import Mathlib
import Optlib


open Set InnerProductSpace Topology Filter InnerProduct

open scoped Pointwise

theorem extracted_formal_statement_170.{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 succ_n := (n + 1).toPNat';
  let e₁' := e₁ ↑succ_n;
  let e₂' := e₂ ↑succ_n;
  let y_k_1 := ADMM.y E₁ E₂ ↑succ_n;
  ⟪-(ContinuousLinearMap.adjoint (OptProblem.A₂ E₁))
            (y_k_1 - admm_kkt.y +
              ((1 - ADMM.τ E₁ E₂ F) * ADMM.ρ E₁ E₂ F) • ((OptProblem.A₁ E₂) e₁' + (OptProblem.A₂ E₁) e₂')),
        e₂'⟫_ℝ =
      ⟪-(ContinuousLinearMap.adjoint (OptProblem.A₂ E₁))
              (y_k_1 + ((1 - ADMM.τ E₁ E₂ F) * ADMM.ρ E₁ E₂ F) • ((OptProblem.A₁ E₂) e₁' + (OptProblem.A₂ E₁) e₂')) +
          (ContinuousLinearMap.adjoint (OptProblem.A₂ E₁)) admm_kkt.y,
        e₂'⟫_ℝ →
    ⟪-((ContinuousLinearMap.adjoint (OptProblem.A₂ E₁)) (-admm_kkt.y) +
            (ContinuousLinearMap.adjoint (OptProblem.A₂ E₁))
              (y_k_1 + ((1 - ADMM.τ E₁ E₂ F) * ADMM.ρ E₁ E₂ F) • ((OptProblem.A₁ E₂) e₁' + (OptProblem.A₂ E₁) e₂'))),
        e₂'⟫_ℝ =
      ⟪-(ContinuousLinearMap.adjoint (OptProblem.A₂ E₁))
              (y_k_1 + ((1 - ADMM.τ E₁ E₂ F) * ADMM.ρ E₁ E₂ F) • ((OptProblem.A₁ E₂) e₁' + (OptProblem.A₂ E₁) e₂')) +
          (ContinuousLinearMap.adjoint (OptProblem.A₂ E₁)) admm_kkt.y,
        e₂'⟫_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
succ_n : ℕ+ := (n + 1).toPNat'
ey' : F := ey ↑succ_n
e₁' : E₁ := e₁ ↑succ_n
e₂' : E₂ := e₂ ↑succ_n
y_k_1 : F := ADMM.y E₁ E₂ ↑succ_n
v_k_1 : E₂ := v ↑succ_n
x_diff : E₂ := ADMM.x₂ E₁ F ↑succ_n - admm_kkt.x₂
⊢ ⟪-(ContinuousLinearMap.adjoint (OptProblem.A₂ E₁))
          (y_k_1 - admm_kkt.y +
            ((1 - ADMM.τ E₁ E₂ F) * ADMM.ρ E₁ E₂ F) • ((OptProblem.A₁ E₂) e₁' + (OptProblem.A₂ E₁) e₂')),
      e₂'⟫_ℝ =
    ⟪-(ContinuousLinearMap.adjoint (OptProblem.A₂ E₁))
            (y_k_1 + ((1 - ADMM.τ E₁ E₂ F) * ADMM.ρ E₁ E₂ F) • ((OptProblem.A₁ E₂) e₁' + (OptProblem.A₂ E₁) e₂')) +
        (ContinuousLinearMap.adjoint (OptProblem.A₂ E₁)) admm_kkt.y,
      e₂'⟫_ℝ

Tactic:
rw [sub_eq_add_neg, add_comm y_k_1, add_assoc, A₂†.map_add]

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
 succ_n : ℕ+ := (n + 1).toPNat'
 ey' : F := ey ↑succ_n
 e₁' : E₁ := e₁ ↑succ_n
 e₂' : E₂ := e₂ ↑succ_n
 y_k_1 : F := ADMM.y E₁ E₂ ↑succ_n
 v_k_1 : E₂ := v ↑succ_n
 x_diff : E₂ := ADMM.x₂ E₁ F ↑succ_n - admm_kkt.x₂
 ⊢ ⟪-((ContinuousLinearMap.adjoint (OptProblem.A₂ E₁)) (-admm_kkt.y) +
           (ContinuousLinearMap.adjoint (OptProblem.A₂ E₁))
             (y_k_1 + ((1 - ADMM.τ E₁ E₂ F) * ADMM.ρ E₁ E₂ F) • ((OptProblem.A₁ E₂) e₁' + (OptProblem.A₂ E₁) e₂'))),
       e₂'⟫_ℝ =
     ⟪-(ContinuousLinearMap.adjoint (OptProblem.A₂ E₁))
             (y_k_1 + ((1 - ADMM.τ E₁ E₂ F) * ADMM.ρ E₁ E₂ F) • ((OptProblem.A₁ E₂) e₁' + (OptProblem.A₂ E₁) e₂')) +
         (ContinuousLinearMap.adjoint (OptProblem.A₂ E₁)) admm_kkt.y,
       e₂'⟫_ℝ]