zzhisthebest/LeanBenchmark huggingface dataset:
import Mathlib
import Optlib


open Set

theorem extracted_formal_statement_55.{u_1} {E : Type u_1} [inst : NormedAddCommGroup E] [inst_1 : InnerProductSpace ℝ E]
  [inst_2 : CompleteSpace E] {f : E → ℝ} {f' : E → E} {xm x₀ : E} {γ : ℕ+ → ℝ} {alg : Nesterov f f' γ x₀}
  (hfun : ConvexOn ℝ univ f) (hg : ∀ (k : ℕ+), γ k = 2 / (↑↑k + 1)) (min : IsMinOn f univ xm) (k : ℕ+)
  (h4 :
    1 / γ k ^ 2 * (f (Nesterov.x f f' γ x₀ ↑k) - f xm) +
        ↑(Nesterov.l f f' γ x₀) / 2 * ‖Nesterov.v f f' γ x₀ ↑k - xm‖ ^ 2 ≤
      (1 - γ k) / γ k ^ 2 * (f (Nesterov.x f f' γ x₀ (↑k - 1)) - f xm) +
        ↑(Nesterov.l f f' γ x₀) / 2 * ‖Nesterov.v f f' γ x₀ (↑k - 1) - xm‖ ^ 2)
  (con : (1 - γ k) / γ k ^ 2 ≤ 1 / γ (k - 1) ^ 2) (this_1 : γ k ^ 2 > 0)
  (this : f xm ≤ f (Nesterov.x f f' γ x₀ (↑k - 1)))
  (h_original_goal :
    (1 - γ k) / γ k ^ 2 * (f (Nesterov.x f f' γ x₀ (↑k - 1)) - f xm) ≤
      1 / γ (k - 1) ^ 2 * (f (Nesterov.x f f' γ x₀ (↑k - 1)) - f xm)) :
  (1 - γ k) / γ k ^ 2 ≤ 1 / γ (k - 1) ^ 2 := sorry

Tactic state before:
E : Type u_1
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
f : E → ℝ
f' : E → E
xm x₀ : E
γ : ℕ+ → ℝ
alg : Nesterov f f' γ x₀
hfun : ConvexOn ℝ univ f
hg : ∀ (k : ℕ+), γ k = 2 / (↑↑k + 1)
min : IsMinOn f univ xm
k : ℕ+
h4 :
  1 / γ k ^ 2 * (f (Nesterov.x f f' γ x₀ ↑k) - f xm) +
      ↑(Nesterov.l f f' γ x₀) / 2 * ‖Nesterov.v f f' γ x₀ ↑k - xm‖ ^ 2 ≤
    (1 - γ k) / γ k ^ 2 * (f (Nesterov.x f f' γ x₀ (↑k - 1)) - f xm) +
      ↑(Nesterov.l f f' γ x₀) / 2 * ‖Nesterov.v f f' γ x₀ (↑k - 1) - xm‖ ^ 2
con : (1 - γ k) / γ k ^ 2 ≤ 1 / γ (k - 1) ^ 2
this✝ : γ k ^ 2 > 0
this : f xm ≤ f (Nesterov.x f f' γ x₀ (↑k - 1))
⊢ (1 - γ k) / γ k ^ 2 * (f (Nesterov.x f f' γ x₀ (↑k - 1)) - f xm) ≤
    1 / γ (k - 1) ^ 2 * (f (Nesterov.x f f' γ x₀ (↑k - 1)) - f xm)

Tactic:
apply mul_le_mul_of_nonneg_right _ (by linarith)

Tactic state after:
[E : Type u_1
 inst✝² : NormedAddCommGroup E
 inst✝¹ : InnerProductSpace ℝ E
 inst✝ : CompleteSpace E
 f : E → ℝ
 f' : E → E
 xm x₀ : E
 γ : ℕ+ → ℝ
 alg : Nesterov f f' γ x₀
 hfun : ConvexOn ℝ univ f
 hg : ∀ (k : ℕ+), γ k = 2 / (↑↑k + 1)
 min : IsMinOn f univ xm
 k : ℕ+
 h4 :
   1 / γ k ^ 2 * (f (Nesterov.x f f' γ x₀ ↑k) - f xm) +
       ↑(Nesterov.l f f' γ x₀) / 2 * ‖Nesterov.v f f' γ x₀ ↑k - xm‖ ^ 2 ≤
     (1 - γ k) / γ k ^ 2 * (f (Nesterov.x f f' γ x₀ (↑k - 1)) - f xm) +
       ↑(Nesterov.l f f' γ x₀) / 2 * ‖Nesterov.v f f' γ x₀ (↑k - 1) - xm‖ ^ 2
 con : (1 - γ k) / γ k ^ 2 ≤ 1 / γ (k - 1) ^ 2
 this✝ : γ k ^ 2 > 0
 this : f xm ≤ f (Nesterov.x f f' γ x₀ (↑k - 1))
 ⊢ (1 - γ k) / γ k ^ 2 ≤ 1 / γ (k - 1) ^ 2]
import Mathlib
import Optlib


open Set

theorem extracted_formal_statement_56.{u_1} {E : Type u_1} [inst : NormedAddCommGroup E] [inst_1 : InnerProductSpace ℝ E]
  [inst_2 : CompleteSpace E] {f : E → ℝ} {f' : E → E} {xm x₀ : E} {γ : ℕ+ → ℝ} {alg : Nesterov f f' γ x₀}
  (hfun : ConvexOn ℝ univ f) (hg : ∀ (k : ℕ+), γ k = 2 / (↑↑k + 1)) (min : IsMinOn f univ xm) (k : ℕ+)
  (h4 :
    1 / γ k ^ 2 * (f (Nesterov.x f f' γ x₀ ↑k) - f xm) +
        ↑(Nesterov.l f f' γ x₀) / 2 * ‖Nesterov.v f f' γ x₀ ↑k - xm‖ ^ 2 ≤
      (1 - γ k) / γ k ^ 2 * (f (Nesterov.x f f' γ x₀ (↑k - 1)) - f xm) +
        ↑(Nesterov.l f f' γ x₀) / 2 * ‖Nesterov.v f f' γ x₀ (↑k - 1) - xm‖ ^ 2)
  (con : (1 - γ k) / γ k ^ 2 ≤ 1 / γ (k - 1) ^ 2) (this_1 : γ k ^ 2 > 0)
  (this : f xm ≤ f (Nesterov.x f f' γ x₀ (↑k - 1))) : (1 - γ k) / γ k ^ 2 ≤ 1 / γ (k - 1) ^ 2 := sorry

Tactic state before:
E : Type u_1
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
f : E → ℝ
f' : E → E
xm x₀ : E
γ : ℕ+ → ℝ
alg : Nesterov f f' γ x₀
hfun : ConvexOn ℝ univ f
hg : ∀ (k : ℕ+), γ k = 2 / (↑↑k + 1)
min : IsMinOn f univ xm
k : ℕ+
h4 :
  1 / γ k ^ 2 * (f (Nesterov.x f f' γ x₀ ↑k) - f xm) +
      ↑(Nesterov.l f f' γ x₀) / 2 * ‖Nesterov.v f f' γ x₀ ↑k - xm‖ ^ 2 ≤
    (1 - γ k) / γ k ^ 2 * (f (Nesterov.x f f' γ x₀ (↑k - 1)) - f xm) +
      ↑(Nesterov.l f f' γ x₀) / 2 * ‖Nesterov.v f f' γ x₀ (↑k - 1) - xm‖ ^ 2
con : (1 - γ k) / γ k ^ 2 ≤ 1 / γ (k - 1) ^ 2
this✝ : γ k ^ 2 > 0
this : f xm ≤ f (Nesterov.x f f' γ x₀ (↑k - 1))
⊢ (1 - γ k) / γ k ^ 2 ≤ 1 / γ (k - 1) ^ 2

Tactic:
exact con

Tactic state after:
import Mathlib
import Optlib


open Set

theorem extracted_formal_statement_57.{u_1} {E : Type u_1} [inst : NormedAddCommGroup E] [inst_1 : InnerProductSpace ℝ E]
  [inst_2 : CompleteSpace E] {f : E → ℝ} {f' : E → E} {xm x₀ : E} {γ : ℕ+ → ℝ} {alg : Nesterov f f' γ x₀}
  (hfun : ConvexOn ℝ univ f) (hg : ∀ (k : ℕ+), γ k = 2 / (↑↑k + 1)) (min : IsMinOn f univ xm)
  (con : ∀ (k : ℕ+), (1 - γ k) / γ k ^ 2 ≤ 1 / γ (k - 1) ^ 2)
  (h4 :
    ∀ (k : ℕ+),
      f (Nesterov.x f f' γ x₀ ↑k) - f xm - (1 - γ k) * (f (Nesterov.x f f' γ x₀ (↑k - 1)) - f xm) ≤
        ↑(Nesterov.l f f' γ x₀) * γ k ^ 2 / 2 *
          (‖Nesterov.v f f' γ x₀ (↑k - 1) - xm‖ ^ 2 - ‖Nesterov.v f f' γ x₀ ↑k - xm‖ ^ 2))
  (h5 :
    ∀ (k : ℕ+),
      1 / γ k ^ 2 * (f (Nesterov.x f f' γ x₀ ↑k) - f xm) +
          ↑(Nesterov.l f f' γ x₀) / 2 * ‖Nesterov.v f f' γ x₀ ↑k - xm‖ ^ 2 ≤
        1 / γ (k - 1) ^ 2 * (f (Nesterov.x f f' γ x₀ (↑k - 1)) - f xm) +
          ↑(Nesterov.l f f' γ x₀) / 2 * ‖Nesterov.v f f' γ x₀ (↑k - 1) - xm‖ ^ 2) :
  1 / γ 1 ^ 2 * (f (Nesterov.x f f' γ x₀ ↑1) - f xm) +
      ↑(Nesterov.l f f' γ x₀) / 2 * ‖Nesterov.v f f' γ x₀ ↑1 - xm‖ ^ 2 ≤
    1 / γ 1 ^ 2 * (f (Nesterov.x f f' γ x₀ 1) - f xm) +
      ↑(Nesterov.l f f' γ x₀) / 2 * ‖Nesterov.v f f' γ x₀ 1 - xm‖ ^ 2 := sorry

Tactic state before:
case hz
E : Type u_1
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
f : E → ℝ
f' : E → E
xm x₀ : E
γ : ℕ+ → ℝ
alg : Nesterov f f' γ x₀
hfun : ConvexOn ℝ univ f
hg : ∀ (k : ℕ+), γ k = 2 / (↑↑k + 1)
min : IsMinOn f univ xm
con : ∀ (k : ℕ+), (1 - γ k) / γ k ^ 2 ≤ 1 / γ (k - 1) ^ 2
h4 :
  ∀ (k : ℕ+),
    f (Nesterov.x f f' γ x₀ ↑k) - f xm - (1 - γ k) * (f (Nesterov.x f f' γ x₀ (↑k - 1)) - f xm) ≤
      ↑(Nesterov.l f f' γ x₀) * γ k ^ 2 / 2 *
        (‖Nesterov.v f f' γ x₀ (↑k - 1) - xm‖ ^ 2 - ‖Nesterov.v f f' γ x₀ ↑k - xm‖ ^ 2)
h5 :
  ∀ (k : ℕ+),
    1 / γ k ^ 2 * (f (Nesterov.x f f' γ x₀ ↑k) - f xm) +
        ↑(Nesterov.l f f' γ x₀) / 2 * ‖Nesterov.v f f' γ x₀ ↑k - xm‖ ^ 2 ≤
      1 / γ (k - 1) ^ 2 * (f (Nesterov.x f f' γ x₀ (↑k - 1)) - f xm) +
        ↑(Nesterov.l f f' γ x₀) / 2 * ‖Nesterov.v f f' γ x₀ (↑k - 1) - xm‖ ^ 2
⊢ 1 / γ 1 ^ 2 * (f (Nesterov.x f f' γ x₀ ↑1) - f xm) +
      ↑(Nesterov.l f f' γ x₀) / 2 * ‖Nesterov.v f f' γ x₀ ↑1 - xm‖ ^ 2 ≤
    1 / γ 1 ^ 2 * (f (Nesterov.x f f' γ x₀ 1) - f xm) + ↑(Nesterov.l f f' γ x₀) / 2 * ‖Nesterov.v f f' γ x₀ 1 - xm‖ ^ 2

Tactic:
simp

Tactic state after:
import Mathlib
import Optlib


open Set

theorem extracted_formal_statement_58.{u_1} {E : Type u_1} [inst : NormedAddCommGroup E] [inst_1 : InnerProductSpace ℝ E]
  [inst_2 : CompleteSpace E] {f : E → ℝ} {f' : E → E} {xm x₀ : E} {γ : ℕ+ → ℝ} {alg : Nesterov f f' γ x₀}
  (hfun : ConvexOn ℝ univ f) (hg : ∀ (k : ℕ+), γ k = 2 / (↑↑k + 1)) (min : IsMinOn f univ xm)
  (con : ∀ (k : ℕ+), (1 - γ k) / γ k ^ 2 ≤ 1 / γ (k - 1) ^ 2)
  (h4 :
    ∀ (k : ℕ+),
      f (Nesterov.x f f' γ x₀ ↑k) - f xm - (1 - γ k) * (f (Nesterov.x f f' γ x₀ (↑k - 1)) - f xm) ≤
        ↑(Nesterov.l f f' γ x₀) * γ k ^ 2 / 2 *
          (‖Nesterov.v f f' γ x₀ (↑k - 1) - xm‖ ^ 2 - ‖Nesterov.v f f' γ x₀ ↑k - xm‖ ^ 2))
  (j : ℕ+)
  (IH :
    1 / γ j ^ 2 * (f (Nesterov.x f f' γ x₀ ↑j) - f xm) +
        ↑(Nesterov.l f f' γ x₀) / 2 * ‖Nesterov.v f f' γ x₀ ↑j - xm‖ ^ 2 ≤
      1 / γ 1 ^ 2 * (f (Nesterov.x f f' γ x₀ 1) - f xm) +
        ↑(Nesterov.l f f' γ x₀) / 2 * ‖Nesterov.v f f' γ x₀ 1 - xm‖ ^ 2)
  (h5 :
    1 / γ (j + 1) ^ 2 * (f (Nesterov.x f f' γ x₀ ↑(j + 1)) - f xm) +
        ↑(Nesterov.l f f' γ x₀) / 2 * ‖Nesterov.v f f' γ x₀ ↑(j + 1) - xm‖ ^ 2 ≤
      1 / γ (j + 1 - 1) ^ 2 * (f (Nesterov.x f f' γ x₀ (↑(j + 1) - 1)) - f xm) +
        ↑(Nesterov.l f f' γ x₀) / 2 * ‖Nesterov.v f f' γ x₀ (↑(j + 1) - 1) - xm‖ ^ 2) :
  ↑(j + 1) - 1 = ↑j := sorry

Tactic state before:
E : Type u_1
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
f : E → ℝ
f' : E → E
xm x₀ : E
γ : ℕ+ → ℝ
alg : Nesterov f f' γ x₀
hfun : ConvexOn ℝ univ f
hg : ∀ (k : ℕ+), γ k = 2 / (↑↑k + 1)
min : IsMinOn f univ xm
con : ∀ (k : ℕ+), (1 - γ k) / γ k ^ 2 ≤ 1 / γ (k - 1) ^ 2
h4 :
  ∀ (k : ℕ+),
    f (Nesterov.x f f' γ x₀ ↑k) - f xm - (1 - γ k) * (f (Nesterov.x f f' γ x₀ (↑k - 1)) - f xm) ≤
      ↑(Nesterov.l f f' γ x₀) * γ k ^ 2 / 2 *
        (‖Nesterov.v f f' γ x₀ (↑k - 1) - xm‖ ^ 2 - ‖Nesterov.v f f' γ x₀ ↑k - xm‖ ^ 2)
j : ℕ+
IH :
  1 / γ j ^ 2 * (f (Nesterov.x f f' γ x₀ ↑j) - f xm) +
      ↑(Nesterov.l f f' γ x₀) / 2 * ‖Nesterov.v f f' γ x₀ ↑j - xm‖ ^ 2 ≤
    1 / γ 1 ^ 2 * (f (Nesterov.x f f' γ x₀ 1) - f xm) + ↑(Nesterov.l f f' γ x₀) / 2 * ‖Nesterov.v f f' γ x₀ 1 - xm‖ ^ 2
h5 :
  1 / γ (j + 1) ^ 2 * (f (Nesterov.x f f' γ x₀ ↑(j + 1)) - f xm) +
      ↑(Nesterov.l f f' γ x₀) / 2 * ‖Nesterov.v f f' γ x₀ ↑(j + 1) - xm‖ ^ 2 ≤
    1 / γ (j + 1 - 1) ^ 2 * (f (Nesterov.x f f' γ x₀ (↑(j + 1) - 1)) - f xm) +
      ↑(Nesterov.l f f' γ x₀) / 2 * ‖Nesterov.v f f' γ x₀ (↑(j + 1) - 1) - xm‖ ^ 2
⊢ ↑(j + 1) - 1 = ↑j

Tactic:
simp

Tactic state after:
import Mathlib
import Optlib


open Set

theorem extracted_formal_statement_59.{u_1} {E : Type u_1} [inst : NormedAddCommGroup E] [inst_1 : InnerProductSpace ℝ E]
  [inst_2 : CompleteSpace E] {f : E → ℝ} {f' : E → E} {xm x₀ : E} {γ : ℕ+ → ℝ} {alg : Nesterov f f' γ x₀}
  (hfun : ConvexOn ℝ univ f) (hg : ∀ (k : ℕ+), γ k = 2 / (↑↑k + 1)) (min : IsMinOn f univ xm)
  (con : ∀ (k : ℕ+), (1 - γ k) / γ k ^ 2 ≤ 1 / γ (k - 1) ^ 2)
  (h4 :
    ∀ (k : ℕ+),
      f (Nesterov.x f f' γ x₀ ↑k) - f xm - (1 - γ k) * (f (Nesterov.x f f' γ x₀ (↑k - 1)) - f xm) ≤
        ↑(Nesterov.l f f' γ x₀) * γ k ^ 2 / 2 *
          (‖Nesterov.v f f' γ x₀ (↑k - 1) - xm‖ ^ 2 - ‖Nesterov.v f f' γ x₀ ↑k - xm‖ ^ 2))
  (j : ℕ+)
  (IH :
    1 / γ j ^ 2 * (f (Nesterov.x f f' γ x₀ ↑j) - f xm) +
        ↑(Nesterov.l f f' γ x₀) / 2 * ‖Nesterov.v f f' γ x₀ ↑j - xm‖ ^ 2 ≤
      1 / γ 1 ^ 2 * (f (Nesterov.x f f' γ x₀ 1) - f xm) +
        ↑(Nesterov.l f f' γ x₀) / 2 * ‖Nesterov.v f f' γ x₀ 1 - xm‖ ^ 2)
  (h5 :
    1 / γ (j + 1) ^ 2 * (f (Nesterov.x f f' γ x₀ ↑(j + 1)) - f xm) +
        ↑(Nesterov.l f f' γ x₀) / 2 * ‖Nesterov.v f f' γ x₀ ↑(j + 1) - xm‖ ^ 2 ≤
      1 / γ (j + 1 - 1) ^ 2 * (f (Nesterov.x f f' γ x₀ (↑(j + 1) - 1)) - f xm) +
        ↑(Nesterov.l f f' γ x₀) / 2 * ‖Nesterov.v f f' γ x₀ (↑(j + 1) - 1) - xm‖ ^ 2)
  (y1 : ↑(j + 1) - 1 = ↑j) : j + 1 - 1 = j := sorry

Tactic state before:
E : Type u_1
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
f : E → ℝ
f' : E → E
xm x₀ : E
γ : ℕ+ → ℝ
alg : Nesterov f f' γ x₀
hfun : ConvexOn ℝ univ f
hg : ∀ (k : ℕ+), γ k = 2 / (↑↑k + 1)
min : IsMinOn f univ xm
con : ∀ (k : ℕ+), (1 - γ k) / γ k ^ 2 ≤ 1 / γ (k - 1) ^ 2
h4 :
  ∀ (k : ℕ+),
    f (Nesterov.x f f' γ x₀ ↑k) - f xm - (1 - γ k) * (f (Nesterov.x f f' γ x₀ (↑k - 1)) - f xm) ≤
      ↑(Nesterov.l f f' γ x₀) * γ k ^ 2 / 2 *
        (‖Nesterov.v f f' γ x₀ (↑k - 1) - xm‖ ^ 2 - ‖Nesterov.v f f' γ x₀ ↑k - xm‖ ^ 2)
j : ℕ+
IH :
  1 / γ j ^ 2 * (f (Nesterov.x f f' γ x₀ ↑j) - f xm) +
      ↑(Nesterov.l f f' γ x₀) / 2 * ‖Nesterov.v f f' γ x₀ ↑j - xm‖ ^ 2 ≤
    1 / γ 1 ^ 2 * (f (Nesterov.x f f' γ x₀ 1) - f xm) + ↑(Nesterov.l f f' γ x₀) / 2 * ‖Nesterov.v f f' γ x₀ 1 - xm‖ ^ 2
h5 :
  1 / γ (j + 1) ^ 2 * (f (Nesterov.x f f' γ x₀ ↑(j + 1)) - f xm) +
      ↑(Nesterov.l f f' γ x₀) / 2 * ‖Nesterov.v f f' γ x₀ ↑(j + 1) - xm‖ ^ 2 ≤
    1 / γ (j + 1 - 1) ^ 2 * (f (Nesterov.x f f' γ x₀ (↑(j + 1) - 1)) - f xm) +
      ↑(Nesterov.l f f' γ x₀) / 2 * ‖Nesterov.v f f' γ x₀ (↑(j + 1) - 1) - xm‖ ^ 2
y1 : ↑(j + 1) - 1 = ↑j
⊢ j + 1 - 1 = j

Tactic:
exact Iff.mp PNat.natPred_inj rfl

Tactic state after:
import Mathlib
import Optlib


open Set

theorem extracted_formal_statement_62.{u_1} {E : Type u_1} [inst : NormedAddCommGroup E] [inst_1 : InnerProductSpace ℝ E]
  [inst_2 : CompleteSpace E] {f : E → ℝ} {f' : E → E} {xm x₀ : E} {γ : ℕ+ → ℝ} {alg : Nesterov f f' γ x₀}
  (hfun : ConvexOn ℝ univ f) (hg : ∀ (k : ℕ+), γ k = 2 / (↑↑k + 1)) (min : IsMinOn f univ xm)
  (con : ∀ (k : ℕ+), (1 - γ k) / γ k ^ 2 ≤ 1 / γ (k - 1) ^ 2)
  (h4 :
    ∀ (k : ℕ+),
      f (Nesterov.x f f' γ x₀ ↑k) - f xm - (1 - γ k) * (f (Nesterov.x f f' γ x₀ (↑k - 1)) - f xm) ≤
        ↑(Nesterov.l f f' γ x₀) * γ k ^ 2 / 2 *
          (‖Nesterov.v f f' γ x₀ (↑k - 1) - xm‖ ^ 2 - ‖Nesterov.v f f' γ x₀ ↑k - xm‖ ^ 2))
  (j : ℕ+)
  (IH :
    1 / γ j ^ 2 * (f (Nesterov.x f f' γ x₀ ↑j) - f xm) +
        ↑(Nesterov.l f f' γ x₀) / 2 * ‖Nesterov.v f f' γ x₀ ↑j - xm‖ ^ 2 ≤
      1 / γ 1 ^ 2 * (f (Nesterov.x f f' γ x₀ 1) - f xm) +
        ↑(Nesterov.l f f' γ x₀) / 2 * ‖Nesterov.v f f' γ x₀ 1 - xm‖ ^ 2)
  (h5 :
    1 / γ (j + 1) ^ 2 * (f (Nesterov.x f f' γ x₀ ↑(j + 1)) - f xm) +
        ↑(Nesterov.l f f' γ x₀) / 2 * ‖Nesterov.v f f' γ x₀ ↑(j + 1) - xm‖ ^ 2 ≤
      1 / γ (j + 1 - 1) ^ 2 * (f (Nesterov.x f f' γ x₀ (↑(j + 1) - 1)) - f xm) +
        ↑(Nesterov.l f f' γ x₀) / 2 * ‖Nesterov.v f f' γ x₀ (↑(j + 1) - 1) - xm‖ ^ 2)
  (y1 : ↑(j + 1) - 1 = ↑j) (y2 : j + 1 - 1 = j) :
  1 / γ j ^ 2 * (f (Nesterov.x f f' γ x₀ ↑j) - f xm) +
      ↑(Nesterov.l f f' γ x₀) / 2 * ‖Nesterov.v f f' γ x₀ ↑j - xm‖ ^ 2 ≤
    1 / γ 1 ^ 2 * (f (Nesterov.x f f' γ x₀ 1) - f xm) +
      ↑(Nesterov.l f f' γ x₀) / 2 * ‖Nesterov.v f f' γ x₀ 1 - xm‖ ^ 2 := sorry

Tactic state before:
E : Type u_1
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
f : E → ℝ
f' : E → E
xm x₀ : E
γ : ℕ+ → ℝ
alg : Nesterov f f' γ x₀
hfun : ConvexOn ℝ univ f
hg : ∀ (k : ℕ+), γ k = 2 / (↑↑k + 1)
min : IsMinOn f univ xm
con : ∀ (k : ℕ+), (1 - γ k) / γ k ^ 2 ≤ 1 / γ (k - 1) ^ 2
h4 :
  ∀ (k : ℕ+),
    f (Nesterov.x f f' γ x₀ ↑k) - f xm - (1 - γ k) * (f (Nesterov.x f f' γ x₀ (↑k - 1)) - f xm) ≤
      ↑(Nesterov.l f f' γ x₀) * γ k ^ 2 / 2 *
        (‖Nesterov.v f f' γ x₀ (↑k - 1) - xm‖ ^ 2 - ‖Nesterov.v f f' γ x₀ ↑k - xm‖ ^ 2)
j : ℕ+
IH :
  1 / γ j ^ 2 * (f (Nesterov.x f f' γ x₀ ↑j) - f xm) +
      ↑(Nesterov.l f f' γ x₀) / 2 * ‖Nesterov.v f f' γ x₀ ↑j - xm‖ ^ 2 ≤
    1 / γ 1 ^ 2 * (f (Nesterov.x f f' γ x₀ 1) - f xm) + ↑(Nesterov.l f f' γ x₀) / 2 * ‖Nesterov.v f f' γ x₀ 1 - xm‖ ^ 2
h5 :
  1 / γ (j + 1) ^ 2 * (f (Nesterov.x f f' γ x₀ ↑(j + 1)) - f xm) +
      ↑(Nesterov.l f f' γ x₀) / 2 * ‖Nesterov.v f f' γ x₀ ↑(j + 1) - xm‖ ^ 2 ≤
    1 / γ (j + 1 - 1) ^ 2 * (f (Nesterov.x f f' γ x₀ (↑(j + 1) - 1)) - f xm) +
      ↑(Nesterov.l f f' γ x₀) / 2 * ‖Nesterov.v f f' γ x₀ (↑(j + 1) - 1) - xm‖ ^ 2
y1 : ↑(j + 1) - 1 = ↑j
y2 : j + 1 - 1 = j
⊢ 1 / γ j ^ 2 * (f (Nesterov.x f f' γ x₀ ↑j) - f xm) +
      ↑(Nesterov.l f f' γ x₀) / 2 * ‖Nesterov.v f f' γ x₀ ↑j - xm‖ ^ 2 ≤
    1 / γ 1 ^ 2 * (f (Nesterov.x f f' γ x₀ 1) - f xm) + ↑(Nesterov.l f f' γ x₀) / 2 * ‖Nesterov.v f f' γ x₀ 1 - xm‖ ^ 2

Tactic:
exact IH

Tactic state after:
import Mathlib
import Optlib


open Set

theorem extracted_formal_statement_66.{u_1} {E : Type u_1} [inst : NormedAddCommGroup E] [inst_1 : InnerProductSpace ℝ E]
  [inst_2 : CompleteSpace E] {f : E → ℝ} {f' : E → E} {xm x₀ : E} {γ : ℕ+ → ℝ} {alg : Nesterov f f' γ x₀}
  (hfun : ConvexOn ℝ univ f) (hg : ∀ (k : ℕ+), γ k = 2 / (↑↑k + 1)) (min : IsMinOn f univ xm)
  (con : ∀ (k : ℕ+), (1 - γ k) / γ k ^ 2 ≤ 1 / γ (k - 1) ^ 2)
  (h5 :
    ∀ (k : ℕ+),
      1 / γ k ^ 2 * (f (Nesterov.x f f' γ x₀ ↑k) - f xm) +
          ↑(Nesterov.l f f' γ x₀) / 2 * ‖Nesterov.v f f' γ x₀ ↑k - xm‖ ^ 2 ≤
        1 / γ (k - 1) ^ 2 * (f (Nesterov.x f f' γ x₀ (↑k - 1)) - f xm) +
          ↑(Nesterov.l f f' γ x₀) / 2 * ‖Nesterov.v f f' γ x₀ (↑k - 1) - xm‖ ^ 2)
  (h6 :
    ∀ (k : ℕ+),
      1 / γ k ^ 2 * (f (Nesterov.x f f' γ x₀ ↑k) - f xm) +
          ↑(Nesterov.l f f' γ x₀) / 2 * ‖Nesterov.v f f' γ x₀ ↑k - xm‖ ^ 2 ≤
        1 / γ 1 ^ 2 * (f (Nesterov.x f f' γ x₀ 1) - f xm) +
          ↑(Nesterov.l f f' γ x₀) / 2 * ‖Nesterov.v f f' γ x₀ 1 - xm‖ ^ 2)
  (h4 :
    f (Nesterov.x f f' γ x₀ 1) - f xm ≤
      ↑(Nesterov.l f f' γ x₀) / 2 * (‖Nesterov.v f f' γ x₀ 0 - xm‖ ^ 2 - ‖Nesterov.v f f' γ x₀ 1 - xm‖ ^ 2)) :
  f (Nesterov.x f f' γ x₀ 1) - f xm ≤
    ↑(Nesterov.l f f' γ x₀) / 2 * (‖Nesterov.v f f' γ x₀ 0 - xm‖ ^ 2 - ‖Nesterov.v f f' γ x₀ 1 - xm‖ ^ 2) := sorry

Tactic state before:
E : Type u_1
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
f : E → ℝ
f' : E → E
xm x₀ : E
γ : ℕ+ → ℝ
alg : Nesterov f f' γ x₀
hfun : ConvexOn ℝ univ f
hg : ∀ (k : ℕ+), γ k = 2 / (↑↑k + 1)
min : IsMinOn f univ xm
con : ∀ (k : ℕ+), (1 - γ k) / γ k ^ 2 ≤ 1 / γ (k - 1) ^ 2
h5 :
  ∀ (k : ℕ+),
    1 / γ k ^ 2 * (f (Nesterov.x f f' γ x₀ ↑k) - f xm) +
        ↑(Nesterov.l f f' γ x₀) / 2 * ‖Nesterov.v f f' γ x₀ ↑k - xm‖ ^ 2 ≤
      1 / γ (k - 1) ^ 2 * (f (Nesterov.x f f' γ x₀ (↑k - 1)) - f xm) +
        ↑(Nesterov.l f f' γ x₀) / 2 * ‖Nesterov.v f f' γ x₀ (↑k - 1) - xm‖ ^ 2
h6 :
  ∀ (k : ℕ+),
    1 / γ k ^ 2 * (f (Nesterov.x f f' γ x₀ ↑k) - f xm) +
        ↑(Nesterov.l f f' γ x₀) / 2 * ‖Nesterov.v f f' γ x₀ ↑k - xm‖ ^ 2 ≤
      1 / γ 1 ^ 2 * (f (Nesterov.x f f' γ x₀ 1) - f xm) +
        ↑(Nesterov.l f f' γ x₀) / 2 * ‖Nesterov.v f f' γ x₀ 1 - xm‖ ^ 2
h4 :
  f (Nesterov.x f f' γ x₀ 1) - f xm ≤
    ↑(Nesterov.l f f' γ x₀) / 2 * (‖Nesterov.v f f' γ x₀ 0 - xm‖ ^ 2 - ‖Nesterov.v f f' γ x₀ 1 - xm‖ ^ 2)
⊢ f (Nesterov.x f f' γ x₀ 1) - f xm ≤
    ↑(Nesterov.l f f' γ x₀) / 2 * (‖Nesterov.v f f' γ x₀ 0 - xm‖ ^ 2 - ‖Nesterov.v f f' γ x₀ 1 - xm‖ ^ 2)

Tactic:
exact h4

Tactic state after:
import Mathlib
import Optlib


open Set

theorem extracted_formal_statement_80 : 0 < 2 ^ 2 := sorry

Tactic state before:
case hb
E : Type u_1
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
f : E → ℝ
f' : E → E
xm x₀ : E
γ : ℕ+ → ℝ
alg : Nesterov f f' γ x₀
hfun : ConvexOn ℝ univ f
hg : ∀ (k : ℕ+), γ k = 2 / (↑↑k + 1)
min : IsMinOn f univ xm
con : ∀ (k : ℕ+), (1 - γ k) / γ k ^ 2 ≤ 1 / γ (k - 1) ^ 2
h4 :
  ∀ (k : ℕ+),
    f (Nesterov.x f f' γ x₀ ↑k) - f xm - (1 - γ k) * (f (Nesterov.x f f' γ x₀ (↑k - 1)) - f xm) ≤
      ↑(Nesterov.l f f' γ x₀) * γ k ^ 2 / 2 *
        (‖Nesterov.v f f' γ x₀ (↑k - 1) - xm‖ ^ 2 - ‖Nesterov.v f f' γ x₀ ↑k - xm‖ ^ 2)
h5 :
  ∀ (k : ℕ+),
    1 / γ k ^ 2 * (f (Nesterov.x f f' γ x₀ ↑k) - f xm) +
        ↑(Nesterov.l f f' γ x₀) / 2 * ‖Nesterov.v f f' γ x₀ ↑k - xm‖ ^ 2 ≤
      1 / γ (k - 1) ^ 2 * (f (Nesterov.x f f' γ x₀ (↑k - 1)) - f xm) +
        ↑(Nesterov.l f f' γ x₀) / 2 * ‖Nesterov.v f f' γ x₀ (↑k - 1) - xm‖ ^ 2
h6 :
  ∀ (k : ℕ+),
    1 / γ k ^ 2 * (f (Nesterov.x f f' γ x₀ ↑k) - f xm) +
        ↑(Nesterov.l f f' γ x₀) / 2 * ‖Nesterov.v f f' γ x₀ ↑k - xm‖ ^ 2 ≤
      1 / γ 1 ^ 2 * (f (Nesterov.x f f' γ x₀ 1) - f xm) +
        ↑(Nesterov.l f f' γ x₀) / 2 * ‖Nesterov.v f f' γ x₀ 1 - xm‖ ^ 2
h7 :
  1 / γ 1 ^ 2 * (f (Nesterov.x f f' γ x₀ 1) - f xm) + ↑(Nesterov.l f f' γ x₀) / 2 * ‖Nesterov.v f f' γ x₀ 1 - xm‖ ^ 2 ≤
    (1 - γ 1) / γ 1 ^ 2 * (f (Nesterov.x f f' γ x₀ 0) - f xm) +
      ↑(Nesterov.l f f' γ x₀) / 2 * ‖Nesterov.v f f' γ x₀ 0 - xm‖ ^ 2
k : ℕ+
h8 :
  1 / γ k ^ 2 * (f (Nesterov.x f f' γ x₀ ↑k) - f xm) +
      ↑(Nesterov.l f f' γ x₀) / 2 * ‖Nesterov.v f f' γ x₀ ↑k - xm‖ ^ 2 ≤
    ↑(Nesterov.l f f' γ x₀) / 2 * ‖x₀ - xm‖ ^ 2
h9 : 1 / (2 / (↑↑k + 1)) ^ 2 * (f (Nesterov.x f f' γ x₀ ↑k) - f xm) ≤ ↑(Nesterov.l f f' γ x₀) / 2 * ‖x₀ - xm‖ ^ 2
h10 :
  ↑(Nesterov.l f f' γ x₀) / 2 * ‖x₀ - xm‖ ^ 2 / (1 / (2 / (↑↑k + 1)) ^ 2) =
    2 * ↑(Nesterov.l f f' γ x₀) / (↑↑k + 1) ^ 2 * ‖x₀ - xm‖ ^ 2
⊢ 0 < 2 ^ 2

Tactic:
simp only [gt_iff_lt, zero_lt_two, pow_pos]

Tactic state after:
import Mathlib
import Optlib


open Set Real

open Set Real PNat

theorem extracted_formal_statement_0.{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_first f h f' x0} {xm : E}
  (minφ : IsMinOn (f + h) univ xm) (k : ℕ) :
  let update2 := Nesterov_first.update2 k;
  Nesterov_first.y f h f' x0 k - Nesterov_first.x f h f' x0 (k + 1) -
        Nesterov_first.t f h f' x0 k • f' (Nesterov_first.y f h f' x0 k) ∈
      SubderivAt (Nesterov_first.t f h f' x0 k • h) (Nesterov_first.x f h f' x0 (k + 1)) →
    ConvexOn ℝ univ (Nesterov_first.t f h f' x0 k • h) := 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_first f h f' x0
xm : E
minφ : IsMinOn (f + h) univ xm
k : ℕ
update2 : prox_prop (Nesterov_first.t f h f' x0 k • h)
  (Nesterov_first.y f h f' x0 k - Nesterov_first.t f h f' x0 k • f' (Nesterov_first.y f h f' x0 k))
  (Nesterov_first.x f h f' x0 (k + 1)) :=
  Nesterov_first.update2 k
⊢ Nesterov_first.y f h f' x0 k - Nesterov_first.x f h f' x0 (k + 1) -
      Nesterov_first.t f h f' x0 k • f' (Nesterov_first.y f h f' x0 k) ∈
    SubderivAt (Nesterov_first.t f h f' x0 k • h) (Nesterov_first.x f h f' x0 (k + 1))

Tactic:
rw [prox_iff_subderiv, sub_right_comm] at update2

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_first f h f' x0
 xm : E
 minφ : IsMinOn (f + h) univ xm
 k : ℕ
 update2 :
   Nesterov_first.y f h f' x0 k - Nesterov_first.x f h f' x0 (k + 1) -
       Nesterov_first.t f h f' x0 k • f' (Nesterov_first.y f h f' x0 k) ∈
     SubderivAt (Nesterov_first.t f h f' x0 k • h) (Nesterov_first.x f h f' x0 (k + 1))
 ⊢ Nesterov_first.y f h f' x0 k - Nesterov_first.x f h f' x0 (k + 1) -
       Nesterov_first.t f h f' x0 k • f' (Nesterov_first.y f h f' x0 k) ∈
     SubderivAt (Nesterov_first.t f h f' x0 k • h) (Nesterov_first.x f h f' x0 (k + 1)),
 case hfun
 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_first f h f' x0
 xm : E
 minφ : IsMinOn (f + h) univ xm
 k : ℕ
 update2 : prox_prop (Nesterov_first.t f h f' x0 k • h)
   (Nesterov_first.y f h f' x0 k - Nesterov_first.t f h f' x0 k • f' (Nesterov_first.y f h f' x0 k))
   (Nesterov_first.x f h f' x0 (k + 1)) :=
   Nesterov_first.update2 k
 ⊢ ConvexOn ℝ univ (Nesterov_first.t f h f' x0 k • h)]
import Mathlib
import Optlib


open Set Real

open Set Real PNat

theorem extracted_formal_statement_1.{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_first f h f' x0} {xm : E}
  (minφ : IsMinOn (f + h) univ xm) (k : ℕ) :
  let update2 := Nesterov_first.update2 k;
  ConvexOn ℝ univ (Nesterov_first.t f h f' x0 k • h) → 0 ≤ Nesterov_first.t f h f' x0 k := sorry

Tactic state before:
case hfun
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_first f h f' x0
xm : E
minφ : IsMinOn (f + h) univ xm
k : ℕ
update2 : prox_prop (Nesterov_first.t f h f' x0 k • h)
  (Nesterov_first.y f h f' x0 k - Nesterov_first.t f h f' x0 k • f' (Nesterov_first.y f h f' x0 k))
  (Nesterov_first.x f h f' x0 (k + 1)) :=
  Nesterov_first.update2 k
⊢ ConvexOn ℝ univ (Nesterov_first.t f h f' x0 k • h)

Tactic:
apply ConvexOn.smul

Tactic state after:
[case hfun.hc
 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_first f h f' x0
 xm : E
 minφ : IsMinOn (f + h) univ xm
 k : ℕ
 update2 : prox_prop (Nesterov_first.t f h f' x0 k • h)
   (Nesterov_first.y f h f' x0 k - Nesterov_first.t f h f' x0 k • f' (Nesterov_first.y f h f' x0 k))
   (Nesterov_first.x f h f' x0 (k + 1)) :=
   Nesterov_first.update2 k
 ⊢ 0 ≤ Nesterov_first.t f h f' x0 k,
 case hfun.hf
 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_first f h f' x0
 xm : E
 minφ : IsMinOn (f + h) univ xm
 k : ℕ
 update2 : prox_prop (Nesterov_first.t f h f' x0 k • h)
   (Nesterov_first.y f h f' x0 k - Nesterov_first.t f h f' x0 k • f' (Nesterov_first.y f h f' x0 k))
   (Nesterov_first.x f h f' x0 (k + 1)) :=
   Nesterov_first.update2 k
 ⊢ ConvexOn ℝ univ h]