theorem extracted_formal_statement_268.{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+)
  (h₄'' :
    ‖(OptProblem.A₂ E₁) admm_kkt.x₂ - (OptProblem.A₂ E₁) (ADMM.x₂ E₁ F ↑n)‖ =
      ‖-((OptProblem.A₂ E₁) (ADMM.x₂ E₁ F ↑n) - (OptProblem.A₂ E₁) admm_kkt.x₂)‖)
  (h_original_goal :
    ‖(OptProblem.A₂ E₁) (ADMM.x₂ E₁ F (↑n + 1) - ADMM.x₂ E₁ F ↑n) -
          (OptProblem.A₂ E₁) (ADMM.x₂ E₁ F (↑n + 1) - admm_kkt.x₂)‖ =
      ‖(OptProblem.A₂ E₁) (ADMM.x₂ E₁ F ↑n - admm_kkt.x₂)‖) :
  ‖(OptProblem.A₂ E₁) admm_kkt.x₂ - (OptProblem.A₂ E₁) (ADMM.x₂ E₁ F ↑n)‖ =
    ‖(OptProblem.A₂ E₁) (ADMM.x₂ E₁ F ↑n) - (OptProblem.A₂ E₁) admm_kkt.x₂‖ := 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+
h₄'' :
  ‖(OptProblem.A₂ E₁) admm_kkt.x₂ - (OptProblem.A₂ E₁) (ADMM.x₂ E₁ F ↑n)‖ =
    ‖-((OptProblem.A₂ E₁) (ADMM.x₂ E₁ F ↑n) - (OptProblem.A₂ E₁) admm_kkt.x₂)‖
⊢ ‖(OptProblem.A₂ E₁) (ADMM.x₂ E₁ F (↑n + 1) - ADMM.x₂ E₁ F ↑n) -
        (OptProblem.A₂ E₁) (ADMM.x₂ E₁ F (↑n + 1) - admm_kkt.x₂)‖ =
    ‖(OptProblem.A₂ E₁) (ADMM.x₂ E₁ F ↑n - admm_kkt.x₂)‖

Tactic: simp only [map_sub, sub_sub_sub_cancel_left]

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+
 h₄'' :
   ‖(OptProblem.A₂ E₁) admm_kkt.x₂ - (OptProblem.A₂ E₁) (ADMM.x₂ E₁ F ↑n)‖ =
     ‖-((OptProblem.A₂ E₁) (ADMM.x₂ E₁ F ↑n) - (OptProblem.A₂ E₁) admm_kkt.x₂)‖
 ⊢ ‖(OptProblem.A₂ E₁) admm_kkt.x₂ - (OptProblem.A₂ E₁) (ADMM.x₂ E₁ F ↑n)‖ =
     ‖(OptProblem.A₂ E₁) (ADMM.x₂ E₁ F ↑n) - (OptProblem.A₂ E₁) admm_kkt.x₂‖]

Relevant lemmas:
sub_sub_sub_cancel_left : ∀ {G : Type u_3} [inst : AddCommGroup G] (a b c : G), c - a - (c - b) = b - a
map_sub : ∀ {G : Type u_7} {H : Type u_8} {F : Type u_9} [inst : FunLike F G H] [inst_9 : AddGroup G]
  [inst_10 : SubtractionMonoid H] [inst_11 : AddMonoidHomClass F G H] (f : F) (a b : G), f (a - b) = f a - f b