theorem extracted_formal_statement_274.{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+)
  (h₄'' :
    ‖(OptProblem.A₂ E₁) admm_kkt.x₂ - (OptProblem.A₂ E₁) (ADMM.x₂ E₁ F ↑n)‖ =
      ‖-((OptProblem.A₂ E₁) (ADMM.x₂ E₁ F ↑n) - (OptProblem.A₂ E₁) admm_kkt.x₂)‖)
  (h₄' :
    ‖(OptProblem.A₂ E₁) (ADMM.x₂ E₁ F (↑n + 1) - ADMM.x₂ E₁ F ↑n) - (OptProblem.A₂ E₁) (e₂ (↑n + 1))‖ =
      ‖(OptProblem.A₂ E₁) (e₂ ↑n)‖)
  (h1 : ADMM.τ E₁ E₂ F ≥ 0) (h2 : ADMM.ρ E₁ E₂ F ≥ 0) : |ADMM.τ E₁ E₂ F| = ADMM.τ E₁ E₂ F := 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
inst✝ : Setting E₁ E₂ F admm admm_kkt
n : ℕ+
h₄'' :
  ‖(OptProblem.A₂ E₁) admm_kkt.x₂ - (OptProblem.A₂ E₁) (ADMM.x₂ E₁ F ↑n)‖ =
    ‖-((OptProblem.A₂ E₁) (ADMM.x₂ E₁ F ↑n) - (OptProblem.A₂ E₁) admm_kkt.x₂)‖
h₄' :
  ‖(OptProblem.A₂ E₁) (ADMM.x₂ E₁ F (↑n + 1) - ADMM.x₂ E₁ F ↑n) - (OptProblem.A₂ E₁) (e₂ (↑n + 1))‖ =
    ‖(OptProblem.A₂ E₁) (e₂ ↑n)‖
h1 : ADMM.τ E₁ E₂ F ≥ 0
h2 : ADMM.ρ E₁ E₂ F ≥ 0
⊢ |ADMM.τ E₁ E₂ F| = ADMM.τ E₁ E₂ F

Tactic: apply abs_eq_self.mpr h1

State after:

Relevant lemmas: