theorem extracted_formal_statement_21.{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) × (Fin n → Ω × Ω)) :
  |2⁻¹ ^ (n + 1) *
        ∑ σ,
          ⨆ 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)| ≤
    ↑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) × (Fin n → Ω × Ω)
⊢ |2⁻¹ ^ (n + 1) *
        ∑ σ,
          ⨆ 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)| ≤
    ↑n * (b + b) + (b + b + C)

Tactic: rw [abs_mul, abs_of_pos (by simp)]
    have :
  |∑ σ : Fin (n + 1) → ({-1, 1} : Finset ℤ),
        ⨆ I : ι,
          ∑ i : Fin n, (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)| ≤
    2 ^ (n + 1) * (n * (b + b) + (b + b + C)) :=
  by
  calc
    _ ≤
        ∑ σ : Fin (n + 1) → ({-1, 1} : Finset ℤ),
          |⨆ I : ι,
              ∑ i : Fin n, (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)| :=
      by apply Finset.abs_sum_le_sum_abs
    _ ≤ ∑ σ : Fin (n + 1) → ({-1, 1} : Finset ℤ), (n * (b + b) + (b + b + C)) :=
      by
      apply Finset.sum_le_sum
      intro σ _
      apply bound_isum h𝓕'
      exact boundedness₁ h𝓕' hC ω.1 (σ n)
    _ = (Finset.univ : Finset (Fin (n + 1) → ({-1, 1} : Finset ℤ))).card • (n * (b + b) + (b + b + C)) :=
      (Finset.sum_const (n * (b + b) + (b + b + C)))
    _ = _ :=
      by
      simp only [Int.reduceNeg, Finset.card_univ, Finset.mem_insert, Finset.mem_singleton, Fintype.card_pi,
        Fintype.card_coe, reduceCtorEq, not_false_eq_true, Finset.card_insert_of_not_mem, Finset.card_singleton,
        Nat.reduceAdd, Finset.prod_const, Fintype.card_fin, smul_add, nsmul_eq_mul, Nat.cast_pow, Nat.cast_ofNat]
      ring_nf
    calc
  _ ≤ 2⁻¹ ^ (n + 1) * (2 ^ (n + 1) * (n * (b + b) + (b + b + C))) := mul_le_mul_of_nonneg_left this (by simp)
  _ = _ := by simp

State after:

Relevant lemmas:
error