-- Module: Mathlib.Probability.Independence.Kernel.Indep
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
s : ι → Set (Set Ω)
S T : Set ι
h_indep : iIndepSets s κ μ
hST : Disjoint S T
t1 t2 : Set Ω
p1 : Finset ι
hp1 : (↑p1 : Set ι) ⊆ S
f1 : ι → Set Ω
ht1_m : ∀ x ∈ p1, f1 x ∈ s x
ht1_eq : t1 = ⋂ x ∈ p1, f1 x
p2 : Finset ι
hp2 : (↑p2 : Set ι) ⊆ T
f2 : ι → Set Ω
ht2_m : ∀ x ∈ p2, f2 x ∈ s x
ht2_eq : t2 = ⋂ x ∈ p2, f2 x
let let₁ let₂ :
  ∀ᵐ (a : α) ∂μ,
    (((κ : (a : α) → Measure Ω) a : Measure Ω) : (a : Set Ω) → ℝ≥0∞) (t1 ∩ t2) =
      HMul.hMul (α := ℝ≥0∞) (β := ℝ≥0∞) ((((κ : (a : α) → Measure Ω) a : Measure Ω) : (a : Set Ω) → ℝ≥0∞) t1 : ℝ≥0∞)
        ((((κ : (a : α) → Measure Ω) a : Measure Ω) : (a : Set Ω) → ℝ≥0∞) t2 : ℝ≥0∞)
let₃ :
  let g : (i : ι) → Set Ω := fun (i : ι) ↦ (if i ∈ p1 then f1 i else univ) ∩ if i ∈ p2 then f2 i else univ;
  ∀
    (h_P_inter :
      ∀ᵐ (a : α) ∂μ,
        Eq (α := ℝ≥0∞) ((((κ : (a : α) → Measure Ω) a : Measure Ω) : (a : Set Ω) → ℝ≥0∞) (t1 ∩ t2) : ℝ≥0∞)
          (∏ n ∈ p1 ∪ p2, (((κ : (a : α) → Measure Ω) a : Measure Ω) : (a : Set Ω) → ℝ≥0∞) (g n) : ℝ≥0∞)),
    ∀ᵐ (a : α) ∂μ,
      (((κ : (a : α) → Measure Ω) a : Measure Ω) : (a : Set Ω) → ℝ≥0∞) (t1 ∩ t2) =
        HMul.hMul (α := ℝ≥0∞) (β := ℝ≥0∞) ((((κ : (a : α) → Measure Ω) a : Measure Ω) : (a : Set Ω) → ℝ≥0∞) t1 : ℝ≥0∞)
          ((((κ : (a : α) → Measure Ω) a : Measure Ω) : (a : Set Ω) → ℝ≥0∞) t2 : ℝ≥0∞)
h :
  let g : (i : ι) → Set Ω := fun (i : ι) ↦ (if i ∈ p1 then f1 i else univ) ∩ if i ∈ p2 then f2 i else univ;
  ∀
    (h_P_inter :
      ∀ᵐ (a : α) ∂μ,
        Eq (α := ℝ≥0∞) ((((κ : (a : α) → Measure Ω) a : Measure Ω) : (a : Set Ω) → ℝ≥0∞) (t1 ∩ t2) : ℝ≥0∞)
          (∏ n ∈ p1 ∪ p2, (((κ : (a : α) → Measure Ω) a : Measure Ω) : (a : Set Ω) → ℝ≥0∞) (g n) : ℝ≥0∞))
    (a : α)
    (h_P_inter :
      Eq (α := ℝ≥0∞) ((((κ : (a : α) → Measure Ω) a : Measure Ω) : (a : Set Ω) → ℝ≥0∞) (t1 ∩ t2) : ℝ≥0∞)
        (∏ n ∈ p1 ∪ p2, (((κ : (a : α) → Measure Ω) a : Measure Ω) : (a : Set Ω) → ℝ≥0∞) (g n) : ℝ≥0∞))
    (ha1 :
      Eq (α := ℝ≥0∞) ((((κ : (a : α) → Measure Ω) a : Measure Ω) : (a : Set Ω) → ℝ≥0∞) (⋂ i ∈ p1, f1 i) : ℝ≥0∞)
        (∏ i ∈ p1, (((κ : (a : α) → Measure Ω) a : Measure Ω) : (a : Set Ω) → ℝ≥0∞) (f1 i) : ℝ≥0∞))
    (ha2 :
      Eq (α := ℝ≥0∞) ((((κ : (a : α) → Measure Ω) a : Measure Ω) : (a : Set Ω) → ℝ≥0∞) (⋂ i ∈ p2, f2 i) : ℝ≥0∞)
        (∏ i ∈ p2, (((κ : (a : α) → Measure Ω) a : Measure Ω) : (a : Set Ω) → ℝ≥0∞) (f2 i) : ℝ≥0∞))
    (h' : IsProbabilityMeasure.{u_2} (α := Ω) (m0 := _mΩ) ((κ : (a : α) → Measure Ω) a)),
    (((κ : (a : α) → Measure Ω) a : Measure Ω) : (a : Set Ω) → ℝ≥0∞) (t1 ∩ t2) =
      HMul.hMul (α := ℝ≥0∞) (β := ℝ≥0∞) ((((κ : (a : α) → Measure Ω) a : Measure Ω) : (a : Set Ω) → ℝ≥0∞) t1 : ℝ≥0∞)
        ((((κ : (a : α) → Measure Ω) a : Measure Ω) : (a : Set Ω) → ℝ≥0∞) t2 : ℝ≥0∞)
h₁ :
  let g : (i : ι) → Set Ω := fun (i : ι) ↦ (if i ∈ p1 then f1 i else univ) ∩ if i ∈ p2 then f2 i else univ;
  ∀
    (h_P_inter :
      ∀ᵐ (a : α) ∂μ,
        Eq (α := ℝ≥0∞) ((((κ : (a : α) → Measure Ω) a : Measure Ω) : (a : Set Ω) → ℝ≥0∞) (t1 ∩ t2) : ℝ≥0∞)
          (∏ n ∈ p1 ∪ p2, (((κ : (a : α) → Measure Ω) a : Measure Ω) : (a : Set Ω) → ℝ≥0∞) (g n) : ℝ≥0∞))
    (a : α)
    (h_P_inter :
      Eq (α := ℝ≥0∞) ((((κ : (a : α) → Measure Ω) a : Measure Ω) : (a : Set Ω) → ℝ≥0∞) (t1 ∩ t2) : ℝ≥0∞)
        (∏ n ∈ p1 ∪ p2, (((κ : (a : α) → Measure Ω) a : Measure Ω) : (a : Set Ω) → ℝ≥0∞) (g n) : ℝ≥0∞))
    (ha1 :
      Eq (α := ℝ≥0∞) ((((κ : (a : α) → Measure Ω) a : Measure Ω) : (a : Set Ω) → ℝ≥0∞) (⋂ i ∈ p1, f1 i) : ℝ≥0∞)
        (∏ i ∈ p1, (((κ : (a : α) → Measure Ω) a : Measure Ω) : (a : Set Ω) → ℝ≥0∞) (f1 i) : ℝ≥0∞))
    (ha2 :
      Eq (α := ℝ≥0∞) ((((κ : (a : α) → Measure Ω) a : Measure Ω) : (a : Set Ω) → ℝ≥0∞) (⋂ i ∈ p2, f2 i) : ℝ≥0∞)
        (∏ i ∈ p2, (((κ : (a : α) → Measure Ω) a : Measure Ω) : (a : Set Ω) → ℝ≥0∞) (f2 i) : ℝ≥0∞))
    (h' : IsProbabilityMeasure.{u_2} (α := Ω) (m0 := _mΩ) ((κ : (a : α) → Measure Ω) a))
    (h_μg :
      ∀ (n : ι),
        (((κ : (a : α) → Measure Ω) a : Measure Ω) : (a : Set Ω) → ℝ≥0∞) (g n) =
          HMul.hMul (α := ℝ≥0∞) (β := ℝ≥0∞)
            (if n ∈ p1 then (((κ : (a : α) → Measure Ω) a : Measure Ω) : (a : Set Ω) → ℝ≥0∞) (f1 n) else 1 : ℝ≥0∞)
            (if n ∈ p2 then (((κ : (a : α) → Measure Ω) a : Measure Ω) : (a : Set Ω) → ℝ≥0∞) (f2 n) else 1 : ℝ≥0∞)),
    (((κ : (a : α) → Measure Ω) a : Measure Ω) : (a : Set Ω) → ℝ≥0∞) (t1 ∩ t2) =
      HMul.hMul (α := ℝ≥0∞) (β := ℝ≥0∞) ((((κ : (a : α) → Measure Ω) a : Measure Ω) : (a : Set Ω) → ℝ≥0∞) t1 : ℝ≥0∞)
        ((((κ : (a : α) → Measure Ω) a : Measure Ω) : (a : Set Ω) → ℝ≥0∞) t2 : ℝ≥0∞)
⊢ ∀ᵐ (a : α) ∂μ,
    (((κ : (a : α) → Measure Ω) a : Measure Ω) : (a : Set Ω) → ℝ≥0∞) (t1 ∩ t2) =
      HMul.hMul (α := ℝ≥0∞) (β := ℝ≥0∞) ((((κ : (a : α) → Measure Ω) a : Measure Ω) : (a : Set Ω) → ℝ≥0∞) t1 : ℝ≥0∞)
        ((((κ : (a : α) → Measure Ω) a : Measure Ω) : (a : Set Ω) → ℝ≥0∞) t2 : ℝ≥0∞)

Tactic:
classical
  let g i := ite (i ∈ p1) (f1 i) Set.univ ∩ ite (i ∈ p2) (f2 i) Set.univ
  have h_P_inter : ∀ᵐ a ∂μ, κ a (t1 ∩ t2) = ∏ n ∈ p1 ∪ p2, κ a (g n) := by
    have hgm : ∀ i ∈ p1 ∪ p2, g i ∈ s i := by
      intro i hi_mem_union
      rw [Finset.mem_union] at hi_mem_union
      rcases hi_mem_union with hi1 | hi2
      · have hi2 : i ∉ p2 := fun hip2 => Set.disjoint_left.mp hST (hp1 hi1) (hp2 hip2)
        simp_rw [g, if_pos hi1, if_neg hi2, Set.inter_univ]
        exact ht1_m i hi1
      · have hi1 : i ∉ p1 := fun hip1 => Set.disjoint_right.mp hST (hp2 hi2) (hp1 hip1)
        simp_rw [g, if_neg hi1, if_pos hi2, Set.univ_inter]
        exact ht2_m i hi2
    have h_p1_inter_p2 :
      ((⋂ x ∈ p1, f1 x) ∩ ⋂ x ∈ p2, f2 x) =
        ⋂ i ∈ p1 ∪ p2, ite (i ∈ p1) (f1 i) Set.univ ∩ ite (i ∈ p2) (f2 i) Set.univ := by
      ext1 x
      simp only [Set.mem_ite_univ_right, Set.mem_inter_iff, Set.mem_iInter, Finset.mem_union]
      exact
        ⟨fun h i _ => ⟨h.1 i, h.2 i⟩, fun h =>
          ⟨fun i hi => (h i (Or.inl hi)).1 hi, fun i hi => (h i (Or.inr hi)).2 hi⟩⟩
    filter_upwards [h_indep _ hgm] with a ha
    rw [ht1_eq, ht2_eq, h_p1_inter_p2, ← ha]
  filter_upwards [h_P_inter, h_indep p1 ht1_m, h_indep p2 ht2_m, h_indep.ae_isProbabilityMeasure]
    with a h_P_inter ha1 ha2 h'
  have h_μg : ∀ n, κ a (g n) = (ite (n ∈ p1) (κ a (f1 n)) 1) * (ite (n ∈ p2) (κ a (f2 n)) 1) := by
    intro n
    dsimp only [g]
    split_ifs with h1 h2
    · exact absurd rfl (Set.disjoint_iff_forall_ne.mp hST (hp1 h1) (hp2 h2))
    all_goals simp only [measure_univ, one_mul, mul_one, Set.inter_univ, Set.univ_inter]
  simp_rw [h_P_inter, h_μg, Finset.prod_mul_distrib,
    Finset.prod_ite_mem (p1 ∪ p2) p1 (fun x ↦ κ a (f1 x)), Finset.union_inter_cancel_left,
    Finset.prod_ite_mem (p1 ∪ p2) p2 (fun x => κ a (f2 x)), Finset.union_inter_cancel_right, ht1_eq,
      ← ha1, ht2_eq, ← ha2]

Premises used: Finset.prod_mul_distrib, Finset.union_inter_cancel_left, Finset.union_inter_cancel_right, ProbabilityTheory.Kernel.iIndepSets.ae_isProbabilityMeasure