theorem extracted_formal_statement_408.{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} (fullrank₁ : Function.Injective ⇑(OptProblem.A₁ E₂))
  (fullrank₂ : Function.Injective ⇑(OptProblem.A₂ E₁)) [inst_9 : Setting E₁ E₂ F admm (admm_kkt₁ fullrank₁ fullrank₂)]
  (hτ : ADMM.τ E₁ E₂ F > 0) (hτ' : ADMM.τ E₁ E₂ F ≥ 0)
  (h1 : ∀ (n : ℕ), 1 / ADMM.ρ E₁ E₂ F * Φ n = (ADMM.ρ E₁ E₂ F)⁻¹ * Φ n)
  (open_Φ :
    ∀ (n : ℕ),
      (ADMM.ρ E₁ E₂ F)⁻¹ * Φ n =
        (ADMM.ρ E₁ E₂ F)⁻¹ ^ 2 * (ADMM.τ E₁ E₂ F)⁻¹ * ‖ey n‖ ^ 2 + ‖(OptProblem.A₂ E₁) (e₂ n)‖ ^ 2 +
          ((1 - ADMM.τ E₁ E₂ F) ⊔ (1 - 1 / ADMM.τ E₁ E₂ F)) *
            ‖(OptProblem.A₁ E₂) (e₁ n) + (OptProblem.A₂ E₁) (e₂ n)‖ ^ 2)
  (open_Φ' :
    ∀ (n : ℕ),
      (ADMM.ρ E₁ E₂ F)⁻¹ ^ 2 * (ADMM.τ E₁ E₂ F)⁻¹ * ‖ey n‖ ^ 2 + ‖(OptProblem.A₂ E₁) (e₂ n)‖ ^ 2 +
          ((1 - ADMM.τ E₁ E₂ F) ⊔ (1 - 1 / ADMM.τ E₁ E₂ F)) *
            ‖(OptProblem.A₁ E₂) (e₁ n) + (OptProblem.A₂ E₁) (e₂ n)‖ ^ 2 =
        ‖(OptProblem.A₂ E₁) (e₂ n)‖ ^ 2 +
          ((ADMM.ρ E₁ E₂ F)⁻¹ ^ 2 * (ADMM.τ E₁ E₂ F)⁻¹ * ‖ey n‖ ^ 2 +
            ((1 - ADMM.τ E₁ E₂ F) ⊔ (1 - 1 / ADMM.τ E₁ E₂ F)) *
              ‖(OptProblem.A₁ E₂) (e₁ n) + (OptProblem.A₂ E₁) (e₂ n)‖ ^ 2))
  (nonneg1 : ∀ (n : ℕ), (ADMM.ρ E₁ E₂ F)⁻¹ ^ 2 * (ADMM.τ E₁ E₂ F)⁻¹ * ‖ey n‖ ^ 2 ≥ 0) (n : ℕ)
  (h3 : (1 - ADMM.τ E₁ E₂ F) ⊔ (1 - 1 / ADMM.τ E₁ E₂ F) ≥ 0)
  (h_original_goal :
    ((1 - ADMM.τ E₁ E₂ F) ⊔ (1 - 1 / ADMM.τ E₁ E₂ F)) * ‖(OptProblem.A₁ E₂) (e₁ n) + (OptProblem.A₂ E₁) (e₂ n)‖ ^ 2 ≥
      0) :
  0 ≤ ‖(OptProblem.A₁ E₂) (e₁ n) + (OptProblem.A₂ E₁) (e₂ 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
fullrank₁ : Function.Injective ⇑(OptProblem.A₁ E₂)
fullrank₂ : Function.Injective ⇑(OptProblem.A₂ E₁)
n✝ : ℕ
inst✝ : Setting E₁ E₂ F admm (admm_kkt₁ fullrank₁ fullrank₂)
hτ : ADMM.τ E₁ E₂ F > 0
hτ' : ADMM.τ E₁ E₂ F ≥ 0
h1 : ∀ (n : ℕ), 1 / ADMM.ρ E₁ E₂ F * Φ n = (ADMM.ρ E₁ E₂ F)⁻¹ * Φ n
open_Φ :
  ∀ (n : ℕ),
    (ADMM.ρ E₁ E₂ F)⁻¹ * Φ n =
      (ADMM.ρ E₁ E₂ F)⁻¹ ^ 2 * (ADMM.τ E₁ E₂ F)⁻¹ * ‖ey n‖ ^ 2 + ‖(OptProblem.A₂ E₁) (e₂ n)‖ ^ 2 +
        ((1 - ADMM.τ E₁ E₂ F) ⊔ (1 - 1 / ADMM.τ E₁ E₂ F)) * ‖(OptProblem.A₁ E₂) (e₁ n) + (OptProblem.A₂ E₁) (e₂ n)‖ ^ 2
open_Φ' :
  ∀ (n : ℕ),
    (ADMM.ρ E₁ E₂ F)⁻¹ ^ 2 * (ADMM.τ E₁ E₂ F)⁻¹ * ‖ey n‖ ^ 2 + ‖(OptProblem.A₂ E₁) (e₂ n)‖ ^ 2 +
        ((1 - ADMM.τ E₁ E₂ F) ⊔ (1 - 1 / ADMM.τ E₁ E₂ F)) *
          ‖(OptProblem.A₁ E₂) (e₁ n) + (OptProblem.A₂ E₁) (e₂ n)‖ ^ 2 =
      ‖(OptProblem.A₂ E₁) (e₂ n)‖ ^ 2 +
        ((ADMM.ρ E₁ E₂ F)⁻¹ ^ 2 * (ADMM.τ E₁ E₂ F)⁻¹ * ‖ey n‖ ^ 2 +
          ((1 - ADMM.τ E₁ E₂ F) ⊔ (1 - 1 / ADMM.τ E₁ E₂ F)) *
            ‖(OptProblem.A₁ E₂) (e₁ n) + (OptProblem.A₂ E₁) (e₂ n)‖ ^ 2)
nonneg1 : ∀ (n : ℕ), (ADMM.ρ E₁ E₂ F)⁻¹ ^ 2 * (ADMM.τ E₁ E₂ F)⁻¹ * ‖ey n‖ ^ 2 ≥ 0
n : ℕ
h3 : (1 - ADMM.τ E₁ E₂ F) ⊔ (1 - 1 / ADMM.τ E₁ E₂ F) ≥ 0
⊢ ((1 - ADMM.τ E₁ E₂ F) ⊔ (1 - 1 / ADMM.τ E₁ E₂ F)) * ‖(OptProblem.A₁ E₂) (e₁ n) + (OptProblem.A₂ E₁) (e₂ n)‖ ^ 2 ≥ 0

Tactic: apply mul_nonneg h3

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
 fullrank₁ : Function.Injective ⇑(OptProblem.A₁ E₂)
 fullrank₂ : Function.Injective ⇑(OptProblem.A₂ E₁)
 n✝ : ℕ
 inst✝ : Setting E₁ E₂ F admm (admm_kkt₁ fullrank₁ fullrank₂)
 hτ : ADMM.τ E₁ E₂ F > 0
 hτ' : ADMM.τ E₁ E₂ F ≥ 0
 h1 : ∀ (n : ℕ), 1 / ADMM.ρ E₁ E₂ F * Φ n = (ADMM.ρ E₁ E₂ F)⁻¹ * Φ n
 open_Φ :
   ∀ (n : ℕ),
     (ADMM.ρ E₁ E₂ F)⁻¹ * Φ n =
       (ADMM.ρ E₁ E₂ F)⁻¹ ^ 2 * (ADMM.τ E₁ E₂ F)⁻¹ * ‖ey n‖ ^ 2 + ‖(OptProblem.A₂ E₁) (e₂ n)‖ ^ 2 +
         ((1 - ADMM.τ E₁ E₂ F) ⊔ (1 - 1 / ADMM.τ E₁ E₂ F)) * ‖(OptProblem.A₁ E₂) (e₁ n) + (OptProblem.A₂ E₁) (e₂ n)‖ ^ 2
 open_Φ' :
   ∀ (n : ℕ),
     (ADMM.ρ E₁ E₂ F)⁻¹ ^ 2 * (ADMM.τ E₁ E₂ F)⁻¹ * ‖ey n‖ ^ 2 + ‖(OptProblem.A₂ E₁) (e₂ n)‖ ^ 2 +
         ((1 - ADMM.τ E₁ E₂ F) ⊔ (1 - 1 / ADMM.τ E₁ E₂ F)) *
           ‖(OptProblem.A₁ E₂) (e₁ n) + (OptProblem.A₂ E₁) (e₂ n)‖ ^ 2 =
       ‖(OptProblem.A₂ E₁) (e₂ n)‖ ^ 2 +
         ((ADMM.ρ E₁ E₂ F)⁻¹ ^ 2 * (ADMM.τ E₁ E₂ F)⁻¹ * ‖ey n‖ ^ 2 +
           ((1 - ADMM.τ E₁ E₂ F) ⊔ (1 - 1 / ADMM.τ E₁ E₂ F)) *
             ‖(OptProblem.A₁ E₂) (e₁ n) + (OptProblem.A₂ E₁) (e₂ n)‖ ^ 2)
 nonneg1 : ∀ (n : ℕ), (ADMM.ρ E₁ E₂ F)⁻¹ ^ 2 * (ADMM.τ E₁ E₂ F)⁻¹ * ‖ey n‖ ^ 2 ≥ 0
 n : ℕ
 h3 : (1 - ADMM.τ E₁ E₂ F) ⊔ (1 - 1 / ADMM.τ E₁ E₂ F) ≥ 0
 ⊢ 0 ≤ ‖(OptProblem.A₁ E₂) (e₁ n) + (OptProblem.A₂ E₁) (e₂ n)‖ ^ 2]

Relevant lemmas:
mul_nonneg : ∀ {α : Type u_3} {a b : α} [inst : MulZeroClass α] [inst_10 : Preorder α] [inst_11 : PosMulMono α],
  0 ≤ a → 0 ≤ b → 0 ≤ a * b