import Mathlib
import Optlib


open Set InnerProductSpace Topology Filter InnerProduct

open scoped Pointwise

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} [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)‖)
  (h₆ :
    ‖ey (↑n + 1) - ey ↑n‖ =
      ADMM.τ E₁ E₂ F * ADMM.ρ E₁ E₂ F *
        ‖(OptProblem.A₁ E₂) (ADMM.x₁ E₂ F (↑n + 1)) + (OptProblem.A₂ E₁) (ADMM.x₂ E₁ F (↑n + 1)) - OptProblem.b E₁ E₂‖)
  (h_original_goal :
    (ADMM.ρ E₁ E₂ F)⁻¹ * ((ADMM.τ E₁ E₂ F)⁻¹ * ADMM.τ E₁ E₂ F * ADMM.ρ E₁ E₂ F * ADMM.τ E₁ E₂ F * ADMM.ρ E₁ E₂ F) =
      ADMM.τ E₁ E₂ F * ADMM.ρ E₁ E₂ F) :
  ADMM.τ E₁ E₂ F ≠ 0 := sorry

Tactic state before:
case h
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)‖
h₆ :
  ‖ey (↑n + 1) - ey ↑n‖ =
    ADMM.τ E₁ E₂ F * ADMM.ρ E₁ E₂ F *
      ‖(OptProblem.A₁ E₂) (ADMM.x₁ E₂ F (↑n + 1)) + (OptProblem.A₂ E₁) (ADMM.x₂ E₁ F (↑n + 1)) - OptProblem.b E₁ E₂‖
⊢ (ADMM.ρ E₁ E₂ F)⁻¹ * ((ADMM.τ E₁ E₂ F)⁻¹ * ADMM.τ E₁ E₂ F * ADMM.ρ E₁ E₂ F * ADMM.τ E₁ E₂ F * ADMM.ρ E₁ E₂ F) =
    ADMM.τ E₁ E₂ F * ADMM.ρ E₁ E₂ F

Tactic:
rw [inv_mul_cancel₀, one_mul]
    repeat rw [← mul_assoc]
    rw [inv_mul_cancel₀, one_mul]
    apply ne_of_gt admm.hrho
    rcases admm.htau with ⟨h₁, _⟩

Tactic state after:
[case h.intro
 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)‖
 h₆ :
   ‖ey (↑n + 1) - ey ↑n‖ =
     ADMM.τ E₁ E₂ F * ADMM.ρ E₁ E₂ F *
       ‖(OptProblem.A₁ E₂) (ADMM.x₁ E₂ F (↑n + 1)) + (OptProblem.A₂ E₁) (ADMM.x₂ E₁ F (↑n + 1)) - OptProblem.b E₁ E₂‖
 h₁ : 0 < ADMM.τ E₁ E₂ F
 right✝ : ADMM.τ E₁ E₂ F < (1 + √5) / 2
 ⊢ ADMM.τ E₁ E₂ F ≠ 0]