theorem extracted_formal_statement_294.{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} (n : ℕ+)
  (norm_abs :
    ‖(OptProblem.A₂ E₁) (ADMM.x₂ E₁ F ↑n - ADMM.x₂ E₁ F (↑n + 1))‖ ^ 2 =
      ‖(OptProblem.A₂ E₁) (ADMM.x₂ E₁ F (↑n + 1) - ADMM.x₂ E₁ F ↑n)‖ ^ 2)
  (this :
    ‖(OptProblem.A₂ E₁) (ADMM.x₂ E₁ F ↑n - ADMM.x₂ E₁ F (↑n + 1))‖ ^ 2 +
          ‖(OptProblem.A₁ E₂) (ADMM.x₁ E₂ F ↑n) + (OptProblem.A₂ E₁) (ADMM.x₂ E₁ F ↑n) - OptProblem.b E₁ E₂‖ ^ 2 -
        2 *
          ⟪(OptProblem.A₂ E₁) (ADMM.x₂ E₁ F (↑n + 1) - ADMM.x₂ E₁ F ↑n),
            (OptProblem.A₁ E₂) (ADMM.x₁ E₂ F ↑n) + (OptProblem.A₂ E₁) (ADMM.x₂ E₁ F ↑n) - OptProblem.b E₁ E₂⟫_ℝ =
      ‖(OptProblem.A₂ E₁) (ADMM.x₂ E₁ F ↑n - ADMM.x₂ E₁ F (↑n + 1))‖ ^ 2 -
          2 *
            ⟪(OptProblem.A₂ E₁) (ADMM.x₂ E₁ F (↑n + 1) - ADMM.x₂ E₁ F ↑n),
              (OptProblem.A₁ E₂) (ADMM.x₁ E₂ F ↑n) + (OptProblem.A₂ E₁) (ADMM.x₂ E₁ F ↑n) - OptProblem.b E₁ E₂⟫_ℝ +
        ‖(OptProblem.A₁ E₂) (ADMM.x₁ E₂ F ↑n) + (OptProblem.A₂ E₁) (ADMM.x₂ E₁ F ↑n) - OptProblem.b E₁ E₂‖ ^ 2) :
  0 ≤
    ‖(OptProblem.A₂ E₁) (ADMM.x₂ E₁ F (↑n + 1) - ADMM.x₂ E₁ F ↑n) -
          ((OptProblem.A₁ E₂) (ADMM.x₁ E₂ F ↑n) + (OptProblem.A₂ E₁) (ADMM.x₂ E₁ F ↑n) - OptProblem.b E₁ E₂)‖ ^
      2 := sorry

State before:
E₁ : Type u_1
E₂ : Type u_2
F : Type u_3
inst✝⁸ : NormedAddCommGroup E₁
inst✝⁷ : InnerProductSpace ℝ E₁
inst✝⁶ : FiniteDimensional ℝ E₁
inst✝⁵ : NormedAddCommGroup E₂
inst✝⁴ : InnerProductSpace ℝ E₂
inst✝³ : FiniteDimensional ℝ E₂
inst✝² : NormedAddCommGroup F
inst✝¹ : InnerProductSpace ℝ F
inst✝ : FiniteDimensional ℝ F
admm : ADMM E₁ E₂ F
n : ℕ+
norm_abs :
  ‖(OptProblem.A₂ E₁) (ADMM.x₂ E₁ F ↑n - ADMM.x₂ E₁ F (↑n + 1))‖ ^ 2 =
    ‖(OptProblem.A₂ E₁) (ADMM.x₂ E₁ F (↑n + 1) - ADMM.x₂ E₁ F ↑n)‖ ^ 2
this :
  ‖(OptProblem.A₂ E₁) (ADMM.x₂ E₁ F ↑n - ADMM.x₂ E₁ F (↑n + 1))‖ ^ 2 +
        ‖(OptProblem.A₁ E₂) (ADMM.x₁ E₂ F ↑n) + (OptProblem.A₂ E₁) (ADMM.x₂ E₁ F ↑n) - OptProblem.b E₁ E₂‖ ^ 2 -
      2 *
        ⟪(OptProblem.A₂ E₁) (ADMM.x₂ E₁ F (↑n + 1) - ADMM.x₂ E₁ F ↑n),
          (OptProblem.A₁ E₂) (ADMM.x₁ E₂ F ↑n) + (OptProblem.A₂ E₁) (ADMM.x₂ E₁ F ↑n) - OptProblem.b E₁ E₂⟫_ℝ =
    ‖(OptProblem.A₂ E₁) (ADMM.x₂ E₁ F ↑n - ADMM.x₂ E₁ F (↑n + 1))‖ ^ 2 -
        2 *
          ⟪(OptProblem.A₂ E₁) (ADMM.x₂ E₁ F (↑n + 1) - ADMM.x₂ E₁ F ↑n),
            (OptProblem.A₁ E₂) (ADMM.x₁ E₂ F ↑n) + (OptProblem.A₂ E₁) (ADMM.x₂ E₁ F ↑n) - OptProblem.b E₁ E₂⟫_ℝ +
      ‖(OptProblem.A₁ E₂) (ADMM.x₁ E₂ F ↑n) + (OptProblem.A₂ E₁) (ADMM.x₂ E₁ F ↑n) - OptProblem.b E₁ E₂‖ ^ 2
⊢ 0 ≤
    ‖(OptProblem.A₂ E₁) (ADMM.x₂ E₁ F (↑n + 1) - ADMM.x₂ E₁ F ↑n) -
          ((OptProblem.A₁ E₂) (ADMM.x₁ E₂ F ↑n) + (OptProblem.A₂ E₁) (ADMM.x₂ E₁ F ↑n) - OptProblem.b E₁ E₂)‖ ^
      2

Tactic: apply pow_two_nonneg

State after:

Relevant lemmas:
pow_two_nonneg : ∀ {α : Type u} [inst : Semiring α] [inst_9 : LinearOrder α] [inst_10 : IsRightCancelAdd α]
  [inst_11 : ZeroLEOneClass α] [inst_12 : ExistsAddOfLE α] [inst_13 : PosMulMono α] [inst_14 : AddLeftStrictMono α]
  (a : α), 0 ≤ a ^ 2