import Mathlib
import Optlib


open Set InnerProductSpace Topology Filter InnerProduct

open scoped Pointwise

theorem extracted_formal_statement_27.{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) :
  let A_e_sum := (OptProblem.A₁ E₂) (e₁ (n + 1)) + (OptProblem.A₂ E₁) (e₂ (n + 1));
  let A_x_sum := -(OptProblem.A₂ E₁) (ADMM.x₂ E₁ F n - ADMM.x₂ E₁ F (n + 1));
  let ey' := ey (n + 1);
  let Ae1 := (OptProblem.A₁ E₂) (e₁ (n + 1));
  ⟪-ey', A_e_sum⟫_ℝ + ⟪-((1 - ADMM.τ E₁ E₂ F) * ADMM.ρ E₁ E₂ F) • A_e_sum, A_e_sum⟫_ℝ +
        ADMM.ρ E₁ E₂ F * ⟪A_x_sum, Ae1⟫_ℝ =
      ⟪-ey' - ((1 - ADMM.τ E₁ E₂ F) * ADMM.ρ E₁ E₂ F) • A_e_sum, A_e_sum⟫_ℝ + ADMM.ρ E₁ E₂ F * ⟪A_x_sum, Ae1⟫_ℝ →
    ADMM.ρ E₁ E₂ F * ⟪A_x_sum, Ae1⟫_ℝ +
        ⟪-ey' + (ADMM.τ E₁ E₂ F * ADMM.ρ E₁ E₂ F - ADMM.ρ E₁ E₂ F) • A_e_sum, A_e_sum⟫_ℝ =
      ADMM.ρ E₁ E₂ F * ⟪A_x_sum, Ae1⟫_ℝ +
        ⟪-ey' - (-(ADMM.τ E₁ E₂ F * ADMM.ρ E₁ E₂ F) + ADMM.ρ E₁ E₂ F) • A_e_sum, A_e_sum⟫_ℝ := sorry

Tactic 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
A_e_sum : F := (OptProblem.A₁ E₂) (e₁ (n + 1)) + (OptProblem.A₂ E₁) (e₂ (n + 1))
A_x_sum : F := -(OptProblem.A₂ E₁) (ADMM.x₂ E₁ F n - ADMM.x₂ E₁ F (n + 1))
ey' : F := ey (n + 1)
Ae1 : F := (OptProblem.A₁ E₂) (e₁ (n + 1))
Ae2 : F := (OptProblem.A₂ E₁) (e₂ (n + 1))
⊢ ⟪-ey', A_e_sum⟫_ℝ + ⟪-((1 - ADMM.τ E₁ E₂ F) * ADMM.ρ E₁ E₂ F) • A_e_sum, A_e_sum⟫_ℝ +
      ADMM.ρ E₁ E₂ F * ⟪A_x_sum, Ae1⟫_ℝ =
    ⟪-ey' - ((1 - ADMM.τ E₁ E₂ F) * ADMM.ρ E₁ E₂ F) • A_e_sum, A_e_sum⟫_ℝ + ADMM.ρ E₁ E₂ F * ⟪A_x_sum, Ae1⟫_ℝ

Tactic:
rw [← add_left]
    ring_nf
    have sub : -ey' + (τ * ρ - ρ) • A_e_sum = -ey' - (-(τ * ρ) + ρ) • A_e_sum :=
  by
  to_theorem
    rw [← sub_eq_zero,
      sub_eq_add_neg]rw [sub_eq_add_neg (G := F) (-ey')
        ((-(τ * ρ) + ρ) • A_e_sum)]rw [← neg_one_smul (R := ℝ) (-ey' + -((-(τ * ρ) + ρ) • A_e_sum))]
  to_theorem
    rw [smul_add (-1) (-ey')
        (-((-(τ * ρ) + ρ) •
            A_e_sum))]rw [neg_smul_neg, neg_smul_neg, one_smul,
      one_smul]rw [add_assoc, add_comm, add_assoc, add_comm ey' ((-(τ * ρ) + ρ) • A_e_sum)]
  to_theorem rw [add_assoc]rw [add_neg_cancel, add_zero]rw [← add_smul (τ * ρ - ρ) (-(τ * ρ) + ρ) (A_e_sum)]
  rw [add_comm (-(τ * ρ)) ρ, ← add_assoc]
  rw [sub_eq_add_neg, add_assoc (τ * ρ) (-ρ) ρ, add_comm (-ρ) ρ, add_neg_cancel, add_zero, add_neg_cancel, zero_smul]

Tactic state after:
[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
 A_e_sum : F := (OptProblem.A₁ E₂) (e₁ (n + 1)) + (OptProblem.A₂ E₁) (e₂ (n + 1))
 A_x_sum : F := -(OptProblem.A₂ E₁) (ADMM.x₂ E₁ F n - ADMM.x₂ E₁ F (n + 1))
 ey' : F := ey (n + 1)
 Ae1 : F := (OptProblem.A₁ E₂) (e₁ (n + 1))
 Ae2 : F := (OptProblem.A₂ E₁) (e₂ (n + 1))
 sub :
   -ey' + (ADMM.τ E₁ E₂ F * ADMM.ρ E₁ E₂ F - ADMM.ρ E₁ E₂ F) • A_e_sum =
     -ey' - (-(ADMM.τ E₁ E₂ F * ADMM.ρ E₁ E₂ F) + ADMM.ρ E₁ E₂ F) • A_e_sum
 ⊢ ADMM.ρ E₁ E₂ F * ⟪A_x_sum, Ae1⟫_ℝ +
       ⟪-ey' + (ADMM.τ E₁ E₂ F * ADMM.ρ E₁ E₂ F - ADMM.ρ E₁ E₂ F) • A_e_sum, A_e_sum⟫_ℝ =
     ADMM.ρ E₁ E₂ F * ⟪A_x_sum, Ae1⟫_ℝ +
       ⟪-ey' - (-(ADMM.τ E₁ E₂ F * ADMM.ρ E₁ E₂ F) + ADMM.ρ E₁ E₂ F) • A_e_sum, A_e_sum⟫_ℝ]