zzhisthebest/LeanBenchmark huggingface dataset:
import Mathlib
import Optlib


open Set

theorem extracted_formal_statement_0.{u_1} {E : Type u_1} [inst : NormedAddCommGroup E] [inst_1 : InnerProductSpace ℝ E]
  [inst_2 : CompleteSpace E] [inst_3 : ProperSpace E] {x₀ : E} {f : E → ℝ} {f' : E → E} {h : E → ℝ}
  {alg : proximal_gradient_method f h f' x₀} (k : ℕ+)
  (h_original_goal :
    f (proximal_gradient_method.x f h f' x₀ ↑k) + h (proximal_gradient_method.x f h f' x₀ ↑k) -
          f (proximal_gradient_method.xm f h f' x₀) -
        h (proximal_gradient_method.xm f h f' x₀) ≤
      1 / (2 * ↑↑k * proximal_gradient_method.t f h f' x₀) * ‖x₀ - proximal_gradient_method.xm f h f' x₀‖ ^ 2) :
  2 * ↑↑k * proximal_gradient_method.t f h f' x₀ *
      (f (proximal_gradient_method.x f h f' x₀ ↑k) + h (proximal_gradient_method.x f h f' x₀ ↑k) -
          f (proximal_gradient_method.xm f h f' x₀) -
        h (proximal_gradient_method.xm f h f' x₀)) ≤
    ‖x₀ - proximal_gradient_method.xm f h f' x₀‖ ^ 2 := sorry

Tactic state before:
E : Type u_1
inst✝³ : NormedAddCommGroup E
inst✝² : InnerProductSpace ℝ E
inst✝¹ : CompleteSpace E
inst✝ : ProperSpace E
x₀ : E
f : E → ℝ
f' : E → E
h : E → ℝ
alg : proximal_gradient_method f h f' x₀
k : ℕ+
⊢ f (proximal_gradient_method.x f h f' x₀ ↑k) + h (proximal_gradient_method.x f h f' x₀ ↑k) -
        f (proximal_gradient_method.xm f h f' x₀) -
      h (proximal_gradient_method.xm f h f' x₀) ≤
    1 / (2 * ↑↑k * proximal_gradient_method.t f h f' x₀) * ‖x₀ - proximal_gradient_method.xm f h f' x₀‖ ^ 2

Tactic:
rw [mul_comm, mul_one_div, le_div_iff₀, mul_comm]

Tactic state after:
[E : Type u_1
 inst✝³ : NormedAddCommGroup E
 inst✝² : InnerProductSpace ℝ E
 inst✝¹ : CompleteSpace E
 inst✝ : ProperSpace E
 x₀ : E
 f : E → ℝ
 f' : E → E
 h : E → ℝ
 alg : proximal_gradient_method f h f' x₀
 k : ℕ+
 ⊢ 2 * ↑↑k * proximal_gradient_method.t f h f' x₀ *
       (f (proximal_gradient_method.x f h f' x₀ ↑k) + h (proximal_gradient_method.x f h f' x₀ ↑k) -
           f (proximal_gradient_method.xm f h f' x₀) -
         h (proximal_gradient_method.xm f h f' x₀)) ≤
     ‖x₀ - proximal_gradient_method.xm f h f' x₀‖ ^ 2,
 E : Type u_1
 inst✝³ : NormedAddCommGroup E
 inst✝² : InnerProductSpace ℝ E
 inst✝¹ : CompleteSpace E
 inst✝ : ProperSpace E
 x₀ : E
 f : E → ℝ
 f' : E → E
 h : E → ℝ
 alg : proximal_gradient_method f h f' x₀
 k : ℕ+
 ⊢ 0 < 2 * ↑↑k * proximal_gradient_method.t f h f' x₀]
import Mathlib
import Optlib


open Set

theorem extracted_formal_statement_1.{u_1} {E : Type u_1} [inst : NormedAddCommGroup E] [inst_1 : InnerProductSpace ℝ E]
  [inst_2 : CompleteSpace E] [inst_3 : ProperSpace E] {x₀ : E} {f : E → ℝ} {f' : E → E} {h : E → ℝ}
  {alg : proximal_gradient_method f h f' x₀} (k : ℕ+)
  (h_original_goal :
    f (proximal_gradient_method.x f h f' x₀ ↑k) + h (proximal_gradient_method.x f h f' x₀ ↑k) -
          f (proximal_gradient_method.xm f h f' x₀) -
        h (proximal_gradient_method.xm f h f' x₀) ≤
      1 / (2 * ↑↑k * proximal_gradient_method.t f h f' x₀) * ‖x₀ - proximal_gradient_method.xm f h f' x₀‖ ^ 2) :
  0 < 2 * ↑↑k * proximal_gradient_method.t f h f' x₀ := sorry

Tactic state before:
E : Type u_1
inst✝³ : NormedAddCommGroup E
inst✝² : InnerProductSpace ℝ E
inst✝¹ : CompleteSpace E
inst✝ : ProperSpace E
x₀ : E
f : E → ℝ
f' : E → E
h : E → ℝ
alg : proximal_gradient_method f h f' x₀
k : ℕ+
⊢ f (proximal_gradient_method.x f h f' x₀ ↑k) + h (proximal_gradient_method.x f h f' x₀ ↑k) -
        f (proximal_gradient_method.xm f h f' x₀) -
      h (proximal_gradient_method.xm f h f' x₀) ≤
    1 / (2 * ↑↑k * proximal_gradient_method.t f h f' x₀) * ‖x₀ - proximal_gradient_method.xm f h f' x₀‖ ^ 2

Tactic:
rw [mul_comm, mul_one_div, le_div_iff₀, mul_comm]

Tactic state after:
[E : Type u_1
 inst✝³ : NormedAddCommGroup E
 inst✝² : InnerProductSpace ℝ E
 inst✝¹ : CompleteSpace E
 inst✝ : ProperSpace E
 x₀ : E
 f : E → ℝ
 f' : E → E
 h : E → ℝ
 alg : proximal_gradient_method f h f' x₀
 k : ℕ+
 ⊢ 2 * ↑↑k * proximal_gradient_method.t f h f' x₀ *
       (f (proximal_gradient_method.x f h f' x₀ ↑k) + h (proximal_gradient_method.x f h f' x₀ ↑k) -
           f (proximal_gradient_method.xm f h f' x₀) -
         h (proximal_gradient_method.xm f h f' x₀)) ≤
     ‖x₀ - proximal_gradient_method.xm f h f' x₀‖ ^ 2,
 E : Type u_1
 inst✝³ : NormedAddCommGroup E
 inst✝² : InnerProductSpace ℝ E
 inst✝¹ : CompleteSpace E
 inst✝ : ProperSpace E
 x₀ : E
 f : E → ℝ
 f' : E → E
 h : E → ℝ
 alg : proximal_gradient_method f h f' x₀
 k : ℕ+
 ⊢ 0 < 2 * ↑↑k * proximal_gradient_method.t f h f' x₀]
import Mathlib
import Optlib


open Set

theorem extracted_formal_statement_2.{u_1} {E : Type u_1} [inst : NormedAddCommGroup E] [inst_1 : InnerProductSpace ℝ E]
  [inst_2 : CompleteSpace E] [inst_3 : ProperSpace E] {x₀ : E} {f : E → ℝ} {f' : E → E} {h : E → ℝ}
  {alg : proximal_gradient_method f h f' x₀} : ContinuousOn (proximal_gradient_method.t f h f' x₀ • h) univ := sorry

Tactic state before:
E : Type u_1
inst✝³ : NormedAddCommGroup E
inst✝² : InnerProductSpace ℝ E
inst✝¹ : CompleteSpace E
inst✝ : ProperSpace E
x₀ : E
f : E → ℝ
f' : E → E
h : E → ℝ
alg : proximal_gradient_method f h f' x₀
k : ℕ+
⊢ ContinuousOn (proximal_gradient_method.t f h f' x₀ • h) univ

Tactic:
apply ContinuousOn.const_smul alg.h₃ alg.t

Tactic state after:
import Mathlib
import Optlib


open Set

theorem extracted_formal_statement_3.{u_1} {E : Type u_1} [inst : NormedAddCommGroup E] [inst_1 : InnerProductSpace ℝ E]
  [inst_2 : CompleteSpace E] [inst_3 : ProperSpace E] {x₀ : E} {f : E → ℝ} {f' : E → E} {h : E → ℝ}
  {alg : proximal_gradient_method f h f' x₀} (th : ContinuousOn (proximal_gradient_method.t f h f' x₀ • h) univ)
  (h_original_goal : ConvexOn ℝ univ (proximal_gradient_method.t f h f' x₀ • h)) :
  0 ≤ proximal_gradient_method.t f h f' x₀ := sorry

Tactic state before:
E : Type u_1
inst✝³ : NormedAddCommGroup E
inst✝² : InnerProductSpace ℝ E
inst✝¹ : CompleteSpace E
inst✝ : ProperSpace E
x₀ : E
f : E → ℝ
f' : E → E
h : E → ℝ
alg : proximal_gradient_method f h f' x₀
k : ℕ+
th : ContinuousOn (proximal_gradient_method.t f h f' x₀ • h) univ
⊢ ConvexOn ℝ univ (proximal_gradient_method.t f h f' x₀ • h)

Tactic:
apply ConvexOn.smul

Tactic state after:
[case hc
 E : Type u_1
 inst✝³ : NormedAddCommGroup E
 inst✝² : InnerProductSpace ℝ E
 inst✝¹ : CompleteSpace E
 inst✝ : ProperSpace E
 x₀ : E
 f : E → ℝ
 f' : E → E
 h : E → ℝ
 alg : proximal_gradient_method f h f' x₀
 k : ℕ+
 th : ContinuousOn (proximal_gradient_method.t f h f' x₀ • h) univ
 ⊢ 0 ≤ proximal_gradient_method.t f h f' x₀,
 case hf
 E : Type u_1
 inst✝³ : NormedAddCommGroup E
 inst✝² : InnerProductSpace ℝ E
 inst✝¹ : CompleteSpace E
 inst✝ : ProperSpace E
 x₀ : E
 f : E → ℝ
 f' : E → E
 h : E → ℝ
 alg : proximal_gradient_method f h f' x₀
 k : ℕ+
 th : ContinuousOn (proximal_gradient_method.t f h f' x₀ • h) univ
 ⊢ ConvexOn ℝ univ h]
import Mathlib
import Optlib


open Set

theorem extracted_formal_statement_4.{u_1} {E : Type u_1} [inst : NormedAddCommGroup E] [inst_1 : InnerProductSpace ℝ E]
  [inst_2 : CompleteSpace E] [inst_3 : ProperSpace E] {x₀ : E} {f : E → ℝ} {f' : E → E} {h : E → ℝ}
  {alg : proximal_gradient_method f h f' x₀} (th : ContinuousOn (proximal_gradient_method.t f h f' x₀ • h) univ)
  (h_original_goal : ConvexOn ℝ univ (proximal_gradient_method.t f h f' x₀ • h)) : ConvexOn ℝ univ h := sorry

Tactic state before:
E : Type u_1
inst✝³ : NormedAddCommGroup E
inst✝² : InnerProductSpace ℝ E
inst✝¹ : CompleteSpace E
inst✝ : ProperSpace E
x₀ : E
f : E → ℝ
f' : E → E
h : E → ℝ
alg : proximal_gradient_method f h f' x₀
k : ℕ+
th : ContinuousOn (proximal_gradient_method.t f h f' x₀ • h) univ
⊢ ConvexOn ℝ univ (proximal_gradient_method.t f h f' x₀ • h)

Tactic:
apply ConvexOn.smul

Tactic state after:
[case hc
 E : Type u_1
 inst✝³ : NormedAddCommGroup E
 inst✝² : InnerProductSpace ℝ E
 inst✝¹ : CompleteSpace E
 inst✝ : ProperSpace E
 x₀ : E
 f : E → ℝ
 f' : E → E
 h : E → ℝ
 alg : proximal_gradient_method f h f' x₀
 k : ℕ+
 th : ContinuousOn (proximal_gradient_method.t f h f' x₀ • h) univ
 ⊢ 0 ≤ proximal_gradient_method.t f h f' x₀,
 case hf
 E : Type u_1
 inst✝³ : NormedAddCommGroup E
 inst✝² : InnerProductSpace ℝ E
 inst✝¹ : CompleteSpace E
 inst✝ : ProperSpace E
 x₀ : E
 f : E → ℝ
 f' : E → E
 h : E → ℝ
 alg : proximal_gradient_method f h f' x₀
 k : ℕ+
 th : ContinuousOn (proximal_gradient_method.t f h f' x₀ • h) univ
 ⊢ ConvexOn ℝ univ h]
import Mathlib
import Optlib


open Set

theorem extracted_formal_statement_5.{u_1} {E : Type u_1} [inst : NormedAddCommGroup E] [inst_1 : InnerProductSpace ℝ E]
  [inst_2 : CompleteSpace E] [inst_3 : ProperSpace E] {x₀ : E} {f : E → ℝ} {f' : E → E} {h : E → ℝ}
  {alg : proximal_gradient_method f h f' x₀} (th : ContinuousOn (proximal_gradient_method.t f h f' x₀ • h) univ) :
  ConvexOn ℝ univ h := sorry

Tactic state before:
case hf
E : Type u_1
inst✝³ : NormedAddCommGroup E
inst✝² : InnerProductSpace ℝ E
inst✝¹ : CompleteSpace E
inst✝ : ProperSpace E
x₀ : E
f : E → ℝ
f' : E → E
h : E → ℝ
alg : proximal_gradient_method f h f' x₀
k : ℕ+
th : ContinuousOn (proximal_gradient_method.t f h f' x₀ • h) univ
⊢ ConvexOn ℝ univ h

Tactic:
apply alg.hconv

Tactic state after:
import Mathlib
import Optlib


open Set

theorem extracted_formal_statement_6.{u_1} {E : Type u_1} [inst : NormedAddCommGroup E] [inst_1 : InnerProductSpace ℝ E]
  [inst_2 : CompleteSpace E] [inst_3 : ProperSpace E] {x₀ : E} {f : E → ℝ} {f' : E → E} {h : E → ℝ}
  {alg : proximal_gradient_method f h f' x₀} (th : ContinuousOn (proximal_gradient_method.t f h f' x₀ • h) univ)
  (th' : ConvexOn ℝ univ (proximal_gradient_method.t f h f' x₀ • h)) :
  let Gt := fun x =>
    (1 / proximal_gradient_method.t f h f' x₀) •
      (x -
        prox_point_c' (proximal_gradient_method.t f h f' x₀ • h) (x - proximal_gradient_method.t f h f' x₀ • f' x) th
          th');
  ∀ (z : E),
    z - proximal_gradient_method.t f h f' x₀ • Gt z =
        prox_point_c' (proximal_gradient_method.t f h f' x₀ • h) (z - proximal_gradient_method.t f h f' x₀ • f' z) th
          th' →
      z -
          proximal_gradient_method.t f h f' x₀ •
            (proximal_gradient_method.t f h f' x₀)⁻¹ •
              (z -
                prox_point_c' (proximal_gradient_method.t f h f' x₀ • h)
                  (z - proximal_gradient_method.t f h f' x₀ • f' z) th th') =
        prox_point_c' (proximal_gradient_method.t f h f' x₀ • h) (z - proximal_gradient_method.t f h f' x₀ • f' z) th
          th' := sorry

Tactic state before:
E : Type u_1
inst✝³ : NormedAddCommGroup E
inst✝² : InnerProductSpace ℝ E
inst✝¹ : CompleteSpace E
inst✝ : ProperSpace E
x₀ : E
f : E → ℝ
f' : E → E
h : E → ℝ
alg : proximal_gradient_method f h f' x₀
k : ℕ+
th : ContinuousOn (proximal_gradient_method.t f h f' x₀ • h) univ
th' : ConvexOn ℝ univ (proximal_gradient_method.t f h f' x₀ • h)
Gt : E → E :=
  fun x =>
    (1 / proximal_gradient_method.t f h f' x₀) •
      (x -
        prox_point_c' (proximal_gradient_method.t f h f' x₀ • h) (x - proximal_gradient_method.t f h f' x₀ • f' x) th
          th')
φ : E → ℝ := fun x => f x + h x
z : E
⊢ z - proximal_gradient_method.t f h f' x₀ • Gt z =
    prox_point_c' (proximal_gradient_method.t f h f' x₀ • h) (z - proximal_gradient_method.t f h f' x₀ • f' z) th th'

Tactic:
simp [Gt]

Tactic state after:
[E : Type u_1
 inst✝³ : NormedAddCommGroup E
 inst✝² : InnerProductSpace ℝ E
 inst✝¹ : CompleteSpace E
 inst✝ : ProperSpace E
 x₀ : E
 f : E → ℝ
 f' : E → E
 h : E → ℝ
 alg : proximal_gradient_method f h f' x₀
 k : ℕ+
 th : ContinuousOn (proximal_gradient_method.t f h f' x₀ • h) univ
 th' : ConvexOn ℝ univ (proximal_gradient_method.t f h f' x₀ • h)
 Gt : E → E :=
   fun x =>
     (1 / proximal_gradient_method.t f h f' x₀) •
       (x -
         prox_point_c' (proximal_gradient_method.t f h f' x₀ • h) (x - proximal_gradient_method.t f h f' x₀ • f' x) th
           th')
 φ : E → ℝ := fun x => f x + h x
 z : E
 ⊢ z -
       proximal_gradient_method.t f h f' x₀ •
         (proximal_gradient_method.t f h f' x₀)⁻¹ •
           (z -
             prox_point_c' (proximal_gradient_method.t f h f' x₀ • h) (z - proximal_gradient_method.t f h f' x₀ • f' z)
               th th') =
     prox_point_c' (proximal_gradient_method.t f h f' x₀ • h) (z - proximal_gradient_method.t f h f' x₀ • f' z) th th']
import Mathlib
import Optlib


open Set

theorem extracted_formal_statement_9.{u_1} {E : Type u_1} [inst : NormedAddCommGroup E] [inst_1 : InnerProductSpace ℝ E]
  [inst_2 : CompleteSpace E] [inst_3 : ProperSpace E] {x₀ : E} {f : E → ℝ} {f' : E → E} {h : E → ℝ}
  {alg : proximal_gradient_method f h f' x₀} (th : ContinuousOn (proximal_gradient_method.t f h f' x₀ • h) univ)
  (th' : ConvexOn ℝ univ (proximal_gradient_method.t f h f' x₀ • h)) :
  let Gt := fun x =>
    (1 / proximal_gradient_method.t f h f' x₀) •
      (x -
        prox_point_c' (proximal_gradient_method.t f h f' x₀ • h) (x - proximal_gradient_method.t f h f' x₀ • f' x) th
          th');
  ∀ (z : E),
    z - proximal_gradient_method.t f h f' x₀ • Gt z =
        Classical.choose
          (prox_well_define_convex (proximal_gradient_method.t f h f' x₀ • h) th th'
            (z - proximal_gradient_method.t f h f' x₀ • f' z)) →
      prox_prop (proximal_gradient_method.t f h f' x₀ • h) (z - proximal_gradient_method.t f h f' x₀ • f' z)
          (z - proximal_gradient_method.t f h f' x₀ • Gt z) →
        prox_prop (proximal_gradient_method.t f h f' x₀ • h) (z - proximal_gradient_method.t f h f' x₀ • f' z)
          (Classical.choose
            (prox_well_define_convex (proximal_gradient_method.t f h f' x₀ • h) th th'
              (z - proximal_gradient_method.t f h f' x₀ • f' z))) := sorry

Tactic state before:
E : Type u_1
inst✝³ : NormedAddCommGroup E
inst✝² : InnerProductSpace ℝ E
inst✝¹ : CompleteSpace E
inst✝ : ProperSpace E
x₀ : E
f : E → ℝ
f' : E → E
h : E → ℝ
alg : proximal_gradient_method f h f' x₀
k : ℕ+
th : ContinuousOn (proximal_gradient_method.t f h f' x₀ • h) univ
th' : ConvexOn ℝ univ (proximal_gradient_method.t f h f' x₀ • h)
Gt : E → E :=
  fun x =>
    (1 / proximal_gradient_method.t f h f' x₀) •
      (x -
        prox_point_c' (proximal_gradient_method.t f h f' x₀ • h) (x - proximal_gradient_method.t f h f' x₀ • f' x) th
          th')
φ : E → ℝ := fun x => f x + h x
z : E
eq1 : z - proximal_gradient_method.t f h f' x₀ • Gt z = Classical.choose ⋯
⊢ prox_prop (proximal_gradient_method.t f h f' x₀ • h) (z - proximal_gradient_method.t f h f' x₀ • f' z)
    (z - proximal_gradient_method.t f h f' x₀ • Gt z)

Tactic:
rw [eq1]

Tactic state after:
[E : Type u_1
 inst✝³ : NormedAddCommGroup E
 inst✝² : InnerProductSpace ℝ E
 inst✝¹ : CompleteSpace E
 inst✝ : ProperSpace E
 x₀ : E
 f : E → ℝ
 f' : E → E
 h : E → ℝ
 alg : proximal_gradient_method f h f' x₀
 k : ℕ+
 th : ContinuousOn (proximal_gradient_method.t f h f' x₀ • h) univ
 th' : ConvexOn ℝ univ (proximal_gradient_method.t f h f' x₀ • h)
 Gt : E → E :=
   fun x =>
     (1 / proximal_gradient_method.t f h f' x₀) •
       (x -
         prox_point_c' (proximal_gradient_method.t f h f' x₀ • h) (x - proximal_gradient_method.t f h f' x₀ • f' x) th
           th')
 φ : E → ℝ := fun x => f x + h x
 z : E
 eq1 : z - proximal_gradient_method.t f h f' x₀ • Gt z = Classical.choose ⋯
 ⊢ prox_prop (proximal_gradient_method.t f h f' x₀ • h) (z - proximal_gradient_method.t f h f' x₀ • f' z)
     (Classical.choose ⋯)]
import Mathlib
import Optlib


open Set

theorem extracted_formal_statement_10.{u_1} {E : Type u_1} [inst : NormedAddCommGroup E] [inst_1 : InnerProductSpace ℝ E]
  [inst_2 : CompleteSpace E] [inst_3 : ProperSpace E] {x₀ : E} {f : E → ℝ} {f' : E → E} {h : E → ℝ}
  {alg : proximal_gradient_method f h f' x₀} (th : ContinuousOn (proximal_gradient_method.t f h f' x₀ • h) univ)
  (th' : ConvexOn ℝ univ (proximal_gradient_method.t f h f' x₀ • h)) :
  let Gt := fun x =>
    (1 / proximal_gradient_method.t f h f' x₀) •
      (x -
        prox_point_c' (proximal_gradient_method.t f h f' x₀ • h) (x - proximal_gradient_method.t f h f' x₀ • f' x) th
          th');
  ∀ (z : E),
    z - proximal_gradient_method.t f h f' x₀ • Gt z =
        Classical.choose
          (prox_well_define_convex (proximal_gradient_method.t f h f' x₀ • h) th th'
            (z - proximal_gradient_method.t f h f' x₀ • f' z)) →
      prox_prop (proximal_gradient_method.t f h f' x₀ • h) (z - proximal_gradient_method.t f h f' x₀ • f' z)
        (Classical.choose
          (prox_well_define_convex (proximal_gradient_method.t f h f' x₀ • h) th th'
            (z - proximal_gradient_method.t f h f' x₀ • f' z))) := sorry

Tactic state before:
E : Type u_1
inst✝³ : NormedAddCommGroup E
inst✝² : InnerProductSpace ℝ E
inst✝¹ : CompleteSpace E
inst✝ : ProperSpace E
x₀ : E
f : E → ℝ
f' : E → E
h : E → ℝ
alg : proximal_gradient_method f h f' x₀
k : ℕ+
th : ContinuousOn (proximal_gradient_method.t f h f' x₀ • h) univ
th' : ConvexOn ℝ univ (proximal_gradient_method.t f h f' x₀ • h)
Gt : E → E :=
  fun x =>
    (1 / proximal_gradient_method.t f h f' x₀) •
      (x -
        prox_point_c' (proximal_gradient_method.t f h f' x₀ • h) (x - proximal_gradient_method.t f h f' x₀ • f' x) th
          th')
φ : E → ℝ := fun x => f x + h x
z : E
eq1 : z - proximal_gradient_method.t f h f' x₀ • Gt z = Classical.choose ⋯
⊢ prox_prop (proximal_gradient_method.t f h f' x₀ • h) (z - proximal_gradient_method.t f h f' x₀ • f' z)
    (Classical.choose ⋯)

Tactic:
apply Classical.choose_spec

Tactic state after:
import Mathlib
import Optlib


open Set

theorem extracted_formal_statement_13.{u_1} {E : Type u_1} [inst : NormedAddCommGroup E] [inst_1 : InnerProductSpace ℝ E]
  [inst_2 : CompleteSpace E] [inst_3 : ProperSpace E] {x₀ : E} {f : E → ℝ} {f' : E → E} {h : E → ℝ}
  {alg : proximal_gradient_method f h f' x₀} (th : ContinuousOn (proximal_gradient_method.t f h f' x₀ • h) univ)
  (th' : ConvexOn ℝ univ (proximal_gradient_method.t f h f' x₀ • h)) :
  let Gt := fun x =>
    (1 / proximal_gradient_method.t f h f' x₀) •
      (x -
        prox_point_c' (proximal_gradient_method.t f h f' x₀ • h) (x - proximal_gradient_method.t f h f' x₀ • f' x) th
          th');
  (∀ (z : E), Gt z - f' z ∈ SubderivAt h (z - proximal_gradient_method.t f h f' x₀ • Gt z)) →
    ∀ (x : E),
      let y := x - proximal_gradient_method.t f h f' x₀ • Gt x;
      f y ≤ f x + inner (f' x) (y - x) + ↑(proximal_gradient_method.L f h f' x₀) / 2 * ‖y - x‖ ^ 2 := sorry

Tactic state before:
E : Type u_1
inst✝³ : NormedAddCommGroup E
inst✝² : InnerProductSpace ℝ E
inst✝¹ : CompleteSpace E
inst✝ : ProperSpace E
x₀ : E
f : E → ℝ
f' : E → E
h : E → ℝ
alg : proximal_gradient_method f h f' x₀
k : ℕ+
th : ContinuousOn (proximal_gradient_method.t f h f' x₀ • h) univ
th' : ConvexOn ℝ univ (proximal_gradient_method.t f h f' x₀ • h)
Gt : E → E :=
  fun x =>
    (1 / proximal_gradient_method.t f h f' x₀) •
      (x -
        prox_point_c' (proximal_gradient_method.t f h f' x₀ • h) (x - proximal_gradient_method.t f h f' x₀ • f' x) th
          th')
φ : E → ℝ := fun x => f x + h x
hG : ∀ (z : E), Gt z - f' z ∈ SubderivAt h (z - proximal_gradient_method.t f h f' x₀ • Gt z)
x : E
y : E := x - proximal_gradient_method.t f h f' x₀ • Gt x
⊢ f y ≤ f x + inner (f' x) (y - x) + ↑(proximal_gradient_method.L f h f' x₀) / 2 * ‖y - x‖ ^ 2

Tactic:
apply lipschitz_continuos_upper_bound' alg.h₁ alg.h₂

Tactic state after:
import Mathlib
import Optlib


open Set

theorem extracted_formal_statement_14.{u_1} {E : Type u_1} [inst : NormedAddCommGroup E] [inst_1 : InnerProductSpace ℝ E]
  [inst_2 : CompleteSpace E] [inst_3 : ProperSpace E] {x₀ : E} {f : E → ℝ} {f' : E → E} {h : E → ℝ}
  {alg : proximal_gradient_method f h f' x₀} (th : ContinuousOn (proximal_gradient_method.t f h f' x₀ • h) univ)
  (th' : ConvexOn ℝ univ (proximal_gradient_method.t f h f' x₀ • h)) :
  let Gt := fun x =>
    (1 / proximal_gradient_method.t f h f' x₀) •
      (x -
        prox_point_c' (proximal_gradient_method.t f h f' x₀ • h) (x - proximal_gradient_method.t f h f' x₀ • f' x) th
          th');
  (∀ (z : E), Gt z - f' z ∈ SubderivAt h (z - proximal_gradient_method.t f h f' x₀ • Gt z)) →
    ∀ (x : E),
      let y := x - proximal_gradient_method.t f h f' x₀ • Gt x;
      f y ≤ f x + inner (f' x) (y - x) + ↑(proximal_gradient_method.L f h f' x₀) / 2 * ‖y - x‖ ^ 2 →
        y - x = -proximal_gradient_method.t f h f' x₀ • Gt x := sorry

Tactic state before:
E : Type u_1
inst✝³ : NormedAddCommGroup E
inst✝² : InnerProductSpace ℝ E
inst✝¹ : CompleteSpace E
inst✝ : ProperSpace E
x₀ : E
f : E → ℝ
f' : E → E
h : E → ℝ
alg : proximal_gradient_method f h f' x₀
k : ℕ+
th : ContinuousOn (proximal_gradient_method.t f h f' x₀ • h) univ
th' : ConvexOn ℝ univ (proximal_gradient_method.t f h f' x₀ • h)
Gt : E → E :=
  fun x =>
    (1 / proximal_gradient_method.t f h f' x₀) •
      (x -
        prox_point_c' (proximal_gradient_method.t f h f' x₀ • h) (x - proximal_gradient_method.t f h f' x₀ • f' x) th
          th')
φ : E → ℝ := fun x => f x + h x
hG : ∀ (z : E), Gt z - f' z ∈ SubderivAt h (z - proximal_gradient_method.t f h f' x₀ • Gt z)
x : E
y : E := x - proximal_gradient_method.t f h f' x₀ • Gt x
ieq1 : f y ≤ f x + inner (f' x) (y - x) + ↑(proximal_gradient_method.L f h f' x₀) / 2 * ‖y - x‖ ^ 2
⊢ y - x = -proximal_gradient_method.t f h f' x₀ • Gt x

Tactic:
simp [Gt, y]

Tactic state after:
import Mathlib
import Optlib


open Set

theorem extracted_formal_statement_21.{u_1} {E : Type u_1} [inst : NormedAddCommGroup E] [inst_1 : InnerProductSpace ℝ E]
  [inst_2 : CompleteSpace E] [inst_3 : ProperSpace E] {x₀ : E} {f : E → ℝ} {f' : E → E} {h : E → ℝ}
  {alg : proximal_gradient_method f h f' x₀} (th : ContinuousOn (proximal_gradient_method.t f h f' x₀ • h) univ)
  (th' : ConvexOn ℝ univ (proximal_gradient_method.t f h f' x₀ • h)) :
  let Gt := fun x =>
    (1 / proximal_gradient_method.t f h f' x₀) •
      (x -
        prox_point_c' (proximal_gradient_method.t f h f' x₀ • h) (x - proximal_gradient_method.t f h f' x₀ • f' x) th
          th');
  (∀ (z : E), Gt z - f' z ∈ SubderivAt h (z - proximal_gradient_method.t f h f' x₀ • Gt z)) →
    (∀ (x : E),
        f (x - proximal_gradient_method.t f h f' x₀ • Gt x) ≤
          f x - proximal_gradient_method.t f h f' x₀ * inner (f' x) (Gt x) +
            proximal_gradient_method.t f h f' x₀ ^ 2 * ↑(proximal_gradient_method.L f h f' x₀) / 2 * ‖Gt x‖ ^ 2) →
      0 ≤ ↑(proximal_gradient_method.L f h f' x₀) := sorry

Tactic state before:
case a0
E : Type u_1
inst✝³ : NormedAddCommGroup E
inst✝² : InnerProductSpace ℝ E
inst✝¹ : CompleteSpace E
inst✝ : ProperSpace E
x₀ : E
f : E → ℝ
f' : E → E
h : E → ℝ
alg : proximal_gradient_method f h f' x₀
k : ℕ+
th : ContinuousOn (proximal_gradient_method.t f h f' x₀ • h) univ
th' : ConvexOn ℝ univ (proximal_gradient_method.t f h f' x₀ • h)
Gt : E → E :=
  fun x =>
    (1 / proximal_gradient_method.t f h f' x₀) •
      (x -
        prox_point_c' (proximal_gradient_method.t f h f' x₀ • h) (x - proximal_gradient_method.t f h f' x₀ • f' x) th
          th')
φ : E → ℝ := fun x => f x + h x
hG : ∀ (z : E), Gt z - f' z ∈ SubderivAt h (z - proximal_gradient_method.t f h f' x₀ • Gt z)
fieq1 :
  ∀ (x : E),
    f (x - proximal_gradient_method.t f h f' x₀ • Gt x) ≤
      f x - proximal_gradient_method.t f h f' x₀ * inner (f' x) (Gt x) +
        proximal_gradient_method.t f h f' x₀ ^ 2 * ↑(proximal_gradient_method.L f h f' x₀) / 2 * ‖Gt x‖ ^ 2
x : E
⊢ 0 ≤ ↑(proximal_gradient_method.L f h f' x₀)

Tactic:
simp

Tactic state after: