theorem extracted_formal_statement_14.{w, v, u} {Z : Type w} {ι : Type v} {f : ι → Z → ℝ} {Ω : Type u} {X : Ω → Z} {n : ℕ}
  [inst : Nonempty ι] {b : ℝ} (h𝓕' : ∀ (I : ι) (z : Z), |f I z| ≤ b) {c : ι → ℝ} {C : ℝ} (hC : ∀ (I : ι), |c I| ≤ C)
  (ω' : Fin n → Ω × Ω) :
  |2⁻¹ ^ n * ∑ σ, ⨆ I, ∑ i, ↑↑(σ i) * (f I (X (ω' i).1) - f I (X (ω' i).2)) + c I| ≤ ↑n * (b + b) + C := sorry

State before:
Z : Type w
ι : Type v
f : ι → Z → ℝ
Ω : Type u
X : Ω → Z
n : ℕ
inst✝ : Nonempty ι
b : ℝ
h𝓕' : ∀ (I : ι) (z : Z), |f I z| ≤ b
c : ι → ℝ
C : ℝ
hC : ∀ (I : ι), |c I| ≤ C
ω' : Fin n → Ω × Ω
⊢ |2⁻¹ ^ n * ∑ σ, ⨆ I, ∑ i, ↑↑(σ i) * (f I (X (ω' i).1) - f I (X (ω' i).2)) + c I| ≤ ↑n * (b + b) + C

Tactic: rw [abs_mul, abs_of_pos (by simp)]
    have :
  |∑ σ : Fin n → ({-1, 1} : Finset ℤ), ⨆ I : ι, ∑ i : Fin n, (σ i) * (f I (X (ω' i).1) - f I (X (ω' i).2)) + c I| ≤
    2 ^ n * (n * (b + b) + C) :=
  by
  calc
    _ ≤
        ∑ σ : Fin n → ({-1, 1} : Finset ℤ),
          |⨆ I : ι, ∑ i : Fin n, (σ i) * (f I (X (ω' i).1) - f I (X (ω' i).2)) + c I| :=
      by apply Finset.abs_sum_le_sum_abs
    _ ≤ ∑ σ : Fin n → ({-1, 1} : Finset ℤ), (n * (b + b) + C) :=
      by
      apply Finset.sum_le_sum
      intro σ _
      apply bound_isum h𝓕' hC
    _ = (Finset.univ : Finset (Fin n → ({-1, 1} : Finset ℤ))).card • (n * (b + b) + C) := by
      exact Finset.sum_const (n * (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 * (2 ^ n * (n * (b + b) + C)) := mul_le_mul_of_nonneg_left this (by simp)
  _ = _ := by simp

State after:

Relevant lemmas:
error