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_7.{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] {H : WithLp 2 (E × F) → ℝ} (h : Differentiable ℝ H)
  (z : WithLp 2 (E × F)) (h_original_goal : HasGradientAt (fun x => H (x, z.2)) (grad_fst H z.2 z.1) z.1) :
  DifferentiableAt ℝ (fun x => H (x, z.2)) z.1 := sorry

Tactic state before:
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
H : WithLp 2 (E × F) → ℝ
h : Differentiable ℝ H
z : WithLp 2 (E × F)
⊢ HasGradientAt (fun x => H (x, z.2)) (grad_fst H z.2 z.1) z.1

Tactic:
apply DifferentiableAt.hasGradientAt

Tactic state after:
[case h
 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
 H : WithLp 2 (E × F) → ℝ
 h : Differentiable ℝ H
 z : WithLp 2 (E × F)
 ⊢ DifferentiableAt ℝ (fun x => H (x, z.2)) z.1]
import Mathlib
import Optlib


open Set Bornology Filter BigOperators Topology

open Set Bornology Filter BigOperators Topology

open BCD

theorem extracted_formal_statement_8.{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] {H : WithLp 2 (E × F) → ℝ} (h : Differentiable ℝ H)
  (z : WithLp 2 (E × F)) : DifferentiableAt ℝ (fun x => H (x, z.2)) z.1 := sorry

Tactic state before:
case h
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
H : WithLp 2 (E × F) → ℝ
h : Differentiable ℝ H
z : WithLp 2 (E × F)
⊢ DifferentiableAt ℝ (fun x => H (x, z.2)) z.1

Tactic:
apply diff_prod₁ h

Tactic state after:
import Mathlib
import Optlib


open Set Bornology Filter BigOperators Topology

open Set Bornology Filter BigOperators Topology

open BCD

theorem extracted_formal_statement_9.{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] {H : WithLp 2 (E × F) → ℝ} (h : Differentiable ℝ H)
  (z : WithLp 2 (E × F)) (h₁ : HasGradientAt (fun x => H (x, z.2)) (grad_fst H z.2 z.1) z.1)
  (h₃ : HasGradientAt H (gradient H z) z) (h_original_goal : HasGradientAt (fun x => H (x, z.2)) (gradient H z).1 z.1)
  ⦃c : ℝ⦄ :
  0 < c → ∀ᶠ (x : E) in 𝓝 z.1, ‖H (x, z.2) - H (z.1, z.2) - inner (gradient H z).1 (x - z.1)‖ ≤ c * ‖x - z.1‖ := sorry

Tactic state before:
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
H : WithLp 2 (E × F) → ℝ
h : Differentiable ℝ H
z : WithLp 2 (E × F)
h₁ : HasGradientAt (fun x => H (x, z.2)) (grad_fst H z.2 z.1) z.1
h₃ : HasGradientAt H (gradient H z) z
⊢ HasGradientAt (fun x => H (x, z.2)) (gradient H z).1 z.1

Tactic:
rw [hasGradientAt_iff_isLittleO, Asymptotics.isLittleO_iff] at h₃ ⊢

Tactic state after:
[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
 H : WithLp 2 (E × F) → ℝ
 h : Differentiable ℝ H
 z : WithLp 2 (E × F)
 h₁ : HasGradientAt (fun x => H (x, z.2)) (grad_fst H z.2 z.1) z.1
 h₃ : ∀ ⦃c : ℝ⦄, 0 < c → ∀ᶠ (x : WithLp 2 (E × F)) in 𝓝 z, ‖H x - H z - inner (gradient H z) (x - z)‖ ≤ c * ‖x - z‖
 ⊢ ∀ ⦃c : ℝ⦄,
     0 < c → ∀ᶠ (x : E) in 𝓝 z.1, ‖H (x, z.2) - H (z.1, z.2) - inner (gradient H z).1 (x - z.1)‖ ≤ c * ‖x - z.1‖]
import Mathlib
import Optlib


open Set Bornology Filter BigOperators Topology

open Set Bornology Filter BigOperators Topology

open BCD

theorem extracted_formal_statement_10.{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] {H : WithLp 2 (E × F) → ℝ} (h : Differentiable ℝ H)
  (z : WithLp 2 (E × F)) (h₁ : HasGradientAt (fun x => H (x, z.2)) (grad_fst H z.2 z.1) z.1) ⦃c : ℝ⦄ (hc : 0 < c)
  (h₃ : ∀ᶠ (x : WithLp 2 (E × F)) in 𝓝 z, ‖H x - H z - inner (gradient H z) (x - z)‖ ≤ c * ‖x - z‖)
  (h₃' :
    ∀ᶠ (x' : E) in 𝓝 z.1,
      ∀ᶠ (y' : F) in 𝓝 z.2, ‖H (x', y') - H z - inner (gradient H z) ((x', y') - z)‖ ≤ c * ‖(x', y') - z‖)
  (h_original_goal :
    ∀ᶠ (x : E) in 𝓝 z.1, ‖H (x, z.2) - H (z.1, z.2) - inner (gradient H z).1 (x - z.1)‖ ≤ c * ‖x - z.1‖) :
  ∃ v ∈ 𝓝 z.1, ∀ y ∈ v, ‖H (y, z.2) - H (z.1, z.2) - inner (gradient H z).1 (y - z.1)‖ ≤ c * ‖y - z.1‖ := sorry

Tactic state before:
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
H : WithLp 2 (E × F) → ℝ
h : Differentiable ℝ H
z : WithLp 2 (E × F)
h₁ : HasGradientAt (fun x => H (x, z.2)) (grad_fst H z.2 z.1) z.1
c : ℝ
hc : 0 < c
h₃ : ∀ᶠ (x : WithLp 2 (E × F)) in 𝓝 z, ‖H x - H z - inner (gradient H z) (x - z)‖ ≤ c * ‖x - z‖
h₃' :
  ∀ᶠ (x' : E) in 𝓝 z.1,
    ∀ᶠ (y' : F) in 𝓝 z.2, ‖H (x', y') - H z - inner (gradient H z) ((x', y') - z)‖ ≤ c * ‖(x', y') - z‖
⊢ ∀ᶠ (x : E) in 𝓝 z.1, ‖H (x, z.2) - H (z.1, z.2) - inner (gradient H z).1 (x - z.1)‖ ≤ c * ‖x - z.1‖

Tactic:
rw [Filter.eventually_iff_exists_mem] at h₃' ⊢

Tactic state after:
[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
 H : WithLp 2 (E × F) → ℝ
 h : Differentiable ℝ H
 z : WithLp 2 (E × F)
 h₁ : HasGradientAt (fun x => H (x, z.2)) (grad_fst H z.2 z.1) z.1
 c : ℝ
 hc : 0 < c
 h₃ : ∀ᶠ (x : WithLp 2 (E × F)) in 𝓝 z, ‖H x - H z - inner (gradient H z) (x - z)‖ ≤ c * ‖x - z‖
 h₃' :
   ∃ v ∈ 𝓝 z.1,
     ∀ y ∈ v, ∀ᶠ (y' : F) in 𝓝 z.2, ‖H (y, y') - H z - inner (gradient H z) ((y, y') - z)‖ ≤ c * ‖(y, y') - z‖
 ⊢ ∃ v ∈ 𝓝 z.1, ∀ y ∈ v, ‖H (y, z.2) - H (z.1, z.2) - inner (gradient H z).1 (y - z.1)‖ ≤ c * ‖y - z.1‖]
import Mathlib
import Optlib


open Set Bornology Filter BigOperators Topology

open Set Bornology Filter BigOperators Topology

open BCD

theorem extracted_formal_statement_11.{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] {H : WithLp 2 (E × F) → ℝ} (h : Differentiable ℝ H)
  (z : WithLp 2 (E × F)) (h₁ : HasGradientAt (fun x => H (x, z.2)) (grad_fst H z.2 z.1) z.1) ⦃c : ℝ⦄ (hc : 0 < c)
  (h₃ : ∀ᶠ (x : WithLp 2 (E × F)) in 𝓝 z, ‖H x - H z - inner (gradient H z) (x - z)‖ ≤ c * ‖x - z‖) (v : Set E)
  (hv1 : v ∈ 𝓝 z.1)
  (hv2 : ∀ y ∈ v, ∀ᶠ (y' : F) in 𝓝 z.2, ‖H (y, y') - H z - inner (gradient H z) ((y, y') - z)‖ ≤ c * ‖(y, y') - z‖) :
  v ∈ 𝓝 z.1 := sorry

Tactic state before:
case h.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
H : WithLp 2 (E × F) → ℝ
h : Differentiable ℝ H
z : WithLp 2 (E × F)
h₁ : HasGradientAt (fun x => H (x, z.2)) (grad_fst H z.2 z.1) z.1
c : ℝ
hc : 0 < c
h₃ : ∀ᶠ (x : WithLp 2 (E × F)) in 𝓝 z, ‖H x - H z - inner (gradient H z) (x - z)‖ ≤ c * ‖x - z‖
v : Set E
hv1 : v ∈ 𝓝 z.1
hv2 : ∀ y ∈ v, ∀ᶠ (y' : F) in 𝓝 z.2, ‖H (y, y') - H z - inner (gradient H z) ((y, y') - z)‖ ≤ c * ‖(y, y') - z‖
⊢ v ∈ 𝓝 z.1

Tactic:
exact hv1

Tactic state after:
import Mathlib
import Optlib


open Set Bornology Filter BigOperators Topology

open Set Bornology Filter BigOperators Topology

open BCD

theorem extracted_formal_statement_12.{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] {H : WithLp 2 (E × F) → ℝ} (h : Differentiable ℝ H)
  (z : WithLp 2 (E × F)) (h₁ : HasGradientAt (fun x => H (x, z.2)) (grad_fst H z.2 z.1) z.1) ⦃c : ℝ⦄ (hc : 0 < c)
  (h₃ : ∀ᶠ (x : WithLp 2 (E × F)) in 𝓝 z, ‖H x - H z - inner (gradient H z) (x - z)‖ ≤ c * ‖x - z‖) (v : Set E)
  (hv1 : v ∈ 𝓝 z.1) (y : E) (yv : y ∈ v)
  (hv2 : ∀ᶠ (y' : F) in 𝓝 z.2, ‖H (y, y') - H z - inner (gradient H z) ((y, y') - z)‖ ≤ c * ‖(y, y') - z‖)
  (this : z = (z.1, z.2)) (hv2' : |H (y, z.2) - H z - inner (gradient H z).1 (y - z.1)| ≤ c * ‖y - z.1‖) :
  ‖H (y, z.2) - H (z.1, z.2) - inner (gradient H z).1 (y - z.1)‖ ≤ c * ‖y - z.1‖ := sorry

Tactic state before:
case h.right
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
H : WithLp 2 (E × F) → ℝ
h : Differentiable ℝ H
z : WithLp 2 (E × F)
h₁ : HasGradientAt (fun x => H (x, z.2)) (grad_fst H z.2 z.1) z.1
c : ℝ
hc : 0 < c
h₃ : ∀ᶠ (x : WithLp 2 (E × F)) in 𝓝 z, ‖H x - H z - inner (gradient H z) (x - z)‖ ≤ c * ‖x - z‖
v : Set E
hv1 : v ∈ 𝓝 z.1
y : E
yv : y ∈ v
hv2 : ∀ᶠ (y' : F) in 𝓝 z.2, ‖H (y, y') - H z - inner (gradient H z) ((y, y') - z)‖ ≤ c * ‖(y, y') - z‖
this : z = (z.1, z.2)
hv2' : |H (y, z.2) - H z - inner (gradient H z).1 (y - z.1)| ≤ c * ‖y - z.1‖
⊢ ‖H (y, z.2) - H (z.1, z.2) - inner (gradient H z).1 (y - z.1)‖ ≤ c * ‖y - z.1‖

Tactic:
exact hv2'

Tactic state after:
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] {H : WithLp 2 (E × F) → ℝ} (h : Differentiable ℝ H)
  (z : WithLp 2 (E × F)) (h₁ : HasGradientAt (fun x => H (x, z.2)) (grad_fst H z.2 z.1) z.1)
  (h₂ : HasGradientAt (fun x => H (x, z.2)) (gradient H z).1 z.1) : (gradient H z).1 = grad_fst H z.2 z.1 := sorry

Tactic state before:
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
H : WithLp 2 (E × F) → ℝ
h : Differentiable ℝ H
z : WithLp 2 (E × F)
h₁ : HasGradientAt (fun x => H (x, z.2)) (grad_fst H z.2 z.1) z.1
h₂ : HasGradientAt (fun x => H (x, z.2)) (gradient H z).1 z.1
⊢ (gradient H z).1 = grad_fst H z.2 z.1

Tactic:
exact HasGradientAt.unique h₂ h₁

Tactic state after:
import Mathlib
import Optlib


open Set Bornology Filter BigOperators Topology

open Set Bornology Filter BigOperators Topology

open BCD

theorem extracted_formal_statement_14.{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] {H : WithLp 2 (E × F) → ℝ} (h : Differentiable ℝ H)
  (z : WithLp 2 (E × F)) (h_original_goal : HasGradientAt (fun y => H (z.1, y)) (grad_snd H z.1 z.2) z.2) :
  DifferentiableAt ℝ (fun y => H (z.1, y)) z.2 := sorry

Tactic state before:
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
H : WithLp 2 (E × F) → ℝ
h : Differentiable ℝ H
z : WithLp 2 (E × F)
⊢ HasGradientAt (fun y => H (z.1, y)) (grad_snd H z.1 z.2) z.2

Tactic:
apply DifferentiableAt.hasGradientAt

Tactic state after:
[case h
 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
 H : WithLp 2 (E × F) → ℝ
 h : Differentiable ℝ H
 z : WithLp 2 (E × F)
 ⊢ DifferentiableAt ℝ (fun y => H (z.1, y)) z.2]
import Mathlib
import Optlib


open Set Bornology Filter BigOperators Topology

open Set Bornology Filter BigOperators Topology

open BCD

theorem extracted_formal_statement_15.{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] {H : WithLp 2 (E × F) → ℝ} (h : Differentiable ℝ H)
  (z : WithLp 2 (E × F)) : DifferentiableAt ℝ (fun y => H (z.1, y)) z.2 := sorry

Tactic state before:
case h
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
H : WithLp 2 (E × F) → ℝ
h : Differentiable ℝ H
z : WithLp 2 (E × F)
⊢ DifferentiableAt ℝ (fun y => H (z.1, y)) z.2

Tactic:
apply diff_prod₂ h

Tactic state after:
import Mathlib
import Optlib


open Set Bornology Filter BigOperators Topology

open Set Bornology Filter BigOperators Topology

open BCD

theorem extracted_formal_statement_16.{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] {H : WithLp 2 (E × F) → ℝ} (h : Differentiable ℝ H)
  (z : WithLp 2 (E × F)) (h₁ : HasGradientAt (fun y => H (z.1, y)) (grad_snd H z.1 z.2) z.2)
  (h₃ : HasGradientAt H (gradient H z) z) (h_original_goal : HasGradientAt (fun y => H (z.1, y)) (gradient H z).2 z.2)
  ⦃c : ℝ⦄ :
  0 < c → ∀ᶠ (x : F) in 𝓝 z.2, ‖H (z.1, x) - H (z.1, z.2) - inner (gradient H z).2 (x - z.2)‖ ≤ c * ‖x - z.2‖ := sorry

Tactic state before:
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
H : WithLp 2 (E × F) → ℝ
h : Differentiable ℝ H
z : WithLp 2 (E × F)
h₁ : HasGradientAt (fun y => H (z.1, y)) (grad_snd H z.1 z.2) z.2
h₃ : HasGradientAt H (gradient H z) z
⊢ HasGradientAt (fun y => H (z.1, y)) (gradient H z).2 z.2

Tactic:
rw [hasGradientAt_iff_isLittleO, Asymptotics.isLittleO_iff] at h₃ ⊢

Tactic state after:
[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
 H : WithLp 2 (E × F) → ℝ
 h : Differentiable ℝ H
 z : WithLp 2 (E × F)
 h₁ : HasGradientAt (fun y => H (z.1, y)) (grad_snd H z.1 z.2) z.2
 h₃ : ∀ ⦃c : ℝ⦄, 0 < c → ∀ᶠ (x : WithLp 2 (E × F)) in 𝓝 z, ‖H x - H z - inner (gradient H z) (x - z)‖ ≤ c * ‖x - z‖
 ⊢ ∀ ⦃c : ℝ⦄,
     0 < c → ∀ᶠ (x : F) in 𝓝 z.2, ‖H (z.1, x) - H (z.1, z.2) - inner (gradient H z).2 (x - z.2)‖ ≤ c * ‖x - z.2‖]
import Mathlib
import Optlib


open Set Bornology Filter BigOperators Topology

open Set Bornology Filter BigOperators Topology

open BCD

theorem extracted_formal_statement_17.{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] {H : WithLp 2 (E × F) → ℝ} (h : Differentiable ℝ H)
  (z : WithLp 2 (E × F)) (h₁ : HasGradientAt (fun y => H (z.1, y)) (grad_snd H z.1 z.2) z.2) ⦃c : ℝ⦄ (hc : 0 < c)
  (h₃ : ∀ᶠ (x : WithLp 2 (E × F)) in 𝓝 z, ‖H x - H z - inner (gradient H z) (x - z)‖ ≤ c * ‖x - z‖)
  (h₃' :
    ∀ᶠ (x' : E) in 𝓝 z.1,
      ∀ᶠ (y' : F) in 𝓝 z.2, ‖H (x', y') - H z - inner (gradient H z) ((x', y') - z)‖ ≤ c * ‖(x', y') - z‖)
  (h₃'' : ∀ᶠ (y' : F) in 𝓝 z.2, ‖H (z.1, y') - H z - inner (gradient H z) ((z.1, y') - z)‖ ≤ c * ‖(z.1, y') - z‖)
  (h_original_goal :
    ∀ᶠ (x : F) in 𝓝 z.2, ‖H (z.1, x) - H (z.1, z.2) - inner (gradient H z).2 (x - z.2)‖ ≤ c * ‖x - z.2‖) :
  ∃ v ∈ 𝓝 z.2, ∀ y ∈ v, ‖H (z.1, y) - H (z.1, z.2) - inner (gradient H z).2 (y - z.2)‖ ≤ c * ‖y - z.2‖ := sorry

Tactic state before:
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
H : WithLp 2 (E × F) → ℝ
h : Differentiable ℝ H
z : WithLp 2 (E × F)
h₁ : HasGradientAt (fun y => H (z.1, y)) (grad_snd H z.1 z.2) z.2
c : ℝ
hc : 0 < c
h₃ : ∀ᶠ (x : WithLp 2 (E × F)) in 𝓝 z, ‖H x - H z - inner (gradient H z) (x - z)‖ ≤ c * ‖x - z‖
h₃' :
  ∀ᶠ (x' : E) in 𝓝 z.1,
    ∀ᶠ (y' : F) in 𝓝 z.2, ‖H (x', y') - H z - inner (gradient H z) ((x', y') - z)‖ ≤ c * ‖(x', y') - z‖
h₃'' : ∀ᶠ (y' : F) in 𝓝 z.2, ‖H (z.1, y') - H z - inner (gradient H z) ((z.1, y') - z)‖ ≤ c * ‖(z.1, y') - z‖
⊢ ∀ᶠ (x : F) in 𝓝 z.2, ‖H (z.1, x) - H (z.1, z.2) - inner (gradient H z).2 (x - z.2)‖ ≤ c * ‖x - z.2‖

Tactic:
rw [Filter.eventually_iff_exists_mem] at h₃'' ⊢

Tactic state after:
[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
 H : WithLp 2 (E × F) → ℝ
 h : Differentiable ℝ H
 z : WithLp 2 (E × F)
 h₁ : HasGradientAt (fun y => H (z.1, y)) (grad_snd H z.1 z.2) z.2
 c : ℝ
 hc : 0 < c
 h₃ : ∀ᶠ (x : WithLp 2 (E × F)) in 𝓝 z, ‖H x - H z - inner (gradient H z) (x - z)‖ ≤ c * ‖x - z‖
 h₃' :
   ∀ᶠ (x' : E) in 𝓝 z.1,
     ∀ᶠ (y' : F) in 𝓝 z.2, ‖H (x', y') - H z - inner (gradient H z) ((x', y') - z)‖ ≤ c * ‖(x', y') - z‖
 h₃'' : ∃ v ∈ 𝓝 z.2, ∀ y ∈ v, ‖H (z.1, y) - H z - inner (gradient H z) ((z.1, y) - z)‖ ≤ c * ‖(z.1, y) - z‖
 ⊢ ∃ v ∈ 𝓝 z.2, ∀ y ∈ v, ‖H (z.1, y) - H (z.1, z.2) - inner (gradient H z).2 (y - z.2)‖ ≤ c * ‖y - z.2‖]
import Mathlib
import Optlib


open Set Bornology Filter BigOperators Topology

open Set Bornology Filter BigOperators Topology

open BCD

theorem extracted_formal_statement_18.{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] {H : WithLp 2 (E × F) → ℝ} (h : Differentiable ℝ H)
  (z : WithLp 2 (E × F)) (h₁ : HasGradientAt (fun y => H (z.1, y)) (grad_snd H z.1 z.2) z.2) ⦃c : ℝ⦄ (hc : 0 < c)
  (h₃ : ∀ᶠ (x : WithLp 2 (E × F)) in 𝓝 z, ‖H x - H z - inner (gradient H z) (x - z)‖ ≤ c * ‖x - z‖)
  (h₃' :
    ∀ᶠ (x' : E) in 𝓝 z.1,
      ∀ᶠ (y' : F) in 𝓝 z.2, ‖H (x', y') - H z - inner (gradient H z) ((x', y') - z)‖ ≤ c * ‖(x', y') - z‖)
  (v : Set F) (hv1 : v ∈ 𝓝 z.2)
  (hv2 : ∀ y ∈ v, ‖H (z.1, y) - H z - inner (gradient H z) ((z.1, y) - z)‖ ≤ c * ‖(z.1, y) - z‖) : v ∈ 𝓝 z.2 := sorry

Tactic state before:
case h.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
H : WithLp 2 (E × F) → ℝ
h : Differentiable ℝ H
z : WithLp 2 (E × F)
h₁ : HasGradientAt (fun y => H (z.1, y)) (grad_snd H z.1 z.2) z.2
c : ℝ
hc : 0 < c
h₃ : ∀ᶠ (x : WithLp 2 (E × F)) in 𝓝 z, ‖H x - H z - inner (gradient H z) (x - z)‖ ≤ c * ‖x - z‖
h₃' :
  ∀ᶠ (x' : E) in 𝓝 z.1,
    ∀ᶠ (y' : F) in 𝓝 z.2, ‖H (x', y') - H z - inner (gradient H z) ((x', y') - z)‖ ≤ c * ‖(x', y') - z‖
v : Set F
hv1 : v ∈ 𝓝 z.2
hv2 : ∀ y ∈ v, ‖H (z.1, y) - H z - inner (gradient H z) ((z.1, y) - z)‖ ≤ c * ‖(z.1, y) - z‖
⊢ v ∈ 𝓝 z.2

Tactic:
exact hv1

Tactic state after:
import Mathlib
import Optlib


open Set Bornology Filter BigOperators Topology

open Set Bornology Filter BigOperators Topology

open BCD

theorem extracted_formal_statement_19.{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] {H : WithLp 2 (E × F) → ℝ} (h : Differentiable ℝ H)
  (z : WithLp 2 (E × F)) (h₁ : HasGradientAt (fun y => H (z.1, y)) (grad_snd H z.1 z.2) z.2) ⦃c : ℝ⦄ (hc : 0 < c)
  (h₃ : ∀ᶠ (x : WithLp 2 (E × F)) in 𝓝 z, ‖H x - H z - inner (gradient H z) (x - z)‖ ≤ c * ‖x - z‖)
  (h₃' :
    ∀ᶠ (x' : E) in 𝓝 z.1,
      ∀ᶠ (y' : F) in 𝓝 z.2, ‖H (x', y') - H z - inner (gradient H z) ((x', y') - z)‖ ≤ c * ‖(x', y') - z‖)
  (v : Set F) (hv1 : v ∈ 𝓝 z.2) (y : F) (yv : y ∈ v) (this : z = (z.1, z.2))
  (hv2 : |H (z.1, y) - H z - inner (gradient H z).2 (y - z.2)| ≤ c * ‖y - z.2‖) :
  ‖H (z.1, y) - H (z.1, z.2) - inner (gradient H z).2 (y - z.2)‖ ≤ c * ‖y - z.2‖ := sorry

Tactic state before:
case h.right
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
H : WithLp 2 (E × F) → ℝ
h : Differentiable ℝ H
z : WithLp 2 (E × F)
h₁ : HasGradientAt (fun y => H (z.1, y)) (grad_snd H z.1 z.2) z.2
c : ℝ
hc : 0 < c
h₃ : ∀ᶠ (x : WithLp 2 (E × F)) in 𝓝 z, ‖H x - H z - inner (gradient H z) (x - z)‖ ≤ c * ‖x - z‖
h₃' :
  ∀ᶠ (x' : E) in 𝓝 z.1,
    ∀ᶠ (y' : F) in 𝓝 z.2, ‖H (x', y') - H z - inner (gradient H z) ((x', y') - z)‖ ≤ c * ‖(x', y') - z‖
v : Set F
hv1 : v ∈ 𝓝 z.2
y : F
yv : y ∈ v
this : z = (z.1, z.2)
hv2 : |H (z.1, y) - H z - inner (gradient H z).2 (y - z.2)| ≤ c * ‖y - z.2‖
⊢ ‖H (z.1, y) - H (z.1, z.2) - inner (gradient H z).2 (y - z.2)‖ ≤ c * ‖y - z.2‖

Tactic:
exact hv2

Tactic state after:
import Mathlib
import Optlib


open Set Bornology Filter BigOperators Topology

open Set Bornology Filter BigOperators Topology

open BCD

theorem extracted_formal_statement_20.{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] {H : WithLp 2 (E × F) → ℝ} (h : Differentiable ℝ H)
  (z : WithLp 2 (E × F)) (h₁ : HasGradientAt (fun y => H (z.1, y)) (grad_snd H z.1 z.2) z.2)
  (h₂ : HasGradientAt (fun y => H (z.1, y)) (gradient H z).2 z.2) : (gradient H z).2 = grad_snd H z.1 z.2 := sorry

Tactic state before:
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
H : WithLp 2 (E × F) → ℝ
h : Differentiable ℝ H
z : WithLp 2 (E × F)
h₁ : HasGradientAt (fun y => H (z.1, y)) (grad_snd H z.1 z.2) z.2
h₂ : HasGradientAt (fun y => H (z.1, y)) (gradient H z).2 z.2
⊢ (gradient H z).2 = grad_snd H z.1 z.2

Tactic:
exact HasGradientAt.unique h₂ h₁

Tactic state after:
import Mathlib
import Optlib


open Set Bornology Filter BigOperators Topology

open Set Bornology Filter BigOperators Topology

open BCD

theorem extracted_formal_statement_21.{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] {H : WithLp 2 (E × F) → ℝ} (h : Differentiable ℝ H)
  (z : WithLp 2 (E × F)) : ((gradient H z).1, (gradient H z).2) = (grad_fst H z.2 z.1, grad_snd H z.1 z.2) := sorry

Tactic state before:
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
H : WithLp 2 (E × F) → ℝ
h : Differentiable ℝ H
z : WithLp 2 (E × F)
⊢ ((gradient H z).1, (gradient H z).2) = (grad_fst H z.2 z.1, grad_snd H z.1 z.2)

Tactic:
rw [← grad_fst_eq h, ← grad_snd_eq h]

Tactic state after: