theorem extracted_formal_statement_239.{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 := (OptProblem.A₁ E₂) (ADMM.x₁ E₂ F ↑n) + (OptProblem.A₂ E₁) (ADMM.x₂ E₁ F ↑n) - OptProblem.b E₁ E₂;
  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;
  (1 - ADMM.τ E₁ E₂ F) * ADMM.ρ E₁ E₂ F * ⟪A₂_x_diff, r_n⟫_ℝ -
            (1 - ADMM.τ E₁ E₂ F) * ADMM.ρ E₁ E₂ F * ⟪A₂_x_diff, r_n⟫_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 =
      M (n + 1) - (1 - ADMM.τ E₁ E₂ F) * ADMM.ρ E₁ E₂ F * ⟪A₂_x_diff, r_n⟫_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 →
    (1 - ADMM.τ E₁ E₂ F) * ADMM.ρ E₁ E₂ F * ⟪A₂_x_diff, r_n⟫_ℝ -
            (1 - ADMM.τ E₁ E₂ F) * ADMM.ρ E₁ E₂ F * ⟪A₂_x_diff, r_n⟫_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 =
      (1 - ADMM.τ E₁ E₂ F) * ADMM.ρ E₁ E₂ F *
              ⟪(OptProblem.A₂ E₁) (ADMM.x₂ E₁ F ↑(n + 1) - ADMM.x₂ E₁ F (n + 1).natPred),
                (OptProblem.A₁ E₂) (ADMM.x₁ E₂ F (n + 1).natPred) + (OptProblem.A₂ E₁) (ADMM.x₂ E₁ F (n + 1).natPred) -
                  OptProblem.b E₁ E₂⟫_ℝ -
            (1 - ADMM.τ E₁ E₂ F) * ADMM.ρ E₁ E₂ F * ⟪A₂_x_diff, r_n⟫_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)
⊢ (1 - ADMM.τ E₁ E₂ F) * ADMM.ρ E₁ E₂ F * ⟪A₂_x_diff, r_n⟫_ℝ -
          (1 - ADMM.τ E₁ E₂ F) * ADMM.ρ E₁ E₂ F * ⟪A₂_x_diff, r_n⟫_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 =
    M (n + 1) - (1 - ADMM.τ E₁ E₂ F) * ADMM.ρ E₁ E₂ F * ⟪A₂_x_diff, r_n⟫_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 [M]

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)
 ⊢ (1 - ADMM.τ E₁ E₂ F) * ADMM.ρ E₁ E₂ F * ⟪A₂_x_diff, r_n⟫_ℝ -
           (1 - ADMM.τ E₁ E₂ F) * ADMM.ρ E₁ E₂ F * ⟪A₂_x_diff, r_n⟫_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 =
     (1 - ADMM.τ E₁ E₂ F) * ADMM.ρ E₁ E₂ F *
             ⟪(OptProblem.A₂ E₁) (ADMM.x₂ E₁ F ↑(n + 1) - ADMM.x₂ E₁ F (n + 1).natPred),
               (OptProblem.A₁ E₂) (ADMM.x₁ E₂ F (n + 1).natPred) + (OptProblem.A₂ E₁) (ADMM.x₂ E₁ F (n + 1).natPred) -
                 OptProblem.b E₁ E₂⟫_ℝ -
           (1 - ADMM.τ E₁ E₂ F) * ADMM.ρ E₁ E₂ F * ⟪A₂_x_diff, r_n⟫_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]

Relevant lemmas: