theorem extracted_formal_statement_2.{u_1, w, u} {Ω : Type u} {Z : Type w} {n : ℕ} {ι : Type u_1} {f : ι → Z → ℝ}
  [inst : Nonempty ι] {X : Ω → Z} {b : ℝ} (hf' : ∀ (i : ι) (z : Z), |f i z| ≤ b) (ω : Fin n → Ω × Ω)
  (h_original_goal :
    (↑(Fintype.card (Signs n)))⁻¹ * ∑ σ, ⨆ i, |∑ k, ↑↑(σ k) * (f i (X (ω k).1) - f i (X (ω k).2))| ≤
      ((↑(Fintype.card (Signs n)))⁻¹ * ∑ σ, ⨆ i, |∑ k, ↑↑(σ k) * f i (X (ω k).1)|) +
        (↑(Fintype.card (Signs n)))⁻¹ * ∑ σ, ⨆ i, |∑ k, ↑↑(σ k) * f i (X (ω k).2)|) :
  0 ≤ (↑(Fintype.card (Signs n)))⁻¹ := sorry

State before:
Ω : Type u
Z : Type w
n : ℕ
ι : Type u_1
f : ι → Z → ℝ
inst✝ : Nonempty ι
X : Ω → Z
b : ℝ
hf' : ∀ (i : ι) (z : Z), |f i z| ≤ b
ω : Fin n → Ω × Ω
⊢ (↑(Fintype.card (Signs n)))⁻¹ * ∑ σ, ⨆ i, |∑ k, ↑↑(σ k) * (f i (X (ω k).1) - f i (X (ω k).2))| ≤
    ((↑(Fintype.card (Signs n)))⁻¹ * ∑ σ, ⨆ i, |∑ k, ↑↑(σ k) * f i (X (ω k).1)|) +
      (↑(Fintype.card (Signs n)))⁻¹ * ∑ σ, ⨆ i, |∑ k, ↑↑(σ k) * f i (X (ω k).2)|

Tactic: rw [← mul_add]
    apply mul_le_mul_of_nonneg_left
    .
  rw [← Finset.sum_add_distrib]
  to_theorem
    apply Finset.sum_le_sum intro σ _ calc
      _ ≤ ⨆ i, |∑ k : Fin n, (σ k : ℝ) * f i (X (ω k).1)| + |∑ k : Fin n, (σ k : ℝ) * f i (X (ω k).2)| :=
        by
        to_theorem
          apply ciSup_mono·
            refine BddAbove.range_add ?B.hf ?B.hg
            rw [bddAbove_def]
            use n * b
            intro y hy
            simp only [Int.reduceNeg, Set.mem_range] at hy
            obtain ⟨y_1, hy'⟩ := hy
            rw [<- hy']
            · apply extracted_1_aux hf'
            · rw [bddAbove_def]
              use n * b
              intro y hy
              simp only [Int.reduceNeg, Set.mem_range] at hy
              obtain ⟨i, hi⟩ := hy
              rw [← hi]
              apply extracted_1_aux hf' intro i
        convert abs_sub _ _
        to_theorem rw [← Finset.sum_sub_distrib]congr ext k
        exact mul_sub_left_distrib ((σ k : ℝ)) (f i (X (ω k).1)) (f i (X (ω k).2))
      _ ≤ _ := by to_theorem apply ciSup_add_le_add_ciSup· apply extracted_1_aux' hf'· apply extracted_1_aux' hf'

State after:
[case a0
 Ω : Type u
 Z : Type w
 n : ℕ
 ι : Type u_1
 f : ι → Z → ℝ
 inst✝ : Nonempty ι
 X : Ω → Z
 b : ℝ
 hf' : ∀ (i : ι) (z : Z), |f i z| ≤ b
 ω : Fin n → Ω × Ω
 ⊢ 0 ≤ (↑(Fintype.card (Signs n)))⁻¹]

Relevant lemmas:
error