import Mathlib
import Optlib


open Set InnerProductSpace Topology Filter InnerProduct

open scoped Pointwise

theorem extracted_formal_statement_13.{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+) :
  let c := (OptProblem.A₂ E₁) (ADMM.x₂ E₁ F n.natPred) - OptProblem.b E₁ E₂;
  ((fun x =>
        ADMM.ρ E₁ E₂ F / 2 *
          ‖(OptProblem.A₁ E₂) x + (OptProblem.A₂ E₁) (ADMM.x₂ E₁ F n.natPred) - OptProblem.b E₁ E₂‖ ^ 2) =
      fun x =>
      ADMM.ρ E₁ E₂ F / 2 *
        ‖(OptProblem.A₁ E₂) x + ((OptProblem.A₂ E₁) (ADMM.x₂ E₁ F n.natPred) - OptProblem.b E₁ E₂)‖ ^ 2) →
    ((fun x => ADMM.ρ E₁ E₂ F / 2 * ‖(OptProblem.A₁ E₂) x + c‖ ^ 2) = fun x =>
        ADMM.ρ E₁ E₂ F / 2 *
          (⟪(OptProblem.A₁ E₂) x, (OptProblem.A₁ E₂) x⟫_ℝ + 2 * ⟪c, (OptProblem.A₁ E₂) x⟫_ℝ + ⟪c, c⟫_ℝ)) →
      ADMM.ρ E₁ E₂ F • (ContinuousLinearMap.adjoint (OptProblem.A₁ E₂)) ((OptProblem.A₁ E₂) (ADMM.x₁ E₂ F ↑n) + c) =
          (ADMM.ρ E₁ E₂ F / 2) •
            2 • (ContinuousLinearMap.adjoint (OptProblem.A₁ E₂)) ((OptProblem.A₁ E₂) (ADMM.x₁ E₂ F ↑n) + c) →
        2 • (ContinuousLinearMap.adjoint (OptProblem.A₁ E₂)) ((OptProblem.A₁ E₂) (ADMM.x₁ E₂ F ↑n) + c) =
            2 • (ContinuousLinearMap.adjoint (OptProblem.A₁ E₂)) ((OptProblem.A₁ E₂) (ADMM.x₁ E₂ F ↑n)) +
              2 • (ContinuousLinearMap.adjoint (OptProblem.A₁ E₂)) c →
          HasGradientAt (fun x => ⟪(OptProblem.A₁ E₂) x, (OptProblem.A₁ E₂) x⟫_ℝ + 2 * ⟪c, (OptProblem.A₁ E₂) x⟫_ℝ)
            (2 • (ContinuousLinearMap.adjoint (OptProblem.A₁ E₂)) ((OptProblem.A₁ E₂) (ADMM.x₁ E₂ F ↑n)) +
              2 • (ContinuousLinearMap.adjoint (OptProblem.A₁ E₂)) c)
            (ADMM.x₁ E₂ F ↑n) := sorry

Tactic state before:
case hf
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+
c : F := (OptProblem.A₂ E₁) (ADMM.x₂ E₁ F n.natPred) - OptProblem.b E₁ E₂
h1 :
  (fun x =>
      ADMM.ρ E₁ E₂ F / 2 *
        ‖(OptProblem.A₁ E₂) x + (OptProblem.A₂ E₁) (ADMM.x₂ E₁ F n.natPred) - OptProblem.b E₁ E₂‖ ^ 2) =
    fun x =>
    ADMM.ρ E₁ E₂ F / 2 * ‖(OptProblem.A₁ E₂) x + ((OptProblem.A₂ E₁) (ADMM.x₂ E₁ F n.natPred) - OptProblem.b E₁ E₂)‖ ^ 2
h2 :
  (fun x => ADMM.ρ E₁ E₂ F / 2 * ‖(OptProblem.A₁ E₂) x + c‖ ^ 2) = fun x =>
    ADMM.ρ E₁ E₂ F / 2 * (⟪(OptProblem.A₁ E₂) x, (OptProblem.A₁ E₂) x⟫_ℝ + 2 * ⟪c, (OptProblem.A₁ E₂) x⟫_ℝ + ⟪c, c⟫_ℝ)
h3 :
  ADMM.ρ E₁ E₂ F • (ContinuousLinearMap.adjoint (OptProblem.A₁ E₂)) ((OptProblem.A₁ E₂) (ADMM.x₁ E₂ F ↑n) + c) =
    (ADMM.ρ E₁ E₂ F / 2) •
      2 • (ContinuousLinearMap.adjoint (OptProblem.A₁ E₂)) ((OptProblem.A₁ E₂) (ADMM.x₁ E₂ F ↑n) + c)
h4 :
  2 • (ContinuousLinearMap.adjoint (OptProblem.A₁ E₂)) ((OptProblem.A₁ E₂) (ADMM.x₁ E₂ F ↑n) + c) =
    2 • (ContinuousLinearMap.adjoint (OptProblem.A₁ E₂)) ((OptProblem.A₁ E₂) (ADMM.x₁ E₂ F ↑n)) +
      2 • (ContinuousLinearMap.adjoint (OptProblem.A₁ E₂)) c
⊢ HasGradientAt (fun x => ⟪(OptProblem.A₁ E₂) x, (OptProblem.A₁ E₂) x⟫_ℝ + 2 * ⟪c, (OptProblem.A₁ E₂) x⟫_ℝ)
    (2 • (ContinuousLinearMap.adjoint (OptProblem.A₁ E₂)) ((OptProblem.A₁ E₂) (ADMM.x₁ E₂ F ↑n)) +
      2 • (ContinuousLinearMap.adjoint (OptProblem.A₁ E₂)) c)
    (ADMM.x₁ E₂ F ↑n)

Tactic:
apply HasGradientAt.add
    · apply Gradient_of_quadratic_forms
    ·
  let a := (fun x => ⟪c, A₁ x⟫)
  show HasGradientAt (fun x ↦ 2 * a x) ((2 : ℝ) • (A₁† c)) (x₁ n)
  apply HasGradientAt.const_mul' 2
  apply inner_gradient

Tactic state after: