import Mathlib
import Optlib


open Set InnerProductSpace Topology Filter InnerProduct

open scoped Pointwise

theorem extracted_formal_statement_69.{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} (n : ℕ+)
  (this :
    IsMinOn
        (fun x =>
          Augmented_Lagrangian_Function E₁ E₂ F ADMM.toOptProblem (ADMM.ρ E₁ E₂ F)
            (x, ADMM.x₂ E₁ F n.natPred, ADMM.y E₁ E₂ n.natPred))
        univ (ADMM.x₁ E₂ F ↑n) ↔
      0 ∈
        SubderivAt
          (fun x =>
            Augmented_Lagrangian_Function E₁ E₂ F ADMM.toOptProblem (ADMM.ρ E₁ E₂ F)
              (x, ADMM.x₂ E₁ F n.natPred, ADMM.y E₁ E₂ n.natPred))
          (ADMM.x₁ E₂ F ↑n))
  (h :
    0 ∈
      SubderivAt (OptProblem.f₁ E₂ F) (ADMM.x₁ E₂ F ↑n) +
          {(ContinuousLinearMap.adjoint (OptProblem.A₁ E₂)) (ADMM.y E₁ E₂ n.natPred)} +
        {ADMM.ρ E₁ E₂ F •
            (ContinuousLinearMap.adjoint (OptProblem.A₁ E₂))
              ((OptProblem.A₁ E₂) (ADMM.x₁ E₂ F ↑n) + (OptProblem.A₂ E₁) (ADMM.x₂ E₁ F n.natPred) -
                OptProblem.b E₁ E₂)})
  (eq : n.natPred + 1 = ↑n) :
  0 ∈
    SubderivAt (OptProblem.f₁ E₂ F) (ADMM.x₁ E₂ F ↑n) +
        {(ContinuousLinearMap.adjoint (OptProblem.A₁ E₂)) (ADMM.y E₁ E₂ n.natPred)} +
      {ADMM.ρ E₁ E₂ F •
          (ContinuousLinearMap.adjoint (OptProblem.A₁ E₂))
            ((OptProblem.A₁ E₂) (ADMM.x₁ E₂ F ↑n) + (OptProblem.A₂ E₁) (ADMM.x₂ E₁ F n.natPred) -
              OptProblem.b E₁ E₂)} := 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 :
  IsMinOn
      (fun x =>
        Augmented_Lagrangian_Function E₁ E₂ F ADMM.toOptProblem (ADMM.ρ E₁ E₂ F)
          (x, ADMM.x₂ E₁ F n.natPred, ADMM.y E₁ E₂ n.natPred))
      univ (ADMM.x₁ E₂ F ↑n) ↔
    0 ∈
      SubderivAt
        (fun x =>
          Augmented_Lagrangian_Function E₁ E₂ F ADMM.toOptProblem (ADMM.ρ E₁ E₂ F)
            (x, ADMM.x₂ E₁ F n.natPred, ADMM.y E₁ E₂ n.natPred))
        (ADMM.x₁ E₂ F ↑n)
h :
  0 ∈
    SubderivAt (OptProblem.f₁ E₂ F) (ADMM.x₁ E₂ F ↑n) +
        {(ContinuousLinearMap.adjoint (OptProblem.A₁ E₂)) (ADMM.y E₁ E₂ n.natPred)} +
      {ADMM.ρ E₁ E₂ F •
          (ContinuousLinearMap.adjoint (OptProblem.A₁ E₂))
            ((OptProblem.A₁ E₂) (ADMM.x₁ E₂ F ↑n) + (OptProblem.A₂ E₁) (ADMM.x₂ E₁ F n.natPred) - OptProblem.b E₁ E₂)}
eq : n.natPred + 1 = ↑n
⊢ 0 ∈
    SubderivAt (OptProblem.f₁ E₂ F) (ADMM.x₁ E₂ F ↑n) +
        {(ContinuousLinearMap.adjoint (OptProblem.A₁ E₂)) (ADMM.y E₁ E₂ n.natPred)} +
      {ADMM.ρ E₁ E₂ F •
          (ContinuousLinearMap.adjoint (OptProblem.A₁ E₂))
            ((OptProblem.A₁ E₂) (ADMM.x₁ E₂ F ↑n) + (OptProblem.A₂ E₁) (ADMM.x₂ E₁ F n.natPred) - OptProblem.b E₁ E₂)}

Tactic:
exact h

Tactic state after: