import Mathlib
import Optlib


open Set InnerProductSpace Topology Filter InnerProduct

open scoped Pointwise

theorem extracted_formal_statement_121.{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+)
  (this : ↑n - 1 + 1 = ↑n) :
  let un := u n;
  un = u n →
    let yn := ADMM.y E₁ E₂ ↑n;
    let yn' := ADMM.y E₁ E₂ (↑n - 1);
    yn = ADMM.y E₁ E₂ ↑n →
      yn' = ADMM.y E₁ E₂ (↑n - 1) →
        let x₁n := ADMM.x₁ E₂ F ↑n;
        let x₂n := ADMM.x₂ E₁ F ↑n;
        let x₂n' := ADMM.x₂ E₁ F (↑n - 1);
        x₁n = ADMM.x₁ E₂ F ↑n →
          x₂n = ADMM.x₂ E₁ F ↑n →
            yn' =
                yn -
                  (ADMM.τ E₁ E₂ F * ADMM.ρ E₁ E₂ F) •
                    ((OptProblem.A₁ E₂) x₁n + (OptProblem.A₂ E₁) x₂n - OptProblem.b E₁ E₂) →
              (OptProblem.A₁ E₂) x₁n - (OptProblem.A₁ E₂) admm_kkt.x₁ = (OptProblem.A₁ E₂) (e₁ ↑n) →
                (OptProblem.A₂ E₁) x₂n - (OptProblem.A₂ E₁) admm_kkt.x₂ = (OptProblem.A₂ E₁) (e₂ ↑n) →
                  -(ContinuousLinearMap.adjoint (OptProblem.A₁ E₂))
                        (yn +
                            ((1 - ADMM.τ E₁ E₂ F) * ADMM.ρ E₁ E₂ F) •
                              ((OptProblem.A₁ E₂) x₁n - (OptProblem.A₁ E₂) admm_kkt.x₁ +
                                ((OptProblem.A₂ E₁) x₂n - (OptProblem.A₂ E₁) admm_kkt.x₂)) +
                          ADMM.ρ E₁ E₂ F • (OptProblem.A₂ E₁) (x₂n' - x₂n)) =
                    -(ContinuousLinearMap.adjoint (OptProblem.A₁ E₂))
                        (yn +
                            ((1 - ADMM.τ E₁ E₂ F) * ADMM.ρ E₁ E₂ F) •
                              ((OptProblem.A₁ E₂) (e₁ ↑n) + (OptProblem.A₂ E₁) (e₂ ↑n)) +
                          ADMM.ρ E₁ E₂ F • (OptProblem.A₂ E₁) (x₂n' - x₂n)) := 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+
this : ↑n - 1 + 1 = ↑n
un : E₁ := u n
h₁ : un = u n
yn : F := ADMM.y E₁ E₂ ↑n
yn' : F := ADMM.y E₁ E₂ (↑n - 1)
h₂ : yn = ADMM.y E₁ E₂ ↑n
h₃ : yn' = ADMM.y E₁ E₂ (↑n - 1)
x₁n : E₁ := ADMM.x₁ E₂ F ↑n
x₂n : E₂ := ADMM.x₂ E₁ F ↑n
x₂n' : E₂ := ADMM.x₂ E₁ F (↑n - 1)
h₄ : x₁n = ADMM.x₁ E₂ F ↑n
h₅ : x₂n = ADMM.x₂ E₁ F ↑n
aux :
  yn' = yn - (ADMM.τ E₁ E₂ F * ADMM.ρ E₁ E₂ F) • ((OptProblem.A₁ E₂) x₁n + (OptProblem.A₂ E₁) x₂n - OptProblem.b E₁ E₂)
h1 : (OptProblem.A₁ E₂) x₁n - (OptProblem.A₁ E₂) admm_kkt.x₁ = (OptProblem.A₁ E₂) (e₁ ↑n)
h2 : (OptProblem.A₂ E₁) x₂n - (OptProblem.A₂ E₁) admm_kkt.x₂ = (OptProblem.A₂ E₁) (e₂ ↑n)
⊢ -(ContinuousLinearMap.adjoint (OptProblem.A₁ E₂))
        (yn +
            ((1 - ADMM.τ E₁ E₂ F) * ADMM.ρ E₁ E₂ F) •
              ((OptProblem.A₁ E₂) x₁n - (OptProblem.A₁ E₂) admm_kkt.x₁ +
                ((OptProblem.A₂ E₁) x₂n - (OptProblem.A₂ E₁) admm_kkt.x₂)) +
          ADMM.ρ E₁ E₂ F • (OptProblem.A₂ E₁) (x₂n' - x₂n)) =
    -(ContinuousLinearMap.adjoint (OptProblem.A₁ E₂))
        (yn + ((1 - ADMM.τ E₁ E₂ F) * ADMM.ρ E₁ E₂ F) • ((OptProblem.A₁ E₂) (e₁ ↑n) + (OptProblem.A₂ E₁) (e₂ ↑n)) +
          ADMM.ρ E₁ E₂ F • (OptProblem.A₂ E₁) (x₂n' - x₂n))

Tactic:
rw [← h1, ← h2]

Tactic state after: