theorem extracted_formal_statement_217.{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 pm1 := 1 / (ADMM.τ E₁ E₂ F * ADMM.ρ E₁ E₂ F);
  let pm2 := (1 - ADMM.τ E₁ E₂ F) * ADMM.ρ E₁ E₂ F;
  pm1 * ⟪ey (n + 1), ey n - ey (n + 1)⟫_ℝ =
      ⟪ey (n + 1), -((OptProblem.A₁ E₂) (e₁ (n + 1)) + (OptProblem.A₂ E₁) (e₂ (n + 1)))⟫_ℝ →
    pm2 *
          ‖(OptProblem.A₁ E₂) (ADMM.x₁ E₂ F (n + 1)) + (OptProblem.A₂ E₁) (ADMM.x₂ E₁ F (n + 1)) - OptProblem.b E₁ E₂‖ ^
            2 =
        pm2 * ‖(OptProblem.A₁ E₂) (e₁ (n + 1)) + (OptProblem.A₂ E₁) (e₂ (n + 1))‖ ^ 2 →
      ADMM.ρ E₁ E₂ F *
          (⟪(OptProblem.A₂ E₁) (ADMM.x₂ E₁ F n - ADMM.x₂ E₁ F (n + 1)), (OptProblem.A₂ E₁) (e₂ (n + 1))⟫_ℝ -
            ⟪(OptProblem.A₂ E₁) (ADMM.x₂ E₁ F n - ADMM.x₂ E₁ F (n + 1)),
              (OptProblem.A₁ E₂) (ADMM.x₁ E₂ F (n + 1)) + (OptProblem.A₂ E₁) (ADMM.x₂ E₁ F (n + 1)) -
                OptProblem.b E₁ E₂⟫_ℝ) =
        ADMM.ρ E₁ E₂ F *
          ⟪(OptProblem.A₂ E₁) (ADMM.x₂ E₁ F n - ADMM.x₂ E₁ F (n + 1)),
            (OptProblem.A₂ E₁) (e₂ (n + 1)) -
              ((OptProblem.A₁ E₂) (ADMM.x₁ E₂ F (n + 1)) + (OptProblem.A₂ E₁) (ADMM.x₂ E₁ F (n + 1)) -
                OptProblem.b E₁ E₂)⟫_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
pm1 : ℝ := 1 / (ADMM.τ E₁ E₂ F * ADMM.ρ E₁ E₂ F)
pm2 : ℝ := (1 - ADMM.τ E₁ E₂ F) * ADMM.ρ E₁ E₂ F
h1 :
  pm1 * ⟪ey (n + 1), ey n - ey (n + 1)⟫_ℝ =
    ⟪ey (n + 1), -((OptProblem.A₁ E₂) (e₁ (n + 1)) + (OptProblem.A₂ E₁) (e₂ (n + 1)))⟫_ℝ
h2 :
  pm2 *
      ‖(OptProblem.A₁ E₂) (ADMM.x₁ E₂ F (n + 1)) + (OptProblem.A₂ E₁) (ADMM.x₂ E₁ F (n + 1)) - OptProblem.b E₁ E₂‖ ^ 2 =
    pm2 * ‖(OptProblem.A₁ E₂) (e₁ (n + 1)) + (OptProblem.A₂ E₁) (e₂ (n + 1))‖ ^ 2
⊢ ADMM.ρ E₁ E₂ F *
      (⟪(OptProblem.A₂ E₁) (ADMM.x₂ E₁ F n - ADMM.x₂ E₁ F (n + 1)), (OptProblem.A₂ E₁) (e₂ (n + 1))⟫_ℝ -
        ⟪(OptProblem.A₂ E₁) (ADMM.x₂ E₁ F n - ADMM.x₂ E₁ F (n + 1)),
          (OptProblem.A₁ E₂) (ADMM.x₁ E₂ F (n + 1)) + (OptProblem.A₂ E₁) (ADMM.x₂ E₁ F (n + 1)) -
            OptProblem.b E₁ E₂⟫_ℝ) =
    ADMM.ρ E₁ E₂ F *
      ⟪(OptProblem.A₂ E₁) (ADMM.x₂ E₁ F n - ADMM.x₂ E₁ F (n + 1)),
        (OptProblem.A₂ E₁) (e₂ (n + 1)) -
          ((OptProblem.A₁ E₂) (ADMM.x₁ E₂ F (n + 1)) + (OptProblem.A₂ E₁) (ADMM.x₂ E₁ F (n + 1)) -
            OptProblem.b E₁ E₂)⟫_ℝ

Tactic: rw [←
  inner_sub_right (A₂ (x₂ (n) - x₂ (n + 1))) (A₂ (e₂ (n + 1))) ((A₁ (x₁ (n + 1)) + A₂ (x₂ (n + 1)) - b))]

State after:

Relevant lemmas:
inner_sub_right : ∀ {𝕜 : Type u_1} {E : Type u_2} [inst : RCLike 𝕜] [inst_10 : SeminormedAddCommGroup E]
  [inst_11 : InnerProductSpace 𝕜 E] (x y z : E), ⟪x, y - z⟫_𝕜 = ⟪x, y⟫_𝕜 - ⟪x, z⟫_𝕜