zzhisthebest/LeanBenchmark huggingface dataset:
import Mathlib
import Optlib


open Set InnerProductSpace Topology Filter InnerProduct

open scoped Pointwise

theorem extracted_formal_statement_38.{u_5, u_4} {α : Type u_4} {β : Type u_5} [inst : NormedAddCommGroup α]
  [inst_1 : NormedAddCommGroup β] [inst_2 : InnerProductSpace ℝ α] [inst_3 : InnerProductSpace ℝ β]
  [inst_4 : CompleteSpace α] [inst_5 : CompleteSpace β] (A : α →L[ℝ] β) (c : β) (x : α) (ε : ℝ) (εpos : ε > 0) (x1 : α)
  (a : ‖x - x1‖ ≤ ε) (h_original_goal : ‖⟪c, A x1 - A x - A (x1 - x)⟫_ℝ‖ ≤ ε * ‖x - x1‖) : 0 ≤ ε * ‖x - x1‖ := sorry

Tactic state before:
case right
α : Type u_4
β : Type u_5
inst✝⁵ : NormedAddCommGroup α
inst✝⁴ : NormedAddCommGroup β
inst✝³ : InnerProductSpace ℝ α
inst✝² : InnerProductSpace ℝ β
inst✝¹ : CompleteSpace α
inst✝ : CompleteSpace β
A : α →L[ℝ] β
c : β
x : α
ε : ℝ
εpos : ε > 0
x1 : α
a✝ : ‖x - x1‖ ≤ ε
⊢ ‖⟪c, A x1 - A x - A (x1 - x)⟫_ℝ‖ ≤ ε * ‖x - x1‖

Tactic:
simp only [map_sub, sub_self, inner_zero_right, norm_zero]

Tactic state after:
[case right
 α : Type u_4
 β : Type u_5
 inst✝⁵ : NormedAddCommGroup α
 inst✝⁴ : NormedAddCommGroup β
 inst✝³ : InnerProductSpace ℝ α
 inst✝² : InnerProductSpace ℝ β
 inst✝¹ : CompleteSpace α
 inst✝ : CompleteSpace β
 A : α →L[ℝ] β
 c : β
 x : α
 ε : ℝ
 εpos : ε > 0
 x1 : α
 a✝ : ‖x - x1‖ ≤ ε
 ⊢ 0 ≤ ε * ‖x - x1‖]
import Mathlib
import Optlib


open Set InnerProductSpace Topology Filter InnerProduct

open scoped Pointwise

theorem extracted_formal_statement_39.{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} (c : F) (x : E₁) :
  HasGradientAt (fun x => ⟪c, (OptProblem.A₁ E₂) x⟫_ℝ) ((ContinuousLinearMap.adjoint (OptProblem.A₁ E₂)) c) x := 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
c : F
⊢ ∀ (x : E₁),
    HasGradientAt (fun x => ⟪c, (OptProblem.A₁ E₂) x⟫_ℝ) ((ContinuousLinearMap.adjoint (OptProblem.A₁ E₂)) c) x

Tactic:
apply inner_gradient

Tactic state after:
import Mathlib
import Optlib


open Set InnerProductSpace Topology Filter InnerProduct

open scoped Pointwise

theorem extracted_formal_statement_40.{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+) (x : E₁)
  (h_original_goal :
    ⟪ADMM.y E₁ E₂ n.natPred,
        (OptProblem.A₁ E₂) x + (OptProblem.A₂ E₁) (ADMM.x₂ E₁ F n.natPred) - OptProblem.b E₁ E₂⟫_ℝ =
      ((fun x => ⟪ADMM.y E₁ E₂ n.natPred, (OptProblem.A₁ E₂) x⟫_ℝ) + fun x =>
          ⟪ADMM.y E₁ E₂ n.natPred, (OptProblem.A₂ E₁) (ADMM.x₂ E₁ F n.natPred) - OptProblem.b E₁ E₂⟫_ℝ)
        x) :
  ⟪ADMM.y E₁ E₂ n.natPred, (OptProblem.A₁ E₂) x + (OptProblem.A₂ E₁) (ADMM.x₂ E₁ F n.natPred) - OptProblem.b E₁ E₂⟫_ℝ =
    ⟪ADMM.y E₁ E₂ n.natPred, (OptProblem.A₁ E₂) x⟫_ℝ +
      ⟪ADMM.y E₁ E₂ n.natPred, (OptProblem.A₂ E₁) (ADMM.x₂ E₁ F n.natPred) - OptProblem.b E₁ E₂⟫_ℝ := 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+
x✝ : E₁
c : F := ADMM.y E₁ E₂ n.natPred
c₁ : ℝ := ⟪ADMM.y E₁ E₂ n.natPred, (OptProblem.A₂ E₁) (ADMM.x₂ E₁ F n.natPred) - OptProblem.b E₁ E₂⟫_ℝ
x : E₁
⊢ ⟪ADMM.y E₁ E₂ n.natPred, (OptProblem.A₁ E₂) x + (OptProblem.A₂ E₁) (ADMM.x₂ E₁ F n.natPred) - OptProblem.b E₁ E₂⟫_ℝ =
    ((fun x => ⟪ADMM.y E₁ E₂ n.natPred, (OptProblem.A₁ E₂) x⟫_ℝ) + fun x =>
        ⟪ADMM.y E₁ E₂ n.natPred, (OptProblem.A₂ E₁) (ADMM.x₂ E₁ F n.natPred) - OptProblem.b E₁ E₂⟫_ℝ)
      x

Tactic:
simp only [Pi.add_apply]

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+
 x✝ : E₁
 c : F := ADMM.y E₁ E₂ n.natPred
 c₁ : ℝ := ⟪ADMM.y E₁ E₂ n.natPred, (OptProblem.A₂ E₁) (ADMM.x₂ E₁ F n.natPred) - OptProblem.b E₁ E₂⟫_ℝ
 x : E₁
 ⊢ ⟪ADMM.y E₁ E₂ n.natPred, (OptProblem.A₁ E₂) x + (OptProblem.A₂ E₁) (ADMM.x₂ E₁ F n.natPred) - OptProblem.b E₁ E₂⟫_ℝ =
     ⟪ADMM.y E₁ E₂ n.natPred, (OptProblem.A₁ E₂) x⟫_ℝ +
       ⟪ADMM.y E₁ E₂ n.natPred, (OptProblem.A₂ E₁) (ADMM.x₂ E₁ F n.natPred) - OptProblem.b E₁ E₂⟫_ℝ]
import Mathlib
import Optlib


open Set InnerProductSpace Topology Filter InnerProduct

open scoped Pointwise

theorem extracted_formal_statement_41.{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+) (x : E₁) :
  ⟪ADMM.y E₁ E₂ n.natPred,
      (OptProblem.A₁ E₂) x + ((OptProblem.A₂ E₁) (ADMM.x₂ E₁ F n.natPred) - OptProblem.b E₁ E₂)⟫_ℝ =
    ⟪ADMM.y E₁ E₂ n.natPred, (OptProblem.A₁ E₂) x⟫_ℝ +
      ⟪ADMM.y E₁ E₂ n.natPred, (OptProblem.A₂ E₁) (ADMM.x₂ E₁ F n.natPred) - OptProblem.b E₁ E₂⟫_ℝ := 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+
x✝ : E₁
c : F := ADMM.y E₁ E₂ n.natPred
c₁ : ℝ := ⟪ADMM.y E₁ E₂ n.natPred, (OptProblem.A₂ E₁) (ADMM.x₂ E₁ F n.natPred) - OptProblem.b E₁ E₂⟫_ℝ
x : E₁
⊢ ⟪ADMM.y E₁ E₂ n.natPred,
      (OptProblem.A₁ E₂) x + ((OptProblem.A₂ E₁) (ADMM.x₂ E₁ F n.natPred) - OptProblem.b E₁ E₂)⟫_ℝ =
    ⟪ADMM.y E₁ E₂ n.natPred, (OptProblem.A₁ E₂) x⟫_ℝ +
      ⟪ADMM.y E₁ E₂ n.natPred, (OptProblem.A₂ E₁) (ADMM.x₂ E₁ F n.natPred) - OptProblem.b E₁ E₂⟫_ℝ

Tactic:
exact
  inner_add_right (ADMM.y E₁ E₂ n.natPred) ((OptProblem.A₁ E₂) x)
    ((OptProblem.A₂ E₁) (ADMM.x₂ E₁ F n.natPred) - OptProblem.b E₁ E₂)

Tactic state after:
import Mathlib
import Optlib


open Set InnerProductSpace Topology Filter InnerProduct

open scoped Pointwise

theorem extracted_formal_statement_43.{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+) (x : E₁) :
  let c := ADMM.y E₁ E₂ n.natPred;
  ((fun x =>
        ⟪ADMM.y E₁ E₂ n.natPred,
          (OptProblem.A₁ E₂) x + (OptProblem.A₂ E₁) (ADMM.x₂ E₁ F n.natPred) - OptProblem.b E₁ E₂⟫_ℝ) =
      (fun x => ⟪ADMM.y E₁ E₂ n.natPred, (OptProblem.A₁ E₂) x⟫_ℝ) + fun x =>
        ⟪ADMM.y E₁ E₂ n.natPred, (OptProblem.A₂ E₁) (ADMM.x₂ E₁ F n.natPred) - OptProblem.b E₁ E₂⟫_ℝ) →
    HasGradientAt (fun x => ⟪c, (OptProblem.A₁ E₂) x⟫_ℝ) ((ContinuousLinearMap.adjoint (OptProblem.A₁ E₂)) c) x := 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+
x : E₁
c : F := ADMM.y E₁ E₂ n.natPred
c₁ : ℝ := ⟪ADMM.y E₁ E₂ n.natPred, (OptProblem.A₂ E₁) (ADMM.x₂ E₁ F n.natPred) - OptProblem.b E₁ E₂⟫_ℝ
this :
  (fun x =>
      ⟪ADMM.y E₁ E₂ n.natPred,
        (OptProblem.A₁ E₂) x + (OptProblem.A₂ E₁) (ADMM.x₂ E₁ F n.natPred) - OptProblem.b E₁ E₂⟫_ℝ) =
    (fun x => ⟪ADMM.y E₁ E₂ n.natPred, (OptProblem.A₁ E₂) x⟫_ℝ) + fun x =>
      ⟪ADMM.y E₁ E₂ n.natPred, (OptProblem.A₂ E₁) (ADMM.x₂ E₁ F n.natPred) - OptProblem.b E₁ E₂⟫_ℝ
⊢ HasGradientAt (fun x => ⟪c, (OptProblem.A₁ E₂) x⟫_ℝ) ((ContinuousLinearMap.adjoint (OptProblem.A₁ E₂)) c) x

Tactic:
apply ADMM_iter_process₁'_eq3_2'_1

Tactic state after:
import Mathlib
import Optlib


open Set InnerProductSpace Topology Filter InnerProduct

open scoped Pointwise

theorem extracted_formal_statement_44.{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+) (x : E₁) (a : x ∈ univ) :
  HasGradientAt
    (fun x =>
      ⟪ADMM.y E₁ E₂ n.natPred,
        (OptProblem.A₁ E₂) x + (OptProblem.A₂ E₁) (ADMM.x₂ E₁ F n.natPred) - OptProblem.b E₁ E₂⟫_ℝ)
    ((ContinuousLinearMap.adjoint (OptProblem.A₁ E₂)) (ADMM.y E₁ E₂ n.natPred)) x := 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+
x : E₁
a✝ : x ∈ univ
⊢ HasGradientAt
    (fun x =>
      ⟪ADMM.y E₁ E₂ n.natPred,
        (OptProblem.A₁ E₂) x + (OptProblem.A₂ E₁) (ADMM.x₂ E₁ F n.natPred) - OptProblem.b E₁ E₂⟫_ℝ)
    ((ContinuousLinearMap.adjoint (OptProblem.A₁ E₂)) (ADMM.y E₁ E₂ n.natPred)) x

Tactic:
apply ADMM_iter_process₁'_eq3_2' n

Tactic state after:
import Mathlib
import Optlib


open Set InnerProductSpace Topology Filter InnerProduct

open scoped Pointwise

theorem extracted_formal_statement_45.{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 =>
          ⟪ADMM.y E₁ E₂ n.natPred,
            (OptProblem.A₁ E₂) x + (OptProblem.A₂ E₁) (ADMM.x₂ E₁ F n.natPred) - OptProblem.b E₁ E₂⟫_ℝ)
        (ADMM.x₁ E₂ F ↑n) =
      {(ContinuousLinearMap.adjoint (OptProblem.A₁ E₂)) (ADMM.y E₁ E₂ n.natPred)}) :
  HasGradientAt
    (fun x =>
      ⟪ADMM.y E₁ E₂ n.natPred,
        (OptProblem.A₁ E₂) x + (OptProblem.A₂ E₁) (ADMM.x₂ E₁ F n.natPred) - OptProblem.b E₁ E₂⟫_ℝ)
    ((ContinuousLinearMap.adjoint (OptProblem.A₁ E₂)) (ADMM.y E₁ E₂ n.natPred)) (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 =>
        ⟪ADMM.y E₁ E₂ n.natPred,
          (OptProblem.A₁ E₂) x + (OptProblem.A₂ E₁) (ADMM.x₂ E₁ F n.natPred) - OptProblem.b E₁ E₂⟫_ℝ)
      (ADMM.x₁ E₂ F ↑n) =
    {(ContinuousLinearMap.adjoint (OptProblem.A₁ E₂)) (ADMM.y E₁ E₂ n.natPred)}

Tactic:
apply SubderivAt_eq_gradient (inner_convex1 n)

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 =>
       ⟪ADMM.y E₁ E₂ n.natPred,
         (OptProblem.A₁ E₂) x + (OptProblem.A₂ E₁) (ADMM.x₂ E₁ F n.natPred) - OptProblem.b E₁ E₂⟫_ℝ)
     ((ContinuousLinearMap.adjoint (OptProblem.A₁ E₂)) (ADMM.y E₁ E₂ n.natPred)) (ADMM.x₁ E₂ F ↑n)]
import Mathlib
import Optlib


open Set InnerProductSpace Topology Filter InnerProduct

open scoped Pointwise

theorem extracted_formal_statement_46.{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 =>
      ⟪ADMM.y E₁ E₂ n.natPred,
        (OptProblem.A₁ E₂) x + (OptProblem.A₂ E₁) (ADMM.x₂ E₁ F n.natPred) - OptProblem.b E₁ E₂⟫_ℝ)
    ((ContinuousLinearMap.adjoint (OptProblem.A₁ E₂)) (ADMM.y E₁ E₂ n.natPred)) (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 =>
      ⟪ADMM.y E₁ E₂ n.natPred,
        (OptProblem.A₁ E₂) x + (OptProblem.A₂ E₁) (ADMM.x₂ E₁ F n.natPred) - OptProblem.b E₁ E₂⟫_ℝ)
    ((ContinuousLinearMap.adjoint (OptProblem.A₁ E₂)) (ADMM.y E₁ E₂ n.natPred)) (ADMM.x₁ E₂ F ↑n)

Tactic:
apply ADMM_iter_process₁'_eq3_2'

Tactic state after:
import Mathlib
import Optlib


open Set InnerProductSpace Topology Filter InnerProduct

open scoped Pointwise

theorem extracted_formal_statement_47.{u_5, u_4} {α : Type u_4} {β : Type u_5} [inst : NormedAddCommGroup α]
  [inst_1 : InnerProductSpace ℝ α] [inst_2 : NormedAddCommGroup β] [inst_3 : InnerProductSpace ℝ β] (A : α →L[ℝ] β)
  (a₁ a₂ : α) (h_original_goal : ‖A (a₁ - a₂)‖ = ‖A (a₂ - a₁)‖) : ‖-(A a₂ - A a₁)‖ = ‖A a₂ - A a₁‖ := sorry

Tactic state before:
α : Type u_4
β : Type u_5
inst✝³ : NormedAddCommGroup α
inst✝² : InnerProductSpace ℝ α
inst✝¹ : NormedAddCommGroup β
inst✝ : InnerProductSpace ℝ β
A : α →L[ℝ] β
a₁ a₂ : α
⊢ ‖A (a₁ - a₂)‖ = ‖A (a₂ - a₁)‖

Tactic:
rw [map_sub, map_sub, ← neg_sub (A a₂) (A a₁)]

Tactic state after:
[α : Type u_4
 β : Type u_5
 inst✝³ : NormedAddCommGroup α
 inst✝² : InnerProductSpace ℝ α
 inst✝¹ : NormedAddCommGroup β
 inst✝ : InnerProductSpace ℝ β
 A : α →L[ℝ] β
 a₁ a₂ : α
 ⊢ ‖-(A a₂ - A a₁)‖ = ‖A a₂ - A a₁‖]
import Mathlib
import Optlib


open Set InnerProductSpace Topology Filter InnerProduct

open scoped Pointwise

theorem extracted_formal_statement_48.{u_5, u_4} {α : Type u_4} {β : Type u_5} [inst : NormedAddCommGroup α]
  [inst_1 : InnerProductSpace ℝ α] [inst_2 : NormedAddCommGroup β] [inst_3 : InnerProductSpace ℝ β] (A : α →L[ℝ] β)
  (a₁ a₂ : α) : ‖-(A a₂ - A a₁)‖ = ‖A a₂ - A a₁‖ := sorry

Tactic state before:
α : Type u_4
β : Type u_5
inst✝³ : NormedAddCommGroup α
inst✝² : InnerProductSpace ℝ α
inst✝¹ : NormedAddCommGroup β
inst✝ : InnerProductSpace ℝ β
A : α →L[ℝ] β
a₁ a₂ : α
⊢ ‖-(A a₂ - A a₁)‖ = ‖A a₂ - A a₁‖

Tactic:
apply norm_neg

Tactic state after:
import Mathlib
import Optlib


open Set InnerProductSpace Topology Filter InnerProduct

open scoped Pointwise

theorem extracted_formal_statement_49.{u_5, u_4} {α : Type u_4} {β : Type u_5} [inst : NormedAddCommGroup α]
  [inst_1 : NormedAddCommGroup β] [inst_2 : InnerProductSpace ℝ α] [inst_3 : InnerProductSpace ℝ β]
  [inst_4 : CompleteSpace α] [inst_5 : CompleteSpace β] (A : α →L[ℝ] β) (s : α) (ε : ℝ) (εpos : ε > 0) (h : 0 < ‖A‖)
  (x : α) (hx : ‖s - x‖ ≤ ε / ‖A‖ ^ 2) : 0 < ‖A‖ ^ 2 := sorry

Tactic state before:
α : Type u_4
β : Type u_5
inst✝⁵ : NormedAddCommGroup α
inst✝⁴ : NormedAddCommGroup β
inst✝³ : InnerProductSpace ℝ α
inst✝² : InnerProductSpace ℝ β
inst✝¹ : CompleteSpace α
inst✝ : CompleteSpace β
A : α →L[ℝ] β
s : α
ε : ℝ
εpos : ε > 0
h : 0 < ‖A‖
x : α
hx : ‖s - x‖ ≤ ε / ‖A‖ ^ 2
⊢ 0 < ‖A‖ ^ 2

Tactic:
apply sq_pos_of_pos h

Tactic state after:
import Mathlib
import Optlib


open Set InnerProductSpace Topology Filter InnerProduct

open scoped Pointwise

theorem extracted_formal_statement_50.{u_5, u_4} {α : Type u_4} {β : Type u_5} [inst : NormedAddCommGroup α]
  [inst_1 : NormedAddCommGroup β] [inst_2 : InnerProductSpace ℝ α] [inst_3 : InnerProductSpace ℝ β]
  [inst_4 : CompleteSpace α] [inst_5 : CompleteSpace β] (A : α →L[ℝ] β) (s : α) (ε : ℝ) (εpos : ε > 0) (h : 0 < ‖A‖)
  (x : α) (hx : ‖s - x‖ ≤ ε / ‖A‖ ^ 2) (hzero : 0 < ‖A‖ ^ 2) : s + (x - s) = x := sorry

Tactic state before:
α : Type u_4
β : Type u_5
inst✝⁵ : NormedAddCommGroup α
inst✝⁴ : NormedAddCommGroup β
inst✝³ : InnerProductSpace ℝ α
inst✝² : InnerProductSpace ℝ β
inst✝¹ : CompleteSpace α
inst✝ : CompleteSpace β
A : α →L[ℝ] β
s : α
ε : ℝ
εpos : ε > 0
h : 0 < ‖A‖
x : α
hx : ‖s - x‖ ≤ ε / ‖A‖ ^ 2
hzero : 0 < ‖A‖ ^ 2
t : α := x - s
⊢ s + (x - s) = x

Tactic:
simp only [add_sub_cancel]

Tactic state after:
import Mathlib
import Optlib


open Set InnerProductSpace Topology Filter InnerProduct

open scoped Pointwise

theorem extracted_formal_statement_51.{u_5, u_4} {α : Type u_4} {β : Type u_5} [inst : NormedAddCommGroup α]
  [inst_1 : NormedAddCommGroup β] [inst_2 : InnerProductSpace ℝ α] [inst_3 : InnerProductSpace ℝ β]
  [inst_4 : CompleteSpace α] [inst_5 : CompleteSpace β] (A : α →L[ℝ] β) (s : α) (ε : ℝ) (εpos : ε > 0) (h : 0 < ‖A‖)
  (x : α) (hx : ‖s - x‖ ≤ ε / ‖A‖ ^ 2) (hzero : 0 < ‖A‖ ^ 2) :
  let t := x - s;
  s + t = x →
    ⟪A (s + t), A (s + t)⟫_ℝ - ⟪A s, A s⟫_ℝ - ⟪2 • (ContinuousLinearMap.adjoint A) (A s), s + t - s⟫_ℝ =
        ⟪A (s + t - s), A (s + t - s)⟫_ℝ →
      ⟪A s + A t, A s + A t⟫_ℝ - ⟪A s, A s⟫_ℝ - ⟪2 • (ContinuousLinearMap.adjoint A) (A s), t⟫_ℝ = ⟪A t, A t⟫_ℝ := sorry

Tactic state before:
α : Type u_4
β : Type u_5
inst✝⁵ : NormedAddCommGroup α
inst✝⁴ : NormedAddCommGroup β
inst✝³ : InnerProductSpace ℝ α
inst✝² : InnerProductSpace ℝ β
inst✝¹ : CompleteSpace α
inst✝ : CompleteSpace β
A : α →L[ℝ] β
s : α
ε : ℝ
εpos : ε > 0
h : 0 < ‖A‖
x : α
hx : ‖s - x‖ ≤ ε / ‖A‖ ^ 2
hzero : 0 < ‖A‖ ^ 2
t : α := x - s
t1 : s + t = x
⊢ ⟪A (s + t), A (s + t)⟫_ℝ - ⟪A s, A s⟫_ℝ - ⟪2 • (ContinuousLinearMap.adjoint A) (A s), s + t - s⟫_ℝ =
    ⟪A (s + t - s), A (s + t - s)⟫_ℝ

Tactic:
simp only [map_add, add_sub_cancel_left]

Tactic state after:
[α : Type u_4
 β : Type u_5
 inst✝⁵ : NormedAddCommGroup α
 inst✝⁴ : NormedAddCommGroup β
 inst✝³ : InnerProductSpace ℝ α
 inst✝² : InnerProductSpace ℝ β
 inst✝¹ : CompleteSpace α
 inst✝ : CompleteSpace β
 A : α →L[ℝ] β
 s : α
 ε : ℝ
 εpos : ε > 0
 h : 0 < ‖A‖
 x : α
 hx : ‖s - x‖ ≤ ε / ‖A‖ ^ 2
 hzero : 0 < ‖A‖ ^ 2
 t : α := x - s
 t1 : s + t = x
 ⊢ ⟪A s + A t, A s + A t⟫_ℝ - ⟪A s, A s⟫_ℝ - ⟪2 • (ContinuousLinearMap.adjoint A) (A s), t⟫_ℝ = ⟪A t, A t⟫_ℝ]
import Mathlib
import Optlib


open Set InnerProductSpace Topology Filter InnerProduct

open scoped Pointwise

theorem extracted_formal_statement_52.{u_5, u_4} {α : Type u_4} {β : Type u_5} [inst : NormedAddCommGroup α]
  [inst_1 : NormedAddCommGroup β] [inst_2 : InnerProductSpace ℝ α] [inst_3 : InnerProductSpace ℝ β]
  [inst_4 : CompleteSpace α] [inst_5 : CompleteSpace β] (A : α →L[ℝ] β) (s : α) (ε : ℝ) (εpos : ε > 0) (h : 0 < ‖A‖)
  (x : α) (hx : ‖s - x‖ ≤ ε / ‖A‖ ^ 2) (hzero : 0 < ‖A‖ ^ 2) :
  let t := x - s;
  s + t = x →
    ⟪A x, A x⟫_ℝ - ⟪A s, A s⟫_ℝ - ⟪2 • (ContinuousLinearMap.adjoint A) (A s), x - s⟫_ℝ = ⟪A (x - s), A (x - s)⟫_ℝ →
      |‖A (x - s)‖ ^ 2| ≤ ε * ‖s - x‖ → ‖A (x - s)‖ ^ 2 ≤ ε * ‖s - x‖ := sorry

Tactic state before:
case h
α : Type u_4
β : Type u_5
inst✝⁵ : NormedAddCommGroup α
inst✝⁴ : NormedAddCommGroup β
inst✝³ : InnerProductSpace ℝ α
inst✝² : InnerProductSpace ℝ β
inst✝¹ : CompleteSpace α
inst✝ : CompleteSpace β
A : α →L[ℝ] β
s : α
ε : ℝ
εpos : ε > 0
h : 0 < ‖A‖
x : α
hx : ‖s - x‖ ≤ ε / ‖A‖ ^ 2
hzero : 0 < ‖A‖ ^ 2
t : α := x - s
t1 : s + t = x
this : ⟪A x, A x⟫_ℝ - ⟪A s, A s⟫_ℝ - ⟪2 • (ContinuousLinearMap.adjoint A) (A s), x - s⟫_ℝ = ⟪A (x - s), A (x - s)⟫_ℝ
⊢ |‖A (x - s)‖ ^ 2| ≤ ε * ‖s - x‖

Tactic:
simp only [abs_pow, abs_norm, ge_iff_le]

Tactic state after:
[case h
 α : Type u_4
 β : Type u_5
 inst✝⁵ : NormedAddCommGroup α
 inst✝⁴ : NormedAddCommGroup β
 inst✝³ : InnerProductSpace ℝ α
 inst✝² : InnerProductSpace ℝ β
 inst✝¹ : CompleteSpace α
 inst✝ : CompleteSpace β
 A : α →L[ℝ] β
 s : α
 ε : ℝ
 εpos : ε > 0
 h : 0 < ‖A‖
 x : α
 hx : ‖s - x‖ ≤ ε / ‖A‖ ^ 2
 hzero : 0 < ‖A‖ ^ 2
 t : α := x - s
 t1 : s + t = x
 this : ⟪A x, A x⟫_ℝ - ⟪A s, A s⟫_ℝ - ⟪2 • (ContinuousLinearMap.adjoint A) (A s), x - s⟫_ℝ = ⟪A (x - s), A (x - s)⟫_ℝ
 ⊢ ‖A (x - s)‖ ^ 2 ≤ ε * ‖s - x‖]
import Mathlib
import Optlib


open Set InnerProductSpace Topology Filter InnerProduct

open scoped Pointwise

theorem extracted_formal_statement_53.{u_5, u_4} {α : Type u_4} {β : Type u_5} [inst : NormedAddCommGroup α]
  [inst_1 : NormedAddCommGroup β] [inst_2 : InnerProductSpace ℝ α] [inst_3 : InnerProductSpace ℝ β]
  [inst_4 : CompleteSpace α] [inst_5 : CompleteSpace β] (A : α →L[ℝ] β) (s : α) (ε : ℝ) (εpos : ε > 0) (h : 0 < ‖A‖)
  (x : α) (hx : ‖s - x‖ ≤ ε / ‖A‖ ^ 2) (hzero : 0 < ‖A‖ ^ 2) :
  let t := x - s;
  s + t = x →
    ⟪A x, A x⟫_ℝ - ⟪A s, A s⟫_ℝ - ⟪2 • (ContinuousLinearMap.adjoint A) (A s), x - s⟫_ℝ = ⟪A (x - s), A (x - s)⟫_ℝ →
      0 = ‖s - x‖ → ‖A‖ ^ 2 * 0 ^ 2 ≤ ε * 0 := sorry

Tactic state before:
case h.calc.step.inl
α : Type u_4
β : Type u_5
inst✝⁵ : NormedAddCommGroup α
inst✝⁴ : NormedAddCommGroup β
inst✝³ : InnerProductSpace ℝ α
inst✝² : InnerProductSpace ℝ β
inst✝¹ : CompleteSpace α
inst✝ : CompleteSpace β
A : α →L[ℝ] β
s : α
ε : ℝ
εpos : ε > 0
h : 0 < ‖A‖
x : α
hx : ‖s - x‖ ≤ ε / ‖A‖ ^ 2
hzero : 0 < ‖A‖ ^ 2
t : α := x - s
t1 : s + t = x
this : ⟪A x, A x⟫_ℝ - ⟪A s, A s⟫_ℝ - ⟪2 • (ContinuousLinearMap.adjoint A) (A s), x - s⟫_ℝ = ⟪A (x - s), A (x - s)⟫_ℝ
h1 : 0 = ‖s - x‖
⊢ ‖A‖ ^ 2 * 0 ^ 2 ≤ ε * 0

Tactic:
simp only [ne_eq, OfNat.ofNat_ne_zero, not_false_eq_true, zero_pow, mul_zero, le_refl]

Tactic state after: