awhecmu/canonical-drafter-extract-mathlib huggingface dataset:
-- Lean 4 proof step from Mathlib.Probability.Independence.Kernel.IndepFun (goal)
Goal:
α : Type u_1
Ω : Type u_2
ι : Type u_3
mα : MeasurableSpace α
mΩ : MeasurableSpace Ω
κ : Kernel α Ω
μ : Measure α
β : ι → Type u_8
m : (i : ι) → MeasurableSpace (β i)
f : (i : ι) → Ω → β i
hf_Indep : iIndepFun (β := β) f κ μ
hf_meas : ∀ (i : ι), Measurable (f i)
i j k : ι
hik : i ≠ k
hjk : j ≠ k
have have₁ have₂ :
  ∀
    (h_right :
      f k =
        (fun (p : (j : ↥{k}) → β (↑j : ι)) ↦
            p (Subtype.mk (p := fun (x : ι) ↦ Membership.mem (γ := Finset ι) {k} x) k ⋯)) ∘
          fun (a : Ω) (j : ↥{k}) ↦ f (↑j : ι) a),
    IndepFun (β := β i × β j) (fun (a : Ω) ↦ (f i a, f j a)) (f k) κ μ
⊢ IndepFun (β := β i × β j) (fun (a : Ω) ↦ (f i a, f j a)) (f k) κ μ
have have₃ : ∀
  (h_right :
    f k =
      (fun (p : (j : ↥{k}) → β (↑j : ι)) ↦
          p (Subtype.mk (p := fun (x : ι) ↦ Membership.mem (γ := Finset ι) {k} x) k sorry)) ∘
        fun (a : Ω) (j : ↥{k}) ↦ f (↑j : ι) a)
  (h_meas_right :
    Measurable (β := β (↑(Subtype.mk (p := fun (x : ι) ↦ Membership.mem (γ := Finset ι) {k} x) k sorry) : ι))
      fun (p : (j : ↥{k}) → β (↑j : ι)) ↦ p (Subtype.mk (p := fun (x : ι) ↦ Membership.mem (γ := Finset ι) {k} x) k sorry)),
  IndepFun (β := β i × β j) (fun (a : Ω) ↦ (f i a, f j a)) (f k) κ μ
Tactic: classical
  have h_right :
      f k = (fun p : ∀ j : ({k} : Finset ι), β j => p ⟨k, Finset.mem_singleton_self k⟩) ∘
        fun a (j : ({k} : Finset ι)) => f j a :=
    rfl
  have h_meas_right : Measurable fun p : ∀ j : ({k} : Finset ι),
      β j => p ⟨k, Finset.mem_singleton_self k⟩ :=
    measurable_pi_apply _
  let s : Finset ι := {i, j}
  have h_left : (fun ω => (f i ω, f j ω)) = (fun p : ∀ l : s, β l =>
      (p ⟨i, Finset.mem_insert_self i _⟩,
        p ⟨j, Finset.mem_insert_of_mem (Finset.mem_singleton_self _)⟩)) ∘
        fun a (j : s) => f j a := by
    ext1 a
    simp only
    constructor
  have h_meas_left : Measurable fun p : ∀ l : s, β l =>
      (p ⟨i, Finset.mem_insert_self i _⟩,
        p ⟨j, Finset.mem_insert_of_mem (Finset.mem_singleton_self _)⟩) :=
    Measurable.prod (measurable_pi_apply _) (measurable_pi_apply _)
  rw [h_left, h_right]
  refine (hf_Indep.indepFun_finset s {k} ?_ hf_meas).comp h_meas_left h_meas_right
  rw [Finset.disjoint_singleton_right]
  simp only [s, Finset.mem_insert, Finset.mem_singleton, not_or]
  exact ⟨hik.symm, hjk.symm⟩
-- Lean 4 proof step from Mathlib.Probability.Independence.Kernel.IndepFun (goal)
Goal:
α : Type u_1
Ω : Type u_2
ι : Type u_3
mα : MeasurableSpace α
mΩ : MeasurableSpace Ω
κ : Kernel α Ω
μ : Measure α
β : ι → Type u_8
m : (i : ι) → MeasurableSpace (β i)
f : (i : ι) → Ω → β i
hf_Indep : iIndepFun (β := β) f κ μ
hf_meas : ∀ (i : ι), AEMeasurable (_m := mΩ) (f i) ((⇑κ : (a : α) → Measure Ω) ∘ₘ μ)
i j k : ι
hik : i ≠ k
hjk : j ≠ k
⊢ IndepFun (β := β i × β j) (fun (a : Ω) ↦ (AEMeasurable.mk (f i) ⋯ a, AEMeasurable.mk (f j) ⋯ a))
    (AEMeasurable.mk (f k) ⋯) κ μ
have refine : iIndepFun (β := β) (fun (i : ι) ↦ AEMeasurable.mk (f i) sorry) κ μ
Tactic: refine iIndepFun.indepFun_prodMk ?_ (fun i ↦ (hf_meas i).measurable_mk) _ _ _ hik hjk
-- Lean 4 proof step from Mathlib.Probability.Independence.Kernel.IndepFun (hypothesis)
Goal:
α : Type u_1
Ω : Type u_2
ι : Type u_3
mα : MeasurableSpace α
mΩ : MeasurableSpace Ω
κ : Kernel α Ω
μ : Measure α
β : ι → Type u_8
m : (i : ι) → MeasurableSpace (β i)
f : (i : ι) → Ω → β i
hf_Indep : iIndepFun (β := β) f κ μ
hf_meas : ∀ (i : ι), AEMeasurable (_m := mΩ) (f i) ((⇑κ : (a : α) → Measure Ω) ∘ₘ μ)
i j k : ι
hik : i ≠ k
hjk : j ≠ k
⊢ IndepFun (β := β i × β j) (fun (a : Ω) ↦ (f i a, f j a)) (f k) κ μ
have h : IndepFun (β := β i × β j) (fun (a : Ω) ↦ (AEMeasurable.mk (f i) sorry a, AEMeasurable.mk (f j) sorry a))
  (AEMeasurable.mk (f k) sorry) κ μ
Tactic: have h : IndepFun (fun a ↦ ((hf_meas i).mk (f i) a, (hf_meas j).mk (f j) a))
      ((hf_meas k).mk (f k)) κ μ := by
    refine iIndepFun.indepFun_prodMk ?_ (fun i ↦ (hf_meas i).measurable_mk) _ _ _ hik hjk
    exact iIndepFun.congr' hf_Indep fun i ↦ Measure.ae_ae_of_ae_comp (hf_meas i).ae_eq_mk
-- Lean 4 proof step from Mathlib.Probability.Independence.Kernel.IndepFun (goal)
Goal:
α : Type u_1
Ω : Type u_2
ι : Type u_3
mα : MeasurableSpace α
mΩ : MeasurableSpace Ω
κ : Kernel α Ω
μ : Measure α
β : ι → Type u_8
m : (i : ι) → MeasurableSpace (β i)
f : (i : ι) → Ω → β i
hf_Indep : iIndepFun (β := β) f κ μ
hf_meas : ∀ (i : ι), AEMeasurable (_m := mΩ) (f i) ((⇑κ : (a : α) → Measure Ω) ∘ₘ μ)
i j k : ι
hik : i ≠ k
hjk : j ≠ k
h :
  IndepFun (β := β i × β j) (fun (a : Ω) ↦ (AEMeasurable.mk (f i) ⋯ a, AEMeasurable.mk (f j) ⋯ a))
    (AEMeasurable.mk (f k) ⋯) κ μ
⊢ IndepFun (β := β i × β j) (fun (a : Ω) ↦ (f i a, f j a)) (f k) κ μ
have refine_1 : ∀ᵐ (a : α) ∂μ,
  Filter.EventuallyEq.{u_2, u_8} (β := β i × β j) (ae (F := Measure Ω) ((κ : (a : α) → Measure Ω) a : Measure Ω))
    (fun (a : Ω) ↦ (AEMeasurable.mk (f i) sorry a, AEMeasurable.mk (f j) sorry a)) fun (a : Ω) ↦ (f i a, f j a)
Tactic: refine IndepFun.congr' h ?_ ?_
-- Lean 4 proof step from Mathlib.Probability.Independence.Kernel.IndepFun (goal)
Goal:
α : Type u_1
Ω : Type u_2
ι : Type u_3
mα : MeasurableSpace α
mΩ : MeasurableSpace Ω
κ : Kernel α Ω
μ : Measure α
β : ι → Type u_8
m : (i : ι) → MeasurableSpace (β i)
f : (i : ι) → Ω → β i
hf_Indep : iIndepFun (β := β) f κ μ
hf_meas : ∀ (i : ι), AEMeasurable (_m := mΩ) (f i) ((⇑κ : (a : α) → Measure Ω) ∘ₘ μ)
i j k : ι
hik : i ≠ k
hjk : j ≠ k
h :
  IndepFun (β := β i × β j) (fun (a : Ω) ↦ (AEMeasurable.mk (f i) ⋯ a, AEMeasurable.mk (f j) ⋯ a))
    (AEMeasurable.mk (f k) ⋯) κ μ
refine_1 :
  ∀ᵐ (a : α) ∂μ,
    Filter.EventuallyEq.{u_2, u_8} (β := β i × β j) (ae (F := Measure Ω) ((κ : (a : α) → Measure Ω) a : Measure Ω))
      (fun (a : Ω) ↦ (AEMeasurable.mk (f i) ⋯ a, AEMeasurable.mk (f j) ⋯ a)) fun (a : Ω) ↦ (f i a, f j a)
⊢ IndepFun (β := β i × β j) (fun (a : Ω) ↦ (f i a, f j a)) (f k) κ μ
have refine_2 : ∀ᵐ (a : α) ∂μ, AEMeasurable.mk (f k) sorry =ᶠ[ae (F := Measure Ω) ((κ : (a : α) → Measure Ω) a : Measure Ω)] f k
Tactic: refine IndepFun.congr' h ?_ ?_
-- Lean 4 proof step from Mathlib.Probability.Independence.Kernel.IndepFun (goal)
Goal:
α : Type u_1
Ω : Type u_2
ι : Type u_3
mα : MeasurableSpace α
mΩ : MeasurableSpace Ω
κ : Kernel α Ω
μ : Measure α
β : ι → Type u_8
m : (i : ι) → MeasurableSpace (β i)
f : (i : ι) → Ω → β i
hf_Indep : iIndepFun (β := β) f κ μ
hf_meas : ∀ (i : ι), AEMeasurable (_m := mΩ) (f i) ((⇑κ : (a : α) → Measure Ω) ∘ₘ μ)
i j k : ι
hik : i ≠ k
hjk : j ≠ k
h :
  IndepFun (β := β i × β j) (fun (a : Ω) ↦ (AEMeasurable.mk (f i) ⋯ a, AEMeasurable.mk (f j) ⋯ a))
    (AEMeasurable.mk (f k) ⋯) κ μ
⊢ ∀ᵐ (a : α) ∂μ,
    Filter.EventuallyEq.{u_2, u_8} (β := β i × β j) (ae (F := Measure Ω) ((κ : (a : α) → Measure Ω) a : Measure Ω))
      (fun (a : Ω) ↦ (AEMeasurable.mk (f i) ⋯ a, AEMeasurable.mk (f j) ⋯ a)) fun (a : Ω) ↦ (f i a, f j a)
have h₁ : ∀ (a : α) (hi : ∀ᵐ (ω : Ω) ∂((κ : (a : α) → Measure Ω) a : Measure Ω), f i ω = AEMeasurable.mk (f i) sorry ω)
  (hj : ∀ᵐ (ω : Ω) ∂((κ : (a : α) → Measure Ω) a : Measure Ω), f j ω = AEMeasurable.mk (f j) sorry ω),
  Filter.EventuallyEq.{u_2, u_8} (β := β i × β j) (ae (F := Measure Ω) ((κ : (a : α) → Measure Ω) a : Measure Ω))
    (fun (a : Ω) ↦ (AEMeasurable.mk (f i) sorry a, AEMeasurable.mk (f j) sorry a)) fun (a : Ω) ↦ (f i a, f j a)
Tactic: filter_upwards [Measure.ae_ae_of_ae_comp (hf_meas i).ae_eq_mk,
      Measure.ae_ae_of_ae_comp (hf_meas j).ae_eq_mk] with a hi hj
-- Lean 4 proof step from Mathlib.Probability.Independence.Kernel.IndepFun (goal)
Goal:
α : Type u_1
Ω : Type u_2
ι : Type u_3
mα : MeasurableSpace α
mΩ : MeasurableSpace Ω
κ : Kernel α Ω
μ : Measure α
β : ι → Type u_8
m : (i : ι) → MeasurableSpace (β i)
f : (i : ι) → Ω → β i
hf_Indep : iIndepFun (β := β) f κ μ
hf_meas : ∀ (i : ι), AEMeasurable (_m := mΩ) (f i) ((⇑κ : (a : α) → Measure Ω) ∘ₘ μ)
i j k : ι
hik : i ≠ k
hjk : j ≠ k
h :
  IndepFun (β := β i × β j) (fun (a : Ω) ↦ (AEMeasurable.mk (f i) ⋯ a, AEMeasurable.mk (f j) ⋯ a))
    (AEMeasurable.mk (f k) ⋯) κ μ
a : α
hi : ∀ᵐ (ω : Ω) ∂((κ : (a : α) → Measure Ω) a : Measure Ω), f i ω = AEMeasurable.mk (f i) ⋯ ω
hj : ∀ᵐ (ω : Ω) ∂((κ : (a : α) → Measure Ω) a : Measure Ω), f j ω = AEMeasurable.mk (f j) ⋯ ω
⊢ Filter.EventuallyEq.{u_2, u_8} (β := β i × β j) (ae (F := Measure Ω) ((κ : (a : α) → Measure Ω) a : Measure Ω))
    (fun (a : Ω) ↦ (AEMeasurable.mk (f i) ⋯ a, AEMeasurable.mk (f j) ⋯ a)) fun (a : Ω) ↦ (f i a, f j a)
have h₁ : ∀ (ω : Ω) (hωi : f i ω = AEMeasurable.mk (f i) sorry ω) (hωj : f j ω = AEMeasurable.mk (f j) sorry ω),
  (AEMeasurable.mk (f i) sorry ω, AEMeasurable.mk (f j) sorry ω) = (f i ω, f j ω)
Tactic: filter_upwards [hi, hj] with ω hωi hωj
-- Lean 4 proof step from Mathlib.Probability.Independence.Kernel.IndepFun (goal)
Goal:
α : Type u_1
Ω : Type u_2
ι : Type u_3
mα : MeasurableSpace α
mΩ : MeasurableSpace Ω
κ : Kernel α Ω
μ : Measure α
β : ι → Type u_8
m : (i : ι) → MeasurableSpace (β i)
f : (i : ι) → Ω → β i
hf_Indep : iIndepFun (β := β) f κ μ
hf_meas : ∀ (i : ι), AEMeasurable (_m := mΩ) (f i) ((⇑κ : (a : α) → Measure Ω) ∘ₘ μ)
i j k : ι
hik : i ≠ k
hjk : j ≠ k
h :
  IndepFun (β := β i × β j) (fun (a : Ω) ↦ (AEMeasurable.mk (f i) ⋯ a, AEMeasurable.mk (f j) ⋯ a))
    (AEMeasurable.mk (f k) ⋯) κ μ
a : α
hi : ∀ᵐ (ω : Ω) ∂((κ : (a : α) → Measure Ω) a : Measure Ω), f i ω = AEMeasurable.mk (f i) ⋯ ω
hj : ∀ᵐ (ω : Ω) ∂((κ : (a : α) → Measure Ω) a : Measure Ω), f j ω = AEMeasurable.mk (f j) ⋯ ω
ω : Ω
hωi : f i ω = AEMeasurable.mk (f i) ⋯ ω
hωj : f j ω = AEMeasurable.mk (f j) ⋯ ω
⊢ (AEMeasurable.mk (f i) ⋯ ω, AEMeasurable.mk (f j) ⋯ ω) = (f i ω, f j ω)
have h₁ : (f i ω, AEMeasurable.mk (f j) sorry ω) = (f i ω, f j ω)
Tactic: rw [← hωi, ← hωj]
-- Lean 4 proof step from Mathlib.Probability.Independence.Kernel.IndepFun (goal)
Goal:
α : Type u_1
Ω : Type u_2
ι : Type u_3
mα : MeasurableSpace α
mΩ : MeasurableSpace Ω
κ : Kernel α Ω
μ : Measure α
β : ι → Type u_8
m : (i : ι) → MeasurableSpace (β i)
f : (i : ι) → Ω → β i
hf_Indep : iIndepFun (β := β) f κ μ
hf_meas : ∀ (i : ι), AEMeasurable (_m := mΩ) (f i) ((⇑κ : (a : α) → Measure Ω) ∘ₘ μ)
i j k : ι
hik : i ≠ k
hjk : j ≠ k
h :
  IndepFun (β := β i × β j) (fun (a : Ω) ↦ (AEMeasurable.mk (f i) ⋯ a, AEMeasurable.mk (f j) ⋯ a))
    (AEMeasurable.mk (f k) ⋯) κ μ
a : α
hi : ∀ᵐ (ω : Ω) ∂((κ : (a : α) → Measure Ω) a : Measure Ω), f i ω = AEMeasurable.mk (f i) ⋯ ω
hj : ∀ᵐ (ω : Ω) ∂((κ : (a : α) → Measure Ω) a : Measure Ω), f j ω = AEMeasurable.mk (f j) ⋯ ω
ω : Ω
hωi : f i ω = AEMeasurable.mk (f i) ⋯ ω
hωj : f j ω = AEMeasurable.mk (f j) ⋯ ω
h₁ : (f i ω, AEMeasurable.mk (f j) ⋯ ω) = (f i ω, f j ω)
⊢ (AEMeasurable.mk (f i) ⋯ ω, AEMeasurable.mk (f j) ⋯ ω) = (f i ω, f j ω)
have h₂ : (f i ω, f j ω) = (f i ω, f j ω)
Tactic: rw [← hωi, ← hωj]
-- Lean 4 proof step from Mathlib.Probability.Independence.Kernel.IndepFun (hypothesis)
Goal:
α : Type u_1
Ω : Type u_2
ι : Type u_3
mα : MeasurableSpace α
mΩ : MeasurableSpace Ω
κ : Kernel α Ω
μ : Measure α
β : ι → Type u_8
m : (i : ι) → MeasurableSpace (β i)
f : (i : ι) → Ω → β i
hf_indep : iIndepFun (β := β) f κ μ
hf_meas : ∀ (i : ι), Measurable (f i)
i j k l : ι
hik : i ≠ k
hil : i ≠ l
hjk : j ≠ k
hjl : j ≠ l
g : (i j : ι) → (v : (x : ↥{i, j}) → β (↑x : ι)) → β i × β j :=
  fun (i j : ι) (v : (x : ↥{i, j}) → β (↑x : ι)) ↦
    (v (Subtype.mk (p := fun (x : ι) ↦ Membership.mem (γ := Finset ι) {i, j} x) i ⋯),
      v (Subtype.mk (p := fun (x : ι) ↦ Membership.mem (γ := Finset ι) {i, j} x) j ⋯))
⊢ IndepFun.{u_1, u_2, u_8, u_8} (β := β i × β j) (γ := β k × β l) (fun (a : Ω) ↦ (f i a, f j a))
    (fun (a : Ω) ↦ (f k a, f l a)) κ μ
have hg : ∀ (i j : ι), Measurable (g i j)
Tactic: have hg (i j : ι) : Measurable (g i j) := by fun_prop
-- Lean 4 proof step from Mathlib.Probability.Independence.Kernel.IndepFun (goal)
Goal:
α : Type u_1
Ω : Type u_2
ι : Type u_3
mα : MeasurableSpace α
mΩ : MeasurableSpace Ω
κ : Kernel α Ω
μ : Measure α
β : ι → Type u_8
m : (i : ι) → MeasurableSpace (β i)
f : (i : ι) → Ω → β i
hf_indep : iIndepFun (β := β) f κ μ
hf_meas : ∀ (i : ι), Measurable (f i)
i j k l : ι
hik : i ≠ k
hil : i ≠ l
hjk : j ≠ k
hjl : j ≠ l
let let₁ let₂ :
  IndepFun.{u_1, u_2, u_8, u_8} (β := β i × β j) (γ := β k × β l) (fun (a : Ω) ↦ (f i a, f j a))
    (fun (a : Ω) ↦ (f k a, f l a)) κ μ
⊢ IndepFun.{u_1, u_2, u_8, u_8} (β := β i × β j) (γ := β k × β l) (fun (a : Ω) ↦ (f i a, f j a))
    (fun (a : Ω) ↦ (f k a, f l a)) κ μ
have let₃ : let g : (i j : ι) → (v : (x : ↥{i, j}) → β (↑x : ι)) → β i × β j := fun (i j : ι) (v : (x : ↥{i, j}) → β (↑x : ι)) ↦
  (v (Subtype.mk (p := fun (x : ι) ↦ Membership.mem (γ := Finset ι) {i, j} x) i sorry),
    v (Subtype.mk (p := fun (x : ι) ↦ Membership.mem (γ := Finset ι) {i, j} x) j sorry));
∀ (hg : ∀ (i j : ι), Measurable (g i j)),
  IndepFun.{u_1, u_2, u_8, u_8} (β := β i × β j) (γ := β k × β l) (fun (a : Ω) ↦ (f i a, f j a))
    (fun (a : Ω) ↦ (f k a, f l a)) κ μ
Tactic: classical
  let g (i j : ι) (v : Π x : ({i, j} : Finset ι), β x) : β i × β j :=
    ⟨v ⟨i, mem_insert_self _ _⟩, v ⟨j, mem_insert_of_mem <| mem_singleton_self _⟩⟩
  have hg (i j : ι) : Measurable (g i j) := by fun_prop
  exact (hf_indep.indepFun_finset {i, j} {k, l} (by aesop) hf_meas).comp (hg i j) (hg k l)
-- Lean 4 proof step from Mathlib.Probability.Independence.Kernel.IndepFun (goal)
Goal:
α : Type u_1
Ω : Type u_2
ι : Type u_3
mα : MeasurableSpace α
mΩ : MeasurableSpace Ω
κ : Kernel α Ω
μ : Measure α
β : ι → Type u_8
m : (i : ι) → MeasurableSpace (β i)
f : (i : ι) → Ω → β i
hf_indep : iIndepFun (β := β) f κ μ
hf_meas : ∀ (i : ι), AEMeasurable (_m := mΩ) (f i) ((⇑κ : (a : α) → Measure Ω) ∘ₘ μ)
i j k l : ι
hik : i ≠ k
hil : i ≠ l
hjk : j ≠ k
hjl : j ≠ l
⊢ IndepFun.{u_1, u_2, u_8, u_8} (β := β i × β j) (γ := β k × β l)
    (fun (a : Ω) ↦ (AEMeasurable.mk (f i) ⋯ a, AEMeasurable.mk (f j) ⋯ a))
    (fun (a : Ω) ↦ (AEMeasurable.mk (f k) ⋯ a, AEMeasurable.mk (f l) ⋯ a)) κ μ
have refine : iIndepFun (β := β) (fun (i : ι) ↦ AEMeasurable.mk (f i) sorry) κ μ
Tactic: refine iIndepFun.indepFun_prodMk_prodMk ?_ (fun i ↦ (hf_meas i).measurable_mk) _ _ _ _ hik hil
      hjk hjl
-- Lean 4 proof step from Mathlib.Probability.Independence.Kernel.IndepFun (hypothesis)
Goal:
α : Type u_1
Ω : Type u_2
ι : Type u_3
mα : MeasurableSpace α
mΩ : MeasurableSpace Ω
κ : Kernel α Ω
μ : Measure α
β : ι → Type u_8
m : (i : ι) → MeasurableSpace (β i)
f : (i : ι) → Ω → β i
hf_indep : iIndepFun (β := β) f κ μ
hf_meas : ∀ (i : ι), AEMeasurable (_m := mΩ) (f i) ((⇑κ : (a : α) → Measure Ω) ∘ₘ μ)
i j k l : ι
hik : i ≠ k
hil : i ≠ l
hjk : j ≠ k
hjl : j ≠ l
⊢ IndepFun.{u_1, u_2, u_8, u_8} (β := β i × β j) (γ := β k × β l) (fun (a : Ω) ↦ (f i a, f j a))
    (fun (a : Ω) ↦ (f k a, f l a)) κ μ
have h : IndepFun.{u_1, u_2, u_8, u_8} (β := β i × β j) (γ := β k × β l)
  (fun (a : Ω) ↦ (AEMeasurable.mk (f i) sorry a, AEMeasurable.mk (f j) sorry a))
  (fun (a : Ω) ↦ (AEMeasurable.mk (f k) sorry a, AEMeasurable.mk (f l) sorry a)) κ μ
Tactic: have h : IndepFun (fun a ↦ ((hf_meas i).mk (f i) a, (hf_meas j).mk (f j) a))
      (fun a ↦ ((hf_meas k).mk (f k) a, (hf_meas l).mk (f l) a)) κ μ := by
    refine iIndepFun.indepFun_prodMk_prodMk ?_ (fun i ↦ (hf_meas i).measurable_mk) _ _ _ _ hik hil
      hjk hjl
    exact iIndepFun.congr' hf_indep fun i ↦ Measure.ae_ae_of_ae_comp (hf_meas i).ae_eq_mk
-- Lean 4 proof step from Mathlib.Probability.Independence.Kernel.IndepFun (goal)
Goal:
α : Type u_1
Ω : Type u_2
ι : Type u_3
mα : MeasurableSpace α
mΩ : MeasurableSpace Ω
κ : Kernel α Ω
μ : Measure α
β : ι → Type u_8
m : (i : ι) → MeasurableSpace (β i)
f : (i : ι) → Ω → β i
hf_indep : iIndepFun (β := β) f κ μ
hf_meas : ∀ (i : ι), AEMeasurable (_m := mΩ) (f i) ((⇑κ : (a : α) → Measure Ω) ∘ₘ μ)
i j k l : ι
hik : i ≠ k
hil : i ≠ l
hjk : j ≠ k
hjl : j ≠ l
h :
  IndepFun.{u_1, u_2, u_8, u_8} (β := β i × β j) (γ := β k × β l)
    (fun (a : Ω) ↦ (AEMeasurable.mk (f i) ⋯ a, AEMeasurable.mk (f j) ⋯ a))
    (fun (a : Ω) ↦ (AEMeasurable.mk (f k) ⋯ a, AEMeasurable.mk (f l) ⋯ a)) κ μ
⊢ IndepFun.{u_1, u_2, u_8, u_8} (β := β i × β j) (γ := β k × β l) (fun (a : Ω) ↦ (f i a, f j a))
    (fun (a : Ω) ↦ (f k a, f l a)) κ μ
have refine_1 : ∀ᵐ (a : α) ∂μ,
  Filter.EventuallyEq.{u_2, u_8} (β := β i × β j) (ae (F := Measure Ω) ((κ : (a : α) → Measure Ω) a : Measure Ω))
    (fun (a : Ω) ↦ (AEMeasurable.mk (f i) sorry a, AEMeasurable.mk (f j) sorry a)) fun (a : Ω) ↦ (f i a, f j a)
Tactic: refine IndepFun.congr' h ?_ ?_
-- Lean 4 proof step from Mathlib.Probability.Independence.Kernel.IndepFun (goal)
Goal:
α : Type u_1
Ω : Type u_2
ι : Type u_3
mα : MeasurableSpace α
mΩ : MeasurableSpace Ω
κ : Kernel α Ω
μ : Measure α
β : ι → Type u_8
m : (i : ι) → MeasurableSpace (β i)
f : (i : ι) → Ω → β i
hf_indep : iIndepFun (β := β) f κ μ
hf_meas : ∀ (i : ι), AEMeasurable (_m := mΩ) (f i) ((⇑κ : (a : α) → Measure Ω) ∘ₘ μ)
i j k l : ι
hik : i ≠ k
hil : i ≠ l
hjk : j ≠ k
hjl : j ≠ l
h :
  IndepFun.{u_1, u_2, u_8, u_8} (β := β i × β j) (γ := β k × β l)
    (fun (a : Ω) ↦ (AEMeasurable.mk (f i) ⋯ a, AEMeasurable.mk (f j) ⋯ a))
    (fun (a : Ω) ↦ (AEMeasurable.mk (f k) ⋯ a, AEMeasurable.mk (f l) ⋯ a)) κ μ
refine_1 :
  ∀ᵐ (a : α) ∂μ,
    Filter.EventuallyEq.{u_2, u_8} (β := β i × β j) (ae (F := Measure Ω) ((κ : (a : α) → Measure Ω) a : Measure Ω))
      (fun (a : Ω) ↦ (AEMeasurable.mk (f i) ⋯ a, AEMeasurable.mk (f j) ⋯ a)) fun (a : Ω) ↦ (f i a, f j a)
⊢ IndepFun.{u_1, u_2, u_8, u_8} (β := β i × β j) (γ := β k × β l) (fun (a : Ω) ↦ (f i a, f j a))
    (fun (a : Ω) ↦ (f k a, f l a)) κ μ
have refine_2 : ∀ᵐ (a : α) ∂μ,
  Filter.EventuallyEq.{u_2, u_8} (β := β k × β l) (ae (F := Measure Ω) ((κ : (a : α) → Measure Ω) a : Measure Ω))
    (fun (a : Ω) ↦ (AEMeasurable.mk (f k) sorry a, AEMeasurable.mk (f l) sorry a)) fun (a : Ω) ↦ (f k a, f l a)
Tactic: refine IndepFun.congr' h ?_ ?_
-- Lean 4 proof step from Mathlib.Probability.Independence.Kernel.IndepFun (goal)
Goal:
α : Type u_1
Ω : Type u_2
ι : Type u_3
mα : MeasurableSpace α
mΩ : MeasurableSpace Ω
κ : Kernel α Ω
μ : Measure α
β : ι → Type u_8
m : (i : ι) → MeasurableSpace (β i)
f : (i : ι) → Ω → β i
hf_indep : iIndepFun (β := β) f κ μ
hf_meas : ∀ (i : ι), AEMeasurable (_m := mΩ) (f i) ((⇑κ : (a : α) → Measure Ω) ∘ₘ μ)
i j k l : ι
hik : i ≠ k
hil : i ≠ l
hjk : j ≠ k
hjl : j ≠ l
h :
  IndepFun.{u_1, u_2, u_8, u_8} (β := β i × β j) (γ := β k × β l)
    (fun (a : Ω) ↦ (AEMeasurable.mk (f i) ⋯ a, AEMeasurable.mk (f j) ⋯ a))
    (fun (a : Ω) ↦ (AEMeasurable.mk (f k) ⋯ a, AEMeasurable.mk (f l) ⋯ a)) κ μ
⊢ ∀ᵐ (a : α) ∂μ,
    Filter.EventuallyEq.{u_2, u_8} (β := β i × β j) (ae (F := Measure Ω) ((κ : (a : α) → Measure Ω) a : Measure Ω))
      (fun (a : Ω) ↦ (AEMeasurable.mk (f i) ⋯ a, AEMeasurable.mk (f j) ⋯ a)) fun (a : Ω) ↦ (f i a, f j a)
have h₁ : ∀ (a : α) (hi : ∀ᵐ (ω : Ω) ∂((κ : (a : α) → Measure Ω) a : Measure Ω), f i ω = AEMeasurable.mk (f i) sorry ω)
  (hj : ∀ᵐ (ω : Ω) ∂((κ : (a : α) → Measure Ω) a : Measure Ω), f j ω = AEMeasurable.mk (f j) sorry ω),
  Filter.EventuallyEq.{u_2, u_8} (β := β i × β j) (ae (F := Measure Ω) ((κ : (a : α) → Measure Ω) a : Measure Ω))
    (fun (a : Ω) ↦ (AEMeasurable.mk (f i) sorry a, AEMeasurable.mk (f j) sorry a)) fun (a : Ω) ↦ (f i a, f j a)
Tactic: filter_upwards [Measure.ae_ae_of_ae_comp (hf_meas i).ae_eq_mk,
      Measure.ae_ae_of_ae_comp (hf_meas j).ae_eq_mk] with a hi hj
-- Lean 4 proof step from Mathlib.Probability.Independence.Kernel.IndepFun (goal)
Goal:
α : Type u_1
Ω : Type u_2
ι : Type u_3
mα : MeasurableSpace α
mΩ : MeasurableSpace Ω
κ : Kernel α Ω
μ : Measure α
β : ι → Type u_8
m : (i : ι) → MeasurableSpace (β i)
f : (i : ι) → Ω → β i
hf_indep : iIndepFun (β := β) f κ μ
hf_meas : ∀ (i : ι), AEMeasurable (_m := mΩ) (f i) ((⇑κ : (a : α) → Measure Ω) ∘ₘ μ)
i j k l : ι
hik : i ≠ k
hil : i ≠ l
hjk : j ≠ k
hjl : j ≠ l
h :
  IndepFun.{u_1, u_2, u_8, u_8} (β := β i × β j) (γ := β k × β l)
    (fun (a : Ω) ↦ (AEMeasurable.mk (f i) ⋯ a, AEMeasurable.mk (f j) ⋯ a))
    (fun (a : Ω) ↦ (AEMeasurable.mk (f k) ⋯ a, AEMeasurable.mk (f l) ⋯ a)) κ μ
a : α
hi : ∀ᵐ (ω : Ω) ∂((κ : (a : α) → Measure Ω) a : Measure Ω), f i ω = AEMeasurable.mk (f i) ⋯ ω
hj : ∀ᵐ (ω : Ω) ∂((κ : (a : α) → Measure Ω) a : Measure Ω), f j ω = AEMeasurable.mk (f j) ⋯ ω
⊢ Filter.EventuallyEq.{u_2, u_8} (β := β i × β j) (ae (F := Measure Ω) ((κ : (a : α) → Measure Ω) a : Measure Ω))
    (fun (a : Ω) ↦ (AEMeasurable.mk (f i) ⋯ a, AEMeasurable.mk (f j) ⋯ a)) fun (a : Ω) ↦ (f i a, f j a)
have h₁ : ∀ (ω : Ω) (hωi : f i ω = AEMeasurable.mk (f i) sorry ω) (hωj : f j ω = AEMeasurable.mk (f j) sorry ω),
  (AEMeasurable.mk (f i) sorry ω, AEMeasurable.mk (f j) sorry ω) = (f i ω, f j ω)
Tactic: filter_upwards [hi, hj] with ω hωi hωj
-- Lean 4 proof step from Mathlib.Probability.Independence.Kernel.IndepFun (goal)
Goal:
α : Type u_1
Ω : Type u_2
ι : Type u_3
mα : MeasurableSpace α
mΩ : MeasurableSpace Ω
κ : Kernel α Ω
μ : Measure α
β : ι → Type u_8
m : (i : ι) → MeasurableSpace (β i)
f : (i : ι) → Ω → β i
hf_indep : iIndepFun (β := β) f κ μ
hf_meas : ∀ (i : ι), AEMeasurable (_m := mΩ) (f i) ((⇑κ : (a : α) → Measure Ω) ∘ₘ μ)
i j k l : ι
hik : i ≠ k
hil : i ≠ l
hjk : j ≠ k
hjl : j ≠ l
h :
  IndepFun.{u_1, u_2, u_8, u_8} (β := β i × β j) (γ := β k × β l)
    (fun (a : Ω) ↦ (AEMeasurable.mk (f i) ⋯ a, AEMeasurable.mk (f j) ⋯ a))
    (fun (a : Ω) ↦ (AEMeasurable.mk (f k) ⋯ a, AEMeasurable.mk (f l) ⋯ a)) κ μ
a : α
hi : ∀ᵐ (ω : Ω) ∂((κ : (a : α) → Measure Ω) a : Measure Ω), f i ω = AEMeasurable.mk (f i) ⋯ ω
hj : ∀ᵐ (ω : Ω) ∂((κ : (a : α) → Measure Ω) a : Measure Ω), f j ω = AEMeasurable.mk (f j) ⋯ ω
ω : Ω
hωi : f i ω = AEMeasurable.mk (f i) ⋯ ω
hωj : f j ω = AEMeasurable.mk (f j) ⋯ ω
⊢ (AEMeasurable.mk (f i) ⋯ ω, AEMeasurable.mk (f j) ⋯ ω) = (f i ω, f j ω)
have h₁ : (f i ω, AEMeasurable.mk (f j) sorry ω) = (f i ω, f j ω)
Tactic: rw [← hωi, ← hωj]
-- Lean 4 proof step from Mathlib.Probability.Independence.Kernel.IndepFun (goal)
Goal:
α : Type u_1
Ω : Type u_2
ι : Type u_3
mα : MeasurableSpace α
mΩ : MeasurableSpace Ω
κ : Kernel α Ω
μ : Measure α
β : ι → Type u_8
m : (i : ι) → MeasurableSpace (β i)
f : (i : ι) → Ω → β i
hf_indep : iIndepFun (β := β) f κ μ
hf_meas : ∀ (i : ι), AEMeasurable (_m := mΩ) (f i) ((⇑κ : (a : α) → Measure Ω) ∘ₘ μ)
i j k l : ι
hik : i ≠ k
hil : i ≠ l
hjk : j ≠ k
hjl : j ≠ l
h :
  IndepFun.{u_1, u_2, u_8, u_8} (β := β i × β j) (γ := β k × β l)
    (fun (a : Ω) ↦ (AEMeasurable.mk (f i) ⋯ a, AEMeasurable.mk (f j) ⋯ a))
    (fun (a : Ω) ↦ (AEMeasurable.mk (f k) ⋯ a, AEMeasurable.mk (f l) ⋯ a)) κ μ
a : α
hi : ∀ᵐ (ω : Ω) ∂((κ : (a : α) → Measure Ω) a : Measure Ω), f i ω = AEMeasurable.mk (f i) ⋯ ω
hj : ∀ᵐ (ω : Ω) ∂((κ : (a : α) → Measure Ω) a : Measure Ω), f j ω = AEMeasurable.mk (f j) ⋯ ω
ω : Ω
hωi : f i ω = AEMeasurable.mk (f i) ⋯ ω
hωj : f j ω = AEMeasurable.mk (f j) ⋯ ω
h₁ : (f i ω, AEMeasurable.mk (f j) ⋯ ω) = (f i ω, f j ω)
⊢ (AEMeasurable.mk (f i) ⋯ ω, AEMeasurable.mk (f j) ⋯ ω) = (f i ω, f j ω)
have h₂ : (f i ω, f j ω) = (f i ω, f j ω)
Tactic: rw [← hωi, ← hωj]
-- Lean 4 proof step from Mathlib.Probability.Independence.Kernel.IndepFun (goal)
Goal:
α : Type u_1
Ω : Type u_2
ι : Type u_3
mα : MeasurableSpace α
mΩ : MeasurableSpace Ω
κ : Kernel α Ω
μ : Measure α
β : ι → Type u_8
m : (i : ι) → MeasurableSpace (β i)
f : (i : ι) → Ω → β i
hf_indep : iIndepFun (β := β) f κ μ
hf_meas : ∀ (i : ι), AEMeasurable (_m := mΩ) (f i) ((⇑κ : (a : α) → Measure Ω) ∘ₘ μ)
i j k l : ι
hik : i ≠ k
hil : i ≠ l
hjk : j ≠ k
hjl : j ≠ l
h :
  IndepFun.{u_1, u_2, u_8, u_8} (β := β i × β j) (γ := β k × β l)
    (fun (a : Ω) ↦ (AEMeasurable.mk (f i) ⋯ a, AEMeasurable.mk (f j) ⋯ a))
    (fun (a : Ω) ↦ (AEMeasurable.mk (f k) ⋯ a, AEMeasurable.mk (f l) ⋯ a)) κ μ
⊢ ∀ᵐ (a : α) ∂μ,
    Filter.EventuallyEq.{u_2, u_8} (β := β k × β l) (ae (F := Measure Ω) ((κ : (a : α) → Measure Ω) a : Measure Ω))
      (fun (a : Ω) ↦ (AEMeasurable.mk (f k) ⋯ a, AEMeasurable.mk (f l) ⋯ a)) fun (a : Ω) ↦ (f k a, f l a)
have h₁ : ∀ (a : α) (hk : ∀ᵐ (ω : Ω) ∂((κ : (a : α) → Measure Ω) a : Measure Ω), f k ω = AEMeasurable.mk (f k) sorry ω)
  (hl : ∀ᵐ (ω : Ω) ∂((κ : (a : α) → Measure Ω) a : Measure Ω), f l ω = AEMeasurable.mk (f l) sorry ω),
  Filter.EventuallyEq.{u_2, u_8} (β := β k × β l) (ae (F := Measure Ω) ((κ : (a : α) → Measure Ω) a : Measure Ω))
    (fun (a : Ω) ↦ (AEMeasurable.mk (f k) sorry a, AEMeasurable.mk (f l) sorry a)) fun (a : Ω) ↦ (f k a, f l a)
Tactic: filter_upwards [Measure.ae_ae_of_ae_comp (hf_meas k).ae_eq_mk,
      Measure.ae_ae_of_ae_comp (hf_meas l).ae_eq_mk] with a hk hl