-- 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
π : ι → Set (Set Ω)
a : ι
S : Finset ι
hp_ind : iIndepSets π κ μ
haS : a ∉ S
t1 t2 : Set Ω
s : Finset ι
hs_mem : s ⊆ S
ft1 : ι → Set Ω
hft1_mem : ∀ x ∈ s, ft1 x ∈ π x
ht1_eq : t1 = ⋂ x ∈ s, ft1 x
ht2_mem_pia : t2 ∈ π a
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 f : (n : ι) → Set Ω := fun (n : ι) ↦ if n = a then t2 else if n ∈ s then ft1 n else univ;
  ∀ (h_f_mem : ∀ n ∈ insert a s, f n ∈ π n),
    ∀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∞)
let₄ :
  let f : (n : ι) → Set Ω := fun (n : ι) ↦ if n = a then t2 else if n ∈ s then ft1 n else univ;
  ∀ (h_f_mem : ∀ n ∈ insert a s, f n ∈ π n) (h_f_mem_pi : ∀ n ∈ s, f n ∈ π n),
    ∀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∞)
let₅ :
  let f : (n : ι) → Set Ω := fun (n : ι) ↦ if n = a then t2 else if n ∈ s then ft1 n else univ;
  ∀ (h_f_mem : ∀ n ∈ insert a s, f n ∈ π n) (h_f_mem_pi : ∀ n ∈ s, f n ∈ π n) (h_t1 : t1 = ⋂ n ∈ s, f n),
    ∀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∞)
let₆ :
  let f : (n : ι) → Set Ω := fun (n : ι) ↦ if n = a then t2 else if n ∈ s then ft1 n else univ;
  ∀ (h_f_mem : ∀ n ∈ insert a s, f n ∈ π n) (h_f_mem_pi : ∀ n ∈ s, f n ∈ π n) (h_t1 : t1 = ⋂ n ∈ s, f n)
    (h_μ_t1 :
      ∀ᵐ (a' : α) ∂μ,
        Eq (α := ℝ≥0∞) ((((κ : (a : α) → Measure Ω) a' : Measure Ω) : (a : Set Ω) → ℝ≥0∞) t1 : ℝ≥0∞)
          (∏ n ∈ s, (((κ : (a : α) → Measure Ω) a' : Measure Ω) : (a : Set Ω) → ℝ≥0∞) (f 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∞)
let₇ :
  let f : (n : ι) → Set Ω := fun (n : ι) ↦ if n = a then t2 else if n ∈ s then ft1 n else univ;
  ∀ (h_f_mem : ∀ n ∈ insert a s, f n ∈ π n) (h_f_mem_pi : ∀ n ∈ s, f n ∈ π n) (h_t1 : t1 = ⋂ n ∈ s, f n)
    (h_μ_t1 :
      ∀ᵐ (a' : α) ∂μ,
        Eq (α := ℝ≥0∞) ((((κ : (a : α) → Measure Ω) a' : Measure Ω) : (a : Set Ω) → ℝ≥0∞) t1 : ℝ≥0∞)
          (∏ n ∈ s, (((κ : (a : α) → Measure Ω) a' : Measure Ω) : (a : Set Ω) → ℝ≥0∞) (f n) : ℝ≥0∞))
    (h_t2 : t2 = f a),
    ∀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∞)
let₈ :
  let f : (n : ι) → Set Ω := fun (n : ι) ↦ if n = a then t2 else if n ∈ s then ft1 n else univ;
  ∀ (h_f_mem : ∀ n ∈ insert a s, f n ∈ π n) (h_f_mem_pi : ∀ n ∈ s, f n ∈ π n) (h_t1 : t1 = ⋂ n ∈ s, f n)
    (h_μ_t1 :
      ∀ᵐ (a' : α) ∂μ,
        Eq (α := ℝ≥0∞) ((((κ : (a : α) → Measure Ω) a' : Measure Ω) : (a : Set Ω) → ℝ≥0∞) t1 : ℝ≥0∞)
          (∏ n ∈ s, (((κ : (a : α) → Measure Ω) a' : Measure Ω) : (a : Set Ω) → ℝ≥0∞) (f n) : ℝ≥0∞))
    (h_t2 : t2 = f a)
    (h_μ_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 ∈ insert a s, (((κ : (a : α) → Measure Ω) a' : Measure Ω) : (a : Set Ω) → ℝ≥0∞) (f 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∞)
let₉ :
  let f : (n : ι) → Set Ω := fun (n : ι) ↦ if n = a then t2 else if n ∈ s then ft1 n else univ;
  ∀ (h_f_mem : ∀ n ∈ insert a s, f n ∈ π n) (h_f_mem_pi : ∀ n ∈ s, f n ∈ π n) (h_t1 : t1 = ⋂ n ∈ s, f n)
    (h_μ_t1 :
      ∀ᵐ (a' : α) ∂μ,
        Eq (α := ℝ≥0∞) ((((κ : (a : α) → Measure Ω) a' : Measure Ω) : (a : Set Ω) → ℝ≥0∞) t1 : ℝ≥0∞)
          (∏ n ∈ s, (((κ : (a : α) → Measure Ω) a' : Measure Ω) : (a : Set Ω) → ℝ≥0∞) (f n) : ℝ≥0∞))
    (h_t2 : t2 = f a)
    (h_μ_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 ∈ insert a s, (((κ : (a : α) → Measure Ω) a' : Measure Ω) : (a : Set Ω) → ℝ≥0∞) (f n) : ℝ≥0∞))
    (has : a ∉ s),
    ∀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 f : (n : ι) → Set Ω := fun (n : ι) ↦ if n = a then t2 else if n ∈ s then ft1 n else univ;
  ∀ (h_f_mem : ∀ n ∈ insert a s, f n ∈ π n) (h_f_mem_pi : ∀ n ∈ s, f n ∈ π n) (h_t1 : t1 = ⋂ n ∈ s, f n)
    (h_μ_t1 :
      ∀ᵐ (a' : α) ∂μ,
        Eq (α := ℝ≥0∞) ((((κ : (a : α) → Measure Ω) a' : Measure Ω) : (a : Set Ω) → ℝ≥0∞) t1 : ℝ≥0∞)
          (∏ n ∈ s, (((κ : (a : α) → Measure Ω) a' : Measure Ω) : (a : Set Ω) → ℝ≥0∞) (f n) : ℝ≥0∞))
    (h_t2 : t2 = f a)
    (h_μ_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 ∈ insert a s, (((κ : (a : α) → Measure Ω) a' : Measure Ω) : (a : Set Ω) → ℝ≥0∞) (f n) : ℝ≥0∞))
    (has : a ∉ s) (a' : α)
    (ha1 :
      Eq (α := ℝ≥0∞) ((((κ : (a : α) → Measure Ω) a' : Measure Ω) : (a : Set Ω) → ℝ≥0∞) t1 : ℝ≥0∞)
        (∏ n ∈ s, (((κ : (a : α) → Measure Ω) a' : Measure Ω) : (a : Set Ω) → ℝ≥0∞) (f n) : ℝ≥0∞))
    (ha2 :
      Eq (α := ℝ≥0∞) ((((κ : (a : α) → Measure Ω) a' : Measure Ω) : (a : Set Ω) → ℝ≥0∞) (t1 ∩ t2) : ℝ≥0∞)
        (∏ n ∈ insert a s, (((κ : (a : α) → Measure Ω) a' : Measure Ω) : (a : Set Ω) → ℝ≥0∞) (f n) : 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 f := fun n => ite (n = a) t2 (ite (n ∈ s) (ft1 n) Set.univ)
  have h_f_mem : ∀ n ∈ insert a s, f n ∈ π n := by
    intro n hn_mem_insert
    dsimp only [f]
    rcases Finset.mem_insert.mp hn_mem_insert with hn_mem | hn_mem
    · simp [hn_mem, ht2_mem_pia]
    · grind
  have h_f_mem_pi : ∀ n ∈ s, f n ∈ π n := fun x hxS => h_f_mem x (by simp [hxS])
  have h_t1 : t1 = ⋂ n ∈ s, f n := by
    suffices h_forall : ∀ n ∈ s, f n = ft1 n by grind
    intro n hnS
    have hn_ne_a : n ≠ a := by rintro rfl; exact haS (hs_mem hnS)
    simp_rw [f, if_pos hnS, if_neg hn_ne_a]
  have h_μ_t1 : ∀ᵐ a' ∂μ, κ a' t1 = ∏ n ∈ s, κ a' (f n) := by
    filter_upwards [hp_ind s h_f_mem_pi] with a' ha'
    rw [h_t1, ← ha']
  have h_t2 : t2 = f a := by simp [f]
  have h_μ_inter : ∀ᵐ a' ∂μ, κ a' (t1 ∩ t2) = ∏ n ∈ insert a s, κ a' (f n) := by
    have h_t1_inter_t2 : t1 ∩ t2 = ⋂ n ∈ insert a s, f n := by
      rw [h_t1, h_t2, Finset.set_biInter_insert, Set.inter_comm]
    filter_upwards [hp_ind (insert a s) h_f_mem] with a' ha'
    rw [h_t1_inter_t2, ← ha']
  have has : a ∉ s := fun has_mem => haS (hs_mem has_mem)
  filter_upwards [h_μ_t1, h_μ_inter] with a' ha1 ha2
  rw [ha2, Finset.prod_insert has, h_t2, mul_comm, ha1]

Premises used: mul_comm