theorem extracted_formal_statement_16.{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)]
  (n : ℕ) (h : ADMM.ρ E₁ E₂ F * ‖(OptProblem.A₂ E₁) (e₂ n)‖ ^ 2 ≥ 0) :
  ((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 := 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)
inst✝ : Setting E₁ E₂ F admm (admm_kkt₁ fullrank₁ fullrank₂)
n : ℕ
h : ADMM.ρ E₁ E₂ F * ‖(OptProblem.A₂ E₁) (e₂ n)‖ ^ 2 ≥ 0
⊢ ((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

Tactic: apply mul_nonneg
    apply mul_nonneg
    linarith [nonneg₃, admm.hrho]
    linarith [admm.hrho]
    linarith [pow_two_nonneg (‖A₁ (e₁ n) + A₂ (e₂ n)‖)]

State after:

Relevant lemmas:
pow_two_nonneg : ∀ {α : Type u} [inst : Semiring α] [inst_10 : LinearOrder α] [inst_11 : IsRightCancelAdd α]
  [inst_12 : ZeroLEOneClass α] [inst_13 : ExistsAddOfLE α] [inst_14 : PosMulMono α] [inst_15 : AddLeftStrictMono α]
  (a : α), 0 ≤ a ^ 2
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