theorem extracted_formal_statement_209.{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 : 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)
  (this :
    (∫ (x : Fin n → Ω × Ω), ⨆ I, ∑ x_1, f I (X (x x_1).1) - ∑ x_1, f I (X (x x_1).2) ∂Measure.pi fun x => μ.prod μ) =
      ∫ (x : Fin n → Ω × Ω),
        (2 ^ n)⁻¹ * ∑ x_1, ⨆ I, ∑ i, ↑↑(x_1 i) * (f I (X (x i).1) - f I (X (x i).2)) ∂Measure.pi fun x => μ.prod μ)
  (h_original_goal :
    (∫ (x : Fin n → Ω × Ω), (fun ω => ⨆ I, ∑ i, (f I (X (ω i).1) - f I (X (ω i).2))) 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)))
          x ∂Measure.pi fun x => μ.prod μ) :
  (∫ (x : Fin n → Ω × Ω), ⨆ I, ∑ x_1, f I (X (x x_1).1) - ∑ x_1, f I (X (x x_1).2) ∂Measure.pi fun x => μ.prod μ) =
    ∫ (x : Fin n → Ω × Ω),
      (2 ^ n)⁻¹ * ∑ σ, ⨆ I, ∑ i, ↑↑(σ i) * (f I (X (x i).1) - f I (X (x i).2)) ∂Measure.pi fun x => μ.prod μ := 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
this :
  (∫ (x : Fin n → Ω × Ω), ⨆ I, ∑ x_1, f I (X (x x_1).1) - ∑ x_1, f I (X (x x_1).2) ∂Measure.pi fun x => μ.prod μ) =
    ∫ (x : Fin n → Ω × Ω),
      (2 ^ n)⁻¹ * ∑ x_1, ⨆ I, ∑ i, ↑↑(x_1 i) * (f I (X (x i).1) - f I (X (x i).2)) ∂Measure.pi fun x => μ.prod μ
⊢ (∫ (x : Fin n → Ω × Ω), (fun ω => ⨆ I, ∑ i, (f I (X (ω i).1) - f I (X (ω i).2))) 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))) x ∂Measure.pi fun x => μ.prod μ

Tactic: simp only [Finset.sum_sub_distrib, inv_pow, Int.reduceNeg]

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
 this :
   (∫ (x : Fin n → Ω × Ω), ⨆ I, ∑ x_1, f I (X (x x_1).1) - ∑ x_1, f I (X (x x_1).2) ∂Measure.pi fun x => μ.prod μ) =
     ∫ (x : Fin n → Ω × Ω),
       (2 ^ n)⁻¹ * ∑ x_1, ⨆ I, ∑ i, ↑↑(x_1 i) * (f I (X (x i).1) - f I (X (x i).2)) ∂Measure.pi fun x => μ.prod μ
 ⊢ (∫ (x : Fin n → Ω × Ω), ⨆ I, ∑ x_1, f I (X (x x_1).1) - ∑ x_1, f I (X (x x_1).2) ∂Measure.pi fun x => μ.prod μ) =
     ∫ (x : Fin n → Ω × Ω),
       (2 ^ n)⁻¹ * ∑ σ, ⨆ I, ∑ i, ↑↑(σ i) * (f I (X (x i).1) - f I (X (x i).2)) ∂Measure.pi fun x => μ.prod μ]

Relevant lemmas:
inv_pow : ∀ {α : Type u_1} [inst : DivisionMonoid α] (a : α) (n : ℕ), a⁻¹ ^ n = (a ^ n)⁻¹
Finset.sum_sub_distrib : ∀ {α : Type u_3} {β : Type u_4} {s : Finset α} {f g : α → β} [inst : SubtractionCommMonoid β],
  ∑ x ∈ s, (f x - g x) = ∑ x ∈ s, f x - ∑ x ∈ s, g x