import Mathlib
import Optlib


open Set InnerProductSpace Topology Filter InnerProduct

open scoped Pointwise

theorem extracted_formal_statement_5.{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.A₁ E₂) (ADMM.x₁ E₂ F ↑n) - OptProblem.b E₁ E₂;
  ((fun x =>
        ADMM.ρ E₁ E₂ F / 2 * ‖(OptProblem.A₁ E₂) (ADMM.x₁ E₂ F ↑n) + (OptProblem.A₂ E₁) x - OptProblem.b E₁ E₂‖ ^ 2) =
      fun x =>
      ADMM.ρ E₁ E₂ F / 2 * ‖(OptProblem.A₂ E₁) x + ((OptProblem.A₁ E₂) (ADMM.x₁ E₂ F ↑n) - OptProblem.b E₁ E₂)‖ ^ 2) →
    ADMM.ρ E₁ E₂ F •
          (ContinuousLinearMap.adjoint (OptProblem.A₂ E₁))
            ((OptProblem.A₁ E₂) (ADMM.x₁ E₂ F ↑n) + (OptProblem.A₂ E₁) (ADMM.x₂ E₁ F ↑n) - OptProblem.b E₁ E₂) =
        ADMM.ρ E₁ E₂ F •
          (ContinuousLinearMap.adjoint (OptProblem.A₂ E₁))
            ((OptProblem.A₂ E₁) (ADMM.x₂ E₁ F ↑n) + ((OptProblem.A₁ E₂) (ADMM.x₁ E₂ F ↑n) - OptProblem.b E₁ E₂)) →
      ((fun x => ADMM.ρ E₁ E₂ F / 2 * ‖(OptProblem.A₂ E₁) x + c‖ ^ 2) = fun x =>
          ADMM.ρ E₁ E₂ F / 2 *
            (⟪(OptProblem.A₂ E₁) x, (OptProblem.A₂ E₁) x⟫_ℝ + 2 * ⟪c, (OptProblem.A₂ E₁) x⟫_ℝ + ⟪c, c⟫_ℝ)) →
        ADMM.ρ E₁ E₂ F • (ContinuousLinearMap.adjoint (OptProblem.A₂ E₁)) ((OptProblem.A₂ E₁) (ADMM.x₂ E₁ F ↑n) + c) =
            (ADMM.ρ E₁ E₂ F / 2) •
              2 • (ContinuousLinearMap.adjoint (OptProblem.A₂ E₁)) ((OptProblem.A₂ E₁) (ADMM.x₂ E₁ F ↑n) + c) →
          HasGradientAt
              (fun x =>
                ADMM.ρ E₁ E₂ F / 2 *
                  (⟪(OptProblem.A₂ E₁) x, (OptProblem.A₂ E₁) x⟫_ℝ + 2 * ⟪c, (OptProblem.A₂ E₁) x⟫_ℝ + ⟪c, c⟫_ℝ))
              ((ADMM.ρ E₁ E₂ F / 2) •
                2 • (ContinuousLinearMap.adjoint (OptProblem.A₂ E₁)) ((OptProblem.A₂ E₁) (ADMM.x₂ E₁ F ↑n) + c))
              (ADMM.x₂ E₁ F ↑n) →
            HasGradientAt (fun x => 2 * ⟪c, (OptProblem.A₂ E₁) x⟫_ℝ)
              (2 • (ContinuousLinearMap.adjoint (OptProblem.A₂ E₁)) c) (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+
c : F := (OptProblem.A₁ E₂) (ADMM.x₁ E₂ F ↑n) - OptProblem.b E₁ E₂
h1 :
  (fun x =>
      ADMM.ρ E₁ E₂ F / 2 * ‖(OptProblem.A₁ E₂) (ADMM.x₁ E₂ F ↑n) + (OptProblem.A₂ E₁) x - OptProblem.b E₁ E₂‖ ^ 2) =
    fun x =>
    ADMM.ρ E₁ E₂ F / 2 * ‖(OptProblem.A₂ E₁) x + ((OptProblem.A₁ E₂) (ADMM.x₁ E₂ F ↑n) - OptProblem.b E₁ E₂)‖ ^ 2
h1' :
  ADMM.ρ E₁ E₂ F •
      (ContinuousLinearMap.adjoint (OptProblem.A₂ E₁))
        ((OptProblem.A₁ E₂) (ADMM.x₁ E₂ F ↑n) + (OptProblem.A₂ E₁) (ADMM.x₂ E₁ F ↑n) - OptProblem.b E₁ E₂) =
    ADMM.ρ E₁ E₂ F •
      (ContinuousLinearMap.adjoint (OptProblem.A₂ E₁))
        ((OptProblem.A₂ E₁) (ADMM.x₂ E₁ F ↑n) + ((OptProblem.A₁ E₂) (ADMM.x₁ E₂ F ↑n) - OptProblem.b E₁ E₂))
h2 :
  (fun x => ADMM.ρ E₁ E₂ F / 2 * ‖(OptProblem.A₂ E₁) x + c‖ ^ 2) = fun x =>
    ADMM.ρ E₁ E₂ F / 2 * (⟪(OptProblem.A₂ E₁) x, (OptProblem.A₂ E₁) x⟫_ℝ + 2 * ⟪c, (OptProblem.A₂ E₁) x⟫_ℝ + ⟪c, c⟫_ℝ)
h3 :
  ADMM.ρ E₁ E₂ F • (ContinuousLinearMap.adjoint (OptProblem.A₂ E₁)) ((OptProblem.A₂ E₁) (ADMM.x₂ E₁ F ↑n) + c) =
    (ADMM.ρ E₁ E₂ F / 2) •
      2 • (ContinuousLinearMap.adjoint (OptProblem.A₂ E₁)) ((OptProblem.A₂ E₁) (ADMM.x₂ E₁ F ↑n) + c)
⊢ HasGradientAt
    (fun x =>
      ADMM.ρ E₁ E₂ F / 2 *
        (⟪(OptProblem.A₂ E₁) x, (OptProblem.A₂ E₁) x⟫_ℝ + 2 * ⟪c, (OptProblem.A₂ E₁) x⟫_ℝ + ⟪c, c⟫_ℝ))
    ((ADMM.ρ E₁ E₂ F / 2) •
      2 • (ContinuousLinearMap.adjoint (OptProblem.A₂ E₁)) ((OptProblem.A₂ E₁) (ADMM.x₂ E₁ F ↑n) + c))
    (ADMM.x₂ E₁ F ↑n)

Tactic:
apply HasGradientAt.const_mul' (ρ / 2)
    apply HasGradientAt.add_const
    have h4 : (2 : ℝ) • A₂† (A₂ (x₂ n) + c) = (2 : ℝ) • A₂† (A₂ (x₂ n)) + (2 : ℝ) • A₂† c := by
  calc
    _ = (2 : ℝ) • (A₂† (A₂ (x₂ n)) + A₂† c) := by simp only [map_add, smul_add]
    _ = _ := by simp only [smul_add]
    rw [h4]
    apply HasGradientAt.add

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