theorem extracted_formal_statement_9.{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)
  (h1 : 0 < ADMM.τ E₁ E₂ F) (right : ADMM.τ E₁ E₂ F < (1 + √5) / 2) (h_original_goal : Φ n ≥ 0) :
  0 ≤
    Ψ n +
      ((1 - ADMM.τ E₁ E₂ F) ⊔ (1 - (ADMM.τ E₁ E₂ F)⁻¹)) * ADMM.ρ E₁ E₂ F *
        ‖(OptProblem.A₁ E₂) (e₁ n) + (OptProblem.A₂ E₁) (e₂ n)‖ ^ 2 := sorry

State before:
case 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
n : ℕ
h1 : 0 < ADMM.τ E₁ E₂ F
right✝ : ADMM.τ E₁ E₂ F < (1 + √5) / 2
⊢ Φ n ≥ 0

Tactic: simp [Φ]
    have h2 : Ψ n ≥ 0 := by
  simp [Ψ]; apply add_nonneg; repeat apply mul_nonneg
  apply inv_nonneg_of_nonneg; apply le_of_lt admm.hrho
  apply inv_nonneg_of_nonneg; apply le_of_lt h1
  apply pow_two_nonneg; apply mul_nonneg; apply le_of_lt admm.hrho; apply pow_two_nonneg
    have h3 : ((max (1 - τ) (1 - 1 / τ)) * ρ) * ‖A₁ ((e₁) n) + A₂ ((e₂) n)‖ ^ 2 ≥ 0 :=
  by
  repeat apply mul_nonneg
  to_theorem apply nonneg₃ apply le_of_lt admm.hrho apply pow_two_nonneg

State after:
[case 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
 n : ℕ
 h1 : 0 < ADMM.τ E₁ E₂ F
 right✝ : ADMM.τ E₁ E₂ F < (1 + √5) / 2
 h2 : Ψ n ≥ 0
 h3 :
   ((1 - ADMM.τ E₁ E₂ F) ⊔ (1 - 1 / ADMM.τ E₁ E₂ F)) * ADMM.ρ E₁ E₂ F *
       ‖(OptProblem.A₁ E₂) (e₁ n) + (OptProblem.A₂ E₁) (e₂ n)‖ ^ 2 ≥
     0
 ⊢ 0 ≤
     Ψ n +
       ((1 - ADMM.τ E₁ E₂ F) ⊔ (1 - (ADMM.τ E₁ E₂ F)⁻¹)) * ADMM.ρ E₁ E₂ F *
         ‖(OptProblem.A₁ E₂) (e₁ n) + (OptProblem.A₂ E₁) (e₂ n)‖ ^ 2]

Relevant lemmas:
error