theorem extracted_formal_statement_23.{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] (r_Φ : ℝ)
  (hΦ : range Φ ⊆ ball 0 r_Φ)
  (h1 :
    ∀ (k : ℕ+),
      (1 ⊓ (1 + 1 / ADMM.τ E₁ E₂ F - ADMM.τ E₁ E₂ F)) * ADMM.ρ E₁ E₂ F *
          ‖(OptProblem.A₁ E₂) (e₁ (↑k + 1)) + (OptProblem.A₂ E₁) (e₂ (↑k + 1))‖ ^ 2 ≤
        (ADMM.τ E₁ E₂ F ⊓ (1 + ADMM.τ E₁ E₂ F - ADMM.τ E₁ E₂ F ^ 2)) * ADMM.ρ E₁ E₂ F *
            ‖(OptProblem.A₂ E₁) (ADMM.x₂ E₁ F ↑k - ADMM.x₂ E₁ F (↑k + 1))‖ ^ 2 +
          (1 ⊓ (1 + 1 / ADMM.τ E₁ E₂ F - ADMM.τ E₁ E₂ F)) * ADMM.ρ E₁ E₂ F *
            ‖(OptProblem.A₁ E₂) (e₁ (↑k + 1)) + (OptProblem.A₂ E₁) (e₂ (↑k + 1))‖ ^ 2)
  (h2 :
    ∀ (k : ℕ+),
      (ADMM.τ E₁ E₂ F ⊓ (1 + ADMM.τ E₁ E₂ F - ADMM.τ E₁ E₂ F ^ 2)) * ADMM.ρ E₁ E₂ F *
            ‖(OptProblem.A₂ E₁) (ADMM.x₂ E₁ F ↑k - ADMM.x₂ E₁ F (↑k + 1))‖ ^ 2 +
          (1 ⊓ (1 + 1 / ADMM.τ E₁ E₂ F - ADMM.τ E₁ E₂ F)) * ADMM.ρ E₁ E₂ F *
            ‖(OptProblem.A₁ E₂) (e₁ (↑k + 1)) + (OptProblem.A₂ E₁) (e₂ (↑k + 1))‖ ^ 2 ≤
        Φ ↑k - Φ (↑k + 1))
  (h3 : ∀ (k : ℕ+), Φ ↑k - Φ (↑k + 1) < 2 * r_Φ)
  (h4 :
    ∀ (k : ℕ+),
      (1 ⊓ (1 + 1 / ADMM.τ E₁ E₂ F - ADMM.τ E₁ E₂ F)) * ADMM.ρ E₁ E₂ F *
          ‖(OptProblem.A₁ E₂) (e₁ (↑k + 1)) + (OptProblem.A₂ E₁) (e₂ (↑k + 1))‖ ^ 2 <
        2 * r_Φ)
  (h5 :
    ∀ (k : ℕ+),
      ‖(OptProblem.A₁ E₂) (e₁ (↑k + 1)) + (OptProblem.A₂ E₁) (e₂ (↑k + 1))‖ <
        √(2 * r_Φ / ((1 ⊓ (1 + 1 / ADMM.τ E₁ E₂ F - ADMM.τ E₁ E₂ F)) * ADMM.ρ E₁ E₂ F)))
  (h5' :
    ∀ (n : ℕ),
      1 < n →
        ‖(OptProblem.A₁ E₂) (e₁ n) + (OptProblem.A₂ E₁) (e₂ n)‖ <
          √(2 * r_Φ / ((1 ⊓ (1 + 1 / ADMM.τ E₁ E₂ F - ADMM.τ E₁ E₂ F)) * ADMM.ρ E₁ E₂ F)))
  (h_n0 :
    ‖(OptProblem.A₁ E₂) (e₁ 0) + (OptProblem.A₂ E₁) (e₂ 0)‖ <
      ‖(OptProblem.A₁ E₂) (e₁ 0) + (OptProblem.A₂ E₁) (e₂ 0)‖ + 1)
  (h_n1 :
    ‖(OptProblem.A₁ E₂) (e₁ 1) + (OptProblem.A₂ E₁) (e₂ 1)‖ <
      ‖(OptProblem.A₁ E₂) (e₁ 1) + (OptProblem.A₂ E₁) (e₂ 1)‖ + 1) :
  let r :=
    √(2 * r_Φ / ((1 ⊓ (1 + 1 / ADMM.τ E₁ E₂ F - ADMM.τ E₁ E₂ F)) * ADMM.ρ E₁ E₂ F)) ⊔
        (‖(OptProblem.A₁ E₂) (e₁ 0) + (OptProblem.A₂ E₁) (e₂ 0)‖ + 1) ⊔
      (‖(OptProblem.A₁ E₂) (e₁ 1) + (OptProblem.A₂ E₁) (e₂ 1)‖ + 1);
  r =
      √(2 * r_Φ / ((1 ⊓ (1 + 1 / ADMM.τ E₁ E₂ F - ADMM.τ E₁ E₂ F)) * ADMM.ρ E₁ E₂ F)) ⊔
          (‖(OptProblem.A₁ E₂) (e₁ 0) + (OptProblem.A₂ E₁) (e₂ 0)‖ + 1) ⊔
        (‖(OptProblem.A₁ E₂) (e₁ 1) + (OptProblem.A₂ E₁) (e₂ 1)‖ + 1) →
    ∀ (n : ℕ),
      (∀ (n : ℕ), ‖(OptProblem.A₁ E₂) (e₁ n) + (OptProblem.A₂ E₁) (e₂ n)‖ < r) →
        dist ((OptProblem.A₁ E₂) (e₁ n) + (OptProblem.A₂ E₁) (e₂ n)) 0 < r →
          (⇑(OptProblem.A₁ E₂) ∘ e₁ + ⇑(OptProblem.A₂ E₁) ∘ e₂) n ∈ ball 0 r →
            ‖(OptProblem.A₁ E₂) (e₁ n) + (OptProblem.A₂ E₁) (e₂ n)‖ < r := sorry

State before:
case h.intro
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
r_Φ : ℝ
hΦ : range Φ ⊆ ball 0 r_Φ
h1 :
  ∀ (k : ℕ+),
    (1 ⊓ (1 + 1 / ADMM.τ E₁ E₂ F - ADMM.τ E₁ E₂ F)) * ADMM.ρ E₁ E₂ F *
        ‖(OptProblem.A₁ E₂) (e₁ (↑k + 1)) + (OptProblem.A₂ E₁) (e₂ (↑k + 1))‖ ^ 2 ≤
      (ADMM.τ E₁ E₂ F ⊓ (1 + ADMM.τ E₁ E₂ F - ADMM.τ E₁ E₂ F ^ 2)) * ADMM.ρ E₁ E₂ F *
          ‖(OptProblem.A₂ E₁) (ADMM.x₂ E₁ F ↑k - ADMM.x₂ E₁ F (↑k + 1))‖ ^ 2 +
        (1 ⊓ (1 + 1 / ADMM.τ E₁ E₂ F - ADMM.τ E₁ E₂ F)) * ADMM.ρ E₁ E₂ F *
          ‖(OptProblem.A₁ E₂) (e₁ (↑k + 1)) + (OptProblem.A₂ E₁) (e₂ (↑k + 1))‖ ^ 2
h2 :
  ∀ (k : ℕ+),
    (ADMM.τ E₁ E₂ F ⊓ (1 + ADMM.τ E₁ E₂ F - ADMM.τ E₁ E₂ F ^ 2)) * ADMM.ρ E₁ E₂ F *
          ‖(OptProblem.A₂ E₁) (ADMM.x₂ E₁ F ↑k - ADMM.x₂ E₁ F (↑k + 1))‖ ^ 2 +
        (1 ⊓ (1 + 1 / ADMM.τ E₁ E₂ F - ADMM.τ E₁ E₂ F)) * ADMM.ρ E₁ E₂ F *
          ‖(OptProblem.A₁ E₂) (e₁ (↑k + 1)) + (OptProblem.A₂ E₁) (e₂ (↑k + 1))‖ ^ 2 ≤
      Φ ↑k - Φ (↑k + 1)
h3 : ∀ (k : ℕ+), Φ ↑k - Φ (↑k + 1) < 2 * r_Φ
h4 :
  ∀ (k : ℕ+),
    (1 ⊓ (1 + 1 / ADMM.τ E₁ E₂ F - ADMM.τ E₁ E₂ F)) * ADMM.ρ E₁ E₂ F *
        ‖(OptProblem.A₁ E₂) (e₁ (↑k + 1)) + (OptProblem.A₂ E₁) (e₂ (↑k + 1))‖ ^ 2 <
      2 * r_Φ
h5 :
  ∀ (k : ℕ+),
    ‖(OptProblem.A₁ E₂) (e₁ (↑k + 1)) + (OptProblem.A₂ E₁) (e₂ (↑k + 1))‖ <
      √(2 * r_Φ / ((1 ⊓ (1 + 1 / ADMM.τ E₁ E₂ F - ADMM.τ E₁ E₂ F)) * ADMM.ρ E₁ E₂ F))
h5' :
  ∀ (n : ℕ),
    1 < n →
      ‖(OptProblem.A₁ E₂) (e₁ n) + (OptProblem.A₂ E₁) (e₂ n)‖ <
        √(2 * r_Φ / ((1 ⊓ (1 + 1 / ADMM.τ E₁ E₂ F - ADMM.τ E₁ E₂ F)) * ADMM.ρ E₁ E₂ F))
h_n0 :
  ‖(OptProblem.A₁ E₂) (e₁ 0) + (OptProblem.A₂ E₁) (e₂ 0)‖ < ‖(OptProblem.A₁ E₂) (e₁ 0) + (OptProblem.A₂ E₁) (e₂ 0)‖ + 1
h_n1 :
  ‖(OptProblem.A₁ E₂) (e₁ 1) + (OptProblem.A₂ E₁) (e₂ 1)‖ < ‖(OptProblem.A₁ E₂) (e₁ 1) + (OptProblem.A₂ E₁) (e₂ 1)‖ + 1
r : ℝ :=
  √(2 * r_Φ / ((1 ⊓ (1 + 1 / ADMM.τ E₁ E₂ F - ADMM.τ E₁ E₂ F)) * ADMM.ρ E₁ E₂ F)) ⊔
      (‖(OptProblem.A₁ E₂) (e₁ 0) + (OptProblem.A₂ E₁) (e₂ 0)‖ + 1) ⊔
    (‖(OptProblem.A₁ E₂) (e₁ 1) + (OptProblem.A₂ E₁) (e₂ 1)‖ + 1)
hr' :
  r =
    √(2 * r_Φ / ((1 ⊓ (1 + 1 / ADMM.τ E₁ E₂ F - ADMM.τ E₁ E₂ F)) * ADMM.ρ E₁ E₂ F)) ⊔
        (‖(OptProblem.A₁ E₂) (e₁ 0) + (OptProblem.A₂ E₁) (e₂ 0)‖ + 1) ⊔
      (‖(OptProblem.A₁ E₂) (e₁ 1) + (OptProblem.A₂ E₁) (e₂ 1)‖ + 1)
n : ℕ
h_n : ∀ (n : ℕ), ‖(OptProblem.A₁ E₂) (e₁ n) + (OptProblem.A₂ E₁) (e₂ n)‖ < r
h6 : dist ((OptProblem.A₁ E₂) (e₁ n) + (OptProblem.A₂ E₁) (e₂ n)) 0 < r
⊢ (⇑(OptProblem.A₁ E₂) ∘ e₁ + ⇑(OptProblem.A₂ E₁) ∘ e₂) n ∈ ball 0 r

Tactic: rw [← Metric.mem_ball] at h6
    simp
    simp at h6

State after:
[case h.intro
 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
 r_Φ : ℝ
 hΦ : range Φ ⊆ ball 0 r_Φ
 h1 :
   ∀ (k : ℕ+),
     (1 ⊓ (1 + 1 / ADMM.τ E₁ E₂ F - ADMM.τ E₁ E₂ F)) * ADMM.ρ E₁ E₂ F *
         ‖(OptProblem.A₁ E₂) (e₁ (↑k + 1)) + (OptProblem.A₂ E₁) (e₂ (↑k + 1))‖ ^ 2 ≤
       (ADMM.τ E₁ E₂ F ⊓ (1 + ADMM.τ E₁ E₂ F - ADMM.τ E₁ E₂ F ^ 2)) * ADMM.ρ E₁ E₂ F *
           ‖(OptProblem.A₂ E₁) (ADMM.x₂ E₁ F ↑k - ADMM.x₂ E₁ F (↑k + 1))‖ ^ 2 +
         (1 ⊓ (1 + 1 / ADMM.τ E₁ E₂ F - ADMM.τ E₁ E₂ F)) * ADMM.ρ E₁ E₂ F *
           ‖(OptProblem.A₁ E₂) (e₁ (↑k + 1)) + (OptProblem.A₂ E₁) (e₂ (↑k + 1))‖ ^ 2
 h2 :
   ∀ (k : ℕ+),
     (ADMM.τ E₁ E₂ F ⊓ (1 + ADMM.τ E₁ E₂ F - ADMM.τ E₁ E₂ F ^ 2)) * ADMM.ρ E₁ E₂ F *
           ‖(OptProblem.A₂ E₁) (ADMM.x₂ E₁ F ↑k - ADMM.x₂ E₁ F (↑k + 1))‖ ^ 2 +
         (1 ⊓ (1 + 1 / ADMM.τ E₁ E₂ F - ADMM.τ E₁ E₂ F)) * ADMM.ρ E₁ E₂ F *
           ‖(OptProblem.A₁ E₂) (e₁ (↑k + 1)) + (OptProblem.A₂ E₁) (e₂ (↑k + 1))‖ ^ 2 ≤
       Φ ↑k - Φ (↑k + 1)
 h3 : ∀ (k : ℕ+), Φ ↑k - Φ (↑k + 1) < 2 * r_Φ
 h4 :
   ∀ (k : ℕ+),
     (1 ⊓ (1 + 1 / ADMM.τ E₁ E₂ F - ADMM.τ E₁ E₂ F)) * ADMM.ρ E₁ E₂ F *
         ‖(OptProblem.A₁ E₂) (e₁ (↑k + 1)) + (OptProblem.A₂ E₁) (e₂ (↑k + 1))‖ ^ 2 <
       2 * r_Φ
 h5 :
   ∀ (k : ℕ+),
     ‖(OptProblem.A₁ E₂) (e₁ (↑k + 1)) + (OptProblem.A₂ E₁) (e₂ (↑k + 1))‖ <
       √(2 * r_Φ / ((1 ⊓ (1 + 1 / ADMM.τ E₁ E₂ F - ADMM.τ E₁ E₂ F)) * ADMM.ρ E₁ E₂ F))
 h5' :
   ∀ (n : ℕ),
     1 < n →
       ‖(OptProblem.A₁ E₂) (e₁ n) + (OptProblem.A₂ E₁) (e₂ n)‖ <
         √(2 * r_Φ / ((1 ⊓ (1 + 1 / ADMM.τ E₁ E₂ F - ADMM.τ E₁ E₂ F)) * ADMM.ρ E₁ E₂ F))
 h_n0 :
   ‖(OptProblem.A₁ E₂) (e₁ 0) + (OptProblem.A₂ E₁) (e₂ 0)‖ < ‖(OptProblem.A₁ E₂) (e₁ 0) + (OptProblem.A₂ E₁) (e₂ 0)‖ + 1
 h_n1 :
   ‖(OptProblem.A₁ E₂) (e₁ 1) + (OptProblem.A₂ E₁) (e₂ 1)‖ < ‖(OptProblem.A₁ E₂) (e₁ 1) + (OptProblem.A₂ E₁) (e₂ 1)‖ + 1
 r : ℝ :=
   √(2 * r_Φ / ((1 ⊓ (1 + 1 / ADMM.τ E₁ E₂ F - ADMM.τ E₁ E₂ F)) * ADMM.ρ E₁ E₂ F)) ⊔
       (‖(OptProblem.A₁ E₂) (e₁ 0) + (OptProblem.A₂ E₁) (e₂ 0)‖ + 1) ⊔
     (‖(OptProblem.A₁ E₂) (e₁ 1) + (OptProblem.A₂ E₁) (e₂ 1)‖ + 1)
 hr' :
   r =
     √(2 * r_Φ / ((1 ⊓ (1 + 1 / ADMM.τ E₁ E₂ F - ADMM.τ E₁ E₂ F)) * ADMM.ρ E₁ E₂ F)) ⊔
         (‖(OptProblem.A₁ E₂) (e₁ 0) + (OptProblem.A₂ E₁) (e₂ 0)‖ + 1) ⊔
       (‖(OptProblem.A₁ E₂) (e₁ 1) + (OptProblem.A₂ E₁) (e₂ 1)‖ + 1)
 n : ℕ
 h_n : ∀ (n : ℕ), ‖(OptProblem.A₁ E₂) (e₁ n) + (OptProblem.A₂ E₁) (e₂ n)‖ < r
 h6 : ‖(OptProblem.A₁ E₂) (e₁ n) + (OptProblem.A₂ E₁) (e₂ n)‖ < r
 ⊢ ‖(OptProblem.A₁ E₂) (e₁ n) + (OptProblem.A₂ E₁) (e₂ n)‖ < r]

Relevant lemmas:
Metric.mem_ball : ∀ {α : Type u} [inst : PseudoMetricSpace α] {x y : α} {ε : ℝ}, y ∈ ball x ε ↔ dist y x < ε