mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.Analysis.Calculus.IteratedDeriv.Lemmas

Goal state before:
pp: case succ
𝕜 : Type u_1
inst✝² : NontriviallyNormedField 𝕜
F : Type u_2
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace 𝕜 F
s : Set 𝕜
h : UniqueDiffOn 𝕜 s
f : 𝕜 → F
c : 𝕜
hs : Set.MapsTo (fun x ↦ c * x) s s
n : ℕ
ih :
  ∀ {x : 𝕜},
    x ∈ s →
      ContDiffOn 𝕜 (↑n) f s → iteratedDerivWithin n (fun x ↦ f (c * x)) s x = c ^ n • iteratedDerivWithin n f s (c * x)
x : 𝕜
hx : x ∈ s
hf : ContDiffOn 𝕜 (↑(n + 1)) f s
hcx : c * x ∈ s
h₀ : Set.EqOn (iteratedDerivWithin n (fun x ↦ f (c * x)) s) (fun x ↦ c ^ n • iteratedDerivWithin n f s (c * x)) s
h₁ : DifferentiableWithinAt 𝕜 (iteratedDerivWithin n f s) s (c * x)
h₂ : DifferentiableWithinAt 𝕜 (fun x ↦ iteratedDerivWithin n f s (c * x)) s x
⊢ c ^ n • derivWithin (iteratedDerivWithin n f s ∘ HMul.hMul c) s x =
    c ^ (n + 1) • derivWithin (iteratedDerivWithin n f s) s (c * x)

Tactic:
derivWithin.scomp x h₁ (by fun_prop) hs,

Goal state after:
case succ
𝕜 : Type u_1
inst✝² : NontriviallyNormedField 𝕜
F : Type u_2
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace 𝕜 F
s : Set 𝕜
h : UniqueDiffOn 𝕜 s
f : 𝕜 → F
c : 𝕜
hs : Set.MapsTo (fun x ↦ c * x) s s
n : ℕ
ih :
  ∀ {x : 𝕜},
    x ∈ s →
      ContDiffOn 𝕜 (↑n) f s → iteratedDerivWithin n (fun x ↦ f (c * x)) s x = c ^ n • iteratedDerivWithin n f s (c * x)
x : 𝕜
hx : x ∈ s
hf : ContDiffOn 𝕜 (↑(n + 1)) f s
hcx : c * x ∈ s
h₀ : Set.EqOn (iteratedDerivWithin n (fun x ↦ f (c * x)) s) (fun x ↦ c ^ n • iteratedDerivWithin n f s (c * x)) s
h₁ : DifferentiableWithinAt 𝕜 (iteratedDerivWithin n f s) s (c * x)
h₂ : DifferentiableWithinAt 𝕜 (fun x ↦ iteratedDerivWithin n f s (c * x)) s x
⊢ c ^ n • derivWithin (HMul.hMul c) s x • derivWithin (iteratedDerivWithin n f s) s (c * x) =
    c ^ (n + 1) • derivWithin (iteratedDerivWithin n f s) s (c * x)
Module: Mathlib.Analysis.Calculus.ContDiff.Operations

Goal state before:
pp: case pos
𝕜 : Type u_1
inst✝⁴ : NontriviallyNormedField 𝕜
ι : Type u_3
inst✝³ : Fintype ι
F' : ι → Type u_5
inst✝² : (i : ι) → NormedAddCommGroup (F' i)
inst✝¹ : (i : ι) → NormedSpace 𝕜 (F' i)
inst✝ : DecidableEq ι
k : ℕ∞ω
x : (i : ι) → F' i
i j : ι
h : j = i
⊢ ContDiff 𝕜 k fun x ↦ ⋯ ▸ x

Tactic:
subst h

Goal state after:
case pos
𝕜 : Type u_1
inst✝⁴ : NontriviallyNormedField 𝕜
ι : Type u_3
inst✝³ : Fintype ι
F' : ι → Type u_5
inst✝² : (i : ι) → NormedAddCommGroup (F' i)
inst✝¹ : (i : ι) → NormedSpace 𝕜 (F' i)
inst✝ : DecidableEq ι
k : ℕ∞ω
x : (i : ι) → F' i
j : ι
⊢ ContDiff 𝕜 k fun x ↦ ⋯ ▸ x
Module: Mathlib.Analysis.Calculus.ContDiff.FaaDiBruno

Goal state before:
pp: case e_6.inr
𝕜 : Type u_1
inst✝⁶ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace 𝕜 E
F : Type u_3
inst✝³ : NormedAddCommGroup F
inst✝² : NormedSpace 𝕜 F
G : Type u_4
inst✝¹ : NormedAddCommGroup G
inst✝ : NormedSpace 𝕜 G
m : ℕ
q : FormalMultilinearSeries 𝕜 F G
p : FormalMultilinearSeries 𝕜 E F
c : OrderedFinpartition m
i : Fin c.length
e : E
v : Fin m → E
j : Fin c.length
hij : j ≠ i
⊢ (p (update c.partSize i (c.partSize i + 1) j))
      (cons e v ∘
        if h : j = i then (fun i_1 ↦ cases 0 (succ ∘ c.emb i) i_1) ∘ Fin.cast ⋯ else succ ∘ c.emb j ∘ Fin.cast ⋯) =
    (update (fun i ↦ p (c.partSize i)) i ((p (c.partSize i + 1)).curryLeft e) j) (v ∘ c.emb j)

Tactic:
simp only [hij, ↓reduceDIte, ne_eq, not_false_eq_true, update_of_ne]

Goal state after:
case e_6.inr
𝕜 : Type u_1
inst✝⁶ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace 𝕜 E
F : Type u_3
inst✝³ : NormedAddCommGroup F
inst✝² : NormedSpace 𝕜 F
G : Type u_4
inst✝¹ : NormedAddCommGroup G
inst✝ : NormedSpace 𝕜 G
m : ℕ
q : FormalMultilinearSeries 𝕜 F G
p : FormalMultilinearSeries 𝕜 E F
c : OrderedFinpartition m
i : Fin c.length
e : E
v : Fin m → E
j : Fin c.length
hij : j ≠ i
⊢ (p (update c.partSize i (c.partSize i + 1) j)) (cons e v ∘ succ ∘ c.emb j ∘ Fin.cast ⋯) =
    (p (c.partSize j)) (v ∘ c.emb j)
Module: Mathlib.Analysis.Calculus.ContDiff.Operations

Goal state before:
pp: 𝕜 : Type u_1
inst✝⁴ : NontriviallyNormedField 𝕜
E : Type uE
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace 𝕜 E
F : Type uF
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace 𝕜 F
s : Set E
f : E → F
hu : UniqueDiffOn 𝕜 s
i : ℕ
hi : ∀ {x : E}, x ∈ s → iteratedFDerivWithin 𝕜 i (-f) s x = -iteratedFDerivWithin 𝕜 i f s x
x : E
hx : x ∈ s
h : Fin (i + 1) → E
⊢ -((fderivWithin 𝕜 (iteratedFDerivWithin 𝕜 i f s) s x) (h 0)) (tail h) = -(iteratedFDerivWithin 𝕜 (i + 1) f s x) h

Tactic:
by
        rw [iteratedFDerivWithin_succ_apply_left]

Goal state after:
Module: Mathlib.Geometry.Convex.Cone.Basic

Goal state before:
pp: 𝕜 : Type u_1
R : Type u_2
G : Type u_3
M : Type u_4
N : Type u_5
O : Type u_6
inst✝⁴ : Field 𝕜
inst✝³ : LinearOrder 𝕜
inst✝² : IsStrictOrderedRing 𝕜
inst✝¹ : AddCommGroup M
inst✝ : Module 𝕜 M
s : Set M
hs : Convex 𝕜 s
⊢ ConvexCone 𝕜 M

Tactic:
apply ConvexCone.mk (⋃ (c : 𝕜) (_ : 0 < c), c • s) <;> simp only [mem_iUnion, mem_smul_set]

Goal state after:
case smul_mem'
𝕜 : Type u_1
R : Type u_2
G : Type u_3
M : Type u_4
N : Type u_5
O : Type u_6
inst✝⁴ : Field 𝕜
inst✝³ : LinearOrder 𝕜
inst✝² : IsStrictOrderedRing 𝕜
inst✝¹ : AddCommGroup M
inst✝ : Module 𝕜 M
s : Set M
hs : Convex 𝕜 s
⊢ ∀ ⦃c : 𝕜⦄, 0 < c → ∀ ⦃x : M⦄, (∃ i, ∃ (_ : 0 < i), ∃ y ∈ s, i • y = x) → ∃ i, ∃ (_ : 0 < i), ∃ y ∈ s, i • y = c • x

case add_mem'
𝕜 : Type u_1
R : Type u_2
G : Type u_3
M : Type u_4
N : Type u_5
O : Type u_6
inst✝⁴ : Field 𝕜
inst✝³ : LinearOrder 𝕜
inst✝² : IsStrictOrderedRing 𝕜
inst✝¹ : AddCommGroup M
inst✝ : Module 𝕜 M
s : Set M
hs : Convex 𝕜 s
⊢ ∀ ⦃x : M⦄,
    (∃ i, ∃ (_ : 0 < i), ∃ y ∈ s, i • y = x) →
      ∀ ⦃y : M⦄, (∃ i, ∃ (_ : 0 < i), ∃ y_1 ∈ s, i • y_1 = y) → ∃ i, ∃ (_ : 0 < i), ∃ y_1 ∈ s, i • y_1 = x + y
Module: Mathlib.Geometry.Convex.Cone.Basic

Goal state before:
pp: 𝕜 : Type u_1
M : Type u_4
inst✝⁴ : Field 𝕜
inst✝³ : LinearOrder 𝕜
inst✝² : IsStrictOrderedRing 𝕜
inst✝¹ : AddCommGroup M
inst✝ : Module 𝕜 M
s : Set M
t : ConvexCone 𝕜 M
⊢ t ∈ {t | s ⊆ ↑t} ↔ t ∈ {t | (convexHull 𝕜) s ⊆ ↑t}

Tactic:
exact ⟨fun h => convexHull_min h t.convex, (subset_convexHull 𝕜 s).trans⟩

Goal state after:
Module: Mathlib.MeasureTheory.Integral.Bochner.VitaliCaratheodory

Goal state before:
pp: case refine_1
α : Type u_1
inst✝³ : TopologicalSpace α
inst✝² : MeasurableSpace α
inst✝¹ : BorelSpace α
μ : Measure α
inst✝ : μ.WeaklyRegular
f : α → ℝ≥0
fint : Integrable (fun x ↦ ↑(f x)) μ
ε : ℝ≥0
εpos : 0 < ↑ε
If : ∫⁻ (x : α), ↑(f x) ∂μ < ∞
g : α → ℝ≥0
gf : ∀ (x : α), g x ≤ f x
gcont : UpperSemicontinuous g
gint : ∫⁻ (x : α), ↑(f x) ∂μ ≤ ∫⁻ (x : α), ↑(g x) ∂μ + ↑ε
Ig : ∫⁻ (x : α), ↑(g x) ∂μ < ∞
⊢ Integrable (fun x ↦ ↑(g x)) μ

Tactic:
refine
      Integrable.mono fint gcont.measurable.coe_nnreal_real.aemeasurable.aestronglyMeasurable ?_
    exact Filter.Eventually.of_forall fun x => by simp [gf x]

Goal state after:
Module: Mathlib.Analysis.Calculus.FDeriv.Measurable

Goal state before:
pp: case neg
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
⊢ ‖f (x + y) - f x - f' y‖ ≤ 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)

Goal state after:
case neg
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‖
Module: Mathlib.Analysis.LocallyConvex.SeparatingDual

Goal state before:
pp: R : Type u_1
V : Type u_2
inst✝⁶ : Ring R
inst✝⁵ : AddCommGroup V
inst✝⁴ : TopologicalSpace V
inst✝³ : TopologicalSpace R
inst✝² : Module R V
inst✝¹ : SeparatingDual R V
inst✝ : T1Space R
⊢ T1Space V

Tactic:
apply t1Space_iff_exists_open.2 (fun x y hxy ↦ ?_)
  rcases exists_separating_of_ne (R := R) hxy with ⟨f, hf⟩
  exact ⟨f ⁻¹' {f y}ᶜ, isOpen_compl_singleton.preimage f.continuous, hf, by simp⟩

Goal state after:
Module: Mathlib.Analysis.LocallyConvex.SeparatingDual

Goal state before:
pp: R : Type u_1
V : Type u_2
inst✝⁹ : Field R
inst✝⁸ : AddCommGroup V
inst✝⁷ : TopologicalSpace R
inst✝⁶ : TopologicalSpace V
inst✝⁵ : IsTopologicalRing R
inst✝⁴ : Module R V
inst✝³ : SeparatingDual R V
W : Type u_3
inst✝² : AddCommGroup W
inst✝¹ : Module R W
inst✝ : FiniteDimensional R W
f : W →ₗ[R] V
⊢ Surjective (⇑f.dualMap ∘ ContinuousLinearMap.toLinearMap) ↔ Injective ⇑f

Tactic:
constructor <;> intro hf

Goal state after:
case mp
R : Type u_1
V : Type u_2
inst✝⁹ : Field R
inst✝⁸ : AddCommGroup V
inst✝⁷ : TopologicalSpace R
inst✝⁶ : TopologicalSpace V
inst✝⁵ : IsTopologicalRing R
inst✝⁴ : Module R V
inst✝³ : SeparatingDual R V
W : Type u_3
inst✝² : AddCommGroup W
inst✝¹ : Module R W
inst✝ : FiniteDimensional R W
f : W →ₗ[R] V
hf : Surjective (⇑f.dualMap ∘ ContinuousLinearMap.toLinearMap)
⊢ Injective ⇑f

case mpr
R : Type u_1
V : Type u_2
inst✝⁹ : Field R
inst✝⁸ : AddCommGroup V
inst✝⁷ : TopologicalSpace R
inst✝⁶ : TopologicalSpace V
inst✝⁵ : IsTopologicalRing R
inst✝⁴ : Module R V
inst✝³ : SeparatingDual R V
W : Type u_3
inst✝² : AddCommGroup W
inst✝¹ : Module R W
inst✝ : FiniteDimensional R W
f : W →ₗ[R] V
hf : Injective ⇑f
⊢ Surjective (⇑f.dualMap ∘ ContinuousLinearMap.toLinearMap)
Module: Mathlib.Analysis.LocallyConvex.Separation

Goal state before:
pp: case inl
E : Type u_2
inst✝⁵ : TopologicalSpace E
inst✝⁴ : AddCommGroup E
inst✝³ : Module ℝ E
t : Set E
inst✝² : IsTopologicalAddGroup E
inst✝¹ : ContinuousSMul ℝ E
inst✝ : LocallyConvexSpace ℝ E
ht₁ : Convex ℝ t
ht₂ : IsClosed t
hs₁ : Convex ℝ ∅
hs₂ : IsCompact ∅
disj : Disjoint ∅ t
⊢ ∃ f u v, (∀ a ∈ ∅, f a < u) ∧ u < v ∧ ∀ b ∈ t, v < f b

pp: case inr
E : Type u_2
inst✝⁵ : TopologicalSpace E
inst✝⁴ : AddCommGroup E
inst✝³ : Module ℝ E
s t : Set E
inst✝² : IsTopologicalAddGroup E
inst✝¹ : ContinuousSMul ℝ E
inst✝ : LocallyConvexSpace ℝ E
hs₁ : Convex ℝ s
hs₂ : IsCompact s
ht₁ : Convex ℝ t
ht₂ : IsClosed t
disj : Disjoint s t
hs : s.Nonempty
⊢ ∃ f u v, (∀ a ∈ s, f a < u) ∧ u < v ∧ ∀ b ∈ t, v < f b

Tactic:
· exact ⟨0, -2, -1, by simp⟩

Goal state after:
case inr
E : Type u_2
inst✝⁵ : TopologicalSpace E
inst✝⁴ : AddCommGroup E
inst✝³ : Module ℝ E
s t : Set E
inst✝² : IsTopologicalAddGroup E
inst✝¹ : ContinuousSMul ℝ E
inst✝ : LocallyConvexSpace ℝ E
hs₁ : Convex ℝ s
hs₂ : IsCompact s
ht₁ : Convex ℝ t
ht₂ : IsClosed t
disj : Disjoint s t
hs : s.Nonempty
⊢ ∃ f u v, (∀ a ∈ s, f a < u) ∧ u < v ∧ ∀ b ∈ t, v < f b
Module: Mathlib.Analysis.Calculus.FDeriv.Measurable

Goal state before:
pp: 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
this : HasFDerivAt f f' x
⊢ x ∈ {x | DifferentiableAt 𝕜 f x ∧ fderiv 𝕜 f x ∈ K}

Tactic:
rw [← this.fderiv] at f'K

Goal state after:
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 : fderiv 𝕜 f x ∈ 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
this : HasFDerivAt f f' x
⊢ x ∈ {x | DifferentiableAt 𝕜 f x ∧ fderiv 𝕜 f x ∈ K}
Module: Mathlib.Analysis.LocallyConvex.Separation

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
inst✝⁹ : TopologicalSpace E
inst✝⁸ : AddCommGroup E
inst✝⁷ : Module ℝ E
s : Set E
inst✝⁶ : RCLike 𝕜
inst✝⁵ : Module 𝕜 E
inst✝⁴ : IsScalarTower ℝ 𝕜 E
inst✝³ : IsTopologicalAddGroup E
inst✝² : ContinuousSMul 𝕜 E
inst✝¹ : LocallyConvexSpace ℝ E
inst✝ : HereditarilyLindelofSpace E
hs₁ : Convex ℝ s
hs₂ : IsClosed s
⊢ ∃ l c, ⋂ n, {x | re ((l n) x) ≤ c n} = s

Tactic:
set ι := Σ (l : StrongDual 𝕜 E), { c : ℝ // ∀ y ∈ s, re (l y) ≤ c }

Goal state after:
𝕜 : Type u_1
E : Type u_2
inst✝⁹ : TopologicalSpace E
inst✝⁸ : AddCommGroup E
inst✝⁷ : Module ℝ E
s : Set E
inst✝⁶ : RCLike 𝕜
inst✝⁵ : Module 𝕜 E
inst✝⁴ : IsScalarTower ℝ 𝕜 E
inst✝³ : IsTopologicalAddGroup E
inst✝² : ContinuousSMul 𝕜 E
inst✝¹ : LocallyConvexSpace ℝ E
inst✝ : HereditarilyLindelofSpace E
hs₁ : Convex ℝ s
hs₂ : IsClosed s
ι : Type (max u_1 u_2) := (l : StrongDual 𝕜 E) × { c // ∀ y ∈ s, re (l y) ≤ c }
⊢ ∃ l c, ⋂ n, {x | re ((l n) x) ≤ c n} = s
Module: Mathlib.Analysis.Calculus.FDeriv.Measurable

Goal state before:
pp: case h₁
F : Type u_1
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace ℝ F
f : ℝ → F
r x : ℝ
hr : 0 < r
ε : ℝ
L₁ L₂ : F
h₁ : x ∈ A f L₁ r ε
h₂ : x ∈ A f L₂ r ε
⊢ ‖f (x + r / 2) - f x - (x + r / 2 - x) • L₂‖ ≤ ε * r

Tactic:
apply le_of_mem_A h₂ <;> simp [(half_pos hr).le]

Goal state after:
Module: Mathlib.MeasureTheory.Integral.Bochner.ContinuousLinearMap

Goal state before:
pp: case inr
X : Type u_1
E : Type u_3
F : Type u_4
inst✝⁷ : MeasurableSpace X
μ : Measure X
𝕜 : Type u_6
inst✝⁶ : RCLike 𝕜
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace 𝕜 E
inst✝³ : NormedAddCommGroup F
inst✝² : NormedSpace ℝ F
inst✝¹ : NormedSpace 𝕜 F
inst✝ : NormedSpace ℝ E
L : E ≃L[𝕜] F
φ : X → E
this : CompleteSpace E ↔ CompleteSpace F
left✝ : ¬CompleteSpace E
right✝ : ¬CompleteSpace F
⊢ ∫ (x : X), L (φ x) ∂μ = L (∫ (x : X), φ x ∂μ)

Tactic:
· simp [integral, *]

Goal state after:
Module: Mathlib.MeasureTheory.Integral.Bochner.ContinuousLinearMap

Goal state before:
pp: X : Type u_1
E : Type u_3
F : Type u_4
inst✝⁷ : MeasurableSpace X
μ : Measure X
𝕜 : Type u_6
inst✝⁶ : RCLike 𝕜
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace 𝕜 E
inst✝³ : NormedAddCommGroup F
inst✝² : NormedSpace ℝ F
inst✝¹ : NormedSpace 𝕜 F
inst✝ : NormedSpace ℝ E
L : E ≃L[𝕜] F
φ : X → E
⊢ ∫ (x : X), L (φ x) ∂μ = L (∫ (x : X), φ x ∂μ)

Tactic:
have : CompleteSpace E ↔ CompleteSpace F :=
    completeSpace_congr (e := L.toEquiv) L.isUniformEmbedding
  obtain ⟨_, _⟩ | ⟨_, _⟩ := iff_iff_and_or_not_and_not.mp this
  · exact L.toContinuousLinearMap.integral_comp_comm' L.antilipschitz _
  · simp [integral, *]

Goal state after:
Module: Mathlib.MeasureTheory.Integral.Bochner.ContinuousLinearMap

Goal state before:
pp: case pos.refine_1
X : Type u_1
E : Type u_3
inst✝² : MeasurableSpace X
μ : Measure X
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
f : X → ℝ≥0
f_meas : Measurable f
g : X → E
hE : CompleteSpace E
hg : Integrable g (μ.withDensity fun x ↦ ↑(f x))
c : E
s : Set X
s_meas : MeasurableSet s
hs : (μ.withDensity fun x ↦ ↑(f x)) s < ∞
⊢ ∫ (x : X) in s, ↑(f x) • c ∂μ = ∫ (a : X) in s, f a • c ∂μ

pp: case pos.refine_1
X : Type u_1
E : Type u_3
inst✝² : MeasurableSpace X
μ : Measure X
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
f : X → ℝ≥0
f_meas : Measurable f
g : X → E
hE : CompleteSpace E
hg : Integrable g (μ.withDensity fun x ↦ ↑(f x))
c : E
s : Set X
s_meas : MeasurableSet s
hs : (μ.withDensity fun x ↦ ↑(f x)) s < ∞
⊢ 0 ≤ ∫ (a : X) in s, ↑(f a) ∂μ

pp: case pos.refine_1.hfi
X : Type u_1
E : Type u_3
inst✝² : MeasurableSpace X
μ : Measure X
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
f : X → ℝ≥0
f_meas : Measurable f
g : X → E
hE : CompleteSpace E
hg : Integrable g (μ.withDensity fun x ↦ ↑(f x))
c : E
s : Set X
s_meas : MeasurableSet s
hs : (μ.withDensity fun x ↦ ↑(f x)) s < ∞
⊢ Integrable (fun x ↦ ↑(f x)) (μ.restrict s)

Tactic:
· rfl

Goal state after:
case pos.refine_1
X : Type u_1
E : Type u_3
inst✝² : MeasurableSpace X
μ : Measure X
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
f : X → ℝ≥0
f_meas : Measurable f
g : X → E
hE : CompleteSpace E
hg : Integrable g (μ.withDensity fun x ↦ ↑(f x))
c : E
s : Set X
s_meas : MeasurableSet s
hs : (μ.withDensity fun x ↦ ↑(f x)) s < ∞
⊢ 0 ≤ ∫ (a : X) in s, ↑(f a) ∂μ

case pos.refine_1.hfi
X : Type u_1
E : Type u_3
inst✝² : MeasurableSpace X
μ : Measure X
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
f : X → ℝ≥0
f_meas : Measurable f
g : X → E
hE : CompleteSpace E
hg : Integrable g (μ.withDensity fun x ↦ ↑(f x))
c : E
s : Set X
s_meas : MeasurableSet s
hs : (μ.withDensity fun x ↦ ↑(f x)) s < ∞
⊢ Integrable (fun x ↦ ↑(f x)) (μ.restrict s)
Module: Mathlib.MeasureTheory.Integral.IntervalIntegral.Basic

Goal state before:
pp: ε : Type u_3
inst✝² : TopologicalSpace ε
inst✝¹ : ENormedAddMonoid ε
inst✝ : PseudoMetrizableSpace ε
f : ℝ → ε
a b : ℝ
μ : Measure ℝ
g : ℝ → ε
hf : IntervalIntegrable f μ a b
h : f =ᵐ[μ.restrict (Ι a b)] g
⊢ IntervalIntegrable g μ a b

Tactic:
by
  rwa [← intervalIntegrable_congr_ae h]

Goal state after:
Module: Mathlib.Analysis.Calculus.FDeriv.Measurable

Goal state before:
pp: F : Type u_1
inst✝⁴ : NormedAddCommGroup F
inst✝³ : NormedSpace ℝ F
f : ℝ → F
inst✝² : CompleteSpace F
inst✝¹ : MeasurableSpace F
inst✝ : BorelSpace F
s : Set F
hs : IsClosed[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] s
⊢ MeasurableSet ((fun x ↦ derivWithin f (Ici x) x) ⁻¹' s)

Tactic:
have :
    (fun x => derivWithin f (Ici x) x) ⁻¹' s =
      { x | DifferentiableWithinAt ℝ f (Ici x) x ∧ derivWithin f (Ici x) x ∈ s } ∪
        { x | ¬DifferentiableWithinAt ℝ f (Ici x) x } ∩ { _x | (0 : F) ∈ s } :=
    Set.ext fun x => mem_preimage.trans derivWithin_mem_iff

Goal state after:
F : Type u_1
inst✝⁴ : NormedAddCommGroup F
inst✝³ : NormedSpace ℝ F
f : ℝ → F
inst✝² : CompleteSpace F
inst✝¹ : MeasurableSpace F
inst✝ : BorelSpace F
s : Set F
hs : IsClosed[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] s
this :
  (fun x ↦ derivWithin f (Ici x) x) ⁻¹' s =
    {x | DifferentiableWithinAt ℝ f (Ici x) x ∧ derivWithin f (Ici x) x ∈ s} ∪
      {x | ¬DifferentiableWithinAt ℝ f (Ici x) x} ∩ {_x | 0 ∈ s}
⊢ MeasurableSet ((fun x ↦ derivWithin f (Ici x) x) ⁻¹' s)
Module: Mathlib.Analysis.Calculus.FDeriv.Measurable

Goal state before:
pp: 𝕜 : Type u_1
inst✝⁶ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace 𝕜 E
inst✝³ : LocallyCompactSpace E
F : Type u_3
inst✝² : NormedAddCommGroup F
inst✝¹ : NormedSpace 𝕜 F
α : Type u_4
inst✝ : TopologicalSpace α
f : α → E → F
r s : ℝ
hf : Continuous[instTopologicalSpaceProd, PseudoMetricSpace.toUniformSpace.toTopologicalSpace] (Function.uncurry f)
L : E →L[𝕜] F
this : ProperSpace E
a : α
x : E
r' : ℝ
hr :
  ∀ (y : E),
    dist y (a, x).2 < r' → ∀ (z : E), dist z (a, x).2 < r' → ‖f (a, x).1 z - f (a, x).1 y - (L z - L y)‖ < s * r
Irr' : r / 2 < r'
Ir'r : r' ≤ r
t : ℝ
hrt : r / 2 < t
htr' : t < r'
t' : ℝ
hrt' : r / 2 < t'
ht't : t' < t
⊢ {p |
      ∃ r',
        (r / 2 < r' ∧ r' ≤ r) ∧
          ∀ (y : E), dist y p.2 < r' → ∀ (z : E), dist z p.2 < r' → ‖f p.1 z - f p.1 y - (L z - L y)‖ < s * r} ∈
    𝓝 (a, x)

Tactic:
obtain ⟨b, b_lt, hb⟩ : ∃ b, b < s * r ∧ ∀ y ∈ closedBall x t, ∀ z ∈ closedBall x t,
      ‖f a z - f a y - (L z - L y)‖ ≤ b := by
    have B : Continuous (fun (p : E × E) ↦ ‖f a p.2 - f a p.1 - (L p.2 - L p.1)‖) := by fun_prop
    have C : (closedBall x t ×ˢ closedBall x t).Nonempty := by simp; linarith
    rcases ((isCompact_closedBall x t).prod (isCompact_closedBall x t)).exists_isMaxOn
      C B.continuousOn with ⟨p, pt, hp⟩
    simp only [mem_prod, mem_closedBall] at pt
    refine ⟨‖f a p.2 - f a p.1 - (L p.2 - L p.1)‖,
      hr p.1 (pt.1.trans_lt htr') p.2 (pt.2.trans_lt htr'), fun y hy z hz ↦ ?_⟩
    have D : (y, z) ∈ closedBall x t ×ˢ closedBall x t := mem_prod.2 ⟨hy, hz⟩
    exact hp D

Goal state after:
𝕜 : Type u_1
inst✝⁶ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace 𝕜 E
inst✝³ : LocallyCompactSpace E
F : Type u_3
inst✝² : NormedAddCommGroup F
inst✝¹ : NormedSpace 𝕜 F
α : Type u_4
inst✝ : TopologicalSpace α
f : α → E → F
r s : ℝ
hf : Continuous[instTopologicalSpaceProd, PseudoMetricSpace.toUniformSpace.toTopologicalSpace] (Function.uncurry f)
L : E →L[𝕜] F
this : ProperSpace E
a : α
x : E
r' : ℝ
hr :
  ∀ (y : E),
    dist y (a, x).2 < r' → ∀ (z : E), dist z (a, x).2 < r' → ‖f (a, x).1 z - f (a, x).1 y - (L z - L y)‖ < s * r
Irr' : r / 2 < r'
Ir'r : r' ≤ r
t : ℝ
hrt : r / 2 < t
htr' : t < r'
t' : ℝ
hrt' : r / 2 < t'
ht't : t' < t
b : ℝ
b_lt : b < s * r
hb : ∀ y ∈ closedBall x t, ∀ z ∈ closedBall x t, ‖f a z - f a y - (L z - L y)‖ ≤ b
⊢ {p |
      ∃ r',
        (r / 2 < r' ∧ r' ≤ r) ∧
          ∀ (y : E), dist y p.2 < r' → ∀ (z : E), dist z p.2 < r' → ‖f p.1 z - f p.1 y - (L z - L y)‖ < s * r} ∈
    𝓝 (a, x)