theorem extracted_formal_statement_45.{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,
          ⨆ 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)) :
  ∑ x,
      ⨆ I,
        ∑ i, ↑↑(Fin.init x i) * (f I (X (ω' i).1) - f I (X (ω' i).2)) +
          (↑↑(x (Fin.last n)) * (f I (X ω.1) - f I (X ω.2)) + c I) =
    ∑ x,
      ⨆ I,
        ∑ x_1, ↑↑(x x_1.castSucc) * (f I (X (ω' x_1).1) - f I (X (ω' x_1).2)) +
          (↑↑(x (Fin.last 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,
        ⨆ 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)

Tactic: simp only [inv_pow, Int.reduceNeg, Fin.natCast_eq_last, Fin.coe_eq_castSucc, mul_eq_mul_left_iff,
  inv_eq_zero, ne_eq, AddLeftCancelMonoid.add_eq_zero, one_ne_zero, and_false, not_false_eq_true, pow_eq_zero_iff,
  OfNat.ofNat_ne_zero, or_false]

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
 ⊢ ∑ x,
       ⨆ I,
         ∑ i, ↑↑(Fin.init x i) * (f I (X (ω' i).1) - f I (X (ω' i).2)) +
           (↑↑(x (Fin.last n)) * (f I (X ω.1) - f I (X ω.2)) + c I) =
     ∑ x,
       ⨆ I,
         ∑ x_1, ↑↑(x x_1.castSucc) * (f I (X (ω' x_1).1) - f I (X (ω' x_1).2)) +
           (↑↑(x (Fin.last n)) * (f I (X ω.1) - f I (X ω.2)) + c I)]

Relevant lemmas:
OfNat.ofNat_ne_zero : ∀ {R : Type u_1} [inst : AddMonoidWithOne R] [inst_1 : CharZero R] (n : ℕ)
  [inst_2 : n.AtLeastTwo], OfNat.ofNat n ≠ 0
inv_pow : ∀ {α : Type u_1} [inst : DivisionMonoid α] (a : α) (n : ℕ), a⁻¹ ^ n = (a ^ n)⁻¹
Fin.coe_eq_castSucc : ∀ {n : ℕ} {a : Fin n}, ↑↑a = a.castSucc
mul_eq_mul_left_iff : ∀ {M₀ : Type u_1} [inst : CancelMonoidWithZero M₀] {a b c : M₀}, a * b = a * c ↔ b = c ∨ a = 0
inv_eq_zero : ∀ {G₀ : Type u_2} [inst : GroupWithZero G₀] {a : G₀}, a⁻¹ = 0 ↔ a = 0
one_ne_zero : ∀ {α : Type u_2} [inst : Zero α] [inst_1 : One α] [inst_2 : NeZero 1], 1 ≠ 0
pow_eq_zero_iff : ∀ {M₀ : Type u_1} [inst : MonoidWithZero M₀] {a : M₀} {n : ℕ} [inst_1 : NoZeroDivisors M₀],
  n ≠ 0 → (a ^ n = 0 ↔ a = 0)
ne_eq : ∀ {α : Sort u_1} (a b : α), (a ≠ b) = ¬a = b
or_false : ∀ (p : Prop), (p ∨ False) = p
not_false_eq_true : (¬False) = True
AddLeftCancelMonoid.add_eq_zero : ∀ {α : Type u} [inst : AddLeftCancelMonoid α] [inst_1 : Subsingleton (AddUnits α)]
  {a b : α}, a + b = 0 ↔ a = 0 ∧ b = 0
and_false : ∀ (p : Prop), (p ∧ False) = False
Fin.natCast_eq_last : ∀ (n : ℕ), ↑n = Fin.last n