theorem extracted_formal_statement_237.{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 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 y_diff := ADMM.y E₁ E₂ (↑n + 1) - ADMM.y E₁ E₂ ↑n;
  (ADMM.τ E₁ E₂ F * ADMM.ρ E₁ E₂ F) • r_n_1 = y_diff →
    (1 - ADMM.τ E₁ E₂ F) * ADMM.ρ E₁ E₂ F * ⟪A₂_x_diff, r_n_1⟫_ℝ +
        ⟪A₂_x_diff, (ADMM.τ E₁ E₂ F * ADMM.ρ E₁ E₂ F) • r_n_1⟫_ℝ =
      (1 - ADMM.τ E₁ E₂ F) * ADMM.ρ E₁ E₂ F * ⟪A₂_x_diff, r_n_1⟫_ℝ + ⟪A₂_x_diff, y_diff⟫_ℝ := 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)
this : (ADMM.τ E₁ E₂ F * ADMM.ρ E₁ E₂ F) • r_n_1 = y_diff
⊢ (1 - ADMM.τ E₁ E₂ F) * ADMM.ρ E₁ E₂ F * ⟪A₂_x_diff, r_n_1⟫_ℝ +
      ⟪A₂_x_diff, (ADMM.τ E₁ E₂ F * ADMM.ρ E₁ E₂ F) • r_n_1⟫_ℝ =
    (1 - ADMM.τ E₁ E₂ F) * ADMM.ρ E₁ E₂ F * ⟪A₂_x_diff, r_n_1⟫_ℝ + ⟪A₂_x_diff, y_diff⟫_ℝ

Tactic: rw [this]

State after:

Relevant lemmas: