theorem extracted_formal_statement_4.{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 *
        ((2⁻¹ ^ n *
            ∑ σ, ⨆ I, ∑ i, ↑↑(σ i) * (f I (X (ω' i).1) - f I (X (ω' i).2)) + (f I (X ω.2) - f I (X ω.1) + c I)) +
          2⁻¹ ^ n *
            ∑ σ, ⨆ I, ∑ i, ↑↑(σ i) * (f I (X (ω' i).1) - f I (X (ω' i).2)) + (f I (X ω.1) - f I (X ω.2) + c I)) =
      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 * (2 ^ n)⁻¹ = (2 ^ (n + 1))⁻¹ := 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 *
      ((2⁻¹ ^ n * ∑ σ, ⨆ I, ∑ i, ↑↑(σ i) * (f I (X (ω' i).1) - f I (X (ω' i).2)) + (f I (X ω.2) - f I (X ω.1) + c I)) +
        2⁻¹ ^ n * ∑ σ, ⨆ I, ∑ i, ↑↑(σ i) * (f I (X (ω' i).1) - f I (X (ω' i).2)) + (f I (X ω.1) - f I (X ω.2) + c I)) =
    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)

Tactic: rw [← mul_add, ← mul_assoc]
    simp only [inv_pow, Int.reduceNeg, Finset.mem_singleton, reduceCtorEq, not_false_eq_true, Finset.sum_insert,
  Int.cast_neg, Int.cast_one, neg_mul, one_mul, neg_sub, Finset.sum_singleton, mul_eq_mul_right_iff]
    left

State after:
[case h
 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 * (2 ^ n)⁻¹ = (2 ^ (n + 1))⁻¹]

Relevant lemmas:
inv_pow : ∀ {α : Type u_1} [inst : DivisionMonoid α] (a : α) (n : ℕ), a⁻¹ ^ n = (a ^ n)⁻¹
mul_add : ∀ {R : Type v} [inst : Mul R] [inst_1 : Add R] [inst_2 : LeftDistribClass R] (a b c : R),
  a * (b + c) = a * b + a * c
neg_sub : ∀ {α : Type u_1} [inst : SubtractionMonoid α] (a b : α), -(a - b) = b - a
mul_assoc : ∀ {G : Type u_1} [inst : Semigroup G] (a b c : G), a * b * c = a * (b * c)
Finset.mem_singleton : ∀ {α : Type u_1} {a b : α}, b ∈ {a} ↔ b = a
Int.cast_neg : ∀ {R : Type u} [inst : AddGroupWithOne R] (n : ℤ), ↑(-n) = -↑n
mul_eq_mul_right_iff : ∀ {M₀ : Type u_1} [inst : CancelMonoidWithZero M₀] {a b c : M₀}, a * c = b * c ↔ a = b ∨ c = 0
not_false_eq_true : (¬False) = True
Int.cast_one : ∀ {R : Type u} [inst : AddGroupWithOne R], ↑1 = 1
neg_mul : ∀ {α : Type u} [inst : Mul α] [inst_1 : HasDistribNeg α] (a b : α), -a * b = -(a * b)
Finset.sum_singleton : ∀ {α : Type u_3} {β : Type u_4} [inst : AddCommMonoid β] (f : α → β) (a : α),
  ∑ x ∈ {a}, f x = f a
Finset.sum_insert : ∀ {α : Type u_3} {β : Type u_4} {s : Finset α} {a : α} {f : α → β} [inst : AddCommMonoid β]
  [inst_1 : DecidableEq α], a ∉ s → ∑ x ∈ insert a s, f x = f a + ∑ x ∈ s, f x
one_mul : ∀ {M : Type u} [inst : MulOneClass M] (a : M), 1 * a = a