theorem extracted_formal_statement_130.{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+)
  (h :
    (ADMM.τ E₁ E₂ F ⊓ (1 + ADMM.τ E₁ E₂ F - ADMM.τ E₁ E₂ F ^ 2)) * ADMM.ρ E₁ E₂ F *
          ‖(OptProblem.A₂ E₁) (ADMM.x₂ E₁ F ↑n - ADMM.x₂ E₁ F (↑n + 1))‖ ^ 2 +
        (1 ⊓ (1 + 1 / ADMM.τ E₁ E₂ F - ADMM.τ E₁ E₂ F)) * ADMM.ρ E₁ E₂ F *
          ‖(OptProblem.A₁ E₂) (e₁ (↑n + 1)) + (OptProblem.A₂ E₁) (e₂ (↑n + 1))‖ ^ 2 ≥
      0) :
  (ADMM.τ E₁ E₂ F ⊓ (1 + ADMM.τ E₁ E₂ F - ADMM.τ E₁ E₂ F ^ 2)) * ADMM.ρ E₁ E₂ F *
        ‖(OptProblem.A₂ E₁) (ADMM.x₂ E₁ F ↑n - ADMM.x₂ E₁ F (↑n + 1))‖ ^ 2 +
      (1 ⊓ (1 + 1 / ADMM.τ E₁ E₂ F - ADMM.τ E₁ E₂ F)) * ADMM.ρ E₁ E₂ F *
        ‖(OptProblem.A₁ E₂) (e₁ (↑n + 1)) + (OptProblem.A₂ E₁) (e₂ (↑n + 1))‖ ^ 2 ≤
    Φ ↑n - Φ (↑n + 1) := sorry

State before:
case a
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+
h :
  (ADMM.τ E₁ E₂ F ⊓ (1 + ADMM.τ E₁ E₂ F - ADMM.τ E₁ E₂ F ^ 2)) * ADMM.ρ E₁ E₂ F *
        ‖(OptProblem.A₂ E₁) (ADMM.x₂ E₁ F ↑n - ADMM.x₂ E₁ F (↑n + 1))‖ ^ 2 +
      (1 ⊓ (1 + 1 / ADMM.τ E₁ E₂ F - ADMM.τ E₁ E₂ F)) * ADMM.ρ E₁ E₂ F *
        ‖(OptProblem.A₁ E₂) (e₁ (↑n + 1)) + (OptProblem.A₂ E₁) (e₂ (↑n + 1))‖ ^ 2 ≥
    0
⊢ (ADMM.τ E₁ E₂ F ⊓ (1 + ADMM.τ E₁ E₂ F - ADMM.τ E₁ E₂ F ^ 2)) * ADMM.ρ E₁ E₂ F *
        ‖(OptProblem.A₂ E₁) (ADMM.x₂ E₁ F ↑n - ADMM.x₂ E₁ F (↑n + 1))‖ ^ 2 +
      (1 ⊓ (1 + 1 / ADMM.τ E₁ E₂ F - ADMM.τ E₁ E₂ F)) * ADMM.ρ E₁ E₂ F *
        ‖(OptProblem.A₁ E₂) (e₁ (↑n + 1)) + (OptProblem.A₂ E₁) (e₂ (↑n + 1))‖ ^ 2 ≤
    Φ ↑n - Φ (↑n + 1)

Tactic: apply Φ_isdescending

State after:

Relevant lemmas: