theorem extracted_formal_statement_4.{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+) :
  let c := -((OptProblem.A₁ E₂) (ADMM.x₁ E₂ F ↑n) - OptProblem.b E₁ E₂);
  ∀ (x : E₂),
    ADMM.ρ E₁ E₂ F / 2 * ‖(OptProblem.A₂ E₁) x + (OptProblem.A₁ E₂) (ADMM.x₁ E₂ F ↑n) - OptProblem.b E₁ E₂‖ ^ 2 =
      ADMM.ρ E₁ E₂ F * ‖(OptProblem.A₂ E₁) x - c‖ ^ 2 / 2 := sorry

State before:
case h
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+
c : F := -((OptProblem.A₁ E₂) (ADMM.x₁ E₂ F ↑n) - OptProblem.b E₁ E₂)
x : E₂
⊢ ADMM.ρ E₁ E₂ F / 2 * ‖(OptProblem.A₂ E₁) x + (OptProblem.A₁ E₂) (ADMM.x₁ E₂ F ↑n) - OptProblem.b E₁ E₂‖ ^ 2 =
    ADMM.ρ E₁ E₂ F * ‖(OptProblem.A₂ E₁) x - c‖ ^ 2 / 2

Tactic: simp only [c]
    rw [sub_neg_eq_add, add_sub]
    ring

State after:

Relevant lemmas:
add_sub : ∀ {G : Type u_3} [inst : SubNegMonoid G] (a b c : G), a + (b - c) = a + b - c
sub_neg_eq_add : ∀ {α : Type u_1} [inst : SubtractionMonoid α] (a b : α), a - -b = a + b