theorem extracted_formal_statement_57.{u_1, w, u} {Ω : Type u} [inst : MeasurableSpace Ω] {Z : Type w} {n : ℕ}
  {ι : Type u_1} {f : ι → Z → ℝ} {μ : Measure Ω} [inst_1 : IsProbabilityMeasure μ] {X : Ω → Z} [inst_2 : Nonempty ι]
  [inst_3 : Countable ι] (hf : ∀ (i : ι), Measurable (f i ∘ X)) {b : ℝ} (hf' : ∀ (i : ι) (z : Z), |f i z| ≤ b)
  (a : (Fin n → Ω) × (Fin n → Ω))
  (h_original_goal : ‖(↑(Fintype.card (Signs n)))⁻¹ * ∑ σ, ⨆ i, |∑ k, ↑↑(σ k) * f i (X (a.2 k))|‖ ≤ ↑n * b) :
  |∑ σ, ⨆ i, |∑ k, ↑↑(σ k) * f i (X (a.2 k))|| ≤ 2 ^ n * (↑n * b) := sorry

State before:
case right.h
Ω : Type u
inst✝³ : MeasurableSpace Ω
Z : Type w
n : ℕ
ι : Type u_1
f : ι → Z → ℝ
μ : Measure Ω
inst✝² : IsProbabilityMeasure μ
X : Ω → Z
inst✝¹ : Nonempty ι
inst✝ : Countable ι
hf : ∀ (i : ι), Measurable (f i ∘ X)
b : ℝ
hf' : ∀ (i : ι) (z : Z), |f i z| ≤ b
a : (Fin n → Ω) × (Fin n → Ω)
⊢ ‖(↑(Fintype.card (Signs n)))⁻¹ * ∑ σ, ⨆ i, |∑ k, ↑↑(σ k) * f i (X (a.2 k))|‖ ≤ ↑n * b

Tactic: simp only [Signs.card, Nat.cast_pow, Nat.cast_ofNat, Int.reduceNeg, norm_mul, norm_inv, norm_pow,
  norm_ofNat, norm_eq_abs]
    rw [inv_mul_le_iff₀]
    have l0 : |(∑ σ : Signs n, (⨆ i, |(∑ k : Fin n, (σ k : ℝ) * f i (X (a.2 k)))|))| ≤ ↑(2 ^ n) * (↑n * b) := by
  calc
    _ ≤ ∑ σ : Signs n, |(⨆ i, |∑ k : Fin n, (σ k : ℝ) * f i (X (a.2 k))|)| := by apply Finset.abs_sum_le_sum_abs
    _ ≤ ∑ σ : Signs n, ↑n * b :=
      by
      to_theorem apply Finset.sum_le_sum intro i hi apply abs_iSup_le
      intro i_1
      to_theorem
        simp only [Int.reduceNeg, abs_abs]have l1 : |∑ k : Fin n, (i k : ℝ) * f i_1 (X (a.2 k))| ≤ ↑n * b := by
          calc
            _ ≤ ∑ k : Fin n, |(i k : ℝ) * f i_1 (X (a.2 k))| := by apply Finset.abs_sum_le_sum_abs
            _ ≤ ∑ k : Fin n, b := by
              apply Finset.sum_le_sum
              intro i_2 hi_2
              rw [abs_mul]
              rw [abs_sigma (i i_2)]
              simp only [one_mul, ge_iff_le]
              exact hf' i_1 (X (a.2 i_2))
            _ ≤ ↑n * b := by simp exact l1
    _ = Fintype.card (Signs n) * (↑n * b) := by simp
    _ = 2 ^ n * (↑n * b) := by simp

State after:
[case right.h
 Ω : Type u
 inst✝³ : MeasurableSpace Ω
 Z : Type w
 n : ℕ
 ι : Type u_1
 f : ι → Z → ℝ
 μ : Measure Ω
 inst✝² : IsProbabilityMeasure μ
 X : Ω → Z
 inst✝¹ : Nonempty ι
 inst✝ : Countable ι
 hf : ∀ (i : ι), Measurable (f i ∘ X)
 b : ℝ
 hf' : ∀ (i : ι) (z : Z), |f i z| ≤ b
 a : (Fin n → Ω) × (Fin n → Ω)
 l0 : |∑ σ, ⨆ i, |∑ k, ↑↑(σ k) * f i (X (a.2 k))|| ≤ 2 ^ n * (↑n * b)
 ⊢ |∑ σ, ⨆ i, |∑ k, ↑↑(σ k) * f i (X (a.2 k))|| ≤ 2 ^ n * (↑n * b),
 case right.h
 Ω : Type u
 inst✝³ : MeasurableSpace Ω
 Z : Type w
 n : ℕ
 ι : Type u_1
 f : ι → Z → ℝ
 μ : Measure Ω
 inst✝² : IsProbabilityMeasure μ
 X : Ω → Z
 inst✝¹ : Nonempty ι
 inst✝ : Countable ι
 hf : ∀ (i : ι), Measurable (f i ∘ X)
 b : ℝ
 hf' : ∀ (i : ι) (z : Z), |f i z| ≤ b
 a : (Fin n → Ω) × (Fin n → Ω)
 ⊢ 0 < 2 ^ n]

Relevant lemmas:
error