-- File: optlib/Optlib/Algorithm/ADMM/Theroem_converge.lean
Theorem statement:
theorem extracted_formal_statement_295.{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} (fullrank₁ : Function.Injective ⇑(OptProblem.A₁ E₂))
  (fullrank₂ : Function.Injective ⇑(OptProblem.A₂ E₁)) [inst_9 : Setting E₁ E₂ F admm admm_kkt] (n : ℕ)
  (h_original_goal :
    ADMM.ρ E₁ E₂ F • -1 • (OptProblem.A₂ E₁) (ADMM.x₂ E₁ F ↑(φ₁ n) - ADMM.x₂ E₁ F (↑(φ₁ n) - 1)) =
      -ADMM.ρ E₁ E₂ F • (OptProblem.A₂ E₁) (ADMM.x₂ E₁ F ↑(φ₁ n) - ADMM.x₂ E₁ F (↑(φ₁ n) - 1))) :
  -(ADMM.ρ E₁ E₂ F • (OptProblem.A₂ E₁) (ADMM.x₂ E₁ F ↑(φ₁ n) - ADMM.x₂ E₁ F (↑(φ₁ n) - 1))) =
    -ADMM.ρ E₁ E₂ F • (OptProblem.A₂ E₁) (ADMM.x₂ E₁ F ↑(φ₁ n) - ADMM.x₂ E₁ F (↑(φ₁ n) - 1)) := sorry

Tactic 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
fullrank₁ : Function.Injective ⇑(OptProblem.A₁ E₂)
fullrank₂ : Function.Injective ⇑(OptProblem.A₂ E₁)
inst✝ : Setting E₁ E₂ F admm admm_kkt
n : ℕ
⊢ ADMM.ρ E₁ E₂ F • -1 • (OptProblem.A₂ E₁) (ADMM.x₂ E₁ F ↑(φ₁ n) - ADMM.x₂ E₁ F (↑(φ₁ n) - 1)) =
    -ADMM.ρ E₁ E₂ F • (OptProblem.A₂ E₁) (ADMM.x₂ E₁ F ↑(φ₁ n) - ADMM.x₂ E₁ F (↑(φ₁ n) - 1))

Tactic: rw [smul_comm, neg_one_smul]

Tactic 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
 fullrank₁ : Function.Injective ⇑(OptProblem.A₁ E₂)
 fullrank₂ : Function.Injective ⇑(OptProblem.A₂ E₁)
 inst✝ : Setting E₁ E₂ F admm admm_kkt
 n : ℕ
 ⊢ -(ADMM.ρ E₁ E₂ F • (OptProblem.A₂ E₁) (ADMM.x₂ E₁ F ↑(φ₁ n) - ADMM.x₂ E₁ F (↑(φ₁ n) - 1))) =
     -ADMM.ρ E₁ E₂ F • (OptProblem.A₂ E₁) (ADMM.x₂ E₁ F ↑(φ₁ n) - ADMM.x₂ E₁ F (↑(φ₁ n) - 1))]

Premises used:
neg_one_smul : ∀ (R : Type u_1) {M : Type u_3} [inst : Ring R] [inst_10 : AddCommGroup M] [inst_11 : Module R M]
  (x : M), -1 • x = -x
SMulCommClass.smul_comm : ∀ {M : Type u_9} {N : Type u_10} {α : Type u_11} {inst : SMul M α} {inst_10 : SMul N α}
  [self : SMulCommClass M N α] (m : M) (n : N) (a : α), m • n • a = n • m • a