zzhisthebest/LeanBenchmark huggingface dataset:
import Mathlib
import Optlib


open Set InnerProductSpace Topology Filter InnerProduct

open scoped Pointwise

theorem extracted_formal_statement_27.{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 * ‖(OptProblem.A₁ E₂) x - c‖ ^ 2 / 2) →
    let f := fun x => ‖(OptProblem.A₁ E₂) x - c‖ ^ 2 / 2;
    ((fun x => ADMM.ρ E₁ E₂ F • ‖(OptProblem.A₁ E₂) x - c‖ ^ 2 / 2) = fun x => ADMM.ρ E₁ E₂ F • f x) →
      let u := fun x => ‖x - c‖ ^ 2 / 2;
      let g := OptProblem.A₁ E₂;
      u ∘ ⇑g = f → ⇑g ⁻¹' univ = univ → Convex ℝ univ → ConvexOn ℝ (⇑g ⁻¹' univ) (u ∘ ⇑g) := 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.A₂ E₁) (ADMM.x₂ E₁ F n.natPred) - OptProblem.b E₁ E₂)
h :
  (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 * ‖(OptProblem.A₁ E₂) x - c‖ ^ 2 / 2
f : E₁ → ℝ := fun x => ‖(OptProblem.A₁ E₂) x - c‖ ^ 2 / 2
h1 : (fun x => ADMM.ρ E₁ E₂ F • ‖(OptProblem.A₁ E₂) x - c‖ ^ 2 / 2) = fun x => ADMM.ρ E₁ E₂ F • f x
u : F → ℝ := fun x => ‖x - c‖ ^ 2 / 2
g : E₁ →L[ℝ] F := OptProblem.A₁ E₂
h2 : u ∘ ⇑g = f
h3 : ⇑g ⁻¹' univ = univ
h4 : Convex ℝ univ
⊢ ConvexOn ℝ (⇑g ⁻¹' univ) (u ∘ ⇑g)

Tactic:
apply ConvexOn.comp_linearMap (convex_of_norm_sq c h4)

Tactic state after:
import Mathlib
import Optlib


open Set InnerProductSpace Topology Filter InnerProduct

open scoped Pointwise

theorem extracted_formal_statement_28.{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);
  ∀ (x : E₂),
    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 * ‖(OptProblem.A₂ E₁) x - c‖ ^ 2 / 2 →
      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 * ‖(OptProblem.A₂ E₁) x - -((OptProblem.A₁ E₂) (ADMM.x₁ E₂ F ↑n) - OptProblem.b E₁ E₂)‖ ^ 2 /
          2 := sorry

Tactic state before:
case h
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)
x : E₂
⊢ 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 * ‖(OptProblem.A₂ E₁) x - c‖ ^ 2 / 2

Tactic:
simp only [c]

Tactic state after:
[case h
 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)
 x : E₂
 ⊢ 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 * ‖(OptProblem.A₂ E₁) x - -((OptProblem.A₁ E₂) (ADMM.x₁ E₂ F ↑n) - OptProblem.b E₁ E₂)‖ ^ 2 / 2]
import Mathlib
import Optlib


open Set InnerProductSpace Topology Filter InnerProduct

open scoped Pointwise

theorem extracted_formal_statement_29.{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 * ‖(OptProblem.A₂ E₁) x - c‖ ^ 2 / 2) →
    let f := fun x => ‖(OptProblem.A₂ E₁) x - c‖ ^ 2 / 2;
    ∀ (x : E₂),
      ADMM.ρ E₁ E₂ F • ‖(OptProblem.A₂ E₁) x - c‖ ^ 2 / 2 = ADMM.ρ E₁ E₂ F • f x →
        ADMM.ρ E₁ E₂ F * ‖(OptProblem.A₂ E₁) x - c‖ ^ 2 / 2 =
          ADMM.ρ E₁ E₂ F * (‖(OptProblem.A₂ E₁) x - c‖ ^ 2 / 2) := sorry

Tactic state before:
case h
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)
h :
  (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 * ‖(OptProblem.A₂ E₁) x - c‖ ^ 2 / 2
f : E₂ → ℝ := fun x => ‖(OptProblem.A₂ E₁) x - c‖ ^ 2 / 2
x : E₂
⊢ ADMM.ρ E₁ E₂ F • ‖(OptProblem.A₂ E₁) x - c‖ ^ 2 / 2 = ADMM.ρ E₁ E₂ F • f x

Tactic:
simp only [f, smul_eq_mul]

Tactic state after:
[case h
 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)
 h :
   (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 * ‖(OptProblem.A₂ E₁) x - c‖ ^ 2 / 2
 f : E₂ → ℝ := fun x => ‖(OptProblem.A₂ E₁) x - c‖ ^ 2 / 2
 x : E₂
 ⊢ ADMM.ρ E₁ E₂ F * ‖(OptProblem.A₂ E₁) x - c‖ ^ 2 / 2 = ADMM.ρ E₁ E₂ F * (‖(OptProblem.A₂ E₁) x - c‖ ^ 2 / 2)]
import Mathlib
import Optlib


open Set InnerProductSpace Topology Filter InnerProduct

open scoped Pointwise

theorem extracted_formal_statement_30.{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 * ‖(OptProblem.A₂ E₁) x - c‖ ^ 2 / 2) →
    let f := fun x => ‖(OptProblem.A₂ E₁) x - c‖ ^ 2 / 2;
    ((fun x => ADMM.ρ E₁ E₂ F • ‖(OptProblem.A₂ E₁) x - c‖ ^ 2 / 2) = fun x => ADMM.ρ E₁ E₂ F • f x) →
      (ConvexOn ℝ univ fun x => ADMM.ρ E₁ E₂ F • f x) → ConvexOn ℝ univ f := 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)
h :
  (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 * ‖(OptProblem.A₂ E₁) x - c‖ ^ 2 / 2
f : E₂ → ℝ := fun x => ‖(OptProblem.A₂ E₁) x - c‖ ^ 2 / 2
h1 : (fun x => ADMM.ρ E₁ E₂ F • ‖(OptProblem.A₂ E₁) x - c‖ ^ 2 / 2) = fun x => ADMM.ρ E₁ E₂ F • f x
⊢ ConvexOn ℝ univ fun x => ADMM.ρ E₁ E₂ F • f x

Tactic:
apply ConvexOn.smul (le_of_lt admm.hrho)

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+
 c : F := -((OptProblem.A₁ E₂) (ADMM.x₁ E₂ F ↑n) - OptProblem.b E₁ E₂)
 h :
   (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 * ‖(OptProblem.A₂ E₁) x - c‖ ^ 2 / 2
 f : E₂ → ℝ := fun x => ‖(OptProblem.A₂ E₁) x - c‖ ^ 2 / 2
 h1 : (fun x => ADMM.ρ E₁ E₂ F • ‖(OptProblem.A₂ E₁) x - c‖ ^ 2 / 2) = fun x => ADMM.ρ E₁ E₂ F • f x
 ⊢ ConvexOn ℝ univ f]
import Mathlib
import Optlib


open Set InnerProductSpace Topology Filter InnerProduct

open scoped Pointwise

theorem extracted_formal_statement_31.{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 * ‖(OptProblem.A₂ E₁) x - c‖ ^ 2 / 2) →
    let f := fun x => ‖(OptProblem.A₂ E₁) x - c‖ ^ 2 / 2;
    ((fun x => ADMM.ρ E₁ E₂ F • ‖(OptProblem.A₂ E₁) x - c‖ ^ 2 / 2) = fun x => ADMM.ρ E₁ E₂ F • f x) →
      ∀ (x : E₂), ((fun x => ‖x - c‖ ^ 2 / 2) ∘ ⇑(OptProblem.A₂ E₁)) x = ‖(OptProblem.A₂ E₁) x - c‖ ^ 2 / 2 := sorry

Tactic state before:
case h
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)
h :
  (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 * ‖(OptProblem.A₂ E₁) x - c‖ ^ 2 / 2
f : E₂ → ℝ := fun x => ‖(OptProblem.A₂ E₁) x - c‖ ^ 2 / 2
h1 : (fun x => ADMM.ρ E₁ E₂ F • ‖(OptProblem.A₂ E₁) x - c‖ ^ 2 / 2) = fun x => ADMM.ρ E₁ E₂ F • f x
u : F → ℝ := fun x => ‖x - c‖ ^ 2 / 2
g : E₂ →L[ℝ] F := OptProblem.A₂ E₁
x : E₂
⊢ ((fun x => ‖x - c‖ ^ 2 / 2) ∘ ⇑(OptProblem.A₂ E₁)) x = ‖(OptProblem.A₂ E₁) x - c‖ ^ 2 / 2

Tactic:
simp only [Function.comp_apply]

Tactic state after:
import Mathlib
import Optlib


open Set InnerProductSpace Topology Filter InnerProduct

open scoped Pointwise

theorem extracted_formal_statement_32.{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 * ‖(OptProblem.A₂ E₁) x - c‖ ^ 2 / 2) →
    let f := fun x => ‖(OptProblem.A₂ E₁) x - c‖ ^ 2 / 2;
    ((fun x => ADMM.ρ E₁ E₂ F • ‖(OptProblem.A₂ E₁) x - c‖ ^ 2 / 2) = fun x => ADMM.ρ E₁ E₂ F • f x) →
      let u := fun x => ‖x - c‖ ^ 2 / 2;
      let g := OptProblem.A₂ E₁;
      u ∘ ⇑g = f → ⇑g ⁻¹' univ = univ := 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)
h :
  (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 * ‖(OptProblem.A₂ E₁) x - c‖ ^ 2 / 2
f : E₂ → ℝ := fun x => ‖(OptProblem.A₂ E₁) x - c‖ ^ 2 / 2
h1 : (fun x => ADMM.ρ E₁ E₂ F • ‖(OptProblem.A₂ E₁) x - c‖ ^ 2 / 2) = fun x => ADMM.ρ E₁ E₂ F • f x
u : F → ℝ := fun x => ‖x - c‖ ^ 2 / 2
g : E₂ →L[ℝ] F := OptProblem.A₂ E₁
h2 : u ∘ ⇑g = f
⊢ ⇑g ⁻¹' univ = univ

Tactic:
simp only [preimage_univ]

Tactic state after:
import Mathlib
import Optlib


open Set InnerProductSpace Topology Filter InnerProduct

open scoped Pointwise

theorem extracted_formal_statement_33.{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 * ‖(OptProblem.A₂ E₁) x - c‖ ^ 2 / 2) →
    let f := fun x => ‖(OptProblem.A₂ E₁) x - c‖ ^ 2 / 2;
    ((fun x => ADMM.ρ E₁ E₂ F • ‖(OptProblem.A₂ E₁) x - c‖ ^ 2 / 2) = fun x => ADMM.ρ E₁ E₂ F • f x) →
      let u := fun x => ‖x - c‖ ^ 2 / 2;
      let g := OptProblem.A₂ E₁;
      u ∘ ⇑g = f → ⇑g ⁻¹' univ = univ → Convex ℝ univ := 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)
h :
  (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 * ‖(OptProblem.A₂ E₁) x - c‖ ^ 2 / 2
f : E₂ → ℝ := fun x => ‖(OptProblem.A₂ E₁) x - c‖ ^ 2 / 2
h1 : (fun x => ADMM.ρ E₁ E₂ F • ‖(OptProblem.A₂ E₁) x - c‖ ^ 2 / 2) = fun x => ADMM.ρ E₁ E₂ F • f x
u : F → ℝ := fun x => ‖x - c‖ ^ 2 / 2
g : E₂ →L[ℝ] F := OptProblem.A₂ E₁
h2 : u ∘ ⇑g = f
h3 : ⇑g ⁻¹' univ = univ
⊢ Convex ℝ univ

Tactic:
apply convex_univ

Tactic state after:
import Mathlib
import Optlib


open Set InnerProductSpace Topology Filter InnerProduct

open scoped Pointwise

theorem extracted_formal_statement_34.{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 * ‖(OptProblem.A₂ E₁) x - c‖ ^ 2 / 2) →
    let f := fun x => ‖(OptProblem.A₂ E₁) x - c‖ ^ 2 / 2;
    ((fun x => ADMM.ρ E₁ E₂ F • ‖(OptProblem.A₂ E₁) x - c‖ ^ 2 / 2) = fun x => ADMM.ρ E₁ E₂ F • f x) →
      let u := fun x => ‖x - c‖ ^ 2 / 2;
      let g := OptProblem.A₂ E₁;
      u ∘ ⇑g = f → ⇑g ⁻¹' univ = univ → Convex ℝ univ → ConvexOn ℝ (⇑g ⁻¹' univ) (u ∘ ⇑g) := 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)
h :
  (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 * ‖(OptProblem.A₂ E₁) x - c‖ ^ 2 / 2
f : E₂ → ℝ := fun x => ‖(OptProblem.A₂ E₁) x - c‖ ^ 2 / 2
h1 : (fun x => ADMM.ρ E₁ E₂ F • ‖(OptProblem.A₂ E₁) x - c‖ ^ 2 / 2) = fun x => ADMM.ρ E₁ E₂ F • f x
u : F → ℝ := fun x => ‖x - c‖ ^ 2 / 2
g : E₂ →L[ℝ] F := OptProblem.A₂ E₁
h2 : u ∘ ⇑g = f
h3 : ⇑g ⁻¹' univ = univ
h4 : Convex ℝ univ
⊢ ConvexOn ℝ (⇑g ⁻¹' univ) (u ∘ ⇑g)

Tactic:
apply ConvexOn.comp_linearMap (convex_of_norm_sq c h4)

Tactic state after:
import Mathlib
import Optlib


open Set InnerProductSpace Topology Filter InnerProduct

open scoped Pointwise

theorem extracted_formal_statement_35.{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+) :
  HasGradientAt (fun x => OptProblem.f₂ E₁ F (ADMM.x₂ E₁ F n.natPred)) 0 (ADMM.x₁ E₂ 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+
⊢ HasGradientAt (fun x => OptProblem.f₂ E₁ F (ADMM.x₂ E₁ F n.natPred)) 0 (ADMM.x₁ E₂ F ↑n)

Tactic:
apply hasGradientAt_const

Tactic state after:
import Mathlib
import Optlib


open Set InnerProductSpace Topology Filter InnerProduct

open scoped Pointwise

theorem extracted_formal_statement_36.{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+)
  (h_original_goal : SubderivAt (fun x => OptProblem.f₂ E₁ F (ADMM.x₂ E₁ F n.natPred)) (ADMM.x₁ E₂ F ↑n) = 0) :
  HasGradientAt (fun x => OptProblem.f₂ E₁ F (ADMM.x₂ E₁ F n.natPred)) 0 (ADMM.x₁ E₂ 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+
⊢ SubderivAt (fun x => OptProblem.f₂ E₁ F (ADMM.x₂ E₁ F n.natPred)) (ADMM.x₁ E₂ F ↑n) = 0

Tactic:
apply SubderivAt_eq_gradient (convexOn_const (f₂ (x₂ n.natPred)) (convex_univ))

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+
 ⊢ HasGradientAt (fun x => OptProblem.f₂ E₁ F (ADMM.x₂ E₁ F n.natPred)) 0 (ADMM.x₁ E₂ F ↑n)]
import Mathlib
import Optlib


open Set InnerProductSpace Topology Filter InnerProduct

open scoped Pointwise

theorem extracted_formal_statement_37.{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+) :
  HasGradientAt (fun x => OptProblem.f₂ E₁ F (ADMM.x₂ E₁ F n.natPred)) 0 (ADMM.x₁ E₂ 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+
⊢ HasGradientAt (fun x => OptProblem.f₂ E₁ F (ADMM.x₂ E₁ F n.natPred)) 0 (ADMM.x₁ E₂ F ↑n)

Tactic:
apply ADMM_iter_process₁'_eq3_1'

Tactic state after: