theorem extracted_formal_statement_44.{w, v, u} {Z : Type w} {ι : Type v} {f : ι → Z → ℝ} {Ω : Type u} {X : Ω → Z} {n : ℕ}
  (ω : Ω × Ω) (ω' : Fin n → Ω × Ω) {c : ι → ℝ}
  (h_original_goal :
    2⁻¹ ^ (n + 1) *
        ∑ σ' ∈ {-1, 1},
          ∑ σ, ⨆ I, ∑ i, ↑↑(σ i) * (f I (X (ω' i).1) - f I (X (ω' i).2)) + (↑σ' * (f I (X ω.1) - f I (X ω.2)) + c I) =
      2⁻¹ ^ (n + 1) *
        ∑ σ,
          ⨆ I,
            ∑ i, ↑↑(σ ↑↑i) * (f I (X (ω' i).1) - f I (X (ω' i).2)) + (↑↑(σ ↑n) * (f I (X ω.1) - f I (X ω.2)) + c I)) :
  2⁻¹ ^ (n + 1) *
      ∑ σ,
        ⨆ I,
          ∑ i, ↑↑(Fin.init σ i) * (f I (X (ω' i).1) - f I (X (ω' i).2)) +
            (↑↑(σ ↑n) * (f I (X ω.1) - f I (X ω.2)) + c I) =
    2⁻¹ ^ (n + 1) *
      ∑ σ,
        ⨆ I,
          ∑ i, ↑↑(σ ↑↑i) * (f I (X (ω' i).1) - f I (X (ω' i).2)) +
            (↑↑(σ ↑n) * (f I (X ω.1) - f I (X ω.2)) + c I) := sorry

State before:
Z : Type w
ι : Type v
f : ι → Z → ℝ
Ω : Type u
X : Ω → Z
n : ℕ
ω : Ω × Ω
ω' : Fin n → Ω × Ω
c : ι → ℝ
⊢ 2⁻¹ ^ (n + 1) *
      ∑ σ' ∈ {-1, 1},
        ∑ σ, ⨆ I, ∑ i, ↑↑(σ i) * (f I (X (ω' i).1) - f I (X (ω' i).2)) + (↑σ' * (f I (X ω.1) - f I (X ω.2)) + c I) =
    2⁻¹ ^ (n + 1) *
      ∑ σ, ⨆ I, ∑ i, ↑↑(σ ↑↑i) * (f I (X (ω' i).1) - f I (X (ω' i).2)) + (↑↑(σ ↑n) * (f I (X ω.1) - f I (X ω.2)) + c I)

Tactic: rw [sigma_eq]

State after:
[Z : Type w
 ι : Type v
 f : ι → Z → ℝ
 Ω : Type u
 X : Ω → Z
 n : ℕ
 ω : Ω × Ω
 ω' : Fin n → Ω × Ω
 c : ι → ℝ
 ⊢ 2⁻¹ ^ (n + 1) *
       ∑ σ,
         ⨆ I,
           ∑ i, ↑↑(Fin.init σ i) * (f I (X (ω' i).1) - f I (X (ω' i).2)) +
             (↑↑(σ ↑n) * (f I (X ω.1) - f I (X ω.2)) + c I) =
     2⁻¹ ^ (n + 1) *
       ∑ σ,
         ⨆ I, ∑ i, ↑↑(σ ↑↑i) * (f I (X (ω' i).1) - f I (X (ω' i).2)) + (↑↑(σ ↑n) * (f I (X ω.1) - f I (X ω.2)) + c I)]

Relevant lemmas:
sigma_eq : ∀ {n : ℕ} (f : ℤ → Signs n → ℝ), ∑ σ' ∈ {-1, 1}, ∑ σ, f σ' σ = ∑ σ, f (↑(σ ↑n)) (Fin.init σ)