theorem extracted_formal_statement_369.{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)) (i : Signs n) (hi : i ∈ Finset.univ) (i_1 : ι)
  (h_original_goal : |(|∑ k, ↑↑(i k) * f i_1 (X (a.2 k))|)| ≤ ↑n * b) :
  |∑ k, ↑↑(i k) * f i_1 (X (a.2 k))| ≤ ↑n * b := sorry

State before:
case h.hf
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)
i : Signs n
hi : i ∈ Finset.univ
i_1 : ι
⊢ |(|∑ k, ↑↑(i k) * f i_1 (X (a.2 k))|)| ≤ ↑n * b

Tactic: simp only [Int.reduceNeg, abs_abs]

State after:
[case h.hf
 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)
 i : Signs n
 hi : i ∈ Finset.univ
 i_1 : ι
 ⊢ |∑ k, ↑↑(i k) * f i_1 (X (a.2 k))| ≤ ↑n * b]

Relevant lemmas:
abs_abs : ∀ {α : Type u_1} [inst : Lattice α] [inst_4 : AddGroup α] [inst_5 : AddLeftMono α] [inst_6 : AddRightMono α]
  (a : α), |(|a|)| = |a|