theorem extracted_formal_statement_188.{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) :
  let μ2n := Measure.pi fun x => μ.prod μ;
  ∫ (x : Ω × Ω),
        (fun ω =>
            2⁻¹ *
              (∫ (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)) + (f I (X ω.2) - f I (X ω.1) + c I))
                    x ∂μ2n +
                ∫ (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)) + (f I (X ω.1) - f I (X ω.2) + c I))
                    x ∂μ2n))
          x ∂μ.prod μ =
      ∫ (x : Ω × Ω),
        (fun ω =>
            ∫ (x : Fin n → Ω × Ω),
              (fun ω' =>
                  2⁻¹ ^ (n + 1) *
                    ∑ σ,
                      ⨆ I,
                        ∑ i, ↑↑(σ ↑↑i) * (f I (X (ω' i).1) - f I (X (ω' i).2)) +
                          (↑↑(σ ↑n) * (f I (X ω.1) - f I (X ω.2)) + c I))
                x ∂μ2n)
          x ∂μ.prod μ →
    (fun x =>
        (fun ω =>
            2⁻¹ *
              (∫ (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)) + (f I (X ω.2) - f I (X ω.1) + c I))
                    x ∂μ2n +
                ∫ (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)) + (f I (X ω.1) - f I (X ω.2) + c I))
                    x ∂μ2n))
          x) =
      fun x =>
      (fun ω =>
          ∫ (x : Fin n → Ω × Ω),
            (fun ω' =>
                2⁻¹ ^ (n + 1) *
                  ∑ σ,
                    ⨆ I,
                      ∑ i, ↑↑(σ ↑↑i) * (f I (X (ω' i).1) - f I (X (ω' i).2)) +
                        (↑↑(σ ↑n) * (f I (X ω.1) - f I (X ω.2)) + c I))
              x ∂μ2n)
        x := 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
⊢ ∫ (x : Ω × Ω),
      (fun ω =>
          2⁻¹ *
            (∫ (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)) + (f I (X ω.2) - f I (X ω.1) + c I))
                  x ∂μ2n +
              ∫ (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)) + (f I (X ω.1) - f I (X ω.2) + c I))
                  x ∂μ2n))
        x ∂μ.prod μ =
    ∫ (x : Ω × Ω),
      (fun ω =>
          ∫ (x : Fin n → Ω × Ω),
            (fun ω' =>
                2⁻¹ ^ (n + 1) *
                  ∑ σ,
                    ⨆ I,
                      ∑ i, ↑↑(σ ↑↑i) * (f I (X (ω' i).1) - f I (X (ω' i).2)) +
                        (↑↑(σ ↑n) * (f I (X ω.1) - f I (X ω.2)) + c I))
              x ∂μ2n)
        x ∂μ.prod μ

Tactic: apply congr_arg

State after:
[case 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
 ⊢ (fun x =>
       (fun ω =>
           2⁻¹ *
             (∫ (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)) + (f I (X ω.2) - f I (X ω.1) + c I))
                   x ∂μ2n +
               ∫ (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)) + (f I (X ω.1) - f I (X ω.2) + c I))
                   x ∂μ2n))
         x) =
     fun x =>
     (fun ω =>
         ∫ (x : Fin n → Ω × Ω),
           (fun ω' =>
               2⁻¹ ^ (n + 1) *
                 ∑ σ,
                   ⨆ I,
                     ∑ i, ↑↑(σ ↑↑i) * (f I (X (ω' i).1) - f I (X (ω' i).2)) +
                       (↑↑(σ ↑n) * (f I (X ω.1) - f I (X ω.2)) + c I))
             x ∂μ2n)
       x]

Relevant lemmas:
congr_arg : ∀ {α : Sort u} {β : Sort v} {a₁ a₂ : α} (f : α → β), a₁ = a₂ → f a₁ = f a₂