theorem extracted_formal_statement_59.{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]
  (htau : 1 < ADMM.τ E₁ E₂ F) (n : ℕ+) :
  let x_diff := 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;
  2 * M (n + 1) = (ADMM.τ E₁ E₂ F - 1) * ADMM.ρ E₁ E₂ F * (-2 * ⟪(OptProblem.A₂ E₁) x_diff, r_n⟫_ℝ) →
    (ADMM.τ E₁ E₂ F - 1) * ADMM.ρ E₁ E₂ F * (-2 * ⟪(OptProblem.A₂ E₁) x_diff, r_n⟫_ℝ) ≤
        (ADMM.τ E₁ E₂ F - 1) * ADMM.ρ E₁ E₂ F *
          (ADMM.τ E₁ E₂ F * ‖(OptProblem.A₂ E₁) x_diff‖ ^ 2 +
            1 / ADMM.τ E₁ E₂ F *
              ‖(OptProblem.A₁ E₂) (ADMM.x₁ E₂ F ↑n) + (OptProblem.A₂ E₁) (ADMM.x₂ E₁ F ↑n) - OptProblem.b E₁ E₂‖ ^ 2) →
      (ADMM.τ E₁ E₂ F - 1) * ADMM.ρ E₁ E₂ F *
            (ADMM.τ E₁ E₂ F * ‖(OptProblem.A₂ E₁) x_diff‖ ^ 2 +
              1 / ADMM.τ E₁ E₂ F *
                ‖(OptProblem.A₁ E₂) (ADMM.x₁ E₂ F ↑n) + (OptProblem.A₂ E₁) (ADMM.x₂ E₁ F ↑n) - OptProblem.b E₁ E₂‖ ^
                  2) =
          (ADMM.τ E₁ E₂ F ^ 2 - ADMM.τ E₁ E₂ F) * ADMM.ρ E₁ E₂ F * ‖(OptProblem.A₂ E₁) x_diff‖ ^ 2 +
            (1 - 1 / ADMM.τ E₁ E₂ F) * ADMM.ρ E₁ E₂ F * ‖r_n‖ ^ 2 →
        (ADMM.τ E₁ E₂ F * ADMM.τ E₁ E₂ F - 1 * ADMM.τ E₁ E₂ F) * ADMM.ρ E₁ E₂ F * ‖(OptProblem.A₂ E₁) x_diff‖ ^ 2 +
            (ADMM.τ E₁ E₂ F * (1 / ADMM.τ E₁ E₂ F) - 1 * (1 / ADMM.τ E₁ E₂ F)) * ADMM.ρ E₁ E₂ F *
              ‖(OptProblem.A₁ E₂) (ADMM.x₁ E₂ F ↑n) + (OptProblem.A₂ E₁) (ADMM.x₂ E₁ F ↑n) - OptProblem.b E₁ E₂‖ ^ 2 =
          (ADMM.τ E₁ E₂ F ^ 2 - ADMM.τ E₁ E₂ F) * ADMM.ρ E₁ E₂ F * ‖(OptProblem.A₂ E₁) x_diff‖ ^ 2 +
            (1 - 1 / ADMM.τ E₁ E₂ F) * ADMM.ρ E₁ E₂ F * ‖r_n‖ ^ 2 := 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
htau : 1 < ADMM.τ E₁ E₂ F
n : ℕ+
x_diff : E₂ := 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
e_sum : F := (OptProblem.A₁ E₂) (e₁ ↑n) + (OptProblem.A₂ E₁) (e₂ ↑n)
e_sum_1 : F := (OptProblem.A₁ E₂) (e₁ (↑n + 1)) + (OptProblem.A₂ E₁) (e₂ (↑n + 1))
a₁ : ℝ := 1 / (ADMM.τ E₁ E₂ F * ADMM.ρ E₁ E₂ F) * ‖ey ↑n‖ ^ 2
a₂ : ℝ := 1 / (ADMM.τ E₁ E₂ F * ADMM.ρ E₁ E₂ F) * ‖ey (↑n + 1)‖ ^ 2
b₁ : ℝ := ADMM.ρ E₁ E₂ F * ‖(OptProblem.A₂ E₁) (e₂ ↑n)‖ ^ 2
b₂ : ℝ := ADMM.ρ E₁ E₂ F * ‖(OptProblem.A₂ E₁) (e₂ (↑n + 1))‖ ^ 2
c₁ : ℝ := (1 - 1 / ADMM.τ E₁ E₂ F) * ADMM.ρ E₁ E₂ F * ‖e_sum‖ ^ 2
c₂ : ℝ := (1 - 1 / ADMM.τ E₁ E₂ F) * ADMM.ρ E₁ E₂ F * ‖e_sum_1‖ ^ 2
d₁ : ℝ := (1 + ADMM.τ E₁ E₂ F - ADMM.τ E₁ E₂ F ^ 2) * ADMM.ρ E₁ E₂ F * ‖(OptProblem.A₂ E₁) x_diff‖ ^ 2
d₂ : ℝ := (1 + 1 / ADMM.τ E₁ E₂ F - ADMM.τ E₁ E₂ F) * ADMM.ρ E₁ E₂ F * ‖r_n_1‖ ^ 2
h1 : 2 * M (n + 1) = (ADMM.τ E₁ E₂ F - 1) * ADMM.ρ E₁ E₂ F * (-2 * ⟪(OptProblem.A₂ E₁) x_diff, r_n⟫_ℝ)
h2 :
  (ADMM.τ E₁ E₂ F - 1) * ADMM.ρ E₁ E₂ F * (-2 * ⟪(OptProblem.A₂ E₁) x_diff, r_n⟫_ℝ) ≤
    (ADMM.τ E₁ E₂ F - 1) * ADMM.ρ E₁ E₂ F *
      (ADMM.τ E₁ E₂ F * ‖(OptProblem.A₂ E₁) x_diff‖ ^ 2 +
        1 / ADMM.τ E₁ E₂ F *
          ‖(OptProblem.A₁ E₂) (ADMM.x₁ E₂ F ↑n) + (OptProblem.A₂ E₁) (ADMM.x₂ E₁ F ↑n) - OptProblem.b E₁ E₂‖ ^ 2)
⊢ (ADMM.τ E₁ E₂ F - 1) * ADMM.ρ E₁ E₂ F *
      (ADMM.τ E₁ E₂ F * ‖(OptProblem.A₂ E₁) x_diff‖ ^ 2 +
        1 / ADMM.τ E₁ E₂ F *
          ‖(OptProblem.A₁ E₂) (ADMM.x₁ E₂ F ↑n) + (OptProblem.A₂ E₁) (ADMM.x₂ E₁ F ↑n) - OptProblem.b E₁ E₂‖ ^ 2) =
    (ADMM.τ E₁ E₂ F ^ 2 - ADMM.τ E₁ E₂ F) * ADMM.ρ E₁ E₂ F * ‖(OptProblem.A₂ E₁) x_diff‖ ^ 2 +
      (1 - 1 / ADMM.τ E₁ E₂ F) * ADMM.ρ E₁ E₂ F * ‖r_n‖ ^ 2

Tactic: rw [mul_add ((τ - 1) * ρ), ← mul_assoc, ← mul_assoc]
    rw [mul_comm (τ - 1) ρ, mul_assoc ρ, mul_assoc ρ (τ - 1)]
    rw [sub_mul τ 1 τ, sub_mul τ 1 (1 / τ), mul_comm ρ, mul_comm ρ]

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
 htau : 1 < ADMM.τ E₁ E₂ F
 n : ℕ+
 x_diff : E₂ := 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
 e_sum : F := (OptProblem.A₁ E₂) (e₁ ↑n) + (OptProblem.A₂ E₁) (e₂ ↑n)
 e_sum_1 : F := (OptProblem.A₁ E₂) (e₁ (↑n + 1)) + (OptProblem.A₂ E₁) (e₂ (↑n + 1))
 a₁ : ℝ := 1 / (ADMM.τ E₁ E₂ F * ADMM.ρ E₁ E₂ F) * ‖ey ↑n‖ ^ 2
 a₂ : ℝ := 1 / (ADMM.τ E₁ E₂ F * ADMM.ρ E₁ E₂ F) * ‖ey (↑n + 1)‖ ^ 2
 b₁ : ℝ := ADMM.ρ E₁ E₂ F * ‖(OptProblem.A₂ E₁) (e₂ ↑n)‖ ^ 2
 b₂ : ℝ := ADMM.ρ E₁ E₂ F * ‖(OptProblem.A₂ E₁) (e₂ (↑n + 1))‖ ^ 2
 c₁ : ℝ := (1 - 1 / ADMM.τ E₁ E₂ F) * ADMM.ρ E₁ E₂ F * ‖e_sum‖ ^ 2
 c₂ : ℝ := (1 - 1 / ADMM.τ E₁ E₂ F) * ADMM.ρ E₁ E₂ F * ‖e_sum_1‖ ^ 2
 d₁ : ℝ := (1 + ADMM.τ E₁ E₂ F - ADMM.τ E₁ E₂ F ^ 2) * ADMM.ρ E₁ E₂ F * ‖(OptProblem.A₂ E₁) x_diff‖ ^ 2
 d₂ : ℝ := (1 + 1 / ADMM.τ E₁ E₂ F - ADMM.τ E₁ E₂ F) * ADMM.ρ E₁ E₂ F * ‖r_n_1‖ ^ 2
 h1 : 2 * M (n + 1) = (ADMM.τ E₁ E₂ F - 1) * ADMM.ρ E₁ E₂ F * (-2 * ⟪(OptProblem.A₂ E₁) x_diff, r_n⟫_ℝ)
 h2 :
   (ADMM.τ E₁ E₂ F - 1) * ADMM.ρ E₁ E₂ F * (-2 * ⟪(OptProblem.A₂ E₁) x_diff, r_n⟫_ℝ) ≤
     (ADMM.τ E₁ E₂ F - 1) * ADMM.ρ E₁ E₂ F *
       (ADMM.τ E₁ E₂ F * ‖(OptProblem.A₂ E₁) x_diff‖ ^ 2 +
         1 / ADMM.τ E₁ E₂ F *
           ‖(OptProblem.A₁ E₂) (ADMM.x₁ E₂ F ↑n) + (OptProblem.A₂ E₁) (ADMM.x₂ E₁ F ↑n) - OptProblem.b E₁ E₂‖ ^ 2)
 ⊢ (ADMM.τ E₁ E₂ F * ADMM.τ E₁ E₂ F - 1 * ADMM.τ E₁ E₂ F) * ADMM.ρ E₁ E₂ F * ‖(OptProblem.A₂ E₁) x_diff‖ ^ 2 +
       (ADMM.τ E₁ E₂ F * (1 / ADMM.τ E₁ E₂ F) - 1 * (1 / ADMM.τ E₁ E₂ F)) * ADMM.ρ E₁ E₂ F *
         ‖(OptProblem.A₁ E₂) (ADMM.x₁ E₂ F ↑n) + (OptProblem.A₂ E₁) (ADMM.x₂ E₁ F ↑n) - OptProblem.b E₁ E₂‖ ^ 2 =
     (ADMM.τ E₁ E₂ F ^ 2 - ADMM.τ E₁ E₂ F) * ADMM.ρ E₁ E₂ F * ‖(OptProblem.A₂ E₁) x_diff‖ ^ 2 +
       (1 - 1 / ADMM.τ E₁ E₂ F) * ADMM.ρ E₁ E₂ F * ‖r_n‖ ^ 2]

Relevant lemmas:
sub_mul : ∀ {α : Type u} [inst : NonUnitalNonAssocRing α] (a b c : α), (a - b) * c = a * c - b * c
mul_add : ∀ {R : Type v} [inst : Mul R] [inst_10 : Add R] [inst_11 : LeftDistribClass R] (a b c : R),
  a * (b + c) = a * b + a * c
mul_comm : ∀ {G : Type u_1} [inst : CommMagma G] (a b : G), a * b = b * a
mul_assoc : ∀ {G : Type u_1} [inst : Semigroup G] (a b c : G), a * b * c = a * (b * c)