theorem extracted_formal_statement_269.{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] [inst_2 : Nonempty ι]
  [inst_3 : Countable ι] {X : Ω → Z} (hf : ∀ (i : ι), Measurable (f i ∘ X)) {b : ℝ}
  (hf' : ∀ (i : ι) (z : Z), |f i z| ≤ b) (ω : (Fin n → Ω) × (Fin n → Ω))
  (h_original_goal : (2 ^ n)⁻¹ * |∑ σ, ⨆ i, |∑ k, ↑↑(σ k) * f i (X (ω.1 k))|| ≤ ↑n * b) :
  |∑ σ, ⨆ i, |∑ k, ↑↑(σ k) * f i (X (ω.1 k))|| ≤ ↑n * b * 2 ^ n := sorry

State before:
case 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
inst✝¹ : Nonempty ι
inst✝ : Countable ι
X : Ω → Z
hf : ∀ (i : ι), Measurable (f i ∘ X)
b : ℝ
hf' : ∀ (i : ι) (z : Z), |f i z| ≤ b
ω : (Fin n → Ω) × (Fin n → Ω)
⊢ (2 ^ n)⁻¹ * |∑ σ, ⨆ i, |∑ k, ↑↑(σ k) * f i (X (ω.1 k))|| ≤ ↑n * b

Tactic: rw [inv_mul_le_iff₀' (by simp)]

State after:
[case 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
 inst✝¹ : Nonempty ι
 inst✝ : Countable ι
 X : Ω → Z
 hf : ∀ (i : ι), Measurable (f i ∘ X)
 b : ℝ
 hf' : ∀ (i : ι) (z : Z), |f i z| ≤ b
 ω : (Fin n → Ω) × (Fin n → Ω)
 ⊢ |∑ σ, ⨆ i, |∑ k, ↑↑(σ k) * f i (X (ω.1 k))|| ≤ ↑n * b * 2 ^ n]

Relevant lemmas:
inv_mul_le_iff₀' : ∀ {G₀ : Type u_2} [inst : CommGroupWithZero G₀] [inst_4 : PartialOrder G₀]
  [inst_5 : ZeroLEOneClass G₀] [inst_6 : PosMulReflectLT G₀] [inst_7 : PosMulMono G₀] {a b c : G₀},
  0 < c → (c⁻¹ * b ≤ a ↔ b ≤ a * c)