theorem extracted_formal_statement_140.{u_3, u_2, u_1} {E₁ : Type u_1} {E₂ : Type u_2} {F : Type u_3}
  [inst : NormedAddCommGroup E₁] [inst_1 : InnerProductSpace ℝ E₁] [inst_2 : FiniteDimensional ℝ E₁]
  [inst_3 : NormedAddCommGroup E₂] [inst_4 : InnerProductSpace ℝ E₂] [inst_5 : FiniteDimensional ℝ E₂]
  [inst_6 : NormedAddCommGroup F] [inst_7 : InnerProductSpace ℝ F] [inst_8 : FiniteDimensional ℝ F]
  {admm : ADMM E₁ E₂ F} {admm_kkt : Existance_of_kkt admm} (n : ℕ)
  (h_original_goal :
    (OptProblem.A₁ E₂) (ADMM.x₁ E₂ F (n + 1)) + (OptProblem.A₂ E₁) (ADMM.x₂ E₁ F (n + 1)) - OptProblem.b E₁ E₂ =
      (1 / (ADMM.τ E₁ E₂ F * ADMM.ρ E₁ E₂ F)) •
        ((ADMM.τ E₁ E₂ F * ADMM.ρ E₁ E₂ F) •
              ((OptProblem.A₁ E₂) (ADMM.x₁ E₂ F (n + 1)) + (OptProblem.A₂ E₁) (ADMM.x₂ E₁ F (n + 1)) -
                OptProblem.b E₁ E₂) +
            ADMM.y E₁ E₂ n -
          ADMM.y E₁ E₂ n)) :
  (OptProblem.A₁ E₂) (ADMM.x₁ E₂ F (n + 1)) + (OptProblem.A₂ E₁) (ADMM.x₂ E₁ F (n + 1)) - OptProblem.b E₁ E₂ =
    ((ADMM.ρ E₁ E₂ F)⁻¹ * (ADMM.τ E₁ E₂ F)⁻¹) •
      (ADMM.τ E₁ E₂ F * ADMM.ρ E₁ E₂ F) •
        ((OptProblem.A₁ E₂) (ADMM.x₁ E₂ F (n + 1)) + (OptProblem.A₂ E₁) (ADMM.x₂ E₁ F (n + 1)) -
          OptProblem.b E₁ E₂) := sorry

State before:
E₁ : Type u_1
E₂ : Type u_2
F : Type u_3
inst✝⁹ : NormedAddCommGroup E₁
inst✝⁸ : InnerProductSpace ℝ E₁
inst✝⁷ : FiniteDimensional ℝ E₁
inst✝⁶ : NormedAddCommGroup E₂
inst✝⁵ : InnerProductSpace ℝ E₂
inst✝⁴ : FiniteDimensional ℝ E₂
inst✝³ : NormedAddCommGroup F
inst✝² : InnerProductSpace ℝ F
inst✝¹ : FiniteDimensional ℝ F
admm : ADMM E₁ E₂ F
admm_kkt : Existance_of_kkt admm
inst✝ : Setting E₁ E₂ F admm admm_kkt
n : ℕ
⊢ (OptProblem.A₁ E₂) (ADMM.x₁ E₂ F (n + 1)) + (OptProblem.A₂ E₁) (ADMM.x₂ E₁ F (n + 1)) - OptProblem.b E₁ E₂ =
    (1 / (ADMM.τ E₁ E₂ F * ADMM.ρ E₁ E₂ F)) •
      ((ADMM.τ E₁ E₂ F * ADMM.ρ E₁ E₂ F) •
            ((OptProblem.A₁ E₂) (ADMM.x₁ E₂ F (n + 1)) + (OptProblem.A₂ E₁) (ADMM.x₂ E₁ F (n + 1)) -
              OptProblem.b E₁ E₂) +
          ADMM.y E₁ E₂ n -
        ADMM.y E₁ E₂ n)

Tactic: simp only [one_div, mul_inv_rev, add_sub_cancel_right]

State after:
[E₁ : Type u_1
 E₂ : Type u_2
 F : Type u_3
 inst✝⁹ : NormedAddCommGroup E₁
 inst✝⁸ : InnerProductSpace ℝ E₁
 inst✝⁷ : FiniteDimensional ℝ E₁
 inst✝⁶ : NormedAddCommGroup E₂
 inst✝⁵ : InnerProductSpace ℝ E₂
 inst✝⁴ : FiniteDimensional ℝ E₂
 inst✝³ : NormedAddCommGroup F
 inst✝² : InnerProductSpace ℝ F
 inst✝¹ : FiniteDimensional ℝ F
 admm : ADMM E₁ E₂ F
 admm_kkt : Existance_of_kkt admm
 inst✝ : Setting E₁ E₂ F admm admm_kkt
 n : ℕ
 ⊢ (OptProblem.A₁ E₂) (ADMM.x₁ E₂ F (n + 1)) + (OptProblem.A₂ E₁) (ADMM.x₂ E₁ F (n + 1)) - OptProblem.b E₁ E₂ =
     ((ADMM.ρ E₁ E₂ F)⁻¹ * (ADMM.τ E₁ E₂ F)⁻¹) •
       (ADMM.τ E₁ E₂ F * ADMM.ρ E₁ E₂ F) •
         ((OptProblem.A₁ E₂) (ADMM.x₁ E₂ F (n + 1)) + (OptProblem.A₂ E₁) (ADMM.x₂ E₁ F (n + 1)) - OptProblem.b E₁ E₂)]

Relevant lemmas:
mul_inv_rev : ∀ {G : Type u_1} [inst : DivisionMonoid G] (a b : G), (a * b)⁻¹ = b⁻¹ * a⁻¹
one_div : ∀ {G : Type u_1} [inst : DivInvMonoid G] (a : G), 1 / a = a⁻¹
add_sub_cancel_right : ∀ {G : Type u_1} [inst : AddGroup G] (a b : G), a + b - b = a