theorem extracted_formal_statement_365.{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 : (2 ^ 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)
⊢ (2 ^ n)⁻¹ * |∑ σ, ⨆ i, |∑ k, ↑↑(σ k) * f i (X (a.2 k))|| ≤ ↑n * b

Tactic: rw [inv_mul_le_iff₀]

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