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)
x : E
hx : Membership.mem (γ := Set E) {x : E | DifferentiableAt 𝕜 f x ∧ fderiv 𝕜 f x ∈ K} x
e : ℕ
this : 0 < (1 / 2) ^ e
R : ℝ
R_pos : R > 0
hR : ∀ r ∈ Ioo 0 R, x ∈ A f (fderiv 𝕜 f x) r ((1 / 2) ^ e)
n : ℕ
hn : (1 / 2) ^ n < R
refine_1 : ∀ (p : ℕ) (hp : p ≥ n) (q : ℕ) (hq : q ≥ n), x ∈ A f (fderiv 𝕜 f x) ((1 / 2) ^ p) ((1 / 2) ^ e)
refine_2 : ∀ (p : ℕ) (hp : p ≥ n) (q : ℕ) (hq : q ≥ n), x ∈ A f (fderiv 𝕜 f x) ((1 / 2) ^ q) ((1 / 2) ^ e)
⊢ ∃ (i : ℕ),
    ∀ (i_1 : ℕ) (i_2 : i_1 ≥ i) (i_3 : ℕ) (i : i_3 ≥ i),
      ∃ (i : E →L[𝕜] F) (_ : i ∈ K), x ∈ A f i ((1 / 2) ^ i_1) ((1 / 2) ^ e) ∧ x ∈ A f i ((1 / 2) ^ i_3) ((1 / 2) ^ e)

Tactic:
refine ⟨n, fun p hp q hq => ⟨fderiv 𝕜 f x, hx.2, ⟨?_, ?_⟩⟩⟩
-- 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)
x : E
hx : Membership.mem (γ := Set E) {x : E | DifferentiableAt 𝕜 f x ∧ fderiv 𝕜 f x ∈ K} x
e : ℕ
this : 0 < (1 / 2) ^ e
R : ℝ
R_pos : R > 0
hR : ∀ r ∈ Ioo 0 R, x ∈ A f (fderiv 𝕜 f x) r ((1 / 2) ^ e)
n : ℕ
hn : (1 / 2) ^ n < R
p : ℕ
hp : p ≥ n
q : ℕ
hq : q ≥ n
⊢ 0 < (1 / 2) ^ p

Tactic:
positivity
-- 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)
x : E
hx : Membership.mem (γ := Set E) {x : E | DifferentiableAt 𝕜 f x ∧ fderiv 𝕜 f x ∈ K} x
e : ℕ
this : 0 < (1 / 2) ^ e
R : ℝ
R_pos : R > 0
hR : ∀ r ∈ Ioo 0 R, x ∈ A f (fderiv 𝕜 f x) r ((1 / 2) ^ e)
n : ℕ
hn : (1 / 2) ^ n < R
p : ℕ
hp : p ≥ n
q : ℕ
hq : q ≥ n
refine_1 : (1 / 2) ^ p ≤ (1 / 2) ^ n
⊢ x ∈ A f (fderiv 𝕜 f x) ((1 / 2) ^ p) ((1 / 2) ^ e)

Tactic:
refine hR _ ⟨by positivity, lt_of_le_of_lt ?_ hn⟩

Premises used: lt_of_le_of_lt, And.intro
-- 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)
x : E
hx : Membership.mem (γ := Set E) {x : E | DifferentiableAt 𝕜 f x ∧ fderiv 𝕜 f x ∈ K} x
e : ℕ
this : 0 < (1 / 2) ^ e
R : ℝ
R_pos : R > 0
hR : ∀ r ∈ Ioo 0 R, x ∈ A f (fderiv 𝕜 f x) r ((1 / 2) ^ e)
n : ℕ
hn : (1 / 2) ^ n < R
p : ℕ
hp : p ≥ n
q : ℕ
hq : q ≥ n
⊢ 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)
x : E
hx : Membership.mem (γ := Set E) {x : E | DifferentiableAt 𝕜 f x ∧ fderiv 𝕜 f x ∈ K} x
e : ℕ
this : 0 < (1 / 2) ^ e
R : ℝ
R_pos : R > 0
hR : ∀ r ∈ Ioo 0 R, x ∈ A f (fderiv 𝕜 f x) r ((1 / 2) ^ e)
n : ℕ
hn : (1 / 2) ^ n < R
p : ℕ
hp : p ≥ n
q : ℕ
hq : q ≥ n
⊢ 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)
x : E
hx : Membership.mem (γ := Set E) {x : E | DifferentiableAt 𝕜 f x ∧ fderiv 𝕜 f x ∈ K} x
e : ℕ
this : 0 < (1 / 2) ^ e
R : ℝ
R_pos : R > 0
hR : ∀ r ∈ Ioo 0 R, x ∈ A f (fderiv 𝕜 f x) r ((1 / 2) ^ e)
n : ℕ
hn : (1 / 2) ^ n < R
p : ℕ
hp : p ≥ n
q : ℕ
hq : q ≥ n
⊢ n ≤ p

Tactic:
assumption
-- 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)
x : E
hx : Membership.mem (γ := Set E) {x : E | DifferentiableAt 𝕜 f x ∧ fderiv 𝕜 f x ∈ K} x
e : ℕ
this : 0 < (1 / 2) ^ e
R : ℝ
R_pos : R > 0
hR : ∀ r ∈ Ioo 0 R, x ∈ A f (fderiv 𝕜 f x) r ((1 / 2) ^ e)
n : ℕ
hn : (1 / 2) ^ n < R
p : ℕ
hp : p ≥ n
q : ℕ
hq : q ≥ n
⊢ (1 / 2) ^ p ≤ (1 / 2) ^ n

Tactic:
exact pow_le_pow_of_le_one (by norm_num) (by norm_num) (by assumption)

Premises used: pow_le_pow_of_le_one
-- 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)
x : E
hx : Membership.mem (γ := Set E) {x : E | DifferentiableAt 𝕜 f x ∧ fderiv 𝕜 f x ∈ K} x
e : ℕ
this : 0 < (1 / 2) ^ e
R : ℝ
R_pos : R > 0
hR : ∀ r ∈ Ioo 0 R, x ∈ A f (fderiv 𝕜 f x) r ((1 / 2) ^ e)
n : ℕ
hn : (1 / 2) ^ n < R
p : ℕ
hp : p ≥ n
q : ℕ
hq : q ≥ n
⊢ 0 < (1 / 2) ^ q

Tactic:
positivity
-- 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)
x : E
hx : Membership.mem (γ := Set E) {x : E | DifferentiableAt 𝕜 f x ∧ fderiv 𝕜 f x ∈ K} x
e : ℕ
this : 0 < (1 / 2) ^ e
R : ℝ
R_pos : R > 0
hR : ∀ r ∈ Ioo 0 R, x ∈ A f (fderiv 𝕜 f x) r ((1 / 2) ^ e)
n : ℕ
hn : (1 / 2) ^ n < R
p : ℕ
hp : p ≥ n
q : ℕ
hq : q ≥ n
refine_2 : (1 / 2) ^ q ≤ (1 / 2) ^ n
⊢ x ∈ A f (fderiv 𝕜 f x) ((1 / 2) ^ q) ((1 / 2) ^ e)

Tactic:
refine hR _ ⟨by positivity, lt_of_le_of_lt ?_ hn⟩

Premises used: lt_of_le_of_lt, And.intro
-- 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)
x : E
hx : Membership.mem (γ := Set E) {x : E | DifferentiableAt 𝕜 f x ∧ fderiv 𝕜 f x ∈ K} x
e : ℕ
this : 0 < (1 / 2) ^ e
R : ℝ
R_pos : R > 0
hR : ∀ r ∈ Ioo 0 R, x ∈ A f (fderiv 𝕜 f x) r ((1 / 2) ^ e)
n : ℕ
hn : (1 / 2) ^ n < R
p : ℕ
hp : p ≥ n
q : ℕ
hq : q ≥ n
⊢ n ≤ q

Tactic:
assumption
-- 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)
x : E
hx : Membership.mem (γ := Set E) {x : E | DifferentiableAt 𝕜 f x ∧ fderiv 𝕜 f x ∈ K} x
e : ℕ
this : 0 < (1 / 2) ^ e
R : ℝ
R_pos : R > 0
hR : ∀ r ∈ Ioo 0 R, x ∈ A f (fderiv 𝕜 f x) r ((1 / 2) ^ e)
n : ℕ
hn : (1 / 2) ^ n < R
p : ℕ
hp : p ≥ n
q : ℕ
hq : q ≥ n
⊢ (1 / 2) ^ q ≤ (1 / 2) ^ n

Tactic:
exact pow_le_pow_of_le_one (by norm_num) (by norm_num) (by assumption)

Premises used: pow_le_pow_of_le_one
-- 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
n : ℕ
⊢ 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
⊢ ∀ {n : ℕ}, 0 < (1 / 2) ^ n

Tactic:
have P : ∀ {n : ℕ}, (0 : ℝ) < (1 / 2) ^ n := fun {n} => pow_pos (by norm_num) n
-- 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
rcases : ∀ (c : 𝕜) (hc : 1 < ‖c‖), D f K ⊆ {x : E | DifferentiableAt 𝕜 f x ∧ fderiv 𝕜 f x ∈ K}
⊢ D f K ⊆ {x : E | DifferentiableAt 𝕜 f x ∧ fderiv 𝕜 f x ∈ K}

Tactic:
rcases NormedField.exists_one_lt_norm 𝕜 with ⟨c, hc⟩

Premises used: NormedField.exists_one_lt_norm
-- 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
e : ℕ
⊢ Membership.mem (γ := Set E)
    (⋃ (n : ℕ), ⋂ (p : ℕ), ⋂ (_ : p ≥ n), ⋂ (q : ℕ), ⋂ (_ : q ≥ n), B f K ((1 / 2) ^ p) ((1 / 2) ^ q) ((1 / 2) ^ e)) x

Tactic:
have := mem_iInter.1 hx e
-- 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
e : ℕ
this :
  Membership.mem (γ := Set E)
    (⋃ (n : ℕ), ⋂ (p : ℕ), ⋂ (_ : p ≥ n), ⋂ (q : ℕ), ⋂ (_ : q ≥ n), B f K ((1 / 2) ^ p) ((1 / 2) ^ q) ((1 / 2) ^ e)) x
rcases :
  ∀ (n : ℕ)
    (hn :
      Membership.mem (γ := Set E)
        (⋂ (p : ℕ), ⋂ (_ : p ≥ n), ⋂ (q : ℕ), ⋂ (_ : q ≥ n), B f K ((1 / 2) ^ p) ((1 / 2) ^ q) ((1 / 2) ^ e)) x),
    ∃ (n : ℕ),
      ∀ (p q : ℕ), n ≤ p → n ≤ q → ∃ L ∈ K, x ∈ A f L ((1 / 2) ^ p) ((1 / 2) ^ e) ∩ A f L ((1 / 2) ^ q) ((1 / 2) ^ e)
⊢ ∃ (n : ℕ),
    ∀ (p q : ℕ), n ≤ p → n ≤ q → ∃ L ∈ K, x ∈ A f L ((1 / 2) ^ p) ((1 / 2) ^ e) ∩ A f L ((1 / 2) ^ q) ((1 / 2) ^ e)

Tactic:
rcases mem_iUnion.1 this with ⟨n, hn⟩

Premises used: Set.mem_iUnion, 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
e : ℕ
this :
  Membership.mem (γ := Set E)
    (⋃ (n : ℕ), ⋂ (p : ℕ), ⋂ (_ : p ≥ n), ⋂ (q : ℕ), ⋂ (_ : q ≥ n), B f K ((1 / 2) ^ p) ((1 / 2) ^ q) ((1 / 2) ^ e)) x
n : ℕ
hn :
  Membership.mem (γ := Set E)
    (⋂ (p : ℕ), ⋂ (_ : p ≥ n), ⋂ (q : ℕ), ⋂ (_ : q ≥ n), B f K ((1 / 2) ^ p) ((1 / 2) ^ q) ((1 / 2) ^ e)) x
⊢ ∃ (n : ℕ),
    ∀ (p q : ℕ), n ≤ p → n ≤ q → ∃ L ∈ K, x ∈ A f L ((1 / 2) ^ p) ((1 / 2) ^ e) ∩ A f L ((1 / 2) ^ q) ((1 / 2) ^ e)

Tactic:
⟨n, fun p q hp hq => ?_⟩

Premises used: Exists.intro
-- 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
e : ℕ
this :
  Membership.mem (γ := Set E)
    (⋃ (n : ℕ), ⋂ (p : ℕ), ⋂ (_ : p ≥ n), ⋂ (q : ℕ), ⋂ (_ : q ≥ n), B f K ((1 / 2) ^ p) ((1 / 2) ^ q) ((1 / 2) ^ e)) x
n : ℕ
hn :
  Membership.mem (γ := Set E)
    (⋂ (p : ℕ), ⋂ (_ : p ≥ n), ⋂ (q : ℕ), ⋂ (_ : q ≥ n), B f K ((1 / 2) ^ p) ((1 / 2) ^ q) ((1 / 2) ^ e)) x
refine :
  ∀ (p q : ℕ) (hp : n ≤ p) (hq : n ≤ q),
    ∃ L ∈ K, x ∈ A f L ((1 / 2) ^ p) ((1 / 2) ^ e) ∩ A f L ((1 / 2) ^ q) ((1 / 2) ^ e)
⊢ ∃ (n : ℕ),
    ∀ (p q : ℕ), n ≤ p → n ≤ q → ∃ L ∈ K, x ∈ A f L ((1 / 2) ^ p) ((1 / 2) ^ e) ∩ A f L ((1 / 2) ^ q) ((1 / 2) ^ e)

Tactic:
refine ⟨n, fun p q hp hq => ?_⟩
-- 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
e : ℕ
this :
  Membership.mem (γ := Set E)
    (⋃ (n : ℕ), ⋂ (p : ℕ), ⋂ (_ : p ≥ n), ⋂ (q : ℕ), ⋂ (_ : q ≥ n), B f K ((1 / 2) ^ p) ((1 / 2) ^ q) ((1 / 2) ^ e)) x
n : ℕ
hn :
  Membership.mem (γ := Set E)
    (⋂ (p : ℕ), ⋂ (_ : p ≥ n), ⋂ (q : ℕ), ⋂ (_ : q ≥ n), B f K ((1 / 2) ^ p) ((1 / 2) ^ q) ((1 / 2) ^ e)) x
p q : ℕ
hp : n ≤ p
hq : n ≤ q
⊢ ∀ (i : ℕ) (i_1 : i ≥ n) (i_2 : ℕ) (i_3 : i_2 ≥ n), x ∈ B f K ((1 / 2) ^ i) ((1 / 2) ^ i_2) ((1 / 2) ^ e)

Tactic:
simp only [mem_iInter] at hn

Premises used: Set.mem_iInter
-- 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
e : ℕ
this :
  Membership.mem (γ := Set E)
    (⋃ (n : ℕ), ⋂ (p : ℕ), ⋂ (_ : p ≥ n), ⋂ (q : ℕ), ⋂ (_ : q ≥ n), B f K ((1 / 2) ^ p) ((1 / 2) ^ q) ((1 / 2) ^ e)) x
n p q : ℕ
hp : n ≤ p
hq : n ≤ q
hn : ∀ (i : ℕ) (i_1 : i ≥ n) (i_2 : ℕ) (i_3 : i_2 ≥ n), x ∈ B f K ((1 / 2) ^ i) ((1 / 2) ^ i_2) ((1 / 2) ^ e)
rcases :
  ∀ (L : E →L[𝕜] F)
    (hL :
      Membership.mem (γ := Set E) (⋃ (_ : L ∈ K), A f L ((1 / 2) ^ p) ((1 / 2) ^ e) ∩ A f L ((1 / 2) ^ q) ((1 / 2) ^ e))
        x),
    ∃ L ∈ K, x ∈ A f L ((1 / 2) ^ p) ((1 / 2) ^ e) ∩ A f L ((1 / 2) ^ q) ((1 / 2) ^ e)
⊢ ∃ L ∈ K, x ∈ A f L ((1 / 2) ^ p) ((1 / 2) ^ e) ∩ A f L ((1 / 2) ^ q) ((1 / 2) ^ e)

Tactic:
rcases mem_iUnion.1 (hn p hp q hq) with ⟨L, hL⟩

Premises used: Set.mem_iUnion, 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
e : ℕ
this :
  Membership.mem (γ := Set E)
    (⋃ (n : ℕ), ⋂ (p : ℕ), ⋂ (_ : p ≥ n), ⋂ (q : ℕ), ⋂ (_ : q ≥ n), B f K ((1 / 2) ^ p) ((1 / 2) ^ q) ((1 / 2) ^ e)) x
n p q : ℕ
hp : n ≤ p
hq : n ≤ q
hn : ∀ (i : ℕ) (i_1 : i ≥ n) (i_2 : ℕ) (i_3 : i_2 ≥ n), x ∈ B f K ((1 / 2) ^ i) ((1 / 2) ^ i_2) ((1 / 2) ^ e)
L : E →L[𝕜] F
hL :
  Membership.mem (γ := Set E) (⋃ (_ : L ∈ K), A f L ((1 / 2) ^ p) ((1 / 2) ^ e) ∩ A f L ((1 / 2) ^ q) ((1 / 2) ^ e)) x
⊢ ∃ L ∈ K, x ∈ A f L ((1 / 2) ^ p) ((1 / 2) ^ e) ∩ A f L ((1 / 2) ^ q) ((1 / 2) ^ e)

Tactic:
⟨L, exists_prop.mp <| mem_iUnion.1 hL⟩

Premises used: Set.mem_iUnion, exists_prop, Exists.intro, 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
e : ℕ
this :
  Membership.mem (γ := Set E)
    (⋃ (n : ℕ), ⋂ (p : ℕ), ⋂ (_ : p ≥ n), ⋂ (q : ℕ), ⋂ (_ : q ≥ n), B f K ((1 / 2) ^ p) ((1 / 2) ^ q) ((1 / 2) ^ e)) x
n p q : ℕ
hp : n ≤ p
hq : n ≤ q
hn : ∀ (i : ℕ) (i_1 : i ≥ n) (i_2 : ℕ) (i_3 : i_2 ≥ n), x ∈ B f K ((1 / 2) ^ i) ((1 / 2) ^ i_2) ((1 / 2) ^ e)
L : E →L[𝕜] F
hL :
  Membership.mem (γ := Set E) (⋃ (_ : L ∈ K), A f L ((1 / 2) ^ p) ((1 / 2) ^ e) ∩ A f L ((1 / 2) ^ q) ((1 / 2) ^ e)) x
⊢ ∃ L ∈ K, x ∈ A f L ((1 / 2) ^ p) ((1 / 2) ^ e) ∩ A f L ((1 / 2) ^ q) ((1 / 2) ^ e)

Tactic:
exact ⟨L, exists_prop.mp <| mem_iUnion.1 hL⟩
-- 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
⊢ ∀ (e : ℕ),
    ∃ (n : ℕ),
      ∀ (p q : ℕ), n ≤ p → n ≤ q → ∃ L ∈ K, x ∈ A f L ((1 / 2) ^ p) ((1 / 2) ^ e) ∩ A f L ((1 / 2) ^ q) ((1 / 2) ^ e)

Tactic:
have :
    ∀ e : ℕ, ∃ n : ℕ, ∀ p q, n ≤ p → n ≤ q →
      ∃ L ∈ K, x ∈ A f L ((1 / 2) ^ p) ((1 / 2) ^ e) ∩ A f L ((1 / 2) ^ q) ((1 / 2) ^ e) := by
    intro e
    have := mem_iInter.1 hx e
    rcases mem_iUnion.1 this with ⟨n, hn⟩
    refine ⟨n, fun p q hp hq => ?_⟩
    simp only [mem_iInter] at hn
    rcases mem_iUnion.1 (hn p hp q hq) with ⟨L, hL⟩
    exact ⟨L, exists_prop.mp <| mem_iUnion.1 hL⟩
-- 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
this :
  ∀ (e : ℕ),
    ∃ (n : ℕ),
      ∀ (p q : ℕ), n ≤ p → n ≤ q → ∃ L ∈ K, x ∈ A f L ((1 / 2) ^ p) ((1 / 2) ^ e) ∩ A f L ((1 / 2) ^ q) ((1 / 2) ^ e)
choose! :
  ∀ (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)),
    Membership.mem (γ := Set E) {x : E | DifferentiableAt 𝕜 f x ∧ fderiv 𝕜 f x ∈ K} x
⊢ Membership.mem (γ := Set E) {x : E | DifferentiableAt 𝕜 f x ∧ fderiv 𝕜 f x ∈ K} x

Tactic:
choose! n L hn using this
-- 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)
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 := max (n e) (n e')
⊢ 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)
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 := max (n e) (n e')
⊢ 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)
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 := max (n e) (n e')
⊢ (1 / 2) ^ e' ≤ (1 / 2) ^ e

Tactic:
have I : ((1 : ℝ) / 2) ^ e' ≤ (1 / 2) ^ e :=
      pow_le_pow_of_le_one (by norm_num) (by norm_num) he'
-- 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)
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 := max (n e) (n e')
I : (1 / 2) ^ e' ≤ (1 / 2) ^ e
⊢ x ∈ A f (L e p q) ((1 / 2) ^ p) ((1 / 2) ^ e)

Tactic:
have I1 : x ∈ A f (L e p q) ((1 / 2) ^ p) ((1 / 2) ^ e) := (hn e p q hp hq).2.1
-- 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)
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 := max (n e) (n e')
I : (1 / 2) ^ e' ≤ (1 / 2) ^ e
I1 : x ∈ A f (L e p q) ((1 / 2) ^ p) ((1 / 2) ^ e)
⊢ x ∈ A f (L e p r) ((1 / 2) ^ p) ((1 / 2) ^ e)

Tactic:
have I2 : x ∈ A f (L e p r) ((1 / 2) ^ p) ((1 / 2) ^ e) := (hn e p r hp (le_max_left _ _)).2.1
-- 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)
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 := max (n e) (n e')
I : (1 / 2) ^ e' ≤ (1 / 2) ^ e
I1 : x ∈ A f (L e p q) ((1 / 2) ^ p) ((1 / 2) ^ e)
I2 : x ∈ A f (L e p r) ((1 / 2) ^ p) ((1 / 2) ^ e)
⊢ ‖L e p q - L e p r‖ ≤ 4 * ‖c‖ * (1 / 2) ^ e

Tactic:
norm_sub_le_of_mem_A hc P P I1 I2

Premises used: FDerivMeasurableAux.norm_sub_le_of_mem_A
-- 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)
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 := max (n e) (n e')
I : (1 / 2) ^ e' ≤ (1 / 2) ^ e
I1 : x ∈ A f (L e p q) ((1 / 2) ^ p) ((1 / 2) ^ e)
I2 : x ∈ A f (L e p r) ((1 / 2) ^ p) ((1 / 2) ^ e)
⊢ ‖L e p q - L e p r‖ ≤ 4 * ‖c‖ * (1 / 2) ^ e

Tactic:
exact norm_sub_le_of_mem_A hc P P I1 I2

Premises used: FDerivMeasurableAux.norm_sub_le_of_mem_A