theorem extracted_formal_statement_44.{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} [inst_9 : Setting E₁ E₂ F admm admm_kkt] (n : ℕ)
  (h_original_goal :
    (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) =
      ey (n + 1) - ey n) :
  (ADMM.τ E₁ E₂ F * (ADMM.ρ E₁ E₂ F * ((ADMM.ρ E₁ E₂ F)⁻¹ * (ADMM.τ E₁ E₂ F)⁻¹))) • (ey (n + 1) - ey n) =
    ey (n + 1) - ey n := 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
⊢ (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) =
    ey (n + 1) - ey n

Tactic: rw [Φ_isdescending_eq1, Φ_isdescending_eq2]
    simp only [one_div, mul_inv_rev]
    rw [smul_smul, mul_assoc]

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
 ⊢ (ADMM.τ E₁ E₂ F * (ADMM.ρ E₁ E₂ F * ((ADMM.ρ E₁ E₂ F)⁻¹ * (ADMM.τ E₁ E₂ F)⁻¹))) • (ey (n + 1) - ey n) =
     ey (n + 1) - ey n]

Relevant lemmas:
mul_inv_rev : ∀ {G : Type u_1} [inst : DivisionMonoid G] (a b : G), (a * b)⁻¹ = b⁻¹ * a⁻¹
smul_smul : ∀ {M : Type u_1} {α : Type u_5} [inst : Monoid M] [inst_10 : MulAction M α] (a₁ a₂ : M) (b : α),
  a₁ • a₂ • b = (a₁ * a₂) • b
one_div : ∀ {G : Type u_1} [inst : DivInvMonoid G] (a : G), 1 / a = a⁻¹
mul_assoc : ∀ {G : Type u_1} [inst : Semigroup G] (a b c : G), a * b * c = a * (b * c)