theorem extracted_formal_statement_143.{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) :
  (1 / (ADMM.τ E₁ E₂ F * ADMM.ρ E₁ E₂ F)) • (ADMM.y E₁ E₂ (n + 1) - ADMM.y E₁ E₂ n) =
    (1 / (ADMM.τ E₁ E₂ F * ADMM.ρ E₁ E₂ F)) • (ADMM.y E₁ E₂ (n + 1) - admm_kkt.y + admm_kkt.y - ADMM.y E₁ E₂ 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
⊢ (1 / (ADMM.τ E₁ E₂ F * ADMM.ρ E₁ E₂ F)) • (ADMM.y E₁ E₂ (n + 1) - ADMM.y E₁ E₂ n) =
    (1 / (ADMM.τ E₁ E₂ F * ADMM.ρ E₁ E₂ F)) • (ADMM.y E₁ E₂ (n + 1) - admm_kkt.y + admm_kkt.y - ADMM.y E₁ E₂ n)

Tactic: rw [sub_add, sub_self, sub_zero]

State after:

Relevant lemmas:
sub_self : ∀ {G : Type u_1} [inst : AddGroup G] (a : G), a - a = 0
sub_add : ∀ {α : Type u_1} [inst : SubtractionCommMonoid α] (a b c : α), a - b + c = a - (b - c)
sub_zero : ∀ {G : Type u_3} [inst : SubNegZeroMonoid G] (a : G), a - 0 = a