zzhisthebest/LeanBenchmark huggingface dataset:
import Mathlib
import Optlib


open Set Bornology Filter BigOperators Topology

open Set Bornology Filter BigOperators Topology

open BCD

theorem extracted_formal_statement_13.{u_2, u_1} {E : Type u_1} {F : Type u_2} [inst : NormedAddCommGroup E]
  [inst_1 : InnerProductSpace ℝ E] [inst_2 : CompleteSpace E] [inst_3 : NormedAddCommGroup F]
  [inst_4 : InnerProductSpace ℝ F] [inst_5 : CompleteSpace F] {f : E → ℝ} {g : F → ℝ} {H : WithLp 2 (E × F) → ℝ}
  {x0 : E} {y0 : F} {l : NNReal}
  (h_1 : ∀ (x y : WithLp 2 (E × F)), ‖grad_fun_comp H x - grad_fun_comp H y‖ ≤ ↑l * ‖x - y‖) (y : F) (x1 x2 : E)
  (h :
    ‖(WithLp.equiv 2 (E × F)).symm (grad_fst H y x1, grad_snd H x1 y) -
          (WithLp.equiv 2 (E × F)).symm (grad_fst H y x2, grad_snd H x2 y)‖ ≤
      ↑l * ‖(x1, y) - (x2, y)‖) :
  ‖grad_fst H y x1 - grad_fst H y x2‖ ≤ ↑l * ‖x1 - x2‖ := sorry

Tactic state before:
case left
E : Type u_1
F : Type u_2
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : InnerProductSpace ℝ E
inst✝³ : CompleteSpace E
inst✝² : NormedAddCommGroup F
inst✝¹ : InnerProductSpace ℝ F
inst✝ : CompleteSpace F
f : E → ℝ
g : F → ℝ
H : WithLp 2 (E × F) → ℝ
x0 : E
y0 : F
l : NNReal
self : BCD f g H l x0 y0
h✝ : ∀ (x y : WithLp 2 (E × F)), ‖grad_fun_comp H x - grad_fun_comp H y‖ ≤ ↑l * ‖x - y‖
y : F
x1 x2 : E
h :
  ‖(WithLp.equiv 2 (E × F)).symm (grad_fst H y x1, grad_snd H x1 y) -
        (WithLp.equiv 2 (E × F)).symm (grad_fst H y x2, grad_snd H x2 y)‖ ≤
    ↑l * ‖(x1, y) - (x2, y)‖
⊢ ‖grad_fst H y x1 - grad_fst H y x2‖ ≤ ↑l * ‖x1 - x2‖

Tactic:
apply le_trans (fst_norm_le_prod_L2 _)at h
    simp at h
    rwa [sub_prod, sub_self, norm_prod_right_zero] at h

Tactic state after: