theorem extracted_formal_statement_125.{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] [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}
  {c : ι → ℝ} {C : ℝ} (hC : ∀ (I : ι), |c I| ≤ C) (ω' : Ω × Ω)
  (h_original_goal :
    |∫ (x : Fin n → Ω × Ω),
          2⁻¹ ^ n *
            ∑ σ,
              ⨆ I,
                ∑ i, ↑↑(σ i) * (f I (X (x i).1) - f I (X (x i).2)) +
                  (f I (X ω'.1) - f I (X ω'.2) + c I) ∂Measure.pi fun x => μ.prod μ| ≤
      ↑n * (b + b) + (b + b + C)) :
  ∀ᵐ (x : Fin n → Ω × Ω) ∂Measure.pi fun x => μ.prod μ,
    |2⁻¹ ^ n * ∑ σ, ⨆ I, ∑ i, ↑↑(σ i) * (f I (X (x i).1) - f I (X (x i).2)) + (f I (X ω'.1) - f I (X ω'.2) + c I)| ≤
      ↑n * (b + b) + (b + b + C) := sorry

State before:
case right.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
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
c : ι → ℝ
C : ℝ
hC : ∀ (I : ι), |c I| ≤ C
ω' : Ω × Ω
⊢ |∫ (x : Fin n → Ω × Ω),
        2⁻¹ ^ n *
          ∑ σ,
            ⨆ I,
              ∑ i, ↑↑(σ i) * (f I (X (x i).1) - f I (X (x i).2)) +
                (f I (X ω'.1) - f I (X ω'.2) + c I) ∂Measure.pi fun x => μ.prod μ| ≤
    ↑n * (b + b) + (b + b + C)

Tactic: apply abs_expectation_le_of_abs_le_const

State after:
[case right.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
 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
 c : ι → ℝ
 C : ℝ
 hC : ∀ (I : ι), |c I| ≤ C
 ω' : Ω × Ω
 ⊢ ∀ᵐ (x : Fin n → Ω × Ω) ∂Measure.pi fun x => μ.prod μ,
     |2⁻¹ ^ n * ∑ σ, ⨆ I, ∑ i, ↑↑(σ i) * (f I (X (x i).1) - f I (X (x i).2)) + (f I (X ω'.1) - f I (X ω'.2) + c I)| ≤
       ↑n * (b + b) + (b + b + C)]

Relevant lemmas:
abs_expectation_le_of_abs_le_const : ∀ {Ω : Type u_1} [inst : MeasurableSpace Ω] {μ : Measure Ω}
  [inst_4 : IsProbabilityMeasure μ] {f : Ω → ℝ} {C : ℝ}, (∀ᵐ (x : Ω) ∂μ, |f x| ≤ C) → |∫ (x : Ω), f x ∂μ| ≤ C