awhecmu/canonical-drafter-extract-mathlib huggingface dataset:
-- Module: Mathlib.Analysis.Calculus.FDeriv.Measurable
Goal:
𝕜 : Type u_1
inst : NontriviallyNormedField 𝕜
E : Type u_2
inst_1 : NormedAddCommGroup E
inst_2 : NormedSpace 𝕜 E
F : Type u_3
inst_3 : NormedAddCommGroup F
inst_4 : NormedSpace 𝕜 F
f : E → F
K : Set (E →L[𝕜] F)
hK : IsComplete K
P : ∀ {n : ℕ}, 0 < (1 / 2) ^ n
c : 𝕜
hc : 1 < ‖c‖
x : E
hx : x ∈ D f K
n : (e : ℕ) → ℕ
L : (e p q : ℕ) → E →L[𝕜] F
hn :
  ∀ (e p q : ℕ),
    n e ≤ p →
      n e ≤ q → L e p q ∈ K ∧ x ∈ A f (L e p q) ((1 / 2) ^ p) ((1 / 2) ^ e) ∩ A f (L e p q) ((1 / 2) ^ q) ((1 / 2) ^ e)
M :
  ∀ (e p q e' p' q' : ℕ),
    n e ≤ p → n e ≤ q → n e' ≤ p' → n e' ≤ q' → e ≤ e' → ‖L e p q - L e' p' q'‖ ≤ 12 * ‖c‖ * (1 / 2) ^ e
L0 : ℕ → E →L[𝕜] F := fun (e : ℕ) ↦ L e (n e) (n e)
this : CauchySeq L0
f' : E →L[𝕜] F
f'K : f' ∈ K
hf' : Tendsto L0 atTop (𝓝 f')
Lf' : ∀ (e p : ℕ), n e ≤ p → ‖L e (n e) p - f'‖ ≤ 12 * ‖c‖ * (1 / 2) ^ e
ε : ℝ
εpos : 0 < ε
pos : 0 < 4 + 12 * ‖c‖
e : ℕ
he : (1 / 2) ^ e < ε / (4 + 12 * ‖c‖)
refine :
  ∀ (y : E) (hy : Membership.mem (γ := Set E) (ball 0 ((1 / 2) ^ (n e + 1))) y),
    ‖HSub.hSub (β := F) (f (x + y) - f x) ((f' : (a : E) → F) y : F)‖ ≤ ε * ‖y‖
⊢ ∃ ε_1 > 0,
    ∀ (y : E),
      Membership.mem (γ := Set E) (ball 0 ε_1) y →
        ‖HSub.hSub (β := F) (f (x + y) - f x) ((f' : (a : E) → F) y : F)‖ ≤ ε * ‖y‖

Tactic:
refine ⟨(1 / 2) ^ (n e + 1), P, fun y hy => ?_⟩
-- Module: Mathlib.Analysis.Calculus.FDeriv.Measurable
Goal:
𝕜 : Type u_1
inst : NontriviallyNormedField 𝕜
E : Type u_2
inst_1 : NormedAddCommGroup E
inst_2 : NormedSpace 𝕜 E
F : Type u_3
inst_3 : NormedAddCommGroup F
inst_4 : NormedSpace 𝕜 F
f : E → F
K : Set (E →L[𝕜] F)
hK : IsComplete K
P : ∀ {n : ℕ}, 0 < (1 / 2) ^ n
c : 𝕜
hc : 1 < ‖c‖
x : E
hx : x ∈ D f K
n : (e : ℕ) → ℕ
L : (e p q : ℕ) → E →L[𝕜] F
hn :
  ∀ (e p q : ℕ),
    n e ≤ p →
      n e ≤ q → L e p q ∈ K ∧ x ∈ A f (L e p q) ((1 / 2) ^ p) ((1 / 2) ^ e) ∩ A f (L e p q) ((1 / 2) ^ q) ((1 / 2) ^ e)
M :
  ∀ (e p q e' p' q' : ℕ),
    n e ≤ p → n e ≤ q → n e' ≤ p' → n e' ≤ q' → e ≤ e' → ‖L e p q - L e' p' q'‖ ≤ 12 * ‖c‖ * (1 / 2) ^ e
L0 : ℕ → E →L[𝕜] F := fun (e : ℕ) ↦ L e (n e) (n e)
this : CauchySeq L0
f' : E →L[𝕜] F
f'K : f' ∈ K
hf' : Tendsto L0 atTop (𝓝 f')
Lf' : ∀ (e p : ℕ), n e ≤ p → ‖L e (n e) p - f'‖ ≤ 12 * ‖c‖ * (1 / 2) ^ e
ε : ℝ
εpos : 0 < ε
pos : 0 < 4 + 12 * ‖c‖
e : ℕ
he : (1 / 2) ^ e < ε / (4 + 12 * ‖c‖)
y : E
hy : Membership.mem (γ := Set E) (ball 0 ((1 / 2) ^ (n e + 1))) y
pos₁ : ∀ (y_pos : y = 0), ‖HSub.hSub (β := F) (f (x + y) - f x) ((f' : (a : E) → F) y : F)‖ ≤ ε * ‖y‖
neg : ∀ (y_pos : ¬y = 0), ‖HSub.hSub (β := F) (f (x + y) - f x) ((f' : (a : E) → F) y : F)‖ ≤ ε * ‖y‖
⊢ ‖HSub.hSub (β := F) (f (x + y) - f x) ((f' : (a : E) → F) y : F)‖ ≤ ε * ‖y‖

Tactic:
by_cases y_pos : y = 0
-- Module: Mathlib.Analysis.Calculus.FDeriv.Measurable
Goal:
𝕜 : Type u_1
inst : NontriviallyNormedField 𝕜
E : Type u_2
inst_1 : NormedAddCommGroup E
inst_2 : NormedSpace 𝕜 E
F : Type u_3
inst_3 : NormedAddCommGroup F
inst_4 : NormedSpace 𝕜 F
f : E → F
K : Set (E →L[𝕜] F)
hK : IsComplete K
P : ∀ {n : ℕ}, 0 < (1 / 2) ^ n
c : 𝕜
hc : 1 < ‖c‖
x : E
hx : x ∈ D f K
n : (e : ℕ) → ℕ
L : (e p q : ℕ) → E →L[𝕜] F
hn :
  ∀ (e p q : ℕ),
    n e ≤ p →
      n e ≤ q → L e p q ∈ K ∧ x ∈ A f (L e p q) ((1 / 2) ^ p) ((1 / 2) ^ e) ∩ A f (L e p q) ((1 / 2) ^ q) ((1 / 2) ^ e)
M :
  ∀ (e p q e' p' q' : ℕ),
    n e ≤ p → n e ≤ q → n e' ≤ p' → n e' ≤ q' → e ≤ e' → ‖L e p q - L e' p' q'‖ ≤ 12 * ‖c‖ * (1 / 2) ^ e
L0 : ℕ → E →L[𝕜] F := fun (e : ℕ) ↦ L e (n e) (n e)
this : CauchySeq L0
f' : E →L[𝕜] F
f'K : f' ∈ K
hf' : Tendsto L0 atTop (𝓝 f')
Lf' : ∀ (e p : ℕ), n e ≤ p → ‖L e (n e) p - f'‖ ≤ 12 * ‖c‖ * (1 / 2) ^ e
ε : ℝ
εpos : 0 < ε
pos : 0 < 4 + 12 * ‖c‖
e : ℕ
he : (1 / 2) ^ e < ε / (4 + 12 * ‖c‖)
y : E
hy : Membership.mem (γ := Set E) (ball 0 ((1 / 2) ^ (n e + 1))) y
y_pos : y = 0
⊢ ‖HSub.hSub (β := F) (f (x + y) - f x) ((f' : (a : E) → F) y : F)‖ ≤ ε * ‖y‖

Tactic:
simp [y_pos]
-- Module: Mathlib.Analysis.Calculus.FDeriv.Measurable
Goal:
𝕜 : Type u_1
inst : NontriviallyNormedField 𝕜
E : Type u_2
inst_1 : NormedAddCommGroup E
inst_2 : NormedSpace 𝕜 E
F : Type u_3
inst_3 : NormedAddCommGroup F
inst_4 : NormedSpace 𝕜 F
f : E → F
K : Set (E →L[𝕜] F)
hK : IsComplete K
P : ∀ {n : ℕ}, 0 < (1 / 2) ^ n
c : 𝕜
hc : 1 < ‖c‖
x : E
hx : x ∈ D f K
n : (e : ℕ) → ℕ
L : (e p q : ℕ) → E →L[𝕜] F
hn :
  ∀ (e p q : ℕ),
    n e ≤ p →
      n e ≤ q → L e p q ∈ K ∧ x ∈ A f (L e p q) ((1 / 2) ^ p) ((1 / 2) ^ e) ∩ A f (L e p q) ((1 / 2) ^ q) ((1 / 2) ^ e)
M :
  ∀ (e p q e' p' q' : ℕ),
    n e ≤ p → n e ≤ q → n e' ≤ p' → n e' ≤ q' → e ≤ e' → ‖L e p q - L e' p' q'‖ ≤ 12 * ‖c‖ * (1 / 2) ^ e
L0 : ℕ → E →L[𝕜] F := fun (e : ℕ) ↦ L e (n e) (n e)
this : CauchySeq L0
f' : E →L[𝕜] F
f'K : f' ∈ K
hf' : Tendsto L0 atTop (𝓝 f')
Lf' : ∀ (e p : ℕ), n e ≤ p → ‖L e (n e) p - f'‖ ≤ 12 * ‖c‖ * (1 / 2) ^ e
ε : ℝ
εpos : 0 < ε
pos : 0 < 4 + 12 * ‖c‖
e : ℕ
he : (1 / 2) ^ e < ε / (4 + 12 * ‖c‖)
y : E
hy : Membership.mem (γ := Set E) (ball 0 ((1 / 2) ^ (n e + 1))) y
y_pos : ¬y = 0
⊢ 0 < ‖y‖

Tactic:
have yzero : 0 < ‖y‖ := norm_pos_iff.mpr y_pos
-- Module: Mathlib.Analysis.Calculus.FDeriv.Measurable
Goal:
𝕜 : Type u_1
inst : NontriviallyNormedField 𝕜
E : Type u_2
inst_1 : NormedAddCommGroup E
inst_2 : NormedSpace 𝕜 E
F : Type u_3
inst_3 : NormedAddCommGroup F
inst_4 : NormedSpace 𝕜 F
f : E → F
K : Set (E →L[𝕜] F)
hK : IsComplete K
P : ∀ {n : ℕ}, 0 < (1 / 2) ^ n
c : 𝕜
hc : 1 < ‖c‖
x : E
hx : x ∈ D f K
n : (e : ℕ) → ℕ
L : (e p q : ℕ) → E →L[𝕜] F
hn :
  ∀ (e p q : ℕ),
    n e ≤ p →
      n e ≤ q → L e p q ∈ K ∧ x ∈ A f (L e p q) ((1 / 2) ^ p) ((1 / 2) ^ e) ∩ A f (L e p q) ((1 / 2) ^ q) ((1 / 2) ^ e)
M :
  ∀ (e p q e' p' q' : ℕ),
    n e ≤ p → n e ≤ q → n e' ≤ p' → n e' ≤ q' → e ≤ e' → ‖L e p q - L e' p' q'‖ ≤ 12 * ‖c‖ * (1 / 2) ^ e
L0 : ℕ → E →L[𝕜] F := fun (e : ℕ) ↦ L e (n e) (n e)
this : CauchySeq L0
f' : E →L[𝕜] F
f'K : f' ∈ K
hf' : Tendsto L0 atTop (𝓝 f')
Lf' : ∀ (e p : ℕ), n e ≤ p → ‖L e (n e) p - f'‖ ≤ 12 * ‖c‖ * (1 / 2) ^ e
ε : ℝ
εpos : 0 < ε
pos : 0 < 4 + 12 * ‖c‖
e : ℕ
he : (1 / 2) ^ e < ε / (4 + 12 * ‖c‖)
y : E
hy : Membership.mem (γ := Set E) (ball 0 ((1 / 2) ^ (n e + 1))) y
y_pos : ¬y = 0
yzero : 0 < ‖y‖
⊢ ‖y‖ < (1 / 2) ^ (n e + 1)

Tactic:
simpa using mem_ball_iff_norm.1 hy

Premises used: mem_ball_iff_norm, Iff.mp
-- Module: Mathlib.Analysis.Calculus.FDeriv.Measurable
Goal:
𝕜 : Type u_1
inst : NontriviallyNormedField 𝕜
E : Type u_2
inst_1 : NormedAddCommGroup E
inst_2 : NormedSpace 𝕜 E
F : Type u_3
inst_3 : NormedAddCommGroup F
inst_4 : NormedSpace 𝕜 F
f : E → F
K : Set (E →L[𝕜] F)
hK : IsComplete K
P : ∀ {n : ℕ}, 0 < (1 / 2) ^ n
c : 𝕜
hc : 1 < ‖c‖
x : E
hx : x ∈ D f K
n : (e : ℕ) → ℕ
L : (e p q : ℕ) → E →L[𝕜] F
hn :
  ∀ (e p q : ℕ),
    n e ≤ p →
      n e ≤ q → L e p q ∈ K ∧ x ∈ A f (L e p q) ((1 / 2) ^ p) ((1 / 2) ^ e) ∩ A f (L e p q) ((1 / 2) ^ q) ((1 / 2) ^ e)
M :
  ∀ (e p q e' p' q' : ℕ),
    n e ≤ p → n e ≤ q → n e' ≤ p' → n e' ≤ q' → e ≤ e' → ‖L e p q - L e' p' q'‖ ≤ 12 * ‖c‖ * (1 / 2) ^ e
L0 : ℕ → E →L[𝕜] F := fun (e : ℕ) ↦ L e (n e) (n e)
this : CauchySeq L0
f' : E →L[𝕜] F
f'K : f' ∈ K
hf' : Tendsto L0 atTop (𝓝 f')
Lf' : ∀ (e p : ℕ), n e ≤ p → ‖L e (n e) p - f'‖ ≤ 12 * ‖c‖ * (1 / 2) ^ e
ε : ℝ
εpos : 0 < ε
pos : 0 < 4 + 12 * ‖c‖
e : ℕ
he : (1 / 2) ^ e < ε / (4 + 12 * ‖c‖)
y : E
hy : Membership.mem (γ := Set E) (ball 0 ((1 / 2) ^ (n e + 1))) y
y_pos : ¬y = 0
yzero : 0 < ‖y‖
⊢ ‖y‖ < (1 / 2) ^ (n e + 1)

Tactic:
have y_lt : ‖y‖ < (1 / 2) ^ (n e + 1) := by simpa using mem_ball_iff_norm.1 hy
-- Module: Mathlib.Analysis.Calculus.FDeriv.Measurable
Goal:
𝕜 : Type u_1
inst : NontriviallyNormedField 𝕜
E : Type u_2
inst_1 : NormedAddCommGroup E
inst_2 : NormedSpace 𝕜 E
F : Type u_3
inst_3 : NormedAddCommGroup F
inst_4 : NormedSpace 𝕜 F
f : E → F
K : Set (E →L[𝕜] F)
hK : IsComplete K
P : ∀ {n : ℕ}, 0 < (1 / 2) ^ n
c : 𝕜
hc : 1 < ‖c‖
x : E
hx : x ∈ D f K
n : (e : ℕ) → ℕ
L : (e p q : ℕ) → E →L[𝕜] F
hn :
  ∀ (e p q : ℕ),
    n e ≤ p →
      n e ≤ q → L e p q ∈ K ∧ x ∈ A f (L e p q) ((1 / 2) ^ p) ((1 / 2) ^ e) ∩ A f (L e p q) ((1 / 2) ^ q) ((1 / 2) ^ e)
M :
  ∀ (e p q e' p' q' : ℕ),
    n e ≤ p → n e ≤ q → n e' ≤ p' → n e' ≤ q' → e ≤ e' → ‖L e p q - L e' p' q'‖ ≤ 12 * ‖c‖ * (1 / 2) ^ e
L0 : ℕ → E →L[𝕜] F := fun (e : ℕ) ↦ L e (n e) (n e)
this : CauchySeq L0
f' : E →L[𝕜] F
f'K : f' ∈ K
hf' : Tendsto L0 atTop (𝓝 f')
Lf' : ∀ (e p : ℕ), n e ≤ p → ‖L e (n e) p - f'‖ ≤ 12 * ‖c‖ * (1 / 2) ^ e
ε : ℝ
εpos : 0 < ε
pos : 0 < 4 + 12 * ‖c‖
e : ℕ
he : (1 / 2) ^ e < ε / (4 + 12 * ‖c‖)
y : E
hy : Membership.mem (γ := Set E) (ball 0 ((1 / 2) ^ (n e + 1))) y
y_pos : ¬y = 0
yzero : 0 < ‖y‖
y_lt : ‖y‖ < (1 / 2) ^ (n e + 1)
⊢ 0 ≤ 1 / 2

Tactic:
norm_num
-- Module: Mathlib.Analysis.Calculus.FDeriv.Measurable
Goal:
𝕜 : Type u_1
inst : NontriviallyNormedField 𝕜
E : Type u_2
inst_1 : NormedAddCommGroup E
inst_2 : NormedSpace 𝕜 E
F : Type u_3
inst_3 : NormedAddCommGroup F
inst_4 : NormedSpace 𝕜 F
f : E → F
K : Set (E →L[𝕜] F)
hK : IsComplete K
P : ∀ {n : ℕ}, 0 < (1 / 2) ^ n
c : 𝕜
hc : 1 < ‖c‖
x : E
hx : x ∈ D f K
n : (e : ℕ) → ℕ
L : (e p q : ℕ) → E →L[𝕜] F
hn :
  ∀ (e p q : ℕ),
    n e ≤ p →
      n e ≤ q → L e p q ∈ K ∧ x ∈ A f (L e p q) ((1 / 2) ^ p) ((1 / 2) ^ e) ∩ A f (L e p q) ((1 / 2) ^ q) ((1 / 2) ^ e)
M :
  ∀ (e p q e' p' q' : ℕ),
    n e ≤ p → n e ≤ q → n e' ≤ p' → n e' ≤ q' → e ≤ e' → ‖L e p q - L e' p' q'‖ ≤ 12 * ‖c‖ * (1 / 2) ^ e
L0 : ℕ → E →L[𝕜] F := fun (e : ℕ) ↦ L e (n e) (n e)
this : CauchySeq L0
f' : E →L[𝕜] F
f'K : f' ∈ K
hf' : Tendsto L0 atTop (𝓝 f')
Lf' : ∀ (e p : ℕ), n e ≤ p → ‖L e (n e) p - f'‖ ≤ 12 * ‖c‖ * (1 / 2) ^ e
ε : ℝ
εpos : 0 < ε
pos : 0 < 4 + 12 * ‖c‖
e : ℕ
he : (1 / 2) ^ e < ε / (4 + 12 * ‖c‖)
y : E
hy : Membership.mem (γ := Set E) (ball 0 ((1 / 2) ^ (n e + 1))) y
y_pos : ¬y = 0
yzero : 0 < ‖y‖
y_lt : ‖y‖ < (1 / 2) ^ (n e + 1)
⊢ 1 / 2 ≤ 1

Tactic:
norm_num
-- Module: Mathlib.Analysis.Calculus.FDeriv.Measurable
Goal:
𝕜 : Type u_1
inst : NontriviallyNormedField 𝕜
E : Type u_2
inst_1 : NormedAddCommGroup E
inst_2 : NormedSpace 𝕜 E
F : Type u_3
inst_3 : NormedAddCommGroup F
inst_4 : NormedSpace 𝕜 F
f : E → F
K : Set (E →L[𝕜] F)
hK : IsComplete K
P : ∀ {n : ℕ}, 0 < (1 / 2) ^ n
c : 𝕜
hc : 1 < ‖c‖
x : E
hx : x ∈ D f K
n : (e : ℕ) → ℕ
L : (e p q : ℕ) → E →L[𝕜] F
hn :
  ∀ (e p q : ℕ),
    n e ≤ p →
      n e ≤ q → L e p q ∈ K ∧ x ∈ A f (L e p q) ((1 / 2) ^ p) ((1 / 2) ^ e) ∩ A f (L e p q) ((1 / 2) ^ q) ((1 / 2) ^ e)
M :
  ∀ (e p q e' p' q' : ℕ),
    n e ≤ p → n e ≤ q → n e' ≤ p' → n e' ≤ q' → e ≤ e' → ‖L e p q - L e' p' q'‖ ≤ 12 * ‖c‖ * (1 / 2) ^ e
L0 : ℕ → E →L[𝕜] F := fun (e : ℕ) ↦ L e (n e) (n e)
this : CauchySeq L0
f' : E →L[𝕜] F
f'K : f' ∈ K
hf' : Tendsto L0 atTop (𝓝 f')
Lf' : ∀ (e p : ℕ), n e ≤ p → ‖L e (n e) p - f'‖ ≤ 12 * ‖c‖ * (1 / 2) ^ e
ε : ℝ
εpos : 0 < ε
pos : 0 < 4 + 12 * ‖c‖
e : ℕ
he : (1 / 2) ^ e < ε / (4 + 12 * ‖c‖)
y : E
hy : Membership.mem (γ := Set E) (ball 0 ((1 / 2) ^ (n e + 1))) y
y_pos : ¬y = 0
yzero : 0 < ‖y‖
y_lt : ‖y‖ < (1 / 2) ^ (n e + 1)
⊢ ‖y‖ ≤ 1

Tactic:
have yone : ‖y‖ ≤ 1 := le_trans y_lt.le (pow_le_one₀ (by norm_num) (by norm_num))
-- Module: Mathlib.Analysis.Calculus.FDeriv.Measurable
Goal:
𝕜 : Type u_1
inst : NontriviallyNormedField 𝕜
E : Type u_2
inst_1 : NormedAddCommGroup E
inst_2 : NormedSpace 𝕜 E
F : Type u_3
inst_3 : NormedAddCommGroup F
inst_4 : NormedSpace 𝕜 F
f : E → F
K : Set (E →L[𝕜] F)
hK : IsComplete K
P : ∀ {n : ℕ}, 0 < (1 / 2) ^ n
c : 𝕜
hc : 1 < ‖c‖
x : E
hx : x ∈ D f K
n : (e : ℕ) → ℕ
L : (e p q : ℕ) → E →L[𝕜] F
hn :
  ∀ (e p q : ℕ),
    n e ≤ p →
      n e ≤ q → L e p q ∈ K ∧ x ∈ A f (L e p q) ((1 / 2) ^ p) ((1 / 2) ^ e) ∩ A f (L e p q) ((1 / 2) ^ q) ((1 / 2) ^ e)
M :
  ∀ (e p q e' p' q' : ℕ),
    n e ≤ p → n e ≤ q → n e' ≤ p' → n e' ≤ q' → e ≤ e' → ‖L e p q - L e' p' q'‖ ≤ 12 * ‖c‖ * (1 / 2) ^ e
L0 : ℕ → E →L[𝕜] F := fun (e : ℕ) ↦ L e (n e) (n e)
this : CauchySeq L0
f' : E →L[𝕜] F
f'K : f' ∈ K
hf' : Tendsto L0 atTop (𝓝 f')
Lf' : ∀ (e p : ℕ), n e ≤ p → ‖L e (n e) p - f'‖ ≤ 12 * ‖c‖ * (1 / 2) ^ e
ε : ℝ
εpos : 0 < ε
pos : 0 < 4 + 12 * ‖c‖
e : ℕ
he : (1 / 2) ^ e < ε / (4 + 12 * ‖c‖)
y : E
hy : Membership.mem (γ := Set E) (ball 0 ((1 / 2) ^ (n e + 1))) y
y_pos : ¬y = 0
yzero : 0 < ‖y‖
y_lt : ‖y‖ < (1 / 2) ^ (n e + 1)
yone : ‖y‖ ≤ 1
⊢ 0 < 1 / 2

Tactic:
norm_num
-- Module: Mathlib.Analysis.Calculus.FDeriv.Measurable
Goal:
𝕜 : Type u_1
inst : NontriviallyNormedField 𝕜
E : Type u_2
inst_1 : NormedAddCommGroup E
inst_2 : NormedSpace 𝕜 E
F : Type u_3
inst_3 : NormedAddCommGroup F
inst_4 : NormedSpace 𝕜 F
f : E → F
K : Set (E →L[𝕜] F)
hK : IsComplete K
P : ∀ {n : ℕ}, 0 < (1 / 2) ^ n
c : 𝕜
hc : 1 < ‖c‖
x : E
hx : x ∈ D f K
n : (e : ℕ) → ℕ
L : (e p q : ℕ) → E →L[𝕜] F
hn :
  ∀ (e p q : ℕ),
    n e ≤ p →
      n e ≤ q → L e p q ∈ K ∧ x ∈ A f (L e p q) ((1 / 2) ^ p) ((1 / 2) ^ e) ∩ A f (L e p q) ((1 / 2) ^ q) ((1 / 2) ^ e)
M :
  ∀ (e p q e' p' q' : ℕ),
    n e ≤ p → n e ≤ q → n e' ≤ p' → n e' ≤ q' → e ≤ e' → ‖L e p q - L e' p' q'‖ ≤ 12 * ‖c‖ * (1 / 2) ^ e
L0 : ℕ → E →L[𝕜] F := fun (e : ℕ) ↦ L e (n e) (n e)
this : CauchySeq L0
f' : E →L[𝕜] F
f'K : f' ∈ K
hf' : Tendsto L0 atTop (𝓝 f')
Lf' : ∀ (e p : ℕ), n e ≤ p → ‖L e (n e) p - f'‖ ≤ 12 * ‖c‖ * (1 / 2) ^ e
ε : ℝ
εpos : 0 < ε
pos : 0 < 4 + 12 * ‖c‖
e : ℕ
he : (1 / 2) ^ e < ε / (4 + 12 * ‖c‖)
y : E
hy : Membership.mem (γ := Set E) (ball 0 ((1 / 2) ^ (n e + 1))) y
y_pos : ¬y = 0
yzero : 0 < ‖y‖
y_lt : ‖y‖ < (1 / 2) ^ (n e + 1)
yone : ‖y‖ ≤ 1
⊢ 1 / 2 < 1

Tactic:
norm_num
-- Module: Mathlib.Analysis.Calculus.FDeriv.Measurable
Goal:
𝕜 : Type u_1
inst : NontriviallyNormedField 𝕜
E : Type u_2
inst_1 : NormedAddCommGroup E
inst_2 : NormedSpace 𝕜 E
F : Type u_3
inst_3 : NormedAddCommGroup F
inst_4 : NormedSpace 𝕜 F
f : E → F
K : Set (E →L[𝕜] F)
hK : IsComplete K
P : ∀ {n : ℕ}, 0 < (1 / 2) ^ n
c : 𝕜
hc : 1 < ‖c‖
x : E
hx : x ∈ D f K
n : (e : ℕ) → ℕ
L : (e p q : ℕ) → E →L[𝕜] F
hn :
  ∀ (e p q : ℕ),
    n e ≤ p →
      n e ≤ q → L e p q ∈ K ∧ x ∈ A f (L e p q) ((1 / 2) ^ p) ((1 / 2) ^ e) ∩ A f (L e p q) ((1 / 2) ^ q) ((1 / 2) ^ e)
M :
  ∀ (e p q e' p' q' : ℕ),
    n e ≤ p → n e ≤ q → n e' ≤ p' → n e' ≤ q' → e ≤ e' → ‖L e p q - L e' p' q'‖ ≤ 12 * ‖c‖ * (1 / 2) ^ e
L0 : ℕ → E →L[𝕜] F := fun (e : ℕ) ↦ L e (n e) (n e)
this : CauchySeq L0
f' : E →L[𝕜] F
f'K : f' ∈ K
hf' : Tendsto L0 atTop (𝓝 f')
Lf' : ∀ (e p : ℕ), n e ≤ p → ‖L e (n e) p - f'‖ ≤ 12 * ‖c‖ * (1 / 2) ^ e
ε : ℝ
εpos : 0 < ε
pos : 0 < 4 + 12 * ‖c‖
e : ℕ
he : (1 / 2) ^ e < ε / (4 + 12 * ‖c‖)
y : E
hy : Membership.mem (γ := Set E) (ball 0 ((1 / 2) ^ (n e + 1))) y
y_pos : ¬y = 0
yzero : 0 < ‖y‖
y_lt : ‖y‖ < (1 / 2) ^ (n e + 1)
yone : ‖y‖ ≤ 1
neg :
  ∀ (k : ℕ) (hk : (1 / 2) ^ (k + 1) < ‖y‖) (h'k : ‖y‖ ≤ (1 / 2) ^ k),
    ‖HSub.hSub (β := F) (f (x + y) - f x) ((f' : (a : E) → F) y : F)‖ ≤ ε * ‖y‖
⊢ ‖HSub.hSub (β := F) (f (x + y) - f x) ((f' : (a : E) → F) y : F)‖ ≤ ε * ‖y‖

Tactic:
obtain ⟨k, hk, h'k⟩ : ∃ k : ℕ, (1 / 2) ^ (k + 1) < ‖y‖ ∧ ‖y‖ ≤ (1 / 2) ^ k :=
      exists_nat_pow_near_of_lt_one yzero yone (by norm_num : (0 : ℝ) < 1 / 2)
        (by norm_num : (1 : ℝ) / 2 < 1)

Premises used: exists_nat_pow_near_of_lt_one, HDiv.hDiv, OfNat.ofNat
-- Module: Mathlib.Analysis.Calculus.FDeriv.Measurable
Goal:
𝕜 : Type u_1
inst : NontriviallyNormedField 𝕜
E : Type u_2
inst_1 : NormedAddCommGroup E
inst_2 : NormedSpace 𝕜 E
F : Type u_3
inst_3 : NormedAddCommGroup F
inst_4 : NormedSpace 𝕜 F
f : E → F
K : Set (E →L[𝕜] F)
hK : IsComplete K
P : ∀ {n : ℕ}, 0 < (1 / 2) ^ n
c : 𝕜
hc : 1 < ‖c‖
x : E
hx : x ∈ D f K
n : (e : ℕ) → ℕ
L : (e p q : ℕ) → E →L[𝕜] F
hn :
  ∀ (e p q : ℕ),
    n e ≤ p →
      n e ≤ q → L e p q ∈ K ∧ x ∈ A f (L e p q) ((1 / 2) ^ p) ((1 / 2) ^ e) ∩ A f (L e p q) ((1 / 2) ^ q) ((1 / 2) ^ e)
M :
  ∀ (e p q e' p' q' : ℕ),
    n e ≤ p → n e ≤ q → n e' ≤ p' → n e' ≤ q' → e ≤ e' → ‖L e p q - L e' p' q'‖ ≤ 12 * ‖c‖ * (1 / 2) ^ e
L0 : ℕ → E →L[𝕜] F := fun (e : ℕ) ↦ L e (n e) (n e)
this : CauchySeq L0
f' : E →L[𝕜] F
f'K : f' ∈ K
hf' : Tendsto L0 atTop (𝓝 f')
Lf' : ∀ (e p : ℕ), n e ≤ p → ‖L e (n e) p - f'‖ ≤ 12 * ‖c‖ * (1 / 2) ^ e
ε : ℝ
εpos : 0 < ε
pos : 0 < 4 + 12 * ‖c‖
e : ℕ
he : (1 / 2) ^ e < ε / (4 + 12 * ‖c‖)
y : E
hy : Membership.mem (γ := Set E) (ball 0 ((1 / 2) ^ (n e + 1))) y
y_pos : ¬y = 0
yzero : 0 < ‖y‖
y_lt : ‖y‖ < (1 / 2) ^ (n e + 1)
yone : ‖y‖ ≤ 1
k : ℕ
hk : (1 / 2) ^ (k + 1) < ‖y‖
h'k : ‖y‖ ≤ (1 / 2) ^ k
⊢ (1 / 2) ^ (k + 1) < (1 / 2) ^ (n e + 1)

Tactic:
have : ((1 : ℝ) / 2) ^ (k + 1) < (1 / 2) ^ (n e + 1) := lt_trans hk y_lt
-- Module: Mathlib.Analysis.Calculus.FDeriv.Measurable
Goal:
𝕜 : Type u_1
inst : NontriviallyNormedField 𝕜
E : Type u_2
inst_1 : NormedAddCommGroup E
inst_2 : NormedSpace 𝕜 E
F : Type u_3
inst_3 : NormedAddCommGroup F
inst_4 : NormedSpace 𝕜 F
f : E → F
K : Set (E →L[𝕜] F)
hK : IsComplete K
P : ∀ {n : ℕ}, 0 < (1 / 2) ^ n
c : 𝕜
hc : 1 < ‖c‖
x : E
hx : x ∈ D f K
n : (e : ℕ) → ℕ
L : (e p q : ℕ) → E →L[𝕜] F
hn :
  ∀ (e p q : ℕ),
    n e ≤ p →
      n e ≤ q → L e p q ∈ K ∧ x ∈ A f (L e p q) ((1 / 2) ^ p) ((1 / 2) ^ e) ∩ A f (L e p q) ((1 / 2) ^ q) ((1 / 2) ^ e)
M :
  ∀ (e p q e' p' q' : ℕ),
    n e ≤ p → n e ≤ q → n e' ≤ p' → n e' ≤ q' → e ≤ e' → ‖L e p q - L e' p' q'‖ ≤ 12 * ‖c‖ * (1 / 2) ^ e
L0 : ℕ → E →L[𝕜] F := fun (e : ℕ) ↦ L e (n e) (n e)
this_1 : CauchySeq L0
f' : E →L[𝕜] F
f'K : f' ∈ K
hf' : Tendsto L0 atTop (𝓝 f')
Lf' : ∀ (e p : ℕ), n e ≤ p → ‖L e (n e) p - f'‖ ≤ 12 * ‖c‖ * (1 / 2) ^ e
ε : ℝ
εpos : 0 < ε
pos : 0 < 4 + 12 * ‖c‖
e : ℕ
he : (1 / 2) ^ e < ε / (4 + 12 * ‖c‖)
y : E
hy : Membership.mem (γ := Set E) (ball 0 ((1 / 2) ^ (n e + 1))) y
y_pos : ¬y = 0
yzero : 0 < ‖y‖
y_lt : ‖y‖ < (1 / 2) ^ (n e + 1)
yone : ‖y‖ ≤ 1
k : ℕ
hk : (1 / 2) ^ (k + 1) < ‖y‖
h'k : ‖y‖ ≤ (1 / 2) ^ k
this : (1 / 2) ^ (k + 1) < (1 / 2) ^ (n e + 1)
⊢ 0 < 1 / 2

Tactic:
norm_num
-- Module: Mathlib.Analysis.Calculus.FDeriv.Measurable
Goal:
𝕜 : Type u_1
inst : NontriviallyNormedField 𝕜
E : Type u_2
inst_1 : NormedAddCommGroup E
inst_2 : NormedSpace 𝕜 E
F : Type u_3
inst_3 : NormedAddCommGroup F
inst_4 : NormedSpace 𝕜 F
f : E → F
K : Set (E →L[𝕜] F)
hK : IsComplete K
P : ∀ {n : ℕ}, 0 < (1 / 2) ^ n
c : 𝕜
hc : 1 < ‖c‖
x : E
hx : x ∈ D f K
n : (e : ℕ) → ℕ
L : (e p q : ℕ) → E →L[𝕜] F
hn :
  ∀ (e p q : ℕ),
    n e ≤ p →
      n e ≤ q → L e p q ∈ K ∧ x ∈ A f (L e p q) ((1 / 2) ^ p) ((1 / 2) ^ e) ∩ A f (L e p q) ((1 / 2) ^ q) ((1 / 2) ^ e)
M :
  ∀ (e p q e' p' q' : ℕ),
    n e ≤ p → n e ≤ q → n e' ≤ p' → n e' ≤ q' → e ≤ e' → ‖L e p q - L e' p' q'‖ ≤ 12 * ‖c‖ * (1 / 2) ^ e
L0 : ℕ → E →L[𝕜] F := fun (e : ℕ) ↦ L e (n e) (n e)
this_1 : CauchySeq L0
f' : E →L[𝕜] F
f'K : f' ∈ K
hf' : Tendsto L0 atTop (𝓝 f')
Lf' : ∀ (e p : ℕ), n e ≤ p → ‖L e (n e) p - f'‖ ≤ 12 * ‖c‖ * (1 / 2) ^ e
ε : ℝ
εpos : 0 < ε
pos : 0 < 4 + 12 * ‖c‖
e : ℕ
he : (1 / 2) ^ e < ε / (4 + 12 * ‖c‖)
y : E
hy : Membership.mem (γ := Set E) (ball 0 ((1 / 2) ^ (n e + 1))) y
y_pos : ¬y = 0
yzero : 0 < ‖y‖
y_lt : ‖y‖ < (1 / 2) ^ (n e + 1)
yone : ‖y‖ ≤ 1
k : ℕ
hk : (1 / 2) ^ (k + 1) < ‖y‖
h'k : ‖y‖ ≤ (1 / 2) ^ k
this : (1 / 2) ^ (k + 1) < (1 / 2) ^ (n e + 1)
⊢ 1 / 2 < 1

Tactic:
norm_num
-- Module: Mathlib.Analysis.Calculus.FDeriv.Measurable
Goal:
𝕜 : Type u_1
inst : NontriviallyNormedField 𝕜
E : Type u_2
inst_1 : NormedAddCommGroup E
inst_2 : NormedSpace 𝕜 E
F : Type u_3
inst_3 : NormedAddCommGroup F
inst_4 : NormedSpace 𝕜 F
f : E → F
K : Set (E →L[𝕜] F)
hK : IsComplete K
P : ∀ {n : ℕ}, 0 < (1 / 2) ^ n
c : 𝕜
hc : 1 < ‖c‖
x : E
hx : x ∈ D f K
n : (e : ℕ) → ℕ
L : (e p q : ℕ) → E →L[𝕜] F
hn :
  ∀ (e p q : ℕ),
    n e ≤ p →
      n e ≤ q → L e p q ∈ K ∧ x ∈ A f (L e p q) ((1 / 2) ^ p) ((1 / 2) ^ e) ∩ A f (L e p q) ((1 / 2) ^ q) ((1 / 2) ^ e)
M :
  ∀ (e p q e' p' q' : ℕ),
    n e ≤ p → n e ≤ q → n e' ≤ p' → n e' ≤ q' → e ≤ e' → ‖L e p q - L e' p' q'‖ ≤ 12 * ‖c‖ * (1 / 2) ^ e
L0 : ℕ → E →L[𝕜] F := fun (e : ℕ) ↦ L e (n e) (n e)
this_1 : CauchySeq L0
f' : E →L[𝕜] F
f'K : f' ∈ K
hf' : Tendsto L0 atTop (𝓝 f')
Lf' : ∀ (e p : ℕ), n e ≤ p → ‖L e (n e) p - f'‖ ≤ 12 * ‖c‖ * (1 / 2) ^ e
ε : ℝ
εpos : 0 < ε
pos : 0 < 4 + 12 * ‖c‖
e : ℕ
he : (1 / 2) ^ e < ε / (4 + 12 * ‖c‖)
y : E
hy : Membership.mem (γ := Set E) (ball 0 ((1 / 2) ^ (n e + 1))) y
y_pos : ¬y = 0
yzero : 0 < ‖y‖
y_lt : ‖y‖ < (1 / 2) ^ (n e + 1)
yone : ‖y‖ ≤ 1
k : ℕ
hk : (1 / 2) ^ (k + 1) < ‖y‖
h'k : ‖y‖ ≤ (1 / 2) ^ k
this : (1 / 2) ^ (k + 1) < (1 / 2) ^ (n e + 1)
⊢ n e + 1 < k + 1

Tactic:
rw [pow_lt_pow_iff_right_of_lt_one₀ (by norm_num : (0 : ℝ) < 1 / 2) (by norm_num)] at this

Premises used: pow_lt_pow_iff_right_of_lt_one₀
-- Module: Mathlib.Analysis.Calculus.FDeriv.Measurable
Goal:
𝕜 : Type u_1
inst : NontriviallyNormedField 𝕜
E : Type u_2
inst_1 : NormedAddCommGroup E
inst_2 : NormedSpace 𝕜 E
F : Type u_3
inst_3 : NormedAddCommGroup F
inst_4 : NormedSpace 𝕜 F
f : E → F
K : Set (E →L[𝕜] F)
hK : IsComplete K
P : ∀ {n : ℕ}, 0 < (1 / 2) ^ n
c : 𝕜
hc : 1 < ‖c‖
x : E
hx : x ∈ D f K
n : (e : ℕ) → ℕ
L : (e p q : ℕ) → E →L[𝕜] F
hn :
  ∀ (e p q : ℕ),
    n e ≤ p →
      n e ≤ q → L e p q ∈ K ∧ x ∈ A f (L e p q) ((1 / 2) ^ p) ((1 / 2) ^ e) ∩ A f (L e p q) ((1 / 2) ^ q) ((1 / 2) ^ e)
M :
  ∀ (e p q e' p' q' : ℕ),
    n e ≤ p → n e ≤ q → n e' ≤ p' → n e' ≤ q' → e ≤ e' → ‖L e p q - L e' p' q'‖ ≤ 12 * ‖c‖ * (1 / 2) ^ e
L0 : ℕ → E →L[𝕜] F := fun (e : ℕ) ↦ L e (n e) (n e)
this_1 : CauchySeq L0
f' : E →L[𝕜] F
f'K : f' ∈ K
hf' : Tendsto L0 atTop (𝓝 f')
Lf' : ∀ (e p : ℕ), n e ≤ p → ‖L e (n e) p - f'‖ ≤ 12 * ‖c‖ * (1 / 2) ^ e
ε : ℝ
εpos : 0 < ε
pos : 0 < 4 + 12 * ‖c‖
e : ℕ
he : (1 / 2) ^ e < ε / (4 + 12 * ‖c‖)
y : E
hy : Membership.mem (γ := Set E) (ball 0 ((1 / 2) ^ (n e + 1))) y
y_pos : ¬y = 0
yzero : 0 < ‖y‖
y_lt : ‖y‖ < (1 / 2) ^ (n e + 1)
yone : ‖y‖ ≤ 1
k : ℕ
hk : (1 / 2) ^ (k + 1) < ‖y‖
h'k : ‖y‖ ≤ (1 / 2) ^ k
this : n e + 1 < k + 1
⊢ n e < k

Tactic:
lia
-- Module: Mathlib.Analysis.Calculus.FDeriv.Measurable
Goal:
𝕜 : Type u_1
inst : NontriviallyNormedField 𝕜
E : Type u_2
inst_1 : NormedAddCommGroup E
inst_2 : NormedSpace 𝕜 E
F : Type u_3
inst_3 : NormedAddCommGroup F
inst_4 : NormedSpace 𝕜 F
f : E → F
K : Set (E →L[𝕜] F)
hK : IsComplete K
P : ∀ {n : ℕ}, 0 < (1 / 2) ^ n
c : 𝕜
hc : 1 < ‖c‖
x : E
hx : x ∈ D f K
n : (e : ℕ) → ℕ
L : (e p q : ℕ) → E →L[𝕜] F
hn :
  ∀ (e p q : ℕ),
    n e ≤ p →
      n e ≤ q → L e p q ∈ K ∧ x ∈ A f (L e p q) ((1 / 2) ^ p) ((1 / 2) ^ e) ∩ A f (L e p q) ((1 / 2) ^ q) ((1 / 2) ^ e)
M :
  ∀ (e p q e' p' q' : ℕ),
    n e ≤ p → n e ≤ q → n e' ≤ p' → n e' ≤ q' → e ≤ e' → ‖L e p q - L e' p' q'‖ ≤ 12 * ‖c‖ * (1 / 2) ^ e
L0 : ℕ → E →L[𝕜] F := fun (e : ℕ) ↦ L e (n e) (n e)
this : CauchySeq L0
f' : E →L[𝕜] F
f'K : f' ∈ K
hf' : Tendsto L0 atTop (𝓝 f')
Lf' : ∀ (e p : ℕ), n e ≤ p → ‖L e (n e) p - f'‖ ≤ 12 * ‖c‖ * (1 / 2) ^ e
ε : ℝ
εpos : 0 < ε
pos : 0 < 4 + 12 * ‖c‖
e : ℕ
he : (1 / 2) ^ e < ε / (4 + 12 * ‖c‖)
y : E
hy : Membership.mem (γ := Set E) (ball 0 ((1 / 2) ^ (n e + 1))) y
y_pos : ¬y = 0
yzero : 0 < ‖y‖
y_lt : ‖y‖ < (1 / 2) ^ (n e + 1)
yone : ‖y‖ ≤ 1
k : ℕ
hk : (1 / 2) ^ (k + 1) < ‖y‖
h'k : ‖y‖ ≤ (1 / 2) ^ k
⊢ n e < k

Tactic:
have k_gt : n e < k := by
      have : ((1 : ℝ) / 2) ^ (k + 1) < (1 / 2) ^ (n e + 1) := lt_trans hk y_lt
      rw [pow_lt_pow_iff_right_of_lt_one₀ (by norm_num : (0 : ℝ) < 1 / 2) (by norm_num)] at this
      lia
-- Module: Mathlib.Analysis.Calculus.FDeriv.Measurable
Goal:
𝕜 : Type u_1
inst : NontriviallyNormedField 𝕜
E : Type u_2
inst_1 : NormedAddCommGroup E
inst_2 : NormedSpace 𝕜 E
F : Type u_3
inst_3 : NormedAddCommGroup F
inst_4 : NormedSpace 𝕜 F
f : E → F
K : Set (E →L[𝕜] F)
hK : IsComplete K
P : ∀ {n : ℕ}, 0 < (1 / 2) ^ n
c : 𝕜
hc : 1 < ‖c‖
x : E
hx : x ∈ D f K
n : (e : ℕ) → ℕ
L : (e p q : ℕ) → E →L[𝕜] F
hn :
  ∀ (e p q : ℕ),
    n e ≤ p →
      n e ≤ q → L e p q ∈ K ∧ x ∈ A f (L e p q) ((1 / 2) ^ p) ((1 / 2) ^ e) ∩ A f (L e p q) ((1 / 2) ^ q) ((1 / 2) ^ e)
M :
  ∀ (e p q e' p' q' : ℕ),
    n e ≤ p → n e ≤ q → n e' ≤ p' → n e' ≤ q' → e ≤ e' → ‖L e p q - L e' p' q'‖ ≤ 12 * ‖c‖ * (1 / 2) ^ e
L0 : ℕ → E →L[𝕜] F := fun (e : ℕ) ↦ L e (n e) (n e)
this : CauchySeq L0
f' : E →L[𝕜] F
f'K : f' ∈ K
hf' : Tendsto L0 atTop (𝓝 f')
Lf' : ∀ (e p : ℕ), n e ≤ p → ‖L e (n e) p - f'‖ ≤ 12 * ‖c‖ * (1 / 2) ^ e
ε : ℝ
εpos : 0 < ε
pos : 0 < 4 + 12 * ‖c‖
e : ℕ
he : (1 / 2) ^ e < ε / (4 + 12 * ‖c‖)
y : E
hy : Membership.mem (γ := Set E) (ball 0 ((1 / 2) ^ (n e + 1))) y
y_pos : ¬y = 0
yzero : 0 < ‖y‖
y_lt : ‖y‖ < (1 / 2) ^ (n e + 1)
yone : ‖y‖ ≤ 1
k : ℕ
hk : (1 / 2) ^ (k + 1) < ‖y‖
h'k : ‖y‖ ≤ (1 / 2) ^ k
k_gt : n e < k
m : ℕ := k - 1
⊢ n e ≤ m

Tactic:
have m_ge : n e ≤ m := Nat.le_sub_one_of_lt k_gt