zzhisthebest/LeanBenchmark huggingface dataset:
import Mathlib
import Optlib


open Set Finset

open InnerProductSpace Set

theorem extracted_formal_statement_29.{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} {alg : Gradient_Descent_fix_stepsize f f' x₀}
  (hfun : ConvexOn ℝ univ f)
  (step₂_1 : Gradient_Descent_fix_stepsize.a f f' x₀ ≤ 1 / ↑(Gradient_Descent_fix_stepsize.l f f' x₀))
  (step₂ : ↑(Gradient_Descent_fix_stepsize.l f f' x₀) ≤ 1 / Gradient_Descent_fix_stepsize.a f f' x₀)
  (this_1 : LipschitzWith (Gradient_Descent_fix_stepsize.l f f' x₀) f')
  (this : Gradient_Descent_fix_stepsize.l f f' x₀ > 0) (x : E)
  (t1 : 1 / (2 * Gradient_Descent_fix_stepsize.a f f' x₀) * (2 * Gradient_Descent_fix_stepsize.a f f' x₀) = 1)
  (h_original_goal :
    ‖x - Gradient_Descent_fix_stepsize.a f f' x₀ • f' x - xm‖ ^ 2 =
      ‖x - xm‖ ^ 2 - 2 * Gradient_Descent_fix_stepsize.a f f' x₀ * ⟪f' x, x - xm⟫_ℝ +
        ‖Gradient_Descent_fix_stepsize.a f f' x₀ • f' x‖ ^ 2) :
  ‖x - xm - Gradient_Descent_fix_stepsize.a f f' x₀ • f' x‖ ^ 2 =
    ‖x - xm‖ ^ 2 - 2 * Gradient_Descent_fix_stepsize.a f f' x₀ * ⟪f' x, x - xm⟫_ℝ +
      ‖Gradient_Descent_fix_stepsize.a f f' x₀ • f' x‖ ^ 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
alg : Gradient_Descent_fix_stepsize f f' x₀
hfun : ConvexOn ℝ univ f
step₂✝ : Gradient_Descent_fix_stepsize.a f f' x₀ ≤ 1 / ↑(Gradient_Descent_fix_stepsize.l f f' x₀)
step₂ : ↑(Gradient_Descent_fix_stepsize.l f f' x₀) ≤ 1 / Gradient_Descent_fix_stepsize.a f f' x₀
k : ℕ
this✝ : LipschitzWith (Gradient_Descent_fix_stepsize.l f f' x₀) f'
this : Gradient_Descent_fix_stepsize.l f f' x₀ > 0
x : E
t1 : 1 / (2 * Gradient_Descent_fix_stepsize.a f f' x₀) * (2 * Gradient_Descent_fix_stepsize.a f f' x₀) = 1
⊢ ‖x - Gradient_Descent_fix_stepsize.a f f' x₀ • f' x - xm‖ ^ 2 =
    ‖x - xm‖ ^ 2 - 2 * Gradient_Descent_fix_stepsize.a f f' x₀ * ⟪f' x, x - xm⟫_ℝ +
      ‖Gradient_Descent_fix_stepsize.a f f' x₀ • f' x‖ ^ 2

Tactic:
rw [sub_right_comm]

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
 alg : Gradient_Descent_fix_stepsize f f' x₀
 hfun : ConvexOn ℝ univ f
 step₂✝ : Gradient_Descent_fix_stepsize.a f f' x₀ ≤ 1 / ↑(Gradient_Descent_fix_stepsize.l f f' x₀)
 step₂ : ↑(Gradient_Descent_fix_stepsize.l f f' x₀) ≤ 1 / Gradient_Descent_fix_stepsize.a f f' x₀
 k : ℕ
 this✝ : LipschitzWith (Gradient_Descent_fix_stepsize.l f f' x₀) f'
 this : Gradient_Descent_fix_stepsize.l f f' x₀ > 0
 x : E
 t1 : 1 / (2 * Gradient_Descent_fix_stepsize.a f f' x₀) * (2 * Gradient_Descent_fix_stepsize.a f f' x₀) = 1
 ⊢ ‖x - xm - Gradient_Descent_fix_stepsize.a f f' x₀ • f' x‖ ^ 2 =
     ‖x - xm‖ ^ 2 - 2 * Gradient_Descent_fix_stepsize.a f f' x₀ * ⟪f' x, x - xm⟫_ℝ +
       ‖Gradient_Descent_fix_stepsize.a f f' x₀ • f' x‖ ^ 2]
import Mathlib
import Optlib


open Set Finset

open InnerProductSpace Set

theorem extracted_formal_statement_30.{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} {alg : Gradient_Descent_fix_stepsize f f' x₀}
  (hfun : ConvexOn ℝ univ f)
  (step₂_1 : Gradient_Descent_fix_stepsize.a f f' x₀ ≤ 1 / ↑(Gradient_Descent_fix_stepsize.l f f' x₀))
  (step₂ : ↑(Gradient_Descent_fix_stepsize.l f f' x₀) ≤ 1 / Gradient_Descent_fix_stepsize.a f f' x₀)
  (this_1 : LipschitzWith (Gradient_Descent_fix_stepsize.l f f' x₀) f')
  (this : Gradient_Descent_fix_stepsize.l f f' x₀ > 0) (x : E)
  (t1 : 1 / (2 * Gradient_Descent_fix_stepsize.a f f' x₀) * (2 * Gradient_Descent_fix_stepsize.a f f' x₀) = 1)
  (h_original_goal :
    ‖x - xm - Gradient_Descent_fix_stepsize.a f f' x₀ • f' x‖ ^ 2 =
      ‖x - xm‖ ^ 2 - 2 * Gradient_Descent_fix_stepsize.a f f' x₀ * ⟪f' x, x - xm⟫_ℝ +
        ‖Gradient_Descent_fix_stepsize.a f f' x₀ • f' x‖ ^ 2) :
  2 * ⟪x - xm, Gradient_Descent_fix_stepsize.a f f' x₀ • f' x⟫_ℝ =
    2 * Gradient_Descent_fix_stepsize.a f f' x₀ * ⟪f' x, x - xm⟫_ℝ := 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
alg : Gradient_Descent_fix_stepsize f f' x₀
hfun : ConvexOn ℝ univ f
step₂✝ : Gradient_Descent_fix_stepsize.a f f' x₀ ≤ 1 / ↑(Gradient_Descent_fix_stepsize.l f f' x₀)
step₂ : ↑(Gradient_Descent_fix_stepsize.l f f' x₀) ≤ 1 / Gradient_Descent_fix_stepsize.a f f' x₀
k : ℕ
this✝ : LipschitzWith (Gradient_Descent_fix_stepsize.l f f' x₀) f'
this : Gradient_Descent_fix_stepsize.l f f' x₀ > 0
x : E
t1 : 1 / (2 * Gradient_Descent_fix_stepsize.a f f' x₀) * (2 * Gradient_Descent_fix_stepsize.a f f' x₀) = 1
⊢ ‖x - xm - Gradient_Descent_fix_stepsize.a f f' x₀ • f' x‖ ^ 2 =
    ‖x - xm‖ ^ 2 - 2 * Gradient_Descent_fix_stepsize.a f f' x₀ * ⟪f' x, x - xm⟫_ℝ +
      ‖Gradient_Descent_fix_stepsize.a f f' x₀ • f' x‖ ^ 2

Tactic:
simp [norm_sub_sq_real (x - xm) _]

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
 alg : Gradient_Descent_fix_stepsize f f' x₀
 hfun : ConvexOn ℝ univ f
 step₂✝ : Gradient_Descent_fix_stepsize.a f f' x₀ ≤ 1 / ↑(Gradient_Descent_fix_stepsize.l f f' x₀)
 step₂ : ↑(Gradient_Descent_fix_stepsize.l f f' x₀) ≤ 1 / Gradient_Descent_fix_stepsize.a f f' x₀
 k : ℕ
 this✝ : LipschitzWith (Gradient_Descent_fix_stepsize.l f f' x₀) f'
 this : Gradient_Descent_fix_stepsize.l f f' x₀ > 0
 x : E
 t1 : 1 / (2 * Gradient_Descent_fix_stepsize.a f f' x₀) * (2 * Gradient_Descent_fix_stepsize.a f f' x₀) = 1
 ⊢ 2 * ⟪x - xm, Gradient_Descent_fix_stepsize.a f f' x₀ • f' x⟫_ℝ =
     2 * Gradient_Descent_fix_stepsize.a f f' x₀ * ⟪f' x, x - xm⟫_ℝ]
import Mathlib
import Optlib


open Set Finset

open InnerProductSpace Set

theorem extracted_formal_statement_31.{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} {alg : Gradient_Descent_fix_stepsize f f' x₀}
  (hfun : ConvexOn ℝ univ f)
  (step₂_1 : Gradient_Descent_fix_stepsize.a f f' x₀ ≤ 1 / ↑(Gradient_Descent_fix_stepsize.l f f' x₀))
  (step₂ : ↑(Gradient_Descent_fix_stepsize.l f f' x₀) ≤ 1 / Gradient_Descent_fix_stepsize.a f f' x₀)
  (this_1 : LipschitzWith (Gradient_Descent_fix_stepsize.l f f' x₀) f')
  (this : Gradient_Descent_fix_stepsize.l f f' x₀ > 0) (x : E)
  (t1 : 1 / (2 * Gradient_Descent_fix_stepsize.a f f' x₀) * (2 * Gradient_Descent_fix_stepsize.a f f' x₀) = 1) :
  ⟪x - xm, Gradient_Descent_fix_stepsize.a f f' x₀ • f' x⟫_ℝ * 2 =
    Gradient_Descent_fix_stepsize.a f f' x₀ * ⟪f' x, x - xm⟫_ℝ * 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
alg : Gradient_Descent_fix_stepsize f f' x₀
hfun : ConvexOn ℝ univ f
step₂✝ : Gradient_Descent_fix_stepsize.a f f' x₀ ≤ 1 / ↑(Gradient_Descent_fix_stepsize.l f f' x₀)
step₂ : ↑(Gradient_Descent_fix_stepsize.l f f' x₀) ≤ 1 / Gradient_Descent_fix_stepsize.a f f' x₀
k : ℕ
this✝ : LipschitzWith (Gradient_Descent_fix_stepsize.l f f' x₀) f'
this : Gradient_Descent_fix_stepsize.l f f' x₀ > 0
x : E
t1 : 1 / (2 * Gradient_Descent_fix_stepsize.a f f' x₀) * (2 * Gradient_Descent_fix_stepsize.a f f' x₀) = 1
⊢ ⟪x - xm, Gradient_Descent_fix_stepsize.a f f' x₀ • f' x⟫_ℝ * 2 =
    Gradient_Descent_fix_stepsize.a f f' x₀ * ⟪f' x, x - xm⟫_ℝ * 2

Tactic:
rw [real_inner_smul_right, real_inner_comm]

Tactic state after:
import Mathlib
import Optlib


open Set Finset

open InnerProductSpace Set

theorem extracted_formal_statement_33.{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} {alg : Gradient_Descent_fix_stepsize f f' x₀}
  (hfun : ConvexOn ℝ univ f)
  (step₂_1 : Gradient_Descent_fix_stepsize.a f f' x₀ ≤ 1 / ↑(Gradient_Descent_fix_stepsize.l f f' x₀))
  (step₂ : ↑(Gradient_Descent_fix_stepsize.l f f' x₀) ≤ 1 / Gradient_Descent_fix_stepsize.a f f' x₀)
  (this_1 : LipschitzWith (Gradient_Descent_fix_stepsize.l f f' x₀) f')
  (this : Gradient_Descent_fix_stepsize.l f f' x₀ > 0) (x : E)
  (t1 : 1 / (2 * Gradient_Descent_fix_stepsize.a f f' x₀) * (2 * Gradient_Descent_fix_stepsize.a f f' x₀) = 1)
  (t2₁ :
    ‖x - Gradient_Descent_fix_stepsize.a f f' x₀ • f' x - xm‖ ^ 2 =
      ‖x - xm‖ ^ 2 - 2 * Gradient_Descent_fix_stepsize.a f f' x₀ * ⟪f' x, x - xm⟫_ℝ +
        ‖Gradient_Descent_fix_stepsize.a f f' x₀ • f' x‖ ^ 2) :
  1 / (2 * Gradient_Descent_fix_stepsize.a f f' x₀) * (2 * Gradient_Descent_fix_stepsize.a f f' x₀) * ⟪f' x, x - xm⟫_ℝ +
      1 / (2 * Gradient_Descent_fix_stepsize.a f f' x₀) * -‖Gradient_Descent_fix_stepsize.a f f' x₀ • f' x‖ ^ 2 =
    ⟪f' x, x - xm⟫_ℝ +
      1 / (2 * Gradient_Descent_fix_stepsize.a f f' x₀) * -‖Gradient_Descent_fix_stepsize.a f f' x₀ • f' x‖ ^ 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
alg : Gradient_Descent_fix_stepsize f f' x₀
hfun : ConvexOn ℝ univ f
step₂✝ : Gradient_Descent_fix_stepsize.a f f' x₀ ≤ 1 / ↑(Gradient_Descent_fix_stepsize.l f f' x₀)
step₂ : ↑(Gradient_Descent_fix_stepsize.l f f' x₀) ≤ 1 / Gradient_Descent_fix_stepsize.a f f' x₀
k : ℕ
this✝ : LipschitzWith (Gradient_Descent_fix_stepsize.l f f' x₀) f'
this : Gradient_Descent_fix_stepsize.l f f' x₀ > 0
x : E
t1 : 1 / (2 * Gradient_Descent_fix_stepsize.a f f' x₀) * (2 * Gradient_Descent_fix_stepsize.a f f' x₀) = 1
t2₁ :
  ‖x - Gradient_Descent_fix_stepsize.a f f' x₀ • f' x - xm‖ ^ 2 =
    ‖x - xm‖ ^ 2 - 2 * Gradient_Descent_fix_stepsize.a f f' x₀ * ⟪f' x, x - xm⟫_ℝ +
      ‖Gradient_Descent_fix_stepsize.a f f' x₀ • f' x‖ ^ 2
⊢ 1 / (2 * Gradient_Descent_fix_stepsize.a f f' x₀) * (2 * Gradient_Descent_fix_stepsize.a f f' x₀) * ⟪f' x, x - xm⟫_ℝ +
      1 / (2 * Gradient_Descent_fix_stepsize.a f f' x₀) * -‖Gradient_Descent_fix_stepsize.a f f' x₀ • f' x‖ ^ 2 =
    ⟪f' x, x - xm⟫_ℝ +
      1 / (2 * Gradient_Descent_fix_stepsize.a f f' x₀) * -‖Gradient_Descent_fix_stepsize.a f f' x₀ • f' x‖ ^ 2

Tactic:
rw [t1, one_mul]

Tactic state after:
import Mathlib
import Optlib


open Set Finset

open InnerProductSpace Set

theorem extracted_formal_statement_38.{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} {alg : Gradient_Descent_fix_stepsize f f' x₀}
  (hfun : ConvexOn ℝ univ f)
  (step₂_1 : Gradient_Descent_fix_stepsize.a f f' x₀ ≤ 1 / ↑(Gradient_Descent_fix_stepsize.l f f' x₀))
  (step₂ : ↑(Gradient_Descent_fix_stepsize.l f f' x₀) ≤ 1 / Gradient_Descent_fix_stepsize.a f f' x₀)
  (this_1 : LipschitzWith (Gradient_Descent_fix_stepsize.l f f' x₀) f')
  (this : Gradient_Descent_fix_stepsize.l f f' x₀ > 0) (x : E)
  (t1 : 1 / (2 * Gradient_Descent_fix_stepsize.a f f' x₀) * (2 * Gradient_Descent_fix_stepsize.a f f' x₀) = 1)
  (t2₁ :
    ‖x - Gradient_Descent_fix_stepsize.a f f' x₀ • f' x - xm‖ ^ 2 =
      ‖x - xm‖ ^ 2 - 2 * Gradient_Descent_fix_stepsize.a f f' x₀ * ⟪f' x, x - xm⟫_ℝ +
        ‖Gradient_Descent_fix_stepsize.a f f' x₀ • f' x‖ ^ 2) :
  Gradient_Descent_fix_stepsize.a f f' x₀ ^ 2 * (Gradient_Descent_fix_stepsize.a f f' x₀)⁻¹ =
    Gradient_Descent_fix_stepsize.a f f' x₀ := sorry

Tactic state before:
case h
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
alg : Gradient_Descent_fix_stepsize f f' x₀
hfun : ConvexOn ℝ univ f
step₂✝ : Gradient_Descent_fix_stepsize.a f f' x₀ ≤ 1 / ↑(Gradient_Descent_fix_stepsize.l f f' x₀)
step₂ : ↑(Gradient_Descent_fix_stepsize.l f f' x₀) ≤ 1 / Gradient_Descent_fix_stepsize.a f f' x₀
k : ℕ
this✝ : LipschitzWith (Gradient_Descent_fix_stepsize.l f f' x₀) f'
this : Gradient_Descent_fix_stepsize.l f f' x₀ > 0
x : E
t1 : 1 / (2 * Gradient_Descent_fix_stepsize.a f f' x₀) * (2 * Gradient_Descent_fix_stepsize.a f f' x₀) = 1
t2₁ :
  ‖x - Gradient_Descent_fix_stepsize.a f f' x₀ • f' x - xm‖ ^ 2 =
    ‖x - xm‖ ^ 2 - 2 * Gradient_Descent_fix_stepsize.a f f' x₀ * ⟪f' x, x - xm⟫_ℝ +
      ‖Gradient_Descent_fix_stepsize.a f f' x₀ • f' x‖ ^ 2
⊢ Gradient_Descent_fix_stepsize.a f f' x₀ ^ 2 * (Gradient_Descent_fix_stepsize.a f f' x₀)⁻¹ =
    Gradient_Descent_fix_stepsize.a f f' x₀

Tactic:
rw [pow_two, mul_self_mul_inv alg.a]

Tactic state after:
import Mathlib
import Optlib


open Set Finset

open InnerProductSpace Set

theorem extracted_formal_statement_39.{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} {alg : Gradient_Descent_fix_stepsize f f' x₀}
  (hfun : ConvexOn ℝ univ f)
  (step₂_1 : Gradient_Descent_fix_stepsize.a f f' x₀ ≤ 1 / ↑(Gradient_Descent_fix_stepsize.l f f' x₀))
  (step₂ : ↑(Gradient_Descent_fix_stepsize.l f f' x₀) ≤ 1 / Gradient_Descent_fix_stepsize.a f f' x₀)
  (this_1 : LipschitzWith (Gradient_Descent_fix_stepsize.l f f' x₀) f')
  (this : Gradient_Descent_fix_stepsize.l f f' x₀ > 0) (x : E)
  (t1 : 1 / (2 * Gradient_Descent_fix_stepsize.a f f' x₀) * (2 * Gradient_Descent_fix_stepsize.a f f' x₀) = 1)
  (t2 :
    ⟪f' x, x - xm⟫_ℝ - Gradient_Descent_fix_stepsize.a f f' x₀ / 2 * ‖f' x‖ ^ 2 =
      1 / (2 * Gradient_Descent_fix_stepsize.a f f' x₀) *
        (‖x - xm‖ ^ 2 - ‖x - Gradient_Descent_fix_stepsize.a f f' x₀ • f' x - xm‖ ^ 2)) :
  f (x - Gradient_Descent_fix_stepsize.a f f' x₀ • f' x) ≤
    f x - Gradient_Descent_fix_stepsize.a f f' x₀ / 2 * ‖f' x‖ ^ 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
alg : Gradient_Descent_fix_stepsize f f' x₀
hfun : ConvexOn ℝ univ f
step₂✝ : Gradient_Descent_fix_stepsize.a f f' x₀ ≤ 1 / ↑(Gradient_Descent_fix_stepsize.l f f' x₀)
step₂ : ↑(Gradient_Descent_fix_stepsize.l f f' x₀) ≤ 1 / Gradient_Descent_fix_stepsize.a f f' x₀
k : ℕ
this✝ : LipschitzWith (Gradient_Descent_fix_stepsize.l f f' x₀) f'
this : Gradient_Descent_fix_stepsize.l f f' x₀ > 0
x : E
t1 : 1 / (2 * Gradient_Descent_fix_stepsize.a f f' x₀) * (2 * Gradient_Descent_fix_stepsize.a f f' x₀) = 1
t2 :
  ⟪f' x, x - xm⟫_ℝ - Gradient_Descent_fix_stepsize.a f f' x₀ / 2 * ‖f' x‖ ^ 2 =
    1 / (2 * Gradient_Descent_fix_stepsize.a f f' x₀) *
      (‖x - xm‖ ^ 2 - ‖x - Gradient_Descent_fix_stepsize.a f f' x₀ • f' x - xm‖ ^ 2)
⊢ f (x - Gradient_Descent_fix_stepsize.a f f' x₀ • f' x) ≤
    f x - Gradient_Descent_fix_stepsize.a f f' x₀ / 2 * ‖f' x‖ ^ 2

Tactic:
exact convex_lipschitz alg.diff this step₂ alg.step₁ alg.smooth x

Tactic state after:
import Mathlib
import Optlib


open Set Finset

open InnerProductSpace Set

theorem extracted_formal_statement_40.{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} {alg : Gradient_Descent_fix_stepsize f f' x₀}
  (hfun : ConvexOn ℝ univ f)
  (step₂_1 : Gradient_Descent_fix_stepsize.a f f' x₀ ≤ 1 / ↑(Gradient_Descent_fix_stepsize.l f f' x₀))
  (step₂ : ↑(Gradient_Descent_fix_stepsize.l f f' x₀) ≤ 1 / Gradient_Descent_fix_stepsize.a f f' x₀)
  (this_1 : LipschitzWith (Gradient_Descent_fix_stepsize.l f f' x₀) f')
  (this : Gradient_Descent_fix_stepsize.l f f' x₀ > 0) (x : E)
  (t1 : 1 / (2 * Gradient_Descent_fix_stepsize.a f f' x₀) * (2 * Gradient_Descent_fix_stepsize.a f f' x₀) = 1)
  (t2 :
    ⟪f' x, x - xm⟫_ℝ - Gradient_Descent_fix_stepsize.a f f' x₀ / 2 * ‖f' x‖ ^ 2 =
      1 / (2 * Gradient_Descent_fix_stepsize.a f f' x₀) *
        (‖x - xm‖ ^ 2 - ‖x - Gradient_Descent_fix_stepsize.a f f' x₀ • f' x - xm‖ ^ 2)) :
  f xm + ⟪f' x, x - xm⟫_ℝ - Gradient_Descent_fix_stepsize.a f f' x₀ / 2 * ‖f' x‖ ^ 2 =
    f xm +
      1 / (2 * Gradient_Descent_fix_stepsize.a f f' x₀) *
        (‖x - xm‖ ^ 2 - ‖x - Gradient_Descent_fix_stepsize.a f f' x₀ • f' x - 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
alg : Gradient_Descent_fix_stepsize f f' x₀
hfun : ConvexOn ℝ univ f
step₂✝ : Gradient_Descent_fix_stepsize.a f f' x₀ ≤ 1 / ↑(Gradient_Descent_fix_stepsize.l f f' x₀)
step₂ : ↑(Gradient_Descent_fix_stepsize.l f f' x₀) ≤ 1 / Gradient_Descent_fix_stepsize.a f f' x₀
k : ℕ
this✝ : LipschitzWith (Gradient_Descent_fix_stepsize.l f f' x₀) f'
this : Gradient_Descent_fix_stepsize.l f f' x₀ > 0
x : E
t1 : 1 / (2 * Gradient_Descent_fix_stepsize.a f f' x₀) * (2 * Gradient_Descent_fix_stepsize.a f f' x₀) = 1
t2 :
  ⟪f' x, x - xm⟫_ℝ - Gradient_Descent_fix_stepsize.a f f' x₀ / 2 * ‖f' x‖ ^ 2 =
    1 / (2 * Gradient_Descent_fix_stepsize.a f f' x₀) *
      (‖x - xm‖ ^ 2 - ‖x - Gradient_Descent_fix_stepsize.a f f' x₀ • f' x - xm‖ ^ 2)
⊢ f xm + ⟪f' x, x - xm⟫_ℝ - Gradient_Descent_fix_stepsize.a f f' x₀ / 2 * ‖f' x‖ ^ 2 =
    f xm +
      1 / (2 * Gradient_Descent_fix_stepsize.a f f' x₀) *
        (‖x - xm‖ ^ 2 - ‖x - Gradient_Descent_fix_stepsize.a f f' x₀ • f' x - xm‖ ^ 2)

Tactic:
rw [add_sub_assoc, t2]

Tactic state after:
import Mathlib
import Optlib


open Set Finset

open InnerProductSpace Set

theorem extracted_formal_statement_43.{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} {alg : Gradient_Descent_fix_stepsize f f' x₀}
  (hfun : ConvexOn ℝ univ f)
  (step₂ : Gradient_Descent_fix_stepsize.a f f' x₀ ≤ 1 / ↑(Gradient_Descent_fix_stepsize.l f f' x₀))
  (h_original_goal : ↑(Gradient_Descent_fix_stepsize.l f f' x₀) ≤ 1 / Gradient_Descent_fix_stepsize.a f f' x₀) :
  0 < ↑(Gradient_Descent_fix_stepsize.l f f' x₀) := 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
alg : Gradient_Descent_fix_stepsize f f' x₀
hfun : ConvexOn ℝ univ f
step₂ : Gradient_Descent_fix_stepsize.a f f' x₀ ≤ 1 / ↑(Gradient_Descent_fix_stepsize.l f f' x₀)
k : ℕ
⊢ ↑(Gradient_Descent_fix_stepsize.l f f' x₀) ≤ 1 / Gradient_Descent_fix_stepsize.a f f' x₀

Tactic:
rw [le_one_div alg.step₁] at step₂

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
 alg : Gradient_Descent_fix_stepsize f f' x₀
 hfun : ConvexOn ℝ univ f
 step₂ : ↑(Gradient_Descent_fix_stepsize.l f f' x₀) ≤ 1 / Gradient_Descent_fix_stepsize.a f f' x₀
 k : ℕ
 ⊢ ↑(Gradient_Descent_fix_stepsize.l f f' x₀) ≤ 1 / Gradient_Descent_fix_stepsize.a f f' x₀,
 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
 alg : Gradient_Descent_fix_stepsize f f' x₀
 hfun : ConvexOn ℝ univ f
 step₂ : Gradient_Descent_fix_stepsize.a f f' x₀ ≤ 1 / ↑(Gradient_Descent_fix_stepsize.l f f' x₀)
 k : ℕ
 ⊢ 0 < ↑(Gradient_Descent_fix_stepsize.l f f' x₀)]