theorem extracted_formal_statement_50.{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) (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 - 1 / ADMM.τ E₁ E₂ F)) * 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
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 - 1 / ADMM.τ E₁ E₂ F)) * ADMM.ρ E₁ E₂ F *
        ‖(OptProblem.A₁ E₂) (e₁ n) + (OptProblem.A₂ E₁) (e₂ n)‖ ^ 2

Tactic: apply add_nonneg h2 h3

State after:

Relevant lemmas:
add_nonneg : ∀ {α : Type u_1} [inst : AddZeroClass α] [inst_10 : Preorder α] [inst_11 : AddLeftMono α] {a b : α},
  0 ≤ a → 0 ≤ b → 0 ≤ a + b