theorem extracted_formal_statement_195.{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)
  (i : Fin (n + 1)) (h : ↑i < n)
  (h_original_goal :
    ↑↑(σ i) *
        (f I (X (if h : ↑i < n then ω.2 (i.castLT h) else ω.1).1) -
          f I (X (if h : ↑i < n then ω.2 (i.castLT h) else ω.1).2)) =
      if h : ↑i < n then ↑↑(σ ↑↑i) * (f I (X (ω.2 (i.castLT h)).1) - f I (X (ω.2 (i.castLT h)).2))
      else ↑↑(σ ↑n) * (f I (X ω.1.1) - f I (X ω.1.2))) :
  ↑↑(σ i) * (f I (X (ω.2 (i.castLT h)).1) - f I (X (ω.2 (i.castLT h)).2)) =
    ↑↑(σ ↑↑i) * (f I (X (ω.2 (i.castLT h)).1) - f I (X (ω.2 (i.castLT h)).2)) := sorry

State before:
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
i : Fin (n + 1)
h : ↑i < n
⊢ ↑↑(σ i) *
      (f I (X (if h : ↑i < n then ω.2 (i.castLT h) else ω.1).1) -
        f I (X (if h : ↑i < n then ω.2 (i.castLT h) else ω.1).2)) =
    if h : ↑i < n then ↑↑(σ ↑↑i) * (f I (X (ω.2 (i.castLT h)).1) - f I (X (ω.2 (i.castLT h)).2))
    else ↑↑(σ ↑n) * (f I (X ω.1.1) - f I (X ω.1.2))

Tactic: rw [dif_pos h, dif_pos h]

State after:
[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
 i : Fin (n + 1)
 h : ↑i < n
 ⊢ ↑↑(σ i) * (f I (X (ω.2 (i.castLT h)).1) - f I (X (ω.2 (i.castLT h)).2)) =
     ↑↑(σ ↑↑i) * (f I (X (ω.2 (i.castLT h)).1) - f I (X (ω.2 (i.castLT h)).2))]

Relevant lemmas:
dif_pos : ∀ {c : Prop} {h : Decidable c} (hc : c) {α : Sort u} {t : c → α} {e : ¬c → α}, dite c t e = t hc