awhecmu/canonical-drafter-extract-mathlib huggingface dataset:
-- Lean 4 proof step from Mathlib.MeasureTheory.Function.ConditionalExpectation.PullOut (goal)
Goal:
Ω : Type u_1
m mΩ : MeasurableSpace Ω
μ : Measure Ω
E : Type u_2
F : Type u_3
G : Type u_4
inst : NormedAddCommGroup E
inst_1 : NormedSpace ℝ E
inst_2 : NormedAddCommGroup F
inst_3 : NormedSpace ℝ F
inst_4 : NormedAddCommGroup G
inst_5 : NormedSpace ℝ G
inst_6 : CompleteSpace G
B : F →L[ℝ] E →L[ℝ] G
inst_7 : CompleteSpace E
f : Ω → F
g : Ω → E
hf : StronglyMeasurable f
hfg :
  Integrable.{u_4, u_1} (ε := G) (fun (ω : Ω) ↦ (((B : (a : F) → E →L[ℝ] G) (f ω) : E →L[ℝ] G) : (a : E) → G) (g ω)) μ
hg : Integrable g μ
hm : m ≤ mΩ
⊢ EventuallyEq.{u_1, u_4} (β := G) (ae μ)
    μ[fun (ω : Ω) ↦ (((B : (a : F) → E →L[ℝ] G) (f ω) : E →L[ℝ] G) : (a : E) → G) (g ω) | m] fun (ω : Ω) ↦
    (((B : (a : F) → E →L[ℝ] G) (f ω) : E →L[ℝ] G) : (a : E) → G) (μ[g | m] ω)
have pos : ∀ (hμm : SigmaFinite (μ.trim hm)),
  EventuallyEq.{u_1, u_4} (β := G) (ae μ)
    μ[fun (ω : Ω) ↦ (((B : (a : F) → E →L[ℝ] G) (f ω) : E →L[ℝ] G) : (a : E) → G) (g ω) | m] fun (ω : Ω) ↦
    (((B : (a : F) → E →L[ℝ] G) (f ω) : E →L[ℝ] G) : (a : E) → G) (μ[g | m] ω)
Tactic: by_cases hμm : SigmaFinite (μ.trim hm)
-- Lean 4 proof step from Mathlib.MeasureTheory.Function.ConditionalExpectation.PullOut (goal)
Goal:
Ω : Type u_1
m mΩ : MeasurableSpace Ω
μ : Measure Ω
E : Type u_2
F : Type u_3
G : Type u_4
inst : NormedAddCommGroup E
inst_1 : NormedSpace ℝ E
inst_2 : NormedAddCommGroup F
inst_3 : NormedSpace ℝ F
inst_4 : NormedAddCommGroup G
inst_5 : NormedSpace ℝ G
inst_6 : CompleteSpace G
B : F →L[ℝ] E →L[ℝ] G
inst_7 : CompleteSpace E
f : Ω → F
g : Ω → E
hf : StronglyMeasurable f
hfg :
  Integrable.{u_4, u_1} (ε := G) (fun (ω : Ω) ↦ (((B : (a : F) → E →L[ℝ] G) (f ω) : E →L[ℝ] G) : (a : E) → G) (g ω)) μ
hg : Integrable g μ
hm : m ≤ mΩ
pos :
  ∀ (hμm : SigmaFinite (μ.trim hm)),
    EventuallyEq.{u_1, u_4} (β := G) (ae μ)
      μ[fun (ω : Ω) ↦ (((B : (a : F) → E →L[ℝ] G) (f ω) : E →L[ℝ] G) : (a : E) → G) (g ω) | m] fun (ω : Ω) ↦
      (((B : (a : F) → E →L[ℝ] G) (f ω) : E →L[ℝ] G) : (a : E) → G) (μ[g | m] ω)
⊢ EventuallyEq.{u_1, u_4} (β := G) (ae μ)
    μ[fun (ω : Ω) ↦ (((B : (a : F) → E →L[ℝ] G) (f ω) : E →L[ℝ] G) : (a : E) → G) (g ω) | m] fun (ω : Ω) ↦
    (((B : (a : F) → E →L[ℝ] G) (f ω) : E →L[ℝ] G) : (a : E) → G) (μ[g | m] ω)
have neg : ∀ (hμm : ¬SigmaFinite (μ.trim hm)),
  EventuallyEq.{u_1, u_4} (β := G) (ae μ)
    μ[fun (ω : Ω) ↦ (((B : (a : F) → E →L[ℝ] G) (f ω) : E →L[ℝ] G) : (a : E) → G) (g ω) | m] fun (ω : Ω) ↦
    (((B : (a : F) → E →L[ℝ] G) (f ω) : E →L[ℝ] G) : (a : E) → G) (μ[g | m] ω)
Tactic: by_cases hμm : SigmaFinite (μ.trim hm)
-- Lean 4 proof step from Mathlib.MeasureTheory.Function.ConditionalExpectation.PullOut (goal)
Goal:
Ω : Type u_1
m mΩ : MeasurableSpace Ω
μ : Measure Ω
E : Type u_2
F : Type u_3
G : Type u_4
inst : NormedAddCommGroup E
inst_1 : NormedSpace ℝ E
inst_2 : NormedAddCommGroup F
inst_3 : NormedSpace ℝ F
inst_4 : NormedAddCommGroup G
inst_5 : NormedSpace ℝ G
inst_6 : CompleteSpace G
B : F →L[ℝ] E →L[ℝ] G
inst_7 : CompleteSpace E
f : Ω → F
g : Ω → E
hf : StronglyMeasurable f
hfg :
  Integrable.{u_4, u_1} (ε := G) (fun (ω : Ω) ↦ (((B : (a : F) → E →L[ℝ] G) (f ω) : E →L[ℝ] G) : (a : E) → G) (g ω)) μ
hg : Integrable g μ
hm : m ≤ mΩ
hμm : SigmaFinite (μ.trim hm)
⊢ EventuallyEq.{u_1, u_4} (β := G) (ae μ)
    μ[fun (ω : Ω) ↦ (((B : (a : F) → E →L[ℝ] G) (f ω) : E →L[ℝ] G) : (a : E) → G) (g ω) | m] fun (ω : Ω) ↦
    (((B : (a : F) → E →L[ℝ] G) (f ω) : E →L[ℝ] G) : (a : E) → G) (μ[g | m] ω)
have pos : ∀ (sets : ℕ → Set Ω)
  (sets_prop :
    ∀ (n : ℕ),
      MeasurableSet (sets n) ∧
        LT.lt (α := ℝ≥0∞) ((μ : (a : Set Ω) → ℝ≥0∞) (sets n) : ℝ≥0∞) (∞ : ℝ≥0∞) ∧ ∀ x ∈ sets n, ‖f x‖ ≤ (↑n : ℝ))
  (h_univ : Eq.{u_1 + 1} (α := Set Ω) (⋃ (i : ℕ), sets i) Set.univ),
  EventuallyEq.{u_1, u_4} (β := G) (ae μ)
    μ[fun (ω : Ω) ↦ (((B : (a : F) → E →L[ℝ] G) (f ω) : E →L[ℝ] G) : (a : E) → G) (g ω) | m] fun (ω : Ω) ↦
    (((B : (a : F) → E →L[ℝ] G) (f ω) : E →L[ℝ] G) : (a : E) → G) (μ[g | m] ω)
Tactic: obtain ⟨sets, sets_prop, h_univ⟩ := hf.exists_spanning_measurableSet_norm_le hm μ
-- Lean 4 proof step from Mathlib.MeasureTheory.Function.ConditionalExpectation.PullOut (hypothesis)
Goal:
Ω : Type u_1
m mΩ : MeasurableSpace Ω
μ : Measure Ω
E : Type u_2
F : Type u_3
G : Type u_4
inst : NormedAddCommGroup E
inst_1 : NormedSpace ℝ E
inst_2 : NormedAddCommGroup F
inst_3 : NormedSpace ℝ F
inst_4 : NormedAddCommGroup G
inst_5 : NormedSpace ℝ G
inst_6 : CompleteSpace G
B : F →L[ℝ] E →L[ℝ] G
inst_7 : CompleteSpace E
f : Ω → F
g : Ω → E
hf : StronglyMeasurable f
hfg :
  Integrable.{u_4, u_1} (ε := G) (fun (ω : Ω) ↦ (((B : (a : F) → E →L[ℝ] G) (f ω) : E →L[ℝ] G) : (a : E) → G) (g ω)) μ
hg : Integrable g μ
hm : m ≤ mΩ
hμm : SigmaFinite (μ.trim hm)
sets : ℕ → Set Ω
sets_prop :
  ∀ (n : ℕ),
    MeasurableSet (sets n) ∧
      LT.lt (α := ℝ≥0∞) ((μ : (a : Set Ω) → ℝ≥0∞) (sets n) : ℝ≥0∞) (∞ : ℝ≥0∞) ∧ ∀ x ∈ sets n, ‖f x‖ ≤ (↑n : ℝ)
h_univ : Eq.{u_1 + 1} (α := Set Ω) (⋃ (i : ℕ), sets i) Set.univ
⊢ EventuallyEq.{u_1, u_4} (β := G) (ae μ)
    μ[fun (ω : Ω) ↦ (((B : (a : F) → E →L[ℝ] G) (f ω) : E →L[ℝ] G) : (a : E) → G) (g ω) | m] fun (ω : Ω) ↦
    (((B : (a : F) → E →L[ℝ] G) (f ω) : E →L[ℝ] G) : (a : E) → G) (μ[g | m] ω)
have sets_prop₁ : (∀ (x : ℕ), MeasurableSet (sets x)) ∧
  (∀ (x : ℕ), LT.lt (α := ℝ≥0∞) ((μ : (a : Set Ω) → ℝ≥0∞) (sets x) : ℝ≥0∞) (∞ : ℝ≥0∞)) ∧
    ∀ (x : ℕ), ∀ x_1 ∈ sets x, ‖f x_1‖ ≤ (↑x : ℝ)
Tactic: simp_rw [forall_and] at sets_prop
Cleared hypotheses: sets_prop
-- Lean 4 proof step from Mathlib.MeasureTheory.Function.ConditionalExpectation.PullOut (hypothesis)
Goal:
Ω : Type u_1
m mΩ : MeasurableSpace Ω
μ : Measure Ω
E : Type u_2
F : Type u_3
G : Type u_4
inst : NormedAddCommGroup E
inst_1 : NormedSpace ℝ E
inst_2 : NormedAddCommGroup F
inst_3 : NormedSpace ℝ F
inst_4 : NormedAddCommGroup G
inst_5 : NormedSpace ℝ G
inst_6 : CompleteSpace G
B : F →L[ℝ] E →L[ℝ] G
inst_7 : CompleteSpace E
f : Ω → F
g : Ω → E
hf : StronglyMeasurable f
hfg :
  Integrable.{u_4, u_1} (ε := G) (fun (ω : Ω) ↦ (((B : (a : F) → E →L[ℝ] G) (f ω) : E →L[ℝ] G) : (a : E) → G) (g ω)) μ
hg : Integrable g μ
hm : m ≤ mΩ
hμm : SigmaFinite (μ.trim hm)
sets : ℕ → Set Ω
h_univ : Eq.{u_1 + 1} (α := Set Ω) (⋃ (i : ℕ), sets i) Set.univ
sets_prop :
  (∀ (x : ℕ), MeasurableSet (sets x)) ∧
    (∀ (x : ℕ), LT.lt (α := ℝ≥0∞) ((μ : (a : Set Ω) → ℝ≥0∞) (sets x) : ℝ≥0∞) (∞ : ℝ≥0∞)) ∧
      ∀ (x : ℕ), ∀ x_1 ∈ sets x, ‖f x_1‖ ≤ (↑x : ℝ)
⊢ EventuallyEq.{u_1, u_4} (β := G) (ae μ)
    μ[fun (ω : Ω) ↦ (((B : (a : F) → E →L[ℝ] G) (f ω) : E →L[ℝ] G) : (a : E) → G) (g ω) | m] fun (ω : Ω) ↦
    (((B : (a : F) → E →L[ℝ] G) (f ω) : E →L[ℝ] G) : (a : E) → G) (μ[g | m] ω)
have h_meas : ∀ (x : ℕ), MeasurableSet (sets x)
Tactic: obtain ⟨h_meas, h_finite, h_norm⟩ := sets_prop
Cleared hypotheses: sets_prop
-- Lean 4 proof step from Mathlib.MeasureTheory.Function.ConditionalExpectation.PullOut (hypothesis)
Goal:
Ω : Type u_1
m mΩ : MeasurableSpace Ω
μ : Measure Ω
E : Type u_2
F : Type u_3
G : Type u_4
inst : NormedAddCommGroup E
inst_1 : NormedSpace ℝ E
inst_2 : NormedAddCommGroup F
inst_3 : NormedSpace ℝ F
inst_4 : NormedAddCommGroup G
inst_5 : NormedSpace ℝ G
inst_6 : CompleteSpace G
B : F →L[ℝ] E →L[ℝ] G
inst_7 : CompleteSpace E
f : Ω → F
g : Ω → E
hf : StronglyMeasurable f
hfg :
  Integrable.{u_4, u_1} (ε := G) (fun (ω : Ω) ↦ (((B : (a : F) → E →L[ℝ] G) (f ω) : E →L[ℝ] G) : (a : E) → G) (g ω)) μ
hg : Integrable g μ
hm : m ≤ mΩ
hμm : SigmaFinite (μ.trim hm)
sets : ℕ → Set Ω
h_univ : Eq.{u_1 + 1} (α := Set Ω) (⋃ (i : ℕ), sets i) Set.univ
sets_prop :
  (∀ (x : ℕ), MeasurableSet (sets x)) ∧
    (∀ (x : ℕ), LT.lt (α := ℝ≥0∞) ((μ : (a : Set Ω) → ℝ≥0∞) (sets x) : ℝ≥0∞) (∞ : ℝ≥0∞)) ∧
      ∀ (x : ℕ), ∀ x_1 ∈ sets x, ‖f x_1‖ ≤ (↑x : ℝ)
h_meas : ∀ (x : ℕ), MeasurableSet (sets x)
⊢ EventuallyEq.{u_1, u_4} (β := G) (ae μ)
    μ[fun (ω : Ω) ↦ (((B : (a : F) → E →L[ℝ] G) (f ω) : E →L[ℝ] G) : (a : E) → G) (g ω) | m] fun (ω : Ω) ↦
    (((B : (a : F) → E →L[ℝ] G) (f ω) : E →L[ℝ] G) : (a : E) → G) (μ[g | m] ω)
have h_finite : ∀ (x : ℕ), LT.lt (α := ℝ≥0∞) ((μ : (a : Set Ω) → ℝ≥0∞) (sets x) : ℝ≥0∞) (∞ : ℝ≥0∞)
Tactic: obtain ⟨h_meas, h_finite, h_norm⟩ := sets_prop
Cleared hypotheses: sets_prop
-- Lean 4 proof step from Mathlib.MeasureTheory.Function.ConditionalExpectation.PullOut (hypothesis)
Goal:
Ω : Type u_1
m mΩ : MeasurableSpace Ω
μ : Measure Ω
E : Type u_2
F : Type u_3
G : Type u_4
inst : NormedAddCommGroup E
inst_1 : NormedSpace ℝ E
inst_2 : NormedAddCommGroup F
inst_3 : NormedSpace ℝ F
inst_4 : NormedAddCommGroup G
inst_5 : NormedSpace ℝ G
inst_6 : CompleteSpace G
B : F →L[ℝ] E →L[ℝ] G
inst_7 : CompleteSpace E
f : Ω → F
g : Ω → E
hf : StronglyMeasurable f
hfg :
  Integrable.{u_4, u_1} (ε := G) (fun (ω : Ω) ↦ (((B : (a : F) → E →L[ℝ] G) (f ω) : E →L[ℝ] G) : (a : E) → G) (g ω)) μ
hg : Integrable g μ
hm : m ≤ mΩ
hμm : SigmaFinite (μ.trim hm)
sets : ℕ → Set Ω
h_univ : Eq.{u_1 + 1} (α := Set Ω) (⋃ (i : ℕ), sets i) Set.univ
sets_prop :
  (∀ (x : ℕ), MeasurableSet (sets x)) ∧
    (∀ (x : ℕ), LT.lt (α := ℝ≥0∞) ((μ : (a : Set Ω) → ℝ≥0∞) (sets x) : ℝ≥0∞) (∞ : ℝ≥0∞)) ∧
      ∀ (x : ℕ), ∀ x_1 ∈ sets x, ‖f x_1‖ ≤ (↑x : ℝ)
h_meas : ∀ (x : ℕ), MeasurableSet (sets x)
h_finite : ∀ (x : ℕ), LT.lt (α := ℝ≥0∞) ((μ : (a : Set Ω) → ℝ≥0∞) (sets x) : ℝ≥0∞) (∞ : ℝ≥0∞)
⊢ EventuallyEq.{u_1, u_4} (β := G) (ae μ)
    μ[fun (ω : Ω) ↦ (((B : (a : F) → E →L[ℝ] G) (f ω) : E →L[ℝ] G) : (a : E) → G) (g ω) | m] fun (ω : Ω) ↦
    (((B : (a : F) → E →L[ℝ] G) (f ω) : E →L[ℝ] G) : (a : E) → G) (μ[g | m] ω)
have h_norm : ∀ (x : ℕ), ∀ x_1 ∈ sets x, ‖f x_1‖ ≤ (↑x : ℝ)
Tactic: obtain ⟨h_meas, h_finite, h_norm⟩ := sets_prop
Cleared hypotheses: sets_prop
-- Lean 4 proof step from Mathlib.MeasureTheory.Function.ConditionalExpectation.PullOut (hypothesis)
Goal:
Ω : Type u_1
m mΩ : MeasurableSpace Ω
μ : Measure Ω
E : Type u_2
F : Type u_3
G : Type u_4
inst : NormedAddCommGroup E
inst_1 : NormedSpace ℝ E
inst_2 : NormedAddCommGroup F
inst_3 : NormedSpace ℝ F
inst_4 : NormedAddCommGroup G
inst_5 : NormedSpace ℝ G
inst_6 : CompleteSpace G
B : F →L[ℝ] E →L[ℝ] G
inst_7 : CompleteSpace E
f : Ω → F
g : Ω → E
hf : StronglyMeasurable f
hfg :
  Integrable.{u_4, u_1} (ε := G) (fun (ω : Ω) ↦ (((B : (a : F) → E →L[ℝ] G) (f ω) : E →L[ℝ] G) : (a : E) → G) (g ω)) μ
hg : Integrable g μ
hm : m ≤ mΩ
hμm : SigmaFinite (μ.trim hm)
sets : ℕ → Set Ω
h_univ : Eq.{u_1 + 1} (α := Set Ω) (⋃ (i : ℕ), sets i) Set.univ
h_meas : ∀ (x : ℕ), MeasurableSet (sets x)
h_finite : ∀ (x : ℕ), LT.lt (α := ℝ≥0∞) ((μ : (a : Set Ω) → ℝ≥0∞) (sets x) : ℝ≥0∞) (∞ : ℝ≥0∞)
h_norm : ∀ (x : ℕ), ∀ x_1 ∈ sets x, ‖f x_1‖ ≤ (↑x : ℝ)
this :
  ∀ (n : ℕ),
    ∀ᵐ (ω : Ω) ∂μ,
      ω ∈ sets n →
        Eq.{u_4 + 1} (α := G)
          (μ[fun (ω : Ω) ↦ (((B : (a : F) → E →L[ℝ] G) (f ω) : E →L[ℝ] G) : (a : E) → G) (g ω) | m] ω : G)
          ((((B : (a : F) → E →L[ℝ] G) (f ω) : E →L[ℝ] G) : (a : E) → G) (μ[g | m] ω) : G)
⊢ EventuallyEq.{u_1, u_4} (β := G) (ae μ)
    μ[fun (ω : Ω) ↦ (((B : (a : F) → E →L[ℝ] G) (f ω) : E →L[ℝ] G) : (a : E) → G) (g ω) | m] fun (ω : Ω) ↦
    (((B : (a : F) → E →L[ℝ] G) (f ω) : E →L[ℝ] G) : (a : E) → G) (μ[g | m] ω)
have this₁ : ∀ᵐ (a : Ω) ∂μ,
  ∀ (i : ℕ),
    a ∈ sets i →
      Eq.{u_4 + 1} (α := G)
        (μ[fun (ω : Ω) ↦ (((B : (a : F) → E →L[ℝ] G) (f ω) : E →L[ℝ] G) : (a : E) → G) (g ω) | m] a : G)
        ((((B : (a : F) → E →L[ℝ] G) (f a) : E →L[ℝ] G) : (a : E) → G) (μ[g | m] a) : G)
Tactic: rw [← ae_all_iff] at this
Cleared hypotheses: this
-- Lean 4 proof step from Mathlib.MeasureTheory.Function.ConditionalExpectation.PullOut (goal)
Goal:
Ω : Type u_1
m mΩ : MeasurableSpace Ω
μ : Measure Ω
E : Type u_2
F : Type u_3
G : Type u_4
inst : NormedAddCommGroup E
inst_1 : NormedSpace ℝ E
inst_2 : NormedAddCommGroup F
inst_3 : NormedSpace ℝ F
inst_4 : NormedAddCommGroup G
inst_5 : NormedSpace ℝ G
inst_6 : CompleteSpace G
B : F →L[ℝ] E →L[ℝ] G
inst_7 : CompleteSpace E
f : Ω → F
g : Ω → E
hf : StronglyMeasurable f
hfg :
  Integrable.{u_4, u_1} (ε := G) (fun (ω : Ω) ↦ (((B : (a : F) → E →L[ℝ] G) (f ω) : E →L[ℝ] G) : (a : E) → G) (g ω)) μ
hg : Integrable g μ
hm : m ≤ mΩ
hμm : SigmaFinite (μ.trim hm)
sets : ℕ → Set Ω
h_univ : Eq.{u_1 + 1} (α := Set Ω) (⋃ (i : ℕ), sets i) Set.univ
h_meas : ∀ (x : ℕ), MeasurableSet (sets x)
h_finite : ∀ (x : ℕ), LT.lt (α := ℝ≥0∞) ((μ : (a : Set Ω) → ℝ≥0∞) (sets x) : ℝ≥0∞) (∞ : ℝ≥0∞)
h_norm : ∀ (x : ℕ), ∀ x_1 ∈ sets x, ‖f x_1‖ ≤ (↑x : ℝ)
this :
  ∀ᵐ (a : Ω) ∂μ,
    ∀ (i : ℕ),
      a ∈ sets i →
        Eq.{u_4 + 1} (α := G)
          (μ[fun (ω : Ω) ↦ (((B : (a : F) → E →L[ℝ] G) (f ω) : E →L[ℝ] G) : (a : E) → G) (g ω) | m] a : G)
          ((((B : (a : F) → E →L[ℝ] G) (f a) : E →L[ℝ] G) : (a : E) → G) (μ[g | m] a) : G)
⊢ EventuallyEq.{u_1, u_4} (β := G) (ae μ)
    μ[fun (ω : Ω) ↦ (((B : (a : F) → E →L[ℝ] G) (f ω) : E →L[ℝ] G) : (a : E) → G) (g ω) | m] fun (ω : Ω) ↦
    (((B : (a : F) → E →L[ℝ] G) (f ω) : E →L[ℝ] G) : (a : E) → G) (μ[g | m] ω)
have h : ∀ (ω : Ω)
  (hω :
    ∀ (i : ℕ),
      ω ∈ sets i →
        Eq.{u_4 + 1} (α := G)
          (μ[fun (ω : Ω) ↦ (((B : (a : F) → E →L[ℝ] G) (f ω) : E →L[ℝ] G) : (a : E) → G) (g ω) | m] ω : G)
          ((((B : (a : F) → E →L[ℝ] G) (f ω) : E →L[ℝ] G) : (a : E) → G) (μ[g | m] ω) : G)),
  Eq.{u_4 + 1} (α := G) (μ[fun (ω : Ω) ↦ (((B : (a : F) → E →L[ℝ] G) (f ω) : E →L[ℝ] G) : (a : E) → G) (g ω) | m] ω : G)
    ((((B : (a : F) → E →L[ℝ] G) (f ω) : E →L[ℝ] G) : (a : E) → G) (μ[g | m] ω) : G)
Tactic: filter_upwards [this] with ω hω
-- Lean 4 proof step from Mathlib.MeasureTheory.Function.ConditionalExpectation.PullOut (goal)
Goal:
Ω : Type u_1
m mΩ : MeasurableSpace Ω
μ : Measure Ω
E : Type u_2
F : Type u_3
G : Type u_4
inst : NormedAddCommGroup E
inst_1 : NormedSpace ℝ E
inst_2 : NormedAddCommGroup F
inst_3 : NormedSpace ℝ F
inst_4 : NormedAddCommGroup G
inst_5 : NormedSpace ℝ G
inst_6 : CompleteSpace G
B : F →L[ℝ] E →L[ℝ] G
inst_7 : CompleteSpace E
f : Ω → F
g : Ω → E
hf : StronglyMeasurable f
hfg :
  Integrable.{u_4, u_1} (ε := G) (fun (ω : Ω) ↦ (((B : (a : F) → E →L[ℝ] G) (f ω) : E →L[ℝ] G) : (a : E) → G) (g ω)) μ
hg : Integrable g μ
hm : m ≤ mΩ
hμm : SigmaFinite (μ.trim hm)
sets : ℕ → Set Ω
h_univ : Eq.{u_1 + 1} (α := Set Ω) (⋃ (i : ℕ), sets i) Set.univ
h_meas : ∀ (x : ℕ), MeasurableSet (sets x)
h_finite : ∀ (x : ℕ), LT.lt (α := ℝ≥0∞) ((μ : (a : Set Ω) → ℝ≥0∞) (sets x) : ℝ≥0∞) (∞ : ℝ≥0∞)
h_norm : ∀ (x : ℕ), ∀ x_1 ∈ sets x, ‖f x_1‖ ≤ (↑x : ℝ)
this :
  ∀ᵐ (a : Ω) ∂μ,
    ∀ (i : ℕ),
      a ∈ sets i →
        Eq.{u_4 + 1} (α := G)
          (μ[fun (ω : Ω) ↦ (((B : (a : F) → E →L[ℝ] G) (f ω) : E →L[ℝ] G) : (a : E) → G) (g ω) | m] a : G)
          ((((B : (a : F) → E →L[ℝ] G) (f a) : E →L[ℝ] G) : (a : E) → G) (μ[g | m] a) : G)
ω : Ω
hω :
  ∀ (i : ℕ),
    ω ∈ sets i →
      Eq.{u_4 + 1} (α := G)
        (μ[fun (ω : Ω) ↦ (((B : (a : F) → E →L[ℝ] G) (f ω) : E →L[ℝ] G) : (a : E) → G) (g ω) | m] ω : G)
        ((((B : (a : F) → E →L[ℝ] G) (f ω) : E →L[ℝ] G) : (a : E) → G) (μ[g | m] ω) : G)
⊢ Membership.mem (γ := Set Ω) (⋃ (i : ℕ), sets i) ω
have rw : Membership.mem (γ := Set Ω) Set.univ ω
Tactic: rw [h_univ]
-- Lean 4 proof step from Mathlib.MeasureTheory.Function.ConditionalExpectation.PullOut (hypothesis)
Goal:
Ω : Type u_1
m mΩ : MeasurableSpace Ω
μ : Measure Ω
E : Type u_2
F : Type u_3
G : Type u_4
inst : NormedAddCommGroup E
inst_1 : NormedSpace ℝ E
inst_2 : NormedAddCommGroup F
inst_3 : NormedSpace ℝ F
inst_4 : NormedAddCommGroup G
inst_5 : NormedSpace ℝ G
inst_6 : CompleteSpace G
B : F →L[ℝ] E →L[ℝ] G
inst_7 : CompleteSpace E
f : Ω → F
g : Ω → E
hf : StronglyMeasurable f
hfg :
  Integrable.{u_4, u_1} (ε := G) (fun (ω : Ω) ↦ (((B : (a : F) → E →L[ℝ] G) (f ω) : E →L[ℝ] G) : (a : E) → G) (g ω)) μ
hg : Integrable g μ
hm : m ≤ mΩ
hμm : SigmaFinite (μ.trim hm)
sets : ℕ → Set Ω
h_univ : Eq.{u_1 + 1} (α := Set Ω) (⋃ (i : ℕ), sets i) Set.univ
h_meas : ∀ (x : ℕ), MeasurableSet (sets x)
h_finite : ∀ (x : ℕ), LT.lt (α := ℝ≥0∞) ((μ : (a : Set Ω) → ℝ≥0∞) (sets x) : ℝ≥0∞) (∞ : ℝ≥0∞)
h_norm : ∀ (x : ℕ), ∀ x_1 ∈ sets x, ‖f x_1‖ ≤ (↑x : ℝ)
this :
  ∀ᵐ (a : Ω) ∂μ,
    ∀ (i : ℕ),
      a ∈ sets i →
        Eq.{u_4 + 1} (α := G)
          (μ[fun (ω : Ω) ↦ (((B : (a : F) → E →L[ℝ] G) (f ω) : E →L[ℝ] G) : (a : E) → G) (g ω) | m] a : G)
          ((((B : (a : F) → E →L[ℝ] G) (f a) : E →L[ℝ] G) : (a : E) → G) (μ[g | m] a) : G)
ω : Ω
hω :
  ∀ (i : ℕ),
    ω ∈ sets i →
      Eq.{u_4 + 1} (α := G)
        (μ[fun (ω : Ω) ↦ (((B : (a : F) → E →L[ℝ] G) (f ω) : E →L[ℝ] G) : (a : E) → G) (g ω) | m] ω : G)
        ((((B : (a : F) → E →L[ℝ] G) (f ω) : E →L[ℝ] G) : (a : E) → G) (μ[g | m] ω) : G)
⊢ ∃ (i : ℕ), ω ∈ sets i
have h_mem : Membership.mem (γ := Set Ω) (⋃ (i : ℕ), sets i) ω
Tactic: have h_mem : ω ∈ ⋃ i, sets i := by rw [h_univ]; exact Set.mem_univ _
-- Lean 4 proof step from Mathlib.MeasureTheory.Function.ConditionalExpectation.PullOut (goal)
Goal:
Ω : Type u_1
m mΩ : MeasurableSpace Ω
μ : Measure Ω
E : Type u_2
F : Type u_3
G : Type u_4
inst : NormedAddCommGroup E
inst_1 : NormedSpace ℝ E
inst_2 : NormedAddCommGroup F
inst_3 : NormedSpace ℝ F
inst_4 : NormedAddCommGroup G
inst_5 : NormedSpace ℝ G
inst_6 : CompleteSpace G
B : F →L[ℝ] E →L[ℝ] G
inst_7 : CompleteSpace E
f : Ω → F
g : Ω → E
hf : StronglyMeasurable f
hfg :
  Integrable.{u_4, u_1} (ε := G) (fun (ω : Ω) ↦ (((B : (a : F) → E →L[ℝ] G) (f ω) : E →L[ℝ] G) : (a : E) → G) (g ω)) μ
hg : Integrable g μ
hm : m ≤ mΩ
hμm : SigmaFinite (μ.trim hm)
sets : ℕ → Set Ω
h_univ : Eq.{u_1 + 1} (α := Set Ω) (⋃ (i : ℕ), sets i) Set.univ
h_meas : ∀ (x : ℕ), MeasurableSet (sets x)
h_finite : ∀ (x : ℕ), LT.lt (α := ℝ≥0∞) ((μ : (a : Set Ω) → ℝ≥0∞) (sets x) : ℝ≥0∞) (∞ : ℝ≥0∞)
h_norm : ∀ (x : ℕ), ∀ x_1 ∈ sets x, ‖f x_1‖ ≤ (↑x : ℝ)
⊢ EventuallyEq.{u_1, u_4} (β := G) (ae μ)
    μ[fun (ω : Ω) ↦ (((B : (a : F) → E →L[ℝ] G) (f ω) : E →L[ℝ] G) : (a : E) → G) (g ω) | m] fun (ω : Ω) ↦
    (((B : (a : F) → E →L[ℝ] G) (f ω) : E →L[ℝ] G) : (a : E) → G) (μ[g | m] ω)
have pos : ∀ (n : ℕ),
  ∀ᵐ (ω : Ω) ∂μ,
    ω ∈ sets n →
      Eq.{u_4 + 1} (α := G)
        (μ[fun (ω : Ω) ↦ (((B : (a : F) → E →L[ℝ] G) (f ω) : E →L[ℝ] G) : (a : E) → G) (g ω) | m] ω : G)
        ((((B : (a : F) → E →L[ℝ] G) (f ω) : E →L[ℝ] G) : (a : E) → G) (μ[g | m] ω) : G)
Tactic: suffices ∀ n, ∀ᵐ ω ∂μ, ω ∈ sets n → (μ[fun ω ↦ B (f ω) (g ω) | m]) ω = B (f ω) (μ[g | m] ω) by
    rw [← ae_all_iff] at this
    filter_upwards [this] with ω hω
    obtain ⟨i, hi⟩ : ∃ i, ω ∈ sets i := by
      have h_mem : ω ∈ ⋃ i, sets i := by rw [h_univ]; exact Set.mem_univ _
      simpa using h_mem
    exact hω i hi
-- Lean 4 proof step from Mathlib.MeasureTheory.Function.ConditionalExpectation.PullOut (goal)
Goal:
Ω : Type u_1
m mΩ : MeasurableSpace Ω
μ : Measure Ω
E : Type u_2
F : Type u_3
G : Type u_4
inst : NormedAddCommGroup E
inst_1 : NormedSpace ℝ E
inst_2 : NormedAddCommGroup F
inst_3 : NormedSpace ℝ F
inst_4 : NormedAddCommGroup G
inst_5 : NormedSpace ℝ G
inst_6 : CompleteSpace G
B : F →L[ℝ] E →L[ℝ] G
inst_7 : CompleteSpace E
f : Ω → F
g : Ω → E
hf : StronglyMeasurable f
hfg :
  Integrable.{u_4, u_1} (ε := G) (fun (ω : Ω) ↦ (((B : (a : F) → E →L[ℝ] G) (f ω) : E →L[ℝ] G) : (a : E) → G) (g ω)) μ
hg : Integrable g μ
hm : m ≤ mΩ
hμm : SigmaFinite (μ.trim hm)
sets : ℕ → Set Ω
h_univ : Eq.{u_1 + 1} (α := Set Ω) (⋃ (i : ℕ), sets i) Set.univ
h_meas : ∀ (x : ℕ), MeasurableSet (sets x)
h_finite : ∀ (x : ℕ), LT.lt (α := ℝ≥0∞) ((μ : (a : Set Ω) → ℝ≥0∞) (sets x) : ℝ≥0∞) (∞ : ℝ≥0∞)
h_norm : ∀ (x : ℕ), ∀ x_1 ∈ sets x, ‖f x_1‖ ≤ (↑x : ℝ)
⊢ ∀ (n : ℕ),
    ∀ᵐ (ω : Ω) ∂μ,
      ω ∈ sets n →
        Eq.{u_4 + 1} (α := G)
          (μ[fun (ω : Ω) ↦ (((B : (a : F) → E →L[ℝ] G) (f ω) : E →L[ℝ] G) : (a : E) → G) (g ω) | m] ω : G)
          ((((B : (a : F) → E →L[ℝ] G) (f ω) : E →L[ℝ] G) : (a : E) → G) (μ[g | m] ω) : G)
have pos : ∀ (n : ℕ),
  ∀ᵐ (x : Ω) ∂μ.restrict (sets n),
    Eq.{u_4 + 1} (α := G)
      (μ[fun (ω : Ω) ↦ (((B : (a : F) → E →L[ℝ] G) (f ω) : E →L[ℝ] G) : (a : E) → G) (g ω) | m] x : G)
      ((((B : (a : F) → E →L[ℝ] G) (f x) : E →L[ℝ] G) : (a : E) → G) (μ[g | m] x) : G)
Tactic: refine fun n ↦ ae_imp_of_ae_restrict ?_
-- Lean 4 proof step from Mathlib.MeasureTheory.Function.ConditionalExpectation.PullOut (goal)
Goal:
Ω : Type u_1
m mΩ : MeasurableSpace Ω
μ : Measure Ω
E : Type u_2
F : Type u_3
G : Type u_4
inst : NormedAddCommGroup E
inst_1 : NormedSpace ℝ E
inst_2 : NormedAddCommGroup F
inst_3 : NormedSpace ℝ F
inst_4 : NormedAddCommGroup G
inst_5 : NormedSpace ℝ G
inst_6 : CompleteSpace G
B : F →L[ℝ] E →L[ℝ] G
inst_7 : CompleteSpace E
f : Ω → F
g : Ω → E
hf : StronglyMeasurable f
hfg :
  Integrable.{u_4, u_1} (ε := G) (fun (ω : Ω) ↦ (((B : (a : F) → E →L[ℝ] G) (f ω) : E →L[ℝ] G) : (a : E) → G) (g ω)) μ
hg : Integrable g μ
hm : m ≤ mΩ
hμm : SigmaFinite (μ.trim hm)
sets : ℕ → Set Ω
h_univ : Eq.{u_1 + 1} (α := Set Ω) (⋃ (i : ℕ), sets i) Set.univ
h_meas : ∀ (x : ℕ), MeasurableSet (sets x)
h_finite : ∀ (x : ℕ), LT.lt (α := ℝ≥0∞) ((μ : (a : Set Ω) → ℝ≥0∞) (sets x) : ℝ≥0∞) (∞ : ℝ≥0∞)
h_norm : ∀ (x : ℕ), ∀ x_1 ∈ sets x, ‖f x_1‖ ≤ (↑x : ℝ)
n : ℕ
this :
  EventuallyEq.{u_1, u_4} (β := G) (ae (μ.restrict (sets n)))
    (μ.restrict (sets n))[fun (ω : Ω) ↦ (((B : (a : F) → E →L[ℝ] G) (f ω) : E →L[ℝ] G) : (a : E) → G) (g ω) | m]
    fun (ω : Ω) ↦ (((B : (a : F) → E →L[ℝ] G) (f ω) : E →L[ℝ] G) : (a : E) → G) ((μ.restrict (sets n))[g | m] ω)
⊢ ∀ᵐ (x : Ω) ∂μ.restrict (sets n),
    Eq.{u_4 + 1} (α := G)
      (μ[fun (ω : Ω) ↦ (((B : (a : F) → E →L[ℝ] G) (f ω) : E →L[ℝ] G) : (a : E) → G) (g ω) | m] x : G)
      ((((B : (a : F) → E →L[ℝ] G) (f x) : E →L[ℝ] G) : (a : E) → G) (μ[g | m] x) : G)
have refine : EventuallyEq.{u_1, u_4} (β := G) (ae (μ.restrict (sets n)))
  (μ.restrict (sets n))[fun (ω : Ω) ↦ (((B : (a : F) → E →L[ℝ] G) (f ω) : E →L[ℝ] G) : (a : E) → G) (g ω) | m]
  fun (x : Ω) ↦ (((B : (a : F) → E →L[ℝ] G) (f x) : E →L[ℝ] G) : (a : E) → G) (μ[g | m] x)
Tactic: refine (condExp_restrict_ae_eq_restrict hm (h_meas n) hfg).symm.trans ?_
-- Lean 4 proof step from Mathlib.MeasureTheory.Function.ConditionalExpectation.PullOut (goal)
Goal:
Ω : Type u_1
m mΩ : MeasurableSpace Ω
μ : Measure Ω
E : Type u_2
F : Type u_3
G : Type u_4
inst : NormedAddCommGroup E
inst_1 : NormedSpace ℝ E
inst_2 : NormedAddCommGroup F
inst_3 : NormedSpace ℝ F
inst_4 : NormedAddCommGroup G
inst_5 : NormedSpace ℝ G
inst_6 : CompleteSpace G
B : F →L[ℝ] E →L[ℝ] G
inst_7 : CompleteSpace E
f : Ω → F
g : Ω → E
hf : StronglyMeasurable f
hfg :
  Integrable.{u_4, u_1} (ε := G) (fun (ω : Ω) ↦ (((B : (a : F) → E →L[ℝ] G) (f ω) : E →L[ℝ] G) : (a : E) → G) (g ω)) μ
hg : Integrable g μ
hm : m ≤ mΩ
hμm : SigmaFinite (μ.trim hm)
sets : ℕ → Set Ω
h_univ : Eq.{u_1 + 1} (α := Set Ω) (⋃ (i : ℕ), sets i) Set.univ
h_meas : ∀ (x : ℕ), MeasurableSet (sets x)
h_finite : ∀ (x : ℕ), LT.lt (α := ℝ≥0∞) ((μ : (a : Set Ω) → ℝ≥0∞) (sets x) : ℝ≥0∞) (∞ : ℝ≥0∞)
h_norm : ∀ (x : ℕ), ∀ x_1 ∈ sets x, ‖f x_1‖ ≤ (↑x : ℝ)
n : ℕ
this :
  EventuallyEq.{u_1, u_4} (β := G) (ae (μ.restrict (sets n)))
    (μ.restrict (sets n))[fun (ω : Ω) ↦ (((B : (a : F) → E →L[ℝ] G) (f ω) : E →L[ℝ] G) : (a : E) → G) (g ω) | m]
    fun (ω : Ω) ↦ (((B : (a : F) → E →L[ℝ] G) (f ω) : E →L[ℝ] G) : (a : E) → G) ((μ.restrict (sets n))[g | m] ω)
⊢ EventuallyEq.{u_1, u_4} (β := G) (ae (μ.restrict (sets n)))
    (μ.restrict (sets n))[fun (ω : Ω) ↦ (((B : (a : F) → E →L[ℝ] G) (f ω) : E →L[ℝ] G) : (a : E) → G) (g ω) | m]
    fun (x : Ω) ↦ (((B : (a : F) → E →L[ℝ] G) (f x) : E →L[ℝ] G) : (a : E) → G) (μ[g | m] x)
have h : ∀ (ω : Ω)
  (hω1 :
    Eq.{u_4 + 1} (α := G)
      ((μ.restrict (sets n))[fun (ω : Ω) ↦ (((B : (a : F) → E →L[ℝ] G) (f ω) : E →L[ℝ] G) : (a : E) → G) (g ω) | m] ω :
        G)
      ((((B : (a : F) → E →L[ℝ] G) (f ω) : E →L[ℝ] G) : (a : E) → G) ((μ.restrict (sets n))[g | m] ω) : G))
  (hω2 : (μ.restrict (sets n))[g | m] ω = μ[g | m] ω),
  Eq.{u_4 + 1} (α := G)
    ((μ.restrict (sets n))[fun (ω : Ω) ↦ (((B : (a : F) → E →L[ℝ] G) (f ω) : E →L[ℝ] G) : (a : E) → G) (g ω) | m] ω : G)
    ((((B : (a : F) → E →L[ℝ] G) (f ω) : E →L[ℝ] G) : (a : E) → G) (μ[g | m] ω) : G)
Tactic: filter_upwards [this, (condExp_restrict_ae_eq_restrict hm (h_meas n) hg)] with ω hω1 hω2