-- Mathlib/Analysis/Calculus/FDeriv/Measurable.lean
theorem D_subset_differentiable_set {K : Set (E →L[𝕜] F)} (hK : IsComplete K) :
    D f K ⊆ { x | DifferentiableAt 𝕜 f x ∧ fderiv 𝕜 f x ∈ K }

Proof state:
𝕜 : Type u_1
inst✝⁴ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace 𝕜 E
F : Type u_3
inst✝¹ : NormedAddCommGroup F
inst✝ : 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 : ℕ → ℕ
L : ℕ → ℕ → 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 => L e (n e) (n e)
this : CauchySeq L0
f' : E →L[𝕜] F
f'K : f' ∈ K
hf' : Tendsto L0 atTop (𝓝 f')
e p : ℕ
hp : n e ≤ p
⊢ ∀ᶠ (c_1 : ℕ) in atTop, ‖L e (n e) p - L0 c_1‖ ≤ 12 * ‖c‖ * (1 / 2) ^ e

Tactic:
rw [eventually_atTop]

Result:
𝕜 : Type u_1
inst✝⁴ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace 𝕜 E
F : Type u_3
inst✝¹ : NormedAddCommGroup F
inst✝ : 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 : ℕ → ℕ
L : ℕ → ℕ → 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 => L e (n e) (n e)
this : CauchySeq L0
f' : E →L[𝕜] F
f'K : f' ∈ K
hf' : Tendsto L0 atTop (𝓝 f')
e p : ℕ
hp : n e ≤ p
⊢ ∃ a, ∀ b ≥ a, ‖L e (n e) p - L0 b‖ ≤ 12 * ‖c‖ * (1 / 2) ^ e

Proof state:
𝕜 : Type u_1
inst✝⁴ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace 𝕜 E
F : Type u_3
inst✝¹ : NormedAddCommGroup F
inst✝ : 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 : ℕ → ℕ
L : ℕ → ℕ → 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)
e p q e' p' q' : ℕ
hp : n e ≤ p
hq : n e ≤ q
hp' : n e' ≤ p'
hq' : n e' ≤ q'
he' : e ≤ e'
r : ℕ := n e ⊔ n e'
I : (1 / 2) ^ e' ≤ (1 / 2) ^ e
J1 : ‖L e p q - L e p r‖ ≤ 4 * ‖c‖ * (1 / 2) ^ e
I1 : x ∈ A f (L e p r) ((1 / 2) ^ r) ((1 / 2) ^ e)
I2 : x ∈ A f (L e' p' r) ((1 / 2) ^ r) ((1 / 2) ^ e')
⊢ ‖L e p r - L e' p' r‖ ≤ 4 * ‖c‖ * (1 / 2) ^ e

Tactic:
exact norm_sub_le_of_mem_A hc P P I1 (A_mono _ _ I I2)

Result:
no goals

Proof state:
case neg.intro.intro
𝕜 : Type u_1
inst✝⁴ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace 𝕜 E
F : Type u_3
inst✝¹ : NormedAddCommGroup F
inst✝ : 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 : ℕ → ℕ
L : ℕ → ℕ → 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 => 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 : y ∈ ball 0 ((1 / 2) ^ (n e + 1))
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
⊢ ‖f (x + y) - f x - f' y‖ ≤ ε * ‖y‖

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
  omega

Result:
case neg.intro.intro
𝕜 : Type u_1
inst✝⁴ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace 𝕜 E
F : Type u_3
inst✝¹ : NormedAddCommGroup F
inst✝ : 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 : ℕ → ℕ
L : ℕ → ℕ → 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 => 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 : y ∈ ball 0 ((1 / 2) ^ (n e + 1))
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
⊢ ‖f (x + y) - f x - f' y‖ ≤ ε * ‖y‖

Proof state:
case h
𝕜 : Type u_1
inst✝⁴ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace 𝕜 E
F : Type u_3
inst✝¹ : NormedAddCommGroup F
inst✝ : 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 : ℕ → ℕ
L : ℕ → ℕ → 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 => 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 : y ∈ ball 0 ((1 / 2) ^ (n e + 1))
y_pos : ¬y = 0
yzero : 0 < ‖y‖
y_lt : ‖y‖ < (1 / 2) ^ (n e + 1)
yone : ‖y‖ ≤ 1
k : ℕ
k_gt : n e < k
m : ℕ := k - 1
h'k : ‖y‖ ≤ (1 / 2) ^ (m + 1)
hk : (1 / 2) ^ (m + 1 + 1) < ‖y‖
m_ge : n e ≤ m
km : k = m + 1
J1 : ‖f (x + y) - f x - (L e (n e) m) (x + y - x)‖ ≤ (1 / 2) ^ e * (1 / 2) ^ m
J2 : ‖f (x + y) - f x - (L e (n e) m) y‖ ≤ 4 * (1 / 2) ^ e * ‖y‖
⊢ ‖L e (n e) m - f'‖ ≤ 12 * ‖c‖ * (1 / 2) ^ e

Tactic:
exact Lf' _ _ m_ge

Result:
no goals