theorem extracted_formal_statement_200.{w, v, u} {Z : Type w} {ι : Type v} {f : ι → Z → ℝ} {Ω : Type u}
  [inst : MeasurableSpace Ω] {X : Ω → Z} {μ : Measure Ω} [inst_1 : IsProbabilityMeasure μ] {n_1 : ℕ}
  [inst_2 : Countable ι] [inst_3 : Nonempty ι] (h𝓕 : ∀ (I : ι), Measurable (f I ∘ X)) {b : ℝ}
  (h𝓕' : ∀ (I : ι) (z : Z), |f I z| ≤ b) (n : ℕ)
  (ih :
    ∀ (c : ι → ℝ),
      (∃ C, ∀ (I : ι), |c I| ≤ C) →
        (∫ (x : Fin n → Ω × Ω),
            (fun ω => ⨆ I, ∑ i, (f I (X (ω i).1) - f I (X (ω i).2)) + c I) x ∂Measure.pi fun x => μ.prod μ) =
          ∫ (x : Fin n → Ω × Ω),
            (fun ω => 2⁻¹ ^ n * ∑ σ, ⨆ I, ∑ i, ↑↑(σ i) * (f I (X (ω i).1) - f I (X (ω i).2)) + c I)
              x ∂Measure.pi fun x => μ.prod μ)
  (c : ι → ℝ) (C : ℝ) (hC : ∀ (I : ι), |c I| ≤ C) (ω : (Ω × Ω) × (Fin n → Ω × Ω)) (σ : Signs (n + 1)) (I : ι)
  (this :
    ∑ i, ↑↑(σ i) * (f I (X (Fin.snoc ω.2 ω.1 i).1) - f I (X (Fin.snoc ω.2 ω.1 i).2)) =
      ∑ i,
        Fin.snoc (fun i => ↑↑(σ ↑↑i) * (f I (X (ω.2 i).1) - f I (X (ω.2 i).2)))
          (↑↑(σ ↑n) * (f I (X ω.1.1) - f I (X ω.1.2))) i) :
  ∑ i, ↑↑(σ ↑↑i) * (f I (X (ω.2 i).1) - f I (X (ω.2 i).2)) + (↑↑(σ ↑n) * (f I (X ω.1.1) - f I (X ω.1.2)) + c I) =
    ∑ i, ↑↑(σ i) * (f I (X (Fin.snoc ω.2 ω.1 i).1) - f I (X (Fin.snoc ω.2 ω.1 i).2)) + c I := sorry

State before:
case h.h.e_a.e_f.h.h
Z : Type w
ι : Type v
f : ι → Z → ℝ
Ω : Type u
inst✝³ : MeasurableSpace Ω
X : Ω → Z
μ : Measure Ω
inst✝² : IsProbabilityMeasure μ
n✝ : ℕ
inst✝¹ : Countable ι
inst✝ : Nonempty ι
h𝓕 : ∀ (I : ι), Measurable (f I ∘ X)
b : ℝ
h𝓕' : ∀ (I : ι) (z : Z), |f I z| ≤ b
n : ℕ
ih :
  ∀ (c : ι → ℝ),
    (∃ C, ∀ (I : ι), |c I| ≤ C) →
      (∫ (x : Fin n → Ω × Ω),
          (fun ω => ⨆ I, ∑ i, (f I (X (ω i).1) - f I (X (ω i).2)) + c I) x ∂Measure.pi fun x => μ.prod μ) =
        ∫ (x : Fin n → Ω × Ω),
          (fun ω => 2⁻¹ ^ n * ∑ σ, ⨆ I, ∑ i, ↑↑(σ i) * (f I (X (ω i).1) - f I (X (ω i).2)) + c I)
            x ∂Measure.pi fun x => μ.prod μ
c : ι → ℝ
C : ℝ
hC : ∀ (I : ι), |c I| ≤ C
μ2n : Measure (Fin n → Ω × Ω) := Measure.pi fun x => μ.prod μ
ω : (Ω × Ω) × (Fin n → Ω × Ω)
σ : Signs (n + 1)
I : ι
this :
  ∑ i, ↑↑(σ i) * (f I (X (Fin.snoc ω.2 ω.1 i).1) - f I (X (Fin.snoc ω.2 ω.1 i).2)) =
    ∑ i,
      Fin.snoc (fun i => ↑↑(σ ↑↑i) * (f I (X (ω.2 i).1) - f I (X (ω.2 i).2)))
        (↑↑(σ ↑n) * (f I (X ω.1.1) - f I (X ω.1.2))) i
⊢ ∑ i, ↑↑(σ ↑↑i) * (f I (X (ω.2 i).1) - f I (X (ω.2 i).2)) + (↑↑(σ ↑n) * (f I (X ω.1.1) - f I (X ω.1.2)) + c I) =
    ∑ i, ↑↑(σ i) * (f I (X (Fin.snoc ω.2 ω.1 i).1) - f I (X (Fin.snoc ω.2 ω.1 i).2)) + c I

Tactic: rw [this, Fin.sum_snoc, add_assoc]

State after:

Relevant lemmas:
add_assoc : ∀ {G : Type u_1} [inst : AddSemigroup G] (a b c : G), a + b + c = a + (b + c)
Fin.sum_snoc : ∀ {β : Type u_2} [inst : AddCommMonoid β] {n : ℕ} (x : β) (f : Fin n → β),
  ∑ i, Fin.snoc f x i = ∑ i, f i + x