theorem extracted_formal_statement_224.{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] :
  let φn := fun n => ‖(OptProblem.A₁ E₂) (e₁ n) + (OptProblem.A₂ E₁) (e₂ n)‖ ^ 2;
  let φnsucc := fun n => φn (n + 2);
  (∀ (n : ℕ), (1 ⊓ (1 + 1 / ADMM.τ E₁ E₂ F - ADMM.τ E₁ E₂ F)) * ADMM.ρ E₁ E₂ F * φnsucc n ≥ 0) →
    (1 ⊓ (1 + 1 / ADMM.τ E₁ E₂ F - ADMM.τ E₁ E₂ F)) * ADMM.ρ E₁ E₂ F ≠ 0 →
      (Summable fun i => (1 ⊓ (1 + 1 / ADMM.τ E₁ E₂ F - ADMM.τ E₁ E₂ F)) * ADMM.ρ E₁ E₂ F * φnsucc i) →
        Summable fun n => Φ (n + 1) - Φ (n + 1 + 1) := 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
φn : ℕ → ℝ := fun n => ‖(OptProblem.A₁ E₂) (e₁ n) + (OptProblem.A₂ E₁) (e₂ n)‖ ^ 2
φnsucc : ℕ → ℝ := fun n => φn (n + 2)
φn_ge_zero' : ∀ (n : ℕ), (1 ⊓ (1 + 1 / ADMM.τ E₁ E₂ F - ADMM.τ E₁ E₂ F)) * ADMM.ρ E₁ E₂ F * φnsucc n ≥ 0
h : (1 ⊓ (1 + 1 / ADMM.τ E₁ E₂ F - ADMM.τ E₁ E₂ F)) * ADMM.ρ E₁ E₂ F ≠ 0
⊢ Summable fun i => (1 ⊓ (1 + 1 / ADMM.τ E₁ E₂ F - ADMM.τ E₁ E₂ F)) * ADMM.ρ E₁ E₂ F * φnsucc i

Tactic: apply summable_of_nonneg_of_le φn_ge_zero' Summable₁'

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
 φn : ℕ → ℝ := fun n => ‖(OptProblem.A₁ E₂) (e₁ n) + (OptProblem.A₂ E₁) (e₂ n)‖ ^ 2
 φnsucc : ℕ → ℝ := fun n => φn (n + 2)
 φn_ge_zero' : ∀ (n : ℕ), (1 ⊓ (1 + 1 / ADMM.τ E₁ E₂ F - ADMM.τ E₁ E₂ F)) * ADMM.ρ E₁ E₂ F * φnsucc n ≥ 0
 h : (1 ⊓ (1 + 1 / ADMM.τ E₁ E₂ F - ADMM.τ E₁ E₂ F)) * ADMM.ρ E₁ E₂ F ≠ 0
 ⊢ Summable fun n => Φ (n + 1) - Φ (n + 1 + 1)]

Relevant lemmas: