mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.Geometry.Manifold.LocalInvariantProperties

Goal state before:
pp: H : Type u_1
M : Type u_2
H' : Type u_3
M' : Type u_4
inst✝⁵ : TopologicalSpace H
inst✝⁴ : TopologicalSpace M
inst✝³ : ChartedSpace H M
inst✝² : TopologicalSpace H'
inst✝¹ : TopologicalSpace M'
inst✝ : ChartedSpace H' M'
G : StructureGroupoid H
G' : StructureGroupoid H'
P : (H → H') → Set H → H → Prop
g : M → M'
s t : Set M
x : M
hG : G.LocalInvariantProp G' P
ht : t ∈ 𝓝[s] x
⊢ ContinuousWithinAt g (s ∩ t) x ∧
      P (↑(chartAt H' (g x)) ∘ g ∘ ↑(chartAt H x).symm) (↑(chartAt H x).symm ⁻¹' (s ∩ t)) (↑(chartAt H x) x) ↔
    ContinuousWithinAt g s x ∧
      P (↑(chartAt H' (g x)) ∘ g ∘ ↑(chartAt H x).symm) (↑(chartAt H x).symm ⁻¹' s) (↑(chartAt H x) x)

Tactic:
continuousWithinAt_inter' ht,

Goal state after:
H : Type u_1
M : Type u_2
H' : Type u_3
M' : Type u_4
inst✝⁵ : TopologicalSpace H
inst✝⁴ : TopologicalSpace M
inst✝³ : ChartedSpace H M
inst✝² : TopologicalSpace H'
inst✝¹ : TopologicalSpace M'
inst✝ : ChartedSpace H' M'
G : StructureGroupoid H
G' : StructureGroupoid H'
P : (H → H') → Set H → H → Prop
g : M → M'
s t : Set M
x : M
hG : G.LocalInvariantProp G' P
ht : t ∈ 𝓝[s] x
⊢ ContinuousWithinAt g s x ∧
      P (↑(chartAt H' (g x)) ∘ g ∘ ↑(chartAt H x).symm) (↑(chartAt H x).symm ⁻¹' (s ∩ t)) (↑(chartAt H x) x) ↔
    ContinuousWithinAt g s x ∧
      P (↑(chartAt H' (g x)) ∘ g ∘ ↑(chartAt H x).symm) (↑(chartAt H x).symm ⁻¹' s) (↑(chartAt H x) x)
Module: Mathlib.MeasureTheory.Function.Jacobian

Goal state before:
pp: E : Type u_1
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace ℝ E
inst✝³ : FiniteDimensional ℝ E
s : Set E
f' : E → E →L[ℝ] E
inst✝² : MeasurableSpace E
inst✝¹ : BorelSpace E
μ : Measure E
inst✝ : μ.IsAddHaarMeasure
hs : MeasurableSet s
f : E ≃ᵐ E
g : E → ℝ
hg : ∀ᵐ (x : E) ∂μ, x ∈ ⇑f '' s → 0 ≤ g x
hg_int : IntegrableOn g (⇑f '' s) μ
hf' : ∀ x ∈ s, HasFDerivWithinAt (⇑f) (f' x) s x
⊢ (Measure.map (⇑f.symm) (μ.withDensity fun x ↦ ENNReal.ofReal (g x))) s =
    ENNReal.ofReal (∫ (x : E) in s, |(f' x).det| * g (f x) ∂μ)

Tactic:
rw [MeasurableEquiv.map_symm,
    MeasurableEmbedding.withDensity_ofReal_comap_apply_eq_integral_abs_det_fderiv_mul μ hs
      f.measurableEmbedding hg hg_int hf']

Goal state after:
Module: Mathlib.Geometry.Manifold.ContMDiff.Defs

Goal state before:
pp: 𝕜 : Type u_1
inst✝¹² : NontriviallyNormedField 𝕜
E : Type u_2
inst✝¹¹ : NormedAddCommGroup E
inst✝¹⁰ : NormedSpace 𝕜 E
H : Type u_3
inst✝⁹ : TopologicalSpace H
I : ModelWithCorners 𝕜 E H
M : Type u_4
inst✝⁸ : TopologicalSpace M
inst✝⁷ : ChartedSpace H M
E' : Type u_5
inst✝⁶ : NormedAddCommGroup E'
inst✝⁵ : NormedSpace 𝕜 E'
H' : Type u_6
inst✝⁴ : TopologicalSpace H'
I' : ModelWithCorners 𝕜 E' H'
M' : Type u_7
inst✝³ : TopologicalSpace M'
inst✝² : ChartedSpace H' M'
f : M → M'
s : Set M
x : M
m n : ℕ∞ω
inst✝¹ : IsManifold I n M
inst✝ : IsManifold I' n M'
hm : m ≤ n
h' : m = ∞ → n = ω
h : ContMDiffWithinAt I I' n f s x
⊢ ∃ u ∈ 𝓝[insert x s] x, u ⊆ insert x s ∧ ContMDiffOn I I' m f u

Tactic:
let ⟨_u, uo, xu, h⟩ := h.contMDiffOn' hm h'

Goal state after:
𝕜 : Type u_1
inst✝¹² : NontriviallyNormedField 𝕜
E : Type u_2
inst✝¹¹ : NormedAddCommGroup E
inst✝¹⁰ : NormedSpace 𝕜 E
H : Type u_3
inst✝⁹ : TopologicalSpace H
I : ModelWithCorners 𝕜 E H
M : Type u_4
inst✝⁸ : TopologicalSpace M
inst✝⁷ : ChartedSpace H M
E' : Type u_5
inst✝⁶ : NormedAddCommGroup E'
inst✝⁵ : NormedSpace 𝕜 E'
H' : Type u_6
inst✝⁴ : TopologicalSpace H'
I' : ModelWithCorners 𝕜 E' H'
M' : Type u_7
inst✝³ : TopologicalSpace M'
inst✝² : ChartedSpace H' M'
f : M → M'
s : Set M
x : M
m n : ℕ∞ω
inst✝¹ : IsManifold I n M
inst✝ : IsManifold I' n M'
hm : m ≤ n
h' : m = ∞ → n = ω
h✝ : ContMDiffWithinAt I I' n f s x
_u : Set M
uo : IsOpen[inst✝⁸] _u
xu : x ∈ _u
h : ContMDiffOn I I' m f (insert x s ∩ _u)
⊢ ∃ u ∈ 𝓝[insert x s] x, u ⊆ insert x s ∧ ContMDiffOn I I' m f u
Module: Mathlib.Geometry.Manifold.MFDeriv.Basic

Goal state before:
pp: case inl
𝕜 : Type u_1
inst✝¹⁰ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝⁹ : NormedAddCommGroup E
inst✝⁸ : NormedSpace 𝕜 E
H : Type u_3
inst✝⁷ : TopologicalSpace H
I : ModelWithCorners 𝕜 E H
M : Type u_4
inst✝⁶ : TopologicalSpace M
inst✝⁵ : ChartedSpace H M
E' : Type u_5
inst✝⁴ : NormedAddCommGroup E'
inst✝³ : NormedSpace 𝕜 E'
H' : Type u_6
inst✝² : TopologicalSpace H'
I' : ModelWithCorners 𝕜 E' H'
M' : Type u_7
inst✝¹ : TopologicalSpace M'
inst✝ : ChartedSpace H' M'
f : M → M'
x : M
s : Set M
⊢ MDiffAt[insert x s] f x ↔ MDiffAt[s] f x

Tactic:
exact mdifferentiableWithinAt_insert_self

Goal state after:
Module: Mathlib.Topology.VectorBundle.Basic

Goal state before:
pp: case pos
R : Type u_1
B : Type u_2
F : Type u_3
E : B → Type u_4
inst✝⁹ : NontriviallyNormedField R
inst✝⁸ : (x : B) → AddCommMonoid (E x)
inst✝⁷ : (x : B) → Module R (E x)
inst✝⁶ : NormedAddCommGroup F
inst✝⁵ : NormedSpace R F
inst✝⁴ : TopologicalSpace B
inst✝³ : TopologicalSpace (TotalSpace F E)
inst✝² : (x : B) → TopologicalSpace (E x)
inst✝¹ : FiberBundle F E
e : Trivialization F TotalSpace.proj
inst✝ : Trivialization.IsLinear R e
b : B
hb : b ∈ e.baseSet
⊢ Continuous[PseudoMetricSpace.toUniformSpace.toTopologicalSpace, inst✝³]
    (TotalSpace.mk b ∘ (Trivialization.symmₗ R e b).toFun)

Tactic:
refine .congr (f := TotalSpace.mk b ∘ e.symm b) ?_ (by simp [hb])

Goal state after:
case pos
R : Type u_1
B : Type u_2
F : Type u_3
E : B → Type u_4
inst✝⁹ : NontriviallyNormedField R
inst✝⁸ : (x : B) → AddCommMonoid (E x)
inst✝⁷ : (x : B) → Module R (E x)
inst✝⁶ : NormedAddCommGroup F
inst✝⁵ : NormedSpace R F
inst✝⁴ : TopologicalSpace B
inst✝³ : TopologicalSpace (TotalSpace F E)
inst✝² : (x : B) → TopologicalSpace (E x)
inst✝¹ : FiberBundle F E
e : Trivialization F TotalSpace.proj
inst✝ : Trivialization.IsLinear R e
b : B
hb : b ∈ e.baseSet
⊢ Continuous[PseudoMetricSpace.toUniformSpace.toTopologicalSpace, inst✝³] (TotalSpace.mk b ∘ e.symm b)
Module: Mathlib.Geometry.Manifold.MFDeriv.Basic

Goal state before:
pp: case neg
𝕜 : Type u_1
inst✝⁵ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace 𝕜 E
H : Type u_3
inst✝² : TopologicalSpace H
I : ModelWithCorners 𝕜 E H
M : Type u_4
inst✝¹ : TopologicalSpace M
inst✝ : ChartedSpace H M
x : M
s t : Set M
y : M
h : s =ᶠ[𝓝[{y}ᶜ] x] t
this : T1Space M
u : Set M
u_mem : u ∈ 𝓝 (↑(extChartAt I x).symm (↑(extChartAt I x) x))
hu : u ∩ {x}ᶜ ⊆ {y | (y ∈ s) = (y ∈ t)}
B : (extChartAt I x).target ∪ (range ↑I)ᶜ ∈ 𝓝 (↑(extChartAt I x) x)
z : E
hz : z ∈ ↑(extChartAt I x).symm ⁻¹' u ∩ ((extChartAt I x).target ∪ (range ↑I)ᶜ)
h'z : z ∈ {↑(extChartAt I x) x}ᶜ
hIz : z ∉ range ↑I
⊢ z ∈ ↑(extChartAt I x).symm ⁻¹' s ∩ range ↑I ↔ z ∈ ↑(extChartAt I x).symm ⁻¹' t ∩ range ↑I

Tactic:
· simp [hIz]

Goal state after:
Module: Mathlib.Geometry.Manifold.MFDeriv.Basic

Goal state before:
pp: 𝕜 : Type u_1
inst✝¹⁰ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝⁹ : NormedAddCommGroup E
inst✝⁸ : NormedSpace 𝕜 E
H : Type u_3
inst✝⁷ : TopologicalSpace H
I : ModelWithCorners 𝕜 E H
M : Type u_4
inst✝⁶ : TopologicalSpace M
inst✝⁵ : ChartedSpace H M
E' : Type u_5
inst✝⁴ : NormedAddCommGroup E'
inst✝³ : NormedSpace 𝕜 E'
H' : Type u_6
inst✝² : TopologicalSpace H'
I' : ModelWithCorners 𝕜 E' H'
M' : Type u_7
inst✝¹ : TopologicalSpace M'
inst✝ : ChartedSpace H' M'
f : M → M'
x : M
s t : Set M
h : s =ᶠ[𝓝 x] t
⊢ MDiffAt[s] f x ↔ MDiffAt[t] f x

Tactic:
simp only [mdifferentiableWithinAt_iff_exists_hasMFDerivWithinAt]
  exact exists_congr fun _ => hasMFDerivWithinAt_congr_set h

Goal state after:
Module: Mathlib.Geometry.Manifold.MFDeriv.SpecificFunctions

Goal state before:
pp: 𝕜 : Type u_1
inst✝¹⁰ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝⁹ : NormedAddCommGroup E
inst✝⁸ : NormedSpace 𝕜 E
H : Type u_3
inst✝⁷ : TopologicalSpace H
I : ModelWithCorners 𝕜 E H
M : Type u_4
inst✝⁶ : TopologicalSpace M
inst✝⁵ : ChartedSpace H M
E' : Type u_5
inst✝⁴ : NormedAddCommGroup E'
inst✝³ : NormedSpace 𝕜 E'
H' : Type u_6
inst✝² : TopologicalSpace H'
I' : ModelWithCorners 𝕜 E' H'
M' : Type u_7
inst✝¹ : TopologicalSpace M'
inst✝ : ChartedSpace H' M'
x₀ : M
y₀ : M'
⊢ ContinuousLinearMap.prod 0 (ContinuousLinearMap.id 𝕜 (TangentSpace I' y₀)) =
    ContinuousLinearMap.inr 𝕜 (TangentSpace I x₀) (TangentSpace I' y₀)

Tactic:
ContinuousLinearMap.inr

Goal state after:
𝕜 : Type u_1
inst✝¹⁰ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝⁹ : NormedAddCommGroup E
inst✝⁸ : NormedSpace 𝕜 E
H : Type u_3
inst✝⁷ : TopologicalSpace H
I : ModelWithCorners 𝕜 E H
M : Type u_4
inst✝⁶ : TopologicalSpace M
inst✝⁵ : ChartedSpace H M
E' : Type u_5
inst✝⁴ : NormedAddCommGroup E'
inst✝³ : NormedSpace 𝕜 E'
H' : Type u_6
inst✝² : TopologicalSpace H'
I' : ModelWithCorners 𝕜 E' H'
M' : Type u_7
inst✝¹ : TopologicalSpace M'
inst✝ : ChartedSpace H' M'
x₀ : M
y₀ : M'
⊢ ContinuousLinearMap.prod 0 (ContinuousLinearMap.id 𝕜 (TangentSpace I' y₀)) =
    ContinuousLinearMap.prod 0 (ContinuousLinearMap.id 𝕜 (TangentSpace I' y₀))
Module: Mathlib.Geometry.Manifold.VectorBundle.Tangent

Goal state before:
pp: 𝕜 : Type u_1
inst✝⁵ : NontriviallyNormedField 𝕜
n : ℕ∞ω
E : Type u_2
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace 𝕜 E
H : Type u_4
inst✝² : TopologicalSpace H
I : ModelWithCorners 𝕜 E H
M : Type u_6
inst✝¹ : TopologicalSpace M
inst✝ : ChartedSpace H M
h : IsManifold I (n + 1) M
this : IsManifold I 1 M
⊢ ContMDiffVectorBundle n E (TangentSpace I) I

Tactic:
have : (tangentBundleCore I M).IsContMDiff I n := tangentBundleCore.isContMDiff

Goal state after:
𝕜 : Type u_1
inst✝⁵ : NontriviallyNormedField 𝕜
n : ℕ∞ω
E : Type u_2
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace 𝕜 E
H : Type u_4
inst✝² : TopologicalSpace H
I : ModelWithCorners 𝕜 E H
M : Type u_6
inst✝¹ : TopologicalSpace M
inst✝ : ChartedSpace H M
h : IsManifold I (n + 1) M
this✝ : IsManifold I 1 M
this : (tangentBundleCore I M).IsContMDiff I n
⊢ ContMDiffVectorBundle n E (TangentSpace I) I
Module: Mathlib.Topology.Compactness.Paracompact

Goal state before:
pp: X : Type v
inst✝ : TopologicalSpace X
ι : X → Sort u_1
p : (x : X) → ι x → Prop
s : (x : X) → ι x → Set X
hb : ∀ (x : X), (𝓝 x).HasBasis (p x) (s x)
h :
  ∀ (f : (x : X) → ι x),
    (∀ (x : X), p x (f x)) →
      ∃ β t, (∀ (b : β), IsOpen[inst✝] (t b)) ∧ ⋃ b, t b = univ ∧ LocallyFinite t ∧ ∀ (b : β), ∃ x, t b ⊆ s x (f x)
α : Type v
S : α → Set X
ho : ∀ (a : α), IsOpen[inst✝] (S a)
hu : ⋃ a, S a = univ
⊢ ∃ β t, (∀ (b : β), IsOpen[inst✝] (t b)) ∧ ⋃ b, t b = univ ∧ LocallyFinite t ∧ ∀ (b : β), ∃ a, t b ⊆ S a

Tactic:
have := fun x ↦ (iUnion_eq_univ_iff.1 hu x).imp fun a ha ↦ (hb _).mem_iff.1 ((ho a).mem_nhds ha)

Goal state after:
X : Type v
inst✝ : TopologicalSpace X
ι : X → Sort u_1
p : (x : X) → ι x → Prop
s : (x : X) → ι x → Set X
hb : ∀ (x : X), (𝓝 x).HasBasis (p x) (s x)
h :
  ∀ (f : (x : X) → ι x),
    (∀ (x : X), p x (f x)) →
      ∃ β t, (∀ (b : β), IsOpen[inst✝] (t b)) ∧ ⋃ b, t b = univ ∧ LocallyFinite t ∧ ∀ (b : β), ∃ x, t b ⊆ s x (f x)
α : Type v
S : α → Set X
ho : ∀ (a : α), IsOpen[inst✝] (S a)
hu : ⋃ a, S a = univ
this : ∀ (x : X), ∃ a i, p x i ∧ s x i ⊆ S a
⊢ ∃ β t, (∀ (b : β), IsOpen[inst✝] (t b)) ∧ ⋃ b, t b = univ ∧ LocallyFinite t ∧ ∀ (b : β), ∃ a, t b ⊆ S a
Module: Mathlib.Geometry.Manifold.VectorBundle.ContMDiffSection

Goal state before:
pp: case refine_1.h
𝕜 : Type u_1
inst✝¹³ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝¹² : NormedAddCommGroup E
inst✝¹¹ : NormedSpace 𝕜 E
H : Type u_3
inst✝¹⁰ : TopologicalSpace H
I : ModelWithCorners 𝕜 E H
M : Type u_4
inst✝⁹ : TopologicalSpace M
inst✝⁸ : ChartedSpace H M
F : Type u_5
inst✝⁷ : NormedAddCommGroup F
inst✝⁶ : NormedSpace 𝕜 F
n : ℕ∞ω
V : M → Type u_6
inst✝⁵ : TopologicalSpace (TotalSpace F V)
inst✝⁴ : (x : M) → TopologicalSpace (V x)
inst✝³ : FiberBundle F V
inst✝² : (x : M) → AddCommGroup (V x)
inst✝¹ : (x : M) → Module 𝕜 (V x)
inst✝ : VectorBundle 𝕜 F V
s t : (x : M) → V x
u : Set M
x₀ : M
e : Trivialization F TotalSpace.proj := ⋯
hs : ContMDiffWithinAt I 𝓘(𝕜, F) n (fun x ↦ (↑e ⟨x, s x⟩).2) u x₀
ht : ContMDiffWithinAt I 𝓘(𝕜, F) n (fun x ↦ (↑e ⟨x, t x⟩).2) u x₀
⊢ ∀ x ∈ e.baseSet, (fun x ↦ (↑e ⟨x, (s + t) x⟩).2) x = ((fun x ↦ (↑e ⟨x, s x⟩).2) + fun x ↦ (↑e ⟨x, t x⟩).2) x

Tactic:
intro x hx
      apply (e.linear 𝕜 hx).1

Goal state after:
Module: Mathlib.Geometry.Manifold.VectorBundle.ContMDiffSection

Goal state before:
pp: case insert
𝕜 : Type u_1
inst✝¹³ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝¹² : NormedAddCommGroup E
inst✝¹¹ : NormedSpace 𝕜 E
H : Type u_3
inst✝¹⁰ : TopologicalSpace H
I : ModelWithCorners 𝕜 E H
M : Type u_4
inst✝⁹ : TopologicalSpace M
inst✝⁸ : ChartedSpace H M
F : Type u_5
inst✝⁷ : NormedAddCommGroup F
inst✝⁶ : NormedSpace 𝕜 F
n : ℕ∞ω
V : M → Type u_6
inst✝⁵ : TopologicalSpace (TotalSpace F V)
inst✝⁴ : (x : M) → TopologicalSpace (V x)
inst✝³ : FiberBundle F V
inst✝² : (x : M) → AddCommGroup (V x)
inst✝¹ : (x : M) → Module 𝕜 (V x)
inst✝ : VectorBundle 𝕜 F V
u : Set M
x₀ : M
ι : Type u_7
t : ι → (x : M) → V x
i : ι
s : Finset ι
hi : i ∉ s
h :
  (∀ i ∈ s, ContMDiffWithinAt I (I.prod 𝓘(𝕜, F)) n (fun x ↦ ⟨x, t i x⟩) u x₀) →
    ContMDiffWithinAt I (I.prod 𝓘(𝕜, F)) n (fun x ↦ ⟨x, ∑ i ∈ s, t i x⟩) u x₀
hs : ∀ i_1 ∈ insert i s, ContMDiffWithinAt I (I.prod 𝓘(𝕜, F)) n (fun x ↦ ⟨x, t i_1 x⟩) u x₀
⊢ ContMDiffWithinAt I (I.prod 𝓘(𝕜, F)) n (fun x ↦ ⟨x, ∑ i ∈ insert i s, t i x⟩) u x₀

Tactic:
simp only [Finset.sum_insert hi]
    apply (hs _ (s.mem_insert_self i)).add_section
    exact h fun i a ↦ hs _ (s.mem_insert_of_mem a)

Goal state after:
Module: Mathlib.Topology.PartitionOfUnity

Goal state before:
pp: case h.refine_1
X : Type v
inst✝² : TopologicalSpace X
inst✝¹ : T2Space X
inst✝ : LocallyCompactSpace X
n : ℕ
t : Set X
s : Fin n → Set X
hs : ∀ (i : Fin n), IsOpen[inst✝²] (s i)
htcp : IsCompact t
hst : t ⊆ ⋃ i, s i
f : PartitionOfUnity (Fin n) X t
hfsub : f.IsSubordinate s
hfcp : ∀ (i : Fin n), HasCompactSupport ⇑(f i)
x : X
hx : x ∈ t
h : ∑ᶠ (i : Fin n), (f.toFun i) x = 1
⊢ ∑ c, (f c) x = 1

Tactic:
rw [finsum_eq_sum (fun i => (f.toFun i) x)
      (Finite.subset finite_univ (subset_univ (support fun i ↦ (f.toFun i) x)))] at h

Goal state after:
case h.refine_1
X : Type v
inst✝² : TopologicalSpace X
inst✝¹ : T2Space X
inst✝ : LocallyCompactSpace X
n : ℕ
t : Set X
s : Fin n → Set X
hs : ∀ (i : Fin n), IsOpen[inst✝²] (s i)
htcp : IsCompact t
hst : t ⊆ ⋃ i, s i
f : PartitionOfUnity (Fin n) X t
hfsub : f.IsSubordinate s
hfcp : ∀ (i : Fin n), HasCompactSupport ⇑(f i)
x : X
hx : x ∈ t
h : ∑ i ∈ ⋯.toFinset, (f.toFun i) x = 1
⊢ ∑ c, (f c) x = 1
Module: Mathlib.MeasureTheory.Function.AEEqOfIntegral

Goal state before:
pp: α : Type u_1
E : Type u_2
𝕜 : Type u_3
m : MeasurableSpace α
μ : Measure α
inst✝² : RCLike 𝕜
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
t : Set E
f : α → E
hf : ∀ (c : StrongDual 𝕜 E), (fun x ↦ c (f x)) =ᵐ[μ] 0
h't : ∀ᵐ (x : α) ∂μ, f x ∈ t
d : Set E
d_count : d.Countable
hd : t ⊆ closure[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] d
this : Encodable ↑d
s : ↑d → StrongDual 𝕜 E
hs : ∀ (x : ↑d), ‖s x‖ ≤ 1 ∧ (s x) ↑x = ↑‖↑x‖
A : ∀ a ∈ t, (∀ (x : ↑d), (s x) a = 0) → a = 0
hfs : ∀ (y : ↑d), ∀ᵐ (x : α) ∂μ, (s y) (f x) = 0
hf' : ∀ᵐ (x : α) ∂μ, ∀ (y : ↑d), (s y) (f x) = 0
x : α
hx : ∀ (y : ↑d), (s y) (f x) = 0
⊢ f x ∈ t → f x = 0 x

Tactic:
h'x

Goal state after:
α : Type u_1
E : Type u_2
𝕜 : Type u_3
m : MeasurableSpace α
μ : Measure α
inst✝² : RCLike 𝕜
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
t : Set E
f : α → E
hf : ∀ (c : StrongDual 𝕜 E), (fun x ↦ c (f x)) =ᵐ[μ] 0
h't : ∀ᵐ (x : α) ∂μ, f x ∈ t
d : Set E
d_count : d.Countable
hd : t ⊆ closure[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] d
this : Encodable ↑d
s : ↑d → StrongDual 𝕜 E
hs : ∀ (x : ↑d), ‖s x‖ ≤ 1 ∧ (s x) ↑x = ↑‖↑x‖
A : ∀ a ∈ t, (∀ (x : ↑d), (s x) a = 0) → a = 0
hfs : ∀ (y : ↑d), ∀ᵐ (x : α) ∂μ, (s y) (f x) = 0
hf' : ∀ᵐ (x : α) ∂μ, ∀ (y : ↑d), (s y) (f x) = 0
x : α
hx : ∀ (y : ↑d), (s y) (f x) = 0
h'x : f x ∈ t
⊢ f x = 0 x
Module: Mathlib.Analysis.Distribution.AEEqOfIntegralContDiff

Goal state before:
pp: E : Type u_1
inst✝¹² : NormedAddCommGroup E
inst✝¹¹ : NormedSpace ℝ E
inst✝¹⁰ : FiniteDimensional ℝ E
F : Type u_2
inst✝⁹ : NormedAddCommGroup F
inst✝⁸ : NormedSpace ℝ F
inst✝⁷ : CompleteSpace F
H : Type u_3
inst✝⁶ : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type u_4
inst✝⁵ : TopologicalSpace M
inst✝⁴ : ChartedSpace H M
inst✝³ : IsManifold I ∞ M
inst✝² : MeasurableSpace M
inst✝¹ : BorelSpace M
inst✝ : T2Space M
f : M → F
μ : Measure M
U : Set M
hU : IsOpen[inst✝⁵] U
hSig : IsSigmaCompact U
hf : LocallyIntegrableOn f U μ
h : ∀ (g : M → ℝ), ContMDiff I 𝓘(ℝ, ℝ) ∞ g → HasCompactSupport g → tsupport g ⊆ U → ∫ (x : M), g x • f x ∂μ = 0
⊢ ∀ᵐ (x : M) ∂μ, x ∈ U → f x = 0

Tactic:
have meas_U := hU.measurableSet
  rw [← ae_restrict_iff' meas_U, ae_restrict_iff_subtype meas_U]
  let U : Opens M := ⟨U, hU⟩
  change ∀ᵐ (x : U) ∂_, _
  have : SigmaCompactSpace U := isSigmaCompact_iff_sigmaCompactSpace.mp hSig
  refine ae_eq_zero_of_integral_contMDiff_smul_eq_zero I ?_ fun g g_smth g_supp ↦ ?_
  · exact (locallyIntegrable_comap meas_U).mpr hf
  specialize h (Subtype.val.extend g 0) (g_smth.extend_zero g_supp)
    (g_supp.extend_zero continuous_subtype_val) ((g_supp.tsupport_extend_zero_subset
      continuous_subtype_val).trans <| Subtype.coe_image_subset _ _)
  rw [← setIntegral_eq_integral_of_forall_compl_eq_zero (s := U) fun x hx ↦ ?_] at h
  · rw [← integral_subtype_comap] at h
    · simp_rw [Subtype.val_injective.extend_apply] at h; exact h
    · exact meas_U
  rw [Function.extend_apply' _ _ _ (mt _ hx)]
  · apply zero_smul
  · rintro ⟨x, rfl⟩; exact x.2

Goal state after:
Module: Mathlib.Order.SuccPred.IntervalSucc

Goal state before:
pp: α : Type u_1
β : Type u_2
inst✝³ : LinearOrder α
inst✝² : SuccOrder α
inst✝¹ : IsSuccArchimedean α
inst✝ : LinearOrder β
f : α → β
a : α
hf : ∀ i ∈ Ici a, f a ≤ f i
b : β
hb : b ∈ Ici (f a)
h2f : b ∉ ⋃ i ∈ Ici a, Ico (f i) (f (succ i))
⊢ BddAbove (f '' Ici a)

Tactic:
use b

Goal state after:
case h
α : Type u_1
β : Type u_2
inst✝³ : LinearOrder α
inst✝² : SuccOrder α
inst✝¹ : IsSuccArchimedean α
inst✝ : LinearOrder β
f : α → β
a : α
hf : ∀ i ∈ Ici a, f a ≤ f i
b : β
hb : b ∈ Ici (f a)
h2f : b ∉ ⋃ i ∈ Ici a, Ico (f i) (f (succ i))
⊢ b ∈ upperBounds (f '' Ici a)
Module: Mathlib.Analysis.Calculus.Rademacher

Goal state before:
pp: E : Type u_1
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace ℝ E
inst✝³ : MeasurableSpace E
inst✝² : BorelSpace E
C : ℝ≥0
f : E → ℝ
μ : Measure E
inst✝¹ : FiniteDimensional ℝ E
inst✝ : μ.IsAddHaarMeasure
hf : LipschitzWith C f
s : Set E
hs : s.Countable
B : Basis ↑(Basis.ofVectorSpaceIndex ℝ E) ℝ E
⊢ ∀ᵐ (x : E) ∂μ, ∃ L, ∀ v ∈ s, HasLineDerivAt ℝ f (L v) x v

Tactic:
have I1 : ∀ᵐ (x : E) ∂μ, ∀ v ∈ s, lineDeriv ℝ f x (∑ i, (B.repr v i) • B i) =
                                  ∑ i, B.repr v i • lineDeriv ℝ f x (B i) :=
    (ae_ball_iff hs).2 (fun v _ ↦ hf.ae_lineDeriv_sum_eq _ _ _)

Goal state after:
E : Type u_1
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace ℝ E
inst✝³ : MeasurableSpace E
inst✝² : BorelSpace E
C : ℝ≥0
f : E → ℝ
μ : Measure E
inst✝¹ : FiniteDimensional ℝ E
inst✝ : μ.IsAddHaarMeasure
hf : LipschitzWith C f
s : Set E
hs : s.Countable
B : Basis ↑(Basis.ofVectorSpaceIndex ℝ E) ℝ E
I1 : ∀ᵐ (x : E) ∂μ, ∀ v ∈ s, lineDeriv ℝ f x (∑ i, (B.repr v) i • B i) = ∑ i, (B.repr v) i • lineDeriv ℝ f x (B i)
⊢ ∀ᵐ (x : E) ∂μ, ∃ L, ∀ v ∈ s, HasLineDerivAt ℝ f (L v) x v
Module: Mathlib.MeasureTheory.Integral.IntervalIntegral.Slope

Goal state before:
pp: case inr
f : ℝ → ℝ
a b c : ℝ
hf : MonotoneOn f (uIcc a (b + c))
hab : a ≤ b
hc✝ : 0 ≤ c
hc : 0 < c
hf' : IntervalIntegrable f volume a (b + c)
fU : ∫ (x : ℝ) in b..b + c, f x ≤ c * f (b + c)
fL : c * f a ≤ ∫ (x : ℝ) in a..a + c, f x
⊢ c⁻¹ * (c * f (b + c) - c * f a) ≤ f (b + c) - f a

Tactic:
field_simp

Goal state after:
case inr
f : ℝ → ℝ
a b c : ℝ
hf : MonotoneOn f (uIcc a (b + c))
hab : a ≤ b
hc✝ : 0 ≤ c
hc : 0 < c
hf' : IntervalIntegrable f volume a (b + c)
fU : ∫ (x : ℝ) in b..b + c, f x ≤ c * f (b + c)
fL : c * f a ≤ ∫ (x : ℝ) in a..a + c, f x
⊢ f (b + c) - f a ≤ f (b + c) - f a
Module: Mathlib.Analysis.Calculus.Rademacher

Goal state before:
pp: E : Type u_1
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace ℝ E
inst✝³ : MeasurableSpace E
inst✝² : BorelSpace E
C : ℝ≥0
f : E → ℝ
μ : Measure E
inst✝¹ : FiniteDimensional ℝ E
inst✝ : μ.IsAddHaarMeasure
hf : LipschitzWith C f
s : Set E
hs : s.Countable
B : Basis ↑(Basis.ofVectorSpaceIndex ℝ E) ℝ E
I1 : ∀ᵐ (x : E) ∂μ, ∀ v ∈ s, lineDeriv ℝ f x (∑ i, (B.repr v) i • B i) = ∑ i, (B.repr v) i • lineDeriv ℝ f x (B i)
I2 : ∀ᵐ (x : E) ∂μ, ∀ v ∈ s, LineDifferentiableAt ℝ f x v
x : E
hx : ∀ v ∈ s, lineDeriv ℝ f x (∑ i, (B.repr v) i • B i) = ∑ i, (B.repr v) i • lineDeriv ℝ f x (B i)
⊢ (∀ v ∈ s, LineDifferentiableAt ℝ f x v) → ∃ L, ∀ v ∈ s, HasLineDerivAt ℝ f (L v) x v

Tactic:
h'x

Goal state after:
E : Type u_1
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace ℝ E
inst✝³ : MeasurableSpace E
inst✝² : BorelSpace E
C : ℝ≥0
f : E → ℝ
μ : Measure E
inst✝¹ : FiniteDimensional ℝ E
inst✝ : μ.IsAddHaarMeasure
hf : LipschitzWith C f
s : Set E
hs : s.Countable
B : Basis ↑(Basis.ofVectorSpaceIndex ℝ E) ℝ E
I1 : ∀ᵐ (x : E) ∂μ, ∀ v ∈ s, lineDeriv ℝ f x (∑ i, (B.repr v) i • B i) = ∑ i, (B.repr v) i • lineDeriv ℝ f x (B i)
I2 : ∀ᵐ (x : E) ∂μ, ∀ v ∈ s, LineDifferentiableAt ℝ f x v
x : E
hx : ∀ v ∈ s, lineDeriv ℝ f x (∑ i, (B.repr v) i • B i) = ∑ i, (B.repr v) i • lineDeriv ℝ f x (B i)
h'x : ∀ v ∈ s, LineDifferentiableAt ℝ f x v
⊢ ∃ L, ∀ v ∈ s, HasLineDerivAt ℝ f (L v) x v
Module: Mathlib.MeasureTheory.Integral.IntervalIntegral.DerivIntegrable

Goal state before:
pp: f : ℝ → ℝ
a b : ℝ
hf : MonotoneOn f (Icc a b)
hab : a ≤ b
G : ℕ → ℝ → ℝ
hGf : ∀ᵐ (x : ℝ), x ∈ uIcc a b → Tendsto (fun n ↦ G n x) atTop (𝓝 (deriv f x))
hG : ∀ (n : ℕ), AEStronglyMeasurable (G n) (volume.restrict (uIcc a b))
hG' : liminf (fun n ↦ ∫⁻ (x : ℝ) in uIcc a b, ‖G n x‖ₑ) atTop ≤ ENNReal.ofReal (f b - f a)
hG'₀ : liminf (fun n ↦ ∫⁻ (x : ℝ) in Icc a b, ‖G n x‖ₑ) atTop ≠ ⊤
integrable_f_deriv : Integrable (deriv f) (volume.restrict (Icc a b))
this : f a ≤ f b
x : ℝ
hx : x ∈ Ioo a b
⊢ 0 ≤ deriv f x

Tactic:
← derivWithin_of_mem_nhds (Icc_mem_nhds hx.left hx.right)

Goal state after:
f : ℝ → ℝ
a b : ℝ
hf : MonotoneOn f (Icc a b)
hab : a ≤ b
G : ℕ → ℝ → ℝ
hGf : ∀ᵐ (x : ℝ), x ∈ uIcc a b → Tendsto (fun n ↦ G n x) atTop (𝓝 (deriv f x))
hG : ∀ (n : ℕ), AEStronglyMeasurable (G n) (volume.restrict (uIcc a b))
hG' : liminf (fun n ↦ ∫⁻ (x : ℝ) in uIcc a b, ‖G n x‖ₑ) atTop ≤ ENNReal.ofReal (f b - f a)
hG'₀ : liminf (fun n ↦ ∫⁻ (x : ℝ) in Icc a b, ‖G n x‖ₑ) atTop ≠ ⊤
integrable_f_deriv : Integrable (deriv f) (volume.restrict (Icc a b))
this : f a ≤ f b
x : ℝ
hx : x ∈ Ioo a b
⊢ 0 ≤ derivWithin f (Icc a b) x
Module: Mathlib.Analysis.Calculus.Taylor

Goal state before:
pp: case e_a
E : Type u_2
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
f : ℝ → E
n : ℕ
s : Set ℝ
x₀ x : ℝ
⊢ ((↑(n + 1)!)⁻¹ * (x - x₀) ^ (n + 1)) • iteratedDerivWithin (n + 1) f s x₀ =
    ((↑n !)⁻¹ * (↑n + 1)⁻¹ * (x - x₀) ^ (n + 1)) • iteratedDerivWithin (n + 1) f s x₀

Tactic:
Nat.factorial_succ,

Goal state after:
case e_a
E : Type u_2
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
f : ℝ → E
n : ℕ
s : Set ℝ
x₀ x : ℝ
⊢ ((↑((n + 1) * n !))⁻¹ * (x - x₀) ^ (n + 1)) • iteratedDerivWithin (n + 1) f s x₀ =
    ((↑n !)⁻¹ * (↑n + 1)⁻¹ * (x - x₀) ^ (n + 1)) • iteratedDerivWithin (n + 1) f s x₀