theorem extracted_formal_statement_250.{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+) :
  let A₂_x_diff := (OptProblem.A₂ E₁) (ADMM.x₂ E₁ F (↑n + 1) - ADMM.x₂ E₁ F ↑n);
  let r_n_1 :=
    (OptProblem.A₁ E₂) (ADMM.x₁ E₂ F (↑n + 1)) + (OptProblem.A₂ E₁) (ADMM.x₂ E₁ F (↑n + 1)) - OptProblem.b E₁ E₂;
  let υ_diff := υ (↑n + 1) - υ ↑n;
  ADMM.ρ E₁ E₂ F * ⟪A₂_x_diff, r_n_1⟫_ℝ = M (n + 1) + ⟪υ_diff, A₂_x_diff⟫_ℝ →
    ⟪υ (↑n + 1) - υ ↑n, (OptProblem.A₂ E₁) (ADMM.x₂ E₁ F (↑n + 1)) - (OptProblem.A₂ E₁) (ADMM.x₂ E₁ F ↑n)⟫_ℝ ≤ 0 →
      ⟪υ (↑n + 1) - υ ↑n, (OptProblem.A₂ E₁) (ADMM.x₂ E₁ F (↑n + 1) - ADMM.x₂ E₁ F ↑n)⟫_ℝ ≤ 0 := 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+
A₂_x_diff : F := (OptProblem.A₂ E₁) (ADMM.x₂ E₁ F (↑n + 1) - ADMM.x₂ E₁ F ↑n)
r_n : F := (OptProblem.A₁ E₂) (ADMM.x₁ E₂ F ↑n) + (OptProblem.A₂ E₁) (ADMM.x₂ E₁ F ↑n) - OptProblem.b E₁ E₂
r_n_1 : F :=
  (OptProblem.A₁ E₂) (ADMM.x₁ E₂ F (↑n + 1)) + (OptProblem.A₂ E₁) (ADMM.x₂ E₁ F (↑n + 1)) - OptProblem.b E₁ E₂
υ_diff : F := υ (↑n + 1) - υ ↑n
y_diff : F := ADMM.y E₁ E₂ (↑n + 1) - ADMM.y E₁ E₂ ↑n
x_diff : E₂ := ADMM.x₂ E₁ F (↑n + 1) - ADMM.x₂ E₁ F ↑n
A₂T : F →L[ℝ] E₂ := ContinuousLinearMap.adjoint (OptProblem.A₂ E₁)
h : ADMM.ρ E₁ E₂ F * ⟪A₂_x_diff, r_n_1⟫_ℝ = M (n + 1) + ⟪υ_diff, A₂_x_diff⟫_ℝ
⊢ ⟪υ (↑n + 1) - υ ↑n, (OptProblem.A₂ E₁) (ADMM.x₂ E₁ F (↑n + 1)) - (OptProblem.A₂ E₁) (ADMM.x₂ E₁ F ↑n)⟫_ℝ ≤ 0

Tactic: rw [← map_sub A₂]

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+
 A₂_x_diff : F := (OptProblem.A₂ E₁) (ADMM.x₂ E₁ F (↑n + 1) - ADMM.x₂ E₁ F ↑n)
 r_n : F := (OptProblem.A₁ E₂) (ADMM.x₁ E₂ F ↑n) + (OptProblem.A₂ E₁) (ADMM.x₂ E₁ F ↑n) - OptProblem.b E₁ E₂
 r_n_1 : F :=
   (OptProblem.A₁ E₂) (ADMM.x₁ E₂ F (↑n + 1)) + (OptProblem.A₂ E₁) (ADMM.x₂ E₁ F (↑n + 1)) - OptProblem.b E₁ E₂
 υ_diff : F := υ (↑n + 1) - υ ↑n
 y_diff : F := ADMM.y E₁ E₂ (↑n + 1) - ADMM.y E₁ E₂ ↑n
 x_diff : E₂ := ADMM.x₂ E₁ F (↑n + 1) - ADMM.x₂ E₁ F ↑n
 A₂T : F →L[ℝ] E₂ := ContinuousLinearMap.adjoint (OptProblem.A₂ E₁)
 h : ADMM.ρ E₁ E₂ F * ⟪A₂_x_diff, r_n_1⟫_ℝ = M (n + 1) + ⟪υ_diff, A₂_x_diff⟫_ℝ
 ⊢ ⟪υ (↑n + 1) - υ ↑n, (OptProblem.A₂ E₁) (ADMM.x₂ E₁ F (↑n + 1) - ADMM.x₂ E₁ F ↑n)⟫_ℝ ≤ 0]

Relevant lemmas:
map_sub : ∀ {G : Type u_7} {H : Type u_8} {F : Type u_9} [inst : FunLike F G H] [inst_10 : AddGroup G]
  [inst_11 : SubtractionMonoid H] [inst_12 : AddMonoidHomClass F G H] (f : F) (a b : G), f (a - b) = f a - f b