ i, g (ω i) - ↑N * ∫ (x : Ω), g x ∂μ|} ≤ ENNReal.ofReal (↑N * ProbabilityTheory.variance g μ / ε ^ 2)","l":78,"f":"Causalean/Stat/Concentration/Matrix/IidSums.lean"},{"n":"Causalean.Stat.Concentration.iid_sum_union_bound","k":"theorem","m":"Causalean.Stat.Concentration.Matrix.InverseUnionBound","a":"Stat/Concentration/Matrix","t":1,"d":"Union bound for iid coordinate sums. For a finite family g : ι → Ω → ℝ such that each g a is square-integrable under μ, evaluated on an iid sample of size N drawn from the product law Measure.pi, and for any positive deviation threshold η, the probability that *some* index a has its sample sum ∑ᵢ g a (ωᵢ) deviate from its mean N·𝔼[g a] by at least η is bounded by the sum of the per-index Chebyshev bounds N·Var[g a]/η².","s":"∀ {N : ℕ} {ι : Type u_1} [inst : Fintype ι] {Ω : Type u_2} [inst_1 : MeasurableSpace Ω] (μ : MeasureTheory.Measure Ω) [MeasureTheory.IsProbabilityMeasure μ] (g : ι → Ω → ℝ), (∀ (a : ι), MeasureTheory.MemLp (g a) 2 μ) → ∀ {η : ℝ}, 0 < η → (MeasureTheory.Measure.pi fun x => μ) {ω | ∃ a, η ≤ |∑ i, g a (ω i) - ↑N * ∫ (x : Ω), g a x ∂μ|} ≤ ∑ a, ENNReal.ofReal (↑N * ProbabilityTheory.variance (g a) μ / η ^ 2)","l":28,"f":"Causalean/Stat/Concentration/Matrix/InverseUnionBound.lean"},{"n":"Causalean.Stat.Concentration.instCoeFunEmpiricalFunctionSpaceForallReal","k":"instance","m":"Causalean.Stat.Concentration.Covering.EmpiricalPseudoMetric","a":"Stat/Concentration/Covering","t":2,"d":"An empirical function-space element evaluates as the function selected by its index.","s":"{𝒳 : Type v} → {n : ℕ} → {ι : Type u} → {F : ι → 𝒳 → ℝ} → {S : Fin n → 𝒳} → CoeFun (Causalean.Stat.Concentration.EmpiricalFunctionSpace F S) fun x => 𝒳 → ℝ","l":142,"f":"Causalean/Stat/Concentration/Covering/EmpiricalPseudoMetric.lean"},{"n":"Causalean.Stat.Concentration.instDistEmpiricalFunctionSpace","k":"instance","m":"Causalean.Stat.Concentration.Covering.EmpiricalPseudoMetric","a":"Stat/Concentration/Covering","t":2,"d":"The empirical function space inherits the empirical distance on the selected functions.","s":"{𝒳 : Type v} → {n : ℕ} → {ι : Type u} → {F : ι → 𝒳 → ℝ} → {S : Fin n → 𝒳} → Dist (Causalean.Stat.Concentration.EmpiricalFunctionSpace F S)","l":153,"f":"Causalean/Stat/Concentration/Covering/EmpiricalPseudoMetric.lean"},{"n":"Causalean.Stat.Concentration.instMeasurableSingletonClassSigns_causalean","k":"instance","m":"Causalean.Stat.Concentration.TailBounds.Massart","a":"Stat/Concentration/TailBounds","t":2,"d":"The finite sign-vector space has measurable singletons under the product measurable structure.","s":"∀ {m : ℕ}, MeasurableSingletonClass (Signs m)","l":34,"f":"Causalean/Stat/Concentration/TailBounds/Massart.lean"},{"n":"Causalean.Stat.Concentration.instNonemptySubtypeIntMemFinsetInsertNegOfNatSingleton_causalean","k":"instance","m":"Causalean.Stat.Concentration.TailBounds.Massart","a":"Stat/Concentration/TailBounds","t":2,"d":"The two-point sign set is nonempty.","s":"Nonempty ↥{-1, 1}","l":29,"f":"Causalean/Stat/Concentration/TailBounds/Massart.lean"},{"n":"Causalean.Stat.Concentration.instPseudoMetricSpaceEmpiricalFunctionSpace","k":"instance","m":"Causalean.Stat.Concentration.Covering.EmpiricalPseudoMetric","a":"Stat/Concentration/Covering","t":2,"d":"The empirical function space is a pseudometric space under sample empirical distance.","s":"{𝒳 : Type v} → {n : ℕ} → {ι : Type u} → {F : ι → 𝒳 → ℝ} → {S : Fin n → 𝒳} → PseudoMetricSpace (Causalean.Stat.Concentration.EmpiricalFunctionSpace F S)","l":159,"f":"Causalean/Stat/Concentration/Covering/EmpiricalPseudoMetric.lean"},{"n":"Causalean.Stat.Concentration.integral_finiteSampleMeasure","k":"theorem","m":"Causalean.Stat.Concentration.Covering.RealValuedVCSubgraph.Empirical","a":"Stat/Concentration/Covering/RealValuedVCSubgraph","t":2,"d":"Integration against the empirical law is the arithmetic average of a measurable real-valued function over the sample.","s":"∀ {𝒳 : Type u} [inst : MeasurableSpace 𝒳] {n : ℕ} (S : Fin n → 𝒳), 0 < n → ∀ {f : 𝒳 → ℝ}, Measurable f → ∫ (x : 𝒳), f x ∂Causalean.Stat.Concentration.finiteSampleMeasure S = 1 / ↑n * ∑ i, f (S i)","l":42,"f":"Causalean/Stat/Concentration/Covering/RealValuedVCSubgraph/Empirical.lean"},{"n":"Causalean.Stat.Concentration.integral_log_div_eq","k":"theorem","m":"Causalean.Stat.Concentration.Covering.SqrtLogIntegral","a":"Stat/Concentration/Covering","t":2,"d":"Exact evaluation of the logarithmic ratio integral on [ε, δ].","s":"∀ {δ ε : ℝ}, 0 < ε → ε ≤ δ → ∫ (x : ℝ) in ε..δ, Real.log (δ / x) = δ - ε - ε * Real.log (δ / ε)","l":90,"f":"Causalean/Stat/Concentration/Covering/SqrtLogIntegral.lean"},{"n":"Causalean.Stat.Concentration.integral_log_div_le","k":"theorem","m":"Causalean.Stat.Concentration.Covering.SqrtLogIntegral","a":"Stat/Concentration/Covering","t":2,"d":"The logarithmic ratio integral over [ε, δ] is at most the interval length.","s":"∀ {δ ε : ℝ}, 0 < ε → ε ≤ δ → ∫ (x : ℝ) in ε..δ, Real.log (δ / x) ≤ δ - ε","l":116,"f":"Causalean/Stat/Concentration/Covering/SqrtLogIntegral.lean"},{"n":"Causalean.Stat.Concentration.integral_sum_pi_eq","k":"theorem","m":"Causalean.Stat.Concentration.Matrix.IidSums","a":"Stat/Concentration/Matrix","t":2,"d":"Expectation of an iid sum. For an iid sample of size N drawn from a probability measure μ (modelled by the product measure Measure.pi), the expectation of the sum ∑ᵢ g(Xᵢ) of a fixed integrable statistic g equals N · 𝔼[g].","s":"∀ {N : ℕ} {Ω : Type u_1} [inst : MeasurableSpace Ω] (μ : MeasureTheory.Measure Ω) [MeasureTheory.IsProbabilityMeasure μ] (g : Ω → ℝ), MeasureTheory.Integrable g μ → (∫ (ω : Fin N → Ω), ∑ i, g (ω i) ∂MeasureTheory.Measure.pi fun x => μ) = ↑N * ∫ (ω : Ω), g ω ∂μ","l":31,"f":"Causalean/Stat/Concentration/Matrix/IidSums.lean"},{"n":"Causalean.Stat.Concentration.intervalIntegrable_log_div","k":"theorem","m":"Causalean.Stat.Concentration.Covering.SqrtLogIntegral","a":"Stat/Concentration/Covering","t":2,"d":"When ε is positive and no larger than δ, the logarithm of the ratio δ divided by x is interval-integrable from ε to δ with respect to Lebesgue measure.","s":"∀ {δ ε : ℝ}, 0 < ε → ε ≤ δ → IntervalIntegrable (fun x => Real.log (δ / x)) MeasureTheory.volume ε δ","l":74,"f":"Causalean/Stat/Concentration/Covering/SqrtLogIntegral.lean"},{"n":"Causalean.Stat.Concentration.intervalIntegrable_sqrt_log_div","k":"theorem","m":"Causalean.Stat.Concentration.Covering.SqrtLogIntegral","a":"Stat/Concentration/Covering","t":2,"d":"When ε is positive and no larger than δ, the square-root log-ratio kernel is integrable on the interval from ε to δ with respect to Lebesgue measure.","s":"∀ {δ ε : ℝ}, 0 < ε → ε ≤ δ → IntervalIntegrable (fun x => √(Real.log (δ / x))) MeasureTheory.volume ε δ","l":48,"f":"Causalean/Stat/Concentration/Covering/SqrtLogIntegral.lean"},{"n":"Causalean.Stat.Concentration.linearParameterClass","k":"def","m":"Causalean.Stat.Concentration.Covering.RealValuedVCSubgraph.Parametric","a":"Stat/Concentration/Covering/RealValuedVCSubgraph","t":1,"d":"Evaluation of a finite-dimensional linear parameter against a fixed feature family.","s":"{𝒳 : Type u} → {K : Type w} → [Fintype K] → (K → 𝒳 → ℝ) → (B : ℝ) → Causalean.Stat.Concentration.CoeffBox K B → 𝒳 → ℝ","l":27,"f":"Causalean/Stat/Concentration/Covering/RealValuedVCSubgraph/Parametric.lean"},{"n":"Causalean.Stat.Concentration.linearParameterClass_hasPolynomialL2Cover","k":"theorem","m":"Causalean.Stat.Concentration.Covering.RealValuedVCSubgraph.Parametric","a":"Stat/Concentration/Covering/RealValuedVCSubgraph","t":1,"d":"Covering certificate for a bounded-coefficient linear class. Given a finite family of measurable real-valued features, each bounded in absolute value by M, where B is a positive coefficient bound and M is positive, the class of linear combinations of the features with each coefficient constrained to [-B,B] carries a uniform polynomial L² covering certificate at envelope |K|·B·M.","s":"∀ {𝒳 : Type u} [inst : MeasurableSpace 𝒳] {K : Type w} [inst_1 : Fintype K] [Nonempty K] (φ : K → 𝒳 → ℝ) {B M : ℝ}, 0 < B → 0 < M → (∀ (k : K), Measurable (φ k)) → (∀ (k : K) (x : 𝒳), |φ k x| ≤ M) → Causalean.Stat.Concentration.HasPolynomialL2Cover (Causalean.Stat.Concentration.linearParameterClass φ B) (↑(Fintype.card K) * B * M)","l":135,"f":"Causalean/Stat/Concentration/Covering/RealValuedVCSubgraph/Parametric.lean"},{"n":"Causalean.Stat.Concentration.linearParameterClass_hasPseudoDimAtMost","k":"theorem","m":"Causalean.Stat.Concentration.Covering.RealValuedVCSubgraph.Parametric","a":"Stat/Concentration/Covering/RealValuedVCSubgraph","t":2,"d":"A finite-dimensional linear class has pseudo-dimension bounded by the number of coordinates (the stated +1 leaves room for the affine threshold).","s":"∀ {𝒳 : Type u} [MeasurableSpace 𝒳] {K : Type w} [inst : Fintype K] (φ : K → 𝒳 → ℝ) (B : ℝ), Causalean.Stat.Concentration.HasPseudoDimAtMost (Causalean.Stat.Concentration.linearParameterClass φ B) (Fintype.card K + 1)","l":33,"f":"Causalean/Stat/Concentration/Covering/RealValuedVCSubgraph/Parametric.lean"},{"n":"Causalean.Stat.Concentration.linear_subRoot","k":"theorem","m":"Causalean.Stat.Concentration.Covering.VCLocalizedRegime","a":"Stat/Concentration/Covering","t":2,"d":"A nonnegative linear function is sub-root.","s":"∀ {C : ℝ}, 0 ≤ C → Causalean.Stat.Concentration.SubRoot fun r => C * r","l":314,"f":"Causalean/Stat/Concentration/Covering/VCLocalizedRegime.lean"},{"n":"Causalean.Stat.Concentration.linftyOp_abs_entry_le","k":"theorem","m":"Causalean.Stat.Concentration.Matrix.InversePerturbation","a":"Stat/Concentration/Matrix","t":2,"d":"Each coefficient norm is dominated by the ℓ∞-operator (max-row-sum) norm.","s":"∀ {α : Type u_1} {β : Type u_2} {R : Type u_3} [inst : Fintype α] [inst_1 : Fintype β] [inst_2 : NormedAddCommGroup R] (A : Matrix α β R) (i : α) (j : β), ‖A i j‖ ≤ ‖A‖","l":56,"f":"Causalean/Stat/Concentration/Matrix/InversePerturbation.lean"},{"n":"Causalean.Stat.Concentration.linftyOp_norm_le_of_entry","k":"theorem","m":"Causalean.Stat.Concentration.Matrix.InversePerturbation","a":"Stat/Concentration/Matrix","t":2,"d":"The ℓ∞-operator norm of an entrywise-η-bounded rectangular matrix is at most its number of columns times η.","s":"∀ {p q : ℕ} {A : Matrix (Fin (p + 1)) (Fin (q + 1)) ℝ} {η : ℝ}, (∀ (i : Fin (p + 1)) (j : Fin (q + 1)), |A i j| ≤ η) → ‖A‖ ≤ ↑(q + 1) * η","l":88,"f":"Causalean/Stat/Concentration/Matrix/InversePerturbation.lean"},{"n":"Causalean.Stat.Concentration.linftyOp_norm_le_of_rowsum","k":"theorem","m":"Causalean.Stat.Concentration.Matrix.InversePerturbation","a":"Stat/Concentration/Matrix","t":2,"d":"A row-sum upper bound for the ℓ∞-operator norm: if every row's coefficient-norm sum is ≤ c, then ‖A‖ ≤ c.","s":"∀ {α : Type u_1} {β : Type u_2} {R : Type u_3} [inst : Fintype α] [inst_1 : Fintype β] [inst_2 : NormedAddCommGroup R] {A : Matrix α β R} {c : ℝ}, 0 ≤ c → (∀ (i : α), ∑ j, ‖A i j‖ ≤ c) → ‖A‖ ≤ c","l":70,"f":"Causalean/Stat/Concentration/Matrix/InversePerturbation.lean"},{"n":"Causalean.Stat.Concentration.lipschitzAt0_neg","k":"theorem","m":"Causalean.Stat.Concentration.Rademacher.Contraction","a":"Stat/Concentration/Rademacher","t":2,"d":"Negating a function that fixes zero preserves the same Lipschitz constant.","s":"∀ (φ : ℝ → ℝ) {L : ℝ}, Causalean.Stat.Concentration.LipschitzAt0 φ L → Causalean.Stat.Concentration.LipschitzAt0 (fun x => -φ x) L","l":522,"f":"Causalean/Stat/Concentration/Rademacher/Contraction.lean"},{"n":"Causalean.Stat.Concentration.localRademacherComplexity","k":"def","m":"Causalean.Stat.Concentration.Rademacher.LocalRademacher","a":"Stat/Concentration/Rademacher","t":1,"d":"The localized Rademacher complexity of a function family F, evaluated at radius r and norm norm. Defined via the *zero-out* re-indexing: each parameter (α, i) : starHullParam ι contributes α • F i if norm (α • F i) ≤ r, and the constant 0 otherwise.","s":"{Ω : Type u_1} → {ι : Type u_2} → {𝒳 : Type u_3} → [inst : MeasurableSpace Ω] → (ι → 𝒳 → ℝ) → ((𝒳 → ℝ) → ℝ) → MeasureTheory.Measure Ω → (Ω → 𝒳) → ℕ → ℝ → ℝ","l":88,"f":"Causalean/Stat/Concentration/Rademacher/LocalRademacher.lean"},{"n":"Causalean.Stat.Concentration.localRademacherComplexity_le_upperBound","k":"theorem","m":"Causalean.Stat.Concentration.Rademacher.LocalRademacher","a":"Stat/Concentration/Rademacher","t":2,"d":"Upper-bound consumption. A RademacherUpperBound ψ directly bounds the localized Rademacher complexity by ψ r for every r ≥ 0 — by definition, since RademacherUpperBound is stated in terms of starHullZeroOut, the same integrand defining localRademacherComplexity.","s":"∀ {Ω : Type u_1} {ι : Type u_2} {𝒳 : Type u_3} [inst : MeasurableSpace Ω] {F : ι → 𝒳 → ℝ} {norm : (𝒳 → ℝ) → ℝ} {μ : MeasureTheory.Measure Ω} {X : Ω → 𝒳} {n : ℕ} {ψ : ℝ → ℝ}, Causalean.Stat.Concentration.RademacherUpperBound F norm μ X n ψ → ∀ {r : ℝ}, 0 ≤ r → Causalean.Stat.Concentration.localRademacherComplexity F norm μ X n r ≤ ψ r","l":267,"f":"Causalean/Stat/Concentration/Rademacher/LocalRademacher.lean"},{"n":"Causalean.Stat.Concentration.localRademacherComplexity_mono_r","k":"theorem","m":"Causalean.Stat.Concentration.Rademacher.LocalRademacher","a":"Stat/Concentration/Rademacher","t":2,"d":"The localized Rademacher complexity is monotone in the radius.","s":"∀ {Ω : Type u_1} {ι : Type u_2} {𝒳 : Type u_3} [inst : MeasurableSpace Ω] (F : ι → 𝒳 → ℝ) (norm : (𝒳 → ℝ) → ℝ) (μ : MeasureTheory.Measure Ω) (X : Ω → 𝒳) (n : ℕ) {r₁ r₂ : ℝ}, r₁ ≤ r₂ → (∀ (S : Fin n → 𝒳) (σ : Signs n), BddAbove (Set.range fun p => |(↑n)⁻¹ * ∑ k, ↑↑(σ k) * Causalean.Stat.Concentration.starHullZeroOut F norm r₂ p (S k)|)) → MeasureTheory.Integrable (fun ω => empiricalRademacherComplexity n (Causalean.Stat.Concentration.starHullZeroOut F norm r₂) (X ∘ ω)) (MeasureTheory.Measure.pi fun x => μ) → Causalean.Stat.Concentration.localRademacherComplexity F norm μ X n r₁ ≤ Causalean.Stat.Concentration.localRademacherComplexity F norm μ X n r₂","l":228,"f":"Causalean/Stat/Concentration/Rademacher/LocalRademacher.lean"},{"n":"Causalean.Stat.Concentration.localRademacherComplexity_nonneg","k":"theorem","m":"Causalean.Stat.Concentration.Rademacher.LocalRademacher","a":"Stat/Concentration/Rademacher","t":2,"d":"The localized Rademacher complexity is non-negative.","s":"∀ {Ω : Type u_1} {ι : Type u_2} {𝒳 : Type u_3} [inst : MeasurableSpace Ω] (F : ι → 𝒳 → ℝ) (norm : (𝒳 → ℝ) → ℝ) (μ : MeasureTheory.Measure Ω) (X : Ω → 𝒳) (n : ℕ) (r : ℝ), 0 ≤ Causalean.Stat.Concentration.localRademacherComplexity F norm μ X n r","l":97,"f":"Causalean/Stat/Concentration/Rademacher/LocalRademacher.lean"},{"n":"Causalean.Stat.Concentration.localRademacher_le_critical_radius","k":"theorem","m":"Causalean.Stat.Concentration.Rademacher.LocalRademacher","a":"Stat/Concentration/Rademacher","t":1,"d":"Localized inequality (headline). Suppose the envelope ψ is sub-root: nonnegative, non-decreasing, and with the ratio ψ r / r non-increasing in r, and ψ upper-bounds the localized population Rademacher complexity of the class F (measured by norm, under the sampling law μ, map X, and sample size n) at every nonnegative radius. Writing δ* := criticalRadius ψ, suppose δ* is positive and it satisfies the fixed-point bound ψ δ* ≤ δ*². Then for every radius r ≥ δ*, the localized Rademacher complexity at radius r is at most r · δ*.","s":"∀ {Ω : Type u_1} {ι : Type u_2} {𝒳 : Type u_3} [inst : MeasurableSpace Ω] {F : ι → 𝒳 → ℝ} {norm : (𝒳 → ℝ) → ℝ} {μ : MeasureTheory.Measure Ω} {X : Ω → 𝒳} {n : ℕ} {ψ : ℝ → ℝ}, Causalean.Stat.Concentration.SubRoot ψ → Causalean.Stat.Concentration.RademacherUpperBound F norm μ X n ψ → ∀ {r : ℝ}, Causalean.Stat.Concentration.criticalRadius ψ ≤ r → 0 < Causalean.Stat.Concentration.criticalRadius ψ → ψ (Causalean.Stat.Concentration.criticalRadius ψ) ≤ Causalean.Stat.Concentration.criticalRadius ψ ^ 2 → Causalean.Stat.Concentration.localRademacherComplexity F norm μ X n r ≤ r * Causalean.Stat.Concentration.criticalRadius ψ","l":281,"f":"Causalean/Stat/Concentration/Rademacher/LocalRademacher.lean"},{"n":"Causalean.Stat.Concentration.localizedVCDudleyHypotheses_of_empiricalRadius","k":"theorem","m":"Causalean.Stat.Concentration.Covering.VCLocalizedRegime","a":"Stat/Concentration/Covering","t":2,"d":"Assemble the deterministic localized Dudley hypotheses for a bounded binary-factored VC class once the samplewise empirical-radius bridge is known.","s":"∀ {ι : Type u} {𝒳 : Type v} {F : ι → 𝒳 → ℝ} {norm : (𝒳 → ℝ) → ℝ} {B : ℝ}, (∀ (i : ι) (x : 𝒳), |F i x| ≤ B) → (∀ {n : ℕ} (S : Fin n → 𝒳) {r : ℝ}, 0 ≤ r → ∀ (p : Causalean.Stat.Concentration.starHullParam ι), Causalean.Stat.Concentration.empiricalNorm S (Causalean.Stat.Concentration.starHullZeroOut F norm r p) ≤ r) → Causalean.Stat.Concentration.LocalizedVCDudleyHypotheses F norm","l":229,"f":"Causalean/Stat/Concentration/Covering/VCLocalizedRegime.lean"},{"n":"Causalean.Stat.Concentration.localized_offset_expectation","k":"theorem","m":"Causalean.Stat.Concentration.UniformDeviation.LocalizedEnvelopeExpectation","a":"Stat/Concentration/UniformDeviation","t":1,"d":"Localized offset expectation. Fix a localized regime R built from measurable losses F i composed with a measurable map X, a confidence level δ in (0,1], a sample size n at least 1, and a positive upper bound ρ on the positive critical radius criticalRadius (R.ψ n), satisfying the sub-root fixed-point bound R.ψ n (criticalRadius (R.ψ n)) ≤ criticalRadius (R.ψ n) ^ 2, with the star-hull Rademacher process almost-surely bounded and integrable at every radius r ≥ ρ and a slack-domination condition bounding the McDiarmid tail by ρ² across the diameter shells up to Rmax. Suppose further that every norm (F i) lies in [0, Rmax], so the sharp deviation bound applies uniformly over the class, that the exponent κ lies strictly between 0 and 1, that the coupling constant A is nonnegative, that the regret radius Δ i is nonnegative for every i, and that the localization radius is dominated by the regret via the margin coupling norm (F i) ≤ A · (Δ i) ^ κ. Then the expectation over the n-fold sample of the supremum over i of the positive part of 2 · |(Pₙ−P)F i| − Δ i / 4 is at most offsetPeelingConstantC (1/8) κ · (16·ρ·A)^{1/(1−κ)} + 10·ρ² + 4·R.b·δ.","s":"∀ {Ω : Type u_1} {ι : Type u_2} {𝒳 : Type u_3} [inst : MeasurableSpace Ω] [inst_1 : MeasurableSpace 𝒳] [Nonempty 𝒳] [Nonempty ι] [Countable ι] (F : ι → 𝒳 → ℝ) (norm : (𝒳 → ℝ) → ℝ) (μ : MeasureTheory.Measure Ω) (X : Ω → 𝒳) [MeasureTheory.IsProbabilityMeasure μ], Measurable X → (∀ (i : ι), Measurable (F i)) → ∀ (R : Causalean.Stat.Concentration.LocalizedRegime Ω ι 𝒳 F norm μ X) {δ : ℝ}, 0 < δ → δ ≤ 1 → ∀ (n : ℕ), 0 < n → ∀ {ρ Rmax : ℝ}, Causalean.Stat.Concentration.criticalRadius (R.ψ n) ≤ ρ → 0 < ρ → 0 < Causalean.Stat.Concentration.criticalRadius (R.ψ n) → R.ψ n (Causalean.Stat.Concentration.criticalRadius (R.ψ n)) ≤ Causalean.Stat.Concentration.criticalRadius (R.ψ n) ^ 2 → (∀ (r : ℝ), ρ ≤ r → ∀ (S : Fin n → 𝒳) (σ : Signs n), BddAbove (Set.range fun p => |(↑n)⁻¹ * ∑ k, ↑↑(σ k) * Causalean.Stat.Concentration.starHullZeroOut F norm r p (S k)|)) → (∀ (r : ℝ), ρ ≤ r → MeasureTheory.Integrable (fun ω => empiricalRademacherComplexity n (Causalean.Stat.Concentration.starHullZeroOut F norm r) (X ∘ ω)) (MeasureTheory.Measure.pi fun x => μ)) → (∀ (K : ℕ), Rmax ≤ ρ * 2 ^ K → R.b * √(2 * Real.log (2 * (↑K + 1) / δ) / ↑n) ≤ ρ ^ 2) → (∀ (i : ι), 0 ≤ norm (F i)) → (∀ (i : ι), norm (F i) ≤ Rmax) → ∀ (Δ : ι → ℝ) (κ A : ℝ), 0 < κ → κ < 1 → 0 ≤ A → (∀ (i : ι), 0 ≤ Δ i) → (∀ (i : ι), norm (F i) ≤ A * Δ i ^ κ) → (∫ (ω : Fin n → Ω), ⨆ i, max 0 (2 * |(↑n)⁻¹ * ∑ k, F i (X (ω k)) - ∫ (x : Ω), (fun ω' => F i (X ω')) x ∂μ| - Δ i / 4) ∂MeasureTheory.Measure.pi fun x => μ) ≤ Causalean.Mathlib.OffsetPeeling.offsetPeelingConstantC (1 / 8) κ * (16 * ρ * A) ^ (1 / (1 - κ)) + 10 * ρ ^ 2 + 4 * R.b * δ","l":255,"f":"Causalean/Stat/Concentration/UniformDeviation/LocalizedEnvelopeExpectation.lean"},{"n":"Causalean.Stat.Concentration.localized_uniform_deviation","k":"theorem","m":"Causalean.Stat.Concentration.UniformDeviation.UniformDeviationLocalized","a":"Stat/Concentration/UniformDeviation","t":1,"d":"Localized uniform deviation. Fix a localized regime R built from measurable losses F i composed with a measurable map X, a confidence level δ in (0,1], and a sample size n at least 1. If the radius r restricting the class to {i : norm (F i) ≤ r} is at least the population critical radius criticalRadius (R.ψ n), itself positive, the envelope satisfies the sub-root fixed-point bound R.ψ n (criticalRadius (R.ψ n)) ≤ criticalRadius (R.ψ n) ^ 2, and the star-hull Rademacher process at radius r is almost-surely bounded and its empirical complexity integrable, then there is a measurable event of probability at least 1 − δ on which, simultaneously for every i with norm (F i) ≤ r, the empirical mean of F i deviates from its population mean by at most 4 · r · criticalRadius (R.ψ n) + R.b · √(2 · log(1/δ) / n).","s":"∀ {Ω : Type u_1} {ι : Type u_2} {𝒳 : Type u_3} [inst : MeasurableSpace Ω] [inst_1 : MeasurableSpace 𝒳] [Nonempty 𝒳] [Nonempty ι] [Countable ι] (F : ι → 𝒳 → ℝ) (norm : (𝒳 → ℝ) → ℝ) (μ : MeasureTheory.Measure Ω) (X : Ω → 𝒳) [MeasureTheory.IsProbabilityMeasure μ], Measurable X → (∀ (i : ι), Measurable (F i)) → ∀ (R : Causalean.Stat.Concentration.LocalizedRegime Ω ι 𝒳 F norm μ X) {δ : ℝ}, 0 < δ → δ ≤ 1 → ∀ (n : ℕ), 0 < n → ∀ {r : ℝ}, Causalean.Stat.Concentration.criticalRadius (R.ψ n) ≤ r → 0 < Causalean.Stat.Concentration.criticalRadius (R.ψ n) → R.ψ n (Causalean.Stat.Concentration.criticalRadius (R.ψ n)) ≤ Causalean.Stat.Concentration.criticalRadius (R.ψ n) ^ 2 → (∀ (S : Fin n → 𝒳) (σ : Signs n), BddAbove (Set.range fun p => |(↑n)⁻¹ * ∑ k, ↑↑(σ k) * Causalean.Stat.Concentration.starHullZeroOut F norm r p (S k)|)) → MeasureTheory.Integrable (fun ω => empiricalRademacherComplexity n (Causalean.Stat.Concentration.starHullZeroOut F norm r) (X ∘ ω)) (MeasureTheory.Measure.pi fun x => μ) → ∃ E, MeasurableSet E ∧ (MeasureTheory.Measure.pi fun x => μ) E ≥ 1 - ENNReal.ofReal δ ∧ ∀ ω ∈ E, ∀ (i : ι), norm (F i) ≤ r → |(↑n)⁻¹ * ∑ k, F i (X (ω k)) - ∫ (x : Ω), (fun ω' => F i (X ω')) x ∂μ| ≤ 4 * r * Causalean.Stat.Concentration.criticalRadius (R.ψ n) + R.b * √(2 * Real.log (1 / δ) / ↑n)","l":99,"f":"Causalean/Stat/Concentration/UniformDeviation/UniformDeviationLocalized.lean"},{"n":"Causalean.Stat.Concentration.localized_uniform_deviation_expectation","k":"theorem","m":"Causalean.Stat.Concentration.UniformDeviation.LocalizedEnvelopeExpectation","a":"Stat/Concentration/UniformDeviation","t":1,"d":"Localized uniform deviation, in expectation. Fix a localized regime R built from measurable losses F i composed with a measurable data map X, and a sample size n at least 1. If the radius r restricting the class to {i : norm (F i) ≤ r} is at least the population critical radius criticalRadius (R.ψ n), itself positive, the envelope R.ψ n satisfies the sub-root self-bounding condition R.ψ n (criticalRadius (R.ψ n)) ≤ criticalRadius (R.ψ n) ^ 2 at that radius, and the empirical Rademacher complexity of the radius-r star-hull is integrable, then the expectation, over the n-fold sample, of the uniform deviation of the radius-r-restricted class is at most 2·r·criticalRadius (R.ψ n).","s":"∀ {Ω : Type u_1} {ι : Type u_2} {𝒳 : Type u_3} [inst : MeasurableSpace Ω] [inst_1 : MeasurableSpace 𝒳] [Nonempty 𝒳] [Nonempty ι] [Countable ι] (F : ι → 𝒳 → ℝ) (norm : (𝒳 → ℝ) → ℝ) (μ : MeasureTheory.Measure Ω) (X : Ω → 𝒳) [MeasureTheory.IsProbabilityMeasure μ], Measurable X → (∀ (i : ι), Measurable (F i)) → ∀ (R : Causalean.Stat.Concentration.LocalizedRegime Ω ι 𝒳 F norm μ X) (n : ℕ), 0 < n → ∀ {r : ℝ}, Causalean.Stat.Concentration.criticalRadius (R.ψ n) ≤ r → 0 < Causalean.Stat.Concentration.criticalRadius (R.ψ n) → R.ψ n (Causalean.Stat.Concentration.criticalRadius (R.ψ n)) ≤ Causalean.Stat.Concentration.criticalRadius (R.ψ n) ^ 2 → MeasureTheory.Integrable (fun ω => empiricalRademacherComplexity n (Causalean.Stat.Concentration.starHullZeroOut F norm r) (X ∘ ω)) (MeasureTheory.Measure.pi fun x => μ) → (∫ (ω : Fin n → Ω), uniformDeviation n (fun i ω' => if norm (F i) ≤ r then F i (X ω') else 0) μ id (id ∘ ω) ∂MeasureTheory.Measure.pi fun x => μ) ≤ 2 * r * Causalean.Stat.Concentration.criticalRadius (R.ψ n)","l":67,"f":"Causalean/Stat/Concentration/UniformDeviation/LocalizedEnvelopeExpectation.lean"},{"n":"Causalean.Stat.Concentration.localized_uniform_deviation_sharp","k":"theorem","m":"Causalean.Stat.Concentration.UniformDeviation.UniformDeviationLocalized","a":"Stat/Concentration/UniformDeviation","t":1,"d":"Sharp localized uniform deviation. Fix a localized regime R built from measurable losses F i composed with a measurable map X, a confidence level δ in (0,1], and a sample size n at least 1. Let ρ be a positive upper bound on the positive population critical radius criticalRadius (R.ψ n), satisfying the sub-root fixed-point bound R.ψ n (criticalRadius (R.ψ n)) ≤ criticalRadius (R.ψ n) ^ 2, and suppose the star-hull Rademacher process is almost-surely bounded and its empirical complexity integrable at every radius r ≥ ρ. If, further, for every peeling level K covering the diameter cap Rmax ≤ ρ · 2^K, the McDiarmid slack R.b · √(2 · log(2(K+1)/δ) / n) at confidence 1 − δ is itself dominated by ρ², then there is a measurable event of probability at least 1 − δ on which, simultaneously for every i with 0 ≤ norm (F i) ≤ Rmax, the empirical mean of F i deviates from its population mean by at most 8 · ρ · norm (F i) + 5 · ρ².","s":"∀ {Ω : Type u_1} {ι : Type u_2} {𝒳 : Type u_3} [inst : MeasurableSpace Ω] [inst_1 : MeasurableSpace 𝒳] [Nonempty 𝒳] [Nonempty ι] [Countable ι] (F : ι → 𝒳 → ℝ) (norm : (𝒳 → ℝ) → ℝ) (μ : MeasureTheory.Measure Ω) (X : Ω → 𝒳) [MeasureTheory.IsProbabilityMeasure μ], Measurable X → (∀ (i : ι), Measurable (F i)) → ∀ (R : Causalean.Stat.Concentration.LocalizedRegime Ω ι 𝒳 F norm μ X) {δ : ℝ}, 0 < δ → δ ≤ 1 → ∀ (n : ℕ), 0 < n → ∀ {ρ Rmax : ℝ}, Causalean.Stat.Concentration.criticalRadius (R.ψ n) ≤ ρ → 0 < ρ → 0 < Causalean.Stat.Concentration.criticalRadius (R.ψ n) → R.ψ n (Causalean.Stat.Concentration.criticalRadius (R.ψ n)) ≤ Causalean.Stat.Concentration.criticalRadius (R.ψ n) ^ 2 → (∀ (r : ℝ), ρ ≤ r → ∀ (S : Fin n → 𝒳) (σ : Signs n), BddAbove (Set.range fun p => |(↑n)⁻¹ * ∑ k, ↑↑(σ k) * Causalean.Stat.Concentration.starHullZeroOut F norm r p (S k)|)) → (∀ (r : ℝ), ρ ≤ r → MeasureTheory.Integrable (fun ω => empiricalRademacherComplexity n (Causalean.Stat.Concentration.starHullZeroOut F norm r) (X ∘ ω)) (MeasureTheory.Measure.pi fun x => μ)) → (∀ (K : ℕ), Rmax ≤ ρ * 2 ^ K → R.b * √(2 * Real.log (2 * (↑K + 1) / δ) / ↑n) ≤ ρ ^ 2) → ∃ E, MeasurableSet E ∧ (MeasureTheory.Measure.pi fun x => μ) E ≥ 1 - ENNReal.ofReal δ ∧ ∀ ω ∈ E, ∀ (i : ι), 0 ≤ norm (F i) → norm (F i) ≤ Rmax → |(↑n)⁻¹ * ∑ k, F i (X (ω k)) - ∫ (x : Ω), (fun ω' => F i (X ω')) x ∂μ| ≤ 8 * ρ * norm (F i) + 5 * ρ ^ 2","l":346,"f":"Causalean/Stat/Concentration/UniformDeviation/UniformDeviationLocalized.lean"},{"n":"Causalean.Stat.Concentration.log_coveringNumber_le","k":"theorem","m":"Causalean.Stat.Concentration.Covering.VCCovering","a":"Stat/Concentration/Covering","t":1,"d":"Logarithmic Sauer-Shelah covering-number bound. Under the binary-factoring hypothesis F i (S j) = φ j (π i (S j)), with the sample-restricted class totally bounded in the empirical pseudometric, a positive covering radius ε, a positive sample size n, and VC dimension at most d for the induced Boolean growth family on the sample, the logarithm of the empirical covering number at radius ε is at most log(d+1) + d·log n.","s":"∀ {𝒳 : Type v} {ι : Type u} {n d : ℕ} [Nonempty ι] {F : ι → 𝒳 → ℝ} {S : Fin n → 𝒳} {π : ι → 𝒳 → Bool} {φ : Fin n → Bool → ℝ}, (∀ (i : ι) (j : Fin n), F i (S j) = φ j (π i (S j))) → ∀ (h' : TotallyBounded Set.univ) {ε : ℝ}, 0 < ε → 0 < n → (Causalean.Stat.Concentration.growthFamily π S).vcDim ≤ d → Real.log ↑(Causalean.Stat.Concentration.coveringNumber h' ε) ≤ Real.log ↑(d + 1) + ↑d * Real.log ↑n","l":228,"f":"Causalean/Stat/Concentration/Covering/VCCovering.lean"},{"n":"Causalean.Stat.Concentration.log_coveringNumber_le_of_card_bound","k":"theorem","m":"Causalean.Stat.Concentration.Covering.VCCovering","a":"Stat/Concentration/Covering","t":2,"d":"A direct cardinality bound on the Boolean growth family gives the same logarithmic bound on the empirical covering number.","s":"∀ {𝒳 : Type v} {ι : Type u} {n : ℕ} [Nonempty ι] {F : ι → 𝒳 → ℝ} {S : Fin n → 𝒳} {π : ι → 𝒳 → Bool} {φ : Fin n → Bool → ℝ}, (∀ (i : ι) (j : Fin n), F i (S j) = φ j (π i (S j))) → ∀ (h' : TotallyBounded Set.univ) {ε : ℝ}, 0 < ε → ∀ {N : ℕ}, (Causalean.Stat.Concentration.growthFamily π S).card ≤ N → Real.log ↑(Causalean.Stat.Concentration.coveringNumber h' ε) ≤ Real.log ↑N","l":189,"f":"Causalean/Stat/Concentration/Covering/VCCovering.lean"},{"n":"Causalean.Stat.Concentration.log_div_nonneg_of_mem_Icc","k":"theorem","m":"Causalean.Stat.Concentration.Covering.SqrtLogIntegral","a":"Stat/Concentration/Covering","t":2,"d":"On the interval [ε, δ], the log-ratio kernel is nonnegative.","s":"∀ {δ ε x : ℝ}, 0 < ε → x ∈ Set.Icc ε δ → 0 ≤ Real.log (δ / x)","l":27,"f":"Causalean/Stat/Concentration/Covering/SqrtLogIntegral.lean"},{"n":"Causalean.Stat.Concentration.log_two_div_nonneg","k":"theorem","m":"Causalean.Stat.Concentration.UniformDeviation.ConfidenceInterval","a":"Stat/Concentration/UniformDeviation","t":2,"d":"For a confidence level between zero and one, the logarithm of twice its inverse is nonnegative.","s":"∀ {δ : ℝ}, 0 < δ → δ ≤ 1 → 0 ≤ Real.log (2 / δ)","l":71,"f":"Causalean/Stat/Concentration/UniformDeviation/ConfidenceInterval.lean"},{"n":"Causalean.Stat.Concentration.log_two_growth_card_le","k":"theorem","m":"Causalean.Stat.Concentration.Covering.VCLocalizedRegime","a":"Stat/Concentration/Covering","t":2,"d":"When a Boolean growth family on a sample has positive cardinality and at most the degree-d polynomial number of patterns in one plus the sample size, twice the logarithm of twice its cardinality is no greater than twice that degree times the logarithm of one plus the sample size, plus two.","s":"∀ {ι : Type u} {𝒳 : Type v} (π : ι → 𝒳 → Bool) (d n : ℕ) (S : Fin n → 𝒳), 0 < (Causalean.Stat.Concentration.growthFamily π S).card → (Causalean.Stat.Concentration.growthFamily π S).card ≤ (n + 1) ^ d → 2 * Real.log (2 * ↑(Causalean.Stat.Concentration.growthFamily π S).card) ≤ 2 * ↑d * Real.log (↑n + 1) + 2","l":413,"f":"Causalean/Stat/Concentration/Covering/VCLocalizedRegime.lean"},{"n":"Causalean.Stat.Concentration.massart_lemma_pmf","k":"theorem","m":"Causalean.Stat.Concentration.TailBounds.Massart","a":"Stat/Concentration/TailBounds","t":1,"d":"Massart's finite-class lemma. Given a nonempty finite subset f of the index set selecting finitely many functions from the class, the empirical Rademacher complexity (without absolute value) of that finite subclass, evaluated at the sample S of size m, is at most the largest per-function coordinate $\\ell^2$-radius $\\sqrt{\\sum_i (F_j(S_i)/m)^2}$ over j ∈ f, times $\\sqrt{2\\log|f|}$.","s":"∀ {Z : Type v} {m : ℕ} {ι : Type u} (F : ι → Z → ℝ) (S : Fin m → Z) (f : Finset ι) (hs : f.Nonempty), empiricalRademacherComplexity_pmf_without_abs m (Causalean.Stat.Concentration.F_on F f) S ≤ (f.sup' hs fun j => √(∑ i, ((↑m)⁻¹ * |F j (S i)|) ^ 2)) * √(2 * Real.log ↑f.card)","l":201,"f":"Causalean/Stat/Concentration/TailBounds/Massart.lean"},{"n":"Causalean.Stat.Concentration.massart_lemma_pmf.sign_mean_zero","k":"theorem","m":"Causalean.Stat.Concentration.TailBounds.Massart","a":"Stat/Concentration/TailBounds","t":2,"d":"A single Rademacher-signed sample value has mean zero.","s":"∀ {Z : Type v} {m : ℕ} (f : Z → ℝ) (S : Fin m → Z) (a : Fin m), ∫ (ω : Signs m), ↑↑(ω a) * f (S a) ∂(signVecPMF m).toMeasure = 0","l":183,"f":"Causalean/Stat/Concentration/TailBounds/Massart.lean"},{"n":"Causalean.Stat.Concentration.matrix_rank_le_of_gram_factor","k":"theorem","m":"Causalean.Stat.Concentration.ProjectionMatrixTail","a":"Stat/Concentration","t":2,"d":"A matrix admitting a Gram factorization through r vectors has rank at most r.","s":"∀ {R : Type u_1} {ι : Type u_2} [Nontrivial R] [inst : CommRing R] [inst_1 : Fintype ι] {r : ℕ} (Pi : Matrix ι ι R) (vectors : Fin r → ι → R), (∀ (i j : ι), Pi i j = ∑ k, vectors k i * vectors k j) → Pi.rank ≤ r","l":29,"f":"Causalean/Stat/Concentration/ProjectionMatrixTail.lean"},{"n":"Causalean.Stat.Concentration.measurablespace_eq","k":"theorem","m":"Causalean.Stat.Concentration.TailBounds.Massart","a":"Stat/Concentration/TailBounds","t":2,"d":"The sign-vector measurable space agrees with the product measurable space.","s":"∀ {m : ℕ}, instMeasurableSpaceSigns m = MeasurableSpace.pi","l":61,"f":"Causalean/Stat/Concentration/TailBounds/Massart.lean"},{"n":"Causalean.Stat.Concentration.measureL2Dist","k":"def","m":"Causalean.Stat.Concentration.Covering.RealValuedVCSubgraph.Basic","a":"Stat/Concentration/Covering/RealValuedVCSubgraph","t":1,"d":"The L²(Q) semidistance is the square root of the integral of the squared pointwise difference.","s":"{𝒳 : Type u} → [inst : MeasurableSpace 𝒳] → MeasureTheory.Measure 𝒳 → (𝒳 → ℝ) → (𝒳 → ℝ) → ℝ","l":39,"f":"Causalean/Stat/Concentration/Covering/RealValuedVCSubgraph/Basic.lean"},{"n":"Causalean.Stat.Concentration.measureL2Dist_finiteSampleMeasure_eq_empiricalDist","k":"theorem","m":"Causalean.Stat.Concentration.Covering.RealValuedVCSubgraph.Empirical","a":"Stat/Concentration/Covering/RealValuedVCSubgraph","t":2,"d":"L² distance under the empirical law agrees exactly with the empirical root-mean-square distance used by the existing Dudley API.","s":"∀ {𝒳 : Type u} [inst : MeasurableSpace 𝒳] {n : ℕ} (S : Fin n → 𝒳), 0 < n → ∀ {f g : 𝒳 → ℝ}, Measurable f → Measurable g → Causalean.Stat.Concentration.measureL2Dist (Causalean.Stat.Concentration.finiteSampleMeasure S) f g = Causalean.Stat.Concentration.empiricalDist S f g","l":57,"f":"Causalean/Stat/Concentration/Covering/RealValuedVCSubgraph/Empirical.lean"},{"n":"Causalean.Stat.Concentration.measureReal_abs_dev_le_two_sided","k":"theorem","m":"Causalean.Stat.Concentration.TailBounds.Hoeffding","a":"Stat/Concentration/TailBounds","t":2,"d":"Generic two-sided assembly: the two-sided deviation event {ε ≤ |T − m|} is covered by the two one-sided events, so its measure is at most the sum of their one-sided bounds. Reused by the Hoeffding and Bernstein two-sided tail bounds.","s":"∀ {Ω : Type u_3} [inst : MeasurableSpace Ω] {μ : MeasureTheory.Measure Ω} [MeasureTheory.IsFiniteMeasure μ] (T : Ω → ℝ) (m Bup Blow ε : ℝ), μ.real {ω | ε ≤ T ω - m} ≤ Bup → μ.real {ω | ε ≤ -T ω + m} ≤ Blow → μ.real {ω | ε ≤ |T ω - m|} ≤ Bup + Blow","l":141,"f":"Causalean/Stat/Concentration/TailBounds/Hoeffding.lean"},{"n":"Causalean.Stat.Concentration.measure_eq","k":"theorem","m":"Causalean.Stat.Concentration.TailBounds.Massart","a":"Stat/Concentration/TailBounds","t":2,"d":"The Rademacher sign-vector law agrees with the product of uniform two-point coordinate laws.","s":"∀ {m : ℕ}, (signVecPMF m).toMeasure ≍ MeasureTheory.Measure.pi fun x => (PMF.uniformOfFintype ↥{-1, 1}).toMeasure","l":70,"f":"Causalean/Stat/Concentration/TailBounds/Massart.lean"},{"n":"Causalean.Stat.Concentration.measure_exists_abs_ge_le","k":"theorem","m":"Causalean.Stat.Concentration.TailBounds.MaximalInequality","a":"Stat/Concentration/TailBounds","t":1,"d":"Maximal inequality (existential form). If every member Y i, i ∈ t, of a finite family indexed by t has a sub-exponential moment-generating function with common parameters (v, b) with respect to μ and ε is nonnegative, then the probability that |Y i| ≥ ε for at least one i ∈ t is at most the union bound card t · 2 exp(−ε² / (2 (v + b ε))).","s":"∀ {Ω : Type u_1} {mΩ : MeasurableSpace Ω} {μ : MeasureTheory.Measure Ω} {ι : Type u_2} (t : Finset ι) (Y : ι → Ω → ℝ) {v b : NNReal}, (∀ i ∈ t, Causalean.Stat.Concentration.HasSubexponentialMGF (Y i) v b μ) → ∀ {ε : ℝ}, 0 ≤ ε → μ.real {ω | ∃ i ∈ t, ε ≤ |Y i ω|} ≤ ↑t.card * (2 * Real.exp (-ε ^ 2 / (2 * (↑v + ↑b * ε))))","l":69,"f":"Causalean/Stat/Concentration/TailBounds/MaximalInequality.lean"},{"n":"Causalean.Stat.Concentration.measure_projection_energy_gt_le","k":"theorem","m":"Causalean.Stat.Concentration.ProjectionMatrixTail","a":"Stat/Concentration","t":1,"d":"Projected bounded-noise tail bound. Let eps : Fin n → Ω → ℝ be coordinate noise terms such that each eps i is measurable, each is bounded by 1 in absolute value almost surely, each has mean zero, and the coordinates are mutually independent. If the n × n matrix Pi is symmetric and idempotent — so it is an orthogonal projection — and has rank at most r, then for any nonnegative t, the squared Euclidean norm of the projected vector Pi · eps exceeds 4 * t ^ 2 with probability at most 5 ^ r * exp (-t ^ 2 / 2).","s":"∀ {Ω : Type u_1} [inst : MeasurableSpace Ω] {n r : ℕ} (P : MeasureTheory.Measure Ω) [MeasureTheory.IsProbabilityMeasure P] (eps : Fin n → Ω → ℝ) (Pi : Matrix (Fin n) (Fin n) ℝ), (∀ (i : Fin n), AEMeasurable (eps i) P) → (∀ (i : Fin n), ∀ᵐ (ω : Ω) ∂P, |eps i ω| ≤ 1) → (∀ (i : Fin n), ∫ (ω : Ω), eps i ω ∂P = 0) → ProbabilityTheory.iIndepFun eps P → Pi.transpose = Pi → Pi * Pi = Pi → Pi.rank ≤ r → ∀ {t : ℝ}, 0 ≤ t → P.real {ω | 4 * t ^ 2 < ∑ i, (∑ j, Pi i j * eps j ω) ^ 2} ≤ 5 ^ r * Real.exp (-t ^ 2 / 2)","l":45,"f":"Causalean/Stat/Concentration/ProjectionMatrixTail.lean"},{"n":"Causalean.Stat.Concentration.measure_sup'_ge_le","k":"theorem","m":"Causalean.Stat.Concentration.TailBounds.MaximalInequality","a":"Stat/Concentration/TailBounds","t":1,"d":"Maximal inequality (sup' form). If t is a nonempty finite index set, every member Y i, i ∈ t, has a sub-exponential moment-generating function with common parameters (v, b) with respect to μ, and ε is nonnegative, then the probability that the pointwise maximum of |Y i| over i ∈ t is at least ε is at most the union bound card t · 2 exp(−ε² / (2 (v + b ε))).","s":"∀ {Ω : Type u_1} {mΩ : MeasurableSpace Ω} {μ : MeasureTheory.Measure Ω} {ι : Type u_2} (t : Finset ι) (ht : t.Nonempty) (Y : ι → Ω → ℝ) {v b : NNReal}, (∀ i ∈ t, Causalean.Stat.Concentration.HasSubexponentialMGF (Y i) v b μ) → ∀ {ε : ℝ}, 0 ≤ ε → μ.real {ω | ε ≤ t.sup' ht fun i => |Y i ω|} ≤ ↑t.card * (2 * Real.exp (-ε ^ 2 / (2 * (↑v + ↑b * ε))))","l":91,"f":"Causalean/Stat/Concentration/TailBounds/MaximalInequality.lean"},{"n":"Causalean.Stat.Concentration.mem_growthFamily_iff","k":"theorem","m":"Causalean.Stat.Concentration.Covering.VCCovering","a":"Stat/Concentration/Covering","t":2,"d":"Membership in the growth family means that some classifier realizes that restriction pattern on the sample.","s":"∀ {𝒳 : Type v} {ι : Type u} {n : ℕ} {π : ι → 𝒳 → Bool} {S : Fin n → 𝒳} {A : Finset (Fin n)}, A ∈ Causalean.Stat.Concentration.growthFamily π S ↔ ∃ i, Causalean.Stat.Concentration.restrictionPattern (π i) S = A","l":43,"f":"Causalean/Stat/Concentration/Covering/VCCovering.lean"},{"n":"Causalean.Stat.Concentration.mem_starHull_self","k":"theorem","m":"Causalean.Stat.Concentration.Rademacher.StarHull","a":"Stat/Concentration/Rademacher","t":2,"d":"Each F i belongs to its own star hull (take α = 1).","s":"∀ {ι : Type u_1} {𝒳 : Type u_2} (F : ι → 𝒳 → ℝ) (i : ι), F i ∈ Causalean.Stat.Concentration.starHull F","l":47,"f":"Causalean/Stat/Concentration/Rademacher/StarHull.lean"},{"n":"Causalean.Stat.Concentration.mgf_eq_of_mem_zero_one","k":"theorem","m":"Causalean.Stat.Concentration.TailBounds.BinomialCount","a":"Stat/Concentration/TailBounds","t":2,"d":"For a statistic that only takes the values zero and one, the moment generating function is exactly 1 + p * (exp s - 1), where p is the probability that the statistic equals one.","s":"∀ {𝒳 : Type u_2} [inst : MeasurableSpace 𝒳] {P : MeasureTheory.Measure 𝒳} [MeasureTheory.IsProbabilityMeasure P] {f : 𝒳 → ℝ}, AEMeasurable f P → (∀ᵐ (x : 𝒳) ∂P, f x = 0 ∨ f x = 1) → ∀ (p s : ℝ), ∫ (x : 𝒳), f x ∂P = p → ProbabilityTheory.mgf f P s = 1 + p * (Real.exp s - 1)","l":120,"f":"Causalean/Stat/Concentration/TailBounds/BinomialCount.lean"},{"n":"Causalean.Stat.Concentration.mgf_le_of_mem_Icc_zero_one","k":"theorem","m":"Causalean.Stat.Concentration.TailBounds.BinomialCount","a":"Stat/Concentration/TailBounds","t":2,"d":"The moment generating function of a measurable statistic taking values in the unit interval is at most exp (mean * (exp s - 1)), where mean is the statistic's population mean. This holds at every tilt s, positive or negative.","s":"∀ {𝒳 : Type u_2} [inst : MeasurableSpace 𝒳] {P : MeasureTheory.Measure 𝒳} [MeasureTheory.IsProbabilityMeasure P] {f : 𝒳 → ℝ}, AEMeasurable f P → (∀ᵐ (x : 𝒳) ∂P, f x ∈ Set.Icc 0 1) → ∀ (s : ℝ), ProbabilityTheory.mgf f P s ≤ Real.exp ((∫ (x : 𝒳), f x ∂P) * (Real.exp s - 1))","l":85,"f":"Causalean/Stat/Concentration/TailBounds/BinomialCount.lean"},{"n":"Causalean.Stat.Concentration.norm_unitInv_sub_unitInv_le","k":"theorem","m":"Causalean.Stat.Concentration.Matrix.Resolvent","a":"Stat/Concentration/Matrix","t":1,"d":"Resolvent inequality. For a seminormed ring R and units a and b of R, the norm of the difference of their inverses is bounded above by the product of the norms of the two inverses and the norm of their difference.","s":"∀ {R : Type u_1} [inst : SeminormedRing R] (a b : Rˣ), ‖↑a⁻¹ - ↑b⁻¹‖ ≤ ‖↑a⁻¹‖ * ‖↑b⁻¹‖ * ‖↑a - ↑b‖","l":27,"f":"Causalean/Stat/Concentration/Matrix/Resolvent.lean"},{"n":"Causalean.Stat.Concentration.partB.mem_incrementPairFinset_repr","k":"theorem","m":"Causalean.Stat.Concentration.Covering.DudleyEntropy","a":"Stat/Concentration/Covering","t":2,"d":"Every increment-pair representative comes from two adjacent chain approximations for some function in the class.","s":"∀ {Z : Type v} {m : ℕ} {ι : Type u} [inst : Nonempty ι] {F : ι → Z → ℝ} {S : Fin m → Z} {c : ℝ} (c_pos : 0 < c) (h' : TotallyBounded Set.univ) (n : ℕ) (j : Fin n), ∀ hk ∈ Causalean.Stat.Concentration.incrementPairFinset✝ c_pos h' n j, ∃ fh, (Causalean.Stat.Concentration.chainApprox✝ c_pos h' fh (↑j + 1), Causalean.Stat.Concentration.chainApprox✝ c_pos h' fh ↑j) = hk","l":1065,"f":"Causalean/Stat/Concentration/Covering/DudleyEntropy.lean"},{"n":"Causalean.Stat.Concentration.patternCover","k":"def","m":"Causalean.Stat.Concentration.Covering.VCCovering","a":"Stat/Concentration/Covering","t":1,"d":"One empirical-function representative for each realized Boolean pattern.","s":"{𝒳 : Type v} → {ι : Type u} → {n : ℕ} → {F : ι → 𝒳 → ℝ} → (ι → 𝒳 → Bool) → (S : Fin n → 𝒳) → Finset (Causalean.Stat.Concentration.EmpiricalFunctionSpace F S)","l":96,"f":"Causalean/Stat/Concentration/Covering/VCCovering.lean"},{"n":"Causalean.Stat.Concentration.patternCover_card_le","k":"theorem","m":"Causalean.Stat.Concentration.Covering.VCCovering","a":"Stat/Concentration/Covering","t":2,"d":"The pattern representative cover has no more elements than the realized Boolean growth family.","s":"∀ {𝒳 : Type v} {ι : Type u} {n : ℕ} {F : ι → 𝒳 → ℝ} (π : ι → 𝒳 → Bool) (S : Fin n → 𝒳), (Causalean.Stat.Concentration.patternCover π S).card ≤ (Causalean.Stat.Concentration.growthFamily π S).card","l":106,"f":"Causalean/Stat/Concentration/Covering/VCCovering.lean"},{"n":"Causalean.Stat.Concentration.patternCover_covers","k":"theorem","m":"Causalean.Stat.Concentration.Covering.VCCovering","a":"Stat/Concentration/Covering","t":2,"d":"The pattern representative cover covers every empirical function at every positive radius when the real class factors through the Boolean pattern.","s":"∀ {𝒳 : Type v} {ι : Type u} {n : ℕ} {F : ι → 𝒳 → ℝ} {S : Fin n → 𝒳} {π : ι → 𝒳 → Bool} {φ : Fin n → Bool → ℝ}, (∀ (i : ι) (j : Fin n), F i (S j) = φ j (π i (S j))) → ∀ {ε : ℝ}, 0 < ε → Set.univ ⊆ ⋃ y ∈ Causalean.Stat.Concentration.patternCover π S, Metric.ball y ε","l":122,"f":"Causalean/Stat/Concentration/Covering/VCCovering.lean"},{"n":"Causalean.Stat.Concentration.per_pair_collision_sum","k":"theorem","m":"Causalean.Stat.Concentration.Covering.HausslerPacking","a":"Stat/Concentration/Covering","t":2,"d":"The weighted sum over all fixed-length coordinate selections that agree under two Boolean patterns equals the corresponding power of the total weight of their agreeing coordinates.","s":"∀ {n m : ℕ} (w : Fin n → ℝ) (a b : Fin n → Bool), (∑ J, (∏ t, w (J t)) * if ∀ (t : Fin m), a (J t) = b (J t) then 1 else 0) = (∑ j with a j = b j, w j) ^ m","l":109,"f":"Causalean/Stat/Concentration/Covering/HausslerPacking.lean"},{"n":"Causalean.Stat.Concentration.pointwise_bound_from_empirical_norm","k":"theorem","m":"Causalean.Stat.Concentration.Covering.DudleyEntropy","a":"Stat/Concentration/Covering","t":2,"d":"If every function has empirical norm at most a given bound on a positive-size sample, then its absolute value at each sampled observation is at most the sample-size square root times that bound.","s":"∀ {Z : Type v} {m : ℕ} {ι : Type u} {F : ι → Z → ℝ} {S : Fin m → Z} {c : ℝ}, (∀ (f : ι), Causalean.Stat.Concentration.empiricalNorm S (F f) ≤ c) → ∀ (i : Fin m), 0 < m → ∀ (f : ι), |F f (S i)| ≤ √↑m * c","l":159,"f":"Causalean/Stat/Concentration/Covering/DudleyEntropy.lean"},{"n":"Causalean.Stat.Concentration.probability_abs_sub_mean_gt_le","k":"theorem","m":"Causalean.Stat.Concentration.Chebyshev","a":"Stat/Concentration","t":1,"d":"For a finite measure Q and a statistic F on the underlying space, if F is square-integrable under Q, the deviation threshold a is positive, F has mean equal to mean under Q, and the variance of F under Q is at most the envelope v, then the probability that F deviates from mean by more than a in absolute value is at most v/a².","s":"∀ {Ω : Type u_1} [inst : MeasurableSpace Ω] (Q : MeasureTheory.Measure Ω) [MeasureTheory.IsFiniteMeasure Q] (F : Ω → ℝ) (mean v a : ℝ), MeasureTheory.MemLp F 2 Q → 0 < a → ∫ (x : Ω), F x ∂Q = mean → ProbabilityTheory.variance F Q ≤ v → (Q {x | a < |F x - mean|}).toReal ≤ v / a ^ 2","l":23,"f":"Causalean/Stat/Concentration/Chebyshev.lean"},{"n":"Causalean.Stat.Concentration.pseudoDim_gives_finite_subgraph_vc","k":"theorem","m":"Causalean.Stat.Concentration.Covering.RealValuedVCSubgraph.Basic","a":"Stat/Concentration/Covering/RealValuedVCSubgraph","t":2,"d":"The real-valued theorem genuinely reuses the existing finite VC combinatorics: its hypothesis gives the exact threshold-trace certificate needed by VCCovering and HausslerPacking.","s":"∀ {𝒳 : Type u} {ι : Type v} {F : ι → 𝒳 → ℝ} {d n : ℕ}, Causalean.Stat.Concentration.HasPseudoDimAtMost F d → ∀ (T : Fin n → 𝒳 × ℝ), (Causalean.Stat.Concentration.growthFamily (Causalean.Stat.Concentration.subgraphClassifier F) T).vcDim ≤ d","l":595,"f":"Causalean/Stat/Concentration/Covering/RealValuedVCSubgraph/Basic.lean"},{"n":"Causalean.Stat.Concentration.rademacherComplexity_congr_ae","k":"theorem","m":"Causalean.Stat.Concentration.Rademacher.Rademacher","a":"Stat/Concentration/Rademacher","t":1,"d":"The population Rademacher complexity is unchanged when each member of the function family is replaced by an almost-everywhere-equal version: if for every index i, f i ∘ sample equals f' i ∘ sample almost everywhere with respect to μ, then the two families f and f' have equal Rademacher complexity.","s":"∀ {Ω : Type u_1} {𝒳 : Type u_2} {ι : Type u_3} [inst : MeasurableSpace Ω] [Countable ι] (n : ℕ) (f f' : ι → 𝒳 → ℝ) (μ : MeasureTheory.Measure Ω) [MeasureTheory.SigmaFinite μ] (sample : Ω → 𝒳), (∀ (i : ι), (fun ω => f i (sample ω)) =ᵐ[μ] fun ω => f' i (sample ω)) → rademacherComplexity n f μ sample = rademacherComplexity n f' μ sample","l":70,"f":"Causalean/Stat/Concentration/Rademacher/Rademacher.lean"},{"n":"Causalean.Stat.Concentration.rademacherComplexity_congr_ae_all","k":"theorem","m":"Causalean.Stat.Concentration.Rademacher.Rademacher","a":"Stat/Concentration/Rademacher","t":2,"d":"The population Rademacher complexity is unchanged under an almost-everywhere agreement of the whole family at once: if almost every ω satisfies f i (sample ω) = f' i (sample ω) for all indices i simultaneously, the two families have equal Rademacher complexity.","s":"∀ {Ω : Type u_1} {𝒳 : Type u_2} {ι : Type u_3} [inst : MeasurableSpace Ω] (n : ℕ) (f f' : ι → 𝒳 → ℝ) (μ : MeasureTheory.Measure Ω) [MeasureTheory.SigmaFinite μ] (sample : Ω → 𝒳), (∀ᵐ (ω : Ω) ∂μ, ∀ (i : ι), f i (sample ω) = f' i (sample ω)) → rademacherComplexity n f μ sample = rademacherComplexity n f' μ sample","l":98,"f":"Causalean/Stat/Concentration/Rademacher/Rademacher.lean"},{"n":"Causalean.Stat.Concentration.rademacherComplexity_zeroOut_le_starHullZeroOut","k":"theorem","m":"Causalean.Stat.Concentration.Rademacher.LocalRademacher","a":"Stat/Concentration/Rademacher","t":2,"d":"Bridge lemma: ι-indexed zero-out class ≤ starHullParam ι-indexed zero-out class.","s":"∀ {Ω : Type u_1} {ι : Type u_2} {𝒳 : Type u_3} [inst : MeasurableSpace Ω] (F : ι → 𝒳 → ℝ) (norm : (𝒳 → ℝ) → ℝ) (μ : MeasureTheory.Measure Ω) (X : Ω → 𝒳) (n : ℕ) {r : ℝ}, (∀ (S : Fin n → 𝒳) (σ : Signs n), BddAbove (Set.range fun p => |(↑n)⁻¹ * ∑ k, ↑↑(σ k) * Causalean.Stat.Concentration.starHullZeroOut F norm r p (S k)|)) → MeasureTheory.Integrable (fun ω => empiricalRademacherComplexity n (Causalean.Stat.Concentration.starHullZeroOut F norm r) (X ∘ ω)) (MeasureTheory.Measure.pi fun x => μ) → rademacherComplexity n (fun i ω => if norm (F i) ≤ r then F i (X ω) else 0) μ id ≤ rademacherComplexity n (Causalean.Stat.Concentration.starHullZeroOut F norm r) μ X","l":313,"f":"Causalean/Stat/Concentration/Rademacher/LocalRademacher.lean"},{"n":"Causalean.Stat.Concentration.rademacher_contraction","k":"theorem","m":"Causalean.Stat.Concentration.Rademacher.Contraction","a":"Stat/Concentration/Rademacher","t":1,"d":"Ledoux–Talagrand contraction principle (signed form). If L is nonnegative and φ : ℝ → ℝ is L-Lipschitz, i.e. |φ x - φ y| ≤ L * |x - y| for all x, y, then composing each function of the family F with φ does not increase the signed (without-abs) empirical Rademacher complexity on the sample S by more than the factor L.","s":"∀ {ι : Type u_1} {𝒳 : Type u_2} [Nonempty ι] [Finite ι] (φ : ℝ → ℝ) {L : ℝ}, 0 ≤ L → (∀ (x y : ℝ), |φ x - φ y| ≤ L * |x - y|) → ∀ (F : ι → 𝒳 → ℝ) (n : ℕ) (S : Fin n → 𝒳), empiricalRademacherComplexity_without_abs n (fun i x => φ (F i x)) S ≤ L * empiricalRademacherComplexity_without_abs n F S","l":501,"f":"Causalean/Stat/Concentration/Rademacher/Contraction.lean"},{"n":"Causalean.Stat.Concentration.rademacher_contraction_abs","k":"theorem","m":"Causalean.Stat.Concentration.Rademacher.Contraction","a":"Stat/Concentration/Rademacher","t":1,"d":"Contraction principle, absolute-value form. If φ fixes 0 and is L-Lipschitz, i.e. φ 0 = 0 and |φ x - φ y| ≤ L * |x - y| for all x, y, then composing each function of the family F with φ multiplies the (absolute-value) empirical Rademacher complexity on the sample S by at most 2 * L.","s":"∀ {ι : Type u_1} {𝒳 : Type u_2} [Nonempty ι] [Finite ι] (φ : ℝ → ℝ) {L : ℝ}, Causalean.Stat.Concentration.LipschitzAt0 φ L → ∀ (F : ι → 𝒳 → ℝ) (n : ℕ) (S : Fin n → 𝒳), empiricalRademacherComplexity n (fun i x => φ (F i x)) S ≤ 2 * L * empiricalRademacherComplexity n F S","l":664,"f":"Causalean/Stat/Concentration/Rademacher/Contraction.lean"},{"n":"Causalean.Stat.Concentration.real_vcSubgraph_dudley_example","k":"theorem","m":"Causalean.Stat.Concentration.Covering.RealValuedVCSubgraph.Empirical","a":"Stat/Concentration/Covering/RealValuedVCSubgraph","t":1,"d":"Direct application to the fixed-sample Dudley bound. For a measurable class of pseudo-dimension at most d, uniformly bounded by a positive envelope U, and a positive scale δ strictly less than U/2, evaluated on a sample S of positive size n, the class's empirical restriction is totally bounded, its covering number at every relative radius ε in (0,1] obeys the same polynomial bound vcSubgraphCoverBound d ε, and consequently its empirical Rademacher complexity without the outer absolute value is at most the Dudley entropy-integral bound 4δ + (12/√n) ∫_δ^(U/2) √(log(coveringNumber x)) dx.","s":"∀ {𝒳 : Type u} [inst : MeasurableSpace 𝒳] {ι : Type v} [Nonempty ι] {F : ι → 𝒳 → ℝ} {d n : ℕ}, (∀ (i : ι), Measurable (F i)) → Causalean.Stat.Concentration.HasPseudoDimAtMost F d → ∀ {U δ : ℝ}, 0 < U → 0 < δ → δ < U / 2 → (∀ (i : ι) (x : 𝒳), |F i x| ≤ U) → ∀ (S : Fin n → 𝒳), 0 < n → ∃ (htot : TotallyBounded Set.univ), (∀ (ε : ℝ), 0 < ε → ε ≤ 1 → Causalean.Stat.Concentration.coveringNumber htot (ε * U) ≤ Causalean.Stat.Concentration.vcSubgraphCoverBound d ε) ∧ empiricalRademacherComplexity_without_abs n F S ≤ 4 * δ + 12 / √↑n * ∫ (x : ℝ) in δ..U / 2, √(Real.log ↑(Causalean.Stat.Concentration.coveringNumber htot x))","l":163,"f":"Causalean/Stat/Concentration/Covering/RealValuedVCSubgraph/Empirical.lean"},{"n":"Causalean.Stat.Concentration.real_vcSubgraph_empirical_coveringNumber_le","k":"theorem","m":"Causalean.Stat.Concentration.Covering.RealValuedVCSubgraph.Empirical","a":"Stat/Concentration/Covering/RealValuedVCSubgraph","t":2,"d":"The empirical covering number used by Dudley is polynomially bounded at every envelope-relative radius, with the same arbitrary-measure constants.","s":"∀ {𝒳 : Type u} [inst : MeasurableSpace 𝒳] {ι : Type v} [inst_1 : Nonempty ι] {F : ι → 𝒳 → ℝ} {d n : ℕ} (hmeas : ∀ (i : ι), Measurable (F i)) (hpdim : Causalean.Stat.Concentration.HasPseudoDimAtMost F d) {U ε : ℝ} (hU : 0 < U), 0 < ε → ε ≤ 1 → ∀ (henvelope : ∀ (i : ι) (x : 𝒳), |F i x| ≤ U) (S : Fin n → 𝒳) (hn : 0 < n), have htot := Causalean.Stat.Concentration.real_vcSubgraph_empirical_totallyBounded hmeas hpdim hU henvelope S hn; Causalean.Stat.Concentration.coveringNumber htot (ε * U) ≤ Causalean.Stat.Concentration.vcSubgraphCoverBound d ε","l":121,"f":"Causalean/Stat/Concentration/Covering/RealValuedVCSubgraph/Empirical.lean"},{"n":"Causalean.Stat.Concentration.real_vcSubgraph_empirical_l2_covering","k":"theorem","m":"Causalean.Stat.Concentration.Covering.RealValuedVCSubgraph.Empirical","a":"Stat/Concentration/Covering/RealValuedVCSubgraph","t":2,"d":"The arbitrary-measure VC-subgraph theorem specializes to the empirical law of every positive-size finite sample.","s":"∀ {𝒳 : Type u} [inst : MeasurableSpace 𝒳] {ι : Type v} [Nonempty ι] {F : ι → 𝒳 → ℝ} {d n : ℕ}, (∀ (i : ι), Measurable (F i)) → Causalean.Stat.Concentration.HasPseudoDimAtMost F d → ∀ {U ε : ℝ}, 0 < U → 0 < ε → ε ≤ 1 → (∀ (i : ι) (x : 𝒳), |F i x| ≤ U) → ∀ (S : Fin n → 𝒳), 0 < n → Causalean.Stat.Concentration.L2CoveringNumberLe (Causalean.Stat.Concentration.finiteSampleMeasure S) F (ε * U) (Causalean.Stat.Concentration.vcSubgraphCoverBound d ε)","l":67,"f":"Causalean/Stat/Concentration/Covering/RealValuedVCSubgraph/Empirical.lean"},{"n":"Causalean.Stat.Concentration.real_vcSubgraph_empirical_totallyBounded","k":"theorem","m":"Causalean.Stat.Concentration.Covering.RealValuedVCSubgraph.Empirical","a":"Stat/Concentration/Covering/RealValuedVCSubgraph","t":2,"d":"A pseudo-dimension certificate makes the existing empirical function space totally bounded on every positive-size finite sample.","s":"∀ {𝒳 : Type u} [inst : MeasurableSpace 𝒳] {ι : Type v} [Nonempty ι] {F : ι → 𝒳 → ℝ} {d n : ℕ}, (∀ (i : ι), Measurable (F i)) → Causalean.Stat.Concentration.HasPseudoDimAtMost F d → ∀ {U : ℝ}, 0 < U → (∀ (i : ι) (x : 𝒳), |F i x| ≤ U) → ∀ (S : Fin n → 𝒳), 0 < n → TotallyBounded Set.univ","l":83,"f":"Causalean/Stat/Concentration/Covering/RealValuedVCSubgraph/Empirical.lean"},{"n":"Causalean.Stat.Concentration.real_vcSubgraph_l2_covering","k":"theorem","m":"Causalean.Stat.Concentration.Covering.RealValuedVCSubgraph.Basic","a":"Stat/Concentration/Covering/RealValuedVCSubgraph","t":1,"d":"Polynomial L²(Q) covering number from a pseudo-dimension bound. For a family of measurable real-valued functions of pseudo-dimension at most d, uniformly bounded by a positive envelope U, and a relative radius ε strictly between 0 and 1, the L²(Q) covering number at radius ε·U is at most vcSubgraphCoverBound d ε, uniformly over every probability measure Q on the domain.","s":"∀ {𝒳 : Type u} {ι : Type v} [inst : MeasurableSpace 𝒳] [Nonempty ι] (F : ι → 𝒳 → ℝ) (d : ℕ), (∀ (i : ι), Measurable (F i)) → Causalean.Stat.Concentration.HasPseudoDimAtMost F d → ∀ {U ε : ℝ}, 0 < U → 0 < ε → ε ≤ 1 → (∀ (i : ι) (x : 𝒳), |F i x| ≤ U) → ∀ (Q : MeasureTheory.Measure 𝒳) [MeasureTheory.IsProbabilityMeasure Q], Causalean.Stat.Concentration.L2CoveringNumberLe Q F (ε * U) (Causalean.Stat.Concentration.vcSubgraphCoverBound d ε)","l":532,"f":"Causalean/Stat/Concentration/Covering/RealValuedVCSubgraph/Basic.lean"},{"n":"Causalean.Stat.Concentration.restrictionPattern","k":"def","m":"Causalean.Stat.Concentration.Covering.VCCovering","a":"Stat/Concentration/Covering","t":1,"d":"The subset of sample coordinates at which a Boolean classifier is true.","s":"{𝒳 : Type v} → {n : ℕ} → (𝒳 → Bool) → (Fin n → 𝒳) → Finset (Fin n)","l":30,"f":"Causalean/Stat/Concentration/Covering/VCCovering.lean"},{"n":"Causalean.Stat.Concentration.restrictionPattern_mem_iff","k":"theorem","m":"Causalean.Stat.Concentration.Covering.VCCovering","a":"Stat/Concentration/Covering","t":2,"d":"Membership in a restriction pattern means the classifier is true at that sample coordinate.","s":"∀ {𝒳 : Type v} {n : ℕ} {p : 𝒳 → Bool} {S : Fin n → 𝒳} {j : Fin n}, j ∈ Causalean.Stat.Concentration.restrictionPattern p S ↔ p (S j) = true","l":51,"f":"Causalean/Stat/Concentration/Covering/VCCovering.lean"},{"n":"Causalean.Stat.Concentration.sampleMean_sub_ge_setEq","k":"theorem","m":"Causalean.Stat.Concentration.TailBounds.Hoeffding","a":"Stat/Concentration/TailBounds","t":2,"d":"For n > 0, the centered sample-mean tail event coincides with the centered-sum tail event: ε ≤ X̄ₙ − m ⟺ n ε ≤ ∑_{i<n} (f (Z i) − m). Reused by both the Hoeffding and Bernstein tail bounds.","s":"∀ {Ω : Type u_1} {X : Type u_2} [inst : MeasurableSpace Ω] [inst_1 : MeasurableSpace X] {μ : MeasureTheory.Measure Ω} {P : MeasureTheory.Measure X} (S : Causalean.Stat.IIDSample Ω X μ P) (f : X → ℝ) (m : ℝ) {n : ℕ}, 0 < n → ∀ (ε : ℝ), {ω | ε ≤ S.sampleMean f n ω - m} = {ω | ↑n * ε ≤ ∑ i ∈ Finset.range n, (f (S.Z i ω) - m)}","l":67,"f":"Causalean/Stat/Concentration/TailBounds/Hoeffding.lean"},{"n":"Causalean.Stat.Concentration.sample_eq_growthFamilyRep_of_pattern","k":"theorem","m":"Causalean.Stat.Concentration.Covering.VCLocalizedRegime","a":"Stat/Concentration/Covering","t":2,"d":"When a function class factorizes samplewise through Boolean labels, any function with a given observed Boolean pattern agrees on the sample with that pattern's chosen representative.","s":"∀ {ι : Type u_2} {𝒳 : Type u_3} {n : ℕ} {F : ι → 𝒳 → ℝ} {π : ι → 𝒳 → Bool} {S : Fin n → 𝒳} {φ : Fin n → Bool → ℝ}, (∀ (i : ι) (j : Fin n), F i (S j) = φ j (π i (S j))) → ∀ (A : ↥(Causalean.Stat.Concentration.growthFamily π S)) {i : ι}, Causalean.Stat.Concentration.restrictionPattern (π i) S = ↑A → ∀ (k : Fin n), F i (S k) = F (Causalean.Stat.Concentration.growthFamilyRep π S A) (S k)","l":755,"f":"Causalean/Stat/Concentration/Covering/VCLocalizedRegime.lean"},{"n":"Causalean.Stat.Concentration.self_log_solve","k":"theorem","m":"Causalean.Stat.Concentration.Covering.HausslerPacking","a":"Stat/Concentration/Covering","t":2,"d":"A self-referential logarithmic inequality implies an explicit linear-log upper bound.","s":"∀ {a L : ℝ} {d : ℕ}, 1 ≤ d → 2 ≤ a → 0 ≤ L → L ≤ ↑d * Real.log (2 + a * L) → L ≤ 1 + 2 * ↑d * Real.log (2 * a * ↑d)","l":542,"f":"Causalean/Stat/Concentration/Covering/HausslerPacking.lean"},{"n":"Causalean.Stat.Concentration.sharp_vc_weightedHamming_packing_card_le","k":"theorem","m":"Causalean.Stat.Concentration.Covering.RealValuedVCSubgraph.Basic","a":"Stat/Concentration/Covering/RealValuedVCSubgraph","t":2,"d":"A finite Boolean VC class with nonnegative coordinate weights has a dimension-free-base polynomial packing bound in weighted Hamming distance.","s":"∀ {n : ℕ} (d : ℕ) (w : Fin n → ℝ), (∀ (j : Fin n), 0 ≤ w j) → ∀ (r ε : ℝ), 0 < ε → ε ≤ r → ∑ j, w j ≤ r ^ 2 → ∀ (P : Finset (Fin n → Bool)), (Finset.image (fun a => {j | a j = true}) P).vcDim ≤ d → (∀ a ∈ P, ∀ b ∈ P, a ≠ b → ε ^ 2 ≤ Causalean.Stat.Concentration.weightedHammingSq w a b) → P.card ≤ ⌈(16 * r ^ 2 / ε ^ 2) ^ (2 * (d + 1))⌉₊","l":160,"f":"Causalean/Stat/Concentration/Covering/RealValuedVCSubgraph/Basic.lean"},{"n":"Causalean.Stat.Concentration.signed_sum_le_empiricalDist","k":"theorem","m":"Causalean.Stat.Concentration.Covering.DudleyEntropy","a":"Stat/Concentration/Covering","t":2,"d":"A signed sum of pointwise differences is at most the sample size times the empirical distance between the two functions.","s":"∀ {Z : Type v} {m : ℕ} {S : Fin m → Z} (f g : Z → ℝ) (σ : Signs m), ∑ i, ↑↑(σ i) * (f (S i) - g (S i)) ≤ ↑m * Causalean.Stat.Concentration.empiricalDist S f g","l":572,"f":"Causalean/Stat/Concentration/Covering/DudleyEntropy.lean"},{"n":"Causalean.Stat.Concentration.splitBound.bddAbove_increment_term","k":"theorem","m":"Causalean.Stat.Concentration.Covering.DudleyEntropy","a":"Stat/Concentration/Covering","t":2,"d":"The sum of chaining increment terms has a finite upper bound.","s":"∀ {Z : Type v} {m : ℕ} {ι : Type u} [Nonempty ι] {F : ι → Z → ℝ} {S : Fin m → Z} {c : ℝ} {c_pos : 0 < c}, (∀ (f : ι), Causalean.Stat.Concentration.empiricalNorm S (F f) ≤ c) → ∀ (h : TotallyBounded Set.univ) (n : ℕ), ¬m = 0 → ∀ (i : Signs m), BddAbove (Set.range fun fh => ∑ x_1, ∑ i_1, ↑↑(i i_1) * (Causalean.Stat.Concentration.chainApprox✝ c_pos h fh (↑x_1 + 1) (S i_1) - Causalean.Stat.Concentration.chainApprox✝ c_pos h fh (↑x_1) (S i_1)))","l":266,"f":"Causalean/Stat/Concentration/Covering/DudleyEntropy.lean"},{"n":"Causalean.Stat.Concentration.splitBound.bddAbove_main_term","k":"theorem","m":"Causalean.Stat.Concentration.Covering.DudleyEntropy","a":"Stat/Concentration/Covering","t":2,"d":"The main remainder term in the Dudley chain has a finite upper bound.","s":"∀ {Z : Type v} {m : ℕ} {ι : Type u} [Nonempty ι] {F : ι → Z → ℝ} {S : Fin m → Z} {c : ℝ} {c_pos : 0 < c}, (∀ (f : ι), Causalean.Stat.Concentration.empiricalNorm S (F f) ≤ c) → ∀ (h : TotallyBounded Set.univ) (n : ℕ), ¬m = 0 → ∀ (i : Signs m), BddAbove (Set.range fun fh => ∑ i_1, ↑↑(i i_1) * (F fh (S i_1) - Causalean.Stat.Concentration.chainApprox✝ c_pos h fh n (S i_1)))","l":217,"f":"Causalean/Stat/Concentration/Covering/DudleyEntropy.lean"},{"n":"Causalean.Stat.Concentration.sqrtLog_integral_le","k":"theorem","m":"Causalean.Stat.Concentration.Covering.SqrtLogIntegral","a":"Stat/Concentration/Covering","t":1,"d":"Dudley entropy-integral evaluation. For a positive lower limit ε that is at most the upper limit δ, the integral of √(log(δ/x)) over [ε,δ] is at most δ - ε.","s":"∀ {δ ε : ℝ}, 0 < ε → ε ≤ δ → ∫ (x : ℝ) in ε..δ, √(Real.log (δ / x)) ≤ δ - ε","l":127,"f":"Causalean/Stat/Concentration/Covering/SqrtLogIntegral.lean"},{"n":"Causalean.Stat.Concentration.sqrt_le_add_one_div_two","k":"theorem","m":"Causalean.Stat.Concentration.Covering.SqrtLogIntegral","a":"Stat/Concentration/Covering","t":2,"d":"For nonnegative t, the square root is bounded by the arithmetic mean of t and 1.","s":"∀ {t : ℝ}, 0 ≤ t → √t ≤ (t + 1) / 2","l":18,"f":"Causalean/Stat/Concentration/Covering/SqrtLogIntegral.lean"},{"n":"Causalean.Stat.Concentration.sqrt_sum_inv_abs_sq_eq_empiricalNorm_div_sqrt","k":"theorem","m":"Causalean.Stat.Concentration.Covering.VCLocalizedRegime","a":"Stat/Concentration/Covering","t":2,"d":"For a positive sample size, the square root of the sum of squared sample-normalized absolute function values equals the empirical norm divided by the square root of the sample size.","s":"∀ {𝒳 : Type u_2} {n : ℕ}, 0 < n → ∀ (S : Fin n → 𝒳) (g : 𝒳 → ℝ), √(∑ k, ((↑n)⁻¹ * |g (S k)|) ^ 2) = Causalean.Stat.Concentration.empiricalNorm S g / √↑n","l":896,"f":"Causalean/Stat/Concentration/Covering/VCLocalizedRegime.lean"},{"n":"Causalean.Stat.Concentration.starHull","k":"def","m":"Causalean.Stat.Concentration.Rademacher.StarHull","a":"Stat/Concentration/Rademacher","t":1,"d":"The star hull of a function family F : ι → 𝒳 → ℝ around the origin. Each element is an α-rescaling of some F i for α ∈ [0, 1].","s":"{ι : Type u_1} → {𝒳 : Type u_2} → (ι → 𝒳 → ℝ) → Set (𝒳 → ℝ)","l":42,"f":"Causalean/Stat/Concentration/Rademacher/StarHull.lean"},{"n":"Causalean.Stat.Concentration.starHullBall","k":"def","m":"Causalean.Stat.Concentration.UniformDeviation.CriticalRadius","a":"Stat/Concentration/UniformDeviation","t":1,"d":"The intersection of the star hull with a norm-ball of radius r.","s":"{ι : Type u_2} → {𝒳 : Type u_3} → (ι → 𝒳 → ℝ) → ((𝒳 → ℝ) → ℝ) → ℝ → Set (𝒳 → ℝ)","l":53,"f":"Causalean/Stat/Concentration/UniformDeviation/CriticalRadius.lean"},{"n":"Causalean.Stat.Concentration.starHullEval","k":"def","m":"Causalean.Stat.Concentration.Rademacher.StarHull","a":"Stat/Concentration/Rademacher","t":1,"d":"The star-hull element associated to a parameter (α, i): pointwise α · F i x. This is the \"evaluation map\" through which all star-hull arguments factor in the new starHullParam substrate.","s":"{ι : Type u_1} → {𝒳 : Type u_2} → (ι → 𝒳 → ℝ) → Causalean.Stat.Concentration.starHullParam ι → 𝒳 → ℝ","l":97,"f":"Causalean/Stat/Concentration/Rademacher/StarHull.lean"},{"n":"Causalean.Stat.Concentration.starHullEval_mem_starHull","k":"theorem","m":"Causalean.Stat.Concentration.Rademacher.StarHull","a":"Stat/Concentration/Rademacher","t":2,"d":"The evaluation map lands inside the (set-level) star hull.","s":"∀ {ι : Type u_1} {𝒳 : Type u_2} (F : ι → 𝒳 → ℝ) (p : Causalean.Stat.Concentration.starHullParam ι), Causalean.Stat.Concentration.starHullEval F p ∈ Causalean.Stat.Concentration.starHull F","l":104,"f":"Causalean/Stat/Concentration/Rademacher/StarHull.lean"},{"n":"Causalean.Stat.Concentration.starHullEval_one","k":"theorem","m":"Causalean.Stat.Concentration.Rademacher.StarHull","a":"Stat/Concentration/Rademacher","t":2,"d":"At parameter (1, i), evaluation recovers F i.","s":"∀ {ι : Type u_1} {𝒳 : Type u_2} (F : ι → 𝒳 → ℝ) (i : ι), Causalean.Stat.Concentration.starHullEval F (⟨1, of_eq_true (Eq.trans Set.mem_Icc._simp_1 (Eq.trans (congr (congrArg And zero_le_one._simp_1) (Std.le_refl._simp_1 1)) (and_self True)))⟩, i) = F i","l":111,"f":"Causalean/Stat/Concentration/Rademacher/StarHull.lean"},{"n":"Causalean.Stat.Concentration.starHullEval_zero","k":"theorem","m":"Causalean.Stat.Concentration.Rademacher.StarHull","a":"Stat/Concentration/Rademacher","t":2,"d":"At parameter (0, i), evaluation is the zero function.","s":"∀ {ι : Type u_1} {𝒳 : Type u_2} (F : ι → 𝒳 → ℝ) (i : ι), Causalean.Stat.Concentration.starHullEval F (⟨0, of_eq_true (Eq.trans Set.mem_Icc._simp_1 (Eq.trans (congr (congrArg And (Std.le_refl._simp_1 0)) zero_le_one._simp_1) (and_self True)))⟩, i) = fun x => 0","l":117,"f":"Causalean/Stat/Concentration/Rademacher/StarHull.lean"},{"n":"Causalean.Stat.Concentration.starHullIndex","k":"def","m":"Causalean.Stat.Concentration.Rademacher.StarHull","a":"Stat/Concentration/Rademacher","t":1,"d":"The star hull re-indexed as a Type (subtype carrier), suitable for the ι-parameter slot in empiricalRademacherComplexity, rademacherComplexity, etc.","s":"{ι : Type u_1} → {𝒳 : Type u_2} → (ι → 𝒳 → ℝ) → Type u_2","l":76,"f":"Causalean/Stat/Concentration/Rademacher/StarHull.lean"},{"n":"Causalean.Stat.Concentration.starHullParam","k":"def","m":"Causalean.Stat.Concentration.Rademacher.StarHull","a":"Stat/Concentration/Rademacher","t":1,"d":"Star-hull parameter type. A pair (α, i) with α ∈ [0, 1] and i : ι parameterises the star-hull element α • F i. This is the preferred index for downstream Rademacher-complexity arguments: it carries the multiplicative parameter explicitly, and monotonicity in α reduces to a per-coordinate scalar inequality.","s":"Type u_1 → Type u_1","l":89,"f":"Causalean/Stat/Concentration/Rademacher/StarHull.lean"},{"n":"Causalean.Stat.Concentration.starHullPatternClass","k":"def","m":"Causalean.Stat.Concentration.Covering.VCLocalizedRegime","a":"Stat/Concentration/Covering","t":1,"d":"The star-hull pattern class assigns each realized Boolean pattern its representative function multiplied by the pattern coefficient.","s":"{ι : Type u_2} → {𝒳 : Type u_3} → {n : ℕ} → (ι → 𝒳 → ℝ) → ((𝒳 → ℝ) → ℝ) → (π : ι → 𝒳 → Bool) → (S : Fin n → 𝒳) → ℝ → ↥(Causalean.Stat.Concentration.growthFamily π S) → 𝒳 → ℝ","l":666,"f":"Causalean/Stat/Concentration/Covering/VCLocalizedRegime.lean"},{"n":"Causalean.Stat.Concentration.starHullPatternCoeff","k":"def","m":"Causalean.Stat.Concentration.Covering.VCLocalizedRegime","a":"Stat/Concentration/Covering","t":1,"d":"The star-hull pattern coefficient is the largest active scale among functions with the same realized Boolean pattern.","s":"{ι : Type u_2} → {𝒳 : Type u_3} → {n : ℕ} → (ι → 𝒳 → ℝ) → ((𝒳 → ℝ) → ℝ) → (π : ι → 𝒳 → Bool) → (S : Fin n → 𝒳) → ℝ → ↥(Causalean.Stat.Concentration.growthFamily π S) → ℝ","l":657,"f":"Causalean/Stat/Concentration/Covering/VCLocalizedRegime.lean"},{"n":"Causalean.Stat.Concentration.starHullPatternCoeff_nonneg","k":"theorem","m":"Causalean.Stat.Concentration.Covering.VCLocalizedRegime","a":"Stat/Concentration/Covering","t":2,"d":"The coefficient assigned to any observed Boolean pattern by the localized star-hull pattern class is nonnegative.","s":"∀ {ι : Type u_2} {𝒳 : Type u_3} {n : ℕ} (F : ι → 𝒳 → ℝ) (norm : (𝒳 → ℝ) → ℝ) (π : ι → 𝒳 → Bool) (S : Fin n → 𝒳) (r : ℝ) (A : ↥(Causalean.Stat.Concentration.growthFamily π S)), 0 ≤ Causalean.Stat.Concentration.starHullPatternCoeff F norm π S r A","l":715,"f":"Causalean/Stat/Concentration/Covering/VCLocalizedRegime.lean"},{"n":"Causalean.Stat.Concentration.starHullZeroOut","k":"def","m":"Causalean.Stat.Concentration.UniformDeviation.CriticalRadius","a":"Stat/Concentration/UniformDeviation","t":1,"d":"The zero-out family on the starHullParam index: each parameter (α, i) is mapped to α • F i if its norm is at most r, and to the zero function otherwise. This is the localized star-hull family used by the local Rademacher-complexity envelope.","s":"{ι : Type u_2} → {𝒳 : Type u_3} → (ι → 𝒳 → ℝ) → ((𝒳 → ℝ) → ℝ) → ℝ → Causalean.Stat.Concentration.starHullParam ι → 𝒳 → ℝ","l":58,"f":"Causalean/Stat/Concentration/UniformDeviation/CriticalRadius.lean"},{"n":"Causalean.Stat.Concentration.starHullZeroOutScaleCoeff","k":"def","m":"Causalean.Stat.Concentration.UniformDeviation.CriticalRadius","a":"Stat/Concentration/UniformDeviation","t":1,"d":"The largest active star-hull scalar for a fixed base index.","s":"{ι : Type u_2} → {𝒳 : Type u_3} → (ι → 𝒳 → ℝ) → ((𝒳 → ℝ) → ℝ) → ℝ → ι → ℝ","l":73,"f":"Causalean/Stat/Concentration/UniformDeviation/CriticalRadius.lean"},{"n":"Causalean.Stat.Concentration.starHullZeroOutScaleCoeff_le_one","k":"theorem","m":"Causalean.Stat.Concentration.UniformDeviation.CriticalRadius","a":"Stat/Concentration/UniformDeviation","t":2,"d":"The star-hull zero-out scale coefficient is at most one.","s":"∀ {ι : Type u_2} {𝒳 : Type u_3} (F : ι → 𝒳 → ℝ) (norm : (𝒳 → ℝ) → ℝ) (r : ℝ) (i : ι), Causalean.Stat.Concentration.starHullZeroOutScaleCoeff F norm r i ≤ 1","l":79,"f":"Causalean/Stat/Concentration/UniformDeviation/CriticalRadius.lean"},{"n":"Causalean.Stat.Concentration.starHullZeroOutScaleCoeff_mul_empiricalNorm_le","k":"theorem","m":"Causalean.Stat.Concentration.Covering.VCLocalizedRegime","a":"Stat/Concentration/Covering","t":2,"d":"Under the localized VC hypotheses, a function's localized scale coefficient times its empirical norm over the sample is at most the localization radius.","s":"∀ {ι : Type u_2} {𝒳 : Type u_3} {n : ℕ} (F : ι → 𝒳 → ℝ) (norm : (𝒳 → ℝ) → ℝ), Causalean.Stat.Concentration.LocalizedVCDudleyHypotheses F norm → ∀ (S : Fin n → 𝒳) {r : ℝ}, 0 ≤ r → ∀ (i : ι), Causalean.Stat.Concentration.starHullZeroOutScaleCoeff F norm r i * Causalean.Stat.Concentration.empiricalNorm S (F i) ≤ r","l":934,"f":"Causalean/Stat/Concentration/Covering/VCLocalizedRegime.lean"},{"n":"Causalean.Stat.Concentration.starHullZeroOutScaleCoeff_nonneg","k":"theorem","m":"Causalean.Stat.Concentration.Covering.VCLocalizedRegime","a":"Stat/Concentration/Covering","t":2,"d":"The largest active scalar in a zeroed star hull is nonnegative because the zero scalar is always available.","s":"∀ {ι : Type u_2} {𝒳 : Type u_3} (F : ι → 𝒳 → ℝ) (norm : (𝒳 → ℝ) → ℝ) (r : ℝ) (i : ι), 0 ≤ Causalean.Stat.Concentration.starHullZeroOutScaleCoeff F norm r i","l":675,"f":"Causalean/Stat/Concentration/Covering/VCLocalizedRegime.lean"},{"n":"Causalean.Stat.Concentration.starHullZeroOut_bddAbove_of_bound","k":"theorem","m":"Causalean.Stat.Concentration.UniformDeviation.CriticalRadius","a":"Stat/Concentration/UniformDeviation","t":2,"d":"The signed empirical average over the zero-out star hull is bounded above by the uniform bound on the base family at the sampled points: if |F i (S_fin k)| ≤ b for every index and sample coordinate, then for any sign vector the family of signed averages indexed by star-hull parameters has b as an upper bound.","s":"∀ {𝒳 : Type u_4} {ι : Type u_5} (F : ι → 𝒳 → ℝ) (norm : (𝒳 → ℝ) → ℝ) {b : ℝ}, 0 ≤ b → ∀ (m : ℕ) (r : ℝ) (S_fin : Fin m → 𝒳), (∀ (i : ι) (k : Fin m), |F i (S_fin k)| ≤ b) → ∀ (σ : Signs m), BddAbove (Set.range fun p => |(↑m)⁻¹ * ∑ k, ↑↑(σ k) * Causalean.Stat.Concentration.starHullZeroOut F norm r p (S_fin k)|)","l":380,"f":"Causalean/Stat/Concentration/UniformDeviation/CriticalRadius.lean"},{"n":"Causalean.Stat.Concentration.starHullZeroOut_empirical_rademacher_le_growthFamily","k":"theorem","m":"Causalean.Stat.Concentration.Covering.VCLocalizedRegime","a":"Stat/Concentration/Covering","t":2,"d":"The empirical Rademacher complexity of a zero-augmented localized star hull is bounded by its radius times a logarithmic factor determined by the number of distinct Boolean patterns in the sample.","s":"∀ {ι : Type u} {𝒳 : Type v} [Nonempty ι] (F : ι → 𝒳 → ℝ) (norm : (𝒳 → ℝ) → ℝ) (π : ι → 𝒳 → Bool), (∀ {m : ℕ} (S : Fin m → 𝒳), ∃ φ, ∀ (i : ι) (j : Fin m), F i (S j) = φ j (π i (S j))) → ∀ {n : ℕ}, 0 < n → Causalean.Stat.Concentration.LocalizedVCDudleyHypotheses F norm → ∀ (S : Fin n → 𝒳) (r : ℝ), 0 ≤ r → empiricalRademacherComplexity n (Causalean.Stat.Concentration.starHullZeroOut F norm r) S ≤ r * √(2 * Real.log (2 * ↑(Causalean.Stat.Concentration.growthFamily π S).card) / ↑n)","l":1059,"f":"Causalean/Stat/Concentration/Covering/VCLocalizedRegime.lean"},{"n":"Causalean.Stat.Concentration.starHullZeroOut_empirical_rademacher_le_patternClass","k":"theorem","m":"Causalean.Stat.Concentration.Covering.VCLocalizedRegime","a":"Stat/Concentration/Covering","t":2,"d":"If function values on every finite sample depend only on Boolean labels, then the empirical Rademacher complexity of the zeroed localized star hull is no greater than that of its finite sample-pattern class.","s":"∀ {ι : Type u_2} {𝒳 : Type u_3} [Nonempty ι] {n : ℕ} (F : ι → 𝒳 → ℝ) (norm : (𝒳 → ℝ) → ℝ) (π : ι → 𝒳 → Bool), (∀ {m : ℕ} (S : Fin m → 𝒳), ∃ φ, ∀ (i : ι) (j : Fin m), F i (S j) = φ j (π i (S j))) → ∀ (S : Fin n → 𝒳) (r : ℝ), empiricalRademacherComplexity n (Causalean.Stat.Concentration.starHullZeroOut F norm r) S ≤ empiricalRademacherComplexity n (Causalean.Stat.Concentration.starHullPatternClass F norm π S r) S","l":771,"f":"Causalean/Stat/Concentration/Covering/VCLocalizedRegime.lean"},{"n":"Causalean.Stat.Concentration.starHullZeroOut_empirical_rademacher_massart_vc","k":"theorem","m":"Causalean.Stat.Concentration.Covering.VCLocalizedRegime","a":"Stat/Concentration/Covering","t":2,"d":"Massart finite-realization bound for the localized star-hull zero-out class under binary trace entropy control.","s":"∀ {ι : Type u} {𝒳 : Type v} (F : ι → 𝒳 → ℝ) (norm : (𝒳 → ℝ) → ℝ) (π : ι → 𝒳 → Bool), (∀ {m : ℕ} (S : Fin m → 𝒳), ∃ φ, ∀ (i : ι) (j : Fin m), F i (S j) = φ j (π i (S j))) → ∀ (d n : ℕ), Causalean.Stat.Concentration.BinaryTraceEntropyControl π d → Causalean.Stat.Concentration.LocalizedVCDudleyHypotheses F norm → ∀ (S : Fin n → 𝒳) (r : ℝ), 0 ≤ r → empiricalRademacherComplexity n (Causalean.Stat.Concentration.starHullZeroOut F norm r) S ≤ r * √((2 * ↑d * Real.log (↑n + 1) + 2) / ↑n)","l":1144,"f":"Causalean/Stat/Concentration/Covering/VCLocalizedRegime.lean"},{"n":"Causalean.Stat.Concentration.starHullZeroOut_inner_sup_eq","k":"theorem","m":"Causalean.Stat.Concentration.UniformDeviation.CriticalRadius","a":"Stat/Concentration/UniformDeviation","t":2,"d":"Supremizing over the star-hull scalar collapses to the largest active coefficient times the base-class inner Rademacher term.","s":"∀ {ι : Type u_2} {𝒳 : Type u_3} (F : ι → 𝒳 → ℝ) (norm : (𝒳 → ℝ) → ℝ) (r : ℝ) {n : ℕ} (ω : Fin n → 𝒳) (σ : Signs n) (i : ι), ⨆ a, |(↑n)⁻¹ * ∑ k, ↑↑(σ k) * Causalean.Stat.Concentration.starHullZeroOut F norm r (a, i) (ω k)| = Causalean.Stat.Concentration.starHullZeroOutScaleCoeff F norm r i * |(↑n)⁻¹ * ∑ k, ↑↑(σ k) * F i (ω k)|","l":122,"f":"Causalean/Stat/Concentration/UniformDeviation/CriticalRadius.lean"},{"n":"Causalean.Stat.Concentration.starHullZeroOut_inner_term_eq","k":"theorem","m":"Causalean.Stat.Concentration.UniformDeviation.CriticalRadius","a":"Stat/Concentration/UniformDeviation","t":2,"d":"The inner Rademacher term for a fixed star-hull scalar factors into that scalar, or zero when the radius test fails.","s":"∀ {ι : Type u_2} {𝒳 : Type u_3} (F : ι → 𝒳 → ℝ) (norm : (𝒳 → ℝ) → ℝ) (r : ℝ) {n : ℕ} (ω : Fin n → 𝒳) (σ : Signs n) (a : ↑(Set.Icc 0 1)) (i : ι), |(↑n)⁻¹ * ∑ k, ↑↑(σ k) * Causalean.Stat.Concentration.starHullZeroOut F norm r (a, i) (ω k)| = (if norm (Causalean.Stat.Concentration.starHullEval F (a, i)) ≤ r then ↑a else 0) * |(↑n)⁻¹ * ∑ k, ↑↑(σ k) * F i (ω k)|","l":94,"f":"Causalean/Stat/Concentration/UniformDeviation/CriticalRadius.lean"},{"n":"Causalean.Stat.Concentration.starHullZeroOut_totallyBounded_of_bounded","k":"theorem","m":"Causalean.Stat.Concentration.Covering.VCLocalizedRegime","a":"Stat/Concentration/Covering","t":2,"d":"A uniformly bounded function class has a totally bounded empirical image on every finite sample after taking its zeroed star hull.","s":"∀ {ι : Type u} {𝒳 : Type v} {F : ι → 𝒳 → ℝ} {norm : (𝒳 → ℝ) → ℝ} {B : ℝ}, (∀ (i : ι) (x : 𝒳), |F i x| ≤ B) → ∀ {n : ℕ} (S : Fin n → 𝒳) {r : ℝ}, TotallyBounded Set.univ","l":140,"f":"Causalean/Stat/Concentration/Covering/VCLocalizedRegime.lean"},{"n":"Causalean.Stat.Concentration.starHull_congr","k":"theorem","m":"Causalean.Stat.Concentration.Rademacher.StarHull","a":"Stat/Concentration/Rademacher","t":2,"d":"Pointwise-equal families have the same star hull.","s":"∀ {ι : Type u_1} {𝒳 : Type u_2} {F G : ι → 𝒳 → ℝ}, (∀ (i : ι) (x : 𝒳), F i x = G i x) → Causalean.Stat.Concentration.starHull F = Causalean.Stat.Concentration.starHull G","l":61,"f":"Causalean/Stat/Concentration/Rademacher/StarHull.lean"},{"n":"Causalean.Stat.Concentration.starHull_smul_mem","k":"theorem","m":"Causalean.Stat.Concentration.Rademacher.StarHull","a":"Stat/Concentration/Rademacher","t":1,"d":"Star-hull rescaling. For any coefficient α satisfying 0 ≤ α and α ≤ 1, the rescaled function α • F i lies in the star hull of the family F.","s":"∀ {ι : Type u_1} {𝒳 : Type u_2} (F : ι → 𝒳 → ℝ) (i : ι) {α : ℝ}, 0 ≤ α → α ≤ 1 → α • F i ∈ Causalean.Stat.Concentration.starHull F","l":69,"f":"Causalean/Stat/Concentration/Rademacher/StarHull.lean"},{"n":"Causalean.Stat.Concentration.subRoot_continuousOn_Ioi","k":"theorem","m":"Causalean.Stat.Concentration.UniformDeviation.CriticalRadius","a":"Stat/Concentration/UniformDeviation","t":2,"d":"Sub-root continuity. If ψ is sub-root, then ψ is continuous on the open ray (0, ∞). The non-increasing ratio condition ψ(r)/r ↘ supplies the linear squeeze bounds, while monotonicity of ψ supplies the opposite side of the squeeze.","s":"∀ {ψ : ℝ → ℝ}, Causalean.Stat.Concentration.SubRoot ψ → ContinuousOn ψ (Set.Ioi 0)","l":220,"f":"Causalean/Stat/Concentration/UniformDeviation/CriticalRadius.lean"},{"n":"Causalean.Stat.Concentration.subRoot_homogeneity","k":"theorem","m":"Causalean.Stat.Concentration.UniformDeviation.CriticalRadius","a":"Stat/Concentration/UniformDeviation","t":1,"d":"Sub-root inequality. If ψ is sub-root and δ* is a positive radius with ψ δ* ≤ δ* ^ 2, then for every radius r at least δ*, ψ r is at most r · δ*.","s":"∀ {ψ : ℝ → ℝ}, Causalean.Stat.Concentration.SubRoot ψ → ∀ {δ_star r : ℝ}, 0 < δ_star → δ_star ≤ r → ψ δ_star ≤ δ_star ^ 2 → ψ r ≤ r * δ_star","l":199,"f":"Causalean/Stat/Concentration/UniformDeviation/CriticalRadius.lean"},{"n":"Causalean.Stat.Concentration.subgraphClassifier","k":"def","m":"Causalean.Stat.Concentration.Covering.RealValuedVCSubgraph.Basic","a":"Stat/Concentration/Covering/RealValuedVCSubgraph","t":1,"d":"The strict subgraph classifier attached to a real-valued function class labels (x,t) precisely when t < f(x).","s":"{𝒳 : Type u} → {ι : Type v} → (ι → 𝒳 → ℝ) → ι → 𝒳 × ℝ → Bool","l":22,"f":"Causalean/Stat/Concentration/Covering/RealValuedVCSubgraph/Basic.lean"},{"n":"Causalean.Stat.Concentration.subsamplePattern","k":"def","m":"Causalean.Stat.Concentration.Covering.HausslerPacking","a":"Stat/Concentration/Covering","t":1,"d":"The set of sampled coordinates, pulled back along a coordinate map J, on which a Boolean vector is true.","s":"{n m : ℕ} → (Fin m → Fin n) → (Fin n → Bool) → Finset (Fin m)","l":24,"f":"Causalean/Stat/Concentration/Covering/HausslerPacking.lean"},{"n":"Causalean.Stat.Concentration.subsamplePattern_eq_iff","k":"theorem","m":"Causalean.Stat.Concentration.Covering.HausslerPacking","a":"Stat/Concentration/Covering","t":2,"d":"Equality of sampled Boolean patterns means that the two patterns agree at every coordinate selected by the sample.","s":"∀ {n m : ℕ} {J : Fin m → Fin n} {a b : Fin n → Bool}, Causalean.Stat.Concentration.subsamplePattern J a = Causalean.Stat.Concentration.subsamplePattern J b ↔ ∀ (t : Fin m), a (J t) = b (J t)","l":80,"f":"Causalean/Stat/Concentration/Covering/HausslerPacking.lean"},{"n":"Causalean.Stat.Concentration.subsamplePattern_mem_iff","k":"theorem","m":"Causalean.Stat.Concentration.Covering.HausslerPacking","a":"Stat/Concentration/Covering","t":2,"d":"A sampled coordinate belongs to a Boolean pattern exactly when the original Boolean vector is true at the coordinate from which it was sampled.","s":"∀ {n m : ℕ} {J : Fin m → Fin n} {a : Fin n → Bool} {i : Fin m}, i ∈ Causalean.Stat.Concentration.subsamplePattern J a ↔ a (J i) = true","l":43,"f":"Causalean/Stat/Concentration/Covering/HausslerPacking.lean"},{"n":"Causalean.Stat.Concentration.subsample_image_vcDim_le","k":"theorem","m":"Causalean.Stat.Concentration.Covering.HausslerPacking","a":"Stat/Concentration/Covering","t":2,"d":"Restricting a Boolean set family along a coordinate map does not increase VC dimension.","s":"∀ {n m : ℕ} (d : ℕ) (J : Fin m → Fin n) (P : Finset (Fin n → Bool)), (Finset.image (fun a => {j | a j = true}) P).vcDim ≤ d → (Finset.image (Causalean.Stat.Concentration.subsamplePattern J) P).vcDim ≤ d","l":230,"f":"Causalean/Stat/Concentration/Covering/HausslerPacking.lean"},{"n":"Causalean.Stat.Concentration.sum_choose_le_succ_mul_pow","k":"theorem","m":"Causalean.Stat.Concentration.Covering.VCCovering","a":"Stat/Concentration/Covering","t":2,"d":"The Sauer-Shelah binomial sum is bounded by the usual polynomial upper bound in sample size and VC dimension.","s":"∀ {n d : ℕ}, 0 < n → ↑(∑ k ≤ d, n.choose k) ≤ ↑(d + 1) * ↑n ^ d","l":212,"f":"Causalean/Stat/Concentration/Covering/VCCovering.lean"},{"n":"Causalean.Stat.Concentration.sum_choose_le_succ_pow","k":"theorem","m":"Causalean.Stat.Concentration.Covering.HausslerPacking","a":"Stat/Concentration/Covering","t":2,"d":"A partial sum of binomial coefficients is no larger than a polynomial power, providing the elementary growth bound used in VC estimates.","s":"∀ (n d : ℕ), ∑ k ≤ d, n.choose k ≤ (n + 1) ^ d","l":51,"f":"Causalean/Stat/Concentration/Covering/HausslerPacking.lean"},{"n":"Causalean.Stat.Concentration.sup_pair_lipschitz_scaled","k":"theorem","m":"Causalean.Stat.Concentration.Rademacher.Contraction","a":"Stat/Concentration/Rademacher","t":2,"d":"For a finite nonempty collection, the sum of the largest values obtained by adding and subtracting a nonnegative multiple of a transformation with a given Lipschitz constant is no greater than the corresponding sum using that linear bound.","s":"∀ {ι : Type u_3} [Nonempty ι] [Finite ι] (φ : ℝ → ℝ) {L c : ℝ}, 0 ≤ c → (∀ (x y : ℝ), |φ x - φ y| ≤ L * |x - y|) → ∀ (a b : ι → ℝ), (⨆ i, a i + c * φ (b i)) + ⨆ i, a i - c * φ (b i) ≤ (⨆ i, a i + c * (L * b i)) + ⨆ i, a i - c * (L * b i)","l":112,"f":"Causalean/Stat/Concentration/Rademacher/Contraction.lean"},{"n":"Causalean.Stat.Concentration.term_le_total_sum_of_nonneg","k":"theorem","m":"Causalean.Stat.Concentration.Covering.DudleyEntropy","a":"Stat/Concentration/Covering","t":2,"d":"A nonnegative term in a finite sum is bounded by the full sum of nonnegative terms.","s":"∀ {α : Type u} [inst : Fintype α] {M : Type u_1} [inst_1 : AddCommMonoid M] [inst_2 : Preorder M] [IsOrderedAddMonoid M] (j : α) (f : α → M), (∀ (j : α), 0 ≤ f j) → f j ≤ ∑ i, f i","l":39,"f":"Causalean/Stat/Concentration/Covering/DudleyEntropy.lean"},{"n":"Causalean.Stat.Concentration.varianceAdaptiveExpectedMaximal_le","k":"theorem","m":"Causalean.Stat.Concentration.VarianceAdaptiveVCExpectedMaximal.ExpectedMaximal","a":"Stat/Concentration/VarianceAdaptiveVCExpectedMaximal","t":1,"d":"Variance-adaptive expected maximal inequality. Let P be a probability measure on the sample space and F a countable family of real-valued functions on it. Suppose the population $L^2$ radius σ is strictly positive and strictly less than the envelope U, the covering-entropy base A is at least Euler's number and the exponent v is at least one, every function in F is measurable and bounded in absolute value by U, each function's population $L^2$ distance from the zero function is at most σ, F has polynomial empirical $L^2$ covering numbers with envelope U, base A, and exponent v, and the sample size n is positive. Then the expected empirical supremum of F over an n-point i.i.d. sample drawn from P is at most the universal constant varianceAdaptiveVCConstant times the variance-adaptive rate σ √(v log(AU/σ)/n) + v U log(AU/σ)/n.","s":"∀ {𝒳 : Type u} [inst : MeasurableSpace 𝒳] {ι : Type v} [Nonempty ι] [Countable ι] (P : MeasureTheory.Measure 𝒳) [MeasureTheory.IsProbabilityMeasure P] (F : ι → 𝒳 → ℝ) {U σ A v : ℝ}, 0 < σ → σ < U → Real.exp 1 ≤ A → 1 ≤ v → (∀ (i : ι), Measurable (F i)) → (∀ (i : ι) (x : 𝒳), |F i x| ≤ U) → (∀ (i : ι), (Causalean.Stat.Concentration.measureL2Dist P (F i) fun x => 0) ≤ σ) → Causalean.Stat.Concentration.HasPolynomialEmpiricalL2Cover F U A v → ∀ (n : ℕ), 0 < n → (∫ (S : Fin n → 𝒳), Causalean.Stat.Concentration.countableEmpiricalSup P F S ∂MeasureTheory.Measure.pi fun x => P) ≤ Causalean.Stat.Concentration.varianceAdaptiveVCConstant * Causalean.Stat.Concentration.vcExpectedMaximalRate U σ A v n","l":21,"f":"Causalean/Stat/Concentration/VarianceAdaptiveVCExpectedMaximal/ExpectedMaximal.lean"},{"n":"Causalean.Stat.Concentration.varianceAdaptiveRademacherComplexity_le","k":"theorem","m":"Causalean.Stat.Concentration.VarianceAdaptiveVCExpectedMaximal.Rademacher","a":"Stat/Concentration/VarianceAdaptiveVCExpectedMaximal","t":1,"d":"Variance-adaptive Rademacher complexity bound. Let P be a probability measure on the sample space and F a countable family of real-valued functions on it. Suppose the population $L^2$ radius σ is strictly positive and strictly less than the envelope U, the covering-entropy base A is at least Euler's number and the exponent v is at least one, every function in F is measurable and bounded in absolute value by U, each function's population $L^2$ distance from the zero function is at most σ, F has polynomial empirical $L^2$ covering numbers with envelope U, base A, and exponent v, and the sample size n is positive. Then the Rademacher complexity of F under P with n i.i.d. observations is at most half the universal constant varianceAdaptiveVCConstant times the variance-adaptive rate σ √(v log(AU/σ)/n) + v U log(AU/σ)/n.","s":"∀ {𝒳 : Type u} [inst : MeasurableSpace 𝒳] {ι : Type v} [Nonempty ι] [Countable ι] (P : MeasureTheory.Measure 𝒳) [MeasureTheory.IsProbabilityMeasure P] (F : ι → 𝒳 → ℝ) {U σ A v : ℝ}, 0 < σ → σ < U → Real.exp 1 ≤ A → 1 ≤ v → (∀ (i : ι), Measurable (F i)) → (∀ (i : ι) (x : 𝒳), |F i x| ≤ U) → (∀ (i : ι), (Causalean.Stat.Concentration.measureL2Dist P (F i) fun x => 0) ≤ σ) → Causalean.Stat.Concentration.HasPolynomialEmpiricalL2Cover F U A v → ∀ (n : ℕ), 0 < n → rademacherComplexity n F P id ≤ Causalean.Stat.Concentration.varianceAdaptiveVCConstant / 2 * Causalean.Stat.Concentration.vcExpectedMaximalRate U σ A v n","l":697,"f":"Causalean/Stat/Concentration/VarianceAdaptiveVCExpectedMaximal/Rademacher.lean"},{"n":"Causalean.Stat.Concentration.varianceAdaptiveVCConstant","k":"def","m":"Causalean.Stat.Concentration.VarianceAdaptiveVCExpectedMaximal.Basic","a":"Stat/Concentration/VarianceAdaptiveVCExpectedMaximal","t":1,"d":"The fixed numerical constant used by the variance-adaptive VC-type expected maximal inequality. Its value is deliberately non-optimized.","s":"ℝ","l":143,"f":"Causalean/Stat/Concentration/VarianceAdaptiveVCExpectedMaximal/Basic.lean"},{"n":"Causalean.Stat.Concentration.variance_sum_pi_eq","k":"theorem","m":"Causalean.Stat.Concentration.Matrix.IidSums","a":"Stat/Concentration/Matrix","t":2,"d":"Variance of an iid sum. For an iid sample of size N from μ (the product measure Measure.pi), the variance of the sum ∑ᵢ g(Xᵢ) of a fixed L² statistic g equals N · Var[g] — the coordinate copies are independent, so cross-covariances vanish.","s":"∀ {N : ℕ} {Ω : Type u_1} [inst : MeasurableSpace Ω] (μ : MeasureTheory.Measure Ω) [MeasureTheory.IsProbabilityMeasure μ] (g : Ω → ℝ), MeasureTheory.MemLp g 2 μ → ProbabilityTheory.variance (fun ω => ∑ i, g (ω i)) (MeasureTheory.Measure.pi fun x => μ) = ↑N * ProbabilityTheory.variance g μ","l":61,"f":"Causalean/Stat/Concentration/Matrix/IidSums.lean"},{"n":"Causalean.Stat.Concentration.vcEntropy_chaining_bound","k":"theorem","m":"Causalean.Stat.Concentration.VarianceAdaptiveVCExpectedMaximal.EntropyChaining","a":"Stat/Concentration/VarianceAdaptiveVCExpectedMaximal","t":1,"d":"Dudley chaining bound for VC-type entropy. Let μ be a probability measure on Ω, g : ι → Ω → ℝ a family of functions, and g0 : ℕ → ι a countable enumeration of the index set. If g has uniform VC-type entropy relative to μ, with envelope U, population $L^2$ radius σ, covering-entropy base A, and exponent v, then there is a universal constant C > 0 such that, for every sample size n ≥ 1, the expectation of the countable empirical-process supremum along the enumeration g0 over the n-fold product of μ is at most C · (σ √(log(U/σ)/n) + U log(U/σ)/n).","s":"∀ {Ω : Type u_1} {ι : Type u_2} [inst : MeasurableSpace Ω] (μ : MeasureTheory.Measure Ω) [MeasureTheory.IsProbabilityMeasure μ] (g : ι → Ω → ℝ) (g0 : ℕ → ι) (U σ A v : ℝ), Causalean.Stat.Concentration.HasVCUniformEntropy μ g U σ A v → ∃ C, 0 < C ∧ ∀ (n : ℕ), 1 ≤ n → (∫⁻ (w : Fin n → Ω), Causalean.Stat.Concentration.countableEmpiricalProcessSup μ g g0 w ∂MeasureTheory.Measure.pi fun x => μ) ≤ ENNReal.ofReal (C * (σ * √(Real.log (U / σ) / ↑n) + U * Real.log (U / σ) / ↑n))","l":172,"f":"Causalean/Stat/Concentration/VarianceAdaptiveVCExpectedMaximal/EntropyChaining.lean"},{"n":"Causalean.Stat.Concentration.vcExpectedMaximalRate","k":"def","m":"Causalean.Stat.Concentration.VarianceAdaptiveVCExpectedMaximal.Basic","a":"Stat/Concentration/VarianceAdaptiveVCExpectedMaximal","t":1,"d":"The variance-adaptive VC-type rate is the sum of a leading σ √(v log(AU/σ)/n) term and a second-order v U log(AU/σ)/n term, with a logarithm normalized to be at least one.","s":"ℝ → ℝ → ℝ → ℝ → ℕ → ℝ","l":135,"f":"Causalean/Stat/Concentration/VarianceAdaptiveVCExpectedMaximal/Basic.lean"},{"n":"Causalean.Stat.Concentration.vcLocalizedEnvelope","k":"theorem","m":"Causalean.Stat.Concentration.Covering.VCLocalizedRegime","a":"Stat/Concentration/Covering","t":2,"d":"Fixed-n finite-VC localized envelope package: sub-root envelope, localized Rademacher upper bound, critical-radius bound by the slope, and squared critical-radius rate.","s":"∀ {Ω : Type u_1} {ι : Type u} {𝒳 : Type v} [inst : MeasurableSpace Ω] (F : ι → 𝒳 → ℝ) (norm : (𝒳 → ℝ) → ℝ) (μ : MeasureTheory.Measure Ω) (X : Ω → 𝒳) [MeasureTheory.IsProbabilityMeasure μ] (K : ℝ) (d n : ℕ), 1 ≤ K → 0 < n → ∀ (Hvc : Causalean.Stat.Concentration.BinaryFactoredVCClass F d), Causalean.Stat.Concentration.LocalizedVCDudleyHypotheses F norm → Causalean.Stat.Concentration.SubRoot (Causalean.Stat.Concentration.vcLocalizedPsi K d n) ∧ Causalean.Stat.Concentration.RademacherUpperBound F norm μ X n (Causalean.Stat.Concentration.vcLocalizedPsi K d n) ∧ Causalean.Stat.Concentration.criticalRadius (Causalean.Stat.Concentration.vcLocalizedPsi K d n) ≤ Causalean.Stat.Concentration.vcLocalizedSlope K d n ∧ Causalean.Stat.Concentration.criticalRadius (Causalean.Stat.Concentration.vcLocalizedPsi K d n) ^ 2 ≤ 36 * ((K * ↑d * Real.log (↑n + 1) + 1) / ↑n)","l":1450,"f":"Causalean/Stat/Concentration/Covering/VCLocalizedRegime.lean"},{"n":"Causalean.Stat.Concentration.vcLocalizedPsi","k":"def","m":"Causalean.Stat.Concentration.Covering.VCLocalizedRegime","a":"Stat/Concentration/Covering","t":1,"d":"The finite-VC localized envelope maps a radius to the slope times that radius.","s":"ℝ → ℕ → ℕ → ℝ → ℝ","l":254,"f":"Causalean/Stat/Concentration/Covering/VCLocalizedRegime.lean"},{"n":"Causalean.Stat.Concentration.vcLocalizedPsi_criticalRadius_fp","k":"theorem","m":"Causalean.Stat.Concentration.Covering.VCUniformDeviation","a":"Stat/Concentration/Covering","t":2,"d":"For the finite-VC localized envelope, its value at the critical radius is no larger than the square of that radius.","s":"∀ {K : ℝ} {d n : ℕ}, 0 ≤ K → 0 < n → Causalean.Stat.Concentration.vcLocalizedPsi K d n (Causalean.Stat.Concentration.criticalRadius (Causalean.Stat.Concentration.vcLocalizedPsi K d n)) ≤ Causalean.Stat.Concentration.criticalRadius (Causalean.Stat.Concentration.vcLocalizedPsi K d n) ^ 2","l":52,"f":"Causalean/Stat/Concentration/Covering/VCUniformDeviation.lean"},{"n":"Causalean.Stat.Concentration.vcLocalizedPsi_subRoot","k":"theorem","m":"Causalean.Stat.Concentration.Covering.VCLocalizedRegime","a":"Stat/Concentration/Covering","t":2,"d":"The finite-VC localized envelope is sub-root.","s":"∀ (K : ℝ) (d n : ℕ), Causalean.Stat.Concentration.SubRoot (Causalean.Stat.Concentration.vcLocalizedPsi K d n)","l":327,"f":"Causalean/Stat/Concentration/Covering/VCLocalizedRegime.lean"},{"n":"Causalean.Stat.Concentration.vcLocalizedRademacherUpperBound","k":"theorem","m":"Causalean.Stat.Concentration.Covering.VCLocalizedRegime","a":"Stat/Concentration/Covering","t":2,"d":"The finite-VC localized envelope upper-bounds population localized Rademacher complexity.","s":"∀ {Ω : Type u_1} {ι : Type u} {𝒳 : Type v} [inst : MeasurableSpace Ω] (F : ι → 𝒳 → ℝ) (norm : (𝒳 → ℝ) → ℝ) (μ : MeasureTheory.Measure Ω) (X : Ω → 𝒳) [MeasureTheory.IsProbabilityMeasure μ] (K : ℝ) (d n : ℕ), 1 ≤ K → ∀ (Hvc : Causalean.Stat.Concentration.BinaryFactoredVCClass F d), Causalean.Stat.Concentration.LocalizedVCDudleyHypotheses F norm → Causalean.Stat.Concentration.RademacherUpperBound F norm μ X n (Causalean.Stat.Concentration.vcLocalizedPsi K d n)","l":1416,"f":"Causalean/Stat/Concentration/Covering/VCLocalizedRegime.lean"},{"n":"Causalean.Stat.Concentration.vcLocalizedRademacherUpperBound_of_card","k":"theorem","m":"Causalean.Stat.Concentration.Covering.VCLocalizedRegime","a":"Stat/Concentration/Covering","t":2,"d":"The growth-cardinality localized envelope upper-bounds population localized Rademacher complexity.","s":"∀ {Ω : Type u_1} {ι : Type u} {𝒳 : Type v} [inst : MeasurableSpace Ω] (F : ι → 𝒳 → ℝ) (norm : (𝒳 → ℝ) → ℝ) (π : ι → 𝒳 → Bool) (μ : MeasureTheory.Measure Ω) (X : Ω → 𝒳) [MeasureTheory.IsProbabilityMeasure μ], (∀ {m : ℕ} (S : Fin m → 𝒳), ∃ φ, ∀ (i : ι) (j : Fin m), F i (S j) = φ j (π i (S j))) → ∀ (K : ℝ) (dPi n : ℕ), 1 ≤ K → (∀ (m : ℕ) (S : Fin m → 𝒳), (Causalean.Stat.Concentration.growthFamily π S).card ≤ (m + 1) ^ dPi) → Causalean.Stat.Concentration.LocalizedVCDudleyHypotheses F norm → Causalean.Stat.Concentration.RademacherUpperBound F norm μ X n (Causalean.Stat.Concentration.vcLocalizedPsi K dPi n)","l":1431,"f":"Causalean/Stat/Concentration/Covering/VCLocalizedRegime.lean"},{"n":"Causalean.Stat.Concentration.vcLocalizedRate_nonneg","k":"theorem","m":"Causalean.Stat.Concentration.Covering.VCLocalizedRegime","a":"Stat/Concentration/Covering","t":2,"d":"The rate inside vcLocalizedSlope is nonnegative when K ≥ 0.","s":"∀ {K : ℝ} {d n : ℕ}, 0 ≤ K → 0 ≤ (K * ↑d * Real.log (↑n + 1) + 1) / ↑n","l":263,"f":"Causalean/Stat/Concentration/Covering/VCLocalizedRegime.lean"},{"n":"Causalean.Stat.Concentration.vcLocalizedRegime","k":"def","m":"Causalean.Stat.Concentration.Covering.VCLocalizedRegime","a":"Stat/Concentration/Covering","t":1,"d":"Build the LocalizedRegime bundle for the localized-deviation theorems from a bounded finite-VC class and the finite-VC localized envelope.","s":"{Ω : Type u_1} → {ι : Type u} → {𝒳 : Type v} → [inst : MeasurableSpace Ω] → (F : ι → 𝒳 → ℝ) → (norm : (𝒳 → ℝ) → ℝ) → (μ : MeasureTheory.Measure Ω) → (X : Ω → 𝒳) → [MeasureTheory.IsProbabilityMeasure μ] → (b : ℝ) → 0 ≤ b → (∀ (i : ι) (ω : Ω), |F i (X ω)| ≤ b) → (K : ℝ) → (d : ℕ) → 1 ≤ K → Causalean.Stat.Concentration.BinaryFactoredVCClass F d → Causalean.Stat.Concentration.LocalizedVCDudleyHypotheses F norm → Causalean.Stat.Concentration.LocalizedRegime Ω ι 𝒳 F norm μ X","l":1471,"f":"Causalean/Stat/Concentration/Covering/VCLocalizedRegime.lean"},{"n":"Causalean.Stat.Concentration.vcLocalizedRegime_of_card","k":"def","m":"Causalean.Stat.Concentration.Covering.VCLocalizedRegime","a":"Stat/Concentration/Covering","t":1,"d":"Build the LocalizedRegime bundle from a direct growth-cardinality bound on the binary trace family.","s":"{Ω : Type u_1} → {ι : Type u} → {𝒳 : Type v} → [inst : MeasurableSpace Ω] → (F : ι → 𝒳 → ℝ) → (norm : (𝒳 → ℝ) → ℝ) → (π : ι → 𝒳 → Bool) → (μ : MeasureTheory.Measure Ω) → (X : Ω → 𝒳) → [MeasureTheory.IsProbabilityMeasure μ] → (b : ℝ) → 0 ≤ b → (∀ (i : ι) (ω : Ω), |F i (X ω)| ≤ b) → (∀ {m : ℕ} (S : Fin m → 𝒳), ∃ φ, ∀ (i : ι) (j : Fin m), F i (S j) = φ j (π i (S j))) → (K : ℝ) → (dPi : ℕ) → 1 ≤ K → (∀ (m : ℕ) (S : Fin m → 𝒳), (Causalean.Stat.Concentration.growthFamily π S).card ≤ (m + 1) ^ dPi) → Causalean.Stat.Concentration.LocalizedVCDudleyHypotheses F norm → Causalean.Stat.Concentration.LocalizedRegime Ω ι 𝒳 F norm μ X","l":1490,"f":"Causalean/Stat/Concentration/Covering/VCLocalizedRegime.lean"},{"n":"Causalean.Stat.Concentration.vcLocalizedSlope","k":"def","m":"Causalean.Stat.Concentration.Covering.VCLocalizedRegime","a":"Stat/Concentration/Covering","t":1,"d":"The finite-VC localized slope is the sample-size dependent coefficient in the linear localized Rademacher envelope.","s":"ℝ → ℕ → ℕ → ℝ","l":246,"f":"Causalean/Stat/Concentration/Covering/VCLocalizedRegime.lean"},{"n":"Causalean.Stat.Concentration.vcLocalizedSlope_nonneg","k":"theorem","m":"Causalean.Stat.Concentration.Covering.VCLocalizedRegime","a":"Stat/Concentration/Covering","t":2,"d":"The finite-VC slope is nonnegative.","s":"∀ (K : ℝ) (d n : ℕ), 0 ≤ Causalean.Stat.Concentration.vcLocalizedSlope K d n","l":279,"f":"Causalean/Stat/Concentration/Covering/VCLocalizedRegime.lean"},{"n":"Causalean.Stat.Concentration.vcLocalizedSlope_pos","k":"theorem","m":"Causalean.Stat.Concentration.Covering.VCLocalizedRegime","a":"Stat/Concentration/Covering","t":2,"d":"If K ≥ 0 and n > 0, the finite-VC slope is strictly positive.","s":"∀ {K : ℝ} {d n : ℕ}, 0 ≤ K → 0 < n → 0 < Causalean.Stat.Concentration.vcLocalizedSlope K d n","l":285,"f":"Causalean/Stat/Concentration/Covering/VCLocalizedRegime.lean"},{"n":"Causalean.Stat.Concentration.vcLocalizedSlope_sq","k":"theorem","m":"Causalean.Stat.Concentration.Covering.VCLocalizedRegime","a":"Stat/Concentration/Covering","t":2,"d":"The exact square of the finite-VC slope.","s":"∀ {K : ℝ} {d n : ℕ}, 0 ≤ K → Causalean.Stat.Concentration.vcLocalizedSlope K d n ^ 2 = 36 * ((K * ↑d * Real.log (↑n + 1) + 1) / ↑n)","l":306,"f":"Causalean/Stat/Concentration/Covering/VCLocalizedRegime.lean"},{"n":"Causalean.Stat.Concentration.vcMaximalLog","k":"def","m":"Causalean.Stat.Concentration.VarianceAdaptiveVCExpectedMaximal.Basic","a":"Stat/Concentration/VarianceAdaptiveVCExpectedMaximal","t":1,"d":"The logarithmic complexity used by the maximal inequality is the log of the larger of Euler's number and the envelope-to-radius ratio A U / σ. This normalization keeps the logarithm at least one.","s":"ℝ → ℝ → ℝ → ℝ","l":129,"f":"Causalean/Stat/Concentration/VarianceAdaptiveVCExpectedMaximal/Basic.lean"},{"n":"Causalean.Stat.Concentration.vcSubgraphCoverBound","k":"def","m":"Causalean.Stat.Concentration.Covering.RealValuedVCSubgraph.Basic","a":"Stat/Concentration/Covering/RealValuedVCSubgraph","t":1,"d":"The explicit polynomial cardinality used by the public VC-subgraph bound. Its constants are universal and intentionally non-optimized.","s":"ℕ → ℝ → ℕ","l":54,"f":"Causalean/Stat/Concentration/Covering/RealValuedVCSubgraph/Basic.lean"},{"n":"Causalean.Stat.Concentration.vc_coveringNumber_le_growth","k":"theorem","m":"Causalean.Stat.Concentration.Covering.VCCovering","a":"Stat/Concentration/Covering","t":2,"d":"The empirical covering number of a binary-factored class is bounded by the number of realized Boolean restriction patterns.","s":"∀ {𝒳 : Type v} {ι : Type u} {n : ℕ} {F : ι → 𝒳 → ℝ} {S : Fin n → 𝒳} {π : ι → 𝒳 → Bool} {φ : Fin n → Bool → ℝ}, (∀ (i : ι) (j : Fin n), F i (S j) = φ j (π i (S j))) → ∀ (h' : TotallyBounded Set.univ) {ε : ℝ}, 0 < ε → Causalean.Stat.Concentration.coveringNumber h' ε ≤ (Causalean.Stat.Concentration.growthFamily π S).card","l":157,"f":"Causalean/Stat/Concentration/Covering/VCCovering.lean"},{"n":"Causalean.Stat.Concentration.vc_coveringNumber_le_sum_choose","k":"theorem","m":"Causalean.Stat.Concentration.Covering.VCCovering","a":"Stat/Concentration/Covering","t":1,"d":"Sauer-Shelah covering-number bound for a binary-factored class. Suppose the real-valued class factors through a Boolean classifier at each sample coordinate: F i (S j) = φ j (π i (S j)), so that the sample-restricted class is totally bounded in the empirical pseudometric, the covering radius ε is positive, and the induced Boolean growth family on the sample has VC dimension at most d. Then the empirical covering number at radius ε is at most the Sauer-Shelah binomial sum ∑_{k≤d} C(n,k).","s":"∀ {𝒳 : Type v} {ι : Type u} {n d : ℕ} {F : ι → 𝒳 → ℝ} {S : Fin n → 𝒳} {π : ι → 𝒳 → Bool} {φ : Fin n → Bool → ℝ}, (∀ (i : ι) (j : Fin n), F i (S j) = φ j (π i (S j))) → ∀ (h' : TotallyBounded Set.univ) {ε : ℝ}, 0 < ε → (Causalean.Stat.Concentration.growthFamily π S).vcDim ≤ d → Causalean.Stat.Concentration.coveringNumber h' ε ≤ ∑ k ≤ d, n.choose k","l":173,"f":"Causalean/Stat/Concentration/Covering/VCCovering.lean"},{"n":"Causalean.Stat.Concentration.vc_localized_deviation_event","k":"theorem","m":"Causalean.Stat.Concentration.Covering.VCUniformDeviation","a":"Stat/Concentration/Covering","t":1,"d":"Finite-VC localized uniform deviation event. For a class of measurable real-valued functions uniformly bounded in absolute value by a nonnegative constant b, with a localization constant K at least 1 and a positive sample size n, suppose the class factors through a binary Boolean family of VC dimension at most d and satisfies the localized Dudley hypotheses relative to the given seminorm. For a failure probability δ in (0,1], writing ρ for the localized slope vcLocalizedSlope K d n, if ρ is at most b and the peeling/log-domination side condition holds at every dyadic scale, then there is a measurable event of probability at least 1 - δ on which every class member i with 0 ≤ norm (F i) ≤ b satisfies the sharp localized deviation bound |n⁻¹ ∑ₖ F i(ωₖ) − 𝔼[F i]| ≤ 8ρ·norm(F i) + 5ρ².","s":"∀ {ι : Type u} {𝒳 : Type v} [inst : MeasurableSpace 𝒳] [Nonempty 𝒳] [Nonempty ι] [Countable ι] (F : ι → 𝒳 → ℝ) (norm : (𝒳 → ℝ) → ℝ) (μ : MeasureTheory.Measure 𝒳) [MeasureTheory.IsProbabilityMeasure μ], (∀ (i : ι), Measurable (F i)) → ∀ (b : ℝ), 0 ≤ b → (∀ (i : ι) (x : 𝒳), |F i x| ≤ b) → ∀ (K : ℝ) (d n : ℕ), 1 ≤ K → 0 < n → ∀ (Hvc : Causalean.Stat.Concentration.BinaryFactoredVCClass F d), Causalean.Stat.Concentration.LocalizedVCDudleyHypotheses F norm → ∀ {δ : ℝ}, 0 < δ → δ ≤ 1 → Causalean.Stat.Concentration.vcLocalizedSlope K d n ≤ b → (∀ (L : ℕ), b ≤ Causalean.Stat.Concentration.vcLocalizedSlope K d n * 2 ^ L → b * √(2 * Real.log (2 * (↑L + 1) / δ) / ↑n) ≤ Causalean.Stat.Concentration.vcLocalizedSlope K d n ^ 2) → ∃ E, MeasurableSet E ∧ (MeasureTheory.Measure.pi fun x => μ) E ≥ 1 - ENNReal.ofReal δ ∧ ∀ ω ∈ E, ∀ (i : ι), 0 ≤ norm (F i) → norm (F i) ≤ b → |(↑n)⁻¹ * ∑ k, F i (ω k) - ∫ (x : 𝒳), (fun x => F i x) x ∂μ| ≤ 8 * Causalean.Stat.Concentration.vcLocalizedSlope K d n * norm (F i) + 5 * Causalean.Stat.Concentration.vcLocalizedSlope K d n ^ 2","l":308,"f":"Causalean/Stat/Concentration/Covering/VCUniformDeviation.lean"},{"n":"Causalean.Stat.Concentration.vc_localized_deviation_event_of_card","k":"theorem","m":"Causalean.Stat.Concentration.Covering.VCUniformDeviation","a":"Stat/Concentration/Covering","t":1,"d":"Growth-cardinality localized uniform deviation event. For a class of measurable real-valued functions uniformly bounded in absolute value by a nonnegative constant b that factors through a Boolean classifier π at every finite sample, for a coordinate transform φ with F i (S j) = φ j (π i (S j)), with a localization constant K at least 1 and a positive sample size n, suppose the induced Boolean growth family satisfies the direct cardinality bound #growthFamily π S ≤ (m+1)^dPi on every finite sample of size m and the class satisfies the localized Dudley hypotheses relative to the given seminorm. For a failure probability δ in (0,1], writing ρ for the localized slope vcLocalizedSlope K dPi n, if ρ is at most b and the peeling/log-domination side condition holds at every dyadic scale, then there is a measurable event of probability at least 1 - δ on which every class member i with 0 ≤ norm (F i) ≤ b satisfies the sharp localized deviation bound |n⁻¹ ∑ₖ F i(ωₖ) − 𝔼[F i]| ≤ 8ρ·norm(F i) + 5ρ².","s":"∀ {ι : Type u} {𝒳 : Type v} [inst : MeasurableSpace 𝒳] [Nonempty 𝒳] [Nonempty ι] [Countable ι] (F : ι → 𝒳 → ℝ) (norm : (𝒳 → ℝ) → ℝ) (π : ι → 𝒳 → Bool) (μ : MeasureTheory.Measure 𝒳) [MeasureTheory.IsProbabilityMeasure μ], (∀ (i : ι), Measurable (F i)) → ∀ (b : ℝ), 0 ≤ b → (∀ (i : ι) (x : 𝒳), |F i x| ≤ b) → (∀ {m : ℕ} (S : Fin m → 𝒳), ∃ φ, ∀ (i : ι) (j : Fin m), F i (S j) = φ j (π i (S j))) → ∀ (K : ℝ) (dPi n : ℕ), 1 ≤ K → 0 < n → (∀ (m : ℕ) (S : Fin m → 𝒳), (Causalean.Stat.Concentration.growthFamily π S).card ≤ (m + 1) ^ dPi) → Causalean.Stat.Concentration.LocalizedVCDudleyHypotheses F norm → ∀ {δ : ℝ}, 0 < δ → δ ≤ 1 → Causalean.Stat.Concentration.vcLocalizedSlope K dPi n ≤ b → (∀ (L : ℕ), b ≤ Causalean.Stat.Concentration.vcLocalizedSlope K dPi n * 2 ^ L → b * √(2 * Real.log (2 * (↑L + 1) / δ) / ↑n) ≤ Causalean.Stat.Concentration.vcLocalizedSlope K dPi n ^ 2) → ∃ E, MeasurableSet E ∧ (MeasureTheory.Measure.pi fun x => μ) E ≥ 1 - ENNReal.ofReal δ ∧ ∀ ω ∈ E, ∀ (i : ι), 0 ≤ norm (F i) → norm (F i) ≤ b → |(↑n)⁻¹ * ∑ k, F i (ω k) - ∫ (x : 𝒳), (fun x => F i x) x ∂μ| ≤ 8 * Causalean.Stat.Concentration.vcLocalizedSlope K dPi n * norm (F i) + 5 * Causalean.Stat.Concentration.vcLocalizedSlope K dPi n ^ 2","l":393,"f":"Causalean/Stat/Concentration/Covering/VCUniformDeviation.lean"},{"n":"Causalean.Stat.Concentration.vc_starHullZeroOut_empirical_rademacher_aemeasurable","k":"theorem","m":"Causalean.Stat.Concentration.Covering.VCUniformDeviation","a":"Stat/Concentration/Covering","t":2,"d":"Measurability bridge for the empirical Rademacher process of the finite-VC localized star hull.","s":"∀ {ι : Type u} {𝒳 : Type v} [inst : MeasurableSpace 𝒳] [Nonempty ι] [Countable ι] (F : ι → 𝒳 → ℝ) (norm : (𝒳 → ℝ) → ℝ), (∀ (i : ι), Measurable (F i)) → ∀ (μ : MeasureTheory.Measure 𝒳) (b r : ℝ), 0 ≤ b → (∀ (i : ι) (x : 𝒳), |F i x| ≤ b) → ∀ (n : ℕ), 0 < n → AEMeasurable (fun ω => empiricalRademacherComplexity n (Causalean.Stat.Concentration.starHullZeroOut F norm r) (id ∘ ω)) (MeasureTheory.Measure.pi fun x => μ)","l":122,"f":"Causalean/Stat/Concentration/Covering/VCUniformDeviation.lean"},{"n":"Causalean.Stat.Concentration.vc_starHullZeroOut_empirical_rademacher_integrable","k":"theorem","m":"Causalean.Stat.Concentration.Covering.VCUniformDeviation","a":"Stat/Concentration/Covering","t":2,"d":"Integrability of the finite-VC localized star-hull empirical Rademacher process follows from the deterministic linear envelope once the residual measurability bridge above is available.","s":"∀ {ι : Type u} {𝒳 : Type v} [inst : MeasurableSpace 𝒳] [Nonempty ι] [Countable ι] (F : ι → 𝒳 → ℝ) (norm : (𝒳 → ℝ) → ℝ) (μ : MeasureTheory.Measure 𝒳) [MeasureTheory.IsProbabilityMeasure μ], (∀ (i : ι), Measurable (F i)) → ∀ (b : ℝ), 0 ≤ b → (∀ (i : ι) (x : 𝒳), |F i x| ≤ b) → ∀ (K : ℝ) (d n : ℕ), 1 ≤ K → 0 < n → ∀ (Hvc : Causalean.Stat.Concentration.BinaryFactoredVCClass F d), Causalean.Stat.Concentration.LocalizedVCDudleyHypotheses F norm → ∀ (r : ℝ), Causalean.Stat.Concentration.vcLocalizedSlope K d n ≤ r → MeasureTheory.Integrable (fun ω => empiricalRademacherComplexity n (Causalean.Stat.Concentration.starHullZeroOut F norm r) (id ∘ ω)) (MeasureTheory.Measure.pi fun x => μ)","l":194,"f":"Causalean/Stat/Concentration/Covering/VCUniformDeviation.lean"},{"n":"Causalean.Stat.Concentration.vc_starHullZeroOut_empirical_rademacher_integrable_of_card","k":"theorem","m":"Causalean.Stat.Concentration.Covering.VCUniformDeviation","a":"Stat/Concentration/Covering","t":2,"d":"Cardinality-bound variant of the integrability bridge for the localized star-hull empirical Rademacher process.","s":"∀ {ι : Type u} {𝒳 : Type v} [inst : MeasurableSpace 𝒳] [Nonempty ι] [Countable ι] (F : ι → 𝒳 → ℝ) (norm : (𝒳 → ℝ) → ℝ) (π : ι → 𝒳 → Bool) (μ : MeasureTheory.Measure 𝒳) [MeasureTheory.IsProbabilityMeasure μ], (∀ (i : ι), Measurable (F i)) → ∀ (b : ℝ), 0 ≤ b → (∀ (i : ι) (x : 𝒳), |F i x| ≤ b) → (∀ {m : ℕ} (S : Fin m → 𝒳), ∃ φ, ∀ (i : ι) (j : Fin m), F i (S j) = φ j (π i (S j))) → ∀ (K : ℝ) (dPi n : ℕ), 1 ≤ K → 0 < n → (∀ (m : ℕ) (S : Fin m → 𝒳), (Causalean.Stat.Concentration.growthFamily π S).card ≤ (m + 1) ^ dPi) → Causalean.Stat.Concentration.LocalizedVCDudleyHypotheses F norm → ∀ (r : ℝ), Causalean.Stat.Concentration.vcLocalizedSlope K dPi n ≤ r → MeasureTheory.Integrable (fun ω => empiricalRademacherComplexity n (Causalean.Stat.Concentration.starHullZeroOut F norm r) (id ∘ ω)) (MeasureTheory.Measure.pi fun x => μ)","l":246,"f":"Causalean/Stat/Concentration/Covering/VCUniformDeviation.lean"},{"n":"Causalean.Stat.Concentration.vc_starHullZeroOut_empirical_rademacher_le_linear","k":"theorem","m":"Causalean.Stat.Concentration.Covering.VCLocalizedRegime","a":"Stat/Concentration/Covering","t":2,"d":"The generic finite-VC/Dudley sample-path bridge for the localized star-hull class.","s":"∀ {ι : Type u} {𝒳 : Type v} (F : ι → 𝒳 → ℝ) (norm : (𝒳 → ℝ) → ℝ) (K : ℝ) (d n : ℕ), 1 ≤ K → ∀ (Hvc : Causalean.Stat.Concentration.BinaryFactoredVCClass F d), Causalean.Stat.Concentration.LocalizedVCDudleyHypotheses F norm → ∀ (S : Fin n → 𝒳) (r : ℝ), 0 ≤ r → empiricalRademacherComplexity n (Causalean.Stat.Concentration.starHullZeroOut F norm r) S ≤ Causalean.Stat.Concentration.vcLocalizedPsi K d n r","l":1289,"f":"Causalean/Stat/Concentration/Covering/VCLocalizedRegime.lean"},{"n":"Causalean.Stat.Concentration.vc_starHullZeroOut_empirical_rademacher_le_linear_of_card","k":"theorem","m":"Causalean.Stat.Concentration.Covering.VCLocalizedRegime","a":"Stat/Concentration/Covering","t":2,"d":"Cardinality-bound sample-path bridge for the localized star-hull class. This is the direct growth-function analogue of vc_starHullZeroOut_empirical_rademacher_le_linear.","s":"∀ {ι : Type u} {𝒳 : Type v} (F : ι → 𝒳 → ℝ) (norm : (𝒳 → ℝ) → ℝ) (π : ι → 𝒳 → Bool), (∀ {m : ℕ} (S : Fin m → 𝒳), ∃ φ, ∀ (i : ι) (j : Fin m), F i (S j) = φ j (π i (S j))) → ∀ (K : ℝ) (dPi n : ℕ), 1 ≤ K → (∀ (m : ℕ) (S : Fin m → 𝒳), (Causalean.Stat.Concentration.growthFamily π S).card ≤ (m + 1) ^ dPi) → Causalean.Stat.Concentration.LocalizedVCDudleyHypotheses F norm → ∀ (S : Fin n → 𝒳) (r : ℝ), 0 ≤ r → empiricalRademacherComplexity n (Causalean.Stat.Concentration.starHullZeroOut F norm r) S ≤ Causalean.Stat.Concentration.vcLocalizedPsi K dPi n r","l":1306,"f":"Causalean/Stat/Concentration/Covering/VCLocalizedRegime.lean"},{"n":"Causalean.Stat.Concentration.vc_starHullZeroOut_population_rademacher_le_linear","k":"theorem","m":"Causalean.Stat.Concentration.Covering.VCLocalizedRegime","a":"Stat/Concentration/Covering","t":2,"d":"The generic finite-VC/Dudley population bridge for the localized star-hull class.","s":"∀ {Ω : Type u_1} {ι : Type u} {𝒳 : Type v} [inst : MeasurableSpace Ω] (F : ι → 𝒳 → ℝ) (norm : (𝒳 → ℝ) → ℝ) (μ : MeasureTheory.Measure Ω) (X : Ω → 𝒳) [MeasureTheory.IsProbabilityMeasure μ] (K : ℝ) (d n : ℕ), 1 ≤ K → ∀ (Hvc : Causalean.Stat.Concentration.BinaryFactoredVCClass F d), Causalean.Stat.Concentration.LocalizedVCDudleyHypotheses F norm → ∀ (r : ℝ), 0 ≤ r → rademacherComplexity n (Causalean.Stat.Concentration.starHullZeroOut F norm r) μ X ≤ Causalean.Stat.Concentration.vcLocalizedPsi K d n r","l":1325,"f":"Causalean/Stat/Concentration/Covering/VCLocalizedRegime.lean"},{"n":"Causalean.Stat.Concentration.vc_starHullZeroOut_population_rademacher_le_linear_of_card","k":"theorem","m":"Causalean.Stat.Concentration.Covering.VCLocalizedRegime","a":"Stat/Concentration/Covering","t":2,"d":"Cardinality-bound population bridge for the localized star-hull class.","s":"∀ {Ω : Type u_1} {ι : Type u} {𝒳 : Type v} [inst : MeasurableSpace Ω] (F : ι → 𝒳 → ℝ) (norm : (𝒳 → ℝ) → ℝ) (π : ι → 𝒳 → Bool) (μ : MeasureTheory.Measure Ω) (X : Ω → 𝒳) [MeasureTheory.IsProbabilityMeasure μ], (∀ {m : ℕ} (S : Fin m → 𝒳), ∃ φ, ∀ (i : ι) (j : Fin m), F i (S j) = φ j (π i (S j))) → ∀ (K : ℝ) (dPi n : ℕ), 1 ≤ K → (∀ (m : ℕ) (S : Fin m → 𝒳), (Causalean.Stat.Concentration.growthFamily π S).card ≤ (m + 1) ^ dPi) → Causalean.Stat.Concentration.LocalizedVCDudleyHypotheses F norm → ∀ (r : ℝ), 0 ≤ r → rademacherComplexity n (Causalean.Stat.Concentration.starHullZeroOut F norm r) μ X ≤ Causalean.Stat.Concentration.vcLocalizedPsi K dPi n r","l":1369,"f":"Causalean/Stat/Concentration/Covering/VCLocalizedRegime.lean"},{"n":"Causalean.Stat.Concentration.vc_weightedHamming_packing_card_le","k":"theorem","m":"Causalean.Stat.Concentration.Covering.HausslerPacking","a":"Stat/Concentration/Covering","t":1,"d":"Haussler ε-packing bound, logarithmic form. Let w be a nonnegative weight on the n sample coordinates with total weight at most r², where ε is positive and at most r. If the Boolean family P, viewed as the sets of coordinates where each member is true, has VC dimension at most d and every two distinct members of P are separated by weighted Hamming distance at least ε² in that weighting, then the logarithm of the cardinality of P is at most 1 + 2d·log(4d·r²/ε²) — a bound depending only on d and the ratio r²/ε², with no dependence on the ambient coordinate count n.","s":"∀ {n : ℕ} (d : ℕ) (w : Fin n → ℝ), (∀ (j : Fin n), 0 ≤ w j) → ∀ (r ε : ℝ), 0 < ε → ε ≤ r → ∑ j, w j ≤ r ^ 2 → ∀ (P : Finset (Fin n → Bool)), (Finset.image (fun a => {j | a j = true}) P).vcDim ≤ d → (∀ a ∈ P, ∀ b ∈ P, a ≠ b → ε ^ 2 ≤ Causalean.Stat.Concentration.weightedHammingSq w a b) → Real.log ↑P.card ≤ 1 + 2 * ↑d * Real.log (4 * ↑d * r ^ 2 / ε ^ 2)","l":616,"f":"Causalean/Stat/Concentration/Covering/HausslerPacking.lean"},{"n":"Causalean.Stat.Concentration.weightedHammingSq","k":"def","m":"Causalean.Stat.Concentration.Covering.HausslerPacking","a":"Stat/Concentration/Covering","t":1,"d":"Weighted Hamming pseudo-distance squared between two Boolean vectors on Fin n, using nonnegative coordinate weights.","s":"{n : ℕ} → (Fin n → ℝ) → (Fin n → Bool) → (Fin n → Bool) → ℝ","l":19,"f":"Causalean/Stat/Concentration/Covering/HausslerPacking.lean"},{"n":"Causalean.Stat.Concentration.weightedHammingSq_nonneg","k":"theorem","m":"Causalean.Stat.Concentration.Covering.HausslerPacking","a":"Stat/Concentration/Covering","t":2,"d":"Nonnegative coordinate weights make the weighted Hamming pseudo-distance nonnegative.","s":"∀ {n : ℕ} (w : Fin n → ℝ), (∀ (j : Fin n), 0 ≤ w j) → ∀ (a b : Fin n → Bool), 0 ≤ Causalean.Stat.Concentration.weightedHammingSq w a b","l":31,"f":"Causalean/Stat/Concentration/Covering/HausslerPacking.lean"},{"n":"Causalean.Stat.Concentration.zero_mem_starHull","k":"theorem","m":"Causalean.Stat.Concentration.Rademacher.StarHull","a":"Stat/Concentration/Rademacher","t":2,"d":"The zero function belongs to the star hull of any nonempty family (take α = 0).","s":"∀ {ι : Type u_1} {𝒳 : Type u_2} [Nonempty ι] (F : ι → 𝒳 → ℝ), (fun x => 0) ∈ Causalean.Stat.Concentration.starHull F","l":53,"f":"Causalean/Stat/Concentration/Rademacher/StarHull.lean"},{"n":"Causalean.Stat.DegenKernel","k":"structure","m":"Causalean.Stat.UStatistic.Variance","a":"Stat/UStatistic","t":1,"d":"Degenerate order-2 kernel. A kernel g : X → X → ℝ is doubly degenerate under the measure P when g is measurable as a function on $X \\times X$, g is symmetric, $g(x,y) = g(y,x)$, integrating g over its second argument against P gives zero for every fixed first argument, and g is square-integrable under the product measure $P \\times P$.","s":"{X : Type u_2} → [inst : MeasurableSpace X] → MeasureTheory.Measure X → (X → X → ℝ) → Prop","l":95,"f":"Causalean/Stat/UStatistic/Variance.lean"},{"n":"Causalean.Stat.DegenKernel.deg'","k":"theorem","m":"Causalean.Stat.UStatistic.Variance","a":"Stat/UStatistic","t":2,"d":"Left degeneracy, from symmetry and right degeneracy.","s":"∀ {X : Type u_2} [inst : MeasurableSpace X] {P : MeasureTheory.Measure X} {g : X → X → ℝ}, Causalean.Stat.DegenKernel P g → ∀ (y : X), ∫ (x : X), g x y ∂P = 0","l":123,"f":"Causalean/Stat/UStatistic/Variance.lean"},{"n":"Causalean.Stat.DegenKernel.integrable","k":"theorem","m":"Causalean.Stat.UStatistic.Variance","a":"Stat/UStatistic","t":2,"d":"g is L¹ on a finite product measure, from L².","s":"∀ {X : Type u_2} [inst : MeasurableSpace X] {P : MeasureTheory.Measure X} {g : X → X → ℝ} [MeasureTheory.IsFiniteMeasure P], Causalean.Stat.DegenKernel P g → MeasureTheory.Integrable (fun p => g p.1 p.2) (P.prod P)","l":116,"f":"Causalean/Stat/UStatistic/Variance.lean"},{"n":"Causalean.Stat.DegenKernel.toOrderDegenKernel","k":"theorem","m":"Causalean.Stat.UStatistic.Variance","a":"Stat/UStatistic","t":1,"d":"Bridge to the order-m theory (m = 2). If the two-argument kernel g : X → X → ℝ is degenerate: measurable, symmetric, square-integrable under P × P, and with zero conditional mean in each coordinate, then the paired order-2 kernel pairKernel g, defined by z ↦ g(z 0, z 1), is completely degenerate in the order-m sense.","s":"∀ {X : Type u_2} [inst : MeasurableSpace X] {P : MeasureTheory.Measure X} {g : X → X → ℝ} [MeasureTheory.SigmaFinite P], Causalean.Stat.DegenKernel P g → Causalean.Stat.OrderDegenKernel P (Causalean.Stat.pairKernel g)","l":128,"f":"Causalean/Stat/UStatistic/Variance.lean"},{"n":"Causalean.Stat.FiniteStratumMarkedRatioMse.armCategoryEvent","k":"def","m":"Causalean.Stat.Sample.FiniteStratumMarkedRatioMse.Basic","a":"Stat/Sample/FiniteStratumMarkedRatioMse","t":1,"d":"The arm/category event contains observations with the requested arm and category labels.","s":"{Omega : Type u_1} → {kappa : Type u_2} → (Omega → kappa) → (Omega → Bool) → Bool → kappa → Set Omega","l":42,"f":"Causalean/Stat/Sample/FiniteStratumMarkedRatioMse/Basic.lean"},{"n":"Causalean.Stat.FiniteStratumMarkedRatioMse.armCategoryMass","k":"def","m":"Causalean.Stat.Sample.FiniteStratumMarkedRatioMse.Basic","a":"Stat/Sample/FiniteStratumMarkedRatioMse","t":1,"d":"The population arm/category mass is the real mass of the joint label event.","s":"{Omega : Type u_1} → {kappa : Type u_2} → [inst : MeasurableSpace Omega] → MeasureTheory.Measure Omega → (Omega → kappa) → (Omega → Bool) → Bool → kappa → ℝ","l":58,"f":"Causalean/Stat/Sample/FiniteStratumMarkedRatioMse/Basic.lean"},{"n":"Causalean.Stat.FiniteStratumMarkedRatioMse.armMarkSum","k":"def","m":"Causalean.Stat.Sample.FiniteStratumMarkedRatioMse.Basic","a":"Stat/Sample/FiniteStratumMarkedRatioMse","t":1,"d":"The empirical arm/category mark sum adds the supported mark over all sample coordinates.","s":"{Omega : Type u_1} → {kappa : Type u_2} → {m : ℕ} → (Omega → kappa) → (Omega → Bool) → (Omega → ℝ) → (Fin m → Omega) → Bool → kappa → ℝ","l":70,"f":"Causalean/Stat/Sample/FiniteStratumMarkedRatioMse/Basic.lean"},{"n":"Causalean.Stat.FiniteStratumMarkedRatioMse.categoryArmCount","k":"def","m":"Causalean.Stat.Sample.FiniteStratumMarkedRatioMse.Basic","a":"Stat/Sample/FiniteStratumMarkedRatioMse","t":1,"d":"The empirical arm/category count is the number of coordinates with the specified Boolean arm and finite category.","s":"{Omega : Type u_1} → {kappa : Type u_2} → [DecidableEq kappa] → {m : ℕ} → (Omega → kappa) → (Omega → Bool) → (Fin m → Omega) → Bool → kappa → ℕ","l":31,"f":"Causalean/Stat/Sample/FiniteStratumMarkedRatioMse/Basic.lean"},{"n":"Causalean.Stat.FiniteStratumMarkedRatioMse.categoryCount","k":"def","m":"Causalean.Stat.Sample.FiniteStratumMarkedRatioMse.Basic","a":"Stat/Sample/FiniteStratumMarkedRatioMse","t":1,"d":"The empirical category count includes both Boolean arms.","s":"{Omega : Type u_1} → {kappa : Type u_2} → [DecidableEq kappa] → {m : ℕ} → (Omega → kappa) → (Omega → Bool) → (Fin m → Omega) → kappa → ℕ","l":37,"f":"Causalean/Stat/Sample/FiniteStratumMarkedRatioMse/Basic.lean"},{"n":"Causalean.Stat.FiniteStratumMarkedRatioMse.categoryEvent","k":"def","m":"Causalean.Stat.Sample.FiniteStratumMarkedRatioMse.Basic","a":"Stat/Sample/FiniteStratumMarkedRatioMse","t":1,"d":"The category event contains observations with the requested category, irrespective of arm.","s":"{Omega : Type u_1} → {kappa : Type u_2} → (Omega → kappa) → kappa → Set Omega","l":48,"f":"Causalean/Stat/Sample/FiniteStratumMarkedRatioMse/Basic.lean"},{"n":"Causalean.Stat.FiniteStratumMarkedRatioMse.categoryMass","k":"def","m":"Causalean.Stat.Sample.FiniteStratumMarkedRatioMse.Basic","a":"Stat/Sample/FiniteStratumMarkedRatioMse","t":1,"d":"The population category mass is the real mass of the category event.","s":"{Omega : Type u_1} → {kappa : Type u_2} → [inst : MeasurableSpace Omega] → MeasureTheory.Measure Omega → (Omega → kappa) → kappa → ℝ","l":53,"f":"Causalean/Stat/Sample/FiniteStratumMarkedRatioMse/Basic.lean"},{"n":"Causalean.Stat.FiniteStratumMarkedRatioMse.fixedStratumArmCenterTarget","k":"def","m":"Causalean.Stat.Sample.FiniteStratumMarkedRatioMse.Basic","a":"Stat/Sample/FiniteStratumMarkedRatioMse","t":1,"d":"The center-weighted population target is the auxiliary form used by the residual decomposition.","s":"{Omega : Type u_1} → {kappa : Type u_2} → [inst : MeasurableSpace Omega] → MeasureTheory.Measure Omega → (Omega → kappa) → Finset kappa → (Bool → kappa → ℝ) → Bool → ℝ","l":126,"f":"Causalean/Stat/Sample/FiniteStratumMarkedRatioMse/Basic.lean"},{"n":"Causalean.Stat.FiniteStratumMarkedRatioMse.fixedStratumArmCenterTarget_eq_target","k":"theorem","m":"Causalean.Stat.Sample.FiniteStratumMarkedRatioMse.Basic","a":"Stat/Sample/FiniteStratumMarkedRatioMse","t":2,"d":"Measurable group and arm labels, square-integrable supported residuals, cellwise residual centering, and positive arm mass in every selected positive-mass category ensure that the center-weighted target equals the population arm target.","s":"∀ {Omega : Type u_1} {kappa : Type u_2} [inst : MeasurableSpace Omega] [Fintype kappa] [DecidableEq kappa] [inst_3 : MeasurableSpace kappa] [MeasurableSingletonClass kappa] (mu : MeasureTheory.Measure Omega) [MeasureTheory.IsProbabilityMeasure mu] (group : Omega → kappa) (arm : Omega → Bool) (Y : Omega → ℝ) (center : Bool → kappa → ℝ) (H : Finset kappa) (a : Bool), Measurable group → Measurable arm → (∀ (k : kappa), MeasureTheory.MemLp (Causalean.Stat.FiniteStratumMarkedRatioMse.supportedArmResidual group arm Y center a k) 2 mu) → (∀ (k : kappa), ∫ (omega : Omega) in Causalean.Stat.FiniteStratumMarkedRatioMse.armCategoryEvent group arm a k, Y omega - center a k ∂mu = 0) → (∀ k ∈ H, 0 < Causalean.Stat.FiniteStratumMarkedRatioMse.categoryMass mu group k → 0 < Causalean.Stat.FiniteStratumMarkedRatioMse.armCategoryMass mu group arm a k) → Causalean.Stat.FiniteStratumMarkedRatioMse.fixedStratumArmCenterTarget mu group H center a = Causalean.Stat.FiniteStratumMarkedRatioMse.fixedStratumArmTarget mu group arm Y H a","l":353,"f":"Causalean/Stat/Sample/FiniteStratumMarkedRatioMse/Basic.lean"},{"n":"Causalean.Stat.FiniteStratumMarkedRatioMse.fixedStratumArmCenteredNoise","k":"def","m":"Causalean.Stat.Sample.FiniteStratumMarkedRatioMse.Basic","a":"Stat/Sample/FiniteStratumMarkedRatioMse","t":1,"d":"The fixed-arm centered noise is the occupancy-weighted sum of totalized cell residual means.","s":"{Omega : Type u_1} → {kappa : Type u_2} → [DecidableEq kappa] → {m : ℕ} → (Omega → kappa) → (Omega → Bool) → (Omega → ℝ) → (Bool → kappa → ℝ) → Finset kappa → Bool → (Fin m → Omega) → ℝ","l":153,"f":"Causalean/Stat/Sample/FiniteStratumMarkedRatioMse/Basic.lean"},{"n":"Causalean.Stat.FiniteStratumMarkedRatioMse.fixedStratumArmMassFluctuation","k":"def","m":"Causalean.Stat.Sample.FiniteStratumMarkedRatioMse.Basic","a":"Stat/Sample/FiniteStratumMarkedRatioMse","t":1,"d":"The empirical-mass fluctuation is the centered sample category weighting of the supplied arm/category centers.","s":"{Omega : Type u_1} → {kappa : Type u_2} → [inst : MeasurableSpace Omega] → [DecidableEq kappa] → {m : ℕ} → MeasureTheory.Measure Omega → (Omega → kappa) → (Omega → Bool) → (Bool → kappa → ℝ) → Finset kappa → Bool → (Fin m → Omega) → ℝ","l":169,"f":"Causalean/Stat/Sample/FiniteStratumMarkedRatioMse/Basic.lean"},{"n":"Causalean.Stat.FiniteStratumMarkedRatioMse.fixedStratumArmMissingRemainder","k":"def","m":"Causalean.Stat.Sample.FiniteStratumMarkedRatioMse.Basic","a":"Stat/Sample/FiniteStratumMarkedRatioMse","t":1,"d":"The fixed-arm missing remainder is the normalized sum of cell centers times category occupancies whose requested arm is absent.","s":"{Omega : Type u_1} → {kappa : Type u_2} → [DecidableEq kappa] → {m : ℕ} → (Omega → kappa) → (Omega → Bool) → (Bool → kappa → ℝ) → Finset kappa → Bool → (Fin m → Omega) → ℝ","l":161,"f":"Causalean/Stat/Sample/FiniteStratumMarkedRatioMse/Basic.lean"},{"n":"Causalean.Stat.FiniteStratumMarkedRatioMse.fixedStratumArmScore","k":"def","m":"Causalean.Stat.Sample.FiniteStratumMarkedRatioMse.Basic","a":"Stat/Sample/FiniteStratumMarkedRatioMse","t":1,"d":"The fixed-set single-arm score weights each zero-safe empirical arm mean by its empirical category occupancy divided by the nominal sample size.","s":"{Omega : Type u_1} → {kappa : Type u_2} → [DecidableEq kappa] → {m : ℕ} → (Omega → kappa) → (Omega → Bool) → (Omega → ℝ) → Finset kappa → Bool → (Fin m → Omega) → ℝ","l":95,"f":"Causalean/Stat/Sample/FiniteStratumMarkedRatioMse/Basic.lean"},{"n":"Causalean.Stat.FiniteStratumMarkedRatioMse.fixedStratumArmScore_sub_centerTarget_decomposition","k":"theorem","m":"Causalean.Stat.Sample.FiniteStratumMarkedRatioMse.Basic","a":"Stat/Sample/FiniteStratumMarkedRatioMse","t":1,"d":"The single-arm empirical score error around a supplied center target is exactly centered ratio noise plus empirical-mass fluctuation minus the explicit missing-arm remainder.","s":"∀ {Omega : Type u_1} {kappa : Type u_2} [inst : MeasurableSpace Omega] [Fintype kappa] [inst_2 : DecidableEq kappa] [inst_3 : MeasurableSpace kappa] [MeasurableSingletonClass kappa] {m : ℕ} (mu : MeasureTheory.Measure Omega) (group : Omega → kappa) (arm : Omega → Bool) (Y : Omega → ℝ) (center : Bool → kappa → ℝ) (H : Finset kappa) (a : Bool) (z : Fin m → Omega), Causalean.Stat.FiniteStratumMarkedRatioMse.fixedStratumArmScore group arm Y H a z - Causalean.Stat.FiniteStratumMarkedRatioMse.fixedStratumArmCenterTarget mu group H center a = Causalean.Stat.FiniteStratumMarkedRatioMse.fixedStratumArmCenteredNoise group arm Y center H a z + Causalean.Stat.FiniteStratumMarkedRatioMse.fixedStratumArmMassFluctuation mu group arm center H a z - Causalean.Stat.FiniteStratumMarkedRatioMse.fixedStratumArmMissingRemainder group arm center H a z","l":319,"f":"Causalean/Stat/Sample/FiniteStratumMarkedRatioMse/Basic.lean"},{"n":"Causalean.Stat.FiniteStratumMarkedRatioMse.fixedStratumArmTarget","k":"def","m":"Causalean.Stat.Sample.FiniteStratumMarkedRatioMse.Basic","a":"Stat/Sample/FiniteStratumMarkedRatioMse","t":1,"d":"The population fixed-set single-arm target weights conditional arm means by population category masses.","s":"{Omega : Type u_1} → {kappa : Type u_2} → [inst : MeasurableSpace Omega] → MeasureTheory.Measure Omega → (Omega → kappa) → (Omega → Bool) → (Omega → ℝ) → Finset kappa → Bool → ℝ","l":111,"f":"Causalean/Stat/Sample/FiniteStratumMarkedRatioMse/Basic.lean"},{"n":"Causalean.Stat.FiniteStratumMarkedRatioMse.fixedStratumMarkedRatio","k":"def","m":"Causalean.Stat.Sample.FiniteStratumMarkedRatioMse.Basic","a":"Stat/Sample/FiniteStratumMarkedRatioMse","t":1,"d":"The fixed-set marked ratio score is the treated single-arm score minus the control single-arm score.","s":"{Omega : Type u_1} → {kappa : Type u_2} → [DecidableEq kappa] → {m : ℕ} → (Omega → kappa) → (Omega → Bool) → (Omega → ℝ) → Finset kappa → (Fin m → Omega) → ℝ","l":103,"f":"Causalean/Stat/Sample/FiniteStratumMarkedRatioMse/Basic.lean"},{"n":"Causalean.Stat.FiniteStratumMarkedRatioMse.fixedStratumMarkedRatio_memLp_two","k":"theorem","m":"Causalean.Stat.Sample.FiniteStratumMarkedRatioMse.Main","a":"Stat/Sample/FiniteStratumMarkedRatioMse","t":2,"d":"Measurable group and arm labels and a measurable mark together with finite second moments for every supported residual cell imply finite second moments for the treated-minus-control fixed-set score.","s":"∀ {Omega : Type u_1} {kappa : Type u_2} [inst : MeasurableSpace Omega] [Fintype kappa] [inst_2 : DecidableEq kappa] [inst_3 : MeasurableSpace kappa] [MeasurableSingletonClass kappa] {m : ℕ} (mu : MeasureTheory.Measure Omega) [MeasureTheory.IsProbabilityMeasure mu] (group : Omega → kappa) (arm : Omega → Bool) (Y : Omega → ℝ) (center : Bool → kappa → ℝ) (H : Finset kappa), Measurable group → Measurable arm → Measurable Y → (∀ (a : Bool) (k : kappa), MeasureTheory.MemLp (Causalean.Stat.FiniteStratumMarkedRatioMse.supportedArmResidual group arm Y center a k) 2 mu) → MeasureTheory.MemLp (Causalean.Stat.FiniteStratumMarkedRatioMse.fixedStratumMarkedRatio group arm Y H) 2 (MeasureTheory.Measure.pi fun x => mu)","l":175,"f":"Causalean/Stat/Sample/FiniteStratumMarkedRatioMse/Main.lean"},{"n":"Causalean.Stat.FiniteStratumMarkedRatioMse.fixedStratumMarkedTarget","k":"def","m":"Causalean.Stat.Sample.FiniteStratumMarkedRatioMse.Basic","a":"Stat/Sample/FiniteStratumMarkedRatioMse","t":1,"d":"The population fixed-set marked target is the treated arm target minus the control arm target.","s":"{Omega : Type u_1} → {kappa : Type u_2} → [inst : MeasurableSpace Omega] → MeasureTheory.Measure Omega → (Omega → kappa) → (Omega → Bool) → (Omega → ℝ) → Finset kappa → ℝ","l":118,"f":"Causalean/Stat/Sample/FiniteStratumMarkedRatioMse/Basic.lean"},{"n":"Causalean.Stat.FiniteStratumMarkedRatioMse.integral_fixedStratumArmCenteredNoise_sq_le","k":"theorem","m":"Causalean.Stat.Sample.FiniteStratumMarkedRatioMse.CenteredNoiseBound","a":"Stat/Sample/FiniteStratumMarkedRatioMse","t":1,"d":"Measurable group and arm labels and a measurable mark, square-integrable supported residuals, cellwise residual centering, the stated cellwise second-moment envelope, a positive overlap margin, and arm mass at least that margin times category mass imply that the occupancy-weighted centered ratio noise has second moment at most twice the squared envelope times selected mass, divided by safe sample size and overlap.","s":"∀ {Omega : Type u_1} {kappa : Type u_2} [inst : MeasurableSpace Omega] [Fintype kappa] [inst_2 : DecidableEq kappa] [inst_3 : MeasurableSpace kappa] [MeasurableSingletonClass kappa] {m : ℕ} (mu : MeasureTheory.Measure Omega) [MeasureTheory.IsProbabilityMeasure mu] (group : Omega → kappa) (arm : Omega → Bool) (Y : Omega → ℝ) (center : Bool → kappa → ℝ) (H : Finset kappa) (a : Bool) (M epsilon : ℝ), Measurable group → Measurable arm → Measurable Y → (∀ (k : kappa), MeasureTheory.MemLp (Causalean.Stat.FiniteStratumMarkedRatioMse.supportedArmResidual group arm Y center a k) 2 mu) → (∀ (k : kappa), ∫ (omega : Omega) in Causalean.Stat.FiniteStratumMarkedRatioMse.armCategoryEvent group arm a k, Y omega - center a k ∂mu = 0) → (∀ (k : kappa), ∫ (omega : Omega) in Causalean.Stat.FiniteStratumMarkedRatioMse.armCategoryEvent group arm a k, (Y omega - center a k) ^ 2 ∂mu ≤ Causalean.Stat.FiniteStratumMarkedRatioMse.armCategoryMass mu group arm a k * M ^ 2) → 0 < epsilon → (∀ (k : kappa), 0 < Causalean.Stat.FiniteStratumMarkedRatioMse.categoryMass mu group k → epsilon * Causalean.Stat.FiniteStratumMarkedRatioMse.categoryMass mu group k ≤ Causalean.Stat.FiniteStratumMarkedRatioMse.armCategoryMass mu group arm a k) → (∫ (z : Fin m → Omega), Causalean.Stat.FiniteStratumMarkedRatioMse.fixedStratumArmCenteredNoise group arm Y center H a z ^ 2 ∂MeasureTheory.Measure.pi fun x => mu) ≤ (2 * M ^ 2 * ∑ k ∈ H, Causalean.Stat.FiniteStratumMarkedRatioMse.categoryMass mu group k) / (Causalean.Stat.FiniteStratumMarkedRatioMse.safeSampleSize m * epsilon)","l":29,"f":"Causalean/Stat/Sample/FiniteStratumMarkedRatioMse/CenteredNoiseBound.lean"},{"n":"Causalean.Stat.FiniteStratumMarkedRatioMse.integral_fixedStratumArmMassFluctuation_sq_le","k":"theorem","m":"Causalean.Stat.Sample.FiniteStratumMarkedRatioMse.Main","a":"Stat/Sample/FiniteStratumMarkedRatioMse","t":2,"d":"Measurable group and arm labels and cell centers bounded in absolute value by the envelope imply that the empirical category-mass fluctuation has second moment at most the squared envelope divided by the safe sample size, including for an empty sample.","s":"∀ {Omega : Type u_1} {kappa : Type u_2} [inst : MeasurableSpace Omega] [Fintype kappa] [inst_2 : DecidableEq kappa] [inst_3 : MeasurableSpace kappa] [MeasurableSingletonClass kappa] {m : ℕ} (mu : MeasureTheory.Measure Omega) [MeasureTheory.IsProbabilityMeasure mu] (group : Omega → kappa) (arm : Omega → Bool) (center : Bool → kappa → ℝ) (H : Finset kappa) (a : Bool) (M : ℝ), Measurable group → Measurable arm → (∀ (k : kappa), |center a k| ≤ M) → (∫ (z : Fin m → Omega), Causalean.Stat.FiniteStratumMarkedRatioMse.fixedStratumArmMassFluctuation mu group arm center H a z ^ 2 ∂MeasureTheory.Measure.pi fun x => mu) ≤ M ^ 2 / Causalean.Stat.FiniteStratumMarkedRatioMse.safeSampleSize m","l":240,"f":"Causalean/Stat/Sample/FiniteStratumMarkedRatioMse/Main.lean"},{"n":"Causalean.Stat.FiniteStratumMarkedRatioMse.integral_fixedStratumArmMissingRemainder_sq_le","k":"theorem","m":"Causalean.Stat.Sample.FiniteStratumMarkedRatioMse.Main","a":"Stat/Sample/FiniteStratumMarkedRatioMse","t":2,"d":"Measurable group and arm labels, cell centers bounded in absolute value by the envelope, a positive overlap margin, and arm mass at least that margin times category mass imply that the normalized aggregate empty-arm remainder has second moment bounded by a parametric diagonal term plus the squared exponentially damped missing-arm envelope.","s":"∀ {Omega : Type u_1} {kappa : Type u_2} [inst : MeasurableSpace Omega] [Fintype kappa] [inst_2 : DecidableEq kappa] [inst_3 : MeasurableSpace kappa] [MeasurableSingletonClass kappa] {m : ℕ} (mu : MeasureTheory.Measure Omega) [MeasureTheory.IsProbabilityMeasure mu] (group : Omega → kappa) (arm : Omega → Bool) (center : Bool → kappa → ℝ) (H : Finset kappa) (a : Bool) (M epsilon : ℝ), Measurable group → Measurable arm → (∀ (k : kappa), |center a k| ≤ M) → 0 < epsilon → (∀ (k : kappa), 0 < Causalean.Stat.FiniteStratumMarkedRatioMse.categoryMass mu group k → epsilon * Causalean.Stat.FiniteStratumMarkedRatioMse.categoryMass mu group k ≤ Causalean.Stat.FiniteStratumMarkedRatioMse.armCategoryMass mu group arm a k) → (∫ (z : Fin m → Omega), Causalean.Stat.FiniteStratumMarkedRatioMse.fixedStratumArmMissingRemainder group arm center H a z ^ 2 ∂MeasureTheory.Measure.pi fun x => mu) ≤ M ^ 2 * (1 / Causalean.Stat.FiniteStratumMarkedRatioMse.safeSampleSize m + Causalean.Stat.FiniteStratumMarkedRatioMse.missingArmExponentialEnvelope mu group m epsilon H ^ 2)","l":402,"f":"Causalean/Stat/Sample/FiniteStratumMarkedRatioMse/Main.lean"},{"n":"Causalean.Stat.FiniteStratumMarkedRatioMse.integral_fixedStratumArmMissingRemainder_sq_le_aux","k":"theorem","m":"Causalean.Stat.Sample.FiniteStratumMarkedRatioMse.MissingBound","a":"Stat/Sample/FiniteStratumMarkedRatioMse","t":1,"d":"Measurable group and arm labels, cell centers bounded in absolute value by the envelope, a positive overlap margin, and arm mass at least that margin times category mass imply that the missing-arm remainder has a boundary-safe diagonal-plus-exponential second-moment bound.","s":"∀ {Omega : Type u_1} {kappa : Type u_2} [inst : MeasurableSpace Omega] [Fintype kappa] [inst_2 : DecidableEq kappa] [inst_3 : MeasurableSpace kappa] [MeasurableSingletonClass kappa] {m : ℕ} (mu : MeasureTheory.Measure Omega) [MeasureTheory.IsProbabilityMeasure mu] (group : Omega → kappa) (arm : Omega → Bool) (center : Bool → kappa → ℝ) (H : Finset kappa) (a : Bool) (M epsilon : ℝ), Measurable group → Measurable arm → (∀ (k : kappa), |center a k| ≤ M) → 0 < epsilon → (∀ (k : kappa), 0 < Causalean.Stat.FiniteStratumMarkedRatioMse.categoryMass mu group k → epsilon * Causalean.Stat.FiniteStratumMarkedRatioMse.categoryMass mu group k ≤ Causalean.Stat.FiniteStratumMarkedRatioMse.armCategoryMass mu group arm a k) → (∫ (z : Fin m → Omega), Causalean.Stat.FiniteStratumMarkedRatioMse.fixedStratumArmMissingRemainder group arm center H a z ^ 2 ∂MeasureTheory.Measure.pi fun x => mu) ≤ M ^ 2 * (1 / Causalean.Stat.FiniteStratumMarkedRatioMse.safeSampleSize m + Causalean.Stat.FiniteStratumMarkedRatioMse.missingArmExponentialEnvelope mu group m epsilon H ^ 2)","l":385,"f":"Causalean/Stat/Sample/FiniteStratumMarkedRatioMse/MissingBound.lean"},{"n":"Causalean.Stat.FiniteStratumMarkedRatioMse.integral_fixedStratumArm_error_sq_le","k":"theorem","m":"Causalean.Stat.Sample.FiniteStratumMarkedRatioMse.Main","a":"Stat/Sample/FiniteStratumMarkedRatioMse","t":2,"d":"Measurable group and arm labels and a measurable mark, square-integrable supported residuals, cellwise residual centering, the stated residual second-moment envelope, bounded cell centers, a positive overlap margin, arm mass at least that margin times category mass, and a deterministic lower bound on every selected category mass imply the boundary-safe one-arm mean-squared-error bound with an inverse-polynomial missing-arm envelope.","s":"∀ {Omega : Type u_1} {kappa : Type u_2} [inst : MeasurableSpace Omega] [Fintype kappa] [inst_2 : DecidableEq kappa] [inst_3 : MeasurableSpace kappa] [MeasurableSingletonClass kappa] {m : ℕ} (mu : MeasureTheory.Measure Omega) [MeasureTheory.IsProbabilityMeasure mu] (group : Omega → kappa) (arm : Omega → Bool) (Y : Omega → ℝ) (center : Bool → kappa → ℝ) (H : Finset kappa) (a : Bool) (M epsilon B : ℝ), Measurable group → Measurable arm → Measurable Y → (∀ (k : kappa), MeasureTheory.MemLp (Causalean.Stat.FiniteStratumMarkedRatioMse.supportedArmResidual group arm Y center a k) 2 mu) → (∀ (k : kappa), ∫ (omega : Omega) in Causalean.Stat.FiniteStratumMarkedRatioMse.armCategoryEvent group arm a k, Y omega - center a k ∂mu = 0) → (∀ (k : kappa), ∫ (omega : Omega) in Causalean.Stat.FiniteStratumMarkedRatioMse.armCategoryEvent group arm a k, (Y omega - center a k) ^ 2 ∂mu ≤ Causalean.Stat.FiniteStratumMarkedRatioMse.armCategoryMass mu group arm a k * M ^ 2) → (∀ (k : kappa), |center a k| ≤ M) → 0 < epsilon → (∀ (k : kappa), 0 < Causalean.Stat.FiniteStratumMarkedRatioMse.categoryMass mu group k → epsilon * Causalean.Stat.FiniteStratumMarkedRatioMse.categoryMass mu group k ≤ Causalean.Stat.FiniteStratumMarkedRatioMse.armCategoryMass mu group arm a k) → (∀ k ∈ H, B ≤ Causalean.Stat.FiniteStratumMarkedRatioMse.categoryMass mu group k) → (∫ (z : Fin m → Omega), (Causalean.Stat.FiniteStratumMarkedRatioMse.fixedStratumArmScore group arm Y H a z - Causalean.Stat.FiniteStratumMarkedRatioMse.fixedStratumArmTarget mu group arm Y H a) ^ 2 ∂MeasureTheory.Measure.pi fun x => mu) ≤ M ^ 2 * ((8 * ∑ k ∈ H, Causalean.Stat.FiniteStratumMarkedRatioMse.categoryMass mu group k) / (Causalean.Stat.FiniteStratumMarkedRatioMse.safeSampleSize m * epsilon) + 6 / Causalean.Stat.FiniteStratumMarkedRatioMse.safeSampleSize m + 4 * Causalean.Stat.FiniteStratumMarkedRatioMse.lowerMassMissingEnvelope mu group m epsilon B H ^ 2)","l":587,"f":"Causalean/Stat/Sample/FiniteStratumMarkedRatioMse/Main.lean"},{"n":"Causalean.Stat.FiniteStratumMarkedRatioMse.integral_fixedStratumArm_error_sq_le_exponential","k":"theorem","m":"Causalean.Stat.Sample.FiniteStratumMarkedRatioMse.Main","a":"Stat/Sample/FiniteStratumMarkedRatioMse","t":2,"d":"Measurable group and arm labels and a measurable mark, square-integrable supported residuals, cellwise residual centering, the stated residual second-moment envelope, bounded cell centers, a positive overlap margin, and arm mass at least that margin times category mass imply a uniform one-arm fixed-set mean-squared-error bound with an explicit exponentially damped missing-arm remainder.","s":"∀ {Omega : Type u_1} {kappa : Type u_2} [inst : MeasurableSpace Omega] [Fintype kappa] [inst_2 : DecidableEq kappa] [inst_3 : MeasurableSpace kappa] [MeasurableSingletonClass kappa] {m : ℕ} (mu : MeasureTheory.Measure Omega) [MeasureTheory.IsProbabilityMeasure mu] (group : Omega → kappa) (arm : Omega → Bool) (Y : Omega → ℝ) (center : Bool → kappa → ℝ) (H : Finset kappa) (a : Bool) (M epsilon : ℝ), Measurable group → Measurable arm → Measurable Y → (∀ (k : kappa), MeasureTheory.MemLp (Causalean.Stat.FiniteStratumMarkedRatioMse.supportedArmResidual group arm Y center a k) 2 mu) → (∀ (k : kappa), ∫ (omega : Omega) in Causalean.Stat.FiniteStratumMarkedRatioMse.armCategoryEvent group arm a k, Y omega - center a k ∂mu = 0) → (∀ (k : kappa), ∫ (omega : Omega) in Causalean.Stat.FiniteStratumMarkedRatioMse.armCategoryEvent group arm a k, (Y omega - center a k) ^ 2 ∂mu ≤ Causalean.Stat.FiniteStratumMarkedRatioMse.armCategoryMass mu group arm a k * M ^ 2) → (∀ (k : kappa), |center a k| ≤ M) → 0 < epsilon → (∀ (k : kappa), 0 < Causalean.Stat.FiniteStratumMarkedRatioMse.categoryMass mu group k → epsilon * Causalean.Stat.FiniteStratumMarkedRatioMse.categoryMass mu group k ≤ Causalean.Stat.FiniteStratumMarkedRatioMse.armCategoryMass mu group arm a k) → (∫ (z : Fin m → Omega), (Causalean.Stat.FiniteStratumMarkedRatioMse.fixedStratumArmScore group arm Y H a z - Causalean.Stat.FiniteStratumMarkedRatioMse.fixedStratumArmTarget mu group arm Y H a) ^ 2 ∂MeasureTheory.Measure.pi fun x => mu) ≤ M ^ 2 * ((8 * ∑ k ∈ H, Causalean.Stat.FiniteStratumMarkedRatioMse.categoryMass mu group k) / (Causalean.Stat.FiniteStratumMarkedRatioMse.safeSampleSize m * epsilon) + 6 / Causalean.Stat.FiniteStratumMarkedRatioMse.safeSampleSize m + 4 * Causalean.Stat.FiniteStratumMarkedRatioMse.missingArmExponentialEnvelope mu group m epsilon H ^ 2)","l":424,"f":"Causalean/Stat/Sample/FiniteStratumMarkedRatioMse/Main.lean"},{"n":"Causalean.Stat.FiniteStratumMarkedRatioMse.integral_fixedStratumMarkedRatio_error_sq_le","k":"theorem","m":"Causalean.Stat.Sample.FiniteStratumMarkedRatioMse.Main","a":"Stat/Sample/FiniteStratumMarkedRatioMse","t":1,"d":"Measurable group and arm labels and a measurable mark, square-integrable supported residuals in both arms, cellwise residual centering, the stated residual second-moment envelope, bounded cell centers, a positive overlap margin, both arm masses at least that margin times category mass, and a deterministic lower bound on every selected category mass imply that the treated-minus-control fixed-set score obeys the boundary-safe mean-squared-error bound at the same parametric and missing-arm scale, up to the universal two-arm factor.","s":"∀ {Omega : Type u_1} {kappa : Type u_2} [inst : MeasurableSpace Omega] [Fintype kappa] [inst_2 : DecidableEq kappa] [inst_3 : MeasurableSpace kappa] [MeasurableSingletonClass kappa] {m : ℕ} (mu : MeasureTheory.Measure Omega) [MeasureTheory.IsProbabilityMeasure mu] (group : Omega → kappa) (arm : Omega → Bool) (Y : Omega → ℝ) (center : Bool → kappa → ℝ) (H : Finset kappa) (M epsilon B : ℝ), Measurable group → Measurable arm → Measurable Y → (∀ (a : Bool) (k : kappa), MeasureTheory.MemLp (Causalean.Stat.FiniteStratumMarkedRatioMse.supportedArmResidual group arm Y center a k) 2 mu) → (∀ (a : Bool) (k : kappa), ∫ (omega : Omega) in Causalean.Stat.FiniteStratumMarkedRatioMse.armCategoryEvent group arm a k, Y omega - center a k ∂mu = 0) → (∀ (a : Bool) (k : kappa), ∫ (omega : Omega) in Causalean.Stat.FiniteStratumMarkedRatioMse.armCategoryEvent group arm a k, (Y omega - center a k) ^ 2 ∂mu ≤ Causalean.Stat.FiniteStratumMarkedRatioMse.armCategoryMass mu group arm a k * M ^ 2) → (∀ (a : Bool) (k : kappa), |center a k| ≤ M) → 0 < epsilon → (∀ (k : kappa), 0 < Causalean.Stat.FiniteStratumMarkedRatioMse.categoryMass mu group k → ∀ (a : Bool), epsilon * Causalean.Stat.FiniteStratumMarkedRatioMse.categoryMass mu group k ≤ Causalean.Stat.FiniteStratumMarkedRatioMse.armCategoryMass mu group arm a k) → (∀ k ∈ H, B ≤ Causalean.Stat.FiniteStratumMarkedRatioMse.categoryMass mu group k) → (∫ (z : Fin m → Omega), (Causalean.Stat.FiniteStratumMarkedRatioMse.fixedStratumMarkedRatio group arm Y H z - Causalean.Stat.FiniteStratumMarkedRatioMse.fixedStratumMarkedTarget mu group arm Y H) ^ 2 ∂MeasureTheory.Measure.pi fun x => mu) ≤ 4 * M ^ 2 * ((8 * ∑ k ∈ H, Causalean.Stat.FiniteStratumMarkedRatioMse.categoryMass mu group k) / (Causalean.Stat.FiniteStratumMarkedRatioMse.safeSampleSize m * epsilon) + 6 / Causalean.Stat.FiniteStratumMarkedRatioMse.safeSampleSize m + 4 * Causalean.Stat.FiniteStratumMarkedRatioMse.lowerMassMissingEnvelope mu group m epsilon B H ^ 2)","l":661,"f":"Causalean/Stat/Sample/FiniteStratumMarkedRatioMse/Main.lean"},{"n":"Causalean.Stat.FiniteStratumMarkedRatioMse.integral_missingArmCount_eq","k":"theorem","m":"Causalean.Stat.Sample.FiniteStratumMarkedRatioMse.MissingMoments","a":"Stat/Sample/FiniteStratumMarkedRatioMse","t":2,"d":"Measurable group and arm labels imply that the expected category occupancy retained only when one arm is absent equals the sample size times the off-arm category mass times the empty-arm probability for the other coordinates.","s":"∀ {Omega : Type u_1} {kappa : Type u_2} [inst : MeasurableSpace Omega] [Fintype kappa] [inst_2 : DecidableEq kappa] [inst_3 : MeasurableSpace kappa] [MeasurableSingletonClass kappa] {m : ℕ} (mu : MeasureTheory.Measure Omega) [MeasureTheory.IsProbabilityMeasure mu] (group : Omega → kappa) (arm : Omega → Bool), Measurable group → Measurable arm → ∀ (a : Bool) (k : kappa), (∫ (z : Fin m → Omega), ↑(Causalean.Stat.FiniteStratumMarkedRatioMse.missingArmCount group arm z a k) ∂MeasureTheory.Measure.pi fun x => mu) = ↑m * (Causalean.Stat.FiniteStratumMarkedRatioMse.categoryMass mu group k - Causalean.Stat.FiniteStratumMarkedRatioMse.armCategoryMass mu group arm a k) * (1 - Causalean.Stat.FiniteStratumMarkedRatioMse.armCategoryMass mu group arm a k) ^ (m - 1)","l":372,"f":"Causalean/Stat/Sample/FiniteStratumMarkedRatioMse/MissingMoments.lean"},{"n":"Causalean.Stat.FiniteStratumMarkedRatioMse.integral_missingArmCount_mul_eq","k":"theorem","m":"Causalean.Stat.Sample.FiniteStratumMarkedRatioMse.MissingMoments","a":"Stat/Sample/FiniteStratumMarkedRatioMse","t":2,"d":"Measurable group and arm labels and distinct categories imply that their missing-arm counts have the exact ordered-pair cross moment obtained by excluding the union of the two arm/category cells.","s":"∀ {Omega : Type u_1} {kappa : Type u_2} [inst : MeasurableSpace Omega] [Fintype kappa] [inst_2 : DecidableEq kappa] [inst_3 : MeasurableSpace kappa] [MeasurableSingletonClass kappa] {m : ℕ} (mu : MeasureTheory.Measure Omega) [MeasureTheory.IsProbabilityMeasure mu] (group : Omega → kappa) (arm : Omega → Bool), Measurable group → Measurable arm → ∀ (a : Bool) {k l : kappa}, k ≠ l → (∫ (z : Fin m → Omega), ↑(Causalean.Stat.FiniteStratumMarkedRatioMse.missingArmCount group arm z a k) * ↑(Causalean.Stat.FiniteStratumMarkedRatioMse.missingArmCount group arm z a l) ∂MeasureTheory.Measure.pi fun x => mu) = ↑(m.descFactorial 2) * (Causalean.Stat.FiniteStratumMarkedRatioMse.categoryMass mu group k - Causalean.Stat.FiniteStratumMarkedRatioMse.armCategoryMass mu group arm a k) * (Causalean.Stat.FiniteStratumMarkedRatioMse.categoryMass mu group l - Causalean.Stat.FiniteStratumMarkedRatioMse.armCategoryMass mu group arm a l) * (1 - Causalean.Stat.FiniteStratumMarkedRatioMse.armCategoryMass mu group arm a k - Causalean.Stat.FiniteStratumMarkedRatioMse.armCategoryMass mu group arm a l) ^ (m - 2)","l":537,"f":"Causalean/Stat/Sample/FiniteStratumMarkedRatioMse/MissingMoments.lean"},{"n":"Causalean.Stat.FiniteStratumMarkedRatioMse.integral_missingArmCount_sq_eq","k":"theorem","m":"Causalean.Stat.Sample.FiniteStratumMarkedRatioMse.MissingMoments","a":"Stat/Sample/FiniteStratumMarkedRatioMse","t":1,"d":"Measurable group and arm labels imply that the exact second moment of one missing-arm category count is the sum of its one-coordinate diagonal and ordered two-coordinate contributions.","s":"∀ {Omega : Type u_1} {kappa : Type u_2} [inst : MeasurableSpace Omega] [Fintype kappa] [inst_2 : DecidableEq kappa] [inst_3 : MeasurableSpace kappa] [MeasurableSingletonClass kappa] {m : ℕ} (mu : MeasureTheory.Measure Omega) [MeasureTheory.IsProbabilityMeasure mu] (group : Omega → kappa) (arm : Omega → Bool), Measurable group → Measurable arm → ∀ (a : Bool) (k : kappa), (∫ (z : Fin m → Omega), ↑(Causalean.Stat.FiniteStratumMarkedRatioMse.missingArmCount group arm z a k) ^ 2 ∂MeasureTheory.Measure.pi fun x => mu) = ↑m * (Causalean.Stat.FiniteStratumMarkedRatioMse.categoryMass mu group k - Causalean.Stat.FiniteStratumMarkedRatioMse.armCategoryMass mu group arm a k) * (1 - Causalean.Stat.FiniteStratumMarkedRatioMse.armCategoryMass mu group arm a k) ^ (m - 1) + ↑(m.descFactorial 2) * (Causalean.Stat.FiniteStratumMarkedRatioMse.categoryMass mu group k - Causalean.Stat.FiniteStratumMarkedRatioMse.armCategoryMass mu group arm a k) ^ 2 * (1 - Causalean.Stat.FiniteStratumMarkedRatioMse.armCategoryMass mu group arm a k) ^ (m - 2)","l":429,"f":"Causalean/Stat/Sample/FiniteStratumMarkedRatioMse/MissingMoments.lean"},{"n":"Causalean.Stat.FiniteStratumMarkedRatioMse.integral_nested_count_sq_mul_totalized_inverse_le","k":"theorem","m":"Causalean.Stat.Sample.FiniteStratumMarkedRatioMse.NestedCountBound","a":"Stat/Sample/FiniteStratumMarkedRatioMse","t":1,"d":"Measurable outer and inner events, nesting of the inner event inside the outer event, a positive overlap margin, and inner-event mass at least that margin times outer-event mass imply that the expected squared outer count times the zero-safe inverse inner count is at most twice sample size times outer-event mass divided by the margin.","s":"∀ {m : ℕ} {Omega : Type u_1} [inst : MeasurableSpace Omega] (mu : MeasureTheory.Measure Omega) [MeasureTheory.IsProbabilityMeasure mu] (C R : Set Omega), MeasurableSet C → MeasurableSet R → R ⊆ C → ∀ (epsilon : ℝ), 0 < epsilon → epsilon * (mu C).toReal ≤ (mu R).toReal → (∫ (z : Fin m → Omega), ↑(Causalean.Stat.FiniteStratumMarkedRatioMse.sampleIndexSet z C).card ^ 2 * if 0 < (Causalean.Stat.FiniteStratumMarkedRatioMse.sampleIndexSet z R).card then (↑(Causalean.Stat.FiniteStratumMarkedRatioMse.sampleIndexSet z R).card)⁻¹ else 0 ∂MeasureTheory.Measure.pi fun x => mu) ≤ 2 * ↑m * (mu C).toReal / epsilon","l":255,"f":"Causalean/Stat/Sample/FiniteStratumMarkedRatioMse/NestedCountBound.lean"},{"n":"Causalean.Stat.FiniteStratumMarkedRatioMse.lowerMassMissingEnvelope","k":"def","m":"Causalean.Stat.Sample.FiniteStratumMarkedRatioMse.MissingMoments","a":"Stat/Sample/FiniteStratumMarkedRatioMse","t":1,"d":"The lower-mass envelope uses inverse-polynomial decay when its denominator is positive and otherwise falls back to total selected mass, so it remains defined for small samples, zero overlap, and a zero mass lower bound.","s":"{Omega : Type u_1} → {kappa : Type u_2} → [inst : MeasurableSpace Omega] → MeasureTheory.Measure Omega → (Omega → kappa) → ℕ → ℝ → ℝ → Finset kappa → ℝ","l":664,"f":"Causalean/Stat/Sample/FiniteStratumMarkedRatioMse/MissingMoments.lean"},{"n":"Causalean.Stat.FiniteStratumMarkedRatioMse.lowerMassMissingEnvelope_eq_of_pos","k":"theorem","m":"Causalean.Stat.Sample.FiniteStratumMarkedRatioMse.MissingMoments","a":"Stat/Sample/FiniteStratumMarkedRatioMse","t":2,"d":"A sample size of at least three, a positive overlap margin, and a positive category-mass lower bound make the boundary-safe lower-mass envelope equal its inverse-polynomial expression.","s":"∀ {Omega : Type u_1} {kappa : Type u_2} [inst : MeasurableSpace Omega] [Fintype kappa] [DecidableEq kappa] [inst_3 : MeasurableSpace kappa] [MeasurableSingletonClass kappa] (mu : MeasureTheory.Measure Omega) (group : Omega → kappa) {m : ℕ} {epsilon B : ℝ} (H : Finset kappa), 3 ≤ m → 0 < epsilon → 0 < B → Causalean.Stat.FiniteStratumMarkedRatioMse.lowerMassMissingEnvelope mu group m epsilon B H = ↑H.card / ((↑(m - 2) / 2 * epsilon) ^ 2 * B)","l":753,"f":"Causalean/Stat/Sample/FiniteStratumMarkedRatioMse/MissingMoments.lean"},{"n":"Causalean.Stat.FiniteStratumMarkedRatioMse.measurable_armMarkSum","k":"theorem","m":"Causalean.Stat.Sample.FiniteStratumMarkedRatioMse.Basic","a":"Stat/Sample/FiniteStratumMarkedRatioMse","t":2,"d":"Measurable group and arm labels and a measurable mark make every empirical arm/category mark sum measurable.","s":"∀ {Omega : Type u_1} {kappa : Type u_2} [inst : MeasurableSpace Omega] [Fintype kappa] [DecidableEq kappa] [inst_3 : MeasurableSpace kappa] [MeasurableSingletonClass kappa] {m : ℕ} (group : Omega → kappa) (arm : Omega → Bool) (Y : Omega → ℝ), Measurable group → Measurable arm → Measurable Y → ∀ (a : Bool) (k : kappa), Measurable fun z => Causalean.Stat.FiniteStratumMarkedRatioMse.armMarkSum group arm Y z a k","l":194,"f":"Causalean/Stat/Sample/FiniteStratumMarkedRatioMse/Basic.lean"},{"n":"Causalean.Stat.FiniteStratumMarkedRatioMse.measurable_categoryArmCount","k":"theorem","m":"Causalean.Stat.Sample.FiniteStratumMarkedRatioMse.Basic","a":"Stat/Sample/FiniteStratumMarkedRatioMse","t":2,"d":"Measurable group and arm labels make every fixed empirical arm/category count measurable on the product sample space.","s":"∀ {Omega : Type u_1} {kappa : Type u_2} [inst : MeasurableSpace Omega] [Fintype kappa] [inst_2 : DecidableEq kappa] [inst_3 : MeasurableSpace kappa] [MeasurableSingletonClass kappa] {m : ℕ} (group : Omega → kappa) (arm : Omega → Bool), Measurable group → Measurable arm → ∀ (a : Bool) (k : kappa), Measurable fun z => Causalean.Stat.FiniteStratumMarkedRatioMse.categoryArmCount group arm z a k","l":179,"f":"Causalean/Stat/Sample/FiniteStratumMarkedRatioMse/Basic.lean"},{"n":"Causalean.Stat.FiniteStratumMarkedRatioMse.measurable_categoryCount","k":"theorem","m":"Causalean.Stat.Sample.FiniteStratumMarkedRatioMse.Basic","a":"Stat/Sample/FiniteStratumMarkedRatioMse","t":2,"d":"Measurable group and arm labels make every fixed empirical category count measurable on the product sample space.","s":"∀ {Omega : Type u_1} {kappa : Type u_2} [inst : MeasurableSpace Omega] [Fintype kappa] [inst_2 : DecidableEq kappa] [inst_3 : MeasurableSpace kappa] [MeasurableSingletonClass kappa] {m : ℕ} (group : Omega → kappa) (arm : Omega → Bool), Measurable group → Measurable arm → ∀ (k : kappa), Measurable fun z => Causalean.Stat.FiniteStratumMarkedRatioMse.categoryCount group arm z k","l":187,"f":"Causalean/Stat/Sample/FiniteStratumMarkedRatioMse/Basic.lean"},{"n":"Causalean.Stat.FiniteStratumMarkedRatioMse.measurable_fixedStratumMarkedRatio","k":"theorem","m":"Causalean.Stat.Sample.FiniteStratumMarkedRatioMse.Basic","a":"Stat/Sample/FiniteStratumMarkedRatioMse","t":2,"d":"Measurable group and arm labels and a measurable mark make the fixed-set marked ratio score measurable.","s":"∀ {Omega : Type u_1} {kappa : Type u_2} [inst : MeasurableSpace Omega] [Fintype kappa] [inst_2 : DecidableEq kappa] [inst_3 : MeasurableSpace kappa] [MeasurableSingletonClass kappa] {m : ℕ} (group : Omega → kappa) (arm : Omega → Bool) (Y : Omega → ℝ) (H : Finset kappa), Measurable group → Measurable arm → Measurable Y → Measurable (Causalean.Stat.FiniteStratumMarkedRatioMse.fixedStratumMarkedRatio group arm Y H)","l":227,"f":"Causalean/Stat/Sample/FiniteStratumMarkedRatioMse/Basic.lean"},{"n":"Causalean.Stat.FiniteStratumMarkedRatioMse.measurable_missingArmCount","k":"theorem","m":"Causalean.Stat.Sample.FiniteStratumMarkedRatioMse.Basic","a":"Stat/Sample/FiniteStratumMarkedRatioMse","t":2,"d":"Measurable group and arm labels make each missing-arm category count measurable.","s":"∀ {Omega : Type u_1} {kappa : Type u_2} [inst : MeasurableSpace Omega] [Fintype kappa] [inst_2 : DecidableEq kappa] [inst_3 : MeasurableSpace kappa] [MeasurableSingletonClass kappa] {m : ℕ} (group : Omega → kappa) (arm : Omega → Bool), Measurable group → Measurable arm → ∀ (a : Bool) (k : kappa), Measurable fun z => Causalean.Stat.FiniteStratumMarkedRatioMse.missingArmCount group arm z a k","l":249,"f":"Causalean/Stat/Sample/FiniteStratumMarkedRatioMse/Basic.lean"},{"n":"Causalean.Stat.FiniteStratumMarkedRatioMse.measurable_totalizedArmMean","k":"theorem","m":"Causalean.Stat.Sample.FiniteStratumMarkedRatioMse.Basic","a":"Stat/Sample/FiniteStratumMarkedRatioMse","t":2,"d":"Measurable group and arm labels and a measurable mark make every totalized empirical arm/category mark mean measurable, including at zero count.","s":"∀ {Omega : Type u_1} {kappa : Type u_2} [inst : MeasurableSpace Omega] [Fintype kappa] [inst_2 : DecidableEq kappa] [inst_3 : MeasurableSpace kappa] [MeasurableSingletonClass kappa] {m : ℕ} (group : Omega → kappa) (arm : Omega → Bool) (Y : Omega → ℝ), Measurable group → Measurable arm → Measurable Y → ∀ (a : Bool) (k : kappa), Measurable fun z => Causalean.Stat.FiniteStratumMarkedRatioMse.totalizedArmMean group arm Y z a k","l":208,"f":"Causalean/Stat/Sample/FiniteStratumMarkedRatioMse/Basic.lean"},{"n":"Causalean.Stat.FiniteStratumMarkedRatioMse.missingArmCount","k":"def","m":"Causalean.Stat.Sample.FiniteStratumMarkedRatioMse.Basic","a":"Stat/Sample/FiniteStratumMarkedRatioMse","t":1,"d":"A missing-arm count is the category occupancy when the requested empirical arm count is zero, and is zero otherwise.","s":"{Omega : Type u_1} → {kappa : Type u_2} → [DecidableEq kappa] → {m : ℕ} → (Omega → kappa) → (Omega → Bool) → (Fin m → Omega) → Bool → kappa → ℕ","l":147,"f":"Causalean/Stat/Sample/FiniteStratumMarkedRatioMse/Basic.lean"},{"n":"Causalean.Stat.FiniteStratumMarkedRatioMse.missingArmExponentialEnvelope","k":"def","m":"Causalean.Stat.Sample.FiniteStratumMarkedRatioMse.MissingMoments","a":"Stat/Sample/FiniteStratumMarkedRatioMse","t":1,"d":"The aggregate exponentially damped missing-arm envelope sums each category mass times the overlap-driven empty-arm decay.","s":"{Omega : Type u_1} → {kappa : Type u_2} → [inst : MeasurableSpace Omega] → MeasureTheory.Measure Omega → (Omega → kappa) → ℕ → ℝ → Finset kappa → ℝ","l":657,"f":"Causalean/Stat/Sample/FiniteStratumMarkedRatioMse/MissingMoments.lean"},{"n":"Causalean.Stat.FiniteStratumMarkedRatioMse.missingArmExponentialEnvelope_le_lowerMass","k":"theorem","m":"Causalean.Stat.Sample.FiniteStratumMarkedRatioMse.MissingMoments","a":"Stat/Sample/FiniteStratumMarkedRatioMse","t":2,"d":"A measurable group label, a positive overlap margin, and a deterministic lower bound on every selected category mass ensure that the exponential missing-arm envelope is at most the boundary-safe lower-mass envelope.","s":"∀ {Omega : Type u_1} {kappa : Type u_2} [inst : MeasurableSpace Omega] [Fintype kappa] [DecidableEq kappa] [inst_3 : MeasurableSpace kappa] [MeasurableSingletonClass kappa] (mu : MeasureTheory.Measure Omega) [MeasureTheory.IsProbabilityMeasure mu] (group : Omega → kappa), Measurable group → ∀ (m : ℕ) (epsilon B : ℝ) (H : Finset kappa), 0 < epsilon → (∀ k ∈ H, B ≤ Causalean.Stat.FiniteStratumMarkedRatioMse.categoryMass mu group k) → Causalean.Stat.FiniteStratumMarkedRatioMse.missingArmExponentialEnvelope mu group m epsilon H ≤ Causalean.Stat.FiniteStratumMarkedRatioMse.lowerMassMissingEnvelope mu group m epsilon B H","l":686,"f":"Causalean/Stat/Sample/FiniteStratumMarkedRatioMse/MissingMoments.lean"},{"n":"Causalean.Stat.FiniteStratumMarkedRatioMse.populationArmMean","k":"def","m":"Causalean.Stat.Sample.FiniteStratumMarkedRatioMse.Basic","a":"Stat/Sample/FiniteStratumMarkedRatioMse","t":1,"d":"The population arm/category mark mean is the cell mark integral divided by its positive cell mass, and is zero on a zero-mass cell.","s":"{Omega : Type u_1} → {kappa : Type u_2} → [inst : MeasurableSpace Omega] → MeasureTheory.Measure Omega → (Omega → kappa) → (Omega → Bool) → (Omega → ℝ) → Bool → kappa → ℝ","l":86,"f":"Causalean/Stat/Sample/FiniteStratumMarkedRatioMse/Basic.lean"},{"n":"Causalean.Stat.FiniteStratumMarkedRatioMse.safeSampleSize","k":"def","m":"Causalean.Stat.Sample.FiniteStratumMarkedRatioMse.Basic","a":"Stat/Sample/FiniteStratumMarkedRatioMse","t":1,"d":"The safe real sample size is one at an empty sample and otherwise equals the ordinary sample size.","s":"ℕ → ℝ","l":27,"f":"Causalean/Stat/Sample/FiniteStratumMarkedRatioMse/Basic.lean"},{"n":"Causalean.Stat.FiniteStratumMarkedRatioMse.sampleIndexSet","k":"def","m":"Causalean.Stat.Sample.FiniteStratumMarkedRatioMse.NestedCountBound","a":"Stat/Sample/FiniteStratumMarkedRatioMse","t":1,"d":"The finite set of product coordinates whose observations belong to a set.","s":"{I : Type u_1} → {Omega : Type u_2} → [Fintype I] → (I → Omega) → Set Omega → Finset I","l":25,"f":"Causalean/Stat/Sample/FiniteStratumMarkedRatioMse/NestedCountBound.lean"},{"n":"Causalean.Stat.FiniteStratumMarkedRatioMse.supportedArmMark","k":"def","m":"Causalean.Stat.Sample.FiniteStratumMarkedRatioMse.Basic","a":"Stat/Sample/FiniteStratumMarkedRatioMse","t":1,"d":"The supported mark equals the real mark on one arm/category cell and zero off that cell.","s":"{Omega : Type u_1} → {kappa : Type u_2} → (Omega → kappa) → (Omega → Bool) → (Omega → ℝ) → Bool → kappa → Omega → ℝ","l":64,"f":"Causalean/Stat/Sample/FiniteStratumMarkedRatioMse/Basic.lean"},{"n":"Causalean.Stat.FiniteStratumMarkedRatioMse.supportedArmResidual","k":"def","m":"Causalean.Stat.Sample.FiniteStratumMarkedRatioMse.Basic","a":"Stat/Sample/FiniteStratumMarkedRatioMse","t":1,"d":"The centered arm/category residual is the mark minus its supplied cell center, supported on that cell.","s":"{Omega : Type u_1} → {kappa : Type u_2} → (Omega → kappa) → (Omega → Bool) → (Omega → ℝ) → (Bool → kappa → ℝ) → Bool → kappa → Omega → ℝ","l":133,"f":"Causalean/Stat/Sample/FiniteStratumMarkedRatioMse/Basic.lean"},{"n":"Causalean.Stat.FiniteStratumMarkedRatioMse.totalizedArmMean","k":"def","m":"Causalean.Stat.Sample.FiniteStratumMarkedRatioMse.Basic","a":"Stat/Sample/FiniteStratumMarkedRatioMse","t":1,"d":"The totalized empirical arm mean is the mark sum divided by its positive arm/category count, and is zero when that count is empty.","s":"{Omega : Type u_1} → {kappa : Type u_2} → [DecidableEq kappa] → {m : ℕ} → (Omega → kappa) → (Omega → Bool) → (Omega → ℝ) → (Fin m → Omega) → Bool → kappa → ℝ","l":77,"f":"Causalean/Stat/Sample/FiniteStratumMarkedRatioMse/Basic.lean"},{"n":"Causalean.Stat.FiniteStratumMarkedRatioMse.totalizedArmResidualMean","k":"def","m":"Causalean.Stat.Sample.FiniteStratumMarkedRatioMse.Basic","a":"Stat/Sample/FiniteStratumMarkedRatioMse","t":1,"d":"The totalized empirical residual mean is zero on an empty arm/category cell and otherwise averages its supported centered residuals.","s":"{Omega : Type u_1} → {kappa : Type u_2} → [DecidableEq kappa] → {m : ℕ} → (Omega → kappa) → (Omega → Bool) → (Omega → ℝ) → (Bool → kappa → ℝ) → (Fin m → Omega) → Bool → kappa → ℝ","l":140,"f":"Causalean/Stat/Sample/FiniteStratumMarkedRatioMse/Basic.lean"},{"n":"Causalean.Stat.GMMProblem","k":"structure","m":"Causalean.Stat.GMM.Setup","a":"Stat/GMM","t":1,"d":"GMM problem. Bundles, over a probability measure on the data space, a moment function evaluated at the parameter truth θ₀, with a self-adjoint weighting operator and a Jacobian of the population moment at θ₀, verified to be its Fréchet derivative. It asserts the population moment vanishes at the truth and that the moment function at the truth is measurable and square-integrable, and packages a moment covariance operator defined as the second moment of the moment vector together with two-sided inverse witnesses for the covariance operator, for the GMM bread GᵀWG, and for the efficient bread GᵀCov⁻¹G.","s":"{E : Type u_1} → {F : Type u_2} → {X : Type u_3} → [inst : NormedAddCommGroup E] → [inst_1 : InnerProductSpace ℝ E] → [FiniteDimensional ℝ E] → [inst : NormedAddCommGroup F] → [inst_2 : InnerProductSpace ℝ F] → [FiniteDimensional ℝ F] → [MeasurableSpace F] → [inst : MeasurableSpace X] → MeasureTheory.Measure X → Type (max (max u_1 u_2) u_3)","l":72,"f":"Causalean/Stat/GMM/Setup.lean"},{"n":"Causalean.Stat.GMMProblem.asympVar","k":"def","m":"Causalean.Stat.GMM.Setup","a":"Stat/GMM","t":1,"d":"The sandwich asymptotic variance (GᵀWG)⁻¹ GᵀW Cov WG (GᵀWG)⁻¹.","s":"{E : Type u_1} → {F : Type u_2} → {X : Type u_3} → [inst : NormedAddCommGroup E] → [inst_1 : InnerProductSpace ℝ E] → [inst_2 : FiniteDimensional ℝ E] → [inst_3 : NormedAddCommGroup F] → [inst_4 : InnerProductSpace ℝ F] → [inst_5 : FiniteDimensional ℝ F] → [inst_6 : MeasurableSpace F] → [inst_7 : MeasurableSpace X] → {P : MeasureTheory.Measure X} → Causalean.Stat.GMMProblem P → E →L[ℝ] E","l":129,"f":"Causalean/Stat/GMM/Setup.lean"},{"n":"Causalean.Stat.GMMProblem.cov_isPositive","k":"theorem","m":"Causalean.Stat.GMM.Setup","a":"Stat/GMM","t":2,"d":"The covariance operator is positive — it is a second moment.","s":"∀ {E : Type u_1} {F : Type u_2} {X : Type u_3} [inst : NormedAddCommGroup E] [inst_1 : InnerProductSpace ℝ E] [inst_2 : FiniteDimensional ℝ E] [inst_3 : NormedAddCommGroup F] [inst_4 : InnerProductSpace ℝ F] [inst_5 : FiniteDimensional ℝ F] [inst_6 : MeasurableSpace F] [inst_7 : MeasurableSpace X] {P : MeasureTheory.Measure X} (prob : Causalean.Stat.GMMProblem P), prob.Cov.IsPositive","l":137,"f":"Causalean/Stat/GMM/Setup.lean"},{"n":"Causalean.Stat.GMMProblem.effVar","k":"def","m":"Causalean.Stat.GMM.Setup","a":"Stat/GMM","t":1,"d":"The efficient asymptotic variance (GᵀCov⁻¹G)⁻¹.","s":"{E : Type u_1} → {F : Type u_2} → {X : Type u_3} → [inst : NormedAddCommGroup E] → [inst_1 : InnerProductSpace ℝ E] → [inst_2 : FiniteDimensional ℝ E] → [inst_3 : NormedAddCommGroup F] → [inst_4 : InnerProductSpace ℝ F] → [inst_5 : FiniteDimensional ℝ F] → [inst_6 : MeasurableSpace F] → [inst_7 : MeasurableSpace X] → {P : MeasureTheory.Measure X} → Causalean.Stat.GMMProblem P → E →L[ℝ] E","l":133,"f":"Causalean/Stat/GMM/Setup.lean"},{"n":"Causalean.Stat.GMMProblem.efficiency","k":"theorem","m":"Causalean.Stat.GMM.Setup","a":"Stat/GMM","t":1,"d":"GMM optimal-weighting theorem (Hansen 1982), statistical form. The sandwich asymptotic variance of a GMM problem with an arbitrary symmetric weighting dominates the efficient variance (GᵀCov⁻¹G)⁻¹ in the Löwner order: asympVar − effVar is a positive operator.","s":"∀ {E : Type u_1} {F : Type u_2} {X : Type u_3} [inst : NormedAddCommGroup E] [inst_1 : InnerProductSpace ℝ E] [inst_2 : FiniteDimensional ℝ E] [inst_3 : NormedAddCommGroup F] [inst_4 : InnerProductSpace ℝ F] [inst_5 : FiniteDimensional ℝ F] [inst_6 : MeasurableSpace F] [inst_7 : MeasurableSpace X] {P : MeasureTheory.Measure X} (prob : Causalean.Stat.GMMProblem P), (prob.asympVar - prob.effVar).IsPositive","l":153,"f":"Causalean/Stat/GMM/Setup.lean"},{"n":"Causalean.Stat.GMMProblem.influence","k":"def","m":"Causalean.Stat.GMM.Setup","a":"Stat/GMM","t":1,"d":"The influence function of the problem.","s":"{E : Type u_1} → {F : Type u_2} → {X : Type u_3} → [inst : NormedAddCommGroup E] → [inst_1 : InnerProductSpace ℝ E] → [inst_2 : FiniteDimensional ℝ E] → [inst_3 : NormedAddCommGroup F] → [inst_4 : InnerProductSpace ℝ F] → [inst_5 : FiniteDimensional ℝ F] → [inst_6 : MeasurableSpace F] → [inst_7 : MeasurableSpace X] → {P : MeasureTheory.Measure X} → Causalean.Stat.GMMProblem P → X → E","l":125,"f":"Causalean/Stat/GMM/Setup.lean"},{"n":"Causalean.Stat.GMMProblem.score","k":"def","m":"Causalean.Stat.GMM.Setup","a":"Stat/GMM","t":1,"d":"The combined score of the problem, ψ(θ,x) = GᵀW g(θ,x).","s":"{E : Type u_1} → {F : Type u_2} → {X : Type u_3} → [inst : NormedAddCommGroup E] → [inst_1 : InnerProductSpace ℝ E] → [inst_2 : FiniteDimensional ℝ E] → [inst_3 : NormedAddCommGroup F] → [inst_4 : InnerProductSpace ℝ F] → [inst_5 : FiniteDimensional ℝ F] → [inst_6 : MeasurableSpace F] → [inst_7 : MeasurableSpace X] → {P : MeasureTheory.Measure X} → Causalean.Stat.GMMProblem P → E → X → E","l":122,"f":"Causalean/Stat/GMM/Setup.lean"},{"n":"Causalean.Stat.GlivenkoCantelli","k":"def","m":"Causalean.Stat.EmpiricalProcess.Basic","a":"Stat/EmpiricalProcess","t":1,"d":"(Weak) Glivenko–Cantelli property. The class f : ι → X → ℝ obeys a uniform law of large numbers for the sample S: for every ε > 0, the probability that some class member's empirical mean deviates from its population mean by at least ε tends to 0.","s":"{Ω : Type u_1} → {X : Type u_2} → {ι : Type u_3} → [inst : MeasurableSpace Ω] → [inst_1 : MeasurableSpace X] → {μ : MeasureTheory.Measure Ω} → {P : MeasureTheory.Measure X} → Causalean.Stat.IIDSample Ω X μ P → (ι → X → ℝ) → Prop","l":75,"f":"Causalean/Stat/EmpiricalProcess/Basic.lean"},{"n":"Causalean.Stat.HasFDerivAt.hasHadamardDirDerivAt","k":"theorem","m":"Causalean.Stat.Inference.HadamardDeriv","a":"Stat/Inference","t":1,"d":"Fréchet implies Hadamard. If the map φ is Fréchet-differentiable at the point θ, with continuous linear derivative L, then φ is Hadamard directionally differentiable at θ, with the same map L serving as its Hadamard directional derivative.","s":"∀ {E : Type u_1} {F : Type u_2} [inst : NormedAddCommGroup E] [inst_1 : NormedSpace ℝ E] [inst_2 : NormedAddCommGroup F] [inst_3 : NormedSpace ℝ F] {φ : E → F} {L : E →L[ℝ] F} {θ : E}, HasFDerivAt φ L θ → Causalean.Stat.HasHadamardDirDerivAt φ (fun x => L x) θ","l":83,"f":"Causalean/Stat/Inference/HadamardDeriv.lean"},{"n":"Causalean.Stat.HasHadamardDirDerivAt","k":"def","m":"Causalean.Stat.Inference.HadamardDeriv","a":"Stat/Inference","t":1,"d":"Hadamard directional differentiability. φ has Hadamard directional derivative φ' at θ if for every direction h, every sequence hₙ → h, and every step sequence tₙ → 0 with tₙ > 0,","s":"{E : Type u_1} → {F : Type u_2} → [inst : NormedAddCommGroup E] → [NormedSpace ℝ E] → [inst : NormedAddCommGroup F] → [NormedSpace ℝ F] → (E → F) → (E → F) → E → Prop","l":68,"f":"Causalean/Stat/Inference/HadamardDeriv.lean"},{"n":"Causalean.Stat.HasIIDSample","k":"def","m":"Causalean.Stat.Sample.PiTransport","a":"Stat/Sample","t":1,"d":"Existence of an i.i.d. sample with a given law: the assertion that some probability space carries an independent, identically distributed sample whose common marginal law is a given measure.","s":"{X : Type u} → [inst : MeasurableSpace X] → MeasureTheory.Measure X → Prop","l":234,"f":"Causalean/Stat/Sample/PiTransport.lean"},{"n":"Causalean.Stat.HasL1Bracketing","k":"def","m":"Causalean.Stat.EmpiricalProcess.Basic","a":"Stat/EmpiricalProcess","t":1,"d":"A real-valued function class has finite L¹(P) brackets of arbitrarily small width.","s":"{X : Type u_2} → {ι : Type u_3} → [inst : MeasurableSpace X] → (ι → X → ℝ) → MeasureTheory.Measure X → Prop","l":120,"f":"Causalean/Stat/EmpiricalProcess/Basic.lean"},{"n":"Causalean.Stat.IIDSample","k":"structure","m":"Causalean.Stat.Sample","a":"Stat/Sample","t":1,"d":"An independent and identically distributed sample with marginal law P, realized as a sequence of sample points given by measurable maps on a single ambient probability space: the family is mutually independent, identically distributed, and the law of each point is the population law P.","s":"(Ω : Type u_1) → (X : Type u_2) → [inst : MeasurableSpace Ω] → [inst_1 : MeasurableSpace X] → MeasureTheory.Measure Ω → MeasureTheory.Measure X → Type (max u_1 u_2)","l":32,"f":"Causalean/Stat/Sample.lean"},{"n":"Causalean.Stat.IIDSample.abs_apply_mul_le_norm_sq","k":"theorem","m":"Causalean.Stat.Inference.VarianceEstimation","a":"Stat/Inference","t":2,"d":"The pointwise product of two functional-evaluations is dominated by a constant times ‖ψ x‖²: |φ (ψ x) * φ' (ψ x)| ≤ (‖φ‖ * ‖φ'‖) * ‖ψ x‖². Used to derive entrywise integrability of the product from the single hypothesis Integrable (fun x => ‖ψ x‖²) P.","s":"∀ {X : Type u_2} {E : Type u_3} [inst : NormedAddCommGroup E] [inst_1 : InnerProductSpace ℝ E] (φ φ' : E →L[ℝ] ℝ) (ψ : X → E) (x : X), |φ (ψ x) * φ' (ψ x)| ≤ ‖φ‖ * ‖φ'‖ * ‖ψ x‖ ^ 2","l":107,"f":"Causalean/Stat/Inference/VarianceEstimation.lean"},{"n":"Causalean.Stat.IIDSample.bootstrapSE","k":"def","m":"Causalean.Stat.Bootstrap.CI","a":"Stat/Bootstrap","t":1,"d":"Bootstrap standard error of √n θ̂: the square root of the bootstrap variance. Since the bootstrap variance of √n (X̄* − X̄) is exactly the empirical variance, bootstrapSE is the bootstrap estimate of the asymptotic standard deviation √(∫ ψ² dP).","s":"{Ω : Type u_1} → {X : Type u_2} → [inst : MeasurableSpace Ω] → [inst_1 : MeasurableSpace X] → {μ : MeasureTheory.Measure Ω} → {P : MeasureTheory.Measure X} → Causalean.Stat.IIDSample Ω X μ P → (X → ℝ) → ℕ → Ω → ℝ","l":57,"f":"Causalean/Stat/Bootstrap/CI.lean"},{"n":"Causalean.Stat.IIDSample.bootstrapSE_tendsto_inProb","k":"theorem","m":"Causalean.Stat.Bootstrap.CI","a":"Stat/Bootstrap","t":1,"d":"Consistency of the bootstrap standard error. Along an i.i.d. sample S, if the influence function ψ is measurable, integrable, square-integrable, and has population mean zero, then the bootstrap standard error of √n θ̂ converges in probability to the asymptotic standard deviation $\\sqrt{\\int \\psi^2\\,dP}$.","s":"∀ {Ω : Type u_1} {X : Type u_2} [inst : MeasurableSpace Ω] [inst_1 : MeasurableSpace X] {μ : MeasureTheory.Measure Ω} {P : MeasureTheory.Measure X} (S : Causalean.Stat.IIDSample Ω X μ P) [MeasureTheory.IsProbabilityMeasure P] {ψ : X → ℝ}, Measurable ψ → MeasureTheory.Integrable (fun ω => ψ (S.Z 0 ω)) μ → MeasureTheory.Integrable (fun ω => ψ (S.Z 0 ω) ^ 2) μ → ∫ (x : X), ψ x ∂P = 0 → Causalean.Stat.Tendsto_inProb (S.bootstrapSE ψ) (fun x => √(∫ (x : X), ψ x ^ 2 ∂P)) μ","l":65,"f":"Causalean/Stat/Bootstrap/CI.lean"},{"n":"Causalean.Stat.IIDSample.bootstrapStudentized","k":"def","m":"Causalean.Stat.Bootstrap.CI","a":"Stat/Bootstrap","t":1,"d":"Bootstrap studentized statistic √n (θ̂ − θ₀) / σ̂ₙ, where σ̂ₙ = bootstrapSE is the bootstrap standard error. The full-sample index family I n = Finset.range n is used.","s":"{Ω : Type u_1} → {X : Type u_2} → [inst : MeasurableSpace Ω] → [inst_1 : MeasurableSpace X] → {μ : MeasureTheory.Measure Ω} → {P : MeasureTheory.Measure X} → (ℕ → Ω → ℝ) → ℝ → Causalean.Stat.IIDSample Ω X μ P → (X → ℝ) → ℕ → Ω → ℝ","l":83,"f":"Causalean/Stat/Bootstrap/CI.lean"},{"n":"Causalean.Stat.IIDSample.bootstrapVar","k":"def","m":"Causalean.Stat.Bootstrap.Variance","a":"Stat/Bootstrap","t":1,"d":"Nonparametric bootstrap variance. The conditional variance of a single multinomial-bootstrap draw ψ(Z*₁) given the first n sample points: the plug-in (empirical) variance of ψ over the empirical distribution P̂ₙ(ω) = (1/n) Σ_{i<n} δ_{Zᵢ ω},","s":"{Ω : Type u_1} → {X : Type u_2} → [inst : MeasurableSpace Ω] → [inst_1 : MeasurableSpace X] → {μ : MeasureTheory.Measure Ω} → {P : MeasureTheory.Measure X} → Causalean.Stat.IIDSample Ω X μ P → (X → ℝ) → ℕ → Ω → ℝ","l":60,"f":"Causalean/Stat/Bootstrap/Variance.lean"},{"n":"Causalean.Stat.IIDSample.bootstrapVar_eq_centered","k":"theorem","m":"Causalean.Stat.Bootstrap.Variance","a":"Stat/Bootstrap","t":1,"d":"Exact bootstrap-variance identity. For an iid sample S, a statistic ψ, a sample size n, and a sample-path outcome ω, the plug-in bootstrap variance of ψ equals the centered empirical second moment of ψ over the first n observations.","s":"∀ {Ω : Type u_1} {X : Type u_2} [inst : MeasurableSpace Ω] [inst_1 : MeasurableSpace X] {μ : MeasureTheory.Measure Ω} {P : MeasureTheory.Measure X} (S : Causalean.Stat.IIDSample Ω X μ P) (ψ : X → ℝ) (n : ℕ) (ω : Ω), S.bootstrapVar ψ n ω = (↑n)⁻¹ * ∑ i ∈ Finset.range n, (ψ (S.Z i ω) - S.sampleMean ψ n ω) ^ 2","l":75,"f":"Causalean/Stat/Bootstrap/Variance.lean"},{"n":"Causalean.Stat.IIDSample.bootstrapVar_nonneg","k":"theorem","m":"Causalean.Stat.Bootstrap.Variance","a":"Stat/Bootstrap","t":2,"d":"The bootstrap variance is nonnegative: it is a centered empirical second moment (bootstrapVar_eq_centered).","s":"∀ {Ω : Type u_1} {X : Type u_2} [inst : MeasurableSpace Ω] [inst_1 : MeasurableSpace X] {μ : MeasureTheory.Measure Ω} {P : MeasureTheory.Measure X} (S : Causalean.Stat.IIDSample Ω X μ P) (ψ : X → ℝ) (n : ℕ) (ω : Ω), 0 ≤ S.bootstrapVar ψ n ω","l":111,"f":"Causalean/Stat/Bootstrap/Variance.lean"},{"n":"Causalean.Stat.IIDSample.bootstrapVar_tendsto_inProb","k":"theorem","m":"Causalean.Stat.Bootstrap.Variance","a":"Stat/Bootstrap","t":1,"d":"Consistency of the bootstrap variance. Along the i.i.d. sample S, if the influence function ψ is measurable, integrable, square-integrable, and has population mean zero, then the bootstrap variance converges in probability to the population second moment $\\int \\psi^2\\,dP$.","s":"∀ {Ω : Type u_1} {X : Type u_2} [inst : MeasurableSpace Ω] [inst_1 : MeasurableSpace X] {μ : MeasureTheory.Measure Ω} {P : MeasureTheory.Measure X} (S : Causalean.Stat.IIDSample Ω X μ P) [MeasureTheory.IsProbabilityMeasure P] {ψ : X → ℝ}, Measurable ψ → MeasureTheory.Integrable (fun ω => ψ (S.Z 0 ω)) μ → MeasureTheory.Integrable (fun ω => ψ (S.Z 0 ω) ^ 2) μ → ∫ (x : X), ψ x ∂P = 0 → Causalean.Stat.Tendsto_inProb (S.bootstrapVar ψ) (fun x => ∫ (x : X), ψ x ^ 2 ∂P) μ","l":120,"f":"Causalean/Stat/Bootstrap/Variance.lean"},{"n":"Causalean.Stat.IIDSample.card_injectiveTuples_image_eq","k":"theorem","m":"Causalean.Stat.UStatistic.OrderM.ExactVariance","a":"Stat/UStatistic/OrderM","t":2,"d":"For an ordered injective m-tuple t, the ordered injective tuples with the same image are exactly its m! reorderings.","s":"∀ {m n : ℕ} {t : Fin m → Fin n}, Function.Injective t → {q ∈ Causalean.Stat.injectiveTuples m n | Finset.image q Finset.univ = Finset.image t Finset.univ}.card = m.factorial","l":328,"f":"Causalean/Stat/UStatistic/OrderM/ExactVariance.lean"},{"n":"Causalean.Stat.IIDSample.card_share_ge_two_le","k":"theorem","m":"Causalean.Stat.UStatistic.OrderM.RemainderSecondMoment","a":"Stat/UStatistic/OrderM","t":2,"d":"The number of ordered pairs of injective tuples of length m drawn from n observations that share at least two sample indices is bounded by a polynomial in n determined by the tuple length.","s":"∀ {m n : ℕ}, {tq ∈ Causalean.Stat.injectiveTuples m n ×ˢ Causalean.Stat.injectiveTuples m n | 2 ≤ (Finset.image tq.1 Finset.univ ∩ Finset.image tq.2 Finset.univ).card}.card ≤ m.choose 2 * m ^ 2 * n ^ (2 * m - 2)","l":413,"f":"Causalean/Stat/UStatistic/OrderM/RemainderSecondMoment.lean"},{"n":"Causalean.Stat.IIDSample.charFun_normalizedSum_finset_eq_range_card","k":"theorem","m":"Causalean.Stat.SampleSplit.PartialFoldCLT","a":"Stat/SampleSplit","t":2,"d":"A normalized finite-set sum has the characteristic function of an equal-size initial block.","s":"∀ {Ω : Type u_1} {X : Type u_2} [inst : MeasurableSpace Ω] [inst_1 : MeasurableSpace X] {μ : MeasureTheory.Measure Ω} {P : MeasureTheory.Measure X} [MeasureTheory.IsProbabilityMeasure μ] (S : Causalean.Stat.IIDSample Ω X μ P) {ψ : X → ℝ}, Measurable ψ → ∀ (s : Finset ℕ) (t : ℝ), MeasureTheory.charFun (MeasureTheory.Measure.map (fun ω => (√↑s.card)⁻¹ * ∑ i ∈ s, ψ (S.Z i ω)) μ) t = MeasureTheory.charFun (MeasureTheory.Measure.map (fun ω => (√↑s.card)⁻¹ * ∑ i ∈ Finset.range s.card, ψ (S.Z i ω)) μ) t","l":83,"f":"Causalean/Stat/SampleSplit/PartialFoldCLT.lean"},{"n":"Causalean.Stat.IIDSample.clt_normalizedFoldB","k":"theorem","m":"Causalean.Stat.SampleSplit.PartialFoldCLT","a":"Stat/SampleSplit","t":1,"d":"Fold-B CLT. Along an i.i.d. sample S under a one-shot sample split split, fix a transform ψ : X → ℝ that is measurable, has population mean zero under P, and is square-integrable; provided the fold-B normalized partial sum is almost-everywhere measurable at every sample size, the fold-B normalized partial sum (1/√|B(n)|) Σ_{i∈B(n)} ψ(Z_i) converges in distribution to the centered Gaussian law with variance ∫ψ²dP.","s":"∀ {Ω : Type u_1} {X : Type u_2} [inst : MeasurableSpace Ω] [inst_1 : MeasurableSpace X] {μ : MeasureTheory.Measure Ω} {P : MeasureTheory.Measure X} [inst_2 : MeasureTheory.IsProbabilityMeasure μ] (S : Causalean.Stat.IIDSample Ω X μ P) (split : Causalean.Stat.OneShotSplit S) {ψ : X → ℝ}, Measurable ψ → ∫ (x : X), ψ x ∂P = 0 → MeasureTheory.Integrable (fun x => ψ x ^ 2) P → ∀ (hSum_meas : ∀ (n : ℕ), AEMeasurable (Causalean.Stat.IsAsymLinear.normalizedSum S ψ split.foldB n) μ), Causalean.Stat.Tendsto_dist (Causalean.Stat.IsAsymLinear.normalizedSum S ψ split.foldB) (Causalean.Stat.gaussianMeasure 0 (∫ (x : X), ψ x ^ 2 ∂P)) μ hSum_meas","l":149,"f":"Causalean/Stat/SampleSplit/PartialFoldCLT.lean"},{"n":"Causalean.Stat.IIDSample.clt_normalizedSum_vec","k":"theorem","m":"Causalean.Stat.CLT.GaussianLimit","a":"Stat/CLT","t":1,"d":"Multivariate CLT against the concrete Gaussian limit. Along the i.i.d. sample S, with ψ measurable and square-integrable, if ψ has population mean zero, $\\int \\psi\\,dP=0$, then the vector normalised sum converges in distribution to gaussianLimit ψ, the centered Gaussian on E with covariance equal to the second-moment operator of ψ; no abstract target or characteristic-function hypothesis remains.","s":"∀ {Ω : Type u_1} {X : Type u_2} [inst : MeasurableSpace Ω] [inst_1 : MeasurableSpace X] {μ : MeasureTheory.Measure Ω} {P : MeasureTheory.Measure X} {E : Type u_3} [inst_2 : NormedAddCommGroup E] [inst_3 : InnerProductSpace ℝ E] [inst_4 : FiniteDimensional ℝ E] [inst_5 : MeasurableSpace E] [inst_6 : BorelSpace E] {ψ : X → E} (hψ : Measurable ψ) (hvar : MeasureTheory.Integrable (fun x => ‖ψ x‖ ^ 2) P) (S : Causalean.Stat.IIDSample Ω X μ P), ∫ (x : X), ψ x ∂P = 0 → Causalean.Stat.Tendsto_dist_vec (Causalean.Stat.IsAsymLinearVec.normalizedSum S ψ fun m => Finset.range m) (Causalean.Stat.gaussianLimit hψ hvar) μ fun n => id (Measurable.aemeasurable (Measurable.const_smul (Finset.measurable_sum ((fun m => Finset.range m) n) fun i x => Measurable.comp hψ (S.meas i)) (√↑((fun m => Finset.range m) n).card)⁻¹))","l":111,"f":"Causalean/Stat/CLT/GaussianLimit.lean"},{"n":"Causalean.Stat.IIDSample.clt_normalizedSum_vec_of_charFun","k":"theorem","m":"Causalean.Stat.CLT.MultivariateCLT","a":"Stat/CLT","t":1,"d":"Multivariate CLT contact. Along the i.i.d. sample S, for an influence function ψ that is measurable, has population mean zero, and is square-integrable, and a target probability measure Q on E whose characteristic function is the Gaussian one $\\exp(-\\tfrac12\\int\\langle t,\\psi\\rangle^2\\,dP)$ at every point t, then the vector normalised sum converges in distribution to Q.","s":"∀ {Ω : Type u_1} {X : Type u_2} [inst : MeasurableSpace Ω] [inst_1 : MeasurableSpace X] {μ : MeasureTheory.Measure Ω} {P : MeasureTheory.Measure X} {E : Type u_3} [inst_2 : NormedAddCommGroup E] [inst_3 : InnerProductSpace ℝ E] [inst_4 : FiniteDimensional ℝ E] [inst_5 : MeasurableSpace E] [inst_6 : BorelSpace E] (S : Causalean.Stat.IIDSample Ω X μ P) {ψ : X → E} (hψ_meas : Measurable ψ), ∫ (x : X), ψ x ∂P = 0 → MeasureTheory.Integrable (fun x => ‖ψ x‖ ^ 2) P → ∀ (Q : MeasureTheory.Measure E) [inst_7 : MeasureTheory.IsProbabilityMeasure Q], (∀ (t : E), MeasureTheory.charFun Q t = Complex.exp (-↑(∫ (x : X), inner ℝ t (ψ x) ^ 2 ∂P) / 2)) → Causalean.Stat.Tendsto_dist_vec (Causalean.Stat.IsAsymLinearVec.normalizedSum S ψ Causalean.Stat.rng✝) Q μ fun n => id (Measurable.aemeasurable (Measurable.const_smul (Finset.measurable_sum (Causalean.Stat.rng✝ n) fun i x => Measurable.comp hψ_meas (S.meas i)) (√↑(Causalean.Stat.rng✝ n).card)⁻¹))","l":239,"f":"Causalean/Stat/CLT/MultivariateCLT.lean"},{"n":"Causalean.Stat.IIDSample.clt_normalized_sum","k":"theorem","m":"Causalean.Stat.CLT.AsymptoticLinearity","a":"Stat/CLT","t":1,"d":"Central limit theorem for normalised sample sums. Along the i.i.d. sample S, if the transform ψ of the observations is measurable, has population mean zero, and is square-integrable, then the normalised partial sum — the sum of the transformed observations over the first n indices, divided by $\\sqrt n$ — converges in distribution to the centred normal law with variance equal to the population second moment $\\int \\psi^2\\,dP$.","s":"∀ {Ω : Type u_1} {X : Type u_2} [inst : MeasurableSpace Ω] [inst_1 : MeasurableSpace X] {μ : MeasureTheory.Measure Ω} {P : MeasureTheory.Measure X} (S : Causalean.Stat.IIDSample Ω X μ P) {ψ : X → ℝ} (hψ_meas : Measurable ψ), ∫ (x : X), ψ x ∂P = 0 → MeasureTheory.Integrable (fun x => ψ x ^ 2) P → Causalean.Stat.Tendsto_dist (Causalean.Stat.IsAsymLinear.normalizedSum S ψ fun m => Finset.range m) (Causalean.Stat.gaussianMeasure 0 (∫ (x : X), ψ x ^ 2 ∂P)) μ fun n => id (Measurable.aemeasurable (Measurable.const_mul (Finset.measurable_sum ((fun m => Finset.range m) n) fun i x => Measurable.comp hψ_meas (S.meas i)) (√↑((fun m => Finset.range m) n).card)⁻¹))","l":121,"f":"Causalean/Stat/CLT/AsymptoticLinearity.lean"},{"n":"Causalean.Stat.IIDSample.crossterm_abs_le_zeta","k":"theorem","m":"Causalean.Stat.UStatistic.OrderM.RemainderSecondMoment","a":"Stat/UStatistic/OrderM","t":2,"d":"The absolute expected product of two kernel evaluations on distinct sample tuples is no larger than the kernel's second moment under the product distribution.","s":"∀ {Ω : Type u_1} {X : Type u_2} [inst : MeasurableSpace Ω] [inst_1 : MeasurableSpace X] {μ : MeasureTheory.Measure Ω} {P : MeasureTheory.Measure X} {m : ℕ} {g : (Fin m → X) → ℝ} (S : Causalean.Stat.IIDSample Ω X μ P), Measurable g → MeasureTheory.Integrable (fun z => g z ^ 2) (MeasureTheory.Measure.pi fun x => P) → ∀ {n : ℕ} {t q : Fin m → Fin n}, Function.Injective t → Function.Injective q → |∫ (ω : Ω), (g fun j => S.Z (↑(t j)) ω) * g fun j => S.Z (↑(q j)) ω ∂μ| ≤ Causalean.Stat.IIDSample.zetaOrder P g","l":298,"f":"Causalean/Stat/UStatistic/OrderM/RemainderSecondMoment.lean"},{"n":"Causalean.Stat.IIDSample.crossterm_eq_zero_of_image_ne","k":"theorem","m":"Causalean.Stat.UStatistic.OrderM.ExactVariance","a":"Stat/UStatistic/OrderM","t":2,"d":"Distinct-image cross term. If two ordered injective m-tuples have different images, complete degeneracy kills the cross expectation: some index of t is absent from q, and integrating that coordinate out gives 0.","s":"∀ {Ω : Type u_1} {X : Type u_2} [inst : MeasurableSpace Ω] [inst_1 : MeasurableSpace X] {μ : MeasureTheory.Measure Ω} {P : MeasureTheory.Measure X} [MeasureTheory.IsProbabilityMeasure P] {m : ℕ} {g : (Fin m → X) → ℝ} (S : Causalean.Stat.IIDSample Ω X μ P), Measurable g → MeasureTheory.Integrable (fun z => g z ^ 2) (MeasureTheory.Measure.pi fun x => P) → (∀ (j : Fin m) (tail : { k // k ≠ j } → X), ∫ (x : X), g (Causalean.Stat.insertCoord j x tail) ∂P = 0) → ∀ {n : ℕ} {t q : Fin m → Fin n}, Function.Injective t → Function.Injective q → Finset.image t Finset.univ ≠ Finset.image q Finset.univ → ∫ (ω : Ω), (g fun j => S.Z (↑(t j)) ω) * g fun j => S.Z (↑(q j)) ω ∂μ = 0","l":155,"f":"Causalean/Stat/UStatistic/OrderM/ExactVariance.lean"},{"n":"Causalean.Stat.IIDSample.crossterm_eq_zero_of_shared_le_one","k":"theorem","m":"Causalean.Stat.UStatistic.OrderM.RemainderSecondMoment","a":"Stat/UStatistic/OrderM","t":1,"d":"Cross-term vanishing. For an i.i.d. sample S, sample size n, and order-m kernel g that is first-order degenerate, let t and q be ordered m-tuples of sample indices that are each injective, and suppose the images of t and q share at most one sample index. Then the expected product of the kernel evaluated along t and along q is zero: zero shared indices give independence with mean zero on each factor, while one shared index reduces, after conditioning on it, to first-order degeneracy of each factor.","s":"∀ {Ω : Type u_1} {X : Type u_2} [inst : MeasurableSpace Ω] [inst_1 : MeasurableSpace X] {μ : MeasureTheory.Measure Ω} {P : MeasureTheory.Measure X} {m : ℕ} [inst_2 : NeZero m] {g : (Fin m → X) → ℝ} (S : Causalean.Stat.IIDSample Ω X μ P), Causalean.Stat.OrderFirstDegenKernel P g → ∀ {n : ℕ} {t q : Fin m → Fin n}, Function.Injective t → Function.Injective q → (Finset.image t Finset.univ ∩ Finset.image q Finset.univ).card ≤ 1 → ∫ (ω : Ω), (g fun j => S.Z (↑(t j)) ω) * g fun j => S.Z (↑(q j)) ω ∂μ = 0","l":278,"f":"Causalean/Stat/UStatistic/OrderM/RemainderSecondMoment.lean"},{"n":"Causalean.Stat.IIDSample.crossterm_eq_zeta_of_image_eq","k":"theorem","m":"Causalean.Stat.UStatistic.OrderM.ExactVariance","a":"Stat/UStatistic/OrderM","t":2,"d":"Equal-image cross term. If two ordered injective m-tuples have the same image, then q is a reordering of t, so by symmetry g(Z_q) = g(Z_t) and the cross expectation is ζ_m = zetaOrder P g.","s":"∀ {Ω : Type u_1} {X : Type u_2} [inst : MeasurableSpace Ω] [inst_1 : MeasurableSpace X] {μ : MeasureTheory.Measure Ω} {P : MeasureTheory.Measure X} {m : ℕ} {g : (Fin m → X) → ℝ} (S : Causalean.Stat.IIDSample Ω X μ P), Measurable g → (∀ (σ : Equiv.Perm (Fin m)) (z : Fin m → X), g (z ∘ ⇑σ) = g z) → ∀ {n : ℕ} {t q : Fin m → Fin n}, Function.Injective t → Function.Injective q → Finset.image t Finset.univ = Finset.image q Finset.univ → ∫ (ω : Ω), (g fun j => S.Z (↑(t j)) ω) * g fun j => S.Z (↑(q j)) ω ∂μ = Causalean.Stat.IIDSample.zetaOrder P g","l":122,"f":"Causalean/Stat/UStatistic/OrderM/ExactVariance.lean"},{"n":"Causalean.Stat.IIDSample.crossterm_zero_of_disjoint","k":"theorem","m":"Causalean.Stat.UStatistic.OrderM.RemainderSecondMoment","a":"Stat/UStatistic/OrderM","t":2,"d":"For a first-order-degenerate kernel, the expected product of terms evaluated on two injective sample-index tuples with disjoint index sets is zero.","s":"∀ {Ω : Type u_1} {X : Type u_2} [inst : MeasurableSpace Ω] [inst_1 : MeasurableSpace X] {μ : MeasureTheory.Measure Ω} {P : MeasureTheory.Measure X} {m : ℕ} [inst_2 : NeZero m] {g : (Fin m → X) → ℝ} (S : Causalean.Stat.IIDSample Ω X μ P) [MeasureTheory.IsFiniteMeasure P], Causalean.Stat.OrderFirstDegenKernel P g → ∀ {n : ℕ} {t q : Fin m → Fin n}, Function.Injective t → Function.Injective q → Disjoint (Finset.image t Finset.univ) (Finset.image q Finset.univ) → ∫ (ω : Ω), (g fun j => S.Z (↑(t j)) ω) * g fun j => S.Z (↑(q j)) ω ∂μ = 0","l":68,"f":"Causalean/Stat/UStatistic/OrderM/RemainderSecondMoment.lean"},{"n":"Causalean.Stat.IIDSample.crossterm_zero_of_shared_one","k":"theorem","m":"Causalean.Stat.UStatistic.OrderM.RemainderSecondMoment","a":"Stat/UStatistic/OrderM","t":2,"d":"For a first-order-degenerate kernel, the expected product of terms evaluated on two injective sample-index tuples sharing exactly one index is zero.","s":"∀ {Ω : Type u_1} {X : Type u_2} [inst : MeasurableSpace Ω] [inst_1 : MeasurableSpace X] {μ : MeasureTheory.Measure Ω} {P : MeasureTheory.Measure X} [MeasureTheory.IsProbabilityMeasure μ] {m : ℕ} [inst_3 : NeZero m] {g : (Fin m → X) → ℝ} (S : Causalean.Stat.IIDSample Ω X μ P) [MeasureTheory.IsFiniteMeasure P], Causalean.Stat.OrderFirstDegenKernel P g → ∀ {n : ℕ} {t q : Fin m → Fin n}, Function.Injective t → Function.Injective q → ∀ {a : Fin n}, Finset.image t Finset.univ ∩ Finset.image q Finset.univ = {a} → ∫ (ω : Ω), (g fun j => S.Z (↑(t j)) ω) * g fun j => S.Z (↑(q j)) ω ∂μ = 0","l":114,"f":"Causalean/Stat/UStatistic/OrderM/RemainderSecondMoment.lean"},{"n":"Causalean.Stat.IIDSample.descFactorial_ge","k":"theorem","m":"Causalean.Stat.UStatistic.OrderM.RemainderSecondMoment","a":"Stat/UStatistic/OrderM","t":2,"d":"When the sample size is at least the order, its falling factorial is at least the sample-size power divided by the order power.","s":"∀ {m n : ℕ}, m ≤ n → ↑n ^ m / ↑m ^ m ≤ ↑(n.descFactorial m)","l":501,"f":"Causalean/Stat/UStatistic/OrderM/RemainderSecondMoment.lean"},{"n":"Causalean.Stat.IIDSample.empProcVec","k":"def","m":"Causalean.Stat.EmpiricalProcess.Equicontinuity.Process","a":"Stat/EmpiricalProcess/Equicontinuity","t":1,"d":"Vector-valued centered empirical process.","s":"{Ω : Type u_1} → {X : Type u_2} → {E : Type u_3} → [inst : MeasurableSpace Ω] → [inst_1 : MeasurableSpace X] → [inst_2 : NormedAddCommGroup E] → [NormedSpace ℝ E] → {μ : MeasureTheory.Measure Ω} → {P : MeasureTheory.Measure X} → Causalean.Stat.IIDSample Ω X μ P → (X → E) → ℕ → Ω → E","l":50,"f":"Causalean/Stat/EmpiricalProcess/Equicontinuity/Process.lean"},{"n":"Causalean.Stat.IIDSample.empProcVec_eq_stochEquicont_gap","k":"theorem","m":"Causalean.Stat.EmpiricalProcess.Equicontinuity.Process","a":"Stat/EmpiricalProcess/Equicontinuity","t":1,"d":"For an i.i.d. sample S, a score function ψ, candidate and true parameter values θ and θ₀, sample size n, and outcome ω, the centered empirical-process gap R_n of StochEquicontAt, evaluated at θ, equals empProcVec of the score difference ψ(θ,·) − ψ(θ₀,·).","s":"∀ {Ω : Type u_1} {X : Type u_2} {E : Type u_3} [inst : MeasurableSpace Ω] [inst_1 : MeasurableSpace X] [inst_2 : NormedAddCommGroup E] [inst_3 : NormedSpace ℝ E] {μ : MeasureTheory.Measure Ω} {P : MeasureTheory.Measure X} (S : Causalean.Stat.IIDSample Ω X μ P) (ψ : E → X → E) (θ θ₀ : E) (n : ℕ) (ω : Ω), S.empProcVec (fun z => ψ θ z - ψ θ₀ z) n ω = (√↑n)⁻¹ • ∑ i ∈ Finset.range n, (ψ θ (S.Z i ω) - ψ θ₀ (S.Z i ω)) - √↑n • ∫ (z : X), ψ θ z - ψ θ₀ z ∂P","l":62,"f":"Causalean/Stat/EmpiricalProcess/Equicontinuity/Process.lean"},{"n":"Causalean.Stat.IIDSample.empProcess","k":"def","m":"Causalean.Stat.Quantile.SampleQuantileBahadur.Oscillation","a":"Stat/Quantile/SampleQuantileBahadur","t":1,"d":"The centered, scaled empirical process Gₙ(y) = √n (F̂ₙ(y) − F(y)).","s":"{Ω : Type u_1} → [inst : MeasurableSpace Ω] → {μ : MeasureTheory.Measure Ω} → {P : MeasureTheory.Measure ℝ} → Causalean.Stat.IIDSample Ω ℝ μ P → ℕ → Ω → ℝ → ℝ","l":95,"f":"Causalean/Stat/Quantile/SampleQuantileBahadur/Oscillation.lean"},{"n":"Causalean.Stat.IIDSample.empProcess_cell_sandwich","k":"theorem","m":"Causalean.Stat.Quantile.SampleQuantileBahadur.Oscillation","a":"Stat/Quantile/SampleQuantileBahadur","t":2,"d":"Pointwise monotone sandwich for the empirical process. Fix n, ω and a bracketing cell a ≤ u ≤ b of local shifts. Writing Gₙ(y) := empProcess n ω y and ya = q₀+a/√n, yb = q₀+b/√n, monotonicity of *both* F̂ₙ and F sandwiches the oscillation at the interior point q₀+u/√n between the two grid-node increments plus the deterministic mesh term:","s":"∀ {Ω : Type u_1} [inst : MeasurableSpace Ω] {μ : MeasureTheory.Measure Ω} {P : MeasureTheory.Measure ℝ} (S : Causalean.Stat.IIDSample Ω ℝ μ P) (n : ℕ) (ω : Ω) {q₀ a b u : ℝ}, a ≤ u → u ≤ b → |S.empProcess n ω (q₀ + u / √↑n) - S.empProcess n ω q₀| ≤ max |S.empProcess n ω (q₀ + a / √↑n) - S.empProcess n ω q₀| |S.empProcess n ω (q₀ + b / √↑n) - S.empProcess n ω q₀| + √↑n * (↑(ProbabilityTheory.cdf P) (q₀ + b / √↑n) - ↑(ProbabilityTheory.cdf P) (q₀ + a / √↑n))","l":324,"f":"Causalean/Stat/Quantile/SampleQuantileBahadur/Oscillation.lean"},{"n":"Causalean.Stat.IIDSample.empProcess_increment_tendsto_zero","k":"theorem","m":"Causalean.Stat.Quantile.SampleQuantileBahadur.Oscillation","a":"Stat/Quantile/SampleQuantileBahadur","t":2,"d":"Fixed local-shift increment. For a fixed local shift u, the empirical process increment between q₀ and q₀ + u/√n vanishes in probability: Gₙ(q₀ + u/√n) − Gₙ(q₀) →ₚ 0. Elementary second-moment / Chebyshev bound, since the increment is a centered Bernoulli sum of variance pₙ(1−pₙ) → 0.","s":"∀ {Ω : Type u_1} [inst : MeasurableSpace Ω] {μ : MeasureTheory.Measure Ω} {P : MeasureTheory.Measure ℝ} [MeasureTheory.IsProbabilityMeasure μ] [MeasureTheory.IsProbabilityMeasure P] (S : Causalean.Stat.IIDSample Ω ℝ μ P) {τ q₀ f₀ : ℝ}, Causalean.Stat.SampleQuantileReg P τ q₀ f₀ → ∀ (u : ℝ), Causalean.Stat.Tendsto_inProb (fun n ω => S.empProcess n ω (q₀ + u / √↑n) - S.empProcess n ω q₀) (fun x => 0) μ","l":104,"f":"Causalean/Stat/Quantile/SampleQuantileBahadur/Oscillation.lean"},{"n":"Causalean.Stat.IIDSample.empProcess_node_max_tendsto_zero","k":"theorem","m":"Causalean.Stat.Quantile.SampleQuantileBahadur.Oscillation","a":"Stat/Quantile/SampleQuantileBahadur","t":2,"d":"Finite grid-node maximum tends to zero. The maximum, over a *finite* nonempty index set s of local shifts v i, of the grid-node increments |Gₙ(q₀+v i/√n) − Gₙ(q₀)| vanishes in probability. A finite union of the L2 limits empProcess_increment_tendsto_zero: μ{ε ≤ maxᵢ |Δᵢ|} ≤ Σᵢ μ{ε ≤ |Δᵢ|}, each summand → 0.","s":"∀ {Ω : Type u_1} [inst : MeasurableSpace Ω] {μ : MeasureTheory.Measure Ω} {P : MeasureTheory.Measure ℝ} [MeasureTheory.IsProbabilityMeasure μ] [MeasureTheory.IsProbabilityMeasure P] (S : Causalean.Stat.IIDSample Ω ℝ μ P) {τ q₀ f₀ : ℝ}, Causalean.Stat.SampleQuantileReg P τ q₀ f₀ → ∀ {ι : Type u_2} (s : Finset ι) (hs : s.Nonempty) (v : ι → ℝ), Causalean.Stat.Tendsto_inProb (fun n ω => s.sup' hs fun i => |S.empProcess n ω (q₀ + v i / √↑n) - S.empProcess n ω q₀|) (fun x => 0) μ","l":411,"f":"Causalean/Stat/Quantile/SampleQuantileBahadur/Oscillation.lean"},{"n":"Causalean.Stat.IIDSample.empProcess_oscillation","k":"theorem","m":"Causalean.Stat.Quantile.SampleQuantileBahadur.Oscillation","a":"Stat/Quantile/SampleQuantileBahadur","t":1,"d":"Local oscillation of the sample-quantile empirical process. Given a SampleQuantileReg regularity bundle hreg for the population $\\tau$-quantile $q_0$ with density $f_0$, and a sequence of random endpoints Un that is bounded in probability, $U_n=O_p(1)$, the empirical process $G_n$, evaluated at the shrinking-window point $q_0+U_n/\\sqrt n$ minus its value at $q_0$, converges to zero in probability as $n\\to\\infty$.","s":"∀ {Ω : Type u_1} [inst : MeasurableSpace Ω] {μ : MeasureTheory.Measure Ω} {P : MeasureTheory.Measure ℝ} [MeasureTheory.IsProbabilityMeasure μ] [MeasureTheory.IsProbabilityMeasure P] (S : Causalean.Stat.IIDSample Ω ℝ μ P) {τ q₀ f₀ : ℝ}, Causalean.Stat.SampleQuantileReg P τ q₀ f₀ → ∀ {Un : ℕ → Ω → ℝ}, Causalean.Stat.IsBigOp Un (fun x => 1) μ → Causalean.Stat.Tendsto_inProb (fun n ω => S.empProcess n ω (q₀ + Un n ω / √↑n) - S.empProcess n ω q₀) (fun x => 0) μ","l":537,"f":"Causalean/Stat/Quantile/SampleQuantileBahadur/Oscillation.lean"},{"n":"Causalean.Stat.IIDSample.empProcess_q0_bigO","k":"theorem","m":"Causalean.Stat.Quantile.SampleQuantileBahadur.Rate","a":"Stat/Quantile/SampleQuantileBahadur","t":2,"d":"Fixed-quantile empirical-process tightness. empProcess(·,q₀) is O_p(1).","s":"∀ {Ω : Type u_1} [inst : MeasurableSpace Ω] {μ : MeasureTheory.Measure Ω} {P : MeasureTheory.Measure ℝ} [MeasureTheory.IsProbabilityMeasure μ] [MeasureTheory.IsProbabilityMeasure P] (S : Causalean.Stat.IIDSample Ω ℝ μ P) {τ q₀ f₀ : ℝ}, Causalean.Stat.SampleQuantileReg P τ q₀ f₀ → Causalean.Stat.IsBigOp (fun n ω => S.empProcess n ω q₀) (fun x => 1) μ","l":95,"f":"Causalean/Stat/Quantile/SampleQuantileBahadur/Rate.lean"},{"n":"Causalean.Stat.IIDSample.empProcess_q0_tendsto_normal","k":"theorem","m":"Causalean.Stat.Quantile.SampleQuantileBahadur.Rate","a":"Stat/Quantile/SampleQuantileBahadur","t":2,"d":"Fixed-quantile empirical-process CLT. empProcess(·,q₀) = √n(F̂ₙ(q₀) − F(q₀)) ⇒ N(0, τ(1−τ)). Restates empiricalCDF_tendsto_normal at y = q₀, using cdf P q₀ = τ.","s":"∀ {Ω : Type u_1} [inst : MeasurableSpace Ω] {μ : MeasureTheory.Measure Ω} {P : MeasureTheory.Measure ℝ} [inst_1 : MeasureTheory.IsProbabilityMeasure μ] [MeasureTheory.IsProbabilityMeasure P] (S : Causalean.Stat.IIDSample Ω ℝ μ P) {τ q₀ f₀ : ℝ}, Causalean.Stat.SampleQuantileReg P τ q₀ f₀ → ∀ (hmeas : ∀ (n : ℕ), AEMeasurable (fun ω => S.empProcess n ω q₀) μ), Causalean.Stat.Tendsto_dist (fun n ω => S.empProcess n ω q₀) (Causalean.Stat.gaussianMeasure 0 (τ * (1 - τ))) μ hmeas","l":63,"f":"Causalean/Stat/Quantile/SampleQuantileBahadur/Rate.lean"},{"n":"Causalean.Stat.IIDSample.empiricalCDF","k":"def","m":"Causalean.Stat.Quantile.EmpiricalCDF","a":"Stat/Quantile","t":1,"d":"The empirical cumulative distribution function F̂ₙ(y) = (1/n) Σ_{i<n} 1{Z_i ≤ y}, the sampleMean of cdfStat y.","s":"{Ω : Type u_1} → [inst : MeasurableSpace Ω] → {μ : MeasureTheory.Measure Ω} → {P : MeasureTheory.Measure ℝ} → Causalean.Stat.IIDSample Ω ℝ μ P → ℝ → ℕ → Ω → ℝ","l":92,"f":"Causalean/Stat/Quantile/EmpiricalCDF.lean"},{"n":"Causalean.Stat.IIDSample.empiricalCDF_monotone","k":"theorem","m":"Causalean.Stat.Quantile.EmpiricalQuantile","a":"Stat/Quantile","t":2,"d":"The empirical cdf is monotone in its real argument y (a sum of monotone lower-ray indicators).","s":"∀ {Ω : Type u_1} [inst : MeasurableSpace Ω] {μ : MeasureTheory.Measure Ω} {P : MeasureTheory.Measure ℝ} (S : Causalean.Stat.IIDSample Ω ℝ μ P) (n : ℕ) (ω : Ω), Monotone fun y => S.empiricalCDF y n ω","l":123,"f":"Causalean/Stat/Quantile/EmpiricalQuantile.lean"},{"n":"Causalean.Stat.IIDSample.empiricalMeasure","k":"def","m":"Causalean.Stat.Quantile.EmpiricalQuantile","a":"Stat/Quantile","t":1,"d":"The empirical measure νₙ = (1/n) Σ_{i<n} δ_{Zᵢ} of an i.i.d. sample.","s":"{Ω : Type u_1} → [inst : MeasurableSpace Ω] → {μ : MeasureTheory.Measure Ω} → {P : MeasureTheory.Measure ℝ} → Causalean.Stat.IIDSample Ω ℝ μ P → ℕ → Ω → MeasureTheory.Measure ℝ","l":57,"f":"Causalean/Stat/Quantile/EmpiricalQuantile.lean"},{"n":"Causalean.Stat.IIDSample.empiricalMeasure_cdf","k":"theorem","m":"Causalean.Stat.Quantile.EmpiricalQuantile","a":"Stat/Quantile","t":1,"d":"cdf bridge. For a positive sample size $n$, the cumulative distribution function of the empirical measure S.empiricalMeasure n ω built from an i.i.d. sample at outcome ω coincides pointwise, at every threshold y, with the empirical cdf S.empiricalCDF y n ω.","s":"∀ {Ω : Type u_1} [inst : MeasurableSpace Ω] {μ : MeasureTheory.Measure Ω} {P : MeasureTheory.Measure ℝ} (S : Causalean.Stat.IIDSample Ω ℝ μ P) {n : ℕ}, 0 < n → ∀ (ω : Ω) (y : ℝ), ↑(ProbabilityTheory.cdf (S.empiricalMeasure n ω)) y = S.empiricalCDF y n ω","l":94,"f":"Causalean/Stat/Quantile/EmpiricalQuantile.lean"},{"n":"Causalean.Stat.IIDSample.empiricalMeasure_isProbabilityMeasure","k":"theorem","m":"Causalean.Stat.Quantile.EmpiricalQuantile","a":"Stat/Quantile","t":2,"d":"For 0 < n the empirical measure is a probability measure.","s":"∀ {Ω : Type u_1} [inst : MeasurableSpace Ω] {μ : MeasureTheory.Measure Ω} {P : MeasureTheory.Measure ℝ} (S : Causalean.Stat.IIDSample Ω ℝ μ P) {n : ℕ}, 0 < n → ∀ (ω : Ω), MeasureTheory.IsProbabilityMeasure (S.empiricalMeasure n ω)","l":62,"f":"Causalean/Stat/Quantile/EmpiricalQuantile.lean"},{"n":"Causalean.Stat.IIDSample.empiricalMeasure_real_Iic","k":"theorem","m":"Causalean.Stat.Quantile.EmpiricalQuantile","a":"Stat/Quantile","t":2,"d":"The empirical measure of the lower ray Iic y is F̂ₙ(y) (as ℝ).","s":"∀ {Ω : Type u_1} [inst : MeasurableSpace Ω] {μ : MeasureTheory.Measure Ω} {P : MeasureTheory.Measure ℝ} (S : Causalean.Stat.IIDSample Ω ℝ μ P) {n : ℕ}, 0 < n → ∀ (ω : Ω) (y : ℝ), (S.empiricalMeasure n ω).real (Set.Iic y) = S.empiricalCDF y n ω","l":75,"f":"Causalean/Stat/Quantile/EmpiricalQuantile.lean"},{"n":"Causalean.Stat.IIDSample.empiricalProcess","k":"def","m":"Causalean.Stat.EmpiricalProcess.Basic","a":"Stat/EmpiricalProcess","t":1,"d":"Empirical process of the class member f i at sample size n: √n · ((1/n) Σ_{k<n} f_i(Z_k) − ∫ f_i dP). The object whose weak limit (a Gaussian process) is the subject of Donsker theory.","s":"{Ω : Type u_1} → {X : Type u_2} → {ι : Type u_3} → [inst : MeasurableSpace Ω] → [inst_1 : MeasurableSpace X] → {μ : MeasureTheory.Measure Ω} → {P : MeasureTheory.Measure X} → Causalean.Stat.IIDSample Ω X μ P → (ι → X → ℝ) → ι → ℕ → Ω → ℝ","l":58,"f":"Causalean/Stat/EmpiricalProcess/Basic.lean"},{"n":"Causalean.Stat.IIDSample.iIndepFun_comp","k":"theorem","m":"Causalean.Stat.Concentration.TailBounds.Hoeffding","a":"Stat/Concentration/TailBounds","t":2,"d":"Independence of the composed family fun i ↦ f ∘ S.Z i.","s":"∀ {Ω : Type u_1} {X : Type u_2} [inst : MeasurableSpace Ω] [inst_1 : MeasurableSpace X] {μ : MeasureTheory.Measure Ω} {P : MeasureTheory.Measure X} (S : Causalean.Stat.IIDSample Ω X μ P) {f : X → ℝ}, Measurable f → ProbabilityTheory.iIndepFun (fun i ω => f (S.Z i ω)) μ","l":60,"f":"Causalean/Stat/Concentration/TailBounds/Hoeffding.lean"},{"n":"Causalean.Stat.IIDSample.integrable_injectiveTuples_sum","k":"theorem","m":"Causalean.Stat.UStatistic.OrderM.Variance","a":"Stat/UStatistic/OrderM","t":2,"d":"The injective-tuple sum of an order-m kernel is integrable.","s":"∀ {Ω : Type u_1} {X : Type u_2} [inst : MeasurableSpace Ω] [inst_1 : MeasurableSpace X] {μ : MeasureTheory.Measure Ω} {P : MeasureTheory.Measure X} {m : ℕ} {g : (Fin m → X) → ℝ} (S : Causalean.Stat.IIDSample Ω X μ P), Measurable g → MeasureTheory.Integrable g (MeasureTheory.Measure.pi fun x => P) → ∀ (n : ℕ), MeasureTheory.Integrable (fun ω => ∑ t ∈ Causalean.Stat.injectiveTuples m n, g fun j => S.Z (↑(t j)) ω) μ","l":428,"f":"Causalean/Stat/UStatistic/OrderM/Variance.lean"},{"n":"Causalean.Stat.IIDSample.integrable_orderKernelTerm","k":"theorem","m":"Causalean.Stat.UStatistic.OrderM.Variance","a":"Stat/UStatistic/OrderM","t":2,"d":"An order-m kernel term along an injective tuple is integrable whenever the kernel is integrable under the product law.","s":"∀ {Ω : Type u_1} {X : Type u_2} [inst : MeasurableSpace Ω] [inst_1 : MeasurableSpace X] {μ : MeasureTheory.Measure Ω} {P : MeasureTheory.Measure X} {m n : ℕ} (S : Causalean.Stat.IIDSample Ω X μ P) {h : (Fin m → X) → ℝ}, Measurable h → MeasureTheory.Integrable h (MeasureTheory.Measure.pi fun x => P) → ∀ {t : Fin m → Fin n}, Function.Injective t → MeasureTheory.Integrable (fun ω => h fun j => S.Z (↑(t j)) ω) μ","l":193,"f":"Causalean/Stat/UStatistic/OrderM/Variance.lean"},{"n":"Causalean.Stat.IIDSample.integrable_orderTerm","k":"theorem","m":"Causalean.Stat.UStatistic.OrderM.Variance","a":"Stat/UStatistic/OrderM","t":2,"d":"An order-m kernel term along an injective tuple is integrable.","s":"∀ {Ω : Type u_1} {X : Type u_2} [inst : MeasurableSpace Ω] [inst_1 : MeasurableSpace X] {μ : MeasureTheory.Measure Ω} {P : MeasureTheory.Measure X} {m n : ℕ} {g : (Fin m → X) → ℝ} (S : Causalean.Stat.IIDSample Ω X μ P), Measurable g → MeasureTheory.Integrable g (MeasureTheory.Measure.pi fun x => P) → ∀ {t : Fin m → Fin n}, Function.Injective t → MeasureTheory.Integrable (fun ω => g fun j => S.Z (↑(t j)) ω) μ","l":371,"f":"Causalean/Stat/UStatistic/OrderM/Variance.lean"},{"n":"Causalean.Stat.IIDSample.integrable_orderTerm_mul","k":"theorem","m":"Causalean.Stat.UStatistic.OrderM.Variance","a":"Stat/UStatistic/OrderM","t":2,"d":"The product of two injective order-m kernel terms is integrable.","s":"∀ {Ω : Type u_1} {X : Type u_2} [inst : MeasurableSpace Ω] [inst_1 : MeasurableSpace X] {μ : MeasureTheory.Measure Ω} {P : MeasureTheory.Measure X} {m n : ℕ} {g : (Fin m → X) → ℝ} (S : Causalean.Stat.IIDSample Ω X μ P), Measurable g → MeasureTheory.Integrable (fun z => g z ^ 2) (MeasureTheory.Measure.pi fun x => P) → ∀ {t q : Fin m → Fin n}, Function.Injective t → Function.Injective q → MeasureTheory.Integrable (fun ω => (g fun j => S.Z (↑(t j)) ω) * g fun j => S.Z (↑(q j)) ω) μ","l":418,"f":"Causalean/Stat/UStatistic/OrderM/Variance.lean"},{"n":"Causalean.Stat.IIDSample.integrable_orderTerm_sq","k":"theorem","m":"Causalean.Stat.UStatistic.OrderM.Variance","a":"Stat/UStatistic/OrderM","t":2,"d":"The square of an order-m kernel term along an injective tuple is integrable.","s":"∀ {Ω : Type u_1} {X : Type u_2} [inst : MeasurableSpace Ω] [inst_1 : MeasurableSpace X] {μ : MeasureTheory.Measure Ω} {P : MeasureTheory.Measure X} {m n : ℕ} {g : (Fin m → X) → ℝ} (S : Causalean.Stat.IIDSample Ω X μ P), Measurable g → MeasureTheory.Integrable (fun z => g z ^ 2) (MeasureTheory.Measure.pi fun x => P) → ∀ {t : Fin m → Fin n}, Function.Injective t → MeasureTheory.Integrable (fun ω => (g fun j => S.Z (↑(t j)) ω) ^ 2) μ","l":379,"f":"Causalean/Stat/UStatistic/OrderM/Variance.lean"},{"n":"Causalean.Stat.IIDSample.integral_comp_eq","k":"theorem","m":"Causalean.Stat.Concentration.TailBounds.Hoeffding","a":"Stat/Concentration/TailBounds","t":2,"d":"The population mean of a statistic equals its sample-point expectation: ∫ ω, f (S.Z i ω) ∂μ = ∫ x, f x ∂P.","s":"∀ {Ω : Type u_1} {X : Type u_2} [inst : MeasurableSpace Ω] [inst_1 : MeasurableSpace X] {μ : MeasureTheory.Measure Ω} {P : MeasureTheory.Measure X} (S : Causalean.Stat.IIDSample Ω X μ P) {f : X → ℝ}, AEMeasurable f P → ∀ (i : ℕ), ∫ (ω : Ω), f (S.Z i ω) ∂μ = ∫ (x : X), f x ∂P","l":47,"f":"Causalean/Stat/Concentration/TailBounds/Hoeffding.lean"},{"n":"Causalean.Stat.IIDSample.integral_injectiveTuples_sum_sq_degen","k":"theorem","m":"Causalean.Stat.UStatistic.OrderM.ExactVariance","a":"Stat/UStatistic/OrderM","t":1,"d":"Exact second moment of the injective-tuple sum. For an i.i.d. sample S and sample size n, if the order-m kernel g is completely degenerate: symmetric, square-integrable, and with zero conditional mean after integrating out any single coordinate, then the second moment of the sum of g over all ordered injective m-tuples drawn from the first n sample indices equals m! · n^{(m)} · ζ_m, where n^{(m)} is the number of such tuples and ζ_m is the kernel's second moment under the m-fold product law.","s":"∀ {Ω : Type u_1} {X : Type u_2} [inst : MeasurableSpace Ω] [inst_1 : MeasurableSpace X] {μ : MeasureTheory.Measure Ω} {P : MeasureTheory.Measure X} [MeasureTheory.IsProbabilityMeasure P] {m : ℕ} [inst_3 : NeZero m] {g : (Fin m → X) → ℝ} (S : Causalean.Stat.IIDSample Ω X μ P), Causalean.Stat.OrderDegenKernel P g → ∀ (n : ℕ), ∫ (ω : Ω), (∑ t ∈ Causalean.Stat.injectiveTuples m n, g fun j => S.Z (↑(t j)) ω) ^ 2 ∂μ = ↑m.factorial * Causalean.Stat.injectiveTupleCount m n * Causalean.Stat.IIDSample.zetaOrder P g","l":368,"f":"Causalean/Stat/UStatistic/OrderM/ExactVariance.lean"},{"n":"Causalean.Stat.IIDSample.integral_injectiveTuples_sum_sq_le_shared_count","k":"theorem","m":"Causalean.Stat.UStatistic.OrderM.RemainderSecondMoment","a":"Stat/UStatistic/OrderM","t":2,"d":"For a first-order-degenerate kernel, the second moment of the unnormalised sum over ordered injective sample tuples is bounded by the number of tuple pairs sharing at least two observations times the kernel's second moment.","s":"∀ {Ω : Type u_1} {X : Type u_2} [inst : MeasurableSpace Ω] [inst_1 : MeasurableSpace X] {μ : MeasureTheory.Measure Ω} {P : MeasureTheory.Measure X} {m : ℕ} [inst_2 : NeZero m] {g : (Fin m → X) → ℝ} (S : Causalean.Stat.IIDSample Ω X μ P) [MeasureTheory.IsFiniteMeasure P], Causalean.Stat.OrderFirstDegenKernel P g → ∀ {n : ℕ}, ∫ (ω : Ω), (∑ t ∈ Causalean.Stat.injectiveTuples m n, g fun j => S.Z (↑(t j)) ω) ^ 2 ∂μ ≤ ↑{tq ∈ Causalean.Stat.injectiveTuples m n ×ˢ Causalean.Stat.injectiveTuples m n | 2 ≤ (Finset.image tq.1 Finset.univ ∩ Finset.image tq.2 Finset.univ).card}.card * Causalean.Stat.IIDSample.zetaOrder P g","l":531,"f":"Causalean/Stat/UStatistic/OrderM/RemainderSecondMoment.lean"},{"n":"Causalean.Stat.IIDSample.integral_offDiag_sum_sq","k":"theorem","m":"Causalean.Stat.UStatistic.Variance","a":"Stat/UStatistic","t":1,"d":"Second moment of the off-diagonal sum. For an i.i.d. sample S and sample size n, if the two-argument kernel g is degenerate, then the second moment of the sum of g(Z_i, Z_j) over all ordered off-diagonal index pairs i ≠ j below n equals 2 · |offDiag| · ζ, where ζ is the kernel's second moment under P × P. The m = 2 case of integral_injectiveTuples_sum_sq_degen, using sum_injectiveTuples_two_eq_offDiag, toOrderDegenKernel, and zeta_eq_zetaOrder (2! = 2, injectiveTupleCount 2 n = |offDiag| = n(n−1)).","s":"∀ {Ω : Type u_1} {X : Type u_2} [inst : MeasurableSpace Ω] [inst_1 : MeasurableSpace X] {μ : MeasureTheory.Measure Ω} {P : MeasureTheory.Measure X} {g : X → X → ℝ} (S : Causalean.Stat.IIDSample Ω X μ P), Causalean.Stat.DegenKernel P g → ∀ (n : ℕ), ∫ (ω : Ω), (∑ p ∈ (Finset.range n).offDiag, g (S.Z p.1 ω) (S.Z p.2 ω)) ^ 2 ∂μ = 2 * ↑(Finset.range n).offDiag.card * Causalean.Stat.IIDSample.zeta P g","l":237,"f":"Causalean/Stat/UStatistic/Variance.lean"},{"n":"Causalean.Stat.IIDSample.integral_orderKernelTerm_eq","k":"theorem","m":"Causalean.Stat.UStatistic.OrderM.Variance","a":"Stat/UStatistic/OrderM","t":2,"d":"The expectation of an injectively indexed order-m kernel term equals the kernel's product-law mean.","s":"∀ {Ω : Type u_1} {X : Type u_2} [inst : MeasurableSpace Ω] [inst_1 : MeasurableSpace X] {μ : MeasureTheory.Measure Ω} {P : MeasureTheory.Measure X} {m n : ℕ} (S : Causalean.Stat.IIDSample Ω X μ P) {h : (Fin m → X) → ℝ}, Measurable h → ∀ {t : Fin m → Fin n}, Function.Injective t → ∫ (ω : Ω), h fun j => S.Z (↑(t j)) ω ∂μ = ∫ (z : Fin m → X), h z ∂MeasureTheory.Measure.pi fun x => P","l":208,"f":"Causalean/Stat/UStatistic/OrderM/Variance.lean"},{"n":"Causalean.Stat.IIDSample.integral_orderKernelTerm_eq_zero_of_uMean_zero","k":"theorem","m":"Causalean.Stat.UStatistic.OrderM.Variance","a":"Stat/UStatistic/OrderM","t":2,"d":"An injectively indexed order-m kernel term has mean zero whenever the kernel has zero product-law mean.","s":"∀ {Ω : Type u_1} {X : Type u_2} [inst : MeasurableSpace Ω] [inst_1 : MeasurableSpace X] {μ : MeasureTheory.Measure Ω} {P : MeasureTheory.Measure X} {m n : ℕ} (S : Causalean.Stat.IIDSample Ω X μ P) {h : (Fin m → X) → ℝ}, Measurable h → ∀ {t : Fin m → Fin n}, Function.Injective t → Causalean.Stat.uMeanOrder h P = 0 → ∫ (ω : Ω), h fun j => S.Z (↑(t j)) ω ∂μ = 0","l":220,"f":"Causalean/Stat/UStatistic/OrderM/Variance.lean"},{"n":"Causalean.Stat.IIDSample.integral_orderTerm_eq_zero","k":"theorem","m":"Causalean.Stat.UStatistic.OrderM.Variance","a":"Stat/UStatistic/OrderM","t":2,"d":"An injectively indexed fully degenerate order-m kernel term has mean zero.","s":"∀ {Ω : Type u_1} {X : Type u_2} [inst : MeasurableSpace Ω] [inst_1 : MeasurableSpace X] {μ : MeasureTheory.Measure Ω} {P : MeasureTheory.Measure X} [MeasureTheory.IsProbabilityMeasure P] {m n : ℕ} {g : (Fin m → X) → ℝ} (S : Causalean.Stat.IIDSample Ω X μ P) [inst_3 : NeZero m], Causalean.Stat.OrderDegenKernel P g → ∀ {t : Fin m → Fin n}, Function.Injective t → ∫ (ω : Ω), g fun j => S.Z (↑(t j)) ω ∂μ = 0","l":328,"f":"Causalean/Stat/UStatistic/OrderM/Variance.lean"},{"n":"Causalean.Stat.IIDSample.integral_orderTerm_eq_zero_of_degenKernel_uMean_zero","k":"theorem","m":"Causalean.Stat.UStatistic.OrderM.Variance","a":"Stat/UStatistic/OrderM","t":2,"d":"An injectively indexed fully degenerate order-m kernel term has mean zero when its product-law mean is zero.","s":"∀ {Ω : Type u_1} {X : Type u_2} [inst : MeasurableSpace Ω] [inst_1 : MeasurableSpace X] {μ : MeasureTheory.Measure Ω} {P : MeasureTheory.Measure X} {m n : ℕ} {g : (Fin m → X) → ℝ} (S : Causalean.Stat.IIDSample Ω X μ P) [inst_2 : NeZero m], Causalean.Stat.OrderDegenKernel P g → ∀ {t : Fin m → Fin n}, Function.Injective t → Causalean.Stat.uMeanOrder g P = 0 → ∫ (ω : Ω), g fun j => S.Z (↑(t j)) ω ∂μ = 0","l":319,"f":"Causalean/Stat/UStatistic/OrderM/Variance.lean"},{"n":"Causalean.Stat.IIDSample.integral_rescaled_eq_zero","k":"theorem","m":"Causalean.Stat.UStatistic.Variance","a":"Stat/UStatistic","t":2,"d":"The rescaled degenerate U-statistic has mean zero.","s":"∀ {Ω : Type u_1} {X : Type u_2} [inst : MeasurableSpace Ω] [inst_1 : MeasurableSpace X] {μ : MeasureTheory.Measure Ω} {P : MeasureTheory.Measure X} {g : X → X → ℝ} (S : Causalean.Stat.IIDSample Ω X μ P), Causalean.Stat.DegenKernel P g → ∀ (n : ℕ), ∫ (ω : Ω), √↑n * Causalean.Stat.uStatistic S g n ω ∂μ = 0","l":296,"f":"Causalean/Stat/UStatistic/Variance.lean"},{"n":"Causalean.Stat.IIDSample.integral_rescaled_order_eq_zero","k":"theorem","m":"Causalean.Stat.UStatistic.OrderM.RemainderNegligible","a":"Stat/UStatistic/OrderM","t":2,"d":"The rescaled higher-order remainder has mean zero.","s":"∀ {Ω : Type u_1} {X : Type u_2} [inst : MeasurableSpace Ω] [inst_1 : MeasurableSpace X] {μ : MeasureTheory.Measure Ω} {P : MeasureTheory.Measure X} {m : ℕ} [inst_2 : NeZero m] {g : (Fin m → X) → ℝ} (S : Causalean.Stat.IIDSample Ω X μ P), Causalean.Stat.OrderFirstDegenKernel P g → ∀ {n : ℕ}, m ≤ n → ∫ (ω : Ω), √↑n * Causalean.Stat.uStatisticOrder S g n ω ∂μ = 0","l":54,"f":"Causalean/Stat/UStatistic/OrderM/RemainderNegligible.lean"},{"n":"Causalean.Stat.IIDSample.integral_rescaled_order_sq_degen","k":"theorem","m":"Causalean.Stat.UStatistic.OrderM.ExactVariance","a":"Stat/UStatistic/OrderM","t":1,"d":"Exact variance of the rescaled degenerate fixed-order U-statistic. For an i.i.d. sample S, if the order-m kernel g is completely degenerate and the sample size is at least m, then the second moment of the √n-rescaled order-m U-statistic of g equals n · m! · ζ_m / n^{(m)}, which specializes to 2ζ/(n−1) when m = 2.","s":"∀ {Ω : Type u_1} {X : Type u_2} [inst : MeasurableSpace Ω] [inst_1 : MeasurableSpace X] {μ : MeasureTheory.Measure Ω} {P : MeasureTheory.Measure X} [MeasureTheory.IsProbabilityMeasure P] {m : ℕ} [inst_3 : NeZero m] {g : (Fin m → X) → ℝ} (S : Causalean.Stat.IIDSample Ω X μ P), Causalean.Stat.OrderDegenKernel P g → ∀ {n : ℕ}, m ≤ n → ∫ (ω : Ω), (√↑n * Causalean.Stat.uStatisticOrder S g n ω) ^ 2 ∂μ = ↑n * ↑m.factorial * Causalean.Stat.IIDSample.zetaOrder P g / Causalean.Stat.injectiveTupleCount m n","l":455,"f":"Causalean/Stat/UStatistic/OrderM/ExactVariance.lean"},{"n":"Causalean.Stat.IIDSample.integral_rescaled_order_sq_le","k":"theorem","m":"Causalean.Stat.UStatistic.OrderM.RemainderSecondMoment","a":"Stat/UStatistic/OrderM","t":1,"d":"L² bound on the rescaled higher-order remainder. For an i.i.d. sample S, if the order-m kernel g is first-order degenerate, then there is a nonnegative constant C, depending only on the order m and the kernel's second moment ζ_m = E[g²], such that the second moment of the √n-rescaled order-m U-statistic of g is at most C/n for every sample size n ≥ m. This is the keystone estimate; it packages cross-term vanishing, the Cauchy–Schwarz bound |E[g_t g_q]| ≤ ζ_m, the O(n^{2m-2}) count of surviving tuple pairs, and the n · (n^{(m)})⁻² normalization.","s":"∀ {Ω : Type u_1} {X : Type u_2} [inst : MeasurableSpace Ω] [inst_1 : MeasurableSpace X] {μ : MeasureTheory.Measure Ω} {P : MeasureTheory.Measure X} {m : ℕ} [inst_2 : NeZero m] {g : (Fin m → X) → ℝ} (S : Causalean.Stat.IIDSample Ω X μ P), Causalean.Stat.OrderFirstDegenKernel P g → ∃ C, 0 ≤ C ∧ ∀ {n : ℕ}, m ≤ n → ∫ (ω : Ω), (√↑n * Causalean.Stat.uStatisticOrder S g n ω) ^ 2 ∂μ ≤ C / ↑n","l":745,"f":"Causalean/Stat/UStatistic/OrderM/RemainderSecondMoment.lean"},{"n":"Causalean.Stat.IIDSample.integral_rescaled_sq","k":"theorem","m":"Causalean.Stat.UStatistic.Variance","a":"Stat/UStatistic","t":1,"d":"L² bound on the rescaled degenerate U-statistic. For an i.i.d. sample S, if the two-argument kernel g is degenerate and the sample size is at least two, then the second moment of the √n-rescaled degenerate U-statistic of g equals 2ζ/(n−1), where ζ is the kernel's second moment under P × P. The m = 2 case of integral_rescaled_order_sq_degen (n · 2! · ζ / n^{(2)} = 2ζ/(n−1)), via uStatisticOrder_two_eq_uStatistic and zeta_eq_zetaOrder.","s":"∀ {Ω : Type u_1} {X : Type u_2} [inst : MeasurableSpace Ω] [inst_1 : MeasurableSpace X] {μ : MeasureTheory.Measure Ω} {P : MeasureTheory.Measure X} {g : X → X → ℝ} (S : Causalean.Stat.IIDSample Ω X μ P), Causalean.Stat.DegenKernel P g → ∀ {n : ℕ}, 2 ≤ n → ∫ (ω : Ω), (√↑n * Causalean.Stat.uStatistic S g n ω) ^ 2 ∂μ = 2 * Causalean.Stat.IIDSample.zeta P g / (↑n - 1)","l":265,"f":"Causalean/Stat/UStatistic/Variance.lean"},{"n":"Causalean.Stat.IIDSample.integral_rescaled_uStatisticOrder_eq_sqrt_mul_uMean","k":"theorem","m":"Causalean.Stat.UStatistic.OrderM.Variance","a":"Stat/UStatistic/OrderM","t":2,"d":"The rescaled fixed-order U-statistic has mean equal to the same rescaling of the product-law kernel mean.","s":"∀ {Ω : Type u_1} {X : Type u_2} [inst : MeasurableSpace Ω] [inst_1 : MeasurableSpace X] {μ : MeasureTheory.Measure Ω} {P : MeasureTheory.Measure X} (S : Causalean.Stat.IIDSample Ω X μ P) {m n : ℕ} {h : (Fin m → X) → ℝ}, Measurable h → MeasureTheory.Integrable h (MeasureTheory.Measure.pi fun x => P) → m ≤ n → ∫ (ω : Ω), √↑n * Causalean.Stat.uStatisticOrder S h n ω ∂μ = √↑n * Causalean.Stat.uMeanOrder h P","l":273,"f":"Causalean/Stat/UStatistic/OrderM/Variance.lean"},{"n":"Causalean.Stat.IIDSample.integral_rescaled_uStatisticOrder_eq_zero_of_degenKernel","k":"theorem","m":"Causalean.Stat.UStatistic.OrderM.Variance","a":"Stat/UStatistic/OrderM","t":2,"d":"A fully degenerate order-m kernel gives a mean-zero rescaled fixed-order U-statistic in the nonempty sampling regime m ≤ n. This is a mean statement only; it does not assert the variance bound or negligibility.","s":"∀ {Ω : Type u_1} {X : Type u_2} [inst : MeasurableSpace Ω] [inst_1 : MeasurableSpace X] {μ : MeasureTheory.Measure Ω} {P : MeasureTheory.Measure X} [MeasureTheory.IsProbabilityMeasure μ] {m n : ℕ} {g : (Fin m → X) → ℝ} (S : Causalean.Stat.IIDSample Ω X μ P) [inst_3 : NeZero m], Causalean.Stat.OrderDegenKernel P g → m ≤ n → ∫ (ω : Ω), √↑n * Causalean.Stat.uStatisticOrder S g n ω ∂μ = 0","l":348,"f":"Causalean/Stat/UStatistic/OrderM/Variance.lean"},{"n":"Causalean.Stat.IIDSample.integral_rescaled_uStatisticOrder_eq_zero_of_degenKernel_uMean_zero","k":"theorem","m":"Causalean.Stat.UStatistic.OrderM.Variance","a":"Stat/UStatistic/OrderM","t":2,"d":"A fully degenerate order-m kernel whose product-law mean is zero gives a mean-zero rescaled fixed-order U-statistic. This is a mean statement only; it does not assert the variance bound or negligibility.","s":"∀ {Ω : Type u_1} {X : Type u_2} [inst : MeasurableSpace Ω] [inst_1 : MeasurableSpace X] {μ : MeasureTheory.Measure Ω} {P : MeasureTheory.Measure X} {m n : ℕ} {g : (Fin m → X) → ℝ} (S : Causalean.Stat.IIDSample Ω X μ P) [MeasureTheory.IsFiniteMeasure P] [inst_3 : NeZero m], Causalean.Stat.OrderDegenKernel P g → m ≤ n → Causalean.Stat.uMeanOrder g P = 0 → ∫ (ω : Ω), √↑n * Causalean.Stat.uStatisticOrder S g n ω ∂μ = 0","l":336,"f":"Causalean/Stat/UStatistic/OrderM/Variance.lean"},{"n":"Causalean.Stat.IIDSample.integral_rescaled_uStatisticOrder_eq_zero_of_uMean_zero","k":"theorem","m":"Causalean.Stat.UStatistic.OrderM.Variance","a":"Stat/UStatistic/OrderM","t":2,"d":"If an order-m kernel has product-law mean zero, then the rescaled fixed-order U-statistic has mean zero.","s":"∀ {Ω : Type u_1} {X : Type u_2} [inst : MeasurableSpace Ω] [inst_1 : MeasurableSpace X] {μ : MeasureTheory.Measure Ω} {P : MeasureTheory.Measure X} (S : Causalean.Stat.IIDSample Ω X μ P) {m n : ℕ} {h : (Fin m → X) → ℝ}, Measurable h → MeasureTheory.Integrable h (MeasureTheory.Measure.pi fun x => P) → m ≤ n → Causalean.Stat.uMeanOrder h P = 0 → ∫ (ω : Ω), √↑n * Causalean.Stat.uStatisticOrder S h n ω ∂μ = 0","l":285,"f":"Causalean/Stat/UStatistic/OrderM/Variance.lean"},{"n":"Causalean.Stat.IIDSample.integral_uStatisticOrder_eq_uMean","k":"theorem","m":"Causalean.Stat.UStatistic.OrderM.Variance","a":"Stat/UStatistic/OrderM","t":2,"d":"The fixed-order U-statistic is unbiased: its expectation is the product-law kernel mean.","s":"∀ {Ω : Type u_1} {X : Type u_2} [inst : MeasurableSpace Ω] [inst_1 : MeasurableSpace X] {μ : MeasureTheory.Measure Ω} {P : MeasureTheory.Measure X} (S : Causalean.Stat.IIDSample Ω X μ P) {m n : ℕ} {h : (Fin m → X) → ℝ}, Measurable h → MeasureTheory.Integrable h (MeasureTheory.Measure.pi fun x => P) → m ≤ n → ∫ (ω : Ω), Causalean.Stat.uStatisticOrder S h n ω ∂μ = Causalean.Stat.uMeanOrder h P","l":229,"f":"Causalean/Stat/UStatistic/OrderM/Variance.lean"},{"n":"Causalean.Stat.IIDSample.integral_uStatisticOrder_eq_zero_of_degenKernel","k":"theorem","m":"Causalean.Stat.UStatistic.OrderM.Variance","a":"Stat/UStatistic/OrderM","t":2,"d":"A fully degenerate order-m kernel gives a mean-zero fixed-order U-statistic in the nonempty sampling regime m ≤ n.","s":"∀ {Ω : Type u_1} {X : Type u_2} [inst : MeasurableSpace Ω] [inst_1 : MeasurableSpace X] {μ : MeasureTheory.Measure Ω} {P : MeasureTheory.Measure X} [MeasureTheory.IsProbabilityMeasure μ] {m n : ℕ} {g : (Fin m → X) → ℝ} (S : Causalean.Stat.IIDSample Ω X μ P) [inst_3 : NeZero m], Causalean.Stat.OrderDegenKernel P g → m ≤ n → ∫ (ω : Ω), Causalean.Stat.uStatisticOrder S g n ω ∂μ = 0","l":308,"f":"Causalean/Stat/UStatistic/OrderM/Variance.lean"},{"n":"Causalean.Stat.IIDSample.integral_uStatisticOrder_eq_zero_of_degenKernel_uMean_zero","k":"theorem","m":"Causalean.Stat.UStatistic.OrderM.Variance","a":"Stat/UStatistic/OrderM","t":2,"d":"A fully degenerate order-m kernel whose product-law mean is zero gives a mean-zero fixed-order U-statistic. The product-law mean-zero assumption is kept explicit here rather than inferred from coordinatewise degeneracy.","s":"∀ {Ω : Type u_1} {X : Type u_2} [inst : MeasurableSpace Ω] [inst_1 : MeasurableSpace X] {μ : MeasureTheory.Measure Ω} {P : MeasureTheory.Measure X} {m n : ℕ} {g : (Fin m → X) → ℝ} (S : Causalean.Stat.IIDSample Ω X μ P) [MeasureTheory.IsFiniteMeasure P] [inst_3 : NeZero m], Causalean.Stat.OrderDegenKernel P g → m ≤ n → Causalean.Stat.uMeanOrder g P = 0 → ∫ (ω : Ω), Causalean.Stat.uStatisticOrder S g n ω ∂μ = 0","l":298,"f":"Causalean/Stat/UStatistic/OrderM/Variance.lean"},{"n":"Causalean.Stat.IIDSample.integral_uStatisticOrder_eq_zero_of_uMean_zero","k":"theorem","m":"Causalean.Stat.UStatistic.OrderM.Variance","a":"Stat/UStatistic/OrderM","t":2,"d":"A fixed-order U-statistic with product-law mean zero has expectation zero.","s":"∀ {Ω : Type u_1} {X : Type u_2} [inst : MeasurableSpace Ω] [inst_1 : MeasurableSpace X] {μ : MeasureTheory.Measure Ω} {P : MeasureTheory.Measure X} (S : Causalean.Stat.IIDSample Ω X μ P) {m n : ℕ} {h : (Fin m → X) → ℝ}, Measurable h → MeasureTheory.Integrable h (MeasureTheory.Measure.pi fun x => P) → m ≤ n → Causalean.Stat.uMeanOrder h P = 0 → ∫ (ω : Ω), Causalean.Stat.uStatisticOrder S h n ω ∂μ = 0","l":262,"f":"Causalean/Stat/UStatistic/OrderM/Variance.lean"},{"n":"Causalean.Stat.IIDSample.map_eq","k":"theorem","m":"Causalean.Stat.Sample","a":"Stat/Sample","t":1,"d":"For an i.i.d. sample S and any sample index i, the pushforward law of the i-th sample point equals the population law P.","s":"∀ {Ω : Type u_1} {X : Type u_2} [inst : MeasurableSpace Ω] [inst_1 : MeasurableSpace X] {μ : MeasureTheory.Measure Ω} {P : MeasureTheory.Measure X} (S : Causalean.Stat.IIDSample Ω X μ P) (i : ℕ), MeasureTheory.Measure.map (S.Z i) μ = P","l":63,"f":"Causalean/Stat/Sample.lean"},{"n":"Causalean.Stat.IIDSample.map_fintype_tuple_eq","k":"theorem","m":"Causalean.Stat.UStatistic.OrderM.Variance","a":"Stat/UStatistic/OrderM","t":2,"d":"The joint law of any finite collection of distinct sample coordinates is the corresponding product law.","s":"∀ {Ω : Type u_1} {X : Type u_2} [inst : MeasurableSpace Ω] [inst_1 : MeasurableSpace X] {μ : MeasureTheory.Measure Ω} {P : MeasureTheory.Measure X} (S : Causalean.Stat.IIDSample Ω X μ P) {ι : Type u_3} [inst_2 : Fintype ι] {n : ℕ} {r : ι → Fin n}, Function.Injective r → MeasureTheory.Measure.map (fun ω i => S.Z (↑(r i)) ω) μ = MeasureTheory.Measure.pi fun x => P","l":36,"f":"Causalean/Stat/UStatistic/OrderM/Variance.lean"},{"n":"Causalean.Stat.IIDSample.map_tuple_eq","k":"theorem","m":"Causalean.Stat.UStatistic.OrderM.Variance","a":"Stat/UStatistic/OrderM","t":1,"d":"For an i.i.d. sample S, sample size n, and order m, if the index map t : Fin m → Fin n is injective, then the joint law of the sample coordinates selected by t is the m-fold product measure P^m.","s":"∀ {Ω : Type u_1} {X : Type u_2} [inst : MeasurableSpace Ω] [inst_1 : MeasurableSpace X] {μ : MeasureTheory.Measure Ω} {P : MeasureTheory.Measure X} (S : Causalean.Stat.IIDSample Ω X μ P) {m n : ℕ} {t : Fin m → Fin n}, Function.Injective t → MeasureTheory.Measure.map (fun ω j => S.Z (↑(t j)) ω) μ = MeasureTheory.Measure.pi fun x => P","l":62,"f":"Causalean/Stat/UStatistic/OrderM/Variance.lean"},{"n":"Causalean.Stat.IIDSample.measurable_empProcVec","k":"theorem","m":"Causalean.Stat.EmpiricalProcess.Equicontinuity.SecondMoment","a":"Stat/EmpiricalProcess/Equicontinuity","t":2,"d":"The vector empirical process is measurable in ω.","s":"∀ {Ω : Type u_1} {X : Type u_2} {E : Type u_3} [inst : MeasurableSpace Ω] [inst_1 : MeasurableSpace X] [inst_2 : NormedAddCommGroup E] [inst_3 : InnerProductSpace ℝ E] [FiniteDimensional ℝ E] {μ : MeasureTheory.Measure Ω} {P : MeasureTheory.Measure X} [inst_5 : MeasurableSpace E] [BorelSpace E] (S : Causalean.Stat.IIDSample Ω X μ P) (f : X → E), Measurable f → ∀ (n : ℕ), Measurable fun ω => S.empProcVec f n ω","l":95,"f":"Causalean/Stat/EmpiricalProcess/Equicontinuity/SecondMoment.lean"},{"n":"Causalean.Stat.IIDSample.measurable_empProcess","k":"theorem","m":"Causalean.Stat.Quantile.SampleQuantileBahadur.Rate","a":"Stat/Quantile/SampleQuantileBahadur","t":2,"d":"Measurability of empProcess(·,y) (a constant times a measurable sample mean minus a constant).","s":"∀ {Ω : Type u_1} [inst : MeasurableSpace Ω] {μ : MeasureTheory.Measure Ω} {P : MeasureTheory.Measure ℝ} (S : Causalean.Stat.IIDSample Ω ℝ μ P) (n : ℕ) (y : ℝ), Measurable fun ω => S.empProcess n ω y","l":54,"f":"Causalean/Stat/Quantile/SampleQuantileBahadur/Rate.lean"},{"n":"Causalean.Stat.IIDSample.measurable_sampleMean","k":"theorem","m":"Causalean.Stat.Limit.WLLN","a":"Stat/Limit","t":2,"d":"S.sampleMean g N is measurable for measurable g: it is a finite sum of g ∘ Z i scaled by a constant.","s":"∀ {Ω : Type u_1} {X : Type u_2} [inst : MeasurableSpace Ω] [inst_1 : MeasurableSpace X] {μ : MeasureTheory.Measure Ω} {P : MeasureTheory.Measure X} (S : Causalean.Stat.IIDSample Ω X μ P) {g : X → ℝ}, Measurable g → ∀ (N : ℕ), Measurable (S.sampleMean g N)","l":43,"f":"Causalean/Stat/Limit/WLLN.lean"},{"n":"Causalean.Stat.IIDSample.measurable_sampleVariance","k":"theorem","m":"Causalean.Stat.Concentration.TailBounds.EmpiricalBernstein","a":"Stat/Concentration/TailBounds","t":2,"d":"The sample variance is a measurable function of ω.","s":"∀ {Ω : Type u_1} {X : Type u_2} [inst : MeasurableSpace Ω] [inst_1 : MeasurableSpace X] {μ : MeasureTheory.Measure Ω} {P : MeasureTheory.Measure X} {f : X → ℝ} (S : Causalean.Stat.IIDSample Ω X μ P), Measurable f → ∀ (n : ℕ), Measurable (S.sampleVariance f n)","l":100,"f":"Causalean/Stat/Concentration/TailBounds/EmpiricalBernstein.lean"},{"n":"Causalean.Stat.IIDSample.memLp_injectiveTuples_sum","k":"theorem","m":"Causalean.Stat.UStatistic.OrderM.Variance","a":"Stat/UStatistic/OrderM","t":2,"d":"The injective-tuple sum of an order-m kernel is in L².","s":"∀ {Ω : Type u_1} {X : Type u_2} [inst : MeasurableSpace Ω] [inst_1 : MeasurableSpace X] {μ : MeasureTheory.Measure Ω} {P : MeasureTheory.Measure X} {m : ℕ} {g : (Fin m → X) → ℝ} (S : Causalean.Stat.IIDSample Ω X μ P), Measurable g → MeasureTheory.Integrable (fun z => g z ^ 2) (MeasureTheory.Measure.pi fun x => P) → ∀ (n : ℕ), MeasureTheory.MemLp (fun ω => ∑ t ∈ Causalean.Stat.injectiveTuples m n, g fun j => S.Z (↑(t j)) ω) 2 μ","l":439,"f":"Causalean/Stat/UStatistic/OrderM/Variance.lean"},{"n":"Causalean.Stat.IIDSample.memLp_orderTerm","k":"theorem","m":"Causalean.Stat.UStatistic.OrderM.Variance","a":"Stat/UStatistic/OrderM","t":2,"d":"Each injective order-m kernel term is in L².","s":"∀ {Ω : Type u_1} {X : Type u_2} [inst : MeasurableSpace Ω] [inst_1 : MeasurableSpace X] {μ : MeasureTheory.Measure Ω} {P : MeasureTheory.Measure X} {m n : ℕ} {g : (Fin m → X) → ℝ} (S : Causalean.Stat.IIDSample Ω X μ P), Measurable g → MeasureTheory.Integrable (fun z => g z ^ 2) (MeasureTheory.Measure.pi fun x => P) → ∀ {t : Fin m → Fin n}, Function.Injective t → MeasureTheory.MemLp (fun ω => g fun j => S.Z (↑(t j)) ω) 2 μ","l":407,"f":"Causalean/Stat/UStatistic/OrderM/Variance.lean"},{"n":"Causalean.Stat.IIDSample.memLp_rescaled","k":"theorem","m":"Causalean.Stat.UStatistic.Variance","a":"Stat/UStatistic","t":2,"d":"The rescaled degenerate U-statistic is in L².","s":"∀ {Ω : Type u_1} {X : Type u_2} [inst : MeasurableSpace Ω] [inst_1 : MeasurableSpace X] {μ : MeasureTheory.Measure Ω} {P : MeasureTheory.Measure X} {g : X → X → ℝ} (S : Causalean.Stat.IIDSample Ω X μ P), Causalean.Stat.DegenKernel P g → ∀ (n : ℕ), MeasureTheory.MemLp (fun ω => √↑n * Causalean.Stat.uStatistic S g n ω) 2 μ","l":285,"f":"Causalean/Stat/UStatistic/Variance.lean"},{"n":"Causalean.Stat.IIDSample.memLp_rescaled_order","k":"theorem","m":"Causalean.Stat.UStatistic.OrderM.Variance","a":"Stat/UStatistic/OrderM","t":1,"d":"For an i.i.d. sample S, order-m kernel g that is measurable and square-integrable under the m-fold product law, and sample size n, the √n-rescaled order-m U-statistic of g is square-integrable.","s":"∀ {Ω : Type u_1} {X : Type u_2} [inst : MeasurableSpace Ω] [inst_1 : MeasurableSpace X] {μ : MeasureTheory.Measure Ω} {P : MeasureTheory.Measure X} {m : ℕ} {g : (Fin m → X) → ℝ} (S : Causalean.Stat.IIDSample Ω X μ P), Measurable g → MeasureTheory.Integrable (fun z => g z ^ 2) (MeasureTheory.Measure.pi fun x => P) → ∀ (n : ℕ), MeasureTheory.MemLp (fun ω => √↑n * Causalean.Stat.uStatisticOrder S g n ω) 2 μ","l":451,"f":"Causalean/Stat/UStatistic/OrderM/Variance.lean"},{"n":"Causalean.Stat.IIDSample.normalizedSum_quantileIF_eq","k":"theorem","m":"Causalean.Stat.Quantile.SampleQuantileBahadur.Linearity","a":"Stat/Quantile/SampleQuantileBahadur","t":2,"d":"The normalized influence-function sum equals −Gₙ(q₀)/f₀:","s":"∀ {Ω : Type u_1} [inst : MeasurableSpace Ω] {μ : MeasureTheory.Measure Ω} {P : MeasureTheory.Measure ℝ} (S : Causalean.Stat.IIDSample Ω ℝ μ P) {τ q₀ f₀ : ℝ}, f₀ ≠ 0 → ↑(ProbabilityTheory.cdf P) q₀ = τ → ∀ (n : ℕ) (ω : Ω), (√↑n)⁻¹ * ∑ i ∈ Finset.range n, Causalean.Stat.quantileIF τ q₀ f₀ (S.Z i ω) = -S.empProcess n ω q₀ / f₀","l":328,"f":"Causalean/Stat/Quantile/SampleQuantileBahadur/Linearity.lean"},{"n":"Causalean.Stat.IIDSample.normalizedSum_vec_charFun_tendsto","k":"theorem","m":"Causalean.Stat.CLT.MultivariateCLT","a":"Stat/CLT","t":2,"d":"Per-direction charFun limit (full-vector form). Convenience wrapper of normalizedSum_vec_charFun_tendsto_of_proj_integrable for the common case where the whole vector ψ is square-integrable: the projection ⟪t,ψ⟫ is then square-integrable by Cauchy–Schwarz, so callers holding the standard full-vector L² condition need not re-establish the per-direction one.","s":"∀ {Ω : Type u_1} {X : Type u_2} [inst : MeasurableSpace Ω] [inst_1 : MeasurableSpace X] {μ : MeasureTheory.Measure Ω} {P : MeasureTheory.Measure X} {E : Type u_3} [inst_2 : NormedAddCommGroup E] [inst_3 : InnerProductSpace ℝ E] [FiniteDimensional ℝ E] [inst_5 : MeasurableSpace E] [BorelSpace E] (S : Causalean.Stat.IIDSample Ω X μ P) {ψ : X → E}, Measurable ψ → MeasureTheory.Integrable (fun x => ‖ψ x‖ ^ 2) P → ∀ (t : E), ∫ (x : X), inner ℝ t (ψ x) ∂P = 0 → Filter.Tendsto (fun n => MeasureTheory.charFun (MeasureTheory.Measure.map (Causalean.Stat.IsAsymLinearVec.normalizedSum S ψ Causalean.Stat.rng✝ n) μ) t) Filter.atTop (nhds (Complex.exp (-↑(∫ (x : X), inner ℝ t (ψ x) ^ 2 ∂P) / 2)))","l":187,"f":"Causalean/Stat/CLT/MultivariateCLT.lean"},{"n":"Causalean.Stat.IIDSample.normalizedSum_vec_charFun_tendsto_of_proj_integrable","k":"theorem","m":"Causalean.Stat.CLT.MultivariateCLT","a":"Stat/CLT","t":2,"d":"Per-direction charFun limit (general core). Assuming only that the scalar projection ⟪t,ψ⟫ in the chosen direction t is square-integrable (weaker than requiring the whole vector ψ to be square-integrable) and mean-zero, the characteristic function of the vector normalised sum, evaluated at t, converges to exp(−½ ∫⟪t,ψ⟫² dP). The full-vector convenience form is normalizedSum_vec_charFun_tendsto below.","s":"∀ {Ω : Type u_1} {X : Type u_2} [inst : MeasurableSpace Ω] [inst_1 : MeasurableSpace X] {μ : MeasureTheory.Measure Ω} {P : MeasureTheory.Measure X} {E : Type u_3} [inst_2 : NormedAddCommGroup E] [inst_3 : InnerProductSpace ℝ E] [FiniteDimensional ℝ E] [inst_5 : MeasurableSpace E] [BorelSpace E] (S : Causalean.Stat.IIDSample Ω X μ P) {ψ : X → E}, Measurable ψ → ∀ (t : E), MeasureTheory.Integrable (fun x => inner ℝ t (ψ x) ^ 2) P → ∫ (x : X), inner ℝ t (ψ x) ∂P = 0 → Filter.Tendsto (fun n => MeasureTheory.charFun (MeasureTheory.Measure.map (Causalean.Stat.IsAsymLinearVec.normalizedSum S ψ Causalean.Stat.rng✝ n) μ) t) Filter.atTop (nhds (Complex.exp (-↑(∫ (x : X), inner ℝ t (ψ x) ^ 2 ∂P) / 2)))","l":102,"f":"Causalean/Stat/CLT/MultivariateCLT.lean"},{"n":"Causalean.Stat.IIDSample.orderDegenerateNegligible_of_firstDegen","k":"theorem","m":"Causalean.Stat.UStatistic.OrderM.RemainderNegligible","a":"Stat/UStatistic/OrderM","t":1,"d":"Negligibility of the higher-order remainder. For an i.i.d. sample S, if the order-m kernel g is first-order degenerate: measurable, square-integrable, and mean zero after integrating out any single coordinate, then the √n-rescaled order-m U-statistic of g converges to zero in probability, i.e. it is o_p(1).","s":"∀ {Ω : Type u_1} {X : Type u_2} [inst : MeasurableSpace Ω] [inst_1 : MeasurableSpace X] {μ : MeasureTheory.Measure Ω} {P : MeasureTheory.Measure X} {m : ℕ} [inst_2 : NeZero m] {g : (Fin m → X) → ℝ} (S : Causalean.Stat.IIDSample Ω X μ P) [MeasureTheory.IsFiniteMeasure P], Causalean.Stat.OrderFirstDegenKernel P g → Causalean.Stat.IsLittleOp (fun n ω => √↑n * Causalean.Stat.uStatisticOrder S g n ω) (fun x => 1) μ","l":66,"f":"Causalean/Stat/UStatistic/OrderM/RemainderNegligible.lean"},{"n":"Causalean.Stat.IIDSample.orderTerm_diag","k":"theorem","m":"Causalean.Stat.UStatistic.OrderM.Variance","a":"Stat/UStatistic/OrderM","t":1,"d":"For an i.i.d. sample S and order-m kernel g that is measurable, if the index map t is injective, then the second moment of the kernel term evaluated along the sample coordinates selected by t equals the kernel's second moment ζ_m under the m-fold product law.","s":"∀ {Ω : Type u_1} {X : Type u_2} [inst : MeasurableSpace Ω] [inst_1 : MeasurableSpace X] {μ : MeasureTheory.Measure Ω} {P : MeasureTheory.Measure X} {m n : ℕ} {g : (Fin m → X) → ℝ} (S : Causalean.Stat.IIDSample Ω X μ P), Measurable g → ∀ {t : Fin m → Fin n}, Function.Injective t → ∫ (ω : Ω), (g fun j => S.Z (↑(t j)) ω) ^ 2 ∂μ = Causalean.Stat.IIDSample.zetaOrder P g","l":393,"f":"Causalean/Stat/UStatistic/OrderM/Variance.lean"},{"n":"Causalean.Stat.IIDSample.permOfImageEq","k":"def","m":"Causalean.Stat.UStatistic.OrderM.ExactVariance","a":"Stat/UStatistic/OrderM","t":1,"d":"Two injective maps from a finite index set with the same image determine a permutation of that index set which reorders the first map into the second.","s":"{m n : ℕ} → {t q : Fin m → Fin n} → Function.Injective t → Function.Injective q → Finset.image t Finset.univ = Finset.image q Finset.univ → Equiv.Perm (Fin m)","l":69,"f":"Causalean/Stat/UStatistic/OrderM/ExactVariance.lean"},{"n":"Causalean.Stat.IIDSample.rescaled_order_normalization_le","k":"theorem","m":"Causalean.Stat.UStatistic.OrderM.RemainderSecondMoment","a":"Stat/UStatistic/OrderM","t":2,"d":"For a nonnegative second-moment bound and a sample size at least the kernel order, the falling-factorial normalization term is bounded by a constant divided by the sample size.","s":"∀ {m : ℕ} [NeZero m] {n : ℕ}, m ≤ n → ∀ {ζ : ℝ}, 0 ≤ ζ → ↑n * (Causalean.Stat.injectiveTupleCount m n)⁻¹ ^ 2 * (↑(m.choose 2) * ↑m ^ 2 * ↑n ^ (2 * m - 2) * ζ) ≤ ↑(m.choose 2) * ↑m ^ (2 * m + 2) * ζ / ↑n","l":604,"f":"Causalean/Stat/UStatistic/OrderM/RemainderSecondMoment.lean"},{"n":"Causalean.Stat.IIDSample.sampleCov_entry_tendsto_inProb","k":"theorem","m":"Causalean.Stat.Inference.VarianceEstimation","a":"Stat/Inference","t":1,"d":"Entrywise covariance-matrix consistency. For an i.i.d. sample S and two continuous linear coordinate functionals φ φ' : E →L[ℝ] ℝ, suppose a vector influence function ψ : X → E is measurable and has square-integrable norm along the sample. Then the empirical mean of the entry product φ(ψ·) · φ'(ψ·) converges in probability to the population integral ∫ x, φ(ψ x) · φ'(ψ x) ∂P.","s":"∀ {Ω : Type u_1} {X : Type u_2} [inst : MeasurableSpace Ω] [inst_1 : MeasurableSpace X] {μ : MeasureTheory.Measure Ω} {P : MeasureTheory.Measure X} {E : Type u_3} [inst_2 : NormedAddCommGroup E] [inst_3 : InnerProductSpace ℝ E] [inst_4 : MeasurableSpace E] [BorelSpace E] (S : Causalean.Stat.IIDSample Ω X μ P) [MeasureTheory.IsProbabilityMeasure P] {ψ : X → E}, Measurable ψ → MeasureTheory.Integrable (fun ω => ‖ψ (S.Z 0 ω)‖ ^ 2) μ → ∀ (φ φ' : E →L[ℝ] ℝ), Causalean.Stat.Tendsto_inProb (S.sampleMean fun x => φ (ψ x) * φ' (ψ x)) (fun x => ∫ (x : X), φ (ψ x) * φ' (ψ x) ∂P) μ","l":126,"f":"Causalean/Stat/Inference/VarianceEstimation.lean"},{"n":"Causalean.Stat.IIDSample.sampleMean","k":"def","m":"Causalean.Stat.Sample","a":"Stat/Sample","t":1,"d":"Sample mean of a real-valued statistic f along the sample's first n points: (1/n) Σ_{i < n} f (Z i ω).","s":"{Ω : Type u_1} → {X : Type u_2} → [inst : MeasurableSpace Ω] → [inst_1 : MeasurableSpace X] → {μ : MeasureTheory.Measure Ω} → {P : MeasureTheory.Measure X} → Causalean.Stat.IIDSample Ω X μ P → (X → ℝ) → ℕ → Ω → ℝ","l":57,"f":"Causalean/Stat/Sample.lean"},{"n":"Causalean.Stat.IIDSample.sampleMean_mul_tendsto_inProb","k":"theorem","m":"Causalean.Stat.Inference.VarianceEstimation","a":"Stat/Inference","t":2,"d":"Empirical mean of a product. For two measurable real-valued statistics g₁, g₂ of an i.i.d. sample whose product is integrable, the empirical mean S.sampleMean (g₁ · g₂) N converges in probability to the population integral ∫ x, g₁ x * g₂ x ∂P. Direct application of the generic WLLN to the product g := fun x => g₁ x * g₂ x.","s":"∀ {Ω : Type u_1} {X : Type u_2} [inst : MeasurableSpace Ω] [inst_1 : MeasurableSpace X] {μ : MeasureTheory.Measure Ω} {P : MeasureTheory.Measure X} (S : Causalean.Stat.IIDSample Ω X μ P) [MeasureTheory.IsProbabilityMeasure P] {g₁ g₂ : X → ℝ}, Measurable g₁ → Measurable g₂ → MeasureTheory.Integrable (fun ω => g₁ (S.Z 0 ω) * g₂ (S.Z 0 ω)) μ → Causalean.Stat.Tendsto_inProb (S.sampleMean fun x => g₁ x * g₂ x) (fun x => ∫ (x : X), g₁ x * g₂ x ∂P) μ","l":62,"f":"Causalean/Stat/Inference/VarianceEstimation.lean"},{"n":"Causalean.Stat.IIDSample.sampleMean_sub_isBigOp","k":"theorem","m":"Causalean.Stat.EmpiricalProcess.CrossFitRate","a":"Stat/EmpiricalProcess","t":1,"d":"Unconditional O_p rate (Lemma A). For an i.i.d. sample S and a statistic f that is measurable and square-integrable under the sampling distribution P, the sample mean over the first n observations, centered at the population mean $\\int f\\,dP$, is stochastically bounded at the rate $\\sqrt{E_P[f^2]/n}$: it is $O_p(n^{-1/2}(E_P[f^2])^{1/2})$:","s":"∀ {Ω : Type u_1} [inst : MeasurableSpace Ω] {μ : MeasureTheory.Measure Ω} {X : Type u_2} [inst_1 : MeasurableSpace X] {P : MeasureTheory.Measure X} (S : Causalean.Stat.IIDSample Ω X μ P) [MeasureTheory.IsProbabilityMeasure μ] [MeasureTheory.IsProbabilityMeasure P] {f : X → ℝ}, Measurable f → MeasureTheory.MemLp f 2 P → Causalean.Stat.IsBigOp (fun n ω => S.sampleMean f n ω - ∫ (x : X), f x ∂P) (fun n => √((∫ (x : X), f x ^ 2 ∂P) / ↑n)) μ","l":702,"f":"Causalean/Stat/EmpiricalProcess/CrossFitRate.lean"},{"n":"Causalean.Stat.IIDSample.sampleMean_sub_meas_ge_le","k":"theorem","m":"Causalean.Stat.EmpiricalProcess.CrossFitRate","a":"Stat/EmpiricalProcess","t":2,"d":"Chebyshev tail for the centered sample mean. For t > 0,","s":"∀ {Ω : Type u_1} [inst : MeasurableSpace Ω] {μ : MeasureTheory.Measure Ω} {X : Type u_2} [inst_1 : MeasurableSpace X] {P : MeasureTheory.Measure X} (S : Causalean.Stat.IIDSample Ω X μ P) [MeasureTheory.IsProbabilityMeasure μ] [MeasureTheory.IsProbabilityMeasure P] {f : X → ℝ}, Measurable f → MeasureTheory.MemLp f 2 P → ∀ {n : ℕ}, 0 < n → ∀ {t : ℝ}, 0 < t → μ {ω | t ≤ |S.sampleMean f n ω - ∫ (x : X), f x ∂P|} ≤ ENNReal.ofReal ((∫ (x : X), f x ^ 2 ∂P) / (↑n * t ^ 2))","l":652,"f":"Causalean/Stat/EmpiricalProcess/CrossFitRate.lean"},{"n":"Causalean.Stat.IIDSample.sampleMean_sub_sq_lintegral_le","k":"theorem","m":"Causalean.Stat.EmpiricalProcess.CrossFitRate","a":"Stat/EmpiricalProcess","t":2,"d":"Centered sample-mean second moment. For an i.i.d. sample and a square-integrable statistic f, the centered sample mean over the first n points has second moment bounded by E_P[f²]/n:","s":"∀ {Ω : Type u_1} [inst : MeasurableSpace Ω] {μ : MeasureTheory.Measure Ω} {X : Type u_2} [inst_1 : MeasurableSpace X] {P : MeasureTheory.Measure X} (S : Causalean.Stat.IIDSample Ω X μ P) [MeasureTheory.IsProbabilityMeasure μ] [MeasureTheory.IsProbabilityMeasure P] {f : X → ℝ}, Measurable f → MeasureTheory.MemLp f 2 P → ∀ {n : ℕ}, 0 < n → ∫⁻ (ω : Ω), ENNReal.ofReal ((S.sampleMean f n ω - ∫ (x : X), f x ∂P) ^ 2) ∂μ ≤ ENNReal.ofReal ((∫ (x : X), f x ^ 2 ∂P) / ↑n)","l":524,"f":"Causalean/Stat/EmpiricalProcess/CrossFitRate.lean"},{"n":"Causalean.Stat.IIDSample.sampleMean_tendsto_inProb","k":"theorem","m":"Causalean.Stat.Limit.WLLN","a":"Stat/Limit","t":1,"d":"Generic weak law of large numbers. For an i.i.d. sample S, suppose a real-valued statistic g is measurable and integrable under the population law P. Then the sample mean S.sampleMean g N converges in probability to the population integral ∫ x, g x ∂P as N → ∞.","s":"∀ {Ω : Type u_1} {X : Type u_2} [inst : MeasurableSpace Ω] [inst_1 : MeasurableSpace X] {μ : MeasureTheory.Measure Ω} {P : MeasureTheory.Measure X} (S : Causalean.Stat.IIDSample Ω X μ P) {g : X → ℝ}, Measurable g → MeasureTheory.Integrable g P → Causalean.Stat.Tendsto_inProb (S.sampleMean g) (fun x => ∫ (x : X), g x ∂P) μ","l":51,"f":"Causalean/Stat/Limit/WLLN.lean"},{"n":"Causalean.Stat.IIDSample.sampleQuantile","k":"def","m":"Causalean.Stat.Quantile.EmpiricalQuantile","a":"Stat/Quantile","t":1,"d":"The sample τ-quantile q̂ₙ(τ) = quantile νₙ τ, the generalized inverse of the empirical cdf.","s":"{Ω : Type u_1} → [inst : MeasurableSpace Ω] → {μ : MeasureTheory.Measure Ω} → {P : MeasureTheory.Measure ℝ} → Causalean.Stat.IIDSample Ω ℝ μ P → ℝ → ℕ → Ω → ℝ","l":106,"f":"Causalean/Stat/Quantile/EmpiricalQuantile.lean"},{"n":"Causalean.Stat.IIDSample.sampleQuantileVec","k":"def","m":"Causalean.Stat.Quantile.SampleQuantileJoint","a":"Stat/Quantile","t":1,"d":"The sample-quantile vector (q̂ₙ(τ₁), …, q̂ₙ(τ_k)).","s":"{Ω : Type u_1} → [inst : MeasurableSpace Ω] → {μ : MeasureTheory.Measure Ω} → {P : MeasureTheory.Measure ℝ} → {k : ℕ} → Causalean.Stat.IIDSample Ω ℝ μ P → (Fin k → ℝ) → ℕ → Ω → EuclideanSpace ℝ (Fin k)","l":67,"f":"Causalean/Stat/Quantile/SampleQuantileJoint.lean"},{"n":"Causalean.Stat.IIDSample.sampleQuantileVec_isAsymLinearVec","k":"theorem","m":"Causalean.Stat.Quantile.SampleQuantileJoint","a":"Stat/Quantile","t":1,"d":"The sample-quantile vector is asymptotically linear with the joint influence function ψ. Given a SampleQuantileReg regularity bundle at every coordinate j: interior level $\\tau_j$, positive density $f_j$ at the population quantile $q_j$, cdf identification, differentiability of the population cdf, and an atomless population, the vector of sample $\\tau$-quantiles is jointly asymptotically linear at the vector of population quantiles, with influence function the joint quantile influence function $\\psi$.","s":"∀ {Ω : Type u_1} [inst : MeasurableSpace Ω] {μ : MeasureTheory.Measure Ω} {P : MeasureTheory.Measure ℝ} {k : ℕ} [MeasureTheory.IsProbabilityMeasure μ] [MeasureTheory.IsProbabilityMeasure P] (S : Causalean.Stat.IIDSample Ω ℝ μ P) {τ q f : Fin k → ℝ}, (∀ (j : Fin k), Causalean.Stat.SampleQuantileReg P (τ j) (q j) (f j)) → Causalean.Stat.IsAsymLinearVec (S.sampleQuantileVec τ) (Causalean.Stat.eucl q) (Causalean.Stat.quantileIFVec τ q f) S fun m => Finset.range m","l":208,"f":"Causalean/Stat/Quantile/SampleQuantileJoint.lean"},{"n":"Causalean.Stat.IIDSample.sampleQuantileVec_tendsto_normal","k":"theorem","m":"Causalean.Stat.Quantile.SampleQuantileJoint","a":"Stat/Quantile","t":1,"d":"Joint asymptotic normality of the sample-quantile vector. Given a SampleQuantileReg bundle at every coordinate j, a candidate limit measure Q on the joint quantile space whose characteristic function at every direction t matches $\\exp(-\\tfrac12\\int\\langle t,\\psi\\rangle^2\\,dP)$, the Gaussian shape determined by the joint influence function $\\psi$, and almost-everywhere measurability of the rescaled estimator sequence at every sample size, then the law of the rescaled sample-quantile vector $\\sqrt n(\\hat q_n(\\tau_\\bullet)-q_\\bullet)$ converges weakly to Q as $n\\to\\infty$.","s":"∀ {Ω : Type u_1} [inst : MeasurableSpace Ω] {μ : MeasureTheory.Measure Ω} {P : MeasureTheory.Measure ℝ} {k : ℕ} [inst_1 : MeasureTheory.IsProbabilityMeasure μ] [MeasureTheory.IsProbabilityMeasure P] (S : Causalean.Stat.IIDSample Ω ℝ μ P) {τ q f : Fin k → ℝ}, (∀ (j : Fin k), Causalean.Stat.SampleQuantileReg P (τ j) (q j) (f j)) → ∀ (Q : MeasureTheory.Measure (EuclideanSpace ℝ (Fin k))) [inst_3 : MeasureTheory.IsProbabilityMeasure Q], (∀ (t : EuclideanSpace ℝ (Fin k)), MeasureTheory.charFun Q t = Complex.exp (-↑(∫ (z : ℝ), inner ℝ t (Causalean.Stat.quantileIFVec τ q f z) ^ 2 ∂P) / 2)) → ∀ (hθn_meas : ∀ (n : ℕ), AEMeasurable (Causalean.Stat.IsAsymLinearVec.rescaledEstimator (S.sampleQuantileVec τ) (Causalean.Stat.eucl q) (fun m => Finset.range m) n) μ), Filter.Tendsto (fun n => ⟨MeasureTheory.Measure.map (Causalean.Stat.IsAsymLinearVec.rescaledEstimator (S.sampleQuantileVec τ) (Causalean.Stat.eucl q) (fun m => Finset.range m) n) μ, MeasureTheory.Measure.isProbabilityMeasure_map (hθn_meas n)⟩) Filter.atTop (nhds ⟨Q, inst_3⟩)","l":307,"f":"Causalean/Stat/Quantile/SampleQuantileJoint.lean"},{"n":"Causalean.Stat.IIDSample.sampleQuantile_atom_bound","k":"theorem","m":"Causalean.Stat.Quantile.EmpiricalQuantile","a":"Stat/Quantile","t":1,"d":"Atom bound. If the population cdf $F$ is continuous, i.e. the population is atomless, the sample size $n$ is positive, and the quantile level $\\tau$ is interior, $0<\\tau<1$, then almost surely the empirical cdf evaluated at the sample $\\tau$-quantile $\\hat q_n(\\tau)$ deviates from $\\tau$ by at most $1/n$.","s":"∀ {Ω : Type u_1} [inst : MeasurableSpace Ω] {μ : MeasureTheory.Measure Ω} {P : MeasureTheory.Measure ℝ} [MeasureTheory.IsProbabilityMeasure μ] (S : Causalean.Stat.IIDSample Ω ℝ μ P), (Continuous fun y => ↑(ProbabilityTheory.cdf P) y) → ∀ {n : ℕ}, 0 < n → ∀ {τ : ℝ}, 0 < τ → τ < 1 → ∀ᵐ (ω : Ω) ∂μ, |S.empiricalCDF (S.sampleQuantile τ n ω) n ω - τ| ≤ (↑n)⁻¹","l":141,"f":"Causalean/Stat/Quantile/EmpiricalQuantile.lean"},{"n":"Causalean.Stat.IIDSample.sampleQuantile_atom_term_tendsto_zero","k":"theorem","m":"Causalean.Stat.Quantile.SampleQuantileBahadur.Linearity","a":"Stat/Quantile/SampleQuantileBahadur","t":2,"d":"The sample-quantile atom term vanishes. √n (F̂ₙ(q̂ₙ) − τ) →ₚ 0. The switching atom bound |F̂ₙ(q̂ₙ) − τ| ≤ 1/n (a.e.) gives |√n(F̂ₙ(q̂ₙ) − τ)| ≤ 1/√n → 0, so the term converges to 0 even a.e.-deterministically.","s":"∀ {Ω : Type u_1} [inst : MeasurableSpace Ω] {μ : MeasureTheory.Measure Ω} {P : MeasureTheory.Measure ℝ} [MeasureTheory.IsProbabilityMeasure μ] (S : Causalean.Stat.IIDSample Ω ℝ μ P) {τ q₀ f₀ : ℝ}, Causalean.Stat.SampleQuantileReg P τ q₀ f₀ → Causalean.Stat.Tendsto_inProb (fun n ω => √↑n * (S.empiricalCDF (S.sampleQuantile τ n ω) n ω - τ)) (fun x => 0) μ","l":88,"f":"Causalean/Stat/Quantile/SampleQuantileBahadur/Linearity.lean"},{"n":"Causalean.Stat.IIDSample.sampleQuantile_inversion","k":"theorem","m":"Causalean.Stat.Quantile.SampleQuantileBahadur.Linearity","a":"Stat/Quantile/SampleQuantileBahadur","t":1,"d":"Sample-quantile inversion identity. Given a SampleQuantileReg regularity bundle hreg — interior quantile level, positive density $f_0$ at the population quantile $q_0$, cdf identification, differentiability of the population cdf at $q_0$, and an atomless population — the empirical process $G_n$ evaluated at the sample quantile $\\hat q_n(\\tau)$, plus $f_0$ times the rescaled deviation $\\sqrt n(\\hat q_n(\\tau)-q_0)$, converges to zero in probability; equivalently $G_n(\\hat q_n(\\tau)) = -f_0\\sqrt n(\\hat q_n(\\tau)-q_0) + o_p(1)$.","s":"∀ {Ω : Type u_1} [inst : MeasurableSpace Ω] {μ : MeasureTheory.Measure Ω} {P : MeasureTheory.Measure ℝ} [MeasureTheory.IsProbabilityMeasure μ] [MeasureTheory.IsProbabilityMeasure P] (S : Causalean.Stat.IIDSample Ω ℝ μ P) {τ q₀ f₀ : ℝ}, Causalean.Stat.SampleQuantileReg P τ q₀ f₀ → Causalean.Stat.Tendsto_inProb (fun n ω => S.empProcess n ω (S.sampleQuantile τ n ω) + f₀ * (√↑n * (S.sampleQuantile τ n ω - q₀))) (fun x => 0) μ","l":286,"f":"Causalean/Stat/Quantile/SampleQuantileBahadur/Linearity.lean"},{"n":"Causalean.Stat.IIDSample.sampleQuantile_isAsymLinear","k":"theorem","m":"Causalean.Stat.Quantile.SampleQuantileBahadur.Linearity","a":"Stat/Quantile/SampleQuantileBahadur","t":1,"d":"Sample-quantile asymptotic linearity (derived). Given a SampleQuantileReg regularity bundle hreg for the population $\\tau$-quantile $q_0$ with density $f_0$, the sample $\\tau$-quantile $\\hat q_n(\\tau)$ is asymptotically linear at $q_0$ with influence function $\\psi_\\tau$, the Bahadur remainder being proved — not assumed — $o_p(1)$.","s":"∀ {Ω : Type u_1} [inst : MeasurableSpace Ω] {μ : MeasureTheory.Measure Ω} {P : MeasureTheory.Measure ℝ} [MeasureTheory.IsProbabilityMeasure μ] [MeasureTheory.IsProbabilityMeasure P] (S : Causalean.Stat.IIDSample Ω ℝ μ P) {τ q₀ f₀ : ℝ}, Causalean.Stat.SampleQuantileReg P τ q₀ f₀ → Causalean.Stat.IsAsymLinear (S.sampleQuantile τ) q₀ (Causalean.Stat.quantileIF τ q₀ f₀) S fun m => Finset.range m","l":382,"f":"Causalean/Stat/Quantile/SampleQuantileBahadur/Linearity.lean"},{"n":"Causalean.Stat.IIDSample.sampleQuantile_le_iff","k":"theorem","m":"Causalean.Stat.Quantile.EmpiricalQuantile","a":"Stat/Quantile","t":1,"d":"Switching relation. For a positive sample size $n$ and an interior quantile level $\\tau\\in(0,1)$, the sample $\\tau$-quantile $\\hat q_n(\\tau)$ is at most a given point x exactly when $\\tau$ is at most the empirical cdf at x.","s":"∀ {Ω : Type u_1} [inst : MeasurableSpace Ω] {μ : MeasureTheory.Measure Ω} {P : MeasureTheory.Measure ℝ} (S : Causalean.Stat.IIDSample Ω ℝ μ P) {n : ℕ}, 0 < n → ∀ (ω : Ω) {τ : ℝ}, 0 < τ → τ < 1 → ∀ (x : ℝ), S.sampleQuantile τ n ω ≤ x ↔ τ ≤ S.empiricalCDF x n ω","l":112,"f":"Causalean/Stat/Quantile/EmpiricalQuantile.lean"},{"n":"Causalean.Stat.IIDSample.sampleQuantile_quantileRegularity","k":"theorem","m":"Causalean.Stat.Quantile.SampleQuantileBahadur.Linearity","a":"Stat/Quantile/SampleQuantileBahadur","t":1,"d":"Sample quantile satisfies QuantileRegularity (derived Bahadur). Given a SampleQuantileReg regularity bundle hreg for the population $\\tau$-quantile $q_0$ with density $f_0$, the sample $\\tau$-quantile sequence itself satisfies the generic QuantileRegularity bundle for these parameters, with the Bahadur remainder now derived rather than assumed.","s":"∀ {Ω : Type u_1} [inst : MeasurableSpace Ω] {μ : MeasureTheory.Measure Ω} {P : MeasureTheory.Measure ℝ} [MeasureTheory.IsProbabilityMeasure μ] [MeasureTheory.IsProbabilityMeasure P] (S : Causalean.Stat.IIDSample Ω ℝ μ P) {τ q₀ f₀ : ℝ}, Causalean.Stat.SampleQuantileReg P τ q₀ f₀ → Causalean.Stat.QuantileRegularity S (S.sampleQuantile τ) τ q₀ f₀","l":436,"f":"Causalean/Stat/Quantile/SampleQuantileBahadur/Linearity.lean"},{"n":"Causalean.Stat.IIDSample.sampleQuantile_rate","k":"theorem","m":"Causalean.Stat.Quantile.SampleQuantileBahadur.Rate","a":"Stat/Quantile/SampleQuantileBahadur","t":1,"d":"Root-n consistency of the sample quantile. Given a SampleQuantileReg regularity bundle hreg for the population $\\tau$-quantile $q_0$ with density $f_0$, the rescaled deviation $\\sqrt n(\\hat q_n(\\tau)-q_0)$ of the sample quantile from the population quantile is bounded in probability, $O_p(1)$.","s":"∀ {Ω : Type u_1} [inst : MeasurableSpace Ω] {μ : MeasureTheory.Measure Ω} {P : MeasureTheory.Measure ℝ} [MeasureTheory.IsProbabilityMeasure μ] [MeasureTheory.IsProbabilityMeasure P] (S : Causalean.Stat.IIDSample Ω ℝ μ P) {τ q₀ f₀ : ℝ}, Causalean.Stat.SampleQuantileReg P τ q₀ f₀ → Causalean.Stat.IsBigOp (fun n ω => √↑n * (S.sampleQuantile τ n ω - q₀)) (fun x => 1) μ","l":128,"f":"Causalean/Stat/Quantile/SampleQuantileBahadur/Rate.lean"},{"n":"Causalean.Stat.IIDSample.sampleQuantile_taylor_remainder_tendsto_zero","k":"theorem","m":"Causalean.Stat.Quantile.SampleQuantileBahadur.Linearity","a":"Stat/Quantile/SampleQuantileBahadur","t":2,"d":"The Taylor remainder vanishes at the sample quantile. Writing R(y) = F(y) − F(q₀) − f₀(y − q₀) for the first-order Taylor remainder of F = cdf P at q₀, the scaled remainder at the sample quantile vanishes in probability: √n · R(q̂ₙ) →ₚ 0. Mirrors the hRn block of deltaMethod_scalar: the driver Un = √n(q̂ₙ − q₀) is O_p(1), consistency q̂ₙ →ₚ q₀ localizes the derivative little-o |R(y)| ≤ η|y − q₀|, and |√n·R(q̂ₙ)| ≤ η|Un|.","s":"∀ {Ω : Type u_1} [inst : MeasurableSpace Ω] {μ : MeasureTheory.Measure Ω} {P : MeasureTheory.Measure ℝ} [MeasureTheory.IsProbabilityMeasure μ] [MeasureTheory.IsProbabilityMeasure P] (S : Causalean.Stat.IIDSample Ω ℝ μ P) {τ q₀ f₀ : ℝ}, Causalean.Stat.SampleQuantileReg P τ q₀ f₀ → Causalean.Stat.Tendsto_inProb (fun n ω => √↑n * (↑(ProbabilityTheory.cdf P) (S.sampleQuantile τ n ω) - ↑(ProbabilityTheory.cdf P) q₀ - f₀ * (S.sampleQuantile τ n ω - q₀))) (fun x => 0) μ","l":145,"f":"Causalean/Stat/Quantile/SampleQuantileBahadur/Linearity.lean"},{"n":"Causalean.Stat.IIDSample.sampleSecondMoment_tendsto_inProb","k":"theorem","m":"Causalean.Stat.Limit.WLLN","a":"Stat/Limit","t":1,"d":"Second-moment consistency. For an i.i.d. sample S, suppose a known influence function ψ is measurable and has square-integrable values under the population law P. Then the empirical second moment S.sampleMean (ψ²) N converges in probability to ∫ x, (ψ x)² ∂P. Direct application of the WLLN to g := ψ².","s":"∀ {Ω : Type u_1} {X : Type u_2} [inst : MeasurableSpace Ω] [inst_1 : MeasurableSpace X] {μ : MeasureTheory.Measure Ω} {P : MeasureTheory.Measure X} (S : Causalean.Stat.IIDSample Ω X μ P) {ψ : X → ℝ}, Measurable ψ → MeasureTheory.Integrable (fun x => ψ x ^ 2) P → Causalean.Stat.Tendsto_inProb (S.sampleMean fun x => ψ x ^ 2) (fun x => ∫ (x : X), ψ x ^ 2 ∂P) μ","l":100,"f":"Causalean/Stat/Limit/WLLN.lean"},{"n":"Causalean.Stat.IIDSample.sampleVariance","k":"def","m":"Causalean.Stat.Concentration.TailBounds.EmpiricalBernstein","a":"Stat/Concentration/TailBounds","t":1,"d":"The sample variance of f over the first n observations, normalised by 1/n: V̂ₙ(ω) = (1/n) ∑_{i<n} (f (Zᵢ ω) − X̄ₙ(ω))², where X̄ₙ = S.sampleMean f n is the sample mean. (This is the biased estimator; the unbiased one would divide by n − 1.)","s":"{Ω : Type u_1} → {X : Type u_2} → [inst : MeasurableSpace Ω] → [inst_1 : MeasurableSpace X] → {μ : MeasureTheory.Measure Ω} → {P : MeasureTheory.Measure X} → Causalean.Stat.IIDSample Ω X μ P → (X → ℝ) → ℕ → Ω → ℝ","l":80,"f":"Causalean/Stat/Concentration/TailBounds/EmpiricalBernstein.lean"},{"n":"Causalean.Stat.IIDSample.sampleVariance_eq","k":"theorem","m":"Causalean.Stat.Concentration.TailBounds.EmpiricalBernstein","a":"Stat/Concentration/TailBounds","t":2,"d":"The classical computational form of the sample variance: empirical second moment minus the square of the empirical mean, V̂ₙ(ω) = (1/n) ∑_{i<n} f(Zᵢ ω)² − X̄ₙ(ω)².","s":"∀ {Ω : Type u_1} {X : Type u_2} [inst : MeasurableSpace Ω] [inst_1 : MeasurableSpace X] {μ : MeasureTheory.Measure Ω} {P : MeasureTheory.Measure X} (S : Causalean.Stat.IIDSample Ω X μ P) (f : X → ℝ) (n : ℕ) (ω : Ω), S.sampleVariance f n ω = (↑n)⁻¹ * ∑ i ∈ Finset.range n, f (S.Z i ω) ^ 2 - S.sampleMean f n ω ^ 2","l":115,"f":"Causalean/Stat/Concentration/TailBounds/EmpiricalBernstein.lean"},{"n":"Causalean.Stat.IIDSample.sampleVariance_nonneg","k":"theorem","m":"Causalean.Stat.Concentration.TailBounds.EmpiricalBernstein","a":"Stat/Concentration/TailBounds","t":2,"d":"The sample variance is nonnegative: it is 1/n times a sum of squares.","s":"∀ {Ω : Type u_1} {X : Type u_2} [inst : MeasurableSpace Ω] [inst_1 : MeasurableSpace X] {μ : MeasureTheory.Measure Ω} {P : MeasureTheory.Measure X} (S : Causalean.Stat.IIDSample Ω X μ P) (f : X → ℝ) (n : ℕ) (ω : Ω), 0 ≤ S.sampleVariance f n ω","l":90,"f":"Causalean/Stat/Concentration/TailBounds/EmpiricalBernstein.lean"},{"n":"Causalean.Stat.IIDSample.sharedPositions_card_eq","k":"theorem","m":"Causalean.Stat.UStatistic.OrderM.RemainderSecondMoment","a":"Stat/UStatistic/OrderM","t":2,"d":"For an injective tuple map, the number of positions whose values occur in a second tuple equals the number of values shared by the two tuple images.","s":"∀ {α : Type u_3} {β : Type u_4} [inst : Fintype α] [inst_1 : DecidableEq β] {t q : α → β}, Function.Injective q → {j | q j ∈ Finset.image t Finset.univ}.card = (Finset.image t Finset.univ ∩ Finset.image q Finset.univ).card","l":386,"f":"Causalean/Stat/UStatistic/OrderM/RemainderSecondMoment.lean"},{"n":"Causalean.Stat.IIDSample.supDeviation","k":"def","m":"Causalean.Stat.EmpiricalProcess.Basic","a":"Stat/EmpiricalProcess","t":1,"d":"Finite-class sup deviation: ⨆ i, |Pₙ f_i − P f_i|. Meaningful as a real-valued statistic when the class ι is finite (otherwise the ⨆ may collapse to 0 on an unbounded family, which is why GlivenkoCantelli is stated existentially instead).","s":"{Ω : Type u_1} → {X : Type u_2} → {ι : Type u_3} → [inst : MeasurableSpace Ω] → [inst_1 : MeasurableSpace X] → {μ : MeasureTheory.Measure Ω} → {P : MeasureTheory.Measure X} → Causalean.Stat.IIDSample Ω X μ P → (ι → X → ℝ) → ℕ → Ω → ℝ","l":65,"f":"Causalean/Stat/EmpiricalProcess/Basic.lean"},{"n":"Causalean.Stat.IIDSample.zeta","k":"def","m":"Causalean.Stat.UStatistic.Variance","a":"Stat/UStatistic","t":1,"d":"ζ = ∬ g² dP dP, the second moment of the kernel.","s":"{X : Type u_2} → [inst : MeasurableSpace X] → MeasureTheory.Measure X → (X → X → ℝ) → ℝ","l":211,"f":"Causalean/Stat/UStatistic/Variance.lean"},{"n":"Causalean.Stat.IIDSample.zetaOrder","k":"def","m":"Causalean.Stat.UStatistic.OrderM.Variance","a":"Stat/UStatistic/OrderM","t":1,"d":"ζ_m = E[g(Z₁,…,Z_m)^2], the second moment of an order-m kernel under the product law P^m.","s":"{X : Type u_2} → [inst : MeasurableSpace X] → MeasureTheory.Measure X → {m : ℕ} → ((Fin m → X) → ℝ) → ℝ","l":360,"f":"Causalean/Stat/UStatistic/OrderM/Variance.lean"},{"n":"Causalean.Stat.IIDSample.zetaOrder_nonneg","k":"theorem","m":"Causalean.Stat.UStatistic.OrderM.Variance","a":"Stat/UStatistic/OrderM","t":2,"d":"ζ_m is nonnegative.","s":"∀ {X : Type u_2} [inst : MeasurableSpace X] {P : MeasureTheory.Measure X} {m : ℕ} {g : (Fin m → X) → ℝ}, 0 ≤ Causalean.Stat.IIDSample.zetaOrder P g","l":470,"f":"Causalean/Stat/UStatistic/OrderM/Variance.lean"},{"n":"Causalean.Stat.IIDSample.zeta_eq_zetaOrder","k":"theorem","m":"Causalean.Stat.UStatistic.Variance","a":"Stat/UStatistic","t":2,"d":"The order-2 second moment ζ equals the order-m second moment ζ_m of the paired kernel: ∬ g² dP dP = ∫ (pairKernel g)² dP². A change of variables along the Fin 2 → X ≃ X × X measure equivalence.","s":"∀ {X : Type u_2} [inst : MeasurableSpace X] {P : MeasureTheory.Measure X} {g : X → X → ℝ} [MeasureTheory.SigmaFinite P], Causalean.Stat.IIDSample.zeta P g = Causalean.Stat.IIDSample.zetaOrder P (Causalean.Stat.pairKernel g)","l":216,"f":"Causalean/Stat/UStatistic/Variance.lean"},{"n":"Causalean.Stat.IIDSample.zeta_nonneg","k":"theorem","m":"Causalean.Stat.UStatistic.Variance","a":"Stat/UStatistic","t":2,"d":"ζ ≥ 0.","s":"∀ {X : Type u_2} [inst : MeasurableSpace X] {P : MeasureTheory.Measure X} {g : X → X → ℝ}, 0 ≤ Causalean.Stat.IIDSample.zeta P g","l":232,"f":"Causalean/Stat/UStatistic/Variance.lean"},{"n":"Causalean.Stat.IndepFun.integrable_finiteSelector_sq","k":"theorem","m":"Causalean.Stat.SampleSplit.FiniteSelector","a":"Stat/SampleSplit","t":2,"d":"If pilot and tail coordinates are independent, the pilot coordinate is measurable, the tail coordinate is measurable, the finite selection rule is measurable, and every fixed branch has integrable squared error under the tail law, the pilot-selected squared error is integrable under the ambient probability law.","s":"∀ {Omega : Type u_1} {Alpha : Type u_2} {Beta : Type u_3} {Iota : Type u_4} [inst : MeasurableSpace Omega] [inst_1 : MeasurableSpace Alpha] [inst_2 : MeasurableSpace Beta] [inst_3 : MeasurableSpace Iota] [Fintype Iota] [MeasurableSingletonClass Iota] {mu : MeasureTheory.Measure Omega} [MeasureTheory.IsProbabilityMeasure mu] {pilot : Omega → Alpha} {tail : Omega → Beta} {select : Alpha → Iota} {err : Iota → Beta → ℝ}, ProbabilityTheory.IndepFun pilot tail mu → Measurable pilot → Measurable tail → Measurable select → (∀ (i : Iota), MeasureTheory.Integrable (fun b => err i b ^ 2) (MeasureTheory.Measure.map tail mu)) → MeasureTheory.Integrable (fun omega => err (select (pilot omega)) (tail omega) ^ 2) mu","l":83,"f":"Causalean/Stat/SampleSplit/FiniteSelector.lean"},{"n":"Causalean.Stat.IndepFun.integral_finiteSelector_sq_eq_sum","k":"theorem","m":"Causalean.Stat.SampleSplit.FiniteSelector","a":"Stat/SampleSplit","t":2,"d":"If pilot and tail coordinates are independent, the pilot coordinate is measurable, the tail coordinate is measurable, the finite selection rule is measurable, the designated pilot event is measurable, and every fixed branch has integrable squared error under the tail law, the selected risk on that event equals the sum of each selector-cell probability times its fixed-branch tail risk.","s":"∀ {Omega : Type u_1} {Alpha : Type u_2} {Beta : Type u_3} {Iota : Type u_4} [inst : MeasurableSpace Omega] [inst_1 : MeasurableSpace Alpha] [inst_2 : MeasurableSpace Beta] [inst_3 : MeasurableSpace Iota] [inst_4 : Fintype Iota] [MeasurableSingletonClass Iota] {mu : MeasureTheory.Measure Omega} [MeasureTheory.IsProbabilityMeasure mu] {pilot : Omega → Alpha} {tail : Omega → Beta} {select : Alpha → Iota} {err : Iota → Beta → ℝ}, ProbabilityTheory.IndepFun pilot tail mu → Measurable pilot → Measurable tail → Measurable select → ∀ {good : Set Alpha}, MeasurableSet good → (∀ (i : Iota), MeasureTheory.Integrable (fun b => err i b ^ 2) (MeasureTheory.Measure.map tail mu)) → ∫ (omega : Omega) in pilot ⁻¹' good, err (select (pilot omega)) (tail omega) ^ 2 ∂mu = ∑ i, (mu (pilot ⁻¹' Causalean.Stat.selectorCell select good i)).toReal * ∫ (b : Beta), err i b ^ 2 ∂MeasureTheory.Measure.map tail mu","l":118,"f":"Causalean/Stat/SampleSplit/FiniteSelector.lean"},{"n":"Causalean.Stat.IndepFun.integral_finiteSelector_sq_le","k":"theorem","m":"Causalean.Stat.SampleSplit.FiniteSelector","a":"Stat/SampleSplit","t":1,"d":"If pilot and tail coordinates are independent, the pilot coordinate is measurable, the tail coordinate is measurable, the finite selection rule is measurable, the designated pilot event is measurable, the common risk bound is nonnegative, every branch selected on that event is eligible, eligible branches have integrable squared tail error, and their fixed-branch risks obey the common bound, the pilot-selected risk on the event obeys that same bound.","s":"∀ {Omega : Type u_1} {Alpha : Type u_2} {Beta : Type u_3} {Iota : Type u_4} [inst : MeasurableSpace Omega] [inst_1 : MeasurableSpace Alpha] [inst_2 : MeasurableSpace Beta] [inst_3 : MeasurableSpace Iota] [Fintype Iota] [MeasurableSingletonClass Iota] {mu : MeasureTheory.Measure Omega} [MeasureTheory.IsProbabilityMeasure mu] {pilot : Omega → Alpha} {tail : Omega → Beta} {select : Alpha → Iota} {err : Iota → Beta → ℝ} {Eligible : Iota → Prop} {V : ℝ}, ProbabilityTheory.IndepFun pilot tail mu → Measurable pilot → Measurable tail → Measurable select → ∀ {good : Set Alpha}, MeasurableSet good → 0 ≤ V → (∀ a ∈ good, Eligible (select a)) → (∀ (i : Iota), Eligible i → MeasureTheory.Integrable (fun b => err i b ^ 2) (MeasureTheory.Measure.map tail mu)) → (∀ (i : Iota), Eligible i → ∫ (b : Beta), err i b ^ 2 ∂MeasureTheory.Measure.map tail mu ≤ V) → ∫ (omega : Omega) in pilot ⁻¹' good, err (select (pilot omega)) (tail omega) ^ 2 ∂mu ≤ V","l":246,"f":"Causalean/Stat/SampleSplit/FiniteSelector.lean"},{"n":"Causalean.Stat.IndepFun.integral_finiteSelector_sq_le_add_bad","k":"theorem","m":"Causalean.Stat.SampleSplit.FiniteSelector","a":"Stat/SampleSplit","t":2,"d":"If pilot and tail coordinates are independent, the pilot coordinate is measurable, the tail coordinate is measurable, the finite selection rule is measurable, all branch errors are measurable, the designated pilot event is measurable, the good-event risk bound is nonnegative, the global squared-error envelope is nonnegative, every branch selected on the good event is eligible, eligible branches have integrable squared tail error, their fixed-branch risks obey the good-event bound, the selected squared error obeys the global envelope, and the bad pilot event has probability at most the given tolerance, the full risk is at most the good-event bound plus the envelope times that tolerance.","s":"∀ {Omega : Type u_1} {Alpha : Type u_2} {Beta : Type u_3} {Iota : Type u_4} [inst : MeasurableSpace Omega] [inst_1 : MeasurableSpace Alpha] [inst_2 : MeasurableSpace Beta] [inst_3 : MeasurableSpace Iota] [Fintype Iota] [MeasurableSingletonClass Iota] {mu : MeasureTheory.Measure Omega} [MeasureTheory.IsProbabilityMeasure mu] {pilot : Omega → Alpha} {tail : Omega → Beta} {select : Alpha → Iota} {err : Iota → Beta → ℝ} {Eligible : Iota → Prop} {V H delta : ℝ}, ProbabilityTheory.IndepFun pilot tail mu → Measurable pilot → Measurable tail → Measurable select → (∀ (i : Iota), Measurable (err i)) → ∀ {good : Set Alpha}, MeasurableSet good → 0 ≤ V → 0 ≤ H → (∀ a ∈ good, Eligible (select a)) → (∀ (i : Iota), Eligible i → MeasureTheory.Integrable (fun b => err i b ^ 2) (MeasureTheory.Measure.map tail mu)) → (∀ (i : Iota), Eligible i → ∫ (b : Beta), err i b ^ 2 ∂MeasureTheory.Measure.map tail mu ≤ V) → (∀ (omega : Omega), err (select (pilot omega)) (tail omega) ^ 2 ≤ H) → mu.real (pilot ⁻¹' goodᶜ) ≤ delta → ∫ (omega : Omega), err (select (pilot omega)) (tail omega) ^ 2 ∂mu ≤ V + H * delta","l":348,"f":"Causalean/Stat/SampleSplit/FiniteSelector.lean"},{"n":"Causalean.Stat.InfluenceFunction","k":"structure","m":"Causalean.Stat.MEstimation.InfluenceFunction","a":"Stat/MEstimation","t":1,"d":"An *influence function* for a parameter θ ∈ E under the law P : Measure X is a measurable, mean-zero, square-integrable function ψ : X → E.","s":"{X : Type u_1} → [inst : MeasurableSpace X] → {E : Type u_2} → [inst_1 : NormedAddCommGroup E] → [InnerProductSpace ℝ E] → [MeasurableSpace E] → MeasureTheory.Measure X → (X → E) → Prop","l":41,"f":"Causalean/Stat/MEstimation/InfluenceFunction.lean"},{"n":"Causalean.Stat.IsAsymLinear","k":"structure","m":"Causalean.Stat.CLT.AsymptoticLinearity","a":"Stat/CLT","t":1,"d":"An estimator is asymptotically linear when its scaled estimation error equals the normalized empirical average of an influence function that is mean zero and square-integrable under the population law, up to a term that is negligible in probability.","s":"{Ω : Type u_1} → {X : Type u_2} → [inst : MeasurableSpace Ω] → [inst_1 : MeasurableSpace X] → {μ : MeasureTheory.Measure Ω} → {P : MeasureTheory.Measure X} → (ℕ → Ω → ℝ) → ℝ → (X → ℝ) → Causalean.Stat.IIDSample Ω X μ P → (ℕ → Finset ℕ) → Prop","l":63,"f":"Causalean/Stat/CLT/AsymptoticLinearity.lean"},{"n":"Causalean.Stat.IsAsymLinear.normalizedSum","k":"def","m":"Causalean.Stat.CLT.AsymptoticLinearity","a":"Stat/CLT","t":1,"d":"The normalised partial sum (1/√|I n|) Σ_{i ∈ I n} ψ(Z_i).","s":"{Ω : Type u_1} → {X : Type u_2} → [inst : MeasurableSpace Ω] → [inst_1 : MeasurableSpace X] → {μ : MeasureTheory.Measure Ω} → {P : MeasureTheory.Measure X} → Causalean.Stat.IIDSample Ω X μ P → (X → ℝ) → (ℕ → Finset ℕ) → ℕ → Ω → ℝ","l":101,"f":"Causalean/Stat/CLT/AsymptoticLinearity.lean"},{"n":"Causalean.Stat.IsAsymLinear.rescaledEstimator","k":"def","m":"Causalean.Stat.CLT.AsymptoticLinearity","a":"Stat/CLT","t":1,"d":"The rescaled estimator √|I n| (θn n − θ₀).","s":"{Ω : Type u_1} → (ℕ → Ω → ℝ) → ℝ → (ℕ → Finset ℕ) → ℕ → Ω → ℝ","l":106,"f":"Causalean/Stat/CLT/AsymptoticLinearity.lean"},{"n":"Causalean.Stat.IsAsymLinear.tendsto_normal","k":"theorem","m":"Causalean.Stat.CLT.AsymptoticLinearity","a":"Stat/CLT","t":1,"d":"Given that θn is asymptotically linear at θ₀ with influence function ψ along the i.i.d. sample S, where ψ is measurable and the rescaled estimator $\\sqrt n(\\theta_n-\\theta_0)$ is a.e. measurable at every sample size, then the rescaled estimator converges in distribution to the centred normal law with variance $\\int \\psi^2\\,dP$.","s":"∀ {Ω : Type u_1} {X : Type u_2} [inst : MeasurableSpace Ω] [inst_1 : MeasurableSpace X] {μ : MeasureTheory.Measure Ω} {P : MeasureTheory.Measure X} {θn : ℕ → Ω → ℝ} {θ₀ : ℝ} {ψ : X → ℝ} {S : Causalean.Stat.IIDSample Ω X μ P}, (Causalean.Stat.IsAsymLinear θn θ₀ ψ S fun m => Finset.range m) → Measurable ψ → ∀ (hθn_meas : ∀ (n : ℕ), AEMeasurable (Causalean.Stat.IsAsymLinear.rescaledEstimator θn θ₀ (fun m => Finset.range m) n) μ), Causalean.Stat.Tendsto_dist (Causalean.Stat.IsAsymLinear.rescaledEstimator θn θ₀ fun m => Finset.range m) (Causalean.Stat.gaussianMeasure 0 (∫ (x : X), ψ x ^ 2 ∂P)) μ hθn_meas","l":384,"f":"Causalean/Stat/CLT/AsymptoticLinearity.lean"},{"n":"Causalean.Stat.IsAsymLinear.tendsto_normal_foldB","k":"theorem","m":"Causalean.Stat.SampleSplit.PartialFoldCLT","a":"Stat/SampleSplit","t":1,"d":"Fold-B asymptotic linearity ⇒ asymptotic normality at rate √|B(n)|. Along an i.i.d. sample S under a one-shot sample split split, if the estimator sequence θn is fold-B asymptotically linear toward θ₀ with influence function ψ that is measurable, and if the rescaled estimator and the fold-B normalized influence-function sum are almost-everywhere measurable at every sample size, then √|B(n)| · (θn n − θ₀) converges in distribution to the centered Gaussian law with variance ∫ψ²dP.","s":"∀ {Ω : Type u_1} {X : Type u_2} [inst : MeasurableSpace Ω] [inst_1 : MeasurableSpace X] {μ : MeasureTheory.Measure Ω} {P : MeasureTheory.Measure X} [inst_2 : MeasureTheory.IsProbabilityMeasure μ] {θn : ℕ → Ω → ℝ} {θ₀ : ℝ} {ψ : X → ℝ} {S : Causalean.Stat.IIDSample Ω X μ P} (split : Causalean.Stat.OneShotSplit S), Causalean.Stat.IsAsymLinear θn θ₀ ψ S split.foldB → Measurable ψ → ∀ (hθn_meas : ∀ (n : ℕ), AEMeasurable (Causalean.Stat.IsAsymLinear.rescaledEstimator θn θ₀ split.foldB n) μ), (∀ (n : ℕ), AEMeasurable (Causalean.Stat.IsAsymLinear.normalizedSum S ψ split.foldB n) μ) → Causalean.Stat.Tendsto_dist (Causalean.Stat.IsAsymLinear.rescaledEstimator θn θ₀ split.foldB) (Causalean.Stat.gaussianMeasure 0 (∫ (x : X), ψ x ^ 2 ∂P)) μ hθn_meas","l":210,"f":"Causalean/Stat/SampleSplit/PartialFoldCLT.lean"},{"n":"Causalean.Stat.IsAsymLinear.tendsto_normal_foldB_sqrt_n","k":"theorem","m":"Causalean.Stat.SampleSplit.PartialFoldCLT","a":"Stat/SampleSplit","t":1,"d":"Conversion to √n-rate under a fixed split ratio. Along an i.i.d. sample S under a one-shot sample split split with a fold-B asymptotically linear estimator sequence θn (h), suppose the split ratio c is strictly positive, the estimation-fold share |B(n)|/n converges to c, the influence function ψ is measurable, and the √n-rescaled estimator is almost-everywhere measurable at every sample size; then √n · (θn n − θ₀) converges in distribution to the centered Gaussian law with variance (∫ψ²dP)/c.","s":"∀ {Ω : Type u_1} {X : Type u_2} [inst : MeasurableSpace Ω] [inst_1 : MeasurableSpace X] {μ : MeasureTheory.Measure Ω} {P : MeasureTheory.Measure X} [inst_2 : MeasureTheory.IsProbabilityMeasure μ] {θn : ℕ → Ω → ℝ} {θ₀ : ℝ} {ψ : X → ℝ} {S : Causalean.Stat.IIDSample Ω X μ P} (split : Causalean.Stat.OneShotSplit S) {c : ℝ}, 0 < c → Filter.Tendsto (fun n => ↑(split.foldB n).card / ↑n) Filter.atTop (nhds c) → Causalean.Stat.IsAsymLinear θn θ₀ ψ S split.foldB → Measurable ψ → ∀ (hθn_meas : ∀ (n : ℕ), AEMeasurable (fun ω => √↑n * (θn n ω - θ₀)) μ), Causalean.Stat.Tendsto_dist (fun n ω => √↑n * (θn n ω - θ₀)) (Causalean.Stat.gaussianMeasure 0 ((∫ (x : X), ψ x ^ 2 ∂P) / c)) μ hθn_meas","l":238,"f":"Causalean/Stat/SampleSplit/PartialFoldCLT.lean"},{"n":"Causalean.Stat.IsAsymLinear.toVec","k":"theorem","m":"Causalean.Stat.CLT.AsymptoticLinearityVec","a":"Stat/CLT","t":2,"d":"Conversely, scalar asymptotic linearity lifts to the vector predicate.","s":"∀ {Ω : Type u_1} {X : Type u_2} [inst : MeasurableSpace Ω] [inst_1 : MeasurableSpace X] {μ : MeasureTheory.Measure Ω} {P : MeasureTheory.Measure X} {θn : ℕ → Ω → ℝ} {θ₀ : ℝ} {ψ : X → ℝ} {S : Causalean.Stat.IIDSample Ω X μ P} {I : ℕ → Finset ℕ}, Causalean.Stat.IsAsymLinear θn θ₀ ψ S I → Causalean.Stat.IsAsymLinearVec θn θ₀ ψ S I","l":129,"f":"Causalean/Stat/CLT/AsymptoticLinearityVec.lean"},{"n":"Causalean.Stat.IsAsymLinearVec","k":"structure","m":"Causalean.Stat.CLT.AsymptoticLinearityVec","a":"Stat/CLT","t":1,"d":"A vector-valued estimator sequence is asymptotically linear at a target value when its scaled estimation error equals the normalized empirical average of an influence function that has Bochner mean zero and finite second moment under the population law, up to a remainder that is negligible in probability, along a chosen family of finite index sets selecting which observations enter each empirical average.","s":"{Ω : Type u_1} → {X : Type u_2} → [inst : MeasurableSpace Ω] → [inst_1 : MeasurableSpace X] → {μ : MeasureTheory.Measure Ω} → {P : MeasureTheory.Measure X} → {E : Type u_3} → [inst_2 : NormedAddCommGroup E] → [NormedSpace ℝ E] → (ℕ → Ω → E) → E → (X → E) → Causalean.Stat.IIDSample Ω X μ P → (ℕ → Finset ℕ) → Prop","l":59,"f":"Causalean/Stat/CLT/AsymptoticLinearityVec.lean"},{"n":"Causalean.Stat.IsAsymLinearVec.normalizedSum","k":"def","m":"Causalean.Stat.CLT.AsymptoticLinearityVec","a":"Stat/CLT","t":1,"d":"Vector normalised partial sum (1/√|I n|) Σ_{i ∈ I n} ψ(Z_i).","s":"{Ω : Type u_1} → {X : Type u_2} → [inst : MeasurableSpace Ω] → [inst_1 : MeasurableSpace X] → {μ : MeasureTheory.Measure Ω} → {P : MeasureTheory.Measure X} → {E : Type u_3} → [inst_2 : NormedAddCommGroup E] → [NormedSpace ℝ E] → Causalean.Stat.IIDSample Ω X μ P → (X → E) → (ℕ → Finset ℕ) → ℕ → Ω → E","l":98,"f":"Causalean/Stat/CLT/AsymptoticLinearityVec.lean"},{"n":"Causalean.Stat.IsAsymLinearVec.rescaledEstimator","k":"def","m":"Causalean.Stat.CLT.AsymptoticLinearityVec","a":"Stat/CLT","t":1,"d":"Vector rescaled estimator √|I n| · (θn n − θ₀).","s":"{Ω : Type u_1} → {E : Type u_3} → [inst : NormedAddCommGroup E] → [NormedSpace ℝ E] → (ℕ → Ω → E) → E → (ℕ → Finset ℕ) → ℕ → Ω → E","l":103,"f":"Causalean/Stat/CLT/AsymptoticLinearityVec.lean"},{"n":"Causalean.Stat.IsAsymLinearVec.tendsto_normal_vec","k":"theorem","m":"Causalean.Stat.CLT.AsymptoticLinearityVec","a":"Stat/CLT","t":1,"d":"Vector asymptotic normality. For a vector-valued estimator sequence θn targeting θ₀ with influence function ψ along the i.i.d. sample S, suppose the remainder between the rescaled estimator and the normalised partial sum is asymptotically negligible (little-o of 1 in norm), the rescaled estimator is a.e. measurable at every sample size, the normalised partial sum is a.e. measurable at every sample size, and the pushforward laws of the normalised partial sum converge to a target probability measure Q on E. Then the pushforward laws of the rescaled estimator likewise converge to Q.","s":"∀ {Ω : Type u_1} {X : Type u_2} [inst : MeasurableSpace Ω] [inst_1 : MeasurableSpace X] {μ : MeasureTheory.Measure Ω} {P : MeasureTheory.Measure X} {E : Type u_3} [inst_2 : NormedAddCommGroup E] [inst_3 : NormedSpace ℝ E] [inst_4 : MeasurableSpace E] [inst_5 : OpensMeasurableSpace E] {θn : ℕ → Ω → E} {θ₀ : E} {ψ : X → E} {S : Causalean.Stat.IIDSample Ω X μ P} {I : ℕ → Finset ℕ} (Q : MeasureTheory.ProbabilityMeasure E), Causalean.Stat.IsLittleOp (fun n ω => ‖√↑(I n).card • (θn n ω - θ₀) - (√↑(I n).card)⁻¹ • ∑ i ∈ I n, ψ (S.Z i ω)‖) (fun x => 1) μ → ∀ (_hθn_meas : ∀ (n : ℕ), AEMeasurable (Causalean.Stat.IsAsymLinearVec.rescaledEstimator θn θ₀ I n) μ) (_hSum_meas : ∀ (n : ℕ), AEMeasurable (Causalean.Stat.IsAsymLinearVec.normalizedSum S ψ I n) μ), Filter.Tendsto (fun n => ⟨MeasureTheory.Measure.map (Causalean.Stat.IsAsymLinearVec.normalizedSum S ψ I n) μ, MeasureTheory.Measure.isProbabilityMeasure_map (_hSum_meas n)⟩) Filter.atTop (nhds Q) → Filter.Tendsto (fun n => ⟨MeasureTheory.Measure.map (Causalean.Stat.IsAsymLinearVec.rescaledEstimator θn θ₀ I n) μ, MeasureTheory.Measure.isProbabilityMeasure_map (_hθn_meas n)⟩) Filter.atTop (nhds Q)","l":153,"f":"Causalean/Stat/CLT/AsymptoticLinearityVec.lean"},{"n":"Causalean.Stat.IsAsymLinearVec.tendsto_normal_vec_clt","k":"theorem","m":"Causalean.Stat.CLT.MultivariateCLT","a":"Stat/CLT","t":1,"d":"Vector asymptotic normality from asymptotic linearity (no CLT hypothesis). Given that θn is vector-asymptotically-linear at θ₀ with influence function ψ along the i.i.d. sample S, where ψ is measurable, a target probability measure Q on E whose characteristic function is the Gaussian one $\\exp(-\\tfrac12\\int\\langle t,\\psi\\rangle^2\\,dP)$ at every point t, and the rescaled estimator is a.e. measurable at every sample size, then the pushforward laws of the rescaled estimator converge to Q.","s":"∀ {Ω : Type u_1} {X : Type u_2} [inst : MeasurableSpace Ω] [inst_1 : MeasurableSpace X] {μ : MeasureTheory.Measure Ω} {P : MeasureTheory.Measure X} {E : Type u_3} [inst_2 : NormedAddCommGroup E] [inst_3 : InnerProductSpace ℝ E] [FiniteDimensional ℝ E] [inst_5 : MeasurableSpace E] [inst_6 : BorelSpace E] {θn : ℕ → Ω → E} {θ₀ : E} {ψ : X → E} {S : Causalean.Stat.IIDSample Ω X μ P}, Causalean.Stat.IsAsymLinearVec θn θ₀ ψ S Causalean.Stat.rng✝ → Measurable ψ → ∀ (Q : MeasureTheory.Measure E) [inst_7 : MeasureTheory.IsProbabilityMeasure Q], (∀ (t : E), MeasureTheory.charFun Q t = Complex.exp (-↑(∫ (x : X), inner ℝ t (ψ x) ^ 2 ∂P) / 2)) → ∀ (hθn_meas : ∀ (n : ℕ), AEMeasurable (Causalean.Stat.IsAsymLinearVec.rescaledEstimator θn θ₀ Causalean.Stat.rng✝ n) μ), Filter.Tendsto (fun n => ⟨MeasureTheory.Measure.map (Causalean.Stat.IsAsymLinearVec.rescaledEstimator θn θ₀ Causalean.Stat.rng✝ n) μ, MeasureTheory.Measure.isProbabilityMeasure_map (hθn_meas n)⟩) Filter.atTop (nhds ⟨Q, inst_7⟩)","l":282,"f":"Causalean/Stat/CLT/MultivariateCLT.lean"},{"n":"Causalean.Stat.IsAsymLinearVec.toScalar","k":"theorem","m":"Causalean.Stat.CLT.AsymptoticLinearityVec","a":"Stat/CLT","t":2,"d":"When the parameter space is ℝ, the vector predicate IsAsymLinearVec unfolds to the scalar IsAsymLinear. The forward direction is a direct field-by-field rewrite using ‖x‖ = |x| on ℝ and scalar • = *.","s":"∀ {Ω : Type u_1} {X : Type u_2} [inst : MeasurableSpace Ω] [inst_1 : MeasurableSpace X] {μ : MeasureTheory.Measure Ω} {P : MeasureTheory.Measure X} {θn : ℕ → Ω → ℝ} {θ₀ : ℝ} {ψ : X → ℝ} {S : Causalean.Stat.IIDSample Ω X μ P} {I : ℕ → Finset ℕ}, Causalean.Stat.IsAsymLinearVec θn θ₀ ψ S I → Causalean.Stat.IsAsymLinear θn θ₀ ψ S I","l":112,"f":"Causalean/Stat/CLT/AsymptoticLinearityVec.lean"},{"n":"Causalean.Stat.IsBigOp","k":"def","m":"Causalean.Stat.Limit.Convergence","a":"Stat/Limit","t":1,"d":"IsBigOp Xn rn μ: the sequence Xn is bounded in probability at rate rn under μ. Matches def:est-stoch-order(1):","s":"{Ω : Type u_1} → [inst : MeasurableSpace Ω] → (ℕ → Ω → ℝ) → (ℕ → ℝ) → MeasureTheory.Measure Ω → Prop","l":125,"f":"Causalean/Stat/Limit/Convergence.lean"},{"n":"Causalean.Stat.IsBigOp.add","k":"theorem","m":"Causalean.Stat.Limit.Convergence","a":"Stat/Limit","t":2,"d":"O_p(rn) + O_p(rn) = O_p(rn).","s":"∀ {Ω : Type u_1} [inst : MeasurableSpace Ω] {Xn Yn : ℕ → Ω → ℝ} {rn : ℕ → ℝ} {μ : MeasureTheory.Measure Ω}, Causalean.Stat.IsBigOp Xn rn μ → Causalean.Stat.IsBigOp Yn rn μ → Causalean.Stat.IsBigOp (fun n ω => Xn n ω + Yn n ω) rn μ","l":496,"f":"Causalean/Stat/Limit/Convergence.lean"},{"n":"Causalean.Stat.IsBigOp.add'","k":"theorem","m":"Causalean.Stat.EmpiricalProcess.CrossFitRate","a":"Stat/EmpiricalProcess","t":2,"d":"Additivity at the sum rate. O_p(rₙ) + O_p(sₙ) = O_p(rₙ + sₙ), for nonnegative rates. (IsBigOp.add is the special case rₙ = sₙ.)","s":"∀ {Ω : Type u_1} [inst : MeasurableSpace Ω] {μ : MeasureTheory.Measure Ω} {Xn Yn : ℕ → Ω → ℝ} {rn sn : ℕ → ℝ}, (∀ (n : ℕ), 0 ≤ rn n) → (∀ (n : ℕ), 0 ≤ sn n) → Causalean.Stat.IsBigOp Xn rn μ → Causalean.Stat.IsBigOp Yn sn μ → Causalean.Stat.IsBigOp (fun n ω => Xn n ω + Yn n ω) (fun n => rn n + sn n) μ","l":137,"f":"Causalean/Stat/EmpiricalProcess/CrossFitRate.lean"},{"n":"Causalean.Stat.IsBigOp.const_mul","k":"theorem","m":"Causalean.Stat.EmpiricalProcess.CrossFitRate","a":"Stat/EmpiricalProcess","t":2,"d":"Constant multiple. If Xₙ = O_p(rₙ) then c · Xₙ = O_p(rₙ) for any fixed scalar c.","s":"∀ {Ω : Type u_1} [inst : MeasurableSpace Ω] {μ : MeasureTheory.Measure Ω} {Xn : ℕ → Ω → ℝ} {rn : ℕ → ℝ} (c : ℝ), Causalean.Stat.IsBigOp Xn rn μ → Causalean.Stat.IsBigOp (fun n ω => c * Xn n ω) rn μ","l":110,"f":"Causalean/Stat/EmpiricalProcess/CrossFitRate.lean"},{"n":"Causalean.Stat.IsBigOp.const_mul_tendsto_zero","k":"theorem","m":"Causalean.Stat.Limit.Convergence","a":"Stat/Limit","t":1,"d":"Degenerate Slutsky. If a real-valued sequence Xn is bounded in probability, O_p(1), under μ, and a deterministic scalar sequence a converges to 0, then the product sequence a n · Xn is o_p(1). Concretely: the product of a sequence converging to 0 with a tight sequence is o_p(1).","s":"∀ {Ω : Type u_2} [inst : MeasurableSpace Ω] {μ : MeasureTheory.Measure Ω} {Xn : ℕ → Ω → ℝ} {a : ℕ → ℝ}, Causalean.Stat.IsBigOp Xn (fun x => 1) μ → Filter.Tendsto a Filter.atTop (nhds 0) → Causalean.Stat.IsLittleOp (fun n ω => a n * Xn n ω) (fun x => 1) μ","l":220,"f":"Causalean/Stat/Limit/Convergence.lean"},{"n":"Causalean.Stat.IsBigOp.const_rate_collapse","k":"theorem","m":"Causalean.Stat.EmpiricalProcess.CrossFitRate","a":"Stat/EmpiricalProcess","t":2,"d":"Collapse a constant rate to 1. For a *fixed* nonnegative N, O_p(fun _ => N) is O_p(fun _ => 1): a constant scale only changes the witness M. Used to normalize the fold-sum O_p bounds to the canonical unit rate consumed by the cross-fitted DML proofs.","s":"∀ {Ω : Type u_1} [inst : MeasurableSpace Ω] {μ : MeasureTheory.Measure Ω} {Xn : ℕ → Ω → ℝ} {N : ℝ}, 0 ≤ N → Causalean.Stat.IsBigOp Xn (fun x => N) μ → Causalean.Stat.IsBigOp Xn (fun x => 1) μ","l":98,"f":"Causalean/Stat/EmpiricalProcess/CrossFitRate.lean"},{"n":"Causalean.Stat.IsBigOp.finset_sum","k":"theorem","m":"Causalean.Stat.EmpiricalProcess.CrossFitRate","a":"Stat/EmpiricalProcess","t":2,"d":"A finite sum of O_p(rₙ) sequences is O_p(rₙ) (same rate; constants absorb).","s":"∀ {Ω : Type u_1} [inst : MeasurableSpace Ω] {μ : MeasureTheory.Measure Ω} {rn : ℕ → ℝ} {ι : Type u_2} (s : Finset ι) {X : ι → ℕ → Ω → ℝ}, (∀ i ∈ s, Causalean.Stat.IsBigOp (X i) rn μ) → Causalean.Stat.IsBigOp (fun n ω => ∑ i ∈ s, X i n ω) rn μ","l":229,"f":"Causalean/Stat/EmpiricalProcess/CrossFitRate.lean"},{"n":"Causalean.Stat.IsBigOp.mono_rate","k":"theorem","m":"Causalean.Stat.EmpiricalProcess.CrossFitRate","a":"Stat/EmpiricalProcess","t":2,"d":"Weaken to a larger rate. O_p(rₙ) with 0 ≤ rₙ ≤ sₙ is O_p(sₙ): a larger envelope is a weaker statement.","s":"∀ {Ω : Type u_1} [inst : MeasurableSpace Ω] {μ : MeasureTheory.Measure Ω} {Xn : ℕ → Ω → ℝ} {rn sn : ℕ → ℝ}, (∀ (n : ℕ), 0 ≤ rn n) → (∀ (n : ℕ), rn n ≤ sn n) → Causalean.Stat.IsBigOp Xn rn μ → Causalean.Stat.IsBigOp Xn sn μ","l":64,"f":"Causalean/Stat/EmpiricalProcess/CrossFitRate.lean"},{"n":"Causalean.Stat.IsBigOp.mul","k":"theorem","m":"Causalean.Stat.EmpiricalProcess.CrossFitRate","a":"Stat/EmpiricalProcess","t":2,"d":"Product rule for stochastic big-O. If Xₙ = O_p(rₙ) and Yₙ = O_p(sₙ) for nonnegative rates, then XₙYₙ = O_p(rₙsₙ).","s":"∀ {Ω : Type u_1} [inst : MeasurableSpace Ω] {μ : MeasureTheory.Measure Ω} {Xn Yn : ℕ → Ω → ℝ} {rn sn : ℕ → ℝ}, (∀ (n : ℕ), 0 ≤ rn n) → (∀ (n : ℕ), 0 ≤ sn n) → Causalean.Stat.IsBigOp Xn rn μ → Causalean.Stat.IsBigOp Yn sn μ → Causalean.Stat.IsBigOp (fun n ω => Xn n ω * Yn n ω) (fun n => rn n * sn n) μ","l":248,"f":"Causalean/Stat/EmpiricalProcess/CrossFitRate.lean"},{"n":"Causalean.Stat.IsBigOp.mul_isLittleOp_one_isLittleOp","k":"theorem","m":"Causalean.Stat.Limit.Convergence","a":"Stat/Limit","t":2,"d":"Slutsky-style product: Xn = O_p(1) and Yn = o_p(1) imply Xn · Yn = o_p(1).","s":"∀ {Ω : Type u_1} [inst : MeasurableSpace Ω] {Xn Yn : ℕ → Ω → ℝ} {μ : MeasureTheory.Measure Ω}, Causalean.Stat.IsBigOp Xn (fun x => 1) μ → Causalean.Stat.IsLittleOp Yn (fun x => 1) μ → Causalean.Stat.IsLittleOp (fun n ω => Xn n ω * Yn n ω) (fun x => 1) μ","l":549,"f":"Causalean/Stat/Limit/Convergence.lean"},{"n":"Causalean.Stat.IsBigOp.of_abs_le","k":"theorem","m":"Causalean.Stat.EmpiricalProcess.CrossFitRate","a":"Stat/EmpiricalProcess","t":2,"d":"If |Xₙ| ≤ |Yₙ| pointwise and Yₙ = O_p(rₙ), then Xₙ = O_p(rₙ).","s":"∀ {Ω : Type u_1} [inst : MeasurableSpace Ω] {μ : MeasureTheory.Measure Ω} {Xn Yn : ℕ → Ω → ℝ} {rn : ℕ → ℝ}, (∀ (n : ℕ) (ω : Ω), |Xn n ω| ≤ |Yn n ω|) → Causalean.Stat.IsBigOp Yn rn μ → Causalean.Stat.IsBigOp Xn rn μ","l":211,"f":"Causalean/Stat/EmpiricalProcess/CrossFitRate.lean"},{"n":"Causalean.Stat.IsBigOp.of_sqEnvelope","k":"theorem","m":"Causalean.Stat.Orthogonality.ConditionalOp","a":"Stat/Orthogonality","t":2,"d":"Square-envelope O_p lemma. If |X_n|² ≤ c_n μ-a.s. and c_n is O_p(rₙ²) with rₙ > 0, then X_n = O_p(rₙ).","s":"∀ {Ω : Type u_1} [inst : MeasurableSpace Ω] {μ : MeasureTheory.Measure Ω} {Xn cn : ℕ → Ω → ℝ} {rn : ℕ → ℝ}, (∀ (n : ℕ), 0 < rn n) → (∀ (n : ℕ), ∀ᵐ (ω : Ω) ∂μ, Xn n ω ^ 2 ≤ cn n ω) → Causalean.Stat.IsBigOp cn (fun n => rn n ^ 2) μ → Causalean.Stat.IsBigOp Xn rn μ","l":59,"f":"Causalean/Stat/Orthogonality/ConditionalOp.lean"},{"n":"Causalean.Stat.IsBigOp.of_sq_lintegral_le","k":"theorem","m":"Causalean.Stat.EmpiricalProcess.CrossFitRate","a":"Stat/EmpiricalProcess","t":2,"d":"Markov second-moment ⇒ O_p. If each Xₙ is μ-a.e.-measurable and its second moment is bounded by a deterministic envelope, ∫⁻ (Xₙ ω)² dμ ≤ Vₙ with 0 ≤ Vₙ, then Xₙ = O_p(√Vₙ).","s":"∀ {Ω : Type u_1} [inst : MeasurableSpace Ω] {μ : MeasureTheory.Measure Ω} {Xn : ℕ → Ω → ℝ} {Vn : ℕ → ℝ}, (∀ (n : ℕ), AEMeasurable (Xn n) μ) → (∀ (n : ℕ), 0 ≤ Vn n) → (∀ (n : ℕ), ∫⁻ (ω : Ω), ENNReal.ofReal (Xn n ω ^ 2) ∂μ ≤ ENNReal.ofReal (Vn n)) → Causalean.Stat.IsBigOp Xn (fun n => √(Vn n)) μ","l":321,"f":"Causalean/Stat/EmpiricalProcess/CrossFitRate.lean"},{"n":"Causalean.Stat.IsBigOp.scale_rate","k":"theorem","m":"Causalean.Stat.EmpiricalProcess.CrossFitRate","a":"Stat/EmpiricalProcess","t":2,"d":"Absorb a positive constant rate factor. O_p(c · rₙ) with c > 0 is O_p(rₙ); the constant is absorbed into the witness M.","s":"∀ {Ω : Type u_1} [inst : MeasurableSpace Ω] {μ : MeasureTheory.Measure Ω} {Xn : ℕ → Ω → ℝ} {rn : ℕ → ℝ} {c : ℝ}, 0 < c → Causalean.Stat.IsBigOp Xn (fun n => c * rn n) μ → Causalean.Stat.IsBigOp Xn rn μ","l":88,"f":"Causalean/Stat/EmpiricalProcess/CrossFitRate.lean"},{"n":"Causalean.Stat.IsBigOp.zero","k":"theorem","m":"Causalean.Stat.EmpiricalProcess.CrossFitRate","a":"Stat/EmpiricalProcess","t":2,"d":"The constant-zero sequence is O_p(rₙ) for any rate.","s":"∀ {Ω : Type u_1} [inst : MeasurableSpace Ω] {μ : MeasureTheory.Measure Ω} {rn : ℕ → ℝ}, Causalean.Stat.IsBigOp (fun x x_1 => 0) rn μ","l":220,"f":"Causalean/Stat/EmpiricalProcess/CrossFitRate.lean"},{"n":"Causalean.Stat.IsCoupling","k":"structure","m":"Causalean.Stat.Coupling.ProductLossMonotoneCoupling.Coupling","a":"Stat/Coupling/ProductLossMonotoneCoupling","t":1,"d":"π is a coupling of μ and ν, i.e. a member of the Fréchet class Π(μ, ν), when it is a probability measure on ℝ × ℝ whose first marginal is μ and whose second marginal is ν.","s":"MeasureTheory.Measure (ℝ × ℝ) → MeasureTheory.Measure ℝ → MeasureTheory.Measure ℝ → Prop","l":32,"f":"Causalean/Stat/Coupling/ProductLossMonotoneCoupling/Coupling.lean"},{"n":"Causalean.Stat.IsLittleOp","k":"def","m":"Causalean.Stat.Limit.Convergence","a":"Stat/Limit","t":1,"d":"IsLittleOp Xn rn μ: the sequence Xn is o_p(rn) under μ. Matches def:est-stoch-order(2):","s":"{Ω : Type u_1} → [inst : MeasurableSpace Ω] → (ℕ → Ω → ℝ) → (ℕ → ℝ) → MeasureTheory.Measure Ω → Prop","l":134,"f":"Causalean/Stat/Limit/Convergence.lean"},{"n":"Causalean.Stat.IsLittleOp.add_eventually_nonneg_rate","k":"theorem","m":"Causalean.Stat.Limit.Convergence","a":"Stat/Limit","t":2,"d":"The sum of two stochastic little-o terms is stochastic little-o for an eventually nonnegative rate.","s":"∀ {Ω : Type u_1} [inst : MeasurableSpace Ω] {Xn Yn : ℕ → Ω → ℝ} {rn : ℕ → ℝ} {μ : MeasureTheory.Measure Ω}, (∀ᶠ (n : ℕ) in Filter.atTop, 0 ≤ rn n) → Causalean.Stat.IsLittleOp Xn rn μ → Causalean.Stat.IsLittleOp Yn rn μ → Causalean.Stat.IsLittleOp (fun n ω => Xn n ω + Yn n ω) rn μ","l":298,"f":"Causalean/Stat/Limit/Convergence.lean"},{"n":"Causalean.Stat.IsLittleOp.add_one","k":"theorem","m":"Causalean.Stat.Limit.Convergence","a":"Stat/Limit","t":2,"d":"The sum of two o_p(1) sequences is o_p(1).","s":"∀ {Ω : Type u_1} [inst : MeasurableSpace Ω] {Xn Yn : ℕ → Ω → ℝ} {μ : MeasureTheory.Measure Ω}, Causalean.Stat.IsLittleOp Xn (fun x => 1) μ → Causalean.Stat.IsLittleOp Yn (fun x => 1) μ → Causalean.Stat.IsLittleOp (fun n ω => Xn n ω + Yn n ω) (fun x => 1) μ","l":386,"f":"Causalean/Stat/Limit/Convergence.lean"},{"n":"Causalean.Stat.IsLittleOp.mul_isBigOp","k":"theorem","m":"Causalean.Stat.Limit.Convergence","a":"Stat/Limit","t":2,"d":"o_p(rn) · O_p(sn) = o_p(rn · sn), assuming positive rates.","s":"∀ {Ω : Type u_1} [inst : MeasurableSpace Ω] {Xn Yn : ℕ → Ω → ℝ} {rn sn : ℕ → ℝ} {μ : MeasureTheory.Measure Ω}, (∀ (n : ℕ), 0 < rn n) → (∀ (n : ℕ), 0 < sn n) → Causalean.Stat.IsLittleOp Xn rn μ → Causalean.Stat.IsBigOp Yn sn μ → Causalean.Stat.IsLittleOp (fun n ω => Xn n ω * Yn n ω) (fun n => rn n * sn n) μ","l":402,"f":"Causalean/Stat/Limit/Convergence.lean"},{"n":"Causalean.Stat.IsLittleOp.of_abs_le_const_mul","k":"theorem","m":"Causalean.Stat.Limit.Convergence","a":"Stat/Limit","t":2,"d":"Domination by a positive constant times a stochastic little-o term preserves the stochastic little-o rate.","s":"∀ {Ω : Type u_1} [inst : MeasurableSpace Ω] {Xn Yn : ℕ → Ω → ℝ} {rn : ℕ → ℝ} {μ : MeasureTheory.Measure Ω} {C : ℝ}, 0 < C → Causalean.Stat.IsLittleOp Yn rn μ → (∀ (n : ℕ) (ω : Ω), |Xn n ω| ≤ C * |Yn n ω|) → Causalean.Stat.IsLittleOp Xn rn μ","l":362,"f":"Causalean/Stat/Limit/Convergence.lean"},{"n":"Causalean.Stat.IsLittleOp.of_abs_le_const_mul_one","k":"theorem","m":"Causalean.Stat.Limit.Convergence","a":"Stat/Limit","t":2,"d":"Domination by a positive constant times an o_p(1) sequence preserves o_p(1).","s":"∀ {Ω : Type u_1} [inst : MeasurableSpace Ω] {Xn Yn : ℕ → Ω → ℝ} {μ : MeasureTheory.Measure Ω} {C : ℝ}, 0 < C → Causalean.Stat.IsLittleOp Yn (fun x => 1) μ → (∀ (n : ℕ) (ω : Ω), |Xn n ω| ≤ C * |Yn n ω|) → Causalean.Stat.IsLittleOp Xn (fun x => 1) μ","l":394,"f":"Causalean/Stat/Limit/Convergence.lean"},{"n":"Causalean.Stat.IsLittleOp.of_eq_on_asymptotic","k":"theorem","m":"Causalean.Stat.Orthogonality.ConditionalOp","a":"Stat/Orthogonality","t":2,"d":"Truncation reduction. If Xn differs from Yn only on a set whose μ-measure tends to 0, and Yn = o_p(rn), then Xn = o_p(rn).","s":"∀ {Ω : Type u_1} [inst : MeasurableSpace Ω] {μ : MeasureTheory.Measure Ω} {Xn Yn : ℕ → Ω → ℝ} {rn : ℕ → ℝ}, Filter.Tendsto (fun n => μ {ω | Xn n ω ≠ Yn n ω}) Filter.atTop (nhds 0) → Causalean.Stat.IsLittleOp Yn rn μ → Causalean.Stat.IsLittleOp Xn rn μ","l":151,"f":"Causalean/Stat/Orthogonality/ConditionalOp.lean"},{"n":"Causalean.Stat.IsLittleOp.of_sqEnvelope","k":"theorem","m":"Causalean.Stat.Orthogonality.ConditionalOp","a":"Stat/Orthogonality","t":1,"d":"Square-envelope o_p lemma. For sequences of random variables X_n, c_n and rates r_n, if each rate r_n is positive, X_n² is bounded above by c_n almost surely, for every n, and c_n is o_p(r_n²), then X_n is o_p(r_n).","s":"∀ {Ω : Type u_1} [inst : MeasurableSpace Ω] {μ : MeasureTheory.Measure Ω} {Xn cn : ℕ → Ω → ℝ} {rn : ℕ → ℝ}, (∀ (n : ℕ), 0 < rn n) → (∀ (n : ℕ), ∀ᵐ (ω : Ω) ∂μ, Xn n ω ^ 2 ≤ cn n ω) → Causalean.Stat.IsLittleOp cn (fun n => rn n ^ 2) μ → Causalean.Stat.IsLittleOp Xn rn μ","l":105,"f":"Causalean/Stat/Orthogonality/ConditionalOp.lean"},{"n":"Causalean.Stat.IsUniqueGlobalMax","k":"def","m":"Causalean.Stat.MEstimation.FinitePoisson","a":"Stat/MEstimation","t":1,"d":"A point is the unique global maximizer of a real-valued criterion.","s":"{A : Type u_1} → (A → ℝ) → A → Prop","l":26,"f":"Causalean/Stat/MEstimation/FinitePoisson.lean"},{"n":"Causalean.Stat.KFoldSplit","k":"structure","m":"Causalean.Stat.SampleSplit.KFold","a":"Stat/SampleSplit","t":1,"d":"K-fold sample split. For an i.i.d. sample S, a schedule assigning each sample size n and fold index k a finite index set fold n k, forming a K-fold cross-fitting scheme in which distinct folds are pairwise disjoint at every sample size, the K folds together cover the full index set $\\{0,\\dots,n-1\\}$, every fold grows without bound as $n \\to \\infty$, and each fold's share of the sample converges to $1/K$.","s":"{Ω : Type u_3} → {X : Type u_4} → [inst : MeasurableSpace Ω] → [inst_1 : MeasurableSpace X] → {μ : MeasureTheory.Measure Ω} → {P : MeasureTheory.Measure X} → Causalean.Stat.IIDSample Ω X μ P → ℕ → Type","l":42,"f":"Causalean/Stat/SampleSplit/KFold.lean"},{"n":"Causalean.Stat.KFoldSplit.fold_centered_sum_isLittleOp_one","k":"theorem","m":"Causalean.Stat.SampleSplit.FoldBEmpiricalProcess","a":"Stat/SampleSplit","t":1,"d":"Per-fold centered evaluation sum is o_p(1) (K-fold). For an i.i.d. sample and a K-fold split, fixing an evaluation fold k, given a family of random functions f n ω : X → ℝ that is jointly measurable in the training-complement data and the outcome, for every n, measurable with respect to the training-complement σ-algebra jointly with the outcome, for every n, square-integrable under the population measure, for every n and ω, and whose $L^2(P)$ norm is $o_p(1)$, the centered per-fold empirical sum of f over the evaluation fold k — its fold sample average minus its population mean, rescaled by the square root of the fold size — is $o_p(1)$.","s":"∀ {Ω : Type u_1} {X : Type u_2} [inst : MeasurableSpace Ω] [inst_1 : MeasurableSpace X] {μ : MeasureTheory.Measure Ω} {P : MeasureTheory.Measure X} [MeasureTheory.IsProbabilityMeasure μ] [MeasureTheory.IsProbabilityMeasure P] (S : Causalean.Stat.IIDSample Ω X μ P) {K : ℕ} (split : Causalean.Stat.KFoldSplit S K) (k : Fin K) (f : ℕ → Ω → X → ℝ), (∀ (n : ℕ), Measurable (Function.uncurry (f n))) → (∀ (n : ℕ), Measurable (Function.uncurry (f n))) → (∀ (n : ℕ) (ω : Ω), MeasureTheory.MemLp (f n ω) 2 P) → Causalean.Stat.IsLittleOp (fun n ω => (MeasureTheory.eLpNorm (f n ω) 2 P).toReal) (fun x => 1) μ → Causalean.Stat.IsLittleOp (fun n ω => (√↑(split.fold n k).card)⁻¹ * ∑ i ∈ split.fold n k, (f n ω (S.Z i ω) - ∫ (x : X), f n ω x ∂P)) (fun x => 1) μ","l":742,"f":"Causalean/Stat/SampleSplit/FoldBEmpiricalProcess.lean"},{"n":"Causalean.Stat.KFoldSplit.fold_disjoint_trainComplement","k":"theorem","m":"Causalean.Stat.SampleSplit.KFold","a":"Stat/SampleSplit","t":2,"d":"The evaluation fold is disjoint from its training complement.","s":"∀ {Ω : Type u_1} {X : Type u_2} [inst : MeasurableSpace Ω] [inst_1 : MeasurableSpace X] {μ : MeasureTheory.Measure Ω} {P : MeasureTheory.Measure X} {S : Causalean.Stat.IIDSample Ω X μ P} {K : ℕ} (split : Causalean.Stat.KFoldSplit S K) (n : ℕ) (k : Fin K), Disjoint (split.fold n k) (split.trainComplement n k)","l":68,"f":"Causalean/Stat/SampleSplit/KFold.lean"},{"n":"Causalean.Stat.KFoldSplit.folds_indep","k":"theorem","m":"Causalean.Stat.SampleSplit.KFold","a":"Stat/SampleSplit","t":1,"d":"Independence of evaluation fold and training complement. For a fixed sample size n and fold index k, the sample sub-tuple indexed by the evaluation fold is independent, under μ, of the sub-tuple indexed by the training complement.","s":"∀ {Ω : Type u_1} {X : Type u_2} [inst : MeasurableSpace Ω] [inst_1 : MeasurableSpace X] {μ : MeasureTheory.Measure Ω} {P : MeasureTheory.Measure X} {S : Causalean.Stat.IIDSample Ω X μ P} {K : ℕ} (split : Causalean.Stat.KFoldSplit S K) (n : ℕ) (k : Fin K), ProbabilityTheory.IndepFun (fun ω i => S.Z (↑i) ω) (fun ω i => S.Z (↑i) ω) μ","l":76,"f":"Causalean/Stat/SampleSplit/KFold.lean"},{"n":"Causalean.Stat.KFoldSplit.sqrtFold_integral_isLittleOp_one","k":"theorem","m":"Causalean.Stat.SampleSplit.FoldBEmpiricalProcess","a":"Stat/SampleSplit","t":1,"d":"Per-fold bias term is o_p(1) under a nonempty K-fold split. For an i.i.d. sample and a K-fold split with a positive number of folds $K$, given a family of random functions f n ω that is square-integrable under the population measure, for every n, ω with $L^2(P)$ norm that is $o_p(n^{-1/2})$, the per-fold bias term $\\sqrt{|{\\rm fold}(n,k)|}\\cdot\\int f(n,\\omega,\\cdot)\\,dP$ at evaluation fold k is $o_p(1)$.","s":"∀ {Ω : Type u_1} {X : Type u_2} [inst : MeasurableSpace Ω] [inst_1 : MeasurableSpace X] {μ : MeasureTheory.Measure Ω} {P : MeasureTheory.Measure X} [MeasureTheory.IsProbabilityMeasure P] (S : Causalean.Stat.IIDSample Ω X μ P) {K : ℕ} (split : Causalean.Stat.KFoldSplit S K) (k : Fin K), 0 < K → ∀ (f : ℕ → Ω → X → ℝ), (∀ (n : ℕ) (ω : Ω), MeasureTheory.MemLp (f n ω) 2 P) → Causalean.Stat.IsLittleOp (fun n ω => (MeasureTheory.eLpNorm (f n ω) 2 P).toReal) (fun n => ↑n ^ (-(1 / 2))) μ → Causalean.Stat.IsLittleOp (fun n ω => √↑(split.fold n k).card * ∫ (x : X), f n ω x ∂P) (fun x => 1) μ","l":777,"f":"Causalean/Stat/SampleSplit/FoldBEmpiricalProcess.lean"},{"n":"Causalean.Stat.KFoldSplit.trainComplement","k":"def","m":"Causalean.Stat.SampleSplit.KFold","a":"Stat/SampleSplit","t":1,"d":"The training complement of fold k at sample size n: {0, …, n-1} \\ fold(n, k).","s":"{Ω : Type u_1} → {X : Type u_2} → [inst : MeasurableSpace Ω] → [inst_1 : MeasurableSpace X] → {μ : MeasureTheory.Measure Ω} → {P : MeasureTheory.Measure X} → {S : Causalean.Stat.IIDSample Ω X μ P} → {K : ℕ} → Causalean.Stat.KFoldSplit S K → ℕ → Fin K → Finset ℕ","l":63,"f":"Causalean/Stat/SampleSplit/KFold.lean"},{"n":"Causalean.Stat.L1Bracketing","k":"structure","m":"Causalean.Stat.EmpiricalProcess.Basic","a":"Stat/EmpiricalProcess","t":1,"d":"A finite L¹(P) ε-bracketing of a real-valued function class consists of finitely many integrable lower and upper endpoints, a common full-measure support on which every class member is sandwiched by its assigned bracket, and an integrated absolute bracket width at most ε.","s":"{X : Type u_2} → {ι : Type u_3} → [inst : MeasurableSpace X] → (ι → X → ℝ) → MeasureTheory.Measure X → ℝ → Type (max u_2 u_3)","l":89,"f":"Causalean/Stat/EmpiricalProcess/Basic.lean"},{"n":"Causalean.Stat.Limit.van_trees_inequality","k":"theorem","m":"Causalean.Stat.Limit.VanTreesInequality","a":"Stat/Limit","t":1,"d":"van Trees inequality (Bayesian Cramér–Rao bound), single parameter. Let a scalar parameter h range over an interval [a, b], drawn from a prior density q that is continuously differentiable on [a, b], nonnegative, positive on the open interior, and vanishing at both endpoints. Given h, data are drawn from a law P h whose score function S h has conditional mean zero and second moment equal to the Fisher information I h; let δ be an estimator of a target ψ that is differentiable with derivative dψ, whose posterior mean is differentiable with derivative equal to its covariance with the score. Assume also that δ, S, and their pointwise products are integrable, that the three quadratic building blocks (δ-ψ)², (δ-ψ)(S+dq/q), (S+dq/q)² are integrable under P h, that the posterior-mean derivative, dψ, dq, and their combination are interval-integrable, that ψ·dq, dq²/q, and I·q are interval-integrable, and that the three h-indexed second-moment integrands are interval-integrable — the standard integrability side conditions for the Bochner/interval-integral manipulations. Assume finally that the total information — prior information plus average experimental information — is positive. Then the Bayes mean-squared error ∫ E_h[(δ − ψ h)²] q(h) dh is at least (∫ ψ'(h) q(h) dh)² / (I_q + ∫ I(h) q(h) dh), where I_q = ∫ q'(h)²/q(h) dh.","s":"∀ {Z : Type u_1} [inst : MeasurableSpace Z] {a b : ℝ}, a ≤ b → ∀ (P : ℝ → MeasureTheory.Measure Z) [∀ (h : ℝ), MeasureTheory.IsProbabilityMeasure (P h)] (δ : Z → ℝ) (ψ dψ q dq I : ℝ → ℝ) (S : ℝ → Z → ℝ), (∀ h ∈ Set.Icc a b, HasDerivAt q (dq h) h) → (∀ h ∈ Set.Icc a b, 0 ≤ q h) → (∀ h ∈ Set.Ioo a b, 0 < q h) → q a = 0 → q b = 0 → (∀ h ∈ Set.Icc a b, HasDerivAt ψ (dψ h) h) → (∀ h ∈ Set.Icc a b, ∫ (z : Z), S h z ∂P h = 0) → (∀ h ∈ Set.Icc a b, ∫ (z : Z), S h z ^ 2 ∂P h = I h) → (∀ h ∈ Set.Icc a b, HasDerivAt (fun h' => ∫ (z : Z), δ z ∂P h') (∫ (z : Z), δ z * S h z ∂P h) h) → (∀ h ∈ Set.Icc a b, MeasureTheory.Integrable δ (P h)) → (∀ h ∈ Set.Icc a b, MeasureTheory.Integrable (S h) (P h)) → (∀ h ∈ Set.Icc a b, MeasureTheory.Integrable (fun z => δ z * S h z) (P h)) → (∀ h ∈ Set.Icc a b, MeasureTheory.Integrable (fun z => S h z * S h z) (P h)) → (∀ h ∈ Set.Icc a b, MeasureTheory.Integrable (fun z => (δ z - ψ h) * (δ z - ψ h)) (P h)) → (∀ h ∈ Set.Icc a b, MeasureTheory.Integrable (fun z => (δ z - ψ h) * (S h z + dq h / q h)) (P h)) → (∀ h ∈ Set.Icc a b, MeasureTheory.Integrable (fun z => (S h z + dq h / q h) * (S h z + dq h / q h)) (P h)) → IntervalIntegrable (fun h => ∫ (z : Z), δ z * S h z ∂P h) MeasureTheory.volume a b → IntervalIntegrable dψ MeasureTheory.volume a b → IntervalIntegrable dq MeasureTheory.volume a b → IntervalIntegrable (fun h => (∫ (z : Z), δ z * S h z ∂P h) * q h + (∫ (z : Z), δ z ∂P h) * dq h) MeasureTheory.volume a b → IntervalIntegrable (fun h => ψ h * dq h) MeasureTheory.volume a b → IntervalIntegrable (fun h => dq h ^ 2 / q h) MeasureTheory.volume a b → IntervalIntegrable (fun h => I h * q h) MeasureTheory.volume a b → IntervalIntegrable (fun h => (∫ (z : Z), (δ z - ψ h) * (δ z - ψ h) ∂P h) * q h) MeasureTheory.volume a b → IntervalIntegrable (fun h => (∫ (z : Z), (δ z - ψ h) * (S h z + dq h / q h) ∂P h) * q h) MeasureTheory.volume a b → IntervalIntegrable (fun h => (∫ (z : Z), (S h z + dq h / q h) * (S h z + dq h / q h) ∂P h) * q h) MeasureTheory.volume a b → 0 < (∫ (h : ℝ) in a..b, dq h ^ 2 / q h) + ∫ (h : ℝ) in a..b, I h * q h → (∫ (h : ℝ) in a..b, dψ h * q h) ^ 2 / ((∫ (h : ℝ) in a..b, dq h ^ 2 / q h) + ∫ (h : ℝ) in a..b, I h * q h) ≤ ∫ (h : ℝ) in a..b, (∫ (z : Z), (δ z - ψ h) ^ 2 ∂P h) * q h","l":48,"f":"Causalean/Stat/Limit/VanTreesInequality.lean"},{"n":"Causalean.Stat.Minimax.leCam_two_point_L1_lower","k":"theorem","m":"Causalean.Stat.Minimax.LeCamTwoPoint","a":"Stat/Minimax","t":1,"d":"Le Cam two-point lower bound on finite Bochner L¹ risk. Fix a strictly positive KL budget C with C at most 1/2. Then there is a universal positive constant c₀ such that, for every sample size, every pair of single-observation laws whose scaled KL divergence is bounded by C and whose n-fold product KL divergence is controlled by a supplied product-KL comparison and a supplied product-form Pinsker bound, every pair of separated real targets, and every measurable estimator with integrable absolute loss under both n-fold product laws, the worst-case Bochner L¹ risk is at least c₀ times the target separation.","s":"∀ {Ω : Type u_1} [inst : MeasurableSpace Ω] (C : ℝ), 0 < C → C ≤ 1 / 2 → ∃ c₀, 0 < c₀ ∧ ∀ (n : ℕ), 1 ≤ n → ∀ (P Q : MeasureTheory.Measure Ω) [MeasureTheory.IsProbabilityMeasure P] [MeasureTheory.IsProbabilityMeasure Q] (θP θQ δ : ℝ), ↑n * (InformationTheory.klDiv P Q).toReal ≤ C → (InformationTheory.klDiv (MeasureTheory.Measure.pi fun x => P) (MeasureTheory.Measure.pi fun x => Q)).toReal ≤ ↑n * (InformationTheory.klDiv P Q).toReal → Causalean.Stat.PinskerBound (MeasureTheory.Measure.pi fun x => P) (MeasureTheory.Measure.pi fun x => Q) → 0 ≤ δ → δ ≤ |θP - θQ| → ∀ (T : (Fin n → Ω) → ℝ), Measurable T → MeasureTheory.Integrable (fun ω => |T ω - θP|) (MeasureTheory.Measure.pi fun x => P) → MeasureTheory.Integrable (fun ω => |T ω - θQ|) (MeasureTheory.Measure.pi fun x => Q) → c₀ * δ ≤ max (∫ (ω : Fin n → Ω), |T ω - θP| ∂MeasureTheory.Measure.pi fun x => P) (∫ (ω : Fin n → Ω), |T ω - θQ| ∂MeasureTheory.Measure.pi fun x => Q)","l":67,"f":"Causalean/Stat/Minimax/LeCamTwoPoint.lean"},{"n":"Causalean.Stat.Minimax.le_cam_two_point_mse","k":"theorem","m":"Causalean.Stat.Minimax.LeCamTwoPoint","a":"Stat/Minimax","t":1,"d":"Le Cam two-point reduction in mean-squared-error form, uniform over a finite KL budget. For every Kullback–Leibler budget K, there is a single positive constant c_K (here exp(−K)/32), chosen before the laws, such that for every pair of probability laws Q₀, Q₁ whose divergence obeys KL(Q₀, Q₁) ≤ K, any measurable estimator with integrable squared loss under both laws has worst-case mean-squared error at least c_K times the squared separation of the two candidate parameter values. The finite-budget hypothesis is encoded as the ℝ≥0∞ inequality klDiv Q₀ Q₁ ≤ ENNReal.ofReal K, which forces a finite divergence (so it cannot be met vacuously by an infinite divergence collapsing under .toReal) and pins c_K to K alone.","s":"∀ (K : ℝ), ∃ cK, 0 < cK ∧ ∀ {S : Type u_1} [inst : MeasurableSpace S] (Q0 Q1 : MeasureTheory.Measure S) [MeasureTheory.IsProbabilityMeasure Q0] [MeasureTheory.IsProbabilityMeasure Q1] (theta0 theta1 : ℝ), InformationTheory.klDiv Q0 Q1 ≤ ENNReal.ofReal K → ∀ (T : S → ℝ), Measurable T → MeasureTheory.Integrable (fun s => (T s - theta0) ^ 2) Q0 → MeasureTheory.Integrable (fun s => (T s - theta1) ^ 2) Q1 → cK * (theta1 - theta0) ^ 2 ≤ max (∫ (s : S), (T s - theta0) ^ 2 ∂Q0) (∫ (s : S), (T s - theta1) ^ 2 ∂Q1)","l":213,"f":"Causalean/Stat/Minimax/LeCamTwoPoint.lean"},{"n":"Causalean.Stat.MomentProblems.BoundedOutcomeEnvelope.Admissible","k":"structure","m":"Causalean.Stat.Nonparametric.MomentProblems.BoundedOutcomeEnvelope.Defs","a":"Stat/Nonparametric/MomentProblems/BoundedOutcomeEnvelope","t":1,"d":"μ is admissible for the envelope at level v: a probability measure a.e. supported in [0,1] with second moment ∫ y² ∂μ = v². These are exactly the laws over which the residual l2ResidualQuadratic is maximized to give ρ(v).","s":"ℝ → MeasureTheory.Measure ℝ → Prop","l":70,"f":"Causalean/Stat/Nonparametric/MomentProblems/BoundedOutcomeEnvelope/Defs.lean"},{"n":"Causalean.Stat.MomentProblems.BoundedOutcomeEnvelope.Admissible.moment2_eq","k":"theorem","m":"Causalean.Stat.Nonparametric.MomentProblems.BoundedOutcomeEnvelope.Defs","a":"Stat/Nonparametric/MomentProblems/BoundedOutcomeEnvelope","t":2,"d":"Second moment of an admissible law in moment-form: moment μ 2 = v².","s":"∀ {v : ℝ} {μ : MeasureTheory.Measure ℝ}, Causalean.Stat.MomentProblems.BoundedOutcomeEnvelope.Admissible v μ → Causalean.Stat.MomentProblems.ResidualQuadratic.MeasureBridge.moment μ 2 = v ^ 2","l":87,"f":"Causalean/Stat/Nonparametric/MomentProblems/BoundedOutcomeEnvelope/Defs.lean"},{"n":"Causalean.Stat.MomentProblems.BoundedOutcomeEnvelope.continuous_envelopeQuartic","k":"theorem","m":"Causalean.Stat.Nonparametric.MomentProblems.BoundedOutcomeEnvelope.QuarticRoot","a":"Stat/Nonparametric/MomentProblems/BoundedOutcomeEnvelope","t":2,"d":"The continuous polynomial t ↦ envelopeQuartic t q.","s":"∀ (q : ℝ), Continuous fun t => Causalean.Stat.MomentProblems.ResidualQuadratic.MomentAlgebra.envelopeQuartic t q","l":63,"f":"Causalean/Stat/Nonparametric/MomentProblems/BoundedOutcomeEnvelope/QuarticRoot.lean"},{"n":"Causalean.Stat.MomentProblems.BoundedOutcomeEnvelope.crossMoment_nonneg","k":"theorem","m":"Causalean.Stat.Nonparametric.MomentProblems.BoundedOutcomeEnvelope.Bounds","a":"Stat/Nonparametric/MomentProblems/BoundedOutcomeEnvelope","t":2,"d":"0 ≤ crossMoment m q m₃ m₄. The certificate cross moment equals the integral ∫ y (1 − y) (y − xᵥ)² ∂μ (expand the degree-4 polynomial and integrate term by term, matching the definition of crossMoment); on [0,1] the integrand is ≥ 0, so the integral is ≥ 0.","s":"∀ {v : ℝ} {μ : MeasureTheory.Measure ℝ}, Causalean.Stat.MomentProblems.BoundedOutcomeEnvelope.Admissible v μ → 0 ≤ Causalean.Stat.MomentProblems.ResidualQuadratic.MomentAlgebra.crossMoment (Causalean.Stat.MomentProblems.ResidualQuadratic.MeasureBridge.moment μ 1) (Causalean.Stat.MomentProblems.ResidualQuadratic.MeasureBridge.moment μ 2) (Causalean.Stat.MomentProblems.ResidualQuadratic.MeasureBridge.moment μ 3) (Causalean.Stat.MomentProblems.ResidualQuadratic.MeasureBridge.moment μ 4)","l":82,"f":"Causalean/Stat/Nonparametric/MomentProblems/BoundedOutcomeEnvelope/Bounds.lean"},{"n":"Causalean.Stat.MomentProblems.BoundedOutcomeEnvelope.envelopeQuartic_deriv_neg","k":"theorem","m":"Causalean.Stat.Nonparametric.MomentProblems.BoundedOutcomeEnvelope.QuarticRoot","a":"Stat/Nonparametric/MomentProblems/BoundedOutcomeEnvelope","t":2,"d":"Strict negativity of the derivative on (v², v). For v ∈ (0,1) and t ∈ (v², v), 4 t³ − 6 t² + 4 v² t − 2 v⁴ < 0. (This is the derivative of envelopeQuartic · (v²).) Proof: nlinarith from v² < t, t < v, 0 < v, v < 1.","s":"∀ (v t : ℝ), 0 < v → v < 1 → v ^ 2 < t → t < v → 4 * t ^ 3 - 6 * t ^ 2 + 4 * v ^ 2 * t - 2 * (v ^ 2) ^ 2 < 0","l":104,"f":"Causalean/Stat/Nonparametric/MomentProblems/BoundedOutcomeEnvelope/QuarticRoot.lean"},{"n":"Causalean.Stat.MomentProblems.BoundedOutcomeEnvelope.envelopeQuartic_neg_at_v","k":"theorem","m":"Causalean.Stat.Nonparametric.MomentProblems.BoundedOutcomeEnvelope.QuarticRoot","a":"Stat/Nonparametric/MomentProblems/BoundedOutcomeEnvelope","t":2,"d":"envelopeQuartic v (v²) = −2 v³ (v − 1)² < 0 for v ∈ (0,1). This is the negative endpoint of the sign change (at t = v = √q) that locates the interior root.","s":"∀ (v : ℝ), 0 < v → v < 1 → Causalean.Stat.MomentProblems.ResidualQuadratic.MomentAlgebra.envelopeQuartic v (v ^ 2) < 0","l":51,"f":"Causalean/Stat/Nonparametric/MomentProblems/BoundedOutcomeEnvelope/QuarticRoot.lean"},{"n":"Causalean.Stat.MomentProblems.BoundedOutcomeEnvelope.envelopeQuartic_pos_at_q","k":"theorem","m":"Causalean.Stat.Nonparametric.MomentProblems.BoundedOutcomeEnvelope.QuarticRoot","a":"Stat/Nonparametric/MomentProblems/BoundedOutcomeEnvelope","t":2,"d":"envelopeQuartic q q = q² (1 − q)² > 0 for q ∈ (0,1). This is the positive endpoint of the sign change that locates the interior root.","s":"∀ (q : ℝ), 0 < q → q < 1 → 0 < Causalean.Stat.MomentProblems.ResidualQuadratic.MomentAlgebra.envelopeQuartic q q","l":41,"f":"Causalean/Stat/Nonparametric/MomentProblems/BoundedOutcomeEnvelope/QuarticRoot.lean"},{"n":"Causalean.Stat.MomentProblems.BoundedOutcomeEnvelope.extremalMeasure","k":"def","m":"Causalean.Stat.Nonparametric.MomentProblems.BoundedOutcomeEnvelope.Attainment","a":"Stat/Nonparametric/MomentProblems/BoundedOutcomeEnvelope","t":1,"d":"The extremal three-point probability law w₀ δ₀ + w₁ δ_{xᵥ} + w₂ δ₁.","s":"ℝ → MeasureTheory.Measure ℝ","l":48,"f":"Causalean/Stat/Nonparametric/MomentProblems/BoundedOutcomeEnvelope/Attainment.lean"},{"n":"Causalean.Stat.MomentProblems.BoundedOutcomeEnvelope.extremalMeasure_admissible","k":"theorem","m":"Causalean.Stat.Nonparametric.MomentProblems.BoundedOutcomeEnvelope.Attainment","a":"Stat/Nonparametric/MomentProblems/BoundedOutcomeEnvelope","t":2,"d":"The extremal law is admissible.","s":"∀ (v : ℝ), 0 < v → v < 1 → Causalean.Stat.MomentProblems.BoundedOutcomeEnvelope.Admissible v (Causalean.Stat.MomentProblems.BoundedOutcomeEnvelope.extremalMeasure v)","l":310,"f":"Causalean/Stat/Nonparametric/MomentProblems/BoundedOutcomeEnvelope/Attainment.lean"},{"n":"Causalean.Stat.MomentProblems.BoundedOutcomeEnvelope.extremalMeasure_isProb","k":"theorem","m":"Causalean.Stat.Nonparametric.MomentProblems.BoundedOutcomeEnvelope.Attainment","a":"Stat/Nonparametric/MomentProblems/BoundedOutcomeEnvelope","t":2,"d":"extremalMeasure v is a probability measure (total mass w₀ + w₁ + w₂ = 1, weights ≥ 0).","s":"∀ (v : ℝ), 0 < v → v < 1 → MeasureTheory.IsProbabilityMeasure (Causalean.Stat.MomentProblems.BoundedOutcomeEnvelope.extremalMeasure v)","l":186,"f":"Causalean/Stat/Nonparametric/MomentProblems/BoundedOutcomeEnvelope/Attainment.lean"},{"n":"Causalean.Stat.MomentProblems.BoundedOutcomeEnvelope.extremalMeasure_moment1","k":"theorem","m":"Causalean.Stat.Nonparametric.MomentProblems.BoundedOutcomeEnvelope.Attainment","a":"Stat/Nonparametric/MomentProblems/BoundedOutcomeEnvelope","t":2,"d":"First moment: ∫ y ∂(extremalMeasure v) = μᵥ. Algebraic: w₁ xᵥ + w₂ = μᵥ.","s":"∀ (v : ℝ), 0 < v → v < 1 → Causalean.Stat.MomentProblems.ResidualQuadratic.MeasureBridge.moment (Causalean.Stat.MomentProblems.BoundedOutcomeEnvelope.extremalMeasure v) 1 = Causalean.Stat.MomentProblems.BoundedOutcomeEnvelope.maximizingRoot v","l":215,"f":"Causalean/Stat/Nonparametric/MomentProblems/BoundedOutcomeEnvelope/Attainment.lean"},{"n":"Causalean.Stat.MomentProblems.BoundedOutcomeEnvelope.extremalMeasure_moment2","k":"theorem","m":"Causalean.Stat.Nonparametric.MomentProblems.BoundedOutcomeEnvelope.Attainment","a":"Stat/Nonparametric/MomentProblems/BoundedOutcomeEnvelope","t":2,"d":"Second moment: ∫ y² ∂(extremalMeasure v) = v². Algebraic: w₁ xᵥ² + w₂ = v².","s":"∀ (v : ℝ), 0 < v → v < 1 → Causalean.Stat.MomentProblems.ResidualQuadratic.MeasureBridge.moment (Causalean.Stat.MomentProblems.BoundedOutcomeEnvelope.extremalMeasure v) 2 = v ^ 2","l":227,"f":"Causalean/Stat/Nonparametric/MomentProblems/BoundedOutcomeEnvelope/Attainment.lean"},{"n":"Causalean.Stat.MomentProblems.BoundedOutcomeEnvelope.extremalMeasure_moment3","k":"theorem","m":"Causalean.Stat.Nonparametric.MomentProblems.BoundedOutcomeEnvelope.Attainment","a":"Stat/Nonparametric/MomentProblems/BoundedOutcomeEnvelope","t":2,"d":"Third moment matches the extremal M₃: w₁ xᵥ³ + w₂ = extremalM3 μᵥ (v²). Pure algebra (field_simp/ring) using xᵥ = extremalMid μᵥ (v²) and the weight formulas.","s":"∀ (v : ℝ), 0 < v → v < 1 → Causalean.Stat.MomentProblems.ResidualQuadratic.MeasureBridge.moment (Causalean.Stat.MomentProblems.BoundedOutcomeEnvelope.extremalMeasure v) 3 = Causalean.Stat.MomentProblems.ResidualQuadratic.MomentAlgebra.extremalM3 (Causalean.Stat.MomentProblems.BoundedOutcomeEnvelope.maximizingRoot v) (v ^ 2)","l":239,"f":"Causalean/Stat/Nonparametric/MomentProblems/BoundedOutcomeEnvelope/Attainment.lean"},{"n":"Causalean.Stat.MomentProblems.BoundedOutcomeEnvelope.extremalMeasure_moment4","k":"theorem","m":"Causalean.Stat.Nonparametric.MomentProblems.BoundedOutcomeEnvelope.Attainment","a":"Stat/Nonparametric/MomentProblems/BoundedOutcomeEnvelope","t":2,"d":"Fourth moment matches the extremal M₄: w₁ xᵥ⁴ + w₂ = extremalM4 μᵥ (v²). Pure algebra.","s":"∀ (v : ℝ), 0 < v → v < 1 → Causalean.Stat.MomentProblems.ResidualQuadratic.MeasureBridge.moment (Causalean.Stat.MomentProblems.BoundedOutcomeEnvelope.extremalMeasure v) 4 = Causalean.Stat.MomentProblems.ResidualQuadratic.MomentAlgebra.extremalM4 (Causalean.Stat.MomentProblems.BoundedOutcomeEnvelope.maximizingRoot v) (v ^ 2)","l":275,"f":"Causalean/Stat/Nonparametric/MomentProblems/BoundedOutcomeEnvelope/Attainment.lean"},{"n":"Causalean.Stat.MomentProblems.BoundedOutcomeEnvelope.extremalMeasure_moment_pow","k":"theorem","m":"Causalean.Stat.Nonparametric.MomentProblems.BoundedOutcomeEnvelope.Attainment","a":"Stat/Nonparametric/MomentProblems/BoundedOutcomeEnvelope","t":2,"d":"Raw moments of the extremal law. For k ≥ 1, ∫ yᵏ ∂(extremalMeasure v) = w₁ xᵥᵏ + w₂ (the mass at 0 contributes nothing, the mass at 1 contributes w₂). Proof: expand via integral_add_measure, integral_smul_measure, integral_dirac, using w_i ≥ 0 to convert ENNReal.ofReal wᵢ back to wᵢ, and (0:ℝ)^k = 0, (1:ℝ)^k = 1.","s":"∀ (v : ℝ), 0 < v → v < 1 → ∀ {k : ℕ}, 1 ≤ k → Causalean.Stat.MomentProblems.ResidualQuadratic.MeasureBridge.moment (Causalean.Stat.MomentProblems.BoundedOutcomeEnvelope.extremalMeasure v) k = Causalean.Stat.MomentProblems.BoundedOutcomeEnvelope.extremalW1 v * Causalean.Stat.MomentProblems.BoundedOutcomeEnvelope.extremalSupp v ^ k + Causalean.Stat.MomentProblems.BoundedOutcomeEnvelope.extremalW2 v","l":155,"f":"Causalean/Stat/Nonparametric/MomentProblems/BoundedOutcomeEnvelope/Attainment.lean"},{"n":"Causalean.Stat.MomentProblems.BoundedOutcomeEnvelope.extremalMeasure_residual","k":"theorem","m":"Causalean.Stat.Nonparametric.MomentProblems.BoundedOutcomeEnvelope.Attainment","a":"Stat/Nonparametric/MomentProblems/BoundedOutcomeEnvelope","t":2,"d":"The extremal law realizes ρ(v). Its residual equals the envelope value: l2ResidualQuadratic (extremalMeasure v) = rhoEnvelope v.","s":"∀ (v : ℝ), 0 < v → v < 1 → Causalean.Stat.MomentProblems.ResidualQuadratic.MeasureBridge.l2ResidualQuadratic (Causalean.Stat.MomentProblems.BoundedOutcomeEnvelope.extremalMeasure v) = Causalean.Stat.MomentProblems.BoundedOutcomeEnvelope.rhoEnvelope v","l":317,"f":"Causalean/Stat/Nonparametric/MomentProblems/BoundedOutcomeEnvelope/Attainment.lean"},{"n":"Causalean.Stat.MomentProblems.BoundedOutcomeEnvelope.extremalMeasure_supp","k":"theorem","m":"Causalean.Stat.Nonparametric.MomentProblems.BoundedOutcomeEnvelope.Attainment","a":"Stat/Nonparametric/MomentProblems/BoundedOutcomeEnvelope","t":2,"d":"extremalMeasure v is a.e. supported in [0,1] (its atoms 0, xᵥ, 1 all lie in [0,1]).","s":"∀ (v : ℝ), 0 < v → v < 1 → ∀ᵐ (y : ℝ) ∂Causalean.Stat.MomentProblems.BoundedOutcomeEnvelope.extremalMeasure v, y ∈ Set.Icc 0 1","l":204,"f":"Causalean/Stat/Nonparametric/MomentProblems/BoundedOutcomeEnvelope/Attainment.lean"},{"n":"Causalean.Stat.MomentProblems.BoundedOutcomeEnvelope.extremalSupp","k":"def","m":"Causalean.Stat.Nonparametric.MomentProblems.BoundedOutcomeEnvelope.Attainment","a":"Stat/Nonparametric/MomentProblems/BoundedOutcomeEnvelope","t":1,"d":"Interior support point xᵥ = extremalMid μᵥ (v²) of the extremal three-point law.","s":"ℝ → ℝ","l":34,"f":"Causalean/Stat/Nonparametric/MomentProblems/BoundedOutcomeEnvelope/Attainment.lean"},{"n":"Causalean.Stat.MomentProblems.BoundedOutcomeEnvelope.extremalSupp_mem","k":"theorem","m":"Causalean.Stat.Nonparametric.MomentProblems.BoundedOutcomeEnvelope.Attainment","a":"Stat/Nonparametric/MomentProblems/BoundedOutcomeEnvelope","t":2,"d":"The interior support point lies in (0,1): xᵥ = (μᵥ² − 2 μᵥ v² + v²)/(2 μᵥ (1 − μᵥ)) ∈ (0,1) for μᵥ ∈ (v², v). Proof: nlinarith from v² < μᵥ < v < 1.","s":"∀ (v : ℝ), 0 < v → v < 1 → Causalean.Stat.MomentProblems.BoundedOutcomeEnvelope.extremalSupp v ∈ Set.Ioo 0 1","l":54,"f":"Causalean/Stat/Nonparametric/MomentProblems/BoundedOutcomeEnvelope/Attainment.lean"},{"n":"Causalean.Stat.MomentProblems.BoundedOutcomeEnvelope.extremalW0","k":"def","m":"Causalean.Stat.Nonparametric.MomentProblems.BoundedOutcomeEnvelope.Attainment","a":"Stat/Nonparametric/MomentProblems/BoundedOutcomeEnvelope","t":1,"d":"Weight w₀ on the point 0.","s":"ℝ → ℝ","l":45,"f":"Causalean/Stat/Nonparametric/MomentProblems/BoundedOutcomeEnvelope/Attainment.lean"},{"n":"Causalean.Stat.MomentProblems.BoundedOutcomeEnvelope.extremalW1","k":"def","m":"Causalean.Stat.Nonparametric.MomentProblems.BoundedOutcomeEnvelope.Attainment","a":"Stat/Nonparametric/MomentProblems/BoundedOutcomeEnvelope","t":1,"d":"Weight w₁ on the interior point xᵥ.","s":"ℝ → ℝ","l":37,"f":"Causalean/Stat/Nonparametric/MomentProblems/BoundedOutcomeEnvelope/Attainment.lean"},{"n":"Causalean.Stat.MomentProblems.BoundedOutcomeEnvelope.extremalW2","k":"def","m":"Causalean.Stat.Nonparametric.MomentProblems.BoundedOutcomeEnvelope.Attainment","a":"Stat/Nonparametric/MomentProblems/BoundedOutcomeEnvelope","t":1,"d":"Weight w₂ on the point 1.","s":"ℝ → ℝ","l":41,"f":"Causalean/Stat/Nonparametric/MomentProblems/BoundedOutcomeEnvelope/Attainment.lean"},{"n":"Causalean.Stat.MomentProblems.BoundedOutcomeEnvelope.extremalW_nonneg","k":"theorem","m":"Causalean.Stat.Nonparametric.MomentProblems.BoundedOutcomeEnvelope.Attainment","a":"Stat/Nonparametric/MomentProblems/BoundedOutcomeEnvelope","t":2,"d":"The three weights are nonnegative for v ∈ (0,1). Proof: w₁ > 0 since μᵥ > v² and xᵥ ∈ (0,1); w₂ ≥ 0 since v² ≥ μᵥ xᵥ; w₀ ≥ 0 since w₁ + w₂ ≤ 1. All via nlinarith from v² < μᵥ < v and extremalSupp_mem.","s":"∀ (v : ℝ), 0 < v → v < 1 → 0 ≤ Causalean.Stat.MomentProblems.BoundedOutcomeEnvelope.extremalW0 v ∧ 0 ≤ Causalean.Stat.MomentProblems.BoundedOutcomeEnvelope.extremalW1 v ∧ 0 ≤ Causalean.Stat.MomentProblems.BoundedOutcomeEnvelope.extremalW2 v","l":81,"f":"Causalean/Stat/Nonparametric/MomentProblems/BoundedOutcomeEnvelope/Attainment.lean"},{"n":"Causalean.Stat.MomentProblems.BoundedOutcomeEnvelope.finiteMoment4_of_admissible","k":"theorem","m":"Causalean.Stat.Nonparametric.MomentProblems.BoundedOutcomeEnvelope.Bounds","a":"Stat/Nonparametric/MomentProblems/BoundedOutcomeEnvelope","t":2,"d":"Every power y ↦ yᵏ (k ≤ 4) is integrable against an admissible law: on [0,1] we have |yᵏ| ≤ 1, and μ is a finite measure, so Integrable.mono' against the constant 1 applies.","s":"∀ {v : ℝ} {μ : MeasureTheory.Measure ℝ}, Causalean.Stat.MomentProblems.BoundedOutcomeEnvelope.Admissible v μ → Causalean.Stat.MomentProblems.ResidualQuadratic.MeasureBridge.FiniteMoment4 μ","l":38,"f":"Causalean/Stat/Nonparametric/MomentProblems/BoundedOutcomeEnvelope/Bounds.lean"},{"n":"Causalean.Stat.MomentProblems.BoundedOutcomeEnvelope.hasDerivAt_envelopeQuartic","k":"theorem","m":"Causalean.Stat.Nonparametric.MomentProblems.BoundedOutcomeEnvelope.QuarticRoot","a":"Stat/Nonparametric/MomentProblems/BoundedOutcomeEnvelope","t":2,"d":"Derivative of the quartic. d/dt envelopeQuartic t q = 4 t³ − 6 t² + 4 q t − 2 q².","s":"∀ (t q : ℝ), HasDerivAt (fun s => Causalean.Stat.MomentProblems.ResidualQuadratic.MomentAlgebra.envelopeQuartic s q) (4 * t ^ 3 - 6 * t ^ 2 + 4 * q * t - 2 * q ^ 2) t","l":68,"f":"Causalean/Stat/Nonparametric/MomentProblems/BoundedOutcomeEnvelope/QuarticRoot.lean"},{"n":"Causalean.Stat.MomentProblems.BoundedOutcomeEnvelope.interior_quartic_exists","k":"theorem","m":"Causalean.Stat.Nonparametric.MomentProblems.BoundedOutcomeEnvelope.QuarticRoot","a":"Stat/Nonparametric/MomentProblems/BoundedOutcomeEnvelope","t":2,"d":"Interior existence. For v ∈ (0,1) and q = v², the quartic has a root strictly inside (v², v). Proof: the sign change envelopeQuartic (v²) (v²) > 0, envelopeQuartic v (v²) < 0 plus continuity, via the intermediate value theorem (intermediate_value_Ioo').","s":"∀ (v : ℝ), 0 < v → v < 1 → ∃ u ∈ Set.Ioo (v ^ 2) v, Causalean.Stat.MomentProblems.ResidualQuadratic.MomentAlgebra.envelopeQuartic u (v ^ 2) = 0","l":87,"f":"Causalean/Stat/Nonparametric/MomentProblems/BoundedOutcomeEnvelope/QuarticRoot.lean"},{"n":"Causalean.Stat.MomentProblems.BoundedOutcomeEnvelope.interior_quartic_unique_root","k":"theorem","m":"Causalean.Stat.Nonparametric.MomentProblems.BoundedOutcomeEnvelope.QuarticRoot","a":"Stat/Nonparametric/MomentProblems/BoundedOutcomeEnvelope","t":1,"d":"Unique interior root of the FOC quartic. For v strictly between 0 and 1 (write q = v²), there is a unique μᵥ ∈ (v², v) with envelopeQuartic μᵥ (v²) = 0. This μᵥ is the envelope maximizer selecting ρ(v). Existence is interior_quartic_exists; uniqueness follows from strict antitonicity (strictAntiOn_envelopeQuartic), whose InjOn forces two roots to coincide.","s":"∀ (v : ℝ), 0 < v → v < 1 → ∃! u, u ∈ Set.Ioo (v ^ 2) v ∧ Causalean.Stat.MomentProblems.ResidualQuadratic.MomentAlgebra.envelopeQuartic u (v ^ 2) = 0","l":137,"f":"Causalean/Stat/Nonparametric/MomentProblems/BoundedOutcomeEnvelope/QuarticRoot.lean"},{"n":"Causalean.Stat.MomentProblems.BoundedOutcomeEnvelope.l2ResidualQuadratic_eq_zero_of_degenerate","k":"theorem","m":"Causalean.Stat.Nonparametric.MomentProblems.BoundedOutcomeEnvelope.Bounds","a":"Stat/Nonparametric/MomentProblems/BoundedOutcomeEnvelope","t":2,"d":"In the degenerate case m² = q the closed-form residual is 0 (the Hankel ratio's denominator m₁² − m₂ vanishes, so momentResidual _ _ _ _ = _ / 0 = 0).","s":"∀ {μ : MeasureTheory.Measure ℝ}, Causalean.Stat.MomentProblems.ResidualQuadratic.MeasureBridge.moment μ 1 ^ 2 = Causalean.Stat.MomentProblems.ResidualQuadratic.MeasureBridge.moment μ 2 → Causalean.Stat.MomentProblems.ResidualQuadratic.MeasureBridge.l2ResidualQuadratic μ = 0","l":124,"f":"Causalean/Stat/Nonparametric/MomentProblems/BoundedOutcomeEnvelope/Bounds.lean"},{"n":"Causalean.Stat.MomentProblems.BoundedOutcomeEnvelope.l2ResidualQuadratic_le_rho","k":"theorem","m":"Causalean.Stat.Nonparametric.MomentProblems.BoundedOutcomeEnvelope.Bounds","a":"Stat/Nonparametric/MomentProblems/BoundedOutcomeEnvelope","t":1,"d":"Measure-level sharp upper bound. For v strictly between 0 and 1 and every admissible probability law μ on [0,1] with second moment v², the closed-form residual is at most the envelope value: l2ResidualQuadratic μ ≤ rhoEnvelope v.","s":"∀ (v : ℝ) (μ : MeasureTheory.Measure ℝ), Causalean.Stat.MomentProblems.BoundedOutcomeEnvelope.Admissible v μ → 0 < v → v < 1 → Causalean.Stat.MomentProblems.ResidualQuadratic.MeasureBridge.l2ResidualQuadratic μ ≤ Causalean.Stat.MomentProblems.BoundedOutcomeEnvelope.rhoEnvelope v","l":132,"f":"Causalean/Stat/Nonparametric/MomentProblems/BoundedOutcomeEnvelope/Bounds.lean"},{"n":"Causalean.Stat.MomentProblems.BoundedOutcomeEnvelope.maximizingRoot","k":"def","m":"Causalean.Stat.Nonparametric.MomentProblems.BoundedOutcomeEnvelope.Defs","a":"Stat/Nonparametric/MomentProblems/BoundedOutcomeEnvelope","t":1,"d":"The envelope maximizer μᵥ: the unique root of envelopeQuartic · (v²) in (v², v), extracted by classical choice from interior_quartic_exists. Outside the admissible range v ∈ (0,1) it is set to 0 (junk value).","s":"ℝ → ℝ","l":32,"f":"Causalean/Stat/Nonparametric/MomentProblems/BoundedOutcomeEnvelope/Defs.lean"},{"n":"Causalean.Stat.MomentProblems.BoundedOutcomeEnvelope.maximizingRoot_mem","k":"theorem","m":"Causalean.Stat.Nonparametric.MomentProblems.BoundedOutcomeEnvelope.Defs","a":"Stat/Nonparametric/MomentProblems/BoundedOutcomeEnvelope","t":2,"d":"maximizingRoot v lies in the open interval (v², v).","s":"∀ (v : ℝ), 0 < v → v < 1 → Causalean.Stat.MomentProblems.BoundedOutcomeEnvelope.maximizingRoot v ∈ Set.Ioo (v ^ 2) v","l":38,"f":"Causalean/Stat/Nonparametric/MomentProblems/BoundedOutcomeEnvelope/Defs.lean"},{"n":"Causalean.Stat.MomentProblems.BoundedOutcomeEnvelope.maximizingRoot_quartic","k":"theorem","m":"Causalean.Stat.Nonparametric.MomentProblems.BoundedOutcomeEnvelope.Defs","a":"Stat/Nonparametric/MomentProblems/BoundedOutcomeEnvelope","t":2,"d":"maximizingRoot v is a root of the FOC quartic at q = v².","s":"∀ (v : ℝ), 0 < v → v < 1 → Causalean.Stat.MomentProblems.ResidualQuadratic.MomentAlgebra.envelopeQuartic (Causalean.Stat.MomentProblems.BoundedOutcomeEnvelope.maximizingRoot v) (v ^ 2) = 0","l":44,"f":"Causalean/Stat/Nonparametric/MomentProblems/BoundedOutcomeEnvelope/Defs.lean"},{"n":"Causalean.Stat.MomentProblems.BoundedOutcomeEnvelope.moment1_sq_le_moment2","k":"theorem","m":"Causalean.Stat.Nonparametric.MomentProblems.BoundedOutcomeEnvelope.Bounds","a":"Stat/Nonparametric/MomentProblems/BoundedOutcomeEnvelope","t":2,"d":"m² ≤ q: the variance is nonnegative. For a probability measure, (∫ y)² ≤ ∫ y² (Cauchy–Schwarz / Jensen).","s":"∀ {v : ℝ} {μ : MeasureTheory.Measure ℝ}, Causalean.Stat.MomentProblems.BoundedOutcomeEnvelope.Admissible v μ → Causalean.Stat.MomentProblems.ResidualQuadratic.MeasureBridge.moment μ 1 ^ 2 ≤ Causalean.Stat.MomentProblems.ResidualQuadratic.MeasureBridge.moment μ 2","l":62,"f":"Causalean/Stat/Nonparametric/MomentProblems/BoundedOutcomeEnvelope/Bounds.lean"},{"n":"Causalean.Stat.MomentProblems.BoundedOutcomeEnvelope.moment2_le_moment1","k":"theorem","m":"Causalean.Stat.Nonparametric.MomentProblems.BoundedOutcomeEnvelope.Bounds","a":"Stat/Nonparametric/MomentProblems/BoundedOutcomeEnvelope","t":2,"d":"q ≤ m: the second moment is at most the first, because y² ≤ y a.e. on [0,1]. Uses integral_mono_ae on the a.e. support bound plus integrability of y, y².","s":"∀ {v : ℝ} {μ : MeasureTheory.Measure ℝ}, Causalean.Stat.MomentProblems.BoundedOutcomeEnvelope.Admissible v μ → Causalean.Stat.MomentProblems.ResidualQuadratic.MeasureBridge.moment μ 2 ≤ Causalean.Stat.MomentProblems.ResidualQuadratic.MeasureBridge.moment μ 1","l":51,"f":"Causalean/Stat/Nonparametric/MomentProblems/BoundedOutcomeEnvelope/Bounds.lean"},{"n":"Causalean.Stat.MomentProblems.BoundedOutcomeEnvelope.residualSet","k":"def","m":"Causalean.Stat.Nonparametric.MomentProblems.BoundedOutcomeEnvelope.Defs","a":"Stat/Nonparametric/MomentProblems/BoundedOutcomeEnvelope","t":1,"d":"The set of residual values realized by admissible laws: { r | ∃ μ, Admissible v μ ∧ r = l2ResidualQuadratic μ }. The main theorem is IsLUB (residualSet v) (rhoEnvelope v).","s":"ℝ → Set ℝ","l":81,"f":"Causalean/Stat/Nonparametric/MomentProblems/BoundedOutcomeEnvelope/Defs.lean"},{"n":"Causalean.Stat.MomentProblems.BoundedOutcomeEnvelope.rhoEnvelope","k":"def","m":"Causalean.Stat.Nonparametric.MomentProblems.BoundedOutcomeEnvelope.Defs","a":"Stat/Nonparametric/MomentProblems/BoundedOutcomeEnvelope","t":1,"d":"The measure-level residual envelope ρ(v) = momentEnvelope μᵥ (v²), evaluated at the maximizing support parameter μᵥ = maximizingRoot v.","s":"ℝ → ℝ","l":50,"f":"Causalean/Stat/Nonparametric/MomentProblems/BoundedOutcomeEnvelope/Defs.lean"},{"n":"Causalean.Stat.MomentProblems.BoundedOutcomeEnvelope.rhoEnvelope_pos","k":"theorem","m":"Causalean.Stat.Nonparametric.MomentProblems.BoundedOutcomeEnvelope.Defs","a":"Stat/Nonparametric/MomentProblems/BoundedOutcomeEnvelope","t":1,"d":"The envelope value ρ(v) is strictly positive for v strictly between 0 and 1: momentEnvelope μᵥ (v²) = (μᵥ − v²)(v² − μᵥ²) / (4 μᵥ (1 − μᵥ)) has all four factors positive when μᵥ ∈ (v², v).","s":"∀ (v : ℝ), 0 < v → v < 1 → 0 < Causalean.Stat.MomentProblems.BoundedOutcomeEnvelope.rhoEnvelope v","l":54,"f":"Causalean/Stat/Nonparametric/MomentProblems/BoundedOutcomeEnvelope/Defs.lean"},{"n":"Causalean.Stat.MomentProblems.BoundedOutcomeEnvelope.rho_envelope_attained","k":"theorem","m":"Causalean.Stat.Nonparametric.MomentProblems.BoundedOutcomeEnvelope.Attainment","a":"Stat/Nonparametric/MomentProblems/BoundedOutcomeEnvelope","t":1,"d":"Envelope attainment. For v strictly between 0 and 1, there is an admissible law whose residual is exactly ρ(v). Together with the upper bound this makes ρ(v) the supremum.","s":"∀ (v : ℝ), 0 < v → v < 1 → ∃ μ, Causalean.Stat.MomentProblems.BoundedOutcomeEnvelope.Admissible v μ ∧ Causalean.Stat.MomentProblems.ResidualQuadratic.MeasureBridge.l2ResidualQuadratic μ = Causalean.Stat.MomentProblems.BoundedOutcomeEnvelope.rhoEnvelope v","l":345,"f":"Causalean/Stat/Nonparametric/MomentProblems/BoundedOutcomeEnvelope/Attainment.lean"},{"n":"Causalean.Stat.MomentProblems.BoundedOutcomeEnvelope.rho_envelope_isLUB","k":"theorem","m":"Causalean.Stat.Nonparametric.MomentProblems.BoundedOutcomeEnvelope.Envelope","a":"Stat/Nonparametric/MomentProblems/BoundedOutcomeEnvelope","t":1,"d":"Measure-level sharp envelope (IsLUB). For v strictly between 0 and 1, rhoEnvelope v is the least upper bound of the set of residuals l2ResidualQuadratic μ over admissible laws μ. Equivalently: the sup over all probability measures on [0,1] with ∫ y² ∂μ = v² of the L² residual of y² on span{1, y} equals the closed form ρ(v), and is attained (by the extremal three-point law).","s":"∀ (v : ℝ), 0 < v → v < 1 → IsLUB (Causalean.Stat.MomentProblems.BoundedOutcomeEnvelope.residualSet v) (Causalean.Stat.MomentProblems.BoundedOutcomeEnvelope.rhoEnvelope v)","l":31,"f":"Causalean/Stat/Nonparametric/MomentProblems/BoundedOutcomeEnvelope/Envelope.lean"},{"n":"Causalean.Stat.MomentProblems.BoundedOutcomeEnvelope.strictAntiOn_envelopeQuartic","k":"theorem","m":"Causalean.Stat.Nonparametric.MomentProblems.BoundedOutcomeEnvelope.QuarticRoot","a":"Stat/Nonparametric/MomentProblems/BoundedOutcomeEnvelope","t":2,"d":"t ↦ envelopeQuartic t (v²) is strictly antitone on Icc (v²) v.","s":"∀ (v : ℝ), 0 < v → v < 1 → StrictAntiOn (fun t => Causalean.Stat.MomentProblems.ResidualQuadratic.MomentAlgebra.envelopeQuartic t (v ^ 2)) (Set.Icc (v ^ 2) v)","l":126,"f":"Causalean/Stat/Nonparametric/MomentProblems/BoundedOutcomeEnvelope/QuarticRoot.lean"},{"n":"Causalean.Stat.MomentProblems.IsGaussianLaw","k":"def","m":"Causalean.Stat.Nonparametric.MomentProblems.Cumulant","a":"Stat/Nonparametric/MomentProblems","t":1,"d":"A probability law on the real line is a Gaussian law when it is a normal distribution: there is a mean and a (possibly zero) variance whose normal distribution is exactly this law. Allowing zero variance means the point masses (Diracs) count as degenerate Gaussians, so the negation \"not a Gaussian law\" is the strongest possible non-Gaussianity requirement.","s":"MeasureTheory.Measure ℝ → Prop","l":34,"f":"Causalean/Stat/Nonparametric/MomentProblems/Cumulant.lean"},{"n":"Causalean.Stat.MomentProblems.ResidualQuadratic.MeasureBridge.FiniteMoment4","k":"structure","m":"Causalean.Stat.Nonparametric.MomentProblems.ResidualQuadratic.MeasureBridge","a":"Stat/Nonparametric/MomentProblems/ResidualQuadratic","t":1,"d":"Finite-fourth-moment hypothesis bundle: integrability of y, of y², of y³, and of y⁴ against μ. For a probability measure with ∫ y⁴ ∂μ < ∞ all four hold, so this is exactly \"μ has a finite fourth moment\".","s":"MeasureTheory.Measure ℝ → Prop","l":57,"f":"Causalean/Stat/Nonparametric/MomentProblems/ResidualQuadratic/MeasureBridge.lean"},{"n":"Causalean.Stat.MomentProblems.ResidualQuadratic.MeasureBridge.iInf_residualQuad","k":"theorem","m":"Causalean.Stat.Nonparametric.MomentProblems.ResidualQuadratic.MeasureBridge","a":"Stat/Nonparametric/MomentProblems/ResidualQuadratic","t":1,"d":"Headline infimum identity. For a probability measure μ on ℝ with a finite fourth moment and first moment squared strictly below the second moment (positive variance), the infimum, over every intercept and slope, of the mean squared residual of fitting b₀ + b₁y to y² equals the closed-form Hankel-determinant residual l2ResidualQuadratic μ, attained at the least-squares-optimal coefficients (optIntercept μ, optSlope μ):","s":"∀ (μ : MeasureTheory.Measure ℝ) [MeasureTheory.IsProbabilityMeasure μ], Causalean.Stat.MomentProblems.ResidualQuadratic.MeasureBridge.FiniteMoment4 μ → Causalean.Stat.MomentProblems.ResidualQuadratic.MeasureBridge.moment μ 1 ^ 2 < Causalean.Stat.MomentProblems.ResidualQuadratic.MeasureBridge.moment μ 2 → ⨅ b₀, ⨅ b₁, Causalean.Stat.MomentProblems.ResidualQuadratic.MeasureBridge.residualQuad μ b₀ b₁ = Causalean.Stat.MomentProblems.ResidualQuadratic.MeasureBridge.l2ResidualQuadratic μ","l":171,"f":"Causalean/Stat/Nonparametric/MomentProblems/ResidualQuadratic/MeasureBridge.lean"},{"n":"Causalean.Stat.MomentProblems.ResidualQuadratic.MeasureBridge.l2ResidualQuadratic","k":"def","m":"Causalean.Stat.Nonparametric.MomentProblems.ResidualQuadratic.MeasureBridge","a":"Stat/Nonparametric/MomentProblems/ResidualQuadratic","t":1,"d":"The closed-form minimal residual r(μ) = (m₄ − m₂²) − (m₃ − m₁ m₂)² / (m₂ − m₁²), expressed as the Hankel-determinant ratio MomentAlgebra.momentResidual of the raw moments m₁, m₂, m₃, m₄.","s":"MeasureTheory.Measure ℝ → ℝ","l":75,"f":"Causalean/Stat/Nonparametric/MomentProblems/ResidualQuadratic/MeasureBridge.lean"},{"n":"Causalean.Stat.MomentProblems.ResidualQuadratic.MeasureBridge.l2ResidualQuadratic_le","k":"theorem","m":"Causalean.Stat.Nonparametric.MomentProblems.ResidualQuadratic.MeasureBridge","a":"Stat/Nonparametric/MomentProblems/ResidualQuadratic","t":2,"d":"Lower bound (infimum). The closed-form residual is a lower bound of the integral objective at every choice of coefficients, when the design is non-degenerate. Together with residualQuad_opt_eq this identifies l2ResidualQuadratic μ as the minimum ⨅ b₀ b₁, ∫ (y² − b₀ − b₁ y)² ∂μ.","s":"∀ (μ : MeasureTheory.Measure ℝ) [MeasureTheory.IsProbabilityMeasure μ], Causalean.Stat.MomentProblems.ResidualQuadratic.MeasureBridge.FiniteMoment4 μ → Causalean.Stat.MomentProblems.ResidualQuadratic.MeasureBridge.moment μ 1 ^ 2 < Causalean.Stat.MomentProblems.ResidualQuadratic.MeasureBridge.moment μ 2 → ∀ (b₀ b₁ : ℝ), Causalean.Stat.MomentProblems.ResidualQuadratic.MeasureBridge.l2ResidualQuadratic μ ≤ Causalean.Stat.MomentProblems.ResidualQuadratic.MeasureBridge.residualQuad μ b₀ b₁","l":152,"f":"Causalean/Stat/Nonparametric/MomentProblems/ResidualQuadratic/MeasureBridge.lean"},{"n":"Causalean.Stat.MomentProblems.ResidualQuadratic.MeasureBridge.l2ResidualQuadratic_nonneg","k":"theorem","m":"Causalean.Stat.Nonparametric.MomentProblems.ResidualQuadratic.MeasureBridge","a":"Stat/Nonparametric/MomentProblems/ResidualQuadratic","t":2,"d":"Non-negativity. The closed-form residual is non-negative (it is the value at the optimum of an integral of a square), for a non-degenerate design.","s":"∀ (μ : MeasureTheory.Measure ℝ) [MeasureTheory.IsProbabilityMeasure μ], Causalean.Stat.MomentProblems.ResidualQuadratic.MeasureBridge.FiniteMoment4 μ → Causalean.Stat.MomentProblems.ResidualQuadratic.MeasureBridge.moment μ 1 ^ 2 < Causalean.Stat.MomentProblems.ResidualQuadratic.MeasureBridge.moment μ 2 → 0 ≤ Causalean.Stat.MomentProblems.ResidualQuadratic.MeasureBridge.l2ResidualQuadratic μ","l":163,"f":"Causalean/Stat/Nonparametric/MomentProblems/ResidualQuadratic/MeasureBridge.lean"},{"n":"Causalean.Stat.MomentProblems.ResidualQuadratic.MeasureBridge.moment","k":"def","m":"Causalean.Stat.Nonparametric.MomentProblems.ResidualQuadratic.MeasureBridge","a":"Stat/Nonparametric/MomentProblems/ResidualQuadratic","t":1,"d":"The raw k-th moment mₖ = ∫ yᵏ ∂μ of a measure μ on ℝ.","s":"MeasureTheory.Measure ℝ → ℕ → ℝ","l":54,"f":"Causalean/Stat/Nonparametric/MomentProblems/ResidualQuadratic/MeasureBridge.lean"},{"n":"Causalean.Stat.MomentProblems.ResidualQuadratic.MeasureBridge.optIntercept","k":"def","m":"Causalean.Stat.Nonparametric.MomentProblems.ResidualQuadratic.MeasureBridge","a":"Stat/Nonparametric/MomentProblems/ResidualQuadratic","t":1,"d":"The least-squares optimal intercept b₀* = (m₁ m₃ − m₂²)/(m₁² − m₂) = m₂ − b₁* m₁.","s":"MeasureTheory.Measure ℝ → ℝ","l":81,"f":"Causalean/Stat/Nonparametric/MomentProblems/ResidualQuadratic/MeasureBridge.lean"},{"n":"Causalean.Stat.MomentProblems.ResidualQuadratic.MeasureBridge.optSlope","k":"def","m":"Causalean.Stat.Nonparametric.MomentProblems.ResidualQuadratic.MeasureBridge","a":"Stat/Nonparametric/MomentProblems/ResidualQuadratic","t":1,"d":"The least-squares optimal slope b₁* = (m₃ − m₁ m₂)/(m₂ − m₁²).","s":"MeasureTheory.Measure ℝ → ℝ","l":86,"f":"Causalean/Stat/Nonparametric/MomentProblems/ResidualQuadratic/MeasureBridge.lean"},{"n":"Causalean.Stat.MomentProblems.ResidualQuadratic.MeasureBridge.residualQuad","k":"def","m":"Causalean.Stat.Nonparametric.MomentProblems.ResidualQuadratic.MeasureBridge","a":"Stat/Nonparametric/MomentProblems/ResidualQuadratic","t":1,"d":"The quadratic regression objective in L²(μ): the mean squared residual of the linear fit b₀ + b₁ y to y², i.e. ∫ (y² − b₀ − b₁ y)² ∂μ.","s":"MeasureTheory.Measure ℝ → ℝ → ℝ → ℝ","l":70,"f":"Causalean/Stat/Nonparametric/MomentProblems/ResidualQuadratic/MeasureBridge.lean"},{"n":"Causalean.Stat.MomentProblems.ResidualQuadratic.MeasureBridge.residualQuad_eq","k":"theorem","m":"Causalean.Stat.Nonparametric.MomentProblems.ResidualQuadratic.MeasureBridge","a":"Stat/Nonparametric/MomentProblems/ResidualQuadratic","t":2,"d":"Bridge (integral → moments). For a probability measure with finite fourth moment, the integral objective equals the moment polynomial: expanding (y² − b₀ − b₁ y)² = y⁴ − 2b₁ y³ + (b₁² − 2b₀) y² + 2b₀b₁ y + b₀² and integrating term by term (using ∫ 1 ∂μ = 1),","s":"∀ (μ : MeasureTheory.Measure ℝ) [MeasureTheory.IsProbabilityMeasure μ], Causalean.Stat.MomentProblems.ResidualQuadratic.MeasureBridge.FiniteMoment4 μ → ∀ (b₀ b₁ : ℝ), Causalean.Stat.MomentProblems.ResidualQuadratic.MeasureBridge.residualQuad μ b₀ b₁ = Causalean.Stat.MomentProblems.ResidualQuadratic.MomentAlgebra.residualQuad (Causalean.Stat.MomentProblems.ResidualQuadratic.MeasureBridge.moment μ 1) (Causalean.Stat.MomentProblems.ResidualQuadratic.MeasureBridge.moment μ 2) (Causalean.Stat.MomentProblems.ResidualQuadratic.MeasureBridge.moment μ 3) (Causalean.Stat.MomentProblems.ResidualQuadratic.MeasureBridge.moment μ 4) b₀ b₁","l":91,"f":"Causalean/Stat/Nonparametric/MomentProblems/ResidualQuadratic/MeasureBridge.lean"},{"n":"Causalean.Stat.MomentProblems.ResidualQuadratic.MeasureBridge.residualQuad_opt_eq","k":"theorem","m":"Causalean.Stat.Nonparametric.MomentProblems.ResidualQuadratic.MeasureBridge","a":"Stat/Nonparametric/MomentProblems/ResidualQuadratic","t":2,"d":"Attainment. At the optimal coefficients (optIntercept μ, optSlope μ) the integral objective attains the closed-form residual l2ResidualQuadratic μ, whenever the design is non-degenerate (m₁² < m₂).","s":"∀ (μ : MeasureTheory.Measure ℝ) [MeasureTheory.IsProbabilityMeasure μ], Causalean.Stat.MomentProblems.ResidualQuadratic.MeasureBridge.FiniteMoment4 μ → Causalean.Stat.MomentProblems.ResidualQuadratic.MeasureBridge.moment μ 1 ^ 2 < Causalean.Stat.MomentProblems.ResidualQuadratic.MeasureBridge.moment μ 2 → Causalean.Stat.MomentProblems.ResidualQuadratic.MeasureBridge.residualQuad μ (Causalean.Stat.MomentProblems.ResidualQuadratic.MeasureBridge.optIntercept μ) (Causalean.Stat.MomentProblems.ResidualQuadratic.MeasureBridge.optSlope μ) = Causalean.Stat.MomentProblems.ResidualQuadratic.MeasureBridge.l2ResidualQuadratic μ","l":142,"f":"Causalean/Stat/Nonparametric/MomentProblems/ResidualQuadratic/MeasureBridge.lean"},{"n":"Causalean.Stat.MomentProblems.ResidualQuadratic.MomentAlgebra.crossMoment","k":"def","m":"Causalean.Stat.Nonparametric.MomentProblems.ResidualQuadratic.MomentAlgebra","a":"Stat/Nonparametric/MomentProblems/ResidualQuadratic","t":1,"d":"The certificate cross moment ∫ y (1 − y) (y − xᵥ)² dμ of a law with moments (1, m, q, m₃, m₄), expanded in the moments (xᵥ = extremalMid m q). It is nonnegative for every law supported in [0,1] — the integral of a nonnegative polynomial — and equals the certificate slack momentEnvelope m q − residualQuad m q m₃ m₄ b₀ b₁.","s":"ℝ → ℝ → ℝ → ℝ → ℝ","l":228,"f":"Causalean/Stat/Nonparametric/MomentProblems/ResidualQuadratic/MomentAlgebra.lean"},{"n":"Causalean.Stat.MomentProblems.ResidualQuadratic.MomentAlgebra.envelopeQuartic","k":"def","m":"Causalean.Stat.Nonparametric.MomentProblems.ResidualQuadratic.MomentAlgebra","a":"Stat/Nonparametric/MomentProblems/ResidualQuadratic","t":1,"d":"The quartic whose stationary root in (q, √q) is the envelope maximizer t = μᵥ.","s":"ℝ → ℝ → ℝ","l":130,"f":"Causalean/Stat/Nonparametric/MomentProblems/ResidualQuadratic/MomentAlgebra.lean"},{"n":"Causalean.Stat.MomentProblems.ResidualQuadratic.MomentAlgebra.extremalCoeff0","k":"def","m":"Causalean.Stat.Nonparametric.MomentProblems.ResidualQuadratic.MomentAlgebra","a":"Stat/Nonparametric/MomentProblems/ResidualQuadratic","t":1,"d":"The intercept b₀ of the certificate's linear fit: the regression coefficient of y² on 1 for the extremal three-point law at first moment m, second moment q.","s":"ℝ → ℝ → ℝ","l":220,"f":"Causalean/Stat/Nonparametric/MomentProblems/ResidualQuadratic/MomentAlgebra.lean"},{"n":"Causalean.Stat.MomentProblems.ResidualQuadratic.MomentAlgebra.extremalCoeff1","k":"def","m":"Causalean.Stat.Nonparametric.MomentProblems.ResidualQuadratic.MomentAlgebra","a":"Stat/Nonparametric/MomentProblems/ResidualQuadratic","t":1,"d":"The slope b₁ of the certificate's linear fit: the regression coefficient of y² on y for the extremal three-point law at first moment m, second moment q.","s":"ℝ → ℝ → ℝ","l":224,"f":"Causalean/Stat/Nonparametric/MomentProblems/ResidualQuadratic/MomentAlgebra.lean"},{"n":"Causalean.Stat.MomentProblems.ResidualQuadratic.MomentAlgebra.extremalM1","k":"def","m":"Causalean.Stat.Nonparametric.MomentProblems.ResidualQuadratic.MomentAlgebra","a":"Stat/Nonparametric/MomentProblems/ResidualQuadratic","t":1,"d":"First moment M₁ = ∫ y of the extremal three-point law: the support parameter t = μᵥ.","s":"ℝ → ℝ","l":117,"f":"Causalean/Stat/Nonparametric/MomentProblems/ResidualQuadratic/MomentAlgebra.lean"},{"n":"Causalean.Stat.MomentProblems.ResidualQuadratic.MomentAlgebra.extremalM3","k":"def","m":"Causalean.Stat.Nonparametric.MomentProblems.ResidualQuadratic.MomentAlgebra","a":"Stat/Nonparametric/MomentProblems/ResidualQuadratic","t":1,"d":"Third moment M₃ = ∫ y³ of the extremal three-point law.","s":"ℝ → ℝ → ℝ","l":120,"f":"Causalean/Stat/Nonparametric/MomentProblems/ResidualQuadratic/MomentAlgebra.lean"},{"n":"Causalean.Stat.MomentProblems.ResidualQuadratic.MomentAlgebra.extremalM4","k":"def","m":"Causalean.Stat.Nonparametric.MomentProblems.ResidualQuadratic.MomentAlgebra","a":"Stat/Nonparametric/MomentProblems/ResidualQuadratic","t":1,"d":"Fourth moment M₄ = ∫ y⁴ of the extremal three-point law.","s":"ℝ → ℝ → ℝ","l":124,"f":"Causalean/Stat/Nonparametric/MomentProblems/ResidualQuadratic/MomentAlgebra.lean"},{"n":"Causalean.Stat.MomentProblems.ResidualQuadratic.MomentAlgebra.extremalMid","k":"def","m":"Causalean.Stat.Nonparametric.MomentProblems.ResidualQuadratic.MomentAlgebra","a":"Stat/Nonparametric/MomentProblems/ResidualQuadratic","t":1,"d":"The interior support point xᵥ = (m² − 2 m q + q) / (2 m (1 − m)) of the extremal three-point law realizing first moment m and second moment q.","s":"ℝ → ℝ → ℝ","l":216,"f":"Causalean/Stat/Nonparametric/MomentProblems/ResidualQuadratic/MomentAlgebra.lean"},{"n":"Causalean.Stat.MomentProblems.ResidualQuadratic.MomentAlgebra.extremalResidual_eq_envelope","k":"theorem","m":"Causalean.Stat.Nonparametric.MomentProblems.ResidualQuadratic.MomentAlgebra","a":"Stat/Nonparametric/MomentProblems/ResidualQuadratic","t":2,"d":"Family residual identity. For every nondegenerate support parameter t (with t ≠ 0, t ≠ 1, and second moment q ≠ t²), the residual variance of regressing y² on {1, y} under the three-point law with moments (t, q, M₃, M₄) equals the closed form momentEnvelope t q. In particular ρ(v) is attained by an admissible law at second moment q = v².","s":"∀ (t q : ℝ), t ≠ 0 → t ≠ 1 → t ^ 2 ≠ q → Causalean.Stat.MomentProblems.ResidualQuadratic.MomentAlgebra.momentResidual (Causalean.Stat.MomentProblems.ResidualQuadratic.MomentAlgebra.extremalM1 t) q (Causalean.Stat.MomentProblems.ResidualQuadratic.MomentAlgebra.extremalM3 t q) (Causalean.Stat.MomentProblems.ResidualQuadratic.MomentAlgebra.extremalM4 t q) = Causalean.Stat.MomentProblems.ResidualQuadratic.MomentAlgebra.momentEnvelope t q","l":133,"f":"Causalean/Stat/Nonparametric/MomentProblems/ResidualQuadratic/MomentAlgebra.lean"},{"n":"Causalean.Stat.MomentProblems.ResidualQuadratic.MomentAlgebra.momentEnvelope","k":"def","m":"Causalean.Stat.Nonparametric.MomentProblems.ResidualQuadratic.MomentAlgebra","a":"Stat/Nonparametric/MomentProblems/ResidualQuadratic","t":1,"d":"The envelope value ρ, as a function of the support parameter t = μᵥ and q = v²: ρ = ((t − q)(q − t²)) / (4 t (1 − t)).","s":"ℝ → ℝ → ℝ","l":113,"f":"Causalean/Stat/Nonparametric/MomentProblems/ResidualQuadratic/MomentAlgebra.lean"},{"n":"Causalean.Stat.MomentProblems.ResidualQuadratic.MomentAlgebra.momentEnvelope_hasDerivAt","k":"theorem","m":"Causalean.Stat.Nonparametric.MomentProblems.ResidualQuadratic.MomentAlgebra","a":"Stat/Nonparametric/MomentProblems/ResidualQuadratic","t":2,"d":"Stationarity / envelope FOC. As a function of the support parameter t, the family residual momentEnvelope t q has derivative envelopeQuartic t q / (4 t² (t − 1)²). Hence the quartic is exactly the numerator of dρ/dt, and the envelope's stationary points are its roots.","s":"∀ (t q : ℝ), t ≠ 0 → t ≠ 1 → HasDerivAt (fun s => Causalean.Stat.MomentProblems.ResidualQuadratic.MomentAlgebra.momentEnvelope s q) (Causalean.Stat.MomentProblems.ResidualQuadratic.MomentAlgebra.envelopeQuartic t q / (4 * t ^ 2 * (t - 1) ^ 2)) t","l":157,"f":"Causalean/Stat/Nonparametric/MomentProblems/ResidualQuadratic/MomentAlgebra.lean"},{"n":"Causalean.Stat.MomentProblems.ResidualQuadratic.MomentAlgebra.momentEnvelope_le_root","k":"theorem","m":"Causalean.Stat.Nonparametric.MomentProblems.ResidualQuadratic.MomentAlgebra","a":"Stat/Nonparametric/MomentProblems/ResidualQuadratic","t":2,"d":"Envelope maximality. Among admissible first moments the envelope momentEnvelope · q is maximized at a quartic root u = μᵥ: if q < u, u² < q, and envelopeQuartic u q = 0, then for every admissible first moment m with q ≤ m, m² < q, one has momentEnvelope m q ≤ momentEnvelope u q.","s":"∀ (m q u : ℝ), 0 < q → q < 1 → q ≤ m → m ^ 2 < q → q < u → u ^ 2 < q → Causalean.Stat.MomentProblems.ResidualQuadratic.MomentAlgebra.envelopeQuartic u q = 0 → Causalean.Stat.MomentProblems.ResidualQuadratic.MomentAlgebra.momentEnvelope m q ≤ Causalean.Stat.MomentProblems.ResidualQuadratic.MomentAlgebra.momentEnvelope u q","l":271,"f":"Causalean/Stat/Nonparametric/MomentProblems/ResidualQuadratic/MomentAlgebra.lean"},{"n":"Causalean.Stat.MomentProblems.ResidualQuadratic.MomentAlgebra.momentEnvelope_stationary_of_quartic","k":"theorem","m":"Causalean.Stat.Nonparametric.MomentProblems.ResidualQuadratic.MomentAlgebra","a":"Stat/Nonparametric/MomentProblems/ResidualQuadratic","t":2,"d":"At a root t of the quartic, the family residual momentEnvelope · q is stationary. The maximizing root t = μᵥ ∈ (q, √q) therefore realizes the envelope value ρ(v).","s":"∀ (t q : ℝ), t ≠ 0 → t ≠ 1 → Causalean.Stat.MomentProblems.ResidualQuadratic.MomentAlgebra.envelopeQuartic t q = 0 → deriv (fun s => Causalean.Stat.MomentProblems.ResidualQuadratic.MomentAlgebra.momentEnvelope s q) t = 0","l":195,"f":"Causalean/Stat/Nonparametric/MomentProblems/ResidualQuadratic/MomentAlgebra.lean"},{"n":"Causalean.Stat.MomentProblems.ResidualQuadratic.MomentAlgebra.momentResidual","k":"def","m":"Causalean.Stat.Nonparametric.MomentProblems.ResidualQuadratic.MomentAlgebra","a":"Stat/Nonparametric/MomentProblems/ResidualQuadratic","t":1,"d":"The closed-form residual variance of regressing y² on {1, y}: the value at the optimal coefficients of residualQuad, equal to the ratio of the two leading Hankel determinants det [[1,m₁,m₂],[m₁,m₂,m₃],[m₂,m₃,m₄]] / det [[1,m₁],[m₁,m₂]].","s":"ℝ → ℝ → ℝ → ℝ → ℝ","l":63,"f":"Causalean/Stat/Nonparametric/MomentProblems/ResidualQuadratic/MomentAlgebra.lean"},{"n":"Causalean.Stat.MomentProblems.ResidualQuadratic.MomentAlgebra.momentResidual_le_envelope","k":"theorem","m":"Causalean.Stat.Nonparametric.MomentProblems.ResidualQuadratic.MomentAlgebra","a":"Stat/Nonparametric/MomentProblems/ResidualQuadratic","t":1,"d":"Conditional moment-level envelope bound. Consider raw moments m, q, m3, m4 of a law and a candidate root u. If the second moment q lies strictly between 0 and 1, if the first moment m is at least q while m² still lies below q, i.e. the design has positive variance, if a supplied cross-moment combination of m, q, m3, m4 is nonnegative, and if u satisfies q < u, u² < q, and solves the envelope's stationarity quartic exactly, then the moment-level residual variance of regressing y² on {1, y} under moments (m, q, m3, m4) is at most the envelope value momentEnvelope u q.","s":"∀ (m q m3 m4 u : ℝ), 0 < q → q < 1 → q ≤ m → m ^ 2 < q → 0 ≤ Causalean.Stat.MomentProblems.ResidualQuadratic.MomentAlgebra.crossMoment m q m3 m4 → q < u → u ^ 2 < q → Causalean.Stat.MomentProblems.ResidualQuadratic.MomentAlgebra.envelopeQuartic u q = 0 → Causalean.Stat.MomentProblems.ResidualQuadratic.MomentAlgebra.momentResidual m q m3 m4 ≤ Causalean.Stat.MomentProblems.ResidualQuadratic.MomentAlgebra.momentEnvelope u q","l":328,"f":"Causalean/Stat/Nonparametric/MomentProblems/ResidualQuadratic/MomentAlgebra.lean"},{"n":"Causalean.Stat.MomentProblems.ResidualQuadratic.MomentAlgebra.momentResidual_le_momentEnvelope","k":"theorem","m":"Causalean.Stat.Nonparametric.MomentProblems.ResidualQuadratic.MomentAlgebra","a":"Stat/Nonparametric/MomentProblems/ResidualQuadratic","t":2,"d":"Sharp per-instance bound. For an admissible moment tuple (1, m, q, m₃, m₄) of a law on [0,1] with positive variance (m² < q) and nonnegative cross moment (0 ≤ crossMoment m q m₃ m₄, automatic for any law on [0,1]), the residual variance of regressing y² on {1, y} is at most the envelope value at the law's own first moment: momentResidual m q m₃ m₄ ≤ momentEnvelope m q.","s":"∀ (m q m3 m4 : ℝ), 0 < m → m < 1 → m ^ 2 < q → 0 ≤ Causalean.Stat.MomentProblems.ResidualQuadratic.MomentAlgebra.crossMoment m q m3 m4 → Causalean.Stat.MomentProblems.ResidualQuadratic.MomentAlgebra.momentResidual m q m3 m4 ≤ Causalean.Stat.MomentProblems.ResidualQuadratic.MomentAlgebra.momentEnvelope m q","l":256,"f":"Causalean/Stat/Nonparametric/MomentProblems/ResidualQuadratic/MomentAlgebra.lean"},{"n":"Causalean.Stat.MomentProblems.ResidualQuadratic.MomentAlgebra.optIntercept","k":"def","m":"Causalean.Stat.Nonparametric.MomentProblems.ResidualQuadratic.MomentAlgebra","a":"Stat/Nonparametric/MomentProblems/ResidualQuadratic","t":1,"d":"The optimal intercept b₀* in the regression of y² on {1, y}.","s":"ℝ → ℝ → ℝ → ℝ","l":69,"f":"Causalean/Stat/Nonparametric/MomentProblems/ResidualQuadratic/MomentAlgebra.lean"},{"n":"Causalean.Stat.MomentProblems.ResidualQuadratic.MomentAlgebra.optSlope","k":"def","m":"Causalean.Stat.Nonparametric.MomentProblems.ResidualQuadratic.MomentAlgebra","a":"Stat/Nonparametric/MomentProblems/ResidualQuadratic","t":1,"d":"The optimal slope b₁* in the regression of y² on {1, y}.","s":"ℝ → ℝ → ℝ → ℝ","l":72,"f":"Causalean/Stat/Nonparametric/MomentProblems/ResidualQuadratic/MomentAlgebra.lean"},{"n":"Causalean.Stat.MomentProblems.ResidualQuadratic.MomentAlgebra.residualQuad","k":"def","m":"Causalean.Stat.Nonparametric.MomentProblems.ResidualQuadratic.MomentAlgebra","a":"Stat/Nonparametric/MomentProblems/ResidualQuadratic","t":1,"d":"The regression objective, in moment coordinates: the second moment about the linear fit b₀ + b₁ y of y², i.e. ∫ (y² − b₀ − b₁ y)² dμ expressed through the raw moments m₁ = ∫ y, m₂ = ∫ y², m₃ = ∫ y³, m₄ = ∫ y⁴ (with m₀ = 1).","s":"ℝ → ℝ → ℝ → ℝ → ℝ → ℝ → ℝ","l":57,"f":"Causalean/Stat/Nonparametric/MomentProblems/ResidualQuadratic/MomentAlgebra.lean"},{"n":"Causalean.Stat.MomentProblems.ResidualQuadratic.MomentAlgebra.residualQuad_extremalCoeff_eq","k":"theorem","m":"Causalean.Stat.Nonparametric.MomentProblems.ResidualQuadratic.MomentAlgebra","a":"Stat/Nonparametric/MomentProblems/ResidualQuadratic","t":2,"d":"Dual-certificate identity (integrated). For the certificate's linear-fit coefficients, the regression objective equals the envelope value minus the cross moment: residualQuad m q m₃ m₄ b₀ b₁ = momentEnvelope m q − crossMoment m q m₃ m₄. This is the moment-level form of the pointwise SOS certificate, a pure algebraic identity (ring) valid for m ≠ 0, m ≠ 1, m² ≠ q.","s":"∀ (m q m3 m4 : ℝ), m ≠ 0 → m ≠ 1 → m ^ 2 ≠ q → Causalean.Stat.MomentProblems.ResidualQuadratic.MomentAlgebra.residualQuad m q m3 m4 (Causalean.Stat.MomentProblems.ResidualQuadratic.MomentAlgebra.extremalCoeff0 m q) (Causalean.Stat.MomentProblems.ResidualQuadratic.MomentAlgebra.extremalCoeff1 m q) = Causalean.Stat.MomentProblems.ResidualQuadratic.MomentAlgebra.momentEnvelope m q - Causalean.Stat.MomentProblems.ResidualQuadratic.MomentAlgebra.crossMoment m q m3 m4","l":236,"f":"Causalean/Stat/Nonparametric/MomentProblems/ResidualQuadratic/MomentAlgebra.lean"},{"n":"Causalean.Stat.MomentProblems.ResidualQuadratic.MomentAlgebra.residualQuad_ge_momentResidual","k":"theorem","m":"Causalean.Stat.Nonparametric.MomentProblems.ResidualQuadratic.MomentAlgebra","a":"Stat/Nonparametric/MomentProblems/ResidualQuadratic","t":2,"d":"The closed-form residual momentResidual is a lower bound for the regression objective at every choice of coefficients, when the design is nondegenerate (m₁² < m₂, positive variance). Together with residualQuad_optimalCoeff this shows momentResidual is the minimum min_{b₀,b₁} ∫ (y² − b₀ − b₁ y)² dμ.","s":"∀ (m1 m2 m3 m4 b0 b1 : ℝ), m1 ^ 2 < m2 → Causalean.Stat.MomentProblems.ResidualQuadratic.MomentAlgebra.momentResidual m1 m2 m3 m4 ≤ Causalean.Stat.MomentProblems.ResidualQuadratic.MomentAlgebra.residualQuad m1 m2 m3 m4 b0 b1","l":88,"f":"Causalean/Stat/Nonparametric/MomentProblems/ResidualQuadratic/MomentAlgebra.lean"},{"n":"Causalean.Stat.MomentProblems.ResidualQuadratic.MomentAlgebra.residualQuad_optimalCoeff","k":"theorem","m":"Causalean.Stat.Nonparametric.MomentProblems.ResidualQuadratic.MomentAlgebra","a":"Stat/Nonparametric/MomentProblems/ResidualQuadratic","t":1,"d":"Attainment at the optimal coefficients. For raw moments m1, m2, m3, m4 of a law with first moment squared strictly below the second moment (positive variance), the regression objective, evaluated at the optimal intercept and slope (optIntercept m1 m2 m3, optSlope m1 m2 m3), equals the closed-form residual momentResidual m1 m2 m3 m4.","s":"∀ (m1 m2 m3 m4 : ℝ), m1 ^ 2 < m2 → Causalean.Stat.MomentProblems.ResidualQuadratic.MomentAlgebra.residualQuad m1 m2 m3 m4 (Causalean.Stat.MomentProblems.ResidualQuadratic.MomentAlgebra.optIntercept m1 m2 m3) (Causalean.Stat.MomentProblems.ResidualQuadratic.MomentAlgebra.optSlope m1 m2 m3) = Causalean.Stat.MomentProblems.ResidualQuadratic.MomentAlgebra.momentResidual m1 m2 m3 m4","l":75,"f":"Causalean/Stat/Nonparametric/MomentProblems/ResidualQuadratic/MomentAlgebra.lean"},{"n":"Causalean.Stat.MomentProblems.ResidualQuadratic.ProjectionResidual.integrable_projResidual","k":"theorem","m":"Causalean.Stat.Nonparametric.MomentProblems.ResidualQuadratic.ProjectionResidual","a":"Stat/Nonparametric/MomentProblems/ResidualQuadratic","t":2,"d":"q is integrable (finite measure + q ∈ L²).","s":"∀ (μ : MeasureTheory.Measure ℝ) [MeasureTheory.IsFiniteMeasure μ], Causalean.Stat.MomentProblems.ResidualQuadratic.MeasureBridge.FiniteMoment4 μ → MeasureTheory.Integrable (Causalean.Stat.MomentProblems.ResidualQuadratic.ProjectionResidual.projResidual μ) μ","l":73,"f":"Causalean/Stat/Nonparametric/MomentProblems/ResidualQuadratic/ProjectionResidual.lean"},{"n":"Causalean.Stat.MomentProblems.ResidualQuadratic.ProjectionResidual.integral_id_mul_projResidual","k":"theorem","m":"Causalean.Stat.Nonparametric.MomentProblems.ResidualQuadratic.ProjectionResidual","a":"Stat/Nonparametric/MomentProblems/ResidualQuadratic","t":2,"d":"Orthogonality to y. ∫ y·q dμ = 0.","s":"∀ (μ : MeasureTheory.Measure ℝ) [MeasureTheory.IsProbabilityMeasure μ], Causalean.Stat.MomentProblems.ResidualQuadratic.MeasureBridge.FiniteMoment4 μ → Causalean.Stat.MomentProblems.ResidualQuadratic.MeasureBridge.moment μ 1 ^ 2 < Causalean.Stat.MomentProblems.ResidualQuadratic.MeasureBridge.moment μ 2 → ∫ (y : ℝ), y * Causalean.Stat.MomentProblems.ResidualQuadratic.ProjectionResidual.projResidual μ y ∂μ = 0","l":110,"f":"Causalean/Stat/Nonparametric/MomentProblems/ResidualQuadratic/ProjectionResidual.lean"},{"n":"Causalean.Stat.MomentProblems.ResidualQuadratic.ProjectionResidual.integral_projResidual","k":"theorem","m":"Causalean.Stat.Nonparametric.MomentProblems.ResidualQuadratic.ProjectionResidual","a":"Stat/Nonparametric/MomentProblems/ResidualQuadratic","t":2,"d":"Orthogonality to the constant. ∫ q dμ = 0.","s":"∀ (μ : MeasureTheory.Measure ℝ) [MeasureTheory.IsProbabilityMeasure μ], Causalean.Stat.MomentProblems.ResidualQuadratic.MeasureBridge.FiniteMoment4 μ → Causalean.Stat.MomentProblems.ResidualQuadratic.MeasureBridge.moment μ 1 ^ 2 < Causalean.Stat.MomentProblems.ResidualQuadratic.MeasureBridge.moment μ 2 → ∫ (y : ℝ), Causalean.Stat.MomentProblems.ResidualQuadratic.ProjectionResidual.projResidual μ y ∂μ = 0","l":78,"f":"Causalean/Stat/Nonparametric/MomentProblems/ResidualQuadratic/ProjectionResidual.lean"},{"n":"Causalean.Stat.MomentProblems.ResidualQuadratic.ProjectionResidual.integral_sq_mul_projResidual","k":"theorem","m":"Causalean.Stat.Nonparametric.MomentProblems.ResidualQuadratic.ProjectionResidual","a":"Stat/Nonparametric/MomentProblems/ResidualQuadratic","t":2,"d":"y²-moment. ∫ y²·q dμ = r.","s":"∀ (μ : MeasureTheory.Measure ℝ) [MeasureTheory.IsProbabilityMeasure μ], Causalean.Stat.MomentProblems.ResidualQuadratic.MeasureBridge.FiniteMoment4 μ → Causalean.Stat.MomentProblems.ResidualQuadratic.MeasureBridge.moment μ 1 ^ 2 < Causalean.Stat.MomentProblems.ResidualQuadratic.MeasureBridge.moment μ 2 → ∫ (y : ℝ), y ^ 2 * Causalean.Stat.MomentProblems.ResidualQuadratic.ProjectionResidual.projResidual μ y ∂μ = Causalean.Stat.MomentProblems.ResidualQuadratic.MeasureBridge.l2ResidualQuadratic μ","l":162,"f":"Causalean/Stat/Nonparametric/MomentProblems/ResidualQuadratic/ProjectionResidual.lean"},{"n":"Causalean.Stat.MomentProblems.ResidualQuadratic.ProjectionResidual.integral_sq_projResidual","k":"theorem","m":"Causalean.Stat.Nonparametric.MomentProblems.ResidualQuadratic.ProjectionResidual","a":"Stat/Nonparametric/MomentProblems/ResidualQuadratic","t":1,"d":"Self L²-norm of the projection residual. For a probability measure μ on ℝ with a finite fourth moment and first moment squared strictly below the second moment (positive variance), the squared L² norm of the projection residual q(y) = y² − (optIntercept μ + optSlope μ · y) equals the closed-form residual l2ResidualQuadratic μ: ∫ q² dμ = l2ResidualQuadratic μ.","s":"∀ (μ : MeasureTheory.Measure ℝ) [MeasureTheory.IsProbabilityMeasure μ], Causalean.Stat.MomentProblems.ResidualQuadratic.MeasureBridge.FiniteMoment4 μ → Causalean.Stat.MomentProblems.ResidualQuadratic.MeasureBridge.moment μ 1 ^ 2 < Causalean.Stat.MomentProblems.ResidualQuadratic.MeasureBridge.moment μ 2 → ∫ (y : ℝ), Causalean.Stat.MomentProblems.ResidualQuadratic.ProjectionResidual.projResidual μ y ^ 2 ∂μ = Causalean.Stat.MomentProblems.ResidualQuadratic.MeasureBridge.l2ResidualQuadratic μ","l":144,"f":"Causalean/Stat/Nonparametric/MomentProblems/ResidualQuadratic/ProjectionResidual.lean"},{"n":"Causalean.Stat.MomentProblems.ResidualQuadratic.ProjectionResidual.memL2_id","k":"theorem","m":"Causalean.Stat.Nonparametric.MomentProblems.ResidualQuadratic.ProjectionResidual","a":"Stat/Nonparametric/MomentProblems/ResidualQuadratic","t":2,"d":"y ↦ y is in L²(μ) when the second moment is finite.","s":"∀ (μ : MeasureTheory.Measure ℝ) [MeasureTheory.IsFiniteMeasure μ], Causalean.Stat.MomentProblems.ResidualQuadratic.MeasureBridge.FiniteMoment4 μ → MeasureTheory.MemLp (fun y => y) 2 μ","l":48,"f":"Causalean/Stat/Nonparametric/MomentProblems/ResidualQuadratic/ProjectionResidual.lean"},{"n":"Causalean.Stat.MomentProblems.ResidualQuadratic.ProjectionResidual.memL2_sq","k":"theorem","m":"Causalean.Stat.Nonparametric.MomentProblems.ResidualQuadratic.ProjectionResidual","a":"Stat/Nonparametric/MomentProblems/ResidualQuadratic","t":2,"d":"y ↦ y² is in L²(μ) when the fourth moment is finite.","s":"∀ (μ : MeasureTheory.Measure ℝ) [MeasureTheory.IsFiniteMeasure μ], Causalean.Stat.MomentProblems.ResidualQuadratic.MeasureBridge.FiniteMoment4 μ → MeasureTheory.MemLp (fun y => y ^ 2) 2 μ","l":54,"f":"Causalean/Stat/Nonparametric/MomentProblems/ResidualQuadratic/ProjectionResidual.lean"},{"n":"Causalean.Stat.MomentProblems.ResidualQuadratic.ProjectionResidual.projResidual","k":"def","m":"Causalean.Stat.Nonparametric.MomentProblems.ResidualQuadratic.ProjectionResidual","a":"Stat/Nonparametric/MomentProblems/ResidualQuadratic","t":1,"d":"The L² projection residual function of y ↦ y² onto span{1, y}: q(y) = y² − (optIntercept μ + optSlope μ · y), i.e. y² minus its least-squares linear fit.","s":"MeasureTheory.Measure ℝ → ℝ → ℝ","l":43,"f":"Causalean/Stat/Nonparametric/MomentProblems/ResidualQuadratic/ProjectionResidual.lean"},{"n":"Causalean.Stat.MomentProblems.ResidualQuadratic.ProjectionResidual.projResidual_memL2","k":"theorem","m":"Causalean.Stat.Nonparametric.MomentProblems.ResidualQuadratic.ProjectionResidual","a":"Stat/Nonparametric/MomentProblems/ResidualQuadratic","t":2,"d":"The residual function q is square-integrable (q ∈ L²(μ)): it is a degree-2 polynomial in y and μ has a finite fourth moment.","s":"∀ (μ : MeasureTheory.Measure ℝ) [MeasureTheory.IsFiniteMeasure μ], Causalean.Stat.MomentProblems.ResidualQuadratic.MeasureBridge.FiniteMoment4 μ → MeasureTheory.MemLp (Causalean.Stat.MomentProblems.ResidualQuadratic.ProjectionResidual.projResidual μ) 2 μ","l":60,"f":"Causalean/Stat/Nonparametric/MomentProblems/ResidualQuadratic/ProjectionResidual.lean"},{"n":"Causalean.Stat.MomentProblems.ScoreProgram.FeasibleScore","k":"structure","m":"Causalean.Stat.Nonparametric.MomentProblems.ScoreProgram.ScoreProgram","a":"Stat/Nonparametric/MomentProblems/ScoreProgram","t":1,"d":"Feasibility of a score. s : ℝ → ℝ is feasible for target x if it lies in L²(μ) and is moment-orthogonal to 1 and y while carrying prescribed y²-moment x: s ∈ L²(μ), ∫ s dμ = 0, ∫ y·s dμ = 0, ∫ y²·s dμ = x.","s":"MeasureTheory.Measure ℝ → ℝ → (ℝ → ℝ) → Prop","l":53,"f":"Causalean/Stat/Nonparametric/MomentProblems/ScoreProgram/ScoreProgram.lean"},{"n":"Causalean.Stat.MomentProblems.ScoreProgram.feasibleScore_cost_lower_bound","k":"theorem","m":"Causalean.Stat.Nonparametric.MomentProblems.ScoreProgram.ScoreProgram","a":"Stat/Nonparametric/MomentProblems/ScoreProgram","t":2,"d":"Lower bound (duality ≤). Every feasible score has cost at least x² / r.","s":"∀ (μ : MeasureTheory.Measure ℝ) [MeasureTheory.IsProbabilityMeasure μ], Causalean.Stat.MomentProblems.ResidualQuadratic.MeasureBridge.FiniteMoment4 μ → Causalean.Stat.MomentProblems.ResidualQuadratic.MeasureBridge.moment μ 1 ^ 2 < Causalean.Stat.MomentProblems.ResidualQuadratic.MeasureBridge.moment μ 2 → 0 < Causalean.Stat.MomentProblems.ResidualQuadratic.MeasureBridge.l2ResidualQuadratic μ → ∀ (x : ℝ) {s : ℝ → ℝ}, Causalean.Stat.MomentProblems.ScoreProgram.FeasibleScore μ x s → x ^ 2 / Causalean.Stat.MomentProblems.ResidualQuadratic.MeasureBridge.l2ResidualQuadratic μ ≤ ∫ (y : ℝ), s y ^ 2 ∂μ","l":123,"f":"Causalean/Stat/Nonparametric/MomentProblems/ScoreProgram/ScoreProgram.lean"},{"n":"Causalean.Stat.MomentProblems.ScoreProgram.optScore","k":"def","m":"Causalean.Stat.Nonparametric.MomentProblems.ScoreProgram.ScoreProgram","a":"Stat/Nonparametric/MomentProblems/ScoreProgram","t":1,"d":"The optimal score s*(y) = (x / r)·q(y), where q is the L² projection residual of y² onto span{1, y} and r = l2ResidualQuadratic μ. This is the minimum-norm feasible score.","s":"MeasureTheory.Measure ℝ → ℝ → ℝ → ℝ","l":70,"f":"Causalean/Stat/Nonparametric/MomentProblems/ScoreProgram/ScoreProgram.lean"},{"n":"Causalean.Stat.MomentProblems.ScoreProgram.optScore_cost","k":"theorem","m":"Causalean.Stat.Nonparametric.MomentProblems.ScoreProgram.ScoreProgram","a":"Stat/Nonparametric/MomentProblems/ScoreProgram","t":2,"d":"Attainment. The optimal score s* achieves cost exactly x² / r: ∫ (s*)² dμ = ∫ (x/r)²·q² dμ = (x/r)²·r = x²/r.","s":"∀ (μ : MeasureTheory.Measure ℝ) [MeasureTheory.IsProbabilityMeasure μ], Causalean.Stat.MomentProblems.ResidualQuadratic.MeasureBridge.FiniteMoment4 μ → Causalean.Stat.MomentProblems.ResidualQuadratic.MeasureBridge.moment μ 1 ^ 2 < Causalean.Stat.MomentProblems.ResidualQuadratic.MeasureBridge.moment μ 2 → 0 < Causalean.Stat.MomentProblems.ResidualQuadratic.MeasureBridge.l2ResidualQuadratic μ → ∀ (x : ℝ), ∫ (y : ℝ), Causalean.Stat.MomentProblems.ScoreProgram.optScore μ x y ^ 2 ∂μ = x ^ 2 / Causalean.Stat.MomentProblems.ResidualQuadratic.MeasureBridge.l2ResidualQuadratic μ","l":108,"f":"Causalean/Stat/Nonparametric/MomentProblems/ScoreProgram/ScoreProgram.lean"},{"n":"Causalean.Stat.MomentProblems.ScoreProgram.optScore_feasible","k":"theorem","m":"Causalean.Stat.Nonparametric.MomentProblems.ScoreProgram.ScoreProgram","a":"Stat/Nonparametric/MomentProblems/ScoreProgram","t":2,"d":"Feasibility of the optimal score. s* = (x/r)·q is a feasible score for target x: it lies in L²(μ) and satisfies the three moment constraints. The y²-moment constraint uses ∫ y²·q = r and (x/r)·r = x (needs r ≠ 0).","s":"∀ (μ : MeasureTheory.Measure ℝ) [MeasureTheory.IsProbabilityMeasure μ], Causalean.Stat.MomentProblems.ResidualQuadratic.MeasureBridge.FiniteMoment4 μ → Causalean.Stat.MomentProblems.ResidualQuadratic.MeasureBridge.moment μ 1 ^ 2 < Causalean.Stat.MomentProblems.ResidualQuadratic.MeasureBridge.moment μ 2 → 0 < Causalean.Stat.MomentProblems.ResidualQuadratic.MeasureBridge.l2ResidualQuadratic μ → ∀ (x : ℝ), Causalean.Stat.MomentProblems.ScoreProgram.FeasibleScore μ x (Causalean.Stat.MomentProblems.ScoreProgram.optScore μ x)","l":79,"f":"Causalean/Stat/Nonparametric/MomentProblems/ScoreProgram/ScoreProgram.lean"},{"n":"Causalean.Stat.MomentProblems.ScoreProgram.scoreCost","k":"def","m":"Causalean.Stat.Nonparametric.MomentProblems.ScoreProgram.ScoreProgram","a":"Stat/Nonparametric/MomentProblems/ScoreProgram","t":1,"d":"The value of the score program: the infimum of ∫ s² dμ over all feasible scores s.","s":"MeasureTheory.Measure ℝ → ℝ → ℝ","l":75,"f":"Causalean/Stat/Nonparametric/MomentProblems/ScoreProgram/ScoreProgram.lean"},{"n":"Causalean.Stat.MomentProblems.ScoreProgram.scoreCost_eq","k":"theorem","m":"Causalean.Stat.Nonparametric.MomentProblems.ScoreProgram.ScoreProgram","a":"Stat/Nonparametric/MomentProblems/ScoreProgram","t":1,"d":"The score-program duality (headline). For a probability measure μ on ℝ with a finite fourth moment and first moment squared strictly below the second moment (positive variance), if moreover the closed-form residual l2ResidualQuadratic μ is positive, then for every target y²-moment x, the value of the constrained minimum-norm score program equals x² / l2ResidualQuadratic μ:","s":"∀ (μ : MeasureTheory.Measure ℝ) [MeasureTheory.IsProbabilityMeasure μ], Causalean.Stat.MomentProblems.ResidualQuadratic.MeasureBridge.FiniteMoment4 μ → Causalean.Stat.MomentProblems.ResidualQuadratic.MeasureBridge.moment μ 1 ^ 2 < Causalean.Stat.MomentProblems.ResidualQuadratic.MeasureBridge.moment μ 2 → 0 < Causalean.Stat.MomentProblems.ResidualQuadratic.MeasureBridge.l2ResidualQuadratic μ → ∀ (x : ℝ), Causalean.Stat.MomentProblems.ScoreProgram.scoreCost μ x = x ^ 2 / Causalean.Stat.MomentProblems.ResidualQuadratic.MeasureBridge.l2ResidualQuadratic μ","l":205,"f":"Causalean/Stat/Nonparametric/MomentProblems/ScoreProgram/ScoreProgram.lean"},{"n":"Causalean.Stat.MomentProblems.TruncatedMomentInterior","k":"def","m":"Causalean.Stat.Nonparametric.MomentProblems.TruncatedCumulantInterior","a":"Stat/Nonparametric/MomentProblems","t":1,"d":"Interior of the truncated cumulant range at order L. There is a cumulant target and a strictly positive radius such that every cumulant sequence within that radius of the target, in orders two through L, is realized by a probability law on the real line that is centered, is not a Gaussian law, and has finite L-th moment.","s":"ℕ → Prop","l":40,"f":"Causalean/Stat/Nonparametric/MomentProblems/TruncatedCumulantInterior.lean"},{"n":"Causalean.Stat.MomentProblems.atomSolve","k":"def","m":"Causalean.Stat.Nonparametric.MomentProblems.SymmetricAtomSolve","a":"Stat/Nonparametric/MomentProblems","t":1,"d":"The weight vector solving the moment-matching system: given a target for the raw moments of orders 0 through L, the unique weights on the symmetric atoms reproducing them.","s":"(L : ℕ) → (Fin (L + 1) → ℝ) → Fin (L + 1) → ℝ","l":126,"f":"Causalean/Stat/Nonparametric/MomentProblems/SymmetricAtomSolve.lean"},{"n":"Causalean.Stat.MomentProblems.atomSolve_mulVec","k":"theorem","m":"Causalean.Stat.Nonparametric.MomentProblems.SymmetricAtomSolve","a":"Stat/Nonparametric/MomentProblems","t":2,"d":"The solve is a left inverse of taking moments: solving for the weights from the moments that a given weight vector produces returns that same weight vector.","s":"∀ (L : ℕ) (q : Fin (L + 1) → ℝ), Causalean.Stat.MomentProblems.atomSolve L ((Causalean.Stat.MomentProblems.atomVandermonde L).transpose.mulVec q) = q","l":149,"f":"Causalean/Stat/Nonparametric/MomentProblems/SymmetricAtomSolve.lean"},{"n":"Causalean.Stat.MomentProblems.atomSolve_spec","k":"theorem","m":"Causalean.Stat.Nonparametric.MomentProblems.SymmetricAtomSolve","a":"Stat/Nonparametric/MomentProblems","t":2,"d":"The solved weights do reproduce the target: the weighted sum of the k-th powers of the atoms is the k-th target moment, for every order k from 0 to L.","s":"∀ (L : ℕ) (b : Fin (L + 1) → ℝ) (k : Fin (L + 1)), ∑ i, Causalean.Stat.MomentProblems.atomSolve L b i * Causalean.Stat.MomentProblems.symmetricAtoms L i ^ ↑k = b k","l":132,"f":"Causalean/Stat/Nonparametric/MomentProblems/SymmetricAtomSolve.lean"},{"n":"Causalean.Stat.MomentProblems.atomSolve_uniformMoments","k":"theorem","m":"Causalean.Stat.Nonparametric.MomentProblems.SymmetricAtomSolve","a":"Stat/Nonparametric/MomentProblems","t":2,"d":"Solving at the moments of the uniform-weight law returns the uniform weights.","s":"∀ (L : ℕ), (Causalean.Stat.MomentProblems.atomSolve L fun k => Causalean.Stat.MomentProblems.uniformMoments L ↑k) = Causalean.Stat.MomentProblems.uniformWeights L","l":159,"f":"Causalean/Stat/Nonparametric/MomentProblems/SymmetricAtomSolve.lean"},{"n":"Causalean.Stat.MomentProblems.atomVandermonde","k":"def","m":"Causalean.Stat.Nonparametric.MomentProblems.SymmetricAtomSolve","a":"Stat/Nonparametric/MomentProblems","t":1,"d":"The Vandermonde matrix of the symmetric atoms: row i lists the powers of the i-th atom.","s":"(L : ℕ) → Matrix (Fin (L + 1)) (Fin (L + 1)) ℝ","l":114,"f":"Causalean/Stat/Nonparametric/MomentProblems/SymmetricAtomSolve.lean"},{"n":"Causalean.Stat.MomentProblems.atomVandermonde_transpose_isUnit","k":"theorem","m":"Causalean.Stat.Nonparametric.MomentProblems.SymmetricAtomSolve","a":"Stat/Nonparametric/MomentProblems","t":2,"d":"The transposed Vandermonde matrix of the symmetric atoms is invertible, because the atoms are pairwise distinct.","s":"∀ (L : ℕ), IsUnit (Causalean.Stat.MomentProblems.atomVandermonde L).transpose.det","l":119,"f":"Causalean/Stat/Nonparametric/MomentProblems/SymmetricAtomSolve.lean"},{"n":"Causalean.Stat.MomentProblems.atomicLaw","k":"def","m":"Causalean.Stat.Nonparametric.MomentProblems.AtomicLaw","a":"Stat/Nonparametric/MomentProblems","t":1,"d":"The finite atomic law determined by a list of real locations and a matching list of weights: the measure that puts each weight as a point mass at its own location.","s":"(n : ℕ) → (Fin n → ℝ) → (Fin n → ℝ) → MeasureTheory.Measure ℝ","l":35,"f":"Causalean/Stat/Nonparametric/MomentProblems/AtomicLaw.lean"},{"n":"Causalean.Stat.MomentProblems.atomicLaw_singleton","k":"theorem","m":"Causalean.Stat.Nonparametric.MomentProblems.AtomicLaw","a":"Stat/Nonparametric/MomentProblems","t":2,"d":"A finite atomic law with distinct locations and nonnegative weights puts exactly its own weight of mass on each single atom.","s":"∀ {n : ℕ} {x p : Fin n → ℝ}, Function.Injective x → (∀ (i : Fin n), 0 ≤ p i) → ∀ (j : Fin n), (Causalean.Stat.MomentProblems.atomicLaw n x p) {x j} = ENNReal.ofReal (p j)","l":81,"f":"Causalean/Stat/Nonparametric/MomentProblems/AtomicLaw.lean"},{"n":"Causalean.Stat.MomentProblems.atomicWeight","k":"def","m":"Causalean.Stat.Nonparametric.MomentProblems.AtomicLaw","a":"Stat/Nonparametric/MomentProblems","t":1,"d":"The total weight that an indexed list of atoms places on a given point of the real line: the sum of the weights of all indices whose location is that point.","s":"(n : ℕ) → (Fin n → ℝ) → (Fin n → ℝ) → ℝ → ℝ","l":30,"f":"Causalean/Stat/Nonparametric/MomentProblems/AtomicLaw.lean"},{"n":"Causalean.Stat.MomentProblems.atomicWeight_apply","k":"theorem","m":"Causalean.Stat.Nonparametric.MomentProblems.AtomicLaw","a":"Stat/Nonparametric/MomentProblems","t":2,"d":"When the atom locations are pairwise distinct, the total weight sitting at one of them is exactly that atom's own weight — no two indices collide.","s":"∀ {n : ℕ} {x p : Fin n → ℝ}, Function.Injective x → ∀ (j : Fin n), Causalean.Stat.MomentProblems.atomicWeight n x p (x j) = p j","l":41,"f":"Causalean/Stat/Nonparametric/MomentProblems/AtomicLaw.lean"},{"n":"Causalean.Stat.MomentProblems.continuous_atomSolve","k":"theorem","m":"Causalean.Stat.Nonparametric.MomentProblems.SymmetricAtomSolve","a":"Stat/Nonparametric/MomentProblems","t":2,"d":"The moment-matching solve depends continuously on the target moments — it is linear.","s":"∀ (L : ℕ), Continuous (Causalean.Stat.MomentProblems.atomSolve L)","l":168,"f":"Causalean/Stat/Nonparametric/MomentProblems/SymmetricAtomSolve.lean"},{"n":"Causalean.Stat.MomentProblems.continuous_cumulantToWeights","k":"theorem","m":"Causalean.Stat.Nonparametric.MomentProblems.SymmetricAtomSolve","a":"Stat/Nonparametric/MomentProblems","t":1,"d":"The cumulant-to-weights map is continuous. For any truncation order L, the map sending a truncated cumulant target to the recovered symmetric-atom weights is continuous, since both the moment↔cumulant inversion and the Vandermonde solve it composes are continuous.","s":"∀ (L : ℕ), Continuous (Causalean.Stat.MomentProblems.cumulantToWeights L)","l":200,"f":"Causalean/Stat/Nonparametric/MomentProblems/SymmetricAtomSolve.lean"},{"n":"Causalean.Stat.MomentProblems.continuous_momFromCum","k":"theorem","m":"Causalean.Stat.Nonparametric.MomentProblems.MomentCumulantInversion","a":"Stat/Nonparametric/MomentProblems","t":1,"d":"Continuity of the inversion. At any fixed order r, the reconstructed moment of that order is a continuous function of the prescribed cumulant sequence, so small perturbations of the target cumulants move the moments only slightly — the key to the openness arguments that use this inversion.","s":"∀ (r : ℕ), Continuous fun c => Causalean.Stat.MomentProblems.momFromCum c r","l":215,"f":"Causalean/Stat/Nonparametric/MomentProblems/MomentCumulantInversion.lean"},{"n":"Causalean.Stat.MomentProblems.continuous_padCumulants","k":"theorem","m":"Causalean.Stat.Nonparametric.MomentProblems.SymmetricAtomSolve","a":"Stat/Nonparametric/MomentProblems","t":2,"d":"Zero-extension of a finite cumulant target is continuous in the target.","s":"∀ (L : ℕ), Continuous (Causalean.Stat.MomentProblems.padCumulants L)","l":184,"f":"Causalean/Stat/Nonparametric/MomentProblems/SymmetricAtomSolve.lean"},{"n":"Causalean.Stat.MomentProblems.cumFromMom","k":"def","m":"Causalean.Stat.Nonparametric.MomentProblems.MomentCumulantInversion","a":"Stat/Nonparametric/MomentProblems","t":1,"d":"The order-r cumulant read off from a moment sequence: the set-partition (Möbius) formula that expresses the r-th cumulant as a signed, factorial-weighted sum over partitions of r slots of products of moments, one moment per block, at the block's size.","s":"ℕ → (ℕ → ℝ) → ℝ","l":31,"f":"Causalean/Stat/Nonparametric/MomentProblems/MomentCumulantInversion.lean"},{"n":"Causalean.Stat.MomentProblems.cumFromMom_congr","k":"theorem","m":"Causalean.Stat.Nonparametric.MomentProblems.MomentCumulantInversion","a":"Stat/Nonparametric/MomentProblems","t":2,"d":"The order-r cumulant depends only on the moments up to order r: changing higher moments leaves it unchanged.","s":"∀ (r : ℕ) {m m' : ℕ → ℝ}, (∀ (k : ℕ), 1 ≤ k → k ≤ r → m k = m' k) → Causalean.Stat.MomentProblems.cumFromMom r m = Causalean.Stat.MomentProblems.cumFromMom r m'","l":130,"f":"Causalean/Stat/Nonparametric/MomentProblems/MomentCumulantInversion.lean"},{"n":"Causalean.Stat.MomentProblems.cumFromMom_eq","k":"theorem","m":"Causalean.Stat.Nonparametric.MomentProblems.MomentCumulantInversion","a":"Stat/Nonparametric/MomentProblems","t":2,"d":"Triangularity. At any positive order the cumulant equals the moment of that same order plus a remainder assembled only from strictly lower moments. This is what makes the moment-to-cumulant map invertible by recursion.","s":"∀ (r : ℕ), 1 ≤ r → ∀ (m : ℕ → ℝ), Causalean.Stat.MomentProblems.cumFromMom r m = m r + Causalean.Stat.MomentProblems.restFromMom r m","l":98,"f":"Causalean/Stat/Nonparametric/MomentProblems/MomentCumulantInversion.lean"},{"n":"Causalean.Stat.MomentProblems.cumFromMom_momFromCum","k":"theorem","m":"Causalean.Stat.Nonparametric.MomentProblems.MomentCumulantInversion","a":"Stat/Nonparametric/MomentProblems","t":1,"d":"Correctness of the inversion. Reading the cumulants back off the reconstructed moment sequence returns the prescribed cumulants, at every order r at least two.","s":"∀ (c : ℕ → ℝ) (r : ℕ), 2 ≤ r → Causalean.Stat.MomentProblems.cumFromMom r (Causalean.Stat.MomentProblems.momFromCum c) = c r","l":192,"f":"Causalean/Stat/Nonparametric/MomentProblems/MomentCumulantInversion.lean"},{"n":"Causalean.Stat.MomentProblems.cumulantToWeights","k":"def","m":"Causalean.Stat.Nonparametric.MomentProblems.SymmetricAtomSolve","a":"Stat/Nonparametric/MomentProblems","t":1,"d":"The cumulant-to-weights map: from a prescribed truncated cumulant target, invert the triangular moment↔cumulant relation to get the target moments, then solve the Vandermonde system for the weights on the symmetric atoms.","s":"(L : ℕ) → (Fin (L + 1) → ℝ) → Fin (L + 1) → ℝ","l":193,"f":"Causalean/Stat/Nonparametric/MomentProblems/SymmetricAtomSolve.lean"},{"n":"Causalean.Stat.MomentProblems.cumulantToWeights_uniformCumulantPoint","k":"theorem","m":"Causalean.Stat.Nonparametric.MomentProblems.SymmetricAtomSolve","a":"Stat/Nonparametric/MomentProblems","t":1,"d":"The base point maps to the uniform weights. For any truncation order L, feeding the cumulants of the uniform-weight law on the symmetric atoms into the cumulant-to-weights map returns exactly the uniform weights — all strictly positive, hence strictly inside the positivity constraints.","s":"∀ (L : ℕ), Causalean.Stat.MomentProblems.cumulantToWeights L (Causalean.Stat.MomentProblems.uniformCumulantPoint L) = Causalean.Stat.MomentProblems.uniformWeights L","l":223,"f":"Causalean/Stat/Nonparametric/MomentProblems/SymmetricAtomSolve.lean"},{"n":"Causalean.Stat.MomentProblems.integral_atomicLaw","k":"theorem","m":"Causalean.Stat.Nonparametric.MomentProblems.AtomicLaw","a":"Stat/Nonparametric/MomentProblems","t":2,"d":"Integrating a function against a finite atomic law with distinct locations and nonnegative weights gives the weighted sum of the function's values at the atoms.","s":"∀ {n : ℕ} {x p : Fin n → ℝ}, Function.Injective x → (∀ (i : Fin n), 0 ≤ p i) → ∀ (f : ℝ → ℝ), ∫ (t : ℝ), f t ∂Causalean.Stat.MomentProblems.atomicLaw n x p = ∑ i, p i * f (x i)","l":49,"f":"Causalean/Stat/Nonparametric/MomentProblems/AtomicLaw.lean"},{"n":"Causalean.Stat.MomentProblems.isProbabilityMeasure_atomicLaw","k":"theorem","m":"Causalean.Stat.Nonparametric.MomentProblems.AtomicLaw","a":"Stat/Nonparametric/MomentProblems","t":1,"d":"A finite atomic law on the real line, built from n pairwise distinct atom locations carrying nonnegative weights that sum to one, is a probability measure.","s":"∀ {n : ℕ} {x p : Fin n → ℝ}, Function.Injective x → (∀ (i : Fin n), 0 ≤ p i) → ∑ i, p i = 1 → MeasureTheory.IsProbabilityMeasure (Causalean.Stat.MomentProblems.atomicLaw n x p)","l":66,"f":"Causalean/Stat/Nonparametric/MomentProblems/AtomicLaw.lean"},{"n":"Causalean.Stat.MomentProblems.jointCumulant","k":"def","m":"Causalean.Stat.Nonparametric.MomentProblems.Cumulant","a":"Stat/Nonparametric/MomentProblems","t":1,"d":"The joint cumulant of two real random variables at bidegree (p, q): the cumulant of the first variable taken p times together with the second taken q times. It is obtained from the mixed moments by the classical set-partition (Möbius) inversion — sum over all partitions of the p + q slots of a signed factorial weight times the product, over the blocks of the partition, of the mixed moment matching how many first-variable and second-variable slots that block contains.","s":"{Ω : Type u_1} → [inst : MeasurableSpace Ω] → MeasureTheory.Measure Ω → (Ω → ℝ) → (Ω → ℝ) → ℕ → ℕ → ℝ","l":41,"f":"Causalean/Stat/Nonparametric/MomentProblems/Cumulant.lean"},{"n":"Causalean.Stat.MomentProblems.memLp_id_atomicLaw","k":"theorem","m":"Causalean.Stat.Nonparametric.MomentProblems.AtomicLaw","a":"Stat/Nonparametric/MomentProblems","t":2,"d":"A finite atomic probability law has finite moments of every order: its support is a bounded finite set, so the identity function is bounded almost everywhere and hence in every Lᵖ.","s":"∀ {n : ℕ} {x p : Fin n → ℝ}, Function.Injective x → (∀ (i : Fin n), 0 ≤ p i) → ∑ i, p i = 1 → ∀ (q : ENNReal), MeasureTheory.MemLp id q (Causalean.Stat.MomentProblems.atomicLaw n x p)","l":92,"f":"Causalean/Stat/Nonparametric/MomentProblems/AtomicLaw.lean"},{"n":"Causalean.Stat.MomentProblems.momFromCum","k":"def","m":"Causalean.Stat.Nonparametric.MomentProblems.MomentCumulantInversion","a":"Stat/Nonparametric/MomentProblems","t":1,"d":"The moment sequence reconstructed from a prescribed cumulant sequence, by inverting the triangular formula: total mass one, mean zero, and at each order at least two the moment is the prescribed cumulant minus the remainder assembled from the already-reconstructed lower moments.","s":"(ℕ → ℝ) → ℕ → ℝ","l":157,"f":"Causalean/Stat/Nonparametric/MomentProblems/MomentCumulantInversion.lean"},{"n":"Causalean.Stat.MomentProblems.momFromCum_congr","k":"theorem","m":"Causalean.Stat.Nonparametric.MomentProblems.MomentCumulantInversion","a":"Stat/Nonparametric/MomentProblems","t":2,"d":"Locality of the inversion. The reconstructed moment of order r depends only on the prescribed cumulants of orders two through r, so truncating the cumulant target beyond r is harmless.","s":"∀ {c c' : ℕ → ℝ} (r : ℕ), (∀ (k : ℕ), 2 ≤ k → k ≤ r → c k = c' k) → Causalean.Stat.MomentProblems.momFromCum c r = Causalean.Stat.MomentProblems.momFromCum c' r","l":199,"f":"Causalean/Stat/Nonparametric/MomentProblems/MomentCumulantInversion.lean"},{"n":"Causalean.Stat.MomentProblems.momFromCum_eq_of_cum","k":"theorem","m":"Causalean.Stat.Nonparametric.MomentProblems.MomentCumulantInversion","a":"Stat/Nonparametric/MomentProblems","t":2,"d":"Uniqueness of the inversion. Any moment sequence with total mass one and mean zero whose cumulants are the prescribed ones is exactly the sequence produced by the recursion.","s":"∀ (m : ℕ → ℝ), m 0 = 1 → m 1 = 0 → ∀ (c : ℕ → ℝ), (∀ (r : ℕ), 2 ≤ r → c r = Causalean.Stat.MomentProblems.cumFromMom r m) → ∀ (r : ℕ), Causalean.Stat.MomentProblems.momFromCum c r = m r","l":240,"f":"Causalean/Stat/Nonparametric/MomentProblems/MomentCumulantInversion.lean"},{"n":"Causalean.Stat.MomentProblems.momFromCum_one","k":"theorem","m":"Causalean.Stat.Nonparametric.MomentProblems.MomentCumulantInversion","a":"Stat/Nonparametric/MomentProblems","t":2,"d":"The reconstructed moment sequence is centered: its mean is zero.","s":"∀ (c : ℕ → ℝ), Causalean.Stat.MomentProblems.momFromCum c 1 = 0","l":172,"f":"Causalean/Stat/Nonparametric/MomentProblems/MomentCumulantInversion.lean"},{"n":"Causalean.Stat.MomentProblems.momFromCum_padCumulants_uniform","k":"theorem","m":"Causalean.Stat.Nonparametric.MomentProblems.SymmetricAtomSolve","a":"Stat/Nonparametric/MomentProblems","t":2,"d":"Inverting the base cumulant target order by order returns the moments of the uniform-weight law.","s":"∀ (L : ℕ) (k : Fin (L + 1)), Causalean.Stat.MomentProblems.momFromCum (Causalean.Stat.MomentProblems.padCumulants L (Causalean.Stat.MomentProblems.uniformCumulantPoint L)) ↑k = Causalean.Stat.MomentProblems.uniformMoments L ↑k","l":214,"f":"Causalean/Stat/Nonparametric/MomentProblems/SymmetricAtomSolve.lean"},{"n":"Causalean.Stat.MomentProblems.momFromCum_succ","k":"theorem","m":"Causalean.Stat.Nonparametric.MomentProblems.MomentCumulantInversion","a":"Stat/Nonparametric/MomentProblems","t":2,"d":"The defining recursion: at every order at least two, the reconstructed moment is the target cumulant of that order minus the remainder built from the lower reconstructed moments.","s":"∀ (c : ℕ → ℝ) (r : ℕ), 2 ≤ r → Causalean.Stat.MomentProblems.momFromCum c r = c r - Causalean.Stat.MomentProblems.restFromMom r (Causalean.Stat.MomentProblems.momFromCum c)","l":176,"f":"Causalean/Stat/Nonparametric/MomentProblems/MomentCumulantInversion.lean"},{"n":"Causalean.Stat.MomentProblems.momFromCum_uniformCumulants","k":"theorem","m":"Causalean.Stat.Nonparametric.MomentProblems.SymmetricAtomSolve","a":"Stat/Nonparametric/MomentProblems","t":2,"d":"Inverting the cumulants of the uniform-weight law returns its moments: the base point is a fixed point of the moment↔cumulant round trip.","s":"∀ (L k : ℕ), Causalean.Stat.MomentProblems.momFromCum (Causalean.Stat.MomentProblems.uniformCumulants L) k = Causalean.Stat.MomentProblems.uniformMoments L k","l":104,"f":"Causalean/Stat/Nonparametric/MomentProblems/SymmetricAtomSolve.lean"},{"n":"Causalean.Stat.MomentProblems.momFromCum_zero","k":"theorem","m":"Causalean.Stat.Nonparametric.MomentProblems.MomentCumulantInversion","a":"Stat/Nonparametric/MomentProblems","t":2,"d":"The reconstructed moment sequence has total mass one.","s":"∀ (c : ℕ → ℝ), Causalean.Stat.MomentProblems.momFromCum c 0 = 1","l":168,"f":"Causalean/Stat/Nonparametric/MomentProblems/MomentCumulantInversion.lean"},{"n":"Causalean.Stat.MomentProblems.not_isGaussianLaw_atomicLaw","k":"theorem","m":"Causalean.Stat.Nonparametric.MomentProblems.AtomicLaw","a":"Stat/Nonparametric/MomentProblems","t":1,"d":"A finite atomic probability law with at least two locations that are pairwise distinct and each carry strictly positive mass is not a Gaussian law: a normal law with positive variance has no point masses at all, and one with zero variance is a single point mass.","s":"∀ {n : ℕ} {x p : Fin n → ℝ}, Function.Injective x → (∀ (i : Fin n), 0 < p i) → 2 ≤ n → ¬Causalean.Stat.MomentProblems.IsGaussianLaw (Causalean.Stat.MomentProblems.atomicLaw n x p)","l":113,"f":"Causalean/Stat/Nonparametric/MomentProblems/AtomicLaw.lean"},{"n":"Causalean.Stat.MomentProblems.padCumulants","k":"def","m":"Causalean.Stat.Nonparametric.MomentProblems.SymmetricAtomSolve","a":"Stat/Nonparametric/MomentProblems","t":1,"d":"Extending a finite cumulant target of orders 0, …, L to a full sequence by zeros beyond order L.","s":"(L : ℕ) → (Fin (L + 1) → ℝ) → ℕ → ℝ","l":179,"f":"Causalean/Stat/Nonparametric/MomentProblems/SymmetricAtomSolve.lean"},{"n":"Causalean.Stat.MomentProblems.restFromMom","k":"def","m":"Causalean.Stat.Nonparametric.MomentProblems.MomentCumulantInversion","a":"Stat/Nonparametric/MomentProblems","t":1,"d":"The lower-order remainder of the order-r cumulant formula: the same sum with the single one-block partition removed. Because every block of a multi-block partition is a proper subset, this depends only on moments of order strictly below r.","s":"ℕ → (ℕ → ℝ) → ℝ","l":39,"f":"Causalean/Stat/Nonparametric/MomentProblems/MomentCumulantInversion.lean"},{"n":"Causalean.Stat.MomentProblems.restFromMom_congr","k":"theorem","m":"Causalean.Stat.Nonparametric.MomentProblems.MomentCumulantInversion","a":"Stat/Nonparametric/MomentProblems","t":2,"d":"The lower-order remainder at order r depends only on the moments strictly below r: changing the order-r moment, or any higher one, leaves it unchanged.","s":"∀ (r : ℕ) {m m' : ℕ → ℝ}, (∀ (k : ℕ), 1 ≤ k → k < r → m k = m' k) → Causalean.Stat.MomentProblems.restFromMom r m = Causalean.Stat.MomentProblems.restFromMom r m'","l":143,"f":"Causalean/Stat/Nonparametric/MomentProblems/MomentCumulantInversion.lean"},{"n":"Causalean.Stat.MomentProblems.sourceCumulant","k":"def","m":"Causalean.Stat.Nonparametric.MomentProblems.Cumulant","a":"Stat/Nonparametric/MomentProblems","t":1,"d":"The cumulant of order r of a single real random variable: the joint cumulant of the variable with itself in which every slot is filled by that one variable. Order two is the variance, order three the centered third moment, order four the excess kurtosis.","s":"{Ω : Type u_1} → [inst : MeasurableSpace Ω] → MeasureTheory.Measure Ω → (Ω → ℝ) → ℕ → ℝ","l":56,"f":"Causalean/Stat/Nonparametric/MomentProblems/Cumulant.lean"},{"n":"Causalean.Stat.MomentProblems.sourceCumulant_eq_cumFromMom","k":"theorem","m":"Causalean.Stat.Nonparametric.MomentProblems.MomentCumulantInversion","a":"Stat/Nonparametric/MomentProblems","t":1,"d":"For a real random variable with law ν and any order r, the order-r cumulant of ν equals the abstract combinatorial cumulant formula evaluated at ν's own raw-moment sequence: the measure-theoretic and combinatorial definitions agree.","s":"∀ (ν : MeasureTheory.Measure ℝ) (r : ℕ), Causalean.Stat.MomentProblems.sourceCumulant ν id r = Causalean.Stat.MomentProblems.cumFromMom r fun k => ∫ (t : ℝ), t ^ k ∂ν","l":79,"f":"Causalean/Stat/Nonparametric/MomentProblems/MomentCumulantInversion.lean"},{"n":"Causalean.Stat.MomentProblems.symmetricAtoms","k":"def","m":"Causalean.Stat.Nonparametric.MomentProblems.SymmetricAtomSolve","a":"Stat/Nonparametric/MomentProblems","t":1,"d":"The L + 1 equally spaced symmetric atoms −L, −L + 2, …, L − 2, L: the i-th atom sits at twice i minus L. They are symmetric about the origin and pairwise distinct.","s":"(L : ℕ) → Fin (L + 1) → ℝ","l":34,"f":"Causalean/Stat/Nonparametric/MomentProblems/SymmetricAtomSolve.lean"},{"n":"Causalean.Stat.MomentProblems.symmetricAtoms_injective","k":"theorem","m":"Causalean.Stat.Nonparametric.MomentProblems.SymmetricAtomSolve","a":"Stat/Nonparametric/MomentProblems","t":2,"d":"The symmetric atoms are pairwise distinct.","s":"∀ (L : ℕ), Function.Injective (Causalean.Stat.MomentProblems.symmetricAtoms L)","l":39,"f":"Causalean/Stat/Nonparametric/MomentProblems/SymmetricAtomSolve.lean"},{"n":"Causalean.Stat.MomentProblems.symmetricAtoms_sum","k":"theorem","m":"Causalean.Stat.Nonparametric.MomentProblems.SymmetricAtomSolve","a":"Stat/Nonparametric/MomentProblems","t":2,"d":"The symmetric atoms sum to zero — they are balanced about the origin.","s":"∀ (L : ℕ), ∑ i, Causalean.Stat.MomentProblems.symmetricAtoms L i = 0","l":47,"f":"Causalean/Stat/Nonparametric/MomentProblems/SymmetricAtomSolve.lean"},{"n":"Causalean.Stat.MomentProblems.truncatedMomentInterior","k":"theorem","m":"Causalean.Stat.Nonparametric.MomentProblems.TruncatedCumulantInterior","a":"Stat/Nonparametric/MomentProblems","t":1,"d":"The truncated cumulant range has nonempty interior. For every truncation order L, there is a cumulant vector, of orders two through L, together with a strictly positive radius, such that every cumulant vector within that radius is the truncated cumulant vector of some centered, non-Gaussian probability law on the real line with finite L-th moment.","s":"∀ (L : ℕ), Causalean.Stat.MomentProblems.TruncatedMomentInterior L","l":82,"f":"Causalean/Stat/Nonparametric/MomentProblems/TruncatedCumulantInterior.lean"},{"n":"Causalean.Stat.MomentProblems.uniformCumulantPoint","k":"def","m":"Causalean.Stat.Nonparametric.MomentProblems.SymmetricAtomSolve","a":"Stat/Nonparametric/MomentProblems","t":1,"d":"The base cumulant target: the cumulants, of orders 0 through L, of the uniform-weight law on the symmetric atoms.","s":"(L : ℕ) → Fin (L + 1) → ℝ","l":209,"f":"Causalean/Stat/Nonparametric/MomentProblems/SymmetricAtomSolve.lean"},{"n":"Causalean.Stat.MomentProblems.uniformCumulants","k":"def","m":"Causalean.Stat.Nonparametric.MomentProblems.SymmetricAtomSolve","a":"Stat/Nonparametric/MomentProblems","t":1,"d":"The cumulant sequence of the uniform-weight law on the symmetric atoms — the base point around which the truncated cumulant range is shown to have interior.","s":"ℕ → ℕ → ℝ","l":99,"f":"Causalean/Stat/Nonparametric/MomentProblems/SymmetricAtomSolve.lean"},{"n":"Causalean.Stat.MomentProblems.uniformMoments","k":"def","m":"Causalean.Stat.Nonparametric.MomentProblems.SymmetricAtomSolve","a":"Stat/Nonparametric/MomentProblems","t":1,"d":"The raw-moment sequence of the uniform-weight law on the symmetric atoms.","s":"ℕ → ℕ → ℝ","l":87,"f":"Causalean/Stat/Nonparametric/MomentProblems/SymmetricAtomSolve.lean"},{"n":"Causalean.Stat.MomentProblems.uniformMoments_one","k":"theorem","m":"Causalean.Stat.Nonparametric.MomentProblems.SymmetricAtomSolve","a":"Stat/Nonparametric/MomentProblems","t":2,"d":"The uniform-weight law is centered.","s":"∀ (L : ℕ), Causalean.Stat.MomentProblems.uniformMoments L 1 = 0","l":95,"f":"Causalean/Stat/Nonparametric/MomentProblems/SymmetricAtomSolve.lean"},{"n":"Causalean.Stat.MomentProblems.uniformMoments_zero","k":"theorem","m":"Causalean.Stat.Nonparametric.MomentProblems.SymmetricAtomSolve","a":"Stat/Nonparametric/MomentProblems","t":2,"d":"The uniform-weight law has total mass one.","s":"∀ (L : ℕ), Causalean.Stat.MomentProblems.uniformMoments L 0 = 1","l":91,"f":"Causalean/Stat/Nonparametric/MomentProblems/SymmetricAtomSolve.lean"},{"n":"Causalean.Stat.MomentProblems.uniformWeights","k":"def","m":"Causalean.Stat.Nonparametric.MomentProblems.SymmetricAtomSolve","a":"Stat/Nonparametric/MomentProblems","t":1,"d":"The uniform weight vector on the L + 1 atoms: each atom carries mass 1 / (L + 1).","s":"(L : ℕ) → Fin (L + 1) → ℝ","l":65,"f":"Causalean/Stat/Nonparametric/MomentProblems/SymmetricAtomSolve.lean"},{"n":"Causalean.Stat.MomentProblems.uniformWeights_mean","k":"theorem","m":"Causalean.Stat.Nonparametric.MomentProblems.SymmetricAtomSolve","a":"Stat/Nonparametric/MomentProblems","t":2,"d":"The uniform-weight law on the symmetric atoms has mean zero.","s":"∀ (L : ℕ), ∑ i, Causalean.Stat.MomentProblems.uniformWeights L i * Causalean.Stat.MomentProblems.symmetricAtoms L i = 0","l":80,"f":"Causalean/Stat/Nonparametric/MomentProblems/SymmetricAtomSolve.lean"},{"n":"Causalean.Stat.MomentProblems.uniformWeights_pos","k":"theorem","m":"Causalean.Stat.Nonparametric.MomentProblems.SymmetricAtomSolve","a":"Stat/Nonparametric/MomentProblems","t":2,"d":"Every uniform weight is strictly positive.","s":"∀ (L : ℕ) (i : Fin (L + 1)), 0 < Causalean.Stat.MomentProblems.uniformWeights L i","l":69,"f":"Causalean/Stat/Nonparametric/MomentProblems/SymmetricAtomSolve.lean"},{"n":"Causalean.Stat.MomentProblems.uniformWeights_sum","k":"theorem","m":"Causalean.Stat.Nonparametric.MomentProblems.SymmetricAtomSolve","a":"Stat/Nonparametric/MomentProblems","t":2,"d":"The uniform weights sum to one, so the uniform-weight atomic law is a probability law.","s":"∀ (L : ℕ), ∑ i, Causalean.Stat.MomentProblems.uniformWeights L i = 1","l":75,"f":"Causalean/Stat/Nonparametric/MomentProblems/SymmetricAtomSolve.lean"},{"n":"Causalean.Stat.NeymanOrthogonal","k":"def","m":"Causalean.Stat.Orthogonality.Orthogonality","a":"Stat/Orthogonality","t":1,"d":"Neyman orthogonality of a moment functional m : H → X → ℝ → ℝ at nuisance value η₀ : H and target value θ₀ : ℝ, with respect to the population measure P : Measure X.","s":"{H : Type u_1} → {X : Type u_2} → [inst : AddCommGroup H] → [Module ℝ H] → [inst : MeasurableSpace X] → (H → X → ℝ → ℝ) → H → ℝ → MeasureTheory.Measure X → Prop","l":35,"f":"Causalean/Stat/Orthogonality/Orthogonality.lean"},{"n":"Causalean.Stat.Nonparametric.HOIF.degenerate_uStatistic_variance","k":"theorem","m":"Causalean.Stat.Nonparametric.HOIF.DegenerateUStatVariance","a":"Stat/Nonparametric/HOIF","t":1,"d":"Exact variance of the degenerate order-2 U-statistic. For an i.i.d. sample S, if the kernel g is symmetric, square-integrable, and doubly degenerate — its conditional expectation given either argument vanishes — and the sample size n is at least 2, then the order-2 U-statistic built from g on n observations has variance Var[Uₙ] = 2ζ / (n(n−1)), where ζ = ∬ g² dP dP.","s":"∀ {Ω : Type u_1} {X : Type u_2} [inst : MeasurableSpace Ω] [inst_1 : MeasurableSpace X] {μ : MeasureTheory.Measure Ω} {P : MeasureTheory.Measure X} [MeasureTheory.IsProbabilityMeasure μ] [MeasureTheory.IsProbabilityMeasure P] {g : X → X → ℝ} (S : Causalean.Stat.IIDSample Ω X μ P), Causalean.Stat.DegenKernel P g → ∀ {n : ℕ}, 2 ≤ n → ProbabilityTheory.variance (Causalean.Stat.uStatistic S g n) μ = 2 * Causalean.Stat.IIDSample.zeta P g / (↑n * (↑n - 1))","l":66,"f":"Causalean/Stat/Nonparametric/HOIF/DegenerateUStatVariance.lean"},{"n":"Causalean.Stat.Nonparametric.HOIF.degenerate_uStatistic_variance_le","k":"theorem","m":"Causalean.Stat.Nonparametric.HOIF.DegenerateUStatVariance","a":"Stat/Nonparametric/HOIF","t":1,"d":"HOIF degenerate U-statistic variance bound O(J/(nh)²). For an i.i.d. sample S, if the kernel g is symmetric, square-integrable, and doubly degenerate, the sample size n is at least 2, the bandwidth h is positive, the projection dimension J is nonnegative, the trace constant C is nonnegative, and the localized, J-dimensional projection kernel g has L²-energy ζ = ∬ g² dP dP bounded by C·J/h², then the degenerate order-2 U-statistic's variance satisfies Var[Uₙ] ≤ 4C·J / (nh)².","s":"∀ {Ω : Type u_1} {X : Type u_2} [inst : MeasurableSpace Ω] [inst_1 : MeasurableSpace X] {μ : MeasureTheory.Measure Ω} {P : MeasureTheory.Measure X} [MeasureTheory.IsProbabilityMeasure μ] [MeasureTheory.IsProbabilityMeasure P] {g : X → X → ℝ} (S : Causalean.Stat.IIDSample Ω X μ P), Causalean.Stat.DegenKernel P g → ∀ {C J h : ℝ} {n : ℕ}, 2 ≤ n → 0 < h → 0 ≤ J → 0 ≤ C → Causalean.Stat.IIDSample.zeta P g ≤ C * J / h ^ 2 → ProbabilityTheory.variance (Causalean.Stat.uStatistic S g n) μ ≤ 4 * C * J / (↑n * h) ^ 2","l":99,"f":"Causalean/Stat/Nonparametric/HOIF/DegenerateUStatVariance.lean"},{"n":"Causalean.Stat.Nonparametric.HOIF.gram","k":"def","m":"Causalean.Stat.Nonparametric.HOIF.ProjectedKernelTrace","a":"Stat/Nonparametric/HOIF","t":1,"d":"The second-moment (Gram) matrix of the basis c: Σ_{kl} = ∫ c(x)_k c(x)_l dP.","s":"{X : Type u_1} → [inst : MeasurableSpace X] → {J : ℕ} → (X → Fin J → ℝ) → MeasureTheory.Measure X → Matrix (Fin J) (Fin J) ℝ","l":49,"f":"Causalean/Stat/Nonparametric/HOIF/ProjectedKernelTrace.lean"},{"n":"Causalean.Stat.Nonparametric.HOIF.gram_symm","k":"theorem","m":"Causalean.Stat.Nonparametric.HOIF.ProjectedKernelTrace","a":"Stat/Nonparametric/HOIF","t":2,"d":"The Gram matrix is symmetric.","s":"∀ {X : Type u_1} [inst : MeasurableSpace X] {P : MeasureTheory.Measure X} {J : ℕ} (c : X → Fin J → ℝ), (Causalean.Stat.Nonparametric.HOIF.gram c P).transpose = Causalean.Stat.Nonparametric.HOIF.gram c P","l":59,"f":"Causalean/Stat/Nonparametric/HOIF/ProjectedKernelTrace.lean"},{"n":"Causalean.Stat.Nonparametric.HOIF.hoif_order_choice_negligible","k":"theorem","m":"Causalean.Stat.Nonparametric.HOIF.ProductRemainder","a":"Stat/Nonparametric/HOIF","t":1,"d":"Rate selection: a high-enough HOIF order makes the squared remainder o(ρ_n). For a squared product-remainder bound of the power-law form K·n^{-a} and a target risk rate ρ_n = n^{-κ}, if the remainder's decay exponent a exceeds the target rate's exponent κ, then the ratio (K·n^{-a}) / n^{-κ} = K·n^{κ-a} tends to 0 as n → ∞: the HOIF estimation bias is asymptotically negligible relative to the target rate.","s":"∀ {K a κ : ℝ}, κ < a → Filter.Tendsto (fun n => K * ↑n ^ (κ - a)) Filter.atTop (nhds 0)","l":82,"f":"Causalean/Stat/Nonparametric/HOIF/ProductRemainder.lean"},{"n":"Causalean.Stat.Nonparametric.HOIF.hoif_projection_risk_bound","k":"theorem","m":"Causalean.Stat.Nonparametric.HOIF.ProjectionRisk","a":"Stat/Nonparametric/HOIF","t":1,"d":"Order-m HOIF projection-risk bound. For an i.i.d. sample S and a projection kernel g, suppose g is symmetric, square-integrable, and doubly degenerate, the sample size n is at least 2, and the bandwidth h is positive, the projection dimension J and trace constant C are nonnegative, and the kernel's L²-energy ζ obeys the trace bound ζ ≤ C·J/h². Suppose further that the first-order variance V1 obeys the rate V1 ≤ Cv1/(nh) and the projection bias² Bsq obeys the rate Bsq ≤ Cb·J^{-4s/d}, and that the order-m estimation remainder R is controlled, over a finite index set T, by a sum of products of m+1 nuisance-error factors: every such factor is nonnegative, every factor is at most the largest nuisance error δ, and the absolute value of R is bounded by that sum of products. If, finally, the estimator's risk decomposes as risk ≤ V1 + Bsq + Var[Uₙ] + R², then the risk is bounded by the explicit sum Cv1/(nh) + Cb·J^{-4s/d} + 4C·J/(nh)² + |T|²·δ^{2(m+1)}.","s":"∀ {Ω : Type u_1} {X : Type u_2} [inst : MeasurableSpace Ω] [inst_1 : MeasurableSpace X] {μ : MeasureTheory.Measure Ω} {P : MeasureTheory.Measure X} [MeasureTheory.IsProbabilityMeasure μ] [MeasureTheory.IsProbabilityMeasure P] {g : X → X → ℝ} (S : Causalean.Stat.IIDSample Ω X μ P), Causalean.Stat.DegenKernel P g → ∀ {ι : Type u_3} (T : Finset ι) (e : ι → ℕ → ℝ) (R : ℝ) {risk V1 Bsq Cv1 Cb C J h s d δ : ℝ} {n m : ℕ}, 2 ≤ n → 0 < h → 0 ≤ J → 0 ≤ C → Causalean.Stat.IIDSample.zeta P g ≤ C * J / h ^ 2 → V1 ≤ Cv1 / (↑n * h) → Bsq ≤ Cb * J ^ (-(4 * s / d)) → (∀ t ∈ T, ∀ k ∈ Finset.range (m + 1), 0 ≤ e t k) → (∀ t ∈ T, ∀ k ∈ Finset.range (m + 1), e t k ≤ δ) → |R| ≤ ∑ t ∈ T, ∏ k ∈ Finset.range (m + 1), e t k → risk ≤ V1 + Bsq + ProbabilityTheory.variance (Causalean.Stat.uStatistic S g n) μ + R ^ 2 → risk ≤ Cv1 / (↑n * h) + Cb * J ^ (-(4 * s / d)) + 4 * C * J / (↑n * h) ^ 2 + ↑T.card ^ 2 * δ ^ (2 * (m + 1))","l":46,"f":"Causalean/Stat/Nonparametric/HOIF/ProjectionRisk.lean"},{"n":"Causalean.Stat.Nonparametric.HOIF.hoif_remainder_sq_le","k":"theorem","m":"Causalean.Stat.Nonparametric.HOIF.ProductRemainder","a":"Stat/Nonparametric/HOIF","t":2,"d":"Squared HOIF product-remainder bound. The order-m HOIF estimation-bias remainder R is dominated by a sum, over a finite index set T of terms, of products of m+1 nuisance-error factors e t k; every factor lies in [0, δ], where δ is the largest nuisance L²-error. Then the remainder's squared contribution obeys R² ≤ |T|²·δ^{2(m+1)}. This is the order-m doubly-robust bound: the higher the order, the higher the power of δ.","s":"∀ {ι : Type u_1} (T : Finset ι) (e : ι → ℕ → ℝ) (m : ℕ) (δ R : ℝ), (∀ t ∈ T, ∀ k ∈ Finset.range (m + 1), 0 ≤ e t k) → (∀ t ∈ T, ∀ k ∈ Finset.range (m + 1), e t k ≤ δ) → |R| ≤ ∑ t ∈ T, ∏ k ∈ Finset.range (m + 1), e t k → R ^ 2 ≤ ↑T.card ^ 2 * δ ^ (2 * (m + 1))","l":45,"f":"Causalean/Stat/Nonparametric/HOIF/ProductRemainder.lean"},{"n":"Causalean.Stat.Nonparametric.HOIF.integral_uStatistic_eq_zero","k":"theorem","m":"Causalean.Stat.Nonparametric.HOIF.DegenerateUStatVariance","a":"Stat/Nonparametric/HOIF","t":2,"d":"Mean of the degenerate U-statistic is zero.","s":"∀ {Ω : Type u_1} {X : Type u_2} [inst : MeasurableSpace Ω] [inst_1 : MeasurableSpace X] {μ : MeasureTheory.Measure Ω} {P : MeasureTheory.Measure X} [MeasureTheory.IsProbabilityMeasure μ] [MeasureTheory.IsProbabilityMeasure P] {g : X → X → ℝ} (S : Causalean.Stat.IIDSample Ω X μ P), Causalean.Stat.DegenKernel P g → ∀ {n : ℕ}, 2 ≤ n → ∫ (ω : Ω), Causalean.Stat.uStatistic S g n ω ∂μ = 0","l":56,"f":"Causalean/Stat/Nonparametric/HOIF/DegenerateUStatVariance.lean"},{"n":"Causalean.Stat.Nonparametric.HOIF.memLp_uStatistic","k":"theorem","m":"Causalean.Stat.Nonparametric.HOIF.DegenerateUStatVariance","a":"Stat/Nonparametric/HOIF","t":2,"d":"The degenerate order-2 U-statistic is in L² (rescaling the √n-version).","s":"∀ {Ω : Type u_1} {X : Type u_2} [inst : MeasurableSpace Ω] [inst_1 : MeasurableSpace X] {μ : MeasureTheory.Measure Ω} {P : MeasureTheory.Measure X} [MeasureTheory.IsProbabilityMeasure μ] [MeasureTheory.IsProbabilityMeasure P] {g : X → X → ℝ} (S : Causalean.Stat.IIDSample Ω X μ P), Causalean.Stat.DegenKernel P g → ∀ {n : ℕ}, 2 ≤ n → MeasureTheory.MemLp (Causalean.Stat.uStatistic S g n) 2 μ","l":42,"f":"Causalean/Stat/Nonparametric/HOIF/DegenerateUStatVariance.lean"},{"n":"Causalean.Stat.Nonparametric.HOIF.prod_le_pow_of_factors_le","k":"theorem","m":"Causalean.Stat.Nonparametric.HOIF.ProductRemainder","a":"Stat/Nonparametric/HOIF","t":2,"d":"A product of m+1 factors each in [0, δ] is at most δ^{m+1}. Each nuisance-error factor of a single HOIF remainder term is nonnegative and bounded by the largest error δ; multiplying m+1 of them keeps the product below δ^{m+1}.","s":"∀ {ι : Type u_1} (e : ι → ℝ) (m : ℕ) (δ : ℝ) (s : Finset ℕ) (idx : ℕ → ι), s.card = m + 1 → (∀ k ∈ s, 0 ≤ e (idx k)) → (∀ k ∈ s, e (idx k) ≤ δ) → ∏ k ∈ s, e (idx k) ≤ δ ^ (m + 1)","l":33,"f":"Causalean/Stat/Nonparametric/HOIF/ProductRemainder.lean"},{"n":"Causalean.Stat.Nonparametric.HOIF.projKernel","k":"def","m":"Causalean.Stat.Nonparametric.HOIF.ProjectedKernelTrace","a":"Stat/Nonparametric/HOIF","t":1,"d":"The projected HOIF kernel g(x,y) = ∑_{k,l} c(x)_k M_{kl} c(y)_l.","s":"{X : Type u_1} → {J : ℕ} → (X → Fin J → ℝ) → Matrix (Fin J) (Fin J) ℝ → X → X → ℝ","l":53,"f":"Causalean/Stat/Nonparametric/HOIF/ProjectedKernelTrace.lean"},{"n":"Causalean.Stat.Nonparametric.HOIF.projKernel_L2_eq_dim","k":"theorem","m":"Causalean.Stat.Nonparametric.HOIF.ProjectedKernelTrace","a":"Stat/Nonparametric/HOIF","t":1,"d":"Projected degenerate-kernel L²-energy equals the dimension: ζ = J. For a feature map c into J-dimensional space, if every coordinate of c is square-integrable under P, the Gram matrix of c under P is invertible, and M is the inverse of that Gram matrix, then the squared L²(P⊗P)-norm of the HOIF projection kernel g(x,y) = ⟨c(x), M·c(y)⟩ equals the projection dimension J.","s":"∀ {X : Type u_1} [inst : MeasurableSpace X] {P : MeasureTheory.Measure X} [MeasureTheory.IsProbabilityMeasure P] {J : ℕ} (c : X → Fin J → ℝ) (M : Matrix (Fin J) (Fin J) ℝ), (∀ (k : Fin J), MeasureTheory.MemLp (fun x => c x k) 2 P) → IsUnit (Causalean.Stat.Nonparametric.HOIF.gram c P).det → M = (Causalean.Stat.Nonparametric.HOIF.gram c P)⁻¹ → ∫ (p : X × X), Causalean.Stat.Nonparametric.HOIF.projKernel c M p.1 p.2 ^ 2 ∂P.prod P = ↑J","l":215,"f":"Causalean/Stat/Nonparametric/HOIF/ProjectedKernelTrace.lean"},{"n":"Causalean.Stat.Nonparametric.HOIF.projKernel_L2_eq_sum","k":"theorem","m":"Causalean.Stat.Nonparametric.HOIF.ProjectedKernelTrace","a":"Stat/Nonparametric/HOIF","t":2,"d":"L²-energy of the projected kernel (expanded form). By independence of the two sample points, the double integral of the squared projected kernel factors into Gram entries:","s":"∀ {X : Type u_1} [inst : MeasurableSpace X] {P : MeasureTheory.Measure X} [MeasureTheory.IsProbabilityMeasure P] {J : ℕ} (c : X → Fin J → ℝ) (M : Matrix (Fin J) (Fin J) ℝ), (∀ (k : Fin J), MeasureTheory.MemLp (fun x => c x k) 2 P) → ∫ (p : X × X), Causalean.Stat.Nonparametric.HOIF.projKernel c M p.1 p.2 ^ 2 ∂P.prod P = ∑ k, ∑ l, ∑ k', ∑ l', M k l * M k' l' * Causalean.Stat.Nonparametric.HOIF.gram c P k k' * Causalean.Stat.Nonparametric.HOIF.gram c P l l'","l":90,"f":"Causalean/Stat/Nonparametric/HOIF/ProjectedKernelTrace.lean"},{"n":"Causalean.Stat.Nonparametric.HOIF.projKernel_degen","k":"theorem","m":"Causalean.Stat.Nonparametric.HOIF.ProjectedKernelTrace","a":"Stat/Nonparametric/HOIF","t":2,"d":"One-sided degeneracy of the projected kernel. If the basis coordinates are centered (∫ c_l dP = 0 for every l), then ∫ g(x, ·) dP = 0 for every x. This records the zero-integral field only; measurability, symmetry, and square-integrability are separate requirements for packaging the kernel as a degenerate U-statistic kernel.","s":"∀ {X : Type u_1} [inst : MeasurableSpace X] {P : MeasureTheory.Measure X} {J : ℕ} (c : X → Fin J → ℝ) (M : Matrix (Fin J) (Fin J) ℝ), (∀ (l : Fin J), MeasureTheory.Integrable (fun x => c x l) P) → (∀ (l : Fin J), ∫ (x : X), c x l ∂P = 0) → ∀ (x : X), ∫ (y : X), Causalean.Stat.Nonparametric.HOIF.projKernel c M x y ∂P = 0","l":66,"f":"Causalean/Stat/Nonparametric/HOIF/ProjectedKernelTrace.lean"},{"n":"Causalean.Stat.Nonparametric.HOIF.sum_collapse_dim","k":"theorem","m":"Causalean.Stat.Nonparametric.HOIF.ProjectedKernelTrace","a":"Stat/Nonparametric/HOIF","t":2,"d":"The 4-fold Gram sum collapses to the projection dimension. With the HOIF weighting M = Σ⁻¹ (Σ = gram c P invertible), the expanded L²-energy equals the projection dimension J, because Σ Σ⁻¹ = Σ⁻¹ Σ = 1.","s":"∀ {J : ℕ} (M S : Matrix (Fin J) (Fin J) ℝ), IsUnit S.det → M = S⁻¹ → ∑ k, ∑ l, ∑ k', ∑ l', M k l * M k' l' * S k k' * S l l' = ↑J","l":184,"f":"Causalean/Stat/Nonparametric/HOIF/ProjectedKernelTrace.lean"},{"n":"Causalean.Stat.Nonparametric.HolderBallStd","k":"def","m":"Causalean.Stat.Nonparametric.Approximation.HolderInterpolation.Defs","a":"Stat/Nonparametric/Approximation/HolderInterpolation","t":1,"d":"The standard multivariate Hölder ball of a given smoothness order and radius on a region: derivatives through the conventional highest order are continuous and bounded, and the highest derivative changes at the Hölder rate set by that order.","s":"{d : ℕ} → ((Fin d → ℝ) → ℝ) → ℝ → ℝ → Set (Fin d → ℝ) → Prop","l":97,"f":"Causalean/Stat/Nonparametric/Approximation/HolderInterpolation/Defs.lean"},{"n":"Causalean.Stat.Nonparametric.KernelOrder","k":"structure","m":"Causalean.Stat.Nonparametric.Approximation.Kernel","a":"Stat/Nonparametric/Approximation","t":1,"d":"A kernel of order at least p: supported in [-1,1], integrable, with unit mass ∫ K = 1 and vanishing moments ∫ uʲ K(u) du = 0 for 1 ≤ j ≤ p. These are the inputs of the classical interior local-polynomial / kernel bias theorem.","s":"(ℝ → ℝ) → ℕ → Prop","l":35,"f":"Causalean/Stat/Nonparametric/Approximation/Kernel.lean"},{"n":"Causalean.Stat.Nonparametric.LocalPolynomial.coordinateDirections","k":"def","m":"Causalean.Stat.Nonparametric.LocalPolynomial.CoordinateDerivative","a":"Stat/Nonparametric/LocalPolynomial","t":1,"d":"The ordered list of standard coordinate directions associated with a bivariate multi-index.","s":"(alpha : Fin 2 → ℕ) → Fin (Causalean.Stat.Nonparametric.LocalPolynomial.coordinateMultiOrder alpha) → EuclideanSpace ℝ (Fin 2)","l":25,"f":"Causalean/Stat/Nonparametric/LocalPolynomial/CoordinateDerivative.lean"},{"n":"Causalean.Stat.Nonparametric.LocalPolynomial.coordinateMultiOrder","k":"def","m":"Causalean.Stat.Nonparametric.LocalPolynomial.CoordinateDerivative","a":"Stat/Nonparametric/LocalPolynomial","t":1,"d":"Total order of a bivariate coordinate multi-index.","s":"(Fin 2 → ℕ) → ℕ","l":22,"f":"Causalean/Stat/Nonparametric/LocalPolynomial/CoordinateDerivative.lean"},{"n":"Causalean.Stat.Nonparametric.LocalPolynomial.coordinatePartial","k":"def","m":"Causalean.Stat.Nonparametric.LocalPolynomial.CoordinateDerivative","a":"Stat/Nonparametric/LocalPolynomial","t":1,"d":"The scalar coordinate partial derivative indexed by alpha.","s":"(EuclideanSpace ℝ (Fin 2) → ℝ) → (Fin 2 → ℕ) → EuclideanSpace ℝ (Fin 2) → ℝ","l":32,"f":"Causalean/Stat/Nonparametric/LocalPolynomial/CoordinateDerivative.lean"},{"n":"Causalean.Stat.Nonparametric.LocalPolynomial.coordinatePartial_abs_le_iteratedFDeriv_norm","k":"theorem","m":"Causalean.Stat.Nonparametric.LocalPolynomial.CoordinateDerivative","a":"Stat/Nonparametric/LocalPolynomial","t":1,"d":"Evaluating the iterated Fréchet derivative of a function f of a bivariate multi-index alpha at a point x, along the standard coordinate directions, cannot increase its operator norm — the resulting scalar coordinate partial is bounded in absolute value by the operator norm of the full iterated derivative.","s":"∀ (f : EuclideanSpace ℝ (Fin 2) → ℝ) (alpha : Fin 2 → ℕ) (x : EuclideanSpace ℝ (Fin 2)), |Causalean.Stat.Nonparametric.LocalPolynomial.coordinatePartial f alpha x| ≤ ‖iteratedFDeriv ℝ (Causalean.Stat.Nonparametric.LocalPolynomial.coordinateMultiOrder alpha) f x‖","l":36,"f":"Causalean/Stat/Nonparametric/LocalPolynomial/CoordinateDerivative.lean"},{"n":"Causalean.Stat.Nonparametric.LocalPolynomial.coordinatePartial_sub_abs_le_iteratedFDeriv_sub_norm","k":"theorem","m":"Causalean.Stat.Nonparametric.LocalPolynomial.CoordinateDerivative","a":"Stat/Nonparametric/LocalPolynomial","t":2,"d":"Differences of scalar coordinate partials are bounded by the operator norm of the corresponding Fréchet-derivative difference.","s":"∀ (f : EuclideanSpace ℝ (Fin 2) → ℝ) (alpha : Fin 2 → ℕ) (x z : EuclideanSpace ℝ (Fin 2)), |Causalean.Stat.Nonparametric.LocalPolynomial.coordinatePartial f alpha x - Causalean.Stat.Nonparametric.LocalPolynomial.coordinatePartial f alpha z| ≤ ‖iteratedFDeriv ℝ (Causalean.Stat.Nonparametric.LocalPolynomial.coordinateMultiOrder alpha) f x - iteratedFDeriv ℝ (Causalean.Stat.Nonparametric.LocalPolynomial.coordinateMultiOrder alpha) f z‖","l":58,"f":"Causalean/Stat/Nonparametric/LocalPolynomial/CoordinateDerivative.lean"},{"n":"Causalean.Stat.Nonparametric.LocalPolynomial.localPolynomial","k":"def","m":"Causalean.Stat.Nonparametric.LocalPolynomial.GramCoercivity","a":"Stat/Nonparametric/LocalPolynomial","t":1,"d":"The polynomial represented by the local-polynomial coefficient vector.","s":"(p : ℕ) → (Fin (p + 1) → ℝ) → Polynomial ℝ","l":22,"f":"Causalean/Stat/Nonparametric/LocalPolynomial/GramCoercivity.lean"},{"n":"Causalean.Stat.Nonparametric.LocalPolynomial.localPolynomial_eq_zero_iff","k":"theorem","m":"Causalean.Stat.Nonparametric.LocalPolynomial.GramCoercivity","a":"Stat/Nonparametric/LocalPolynomial","t":2,"d":"The coefficient polynomial vanishes only when every coefficient does.","s":"∀ (p : ℕ) (v : Fin (p + 1) → ℝ), Causalean.Stat.Nonparametric.LocalPolynomial.localPolynomial p v = 0 ↔ v = 0","l":35,"f":"Causalean/Stat/Nonparametric/LocalPolynomial/GramCoercivity.lean"},{"n":"Causalean.Stat.Nonparametric.LocalPolynomial.localPolynomial_eval","k":"theorem","m":"Causalean.Stat.Nonparametric.LocalPolynomial.GramCoercivity","a":"Stat/Nonparametric/LocalPolynomial","t":2,"d":"Evaluation of the coefficient polynomial is the dot product with the monomial basis.","s":"∀ (p : ℕ) (v : Fin (p + 1) → ℝ) (u : ℝ), Polynomial.eval u (Causalean.Stat.Nonparametric.LocalPolynomial.localPolynomial p v) = ∑ i, v i * u ^ ↑i","l":27,"f":"Causalean/Stat/Nonparametric/LocalPolynomial/GramCoercivity.lean"},{"n":"Causalean.Stat.Nonparametric.LocalPolynomial.radialPolynomialEnergy","k":"def","m":"Causalean.Stat.Nonparametric.LocalPolynomial.GramCoercivity","a":"Stat/Nonparametric/LocalPolynomial","t":1,"d":"The fixed radial energy used in the polar-sector lower bound.","s":"(p : ℕ) → (Fin (p + 1) → ℝ) → ℝ","l":56,"f":"Causalean/Stat/Nonparametric/LocalPolynomial/GramCoercivity.lean"},{"n":"Causalean.Stat.Nonparametric.LocalPolynomial.radialPolynomialEnergy_coercive","k":"theorem","m":"Causalean.Stat.Nonparametric.LocalPolynomial.GramCoercivity","a":"Stat/Nonparametric/LocalPolynomial","t":1,"d":"For a polynomial degree bound p, there is a positive constant such that the radial polynomial energy of any coefficient vector is bounded below by that constant times the sum of the squared coefficients: on the Euclidean unit sphere, radial polynomial energy has a positive minimum, and homogeneity packages this as a coercive lower bound for all coefficient vectors.","s":"∀ (p : ℕ), ∃ c, 0 < c ∧ ∀ (v : Fin (p + 1) → ℝ), c * ∑ i, v i ^ 2 ≤ Causalean.Stat.Nonparametric.LocalPolynomial.radialPolynomialEnergy p v","l":151,"f":"Causalean/Stat/Nonparametric/LocalPolynomial/GramCoercivity.lean"},{"n":"Causalean.Stat.Nonparametric.LocalPolynomial.radialPolynomialEnergy_continuous","k":"theorem","m":"Causalean.Stat.Nonparametric.LocalPolynomial.GramCoercivity","a":"Stat/Nonparametric/LocalPolynomial","t":2,"d":"The radial energy is a continuous quadratic function of its coefficient vector.","s":"∀ (p : ℕ), Continuous (Causalean.Stat.Nonparametric.LocalPolynomial.radialPolynomialEnergy p)","l":128,"f":"Causalean/Stat/Nonparametric/LocalPolynomial/GramCoercivity.lean"},{"n":"Causalean.Stat.Nonparametric.LocalPolynomial.radialPolynomialEnergy_eq_integral","k":"theorem","m":"Causalean.Stat.Nonparametric.LocalPolynomial.GramCoercivity","a":"Stat/Nonparametric/LocalPolynomial","t":2,"d":"The explicit moment matrix is exactly the weighted squared-polynomial integral on the unit interval.","s":"∀ (p : ℕ) (v : Fin (p + 1) → ℝ), Causalean.Stat.Nonparametric.LocalPolynomial.radialPolynomialEnergy p v = ∫ (u : ℝ) in 0..1, (∑ i, v i * u ^ ↑i) ^ 2 * u","l":63,"f":"Causalean/Stat/Nonparametric/LocalPolynomial/GramCoercivity.lean"},{"n":"Causalean.Stat.Nonparametric.LocalPolynomial.radialPolynomialEnergy_pos","k":"theorem","m":"Causalean.Stat.Nonparametric.LocalPolynomial.GramCoercivity","a":"Stat/Nonparametric/LocalPolynomial","t":2,"d":"A nonzero coefficient vector has strictly positive radial energy.","s":"∀ (p : ℕ) {v : Fin (p + 1) → ℝ}, v ≠ 0 → 0 < Causalean.Stat.Nonparametric.LocalPolynomial.radialPolynomialEnergy p v","l":90,"f":"Causalean/Stat/Nonparametric/LocalPolynomial/GramCoercivity.lean"},{"n":"Causalean.Stat.Nonparametric.LocalPolynomial.radialPolynomialEnergy_smul","k":"theorem","m":"Causalean.Stat.Nonparametric.LocalPolynomial.GramCoercivity","a":"Stat/Nonparametric/LocalPolynomial","t":2,"d":"Radial energy is homogeneous of degree two in the coefficient vector.","s":"∀ (p : ℕ) (a : ℝ) (v : Fin (p + 1) → ℝ), Causalean.Stat.Nonparametric.LocalPolynomial.radialPolynomialEnergy p (a • v) = a ^ 2 * Causalean.Stat.Nonparametric.LocalPolynomial.radialPolynomialEnergy p v","l":136,"f":"Causalean/Stat/Nonparametric/LocalPolynomial/GramCoercivity.lean"},{"n":"Causalean.Stat.Nonparametric.LocalPolynomial.signedRadialPolynomialEnergy_coercive","k":"theorem","m":"Causalean.Stat.Nonparametric.LocalPolynomial.GramCoercivity","a":"Stat/Nonparametric/LocalPolynomial","t":2,"d":"The same coercivity constant works in both signed-distance orientations. The negative orientation merely changes coefficient i by the sign (-1)^i, which preserves the sum of coefficient squares.","s":"∀ (p : ℕ), ∃ c, 0 < c ∧ ∀ (t : Bool) (v : Fin (p + 1) → ℝ), c * ∑ i, v i ^ 2 ≤ ∫ (u : ℝ) in 0..1, (∑ i, v i * (if t = true then u else -u) ^ ↑i) ^ 2 * u","l":215,"f":"Causalean/Stat/Nonparametric/LocalPolynomial/GramCoercivity.lean"},{"n":"Causalean.Stat.Nonparametric.SeriesSieve.jacksonSeriesBias_le","k":"theorem","m":"Causalean.Stat.Nonparametric.SeriesSieve.PredictionRate","a":"Stat/Nonparametric/SeriesSieve","t":1,"d":"Jackson bias rate for the series least-squares objective. If the noise-free least-squares objective A := lstsqObjective Φ w f c0 obeys the squared Jackson best-approximation bound A ≤ (∑ᵢ wᵢ)·(C_J·J^{−s/d})² — the shape produced by seriesApprox_le_of_sup with sup-error δ = C_J·J^{−s/d} — then the same objective satisfies the explicit doubled-exponent bias rate A ≤ ((∑ᵢ wᵢ)·C_J²) · J^{−2s/d}:","s":"∀ {N : ℕ} {ι : Type u_1} [inst : Fintype ι] {Φ : Fin N → ι → ℝ} {w f : Fin N → ℝ} {c0 : ι → ℝ} {s d C_J : ℝ} {J : ℕ}, Causalean.Stat.Nonparametric.lstsqObjective Φ w f c0 ≤ (∑ i, w i) * (C_J * ↑J ^ (-(s / d))) ^ 2 → Causalean.Stat.Nonparametric.lstsqObjective Φ w f c0 ≤ (∑ i, w i) * C_J ^ 2 * ↑J ^ (-(2 * s / d))","l":44,"f":"Causalean/Stat/Nonparametric/SeriesSieve/PredictionRate.lean"},{"n":"Causalean.Stat.Nonparametric.SeriesSieve.seriesEffectiveDoF_le","k":"theorem","m":"Causalean.Stat.Nonparametric.SeriesSieve.PredictionRate","a":"Stat/Nonparametric/SeriesSieve","t":1,"d":"Effective degrees of freedom is controlled by J/N. If the sample size N is positive, the least-squares weights are normalized as wᵢ = 1/N, and the hat map a obeys the Frobenius/trace bound ∑ᵢ ∑ₖ aᵢₖ² ≤ Cvar·J, then the weighted coefficient sum — the effective degrees of freedom V of the oracle inequality — obeys the bound Cvar · J / N:","s":"∀ {N : ℕ} {a : Fin N → Fin N → ℝ} {w : Fin N → ℝ} {Cvar : ℝ} {J : ℕ}, 0 < N → (∀ (i : Fin N), w i = 1 / ↑N) → ∑ i, ∑ k, a i k ^ 2 ≤ Cvar * ↑J → ∑ i, w i * ∑ k, a i k ^ 2 ≤ Cvar * ↑J / ↑N","l":72,"f":"Causalean/Stat/Nonparametric/SeriesSieve/PredictionRate.lean"},{"n":"Causalean.Stat.Nonparametric.SeriesSieve.seriesLS_prediction_rate","k":"theorem","m":"Causalean.Stat.Nonparametric.SeriesSieve.PredictionRate","a":"Stat/Nonparametric/SeriesSieve","t":1,"d":"Series / sieve least-squares prediction rate O(J^{−2s/d} + J/N). Assume a positive sample size; the noise-free projection c0 has residual orthogonal to every design column; the data fit chat ω differs from c0 by a deterministic linear image a of a noise family ε that is square-integrable, mean zero, and spherical with scale σ; the noise-free objective obeys the squared Jackson best-approximation bound at rate s/d with constant C_J; and the weights are normalized wᵢ = 1/N with the hat map obeying the Frobenius/trace bound ∑ᵢ ∑ₖ aᵢₖ² ≤ Cvar·J. Then the expected weighted quadratic prediction error of the fitted series coefficients obeys the standard series/sieve rate:","s":"∀ {Ω : Type u_1} {N : ℕ} {ι : Type u_2} [inst : Fintype ι] [inst_1 : MeasurableSpace Ω] {μ : MeasureTheory.Measure Ω} [MeasureTheory.IsProbabilityMeasure μ] {Φ : Fin N → ι → ℝ} {w f : Fin N → ℝ} {c0 : ι → ℝ} {chat : Ω → ι → ℝ} {ε : Fin N → Ω → ℝ} {a : Fin N → Fin N → ℝ} {s d C_J Cvar σ : ℝ} {J : ℕ}, 0 < N → (∀ (k : ι), ∑ i, w i * Causalean.Stat.Nonparametric.lstsqResidual Φ f c0 i * Φ i k = 0) → (∀ (ω : Ω) (i : Fin N), ∑ j, (c0 j - chat ω j) * Φ i j = ∑ k, a i k * ε k ω) → (∀ (k : Fin N), MeasureTheory.MemLp (ε k) 2 μ) → (∀ (k : Fin N), ∫ (ω : Ω), ε k ω ∂μ = 0) → Causalean.GaussMarkov.SphericalFamily ε μ σ → Causalean.Stat.Nonparametric.lstsqObjective Φ w f c0 ≤ (∑ i, w i) * (C_J * ↑J ^ (-(s / d))) ^ 2 → (∀ (i : Fin N), w i = 1 / ↑N) → ∑ i, ∑ k, a i k ^ 2 ≤ Cvar * ↑J → ∫ (ω : Ω), Causalean.Stat.Nonparametric.lstsqObjective Φ w f (chat ω) ∂μ ≤ (∑ i, w i) * C_J ^ 2 * ↑J ^ (-(2 * s / d)) + σ ^ 2 * (Cvar * ↑J / ↑N)","l":98,"f":"Causalean/Stat/Nonparametric/SeriesSieve/PredictionRate.lean"},{"n":"Causalean.Stat.Nonparametric.cellBase","k":"def","m":"Causalean.Stat.Nonparametric.SeriesSieve.Jackson","a":"Stat/Nonparametric/SeriesSieve","t":1,"d":"The left endpoint lo + (cellIdx x)·δ of the cell containing x. The piecewise-Taylor approximant expands f around this base point.","s":"ℝ → ℝ → ℕ → ℝ → ℝ","l":47,"f":"Causalean/Stat/Nonparametric/SeriesSieve/Jackson.lean"},{"n":"Causalean.Stat.Nonparametric.cellBase_dist","k":"theorem","m":"Causalean.Stat.Nonparametric.SeriesSieve.Jackson","a":"Stat/Nonparametric/SeriesSieve","t":2,"d":"x is within one cell width to the right of its base point: 0 ≤ x − base ≤ δ.","s":"∀ {lo hi : ℝ} {J : ℕ}, lo < hi → 0 < J → ∀ {x : ℝ}, x ∈ Set.Icc lo hi → 0 ≤ x - Causalean.Stat.Nonparametric.cellBase lo hi J x ∧ x - Causalean.Stat.Nonparametric.cellBase lo hi J x ≤ Causalean.Stat.Nonparametric.cellWidth lo hi J","l":99,"f":"Causalean/Stat/Nonparametric/SeriesSieve/Jackson.lean"},{"n":"Causalean.Stat.Nonparametric.cellBase_mem","k":"theorem","m":"Causalean.Stat.Nonparametric.SeriesSieve.Jackson","a":"Stat/Nonparametric/SeriesSieve","t":2,"d":"The cell base point lies inside the window [lo, hi].","s":"∀ {lo hi : ℝ} {J : ℕ}, lo < hi → 0 < J → ∀ {x : ℝ}, Causalean.Stat.Nonparametric.cellBase lo hi J x ∈ Set.Icc lo hi","l":66,"f":"Causalean/Stat/Nonparametric/SeriesSieve/Jackson.lean"},{"n":"Causalean.Stat.Nonparametric.cellIdx","k":"def","m":"Causalean.Stat.Nonparametric.SeriesSieve.Jackson","a":"Stat/Nonparametric/SeriesSieve","t":1,"d":"The cell index of x in a uniform J-cell partition of [lo, hi]: the floor ⌊(x − lo)/δ⌋, clamped to the last cell J − 1 (so the right endpoint hi lands in the final cell).","s":"ℝ → ℝ → ℕ → ℝ → ℕ","l":41,"f":"Causalean/Stat/Nonparametric/SeriesSieve/Jackson.lean"},{"n":"Causalean.Stat.Nonparametric.cellWidth","k":"def","m":"Causalean.Stat.Nonparametric.SeriesSieve.Jackson","a":"Stat/Nonparametric/SeriesSieve","t":1,"d":"The width δ = (hi − lo)/J of a uniform J-cell partition of [lo, hi].","s":"ℝ → ℝ → ℕ → ℝ","l":38,"f":"Causalean/Stat/Nonparametric/SeriesSieve/Jackson.lean"},{"n":"Causalean.Stat.Nonparametric.cellWidth_pos","k":"theorem","m":"Causalean.Stat.Nonparametric.SeriesSieve.Jackson","a":"Stat/Nonparametric/SeriesSieve","t":2,"d":"The cell width is positive on a nondegenerate window.","s":"∀ {lo hi : ℝ} {J : ℕ}, lo < hi → 0 < J → 0 < Causalean.Stat.Nonparametric.cellWidth lo hi J","l":58,"f":"Causalean/Stat/Nonparametric/SeriesSieve/Jackson.lean"},{"n":"Causalean.Stat.Nonparametric.designMatrix","k":"def","m":"Causalean.Stat.Nonparametric.LocalPoly.Weights","a":"Stat/Nonparametric/LocalPoly","t":1,"d":"The weighted design moment matrix M_{jk} = ∑ᵢ wᵢ xᵢʲ xᵢᵏ of a degree-p local-polynomial fit with design points xᵢ (typically aᵢ − t) and weights wᵢ.","s":"(p : ℕ) → {N : ℕ} → (Fin N → ℝ) → (Fin N → ℝ) → Matrix (Fin (p + 1)) (Fin (p + 1)) ℝ","l":39,"f":"Causalean/Stat/Nonparametric/LocalPoly/Weights.lean"},{"n":"Causalean.Stat.Nonparametric.designMatrix_isHermitian","k":"theorem","m":"Causalean.Stat.Nonparametric.LocalPoly.DesignMatrixPosDef","a":"Stat/Nonparametric/LocalPoly","t":2,"d":"The design moment matrix is symmetric.","s":"∀ {N p : ℕ} (x w : Fin N → ℝ), (Causalean.Stat.Nonparametric.designMatrix p x w).IsHermitian","l":62,"f":"Causalean/Stat/Nonparametric/LocalPoly/DesignMatrixPosDef.lean"},{"n":"Causalean.Stat.Nonparametric.designMatrix_isUnit_det","k":"theorem","m":"Causalean.Stat.Nonparametric.LocalPoly.DesignMatrixPosDef","a":"Stat/Nonparametric/LocalPoly","t":1,"d":"Invertibility of the design moment matrix from design non-degeneracy. If the design weights w are nonnegative and the design is non-degenerate — no nonzero coefficient vector v yields a degree-p polynomial ∑ⱼ vⱼ xᵢʲ vanishing at every positively weighted design point, then the design moment matrix's determinant is a unit, i.e. the matrix is invertible: a positive definite matrix has a unit determinant, discharging the IsUnit (designMatrix p x w).det hypothesis used throughout the local-polynomial analysis.","s":"∀ {N p : ℕ} {x w : Fin N → ℝ}, (∀ (i : Fin N), 0 ≤ w i) → (∀ (v : Fin (p + 1) → ℝ), v ≠ 0 → ∃ i, 0 < w i ∧ ∑ j, v j * x i ^ ↑j ≠ 0) → IsUnit (Causalean.Stat.Nonparametric.designMatrix p x w).det","l":102,"f":"Causalean/Stat/Nonparametric/LocalPoly/DesignMatrixPosDef.lean"},{"n":"Causalean.Stat.Nonparametric.designMatrix_posDef","k":"theorem","m":"Causalean.Stat.Nonparametric.LocalPoly.DesignMatrixPosDef","a":"Stat/Nonparametric/LocalPoly","t":1,"d":"The design moment matrix is positive definite under a non-degeneracy condition on the design. If the design weights w are nonnegative and no nonzero coefficient vector v yields a degree-p polynomial ∑ⱼ vⱼ xᵢʲ that vanishes at every positively weighted design point, then the design moment matrix is positive definite. (Implied by the existence of p+1 distinct design points with positive weight, via Vandermonde.)","s":"∀ {N p : ℕ} {x w : Fin N → ℝ}, (∀ (i : Fin N), 0 ≤ w i) → (∀ (v : Fin (p + 1) → ℝ), v ≠ 0 → ∃ i, 0 < w i ∧ ∑ j, v j * x i ^ ↑j ≠ 0) → (Causalean.Stat.Nonparametric.designMatrix p x w).PosDef","l":81,"f":"Causalean/Stat/Nonparametric/LocalPoly/DesignMatrixPosDef.lean"},{"n":"Causalean.Stat.Nonparametric.designMatrix_posSemidef","k":"theorem","m":"Causalean.Stat.Nonparametric.LocalPoly.DesignMatrixPosDef","a":"Stat/Nonparametric/LocalPoly","t":2,"d":"The design moment matrix is positive semidefinite when the weights are nonnegative (a sum of weighted rank-one squares).","s":"∀ {N p : ℕ} {x w : Fin N → ℝ}, (∀ (i : Fin N), 0 ≤ w i) → (Causalean.Stat.Nonparametric.designMatrix p x w).PosSemidef","l":69,"f":"Causalean/Stat/Nonparametric/LocalPoly/DesignMatrixPosDef.lean"},{"n":"Causalean.Stat.Nonparametric.designMatrix_quadForm","k":"theorem","m":"Causalean.Stat.Nonparametric.LocalPoly.DesignMatrixPosDef","a":"Stat/Nonparametric/LocalPoly","t":2,"d":"Gram quadratic form of the design moment matrix. vᵀ M v = ∑ᵢ wᵢ (∑ⱼ vⱼ xᵢʲ)² — the design moment matrix is the weighted Gram matrix of the monomial feature vectors.","s":"∀ {N p : ℕ} (x w : Fin N → ℝ) (v : Fin (p + 1) → ℝ), v ⬝ᵥ (Causalean.Stat.Nonparametric.designMatrix p x w).mulVec v = ∑ i, w i * (∑ j, v j * x i ^ ↑j) ^ 2","l":31,"f":"Causalean/Stat/Nonparametric/LocalPoly/DesignMatrixPosDef.lean"},{"n":"Causalean.Stat.Nonparametric.entry00_eq_quadForm","k":"theorem","m":"Causalean.Stat.Nonparametric.LocalPoly.EstimatorRisk.SquareCompletion","a":"Stat/Nonparametric/LocalPoly/EstimatorRisk","t":2,"d":"The (0,0) entry of a matrix as a quadratic form at the first basis vector: A₀₀ = e₀ᵀ A e₀ with e₀ = Pi.single 0 1.","s":"∀ {p : ℕ} (A : Matrix (Fin (p + 1)) (Fin (p + 1)) ℝ), A 0 0 = Pi.single 0 1 ⬝ᵥ A.mulVec (Pi.single 0 1)","l":82,"f":"Causalean/Stat/Nonparametric/LocalPoly/EstimatorRisk/SquareCompletion.lean"},{"n":"Causalean.Stat.Nonparametric.entry00_le_of_quadForm_sandwich","k":"theorem","m":"Causalean.Stat.Nonparametric.LocalPoly.EstimatorRisk.SquareCompletion","a":"Stat/Nonparametric/LocalPoly/EstimatorRisk","t":2,"d":"Loewner sandwich bound on the (0,0) entry. If A ⪯ C·B in the Loewner order on quadratic forms (wᵀ A w ≤ C·wᵀ B w for all w), then the top weight entries satisfy A₀₀ ≤ C·B₀₀. (Just evaluate the quadratic forms at the first basis vector.)","s":"∀ {p : ℕ} {A B : Matrix (Fin (p + 1)) (Fin (p + 1)) ℝ} {C : ℝ}, (∀ (w : Fin (p + 1) → ℝ), w ⬝ᵥ A.mulVec w ≤ C * w ⬝ᵥ B.mulVec w) → A 0 0 ≤ C * B 0 0","l":137,"f":"Causalean/Stat/Nonparametric/LocalPoly/EstimatorRisk/SquareCompletion.lean"},{"n":"Causalean.Stat.Nonparametric.equivKernelWeight","k":"def","m":"Causalean.Stat.Nonparametric.LocalPoly.Weights","a":"Stat/Nonparametric/LocalPoly","t":1,"d":"The local-polynomial equivalent-kernel weight Sᵢ = ∑ₖ (M⁻¹)₀ₖ wᵢ xᵢᵏ extracting the fitted intercept: the degree-p WLS intercept equals ∑ᵢ Sᵢ Yᵢ.","s":"ℕ → {N : ℕ} → (Fin N → ℝ) → (Fin N → ℝ) → Fin N → ℝ","l":45,"f":"Causalean/Stat/Nonparametric/LocalPoly/Weights.lean"},{"n":"Causalean.Stat.Nonparametric.equivKernelWeight_abs_sum_sq_le","k":"theorem","m":"Causalean.Stat.Nonparametric.LocalPoly.Weights","a":"Stat/Nonparametric/LocalPoly","t":2,"d":"Cauchy–Schwarz leverage bound for the equivalent kernel. The ℓ¹ leverage of the equivalent-kernel weights is controlled by the product of the (0,0) entries of the design moment matrix and its inverse: (∑ᵢ |Sᵢ|)² ≤ M₀₀ · (M⁻¹)₀₀, where M₀₀ = ∑ᵢ wᵢ is the total weight. (Cauchy–Schwarz on Sᵢ = wᵢ gᵢ split as √wᵢ · √wᵢ gᵢ, using ∑ᵢ wᵢ gᵢ² = (M⁻¹)₀₀.) Together with equivKernelWeight_sq_sum_le this reduces *both* the bias leverage ∑ᵢ|Sᵢ| and the variance leverage ∑ᵢ Sᵢ² to the design quantities M₀₀ and (M⁻¹)₀₀.","s":"∀ {N p : ℕ} {x w : Fin N → ℝ}, IsUnit (Causalean.Stat.Nonparametric.designMatrix p x w).det → (∀ (i : Fin N), 0 ≤ w i) → (∑ i, |Causalean.Stat.Nonparametric.equivKernelWeight p x w i|) ^ 2 ≤ Causalean.Stat.Nonparametric.designMatrix p x w 0 0 * (Causalean.Stat.Nonparametric.designMatrix p x w)⁻¹ 0 0","l":318,"f":"Causalean/Stat/Nonparametric/LocalPoly/Weights.lean"},{"n":"Causalean.Stat.Nonparametric.equivKernelWeight_reproduces","k":"theorem","m":"Causalean.Stat.Nonparametric.LocalPoly.Weights","a":"Stat/Nonparametric/LocalPoly","t":1,"d":"Polynomial reproduction of the equivalent kernel. If the weighted design moment matrix is invertible, then the local-polynomial equivalent-kernel weights reproduce polynomials up to degree p: ∑ᵢ Sᵢ xᵢᵐ = [m = 0] for every m ≤ p. This discharges the reproduction hypothesis of linearSmoother_bias_of_reproduces.","s":"∀ {N p : ℕ} {x w : Fin N → ℝ}, IsUnit (Causalean.Stat.Nonparametric.designMatrix p x w).det → ∀ m ≤ p, ∑ i, Causalean.Stat.Nonparametric.equivKernelWeight p x w i * x i ^ m = if m = 0 then 1 else 0","l":50,"f":"Causalean/Stat/Nonparametric/LocalPoly/Weights.lean"},{"n":"Causalean.Stat.Nonparametric.equivKernelWeight_sq_sum_le","k":"theorem","m":"Causalean.Stat.Nonparametric.LocalPoly.Weights","a":"Stat/Nonparametric/LocalPoly","t":2,"d":"Leverage bound for the equivalent kernel. With nonnegative weights bounded by W, the local-polynomial equivalent-kernel weights satisfy ∑ᵢ Sᵢ² ≤ W · (M⁻¹)₀₀. Combined with linearSmoother_variance_le, this reduces the interior O((Nh)^{−1/2}) stochastic-error rate to the single concentration bound (M⁻¹)₀₀ = O(1/(Nh)).","s":"∀ {N p : ℕ} {x w : Fin N → ℝ} {W : ℝ}, IsUnit (Causalean.Stat.Nonparametric.designMatrix p x w).det → (∀ (i : Fin N), 0 ≤ w i) → (∀ (i : Fin N), w i ≤ W) → ∑ i, Causalean.Stat.Nonparametric.equivKernelWeight p x w i ^ 2 ≤ W * (Causalean.Stat.Nonparametric.designMatrix p x w)⁻¹ 0 0","l":266,"f":"Causalean/Stat/Nonparametric/LocalPoly/Weights.lean"},{"n":"Causalean.Stat.Nonparametric.equivKernel_weighted_sq_sum","k":"theorem","m":"Causalean.Stat.Nonparametric.LocalPoly.Weights","a":"Stat/Nonparametric/LocalPoly","t":2,"d":"Leverage identity for the equivalent kernel. Writing the equivalent-kernel weight as Sᵢ = wᵢ · gᵢ with gᵢ = ∑ₖ (M⁻¹)₀ₖ xᵢᵏ the unweighted kernel, the weighted leverage equals the (0,0) entry of the inverse moment matrix: ∑ᵢ wᵢ gᵢ² = (M⁻¹)₀₀. (Algebraically e₀ᵀ M⁻¹ M M⁻¹ e₀ = e₀ᵀ M⁻¹ e₀.)","s":"∀ {N p : ℕ} {x w : Fin N → ℝ}, IsUnit (Causalean.Stat.Nonparametric.designMatrix p x w).det → ∑ i, w i * (∑ k, (Causalean.Stat.Nonparametric.designMatrix p x w)⁻¹ 0 k * x i ^ ↑k) ^ 2 = (Causalean.Stat.Nonparametric.designMatrix p x w)⁻¹ 0 0","l":187,"f":"Causalean/Stat/Nonparametric/LocalPoly/Weights.lean"},{"n":"Causalean.Stat.Nonparametric.estimatorBias_unconditional","k":"theorem","m":"Causalean.Stat.Nonparametric.LocalPoly.EstimatorRisk.Unconditional","a":"Stat/Nonparametric/LocalPoly/EstimatorRisk","t":1,"d":"Unconditional bias of a truncated estimator. Let est be an integrable statistic bounded by M in absolute value estimating a target θ with |θ| ≤ M, relative to a sub-σ-algebra m of the ambient σ-algebra. If G is an m-measurable good-design event on which, almost everywhere, the conditional bias |𝔼[est | m] − θ| is at most a nonnegative constant B, then the full-sample-law bias obeys |𝔼[est] − θ| ≤ B + 2M·μ(Gᶜ), the conditional bias B plus a truncation tail proportional to the bad-design probability.","s":"∀ {Ω : Type u_1} {m m0 : MeasurableSpace Ω} {μ : MeasureTheory.Measure Ω} [MeasureTheory.IsProbabilityMeasure μ] {est : Ω → ℝ} {θ B M : ℝ} {G : Set Ω}, m ≤ m0 → MeasureTheory.Integrable est μ → (∀ (ω : Ω), |est ω| ≤ M) → |θ| ≤ M → 0 ≤ B → MeasurableSet G → (∀ᵐ (ω : Ω) ∂μ, ω ∈ G → |μ[est | m] ω - θ| ≤ B) → |∫ (ω : Ω), est ω ∂μ - θ| ≤ B + 2 * M * (μ Gᶜ).toReal","l":82,"f":"Causalean/Stat/Nonparametric/LocalPoly/EstimatorRisk/Unconditional.lean"},{"n":"Causalean.Stat.Nonparametric.estimatorStochL2_unconditional","k":"theorem","m":"Causalean.Stat.Nonparametric.LocalPoly.EstimatorRisk.Unconditional","a":"Stat/Nonparametric/LocalPoly/EstimatorRisk","t":1,"d":"Unconditional stochastic L² error of a truncated estimator. The √· form of estimatorVariance_unconditional: for an L² statistic est bounded by M in absolute value estimating θ with |θ| ≤ M relative to a sub-σ-algebra m of the ambient σ-algebra, and an m-measurable good-design event G on which, almost everywhere, the conditional variance is at most a nonnegative rate Vrate and the squared conditional bias is at most a nonnegative constant Bsq, the full-sample-law stochastic L² error obeys √Var(est) ≤ √(Vrate + Bsq + 5M²·μ(Gᶜ)). With Vrate = Cvar²·(Nh)⁻¹, Bsq = Cbias²·h^{2β} and a negligible truncation tail (5M²·μ(Gᶜ) driven to o((Nh)⁻¹) by designMatrix_inv_concentration), this gives the full-sample stochastic L² rate up to the bias and the negligible tail.","s":"∀ {Ω : Type u_1} {m m0 : MeasurableSpace Ω} {μ : MeasureTheory.Measure Ω} [MeasureTheory.IsProbabilityMeasure μ] {est : Ω → ℝ} {θ Bsq Vrate M : ℝ} {G : Set Ω}, m ≤ m0 → MeasureTheory.MemLp est 2 μ → (∀ (ω : Ω), |est ω| ≤ M) → |θ| ≤ M → 0 ≤ Vrate → 0 ≤ Bsq → MeasurableSet G → (∀ᵐ (ω : Ω) ∂μ, ω ∈ G → ProbabilityTheory.condVar m est μ ω ≤ Vrate) → (∀ᵐ (ω : Ω) ∂μ, ω ∈ G → (μ[est | m] ω - θ) ^ 2 ≤ Bsq) → √(ProbabilityTheory.variance est μ) ≤ √(Vrate + Bsq + 5 * M ^ 2 * (μ Gᶜ).toReal)","l":189,"f":"Causalean/Stat/Nonparametric/LocalPoly/EstimatorRisk/Unconditional.lean"},{"n":"Causalean.Stat.Nonparametric.estimatorVariance_unconditional","k":"theorem","m":"Causalean.Stat.Nonparametric.LocalPoly.EstimatorRisk.Unconditional","a":"Stat/Nonparametric/LocalPoly/EstimatorRisk","t":1,"d":"Unconditional variance of a truncated estimator (law of total variance). Let est be an L² statistic bounded by M in absolute value estimating a target θ with |θ| ≤ M, relative to a sub-σ-algebra m of the ambient σ-algebra. Let G be an m-measurable good-design event on which, almost everywhere, the conditional variance Var[est | m] is at most a nonnegative rate Vrate and the squared conditional bias (𝔼[est | m] − θ)² is at most a nonnegative constant Bsq. Then the full-sample-law variance obeys Var(est) ≤ Vrate + Bsq + 5M²·μ(Gᶜ), the within-design variance rate plus the squared bias plus a truncation tail.","s":"∀ {Ω : Type u_1} {m m0 : MeasurableSpace Ω} {μ : MeasureTheory.Measure Ω} [MeasureTheory.IsProbabilityMeasure μ] {est : Ω → ℝ} {θ Bsq Vrate M : ℝ} {G : Set Ω}, m ≤ m0 → MeasureTheory.MemLp est 2 μ → (∀ (ω : Ω), |est ω| ≤ M) → |θ| ≤ M → 0 ≤ Vrate → 0 ≤ Bsq → MeasurableSet G → (∀ᵐ (ω : Ω) ∂μ, ω ∈ G → ProbabilityTheory.condVar m est μ ω ≤ Vrate) → (∀ᵐ (ω : Ω) ∂μ, ω ∈ G → (μ[est | m] ω - θ) ^ 2 ≤ Bsq) → ProbabilityTheory.variance est μ ≤ Vrate + Bsq + 5 * M ^ 2 * (μ Gᶜ).toReal","l":117,"f":"Causalean/Stat/Nonparametric/LocalPoly/EstimatorRisk/Unconditional.lean"},{"n":"Causalean.Stat.Nonparametric.exists_moment_cancelling_kernel_1d","k":"theorem","m":"Causalean.Stat.Nonparametric.Approximation.HolderInterpolation.Kernel","a":"Stat/Nonparametric/Approximation/HolderInterpolation","t":1,"d":"For any prescribed finite order m, a compactly supported one-dimensional kernel can be chosen to have unit total mass while cancelling every polynomial moment through that order.","s":"∀ (m : ℕ), ∃ k, Continuous k ∧ (∀ (u : ℝ), 1 < |u| → k u = 0) ∧ ∫ (u : ℝ) in Set.Icc (-1) 1, k u = 1 ∧ ∀ (j : ℕ), 1 ≤ j → j ≤ m → ∫ (u : ℝ) in Set.Icc (-1) 1, u ^ j * k u = 0","l":158,"f":"Causalean/Stat/Nonparametric/Approximation/HolderInterpolation/Kernel.lean"},{"n":"Causalean.Stat.Nonparametric.expected_weighted_sq_image_spherical","k":"theorem","m":"Causalean.Stat.Nonparametric.SeriesSieve.Prediction","a":"Stat/Nonparametric/SeriesSieve","t":2,"d":"Expected weighted quadratic form of a deterministic linear image of spherical mean-zero noise. If ε is a spherical mean-zero square-integrable family with scale σ, and a is a deterministic matrix of coefficients, then the linear image Dᵢ(ω) = ∑ₖ aᵢₖ εₖ(ω) has expected weighted sum of squares","s":"∀ {Ω : Type u_1} {N : ℕ} [inst : MeasurableSpace Ω] {μ : MeasureTheory.Measure Ω} [MeasureTheory.IsProbabilityMeasure μ] {ε : Fin N → Ω → ℝ} {a : Fin N → Fin N → ℝ} {w : Fin N → ℝ} {σ : ℝ}, (∀ (k : Fin N), MeasureTheory.MemLp (ε k) 2 μ) → (∀ (k : Fin N), ∫ (ω : Ω), ε k ω ∂μ = 0) → Causalean.GaussMarkov.SphericalFamily ε μ σ → ∫ (ω : Ω), ∑ i, w i * (∑ k, a i k * ε k ω) ^ 2 ∂μ = σ ^ 2 * ∑ i, w i * ∑ k, a i k ^ 2","l":87,"f":"Causalean/Stat/Nonparametric/SeriesSieve/Prediction.lean"},{"n":"Causalean.Stat.Nonparametric.holderDerivOrder","k":"def","m":"Causalean.Stat.Nonparametric.Approximation.HolderTaylor","a":"Stat/Nonparametric/Approximation","t":1,"d":"The standard Hölder derivative order for smoothness β: the largest natural number strictly below β. For positive noninteger β this is ⌊β⌋; for a positive integer β = m this is m - 1.","s":"ℝ → ℕ","l":36,"f":"Causalean/Stat/Nonparametric/Approximation/HolderTaylor.lean"},{"n":"Causalean.Stat.Nonparametric.holderDerivOrder_lt","k":"theorem","m":"Causalean.Stat.Nonparametric.Approximation.HolderTaylor","a":"Stat/Nonparametric/Approximation","t":2,"d":"","s":"∀ {β : ℝ}, 0 < β → ↑(Causalean.Stat.Nonparametric.holderDerivOrder β) < β","l":41,"f":"Causalean/Stat/Nonparametric/Approximation/HolderTaylor.lean"},{"n":"Causalean.Stat.Nonparametric.holder_point_l1_interpolation","k":"theorem","m":"Causalean.Stat.Nonparametric.Approximation.HolderInterpolation","a":"Stat/Nonparametric/Approximation/HolderInterpolation","t":1,"d":"Hölder pointwise ⟹ local L¹ mass interpolation. Fix a point x0 in d-dimensional Euclidean space. If the Hölder exponent γ is positive, the Hölder constant M is positive, the neighbourhood radius r is positive, and the sup-norm ball of radius r around x0 is contained in the domain S, then there is a constant c_H > 0, depending only on γ, d, M, r and uniform over the Hölder ball, such that every function g in the standard Hölder ball of exponent γ, constant M, and domain S satisfies c_H · |g(x0)|^{1 + d/γ} ≤ ∫_{supBall x0 r} |g|.","s":"∀ {d : ℕ} {γ M r : ℝ} {x0 : Fin d → ℝ} {S : Set (Fin d → ℝ)}, 0 < γ → 0 < M → 0 < r → Causalean.Stat.Nonparametric.supBall x0 r ⊆ S → ∃ cH, 0 < cH ∧ ∀ (g : (Fin d → ℝ) → ℝ), Causalean.Stat.Nonparametric.HolderBallStd g γ M S → cH * |g x0| ^ (1 + ↑d / γ) ≤ ∫ (x : Fin d → ℝ) in Causalean.Stat.Nonparametric.supBall x0 r, |g x|","l":795,"f":"Causalean/Stat/Nonparametric/Approximation/HolderInterpolation.lean"},{"n":"Causalean.Stat.Nonparametric.holder_taylor_monomial_approx","k":"theorem","m":"Causalean.Stat.Nonparametric.Approximation.HolderTaylorMonomial","a":"Stat/Nonparametric/Approximation","t":1,"d":"Local monomial approximation in a Hölder ball. Fix a centre x0 in d-dimensional Euclidean space and a finite family expo of exponent multi-indices. If the smoothness index β is positive, the Hölder constant L is positive, the neighbourhood radius r is positive, the closed sup-norm cube of radius r around x0 is contained in the domain S, and expo lists every exponent multi-index of total degree up to the Taylor order, then there is a constant C_b ≥ 0, depending only on β and d, such that every function f in the standard Hölder ball of exponent β, constant L, and domain S is approximated near x0, at any bandwidth h ∈ (0, r), by a monomial combination in the expo basis with error at most C_b · L · h^β, uniformly over the unit cube of rescaled directions.","s":"∀ {d p : ℕ} {β L r : ℝ} {x0 : Fin d → ℝ} {S : Set (Fin d → ℝ)}, 0 < β → 0 < L → 0 < r → {x | ∀ (i : Fin d), |x i - x0 i| ≤ r} ⊆ S → ∀ (expo : Fin p → Fin d → ℕ), (∀ (e : Fin d → ℕ), ∑ j, e j ≤ ⌈β⌉₊ - 1 → ∃ k, expo k = e) → ∃ Cb, 0 ≤ Cb ∧ ∀ (f : (Fin d → ℝ) → ℝ), Causalean.Stat.Nonparametric.HolderBallStd f β L S → ∀ (h : ℝ), 0 < h → h < r → ∃ θ, ∀ (u : Fin d → ℝ), (∀ (j : Fin d), |u j| ≤ 1) → |f (x0 + h • u) - ∑ k, θ k * ∏ j, u j ^ expo k j| ≤ Cb * L * h ^ β","l":306,"f":"Causalean/Stat/Nonparametric/Approximation/HolderTaylorMonomial.lean"},{"n":"Causalean.Stat.Nonparametric.holder_taylor_remainder","k":"theorem","m":"Causalean.Stat.Nonparametric.Approximation.HolderTaylor","a":"Stat/Nonparametric/Approximation","t":1,"d":"Hölder–Taylor remainder bound under the standard Hölder convention. Let p denote the largest natural number strictly below the smoothness index β. If β is positive, the Hölder constant M is nonnegative, the expansion point t lies in a window [lo,hi], the evaluation point a lies in the same window, f is p times continuously differentiable, and its p-th derivative is (β − p)-Hölder with constant M on the window, then the error of the degree-p Taylor approximation of f at t, evaluated at a, is bounded by (M / p!) · |a − t|^β. For positive integer β = m, this uses derivative order m - 1 and Hölder exponent 1.","s":"∀ {f : ℝ → ℝ} {M β lo hi t a : ℝ}, 0 < β → 0 ≤ M → t ∈ Set.Icc lo hi → a ∈ Set.Icc lo hi → ContDiff ℝ (↑(Causalean.Stat.Nonparametric.holderDerivOrder β)) f → (∀ x ∈ Set.Icc lo hi, ∀ y ∈ Set.Icc lo hi, |iteratedDeriv (Causalean.Stat.Nonparametric.holderDerivOrder β) f x - iteratedDeriv (Causalean.Stat.Nonparametric.holderDerivOrder β) f y| ≤ M * |x - y| ^ (β - ↑(Causalean.Stat.Nonparametric.holderDerivOrder β))) → |f a - Causalean.Stat.Nonparametric.taylorPoly (Causalean.Stat.Nonparametric.holderDerivOrder β) f t a| ≤ M / ↑(Causalean.Stat.Nonparametric.holderDerivOrder β).factorial * |a - t| ^ β","l":190,"f":"Causalean/Stat/Nonparametric/Approximation/HolderTaylor.lean"},{"n":"Causalean.Stat.Nonparametric.holder_taylor_remainder_of_lt","k":"theorem","m":"Causalean.Stat.Nonparametric.Approximation.HolderTaylor","a":"Stat/Nonparametric/Approximation","t":2,"d":"Hölder-Taylor remainder when the evaluation point lies to the right of the base.","s":"∀ {f : ℝ → ℝ} {M β lo hi t a : ℝ} {p : ℕ}, 0 ≤ M → t ∈ Set.Icc lo hi → a ∈ Set.Icc lo hi → ContDiff ℝ (↑p) f → (∀ x ∈ Set.Icc lo hi, ∀ y ∈ Set.Icc lo hi, |iteratedDeriv p f x - iteratedDeriv p f y| ≤ M * |x - y| ^ (β - ↑p)) → 0 < p → ↑p ≤ β → t < a → |f a - Causalean.Stat.Nonparametric.taylorPoly p f t a| ≤ M / ↑p.factorial * |a - t| ^ β","l":117,"f":"Causalean/Stat/Nonparametric/Approximation/HolderTaylor.lean"},{"n":"Causalean.Stat.Nonparametric.intMomentMatrix","k":"def","m":"Causalean.Stat.Nonparametric.LocalPoly.Rate.IntegralMoment","a":"Stat/Nonparametric/LocalPoly/Rate","t":1,"d":"The integral (population) moment matrix S_{jk} = ∫ (a − t)^j (a − t)^k dν of the centered monomials against a measure ν.","s":"(p : ℕ) → MeasureTheory.Measure ℝ → ℝ → Matrix (Fin (p + 1)) (Fin (p + 1)) ℝ","l":36,"f":"Causalean/Stat/Nonparametric/LocalPoly/Rate/IntegralMoment.lean"},{"n":"Causalean.Stat.Nonparametric.intMomentMatrix_isHermitian","k":"theorem","m":"Causalean.Stat.Nonparametric.LocalPoly.Rate.IntegralMoment","a":"Stat/Nonparametric/LocalPoly/Rate","t":2,"d":"The integral moment matrix is symmetric.","s":"∀ {p : ℕ} {ν : MeasureTheory.Measure ℝ} {t : ℝ}, (Causalean.Stat.Nonparametric.intMomentMatrix p ν t).IsHermitian","l":98,"f":"Causalean/Stat/Nonparametric/LocalPoly/Rate/IntegralMoment.lean"},{"n":"Causalean.Stat.Nonparametric.intMomentMatrix_isUnit_det","k":"theorem","m":"Causalean.Stat.Nonparametric.LocalPoly.Rate.IntegralMoment","a":"Stat/Nonparametric/LocalPoly/Rate","t":2,"d":"Invertibility of the integral moment matrix from design non-degeneracy: a positive definite matrix has a unit determinant, discharging the IsUnit S.det hypothesis for the population shape matrix.","s":"∀ {p : ℕ} {ν : MeasureTheory.Measure ℝ} {t : ℝ}, (∀ (j k : Fin (p + 1)), MeasureTheory.Integrable (fun a => (a - t) ^ ↑j * (a - t) ^ ↑k) ν) → (∀ (v : Fin (p + 1) → ℝ), v ≠ 0 → ¬∀ᵐ (a : ℝ) ∂ν, ∑ j, v j * (a - t) ^ ↑j = 0) → IsUnit (Causalean.Stat.Nonparametric.intMomentMatrix p ν t).det","l":157,"f":"Causalean/Stat/Nonparametric/LocalPoly/Rate/IntegralMoment.lean"},{"n":"Causalean.Stat.Nonparametric.intMomentMatrix_posDef","k":"theorem","m":"Causalean.Stat.Nonparametric.LocalPoly.Rate.IntegralMoment","a":"Stat/Nonparametric/LocalPoly/Rate","t":1,"d":"The integral moment matrix is positive definite whenever every centered-monomial product (a−t)^j·(a−t)^k is integrable against the design measure ν and the design law is non-degenerate: no nonzero coefficient vector v makes the centered polynomial ∑ⱼ vⱼ (a−t)^j vanish ν-almost everywhere. Then the integral moment matrix intMomentMatrix p ν t is positive definite.","s":"∀ {p : ℕ} {ν : MeasureTheory.Measure ℝ} {t : ℝ}, (∀ (j k : Fin (p + 1)), MeasureTheory.Integrable (fun a => (a - t) ^ ↑j * (a - t) ^ ↑k) ν) → (∀ (v : Fin (p + 1) → ℝ), v ≠ 0 → ¬∀ᵐ (a : ℝ) ∂ν, ∑ j, v j * (a - t) ^ ↑j = 0) → (Causalean.Stat.Nonparametric.intMomentMatrix p ν t).PosDef","l":115,"f":"Causalean/Stat/Nonparametric/LocalPoly/Rate/IntegralMoment.lean"},{"n":"Causalean.Stat.Nonparametric.intMomentMatrix_posSemidef","k":"theorem","m":"Causalean.Stat.Nonparametric.LocalPoly.Rate.IntegralMoment","a":"Stat/Nonparametric/LocalPoly/Rate","t":2,"d":"The integral moment matrix is positive semidefinite (a Gram matrix of an L² family).","s":"∀ {p : ℕ} {ν : MeasureTheory.Measure ℝ} {t : ℝ}, (∀ (j k : Fin (p + 1)), MeasureTheory.Integrable (fun a => (a - t) ^ ↑j * (a - t) ^ ↑k) ν) → (Causalean.Stat.Nonparametric.intMomentMatrix p ν t).PosSemidef","l":105,"f":"Causalean/Stat/Nonparametric/LocalPoly/Rate/IntegralMoment.lean"},{"n":"Causalean.Stat.Nonparametric.intMomentMatrix_quadForm","k":"theorem","m":"Causalean.Stat.Nonparametric.LocalPoly.Rate.IntegralMoment","a":"Stat/Nonparametric/LocalPoly/Rate","t":2,"d":"Gram quadratic form of the integral moment matrix. vᵀ S v = ∫ (∑ⱼ vⱼ (a−t)^j)² dν — the population moment matrix is the integral Gram matrix of the centered monomial features.","s":"∀ {p : ℕ} {ν : MeasureTheory.Measure ℝ} {t : ℝ}, (∀ (j k : Fin (p + 1)), MeasureTheory.Integrable (fun a => (a - t) ^ ↑j * (a - t) ^ ↑k) ν) → ∀ (v : Fin (p + 1) → ℝ), v ⬝ᵥ (Causalean.Stat.Nonparametric.intMomentMatrix p ν t).mulVec v = ∫ (a : ℝ), (∑ j, v j * (a - t) ^ ↑j) ^ 2 ∂ν","l":42,"f":"Causalean/Stat/Nonparametric/LocalPoly/Rate/IntegralMoment.lean"},{"n":"Causalean.Stat.Nonparametric.integrable_of_abs_le_const_mul_kernel","k":"theorem","m":"Causalean.Stat.Nonparametric.Approximation.Kernel","a":"Stat/Nonparametric/Approximation","t":2,"d":"If a function F is dominated in absolute value by a constant multiple C·|K| of an integrable kernel K, then F is itself integrable. This is the integrability workhorse for kernel-smoothing arguments: every factor multiplying K (a monomial, a Taylor remainder) is bounded by |K| on the kernel's support and vanishes off it.","s":"∀ {K F : ℝ → ℝ} {C : ℝ}, MeasureTheory.Integrable K MeasureTheory.volume → MeasureTheory.AEStronglyMeasurable F MeasureTheory.volume → (∀ (v : ℝ), |F v| ≤ C * |K v|) → MeasureTheory.Integrable F MeasureTheory.volume","l":55,"f":"Causalean/Stat/Nonparametric/Approximation/Kernel.lean"},{"n":"Causalean.Stat.Nonparametric.inv00_diag_conj","k":"theorem","m":"Causalean.Stat.Nonparametric.LocalPoly.Rate.Conjugation","a":"Stat/Nonparametric/LocalPoly/Rate","t":1,"d":"Leverage entry under diagonal-conjugation scaling. If S factors as κ • (D · T · D) for a nonzero scalar κ with D = diagonal d an invertible diagonal matrix (every d i nonzero) whose first entry is 1, and T invertible, then S is invertible and its (0,0) inverse entry collapses to (S⁻¹)₀₀ = κ⁻¹ · (T⁻¹)₀₀: the diagonal bandwidth powers cancel at the intercept because D₀₀ = 1.","s":"∀ {p : ℕ} {κ : ℝ}, κ ≠ 0 → ∀ {d : Fin (p + 1) → ℝ}, (∀ (i : Fin (p + 1)), d i ≠ 0) → d 0 = 1 → ∀ {T S : Matrix (Fin (p + 1)) (Fin (p + 1)) ℝ}, IsUnit T.det → S = κ • (Matrix.diagonal d * T * Matrix.diagonal d) → IsUnit S.det ∧ S⁻¹ 0 0 = κ⁻¹ * T⁻¹ 0 0","l":80,"f":"Causalean/Stat/Nonparametric/LocalPoly/Rate/Conjugation.lean"},{"n":"Causalean.Stat.Nonparametric.inv00_eq_quadForm","k":"theorem","m":"Causalean.Stat.Nonparametric.LocalPoly.EstimatorRisk.SquareCompletion","a":"Stat/Nonparametric/LocalPoly/EstimatorRisk","t":2,"d":"The (0,0) entry of an inverse as an inverse quadratic form at the first basis vector: (A⁻¹)₀₀ = e₀ᵀ A⁻¹ e₀ with e₀ = Pi.single 0 1.","s":"∀ {p : ℕ} (A : Matrix (Fin (p + 1)) (Fin (p + 1)) ℝ), A⁻¹ 0 0 = Pi.single 0 1 ⬝ᵥ A⁻¹.mulVec (Pi.single 0 1)","l":76,"f":"Causalean/Stat/Nonparametric/LocalPoly/EstimatorRisk/SquareCompletion.lean"},{"n":"Causalean.Stat.Nonparametric.inv00_le_of_quadForm_sandwich","k":"theorem","m":"Causalean.Stat.Nonparametric.LocalPoly.EstimatorRisk.SquareCompletion","a":"Stat/Nonparametric/LocalPoly/EstimatorRisk","t":2,"d":"Loewner sandwich bound on the (0,0) inverse entry. If A and B are positive definite and A dominates c·B in the Loewner order on quadratic forms (c·(wᵀ B w) ≤ wᵀ A w for all w) with c > 0, then the intercept leverage entries satisfy","s":"∀ {p : ℕ} {A B : Matrix (Fin (p + 1)) (Fin (p + 1)) ℝ}, A.PosDef → B.PosDef → ∀ {c : ℝ}, 0 < c → (∀ (w : Fin (p + 1) → ℝ), c * w ⬝ᵥ B.mulVec w ≤ w ⬝ᵥ A.mulVec w) → A⁻¹ 0 0 ≤ B⁻¹ 0 0 / c","l":88,"f":"Causalean/Stat/Nonparametric/LocalPoly/EstimatorRisk/SquareCompletion.lean"},{"n":"Causalean.Stat.Nonparametric.inv_diag_conj","k":"theorem","m":"Causalean.Stat.Nonparametric.LocalPoly.Rate.Conjugation","a":"Stat/Nonparametric/LocalPoly/Rate","t":2,"d":"Diagonal-conjugation inverse formula. If S = κ • (D · T · D) with κ ≠ 0 and D = diagonal d an invertible diagonal matrix (d i ≠ 0), T invertible, then S is invertible with S⁻¹ = κ⁻¹ • (D⁻¹ · T⁻¹ · D⁻¹) where D⁻¹ = diagonal (fun i => (d i)⁻¹).","s":"∀ {p : ℕ} {κ : ℝ}, κ ≠ 0 → ∀ {d : Fin (p + 1) → ℝ}, (∀ (i : Fin (p + 1)), d i ≠ 0) → ∀ {T S : Matrix (Fin (p + 1)) (Fin (p + 1)) ℝ}, IsUnit T.det → S = κ • (Matrix.diagonal d * T * Matrix.diagonal d) → IsUnit S.det ∧ S⁻¹ = κ⁻¹ • ((Matrix.diagonal fun i => (d i)⁻¹) * T⁻¹ * Matrix.diagonal fun i => (d i)⁻¹)","l":36,"f":"Causalean/Stat/Nonparametric/LocalPoly/Rate/Conjugation.lean"},{"n":"Causalean.Stat.Nonparametric.isCompact_cube","k":"theorem","m":"Causalean.Stat.Nonparametric.Approximation.HolderInterpolation.Defs","a":"Stat/Nonparametric/Approximation/HolderInterpolation","t":2,"d":"An axis-aligned box over a finite index type is compact: it is a finite product of closed bounded intervals.","s":"∀ {ι : Type u_1} [Finite ι] (a b : ℝ), IsCompact (Set.univ.pi fun x => Set.Icc a b)","l":90,"f":"Causalean/Stat/Nonparametric/Approximation/HolderInterpolation/Defs.lean"},{"n":"Causalean.Stat.Nonparametric.isCompact_supBall","k":"theorem","m":"Causalean.Stat.Nonparametric.Approximation.HolderInterpolation.Defs","a":"Stat/Nonparametric/Approximation/HolderInterpolation","t":1,"d":"A closed sup-norm ball — the axis-aligned cube, or box, of half-width r centred at a point x0 in d-dimensional space — is compact, being a finite product of closed bounded intervals.","s":"∀ {d : ℕ} (x0 : Fin d → ℝ) (r : ℝ), IsCompact (Causalean.Stat.Nonparametric.supBall x0 r)","l":56,"f":"Causalean/Stat/Nonparametric/Approximation/HolderInterpolation/Defs.lean"},{"n":"Causalean.Stat.Nonparametric.kernelDensity_lower_dom","k":"theorem","m":"Causalean.Stat.Nonparametric.LocalPoly.EstimatorRisk.DensityConstants","a":"Stat/Nonparametric/LocalPoly/EstimatorRisk","t":2,"d":"Pointwise weight domination from the design-density window bound. If the kernel K is nonnegative and supported in [-1,1] and the design density p obeys cDesign ≤ p(a) for every a within bandwidth h of t, then cDesign · K(u) ≤ K(u) · p(t + h·u) for every u: on the support |u| ≤ 1 the argument t + h·u lies in the window, off the support both sides vanish.","s":"∀ {K pdens : ℝ → ℝ} {t h cDesign : ℝ}, 0 < h → (∀ (u : ℝ), 0 ≤ K u) → (∀ (u : ℝ), 1 < |u| → K u = 0) → (∀ (a : ℝ), |a - t| ≤ h → cDesign ≤ pdens a) → ∀ (u : ℝ), cDesign * K u ≤ K u * pdens (t + h * u)","l":222,"f":"Causalean/Stat/Nonparametric/LocalPoly/EstimatorRisk/DensityConstants.lean"},{"n":"Causalean.Stat.Nonparametric.kernelDensity_upper_dom","k":"theorem","m":"Causalean.Stat.Nonparametric.LocalPoly.EstimatorRisk.DensityConstants","a":"Stat/Nonparametric/LocalPoly/EstimatorRisk","t":2,"d":"Pointwise weight domination, upper side. Same hypotheses with an upper density bound p(a) ≤ CDesign on the window give K(u) · p(t + h·u) ≤ CDesign · K(u) for every u.","s":"∀ {K pdens : ℝ → ℝ} {t h CDesign : ℝ}, 0 < h → (∀ (u : ℝ), 0 ≤ K u) → (∀ (u : ℝ), 1 < |u| → K u = 0) → (∀ (a : ℝ), |a - t| ≤ h → pdens a ≤ CDesign) → ∀ (u : ℝ), K u * pdens (t + h * u) ≤ CDesign * K u","l":245,"f":"Causalean/Stat/Nonparametric/LocalPoly/EstimatorRisk/DensityConstants.lean"},{"n":"Causalean.Stat.Nonparametric.kernelSmoothingBias","k":"def","m":"Causalean.Stat.Nonparametric.Approximation.Kernel","a":"Stat/Nonparametric/Approximation","t":1,"d":"Population kernel smoothing bias of estimating the value f t of a regression function f at point t with bandwidth h and kernel K: ∫ h⁻¹ K((u−t)/h) (f u − f t) du.","s":"(ℝ → ℝ) → (ℝ → ℝ) → ℝ → ℝ → ℝ","l":49,"f":"Causalean/Stat/Nonparametric/Approximation/Kernel.lean"},{"n":"Causalean.Stat.Nonparametric.kernelSmoothingBias_bound","k":"theorem","m":"Causalean.Stat.Nonparametric.Approximation.Kernel","a":"Stat/Nonparametric/Approximation","t":1,"d":"Interior kernel smoothing bias is O(h^β). Let p denote the largest natural number strictly below the smoothness index β. If β is positive, the Hölder constant M is nonnegative, the bandwidth h is positive, the kernel K has order p, the regression function f is p times continuously differentiable, and its p-th derivative is (β−p)-Hölder with constant M on the window [t−h, t+h], then the kernel smoothing bias of f at t with bandwidth h is bounded by (M/p! · ∫|K|)·h^β. For positive integer β = m, p = m - 1 and the Hölder exponent is 1. (Fan–Gijbels 1996 §3.1–3.3; Tsybakov 2009 Ch. 1.)","s":"∀ {f K : ℝ → ℝ} {β M t h : ℝ}, 0 < β → 0 ≤ M → 0 < h → Causalean.Stat.Nonparametric.KernelOrder K (Causalean.Stat.Nonparametric.holderDerivOrder β) → ContDiff ℝ (↑(Causalean.Stat.Nonparametric.holderDerivOrder β)) f → (∀ x ∈ Set.Icc (t - h) (t + h), ∀ y ∈ Set.Icc (t - h) (t + h), |iteratedDeriv (Causalean.Stat.Nonparametric.holderDerivOrder β) f x - iteratedDeriv (Causalean.Stat.Nonparametric.holderDerivOrder β) f y| ≤ M * |x - y| ^ (β - ↑(Causalean.Stat.Nonparametric.holderDerivOrder β))) → |Causalean.Stat.Nonparametric.kernelSmoothingBias f K t h| ≤ (M / ↑(Causalean.Stat.Nonparametric.holderDerivOrder β).factorial * ∫ (u : ℝ), |K u|) * h ^ β","l":91,"f":"Causalean/Stat/Nonparametric/Approximation/Kernel.lean"},{"n":"Causalean.Stat.Nonparametric.kernelSmoothingBias_changeOfVar","k":"theorem","m":"Causalean.Stat.Nonparametric.Approximation.Kernel","a":"Stat/Nonparametric/Approximation","t":2,"d":"Change of variables for the kernel smoothing bias. Substituting u = t + h v (h > 0) turns the bias integral into ∫ K(v) (f(t+hv) − f t) dv: the h⁻¹ prefactor cancels the Jacobian h, and (u−t)/h = v.","s":"∀ (f K : ℝ → ℝ) (t h : ℝ), 0 < h → Causalean.Stat.Nonparametric.kernelSmoothingBias f K t h = ∫ (v : ℝ), K v * (f (t + h * v) - f t)","l":66,"f":"Causalean/Stat/Nonparametric/Approximation/Kernel.lean"},{"n":"Causalean.Stat.Nonparametric.l1_lower_of_bias_bound","k":"theorem","m":"Causalean.Stat.Nonparametric.Approximation.HolderInterpolation","a":"Stat/Nonparametric/Approximation/HolderInterpolation","t":2,"d":"The bandwidth optimization converts a lower bound on a kernel-smoothed signal into the Hölder interpolation rate: local absolute mass grows at least as the pointwise signal to the power one plus dimension divided by smoothness.","s":"∀ {d : ℕ} {γ Ksup Δ Ival cstar h : ℝ}, 0 < γ → 0 < Ksup → 0 < Δ → 0 < cstar → h = cstar * Δ ^ (1 / γ) → 3 * Δ / 4 ≤ Ksup * h⁻¹ ^ d * Ival → 3 / (4 * Ksup) * cstar ^ d * Δ ^ (1 + ↑d / γ) ≤ Ival","l":47,"f":"Causalean/Stat/Nonparametric/Approximation/HolderInterpolation.lean"},{"n":"Causalean.Stat.Nonparametric.linearSmoother_bias_of_reproduces","k":"theorem","m":"Causalean.Stat.Nonparametric.LeastSquares.SmootherBias","a":"Stat/Nonparametric/LeastSquares","t":2,"d":"Bias of a polynomial-reproducing linear smoother. If the weights Sᵢ reproduce polynomials up to degree p = holderDerivOrder β at t (∑ᵢ Sᵢ (aᵢ − t)ᵏ = [k = 0] for k ≤ p), the design points aᵢ and t lie in a window [lo, hi], and f is β-Hölder there (p-times continuously differentiable with (β−p)-Hölder top derivative, constant M), then the smoother's bias is controlled by the weighted spread of the design: |∑ᵢ Sᵢ f(aᵢ) − f t| ≤ (M/p!) · ∑ᵢ |Sᵢ| · |aᵢ − t|^β. (Tsybakov 2009, Ch. 1.)","s":"∀ {f : ℝ → ℝ} {β M lo hi t : ℝ} {N : ℕ} {a S : Fin N → ℝ}, 0 < β → 0 ≤ M → t ∈ Set.Icc lo hi → (∀ (i : Fin N), a i ∈ Set.Icc lo hi) → ContDiff ℝ (↑(Causalean.Stat.Nonparametric.holderDerivOrder β)) f → (∀ x ∈ Set.Icc lo hi, ∀ y ∈ Set.Icc lo hi, |iteratedDeriv (Causalean.Stat.Nonparametric.holderDerivOrder β) f x - iteratedDeriv (Causalean.Stat.Nonparametric.holderDerivOrder β) f y| ≤ M * |x - y| ^ (β - ↑(Causalean.Stat.Nonparametric.holderDerivOrder β))) → (∀ k ≤ Causalean.Stat.Nonparametric.holderDerivOrder β, ∑ i, S i * (a i - t) ^ k = if k = 0 then 1 else 0) → |∑ i, S i * f (a i) - f t| ≤ M / ↑(Causalean.Stat.Nonparametric.holderDerivOrder β).factorial * ∑ i, |S i| * |a i - t| ^ β","l":65,"f":"Causalean/Stat/Nonparametric/LeastSquares/SmootherBias.lean"},{"n":"Causalean.Stat.Nonparametric.linearSmoother_bias_window","k":"theorem","m":"Causalean.Stat.Nonparametric.LeastSquares.SmootherBias","a":"Stat/Nonparametric/LeastSquares","t":1,"d":"Interior O(h^β) bias of a polynomial-reproducing linear smoother. Let p denote the largest natural number strictly below the smoothness index β. If β is positive, the Hölder constant M is nonnegative, the target point t lies in a window [lo,hi], every design point aᵢ lies in the same window, every design point is within bandwidth h of t, f is p times continuously differentiable, its p-th derivative is (β−p)-Hölder with constant M on the window, and the smoother weights S reproduce polynomials of degree up to p at t, then the smoother's bias collapses to the standard rate |∑ᵢ Sᵢ f(aᵢ) − f t| ≤ (M/p!) · (∑ᵢ |Sᵢ|) · h^β.","s":"∀ {f : ℝ → ℝ} {β M lo hi t h : ℝ} {N : ℕ} {a S : Fin N → ℝ}, 0 < β → 0 ≤ M → t ∈ Set.Icc lo hi → (∀ (i : Fin N), a i ∈ Set.Icc lo hi) → (∀ (i : Fin N), |a i - t| ≤ h) → ContDiff ℝ (↑(Causalean.Stat.Nonparametric.holderDerivOrder β)) f → (∀ x ∈ Set.Icc lo hi, ∀ y ∈ Set.Icc lo hi, |iteratedDeriv (Causalean.Stat.Nonparametric.holderDerivOrder β) f x - iteratedDeriv (Causalean.Stat.Nonparametric.holderDerivOrder β) f y| ≤ M * |x - y| ^ (β - ↑(Causalean.Stat.Nonparametric.holderDerivOrder β))) → (∀ k ≤ Causalean.Stat.Nonparametric.holderDerivOrder β, ∑ i, S i * (a i - t) ^ k = if k = 0 then 1 else 0) → |∑ i, S i * f (a i) - f t| ≤ (M / ↑(Causalean.Stat.Nonparametric.holderDerivOrder β).factorial * ∑ i, |S i|) * h ^ β","l":105,"f":"Causalean/Stat/Nonparametric/LeastSquares/SmootherBias.lean"},{"n":"Causalean.Stat.Nonparametric.linearSmoother_reproduces_taylorPoly","k":"theorem","m":"Causalean.Stat.Nonparametric.LeastSquares.SmootherBias","a":"Stat/Nonparametric/LeastSquares","t":2,"d":"A linear smoother whose weights reproduce polynomials up to degree p (∑ᵢ Sᵢ (aᵢ − t)ᵏ = [k = 0] for k ≤ p) reproduces the degree-p Taylor polynomial of f at t exactly: ∑ᵢ Sᵢ · T_p(aᵢ; t) = f t. Only the constant term of the Taylor polynomial survives the reproduction identities.","s":"∀ {f : ℝ → ℝ} {t : ℝ} {N : ℕ} {a S : Fin N → ℝ} (p : ℕ), (∀ k ≤ p, ∑ i, S i * (a i - t) ^ k = if k = 0 then 1 else 0) → ∑ i, S i * Causalean.Stat.Nonparametric.taylorPoly p f t (a i) = f t","l":37,"f":"Causalean/Stat/Nonparametric/LeastSquares/SmootherBias.lean"},{"n":"Causalean.Stat.Nonparametric.linearSmoother_variance_le","k":"theorem","m":"Causalean.Stat.Nonparametric.LeastSquares.SmootherVariance","a":"Stat/Nonparametric/LeastSquares","t":1,"d":"Stochastic-error bound for a fixed-weight linear smoother. If each response Yᵢ is square-integrable, the responses form a spherical family with common scale σ — each has variance σ², and distinct responses are uncorrelated, and the sum of squared smoother weights, ∑ᵢ Sᵢ², is bounded by V, then the variance of the linear smoother ∑ᵢ Sᵢ Yᵢ is at most σ² V.","s":"∀ {Ω : Type u_1} {N : ℕ} [inst : MeasurableSpace Ω] {μ : MeasureTheory.Measure Ω} [MeasureTheory.IsProbabilityMeasure μ] {Y : Fin N → Ω → ℝ} {S : Fin N → ℝ} {σ V : ℝ}, (∀ (i : Fin N), MeasureTheory.MemLp (Y i) 2 μ) → Causalean.GaussMarkov.SphericalFamily Y μ σ → ∑ i, S i ^ 2 ≤ V → ProbabilityTheory.variance (fun ω => ∑ i, S i * Y i ω) μ ≤ σ ^ 2 * V","l":49,"f":"Causalean/Stat/Nonparametric/LeastSquares/SmootherVariance.lean"},{"n":"Causalean.Stat.Nonparametric.linearSmoother_variance_spherical","k":"theorem","m":"Causalean.Stat.Nonparametric.LeastSquares.SmootherVariance","a":"Stat/Nonparametric/LeastSquares","t":2,"d":"Variance of a fixed-weight linear smoother under spherical errors. If Y is a spherical random family with scale σ (each cell has variance σ², distinct cells are uncorrelated) and each Yᵢ is L², then the linear smoother with deterministic weights S has variance Var[∑ᵢ Sᵢ Yᵢ] = σ² · ∑ᵢ Sᵢ².","s":"∀ {Ω : Type u_1} {N : ℕ} [inst : MeasurableSpace Ω] {μ : MeasureTheory.Measure Ω} [MeasureTheory.IsProbabilityMeasure μ] {Y : Fin N → Ω → ℝ} {S : Fin N → ℝ} {σ : ℝ}, (∀ (i : Fin N), MeasureTheory.MemLp (Y i) 2 μ) → Causalean.GaussMarkov.SphericalFamily Y μ σ → ProbabilityTheory.variance (fun ω => ∑ i, S i * Y i ω) μ = σ ^ 2 * ∑ i, S i ^ 2","l":32,"f":"Causalean/Stat/Nonparametric/LeastSquares/SmootherVariance.lean"},{"n":"Causalean.Stat.Nonparametric.localPoly_density_inv00_rate","k":"theorem","m":"Causalean.Stat.Nonparametric.LocalPoly.EstimatorRisk.DensityLeverage","a":"Stat/Nonparametric/LocalPoly/EstimatorRisk","t":1,"d":"Explicit Θ(1/(Nh)) leverage rate from density + kernel-moment constants. Fix a bandwidth h > 0, a sample size N with N > 0, and a design-density lower bound cDesign > 0. Let the kernel K be nonnegative and supported in [-1,1], with both centered-monomial integrands — against the shape weight K·p(t+h·) and against the pure kernel K — integrable, the pure kernel-moment matrix G positive definite, and the kernel shape matrix T positive definite; suppose the design density obeys cDesign ≤ p on the window |a − t| ≤ h. On the good design event where the population matrix's inverse row sums are bounded by a nonnegative constant c, the empirical moment matrix M lies entrywise within a nonnegative scale η of the population matrix S = popDesignMatrix p N K pdens t h, and where the regime constants are small enough (c·(p+1)·η ≤ 1/2 and 2c²(p+1)η is at most the explicit density + kernel-moment constant cInv/(Nh)), then the empirical moment matrix M is invertible and its intercept leverage obeys the explicit interior rate (M⁻¹)₀₀ ≤ 2·cInv/(Nh), with cInv = (G⁻¹)₀₀/cDesign an explicit density + kernel-moment constant — no S-level invertibility or leverage assumption remains.","s":"∀ {p N : ℕ} {h cDesign η c t : ℝ} {K pdens : ℝ → ℝ} {M : Matrix (Fin (p + 1)) (Fin (p + 1)) ℝ}, 0 < h → 0 < ↑N → 0 < cDesign → (∀ (u : ℝ), 0 ≤ K u) → (∀ (u : ℝ), 1 < |u| → K u = 0) → (∀ (j k : Fin (p + 1)), MeasureTheory.Integrable (fun u => K u * pdens (t + h * u) * (u ^ ↑j * u ^ ↑k)) MeasureTheory.volume) → (∀ (j k : Fin (p + 1)), MeasureTheory.Integrable (fun u => K u * (u ^ ↑j * u ^ ↑k)) MeasureTheory.volume) → (Causalean.Stat.Nonparametric.weightMomentMatrix p K).PosDef → (Causalean.Stat.Nonparametric.weightMomentMatrix p fun u => K u * pdens (t + h * u)).PosDef → (∀ (a : ℝ), |a - t| ≤ h → cDesign ≤ pdens a) → 0 ≤ c → 0 ≤ η → (∀ (i : Fin (p + 1)), ∑ j, |(Causalean.Stat.Nonparametric.popDesignMatrix p N K pdens t h)⁻¹ i j| ≤ c) → (∀ (j k : Fin (p + 1)), |M j k - Causalean.Stat.Nonparametric.popDesignMatrix p N K pdens t h j k| ≤ η) → c * (↑(p + 1) * η) ≤ 1 / 2 → 2 * c ^ 2 * (↑(p + 1) * η) ≤ (Causalean.Stat.Nonparametric.weightMomentMatrix p K)⁻¹ 0 0 / cDesign / (↑N * h) → IsUnit M.det ∧ M⁻¹ 0 0 ≤ 2 * ((Causalean.Stat.Nonparametric.weightMomentMatrix p K)⁻¹ 0 0 / cDesign / (↑N * h))","l":44,"f":"Causalean/Stat/Nonparametric/LocalPoly/EstimatorRisk/DensityLeverage.lean"},{"n":"Causalean.Stat.Nonparametric.localPoly_density_leverage_bound","k":"theorem","m":"Causalean.Stat.Nonparametric.LocalPoly.EstimatorRisk.DensityLeverage","a":"Stat/Nonparametric/LocalPoly/EstimatorRisk","t":1,"d":"Bandwidth-free density bound on the local-polynomial leverage product. Fix a bandwidth h > 0, a sample size N > 0, and density-window bounds 0 < cDesign with cDesign ≤ CDesign. Let the kernel K be nonnegative and supported in [-1,1], with both centered-monomial integrands integrable, the pure kernel-moment matrix G positive definite, and the kernel shape matrix T positive definite; suppose the design density obeys the lower window bound cDesign ≤ p and the upper window bound p ≤ CDesign on |a − t| ≤ h. On the good design event — the population matrix's inverse row sums bounded by a nonnegative c, the empirical moment matrix M entrywise within a nonnegative scale η of the population matrix, with the regime constants small (c·(p+1)·η ≤ 1/2), the perturbation bound 2c²(p+1)η ≤ cInv/(Nh), and η at most Nh — and given the total weight M₀₀ is nonnegative and the inverse leverage (M⁻¹)₀₀ is nonnegative, then the geometric mean of the total weight and the inverse leverage is bounded by the bandwidth-free density constant √(M₀₀·(M⁻¹)₀₀) ≤ √(2·cInv·(cTop+1)), with cInv = (G⁻¹)₀₀/cDesign and cTop = CDesign·G₀₀. Via equivKernelWeight_abs_sum_sq_le this controls the ℓ¹ bias leverage ∑ᵢ|Sᵢ| by a bandwidth-free constant.","s":"∀ {p N : ℕ} {h cDesign CDesign η c t : ℝ} {K pdens : ℝ → ℝ} {M : Matrix (Fin (p + 1)) (Fin (p + 1)) ℝ}, 0 < h → 0 < ↑N → 0 < cDesign → cDesign ≤ CDesign → (∀ (u : ℝ), 0 ≤ K u) → (∀ (u : ℝ), 1 < |u| → K u = 0) → (∀ (j k : Fin (p + 1)), MeasureTheory.Integrable (fun u => K u * pdens (t + h * u) * (u ^ ↑j * u ^ ↑k)) MeasureTheory.volume) → (∀ (j k : Fin (p + 1)), MeasureTheory.Integrable (fun u => K u * (u ^ ↑j * u ^ ↑k)) MeasureTheory.volume) → (Causalean.Stat.Nonparametric.weightMomentMatrix p K).PosDef → (Causalean.Stat.Nonparametric.weightMomentMatrix p fun u => K u * pdens (t + h * u)).PosDef → (∀ (a : ℝ), |a - t| ≤ h → cDesign ≤ pdens a) → (∀ (a : ℝ), |a - t| ≤ h → pdens a ≤ CDesign) → 0 ≤ c → 0 ≤ η → (∀ (i : Fin (p + 1)), ∑ j, |(Causalean.Stat.Nonparametric.popDesignMatrix p N K pdens t h)⁻¹ i j| ≤ c) → (∀ (j k : Fin (p + 1)), |M j k - Causalean.Stat.Nonparametric.popDesignMatrix p N K pdens t h j k| ≤ η) → c * (↑(p + 1) * η) ≤ 1 / 2 → 2 * c ^ 2 * (↑(p + 1) * η) ≤ (Causalean.Stat.Nonparametric.weightMomentMatrix p K)⁻¹ 0 0 / cDesign / (↑N * h) → η ≤ ↑N * h → 0 ≤ M 0 0 → 0 ≤ M⁻¹ 0 0 → √(M 0 0 * M⁻¹ 0 0) ≤ √(2 * ((Causalean.Stat.Nonparametric.weightMomentMatrix p K)⁻¹ 0 0 / cDesign) * (CDesign * Causalean.Stat.Nonparametric.weightMomentMatrix p K 0 0 + 1))","l":111,"f":"Causalean/Stat/Nonparametric/LocalPoly/EstimatorRisk/DensityLeverage.lean"},{"n":"Causalean.Stat.Nonparametric.localPoly_estimatorBias_window","k":"theorem","m":"Causalean.Stat.Nonparametric.LocalPoly.EstimatorRisk.EstimatorRisk","a":"Stat/Nonparametric/LocalPoly/EstimatorRisk","t":1,"d":"Interior local-polynomial estimator bias ≤ Cbias · h^β. Let ĉ₀ = c 0 be the degree- p = holderDerivOrder β weighted least-squares intercept fit — the coefficient vector c minimizing the weighted sum of squared residuals — of a β-Hölder function f (exponent β > 0) with Hölder constant MH ≥ 0 bounding the highest derivative's modulus of continuity at noise-free responses, where the weights are nonnegative, the target point t and every design point a i lie in a common interval [lo, hi], every design point is within bandwidth h ≥ 0 of t, and the design moment matrix is invertible. If the bandwidth-free leverage bound √(M₀₀·(M⁻¹)₀₀) ≤ L holds for some L ≥ 0 — supplied by localPoly_density_leverage_bound — then the conditional bias obeys |ĉ₀ − f(t)| ≤ (MH/p!) · L · h^β, the standard local-polynomial bias rate with constant Cbias = (MH/p!) · L.","s":"∀ {N : ℕ} {β MH lo hi t h L : ℝ} {a w : Fin N → ℝ} {f : ℝ → ℝ} {c : Fin (Causalean.Stat.Nonparametric.holderDerivOrder β + 1) → ℝ}, 0 < β → 0 ≤ MH → 0 ≤ h → (∀ (i : Fin N), 0 ≤ w i) → t ∈ Set.Icc lo hi → (∀ (i : Fin N), a i ∈ Set.Icc lo hi) → (∀ (i : Fin N), |a i - t| ≤ h) → ContDiff ℝ (↑(Causalean.Stat.Nonparametric.holderDerivOrder β)) f → (∀ x ∈ Set.Icc lo hi, ∀ y ∈ Set.Icc lo hi, |iteratedDeriv (Causalean.Stat.Nonparametric.holderDerivOrder β) f x - iteratedDeriv (Causalean.Stat.Nonparametric.holderDerivOrder β) f y| ≤ MH * |x - y| ^ (β - ↑(Causalean.Stat.Nonparametric.holderDerivOrder β))) → IsUnit (Causalean.Stat.Nonparametric.designMatrix (Causalean.Stat.Nonparametric.holderDerivOrder β) (fun i => a i - t) w).det → (∀ (c' : Fin (Causalean.Stat.Nonparametric.holderDerivOrder β + 1) → ℝ), ∑ i, w i * (f (a i) - ∑ j, c j * (a i - t) ^ ↑j) ^ 2 ≤ ∑ i, w i * (f (a i) - ∑ j, c' j * (a i - t) ^ ↑j) ^ 2) → √(Causalean.Stat.Nonparametric.designMatrix (Causalean.Stat.Nonparametric.holderDerivOrder β) (fun i => a i - t) w 0 0 * (Causalean.Stat.Nonparametric.designMatrix (Causalean.Stat.Nonparametric.holderDerivOrder β) (fun i => a i - t) w)⁻¹ 0 0) ≤ L → |c 0 - f t| ≤ MH / ↑(Causalean.Stat.Nonparametric.holderDerivOrder β).factorial * L * h ^ β","l":42,"f":"Causalean/Stat/Nonparametric/LocalPoly/EstimatorRisk/EstimatorRisk.lean"},{"n":"Causalean.Stat.Nonparametric.localPoly_estimatorStochL2","k":"theorem","m":"Causalean.Stat.Nonparametric.LocalPoly.EstimatorRisk.EstimatorRisk","a":"Stat/Nonparametric/LocalPoly/EstimatorRisk","t":1,"d":"Interior local-polynomial estimator stochastic L² error ≤ Cvar · (Nh)^{-1/2}. The degree-p local-polynomial equivalent-kernel smoother ĉ₀ = ∑ᵢ Sᵢ Yᵢ, applied to a family Y of square-integrable responses that is spherical with common scale σ, with an invertible design moment matrix and nonnegative weights bounded by a constant W ≥ 0, satisfies, on the good design event where the intercept leverage obeys the rate (M⁻¹)₀₀ ≤ 2·cInv/Nh for a nonnegative constant cInv and a positive scale Nh, the conditional stochastic L² error bound √Var(ĉ₀) ≤ √(2σ²·W·cInv) · (Nh)^{-1/2}, the standard interior stochastic rate with constant Cvar = √(2σ²·W·cInv).","s":"∀ {Ω : Type u_1} {N p : ℕ} [inst : MeasurableSpace Ω] {μ : MeasureTheory.Measure Ω} [MeasureTheory.IsProbabilityMeasure μ] {x w : Fin N → ℝ} {Y : Fin N → Ω → ℝ} {σ W cInv Nh : ℝ}, (∀ (i : Fin N), MeasureTheory.MemLp (Y i) 2 μ) → Causalean.GaussMarkov.SphericalFamily Y μ σ → IsUnit (Causalean.Stat.Nonparametric.designMatrix p x w).det → (∀ (i : Fin N), 0 ≤ w i) → (∀ (i : Fin N), w i ≤ W) → 0 ≤ W → 0 ≤ cInv → 0 < Nh → (Causalean.Stat.Nonparametric.designMatrix p x w)⁻¹ 0 0 ≤ 2 * (cInv / Nh) → √(ProbabilityTheory.variance (fun ω => ∑ i, Causalean.Stat.Nonparametric.equivKernelWeight p x w i * Y i ω) μ) ≤ √(2 * σ ^ 2 * W * cInv) * √Nh⁻¹","l":86,"f":"Causalean/Stat/Nonparametric/LocalPoly/EstimatorRisk/EstimatorRisk.lean"},{"n":"Causalean.Stat.Nonparametric.localPoly_intercept_bias","k":"theorem","m":"Causalean.Stat.Nonparametric.LocalPoly.Bias","a":"Stat/Nonparametric/LocalPoly","t":1,"d":"Interior local-polynomial bias is O(h^β). Let p denote the largest natural number strictly below the smoothness index β. If β is positive, the Hölder constant M is nonnegative, the design weights w are nonnegative, the target point t lies in a window [lo,hi], every design point aᵢ lies in the same window, every design point is within bandwidth h of t, the regression function f is p times continuously differentiable, its p-th derivative is (β−p)-Hölder with constant M on the window, the weighted design moment matrix is invertible, and c globally minimizes the weighted degree-p least-squares objective at noise-free responses f(aᵢ), then the fitted intercept c 0 estimates f t with bias |c 0 − f t| ≤ (M/p!)·(∑ᵢ |Sᵢ|)·h^β, where Sᵢ are the local-polynomial equivalent-kernel weights.","s":"∀ {N : ℕ} {β M lo hi t h : ℝ} {a w : Fin N → ℝ} {f : ℝ → ℝ} {c : Fin (Causalean.Stat.Nonparametric.holderDerivOrder β + 1) → ℝ}, 0 < β → 0 ≤ M → (∀ (i : Fin N), 0 ≤ w i) → t ∈ Set.Icc lo hi → (∀ (i : Fin N), a i ∈ Set.Icc lo hi) → (∀ (i : Fin N), |a i - t| ≤ h) → ContDiff ℝ (↑(Causalean.Stat.Nonparametric.holderDerivOrder β)) f → (∀ x ∈ Set.Icc lo hi, ∀ y ∈ Set.Icc lo hi, |iteratedDeriv (Causalean.Stat.Nonparametric.holderDerivOrder β) f x - iteratedDeriv (Causalean.Stat.Nonparametric.holderDerivOrder β) f y| ≤ M * |x - y| ^ (β - ↑(Causalean.Stat.Nonparametric.holderDerivOrder β))) → IsUnit (Causalean.Stat.Nonparametric.designMatrix (Causalean.Stat.Nonparametric.holderDerivOrder β) (fun i => a i - t) w).det → (∀ (c' : Fin (Causalean.Stat.Nonparametric.holderDerivOrder β + 1) → ℝ), ∑ i, w i * (f (a i) - ∑ j, c j * (a i - t) ^ ↑j) ^ 2 ≤ ∑ i, w i * (f (a i) - ∑ j, c' j * (a i - t) ^ ↑j) ^ 2) → |c 0 - f t| ≤ (M / ↑(Causalean.Stat.Nonparametric.holderDerivOrder β).factorial * ∑ i, |Causalean.Stat.Nonparametric.equivKernelWeight (Causalean.Stat.Nonparametric.holderDerivOrder β) (fun i => a i - t) w i|) * h ^ β","l":38,"f":"Causalean/Stat/Nonparametric/LocalPoly/Bias.lean"},{"n":"Causalean.Stat.Nonparametric.localPoly_intercept_variance_le","k":"theorem","m":"Causalean.Stat.Nonparametric.LocalPoly.SmootherVariance","a":"Stat/Nonparametric/LocalPoly","t":1,"d":"Variance of the interior local-polynomial estimator. The degree-p local-polynomial equivalent-kernel smoother ∑ᵢ Sᵢ Yᵢ, applied to a family Y of square-integrable responses that is spherical with common scale σ, with an invertible design moment matrix and nonnegative weights bounded above by a constant W, has variance Var[∑ᵢ Sᵢ Yᵢ] ≤ σ² · W · (M⁻¹)₀₀. This reduces the interior O((Nh)^{−1/2}) stochastic-error rate to the single design-concentration bound (M⁻¹)₀₀ = O(1/(Nh)).","s":"∀ {Ω : Type u_1} {N p : ℕ} [inst : MeasurableSpace Ω] {μ : MeasureTheory.Measure Ω} [MeasureTheory.IsProbabilityMeasure μ] {x w : Fin N → ℝ} {Y : Fin N → Ω → ℝ} {σ W : ℝ}, (∀ (i : Fin N), MeasureTheory.MemLp (Y i) 2 μ) → Causalean.GaussMarkov.SphericalFamily Y μ σ → IsUnit (Causalean.Stat.Nonparametric.designMatrix p x w).det → (∀ (i : Fin N), 0 ≤ w i) → (∀ (i : Fin N), w i ≤ W) → ProbabilityTheory.variance (fun ω => ∑ i, Causalean.Stat.Nonparametric.equivKernelWeight p x w i * Y i ω) μ ≤ σ ^ 2 * (W * (Causalean.Stat.Nonparametric.designMatrix p x w)⁻¹ 0 0)","l":25,"f":"Causalean/Stat/Nonparametric/LocalPoly/SmootherVariance.lean"},{"n":"Causalean.Stat.Nonparametric.localPoly_inv00_rate","k":"theorem","m":"Causalean.Stat.Nonparametric.LocalPoly.Rate","a":"Stat/Nonparametric/LocalPoly/Rate","t":1,"d":"Explicit Θ(1/(Nh)) rate for the local-polynomial leverage. On a good design event with positive scale Nh where the population moment matrix S is invertible, its inverse row sums are bounded by a nonnegative constant c, the empirical moment matrix M lies entrywise within a nonnegative perturbation scale η of S, the perturbation is small relative to the dimension: c·(p+1)·η ≤ 1/2, and the population intercept leverage sits at the Θ(Nh) scale: (S⁻¹)₀₀ ≤ cInv/(Nh) and 2c²(p+1)η ≤ cInv/(Nh), then the empirical moment matrix M is invertible and its intercept leverage obeys the explicit interior rate (M⁻¹)₀₀ ≤ 2·cInv/(Nh). This is the variance-rate capstone for the local-polynomial upper bound: combined with localPoly_intercept_variance_le it yields the O((Nh)^{-1/2}) stochastic error.","s":"∀ {p N : ℕ} {h c cInv η : ℝ} {S M : Matrix (Fin (p + 1)) (Fin (p + 1)) ℝ}, 0 < ↑N * h → IsUnit S.det → 0 ≤ c → 0 ≤ η → (∀ (i : Fin (p + 1)), ∑ j, |S⁻¹ i j| ≤ c) → (∀ (j k : Fin (p + 1)), |M j k - S j k| ≤ η) → c * (↑(p + 1) * η) ≤ 1 / 2 → S⁻¹ 0 0 ≤ cInv / (↑N * h) → 2 * c ^ 2 * (↑(p + 1) * η) ≤ cInv / (↑N * h) → IsUnit M.det ∧ M⁻¹ 0 0 ≤ 2 * (cInv / (↑N * h))","l":77,"f":"Causalean/Stat/Nonparametric/LocalPoly/Rate.lean"},{"n":"Causalean.Stat.Nonparametric.localPoly_leverage_bound","k":"theorem","m":"Causalean.Stat.Nonparametric.LocalPoly.Rate","a":"Stat/Nonparametric/LocalPoly/Rate","t":1,"d":"Bandwidth-free bound on the local-polynomial leverage product. On the same good design event with positive scale Nh, where the population moment matrix S is invertible, its inverse row sums are bounded by a nonnegative constant c, the empirical moment matrix M lies entrywise within a nonnegative perturbation scale η that is small relative to the dimension (c·(p+1)·η ≤ 1/2) and at most Nh, with nonnegative density constants cInv and cTop such that the population intercept leverage and perturbation obey (S⁻¹)₀₀ ≤ cInv/(Nh) and 2c²(p+1)η ≤ cInv/(Nh), the population top weight obeys S₀₀ ≤ cTop·(Nh), and the empirical top weight M₀₀ and inverse leverage (M⁻¹)₀₀ are both nonnegative, the geometric mean of the total weight and the inverse leverage is bounded by the bandwidth-free constant √(M₀₀·(M⁻¹)₀₀) ≤ √(2·cInv·(cTop+1)). The Θ(Nh) growth of M₀₀ ≤ (cTop+1)·(Nh) exactly cancels the Θ(1/(Nh)) decay of (M⁻¹)₀₀. Via equivKernelWeight_abs_sum_sq_le ((∑ᵢ|Sᵢ|)² ≤ M₀₀·(M⁻¹)₀₀) this controls the ℓ¹ bias leverage ∑ᵢ|Sᵢ| by a bandwidth-free constant, the second leverage capstone used by the upper-bound analysis.","s":"∀ {p N : ℕ} {h c cInv cTop η : ℝ} {S M : Matrix (Fin (p + 1)) (Fin (p + 1)) ℝ}, 0 < ↑N * h → IsUnit S.det → 0 ≤ c → 0 ≤ η → 0 ≤ cInv → 0 ≤ cTop → (∀ (i : Fin (p + 1)), ∑ j, |S⁻¹ i j| ≤ c) → (∀ (j k : Fin (p + 1)), |M j k - S j k| ≤ η) → c * (↑(p + 1) * η) ≤ 1 / 2 → S⁻¹ 0 0 ≤ cInv / (↑N * h) → 2 * c ^ 2 * (↑(p + 1) * η) ≤ cInv / (↑N * h) → S 0 0 ≤ cTop * (↑N * h) → η ≤ ↑N * h → 0 ≤ M 0 0 → 0 ≤ M⁻¹ 0 0 → √(M 0 0 * M⁻¹ 0 0) ≤ √(2 * cInv * (cTop + 1))","l":102,"f":"Causalean/Stat/Nonparametric/LocalPoly/Rate.lean"},{"n":"Causalean.Stat.Nonparametric.lstsqObjective","k":"def","m":"Causalean.Stat.Nonparametric.LeastSquares.NormalEquations","a":"Stat/Nonparametric/LeastSquares","t":1,"d":"The weighted sum of squares ∑ᵢ wᵢ rᵢ(c)².","s":"{N : ℕ} → {ι : Type u_1} → [Fintype ι] → (Fin N → ι → ℝ) → (Fin N → ℝ) → (Fin N → ℝ) → (ι → ℝ) → ℝ","l":44,"f":"Causalean/Stat/Nonparametric/LeastSquares/NormalEquations.lean"},{"n":"Causalean.Stat.Nonparametric.lstsqResidual","k":"def","m":"Causalean.Stat.Nonparametric.LeastSquares.NormalEquations","a":"Stat/Nonparametric/LeastSquares","t":1,"d":"The weighted residual at coefficient vector c against design matrix Φ: rᵢ(c) = Yᵢ − ∑ⱼ cⱼ Φᵢⱼ.","s":"{N : ℕ} → {ι : Type u_1} → [Fintype ι] → (Fin N → ι → ℝ) → (Fin N → ℝ) → (ι → ℝ) → Fin N → ℝ","l":38,"f":"Causalean/Stat/Nonparametric/LeastSquares/NormalEquations.lean"},{"n":"Causalean.Stat.Nonparametric.lstsq_normal_equations","k":"theorem","m":"Causalean.Stat.Nonparametric.LeastSquares.NormalEquations","a":"Stat/Nonparametric/LeastSquares","t":1,"d":"Normal equations for weighted linear least squares. If the weights w are nonnegative and the coefficient vector c globally minimizes the weighted sum of squares ∑ᵢ wᵢ (Yᵢ − ∑ⱼ cⱼ Φᵢⱼ)² over all coefficient vectors, then the weighted residual is orthogonal to every design column: ∑ᵢ wᵢ (Yᵢ − ∑ⱼ cⱼ Φᵢⱼ) Φᵢₖ = 0 for each basis index k.","s":"∀ {N : ℕ} {ι : Type u_1} [inst : Fintype ι] {Φ : Fin N → ι → ℝ} {w Y : Fin N → ℝ} {c : ι → ℝ}, (∀ (i : Fin N), 0 ≤ w i) → (∀ (c' : ι → ℝ), Causalean.Stat.Nonparametric.lstsqObjective Φ w Y c ≤ Causalean.Stat.Nonparametric.lstsqObjective Φ w Y c') → ∀ (k : ι), ∑ i, w i * Causalean.Stat.Nonparametric.lstsqResidual Φ Y c i * Φ i k = 0","l":77,"f":"Causalean/Stat/Nonparametric/LeastSquares/NormalEquations.lean"},{"n":"Causalean.Stat.Nonparametric.lstsq_objective_le_of_orthogonal","k":"theorem","m":"Causalean.Stat.Nonparametric.LeastSquares.NormalEquations","a":"Stat/Nonparametric/LeastSquares","t":2,"d":"Optimality of the orthogonal least-squares fit. With nonnegative weights, a residual orthogonal to every design column attains the minimal weighted sum of squares: SSE(c) ≤ SSE(c') for every c'. (Immediate from lstsq_pythagoras, the excess term being a nonnegative weighted sum of squares.)","s":"∀ {N : ℕ} {ι : Type u_1} [inst : Fintype ι] {Φ : Fin N → ι → ℝ} {w Y : Fin N → ℝ} {c : ι → ℝ}, (∀ (i : Fin N), 0 ≤ w i) → (∀ (k : ι), ∑ i, w i * Causalean.Stat.Nonparametric.lstsqResidual Φ Y c i * Φ i k = 0) → ∀ (c' : ι → ℝ), Causalean.Stat.Nonparametric.lstsqObjective Φ w Y c ≤ Causalean.Stat.Nonparametric.lstsqObjective Φ w Y c'","l":230,"f":"Causalean/Stat/Nonparametric/LeastSquares/NormalEquations.lean"},{"n":"Causalean.Stat.Nonparametric.lstsq_pythagoras","k":"theorem","m":"Causalean.Stat.Nonparametric.LeastSquares.NormalEquations","a":"Stat/Nonparametric/LeastSquares","t":2,"d":"Pythagorean decomposition for weighted least squares. If the residual at c is orthogonal to every design column (∑ᵢ wᵢ rᵢ(c) Φᵢₖ = 0 for all k — e.g. c is the least-squares minimizer, by lstsq_normal_equations), then for any coefficient vector c' the weighted sum of squares splits exactly as SSE(c') = SSE(c) + ∑ᵢ wᵢ (∑ⱼ (cⱼ − c'ⱼ) Φᵢⱼ)²: the orthogonality kills the cross term, so the excess error is the weighted norm of the fitted-value difference.","s":"∀ {N : ℕ} {ι : Type u_1} [inst : Fintype ι] {Φ : Fin N → ι → ℝ} {w Y : Fin N → ℝ} {c : ι → ℝ}, (∀ (k : ι), ∑ i, w i * Causalean.Stat.Nonparametric.lstsqResidual Φ Y c i * Φ i k = 0) → ∀ (c' : ι → ℝ), Causalean.Stat.Nonparametric.lstsqObjective Φ w Y c' = Causalean.Stat.Nonparametric.lstsqObjective Φ w Y c + ∑ i, w i * (∑ j, (c j - c' j) * Φ i j) ^ 2","l":160,"f":"Causalean/Stat/Nonparametric/LeastSquares/NormalEquations.lean"},{"n":"Causalean.Stat.Nonparametric.measurableSet_supBall","k":"theorem","m":"Causalean.Stat.Nonparametric.Approximation.HolderInterpolation.Defs","a":"Stat/Nonparametric/Approximation/HolderInterpolation","t":2,"d":"A closed sup-norm ball (an axis-aligned cube, or box) is a measurable set.","s":"∀ {d : ℕ} (x0 : Fin d → ℝ) (r : ℝ), MeasurableSet (Causalean.Stat.Nonparametric.supBall x0 r)","l":63,"f":"Causalean/Stat/Nonparametric/Approximation/HolderInterpolation/Defs.lean"},{"n":"Causalean.Stat.Nonparametric.mem_cube","k":"theorem","m":"Causalean.Stat.Nonparametric.Approximation.HolderInterpolation.Defs","a":"Stat/Nonparametric/Approximation/HolderInterpolation","t":2,"d":"Membership in an axis-aligned box: a point lies in it exactly when each of its coordinates lies between a and b.","s":"∀ {ι : Type u_1} {a b : ℝ} {x : ι → ℝ}, (x ∈ Set.univ.pi fun x => Set.Icc a b) ↔ ∀ (i : ι), a ≤ x i ∧ x i ≤ b","l":84,"f":"Causalean/Stat/Nonparametric/Approximation/HolderInterpolation/Defs.lean"},{"n":"Causalean.Stat.Nonparametric.mem_supBall_self","k":"theorem","m":"Causalean.Stat.Nonparametric.Approximation.HolderInterpolation.Defs","a":"Stat/Nonparametric/Approximation/HolderInterpolation","t":2,"d":"A closed sup-norm ball (axis-aligned cube) of nonnegative half-width contains its own centre.","s":"∀ {d : ℕ} (x0 : Fin d → ℝ) {r : ℝ}, 0 ≤ r → x0 ∈ Causalean.Stat.Nonparametric.supBall x0 r","l":68,"f":"Causalean/Stat/Nonparametric/Approximation/HolderInterpolation/Defs.lean"},{"n":"Causalean.Stat.Nonparametric.nondegenerate_of_distinct_points","k":"theorem","m":"Causalean.Stat.Nonparametric.LocalPoly.DesignMatrixPosDef","a":"Stat/Nonparametric/LocalPoly","t":1,"d":"Design non-degeneracy from distinct positively-weighted points (Vandermonde). For a set S of design indices, if every index in S carries a positive weight and the design points xᵢ take at least p+1 distinct values on S, then the design non-degeneracy condition holds: no nonzero coefficient vector v yields a degree-p polynomial ∑ⱼ vⱼ xᵢʲ that vanishes at every positively weighted design point.","s":"∀ {N p : ℕ} {x w : Fin N → ℝ} (S : Finset (Fin N)), (∀ i ∈ S, 0 < w i) → p + 1 ≤ (Finset.image x S).card → ∀ (v : Fin (p + 1) → ℝ), v ≠ 0 → ∃ i, 0 < w i ∧ ∑ j, v j * x i ^ ↑j ≠ 0","l":115,"f":"Causalean/Stat/Nonparametric/LocalPoly/DesignMatrixPosDef.lean"},{"n":"Causalean.Stat.Nonparametric.piecewiseTaylorApprox","k":"def","m":"Causalean.Stat.Nonparametric.SeriesSieve.Jackson","a":"Stat/Nonparametric/SeriesSieve","t":1,"d":"The piecewise-Taylor sieve approximant: on the cell containing x, the value of the degree-p Taylor polynomial of f expanded at the cell's left endpoint. This is a piecewise polynomial of degree ≤ p on the uniform J-cell partition of [lo, hi].","s":"ℕ → (ℝ → ℝ) → ℝ → ℝ → ℕ → ℝ → ℝ","l":52,"f":"Causalean/Stat/Nonparametric/SeriesSieve/Jackson.lean"},{"n":"Causalean.Stat.Nonparametric.piecewiseTaylor_sup_approx","k":"theorem","m":"Causalean.Stat.Nonparametric.SeriesSieve.Jackson","a":"Stat/Nonparametric/SeriesSieve","t":2,"d":"Piecewise-Taylor approximation bound under the standard Hölder convention. If f has p = holderDerivOrder β continuous derivatives and its p-th derivative obeys a Hölder bound with exponent β - p on the window, then the corresponding piecewise-Taylor approximant on a uniform J-cell partition has pointwise error at most (M / p!)·((hi − lo)/J)^β. For positive integer β = m, this uses derivative order m - 1 and Hölder exponent 1.","s":"∀ {f : ℝ → ℝ} {M β lo hi : ℝ} {J : ℕ}, 0 < β → 0 ≤ M → lo < hi → 0 < J → ContDiff ℝ (↑(Causalean.Stat.Nonparametric.holderDerivOrder β)) f → (∀ x ∈ Set.Icc lo hi, ∀ y ∈ Set.Icc lo hi, |iteratedDeriv (Causalean.Stat.Nonparametric.holderDerivOrder β) f x - iteratedDeriv (Causalean.Stat.Nonparametric.holderDerivOrder β) f y| ≤ M * |x - y| ^ (β - ↑(Causalean.Stat.Nonparametric.holderDerivOrder β))) → ∀ {x : ℝ}, x ∈ Set.Icc lo hi → |f x - Causalean.Stat.Nonparametric.piecewiseTaylorApprox (Causalean.Stat.Nonparametric.holderDerivOrder β) f lo hi J x| ≤ M / ↑(Causalean.Stat.Nonparametric.holderDerivOrder β).factorial * ((hi - lo) / ↑J) ^ β","l":167,"f":"Causalean/Stat/Nonparametric/SeriesSieve/Jackson.lean"},{"n":"Causalean.Stat.Nonparametric.piecewiseTaylor_sup_approx_rate","k":"theorem","m":"Causalean.Stat.Nonparametric.SeriesSieve.Jackson","a":"Stat/Nonparametric/SeriesSieve","t":1,"d":"Jackson rate, J^{−β} form. Fix a Hölder exponent β > 0, a nonnegative Hölder constant M, and a nondegenerate window [lo, hi] with lo < hi subdivided into a positive number J of uniform cells. If f has p = holderDerivOrder β continuous derivatives on the window and its p-th derivative obeys the Hölder bound |f^(p) x − f^(p) y| ≤ M · |x − y|^(β − p) for all x, y in the window, then at any evaluation point x in the window, the piecewise-Taylor approximant on the uniform J-cell partition has pointwise error |f x − g x| ≤ C · J^{−β}, with constant C = (M / p!) · (hi − lo)^β independent of J.","s":"∀ {f : ℝ → ℝ} {M β lo hi : ℝ} {J : ℕ}, 0 < β → 0 ≤ M → lo < hi → 0 < J → ContDiff ℝ (↑(Causalean.Stat.Nonparametric.holderDerivOrder β)) f → (∀ x ∈ Set.Icc lo hi, ∀ y ∈ Set.Icc lo hi, |iteratedDeriv (Causalean.Stat.Nonparametric.holderDerivOrder β) f x - iteratedDeriv (Causalean.Stat.Nonparametric.holderDerivOrder β) f y| ≤ M * |x - y| ^ (β - ↑(Causalean.Stat.Nonparametric.holderDerivOrder β))) → ∀ {x : ℝ}, x ∈ Set.Icc lo hi → |f x - Causalean.Stat.Nonparametric.piecewiseTaylorApprox (Causalean.Stat.Nonparametric.holderDerivOrder β) f lo hi J x| ≤ M / ↑(Causalean.Stat.Nonparametric.holderDerivOrder β).factorial * (hi - lo) ^ β * ↑J ^ (-β)","l":203,"f":"Causalean/Stat/Nonparametric/SeriesSieve/Jackson.lean"},{"n":"Causalean.Stat.Nonparametric.popDesignMatrix","k":"def","m":"Causalean.Stat.Nonparametric.LocalPoly.EstimatorRisk.Factorization","a":"Stat/Nonparametric/LocalPoly/EstimatorRisk","t":1,"d":"The population design moment matrix S_{jk} = N · ∫ K((a−t)/h) (a−t)^{j+k} p(a) da.","s":"(p : ℕ) → ℕ → (ℝ → ℝ) → (ℝ → ℝ) → ℝ → ℝ → Matrix (Fin (p + 1)) (Fin (p + 1)) ℝ","l":82,"f":"Causalean/Stat/Nonparametric/LocalPoly/EstimatorRisk/Factorization.lean"},{"n":"Causalean.Stat.Nonparametric.popDesignMatrix_factor","k":"theorem","m":"Causalean.Stat.Nonparametric.LocalPoly.EstimatorRisk.Factorization","a":"Stat/Nonparametric/LocalPoly/EstimatorRisk","t":1,"d":"Diagonal-conjugation factorization of the population moment matrix. For a positive bandwidth h > 0, writing T for the kernel shape matrix weightMomentMatrix p (fun u => K u · p(t+h·u)) and D for the diagonal matrix diagonal (fun j => h^j), the population design moment matrix factors as popDesignMatrix p N K pdens t h = (N·h) • (D · T · D). This is the literal S = (Nh)·(D T D) hypothesis of population_scaling_of_conj, proved by the single-entry change of variables popMomentEntry_changeOfVar.","s":"∀ {p : ℕ} (N : ℕ) (K pdens : ℝ → ℝ) (t h : ℝ), 0 < h → Causalean.Stat.Nonparametric.popDesignMatrix p N K pdens t h = (↑N * h) • (((Matrix.diagonal fun j => h ^ ↑j) * Causalean.Stat.Nonparametric.weightMomentMatrix p fun u => K u * pdens (t + h * u)) * Matrix.diagonal fun j => h ^ ↑j)","l":88,"f":"Causalean/Stat/Nonparametric/LocalPoly/EstimatorRisk/Factorization.lean"},{"n":"Causalean.Stat.Nonparametric.popMomentEntry_changeOfVar","k":"theorem","m":"Causalean.Stat.Nonparametric.LocalPoly.EstimatorRisk.Factorization","a":"Stat/Nonparametric/LocalPoly/EstimatorRisk","t":2,"d":"Change of variables for a single moment entry. For h > 0, ∫ a, K((a−t)/h) · (a−t)^m · p(a) da = h^{m+1} · ∫ u, K(u) · u^m · p(t+h·u) du: the substitution a = t + h·u contributes the Jacobian h and turns (a−t)^m into (h·u)^m = h^m u^m.","s":"∀ (K pdens : ℝ → ℝ) (t h : ℝ), 0 < h → ∀ (m : ℕ), ∫ (a : ℝ), K ((a - t) / h) * (a - t) ^ m * pdens a = h ^ (m + 1) * ∫ (u : ℝ), K u * u ^ m * pdens (t + h * u)","l":38,"f":"Causalean/Stat/Nonparametric/LocalPoly/EstimatorRisk/Factorization.lean"},{"n":"Causalean.Stat.Nonparametric.population_scaling_of_conj","k":"theorem","m":"Causalean.Stat.Nonparametric.LocalPoly.Rate","a":"Stat/Nonparametric/LocalPoly/Rate","t":2,"d":"Θ(Nh) leverage scaling from the change-of-variables factorization. If the population moment matrix factors as S = (Nh)·D·T·D with D = diagonal (fun j => h^j) (so D₀₀ = 1) and a bandwidth-free shape matrix T that is invertible with (T⁻¹)₀₀ ≤ cInv and T₀₀ ≤ cTop, then S is invertible and its intercept leverage scales as (S⁻¹)₀₀ ≤ cInv/(Nh) while its top weight scales as S₀₀ ≤ cTop·(Nh). This discharges the leverage-scaling hypotheses of the rate capstones from the bandwidth-free shape matrix T.","s":"∀ {p N : ℕ} {h cInv cTop : ℝ}, 0 < h → 0 < ↑N → ∀ {T S : Matrix (Fin (p + 1)) (Fin (p + 1)) ℝ}, IsUnit T.det → T⁻¹ 0 0 ≤ cInv → T 0 0 ≤ cTop → S = (↑N * h) • ((Matrix.diagonal fun j => h ^ ↑j) * T * Matrix.diagonal fun j => h ^ ↑j) → IsUnit S.det ∧ S⁻¹ 0 0 ≤ cInv / (↑N * h) ∧ S 0 0 ≤ cTop * (↑N * h)","l":41,"f":"Causalean/Stat/Nonparametric/LocalPoly/Rate.lean"},{"n":"Causalean.Stat.Nonparametric.prodKernel","k":"def","m":"Causalean.Stat.Nonparametric.Approximation.HolderInterpolation.Defs","a":"Stat/Nonparametric/Approximation/HolderInterpolation","t":1,"d":"The multivariate product kernel obtained by multiplying the same one-dimensional kernel across all covariate coordinates.","s":"(ℝ → ℝ) → (d : ℕ) → (Fin d → ℝ) → ℝ","l":108,"f":"Causalean/Stat/Nonparametric/Approximation/HolderInterpolation/Defs.lean"},{"n":"Causalean.Stat.Nonparametric.prodKernel_abs_le","k":"theorem","m":"Causalean.Stat.Nonparametric.Approximation.HolderInterpolation.Kernel","a":"Stat/Nonparametric/Approximation/HolderInterpolation","t":2,"d":"A product kernel is uniformly bounded by the coordinatewise bound raised to the dimension, whenever its one-dimensional factor has that bound.","s":"∀ {d : ℕ} {k : ℝ → ℝ} {B : ℝ}, (∀ (u : ℝ), |k u| ≤ B) → ∀ (u : Fin d → ℝ), |Causalean.Stat.Nonparametric.prodKernel k d u| ≤ B ^ d","l":211,"f":"Causalean/Stat/Nonparametric/Approximation/HolderInterpolation/Kernel.lean"},{"n":"Causalean.Stat.Nonparametric.prodKernel_integral","k":"theorem","m":"Causalean.Stat.Nonparametric.Approximation.HolderInterpolation.Kernel","a":"Stat/Nonparametric/Approximation/HolderInterpolation","t":2,"d":"The total mass of a product kernel factorizes into the product of the one-dimensional masses, so a unit-mass factor yields a unit-mass multivariate kernel.","s":"∀ {d : ℕ} (k : ℝ → ℝ), ∫ (u : Fin d → ℝ), Causalean.Stat.Nonparametric.prodKernel k d u = (∫ (t : ℝ), k t) ^ d","l":224,"f":"Causalean/Stat/Nonparametric/Approximation/HolderInterpolation/Kernel.lean"},{"n":"Causalean.Stat.Nonparametric.prodKernel_moment","k":"theorem","m":"Causalean.Stat.Nonparametric.Approximation.HolderInterpolation.Kernel","a":"Stat/Nonparametric/Approximation/HolderInterpolation","t":2,"d":"Each polynomial moment of a product kernel factorizes into the corresponding one-dimensional moments, allowing any cancelled coordinate moment to cancel the entire multivariate moment.","s":"∀ {d : ℕ} (k : ℝ → ℝ) (ν : Fin d → ℕ), ∫ (u : Fin d → ℝ), (∏ i, u i ^ ν i) * Causalean.Stat.Nonparametric.prodKernel k d u = ∏ i, ∫ (t : ℝ), t ^ ν i * k t","l":232,"f":"Causalean/Stat/Nonparametric/Approximation/HolderInterpolation/Kernel.lean"},{"n":"Causalean.Stat.Nonparametric.seriesApprox_le_of_sup","k":"theorem","m":"Causalean.Stat.Nonparametric.SeriesSieve.Prediction","a":"Stat/Nonparametric/SeriesSieve","t":2,"d":"Empirical approximation error is controlled by the best uniform approximant. If the least-squares fit c to target values f has residual orthogonal to every design column (lstsq_normal_equations) and some comparator cstar approximates f uniformly within δ at the design points (|fᵢ − ∑ⱼ cstarⱼ Φᵢⱼ| ≤ δ), then the empirical fit error is at most (∑ᵢ wᵢ)·δ². With the Jackson rate δ = C·J^{−s/d} this is the empirical best-approximation half of the series prediction rate.","s":"∀ {N : ℕ} {ι : Type u_1} [inst : Fintype ι] {Φ : Fin N → ι → ℝ} {w f : Fin N → ℝ} {c cstar : ι → ℝ} {δ : ℝ}, (∀ (i : Fin N), 0 ≤ w i) → (∀ (k : ι), ∑ i, w i * Causalean.Stat.Nonparametric.lstsqResidual Φ f c i * Φ i k = 0) → (∀ (i : Fin N), |f i - ∑ j, cstar j * Φ i j| ≤ δ) → Causalean.Stat.Nonparametric.lstsqObjective Φ w f c ≤ (∑ i, w i) * δ ^ 2","l":41,"f":"Causalean/Stat/Nonparametric/SeriesSieve/Prediction.lean"},{"n":"Causalean.Stat.Nonparametric.seriesLS_expected_prediction_le","k":"theorem","m":"Causalean.Stat.Nonparametric.SeriesSieve.Prediction","a":"Stat/Nonparametric/SeriesSieve","t":1,"d":"Conditional oracle inequality for series least-squares prediction. Suppose the noise-free projection coefficients c0 have residual orthogonal to every design column, so that the noise-free least-squares objective is bounded by A, and suppose the data fit chat ω differs from c0, at every design row i, by a deterministic linear image ∑ₖ a i k · ε k ω of a noise family ε that is square-integrable, mean zero, and spherical with scale σ, with the weighted trace sum ∑ᵢ wᵢ ∑ₖ aᵢₖ² bounded by V. Then the expected weighted quadratic prediction error of the fitted series coefficients against the target values f is at most A + σ² V:","s":"∀ {Ω : Type u_1} {N : ℕ} {ι : Type u_2} [inst : Fintype ι] [inst_1 : MeasurableSpace Ω] {μ : MeasureTheory.Measure Ω} [MeasureTheory.IsProbabilityMeasure μ] {Φ : Fin N → ι → ℝ} {w f : Fin N → ℝ} {c0 : ι → ℝ} {chat : Ω → ι → ℝ} {ε : Fin N → Ω → ℝ} {a : Fin N → Fin N → ℝ} {σ A V : ℝ}, (∀ (k : ι), ∑ i, w i * Causalean.Stat.Nonparametric.lstsqResidual Φ f c0 i * Φ i k = 0) → Causalean.Stat.Nonparametric.lstsqObjective Φ w f c0 ≤ A → (∀ (ω : Ω) (i : Fin N), ∑ j, (c0 j - chat ω j) * Φ i j = ∑ k, a i k * ε k ω) → (∀ (k : Fin N), MeasureTheory.MemLp (ε k) 2 μ) → (∀ (k : Fin N), ∫ (ω : Ω), ε k ω ∂μ = 0) → Causalean.GaussMarkov.SphericalFamily ε μ σ → ∑ i, w i * ∑ k, a i k ^ 2 ≤ V → ∫ (ω : Ω), Causalean.Stat.Nonparametric.lstsqObjective Φ w f (chat ω) ∂μ ≤ A + σ ^ 2 * V","l":171,"f":"Causalean/Stat/Nonparametric/SeriesSieve/Prediction.lean"},{"n":"Causalean.Stat.Nonparametric.seriesLS_prediction_decomp","k":"theorem","m":"Causalean.Stat.Nonparametric.SeriesSieve.Prediction","a":"Stat/Nonparametric/SeriesSieve","t":2,"d":"Pythagorean decomposition of the series prediction error. If the noise-free projection coefficient c0 (least-squares fit to the true values f) has residual orthogonal to every design column, then for *any* coefficient vector chat the prediction error of Φ·chat against the truth f splits exactly into the approximation error of the projection plus the squared fitted-value gap:","s":"∀ {N : ℕ} {ι : Type u_1} [inst : Fintype ι] {Φ : Fin N → ι → ℝ} {w f : Fin N → ℝ} {c0 chat : ι → ℝ}, (∀ (k : ι), ∑ i, w i * Causalean.Stat.Nonparametric.lstsqResidual Φ f c0 i * Φ i k = 0) → Causalean.Stat.Nonparametric.lstsqObjective Φ w f chat = Causalean.Stat.Nonparametric.lstsqObjective Φ w f c0 + ∑ i, w i * (∑ j, (c0 j - chat j) * Φ i j) ^ 2","l":70,"f":"Causalean/Stat/Nonparametric/SeriesSieve/Prediction.lean"},{"n":"Causalean.Stat.Nonparametric.supBall","k":"def","m":"Causalean.Stat.Nonparametric.Approximation.HolderInterpolation.Defs","a":"Stat/Nonparametric/Approximation/HolderInterpolation","t":1,"d":"The closed coordinatewise neighbourhood of radius r around a point: the cube containing exactly the covariate values whose every coordinate is within r of the corresponding coordinate of the centre.","s":"{d : ℕ} → (Fin d → ℝ) → ℝ → Set (Fin d → ℝ)","l":40,"f":"Causalean/Stat/Nonparametric/Approximation/HolderInterpolation/Defs.lean"},{"n":"Causalean.Stat.Nonparametric.supBall_eq_pi","k":"theorem","m":"Causalean.Stat.Nonparametric.Approximation.HolderInterpolation.Defs","a":"Stat/Nonparametric/Approximation/HolderInterpolation","t":2,"d":"A closed sup-norm ball — the axis-aligned cube of half-width r centred at a point — is exactly the product of the coordinate intervals of radius r around the centre's coordinates.","s":"∀ {d : ℕ} (x0 : Fin d → ℝ) (r : ℝ), Causalean.Stat.Nonparametric.supBall x0 r = Set.univ.pi fun i => Set.Icc (x0 i - r) (x0 i + r)","l":46,"f":"Causalean/Stat/Nonparametric/Approximation/HolderInterpolation/Defs.lean"},{"n":"Causalean.Stat.Nonparametric.taylorPoly","k":"def","m":"Causalean.Stat.Nonparametric.Approximation.HolderTaylor","a":"Stat/Nonparametric/Approximation","t":1,"d":"Degree-p Taylor polynomial of f at base point t, evaluated at a: ∑_{k ≤ p} f⁽ᵏ⁾(t)/k! · (a − t)^k.","s":"ℕ → (ℝ → ℝ) → ℝ → ℝ → ℝ","l":48,"f":"Causalean/Stat/Nonparametric/Approximation/HolderTaylor.lean"},{"n":"Causalean.Stat.Nonparametric.taylorPoly_eval_base","k":"theorem","m":"Causalean.Stat.Nonparametric.Approximation.HolderTaylor","a":"Stat/Nonparametric/Approximation","t":2,"d":"The Taylor polynomial of f at base point t, evaluated at t itself, is f t (every positive-degree term carries a (t − t)^k = 0 factor).","s":"∀ (p : ℕ) (f : ℝ → ℝ) (t : ℝ), Causalean.Stat.Nonparametric.taylorPoly p f t t = f t","l":53,"f":"Causalean/Stat/Nonparametric/Approximation/HolderTaylor.lean"},{"n":"Causalean.Stat.Nonparametric.taylorPoly_reflect","k":"theorem","m":"Causalean.Stat.Nonparametric.Approximation.HolderTaylor","a":"Stat/Nonparametric/Approximation","t":2,"d":"Reflecting across the base point preserves the Taylor polynomial value at the reflected evaluation point.","s":"∀ (p : ℕ) (f : ℝ → ℝ) (t a : ℝ), Causalean.Stat.Nonparametric.taylorPoly p (fun x => f (2 * t - x)) t (2 * t - a) = Causalean.Stat.Nonparametric.taylorPoly p f t a","l":92,"f":"Causalean/Stat/Nonparametric/Approximation/HolderTaylor.lean"},{"n":"Causalean.Stat.Nonparametric.taylorPoly_succ","k":"theorem","m":"Causalean.Stat.Nonparametric.Approximation.HolderTaylor","a":"Stat/Nonparametric/Approximation","t":2,"d":"Split the last term off the Taylor polynomial.","s":"∀ (n : ℕ) (f : ℝ → ℝ) (t a : ℝ), Causalean.Stat.Nonparametric.taylorPoly (n + 1) f t a = Causalean.Stat.Nonparametric.taylorPoly n f t a + iteratedDeriv (n + 1) f t / ↑(n + 1).factorial * (a - t) ^ (n + 1)","l":84,"f":"Causalean/Stat/Nonparametric/Approximation/HolderTaylor.lean"},{"n":"Causalean.Stat.Nonparametric.taylorWithinEval_eq_taylorPoly","k":"theorem","m":"Causalean.Stat.Nonparametric.Approximation.HolderTaylor","a":"Stat/Nonparametric/Approximation","t":2,"d":"On an interval with distinct endpoints, the Taylor polynomial computed from derivatives restricted to the interval equals the usual Taylor polynomial computed from ordinary derivatives.","s":"∀ {f : ℝ → ℝ} {p n : ℕ} {x₀ x : ℝ}, ContDiff ℝ (↑p) f → n ≤ p → x₀ < x → taylorWithinEval f n (Set.Icc x₀ x) x₀ x = Causalean.Stat.Nonparametric.taylorPoly n f x₀ x","l":64,"f":"Causalean/Stat/Nonparametric/Approximation/HolderTaylor.lean"},{"n":"Causalean.Stat.Nonparametric.top00_diag_conj","k":"theorem","m":"Causalean.Stat.Nonparametric.LocalPoly.Rate.Conjugation","a":"Stat/Nonparametric/LocalPoly/Rate","t":2,"d":"Top entry under diagonal-conjugation scaling. Under the same factorization S = κ • (D · T · D) with d 0 = 1, the (0,0) entry of S itself is S₀₀ = κ · T₀₀.","s":"∀ {p : ℕ} {κ : ℝ} {d : Fin (p + 1) → ℝ}, d 0 = 1 → ∀ {T S : Matrix (Fin (p + 1)) (Fin (p + 1)) ℝ}, S = κ • (Matrix.diagonal d * T * Matrix.diagonal d) → S 0 0 = κ * T 0 0","l":95,"f":"Causalean/Stat/Nonparametric/LocalPoly/Rate/Conjugation.lean"},{"n":"Causalean.Stat.Nonparametric.two_dotProduct_sub_quadForm_le_inv","k":"theorem","m":"Causalean.Stat.Nonparametric.LocalPoly.EstimatorRisk.SquareCompletion","a":"Stat/Nonparametric/LocalPoly/EstimatorRisk","t":1,"d":"Completing-the-square inequality for a positive definite quadratic form. For a positive definite matrix A and any vectors e, w, the affine functional 2 eᵀw − wᵀ A w is bounded above by the inverse quadratic form eᵀ A⁻¹ e, with equality at w = A⁻¹ e. This is the elementary identity eᵀ A⁻¹ e − (2 eᵀw − wᵀ A w) = (w − A⁻¹ e)ᵀ A (w − A⁻¹ e) ≥ 0.","s":"∀ {p : ℕ} {A : Matrix (Fin (p + 1)) (Fin (p + 1)) ℝ}, A.PosDef → ∀ (e w : Fin (p + 1) → ℝ), 2 * e ⬝ᵥ w - w ⬝ᵥ A.mulVec w ≤ e ⬝ᵥ A⁻¹.mulVec e","l":38,"f":"Causalean/Stat/Nonparametric/LocalPoly/EstimatorRisk/SquareCompletion.lean"},{"n":"Causalean.Stat.Nonparametric.volume_supBall","k":"theorem","m":"Causalean.Stat.Nonparametric.Approximation.HolderInterpolation.Defs","a":"Stat/Nonparametric/Approximation/HolderInterpolation","t":1,"d":"The Lebesgue volume of a closed sup-norm ball — the axis-aligned cube of side 2r centred at x0 in d dimensions — equals the d-th power of the side length, (2r)^d. No sign restriction on r is needed: a negative half-width gives an empty cube and a zero right-hand side.","s":"∀ {d : ℕ} (x0 : Fin d → ℝ) (r : ℝ), MeasureTheory.volume (Causalean.Stat.Nonparametric.supBall x0 r) = ENNReal.ofReal (2 * r) ^ d","l":74,"f":"Causalean/Stat/Nonparametric/Approximation/HolderInterpolation/Defs.lean"},{"n":"Causalean.Stat.Nonparametric.weightMomentMatrix","k":"def","m":"Causalean.Stat.Nonparametric.LocalPoly.EstimatorRisk.DensityConstants","a":"Stat/Nonparametric/LocalPoly/EstimatorRisk","t":1,"d":"The weight-W moment matrix (weightMomentMatrix p W)_{jk} = ∫ W(u) · u^j · u^k du of the centered monomials against the Lebesgue weight W. With W = K · (p ∘ (t + h·))  this is the kernel shape matrix T; with W = K it is the pure kernel-moment matrix G.","s":"(p : ℕ) → (ℝ → ℝ) → Matrix (Fin (p + 1)) (Fin (p + 1)) ℝ","l":39,"f":"Causalean/Stat/Nonparametric/LocalPoly/EstimatorRisk/DensityConstants.lean"},{"n":"Causalean.Stat.Nonparametric.weightMomentMatrix_isHermitian","k":"theorem","m":"Causalean.Stat.Nonparametric.LocalPoly.EstimatorRisk.DensityConstants","a":"Stat/Nonparametric/LocalPoly/EstimatorRisk","t":2,"d":"The weight moment matrix is symmetric.","s":"∀ {p : ℕ} {W : ℝ → ℝ}, (Causalean.Stat.Nonparametric.weightMomentMatrix p W).IsHermitian","l":104,"f":"Causalean/Stat/Nonparametric/LocalPoly/EstimatorRisk/DensityConstants.lean"},{"n":"Causalean.Stat.Nonparametric.weightMomentMatrix_quadForm","k":"theorem","m":"Causalean.Stat.Nonparametric.LocalPoly.EstimatorRisk.DensityConstants","a":"Stat/Nonparametric/LocalPoly/EstimatorRisk","t":2,"d":"Gram quadratic form of the weight moment matrix. vᵀ (weightMomentMatrix p W) v = ∫ W(u) (∑ⱼ vⱼ uʲ)² du.","s":"∀ {p : ℕ} {W : ℝ → ℝ}, (∀ (j k : Fin (p + 1)), MeasureTheory.Integrable (fun u => W u * (u ^ ↑j * u ^ ↑k)) MeasureTheory.volume) → ∀ (v : Fin (p + 1) → ℝ), v ⬝ᵥ (Causalean.Stat.Nonparametric.weightMomentMatrix p W).mulVec v = ∫ (u : ℝ), W u * (∑ j, v j * u ^ ↑j) ^ 2","l":46,"f":"Causalean/Stat/Nonparametric/LocalPoly/EstimatorRisk/DensityConstants.lean"},{"n":"Causalean.Stat.Nonparametric.weightMomentMatrix_quadForm_sandwich","k":"theorem","m":"Causalean.Stat.Nonparametric.LocalPoly.EstimatorRisk.DensityConstants","a":"Stat/Nonparametric/LocalPoly/EstimatorRisk","t":1,"d":"Loewner sandwich on quadratic forms from pointwise weight domination. Let WT and WG be weight functions and c a nonnegative scalar. If every centered-monomial integrand built from WT is integrable and the same holds for WG, and if the weights obey the pointwise domination c · WG(u) ≤ WT(u) for every u, then for every coefficient vector v the shape-matrix quadratic form dominates c times the pure-matrix quadratic form: c · (vᵀ G v) ≤ vᵀ T v, where T = weightMomentMatrix p WT and G = weightMomentMatrix p WG.","s":"∀ {p : ℕ} {WT WG : ℝ → ℝ} {c : ℝ}, 0 ≤ c → (∀ (j k : Fin (p + 1)), MeasureTheory.Integrable (fun u => WT u * (u ^ ↑j * u ^ ↑k)) MeasureTheory.volume) → (∀ (j k : Fin (p + 1)), MeasureTheory.Integrable (fun u => WG u * (u ^ ↑j * u ^ ↑k)) MeasureTheory.volume) → (∀ (u : ℝ), c * WG u ≤ WT u) → ∀ (v : Fin (p + 1) → ℝ), c * v ⬝ᵥ (Causalean.Stat.Nonparametric.weightMomentMatrix p WG).mulVec v ≤ v ⬝ᵥ (Causalean.Stat.Nonparametric.weightMomentMatrix p WT).mulVec v","l":111,"f":"Causalean/Stat/Nonparametric/LocalPoly/EstimatorRisk/DensityConstants.lean"},{"n":"Causalean.Stat.Nonparametric.weightMomentMatrix_quadForm_sandwich_upper","k":"theorem","m":"Causalean.Stat.Nonparametric.LocalPoly.EstimatorRisk.DensityConstants","a":"Stat/Nonparametric/LocalPoly/EstimatorRisk","t":2,"d":"Loewner sandwich on quadratic forms, upper direction. If W_T(u) ≤ C · W_G(u) pointwise (C ≥ 0) and both monomial integrands are integrable, then vᵀ T v ≤ C · (vᵀ G v) for every v, where T = weightMomentMatrix p W_T, G = weightMomentMatrix p W_G.","s":"∀ {p : ℕ} {WT WG : ℝ → ℝ} {C : ℝ}, 0 ≤ C → (∀ (j k : Fin (p + 1)), MeasureTheory.Integrable (fun u => WT u * (u ^ ↑j * u ^ ↑k)) MeasureTheory.volume) → (∀ (j k : Fin (p + 1)), MeasureTheory.Integrable (fun u => WG u * (u ^ ↑j * u ^ ↑k)) MeasureTheory.volume) → (∀ (u : ℝ), WT u ≤ C * WG u) → ∀ (v : Fin (p + 1) → ℝ), v ⬝ᵥ (Causalean.Stat.Nonparametric.weightMomentMatrix p WT).mulVec v ≤ C * v ⬝ᵥ (Causalean.Stat.Nonparametric.weightMomentMatrix p WG).mulVec v","l":168,"f":"Causalean/Stat/Nonparametric/LocalPoly/EstimatorRisk/DensityConstants.lean"},{"n":"Causalean.Stat.Nonparametric.wls_intercept_eq_equivKernelSmoother","k":"theorem","m":"Causalean.Stat.Nonparametric.LocalPoly.Weights","a":"Stat/Nonparametric/LocalPoly","t":2,"d":"The local-polynomial WLS intercept is the equivalent-kernel linear smoother. If the weighted design moment matrix is invertible and c minimizes the weighted sum of squares, then the fitted intercept c 0 equals the linear smoother ∑ᵢ Sᵢ Yᵢ with the equivalent-kernel weights Sᵢ = equivKernelWeight p x w i. Combined with equivKernelWeight_reproduces and linearSmoother_bias_of_reproduces, this yields the interior local-polynomial bias estimate.","s":"∀ {N p : ℕ} {x w Y : Fin N → ℝ} {c : Fin (p + 1) → ℝ}, (∀ (i : Fin N), 0 ≤ w i) → IsUnit (Causalean.Stat.Nonparametric.designMatrix p x w).det → (∀ (c' : Fin (p + 1) → ℝ), ∑ i, w i * (Y i - ∑ j, c j * x i ^ ↑j) ^ 2 ≤ ∑ i, w i * (Y i - ∑ j, c' j * x i ^ ↑j) ^ 2) → c 0 = ∑ i, Causalean.Stat.Nonparametric.equivKernelWeight p x w i * Y i","l":99,"f":"Causalean/Stat/Nonparametric/LocalPoly/Weights.lean"},{"n":"Causalean.Stat.Nonparametric.wls_normal_equations","k":"theorem","m":"Causalean.Stat.Nonparametric.LeastSquares.NormalEquations","a":"Stat/Nonparametric/LeastSquares","t":2,"d":"Normal equations for weighted polynomial least squares (local-polynomial design Φᵢⱼ = (xᵢ)ʲ). The weighted least-squares minimizer of ∑ᵢ wᵢ (Yᵢ − ∑ⱼ cⱼ xᵢʲ)² has residual orthogonal to every design monomial: ∑ᵢ wᵢ (Yᵢ − ∑ⱼ cⱼ xᵢʲ) xᵢᵏ = 0.","s":"∀ {N p : ℕ} {x w Y : Fin N → ℝ} {c : Fin (p + 1) → ℝ}, (∀ (i : Fin N), 0 ≤ w i) → (∀ (c' : Fin (p + 1) → ℝ), ∑ i, w i * (Y i - ∑ j, c j * x i ^ ↑j) ^ 2 ≤ ∑ i, w i * (Y i - ∑ j, c' j * x i ^ ↑j) ^ 2) → ∀ (k : Fin (p + 1)), ∑ i, w i * (Y i - ∑ j, c j * x i ^ ↑j) * x i ^ ↑k = 0","l":140,"f":"Causalean/Stat/Nonparametric/LeastSquares/NormalEquations.lean"},{"n":"Causalean.Stat.OneShotSplit","k":"structure","m":"Causalean.Stat.SampleSplit.OneShot","a":"Stat/SampleSplit","t":1,"d":"One-shot sample split. For an i.i.d. sample S, a fold-size schedule n₁ : ℕ → ℕ splitting the sample of size n into a nuisance fold $A(n) = \\{0,\\dots,n_1(n)-1\\}$ and an estimation fold $B(n) = \\{n_1(n),\\dots,n-1\\}$, where the nuisance fold never exceeds the full sample, $n_1(n) \\le n$, the nuisance fold grows without bound, and the estimation fold also grows without bound, $n - n_1(n) \\to \\infty$.","s":"{Ω : Type u_3} → {X : Type u_4} → [inst : MeasurableSpace Ω] → [inst_1 : MeasurableSpace X] → {μ : MeasureTheory.Measure Ω} → {P : MeasureTheory.Measure X} → Causalean.Stat.IIDSample Ω X μ P → Type","l":37,"f":"Causalean/Stat/SampleSplit/OneShot.lean"},{"n":"Causalean.Stat.OneShotSplit.card_foldB","k":"theorem","m":"Causalean.Stat.SampleSplit.PartialFoldCLT","a":"Stat/SampleSplit","t":2,"d":"The size of the estimation fold is the sample size minus the split point.","s":"∀ {Ω : Type u_1} {X : Type u_2} [inst : MeasurableSpace Ω] [inst_1 : MeasurableSpace X] {μ : MeasureTheory.Measure Ω} {P : MeasureTheory.Measure X} {S : Causalean.Stat.IIDSample Ω X μ P} (split : Causalean.Stat.OneShotSplit S) (n : ℕ), (split.foldB n).card = n - split.n₁ n","l":73,"f":"Causalean/Stat/SampleSplit/PartialFoldCLT.lean"},{"n":"Causalean.Stat.OneShotSplit.foldA","k":"def","m":"Causalean.Stat.SampleSplit.OneShot","a":"Stat/SampleSplit","t":1,"d":"Nuisance-fold index set at horizon n: A(n) := {0, …, n₁(n)−1}.","s":"{Ω : Type u_1} → {X : Type u_2} → [inst : MeasurableSpace Ω] → [inst_1 : MeasurableSpace X] → {μ : MeasureTheory.Measure Ω} → {P : MeasureTheory.Measure X} → {S : Causalean.Stat.IIDSample Ω X μ P} → Causalean.Stat.OneShotSplit S → ℕ → Finset ℕ","l":55,"f":"Causalean/Stat/SampleSplit/OneShot.lean"},{"n":"Causalean.Stat.OneShotSplit.foldA_disjoint_foldB","k":"theorem","m":"Causalean.Stat.SampleSplit.OneShot","a":"Stat/SampleSplit","t":2,"d":"Fold-A and fold-B index sets are disjoint.","s":"∀ {Ω : Type u_1} {X : Type u_2} [inst : MeasurableSpace Ω] [inst_1 : MeasurableSpace X] {μ : MeasureTheory.Measure Ω} {P : MeasureTheory.Measure X} {S : Causalean.Stat.IIDSample Ω X μ P} (split : Causalean.Stat.OneShotSplit S) (n : ℕ), Disjoint (split.foldA n) (split.foldB n)","l":75,"f":"Causalean/Stat/SampleSplit/OneShot.lean"},{"n":"Causalean.Stat.OneShotSplit.foldB","k":"def","m":"Causalean.Stat.SampleSplit.OneShot","a":"Stat/SampleSplit","t":1,"d":"Estimation-fold index set at horizon n: B(n) := {n₁(n), …, n−1}.","s":"{Ω : Type u_1} → {X : Type u_2} → [inst : MeasurableSpace Ω] → [inst_1 : MeasurableSpace X] → {μ : MeasureTheory.Measure Ω} → {P : MeasureTheory.Measure X} → {S : Causalean.Stat.IIDSample Ω X μ P} → Causalean.Stat.OneShotSplit S → ℕ → Finset ℕ","l":58,"f":"Causalean/Stat/SampleSplit/OneShot.lean"},{"n":"Causalean.Stat.OneShotSplit.foldB_card","k":"theorem","m":"Causalean.Stat.SampleSplit.OneShot","a":"Stat/SampleSplit","t":2,"d":"Cardinality of the estimation fold: |B(n)| = n - n₁(n).","s":"∀ {Ω : Type u_1} {X : Type u_2} [inst : MeasurableSpace Ω] [inst_1 : MeasurableSpace X] {μ : MeasureTheory.Measure Ω} {P : MeasureTheory.Measure X} {S : Causalean.Stat.IIDSample Ω X μ P} (split : Causalean.Stat.OneShotSplit S) (n : ℕ), (split.foldB n).card = n - split.n₁ n","l":62,"f":"Causalean/Stat/SampleSplit/OneShot.lean"},{"n":"Causalean.Stat.OneShotSplit.foldB_card_tendsto","k":"theorem","m":"Causalean.Stat.SampleSplit.OneShot","a":"Stat/SampleSplit","t":2,"d":"The estimation-fold cardinality grows to infinity.","s":"∀ {Ω : Type u_1} {X : Type u_2} [inst : MeasurableSpace Ω] [inst_1 : MeasurableSpace X] {μ : MeasureTheory.Measure Ω} {P : MeasureTheory.Measure X} {S : Causalean.Stat.IIDSample Ω X μ P} (split : Causalean.Stat.OneShotSplit S), Filter.Tendsto (fun n => (split.foldB n).card) Filter.atTop Filter.atTop","l":70,"f":"Causalean/Stat/SampleSplit/OneShot.lean"},{"n":"Causalean.Stat.OneShotSplit.foldB_eq_Ico","k":"theorem","m":"Causalean.Stat.SampleSplit.PartialFoldCLT","a":"Stat/SampleSplit","t":2,"d":"The estimation fold of a one-shot split is the interval from the split point to the sample size.","s":"∀ {Ω : Type u_1} {X : Type u_2} [inst : MeasurableSpace Ω] [inst_1 : MeasurableSpace X] {μ : MeasureTheory.Measure Ω} {P : MeasureTheory.Measure X} {S : Causalean.Stat.IIDSample Ω X μ P} (split : Causalean.Stat.OneShotSplit S) (n : ℕ), split.foldB n = Finset.Ico (split.n₁ n) n","l":65,"f":"Causalean/Stat/SampleSplit/PartialFoldCLT.lean"},{"n":"Causalean.Stat.OneShotSplit.foldB_sampleMean_tendsto_inProb","k":"theorem","m":"Causalean.Stat.SampleSplit.FoldBWLLN","a":"Stat/SampleSplit","t":1,"d":"Fold-B weak law of large numbers. For an i.i.d. sample and a one-shot split into a nuisance fold and an estimation fold, given a measurable statistic g that is square-integrable under the population measure, the estimation-fold sample average of g converges in probability to the population integral $\\int g\\,dP$ as the sample size grows.","s":"∀ {Ω : Type u_1} {X : Type u_2} [inst : MeasurableSpace Ω] [inst_1 : MeasurableSpace X] {μ : MeasureTheory.Measure Ω} {P : MeasureTheory.Measure X} [MeasureTheory.IsProbabilityMeasure μ] [MeasureTheory.IsProbabilityMeasure P] (S : Causalean.Stat.IIDSample Ω X μ P) (split : Causalean.Stat.OneShotSplit S) {g : X → ℝ}, Measurable g → MeasureTheory.MemLp g 2 P → Causalean.Stat.Tendsto_inProb (fun n ω => (↑(split.foldB n).card)⁻¹ * ∑ i ∈ split.foldB n, g (S.Z i ω)) (fun x => ∫ (x : X), g x ∂P) μ","l":85,"f":"Causalean/Stat/SampleSplit/FoldBWLLN.lean"},{"n":"Causalean.Stat.OneShotSplit.folds_indep","k":"theorem","m":"Causalean.Stat.SampleSplit.OneShot","a":"Stat/SampleSplit","t":1,"d":"Independence of folds. For a fixed sample size n, the sample sub-tuple indexed by the nuisance fold A(n) is independent, under μ, of the sub-tuple indexed by the estimation fold B(n).","s":"∀ {Ω : Type u_1} {X : Type u_2} [inst : MeasurableSpace Ω] [inst_1 : MeasurableSpace X] {μ : MeasureTheory.Measure Ω} {P : MeasureTheory.Measure X} {S : Causalean.Stat.IIDSample Ω X μ P} (split : Causalean.Stat.OneShotSplit S) (n : ℕ), ProbabilityTheory.IndepFun (fun ω i => S.Z (↑i) ω) (fun ω i => S.Z (↑i) ω) μ","l":85,"f":"Causalean/Stat/SampleSplit/OneShot.lean"},{"n":"Causalean.Stat.OneShotSplit.integral_finiteSelector_sq_le","k":"theorem","m":"Causalean.Stat.SampleSplit.FiniteSelector","a":"Stat/SampleSplit","t":2,"d":"Given an iid one-shot sample split, a sample horizon, a measurable finite branch selector on the pilot fold, a measurable good pilot event, a nonnegative common risk bound, eligibility of every branch selected on the good event, integrability of every eligible branch's squared error under the tail-fold law, and the common fixed-branch risk bound, the selected squared risk on the good event is at most that common bound.","s":"∀ {Omega : Type u_1} {Iota : Type u_4} {X : Type u_5} [inst : MeasurableSpace Omega] [inst_1 : MeasurableSpace Iota] [inst_2 : MeasurableSpace X] {mu : MeasureTheory.Measure Omega} {P : MeasureTheory.Measure X} [Fintype Iota] [MeasurableSingletonClass Iota] {S : Causalean.Stat.IIDSample Omega X mu P} (split : Causalean.Stat.OneShotSplit S) (n : ℕ) {select : (↥(split.foldA n) → X) → Iota} {err : Iota → (↥(split.foldB n) → X) → ℝ} {Eligible : Iota → Prop} {V : ℝ}, Measurable select → ∀ {good : Set (↥(split.foldA n) → X)}, MeasurableSet good → 0 ≤ V → (∀ a ∈ good, Eligible (select a)) → (∀ (i : Iota), Eligible i → MeasureTheory.Integrable (fun b => err i b ^ 2) (MeasureTheory.Measure.map (fun omega j => S.Z (↑j) omega) mu)) → (∀ (i : Iota), Eligible i → ∫ (b : ↥(split.foldB n) → X), err i b ^ 2 ∂MeasureTheory.Measure.map (fun omega j => S.Z (↑j) omega) mu ≤ V) → ∫ (omega : Omega) in (fun omega j => S.Z (↑j) omega) ⁻¹' good, (err (select fun j => S.Z (↑j) omega) fun j => S.Z (↑j) omega) ^ 2 ∂mu ≤ V","l":417,"f":"Causalean/Stat/SampleSplit/FiniteSelector.lean"},{"n":"Causalean.Stat.OrderDegenKernel","k":"structure","m":"Causalean.Stat.UStatistic.OrderM.Hajek","a":"Stat/UStatistic/OrderM","t":1,"d":"Fully degenerate order-m kernel. A kernel g on m-tuples over X is completely degenerate under the product measure $P^{\\otimes m}$ when g is measurable, g is invariant under permuting its m coordinates, integrating g over any one coordinate against P gives zero, whichever coordinate and values are held fixed for the rest, and g is square-integrable under $P^{\\otimes m}$.","s":"{X : Type u_2} → [inst : MeasurableSpace X] → MeasureTheory.Measure X → {m : ℕ} → [NeZero m] → ((Fin m → X) → ℝ) → Prop","l":62,"f":"Causalean/Stat/UStatistic/OrderM/Hajek.lean"},{"n":"Causalean.Stat.OrderDegenKernel.integrable","k":"theorem","m":"Causalean.Stat.UStatistic.OrderM.Variance","a":"Stat/UStatistic/OrderM","t":2,"d":"A square-integrable order-m degenerate kernel is integrable under the product law.","s":"∀ {X : Type u_2} [inst : MeasurableSpace X] {P : MeasureTheory.Measure X} {m : ℕ} [inst_1 : NeZero m] {g : (Fin m → X) → ℝ} [MeasureTheory.IsFiniteMeasure P], Causalean.Stat.OrderDegenKernel P g → MeasureTheory.Integrable g (MeasureTheory.Measure.pi fun x => P)","l":82,"f":"Causalean/Stat/UStatistic/OrderM/Variance.lean"},{"n":"Causalean.Stat.OrderDegenKernel.integral_eq_zero","k":"theorem","m":"Causalean.Stat.UStatistic.OrderM.Variance","a":"Stat/UStatistic/OrderM","t":2,"d":"A fully degenerate order-m kernel has zero product-law mean.","s":"∀ {X : Type u_2} [inst : MeasurableSpace X] {P : MeasureTheory.Measure X} [MeasureTheory.IsProbabilityMeasure P] {m : ℕ} [inst_2 : NeZero m] {g : (Fin m → X) → ℝ}, Causalean.Stat.OrderDegenKernel P g → Causalean.Stat.uMeanOrder g P = 0","l":89,"f":"Causalean/Stat/UStatistic/OrderM/Variance.lean"},{"n":"Causalean.Stat.OrderDegenerateNegligible","k":"def","m":"Causalean.Stat.UStatistic.OrderM.Hajek","a":"Stat/UStatistic/OrderM","t":1,"d":"The higher-order order-m Hájek remainder is negligible at the √n scale.","s":"{Ω : Type u_1} → {X : Type u_2} → [inst : MeasurableSpace Ω] → [inst_1 : MeasurableSpace X] → {μ : MeasureTheory.Measure Ω} → {P : MeasureTheory.Measure X} → Causalean.Stat.IIDSample Ω X μ P → {m : ℕ} → [NeZero m] → ((Fin m → X) → ℝ) → Prop","l":83,"f":"Causalean/Stat/UStatistic/OrderM/Hajek.lean"},{"n":"Causalean.Stat.OrderFirstDegenKernel","k":"structure","m":"Causalean.Stat.UStatistic.OrderM.FirstDegenKernel","a":"Stat/UStatistic/OrderM","t":1,"d":"First-order degenerate order-m kernel. A kernel g on m-tuples over X, together with the population measure P, is first-order degenerate when g is measurable, its first Hoeffding projection vanishes in every coordinate — integrating g over the other $m-1$ coordinates against the product measure leaves zero, whichever coordinate and value are held fixed, and g is square-integrable under the product measure $P^{\\otimes m}$.","s":"{X : Type u_2} → [inst : MeasurableSpace X] → MeasureTheory.Measure X → {m : ℕ} → [NeZero m] → ((Fin m → X) → ℝ) → Prop","l":44,"f":"Causalean/Stat/UStatistic/OrderM/FirstDegenKernel.lean"},{"n":"Causalean.Stat.OrderFirstDegenKernel.integrable","k":"theorem","m":"Causalean.Stat.UStatistic.OrderM.FirstDegenKernel","a":"Stat/UStatistic/OrderM","t":2,"d":"A first-order degenerate square-integrable kernel is integrable under the product law.","s":"∀ {X : Type u_2} [inst : MeasurableSpace X] {P : MeasureTheory.Measure X} {m : ℕ} [inst_1 : NeZero m] {g : (Fin m → X) → ℝ} [MeasureTheory.IsFiniteMeasure P], Causalean.Stat.OrderFirstDegenKernel P g → MeasureTheory.Integrable g (MeasureTheory.Measure.pi fun x => P)","l":65,"f":"Causalean/Stat/UStatistic/OrderM/FirstDegenKernel.lean"},{"n":"Causalean.Stat.OrderFirstDegenKernel.integral_eq_zero","k":"theorem","m":"Causalean.Stat.UStatistic.OrderM.FirstDegenKernel","a":"Stat/UStatistic/OrderM","t":1,"d":"Population mean of a first-order degenerate kernel is zero. If the order-m kernel g is first-order degenerate: measurable, square-integrable under the m-fold product law, and with zero mean after integrating out all but any single coordinate, then the population mean of g under the m-fold product law is zero.","s":"∀ {X : Type u_2} [inst : MeasurableSpace X] {P : MeasureTheory.Measure X} {m : ℕ} [inst_1 : NeZero m] {g : (Fin m → X) → ℝ} [MeasureTheory.IsFiniteMeasure P], Causalean.Stat.OrderFirstDegenKernel P g → Causalean.Stat.uMeanOrder g P = 0","l":73,"f":"Causalean/Stat/UStatistic/OrderM/FirstDegenKernel.lean"},{"n":"Causalean.Stat.PartialMatching","k":"structure","m":"Causalean.Stat.UStatistic.OrderM.PartialMatching","a":"Stat/UStatistic/OrderM","t":1,"d":"A partial matching between ordered coordinate sets of sizes r and s selects a subset from each side and pairs the selected coordinates bijectively.","s":"ℕ → ℕ → Type","l":25,"f":"Causalean/Stat/UStatistic/OrderM/PartialMatching.lean"},{"n":"Causalean.Stat.PartialMatching.MergedIndex","k":"def","m":"Causalean.Stat.UStatistic.OrderM.PartialMatching","a":"Stat/UStatistic/OrderM","t":1,"d":"The merged coordinate set retains every left coordinate and only the unmatched right coordinates.","s":"{r s : ℕ} → Causalean.Stat.PartialMatching r s → Type","l":78,"f":"Causalean/Stat/UStatistic/OrderM/PartialMatching.lean"},{"n":"Causalean.Stat.PartialMatching.empty","k":"def","m":"Causalean.Stat.UStatistic.OrderM.PartialMatching","a":"Stat/UStatistic/OrderM","t":1,"d":"The empty partial matching between two coordinate sets pairs no coordinates.","s":"(r s : ℕ) → Causalean.Stat.PartialMatching r s","l":47,"f":"Causalean/Stat/UStatistic/OrderM/PartialMatching.lean"},{"n":"Causalean.Stat.PartialMatching.empty_size","k":"theorem","m":"Causalean.Stat.UStatistic.OrderM.PartialMatching","a":"Stat/UStatistic/OrderM","t":2,"d":"For coordinate-set sizes r and s, the empty partial matching has no pairs.","s":"∀ (r s : ℕ), (Causalean.Stat.PartialMatching.empty r s).size = 0","l":57,"f":"Causalean/Stat/UStatistic/OrderM/PartialMatching.lean"},{"n":"Causalean.Stat.PartialMatching.eq_empty_of_size_eq_zero","k":"theorem","m":"Causalean.Stat.UStatistic.OrderM.PartialMatching","a":"Stat/UStatistic/OrderM","t":2,"d":"A partial matching with no pairs is the empty partial matching.","s":"∀ {r s : ℕ} (M : Causalean.Stat.PartialMatching r s), M.size = 0 → M = Causalean.Stat.PartialMatching.empty r s","l":62,"f":"Causalean/Stat/UStatistic/OrderM/PartialMatching.lean"},{"n":"Causalean.Stat.PartialMatching.leftInjection","k":"def","m":"Causalean.Stat.UStatistic.OrderM.PartialMatching","a":"Stat/UStatistic/OrderM","t":1,"d":"A left coordinate occupies its own position in the merged coordinate set.","s":"{r s : ℕ} → (M : Causalean.Stat.PartialMatching r s) → Fin r → M.MergedIndex","l":83,"f":"Causalean/Stat/UStatistic/OrderM/PartialMatching.lean"},{"n":"Causalean.Stat.PartialMatching.mergedIndex_card","k":"theorem","m":"Causalean.Stat.UStatistic.OrderM.PartialMatching","a":"Stat/UStatistic/OrderM","t":2,"d":"For a partial matching between coordinate sets of sizes r and s, the merged coordinate set contains r + s minus the matching size coordinates.","s":"∀ {r s : ℕ} (M : Causalean.Stat.PartialMatching r s), Fintype.card M.MergedIndex = r + s - M.size","l":95,"f":"Causalean/Stat/UStatistic/OrderM/PartialMatching.lean"},{"n":"Causalean.Stat.PartialMatching.rightInjection","k":"def","m":"Causalean.Stat.UStatistic.OrderM.PartialMatching","a":"Stat/UStatistic/OrderM","t":1,"d":"A right coordinate shares the position of its matched left coordinate, or occupies a separate position when it is unmatched.","s":"{r s : ℕ} → (M : Causalean.Stat.PartialMatching r s) → Fin s → M.MergedIndex","l":87,"f":"Causalean/Stat/UStatistic/OrderM/PartialMatching.lean"},{"n":"Causalean.Stat.PartialMatching.right_card","k":"theorem","m":"Causalean.Stat.UStatistic.OrderM.PartialMatching","a":"Stat/UStatistic/OrderM","t":2,"d":"Given a partial matching, its selected right subset has the same number of coordinates as the matching.","s":"∀ {r s : ℕ} (M : Causalean.Stat.PartialMatching r s), M.right.card = M.size","l":42,"f":"Causalean/Stat/UStatistic/OrderM/PartialMatching.lean"},{"n":"Causalean.Stat.PartialMatching.size","k":"def","m":"Causalean.Stat.UStatistic.OrderM.PartialMatching","a":"Stat/UStatistic/OrderM","t":1,"d":"The size of a partial matching is its number of paired coordinates.","s":"{r s : ℕ} → Causalean.Stat.PartialMatching r s → ℕ","l":39,"f":"Causalean/Stat/UStatistic/OrderM/PartialMatching.lean"},{"n":"Causalean.Stat.PartialMatching.size_le_min","k":"theorem","m":"Causalean.Stat.UStatistic.OrderM.PartialMatching","a":"Stat/UStatistic/OrderM","t":2,"d":"A partial matching has at most the smaller coordinate-set size many pairs.","s":"∀ {r s : ℕ} (M : Causalean.Stat.PartialMatching r s), M.size ≤ min r s","l":198,"f":"Causalean/Stat/UStatistic/OrderM/PartialMatching.lean"},{"n":"Causalean.Stat.PinskerBound","k":"def","m":"Causalean.Stat.Minimax.Pinsker","a":"Stat/Minimax","t":1,"d":"Pinsker-type control of total variation by KL divergence: tvDist μ ν ≤ √((klDiv μ ν).toReal / 2).","s":"{Ω : Type u_1} → {mΩ : MeasurableSpace Ω} → MeasureTheory.Measure Ω → MeasureTheory.Measure Ω → Prop","l":243,"f":"Causalean/Stat/Minimax/Pinsker.lean"},{"n":"Causalean.Stat.PolynomialTail.PolyTail","k":"structure","m":"Causalean.Stat.PolynomialTail.Basic","a":"Stat/PolynomialTail","t":1,"d":"Polynomial lower tail. For a measurable [0,1]-valued function U, says that the lower CDF t ↦ P{U ≤ t} is squeezed between cm·tᵏ and cp·tᵏ on the window (0, t₀], with a positive exponent κ, a window endpoint t₀ strictly between 0 and 1, and constants with 0 < cm < cp. This is the sole distributional input to the inverse-moment asymptotics.","s":"{Ω : Type u_1} → [inst : MeasurableSpace Ω] → MeasureTheory.Measure Ω → (Ω → ℝ) → ℝ → ℝ → ℝ → ℝ → Prop","l":58,"f":"Causalean/Stat/PolynomialTail/Basic.lean"},{"n":"Causalean.Stat.PolynomialTail.PolyTail.cp_pos","k":"theorem","m":"Causalean.Stat.PolynomialTail.Basic","a":"Stat/PolynomialTail","t":2,"d":"The upper constant is positive.","s":"∀ {Ω : Type u_1} [inst : MeasurableSpace Ω] {P : MeasureTheory.Measure Ω} {U : Ω → ℝ} {κ t₀ cm cp : ℝ}, Causalean.Stat.PolynomialTail.PolyTail P U κ t₀ cm cp → 0 < cp","l":84,"f":"Causalean/Stat/PolynomialTail/Basic.lean"},{"n":"Causalean.Stat.PolynomialTail.PolyTail.t0_mem","k":"theorem","m":"Causalean.Stat.PolynomialTail.Basic","a":"Stat/PolynomialTail","t":2,"d":"The window endpoint is positive and < 1 packaged together.","s":"∀ {Ω : Type u_1} [inst : MeasurableSpace Ω] {P : MeasureTheory.Measure Ω} {U : Ω → ℝ} {κ t₀ cm cp : ℝ}, Causalean.Stat.PolynomialTail.PolyTail P U κ t₀ cm cp → 0 < t₀ ∧ t₀ < 1","l":87,"f":"Causalean/Stat/PolynomialTail/Basic.lean"},{"n":"Causalean.Stat.PolynomialTail.TailSetup","k":"structure","m":"Causalean.Stat.PolynomialTail.Basic","a":"Stat/PolynomialTail","t":1,"d":"Tail setup. Bundles the structural hypotheses on U: measurability and 0 < U ≤ 1 almost surely. IsProbabilityMeasure P is required separately as a typeclass on the theorems that need it.","s":"{Ω : Type u_1} → [inst : MeasurableSpace Ω] → MeasureTheory.Measure Ω → (Ω → ℝ) → Prop","l":95,"f":"Causalean/Stat/PolynomialTail/Basic.lean"},{"n":"Causalean.Stat.PolynomialTail.integrableOn_rpow_neg_window","k":"theorem","m":"Causalean.Stat.PolynomialTail.TailIntegralBounds","a":"Stat/PolynomialTail","t":2,"d":"t^{-κ} is integrable on the window (t₀⁻¹, λ⁻¹].","s":"∀ {Ω : Type u_1} [inst : MeasurableSpace Ω] {P : MeasureTheory.Measure Ω} {U : Ω → ℝ} {κ t₀ cm cp lam : ℝ}, Causalean.Stat.PolynomialTail.PolyTail P U κ t₀ cm cp → 0 < lam → lam ≤ t₀ → MeasureTheory.IntegrableOn (fun t => t ^ (-κ)) (Set.Ioc t₀⁻¹ lam⁻¹) MeasureTheory.volume","l":70,"f":"Causalean/Stat/PolynomialTail/TailIntegralBounds.lean"},{"n":"Causalean.Stat.PolynomialTail.integrableOn_tailIntegrand","k":"theorem","m":"Causalean.Stat.PolynomialTail.LayerCakeReduction","a":"Stat/PolynomialTail","t":2,"d":"The tail integrand is integrable on Ioc a b.","s":"∀ {Ω : Type u_1} [inst : MeasurableSpace Ω] {P : MeasureTheory.Measure Ω} {U : Ω → ℝ} [MeasureTheory.IsProbabilityMeasure P] (a b : ℝ), MeasureTheory.IntegrableOn (fun t => P.real {ω | U ω ≤ t⁻¹}) (Set.Ioc a b) MeasureTheory.volume","l":75,"f":"Causalean/Stat/PolynomialTail/LayerCakeReduction.lean"},{"n":"Causalean.Stat.PolynomialTail.integrable_invMomentI_integrand","k":"theorem","m":"Causalean.Stat.PolynomialTail.Basic","a":"Stat/PolynomialTail","t":2,"d":"The I-integrand is integrable (bounded a.e. on a probability space).","s":"∀ {Ω : Type u_1} [inst : MeasurableSpace Ω] {P : MeasureTheory.Measure Ω} {U : Ω → ℝ} {lam : ℝ} [MeasureTheory.IsProbabilityMeasure P], Causalean.Stat.PolynomialTail.TailSetup P U → 0 < lam → MeasureTheory.Integrable (fun ω => U ω / max (U ω) lam ^ 2) P","l":204,"f":"Causalean/Stat/PolynomialTail/Basic.lean"},{"n":"Causalean.Stat.PolynomialTail.integrable_invMomentJ_integrand","k":"theorem","m":"Causalean.Stat.PolynomialTail.Basic","a":"Stat/PolynomialTail","t":2,"d":"The J-integrand is integrable.","s":"∀ {Ω : Type u_1} [inst : MeasurableSpace Ω] {P : MeasureTheory.Measure Ω} {U : Ω → ℝ} {lam : ℝ} [MeasureTheory.IsProbabilityMeasure P], Causalean.Stat.PolynomialTail.TailSetup P U → 0 < lam → MeasureTheory.Integrable (fun ω => (max (U ω) lam)⁻¹) P","l":214,"f":"Causalean/Stat/PolynomialTail/Basic.lean"},{"n":"Causalean.Stat.PolynomialTail.integrable_trimWeight","k":"theorem","m":"Causalean.Stat.PolynomialTail.Basic","a":"Stat/PolynomialTail","t":2,"d":"The leftover weight is integrable.","s":"∀ {Ω : Type u_1} [inst : MeasurableSpace Ω] {P : MeasureTheory.Measure Ω} {U : Ω → ℝ} {lam : ℝ} [MeasureTheory.IsProbabilityMeasure P], Causalean.Stat.PolynomialTail.TailSetup P U → 0 < lam → MeasureTheory.Integrable (Causalean.Stat.PolynomialTail.trimWeight U lam) P","l":224,"f":"Causalean/Stat/PolynomialTail/Basic.lean"},{"n":"Causalean.Stat.PolynomialTail.integral_inv_neg_Ioc","k":"theorem","m":"Causalean.Stat.PolynomialTail.PowerIntegral","a":"Stat/PolynomialTail","t":2,"d":"Power integral, κ = 1. ∫ t in (a,b], t^{-1} = log b − log a.","s":"∀ {a b : ℝ}, 0 < a → a ≤ b → ∫ (t : ℝ) in Set.Ioc a b, t ^ (-1) = Real.log b - Real.log a","l":54,"f":"Causalean/Stat/PolynomialTail/PowerIntegral.lean"},{"n":"Causalean.Stat.PolynomialTail.integral_rpow_neg_Ioc","k":"theorem","m":"Causalean.Stat.PolynomialTail.PowerIntegral","a":"Stat/PolynomialTail","t":1,"d":"Power integral, κ ≠ 1. If the exponent κ is not equal to 1, a is positive, and a is at most b, then the interval integral of t ↦ t^{-κ} over (a, b] equals (b^{1-κ} − a^{1-κ})/(1−κ).","s":"∀ {κ : ℝ}, κ ≠ 1 → ∀ {a b : ℝ}, 0 < a → a ≤ b → ∫ (t : ℝ) in Set.Ioc a b, t ^ (-κ) = (b ^ (1 - κ) - a ^ (1 - κ)) / (1 - κ)","l":45,"f":"Causalean/Stat/PolynomialTail/PowerIntegral.lean"},{"n":"Causalean.Stat.PolynomialTail.invMomentI","k":"def","m":"Causalean.Stat.PolynomialTail.Basic","a":"Stat/PolynomialTail","t":1,"d":"I P U λ = ∫ U / (max U λ)². Equals 1/U on {U ≥ λ} and U/λ² on {U < λ}; the truncated inverse *second moment*.","s":"{Ω : Type u_1} → [inst : MeasurableSpace Ω] → MeasureTheory.Measure Ω → (Ω → ℝ) → ℝ → ℝ","l":111,"f":"Causalean/Stat/PolynomialTail/Basic.lean"},{"n":"Causalean.Stat.PolynomialTail.invMomentI_antitone","k":"theorem","m":"Causalean.Stat.PolynomialTail.MomentIBounds","a":"Stat/PolynomialTail","t":2,"d":"I is antitone in λ. As λ decreases, max U λ decreases, so the integrand U/(max U λ)² increases. Hence I P U λ₂ ≤ I P U λ₁ when λ₁ ≤ λ₂.","s":"∀ {Ω : Type u_1} [inst : MeasurableSpace Ω] {P : MeasureTheory.Measure Ω} {U : Ω → ℝ} [MeasureTheory.IsProbabilityMeasure P], Causalean.Stat.PolynomialTail.TailSetup P U → ∀ {l1 l2 : ℝ}, 0 < l1 → l1 ≤ l2 → Causalean.Stat.PolynomialTail.invMomentI P U l2 ≤ Causalean.Stat.PolynomialTail.invMomentI P U l1","l":103,"f":"Causalean/Stat/PolynomialTail/MomentIBounds.lean"},{"n":"Causalean.Stat.PolynomialTail.invMomentI_bounds_eq","k":"theorem","m":"Causalean.Stat.PolynomialTail.MomentIBounds","a":"Stat/PolynomialTail","t":1,"d":"Regime κ = 1. Under the polynomial lower-tail hypothesis PolyTail P U κ t₀ cm cp with U measurable and almost surely in (0,1] and tail exponent exactly 1, there exist constants 0 < a ≤ A such that the truncated inverse second moment I P U λ is squeezed between a·log(1/λ) and A·log(1/λ) + A, uniformly for every λ ∈ (0, t₀]. Upper: I ≤ J; lower: I = J − (J−I) ≥ J − cp, uniformized with the constant shell floor.","s":"∀ {Ω : Type u_1} [inst : MeasurableSpace Ω] {P : MeasureTheory.Measure Ω} {U : Ω → ℝ} {κ t₀ cm cp : ℝ} [MeasureTheory.IsProbabilityMeasure P], Causalean.Stat.PolynomialTail.PolyTail P U κ t₀ cm cp → Causalean.Stat.PolynomialTail.TailSetup P U → κ = 1 → ∃ a A, 0 < a ∧ a ≤ A ∧ ∀ (lam : ℝ), 0 < lam → lam ≤ t₀ → a * Real.log (1 / lam) ≤ Causalean.Stat.PolynomialTail.invMomentI P U lam ∧ Causalean.Stat.PolynomialTail.invMomentI P U lam ≤ A * Real.log (1 / lam) + A","l":214,"f":"Causalean/Stat/PolynomialTail/MomentIBounds.lean"},{"n":"Causalean.Stat.PolynomialTail.invMomentI_bounds_gt","k":"theorem","m":"Causalean.Stat.PolynomialTail.MomentIBounds","a":"Stat/PolynomialTail","t":1,"d":"Regime κ > 1. Under the polynomial lower-tail hypothesis PolyTail P U κ t₀ cm cp with U measurable and almost surely in (0,1] and tail exponent strictly above 1, there exist constants 0 < a ≤ A such that the truncated inverse second moment I P U λ is bounded between a and A, uniformly for every λ ∈ (0, t₀]: the inverse second moment does not blow up. Lower: I is antitone, so I(λ) ≥ I(t₀) ≥ (shell at t₀), a positive constant; upper: I ≤ J.","s":"∀ {Ω : Type u_1} [inst : MeasurableSpace Ω] {P : MeasureTheory.Measure Ω} {U : Ω → ℝ} {κ t₀ cm cp : ℝ} [MeasureTheory.IsProbabilityMeasure P], Causalean.Stat.PolynomialTail.PolyTail P U κ t₀ cm cp → Causalean.Stat.PolynomialTail.TailSetup P U → 1 < κ → ∃ a A, 0 < a ∧ a ≤ A ∧ ∀ (lam : ℝ), 0 < lam → lam ≤ t₀ → a ≤ Causalean.Stat.PolynomialTail.invMomentI P U lam ∧ Causalean.Stat.PolynomialTail.invMomentI P U lam ≤ A","l":272,"f":"Causalean/Stat/PolynomialTail/MomentIBounds.lean"},{"n":"Causalean.Stat.PolynomialTail.invMomentI_bounds_lt","k":"theorem","m":"Causalean.Stat.PolynomialTail.MomentIBounds","a":"Stat/PolynomialTail","t":1,"d":"Regime 0 < κ < 1. Under the polynomial lower-tail hypothesis PolyTail P U κ t₀ cm cp with U measurable and almost surely in (0,1] and tail exponent strictly between 0 and 1, there exist constants 0 < a ≤ A such that the truncated inverse second moment I P U λ is squeezed between a·λ^{κ-1} and A·λ^{κ-1}, uniformly for every λ ∈ (0, t₀]. Lower: shell; upper: I ≤ J.","s":"∀ {Ω : Type u_1} [inst : MeasurableSpace Ω] {P : MeasureTheory.Measure Ω} {U : Ω → ℝ} {κ t₀ cm cp : ℝ} [MeasureTheory.IsProbabilityMeasure P], Causalean.Stat.PolynomialTail.PolyTail P U κ t₀ cm cp → Causalean.Stat.PolynomialTail.TailSetup P U → κ < 1 → ∃ a A, 0 < a ∧ a ≤ A ∧ ∀ (lam : ℝ), 0 < lam → lam ≤ t₀ → a * lam ^ (κ - 1) ≤ Causalean.Stat.PolynomialTail.invMomentI P U lam ∧ Causalean.Stat.PolynomialTail.invMomentI P U lam ≤ A * lam ^ (κ - 1)","l":190,"f":"Causalean/Stat/PolynomialTail/MomentIBounds.lean"},{"n":"Causalean.Stat.PolynomialTail.invMomentI_ge_shell","k":"theorem","m":"Causalean.Stat.PolynomialTail.MomentIBounds","a":"Stat/PolynomialTail","t":2,"d":"Shell lower bound on I. (ρ·cm/2)·λ^{κ-1} ≤ I P U λ, with ρ = (cm/(2cp))^{1/κ}. Valid for every λ ∈ (0, t₀].","s":"∀ {Ω : Type u_1} [inst : MeasurableSpace Ω] {P : MeasureTheory.Measure Ω} {U : Ω → ℝ} {κ t₀ cm cp lam : ℝ} [MeasureTheory.IsProbabilityMeasure P], Causalean.Stat.PolynomialTail.PolyTail P U κ t₀ cm cp → Causalean.Stat.PolynomialTail.TailSetup P U → 0 < lam → lam ≤ t₀ → cm / 2 * (cm / (2 * cp)) ^ (1 / κ) * lam ^ (κ - 1) ≤ Causalean.Stat.PolynomialTail.invMomentI P U lam","l":118,"f":"Causalean/Stat/PolynomialTail/MomentIBounds.lean"},{"n":"Causalean.Stat.PolynomialTail.invMomentI_integrand_mem","k":"theorem","m":"Causalean.Stat.PolynomialTail.Basic","a":"Stat/PolynomialTail","t":2,"d":"On {0 < U ≤ 1} the I-integrand is squeezed in [0, 1/λ²].","s":"∀ {lam : ℝ}, 0 < lam → ∀ {x : ℝ}, 0 < x → x ≤ 1 → 0 ≤ x / max x lam ^ 2 ∧ x / max x lam ^ 2 ≤ (lam ^ 2)⁻¹","l":177,"f":"Causalean/Stat/PolynomialTail/Basic.lean"},{"n":"Causalean.Stat.PolynomialTail.invMomentI_isTheta_gt","k":"theorem","m":"Causalean.Stat.PolynomialTail.MomentIBounds","a":"Stat/PolynomialTail","t":1,"d":"IsTheta for I (κ > 1). Under the polynomial lower-tail hypothesis with U measurable and almost surely in (0,1] and tail exponent strictly above 1, the truncated inverse second moment I P U λ is Θ(1) — bounded — as λ → 0⁺.","s":"∀ {Ω : Type u_1} [inst : MeasurableSpace Ω] {P : MeasureTheory.Measure Ω} {U : Ω → ℝ} {κ t₀ cm cp : ℝ} [MeasureTheory.IsProbabilityMeasure P], Causalean.Stat.PolynomialTail.PolyTail P U κ t₀ cm cp → Causalean.Stat.PolynomialTail.TailSetup P U → 1 < κ → (fun lam => Causalean.Stat.PolynomialTail.invMomentI P U lam) =Θ[nhdsWithin 0 (Set.Ioi 0)] fun x => 1","l":355,"f":"Causalean/Stat/PolynomialTail/MomentIBounds.lean"},{"n":"Causalean.Stat.PolynomialTail.invMomentI_isTheta_lt","k":"theorem","m":"Causalean.Stat.PolynomialTail.MomentIBounds","a":"Stat/PolynomialTail","t":1,"d":"IsTheta for I (κ < 1). Under the polynomial lower-tail hypothesis with U measurable and almost surely in (0,1] and tail exponent strictly between 0 and 1, the truncated inverse second moment I P U λ is Θ(λ^{κ-1}) as λ → 0⁺.","s":"∀ {Ω : Type u_1} [inst : MeasurableSpace Ω] {P : MeasureTheory.Measure Ω} {U : Ω → ℝ} {κ t₀ cm cp : ℝ} [MeasureTheory.IsProbabilityMeasure P], Causalean.Stat.PolynomialTail.PolyTail P U κ t₀ cm cp → Causalean.Stat.PolynomialTail.TailSetup P U → κ < 1 → (fun lam => Causalean.Stat.PolynomialTail.invMomentI P U lam) =Θ[nhdsWithin 0 (Set.Ioi 0)] fun lam => lam ^ (κ - 1)","l":331,"f":"Causalean/Stat/PolynomialTail/MomentIBounds.lean"},{"n":"Causalean.Stat.PolynomialTail.invMomentI_le_invMomentJ","k":"theorem","m":"Causalean.Stat.PolynomialTail.MomentIBounds","a":"Stat/PolynomialTail","t":2,"d":"I ≤ J. Pointwise U/(max U λ)² ≤ (max U λ)⁻¹ (since U ≤ max U λ).","s":"∀ {Ω : Type u_1} [inst : MeasurableSpace Ω] {P : MeasureTheory.Measure Ω} {U : Ω → ℝ} {lam : ℝ} [MeasureTheory.IsProbabilityMeasure P], Causalean.Stat.PolynomialTail.TailSetup P U → 0 < lam → Causalean.Stat.PolynomialTail.invMomentI P U lam ≤ Causalean.Stat.PolynomialTail.invMomentJ P U lam","l":45,"f":"Causalean/Stat/PolynomialTail/MomentIBounds.lean"},{"n":"Causalean.Stat.PolynomialTail.invMomentJ","k":"def","m":"Causalean.Stat.PolynomialTail.Basic","a":"Stat/PolynomialTail","t":1,"d":"J P U λ = ∫ (max U λ)⁻¹. Equals 1/U on {U ≥ λ} and 1/λ on {U < λ}; the truncated inverse *first moment*.","s":"{Ω : Type u_1} → [inst : MeasurableSpace Ω] → MeasureTheory.Measure Ω → (Ω → ℝ) → ℝ → ℝ","l":116,"f":"Causalean/Stat/PolynomialTail/Basic.lean"},{"n":"Causalean.Stat.PolynomialTail.invMomentJ_bounds_eq","k":"theorem","m":"Causalean.Stat.PolynomialTail.MomentJBounds","a":"Stat/PolynomialTail","t":1,"d":"Regime κ = 1. Under the polynomial lower-tail hypothesis PolyTail P U κ t₀ cm cp with U measurable and almost surely in (0,1] and tail exponent exactly 1, there exist constants 0 < a ≤ A such that the truncated inverse first moment J P U λ is squeezed between a·log(1/λ) and A·log(1/λ) + A, uniformly for every λ ∈ (0, t₀].","s":"∀ {Ω : Type u_1} [inst : MeasurableSpace Ω] {P : MeasureTheory.Measure Ω} {U : Ω → ℝ} {κ t₀ cm cp : ℝ} [MeasureTheory.IsProbabilityMeasure P], Causalean.Stat.PolynomialTail.PolyTail P U κ t₀ cm cp → Causalean.Stat.PolynomialTail.TailSetup P U → κ = 1 → ∃ a A, 0 < a ∧ a ≤ A ∧ ∀ (lam : ℝ), 0 < lam → lam ≤ t₀ → a * Real.log (1 / lam) ≤ Causalean.Stat.PolynomialTail.invMomentJ P U lam ∧ Causalean.Stat.PolynomialTail.invMomentJ P U lam ≤ A * Real.log (1 / lam) + A","l":126,"f":"Causalean/Stat/PolynomialTail/MomentJBounds.lean"},{"n":"Causalean.Stat.PolynomialTail.invMomentJ_bounds_gt","k":"theorem","m":"Causalean.Stat.PolynomialTail.MomentJBounds","a":"Stat/PolynomialTail","t":1,"d":"Regime κ > 1. Under the polynomial lower-tail hypothesis PolyTail P U κ t₀ cm cp with U measurable and almost surely in (0,1] and tail exponent strictly above 1, there exist constants 0 < a ≤ A such that the truncated inverse first moment J P U λ is bounded between a and A, uniformly for every λ ∈ (0, t₀]: the inverse first moment does not blow up.","s":"∀ {Ω : Type u_1} [inst : MeasurableSpace Ω] {P : MeasureTheory.Measure Ω} {U : Ω → ℝ} {κ t₀ cm cp : ℝ} [MeasureTheory.IsProbabilityMeasure P], Causalean.Stat.PolynomialTail.PolyTail P U κ t₀ cm cp → Causalean.Stat.PolynomialTail.TailSetup P U → 1 < κ → ∃ a A, 0 < a ∧ a ≤ A ∧ ∀ (lam : ℝ), 0 < lam → lam ≤ t₀ → a ≤ Causalean.Stat.PolynomialTail.invMomentJ P U lam ∧ Causalean.Stat.PolynomialTail.invMomentJ P U lam ≤ A","l":173,"f":"Causalean/Stat/PolynomialTail/MomentJBounds.lean"},{"n":"Causalean.Stat.PolynomialTail.invMomentJ_bounds_lt","k":"theorem","m":"Causalean.Stat.PolynomialTail.MomentJBounds","a":"Stat/PolynomialTail","t":1,"d":"Regime 0 < κ < 1. Under the polynomial lower-tail hypothesis PolyTail P U κ t₀ cm cp with U measurable and almost surely in (0,1] and tail exponent strictly between 0 and 1, there exist constants 0 < a ≤ A, depending only on κ, cm, cp, t₀, such that the truncated inverse first moment J P U λ is squeezed between a·λ^{κ-1} and A·λ^{κ-1}, uniformly for every λ ∈ (0, t₀].","s":"∀ {Ω : Type u_1} [inst : MeasurableSpace Ω] {P : MeasureTheory.Measure Ω} {U : Ω → ℝ} {κ t₀ cm cp : ℝ} [MeasureTheory.IsProbabilityMeasure P], Causalean.Stat.PolynomialTail.PolyTail P U κ t₀ cm cp → Causalean.Stat.PolynomialTail.TailSetup P U → κ < 1 → ∃ a A, 0 < a ∧ a ≤ A ∧ ∀ (lam : ℝ), 0 < lam → lam ≤ t₀ → a * lam ^ (κ - 1) ≤ Causalean.Stat.PolynomialTail.invMomentJ P U lam ∧ Causalean.Stat.PolynomialTail.invMomentJ P U lam ≤ A * lam ^ (κ - 1)","l":65,"f":"Causalean/Stat/PolynomialTail/MomentJBounds.lean"},{"n":"Causalean.Stat.PolynomialTail.invMomentJ_eq_one_add_tailIntegral","k":"theorem","m":"Causalean.Stat.PolynomialTail.LayerCakeReduction","a":"Stat/PolynomialTail","t":1,"d":"Master layer-cake reduction. For a variable U that is measurable and almost surely strictly positive and at most 1, and a threshold λ that is positive and strictly below 1, the truncated inverse first moment J P U λ equals 1 plus the threshold tail integral tailIntegral P U λ:","s":"∀ {Ω : Type u_1} [inst : MeasurableSpace Ω] {P : MeasureTheory.Measure Ω} {U : Ω → ℝ} {lam : ℝ} [MeasureTheory.IsProbabilityMeasure P], Causalean.Stat.PolynomialTail.TailSetup P U → 0 < lam → lam < 1 → Causalean.Stat.PolynomialTail.invMomentJ P U lam = 1 + Causalean.Stat.PolynomialTail.tailIntegral P U lam","l":86,"f":"Causalean/Stat/PolynomialTail/LayerCakeReduction.lean"},{"n":"Causalean.Stat.PolynomialTail.invMomentJ_integrand_mem","k":"theorem","m":"Causalean.Stat.PolynomialTail.Basic","a":"Stat/PolynomialTail","t":2,"d":"The J-integrand is squeezed in [0, 1/λ].","s":"∀ {lam : ℝ}, 0 < lam → ∀ (x : ℝ), 0 ≤ (max x lam)⁻¹ ∧ (max x lam)⁻¹ ≤ lam⁻¹","l":187,"f":"Causalean/Stat/PolynomialTail/Basic.lean"},{"n":"Causalean.Stat.PolynomialTail.invMomentJ_isTheta_gt","k":"theorem","m":"Causalean.Stat.PolynomialTail.MomentIBounds","a":"Stat/PolynomialTail","t":1,"d":"IsTheta for J (κ > 1). Under the polynomial lower-tail hypothesis with U measurable and almost surely in (0,1] and tail exponent strictly above 1, the truncated inverse first moment J P U λ is Θ(1) — bounded — as λ → 0⁺.","s":"∀ {Ω : Type u_1} [inst : MeasurableSpace Ω] {P : MeasureTheory.Measure Ω} {U : Ω → ℝ} {κ t₀ cm cp : ℝ} [MeasureTheory.IsProbabilityMeasure P], Causalean.Stat.PolynomialTail.PolyTail P U κ t₀ cm cp → Causalean.Stat.PolynomialTail.TailSetup P U → 1 < κ → (fun lam => Causalean.Stat.PolynomialTail.invMomentJ P U lam) =Θ[nhdsWithin 0 (Set.Ioi 0)] fun x => 1","l":343,"f":"Causalean/Stat/PolynomialTail/MomentIBounds.lean"},{"n":"Causalean.Stat.PolynomialTail.invMomentJ_isTheta_lt","k":"theorem","m":"Causalean.Stat.PolynomialTail.MomentIBounds","a":"Stat/PolynomialTail","t":1,"d":"IsTheta for J (κ < 1). Under the polynomial lower-tail hypothesis with U measurable and almost surely in (0,1] and tail exponent strictly between 0 and 1, the truncated inverse first moment J P U λ is Θ(λ^{κ-1}) as λ → 0⁺.","s":"∀ {Ω : Type u_1} [inst : MeasurableSpace Ω] {P : MeasureTheory.Measure Ω} {U : Ω → ℝ} {κ t₀ cm cp : ℝ} [MeasureTheory.IsProbabilityMeasure P], Causalean.Stat.PolynomialTail.PolyTail P U κ t₀ cm cp → Causalean.Stat.PolynomialTail.TailSetup P U → κ < 1 → (fun lam => Causalean.Stat.PolynomialTail.invMomentJ P U lam) =Θ[nhdsWithin 0 (Set.Ioi 0)] fun lam => lam ^ (κ - 1)","l":319,"f":"Causalean/Stat/PolynomialTail/MomentIBounds.lean"},{"n":"Causalean.Stat.PolynomialTail.invMomentJ_sub_invMomentI_le","k":"theorem","m":"Causalean.Stat.PolynomialTail.MomentIBounds","a":"Stat/PolynomialTail","t":2,"d":"J − I is controlled by the tail mass. J P U λ − I P U λ ≤ cp·λ^{κ-1}. Pointwise the gap equals wλ/(max U λ) ≤ λ⁻¹·wλ, and ∫ wλ ≤ P{U<λ} ≤ cp·λᵏ.","s":"∀ {Ω : Type u_1} [inst : MeasurableSpace Ω] {P : MeasureTheory.Measure Ω} {U : Ω → ℝ} {κ t₀ cm cp lam : ℝ} [MeasureTheory.IsProbabilityMeasure P], Causalean.Stat.PolynomialTail.PolyTail P U κ t₀ cm cp → Causalean.Stat.PolynomialTail.TailSetup P U → 0 < lam → lam ≤ t₀ → Causalean.Stat.PolynomialTail.invMomentJ P U lam - Causalean.Stat.PolynomialTail.invMomentI P U lam ≤ cp * lam ^ (κ - 1)","l":61,"f":"Causalean/Stat/PolynomialTail/MomentIBounds.lean"},{"n":"Causalean.Stat.PolynomialTail.inv_rpow_eq_rpow_neg","k":"theorem","m":"Causalean.Stat.PolynomialTail.TailIntegralBounds","a":"Stat/PolynomialTail","t":2,"d":"(t⁻¹)^κ = t^{-κ} for t > 0.","s":"∀ {t : ℝ}, 0 < t → ∀ (κ : ℝ), t⁻¹ ^ κ = t ^ (-κ)","l":48,"f":"Causalean/Stat/PolynomialTail/TailIntegralBounds.lean"},{"n":"Causalean.Stat.PolynomialTail.isTheta_of_two_sided","k":"theorem","m":"Causalean.Stat.PolynomialTail.MomentIBounds","a":"Stat/PolynomialTail","t":2,"d":"Generic two-sided-bound ⟹ IsTheta packager on 𝓝[>] 0.","s":"∀ {f g : ℝ → ℝ} {t₀ a A : ℝ}, 0 < t₀ → 0 < a → (∀ (lam : ℝ), 0 < lam → lam ≤ t₀ → 0 < g lam) → (∀ (lam : ℝ), 0 < lam → lam ≤ t₀ → 0 ≤ f lam) → (∀ (lam : ℝ), 0 < lam → lam ≤ t₀ → a * g lam ≤ f lam ∧ f lam ≤ A * g lam) → f =Θ[nhdsWithin 0 (Set.Ioi 0)] g","l":298,"f":"Causalean/Stat/PolynomialTail/MomentIBounds.lean"},{"n":"Causalean.Stat.PolynomialTail.le_truncDen","k":"theorem","m":"Causalean.Stat.PolynomialTail.Basic","a":"Stat/PolynomialTail","t":2,"d":"max U λ ≥ λ (pointwise, no hypotheses).","s":"∀ {Ω : Type u_1} {U : Ω → ℝ} {lam : ℝ} (ω : Ω), lam ≤ Causalean.Stat.PolynomialTail.truncDen U lam ω","l":152,"f":"Causalean/Stat/PolynomialTail/Basic.lean"},{"n":"Causalean.Stat.PolynomialTail.measurable_invMomentI_integrand","k":"theorem","m":"Causalean.Stat.PolynomialTail.Basic","a":"Stat/PolynomialTail","t":2,"d":"The inverse second-moment integrand is measurable when U is measurable.","s":"∀ {Ω : Type u_1} [inst : MeasurableSpace Ω] {U : Ω → ℝ} {lam : ℝ}, Measurable U → Measurable fun ω => U ω / max (U ω) lam ^ 2","l":134,"f":"Causalean/Stat/PolynomialTail/Basic.lean"},{"n":"Causalean.Stat.PolynomialTail.measurable_invMomentJ_integrand","k":"theorem","m":"Causalean.Stat.PolynomialTail.Basic","a":"Stat/PolynomialTail","t":2,"d":"The inverse first-moment integrand is measurable when U is measurable.","s":"∀ {Ω : Type u_1} [inst : MeasurableSpace Ω] {U : Ω → ℝ} {lam : ℝ}, Measurable U → Measurable fun ω => (max (U ω) lam)⁻¹","l":139,"f":"Causalean/Stat/PolynomialTail/Basic.lean"},{"n":"Causalean.Stat.PolynomialTail.measurable_tailIntegrand","k":"theorem","m":"Causalean.Stat.PolynomialTail.LayerCakeReduction","a":"Stat/PolynomialTail","t":2,"d":"The integrand t ↦ P{U ≤ t⁻¹} is measurable (monotone CDF ∘ inversion).","s":"∀ {Ω : Type u_1} [inst : MeasurableSpace Ω] {P : MeasureTheory.Measure Ω} {U : Ω → ℝ} [MeasureTheory.IsFiniteMeasure P], Measurable fun t => P.real {ω | U ω ≤ t⁻¹}","l":59,"f":"Causalean/Stat/PolynomialTail/LayerCakeReduction.lean"},{"n":"Causalean.Stat.PolynomialTail.measurable_trimWeight","k":"theorem","m":"Causalean.Stat.PolynomialTail.Basic","a":"Stat/PolynomialTail","t":2,"d":"The leftover trimming weight is measurable when U is measurable.","s":"∀ {Ω : Type u_1} [inst : MeasurableSpace Ω] {U : Ω → ℝ} {lam : ℝ}, Measurable U → Measurable (Causalean.Stat.PolynomialTail.trimWeight U lam)","l":144,"f":"Causalean/Stat/PolynomialTail/Basic.lean"},{"n":"Causalean.Stat.PolynomialTail.measurable_truncDen","k":"theorem","m":"Causalean.Stat.PolynomialTail.Basic","a":"Stat/PolynomialTail","t":2,"d":"The truncated denominator is measurable when U is measurable.","s":"∀ {Ω : Type u_1} [inst : MeasurableSpace Ω] {U : Ω → ℝ} {lam : ℝ}, Measurable U → Measurable (Causalean.Stat.PolynomialTail.truncDen U lam)","l":129,"f":"Causalean/Stat/PolynomialTail/Basic.lean"},{"n":"Causalean.Stat.PolynomialTail.measureReal_lt_le","k":"theorem","m":"Causalean.Stat.PolynomialTail.Basic","a":"Stat/PolynomialTail","t":1,"d":"Mass below λ. Under the polynomial lower-tail hypothesis packaged in h, for a threshold λ that is positive and at most the window endpoint t₀, the open lower level set {U < λ} has P-mass at most cp · λᵏ. Immediate from monotonicity and the upper tail bound at t = λ.","s":"∀ {Ω : Type u_1} [inst : MeasurableSpace Ω] {P : MeasureTheory.Measure Ω} {U : Ω → ℝ} {lam κ t₀ cm cp : ℝ} [MeasureTheory.IsFiniteMeasure P], Causalean.Stat.PolynomialTail.PolyTail P U κ t₀ cm cp → 0 < lam → lam ≤ t₀ → P.real {ω | U ω < lam} ≤ cp * lam ^ κ","l":164,"f":"Causalean/Stat/PolynomialTail/Basic.lean"},{"n":"Causalean.Stat.PolynomialTail.monotone_cdf","k":"theorem","m":"Causalean.Stat.PolynomialTail.LayerCakeReduction","a":"Stat/PolynomialTail","t":2,"d":"The lower CDF s ↦ P{U ≤ s} is monotone.","s":"∀ {Ω : Type u_1} [inst : MeasurableSpace Ω] {P : MeasureTheory.Measure Ω} {U : Ω → ℝ} [MeasureTheory.IsFiniteMeasure P], Monotone fun s => P.real {ω | U ω ≤ s}","l":54,"f":"Causalean/Stat/PolynomialTail/LayerCakeReduction.lean"},{"n":"Causalean.Stat.PolynomialTail.powInt_eq_one","k":"theorem","m":"Causalean.Stat.PolynomialTail.MomentJBounds","a":"Stat/PolynomialTail","t":2,"d":"PowInt = log(1/λ) − log(1/t₀) when κ = 1.","s":"∀ {Ω : Type u_1} [inst : MeasurableSpace Ω] {P : MeasureTheory.Measure Ω} {U : Ω → ℝ} {κ t₀ cm cp lam : ℝ}, Causalean.Stat.PolynomialTail.PolyTail P U κ t₀ cm cp → κ = 1 → 0 < lam → lam ≤ t₀ → ∫ (t : ℝ) in Set.Ioc t₀⁻¹ lam⁻¹, t ^ (-κ) = Real.log (1 / lam) - Real.log (1 / t₀)","l":55,"f":"Causalean/Stat/PolynomialTail/MomentJBounds.lean"},{"n":"Causalean.Stat.PolynomialTail.powInt_ne_one","k":"theorem","m":"Causalean.Stat.PolynomialTail.MomentJBounds","a":"Stat/PolynomialTail","t":2,"d":"PowInt = (λ^{κ-1} − t₀^{κ-1})/(1−κ) when κ ≠ 1.","s":"∀ {Ω : Type u_1} [inst : MeasurableSpace Ω] {P : MeasureTheory.Measure Ω} {U : Ω → ℝ} {κ t₀ cm cp lam : ℝ}, Causalean.Stat.PolynomialTail.PolyTail P U κ t₀ cm cp → κ ≠ 1 → 0 < lam → lam ≤ t₀ → ∫ (t : ℝ) in Set.Ioc t₀⁻¹ lam⁻¹, t ^ (-κ) = (lam ^ (κ - 1) - t₀ ^ (κ - 1)) / (1 - κ)","l":46,"f":"Causalean/Stat/PolynomialTail/MomentJBounds.lean"},{"n":"Causalean.Stat.PolynomialTail.tailIntegral","k":"def","m":"Causalean.Stat.PolynomialTail.LayerCakeReduction","a":"Stat/PolynomialTail","t":1,"d":"The threshold (\"tail\") integral ∫ t in (1, λ⁻¹], P{U ≤ t⁻¹} dt.","s":"{Ω : Type u_1} → [inst : MeasurableSpace Ω] → MeasureTheory.Measure Ω → (Ω → ℝ) → ℝ → ℝ","l":50,"f":"Causalean/Stat/PolynomialTail/LayerCakeReduction.lean"},{"n":"Causalean.Stat.PolynomialTail.tailIntegral_ge","k":"theorem","m":"Causalean.Stat.PolynomialTail.TailIntegralBounds","a":"Stat/PolynomialTail","t":1,"d":"Lower bound on the tail integral. Under the polynomial lower-tail hypothesis PolyTail P U κ t₀ cm cp, for any threshold λ that is positive and at most the window endpoint t₀, cm times the power comparison integral over (t₀⁻¹, λ⁻¹] is a lower bound for the layer-cake tail integral tailIntegral P U λ:","s":"∀ {Ω : Type u_1} [inst : MeasurableSpace Ω] {P : MeasureTheory.Measure Ω} {U : Ω → ℝ} {κ t₀ cm cp lam : ℝ} [MeasureTheory.IsProbabilityMeasure P], Causalean.Stat.PolynomialTail.PolyTail P U κ t₀ cm cp → 0 < lam → lam ≤ t₀ → cm * ∫ (t : ℝ) in Set.Ioc t₀⁻¹ lam⁻¹, t ^ (-κ) ≤ Causalean.Stat.PolynomialTail.tailIntegral P U lam","l":122,"f":"Causalean/Stat/PolynomialTail/TailIntegralBounds.lean"},{"n":"Causalean.Stat.PolynomialTail.tailIntegral_le","k":"theorem","m":"Causalean.Stat.PolynomialTail.TailIntegralBounds","a":"Stat/PolynomialTail","t":1,"d":"Upper bound on the tail integral. Under the polynomial lower-tail hypothesis PolyTail P U κ t₀ cm cp, for any threshold λ that is positive and at most the window endpoint t₀, the layer-cake tail integral tailIntegral P U λ is bounded by a constant pre-window contribution t₀⁻¹ − 1 plus cp times the power comparison integral over (t₀⁻¹, λ⁻¹]:","s":"∀ {Ω : Type u_1} [inst : MeasurableSpace Ω] {P : MeasureTheory.Measure Ω} {U : Ω → ℝ} {κ t₀ cm cp lam : ℝ} [MeasureTheory.IsProbabilityMeasure P], Causalean.Stat.PolynomialTail.PolyTail P U κ t₀ cm cp → 0 < lam → lam ≤ t₀ → Causalean.Stat.PolynomialTail.tailIntegral P U lam ≤ t₀⁻¹ - 1 + cp * ∫ (t : ℝ) in Set.Ioc t₀⁻¹ lam⁻¹, t ^ (-κ)","l":83,"f":"Causalean/Stat/PolynomialTail/TailIntegralBounds.lean"},{"n":"Causalean.Stat.PolynomialTail.tailIntegral_nonneg","k":"theorem","m":"Causalean.Stat.PolynomialTail.LayerCakeReduction","a":"Stat/PolynomialTail","t":2,"d":"tailIntegral is nonnegative.","s":"∀ {Ω : Type u_1} [inst : MeasurableSpace Ω] {P : MeasureTheory.Measure Ω} {U : Ω → ℝ} {lam : ℝ} [MeasureTheory.IsProbabilityMeasure P], 0 ≤ Causalean.Stat.PolynomialTail.tailIntegral P U lam","l":145,"f":"Causalean/Stat/PolynomialTail/LayerCakeReduction.lean"},{"n":"Causalean.Stat.PolynomialTail.tailIntegrand_le_one","k":"theorem","m":"Causalean.Stat.PolynomialTail.LayerCakeReduction","a":"Stat/PolynomialTail","t":2,"d":"The integrand is bounded by 1.","s":"∀ {Ω : Type u_1} [inst : MeasurableSpace Ω] {P : MeasureTheory.Measure Ω} {U : Ω → ℝ} [MeasureTheory.IsProbabilityMeasure P] (t : ℝ), P.real {ω | U ω ≤ t⁻¹} ≤ 1","l":68,"f":"Causalean/Stat/PolynomialTail/LayerCakeReduction.lean"},{"n":"Causalean.Stat.PolynomialTail.tailIntegrand_lower_window","k":"theorem","m":"Causalean.Stat.PolynomialTail.TailIntegralBounds","a":"Stat/PolynomialTail","t":2,"d":"Lower sandwich of the tail integrand on the polynomial window.","s":"∀ {Ω : Type u_1} [inst : MeasurableSpace Ω] {P : MeasureTheory.Measure Ω} {U : Ω → ℝ} {κ t₀ cm cp lam : ℝ} [MeasureTheory.IsFiniteMeasure P], Causalean.Stat.PolynomialTail.PolyTail P U κ t₀ cm cp → ∀ {t : ℝ}, t ∈ Set.Ioc t₀⁻¹ lam⁻¹ → cm * t ^ (-κ) ≤ P.real {ω | U ω ≤ t⁻¹}","l":61,"f":"Causalean/Stat/PolynomialTail/TailIntegralBounds.lean"},{"n":"Causalean.Stat.PolynomialTail.tailIntegrand_nonneg","k":"theorem","m":"Causalean.Stat.PolynomialTail.LayerCakeReduction","a":"Stat/PolynomialTail","t":2,"d":"The integrand is nonnegative.","s":"∀ {Ω : Type u_1} [inst : MeasurableSpace Ω] {P : MeasureTheory.Measure Ω} {U : Ω → ℝ} (t : ℝ), 0 ≤ P.real {ω | U ω ≤ t⁻¹}","l":64,"f":"Causalean/Stat/PolynomialTail/LayerCakeReduction.lean"},{"n":"Causalean.Stat.PolynomialTail.tailIntegrand_upper_window","k":"theorem","m":"Causalean.Stat.PolynomialTail.TailIntegralBounds","a":"Stat/PolynomialTail","t":2,"d":"Upper sandwich of the tail integrand on the polynomial window.","s":"∀ {Ω : Type u_1} [inst : MeasurableSpace Ω] {P : MeasureTheory.Measure Ω} {U : Ω → ℝ} {κ t₀ cm cp lam : ℝ} [MeasureTheory.IsFiniteMeasure P], Causalean.Stat.PolynomialTail.PolyTail P U κ t₀ cm cp → ∀ {t : ℝ}, t ∈ Set.Ioc t₀⁻¹ lam⁻¹ → P.real {ω | U ω ≤ t⁻¹} ≤ cp * t ^ (-κ)","l":52,"f":"Causalean/Stat/PolynomialTail/TailIntegralBounds.lean"},{"n":"Causalean.Stat.PolynomialTail.trimWeight","k":"def","m":"Causalean.Stat.PolynomialTail.Basic","a":"Stat/PolynomialTail","t":1,"d":"The leftover trimming weight wλ ω = 1 − U/(max U λ) ∈ [0,1], supported on {U < λ}.","s":"{Ω : Type u_1} → (Ω → ℝ) → ℝ → Ω → ℝ","l":121,"f":"Causalean/Stat/PolynomialTail/Basic.lean"},{"n":"Causalean.Stat.PolynomialTail.trimWeight_bias_bound","k":"theorem","m":"Causalean.Stat.PolynomialTail.TrimBias","a":"Stat/PolynomialTail","t":1,"d":"General trimming-bias bound. Under the polynomial lower-tail hypothesis PolyTail P U κ t₀ cm cp with U measurable and almost surely in (0,1], fix a threshold λ that is positive and at most the window endpoint t₀. If the envelope constant Cβ is nonnegative, g is measurable, and g obeys the Hölder envelope |g| ≤ Cβ · λ^β on the event {U < λ}, then the integrated bias |∫ wλ · g dP| is at most Cβ · cp · λ^{κ+β}:","s":"∀ {Ω : Type u_1} [inst : MeasurableSpace Ω] {P : MeasureTheory.Measure Ω} {U : Ω → ℝ} {κ t₀ cm cp lam β : ℝ} [MeasureTheory.IsProbabilityMeasure P], Causalean.Stat.PolynomialTail.PolyTail P U κ t₀ cm cp → Causalean.Stat.PolynomialTail.TailSetup P U → 0 < lam → lam ≤ t₀ → ∀ {g : Ω → ℝ} {Cβ : ℝ}, 0 ≤ Cβ → Measurable g → (∀ (ω : Ω), U ω < lam → |g ω| ≤ Cβ * lam ^ β) → |∫ (ω : Ω), Causalean.Stat.PolynomialTail.trimWeight U lam ω * g ω ∂P| ≤ Cβ * cp * lam ^ (κ + β)","l":43,"f":"Causalean/Stat/PolynomialTail/TrimBias.lean"},{"n":"Causalean.Stat.PolynomialTail.trimWeight_mem","k":"theorem","m":"Causalean.Stat.PolynomialTail.Basic","a":"Stat/PolynomialTail","t":2,"d":"The leftover weight lies in [0,1] whenever U > 0.","s":"∀ {lam : ℝ}, 0 < lam → ∀ {x : ℝ}, 0 < x → 0 ≤ 1 - x / max x lam ∧ 1 - x / max x lam ≤ 1","l":195,"f":"Causalean/Stat/PolynomialTail/Basic.lean"},{"n":"Causalean.Stat.PolynomialTail.truncDen","k":"def","m":"Causalean.Stat.PolynomialTail.Basic","a":"Stat/PolynomialTail","t":1,"d":"The truncated denominator max U λ shared by the inverse-moment integrands.","s":"{Ω : Type u_1} → (Ω → ℝ) → ℝ → Ω → ℝ","l":108,"f":"Causalean/Stat/PolynomialTail/Basic.lean"},{"n":"Causalean.Stat.PolynomialTail.truncDen_pos","k":"theorem","m":"Causalean.Stat.PolynomialTail.Basic","a":"Stat/PolynomialTail","t":2,"d":"For 0 < λ, the denominator is positive.","s":"∀ {Ω : Type u_1} {U : Ω → ℝ} {lam : ℝ}, 0 < lam → ∀ (ω : Ω), 0 < Causalean.Stat.PolynomialTail.truncDen U lam ω","l":156,"f":"Causalean/Stat/PolynomialTail/Basic.lean"},{"n":"Causalean.Stat.PolynomialTail.zero_notMem_uIcc","k":"theorem","m":"Causalean.Stat.PolynomialTail.PowerIntegral","a":"Stat/PolynomialTail","t":2,"d":"0 ∉ [[a,b]] when 0 < a ≤ b.","s":"∀ {a b : ℝ}, 0 < a → a ≤ b → 0 ∉ Set.uIcc a b","l":39,"f":"Causalean/Stat/PolynomialTail/PowerIntegral.lean"},{"n":"Causalean.Stat.Privacy.approx_dp_postprocess","k":"theorem","m":"Causalean.Stat.Privacy.LaplaceMechanism","a":"Stat/Privacy","t":1,"d":"Post-processing preserves approximate differential privacy. Given a vector-valued mechanism M obeying an $(\\varepsilon,\\delta)$-approximate DP bound between the releases M d and M d' for every measurable event, with a nonnegative failure allowance $\\delta$, post-processing the release by any measurable scalar summary f again satisfies the same $(\\varepsilon,\\delta)$-approximate DP bound, now between the f-summaries of M d and M d'.","s":"∀ {D : Type u_1} {ι : Type u_2} (M : D → MeasureTheory.Measure (ι → ℝ)) (d d' : D) (ε δ : ℝ), (∀ (s : Set (ι → ℝ)), MeasurableSet s → (M d).real s ≤ Real.exp ε * (M d').real s + δ) → 0 ≤ δ → ∀ (f : (ι → ℝ) → ℝ), Measurable f → ∀ (s : Set ℝ), MeasurableSet s → (MeasureTheory.Measure.map f (M d)).real s ≤ Real.exp ε * (MeasureTheory.Measure.map f (M d')).real s + δ","l":469,"f":"Causalean/Stat/Privacy/LaplaceMechanism.lean"},{"n":"Causalean.Stat.Privacy.laplaceMeasure","k":"def","m":"Causalean.Stat.Privacy.LaplaceMechanism","a":"Stat/Privacy","t":1,"d":"The centered Laplace distribution is obtained by weighting Lebesgue measure with the centered Laplace density at the chosen scale.","s":"ℝ → MeasureTheory.Measure ℝ","l":33,"f":"Causalean/Stat/Privacy/LaplaceMechanism.lean"},{"n":"Causalean.Stat.Privacy.laplaceMeasure_integrable_abs","k":"theorem","m":"Causalean.Stat.Privacy.LaplaceMechanism","a":"Stat/Privacy","t":2,"d":"The absolute value of a centered Laplace draw has a finite expectation at every positive scale.","s":"∀ (b : ℝ), 0 < b → MeasureTheory.Integrable (fun x => |x|) (Causalean.Stat.Privacy.laplaceMeasure b)","l":528,"f":"Causalean/Stat/Privacy/LaplaceMechanism.lean"},{"n":"Causalean.Stat.Privacy.laplaceMeasure_integral_abs","k":"theorem","m":"Causalean.Stat.Privacy.LaplaceMechanism","a":"Stat/Privacy","t":2,"d":"The expected absolute value of a centered Laplace draw equals its positive scale.","s":"∀ (b : ℝ), 0 < b → ∫ (x : ℝ), |x| ∂Causalean.Stat.Privacy.laplaceMeasure b = b","l":538,"f":"Causalean/Stat/Privacy/LaplaceMechanism.lean"},{"n":"Causalean.Stat.Privacy.laplaceMeasure_isProbabilityMeasure","k":"theorem","m":"Causalean.Stat.Privacy.LaplaceMechanism","a":"Stat/Privacy","t":2,"d":"At every positive scale, the centered Laplace measure has total mass one and therefore defines a probability distribution.","s":"∀ (b : ℝ), 0 < b → MeasureTheory.IsProbabilityMeasure (Causalean.Stat.Privacy.laplaceMeasure b)","l":105,"f":"Causalean/Stat/Privacy/LaplaceMechanism.lean"},{"n":"Causalean.Stat.Privacy.laplaceMech","k":"def","m":"Causalean.Stat.Privacy.LaplaceMechanism","a":"Stat/Privacy","t":1,"d":"The scalar Laplace mechanism releases a real-valued query after adding independent centered Laplace noise at the chosen scale.","s":"{D : Type u_1} → ℝ → (D → ℝ) → D → MeasureTheory.Measure ℝ","l":149,"f":"Causalean/Stat/Privacy/LaplaceMechanism.lean"},{"n":"Causalean.Stat.Privacy.laplaceMechPi","k":"def","m":"Causalean.Stat.Privacy.LaplaceMechanism","a":"Stat/Privacy","t":1,"d":"The finite-dimensional Laplace mechanism releases a vector-valued query after adding mutually independent centered Laplace noise to its coordinates.","s":"{D : Type u_1} → {ι : Type u_2} → [Fintype ι] → ℝ → (D → ι → ℝ) → D → MeasureTheory.Measure (ι → ℝ)","l":247,"f":"Causalean/Stat/Privacy/LaplaceMechanism.lean"},{"n":"Causalean.Stat.Privacy.laplaceMechPi_isProbabilityMeasure","k":"theorem","m":"Causalean.Stat.Privacy.LaplaceMechanism","a":"Stat/Privacy","t":2,"d":"Adding independent positive-scale Laplace noise in finitely many coordinates produces a probability distribution for the vector release under every dataset.","s":"∀ {D : Type u_1} {ι : Type u_2} [inst : Fintype ι] (b : ℝ), 0 < b → ∀ (q : D → ι → ℝ) (d : D), MeasureTheory.IsProbabilityMeasure (Causalean.Stat.Privacy.laplaceMechPi b q d)","l":253,"f":"Causalean/Stat/Privacy/LaplaceMechanism.lean"},{"n":"Causalean.Stat.Privacy.laplaceMechPi_pure_dp","k":"theorem","m":"Causalean.Stat.Privacy.LaplaceMechanism","a":"Stat/Privacy","t":1,"d":"Vector Laplace mechanism is purely differentially private. Given a positive $\\ell^1$-sensitivity bound $\\Delta$ and a positive privacy level $\\varepsilon$ such that the coordinatewise absolute differences of the query q sum to at most $\\Delta$ between any pair of adjacent datasets, releasing q after adding independent Laplace noise of scale $\\Delta/\\varepsilon$ to each coordinate satisfies pure $\\varepsilon$-differential privacy: for every adjacent pair, the probability of any measurable event under one release is at most $e^\\varepsilon$ times its probability under the other.","s":"∀ {D : Type u_1} {ι : Type u_2} [inst : Fintype ι] (Adj : D → D → Prop) (q : D → ι → ℝ) {Δ ε : ℝ}, 0 < Δ → 0 < ε → (∀ (d d' : D), Adj d d' → ∑ i, |q d i - q d' i| ≤ Δ) → ∀ (d d' : D), Adj d d' → ∀ (s : Set (ι → ℝ)), MeasurableSet s → (Causalean.Stat.Privacy.laplaceMechPi (Δ / ε) q d).real s ≤ Real.exp ε * (Causalean.Stat.Privacy.laplaceMechPi (Δ / ε) q d').real s","l":373,"f":"Causalean/Stat/Privacy/LaplaceMechanism.lean"},{"n":"Causalean.Stat.Privacy.laplaceMech_isProbabilityMeasure","k":"theorem","m":"Causalean.Stat.Privacy.LaplaceMechanism","a":"Stat/Privacy","t":2,"d":"Adding positive-scale Laplace noise to a scalar query produces a probability distribution for the release under every dataset.","s":"∀ {D : Type u_1} (b : ℝ), 0 < b → ∀ (q : D → ℝ) (d : D), MeasureTheory.IsProbabilityMeasure (Causalean.Stat.Privacy.laplaceMech b q d)","l":180,"f":"Causalean/Stat/Privacy/LaplaceMechanism.lean"},{"n":"Causalean.Stat.Privacy.laplaceMech_pure_dp","k":"theorem","m":"Causalean.Stat.Privacy.LaplaceMechanism","a":"Stat/Privacy","t":1,"d":"Scalar Laplace mechanism is purely differentially private. Given a positive sensitivity bound $\\Delta$ and a positive privacy level $\\varepsilon$ such that the query q changes by at most $\\Delta$ between any pair of adjacent datasets, releasing q after adding independent Laplace noise of scale $\\Delta/\\varepsilon$ satisfies pure $\\varepsilon$-differential privacy: for every adjacent pair, the probability of any measurable event under one release is at most $e^\\varepsilon$ times its probability under the other.","s":"∀ {D : Type u_1} (Adj : D → D → Prop) (q : D → ℝ) {Δ ε : ℝ}, 0 < Δ → 0 < ε → (∀ (d d' : D), Adj d d' → |q d - q d'| ≤ Δ) → ∀ (d d' : D), Adj d d' → ∀ (s : Set ℝ), MeasurableSet s → (Causalean.Stat.Privacy.laplaceMech (Δ / ε) q d).real s ≤ Real.exp ε * (Causalean.Stat.Privacy.laplaceMech (Δ / ε) q d').real s","l":189,"f":"Causalean/Stat/Privacy/LaplaceMechanism.lean"},{"n":"Causalean.Stat.Privacy.laplacePDF","k":"def","m":"Causalean.Stat.Privacy.LaplaceMechanism","a":"Stat/Privacy","t":1,"d":"The centered Laplace density at a real value decays exponentially with its absolute distance from zero and is normalized by twice its positive scale.","s":"ℝ → ℝ → ℝ","l":29,"f":"Causalean/Stat/Privacy/LaplaceMechanism.lean"},{"n":"Causalean.Stat.Privacy.laplacePDF_shift_le","k":"theorem","m":"Causalean.Stat.Privacy.LaplaceMechanism","a":"Stat/Privacy","t":2,"d":"Moving the center of a positive-scale Laplace density changes its value at any observation by at most an exponential factor determined by the distance between the old and new centers divided by the scale.","s":"∀ (b u v z : ℝ), 0 < b → Causalean.Stat.Privacy.laplacePDF b (z - u) ≤ Real.exp (|u - v| / b) * Causalean.Stat.Privacy.laplacePDF b (z - v)","l":124,"f":"Causalean/Stat/Privacy/LaplaceMechanism.lean"},{"n":"Causalean.Stat.Privacy.laplacePi_integrable_euclidean_norm","k":"theorem","m":"Causalean.Stat.Privacy.LaplaceMechanism","a":"Stat/Privacy","t":2,"d":"The Euclidean norm of finitely many independent centered Laplace draws has a finite expectation whenever their common scale is positive.","s":"∀ {ι : Type u_1} [inst : Fintype ι] (b : ℝ), 0 < b → MeasureTheory.Integrable (fun w => √(∑ i, w i ^ 2)) (MeasureTheory.Measure.pi fun x => Causalean.Stat.Privacy.laplaceMeasure b)","l":616,"f":"Causalean/Stat/Privacy/LaplaceMechanism.lean"},{"n":"Causalean.Stat.Privacy.laplacePi_integral_euclidean_norm_le","k":"theorem","m":"Causalean.Stat.Privacy.LaplaceMechanism","a":"Stat/Privacy","t":2,"d":"The expected Euclidean norm of a finite vector of independent centered Laplace draws is at most the number of coordinates times their common positive scale.","s":"∀ {ι : Type u_1} [inst : Fintype ι] (b : ℝ), 0 < b → (∫ (w : ι → ℝ), √(∑ i, w i ^ 2) ∂MeasureTheory.Measure.pi fun x => Causalean.Stat.Privacy.laplaceMeasure b) ≤ ↑(Fintype.card ι) * b","l":579,"f":"Causalean/Stat/Privacy/LaplaceMechanism.lean"},{"n":"Causalean.Stat.Privacy.pure_dp_implies_approx_dp","k":"theorem","m":"Causalean.Stat.Privacy.LaplaceMechanism","a":"Stat/Privacy","t":1,"d":"Pure DP implies approximate DP. Given a mechanism M satisfying pure $\\varepsilon$-differential privacy between releases M d and M d', i.e. the probability of every measurable event under one release is at most $e^\\varepsilon$ times its probability under the other, adding any nonnegative failure allowance $\\delta$ to the bound still yields a valid $(\\varepsilon,\\delta)$-approximate differential-privacy guarantee between M d and M d'.","s":"∀ {D : Sort u_2} {α : Type u_1} [inst : MeasurableSpace α] (M : D → MeasureTheory.Measure α) (d d' : D) (ε δ : ℝ), (∀ (s : Set α), MeasurableSet s → (M d).real s ≤ Real.exp ε * (M d').real s) → 0 ≤ δ → ∀ (s : Set α), MeasurableSet s → (M d).real s ≤ Real.exp ε * (M d').real s + δ","l":437,"f":"Causalean/Stat/Privacy/LaplaceMechanism.lean"},{"n":"Causalean.Stat.Privacy.pure_dp_postprocess","k":"theorem","m":"Causalean.Stat.Privacy.LaplaceMechanism","a":"Stat/Privacy","t":1,"d":"Post-processing preserves pure differential privacy. Given a vector-valued mechanism M satisfying pure $\\varepsilon$-differential privacy between the releases M d and M d', post-processing the release by any measurable scalar summary f again satisfies pure $\\varepsilon$-differential privacy, now between the f-summaries of M d and M d'.","s":"∀ {D : Type u_1} {ι : Type u_2} (M : D → MeasureTheory.Measure (ι → ℝ)) (d d' : D) (ε : ℝ), (∀ (s : Set (ι → ℝ)), MeasurableSet s → (M d).real s ≤ Real.exp ε * (M d').real s) → ∀ (f : (ι → ℝ) → ℝ), Measurable f → ∀ (s : Set ℝ), MeasurableSet s → (MeasureTheory.Measure.map f (M d)).real s ≤ Real.exp ε * (MeasureTheory.Measure.map f (M d')).real s","l":452,"f":"Causalean/Stat/Privacy/LaplaceMechanism.lean"},{"n":"Causalean.Stat.QuantileRegularity","k":"structure","m":"Causalean.Stat.Quantile.SampleQuantile","a":"Stat/Quantile","t":1,"d":"Quantile-estimator regularity. Bundles the analytic and empirical-process hypotheses under which a quantile-estimator sequence $q̂_n$ is $\\sqrt n$-asymptotically linear for the $\\tau$-quantile $q_0$ of $P$ with density $f_0$: the level lies in the open unit interval $\\tau \\in (0,1)$, the density at the quantile is positive, $q_0$ is indeed the population $\\tau$-quantile, i.e. the cdf satisfies $F(q_0) = \\tau$, the cdf is differentiable at $q_0$ with derivative $f_0$, so that $\\tau(1-\\tau)/f_0^2$ is the genuine asymptotic variance, and the rescaled estimator matches the normalized influence-function sum up to a term vanishing in probability — the Bahadur remainder.","s":"{Ω : Type u_1} → [inst : MeasurableSpace Ω] → {μ : MeasureTheory.Measure Ω} → {P : MeasureTheory.Measure ℝ} → Causalean.Stat.IIDSample Ω ℝ μ P → (ℕ → Ω → ℝ) → ℝ → ℝ → ℝ → Prop","l":137,"f":"Causalean/Stat/Quantile/SampleQuantile.lean"},{"n":"Causalean.Stat.QuantileRegularity.isAsymLinear","k":"theorem","m":"Causalean.Stat.Quantile.SampleQuantile","a":"Stat/Quantile","t":1,"d":"Regularity implies asymptotic linearity. Given a QuantileRegularity witness h for the estimator sequence qn, the estimator is asymptotically linear at the quantile q₀ with influence function ψ_τ.","s":"∀ {Ω : Type u_1} [inst : MeasurableSpace Ω] {μ : MeasureTheory.Measure Ω} {P : MeasureTheory.Measure ℝ} [MeasureTheory.IsProbabilityMeasure P] {S : Causalean.Stat.IIDSample Ω ℝ μ P} {qn : ℕ → Ω → ℝ} {τ q₀ f₀ : ℝ}, Causalean.Stat.QuantileRegularity S qn τ q₀ f₀ → Causalean.Stat.IsAsymLinear qn q₀ (Causalean.Stat.quantileIF τ q₀ f₀) S fun m => Finset.range m","l":167,"f":"Causalean/Stat/Quantile/SampleQuantile.lean"},{"n":"Causalean.Stat.QuantileRegularity.tendsto_normal","k":"theorem","m":"Causalean.Stat.Quantile.SampleQuantile","a":"Stat/Quantile","t":1,"d":"Sample-quantile asymptotic normality. Given a QuantileRegularity witness h for the estimator sequence qn — interior level $\\tau\\in(0,1)$, positive density $f_0$ at the population quantile $q_0$, cdf identification, and the exposed Bahadur/Donsker remainder — provided the rescaled estimator is almost-everywhere measurable at each sample size and the normalized influence-function sum is almost-everywhere measurable at each sample size, then $\\sqrt n\\,(\\hat q_n-q_0)$ converges in distribution to the centered Gaussian law with variance $\\tau(1-\\tau)/f_0^2$.","s":"∀ {Ω : Type u_1} [inst : MeasurableSpace Ω] {μ : MeasureTheory.Measure Ω} {P : MeasureTheory.Measure ℝ} [inst_1 : MeasureTheory.IsProbabilityMeasure μ] [MeasureTheory.IsProbabilityMeasure P] {S : Causalean.Stat.IIDSample Ω ℝ μ P} {qn : ℕ → Ω → ℝ} {τ q₀ f₀ : ℝ}, Causalean.Stat.QuantileRegularity S qn τ q₀ f₀ → ∀ (hθn_meas : ∀ (n : ℕ), AEMeasurable (Causalean.Stat.IsAsymLinear.rescaledEstimator qn q₀ (fun m => Finset.range m) n) μ), (∀ (n : ℕ), AEMeasurable (Causalean.Stat.IsAsymLinear.normalizedSum S (Causalean.Stat.quantileIF τ q₀ f₀) (fun m => Finset.range m) n) μ) → Causalean.Stat.Tendsto_dist (Causalean.Stat.IsAsymLinear.rescaledEstimator qn q₀ fun m => Finset.range m) (Causalean.Stat.gaussianMeasure 0 (τ * (1 - τ) / f₀ ^ 2)) μ hθn_meas","l":177,"f":"Causalean/Stat/Quantile/SampleQuantile.lean"},{"n":"Causalean.Stat.SampleQuantileReg","k":"structure","m":"Causalean.Stat.Quantile.SampleQuantileBahadur.Oscillation","a":"Stat/Quantile/SampleQuantileBahadur","t":1,"d":"Regularity for sample-quantile asymptotics. Bundles the hypotheses on the population cdf $F$ of $P$ under which the empirical sample quantile admits a Bahadur representation at level $\\tau$: the level lies in the open unit interval $\\tau \\in (0,1)$, the density $f_0$ at the quantile is positive, $q_0$ is the population $\\tau$-quantile, i.e. $F(q_0) = \\tau$, $F$ is differentiable at $q_0$ with derivative $f_0$, and $F$ is continuous, so the population has no atoms.","s":"MeasureTheory.Measure ℝ → ℝ → ℝ → ℝ → Prop","l":70,"f":"Causalean/Stat/Quantile/SampleQuantileBahadur/Oscillation.lean"},{"n":"Causalean.Stat.StochEquicontAt","k":"def","m":"Causalean.Stat.EmpiricalProcess.Equicontinuity.StochEquicont","a":"Stat/EmpiricalProcess/Equicontinuity","t":1,"d":"Asymptotic equicontinuity of the score family at θ₀ along the sequence θn.","s":"{X : Type u_1} → {E : Type u_2} → [inst : MeasurableSpace X] → [inst_1 : NormedAddCommGroup E] → [NormedSpace ℝ E] → (E → X → E) → E → (P : MeasureTheory.Measure X) → {Ω : Type u_3} → [inst_3 : MeasurableSpace Ω] → (μ : MeasureTheory.Measure Ω) → Causalean.Stat.IIDSample Ω X μ P → (ℕ → Ω → E) → Prop","l":37,"f":"Causalean/Stat/EmpiricalProcess/Equicontinuity/StochEquicont.lean"},{"n":"Causalean.Stat.Tendsto_L2","k":"def","m":"Causalean.Stat.Limit.Convergence","a":"Stat/Limit","t":1,"d":"Tendsto_L2 Xn X μ is convergence in L²(μ) of Xn to X.","s":"{Ω : Type u_1} → [inst : MeasurableSpace Ω] → (ℕ → Ω → ℝ) → (Ω → ℝ) → MeasureTheory.Measure Ω → Prop","l":60,"f":"Causalean/Stat/Limit/Convergence.lean"},{"n":"Causalean.Stat.Tendsto_dist","k":"def","m":"Causalean.Stat.Limit.Convergence","a":"Stat/Limit","t":1,"d":"Convergence in distribution means that the laws of the random variables converge weakly to a specified probability law on the real line.","s":"{Ω : Type u_1} → [inst : MeasurableSpace Ω] → (Xn : ℕ → Ω → ℝ) → (Q : MeasureTheory.Measure ℝ) → (μ : MeasureTheory.Measure Ω) → [MeasureTheory.IsProbabilityMeasure μ] → [MeasureTheory.IsProbabilityMeasure Q] → (∀ (n : ℕ), AEMeasurable (Xn n) μ) → Prop","l":71,"f":"Causalean/Stat/Limit/Convergence.lean"},{"n":"Causalean.Stat.Tendsto_dist.add_isLittleOp_one","k":"theorem","m":"Causalean.Stat.CLT.AsymptoticLinearity","a":"Stat/CLT","t":2,"d":"If Xn ⇒ Q in distribution and Yn − Xn = o_p(1), then Yn ⇒ Q.","s":"∀ {Ω : Type u_1} [inst : MeasurableSpace Ω] {μ : MeasureTheory.Measure Ω} [inst_1 : MeasureTheory.IsProbabilityMeasure μ] {Xn Yn : ℕ → Ω → ℝ} {Q : MeasureTheory.Measure ℝ} [inst_2 : MeasureTheory.IsProbabilityMeasure Q] (hXn : ∀ (n : ℕ), AEMeasurable (Xn n) μ) (hYn : ∀ (n : ℕ), AEMeasurable (Yn n) μ), Causalean.Stat.Tendsto_dist Xn Q μ hXn → Causalean.Stat.IsLittleOp (fun n ω => Yn n ω - Xn n ω) (fun x => 1) μ → Causalean.Stat.Tendsto_dist Yn Q μ hYn","l":221,"f":"Causalean/Stat/CLT/AsymptoticLinearity.lean"},{"n":"Causalean.Stat.Tendsto_dist.congr_ae","k":"theorem","m":"Causalean.Stat.CLT.AsymptoticLinearity","a":"Stat/CLT","t":2,"d":"Convergence in distribution is invariant under eventual a.e. equality of the random variables.","s":"∀ {Ω : Type u_1} [inst : MeasurableSpace Ω] {μ : MeasureTheory.Measure Ω} [inst_1 : MeasureTheory.IsProbabilityMeasure μ] {Xn Yn : ℕ → Ω → ℝ} {Q : MeasureTheory.Measure ℝ} [inst_2 : MeasureTheory.IsProbabilityMeasure Q] (hXn : ∀ (n : ℕ), AEMeasurable (Xn n) μ) (hYn : ∀ (n : ℕ), AEMeasurable (Yn n) μ), Causalean.Stat.Tendsto_dist Xn Q μ hXn → (∀ᶠ (n : ℕ) in Filter.atTop, Xn n =ᵐ[μ] Yn n) → Causalean.Stat.Tendsto_dist Yn Q μ hYn","l":329,"f":"Causalean/Stat/CLT/AsymptoticLinearity.lean"},{"n":"Causalean.Stat.Tendsto_dist.const_mul_tendsto","k":"theorem","m":"Causalean.Stat.Limit.Convergence","a":"Stat/Limit","t":2,"d":"Deterministic-scalar Slutsky for the project's measure-level Tendsto_dist wrapper.","s":"∀ {Ω : Type u_2} [inst : MeasurableSpace Ω] {μ : MeasureTheory.Measure Ω} [inst_1 : MeasureTheory.IsProbabilityMeasure μ] {Xn : ℕ → Ω → ℝ} {Q : MeasureTheory.Measure ℝ} [inst_2 : MeasureTheory.IsProbabilityMeasure Q] {a : ℕ → ℝ} {a₀ : ℝ} (hXn : ∀ (n : ℕ), AEMeasurable (Xn n) μ), Causalean.Stat.Tendsto_dist Xn Q μ hXn → Filter.Tendsto a Filter.atTop (nhds a₀) → Filter.Tendsto (fun n => ⟨MeasureTheory.Measure.map (fun ω => a n * Xn n ω) μ, MeasureTheory.Measure.isProbabilityMeasure_map (AEMeasurable.comp_aemeasurable (Measurable.aemeasurable (Measurable.mul measurable_const measurable_id)) (hXn n))⟩) Filter.atTop (nhds ⟨MeasureTheory.Measure.map (fun x => a₀ * x) Q, MeasureTheory.Measure.isProbabilityMeasure_map (Measurable.aemeasurable (Measurable.mul measurable_const measurable_id))⟩)","l":84,"f":"Causalean/Stat/Limit/Convergence.lean"},{"n":"Causalean.Stat.Tendsto_dist.const_mul_tendsto_gaussian","k":"theorem","m":"Causalean.Stat.CLT.AsymptoticLinearity","a":"Stat/CLT","t":2,"d":"Deterministic-scalar Slutsky for Gaussian limits, phrased for the project's measure-level Tendsto_dist wrapper.","s":"∀ {Ω : Type u_1} [inst : MeasurableSpace Ω] {μ : MeasureTheory.Measure Ω} [inst_1 : MeasureTheory.IsProbabilityMeasure μ] {Xn : ℕ → Ω → ℝ} {a : ℕ → ℝ} {a₀ v : ℝ} (hXn : ∀ (n : ℕ), AEMeasurable (Xn n) μ), Causalean.Stat.Tendsto_dist Xn (Causalean.Stat.gaussianMeasure 0 v) μ hXn → Filter.Tendsto a Filter.atTop (nhds a₀) → Causalean.Stat.Tendsto_dist (fun n ω => a n * Xn n ω) (Causalean.Stat.gaussianMeasure 0 (a₀ ^ 2 * v)) μ fun n => AEMeasurable.comp_aemeasurable (Measurable.aemeasurable (Measurable.mul measurable_const measurable_id)) (hXn n)","l":345,"f":"Causalean/Stat/CLT/AsymptoticLinearity.lean"},{"n":"Causalean.Stat.Tendsto_dist.div_tendsto_inProb_gaussian","k":"theorem","m":"Causalean.Stat.Inference.Studentize","a":"Stat/Inference","t":1,"d":"Generic studentized convergence. Let Xn be a real-valued rescaled-estimator sequence and σ_hat a variance-estimator sequence, with σ₀ a positive scale. Suppose Xn is measurable at every sample size and it converges in distribution to the project's Gaussian law with mean zero and variance σ₀², that σ_hat converges in probability to σ₀, and the studentized ratio Xn / σ_hat is measurable at every sample size. Then the studentized ratio Xn / σ_hat converges in distribution to the standard Gaussian law.","s":"∀ {Ω : Type u_1} [inst : MeasurableSpace Ω] {μ : MeasureTheory.Measure Ω} [inst_1 : MeasureTheory.IsProbabilityMeasure μ] {Xn σ_hat : ℕ → Ω → ℝ} {σ₀ : ℝ}, 0 < σ₀ → ∀ (hXn : ∀ (n : ℕ), AEMeasurable (Xn n) μ), Causalean.Stat.Tendsto_dist Xn (Causalean.Stat.gaussianMeasure 0 (σ₀ ^ 2)) μ hXn → Causalean.Stat.Tendsto_inProb σ_hat (fun x => σ₀) μ → ∀ (hdiv : ∀ (n : ℕ), AEMeasurable (fun ω => Xn n ω / σ_hat n ω) μ), Causalean.Stat.Tendsto_dist (fun n ω => Xn n ω / σ_hat n ω) (Causalean.Stat.gaussianMeasure 0 1) μ hdiv","l":101,"f":"Causalean/Stat/Inference/Studentize.lean"},{"n":"Causalean.Stat.Tendsto_dist.limsup_measure_closed_le","k":"theorem","m":"Causalean.Stat.Quantile.SampleQuantileBahadur.Rate","a":"Stat/Quantile/SampleQuantileBahadur","t":2,"d":"Portmanteau (closed-set limsup). If Xn ⇒ Q in distribution, then for every closed F, limsup μ{ω | Xn n ω ∈ F} ≤ Q F. Mirrors the closed-set half of Tendsto_dist.tightness.","s":"∀ {Ω : Type u_2} [inst : MeasurableSpace Ω] {μ : MeasureTheory.Measure Ω} [inst_1 : MeasureTheory.IsProbabilityMeasure μ] {Xn : ℕ → Ω → ℝ} {Q : MeasureTheory.Measure ℝ} [inst_2 : MeasureTheory.IsProbabilityMeasure Q] (hXn : ∀ (n : ℕ), AEMeasurable (Xn n) μ), Causalean.Stat.Tendsto_dist Xn Q μ hXn → ∀ {F : Set ℝ}, IsClosed F → Filter.limsup (fun n => μ {ω | Xn n ω ∈ F}) Filter.atTop ≤ Q F","l":102,"f":"Causalean/Stat/Quantile/SampleQuantileBahadur/Rate.lean"},{"n":"Causalean.Stat.Tendsto_dist.tendsto_measure_of_null_frontier","k":"theorem","m":"Causalean.Stat.Inference.Studentize","a":"Stat/Inference","t":2,"d":"Continuity-set form of portmanteau for the project's Tendsto_dist wrapper.","s":"∀ {Ω : Type u_1} [inst : MeasurableSpace Ω] {μ : MeasureTheory.Measure Ω} [inst_1 : MeasureTheory.IsProbabilityMeasure μ] {Xn : ℕ → Ω → ℝ} {Q : MeasureTheory.Measure ℝ} [inst_2 : MeasureTheory.IsProbabilityMeasure Q] (hXn : ∀ (n : ℕ), AEMeasurable (Xn n) μ), Causalean.Stat.Tendsto_dist Xn Q μ hXn → ∀ {E : Set ℝ}, Q (frontier E) = 0 → Filter.Tendsto (fun n => ((MeasureTheory.Measure.map (Xn n) μ) E).toReal) Filter.atTop (nhds (Q E).toReal)","l":53,"f":"Causalean/Stat/Inference/Studentize.lean"},{"n":"Causalean.Stat.Tendsto_dist.tightness","k":"theorem","m":"Causalean.Stat.Limit.Convergence","a":"Stat/Limit","t":1,"d":"Tightness from convergence in distribution. Suppose a real-valued sequence Xn is measurable at every sample size and it converges in distribution under μ to a probability measure Q on ℝ. Then Xn is bounded in probability, O_p(1). Standard fact: any single tight limit gives a tight sequence (Prokhorov).","s":"∀ {Ω : Type u_2} [inst : MeasurableSpace Ω] {μ : MeasureTheory.Measure Ω} [inst_1 : MeasureTheory.IsProbabilityMeasure μ] {Xn : ℕ → Ω → ℝ} {Q : MeasureTheory.Measure ℝ} [inst_2 : MeasureTheory.IsProbabilityMeasure Q] (hXn : ∀ (n : ℕ), AEMeasurable (Xn n) μ), Causalean.Stat.Tendsto_dist Xn Q μ hXn → Causalean.Stat.IsBigOp Xn (fun x => 1) μ","l":150,"f":"Causalean/Stat/Limit/Convergence.lean"},{"n":"Causalean.Stat.Tendsto_dist.wald_coverage","k":"theorem","m":"Causalean.Stat.Inference.Studentize","a":"Stat/Inference","t":1,"d":"Generic Wald asymptotic coverage. Suppose a studentized statistic sequence Sn is measurable at every sample size and converges in distribution to the standard Gaussian law, and fix a positive half-width z. If a coverage-probability sequence coverProb is asymptotically equivalent to the studentized-interval event {Sn ∈ [-z, z]}, then coverProb converges to the standard-Gaussian probability of [-z, z].","s":"∀ {Ω : Type u_1} [inst : MeasurableSpace Ω] {μ : MeasureTheory.Measure Ω} [inst_1 : MeasureTheory.IsProbabilityMeasure μ] {Sn : ℕ → Ω → ℝ} (hSn : ∀ (n : ℕ), AEMeasurable (Sn n) μ), Causalean.Stat.Tendsto_dist Sn (Causalean.Stat.gaussianMeasure 0 1) μ hSn → ∀ {z : ℝ}, 0 < z → ∀ (coverProb : ℕ → ℝ), Filter.Tendsto (fun n => coverProb n - (μ {ω | Sn n ω ∈ Set.Icc (-z) z}).toReal) Filter.atTop (nhds 0) → Filter.Tendsto coverProb Filter.atTop (nhds ((Causalean.Stat.gaussianMeasure 0 1) (Set.Icc (-z) z)).toReal)","l":159,"f":"Causalean/Stat/Inference/Studentize.lean"},{"n":"Causalean.Stat.Tendsto_dist.wald_coverage_Iic","k":"theorem","m":"Causalean.Stat.Inference.WaldVec","a":"Stat/Inference","t":1,"d":"Wald / confidence-ellipsoid asymptotic coverage. Suppose the scalar Wald statistic sequence Wₙ is measurable at every sample size and converges in distribution to a limit law χ, and that χ gives zero mass to the boundary frontier (Iic c) = {c}. If a real sequence coverProb is asymptotically equivalent to the ellipsoid event {ω | Wₙ ω ≤ c}, then coverProb converges to χ(Iic c).","s":"∀ {Ω : Type u_1} [inst : MeasurableSpace Ω] {μ : MeasureTheory.Measure Ω} [inst_1 : MeasureTheory.IsProbabilityMeasure μ] {Wn : ℕ → Ω → ℝ} (hWn : ∀ (n : ℕ), AEMeasurable (Wn n) μ) {χ : MeasureTheory.Measure ℝ} [inst_2 : MeasureTheory.IsProbabilityMeasure χ], Causalean.Stat.Tendsto_dist Wn χ μ hWn → ∀ {c : ℝ}, χ (frontier (Set.Iic c)) = 0 → ∀ (coverProb : ℕ → ℝ), Filter.Tendsto (fun n => coverProb n - (μ {ω | Wn n ω ≤ c}).toReal) Filter.atTop (nhds 0) → Filter.Tendsto coverProb Filter.atTop (nhds (χ (Set.Iic c)).toReal)","l":70,"f":"Causalean/Stat/Inference/WaldVec.lean"},{"n":"Causalean.Stat.Tendsto_dist.wald_coverage_Iic_of_noAtoms","k":"theorem","m":"Causalean.Stat.Inference.WaldVec","a":"Stat/Inference","t":1,"d":"Wald / ellipsoid coverage with an atomless limit. Suppose the scalar Wald statistic sequence Wₙ is measurable at every sample size and converges in distribution to a limit law χ with no atoms (which holds for any non-degenerate χ²_d, and more generally for any continuous limit). If a real sequence coverProb is asymptotically equivalent to the ellipsoid event {ω | Wₙ ω ≤ c}, then coverProb converges to χ(Iic c). The boundary-null hypothesis of wald_coverage_Iic is automatic here: frontier (Iic c) = {c} and χ {c} = 0.","s":"∀ {Ω : Type u_1} [inst : MeasurableSpace Ω] {μ : MeasureTheory.Measure Ω} [inst_1 : MeasureTheory.IsProbabilityMeasure μ] {Wn : ℕ → Ω → ℝ} (hWn : ∀ (n : ℕ), AEMeasurable (Wn n) μ) {χ : MeasureTheory.Measure ℝ} [inst_2 : MeasureTheory.IsProbabilityMeasure χ] [MeasureTheory.NullSingletonClass χ], Causalean.Stat.Tendsto_dist Wn χ μ hWn → ∀ (c : ℝ) (coverProb : ℕ → ℝ), Filter.Tendsto (fun n => coverProb n - (μ {ω | Wn n ω ≤ c}).toReal) Filter.atTop (nhds 0) → Filter.Tendsto coverProb Filter.atTop (nhds (χ (Set.Iic c)).toReal)","l":110,"f":"Causalean/Stat/Inference/WaldVec.lean"},{"n":"Causalean.Stat.Tendsto_dist.wald_coverage_chiSq","k":"theorem","m":"Causalean.Stat.Inference.ChiSquaredWald","a":"Stat/Inference","t":1,"d":"χ²-coverage of the Wald confidence ellipsoid. Suppose the Wald statistic sequence Wₙ is measurable at every sample size, d is a positive-integer degrees-of-freedom parameter, and Wₙ converges in distribution to the χ²_d law. If a coverage-probability sequence coverProb is asymptotically equivalent to the ellipsoid event {Wₙ ≤ c}, then coverProb converges to the χ²_d probability of (-∞, c].","s":"∀ {Ω : Type u_3} [inst : MeasurableSpace Ω] {ν : MeasureTheory.Measure Ω} [inst_1 : MeasureTheory.IsProbabilityMeasure ν] {Wn : ℕ → Ω → ℝ} (hWn : ∀ (n : ℕ), AEMeasurable (Wn n) ν) {d : ℕ}, 1 ≤ d → Causalean.Stat.Tendsto_dist Wn (Causalean.Stat.chiSqDist d) ν hWn → ∀ (c : ℝ) (coverProb : ℕ → ℝ), Filter.Tendsto (fun n => coverProb n - (ν {ω | Wn n ω ≤ c}).toReal) Filter.atTop (nhds 0) → Filter.Tendsto coverProb Filter.atTop (nhds ((Causalean.Stat.chiSqDist d) (Set.Iic c)).toReal)","l":166,"f":"Causalean/Stat/Inference/ChiSquaredWald.lean"},{"n":"Causalean.Stat.Tendsto_dist_vec","k":"def","m":"Causalean.Stat.Limit.ConvergenceVec","a":"Stat/Limit","t":1,"d":"Tendsto_dist_vec Xn Q μ hXn is convergence in distribution of an E-valued sequence Xn : ℕ → Ω → E to a probability measure Q : Measure E, formulated as weak convergence of pushforward measures.","s":"{Ω : Type u_1} → {E : Type u_2} → [inst : MeasurableSpace Ω] → [inst_1 : PseudoMetricSpace E] → [inst_2 : MeasurableSpace E] → [OpensMeasurableSpace E] → (Xn : ℕ → Ω → E) → (Q : MeasureTheory.Measure E) → (μ : MeasureTheory.Measure Ω) → [MeasureTheory.IsProbabilityMeasure μ] → [MeasureTheory.IsProbabilityMeasure Q] → (∀ (n : ℕ), AEMeasurable (Xn n) μ) → Prop","l":49,"f":"Causalean/Stat/Limit/ConvergenceVec.lean"},{"n":"Causalean.Stat.Tendsto_dist_vec.add_isLittleOp_one","k":"theorem","m":"Causalean.Stat.Limit.ConvergenceVec","a":"Stat/Limit","t":1,"d":"Vector Slutsky absorption. Suppose Xn and Yn are E-valued sequences that are each measurable at every sample size, Xn converges in distribution to a probability measure Q on E, and the norm of the perturbation ‖Yn − Xn‖ is o_p(1). Then Yn also converges in distribution to Q.","s":"∀ {Ω : Type u_1} {E : Type u_2} [inst : MeasurableSpace Ω] {μ : MeasureTheory.Measure Ω} [inst_1 : MeasureTheory.IsProbabilityMeasure μ] [inst_2 : NormedAddCommGroup E] [inst_3 : MeasurableSpace E] [inst_4 : OpensMeasurableSpace E] {Xn Yn : ℕ → Ω → E} {Q : MeasureTheory.Measure E} [inst_5 : MeasureTheory.IsProbabilityMeasure Q] (hXn : ∀ (n : ℕ), AEMeasurable (Xn n) μ) (hYn : ∀ (n : ℕ), AEMeasurable (Yn n) μ), Causalean.Stat.Tendsto_dist_vec Xn Q μ hXn → Causalean.Stat.IsLittleOp (fun n ω => ‖Yn n ω - Xn n ω‖) (fun x => 1) μ → Causalean.Stat.Tendsto_dist_vec Yn Q μ hYn","l":73,"f":"Causalean/Stat/Limit/ConvergenceVec.lean"},{"n":"Causalean.Stat.Tendsto_dist_vec.congr_ae","k":"theorem","m":"Causalean.Stat.Limit.ConvergenceVec","a":"Stat/Limit","t":2,"d":"Vector convergence in distribution is invariant under eventual a.e. equality of the random variables. Vector analogue of Tendsto_dist.congr_ae.","s":"∀ {Ω : Type u_1} {E : Type u_2} [inst : MeasurableSpace Ω] {μ : MeasureTheory.Measure Ω} [inst_1 : MeasureTheory.IsProbabilityMeasure μ] [inst_2 : PseudoMetricSpace E] [inst_3 : MeasurableSpace E] [inst_4 : OpensMeasurableSpace E] {Xn Yn : ℕ → Ω → E} {Q : MeasureTheory.Measure E} [inst_5 : MeasureTheory.IsProbabilityMeasure Q] (hXn : ∀ (n : ℕ), AEMeasurable (Xn n) μ) (hYn : ∀ (n : ℕ), AEMeasurable (Yn n) μ), Causalean.Stat.Tendsto_dist_vec Xn Q μ hXn → (∀ᶠ (n : ℕ) in Filter.atTop, Xn n =ᵐ[μ] Yn n) → Causalean.Stat.Tendsto_dist_vec Yn Q μ hYn","l":216,"f":"Causalean/Stat/Limit/ConvergenceVec.lean"},{"n":"Causalean.Stat.Tendsto_dist_vec.map_continuous","k":"theorem","m":"Causalean.Stat.Limit.ConvergenceVec","a":"Stat/Limit","t":1,"d":"Pushforward of Tendsto_dist_vec under a continuous map. Suppose Xn is an E-valued sequence, measurable at every sample size, converging in distribution to a probability measure Q on E, and g : E → F is continuous. Then the composed sequence g ∘ Xn converges in distribution to the pushforward measure Q.map g.","s":"∀ {Ω : Type u_1} {E : Type u_2} {F : Type u_3} [inst : MeasurableSpace Ω] {μ : MeasureTheory.Measure Ω} [inst_1 : MeasureTheory.IsProbabilityMeasure μ] [inst_2 : PseudoMetricSpace E] [inst_3 : MeasurableSpace E] [inst_4 : OpensMeasurableSpace E] [inst_5 : PseudoMetricSpace F] [inst_6 : MeasurableSpace F] [inst_7 : BorelSpace F] {Xn : ℕ → Ω → E} {Q : MeasureTheory.Measure E} [inst_8 : MeasureTheory.IsProbabilityMeasure Q] {g : E → F} (hg : Continuous g) (hXn : ∀ (n : ℕ), AEMeasurable (Xn n) μ), Causalean.Stat.Tendsto_dist_vec Xn Q μ hXn → Filter.Tendsto (fun n => ⟨MeasureTheory.Measure.map (fun ω => g (Xn n ω)) μ, MeasureTheory.Measure.isProbabilityMeasure_map (AEMeasurable.comp_aemeasurable (Measurable.aemeasurable (Continuous.measurable hg)) (hXn n))⟩) Filter.atTop (nhds ⟨MeasureTheory.Measure.map g Q, MeasureTheory.Measure.isProbabilityMeasure_map (Measurable.aemeasurable (Continuous.measurable hg))⟩)","l":234,"f":"Causalean/Stat/Limit/ConvergenceVec.lean"},{"n":"Causalean.Stat.Tendsto_dist_vec.of_charFun_tendsto","k":"theorem","m":"Causalean.Stat.CLT.MultivariateCLT","a":"Stat/CLT","t":1,"d":"Cramér–Wold / Lévy continuity wrapper. For an E-valued sequence Xn, if each Xn n is a.e. measurable and the characteristic functions of Xn n converge pointwise, at every point t, to the characteristic function of a probability measure Q, then the sequence Xn converges in distribution to Q.","s":"∀ {Ω : Type u_1} [inst : MeasurableSpace Ω] {μ : MeasureTheory.Measure Ω} {E : Type u_3} [inst_1 : NormedAddCommGroup E] [inst_2 : InnerProductSpace ℝ E] [FiniteDimensional ℝ E] [inst_4 : MeasurableSpace E] [inst_5 : BorelSpace E] [inst_6 : MeasureTheory.IsProbabilityMeasure μ] {Xn : ℕ → Ω → E} {Q : MeasureTheory.Measure E} [inst_7 : MeasureTheory.IsProbabilityMeasure Q] (hXn : ∀ (n : ℕ), AEMeasurable (Xn n) μ), (∀ (t : E), Filter.Tendsto (fun n => MeasureTheory.charFun (MeasureTheory.Measure.map (Xn n) μ) t) Filter.atTop (nhds (MeasureTheory.charFun Q t))) → Causalean.Stat.Tendsto_dist_vec Xn Q μ hXn","l":219,"f":"Causalean/Stat/CLT/MultivariateCLT.lean"},{"n":"Causalean.Stat.Tendsto_inProb","k":"def","m":"Causalean.Stat.Limit.Convergence","a":"Stat/Limit","t":1,"d":"A sequence of real random variables converges in probability to a limit when, for every positive tolerance, the probability that the absolute error exceeds that tolerance tends to zero.","s":"{Ω : Type u_1} → [inst : MeasurableSpace Ω] → (ℕ → Ω → ℝ) → (Ω → ℝ) → MeasureTheory.Measure Ω → Prop","l":49,"f":"Causalean/Stat/Limit/Convergence.lean"},{"n":"Causalean.Stat.Tendsto_inProb.add_zero_zero","k":"theorem","m":"Causalean.Stat.Quantile.SampleQuantileBahadur.Linearity","a":"Stat/Quantile/SampleQuantileBahadur","t":2,"d":"Sum of in-probability limits. If Xn →ₚ 0 and Yn →ₚ 0 then Xn + Yn →ₚ 0. Union bound: {ε ≤ |Xn+Yn|} ⊆ {ε/2 ≤ |Xn|} ∪ {ε/2 ≤ |Yn|}.","s":"∀ {Ω : Type u_1} [inst : MeasurableSpace Ω] {μ : MeasureTheory.Measure Ω} {Xn Yn : ℕ → Ω → ℝ}, Causalean.Stat.Tendsto_inProb Xn (fun x => 0) μ → Causalean.Stat.Tendsto_inProb Yn (fun x => 0) μ → Causalean.Stat.Tendsto_inProb (fun n ω => Xn n ω + Yn n ω) (fun x => 0) μ","l":41,"f":"Causalean/Stat/Quantile/SampleQuantileBahadur/Linearity.lean"},{"n":"Causalean.Stat.Tendsto_inProb.comp_continuousAt","k":"theorem","m":"Causalean.Stat.Limit.ContinuousMapping","a":"Stat/Limit","t":1,"d":"Continuous mapping for convergence in probability at a point. If a real-valued sequence Yn converges in probability to a point c under μ, and a function g is continuous at c, then the composed sequence g ∘ Yn converges in probability to g c. Generalizes Tendsto_inProb.inv (the case g = fun x => 1/x at a nonzero c).","s":"∀ {Ω : Type u_1} [inst : MeasurableSpace Ω] {Yn : ℕ → Ω → ℝ} {c : ℝ} {μ : MeasureTheory.Measure Ω} {g : ℝ → ℝ}, ContinuousAt g c → Causalean.Stat.Tendsto_inProb Yn (fun x => c) μ → Causalean.Stat.Tendsto_inProb (fun n ω => g (Yn n ω)) (fun x => g c) μ","l":40,"f":"Causalean/Stat/Limit/ContinuousMapping.lean"},{"n":"Causalean.Stat.Tendsto_inProb.const_mul_zero","k":"theorem","m":"Causalean.Stat.Quantile.SampleQuantileBahadur.Linearity","a":"Stat/Quantile/SampleQuantileBahadur","t":2,"d":"Scalar multiple preserves in-probability convergence to 0.","s":"∀ {Ω : Type u_1} [inst : MeasurableSpace Ω] {μ : MeasureTheory.Measure Ω} {Xn : ℕ → Ω → ℝ} (c : ℝ), Causalean.Stat.Tendsto_inProb Xn (fun x => 0) μ → Causalean.Stat.Tendsto_inProb (fun n ω => c * Xn n ω) (fun x => 0) μ","l":351,"f":"Causalean/Stat/Quantile/SampleQuantileBahadur/Linearity.lean"},{"n":"Causalean.Stat.Tendsto_inProb.inv","k":"theorem","m":"Causalean.Stat.Limit.ContinuousMapping","a":"Stat/Limit","t":2,"d":"Reciprocal continuity for convergence in probability at a nonzero constant: if Yn →_p Y₀ with Y₀ ≠ 0, then 1 / Yn →_p 1 / Y₀.","s":"∀ {Ω : Type u_1} [inst : MeasurableSpace Ω] {Yn : ℕ → Ω → ℝ} {Y₀ : ℝ} {μ : MeasureTheory.Measure Ω}, Causalean.Stat.Tendsto_inProb Yn (fun x => Y₀) μ → Y₀ ≠ 0 → Causalean.Stat.Tendsto_inProb (fun n ω => 1 / Yn n ω) (fun x => 1 / Y₀) μ","l":65,"f":"Causalean/Stat/Limit/ContinuousMapping.lean"},{"n":"Causalean.Stat.Tendsto_inProb.isBigOp_one","k":"theorem","m":"Causalean.Stat.Limit.ContinuousMapping","a":"Stat/Limit","t":2,"d":"In-probability tightness. A sequence converging in probability to a constant is bounded in probability: Xₙ →ₚ c ⟹ Xₙ = O_p(1).","s":"∀ {Ω : Type u_1} [inst : MeasurableSpace Ω] {Xn : ℕ → Ω → ℝ} {c : ℝ} {μ : MeasureTheory.Measure Ω}, Causalean.Stat.Tendsto_inProb Xn (fun x => c) μ → Causalean.Stat.IsBigOp Xn (fun x => 1) μ","l":157,"f":"Causalean/Stat/Limit/ContinuousMapping.lean"},{"n":"Causalean.Stat.Tendsto_inProb.isLittleOp_one","k":"theorem","m":"Causalean.Stat.Limit.ContinuousMapping","a":"Stat/Limit","t":2,"d":"Convergence in probability to zero implies o_p(1): if Yn →_p 0, then Yn is IsLittleOp of the constant-one rate.","s":"∀ {Ω : Type u_1} [inst : MeasurableSpace Ω] {Yn : ℕ → Ω → ℝ} {μ : MeasureTheory.Measure Ω}, Causalean.Stat.Tendsto_inProb Yn (fun x => 0) μ → Causalean.Stat.IsLittleOp Yn (fun x => 1) μ","l":139,"f":"Causalean/Stat/Limit/ContinuousMapping.lean"},{"n":"Causalean.Stat.Tendsto_inProb.matrix_comp_continuousAt","k":"theorem","m":"Causalean.Stat.Limit.ContinuousMapping","a":"Stat/Limit","t":2,"d":"Matrix continuous mapping in probability. If every entry of the random matrix Mₙ converges in probability to the corresponding entry of M₀, and g is continuous at M₀, then g(Mₙ) →ₚ g(M₀).","s":"∀ {Ω : Type u_1} {K : Type u_2} [inst : MeasurableSpace Ω] [Fintype K] {μ : MeasureTheory.Measure Ω} {Mn : ℕ → Ω → Matrix K K ℝ} {M₀ : Matrix K K ℝ} {g : Matrix K K ℝ → ℝ}, ContinuousAt g M₀ → (∀ (i j : K), Causalean.Stat.Tendsto_inProb (fun n ω => Mn n ω i j) (fun x => M₀ i j) μ) → Causalean.Stat.Tendsto_inProb (fun n ω => g (Mn n ω)) (fun x => g M₀) μ","l":249,"f":"Causalean/Stat/Limit/ContinuousMapping.lean"},{"n":"Causalean.Stat.Tendsto_inProb.of_isLittleOp_one","k":"theorem","m":"Causalean.Stat.Quantile.SampleQuantileBahadur.Linearity","a":"Stat/Quantile/SampleQuantileBahadur","t":2,"d":"Little-o in probability implies convergence in probability. An IsLittleOp _ 1 sequence converges to 0 in probability. Both unwind to the same μ{· < |·|} → 0 statement up to a harmless </≤ slack (handled with ε/2).","s":"∀ {Ω : Type u_1} [inst : MeasurableSpace Ω] {μ : MeasureTheory.Measure Ω} {Xn : ℕ → Ω → ℝ}, Causalean.Stat.IsLittleOp Xn (fun x => 1) μ → Causalean.Stat.Tendsto_inProb Xn (fun x => 0) μ","l":69,"f":"Causalean/Stat/Quantile/SampleQuantileBahadur/Linearity.lean"},{"n":"Causalean.Stat.Tendsto_inProb.pi_comp_continuousAt","k":"theorem","m":"Causalean.Stat.Limit.ContinuousMapping","a":"Stat/Limit","t":2,"d":"Finite-Pi continuous mapping in probability. If every coordinate of a finite-dimensional random vector converges in probability to the corresponding constant coordinate, and g is continuous at the limiting vector, then g(Yₙ) →ₚ g(c).","s":"∀ {Ω : Type u_1} {ι : Type u_2} [inst : MeasurableSpace Ω] [Fintype ι] {μ : MeasureTheory.Measure Ω} {Yn : ℕ → Ω → ι → ℝ} {c : ι → ℝ} {g : (ι → ℝ) → ℝ}, ContinuousAt g c → (∀ (i : ι), Causalean.Stat.Tendsto_inProb (fun n ω => Yn n ω i) (fun x => c i) μ) → Causalean.Stat.Tendsto_inProb (fun n ω => g (Yn n ω)) (fun x => g c) μ","l":188,"f":"Causalean/Stat/Limit/ContinuousMapping.lean"},{"n":"Causalean.Stat.Tendsto_inProb.sqrt","k":"theorem","m":"Causalean.Stat.Inference.VarianceEstimation","a":"Stat/Inference","t":2,"d":"Square root preserves convergence in probability. If Vn →ₚ v₀ under μ, then √Vn →ₚ √v₀. Continuous mapping with the (everywhere continuous) Real.sqrt, via Tendsto_inProb.comp_continuousAt.","s":"∀ {Ω : Type u_3} [inst : MeasurableSpace Ω] {Vn : ℕ → Ω → ℝ} {v₀ : ℝ} {μ : MeasureTheory.Measure Ω}, Causalean.Stat.Tendsto_inProb Vn (fun x => v₀) μ → Causalean.Stat.Tendsto_inProb (fun n ω => √(Vn n ω)) (fun x => √v₀) μ","l":171,"f":"Causalean/Stat/Inference/VarianceEstimation.lean"},{"n":"Causalean.Stat.Tendsto_inProb.sub","k":"theorem","m":"Causalean.Stat.Limit.ContinuousMapping","a":"Stat/Limit","t":2,"d":"Difference of convergent-in-probability sequences. If Xn →_p a and Yn →_p b (both to constant limits), then Xn − Yn →_p a − b. Standard ε/2-union-bound argument; the analogue of TendstoInMeasure.sub, which Mathlib does not currently provide for the constant-limit case.","s":"∀ {Ω : Type u_1} [inst : MeasurableSpace Ω] {Xn Yn : ℕ → Ω → ℝ} {a b : ℝ} {μ : MeasureTheory.Measure Ω}, Causalean.Stat.Tendsto_inProb Xn (fun x => a) μ → Causalean.Stat.Tendsto_inProb Yn (fun x => b) μ → Causalean.Stat.Tendsto_inProb (fun n ω => Xn n ω - Yn n ω) (fun x => a - b) μ","l":99,"f":"Causalean/Stat/Limit/ContinuousMapping.lean"},{"n":"Causalean.Stat.Tendsto_inProb.sub_const","k":"theorem","m":"Causalean.Stat.Limit.ContinuousMapping","a":"Stat/Limit","t":2,"d":"Center a convergence-in-probability statement around a constant limit: if Yn →_p Y₀, then Yn - Y₀ →_p 0.","s":"∀ {Ω : Type u_1} [inst : MeasurableSpace Ω] {Yn : ℕ → Ω → ℝ} {Y₀ : ℝ} {μ : MeasureTheory.Measure Ω}, Causalean.Stat.Tendsto_inProb Yn (fun x => Y₀) μ → Causalean.Stat.Tendsto_inProb (fun n ω => Yn n ω - Y₀) (fun x => 0) μ","l":88,"f":"Causalean/Stat/Limit/ContinuousMapping.lean"},{"n":"Causalean.Stat.UniformlyBounded","k":"def","m":"Causalean.Stat.Minimax.MarkovKernelTransport","a":"Stat/Minimax","t":1,"d":"A real-valued function is uniformly bounded when one finite nonnegative constant bounds its absolute value at every input.","s":"{A : Type u_1} → (A → ℝ) → Prop","l":46,"f":"Causalean/Stat/Minimax/MarkovKernelTransport.lean"},{"n":"Causalean.Stat.ZEstimatorRegularity","k":"structure","m":"Causalean.Stat.MEstimation.ZEstimator","a":"Stat/MEstimation","t":1,"d":"Regularity conditions for the Z-estimator central limit theorem. Bundles, for a score function and target parameter θ₀ under a sampling law, the population identification condition that the score vanishes in mean at the truth, a Jacobian of the population score at θ₀ together with a witnessed inverse, finite variance of the score at the truth, measurability of the score at every parameter value, continuity of the population score map at θ₀, local integrability of the score on a neighborhood of θ₀, and an integrable L² envelope bounding local score differences.","s":"{X : Type u_2} → [inst : MeasurableSpace X] → {E : Type u_3} → [inst_1 : NormedAddCommGroup E] → [InnerProductSpace ℝ E] → [MeasurableSpace E] → (E → X → E) → E → MeasureTheory.Measure X → Type u_3","l":45,"f":"Causalean/Stat/MEstimation/ZEstimator.lean"},{"n":"Causalean.Stat.abs_cdfIF_le_one","k":"theorem","m":"Causalean.Stat.Quantile.EmpiricalCDF","a":"Stat/Quantile","t":2,"d":"cdfIF is bounded by 1 in absolute value.","s":"∀ {P : MeasureTheory.Measure ℝ} [MeasureTheory.IsProbabilityMeasure P] (y z : ℝ), |Causalean.Stat.cdfIF P y z| ≤ 1","l":116,"f":"Causalean/Stat/Quantile/EmpiricalCDF.lean"},{"n":"Causalean.Stat.abs_empiricalMass_le_one","k":"theorem","m":"Causalean.Stat.Sample.EmpiricalMass","a":"Stat/Sample","t":2,"d":"The absolute empirical mass of any point is at most one, including for the empty sample.","s":"∀ {𝒳 : Type u_1} {N : ℕ} (sample : Fin N → 𝒳) (x : 𝒳), |Causalean.Stat.empiricalMass sample x| ≤ 1","l":30,"f":"Causalean/Stat/Sample/EmpiricalMass.lean"},{"n":"Causalean.Stat.abs_integral_le_eLpNorm_two","k":"theorem","m":"Causalean.Stat.Orthogonality.ConditionalOp","a":"Stat/Orthogonality","t":2,"d":"Constant-case Cauchy–Schwarz. If ν is a probability measure (or, more generally, satisfies ν univ ≤ 1) and f ∈ L²(ν), then |∫ f dν| ≤ ‖f‖₂.","s":"∀ {X : Type u_2} [inst : MeasurableSpace X] {ν : MeasureTheory.Measure X} [MeasureTheory.IsProbabilityMeasure ν] {f : X → ℝ}, MeasureTheory.MemLp f 2 ν → |∫ (x : X), f x ∂ν| ≤ (MeasureTheory.eLpNorm f 2 ν).toReal","l":306,"f":"Causalean/Stat/Orthogonality/ConditionalOp.lean"},{"n":"Causalean.Stat.abs_kernelMean_le","k":"theorem","m":"Causalean.Stat.Minimax.MarkovKernelTransport","a":"Stat/Minimax","t":2,"d":"If a proposed bound is nonnegative and bounds the estimator in absolute value at every output, then the kernel mean obeys the same absolute bound at every input when each kernel output is a probability distribution.","s":"∀ {X : Type uX} {Y : Type uY} [inst : MeasurableSpace X] [inst_1 : MeasurableSpace Y] (K : ProbabilityTheory.Kernel X Y) [ProbabilityTheory.IsMarkovKernel K] {T : Y → ℝ} {M : ℝ}, 0 ≤ M → (∀ (y : Y), |T y| ≤ M) → ∀ (x : X), |Causalean.Stat.kernelMean K T x| ≤ M","l":62,"f":"Causalean/Stat/Minimax/MarkovKernelTransport.lean"},{"n":"Causalean.Stat.abs_measureReal_sub_le_one","k":"theorem","m":"Causalean.Stat.Minimax.TotalVariation","a":"Stat/Minimax","t":2,"d":"Every term of the supremum defining tvDist is bounded by 1.","s":"∀ {Ω : Type u_1} {mΩ : MeasurableSpace Ω} {μ ν : MeasureTheory.Measure Ω} [MeasureTheory.IsProbabilityMeasure μ] [MeasureTheory.IsProbabilityMeasure ν] (A : Set Ω), |μ.real A - ν.real A| ≤ 1","l":54,"f":"Causalean/Stat/Minimax/TotalVariation.lean"},{"n":"Causalean.Stat.abs_measureReal_sub_le_tvDist","k":"theorem","m":"Causalean.Stat.Minimax.TotalVariation","a":"Stat/Minimax","t":2,"d":"For a measurable set A, the gap |μ.real A − ν.real A| is at most tvDist μ ν.","s":"∀ {Ω : Type u_1} {mΩ : MeasurableSpace Ω} {μ ν : MeasureTheory.Measure Ω} [MeasureTheory.IsProbabilityMeasure μ] [MeasureTheory.IsProbabilityMeasure ν] {A : Set Ω}, MeasurableSet A → |μ.real A - ν.real A| ≤ Causalean.Stat.tvDist μ ν","l":74,"f":"Causalean/Stat/Minimax/TotalVariation.lean"},{"n":"Causalean.Stat.abs_setIntegral_le_half_integral_abs_of_integral_eq_zero","k":"theorem","m":"Causalean.Stat.Minimax.Scheffe","a":"Stat/Minimax","t":2,"d":"For an integrable function f whose integral vanishes, the integral over any measurable set is bounded in absolute value by half the L¹ norm of f.","s":"∀ {Ω : Type u_1} {mΩ : MeasurableSpace Ω} {ν : MeasureTheory.Measure Ω} {f : Ω → ℝ}, MeasureTheory.Integrable f ν → ∫ (x : Ω), f x ∂ν = 0 → ∀ {A : Set Ω}, MeasurableSet A → |∫ (x : Ω) in A, f x ∂ν| ≤ 1 / 2 * ∫ (x : Ω), |f x| ∂ν","l":50,"f":"Causalean/Stat/Minimax/Scheffe.lean"},{"n":"Causalean.Stat.abs_signedTail_le_one","k":"theorem","m":"Causalean.Stat.Coupling.ProductLossMonotoneCoupling.TailIntegral","a":"Stat/Coupling/ProductLossMonotoneCoupling","t":2,"d":"The signed tail indicator is bounded by 1 in absolute value.","s":"∀ (a s : ℝ), |Causalean.Stat.signedTail a s| ≤ 1","l":82,"f":"Causalean/Stat/Coupling/ProductLossMonotoneCoupling/TailIntegral.lean"},{"n":"Causalean.Stat.acceptanceRegion","k":"def","m":"Causalean.Stat.Minimax.Fano","a":"Stat/Minimax","t":1,"d":"The acceptance region of an estimator for hypothesis value θ: the points it places strictly within s of θ.","s":"{Ω : Type u_1} → {Θ : Type u_2} → [PseudoMetricSpace Θ] → (Ω → Θ) → Θ → ℝ → Set Ω","l":61,"f":"Causalean/Stat/Minimax/Fano.lean"},{"n":"Causalean.Stat.acceptanceRegion_compl","k":"theorem","m":"Causalean.Stat.Minimax.Fano","a":"Stat/Minimax","t":2,"d":"The acceptance region is the complement of the error region.","s":"∀ {Ω : Type u_1} {Θ : Type u_2} [inst : PseudoMetricSpace Θ] (est : Ω → Θ) (θ : Θ) (s : ℝ), (Causalean.Stat.acceptanceRegion est θ s)ᶜ = {ω | s ≤ dist (est ω) θ}","l":75,"f":"Causalean/Stat/Minimax/Fano.lean"},{"n":"Causalean.Stat.acceptanceRegion_disjoint","k":"theorem","m":"Causalean.Stat.Minimax.Fano","a":"Stat/Minimax","t":2,"d":"Disjointness of acceptance regions. If two target values are 2s-separated, their acceptance regions are disjoint: a point within s of both would force the targets within 2s of each other.","s":"∀ {Ω : Type u_1} {Θ : Type u_2} [inst : PseudoMetricSpace Θ] {est : Ω → Θ} {θ₀ θ₁ : Θ} {s : ℝ}, 2 * s ≤ dist θ₀ θ₁ → Disjoint (Causalean.Stat.acceptanceRegion est θ₀ s) (Causalean.Stat.acceptanceRegion est θ₁ s)","l":81,"f":"Causalean/Stat/Minimax/Fano.lean"},{"n":"Causalean.Stat.acceptanceRegion_pairwiseDisjoint","k":"theorem","m":"Causalean.Stat.Minimax.Fano","a":"Stat/Minimax","t":2,"d":"Pairwise disjointness of the acceptance regions of a 2s-separated family.","s":"∀ {Ω : Type u_1} {Θ : Type u_2} [inst : PseudoMetricSpace Θ] {ι : Type u_3} {est : Ω → Θ} {θ : ι → Θ} {s : ℝ}, (∀ (i k : ι), i ≠ k → 2 * s ≤ dist (θ i) (θ k)) → Pairwise (Function.onFun Disjoint fun i => Causalean.Stat.acceptanceRegion est (θ i) s)","l":98,"f":"Causalean/Stat/Minimax/Fano.lean"},{"n":"Causalean.Stat.adjoint_inv_self","k":"theorem","m":"Causalean.Stat.GMM.VarianceAlgebra","a":"Stat/GMM","t":2,"d":"The adjoint (self-adjointness) of a right inverse of a self-adjoint operator: if adjoint M = M and M ∘L N = id, then adjoint N = N. (adjoint N is then a left inverse of M, and in finite dimension a one-sided inverse of an operator that already has a two-sided one is unique.) Shared with OverID.lean.","s":"∀ {E : Type u_1} [inst : NormedAddCommGroup E] [inst_1 : InnerProductSpace ℝ E] [inst_2 : FiniteDimensional ℝ E] {M N : E →L[ℝ] E}, ContinuousLinearMap.adjoint M = M → M ∘SL N = ContinuousLinearMap.id ℝ E → ContinuousLinearMap.adjoint N = N","l":83,"f":"Causalean/Stat/GMM/VarianceAlgebra.lean"},{"n":"Causalean.Stat.aemeasurable_quantile_unifOI","k":"theorem","m":"Causalean.Stat.Coupling.ProductLossMonotoneCoupling.PIT","a":"Stat/Coupling/ProductLossMonotoneCoupling","t":2,"d":"The quantile function is a.e.-measurable with respect to the uniform measure on (0,1); it is monotone there, and a monotone function is measurable.","s":"∀ (μ : MeasureTheory.Measure ℝ), AEMeasurable (Causalean.Stat.quantile μ) Causalean.Stat.unifOI","l":56,"f":"Causalean/Stat/Coupling/ProductLossMonotoneCoupling/PIT.lean"},{"n":"Causalean.Stat.affineInversionSet","k":"def","m":"Causalean.Stat.Inference.AffineInversion","a":"Stat/Inference","t":1,"d":"The subset of a parameter region accepted by a scalar affine inequality.","s":"Set ℝ → ℝ → ℝ → ℝ → Set ℝ","l":24,"f":"Causalean/Stat/Inference/AffineInversion.lean"},{"n":"Causalean.Stat.affineInversionSet_restrictedVolume_le","k":"theorem","m":"Causalean.Stat.Inference.AffineInversion","a":"Stat/Inference","t":2,"d":"Affine inversion has restricted volume at most the parameter-region volume and at most twice the radius divided by the nonzero slope.","s":"∀ (region : Set ℝ), MeasureTheory.volume region ≠ ⊤ → ∀ (A B r : ℝ), B ≠ 0 → 0 ≤ r → Causalean.Stat.restrictedSetVolume region (Causalean.Stat.affineInversionSet region A B r) ≤ min (MeasureTheory.volume region).toReal (2 * r / |B|)","l":28,"f":"Causalean/Stat/Inference/AffineInversion.lean"},{"n":"Causalean.Stat.affineInversionSet_restrictedVolume_le_region","k":"theorem","m":"Causalean.Stat.Inference.AffineInversion","a":"Stat/Inference","t":2,"d":"Affine inversion is always bounded by the volume of its parameter region.","s":"∀ (region : Set ℝ), MeasureTheory.volume region ≠ ⊤ → ∀ (A B r : ℝ), Causalean.Stat.restrictedSetVolume region (Causalean.Stat.affineInversionSet region A B r) ≤ (MeasureTheory.volume region).toReal","l":64,"f":"Causalean/Stat/Inference/AffineInversion.lean"},{"n":"Causalean.Stat.affinePullbackEstimator","k":"def","m":"Causalean.Stat.Minimax.MinimaxRisk","a":"Stat/Minimax","t":1,"d":"The affine pullback estimator first applies the deterministic observation rule, then subtracts the affine offset from the target estimator and divides by the affine slope.","s":"{X : Type uX} → {Y : Type uY} → (X → Y) → ℝ → ℝ → (Y → ℝ) → X → ℝ","l":183,"f":"Causalean/Stat/Minimax/MinimaxRisk.lean"},{"n":"Causalean.Stat.affine_sqLoss_pullback_identity","k":"theorem","m":"Causalean.Stat.Minimax.MinimaxRisk","a":"Stat/Minimax","t":2,"d":"If the affine slope is nonzero, then the squared error of a target estimator after deterministic observation equals the squared error of its affine pullback multiplied by the squared slope, point by point.","s":"∀ {X : Type uX} {Y : Type uY} {phi : X → Y} {a b theta : ℝ} {targetEst : Y → ℝ}, a ≠ 0 → ∀ (z : X), a ^ 2 * (Causalean.Stat.affinePullbackEstimator phi a b targetEst z - theta) ^ 2 = (targetEst (phi z) - (a * theta + b)) ^ 2","l":199,"f":"Causalean/Stat/Minimax/MinimaxRisk.lean"},{"n":"Causalean.Stat.armGroupEvent","k":"def","m":"Causalean.Stat.Sample.OccupancyWeightedMean.Basic","a":"Stat/Sample/OccupancyWeightedMean","t":1,"d":"The arm/group event consists of observations having the requested finite group label and Boolean arm label.","s":"{Omega : Type u_1} → {kappa : Type u_2} → (Omega → kappa) → (Omega → Bool) → Bool → kappa → Set Omega","l":29,"f":"Causalean/Stat/Sample/OccupancyWeightedMean/Basic.lean"},{"n":"Causalean.Stat.armGroupResidual","k":"def","m":"Causalean.Stat.Sample.OccupancyWeightedMean.Basic","a":"Stat/Sample/OccupancyWeightedMean","t":1,"d":"The residual at an arm/group label is the outcome minus its supplied arm/group center.","s":"{Omega : Type u_1} → {kappa : Type u_2} → (Omega → ℝ) → (Bool → kappa → ℝ) → Bool → kappa → Omega → ℝ","l":40,"f":"Causalean/Stat/Sample/OccupancyWeightedMean/Basic.lean"},{"n":"Causalean.Stat.armResidualMean","k":"def","m":"Causalean.Stat.Sample.OccupancyWeightedMean.Basic","a":"Stat/Sample/OccupancyWeightedMean","t":1,"d":"The totalized arm residual mean is the cell residual sum divided by its count when that count is positive, and zero when the count is zero.","s":"{Omega : Type u_1} → {kappa : Type u_2} → [DecidableEq kappa] → {n : ℕ} → (Omega → kappa) → (Omega → Bool) → (Omega → ℝ) → (Bool → kappa → ℝ) → (Fin n → Omega) → Bool → kappa → ℝ","l":91,"f":"Causalean/Stat/Sample/OccupancyWeightedMean/Basic.lean"},{"n":"Causalean.Stat.armResidualSum","k":"def","m":"Causalean.Stat.Sample.OccupancyWeightedMean.Basic","a":"Stat/Sample/OccupancyWeightedMean","t":1,"d":"The residual sum in an arm/group cell adds only coordinates belonging to that cell.","s":"{Omega : Type u_1} → {kappa : Type u_2} → {n : ℕ} → (Omega → kappa) → (Omega → Bool) → (Omega → ℝ) → (Bool → kappa → ℝ) → (Fin n → Omega) → Bool → kappa → ℝ","l":84,"f":"Causalean/Stat/Sample/OccupancyWeightedMean/Basic.lean"},{"n":"Causalean.Stat.assouad_average","k":"theorem","m":"Causalean.Stat.Minimax.Assouad","a":"Stat/Minimax","t":1,"d":"Assouad's lemma (average form). Assume each coordinate's decoded-bit event is measurable. If every hypercube vertex's law is within total variation β of each of its d neighbouring vertices' laws, then the average Hamming risk over the cube is at least (d / 2)(1 − β). Choosing the dimension d large and the per-coordinate divergence β small forces a large number of mis-decoded coordinates.","s":"∀ {Ω : Type u_1} {mΩ : MeasurableSpace Ω} {d : ℕ} (P : (Fin d → Bool) → MeasureTheory.Measure Ω) [∀ (τ : Fin d → Bool), MeasureTheory.IsProbabilityMeasure (P τ)] (est : Ω → Fin d → Bool), (∀ (j : Fin d) (b : Bool), MeasurableSet {ω | est ω j = b}) → ∀ {β : ℝ}, (∀ (j : Fin d) (τ : Fin d → Bool), Causalean.Stat.tvDist (P τ) (P (Causalean.Stat.flipBit j τ)) ≤ β) → ↑d / 2 * (1 - β) ≤ (∑ τ, Causalean.Stat.hammingRisk P est τ) / ↑(Fintype.card (Fin d → Bool))","l":147,"f":"Causalean/Stat/Minimax/Assouad.lean"},{"n":"Causalean.Stat.assouad_exists","k":"theorem","m":"Causalean.Stat.Minimax.Assouad","a":"Stat/Minimax","t":1,"d":"Assouad's lemma (existence form). Under the same hypotheses — each coordinate's decoded-bit event is measurable and every vertex's law is within total variation β of each of its d neighbours — some vertex τ forces Hamming risk at least (d / 2)(1 − β): no cube estimator can decode every vertex's coordinates reliably when neighbouring laws are statistically close.","s":"∀ {Ω : Type u_1} {mΩ : MeasurableSpace Ω} {d : ℕ} (P : (Fin d → Bool) → MeasureTheory.Measure Ω) [∀ (τ : Fin d → Bool), MeasureTheory.IsProbabilityMeasure (P τ)] (est : Ω → Fin d → Bool), (∀ (j : Fin d) (b : Bool), MeasurableSet {ω | est ω j = b}) → ∀ {β : ℝ}, (∀ (j : Fin d) (τ : Fin d → Bool), Causalean.Stat.tvDist (P τ) (P (Causalean.Stat.flipBit j τ)) ≤ β) → ∃ τ, ↑d / 2 * (1 - β) ≤ Causalean.Stat.hammingRisk P est τ","l":186,"f":"Causalean/Stat/Minimax/Assouad.lean"},{"n":"Causalean.Stat.bddAbove_tvDist_range","k":"theorem","m":"Causalean.Stat.Minimax.TotalVariation","a":"Stat/Minimax","t":2,"d":"The family defining tvDist is bounded above (by 1).","s":"∀ {Ω : Type u_1} {mΩ : MeasurableSpace Ω} {μ ν : MeasureTheory.Measure Ω} [MeasureTheory.IsProbabilityMeasure μ] [MeasureTheory.IsProbabilityMeasure ν], BddAbove (Set.range fun A => |μ.real ↑A - ν.real ↑A|)","l":67,"f":"Causalean/Stat/Minimax/TotalVariation.lean"},{"n":"Causalean.Stat.bddBelow_quantileSet","k":"theorem","m":"Causalean.Stat.Quantile.Quantile","a":"Stat/Quantile","t":2,"d":"For 0 < τ, the super-level set is bounded below: since cdf μ → 0 at -∞, any point where the cdf already drops below τ is a lower bound.","s":"∀ {μ : MeasureTheory.Measure ℝ} {τ : ℝ}, 0 < τ → BddBelow (Causalean.Stat.quantileSet μ τ)","l":59,"f":"Causalean/Stat/Quantile/Quantile.lean"},{"n":"Causalean.Stat.bddBelow_range_worstCaseRisk","k":"theorem","m":"Causalean.Stat.Minimax.MinimaxValue","a":"Stat/Minimax","t":2,"d":"For a nonnegative risk the worst-case risks are bounded below by zero, which is the side condition the infimum over estimators needs.","s":"∀ {E : Sort u_1} {Θ : Sort u_3} {risk : E → Θ → ℝ}, (∀ (e : E) (θ : Θ), 0 ≤ risk e θ) → BddBelow (Set.range (Causalean.Stat.worstCaseRisk risk))","l":100,"f":"Causalean/Stat/Minimax/MinimaxValue.lean"},{"n":"Causalean.Stat.bootstrap_studentized_tendsto","k":"theorem","m":"Causalean.Stat.Bootstrap.CI","a":"Stat/Bootstrap","t":1,"d":"Bootstrap studentized CLT. Let θ̂ₙ be asymptotically linear at θ₀ with influence function ψ along the i.i.d. sample S, where ψ is measurable, integrable, and square-integrable; suppose further that the influence function is nondegenerate, $\\int \\psi^2\\,dP > 0$, the rescaled estimator is a.e. measurable at every sample size, and the bootstrap studentized statistic is a.e. measurable at every sample size. Then the bootstrap-studentized statistic $\\sqrt n(\\hat\\theta_n-\\theta_0)/\\hat\\sigma_n$ converges in distribution to the standard normal law, where $\\hat\\sigma_n$ is the bootstrap standard error.","s":"∀ {Ω : Type u_1} {X : Type u_2} [inst : MeasurableSpace Ω] [inst_1 : MeasurableSpace X] {μ : MeasureTheory.Measure Ω} {P : MeasureTheory.Measure X} [inst_2 : MeasureTheory.IsProbabilityMeasure μ] [MeasureTheory.IsProbabilityMeasure P] {θn : ℕ → Ω → ℝ} {θ₀ : ℝ} {ψ : X → ℝ} {S : Causalean.Stat.IIDSample Ω X μ P}, (Causalean.Stat.IsAsymLinear θn θ₀ ψ S fun m => Finset.range m) → Measurable ψ → MeasureTheory.Integrable (fun ω => ψ (S.Z 0 ω)) μ → MeasureTheory.Integrable (fun ω => ψ (S.Z 0 ω) ^ 2) μ → 0 < ∫ (x : X), ψ x ^ 2 ∂P → (∀ (n : ℕ), AEMeasurable (Causalean.Stat.IsAsymLinear.rescaledEstimator θn θ₀ (fun m => Finset.range m) n) μ) → ∀ (hStud_meas : ∀ (n : ℕ), AEMeasurable (Causalean.Stat.IIDSample.bootstrapStudentized θn θ₀ S ψ n) μ), Causalean.Stat.Tendsto_dist (Causalean.Stat.IIDSample.bootstrapStudentized θn θ₀ S ψ) (Causalean.Stat.gaussianMeasure 0 1) μ hStud_meas","l":99,"f":"Causalean/Stat/Bootstrap/CI.lean"},{"n":"Causalean.Stat.bootstrap_wald_coverage","k":"theorem","m":"Causalean.Stat.Bootstrap.CI","a":"Stat/Bootstrap","t":1,"d":"Bootstrap Wald asymptotic coverage. Under the asymptotic-linearity hypothesis, measurability, integrability, and square-integrability of ψ, influence-function nondegeneracy, and measurability of the rescaled estimator and of the studentized statistic at every sample size — the hypotheses of bootstrap_studentized_tendsto — fix a positive critical value z and a coverage-probability sequence coverProb that asymptotically tracks the studentized interval's true coverage event; then coverProb converges to the standard normal probability of the interval [-z, z], so the bootstrap studentized interval has asymptotic $N(0,1)$-coverage.","s":"∀ {Ω : Type u_1} {X : Type u_2} [inst : MeasurableSpace Ω] [inst_1 : MeasurableSpace X] {μ : MeasureTheory.Measure Ω} {P : MeasureTheory.Measure X} [MeasureTheory.IsProbabilityMeasure μ] [MeasureTheory.IsProbabilityMeasure P] {θn : ℕ → Ω → ℝ} {θ₀ : ℝ} {ψ : X → ℝ} {S : Causalean.Stat.IIDSample Ω X μ P}, (Causalean.Stat.IsAsymLinear θn θ₀ ψ S fun m => Finset.range m) → Measurable ψ → MeasureTheory.Integrable (fun ω => ψ (S.Z 0 ω)) μ → MeasureTheory.Integrable (fun ω => ψ (S.Z 0 ω) ^ 2) μ → 0 < ∫ (x : X), ψ x ^ 2 ∂P → (∀ (n : ℕ), AEMeasurable (Causalean.Stat.IsAsymLinear.rescaledEstimator θn θ₀ (fun m => Finset.range m) n) μ) → (∀ (n : ℕ), AEMeasurable (Causalean.Stat.IIDSample.bootstrapStudentized θn θ₀ S ψ n) μ) → ∀ {z : ℝ}, 0 < z → ∀ (coverProb : ℕ → ℝ), Filter.Tendsto (fun n => coverProb n - (μ {ω | Causalean.Stat.IIDSample.bootstrapStudentized θn θ₀ S ψ n ω ∈ Set.Icc (-z) z}).toReal) Filter.atTop (nhds 0) → Filter.Tendsto coverProb Filter.atTop (nhds ((Causalean.Stat.gaussianMeasure 0 1) (Set.Icc (-z) z)).toReal)","l":139,"f":"Causalean/Stat/Bootstrap/CI.lean"},{"n":"Causalean.Stat.bretagnolle_huber_affinity","k":"theorem","m":"Causalean.Stat.Minimax.BretagnolleHuber","a":"Stat/Minimax","t":1,"d":"Bretagnolle–Huber inequality. For probability measures μ, ν such that μ is absolutely continuous with respect to ν and their Kullback–Leibler divergence is finite, the two-point testing affinity 1 − tvDist μ ν is at least ½·exp(-KL(μ‖ν)):","s":"∀ {Ω : Type u_1} {mΩ : MeasurableSpace Ω} (μ ν : MeasureTheory.Measure Ω) [MeasureTheory.IsProbabilityMeasure μ] [MeasureTheory.IsProbabilityMeasure ν], μ.AbsolutelyContinuous ν → InformationTheory.klDiv μ ν ≠ ⊤ → 1 / 2 * Real.exp (-(InformationTheory.klDiv μ ν).toReal) ≤ 1 - Causalean.Stat.tvDist μ ν","l":362,"f":"Causalean/Stat/Minimax/BretagnolleHuber.lean"},{"n":"Causalean.Stat.card_mul_exp_neg_log_eq_rpow","k":"theorem","m":"Causalean.Stat.Minimax.CoordinatewiseOverlap","a":"Stat/Minimax","t":2,"d":"The exponential KL-overlap floor has the expected power-law scaling after multiplication by the number of coordinates.","s":"∀ {M : ℕ}, 1 ≤ M → ∀ (κ : ℝ), ↑M * (1 / 2 * Real.exp (-(κ * Real.log ↑M))) = ↑M ^ (1 - κ) / 2","l":308,"f":"Causalean/Stat/Minimax/CoordinatewiseOverlap.lean"},{"n":"Causalean.Stat.card_partialMatchingsOfSize","k":"theorem","m":"Causalean.Stat.UStatistic.OrderM.PartialMatching","a":"Stat/UStatistic/OrderM","t":1,"d":"For coordinate-set sizes r and s and matching size h, the number of partial matchings is the product of the two subset counts and the number of permutations of h objects.","s":"∀ (r s h : ℕ), (Causalean.Stat.partialMatchingsOfSize r s h).card = r.choose h * s.choose h * h.factorial","l":154,"f":"Causalean/Stat/UStatistic/OrderM/PartialMatching.lean"},{"n":"Causalean.Stat.categoryIndicator","k":"def","m":"Causalean.Stat.SampleSplit.FiniteCategoryPilot","a":"Stat/SampleSplit","t":1,"d":"A category indicator is one when an observation has the designated label and zero otherwise.","s":"{X : Type u_2} → {Iota : Type u_3} → [DecidableEq Iota] → (X → Iota) → Iota → X → ℝ","l":39,"f":"Causalean/Stat/SampleSplit/FiniteCategoryPilot.lean"},{"n":"Causalean.Stat.categoryMass","k":"def","m":"Causalean.Stat.SampleSplit.FiniteCategoryPilot","a":"Stat/SampleSplit","t":1,"d":"A category's population mass is the probability that the observation's label equals that category, represented as a real number.","s":"{X : Type u_2} → {Iota : Type u_3} → [inst : MeasurableSpace X] → MeasureTheory.Measure X → (X → Iota) → Iota → ℝ","l":34,"f":"Causalean/Stat/SampleSplit/FiniteCategoryPilot.lean"},{"n":"Causalean.Stat.cdfIF","k":"def","m":"Causalean.Stat.Quantile.EmpiricalCDF","a":"Stat/Quantile","t":1,"d":"The fixed-y influence function of the empirical cdf: cdfIF P y z = 1{z ≤ y} − F(y).","s":"MeasureTheory.Measure ℝ → ℝ → ℝ → ℝ","l":100,"f":"Causalean/Stat/Quantile/EmpiricalCDF.lean"},{"n":"Causalean.Stat.cdfIF_mean_zero","k":"theorem","m":"Causalean.Stat.Quantile.EmpiricalCDF","a":"Stat/Quantile","t":2,"d":"cdfIF has mean zero under P.","s":"∀ {P : MeasureTheory.Measure ℝ} [MeasureTheory.IsProbabilityMeasure P] (y : ℝ), ∫ (z : ℝ), Causalean.Stat.cdfIF P y z ∂P = 0","l":109,"f":"Causalean/Stat/Quantile/EmpiricalCDF.lean"},{"n":"Causalean.Stat.cdfIF_sq_integrable","k":"theorem","m":"Causalean.Stat.Quantile.EmpiricalCDF","a":"Stat/Quantile","t":2,"d":"cdfIF is square-integrable (it is bounded).","s":"∀ {P : MeasureTheory.Measure ℝ} [MeasureTheory.IsProbabilityMeasure P] (y : ℝ), MeasureTheory.Integrable (fun z => Causalean.Stat.cdfIF P y z ^ 2) P","l":130,"f":"Causalean/Stat/Quantile/EmpiricalCDF.lean"},{"n":"Causalean.Stat.cdfIF_variance","k":"theorem","m":"Causalean.Stat.Quantile.EmpiricalCDF","a":"Stat/Quantile","t":1,"d":"Variance of the empirical-cdf influence function. At a fixed threshold y, the population second moment of the cdf influence function equals F(y)(1 − F(y)), the Bernoulli variance of the indicator event {Z ≤ y}.","s":"∀ {P : MeasureTheory.Measure ℝ} [MeasureTheory.IsProbabilityMeasure P] (y : ℝ), ∫ (z : ℝ), Causalean.Stat.cdfIF P y z ^ 2 ∂P = ↑(ProbabilityTheory.cdf P) y * (1 - ↑(ProbabilityTheory.cdf P) y)","l":139,"f":"Causalean/Stat/Quantile/EmpiricalCDF.lean"},{"n":"Causalean.Stat.cdfStat","k":"def","m":"Causalean.Stat.Quantile.EmpiricalCDF","a":"Stat/Quantile","t":1,"d":"The indicator statistic 1{· ≤ y} : ℝ → ℝ of the lower-ray event.","s":"ℝ → ℝ → ℝ","l":54,"f":"Causalean/Stat/Quantile/EmpiricalCDF.lean"},{"n":"Causalean.Stat.cdfStat_le_one","k":"theorem","m":"Causalean.Stat.Quantile.EmpiricalCDF","a":"Stat/Quantile","t":2,"d":"The lower-ray indicator is bounded above by one.","s":"∀ (y z : ℝ), Causalean.Stat.cdfStat y z ≤ 1","l":67,"f":"Causalean/Stat/Quantile/EmpiricalCDF.lean"},{"n":"Causalean.Stat.cdfStat_nonneg","k":"theorem","m":"Causalean.Stat.Quantile.EmpiricalCDF","a":"Stat/Quantile","t":2,"d":"The lower-ray indicator is nonnegative.","s":"∀ (y z : ℝ), 0 ≤ Causalean.Stat.cdfStat y z","l":62,"f":"Causalean/Stat/Quantile/EmpiricalCDF.lean"},{"n":"Causalean.Stat.cdfStat_sq","k":"theorem","m":"Causalean.Stat.Quantile.EmpiricalCDF","a":"Stat/Quantile","t":2,"d":"The indicator is idempotent: (1{z ≤ y})² = 1{z ≤ y}.","s":"∀ (y z : ℝ), Causalean.Stat.cdfStat y z ^ 2 = Causalean.Stat.cdfStat y z","l":72,"f":"Causalean/Stat/Quantile/EmpiricalCDF.lean"},{"n":"Causalean.Stat.cdf_increment_sqrt_tendsto","k":"theorem","m":"Causalean.Stat.Quantile.SampleQuantileBahadur.Oscillation","a":"Stat/Quantile/SampleQuantileBahadur","t":2,"d":"Taylor increment. √n (F(q₀+a/√n) − F(q₀)) → f₀·a from HasDerivAt F f₀ q₀. Used by both the L3 oscillation mesh term and the L4 root-n rate (in Rate.lean).","s":"∀ {P : MeasureTheory.Measure ℝ} {τ q₀ f₀ : ℝ}, Causalean.Stat.SampleQuantileReg P τ q₀ f₀ → ∀ (a : ℝ), Filter.Tendsto (fun n => √↑n * (↑(ProbabilityTheory.cdf P) (q₀ + a / √↑n) - ↑(ProbabilityTheory.cdf P) q₀)) Filter.atTop (nhds (f₀ * a))","l":502,"f":"Causalean/Stat/Quantile/SampleQuantileBahadur/Oscillation.lean"},{"n":"Causalean.Stat.cellMoment","k":"def","m":"Causalean.Stat.Sample.CollisionEstimator","a":"Stat/Sample","t":1,"d":"A cell moment is the sample average of a statistic within one projected cell, rescaled by that cell's inverse population frequency.","s":"{𝒳 : Type u_1} → {Ω : Type u_2} → (𝒳 → ℝ) → (Ω → 𝒳) → {n : ℕ} → (Fin n → Ω) → (Ω → ℝ) → 𝒳 → ℝ","l":36,"f":"Causalean/Stat/Sample/CollisionEstimator.lean"},{"n":"Causalean.Stat.centeredCrossMoment","k":"def","m":"Causalean.Stat.UStatistic.OrderM.MixedOrderCovariance","a":"Stat/UStatistic/OrderM","t":1,"d":"The centered cross moment subtracts the product of two means from the mean of their product.","s":"{Ω : Type u_1} → [inst : MeasurableSpace Ω] → MeasureTheory.Measure Ω → (Ω → ℝ) → (Ω → ℝ) → ℝ","l":534,"f":"Causalean/Stat/UStatistic/OrderM/MixedOrderCovariance.lean"},{"n":"Causalean.Stat.centeredCrossMoment_normalizedOrderedProductStatistic","k":"theorem","m":"Causalean.Stat.UStatistic.OrderM.MixedOrderCovariance","a":"Stat/UStatistic/OrderM","t":1,"d":"Under a probability population law, for an i.i.d. sample, statistic orders and a sample size, if the first order and the second order do not exceed the sample size, then for two coordinate-function families, when their product kernels are measurable, their product kernels are integrable, every merged kernel is measurable, and every merged kernel is integrable, their centered cross moment equals an explicit disjoint normalization correction plus the merged moments from every positive-size partial matching.","s":"∀ {Ω : Type u_1} {X : Type u_2} [inst : MeasurableSpace Ω] [inst_1 : MeasurableSpace X] {μ : MeasureTheory.Measure Ω} {P : MeasureTheory.Measure X} [MeasureTheory.IsProbabilityMeasure P] (S : Causalean.Stat.IIDSample Ω X μ P) {r s n : ℕ}, r ≤ n → s ≤ n → ∀ (f : Fin r → X → ℝ) (g : Fin s → X → ℝ), Measurable (Causalean.Stat.orderedProductKernel f) → Measurable (Causalean.Stat.orderedProductKernel g) → MeasureTheory.Integrable (Causalean.Stat.orderedProductKernel f) (MeasureTheory.Measure.pi fun x => P) → MeasureTheory.Integrable (Causalean.Stat.orderedProductKernel g) (MeasureTheory.Measure.pi fun x => P) → (∀ (M : Causalean.Stat.PartialMatching r s), Measurable (Causalean.Stat.mergedProductKernel f g M)) → (∀ (M : Causalean.Stat.PartialMatching r s), MeasureTheory.Integrable (Causalean.Stat.mergedProductKernel f g M) (MeasureTheory.Measure.pi fun x => P)) → Causalean.Stat.centeredCrossMoment μ (Causalean.Stat.normalizedOrderedProductStatistic S f n) (Causalean.Stat.normalizedOrderedProductStatistic S g n) = (Causalean.Stat.matchingNormalization n (Causalean.Stat.PartialMatching.empty r s) - 1) * Causalean.Stat.orderedProductMean P f * Causalean.Stat.orderedProductMean P g + ∑ h ∈ Finset.range (min r s + 1) with 0 < h, ∑ M ∈ Causalean.Stat.partialMatchingsOfSize r s h, Causalean.Stat.matchingNormalization n M * Causalean.Stat.mergedProductMoment P f g M","l":585,"f":"Causalean/Stat/UStatistic/OrderM/MixedOrderCovariance.lean"},{"n":"Causalean.Stat.charFun_isGaussian_centered","k":"theorem","m":"Causalean.Stat.CLT.GaussianCharFunBridge","a":"Stat/CLT","t":1,"d":"Centered-Gaussian characteristic function. For a Gaussian measure Q on E with mean zero, its characteristic function at any point t equals $\\exp(-\\tfrac12\\,\\mathrm{covarianceBilin}\\ Q\\ t\\ t)$.","s":"∀ {E : Type u_1} [inst : NormedAddCommGroup E] [inst_1 : InnerProductSpace ℝ E] [inst_2 : MeasurableSpace E] [inst_3 : BorelSpace E] [SecondCountableTopology E] [CompleteSpace E] (Q : MeasureTheory.Measure E) [ProbabilityTheory.IsGaussian Q], ∫ (x : E), x ∂Q = 0 → ∀ (t : E), MeasureTheory.charFun Q t = Complex.exp (-↑(((ProbabilityTheory.covarianceBilin Q) t) t) / 2)","l":50,"f":"Causalean/Stat/CLT/GaussianCharFunBridge.lean"},{"n":"Causalean.Stat.charFun_isGaussian_of_cov_eq","k":"theorem","m":"Causalean.Stat.CLT.GaussianCharFunBridge","a":"Stat/CLT","t":1,"d":"Bridge to the abstract CLT target hQ. For a centered Gaussian measure Q on E with mean zero whose covariance bilinear form at every t equals the influence-function second moment $\\int \\langle t,\\psi(x)\\rangle^2\\,dP$, its characteristic function at t equals $\\exp(-\\tfrac12\\int \\langle t,\\psi(x)\\rangle^2\\,dP)$ — the exact target shape consumed as the hypothesis hQ by IIDSample.clt_normalizedSum_vec_of_charFun and friends.","s":"∀ {E : Type u_1} [inst : NormedAddCommGroup E] [inst_1 : InnerProductSpace ℝ E] [inst_2 : MeasurableSpace E] [inst_3 : BorelSpace E] [SecondCountableTopology E] [CompleteSpace E] {X : Type u_2} [inst_6 : MeasurableSpace X] {P : MeasureTheory.Measure X} {ψ : X → E} (Q : MeasureTheory.Measure E) [ProbabilityTheory.IsGaussian Q], ∫ (x : E), x ∂Q = 0 → (∀ (t : E), ((ProbabilityTheory.covarianceBilin Q) t) t = ∫ (x : X), inner ℝ t (ψ x) ^ 2 ∂P) → ∀ (t : E), MeasureTheory.charFun Q t = Complex.exp (-↑(∫ (x : X), inner ℝ t (ψ x) ^ 2 ∂P) / 2)","l":64,"f":"Causalean/Stat/CLT/GaussianCharFunBridge.lean"},{"n":"Causalean.Stat.chiSqDist","k":"def","m":"Causalean.Stat.CLT.ChiSquared","a":"Stat/CLT","t":1,"d":"The chi-squared distribution with d degrees of freedom: the law of the squared norm of the standard d-dimensional Gaussian.","s":"ℕ → MeasureTheory.Measure ℝ","l":48,"f":"Causalean/Stat/CLT/ChiSquared.lean"},{"n":"Causalean.Stat.chiSqDiv","k":"def","m":"Causalean.Stat.Minimax.ChiSquared","a":"Stat/Minimax","t":1,"d":"The finite real χ² functional ∫ (dμ/dν − 1)² dν of μ relative to ν.","s":"{Ω : Type u_1} → {mΩ : MeasurableSpace Ω} → MeasureTheory.Measure Ω → MeasureTheory.Measure Ω → ℝ","l":53,"f":"Causalean/Stat/Minimax/ChiSquared.lean"},{"n":"Causalean.Stat.chiSqDiv_eq","k":"theorem","m":"Causalean.Stat.Minimax.ChiSquared","a":"Stat/Minimax","t":2,"d":"Expansion of the χ²-divergence: χ²(μ‖ν) = ∫ (dμ/dν)² ∂ν − 1 when μ ≪ ν.","s":"∀ {Ω : Type u_1} {mΩ : MeasurableSpace Ω} {μ ν : MeasureTheory.Measure Ω} [MeasureTheory.IsProbabilityMeasure μ] [MeasureTheory.IsProbabilityMeasure ν], μ.AbsolutelyContinuous ν → MeasureTheory.Integrable (fun x => ((μ.rnDeriv ν x).toReal - 1) ^ 2) ν → Causalean.Stat.chiSqDiv μ ν = ∫ (x : Ω), (μ.rnDeriv ν x).toReal ^ 2 ∂ν - 1","l":114,"f":"Causalean/Stat/Minimax/ChiSquared.lean"},{"n":"Causalean.Stat.chiSqDiv_map_measurableEquiv","k":"theorem","m":"Causalean.Stat.Minimax.ChiSquared","a":"Stat/Minimax","t":2,"d":"χ²-divergence is invariant under a measurable equivalence. Pushing both measures forward through e : Ω ≃ᵐ Ω' leaves the χ²-divergence unchanged.","s":"∀ {Ω : Type u_4} {Ω' : Type u_5} [inst : MeasurableSpace Ω] [inst_1 : MeasurableSpace Ω'] (e : Ω ≃ᵐ Ω') (μ ν : MeasureTheory.Measure Ω) [MeasureTheory.SigmaFinite μ] [MeasureTheory.SigmaFinite ν], Causalean.Stat.chiSqDiv (MeasureTheory.Measure.map (⇑e) μ) (MeasureTheory.Measure.map (⇑e) ν) = Causalean.Stat.chiSqDiv μ ν","l":243,"f":"Causalean/Stat/Minimax/ChiSquared.lean"},{"n":"Causalean.Stat.chiSqDiv_nonneg","k":"theorem","m":"Causalean.Stat.Minimax.ChiSquared","a":"Stat/Minimax","t":2,"d":"The χ²-divergence is nonnegative: its integrand is a square.","s":"∀ {Ω : Type u_1} {mΩ : MeasurableSpace Ω} {μ ν : MeasureTheory.Measure Ω}, 0 ≤ Causalean.Stat.chiSqDiv μ ν","l":57,"f":"Causalean/Stat/Minimax/ChiSquared.lean"},{"n":"Causalean.Stat.chiSqDiv_prod","k":"theorem","m":"Causalean.Stat.Minimax.ChiSquared","a":"Stat/Minimax","t":2,"d":"Binary tensorization of the χ²-divergence. For probability measures with μ₁ ≪ ν₁ and μ₂ ≪ ν₂ and integrable squared density deviations, the χ²-divergence tensorizes multiplicatively: 1 + χ²(μ₁⊗μ₂ ‖ ν₁⊗ν₂) = (1 + χ²(μ₁‖ν₁))·(1 + χ²(μ₂‖ν₂)).","s":"∀ {α : Type u_2} {mα : MeasurableSpace α} {β : Type u_3} {mβ : MeasurableSpace β} (μ₁ ν₁ : MeasureTheory.Measure α) (μ₂ ν₂ : MeasureTheory.Measure β) [MeasureTheory.IsProbabilityMeasure μ₁] [MeasureTheory.IsProbabilityMeasure ν₁] [MeasureTheory.IsProbabilityMeasure μ₂] [MeasureTheory.IsProbabilityMeasure ν₂], μ₁.AbsolutelyContinuous ν₁ → μ₂.AbsolutelyContinuous ν₂ → MeasureTheory.Integrable (fun x => ((μ₁.rnDeriv ν₁ x).toReal - 1) ^ 2) ν₁ → MeasureTheory.Integrable (fun y => ((μ₂.rnDeriv ν₂ y).toReal - 1) ^ 2) ν₂ → 1 + Causalean.Stat.chiSqDiv (μ₁.prod μ₂) (ν₁.prod ν₂) = (1 + Causalean.Stat.chiSqDiv μ₁ ν₁) * (1 + Causalean.Stat.chiSqDiv μ₂ ν₂)","l":180,"f":"Causalean/Stat/Minimax/ChiSquared.lean"},{"n":"Causalean.Stat.chiSqDiv_prod_ancillary","k":"theorem","m":"Causalean.Stat.Minimax.ChiSquared","a":"Stat/Minimax","t":2,"d":"Ancillary product factor leaves the χ²-divergence unchanged. Tensoring both measures with a *common* probability measure ρ (an ancillary coordinate, whose law is the same under μ and ν) does not change the χ²-divergence: χ²(μ⊗ρ ‖ ν⊗ρ) = χ²(μ‖ν). This is the formal content of \"an ancillary observation carries no information\": its contribution is 1 + χ²(ρ‖ρ) = 1. It is the bridge from the finite-cell lower bound to the continuous-covariate one (the within-cell position is ancillary).","s":"∀ {α : Type u_2} {β : Type u_3} [inst : MeasurableSpace α] [inst_1 : MeasurableSpace β] (μ ν : MeasureTheory.Measure α) (ρ : MeasureTheory.Measure β) [MeasureTheory.IsProbabilityMeasure μ] [MeasureTheory.IsProbabilityMeasure ν] [MeasureTheory.IsProbabilityMeasure ρ], μ.AbsolutelyContinuous ν → MeasureTheory.Integrable (fun x => ((μ.rnDeriv ν x).toReal - 1) ^ 2) ν → Causalean.Stat.chiSqDiv (μ.prod ρ) (ν.prod ρ) = Causalean.Stat.chiSqDiv μ ν","l":440,"f":"Causalean/Stat/Minimax/ChiSquared.lean"},{"n":"Causalean.Stat.chiSqDiv_self","k":"theorem","m":"Causalean.Stat.Minimax.ChiSquared","a":"Stat/Minimax","t":2,"d":"The χ²-divergence of a (sigma-finite) measure against itself is zero.","s":"∀ {Ω : Type u_2} [inst : MeasurableSpace Ω] (ρ : MeasureTheory.Measure Ω) [MeasureTheory.SigmaFinite ρ], Causalean.Stat.chiSqDiv ρ ρ = 0","l":257,"f":"Causalean/Stat/Minimax/ChiSquared.lean"},{"n":"Causalean.Stat.classCoverage_liminf","k":"theorem","m":"Causalean.Stat.Minimax.HonestConfidenceSet","a":"Stat/Minimax","t":2,"d":"Pointwise coverage rows with a vanishing uniform error imply asymptotic uniform coverage over any eventually inhabited sequence of model classes.","s":"∀ {Model : Type u_1} (cls : ℕ → Model → Prop) (coverage : ℕ → Model → ℝ) (alpha : ℝ) (delta : ℕ → ℝ), Filter.Tendsto delta Filter.atTop (nhds 0) → (∀ᶠ (n : ℕ) in Filter.atTop, ∃ P, cls n P) → (∀ (n : ℕ) (P : Model), cls n P → 0 ≤ coverage n P ∧ coverage n P ≤ 1) → (∀ (n : ℕ) (P : Model), cls n P → 1 - alpha - delta n ≤ coverage n P) → 1 - alpha ≤ Filter.liminf (fun n => ⨅ P, coverage n ↑P) Filter.atTop","l":169,"f":"Causalean/Stat/Minimax/HonestConfidenceSet.lean"},{"n":"Causalean.Stat.classFrontierRisk","k":"def","m":"Causalean.Stat.Minimax.HonestConfidenceSet","a":"Stat/Minimax","t":1,"d":"The asymptotic frontier risk of an arbitrary model class is the limsup of the worst expected length among models above a strength threshold.","s":"{Model : Type u_1} → (ℕ → Model → Prop) → (ℕ → Model → ℝ) → (ℕ → Model → ℝ) → ℝ → ℝ","l":223,"f":"Causalean/Stat/Minimax/HonestConfidenceSet.lean"},{"n":"Causalean.Stat.classFrontierRisk_le","k":"theorem","m":"Causalean.Stat.Minimax.HonestConfidenceSet","a":"Stat/Minimax","t":2,"d":"A pointwise capped inverse-square-root expected-length bound passes through both the class supremum and asymptotic limsup at the threshold value.","s":"∀ {Model : Type u_1} (cls : ℕ → Model → Prop) (strength expectedLength : ℕ → Model → ℝ) (C0 t0 : ℝ), 0 ≤ C0 → 0 < t0 → (∀ (n : ℕ) (P : Model), cls n P → 0 ≤ expectedLength n P) → (∀ (n : ℕ) (P : Model), cls n P → expectedLength n P ≤ C0 * min 1 (strength n P ^ (-1 / 2))) → Causalean.Stat.classFrontierRisk cls strength expectedLength t0 ≤ C0 * min 1 (t0 ^ (-1 / 2))","l":232,"f":"Causalean/Stat/Minimax/HonestConfidenceSet.lean"},{"n":"Causalean.Stat.collisionKernel","k":"def","m":"Causalean.Stat.Sample.CollisionEstimator","a":"Stat/Sample","t":1,"d":"The collision kernel contributes the inverse mass of a cell when two observations have the same label and zero otherwise.","s":"{𝒳 : Type u_1} → (𝒳 → ℝ) → 𝒳 → 𝒳 → ℝ","l":22,"f":"Causalean/Stat/Sample/CollisionEstimator.lean"},{"n":"Causalean.Stat.collisionScale","k":"def","m":"Causalean.Stat.Sample.CollisionEstimator","a":"Stat/Sample","t":1,"d":"The collision scale averages inverse-frequency collisions over distinct ordered pairs in a sample.","s":"{𝒳 : Type u_1} → (𝒳 → ℝ) → {N : ℕ} → (Fin N → 𝒳) → ℝ","l":29,"f":"Causalean/Stat/Sample/CollisionEstimator.lean"},{"n":"Causalean.Stat.comonotoneCoupling","k":"def","m":"Causalean.Stat.Coupling.ProductLossMonotoneCoupling.Coupling","a":"Stat/Coupling/ProductLossMonotoneCoupling","t":1,"d":"The comonotone (quantile) coupling of μ and ν: the pushforward of the uniform measure on (0,1) under u ↦ (quantile μ u, quantile ν u). Both coordinates are driven by the *same* uniform draw, giving the maximal positive dependence within Π(μ, ν).","s":"MeasureTheory.Measure ℝ → MeasureTheory.Measure ℝ → MeasureTheory.Measure (ℝ × ℝ)","l":43,"f":"Causalean/Stat/Coupling/ProductLossMonotoneCoupling/Coupling.lean"},{"n":"Causalean.Stat.compressedCoordinateLaw","k":"def","m":"Causalean.Stat.Minimax.CoordinatewiseOverlap","a":"Stat/Minimax","t":1,"d":"The law of a compressed local observation.","s":"{Z : Type u_1} → {S : Type u_2} → [inst : MeasurableSpace Z] → [inst_1 : MeasurableSpace S] → (Z → S) → MeasureTheory.Measure Z → MeasureTheory.Measure S","l":23,"f":"Causalean/Stat/Minimax/CoordinatewiseOverlap.lean"},{"n":"Causalean.Stat.compressedCoordinateLaw_klDiv_le","k":"theorem","m":"Causalean.Stat.Minimax.CoordinatewiseOverlap","a":"Stat/Minimax","t":2,"d":"Measurable finite-coordinate compression cannot increase KL divergence.","s":"∀ {Z : Type u_1} {S : Type u_2} [inst : MeasurableSpace Z] [inst_1 : MeasurableSpace S] (compress : Z → S), Measurable compress → ∀ (μ ν : MeasureTheory.Measure Z) [MeasureTheory.IsFiniteMeasure μ] [MeasureTheory.IsFiniteMeasure ν], InformationTheory.klDiv (Causalean.Stat.compressedCoordinateLaw compress μ) (Causalean.Stat.compressedCoordinateLaw compress ν) ≤ InformationTheory.klDiv μ ν","l":29,"f":"Causalean/Stat/Minimax/CoordinatewiseOverlap.lean"},{"n":"Causalean.Stat.compressionCoupling","k":"def","m":"Causalean.Stat.Minimax.MaximalCoupling","a":"Stat/Minimax","t":1,"d":"A maximal coupling of compressed laws, lifted through the two regular conditional distributions back to the original observations.","s":"{Z : Type u_1} → {S : Type u_2} → [inst : MeasurableSpace Z] → [StandardBorelSpace Z] → [inst_2 : MeasurableSpace S] → [StandardBorelSpace S] → (Q0 Q1 : MeasureTheory.Measure Z) → [MeasureTheory.IsProbabilityMeasure Q0] → [MeasureTheory.IsProbabilityMeasure Q1] → (compress : Z → S) → Measurable compress → MeasureTheory.Measure (Z × Z)","l":397,"f":"Causalean/Stat/Minimax/MaximalCoupling.lean"},{"n":"Causalean.Stat.compressionCoupling_equal_compression_mass_ge","k":"theorem","m":"Causalean.Stat.Minimax.MaximalCoupling","a":"Stat/Minimax","t":2,"d":"In the lifted coupling, the compressed observations agree with probability at least one minus the total variation of their compressed laws.","s":"∀ {Z : Type u_1} {S : Type u_2} [inst : MeasurableSpace Z] [inst_1 : StandardBorelSpace Z] [inst_2 : MeasurableSpace S] [inst_3 : StandardBorelSpace S] (Q0 Q1 : MeasureTheory.Measure Z) [inst_4 : MeasureTheory.IsProbabilityMeasure Q0] [inst_5 : MeasureTheory.IsProbabilityMeasure Q1] (compress : Z → S) (hcompress : Measurable compress), ENNReal.ofReal (1 - Causalean.Stat.tvDist (MeasureTheory.Measure.map compress Q0) (MeasureTheory.Measure.map compress Q1)) ≤ (Causalean.Stat.compressionCoupling Q0 Q1 compress hcompress) {p | compress p.1 = compress p.2}","l":559,"f":"Causalean/Stat/Minimax/MaximalCoupling.lean"},{"n":"Causalean.Stat.compressionCoupling_map_compress_pair","k":"theorem","m":"Causalean.Stat.Minimax.MaximalCoupling","a":"Stat/Minimax","t":2,"d":"Compressing both coordinates of the lifted coupling recovers the maximal coupling of the compressed laws.","s":"∀ {Z : Type u_1} {S : Type u_2} [inst : MeasurableSpace Z] [inst_1 : StandardBorelSpace Z] [inst_2 : MeasurableSpace S] [inst_3 : StandardBorelSpace S] (Q0 Q1 : MeasureTheory.Measure Z) [inst_4 : MeasureTheory.IsProbabilityMeasure Q0] [inst_5 : MeasureTheory.IsProbabilityMeasure Q1] (compress : Z → S) (hcompress : Measurable compress) [inst_6 : MeasureTheory.IsProbabilityMeasure (MeasureTheory.Measure.map compress Q0)] [inst_7 : MeasureTheory.IsProbabilityMeasure (MeasureTheory.Measure.map compress Q1)], MeasureTheory.Measure.map (Prod.map compress compress) (Causalean.Stat.compressionCoupling Q0 Q1 compress hcompress) = Causalean.Stat.maximalCoupling (MeasureTheory.Measure.map compress Q0) (MeasureTheory.Measure.map compress Q1)","l":490,"f":"Causalean/Stat/Minimax/MaximalCoupling.lean"},{"n":"Causalean.Stat.compressionCoupling_map_fst","k":"theorem","m":"Causalean.Stat.Minimax.MaximalCoupling","a":"Stat/Minimax","t":2,"d":"The first marginal of the lifted compression coupling is the first raw law.","s":"∀ {Z : Type u_1} {S : Type u_2} [inst : MeasurableSpace Z] [inst_1 : StandardBorelSpace Z] [inst_2 : MeasurableSpace S] [inst_3 : StandardBorelSpace S] (Q0 Q1 : MeasureTheory.Measure Z) [inst_4 : MeasureTheory.IsProbabilityMeasure Q0] [inst_5 : MeasureTheory.IsProbabilityMeasure Q1] (compress : Z → S) (hcompress : Measurable compress), MeasureTheory.Measure.map Prod.fst (Causalean.Stat.compressionCoupling Q0 Q1 compress hcompress) = Q0","l":438,"f":"Causalean/Stat/Minimax/MaximalCoupling.lean"},{"n":"Causalean.Stat.compressionCoupling_map_snd","k":"theorem","m":"Causalean.Stat.Minimax.MaximalCoupling","a":"Stat/Minimax","t":2,"d":"The second marginal of the lifted compression coupling is the second raw law.","s":"∀ {Z : Type u_1} {S : Type u_2} [inst : MeasurableSpace Z] [inst_1 : StandardBorelSpace Z] [inst_2 : MeasurableSpace S] [inst_3 : StandardBorelSpace S] (Q0 Q1 : MeasureTheory.Measure Z) [inst_4 : MeasureTheory.IsProbabilityMeasure Q0] [inst_5 : MeasureTheory.IsProbabilityMeasure Q1] (compress : Z → S) (hcompress : Measurable compress), MeasureTheory.Measure.map Prod.snd (Causalean.Stat.compressionCoupling Q0 Q1 compress hcompress) = Q1","l":464,"f":"Causalean/Stat/Minimax/MaximalCoupling.lean"},{"n":"Causalean.Stat.consistent_lt_norm_of_le_dist","k":"theorem","m":"Causalean.Stat.MEstimation.ExtremumConsistency","a":"Stat/MEstimation","t":2,"d":"Format bridge. On a normed group dist x θ₀ = ‖x − θ₀‖, so the consistency statement produced by mEstimator_consistent_of_glivenkoCantelli (phrased with ε ≤ dist (θn n) θ₀) implies the strictly-larger-radius form ε < ‖θn n − θ₀‖ consumed by the CLT layer. {ε < ‖·‖} ⊆ {ε ≤ dist}, so the measures are squeezed to 0.","s":"∀ {Ω : Type u_1} [inst : MeasurableSpace Ω] {μ : MeasureTheory.Measure Ω} {E : Type u_3} [inst_1 : NormedAddCommGroup E] (θn : ℕ → Ω → E) (θ₀ : E), (∀ (ε : ℝ), 0 < ε → Filter.Tendsto (fun n => μ {ω | ε ≤ dist (θn n ω) θ₀}) Filter.atTop (nhds 0)) → ∀ ε > 0, Filter.Tendsto (fun n => μ {ω | ε < ‖θn n ω - θ₀‖}) Filter.atTop (nhds 0)","l":67,"f":"Causalean/Stat/MEstimation/ExtremumConsistency.lean"},{"n":"Causalean.Stat.continuous_maxDirDeriv","k":"theorem","m":"Causalean.Stat.Inference.HadamardDeriv","a":"Stat/Inference","t":2,"d":"The directional derivative of the maximum functional is continuous.","s":"∀ (a b : ℝ), Continuous (Causalean.Stat.maxDirDeriv a b)","l":165,"f":"Causalean/Stat/Inference/HadamardDeriv.lean"},{"n":"Causalean.Stat.continuous_minDirDeriv","k":"theorem","m":"Causalean.Stat.Inference.HadamardDeriv","a":"Stat/Inference","t":2,"d":"The directional derivative of the minimum functional is continuous.","s":"∀ (a b : ℝ), Continuous (Causalean.Stat.minDirDeriv a b)","l":173,"f":"Causalean/Stat/Inference/HadamardDeriv.lean"},{"n":"Causalean.Stat.coordinateOverlap","k":"def","m":"Causalean.Stat.Minimax.CoordinatewiseOverlap","a":"Stat/Minimax","t":1,"d":"The common-part overlap of the two compressed laws at coordinate j.","s":"{M : ℕ} → {Z : Fin M → Type u_1} → {S : Fin M → Type u_2} → [inst : (j : Fin M) → MeasurableSpace (Z j)] → [(j : Fin M) → MeasurableSpace (S j)] → ((j : Fin M) → Bool → MeasureTheory.Measure (Z j)) → ((j : Fin M) → Z j → S j) → Fin M → ℝ","l":57,"f":"Causalean/Stat/Minimax/CoordinatewiseOverlap.lean"},{"n":"Causalean.Stat.coordinateOverlap_product_le_of_nonnegative_kl","k":"theorem","m":"Causalean.Stat.Minimax.CoordinatewiseOverlap","a":"Stat/Minimax","t":2,"d":"Under a nonnegative logarithmic KL budget, the product of coordinatewise total-variation factors has the finite-M exponential bound.","s":"∀ {M : ℕ}, 1 ≤ M → ∀ {Z : Fin M → Type u_1} {S : Fin M → Type u_2} [inst : (j : Fin M) → MeasurableSpace (Z j)] [inst_1 : (j : Fin M) → MeasurableSpace (S j)] (Q : (j : Fin M) → Bool → MeasureTheory.Measure (Z j)) [∀ (j : Fin M) (b : Bool), MeasureTheory.IsProbabilityMeasure (Q j b)] (compress : (j : Fin M) → Z j → S j), (∀ (j : Fin M), Measurable (compress j)) → ∀ {κ : ℝ}, 0 ≤ κ → (∀ (j : Fin M), InformationTheory.klDiv (Causalean.Stat.compressedCoordinateLaw (compress j) (Q j false)) (Causalean.Stat.compressedCoordinateLaw (compress j) (Q j true)) ≤ ENNReal.ofReal (κ * Real.log ↑M)) → ∏ j, (1 - Causalean.Stat.coordinateOverlap Q compress j) ≤ Real.exp (-↑M ^ (1 - κ) / 2)","l":323,"f":"Causalean/Stat/Minimax/CoordinatewiseOverlap.lean"},{"n":"Causalean.Stat.coordinatewiseSuccessProbability","k":"def","m":"Causalean.Stat.Minimax.CoordinatewiseOverlap","a":"Stat/Minimax","t":1,"d":"The average probability that every decentralized decoder recovers its bit under the uniform hypercube prior and the conditionally independent product experiment.","s":"{M : ℕ} → {Z : Fin M → Type u_1} → {S : Fin M → Type u_2} → {A : Type u_3} → [inst : (j : Fin M) → MeasurableSpace (Z j)] → [(j : Fin M) → MeasurableSpace (S j)] → [inst_2 : MeasurableSpace A] → (Q : (j : Fin M) → Bool → MeasureTheory.Measure (Z j)) → (R : MeasureTheory.Measure A) → [∀ (j : Fin M) (b : Bool), MeasureTheory.IsProbabilityMeasure (Q j b)] → [MeasureTheory.IsProbabilityMeasure R] → ((j : Fin M) → Z j → S j) → ((j : Fin M) → S j → ((k : Fin M) → Z k) → A → Bool) → ENNReal","l":40,"f":"Causalean/Stat/Minimax/CoordinatewiseOverlap.lean"},{"n":"Causalean.Stat.coordinatewise_overlap_direct_product","k":"theorem","m":"Causalean.Stat.Minimax.CoordinatewiseOverlap","a":"Stat/Minimax","t":1,"d":"Coordinatewise-overlap direct-product bound. In a conditionally independent hypercube experiment with at least one coordinate, per-coordinate candidate laws Q j, indexed by a bit, a per-coordinate compression map compress that is measurable at every coordinate, and per-coordinate decoders decoder built from the compressed local summary, the other coordinates' raw data, and shared randomness that are jointly measurable and depend on the raw sample at coordinate j only through its compressed summary, not directly on the raw value at j, then decentralized coordinate decoders cannot on average be correct more often than the common-part product bound built from the per-coordinate total-variation overlaps; and if every compressed adjacent KL divergence is at most κ log M, the displayed finite-M simultaneous-error certificate follows.","s":"∀ {M : ℕ}, 1 ≤ M → ∀ {Z : Fin M → Type u_1} {S : Fin M → Type u_2} {A : Type u_3} [inst : (j : Fin M) → MeasurableSpace (Z j)] [∀ (j : Fin M), StandardBorelSpace (Z j)] [inst_2 : (j : Fin M) → MeasurableSpace (S j)] [∀ (j : Fin M), StandardBorelSpace (S j)] [inst_4 : MeasurableSpace A] [StandardBorelSpace A] (Q : (j : Fin M) → Bool → MeasureTheory.Measure (Z j)) (R : MeasureTheory.Measure A) [inst_6 : ∀ (j : Fin M) (b : Bool), MeasureTheory.IsProbabilityMeasure (Q j b)] [inst_7 : MeasureTheory.IsProbabilityMeasure R] (compress : (j : Fin M) → Z j → S j), (∀ (j : Fin M), Measurable (compress j)) → ∀ (decoder : (j : Fin M) → S j → ((k : Fin M) → Z k) → A → Bool), (∀ (j : Fin M), Measurable fun p => decoder j p.1 p.2.1 p.2.2) → (∀ (j : Fin M) (s : S j) (z z' : (k : Fin M) → Z k) (a : A), (∀ (k : Fin M), k ≠ j → z k = z' k) → decoder j s z a = decoder j s z' a) → Causalean.Stat.coordinatewiseSuccessProbability Q R compress decoder ≤ ENNReal.ofReal (1 / 2 * (1 + ∏ j, (1 - Causalean.Stat.coordinateOverlap Q compress j))) ∧ ∀ κ < 1, (∀ (j : Fin M), InformationTheory.klDiv (Causalean.Stat.compressedCoordinateLaw (compress j) (Q j false)) (Causalean.Stat.compressedCoordinateLaw (compress j) (Q j true)) ≤ ENNReal.ofReal (κ * Real.log ↑M)) → ENNReal.ofReal (1 / 2 * (1 - Real.exp (-↑M ^ (1 - κ) / 2))) ≤ 1 - Causalean.Stat.coordinatewiseSuccessProbability Q R compress decoder","l":389,"f":"Causalean/Stat/Minimax/CoordinatewiseOverlap.lean"},{"n":"Causalean.Stat.coordinatewise_overlap_direct_product_asymptotic","k":"theorem","m":"Causalean.Stat.Minimax.CoordinatewiseOverlap","a":"Stat/Minimax","t":2,"d":"Along any sequence M_n → ∞ and for fixed κ < 1, the finite direct-product certificate tends to one half.","s":"∀ (Mseq : ℕ → ℕ), ∀ κ < 1, Filter.Tendsto Mseq Filter.atTop Filter.atTop → Filter.Tendsto (fun n => 1 / 2 * (1 - Real.exp (-↑(Mseq n) ^ (1 - κ) / 2))) Filter.atTop (nhds (1 / 2))","l":723,"f":"Causalean/Stat/Minimax/CoordinatewiseOverlap.lean"},{"n":"Causalean.Stat.countermonotoneCoupling","k":"def","m":"Causalean.Stat.Coupling.ProductLossMonotoneCoupling.Coupling","a":"Stat/Coupling/ProductLossMonotoneCoupling","t":1,"d":"The countermonotone coupling of μ and ν: the pushforward of the uniform measure on (0,1) under u ↦ (quantile μ u, quantile ν (1 - u)). The reflection u ↦ 1 - u on the second coordinate produces the maximal negative dependence within Π(μ, ν).","s":"MeasureTheory.Measure ℝ → MeasureTheory.Measure ℝ → MeasureTheory.Measure (ℝ × ℝ)","l":50,"f":"Causalean/Stat/Coupling/ProductLossMonotoneCoupling/Coupling.lean"},{"n":"Causalean.Stat.countermonotone_le_jointCdf","k":"theorem","m":"Causalean.Stat.Coupling.ProductLossMonotoneCoupling.FrechetHoeffdingAttainment","a":"Stat/Coupling/ProductLossMonotoneCoupling","t":2,"d":"Corollary: the countermonotone joint cdf is dominated pointwise by any other coupling's joint cdf.","s":"∀ {π : MeasureTheory.Measure (ℝ × ℝ)} {μ ν : MeasureTheory.Measure ℝ}, Causalean.Stat.IsCoupling π μ ν → ∀ [MeasureTheory.IsProbabilityMeasure μ] [MeasureTheory.IsProbabilityMeasure ν] (x y : ℝ), Causalean.Stat.jointCdf (Causalean.Stat.countermonotoneCoupling μ ν) x y ≤ Causalean.Stat.jointCdf π x y","l":218,"f":"Causalean/Stat/Coupling/ProductLossMonotoneCoupling/FrechetHoeffdingAttainment.lean"},{"n":"Causalean.Stat.countermonotone_le_product_expectation","k":"theorem","m":"Causalean.Stat.Coupling.ProductLossMonotoneCoupling.Optimality","a":"Stat/Coupling/ProductLossMonotoneCoupling","t":1,"d":"Lower optimality. For any coupling π of μ and ν, where both marginals have finite second moment, the expectation of the coordinate product under π is at least its expectation under the countermonotone coupling of μ and ν:","s":"∀ {π : MeasureTheory.Measure (ℝ × ℝ)} {μ ν : MeasureTheory.Measure ℝ}, Causalean.Stat.IsCoupling π μ ν → ∀ [MeasureTheory.IsProbabilityMeasure μ] [MeasureTheory.IsProbabilityMeasure ν], MeasureTheory.MemLp (fun x => x) 2 μ → MeasureTheory.MemLp (fun y => y) 2 ν → ∫ (p : ℝ × ℝ), p.1 * p.2 ∂Causalean.Stat.countermonotoneCoupling μ ν ≤ ∫ (p : ℝ × ℝ), p.1 * p.2 ∂π","l":69,"f":"Causalean/Stat/Coupling/ProductLossMonotoneCoupling/Optimality.lean"},{"n":"Causalean.Stat.coupledDecoderGood","k":"def","m":"Causalean.Stat.Minimax.CoordinatewiseOverlap","a":"Stat/Minimax","t":1,"d":"Simultaneous correctness of all decentralized decoders on a coupled raw sample selected by omega.","s":"{M : ℕ} → {Z : Fin M → Type u_1} → {S : Fin M → Type u_2} → {A : Type u_3} → ((j : Fin M) → Z j → S j) → ((j : Fin M) → S j → ((k : Fin M) → Z k) → A → Bool) → (Fin M → Bool) → ((j : Fin M) → Z j × Z j) → A → Prop","l":75,"f":"Causalean/Stat/Minimax/CoordinatewiseOverlap.lean"},{"n":"Causalean.Stat.coupledDecoderGood_count_le_half","k":"theorem","m":"Causalean.Stat.Minimax.CoordinatewiseOverlap","a":"Stat/Minimax","t":2,"d":"Once one coupled coordinate has equal compressions, at most half of the hypercube vertices can be simultaneously decoded correctly.","s":"∀ {M : ℕ} {Z : Fin M → Type u_1} {S : Fin M → Type u_2} {A : Type u_3} (compress : (j : Fin M) → Z j → S j) (decoder : (j : Fin M) → S j → ((k : Fin M) → Z k) → A → Bool), (∀ (j : Fin M) (s : S j) (z z' : (k : Fin M) → Z k) (a : A), (∀ (k : Fin M), k ≠ j → z k = z' k) → decoder j s z a = decoder j s z' a) → ∀ (z : (j : Fin M) → Z j × Z j) (a : A) (j : Fin M), compress j (z j).1 = compress j (z j).2 → ∑ omega, Causalean.Stat.coupledGoodIndicator compress decoder omega z a ≤ 2 ^ M / 2","l":128,"f":"Causalean/Stat/Minimax/CoordinatewiseOverlap.lean"},{"n":"Causalean.Stat.coupledDecoderGood_flip_exclusive","k":"theorem","m":"Causalean.Stat.Minimax.CoordinatewiseOverlap","a":"Stat/Minimax","t":2,"d":"If the two compressed versions agree at coordinate j, simultaneous correctness is impossible at both endpoints of the corresponding cube edge.","s":"∀ {M : ℕ} {Z : Fin M → Type u_1} {S : Fin M → Type u_2} {A : Type u_3} (compress : (j : Fin M) → Z j → S j) (decoder : (j : Fin M) → S j → ((k : Fin M) → Z k) → A → Bool), (∀ (j : Fin M) (s : S j) (z z' : (k : Fin M) → Z k) (a : A), (∀ (k : Fin M), k ≠ j → z k = z' k) → decoder j s z a = decoder j s z' a) → ∀ (omega : Fin M → Bool) (z : (j : Fin M) → Z j × Z j) (a : A) (j : Fin M), compress j (z j).1 = compress j (z j).2 → ¬(Causalean.Stat.coupledDecoderGood compress decoder omega z a ∧ Causalean.Stat.coupledDecoderGood compress decoder (Causalean.Stat.flipBit j omega) z a)","l":97,"f":"Causalean/Stat/Minimax/CoordinatewiseOverlap.lean"},{"n":"Causalean.Stat.coupledGoodIndicator","k":"def","m":"Causalean.Stat.Minimax.CoordinatewiseOverlap","a":"Stat/Minimax","t":1,"d":"The ENNReal indicator of simultaneous decoder correctness.","s":"{M : ℕ} → {Z : Fin M → Type u_1} → {S : Fin M → Type u_2} → {A : Type u_3} → ((j : Fin M) → Z j → S j) → ((j : Fin M) → S j → ((k : Fin M) → Z k) → A → Bool) → (Fin M → Bool) → ((j : Fin M) → Z j × Z j) → A → ENNReal","l":87,"f":"Causalean/Stat/Minimax/CoordinatewiseOverlap.lean"},{"n":"Causalean.Stat.coupling_fst_memLp","k":"theorem","m":"Causalean.Stat.Coupling.ProductLossMonotoneCoupling.HoeffdingFubini","a":"Stat/Coupling/ProductLossMonotoneCoupling","t":2,"d":"The first coordinate is in L²(π) when μ = π.map Prod.fst has a second moment.","s":"∀ {π : MeasureTheory.Measure (ℝ × ℝ)} {μ ν : MeasureTheory.Measure ℝ}, Causalean.Stat.IsCoupling π μ ν → MeasureTheory.MemLp (fun x => x) 2 μ → MeasureTheory.MemLp (fun p => p.1) 2 π","l":46,"f":"Causalean/Stat/Coupling/ProductLossMonotoneCoupling/HoeffdingFubini.lean"},{"n":"Causalean.Stat.coupling_integrable_mul","k":"theorem","m":"Causalean.Stat.Coupling.ProductLossMonotoneCoupling.HoeffdingFubini","a":"Stat/Coupling/ProductLossMonotoneCoupling","t":2,"d":"XY ∈ L¹(π) by Cauchy–Schwarz from the two L² marginals.","s":"∀ {π : MeasureTheory.Measure (ℝ × ℝ)} {μ ν : MeasureTheory.Measure ℝ}, Causalean.Stat.IsCoupling π μ ν → MeasureTheory.MemLp (fun x => x) 2 μ → MeasureTheory.MemLp (fun y => y) 2 ν → MeasureTheory.Integrable (fun p => p.1 * p.2) π","l":80,"f":"Causalean/Stat/Coupling/ProductLossMonotoneCoupling/HoeffdingFubini.lean"},{"n":"Causalean.Stat.coupling_integral_fst","k":"theorem","m":"Causalean.Stat.Coupling.ProductLossMonotoneCoupling.HoeffdingFubini","a":"Stat/Coupling/ProductLossMonotoneCoupling","t":2,"d":"E_π[X] = E_μ[id]: the first moment of a coupling is that of its first marginal.","s":"∀ {π : MeasureTheory.Measure (ℝ × ℝ)} {μ ν : MeasureTheory.Measure ℝ}, Causalean.Stat.IsCoupling π μ ν → MeasureTheory.MemLp (fun x => x) 2 μ → ∫ (p : ℝ × ℝ), p.1 ∂π = ∫ (x : ℝ), x ∂μ","l":62,"f":"Causalean/Stat/Coupling/ProductLossMonotoneCoupling/HoeffdingFubini.lean"},{"n":"Causalean.Stat.coupling_integral_snd","k":"theorem","m":"Causalean.Stat.Coupling.ProductLossMonotoneCoupling.HoeffdingFubini","a":"Stat/Coupling/ProductLossMonotoneCoupling","t":2,"d":"E_π[Y] = E_ν[id]: the first moment of a coupling is that of its second marginal.","s":"∀ {π : MeasureTheory.Measure (ℝ × ℝ)} {μ ν : MeasureTheory.Measure ℝ}, Causalean.Stat.IsCoupling π μ ν → MeasureTheory.MemLp (fun y => y) 2 ν → ∫ (p : ℝ × ℝ), p.2 ∂π = ∫ (y : ℝ), y ∂ν","l":71,"f":"Causalean/Stat/Coupling/ProductLossMonotoneCoupling/HoeffdingFubini.lean"},{"n":"Causalean.Stat.coupling_snd_memLp","k":"theorem","m":"Causalean.Stat.Coupling.ProductLossMonotoneCoupling.HoeffdingFubini","a":"Stat/Coupling/ProductLossMonotoneCoupling","t":2,"d":"The second coordinate is in L²(π) when ν = π.map Prod.snd has a second moment.","s":"∀ {π : MeasureTheory.Measure (ℝ × ℝ)} {μ ν : MeasureTheory.Measure ℝ}, Causalean.Stat.IsCoupling π μ ν → MeasureTheory.MemLp (fun y => y) 2 ν → MeasureTheory.MemLp (fun p => p.2) 2 π","l":54,"f":"Causalean/Stat/Coupling/ProductLossMonotoneCoupling/HoeffdingFubini.lean"},{"n":"Causalean.Stat.coverageInfOrOne","k":"def","m":"Causalean.Stat.Minimax.HonestConfidenceSet","a":"Stat/Minimax","t":1,"d":"Worst-case coverage is the ordinary infimum when the model class is nonempty, but is defined as one when the class is empty. A real-valued infimum over an empty index would otherwise equal zero and misleadingly signal coverage failure for a vacuous model class.","s":"{ι : Sort u_1} → (ι → ℝ) → ℝ","l":285,"f":"Causalean/Stat/Minimax/HonestConfidenceSet.lean"},{"n":"Causalean.Stat.coverageInfOrOne_of_isEmpty","k":"theorem","m":"Causalean.Stat.Minimax.HonestConfidenceSet","a":"Stat/Minimax","t":2,"d":"On an empty model class, worst-case coverage with the empty-class convention is one, expressing that the coverage requirement is vacuously satisfied.","s":"∀ {ι : Sort u_1} [IsEmpty ι] (f : ι → ℝ), Causalean.Stat.coverageInfOrOne f = 1","l":303,"f":"Causalean/Stat/Minimax/HonestConfidenceSet.lean"},{"n":"Causalean.Stat.coverageInfOrOne_of_nonempty","k":"theorem","m":"Causalean.Stat.Minimax.HonestConfidenceSet","a":"Stat/Minimax","t":2,"d":"On a nonempty model class, worst-case coverage with the empty-class convention is the ordinary infimum of coverage across the class.","s":"∀ {ι : Sort u_1} [Nonempty ι] (f : ι → ℝ), Causalean.Stat.coverageInfOrOne f = ⨅ i, f i","l":296,"f":"Causalean/Stat/Minimax/HonestConfidenceSet.lean"},{"n":"Causalean.Stat.coverage_tv_expectedRestrictedVolume_lower","k":"theorem","m":"Causalean.Stat.Minimax.HonestConfidenceSet","a":"Stat/Minimax","t":1,"d":"Coverage-to-expected-restricted-volume bound. For a family of laws Q u indexed by u : ℝ, a random set C, a subset I of a parameter region region, and a reference point reference, suppose every Q u is a probability measure, C covers u with probability at least coverage, for every u in I, Q u is within total variation tv of the reference law Q reference, for every u in I, the graph {(ω, u) | u ∈ C ω} is measurable, region is measurable, region has finite Lebesgue volume, I is measurable, and I is contained in region. Then the expected restricted volume of C under the reference law Q reference is at least (volume I) · (coverage − tv).","s":"∀ {Ω : Type u_1} [inst : MeasurableSpace Ω] (Q : ℝ → MeasureTheory.Measure Ω) (C : Ω → Set ℝ) (region I : Set ℝ) (reference coverage tv : ℝ), (∀ (u : ℝ), MeasureTheory.IsProbabilityMeasure (Q u)) → (∀ u ∈ I, coverage ≤ ((Q u) {ω | u ∈ C ω}).toReal) → (∀ u ∈ I, Causalean.Stat.tvDist (Q u) (Q reference) ≤ tv) → MeasurableSet {p | p.2 ∈ C p.1} → MeasurableSet region → MeasureTheory.volume region ≠ ⊤ → MeasurableSet I → I ⊆ region → (MeasureTheory.volume I).toReal * (coverage - tv) ≤ ∫ (ω : Ω), Causalean.Stat.restrictedSetVolume region (C ω) ∂Q reference","l":94,"f":"Causalean/Stat/Minimax/HonestConfidenceSet.lean"},{"n":"Causalean.Stat.crossAverage","k":"def","m":"Causalean.Stat.Sample.CollisionEstimator","a":"Stat/Sample","t":1,"d":"A cross-sample average evaluates source-sample cell moments at every target observation and averages the resulting values over the target sample.","s":"{𝒳 : Type u_1} → {Ω : Type u_2} → (𝒳 → ℝ) → (Ω → 𝒳) → {n N : ℕ} → (Fin n → Ω) → (Fin N → 𝒳) → (Ω → ℝ) → ℝ","l":45,"f":"Causalean/Stat/Sample/CollisionEstimator.lean"},{"n":"Causalean.Stat.cumulative_risk_engine","k":"theorem","m":"Causalean.Stat.Limit.SequentialCumulativeRisk","a":"Stat/Limit","t":2,"d":"Sequential-experiment cumulative-risk engine.","s":"∀ (J d L Iq : ℝ), 0 < J → d ≠ 0 → 0 ≤ L → 0 ≤ Iq → ∀ (b B : ℕ → ℝ), (∀ (n : ℕ), B n = ∑ t ∈ Finset.Icc 1 n, b t) → (∀ (t : ℕ), 1 ≤ t → d ^ 2 / 4 / (Iq + 5 * J / 4 * ↑t + L * √(↑t * B (t - 1))) ≤ b t) → ∃ T₀, ∀ (T : ℕ), T₀ ≤ T → d ^ 2 / (32 * J) * Real.log ↑T ≤ B T","l":144,"f":"Causalean/Stat/Limit/SequentialCumulativeRisk.lean"},{"n":"Causalean.Stat.decode_ne_flip_compl","k":"theorem","m":"Causalean.Stat.Minimax.Assouad","a":"Stat/Minimax","t":2,"d":"The j-th decoding-error set at the flipped vertex is the complement of the one at τ: the estimator's j-th bit either matches τ j or its flip, never both.","s":"∀ {Ω : Type u_1} {d : ℕ} (est : Ω → Fin d → Bool) (j : Fin d) (τ : Fin d → Bool), {ω | est ω j ≠ Causalean.Stat.flipBit j τ j} = {ω | est ω j ≠ τ j}ᶜ","l":105,"f":"Causalean/Stat/Minimax/Assouad.lean"},{"n":"Causalean.Stat.deltaMethod","k":"theorem","m":"Causalean.Stat.Inference.DeltaMethod","a":"Stat/Inference","t":1,"d":"Multivariate delta method. Let Tn be a sequence of E-valued estimators of a target t₀, and let g : E → F be Fréchet-differentiable at t₀ with derivative Dg. Given that the rescaled deviations √n • (Tn − t₀) are measurable at every sample size and that their images √n • (g(Tn) − g(t₀)) are measurable at every sample size, if the laws of √n • (Tn − t₀) converge weakly to a probability measure Q on E, then the laws of √n • (g(Tn) − g(t₀)) converge weakly to the pushforward of Q along the linear map Dg.","s":"∀ {Ω : Type u_1} [inst : MeasurableSpace Ω] {μ : MeasureTheory.Measure Ω} [inst_1 : MeasureTheory.IsProbabilityMeasure μ] {E : Type u_2} {F : Type u_3} [inst_2 : NormedAddCommGroup E] [inst_3 : NormedSpace ℝ E] [inst_4 : MeasurableSpace E] [inst_5 : BorelSpace E] [inst_6 : NormedAddCommGroup F] [inst_7 : NormedSpace ℝ F] [inst_8 : MeasurableSpace F] [inst_9 : BorelSpace F] (Tn : ℕ → Ω → E) (t₀ : E) (g : E → F) (Dg : E →L[ℝ] F) (Q : MeasureTheory.ProbabilityMeasure E) (hTn : ∀ (n : ℕ), AEMeasurable (fun ω => √↑n • (Tn n ω - t₀)) μ) (hgTn : ∀ (n : ℕ), AEMeasurable (fun ω => √↑n • (g (Tn n ω) - g t₀)) μ), HasFDerivAt g Dg t₀ → Filter.Tendsto (fun n => ⟨MeasureTheory.Measure.map (fun ω => √↑n • (Tn n ω - t₀)) μ, MeasureTheory.Measure.isProbabilityMeasure_map (hTn n)⟩) Filter.atTop (nhds Q) → Filter.Tendsto (fun n => ⟨MeasureTheory.Measure.map (fun ω => √↑n • (g (Tn n ω) - g t₀)) μ, MeasureTheory.Measure.isProbabilityMeasure_map (hgTn n)⟩) Filter.atTop (nhds ⟨MeasureTheory.Measure.map ⇑Dg ↑Q, MeasureTheory.Measure.isProbabilityMeasure_map (Measurable.aemeasurable (Continuous.measurable (ContinuousLinearMap.continuous Dg)))⟩)","l":263,"f":"Causalean/Stat/Inference/DeltaMethod.lean"},{"n":"Causalean.Stat.deltaMethod_max_tie","k":"theorem","m":"Causalean.Stat.Inference.FunctionalDelta","a":"Stat/Inference","t":1,"d":"Directional delta method for max at a tie. Let ân, b̂n be two real-valued estimator sequences of a common value a. Suppose the joint rescaled deviation √n • ((ân, b̂n) − (a, a)) is measurable at every sample size and it converges in distribution to a probability measure Q on ℝ × ℝ. Then the rescaled deviation of the pointwise maximum, √n · (max(ân, b̂n) − a), converges in distribution to the pushforward of Q under the map (x, y) ↦ max(x, y).","s":"∀ {Ω : Type u_1} [inst : MeasurableSpace Ω] {μ : MeasureTheory.Measure Ω} [inst_1 : MeasureTheory.IsProbabilityMeasure μ] (an bn : ℕ → Ω → ℝ) (a : ℝ) (Q : MeasureTheory.Measure (ℝ × ℝ)) [inst_2 : MeasureTheory.IsProbabilityMeasure Q] (hSn_meas : ∀ (n : ℕ), AEMeasurable (fun ω => √↑n • ((an n ω, bn n ω) - (a, a))) μ), Causalean.Stat.Tendsto_dist_vec (fun n ω => √↑n • ((an n ω, bn n ω) - (a, a))) Q μ hSn_meas → Causalean.Stat.Tendsto_dist (fun n ω => √↑n * (max (an n ω) (bn n ω) - a)) (MeasureTheory.Measure.map (fun z => max z.1 z.2) Q) μ (Causalean.Stat.maxStat_aemeasurable an bn a hSn_meas)","l":165,"f":"Causalean/Stat/Inference/FunctionalDelta.lean"},{"n":"Causalean.Stat.deltaMethod_min_tie","k":"theorem","m":"Causalean.Stat.Inference.FunctionalDelta","a":"Stat/Inference","t":1,"d":"Directional delta method for min at a tie. Let ân, b̂n be two real-valued estimator sequences of a common value a. Suppose the joint rescaled deviation √n • ((ân, b̂n) − (a, a)) is measurable at every sample size and it converges in distribution to a probability measure Q on ℝ × ℝ. Then the rescaled deviation of the pointwise minimum, √n · (min(ân, b̂n) − a), converges in distribution to the pushforward of Q under the map (x, y) ↦ min(x, y).","s":"∀ {Ω : Type u_1} [inst : MeasurableSpace Ω] {μ : MeasureTheory.Measure Ω} [inst_1 : MeasureTheory.IsProbabilityMeasure μ] (an bn : ℕ → Ω → ℝ) (a : ℝ) (Q : MeasureTheory.Measure (ℝ × ℝ)) [inst_2 : MeasureTheory.IsProbabilityMeasure Q] (hSn_meas : ∀ (n : ℕ), AEMeasurable (fun ω => √↑n • ((an n ω, bn n ω) - (a, a))) μ), Causalean.Stat.Tendsto_dist_vec (fun n ω => √↑n • ((an n ω, bn n ω) - (a, a))) Q μ hSn_meas → Causalean.Stat.Tendsto_dist (fun n ω => √↑n * (min (an n ω) (bn n ω) - a)) (MeasureTheory.Measure.map (fun z => min z.1 z.2) Q) μ (Causalean.Stat.minStat_aemeasurable an bn a hSn_meas)","l":202,"f":"Causalean/Stat/Inference/FunctionalDelta.lean"},{"n":"Causalean.Stat.deltaMethod_ratio","k":"theorem","m":"Causalean.Stat.Inference.RatioDeltaMethod","a":"Stat/Inference","t":1,"d":"Ratio / quotient delta method. Let t₀ = (a, b) with b nonzero and let Tn n ω = (N̂ₙ, D̂ₙ) be a bivariate estimator sequence of t₀. Given that the rescaled deviation √n • (Tn − t₀) is measurable at every sample size and that the rescaled ratio √n • (Tn 0 / Tn 1 − a/b) is measurable at every sample size, if the laws of the rescaled deviation converge weakly to a probability measure Q on EuclideanSpace ℝ (Fin 2), then the laws of the rescaled ratio √n • (Tn 0 / Tn 1 − a/b) converge weakly to the pushforward of Q along the ratio derivative ratioDeriv t₀.","s":"∀ {Ω : Type u_1} [inst : MeasurableSpace Ω] {μ : MeasureTheory.Measure Ω} [inst_1 : MeasureTheory.IsProbabilityMeasure μ] (Tn : ℕ → Ω → EuclideanSpace ℝ (Fin 2)) (t₀ : EuclideanSpace ℝ (Fin 2)), t₀.ofLp 1 ≠ 0 → ∀ (Q : MeasureTheory.ProbabilityMeasure (EuclideanSpace ℝ (Fin 2))) (hTn : ∀ (n : ℕ), AEMeasurable (fun ω => √↑n • (Tn n ω - t₀)) μ) (hgTn : ∀ (n : ℕ), AEMeasurable (fun ω => √↑n • ((Tn n ω).ofLp 0 / (Tn n ω).ofLp 1 - t₀.ofLp 0 / t₀.ofLp 1)) μ), Filter.Tendsto (fun n => ⟨MeasureTheory.Measure.map (fun ω => √↑n • (Tn n ω - t₀)) μ, MeasureTheory.Measure.isProbabilityMeasure_map (hTn n)⟩) Filter.atTop (nhds Q) → Filter.Tendsto (fun n => ⟨MeasureTheory.Measure.map (fun ω => √↑n • ((Tn n ω).ofLp 0 / (Tn n ω).ofLp 1 - t₀.ofLp 0 / t₀.ofLp 1)) μ, MeasureTheory.Measure.isProbabilityMeasure_map (hgTn n)⟩) Filter.atTop (nhds ⟨MeasureTheory.Measure.map ⇑(Causalean.Stat.ratioDeriv t₀) ↑Q, MeasureTheory.Measure.isProbabilityMeasure_map (Measurable.aemeasurable (Continuous.measurable (ContinuousLinearMap.continuous (Causalean.Stat.ratioDeriv t₀))))⟩)","l":99,"f":"Causalean/Stat/Inference/RatioDeltaMethod.lean"},{"n":"Causalean.Stat.deltaMethod_scalar","k":"theorem","m":"Causalean.Stat.Inference.DeltaMethod","a":"Stat/Inference","t":1,"d":"Scalar delta method. For a real-valued estimator sequence Tn of a target t₀, suppose the rescaled deviation √n(Tn − t₀) is measurable at every sample size and the rescaled image √n(g(Tn) − g(t₀)) is measurable at every sample size, where g is differentiable at t₀ with derivative g'. If the rescaled deviation converges in distribution to the project's Gaussian law with mean zero and variance σ², then the rescaled image converges in distribution to the same Gaussian wrapper with variance g'² · σ².","s":"∀ {Ω : Type u_1} [inst : MeasurableSpace Ω] {μ : MeasureTheory.Measure Ω} [inst_1 : MeasureTheory.IsProbabilityMeasure μ] (Tn : ℕ → Ω → ℝ) (t₀ : ℝ) (g : ℝ → ℝ) (g' σsq : ℝ) (hTn : ∀ (n : ℕ), AEMeasurable (fun ω => √↑n * (Tn n ω - t₀)) μ) (hgTn : ∀ (n : ℕ), AEMeasurable (fun ω => √↑n * (g (Tn n ω) - g t₀)) μ), HasDerivAt g g' t₀ → Causalean.Stat.Tendsto_dist (fun n ω => √↑n * (Tn n ω - t₀)) (Causalean.Stat.gaussianMeasure 0 σsq) μ hTn → Causalean.Stat.Tendsto_dist (fun n ω => √↑n * (g (Tn n ω) - g t₀)) (Causalean.Stat.gaussianMeasure 0 (g' ^ 2 * σsq)) μ hgTn","l":50,"f":"Causalean/Stat/Inference/DeltaMethod.lean"},{"n":"Causalean.Stat.densityAffinity","k":"def","m":"Causalean.Stat.Minimax.HellingerAffinity","a":"Stat/Minimax","t":1,"d":"The Hellinger (Bhattacharyya) affinity of two nonnegative densities taken against one common dominating measure: the integral of the square root of their pointwise product. It equals one when the two densities agree almost everywhere and falls toward zero as the two laws separate, so it measures how hard the two laws are to tell apart.","s":"{α : Type u_1} → [inst : MeasurableSpace α] → MeasureTheory.Measure α → (α → ℝ) → (α → ℝ) → ℝ","l":53,"f":"Causalean/Stat/Minimax/HellingerAffinity.lean"},{"n":"Causalean.Stat.densityAffinity_pi","k":"theorem","m":"Causalean.Stat.Minimax.HellingerAffinity","a":"Stat/Minimax","t":2,"d":"Affinity tensorizes. On a finite product of σ-finite measure spaces, the affinity of two densities that each factor coordinatewise is the product of the coordinate affinities. This is what makes an n-fold product construction tractable: a single coordinate defect computation is enough.","s":"∀ {ι : Type u_1} [inst : Fintype ι] {E : ι → Type u_2} [inst_1 : (i : ι) → MeasurableSpace (E i)] (μ : (i : ι) → MeasureTheory.Measure (E i)) [∀ (i : ι), MeasureTheory.SigmaFinite (μ i)] (f g : (i : ι) → E i → ℝ), (∀ (i : ι) (u : E i), 0 ≤ f i u) → (∀ (i : ι) (u : E i), 0 ≤ g i u) → (Causalean.Stat.densityAffinity (MeasureTheory.Measure.pi μ) (fun x => ∏ i, f i (x i)) fun x => ∏ i, g i (x i)) = ∏ i, ∫ (u : E i), √(f i u * g i u) ∂μ i","l":321,"f":"Causalean/Stat/Minimax/HellingerAffinity.lean"},{"n":"Causalean.Stat.empProcVec_atEstimator_tendsto_zero","k":"theorem","m":"Causalean.Stat.EmpiricalProcess.Equicontinuity.Modulus","a":"Stat/EmpiricalProcess/Equicontinuity","t":1,"d":"Empirical process at the estimator vanishes. If the score family ψ is asymptotically equicontinuous at θ₀ along the i.i.d. sample S and θn is a sequence of estimators consistent for θ₀, then for any fixed tolerance ε > 0, the centered empirical process of the score gap ψ(θn,·) − ψ(θ₀,·), evaluated at the random estimator θn, namely Gₙ(ψ(θn,·) − ψ(θ₀,·)), converges to 0 in probability — unconditionally, with no restriction to a shrinking neighborhood of θ₀.","s":"∀ {Ω : Type u_1} {X : Type u_2} [inst : MeasurableSpace Ω] [inst_1 : MeasurableSpace X] {μ : MeasureTheory.Measure Ω} {P : MeasureTheory.Measure X} {E : Type u_3} [inst_2 : NormedAddCommGroup E] [inst_3 : InnerProductSpace ℝ E] (ψ : E → X → E) (θ₀ : E) (S : Causalean.Stat.IIDSample Ω X μ P) (θn : ℕ → Ω → E), Causalean.Stat.AsymptoticEquicont ψ θ₀ P μ S → (∀ ε > 0, Filter.Tendsto (fun n => μ {ω | ε < ‖θn n ω - θ₀‖}) Filter.atTop (nhds 0)) → ∀ (ε : ℝ), 0 < ε → Filter.Tendsto (fun n => μ {ω | ε < ‖S.empProcVec (fun z => ψ (θn n ω) z - ψ θ₀ z) n ω‖}) Filter.atTop (nhds 0)","l":94,"f":"Causalean/Stat/EmpiricalProcess/Equicontinuity/Modulus.lean"},{"n":"Causalean.Stat.empProcVec_chebyshev","k":"theorem","m":"Causalean.Stat.EmpiricalProcess.Equicontinuity.SecondMoment","a":"Stat/EmpiricalProcess/Equicontinuity","t":1,"d":"Chebyshev bound for the vector empirical process. For an i.i.d. sample S and a function f that is measurable and square-integrable under P, and for any tolerance ε > 0, the probability that the centered empirical process Gₙ(f) exceeds ε in norm at sample size n is at most $(\\int\\|f\\|^2\\,dP)/\\varepsilon^2$:","s":"∀ {Ω : Type u_1} {X : Type u_2} {E : Type u_3} [inst : MeasurableSpace Ω] [inst_1 : MeasurableSpace X] [inst_2 : NormedAddCommGroup E] [inst_3 : InnerProductSpace ℝ E] [FiniteDimensional ℝ E] {μ : MeasureTheory.Measure Ω} {P : MeasureTheory.Measure X} [MeasureTheory.IsProbabilityMeasure μ] [MeasureTheory.IsProbabilityMeasure P] [inst_7 : MeasurableSpace E] [BorelSpace E] (S : Causalean.Stat.IIDSample Ω X μ P) (f : X → E), Measurable f → MeasureTheory.MemLp f 2 P → ∀ (n : ℕ) {ε : ℝ}, 0 < ε → μ {ω | ε < ‖S.empProcVec f n ω‖} ≤ ENNReal.ofReal (∫ (x : X), ‖f x‖ ^ 2 ∂P) / ENNReal.ofReal (ε ^ 2)","l":215,"f":"Causalean/Stat/EmpiricalProcess/Equicontinuity/SecondMoment.lean"},{"n":"Causalean.Stat.empProcVec_isLittleOp_of_L2","k":"theorem","m":"Causalean.Stat.EmpiricalProcess.Equicontinuity.Modulus","a":"Stat/EmpiricalProcess/Equicontinuity","t":1,"d":"Deterministic-curve equicontinuity (non-vacuousness witness). Suppose that for every parameter value θ, the score gap ψ(θ,·) − ψ(θ₀,·) is measurable and square-integrable under P. Along a deterministic parameter sequence θn whose score perturbation shrinks in L²(P): $\\int\\|\\psi(\\theta_n,\\cdot)-\\psi(\\theta_0,\\cdot)\\|^2\\,dP \\to 0$, the centered empirical-process gap Gₙ(ψ(θn,·) − ψ(θ₀,·)) is $o_p(1)$, with no chaining and no consistency hypothesis: it is a direct consequence of the uniform-in-n Chebyshev bound empProcVec_chebyshev.","s":"∀ {Ω : Type u_1} {X : Type u_2} [inst : MeasurableSpace Ω] [inst_1 : MeasurableSpace X] {μ : MeasureTheory.Measure Ω} {P : MeasureTheory.Measure X} {E : Type u_3} [inst_2 : NormedAddCommGroup E] [inst_3 : InnerProductSpace ℝ E] [FiniteDimensional ℝ E] [inst_5 : MeasurableSpace E] [BorelSpace E] [MeasureTheory.IsProbabilityMeasure μ] [MeasureTheory.IsProbabilityMeasure P] (ψ : E → X → E) (θ₀ : E) (θn : ℕ → E) (S : Causalean.Stat.IIDSample Ω X μ P), (∀ (θ : E), Measurable fun x => ψ θ x - ψ θ₀ x) → (∀ (θ : E), MeasureTheory.MemLp (fun x => ψ θ x - ψ θ₀ x) 2 P) → Filter.Tendsto (fun n => ∫ (x : X), ‖ψ (θn n) x - ψ θ₀ x‖ ^ 2 ∂P) Filter.atTop (nhds 0) → Causalean.Stat.IsLittleOp (fun n ω => ‖S.empProcVec (fun z => ψ (θn n) z - ψ θ₀ z) n ω‖) (fun x => 1) μ","l":180,"f":"Causalean/Stat/EmpiricalProcess/Equicontinuity/Modulus.lean"},{"n":"Causalean.Stat.empProcVec_sq_lintegral_le","k":"theorem","m":"Causalean.Stat.EmpiricalProcess.Equicontinuity.SecondMoment","a":"Stat/EmpiricalProcess/Equicontinuity","t":1,"d":"Second-moment bound for the vector empirical process (variance identity at the empirical scale). For an i.i.d. sample S and a function f that is measurable and square-integrable under P, the second moment of the centered empirical process Gₙ(f) at sample size n is bounded by the population second moment of f:","s":"∀ {Ω : Type u_1} {X : Type u_2} {E : Type u_3} [inst : MeasurableSpace Ω] [inst_1 : MeasurableSpace X] [inst_2 : NormedAddCommGroup E] [inst_3 : InnerProductSpace ℝ E] [FiniteDimensional ℝ E] {μ : MeasureTheory.Measure Ω} {P : MeasureTheory.Measure X} [MeasureTheory.IsProbabilityMeasure μ] [MeasureTheory.IsProbabilityMeasure P] [inst_7 : MeasurableSpace E] [BorelSpace E] (S : Causalean.Stat.IIDSample Ω X μ P) (f : X → E), Measurable f → MeasureTheory.MemLp f 2 P → ∀ (n : ℕ), ∫⁻ (ω : Ω), ENNReal.ofReal (‖S.empProcVec f n ω‖ ^ 2) ∂μ ≤ ENNReal.ofReal (∫ (x : X), ‖f x‖ ^ 2 ∂P)","l":104,"f":"Causalean/Stat/EmpiricalProcess/Equicontinuity/SecondMoment.lean"},{"n":"Causalean.Stat.empiricalCDF_isAsymLinear","k":"theorem","m":"Causalean.Stat.Quantile.EmpiricalCDF","a":"Stat/Quantile","t":1,"d":"Asymptotic linearity of the empirical cdf. For an i.i.d. real sample S at a fixed point y, the empirical cdf F̂ₙ(y) is asymptotically linear at F(y) with influence function cdfIF P y and an identically-zero remainder.","s":"∀ {Ω : Type u_1} [inst : MeasurableSpace Ω] {μ : MeasureTheory.Measure Ω} {P : MeasureTheory.Measure ℝ} [MeasureTheory.IsProbabilityMeasure P] (S : Causalean.Stat.IIDSample Ω ℝ μ P) (y : ℝ), Causalean.Stat.IsAsymLinear (S.empiricalCDF y) (↑(ProbabilityTheory.cdf P) y) (Causalean.Stat.cdfIF P y) S fun m => Finset.range m","l":202,"f":"Causalean/Stat/Quantile/EmpiricalCDF.lean"},{"n":"Causalean.Stat.empiricalCDF_tendsto_inProb","k":"theorem","m":"Causalean.Stat.Quantile.EmpiricalCDF","a":"Stat/Quantile","t":1,"d":"Empirical-cdf consistency (WLLN). For an i.i.d. real sample S at a fixed point y, the empirical cdf F̂ₙ(y) converges to F(y) in probability.","s":"∀ {Ω : Type u_1} [inst : MeasurableSpace Ω] {μ : MeasureTheory.Measure Ω} {P : MeasureTheory.Measure ℝ} [MeasureTheory.IsProbabilityMeasure P] (S : Causalean.Stat.IIDSample Ω ℝ μ P) (y : ℝ), Causalean.Stat.Tendsto_inProb (S.empiricalCDF y) (fun x => ↑(ProbabilityTheory.cdf P) y) μ","l":240,"f":"Causalean/Stat/Quantile/EmpiricalCDF.lean"},{"n":"Causalean.Stat.empiricalCDF_tendsto_normal","k":"theorem","m":"Causalean.Stat.Quantile.EmpiricalCDF","a":"Stat/Quantile","t":1,"d":"Empirical-cdf central limit theorem. For an i.i.d. real sample and a threshold y, provided the rescaled empirical-cdf estimator $\\sqrt n\\,(\\hat F_n(y)-F(y))$ is almost-everywhere measurable at every sample size, the rescaled deviation converges in distribution, as $n\\to \\infty$, to the centered Gaussian law with variance $F(y)(1-F(y))$, where $F$ is the population cdf and $\\hat F_n$ the empirical cdf.","s":"∀ {Ω : Type u_1} [inst : MeasurableSpace Ω] {μ : MeasureTheory.Measure Ω} {P : MeasureTheory.Measure ℝ} [inst_1 : MeasureTheory.IsProbabilityMeasure μ] [MeasureTheory.IsProbabilityMeasure P] (S : Causalean.Stat.IIDSample Ω ℝ μ P) (y : ℝ) (hθn_meas : ∀ (n : ℕ), AEMeasurable (Causalean.Stat.IsAsymLinear.rescaledEstimator (S.empiricalCDF y) (↑(ProbabilityTheory.cdf P) y) (fun m => Finset.range m) n) μ), Causalean.Stat.Tendsto_dist (Causalean.Stat.IsAsymLinear.rescaledEstimator (S.empiricalCDF y) (↑(ProbabilityTheory.cdf P) y) fun m => Finset.range m) (Causalean.Stat.gaussianMeasure 0 (↑(ProbabilityTheory.cdf P) y * (1 - ↑(ProbabilityTheory.cdf P) y))) μ hθn_meas","l":219,"f":"Causalean/Stat/Quantile/EmpiricalCDF.lean"},{"n":"Causalean.Stat.empiricalKishDispersion","k":"def","m":"Causalean.Stat.Sample.EffectiveSampleSize","a":"Stat/Sample","t":1,"d":"Empirical Kish dispersion is the sample average of the squared observation-level weights.","s":"{Ω : Type u_1} → (Ω → ℝ) → (n : ℕ) → (Fin n → Ω) → ℝ","l":26,"f":"Causalean/Stat/Sample/EffectiveSampleSize.lean"},{"n":"Causalean.Stat.empiricalKishDispersion_lower_tail_le","k":"theorem","m":"Causalean.Stat.Sample.EffectiveSampleSize","a":"Stat/Sample","t":1,"d":"Lower-tail bound for empirical Kish dispersion. Given a positive sample size $n$, a positive population Kish dispersion $\\kappa$, the empirical Kish dispersion is square-integrable under the sampling measure Q, its expectation under Q equals $\\kappa$, and its variance under Q is at most $4k^2\\kappa/n$ for a weight envelope $k$, then the probability that the empirical Kish dispersion falls below half its mean $\\kappa/2$ is at most $16k^2/(n\\kappa)$.","s":"∀ {Ω : Type u_1} [inst : MeasurableSpace Ω] (n : ℕ) (Q : MeasureTheory.Measure (Fin n → Ω)) [MeasureTheory.IsProbabilityMeasure Q] (g : Ω → ℝ) (k kappa : ℝ), 0 < n → 0 < kappa → MeasureTheory.MemLp (Causalean.Stat.empiricalKishDispersion g n) 2 Q → ∫ (sample : Fin n → Ω), Causalean.Stat.empiricalKishDispersion g n sample ∂Q = kappa → ProbabilityTheory.variance (Causalean.Stat.empiricalKishDispersion g n) Q ≤ 4 * k ^ 2 * kappa / ↑n → (Q {sample | Causalean.Stat.empiricalKishDispersion g n sample < kappa / 2}).toReal ≤ 16 * k ^ 2 / (↑n * kappa)","l":108,"f":"Causalean/Stat/Sample/EffectiveSampleSize.lean"},{"n":"Causalean.Stat.empiricalKishDispersion_mean","k":"theorem","m":"Causalean.Stat.Sample.EffectiveSampleSize","a":"Stat/Sample","t":1,"d":"Expected empirical Kish dispersion. Given a positive sample size $n$ and an integrable squared weight statistic $g^2$ under the population measure, the expectation of the empirical Kish dispersion — the sample average of the squared observation-level weights — under the $n$-fold product sampling measure equals the population second moment $\\int g^2\\,d\\mu$.","s":"∀ {Ω : Type u_1} [inst : MeasurableSpace Ω] (μ : MeasureTheory.Measure Ω) [MeasureTheory.IsProbabilityMeasure μ] (g : Ω → ℝ) (n : ℕ), 0 < n → MeasureTheory.Integrable (fun o => g o ^ 2) μ → (∫ (sample : Fin n → Ω), Causalean.Stat.empiricalKishDispersion g n sample ∂MeasureTheory.Measure.pi fun x => μ) = ∫ (o : Ω), g o ^ 2 ∂μ","l":43,"f":"Causalean/Stat/Sample/EffectiveSampleSize.lean"},{"n":"Causalean.Stat.empiricalKishDispersion_variance_le","k":"theorem","m":"Causalean.Stat.Sample.EffectiveSampleSize","a":"Stat/Sample","t":2,"d":"If the fourth power of a weight is bounded by four times a squared envelope times its second power, empirical Kish dispersion has variance at most four times the squared envelope and the population second moment, divided by sample size.","s":"∀ {Ω : Type u_1} [inst : MeasurableSpace Ω] (μ : MeasureTheory.Measure Ω) [MeasureTheory.IsProbabilityMeasure μ] (g : Ω → ℝ) (n : ℕ) (k kappa : ℝ), 0 < n → MeasureTheory.MemLp (fun o => g o ^ 2) 2 μ → ∫ (o : Ω), g o ^ 2 ∂μ = kappa → (∀ᵐ (o : Ω) ∂μ, g o ^ 4 ≤ 4 * k ^ 2 * g o ^ 2) → ProbabilityTheory.variance (fun sample => Causalean.Stat.empiricalKishDispersion g n sample) (MeasureTheory.Measure.pi fun x => μ) ≤ 4 * k ^ 2 * kappa / ↑n","l":60,"f":"Causalean/Stat/Sample/EffectiveSampleSize.lean"},{"n":"Causalean.Stat.empiricalMass","k":"def","m":"Causalean.Stat.Sample.EmpiricalMass","a":"Stat/Sample","t":1,"d":"The empirical mass of a point is the fraction of sample observations equal to that point.","s":"{𝒳 : Type u_1} → {N : ℕ} → (Fin N → 𝒳) → 𝒳 → ℝ","l":24,"f":"Causalean/Stat/Sample/EmpiricalMass.lean"},{"n":"Causalean.Stat.empiricalScoreDiff_isLittleOp_sqrt","k":"theorem","m":"Causalean.Stat.MEstimation.EmpiricalExpansion","a":"Stat/MEstimation","t":2,"d":"(b) Empirical-process score difference is o_p(√n).","s":"∀ {Ω : Type u_1} {X : Type u_2} [inst : MeasurableSpace Ω] [inst_1 : MeasurableSpace X] {μ : MeasureTheory.Measure Ω} {E : Type u_3} [inst_2 : NormedAddCommGroup E] [inst_3 : InnerProductSpace ℝ E] [inst_4 : MeasurableSpace E] (ψ : E → X → E) (θ₀ : E) (P : MeasureTheory.Measure X) (_reg : Causalean.Stat.ZEstimatorRegularity ψ θ₀ P) [MeasureTheory.IsProbabilityMeasure μ] (S : Causalean.Stat.IIDSample Ω X μ P) (θn : ℕ → Ω → E), (∀ ε > 0, Filter.Tendsto (fun n => μ {ω | ε < ‖θn n ω - θ₀‖}) Filter.atTop (nhds 0)) → Causalean.Stat.StochEquicontAt ψ θ₀ P μ S θn → Causalean.Stat.IsLittleOp (fun n ω => ‖(√↑n)⁻¹ • ∑ i ∈ Finset.range n, (ψ (θn n ω) (S.Z i ω) - ψ θ₀ (S.Z i ω)) - √↑n • ∫ (z : X), ψ (θn n ω) z - ψ θ₀ z ∂P‖) (fun x => 1) μ","l":348,"f":"Causalean/Stat/MEstimation/EmpiricalExpansion.lean"},{"n":"Causalean.Stat.emptyMatchingNormalization_sub_one_le","k":"theorem","m":"Causalean.Stat.UStatistic.OrderM.MixedOrderBounds","a":"Stat/UStatistic/OrderM","t":2,"d":"For sample size n, orders r and s, and common order bound R, if the first order is at most R, the second order is at most R, and the sample size is at least four times R squared, the empty partial matching's normalization differs from one by at most twice R squared divided by n.","s":"∀ {n r s R : ℕ}, r ≤ R → s ≤ R → 4 * R ^ 2 ≤ n → |Causalean.Stat.matchingNormalization n (Causalean.Stat.PartialMatching.empty r s) - 1| ≤ 2 * ↑R ^ 2 / ↑n","l":280,"f":"Causalean/Stat/UStatistic/OrderM/MixedOrderBounds.lean"},{"n":"Causalean.Stat.ennreal_error_lower_bound_of_success_upper_bound","k":"theorem","m":"Causalean.Stat.Minimax.CoordinatewiseOverlap","a":"Stat/Minimax","t":2,"d":"A real-valued product bound below one converts an ENNReal simultaneous success upper bound into the complementary error lower bound.","s":"∀ {s : ENNReal} {p e : ℝ}, 0 ≤ p → e ≤ 1 → p ≤ e → s ≤ ENNReal.ofReal (1 / 2 * (1 + p)) → ENNReal.ofReal (1 / 2 * (1 - e)) ≤ 1 - s","l":365,"f":"Causalean/Stat/Minimax/CoordinatewiseOverlap.lean"},{"n":"Causalean.Stat.equivFintype","k":"def","m":"Causalean.Stat.UStatistic.OrderM.PartialMatching","a":"Stat/UStatistic/OrderM","t":1,"d":"The equivalences between two finite sets form a finite collection.","s":"{α : Type u_1} → {β : Type u_2} → [Fintype α] → [Fintype β] → Fintype (α ≃ β)","l":120,"f":"Causalean/Stat/UStatistic/OrderM/PartialMatching.lean"},{"n":"Causalean.Stat.eucl","k":"def","m":"Causalean.Stat.Quantile.SampleQuantileJoint","a":"Stat/Quantile","t":1,"d":"Pack a coordinate function into Euclidean space (the PiLp 2 synonym).","s":"{k : ℕ} → (Fin k → ℝ) → EuclideanSpace ℝ (Fin k)","l":56,"f":"Causalean/Stat/Quantile/SampleQuantileJoint.lean"},{"n":"Causalean.Stat.eucl_apply","k":"theorem","m":"Causalean.Stat.Quantile.SampleQuantileJoint","a":"Stat/Quantile","t":2,"d":"Coordinate access for eucl: (eucl v) j = v j.","s":"∀ {k : ℕ} (v : Fin k → ℝ) (j : Fin k), (Causalean.Stat.eucl v).ofLp j = v j","l":134,"f":"Causalean/Stat/Quantile/SampleQuantileJoint.lean"},{"n":"Causalean.Stat.event_pullback_along_iidSample","k":"theorem","m":"Causalean.Stat.Sample.PiTransport","a":"Stat/Sample","t":1,"d":"Event transport along an IID sample (Fin n version). Given a measurable event E in the space of length-n outcome tuples whose product-measure probability under n independent copies of the population law is at least $1-\\delta$, the pullback of E along the joint observable built from the first n coordinates of the i.i.d. sample is a measurable event on the underlying sample space, with probability at least $1-\\delta$.","s":"∀ {Ω : Type u_1} {X : Type u_2} [inst : MeasurableSpace Ω] [inst_1 : MeasurableSpace X] {μ : MeasureTheory.Measure Ω} {P : MeasureTheory.Measure X} (S : Causalean.Stat.IIDSample Ω X μ P) (n : ℕ) {E : Set (Fin n → X)}, MeasurableSet E → ∀ {δ : ℝ}, (MeasureTheory.Measure.pi fun x => P) E ≥ 1 - ENNReal.ofReal δ → have Ψ := fun ω k => S.Z (↑k) ω; have E' := Ψ ⁻¹' E; MeasurableSet E' ∧ μ E' ≥ 1 - ENNReal.ofReal δ","l":90,"f":"Causalean/Stat/Sample/PiTransport.lean"},{"n":"Causalean.Stat.exists_grid_bracket","k":"theorem","m":"Causalean.Stat.Quantile.SampleQuantileBahadur.Oscillation","a":"Stat/Quantile/SampleQuantileBahadur","t":2,"d":"Uniform grid bracketing. For the uniform mesh u_k = −M + 2M·k/K on [−M, M] (K ≥ 1, M > 0), any x ∈ [−M, M] lies in some cell [u_j, u_{j+1}] with j < K. Pure real-arithmetic fact (floor of the scaled coordinate, clamped to the last cell).","s":"∀ {M : ℝ}, 0 < M → ∀ {K : ℕ}, 1 ≤ K → ∀ {x : ℝ}, -M ≤ x → x ≤ M → ∃ j < K, -M + 2 * M * ↑j / ↑K ≤ x ∧ x ≤ -M + 2 * M * (↑j + 1) / ↑K","l":466,"f":"Causalean/Stat/Quantile/SampleQuantileBahadur/Oscillation.lean"},{"n":"Causalean.Stat.exists_real_ge_mixture","k":"theorem","m":"Causalean.Stat.Minimax.Mixture","a":"Stat/Minimax","t":2,"d":"Witness extraction. Some component carries at least the mixture's .real-mass: since the mixture is a weighted average (weights summing to 1), its mass on A is at most the maximal component mass, attained over the finite index.","s":"∀ {Ω : Type u_1} {mΩ : MeasurableSpace Ω} {ι : Type u_2} [inst : Fintype ι] [Nonempty ι] (w : ι → ENNReal), ∑ i, w i = 1 → ∀ (P : ι → MeasureTheory.Measure Ω) [∀ (i : ι), MeasureTheory.IsProbabilityMeasure (P i)] (A : Set Ω), ∃ i, (Causalean.Stat.mixture w P).real A ≤ (P i).real A","l":81,"f":"Causalean/Stat/Minimax/Mixture.lean"},{"n":"Causalean.Stat.expCellUpdatedMean","k":"def","m":"Causalean.Stat.MEstimation.FinitePoissonDerivative","a":"Stat/MEstimation","t":1,"d":"Replace one cell mean by a positive exponential path.","s":"{I : Type u_1} → [DecidableEq I] → (I → ℝ) → I → ℝ → ℝ → I → ℝ","l":24,"f":"Causalean/Stat/MEstimation/FinitePoissonDerivative.lean"},{"n":"Causalean.Stat.exp_neg_half_klDiv_le_bhattacharyya","k":"theorem","m":"Causalean.Stat.Minimax.BretagnolleHuber","a":"Stat/Minimax","t":2,"d":"Jensen / Bhattacharyya step. exp(-½·KL) ≤ ∫ √p ∂ν.","s":"∀ {Ω : Type u_1} {mΩ : MeasurableSpace Ω} (μ ν : MeasureTheory.Measure Ω) [MeasureTheory.IsProbabilityMeasure μ] [MeasureTheory.IsProbabilityMeasure ν], μ.AbsolutelyContinuous ν → MeasureTheory.Integrable (MeasureTheory.llr μ ν) μ → Real.exp (-(1 / 2) * (InformationTheory.klDiv μ ν).toReal) ≤ ∫ (x : Ω), √(μ.rnDeriv ν x).toReal ∂ν","l":303,"f":"Causalean/Stat/Minimax/BretagnolleHuber.lean"},{"n":"Causalean.Stat.expectedRestrictedVolume_affineInversion_frontier_le","k":"theorem","m":"Causalean.Stat.Inference.AffineInversion","a":"Stat/Inference","t":1,"d":"Affine-inversion frontier bound over a finite-volume region. Fix a probability space (Ω, Q), real-valued functions A, B, K on it, and a region of the line with finite Lebesgue measure. Suppose a nonnegative radius scale L, a positive centering level mu and a positive sample size n; suppose K is pointwise nonnegative and Q-integrable with Q-mean at most Kbar, and that the Q-probability that B deviates from mu by more than mu/2 is at most q. Suppose further a positive scale kappa, a nonnegative inflation factor inflation and nonnegative slack Y, with Kbar controlled by inflation · kappa and the bad-event contribution (vol region) · q controlled by Y / t, where t is defined as n · mu² / kappa. Then the Q-expected restricted volume of the affine-inversion set built from A, B and the shrinking radius L · √(K/n) is at most max(vol region, 4·√inflation·L + Y) · min(1, t^(-1/2)).","s":"∀ {Ω : Type u_1} [inst : MeasurableSpace Ω] (Q : MeasureTheory.Measure Ω) [MeasureTheory.IsProbabilityMeasure Q] (region : Set ℝ), MeasureTheory.volume region ≠ ⊤ → ∀ (A B K : Ω → ℝ) (n : ℕ) (L mu Kbar q kappa inflation Y t : ℝ), 0 ≤ L → 0 < mu → 0 < n → (∀ (w : Ω), 0 ≤ K w) → MeasureTheory.Integrable K Q → ∫ (w : Ω), K w ∂Q ≤ Kbar → (Q {w | mu / 2 < |B w - mu|}).toReal ≤ q → 0 < kappa → 0 ≤ inflation → 0 ≤ Y → Kbar ≤ inflation * kappa → (MeasureTheory.volume region).toReal * q ≤ Y / t → t = ↑n * mu ^ 2 / kappa → ∫ (w : Ω), Causalean.Stat.restrictedSetVolume region (Causalean.Stat.affineInversionSet region (A w) (B w) (L * √(K w / ↑n))) ∂Q ≤ max (MeasureTheory.volume region).toReal (4 * √inflation * L + Y) * min 1 (t ^ (-1 / 2))","l":293,"f":"Causalean/Stat/Inference/AffineInversion.lean"},{"n":"Causalean.Stat.expectedRestrictedVolume_affineInversion_le","k":"theorem","m":"Causalean.Stat.Inference.AffineInversion","a":"Stat/Inference","t":2,"d":"Expected restricted volume for affine inversion is controlled by the mean radius and by the probability that the random slope is less than half its positive target value.","s":"∀ {Ω : Type u_1} [inst : MeasurableSpace Ω] (Q : MeasureTheory.Measure Ω) [MeasureTheory.IsProbabilityMeasure Q] (region : Set ℝ), MeasureTheory.volume region ≠ ⊤ → ∀ (A B K : Ω → ℝ) (n : ℕ) (L mu Kbar q : ℝ), 0 ≤ L → 0 < mu → 0 < n → (∀ (w : Ω), 0 ≤ K w) → MeasureTheory.Integrable K Q → ∫ (w : Ω), K w ∂Q ≤ Kbar → (Q {w | mu / 2 < |B w - mu|}).toReal ≤ q → ∫ (w : Ω), Causalean.Stat.restrictedSetVolume region (Causalean.Stat.affineInversionSet region (A w) (B w) (L * √(K w / ↑n))) ∂Q ≤ 4 * L * √(Kbar / ↑n) / mu + (MeasureTheory.volume region).toReal * q","l":88,"f":"Causalean/Stat/Inference/AffineInversion.lean"},{"n":"Causalean.Stat.expected_restrictedSetVolume_eq_integral_inclusion","k":"theorem","m":"Causalean.Stat.Minimax.HonestConfidenceSet","a":"Stat/Minimax","t":2,"d":"The expected restricted volume of a jointly measurable random set equals the integral, over the parameter region, of its pointwise inclusion probabilities.","s":"∀ {Ω : Type u_1} [inst : MeasurableSpace Ω] (Q : MeasureTheory.Measure Ω) [MeasureTheory.IsFiniteMeasure Q] (C : Ω → Set ℝ) (region : Set ℝ), MeasurableSet {p | p.2 ∈ C p.1} → MeasurableSet region → MeasureTheory.volume region ≠ ⊤ → ∫ (ω : Ω), Causalean.Stat.restrictedSetVolume region (C ω) ∂Q = ∫ (u : ℝ) in region, (Q {ω | u ∈ C ω}).toReal","l":31,"f":"Causalean/Stat/Minimax/HonestConfidenceSet.lean"},{"n":"Causalean.Stat.factorialDisjointCorrection_le","k":"theorem","m":"Causalean.Stat.UStatistic.OrderM.MixedOrderBounds","a":"Stat/UStatistic/OrderM","t":2,"d":"For sample size n, orders r and s, and common order bound R, if the first order is at most R, the second order is at most R, and the sample size is at least four times R squared, the absolute disjoint normalization correction is at most twice R squared divided by n.","s":"∀ {n r s R : ℕ}, r ≤ R → s ≤ R → 4 * R ^ 2 ≤ n → |↑(n.descFactorial (r + s)) / (↑(n.descFactorial r) * ↑(n.descFactorial s)) - 1| ≤ 2 * ↑R ^ 2 / ↑n","l":249,"f":"Causalean/Stat/UStatistic/OrderM/MixedOrderBounds.lean"},{"n":"Causalean.Stat.factorialMatchingRatio_le","k":"theorem","m":"Causalean.Stat.UStatistic.OrderM.MixedOrderBounds","a":"Stat/UStatistic/OrderM","t":1,"d":"For sample size n, orders r and s, common order bound R, and overlap size h, if the first order is at most R, the second order is at most R, the sample size is at least four times R squared, and the overlap fits inside both orders, the mixed falling-factorial ratio is at most exp(1) divided by n to the overlap size.","s":"∀ {n r s R h : ℕ}, r ≤ R → s ≤ R → 4 * R ^ 2 ≤ n → h ≤ min r s → ↑(n.descFactorial (r + s - h)) / (↑(n.descFactorial r) * ↑(n.descFactorial s)) ≤ Real.exp 1 / ↑n ^ h","l":217,"f":"Causalean/Stat/UStatistic/OrderM/MixedOrderBounds.lean"},{"n":"Causalean.Stat.fano_average_error","k":"theorem","m":"Causalean.Stat.Minimax.Fano","a":"Stat/Minimax","t":1,"d":"Fano average-error lower bound. For a measurable estimator est and a family of parameter values that are pairwise 2s-separated, the average probability of error over the N = card ι hypotheses is at least 1 − (1 + ∑ᵢ tvDist (P i₀) (P i)) / N.","s":"∀ {Ω : Type u_1} {mΩ : MeasurableSpace Ω} {Θ : Type u_2} [inst : PseudoMetricSpace Θ] [inst_1 : MeasurableSpace Θ] [OpensMeasurableSpace Θ] {ι : Type u_3} [inst_3 : Fintype ι] (P : ι → MeasureTheory.Measure Ω) [∀ (i : ι), MeasureTheory.IsProbabilityMeasure (P i)] {est : Ω → Θ}, Measurable est → ∀ {θ : ι → Θ} {s : ℝ}, (∀ (i k : ι), i ≠ k → 2 * s ≤ dist (θ i) (θ k)) → ∀ (i₀ : ι), 1 - (1 + ∑ i, Causalean.Stat.tvDist (P i₀) (P i)) / ↑(Fintype.card ι) ≤ (∑ i, (P i).real {ω | s ≤ dist (est ω) (θ i)}) / ↑(Fintype.card ι)","l":156,"f":"Causalean/Stat/Minimax/Fano.lean"},{"n":"Causalean.Stat.fano_exists_error","k":"theorem","m":"Causalean.Stat.Minimax.Fano","a":"Stat/Minimax","t":1,"d":"Fano existence-of-bad-hypothesis bound (uniform β). For a measurable estimator est and a family of parameter values that are pairwise 2s-separated, if every hypothesis's law is within total variation β of the reference P i₀, then some hypothesis has error probability at least 1 − 1/N − β. This is the directly usable minimax statement: choosing the number of hypotheses N large and the divergence β small forces error.","s":"∀ {Ω : Type u_1} {mΩ : MeasurableSpace Ω} {Θ : Type u_2} [inst : PseudoMetricSpace Θ] [inst_1 : MeasurableSpace Θ] [OpensMeasurableSpace Θ] {ι : Type u_3} [inst_3 : Fintype ι] (P : ι → MeasureTheory.Measure Ω) [∀ (i : ι), MeasureTheory.IsProbabilityMeasure (P i)] {est : Ω → Θ}, Measurable est → ∀ {θ : ι → Θ} {s : ℝ}, (∀ (i k : ι), i ≠ k → 2 * s ≤ dist (θ i) (θ k)) → ∀ (i₀ : ι) {β : ℝ}, (∀ (i : ι), Causalean.Stat.tvDist (P i₀) (P i) ≤ β) → ∃ i, 1 - 1 / ↑(Fintype.card ι) - β ≤ (P i).real {ω | s ≤ dist (est ω) (θ i)}","l":185,"f":"Causalean/Stat/Minimax/Fano.lean"},{"n":"Causalean.Stat.fiber_integral_pi","k":"theorem","m":"Causalean.Stat.Coupling.ProductLossMonotoneCoupling.HoeffdingFubini","a":"Stat/Coupling/ProductLossMonotoneCoupling","t":1,"d":"Fibre integral. For a coupling π of two probability measures, the π-integral of the product of the two signed tail indicators at thresholds s and t equals the joint survival function of π at (s, t), adjusted by cross terms built from the two marginal survival functions and the sign indicators of s and t.","s":"∀ {π : MeasureTheory.Measure (ℝ × ℝ)} {μ ν : MeasureTheory.Measure ℝ}, Causalean.Stat.IsCoupling π μ ν → ∀ (s t : ℝ), ∫ (p : ℝ × ℝ), Causalean.Stat.signedTail p.1 s * Causalean.Stat.signedTail p.2 t ∂π = Causalean.Stat.jointSurv π s t - Causalean.Stat.tailInd 0 t * Causalean.Stat.survFst π s - Causalean.Stat.tailInd 0 s * Causalean.Stat.survSnd π t + Causalean.Stat.tailInd 0 s * Causalean.Stat.tailInd 0 t","l":134,"f":"Causalean/Stat/Coupling/ProductLossMonotoneCoupling/HoeffdingFubini.lean"},{"n":"Causalean.Stat.fiber_sub_mean_prod","k":"theorem","m":"Causalean.Stat.Coupling.ProductLossMonotoneCoupling.Hoeffding","a":"Stat/Coupling/ProductLossMonotoneCoupling","t":2,"d":"Pointwise cancellation: the fibre integral minus the product of the centred marginal survival functions is exactly the Fréchet gap H_π - F·G.","s":"∀ {π : MeasureTheory.Measure (ℝ × ℝ)} {μ ν : MeasureTheory.Measure ℝ}, Causalean.Stat.IsCoupling π μ ν → ∀ (q : ℝ × ℝ), ∫ (p : ℝ × ℝ), Causalean.Stat.signedTail p.1 q.1 * Causalean.Stat.signedTail p.2 q.2 ∂π - (Causalean.Stat.survFst π q.1 - Causalean.Stat.tailInd 0 q.1) * (Causalean.Stat.survSnd π q.2 - Causalean.Stat.tailInd 0 q.2) = Causalean.Stat.jointCdf π q.1 q.2 - ↑(ProbabilityTheory.cdf μ) q.1 * ↑(ProbabilityTheory.cdf ν) q.2","l":45,"f":"Causalean/Stat/Coupling/ProductLossMonotoneCoupling/Hoeffding.lean"},{"n":"Causalean.Stat.finProductKernel","k":"def","m":"Causalean.Stat.Minimax.MarkovKernelTransport","a":"Stat/Minimax","t":1,"d":"The finite product kernel applies one Markov kernel independently to each coordinate, with the unique point-mass kernel on the empty product.","s":"{X : Type uX} → {Y : Type uY} → [inst : MeasurableSpace X] → [inst_1 : MeasurableSpace Y] → (n : ℕ) → ProbabilityTheory.Kernel X Y → ProbabilityTheory.Kernel (Fin n → X) (Fin n → Y)","l":246,"f":"Causalean/Stat/Minimax/MarkovKernelTransport.lean"},{"n":"Causalean.Stat.finProductKernel_apply","k":"theorem","m":"Causalean.Stat.Minimax.MarkovKernelTransport","a":"Stat/Minimax","t":2,"d":"Each fibre of the finite product kernel equals the independent product of its coordinate output laws, including the unique empty product fibre.","s":"∀ {X : Type uX} {Y : Type uY} [inst : MeasurableSpace X] [inst_1 : MeasurableSpace Y] (n : ℕ) (K : ProbabilityTheory.Kernel X Y) [ProbabilityTheory.IsMarkovKernel K] (x : Fin n → X), (Causalean.Stat.finProductKernel n K) x = MeasureTheory.Measure.pi fun i => K (x i)","l":257,"f":"Causalean/Stat/Minimax/MarkovKernelTransport.lean"},{"n":"Causalean.Stat.finProductKernel_comp_pi","k":"theorem","m":"Causalean.Stat.Minimax.MarkovKernelTransport","a":"Stat/Minimax","t":2,"d":"Passing an independent finite product law through the coordinatewise product kernel produces the product of the one-coordinate garbled law, including when there are no coordinates.","s":"∀ {X : Type uX} {Y : Type uY} [inst : MeasurableSpace X] [inst_1 : MeasurableSpace Y] (n : ℕ) (P : MeasureTheory.Measure X) [MeasureTheory.IsProbabilityMeasure P] (K : ProbabilityTheory.Kernel X Y) [ProbabilityTheory.IsMarkovKernel K], (MeasureTheory.Measure.pi fun x => P).bind ⇑(Causalean.Stat.finProductKernel n K) = MeasureTheory.Measure.pi fun x => P.bind ⇑K","l":284,"f":"Causalean/Stat/Minimax/MarkovKernelTransport.lean"},{"n":"Causalean.Stat.finiteCategoryPilotGood","k":"def","m":"Causalean.Stat.SampleSplit.FiniteCategoryPilot","a":"Stat/SampleSplit","t":1,"d":"The finite-category pilot good event requires every selected category to have at least the lower population-mass band and every rejected category to have at most the upper band.","s":"{Omega : Type u_1} → {X : Type u_2} → {Iota : Type u_3} → [inst : MeasurableSpace Omega] → [inst_1 : MeasurableSpace X] → {mu : MeasureTheory.Measure Omega} → {P : MeasureTheory.Measure X} → [Fintype Iota] → [DecidableEq Iota] → Causalean.Stat.IIDSample Omega X mu P → (X → Iota) → Finset ℕ → ℝ → ℝ → ℝ → Set Omega","l":325,"f":"Causalean/Stat/SampleSplit/FiniteCategoryPilot.lean"},{"n":"Causalean.Stat.finiteCategoryPilot_bad_probability","k":"theorem","m":"Causalean.Stat.SampleSplit.FiniteCategoryPilot","a":"Stat/SampleSplit","t":1,"d":"Given an iid sample, a measurable category label, a finite pilot block, a positive selection threshold, a lower population-mass band, an upper population-mass band, an upper-tail exponential tilt, a lower-tail exponential tilt, nonnegativity of the upper-tail tilt, and nonpositivity of the lower-tail tilt, failure of the simultaneous category-mass sandwich has probability at most the number of categories times the sum of the two explicit Chernoff tails. The bound also covers empty blocks and empty category types.","s":"∀ {Omega : Type u_1} {X : Type u_2} {Iota : Type u_3} [inst : MeasurableSpace Omega] [inst_1 : MeasurableSpace X] [inst_2 : MeasurableSpace Iota] {mu : MeasureTheory.Measure Omega} {P : MeasureTheory.Measure X} [inst_3 : Fintype Iota] [inst_4 : DecidableEq Iota] [MeasurableSingletonClass Iota] (S : Causalean.Stat.IIDSample Omega X mu P) {label : X → Iota}, Measurable label → ∀ (block : Finset ℕ) {t : ℝ}, 0 < t → ∀ (lowerBand upperBand sUpper sLower : ℝ), 0 ≤ sUpper → sLower ≤ 0 → mu.real (Causalean.Stat.finiteCategoryPilotGood S label block t lowerBand upperBand)ᶜ ≤ ↑(Fintype.card Iota) * (Real.exp (-sUpper * t + ↑block.card * (lowerBand * (Real.exp sUpper - 1))) + Real.exp (-sLower * t + ↑block.card * (upperBand * (Real.exp sLower - 1))))","l":371,"f":"Causalean/Stat/SampleSplit/FiniteCategoryPilot.lean"},{"n":"Causalean.Stat.finiteCategoryPilot_bad_probability_log_two","k":"theorem","m":"Causalean.Stat.SampleSplit.FiniteCategoryPilot","a":"Stat/SampleSplit","t":2,"d":"Given an iid sample, a measurable category label, a finite pilot block, a positive selection threshold, a lower population-mass band, and an upper population-mass band, the simultaneous category-mass sandwich failure probability obeys the explicit bound obtained from opposite logarithmic tilts of magnitude log two.","s":"∀ {Omega : Type u_1} {X : Type u_2} {Iota : Type u_3} [inst : MeasurableSpace Omega] [inst_1 : MeasurableSpace X] [inst_2 : MeasurableSpace Iota] {mu : MeasureTheory.Measure Omega} {P : MeasureTheory.Measure X} [inst_3 : Fintype Iota] [inst_4 : DecidableEq Iota] [MeasurableSingletonClass Iota] (S : Causalean.Stat.IIDSample Omega X mu P) {label : X → Iota}, Measurable label → ∀ (block : Finset ℕ) {t : ℝ}, 0 < t → ∀ (lowerBand upperBand : ℝ), mu.real (Causalean.Stat.finiteCategoryPilotGood S label block t lowerBand upperBand)ᶜ ≤ ↑(Fintype.card Iota) * (Real.exp (-Real.log 2 * t + ↑block.card * lowerBand) + Real.exp (Real.log 2 * t - ↑block.card * upperBand / 2))","l":470,"f":"Causalean/Stat/SampleSplit/FiniteCategoryPilot.lean"},{"n":"Causalean.Stat.finiteInjectiveTuples","k":"def","m":"Causalean.Stat.UStatistic.OrderM.Basic","a":"Stat/UStatistic/OrderM","t":1,"d":"The injective assignments send each coordinate in a finite family to a distinct observation among the first n sample positions.","s":"(ι : Type u_3) → [Fintype ι] → (n : ℕ) → Finset (ι → Fin n)","l":425,"f":"Causalean/Stat/UStatistic/OrderM/Basic.lean"},{"n":"Causalean.Stat.finiteInjectiveTuples_card","k":"theorem","m":"Causalean.Stat.UStatistic.OrderM.Basic","a":"Stat/UStatistic/OrderM","t":2,"d":"For a finite coordinate family and sample size n, the number of injective sample assignments is the falling factorial of n with length equal to the number of coordinates.","s":"∀ (ι : Type u_3) [inst : Fintype ι] (n : ℕ), (Causalean.Stat.finiteInjectiveTuples ι n).card = n.descFactorial (Fintype.card ι)","l":432,"f":"Causalean/Stat/UStatistic/OrderM/Basic.lean"},{"n":"Causalean.Stat.finitePoissonObjective","k":"def","m":"Causalean.Stat.MEstimation.FinitePoisson","a":"Stat/MEstimation","t":1,"d":"A finite positive-mean Poisson criterion composed with a linear design.","s":"{E : Type u_1} → {I : Type u_2} → [inst : AddCommGroup E] → [inst_1 : Module ℝ E] → [Fintype I] → (I → ℝ) → (I → ℝ) → (E →ₗ[ℝ] I → ℝ) → E → ℝ","l":77,"f":"Causalean/Stat/MEstimation/FinitePoisson.lean"},{"n":"Causalean.Stat.finitePoissonObjective_argmax_continuousAt_mean","k":"theorem","m":"Causalean.Stat.MEstimation.FinitePoissonConsistency","a":"Stat/MEstimation","t":1,"d":"Continuity of the finite Poisson maximizer in the means. If every cell weight is strictly positive and the base mean vector m₀ has every entry strictly positive, and the linear design map is injective, then the selected maximizer of the finite Poisson criterion is continuous, as a function of the mean vector, at m₀.","s":"∀ {E : Type u_1} {I : Type u_2} [inst : NormedAddCommGroup E] [inst_1 : NormedSpace ℝ E] [FiniteDimensional ℝ E] [inst_3 : Fintype I] [Nonempty I] (q : I → ℝ) (A : E →ₗ[ℝ] I → ℝ) (m₀ : I → ℝ), (∀ (i : I), 0 < q i) → (∀ (i : I), 0 < m₀ i) → Function.Injective ⇑A → ContinuousAt (fun m => Causalean.Stat.maximizerOrZero (Causalean.Stat.finitePoissonObjective q m A)) m₀","l":238,"f":"Causalean/Stat/MEstimation/FinitePoissonConsistency.lean"},{"n":"Causalean.Stat.finitePoissonObjective_argmax_tendsto","k":"theorem","m":"Causalean.Stat.MEstimation.FinitePoissonConsistency","a":"Stat/MEstimation","t":1,"d":"Stability of finite Poisson maximizers under convergence of weights and means. Given a sequence of finite Poisson criteria with weights qN N and means mN N, suppose the limiting weights are strictly positive, the limiting means are strictly positive, the linear design map is injective, the weights qN N converge cellwise to the limiting weights q, and the means mN N converge cellwise to the limiting means m. If argmax N maximizes the N-th criterion, eventually in N, and limitArgmax is the unique global maximizer of the limiting criterion, then the maximizer sequence argmax converges to limitArgmax.","s":"∀ {E : Type u_1} {I : Type u_2} [inst : NormedAddCommGroup E] [inst_1 : NormedSpace ℝ E] [FiniteDimensional ℝ E] [SecondCountableTopology E] [inst_4 : Fintype I] (qN mN : ℕ → I → ℝ) (q m : I → ℝ) (A : E →ₗ[ℝ] I → ℝ) (argmax : ℕ → E) (limitArgmax : E), (∀ (i : I), 0 < q i) → (∀ (i : I), 0 < m i) → Function.Injective ⇑A → (∀ (i : I), Filter.Tendsto (fun N => qN N i) Filter.atTop (nhds (q i))) → (∀ (i : I), Filter.Tendsto (fun N => mN N i) Filter.atTop (nhds (m i))) → (∀ᶠ (N : ℕ) in Filter.atTop, ∀ (y : E), Causalean.Stat.finitePoissonObjective (qN N) (mN N) A y ≤ Causalean.Stat.finitePoissonObjective (qN N) (mN N) A (argmax N)) → Causalean.Stat.IsUniqueGlobalMax (Causalean.Stat.finitePoissonObjective q m A) limitArgmax → Filter.Tendsto argmax Filter.atTop (nhds limitArgmax)","l":195,"f":"Causalean/Stat/MEstimation/FinitePoissonConsistency.lean"},{"n":"Causalean.Stat.finitePoissonObjective_continuous","k":"theorem","m":"Causalean.Stat.MEstimation.FinitePoisson","a":"Stat/MEstimation","t":2,"d":"The finite Poisson criterion is continuous in its parameter whenever the linear design acts on a finite-dimensional normed space.","s":"∀ {E : Type u_1} {I : Type u_2} [inst : NormedAddCommGroup E] [inst_1 : NormedSpace ℝ E] [FiniteDimensional ℝ E] [inst_3 : Fintype I] (q m : I → ℝ) (A : E →ₗ[ℝ] I → ℝ), Continuous (Causalean.Stat.finitePoissonObjective q m A)","l":83,"f":"Causalean/Stat/MEstimation/FinitePoisson.lean"},{"n":"Causalean.Stat.finitePoissonObjective_eventually_common_compact","k":"theorem","m":"Causalean.Stat.MEstimation.FinitePoissonConsistency","a":"Stat/MEstimation","t":2,"d":"Convergent positive coefficients and a fixed injective design put any eventual sequence of global maximizers in one common compact ball.","s":"∀ {E : Type u_1} {I : Type u_2} [inst : NormedAddCommGroup E] [inst_1 : NormedSpace ℝ E] [FiniteDimensional ℝ E] [inst_3 : Fintype I] (qN mN : ℕ → I → ℝ) (q m : I → ℝ) (A : E →ₗ[ℝ] I → ℝ) (argmax : ℕ → E), (∀ (i : I), 0 < q i) → (∀ (i : I), 0 < m i) → Function.Injective ⇑A → (∀ (i : I), Filter.Tendsto (fun N => qN N i) Filter.atTop (nhds (q i))) → (∀ (i : I), Filter.Tendsto (fun N => mN N i) Filter.atTop (nhds (m i))) → (∀ᶠ (N : ℕ) in Filter.atTop, ∀ (y : E), Causalean.Stat.finitePoissonObjective (qN N) (mN N) A y ≤ Causalean.Stat.finitePoissonObjective (qN N) (mN N) A (argmax N)) → ∃ K, IsCompact K ∧ ∀ᶠ (N : ℕ) in Filter.atTop, argmax N ∈ K","l":65,"f":"Causalean/Stat/MEstimation/FinitePoissonConsistency.lean"},{"n":"Causalean.Stat.finitePoissonObjective_exists_unique_max","k":"theorem","m":"Causalean.Stat.MEstimation.FinitePoisson","a":"Stat/MEstimation","t":1,"d":"Existence and uniqueness of the finite Poisson pseudo-true parameter. If every cell weight is strictly positive, every cell mean is strictly positive, and the linear design map is injective, then the finite Poisson criterion attains its supremum over the parameter space at a unique point.","s":"∀ {E : Type u_1} {I : Type u_2} [inst : NormedAddCommGroup E] [inst_1 : NormedSpace ℝ E] [FiniteDimensional ℝ E] [inst_3 : Fintype I] [Nonempty I] (q m : I → ℝ) (A : E →ₗ[ℝ] I → ℝ), (∀ (i : I), 0 < q i) → (∀ (i : I), 0 < m i) → Function.Injective ⇑A → ∃! x, ∀ (y : E), Causalean.Stat.finitePoissonObjective q m A y ≤ Causalean.Stat.finitePoissonObjective q m A x","l":92,"f":"Causalean/Stat/MEstimation/FinitePoisson.lean"},{"n":"Causalean.Stat.finitePoissonObjective_expCell_argmax_snd_hasDerivAt","k":"theorem","m":"Causalean.Stat.MEstimation.FinitePoissonDerivative","a":"Stat/MEstimation","t":1,"d":"Derivative of the finite Poisson maximizer under one exponentially perturbed cell mean. Suppose every cell weight is strictly positive, every base cell mean is strictly positive, the perturbation scale B is strictly positive, and the linear design map is injective; consider replacing cell j's mean by the exponential path x ↦ B · exp x. If betaDot is the unique value solving the score equation linearized at the maximizer for base point x₀, for every perturbation direction, then the second (scalar) coordinate of the maximizer, as a function of x, has derivative betaDot at x₀.","s":"∀ {U : Type u_1} {I : Type u_2} [inst : NormedAddCommGroup U] [inst_1 : NormedSpace ℝ U] [FiniteDimensional ℝ U] [inst_3 : Fintype I] [inst_4 : DecidableEq I] [Nonempty I] (q m : I → ℝ) (A : U × ℝ →ₗ[ℝ] I → ℝ) (j : I) (B x₀ betaDot : ℝ), (∀ (i : I), 0 < q i) → (∀ (i : I), 0 < m i) → 0 < B → Function.Injective ⇑A → (∀ (v : U × ℝ), (∀ (d : U × ℝ), ∑ i, q i * A d i * ((if i = j then B * Real.exp x₀ else 0) - Real.exp (A (Causalean.Stat.maximizerOrZero (Causalean.Stat.finitePoissonObjective q (Causalean.Stat.expCellUpdatedMean m j B x₀) A)) i) * A v i) = 0) → v.2 = betaDot) → HasDerivAt (fun x => (Causalean.Stat.maximizerOrZero (Causalean.Stat.finitePoissonObjective q (Causalean.Stat.expCellUpdatedMean m j B x) A)).2) betaDot x₀","l":30,"f":"Causalean/Stat/MEstimation/FinitePoissonDerivative.lean"},{"n":"Causalean.Stat.finitePoissonObjective_isMax_of_score","k":"theorem","m":"Causalean.Stat.MEstimation.FinitePoisson","a":"Stat/MEstimation","t":2,"d":"For a nonnegative weighted Poisson objective, vanishing of every directional score is sufficient for global maximality.","s":"∀ {E : Type u_1} {I : Type u_2} [inst : AddCommGroup E] [inst_1 : Module ℝ E] [inst_2 : Fintype I] (q m : I → ℝ) (A : E →ₗ[ℝ] I → ℝ) (x : E), (∀ (i : I), 0 ≤ q i) → (∀ (d : E), ∑ i, q i * A d i * (m i - Real.exp (A x i)) = 0) → ∀ (y : E), Causalean.Stat.finitePoissonObjective q m A y ≤ Causalean.Stat.finitePoissonObjective q m A x","l":294,"f":"Causalean/Stat/MEstimation/FinitePoisson.lean"},{"n":"Causalean.Stat.finitePoissonObjective_score","k":"theorem","m":"Causalean.Stat.MEstimation.FinitePoisson","a":"Stat/MEstimation","t":2,"d":"Every directional score vanishes at a global maximizer.","s":"∀ {E : Type u_1} {I : Type u_2} [inst : NormedAddCommGroup E] [inst_1 : NormedSpace ℝ E] [inst_2 : Fintype I] (q m : I → ℝ) (A : E →ₗ[ℝ] I → ℝ) (x d : E), (∀ (y : E), Causalean.Stat.finitePoissonObjective q m A y ≤ Causalean.Stat.finitePoissonObjective q m A x) → ∑ i, q i * A d i * (m i - Real.exp (A x i)) = 0","l":257,"f":"Causalean/Stat/MEstimation/FinitePoisson.lean"},{"n":"Causalean.Stat.finitePoissonObjective_snd_sign_of_nuisance_score","k":"theorem","m":"Causalean.Stat.MEstimation.FinitePoissonSign","a":"Stat/MEstimation","t":1,"d":"Sign of the selected scalar coefficient from the nuisance score. Suppose every cell weight is strictly positive, every cell mean is strictly positive, the linear design map is injective, and the conditional nuisance fit u₀, with the scalar coordinate held at zero, clears every nuisance-direction score. Then the selected scalar coefficient of the finite Poisson maximizer has exactly the sign of the remaining scalar score: it is negative, zero, or positive exactly when the scalar score is.","s":"∀ {U : Type u_1} {I : Type u_2} [inst : NormedAddCommGroup U] [inst_1 : NormedSpace ℝ U] [FiniteDimensional ℝ U] [inst_3 : Fintype I] [Nonempty I] (q m : I → ℝ) (A : U × ℝ →ₗ[ℝ] I → ℝ) (u₀ : U), (∀ (i : I), 0 < q i) → (∀ (i : I), 0 < m i) → Function.Injective ⇑A → (∀ (u : U), ∑ i, q i * A (u, 0) i * (m i - Real.exp (A (u₀, 0) i)) = 0) → have beta := (Causalean.Stat.maximizerOrZero (Causalean.Stat.finitePoissonObjective q m A)).2; have scalarScore := ∑ i, q i * A (0, 1) i * (m i - Real.exp (A (u₀, 0) i)); (beta < 0 ↔ scalarScore < 0) ∧ (beta = 0 ↔ scalarScore = 0) ∧ (0 < beta ↔ 0 < scalarScore)","l":21,"f":"Causalean/Stat/MEstimation/FinitePoissonSign.lean"},{"n":"Causalean.Stat.finitePoissonObjective_tendstoUniformlyOn","k":"theorem","m":"Causalean.Stat.MEstimation.FinitePoissonConsistency","a":"Stat/MEstimation","t":2,"d":"Pointwise convergence of the finitely many weights and means gives uniform convergence of the finite Poisson objective on every compact parameter set.","s":"∀ {E : Type u_1} {I : Type u_2} [inst : NormedAddCommGroup E] [inst_1 : NormedSpace ℝ E] [FiniteDimensional ℝ E] [inst_3 : Fintype I] (qN mN : ℕ → I → ℝ) (q m : I → ℝ) (A : E →ₗ[ℝ] I → ℝ) (K : Set E), IsCompact K → (∀ (i : I), Filter.Tendsto (fun N => qN N i) Filter.atTop (nhds (q i))) → (∀ (i : I), Filter.Tendsto (fun N => mN N i) Filter.atTop (nhds (m i))) → TendstoUniformlyOn (fun N => Causalean.Stat.finitePoissonObjective (qN N) (mN N) A) (Causalean.Stat.finitePoissonObjective q m A) Filter.atTop K","l":26,"f":"Causalean/Stat/MEstimation/FinitePoissonConsistency.lean"},{"n":"Causalean.Stat.finite_measure_halfline_tails_small","k":"theorem","m":"Causalean.Stat.CLT.GaussianTail","a":"Stat/CLT","t":1,"d":"Finite-measure half-line tail control. For a finite measure Q on the real line and a positive tolerance ε, there is a positive cutoff R such that both the lower half-line tail $Q((-\\infty,-R])$ and the upper half-line tail $Q([R,\\infty))$ are at most ε.","s":"∀ (Q : MeasureTheory.Measure ℝ) [MeasureTheory.IsFiniteMeasure Q] {ε : ℝ}, 0 < ε → ∃ R, 0 < R ∧ Q (Set.Iic (-R)) ≤ ENNReal.ofReal ε ∧ Q (Set.Ici R) ≤ ENNReal.ofReal ε","l":38,"f":"Causalean/Stat/CLT/GaussianTail.lean"},{"n":"Causalean.Stat.finite_one_add_chiSqDiv","k":"theorem","m":"Causalean.Stat.Minimax.ChiSquaredFinite","a":"Stat/Minimax","t":1,"d":"Finite χ²-divergence formula. On a finite sample space, for probability measures μ, ν with μ absolutely continuous with respect to ν, the (shifted) χ²-divergence is the explicit sum of squared point-mass ratios: 1 + χ²(μ‖ν) = ∑ₓ (μ{x})² / (ν{x}). Combines chiSqDiv_eq, the finite integral formula integral_fintype, and the discrete RN bridge above.","s":"∀ {Ω : Type u_1} [inst : MeasurableSpace Ω] [inst_1 : Fintype Ω] [MeasurableSingletonClass Ω] (μ ν : MeasureTheory.Measure Ω) [MeasureTheory.IsProbabilityMeasure μ] [MeasureTheory.IsProbabilityMeasure ν], μ.AbsolutelyContinuous ν → 1 + Causalean.Stat.chiSqDiv μ ν = ∑ x, μ.real {x} ^ 2 / ν.real {x}","l":51,"f":"Causalean/Stat/Minimax/ChiSquaredFinite.lean"},{"n":"Causalean.Stat.finsetFintype","k":"def","m":"Causalean.Stat.UStatistic.OrderM.PartialMatching","a":"Stat/UStatistic/OrderM","t":1,"d":"The finite subsets of a finite set form a finite collection.","s":"{α : Type u_1} → [Fintype α] → Fintype (Finset α)","l":131,"f":"Causalean/Stat/UStatistic/OrderM/PartialMatching.lean"},{"n":"Causalean.Stat.flipBit","k":"def","m":"Causalean.Stat.Minimax.Assouad","a":"Stat/Minimax","t":1,"d":"Flip the j-th coordinate of a cube vertex τ : Fin d → Bool.","s":"{d : ℕ} → Fin d → (Fin d → Bool) → Fin d → Bool","l":65,"f":"Causalean/Stat/Minimax/Assouad.lean"},{"n":"Causalean.Stat.flipBit_involutive","k":"theorem","m":"Causalean.Stat.Minimax.Assouad","a":"Stat/Minimax","t":2,"d":"Flipping the same coordinate twice is the identity.","s":"∀ {d : ℕ} (j : Fin d), Function.Involutive (Causalean.Stat.flipBit j)","l":73,"f":"Causalean/Stat/Minimax/Assouad.lean"},{"n":"Causalean.Stat.flipBit_self","k":"theorem","m":"Causalean.Stat.Minimax.Assouad","a":"Stat/Minimax","t":2,"d":"Flipping a coordinate changes that coordinate to the opposite Boolean value.","s":"∀ {d : ℕ} (j : Fin d) (τ : Fin d → Bool), Causalean.Stat.flipBit j τ j = !τ j","l":69,"f":"Causalean/Stat/Minimax/Assouad.lean"},{"n":"Causalean.Stat.flipPerm","k":"def","m":"Causalean.Stat.Minimax.Assouad","a":"Stat/Minimax","t":1,"d":"The coordinate flip as a permutation of the cube, used to reindex sums.","s":"{d : ℕ} → Fin d → Equiv.Perm (Fin d → Bool)","l":81,"f":"Causalean/Stat/Minimax/Assouad.lean"},{"n":"Causalean.Stat.flipPerm_apply","k":"theorem","m":"Causalean.Stat.Minimax.Assouad","a":"Stat/Minimax","t":2,"d":"The coordinate-flip permutation acts by flipping that coordinate.","s":"∀ {d : ℕ} (j : Fin d) (τ : Fin d → Bool), (Causalean.Stat.flipPerm j) τ = Causalean.Stat.flipBit j τ","l":85,"f":"Causalean/Stat/Minimax/Assouad.lean"},{"n":"Causalean.Stat.foldB_centered_sum_isLittleOp_one","k":"theorem","m":"Causalean.Stat.SampleSplit.FoldBEmpiricalProcess","a":"Stat/SampleSplit","t":1,"d":"Centered fold-B empirical sum is o_p(1) from L²-rate o_p(1). For an i.i.d. sample and a one-shot split into a nuisance fold and an estimation fold, given a family of random functions f n ω : X → ℝ that is jointly measurable in the training data and the outcome, for every n, measurable with respect to the fold-A σ-algebra jointly with the outcome, for every n, square-integrable under the population measure, for every n and ω, and whose $L^2(P)$ norm is $o_p(1)$ under the sampling measure, the centered fold-B empirical sum of f — the estimation-fold sample average of f n ω minus its population mean, rescaled by $\\sqrt{|B(n)|}$ — is $o_p(1)$.","s":"∀ {Ω : Type u_1} {X : Type u_2} [inst : MeasurableSpace Ω] [inst_1 : MeasurableSpace X] {μ : MeasureTheory.Measure Ω} {P : MeasureTheory.Measure X} [MeasureTheory.IsProbabilityMeasure μ] [MeasureTheory.IsProbabilityMeasure P] (S : Causalean.Stat.IIDSample Ω X μ P) (split : Causalean.Stat.OneShotSplit S) (f : ℕ → Ω → X → ℝ), (∀ (n : ℕ), Measurable (Function.uncurry (f n))) → (∀ (n : ℕ), Measurable (Function.uncurry (f n))) → (∀ (n : ℕ) (ω : Ω), MeasureTheory.MemLp (f n ω) 2 P) → Causalean.Stat.IsLittleOp (fun n ω => (MeasureTheory.eLpNorm (f n ω) 2 P).toReal) (fun x => 1) μ → Causalean.Stat.IsLittleOp (fun n ω => (√↑(split.foldB n).card)⁻¹ * ∑ i ∈ split.foldB n, (f n ω (S.Z i ω) - ∫ (x : X), f n ω x ∂P)) (fun x => 1) μ","l":629,"f":"Causalean/Stat/SampleSplit/FoldBEmpiricalProcess.lean"},{"n":"Causalean.Stat.forall_estimator_exists_sqRisk_ge_of_deterministic_affine_transport","k":"theorem","m":"Causalean.Stat.Minimax.MinimaxRisk","a":"Stat/Minimax","t":2,"d":"Suppose the affine slope is nonzero, the observation rule is measurable, and each target-experiment law is the pushforward of its corresponding source law. If every measurable source estimator has squared risk at least a fixed level for some parameter, then every measurable target estimator has squared risk at least that level multiplied by the squared affine slope for some parameter, where the target parameter is transformed by the same affine map.","s":"∀ {X : Type uX} {Y : Type uY} {Iota : Type uI} [inst : MeasurableSpace X] [inst_1 : MeasurableSpace Y] (P : Iota → MeasureTheory.Measure X) (Q : Iota → MeasureTheory.Measure Y) (theta : Iota → ℝ) (phi : X → Y) (a b L : ℝ), a ≠ 0 → Measurable phi → (∀ (j : Iota), Q j = MeasureTheory.Measure.map phi (P j)) → (∀ (sourceEst : X → ℝ), Measurable sourceEst → ∃ j, L ≤ Causalean.Stat.sqRisk (P j) sourceEst (theta j)) → ∀ (targetEst : Y → ℝ), Measurable targetEst → ∃ j, a ^ 2 * L ≤ Causalean.Stat.sqRisk (Q j) targetEst (a * theta j + b)","l":237,"f":"Causalean/Stat/Minimax/MinimaxRisk.lean"},{"n":"Causalean.Stat.forall_estimator_exists_sqRisk_ge_of_deterministic_affine_transport_pi","k":"theorem","m":"Causalean.Stat.Minimax.MinimaxRisk","a":"Stat/Minimax","t":2,"d":"Suppose every source law is a probability law, the affine slope is nonzero, the observation rule is measurable, and each target marginal law is the pushforward of its corresponding source marginal. If every measurable estimator based on the finite source product experiment has squared risk at least a fixed level for some parameter, then every measurable estimator based on the corresponding target product experiment has squared risk at least that level multiplied by the squared affine slope for some parameter, including when the sample has no coordinates.","s":"∀ {X : Type uX} {Y : Type uY} {Iota : Type uI} [inst : MeasurableSpace X] [inst_1 : MeasurableSpace Y] (n : ℕ) (P : Iota → MeasureTheory.Measure X) (Q : Iota → MeasureTheory.Measure Y) [∀ (j : Iota), MeasureTheory.IsProbabilityMeasure (P j)] (theta : Iota → ℝ) (phi : X → Y) (a b L : ℝ), a ≠ 0 → Measurable phi → (∀ (j : Iota), Q j = MeasureTheory.Measure.map phi (P j)) → (∀ (sourceEst : (Fin n → X) → ℝ), Measurable sourceEst → ∃ j, L ≤ Causalean.Stat.sqRisk (MeasureTheory.Measure.pi fun x => P j) sourceEst (theta j)) → ∀ (targetEst : (Fin n → Y) → ℝ), Measurable targetEst → ∃ j, a ^ 2 * L ≤ Causalean.Stat.sqRisk (MeasureTheory.Measure.pi fun x => Q j) targetEst (a * theta j + b)","l":264,"f":"Causalean/Stat/Minimax/MinimaxRisk.lean"},{"n":"Causalean.Stat.forall_estimator_exists_sqRisk_ge_of_kernel_affine_transport","k":"theorem","m":"Causalean.Stat.Minimax.MarkovKernelTransport","a":"Stat/Minimax","t":2,"d":"Suppose the affine slope is nonzero, each target law is obtained by passing its source law through the common Markov kernel, and every measurable uniformly bounded source estimator incurs squared risk at least a fixed level for some parameter index. Then every measurable uniformly bounded target estimator incurs at least the source level multiplied by the squared slope for some parameter index, with the target parameter transformed by the same affine map.","s":"∀ {X : Type uX} {Y : Type uY} {Iota : Type uI} [inst : MeasurableSpace X] [inst_1 : MeasurableSpace Y] (P : Iota → MeasureTheory.Measure X) (Q : Iota → MeasureTheory.Measure Y) [∀ (j : Iota), MeasureTheory.IsProbabilityMeasure (P j)] (K : ProbabilityTheory.Kernel X Y) [ProbabilityTheory.IsMarkovKernel K] (theta : Iota → ℝ) (a b L : ℝ), a ≠ 0 → (∀ (j : Iota), Q j = (P j).bind ⇑K) → (∀ (sourceEst : X → ℝ), Measurable sourceEst → Causalean.Stat.UniformlyBounded sourceEst → ∃ j, L ≤ Causalean.Stat.sqRisk (P j) sourceEst (theta j)) → ∀ (targetEst : Y → ℝ), Measurable targetEst → Causalean.Stat.UniformlyBounded targetEst → ∃ j, a ^ 2 * L ≤ Causalean.Stat.sqRisk (Q j) targetEst (a * theta j + b)","l":214,"f":"Causalean/Stat/Minimax/MarkovKernelTransport.lean"},{"n":"Causalean.Stat.forall_estimator_exists_sqRisk_ge_of_kernel_affine_transport_pi","k":"theorem","m":"Causalean.Stat.Minimax.MarkovKernelTransport","a":"Stat/Minimax","t":1,"d":"Suppose the affine slope is nonzero, each one-coordinate target law is obtained by applying the common Markov kernel to its source law, and every measurable uniformly bounded estimator on the finite source product incurs squared risk at least a fixed level for some parameter index. Then every measurable uniformly bounded estimator on the target product incurs at least the source level multiplied by the squared slope for some parameter index, including when the sample has no coordinates.","s":"∀ {X : Type uX} {Y : Type uY} {Iota : Type uI} [inst : MeasurableSpace X] [inst_1 : MeasurableSpace Y] (n : ℕ) (P : Iota → MeasureTheory.Measure X) (Q : Iota → MeasureTheory.Measure Y) [∀ (j : Iota), MeasureTheory.IsProbabilityMeasure (P j)] (K : ProbabilityTheory.Kernel X Y) [ProbabilityTheory.IsMarkovKernel K] (theta : Iota → ℝ) (a b L : ℝ), a ≠ 0 → (∀ (j : Iota), Q j = (P j).bind ⇑K) → (∀ (sourceEst : (Fin n → X) → ℝ), Measurable sourceEst → Causalean.Stat.UniformlyBounded sourceEst → ∃ j, L ≤ Causalean.Stat.sqRisk (MeasureTheory.Measure.pi fun x => P j) sourceEst (theta j)) → ∀ (targetEst : (Fin n → Y) → ℝ), Measurable targetEst → Causalean.Stat.UniformlyBounded targetEst → ∃ j, a ^ 2 * L ≤ Causalean.Stat.sqRisk (MeasureTheory.Measure.pi fun x => Q j) targetEst (a * theta j + b)","l":364,"f":"Causalean/Stat/Minimax/MarkovKernelTransport.lean"},{"n":"Causalean.Stat.frechet_hoeffding_lower","k":"theorem","m":"Causalean.Stat.Coupling.ProductLossMonotoneCoupling.FrechetHoeffding","a":"Stat/Coupling/ProductLossMonotoneCoupling","t":1,"d":"Fréchet–Hoeffding lower bound. For a coupling π of the probability measures μ and ν and reals x and y, the joint cdf of π at (x, y) is bounded below by the maximum of 0 and the sum of the marginal cdfs of μ at x and ν at y, minus 1.","s":"∀ {π : MeasureTheory.Measure (ℝ × ℝ)} {μ ν : MeasureTheory.Measure ℝ}, Causalean.Stat.IsCoupling π μ ν → ∀ (x y : ℝ), max (↑(ProbabilityTheory.cdf μ) x + ↑(ProbabilityTheory.cdf ν) y - 1) 0 ≤ Causalean.Stat.jointCdf π x y","l":98,"f":"Causalean/Stat/Coupling/ProductLossMonotoneCoupling/FrechetHoeffding.lean"},{"n":"Causalean.Stat.frechet_hoeffding_upper","k":"theorem","m":"Causalean.Stat.Coupling.ProductLossMonotoneCoupling.FrechetHoeffding","a":"Stat/Coupling/ProductLossMonotoneCoupling","t":1,"d":"Fréchet–Hoeffding upper bound. For a coupling π of the probability measures μ and ν and reals x and y, the joint cdf of π at (x, y) is bounded above by the smaller of the marginal cdfs of μ at x and ν at y.","s":"∀ {π : MeasureTheory.Measure (ℝ × ℝ)} {μ ν : MeasureTheory.Measure ℝ}, Causalean.Stat.IsCoupling π μ ν → ∀ (x y : ℝ), Causalean.Stat.jointCdf π x y ≤ min (↑(ProbabilityTheory.cdf μ) x) (↑(ProbabilityTheory.cdf ν) y)","l":40,"f":"Causalean/Stat/Coupling/ProductLossMonotoneCoupling/FrechetHoeffding.lean"},{"n":"Causalean.Stat.gaussianLimit","k":"def","m":"Causalean.Stat.CLT.GaussianLimit","a":"Stat/CLT","t":1,"d":"The limiting Gaussian law of the multivariate CLT: the centered Gaussian on E with covariance the second-moment operator of ψ, realised as (stdGaussian E).map √Σ.","s":"{X : Type u_2} → [inst : MeasurableSpace X] → {P : MeasureTheory.Measure X} → {E : Type u_3} → [inst_1 : NormedAddCommGroup E] → [inst_2 : InnerProductSpace ℝ E] → [FiniteDimensional ℝ E] → [inst_4 : MeasurableSpace E] → [BorelSpace E] → {ψ : X → E} → Measurable ψ → MeasureTheory.Integrable (fun x => ‖ψ x‖ ^ 2) P → MeasureTheory.Measure E","l":60,"f":"Causalean/Stat/CLT/GaussianLimit.lean"},{"n":"Causalean.Stat.gaussianLimit_charFun","k":"theorem","m":"Causalean.Stat.CLT.GaussianLimit","a":"Stat/CLT","t":2,"d":"The characteristic function of the limiting Gaussian is exactly the abstract target exp(−½ ∫⟪t,ψ⟫² dP) of MultivariateCLT.","s":"∀ {X : Type u_2} [inst : MeasurableSpace X] {P : MeasureTheory.Measure X} {E : Type u_3} [inst_1 : NormedAddCommGroup E] [inst_2 : InnerProductSpace ℝ E] [inst_3 : FiniteDimensional ℝ E] [inst_4 : MeasurableSpace E] [inst_5 : BorelSpace E] {ψ : X → E} (hψ : Measurable ψ) (hvar : MeasureTheory.Integrable (fun x => ‖ψ x‖ ^ 2) P) (t : E), MeasureTheory.charFun (Causalean.Stat.gaussianLimit hψ hvar) t = Complex.exp (-↑(∫ (x : X), inner ℝ t (ψ x) ^ 2 ∂P) / 2)","l":101,"f":"Causalean/Stat/CLT/GaussianLimit.lean"},{"n":"Causalean.Stat.gaussianLimit_covarianceBilin","k":"theorem","m":"Causalean.Stat.CLT.GaussianLimit","a":"Stat/CLT","t":2,"d":"The covariance form of the limiting Gaussian recovers the asymptotic-variance integral ∫⟪s,ψ⟫⟪t,ψ⟫ dP.","s":"∀ {X : Type u_2} [inst : MeasurableSpace X] {P : MeasureTheory.Measure X} {E : Type u_3} [inst_1 : NormedAddCommGroup E] [inst_2 : InnerProductSpace ℝ E] [inst_3 : FiniteDimensional ℝ E] [inst_4 : MeasurableSpace E] [inst_5 : BorelSpace E] {ψ : X → E} (hψ : Measurable ψ) (hvar : MeasureTheory.Integrable (fun x => ‖ψ x‖ ^ 2) P) (s t : E), ((ProbabilityTheory.covarianceBilin (Causalean.Stat.gaussianLimit hψ hvar)) s) t = ∫ (x : X), inner ℝ s (ψ x) * inner ℝ t (ψ x) ∂P","l":78,"f":"Causalean/Stat/CLT/GaussianLimit.lean"},{"n":"Causalean.Stat.gaussianLimit_mean","k":"theorem","m":"Causalean.Stat.CLT.GaussianLimit","a":"Stat/CLT","t":2,"d":"The limiting Gaussian is centered.","s":"∀ {X : Type u_2} [inst : MeasurableSpace X] {P : MeasureTheory.Measure X} {E : Type u_3} [inst_1 : NormedAddCommGroup E] [inst_2 : InnerProductSpace ℝ E] [inst_3 : FiniteDimensional ℝ E] [inst_4 : MeasurableSpace E] [inst_5 : BorelSpace E] {ψ : X → E} (hψ : Measurable ψ) (hvar : MeasureTheory.Integrable (fun x => ‖ψ x‖ ^ 2) P), ∫ (x : E), x ∂Causalean.Stat.gaussianLimit hψ hvar = 0","l":70,"f":"Causalean/Stat/CLT/GaussianLimit.lean"},{"n":"Causalean.Stat.gaussianLimit_waldForm_map","k":"theorem","m":"Causalean.Stat.Inference.ChiSquaredWald","a":"Stat/Inference","t":1,"d":"χ² identification of the Wald quadratic form. When the linear map induced by the asymptotic-variance operator Σ is injective, i.e. Σ is non-degenerate, the multivariate-CLT Gaussian limit, pushed through the Wald quadratic form S ↦ ⟪S, Σ⁻¹ S⟫, is exactly the χ²_d distribution, with d the dimension of the ambient space.","s":"∀ {X : Type u_1} [inst : MeasurableSpace X] {P : MeasureTheory.Measure X} {E : Type u_2} [inst_1 : NormedAddCommGroup E] [inst_2 : InnerProductSpace ℝ E] [inst_3 : FiniteDimensional ℝ E] [inst_4 : MeasurableSpace E] [inst_5 : BorelSpace E] {ψ : X → E} (hψ : Measurable ψ) (hvar : MeasureTheory.Integrable (fun x => ‖ψ x‖ ^ 2) P) (hinj : Function.Injective ⇑(Causalean.Stat.secondMomentLM hψ hvar)), MeasureTheory.Measure.map (fun S => inner ℝ S ((Causalean.Stat.secondMomentInv hψ hvar hinj) S)) (Causalean.Stat.gaussianLimit hψ hvar) = Causalean.Stat.chiSqDist (Module.finrank ℝ E)","l":143,"f":"Causalean/Stat/Inference/ChiSquaredWald.lean"},{"n":"Causalean.Stat.gaussianMeasure","k":"def","m":"Causalean.Stat.CLT.AsymptoticLinearity","a":"Stat/CLT","t":1,"d":"Gaussian measure on ℝ with mean m and variance v. Real-typed wrapper around ProbabilityTheory.gaussianReal; if v < 0 the variance is clipped to 0 and the measure degenerates to Measure.dirac m.","s":"ℝ → ℝ → MeasureTheory.Measure ℝ","l":49,"f":"Causalean/Stat/CLT/AsymptoticLinearity.lean"},{"n":"Causalean.Stat.gaussianMeasure_zero_one_frontier_Icc","k":"theorem","m":"Causalean.Stat.Inference.Studentize","a":"Stat/Inference","t":2,"d":"The standard normal gives zero mass to the boundary of a symmetric closed interval.","s":"∀ {z : ℝ}, 0 < z → (Causalean.Stat.gaussianMeasure 0 1) (frontier (Set.Icc (-z) z)) = 0","l":82,"f":"Causalean/Stat/Inference/Studentize.lean"},{"n":"Causalean.Stat.gaussianMeasure_zero_one_singleton","k":"theorem","m":"Causalean.Stat.Inference.Studentize","a":"Stat/Inference","t":2,"d":"The standard normal has no atom at any real point.","s":"∀ (x : ℝ), (Causalean.Stat.gaussianMeasure 0 1) {x} = 0","l":74,"f":"Causalean/Stat/Inference/Studentize.lean"},{"n":"Causalean.Stat.gaussian_tail_small","k":"theorem","m":"Causalean.Stat.CLT.GaussianTail","a":"Stat/CLT","t":2,"d":"Compatibility alias for probability-measure half-line tail control.","s":"∀ (Q : MeasureTheory.Measure ℝ) [MeasureTheory.IsProbabilityMeasure Q] {ε : ℝ}, 0 < ε → ∃ R, 0 < R ∧ Q (Set.Iic (-R)) ≤ ENNReal.ofReal ε ∧ Q (Set.Ici R) ≤ ENNReal.ofReal ε","l":94,"f":"Causalean/Stat/CLT/GaussianTail.lean"},{"n":"Causalean.Stat.gaussian_tail_small_gaussian","k":"theorem","m":"Causalean.Stat.CLT.GaussianTail","a":"Stat/CLT","t":2,"d":"Both symmetric half-line tails of a Gaussian distribution can be made smaller than any positive tolerance by choosing a sufficiently large positive cutoff.","s":"∀ {m v ε : ℝ}, 0 < ε → ∃ R, 0 < R ∧ (Causalean.Stat.gaussianMeasure m v) (Set.Iic (-R)) ≤ ENNReal.ofReal ε ∧ (Causalean.Stat.gaussianMeasure m v) (Set.Ici R) ≤ ENNReal.ofReal ε","l":102,"f":"Causalean/Stat/CLT/GaussianTail.lean"},{"n":"Causalean.Stat.glivenkoCantelli_of_fintype","k":"theorem","m":"Causalean.Stat.EmpiricalProcess.GlivenkoCantelli","a":"Stat/EmpiricalProcess","t":1,"d":"A finite class of integrable functions is Glivenko–Cantelli. Consider a finite family of real-valued functions f i on the sample space, observed along an i.i.d. sample S drawn from a probability distribution P. If every f i is measurable and every f i is integrable with respect to P, then the worst-case gap between the empirical mean and the population mean of f i, taken over all indices i, converges to zero in probability as the sample size grows.","s":"∀ {Ω : Type u_1} {X : Type u_2} {ι : Type u_3} [inst : MeasurableSpace Ω] [inst_1 : MeasurableSpace X] {μ : MeasureTheory.Measure Ω} {P : MeasureTheory.Measure X} [Finite ι] [MeasureTheory.IsProbabilityMeasure P] (S : Causalean.Stat.IIDSample Ω X μ P) (f : ι → X → ℝ), (∀ (i : ι), Measurable (f i)) → (∀ (i : ι), MeasureTheory.Integrable (f i) P) → Causalean.Stat.GlivenkoCantelli S f","l":102,"f":"Causalean/Stat/EmpiricalProcess/GlivenkoCantelli.lean"},{"n":"Causalean.Stat.glivenkoCantelli_of_hasL1Bracketing","k":"theorem","m":"Causalean.Stat.EmpiricalProcess.GlivenkoCantelli","a":"Stat/EmpiricalProcess","t":1,"d":"A class with finite L¹(P) brackets of arbitrarily small width is Glivenko-Cantelli. Consider a family of real-valued functions f i on the sample space, observed along an i.i.d. sample S drawn from a probability distribution P. If every f i is measurable and for every target width the family can be covered by finitely many upper/lower bracket pairs, each integrable and each sandwiching its assigned member almost everywhere with L¹(P)-gap between the bracket endpoints at most that width, then the worst-case gap between the empirical mean and the population mean of f i, taken over all indices i, converges to zero in probability as the sample size grows.","s":"∀ {Ω : Type u_1} {X : Type u_2} {ι : Type u_3} [inst : MeasurableSpace Ω] [inst_1 : MeasurableSpace X] {μ : MeasureTheory.Measure Ω} {P : MeasureTheory.Measure X} [MeasureTheory.IsProbabilityMeasure P] (S : Causalean.Stat.IIDSample Ω X μ P) (f : ι → X → ℝ), (∀ (i : ι), Measurable (f i)) → Causalean.Stat.HasL1Bracketing f P → Causalean.Stat.GlivenkoCantelli S f","l":139,"f":"Causalean/Stat/EmpiricalProcess/GlivenkoCantelli.lean"},{"n":"Causalean.Stat.gmmBread","k":"def","m":"Causalean.Stat.GMM.VarianceAlgebra","a":"Stat/GMM","t":1,"d":"The bread operator GᵀWG : E →L E of a GMM problem with Jacobian G and weighting W. Symmetric whenever W is, and invertible whenever G has full column rank and W ≻ 0.","s":"{E : Type u_1} → {F : Type u_2} → [inst : NormedAddCommGroup E] → [inst_1 : InnerProductSpace ℝ E] → [FiniteDimensional ℝ E] → [inst_3 : NormedAddCommGroup F] → [inst_4 : InnerProductSpace ℝ F] → [FiniteDimensional ℝ F] → (E →L[ℝ] F) → (F →L[ℝ] F) → E →L[ℝ] E","l":71,"f":"Causalean/Stat/GMM/VarianceAlgebra.lean"},{"n":"Causalean.Stat.gmmHatMatrix","k":"def","m":"Causalean.Stat.GMM.OverID","a":"Stat/GMM","t":1,"d":"The hat matrix H = G (GᵀG)⁻¹ Gᵀ : F →L F of a whitened GMM problem with Jacobian G and bread inverse effInv (a two-sided inverse of GᵀG). It is the orthogonal projection onto range G.","s":"{E : Type u_1} → {F : Type u_2} → [inst : NormedAddCommGroup E] → [inst_1 : InnerProductSpace ℝ E] → [FiniteDimensional ℝ E] → [inst_3 : NormedAddCommGroup F] → [inst_4 : InnerProductSpace ℝ F] → [FiniteDimensional ℝ F] → (E →L[ℝ] F) → (E →L[ℝ] E) → F →L[ℝ] F","l":62,"f":"Causalean/Stat/GMM/OverID.lean"},{"n":"Causalean.Stat.gmmIF","k":"def","m":"Causalean.Stat.GMM.Setup","a":"Stat/GMM","t":1,"d":"The GMM influence function −(GᵀWG)⁻¹ GᵀW g(θ₀,·).","s":"{E : Type u_1} → {F : Type u_2} → {X : Type u_3} → [inst : NormedAddCommGroup E] → [inst_1 : InnerProductSpace ℝ E] → [FiniteDimensional ℝ E] → [inst_3 : NormedAddCommGroup F] → [inst_4 : InnerProductSpace ℝ F] → [FiniteDimensional ℝ F] → (E →L[ℝ] F) → (F →L[ℝ] F) → (E →L[ℝ] E) → (E → X → F) → E → X → E","l":67,"f":"Causalean/Stat/GMM/Setup.lean"},{"n":"Causalean.Stat.gmmResidualMaker","k":"def","m":"Causalean.Stat.GMM.OverID","a":"Stat/GMM","t":1,"d":"The residual maker M = I − H : F →L F of a whitened GMM problem. It is the orthogonal projection onto the orthogonal complement of range G; the J-statistic's limit law is ‖M w‖² for w a standard Gaussian on F.","s":"{E : Type u_1} → {F : Type u_2} → [inst : NormedAddCommGroup E] → [inst_1 : InnerProductSpace ℝ E] → [FiniteDimensional ℝ E] → [inst_3 : NormedAddCommGroup F] → [inst_4 : InnerProductSpace ℝ F] → [FiniteDimensional ℝ F] → (E →L[ℝ] F) → (E →L[ℝ] E) → F →L[ℝ] F","l":68,"f":"Causalean/Stat/GMM/OverID.lean"},{"n":"Causalean.Stat.gmmResidualMaker_finrank_range","k":"theorem","m":"Causalean.Stat.GMM.OverID","a":"Stat/GMM","t":2,"d":"The rank of the residual maker's range is m − k (finrank F − finrank E), the Hansen–Sargan degrees of freedom.","s":"∀ {E : Type u_1} {F : Type u_2} [inst : NormedAddCommGroup E] [inst_1 : InnerProductSpace ℝ E] [inst_2 : FiniteDimensional ℝ E] [inst_3 : NormedAddCommGroup F] [inst_4 : InnerProductSpace ℝ F] [inst_5 : FiniteDimensional ℝ F] {G : E →L[ℝ] F} {effInv : E →L[ℝ] E}, effInv ∘SL ContinuousLinearMap.adjoint G ∘SL G = ContinuousLinearMap.id ℝ E → (ContinuousLinearMap.adjoint G ∘SL G) ∘SL effInv = ContinuousLinearMap.id ℝ E → Module.finrank ℝ ↥(↑(Causalean.Stat.gmmResidualMaker G effInv)).range = Module.finrank ℝ F - Module.finrank ℝ E","l":135,"f":"Causalean/Stat/GMM/OverID.lean"},{"n":"Causalean.Stat.gmmResidualMaker_idempotent","k":"theorem","m":"Causalean.Stat.GMM.OverID","a":"Stat/GMM","t":2,"d":"The residual maker M = I − H is idempotent: M ∘L M = M.","s":"∀ {E : Type u_1} {F : Type u_2} [inst : NormedAddCommGroup E] [inst_1 : InnerProductSpace ℝ E] [inst_2 : FiniteDimensional ℝ E] [inst_3 : NormedAddCommGroup F] [inst_4 : InnerProductSpace ℝ F] [inst_5 : FiniteDimensional ℝ F] {G : E →L[ℝ] F} {effInv : E →L[ℝ] E}, (ContinuousLinearMap.adjoint G ∘SL G) ∘SL effInv = ContinuousLinearMap.id ℝ E → Causalean.Stat.gmmResidualMaker G effInv ∘SL Causalean.Stat.gmmResidualMaker G effInv = Causalean.Stat.gmmResidualMaker G effInv","l":124,"f":"Causalean/Stat/GMM/OverID.lean"},{"n":"Causalean.Stat.gmmResidualMaker_isSelfAdjoint","k":"theorem","m":"Causalean.Stat.GMM.OverID","a":"Stat/GMM","t":2,"d":"The residual maker M = I − H is self-adjoint.","s":"∀ {E : Type u_1} {F : Type u_2} [inst : NormedAddCommGroup E] [inst_1 : InnerProductSpace ℝ E] [inst_2 : FiniteDimensional ℝ E] [inst_3 : NormedAddCommGroup F] [inst_4 : InnerProductSpace ℝ F] [inst_5 : FiniteDimensional ℝ F] {G : E →L[ℝ] F} {effInv : E →L[ℝ] E}, (ContinuousLinearMap.adjoint G ∘SL G) ∘SL effInv = ContinuousLinearMap.id ℝ E → IsSelfAdjoint (Causalean.Stat.gmmResidualMaker G effInv)","l":114,"f":"Causalean/Stat/GMM/OverID.lean"},{"n":"Causalean.Stat.gmmSandwich","k":"def","m":"Causalean.Stat.GMM.VarianceAlgebra","a":"Stat/GMM","t":1,"d":"The sandwich asymptotic variance (GᵀWG)⁻¹ GᵀW Σ WG (GᵀWG)⁻¹, given a two-sided inverse breadInv of GᵀWG.","s":"{E : Type u_1} → {F : Type u_2} → [inst : NormedAddCommGroup E] → [inst_1 : InnerProductSpace ℝ E] → [FiniteDimensional ℝ E] → [inst_3 : NormedAddCommGroup F] → [inst_4 : InnerProductSpace ℝ F] → [FiniteDimensional ℝ F] → (E →L[ℝ] F) → (F →L[ℝ] F) → (F →L[ℝ] F) → (E →L[ℝ] E) → E →L[ℝ] E","l":77,"f":"Causalean/Stat/GMM/VarianceAlgebra.lean"},{"n":"Causalean.Stat.gmmScore","k":"def","m":"Causalean.Stat.GMM.Setup","a":"Stat/GMM","t":1,"d":"The combined GMM score ψ(θ,x) = GᵀW g(θ,x) : E. Setting its empirical mean to zero is the first-order condition of the GMM criterion ḡ(θ)ᵀ W ḡ(θ) (after fixing the Jacobian weight at its population value); the GMM estimator is the resulting Z-estimator.","s":"{E : Type u_1} → {F : Type u_2} → {X : Type u_3} → [inst : NormedAddCommGroup E] → [inst_1 : InnerProductSpace ℝ E] → [FiniteDimensional ℝ E] → [inst_3 : NormedAddCommGroup F] → [inst_4 : InnerProductSpace ℝ F] → [FiniteDimensional ℝ F] → (E →L[ℝ] F) → (F →L[ℝ] F) → (E → X → F) → E → X → E","l":59,"f":"Causalean/Stat/GMM/Setup.lean"},{"n":"Causalean.Stat.gmm_asymptotically_linear","k":"theorem","m":"Causalean.Stat.GMM.AsymptoticNormality","a":"Stat/GMM","t":1,"d":"GMM asymptotic linearity. Consider a GMM problem prob (moment function, weight matrix, and target parameter θ₀) together with Z-estimator regularity conditions reg for its combined score, evaluated along an i.i.d. sample S. If the Jacobian inverse of the combined score's population expectation at θ₀, reg.J₀_inv, coincides with the GMM problem's \"bread\" inverse prob.breadInv, the estimator sequence θn is consistent for θ₀, the combined score is stochastically equicontinuous at θ₀ along θn, θn converges to θ₀ at the parametric rate, i.e. ‖θn n − θ₀‖ is Oₚ(n^(-1/2)), and θn eventually and almost surely solves the sample moment equation ∑ᵢ score(θn n, Zᵢ) = 0, then θn admits the usual GMM asymptotic-linear expansion around θ₀.","s":"∀ {Ω : Type u_1} {X : Type u_2} [inst : MeasurableSpace Ω] [inst_1 : MeasurableSpace X] {μ : MeasureTheory.Measure Ω} {P : MeasureTheory.Measure X} {E : Type u_3} {F : Type u_4} [inst_2 : NormedAddCommGroup E] [inst_3 : InnerProductSpace ℝ E] [inst_4 : FiniteDimensional ℝ E] [inst_5 : MeasurableSpace E] [BorelSpace E] [inst_7 : NormedAddCommGroup F] [inst_8 : InnerProductSpace ℝ F] [inst_9 : FiniteDimensional ℝ F] [inst_10 : MeasurableSpace F] [MeasureTheory.IsProbabilityMeasure μ] (prob : Causalean.Stat.GMMProblem P) (reg : Causalean.Stat.ZEstimatorRegularity prob.score prob.θ₀ P), reg.J₀_inv = prob.breadInv → ∀ (S : Causalean.Stat.IIDSample Ω X μ P) (θn : ℕ → Ω → E), (∀ ε > 0, Filter.Tendsto (fun n => μ {ω | ε < ‖θn n ω - prob.θ₀‖}) Filter.atTop (nhds 0)) → Causalean.Stat.StochEquicontAt prob.score prob.θ₀ P μ S θn → Causalean.Stat.IsBigOp (fun n ω => ‖θn n ω - prob.θ₀‖) (fun n => (√↑n)⁻¹) μ → (∀ᶠ (n : ℕ) in Filter.atTop, ∀ᵐ (ω : Ω) ∂μ, ∑ i ∈ Finset.range n, prob.score (θn n ω) (S.Z i ω) = 0) → Causalean.Stat.IsAsymLinearVec θn prob.θ₀ prob.influence S fun n => Finset.range n","l":67,"f":"Causalean/Stat/GMM/AsymptoticNormality.lean"},{"n":"Causalean.Stat.gmm_asymptotically_linear_of_donsker","k":"theorem","m":"Causalean.Stat.GMM.AsymptoticNormality","a":"Stat/GMM","t":2,"d":"GMM asymptotic linearity with the equicontinuity hypothesis discharged. The GMM analogue of zEstimator_clt_of_donsker: gmm_asymptotically_linear with the opaque modulus hStochEquicont replaced by the class-level Donsker property AsymptoticEquicont prob.score prob.θ₀ P μ S, reconstructed via stochEquicontAt_of_asymptoticEquicont from hAEC and consistency.","s":"∀ {Ω : Type u_1} {X : Type u_2} [inst : MeasurableSpace Ω] [inst_1 : MeasurableSpace X] {μ : MeasureTheory.Measure Ω} {P : MeasureTheory.Measure X} {E : Type u_3} {F : Type u_4} [inst_2 : NormedAddCommGroup E] [inst_3 : InnerProductSpace ℝ E] [inst_4 : FiniteDimensional ℝ E] [inst_5 : MeasurableSpace E] [BorelSpace E] [inst_7 : NormedAddCommGroup F] [inst_8 : InnerProductSpace ℝ F] [inst_9 : FiniteDimensional ℝ F] [inst_10 : MeasurableSpace F] [MeasureTheory.IsProbabilityMeasure μ] (prob : Causalean.Stat.GMMProblem P) (reg : Causalean.Stat.ZEstimatorRegularity prob.score prob.θ₀ P), reg.J₀_inv = prob.breadInv → ∀ (S : Causalean.Stat.IIDSample Ω X μ P) (θn : ℕ → Ω → E), (∀ ε > 0, Filter.Tendsto (fun n => μ {ω | ε < ‖θn n ω - prob.θ₀‖}) Filter.atTop (nhds 0)) → Causalean.Stat.AsymptoticEquicont prob.score prob.θ₀ P μ S → Causalean.Stat.IsBigOp (fun n ω => ‖θn n ω - prob.θ₀‖) (fun n => (√↑n)⁻¹) μ → (∀ᶠ (n : ℕ) in Filter.atTop, ∀ᵐ (ω : Ω) ∂μ, ∑ i ∈ Finset.range n, prob.score (θn n ω) (S.Z i ω) = 0) → Causalean.Stat.IsAsymLinearVec θn prob.θ₀ prob.influence S fun n => Finset.range n","l":176,"f":"Causalean/Stat/GMM/AsymptoticNormality.lean"},{"n":"Causalean.Stat.gmm_asymptotically_linear_of_extremum","k":"theorem","m":"Causalean.Stat.GMM.AsymptoticNormality","a":"Stat/GMM","t":2,"d":"GMM asymptotic linearity from extremum primitives. The GMM analogue of zEstimator_clt_of_extremum: gmm_asymptotically_linear with the consistency hypothesis discharged from a Glivenko–Cantelli GMM criterion m with a well-separated population maximum at θ₀ of which θn is a sample maximiser. The classical instance is m θ = −ḡ_n(θ)ᵀ W ḡ_n(θ) (the GMM objective), whose score is prob.score.","s":"∀ {Ω : Type u_1} {X : Type u_2} [inst : MeasurableSpace Ω] [inst_1 : MeasurableSpace X] {μ : MeasureTheory.Measure Ω} {P : MeasureTheory.Measure X} {E : Type u_3} {F : Type u_4} [inst_2 : NormedAddCommGroup E] [inst_3 : InnerProductSpace ℝ E] [inst_4 : FiniteDimensional ℝ E] [inst_5 : MeasurableSpace E] [BorelSpace E] [inst_7 : NormedAddCommGroup F] [inst_8 : InnerProductSpace ℝ F] [inst_9 : FiniteDimensional ℝ F] [inst_10 : MeasurableSpace F] [MeasureTheory.IsProbabilityMeasure μ] (prob : Causalean.Stat.GMMProblem P) (reg : Causalean.Stat.ZEstimatorRegularity prob.score prob.θ₀ P), reg.J₀_inv = prob.breadInv → ∀ (S : Causalean.Stat.IIDSample Ω X μ P) (θn : ℕ → Ω → E) (m : E → X → ℝ), Causalean.Stat.GlivenkoCantelli S m → (∀ (n : ℕ) (ω : Ω), S.sampleMean (m prob.θ₀) n ω ≤ S.sampleMean (m (θn n ω)) n ω) → (∀ (ε : ℝ), 0 < ε → ∃ η, 0 < η ∧ ∀ (θ : E), ε ≤ dist θ prob.θ₀ → ∫ (x : X), m θ x ∂P + η ≤ ∫ (x : X), m prob.θ₀ x ∂P) → Causalean.Stat.StochEquicontAt prob.score prob.θ₀ P μ S θn → Causalean.Stat.IsBigOp (fun n ω => ‖θn n ω - prob.θ₀‖) (fun n => (√↑n)⁻¹) μ → (∀ᶠ (n : ℕ) in Filter.atTop, ∀ᵐ (ω : Ω) ∂μ, ∑ i ∈ Finset.range n, prob.score (θn n ω) (S.Z i ω) = 0) → Causalean.Stat.IsAsymLinearVec θn prob.θ₀ prob.influence S fun n => Finset.range n","l":146,"f":"Causalean/Stat/GMM/AsymptoticNormality.lean"},{"n":"Causalean.Stat.gmm_asymptotically_linear_of_extremum_donsker","k":"theorem","m":"Causalean.Stat.GMM.AsymptoticNormality","a":"Stat/GMM","t":2,"d":"GMM asymptotic linearity from primitive conditions: both opaque hypotheses discharged. The GMM analogue of zEstimator_clt_of_extremum_donsker: neither hConsistent nor hStochEquicont is assumed. Consistency is derived from the Glivenko–Cantelli GMM criterion m with well-separated maximum, then fed to both the linearisation and the StochEquicontAt reduction applied to hAEC.","s":"∀ {Ω : Type u_1} {X : Type u_2} [inst : MeasurableSpace Ω] [inst_1 : MeasurableSpace X] {μ : MeasureTheory.Measure Ω} {P : MeasureTheory.Measure X} {E : Type u_3} {F : Type u_4} [inst_2 : NormedAddCommGroup E] [inst_3 : InnerProductSpace ℝ E] [inst_4 : FiniteDimensional ℝ E] [inst_5 : MeasurableSpace E] [BorelSpace E] [inst_7 : NormedAddCommGroup F] [inst_8 : InnerProductSpace ℝ F] [inst_9 : FiniteDimensional ℝ F] [inst_10 : MeasurableSpace F] [MeasureTheory.IsProbabilityMeasure μ] (prob : Causalean.Stat.GMMProblem P) (reg : Causalean.Stat.ZEstimatorRegularity prob.score prob.θ₀ P), reg.J₀_inv = prob.breadInv → ∀ (S : Causalean.Stat.IIDSample Ω X μ P) (θn : ℕ → Ω → E) (m : E → X → ℝ), Causalean.Stat.GlivenkoCantelli S m → (∀ (n : ℕ) (ω : Ω), S.sampleMean (m prob.θ₀) n ω ≤ S.sampleMean (m (θn n ω)) n ω) → (∀ (ε : ℝ), 0 < ε → ∃ η, 0 < η ∧ ∀ (θ : E), ε ≤ dist θ prob.θ₀ → ∫ (x : X), m θ x ∂P + η ≤ ∫ (x : X), m prob.θ₀ x ∂P) → Causalean.Stat.AsymptoticEquicont prob.score prob.θ₀ P μ S → Causalean.Stat.IsBigOp (fun n ω => ‖θn n ω - prob.θ₀‖) (fun n => (√↑n)⁻¹) μ → (∀ᶠ (n : ℕ) in Filter.atTop, ∀ᵐ (ω : Ω) ∂μ, ∑ i ∈ Finset.range n, prob.score (θn n ω) (S.Z i ω) = 0) → Causalean.Stat.IsAsymLinearVec θn prob.θ₀ prob.influence S fun n => Finset.range n","l":199,"f":"Causalean/Stat/GMM/AsymptoticNormality.lean"},{"n":"Causalean.Stat.gmm_efficiency","k":"theorem","m":"Causalean.Stat.GMM.VarianceAlgebra","a":"Stat/GMM","t":1,"d":"GMM optimal-weighting theorem (Hansen 1982, Theorem 3.2). For a Jacobian G, a self-adjoint weighting operator W, and a positive-semidefinite covariance operator Cov admitting a two-sided inverse CovInv, suppose further that the \"bread\" GᵀWG has a two-sided inverse breadInv and that the efficient bread Gᵀ CovInv G has a two-sided inverse effInv. Then the sandwich asymptotic variance of the GMM estimator with weighting W dominates the efficient (optimally-weighted) asymptotic variance in the Löwner order, i.e. their difference is a positive-semidefinite operator.","s":"∀ {E : Type u_1} {F : Type u_2} [inst : NormedAddCommGroup E] [inst_1 : InnerProductSpace ℝ E] [inst_2 : FiniteDimensional ℝ E] [inst_3 : NormedAddCommGroup F] [inst_4 : InnerProductSpace ℝ F] [inst_5 : FiniteDimensional ℝ F] (G : E →L[ℝ] F) (W Cov CovInv : F →L[ℝ] F), ContinuousLinearMap.adjoint W = W → Cov.IsPositive → CovInv ∘SL Cov = ContinuousLinearMap.id ℝ F → Cov ∘SL CovInv = ContinuousLinearMap.id ℝ F → ∀ (breadInv : E →L[ℝ] E), breadInv ∘SL Causalean.Stat.gmmBread G W = ContinuousLinearMap.id ℝ E → Causalean.Stat.gmmBread G W ∘SL breadInv = ContinuousLinearMap.id ℝ E → ∀ (effInv : E →L[ℝ] E), effInv ∘SL Causalean.Stat.gmmBread G CovInv = ContinuousLinearMap.id ℝ E → Causalean.Stat.gmmBread G CovInv ∘SL effInv = ContinuousLinearMap.id ℝ E → (Causalean.Stat.gmmSandwich G W Cov breadInv - effInv).IsPositive","l":106,"f":"Causalean/Stat/GMM/VarianceAlgebra.lean"},{"n":"Causalean.Stat.gmm_tendsto_normal_vec","k":"theorem","m":"Causalean.Stat.GMM.AsymptoticNormality","a":"Stat/GMM","t":2,"d":"GMM asymptotic normality from a supplied vector CLT contact. Combining gmm_asymptotically_linear with IsAsymLinearVec.tendsto_normal_vec, the rescaled GMM estimator converges to the caller-supplied vector CLT target Q. The theorem is parametric in the limiting law: the concrete Gaussian or sandwich identification is supplied through hCLT and the target law Q.","s":"∀ {Ω : Type u_1} {X : Type u_2} [inst : MeasurableSpace Ω] [inst_1 : MeasurableSpace X] {μ : MeasureTheory.Measure Ω} {P : MeasureTheory.Measure X} {E : Type u_3} {F : Type u_4} [inst_2 : NormedAddCommGroup E] [inst_3 : InnerProductSpace ℝ E] [inst_4 : FiniteDimensional ℝ E] [inst_5 : MeasurableSpace E] [inst_6 : BorelSpace E] [inst_7 : NormedAddCommGroup F] [inst_8 : InnerProductSpace ℝ F] [inst_9 : FiniteDimensional ℝ F] [inst_10 : MeasurableSpace F] [inst_11 : MeasureTheory.IsProbabilityMeasure μ] (prob : Causalean.Stat.GMMProblem P) (reg : Causalean.Stat.ZEstimatorRegularity prob.score prob.θ₀ P), reg.J₀_inv = prob.breadInv → ∀ (S : Causalean.Stat.IIDSample Ω X μ P) (θn : ℕ → Ω → E), (∀ ε > 0, Filter.Tendsto (fun n => μ {ω | ε < ‖θn n ω - prob.θ₀‖}) Filter.atTop (nhds 0)) → Causalean.Stat.StochEquicontAt prob.score prob.θ₀ P μ S θn → Causalean.Stat.IsBigOp (fun n ω => ‖θn n ω - prob.θ₀‖) (fun n => (√↑n)⁻¹) μ → (∀ᶠ (n : ℕ) in Filter.atTop, ∀ᵐ (ω : Ω) ∂μ, ∑ i ∈ Finset.range n, prob.score (θn n ω) (S.Z i ω) = 0) → ∀ (Q : MeasureTheory.ProbabilityMeasure E) (hθn_meas : ∀ (n : ℕ), AEMeasurable (Causalean.Stat.IsAsymLinearVec.rescaledEstimator θn prob.θ₀ (fun m => Finset.range m) n) μ) (hSum_meas : ∀ (n : ℕ), AEMeasurable (Causalean.Stat.IsAsymLinearVec.normalizedSum S prob.influence (fun m => Finset.range m) n) μ), Filter.Tendsto (fun n => ⟨MeasureTheory.Measure.map (Causalean.Stat.IsAsymLinearVec.normalizedSum S prob.influence (fun m => Finset.range m) n) μ, MeasureTheory.Measure.isProbabilityMeasure_map (hSum_meas n)⟩) Filter.atTop (nhds Q) → Filter.Tendsto (fun n => ⟨MeasureTheory.Measure.map (Causalean.Stat.IsAsymLinearVec.rescaledEstimator θn prob.θ₀ (fun m => Finset.range m) n) μ, MeasureTheory.Measure.isProbabilityMeasure_map (hθn_meas n)⟩) Filter.atTop (nhds Q)","l":108,"f":"Causalean/Stat/GMM/AsymptoticNormality.lean"},{"n":"Causalean.Stat.groupArmCount","k":"def","m":"Causalean.Stat.Sample.OccupancyWeightedMean.Basic","a":"Stat/Sample/OccupancyWeightedMean","t":1,"d":"The arm/group count is the number of sample coordinates with both the requested group and requested arm.","s":"{Omega : Type u_1} → {kappa : Type u_2} → [DecidableEq kappa] → {n : ℕ} → (Omega → kappa) → (Omega → Bool) → (Fin n → Omega) → Bool → kappa → ℕ","l":59,"f":"Causalean/Stat/Sample/OccupancyWeightedMean/Basic.lean"},{"n":"Causalean.Stat.groupCount","k":"def","m":"Causalean.Stat.Sample.OccupancyWeightedMean.Basic","a":"Stat/Sample/OccupancyWeightedMean","t":1,"d":"The group count is the sum of its control and treated arm counts.","s":"{Omega : Type u_1} → {kappa : Type u_2} → [DecidableEq kappa] → {n : ℕ} → (Omega → kappa) → (Omega → Bool) → (Fin n → Omega) → kappa → ℕ","l":65,"f":"Causalean/Stat/Sample/OccupancyWeightedMean/Basic.lean"},{"n":"Causalean.Stat.groupEvent","k":"def","m":"Causalean.Stat.Sample.OccupancyWeightedMean.Basic","a":"Stat/Sample/OccupancyWeightedMean","t":1,"d":"The group event consists of observations having the requested finite group label, irrespective of arm.","s":"{Omega : Type u_1} → {kappa : Type u_2} → (Omega → kappa) → kappa → Set Omega","l":35,"f":"Causalean/Stat/Sample/OccupancyWeightedMean/Basic.lean"},{"n":"Causalean.Stat.half_integral_abs_rnDeriv_sub_le_tvDist","k":"theorem","m":"Causalean.Stat.Minimax.CoordinatewiseOverlap","a":"Stat/Minimax","t":2,"d":"For two probability measures dominated by a common finite measure, half the L¹ distance between their Radon--Nikodym densities is bounded by total variation. This is the reverse Scheffé inequality needed to construct the common submeasure in the maximal-coupling argument.","s":"∀ {Ω : Type u_1} [inst : MeasurableSpace Ω] (μ ν ξ : MeasureTheory.Measure Ω) [MeasureTheory.IsProbabilityMeasure μ] [MeasureTheory.IsProbabilityMeasure ν] [MeasureTheory.IsFiniteMeasure ξ], μ.AbsolutelyContinuous ξ → ν.AbsolutelyContinuous ξ → 1 / 2 * ∫ (x : Ω), |(μ.rnDeriv ξ x).toReal - (ν.rnDeriv ξ x).toReal| ∂ξ ≤ Causalean.Stat.tvDist μ ν","l":170,"f":"Causalean/Stat/Minimax/CoordinatewiseOverlap.lean"},{"n":"Causalean.Stat.half_one_sub_tvDist_le_max_error","k":"theorem","m":"Causalean.Stat.Minimax.LeCam","a":"Stat/Minimax","t":1,"d":"Le Cam two-point bound (max form). For a measurable estimator est, under 2s-separation of θ₀, θ₁, the worst-case error probability is at least ½ (1 − tvDist P₀ P₁).","s":"∀ {Ω : Type u_1} {mΩ : MeasurableSpace Ω} {P₀ P₁ : MeasureTheory.Measure Ω} [MeasureTheory.IsProbabilityMeasure P₀] [MeasureTheory.IsProbabilityMeasure P₁] {Θ : Type u_2} [inst : PseudoMetricSpace Θ] [inst_1 : MeasurableSpace Θ] [OpensMeasurableSpace Θ] {est : Ω → Θ}, Measurable est → ∀ {θ₀ θ₁ : Θ} {s : ℝ}, 2 * s ≤ dist θ₀ θ₁ → (1 - Causalean.Stat.tvDist P₀ P₁) / 2 ≤ max (P₀.real {ω | s ≤ dist (est ω) θ₀}) (P₁.real {ω | s ≤ dist (est ω) θ₁})","l":73,"f":"Causalean/Stat/Minimax/LeCam.lean"},{"n":"Causalean.Stat.hammingRisk","k":"def","m":"Causalean.Stat.Minimax.Assouad","a":"Stat/Minimax","t":1,"d":"The expected Hamming risk of a cube estimator at vertex τ: the expected number of coordinates it decodes incorrectly.","s":"{Ω : Type u_1} → {mΩ : MeasurableSpace Ω} → {d : ℕ} → ((Fin d → Bool) → MeasureTheory.Measure Ω) → (Ω → Fin d → Bool) → (Fin d → Bool) → ℝ","l":89,"f":"Causalean/Stat/Minimax/Assouad.lean"},{"n":"Causalean.Stat.hammingRisk_pair_ge","k":"theorem","m":"Causalean.Stat.Minimax.Assouad","a":"Stat/Minimax","t":2,"d":"Per-coordinate pair bound. For each vertex τ, the coordinate-j error mass at τ plus the one at its j-flip is at least 1 − tvDist (P τ) (P (flip j τ)). This is the two-point testing bound applied to the j-th decoded bit.","s":"∀ {Ω : Type u_1} {mΩ : MeasurableSpace Ω} {d : ℕ} (P : (Fin d → Bool) → MeasureTheory.Measure Ω) [∀ (τ : Fin d → Bool), MeasureTheory.IsProbabilityMeasure (P τ)] (est : Ω → Fin d → Bool), (∀ (j : Fin d) (b : Bool), MeasurableSet {ω | est ω j = b}) → ∀ (j : Fin d) (τ : Fin d → Bool), 1 - Causalean.Stat.tvDist (P τ) (P (Causalean.Stat.flipBit j τ)) ≤ (P τ).real {ω | est ω j ≠ τ j} + (P (Causalean.Stat.flipBit j τ)).real {ω | est ω j ≠ Causalean.Stat.flipBit j τ j}","l":114,"f":"Causalean/Stat/Minimax/Assouad.lean"},{"n":"Causalean.Stat.harmonic_sum_ge_log_sub_log","k":"theorem","m":"Causalean.Stat.Limit.SequentialCumulativeRisk","a":"Stat/Limit","t":2,"d":"Harmonic-tail lower bound: for natural numbers with 1 ≤ m, ∑_{t=m}^{n} 1/t ≥ log(n+1) - log m. Proof by telescoping the pointwise bound 1/t ≥ log(t+1) - log t.","s":"∀ (m n : ℕ), 1 ≤ m → Real.log (↑n + 1) - Real.log ↑m ≤ ∑ t ∈ Finset.Icc m n, 1 / ↑t","l":60,"f":"Causalean/Stat/Limit/SequentialCumulativeRisk.lean"},{"n":"Causalean.Stat.hasFDerivAt_ratio","k":"theorem","m":"Causalean.Stat.Inference.RatioDeltaMethod","a":"Stat/Inference","t":2,"d":"Fréchet derivative of the ratio map. On EuclideanSpace ℝ (Fin 2), the map v ↦ v 0 / v 1 is Fréchet-differentiable at any t₀ whose second coordinate is nonzero, with derivative ratioDeriv t₀.","s":"∀ {t₀ : EuclideanSpace ℝ (Fin 2)}, t₀.ofLp 1 ≠ 0 → HasFDerivAt (fun v => v.ofLp 0 / v.ofLp 1) (Causalean.Stat.ratioDeriv t₀) t₀","l":62,"f":"Causalean/Stat/Inference/RatioDeltaMethod.lean"},{"n":"Causalean.Stat.hasHadamardDirDerivAt_max","k":"theorem","m":"Causalean.Stat.Inference.HadamardDeriv","a":"Stat/Inference","t":2,"d":"max is Hadamard directionally differentiable. At (a, b) the derivative is maxDirDeriv a b (linear off the diagonal, sublinear at a tie).","s":"∀ (a b : ℝ), Causalean.Stat.HasHadamardDirDerivAt (fun z => max z.1 z.2) (Causalean.Stat.maxDirDeriv a b) (a, b)","l":181,"f":"Causalean/Stat/Inference/HadamardDeriv.lean"},{"n":"Causalean.Stat.hasHadamardDirDerivAt_min","k":"theorem","m":"Causalean.Stat.Inference.HadamardDeriv","a":"Stat/Inference","t":2,"d":"min is Hadamard directionally differentiable. At (a, b) the derivative is minDirDeriv a b.","s":"∀ (a b : ℝ), Causalean.Stat.HasHadamardDirDerivAt (fun z => min z.1 z.2) (Causalean.Stat.minDirDeriv a b) (a, b)","l":254,"f":"Causalean/Stat/Inference/HadamardDeriv.lean"},{"n":"Causalean.Stat.hasIIDSample_iff_isProbabilityMeasure","k":"theorem","m":"Causalean.Stat.Sample.PiTransport","a":"Stat/Sample","t":2,"d":"The existence of an i.i.d. sample with a prescribed common marginal is equivalent to that marginal being a probability measure.","s":"∀ {X : Type u} [inst : MeasurableSpace X] (P : MeasureTheory.Measure X), Causalean.Stat.HasIIDSample P ↔ MeasureTheory.IsProbabilityMeasure P","l":248,"f":"Causalean/Stat/Sample/PiTransport.lean"},{"n":"Causalean.Stat.hasIIDSample_of_isProbabilityMeasure","k":"theorem","m":"Causalean.Stat.Sample.PiTransport","a":"Stat/Sample","t":2,"d":"An i.i.d. sample with a given law always exists: every probability measure is the common marginal of some independent, identically distributed sample, realised on the infinite product space by the coordinate projections.","s":"∀ {X : Type u} [inst : MeasurableSpace X] (P : MeasureTheory.Measure X) [MeasureTheory.IsProbabilityMeasure P], Causalean.Stat.HasIIDSample P","l":241,"f":"Causalean/Stat/Sample/PiTransport.lean"},{"n":"Causalean.Stat.hellingerSqDensity","k":"def","m":"Causalean.Stat.Minimax.HellingerAffinity","a":"Stat/Minimax","t":1,"d":"The squared Hellinger distance between two nonnegative densities against one common dominating measure, in the unhalved convention: the integral of the squared difference of their pointwise square roots.","s":"{α : Type u_1} → [inst : MeasurableSpace α] → MeasureTheory.Measure α → (α → ℝ) → (α → ℝ) → ℝ","l":62,"f":"Causalean/Stat/Minimax/HellingerAffinity.lean"},{"n":"Causalean.Stat.hellingerSqDensity_eq_two_mul_one_sub_affinity","k":"theorem","m":"Causalean.Stat.Minimax.HellingerAffinity","a":"Stat/Minimax","t":2,"d":"For two integrable nonnegative densities that each integrate to one, the unhalved squared Hellinger distance is exactly twice the affinity defect, i.e. twice one minus the affinity. This is the algebraic identity that lets an affinity computation be read as a Hellinger distance and back.","s":"∀ {α : Type u_1} [inst : MeasurableSpace α] (μ : MeasureTheory.Measure α) (f g : α → ℝ), MeasureTheory.Integrable f μ → MeasureTheory.Integrable g μ → 0 ≤ f → 0 ≤ g → ∫ (x : α), f x ∂μ = 1 → ∫ (x : α), g x ∂μ = 1 → Causalean.Stat.hellingerSqDensity μ f g = 2 * (1 - Causalean.Stat.densityAffinity μ f g)","l":70,"f":"Causalean/Stat/Minimax/HellingerAffinity.lean"},{"n":"Causalean.Stat.hoeffding_cov_identity","k":"theorem","m":"Causalean.Stat.Coupling.ProductLossMonotoneCoupling.Hoeffding","a":"Stat/Coupling/ProductLossMonotoneCoupling","t":1,"d":"Hoeffding's covariance identity. For a coupling π of μ and ν, where both μ and ν have finite second moment (are L²), the covariance of the coordinates under π equals the double integral, over the plane, of the Fréchet gap between the joint and product cumulative distribution functions, H_π - F·G:","s":"∀ {π : MeasureTheory.Measure (ℝ × ℝ)} {μ ν : MeasureTheory.Measure ℝ}, Causalean.Stat.IsCoupling π μ ν → MeasureTheory.MemLp (fun x => x) 2 μ → MeasureTheory.MemLp (fun y => y) 2 ν → ∫ (p : ℝ × ℝ), p.1 * p.2 ∂π - (∫ (x : ℝ), x ∂μ) * ∫ (y : ℝ), y ∂ν = ∫ (x : ℝ) (y : ℝ), Causalean.Stat.jointCdf π x y - ↑(ProbabilityTheory.cdf μ) x * ↑(ProbabilityTheory.cdf ν) y","l":101,"f":"Causalean/Stat/Coupling/ProductLossMonotoneCoupling/Hoeffding.lean"},{"n":"Causalean.Stat.hoeffding_cov_identity_prod","k":"theorem","m":"Causalean.Stat.Coupling.ProductLossMonotoneCoupling.Hoeffding","a":"Stat/Coupling/ProductLossMonotoneCoupling","t":2,"d":"Hoeffding's covariance identity, product form. For a coupling π of two L² probability measures μ, ν,","s":"∀ {π : MeasureTheory.Measure (ℝ × ℝ)} {μ ν : MeasureTheory.Measure ℝ}, Causalean.Stat.IsCoupling π μ ν → MeasureTheory.MemLp (fun x => x) 2 μ → MeasureTheory.MemLp (fun y => y) 2 ν → ∫ (p : ℝ × ℝ), p.1 * p.2 ∂π - (∫ (x : ℝ), x ∂μ) * ∫ (y : ℝ), y ∂ν = ∫ (q : ℝ × ℝ), Causalean.Stat.jointCdf π q.1 q.2 - ↑(ProbabilityTheory.cdf μ) q.1 * ↑(ProbabilityTheory.cdf ν) q.2 ∂MeasureTheory.volume.prod MeasureTheory.volume","l":79,"f":"Causalean/Stat/Coupling/ProductLossMonotoneCoupling/Hoeffding.lean"},{"n":"Causalean.Stat.hoeffding_decomp","k":"theorem","m":"Causalean.Stat.UStatistic.Basic","a":"Stat/UStatistic","t":1,"d":"Hoeffding decomposition (pointwise identity). For a kernel h, population law P, and points x and y, the kernel value decomposes as h(x, y) = θ + h₁(x) + h₁(y) + g(x, y), the population mean plus the two first-order Hoeffding projections plus the degenerate second-order residual.","s":"∀ {X : Type u_2} [inst : MeasurableSpace X] (h : X → X → ℝ) (P : MeasureTheory.Measure X) (x y : X), h x y = Causalean.Stat.uMean h P + Causalean.Stat.uProj h P x + Causalean.Stat.uProj h P y + Causalean.Stat.uDegen h P x y","l":74,"f":"Causalean/Stat/UStatistic/Basic.lean"},{"n":"Causalean.Stat.hoeffding_decomp_order","k":"theorem","m":"Causalean.Stat.UStatistic.OrderM.Basic","a":"Stat/UStatistic/OrderM","t":1,"d":"For an order-m kernel h, population law P, and an m-tuple of points z, the kernel value decomposes as the population mean plus the sum of the m coordinatewise first Hoeffding projections plus the degenerate higher-order residual kernel evaluated at z.","s":"∀ {X : Type u_2} [inst : MeasurableSpace X] {m : ℕ} [inst_1 : NeZero m] (h : (Fin m → X) → ℝ) (P : MeasureTheory.Measure X) (z : Fin m → X), h z = Causalean.Stat.uMeanOrder h P + ∑ j, Causalean.Stat.uProjOrderAt j h P (z j) + Causalean.Stat.uDegenOrder h P z","l":122,"f":"Causalean/Stat/UStatistic/OrderM/Basic.lean"},{"n":"Causalean.Stat.iidSample_finN_measurable","k":"theorem","m":"Causalean.Stat.Sample.PiTransport","a":"Stat/Sample","t":2,"d":"The joint observable Ψ ω k = S.Z k ω is measurable Ω → (Fin n → X).","s":"∀ {Ω : Type u_1} {X : Type u_2} [inst : MeasurableSpace Ω] [inst_1 : MeasurableSpace X] {μ : MeasureTheory.Measure Ω} {P : MeasureTheory.Measure X} (S : Causalean.Stat.IIDSample Ω X μ P) (n : ℕ), Measurable fun ω k => S.Z (↑k) ω","l":84,"f":"Causalean/Stat/Sample/PiTransport.lean"},{"n":"Causalean.Stat.iidSample_finN_pushforward","k":"theorem","m":"Causalean.Stat.Sample.PiTransport","a":"Stat/Sample","t":1,"d":"For an i.i.d. sample S and a fixed horizon n, the joint map of the first n sample points pushes μ forward to the product measure on Fin n → X.","s":"∀ {Ω : Type u_1} {X : Type u_2} [inst : MeasurableSpace Ω] [inst_1 : MeasurableSpace X] {μ : MeasureTheory.Measure Ω} {P : MeasureTheory.Measure X} (S : Causalean.Stat.IIDSample Ω X μ P) (n : ℕ), MeasureTheory.Measure.map (fun ω k => S.Z (↑k) ω) μ = MeasureTheory.Measure.pi fun x => P","l":66,"f":"Causalean/Stat/Sample/PiTransport.lean"},{"n":"Causalean.Stat.iidSample_infinitePi","k":"def","m":"Causalean.Stat.Sample.PiTransport","a":"Stat/Sample","t":1,"d":"The coordinate projections on an infinite product probability space form an i.i.d. sample with the common marginal law.","s":"{X : Type u_2} → [inst : MeasurableSpace X] → (P : MeasureTheory.Measure X) → [MeasureTheory.IsProbabilityMeasure P] → Causalean.Stat.IIDSample (ℕ → X) X (MeasureTheory.Measure.infinitePi fun x => P) P","l":52,"f":"Causalean/Stat/Sample/PiTransport.lean"},{"n":"Causalean.Stat.iid_two_point_lower_bound","k":"theorem","m":"Causalean.Stat.Minimax.MinimaxRisk","a":"Stat/Minimax","t":1,"d":"n-sample structure-agnostic two-point bound. Given two single-observation laws P₀, P₁ and a real functional τ such that the functional values τ P₀, τ P₁ are 2s-separated, then for any measurable estimator est built from n i.i.d. samples (data law Measure.pi (fun _ ↦ Pⱼ)), the worst-case miss probability, using τ P₀ and τ P₁ as the two parameters, is at least ½(1 − tvDist) between the two n-fold product laws.","s":"∀ {S : Type u_2} [inst : MeasurableSpace S] (P₀ P₁ : MeasureTheory.Measure S) [MeasureTheory.IsProbabilityMeasure P₀] [MeasureTheory.IsProbabilityMeasure P₁] (τ : MeasureTheory.Measure S → ℝ) (n : ℕ) {s : ℝ}, 2 * s ≤ |τ P₀ - τ P₁| → ∀ {est : (Fin n → S) → ℝ}, Measurable est → (1 - Causalean.Stat.tvDist (MeasureTheory.Measure.pi fun x => P₀) (MeasureTheory.Measure.pi fun x => P₁)) / 2 ≤ max ((MeasureTheory.Measure.pi fun x => P₀).real {x | s ≤ |est x - τ P₀|}) ((MeasureTheory.Measure.pi fun x => P₁).real {x | s ≤ |est x - τ P₁|})","l":82,"f":"Causalean/Stat/Minimax/MinimaxRisk.lean"},{"n":"Causalean.Stat.injectiveSubtypeEquivEmbedding","k":"def","m":"Causalean.Stat.UStatistic.OrderM.Basic","a":"Stat/UStatistic/OrderM","t":1,"d":"Injective functions Fin m → Fin n as elements of the finite embedding type.","s":"(m n : ℕ) → { t // Function.Injective t } ≃ (Fin m ↪ Fin n)","l":57,"f":"Causalean/Stat/UStatistic/OrderM/Basic.lean"},{"n":"Causalean.Stat.injectiveTupleCount","k":"def","m":"Causalean.Stat.UStatistic.OrderM.Basic","a":"Stat/UStatistic/OrderM","t":1,"d":"The number of ordered injective m-tuples from the first n sample indices.","s":"ℕ → ℕ → ℝ","l":53,"f":"Causalean/Stat/UStatistic/OrderM/Basic.lean"},{"n":"Causalean.Stat.injectiveTupleCount_eq_descFactorial","k":"theorem","m":"Causalean.Stat.UStatistic.OrderM.Basic","a":"Stat/UStatistic/OrderM","t":2,"d":"The real-valued ordered injective tuple count is the falling factorial.","s":"∀ (m n : ℕ), Causalean.Stat.injectiveTupleCount m n = ↑(n.descFactorial m)","l":77,"f":"Causalean/Stat/UStatistic/OrderM/Basic.lean"},{"n":"Causalean.Stat.injectiveTupleCount_ne_zero","k":"theorem","m":"Causalean.Stat.UStatistic.OrderM.Hajek","a":"Stat/UStatistic/OrderM","t":2,"d":"If m ≤ n, there is at least one ordered injective m-tuple in Fin n.","s":"∀ {m n : ℕ}, m ≤ n → Causalean.Stat.injectiveTupleCount m n ≠ 0","l":97,"f":"Causalean/Stat/UStatistic/OrderM/Hajek.lean"},{"n":"Causalean.Stat.injectiveTupleCount_two_eq_mul_sub_one","k":"theorem","m":"Causalean.Stat.UStatistic.Variance","a":"Stat/UStatistic","t":2,"d":"For a sample of size at least two, the number of ordered injective pairs is the sample size times one less than the sample size.","s":"∀ {n : ℕ}, 2 ≤ n → Causalean.Stat.injectiveTupleCount 2 n = ↑n * (↑n - 1)","l":71,"f":"Causalean/Stat/UStatistic/Variance.lean"},{"n":"Causalean.Stat.injectiveTupleCount_two_eq_offDiag_card","k":"theorem","m":"Causalean.Stat.UStatistic.Variance","a":"Stat/UStatistic","t":2,"d":"The number of ordered injective pairs from a sample of size n equals the number of off-diagonal ordered pairs drawn from that sample.","s":"∀ (n : ℕ), Causalean.Stat.injectiveTupleCount 2 n = ↑(Finset.range n).offDiag.card","l":50,"f":"Causalean/Stat/UStatistic/Variance.lean"},{"n":"Causalean.Stat.injectiveTuples","k":"def","m":"Causalean.Stat.UStatistic.OrderM.Basic","a":"Stat/UStatistic/OrderM","t":1,"d":"Ordered injective m-tuples from the first n sample indices.","s":"(m n : ℕ) → Finset (Fin m → Fin n)","l":48,"f":"Causalean/Stat/UStatistic/OrderM/Basic.lean"},{"n":"Causalean.Stat.injectiveTuples_card_eq_descFactorial","k":"theorem","m":"Causalean.Stat.UStatistic.OrderM.Basic","a":"Stat/UStatistic/OrderM","t":2,"d":"The ordered injective tuple count is the falling factorial n (n-1) ....","s":"∀ (m n : ℕ), (Causalean.Stat.injectiveTuples m n).card = n.descFactorial m","l":65,"f":"Causalean/Stat/UStatistic/OrderM/Basic.lean"},{"n":"Causalean.Stat.injectiveTuples_fiber_card_eq","k":"theorem","m":"Causalean.Stat.UStatistic.OrderM.Hajek","a":"Stat/UStatistic/OrderM","t":2,"d":"Fibres of a fixed coordinate map on ordered injective tuples have the same cardinality. The bijection composes tuples with the codomain transposition swapping the two fibre values.","s":"∀ {m n : ℕ} (j : Fin m) (y y' : Fin n), {t ∈ Causalean.Stat.injectiveTuples m n | t j = y}.card = {t ∈ Causalean.Stat.injectiveTuples m n | t j = y'}.card","l":112,"f":"Causalean/Stat/UStatistic/OrderM/Hajek.lean"},{"n":"Causalean.Stat.inner_normalizedSum","k":"theorem","m":"Causalean.Stat.CLT.MultivariateCLT","a":"Stat/CLT","t":2,"d":"The inner product of t with the vector normalised sum is the scalar normalised sum of the projected influence function x ↦ ⟪t, ψ x⟫.","s":"∀ {Ω : Type u_1} {X : Type u_2} [inst : MeasurableSpace Ω] [inst_1 : MeasurableSpace X] {μ : MeasureTheory.Measure Ω} {P : MeasureTheory.Measure X} {E : Type u_3} [inst_2 : NormedAddCommGroup E] [inst_3 : InnerProductSpace ℝ E] (S : Causalean.Stat.IIDSample Ω X μ P) (ψ : X → E) (t : E) (I : ℕ → Finset ℕ) (n : ℕ) (ω : Ω), inner ℝ t (Causalean.Stat.IsAsymLinearVec.normalizedSum S ψ I n ω) = Causalean.Stat.IsAsymLinear.normalizedSum S (fun x => inner ℝ t (ψ x)) I n ω","l":91,"f":"Causalean/Stat/CLT/MultivariateCLT.lean"},{"n":"Causalean.Stat.insertCoord","k":"def","m":"Causalean.Stat.UStatistic.OrderM.Basic","a":"Stat/UStatistic/OrderM","t":1,"d":"Insert one distinguished coordinate into the remaining coordinates.","s":"{X : Type u_2} → {m : ℕ} → (j : Fin m) → X → ({ k // k ≠ j } → X) → Fin m → X","l":95,"f":"Causalean/Stat/UStatistic/OrderM/Basic.lean"},{"n":"Causalean.Stat.instIsGaussianGaussianLimit","k":"instance","m":"Causalean.Stat.CLT.GaussianLimit","a":"Stat/CLT","t":2,"d":"The limiting law is Gaussian.","s":"∀ {X : Type u_2} [inst : MeasurableSpace X] {P : MeasureTheory.Measure X} {E : Type u_3} [inst_1 : NormedAddCommGroup E] [inst_2 : InnerProductSpace ℝ E] [inst_3 : FiniteDimensional ℝ E] [inst_4 : MeasurableSpace E] [inst_5 : BorelSpace E] {ψ : X → E} (hψ : Measurable ψ) (hvar : MeasureTheory.Integrable (fun x => ‖ψ x‖ ^ 2) P), ProbabilityTheory.IsGaussian (Causalean.Stat.gaussianLimit hψ hvar)","l":66,"f":"Causalean/Stat/CLT/GaussianLimit.lean"},{"n":"Causalean.Stat.instIsMarkovKernelFinProductKernel","k":"instance","m":"Causalean.Stat.Minimax.MarkovKernelTransport","a":"Stat/Minimax","t":2,"d":"The independent finite product of a Markov kernel is itself a Markov kernel, including the zero-coordinate product.","s":"∀ {X : Type uX} {Y : Type uY} [inst : MeasurableSpace X] [inst_1 : MeasurableSpace Y] (n : ℕ) (K : ProbabilityTheory.Kernel X Y) [ProbabilityTheory.IsMarkovKernel K], ProbabilityTheory.IsMarkovKernel (Causalean.Stat.finProductKernel n K)","l":276,"f":"Causalean/Stat/Minimax/MarkovKernelTransport.lean"},{"n":"Causalean.Stat.instIsProbabilityMeasureGaussianMeasure","k":"instance","m":"Causalean.Stat.CLT.AsymptoticLinearity","a":"Stat/CLT","t":2,"d":"gaussianMeasure m v is a probability measure for every real m, v.","s":"∀ (m v : ℝ), MeasureTheory.IsProbabilityMeasure (Causalean.Stat.gaussianMeasure m v)","l":55,"f":"Causalean/Stat/CLT/AsymptoticLinearity.lean"},{"n":"Causalean.Stat.instIsProbabilityMeasureRealChiSqDist","k":"instance","m":"Causalean.Stat.CLT.ChiSquared","a":"Stat/CLT","t":2,"d":"The chi-squared distribution is a probability law.","s":"∀ (d : ℕ), MeasureTheory.IsProbabilityMeasure (Causalean.Stat.chiSqDist d)","l":53,"f":"Causalean/Stat/CLT/ChiSquared.lean"},{"n":"Causalean.Stat.instIsProbabilityMeasure_map_max","k":"instance","m":"Causalean.Stat.Inference.FunctionalDelta","a":"Stat/Inference","t":2,"d":"The maximum of a two-dimensional probability limit is itself a probability law.","s":"∀ (Q : MeasureTheory.Measure (ℝ × ℝ)) [MeasureTheory.IsProbabilityMeasure Q], MeasureTheory.IsProbabilityMeasure (MeasureTheory.Measure.map (fun z => max z.1 z.2) Q)","l":90,"f":"Causalean/Stat/Inference/FunctionalDelta.lean"},{"n":"Causalean.Stat.instIsProbabilityMeasure_map_min","k":"instance","m":"Causalean.Stat.Inference.FunctionalDelta","a":"Stat/Inference","t":2,"d":"The minimum of a two-dimensional probability limit is itself a probability law.","s":"∀ (Q : MeasureTheory.Measure (ℝ × ℝ)) [MeasureTheory.IsProbabilityMeasure Q], MeasureTheory.IsProbabilityMeasure (MeasureTheory.Measure.map (fun z => min z.1 z.2) Q)","l":97,"f":"Causalean/Stat/Inference/FunctionalDelta.lean"},{"n":"Causalean.Stat.instIsProbabilityMeasure_unifOI","k":"instance","m":"Causalean.Stat.Coupling.ProductLossMonotoneCoupling.PIT","a":"Stat/Coupling/ProductLossMonotoneCoupling","t":2,"d":"unifOI is a probability measure: volume (Ioo 0 1) = 1.","s":"MeasureTheory.IsProbabilityMeasure Causalean.Stat.unifOI","l":42,"f":"Causalean/Stat/Coupling/ProductLossMonotoneCoupling/PIT.lean"},{"n":"Causalean.Stat.integrableOn_finiteSelector_sq_of_eligible","k":"theorem","m":"Causalean.Stat.SampleSplit.FiniteSelector","a":"Stat/SampleSplit","t":2,"d":"If the pilot coordinate is measurable, the tail coordinate is measurable, the finite selection rule is measurable, the designated pilot event is measurable, every branch selected there is eligible, and eligible branches have integrable squared error under the tail law, the selected squared error is integrable on that pilot event.","s":"∀ {Omega : Type u_1} {Alpha : Type u_2} {Beta : Type u_3} {Iota : Type u_4} [inst : MeasurableSpace Omega] [inst_1 : MeasurableSpace Alpha] [inst_2 : MeasurableSpace Beta] [inst_3 : MeasurableSpace Iota] [Fintype Iota] [MeasurableSingletonClass Iota] {mu : MeasureTheory.Measure Omega} {pilot : Omega → Alpha} {tail : Omega → Beta} {select : Alpha → Iota} {err : Iota → Beta → ℝ} {Eligible : Iota → Prop}, Measurable pilot → Measurable tail → Measurable select → ∀ {good : Set Alpha}, MeasurableSet good → (∀ a ∈ good, Eligible (select a)) → (∀ (i : Iota), Eligible i → MeasureTheory.Integrable (fun b => err i b ^ 2) (MeasureTheory.Measure.map tail mu)) → MeasureTheory.IntegrableOn (fun omega => err (select (pilot omega)) (tail omega) ^ 2) (pilot ⁻¹' good) mu","l":195,"f":"Causalean/Stat/SampleSplit/FiniteSelector.lean"},{"n":"Causalean.Stat.integrable_bigPhi","k":"theorem","m":"Causalean.Stat.Coupling.ProductLossMonotoneCoupling.HoeffdingFubiniIntegrability","a":"Stat/Coupling/ProductLossMonotoneCoupling","t":1,"d":"The key integrability. Let π be a coupling of μ and ν, where both marginals have finite second moment. Then the product tail kernel Φ q p = signedTail p.1 q.1 * signedTail p.2 q.2 is integrable for the product of Lebesgue measure on the plane with π, i.e. on (volume ⊗ volume) ⊗ π.","s":"∀ {π : MeasureTheory.Measure (ℝ × ℝ)} {μ ν : MeasureTheory.Measure ℝ}, Causalean.Stat.IsCoupling π μ ν → MeasureTheory.MemLp (fun x => x) 2 μ → MeasureTheory.MemLp (fun y => y) 2 ν → MeasureTheory.Integrable (fun z => Causalean.Stat.signedTail z.2.1 z.1.1 * Causalean.Stat.signedTail z.2.2 z.1.2) ((MeasureTheory.volume.prod MeasureTheory.volume).prod π)","l":58,"f":"Causalean/Stat/Coupling/ProductLossMonotoneCoupling/HoeffdingFubiniIntegrability.lean"},{"n":"Causalean.Stat.integrable_cdfStat","k":"theorem","m":"Causalean.Stat.Quantile.EmpiricalCDF","a":"Stat/Quantile","t":2,"d":"cdfStat y is integrable (bounded by 1 on a finite measure).","s":"∀ {P : MeasureTheory.Measure ℝ} [MeasureTheory.IsProbabilityMeasure P] (y : ℝ), MeasureTheory.Integrable (Causalean.Stat.cdfStat y) P","l":77,"f":"Causalean/Stat/Quantile/EmpiricalCDF.lean"},{"n":"Causalean.Stat.integrable_fiber","k":"theorem","m":"Causalean.Stat.Coupling.ProductLossMonotoneCoupling.HoeffdingFubiniIntegrability","a":"Stat/Coupling/ProductLossMonotoneCoupling","t":2,"d":"The fibre q ↦ ∫ p, Φ q p ∂π is integrable on ℝ × ℝ, by Integrable.integral_prod_left applied to integrable_bigPhi.","s":"∀ {π : MeasureTheory.Measure (ℝ × ℝ)} {μ ν : MeasureTheory.Measure ℝ}, Causalean.Stat.IsCoupling π μ ν → MeasureTheory.MemLp (fun x => x) 2 μ → MeasureTheory.MemLp (fun y => y) 2 ν → MeasureTheory.Integrable (fun q => ∫ (p : ℝ × ℝ), Causalean.Stat.signedTail p.1 q.1 * Causalean.Stat.signedTail p.2 q.2 ∂π) (MeasureTheory.volume.prod MeasureTheory.volume)","l":172,"f":"Causalean/Stat/Coupling/ProductLossMonotoneCoupling/HoeffdingFubiniIntegrability.lean"},{"n":"Causalean.Stat.integrable_frechet_gap","k":"theorem","m":"Causalean.Stat.Coupling.ProductLossMonotoneCoupling.Hoeffding","a":"Stat/Coupling/ProductLossMonotoneCoupling","t":2,"d":"The Fréchet gap H_π - F·G is integrable on ℝ × ℝ for an L² coupling; this is the integrability side-condition consumed by hoeffding_cov_identity and by the monotone comparison of double integrals in Optimality.lean.","s":"∀ {π : MeasureTheory.Measure (ℝ × ℝ)} {μ ν : MeasureTheory.Measure ℝ}, Causalean.Stat.IsCoupling π μ ν → MeasureTheory.MemLp (fun x => x) 2 μ → MeasureTheory.MemLp (fun y => y) 2 ν → MeasureTheory.Integrable (fun p => Causalean.Stat.jointCdf π p.1 p.2 - ↑(ProbabilityTheory.cdf μ) p.1 * ↑(ProbabilityTheory.cdf ν) p.2) (MeasureTheory.volume.prod MeasureTheory.volume)","l":60,"f":"Causalean/Stat/Coupling/ProductLossMonotoneCoupling/Hoeffding.lean"},{"n":"Causalean.Stat.integrable_inner_smul","k":"theorem","m":"Causalean.Stat.CLT.SecondMomentOperator","a":"Stat/CLT","t":2,"d":"The integrand x ↦ ⟪t, ψ x⟫ • ψ x is integrable when ‖ψ‖² is, by the Cauchy–Schwarz bound ‖⟪t,ψ⟫ • ψ‖ ≤ ‖t‖ ‖ψ‖².","s":"∀ {X : Type u_1} {E : Type u_2} [inst : MeasurableSpace X] {P : MeasureTheory.Measure X} [inst_1 : NormedAddCommGroup E] [inst_2 : InnerProductSpace ℝ E] [inst_3 : MeasurableSpace E] [OpensMeasurableSpace E] [SecondCountableTopology E] [MeasurableSMul₂ ℝ E] {ψ : X → E}, Measurable ψ → MeasureTheory.Integrable (fun x => ‖ψ x‖ ^ 2) P → ∀ (t : E), MeasureTheory.Integrable (fun x => inner ℝ t (ψ x) • ψ x) P","l":56,"f":"Causalean/Stat/CLT/SecondMomentOperator.lean"},{"n":"Causalean.Stat.integrable_signedTail","k":"theorem","m":"Causalean.Stat.Coupling.ProductLossMonotoneCoupling.TailIntegral","a":"Stat/Coupling/ProductLossMonotoneCoupling","t":2,"d":"signedTail a is Lebesgue integrable: it is a bounded function supported on a bounded interval.","s":"∀ (a : ℝ), MeasureTheory.Integrable (Causalean.Stat.signedTail a) MeasureTheory.volume","l":87,"f":"Causalean/Stat/Coupling/ProductLossMonotoneCoupling/TailIntegral.lean"},{"n":"Causalean.Stat.integrable_signedTail_prod","k":"theorem","m":"Causalean.Stat.Coupling.ProductLossMonotoneCoupling.TailIntegral","a":"Stat/Coupling/ProductLossMonotoneCoupling","t":2,"d":"The tensor product (s, t) ↦ signedTail x s * signedTail y t is integrable on ℝ × ℝ for Lebesgue×Lebesgue.","s":"∀ (x y : ℝ), MeasureTheory.Integrable (fun q => Causalean.Stat.signedTail x q.1 * Causalean.Stat.signedTail y q.2) (MeasureTheory.volume.prod MeasureTheory.volume)","l":178,"f":"Causalean/Stat/Coupling/ProductLossMonotoneCoupling/TailIntegral.lean"},{"n":"Causalean.Stat.integrable_sqrt_rnDeriv","k":"theorem","m":"Causalean.Stat.Minimax.BretagnolleHuber","a":"Stat/Minimax","t":2,"d":"√p is ν-integrable (dominated by (p+1)/2).","s":"∀ {Ω : Type u_1} {mΩ : MeasurableSpace Ω} (μ ν : MeasureTheory.Measure Ω) [MeasureTheory.IsProbabilityMeasure μ] [MeasureTheory.IsProbabilityMeasure ν], MeasureTheory.Integrable (fun x => √(μ.rnDeriv ν x).toReal) ν","l":106,"f":"Causalean/Stat/Minimax/BretagnolleHuber.lean"},{"n":"Causalean.Stat.integrable_survFst_sub","k":"theorem","m":"Causalean.Stat.Coupling.ProductLossMonotoneCoupling.HoeffdingFubiniIntegrability","a":"Stat/Coupling/ProductLossMonotoneCoupling","t":2,"d":"The centred marginal survival function s ↦ SX s - 𝟙{s<0} is Lebesgue integrable (it is E[signedTail X ·], integrable by Integrable.integral_prod_left on integrable_tail_fst_prod).","s":"∀ {π : MeasureTheory.Measure (ℝ × ℝ)} {μ ν : MeasureTheory.Measure ℝ}, Causalean.Stat.IsCoupling π μ ν → MeasureTheory.MemLp (fun x => x) 2 μ → MeasureTheory.Integrable (fun s => Causalean.Stat.survFst π s - Causalean.Stat.tailInd 0 s) MeasureTheory.volume","l":155,"f":"Causalean/Stat/Coupling/ProductLossMonotoneCoupling/HoeffdingFubiniIntegrability.lean"},{"n":"Causalean.Stat.integrable_survSnd_sub","k":"theorem","m":"Causalean.Stat.Coupling.ProductLossMonotoneCoupling.HoeffdingFubiniIntegrability","a":"Stat/Coupling/ProductLossMonotoneCoupling","t":2,"d":"The centred marginal survival function t ↦ SY t - 𝟙{t<0} is Lebesgue integrable.","s":"∀ {π : MeasureTheory.Measure (ℝ × ℝ)} {μ ν : MeasureTheory.Measure ℝ}, Causalean.Stat.IsCoupling π μ ν → MeasureTheory.MemLp (fun y => y) 2 ν → MeasureTheory.Integrable (fun t => Causalean.Stat.survSnd π t - Causalean.Stat.tailInd 0 t) MeasureTheory.volume","l":164,"f":"Causalean/Stat/Coupling/ProductLossMonotoneCoupling/HoeffdingFubiniIntegrability.lean"},{"n":"Causalean.Stat.integrable_tail_fst_prod","k":"theorem","m":"Causalean.Stat.Coupling.ProductLossMonotoneCoupling.HoeffdingFubiniIntegrability","a":"Stat/Coupling/ProductLossMonotoneCoupling","t":2,"d":"The single-coordinate tail kernel is integrable on volume ⊗ π. Domination: ∫ s, |signedTail p.1 s| ds = |p.1|, which is π-integrable since X ∈ L²(π) and π is finite.","s":"∀ {π : MeasureTheory.Measure (ℝ × ℝ)} {μ ν : MeasureTheory.Measure ℝ}, Causalean.Stat.IsCoupling π μ ν → MeasureTheory.MemLp (fun x => x) 2 μ → MeasureTheory.Integrable (fun z => Causalean.Stat.signedTail z.2.1 z.1) (MeasureTheory.volume.prod π)","l":25,"f":"Causalean/Stat/Coupling/ProductLossMonotoneCoupling/HoeffdingFubiniIntegrability.lean"},{"n":"Causalean.Stat.integrable_tail_snd_prod","k":"theorem","m":"Causalean.Stat.Coupling.ProductLossMonotoneCoupling.HoeffdingFubiniIntegrability","a":"Stat/Coupling/ProductLossMonotoneCoupling","t":2,"d":"The single-coordinate tail kernel is integrable on volume ⊗ π (second coordinate).","s":"∀ {π : MeasureTheory.Measure (ℝ × ℝ)} {μ ν : MeasureTheory.Measure ℝ}, Causalean.Stat.IsCoupling π μ ν → MeasureTheory.MemLp (fun y => y) 2 ν → MeasureTheory.Integrable (fun z => Causalean.Stat.signedTail z.2.2 z.1) (MeasureTheory.volume.prod π)","l":42,"f":"Causalean/Stat/Coupling/ProductLossMonotoneCoupling/HoeffdingFubiniIntegrability.lean"},{"n":"Causalean.Stat.integral_abs_mul_le_eLpNorm_mul_eLpNorm","k":"theorem","m":"Causalean.Stat.Orthogonality.ConditionalOp","a":"Stat/Orthogonality","t":1,"d":"Cauchy–Schwarz for products of L²-functions. If f is square-integrable against the measure ν and g is square-integrable against ν, then the integral of |f · g| against ν is at most the product of the L² norms ‖f‖₂ · ‖g‖₂. Stated for the absolute value of the product because that is the form consumed by the AIPW remainder bound.","s":"∀ {X : Type u_2} [inst : MeasurableSpace X] {ν : MeasureTheory.Measure X} {f g : X → ℝ}, MeasureTheory.MemLp f 2 ν → MeasureTheory.MemLp g 2 ν → ∫ (x : X), |f x * g x| ∂ν ≤ (MeasureTheory.eLpNorm f 2 ν).toReal * (MeasureTheory.eLpNorm g 2 ν).toReal","l":321,"f":"Causalean/Stat/Orthogonality/ConditionalOp.lean"},{"n":"Causalean.Stat.integral_abs_mul_op_of_eLpNorm_op","k":"theorem","m":"Causalean.Stat.Orthogonality.ConditionalOp","a":"Stat/Orthogonality","t":1,"d":"L² product rate ⇒ integrated absolute-product rate. For a random family f_n and a fixed function g, suppose each f_n(ω) is square-integrable against ν, for every n and ω, g is square-integrable against ν, the rates s_n are positive, and the L² norm ‖f_n(ω)‖₂ is o_p(s_n). Then the integrated absolute product ∫ |f_n(ω) x · g x| dν is o_p(s_n · ‖g‖₂).","s":"∀ {Ω : Type u_1} [inst : MeasurableSpace Ω] {μ : MeasureTheory.Measure Ω} {X : Type u_2} [inst_1 : MeasurableSpace X] {ν : MeasureTheory.Measure X} {fn : ℕ → Ω → X → ℝ} {g : X → ℝ}, (∀ (n : ℕ) (ω : Ω), MeasureTheory.MemLp (fn n ω) 2 ν) → MeasureTheory.MemLp g 2 ν → ∀ {sn : ℕ → ℝ}, (∀ (n : ℕ), 0 < sn n) → Causalean.Stat.IsLittleOp (fun n ω => (MeasureTheory.eLpNorm (fn n ω) 2 ν).toReal) sn μ → Causalean.Stat.IsLittleOp (fun n ω => ∫ (x : X), |fn n ω x * g x| ∂ν) (fun n => sn n * (MeasureTheory.eLpNorm g 2 ν).toReal) μ","l":385,"f":"Causalean/Stat/Orthogonality/ConditionalOp.lean"},{"n":"Causalean.Stat.integral_abs_signedTail","k":"theorem","m":"Causalean.Stat.Coupling.ProductLossMonotoneCoupling.TailIntegral","a":"Stat/Coupling/ProductLossMonotoneCoupling","t":2,"d":"The L¹ norm of the signed tail indicator is |a|: the integrand is ±1 on an interval of length |a|.","s":"∀ (a : ℝ), ∫ (s : ℝ), |Causalean.Stat.signedTail a s| = |a|","l":156,"f":"Causalean/Stat/Coupling/ProductLossMonotoneCoupling/TailIntegral.lean"},{"n":"Causalean.Stat.integral_categoryIndicator","k":"theorem","m":"Causalean.Stat.SampleSplit.FiniteCategoryPilot","a":"Stat/SampleSplit","t":2,"d":"Given a measurable category label and a fixed category, the mean of its zero-one indicator under a probability law equals the category's population mass.","s":"∀ {X : Type u_2} {Iota : Type u_3} [inst : MeasurableSpace X] [inst_1 : MeasurableSpace Iota] {P : MeasureTheory.Measure X} [inst_2 : DecidableEq Iota] [MeasurableSingletonClass Iota] [MeasureTheory.IsProbabilityMeasure P] {label : X → Iota}, Measurable label → ∀ (k : Iota), ∫ (x : X), Causalean.Stat.categoryIndicator label k x ∂P = Causalean.Stat.categoryMass P label k","l":53,"f":"Causalean/Stat/SampleSplit/FiniteCategoryPilot.lean"},{"n":"Causalean.Stat.integral_cdfStat","k":"theorem","m":"Causalean.Stat.Quantile.EmpiricalCDF","a":"Stat/Quantile","t":2,"d":"The integral of the indicator statistic is the cdf value F(y).","s":"∀ {P : MeasureTheory.Measure ℝ} [MeasureTheory.IsProbabilityMeasure P] (y : ℝ), ∫ (z : ℝ), Causalean.Stat.cdfStat y z ∂P = ↑(ProbabilityTheory.cdf P) y","l":84,"f":"Causalean/Stat/Quantile/EmpiricalCDF.lean"},{"n":"Causalean.Stat.integral_comp_finCoordinatewise","k":"theorem","m":"Causalean.Stat.Sample.PiTransport","a":"Stat/Sample","t":2,"d":"Under a common probability law, if the observation rule is measurable and the real-valued statistic of the recoded sample is measurable, then its expectation after coordinatewise recoding equals its expectation under the product of the recoded marginal law.","s":"∀ {X : Type uX} {Y : Type uY} [inst : MeasurableSpace X] [inst_1 : MeasurableSpace Y] (n : ℕ) (mu : MeasureTheory.Measure X) [MeasureTheory.IsProbabilityMeasure mu] {phi : X → Y}, Measurable phi → ∀ (g : (Fin n → Y) → ℝ), Measurable g → (∫ (z : Fin n → X), g fun i => phi (z i) ∂MeasureTheory.Measure.pi fun x => mu) = ∫ (y : Fin n → Y), g y ∂MeasureTheory.Measure.pi fun x => MeasureTheory.Measure.map phi mu","l":300,"f":"Causalean/Stat/Sample/PiTransport.lean"},{"n":"Causalean.Stat.integral_comp_pi_restrict","k":"theorem","m":"Causalean.Stat.Sample.PiTransport","a":"Stat/Sample","t":2,"d":"Averaging a function of the sub-index coordinates over the whole i.i.d. product sample is the same as averaging it over an i.i.d. product sample indexed by the sub-index alone.","s":"∀ {ι : Type u_3} [inst : Fintype ι] {X : Type u_4} {E : Type u_5} [inst_1 : MeasurableSpace X] [inst_2 : NormedAddCommGroup E] [inst_3 : NormedSpace ℝ E] (μ : MeasureTheory.Measure X) [MeasureTheory.IsProbabilityMeasure μ] (p : ι → Prop) [inst_5 : DecidablePred p] (g : (Subtype p → X) → E), (∫ (s : ι → X), g fun i => s ↑i ∂MeasureTheory.Measure.pi fun x => μ) = ∫ (z : Subtype p → X), g z ∂MeasureTheory.Measure.pi fun x => μ","l":162,"f":"Causalean/Stat/Sample/PiTransport.lean"},{"n":"Causalean.Stat.integral_comp_pi_restrict_finset","k":"theorem","m":"Causalean.Stat.Sample.PiTransport","a":"Stat/Sample","t":2,"d":"Averaging a function of the coordinates in a finite index set over the whole i.i.d. product sample is the same as averaging it over an i.i.d. product sample indexed by that finite set.","s":"∀ {ι : Type u_3} [inst : Fintype ι] {X : Type u_4} {E : Type u_5} [inst_1 : MeasurableSpace X] [inst_2 : NormedAddCommGroup E] [inst_3 : NormedSpace ℝ E] (μ : MeasureTheory.Measure X) [MeasureTheory.IsProbabilityMeasure μ] (S : Finset ι) (g : (↥S → X) → E), (∫ (s : ι → X), g fun i => s ↑i ∂MeasureTheory.Measure.pi fun x => μ) = ∫ (z : ↥S → X), g z ∂MeasureTheory.Measure.pi fun x => μ","l":214,"f":"Causalean/Stat/Sample/PiTransport.lean"},{"n":"Causalean.Stat.integral_designWeight_residual_cross_coordinates_eq_zero","k":"theorem","m":"Causalean.Stat.Sample.OccupancyWeightedMean.MomentBounds","a":"Stat/Sample/OccupancyWeightedMean","t":2,"d":"If group labels, arm labels, and outcomes are measurable, every supported residual has a finite second moment, each residual is centered within its arm/group cell, and the two sample coordinates differ, any finite-design weight times their two supported residuals has product-law integral zero.","s":"∀ {Omega : Type u_1} {kappa : Type u_2} [inst : MeasurableSpace Omega] [Fintype kappa] [DecidableEq kappa] [inst_3 : MeasurableSpace kappa] [MeasurableSingletonClass kappa] {n : ℕ} (mu : MeasureTheory.Measure Omega) [MeasureTheory.IsProbabilityMeasure mu] (group : Omega → kappa) (arm : Omega → Bool) (Y : Omega → ℝ) (center : Bool → kappa → ℝ), Measurable group → Measurable arm → Measurable Y → (∀ (a : Bool) (k : kappa), MeasureTheory.MemLp (Causalean.Stat.supportedArmGroupResidual group arm Y center a k) 2 mu) → (∀ (a : Bool) (k : kappa), ∫ (omega : Omega) in Causalean.Stat.armGroupEvent group arm a k, Causalean.Stat.armGroupResidual Y center a k omega ∂mu = 0) → ∀ (W : (Fin n → kappa × Bool) → ℝ) (i j : Fin n), i ≠ j → ∀ (a b : Bool) (k l : kappa), (∫ (z : Fin n → Omega), W (Causalean.Stat.sampleDesign group arm z) * Causalean.Stat.supportedArmGroupResidual group arm Y center a k (z i) * Causalean.Stat.supportedArmGroupResidual group arm Y center b l (z j) ∂MeasureTheory.Measure.pi fun x => mu) = 0","l":225,"f":"Causalean/Stat/Sample/OccupancyWeightedMean/MomentBounds.lean"},{"n":"Causalean.Stat.integral_designWeight_residual_sq_le_indicator","k":"theorem","m":"Causalean.Stat.Sample.OccupancyWeightedMean.MomentBounds","a":"Stat/Sample/OccupancyWeightedMean","t":2,"d":"If group labels, arm labels, and outcomes are measurable, every supported residual has a finite second moment, each cell's residual second moment obeys the stated envelope, and the design weight is nonnegative, the weighted residual square at one sample coordinate is bounded in expectation by the corresponding weighted cell indicator envelope.","s":"∀ {Omega : Type u_1} {kappa : Type u_2} [inst : MeasurableSpace Omega] [Fintype kappa] [DecidableEq kappa] [inst_3 : MeasurableSpace kappa] [MeasurableSingletonClass kappa] {n : ℕ} (mu : MeasureTheory.Measure Omega) [MeasureTheory.IsProbabilityMeasure mu] (group : Omega → kappa) (arm : Omega → Bool) (Y : Omega → ℝ) (center : Bool → kappa → ℝ) (V : ℝ), Measurable group → Measurable arm → Measurable Y → (∀ (a : Bool) (k : kappa), MeasureTheory.MemLp (Causalean.Stat.supportedArmGroupResidual group arm Y center a k) 2 mu) → (∀ (a : Bool) (k : kappa), ∫ (omega : Omega) in Causalean.Stat.armGroupEvent group arm a k, Causalean.Stat.armGroupResidual Y center a k omega ^ 2 ∂mu ≤ (mu (Causalean.Stat.armGroupEvent group arm a k)).toReal * V ^ 2) → ∀ (W : (Fin n → kappa × Bool) → ℝ), (∀ (d : Fin n → kappa × Bool), 0 ≤ W d) → ∀ (i : Fin n) (a : Bool) (k : kappa), (∫ (z : Fin n → Omega), W (Causalean.Stat.sampleDesign group arm z) * Causalean.Stat.supportedArmGroupResidual group arm Y center a k (z i) ^ 2 ∂MeasureTheory.Measure.pi fun x => mu) ≤ V ^ 2 * ∫ (z : Fin n → Omega), W (Causalean.Stat.sampleDesign group arm z) * (Causalean.Stat.armGroupEvent group arm a k).indicator (fun x => 1) (z i) ∂MeasureTheory.Measure.pi fun x => mu","l":317,"f":"Causalean/Stat/Sample/OccupancyWeightedMean/MomentBounds.lean"},{"n":"Causalean.Stat.integral_empiricalMass","k":"theorem","m":"Causalean.Stat.Sample.EmpiricalMass","a":"Stat/Sample","t":1,"d":"Expected empirical mass. Given a positive sample size $m$ and a measurable singleton {a}, the expectation, under the $m$-fold product sampling measure, of the empirical mass of the point a — the fraction of sample observations equal to a — equals the population probability of {a}.","s":"∀ {𝒳 : Type u_1} [inst : MeasurableSpace 𝒳] (μ : MeasureTheory.Measure 𝒳) [MeasureTheory.IsProbabilityMeasure μ] {m : ℕ}, 0 < m → ∀ (a : 𝒳), MeasurableSet {a} → (∫ (sample : Fin m → 𝒳), Causalean.Stat.empiricalMass sample a ∂MeasureTheory.Measure.pi fun x => μ) = μ.real {a}","l":58,"f":"Causalean/Stat/Sample/EmpiricalMass.lean"},{"n":"Causalean.Stat.integral_empiricalMass_sq","k":"theorem","m":"Causalean.Stat.Sample.EmpiricalMass","a":"Stat/Sample","t":1,"d":"Second moment of the empirical mass. Given a positive sample size $m$ and a measurable singleton {a}, the second moment, under the $m$-fold product sampling measure, of the empirical mass of the point a equals the squared population probability of {a} plus the usual binomial sampling correction $(1/m)(\\mu(\\{a\\})-\\mu(\\{a\\})^2)$.","s":"∀ {𝒳 : Type u_1} [inst : MeasurableSpace 𝒳] (μ : MeasureTheory.Measure 𝒳) [MeasureTheory.IsProbabilityMeasure μ] {m : ℕ}, 0 < m → ∀ (a : 𝒳), MeasurableSet {a} → (∫ (sample : Fin m → 𝒳), Causalean.Stat.empiricalMass sample a ^ 2 ∂MeasureTheory.Measure.pi fun x => μ) = μ.real {a} ^ 2 + (↑m)⁻¹ * (μ.real {a} - μ.real {a} ^ 2)","l":94,"f":"Causalean/Stat/Sample/EmpiricalMass.lean"},{"n":"Causalean.Stat.integral_kernelMean_eq_integral_comp","k":"theorem","m":"Causalean.Stat.Minimax.MarkovKernelTransport","a":"Stat/Minimax","t":2,"d":"If the target estimator is measurable and uniformly bounded, then its expectation after taking the kernel mean under a source probability law equals its expectation under the garbled law.","s":"∀ {X : Type uX} {Y : Type uY} [inst : MeasurableSpace X] [inst_1 : MeasurableSpace Y] (P : MeasureTheory.Measure X) [MeasureTheory.IsProbabilityMeasure P] (K : ProbabilityTheory.Kernel X Y) [ProbabilityTheory.IsMarkovKernel K] {T : Y → ℝ}, Measurable T → Causalean.Stat.UniformlyBounded T → ∫ (x : X), Causalean.Stat.kernelMean K T x ∂P = ∫ (y : Y), T y ∂P.bind ⇑K","l":112,"f":"Causalean/Stat/Minimax/MarkovKernelTransport.lean"},{"n":"Causalean.Stat.integral_le_sSup_range_of_isProbabilityMeasure","k":"theorem","m":"Causalean.Stat.Minimax.MinimaxRisk","a":"Stat/Minimax","t":2,"d":"Bayes risk never exceeds worst-case risk. Averaging an integrable risk function over the parameter space against any prior *probability* distribution gives at most the supremum of that risk over the parameter space (assuming the risk is bounded above). This is the step that lets a minimax lower bound be certified by exhibiting a single prior and bounding its average risk from below.","s":"∀ {Θ : Type u_2} [inst : MeasurableSpace Θ] (π : MeasureTheory.Measure Θ) [MeasureTheory.IsProbabilityMeasure π] (risk : Θ → ℝ), MeasureTheory.Integrable risk π → BddAbove (Set.range risk) → ∫ (θ : Θ), risk θ ∂π ≤ sSup (Set.range risk)","l":137,"f":"Causalean/Stat/Minimax/MinimaxRisk.lean"},{"n":"Causalean.Stat.integral_min_le_one_sub_tvDist","k":"theorem","m":"Causalean.Stat.Minimax.BretagnolleHuber","a":"Stat/Minimax","t":2,"d":"Affinity lower bound. ∫ min(p,1) ∂ν ≤ 1 - tvDist μ ν. Derived from Scheffé's ≤ inequality together with ∫ min(p,1) ∂ν = 1 - ½∫|p-1|∂ν.","s":"∀ {Ω : Type u_1} {mΩ : MeasurableSpace Ω} (μ ν : MeasureTheory.Measure Ω) [MeasureTheory.IsProbabilityMeasure μ] [MeasureTheory.IsProbabilityMeasure ν], μ.AbsolutelyContinuous ν → ∫ (x : Ω), min (μ.rnDeriv ν x).toReal 1 ∂ν ≤ 1 - Causalean.Stat.tvDist μ ν","l":121,"f":"Causalean/Stat/Minimax/BretagnolleHuber.lean"},{"n":"Causalean.Stat.integral_norm_signedTail_prod","k":"theorem","m":"Causalean.Stat.Coupling.ProductLossMonotoneCoupling.TailIntegral","a":"Stat/Coupling/ProductLossMonotoneCoupling","t":2,"d":"The L¹ norm of the product tail representation is |x| * |y|. This is the domination bound that makes the Fubini swap against an L² coupling valid: its π-integral is E|XY| < ∞ by Cauchy–Schwarz.","s":"∀ (x y : ℝ), ∫ (q : ℝ × ℝ), ‖Causalean.Stat.signedTail x q.1 * Causalean.Stat.signedTail y q.2‖ ∂MeasureTheory.volume.prod MeasureTheory.volume = |x| * |y|","l":193,"f":"Causalean/Stat/Coupling/ProductLossMonotoneCoupling/TailIntegral.lean"},{"n":"Causalean.Stat.integral_normalizedFiniteKernelStatistic","k":"theorem","m":"Causalean.Stat.UStatistic.OrderM.Variance","a":"Stat/UStatistic/OrderM","t":2,"d":"For an i.i.d. sample, a finite coordinate family, a kernel, and sample size n, if the number of coordinates does not exceed the sample size, the kernel is measurable, and the kernel is integrable under the product law, the expected normalized statistic equals the kernel's product-law mean.","s":"∀ {Ω : Type u_1} {X : Type u_2} [inst : MeasurableSpace Ω] [inst_1 : MeasurableSpace X] {μ : MeasureTheory.Measure Ω} {P : MeasureTheory.Measure X} (S : Causalean.Stat.IIDSample Ω X μ P) {ι : Type u_3} [inst_2 : Fintype ι] {k : (ι → X) → ℝ} {n : ℕ}, Fintype.card ι ≤ n → Measurable k → MeasureTheory.Integrable k (MeasureTheory.Measure.pi fun x => P) → ∫ (ω : Ω), Causalean.Stat.normalizedFiniteKernelStatistic S k n ω ∂μ = ∫ (z : ι → X), k z ∂MeasureTheory.Measure.pi fun x => P","l":488,"f":"Causalean/Stat/UStatistic/OrderM/Variance.lean"},{"n":"Causalean.Stat.integral_normalizedOrderedProductStatistic_mul","k":"theorem","m":"Causalean.Stat.UStatistic.OrderM.MixedOrderCovariance","a":"Stat/UStatistic/OrderM","t":2,"d":"For an i.i.d. sample, statistic orders and a sample size, if the first order and the second order do not exceed the sample size, then for two coordinate-function families, when every merged kernel is measurable and integrable under its product law, the expected product of their normalized statistics is the partial-matching sum of merged product-law moments.","s":"∀ {Ω : Type u_1} {X : Type u_2} [inst : MeasurableSpace Ω] [inst_1 : MeasurableSpace X] {μ : MeasureTheory.Measure Ω} {P : MeasureTheory.Measure X} (S : Causalean.Stat.IIDSample Ω X μ P) {r s n : ℕ}, r ≤ n → s ≤ n → ∀ (f : Fin r → X → ℝ) (g : Fin s → X → ℝ), (∀ (M : Causalean.Stat.PartialMatching r s), Measurable (Causalean.Stat.mergedProductKernel f g M)) → (∀ (M : Causalean.Stat.PartialMatching r s), MeasureTheory.Integrable (Causalean.Stat.mergedProductKernel f g M) (MeasureTheory.Measure.pi fun x => P)) → ∫ (ω : Ω), Causalean.Stat.normalizedOrderedProductStatistic S f n ω * Causalean.Stat.normalizedOrderedProductStatistic S g n ω ∂μ = ∑ h ∈ Finset.range (min r s + 1), ∑ M ∈ Causalean.Stat.partialMatchingsOfSize r s h, Causalean.Stat.matchingNormalization n M * Causalean.Stat.mergedProductMoment P f g M","l":479,"f":"Causalean/Stat/UStatistic/OrderM/MixedOrderCovariance.lean"},{"n":"Causalean.Stat.integral_occupancyDesignVarianceFactor_le_reciprocal","k":"theorem","m":"Causalean.Stat.Sample.OccupancyWeightedMean.MomentBounds","a":"Stat/Sample/OccupancyWeightedMean","t":1,"d":"If group and arm labels are measurable, the overlap margin is positive and at most one half, and both arms receive at least that share in every positive-mass group, the expected design variance factor is at most sixteen divided by the squared margin times one minus the margin, multiplied by expected reciprocal usable occupancy. This includes zero-mass groups, empty samples, empty group types, and every zero-count boundary.","s":"∀ {Omega : Type u_1} {kappa : Type u_2} [inst : MeasurableSpace Omega] [inst_1 : Fintype kappa] [inst_2 : DecidableEq kappa] [inst_3 : MeasurableSpace kappa] [MeasurableSingletonClass kappa] {n : ℕ} (mu : MeasureTheory.Measure Omega) [MeasureTheory.IsProbabilityMeasure mu] (group : Omega → kappa) (arm : Omega → Bool), Measurable group → Measurable arm → ∀ (epsilon : ℝ), 0 < epsilon → epsilon ≤ 1 / 2 → (∀ (k : kappa), 0 < (mu (Causalean.Stat.groupEvent group k)).toReal → ∀ (a : Bool), epsilon * (mu (Causalean.Stat.groupEvent group k)).toReal ≤ (mu (Causalean.Stat.armGroupEvent group arm a k)).toReal) → (∫ (z : Fin n → Omega), Causalean.Stat.occupancyDesignVarianceFactor group arm z ∂MeasureTheory.Measure.pi fun x => mu) ≤ 16 / (epsilon ^ 2 * (1 - epsilon)) * ∫ (z : Fin n → Omega), Causalean.Stat.inverseUsableGroupTotal group arm z ∂MeasureTheory.Measure.pi fun x => mu","l":443,"f":"Causalean/Stat/Sample/OccupancyWeightedMean/MomentBounds.lean"},{"n":"Causalean.Stat.integral_occupancyWeightedResidual_sq_le_reciprocal","k":"theorem","m":"Causalean.Stat.Sample.OccupancyWeightedMean.Variance","a":"Stat/Sample/OccupancyWeightedMean","t":1,"d":"If group labels, arm labels, and outcomes are measurable, every supported residual has a finite second moment, residuals are centered within every arm/group cell, their cellwise second moments obey the stated envelope, the overlap margin is positive and at most one half, and both arms receive at least that share in every positive-mass group, the squared occupancy-weighted residual has product-law expectation at most sixteen divided by the squared margin times one minus the margin, multiplied by the squared envelope and expected reciprocal usable occupancy.","s":"∀ {Omega : Type u_1} {kappa : Type u_2} [inst : MeasurableSpace Omega] [inst_1 : Fintype kappa] [inst_2 : DecidableEq kappa] [inst_3 : MeasurableSpace kappa] [MeasurableSingletonClass kappa] {n : ℕ} (mu : MeasureTheory.Measure Omega) [MeasureTheory.IsProbabilityMeasure mu] (group : Omega → kappa) (arm : Omega → Bool) (Y : Omega → ℝ) (center : Bool → kappa → ℝ) (V epsilon : ℝ), Measurable group → Measurable arm → Measurable Y → (∀ (a : Bool) (k : kappa), MeasureTheory.MemLp (Causalean.Stat.supportedArmGroupResidual group arm Y center a k) 2 mu) → (∀ (a : Bool) (k : kappa), ∫ (omega : Omega) in Causalean.Stat.armGroupEvent group arm a k, Causalean.Stat.armGroupResidual Y center a k omega ∂mu = 0) → (∀ (a : Bool) (k : kappa), ∫ (omega : Omega) in Causalean.Stat.armGroupEvent group arm a k, Causalean.Stat.armGroupResidual Y center a k omega ^ 2 ∂mu ≤ (mu (Causalean.Stat.armGroupEvent group arm a k)).toReal * V ^ 2) → 0 < epsilon → epsilon ≤ 1 / 2 → (∀ (k : kappa), 0 < (mu (Causalean.Stat.groupEvent group k)).toReal → ∀ (a : Bool), epsilon * (mu (Causalean.Stat.groupEvent group k)).toReal ≤ (mu (Causalean.Stat.armGroupEvent group arm a k)).toReal) → (∫ (z : Fin n → Omega), Causalean.Stat.occupancyWeightedResidual mu group arm Y center z ^ 2 ∂MeasureTheory.Measure.pi fun x => mu) ≤ 16 / (epsilon ^ 2 * (1 - epsilon)) * V ^ 2 * ∫ (z : Fin n → Omega), if 0 < Causalean.Stat.usableGroupTotal group arm z then (↑(Causalean.Stat.usableGroupTotal group arm z))⁻¹ else 0 ∂MeasureTheory.Measure.pi fun x => mu","l":264,"f":"Causalean/Stat/Sample/OccupancyWeightedMean/Variance.lean"},{"n":"Causalean.Stat.integral_op_of_eLpNorm_op","k":"theorem","m":"Causalean.Stat.Orthogonality.ConditionalOp","a":"Stat/Orthogonality","t":2,"d":"L² rate ⇒ integrated bias rate (probability measure case). If eLpNorm (f n ω) 2 ν = o_p(rₙ) and ν is a probability measure, then (∫ x, f n ω x ∂ν) = o_p(rₙ).","s":"∀ {Ω : Type u_1} [inst : MeasurableSpace Ω] {μ : MeasureTheory.Measure Ω} {X : Type u_2} [inst_1 : MeasurableSpace X] {ν : MeasureTheory.Measure X} [MeasureTheory.IsProbabilityMeasure ν] {fn : ℕ → Ω → X → ℝ}, (∀ (n : ℕ) (ω : Ω), MeasureTheory.MemLp (fn n ω) 2 ν) → ∀ {rn : ℕ → ℝ}, (∀ (n : ℕ), 0 < rn n) → Causalean.Stat.IsLittleOp (fun n ω => (MeasureTheory.eLpNorm (fn n ω) 2 ν).toReal) rn μ → Causalean.Stat.IsLittleOp (fun n ω => ∫ (x : X), fn n ω x ∂ν) rn μ","l":359,"f":"Causalean/Stat/Orthogonality/ConditionalOp.lean"},{"n":"Causalean.Stat.integral_pi_eval_eq","k":"theorem","m":"Causalean.Stat.UStatistic.OrderM.Basic","a":"Stat/UStatistic/OrderM","t":2,"d":"Integrating a function of one coordinate under a finite product law recovers the one-dimensional integral.","s":"∀ {X : Type u_2} [inst : MeasurableSpace X] {P : MeasureTheory.Measure X} [MeasureTheory.IsProbabilityMeasure P] {ι : Type u_3} [inst_2 : Fintype ι] {E : Type u_4} [inst_3 : NormedAddCommGroup E] [inst_4 : NormedSpace ℝ E] (i : ι) {f : X → E}, MeasureTheory.Integrable f P → (∫ (z : ι → X), f (z i) ∂MeasureTheory.Measure.pi fun x => P) = ∫ (x : X), f x ∂P","l":182,"f":"Causalean/Stat/UStatistic/OrderM/Basic.lean"},{"n":"Causalean.Stat.integral_prod_eq_integral_fiber","k":"theorem","m":"Causalean.Stat.Coupling.ProductLossMonotoneCoupling.HoeffdingFubiniIntegrability","a":"Stat/Coupling/ProductLossMonotoneCoupling","t":1,"d":"First swap. For a coupling π of μ and ν, where both marginals have finite second moment, the expectation E_π[XY] equals the Lebesgue double integral, over the plane, of the fibre integrals ∫ p, Φ q p ∂π, i.e. ∫ p, p.1 * p.2 ∂π = ∫ q, (∫ p, Φ q p ∂π) dq.","s":"∀ {π : MeasureTheory.Measure (ℝ × ℝ)} {μ ν : MeasureTheory.Measure ℝ}, Causalean.Stat.IsCoupling π μ ν → MeasureTheory.MemLp (fun x => x) 2 μ → MeasureTheory.MemLp (fun y => y) 2 ν → ∫ (p : ℝ × ℝ), p.1 * p.2 ∂π = ∫ (q : ℝ × ℝ), ∫ (p : ℝ × ℝ), Causalean.Stat.signedTail p.1 q.1 * Causalean.Stat.signedTail p.2 q.2 ∂π ∂MeasureTheory.volume.prod MeasureTheory.volume","l":91,"f":"Causalean/Stat/Coupling/ProductLossMonotoneCoupling/HoeffdingFubiniIntegrability.lean"},{"n":"Causalean.Stat.integral_signedTail","k":"theorem","m":"Causalean.Stat.Coupling.ProductLossMonotoneCoupling.TailIntegral","a":"Stat/Coupling/ProductLossMonotoneCoupling","t":2,"d":"Signed tail representation. ∫ s, (𝟙{s < a} - 𝟙{s < 0}) ds = a.","s":"∀ (a : ℝ), ∫ (s : ℝ), Causalean.Stat.signedTail a s = a","l":101,"f":"Causalean/Stat/Coupling/ProductLossMonotoneCoupling/TailIntegral.lean"},{"n":"Causalean.Stat.integral_signedTail_fst","k":"theorem","m":"Causalean.Stat.Coupling.ProductLossMonotoneCoupling.HoeffdingFubini","a":"Stat/Coupling/ProductLossMonotoneCoupling","t":2,"d":"The π-integral of the signed tail indicator of the first coordinate is the marginal survival function minus the constant 𝟙{s<0}: ∫ p, (𝟙{s < p.1} - 𝟙{s < 0}) ∂π = SX s - 𝟙{s < 0}. Uses ∫ p, 𝟙{s < p.1} ∂π = π.real (Prod.fst ⁻¹' Ioi s) and π univ = 1.","s":"∀ {π : MeasureTheory.Measure (ℝ × ℝ)} {μ ν : MeasureTheory.Measure ℝ}, Causalean.Stat.IsCoupling π μ ν → ∀ (s : ℝ), ∫ (p : ℝ × ℝ), Causalean.Stat.signedTail p.1 s ∂π = Causalean.Stat.survFst π s - Causalean.Stat.tailInd 0 s","l":89,"f":"Causalean/Stat/Coupling/ProductLossMonotoneCoupling/HoeffdingFubini.lean"},{"n":"Causalean.Stat.integral_signedTail_prod","k":"theorem","m":"Causalean.Stat.Coupling.ProductLossMonotoneCoupling.TailIntegral","a":"Stat/Coupling/ProductLossMonotoneCoupling","t":1,"d":"Product tail representation. For any two reals x and y, their product x * y equals the two-dimensional Lebesgue integral of the product of their signed tail indicators, ∫∫ (𝟙{s<x} - 𝟙{s<0})(𝟙{t<y} - 𝟙{t<0}) ds dt.","s":"∀ (x y : ℝ), ∫ (q : ℝ × ℝ), Causalean.Stat.signedTail x q.1 * Causalean.Stat.signedTail y q.2 ∂MeasureTheory.volume.prod MeasureTheory.volume = x * y","l":185,"f":"Causalean/Stat/Coupling/ProductLossMonotoneCoupling/TailIntegral.lean"},{"n":"Causalean.Stat.integral_signedTail_snd","k":"theorem","m":"Causalean.Stat.Coupling.ProductLossMonotoneCoupling.HoeffdingFubini","a":"Stat/Coupling/ProductLossMonotoneCoupling","t":2,"d":"The π-integral of the signed tail indicator of the second coordinate.","s":"∀ {π : MeasureTheory.Measure (ℝ × ℝ)} {μ ν : MeasureTheory.Measure ℝ}, Causalean.Stat.IsCoupling π μ ν → ∀ (t : ℝ), ∫ (p : ℝ × ℝ), Causalean.Stat.signedTail p.2 t ∂π = Causalean.Stat.survSnd π t - Causalean.Stat.tailInd 0 t","l":113,"f":"Causalean/Stat/Coupling/ProductLossMonotoneCoupling/HoeffdingFubini.lean"},{"n":"Causalean.Stat.interiorIndicator","k":"def","m":"Causalean.Stat.Sample.OccupancyWeightedMean.BinomialDesign","a":"Stat/Sample/OccupancyWeightedMean","t":1,"d":"The binomial interior indicator is one when both success and failure counts are positive, and zero at either endpoint.","s":"ℕ → ℕ → ℝ","l":100,"f":"Causalean/Stat/Sample/OccupancyWeightedMean/BinomialDesign.lean"},{"n":"Causalean.Stat.inverseSqrtCap_anti","k":"theorem","m":"Causalean.Stat.Minimax.HonestConfidenceSet","a":"Stat/Minimax","t":2,"d":"The capped inverse-square-root rate is antitone on positive strengths.","s":"∀ {t0 t : ℝ}, 0 < t0 → t0 ≤ t → min 1 (t ^ (-1 / 2)) ≤ min 1 (t0 ^ (-1 / 2))","l":217,"f":"Causalean/Stat/Minimax/HonestConfidenceSet.lean"},{"n":"Causalean.Stat.inverseStrength_to_frontier","k":"theorem","m":"Causalean.Stat.Inference.AffineInversion","a":"Stat/Inference","t":2,"d":"An inverse-root plus inverse-strength bound, capped by a nonnegative region volume, is bounded by the compact inverse-square-root frontier form.","s":"∀ {cap A B t : ℝ}, 0 ≤ B → 0 < t → min cap (A / √t + B / t) ≤ max cap (A + B) * min 1 (t ^ (-1 / 2))","l":238,"f":"Causalean/Stat/Inference/AffineInversion.lean"},{"n":"Causalean.Stat.inverseTwoCounts","k":"def","m":"Causalean.Stat.Sample.OccupancyWeightedMean.BinomialDesign","a":"Stat/Sample/OccupancyWeightedMean","t":1,"d":"The two-arm inverse-count contribution is the sum of the inverse success and failure counts when both are positive, and zero at either endpoint.","s":"ℕ → ℕ → ℝ","l":93,"f":"Causalean/Stat/Sample/OccupancyWeightedMean/BinomialDesign.lean"},{"n":"Causalean.Stat.inverseUsableGroupTotal","k":"def","m":"Causalean.Stat.Sample.OccupancyWeightedMean.Basic","a":"Stat/Sample/OccupancyWeightedMean","t":1,"d":"The inverse usable occupancy is the reciprocal of the usable-group total when positive, and zero at the empirical zero boundary.","s":"{Omega : Type u_1} → {kappa : Type u_2} → [Fintype kappa] → [DecidableEq kappa] → {n : ℕ} → (Omega → kappa) → (Omega → Bool) → (Fin n → Omega) → ℝ","l":118,"f":"Causalean/Stat/Sample/OccupancyWeightedMean/Basic.lean"},{"n":"Causalean.Stat.isBigOp_centered_crossFit_sum","k":"theorem","m":"Causalean.Stat.EmpiricalProcess.CrossFitRate","a":"Stat/EmpiricalProcess","t":1,"d":"Cross-fit empirical-increment rate. Given observations W i, each of which is measurable, grouped into evaluation folds s n, each nonempty, and training σ-algebras m_A n, each contained in the ambient σ-algebra on the sample space such that the training σ-algebra m_A n is independent of the observations indexed by the fold s n and those fold observations are, conditionally, i.i.d. draws from P: for a score g n that viewed jointly in the sample point and its argument is measurable with respect to the training σ-algebra m_A n (the cross-fitting case of a fixed integrand evaluated at a nuisance estimated on the other folds) and is square-integrable under P at every sample point, and for any deterministic, nonnegative sequence Vn that dominates the average, over the training draw, of the squared L²(P)-norm of g n, the centered and rescaled evaluation-fold average of g n is stochastically bounded at the rate $\\sqrt{V_n}$.","s":"∀ {Ω : Type u_2} {X : Type u_3} [mΩ : MeasurableSpace Ω] [mX : MeasurableSpace X] {μ : MeasureTheory.Measure Ω} {P : MeasureTheory.Measure X} [MeasureTheory.IsProbabilityMeasure μ] [MeasureTheory.IsProbabilityMeasure P] (W : ℕ → Ω → X), (∀ (i : ℕ), Measurable (W i)) → ∀ (s : ℕ → Finset ℕ), (∀ (n : ℕ), 0 < (s n).card) → ∀ (m_A : ℕ → MeasurableSpace Ω), (∀ (n : ℕ), m_A n ≤ mΩ) → (∀ (n : ℕ), ProbabilityTheory.Indep (m_A n) (MeasurableSpace.comap (fun ω i => W (↑i) ω) inferInstance) μ) → (∀ (n : ℕ), MeasureTheory.Measure.map (fun ω i => W (↑i) ω) μ = MeasureTheory.Measure.pi fun x => P) → ∀ (g : ℕ → Ω → X → ℝ), (∀ (n : ℕ), Measurable (Function.uncurry (g n))) → (∀ (n : ℕ) (ω : Ω), MeasureTheory.MemLp (g n ω) 2 P) → ∀ {Vn : ℕ → ℝ}, (∀ (n : ℕ), 0 ≤ Vn n) → (∀ (n : ℕ), ∫⁻ (ω : Ω), ENNReal.ofReal ((MeasureTheory.eLpNorm (g n ω) 2 P).toReal ^ 2) ∂μ ≤ ENNReal.ofReal (Vn n)) → Causalean.Stat.IsBigOp (fun n ω => (√↑(s n).card)⁻¹ * ∑ i ∈ s n, (g n ω (W i ω) - ∫ (x : X), g n ω x ∂P)) (fun n => √(Vn n)) μ","l":450,"f":"Causalean/Stat/EmpiricalProcess/CrossFitRate.lean"},{"n":"Causalean.Stat.isCoupling_comonotoneCoupling","k":"theorem","m":"Causalean.Stat.Coupling.ProductLossMonotoneCoupling.Coupling","a":"Stat/Coupling/ProductLossMonotoneCoupling","t":1,"d":"For probability measures μ and ν on the reals, the comonotone coupling of μ and ν is indeed a coupling of the pair, i.e. its two marginals recover μ and ν.","s":"∀ (μ ν : MeasureTheory.Measure ℝ) [MeasureTheory.IsProbabilityMeasure μ] [MeasureTheory.IsProbabilityMeasure ν], Causalean.Stat.IsCoupling (Causalean.Stat.comonotoneCoupling μ ν) μ ν","l":86,"f":"Causalean/Stat/Coupling/ProductLossMonotoneCoupling/Coupling.lean"},{"n":"Causalean.Stat.isCoupling_countermonotoneCoupling","k":"theorem","m":"Causalean.Stat.Coupling.ProductLossMonotoneCoupling.Coupling","a":"Stat/Coupling/ProductLossMonotoneCoupling","t":1,"d":"For probability measures μ and ν on the reals, the countermonotone coupling of μ and ν is indeed a coupling of the pair, i.e. its two marginals recover μ and ν.","s":"∀ (μ ν : MeasureTheory.Measure ℝ) [MeasureTheory.IsProbabilityMeasure μ] [MeasureTheory.IsProbabilityMeasure ν], Causalean.Stat.IsCoupling (Causalean.Stat.countermonotoneCoupling μ ν) μ ν","l":119,"f":"Causalean/Stat/Coupling/ProductLossMonotoneCoupling/Coupling.lean"},{"n":"Causalean.Stat.isLittleOp_abs","k":"theorem","m":"Causalean.Stat.Quantile.SampleQuantileJoint","a":"Stat/Quantile","t":2,"d":"The absolute value of an o_p(1) sequence is o_p(1) (the threshold events coincide).","s":"∀ {Ω : Type u_1} [inst : MeasurableSpace Ω] {μ : MeasureTheory.Measure Ω} {R : ℕ → Ω → ℝ}, Causalean.Stat.IsLittleOp R (fun x => 1) μ → Causalean.Stat.IsLittleOp (fun n ω => |R n ω|) (fun x => 1) μ","l":200,"f":"Causalean/Stat/Quantile/SampleQuantileJoint.lean"},{"n":"Causalean.Stat.isLittleOp_finset_sum_one","k":"theorem","m":"Causalean.Stat.Quantile.SampleQuantileJoint","a":"Stat/Quantile","t":2,"d":"A finite sum of o_p(1) sequences is o_p(1).","s":"∀ {Ω : Type u_1} [inst : MeasurableSpace Ω] {μ : MeasureTheory.Measure Ω} {ι : Type u_2} (s : Finset ι) (g : ι → ℕ → Ω → ℝ), (∀ i ∈ s, Causalean.Stat.IsLittleOp (g i) (fun x => 1) μ) → Causalean.Stat.IsLittleOp (fun n ω => ∑ i ∈ s, g i n ω) (fun x => 1) μ","l":177,"f":"Causalean/Stat/Quantile/SampleQuantileJoint.lean"},{"n":"Causalean.Stat.isLittleOp_of_abs_le_const_mul_one","k":"theorem","m":"Causalean.Stat.Quantile.SampleQuantileJoint","a":"Stat/Quantile","t":2,"d":"Domination: if |Xn| ≤ C·|Yn| with Yn o_p(1) and C > 0, then Xn is o_p(1).","s":"∀ {Ω : Type u_1} [inst : MeasurableSpace Ω] {μ : MeasureTheory.Measure Ω} {X Y : ℕ → Ω → ℝ} {C : ℝ}, 0 < C → Causalean.Stat.IsLittleOp Y (fun x => 1) μ → (∀ (n : ℕ) (ω : Ω), |X n ω| ≤ C * |Y n ω|) → Causalean.Stat.IsLittleOp X (fun x => 1) μ","l":191,"f":"Causalean/Stat/Quantile/SampleQuantileJoint.lean"},{"n":"Causalean.Stat.isLittleOp_of_eventuallyEq","k":"theorem","m":"Causalean.Stat.UStatistic.Hajek","a":"Stat/UStatistic","t":1,"d":"Transfer of o_p along eventual equality. For sequences of random variables f, g : ℕ → Ω → ℝ and a rate r : ℕ → ℝ, if g is o_p(r) under μ and f n = g n for all sufficiently large n, then f is also o_p(r) under μ.","s":"∀ {Ω : Type u_1} [inst : MeasurableSpace Ω] {μ : MeasureTheory.Measure Ω} {f g : ℕ → Ω → ℝ} {r : ℕ → ℝ}, Causalean.Stat.IsLittleOp g r μ → (∀ᶠ (n : ℕ) in Filter.atTop, f n = g n) → Causalean.Stat.IsLittleOp f r μ","l":42,"f":"Causalean/Stat/UStatistic/Hajek.lean"},{"n":"Causalean.Stat.isLittleOp_one_of_measure_ne_tendsto_zero","k":"theorem","m":"Causalean.Stat.Inference.FunctionalDelta","a":"Stat/Inference","t":2,"d":"If Yₙ and Xₙ agree except on an event of vanishing probability, then Yₙ − Xₙ = o_p(1). This is the device that handles the *off-diagonal* case of the directional delta method (where the lattice functional locally selects one coordinate, so the rescaled image equals that coordinate's marginal except when the estimated ordering is wrong — an event whose probability tends to 0).","s":"∀ {Ω : Type u_1} [inst : MeasurableSpace Ω] {μ : MeasureTheory.Measure Ω} {Xn Yn : ℕ → Ω → ℝ}, Filter.Tendsto (fun n => μ {ω | Yn n ω ≠ Xn n ω}) Filter.atTop (nhds 0) → Causalean.Stat.IsLittleOp (fun n ω => Yn n ω - Xn n ω) (fun x => 1) μ","l":66,"f":"Causalean/Stat/Inference/FunctionalDelta.lean"},{"n":"Causalean.Stat.isLittleOp_zero","k":"theorem","m":"Causalean.Stat.Quantile.EmpiricalCDF","a":"Stat/Quantile","t":2,"d":"o_p-triviality of the zero sequence at the constant rate rₙ = 1.","s":"∀ {Ω : Type u_1} [inst : MeasurableSpace Ω] {μ : MeasureTheory.Measure Ω}, Causalean.Stat.IsLittleOp (fun x x_1 => 0) (fun x => 1) μ","l":165,"f":"Causalean/Stat/Quantile/EmpiricalCDF.lean"},{"n":"Causalean.Stat.isLittleOp_zero_one'","k":"theorem","m":"Causalean.Stat.Quantile.SampleQuantileJoint","a":"Stat/Quantile","t":2,"d":"The zero sequence is o_p(1).","s":"∀ {Ω : Type u_1} [inst : MeasurableSpace Ω] {μ : MeasureTheory.Measure Ω}, Causalean.Stat.IsLittleOp (fun x x_1 => 0) (fun x => 1) μ","l":162,"f":"Causalean/Stat/Quantile/SampleQuantileJoint.lean"},{"n":"Causalean.Stat.jStatistic_chiSq","k":"theorem","m":"Causalean.Stat.GMM.OverID","a":"Stat/GMM","t":1,"d":"Headline: the whitened GMM J-statistic limit law is χ²_{m−k}. Under a standard Gaussian random vector on the moment space, the distribution of the whitened GMM J-statistic — the squared norm of the residual-maker projection of that Gaussian vector — is the chi-squared law with finrank F − finrank E = m − k degrees of freedom, the Hansen–Sargan over-identification test statistic.","s":"∀ {E : Type u_1} {F : Type u_2} [inst : NormedAddCommGroup E] [inst_1 : InnerProductSpace ℝ E] [inst_2 : FiniteDimensional ℝ E] [inst_3 : NormedAddCommGroup F] [inst_4 : InnerProductSpace ℝ F] [inst_5 : FiniteDimensional ℝ F] [inst_6 : MeasurableSpace F] [BorelSpace F] {G : E →L[ℝ] F} {effInv : E →L[ℝ] E}, effInv ∘SL ContinuousLinearMap.adjoint G ∘SL G = ContinuousLinearMap.id ℝ E → (ContinuousLinearMap.adjoint G ∘SL G) ∘SL effInv = ContinuousLinearMap.id ℝ E → MeasureTheory.Measure.map (fun w => ‖(Causalean.Stat.gmmResidualMaker G effInv) w‖ ^ 2) (Causalean.Mathlib.stdGaussian F) = Causalean.Stat.chiSqDist (Module.finrank ℝ F - Module.finrank ℝ E)","l":205,"f":"Causalean/Stat/GMM/OverID.lean"},{"n":"Causalean.Stat.jointCdf","k":"def","m":"Causalean.Stat.Coupling.ProductLossMonotoneCoupling.FrechetHoeffding","a":"Stat/Coupling/ProductLossMonotoneCoupling","t":1,"d":"The joint cdf of a measure π on ℝ × ℝ: the mass of the lower-left quadrant Iic x ×ˢ Iic y, as a real number.","s":"MeasureTheory.Measure (ℝ × ℝ) → ℝ → ℝ → ℝ","l":35,"f":"Causalean/Stat/Coupling/ProductLossMonotoneCoupling/FrechetHoeffding.lean"},{"n":"Causalean.Stat.jointCdf_comonotoneCoupling","k":"theorem","m":"Causalean.Stat.Coupling.ProductLossMonotoneCoupling.FrechetHoeffdingAttainment","a":"Stat/Coupling/ProductLossMonotoneCoupling","t":1,"d":"Comonotone attainment. The joint cdf of the comonotone coupling of two probability measures μ and ν, evaluated at a point (x, y), equals the Fréchet–Hoeffding upper bound min (cdf μ x) (cdf ν y).","s":"∀ (μ ν : MeasureTheory.Measure ℝ) [MeasureTheory.IsProbabilityMeasure μ] [MeasureTheory.IsProbabilityMeasure ν] (x y : ℝ), Causalean.Stat.jointCdf (Causalean.Stat.comonotoneCoupling μ ν) x y = min (↑(ProbabilityTheory.cdf μ) x) (↑(ProbabilityTheory.cdf ν) y)","l":26,"f":"Causalean/Stat/Coupling/ProductLossMonotoneCoupling/FrechetHoeffdingAttainment.lean"},{"n":"Causalean.Stat.jointCdf_countermonotoneCoupling","k":"theorem","m":"Causalean.Stat.Coupling.ProductLossMonotoneCoupling.FrechetHoeffdingAttainment","a":"Stat/Coupling/ProductLossMonotoneCoupling","t":1,"d":"Countermonotone attainment. The joint cdf of the countermonotone coupling of two probability measures μ and ν, evaluated at a point (x, y), equals the Fréchet–Hoeffding lower bound max (cdf μ x + cdf ν y - 1) 0.","s":"∀ (μ ν : MeasureTheory.Measure ℝ) [MeasureTheory.IsProbabilityMeasure μ] [MeasureTheory.IsProbabilityMeasure ν] (x y : ℝ), Causalean.Stat.jointCdf (Causalean.Stat.countermonotoneCoupling μ ν) x y = max (↑(ProbabilityTheory.cdf μ) x + ↑(ProbabilityTheory.cdf ν) y - 1) 0","l":98,"f":"Causalean/Stat/Coupling/ProductLossMonotoneCoupling/FrechetHoeffdingAttainment.lean"},{"n":"Causalean.Stat.jointCdf_le_comonotone","k":"theorem","m":"Causalean.Stat.Coupling.ProductLossMonotoneCoupling.FrechetHoeffdingAttainment","a":"Stat/Coupling/ProductLossMonotoneCoupling","t":2,"d":"Corollary: for any coupling π of (μ, ν), its joint cdf is dominated pointwise by the comonotone joint cdf. This is the pointwise inequality that the covariance identity turns into optimality of E[XY].","s":"∀ {π : MeasureTheory.Measure (ℝ × ℝ)} {μ ν : MeasureTheory.Measure ℝ}, Causalean.Stat.IsCoupling π μ ν → ∀ [MeasureTheory.IsProbabilityMeasure μ] [MeasureTheory.IsProbabilityMeasure ν] (x y : ℝ), Causalean.Stat.jointCdf π x y ≤ Causalean.Stat.jointCdf (Causalean.Stat.comonotoneCoupling μ ν) x y","l":210,"f":"Causalean/Stat/Coupling/ProductLossMonotoneCoupling/FrechetHoeffdingAttainment.lean"},{"n":"Causalean.Stat.jointSurv","k":"def","m":"Causalean.Stat.Coupling.ProductLossMonotoneCoupling.Survival","a":"Stat/Coupling/ProductLossMonotoneCoupling","t":1,"d":"The joint survival function of π: S s t = π (Ioi s ×ˢ Ioi t), the mass of the open upper-right quadrant.","s":"MeasureTheory.Measure (ℝ × ℝ) → ℝ → ℝ → ℝ","l":42,"f":"Causalean/Stat/Coupling/ProductLossMonotoneCoupling/Survival.lean"},{"n":"Causalean.Stat.jointSurv_eq","k":"theorem","m":"Causalean.Stat.Coupling.ProductLossMonotoneCoupling.Survival","a":"Stat/Coupling/ProductLossMonotoneCoupling","t":2,"d":"Inclusion–exclusion. The joint survival function of a coupling is S s t = 1 - F s - G t + H_π s t. Proof: Ioi s ×ˢ Ioi t is the complement of (Prod.fst ⁻¹' Iic s) ∪ (Prod.snd ⁻¹' Iic t), and the mass of that union is F s + G t - H_π s t by measureReal_union_add_inter, the intersection being the lower-left quadrant Iic s ×ˢ Iic t.","s":"∀ {π : MeasureTheory.Measure (ℝ × ℝ)} {μ ν : MeasureTheory.Measure ℝ}, Causalean.Stat.IsCoupling π μ ν → ∀ (s t : ℝ), Causalean.Stat.jointSurv π s t = 1 - ↑(ProbabilityTheory.cdf μ) s - ↑(ProbabilityTheory.cdf ν) t + Causalean.Stat.jointCdf π s t","l":112,"f":"Causalean/Stat/Coupling/ProductLossMonotoneCoupling/Survival.lean"},{"n":"Causalean.Stat.kernelAffinePullback","k":"def","m":"Causalean.Stat.Minimax.MarkovKernelTransport","a":"Stat/Minimax","t":1,"d":"The affine kernel pullback averages a target estimator over the kernel, subtracts the target offset, and divides by the target slope.","s":"{X : Type uX} → {Y : Type uY} → [inst : MeasurableSpace X] → [inst_1 : MeasurableSpace Y] → ProbabilityTheory.Kernel X Y → ℝ → ℝ → (Y → ℝ) → X → ℝ","l":82,"f":"Causalean/Stat/Minimax/MarkovKernelTransport.lean"},{"n":"Causalean.Stat.kernelMean","k":"def","m":"Causalean.Stat.Minimax.MarkovKernelTransport","a":"Stat/Minimax","t":1,"d":"The kernel mean of an estimator is its expectation under the output distribution selected by each input to the kernel.","s":"{X : Type uX} → {Y : Type uY} → [inst : MeasurableSpace X] → [inst_1 : MeasurableSpace Y] → ProbabilityTheory.Kernel X Y → (Y → ℝ) → X → ℝ","l":51,"f":"Causalean/Stat/Minimax/MarkovKernelTransport.lean"},{"n":"Causalean.Stat.klForm_two_point_lower_bound","k":"theorem","m":"Causalean.Stat.Minimax.Pinsker","a":"Stat/Minimax","t":1,"d":"KL-form Le Cam two-point lower bound (unconditional). For probability measures P₀ and P₁ and two points θ₀, θ₁ in a pseudometric parameter space, if P₀ is absolutely continuous with respect to P₁ and their Kullback–Leibler divergence is finite, while est is a measurable estimator of the parameter and θ₀ and θ₁ are separated by at least 2s, then the worse of the two error probabilities — that est misses θ₀ by at least s under P₀, or misses θ₁ by at least s under P₁ — is at least (1 - √(klDiv(P₀,P₁)/2))/2.","s":"∀ {Ω : Type u_1} {mΩ : MeasurableSpace Ω} {P₀ P₁ : MeasureTheory.Measure Ω} [MeasureTheory.IsProbabilityMeasure P₀] [MeasureTheory.IsProbabilityMeasure P₁] {Θ : Type u_2} [inst : PseudoMetricSpace Θ] [inst_1 : MeasurableSpace Θ] [OpensMeasurableSpace Θ], P₀.AbsolutelyContinuous P₁ → InformationTheory.klDiv P₀ P₁ ≠ ⊤ → ∀ {est : Ω → Θ}, Measurable est → ∀ {θ₀ θ₁ : Θ} {s : ℝ}, 2 * s ≤ dist θ₀ θ₁ → (1 - √((InformationTheory.klDiv P₀ P₁).toReal / 2)) / 2 ≤ max (P₀.real {ω | s ≤ dist (est ω) θ₀}) (P₁.real {ω | s ≤ dist (est ω) θ₁})","l":489,"f":"Causalean/Stat/Minimax/Pinsker.lean"},{"n":"Causalean.Stat.klForm_two_point_lower_bound_of_pinsker","k":"theorem","m":"Causalean.Stat.Minimax.Pinsker","a":"Stat/Minimax","t":1,"d":"KL-form Le Cam two-point lower bound. Given a Pinsker-type bound h controlling the total variation distance between P₀ and P₁ by their Kullback–Leibler divergence, if est is a measurable estimator of the parameter and the parameter values θ₀ and θ₁ are separated by at least 2s, then the worse of the two error probabilities — that est misses θ₀ by at least s under P₀, or misses θ₁ by at least s under P₁ — is at least (1 - √(klDiv(P₀,P₁)/2))/2.","s":"∀ {Ω : Type u_1} {mΩ : MeasurableSpace Ω} {P₀ P₁ : MeasureTheory.Measure Ω} [MeasureTheory.IsProbabilityMeasure P₀] [MeasureTheory.IsProbabilityMeasure P₁] {Θ : Type u_2} [inst : PseudoMetricSpace Θ] [inst_1 : MeasurableSpace Θ] [OpensMeasurableSpace Θ], Causalean.Stat.PinskerBound P₀ P₁ → ∀ {est : Ω → Θ}, Measurable est → ∀ {θ₀ θ₁ : Θ} {s : ℝ}, 2 * s ≤ dist θ₀ θ₁ → (1 - √((InformationTheory.klDiv P₀ P₁).toReal / 2)) / 2 ≤ max (P₀.real {ω | s ≤ dist (est ω) θ₀}) (P₁.real {ω | s ≤ dist (est ω) θ₁})","l":463,"f":"Causalean/Stat/Minimax/Pinsker.lean"},{"n":"Causalean.Stat.klFun_lower_bound","k":"theorem","m":"Causalean.Stat.Minimax.Pinsker","a":"Stat/Minimax","t":2,"d":"Pinsker scalar inequality. For x ≥ 0, (3/2) * (x - 1)^2 / (x + 2) ≤ klFun x. This is the elementary bound that powers Pinsker's inequality through Cauchy–Schwarz.","s":"∀ {x : ℝ}, 0 ≤ x → 3 / 2 * (x - 1) ^ 2 / (x + 2) ≤ InformationTheory.klFun x","l":229,"f":"Causalean/Stat/Minimax/Pinsker.lean"},{"n":"Causalean.Stat.klFun_mul_lower_aux","k":"theorem","m":"Causalean.Stat.Minimax.Pinsker","a":"Stat/Minimax","t":2,"d":"Auxiliary: pinskerPhi x ≥ 0 for x ≥ 0, i.e. (x + 2) * klFun x - (3/2) * (x - 1)^2 ≥ 0.","s":"∀ {x : ℝ}, 0 ≤ x → 0 ≤ (x + 2) * InformationTheory.klFun x - 3 / 2 * (x - 1) ^ 2","l":209,"f":"Causalean/Stat/Minimax/Pinsker.lean"},{"n":"Causalean.Stat.le_cam_two_point_chisq","k":"theorem","m":"Causalean.Stat.Minimax.ChiSquared","a":"Stat/Minimax","t":1,"d":"Le Cam two-point testing floor under a χ²-budget (uniform-constant form). Two statements share this structure: for every χ²-budget C ≥ 0 there is a single floor constant c > 0, fixed before the laws are chosen, such that every pair of probability laws with Q-integrable squared density deviation and χ²-divergence at most C has combined testing error P(Aᶜ) + Q(A) at least c on every measurable test region A; and for every per-observation χ²-budget c0 ≥ 0 there is a single floor c' > 0 such that whenever the single-observation χ²-divergence is at most c0/m, the m-fold i.i.d. product experiment retains that same constant testing floor c', by the product χ²-identity 1 + χ²(P^{⊗m}‖Q^{⊗m}) = (1 + χ²)^m ≤ exp(c0).","s":"(∀ (Cchi : ℝ), 0 ≤ Cchi → ∃ c, 0 < c ∧ ∀ {Ω : Type} [inst : MeasurableSpace Ω] (Pp Pm : MeasureTheory.Measure Ω) [MeasureTheory.IsProbabilityMeasure Pp] [MeasureTheory.IsProbabilityMeasure Pm], Pp.AbsolutelyContinuous Pm → MeasureTheory.Integrable (fun x => ((Pp.rnDeriv Pm x).toReal - 1) ^ 2) Pm → Causalean.Stat.chiSqDiv Pp Pm ≤ Cchi → ∀ (A : Set Ω), MeasurableSet A → Pp.real Aᶜ + Pm.real A ≥ c) ∧ ∀ (c0 : ℝ), 0 ≤ c0 → ∃ c', 0 < c' ∧ ∀ {Ω : Type} [inst : MeasurableSpace Ω] (Pp Pm : MeasureTheory.Measure Ω) [MeasureTheory.IsProbabilityMeasure Pp] [MeasureTheory.IsProbabilityMeasure Pm], Pp.AbsolutelyContinuous Pm → MeasureTheory.Integrable (fun x => ((Pp.rnDeriv Pm x).toReal - 1) ^ 2) Pm → ∀ (m : ℕ), 0 < m → Causalean.Stat.chiSqDiv Pp Pm ≤ c0 / ↑m → ∀ (A : Set (Fin m → Ω)), MeasurableSet A → (MeasureTheory.Measure.pi fun x => Pp).real Aᶜ + (MeasureTheory.Measure.pi fun x => Pm).real A ≥ c'","l":615,"f":"Causalean/Stat/Minimax/ChiSquared.lean"},{"n":"Causalean.Stat.le_cdf_of_quantile_le","k":"theorem","m":"Causalean.Stat.Quantile.Quantile","a":"Stat/Quantile","t":2,"d":"Galois connection (one direction). If the quantile lies at or below x, then the cdf has already reached level τ at x.","s":"∀ {μ : MeasureTheory.Measure ℝ} {τ x : ℝ}, τ < 1 → Causalean.Stat.quantile μ τ ≤ x → τ ≤ ↑(ProbabilityTheory.cdf μ) x","l":94,"f":"Causalean/Stat/Quantile/Quantile.lean"},{"n":"Causalean.Stat.le_cdf_quantile","k":"theorem","m":"Causalean.Stat.Quantile.Quantile","a":"Stat/Quantile","t":2,"d":"Key membership lemma. For interior τ ∈ (0,1), the quantile lands in the super-level set: τ ≤ cdf μ (quantile μ τ). This is where right-continuity of the cdf is used.","s":"∀ {μ : MeasureTheory.Measure ℝ} {τ : ℝ}, τ < 1 → τ ≤ ↑(ProbabilityTheory.cdf μ) (Causalean.Stat.quantile μ τ)","l":74,"f":"Causalean/Stat/Quantile/Quantile.lean"},{"n":"Causalean.Stat.le_minimaxValue","k":"theorem","m":"Causalean.Stat.Minimax.MinimaxValue","a":"Stat/Minimax","t":2,"d":"A bound uniform over admissible estimators is a lower bound. If every admissible estimator has worst-case risk at least c, the minimax value is at least c. This is the converse half of a minimax rate; it needs at least one admissible estimator to exist, since otherwise the minimax value is zero by convention.","s":"∀ {E : Sort u_1} {Θ : Sort u_3} [Nonempty E] {risk : E → Θ → ℝ} {c : ℝ}, (∀ (e : E), c ≤ Causalean.Stat.worstCaseRisk risk e) → c ≤ Causalean.Stat.minimaxValue risk","l":141,"f":"Causalean/Stat/Minimax/MinimaxValue.lean"},{"n":"Causalean.Stat.le_minimaxValue_of_two_point","k":"theorem","m":"Causalean.Stat.Minimax.MinimaxValue","a":"Stat/Minimax","t":1,"d":"Two-point reduction to the minimax value. Fix two models θ₀ and θ₁ from the parameter class. If every estimator's risk is bounded above across the parameter class and for each estimator, the larger of its risks at the two fixed models is at least c, then the minimax value of the problem is at least c.","s":"∀ {E : Sort u_1} {Θ : Sort u_3} [Nonempty E] {risk : E → Θ → ℝ} {c : ℝ} (θ₀ θ₁ : Θ), (∀ (e : E), BddAbove (Set.range (risk e))) → (∀ (e : E), c ≤ max (risk e θ₀) (risk e θ₁)) → c ≤ Causalean.Stat.minimaxValue risk","l":150,"f":"Causalean/Stat/Minimax/MinimaxValue.lean"},{"n":"Causalean.Stat.le_worstCaseRisk","k":"theorem","m":"Causalean.Stat.Minimax.MinimaxValue","a":"Stat/Minimax","t":2,"d":"The risk at any single model in the class is at most the estimator's worst-case risk, provided that estimator's risk is bounded over the class.","s":"∀ {E : Sort u_1} {Θ : Sort u_3} {risk : E → Θ → ℝ} {e : E}, BddAbove (Set.range (risk e)) → ∀ (θ : Θ), risk e θ ≤ Causalean.Stat.worstCaseRisk risk e","l":112,"f":"Causalean/Stat/Minimax/MinimaxValue.lean"},{"n":"Causalean.Stat.lintegral_ofReal_tendsto_zero_of_bdd_isLittleOp","k":"theorem","m":"Causalean.Stat.Orthogonality.ConditionalOp","a":"Stat/Orthogonality","t":2,"d":"Bounded convergence in probability ⇒ L¹ convergence. If Xn ω ∈ [0, M] with M ≥ 0 deterministic, each Xn n is measurable, and Xn = o_p(1) under μ (with μ a probability measure), then ∫⁻ ω, ENNReal.ofReal (Xn n ω) ∂μ → 0 in ℝ≥0∞.","s":"∀ {Ω : Type u_1} [inst : MeasurableSpace Ω] {μ : MeasureTheory.Measure Ω} [MeasureTheory.IsProbabilityMeasure μ] {Xn : ℕ → Ω → ℝ} {M : ℝ}, 0 ≤ M → (∀ (n : ℕ), Measurable (Xn n)) → (∀ (n : ℕ) (ω : Ω), 0 ≤ Xn n ω) → (∀ (n : ℕ) (ω : Ω), Xn n ω ≤ M) → Causalean.Stat.IsLittleOp Xn (fun x => 1) μ → Filter.Tendsto (fun n => ∫⁻ (ω : Ω), ENNReal.ofReal (Xn n ω) ∂μ) Filter.atTop (nhds 0)","l":188,"f":"Causalean/Stat/Orthogonality/ConditionalOp.lean"},{"n":"Causalean.Stat.localStochasticExpansion","k":"theorem","m":"Causalean.Stat.MEstimation.EmpiricalExpansion","a":"Stat/MEstimation","t":1,"d":"Local stochastic expansion for Z-estimators. For a criterion function ψ with population Fréchet-Jacobian reg.J₀ at θ₀, an i.i.d. sample S, and an estimator sequence θn, suppose θn is consistent for θ₀, the centered empirical process of ψ is stochastically equicontinuous at θ₀ along θn, and θn approaches θ₀ at the parametric rate, ‖θn − θ₀‖ = O_p(1/√n). Then the empirical average score difference (1/√n) Σᵢ (ψ(θn,Zᵢ) − ψ(θ₀,Zᵢ)) agrees with its linearization √n · reg.J₀(θn − θ₀) up to an o_p(1) remainder in norm.","s":"∀ {Ω : Type u_1} {X : Type u_2} [inst : MeasurableSpace Ω] [inst_1 : MeasurableSpace X] {μ : MeasureTheory.Measure Ω} {E : Type u_3} [inst_2 : NormedAddCommGroup E] [inst_3 : InnerProductSpace ℝ E] [inst_4 : MeasurableSpace E] (ψ : E → X → E) (θ₀ : E) (P : MeasureTheory.Measure X) (reg : Causalean.Stat.ZEstimatorRegularity ψ θ₀ P) [MeasureTheory.IsProbabilityMeasure μ] (S : Causalean.Stat.IIDSample Ω X μ P) (θn : ℕ → Ω → E), (∀ ε > 0, Filter.Tendsto (fun n => μ {ω | ε < ‖θn n ω - θ₀‖}) Filter.atTop (nhds 0)) → Causalean.Stat.StochEquicontAt ψ θ₀ P μ S θn → Causalean.Stat.IsBigOp (fun n ω => ‖θn n ω - θ₀‖) (fun n => (√↑n)⁻¹) μ → Causalean.Stat.IsLittleOp (fun n ω => ‖(√↑n)⁻¹ • ∑ i ∈ Finset.range n, (ψ (θn n ω) (S.Z i ω) - ψ θ₀ (S.Z i ω)) - √↑n • reg.J₀ (θn n ω - θ₀)‖) (fun x => 1) μ","l":495,"f":"Causalean/Stat/MEstimation/EmpiricalExpansion.lean"},{"n":"Causalean.Stat.log_add_one_sub_log_le_inv","k":"theorem","m":"Causalean.Stat.Limit.SequentialCumulativeRisk","a":"Stat/Limit","t":2,"d":"Elementary per-term bound log(x+1) - log x ≤ 1/x for x > 0, i.e. log(1 + 1/x) ≤ 1/x. This is the pointwise ingredient of the harmonic-tail lower bound.","s":"∀ {x : ℝ}, 0 < x → Real.log (x + 1) - Real.log x ≤ 1 / x","l":45,"f":"Causalean/Stat/Limit/SequentialCumulativeRisk.lean"},{"n":"Causalean.Stat.log_lin_log_le_half_log","k":"theorem","m":"Causalean.Stat.Limit.SequentialCumulativeRisk","a":"Stat/Limit","t":2,"d":"Sub-logarithmic growth of log (c·log T + 1): for any c > 0 there is a threshold N beyond which log (c · log T + 1) ≤ (1/2) · log T. This is the log log T = o(log T) book-keeping that lets the harmonic tail dominate the t_* = Θ(log T) cutoff. (Route: c·log T + 1 ≤ (log T)²/16 ≤ exp((log T)/2) for large T, then take log.)","s":"∀ {c : ℝ}, 0 < c → ∃ N, ∀ (T : ℕ), N ≤ T → Real.log (c * Real.log ↑T + 1) ≤ 1 / 2 * Real.log ↑T","l":127,"f":"Causalean/Stat/Limit/SequentialCumulativeRisk.lean"},{"n":"Causalean.Stat.mEstimator_consistent_of_bracketing","k":"theorem","m":"Causalean.Stat.EmpiricalProcess.MEstimatorConsistency","a":"Stat/EmpiricalProcess","t":1,"d":"Bracketing corollary (the econometrician's headline). Let m be a criterion function of a parameter ranging over a pseudo-metric space Θ, observed along an i.i.d. sample S drawn from P. If each m(θ,·) is measurable, the criterion class admits, for every target width, a finite collection of integrable upper/lower bracket functions sandwiching the class members almost everywhere with L¹(P)-gap at most that width, the estimator sequence thetaHat attains a sample-objective value at every sample size and outcome that is at least as large as the sample objective at θ₀, and the population objective has a well-separated maximum at θ₀, meaning that for every ε>0 there is a gap η>0 such that the objective at any θ at distance at least ε from θ₀ falls short of the objective at θ₀ by at least η, then thetaHat is consistent for θ₀: for every ε>0 the probability that thetaHat n lies at distance at least ε from θ₀ tends to zero as the sample size n grows.","s":"∀ {Ω : Type u_1} {X : Type u_2} [inst : MeasurableSpace Ω] [inst_1 : MeasurableSpace X] {μ : MeasureTheory.Measure Ω} {P : MeasureTheory.Measure X} {Θ : Type u_3} [inst_2 : PseudoMetricSpace Θ] [MeasureTheory.IsProbabilityMeasure P] (S : Causalean.Stat.IIDSample Ω X μ P) (m : Θ → X → ℝ) (θ₀ : Θ) (thetaHat : ℕ → Ω → Θ), (∀ (θ : Θ), Measurable (m θ)) → Causalean.Stat.HasL1Bracketing m P → (∀ (n : ℕ) (ω : Ω), S.sampleMean (m θ₀) n ω ≤ S.sampleMean (m (thetaHat n ω)) n ω) → (∀ (ε : ℝ), 0 < ε → ∃ η, 0 < η ∧ ∀ (θ : Θ), ε ≤ dist θ θ₀ → ∫ (x : X), m θ x ∂P + η ≤ ∫ (x : X), m θ₀ x ∂P) → ∀ (ε : ℝ), 0 < ε → Filter.Tendsto (fun n => μ {ω | ε ≤ dist (thetaHat n ω) θ₀}) Filter.atTop (nhds 0)","l":94,"f":"Causalean/Stat/EmpiricalProcess/MEstimatorConsistency.lean"},{"n":"Causalean.Stat.mEstimator_consistent_of_glivenkoCantelli","k":"theorem","m":"Causalean.Stat.EmpiricalProcess.MEstimatorConsistency","a":"Stat/EmpiricalProcess","t":1,"d":"Consistency of extremum estimators (Newey–McFadden 1994, Thm 2.1). Let m be a criterion function of a parameter ranging over a pseudo-metric space Θ, with population objective M(θ) equal to the expectation of m(θ,·) under P and sample objective the empirical mean of m(θ,·) along an i.i.d. sample S. If the criterion class {m(θ,·) : θ ∈ Θ} obeys the Glivenko–Cantelli uniform law, so the worst-case gap between the sample and population objectives vanishes in probability, the estimator sequence thetaHat attains a sample-objective value at every sample size and outcome that is at least as large as the sample objective at θ₀, and the population objective has a well-separated maximum at θ₀, meaning that for every ε>0 there is a gap η>0 such that the objective at any θ at distance at least ε from θ₀ falls short of the objective at θ₀ by at least η, then thetaHat is consistent for θ₀: for every ε>0 the probability that thetaHat n lies at distance at least ε from θ₀ tends to zero as the sample size n grows.","s":"∀ {Ω : Type u_1} {X : Type u_2} [inst : MeasurableSpace Ω] [inst_1 : MeasurableSpace X] {μ : MeasureTheory.Measure Ω} {P : MeasureTheory.Measure X} {Θ : Type u_3} [inst_2 : PseudoMetricSpace Θ] (S : Causalean.Stat.IIDSample Ω X μ P) (m : Θ → X → ℝ) (θ₀ : Θ) (thetaHat : ℕ → Ω → Θ), Causalean.Stat.GlivenkoCantelli S m → (∀ (n : ℕ) (ω : Ω), S.sampleMean (m θ₀) n ω ≤ S.sampleMean (m (thetaHat n ω)) n ω) → (∀ (ε : ℝ), 0 < ε → ∃ η, 0 < η ∧ ∀ (θ : Θ), ε ≤ dist θ θ₀ → ∫ (x : X), m θ x ∂P + η ≤ ∫ (x : X), m θ₀ x ∂P) → ∀ (ε : ℝ), 0 < ε → Filter.Tendsto (fun n => μ {ω | ε ≤ dist (thetaHat n ω) θ₀}) Filter.atTop (nhds 0)","l":49,"f":"Causalean/Stat/EmpiricalProcess/MEstimatorConsistency.lean"},{"n":"Causalean.Stat.map_one_sub_unifOI","k":"theorem","m":"Causalean.Stat.Coupling.ProductLossMonotoneCoupling.Coupling","a":"Stat/Coupling/ProductLossMonotoneCoupling","t":2,"d":"The reflection u ↦ 1 - u preserves the uniform measure on (0,1): unifOI.map (fun u => 1 - u) = unifOI.","s":"MeasureTheory.Measure.map (fun u => 1 - u) Causalean.Stat.unifOI = Causalean.Stat.unifOI","l":57,"f":"Causalean/Stat/Coupling/ProductLossMonotoneCoupling/Coupling.lean"},{"n":"Causalean.Stat.map_pi_finCoordinatewise","k":"theorem","m":"Causalean.Stat.Sample.PiTransport","a":"Stat/Sample","t":2,"d":"Under a common probability law, applying a measurable observation rule coordinate by coordinate turns the finite product law into the finite product of the recoded marginal law.","s":"∀ {X : Type uX} {Y : Type uY} [inst : MeasurableSpace X] [inst_1 : MeasurableSpace Y] (n : ℕ) (mu : MeasureTheory.Measure X) [MeasureTheory.IsProbabilityMeasure mu] {phi : X → Y}, Measurable phi → MeasureTheory.Measure.map (fun z i => phi (z i)) (MeasureTheory.Measure.pi fun x => mu) = MeasureTheory.Measure.pi fun x => MeasureTheory.Measure.map phi mu","l":288,"f":"Causalean/Stat/Sample/PiTransport.lean"},{"n":"Causalean.Stat.map_pi_restrict","k":"theorem","m":"Causalean.Stat.Sample.PiTransport","a":"Stat/Sample","t":2,"d":"The law of the sub-index coordinates of an i.i.d. product sample is the product law over that sub-index; this is the pushforward packaging of the measure-preserving statement.","s":"∀ {ι : Type u_3} [inst : Fintype ι] {X : Type u_4} [inst_1 : MeasurableSpace X] (μ : MeasureTheory.Measure X) [MeasureTheory.IsProbabilityMeasure μ] (p : ι → Prop) [inst_3 : DecidablePred p], MeasureTheory.Measure.map (fun s i => s ↑i) (MeasureTheory.Measure.pi fun x => μ) = MeasureTheory.Measure.pi fun x => μ","l":154,"f":"Causalean/Stat/Sample/PiTransport.lean"},{"n":"Causalean.Stat.map_pi_restrict_finset","k":"theorem","m":"Causalean.Stat.Sample.PiTransport","a":"Stat/Sample","t":2,"d":"The law of the coordinates in a finite index set of an i.i.d. product sample is the product law over that finite set; this is the pushforward packaging of the measure-preserving statement.","s":"∀ {ι : Type u_3} [inst : Fintype ι] {X : Type u_4} [inst_1 : MeasurableSpace X] (μ : MeasureTheory.Measure X) [MeasureTheory.IsProbabilityMeasure μ] (S : Finset ι), MeasureTheory.Measure.map (fun s i => s ↑i) (MeasureTheory.Measure.pi fun x => μ) = MeasureTheory.Measure.pi fun x => μ","l":206,"f":"Causalean/Stat/Sample/PiTransport.lean"},{"n":"Causalean.Stat.map_weighted_sum_gaussian","k":"theorem","m":"Causalean.Stat.Bootstrap.GaussianMultiplier","a":"Stat/Bootstrap","t":2,"d":"Core multiplier lemma. Let ξ : Fin n → Ω → ℝ be a mutually independent family of measurable random variables, each with law gaussianReal 0 1 (standard Gaussian). Then for any fixed weights a : Fin n → ℝ, the weighted sum ω ↦ ∑ i, a i * ξ i ω has law","s":"∀ {Ω : Type u_1} [inst : MeasurableSpace Ω] {μ : MeasureTheory.Measure Ω} [MeasureTheory.IsProbabilityMeasure μ] {n : ℕ} (ξ : Fin n → Ω → ℝ), ProbabilityTheory.iIndepFun ξ μ → (∀ (i : Fin n), Measurable (ξ i)) → (∀ (i : Fin n), MeasureTheory.Measure.map (ξ i) μ = ProbabilityTheory.gaussianReal 0 1) → ∀ (a : Fin n → ℝ), MeasureTheory.Measure.map (fun ω => ∑ i, a i * ξ i ω) μ = ProbabilityTheory.gaussianReal 0 ⟨∑ i, a i ^ 2, Finset.sum_nonneg fun i x => Even.pow_nonneg (even_two_mul 1) (a i)⟩","l":53,"f":"Causalean/Stat/Bootstrap/GaussianMultiplier.lean"},{"n":"Causalean.Stat.matchingNormalization","k":"def","m":"Causalean.Stat.UStatistic.OrderM.MixedOrderCovariance","a":"Stat/UStatistic/OrderM","t":1,"d":"The normalization of a partial matching is the falling factorial for its number of distinct observations divided by the two marginal falling factorials.","s":"ℕ → {r s : ℕ} → Causalean.Stat.PartialMatching r s → ℝ","l":41,"f":"Causalean/Stat/UStatistic/OrderM/MixedOrderCovariance.lean"},{"n":"Causalean.Stat.matchingNormalization_le","k":"theorem","m":"Causalean.Stat.UStatistic.OrderM.MixedOrderBounds","a":"Stat/UStatistic/OrderM","t":2,"d":"For sample size n, orders r and s, common order bound R, and overlap size h, a partial matching has normalization at most exp(1) divided by n to the overlap size when the first order is at most R, the second order is at most R, the sample size is at least four times R squared, and the matching has size h as claimed.","s":"∀ {n r s R h : ℕ} {M : Causalean.Stat.PartialMatching r s}, r ≤ R → s ≤ R → 4 * R ^ 2 ≤ n → M ∈ Causalean.Stat.partialMatchingsOfSize r s h → Causalean.Stat.matchingNormalization n M ≤ Real.exp 1 / ↑n ^ h","l":234,"f":"Causalean/Stat/UStatistic/OrderM/MixedOrderBounds.lean"},{"n":"Causalean.Stat.matchingNormalization_of_mem","k":"theorem","m":"Causalean.Stat.UStatistic.OrderM.MixedOrderCovariance","a":"Stat/UStatistic/OrderM","t":2,"d":"Given sample size n, statistic orders r and s, and overlap size h, a partial matching belonging to the size-h family, its normalization has the falling factorial of length r + s - h in the numerator.","s":"∀ {n r s h : ℕ} {M : Causalean.Stat.PartialMatching r s}, M ∈ Causalean.Stat.partialMatchingsOfSize r s h → Causalean.Stat.matchingNormalization n M = ↑(n.descFactorial (r + s - h)) / (↑(n.descFactorial r) * ↑(n.descFactorial s))","l":411,"f":"Causalean/Stat/UStatistic/OrderM/MixedOrderCovariance.lean"},{"n":"Causalean.Stat.maxDirDeriv","k":"def","m":"Causalean.Stat.Inference.HadamardDeriv","a":"Stat/Inference","t":1,"d":"The directional derivative of max : ℝ × ℝ → ℝ at (a, b): fst if b < a, snd if a < b, and max itself at a tie a = b (where it is sublinear, not linear).","s":"ℝ → ℝ → ℝ × ℝ → ℝ","l":155,"f":"Causalean/Stat/Inference/HadamardDeriv.lean"},{"n":"Causalean.Stat.maxStat_aemeasurable","k":"theorem","m":"Causalean.Stat.Inference.FunctionalDelta","a":"Stat/Inference","t":2,"d":"The image statistic √n (max âₙ b̂ₙ − a) is AEMeasurable for every n, *derived* from the joint rescaled deviation's measurability via sqrt_mul_max_sub — so deltaMethod_max_tie need not assume it.","s":"∀ {Ω : Type u_1} [inst : MeasurableSpace Ω] {μ : MeasureTheory.Measure Ω} (an bn : ℕ → Ω → ℝ) (a : ℝ), (∀ (n : ℕ), AEMeasurable (fun ω => √↑n • ((an n ω, bn n ω) - (a, a))) μ) → ∀ (n : ℕ), AEMeasurable (fun ω => √↑n * (max (an n ω) (bn n ω) - a)) μ","l":141,"f":"Causalean/Stat/Inference/FunctionalDelta.lean"},{"n":"Causalean.Stat.maximalCoupling","k":"def","m":"Causalean.Stat.Minimax.MaximalCoupling","a":"Stat/Minimax","t":1,"d":"A maximal coupling of two laws on the same standard Borel space.","s":"{X : Type u_1} → [inst : MeasurableSpace X] → (mu nu : MeasureTheory.Measure X) → [MeasureTheory.IsProbabilityMeasure mu] → [MeasureTheory.IsProbabilityMeasure nu] → MeasureTheory.Measure (X × X)","l":216,"f":"Causalean/Stat/Minimax/MaximalCoupling.lean"},{"n":"Causalean.Stat.maximalCoupling_eq_mass_ge","k":"theorem","m":"Causalean.Stat.Minimax.MaximalCoupling","a":"Stat/Minimax","t":1,"d":"For two probability measures mu and nu on a standard Borel space X, the two coordinates of their maximal coupling agree with probability at least one minus their total variation distance.","s":"∀ {X : Type u_1} [inst : MeasurableSpace X] [MeasurableEq X] (mu nu : MeasureTheory.Measure X) [inst_2 : MeasureTheory.IsProbabilityMeasure mu] [inst_3 : MeasureTheory.IsProbabilityMeasure nu], ENNReal.ofReal (1 - Causalean.Stat.tvDist mu nu) ≤ (Causalean.Stat.maximalCoupling mu nu) {p | p.1 = p.2}","l":358,"f":"Causalean/Stat/Minimax/MaximalCoupling.lean"},{"n":"Causalean.Stat.maximalCoupling_map_fst","k":"theorem","m":"Causalean.Stat.Minimax.MaximalCoupling","a":"Stat/Minimax","t":2,"d":"The first marginal of the maximal coupling is the first law.","s":"∀ {X : Type u_1} [inst : MeasurableSpace X] (mu nu : MeasureTheory.Measure X) [inst_1 : MeasureTheory.IsProbabilityMeasure mu] [inst_2 : MeasureTheory.IsProbabilityMeasure nu], MeasureTheory.Measure.map Prod.fst (Causalean.Stat.maximalCoupling mu nu) = mu","l":279,"f":"Causalean/Stat/Minimax/MaximalCoupling.lean"},{"n":"Causalean.Stat.maximalCoupling_map_snd","k":"theorem","m":"Causalean.Stat.Minimax.MaximalCoupling","a":"Stat/Minimax","t":2,"d":"The second marginal of the maximal coupling is the second law.","s":"∀ {X : Type u_1} [inst : MeasurableSpace X] (mu nu : MeasureTheory.Measure X) [inst_1 : MeasureTheory.IsProbabilityMeasure mu] [inst_2 : MeasureTheory.IsProbabilityMeasure nu], MeasureTheory.Measure.map Prod.snd (Causalean.Stat.maximalCoupling mu nu) = nu","l":312,"f":"Causalean/Stat/Minimax/MaximalCoupling.lean"},{"n":"Causalean.Stat.maximizerOrZero","k":"def","m":"Causalean.Stat.MEstimation.FinitePoisson","a":"Stat/MEstimation","t":1,"d":"A total choice of a global maximizer, defaulting to zero when none exists.","s":"{A : Type u_1} → [Zero A] → (A → ℝ) → A","l":30,"f":"Causalean/Stat/MEstimation/FinitePoisson.lean"},{"n":"Causalean.Stat.mean_fst_tail","k":"theorem","m":"Causalean.Stat.Coupling.ProductLossMonotoneCoupling.HoeffdingFubiniIntegrability","a":"Stat/Coupling/ProductLossMonotoneCoupling","t":2,"d":"Tail formula for the mean. E[X] = ∫ s, (SX s - 𝟙{s<0}) ds, by the same swap on volume ⊗ π using integral_signedTail pointwise.","s":"∀ {π : MeasureTheory.Measure (ℝ × ℝ)} {μ ν : MeasureTheory.Measure ℝ}, Causalean.Stat.IsCoupling π μ ν → MeasureTheory.MemLp (fun x => x) 2 μ → ∫ (x : ℝ), x ∂μ = ∫ (s : ℝ), Causalean.Stat.survFst π s - Causalean.Stat.tailInd 0 s","l":118,"f":"Causalean/Stat/Coupling/ProductLossMonotoneCoupling/HoeffdingFubiniIntegrability.lean"},{"n":"Causalean.Stat.mean_snd_tail","k":"theorem","m":"Causalean.Stat.Coupling.ProductLossMonotoneCoupling.HoeffdingFubiniIntegrability","a":"Stat/Coupling/ProductLossMonotoneCoupling","t":2,"d":"Tail formula for the mean (second coordinate).","s":"∀ {π : MeasureTheory.Measure (ℝ × ℝ)} {μ ν : MeasureTheory.Measure ℝ}, Causalean.Stat.IsCoupling π μ ν → MeasureTheory.MemLp (fun y => y) 2 ν → ∫ (y : ℝ), y ∂ν = ∫ (t : ℝ), Causalean.Stat.survSnd π t - Causalean.Stat.tailInd 0 t","l":137,"f":"Causalean/Stat/Coupling/ProductLossMonotoneCoupling/HoeffdingFubiniIntegrability.lean"},{"n":"Causalean.Stat.measurableEqOfStandardBorel","k":"def","m":"Causalean.Stat.Minimax.MaximalCoupling","a":"Stat/Minimax","t":1,"d":"Equality is measurable on every standard Borel space.","s":"∀ (X : Type u_1) [inst : MeasurableSpace X] [StandardBorelSpace X], MeasurableEq X","l":21,"f":"Causalean/Stat/Minimax/MaximalCoupling.lean"},{"n":"Causalean.Stat.measurableSet_acceptanceRegion","k":"theorem","m":"Causalean.Stat.Minimax.Fano","a":"Stat/Minimax","t":2,"d":"A measurable estimator has a measurable acceptance region around any target.","s":"∀ {Ω : Type u_1} {mΩ : MeasurableSpace Ω} {Θ : Type u_2} [inst : PseudoMetricSpace Θ] [inst_1 : MeasurableSpace Θ] [OpensMeasurableSpace Θ] {est : Ω → Θ}, Measurable est → ∀ (θ : Θ) (s : ℝ), MeasurableSet (Causalean.Stat.acceptanceRegion est θ s)","l":66,"f":"Causalean/Stat/Minimax/Fano.lean"},{"n":"Causalean.Stat.measurableSet_armGroupEvent","k":"theorem","m":"Causalean.Stat.Sample.OccupancyWeightedMean.Basic","a":"Stat/Sample/OccupancyWeightedMean","t":2,"d":"Measurable group and arm labels have measurable joint arm/group fibers.","s":"∀ {Omega : Type u_1} {kappa : Type u_2} [inst : MeasurableSpace Omega] [Fintype kappa] [DecidableEq kappa] [inst_3 : MeasurableSpace kappa] [MeasurableSingletonClass kappa] (group : Omega → kappa) (arm : Omega → Bool), Measurable group → Measurable arm → ∀ (a : Bool) (k : kappa), MeasurableSet (Causalean.Stat.armGroupEvent group arm a k)","l":147,"f":"Causalean/Stat/Sample/OccupancyWeightedMean/Basic.lean"},{"n":"Causalean.Stat.measurableSet_decode_ne","k":"theorem","m":"Causalean.Stat.Minimax.Assouad","a":"Stat/Minimax","t":2,"d":"Coordinate-decoding error sets are measurable, being complements of the measurable level sets {ω | est ω j = b}.","s":"∀ {Ω : Type u_1} {mΩ : MeasurableSpace Ω} {d : ℕ} (est : Ω → Fin d → Bool), (∀ (j : Fin d) (b : Bool), MeasurableSet {ω | est ω j = b}) → ∀ (j : Fin d) (b : Bool), MeasurableSet {ω | est ω j ≠ b}","l":98,"f":"Causalean/Stat/Minimax/Assouad.lean"},{"n":"Causalean.Stat.measurableSet_error","k":"theorem","m":"Causalean.Stat.Minimax.LeCam","a":"Stat/Minimax","t":2,"d":"The error region {ω | s ≤ dist (est ω) θ} of an estimator is measurable.","s":"∀ {Ω : Type u_1} {mΩ : MeasurableSpace Ω} {Θ : Type u_2} [inst : PseudoMetricSpace Θ] [inst_1 : MeasurableSpace Θ] [OpensMeasurableSpace Θ] {est : Ω → Θ}, Measurable est → ∀ (θ : Θ) (s : ℝ), MeasurableSet {ω | s ≤ dist (est ω) θ}","l":40,"f":"Causalean/Stat/Minimax/LeCam.lean"},{"n":"Causalean.Stat.measurableSet_finiteCategoryPilotGood","k":"theorem","m":"Causalean.Stat.SampleSplit.FiniteCategoryPilot","a":"Stat/SampleSplit","t":2,"d":"Given an iid sample, a measurable category label, a finite pilot block, a selection threshold, a lower mass band, and an upper mass band, the simultaneous finite-category pilot good event is measurable.","s":"∀ {Omega : Type u_1} {X : Type u_2} {Iota : Type u_3} [inst : MeasurableSpace Omega] [inst_1 : MeasurableSpace X] [inst_2 : MeasurableSpace Iota] {mu : MeasureTheory.Measure Omega} {P : MeasureTheory.Measure X} [inst_3 : Fintype Iota] [inst_4 : DecidableEq Iota] [MeasurableSingletonClass Iota] (S : Causalean.Stat.IIDSample Omega X mu P) {label : X → Iota}, Measurable label → ∀ (block : Finset ℕ) (t lowerBand upperBand : ℝ), MeasurableSet (Causalean.Stat.finiteCategoryPilotGood S label block t lowerBand upperBand)","l":337,"f":"Causalean/Stat/SampleSplit/FiniteCategoryPilot.lean"},{"n":"Causalean.Stat.measurableSet_groupEvent","k":"theorem","m":"Causalean.Stat.Sample.OccupancyWeightedMean.Basic","a":"Stat/Sample/OccupancyWeightedMean","t":2,"d":"A measurable group label has measurable group fibers.","s":"∀ {Omega : Type u_1} {kappa : Type u_2} [inst : MeasurableSpace Omega] [Fintype kappa] [DecidableEq kappa] [inst_3 : MeasurableSpace kappa] [MeasurableSingletonClass kappa] (group : Omega → kappa), Measurable group → ∀ (k : kappa), MeasurableSet (Causalean.Stat.groupEvent group k)","l":141,"f":"Causalean/Stat/Sample/OccupancyWeightedMean/Basic.lean"},{"n":"Causalean.Stat.measurableSet_selectorCell","k":"theorem","m":"Causalean.Stat.SampleSplit.FiniteSelector","a":"Stat/SampleSplit","t":2,"d":"When the selection rule is measurable, the designated pilot event is measurable, and a branch is fixed, its selector cell is measurable.","s":"∀ {Alpha : Type u_2} {Iota : Type u_4} [inst : MeasurableSpace Alpha] [inst_1 : MeasurableSpace Iota] [MeasurableSingletonClass Iota] {select : Alpha → Iota}, Measurable select → ∀ {good : Set Alpha}, MeasurableSet good → ∀ (i : Iota), MeasurableSet (Causalean.Stat.selectorCell select good i)","l":40,"f":"Causalean/Stat/SampleSplit/FiniteSelector.lean"},{"n":"Causalean.Stat.measurableSet_usableGroup","k":"theorem","m":"Causalean.Stat.Sample.OccupancyWeightedMean.Basic","a":"Stat/Sample/OccupancyWeightedMean","t":2,"d":"Measurable group and arm labels make the event that a fixed empirical group has both arms represented measurable.","s":"∀ {Omega : Type u_1} {kappa : Type u_2} [inst : MeasurableSpace Omega] [Fintype kappa] [inst_2 : DecidableEq kappa] [inst_3 : MeasurableSpace kappa] [MeasurableSingletonClass kappa] {n : ℕ} (group : Omega → kappa) (arm : Omega → Bool), Measurable group → Measurable arm → ∀ (k : kappa), MeasurableSet {z | Causalean.Stat.usableGroup group arm z k}","l":193,"f":"Causalean/Stat/Sample/OccupancyWeightedMean/Basic.lean"},{"n":"Causalean.Stat.measurable_affinePullbackEstimator","k":"theorem","m":"Causalean.Stat.Minimax.MinimaxRisk","a":"Stat/Minimax","t":2,"d":"If the observation rule is measurable and the target estimator is measurable, then undoing an affine change after pulling the estimator back through the observation rule is measurable.","s":"∀ {X : Type uX} {Y : Type uY} [inst : MeasurableSpace X] [inst_1 : MeasurableSpace Y] {phi : X → Y} {a b : ℝ} {targetEst : Y → ℝ}, Measurable phi → Measurable targetEst → Measurable (Causalean.Stat.affinePullbackEstimator phi a b targetEst)","l":189,"f":"Causalean/Stat/Minimax/MinimaxRisk.lean"},{"n":"Causalean.Stat.measurable_armGroupResidual","k":"theorem","m":"Causalean.Stat.Sample.OccupancyWeightedMean.Basic","a":"Stat/Sample/OccupancyWeightedMean","t":2,"d":"A measurable outcome makes each arm/group-centered residual measurable.","s":"∀ {Omega : Type u_1} {kappa : Type u_2} [inst : MeasurableSpace Omega] [Fintype kappa] [DecidableEq kappa] [inst_3 : MeasurableSpace kappa] [MeasurableSingletonClass kappa] (Y : Omega → ℝ) (center : Bool → kappa → ℝ), Measurable Y → ∀ (a : Bool) (k : kappa), Measurable (Causalean.Stat.armGroupResidual Y center a k)","l":216,"f":"Causalean/Stat/Sample/OccupancyWeightedMean/Basic.lean"},{"n":"Causalean.Stat.measurable_armResidualMean","k":"theorem","m":"Causalean.Stat.Sample.OccupancyWeightedMean.Basic","a":"Stat/Sample/OccupancyWeightedMean","t":2,"d":"Measurable group labels, arm labels, and outcomes make each zero-safe arm/group residual mean measurable on the finite product sample space.","s":"∀ {Omega : Type u_1} {kappa : Type u_2} [inst : MeasurableSpace Omega] [Fintype kappa] [inst_2 : DecidableEq kappa] [inst_3 : MeasurableSpace kappa] [MeasurableSingletonClass kappa] {n : ℕ} (group : Omega → kappa) (arm : Omega → Bool) (Y : Omega → ℝ) (center : Bool → kappa → ℝ), Measurable group → Measurable arm → Measurable Y → ∀ (a : Bool) (k : kappa), Measurable fun z => Causalean.Stat.armResidualMean group arm Y center z a k","l":252,"f":"Causalean/Stat/Sample/OccupancyWeightedMean/Basic.lean"},{"n":"Causalean.Stat.measurable_armResidualSum","k":"theorem","m":"Causalean.Stat.Sample.OccupancyWeightedMean.Basic","a":"Stat/Sample/OccupancyWeightedMean","t":2,"d":"Measurable group labels, arm labels, and outcomes make each arm/group residual sum measurable on the finite product sample space.","s":"∀ {Omega : Type u_1} {kappa : Type u_2} [inst : MeasurableSpace Omega] [Fintype kappa] [DecidableEq kappa] [inst_3 : MeasurableSpace kappa] [MeasurableSingletonClass kappa] {n : ℕ} (group : Omega → kappa) (arm : Omega → Bool) (Y : Omega → ℝ) (center : Bool → kappa → ℝ), Measurable group → Measurable arm → Measurable Y → ∀ (a : Bool) (k : kappa), Measurable fun z => Causalean.Stat.armResidualSum group arm Y center z a k","l":236,"f":"Causalean/Stat/Sample/OccupancyWeightedMean/Basic.lean"},{"n":"Causalean.Stat.measurable_categoryIndicator","k":"theorem","m":"Causalean.Stat.SampleSplit.FiniteCategoryPilot","a":"Stat/SampleSplit","t":2,"d":"Given a measurable category label and a fixed category, the corresponding zero-one category indicator is measurable.","s":"∀ {X : Type u_2} {Iota : Type u_3} [inst : MeasurableSpace X] [inst_1 : MeasurableSpace Iota] [inst_2 : DecidableEq Iota] [MeasurableSingletonClass Iota] {label : X → Iota}, Measurable label → ∀ (k : Iota), Measurable (Causalean.Stat.categoryIndicator label k)","l":44,"f":"Causalean/Stat/SampleSplit/FiniteCategoryPilot.lean"},{"n":"Causalean.Stat.measurable_cdfIF","k":"theorem","m":"Causalean.Stat.Quantile.EmpiricalCDF","a":"Stat/Quantile","t":2,"d":"The empirical-cdf influence function is measurable.","s":"∀ {P : MeasureTheory.Measure ℝ} (y : ℝ), Measurable (Causalean.Stat.cdfIF P y)","l":105,"f":"Causalean/Stat/Quantile/EmpiricalCDF.lean"},{"n":"Causalean.Stat.measurable_cdfStat","k":"theorem","m":"Causalean.Stat.Quantile.EmpiricalCDF","a":"Stat/Quantile","t":2,"d":"The lower-ray indicator is measurable.","s":"∀ (y : ℝ), Measurable (Causalean.Stat.cdfStat y)","l":58,"f":"Causalean/Stat/Quantile/EmpiricalCDF.lean"},{"n":"Causalean.Stat.measurable_finCoordinatewise","k":"theorem","m":"Causalean.Stat.Sample.PiTransport","a":"Stat/Sample","t":2,"d":"Applying a measurable observation rule separately to every position of a finite sample produces a measurable recoded sample.","s":"∀ {X : Type uX} {Y : Type uY} [inst : MeasurableSpace X] [inst_1 : MeasurableSpace Y] (n : ℕ) {phi : X → Y}, Measurable phi → Measurable fun z i => phi (z i)","l":281,"f":"Causalean/Stat/Sample/PiTransport.lean"},{"n":"Causalean.Stat.measurable_finiteSelector_apply","k":"theorem","m":"Causalean.Stat.SampleSplit.FiniteSelector","a":"Stat/SampleSplit","t":2,"d":"When finite branch selection is measurable and each branch-specific error is measurable, evaluating the selected error on a pilot-tail pair is measurable.","s":"∀ {Alpha : Type u_2} {Beta : Type u_3} {Iota : Type u_4} [inst : MeasurableSpace Alpha] [inst_1 : MeasurableSpace Beta] [inst_2 : MeasurableSpace Iota] [Fintype Iota] [MeasurableSingletonClass Iota] {select : Alpha → Iota}, Measurable select → ∀ {err : Iota → Beta → ℝ}, (∀ (i : Iota), Measurable (err i)) → Measurable fun z => err (select z.1) z.2","l":49,"f":"Causalean/Stat/SampleSplit/FiniteSelector.lean"},{"n":"Causalean.Stat.measurable_finiteSelector_sq","k":"theorem","m":"Causalean.Stat.SampleSplit.FiniteSelector","a":"Stat/SampleSplit","t":2,"d":"If the pilot coordinate is measurable, the tail coordinate is measurable, the finite selection rule is measurable, and every branch error is measurable, the selected squared error is measurable.","s":"∀ {Omega : Type u_1} {Alpha : Type u_2} {Beta : Type u_3} {Iota : Type u_4} [inst : MeasurableSpace Omega] [inst_1 : MeasurableSpace Alpha] [inst_2 : MeasurableSpace Beta] [inst_3 : MeasurableSpace Iota] [Fintype Iota] [MeasurableSingletonClass Iota] {pilot : Omega → Alpha} {tail : Omega → Beta} {select : Alpha → Iota} {err : Iota → Beta → ℝ}, Measurable pilot → Measurable tail → Measurable select → (∀ (i : Iota), Measurable (err i)) → Measurable fun omega => err (select (pilot omega)) (tail omega) ^ 2","l":70,"f":"Causalean/Stat/SampleSplit/FiniteSelector.lean"},{"n":"Causalean.Stat.measurable_groupArmCount","k":"theorem","m":"Causalean.Stat.Sample.OccupancyWeightedMean.Basic","a":"Stat/Sample/OccupancyWeightedMean","t":2,"d":"Measurable group and arm labels make each fixed arm/group count measurable on a finite product sample.","s":"∀ {Omega : Type u_1} {kappa : Type u_2} [inst : MeasurableSpace Omega] [Fintype kappa] [inst_2 : DecidableEq kappa] [inst_3 : MeasurableSpace kappa] [MeasurableSingletonClass kappa] {n : ℕ} (group : Omega → kappa) (arm : Omega → Bool), Measurable group → Measurable arm → ∀ (a : Bool) (k : kappa), Measurable fun z => Causalean.Stat.groupArmCount group arm z a k","l":169,"f":"Causalean/Stat/Sample/OccupancyWeightedMean/Basic.lean"},{"n":"Causalean.Stat.measurable_groupCount","k":"theorem","m":"Causalean.Stat.Sample.OccupancyWeightedMean.Basic","a":"Stat/Sample/OccupancyWeightedMean","t":2,"d":"Measurable group and arm labels make each fixed group count measurable on the finite product sample space.","s":"∀ {Omega : Type u_1} {kappa : Type u_2} [inst : MeasurableSpace Omega] [Fintype kappa] [inst_2 : DecidableEq kappa] [inst_3 : MeasurableSpace kappa] [MeasurableSingletonClass kappa] {n : ℕ} (group : Omega → kappa) (arm : Omega → Bool), Measurable group → Measurable arm → ∀ (k : kappa), Measurable fun z => Causalean.Stat.groupCount group arm z k","l":181,"f":"Causalean/Stat/Sample/OccupancyWeightedMean/Basic.lean"},{"n":"Causalean.Stat.measurable_inverseUsableGroupTotal","k":"theorem","m":"Causalean.Stat.Sample.OccupancyWeightedMean.Basic","a":"Stat/Sample/OccupancyWeightedMean","t":2,"d":"Measurable group and arm labels make the zero-safe reciprocal usable occupancy measurable on the finite product sample space.","s":"∀ {Omega : Type u_1} {kappa : Type u_2} [inst : MeasurableSpace Omega] [inst_1 : Fintype kappa] [inst_2 : DecidableEq kappa] [inst_3 : MeasurableSpace kappa] [MeasurableSingletonClass kappa] {n : ℕ} (group : Omega → kappa) (arm : Omega → Bool), Measurable group → Measurable arm → Measurable (Causalean.Stat.inverseUsableGroupTotal group arm)","l":310,"f":"Causalean/Stat/Sample/OccupancyWeightedMean/Basic.lean"},{"n":"Causalean.Stat.measurable_kernelAffinePullback","k":"theorem","m":"Causalean.Stat.Minimax.MarkovKernelTransport","a":"Stat/Minimax","t":2,"d":"Measurability of a target estimator ensures that its affine kernel pullback is measurable on the source experiment.","s":"∀ {X : Type uX} {Y : Type uY} [inst : MeasurableSpace X] [inst_1 : MeasurableSpace Y] (K : ProbabilityTheory.Kernel X Y) {a b : ℝ} {targetEst : Y → ℝ}, Measurable targetEst → Measurable (Causalean.Stat.kernelAffinePullback K a b targetEst)","l":88,"f":"Causalean/Stat/Minimax/MarkovKernelTransport.lean"},{"n":"Causalean.Stat.measurable_kernelMean","k":"theorem","m":"Causalean.Stat.Minimax.MarkovKernelTransport","a":"Stat/Minimax","t":2,"d":"Measurability of a real-valued estimator ensures that its expectation under each kernel output distribution varies measurably with the kernel input.","s":"∀ {X : Type uX} {Y : Type uY} [inst : MeasurableSpace X] [inst_1 : MeasurableSpace Y] (K : ProbabilityTheory.Kernel X Y) {T : Y → ℝ}, Measurable T → Measurable (Causalean.Stat.kernelMean K T)","l":56,"f":"Causalean/Stat/Minimax/MarkovKernelTransport.lean"},{"n":"Causalean.Stat.measurable_occupancyDesignVarianceFactor","k":"theorem","m":"Causalean.Stat.Sample.OccupancyWeightedMean.Basic","a":"Stat/Sample/OccupancyWeightedMean","t":2,"d":"Measurable group and arm labels make the zero-safe occupancy design variance factor measurable on the finite product sample space.","s":"∀ {Omega : Type u_1} {kappa : Type u_2} [inst : MeasurableSpace Omega] [inst_1 : Fintype kappa] [inst_2 : DecidableEq kappa] [inst_3 : MeasurableSpace kappa] [MeasurableSingletonClass kappa] {n : ℕ} (group : Omega → kappa) (arm : Omega → Bool), Measurable group → Measurable arm → Measurable (Causalean.Stat.occupancyDesignVarianceFactor group arm)","l":327,"f":"Causalean/Stat/Sample/OccupancyWeightedMean/Basic.lean"},{"n":"Causalean.Stat.measurable_occupancyWeightedResidual","k":"theorem","m":"Causalean.Stat.Sample.OccupancyWeightedMean.Basic","a":"Stat/Sample/OccupancyWeightedMean","t":1,"d":"Measurable group labels, arm labels, and outcomes make the zero-safe occupancy-weighted residual measurable on the finite product sample space.","s":"∀ {Omega : Type u_1} {kappa : Type u_2} [inst : MeasurableSpace Omega] [inst_1 : Fintype kappa] [inst_2 : DecidableEq kappa] [inst_3 : MeasurableSpace kappa] [MeasurableSingletonClass kappa] {n : ℕ} (mu : MeasureTheory.Measure Omega) (group : Omega → kappa) (arm : Omega → Bool) (Y : Omega → ℝ) (center : Bool → kappa → ℝ), Measurable group → Measurable arm → Measurable Y → Measurable (Causalean.Stat.occupancyWeightedResidual mu group arm Y center)","l":273,"f":"Causalean/Stat/Sample/OccupancyWeightedMean/Basic.lean"},{"n":"Causalean.Stat.measurable_pilotCategoryCount","k":"theorem","m":"Causalean.Stat.SampleSplit.FiniteCategoryPilot","a":"Stat/SampleSplit","t":2,"d":"Given an iid sample, a measurable category label, a finite coordinate block, and a fixed category, the pilot count of that category is measurable.","s":"∀ {Omega : Type u_1} {X : Type u_2} {Iota : Type u_3} [inst : MeasurableSpace Omega] [inst_1 : MeasurableSpace X] [inst_2 : MeasurableSpace Iota] {mu : MeasureTheory.Measure Omega} {P : MeasureTheory.Measure X} [inst_3 : DecidableEq Iota] [MeasurableSingletonClass Iota] (S : Causalean.Stat.IIDSample Omega X mu P) {label : X → Iota}, Measurable label → ∀ (block : Finset ℕ) (k : Iota), Measurable (Causalean.Stat.pilotCategoryCount S label block k)","l":73,"f":"Causalean/Stat/SampleSplit/FiniteCategoryPilot.lean"},{"n":"Causalean.Stat.measurable_quantileIF","k":"theorem","m":"Causalean.Stat.Quantile.SampleQuantile","a":"Stat/Quantile","t":2,"d":"The sample-quantile influence function is measurable.","s":"∀ (τ q₀ f₀ : ℝ), Measurable (Causalean.Stat.quantileIF τ q₀ f₀)","l":74,"f":"Causalean/Stat/Quantile/SampleQuantile.lean"},{"n":"Causalean.Stat.measurable_quantileIFVec","k":"theorem","m":"Causalean.Stat.Quantile.SampleQuantileJoint","a":"Stat/Quantile","t":2,"d":"The joint quantile influence function is measurable.","s":"∀ {k : ℕ} (τ q f : Fin k → ℝ), Measurable (Causalean.Stat.quantileIFVec τ q f)","l":72,"f":"Causalean/Stat/Quantile/SampleQuantileJoint.lean"},{"n":"Causalean.Stat.measurable_sampleDesign","k":"theorem","m":"Causalean.Stat.Sample.OccupancyWeightedMean.Basic","a":"Stat/Sample/OccupancyWeightedMean","t":2,"d":"Measurable group and arm labels make the coordinatewise finite design measurable.","s":"∀ {Omega : Type u_1} {kappa : Type u_2} [inst : MeasurableSpace Omega] [Fintype kappa] [DecidableEq kappa] [inst_3 : MeasurableSpace kappa] [MeasurableSingletonClass kappa] {n : ℕ} (group : Omega → kappa) (arm : Omega → Bool), Measurable group → Measurable arm → Measurable (Causalean.Stat.sampleDesign group arm)","l":157,"f":"Causalean/Stat/Sample/OccupancyWeightedMean/Basic.lean"},{"n":"Causalean.Stat.measurable_signedTail","k":"theorem","m":"Causalean.Stat.Coupling.ProductLossMonotoneCoupling.TailIntegral","a":"Stat/Coupling/ProductLossMonotoneCoupling","t":2,"d":"signedTail a is measurable in the tail variable s.","s":"∀ (a : ℝ), Measurable (Causalean.Stat.signedTail a)","l":60,"f":"Causalean/Stat/Coupling/ProductLossMonotoneCoupling/TailIntegral.lean"},{"n":"Causalean.Stat.measurable_signedTail_uncurry","k":"theorem","m":"Causalean.Stat.Coupling.ProductLossMonotoneCoupling.TailIntegral","a":"Stat/Coupling/ProductLossMonotoneCoupling","t":2,"d":"signedTail is jointly measurable in (a, s), since {z | z.2 < z.1} is an open (hence measurable) subset of ℝ × ℝ.","s":"Measurable fun z => Causalean.Stat.signedTail z.1 z.2","l":66,"f":"Causalean/Stat/Coupling/ProductLossMonotoneCoupling/TailIntegral.lean"},{"n":"Causalean.Stat.measurable_supportedArmGroupResidual","k":"theorem","m":"Causalean.Stat.Sample.OccupancyWeightedMean.Basic","a":"Stat/Sample/OccupancyWeightedMean","t":2,"d":"Measurable group labels, arm labels, and outcomes make each residual restricted to its own arm/group cell measurable.","s":"∀ {Omega : Type u_1} {kappa : Type u_2} [inst : MeasurableSpace Omega] [Fintype kappa] [DecidableEq kappa] [inst_3 : MeasurableSpace kappa] [MeasurableSingletonClass kappa] (group : Omega → kappa) (arm : Omega → Bool) (Y : Omega → ℝ) (center : Bool → kappa → ℝ), Measurable group → Measurable arm → Measurable Y → ∀ (a : Bool) (k : kappa), Measurable (Causalean.Stat.supportedArmGroupResidual group arm Y center a k)","l":225,"f":"Causalean/Stat/Sample/OccupancyWeightedMean/Basic.lean"},{"n":"Causalean.Stat.measurable_usableGroupTotal","k":"theorem","m":"Causalean.Stat.Sample.OccupancyWeightedMean.Basic","a":"Stat/Sample/OccupancyWeightedMean","t":2,"d":"Measurable group and arm labels make the total occupancy in empirically usable groups measurable.","s":"∀ {Omega : Type u_1} {kappa : Type u_2} [inst : MeasurableSpace Omega] [inst_1 : Fintype kappa] [inst_2 : DecidableEq kappa] [inst_3 : MeasurableSpace kappa] [MeasurableSingletonClass kappa] {n : ℕ} (group : Omega → kappa) (arm : Omega → Bool), Measurable group → Measurable arm → Measurable fun z => Causalean.Stat.usableGroupTotal group arm z","l":204,"f":"Causalean/Stat/Sample/OccupancyWeightedMean/Basic.lean"},{"n":"Causalean.Stat.measurePreserving_pi_restrict","k":"theorem","m":"Causalean.Stat.Sample.PiTransport","a":"Stat/Sample","t":2,"d":"Dropping the coordinates outside a decidable sub-index of an i.i.d. product sample leaves the i.i.d. product sample over that sub-index.","s":"∀ {ι : Type u_3} [inst : Fintype ι] {X : Type u_4} [inst_1 : MeasurableSpace X] (μ : MeasureTheory.Measure X) [MeasureTheory.IsProbabilityMeasure μ] (p : ι → Prop) [inst_3 : DecidablePred p], MeasureTheory.MeasurePreserving (fun s i => s ↑i) (MeasureTheory.Measure.pi fun x => μ) (MeasureTheory.Measure.pi fun x => μ)","l":146,"f":"Causalean/Stat/Sample/PiTransport.lean"},{"n":"Causalean.Stat.measurePreserving_pi_restrict_dep","k":"theorem","m":"Causalean.Stat.Sample.PiTransport","a":"Stat/Sample","t":2,"d":"Dropping the coordinates outside a decidable sub-index of a product of probability measures leaves the product measure over that sub-index: the retained coordinates carry exactly their own product law, with no trace of the discarded ones.","s":"∀ {ι : Type u_3} [inst : Fintype ι] {X : ι → Type u_4} [inst_1 : (i : ι) → MeasurableSpace (X i)] (μ : (i : ι) → MeasureTheory.Measure (X i)) [∀ (i : ι), MeasureTheory.IsProbabilityMeasure (μ i)] (p : ι → Prop) [inst_3 : DecidablePred p], MeasureTheory.MeasurePreserving (fun s i => s ↑i) (MeasureTheory.Measure.pi μ) (MeasureTheory.Measure.pi fun i => μ ↑i)","l":129,"f":"Causalean/Stat/Sample/PiTransport.lean"},{"n":"Causalean.Stat.measurePreserving_pi_restrict_finset","k":"theorem","m":"Causalean.Stat.Sample.PiTransport","a":"Stat/Sample","t":2,"d":"Dropping the coordinates outside a finite set of indices of an i.i.d. product sample leaves the i.i.d. product sample indexed by that finite set.","s":"∀ {ι : Type u_3} [inst : Fintype ι] {X : Type u_4} [inst_1 : MeasurableSpace X] (μ : MeasureTheory.Measure X) [MeasureTheory.IsProbabilityMeasure μ] (S : Finset ι), MeasureTheory.MeasurePreserving (fun s i => s ↑i) (MeasureTheory.Measure.pi fun x => μ) (MeasureTheory.Measure.pi fun x => μ)","l":197,"f":"Causalean/Stat/Sample/PiTransport.lean"},{"n":"Causalean.Stat.measureReal_sub_eq_setIntegral_rnDeriv_sub_one","k":"theorem","m":"Causalean.Stat.Minimax.Scheffe","a":"Stat/Minimax","t":2,"d":"With density p = dμ/dν and μ ≪ ν, the signed gap μ.real A − ν.real A equals ∫ x in A, (p x − 1) ∂ν.","s":"∀ {Ω : Type u_1} {mΩ : MeasurableSpace Ω} {μ ν : MeasureTheory.Measure Ω} [MeasureTheory.IsProbabilityMeasure μ] [MeasureTheory.IsProbabilityMeasure ν], μ.AbsolutelyContinuous ν → ∀ {A : Set Ω}, MeasurableSet A → μ.real A - ν.real A = ∫ (x : Ω) in A, (μ.rnDeriv ν x).toReal - 1 ∂ν","l":94,"f":"Causalean/Stat/Minimax/Scheffe.lean"},{"n":"Causalean.Stat.measureReal_sub_le_tvDist","k":"theorem","m":"Causalean.Stat.Minimax.TotalVariation","a":"Stat/Minimax","t":2,"d":"The signed gap ν.real A − μ.real A is at most tvDist μ ν.","s":"∀ {Ω : Type u_1} {mΩ : MeasurableSpace Ω} {μ ν : MeasureTheory.Measure Ω} [MeasureTheory.IsProbabilityMeasure μ] [MeasureTheory.IsProbabilityMeasure ν] {A : Set Ω}, MeasurableSet A → ν.real A - μ.real A ≤ Causalean.Stat.tvDist μ ν","l":79,"f":"Causalean/Stat/Minimax/TotalVariation.lean"},{"n":"Causalean.Stat.measure_eq_of_tvDist_eq_zero","k":"theorem","m":"Causalean.Stat.Minimax.MaximalCoupling","a":"Stat/Minimax","t":2,"d":"Probability measures at total-variation distance zero are equal.","s":"∀ {X : Type u_1} [inst : MeasurableSpace X] (mu nu : MeasureTheory.Measure X) [MeasureTheory.IsProbabilityMeasure mu] [MeasureTheory.IsProbabilityMeasure nu], Causalean.Stat.tvDist mu nu = 0 → mu = nu","l":200,"f":"Causalean/Stat/Minimax/MaximalCoupling.lean"},{"n":"Causalean.Stat.mem_partialMatchingsOfSize","k":"theorem","m":"Causalean.Stat.UStatistic.OrderM.PartialMatching","a":"Stat/UStatistic/OrderM","t":2,"d":"A partial matching belongs to the family with h pairs exactly when its matching size is h.","s":"∀ {r s h : ℕ} (M : Causalean.Stat.PartialMatching r s), M ∈ Causalean.Stat.partialMatchingsOfSize r s h ↔ M.size = h","l":148,"f":"Causalean/Stat/UStatistic/OrderM/PartialMatching.lean"},{"n":"Causalean.Stat.mergedProductKernel","k":"def","m":"Causalean.Stat.UStatistic.OrderM.MixedOrderCovariance","a":"Stat/UStatistic/OrderM","t":1,"d":"The merged product kernel evaluates matched left and right factors on the same observation and unmatched factors on separate observations.","s":"{X : Type u_2} → {r s : ℕ} → (Fin r → X → ℝ) → (Fin s → X → ℝ) → (M : Causalean.Stat.PartialMatching r s) → (M.MergedIndex → X) → ℝ","l":26,"f":"Causalean/Stat/UStatistic/OrderM/MixedOrderCovariance.lean"},{"n":"Causalean.Stat.mergedProductMoment","k":"def","m":"Causalean.Stat.UStatistic.OrderM.MixedOrderCovariance","a":"Stat/UStatistic/OrderM","t":1,"d":"The merged product moment is the expectation of a merged product kernel under independent copies indexed by its merged coordinates.","s":"{X : Type u_2} → [inst : MeasurableSpace X] → {r s : ℕ} → MeasureTheory.Measure X → (Fin r → X → ℝ) → (Fin s → X → ℝ) → Causalean.Stat.PartialMatching r s → ℝ","l":35,"f":"Causalean/Stat/UStatistic/OrderM/MixedOrderCovariance.lean"},{"n":"Causalean.Stat.mergedProductMoment_empty","k":"theorem","m":"Causalean.Stat.UStatistic.OrderM.MixedOrderCovariance","a":"Stat/UStatistic/OrderM","t":2,"d":"Under a probability law, for two coordinate-function families of orders r and s, if the first product kernel is measurable, the second product kernel is measurable, the first product kernel is integrable, and the second product kernel is integrable, the empty matching's merged moment factors into the two separate product-law means.","s":"∀ {X : Type u_2} [inst : MeasurableSpace X] {P : MeasureTheory.Measure X} [MeasureTheory.IsProbabilityMeasure P] {r s : ℕ} (f : Fin r → X → ℝ) (g : Fin s → X → ℝ), Measurable (Causalean.Stat.orderedProductKernel f) → Measurable (Causalean.Stat.orderedProductKernel g) → MeasureTheory.Integrable (Causalean.Stat.orderedProductKernel f) (MeasureTheory.Measure.pi fun x => P) → MeasureTheory.Integrable (Causalean.Stat.orderedProductKernel g) (MeasureTheory.Measure.pi fun x => P) → Causalean.Stat.mergedProductMoment P f g (Causalean.Stat.PartialMatching.empty r s) = Causalean.Stat.orderedProductMean P f * Causalean.Stat.orderedProductMean P g","l":539,"f":"Causalean/Stat/UStatistic/OrderM/MixedOrderCovariance.lean"},{"n":"Causalean.Stat.minDirDeriv","k":"def","m":"Causalean.Stat.Inference.HadamardDeriv","a":"Stat/Inference","t":1,"d":"The directional derivative of min : ℝ × ℝ → ℝ at (a, b).","s":"ℝ → ℝ → ℝ × ℝ → ℝ","l":161,"f":"Causalean/Stat/Inference/HadamardDeriv.lean"},{"n":"Causalean.Stat.minStat_aemeasurable","k":"theorem","m":"Causalean.Stat.Inference.FunctionalDelta","a":"Stat/Inference","t":2,"d":"The image statistic √n (min âₙ b̂ₙ − a) is AEMeasurable for every n, derived likewise via sqrt_mul_min_sub.","s":"∀ {Ω : Type u_1} [inst : MeasurableSpace Ω] {μ : MeasureTheory.Measure Ω} (an bn : ℕ → Ω → ℝ) (a : ℝ), (∀ (n : ℕ), AEMeasurable (fun ω => √↑n • ((an n ω, bn n ω) - (a, a))) μ) → ∀ (n : ℕ), AEMeasurable (fun ω => √↑n * (min (an n ω) (bn n ω) - a)) μ","l":153,"f":"Causalean/Stat/Inference/FunctionalDelta.lean"},{"n":"Causalean.Stat.minimaxValue","k":"def","m":"Causalean.Stat.Minimax.MinimaxValue","a":"Stat/Minimax","t":1,"d":"The minimax risk of a decision problem: the smallest worst-case risk that any admissible estimator achieves. Admissibility is expressed by the choice of the estimator index type, so instantiating it at a subtype restricts the infimum to the estimators satisfying that subtype's condition.","s":"{E : Sort u_1} → {Θ : Sort u_3} → (E → Θ → ℝ) → ℝ","l":53,"f":"Causalean/Stat/Minimax/MinimaxValue.lean"},{"n":"Causalean.Stat.minimaxValue_eq_sInf_range","k":"theorem","m":"Causalean.Stat.Minimax.MinimaxValue","a":"Stat/Minimax","t":2,"d":"The minimax value is the infimum of the set of worst-case risks the admissible estimators attain. Rewriting handle for arguments phrased on that set.","s":"∀ {E : Sort u_1} {Θ : Sort u_3} (risk : E → Θ → ℝ), Causalean.Stat.minimaxValue risk = sInf (Set.range (Causalean.Stat.worstCaseRisk risk))","l":251,"f":"Causalean/Stat/Minimax/MinimaxValue.lean"},{"n":"Causalean.Stat.minimaxValue_le_minimaxValue","k":"theorem","m":"Causalean.Stat.Minimax.MinimaxValue","a":"Stat/Minimax","t":2,"d":"Comparison of two minimax problems. If every admissible estimator of the second problem is matched by an estimator of the first whose worst-case risk is no larger, the first minimax value is at most the second. Both restricting the model class and enlarging the estimator family are instances; the boundedness hypothesis is on the first problem's worst-case risks, and the second problem must have at least one estimator.","s":"∀ {E : Sort u_1} {E' : Sort u_2} {Θ : Sort u_3} {Θ' : Sort u_4} [Nonempty E'] {risk : E → Θ → ℝ} {risk' : E' → Θ' → ℝ}, BddBelow (Set.range (Causalean.Stat.worstCaseRisk risk)) → (∀ (e' : E'), ∃ e, Causalean.Stat.worstCaseRisk risk e ≤ Causalean.Stat.worstCaseRisk risk' e') → Causalean.Stat.minimaxValue risk ≤ Causalean.Stat.minimaxValue risk'","l":193,"f":"Causalean/Stat/Minimax/MinimaxValue.lean"},{"n":"Causalean.Stat.minimaxValue_le_worstCaseRisk","k":"theorem","m":"Causalean.Stat.Minimax.MinimaxValue","a":"Stat/Minimax","t":2,"d":"Any admissible estimator is an upper bound. Exhibiting one estimator and bounding its worst-case risk bounds the minimax value, which is how achievability half of a minimax rate is certified. The hypothesis rules out a worst-case risk that decreases without bound across the estimator family.","s":"∀ {E : Sort u_1} {Θ : Sort u_3} {risk : E → Θ → ℝ}, BddBelow (Set.range (Causalean.Stat.worstCaseRisk risk)) → ∀ (e : E), Causalean.Stat.minimaxValue risk ≤ Causalean.Stat.worstCaseRisk risk e","l":125,"f":"Causalean/Stat/Minimax/MinimaxValue.lean"},{"n":"Causalean.Stat.minimaxValue_le_worstCaseRisk_of_nonneg","k":"theorem","m":"Causalean.Stat.Minimax.MinimaxValue","a":"Stat/Minimax","t":2,"d":"For a nonnegative risk, any admissible estimator is an upper bound on the minimax value with no further side condition, since zero already bounds the worst-case risks below.","s":"∀ {E : Sort u_1} {Θ : Sort u_3} {risk : E → Θ → ℝ}, (∀ (e : E) (θ : Θ), 0 ≤ risk e θ) → ∀ (e : E), Causalean.Stat.minimaxValue risk ≤ Causalean.Stat.worstCaseRisk risk e","l":134,"f":"Causalean/Stat/Minimax/MinimaxValue.lean"},{"n":"Causalean.Stat.minimaxValue_mono_class","k":"theorem","m":"Causalean.Stat.Minimax.MinimaxValue","a":"Stat/Minimax","t":2,"d":"Restricting the model class cannot raise the minimax value. If each model of the smaller class sits, through the map, inside the larger class with no larger risk, then the minimax value over the smaller class is at most the minimax value over the larger one — the same estimators face a weaker adversary. This is the standard step that transfers a published converse proved on a convenient subclass to the full class.","s":"∀ {E : Sort u_1} {Θ : Sort u_3} {Θ' : Sort u_4} [Nonempty E] [Nonempty Θ] {risk : E → Θ → ℝ} {risk' : E → Θ' → ℝ} (φ : Θ → Θ'), BddBelow (Set.range (Causalean.Stat.worstCaseRisk risk)) → (∀ (e : E), BddAbove (Set.range (risk' e))) → (∀ (e : E) (θ : Θ), risk e θ ≤ risk' e (φ θ)) → Causalean.Stat.minimaxValue risk ≤ Causalean.Stat.minimaxValue risk'","l":206,"f":"Causalean/Stat/Minimax/MinimaxValue.lean"},{"n":"Causalean.Stat.minimaxValue_mono_class_of_nonneg","k":"theorem","m":"Causalean.Stat.Minimax.MinimaxValue","a":"Stat/Minimax","t":1,"d":"For a map φ embedding the parameter class of the first problem into that of the second, if the first problem's risk is nonnegative, the second problem's risk is nonnegative, each estimator's risk is bounded above across the second parameter class, and the first risk at any model is dominated by the second risk at that model's image under φ, then the minimax value of the first problem is at most that of the second.","s":"∀ {E : Sort u_1} {Θ : Sort u_3} {Θ' : Sort u_4} [Nonempty E] {risk : E → Θ → ℝ} {risk' : E → Θ' → ℝ} (φ : Θ → Θ'), (∀ (e : E) (θ : Θ), 0 ≤ risk e θ) → (∀ (e : E) (θ' : Θ'), 0 ≤ risk' e θ') → (∀ (e : E), BddAbove (Set.range (risk' e))) → (∀ (e : E) (θ : Θ), risk e θ ≤ risk' e (φ θ)) → Causalean.Stat.minimaxValue risk ≤ Causalean.Stat.minimaxValue risk'","l":220,"f":"Causalean/Stat/Minimax/MinimaxValue.lean"},{"n":"Causalean.Stat.minimaxValue_nonneg","k":"theorem","m":"Causalean.Stat.Minimax.MinimaxValue","a":"Stat/Minimax","t":2,"d":"The minimax value of a nonnegative-risk problem is nonnegative, with no boundedness or nonemptiness caveat. Squared-error, absolute-error and regret losses all qualify.","s":"∀ {E : Sort u_1} {Θ : Sort u_3} {risk : E → Θ → ℝ}, (∀ (e : E) (θ : Θ), 0 ≤ risk e θ) → 0 ≤ Causalean.Stat.minimaxValue risk","l":94,"f":"Causalean/Stat/Minimax/MinimaxValue.lean"},{"n":"Causalean.Stat.minimaxValue_of_isEmpty_estimators","k":"theorem","m":"Causalean.Stat.Minimax.MinimaxValue","a":"Stat/Minimax","t":2,"d":"When no estimator is admissible the minimax value is zero, so no positive lower bound on the minimax value can hold vacuously.","s":"∀ {E : Sort u_1} {Θ : Sort u_3} [IsEmpty E] (risk : E → Θ → ℝ), Causalean.Stat.minimaxValue risk = 0","l":70,"f":"Causalean/Stat/Minimax/MinimaxValue.lean"},{"n":"Causalean.Stat.minimaxValue_subtype_eq_sInf_setOf","k":"theorem","m":"Causalean.Stat.Minimax.MinimaxValue","a":"Stat/Minimax","t":2,"d":"The minimax value over an admissible-estimator condition and a model-class condition, written with set comprehensions instead of subtypes: the infimum of the set of worst-case risks attained by admissible estimators, each of which is the supremum of the set of risks attained on the class. Runs that spell their minimax risk this way rewrite with this lemma and then use the rest of this file.","s":"∀ {A : Type u_5} {B : Type u_6} (Adm : A → Prop) (Cls : B → Prop) (risk : A → B → ℝ), (Causalean.Stat.minimaxValue fun e θ => risk ↑e ↑θ) = sInf {r | ∃ e, Adm e ∧ r = sSup {q | ∃ θ, Cls θ ∧ q = risk e θ}}","l":292,"f":"Causalean/Stat/Minimax/MinimaxValue.lean"},{"n":"Causalean.Stat.minimaxValue_subtype_eq_sInf_worstCaseRisk","k":"theorem","m":"Causalean.Stat.Minimax.MinimaxValue","a":"Stat/Minimax","t":2,"d":"The minimax value over the estimators satisfying an admissibility condition, written as the infimum of the set of worst-case risks the admissible estimators attain. This is the bridge for developments that state admissibility by a predicate rather than by a subtype while leaving the inner supremum as it stands.","s":"∀ {Θ : Sort u_3} {A : Type u_5} {Adm : A → Prop} (risk : A → Θ → ℝ), (Causalean.Stat.minimaxValue fun e => risk ↑e) = sInf {r | ∃ e, Adm e ∧ r = Causalean.Stat.worstCaseRisk risk e}","l":259,"f":"Causalean/Stat/Minimax/MinimaxValue.lean"},{"n":"Causalean.Stat.mixture","k":"def","m":"Causalean.Stat.Minimax.Mixture","a":"Stat/Minimax","t":1,"d":"Finite mixture of measures with weights w : ι → ℝ≥0∞.","s":"{Ω : Type u_1} → {mΩ : MeasurableSpace Ω} → {ι : Type u_2} → [Fintype ι] → (ι → ENNReal) → (ι → MeasureTheory.Measure Ω) → MeasureTheory.Measure Ω","l":32,"f":"Causalean/Stat/Minimax/Mixture.lean"},{"n":"Causalean.Stat.mixtureReal_le","k":"theorem","m":"Causalean.Stat.Minimax.Mixture","a":"Stat/Minimax","t":1,"d":"Domination. If the mixture weights w sum to 1, each component measure P i is a probability measure, and every component assigns .real-mass at most B to the set A, then the mixture measure also assigns .real-mass at most B to A.","s":"∀ {Ω : Type u_1} {mΩ : MeasurableSpace Ω} {ι : Type u_2} [inst : Fintype ι] (w : ι → ENNReal), ∑ i, w i = 1 → ∀ (P : ι → MeasureTheory.Measure Ω) [∀ (i : ι), MeasureTheory.IsProbabilityMeasure (P i)] (A : Set Ω) (B : ℝ), (∀ (i : ι), (P i).real A ≤ B) → (Causalean.Stat.mixture w P).real A ≤ B","l":51,"f":"Causalean/Stat/Minimax/Mixture.lean"},{"n":"Causalean.Stat.mixture_apply","k":"theorem","m":"Causalean.Stat.Minimax.Mixture","a":"Stat/Minimax","t":2,"d":"Evaluation: the mixture's mass on a set is the weighted sum of the parts' masses.","s":"∀ {Ω : Type u_1} {mΩ : MeasurableSpace Ω} {ι : Type u_2} [inst : Fintype ι] (w : ι → ENNReal) (P : ι → MeasureTheory.Measure Ω) (A : Set Ω), (Causalean.Stat.mixture w P) A = ∑ i, w i * (P i) A","l":36,"f":"Causalean/Stat/Minimax/Mixture.lean"},{"n":"Causalean.Stat.mixture_isProbabilityMeasure","k":"theorem","m":"Causalean.Stat.Minimax.Mixture","a":"Stat/Minimax","t":2,"d":"If the weights sum to 1 and each part is a probability measure, the mixture is one.","s":"∀ {Ω : Type u_1} {mΩ : MeasurableSpace Ω} {ι : Type u_2} [inst : Fintype ι] (w : ι → ENNReal), ∑ i, w i = 1 → ∀ (P : ι → MeasureTheory.Measure Ω) [∀ (i : ι), MeasureTheory.IsProbabilityMeasure (P i)], MeasureTheory.IsProbabilityMeasure (Causalean.Stat.mixture w P)","l":42,"f":"Causalean/Stat/Minimax/Mixture.lean"},{"n":"Causalean.Stat.monotoneOn_quantile","k":"theorem","m":"Causalean.Stat.Coupling.ProductLossMonotoneCoupling.PIT","a":"Stat/Coupling/ProductLossMonotoneCoupling","t":2,"d":"The quantile function is monotone on the open unit interval (0,1) (immediate from quantile_mono, whose hypotheses 0 < τ, τ' < 1 hold throughout the interior).","s":"∀ (μ : MeasureTheory.Measure ℝ), MonotoneOn (Causalean.Stat.quantile μ) (Set.Ioo 0 1)","l":48,"f":"Causalean/Stat/Coupling/ProductLossMonotoneCoupling/PIT.lean"},{"n":"Causalean.Stat.mse_integrable_of_estimator_bound","k":"theorem","m":"Causalean.Stat.Minimax.MinimaxRisk","a":"Stat/Minimax","t":2,"d":"Squared-loss integrability for a truncated estimator. If a measurable estimator T takes values in the bounded interval [-M, M] (with M ≥ 0), then under any finite measure Q its squared loss (T − θ)² against an arbitrary target θ is integrable, because it is bounded by the constant (M + |θ|)². This is the routine integrability bookkeeping needed before the worst-case squared risk of a truncated estimator can be compared in a two-point lower bound.","s":"∀ {S : Type u_2} [inst : MeasurableSpace S] (Q : MeasureTheory.Measure S) [MeasureTheory.IsFiniteMeasure Q] (T : S → ℝ), Measurable T → ∀ {M theta : ℝ}, 0 ≤ M → (∀ (s : S), T s ∈ Set.Icc (-M) M) → MeasureTheory.Integrable (fun s => (T s - theta) ^ 2) Q","l":116,"f":"Causalean/Stat/Minimax/MinimaxRisk.lean"},{"n":"Causalean.Stat.mul_exp_neg_half_log","k":"theorem","m":"Causalean.Stat.Minimax.BretagnolleHuber","a":"Stat/Minimax","t":2,"d":"For y ≥ 0, y * exp(-(1/2)·log y) = √y.","s":"∀ {y : ℝ}, 0 ≤ y → y * Real.exp (-(1 / 2) * Real.log y) = √y","l":77,"f":"Causalean/Stat/Minimax/BretagnolleHuber.lean"},{"n":"Causalean.Stat.multiplierBootstrap_law","k":"theorem","m":"Causalean.Stat.Bootstrap.GaussianMultiplier","a":"Stat/Bootstrap","t":1,"d":"Exact law of the √n-scaled multiplier-bootstrap mean. Fix a sample size n, data x : Fin n → ℝ, and a multiplier family ξ. If ξ is independent across coordinates, each ξ i is measurable, and each ξ i has the standard Gaussian law, then the scaled multiplier-bootstrap statistic $n^{-1/2}\\sum_i (x_i-\\bar x)\\,\\xi_i$ has exactly the centered Gaussian law with variance equal to the sample variance $n^{-1}\\sum_i(x_i-\\bar x)^2$.","s":"∀ {Ω : Type u_1} [inst : MeasurableSpace Ω] {μ : MeasureTheory.Measure Ω} [MeasureTheory.IsProbabilityMeasure μ] {n : ℕ} (ξ : Fin n → Ω → ℝ), ProbabilityTheory.iIndepFun ξ μ → (∀ (i : Fin n), Measurable (ξ i)) → (∀ (i : Fin n), MeasureTheory.Measure.map (ξ i) μ = ProbabilityTheory.gaussianReal 0 1) → ∀ (x : Fin n → ℝ), MeasureTheory.Measure.map (fun ω => (√↑n)⁻¹ * ∑ i, (x i - (↑n)⁻¹ * ∑ j, x j) * ξ i ω) μ = ProbabilityTheory.gaussianReal 0 ⟨(↑n)⁻¹ * ∑ i, (x i - (↑n)⁻¹ * ∑ j, x j) ^ 2, mul_nonneg (inv_nonneg_of_nonneg (Nat.cast_nonneg' n)) (Finset.sum_nonneg fun i x_1 => Even.pow_nonneg (even_two_mul 1) (x i - (↑n)⁻¹ * ∑ j, x j))⟩","l":128,"f":"Causalean/Stat/Bootstrap/GaussianMultiplier.lean"},{"n":"Causalean.Stat.noAtoms_chiSqDist","k":"theorem","m":"Causalean.Stat.CLT.ChiSquared","a":"Stat/CLT","t":2,"d":"χ²_d has no atoms for d ≥ 1.","s":"∀ {d : ℕ}, 1 ≤ d → MeasureTheory.NullSingletonClass (Causalean.Stat.chiSqDist d)","l":164,"f":"Causalean/Stat/CLT/ChiSquared.lean"},{"n":"Causalean.Stat.noAtoms_pi_normSq","k":"theorem","m":"Causalean.Stat.CLT.ChiSquared","a":"Stat/CLT","t":2,"d":"The law of the sum of n ≥ 1 squared independent standard normals has no atoms. The squared first coordinate already has an atomless law (its preimage under squaring is finite, and gaussianReal is atomless), and adding the independent remainder preserves atomlessness via Fubini.","s":"∀ {n : ℕ}, 1 ≤ n → MeasureTheory.NullSingletonClass (MeasureTheory.Measure.map (fun w => ∑ i, w i ^ 2) (MeasureTheory.Measure.pi fun x => ProbabilityTheory.gaussianReal 0 1))","l":107,"f":"Causalean/Stat/CLT/ChiSquared.lean"},{"n":"Causalean.Stat.nonempty_quantileSet","k":"theorem","m":"Causalean.Stat.Quantile.Quantile","a":"Stat/Quantile","t":2,"d":"For τ < 1, the super-level set is nonempty: since cdf μ → 1 at +∞, some point has cdf above τ.","s":"∀ {μ : MeasureTheory.Measure ℝ} {τ : ℝ}, τ < 1 → (Causalean.Stat.quantileSet μ τ).Nonempty","l":68,"f":"Causalean/Stat/Quantile/Quantile.lean"},{"n":"Causalean.Stat.norm_eucl_le_sum_abs","k":"theorem","m":"Causalean.Stat.Quantile.SampleQuantileJoint","a":"Stat/Quantile","t":2,"d":"The Euclidean norm of a packed vector is bounded by the ℓ¹ norm of its coordinates: ‖eucl v‖ ≤ ∑ j, |v j|.","s":"∀ {k : ℕ} (v : Fin k → ℝ), ‖Causalean.Stat.eucl v‖ ≤ ∑ j, |v j|","l":145,"f":"Causalean/Stat/Quantile/SampleQuantileJoint.lean"},{"n":"Causalean.Stat.norm_sq_eucl","k":"theorem","m":"Causalean.Stat.Quantile.SampleQuantileJoint","a":"Stat/Quantile","t":2,"d":"Euclidean norm of a packed vector, squared, is the coordinatewise sum of squares.","s":"∀ {k : ℕ} (v : Fin k → ℝ), ‖Causalean.Stat.eucl v‖ ^ 2 = ∑ j, v j ^ 2","l":137,"f":"Causalean/Stat/Quantile/SampleQuantileJoint.lean"},{"n":"Causalean.Stat.normalizedFiniteKernelStatistic","k":"def","m":"Causalean.Stat.UStatistic.OrderM.Basic","a":"Stat/UStatistic/OrderM","t":1,"d":"The normalized finite-kernel statistic averages a kernel over every injective assignment of its finite coordinate family to sample positions.","s":"{Ω : Type u_1} → {X : Type u_2} → [inst : MeasurableSpace Ω] → [inst_1 : MeasurableSpace X] → {μ : MeasureTheory.Measure Ω} → {P : MeasureTheory.Measure X} → Causalean.Stat.IIDSample Ω X μ P → {ι : Type u_3} → [Fintype ι] → ((ι → X) → ℝ) → ℕ → Ω → ℝ","l":450,"f":"Causalean/Stat/UStatistic/OrderM/Basic.lean"},{"n":"Causalean.Stat.normalizedFiniteKernelStatistic_fin_eq_uStatisticOrder","k":"theorem","m":"Causalean.Stat.UStatistic.OrderM.Basic","a":"Stat/UStatistic/OrderM","t":2,"d":"For an i.i.d. sample, an order-r kernel, and sample size n, the normalized finite-kernel statistic agrees with the existing fixed-order U-statistic.","s":"∀ {Ω : Type u_1} {X : Type u_2} [inst : MeasurableSpace Ω] [inst_1 : MeasurableSpace X] {μ : MeasureTheory.Measure Ω} {P : MeasureTheory.Measure X} (S : Causalean.Stat.IIDSample Ω X μ P) {r : ℕ} (k : (Fin r → X) → ℝ) (n : ℕ), Causalean.Stat.normalizedFiniteKernelStatistic S k n = Causalean.Stat.uStatisticOrder S k n","l":469,"f":"Causalean/Stat/UStatistic/OrderM/Basic.lean"},{"n":"Causalean.Stat.normalizedOrderedProductStatistic","k":"def","m":"Causalean.Stat.UStatistic.OrderM.Basic","a":"Stat/UStatistic/OrderM","t":1,"d":"The normalized ordered-product statistic averages coordinatewise products over injective ordered tuples and divides by the corresponding falling factorial.","s":"{Ω : Type u_1} → {X : Type u_2} → [inst : MeasurableSpace Ω] → [inst_1 : MeasurableSpace X] → {μ : MeasureTheory.Measure Ω} → {P : MeasureTheory.Measure X} → Causalean.Stat.IIDSample Ω X μ P → {r : ℕ} → (Fin r → X → ℝ) → ℕ → Ω → ℝ","l":462,"f":"Causalean/Stat/UStatistic/OrderM/Basic.lean"},{"n":"Causalean.Stat.normalizedOrderedProductStatistic_eq_uStatisticOrder","k":"theorem","m":"Causalean.Stat.UStatistic.OrderM.Basic","a":"Stat/UStatistic/OrderM","t":2,"d":"For an i.i.d. sample, a family of order-r coordinate functions, and sample size n, the normalized ordered-product statistic is the existing fixed-order U-statistic applied to their product kernel.","s":"∀ {Ω : Type u_1} {X : Type u_2} [inst : MeasurableSpace Ω] [inst_1 : MeasurableSpace X] {μ : MeasureTheory.Measure Ω} {P : MeasureTheory.Measure X} (S : Causalean.Stat.IIDSample Ω X μ P) {r : ℕ} (f : Fin r → X → ℝ) (n : ℕ), Causalean.Stat.normalizedOrderedProductStatistic S f n = Causalean.Stat.uStatisticOrder S (Causalean.Stat.orderedProductKernel f) n","l":487,"f":"Causalean/Stat/UStatistic/OrderM/Basic.lean"},{"n":"Causalean.Stat.normalizedOrderedProductStatistic_mul_expansion","k":"theorem","m":"Causalean.Stat.UStatistic.OrderM.MixedOrderCovariance","a":"Stat/UStatistic/OrderM","t":2,"d":"For an i.i.d. sample, statistic orders and a sample size, if the first order does not exceed the sample size and the second order does not exceed the sample size, then the pointwise product of the statistics built from two coordinate-function families equals the sum of normalized merged-kernel statistics over all partial matchings, grouped by overlap size.","s":"∀ {Ω : Type u_1} {X : Type u_2} [inst : MeasurableSpace Ω] [inst_1 : MeasurableSpace X] {μ : MeasureTheory.Measure Ω} {P : MeasureTheory.Measure X} (S : Causalean.Stat.IIDSample Ω X μ P) {r s n : ℕ}, r ≤ n → s ≤ n → ∀ (f : Fin r → X → ℝ) (g : Fin s → X → ℝ), (fun ω => Causalean.Stat.normalizedOrderedProductStatistic S f n ω * Causalean.Stat.normalizedOrderedProductStatistic S g n ω) = fun ω => ∑ h ∈ Finset.range (min r s + 1), ∑ M ∈ Causalean.Stat.partialMatchingsOfSize r s h, Causalean.Stat.matchingNormalization n M * Causalean.Stat.normalizedFiniteKernelStatistic S (Causalean.Stat.mergedProductKernel f g M) n ω","l":422,"f":"Causalean/Stat/UStatistic/OrderM/MixedOrderCovariance.lean"},{"n":"Causalean.Stat.occupancyDesignVarianceFactor","k":"def","m":"Causalean.Stat.Sample.OccupancyWeightedMean.Basic","a":"Stat/Sample/OccupancyWeightedMean","t":1,"d":"The design variance factor is the usable-group sum of squared occupancy weights times the two reciprocal arm counts, totalized to zero when no group is usable.","s":"{Omega : Type u_1} → {kappa : Type u_2} → [Fintype kappa] → [DecidableEq kappa] → {n : ℕ} → (Omega → kappa) → (Omega → Bool) → (Fin n → Omega) → ℝ","l":126,"f":"Causalean/Stat/Sample/OccupancyWeightedMean/Basic.lean"},{"n":"Causalean.Stat.occupancyWeightedResidual","k":"def","m":"Causalean.Stat.Sample.OccupancyWeightedMean.Basic","a":"Stat/Sample/OccupancyWeightedMean","t":1,"d":"The occupancy-weighted residual statistic averages, over usable groups, each group occupancy times the treated-minus-control residual mean. It is zero when no group is usable. The law argument fixes the intended public API but does not alter this sample statistic.","s":"{Omega : Type u_1} → {kappa : Type u_2} → [inst : MeasurableSpace Omega] → [Fintype kappa] → [DecidableEq kappa] → {n : ℕ} → MeasureTheory.Measure Omega → (Omega → kappa) → (Omega → Bool) → (Omega → ℝ) → (Bool → kappa → ℝ) → (Fin n → Omega) → ℝ","l":101,"f":"Causalean/Stat/Sample/OccupancyWeightedMean/Basic.lean"},{"n":"Causalean.Stat.occupancyWeightedResidual_memLp_two","k":"theorem","m":"Causalean.Stat.Sample.OccupancyWeightedMean.Variance","a":"Stat/Sample/OccupancyWeightedMean","t":2,"d":"If group labels, arm labels, and outcomes are measurable and every arm/group-supported residual has a finite second moment, the zero-safe occupancy-weighted residual has a finite second moment under every finite independent product sample.","s":"∀ {Omega : Type u_1} {kappa : Type u_2} [inst : MeasurableSpace Omega] [inst_1 : Fintype kappa] [inst_2 : DecidableEq kappa] [inst_3 : MeasurableSpace kappa] [MeasurableSingletonClass kappa] {n : ℕ} (mu : MeasureTheory.Measure Omega) [MeasureTheory.IsProbabilityMeasure mu] (group : Omega → kappa) (arm : Omega → Bool) (Y : Omega → ℝ) (center : Bool → kappa → ℝ), Measurable group → Measurable arm → Measurable Y → (∀ (a : Bool) (k : kappa), MeasureTheory.MemLp (Causalean.Stat.supportedArmGroupResidual group arm Y center a k) 2 mu) → MeasureTheory.MemLp (Causalean.Stat.occupancyWeightedResidual mu group arm Y center) 2 (MeasureTheory.Measure.pi fun x => mu)","l":156,"f":"Causalean/Stat/Sample/OccupancyWeightedMean/Variance.lean"},{"n":"Causalean.Stat.oneShot_iid","k":"theorem","m":"Causalean.Stat.SampleSplit.FoldBEmpiricalProcess","a":"Stat/SampleSplit","t":2,"d":"OneShot evaluation-fold i.i.d. product law. Public because the orthogonal-learning modulus chain consumes it as the joint-law bridge between fold B and Measure.pi (see Estimation/OrthogonalLearning/LocalEmpProcess/Rademacher.lean).","s":"∀ {Ω : Type u_1} {X : Type u_2} [inst : MeasurableSpace Ω] [inst_1 : MeasurableSpace X] {μ : MeasureTheory.Measure Ω} {P : MeasureTheory.Measure X} [MeasureTheory.IsProbabilityMeasure μ] (S : Causalean.Stat.IIDSample Ω X μ P) (split : Causalean.Stat.OneShotSplit S) (n : ℕ), MeasureTheory.Measure.map (fun ω i => S.Z (↑i) ω) μ = MeasureTheory.Measure.pi fun x => P","l":319,"f":"Causalean/Stat/SampleSplit/FoldBEmpiricalProcess.lean"},{"n":"Causalean.Stat.one_add_chiSqDiv_pi_iid","k":"theorem","m":"Causalean.Stat.Minimax.ChiSquared","a":"Stat/Minimax","t":1,"d":"n-fold i.i.d. tensorization of the χ²-divergence on a finite sample space. For probability measures μ, ν on a finite space Ω with μ absolutely continuous with respect to ν, the χ²-divergence of the n-fold i.i.d. product laws satisfies 1 + χ²(μ^⊗n ‖ ν^⊗n) = (1 + χ²(μ‖ν))^n.","s":"∀ {Ω : Type u_2} [inst : MeasurableSpace Ω] [Fintype Ω] [MeasurableSingletonClass Ω] (μ ν : MeasureTheory.Measure Ω) [MeasureTheory.IsProbabilityMeasure μ] [MeasureTheory.IsProbabilityMeasure ν], μ.AbsolutelyContinuous ν → ∀ (n : ℕ), 1 + Causalean.Stat.chiSqDiv (MeasureTheory.Measure.pi fun x => μ) (MeasureTheory.Measure.pi fun x => ν) = (1 + Causalean.Stat.chiSqDiv μ ν) ^ n","l":290,"f":"Causalean/Stat/Minimax/ChiSquared.lean"},{"n":"Causalean.Stat.one_add_chiSqDiv_pi_iid_general","k":"theorem","m":"Causalean.Stat.Minimax.ChiSquared","a":"Stat/Minimax","t":1,"d":"n-fold i.i.d. tensorization of the χ²-divergence on a GENERAL measurable space. For probability measures μ, ν on any measurable space S such that μ is absolutely continuous with respect to ν and the single-sample squared density deviation (dμ/dν − 1)² is ν-integrable, the χ²-divergence of the n-fold i.i.d. product tensorizes multiplicatively: 1 + χ²(μ^⊗n ‖ ν^⊗n) = (1 + χ²(μ‖ν))^n.","s":"∀ {S : Type u_2} [inst : MeasurableSpace S] (μ ν : MeasureTheory.Measure S) [MeasureTheory.IsProbabilityMeasure μ] [MeasureTheory.IsProbabilityMeasure ν], μ.AbsolutelyContinuous ν → MeasureTheory.Integrable (fun x => ((μ.rnDeriv ν x).toReal - 1) ^ 2) ν → ∀ (n : ℕ), 1 + Causalean.Stat.chiSqDiv (MeasureTheory.Measure.pi fun x => μ) (MeasureTheory.Measure.pi fun x => ν) = (1 + Causalean.Stat.chiSqDiv μ ν) ^ n","l":407,"f":"Causalean/Stat/Minimax/ChiSquared.lean"},{"n":"Causalean.Stat.one_le_secondMoment_of_mean_one","k":"theorem","m":"Causalean.Stat.Sample.EffectiveSampleSize","a":"Stat/Sample","t":2,"d":"A mean-one square-integrable weight has second moment at least one, so its Kish design effect cannot improve on an equally weighted sample.","s":"∀ {𝒳 : Type u_1} [inst : MeasurableSpace 𝒳] (μ : MeasureTheory.Measure 𝒳) [MeasureTheory.IsProbabilityMeasure μ] (w : 𝒳 → ℝ), MeasureTheory.MemLp w 2 μ → ∫ (x : 𝒳), w x ∂μ = 1 → 1 ≤ ∫ (x : 𝒳), w x ^ 2 ∂μ","l":31,"f":"Causalean/Stat/Sample/EffectiveSampleSize.lean"},{"n":"Causalean.Stat.one_sub_prod_le_sum","k":"theorem","m":"Causalean.Stat.Minimax.HellingerAffinity","a":"Stat/Minimax","t":2,"d":"For finitely many numbers in the unit interval, the amount by which their product falls short of one is at most the total shortfall of the individual factors. Applied to coordinate affinities this is the union-bound step that turns a tensorized affinity into a sum of coordinate defects.","s":"∀ {ι : Type u_1} [inst : Fintype ι] (a : ι → ℝ), (∀ (i : ι), 0 ≤ a i) → (∀ (i : ι), a i ≤ 1) → 1 - ∏ i, a i ≤ ∑ i, (1 - a i)","l":292,"f":"Causalean/Stat/Minimax/HellingerAffinity.lean"},{"n":"Causalean.Stat.one_sub_tvDist_le_error_sum","k":"theorem","m":"Causalean.Stat.Minimax.LeCam","a":"Stat/Minimax","t":1,"d":"Le Cam two-point bound (summed form). For a measurable estimator est, if the parameter values θ₀, θ₁ are 2s-separated, then the two error probabilities sum to at least 1 − tvDist P₀ P₁.","s":"∀ {Ω : Type u_1} {mΩ : MeasurableSpace Ω} {P₀ P₁ : MeasureTheory.Measure Ω} [MeasureTheory.IsProbabilityMeasure P₀] [MeasureTheory.IsProbabilityMeasure P₁] {Θ : Type u_2} [inst : PseudoMetricSpace Θ] [inst_1 : MeasurableSpace Θ] [OpensMeasurableSpace Θ] {est : Ω → Θ}, Measurable est → ∀ {θ₀ θ₁ : Θ} {s : ℝ}, 2 * s ≤ dist θ₀ θ₁ → 1 - Causalean.Stat.tvDist P₀ P₁ ≤ P₀.real {ω | s ≤ dist (est ω) θ₀} + P₁.real {ω | s ≤ dist (est ω) θ₁}","l":47,"f":"Causalean/Stat/Minimax/LeCam.lean"},{"n":"Causalean.Stat.one_sub_tvDist_le_test","k":"theorem","m":"Causalean.Stat.Minimax.TotalVariation","a":"Stat/Minimax","t":1,"d":"Le Cam testing bound. For probability measures μ and ν on the same space, any measurable rejection region A yields a total testing error — the probability of A under μ plus the probability of the complement of A under ν — that is at least 1 − tvDist(μ,ν).","s":"∀ {Ω : Type u_1} {mΩ : MeasurableSpace Ω} {μ ν : MeasureTheory.Measure Ω} [MeasureTheory.IsProbabilityMeasure μ] [MeasureTheory.IsProbabilityMeasure ν] {A : Set Ω}, MeasurableSet A → 1 - Causalean.Stat.tvDist μ ν ≤ μ.real A + ν.real Aᶜ","l":101,"f":"Causalean/Stat/Minimax/TotalVariation.lean"},{"n":"Causalean.Stat.orderDegenerateNegligible_of_residual","k":"theorem","m":"Causalean.Stat.UStatistic.OrderM.RemainderNegligible","a":"Stat/UStatistic/OrderM","t":1,"d":"The higher-order remainder of a fixed-order U-statistic is negligible. For an i.i.d. sample S and order-m kernel h, write g for the higher-order Hájek residual of h. If g is measurable and square-integrable under the m-fold product law, and if for every coordinate integrating h over the remaining m − 1 coordinates yields an integrable function of that coordinate with the same population mean uMeanOrder h P in every coordinate and h remains integrable in the remaining coordinates for every fixed value of that coordinate, then the √n-rescaled higher-order residual U-statistic √n · Gₙ converges to zero in probability, i.e. it is o_p(1). This discharges the OrderDegenerateNegligible hypothesis consumed by the order-m CLT uStatisticOrder_clt (Causalean.Stat.UStatistic.OrderM.CLT).","s":"∀ {Ω : Type u_3} {X : Type u_4} [inst : MeasurableSpace Ω] [inst_1 : MeasurableSpace X] {μ : MeasureTheory.Measure Ω} {P : MeasureTheory.Measure X} (S : Causalean.Stat.IIDSample Ω X μ P) {m : ℕ} [inst_2 : NeZero m] (h : (Fin m → X) → ℝ), Measurable (Causalean.Stat.uDegenOrder h P) → MeasureTheory.Integrable (fun z => Causalean.Stat.uDegenOrder h P z ^ 2) (MeasureTheory.Measure.pi fun x => P) → (∀ (j : Fin m), MeasureTheory.Integrable (fun x => ∫ (tail : { k // k ≠ j } → X), h (Causalean.Stat.insertCoord j x tail) ∂MeasureTheory.Measure.pi fun x => P) P) → (∀ (j : Fin m), ∫ (x : X), ∫ (tail : { k // k ≠ j } → X), h (Causalean.Stat.insertCoord j x tail) ∂MeasureTheory.Measure.pi fun x => P ∂P = Causalean.Stat.uMeanOrder h P) → (∀ (j : Fin m) (x : X), MeasureTheory.Integrable (fun tail => h (Causalean.Stat.insertCoord j x tail)) (MeasureTheory.Measure.pi fun x => P)) → Causalean.Stat.OrderDegenerateNegligible S h","l":130,"f":"Causalean/Stat/UStatistic/OrderM/RemainderNegligible.lean"},{"n":"Causalean.Stat.orderedProductKernel","k":"def","m":"Causalean.Stat.UStatistic.OrderM.Basic","a":"Stat/UStatistic/OrderM","t":1,"d":"The ordered-product kernel multiplies one real-valued coordinate function for every position in an ordered tuple.","s":"{X : Type u_2} → {r : ℕ} → (Fin r → X → ℝ) → (Fin r → X) → ℝ","l":457,"f":"Causalean/Stat/UStatistic/OrderM/Basic.lean"},{"n":"Causalean.Stat.orderedProductMean","k":"def","m":"Causalean.Stat.UStatistic.OrderM.MixedOrderCovariance","a":"Stat/UStatistic/OrderM","t":1,"d":"The ordered-product mean is the expectation of a coordinatewise product under independent draws from the population law.","s":"{X : Type u_2} → [inst : MeasurableSpace X] → {r : ℕ} → MeasureTheory.Measure X → (Fin r → X → ℝ) → ℝ","l":528,"f":"Causalean/Stat/UStatistic/OrderM/MixedOrderCovariance.lean"},{"n":"Causalean.Stat.overlap_ge_exp_neg_klBudget","k":"theorem","m":"Causalean.Stat.Minimax.CoordinatewiseOverlap","a":"Stat/Minimax","t":2,"d":"A nonnegative finite KL budget yields the corresponding Bretagnolle--Huber lower bound on testing overlap.","s":"∀ {Ω : Type u_1} [inst : MeasurableSpace Ω] (μ ν : MeasureTheory.Measure Ω) [MeasureTheory.IsProbabilityMeasure μ] [MeasureTheory.IsProbabilityMeasure ν] {B : ℝ}, 0 ≤ B → InformationTheory.klDiv μ ν ≤ ENNReal.ofReal B → 1 / 2 * Real.exp (-B) ≤ 1 - Causalean.Stat.tvDist μ ν","l":235,"f":"Causalean/Stat/Minimax/CoordinatewiseOverlap.lean"},{"n":"Causalean.Stat.pairKernel","k":"def","m":"Causalean.Stat.UStatistic.OrderM.Basic","a":"Stat/UStatistic/OrderM","t":1,"d":"Encode a binary kernel as a kernel on Fin 2 → X.","s":"{X : Type u_2} → (X → X → ℝ) → (Fin 2 → X) → ℝ","l":320,"f":"Causalean/Stat/UStatistic/OrderM/Basic.lean"},{"n":"Causalean.Stat.partialMatchingEquivSigma","k":"def","m":"Causalean.Stat.UStatistic.OrderM.PartialMatching","a":"Stat/UStatistic/OrderM","t":1,"d":"Partial matchings are represented exactly by a selected subset on each side together with a bijection between those subsets.","s":"(r s : ℕ) → Causalean.Stat.PartialMatching r s ≃ (left : Finset (Fin r)) × (right : Finset (Fin s)) × (↑↑left ≃ ↑↑right)","l":109,"f":"Causalean/Stat/UStatistic/OrderM/PartialMatching.lean"},{"n":"Causalean.Stat.partialMatchingFintype","k":"instance","m":"Causalean.Stat.UStatistic.OrderM.PartialMatching","a":"Stat/UStatistic/OrderM","t":2,"d":"The collection of partial matchings between two finite coordinate sets is finite.","s":"(r s : ℕ) → Fintype (Causalean.Stat.PartialMatching r s)","l":137,"f":"Causalean/Stat/UStatistic/OrderM/PartialMatching.lean"},{"n":"Causalean.Stat.partialMatchingsOfSize","k":"def","m":"Causalean.Stat.UStatistic.OrderM.PartialMatching","a":"Stat/UStatistic/OrderM","t":1,"d":"The fixed-size matching family consists of all partial matchings with exactly the prescribed number of pairs.","s":"(r s : ℕ) → ℕ → Finset (Causalean.Stat.PartialMatching r s)","l":142,"f":"Causalean/Stat/UStatistic/OrderM/PartialMatching.lean"},{"n":"Causalean.Stat.pi_iid_absolutelyContinuous","k":"theorem","m":"Causalean.Stat.Minimax.ChiSquared","a":"Stat/Minimax","t":2,"d":"The n-fold product of μ is absolutely continuous w.r.t. that of ν whenever μ ≪ ν (for sigma-finite factors). Proved by induction via the piFinSuccAbove equivalence and the binary AbsolutelyContinuous.prod.","s":"∀ {Ω : Type u_2} [inst : MeasurableSpace Ω] (μ ν : MeasureTheory.Measure Ω) [MeasureTheory.SigmaFinite μ] [MeasureTheory.SigmaFinite ν], μ.AbsolutelyContinuous ν → ∀ (n : ℕ), (MeasureTheory.Measure.pi fun x => μ).AbsolutelyContinuous (MeasureTheory.Measure.pi fun x => ν)","l":265,"f":"Causalean/Stat/Minimax/ChiSquared.lean"},{"n":"Causalean.Stat.pi_iid_integrable_sq_dev","k":"theorem","m":"Causalean.Stat.Minimax.ChiSquared","a":"Stat/Minimax","t":2,"d":"Integrability of the squared density deviation (dμ/dν − 1)² propagates from one sample to the n-fold i.i.d. product (d(μ^⊗n)/d(ν^⊗n) − 1)². On a general (possibly continuous) measurable space this is the side-condition that makes chiSqDiv_prod applicable inside the tensorization induction — it is free on a finite space (Integrable.of_finite) but must be derived here, by L²(ν)-tensorization of the single-sample density dμ/dν.","s":"∀ {S : Type u_2} [inst : MeasurableSpace S] (μ ν : MeasureTheory.Measure S) [MeasureTheory.IsProbabilityMeasure μ] [MeasureTheory.IsProbabilityMeasure ν], μ.AbsolutelyContinuous ν → MeasureTheory.Integrable (fun x => ((μ.rnDeriv ν x).toReal - 1) ^ 2) ν → ∀ (n : ℕ), MeasureTheory.Integrable (fun x => (((MeasureTheory.Measure.pi fun x => μ).rnDeriv (MeasureTheory.Measure.pi fun x => ν) x).toReal - 1) ^ 2) (MeasureTheory.Measure.pi fun x => ν)","l":320,"f":"Causalean/Stat/Minimax/ChiSquared.lean"},{"n":"Causalean.Stat.pi_real_singleton","k":"theorem","m":"Causalean.Stat.Minimax.ChiSquaredFinite","a":"Stat/Minimax","t":2,"d":"Real-valued product point mass. Under the product measure Measure.pi μ of a family of probability measures, the mass of a single point factorizes as the product of the marginal point masses: (Measure.pi μ).real {ω} = ∏ᵢ (μ i).real {ωᵢ}. Rewrites the singleton as Set.univ.pi (fun i => {ωᵢ}), applies Measure.pi_pi, then distributes toReal over the finite product.","s":"∀ {ι : Type u_1} [inst : Fintype ι] {Ω : ι → Type u_2} [inst_1 : (i : ι) → MeasurableSpace (Ω i)] [∀ (i : ι), MeasurableSingletonClass (Ω i)] (μ : (i : ι) → MeasureTheory.Measure (Ω i)) [∀ (i : ι), MeasureTheory.IsProbabilityMeasure (μ i)] (ω : (i : ι) → Ω i), (MeasureTheory.Measure.pi μ).real {ω} = ∏ i, (μ i).real {ω i}","l":81,"f":"Causalean/Stat/Minimax/ChiSquaredFinite.lean"},{"n":"Causalean.Stat.pilotCategoryCount","k":"def","m":"Causalean.Stat.SampleSplit.FiniteCategoryPilot","a":"Stat/SampleSplit","t":1,"d":"A pilot category count is the number of coordinates in a chosen finite block whose observed label equals the designated category.","s":"{Omega : Type u_1} → {X : Type u_2} → {Iota : Type u_3} → [inst : MeasurableSpace Omega] → [inst_1 : MeasurableSpace X] → {mu : MeasureTheory.Measure Omega} → {P : MeasureTheory.Measure X} → Causalean.Stat.IIDSample Omega X mu P → [DecidableEq Iota] → (X → Iota) → Finset ℕ → Iota → Omega → ℕ","l":66,"f":"Causalean/Stat/SampleSplit/FiniteCategoryPilot.lean"},{"n":"Causalean.Stat.pilotCategoryCount_cast_eq_sum","k":"theorem","m":"Causalean.Stat.SampleSplit.FiniteCategoryPilot","a":"Stat/SampleSplit","t":2,"d":"Given an iid sample, a category label, a finite coordinate block, a category, and a sample outcome, the real-valued pilot count equals the sum of category indicators over the block.","s":"∀ {Omega : Type u_1} {X : Type u_2} {Iota : Type u_3} [inst : MeasurableSpace Omega] [inst_1 : MeasurableSpace X] [MeasurableSpace Iota] {mu : MeasureTheory.Measure Omega} {P : MeasureTheory.Measure X} [inst_3 : DecidableEq Iota] (S : Causalean.Stat.IIDSample Omega X mu P) (label : X → Iota) (block : Finset ℕ) (k : Iota) (omega : Omega), ↑(Causalean.Stat.pilotCategoryCount S label block k omega) = ∑ j ∈ block, Causalean.Stat.categoryIndicator label k (S.Z j omega)","l":107,"f":"Causalean/Stat/SampleSplit/FiniteCategoryPilot.lean"},{"n":"Causalean.Stat.pilotCategoryCount_lower_tail_of_tilt","k":"theorem","m":"Causalean.Stat.SampleSplit.FiniteCategoryPilot","a":"Stat/SampleSplit","t":2,"d":"Given an iid sample, a measurable category label, a finite pilot block, a fixed category, a lower bound on that category's population mass, an exponential tilt, and nonpositivity of the tilt, the probability that the category count is at most the specified level is bounded by the corresponding Chernoff exponent.","s":"∀ {Omega : Type u_1} {X : Type u_2} {Iota : Type u_3} [inst : MeasurableSpace Omega] [inst_1 : MeasurableSpace X] [inst_2 : MeasurableSpace Iota] {mu : MeasureTheory.Measure Omega} {P : MeasureTheory.Measure X} [inst_3 : DecidableEq Iota] [MeasurableSingletonClass Iota] (S : Causalean.Stat.IIDSample Omega X mu P) {label : X → Iota}, Measurable label → ∀ (block : Finset ℕ) (k : Iota) {p a : ℝ}, p ≤ Causalean.Stat.categoryMass P label k → ∀ s ≤ 0, mu.real {omega | ↑(Causalean.Stat.pilotCategoryCount S label block k omega) ≤ a} ≤ Real.exp (-s * a + ↑block.card * (p * (Real.exp s - 1)))","l":221,"f":"Causalean/Stat/SampleSplit/FiniteCategoryPilot.lean"},{"n":"Causalean.Stat.pilotCategoryCount_upper_tail_of_tilt","k":"theorem","m":"Causalean.Stat.SampleSplit.FiniteCategoryPilot","a":"Stat/SampleSplit","t":2,"d":"Given an iid sample, a measurable category label, a finite pilot block, a fixed category, an upper bound on that category's population mass, an exponential tilt, and nonnegativity of the tilt, the probability that the category count exceeds the specified level is bounded by the corresponding Chernoff exponent.","s":"∀ {Omega : Type u_1} {X : Type u_2} {Iota : Type u_3} [inst : MeasurableSpace Omega] [inst_1 : MeasurableSpace X] [inst_2 : MeasurableSpace Iota] {mu : MeasureTheory.Measure Omega} {P : MeasureTheory.Measure X} [inst_3 : DecidableEq Iota] [MeasurableSingletonClass Iota] (S : Causalean.Stat.IIDSample Omega X mu P) {label : X → Iota}, Measurable label → ∀ (block : Finset ℕ) (k : Iota) {p a : ℝ}, Causalean.Stat.categoryMass P label k ≤ p → ∀ (s : ℝ), 0 ≤ s → mu.real {omega | a < ↑(Causalean.Stat.pilotCategoryCount S label block k omega)} ≤ Real.exp (-s * a + ↑block.card * (p * (Real.exp s - 1)))","l":119,"f":"Causalean/Stat/SampleSplit/FiniteCategoryPilot.lean"},{"n":"Causalean.Stat.pilotSelected","k":"def","m":"Causalean.Stat.SampleSplit.FiniteCategoryPilot","a":"Stat/SampleSplit","t":1,"d":"Pilot thresholding selects exactly the categories whose finite-block count is strictly larger than the threshold.","s":"{Omega : Type u_1} → {X : Type u_2} → {Iota : Type u_3} → [inst : MeasurableSpace Omega] → [inst_1 : MeasurableSpace X] → {mu : MeasureTheory.Measure Omega} → {P : MeasureTheory.Measure X} → [Fintype Iota] → [DecidableEq Iota] → Causalean.Stat.IIDSample Omega X mu P → (X → Iota) → Finset ℕ → ℝ → Omega → Finset Iota","l":318,"f":"Causalean/Stat/SampleSplit/FiniteCategoryPilot.lean"},{"n":"Causalean.Stat.pinskerBound_of_ac_of_ne_top","k":"theorem","m":"Causalean.Stat.Minimax.Pinsker","a":"Stat/Minimax","t":1,"d":"Pinsker's inequality (unconditional). For probability measures μ and ν on the same space, if μ is absolutely continuous with respect to ν and their Kullback–Leibler divergence is finite, then the total variation distance between μ and ν is at most the square root of half their Kullback–Leibler divergence: tvDist μ ν ≤ √(klDiv(μ,ν)/2).","s":"∀ {Ω : Type u_1} {mΩ : MeasurableSpace Ω} (μ ν : MeasureTheory.Measure Ω) [MeasureTheory.IsProbabilityMeasure μ] [MeasureTheory.IsProbabilityMeasure ν], μ.AbsolutelyContinuous ν → InformationTheory.klDiv μ ν ≠ ⊤ → Causalean.Stat.PinskerBound μ ν","l":286,"f":"Causalean/Stat/Minimax/Pinsker.lean"},{"n":"Causalean.Stat.pinskerBound_pi_iid","k":"theorem","m":"Causalean.Stat.Minimax.Pinsker","a":"Stat/Minimax","t":1,"d":"Pinsker's inequality for i.i.d. finite products (unconditional). For one-sample probability measures μ and ν, if μ is absolutely continuous with respect to ν and the log-likelihood ratio of μ against ν is integrable under μ, then the n-fold product measures μ^{⊗n} and ν^{⊗n} satisfy Pinsker's inequality: their total variation distance is at most the square root of half their Kullback–Leibler divergence.","s":"∀ {α : Type u_2} [inst : MeasurableSpace α] (μ ν : MeasureTheory.Measure α) [MeasureTheory.IsProbabilityMeasure μ] [MeasureTheory.IsProbabilityMeasure ν], μ.AbsolutelyContinuous ν → MeasureTheory.Integrable (MeasureTheory.llr μ ν) μ → ∀ (n : ℕ), Causalean.Stat.PinskerBound (MeasureTheory.Measure.pi fun x => μ) (MeasureTheory.Measure.pi fun x => ν)","l":440,"f":"Causalean/Stat/Minimax/Pinsker.lean"},{"n":"Causalean.Stat.poissonCell_linear_coercive_bound","k":"theorem","m":"Causalean.Stat.MEstimation.FinitePoisson","a":"Stat/MEstimation","t":2,"d":"A positive-mean Poisson cell log likelihood is bounded above by a linearly coercive function. This is the elementary tail estimate used in the finite Poisson maximizer existence argument.","s":"∀ (m x : ℝ), 0 < m → m * x - Real.exp x ≤ (m + 1) ^ 2 / 2 - min m 1 * |x|","l":37,"f":"Causalean/Stat/MEstimation/FinitePoisson.lean"},{"n":"Causalean.Stat.poissonCell_strictConcave_midpoint","k":"theorem","m":"Causalean.Stat.MEstimation.FinitePoisson","a":"Stat/MEstimation","t":2,"d":"Strict midpoint concavity of a positive-weight Poisson cell whenever the two linear predictors differ.","s":"∀ (q m x y : ℝ), 0 < q → x ≠ y → (q * (m * x - Real.exp x) + q * (m * y - Real.exp y)) / 2 < q * (m * ((x + y) / 2) - Real.exp ((x + y) / 2))","l":58,"f":"Causalean/Stat/MEstimation/FinitePoisson.lean"},{"n":"Causalean.Stat.populationScoreDiff_eq_jacobian_plus_remainder","k":"theorem","m":"Causalean.Stat.MEstimation.EmpiricalExpansion","a":"Stat/MEstimation","t":2,"d":"(c) Population Fréchet expansion of the score difference.","s":"∀ {X : Type u_2} [inst : MeasurableSpace X] {E : Type u_3} [inst_1 : NormedAddCommGroup E] [inst_2 : InnerProductSpace ℝ E] [inst_3 : MeasurableSpace E] (ψ : E → X → E) (θ₀ : E) (P : MeasureTheory.Measure X) (reg : Causalean.Stat.ZEstimatorRegularity ψ θ₀ P), (fun θ => ∫ (z : X), ψ θ z - ψ θ₀ z ∂P - reg.J₀ (θ - θ₀)) =o[nhds θ₀] fun θ => ‖θ - θ₀‖","l":455,"f":"Causalean/Stat/MEstimation/EmpiricalExpansion.lean"},{"n":"Causalean.Stat.prod_one_sub_le_exp_neg_card_mul","k":"theorem","m":"Causalean.Stat.Minimax.CoordinatewiseOverlap","a":"Stat/Minimax","t":2,"d":"A common coordinatewise overlap floor c bounds the complementary product by exp (-M * c).","s":"∀ {M : ℕ} (ρ : Fin M → ℝ) (c : ℝ), (∀ (j : Fin M), ρ j ≤ 1) → (∀ (j : Fin M), c ≤ ρ j) → ∏ j, (1 - ρ j) ≤ Real.exp (-↑M * c)","l":276,"f":"Causalean/Stat/Minimax/CoordinatewiseOverlap.lean"},{"n":"Causalean.Stat.prod_one_sub_le_exp_neg_sum","k":"theorem","m":"Causalean.Stat.Minimax.CoordinatewiseOverlap","a":"Stat/Minimax","t":2,"d":"A product of complementary overlap probabilities is bounded by the exponential of minus their sum.","s":"∀ {M : ℕ} (ρ : Fin M → ℝ), (∀ (j : Fin M), ρ j ≤ 1) → ∏ j, (1 - ρ j) ≤ Real.exp (-∑ j, ρ j)","l":257,"f":"Causalean/Stat/Minimax/CoordinatewiseOverlap.lean"},{"n":"Causalean.Stat.product_expectation_comonotoneCoupling","k":"theorem","m":"Causalean.Stat.Coupling.ProductLossMonotoneCoupling.Optimality","a":"Stat/Coupling/ProductLossMonotoneCoupling","t":1,"d":"Closed form of the optimum. For two probability measures μ and ν, each with finite second moment, the expectation of the coordinate product under the comonotone (quantile) coupling of μ and ν equals the integral, over (0,1), of the product of their quantile functions:","s":"∀ (μ ν : MeasureTheory.Measure ℝ) [MeasureTheory.IsProbabilityMeasure μ] [MeasureTheory.IsProbabilityMeasure ν], MeasureTheory.MemLp (fun x => x) 2 μ → MeasureTheory.MemLp (fun y => y) 2 ν → ∫ (p : ℝ × ℝ), p.1 * p.2 ∂Causalean.Stat.comonotoneCoupling μ ν = ∫ (u : ℝ) in Set.Ioo 0 1, Causalean.Stat.quantile μ u * Causalean.Stat.quantile ν u","l":99,"f":"Causalean/Stat/Coupling/ProductLossMonotoneCoupling/Optimality.lean"},{"n":"Causalean.Stat.product_expectation_le_comonotone","k":"theorem","m":"Causalean.Stat.Coupling.ProductLossMonotoneCoupling.Optimality","a":"Stat/Coupling/ProductLossMonotoneCoupling","t":1,"d":"Upper optimality. For any coupling π of μ and ν, where both marginals have finite second moment, the expectation of the coordinate product under π is at most its expectation under the comonotone (quantile) coupling of μ and ν:","s":"∀ {π : MeasureTheory.Measure (ℝ × ℝ)} {μ ν : MeasureTheory.Measure ℝ}, Causalean.Stat.IsCoupling π μ ν → ∀ [MeasureTheory.IsProbabilityMeasure μ] [MeasureTheory.IsProbabilityMeasure ν], MeasureTheory.MemLp (fun x => x) 2 μ → MeasureTheory.MemLp (fun y => y) 2 ν → ∫ (p : ℝ × ℝ), p.1 * p.2 ∂π ≤ ∫ (p : ℝ × ℝ), p.1 * p.2 ∂Causalean.Stat.comonotoneCoupling μ ν","l":36,"f":"Causalean/Stat/Coupling/ProductLossMonotoneCoupling/Optimality.lean"},{"n":"Causalean.Stat.quantile","k":"def","m":"Causalean.Stat.Quantile.Quantile","a":"Stat/Quantile","t":1,"d":"The (lower) quantile function: the left-continuous generalized inverse of the cdf, quantile μ τ = inf {x : ℝ | τ ≤ cdf μ x}.","s":"MeasureTheory.Measure ℝ → ℝ → ℝ","l":48,"f":"Causalean/Stat/Quantile/Quantile.lean"},{"n":"Causalean.Stat.quantileIF","k":"def","m":"Causalean.Stat.Quantile.SampleQuantile","a":"Stat/Quantile","t":1,"d":"The sample-quantile influence function ψ_τ(z) = (τ − 1{z ≤ q₀}) / f₀.","s":"ℝ → ℝ → ℝ → ℝ → ℝ","l":69,"f":"Causalean/Stat/Quantile/SampleQuantile.lean"},{"n":"Causalean.Stat.quantileIFVec","k":"def","m":"Causalean.Stat.Quantile.SampleQuantileJoint","a":"Stat/Quantile","t":1,"d":"The joint quantile influence function ψ(z)_j = (τⱼ − 1{z ≤ qⱼ}) / fⱼ, valued in EuclideanSpace ℝ (Fin k).","s":"{k : ℕ} → (Fin k → ℝ) → (Fin k → ℝ) → (Fin k → ℝ) → ℝ → EuclideanSpace ℝ (Fin k)","l":62,"f":"Causalean/Stat/Quantile/SampleQuantileJoint.lean"},{"n":"Causalean.Stat.quantileIF_cross","k":"theorem","m":"Causalean.Stat.Quantile.SampleQuantileJoint","a":"Stat/Quantile","t":1,"d":"Covariance entry. Given qⱼ is the population τⱼ-quantile: $F(q_j)=\\tau_j$ and qₗ is the population τₗ-quantile: $F(q_l)=\\tau_l$, the cross-moment of the two quantile influence functions $\\psi_{\\tau_j}$ and $\\psi_{\\tau_l}$ under the population measure equals $(\\min(\\tau_j,\\tau_l)-\\tau_j\\tau_l)/(f_jf_l)$.","s":"∀ {P : MeasureTheory.Measure ℝ} [MeasureTheory.IsProbabilityMeasure P] {τj qj fj τl ql fl : ℝ}, ↑(ProbabilityTheory.cdf P) qj = τj → ↑(ProbabilityTheory.cdf P) ql = τl → ∫ (z : ℝ), Causalean.Stat.quantileIF τj qj fj z * Causalean.Stat.quantileIF τl ql fl z ∂P = (min τj τl - τj * τl) / (fj * fl)","l":80,"f":"Causalean/Stat/Quantile/SampleQuantileJoint.lean"},{"n":"Causalean.Stat.quantileIF_mean_zero","k":"theorem","m":"Causalean.Stat.Quantile.SampleQuantile","a":"Stat/Quantile","t":2,"d":"The quantile influence function has mean zero under P, given that q₀ is the population τ-quantile (F(q₀) = τ).","s":"∀ {P : MeasureTheory.Measure ℝ} [MeasureTheory.IsProbabilityMeasure P] {τ q₀ f₀ : ℝ}, ↑(ProbabilityTheory.cdf P) q₀ = τ → ∫ (z : ℝ), Causalean.Stat.quantileIF τ q₀ f₀ z ∂P = 0","l":78,"f":"Causalean/Stat/Quantile/SampleQuantile.lean"},{"n":"Causalean.Stat.quantileIF_sq_integrable","k":"theorem","m":"Causalean.Stat.Quantile.SampleQuantile","a":"Stat/Quantile","t":2,"d":"The quantile influence function is square-integrable (it is bounded).","s":"∀ {P : MeasureTheory.Measure ℝ} [MeasureTheory.IsProbabilityMeasure P] {τ q₀ f₀ : ℝ}, MeasureTheory.Integrable (fun z => Causalean.Stat.quantileIF τ q₀ f₀ z ^ 2) P","l":88,"f":"Causalean/Stat/Quantile/SampleQuantile.lean"},{"n":"Causalean.Stat.quantileIF_variance","k":"theorem","m":"Causalean.Stat.Quantile.SampleQuantile","a":"Stat/Quantile","t":1,"d":"Variance of the quantile influence function. Provided q₀ is the population τ-quantile, i.e. the population cdf satisfies $F(q_0)=\\tau$, the second moment of the influence function $\\psi_\\tau$ under the population measure equals $\\tau(1-\\tau)/f_0^2$, the classical sample-quantile asymptotic variance.","s":"∀ {P : MeasureTheory.Measure ℝ} [MeasureTheory.IsProbabilityMeasure P] {τ q₀ f₀ : ℝ}, ↑(ProbabilityTheory.cdf P) q₀ = τ → ∫ (z : ℝ), Causalean.Stat.quantileIF τ q₀ f₀ z ^ 2 ∂P = τ * (1 - τ) / f₀ ^ 2","l":109,"f":"Causalean/Stat/Quantile/SampleQuantile.lean"},{"n":"Causalean.Stat.quantileSet","k":"def","m":"Causalean.Stat.Quantile.Quantile","a":"Stat/Quantile","t":1,"d":"The defining super-level set of the cdf, {x | τ ≤ cdf μ x}.","s":"MeasureTheory.Measure ℝ → ℝ → Set ℝ","l":45,"f":"Causalean/Stat/Quantile/Quantile.lean"},{"n":"Causalean.Stat.quantileSet_up_closed","k":"theorem","m":"Causalean.Stat.Quantile.Quantile","a":"Stat/Quantile","t":2,"d":"The super-level set is up-closed (monotonicity of the cdf).","s":"∀ {μ : MeasureTheory.Measure ℝ} {τ x x' : ℝ}, x ∈ Causalean.Stat.quantileSet μ τ → x ≤ x' → x' ∈ Causalean.Stat.quantileSet μ τ","l":54,"f":"Causalean/Stat/Quantile/Quantile.lean"},{"n":"Causalean.Stat.quantile_le_iff","k":"theorem","m":"Causalean.Stat.Quantile.Quantile","a":"Stat/Quantile","t":1,"d":"Quantile / cdf Galois connection. For an interior probability level $\\tau\\in(0,1)$, the quantile of a real measure at level $\\tau$ is at most a point x exactly when $\\tau$ is at most the cdf of that measure at x.","s":"∀ {μ : MeasureTheory.Measure ℝ} {τ x : ℝ}, 0 < τ → τ < 1 → (Causalean.Stat.quantile μ τ ≤ x ↔ τ ≤ ↑(ProbabilityTheory.cdf μ) x)","l":106,"f":"Causalean/Stat/Quantile/Quantile.lean"},{"n":"Causalean.Stat.quantile_le_of_le_cdf","k":"theorem","m":"Causalean.Stat.Quantile.Quantile","a":"Stat/Quantile","t":2,"d":"Galois connection (other direction). If the cdf reaches τ at x, then the quantile is at or below x.","s":"∀ {μ : MeasureTheory.Measure ℝ} {τ x : ℝ}, 0 < τ → τ ≤ ↑(ProbabilityTheory.cdf μ) x → Causalean.Stat.quantile μ τ ≤ x","l":100,"f":"Causalean/Stat/Quantile/Quantile.lean"},{"n":"Causalean.Stat.quantile_map_uniform","k":"theorem","m":"Causalean.Stat.Coupling.ProductLossMonotoneCoupling.PIT","a":"Stat/Coupling/ProductLossMonotoneCoupling","t":1,"d":"Probability integral transform. For a Borel probability measure μ on ℝ, the pushforward of the uniform distribution on (0,1) under μ's quantile function equals μ itself.","s":"∀ (μ : MeasureTheory.Measure ℝ) [MeasureTheory.IsProbabilityMeasure μ], MeasureTheory.Measure.map (Causalean.Stat.quantile μ) Causalean.Stat.unifOI = μ","l":85,"f":"Causalean/Stat/Coupling/ProductLossMonotoneCoupling/PIT.lean"},{"n":"Causalean.Stat.quantile_mono","k":"theorem","m":"Causalean.Stat.Quantile.Quantile","a":"Stat/Quantile","t":2,"d":"The quantile function is monotone in the probability level τ on (0,1).","s":"∀ {μ : MeasureTheory.Measure ℝ} {τ τ' : ℝ}, 0 < τ → τ' < 1 → τ ≤ τ' → Causalean.Stat.quantile μ τ ≤ Causalean.Stat.quantile μ τ'","l":113,"f":"Causalean/Stat/Quantile/Quantile.lean"},{"n":"Causalean.Stat.ratioDeriv","k":"def","m":"Causalean.Stat.Inference.RatioDeltaMethod","a":"Stat/Inference","t":1,"d":"The Fréchet derivative of v ↦ v 0 / v 1 at t₀, in closed form via smulRight on the coordinate projections EuclideanSpace.proj 0, EuclideanSpace.proj 1: ratioDeriv t₀ = (1/b) • proj₀ − (a / b²) • proj₁ with (a, b) = (t₀ 0, t₀ 1). This is the gradient of x/y at (a, b) paired against the coordinate functionals. (This Mathlib has no bundled HasFDerivAt.div, so we give the closed form explicitly and verify it via the product/inverse rules.)","s":"EuclideanSpace ℝ (Fin 2) → EuclideanSpace ℝ (Fin 2) →L[ℝ] ℝ","l":50,"f":"Causalean/Stat/Inference/RatioDeltaMethod.lean"},{"n":"Causalean.Stat.real_two_point_lower_bound","k":"theorem","m":"Causalean.Stat.Minimax.MinimaxRisk","a":"Stat/Minimax","t":1,"d":"Le Cam two-point bound for a real-valued parameter. If two candidate values θ₀, θ₁ : ℝ are 2s-separated, then for any measurable estimator est : Ω → ℝ, the worst-case probability of missing the truth by ≥ s is at least ½(1 − tvDist P₀ P₁). Specialization of half_one_sub_tvDist_le_max_error to Θ = ℝ with dist a b = |a − b|.","s":"∀ {Ω : Type u_1} {mΩ : MeasurableSpace Ω} {P₀ P₁ : MeasureTheory.Measure Ω} [MeasureTheory.IsProbabilityMeasure P₀] [MeasureTheory.IsProbabilityMeasure P₁] {est : Ω → ℝ}, Measurable est → ∀ {θ₀ θ₁ s : ℝ}, 2 * s ≤ |θ₀ - θ₁| → (1 - Causalean.Stat.tvDist P₀ P₁) / 2 ≤ max (P₀.real {ω | s ≤ |est ω - θ₀|}) (P₁.real {ω | s ≤ |est ω - θ₁|})","l":56,"f":"Causalean/Stat/Minimax/MinimaxRisk.lean"},{"n":"Causalean.Stat.rescaledEmpiricalCDF_eq_normalizedSum","k":"theorem","m":"Causalean.Stat.Quantile.EmpiricalCDF","a":"Stat/Quantile","t":2,"d":"Key identity. The rescaled empirical-cdf deviation equals the normalized influence-function sum: √n (F̂ₙ(y) − F(y)) = (1/√n) Σ cdfIF(Z_i).","s":"∀ {Ω : Type u_1} [inst : MeasurableSpace Ω] {μ : MeasureTheory.Measure Ω} {P : MeasureTheory.Measure ℝ} (S : Causalean.Stat.IIDSample Ω ℝ μ P) (y : ℝ) (n : ℕ) (ω : Ω), √↑(Finset.range n).card * (S.empiricalCDF y n ω - ↑(ProbabilityTheory.cdf P) y) = (√↑(Finset.range n).card)⁻¹ * ∑ i ∈ Finset.range n, Causalean.Stat.cdfIF P y (S.Z i ω)","l":175,"f":"Causalean/Stat/Quantile/EmpiricalCDF.lean"},{"n":"Causalean.Stat.restrictedSetVolume","k":"def","m":"Causalean.Stat.Minimax.HonestConfidenceSet","a":"Stat/Minimax","t":1,"d":"The volume of a set after restriction to a parameter region.","s":"Set ℝ → Set ℝ → ℝ","l":27,"f":"Causalean/Stat/Minimax/HonestConfidenceSet.lean"},{"n":"Causalean.Stat.rnCommonPart","k":"def","m":"Causalean.Stat.Minimax.MaximalCoupling","a":"Stat/Minimax","t":1,"d":"The common submeasure obtained by taking the pointwise minimum of two Radon--Nikodym densities against a finite dominating measure.","s":"{X : Type u_1} → [inst : MeasurableSpace X] → MeasureTheory.Measure X → MeasureTheory.Measure X → MeasureTheory.Measure X → MeasureTheory.Measure X","l":109,"f":"Causalean/Stat/Minimax/MaximalCoupling.lean"},{"n":"Causalean.Stat.rnCommonPart_le_left","k":"theorem","m":"Causalean.Stat.Minimax.MaximalCoupling","a":"Stat/Minimax","t":2,"d":"The RN common part is dominated by its first law.","s":"∀ {X : Type u_1} [inst : MeasurableSpace X] (mu nu xi : MeasureTheory.Measure X) [MeasureTheory.IsFiniteMeasure mu] [MeasureTheory.IsFiniteMeasure xi], mu.AbsolutelyContinuous xi → Causalean.Stat.rnCommonPart mu nu xi ≤ mu","l":118,"f":"Causalean/Stat/Minimax/MaximalCoupling.lean"},{"n":"Causalean.Stat.rnCommonPart_le_right","k":"theorem","m":"Causalean.Stat.Minimax.MaximalCoupling","a":"Stat/Minimax","t":2,"d":"The RN common part is dominated by its second law.","s":"∀ {X : Type u_1} [inst : MeasurableSpace X] (mu nu xi : MeasureTheory.Measure X) [MeasureTheory.IsFiniteMeasure nu] [MeasureTheory.IsFiniteMeasure xi], nu.AbsolutelyContinuous xi → Causalean.Stat.rnCommonPart mu nu xi ≤ nu","l":130,"f":"Causalean/Stat/Minimax/MaximalCoupling.lean"},{"n":"Causalean.Stat.rnCommonPart_mass_eq_one_sub_tvDist","k":"theorem","m":"Causalean.Stat.Minimax.MaximalCoupling","a":"Stat/Minimax","t":2,"d":"The mass of the RN common part is exactly one minus total variation.","s":"∀ {X : Type u_1} [inst : MeasurableSpace X] (mu nu xi : MeasureTheory.Measure X) [MeasureTheory.IsProbabilityMeasure mu] [MeasureTheory.IsProbabilityMeasure nu] [MeasureTheory.IsFiniteMeasure xi], mu.AbsolutelyContinuous xi → nu.AbsolutelyContinuous xi → (Causalean.Stat.rnCommonPart mu nu xi) Set.univ = ENNReal.ofReal (1 - Causalean.Stat.tvDist mu nu)","l":142,"f":"Causalean/Stat/Minimax/MaximalCoupling.lean"},{"n":"Causalean.Stat.rnDeriv_mul_measure_singleton","k":"theorem","m":"Causalean.Stat.Minimax.ChiSquaredFinite","a":"Stat/Minimax","t":2,"d":"Discrete Radon–Nikodym bridge. On a measurable singleton {x}, the Radon–Nikodym density scaled by the base mass recovers the numerator mass: ν {x} · (dμ/dν)(x) = μ {x}, for μ ≪ ν. Proved by integrating the density over {x} (via setLIntegral_rnDeriv) and collapsing the singleton integral with restrict_singleton/lintegral_dirac'.","s":"∀ {Ω : Type u_1} [inst : MeasurableSpace Ω] [MeasurableSingletonClass Ω] (μ ν : MeasureTheory.Measure Ω) [MeasureTheory.SFinite ν] [μ.HaveLebesgueDecomposition ν], μ.AbsolutelyContinuous ν → ∀ (x : Ω), ν {x} * μ.rnDeriv ν x = μ {x}","l":35,"f":"Causalean/Stat/Minimax/ChiSquaredFinite.lean"},{"n":"Causalean.Stat.rnDeriv_prod_eq","k":"theorem","m":"Causalean.Stat.Minimax.ChiSquared","a":"Stat/Minimax","t":2,"d":"Product density factorization. For μ₁ ≪ ν₁ and μ₂ ≪ ν₂, the Radon–Nikodym density of the product is (a.e.) the product of the marginal densities.","s":"∀ {α : Type u_2} {mα : MeasurableSpace α} {β : Type u_3} {mβ : MeasurableSpace β} (μ₁ ν₁ : MeasureTheory.Measure α) (μ₂ ν₂ : MeasureTheory.Measure β) [MeasureTheory.SigmaFinite ν₁] [MeasureTheory.SigmaFinite ν₂] [MeasureTheory.SFinite ν₂] [μ₁.HaveLebesgueDecomposition ν₁] [μ₂.HaveLebesgueDecomposition ν₂], μ₁.AbsolutelyContinuous ν₁ → μ₂.AbsolutelyContinuous ν₂ → (μ₁.prod μ₂).rnDeriv (ν₁.prod ν₂) =ᵐ[ν₁.prod ν₂] fun z => μ₁.rnDeriv ν₁ z.1 * μ₂.rnDeriv ν₂ z.2","l":154,"f":"Causalean/Stat/Minimax/ChiSquared.lean"},{"n":"Causalean.Stat.rnDeriv_setIntegral_le_sqrt_chi","k":"theorem","m":"Causalean.Stat.Minimax.ChiSquared","a":"Stat/Minimax","t":2,"d":"Cauchy–Schwarz mass transfer under a χ²-budget. For probability measures P ≪ Q whose squared density deviation (dP/dQ − 1)² is Q-integrable, and whose χ²-divergence is at most C, the P-mass of any measurable set A is controlled by its Q-mass through P(A) ≤ √((C + 1)·Q(A)). This is the Cauchy–Schwarz step underlying the two-point testing floor: a set that is small under the reference measure Q cannot be large under P when the χ²-budget is finite.","s":"∀ {Ω : Type u_2} [inst : MeasurableSpace Ω] (P Q : MeasureTheory.Measure Ω) [MeasureTheory.IsProbabilityMeasure P] [MeasureTheory.IsProbabilityMeasure Q], P.AbsolutelyContinuous Q → MeasureTheory.Integrable (fun x => ((P.rnDeriv Q x).toReal - 1) ^ 2) Q → ∀ {C : ℝ}, Causalean.Stat.chiSqDiv P Q ≤ C → ∀ {A : Set Ω}, MeasurableSet A → P.real A ≤ √((C + 1) * Q.real A)","l":461,"f":"Causalean/Stat/Minimax/ChiSquared.lean"},{"n":"Causalean.Stat.sampleDesign","k":"def","m":"Causalean.Stat.Sample.OccupancyWeightedMean.Basic","a":"Stat/Sample/OccupancyWeightedMean","t":1,"d":"The finite design records only the group and arm label of each sample coordinate.","s":"{Omega : Type u_1} → {kappa : Type u_2} → {n : ℕ} → (Omega → kappa) → (Omega → Bool) → (Fin n → Omega) → Fin n → kappa × Bool","l":53,"f":"Causalean/Stat/Sample/OccupancyWeightedMean/Basic.lean"},{"n":"Causalean.Stat.score_diff_L2_isLittleOp_sqrt","k":"theorem","m":"Causalean.Stat.MEstimation.EmpiricalExpansion","a":"Stat/MEstimation","t":2,"d":"(a) Score difference is o_p(1) in mean square.","s":"∀ {Ω : Type u_1} {X : Type u_2} [inst : MeasurableSpace Ω] [inst_1 : MeasurableSpace X] {μ : MeasureTheory.Measure Ω} {E : Type u_3} [inst_2 : NormedAddCommGroup E] [inst_3 : InnerProductSpace ℝ E] [inst_4 : MeasurableSpace E] (ψ : E → X → E) (θ₀ : E) (P : MeasureTheory.Measure X) (reg : Causalean.Stat.ZEstimatorRegularity ψ θ₀ P) [MeasureTheory.IsProbabilityMeasure μ] (S : Causalean.Stat.IIDSample Ω X μ P) (θn : ℕ → Ω → E), (∀ ε > 0, Filter.Tendsto (fun n => μ {ω | ε < ‖θn n ω - θ₀‖}) Filter.atTop (nhds 0)) → Causalean.Stat.IsLittleOp (fun n ω => (↑n)⁻¹ * ∑ i ∈ Finset.range n, ‖ψ (θn n ω) (S.Z i ω) - ψ θ₀ (S.Z i ω)‖ ^ 2) (fun x => 1) μ","l":85,"f":"Causalean/Stat/MEstimation/EmpiricalExpansion.lean"},{"n":"Causalean.Stat.secondMomentInv","k":"def","m":"Causalean.Stat.Inference.ChiSquaredWald","a":"Stat/Inference","t":1,"d":"The inverse Σ⁻¹ = (√Σ)⁻¹ ∘ (√Σ)⁻¹ of the second-moment operator, as a continuous linear map (continuity is automatic in finite dimension).","s":"{X : Type u_1} → [inst : MeasurableSpace X] → {P : MeasureTheory.Measure X} → {E : Type u_2} → [inst_1 : NormedAddCommGroup E] → [inst_2 : InnerProductSpace ℝ E] → [inst_3 : FiniteDimensional ℝ E] → [inst_4 : MeasurableSpace E] → [inst_5 : BorelSpace E] → {ψ : X → E} → (hψ : Measurable ψ) → (hvar : MeasureTheory.Integrable (fun x => ‖ψ x‖ ^ 2) P) → Function.Injective ⇑(Causalean.Stat.secondMomentLM hψ hvar) → E →L[ℝ] E","l":87,"f":"Causalean/Stat/Inference/ChiSquaredWald.lean"},{"n":"Causalean.Stat.secondMomentInv_secondMomentLM","k":"theorem","m":"Causalean.Stat.Inference.ChiSquaredWald","a":"Stat/Inference","t":2,"d":"Σ⁻¹ is a left inverse of Σ.","s":"∀ {X : Type u_1} [inst : MeasurableSpace X] {P : MeasureTheory.Measure X} {E : Type u_2} [inst_1 : NormedAddCommGroup E] [inst_2 : InnerProductSpace ℝ E] [inst_3 : FiniteDimensional ℝ E] [inst_4 : MeasurableSpace E] [inst_5 : BorelSpace E] {ψ : X → E} (hψ : Measurable ψ) (hvar : MeasureTheory.Integrable (fun x => ‖ψ x‖ ^ 2) P) (hinj : Function.Injective ⇑(Causalean.Stat.secondMomentLM hψ hvar)) (x : E), (Causalean.Stat.secondMomentInv hψ hvar hinj) ((Causalean.Stat.secondMomentLM hψ hvar) x) = x","l":99,"f":"Causalean/Stat/Inference/ChiSquaredWald.lean"},{"n":"Causalean.Stat.secondMomentLM","k":"def","m":"Causalean.Stat.CLT.SecondMomentOperator","a":"Stat/CLT","t":1,"d":"The second-moment operator Σ t = ∫ ⟪t, ψ x⟫ • ψ x ∂P, as a linear map.","s":"{X : Type u_1} → {E : Type u_2} → [inst : MeasurableSpace X] → {P : MeasureTheory.Measure X} → [inst_1 : NormedAddCommGroup E] → [inst_2 : InnerProductSpace ℝ E] → [inst_3 : MeasurableSpace E] → [OpensMeasurableSpace E] → [SecondCountableTopology E] → [MeasurableSMul₂ ℝ E] → {ψ : X → E} → Measurable ψ → MeasureTheory.Integrable (fun x => ‖ψ x‖ ^ 2) P → E →ₗ[ℝ] E","l":71,"f":"Causalean/Stat/CLT/SecondMomentOperator.lean"},{"n":"Causalean.Stat.secondMomentLM_inner","k":"theorem","m":"Causalean.Stat.CLT.SecondMomentOperator","a":"Stat/CLT","t":1,"d":"For vectors t and s, the inner product of the second-moment operator applied to t with s equals the expectation under P of the product of the inner products ⟪t, ψ⟩ and ⟪s, ψ⟩.","s":"∀ {X : Type u_1} {E : Type u_2} [inst : MeasurableSpace X] {P : MeasureTheory.Measure X} [inst_1 : NormedAddCommGroup E] [inst_2 : InnerProductSpace ℝ E] [inst_3 : MeasurableSpace E] [inst_4 : OpensMeasurableSpace E] [inst_5 : SecondCountableTopology E] [inst_6 : MeasurableSMul₂ ℝ E] {ψ : X → E} (hψ : Measurable ψ) (hvar : MeasureTheory.Integrable (fun x => ‖ψ x‖ ^ 2) P) [CompleteSpace E] (t s : E), inner ℝ ((Causalean.Stat.secondMomentLM hψ hvar) t) s = ∫ (x : X), inner ℝ t (ψ x) * inner ℝ s (ψ x) ∂P","l":85,"f":"Causalean/Stat/CLT/SecondMomentOperator.lean"},{"n":"Causalean.Stat.secondMomentLM_isPositive","k":"theorem","m":"Causalean.Stat.CLT.SecondMomentOperator","a":"Stat/CLT","t":1,"d":"The second-moment operator is a positive linear operator, hence symmetric and self-adjoint.","s":"∀ {X : Type u_1} {E : Type u_2} [inst : MeasurableSpace X] {P : MeasureTheory.Measure X} [inst_1 : NormedAddCommGroup E] [inst_2 : InnerProductSpace ℝ E] [inst_3 : MeasurableSpace E] [inst_4 : OpensMeasurableSpace E] [inst_5 : SecondCountableTopology E] [inst_6 : MeasurableSMul₂ ℝ E] {ψ : X → E} (hψ : Measurable ψ) (hvar : MeasureTheory.Integrable (fun x => ‖ψ x‖ ^ 2) P) [CompleteSpace E], (Causalean.Stat.secondMomentLM hψ hvar).IsPositive","l":96,"f":"Causalean/Stat/CLT/SecondMomentOperator.lean"},{"n":"Causalean.Stat.secondMomentLM_secondMomentInv","k":"theorem","m":"Causalean.Stat.Inference.ChiSquaredWald","a":"Stat/Inference","t":2,"d":"Σ⁻¹ is a right inverse of Σ.","s":"∀ {X : Type u_1} [inst : MeasurableSpace X] {P : MeasureTheory.Measure X} {E : Type u_2} [inst_1 : NormedAddCommGroup E] [inst_2 : InnerProductSpace ℝ E] [inst_3 : FiniteDimensional ℝ E] [inst_4 : MeasurableSpace E] [inst_5 : BorelSpace E] {ψ : X → E} (hψ : Measurable ψ) (hvar : MeasureTheory.Integrable (fun x => ‖ψ x‖ ^ 2) P) (hinj : Function.Injective ⇑(Causalean.Stat.secondMomentLM hψ hvar)) (x : E), (Causalean.Stat.secondMomentLM hψ hvar) ((Causalean.Stat.secondMomentInv hψ hvar hinj) x) = x","l":110,"f":"Causalean/Stat/Inference/ChiSquaredWald.lean"},{"n":"Causalean.Stat.selectCoupledRaw","k":"def","m":"Causalean.Stat.Minimax.CoordinatewiseOverlap","a":"Stat/Minimax","t":1,"d":"Select the raw coordinate corresponding to a hypercube vertex from a coordinatewise pair coupling.","s":"{M : ℕ} → {Z : Fin M → Type u_1} → (Fin M → Bool) → ((j : Fin M) → Z j × Z j) → (j : Fin M) → Z j","l":67,"f":"Causalean/Stat/Minimax/CoordinatewiseOverlap.lean"},{"n":"Causalean.Stat.selectorCell","k":"def","m":"Causalean.Stat.SampleSplit.FiniteSelector","a":"Stat/SampleSplit","t":1,"d":"The selector cell for a branch is the part of a designated pilot event on which a finite-valued selection rule chooses that branch.","s":"{Alpha : Type u_2} → {Iota : Type u_4} → (Alpha → Iota) → Set Alpha → Iota → Set Alpha","l":35,"f":"Causalean/Stat/SampleSplit/FiniteSelector.lean"},{"n":"Causalean.Stat.sequential_cumulative_risk_regret","k":"theorem","m":"Causalean.Stat.Limit.SequentialCumulativeRisk","a":"Stat/Limit","t":1,"d":"Radius-monotone cumulative Neyman-regret lower bound (lem:local-neighborhood-cumulative-risk). Fix a per-round Fisher information scale J > 0, an oracle-sensitivity d ≠ 0, a nonnegative feedback constant L, and a nonnegative prior information Iq. Let b be a per-round mean-squared-error sequence with running totals B defined by B n = ∑_{t≤n} b t, satisfying the predictable van Trees / Bayes–Cramér–Rao recursion b t ≥ (d²/4) / (Iq + (5J/4)·t + L·√(t·B(t-1))) for every round t ≥ 1. Suppose further the Neyman-gap quadratic conversion 2S²·B T ≤ RB T for every T (the Bayes-average cumulative regret RB dominates twice the scaled cumulative Bayes MSE) and the worst-case cumulative regret Rsup dominates the Bayes average, RB T ≤ Rsup T for every T. Then there is a threshold T₀ beyond which the worst-case cumulative regret grows at least logarithmically: Rsup T ≥ (S² d² / (16J)) · log T for all T ≥ T₀, with universal constant 1/16.","s":"∀ (S J d L Iq : ℝ), 0 < J → d ≠ 0 → 0 ≤ L → 0 ≤ Iq → ∀ (b B RB Rsup : ℕ → ℝ), (∀ (n : ℕ), B n = ∑ t ∈ Finset.Icc 1 n, b t) → (∀ (t : ℕ), 1 ≤ t → d ^ 2 / 4 / (Iq + 5 * J / 4 * ↑t + L * √(↑t * B (t - 1))) ≤ b t) → (∀ (T : ℕ), 2 * S ^ 2 * B T ≤ RB T) → (∀ (T : ℕ), RB T ≤ Rsup T) → ∃ T₀, ∀ (T : ℕ), T₀ ≤ T → S ^ 2 * d ^ 2 / (16 * J) * Real.log ↑T ≤ Rsup T","l":387,"f":"Causalean/Stat/Limit/SequentialCumulativeRisk.lean"},{"n":"Causalean.Stat.signedTail","k":"def","m":"Causalean.Stat.Coupling.ProductLossMonotoneCoupling.TailIntegral","a":"Stat/Coupling/ProductLossMonotoneCoupling","t":1,"d":"signedTail a s = 𝟙{s < a} - 𝟙{s < 0}, the *signed tail indicator* of a. As a function of s it is +1 on [0, a) when a ≥ 0, -1 on [a, 0) when a < 0, and 0 elsewhere; its Lebesgue integral is exactly a.","s":"ℝ → ℝ → ℝ","l":40,"f":"Causalean/Stat/Coupling/ProductLossMonotoneCoupling/TailIntegral.lean"},{"n":"Causalean.Stat.signedTail_eq_indicator_sub","k":"theorem","m":"Causalean.Stat.Coupling.ProductLossMonotoneCoupling.TailIntegral","a":"Stat/Coupling/ProductLossMonotoneCoupling","t":2,"d":"signedTail a is, as a function of s, the difference of the indicators of the two intervals Ico 0 a and Ico a 0 (at most one of which is nonempty). This is the normal form used to compute its integral and to see it is integrable.","s":"∀ (a : ℝ), Causalean.Stat.signedTail a = ((Set.Ico 0 a).indicator fun x => 1) - (Set.Ico a 0).indicator fun x => 1","l":49,"f":"Causalean/Stat/Coupling/ProductLossMonotoneCoupling/TailIntegral.lean"},{"n":"Causalean.Stat.sqLoss_kernelMean_le","k":"theorem","m":"Causalean.Stat.Minimax.MarkovKernelTransport","a":"Stat/Minimax","t":2,"d":"If the target estimator is measurable and uniformly bounded, then the squared error of its kernel mean is no greater than the kernel average of its squared error at each source observation.","s":"∀ {X : Type uX} {Y : Type uY} [inst : MeasurableSpace X] [inst_1 : MeasurableSpace Y] (K : ProbabilityTheory.Kernel X Y) [ProbabilityTheory.IsMarkovKernel K] {T : Y → ℝ}, Measurable T → Causalean.Stat.UniformlyBounded T → ∀ (c : ℝ) (x : X), (Causalean.Stat.kernelMean K T x - c) ^ 2 ≤ ∫ (y : Y), (T y - c) ^ 2 ∂K x","l":126,"f":"Causalean/Stat/Minimax/MarkovKernelTransport.lean"},{"n":"Causalean.Stat.sqRisk","k":"def","m":"Causalean.Stat.Minimax.MinimaxRisk","a":"Stat/Minimax","t":1,"d":"The squared risk of a real-valued estimator is its expected squared error around a specified real target.","s":"{X : Type uX} → [inst : MeasurableSpace X] → MeasureTheory.Measure X → (X → ℝ) → ℝ → ℝ","l":178,"f":"Causalean/Stat/Minimax/MinimaxRisk.lean"},{"n":"Causalean.Stat.sqRisk_finProductKernel_affinePullback_le","k":"theorem","m":"Causalean.Stat.Minimax.MarkovKernelTransport","a":"Stat/Minimax","t":2,"d":"If the affine slope is nonzero, a finite-sample target estimator is measurable, and that estimator is uniformly bounded, then the source-product risk of its affine product-kernel pullback, multiplied by the squared slope, is at most its target-product risk.","s":"∀ {X : Type uX} {Y : Type uY} [inst : MeasurableSpace X] [inst_1 : MeasurableSpace Y] (n : ℕ) (P : MeasureTheory.Measure X) [MeasureTheory.IsProbabilityMeasure P] (K : ProbabilityTheory.Kernel X Y) [ProbabilityTheory.IsMarkovKernel K] {a b theta : ℝ}, a ≠ 0 → ∀ {targetEst : (Fin n → Y) → ℝ}, Measurable targetEst → Causalean.Stat.UniformlyBounded targetEst → a ^ 2 * Causalean.Stat.sqRisk (MeasureTheory.Measure.pi fun x => P) (Causalean.Stat.kernelAffinePullback (Causalean.Stat.finProductKernel n K) a b targetEst) theta ≤ Causalean.Stat.sqRisk (MeasureTheory.Measure.pi fun x => P.bind ⇑K) targetEst (a * theta + b)","l":339,"f":"Causalean/Stat/Minimax/MarkovKernelTransport.lean"},{"n":"Causalean.Stat.sqRisk_kernelAffinePullback_le_comp","k":"theorem","m":"Causalean.Stat.Minimax.MarkovKernelTransport","a":"Stat/Minimax","t":2,"d":"If the affine slope is nonzero, the target estimator is measurable, and the target estimator is uniformly bounded, then the source risk of the affine kernel pullback, multiplied by the squared slope, is at most the target risk under the garbled law.","s":"∀ {X : Type uX} {Y : Type uY} [inst : MeasurableSpace X] [inst_1 : MeasurableSpace Y] (P : MeasureTheory.Measure X) [MeasureTheory.IsProbabilityMeasure P] (K : ProbabilityTheory.Kernel X Y) [ProbabilityTheory.IsMarkovKernel K] {a b theta : ℝ}, a ≠ 0 → ∀ {targetEst : Y → ℝ}, Measurable targetEst → Causalean.Stat.UniformlyBounded targetEst → a ^ 2 * Causalean.Stat.sqRisk P (Causalean.Stat.kernelAffinePullback K a b targetEst) theta ≤ Causalean.Stat.sqRisk (P.bind ⇑K) targetEst (a * theta + b)","l":187,"f":"Causalean/Stat/Minimax/MarkovKernelTransport.lean"},{"n":"Causalean.Stat.sqRisk_kernelMean_le_comp","k":"theorem","m":"Causalean.Stat.Minimax.MarkovKernelTransport","a":"Stat/Minimax","t":2,"d":"If the target estimator is measurable and uniformly bounded, then Rao--Blackwellizing it through a Markov kernel cannot increase squared risk: the source-law risk of the kernel mean is at most the target-law risk after garbling.","s":"∀ {X : Type uX} {Y : Type uY} [inst : MeasurableSpace X] [inst_1 : MeasurableSpace Y] (P : MeasureTheory.Measure X) [MeasureTheory.IsProbabilityMeasure P] (K : ProbabilityTheory.Kernel X Y) [ProbabilityTheory.IsMarkovKernel K] {T : Y → ℝ}, Measurable T → Causalean.Stat.UniformlyBounded T → ∀ (c : ℝ), Causalean.Stat.sqRisk P (Causalean.Stat.kernelMean K T) c ≤ Causalean.Stat.sqRisk (P.bind ⇑K) T c","l":154,"f":"Causalean/Stat/Minimax/MarkovKernelTransport.lean"},{"n":"Causalean.Stat.sqRisk_map_affinePullback","k":"theorem","m":"Causalean.Stat.Minimax.MinimaxRisk","a":"Stat/Minimax","t":2,"d":"If the affine slope is nonzero, the deterministic observation rule is measurable, and the target estimator is measurable, then its squared risk under the pushed-forward law equals the pullback estimator's squared risk multiplied by the squared slope.","s":"∀ {X : Type uX} {Y : Type uY} [inst : MeasurableSpace X] [inst_1 : MeasurableSpace Y] {law : MeasureTheory.Measure X} {phi : X → Y} {a b theta : ℝ} {targetEst : Y → ℝ}, a ≠ 0 → Measurable phi → Measurable targetEst → a ^ 2 * Causalean.Stat.sqRisk law (Causalean.Stat.affinePullbackEstimator phi a b targetEst) theta = Causalean.Stat.sqRisk (MeasureTheory.Measure.map phi law) targetEst (a * theta + b)","l":211,"f":"Causalean/Stat/Minimax/MinimaxRisk.lean"},{"n":"Causalean.Stat.sq_bhattacharyya_le_two_mul_integral_min","k":"theorem","m":"Causalean.Stat.Minimax.BretagnolleHuber","a":"Stat/Minimax","t":2,"d":"Cauchy–Schwarz / Bhattacharyya step. (∫ √p ∂ν)² ≤ 2·∫ min(p,1) ∂ν.","s":"∀ {Ω : Type u_1} {mΩ : MeasurableSpace Ω} (μ ν : MeasureTheory.Measure Ω) [MeasureTheory.IsProbabilityMeasure μ] [MeasureTheory.IsProbabilityMeasure ν], μ.AbsolutelyContinuous ν → (∫ (x : Ω), √(μ.rnDeriv ν x).toReal ∂ν) ^ 2 ≤ 2 * ∫ (x : Ω), min (μ.rnDeriv ν x).toReal 1 ∂ν","l":159,"f":"Causalean/Stat/Minimax/BretagnolleHuber.lean"},{"n":"Causalean.Stat.sqrtFoldB_integral_isLittleOp_one","k":"theorem","m":"Causalean.Stat.SampleSplit.FoldBEmpiricalProcess","a":"Stat/SampleSplit","t":1,"d":"Bias term is o_p(1) under fixed-ratio split. Given a positive limiting fold-B sampling ratio $c$ with the fold-B fraction $|B(n)|/n$ converging to $c$, and a family of random functions f n ω that is square-integrable under the population measure, for every n, ω with $L^2(P)$ norm that is $o_p(n^{-1/2})$ under the sampling measure, the bias term $\\sqrt{|B(n)|}\\cdot\\int f(n,\\omega,\\cdot)\\,dP$ is $o_p(1)$.","s":"∀ {Ω : Type u_1} {X : Type u_2} [inst : MeasurableSpace Ω] [inst_1 : MeasurableSpace X] {μ : MeasureTheory.Measure Ω} {P : MeasureTheory.Measure X} [MeasureTheory.IsProbabilityMeasure P] (S : Causalean.Stat.IIDSample Ω X μ P) (split : Causalean.Stat.OneShotSplit S) {c : ℝ}, 0 < c → Filter.Tendsto (fun n => ↑(split.foldB n).card / ↑n) Filter.atTop (nhds c) → ∀ (f : ℕ → Ω → X → ℝ), (∀ (n : ℕ) (ω : Ω), MeasureTheory.MemLp (f n ω) 2 P) → Causalean.Stat.IsLittleOp (fun n ω => (MeasureTheory.eLpNorm (f n ω) 2 P).toReal) (fun n => ↑n ^ (-(1 / 2))) μ → Causalean.Stat.IsLittleOp (fun n ω => √↑(split.foldB n).card * ∫ (x : X), f n ω x ∂P) (fun x => 1) μ","l":706,"f":"Causalean/Stat/SampleSplit/FoldBEmpiricalProcess.lean"},{"n":"Causalean.Stat.sqrt_le_half_add_one","k":"theorem","m":"Causalean.Stat.Minimax.BretagnolleHuber","a":"Stat/Minimax","t":2,"d":"√y ≤ (y + 1)/2 for y ≥ 0 (AM–GM); used to dominate √p by an integrable function.","s":"∀ {y : ℝ}, 0 ≤ y → √y ≤ (y + 1) / 2","l":91,"f":"Causalean/Stat/Minimax/BretagnolleHuber.lean"},{"n":"Causalean.Stat.sqrt_mul_max_sub","k":"theorem","m":"Causalean.Stat.Inference.FunctionalDelta","a":"Stat/Inference","t":2,"d":"The tie identity for max. √n (max âₙ b̂ₙ − a) = max (√n(âₙ−a)) (√n(b̂ₙ−a)) (no remainder).","s":"∀ {Ω : Type u_1} (an bn : ℕ → Ω → ℝ) (a : ℝ) (n : ℕ) (ω : Ω), √↑n * (max (an n ω) (bn n ω) - a) = (fun z => max z.1 z.2) (√↑n • ((an n ω, bn n ω) - (a, a)))","l":112,"f":"Causalean/Stat/Inference/FunctionalDelta.lean"},{"n":"Causalean.Stat.sqrt_mul_min_sub","k":"theorem","m":"Causalean.Stat.Inference.FunctionalDelta","a":"Stat/Inference","t":2,"d":"The tie identity for min. Companion to sqrt_mul_max_sub.","s":"∀ {Ω : Type u_1} (an bn : ℕ → Ω → ℝ) (a : ℝ) (n : ℕ) (ω : Ω), √↑n * (min (an n ω) (bn n ω) - a) = (fun z => min z.1 z.2) (√↑n • ((an n ω, bn n ω) - (a, a)))","l":127,"f":"Causalean/Stat/Inference/FunctionalDelta.lean"},{"n":"Causalean.Stat.sqrt_var_tendsto_inProb","k":"theorem","m":"Causalean.Stat.Inference.VarianceEstimation","a":"Stat/Inference","t":1,"d":"Standard-error consistency from variance consistency. Fix a positive scale σ₀. If a variance-estimator sequence varhat converges in probability to σ₀², then the standard-error estimator √varhat converges in probability to σ₀. This is exactly the σ̂ →ₚ σ₀ input required by the generic studentized CLT Tendsto_dist.div_tendsto_inProb_gaussian; callers feed fun N ω => Real.sqrt (varhat N ω) to it.","s":"∀ {Ω : Type u_3} [inst : MeasurableSpace Ω] {varhat : ℕ → Ω → ℝ} {σ₀ : ℝ} {μ : MeasureTheory.Measure Ω}, 0 < σ₀ → Causalean.Stat.Tendsto_inProb varhat (fun x => σ₀ ^ 2) μ → Causalean.Stat.Tendsto_inProb (fun n ω => √(varhat n ω)) (fun x => σ₀) μ","l":180,"f":"Causalean/Stat/Inference/VarianceEstimation.lean"},{"n":"Causalean.Stat.stdGaussian_map_linearIsometryEquiv","k":"theorem","m":"Causalean.Stat.CLT.ChiSquared","a":"Stat/CLT","t":2,"d":"stdGaussian is invariant under linear isometry equivalences. Pushing the standard Gaussian forward along an inner-product isometry yields the standard Gaussian on the target. Proved by matching characteristic functions: both sides are centered Gaussians whose covariance form is the inner product (preserved by the isometry).","s":"∀ {E : Type u_1} [inst : NormedAddCommGroup E] [inst_1 : InnerProductSpace ℝ E] [inst_2 : FiniteDimensional ℝ E] [inst_3 : MeasurableSpace E] [BorelSpace E] {F : Type u_2} [inst_5 : NormedAddCommGroup F] [inst_6 : InnerProductSpace ℝ F] [inst_7 : FiniteDimensional ℝ F] [inst_8 : MeasurableSpace F] [BorelSpace F] (ι : E ≃ₗᵢ[ℝ] F), MeasureTheory.Measure.map (⇑ι) (Causalean.Mathlib.stdGaussian E) = Causalean.Mathlib.stdGaussian F","l":61,"f":"Causalean/Stat/CLT/ChiSquared.lean"},{"n":"Causalean.Stat.stdGaussian_map_normSq","k":"theorem","m":"Causalean.Stat.CLT.ChiSquared","a":"Stat/CLT","t":1,"d":"The law of the squared norm under the standard Gaussian distribution on a finite-dimensional real inner product space equals the chi-squared distribution whose degrees of freedom is the space's dimension.","s":"∀ {E : Type u_1} [inst : NormedAddCommGroup E] [inst_1 : InnerProductSpace ℝ E] [inst_2 : FiniteDimensional ℝ E] [inst_3 : MeasurableSpace E] [BorelSpace E], MeasureTheory.Measure.map (fun x => ‖x‖ ^ 2) (Causalean.Mathlib.stdGaussian E) = Causalean.Stat.chiSqDist (Module.finrank ℝ E)","l":92,"f":"Causalean/Stat/CLT/ChiSquared.lean"},{"n":"Causalean.Stat.stdGaussian_map_normSq_orthogonalProjection","k":"theorem","m":"Causalean.Stat.CLT.ChiSquaredProjection","a":"Stat/CLT","t":1,"d":"For a continuous linear self-map P of a finite-dimensional real inner-product space that is self-adjoint and idempotent, i.e. P composed with itself equals P — so that P is an orthogonal projection — the law of $\\|Pz\\|^2$ under the standard Gaussian on the space equals the chi-squared distribution with degrees of freedom equal to the dimension of the range of P.","s":"∀ {E : Type u_1} [inst : NormedAddCommGroup E] [inst_1 : InnerProductSpace ℝ E] [inst_2 : FiniteDimensional ℝ E] [inst_3 : MeasurableSpace E] [BorelSpace E] (P : E →L[ℝ] E), IsSelfAdjoint P → P ∘SL P = P → MeasureTheory.Measure.map (fun z => ‖P z‖ ^ 2) (Causalean.Mathlib.stdGaussian E) = Causalean.Stat.chiSqDist (Module.finrank ℝ ↥(↑P).range)","l":65,"f":"Causalean/Stat/CLT/ChiSquaredProjection.lean"},{"n":"Causalean.Stat.stochEquicontAt_of_asymptoticEquicont","k":"theorem","m":"Causalean.Stat.EmpiricalProcess.Equicontinuity.Modulus","a":"Stat/EmpiricalProcess/Equicontinuity","t":1,"d":"Reduction: StochEquicontAt from class-level equicontinuity + consistency. If the score family ψ is asymptotically equicontinuous at θ₀ along the i.i.d. sample S and θn is a sequence of estimators consistent for θ₀, then the pair (ψ, θ₀) satisfies the stochastic-equicontinuity-at-the-estimator condition StochEquicontAt along S and θn.","s":"∀ {Ω : Type u_1} {X : Type u_2} [inst : MeasurableSpace Ω] [inst_1 : MeasurableSpace X] {μ : MeasureTheory.Measure Ω} {P : MeasureTheory.Measure X} {E : Type u_3} [inst_2 : NormedAddCommGroup E] [inst_3 : InnerProductSpace ℝ E] (ψ : E → X → E) (θ₀ : E) (S : Causalean.Stat.IIDSample Ω X μ P) (θn : ℕ → Ω → E), Causalean.Stat.AsymptoticEquicont ψ θ₀ P μ S → (∀ ε > 0, Filter.Tendsto (fun n => μ {ω | ε < ‖θn n ω - θ₀‖}) Filter.atTop (nhds 0)) → Causalean.Stat.StochEquicontAt ψ θ₀ P μ S θn","l":150,"f":"Causalean/Stat/EmpiricalProcess/Equicontinuity/Modulus.lean"},{"n":"Causalean.Stat.sum_bernoulli_local_variance_le_share","k":"theorem","m":"Causalean.Stat.Sample.OccupancyWeightedMean.BinomialDesign","a":"Stat/Sample/OccupancyWeightedMean","t":1,"d":"If the overlap margin is positive and one group's success probability lies between the margin and one minus the margin, that group's expected inverse-arm variance contribution is controlled by its expected share of reciprocal usable occupancy.","s":"∀ {ι : Type u_1} [inst : Fintype ι] [inst_1 : DecidableEq ι] (r : ℕ) (p epsilon : ℝ), 0 < epsilon → epsilon ≤ p → p ≤ 1 - epsilon → ∑ b, (∏ i, if b i = true then p else 1 - p) * (↑(Fintype.card ι) ^ 2 * (↑(r + Fintype.card ι))⁻¹ ^ 2 * Causalean.Stat.inverseTwoCounts (Fintype.card ι) {i | b i = true}.card) ≤ 4 / (epsilon ^ 2 * (1 - epsilon)) * ∑ b, (∏ i, if b i = true then p else 1 - p) * (↑(Fintype.card ι) * (↑(r + Fintype.card ι))⁻¹ ^ 2 * Causalean.Stat.interiorIndicator (Fintype.card ι) {i | b i = true}.card)","l":105,"f":"Causalean/Stat/Sample/OccupancyWeightedMean/BinomialDesign.lean"},{"n":"Causalean.Stat.sum_correct_le","k":"theorem","m":"Causalean.Stat.Minimax.Fano","a":"Stat/Minimax","t":2,"d":"Heart of Fano. The total probability of correct recovery, summed over the family, is at most 1 + ∑ᵢ tvDist (P i₀) (P i): comparing each Pᵢ(A i) to the reference P i₀(A i) costs one tvDist, and the reference masses sum to ≤ 1.","s":"∀ {Ω : Type u_1} {mΩ : MeasurableSpace Ω} {Θ : Type u_2} [inst : PseudoMetricSpace Θ] [inst_1 : MeasurableSpace Θ] [OpensMeasurableSpace Θ] {ι : Type u_3} [inst_3 : Fintype ι] (P : ι → MeasureTheory.Measure Ω) [∀ (i : ι), MeasureTheory.IsProbabilityMeasure (P i)] {est : Ω → Θ}, Measurable est → ∀ {θ : ι → Θ} {s : ℝ}, (∀ (i k : ι), i ≠ k → 2 * s ≤ dist (θ i) (θ k)) → ∀ (i₀ : ι), ∑ i, (P i).real (Causalean.Stat.acceptanceRegion est (θ i) s) ≤ 1 + ∑ i, Causalean.Stat.tvDist (P i₀) (P i)","l":133,"f":"Causalean/Stat/Minimax/Fano.lean"},{"n":"Causalean.Stat.sum_decode_ge","k":"theorem","m":"Causalean.Stat.Minimax.Assouad","a":"Stat/Minimax","t":2,"d":"Per-coordinate lower bound. Summed over the cube, the coordinate-j error mass is at least half the cube-sum of 1 − tvDist (P τ) (P (flip j τ)), via pairing each vertex with its j-flip (an involution that preserves the sum).","s":"∀ {Ω : Type u_1} {mΩ : MeasurableSpace Ω} {d : ℕ} (P : (Fin d → Bool) → MeasureTheory.Measure Ω) [∀ (τ : Fin d → Bool), MeasureTheory.IsProbabilityMeasure (P τ)] (est : Ω → Fin d → Bool), (∀ (j : Fin d) (b : Bool), MeasurableSet {ω | est ω j = b}) → ∀ (j : Fin d), ∑ τ, (1 - Causalean.Stat.tvDist (P τ) (P (Causalean.Stat.flipBit j τ))) ≤ 2 * ∑ τ, (P τ).real {ω | est ω j ≠ τ j}","l":126,"f":"Causalean/Stat/Minimax/Assouad.lean"},{"n":"Causalean.Stat.sum_injectiveTuples_apply_eq_range","k":"theorem","m":"Causalean.Stat.UStatistic.OrderM.Hajek","a":"Stat/UStatistic/OrderM","t":2,"d":"For each coordinate of an ordered injective tuple, every sample index appears equally often.","s":"∀ {m n : ℕ}, m ≤ n → ∀ (j : Fin m) (f : ℕ → ℝ), ∑ t ∈ Causalean.Stat.injectiveTuples m n, f ↑(t j) = Causalean.Stat.injectiveTupleCount m n / ↑n * ∑ i ∈ Finset.range n, f i","l":146,"f":"Causalean/Stat/UStatistic/OrderM/Hajek.lean"},{"n":"Causalean.Stat.sum_injectiveTuples_two_eq_offDiag","k":"theorem","m":"Causalean.Stat.UStatistic.OrderM.Basic","a":"Stat/UStatistic/OrderM","t":2,"d":"Ordered injective Fin 2 tuples are the same data as off-diagonal ordered pairs.","s":"∀ {Ω : Type u_1} {X : Type u_2} [inst : MeasurableSpace Ω] [inst_1 : MeasurableSpace X] {μ : MeasureTheory.Measure Ω} {P : MeasureTheory.Measure X} (S : Causalean.Stat.IIDSample Ω X μ P) (h : X → X → ℝ) (n : ℕ) (ω : Ω), (∑ t ∈ Causalean.Stat.injectiveTuples 2 n, Causalean.Stat.pairKernel h fun j => S.Z (↑(t j)) ω) = ∑ p ∈ (Finset.range n).offDiag, h (S.Z p.1 ω) (S.Z p.2 ω)","l":324,"f":"Causalean/Stat/UStatistic/OrderM/Basic.lean"},{"n":"Causalean.Stat.sum_jointWeight_variance_le_inverse","k":"theorem","m":"Causalean.Stat.Sample.OccupancyWeightedMean.FiniteDesign","a":"Stat/Sample/OccupancyWeightedMean","t":1,"d":"If all joint group/arm masses are nonnegative, their two arm masses sum to the group mass, the overlap margin is positive and at most one half, and each positive-mass group gives both arms at least the overlap share, the joint-design average of the occupancy variance factor is bounded by a margin-dependent multiple of average reciprocal usable occupancy. Zero-mass groups contribute zero before any conditional arm probability is formed.","s":"∀ {n : ℕ} {κ : Type u_1} [inst : Fintype κ] [inst_1 : DecidableEq κ] (q : κ → Bool → ℝ) (g : κ → ℝ) (epsilon : ℝ), (∀ (k : κ) (a : Bool), 0 ≤ q k a) → (∀ (k : κ), q k false + q k true = g k) → 0 < epsilon → epsilon ≤ 1 / 2 → (∀ (k : κ), 0 < g k → ∀ (a : Bool), epsilon * g k ≤ q k a) → ∑ d, (∏ i, q (d i).1 (d i).2) * Causalean.Stat.occupancyDesignVarianceFactor Prod.fst Prod.snd d ≤ 4 / (epsilon ^ 2 * (1 - epsilon)) * ∑ d, (∏ i, q (d i).1 (d i).2) * Causalean.Stat.inverseUsableGroupTotal Prod.fst Prod.snd d","l":504,"f":"Causalean/Stat/Sample/OccupancyWeightedMean/FiniteDesign.lean"},{"n":"Causalean.Stat.sum_partialMatchingsOfSize","k":"theorem","m":"Causalean.Stat.UStatistic.OrderM.PartialMatching","a":"Stat/UStatistic/OrderM","t":2,"d":"Given a quantity assigned to each partial matching, summing it over all matchings equals summing first by matching size through the smaller coordinate-set size.","s":"∀ {r s : ℕ} {α : Type u_1} [inst : AddCommMonoid α] (F : Causalean.Stat.PartialMatching r s → α), ∑ M, F M = ∑ h ∈ Finset.range (min r s + 1), ∑ M ∈ Causalean.Stat.partialMatchingsOfSize r s h, F M","l":206,"f":"Causalean/Stat/UStatistic/OrderM/PartialMatching.lean"},{"n":"Causalean.Stat.sum_refReal_acceptance_le","k":"theorem","m":"Causalean.Stat.Minimax.Fano","a":"Stat/Minimax","t":2,"d":"A reference hypothesis assigns total mass ≤ 1 across the disjoint acceptance regions, since their union has measure ≤ 1.","s":"∀ {Ω : Type u_1} {mΩ : MeasurableSpace Ω} {Θ : Type u_2} [inst : PseudoMetricSpace Θ] [inst_1 : MeasurableSpace Θ] [OpensMeasurableSpace Θ] {ι : Type u_3} [inst_3 : Fintype ι] (P : ι → MeasureTheory.Measure Ω) [∀ (i : ι), MeasureTheory.IsProbabilityMeasure (P i)] {est : Ω → Θ}, Measurable est → ∀ {θ : ι → Θ} {s : ℝ}, (∀ (i k : ι), i ≠ k → 2 * s ≤ dist (θ i) (θ k)) → ∀ (i₀ : ι), ∑ i, (P i₀).real (Causalean.Stat.acceptanceRegion est (θ i) s) ≤ 1","l":106,"f":"Causalean/Stat/Minimax/Fano.lean"},{"n":"Causalean.Stat.supportedArmGroupResidual","k":"def","m":"Causalean.Stat.Sample.OccupancyWeightedMean.Basic","a":"Stat/Sample/OccupancyWeightedMean","t":1,"d":"The supported residual equals the arm/group residual on its own label event and is zero elsewhere.","s":"{Omega : Type u_1} → {kappa : Type u_2} → (Omega → kappa) → (Omega → Bool) → (Omega → ℝ) → (Bool → kappa → ℝ) → Bool → kappa → Omega → ℝ","l":46,"f":"Causalean/Stat/Sample/OccupancyWeightedMean/Basic.lean"},{"n":"Causalean.Stat.supportedArmGroupResidual_mul_eq_zero_of_ne","k":"theorem","m":"Causalean.Stat.Sample.OccupancyWeightedMean.MomentBounds","a":"Stat/Sample/OccupancyWeightedMean","t":2,"d":"When two arm/group labels differ, their supported residuals have zero pointwise product because their supports are disjoint.","s":"∀ {Omega : Type u_1} {kappa : Type u_2} [MeasurableSpace Omega] [Fintype kappa] [DecidableEq kappa] [inst : MeasurableSpace kappa] [MeasurableSingletonClass kappa] (group : Omega → kappa) (arm : Omega → Bool) (Y : Omega → ℝ) (center : Bool → kappa → ℝ) (a b : Bool) (k l : kappa), (a, k) ≠ (b, l) → ∀ (omega : Omega), Causalean.Stat.supportedArmGroupResidual group arm Y center a k omega * Causalean.Stat.supportedArmGroupResidual group arm Y center b l omega = 0","l":28,"f":"Causalean/Stat/Sample/OccupancyWeightedMean/MomentBounds.lean"},{"n":"Causalean.Stat.survFst","k":"def","m":"Causalean.Stat.Coupling.ProductLossMonotoneCoupling.Survival","a":"Stat/Coupling/ProductLossMonotoneCoupling","t":1,"d":"The first marginal survival function of π: SX s = π{p | s < p.1}.","s":"MeasureTheory.Measure (ℝ × ℝ) → ℝ → ℝ","l":34,"f":"Causalean/Stat/Coupling/ProductLossMonotoneCoupling/Survival.lean"},{"n":"Causalean.Stat.survFst_eq","k":"theorem","m":"Causalean.Stat.Coupling.ProductLossMonotoneCoupling.Survival","a":"Stat/Coupling/ProductLossMonotoneCoupling","t":2,"d":"The first marginal survival function of a coupling is 1 - F, where F = cdf μ. Proof: Prod.fst ⁻¹' Ioi s is the complement of Prod.fst ⁻¹' Iic s, whose π-mass is μ (Iic s) = F s by h.map_fst.","s":"∀ {π : MeasureTheory.Measure (ℝ × ℝ)} {μ ν : MeasureTheory.Measure ℝ}, Causalean.Stat.IsCoupling π μ ν → ∀ (s : ℝ), Causalean.Stat.survFst π s = 1 - ↑(ProbabilityTheory.cdf μ) s","l":47,"f":"Causalean/Stat/Coupling/ProductLossMonotoneCoupling/Survival.lean"},{"n":"Causalean.Stat.survSnd","k":"def","m":"Causalean.Stat.Coupling.ProductLossMonotoneCoupling.Survival","a":"Stat/Coupling/ProductLossMonotoneCoupling","t":1,"d":"The second marginal survival function of π: SY t = π{p | t < p.2}.","s":"MeasureTheory.Measure (ℝ × ℝ) → ℝ → ℝ","l":38,"f":"Causalean/Stat/Coupling/ProductLossMonotoneCoupling/Survival.lean"},{"n":"Causalean.Stat.survSnd_eq","k":"theorem","m":"Causalean.Stat.Coupling.ProductLossMonotoneCoupling.Survival","a":"Stat/Coupling/ProductLossMonotoneCoupling","t":2,"d":"The second marginal survival function of a coupling is 1 - G, where G = cdf ν.","s":"∀ {π : MeasureTheory.Measure (ℝ × ℝ)} {μ ν : MeasureTheory.Measure ℝ}, Causalean.Stat.IsCoupling π μ ν → ∀ (t : ℝ), Causalean.Stat.survSnd π t = 1 - ↑(ProbabilityTheory.cdf ν) t","l":80,"f":"Causalean/Stat/Coupling/ProductLossMonotoneCoupling/Survival.lean"},{"n":"Causalean.Stat.surv_gap_eq","k":"theorem","m":"Causalean.Stat.Coupling.ProductLossMonotoneCoupling.Survival","a":"Stat/Coupling/ProductLossMonotoneCoupling","t":1,"d":"Survival gap = cdf gap. For a coupling π of two probability measures μ and ν, the gap between the joint and product-of-marginals survival functions of π at (s, t) equals the corresponding gap between the joint and product-of-marginals cumulative distribution functions.","s":"∀ {π : MeasureTheory.Measure (ℝ × ℝ)} {μ ν : MeasureTheory.Measure ℝ}, Causalean.Stat.IsCoupling π μ ν → ∀ (s t : ℝ), Causalean.Stat.jointSurv π s t - Causalean.Stat.survFst π s * Causalean.Stat.survSnd π t = Causalean.Stat.jointCdf π s t - ↑(ProbabilityTheory.cdf μ) s * ↑(ProbabilityTheory.cdf ν) t","l":172,"f":"Causalean/Stat/Coupling/ProductLossMonotoneCoupling/Survival.lean"},{"n":"Causalean.Stat.tailInd","k":"def","m":"Causalean.Stat.Coupling.ProductLossMonotoneCoupling.TailIntegral","a":"Stat/Coupling/ProductLossMonotoneCoupling","t":1,"d":"tailInd a s is the indicator 𝟙{s < a}, i.e. 1 when s < a and 0 otherwise, written as the indicator function of the ray Iio a.","s":"ℝ → ℝ → ℝ","l":36,"f":"Causalean/Stat/Coupling/ProductLossMonotoneCoupling/TailIntegral.lean"},{"n":"Causalean.Stat.tailInd_apply","k":"theorem","m":"Causalean.Stat.Coupling.ProductLossMonotoneCoupling.TailIntegral","a":"Stat/Coupling/ProductLossMonotoneCoupling","t":2,"d":"The tail indicator equals one below the threshold and zero at or above it.","s":"∀ (a s : ℝ), Causalean.Stat.tailInd a s = if s < a then 1 else 0","l":45,"f":"Causalean/Stat/Coupling/ProductLossMonotoneCoupling/TailIntegral.lean"},{"n":"Causalean.Stat.tendsto_argmax_of_eventually_mem_compact","k":"theorem","m":"Causalean.Stat.MEstimation.ArgmaxStability","a":"Stat/MEstimation","t":1,"d":"Deterministic argmax stability. Let criterion : ℕ → E → ℝ be a sequence of objective functions with pointwise limit limitCriterion, over a compact set K. Suppose the limiting maximizer limitArgmax lies in K, limitCriterion is continuous on K, limitArgmax maximizes limitCriterion over K, and it is the unique maximizer over K, up to equal criterion values. If criterion converges to limitCriterion uniformly on K, each argmax N eventually lies in K, and argmax N eventually maximizes criterion N over K, then the sequence argmax converges to limitArgmax.","s":"∀ {E : Type u_1} [inst : PseudoMetricSpace E] [SecondCountableTopology E] (criterion : ℕ → E → ℝ) (limitCriterion : E → ℝ) (argmax : ℕ → E) (limitArgmax : E) (K : Set E), IsCompact K → limitArgmax ∈ K → ContinuousOn limitCriterion K → (∀ y ∈ K, limitCriterion y ≤ limitCriterion limitArgmax) → TendstoUniformlyOn criterion limitCriterion Filter.atTop K → (∀ᶠ (N : ℕ) in Filter.atTop, argmax N ∈ K) → (∀ᶠ (N : ℕ) in Filter.atTop, ∀ y ∈ K, criterion N y ≤ criterion N (argmax N)) → (∀ y ∈ K, limitCriterion y = limitCriterion limitArgmax → y = limitArgmax) → Filter.Tendsto argmax Filter.atTop (nhds limitArgmax)","l":23,"f":"Causalean/Stat/MEstimation/ArgmaxStability.lean"},{"n":"Causalean.Stat.testing_error_lower_of_chi","k":"theorem","m":"Causalean.Stat.Minimax.ChiSquared","a":"Stat/Minimax","t":1,"d":"Two-point testing-error floor from a finite χ²-budget. For probability measures P, Q such that P is absolutely continuous with respect to Q and the squared density deviation (dP/dQ − 1)² is Q-integrable, if C is nonnegative and the χ²-divergence chiSqDiv P Q is at most C, then for every measurable test region A, the combined testing error is at least 1/(4(C + 1)): P(Aᶜ) + Q(A) ≥ 1/(4(C + 1)). This is the positive two-point testing floor that powers χ²-budget minimax lower bounds — no test can separate P from Q better than this when their χ²-divergence is bounded.","s":"∀ {Ω : Type u_2} [inst : MeasurableSpace Ω] (P Q : MeasureTheory.Measure Ω) [MeasureTheory.IsProbabilityMeasure P] [MeasureTheory.IsProbabilityMeasure Q], P.AbsolutelyContinuous Q → MeasureTheory.Integrable (fun x => ((P.rnDeriv Q x).toReal - 1) ^ 2) Q → ∀ {C : ℝ}, 0 ≤ C → Causalean.Stat.chiSqDiv P Q ≤ C → ∀ {A : Set Ω}, MeasurableSet A → P.real Aᶜ + Q.real A ≥ 1 / (4 * (C + 1))","l":557,"f":"Causalean/Stat/Minimax/ChiSquared.lean"},{"n":"Causalean.Stat.tvDist","k":"def","m":"Causalean.Stat.Minimax.TotalVariation","a":"Stat/Minimax","t":1,"d":"The statistical total variation distance between two measures: the supremum, over measurable sets A, of the gap |μ.real A − ν.real A|.","s":"{Ω : Type u_1} → {mΩ : MeasurableSpace Ω} → MeasureTheory.Measure Ω → MeasureTheory.Measure Ω → ℝ","l":47,"f":"Causalean/Stat/Minimax/TotalVariation.lean"},{"n":"Causalean.Stat.tvDist_eq_half_integral_abs_rnDeriv_sub","k":"theorem","m":"Causalean.Stat.Minimax.MaximalCoupling","a":"Stat/Minimax","t":2,"d":"Scheffé's identity with both probability laws dominated by an arbitrary finite reference measure.","s":"∀ {X : Type u_1} [inst : MeasurableSpace X] (mu nu xi : MeasureTheory.Measure X) [MeasureTheory.IsProbabilityMeasure mu] [MeasureTheory.IsProbabilityMeasure nu] [MeasureTheory.IsFiniteMeasure xi], mu.AbsolutelyContinuous xi → nu.AbsolutelyContinuous xi → Causalean.Stat.tvDist mu nu = 1 / 2 * ∫ (x : X), |(mu.rnDeriv xi x).toReal - (nu.rnDeriv xi x).toReal| ∂xi","l":34,"f":"Causalean/Stat/Minimax/MaximalCoupling.lean"},{"n":"Causalean.Stat.tvDist_eq_zero_of_klBudget_nonpos","k":"theorem","m":"Causalean.Stat.Minimax.CoordinatewiseOverlap","a":"Stat/Minimax","t":2,"d":"A nonpositive real KL budget forces two probability measures to coincide, and hence forces their total variation distance to vanish.","s":"∀ {Ω : Type u_1} [inst : MeasurableSpace Ω] (μ ν : MeasureTheory.Measure Ω) [MeasureTheory.IsProbabilityMeasure μ] [MeasureTheory.IsProbabilityMeasure ν] {B : ℝ}, B ≤ 0 → InformationTheory.klDiv μ ν ≤ ENNReal.ofReal B → Causalean.Stat.tvDist μ ν = 0","l":291,"f":"Causalean/Stat/Minimax/CoordinatewiseOverlap.lean"},{"n":"Causalean.Stat.tvDist_integral_le_of_abs_le_ae","k":"theorem","m":"Causalean.Stat.Minimax.TotalVariation","a":"Stat/Minimax","t":2,"d":"A measurable statistic bounded in absolute value by M almost surely under both laws has an expectation gap no larger than 2M times their total-variation distance.","s":"∀ {Ω : Type u_1} {mΩ : MeasurableSpace Ω} (μ ν : MeasureTheory.Measure Ω) [MeasureTheory.IsProbabilityMeasure μ] [MeasureTheory.IsProbabilityMeasure ν] (f : Ω → ℝ), Measurable f → ∀ (M : ℝ), 0 ≤ M → (∀ᵐ (x : Ω) ∂μ, |f x| ≤ M) → (∀ᵐ (x : Ω) ∂ν, |f x| ≤ M) → |∫ (x : Ω), f x ∂μ - ∫ (x : Ω), f x ∂ν| ≤ 2 * M * Causalean.Stat.tvDist μ ν","l":220,"f":"Causalean/Stat/Minimax/TotalVariation.lean"},{"n":"Causalean.Stat.tvDist_integral_le_of_range_ae","k":"theorem","m":"Causalean.Stat.Minimax.TotalVariation","a":"Stat/Minimax","t":2,"d":"A measurable statistic that lies almost surely in an interval of width c under both laws has an expectation gap no larger than c times their total-variation distance.","s":"∀ {Ω : Type u_1} {mΩ : MeasurableSpace Ω} (μ ν : MeasureTheory.Measure Ω) [MeasureTheory.IsProbabilityMeasure μ] [MeasureTheory.IsProbabilityMeasure ν] (f : Ω → ℝ), Measurable f → ∀ (a c : ℝ), 0 ≤ c → (∀ᵐ (x : Ω) ∂μ, f x ∈ Set.Icc a (a + c)) → (∀ᵐ (x : Ω) ∂ν, f x ∈ Set.Icc a (a + c)) → |∫ (x : Ω), f x ∂μ - ∫ (x : Ω), f x ∂ν| ≤ Causalean.Stat.tvDist μ ν * c","l":193,"f":"Causalean/Stat/Minimax/TotalVariation.lean"},{"n":"Causalean.Stat.tvDist_integral_range","k":"theorem","m":"Causalean.Stat.Minimax.TotalVariation","a":"Stat/Minimax","t":2,"d":"The expectation gap of a measurable statistic confined to an interval of width c is at most that width times the total-variation distance between the two probability laws.","s":"∀ {Ω : Type u_1} {mΩ : MeasurableSpace Ω} (μ ν : MeasureTheory.Measure Ω) [MeasureTheory.IsProbabilityMeasure μ] [MeasureTheory.IsProbabilityMeasure ν] (f : Ω → ℝ), Measurable f → ∀ (a c : ℝ), 0 ≤ c → (∀ (x : Ω), f x ∈ Set.Icc a (a + c)) → |∫ (x : Ω), f x ∂μ - ∫ (x : Ω), f x ∂ν| ≤ Causalean.Stat.tvDist μ ν * c","l":115,"f":"Causalean/Stat/Minimax/TotalVariation.lean"},{"n":"Causalean.Stat.tvDist_le_half_integral_abs_rnDeriv","k":"theorem","m":"Causalean.Stat.Minimax.Scheffe","a":"Stat/Minimax","t":1,"d":"Scheffé's inequality (≤ direction). For probability measures μ and ν on the same space with μ absolutely continuous with respect to ν, the total variation distance between μ and ν is at most half the L¹(ν)-distance of the Radon–Nikodym density dμ/dν to the constant 1.","s":"∀ {Ω : Type u_1} {mΩ : MeasurableSpace Ω} (μ ν : MeasureTheory.Measure Ω) [MeasureTheory.IsProbabilityMeasure μ] [MeasureTheory.IsProbabilityMeasure ν], μ.AbsolutelyContinuous ν → Causalean.Stat.tvDist μ ν ≤ 1 / 2 * ∫ (x : Ω), |(μ.rnDeriv ν x).toReal - 1| ∂ν","l":109,"f":"Causalean/Stat/Minimax/Scheffe.lean"},{"n":"Causalean.Stat.tvDist_le_half_sqrt_chiSqDiv","k":"theorem","m":"Causalean.Stat.Minimax.ChiSquared","a":"Stat/Minimax","t":1,"d":"Cauchy–Schwarz on Scheffé. For probability measures μ, ν with μ absolutely continuous with respect to ν and whose squared Radon–Nikodym density deviation (dμ/dν − 1)² is ν-integrable, the total variation distance tvDist μ ν is bounded by half the square root of the χ²-divergence chiSqDiv μ ν.","s":"∀ {Ω : Type u_1} {mΩ : MeasurableSpace Ω} (μ ν : MeasureTheory.Measure Ω) [MeasureTheory.IsProbabilityMeasure μ] [MeasureTheory.IsProbabilityMeasure ν], μ.AbsolutelyContinuous ν → MeasureTheory.Integrable (fun x => ((μ.rnDeriv ν x).toReal - 1) ^ 2) ν → Causalean.Stat.tvDist μ ν ≤ 1 / 2 * √(Causalean.Stat.chiSqDiv μ ν)","l":61,"f":"Causalean/Stat/Minimax/ChiSquared.lean"},{"n":"Causalean.Stat.tvDist_le_one","k":"theorem","m":"Causalean.Stat.Minimax.TotalVariation","a":"Stat/Minimax","t":2,"d":"Total variation distance between probability measures is at most one.","s":"∀ {Ω : Type u_1} {mΩ : MeasurableSpace Ω} {μ ν : MeasureTheory.Measure Ω} [MeasureTheory.IsProbabilityMeasure μ] [MeasureTheory.IsProbabilityMeasure ν], Causalean.Stat.tvDist μ ν ≤ 1","l":90,"f":"Causalean/Stat/Minimax/TotalVariation.lean"},{"n":"Causalean.Stat.tvDist_le_sqrt_two_mul_one_sub_affinity","k":"theorem","m":"Causalean.Stat.Minimax.HellingerAffinity","a":"Stat/Minimax","t":1,"d":"Cauchy–Schwarz on the Hellinger affinity. For a dominating measure μ and functions f, g such that f is μ-integrable, g is μ-integrable, f is pointwise nonnegative, g is pointwise nonnegative, f integrates to 1 against μ, and g integrates to 1 against μ — so that f dμ and g dμ are probability densities — the total variation distance between the two weighted laws is at most the square root of twice the affinity defect, √(2(1 − densityAffinity μ f g)). This is the Cauchy–Schwarz half of the standard total-variation–Hellinger comparison, and it is what converts an affinity computation into a Le Cam two-point bound.","s":"∀ {α : Type u_1} [inst : MeasurableSpace α] (μ : MeasureTheory.Measure α) (f g : α → ℝ), MeasureTheory.Integrable f μ → MeasureTheory.Integrable g μ → 0 ≤ f → 0 ≤ g → ∫ (x : α), f x ∂μ = 1 → ∫ (x : α), g x ∂μ = 1 → Causalean.Stat.tvDist (μ.withDensity fun x => ENNReal.ofReal (f x)) (μ.withDensity fun x => ENNReal.ofReal (g x)) ≤ √(2 * (1 - Causalean.Stat.densityAffinity μ f g))","l":120,"f":"Causalean/Stat/Minimax/HellingerAffinity.lean"},{"n":"Causalean.Stat.tvDist_nonneg","k":"theorem","m":"Causalean.Stat.Minimax.TotalVariation","a":"Stat/Minimax","t":2,"d":"Total variation distance between probability measures is nonnegative.","s":"∀ {Ω : Type u_1} {mΩ : MeasurableSpace Ω} {μ ν : MeasureTheory.Measure Ω} [MeasureTheory.IsProbabilityMeasure μ] [MeasureTheory.IsProbabilityMeasure ν], 0 ≤ Causalean.Stat.tvDist μ ν","l":85,"f":"Causalean/Stat/Minimax/TotalVariation.lean"},{"n":"Causalean.Stat.tvDist_symm","k":"theorem","m":"Causalean.Stat.Minimax.TotalVariation","a":"Stat/Minimax","t":2,"d":"tvDist is symmetric.","s":"∀ {Ω : Type u_1} {mΩ : MeasurableSpace Ω} (μ ν : MeasureTheory.Measure Ω), Causalean.Stat.tvDist μ ν = Causalean.Stat.tvDist ν μ","l":94,"f":"Causalean/Stat/Minimax/TotalVariation.lean"},{"n":"Causalean.Stat.two_point_lower_bound_of_chiSqDiv_le","k":"theorem","m":"Causalean.Stat.Minimax.MinimaxRisk","a":"Stat/Minimax","t":1,"d":"χ²-form two-point lower bound. For a measurable estimator est, if two candidate values θ₀, θ₁ are 2s-separated, P₀ is absolutely continuous with respect to P₁, the squared density deviation (dP₀/dP₁ − 1)² is P₁-integrable, and the χ²-divergence chiSqDiv P₀ P₁ is at most c, then the worst-case miss probability is at least (1 − ½√c)/2, via tvDist ≤ ½√χ². Since chiSqDiv tensorizes over i.i.d. samples (chiSqDiv_prod) and is computable for explicit families, this is the form used to certify minimax rates.","s":"∀ {Ω : Type u_1} {mΩ : MeasurableSpace Ω} {P₀ P₁ : MeasureTheory.Measure Ω} [MeasureTheory.IsProbabilityMeasure P₀] [MeasureTheory.IsProbabilityMeasure P₁] {est : Ω → ℝ}, Measurable est → ∀ {θ₀ θ₁ s : ℝ}, 2 * s ≤ |θ₀ - θ₁| → P₀.AbsolutelyContinuous P₁ → MeasureTheory.Integrable (fun x => ((P₀.rnDeriv P₁ x).toReal - 1) ^ 2) P₁ → ∀ {c : ℝ}, Causalean.Stat.chiSqDiv P₀ P₁ ≤ c → (1 - 1 / 2 * √c) / 2 ≤ max (P₀.real {ω | s ≤ |est ω - θ₀|}) (P₁.real {ω | s ≤ |est ω - θ₁|})","l":98,"f":"Causalean/Stat/Minimax/MinimaxRisk.lean"},{"n":"Causalean.Stat.two_point_lower_bound_of_tvDist_le","k":"theorem","m":"Causalean.Stat.Minimax.MinimaxRisk","a":"Stat/Minimax","t":1,"d":"Le Cam two-point bound with an explicit total-variation bound. If two candidate values θ₀, θ₁ : ℝ are 2s-separated, est : Ω → ℝ is a measurable estimator, and the total variation distance tvDist P₀ P₁ is at most c, then the worst-case probability of missing the truth by ≥ s is at least (1 − c)/2. Variant of real_two_point_lower_bound with an explicit total-variation upper bound.","s":"∀ {Ω : Type u_1} {mΩ : MeasurableSpace Ω} {P₀ P₁ : MeasureTheory.Measure Ω} [MeasureTheory.IsProbabilityMeasure P₀] [MeasureTheory.IsProbabilityMeasure P₁] {est : Ω → ℝ}, Measurable est → ∀ {θ₀ θ₁ s c : ℝ}, 2 * s ≤ |θ₀ - θ₁| → Causalean.Stat.tvDist P₀ P₁ ≤ c → (1 - c) / 2 ≤ max (P₀.real {ω | s ≤ |est ω - θ₀|}) (P₁.real {ω | s ≤ |est ω - θ₁|})","l":69,"f":"Causalean/Stat/Minimax/MinimaxRisk.lean"},{"n":"Causalean.Stat.uDegen","k":"def","m":"Causalean.Stat.UStatistic.Basic","a":"Stat/UStatistic","t":1,"d":"Degenerate second-order kernel: g(x, y) = h(x, y) − θ − h₁(x) − h₁(y).","s":"{X : Type u_2} → [inst : MeasurableSpace X] → (X → X → ℝ) → MeasureTheory.Measure X → X → X → ℝ","l":68,"f":"Causalean/Stat/UStatistic/Basic.lean"},{"n":"Causalean.Stat.uDegenOrder","k":"def","m":"Causalean.Stat.UStatistic.OrderM.Basic","a":"Stat/UStatistic/OrderM","t":1,"d":"Higher-order residual kernel after removing the mean and all first Hoeffding projection terms.","s":"{X : Type u_2} → [inst : MeasurableSpace X] → {m : ℕ} → [NeZero m] → ((Fin m → X) → ℝ) → MeasureTheory.Measure X → (Fin m → X) → ℝ","l":116,"f":"Causalean/Stat/UStatistic/OrderM/Basic.lean"},{"n":"Causalean.Stat.uDegenOrder_integral_tail_eq_zero","k":"theorem","m":"Causalean.Stat.UStatistic.OrderM.Basic","a":"Stat/UStatistic/OrderM","t":2,"d":"The first-order Hoeffding residual has zero conditional mean in each coordinate after integrating over all other coordinates, provided the usual finite-product/Fubini identities and slice integrability assumptions hold.","s":"∀ {X : Type u_2} [inst : MeasurableSpace X] {P : MeasureTheory.Measure X} [MeasureTheory.IsProbabilityMeasure P] {m : ℕ} [inst_2 : NeZero m] {h : (Fin m → X) → ℝ}, (∀ (j : Fin m), MeasureTheory.Integrable (fun x => ∫ (tail : { k // k ≠ j } → X), h (Causalean.Stat.insertCoord j x tail) ∂MeasureTheory.Measure.pi fun x => P) P) → (∀ (j : Fin m), ∫ (x : X), ∫ (tail : { k // k ≠ j } → X), h (Causalean.Stat.insertCoord j x tail) ∂MeasureTheory.Measure.pi fun x => P ∂P = Causalean.Stat.uMeanOrder h P) → (∀ (j : Fin m) (x : X), MeasureTheory.Integrable (fun tail => h (Causalean.Stat.insertCoord j x tail)) (MeasureTheory.Measure.pi fun x => P)) → ∀ (j : Fin m) (x : X), (∫ (tail : { k // k ≠ j } → X), Causalean.Stat.uDegenOrder h P (Causalean.Stat.insertCoord j x tail) ∂MeasureTheory.Measure.pi fun x => P) = 0","l":198,"f":"Causalean/Stat/UStatistic/OrderM/Basic.lean"},{"n":"Causalean.Stat.uDegenOrder_pairKernel_of_symm","k":"theorem","m":"Causalean.Stat.UStatistic.OrderM.OrderTwo","a":"Stat/UStatistic/OrderM","t":2,"d":"The order-2 degenerate residual of pairKernel h is the order-2 degenerate kernel uDegen h P, evaluated at the two coordinates.","s":"∀ {X : Type u_2} [inst : MeasurableSpace X] {P : MeasureTheory.Measure X} [MeasureTheory.IsProbabilityMeasure P] {h : X → X → ℝ}, (∀ (x y : X), h x y = h y x) → MeasureTheory.Integrable (fun p => h p.1 p.2) (P.prod P) → (∀ (x : X), MeasureTheory.Integrable (fun y => h x y) P) → Causalean.Stat.uDegenOrder (Causalean.Stat.pairKernel h) P = fun z => Causalean.Stat.uDegen h P (z 0) (z 1)","l":150,"f":"Causalean/Stat/UStatistic/OrderM/OrderTwo.lean"},{"n":"Causalean.Stat.uDegen_integral_right_eq_zero","k":"theorem","m":"Causalean.Stat.UStatistic.Basic","a":"Stat/UStatistic","t":2,"d":"Degeneracy of the second-order kernel. For every fixed x, ∫ g(x, ·) dP = 0. This is the defining property that makes the U-statistic's quadratic remainder negligible at the √n scale.","s":"∀ {X : Type u_2} [inst : MeasurableSpace X] {P : MeasureTheory.Measure X} [MeasureTheory.IsProbabilityMeasure P] {h : X → X → ℝ}, MeasureTheory.Integrable (fun x => ∫ (y : X), h x y ∂P) P → ∀ (x : X), MeasureTheory.Integrable (fun y => h x y) P → ∫ (y : X), Causalean.Stat.uDegen h P x y ∂P = 0","l":99,"f":"Causalean/Stat/UStatistic/Basic.lean"},{"n":"Causalean.Stat.uInfluenceOrder","k":"def","m":"Causalean.Stat.UStatistic.OrderM.Hajek","a":"Stat/UStatistic/OrderM","t":1,"d":"The first-order influence function of an ordered fixed-order kernel: the sum of its coordinatewise first Hoeffding projections. For a symmetric kernel this is m times the usual first projection.","s":"{X : Type u_2} → [inst : MeasurableSpace X] → {m : ℕ} → ((Fin m → X) → ℝ) → MeasureTheory.Measure X → X → ℝ","l":41,"f":"Causalean/Stat/UStatistic/OrderM/Hajek.lean"},{"n":"Causalean.Stat.uInfluenceOrder_eq_card_mul_of_common_projection","k":"theorem","m":"Causalean.Stat.UStatistic.OrderM.Hajek","a":"Stat/UStatistic/OrderM","t":2,"d":"If all coordinatewise first projections agree with a common projection, then the order-m influence function is m times that projection.","s":"∀ {X : Type u_2} [inst : MeasurableSpace X] {P : MeasureTheory.Measure X} {m : ℕ} {h : (Fin m → X) → ℝ} {φ : X → ℝ}, (∀ (j : Fin m) (x : X), Causalean.Stat.uProjOrderAt j h P x = φ x) → Causalean.Stat.uInfluenceOrder h P = fun x => ↑m * φ x","l":48,"f":"Causalean/Stat/UStatistic/OrderM/Hajek.lean"},{"n":"Causalean.Stat.uInfluenceOrder_integrable","k":"theorem","m":"Causalean.Stat.UStatistic.OrderM.Basic","a":"Stat/UStatistic/OrderM","t":2,"d":"The summed first-order influence function is integrable if every coordinatewise first projection is integrable.","s":"∀ {X : Type u_2} [inst : MeasurableSpace X] {P : MeasureTheory.Measure X} {m : ℕ} {h : (Fin m → X) → ℝ}, (∀ (j : Fin m), MeasureTheory.Integrable (Causalean.Stat.uProjOrderAt j h P) P) → MeasureTheory.Integrable (fun x => ∑ j, Causalean.Stat.uProjOrderAt j h P x) P","l":165,"f":"Causalean/Stat/UStatistic/OrderM/Basic.lean"},{"n":"Causalean.Stat.uInfluenceOrder_integral_eq_zero","k":"theorem","m":"Causalean.Stat.UStatistic.OrderM.Basic","a":"Stat/UStatistic/OrderM","t":2,"d":"The summed first-order influence function is centered if every coordinatewise first projection is centered.","s":"∀ {X : Type u_2} [inst : MeasurableSpace X] {P : MeasureTheory.Measure X} {m : ℕ} {h : (Fin m → X) → ℝ}, (∀ (j : Fin m), MeasureTheory.Integrable (Causalean.Stat.uProjOrderAt j h P) P) → (∀ (j : Fin m), ∫ (x : X), Causalean.Stat.uProjOrderAt j h P x ∂P = 0) → ∫ (x : X), ∑ j, Causalean.Stat.uProjOrderAt j h P x ∂P = 0","l":173,"f":"Causalean/Stat/UStatistic/OrderM/Basic.lean"},{"n":"Causalean.Stat.uInfluenceOrder_pairKernel_of_symm","k":"theorem","m":"Causalean.Stat.UStatistic.OrderM.OrderTwo","a":"Stat/UStatistic/OrderM","t":2,"d":"The order-2 influence function of pairKernel h is 2 · h₁, matching the order-2 CLT influence function ψ = 2 · uProj h P.","s":"∀ {X : Type u_2} [inst : MeasurableSpace X] {P : MeasureTheory.Measure X} [MeasureTheory.IsProbabilityMeasure P] {h : X → X → ℝ}, (∀ (x y : X), h x y = h y x) → MeasureTheory.Integrable (fun p => h p.1 p.2) (P.prod P) → (∀ (x : X), MeasureTheory.Integrable (fun y => h x y) P) → Causalean.Stat.uInfluenceOrder (Causalean.Stat.pairKernel h) P = fun x => 2 * Causalean.Stat.uProj h P x","l":138,"f":"Causalean/Stat/UStatistic/OrderM/OrderTwo.lean"},{"n":"Causalean.Stat.uMean","k":"def","m":"Causalean.Stat.UStatistic.Basic","a":"Stat/UStatistic","t":1,"d":"Population mean of the kernel: θ = ∬ h dP dP.","s":"{X : Type u_2} → [inst : MeasurableSpace X] → (X → X → ℝ) → MeasureTheory.Measure X → ℝ","l":60,"f":"Causalean/Stat/UStatistic/Basic.lean"},{"n":"Causalean.Stat.uMeanOrder","k":"def","m":"Causalean.Stat.UStatistic.OrderM.Basic","a":"Stat/UStatistic/OrderM","t":1,"d":"Population mean of an order-m kernel under the product law.","s":"{X : Type u_2} → [inst : MeasurableSpace X] → {m : ℕ} → ((Fin m → X) → ℝ) → MeasureTheory.Measure X → ℝ","l":91,"f":"Causalean/Stat/UStatistic/OrderM/Basic.lean"},{"n":"Causalean.Stat.uMeanOrder_pairKernel","k":"theorem","m":"Causalean.Stat.UStatistic.OrderM.OrderTwo","a":"Stat/UStatistic/OrderM","t":2,"d":"The order-2 population mean of pairKernel h is the order-2 mean of h. Requires product-integrability of h so that the joint integral over Fin 2 → X agrees (via Fubini) with the iterated integral defining uMean.","s":"∀ {X : Type u_2} [inst : MeasurableSpace X] {P : MeasureTheory.Measure X} {h : X → X → ℝ} [MeasureTheory.SigmaFinite P], MeasureTheory.Integrable (fun p => h p.1 p.2) (P.prod P) → Causalean.Stat.uMeanOrder (Causalean.Stat.pairKernel h) P = Causalean.Stat.uMean h P","l":55,"f":"Causalean/Stat/UStatistic/OrderM/OrderTwo.lean"},{"n":"Causalean.Stat.uProj","k":"def","m":"Causalean.Stat.UStatistic.Basic","a":"Stat/UStatistic","t":1,"d":"First Hoeffding projection (centered): h₁(x) = ∫ h(x, ·) dP − θ.","s":"{X : Type u_2} → [inst : MeasurableSpace X] → (X → X → ℝ) → MeasureTheory.Measure X → X → ℝ","l":64,"f":"Causalean/Stat/UStatistic/Basic.lean"},{"n":"Causalean.Stat.uProjOrder","k":"def","m":"Causalean.Stat.UStatistic.OrderM.Basic","a":"Stat/UStatistic/OrderM","t":1,"d":"First Hoeffding projection of a positive-order kernel, using coordinate 0 as the distinguished coordinate.","s":"{X : Type u_2} → [inst : MeasurableSpace X] → {m : ℕ} → [NeZero m] → ((Fin m → X) → ℝ) → MeasureTheory.Measure X → X → ℝ","l":110,"f":"Causalean/Stat/UStatistic/OrderM/Basic.lean"},{"n":"Causalean.Stat.uProjOrderAt","k":"def","m":"Causalean.Stat.UStatistic.OrderM.Basic","a":"Stat/UStatistic/OrderM","t":1,"d":"First Hoeffding projection of an order-m kernel, centred at its population mean. The distinguished coordinate is supplied explicitly; for symmetric kernels all choices agree.","s":"{X : Type u_2} → [inst : MeasurableSpace X] → {m : ℕ} → Fin m → ((Fin m → X) → ℝ) → MeasureTheory.Measure X → X → ℝ","l":100,"f":"Causalean/Stat/UStatistic/OrderM/Basic.lean"},{"n":"Causalean.Stat.uProjOrderAt_integrable","k":"theorem","m":"Causalean.Stat.UStatistic.OrderM.Basic","a":"Stat/UStatistic/OrderM","t":2,"d":"The coordinatewise first projection is integrable whenever the corresponding slice-averaged kernel is integrable.","s":"∀ {X : Type u_2} [inst : MeasurableSpace X] {P : MeasureTheory.Measure X} [MeasureTheory.IsFiniteMeasure P] {m : ℕ} (j : Fin m) {h : (Fin m → X) → ℝ}, MeasureTheory.Integrable (fun x => ∫ (tail : { k // k ≠ j } → X), h (Causalean.Stat.insertCoord j x tail) ∂MeasureTheory.Measure.pi fun x => P) P → MeasureTheory.Integrable (Causalean.Stat.uProjOrderAt j h P) P","l":135,"f":"Causalean/Stat/UStatistic/OrderM/Basic.lean"},{"n":"Causalean.Stat.uProjOrderAt_integral_eq_zero","k":"theorem","m":"Causalean.Stat.UStatistic.OrderM.Basic","a":"Stat/UStatistic/OrderM","t":2,"d":"The coordinatewise first projection integrates to zero once the slice-averaged representation of the population mean is available. The additional equality is the standard finite-product/Fubini identity for the chosen coordinate.","s":"∀ {X : Type u_2} [inst : MeasurableSpace X] {P : MeasureTheory.Measure X} [MeasureTheory.IsProbabilityMeasure P] {m : ℕ} (j : Fin m) {h : (Fin m → X) → ℝ}, MeasureTheory.Integrable (fun x => ∫ (tail : { k // k ≠ j } → X), h (Causalean.Stat.insertCoord j x tail) ∂MeasureTheory.Measure.pi fun x => P) P → ∫ (x : X), ∫ (tail : { k // k ≠ j } → X), h (Causalean.Stat.insertCoord j x tail) ∂MeasureTheory.Measure.pi fun x => P ∂P = Causalean.Stat.uMeanOrder h P → ∫ (x : X), Causalean.Stat.uProjOrderAt j h P x ∂P = 0","l":146,"f":"Causalean/Stat/UStatistic/OrderM/Basic.lean"},{"n":"Causalean.Stat.uProjOrderAt_pairKernel_of_symm","k":"theorem","m":"Causalean.Stat.UStatistic.OrderM.OrderTwo","a":"Stat/UStatistic/OrderM","t":2,"d":"Each coordinate first projection of pairKernel h equals the order-2 first projection of h, for a symmetric kernel. (For j = 0 no symmetry is needed; for j = 1 it is used to swap the integration coordinate.)","s":"∀ {X : Type u_2} [inst : MeasurableSpace X] {P : MeasureTheory.Measure X} [MeasureTheory.IsProbabilityMeasure P] {h : X → X → ℝ}, (∀ (x y : X), h x y = h y x) → MeasureTheory.Integrable (fun p => h p.1 p.2) (P.prod P) → (∀ (x : X), MeasureTheory.Integrable (fun y => h x y) P) → ∀ (j : Fin 2), Causalean.Stat.uProjOrderAt j (Causalean.Stat.pairKernel h) P = Causalean.Stat.uProj h P","l":72,"f":"Causalean/Stat/UStatistic/OrderM/OrderTwo.lean"},{"n":"Causalean.Stat.uProj_integrable","k":"theorem","m":"Causalean.Stat.UStatistic.Basic","a":"Stat/UStatistic","t":2,"d":"The first projection is integrable whenever x ↦ ∫ h(x, ·) dP is.","s":"∀ {X : Type u_2} [inst : MeasurableSpace X] {P : MeasureTheory.Measure X} [MeasureTheory.IsFiniteMeasure P] {h : X → X → ℝ}, MeasureTheory.Integrable (fun x => ∫ (y : X), h x y ∂P) P → MeasureTheory.Integrable (Causalean.Stat.uProj h P) P","l":92,"f":"Causalean/Stat/UStatistic/Basic.lean"},{"n":"Causalean.Stat.uProj_integral_eq_zero","k":"theorem","m":"Causalean.Stat.UStatistic.Basic","a":"Stat/UStatistic","t":2,"d":"The first projection integrates to zero: ∫ h₁ dP = 0.","s":"∀ {X : Type u_2} [inst : MeasurableSpace X] {P : MeasureTheory.Measure X} [MeasureTheory.IsProbabilityMeasure P] {h : X → X → ℝ}, MeasureTheory.Integrable (fun x => ∫ (y : X), h x y ∂P) P → ∫ (x : X), Causalean.Stat.uProj h P x ∂P = 0","l":84,"f":"Causalean/Stat/UStatistic/Basic.lean"},{"n":"Causalean.Stat.uRemainderOrder","k":"def","m":"Causalean.Stat.UStatistic.OrderM.Hajek","a":"Stat/UStatistic/OrderM","t":1,"d":"The higher-order remainder statistic for an order-m U-statistic.","s":"{Ω : Type u_1} → {X : Type u_2} → [inst : MeasurableSpace Ω] → [inst_1 : MeasurableSpace X] → {μ : MeasureTheory.Measure Ω} → {P : MeasureTheory.Measure X} → Causalean.Stat.IIDSample Ω X μ P → {m : ℕ} → [NeZero m] → ((Fin m → X) → ℝ) → ℕ → Ω → ℝ","l":57,"f":"Causalean/Stat/UStatistic/OrderM/Hajek.lean"},{"n":"Causalean.Stat.uStatistic","k":"def","m":"Causalean.Stat.UStatistic.Basic","a":"Stat/UStatistic","t":1,"d":"The order-2 U-statistic with kernel h over the first n sample points: (n(n−1))⁻¹ Σ_{(i,j) ∈ offDiag (range n)} h(Zᵢ, Zⱼ). The off-diagonal sum ranges over ordered pairs i ≠ j; for a symmetric kernel this is the usual (n choose 2)⁻¹ Σ_{i<j}.","s":"{Ω : Type u_1} → {X : Type u_2} → [inst : MeasurableSpace Ω] → [inst_1 : MeasurableSpace X] → {μ : MeasureTheory.Measure Ω} → {P : MeasureTheory.Measure X} → Causalean.Stat.IIDSample Ω X μ P → (X → X → ℝ) → ℕ → Ω → ℝ","l":51,"f":"Causalean/Stat/UStatistic/Basic.lean"},{"n":"Causalean.Stat.uStatisticOrder","k":"def","m":"Causalean.Stat.UStatistic.OrderM.Basic","a":"Stat/UStatistic/OrderM","t":1,"d":"The fixed-order U-statistic averages a kernel over ordered injective sample tuples from the first n observations.","s":"{Ω : Type u_1} → {X : Type u_2} → [inst : MeasurableSpace Ω] → [inst_1 : MeasurableSpace X] → {μ : MeasureTheory.Measure Ω} → {P : MeasureTheory.Measure X} → Causalean.Stat.IIDSample Ω X μ P → {m : ℕ} → ((Fin m → X) → ℝ) → ℕ → Ω → ℝ","l":84,"f":"Causalean/Stat/UStatistic/OrderM/Basic.lean"},{"n":"Causalean.Stat.uStatisticOrder_clt","k":"theorem","m":"Causalean.Stat.UStatistic.OrderM.CLT","a":"Stat/UStatistic/OrderM","t":1,"d":"Fixed-order U-statistic CLT. For an i.i.d. sample S and an order-m kernel h, write ψ for the summed coordinatewise first Hoeffding projection of h. If ψ is measurable, has population mean zero, and is square-integrable, if the higher-order Hájek remainder of the order-m U-statistic is negligible at the √n scale, and if the √n-rescaled U-statistic is almost-everywhere measurable at every sample size, then the √n-rescaled U-statistic converges in distribution to the centered Gaussian law with variance ∫ψ²dP.","s":"∀ {Ω : Type u_1} {X : Type u_2} [inst : MeasurableSpace Ω] [inst_1 : MeasurableSpace X] {μ : MeasureTheory.Measure Ω} {P : MeasureTheory.Measure X} [inst_2 : MeasureTheory.IsProbabilityMeasure μ] (S : Causalean.Stat.IIDSample Ω X μ P) {m : ℕ} [inst_3 : NeZero m] (h : (Fin m → X) → ℝ), Measurable (Causalean.Stat.uInfluenceOrder h P) → ∫ (x : X), Causalean.Stat.uInfluenceOrder h P x ∂P = 0 → MeasureTheory.Integrable (fun x => Causalean.Stat.uInfluenceOrder h P x ^ 2) P → Causalean.Stat.OrderDegenerateNegligible S h → ∀ (hθn_meas : ∀ (n : ℕ), AEMeasurable (Causalean.Stat.IsAsymLinear.rescaledEstimator (Causalean.Stat.uStatisticOrder S h) (Causalean.Stat.uMeanOrder h P) (fun r => Finset.range r) n) μ), Causalean.Stat.Tendsto_dist (Causalean.Stat.IsAsymLinear.rescaledEstimator (Causalean.Stat.uStatisticOrder S h) (Causalean.Stat.uMeanOrder h P) fun r => Finset.range r) (Causalean.Stat.gaussianMeasure 0 (∫ (x : X), Causalean.Stat.uInfluenceOrder h P x ^ 2 ∂P)) μ hθn_meas","l":52,"f":"Causalean/Stat/UStatistic/OrderM/CLT.lean"},{"n":"Causalean.Stat.uStatisticOrder_clt_of_regular","k":"theorem","m":"Causalean.Stat.UStatistic.OrderM.CLT","a":"Stat/UStatistic/OrderM","t":1,"d":"Fixed-order U-statistic CLT (end-to-end). For an i.i.d. sample S and an order-m kernel h, write g for the higher-order Hájek residual of h and ψ for the summed coordinatewise first Hoeffding projection of h. If the residual g is measurable and square-integrable under the m-fold product law, if for every coordinate integrating h over the remaining m − 1 coordinates yields an integrable function of that coordinate with the same population mean uMeanOrder h P in every coordinate and h remains integrable in the remaining coordinates for every fixed value of that coordinate, and if ψ is measurable, mean zero, square-integrable, and the √n-rescaled U-statistic is almost-everywhere measurable at every sample size, then the √n-rescaled order-m U-statistic converges in distribution to the centered Gaussian law with variance ∫ψ²dP.","s":"∀ {Ω : Type u_3} {X : Type u_4} [inst : MeasurableSpace Ω] [inst_1 : MeasurableSpace X] {μ : MeasureTheory.Measure Ω} {P : MeasureTheory.Measure X} [inst_2 : MeasureTheory.IsProbabilityMeasure μ] (S : Causalean.Stat.IIDSample Ω X μ P) {m : ℕ} [inst_3 : NeZero m] (h : (Fin m → X) → ℝ), Measurable (Causalean.Stat.uDegenOrder h P) → MeasureTheory.Integrable (fun z => Causalean.Stat.uDegenOrder h P z ^ 2) (MeasureTheory.Measure.pi fun x => P) → (∀ (j : Fin m), MeasureTheory.Integrable (fun x => ∫ (tail : { k // k ≠ j } → X), h (Causalean.Stat.insertCoord j x tail) ∂MeasureTheory.Measure.pi fun x => P) P) → (∀ (j : Fin m), ∫ (x : X), ∫ (tail : { k // k ≠ j } → X), h (Causalean.Stat.insertCoord j x tail) ∂MeasureTheory.Measure.pi fun x => P ∂P = Causalean.Stat.uMeanOrder h P) → (∀ (j : Fin m) (x : X), MeasureTheory.Integrable (fun tail => h (Causalean.Stat.insertCoord j x tail)) (MeasureTheory.Measure.pi fun x => P)) → Measurable (Causalean.Stat.uInfluenceOrder h P) → ∫ (x : X), Causalean.Stat.uInfluenceOrder h P x ∂P = 0 → MeasureTheory.Integrable (fun x => Causalean.Stat.uInfluenceOrder h P x ^ 2) P → ∀ (hθn_meas : ∀ (n : ℕ), AEMeasurable (Causalean.Stat.IsAsymLinear.rescaledEstimator (Causalean.Stat.uStatisticOrder S h) (Causalean.Stat.uMeanOrder h P) (fun r => Finset.range r) n) μ), Causalean.Stat.Tendsto_dist (Causalean.Stat.IsAsymLinear.rescaledEstimator (Causalean.Stat.uStatisticOrder S h) (Causalean.Stat.uMeanOrder h P) fun r => Finset.range r) (Causalean.Stat.gaussianMeasure 0 (∫ (x : X), Causalean.Stat.uInfluenceOrder h P x ^ 2 ∂P)) μ hθn_meas","l":80,"f":"Causalean/Stat/UStatistic/OrderM/CLT.lean"},{"n":"Causalean.Stat.uStatisticOrder_isAsymLinear","k":"theorem","m":"Causalean.Stat.UStatistic.OrderM.Hajek","a":"Stat/UStatistic/OrderM","t":1,"d":"Fixed-order U-statistic asymptotic linearity. For an i.i.d. sample S and an order-m kernel h, write ψ for the summed coordinatewise first Hoeffding projection of h. If ψ has population mean zero and is square-integrable, and if the higher-order Hájek remainder of the order-m U-statistic is negligible at the √n scale, then the order-m U-statistic is asymptotically linear toward its population mean uMeanOrder h P, with influence function ψ.","s":"∀ {Ω : Type u_1} {X : Type u_2} [inst : MeasurableSpace Ω] [inst_1 : MeasurableSpace X] {μ : MeasureTheory.Measure Ω} {P : MeasureTheory.Measure X} (S : Causalean.Stat.IIDSample Ω X μ P) {m : ℕ} [inst_2 : NeZero m] (h : (Fin m → X) → ℝ), ∫ (x : X), Causalean.Stat.uInfluenceOrder h P x ∂P = 0 → MeasureTheory.Integrable (fun x => Causalean.Stat.uInfluenceOrder h P x ^ 2) P → Causalean.Stat.OrderDegenerateNegligible S h → Causalean.Stat.IsAsymLinear (Causalean.Stat.uStatisticOrder S h) (Causalean.Stat.uMeanOrder h P) (Causalean.Stat.uInfluenceOrder h P) S fun r => Finset.range r","l":284,"f":"Causalean/Stat/UStatistic/OrderM/Hajek.lean"},{"n":"Causalean.Stat.uStatisticOrder_remainder_eq","k":"theorem","m":"Causalean.Stat.UStatistic.OrderM.Hajek","a":"Stat/UStatistic/OrderM","t":2,"d":"The Hájek remainder in IsAsymLinear form is the rescaled higher-order residual U-statistic.","s":"∀ {Ω : Type u_1} {X : Type u_2} [inst : MeasurableSpace Ω] [inst_1 : MeasurableSpace X] {μ : MeasureTheory.Measure Ω} {P : MeasureTheory.Measure X} (S : Causalean.Stat.IIDSample Ω X μ P) {m : ℕ} [inst_2 : NeZero m] (h : (Fin m → X) → ℝ) {n : ℕ}, m ≤ n → (fun ω => √↑(Finset.range n).card * (Causalean.Stat.uStatisticOrder S h n ω - Causalean.Stat.uMeanOrder h P) - (√↑(Finset.range n).card)⁻¹ * ∑ i ∈ Finset.range n, Causalean.Stat.uInfluenceOrder h P (S.Z i ω)) = fun ω => √↑n * Causalean.Stat.uRemainderOrder S h n ω","l":252,"f":"Causalean/Stat/UStatistic/OrderM/Hajek.lean"},{"n":"Causalean.Stat.uStatisticOrder_sub_uMean_eq","k":"theorem","m":"Causalean.Stat.UStatistic.OrderM.Hajek","a":"Stat/UStatistic/OrderM","t":1,"d":"Hájek decomposition for a fixed-order U-statistic. For an i.i.d. sample S, order-m kernel h, and sample outcome ω, if the sample size is at least m, then the order-m U-statistic centered at its population mean decomposes exactly as the average of the first-order influence function over the first n sample points plus the higher-order Hájek remainder statistic.","s":"∀ {Ω : Type u_1} {X : Type u_2} [inst : MeasurableSpace Ω] [inst_1 : MeasurableSpace X] {μ : MeasureTheory.Measure Ω} {P : MeasureTheory.Measure X} (S : Causalean.Stat.IIDSample Ω X μ P) {m : ℕ} [inst_2 : NeZero m] (h : (Fin m → X) → ℝ) {n : ℕ}, m ≤ n → ∀ (ω : Ω), Causalean.Stat.uStatisticOrder S h n ω - Causalean.Stat.uMeanOrder h P = (↑n)⁻¹ * ∑ i ∈ Finset.range n, Causalean.Stat.uInfluenceOrder h P (S.Z i ω) + Causalean.Stat.uRemainderOrder S h n ω","l":197,"f":"Causalean/Stat/UStatistic/OrderM/Hajek.lean"},{"n":"Causalean.Stat.uStatisticOrder_two_eq_uStatistic","k":"theorem","m":"Causalean.Stat.UStatistic.OrderM.Basic","a":"Stat/UStatistic/OrderM","t":2,"d":"The order-2 fixed-order statistic agrees with the existing ordered off-diagonal U-statistic for the corresponding pair kernel.","s":"∀ {Ω : Type u_1} {X : Type u_2} [inst : MeasurableSpace Ω] [inst_1 : MeasurableSpace X] {μ : MeasureTheory.Measure Ω} {P : MeasureTheory.Measure X} (S : Causalean.Stat.IIDSample Ω X μ P) (h : X → X → ℝ) (n : ℕ), Causalean.Stat.uStatisticOrder S (Causalean.Stat.pairKernel h) n = Causalean.Stat.uStatistic S h n","l":362,"f":"Causalean/Stat/UStatistic/OrderM/Basic.lean"},{"n":"Causalean.Stat.uStatistic_clt_of_symmetric_via_orderM","k":"theorem","m":"Causalean.Stat.UStatistic.OrderM.OrderTwo","a":"Stat/UStatistic/OrderM","t":1,"d":"Order-2 U-statistic CLT, derived from the fixed-order-m CLT. For an i.i.d. sample S and a two-argument kernel h : X → X → ℝ that is symmetric and jointly measurable, suppose the order-2 Hájek residual of h is square-integrable under the product law P × P, the row integral x ↦ ∫h(x,y)dP(y) is integrable and each row y ↦ h(x,y) is itself integrable for every x, the first Hoeffding projection of h is square-integrable, and the √n-rescaled U-statistic is almost-everywhere measurable at every sample size. Then the √n-rescaled order-2 U-statistic converges in distribution to the centered Gaussian law with variance 4ζ₁, where ζ₁ is the variance of the first Hoeffding projection.","s":"∀ {Ω : Type u_3} {X : Type u_4} [inst : MeasurableSpace Ω] [inst_1 : MeasurableSpace X] {μ : MeasureTheory.Measure Ω} {P : MeasureTheory.Measure X} (S : Causalean.Stat.IIDSample Ω X μ P) (h : X → X → ℝ), (∀ (x y : X), h x y = h y x) → (Measurable fun p => h p.1 p.2) → MeasureTheory.Integrable (fun p => Causalean.Stat.uDegen h P p.1 p.2 ^ 2) (P.prod P) → MeasureTheory.Integrable (fun x => ∫ (y : X), h x y ∂P) P → (∀ (x : X), MeasureTheory.Integrable (fun y => h x y) P) → MeasureTheory.Integrable (fun x => Causalean.Stat.uProj h P x ^ 2) P → ∀ (hθn_meas : ∀ (n : ℕ), AEMeasurable (Causalean.Stat.IsAsymLinear.rescaledEstimator (Causalean.Stat.uStatistic S h) (Causalean.Stat.uMean h P) (fun m => Finset.range m) n) μ), Causalean.Stat.Tendsto_dist (Causalean.Stat.IsAsymLinear.rescaledEstimator (Causalean.Stat.uStatistic S h) (Causalean.Stat.uMean h P) fun m => Finset.range m) (Causalean.Stat.gaussianMeasure 0 (∫ (x : X), (fun x => 2 * Causalean.Stat.uProj h P x) x ^ 2 ∂P)) μ hθn_meas","l":173,"f":"Causalean/Stat/UStatistic/OrderM/OrderTwo.lean"},{"n":"Causalean.Stat.unifOI","k":"def","m":"Causalean.Stat.Coupling.ProductLossMonotoneCoupling.PIT","a":"Stat/Coupling/ProductLossMonotoneCoupling","t":1,"d":"The uniform probability measure on the open unit interval (0,1), realised as Lebesgue measure restricted to Ioo 0 1.","s":"MeasureTheory.Measure ℝ","l":38,"f":"Causalean/Stat/Coupling/ProductLossMonotoneCoupling/PIT.lean"},{"n":"Causalean.Stat.uniformlyBounded_kernelAffinePullback","k":"theorem","m":"Causalean.Stat.Minimax.MarkovKernelTransport","a":"Stat/Minimax","t":2,"d":"If the affine slope is nonzero and the target estimator is uniformly bounded, then the affine kernel pullback is uniformly bounded on the source experiment.","s":"∀ {X : Type uX} {Y : Type uY} [inst : MeasurableSpace X] [inst_1 : MeasurableSpace Y] (K : ProbabilityTheory.Kernel X Y) [ProbabilityTheory.IsMarkovKernel K] {a b : ℝ}, a ≠ 0 → ∀ {targetEst : Y → ℝ}, Causalean.Stat.UniformlyBounded targetEst → Causalean.Stat.UniformlyBounded (Causalean.Stat.kernelAffinePullback K a b targetEst)","l":95,"f":"Causalean/Stat/Minimax/MarkovKernelTransport.lean"},{"n":"Causalean.Stat.uniformlyBounded_kernelMean","k":"theorem","m":"Causalean.Stat.Minimax.MarkovKernelTransport","a":"Stat/Minimax","t":2,"d":"Uniform boundedness of an estimator implies that averaging it against a Markov kernel is uniformly bounded by the same witness.","s":"∀ {X : Type uX} {Y : Type uY} [inst : MeasurableSpace X] [inst_1 : MeasurableSpace Y] (K : ProbabilityTheory.Kernel X Y) [ProbabilityTheory.IsMarkovKernel K] {T : Y → ℝ}, Causalean.Stat.UniformlyBounded T → Causalean.Stat.UniformlyBounded (Causalean.Stat.kernelMean K T)","l":74,"f":"Causalean/Stat/Minimax/MarkovKernelTransport.lean"},{"n":"Causalean.Stat.uniqueGlobalMax_maximizerOrZero","k":"theorem","m":"Causalean.Stat.MEstimation.FinitePoisson","a":"Stat/MEstimation","t":2,"d":"A criterion with a unique global maximum is maximized uniquely by its total maximizer selector.","s":"∀ {E : Type u_1} [inst : Zero E] (f : E → ℝ), (∃! x, ∀ (y : E), f y ≤ f x) → Causalean.Stat.IsUniqueGlobalMax f (Causalean.Stat.maximizerOrZero f)","l":326,"f":"Causalean/Stat/MEstimation/FinitePoisson.lean"},{"n":"Causalean.Stat.usableGroup","k":"def","m":"Causalean.Stat.Sample.OccupancyWeightedMean.Basic","a":"Stat/Sample/OccupancyWeightedMean","t":1,"d":"A sample group is usable exactly when both of its empirical arm counts are positive.","s":"{Omega : Type u_1} → {kappa : Type u_2} → [DecidableEq kappa] → {n : ℕ} → (Omega → kappa) → (Omega → Bool) → (Fin n → Omega) → kappa → Prop","l":70,"f":"Causalean/Stat/Sample/OccupancyWeightedMean/Basic.lean"},{"n":"Causalean.Stat.usableGroupTotal","k":"def","m":"Causalean.Stat.Sample.OccupancyWeightedMean.Basic","a":"Stat/Sample/OccupancyWeightedMean","t":1,"d":"The usable-group total is the total empirical occupancy of groups having both arms represented.","s":"{Omega : Type u_1} → {kappa : Type u_2} → [Fintype kappa] → [DecidableEq kappa] → {n : ℕ} → (Omega → kappa) → (Omega → Bool) → (Fin n → Omega) → ℕ","l":77,"f":"Causalean/Stat/Sample/OccupancyWeightedMean/Basic.lean"},{"n":"Causalean.Stat.variance_offDiag_kernel_le","k":"theorem","m":"Causalean.Stat.UStatistic.Variance","a":"Stat/UStatistic","t":2,"d":"A bounded kernel's off-diagonal average over an independent sample has variance at most thirty-two times the squared kernel bound divided by the sample size.","s":"∀ {X : Type u_3} [inst : MeasurableSpace X] (μ : MeasureTheory.Measure X) [MeasureTheory.IsProbabilityMeasure μ] (N : ℕ) (kernel : X → X → ℝ), Measurable (Function.uncurry kernel) → ∀ (M : ℝ), (∀ (x y : X), |kernel x y| ≤ M) → ProbabilityTheory.variance (fun target => (↑N * ↑(N - 1))⁻¹ * ∑ i, ∑ j, if i ≠ j then kernel (target i) (target j) else 0) (MeasureTheory.Measure.pi fun x => μ) ≤ 32 * M ^ 2 / ↑N","l":316,"f":"Causalean/Stat/UStatistic/Variance.lean"},{"n":"Causalean.Stat.worstCaseRisk","k":"def","m":"Causalean.Stat.Minimax.MinimaxValue","a":"Stat/Minimax","t":1,"d":"The worst-case risk of a single estimator: the largest risk it incurs anywhere in the model class. The estimator is held fixed and the supremum runs over the class.","s":"{E : Sort u_1} → {Θ : Sort u_3} → (E → Θ → ℝ) → E → ℝ","l":45,"f":"Causalean/Stat/Minimax/MinimaxValue.lean"},{"n":"Causalean.Stat.worstCaseRisk_eq_sSup_range","k":"theorem","m":"Causalean.Stat.Minimax.MinimaxValue","a":"Stat/Minimax","t":2,"d":"The worst-case risk is the supremum of the set of risks the estimator attains across the class. Rewriting handle for arguments phrased on the set of attained risks.","s":"∀ {E : Sort u_1} {Θ : Sort u_3} (risk : E → Θ → ℝ) (e : E), Causalean.Stat.worstCaseRisk risk e = sSup (Set.range (risk e))","l":245,"f":"Causalean/Stat/Minimax/MinimaxValue.lean"},{"n":"Causalean.Stat.worstCaseRisk_le","k":"theorem","m":"Causalean.Stat.Minimax.MinimaxValue","a":"Stat/Minimax","t":2,"d":"A bound holding at every model in a nonempty class bounds the worst-case risk.","s":"∀ {E : Sort u_1} {Θ : Sort u_3} [Nonempty Θ] {risk : E → Θ → ℝ} {e : E} {c : ℝ}, (∀ (θ : Θ), risk e θ ≤ c) → Causalean.Stat.worstCaseRisk risk e ≤ c","l":119,"f":"Causalean/Stat/Minimax/MinimaxValue.lean"},{"n":"Causalean.Stat.worstCaseRisk_mono_class","k":"theorem","m":"Causalean.Stat.Minimax.MinimaxValue","a":"Stat/Minimax","t":2,"d":"Comparing two model classes at a fixed estimator: if every model of the first class is matched, through the map, by a model of the second class whose risk is at least as large, the worst-case risk over the first class is at most that over the second. The second class must bound the estimator's risk, and the first must be nonempty; see worstCaseRisk_mono_class_of_nonneg for the version that drops the nonemptiness.","s":"∀ {E : Sort u_1} {Θ : Sort u_3} {Θ' : Sort u_4} [Nonempty Θ] {risk : E → Θ → ℝ} {risk' : E → Θ' → ℝ} {e : E} (φ : Θ → Θ'), BddAbove (Set.range (risk' e)) → (∀ (θ : Θ), risk e θ ≤ risk' e (φ θ)) → Causalean.Stat.worstCaseRisk risk e ≤ Causalean.Stat.worstCaseRisk risk' e","l":169,"f":"Causalean/Stat/Minimax/MinimaxValue.lean"},{"n":"Causalean.Stat.worstCaseRisk_mono_class_of_nonneg","k":"theorem","m":"Causalean.Stat.Minimax.MinimaxValue","a":"Stat/Minimax","t":2,"d":"Version of worstCaseRisk_mono_class for a nonnegative larger risk: no nonemptiness assumption on the smaller class is needed, because an empty class contributes worst-case risk zero, which the larger nonnegative worst-case risk already dominates.","s":"∀ {E : Sort u_1} {Θ : Sort u_3} {Θ' : Sort u_4} {risk : E → Θ → ℝ} {risk' : E → Θ' → ℝ} {e : E} (φ : Θ → Θ'), BddAbove (Set.range (risk' e)) → (∀ (θ' : Θ'), 0 ≤ risk' e θ') → (∀ (θ : Θ), risk e θ ≤ risk' e (φ θ)) → Causalean.Stat.worstCaseRisk risk e ≤ Causalean.Stat.worstCaseRisk risk' e","l":181,"f":"Causalean/Stat/Minimax/MinimaxValue.lean"},{"n":"Causalean.Stat.worstCaseRisk_nonneg","k":"theorem","m":"Causalean.Stat.Minimax.MinimaxValue","a":"Stat/Minimax","t":2,"d":"A worst-case risk is nonnegative as soon as the risk is, with no boundedness or nonemptiness caveat: an empty class or an unbounded risk both give the value zero.","s":"∀ {E : Sort u_1} {Θ : Sort u_3} {risk : E → Θ → ℝ} {e : E}, (∀ (θ : Θ), 0 ≤ risk e θ) → 0 ≤ Causalean.Stat.worstCaseRisk risk e","l":88,"f":"Causalean/Stat/Minimax/MinimaxValue.lean"},{"n":"Causalean.Stat.worstCaseRisk_of_isEmpty_class","k":"theorem","m":"Causalean.Stat.Minimax.MinimaxValue","a":"Stat/Minimax","t":2,"d":"Over an empty model class the worst-case risk of every estimator is zero.","s":"∀ {E : Sort u_1} {Θ : Sort u_3} [IsEmpty Θ] (risk : E → Θ → ℝ) (e : E), Causalean.Stat.worstCaseRisk risk e = 0","l":64,"f":"Causalean/Stat/Minimax/MinimaxValue.lean"},{"n":"Causalean.Stat.worstCaseRisk_of_unique","k":"theorem","m":"Causalean.Stat.Minimax.MinimaxValue","a":"Stat/Minimax","t":2,"d":"When the model class consists of a single law, the worst-case risk is just the risk at that law. This covers decision problems that minimise a functional of the estimator alone, with no adversarial choice of model.","s":"∀ {E : Sort u_1} {Θ : Sort u_3} [inst : Unique Θ] (risk : E → Θ → ℝ) (e : E), Causalean.Stat.worstCaseRisk risk e = risk e default","l":77,"f":"Causalean/Stat/Minimax/MinimaxValue.lean"},{"n":"Causalean.Stat.worstCaseRisk_subtype_eq_sSup_setOf","k":"theorem","m":"Causalean.Stat.Minimax.MinimaxValue","a":"Stat/Minimax","t":2,"d":"The worst-case risk over a class carved out by a membership condition, written as the supremum of the set of risks attained on the class. This is the bridge to the spelling that states the class by a predicate rather than by a subtype.","s":"∀ {A : Type u_5} {B : Type u_6} {Cls : B → Prop} (risk : A → B → ℝ) (e : A), Causalean.Stat.worstCaseRisk (fun a θ => risk a ↑θ) e = sSup {q | ∃ θ, Cls θ ∧ q = risk e θ}","l":276,"f":"Causalean/Stat/Minimax/MinimaxValue.lean"},{"n":"Causalean.Stat.zEstimator_clt","k":"theorem","m":"Causalean.Stat.MEstimation.ZEstimatorCLT","a":"Stat/MEstimation","t":1,"d":"Z-estimator asymptotic linearity. An estimator sequence that solves the empirical estimating equation eventually, almost surely, is consistent at the target parameter, converges to the target at the parametric $\\sqrt n$-rate, and whose score process is stochastically equicontinuous at the target parameter along the estimator sequence, admits the influence-function representation obtained by applying the negative inverse Jacobian to the target score.","s":"∀ {Ω : Type u_1} {X : Type u_2} [inst : MeasurableSpace Ω] [inst_1 : MeasurableSpace X] {μ : MeasureTheory.Measure Ω} {E : Type u_3} [inst_2 : NormedAddCommGroup E] [inst_3 : InnerProductSpace ℝ E] [FiniteDimensional ℝ E] [inst_5 : MeasurableSpace E] [BorelSpace E] (ψ : E → X → E) (θ₀ : E) (P : MeasureTheory.Measure X) (reg : Causalean.Stat.ZEstimatorRegularity ψ θ₀ P) [MeasureTheory.IsProbabilityMeasure μ] (S : Causalean.Stat.IIDSample Ω X μ P) (θn : ℕ → Ω → E), (∀ ε > 0, Filter.Tendsto (fun n => μ {ω | ε < ‖θn n ω - θ₀‖}) Filter.atTop (nhds 0)) → Causalean.Stat.StochEquicontAt ψ θ₀ P μ S θn → Causalean.Stat.IsBigOp (fun n ω => ‖θn n ω - θ₀‖) (fun n => (√↑n)⁻¹) μ → (∀ᶠ (n : ℕ) in Filter.atTop, ∀ᵐ (ω : Ω) ∂μ, ∑ i ∈ Finset.range n, ψ (θn n ω) (S.Z i ω) = 0) → Causalean.Stat.IsAsymLinearVec θn θ₀ (fun z => -reg.J₀_inv (ψ θ₀ z)) S fun n => Finset.range n","l":63,"f":"Causalean/Stat/MEstimation/ZEstimatorCLT.lean"},{"n":"Causalean.Stat.zEstimator_clt_of_donsker","k":"theorem","m":"Causalean.Stat.MEstimation.ExtremumConsistency","a":"Stat/MEstimation","t":1,"d":"Z-estimator CLT with the equicontinuity hypothesis discharged. Identical conclusion to zEstimator_clt, but the estimator-sequence-specific equicontinuity modulus is replaced by a class-level Donsker condition: if the estimator converges in probability to the target parameter, the score family is asymptotically equicontinuous at the target parameter — a property of the family alone, independent of the particular estimator sequence, the estimator converges to the target at the parametric $\\sqrt n$-rate, and the estimator solves the empirical estimating equation eventually, almost surely, then the estimator is asymptotically linear at the target parameter, with influence function minus the inverse Jacobian applied to the score at the target.","s":"∀ {Ω : Type u_1} {X : Type u_2} [inst : MeasurableSpace Ω] [inst_1 : MeasurableSpace X] {μ : MeasureTheory.Measure Ω} {E : Type u_3} [inst_2 : NormedAddCommGroup E] [inst_3 : InnerProductSpace ℝ E] [FiniteDimensional ℝ E] [inst_5 : MeasurableSpace E] [BorelSpace E] (ψ : E → X → E) (θ₀ : E) (P : MeasureTheory.Measure X) (reg : Causalean.Stat.ZEstimatorRegularity ψ θ₀ P) [MeasureTheory.IsProbabilityMeasure μ] (S : Causalean.Stat.IIDSample Ω X μ P) (θn : ℕ → Ω → E), (∀ ε > 0, Filter.Tendsto (fun n => μ {ω | ε < ‖θn n ω - θ₀‖}) Filter.atTop (nhds 0)) → Causalean.Stat.AsymptoticEquicont ψ θ₀ P μ S → Causalean.Stat.IsBigOp (fun n ω => ‖θn n ω - θ₀‖) (fun n => (√↑n)⁻¹) μ → (∀ᶠ (n : ℕ) in Filter.atTop, ∀ᵐ (ω : Ω) ∂μ, ∑ i ∈ Finset.range n, ψ (θn n ω) (S.Z i ω) = 0) → Causalean.Stat.IsAsymLinearVec θn θ₀ (fun z => -reg.J₀_inv (ψ θ₀ z)) S fun n => Finset.range n","l":126,"f":"Causalean/Stat/MEstimation/ExtremumConsistency.lean"},{"n":"Causalean.Stat.zEstimator_clt_of_extremum","k":"theorem","m":"Causalean.Stat.MEstimation.ExtremumConsistency","a":"Stat/MEstimation","t":1,"d":"Z-estimator CLT from extremum primitives. Identical conclusion to zEstimator_clt, but with consistency derived rather than assumed: given an auxiliary criterion function m, suppose the criterion class indexed by the parameter obeys a uniform law of large numbers, i.e. is Glivenko–Cantelli, the estimator sample-maximises the empirical criterion at every sample size, and the population criterion has a well-separated maximum at the target parameter. Then, provided the score process is stochastically equicontinuous at the target parameter along the estimator sequence, the estimator converges to the target at the parametric $\\sqrt n$-rate, and the estimator solves the empirical estimating equation eventually, almost surely, the estimator is asymptotically linear at the target parameter, with influence function minus the inverse Jacobian applied to the score at the target.","s":"∀ {Ω : Type u_1} {X : Type u_2} [inst : MeasurableSpace Ω] [inst_1 : MeasurableSpace X] {μ : MeasureTheory.Measure Ω} {E : Type u_3} [inst_2 : NormedAddCommGroup E] [inst_3 : InnerProductSpace ℝ E] [FiniteDimensional ℝ E] [inst_5 : MeasurableSpace E] [BorelSpace E] (ψ : E → X → E) (θ₀ : E) (P : MeasureTheory.Measure X) (reg : Causalean.Stat.ZEstimatorRegularity ψ θ₀ P) [MeasureTheory.IsProbabilityMeasure μ] (S : Causalean.Stat.IIDSample Ω X μ P) (θn : ℕ → Ω → E) (m : E → X → ℝ), Causalean.Stat.GlivenkoCantelli S m → (∀ (n : ℕ) (ω : Ω), S.sampleMean (m θ₀) n ω ≤ S.sampleMean (m (θn n ω)) n ω) → (∀ (ε : ℝ), 0 < ε → ∃ η, 0 < η ∧ ∀ (θ : E), ε ≤ dist θ θ₀ → ∫ (x : X), m θ x ∂P + η ≤ ∫ (x : X), m θ₀ x ∂P) → Causalean.Stat.StochEquicontAt ψ θ₀ P μ S θn → Causalean.Stat.IsBigOp (fun n ω => ‖θn n ω - θ₀‖) (fun n => (√↑n)⁻¹) μ → (∀ᶠ (n : ℕ) in Filter.atTop, ∀ᵐ (ω : Ω) ∂μ, ∑ i ∈ Finset.range n, ψ (θn n ω) (S.Z i ω) = 0) → Causalean.Stat.IsAsymLinearVec θn θ₀ (fun z => -reg.J₀_inv (ψ θ₀ z)) S fun n => Finset.range n","l":88,"f":"Causalean/Stat/MEstimation/ExtremumConsistency.lean"},{"n":"Causalean.Stat.zEstimator_clt_of_extremum_donsker","k":"theorem","m":"Causalean.Stat.MEstimation.ExtremumConsistency","a":"Stat/MEstimation","t":1,"d":"Z-estimator CLT from primitive conditions: both opaque hypotheses discharged. Combines zEstimator_clt_of_extremum and zEstimator_clt_of_donsker: if the criterion class indexed by the parameter obeys a uniform law of large numbers, the estimator sample-maximises the empirical criterion at every sample size, the population criterion has a well-separated maximum at the target parameter, the score family is asymptotically equicontinuous at the target parameter, the estimator converges to the target at the parametric $\\sqrt n$-rate, and the estimator solves the empirical estimating equation eventually, almost surely, then the estimator is asymptotically linear at the target parameter, with influence function minus the inverse Jacobian applied to the score at the target.","s":"∀ {Ω : Type u_1} {X : Type u_2} [inst : MeasurableSpace Ω] [inst_1 : MeasurableSpace X] {μ : MeasureTheory.Measure Ω} {E : Type u_3} [inst_2 : NormedAddCommGroup E] [inst_3 : InnerProductSpace ℝ E] [FiniteDimensional ℝ E] [inst_5 : MeasurableSpace E] [BorelSpace E] (ψ : E → X → E) (θ₀ : E) (P : MeasureTheory.Measure X) (reg : Causalean.Stat.ZEstimatorRegularity ψ θ₀ P) [MeasureTheory.IsProbabilityMeasure μ] (S : Causalean.Stat.IIDSample Ω X μ P) (θn : ℕ → Ω → E) (m : E → X → ℝ), Causalean.Stat.GlivenkoCantelli S m → (∀ (n : ℕ) (ω : Ω), S.sampleMean (m θ₀) n ω ≤ S.sampleMean (m (θn n ω)) n ω) → (∀ (ε : ℝ), 0 < ε → ∃ η, 0 < η ∧ ∀ (θ : E), ε ≤ dist θ θ₀ → ∫ (x : X), m θ x ∂P + η ≤ ∫ (x : X), m θ₀ x ∂P) → Causalean.Stat.AsymptoticEquicont ψ θ₀ P μ S → Causalean.Stat.IsBigOp (fun n ω => ‖θn n ω - θ₀‖) (fun n => (√↑n)⁻¹) μ → (∀ᶠ (n : ℕ) in Filter.atTop, ∀ᵐ (ω : Ω) ∂μ, ∑ i ∈ Finset.range n, ψ (θn n ω) (S.Z i ω) = 0) → Causalean.Stat.IsAsymLinearVec θn θ₀ (fun z => -reg.J₀_inv (ψ θ₀ z)) S fun n => Finset.range n","l":158,"f":"Causalean/Stat/MEstimation/ExtremumConsistency.lean"},{"n":"Causalean.SteinMethod.DepGraph","k":"structure","m":"Causalean.Mathlib.Probability.SteinMethod.DepGraphCLT","a":"Mathlib/Probability/SteinMethod","t":1,"d":"A dependency graph for a family of real-valued random variables X on a probability space with law μ: bundles an adjacency relation on the index set that is reflexive and symmetric, together with measurability of each variable in the family and the defining property that any two index sets joined by no edge carry independent variable tuples. The closed neighborhood of an index is the set of indices adjacent to it.","s":"{Ω : Type u_1} → [inst : MeasurableSpace Ω] → {ι : Type u_2} → (ι → Ω → ℝ) → MeasureTheory.Measure Ω → Type u_2","l":45,"f":"Causalean/Mathlib/Probability/SteinMethod/DepGraphCLT.lean"},{"n":"Causalean.SteinMethod.DepGraph.abs_cov_locProd_le","k":"theorem","m":"Causalean.Mathlib.Probability.SteinMethod.DepGraphCLT","a":"Mathlib/Probability/SteinMethod","t":2,"d":"If every variable is bounded in absolute value by a nonnegative constant and each closed dependency neighborhood has at most m members, the absolute covariance of any two localized products is at most twice the square of m times that bound squared.","s":"∀ {Ω : Type u_1} [inst : MeasurableSpace Ω] {μ : MeasureTheory.Measure Ω} [MeasureTheory.IsProbabilityMeasure μ] {ι : Type u_2} [inst_2 : Fintype ι] {X : ι → Ω → ℝ} (D : Causalean.SteinMethod.DepGraph X μ) {B : ℝ}, 0 ≤ B → (∀ (i : ι) (ω : Ω), |X i ω| ≤ B) → ∀ {m : ℕ}, (∀ (i : ι), (D.nbhd i).card ≤ m) → ∀ (i j : ι), |ProbabilityTheory.covariance (fun ω => X i ω * ∑ k ∈ D.nbhd i, X k ω) (fun ω => X j ω * ∑ k ∈ D.nbhd j, X k ω) μ| ≤ 2 * (↑m * B ^ 2) ^ 2","l":206,"f":"Causalean/Mathlib/Probability/SteinMethod/DepGraphCLT.lean"},{"n":"Causalean.SteinMethod.DepGraph.abs_locProd_le","k":"theorem","m":"Causalean.Mathlib.Probability.SteinMethod.DepGraphCLT","a":"Mathlib/Probability/SteinMethod","t":2,"d":"If each summand has absolute value at most B and each neighborhood has at most m members, then a summand times its neighborhood sum has absolute value at most m times B squared.","s":"∀ {Ω : Type u_1} [inst : MeasurableSpace Ω] {μ : MeasureTheory.Measure Ω} {ι : Type u_2} [inst_1 : Fintype ι] {X : ι → Ω → ℝ} (D : Causalean.SteinMethod.DepGraph X μ) {B : ℝ}, 0 ≤ B → (∀ (i : ι) (ω : Ω), |X i ω| ≤ B) → ∀ {m : ℕ}, (∀ (i : ι), (D.nbhd i).card ≤ m) → ∀ (i : ι) (ω : Ω), |X i ω * ∑ k ∈ D.nbhd i, X k ω| ≤ ↑m * B ^ 2","l":110,"f":"Causalean/Mathlib/Probability/SteinMethod/DepGraphCLT.lean"},{"n":"Causalean.SteinMethod.DepGraph.abs_nbhdSum_le","k":"theorem","m":"Causalean.Mathlib.Probability.SteinMethod.DepGraphCLT","a":"Mathlib/Probability/SteinMethod","t":2,"d":"If each summand is bounded in absolute value by B and every closed neighborhood has at most m indices, then the absolute value of each neighborhood sum is at most m times B.","s":"∀ {Ω : Type u_1} [inst : MeasurableSpace Ω] {μ : MeasureTheory.Measure Ω} {ι : Type u_2} [inst_1 : Fintype ι] {X : ι → Ω → ℝ} (D : Causalean.SteinMethod.DepGraph X μ) {B : ℝ}, 0 ≤ B → (∀ (i : ι) (ω : Ω), |X i ω| ≤ B) → ∀ {m : ℕ} (i : ι), (D.nbhd i).card ≤ m → ∀ (ω : Ω), |∑ k ∈ D.nbhd i, X k ω| ≤ ↑m * B","l":97,"f":"Causalean/Mathlib/Probability/SteinMethod/DepGraphCLT.lean"},{"n":"Causalean.SteinMethod.DepGraph.cov_mul_nbhd_eq_zero","k":"theorem","m":"Causalean.Mathlib.Probability.SteinMethod.DepGraphCLT","a":"Mathlib/Probability/SteinMethod","t":2,"d":"Covariance vanishing for separated indices. If Nᵢ and Nⱼ have no edges between them, the localized products Xᵢ·Tᵢ and Xⱼ·Tⱼ are uncorrelated.","s":"∀ {Ω : Type u_1} [inst : MeasurableSpace Ω] {μ : MeasureTheory.Measure Ω} {ι : Type u_2} [inst_1 : Fintype ι] {X : ι → Ω → ℝ} (D : Causalean.SteinMethod.DepGraph X μ) {i j : ι}, (∀ a ∈ D.nbhd i, ∀ b ∈ D.nbhd j, ¬D.G a b) → ∫ (x : Ω), (fun ω => (X i ω * ∑ k ∈ D.nbhd i, X k ω) * (X j ω * ∑ k ∈ D.nbhd j, X k ω)) x ∂μ = (∫ (x : Ω), (fun ω => X i ω * ∑ k ∈ D.nbhd i, X k ω) x ∂μ) * ∫ (x : Ω), (fun ω => X j ω * ∑ k ∈ D.nbhd j, X k ω) x ∂μ","l":154,"f":"Causalean/Mathlib/Probability/SteinMethod/DepGraphCLT.lean"},{"n":"Causalean.SteinMethod.DepGraph.indepFun_leaveOut","k":"theorem","m":"Causalean.Mathlib.Probability.SteinMethod.DepGraphCLT","a":"Mathlib/Probability/SteinMethod","t":2,"d":"Leave-out independence (the stein_cdf_clt hypothesis hindep).","s":"∀ {Ω : Type u_1} [inst : MeasurableSpace Ω] {μ : MeasureTheory.Measure Ω} {ι : Type u_2} [inst_1 : Fintype ι] [inst_2 : DecidableEq ι] {X : ι → Ω → ℝ} (D : Causalean.SteinMethod.DepGraph X μ) (i : ι), ProbabilityTheory.IndepFun (X i) (fun ω => ∑ j ∈ Finset.univ \\ D.nbhd i, X j ω) μ","l":123,"f":"Causalean/Mathlib/Probability/SteinMethod/DepGraphCLT.lean"},{"n":"Causalean.SteinMethod.DepGraph.memLp_locProd","k":"theorem","m":"Causalean.Mathlib.Probability.SteinMethod.DepGraphCLT","a":"Mathlib/Probability/SteinMethod","t":2,"d":"If every variable is bounded in absolute value by a nonnegative constant and each closed dependency neighborhood has bounded size, each variable times its neighborhood sum has a finite second moment.","s":"∀ {Ω : Type u_1} [inst : MeasurableSpace Ω] {μ : MeasureTheory.Measure Ω} [MeasureTheory.IsProbabilityMeasure μ] {ι : Type u_2} [inst_2 : Fintype ι] {X : ι → Ω → ℝ} (D : Causalean.SteinMethod.DepGraph X μ) {B : ℝ}, 0 ≤ B → (∀ (i : ι) (ω : Ω), |X i ω| ≤ B) → ∀ {m : ℕ}, (∀ (i : ι), (D.nbhd i).card ≤ m) → ∀ (i : ι), MeasureTheory.MemLp (fun ω => X i ω * ∑ k ∈ D.nbhd i, X k ω) 2 μ","l":195,"f":"Causalean/Mathlib/Probability/SteinMethod/DepGraphCLT.lean"},{"n":"Causalean.SteinMethod.DepGraph.mem_nbhd_iff","k":"theorem","m":"Causalean.Mathlib.Probability.SteinMethod.DepGraphCLT","a":"Mathlib/Probability/SteinMethod","t":2,"d":"Membership in the neighborhood is exactly adjacency.","s":"∀ {Ω : Type u_1} [inst : MeasurableSpace Ω] {μ : MeasureTheory.Measure Ω} {ι : Type u_2} [inst_1 : Fintype ι] {X : ι → Ω → ℝ} (D : Causalean.SteinMethod.DepGraph X μ) {i j : ι}, j ∈ D.nbhd i ↔ D.G i j","l":75,"f":"Causalean/Mathlib/Probability/SteinMethod/DepGraphCLT.lean"},{"n":"Causalean.SteinMethod.DepGraph.nbhd","k":"def","m":"Causalean.Mathlib.Probability.SteinMethod.DepGraphCLT","a":"Mathlib/Probability/SteinMethod","t":1,"d":"The closed dependency neighborhood N i = {j | G i j}.","s":"{Ω : Type u_1} → [inst : MeasurableSpace Ω] → {μ : MeasureTheory.Measure Ω} → {ι : Type u_2} → [Fintype ι] → {X : ι → Ω → ℝ} → Causalean.SteinMethod.DepGraph X μ → ι → Finset ι","l":70,"f":"Causalean/Mathlib/Probability/SteinMethod/DepGraphCLT.lean"},{"n":"Causalean.SteinMethod.DepGraph.self_mem_nbhd","k":"theorem","m":"Causalean.Mathlib.Probability.SteinMethod.DepGraphCLT","a":"Mathlib/Probability/SteinMethod","t":2,"d":"Each index is in its own neighborhood.","s":"∀ {Ω : Type u_1} [inst : MeasurableSpace Ω] {μ : MeasureTheory.Measure Ω} {ι : Type u_2} [inst_1 : Fintype ι] {X : ι → Ω → ℝ} (D : Causalean.SteinMethod.DepGraph X μ) (i : ι), i ∈ D.nbhd i","l":81,"f":"Causalean/Mathlib/Probability/SteinMethod/DepGraphCLT.lean"},{"n":"Causalean.SteinMethod.DepGraph.sum_E_nbhd_sq_le","k":"theorem","m":"Causalean.Mathlib.Probability.SteinMethod.DepGraphCLT","a":"Mathlib/Probability/SteinMethod","t":2,"d":"Negligibility bound (herr2). For any finite neighborhoods of size at most m, measurable summands bounded by B satisfy ∑ᵢ E[|Xᵢ|·Tᵢ²] ≤ m²·N·B³.","s":"∀ {Ω : Type u_1} [inst : MeasurableSpace Ω] {μ : MeasureTheory.Measure Ω} [MeasureTheory.IsProbabilityMeasure μ] {ι : Type u_2} [inst_2 : Fintype ι] {X : ι → Ω → ℝ} (N : ι → Finset ι), (∀ (i : ι), Measurable (X i)) → ∀ {B : ℝ}, 0 ≤ B → (∀ (i : ι) (ω : Ω), |X i ω| ≤ B) → ∀ {m : ℕ}, (∀ (i : ι), (N i).card ≤ m) → ∑ i, ∫ (ω : Ω), |X i ω| * (∑ k ∈ N i, X k ω) ^ 2 ∂μ ≤ ↑m ^ 2 * ↑(Fintype.card ι) * B ^ 3","l":337,"f":"Causalean/Mathlib/Probability/SteinMethod/DepGraphCLT.lean"},{"n":"Causalean.SteinMethod.DepGraph.var_nbhd_prod_le","k":"theorem","m":"Causalean.Mathlib.Probability.SteinMethod.DepGraphCLT","a":"Mathlib/Probability/SteinMethod","t":2,"d":"Pair-counting variance bound (herr1). With bounded summands |Xᵢ| ≤ B and degree ≤ m, the covariance double sum collapses to the ≤ N·m³ pairs at graph distance at most three, each bounded by 2(mB²)², giving Var(∑ᵢ Xᵢ·Tᵢ) ≤ 2·m⁵·N·B⁴.","s":"∀ {Ω : Type u_1} [inst : MeasurableSpace Ω] {μ : MeasureTheory.Measure Ω} [MeasureTheory.IsProbabilityMeasure μ] {ι : Type u_2} [inst_2 : Fintype ι] {X : ι → Ω → ℝ} (D : Causalean.SteinMethod.DepGraph X μ) {B : ℝ}, 0 ≤ B → (∀ (i : ι) (ω : Ω), |X i ω| ≤ B) → ∀ {m : ℕ}, (∀ (i : ι), (D.nbhd i).card ≤ m) → ProbabilityTheory.variance (fun ω => ∑ i, X i ω * ∑ k ∈ D.nbhd i, X k ω) μ ≤ 2 * ↑m ^ 5 * ↑(Fintype.card ι) * B ^ 4","l":255,"f":"Causalean/Mathlib/Probability/SteinMethod/DepGraphCLT.lean"},{"n":"Causalean.SteinMethod.abs_gExpect_le","k":"theorem","m":"Causalean.Mathlib.Probability.SteinMethod.Solution","a":"Mathlib/Probability/SteinMethod","t":2,"d":"The absolute value of the standard-normal expectation of a uniformly bounded function is at most the same bound. This is useful throughout Stein-method estimates.","s":"∀ {h : ℝ → ℝ} {C : ℝ}, (∀ (x : ℝ), |h x| ≤ C) → |Causalean.SteinMethod.gExpect h| ≤ C","l":56,"f":"Causalean/Mathlib/Probability/SteinMethod/Solution.lean"},{"n":"Causalean.SteinMethod.abs_integral_mul_le_sqrt","k":"theorem","m":"Causalean.Mathlib.Probability.SteinMethod.DependencyCLT","a":"Mathlib/Probability/SteinMethod","t":2,"d":"For square-integrable real-valued functions, the absolute value of their product integral is at most the product of the square roots of their two squared integrals.","s":"∀ {Ω : Type u_1} [inst : MeasurableSpace Ω] {μ : MeasureTheory.Measure Ω} (f g : Ω → ℝ), MeasureTheory.MemLp f 2 μ → MeasureTheory.MemLp g 2 μ → |∫ (ω : Ω), f ω * g ω ∂μ| ≤ √(∫ (ω : Ω), f ω ^ 2 ∂μ) * √(∫ (ω : Ω), g ω ^ 2 ∂μ)","l":55,"f":"Causalean/Mathlib/Probability/SteinMethod/DependencyCLT.lean"},{"n":"Causalean.SteinMethod.abs_le_exp_sq_div_four","k":"theorem","m":"Causalean.Mathlib.Probability.SteinMethod.Bounds","a":"Mathlib/Probability/SteinMethod","t":2,"d":"The absolute value of any real number is at most the exponential of one quarter of its square.","s":"∀ (x : ℝ), |x| ≤ Real.exp (x ^ 2 / 4)","l":123,"f":"Causalean/Mathlib/Probability/SteinMethod/Bounds.lean"},{"n":"Causalean.SteinMethod.bounded_degree_dependency_clt","k":"theorem","m":"Causalean.Mathlib.Probability.SteinMethod.StandardizedDepGraphCLT","a":"Mathlib/Probability/SteinMethod","t":1,"d":"Bounded-degree dependency-graph CLT (raw, variance-floor form). Fix, for each n, a probability measure μ n, triangular-array summands X n i, and a dependency graph Dep n on the index set, with every dependency neighborhood of size at most a fixed bound Dmax. Suppose the summands are uniformly bounded in absolute value by a nonnegative constant M and mean zero, with the total variance v n satisfying, for a strictly positive constant c, a floor v n ≥ c · card (ι n) eventually in n; suppose also the index-set cardinality diverges to infinity. Then for every threshold s, the CDF of the standardized sum depSum (X n) / √(v n) under μ n at s converges, as n → ∞, to the standard-normal CDF at s. The index-size divergence together with the variance floor forces the Lyapunov ratio card (ι n) · (M / √(v n))³ → 0 that drives the underlying Stein bound.","s":"∀ {Ω : ℕ → Type u_1} [inst : (n : ℕ) → MeasurableSpace (Ω n)] (μ : (n : ℕ) → MeasureTheory.Measure (Ω n)) [∀ (n : ℕ), MeasureTheory.IsProbabilityMeasure (μ n)] {ι : ℕ → Type u_2} [inst_2 : (n : ℕ) → Fintype (ι n)] (X : (n : ℕ) → ι n → Ω n → ℝ) (Dep : (n : ℕ) → Causalean.SteinMethod.DepGraph (X n) (μ n)) (Dmax : ℕ), (∀ (n : ℕ) (i : ι n), ((Dep n).nbhd i).card ≤ Dmax) → ∀ (M : ℝ), 0 ≤ M → (∀ (n : ℕ) (i : ι n) (ω : Ω n), |X n i ω| ≤ M) → (∀ (n : ℕ) (i : ι n), ∫ (ω : Ω n), X n i ω ∂μ n = 0) → ∀ (v : ℕ → ℝ), (∀ (n : ℕ), ∫ (ω : Ω n), Causalean.SteinMethod.depSum (X n) ω ^ 2 ∂μ n = v n) → ∀ (c : ℝ), 0 < c → (∀ᶠ (n : ℕ) in Filter.atTop, c * ↑(Fintype.card (ι n)) ≤ v n) → Filter.Tendsto (fun n => Fintype.card (ι n)) Filter.atTop Filter.atTop → ∀ (s : ℝ), Filter.Tendsto (fun n => (MeasureTheory.Measure.map (fun ω => Causalean.SteinMethod.depSum (X n) ω / √(v n)) (μ n)).real (Set.Iic s)) Filter.atTop (nhds ((ProbabilityTheory.gaussianReal 0 1).real (Set.Iic s)))","l":172,"f":"Causalean/Mathlib/Probability/SteinMethod/StandardizedDepGraphCLT.lean"},{"n":"Causalean.SteinMethod.bounded_degree_dependency_clt_eventually_bounded","k":"theorem","m":"Causalean.Mathlib.Probability.SteinMethod.StandardizedDepGraphCLT","a":"Mathlib/Probability/SteinMethod","t":2,"d":"Eventual-bound variant of bounded_degree_dependency_clt: only the tail of a triangular array affects the limiting CDF, so the uniform summand bound need only hold after a finite prefix.","s":"∀ {Ω : ℕ → Type u_1} [inst : (n : ℕ) → MeasurableSpace (Ω n)] (μ : (n : ℕ) → MeasureTheory.Measure (Ω n)) [∀ (n : ℕ), MeasureTheory.IsProbabilityMeasure (μ n)] {ι : ℕ → Type u_2} [inst_2 : (n : ℕ) → Fintype (ι n)] (X : (n : ℕ) → ι n → Ω n → ℝ) (Dep : (n : ℕ) → Causalean.SteinMethod.DepGraph (X n) (μ n)) (Dmax : ℕ), (∀ (n : ℕ) (i : ι n), ((Dep n).nbhd i).card ≤ Dmax) → ∀ (M : ℝ), 0 ≤ M → (∀ᶠ (n : ℕ) in Filter.atTop, ∀ (i : ι n) (ω : Ω n), |X n i ω| ≤ M) → (∀ (n : ℕ) (i : ι n), ∫ (ω : Ω n), X n i ω ∂μ n = 0) → ∀ (v : ℕ → ℝ), (∀ (n : ℕ), ∫ (ω : Ω n), Causalean.SteinMethod.depSum (X n) ω ^ 2 ∂μ n = v n) → ∀ (c : ℝ), 0 < c → (∀ᶠ (n : ℕ) in Filter.atTop, c * ↑(Fintype.card (ι n)) ≤ v n) → Filter.Tendsto (fun n => Fintype.card (ι n)) Filter.atTop Filter.atTop → ∀ (s : ℝ), Filter.Tendsto (fun n => (MeasureTheory.Measure.map (fun ω => Causalean.SteinMethod.depSum (X n) ω / √(v n)) (μ n)).real (Set.Iic s)) Filter.atTop (nhds ((ProbabilityTheory.gaussianReal 0 1).real (Set.Iic s)))","l":228,"f":"Causalean/Mathlib/Probability/SteinMethod/StandardizedDepGraphCLT.lean"},{"n":"Causalean.SteinMethod.bounded_degree_dependency_clt_of_variance_floor_all","k":"theorem","m":"Causalean.Mathlib.Probability.SteinMethod.StandardizedDepGraphCLT","a":"Mathlib/Probability/SteinMethod","t":2,"d":"Dependency-graph CLT wrapper when the linear variance floor holds at every index. The public theorem below removes this all-index convenience by shifting to a tail.","s":"∀ {Ω : ℕ → Type u_1} [inst : (n : ℕ) → MeasurableSpace (Ω n)] (μ : (n : ℕ) → MeasureTheory.Measure (Ω n)) [∀ (n : ℕ), MeasureTheory.IsProbabilityMeasure (μ n)] {ι : ℕ → Type u_2} [inst_2 : (n : ℕ) → Fintype (ι n)] (X : (n : ℕ) → ι n → Ω n → ℝ) (Dep : (n : ℕ) → Causalean.SteinMethod.DepGraph (X n) (μ n)) (Dmax : ℕ), (∀ (n : ℕ) (i : ι n), ((Dep n).nbhd i).card ≤ Dmax) → ∀ (M : ℝ), 0 ≤ M → (∀ (n : ℕ) (i : ι n) (ω : Ω n), |X n i ω| ≤ M) → (∀ (n : ℕ) (i : ι n), ∫ (ω : Ω n), X n i ω ∂μ n = 0) → ∀ (v : ℕ → ℝ), (∀ (n : ℕ), ∫ (ω : Ω n), Causalean.SteinMethod.depSum (X n) ω ^ 2 ∂μ n = v n) → ∀ (c : ℝ), 0 < c → (∀ (n : ℕ), c * ↑(Fintype.card (ι n)) ≤ v n) → (∀ (n : ℕ), 0 < Fintype.card (ι n)) → Filter.Tendsto (fun n => Fintype.card (ι n)) Filter.atTop Filter.atTop → ∀ (s : ℝ), Filter.Tendsto (fun n => (MeasureTheory.Measure.map (fun ω => Causalean.SteinMethod.depSum (X n) ω / √(v n)) (μ n)).real (Set.Iic s)) Filter.atTop (nhds ((ProbabilityTheory.gaussianReal 0 1).real (Set.Iic s)))","l":57,"f":"Causalean/Mathlib/Probability/SteinMethod/StandardizedDepGraphCLT.lean"},{"n":"Causalean.SteinMethod.cdf_tendsto_of_charFun_tendsto","k":"theorem","m":"Causalean.Mathlib.Probability.SteinMethod.CLT","a":"Mathlib/Probability/SteinMethod","t":2,"d":"CDF convergence from characteristic-function convergence. A sequence of real probability laws whose characteristic functions converge to those of an atomless target law has convergent CDF values at every threshold.","s":"∀ (lawn : ℕ → MeasureTheory.ProbabilityMeasure ℝ) (ν : MeasureTheory.ProbabilityMeasure ℝ) [MeasureTheory.NullSingletonClass ↑ν], (∀ (t : ℝ), Filter.Tendsto (fun n => MeasureTheory.charFun (↑(lawn n)) t) Filter.atTop (nhds (MeasureTheory.charFun (↑ν) t))) → ∀ (s : ℝ), Filter.Tendsto (fun n => (↑(lawn n)).real (Set.Iic s)) Filter.atTop (nhds ((↑ν).real (Set.Iic s)))","l":42,"f":"Causalean/Mathlib/Probability/SteinMethod/CLT.lean"},{"n":"Causalean.SteinMethod.charFun_map_eq_cos_sin","k":"theorem","m":"Causalean.Mathlib.Probability.SteinMethod.CLT","a":"Mathlib/Probability/SteinMethod","t":2,"d":"A measurable real random variable under a finite measure has a characteristic function whose real and imaginary components are the corresponding cosine and sine integrals.","s":"∀ {Ω : Type u_1} [inst : MeasurableSpace Ω] (ν : MeasureTheory.Measure Ω) [MeasureTheory.IsFiniteMeasure ν] (W : Ω → ℝ), Measurable W → ∀ (t : ℝ), MeasureTheory.charFun (MeasureTheory.Measure.map W ν) t = ↑(∫ (ω : Ω), Real.cos (t * W ω) ∂ν) + ↑(∫ (ω : Ω), Real.sin (t * W ω) ∂ν) * Complex.I","l":76,"f":"Causalean/Mathlib/Probability/SteinMethod/CLT.lean"},{"n":"Causalean.SteinMethod.depGraph_div_const","k":"def","m":"Causalean.Mathlib.Probability.SteinMethod.StandardizedDepGraphCLT","a":"Mathlib/Probability/SteinMethod","t":1,"d":"Dividing every summand in a dependency graph by the same deterministic constant preserves the graph and transfers the independence field by measurable post-composition.","s":"{Ω : Type u_1} → {ι : Type u_2} → [inst : MeasurableSpace Ω] → [Fintype ι] → {μ : MeasureTheory.Measure Ω} → {X : ι → Ω → ℝ} → Causalean.SteinMethod.DepGraph X μ → (s : ℝ) → Causalean.SteinMethod.DepGraph (fun i ω => X i ω / s) μ","l":29,"f":"Causalean/Mathlib/Probability/SteinMethod/StandardizedDepGraphCLT.lean"},{"n":"Causalean.SteinMethod.depSum","k":"def","m":"Causalean.Mathlib.Probability.SteinMethod.DependencyCLT","a":"Mathlib/Probability/SteinMethod","t":1,"d":"The standardized sum W = ∑ᵢ Xᵢ.","s":"{Ω : Type u_1} → {ι : Type u_2} → [Fintype ι] → (ι → Ω → ℝ) → Ω → ℝ","l":48,"f":"Causalean/Mathlib/Probability/SteinMethod/DependencyCLT.lean"},{"n":"Causalean.SteinMethod.gExpect","k":"def","m":"Causalean.Mathlib.Probability.SteinMethod.Solution","a":"Mathlib/Probability/SteinMethod","t":1,"d":"Expectation of h under the standard normal law, E[h(Z)].","s":"(ℝ → ℝ) → ℝ","l":40,"f":"Causalean/Mathlib/Probability/SteinMethod/Solution.lean"},{"n":"Causalean.SteinMethod.nbhdSum","k":"def","m":"Causalean.Mathlib.Probability.SteinMethod.DependencyCLT","a":"Mathlib/Probability/SteinMethod","t":1,"d":"The neighborhood sum Tᵢ = ∑_{j∈Nᵢ} Xⱼ.","s":"{Ω : Type u_1} → {ι : Type u_2} → (ι → Ω → ℝ) → (ι → Finset ι) → ι → Ω → ℝ","l":51,"f":"Causalean/Mathlib/Probability/SteinMethod/DependencyCLT.lean"},{"n":"Causalean.SteinMethod.steinSol","k":"def","m":"Causalean.Mathlib.Probability.SteinMethod.Solution","a":"Mathlib/Probability/SteinMethod","t":1,"d":"The bounded solution of the Stein equation f' − w·f = h − E[h(Z)].","s":"(ℝ → ℝ) → ℝ → ℝ","l":43,"f":"Causalean/Mathlib/Probability/SteinMethod/Solution.lean"},{"n":"Causalean.SteinMethod.steinSol_abs_le","k":"theorem","m":"Causalean.Mathlib.Probability.SteinMethod.Bounds","a":"Mathlib/Probability/SteinMethod","t":1,"d":"For a real-valued test function h that is bounded in absolute value by a constant C, has derivative bounded in absolute value by a constant L, and is differentiable everywhere, the Stein equation's solution steinSol h, evaluated at any point w, is bounded in absolute value by the derivative bound L.","s":"∀ (h : ℝ → ℝ) {C L : ℝ}, (∀ (x : ℝ), |h x| ≤ C) → (∀ (x : ℝ), |deriv h x| ≤ L) → Differentiable ℝ h → ∀ (w : ℝ), |Causalean.SteinMethod.steinSol h w| ≤ L","l":484,"f":"Causalean/Mathlib/Probability/SteinMethod/Bounds.lean"},{"n":"Causalean.SteinMethod.steinSol_deriv_abs_le","k":"theorem","m":"Causalean.Mathlib.Probability.SteinMethod.Bounds","a":"Mathlib/Probability/SteinMethod","t":2,"d":"Sup bound on the derivative of the Stein solution in terms of the h-derivative bound.","s":"∀ (h : ℝ → ℝ) {C L : ℝ}, (∀ (x : ℝ), |h x| ≤ C) → (∀ (x : ℝ), |deriv h x| ≤ L) → Differentiable ℝ h → ∀ (w : ℝ), |deriv (Causalean.SteinMethod.steinSol h) w| ≤ 2 * L","l":772,"f":"Causalean/Mathlib/Probability/SteinMethod/Bounds.lean"},{"n":"Causalean.SteinMethod.steinSol_deriv_continuous","k":"theorem","m":"Causalean.Mathlib.Probability.SteinMethod.DependencyCLT","a":"Mathlib/Probability/SteinMethod","t":2,"d":"A continuous differentiable test function with bounded values and bounded derivative has a Stein solution whose derivative is continuous.","s":"∀ (h : ℝ → ℝ), Continuous h → ∀ {C L : ℝ}, 0 ≤ L → (∀ (x : ℝ), |h x| ≤ C) → (∀ (x : ℝ), |deriv h x| ≤ L) → Differentiable ℝ h → Continuous (deriv (Causalean.SteinMethod.steinSol h))","l":81,"f":"Causalean/Mathlib/Probability/SteinMethod/DependencyCLT.lean"},{"n":"Causalean.SteinMethod.steinSol_deriv_hasDerivAt","k":"theorem","m":"Causalean.Mathlib.Probability.SteinMethod.Bounds","a":"Mathlib/Probability/SteinMethod","t":2,"d":"For a bounded differentiable test function, the derivative of its standard-normal Stein solution is differentiable at each point, with derivative equal to the solution value plus the point times its first derivative plus the derivative of the test function.","s":"∀ (h : ℝ → ℝ) {C : ℝ}, (∀ (x : ℝ), |h x| ≤ C) → Differentiable ℝ h → ∀ (w : ℝ), HasDerivAt (deriv (Causalean.SteinMethod.steinSol h)) (Causalean.SteinMethod.steinSol h w + (w * deriv (Causalean.SteinMethod.steinSol h) w + deriv h w)) w","l":958,"f":"Causalean/Mathlib/Probability/SteinMethod/Bounds.lean"},{"n":"Causalean.SteinMethod.steinSol_deriv_lipschitz","k":"theorem","m":"Causalean.Mathlib.Probability.SteinMethod.Bounds","a":"Mathlib/Probability/SteinMethod","t":2,"d":"Lipschitz bound on the derivative of the Stein solution (equivalent to ‖f_h''‖ ≤ 2L; Chen–Goldstein–Shao Lemma 2.5). The form f_h' is 2L-Lipschitz is what the second-order Taylor step in the local-dependence Stein bound consumes.","s":"∀ (h : ℝ → ℝ) {C L : ℝ}, (∀ (x : ℝ), |h x| ≤ C) → (∀ (x : ℝ), |deriv h x| ≤ L) → Differentiable ℝ h → ∀ (u v : ℝ), |deriv (Causalean.SteinMethod.steinSol h) u - deriv (Causalean.SteinMethod.steinSol h) v| ≤ 2 * L * |u - v|","l":1003,"f":"Causalean/Mathlib/Probability/SteinMethod/Bounds.lean"},{"n":"Causalean.SteinMethod.steinSol_hasDerivAt","k":"theorem","m":"Causalean.Mathlib.Probability.SteinMethod.Solution","a":"Mathlib/Probability/SteinMethod","t":1,"d":"The Stein equation. For a continuous test function h that is bounded in absolute value by a constant C, the Stein solution steinSol h is differentiable at every point w, with derivative w·steinSol h w + (h w − E[h(Z)]) for a standard normal Z, i.e. it solves the Stein equation f'(w) − w·f(w) = h(w) − E[h(Z)].","s":"∀ (h : ℝ → ℝ), Continuous h → ∀ {C : ℝ}, (∀ (x : ℝ), |h x| ≤ C) → ∀ (w : ℝ), HasDerivAt (Causalean.SteinMethod.steinSol h) (w * Causalean.SteinMethod.steinSol h w + (h w - Causalean.SteinMethod.gExpect h)) w","l":86,"f":"Causalean/Mathlib/Probability/SteinMethod/Solution.lean"},{"n":"Causalean.SteinMethod.steinSol_secondDeriv_abs_le","k":"theorem","m":"Causalean.Mathlib.Probability.SteinMethod.Bounds","a":"Mathlib/Probability/SteinMethod","t":2,"d":"The absolute second derivative of the standard-normal Stein solution at each point is at most twice the uniform bound on the derivative of the test function.","s":"∀ (h : ℝ → ℝ) {C L : ℝ}, (∀ (x : ℝ), |h x| ≤ C) → (∀ (x : ℝ), |deriv h x| ≤ L) → Differentiable ℝ h → ∀ (w : ℝ), |Causalean.SteinMethod.steinSol h w + (w * deriv (Causalean.SteinMethod.steinSol h) w + deriv h w)| ≤ 2 * L","l":982,"f":"Causalean/Mathlib/Probability/SteinMethod/Bounds.lean"},{"n":"Causalean.SteinMethod.steinSol_stein_eq","k":"theorem","m":"Causalean.Mathlib.Probability.SteinMethod.Solution","a":"Mathlib/Probability/SteinMethod","t":2,"d":"The Stein equation in subtractive form.","s":"∀ (h : ℝ → ℝ), Continuous h → ∀ {C : ℝ}, (∀ (x : ℝ), |h x| ≤ C) → ∀ (w : ℝ), deriv (Causalean.SteinMethod.steinSol h) w - w * Causalean.SteinMethod.steinSol h w = h w - Causalean.SteinMethod.gExpect h","l":138,"f":"Causalean/Mathlib/Probability/SteinMethod/Solution.lean"},{"n":"Causalean.SteinMethod.steinSol_taylor_right","k":"theorem","m":"Causalean.Mathlib.Probability.SteinMethod.DependencyCLT","a":"Mathlib/Probability/SteinMethod","t":2,"d":"The standard-normal Stein solution has a first-order Taylor error that grows no faster than the test function's derivative bound times the squared step size.","s":"∀ (h : ℝ → ℝ) {C L : ℝ}, (∀ (x : ℝ), |h x| ≤ C) → (∀ (x : ℝ), |deriv h x| ≤ L) → Differentiable ℝ h → ∀ (a t : ℝ), |Causalean.SteinMethod.steinSol h (a + t) - Causalean.SteinMethod.steinSol h a - t * deriv (Causalean.SteinMethod.steinSol h) (a + t)| ≤ L * t ^ 2","l":92,"f":"Causalean/Mathlib/Probability/SteinMethod/DependencyCLT.lean"},{"n":"Causalean.SteinMethod.stein_cdf_clt","k":"theorem","m":"Causalean.Mathlib.Probability.SteinMethod.CLT","a":"Mathlib/Probability/SteinMethod","t":1,"d":"The bounded local-dependence CLT (CDF form). Fix, for each sample size n, a probability measure μ n, triangular-array summands X n i, and an index-dependent neighborhood set N n i for each i, all jointly measurable. Suppose the summands are uniformly bounded by a nonnegative constant sequence B n, mean zero, and each summand is independent of the sum of the summands outside its neighborhood; suppose the standardized sum has unit variance, and that the variance of the neighborhood-weighted cross term and the aggregate third-absolute-moment error term both tend to zero as n → ∞. Then for every threshold s, the CDF of the dependency sum under μ n at s converges, as n → ∞, to the standard-normal CDF at s.","s":"∀ {Ω : ℕ → Type u_1} [inst : (n : ℕ) → MeasurableSpace (Ω n)] (μ : (n : ℕ) → MeasureTheory.Measure (Ω n)) [∀ (n : ℕ), MeasureTheory.IsProbabilityMeasure (μ n)] {ι : ℕ → Type u_2} [inst_2 : (n : ℕ) → Fintype (ι n)] [inst_3 : (n : ℕ) → DecidableEq (ι n)] (X : (n : ℕ) → ι n → Ω n → ℝ) (N : (n : ℕ) → ι n → Finset (ι n)), (∀ (n : ℕ) (i : ι n), Measurable (X n i)) → ∀ (B : ℕ → ℝ), (∀ (n : ℕ), 0 ≤ B n) → (∀ (n : ℕ) (i : ι n) (ω : Ω n), |X n i ω| ≤ B n) → (∀ (n : ℕ) (i : ι n), ∫ (ω : Ω n), X n i ω ∂μ n = 0) → (∀ (n : ℕ) (i : ι n), ProbabilityTheory.IndepFun (X n i) (fun ω => ∑ j ∈ Finset.univ \\ N n i, X n j ω) (μ n)) → (∀ (n : ℕ), ∫ (ω : Ω n), Causalean.SteinMethod.depSum (X n) ω ^ 2 ∂μ n = 1) → Filter.Tendsto (fun n => ProbabilityTheory.variance (fun ω => ∑ i, X n i ω * Causalean.SteinMethod.nbhdSum (X n) (N n) i ω) (μ n)) Filter.atTop (nhds 0) → Filter.Tendsto (fun n => ∑ i, ∫ (ω : Ω n), |X n i ω| * Causalean.SteinMethod.nbhdSum (X n) (N n) i ω ^ 2 ∂μ n) Filter.atTop (nhds 0) → ∀ (s : ℝ), Filter.Tendsto (fun n => (MeasureTheory.Measure.map (Causalean.SteinMethod.depSum (X n)) (μ n)).real (Set.Iic s)) Filter.atTop (nhds ((ProbabilityTheory.gaussianReal 0 1).real (Set.Iic s)))","l":158,"f":"Causalean/Mathlib/Probability/SteinMethod/CLT.lean"},{"n":"Causalean.SteinMethod.stein_cdf_clt_of_depGraph","k":"theorem","m":"Causalean.Mathlib.Probability.SteinMethod.DepGraphCLT","a":"Mathlib/Probability/SteinMethod","t":1,"d":"The bounded dependency-graph CLT. Fix, for each n, a probability measure μ n, triangular-array summands X n i, and a dependency graph D n on the index set recording which pairs of summands may be dependent, with every dependency neighborhood of size at most m. Suppose the summands are uniformly bounded in absolute value by a constant sequence B n that tends to zero, with the product of the index-set cardinality and the cube of the bound also tending to zero, and suppose each summand is mean zero with the standardized sum having unit total variance. Then for every threshold s, the CDF of the dependency sum under μ n at s converges, as n → ∞, to the standard-normal CDF at s. The two Stein negligibility limits are derived internally.","s":"∀ {Ω : ℕ → Type u_3} [inst : (n : ℕ) → MeasurableSpace (Ω n)] (μ : (n : ℕ) → MeasureTheory.Measure (Ω n)) [∀ (n : ℕ), MeasureTheory.IsProbabilityMeasure (μ n)] {ι : ℕ → Type u_4} [inst_2 : (n : ℕ) → Fintype (ι n)] (X : (n : ℕ) → ι n → Ω n → ℝ) (D : (n : ℕ) → Causalean.SteinMethod.DepGraph (X n) (μ n)) (m : ℕ), (∀ (n : ℕ) (i : ι n), ((D n).nbhd i).card ≤ m) → ∀ (B : ℕ → ℝ), (∀ (n : ℕ), 0 ≤ B n) → (∀ (n : ℕ) (i : ι n) (ω : Ω n), |X n i ω| ≤ B n) → Filter.Tendsto B Filter.atTop (nhds 0) → Filter.Tendsto (fun n => ↑(Fintype.card (ι n)) * B n ^ 3) Filter.atTop (nhds 0) → (∀ (n : ℕ) (i : ι n), ∫ (ω : Ω n), X n i ω ∂μ n = 0) → (∀ (n : ℕ), ∫ (ω : Ω n), Causalean.SteinMethod.depSum (X n) ω ^ 2 ∂μ n = 1) → ∀ (s : ℝ), Filter.Tendsto (fun n => (MeasureTheory.Measure.map (Causalean.SteinMethod.depSum (X n)) (μ n)).real (Set.Iic s)) Filter.atTop (nhds ((ProbabilityTheory.gaussianReal 0 1).real (Set.Iic s)))","l":391,"f":"Causalean/Mathlib/Probability/SteinMethod/DepGraphCLT.lean"},{"n":"Causalean.SteinMethod.stein_expect_tendsto","k":"theorem","m":"Causalean.Mathlib.Probability.SteinMethod.CLT","a":"Mathlib/Probability/SteinMethod","t":2,"d":"For uniformly bounded, mean-zero locally dependent sums with unit variance whose Stein error terms vanish, expectations of a differentiable test function converge to its standard-normal expectation.","s":"∀ {Ω : ℕ → Type u_1} [inst : (n : ℕ) → MeasurableSpace (Ω n)] (μ : (n : ℕ) → MeasureTheory.Measure (Ω n)) [∀ (n : ℕ), MeasureTheory.IsProbabilityMeasure (μ n)] {ι : ℕ → Type u_2} [inst_2 : (n : ℕ) → Fintype (ι n)] [inst_3 : (n : ℕ) → DecidableEq (ι n)] (X : (n : ℕ) → ι n → Ω n → ℝ) (N : (n : ℕ) → ι n → Finset (ι n)), (∀ (n : ℕ) (i : ι n), Measurable (X n i)) → ∀ (B : ℕ → ℝ), (∀ (n : ℕ), 0 ≤ B n) → (∀ (n : ℕ) (i : ι n) (ω : Ω n), |X n i ω| ≤ B n) → (∀ (n : ℕ) (i : ι n), ∫ (ω : Ω n), X n i ω ∂μ n = 0) → (∀ (n : ℕ) (i : ι n), ProbabilityTheory.IndepFun (X n i) (fun ω => ∑ j ∈ Finset.univ \\ N n i, X n j ω) (μ n)) → (∀ (n : ℕ), ∫ (ω : Ω n), Causalean.SteinMethod.depSum (X n) ω ^ 2 ∂μ n = 1) → Filter.Tendsto (fun n => ProbabilityTheory.variance (fun ω => ∑ i, X n i ω * Causalean.SteinMethod.nbhdSum (X n) (N n) i ω) (μ n)) Filter.atTop (nhds 0) → Filter.Tendsto (fun n => ∑ i, ∫ (ω : Ω n), |X n i ω| * Causalean.SteinMethod.nbhdSum (X n) (N n) i ω ^ 2 ∂μ n) Filter.atTop (nhds 0) → ∀ (h : ℝ → ℝ) {C L : ℝ}, (∀ (x : ℝ), |h x| ≤ C) → (∀ (x : ℝ), |deriv h x| ≤ L) → Differentiable ℝ h → Filter.Tendsto (fun n => ∫ (ω : Ω n), h (Causalean.SteinMethod.depSum (X n) ω) ∂μ n) Filter.atTop (nhds (Causalean.SteinMethod.gExpect h))","l":111,"f":"Causalean/Mathlib/Probability/SteinMethod/CLT.lean"},{"n":"Causalean.SteinMethod.stein_local_dependence_bound","k":"theorem","m":"Causalean.Mathlib.Probability.SteinMethod.DependencyCLT","a":"Mathlib/Probability/SteinMethod","t":1,"d":"The local-dependence Stein bound. For jointly measurable summands X i uniformly bounded in absolute value by a nonnegative constant B, each mean zero and independent of the sum of the summands outside its neighborhood set N i, with the standardized sum having unit variance, and for a test function h that is bounded in absolute value by C, has derivative bounded in absolute value by L, and is differentiable, the deviation of the expected test-function value of the local-dependence sum from its standard-normal expectation is at most 2L·√(Var(∑ᵢ Xᵢ·nbhdSumᵢ)) + L·∑ᵢ E[|Xᵢ|·nbhdSumᵢ²].","s":"∀ {Ω : Type u_1} [inst : MeasurableSpace Ω] {μ : MeasureTheory.Measure Ω} [MeasureTheory.IsProbabilityMeasure μ] {ι : Type u_2} [inst_2 : Fintype ι] [inst_3 : DecidableEq ι] (X : ι → Ω → ℝ) (N : ι → Finset ι), (∀ (i : ι), Measurable (X i)) → ∀ {B : ℝ}, 0 ≤ B → (∀ (i : ι) (ω : Ω), |X i ω| ≤ B) → (∀ (i : ι), ∫ (ω : Ω), X i ω ∂μ = 0) → (∀ (i : ι), ProbabilityTheory.IndepFun (X i) (fun ω => ∑ j ∈ Finset.univ \\ N i, X j ω) μ) → ∫ (ω : Ω), Causalean.SteinMethod.depSum X ω ^ 2 ∂μ = 1 → ∀ (h : ℝ → ℝ) {C L : ℝ}, (∀ (x : ℝ), |h x| ≤ C) → (∀ (x : ℝ), |deriv h x| ≤ L) → Differentiable ℝ h → |∫ (ω : Ω), h (Causalean.SteinMethod.depSum X ω) ∂μ - Causalean.SteinMethod.gExpect h| ≤ 2 * L * √(ProbabilityTheory.variance (fun ω => ∑ i, X i ω * Causalean.SteinMethod.nbhdSum X N i ω) μ) + L * ∑ i, ∫ (ω : Ω), |X i ω| * Causalean.SteinMethod.nbhdSum X N i ω ^ 2 ∂μ","l":157,"f":"Causalean/Mathlib/Probability/SteinMethod/DependencyCLT.lean"},{"n":"Causalean.ValuesOn","k":"def","m":"Causalean.Mathlib.MeasureTheory.FinsetValues","a":"Mathlib/MeasureTheory","t":1,"d":"A value assignment over a finite node set gives one value in the appropriate space for each node in the set.","s":"{M : Type u_1} → Finset M → (M → Type u_2) → Type (max u_2 u_1)","l":30,"f":"Causalean/Mathlib/MeasureTheory/FinsetValues.lean"},{"n":"Causalean.aeEq_restrict_of_indicator_aeEq","k":"theorem","m":"Causalean.Mathlib.CondIndep.AELift","a":"Mathlib/CondIndep","t":2,"d":"Recover a μ.restrict s a.e.-equality from a global equality of s-indicators.","s":"∀ {Ω : Type u_1} {mΩ : MeasurableSpace Ω} {μ : MeasureTheory.Measure Ω} {s : Set Ω}, MeasurableSet s → ∀ {β : Type u_2} [inst : Zero β] {f g : Ω → β}, s.indicator f =ᵐ[μ] s.indicator g → f =ᵐ[μ.restrict s] g","l":57,"f":"Causalean/Mathlib/CondIndep/AELift.lean"},{"n":"Causalean.ae_eq_of_ae_eq_restrict_arm","k":"theorem","m":"Causalean.Mathlib.CondIndep.AELift","a":"Mathlib/CondIndep","t":1,"d":"Single-arm a.e.-equality lift for σ-measurable functions. For a sub-σ-algebra m such that m is coarser than the ambient σ-algebra, if f and g agree on an m-measurable set, they agree almost everywhere on the restriction of the measure to a set E, and the conditional probability of E given m is positive almost everywhere — every m-measurable set overlapping E only on a null set is itself null, then f and g agree almost everywhere on the whole space.","s":"∀ {Ω : Type u_1} {mΩ : MeasurableSpace Ω}, ∀ m ≤ mΩ, ∀ {μ : MeasureTheory.Measure Ω} {E : Set Ω} {β : Type u_2} {f g : Ω → β}, MeasurableSet {ω | f ω = g ω} → f =ᵐ[μ.restrict E] g → (∀ (s : Set Ω), MeasurableSet s → μ (s ∩ E) = 0 → μ s = 0) → f =ᵐ[μ] g","l":85,"f":"Causalean/Mathlib/CondIndep/AELift.lean"},{"n":"Causalean.ae_le_YofA_of_ae_le_Y","k":"theorem","m":"Causalean.Mathlib.CondIndep.AELift","a":"Mathlib/CondIndep","t":1,"d":"Y(a) inherits Y's a.e. upper bound under latent exchangeability and consistency. For a sub-σ-algebra σ_UX coarser than the ambient σ-algebra, a measurable treatment map A, a measurable potential outcome Ya, and a treatment value a, if Ya is conditionally independent of A given σ_UX — latent exchangeability, the observed outcome Y agrees with Ya almost everywhere on the restriction of the measure to the event {A = a} — consistency, the conditional probability of {A = a} given σ_UX is positive almost everywhere — overlap, and Y is at most M almost everywhere, then Ya is at most M almost everywhere as well.","s":"∀ {Ω : Type u_1} {mΩ : MeasurableSpace Ω} [inst : StandardBorelSpace Ω] {μ : MeasureTheory.Measure Ω} [inst_1 : MeasureTheory.IsFiniteMeasure μ] {σ_UX : MeasurableSpace Ω} (_hσUX : σ_UX ≤ mΩ) {α : Type u_2} [inst_2 : MeasurableSpace α] [MeasurableSingletonClass α] {A : Ω → α} {Y Ya : Ω → ℝ}, Measurable A → Measurable Ya → ∀ (a : α), ProbabilityTheory.CondIndepFun σ_UX _hσUX Ya A μ → Y =ᵐ[μ.restrict {ω | A ω = a}] Ya → (∀ (s : Set Ω), MeasurableSet s → μ (s ∩ {ω | A ω = a}) = 0 → μ s = 0) → ∀ {M : ℝ}, (∀ᵐ (ω : Ω) ∂μ, Y ω ≤ M) → ∀ᵐ (ω : Ω) ∂μ, Ya ω ≤ M","l":125,"f":"Causalean/Mathlib/CondIndep/AELift.lean"},{"n":"Causalean.ae_le_YofA_of_ae_le_Y_below","k":"theorem","m":"Causalean.Mathlib.CondIndep.AELift","a":"Mathlib/CondIndep","t":1,"d":"Y(a) inherits Y's a.e. lower bound under latent exchangeability and consistency. For a sub-σ-algebra σ_UX coarser than the ambient σ-algebra, a measurable treatment map A, a measurable potential outcome Ya, and a treatment value a, if Ya is conditionally independent of A given σ_UX — latent exchangeability, the observed outcome Y agrees with Ya almost everywhere on the restriction of the measure to the event {A = a} — consistency, the conditional probability of {A = a} given σ_UX is positive almost everywhere — overlap, and Y is at least M almost everywhere, then Ya is at least M almost everywhere as well.","s":"∀ {Ω : Type u_1} {mΩ : MeasurableSpace Ω} [inst : StandardBorelSpace Ω] {μ : MeasureTheory.Measure Ω} [inst_1 : MeasureTheory.IsFiniteMeasure μ] {σ_UX : MeasurableSpace Ω} (_hσUX : σ_UX ≤ mΩ) {α : Type u_2} [inst_2 : MeasurableSpace α] [MeasurableSingletonClass α] {A : Ω → α} {Y Ya : Ω → ℝ}, Measurable A → Measurable Ya → ∀ (a : α), ProbabilityTheory.CondIndepFun σ_UX _hσUX Ya A μ → Y =ᵐ[μ.restrict {ω | A ω = a}] Ya → (∀ (s : Set Ω), MeasurableSet s → μ (s ∩ {ω | A ω = a}) = 0 → μ s = 0) → ∀ {M : ℝ}, (∀ᵐ (ω : Ω) ∂μ, M ≤ Y ω) → ∀ᵐ (ω : Ω) ∂μ, M ≤ Ya ω","l":221,"f":"Causalean/Mathlib/CondIndep/AELift.lean"},{"n":"Causalean.ae_pos_condExp_indicator_of_le","k":"theorem","m":"Causalean.Mathlib.CondIndep.Integrability","a":"Mathlib/CondIndep","t":1,"d":"σ-projection of indicator-conditional positivity. For a sub-σ-algebra m₁ coarser than the ambient σ-algebra, a measurable event E, and an overlap condition — every m₁-measurable set that meets E only on a null set is itself null, then the conditional probability of E given m₁ is strictly positive almost everywhere.","s":"∀ {Ω : Type u_1} {mΩ : MeasurableSpace Ω} {μ : MeasureTheory.Measure Ω} [MeasureTheory.IsFiniteMeasure μ] {m₁ : MeasurableSpace Ω}, m₁ ≤ mΩ → ∀ {E : Set Ω}, MeasurableSet E → (∀ (s : Set Ω), MeasurableSet s → μ (s ∩ E) = 0 → μ s = 0) → ∀ᵐ (ω : Ω) ∂μ, 0 < μ[E.indicator fun x => 1 | m₁] ω","l":36,"f":"Causalean/Mathlib/CondIndep/Integrability.lean"},{"n":"Causalean.comap_eq_bot_of_subsingleton","k":"theorem","m":"Causalean.Mathlib.Indep","a":"Mathlib","t":2,"d":"The comap σ-algebra of a function into a Subsingleton codomain is the trivial σ-algebra ⊥. Every measurable set in the codomain is either empty or the full space, so every preimage is either ∅ or univ.","s":"∀ {α : Type u_1} {β : Type u_2} [inst : MeasurableSpace β] [Subsingleton β] (f : α → β), MeasurableSpace.comap f inferInstance = ⊥","l":402,"f":"Causalean/Mathlib/Indep.lean"},{"n":"Causalean.comap_valuesProjection_le","k":"theorem","m":"Causalean.Mathlib.MeasureTheory.FinsetValues","a":"Mathlib/MeasureTheory","t":1,"d":"For value assignments over a finite node set, if a finite node set W is a subset of a larger finite node set I, then the σ-algebra pulled back, via the coordinate restriction to W, from the measurable space on value assignments over W is a sub-σ-algebra of the ambient measurable space on value assignments over I.","s":"∀ {M : Type u_1} {I W : Finset M} {Ω' : M → Type u_2} [inst : (n : M) → MeasurableSpace (Ω' n)] (hW : W ⊆ I), MeasurableSpace.comap (Causalean.valuesProjection hW) inferInstance ≤ inferInstance","l":57,"f":"Causalean/Mathlib/MeasureTheory/FinsetValues.lean"},{"n":"Causalean.comap_valuesProjection_mono","k":"theorem","m":"Causalean.SCM.Do.ValuesProjectionCI","a":"SCM/Do","t":2,"d":"The comap σ-algebra is monotone in the projected set: if A ⊆ B ⊆ I, then σ(π_A) ≤ σ(π_B) on ValuesOn I.","s":"∀ {M : Type u_1} {I A B : Finset M} {Ω' : M → Type u_2} [inst : (n : M) → MeasurableSpace (Ω' n)] (hAB : A ⊆ B) (hBI : B ⊆ I), MeasurableSpace.comap (Causalean.valuesProjection (LE.le.trans hAB hBI)) inferInstance ≤ MeasurableSpace.comap (Causalean.valuesProjection hBI) inferInstance","l":38,"f":"Causalean/SCM/Do/ValuesProjectionCI.lean"},{"n":"Causalean.comap_valuesProjection_union_eq_sup","k":"theorem","m":"Causalean.SCM.Do.ValuesProjectionCI","a":"SCM/Do","t":2,"d":"The σ-algebra generated by the projection to A ∪ B is the supremum of the σ-algebras generated by the separate projections to A and to B.","s":"∀ {M : Type u_1} [inst : DecidableEq M] {I A B : Finset M} {Ω' : M → Type u_2} [inst_1 : (n : M) → MeasurableSpace (Ω' n)] (hA : A ⊆ I) (hB : B ⊆ I), MeasurableSpace.comap (Causalean.valuesProjection (Finset.union_subset hA hB)) inferInstance = MeasurableSpace.comap (Causalean.valuesProjection hA) inferInstance ⊔ MeasurableSpace.comap (Causalean.valuesProjection hB) inferInstance","l":95,"f":"Causalean/SCM/Do/ValuesProjectionCI.lean"},{"n":"Causalean.condDistrib_comp_right_measurableEquiv","k":"theorem","m":"Causalean.Mathlib.CondDistrib","a":"Mathlib","t":2,"d":"Push-forward invariance of condDistrib under a measurable equivalence of the conditioning variable.","s":"∀ {α : Type u_1} {β : Type u_2} {Ω : Type u_3} {γ : Type u_4} {mα : MeasurableSpace α} {mβ : MeasurableSpace β} {mΩ : MeasurableSpace Ω} [inst : StandardBorelSpace Ω] [inst_1 : Nonempty Ω] {mγ : MeasurableSpace γ} (μ : MeasureTheory.Measure γ) [inst_2 : MeasureTheory.IsFiniteMeasure μ] {Y : γ → Ω} {X : γ → α} (e : α ≃ᵐ β), Measurable Y → Measurable X → ∀ᵐ (x : α) ∂MeasureTheory.Measure.map X μ, (ProbabilityTheory.condDistrib Y (⇑e ∘ X) μ) (e x) = (ProbabilityTheory.condDistrib Y X μ) x","l":93,"f":"Causalean/Mathlib/CondDistrib.lean"},{"n":"Causalean.condDistrib_map_comp","k":"theorem","m":"Causalean.Mathlib.CondDistrib","a":"Mathlib","t":1,"d":"Reparameterization of condDistrib through a pushforward. For a measurable map φ from the sample space Ω to Ω', a measurable outcome map g, and a measurable conditioning map f, the conditional distribution of g given f, computed under the pushforward of μ by φ, agrees almost everywhere on the f-marginal with the conditional distribution of the pullbacks g ∘ φ given f ∘ φ, computed under μ directly.","s":"∀ {Ω : Type u_1} {Ω' : Type u_2} {𝒳 : Type u_3} {𝒴 : Type u_4} [inst : MeasurableSpace Ω] [inst_1 : MeasurableSpace Ω'] [inst_2 : MeasurableSpace 𝒳] [inst_3 : MeasurableSpace 𝒴] [inst_4 : StandardBorelSpace 𝒴] [inst_5 : Nonempty 𝒴] (μ : MeasureTheory.Measure Ω) [inst_6 : MeasureTheory.IsFiniteMeasure μ] {φ : Ω → Ω'} {g : Ω' → 𝒴} {f : Ω' → 𝒳}, Measurable φ → Measurable g → Measurable f → ⇑(ProbabilityTheory.condDistrib g f (MeasureTheory.Measure.map φ μ)) =ᵐ[MeasureTheory.Measure.map f (MeasureTheory.Measure.map φ μ)] ⇑(ProbabilityTheory.condDistrib (g ∘ φ) (f ∘ φ) μ)","l":262,"f":"Causalean/Mathlib/CondDistrib.lean"},{"n":"Causalean.condExp_indicator_aeEq_zero","k":"theorem","m":"Causalean.Mathlib.CondIndep.AELift","a":"Mathlib/CondIndep","t":2,"d":"If an integrable function vanishes after restriction by an m-measurable indicator, so does its conditional expectation.","s":"∀ {Ω : Type u_1} {mΩ : MeasurableSpace Ω} {μ : MeasureTheory.Measure Ω} {m : MeasurableSpace Ω} {s : Set Ω}, MeasurableSet s → ∀ {E : Type u_2} [inst : NormedAddCommGroup E] [inst_1 : NormedSpace ℝ E] [CompleteSpace E] {f : Ω → E}, MeasureTheory.Integrable f μ → s.indicator f =ᵐ[μ] 0 → s.indicator μ[f | m] =ᵐ[μ] 0","l":70,"f":"Causalean/Mathlib/CondIndep/AELift.lean"},{"n":"Causalean.condExp_mul_of_condIndep","k":"theorem","m":"Causalean.Mathlib.CondIndep.CondExp","a":"Mathlib/CondIndep","t":1,"d":"Conditional-independence factorization of a product's conditional expectation. For a sub-σ-algebra m coarser than the ambient σ-algebra, measurable maps f and g that are conditionally independent given m, and measurable real-valued functions u and v such that u ∘ f, v ∘ g, and their pointwise product are all integrable, then the conditional expectation, given m, of the product (u ∘ f)·(v ∘ g) equals the product of the separate conditional expectations of u ∘ f and v ∘ g given m.","s":"∀ {Ω : Type u_1} {α : Type u_2} {β : Type u_3} {m mΩ : MeasurableSpace Ω} (hm : m ≤ mΩ) [inst : MeasurableSpace α] [inst_1 : MeasurableSpace β] [inst_2 : StandardBorelSpace Ω] {μ : MeasureTheory.Measure Ω} [inst_3 : MeasureTheory.IsFiniteMeasure μ] {f : Ω → α} {g : Ω → β}, Measurable f → Measurable g → ProbabilityTheory.CondIndepFun m hm f g μ → ∀ {u : α → ℝ} {v : β → ℝ}, Measurable u → Measurable v → MeasureTheory.Integrable (fun ω => u (f ω)) μ → MeasureTheory.Integrable (fun ω => v (g ω)) μ → MeasureTheory.Integrable (fun ω => u (f ω) * v (g ω)) μ → μ[fun ω => u (f ω) * v (g ω) | m] =ᵐ[μ] μ[fun ω => u (f ω) | m] * μ[fun ω => v (g ω) | m]","l":436,"f":"Causalean/Mathlib/CondIndep/CondExp.lean"},{"n":"Causalean.condExp_sup_comap_eq_of_condIndep","k":"theorem","m":"Causalean.Mathlib.CondIndep.CondExp","a":"Mathlib/CondIndep","t":2,"d":"Drop-of-conditioning for conditional expectation under conditional independence.","s":"∀ {Ω : Type u_1} {α : Type u_2} {β : Type u_3} {m mΩ : MeasurableSpace Ω} (hm : m ≤ mΩ) [inst : MeasurableSpace α] [inst_1 : MeasurableSpace β] [inst_2 : StandardBorelSpace Ω] {μ : MeasureTheory.Measure Ω} [inst_3 : MeasureTheory.IsFiniteMeasure μ] {f : Ω → α} {g : Ω → β}, Measurable f → Measurable g → ProbabilityTheory.CondIndepFun m hm g f μ → ∀ {h : β → ℝ}, Measurable h → MeasureTheory.Integrable (fun ω => h (g ω)) μ → μ[fun ω => h (g ω) | m ⊔ MeasurableSpace.comap f inferInstance] =ᵐ[μ] μ[fun ω => h (g ω) | m]","l":406,"f":"Causalean/Mathlib/CondIndep/CondExp.lean"},{"n":"Causalean.condExp_sup_comap_eq_of_condIndep_comap","k":"theorem","m":"Causalean.Mathlib.CondIndep.CondExp","a":"Mathlib/CondIndep","t":2,"d":"If g is conditionally independent of f given m, conditioning a σ(g)-measurable integrable real function on m ⊔ σ(f) is the same as conditioning it on m, up to μ-a.e. equality.","s":"∀ {Ω : Type u_1} {α : Type u_2} {β : Type u_3} {m mΩ : MeasurableSpace Ω} (hm : m ≤ mΩ) [inst : MeasurableSpace α] [inst_1 : MeasurableSpace β] [inst_2 : StandardBorelSpace Ω] {μ : MeasureTheory.Measure Ω} [inst_3 : MeasureTheory.IsFiniteMeasure μ] {f : Ω → α} {g : Ω → β}, Measurable f → Measurable g → ProbabilityTheory.CondIndepFun m hm g f μ → ∀ {Y : Ω → ℝ}, Measurable Y → MeasureTheory.Integrable Y μ → μ[Y | m ⊔ MeasurableSpace.comap f inferInstance] =ᵐ[μ] μ[Y | m]","l":312,"f":"Causalean/Mathlib/CondIndep/CondExp.lean"},{"n":"Causalean.condIndepFun_bot_of_indepFun","k":"theorem","m":"Causalean.Mathlib.Indep","a":"Mathlib","t":2,"d":"Bridge: if μ is a probability measure on a standard Borel space Ω, then unconditional IndepFun f g μ lifts to CondIndepFun ⊥ bot_le f g μ — conditional independence given the trivial σ-algebra.","s":"∀ {Ω : Type u_1} [inst : MeasurableSpace Ω] [inst_1 : StandardBorelSpace Ω] {β : Type u_2} {γ : Type u_3} [inst_2 : MeasurableSpace β] [inst_3 : MeasurableSpace γ] {f : Ω → β} {g : Ω → γ}, Measurable f → Measurable g → ∀ {μ : MeasureTheory.Measure Ω} [inst_4 : MeasureTheory.IsProbabilityMeasure μ], ProbabilityTheory.IndepFun f g μ → ProbabilityTheory.CondIndepFun ⊥ bot_le f g μ","l":180,"f":"Causalean/Mathlib/Indep.lean"},{"n":"Causalean.condIndepFun_contraction_of_prodMk","k":"theorem","m":"Causalean.Mathlib.CondIndep.CondExp","a":"Mathlib/CondIndep","t":1,"d":"Contraction for conditional independence via a product right side. For a sub-σ-algebra m coarser than the ambient σ-algebra and measurable maps X, Y, and W, if X is conditionally independent of Y given the enlarged σ-algebra m joined with the σ-algebra generated by W and X is conditionally independent of W given m, then X is conditionally independent of the pair (Y, W) given m.","s":"∀ {Ω : Type u_1} {α : Type u_2} {β : Type u_3} {γ : Type u_4} [inst : MeasurableSpace α] [inst_1 : MeasurableSpace β] [inst_2 : MeasurableSpace γ] {m mΩ : MeasurableSpace Ω} (hm : m ≤ mΩ) [inst_3 : StandardBorelSpace Ω] {μ : MeasureTheory.Measure Ω} [inst_4 : MeasureTheory.IsFiniteMeasure μ] {X : Ω → α} {Y : Ω → β} {W : Ω → γ}, Measurable X → Measurable Y → ∀ (hW : Measurable W), ProbabilityTheory.CondIndepFun (m ⊔ MeasurableSpace.comap W inferInstance) (sup_le hm (Measurable.comap_le hW)) X Y μ → ProbabilityTheory.CondIndepFun m hm X W μ → ProbabilityTheory.CondIndepFun m hm X (fun ω => (Y ω, W ω)) μ","l":801,"f":"Causalean/Mathlib/CondIndep/CondExp.lean"},{"n":"Causalean.condIndepFun_of_indepFun_indep","k":"theorem","m":"Causalean.Mathlib.Indep","a":"Mathlib","t":2,"d":"If X and Y are independent, and the joint (X, Y) is independent of the conditioning variable Z, then X and Y are conditionally independent given the σ-algebra generated by Z.","s":"∀ {Ω : Type u_1} {β : Type u_2} {γ : Type u_3} {δ : Type u_4} [inst : MeasurableSpace Ω] [inst_1 : StandardBorelSpace Ω] [inst_2 : MeasurableSpace β] [inst_3 : MeasurableSpace γ] [inst_4 : MeasurableSpace δ] {μ : MeasureTheory.Measure Ω} [inst_5 : MeasureTheory.IsProbabilityMeasure μ] {X : Ω → β} {Y : Ω → γ} {Z : Ω → δ}, Measurable X → Measurable Y → ∀ (_hZ : Measurable Z), ProbabilityTheory.IndepFun X Y μ → ProbabilityTheory.Indep (MeasurableSpace.comap (fun ω => (X ω, Y ω)) inferInstance) (MeasurableSpace.comap Z inferInstance) μ → ProbabilityTheory.CondIndepFun (MeasurableSpace.comap Z inferInstance) (Measurable.comap_le _hZ) X Y μ","l":448,"f":"Causalean/Mathlib/Indep.lean"},{"n":"Causalean.condIndepFun_pi_cond_residual_of_disjoint","k":"theorem","m":"Causalean.Mathlib.Indep","a":"Mathlib","t":1,"d":"For a finite index set, a family of standard Borel probability spaces indexed by it, and finite subsets S0, T0, U of the index set, if S0 and T0 are disjoint from each other and S0 is disjoint from U, then under the product probability measure, the pair consisting of the U-coordinates and the S0-coordinates is conditionally independent of the pair consisting of the U-coordinates and the T0-coordinates, given the σ-algebra generated by the U-coordinates.","s":"∀ {ι : Type u_1} [inst : Finite ι] {Ω : ι → Type u_2} [inst_1 : (i : ι) → MeasurableSpace (Ω i)] [inst_2 : ∀ (i : ι), StandardBorelSpace (Ω i)] (μ : (i : ι) → MeasureTheory.Measure (Ω i)) [inst_3 : ∀ (i : ι), MeasureTheory.IsProbabilityMeasure (μ i)] {S0 T0 U : Finset ι}, Disjoint S0 T0 → Disjoint S0 U → ProbabilityTheory.CondIndepFun (MeasurableSpace.comap (Causalean.finsetCoordProj U) inferInstance) (Measurable.comap_le (Causalean.measurable_finsetCoordProj U)) (fun x => (Causalean.finsetCoordProj U x, Causalean.finsetCoordProj S0 x)) (fun x => (Causalean.finsetCoordProj U x, Causalean.finsetCoordProj T0 x)) (MeasureTheory.Measure.pi μ)","l":212,"f":"Causalean/Mathlib/Indep.lean"},{"n":"Causalean.condIndepFun_pi_of_inter_subset","k":"theorem","m":"Causalean.Mathlib.Indep","a":"Mathlib","t":1,"d":"Under a finite product probability measure, if the overlap S ∩ T is contained in the conditioning block U, then the coordinate tuple on S is conditionally independent of the coordinate tuple on T, given the coordinate tuple on U.","s":"∀ {ι : Type u_1} [inst : DecidableEq ι] [inst_1 : Finite ι] {Ω : ι → Type u_2} [inst_2 : (i : ι) → MeasurableSpace (Ω i)] [inst_3 : ∀ (i : ι), StandardBorelSpace (Ω i)] (μ : (i : ι) → MeasureTheory.Measure (Ω i)) [inst_4 : ∀ (i : ι), MeasureTheory.IsProbabilityMeasure (μ i)] {S T U : Finset ι}, S ∩ T ⊆ U → ProbabilityTheory.CondIndepFun (MeasurableSpace.comap (Causalean.finsetCoordProj U) inferInstance) (Measurable.comap_le (Causalean.measurable_finsetCoordProj U)) (Causalean.finsetCoordProj S) (Causalean.finsetCoordProj T) (MeasureTheory.Measure.pi μ)","l":339,"f":"Causalean/Mathlib/Indep.lean"},{"n":"Causalean.condIndepFun_prodMk_of_measurable_left","k":"theorem","m":"Causalean.Mathlib.CondIndep.CondExp","a":"Mathlib/CondIndep","t":2,"d":"Extension of CondIndepFun by an m-measurable function.","s":"∀ {Ω : Type u_1} {α : Type u_2} {β : Type u_3} {γ : Type u_4} [inst : MeasurableSpace α] [inst_1 : MeasurableSpace β] [inst_2 : MeasurableSpace γ] {m mΩ : MeasurableSpace Ω} (hm : m ≤ mΩ) [inst_3 : StandardBorelSpace Ω] {μ : MeasureTheory.Measure Ω} [inst_4 : MeasureTheory.IsFiniteMeasure μ] {W : Ω → α} {X : Ω → β} {Z : Ω → γ}, Measurable W → Measurable Z → Measurable X → ProbabilityTheory.CondIndepFun m hm W Z μ → ProbabilityTheory.CondIndepFun m hm (fun ω => (W ω, X ω)) Z μ","l":668,"f":"Causalean/Mathlib/CondIndep/CondExp.lean"},{"n":"Causalean.condIndepFun_weak_union_of_prodMk","k":"theorem","m":"Causalean.Mathlib.CondIndep.CondExp","a":"Mathlib/CondIndep","t":1,"d":"Weak union for conditional independence. For a sub-σ-algebra m coarser than the ambient σ-algebra and measurable maps W, V, and A, if W is conditionally independent of the pair (V, A) given m, then W is conditionally independent of V given the enlarged σ-algebra m joined with the σ-algebra generated by A.","s":"∀ {Ω : Type u_1} {α : Type u_2} {β : Type u_3} {γ : Type u_4} [inst : MeasurableSpace α] [inst_1 : MeasurableSpace β] [inst_2 : MeasurableSpace γ] {m mΩ : MeasurableSpace Ω} (hm : m ≤ mΩ) [inst_3 : StandardBorelSpace Ω] {μ : MeasureTheory.Measure Ω} [inst_4 : MeasureTheory.IsFiniteMeasure μ] {W : Ω → α} {V : Ω → β} {A : Ω → γ}, Measurable W → Measurable V → ∀ (hA : Measurable A), ProbabilityTheory.CondIndepFun m hm W (fun ω => (V ω, A ω)) μ → ProbabilityTheory.CondIndepFun (m ⊔ MeasurableSpace.comap A inferInstance) (sup_le hm (Measurable.comap_le hA)) W V μ","l":536,"f":"Causalean/Mathlib/CondIndep/CondExp.lean"},{"n":"Causalean.condIndep_valuesProjection_contraction","k":"theorem","m":"Causalean.SCM.Do.ValuesProjectionCI","a":"SCM/Do","t":1,"d":"Contraction for coordinate projections. Let X, Y, W, Z be subsets of a common finite index set I, with X, Y, W, and Z all contained in I, and let μ be a finite measure on the resulting finite-product value space indexed by I. If X is conditionally independent of Y given the union of Z and W and X is conditionally independent of W given Z, then X is conditionally independent of the union of Y and W given Z, where independence is always of the corresponding coordinate projections under μ.","s":"∀ {M : Type uM} [inst : DecidableEq M] {I X Y W Z : Finset M} {Ω : M → Type uΩ} [inst_1 : (n : M) → MeasurableSpace (Ω n)] [inst_2 : StandardBorelSpace (Causalean.ValuesOn I Ω)] (hX : X ⊆ I) (hY : Y ⊆ I) (hW : W ⊆ I) (hZ : Z ⊆ I) {μ : MeasureTheory.Measure (Causalean.ValuesOn I Ω)} [inst_3 : MeasureTheory.IsFiniteMeasure μ], ProbabilityTheory.CondIndepFun (MeasurableSpace.comap (Causalean.valuesProjection (Finset.union_subset hZ hW)) inferInstance) (Causalean.comap_valuesProjection_le (Finset.union_subset hZ hW)) (Causalean.valuesProjection hX) (Causalean.valuesProjection hY) μ → ProbabilityTheory.CondIndepFun (MeasurableSpace.comap (Causalean.valuesProjection hZ) inferInstance) (Causalean.comap_valuesProjection_le hZ) (Causalean.valuesProjection hX) (Causalean.valuesProjection hW) μ → ProbabilityTheory.CondIndepFun (MeasurableSpace.comap (Causalean.valuesProjection hZ) inferInstance) (Causalean.comap_valuesProjection_le hZ) (Causalean.valuesProjection hX) (Causalean.valuesProjection (Finset.union_subset hY hW)) μ","l":206,"f":"Causalean/SCM/Do/ValuesProjectionCI.lean"},{"n":"Causalean.condIndep_valuesProjection_weak_union","k":"theorem","m":"Causalean.SCM.Do.ValuesProjectionCI","a":"SCM/Do","t":1,"d":"Weak union for coordinate projections. Let X, Y, W, Z be subsets of a finite index set I, with X, the union of Y and W, and the union of Z and W all contained in I, and let μ be a finite measure on the finite-product value space indexed by I. If X is conditionally independent of the union of Y and W given Z, then X is conditionally independent of Y given the union of Z and W, where independence is always of the corresponding coordinate projections under μ.","s":"∀ {M : Type uM} [inst : DecidableEq M] {I X Y W Z : Finset M} {Ω : M → Type uΩ} [inst_1 : (n : M) → MeasurableSpace (Ω n)] [inst_2 : StandardBorelSpace (Causalean.ValuesOn I Ω)] (hX : X ⊆ I) (hYW : Y ∪ W ⊆ I) (hZW : Z ∪ W ⊆ I) {μ : MeasureTheory.Measure (Causalean.ValuesOn I Ω)} [inst_3 : MeasureTheory.IsFiniteMeasure μ], ProbabilityTheory.CondIndepFun (MeasurableSpace.comap (Causalean.valuesProjection (LE.le.trans Finset.subset_union_left hZW)) inferInstance) (Causalean.comap_valuesProjection_le (LE.le.trans Finset.subset_union_left hZW)) (Causalean.valuesProjection hX) (Causalean.valuesProjection hYW) μ → ProbabilityTheory.CondIndepFun (MeasurableSpace.comap (Causalean.valuesProjection hZW) inferInstance) (Causalean.comap_valuesProjection_le hZW) (Causalean.valuesProjection hX) (Causalean.valuesProjection (LE.le.trans Finset.subset_union_left hYW)) μ","l":131,"f":"Causalean/SCM/Do/ValuesProjectionCI.lean"},{"n":"Causalean.edgeDiff","k":"def","m":"Causalean.Graph.MarkovEquiv.Decompose","a":"Graph/MarkovEquiv","t":1,"d":"The directed edges of G₁ that are absent from G₂.","s":"{V : Type u_1} → [inst : DecidableEq V] → [inst_1 : Fintype V] → Causalean.DAG V → Causalean.DAG V → Finset (V × V)","l":56,"f":"Causalean/Graph/MarkovEquiv/Decompose.lean"},{"n":"Causalean.edgeDiffCount","k":"def","m":"Causalean.Graph.MarkovEquiv.Decompose","a":"Graph/MarkovEquiv","t":1,"d":"Number of directed edges of G₁ absent from G₂.","s":"{V : Type u_1} → [inst : DecidableEq V] → [inst_1 : Fintype V] → Causalean.DAG V → Causalean.DAG V → ℕ","l":60,"f":"Causalean/Graph/MarkovEquiv/Decompose.lean"},{"n":"Causalean.edgeDiffCount_flipEdge_lt","k":"theorem","m":"Causalean.Graph.MarkovEquiv.Decompose","a":"Graph/MarkovEquiv","t":2,"d":"Flipping a covered, oppositely oriented edge strictly decreases the edge-difference count (it fixes exactly the pair (a,b) and changes nothing else).","s":"∀ {V : Type u_1} [inst : DecidableEq V] [inst_1 : Fintype V] {G₁ G₂ : Causalean.DAG V} {a b : V} (hcov : G₁.IsCoveredEdge a b), G₂.edge b a → Causalean.edgeDiffCount (Causalean.DAG.flipEdge hcov) G₂ < Causalean.edgeDiffCount G₁ G₂","l":178,"f":"Causalean/Graph/MarkovEquiv/Decompose.lean"},{"n":"Causalean.exists_covered_reversed_edge","k":"theorem","m":"Causalean.Graph.MarkovEquiv.Decompose","a":"Graph/MarkovEquiv","t":1,"d":"AMP Lemma 3.2 (existence). For DAGs G₁ and G₂ on the same vertex set, suppose G₁ and G₂ have the same skeleton (the same underlying undirected adjacency), the same immoralities (unshielded colliders), and there is a pair a₀, b₀ with an edge a₀ → b₀ in G₁ that appears reversed as b₀ → a₀ in G₂. Then there is a *covered* edge a → b in G₁ that is likewise reversed to b → a in G₂. The proof chooses a head of a reversed edge that is minimal in the first graph's topological order, then a tail into that head that is maximal among reversed tails; the skeleton and immorality hypotheses force this edge to have the same non-tail parents at both endpoints.","s":"∀ {V : Type u_1} [inst : DecidableEq V] [inst_1 : Fintype V] {G₁ G₂ : Causalean.DAG V}, Causalean.SameSkeleton G₁ G₂ → Causalean.SameImmoralities G₁ G₂ → ∀ {a₀ b₀ : V}, G₁.edge a₀ b₀ → G₂.edge b₀ a₀ → ∃ a b, G₁.edge a b ∧ G₂.edge b a ∧ G₁.IsCoveredEdge a b","l":79,"f":"Causalean/Graph/MarkovEquiv/Decompose.lean"},{"n":"Causalean.exists_half_net_card_le_five_pow_finrank","k":"theorem","m":"Causalean.Stat.Concentration.FiniteDimensionalNet","a":"Stat/Concentration","t":1,"d":"For a finite-dimensional real normed space E, there exists a finite set of unit vectors, of size at most 5 raised to the dimension of E, that forms a half-net of the unit sphere: every unit vector lies within distance 1/2 of some point in the set.","s":"∀ (E : Type u_1) [inst : NormedAddCommGroup E] [inst_1 : NormedSpace ℝ E] [FiniteDimensional ℝ E], ∃ N, (∀ v ∈ N, ‖v‖ = 1) ∧ (∀ (x : E), ‖x‖ = 1 → ∃ v ∈ N, ‖x - v‖ ≤ 1 / 2) ∧ N.card ≤ 5 ^ Module.finrank ℝ E","l":160,"f":"Causalean/Stat/Concentration/FiniteDimensionalNet.lean"},{"n":"Causalean.exists_internal_net_card_le","k":"theorem","m":"Causalean.Stat.Concentration.FiniteDimensionalNet","a":"Stat/Concentration","t":2,"d":"Every subset of the closed unit ball of a finite-dimensional real normed space admits an internal net at any prescribed accuracy v: finitely many points, all taken from the set itself, such that every point of the set is within distance v of one of them, and the number of net points is at most (1 + 2 / v) raised to the dimension of the space.","s":"∀ (E : Type u_1) [inst : NormedAddCommGroup E] [inst_1 : NormedSpace ℝ E] [FiniteDimensional ℝ E] (A : Set E), (∀ x ∈ A, ‖x‖ ≤ 1) → ∀ {v : ℝ}, 0 < v → ∃ N, (∀ y ∈ N, y ∈ A) ∧ (∀ x ∈ A, ∃ y ∈ N, ‖x - y‖ ≤ v) ∧ ↑N.card ≤ (1 + 2 / v) ^ Module.finrank ℝ E","l":29,"f":"Causalean/Stat/Concentration/FiniteDimensionalNet.lean"},{"n":"Causalean.exists_unit_ball_net_card_le","k":"theorem","m":"Causalean.Stat.Concentration.FiniteDimensionalNet","a":"Stat/Concentration","t":2,"d":"The closed unit ball of a finite-dimensional real normed space has, at every accuracy v, a finite net of at most (1 + 2 / v) raised to the dimension of the space many points.","s":"∀ (E : Type u_1) [inst : NormedAddCommGroup E] [inst_1 : NormedSpace ℝ E] [FiniteDimensional ℝ E] {v : ℝ}, 0 < v → ∃ N, (∀ (x : E), ‖x‖ ≤ 1 → ∃ y ∈ N, ‖x - y‖ ≤ v) ∧ ↑N.card ≤ (1 + 2 / v) ^ Module.finrank ℝ E","l":148,"f":"Causalean/Stat/Concentration/FiniteDimensionalNet.lean"},{"n":"Causalean.finsetCoordProj","k":"def","m":"Causalean.Mathlib.Indep","a":"Mathlib","t":1,"d":"Tuple projection to a finite coordinate block of a dependent product.","s":"{ι : Type u_1} → {Ω : ι → Type u_2} → (S : Finset ι) → ((i : ι) → Ω i) → (i : ↥S) → Ω ↑i","l":88,"f":"Causalean/Mathlib/Indep.lean"},{"n":"Causalean.finsetCoordProjFromCondResidual","k":"def","m":"Causalean.Mathlib.Indep","a":"Mathlib","t":1,"d":"Reassemble coordinates on S from coordinates on U and on the residual block S \\ U.","s":"{ι : Type u_1} → [inst : DecidableEq ι] → {Ω : ι → Type u_2} → (S U : Finset ι) → ((i : ↥U) → Ω ↑i) × ((i : ↥(S \\ U)) → Ω ↑i) → (i : ↥S) → Ω ↑i","l":104,"f":"Causalean/Mathlib/Indep.lean"},{"n":"Causalean.finsetCoordProjFromCondResidual_comp","k":"theorem","m":"Causalean.Mathlib.Indep","a":"Mathlib","t":2,"d":"Reassembling π_U with the residual projection π_{S \\ U} gives π_S.","s":"∀ {ι : Type u_1} [inst : DecidableEq ι] {Ω : ι → Type u_2} (S U : Finset ι), (Causalean.finsetCoordProjFromCondResidual S U ∘ fun x => (Causalean.finsetCoordProj U x, Causalean.finsetCoordProj (S \\ U) x)) = Causalean.finsetCoordProj S","l":132,"f":"Causalean/Mathlib/Indep.lean"},{"n":"Causalean.finsetCoordProjPairFromUnion","k":"def","m":"Causalean.Mathlib.Indep","a":"Mathlib","t":1,"d":"Extract two sub-blocks from the tuple on their union.","s":"{ι : Type u_1} → [inst : DecidableEq ι] → {Ω : ι → Type u_2} → (A B : Finset ι) → ((i : ↥(A ∪ B)) → Ω ↑i) → ((i : ↥A) → Ω ↑i) × ((i : ↥B) → Ω ↑i)","l":143,"f":"Causalean/Mathlib/Indep.lean"},{"n":"Causalean.finsetCoordProjPairFromUnion_comp","k":"theorem","m":"Causalean.Mathlib.Indep","a":"Mathlib","t":2,"d":"Projecting to A ∪ B and then extracting the two sub-blocks gives the pair of direct projections to A and B.","s":"∀ {ι : Type u_1} [inst : DecidableEq ι] {Ω : ι → Type u_2} (A B : Finset ι), Causalean.finsetCoordProjPairFromUnion A B ∘ Causalean.finsetCoordProj (A ∪ B) = fun x => (Causalean.finsetCoordProj A x, Causalean.finsetCoordProj B x)","l":168,"f":"Causalean/Mathlib/Indep.lean"},{"n":"Causalean.half_norm_lt_inner_of_mem_net","k":"theorem","m":"Causalean.Stat.Concentration.SubGaussianNorm","a":"Stat/Concentration","t":2,"d":"Every nonzero vector in a subspace has a point in a finite half-net whose inner product with that vector is at least half of its norm.","s":"∀ {E : Type u_1} [inst : NormedAddCommGroup E] [inst_1 : InnerProductSpace ℝ E] {N : Finset E} {z : E} (V : Submodule ℝ E), (∀ x ∈ V, ‖x‖ = 1 → ∃ v ∈ N, ‖x - v‖ ≤ 1 / 2) → z ∈ V → z ≠ 0 → ∃ v ∈ N, ‖z‖ / 2 ≤ inner ℝ v z","l":23,"f":"Causalean/Stat/Concentration/SubGaussianNorm.lean"},{"n":"Causalean.indepFun_of_map","k":"theorem","m":"Causalean.Mathlib.Indep","a":"Mathlib","t":2,"d":"Pushforward bridge for IndepFun: if X ∘ φ ⟂ᵢ Y ∘ φ under ν, then X ⟂ᵢ Y under ν.map φ. Analogue of LocalMarkov.condIndepFun_of_map at the IndepFun level.","s":"∀ {α : Type u_1} {β : Type u_2} {γ : Type u_3} {δ : Type u_4} [inst : MeasurableSpace α] [inst_1 : MeasurableSpace β] [inst_2 : MeasurableSpace γ] [inst_3 : MeasurableSpace δ] {ν : MeasureTheory.Measure α} {φ : α → β}, AEMeasurable φ ν → ∀ {X : β → γ}, Measurable X → ∀ {Y : β → δ}, Measurable Y → ProbabilityTheory.IndepFun (X ∘ φ) (Y ∘ φ) ν → ProbabilityTheory.IndepFun X Y (MeasureTheory.Measure.map φ ν)","l":425,"f":"Causalean/Mathlib/Indep.lean"},{"n":"Causalean.indepFun_pi_of_disjoint","k":"theorem","m":"Causalean.Mathlib.Indep","a":"Mathlib","t":2,"d":"Coordinate-tuple projections at disjoint Finsets are IndepFun under Measure.pi of a family of probability measures. This is the binary aggregation of iIndepFun_pi via iIndepFun.indepFun_finset.","s":"∀ {ι : Type u_1} [inst : Finite ι] {Ω : ι → Type u_2} [inst_1 : (i : ι) → MeasurableSpace (Ω i)] (μ : (i : ι) → MeasureTheory.Measure (Ω i)) [∀ (i : ι), MeasureTheory.IsProbabilityMeasure (μ i)] {S T : Finset ι}, Disjoint S T → ProbabilityTheory.IndepFun (fun x i => x ↑i) (fun x i => x ↑i) (MeasureTheory.Measure.pi μ)","l":71,"f":"Causalean/Mathlib/Indep.lean"},{"n":"Causalean.indicator_aeEq_of_aeEq_restrict","k":"theorem","m":"Causalean.Mathlib.CondIndep.AELift","a":"Mathlib/CondIndep","t":2,"d":"Push an a.e.-equality under μ.restrict s to a global equality of s-indicators.","s":"∀ {Ω : Type u_1} {mΩ : MeasurableSpace Ω} {μ : MeasureTheory.Measure Ω} {s : Set Ω}, MeasurableSet s → ∀ {β : Type u_2} [inst : Zero β] {f g : Ω → β}, f =ᵐ[μ.restrict s] g → s.indicator f =ᵐ[μ] s.indicator g","l":41,"f":"Causalean/Mathlib/CondIndep/AELift.lean"},{"n":"Causalean.initialSWIG","k":"def","m":"Causalean.Graph.SWIG","a":"Graph","t":1,"d":"The initial SWIG DAG with no intervention targets. All edges stay between random nodes; all fixed nodes are isolated. This is the DAG used by a standard causal model.","s":"{N : Type u_1} → [inst : DecidableEq N] → [inst_1 : Fintype N] → Causalean.DAG N → Causalean.DAG (Causalean.SWIGNode N)","l":333,"f":"Causalean/Graph/SWIG.lean"},{"n":"Causalean.initialSWIG_fixed_isolated","k":"theorem","m":"Causalean.Graph.SWIG","a":"Graph","t":1,"d":"For any base DAG G and any node n, the fixed copy of n has no parents in the initial SWIG of G (the SWIG with no intervention targets).","s":"∀ {N : Type u_1} [inst : DecidableEq N] [inst_1 : Fintype N] (G : Causalean.DAG N) (n : N), (Causalean.initialSWIG G).parents (Causalean.SWIGNode.fixed n) = ∅","l":345,"f":"Causalean/Graph/SWIG.lean"},{"n":"Causalean.initialSWIG_random_edge","k":"theorem","m":"Causalean.Graph.SWIG","a":"Graph","t":1,"d":"For any base DAG G and any nodes u, v, in the initial SWIG of G (the SWIG with no intervention targets), the random copies of u and v are joined by an edge exactly when u and v are joined by an edge in G.","s":"∀ {N : Type u_1} [inst : DecidableEq N] [inst_1 : Fintype N] (G : Causalean.DAG N) (u v : N), (Causalean.initialSWIG G).edge (Causalean.SWIGNode.random u) (Causalean.SWIGNode.random v) ↔ G.edge u v","l":338,"f":"Causalean/Graph/SWIG.lean"},{"n":"Causalean.instDecidableEqEdgeType","k":"instance","m":"Causalean.SCM.Model.EdgeType","a":"SCM/Model","t":2,"d":"","s":"DecidableEq Causalean.EdgeType","l":66,"f":"Causalean/SCM/Model/EdgeType.lean"},{"n":"Causalean.instDecidableEqMonotonicityKind","k":"instance","m":"Causalean.SCM.Model.EdgeType","a":"SCM/Model","t":2,"d":"","s":"DecidableEq Causalean.MonotonicityKind","l":44,"f":"Causalean/SCM/Model/EdgeType.lean"},{"n":"Causalean.instDecidableEqSWIGNode","k":"instance","m":"Causalean.Graph.SWIG","a":"Graph","t":2,"d":"","s":"{N : Type u_1} → [DecidableEq N] → DecidableEq (Causalean.SWIGNode N)","l":88,"f":"Causalean/Graph/SWIG.lean"},{"n":"Causalean.instDecidableSameImmoralities","k":"instance","m":"Causalean.Graph.MarkovEquiv.Defs","a":"Graph/MarkovEquiv","t":2,"d":"Whether two finite DAGs have the same v-structures is decidable.","s":"{V : Type u_1} → [inst : DecidableEq V] → [inst_1 : Fintype V] → (G₁ G₂ : Causalean.DAG V) → Decidable (Causalean.SameImmoralities G₁ G₂)","l":73,"f":"Causalean/Graph/MarkovEquiv/Defs.lean"},{"n":"Causalean.instDecidableSameSkeleton","k":"instance","m":"Causalean.Graph.MarkovEquiv.Defs","a":"Graph/MarkovEquiv","t":2,"d":"Whether two finite DAGs have the same skeleton is decidable.","s":"{V : Type u_1} → [inst : DecidableEq V] → [inst_1 : Fintype V] → (G₁ G₂ : Causalean.DAG V) → Decidable (Causalean.SameSkeleton G₁ G₂)","l":69,"f":"Causalean/Graph/MarkovEquiv/Defs.lean"},{"n":"Causalean.instMeasurableSpaceSwigΩ","k":"instance","m":"Causalean.Graph.SWIG","a":"Graph","t":2,"d":"The shared SWIG value-space family inherits measurable spaces from the base variables.","s":"{N : Type u_1} → (Ω : N → Type u_2) → [(n : N) → MeasurableSpace (Ω n)] → (sn : Causalean.SWIGNode N) → MeasurableSpace (Causalean.swigΩ Ω sn)","l":135,"f":"Causalean/Graph/SWIG.lean"},{"n":"Causalean.instNonemptySwigΩ","k":"instance","m":"Causalean.Graph.SWIG","a":"Graph","t":2,"d":"The shared SWIG value-space family is nonempty whenever each base value space is nonempty.","s":"∀ {N : Type u_1} (Ω : N → Type u_2) [∀ (n : N), Nonempty (Ω n)] (sn : Causalean.SWIGNode N), Nonempty (Causalean.swigΩ Ω sn)","l":167,"f":"Causalean/Graph/SWIG.lean"},{"n":"Causalean.instReprEdgeType","k":"instance","m":"Causalean.SCM.Model.EdgeType","a":"SCM/Model","t":2,"d":"","s":"Repr Causalean.EdgeType","l":66,"f":"Causalean/SCM/Model/EdgeType.lean"},{"n":"Causalean.instReprMonotonicityKind","k":"instance","m":"Causalean.SCM.Model.EdgeType","a":"SCM/Model","t":2,"d":"","s":"Repr Causalean.MonotonicityKind","l":44,"f":"Causalean/SCM/Model/EdgeType.lean"},{"n":"Causalean.instReprSWIGNode","k":"instance","m":"Causalean.Graph.SWIG","a":"Graph","t":2,"d":"","s":"{N : Type u_1} → [Repr N] → Repr (Causalean.SWIGNode N)","l":88,"f":"Causalean/Graph/SWIG.lean"},{"n":"Causalean.instStandardBorelSpaceSwigΩ","k":"instance","m":"Causalean.Graph.SWIG","a":"Graph","t":2,"d":"The shared SWIG value-space family inherits standard Borel spaces from the base variables.","s":"∀ {N : Type u_1} (Ω : N → Type u_2) [inst : (n : N) → MeasurableSpace (Ω n)] [∀ (n : N), StandardBorelSpace (Ω n)] (sn : Causalean.SWIGNode N), StandardBorelSpace (Causalean.swigΩ Ω sn)","l":160,"f":"Causalean/Graph/SWIG.lean"},{"n":"Causalean.integrableOn_of_condExp_indicator_mul","k":"theorem","m":"Causalean.Mathlib.CondIndep.Integrability","a":"Mathlib/CondIndep","t":1,"d":"Integrability on a stratum from a conditional-expectation indicator identity. For a sub-σ-algebra m coarser than the ambient σ-algebra, measurable events B and C, and an m-measurable real-valued function g that is almost everywhere nonnegative, if the conditional expectation of the indicator of B given m, multiplied pointwise by g, equals almost everywhere the conditional expectation of the indicator of C given m, then g is integrable on B (moreover ∫_B g dμ ≤ μ C).","s":"∀ {Ω : Type u_1} {mΩ : MeasurableSpace Ω} {μ : MeasureTheory.Measure Ω} [MeasureTheory.IsFiniteMeasure μ] {m : MeasurableSpace Ω}, m ≤ mΩ → ∀ {B C : Set Ω}, MeasurableSet B → MeasurableSet C → ∀ {g : Ω → ℝ}, Measurable g → 0 ≤ᵐ[μ] g → (fun ω => μ[B.indicator fun x => 1 | m] ω * g ω) =ᵐ[μ] μ[C.indicator fun x => 1 | m] → MeasureTheory.IntegrableOn g B μ","l":105,"f":"Causalean/Mathlib/CondIndep/Integrability.lean"},{"n":"Causalean.iotaMap","k":"def","m":"Causalean.Graph.SWIG","a":"Graph","t":1,"d":"The injection ι mapping each fixed intervention parameter to its random counterpart. In the SWIG, ι(fixed d) = random d.","s":"{N : Type u_1} → Causalean.SWIGNode N → Causalean.SWIGNode N","l":259,"f":"Causalean/Graph/SWIG.lean"},{"n":"Causalean.iotaMap_fixed","k":"theorem","m":"Causalean.Graph.SWIG","a":"Graph","t":2,"d":"The link map sends the fixed copy of a base variable to its random copy.","s":"∀ {N : Type u_1} (n : N), Causalean.iotaMap (Causalean.SWIGNode.fixed n) = Causalean.SWIGNode.random n","l":269,"f":"Causalean/Graph/SWIG.lean"},{"n":"Causalean.map_compProd_prodMap_left_eq_compProd_comap","k":"theorem","m":"Causalean.Mathlib.CondDistrib","a":"Mathlib","t":1,"d":"Pushforward of compProd through a measurable equivalence on the first coordinate. For a measurable equivalence e between the first-coordinate spaces, an s-finite measure ν on the source first coordinate, and an s-finite kernel κ from that coordinate to a second space, pushing the composed-product measure of ν and κ forward through e on the first coordinate (identity on the second) equals the composed-product measure of the pushed-forward ν and κ transported back along e's inverse.","s":"∀ {β : Type u_1} {β' : Type u_2} {γ : Type u_3} [inst : MeasurableSpace β] [inst_1 : MeasurableSpace β'] [inst_2 : MeasurableSpace γ] (ν : MeasureTheory.Measure β) [MeasureTheory.SFinite ν] (e : β ≃ᵐ β') (κ : ProbabilityTheory.Kernel β γ) [ProbabilityTheory.IsSFiniteKernel κ], MeasureTheory.Measure.map (Prod.map (⇑e) id) (ν.compProd κ) = (MeasureTheory.Measure.map (⇑e) ν).compProd (κ.comap (⇑e.symm) (MeasurableEquiv.measurable e.symm))","l":58,"f":"Causalean/Mathlib/CondDistrib.lean"},{"n":"Causalean.markovEquiv_iff_sameSkeleton_sameImmoralities","k":"theorem","m":"Causalean.Graph.MarkovEquiv","a":"Graph/MarkovEquiv","t":1,"d":"Verma–Pearl (1990). For two DAGs G₁, G₂ on the same vertex set, they are Markov equivalent — they declare exactly the same d-separations, hence impose the same conditional-independence constraints — if and only if they have the same skeleton and the same v-structures (immoralities).","s":"∀ {V : Type u_1} [inst : DecidableEq V] [inst_1 : Fintype V] (G₁ G₂ : Causalean.DAG V), Causalean.MarkovEquiv G₁ G₂ ↔ Causalean.SameSkeleton G₁ G₂ ∧ Causalean.SameImmoralities G₁ G₂","l":51,"f":"Causalean/Graph/MarkovEquiv.lean"},{"n":"Causalean.markovEquiv_of_sameSkeleton_sameImmoralities","k":"theorem","m":"Causalean.Graph.MarkovEquiv.Decompose","a":"Graph/MarkovEquiv","t":1,"d":"Verma–Pearl hard direction (covered-edge route). For DAGs G₁ and G₂ on the same vertex set, if G₁ and G₂ have the same skeleton and the same immoralities, then G₁ and G₂ are Markov equivalent: they license exactly the same d-separation statements — proven via AMP Lemma 3.2 (covered-edge reversals), independent of the moralization/ancestral kernel.","s":"∀ {V : Type u_1} [inst : DecidableEq V] [inst_1 : Fintype V] {G₁ G₂ : Causalean.DAG V}, Causalean.SameSkeleton G₁ G₂ → Causalean.SameImmoralities G₁ G₂ → Causalean.MarkovEquiv G₁ G₂","l":228,"f":"Causalean/Graph/MarkovEquiv/Decompose.lean"},{"n":"Causalean.markovEquiv_of_same_edge","k":"theorem","m":"Causalean.Graph.MarkovEquiv.Decompose","a":"Graph/MarkovEquiv","t":2,"d":"Edge-congruence for Markov equivalence. DAGs with the same directed-edge relation are Markov equivalent (d-separation depends only on the edge relation).","s":"∀ {V : Type u_1} [inst : DecidableEq V] [inst_1 : Fintype V] {G₁ G₂ : Causalean.DAG V}, (∀ (u w : V), G₁.edge u w ↔ G₂.edge u w) → Causalean.MarkovEquiv G₁ G₂","l":40,"f":"Causalean/Graph/MarkovEquiv/Decompose.lean"},{"n":"Causalean.measurable_finsetCoordProj","k":"theorem","m":"Causalean.Mathlib.Indep","a":"Mathlib","t":2,"d":"finsetCoordProj is measurable.","s":"∀ {ι : Type u_1} {Ω : ι → Type u_2} [inst : (i : ι) → MeasurableSpace (Ω i)] (S : Finset ι), Measurable (Causalean.finsetCoordProj S)","l":95,"f":"Causalean/Mathlib/Indep.lean"},{"n":"Causalean.measurable_finsetCoordProjFromCondResidual","k":"theorem","m":"Causalean.Mathlib.Indep","a":"Mathlib","t":2,"d":"The residual reassembly map is measurable.","s":"∀ {ι : Type u_1} [inst : DecidableEq ι] {Ω : ι → Type u_2} [inst_1 : (i : ι) → MeasurableSpace (Ω i)] (S U : Finset ι), Measurable (Causalean.finsetCoordProjFromCondResidual S U)","l":118,"f":"Causalean/Mathlib/Indep.lean"},{"n":"Causalean.measurable_finsetCoordProjPairFromUnion","k":"theorem","m":"Causalean.Mathlib.Indep","a":"Mathlib","t":2,"d":"The union sub-block extraction map is measurable.","s":"∀ {ι : Type u_1} [inst : DecidableEq ι] {Ω : ι → Type u_2} [inst_1 : (i : ι) → MeasurableSpace (Ω i)] (A B : Finset ι), Measurable (Causalean.finsetCoordProjPairFromUnion A B)","l":153,"f":"Causalean/Mathlib/Indep.lean"},{"n":"Causalean.measurable_valuesProjection","k":"theorem","m":"Causalean.Mathlib.MeasureTheory.FinsetValues","a":"Mathlib/MeasureTheory","t":1,"d":"For value assignments over a finite node set, if a finite node set J is a subset of a larger finite node set I, then restricting a value assignment over I to its coordinates in J is a measurable map.","s":"∀ {M : Type u_1} {I J : Finset M} {Ω' : M → Type u_2} [inst : (n : M) → MeasurableSpace (Ω' n)] (hJI : J ⊆ I), Measurable (Causalean.valuesProjection hJI)","l":46,"f":"Causalean/Mathlib/MeasureTheory/FinsetValues.lean"},{"n":"Causalean.measurable_valuesProjection_union_sup","k":"theorem","m":"Causalean.SCM.Do.ValuesProjectionCI","a":"SCM/Do","t":2,"d":"The projection to A ∪ B is measurable for the supremum of the separate projection σ-algebras.","s":"∀ {M : Type u_1} [inst : DecidableEq M] {I A B : Finset M} {Ω' : M → Type u_2} [inst_1 : (n : M) → MeasurableSpace (Ω' n)] (hA : A ⊆ I) (hB : B ⊆ I), Measurable (Causalean.valuesProjection (Finset.union_subset hA hB))","l":50,"f":"Causalean/SCM/Do/ValuesProjectionCI.lean"},{"n":"Causalean.measurable_valuesUnionMk","k":"theorem","m":"Causalean.Mathlib.MeasureTheory.FinsetValues","a":"Mathlib/MeasureTheory","t":2,"d":"Combining assignments is jointly measurable in both input assignments.","s":"∀ {M : Type u_1} [inst : DecidableEq M] {Ω : M → Type u_2} [inst_1 : (n : M) → MeasurableSpace (Ω n)] {A B : Finset M}, Measurable fun p => Causalean.valuesUnionMk p.1 p.2","l":167,"f":"Causalean/Mathlib/MeasureTheory/FinsetValues.lean"},{"n":"Causalean.measurable_valuesUnionMk_right","k":"theorem","m":"Causalean.Mathlib.MeasureTheory.FinsetValues","a":"Mathlib/MeasureTheory","t":2,"d":"Combining assignments is measurable in the second assignment with the first held fixed.","s":"∀ {M : Type u_1} [inst : DecidableEq M] {Ω : M → Type u_2} [inst_1 : (n : M) → MeasurableSpace (Ω n)] {A B : Finset M} (a : Causalean.ValuesOn A Ω), Measurable fun b => Causalean.valuesUnionMk a b","l":142,"f":"Causalean/Mathlib/MeasureTheory/FinsetValues.lean"},{"n":"Causalean.measurePreserving_valuesEquivOfEq","k":"theorem","m":"Causalean.Mathlib.MeasureTheory.FinsetValues","a":"Mathlib/MeasureTheory","t":1,"d":"Given two propositionally equal finite node sets I and J and a family μ of measures, one per coordinate of I, the canonical measurable equivalence between value assignments over I and over J carries the product measure Measure.pi μ to the product measure built from μ re-indexed along J through the equality.","s":"∀ {M : Type u_1} {I J : Finset M} {Ω : M → Type u_2} [inst : (n : M) → MeasurableSpace (Ω n)] (h : I = J) (μ : (i : ↥I) → MeasureTheory.Measure (Ω ↑i)), MeasureTheory.MeasurePreserving (⇑(Causalean.valuesEquivOfEq h)) (MeasureTheory.Measure.pi μ) (MeasureTheory.Measure.pi fun j => μ ⟨↑j, (fun h_1 => Eq.ndrec (motive := fun {J} => ∀ (j : ↥J), ↑j ∈ J → ↑j ∈ I) (fun j h => h) h j h_1) j.property⟩)","l":87,"f":"Causalean/Mathlib/MeasureTheory/FinsetValues.lean"},{"n":"Causalean.measure_eq_bind_marginal_condDistrib","k":"theorem","m":"Causalean.Mathlib.CondDistrib","a":"Mathlib","t":1,"d":"Measure-theoretic chain rule / disintegration (Mathlib gap). For a finite measure μ on a product space, μ equals the composition obtained by first drawing the second coordinate from its marginal distribution and then drawing the first coordinate from its regular conditional distribution given that second coordinate.","s":"∀ {β : Type u_1} {γ : Type u_2} {mβ : MeasurableSpace β} {mγ : MeasurableSpace γ} [inst : StandardBorelSpace β] [inst_1 : Nonempty β] (μ : MeasureTheory.Measure (β × γ)) [inst_2 : MeasureTheory.IsFiniteMeasure μ], μ = (MeasureTheory.Measure.map Prod.snd μ).bind fun c => MeasureTheory.Measure.map (fun b => (b, c)) ((ProbabilityTheory.condDistrib Prod.fst Prod.snd μ) c)","l":169,"f":"Causalean/Mathlib/CondDistrib.lean"},{"n":"Causalean.measure_norm_gt_le_five_pow_finrank","k":"theorem","m":"Causalean.Stat.Concentration.SubGaussianNorm","a":"Stat/Concentration","t":1,"d":"Dimension-only sub-Gaussian norm tail. Let V be a finite-dimensional subspace of E. If Z lies in V almost surely under P and every unit vector v in V has the scalar projection ⟪v, Z⟫ sub-Gaussian with parameter 1 under P, then for any nonnegative t, the probability that ‖Z‖ exceeds 2 * t is at most 5 raised to the dimension of V, times exp (-t ^ 2 / 2).","s":"∀ {Ω : Type u_1} {E : Type u_2} [inst : MeasurableSpace Ω] [inst_1 : NormedAddCommGroup E] [inst_2 : InnerProductSpace ℝ E] (P : MeasureTheory.Measure Ω) [MeasureTheory.IsFiniteMeasure P] (Z : Ω → E) (V : Submodule ℝ E) [FiniteDimensional ℝ ↥V], (∀ᵐ (ω : Ω) ∂P, Z ω ∈ V) → (∀ v ∈ V, ‖v‖ = 1 → ProbabilityTheory.HasSubgaussianMGF (fun ω => inner ℝ v (Z ω)) 1 P) → ∀ {t : ℝ}, 0 ≤ t → P.real {ω | 2 * t < ‖Z ω‖} ≤ ↑(5 ^ Module.finrank ℝ ↥V) * Real.exp (-t ^ 2 / 2)","l":87,"f":"Causalean/Stat/Concentration/SubGaussianNorm.lean"},{"n":"Causalean.measure_norm_gt_le_of_half_net","k":"theorem","m":"Causalean.Stat.Concentration.SubGaussianNorm","a":"Stat/Concentration","t":2,"d":"If a finite half-net controls every scalar projection with unit sub-Gaussian parameter, then the norm has a Gaussian upper tail. The support condition is only required almost everywhere.","s":"∀ {Ω : Type u_1} {E : Type u_2} [inst : MeasurableSpace Ω] [inst_1 : NormedAddCommGroup E] [inst_2 : InnerProductSpace ℝ E] (P : MeasureTheory.Measure Ω) [MeasureTheory.IsFiniteMeasure P] (Z : Ω → E) (V : Submodule ℝ E) (N : Finset E), (∀ x ∈ V, ‖x‖ = 1 → ∃ v ∈ N, ‖x - v‖ ≤ 1 / 2) → (∀ᵐ (ω : Ω) ∂P, Z ω ∈ V) → (∀ v ∈ N, ProbabilityTheory.HasSubgaussianMGF (fun ω => inner ℝ v (Z ω)) 1 P) → ∀ {t : ℝ}, 0 ≤ t → P.real {ω | 2 * t < ‖Z ω‖} ≤ ↑N.card * Real.exp (-t ^ 2 / 2)","l":52,"f":"Causalean/Stat/Concentration/SubGaussianNorm.lean"},{"n":"Causalean.moralAdj_congr","k":"theorem","m":"Causalean.Graph.MarkovEquiv.Moralization","a":"Graph/MarkovEquiv","t":2,"d":"Moral adjacency is a skeleton + v-structure invariant. Two DAGs with the same skeleton and the same v-structures induce the same moral adjacency on any ground set: a shielded pair is moral-adjacent via the shared skeleton edge, and an unshielded married pair is exactly the apex of a shared immorality.","s":"∀ {V : Type u_1} [inst : DecidableEq V] [inst_1 : Fintype V] {G₁ G₂ : Causalean.DAG V}, Causalean.SameSkeleton G₁ G₂ → Causalean.SameImmoralities G₁ G₂ → ∀ (S : Finset V) (u v : V), G₁.MoralAdj S u v ↔ G₂.MoralAdj S u v","l":526,"f":"Causalean/Graph/MarkovEquiv/Moralization.lean"},{"n":"Causalean.moralConn_congr","k":"theorem","m":"Causalean.Graph.MarkovEquiv.Moralization","a":"Graph/MarkovEquiv","t":2,"d":"Moral connectivity agrees across DAGs with the same skeleton and v-structures, for a fixed ground set S. (The ancestral sets used by MoralSep still differ between the graphs; that reconciliation is moralSep_congr.)","s":"∀ {V : Type u_1} [inst : DecidableEq V] [inst_1 : Fintype V] {G₁ G₂ : Causalean.DAG V}, Causalean.SameSkeleton G₁ G₂ → Causalean.SameImmoralities G₁ G₂ → ∀ (S Z : Finset V) (u v : V), G₁.MoralConn S Z u v ↔ G₂.MoralConn S Z u v","l":564,"f":"Causalean/Graph/MarkovEquiv/Moralization.lean"},{"n":"Causalean.moralStep_congr","k":"theorem","m":"Causalean.Graph.MarkovEquiv.Moralization","a":"Graph/MarkovEquiv","t":2,"d":"Moral steps agree across DAGs with the same skeleton and v-structures (same ground set, same conditioning set).","s":"∀ {V : Type u_1} [inst : DecidableEq V] [inst_1 : Fintype V] {G₁ G₂ : Causalean.DAG V}, Causalean.SameSkeleton G₁ G₂ → Causalean.SameImmoralities G₁ G₂ → ∀ (S Z : Finset V) (u v : V), G₁.MoralStep S Z u v ↔ G₂.MoralStep S Z u v","l":556,"f":"Causalean/Graph/MarkovEquiv/Moralization.lean"},{"n":"Causalean.sameImmoralities_of_markovEquiv","k":"theorem","m":"Causalean.Graph.MarkovEquiv.Readoff","a":"Graph/MarkovEquiv","t":2,"d":"Markov equivalence ⇒ same v-structures.","s":"∀ {V : Type u_1} [inst : DecidableEq V] [inst_1 : Fintype V] {G₁ G₂ : Causalean.DAG V}, Causalean.MarkovEquiv G₁ G₂ → Causalean.SameImmoralities G₁ G₂","l":351,"f":"Causalean/Graph/MarkovEquiv/Readoff.lean"},{"n":"Causalean.sameSkeleton_of_markovEquiv","k":"theorem","m":"Causalean.Graph.MarkovEquiv.Readoff","a":"Graph/MarkovEquiv","t":2,"d":"Markov equivalence ⇒ same skeleton.","s":"∀ {V : Type u_1} [inst : DecidableEq V] [inst_1 : Fintype V] {G₁ G₂ : Causalean.DAG V}, Causalean.MarkovEquiv G₁ G₂ → Causalean.SameSkeleton G₁ G₂","l":334,"f":"Causalean/Graph/MarkovEquiv/Readoff.lean"},{"n":"Causalean.sameSkeleton_sameImmoralities_of_markovEquiv","k":"theorem","m":"Causalean.Graph.MarkovEquiv.Readoff","a":"Graph/MarkovEquiv","t":1,"d":"Easy direction of Verma–Pearl. For two DAGs G₁, G₂, if they are Markov equivalent then they share the same skeleton and the same v-structures (immoralities).","s":"∀ {V : Type u_1} [inst : DecidableEq V] [inst_1 : Fintype V] {G₁ G₂ : Causalean.DAG V}, Causalean.MarkovEquiv G₁ G₂ → Causalean.SameSkeleton G₁ G₂ ∧ Causalean.SameImmoralities G₁ G₂","l":384,"f":"Causalean/Graph/MarkovEquiv/Readoff.lean"},{"n":"Causalean.same_edge_of_edgeDiff_empty","k":"theorem","m":"Causalean.Graph.MarkovEquiv.Decompose","a":"Graph/MarkovEquiv","t":2,"d":"With a one-way skeleton inclusion, an empty edge-difference forces equal edge relations.","s":"∀ {V : Type u_1} [inst : DecidableEq V] [inst_1 : Fintype V] {G₁ G₂ : Causalean.DAG V}, (∀ (u w : V), G₂.UAdj u w → G₁.UAdj u w) → Causalean.edgeDiff G₁ G₂ = ∅ → ∀ (u w : V), G₁.edge u w ↔ G₂.edge u w","l":63,"f":"Causalean/Graph/MarkovEquiv/Decompose.lean"},{"n":"Causalean.setIntegral_condExp_indep_indicator","k":"theorem","m":"Causalean.Mathlib.CondIndep.CondExp","a":"Mathlib/CondIndep","t":2,"d":"Under conditional independence of two variables given a σ-algebra, integrating the conditional expectation of a constant times an event indicator over the intersection of a conditioning event and an event determined by one variable equals integrating that indicator directly.","s":"∀ {Ω : Type u_1} {α : Type u_2} {β : Type u_3} {m mΩ : MeasurableSpace Ω} (hm : m ≤ mΩ) [inst : MeasurableSpace α] [inst_1 : MeasurableSpace β] [inst_2 : StandardBorelSpace Ω] {μ : MeasureTheory.Measure Ω} [inst_3 : MeasureTheory.IsFiniteMeasure μ] {f : Ω → α} {g : Ω → β}, Measurable f → Measurable g → ProbabilityTheory.CondIndepFun m hm g f μ → ∀ {S A F : Set Ω}, MeasurableSet S → MeasurableSet A → MeasurableSet F → ∀ (c : ℝ), ∫ (x : Ω) in A ∩ F, μ[S.indicator fun x => c | m] x ∂μ = ∫ (x : Ω) in A ∩ F, S.indicator (fun x => c) x ∂μ","l":183,"f":"Causalean/Mathlib/CondIndep/CondExp.lean"},{"n":"Causalean.setIntegral_eq_setIntegral_mul_of_likelihoodRatio_swap","k":"theorem","m":"Causalean.Mathlib.LikelihoodRatioSwap","a":"Mathlib","t":1,"d":"Arm-swap via likelihood ratio. For a sub-σ-algebra m of the ambient σ-algebra on Ω, a finite measure μ, and measurable sets s and t, suppose the product f · L is almost-everywhere strongly measurable relative to m, the integrand f itself is almost-everywhere strongly measurable relative to m, f restricted to the source set t is integrable, the product f · L restricted to the target set s is integrable, and the likelihood ratio L satisfies the change-of-measure relation: the conditional probability of s given m, times L, agrees μ-almost everywhere with the conditional probability of t given m. Then the integral of f over t equals the integral of f · L over s: ∫ₜ f dμ = ∫ₛ f · L dμ.","s":"∀ {Ω : Type u_1} {m mΩ : MeasurableSpace Ω}, m ≤ mΩ → ∀ {μ : MeasureTheory.Measure Ω} [MeasureTheory.IsFiniteMeasure μ] (s t : Set Ω), MeasurableSet s → MeasurableSet t → ∀ {L f : Ω → ℝ}, MeasureTheory.AEStronglyMeasurable (fun ω => f ω * L ω) μ → MeasureTheory.AEStronglyMeasurable f μ → MeasureTheory.IntegrableOn f t μ → MeasureTheory.IntegrableOn (fun ω => f ω * L ω) s μ → (fun ω => μ[s.indicator fun x => 1 | m] ω * L ω) =ᵐ[μ] μ[t.indicator fun x => 1 | m] → ∫ (ω : Ω) in t, f ω ∂μ = ∫ (ω : Ω) in s, f ω * L ω ∂μ","l":22,"f":"Causalean/Mathlib/LikelihoodRatioSwap.lean"},{"n":"Causalean.swigDAG","k":"def","m":"Causalean.Graph.SWIG","a":"Graph","t":1,"d":"The SWIG DAG: the DAG on SWIGNode N constructed by node-splitting.","s":"{N : Type u_1} → [inst : DecidableEq N] → [inst_1 : Fintype N] → Causalean.DAG N → Finset N → Causalean.DAG (Causalean.SWIGNode N)","l":244,"f":"Causalean/Graph/SWIG.lean"},{"n":"Causalean.swigEdge","k":"def","m":"Causalean.Graph.SWIG","a":"Graph","t":1,"d":"The edge relation in a SWIG.","s":"{N : Type u_1} → [inst : DecidableEq N] → [inst_1 : Fintype N] → Causalean.DAG N → Finset N → Causalean.SWIGNode N → Causalean.SWIGNode N → Prop","l":179,"f":"Causalean/Graph/SWIG.lean"},{"n":"Causalean.swigEdge_decidable","k":"instance","m":"Causalean.Graph.SWIG","a":"Graph","t":2,"d":"The SWIG edge relation is decidable whenever the base variables and target set are finite.","s":"{N : Type u_1} → [inst : DecidableEq N] → [inst_1 : Fintype N] → (G : Causalean.DAG N) → (targets : Finset N) → DecidableRel (Causalean.swigEdge G targets)","l":195,"f":"Causalean/Graph/SWIG.lean"},{"n":"Causalean.swigTopo","k":"def","m":"Causalean.Graph.SWIG","a":"Graph","t":1,"d":"Topological order for the SWIG.","s":"{N : Type u_1} → [inst : DecidableEq N] → [inst_1 : Fintype N] → Causalean.DAG N → Causalean.SWIGNode N → ℕ","l":205,"f":"Causalean/Graph/SWIG.lean"},{"n":"Causalean.swigTopo_lt","k":"theorem","m":"Causalean.Graph.SWIG","a":"Graph","t":2,"d":"Every SWIG edge points from a lower to a higher position in the interleaved topological order.","s":"∀ {N : Type u_1} [inst : DecidableEq N] [inst_1 : Fintype N] (G : Causalean.DAG N) (targets : Finset N) (u v : Causalean.SWIGNode N), Causalean.swigEdge G targets u v → Causalean.swigTopo G u < Causalean.swigTopo G v","l":217,"f":"Causalean/Graph/SWIG.lean"},{"n":"Causalean.swig_fixed_are_roots","k":"theorem","m":"Causalean.Graph.SWIG","a":"Graph","t":1,"d":"For any base DAG G, any set of intervention targets, and any node n, the fixed copy of n has no parents in the single-world intervention graph built from G and targets.","s":"∀ {N : Type u_1} [inst : DecidableEq N] [inst_1 : Fintype N] (G : Causalean.DAG N) (targets : Finset N) (n : N), (Causalean.swigDAG G targets).parents (Causalean.SWIGNode.fixed n) = ∅","l":276,"f":"Causalean/Graph/SWIG.lean"},{"n":"Causalean.swig_random_root_of_root","k":"theorem","m":"Causalean.Graph.SWIG","a":"Graph","t":2,"d":"If n is a root in G, then random n is a root in the SWIG.","s":"∀ {N : Type u_1} [inst : DecidableEq N] [inst_1 : Fintype N] (G : Causalean.DAG N) (targets : Finset N) (n : N), G.parents n = ∅ → (Causalean.swigDAG G targets).parents (Causalean.SWIGNode.random n) = ∅","l":311,"f":"Causalean/Graph/SWIG.lean"},{"n":"Causalean.swig_target_parents","k":"theorem","m":"Causalean.Graph.SWIG","a":"Graph","t":1,"d":"For any base DAG G, any set of intervention targets, and any node d, the parents of the random copy of d in the single-world intervention graph are exactly the copies of d's original parents in G, each represented by its random version if it is not a target and by its fixed version if it is.","s":"∀ {N : Type u_1} [inst : DecidableEq N] [inst_1 : Fintype N] (G : Causalean.DAG N) (targets : Finset N) (d : N) (x : Causalean.SWIGNode N), x ∈ (Causalean.swigDAG G targets).parents (Causalean.SWIGNode.random d) ↔ ∃ p, G.edge p d ∧ x = Causalean.SWIGNode.random p ∧ p ∉ targets ∨ G.edge p d ∧ x = Causalean.SWIGNode.fixed p ∧ p ∈ targets","l":286,"f":"Causalean/Graph/SWIG.lean"},{"n":"Causalean.swigΩ","k":"def","m":"Causalean.Graph.SWIG","a":"Graph","t":1,"d":"Value-space family for the SWIG model. Both random and fixed versions of a node share the same value space as the original node: swigΩ(.random n) = Ω n and swigΩ(.fixed n) = Ω n. This matches the tex requirement X_d = X_{ι(d)}. Declared as abbrev so that swigΩ Ω (.random n) reduces to Ω n during type class synthesis.","s":"{N : Type u_1} → (N → Type u_2) → Causalean.SWIGNode N → Type u_2","l":125,"f":"Causalean/Graph/SWIG.lean"},{"n":"Causalean.valuesEquivOfEq","k":"def","m":"Causalean.Mathlib.MeasureTheory.FinsetValues","a":"Mathlib/MeasureTheory","t":1,"d":"Value assignments over propositionally equal finite node sets are canonically measurably equivalent.","s":"{M : Type u_1} → {I J : Finset M} → {Ω : M → Type u_2} → [inst : (n : M) → MeasurableSpace (Ω n)] → I = J → Causalean.ValuesOn I Ω ≃ᵐ Causalean.ValuesOn J Ω","l":71,"f":"Causalean/Mathlib/MeasureTheory/FinsetValues.lean"},{"n":"Causalean.valuesProjection","k":"def","m":"Causalean.Mathlib.MeasureTheory.FinsetValues","a":"Mathlib/MeasureTheory","t":1,"d":"A value assignment can be restricted from a larger finite node set to a smaller one.","s":"{M : Type u_1} → {I J : Finset M} → {Ω : M → Type u_2} → [(n : M) → MeasurableSpace (Ω n)] → J ⊆ I → Causalean.ValuesOn I Ω → Causalean.ValuesOn J Ω","l":36,"f":"Causalean/Mathlib/MeasureTheory/FinsetValues.lean"},{"n":"Causalean.valuesProjection_comp","k":"theorem","m":"Causalean.SCM.Do.ValuesProjectionCI","a":"SCM/Do","t":2,"d":"Composition identity for valuesProjection: restricting to K ⊆ J ⊆ I in one step equals composing the two restrictions.","s":"∀ {M : Type u_1} {I J K : Finset M} {Ω' : M → Type u_2} [inst : (n : M) → MeasurableSpace (Ω' n)] (hKJ : K ⊆ J) (hJI : J ⊆ I), Causalean.valuesProjection (LE.le.trans hKJ hJI) = Causalean.valuesProjection hKJ ∘ Causalean.valuesProjection hJI","l":28,"f":"Causalean/SCM/Do/ValuesProjectionCI.lean"},{"n":"Causalean.valuesUnionMk","k":"def","m":"Causalean.Mathlib.MeasureTheory.FinsetValues","a":"Mathlib/MeasureTheory","t":1,"d":"This map combines two coordinate assignments into one assignment on their union.","s":"{M : Type u_1} → [inst : DecidableEq M] → {Ω : M → Type u_2} → [(n : M) → MeasurableSpace (Ω n)] → {A B : Finset M} → Causalean.ValuesOn A Ω → Causalean.ValuesOn B Ω → Causalean.ValuesOn (A ∪ B) Ω","l":113,"f":"Causalean/Mathlib/MeasureTheory/FinsetValues.lean"},{"n":"Causalean.valuesUnionMk_apply_left","k":"theorem","m":"Causalean.Mathlib.MeasureTheory.FinsetValues","a":"Mathlib/MeasureTheory","t":2,"d":"Projecting a union assignment to a coordinate from the first input returns that value.","s":"∀ {M : Type u_1} [inst : DecidableEq M] {Ω : M → Type u_2} [inst_1 : (n : M) → MeasurableSpace (Ω n)] {A B : Finset M} (a : Causalean.ValuesOn A Ω) (b : Causalean.ValuesOn B Ω) {v : M} (hA : v ∈ A), Causalean.valuesUnionMk a b ⟨v, Finset.mem_union_left B hA⟩ = a ⟨v, hA⟩","l":124,"f":"Causalean/Mathlib/MeasureTheory/FinsetValues.lean"},{"n":"Causalean.valuesUnionMk_apply_right","k":"theorem","m":"Causalean.Mathlib.MeasureTheory.FinsetValues","a":"Mathlib/MeasureTheory","t":2,"d":"Projecting a union assignment outside the first input returns the second input's value.","s":"∀ {M : Type u_1} [inst : DecidableEq M] {Ω : M → Type u_2} [inst_1 : (n : M) → MeasurableSpace (Ω n)] {A B : Finset M} (a : Causalean.ValuesOn A Ω) (b : Causalean.ValuesOn B Ω) {v : M} (hv : v ∈ A ∪ B) (hA : v ∉ A), Causalean.valuesUnionMk a b ⟨v, hv⟩ = b ⟨v, Or.resolve_left (Finset.mem_union.mp hv) hA⟩","l":133,"f":"Causalean/Mathlib/MeasureTheory/FinsetValues.lean"},{"n":"Causalean.valuesUnionMk_projection_comp","k":"theorem","m":"Causalean.SCM.Do.ValuesProjectionCI","a":"SCM/Do","t":2,"d":"Projecting to A and B and then assembling the union block recovers the direct projection to A ∪ B.","s":"∀ {M : Type u_1} [inst : DecidableEq M] {I A B : Finset M} {Ω' : M → Type u_2} [inst_1 : (n : M) → MeasurableSpace (Ω' n)] (hA : A ⊆ I) (hB : B ⊆ I), (fun ξ => Causalean.valuesUnionMk (Causalean.valuesProjection hA ξ) (Causalean.valuesProjection hB ξ)) = Causalean.valuesProjection (Finset.union_subset hA hB)","l":116,"f":"Causalean/SCM/Do/ValuesProjectionCI.lean"},{"n":"LinearMap.IsPositive.apply_eigBasis","k":"theorem","m":"Causalean.Mathlib.OperatorSqrt","a":"Mathlib","t":2,"d":"Applying the operator to an eigenbasis vector scales that vector by its indexed eigenvalue.","s":"∀ {E : Type u_1} [inst : NormedAddCommGroup E] [inst_1 : InnerProductSpace ℝ E] [inst_2 : FiniteDimensional ℝ E] {T : E →ₗ[ℝ] E} (hT : T.IsPositive) (i : Fin (Module.finrank ℝ E)), T (hT.eigBasis i) = hT.eigVals i • hT.eigBasis i","l":43,"f":"Causalean/Mathlib/OperatorSqrt.lean"},{"n":"LinearMap.IsPositive.eigVals_nonneg","k":"theorem","m":"Causalean.Mathlib.OperatorSqrt","a":"Mathlib","t":2,"d":"Eigenvalues of a positive operator are nonnegative.","s":"∀ {E : Type u_1} [inst : NormedAddCommGroup E] [inst_1 : InnerProductSpace ℝ E] [inst_2 : FiniteDimensional ℝ E] {T : E →ₗ[ℝ] E} (hT : T.IsPositive) (i : Fin (Module.finrank ℝ E)), 0 ≤ hT.eigVals i","l":50,"f":"Causalean/Mathlib/OperatorSqrt.lean"},{"n":"LinearMap.IsPositive.posSqrtCLM_adjoint","k":"theorem","m":"Causalean.Mathlib.OperatorSqrt","a":"Mathlib","t":2,"d":"Adjoint form of self-adjointness.","s":"∀ {E : Type u_1} [inst : NormedAddCommGroup E] [inst_1 : InnerProductSpace ℝ E] [inst_2 : FiniteDimensional ℝ E] {T : E →ₗ[ℝ] E} (hT : T.IsPositive), ContinuousLinearMap.adjoint hT.posSqrtCLM = hT.posSqrtCLM","l":123,"f":"Causalean/Mathlib/OperatorSqrt.lean"},{"n":"LinearMap.IsPositive.posSqrtCLM_apply","k":"theorem","m":"Causalean.Mathlib.OperatorSqrt","a":"Mathlib","t":2,"d":"The continuous-linear-map packaging of the positive square root has the same pointwise action as the linear-map square root.","s":"∀ {E : Type u_1} [inst : NormedAddCommGroup E] [inst_1 : InnerProductSpace ℝ E] [inst_2 : FiniteDimensional ℝ E] {T : E →ₗ[ℝ] E} (hT : T.IsPositive) (x : E), hT.posSqrtCLM x = hT.posSqrt x","l":113,"f":"Causalean/Mathlib/OperatorSqrt.lean"},{"n":"LinearMap.IsPositive.posSqrtCLM_comp_self","k":"theorem","m":"Causalean.Mathlib.OperatorSqrt","a":"Mathlib","t":1,"d":"Composing the continuous positive square root of T with itself recovers T, packaged as a continuous linear map.","s":"∀ {E : Type u_1} [inst : NormedAddCommGroup E] [inst_1 : InnerProductSpace ℝ E] [inst_2 : FiniteDimensional ℝ E] {T : E →ₗ[ℝ] E} (hT : T.IsPositive), hT.posSqrtCLM ∘SL hT.posSqrtCLM = LinearMap.toContinuousLinearMap T","l":128,"f":"Causalean/Mathlib/OperatorSqrt.lean"},{"n":"LinearMap.IsPositive.posSqrtCLM_isSelfAdjoint","k":"theorem","m":"Causalean.Mathlib.OperatorSqrt","a":"Mathlib","t":2,"d":"The continuous square root is self-adjoint.","s":"∀ {E : Type u_1} [inst : NormedAddCommGroup E] [inst_1 : InnerProductSpace ℝ E] [inst_2 : FiniteDimensional ℝ E] {T : E →ₗ[ℝ] E} (hT : T.IsPositive), IsSelfAdjoint hT.posSqrtCLM","l":118,"f":"Causalean/Mathlib/OperatorSqrt.lean"},{"n":"LinearMap.IsPositive.posSqrt_apply_eigBasis","k":"theorem","m":"Causalean.Mathlib.OperatorSqrt","a":"Mathlib","t":2,"d":"The positive square root sends each eigenbasis vector to the same vector scaled by the square root of its eigenvalue.","s":"∀ {E : Type u_1} [inst : NormedAddCommGroup E] [inst_1 : InnerProductSpace ℝ E] [inst_2 : FiniteDimensional ℝ E] {T : E →ₗ[ℝ] E} (hT : T.IsPositive) (i : Fin (Module.finrank ℝ E)), hT.posSqrt (hT.eigBasis i) = √(hT.eigVals i) • hT.eigBasis i","l":65,"f":"Causalean/Mathlib/OperatorSqrt.lean"},{"n":"LinearMap.IsPositive.posSqrt_inner","k":"theorem","m":"Causalean.Mathlib.OperatorSqrt","a":"Mathlib","t":2,"d":"Inner-product form of posSqrt: a symmetric weighted sum over the eigenbasis.","s":"∀ {E : Type u_1} [inst : NormedAddCommGroup E] [inst_1 : InnerProductSpace ℝ E] [inst_2 : FiniteDimensional ℝ E] {T : E →ₗ[ℝ] E} (hT : T.IsPositive) (x y : E), inner ℝ (hT.posSqrt x) y = ∑ i, √(hT.eigVals i) * inner ℝ (hT.eigBasis i) x * inner ℝ (hT.eigBasis i) y","l":78,"f":"Causalean/Mathlib/OperatorSqrt.lean"},{"n":"LinearMap.IsPositive.posSqrt_isPositive","k":"theorem","m":"Causalean.Mathlib.OperatorSqrt","a":"Mathlib","t":2,"d":"posSqrt is itself a positive operator.","s":"∀ {E : Type u_1} [inst : NormedAddCommGroup E] [inst_1 : InnerProductSpace ℝ E] [inst_2 : FiniteDimensional ℝ E] {T : E →ₗ[ℝ] E} (hT : T.IsPositive), hT.posSqrt.IsPositive","l":101,"f":"Causalean/Mathlib/OperatorSqrt.lean"},{"n":"LinearMap.IsPositive.posSqrt_isSymmetric","k":"theorem","m":"Causalean.Mathlib.OperatorSqrt","a":"Mathlib","t":2,"d":"posSqrt is symmetric.","s":"∀ {E : Type u_1} [inst : NormedAddCommGroup E] [inst_1 : InnerProductSpace ℝ E] [inst_2 : FiniteDimensional ℝ E] {T : E →ₗ[ℝ] E} (hT : T.IsPositive), hT.posSqrt.IsSymmetric","l":94,"f":"Causalean/Mathlib/OperatorSqrt.lean"},{"n":"LinearMap.IsPositive.posSqrt_mul_self","k":"theorem","m":"Causalean.Mathlib.OperatorSqrt","a":"Mathlib","t":1,"d":"Composing the positive square root of T with itself recovers T.","s":"∀ {E : Type u_1} [inst : NormedAddCommGroup E] [inst_1 : InnerProductSpace ℝ E] [inst_2 : FiniteDimensional ℝ E] {T : E →ₗ[ℝ] E} (hT : T.IsPositive), hT.posSqrt ∘ₗ hT.posSqrt = T","l":71,"f":"Causalean/Mathlib/OperatorSqrt.lean"},{"n":"MeasureTheory.AnalyticSet.isCapacitable","k":"theorem","m":"Causalean.Mathlib.MeasureTheory.AnalyticSetUniversalMeasurability.Capacity","a":"Mathlib/MeasureTheory/AnalyticSetUniversalMeasurability","t":1,"d":"Choquet's capacitability theorem. On a Polish sample space, if a set s is analytic — the continuous image of a Polish space, or empty, then s is capacitable for every Choquet capacity c: the capacity of s equals the supremum, over the compact subsets of s, of their capacities.","s":"∀ {Ω : Type u_1} [inst : TopologicalSpace Ω] [PolishSpace Ω] {c : MeasureTheory.ChoquetCapacity Ω} {s : Set Ω}, MeasureTheory.AnalyticSet s → c.IsCapacitable s","l":206,"f":"Causalean/Mathlib/MeasureTheory/AnalyticSetUniversalMeasurability/Capacity.lean"},{"n":"MeasureTheory.AnalyticSet.nullMeasurableSet","k":"theorem","m":"Causalean.Mathlib.MeasureTheory.AnalyticSetUniversalMeasurability.UniversalMeasurability","a":"Mathlib/MeasureTheory/AnalyticSetUniversalMeasurability","t":1,"d":"On a Polish sample space equipped with its Borel σ-algebra, if a set s is analytic, then s is null-measurable for every finite Borel measure μ: it becomes measurable after completing μ, so it differs from an ordinary measurable event only on a μ-null set.","s":"∀ {Ω : Type u_1} [inst : TopologicalSpace Ω] [inst_1 : MeasurableSpace Ω] [PolishSpace Ω] [BorelSpace Ω] {s : Set Ω}, MeasureTheory.AnalyticSet s → ∀ (μ : MeasureTheory.Measure Ω) [MeasureTheory.IsFiniteMeasure μ], MeasureTheory.NullMeasurableSet s μ","l":71,"f":"Causalean/Mathlib/MeasureTheory/AnalyticSetUniversalMeasurability/UniversalMeasurability.lean"},{"n":"MeasureTheory.ChoquetCapacity","k":"structure","m":"Causalean.Mathlib.MeasureTheory.AnalyticSetUniversalMeasurability.Capacity","a":"Mathlib/MeasureTheory/AnalyticSetUniversalMeasurability","t":1,"d":"A Choquet capacity on a topological sample space bundles an extended-real-valued set function that is monotone under set inclusion, continuous from below along increasing sequences of sets, finite on every compact set, and right-continuous at compact sets: any strict upper bound on a compact set's value is already an upper bound on some open neighborhood of that compact set.","s":"(Ω : Type u_2) → [TopologicalSpace Ω] → Type u_2","l":36,"f":"Causalean/Mathlib/MeasureTheory/AnalyticSetUniversalMeasurability/Capacity.lean"},{"n":"MeasureTheory.ChoquetCapacity.IsCapacitable","k":"def","m":"Causalean.Mathlib.MeasureTheory.AnalyticSetUniversalMeasurability.Capacity","a":"Mathlib/MeasureTheory/AnalyticSetUniversalMeasurability","t":1,"d":"A set is capacitable when its capacity is the supremum of the capacities of its compact subsets.","s":"{Ω : Type u_1} → [inst : TopologicalSpace Ω] → MeasureTheory.ChoquetCapacity Ω → Set Ω → Prop","l":77,"f":"Causalean/Mathlib/MeasureTheory/AnalyticSetUniversalMeasurability/Capacity.lean"},{"n":"MeasureTheory.ChoquetCapacity.IsCapacitable.nullMeasurableSet","k":"theorem","m":"Causalean.Mathlib.MeasureTheory.AnalyticSetUniversalMeasurability.UniversalMeasurability","a":"Mathlib/MeasureTheory/AnalyticSetUniversalMeasurability","t":2,"d":"A set capacitable for the outer measure of a finite measure is null-measurable for that measure.","s":"∀ {Ω : Type u_1} [inst : TopologicalSpace Ω] [inst_1 : MeasurableSpace Ω] [inst_2 : PolishSpace Ω] [inst_3 : BorelSpace Ω] {μ : MeasureTheory.Measure Ω} [inst_4 : MeasureTheory.IsFiniteMeasure μ] {s : Set Ω}, μ.toChoquetCapacity.IsCapacitable s → MeasureTheory.NullMeasurableSet s μ","l":28,"f":"Causalean/Mathlib/MeasureTheory/AnalyticSetUniversalMeasurability/UniversalMeasurability.lean"},{"n":"MeasureTheory.ChoquetCapacity.exists_isOpen_superset_lt","k":"theorem","m":"Causalean.Mathlib.MeasureTheory.AnalyticSetUniversalMeasurability.Capacity","a":"Mathlib/MeasureTheory/AnalyticSetUniversalMeasurability","t":2,"d":"A strict upper bound for the capacity of a compact set also bounds some open neighborhood.","s":"∀ {Ω : Type u_1} [inst : TopologicalSpace Ω] (c : MeasureTheory.ChoquetCapacity Ω) {K : Set Ω}, IsCompact K → ∀ {a : ENNReal}, c.toFun K < a → ∃ U, K ⊆ U ∧ IsOpen U ∧ c.toFun U < a","l":72,"f":"Causalean/Mathlib/MeasureTheory/AnalyticSetUniversalMeasurability/Capacity.lean"},{"n":"MeasureTheory.ChoquetCapacity.isCompact_lt_top","k":"theorem","m":"Causalean.Mathlib.MeasureTheory.AnalyticSetUniversalMeasurability.Capacity","a":"Mathlib/MeasureTheory/AnalyticSetUniversalMeasurability","t":2,"d":"Capacity values on compact sets are finite.","s":"∀ {Ω : Type u_1} [inst : TopologicalSpace Ω] (c : MeasureTheory.ChoquetCapacity Ω) {K : Set Ω}, IsCompact K → c.toFun K < ⊤","l":67,"f":"Causalean/Mathlib/MeasureTheory/AnalyticSetUniversalMeasurability/Capacity.lean"},{"n":"MeasureTheory.ChoquetCapacity.mono","k":"theorem","m":"Causalean.Mathlib.MeasureTheory.AnalyticSetUniversalMeasurability.Capacity","a":"Mathlib/MeasureTheory/AnalyticSetUniversalMeasurability","t":2,"d":"Capacity values are monotone under inclusion.","s":"∀ {Ω : Type u_1} [inst : TopologicalSpace Ω] (c : MeasureTheory.ChoquetCapacity Ω) {s t : Set Ω}, s ⊆ t → c.toFun s ≤ c.toFun t","l":63,"f":"Causalean/Mathlib/MeasureTheory/AnalyticSetUniversalMeasurability/Capacity.lean"},{"n":"MeasureTheory.Measure.eq_of_rnDeriv_eq","k":"theorem","m":"Causalean.Mathlib.MeasureTheory.EqOfRnDerivEq","a":"Mathlib/MeasureTheory","t":1,"d":"A measure is determined by its Radon–Nikodym derivative against a fixed reference. If the measure μ is absolutely continuous with respect to a σ-finite reference measure ρ, the measure ν is likewise absolutely continuous with respect to ρ, and their Radon–Nikodym densities dμ/dρ and dν/dρ agree ρ-almost everywhere, then μ and ν are equal as measures.","s":"∀ {α : Type u_1} {m : MeasurableSpace α} {μ ν ρ : MeasureTheory.Measure α} [MeasureTheory.SigmaFinite ρ], μ.AbsolutelyContinuous ρ → ν.AbsolutelyContinuous ρ → μ.rnDeriv ρ =ᵐ[ρ] ν.rnDeriv ρ → μ = ν","l":26,"f":"Causalean/Mathlib/MeasureTheory/EqOfRnDerivEq.lean"},{"n":"MeasureTheory.Measure.toChoquetCapacity","k":"def","m":"Causalean.Mathlib.MeasureTheory.AnalyticSetUniversalMeasurability.Capacity","a":"Mathlib/MeasureTheory/AnalyticSetUniversalMeasurability","t":1,"d":"The outer-measure set function associated with a finite Borel measure on a Polish space is a Choquet capacity.","s":"{Ω : Type u_1} → [inst : TopologicalSpace Ω] → [PolishSpace Ω] → [inst_2 : MeasurableSpace Ω] → [BorelSpace Ω] → (μ : MeasureTheory.Measure Ω) → [MeasureTheory.IsFiniteMeasure μ] → MeasureTheory.ChoquetCapacity Ω","l":298,"f":"Causalean/Mathlib/MeasureTheory/AnalyticSetUniversalMeasurability/Capacity.lean"},{"n":"MeasureTheory.Measure.toChoquetCapacity_apply","k":"theorem","m":"Causalean.Mathlib.MeasureTheory.AnalyticSetUniversalMeasurability.Capacity","a":"Mathlib/MeasureTheory/AnalyticSetUniversalMeasurability","t":2,"d":"The capacity induced by a finite measure assigns every event exactly that measure's outer mass.","s":"∀ {Ω : Type u_1} [inst : TopologicalSpace Ω] [inst_1 : PolishSpace Ω] [inst_2 : MeasurableSpace Ω] [inst_3 : BorelSpace Ω] (μ : MeasureTheory.Measure Ω) [inst_4 : MeasureTheory.IsFiniteMeasure μ] (s : Set Ω), μ.toChoquetCapacity.toFun s = μ s","l":308,"f":"Causalean/Mathlib/MeasureTheory/AnalyticSetUniversalMeasurability/Capacity.lean"},{"n":"MeasureTheory.ProbabilityMeasure.map_mul_eq_map_prod_dirac","k":"theorem","m":"Causalean.Mathlib.ConvergenceInDistribution","a":"Mathlib","t":2,"d":"Multiplying a deterministic real value and a random draw has the same distribution as scaling that random draw by the deterministic value.","s":"∀ (c : ℝ) (ν : MeasureTheory.ProbabilityMeasure ℝ), ((MeasureTheory.diracProba c).prod ν).map (Measurable.aemeasurable (Measurable.fun_mul (Measurable.fst measurable_id') (Measurable.snd measurable_id'))) = ν.map (Measurable.aemeasurable (Measurable.mul measurable_const measurable_id))","l":84,"f":"Causalean/Mathlib/ConvergenceInDistribution.lean"},{"n":"MeasureTheory.ProbabilityMeasure.tendsto_map_mul_of_tendsto","k":"theorem","m":"Causalean.Mathlib.ConvergenceInDistribution","a":"Mathlib","t":1,"d":"Measure-level deterministic-scalar Slutsky theorem for weak convergence. For a filter l along which a family of probability measures on the reals converges weakly to a limit measure ν and a family of real scalars converges to a limit a₀, then the pushforwards of the measures by scalar multiplication x ↦ aᵢ·x converge weakly to the pushforward of ν by x ↦ a₀·x.","s":"∀ {ι : Type u_3} {l : Filter ι} {νs : ι → MeasureTheory.ProbabilityMeasure ℝ} {ν : MeasureTheory.ProbabilityMeasure ℝ} {a : ι → ℝ} {a₀ : ℝ}, Filter.Tendsto νs l (nhds ν) → Filter.Tendsto a l (nhds a₀) → Filter.Tendsto (fun i => (νs i).map (Measurable.aemeasurable (Measurable.mul measurable_const measurable_id))) l (nhds (ν.map (Measurable.aemeasurable (Measurable.mul measurable_const measurable_id))))","l":102,"f":"Causalean/Mathlib/ConvergenceInDistribution.lean"},{"n":"MeasureTheory.TendstoInDistribution.const_mul_of_tendsto_const","k":"theorem","m":"Causalean.Mathlib.ConvergenceInDistribution","a":"Mathlib","t":1,"d":"Deterministic-scalar Slutsky theorem for random variables. If a sequence of random variables X n converges in distribution to Z, all under the same probability measure μ and a sequence of deterministic real scalars a n converges to a limit a₀, then the scaled sequence a n · X n converges in distribution to a₀ · Z.","s":"∀ {Ω : Type u_1} {ι : Type u_2} [inst : MeasurableSpace Ω] {μ : MeasureTheory.Measure Ω} {l : Filter ι} [inst_1 : MeasureTheory.IsProbabilityMeasure μ] [l.IsCountablyGenerated] {X : ι → Ω → ℝ} {Z : Ω → ℝ} {a : ι → ℝ} {a₀ : ℝ}, MeasureTheory.TendstoInDistribution X l Z (fun x => μ) μ → Filter.Tendsto a l (nhds a₀) → MeasureTheory.TendstoInDistribution (fun n ω => a n * X n ω) l (fun ω => a₀ * Z ω) (fun x => μ) μ","l":62,"f":"Causalean/Mathlib/ConvergenceInDistribution.lean"},{"n":"MeasureTheory.UpperSemianalytic","k":"def","m":"Causalean.Mathlib.MeasureTheory.AnalyticSetUniversalMeasurability.UpperSemianalytic","a":"Mathlib/MeasureTheory/AnalyticSetUniversalMeasurability","t":1,"d":"An extended-nonnegative function is upper-semi-analytic when every strict superlevel set is analytic.","s":"{Ω : Type u_1} → [TopologicalSpace Ω] → (Ω → ENNReal) → Prop","l":125,"f":"Causalean/Mathlib/MeasureTheory/AnalyticSetUniversalMeasurability/UpperSemianalytic.lean"},{"n":"MeasureTheory.UpperSemianalytic.lintegral_completion_eq_outerLIntegral","k":"theorem","m":"Causalean.Mathlib.MeasureTheory.AnalyticSetUniversalMeasurability.UpperSemianalytic","a":"Mathlib/MeasureTheory/AnalyticSetUniversalMeasurability","t":1,"d":"On a Polish sample space equipped with its Borel σ-algebra and a finite measure μ, if f is upper-semi-analytic — every strict superlevel set {ω | a < f ω} is analytic, then the lower Lebesgue integral of f against the completion of μ equals the outer integral of f with respect to μ, i.e. the infimum of the lower integrals of all measurable pointwise majorants of f.","s":"∀ {Ω : Type u_1} [inst : TopologicalSpace Ω] [inst_1 : MeasurableSpace Ω] [PolishSpace Ω] [BorelSpace Ω] {f : Ω → ENNReal}, MeasureTheory.UpperSemianalytic f → ∀ (μ : MeasureTheory.Measure Ω) [MeasureTheory.IsFiniteMeasure μ], ∫⁻ (ω : MeasureTheory.NullMeasurableSpace Ω μ), f ω ∂μ.completion = MeasureTheory.outerLIntegral μ f","l":149,"f":"Causalean/Mathlib/MeasureTheory/AnalyticSetUniversalMeasurability/UpperSemianalytic.lean"},{"n":"MeasureTheory.UpperSemianalytic.measurable_completion","k":"theorem","m":"Causalean.Mathlib.MeasureTheory.AnalyticSetUniversalMeasurability.UpperSemianalytic","a":"Mathlib/MeasureTheory/AnalyticSetUniversalMeasurability","t":2,"d":"An upper-semi-analytic extended-nonnegative loss is measurable once a finite Borel sampling measure is completed, so it is available to ordinary completed-measure integration.","s":"∀ {Ω : Type u_1} [inst : TopologicalSpace Ω] [inst_1 : MeasurableSpace Ω] [PolishSpace Ω] [BorelSpace Ω] {f : Ω → ENNReal}, MeasureTheory.UpperSemianalytic f → ∀ (μ : MeasureTheory.Measure Ω) [MeasureTheory.IsFiniteMeasure μ], Measurable f","l":142,"f":"Causalean/Mathlib/MeasureTheory/AnalyticSetUniversalMeasurability/UpperSemianalytic.lean"},{"n":"MeasureTheory.UpperSemianalytic.nullMeasurable","k":"theorem","m":"Causalean.Mathlib.MeasureTheory.AnalyticSetUniversalMeasurability.UpperSemianalytic","a":"Mathlib/MeasureTheory/AnalyticSetUniversalMeasurability","t":2,"d":"An upper-semi-analytic extended-nonnegative function is null-measurable for every finite Borel measure.","s":"∀ {Ω : Type u_1} [inst : TopologicalSpace Ω] [inst_1 : MeasurableSpace Ω] [PolishSpace Ω] [BorelSpace Ω] {f : Ω → ENNReal}, MeasureTheory.UpperSemianalytic f → ∀ (μ : MeasureTheory.Measure Ω) [MeasureTheory.IsFiniteMeasure μ], MeasureTheory.NullMeasurable f μ","l":132,"f":"Causalean/Mathlib/MeasureTheory/AnalyticSetUniversalMeasurability/UpperSemianalytic.lean"},{"n":"MeasureTheory.compProd_eq_prod_withDensity_fiber","k":"theorem","m":"Causalean.Mathlib.MeasureTheory.RnDerivCompProdSigmaFinite","a":"Mathlib/MeasureTheory","t":2,"d":"A composition product is the product reference measure weighted by the fibrewise Radon–Nikodym derivative whenever the fibres are almost everywhere dominated by the reference measure and that derivative is almost everywhere measurable.","s":"∀ {α : Type u_1} {β : Type u_2} [inst : MeasurableSpace α] [inst_1 : MeasurableSpace β] (μ : MeasureTheory.Measure α) (ρ : MeasureTheory.Measure β) (κ : ProbabilityTheory.Kernel α β) [MeasureTheory.SFinite μ] [MeasureTheory.SigmaFinite ρ] [ProbabilityTheory.IsSFiniteKernel κ], (∀ᵐ (a : α) ∂μ, (κ a).AbsolutelyContinuous ρ) → AEMeasurable (fun p => (κ p.1).rnDeriv ρ p.2) (μ.prod ρ) → μ.compProd κ = (μ.prod ρ).withDensity fun p => (κ p.1).rnDeriv ρ p.2","l":47,"f":"Causalean/Mathlib/MeasureTheory/RnDerivCompProdSigmaFinite.lean"},{"n":"MeasureTheory.condExp_setIndicator_condExp_of_le","k":"theorem","m":"Causalean.Mathlib.MeasureTheory.CondExpIndicatorTower","a":"Mathlib/MeasureTheory","t":1,"d":"Conditioning a masked outcome equals conditioning its inner regression. Given a tower of σ-algebras m ≤ m' ≤ m0 on the sample space, with the trim of the measure μ to m' σ-finite, a set s measurable with respect to the finer σ-algebra m', and an integrable function f, the conditional expectation given m of the masked outcome 1_s · f agrees μ-almost everywhere with the conditional expectation given m of f — first replaced by its conditional expectation given m', then masked by s.","s":"∀ {Ω : Type u_1} {m m' m0 : MeasurableSpace Ω} {μ : MeasureTheory.Measure Ω} {E : Type u_2} [inst : NormedAddCommGroup E] [inst_1 : NormedSpace ℝ E] [CompleteSpace E], m ≤ m' → ∀ (hm' : m' ≤ m0) [MeasureTheory.SigmaFinite (μ.trim hm')] {s : Set Ω}, MeasurableSet s → ∀ {f : Ω → E}, MeasureTheory.Integrable f μ → μ[s.indicator f | m] =ᵐ[μ] μ[s.indicator μ[f | m'] | m]","l":39,"f":"Causalean/Mathlib/MeasureTheory/CondExpIndicatorTower.lean"},{"n":"MeasureTheory.instCoeFunChoquetCapacityForallSetENNReal","k":"instance","m":"Causalean.Mathlib.MeasureTheory.AnalyticSetUniversalMeasurability.Capacity","a":"Mathlib/MeasureTheory/AnalyticSetUniversalMeasurability","t":2,"d":"A Choquet capacity coerces to its underlying extended-nonnegative set function.","s":"{Ω : Type u_1} → [inst : TopologicalSpace Ω] → CoeFun (MeasureTheory.ChoquetCapacity Ω) fun x => Set Ω → ENNReal","l":57,"f":"Causalean/Mathlib/MeasureTheory/AnalyticSetUniversalMeasurability/Capacity.lean"},{"n":"MeasureTheory.lintegral_completion_eq_of_measurable","k":"theorem","m":"Causalean.Mathlib.MeasureTheory.AnalyticSetUniversalMeasurability.UpperSemianalytic","a":"Mathlib/MeasureTheory/AnalyticSetUniversalMeasurability","t":2,"d":"A measurable extended-nonnegative function has the same integral before and after completion of the measure.","s":"∀ {Ω : Type u_1} [TopologicalSpace Ω] [inst : MeasurableSpace Ω] (μ : MeasureTheory.Measure Ω) {f : Ω → ENNReal}, Measurable f → ∫⁻ (ω : MeasureTheory.NullMeasurableSpace Ω μ), f ω ∂μ.completion = ∫⁻ (ω : Ω), f ω ∂μ","l":98,"f":"Causalean/Mathlib/MeasureTheory/AnalyticSetUniversalMeasurability/UpperSemianalytic.lean"},{"n":"MeasureTheory.lintegral_le_lintegral_completion","k":"theorem","m":"Causalean.Mathlib.MeasureTheory.AnalyticSetUniversalMeasurability.UpperSemianalytic","a":"Mathlib/MeasureTheory/AnalyticSetUniversalMeasurability","t":2,"d":"Completing a measure can only increase the lower integral of an arbitrary extended-nonnegative function.","s":"∀ {Ω : Type u_1} [TopologicalSpace Ω] [inst : MeasurableSpace Ω] (μ : MeasureTheory.Measure Ω) (f : Ω → ENNReal), ∫⁻ (ω : Ω), f ω ∂μ ≤ ∫⁻ (ω : MeasureTheory.NullMeasurableSpace Ω μ), f ω ∂μ.completion","l":66,"f":"Causalean/Mathlib/MeasureTheory/AnalyticSetUniversalMeasurability/UpperSemianalytic.lean"},{"n":"MeasureTheory.lintegral_le_outerLIntegral","k":"theorem","m":"Causalean.Mathlib.MeasureTheory.AnalyticSetUniversalMeasurability.UpperSemianalytic","a":"Mathlib/MeasureTheory/AnalyticSetUniversalMeasurability","t":2,"d":"The lower Lebesgue integral is bounded by the outer integral.","s":"∀ {Ω : Type u_1} [TopologicalSpace Ω] [inst : MeasurableSpace Ω] (μ : MeasureTheory.Measure Ω) (f : Ω → ENNReal), ∫⁻ (ω : Ω), f ω ∂μ ≤ MeasureTheory.outerLIntegral μ f","l":43,"f":"Causalean/Mathlib/MeasureTheory/AnalyticSetUniversalMeasurability/UpperSemianalytic.lean"},{"n":"MeasureTheory.lintegral_le_outerLIntegral_of_measurable_le","k":"theorem","m":"Causalean.Mathlib.MeasureTheory.AnalyticSetUniversalMeasurability.UpperSemianalytic","a":"Mathlib/MeasureTheory/AnalyticSetUniversalMeasurability","t":2,"d":"A pointwise lower bound may be integrated before comparison with an outer integral, without measurability of either function.","s":"∀ {Ω : Type u_1} [TopologicalSpace Ω] [inst : MeasurableSpace Ω] {μ : MeasureTheory.Measure Ω} {f g : Ω → ENNReal}, f ≤ g → ∫⁻ (ω : Ω), f ω ∂μ ≤ MeasureTheory.outerLIntegral μ g","l":50,"f":"Causalean/Mathlib/MeasureTheory/AnalyticSetUniversalMeasurability/UpperSemianalytic.lean"},{"n":"MeasureTheory.outerLIntegral","k":"def","m":"Causalean.Mathlib.MeasureTheory.AnalyticSetUniversalMeasurability.UpperSemianalytic","a":"Mathlib/MeasureTheory/AnalyticSetUniversalMeasurability","t":1,"d":"The outer integral of an extended-nonnegative function is the infimum of the lower integrals of all measurable pointwise majorants.","s":"{Ω : Type u_1} → [inst : MeasurableSpace Ω] → MeasureTheory.Measure Ω → (Ω → ENNReal) → ENNReal","l":31,"f":"Causalean/Mathlib/MeasureTheory/AnalyticSetUniversalMeasurability/UpperSemianalytic.lean"},{"n":"MeasureTheory.outerLIntegral_eq_lintegral_of_measurable","k":"theorem","m":"Causalean.Mathlib.MeasureTheory.AnalyticSetUniversalMeasurability.UpperSemianalytic","a":"Mathlib/MeasureTheory/AnalyticSetUniversalMeasurability","t":2,"d":"Outer integration agrees with Lebesgue integration for measurable functions.","s":"∀ {Ω : Type u_1} [TopologicalSpace Ω] [inst : MeasurableSpace Ω] {μ : MeasureTheory.Measure Ω} {f : Ω → ENNReal}, Measurable f → MeasureTheory.outerLIntegral μ f = ∫⁻ (ω : Ω), f ω ∂μ","l":57,"f":"Causalean/Mathlib/MeasureTheory/AnalyticSetUniversalMeasurability/UpperSemianalytic.lean"},{"n":"MeasureTheory.outerLIntegral_mono","k":"theorem","m":"Causalean.Mathlib.MeasureTheory.AnalyticSetUniversalMeasurability.UpperSemianalytic","a":"Mathlib/MeasureTheory/AnalyticSetUniversalMeasurability","t":2,"d":"Outer integration is monotone in its extended-nonnegative integrand.","s":"∀ {Ω : Type u_1} [TopologicalSpace Ω] [inst : MeasurableSpace Ω] {μ : MeasureTheory.Measure Ω} {f g : Ω → ENNReal}, f ≤ g → MeasureTheory.outerLIntegral μ f ≤ MeasureTheory.outerLIntegral μ g","l":36,"f":"Causalean/Mathlib/MeasureTheory/AnalyticSetUniversalMeasurability/UpperSemianalytic.lean"},{"n":"MeasureTheory.restrict_range_eq_of_forall_image","k":"theorem","m":"Causalean.Mathlib.MeasurableEmbeddingExtras","a":"Mathlib","t":1,"d":"If F is a measurable embedding of one measurable space into another and two measures μ and ν on the codomain assign the same mass to the image F '' A of every measurable set A in the domain, then restricting μ and ν to the range of F yields identical measures.","s":"∀ {α : Type u_1} {β : Type u_2} {x : MeasurableSpace α} {x_1 : MeasurableSpace β} {F : α → β}, MeasurableEmbedding F → ∀ (μ ν : MeasureTheory.Measure β), (∀ (A : Set α), MeasurableSet A → μ (F '' A) = ν (F '' A)) → μ.restrict (Set.range F) = ν.restrict (Set.range F)","l":24,"f":"Causalean/Mathlib/MeasurableEmbeddingExtras.lean"},{"n":"MeasureTheory.rnDeriv_compProd_prod_sigmaFinite","k":"theorem","m":"Causalean.Mathlib.MeasureTheory.RnDerivCompProdSigmaFinite","a":"Mathlib/MeasureTheory","t":1,"d":"σ-finite-reference Radon–Nikodym derivative of a composition-product. Fix an s-finite measure μ, a σ-finite base reference ν, a σ-finite fibre reference ρ, and an s-finite kernel κ from the base space to the fibre space, together with a candidate base density f. If μ is absolutely continuous with respect to ν, μ-almost every fibre κ a is absolutely continuous with respect to ρ, the raw per-slice fibre derivative (a, b) ↦ (dκ_a/dρ)(b) is almost-everywhere measurable with respect to the product reference ν × ρ, and f agrees ν-almost everywhere with the Radon–Nikodym derivative dμ/dν, then the Radon–Nikodym derivative of the composition-product μ ⊗ₘ κ with respect to the product reference ν × ρ agrees, ν × ρ-almost everywhere, with the pointwise product (a, b) ↦ f a · (dκ_a/dρ)(b) of the base density and the fibre derivative.","s":"∀ {α : Type u_1} {β : Type u_2} [inst : MeasurableSpace α] [inst_1 : MeasurableSpace β] (μ ν : MeasureTheory.Measure α) (ρ : MeasureTheory.Measure β) (κ : ProbabilityTheory.Kernel α β) (f : α → ENNReal) [MeasureTheory.SigmaFinite ν] [MeasureTheory.SigmaFinite ρ] [ProbabilityTheory.IsSFiniteKernel κ], μ.AbsolutelyContinuous ν → (∀ᵐ (a : α) ∂μ, (κ a).AbsolutelyContinuous ρ) → AEMeasurable (fun p => (κ p.1).rnDeriv ρ p.2) (ν.prod ρ) → μ.rnDeriv ν =ᵐ[ν] f → (μ.compProd κ).rnDeriv (ν.prod ρ) =ᵐ[ν.prod ρ] fun p => f p.1 * (κ p.1).rnDeriv ρ p.2","l":70,"f":"Causalean/Mathlib/MeasureTheory/RnDerivCompProdSigmaFinite.lean"},{"n":"MeasureTheory.tendstoInMeasure_const_of_tendsto","k":"theorem","m":"Causalean.Mathlib.ConvergenceInDistribution","a":"Mathlib","t":2,"d":"Alias for tendstoInMeasure_const_of_tendsto_real whose name reflects that the result holds in any seminormed additive commutative group.","s":"∀ {Ω : Type u_1} {ι : Type u_2} [inst : MeasurableSpace Ω] {μ : MeasureTheory.Measure Ω} {l : Filter ι} {E : Type u_3} [inst_1 : SeminormedAddCommGroup E] {a : ι → E} {a₀ : E}, Filter.Tendsto a l (nhds a₀) → MeasureTheory.TendstoInMeasure μ (fun n x => a n) l fun x => a₀","l":54,"f":"Causalean/Mathlib/ConvergenceInDistribution.lean"},{"n":"MeasureTheory.tendstoInMeasure_const_of_tendsto_real","k":"theorem","m":"Causalean.Mathlib.ConvergenceInDistribution","a":"Mathlib","t":2,"d":"A deterministic sequence in a seminormed additive commutative group that converges in the usual topological sense also converges in measure when regarded as a sequence of constant random variables.","s":"∀ {Ω : Type u_1} {ι : Type u_2} [inst : MeasurableSpace Ω] {μ : MeasureTheory.Measure Ω} {l : Filter ι} {E : Type u_3} [inst_1 : SeminormedAddCommGroup E] {a : ι → E} {a₀ : E}, Filter.Tendsto a l (nhds a₀) → MeasureTheory.TendstoInMeasure μ (fun n x => a n) l fun x => a₀","l":33,"f":"Causalean/Mathlib/ConvergenceInDistribution.lean"},{"n":"ProbabilityTheory.IndepFun.integral_restrict_preimage_eq_mul","k":"theorem","m":"Causalean.Mathlib.IndepIntegral","a":"Mathlib","t":1,"d":"Drop of conditioning. For random elements f and g that are independent, with f almost-everywhere measurable and g almost-everywhere measurable, given a measurable set E whose preimage f ⁻¹' E is measurable, and a function h that is almost-everywhere strongly measurable with respect to the pushforward measure of g, then restricting the integral of h ∘ g to the event f ⁻¹' E equals the probability of that event times the unrestricted integral of h ∘ g.","s":"∀ {Ω : Type u_1} {α : Type u_2} {β : Type u_3} [inst : MeasurableSpace Ω] [inst_1 : MeasurableSpace α] [inst_2 : MeasurableSpace β] {μ : MeasureTheory.Measure Ω} {f : Ω → α} {g : Ω → β}, ProbabilityTheory.IndepFun f g μ → AEMeasurable f μ → AEMeasurable g μ → ∀ {E : Set α}, MeasurableSet E → MeasurableSet (f ⁻¹' E) → ∀ {h : β → ℝ}, MeasureTheory.AEStronglyMeasurable h (MeasureTheory.Measure.map g μ) → ∫ (ω : Ω) in f ⁻¹' E, h (g ω) ∂μ = (μ (f ⁻¹' E)).toReal * ∫ (ω : Ω), h (g ω) ∂μ","l":23,"f":"Causalean/Mathlib/IndepIntegral.lean"},{"n":"ProbabilityTheory.condDistrib_map_of_condDistrib_fst_eq","k":"theorem","m":"Causalean.Mathlib.CondDistribWitness","a":"Mathlib","t":1,"d":"Witness-kernel form of a conditional distribution under conditional independence. For measurable maps X, Z, and C and a jointly measurable structural-response function h, if the conditional distribution of C given the pair (X,Z) agrees almost everywhere with the conditional distribution of C given Z alone — conditional independence of C from X given Z, then the conditional distribution of the outcome h(X,Z,C) given (X,Z) agrees almost everywhere with the pushforward, by the map h x z, of the conditional distribution of C given Z.","s":"∀ {Ω : Type u_1} {𝒳 : Type u_2} {𝒵 : Type u_3} {𝒞 : Type u_4} {𝒴 : Type u_5} [inst : MeasurableSpace Ω] [inst_1 : MeasurableSpace 𝒳] [inst_2 : MeasurableSpace 𝒵] [inst_3 : MeasurableSpace 𝒞] [inst_4 : StandardBorelSpace 𝒞] [inst_5 : Nonempty 𝒞] [inst_6 : MeasurableSpace 𝒴] [inst_7 : StandardBorelSpace 𝒴] [inst_8 : Nonempty 𝒴] (μ : MeasureTheory.Measure Ω) [inst_9 : MeasureTheory.IsFiniteMeasure μ] {X : Ω → 𝒳} {Z : Ω → 𝒵} {C : Ω → 𝒞} {h : 𝒳 → 𝒵 → 𝒞 → 𝒴}, Measurable X → Measurable Z → Measurable C → (Measurable fun p => h p.1.1 p.1.2 p.2) → ((fun p => (ProbabilityTheory.condDistrib C (fun ω => (X ω, Z ω)) μ) p) =ᵐ[MeasureTheory.Measure.map (fun ω => (X ω, Z ω)) μ] fun p => (ProbabilityTheory.condDistrib C Z μ) p.2) → (fun p => (ProbabilityTheory.condDistrib (fun ω => h (X ω) (Z ω) (C ω)) (fun ω => (X ω, Z ω)) μ) p) =ᵐ[MeasureTheory.Measure.map (fun ω => (X ω, Z ω)) μ] fun p => MeasureTheory.Measure.map (h p.1 p.2) ((ProbabilityTheory.condDistrib C Z μ) p.2)","l":101,"f":"Causalean/Mathlib/CondDistribWitness.lean"},{"n":"ProbabilityTheory.condDistrib_map_of_funext","k":"theorem","m":"Causalean.Mathlib.CondDistribWitness","a":"Mathlib","t":1,"d":"No-treatment witness-kernel form of a conditional distribution. For measurable maps Z and C and a jointly measurable response function H, the conditional distribution of the outcome H(Z,C) given Z agrees almost everywhere with the pushforward, by the map H z, of the conditional distribution of C given Z.","s":"∀ {Ω : Type u_1} {𝒵 : Type u_3} {𝒞 : Type u_4} {𝒴 : Type u_5} [inst : MeasurableSpace Ω] [inst_1 : MeasurableSpace 𝒵] [inst_2 : MeasurableSpace 𝒞] [inst_3 : StandardBorelSpace 𝒞] [inst_4 : Nonempty 𝒞] [inst_5 : MeasurableSpace 𝒴] [inst_6 : StandardBorelSpace 𝒴] [inst_7 : Nonempty 𝒴] (μ : MeasureTheory.Measure Ω) [inst_8 : MeasureTheory.IsFiniteMeasure μ] {Z : Ω → 𝒵} {C : Ω → 𝒞} {H : 𝒵 → 𝒞 → 𝒴}, Measurable Z → Measurable C → (Measurable fun p => H p.1 p.2) → (fun z => (ProbabilityTheory.condDistrib (fun ω => H (Z ω) (C ω)) Z μ) z) =ᵐ[MeasureTheory.Measure.map Z μ] fun z => MeasureTheory.Measure.map (H z) ((ProbabilityTheory.condDistrib C Z μ) z)","l":178,"f":"Causalean/Mathlib/CondDistribWitness.lean"},{"n":"ProbabilityTheory.instIsMarkovKernel_witnessKernel","k":"instance","m":"Causalean.Mathlib.CondDistribWitness","a":"Mathlib","t":2,"d":"The witness kernel is a Markov kernel when the base measure is finite.","s":"∀ {Ω : Type u_1} {𝒳 : Type u_2} {𝒵 : Type u_3} {𝒞 : Type u_4} {𝒴 : Type u_5} [inst : MeasurableSpace Ω] [inst_1 : MeasurableSpace 𝒳] [inst_2 : MeasurableSpace 𝒵] [inst_3 : MeasurableSpace 𝒞] [inst_4 : StandardBorelSpace 𝒞] [inst_5 : Nonempty 𝒞] [inst_6 : MeasurableSpace 𝒴] (μ : MeasureTheory.Measure Ω) [inst_7 : MeasureTheory.IsFiniteMeasure μ] {Z : Ω → 𝒵} {C : Ω → 𝒞} {h : 𝒳 → 𝒵 → 𝒞 → 𝒴} (hh : Measurable fun p => h p.1.1 p.1.2 p.2), ProbabilityTheory.IsMarkovKernel (ProbabilityTheory.witnessKernel μ hh)","l":89,"f":"Causalean/Mathlib/CondDistribWitness.lean"},{"n":"ProbabilityTheory.witnessKernel","k":"def","m":"Causalean.Mathlib.CondDistribWitness","a":"Mathlib","t":1,"d":"For each treatment-covariate pair, this kernel gives the conditional law of the outcome obtained by drawing the residual coordinate from its conditional law given the covariates and then applying the structural response function.","s":"{Ω : Type u_1} → {𝒳 : Type u_2} → {𝒵 : Type u_3} → {𝒞 : Type u_4} → {𝒴 : Type u_5} → [inst : MeasurableSpace Ω] → [inst_1 : MeasurableSpace 𝒳] → [inst_2 : MeasurableSpace 𝒵] → [inst_3 : MeasurableSpace 𝒞] → [StandardBorelSpace 𝒞] → [Nonempty 𝒞] → [inst_6 : MeasurableSpace 𝒴] → (μ : MeasureTheory.Measure Ω) → [MeasureTheory.IsFiniteMeasure μ] → {Z : Ω → 𝒵} → {C : Ω → 𝒞} → {h : 𝒳 → 𝒵 → 𝒞 → 𝒴} → (Measurable fun p => h p.1.1 p.1.2 p.2) → ProbabilityTheory.Kernel (𝒳 × 𝒵) 𝒴","l":45,"f":"Causalean/Mathlib/CondDistribWitness.lean"},{"n":"ProbabilityTheory.witnessKernel_apply","k":"theorem","m":"Causalean.Mathlib.CondDistribWitness","a":"Mathlib","t":2,"d":"Evaluating the witness kernel at a conditioning pair gives the residual conditional law pushed through the corresponding structural slice.","s":"∀ {Ω : Type u_1} {𝒳 : Type u_2} {𝒵 : Type u_3} {𝒞 : Type u_4} {𝒴 : Type u_5} [inst : MeasurableSpace Ω] [inst_1 : MeasurableSpace 𝒳] [inst_2 : MeasurableSpace 𝒵] [inst_3 : MeasurableSpace 𝒞] [inst_4 : StandardBorelSpace 𝒞] [inst_5 : Nonempty 𝒞] [inst_6 : MeasurableSpace 𝒴] (μ : MeasureTheory.Measure Ω) [inst_7 : MeasureTheory.IsFiniteMeasure μ] (Z : Ω → 𝒵) (C : Ω → 𝒞) {h : 𝒳 → 𝒵 → 𝒞 → 𝒴} (hh : Measurable fun p => h p.1.1 p.1.2 p.2) (p : 𝒳 × 𝒵), (ProbabilityTheory.witnessKernel μ hh) p = MeasureTheory.Measure.map (h p.1 p.2) ((ProbabilityTheory.condDistrib C Z μ) p.2)","l":81,"f":"Causalean/Mathlib/CondDistribWitness.lean"}]}