import Mathlib
import Optlib


open Set InnerProductSpace Topology Filter InnerProduct

open scoped Pointwise

theorem extracted_formal_statement_194.{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)
  (sub :
    ADMM.ρ E₁ E₂ F • ((OptProblem.A₂ E₁) (ADMM.x₂ E₁ F (n + 1)) - (OptProblem.A₂ E₁) (ADMM.x₂ E₁ F n)) =
      -1 • ADMM.ρ E₁ E₂ F • ((OptProblem.A₂ E₁) (ADMM.x₂ E₁ F n) - (OptProblem.A₂ E₁) (ADMM.x₂ E₁ F (n + 1)))) :
  ⟪-ey (n + 1) -
          (((1 - ADMM.τ E₁ E₂ F) * ADMM.ρ E₁ E₂ F) • (OptProblem.A₁ E₂) (e₁ (n + 1)) +
            ((1 - ADMM.τ E₁ E₂ F) * ADMM.ρ E₁ E₂ F) • (OptProblem.A₂ E₁) (e₂ (n + 1))) +
        -1 • ADMM.ρ E₁ E₂ F • ((OptProblem.A₂ E₁) (ADMM.x₂ E₁ F n) - (OptProblem.A₂ E₁) (ADMM.x₂ E₁ F (n + 1))),
      (OptProblem.A₁ E₂) (e₁ (n + 1))⟫_ℝ =
    ⟪-ey (n + 1) -
          (((1 - ADMM.τ E₁ E₂ F) * ADMM.ρ E₁ E₂ F) • (OptProblem.A₁ E₂) (e₁ (n + 1)) +
            ((1 - ADMM.τ E₁ E₂ F) * ADMM.ρ E₁ E₂ F) • (OptProblem.A₂ E₁) (e₂ (n + 1))) +
        -1 • ADMM.ρ E₁ E₂ F • ((OptProblem.A₂ E₁) (ADMM.x₂ E₁ F n) - (OptProblem.A₂ E₁) (ADMM.x₂ E₁ F (n + 1))),
      (OptProblem.A₁ E₂) (e₁ (n + 1))⟫_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))
sub :
  ADMM.ρ E₁ E₂ F • ((OptProblem.A₂ E₁) (ADMM.x₂ E₁ F (n + 1)) - (OptProblem.A₂ E₁) (ADMM.x₂ E₁ F n)) =
    -1 • ADMM.ρ E₁ E₂ F • ((OptProblem.A₂ E₁) (ADMM.x₂ E₁ F n) - (OptProblem.A₂ E₁) (ADMM.x₂ E₁ F (n + 1)))
⊢ ⟪-ey (n + 1) -
          (((1 - ADMM.τ E₁ E₂ F) * ADMM.ρ E₁ E₂ F) • (OptProblem.A₁ E₂) (e₁ (n + 1)) +
            ((1 - ADMM.τ E₁ E₂ F) * ADMM.ρ E₁ E₂ F) • (OptProblem.A₂ E₁) (e₂ (n + 1))) +
        -1 • ADMM.ρ E₁ E₂ F • ((OptProblem.A₂ E₁) (ADMM.x₂ E₁ F n) - (OptProblem.A₂ E₁) (ADMM.x₂ E₁ F (n + 1))),
      (OptProblem.A₁ E₂) (e₁ (n + 1))⟫_ℝ =
    ⟪-ey (n + 1) -
          (((1 - ADMM.τ E₁ E₂ F) * ADMM.ρ E₁ E₂ F) • (OptProblem.A₁ E₂) (e₁ (n + 1)) +
            ((1 - ADMM.τ E₁ E₂ F) * ADMM.ρ E₁ E₂ F) • (OptProblem.A₂ E₁) (e₂ (n + 1))) +
        -1 • ADMM.ρ E₁ E₂ F • ((OptProblem.A₂ E₁) (ADMM.x₂ E₁ F n) - (OptProblem.A₂ E₁) (ADMM.x₂ E₁ F (n + 1))),
      (OptProblem.A₁ E₂) (e₁ (n + 1))⟫_ℝ

Tactic:
simp only [Int.reduceNeg, neg_smul, one_smul]

Tactic state after: