PO.ID.Exact.Heckman­Roy

lemma measurableSet_intervalComplierEvent (z₀ z₁ : α) : MeasurableSet (S.intervalComplierEvent z₀ z₁) := by unfold intervalComplierEvent exact (measurableSet_lt measurable_const S.measurable_factualU).inter (measurableSet_le S.measurable_factualU measurable_const)

lemma interval_indicator_sub (z₀ z₁ : α) (hpz : S.p z₀ ≤ S.p z₁) (ω : P.Ω) : (if S.factualU ω ≤ S.p z₁ then (1:ℝ) else 0) - (if S.factualU ω ≤ S.p z₀ then (1:ℝ) else 0) = (S.intervalComplierEvent z₀ z₁).indicator (fun _ => (1:ℝ)) ω := by -- Pure pointwise indicator algebra: case-split on `U ω ≤ p z₀` and -- `U ω ≤ p z₁`, then unfold `intervalComplierEvent`. unfold intervalComplierEvent by_cases h₁ : S.factualU ω ≤ S.p z₁ · by_cases h₀ : S.factualU ω ≤ S.p z₀ · have hnC : ω ∉ {ω | S.p z₀ < S.factualU ω ∧ S.factualU ω ≤ S.p z₁} := by intro hω exact (not_lt_of_ge h₀) hω.1 rw [Set.indicator_of_notMem hnC] simp [h₀, h₁] · have h₀' : S.p z₀ < S.factualU ω := lt_of_not_ge h₀ have hC : ω ∈ {ω | S.p z₀ < S.factualU ω ∧ S.factualU ω ≤ S.p z₁} := ⟨h₀', h₁⟩ simp [h₀, h₁, Set.indicator_of_mem hC] · have h₁' : S.p z₁ < S.factualU ω := lt_of_not_ge h₁ have h₀ : ¬ S.factualU ω ≤ S.p z₀ := fun h => (not_le_of_gt h₁') (le_trans h hpz) have hnC : ω ∉ {ω | S.p z₀ < S.factualU ω ∧ S.factualU ω ≤ S.p z₁} := by intro hω exact (not_le_of_gt h₁') hω.2 rw [Set.indicator_of_notMem hnC] simp [h₀, h₁]

lemma complier_measure (hA : S.Assumptions) (z₀ z₁ : α) (hpz : S.p z₀ < S.p z₁) : (P.μ (S.intervalComplierEvent z₀ z₁)).toReal = S.p z₁ - S.p z₀ := by -- Evaluate the two threshold sets with `uniformU`, subtract the lower set -- from the upper set, and convert the finite ENNReal difference to a real -- difference. `S.hp_mem z₀`, `S.hp_mem z₁`, and `hpz` discharge the -- `q ∈ [0,1]` side conditions. have hC_eq : S.intervalComplierEvent z₀ z₁ = {ω | S.factualU ω ≤ S.p z₁} \ {ω | S.factualU ω ≤ S.p z₀} := by ext ω constructor · rintro ⟨h₀, h₁⟩ refine ⟨h₁, ?_⟩ intro h exact (lt_irrefl _ (lt_of_lt_of_le h₀ h)) · rintro ⟨h₁, h₀⟩ refine ⟨?_, h₁⟩ exact lt_of_not_ge h₀ have hp0 : S.p z₀ ∈ Icc (0 : ℝ) 1 := S.hp_mem z₀ have hp1 : S.p z₁ ∈ Icc (0 : ℝ) 1 := S.hp_mem z₁ have h_sub : {ω | S.factualU ω ≤ S.p z₀} ⊆ {ω | S.factualU ω ≤ S.p z₁} := fun _ h => le_trans h (le_of_lt hpz) have hμ0 : P.μ {ω | S.factualU ω ≤ S.p z₀} = ofReal (S.p z₀) := hA.uniformU _ hp0 have hμ1 : P.μ {ω | S.factualU ω ≤ S.p z₁} = ofReal (S.p z₁) := hA.uniformU _ hp1 have hμ0_finite : P.μ {ω | S.factualU ω ≤ S.p z₀} ≠ ⊤ := by rw [hμ0] exact ENNReal.ofReal_ne_top rw [hC_eq, MeasureTheory.measure_diff h_sub (measurableSet_le S.measurable_factualU measurable_const).nullMeasurableSet hμ0_finite, hμ1, hμ0] rw [ENNReal.toReal_sub_of_le] · simp [ENNReal.toReal_ofReal hp1.1, ENNReal.toReal_ofReal hp0.1] · exact ENNReal.ofReal_le_ofReal (le_of_lt hpz) · exact ENNReal.ofReal_ne_top

theorem first_stage_identity (hA : S.Assumptions) (z₀ z₁ : α) (hZ0 : 0 < (P.μ (S.zEvent z₀)).toReal) (hZ1 : 0 < (P.μ (S.zEvent z₁)).toReal) (hpz : S.p z₀ < S.p z₁) : S.condExpDZ z₁ - S.condExpDZ z₀ = S.p z₁ - S.p z₀ := by classical have hμne_zero : ∀ z, 0 < (P.μ (S.zEvent z)).toReal → P.μ (S.zVar.event z) ≠ 0 := fun z hZ h => absurd hZ (by simp [show S.zEvent z = S.zVar.event z from rfl, h]) have hμne_top : ∀ z, P.μ (S.zVar.event z) ≠ ⊤ := fun _ => measure_ne_top _ _ have hDofZ_to_indicator : ∀ z, ∀ᵐ ω ∂P.μ, ((S.DofZ z ω).toNat : ℝ) = (if S.factualU ω ≤ S.p z then (1:ℝ) else 0) := by intro z filter_upwards [hA.thresholdCrossing z] with ω hω by_cases hu : S.factualU ω ≤ S.p z · have hD : S.DofZ z ω = true := hω.mpr hu simp [hD, hu] · have hD : S.DofZ z ω = false := by cases hd : S.DofZ z ω · rfl · exact absurd (hω.mp hd) hu simp [hD, hu] have hCE : ∀ z (_hZ : 0 < (P.μ (S.zEvent z)).toReal), S.condExpDZ z = ∫ ω, (if S.factualU ω ≤ S.p z then (1:ℝ) else 0) ∂P.μ := by intro z hZ let getU : (∀ i : Fin S.cfBundle.n, S.cfBundle.type i) → ℝ := fun f => by exact f (0 : Fin 3) have hgetU_meas : Measurable getU := by -- Instance search no longer unfolds `cfBundle` to see `n = 3`, so supply -- the coordinate measurable-space family at index type `Fin 3` directly. let _ : ∀ i : Fin 3, MeasurableSpace (S.cfBundle.type i) := fun i => S.cfBundle.inst i change Measurable fun f : ∀ i : Fin S.cfBundle.n, S.cfBundle.type i => f (0 : Fin 3) exact measurable_pi_apply (0 : Fin 3) have hgetU_joint : ∀ ω, getU (S.cfBundle.jointValue ω) = S.factualU ω := by intro ω rfl let h_proj : (∀ i : Fin S.cfBundle.n, S.cfBundle.type i) → ℝ := fun f => if getU f ≤ S.p z then (1:ℝ) else 0 have hh_meas : Measurable h_proj := by refine Measurable.ite ?_ measurable_const measurable_const exact measurableSet_le hgetU_meas measurable_const have h_cons : ∀ ω ∈ S.zVar.event z, (if S.factualU ω ≤ S.p z then (1:ℝ) else 0) = h_proj (S.cfBundle.jointValue ω) := by intro ω _ dsimp [h_proj] rw [hgetU_joint] have hcond_d : S.condExpDZ z = eventCondExp P.μ (S.zVar.event z) (fun ω => ((S.DofZ z ω).toNat : ℝ)) := by unfold POHeckmanRoySystem.condExpDZ eventCondExp congr 1 refine MeasureTheory.setIntegral_congr_fun (S.measurableSet_zEvent z) ?_ intro ω hω exact congrArg (fun d : Bool => ((d.toNat : ℝ))) (S.DofZ_eq_factualD_on_zEvent hA z hω).symm have hcond_ind : eventCondExp P.μ (S.zVar.event z) (fun ω => ((S.DofZ z ω).toNat : ℝ)) = eventCondExp P.μ (S.zVar.event z) (fun ω => if S.factualU ω ≤ S.p z then (1:ℝ) else 0) := by unfold eventCondExp congr 1 refine MeasureTheory.setIntegral_congr_ae (S.measurableSet_zEvent z) ?_ filter_upwards [hDofZ_to_indicator z] with ω hω _ exact hω rw [hcond_d, hcond_ind, POSystem.eventCondExp_of_consistency_IndepCF hA.instrumentIndep (a := S.zVar) hh_meas (MeasurableSet.singleton z) (ae_restrict_of_forall_mem (S.measurableSet_zEvent z) h_cons) (hμne_zero z hZ) (hμne_top z)] refine MeasureTheory.integral_congr_ae (Filter.Eventually.of_forall ?_) intro ω dsimp [h_proj] rw [hgetU_joint] rw [hCE z₁ hZ1, hCE z₀ hZ0] have hbdd : ∀ z, ∀ ω, |if S.factualU ω ≤ S.p z then (1:ℝ) else 0| ≤ 1 := by intro z ω by_cases h : S.factualU ω ≤ S.p z <;> simp [h] have hint : ∀ z, Integrable (fun ω => if S.factualU ω ≤ S.p z then (1:ℝ) else 0) P.μ := by intro z refine (MeasureTheory.integrable_const (1:ℝ)).mono' ?_ (Filter.Eventually.of_forall (hbdd z)) exact (Measurable.ite (measurableSet_le S.measurable_factualU measurable_const) measurable_const measurable_const).aestronglyMeasurable rw [← MeasureTheory.integral_sub (hint z₁) (hint z₀)] rw [MeasureTheory.integral_congr_ae (Filter.Eventually.of_forall (S.interval_indicator_sub z₀ z₁ (le_of_lt hpz)))] rw [MeasureTheory.integral_indicator_const (1:ℝ) (S.measurableSet_intervalComplierEvent z₀ z₁)] simpa [MeasureTheory.measureReal_def] using S.complier_measure hA z₀ z₁ hpz

theorem reduced_form_identity (hA : S.Assumptions) (z₀ z₁ : α) (hZ0 : 0 < (P.μ (S.zEvent z₀)).toReal) (hZ1 : 0 < (P.μ (S.zEvent z₁)).toReal) (hY1 : Integrable (S.YofD true) P.μ) (hY0 : Integrable (S.YofD false) P.μ) : S.condExpYZ z₁ - S.condExpYZ z₀ = ∫ ω, (S.YofDofZ z₁ ω - S.YofDofZ z₀ ω) ∂P.μ := by have hYDZ_meas : ∀ z, Measurable (S.YofDofZ z) := fun z => S.measurable_YofDofZ z have hYDZ_bdd : ∀ z, ∀ ω, |S.YofDofZ z ω| ≤ |S.YofD true ω| + |S.YofD false ω| := fun z ω => by have h1 := abs_nonneg (S.YofD true ω) have h0 := abs_nonneg (S.YofD false ω) unfold YofDofZ cases S.DofZ z ω <;> simp [h1, h0] have hYDZ_int : ∀ z, Integrable (S.YofDofZ z) P.μ := fun z => (hY1.norm.add hY0.norm).mono' (hYDZ_meas z).aestronglyMeasurable (Filter.Eventually.of_forall (hYDZ_bdd z)) have hμne_zero : ∀ z, 0 < (P.μ (S.zEvent z)).toReal → P.μ (S.zVar.event z) ≠ 0 := fun z hZ h => absurd hZ (by simp [show S.zEvent z = S.zVar.event z from rfl, h]) have hμne_top : ∀ z, P.μ (S.zVar.event z) ≠ ⊤ := fun _ => measure_ne_top _ _ have h_factualY_ae : ∀ z, ∀ᵐ ω ∂P.μ, (ω ∈ S.zEvent z) → S.factualY ω = (if S.factualU ω ≤ S.p z then S.YofD true ω else S.YofD false ω) := by intro z filter_upwards [hA.thresholdCrossing z] with ω hth hω rw [S.factualY_eq_YofD_factualD hA ω, ← S.DofZ_eq_factualD_on_zEvent hA z hω] by_cases hu : S.factualU ω ≤ S.p z · have hD : S.DofZ z ω = true := hth.mpr hu simp [hD, hu] · have hD : S.DofZ z ω = false := by cases hd : S.DofZ z ω · rfl · exact absurd (hth.mp hd) hu simp [hD, hu] have hCE : ∀ z (_hZ : 0 < (P.μ (S.zEvent z)).toReal), S.condExpYZ z = ∫ ω, (if S.factualU ω ≤ S.p z then S.YofD true ω else S.YofD false ω) ∂P.μ := by intro z hZ let getU : (∀ i : Fin S.cfBundle.n, S.cfBundle.type i) → ℝ := fun f => by exact f (0 : Fin 3) have hgetU_meas : Measurable getU := by -- Instance search no longer unfolds `cfBundle` to see `n = 3`, so supply -- the coordinate measurable-space family at index type `Fin 3` directly. let _ : ∀ i : Fin 3, MeasurableSpace (S.cfBundle.type i) := fun i => S.cfBundle.inst i change Measurable fun f : ∀ i : Fin S.cfBundle.n, S.cfBundle.type i => f (0 : Fin 3) exact measurable_pi_apply (0 : Fin 3) have hgetU_joint : ∀ ω, getU (S.cfBundle.jointValue ω) = S.factualU ω := by intro ω rfl let h_proj : (∀ i : Fin S.cfBundle.n, S.cfBundle.type i) → ℝ := fun f => if getU f ≤ S.p z then ((f (1 : Fin 3)) : ℝ) else ((f (2 : Fin 3)) : ℝ) have hh_meas : Measurable h_proj := by let _ : ∀ i : Fin 3, MeasurableSpace (S.cfBundle.type i) := fun i => S.cfBundle.inst i change Measurable fun f : ∀ i : Fin S.cfBundle.n, S.cfBundle.type i => if getU f ≤ S.p z then ((f (1 : Fin 3)) : ℝ) else ((f (2 : Fin 3)) : ℝ) refine Measurable.ite ?_ ?_ ?_ · exact measurableSet_le hgetU_meas measurable_const · exact measurable_pi_apply (1 : Fin 3) · exact measurable_pi_apply (2 : Fin 3) have h_cons : ∀ ω ∈ S.zVar.event z, (if S.factualU ω ≤ S.p z then S.YofD true ω else S.YofD false ω) = h_proj (S.cfBundle.jointValue ω) := by intro ω _ dsimp [h_proj] rw [hgetU_joint] have hJV1 : (S.cfBundle.jointValue ω (1 : Fin 3) : ℝ) = S.YofD true ω := rfl have hJV2 : (S.cfBundle.jointValue ω (2 : Fin 3) : ℝ) = S.YofD false ω := rfl rw [hJV1, hJV2] have hbridge : S.condExpYZ z = eventCondExp P.μ (S.zVar.event z) (fun ω => if S.factualU ω ≤ S.p z then S.YofD true ω else S.YofD false ω) := by unfold POHeckmanRoySystem.condExpYZ eventCondExp change (∫ ω in S.zEvent z, S.factualY ω ∂P.μ) / (P.μ (S.zEvent z)).toReal = (∫ ω in S.zVar.event z, (if S.factualU ω ≤ S.p z then S.YofD true ω else S.YofD false ω) ∂P.μ) / (P.μ (S.zVar.event z)).toReal have hzev : S.zEvent z = S.zVar.event z := rfl rw [hzev] congr 1 refine MeasureTheory.setIntegral_congr_ae (S.measurableSet_zEvent z) ?_ filter_upwards [h_factualY_ae z] with ω hω hin rw [show S.zEvent z = S.zVar.event z from rfl] at hω exact hω hin rw [hbridge, POSystem.eventCondExp_of_consistency_IndepCF hA.instrumentIndep (a := S.zVar) hh_meas (MeasurableSet.singleton z) (ae_restrict_of_forall_mem (S.measurableSet_zEvent z) h_cons) (hμne_zero z hZ) (hμne_top z)] refine MeasureTheory.integral_congr_ae (Filter.Eventually.of_forall ?_) intro ω dsimp [h_proj] rw [hgetU_joint] have hJV1 : (S.cfBundle.jointValue ω (1 : Fin 3) : ℝ) = S.YofD true ω := rfl have hJV2 : (S.cfBundle.jointValue ω (2 : Fin 3) : ℝ) = S.YofD false ω := rfl rw [hJV1, hJV2] have hAlt_eq : ∀ z, (∫ ω, (if S.factualU ω ≤ S.p z then S.YofD true ω else S.YofD false ω) ∂P.μ) = ∫ ω, S.YofDofZ z ω ∂P.μ := by intro z refine MeasureTheory.integral_congr_ae ?_ filter_upwards [hA.thresholdCrossing z] with ω hth unfold YofDofZ by_cases hu : S.factualU ω ≤ S.p z · have hD : S.DofZ z ω = true := hth.mpr hu simp [hD, hu] · have hD : S.DofZ z ω = false := by cases hd : S.DofZ z ω · rfl · exact absurd (hth.mp hd) hu simp [hD, hu] rw [hCE z₁ hZ1, hCE z₀ hZ0, hAlt_eq z₁, hAlt_eq z₀] rw [← MeasureTheory.integral_sub (hYDZ_int z₁) (hYDZ_int z₀)]

theorem pointwise_threshold_identity (hA : S.Assumptions) (z₀ z₁ : α) (hpz : S.p z₀ ≤ S.p z₁) : ∀ᵐ 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 _ => (1:ℝ)) ω := by -- Combine threshold crossing at `z₀` and `z₁`; then unfold `YofDofZ` and -- case-split on `U ω ≤ p z₀` and `U ω ≤ p z₁`, using -- `intervalComplierEvent` membership in each case. filter_upwards [hA.thresholdCrossing z₀, hA.thresholdCrossing z₁] with ω h₀ h₁ unfold YofDofZ intervalComplierEvent by_cases hu1 : S.factualU ω ≤ S.p z₁ · by_cases hu0 : S.factualU ω ≤ S.p z₀ · have hD0 : S.DofZ z₀ ω = true := h₀.mpr hu0 have hD1 : S.DofZ z₁ ω = true := h₁.mpr hu1 have hnC : ω ∉ ({ω | S.p z₀ < S.factualU ω ∧ S.factualU ω ≤ S.p z₁} : Set P.Ω) := by intro hω exact (not_lt_of_ge hu0) hω.1 simp [hD0, hD1, Set.indicator_of_notMem hnC] · have hu0' : S.p z₀ < S.factualU ω := lt_of_not_ge hu0 have hD0 : S.DofZ z₀ ω = false := by cases hd : S.DofZ z₀ ω · rfl · exact absurd (h₀.mp hd) hu0 have hD1 : S.DofZ z₁ ω = true := h₁.mpr hu1 have hC : ω ∈ ({ω | S.p z₀ < S.factualU ω ∧ S.factualU ω ≤ S.p z₁} : Set P.Ω) := ⟨hu0', hu1⟩ simp [hD0, hD1, Set.indicator_of_mem hC] · have hu0 : ¬ S.factualU ω ≤ S.p z₀ := fun h => hu1 (le_trans h hpz) have hD0 : S.DofZ z₀ ω = false := by cases hd : S.DofZ z₀ ω · rfl · exact absurd (h₀.mp hd) hu0 have hD1 : S.DofZ z₁ ω = false := by cases hd : S.DofZ z₁ ω · rfl · exact absurd (h₁.mp hd) hu1 have hnC : ω ∉ ({ω | S.p z₀ < S.factualU ω ∧ S.factualU ω ≤ S.p z₁} : Set P.Ω) := by intro hω exact hu1 hω.2 simp [hD0, hD1, Set.indicator_of_notMem hnC]

theorem event_conditioning_identity (z₀ z₁ : α) : ∫ ω, (S.YofD true ω - S.YofD false ω) * (S.intervalComplierEvent z₀ z₁).indicator (fun _ => (1:ℝ)) ω ∂P.μ = (P.μ (S.intervalComplierEvent z₀ z₁)).toReal * S.LATE z₀ z₁ := by -- Same event-conditioning algebra as binary LATE, with the interval -- complier event in place of the binary complier event. unfold LATE have hC : MeasurableSet (S.intervalComplierEvent z₀ z₁) := S.measurableSet_intervalComplierEvent z₀ z₁ have h_rw : (fun ω => (S.YofD true ω - S.YofD false ω) * (S.intervalComplierEvent z₀ z₁).indicator (fun _ => (1:ℝ)) ω) = (S.intervalComplierEvent z₀ z₁).indicator (fun ω => S.YofD true ω - S.YofD false ω) := by funext ω by_cases hω : ω ∈ S.intervalComplierEvent z₀ z₁ · simp [Set.indicator_of_mem hω] · simp [Set.indicator_of_notMem hω] rw [h_rw, MeasureTheory.integral_indicator hC] by_cases hμ : (P.μ (S.intervalComplierEvent z₀ z₁)).toReal = 0 · rw [hμ, zero_mul] have hμ0 : P.μ (S.intervalComplierEvent z₀ z₁) = 0 := by have hne : P.μ (S.intervalComplierEvent z₀ z₁) ≠ ⊤ := measure_ne_top _ _ exact (ENNReal.toReal_eq_zero_iff _).mp hμ |>.resolve_right hne have hrest : P.μ.restrict (S.intervalComplierEvent z₀ z₁) = 0 := by rw [MeasureTheory.Measure.restrict_eq_zero]; exact hμ0 simp [hrest] · field_simp