import Mathlib
import Optlib


open Set InnerProductSpace Topology Filter InnerProduct

open scoped Pointwise

theorem extracted_formal_statement_163.{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 block :=
    -ey (n + 1) -
        ((1 - ADMM.τ E₁ E₂ F) * ADMM.ρ E₁ E₂ F) • ((OptProblem.A₁ E₂) (e₁ (n + 1)) + (OptProblem.A₂ E₁) (e₂ (n + 1))) -
      ADMM.ρ E₁ E₂ F • (OptProblem.A₂ E₁) (ADMM.x₂ E₁ F n - ADMM.x₂ E₁ F (n + 1));
  let block₁ :=
    ADMM.y E₁ E₂ (n + 1) +
        ((1 - ADMM.τ E₁ E₂ F) * ADMM.ρ E₁ E₂ F) • ((OptProblem.A₁ E₂) (e₁ (n + 1)) + (OptProblem.A₂ E₁) (e₂ (n + 1))) +
      ADMM.ρ E₁ E₂ F • (OptProblem.A₂ E₁) (ADMM.x₂ E₁ F n - ADMM.x₂ E₁ F (n + 1));
  -block = block₁ - admm_kkt.y →
    -(ADMM.ρ E₁ E₂ F •
          ((ContinuousLinearMap.adjoint (OptProblem.A₁ E₂)) ((OptProblem.A₂ E₁) (ADMM.x₂ E₁ F n)) -
            (ContinuousLinearMap.adjoint (OptProblem.A₁ E₂)) ((OptProblem.A₂ E₁) (ADMM.x₂ E₁ F (n + 1))))) =
      ADMM.ρ E₁ E₂ F • (ContinuousLinearMap.adjoint (OptProblem.A₁ E₂)) ((OptProblem.A₂ E₁) (ADMM.x₂ E₁ F (n + 1))) -
        ADMM.ρ E₁ E₂ F • (ContinuousLinearMap.adjoint (OptProblem.A₁ E₂)) ((OptProblem.A₂ E₁) (ADMM.x₂ E₁ F n)) := 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
Ae1 : F := (OptProblem.A₁ E₂) (e₁ (n + 1))
e' : E₁ := e₁ (n + 1)
block : F :=
  -ey (n + 1) -
      ((1 - ADMM.τ E₁ E₂ F) * ADMM.ρ E₁ E₂ F) • ((OptProblem.A₁ E₂) (e₁ (n + 1)) + (OptProblem.A₂ E₁) (e₂ (n + 1))) -
    ADMM.ρ E₁ E₂ F • (OptProblem.A₂ E₁) (ADMM.x₂ E₁ F n - ADMM.x₂ E₁ F (n + 1))
u' : E₁ :=
  -(ContinuousLinearMap.adjoint (OptProblem.A₁ E₂))
      (ADMM.y E₁ E₂ (n + 1) +
          ((1 - ADMM.τ E₁ E₂ F) * ADMM.ρ E₁ E₂ F) •
            ((OptProblem.A₁ E₂) (e₁ (n + 1)) + (OptProblem.A₂ E₁) (e₂ (n + 1))) +
        ADMM.ρ E₁ E₂ F • (OptProblem.A₂ E₁) (ADMM.x₂ E₁ F n - ADMM.x₂ E₁ F (n + 1)))
x_diff : E₁ := ADMM.x₁ E₂ F (n + 1) - admm_kkt.x₁
succ_n : ℕ+ := (n + 1).toPNat'
block₁ : F :=
  ADMM.y E₁ E₂ (n + 1) +
      ((1 - ADMM.τ E₁ E₂ F) * ADMM.ρ E₁ E₂ F) • ((OptProblem.A₁ E₂) (e₁ (n + 1)) + (OptProblem.A₂ E₁) (e₂ (n + 1))) +
    ADMM.ρ E₁ E₂ F • (OptProblem.A₂ E₁) (ADMM.x₂ E₁ F n - ADMM.x₂ E₁ F (n + 1))
split_block : -block = block₁ - admm_kkt.y
⊢ -(ADMM.ρ E₁ E₂ F •
        ((ContinuousLinearMap.adjoint (OptProblem.A₁ E₂)) ((OptProblem.A₂ E₁) (ADMM.x₂ E₁ F n)) -
          (ContinuousLinearMap.adjoint (OptProblem.A₁ E₂)) ((OptProblem.A₂ E₁) (ADMM.x₂ E₁ F (n + 1))))) =
    ADMM.ρ E₁ E₂ F • (ContinuousLinearMap.adjoint (OptProblem.A₁ E₂)) ((OptProblem.A₂ E₁) (ADMM.x₂ E₁ F (n + 1))) -
      ADMM.ρ E₁ E₂ F • (ContinuousLinearMap.adjoint (OptProblem.A₁ E₂)) ((OptProblem.A₂ E₁) (ADMM.x₂ E₁ F n))

Tactic:
rw [← smul_neg, neg_sub, smul_sub]

Tactic state after: