zzhisthebest/LeanBenchmark huggingface dataset:
import Mathlib
import Optlib


open Set Real

open Set Real PNat

theorem extracted_formal_statement_132 (k : ℕ) (hk : k ≠ 0) (h_original_goal : 2 / (1 + ↑k) ≤ 1) : 2 ≤ 1 * (1 + ↑k) := sorry

Tactic state before:
case neg
E : Type u_1
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
f✝ h✝ : E → ℝ
f'✝ : E → E
x0✝ : E
f h : E → ℝ
f' : E → E
x0 : E
p : Nesterov_second_fix_stepsize f h f' x0
k : ℕ
hk : k ≠ 0
⊢ 2 / (1 + ↑k) ≤ 1

Tactic:
rw [div_le_iff₀ (by positivity)]

Tactic state after:
[case neg
 E : Type u_1
 inst✝² : NormedAddCommGroup E
 inst✝¹ : InnerProductSpace ℝ E
 inst✝ : CompleteSpace E
 f✝ h✝ : E → ℝ
 f'✝ : E → E
 x0✝ : E
 f h : E → ℝ
 f' : E → E
 x0 : E
 p : Nesterov_second_fix_stepsize f h f' x0
 k : ℕ
 hk : k ≠ 0
 ⊢ 2 ≤ 1 * (1 + ↑k)]
import Mathlib
import Optlib


open Set Real

open Set Real PNat

theorem extracted_formal_statement_133 (k : ℕ) (hk : k ≠ 0) (h_original_goal : 2 ≤ 1 * (1 + ↑k)) : 2 ≤ 1 + ↑k := sorry

Tactic state before:
case neg
E : Type u_1
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
f✝ h✝ : E → ℝ
f'✝ : E → E
x0✝ : E
f h : E → ℝ
f' : E → E
x0 : E
p : Nesterov_second_fix_stepsize f h f' x0
k : ℕ
hk : k ≠ 0
⊢ 2 ≤ 1 * (1 + ↑k)

Tactic:
simp [hk]

Tactic state after:
[case neg
 E : Type u_1
 inst✝² : NormedAddCommGroup E
 inst✝¹ : InnerProductSpace ℝ E
 inst✝ : CompleteSpace E
 f✝ h✝ : E → ℝ
 f'✝ : E → E
 x0✝ : E
 f h : E → ℝ
 f' : E → E
 x0 : E
 p : Nesterov_second_fix_stepsize f h f' x0
 k : ℕ
 hk : k ≠ 0
 ⊢ 2 ≤ 1 + ↑k]
import Mathlib
import Optlib


open Set Real

open Set Real PNat

theorem extracted_formal_statement_134 (k : ℕ) (hk : 1 ≤ k) : ↑k ≥ 1 := sorry

Tactic state before:
E : Type u_1
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
f✝ h✝ : E → ℝ
f'✝ : E → E
x0✝ : E
f h : E → ℝ
f' : E → E
x0 : E
p : Nesterov_second_fix_stepsize f h f' x0
k : ℕ
hk : 1 ≤ k
⊢ ↑k ≥ 1

Tactic:
simp [hk]

Tactic state after:
import Mathlib
import Optlib


open Set Real

open Set Real PNat

theorem extracted_formal_statement_136.{u_1} {E : Type u_1} [inst : NormedAddCommGroup E] [inst_1 : InnerProductSpace ℝ E]
  [inst_2 : CompleteSpace E] {f h : E → ℝ} {f' : E → E} {x0 : E} {alg : Nesterov_second_fix_stepsize f h f' x0} {xm : E}
  (minφ : IsMinOn (f + h) univ xm) (k : ℕ)
  (h1 :
    f (Nesterov_second_fix_stepsize.x f h f' x0 (k + 1)) + h (Nesterov_second_fix_stepsize.x f h f' x0 (k + 1)) - f xm -
        h xm =
      f (Nesterov_second.x f h f' x0 (k + 1)) + h (Nesterov_second.x f h f' x0 (k + 1)) - f xm - h xm)
  (h2 :
    Nesterov_second_fix_stepsize.γ f h f' x0 (k + 1) ^ 2 / (2 * Nesterov_second_fix_stepsize.t f h f' x0 (k + 1)) *
        ‖x0 - xm‖ ^ 2 =
      Nesterov_second.γ f h f' x0 (k + 1) ^ 2 / (2 * Nesterov_second.t f h f' x0 (k + 1)) * ‖x0 - xm‖ ^ 2) :
  f (Nesterov_second.x f h f' x0 (k + 1)) + h (Nesterov_second.x f h f' x0 (k + 1)) - f xm - h xm ≤
    Nesterov_second.γ f h f' x0 (k + 1) ^ 2 / (2 * Nesterov_second.t f h f' x0 (k + 1)) * ‖x0 - xm‖ ^ 2 := sorry

Tactic state before:
E : Type u_1
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
f h : E → ℝ
f' : E → E
x0 : E
alg : Nesterov_second_fix_stepsize f h f' x0
xm : E
minφ : IsMinOn (f + h) univ xm
k : ℕ
h1 :
  f (Nesterov_second_fix_stepsize.x f h f' x0 (k + 1)) + h (Nesterov_second_fix_stepsize.x f h f' x0 (k + 1)) - f xm -
      h xm =
    f (Nesterov_second.x f h f' x0 (k + 1)) + h (Nesterov_second.x f h f' x0 (k + 1)) - f xm - h xm
h2 :
  Nesterov_second_fix_stepsize.γ f h f' x0 (k + 1) ^ 2 / (2 * Nesterov_second_fix_stepsize.t f h f' x0 (k + 1)) *
      ‖x0 - xm‖ ^ 2 =
    Nesterov_second.γ f h f' x0 (k + 1) ^ 2 / (2 * Nesterov_second.t f h f' x0 (k + 1)) * ‖x0 - xm‖ ^ 2
⊢ f (Nesterov_second.x f h f' x0 (k + 1)) + h (Nesterov_second.x f h f' x0 (k + 1)) - f xm - h xm ≤
    Nesterov_second.γ f h f' x0 (k + 1) ^ 2 / (2 * Nesterov_second.t f h f' x0 (k + 1)) * ‖x0 - xm‖ ^ 2

Tactic:
apply Nesterov_second_convergence minφ

Tactic state after:
import Mathlib
import Optlib


open Set Real

open Set Real PNat

theorem extracted_formal_statement_137.{u_1} {E : Type u_1} [inst : NormedAddCommGroup E] [inst_1 : InnerProductSpace ℝ E]
  [inst_2 : CompleteSpace E] {f h : E → ℝ} {f' : E → E} {x0 : E} {alg : Nesterov_second_fix_stepsize f h f' x0} {xm : E}
  (minφ : IsMinOn (f + h) univ xm) (k : ℕ)
  (h_original_goal :
    Nesterov_second_fix_stepsize.γ f h f' x0 (k + 1) ^ 2 / (2 * Nesterov_second_fix_stepsize.t f h f' x0 (k + 1)) *
        ‖x0 - xm‖ ^ 2 ≤
      2 * ↑(Nesterov_second_fix_stepsize.l f h f' x0) / (↑k + 2) ^ 2 * ‖x0 - xm‖ ^ 2) :
  Nesterov_second_fix_stepsize.γ f h f' x0 (k + 1) ^ 2 / (2 * Nesterov_second_fix_stepsize.t f h f' x0 (k + 1)) ≤
    2 * ↑(Nesterov_second_fix_stepsize.l f h f' x0) / (↑k + 2) ^ 2 := sorry

Tactic state before:
E : Type u_1
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
f h : E → ℝ
f' : E → E
x0 : E
alg : Nesterov_second_fix_stepsize f h f' x0
xm : E
minφ : IsMinOn (f + h) univ xm
k : ℕ
⊢ Nesterov_second_fix_stepsize.γ f h f' x0 (k + 1) ^ 2 / (2 * Nesterov_second_fix_stepsize.t f h f' x0 (k + 1)) *
      ‖x0 - xm‖ ^ 2 ≤
    2 * ↑(Nesterov_second_fix_stepsize.l f h f' x0) / (↑k + 2) ^ 2 * ‖x0 - xm‖ ^ 2

Tactic:
apply mul_le_mul_of_nonneg_right _ (sq_nonneg _)

Tactic state after:
[E : Type u_1
 inst✝² : NormedAddCommGroup E
 inst✝¹ : InnerProductSpace ℝ E
 inst✝ : CompleteSpace E
 f h : E → ℝ
 f' : E → E
 x0 : E
 alg : Nesterov_second_fix_stepsize f h f' x0
 xm : E
 minφ : IsMinOn (f + h) univ xm
 k : ℕ
 ⊢ Nesterov_second_fix_stepsize.γ f h f' x0 (k + 1) ^ 2 / (2 * Nesterov_second_fix_stepsize.t f h f' x0 (k + 1)) ≤
     2 * ↑(Nesterov_second_fix_stepsize.l f h f' x0) / (↑k + 2) ^ 2]
import Mathlib
import Optlib


open Set Real

open Set Real PNat

theorem extracted_formal_statement_138.{u_1} {E : Type u_1} [inst : NormedAddCommGroup E] [inst_1 : InnerProductSpace ℝ E]
  [inst_2 : CompleteSpace E] {f h : E → ℝ} {f' : E → E} {x0 : E} {alg : Nesterov_second_fix_stepsize f h f' x0} {xm : E}
  (minφ : IsMinOn (f + h) univ xm) (k : ℕ)
  (h_original_goal :
    Nesterov_second_fix_stepsize.γ f h f' x0 (k + 1) ^ 2 / (2 * Nesterov_second_fix_stepsize.t f h f' x0 (k + 1)) ≤
      2 * ↑(Nesterov_second_fix_stepsize.l f h f' x0) / (↑k + 2) ^ 2) :
  (if k + 1 = 0 then 1 / 2 else 2 / (1 + ↑(k + 1))) ^ 2 / (2 * (1 / ↑(Nesterov_second_fix_stepsize.l f h f' x0))) ≤
    2 * ↑(Nesterov_second_fix_stepsize.l f h f' x0) / (↑k + 2) ^ 2 := sorry

Tactic state before:
E : Type u_1
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
f h : E → ℝ
f' : E → E
x0 : E
alg : Nesterov_second_fix_stepsize f h f' x0
xm : E
minφ : IsMinOn (f + h) univ xm
k : ℕ
⊢ Nesterov_second_fix_stepsize.γ f h f' x0 (k + 1) ^ 2 / (2 * Nesterov_second_fix_stepsize.t f h f' x0 (k + 1)) ≤
    2 * ↑(Nesterov_second_fix_stepsize.l f h f' x0) / (↑k + 2) ^ 2

Tactic:
rw [alg.γeq (k + 1), alg.teq (k + 1)]

Tactic state after:
[E : Type u_1
 inst✝² : NormedAddCommGroup E
 inst✝¹ : InnerProductSpace ℝ E
 inst✝ : CompleteSpace E
 f h : E → ℝ
 f' : E → E
 x0 : E
 alg : Nesterov_second_fix_stepsize f h f' x0
 xm : E
 minφ : IsMinOn (f + h) univ xm
 k : ℕ
 ⊢ (if k + 1 = 0 then 1 / 2 else 2 / (1 + ↑(k + 1))) ^ 2 / (2 * (1 / ↑(Nesterov_second_fix_stepsize.l f h f' x0))) ≤
     2 * ↑(Nesterov_second_fix_stepsize.l f h f' x0) / (↑k + 2) ^ 2]
import Mathlib
import Optlib


open Set Real

open Set Real PNat

theorem extracted_formal_statement_139.{u_1} {E : Type u_1} [inst : NormedAddCommGroup E] [inst_1 : InnerProductSpace ℝ E]
  [inst_2 : CompleteSpace E] {f h : E → ℝ} {f' : E → E} {x0 : E} {alg : Nesterov_second_fix_stepsize f h f' x0} {xm : E}
  (minφ : IsMinOn (f + h) univ xm) (k : ℕ)
  (h_original_goal :
    2 ^ 2 * ↑(Nesterov_second_fix_stepsize.l f h f' x0) / ((1 + (↑k + 1)) ^ 2 * 2) ≤
      2 * ↑(Nesterov_second_fix_stepsize.l f h f' x0) / (↑k + 2) ^ 2) :
  2 * 2 * ↑(Nesterov_second_fix_stepsize.l f h f' x0) / ((↑k + 1 + 1) ^ 2 * 2) ≤
    2 * ↑(Nesterov_second_fix_stepsize.l f h f' x0) / (↑k + 2) ^ 2 := sorry

Tactic state before:
E : Type u_1
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
f h : E → ℝ
f' : E → E
x0 : E
alg : Nesterov_second_fix_stepsize f h f' x0
xm : E
minφ : IsMinOn (f + h) univ xm
k : ℕ
⊢ 2 ^ 2 * ↑(Nesterov_second_fix_stepsize.l f h f' x0) / ((1 + (↑k + 1)) ^ 2 * 2) ≤
    2 * ↑(Nesterov_second_fix_stepsize.l f h f' x0) / (↑k + 2) ^ 2

Tactic:
rw [pow_two, add_comm]

Tactic state after:
[E : Type u_1
 inst✝² : NormedAddCommGroup E
 inst✝¹ : InnerProductSpace ℝ E
 inst✝ : CompleteSpace E
 f h : E → ℝ
 f' : E → E
 x0 : E
 alg : Nesterov_second_fix_stepsize f h f' x0
 xm : E
 minφ : IsMinOn (f + h) univ xm
 k : ℕ
 ⊢ 2 * 2 * ↑(Nesterov_second_fix_stepsize.l f h f' x0) / ((↑k + 1 + 1) ^ 2 * 2) ≤
     2 * ↑(Nesterov_second_fix_stepsize.l f h f' x0) / (↑k + 2) ^ 2]
import Mathlib
import Optlib


open Set Real

open Set Real PNat

theorem extracted_formal_statement_140.{u_1} {E : Type u_1} [inst : NormedAddCommGroup E] [inst_1 : InnerProductSpace ℝ E]
  [inst_2 : CompleteSpace E] {f h : E → ℝ} {f' : E → E} {x0 : E} {alg : Nesterov_second_fix_stepsize f h f' x0} {xm : E}
  (minφ : IsMinOn (f + h) univ xm) (k : ℕ)
  (h_original_goal :
    2 * 2 * ↑(Nesterov_second_fix_stepsize.l f h f' x0) / ((↑k + 1 + 1) ^ 2 * 2) ≤
      2 * ↑(Nesterov_second_fix_stepsize.l f h f' x0) / (↑k + 2) ^ 2) :
  2 * 2 * ↑(Nesterov_second_fix_stepsize.l f h f' x0) / (↑k + 2) ^ 2 / 2 ≤
    2 * ↑(Nesterov_second_fix_stepsize.l f h f' x0) / (↑k + 2) ^ 2 := sorry

Tactic state before:
E : Type u_1
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
f h : E → ℝ
f' : E → E
x0 : E
alg : Nesterov_second_fix_stepsize f h f' x0
xm : E
minφ : IsMinOn (f + h) univ xm
k : ℕ
⊢ 2 * 2 * ↑(Nesterov_second_fix_stepsize.l f h f' x0) / ((↑k + 1 + 1) ^ 2 * 2) ≤
    2 * ↑(Nesterov_second_fix_stepsize.l f h f' x0) / (↑k + 2) ^ 2

Tactic:
rw [add_assoc, one_add_one_eq_two, ← div_div]

Tactic state after:
[E : Type u_1
 inst✝² : NormedAddCommGroup E
 inst✝¹ : InnerProductSpace ℝ E
 inst✝ : CompleteSpace E
 f h : E → ℝ
 f' : E → E
 x0 : E
 alg : Nesterov_second_fix_stepsize f h f' x0
 xm : E
 minφ : IsMinOn (f + h) univ xm
 k : ℕ
 ⊢ 2 * 2 * ↑(Nesterov_second_fix_stepsize.l f h f' x0) / (↑k + 2) ^ 2 / 2 ≤
     2 * ↑(Nesterov_second_fix_stepsize.l f h f' x0) / (↑k + 2) ^ 2]
import Mathlib
import Optlib


open Set Real

open Set Real PNat

theorem extracted_formal_statement_141.{u_1} {E : Type u_1} [inst : NormedAddCommGroup E] [inst_1 : InnerProductSpace ℝ E]
  [inst_2 : CompleteSpace E] {f h : E → ℝ} {f' : E → E} {x0 : E} {alg : Nesterov_second_fix_stepsize f h f' x0} {xm : E}
  (minφ : IsMinOn (f + h) univ xm) (k : ℕ)
  (h_original_goal :
    2 * 2 * ↑(Nesterov_second_fix_stepsize.l f h f' x0) / (↑k + 2) ^ 2 / 2 ≤
      2 * ↑(Nesterov_second_fix_stepsize.l f h f' x0) / (↑k + 2) ^ 2) :
  2 * 2 * ↑(Nesterov_second_fix_stepsize.l f h f' x0) / (↑k + 2) ^ 2 / 2 =
    2 * ↑(Nesterov_second_fix_stepsize.l f h f' x0) / (↑k + 2) ^ 2 := sorry

Tactic state before:
E : Type u_1
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
f h : E → ℝ
f' : E → E
x0 : E
alg : Nesterov_second_fix_stepsize f h f' x0
xm : E
minφ : IsMinOn (f + h) univ xm
k : ℕ
⊢ 2 * 2 * ↑(Nesterov_second_fix_stepsize.l f h f' x0) / (↑k + 2) ^ 2 / 2 ≤
    2 * ↑(Nesterov_second_fix_stepsize.l f h f' x0) / (↑k + 2) ^ 2

Tactic:
apply le_of_eq

Tactic state after:
[case hab
 E : Type u_1
 inst✝² : NormedAddCommGroup E
 inst✝¹ : InnerProductSpace ℝ E
 inst✝ : CompleteSpace E
 f h : E → ℝ
 f' : E → E
 x0 : E
 alg : Nesterov_second_fix_stepsize f h f' x0
 xm : E
 minφ : IsMinOn (f + h) univ xm
 k : ℕ
 ⊢ 2 * 2 * ↑(Nesterov_second_fix_stepsize.l f h f' x0) / (↑k + 2) ^ 2 / 2 =
     2 * ↑(Nesterov_second_fix_stepsize.l f h f' x0) / (↑k + 2) ^ 2]