theorem extracted_formal_statement_142.{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) (htau1 : (ADMM.τ E₁ E₂ F)⁻¹ * ADMM.τ E₁ E₂ F = 1)
  (hrho1 : (ADMM.ρ E₁ E₂ F)⁻¹ * ADMM.ρ E₁ E₂ F = 1) :
  (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
htau1 : (ADMM.τ E₁ E₂ F)⁻¹ * ADMM.τ E₁ E₂ F = 1
hrho1 : (ADMM.ρ E₁ E₂ F)⁻¹ * ADMM.ρ E₁ E₂ F = 1
⊢ (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)

Tactic: rw [htau1, one_mul, hrho1, one_smul]

State after:

Relevant lemmas:
one_smul : ∀ (M : Type u_1) {α : Type u_5} [inst : Monoid M] [inst_9 : MulAction M α] (b : α), 1 • b = b
one_mul : ∀ {M : Type u} [inst : MulOneClass M] (a : M), 1 * a = a