mathlib-initiative/mathlib-tactics huggingface dataset:
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
f' : E → E →L[ℝ] E
inst✝² : MeasurableSpace E
inst✝¹ : BorelSpace E
μ : Measure E
inst✝ : μ.IsAddHaarMeasure
hs : MeasurableSet s
hf' : ∀ x ∈ s, HasFDerivWithinAt f (f' x) s x
hf : InjOn f s
u : E → E
u_meas : Measurable u
uf : EqOn u f s
u' : ∀ x ∈ s, HasFDerivWithinAt u (f' x) s x
F : ↑s → E := u ∘ Subtype.val
hF : F = u ∘ Subtype.val
⊢ Measure.map u
      (Measure.map Subtype.val (Measure.comap Subtype.val (μ.withDensity fun x ↦ ENNReal.ofReal |(f' x).det|))) =
    μ.restrict (u '' s)

Tactic:
map_comap_subtype_coe hs,

Goal state after:
E : Type u_1
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace ℝ E
inst✝³ : FiniteDimensional ℝ E
s : Set E
f : E → E
f' : E → E →L[ℝ] E
inst✝² : MeasurableSpace E
inst✝¹ : BorelSpace E
μ : Measure E
inst✝ : μ.IsAddHaarMeasure
hs : MeasurableSet s
hf' : ∀ x ∈ s, HasFDerivWithinAt f (f' x) s x
hf : InjOn f s
u : E → E
u_meas : Measurable u
uf : EqOn u f s
u' : ∀ x ∈ s, HasFDerivWithinAt u (f' x) s x
F : ↑s → E := u ∘ Subtype.val
hF : F = u ∘ Subtype.val
⊢ Measure.map u ((μ.withDensity fun x ↦ ENNReal.ofReal |(f' x).det|).restrict s) = μ.restrict (u '' s)
Module: Mathlib.Geometry.Manifold.MFDeriv.Defs

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
H : Type u_3
inst✝³ : TopologicalSpace H
I : ModelWithCorners 𝕜 E H
E' : Type u_5
inst✝² : NormedAddCommGroup E'
inst✝¹ : NormedSpace 𝕜 E'
H' : Type u_6
inst✝ : TopologicalSpace H'
I' : ModelWithCorners 𝕜 E' H'
s : Set H
x : H
f : H → H'
e : OpenPartialHomeomorph H H
he : e ∈ contDiffGroupoid 1 I
hx : x ∈ e.source
h : DifferentiableWithinAt 𝕜 (↑I' ∘ f ∘ ↑I.symm) (↑I.symm ⁻¹' s ∩ range ↑I) ((↑I ∘ ↑e.symm ∘ ↑I.symm) (↑I (↑e x)))
this✝ : ↑I x = (↑I ∘ ↑e.symm ∘ ↑I.symm) (↑I (↑e x))
this : ↑I (↑e x) ∈ ↑I.symm ⁻¹' e.target ∩ range ↑I
⊢ DifferentiableWithinAt 𝕜 (↑I' ∘ (f ∘ ↑e.symm) ∘ ↑I.symm) (↑I.symm ⁻¹' ↑e.symm ⁻¹' s ∩ range ↑I) (↑I (↑e x))

Tactic:
have := (mem_groupoid_of_pregroupoid.2 he).2.contDiffWithinAt this

Goal state after:
𝕜 : Type u_1
inst✝⁶ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace 𝕜 E
H : Type u_3
inst✝³ : TopologicalSpace H
I : ModelWithCorners 𝕜 E H
E' : Type u_5
inst✝² : NormedAddCommGroup E'
inst✝¹ : NormedSpace 𝕜 E'
H' : Type u_6
inst✝ : TopologicalSpace H'
I' : ModelWithCorners 𝕜 E' H'
s : Set H
x : H
f : H → H'
e : OpenPartialHomeomorph H H
he : e ∈ contDiffGroupoid 1 I
hx : x ∈ e.source
h : DifferentiableWithinAt 𝕜 (↑I' ∘ f ∘ ↑I.symm) (↑I.symm ⁻¹' s ∩ range ↑I) ((↑I ∘ ↑e.symm ∘ ↑I.symm) (↑I (↑e x)))
this✝¹ : ↑I x = (↑I ∘ ↑e.symm ∘ ↑I.symm) (↑I (↑e x))
this✝ : ↑I (↑e x) ∈ ↑I.symm ⁻¹' e.target ∩ range ↑I
this : ContDiffWithinAt 𝕜 1 (↑I ∘ ↑e.symm ∘ ↑I.symm) (↑I.symm ⁻¹' e.target ∩ range ↑I) (↑I (↑e x))
⊢ DifferentiableWithinAt 𝕜 (↑I' ∘ (f ∘ ↑e.symm) ∘ ↑I.symm) (↑I.symm ⁻¹' ↑e.symm ⁻¹' s ∩ range ↑I) (↑I (↑e 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
f' : E → E →L[ℝ] E
inst✝² : MeasurableSpace E
inst✝¹ : BorelSpace E
μ : Measure E
inst✝ : μ.IsAddHaarMeasure
hs : MeasurableSet s
hf' : ∀ x ∈ s, HasFDerivWithinAt f (f' x) s x
hf : InjOn f s
g : E → ℝ≥0∞
⊢ ∫⁻ (a : E),
      (g ∘ f)
        a ∂Measure.map Subtype.val (Measure.comap Subtype.val (μ.withDensity fun x ↦ ENNReal.ofReal |(f' x).det|)) =
    ∫⁻ (x : E) in s, ENNReal.ofReal |(f' x).det| * (g ∘ f) x ∂μ

Tactic:
map_comap_subtype_coe hs,

Goal state after:
E : Type u_1
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace ℝ E
inst✝³ : FiniteDimensional ℝ E
s : Set E
f : E → E
f' : E → E →L[ℝ] E
inst✝² : MeasurableSpace E
inst✝¹ : BorelSpace E
μ : Measure E
inst✝ : μ.IsAddHaarMeasure
hs : MeasurableSet s
hf' : ∀ x ∈ s, HasFDerivWithinAt f (f' x) s x
hf : InjOn f s
g : E → ℝ≥0∞
⊢ (∫⁻ (a : E) in s, (g ∘ f) a ∂μ.withDensity fun x ↦ ENNReal.ofReal |(f' x).det|) =
    ∫⁻ (x : E) in s, ENNReal.ofReal |(f' x).det| * (g ∘ f) x ∂μ
Module: Mathlib.MeasureTheory.Function.Jacobian

Goal state before:
pp: E : Type u_1
F : Type u_2
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : NormedSpace ℝ E
inst✝⁵ : FiniteDimensional ℝ E
inst✝⁴ : NormedAddCommGroup F
inst✝³ : NormedSpace ℝ F
s : Set E
f : E → E
f' : E → E →L[ℝ] E
inst✝² : MeasurableSpace E
inst✝¹ : BorelSpace E
μ : Measure E
inst✝ : μ.IsAddHaarMeasure
hs : MeasurableSet s
hf' : ∀ x ∈ s, HasFDerivWithinAt f (f' x) s x
hf : InjOn f s
g : E → F
⊢ Integrable (g ∘ f)
      (Measure.map Subtype.val (Measure.comap Subtype.val (μ.withDensity fun x ↦ ↑|(f' x).det|.toNNReal))) ↔
    IntegrableOn (fun x ↦ |(f' x).det| • g (f x)) s μ

Tactic:
map_comap_subtype_coe hs,

Goal state after:
E : Type u_1
F : Type u_2
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : NormedSpace ℝ E
inst✝⁵ : FiniteDimensional ℝ E
inst✝⁴ : NormedAddCommGroup F
inst✝³ : NormedSpace ℝ F
s : Set E
f : E → E
f' : E → E →L[ℝ] E
inst✝² : MeasurableSpace E
inst✝¹ : BorelSpace E
μ : Measure E
inst✝ : μ.IsAddHaarMeasure
hs : MeasurableSet s
hf' : ∀ x ∈ s, HasFDerivWithinAt f (f' x) s x
hf : InjOn f s
g : E → F
⊢ Integrable (g ∘ f) ((μ.withDensity fun x ↦ ↑|(f' x).det|.toNNReal).restrict s) ↔
    IntegrableOn (fun x ↦ |(f' x).det| • g (f x)) s μ
Module: Mathlib.MeasureTheory.Function.Jacobian

Goal state before:
pp: E : Type u_1
F : Type u_2
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : NormedSpace ℝ E
inst✝⁵ : FiniteDimensional ℝ E
inst✝⁴ : NormedAddCommGroup F
inst✝³ : NormedSpace ℝ F
s : Set E
f : E → E
f' : E → E →L[ℝ] E
inst✝² : MeasurableSpace E
inst✝¹ : BorelSpace E
μ : Measure E
inst✝ : μ.IsAddHaarMeasure
hs : MeasurableSet s
hf' : ∀ x ∈ s, HasFDerivWithinAt f (f' x) s x
hf : InjOn f s
g : E → F
⊢ ∫ (y : E),
      (g ∘ f) y ∂Measure.map Subtype.val (Measure.comap Subtype.val (μ.withDensity fun x ↦ ↑|(f' x).det|.toNNReal)) =
    ∫ (x : E) in s, |(f' x).det| • (g ∘ f) x ∂μ

Tactic:
map_comap_subtype_coe hs,

Goal state after:
E : Type u_1
F : Type u_2
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : NormedSpace ℝ E
inst✝⁵ : FiniteDimensional ℝ E
inst✝⁴ : NormedAddCommGroup F
inst✝³ : NormedSpace ℝ F
s : Set E
f : E → E
f' : E → E →L[ℝ] E
inst✝² : MeasurableSpace E
inst✝¹ : BorelSpace E
μ : Measure E
inst✝ : μ.IsAddHaarMeasure
hs : MeasurableSet s
hf' : ∀ x ∈ s, HasFDerivWithinAt f (f' x) s x
hf : InjOn f s
g : E → F
⊢ (∫ (y : E) in s, (g ∘ f) y ∂μ.withDensity fun x ↦ ↑|(f' x).det|.toNNReal) =
    ∫ (x : E) in s, |(f' x).det| • (g ∘ f) x ∂μ
Module: Mathlib.Geometry.Manifold.ContMDiff.Atlas

Goal state before:
pp: 𝕜 : Type u_1
inst✝⁸ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : NormedSpace 𝕜 E
H : Type u_3
inst✝⁵ : TopologicalSpace H
I : ModelWithCorners 𝕜 E H
M : Type u_4
inst✝⁴ : TopologicalSpace M
inst✝³ : ChartedSpace H M
n : ℕ∞ω
M' : Type u_5
inst✝² : TopologicalSpace M'
inst✝¹ : IsManifold I n M
inst✝ : ChartedSpace H M'
IsM' : IsManifold I n M'
f : OpenPartialHomeomorph M M'
h : LiftPropOn (contDiffGroupoid n I).IsLocalStructomorphWithinAt (↑f) f.source
X : M'
hX : X ∈ f.symm.source
x : M := ↑f.symm X
hx : x ∈ f.source
c : OpenPartialHomeomorph M H := chartAt H x
c' : OpenPartialHomeomorph M' H := chartAt H X
hxf :
  (contDiffGroupoid n I).IsLocalStructomorphWithinAt (↑(chartAt H (↑f x)) ∘ ↑f ∘ ↑(chartAt H x).symm)
    (↑(chartAt H x).symm ⁻¹' f.source) (↑(chartAt H x) x)
h2x : ↑(chartAt H X) X ∈ ↑(chartAt H X).symm ⁻¹' f.symm.source
⊢ c' = chartAt H (↑f x)

Tactic:
simp only [x, c', f.right_inv hX]

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

Goal state before:
pp: case hs
𝕜 : Type u_1
inst✝⁸ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : NormedSpace 𝕜 E
H : Type u_3
inst✝⁵ : TopologicalSpace H
I : ModelWithCorners 𝕜 E H
M : Type u_4
inst✝⁴ : TopologicalSpace M
inst✝³ : ChartedSpace H M
n : ℕ∞ω
M' : Type u_5
inst✝² : TopologicalSpace M'
inst✝¹ : IsManifold I n M
inst✝ : ChartedSpace H M'
IsM' : IsManifold I n M'
f : OpenPartialHomeomorph M M'
h : LiftPropOn (contDiffGroupoid n I).IsLocalStructomorphWithinAt (↑f) f.source
X : M'
hX : X ∈ f.symm.source
x : M := ↑f.symm X
hx : x ∈ f.source
c : OpenPartialHomeomorph M H := chartAt H x
c' : OpenPartialHomeomorph M' H := chartAt H X
hxf :
  (contDiffGroupoid n I).IsLocalStructomorphWithinAt (↑(c.symm ≫ₕ f ≫ₕ c')) (↑(chartAt H x).symm ⁻¹' f.source)
    (↑(chartAt H x) x)
h2x : ↑(chartAt H X) X ∈ ↑(chartAt H X).symm ⁻¹' f.symm.source
h1 : c' = chartAt H (↑f x)
h2 : ↑c' ∘ ↑f ∘ ↑c.symm = ↑(c.symm ≫ₕ f ≫ₕ c')
hcx : ↑c x ∈ ↑c.symm ⁻¹' f.source
⊢ (c.symm ≫ₕ f ≫ₕ c').source ⊆ ↑(chartAt H x).symm ⁻¹' f.source

Tactic:
dsimp [x, c]

Goal state after:
case hs
𝕜 : Type u_1
inst✝⁸ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : NormedSpace 𝕜 E
H : Type u_3
inst✝⁵ : TopologicalSpace H
I : ModelWithCorners 𝕜 E H
M : Type u_4
inst✝⁴ : TopologicalSpace M
inst✝³ : ChartedSpace H M
n : ℕ∞ω
M' : Type u_5
inst✝² : TopologicalSpace M'
inst✝¹ : IsManifold I n M
inst✝ : ChartedSpace H M'
IsM' : IsManifold I n M'
f : OpenPartialHomeomorph M M'
h : LiftPropOn (contDiffGroupoid n I).IsLocalStructomorphWithinAt (↑f) f.source
X : M'
hX : X ∈ f.symm.source
x : M := ↑f.symm X
hx : x ∈ f.source
c : OpenPartialHomeomorph M H := chartAt H x
c' : OpenPartialHomeomorph M' H := chartAt H X
hxf :
  (contDiffGroupoid n I).IsLocalStructomorphWithinAt (↑(c.symm ≫ₕ f ≫ₕ c')) (↑(chartAt H x).symm ⁻¹' f.source)
    (↑(chartAt H x) x)
h2x : ↑(chartAt H X) X ∈ ↑(chartAt H X).symm ⁻¹' f.symm.source
h1 : c' = chartAt H (↑f x)
h2 : ↑c' ∘ ↑f ∘ ↑c.symm = ↑(c.symm ≫ₕ f ≫ₕ c')
hcx : ↑c x ∈ ↑c.symm ⁻¹' f.source
⊢ (chartAt H (↑f.symm X)).target ∩
      (↑(chartAt H (↑f.symm X)).symm ⁻¹' f.source ∩ ↑(chartAt H (↑f.symm X)).symm ⁻¹' ↑f ⁻¹' c'.source) ⊆
    ↑(chartAt H (↑f.symm X)).symm ⁻¹' f.source
Module: Mathlib.Topology.VectorBundle.Basic

Goal state before:
pp: 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
⊢ Continuous[inst✝² b, PseudoMetricSpace.toUniformSpace.toTopologicalSpace] ⇑(Trivialization.linearMapAt R e b)

Tactic:
rw [e.coe_linearMapAt b]

Goal state after:
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
⊢ Continuous[inst✝² b, PseudoMetricSpace.toUniformSpace.toTopologicalSpace] fun y ↦
    if b ∈ e.baseSet then (↑e { proj := b, snd := y }).2 else 0
Module: Mathlib.Geometry.Manifold.VectorBundle.Tangent

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
H : Type u_4
inst✝³ : TopologicalSpace H
I : ModelWithCorners 𝕜 E H
M : Type u_6
inst✝² : TopologicalSpace M
inst✝¹ : ChartedSpace H M
inst✝ : IsManifold I 1 M
w x y z : M
v : E
h : z ∈ (extChartAt I w).source ∩ (extChartAt I x).source ∩ (extChartAt I y).source
⊢ (tangentCoordChange I x y z) ((tangentCoordChange I w x z) v) = (tangentCoordChange I w y z) v

Tactic:
apply (tangentBundleCore I M).coordChange_comp
  simp only [tangentBundleCore_baseSet, coe_achart, ← extChartAt_source I]
  exact h

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

Goal state before:
pp: case refine_1
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 n M
i j : ↑(atlas H M)
x : E
hx : x ∈ ((↑i).extend I).symm.source ∩ ↑((↑i).extend I).symm ⁻¹' (((↑i).extend I).symm.target ∩ ((↑j).extend I).source)
⊢ ((tangentBundleCore I M).coordChange i j ∘ ↑((↑i).extend I).symm) x =
    fderivWithin 𝕜 (↑((↑j).extend I) ∘ ↑((↑i).extend I).symm) (range ↑I) x

Tactic:
simp_rw [Function.comp_apply, tangentBundleCore_coordChange, (i.1.extend I).right_inv hx.1]

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

Goal state before:
pp: 𝕜 : Type u_1
inst✝⁷ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝⁶ : NormedAddCommGroup E
inst✝⁵ : NormedSpace 𝕜 E
H : Type u_3
inst✝⁴ : TopologicalSpace H
I : ModelWithCorners 𝕜 E H
M : Type u_4
inst✝³ : TopologicalSpace M
inst✝² : ChartedSpace H M
E' : Type u_5
inst✝¹ : NormedAddCommGroup E'
inst✝ : NormedSpace 𝕜 E'
z : M
f : M → E'
f' : TangentSpace I z →L[𝕜] E'
hf : HasMFDerivAt% (-f) z (-f')
⊢ HasMFDerivAt% f z f'

Tactic:
by convert! hf.neg <;> rw [neg_neg]

Goal state after:
Module: Mathlib.Geometry.Manifold.Diffeomorph

Goal state before:
pp: 𝕜 : Type u_1
inst✝¹⁸ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝¹⁷ : NormedAddCommGroup E
inst✝¹⁶ : NormedSpace 𝕜 E
E' : Type u_3
inst✝¹⁵ : NormedAddCommGroup E'
inst✝¹⁴ : NormedSpace 𝕜 E'
F : Type u_4
inst✝¹³ : NormedAddCommGroup F
inst✝¹² : NormedSpace 𝕜 F
H : Type u_5
inst✝¹¹ : TopologicalSpace H
H' : Type u_6
inst✝¹⁰ : TopologicalSpace H'
G : Type u_7
inst✝⁹ : TopologicalSpace G
G' : Type u_8
inst✝⁸ : TopologicalSpace G'
I : ModelWithCorners 𝕜 E H
I' : ModelWithCorners 𝕜 E' H'
J : ModelWithCorners 𝕜 F G
J' : ModelWithCorners 𝕜 F G'
M : Type u_9
inst✝⁷ : TopologicalSpace M
inst✝⁶ : ChartedSpace H M
M' : Type u_10
inst✝⁵ : TopologicalSpace M'
inst✝⁴ : ChartedSpace H' M'
N : Type u_11
inst✝³ : TopologicalSpace N
inst✝² : ChartedSpace G N
N' : Type u_12
inst✝¹ : TopologicalSpace N'
inst✝ : ChartedSpace G' N'
n : ℕ∞ω
e✝ e : E ≃L[𝕜] F
x : M
⊢ ↑(extChartAt I x).symm (↑(extChartAt I x) x) ∈
    ⇑(Equiv.refl M) ⁻¹' (extChartAt (I.transContinuousLinearEquiv e) ((Equiv.refl M) x)).source

Tactic:
by simp only [mfld_simps]

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

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
H : Type u_3
inst✝³ : TopologicalSpace H
I : ModelWithCorners 𝕜 E H
M : Type u_4
inst✝² : TopologicalSpace M
inst✝¹ : ChartedSpace H M
inst✝ : IsManifold I 1 M
x : M
⊢ mfderiv[range ↑I] ↑(extChartAt I x).symm (↑(extChartAt I x) x) =
    ContinuousLinearMap.id 𝕜 (TangentSpace 𝓘(𝕜, E) (↑(extChartAt I x) x))

Tactic:
have hcomp := mfderivWithin_extChartAt_symm_comp_mfderiv_extChartAt' (I := I)
    (mem_extChartAt_source x)

Goal state after:
𝕜 : Type u_1
inst✝⁶ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace 𝕜 E
H : Type u_3
inst✝³ : TopologicalSpace H
I : ModelWithCorners 𝕜 E H
M : Type u_4
inst✝² : TopologicalSpace M
inst✝¹ : ChartedSpace H M
inst✝ : IsManifold I 1 M
x : M
hcomp :
  mfderiv[range ↑I] ↑(extChartAt I x).symm (↑(extChartAt I x) x) ∘SL mfderiv% ↑(extChartAt I x) x =
    ContinuousLinearMap.id 𝕜 (TangentSpace I x)
⊢ mfderiv[range ↑I] ↑(extChartAt I x).symm (↑(extChartAt I x) x) =
    ContinuousLinearMap.id 𝕜 (TangentSpace 𝓘(𝕜, E) (↑(extChartAt I x) x))
Module: Mathlib.Geometry.Manifold.VectorBundle.ContMDiffSection

Goal state before:
pp: 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
f : M → 𝕜
s : (x : M) → V x
u : Set M
x₀ : M
hf : ContMDiffWithinAt I 𝓘(𝕜, 𝕜) n f u x₀
hs : ContMDiffWithinAt I (I.prod 𝓘(𝕜, F)) n (fun x ↦ ⟨x, s x⟩) u x₀
⊢ ContMDiffWithinAt I (I.prod 𝓘(𝕜, F)) n (fun x ↦ ⟨x, (f • s) x⟩) u x₀

Tactic:
rw [contMDiffWithinAt_section] at hs ⊢
  set e := trivializationAt F V x₀
  refine (hf.smul hs).congr_of_eventuallyEq ?_ ?_
  · apply eventually_of_mem (U := e.baseSet)
    · exact mem_nhdsWithin_of_mem_nhds <|
        (e.open_baseSet.mem_nhds <| mem_baseSet_trivializationAt F V x₀)
    · intro x hx
      apply (e.linear 𝕜 hx).2
  · apply (e.linear 𝕜 (FiberBundle.mem_baseSet_trivializationAt' x₀)).2

Goal state after:
Module: Mathlib.Topology.PartitionOfUnity

Goal state before:
pp: case hf
ι : Type u
X : Type v
inst✝ : TopologicalSpace X
s : Set X
f : BumpCovering ι X s
x : X
hx : x ∈ s
⊢ HasFiniteMulSupport fun i ↦ 1 - (f i) x

Tactic:
HasFiniteMulSupport,

Goal state after:
case hf
ι : Type u
X : Type v
inst✝ : TopologicalSpace X
s : Set X
f : BumpCovering ι X s
x : X
hx : x ∈ s
⊢ (mulSupport fun i ↦ 1 - (f i) x).Finite
Module: Mathlib.Geometry.Manifold.PartitionOfUnity

Goal state before:
pp: ι : Type uι
E : Type uE
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℝ E
F : Type uF
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace ℝ F
n : ℕ∞
s : Set E
f : SmoothPartitionOfUnity ι 𝓘(ℝ, E) E s
x₀ : E
g : ι → E → F
hφ : ∀ (i : ι), x₀ ∈ tsupport ⇑(f i) → ContDiffAt ℝ (↑n) (g i) x₀
⊢ ContDiffAt ℝ (↑n) (fun x ↦ ∑ᶠ (i : ι), (f i) x • g i x) x₀

Tactic:
by
  simp only [← contMDiffAt_iff_contDiffAt] at *
  exact f.contMDiffAt_finsum hφ

Goal state after:
Module: Mathlib.Analysis.Distribution.AEEqOfIntegralContDiff

Goal state before:
pp: case refine_1
E : Type u_1
inst✝¹³ : NormedAddCommGroup E
inst✝¹² : NormedSpace ℝ E
inst✝¹¹ : FiniteDimensional ℝ E
F : Type u_2
inst✝¹⁰ : NormedAddCommGroup F
inst✝⁹ : NormedSpace ℝ F
inst✝⁸ : CompleteSpace F
H : Type u_3
inst✝⁷ : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type u_4
inst✝⁶ : TopologicalSpace M
inst✝⁵ : ChartedSpace H M
inst✝⁴ : IsManifold I ∞ M
inst✝³ : MeasurableSpace M
inst✝² : BorelSpace M
inst✝¹ : T2Space M
f : M → F
μ : Measure M
inst✝ : SigmaCompactSpace M
hf : LocallyIntegrable f μ
h : ∀ (g : M → ℝ), ContMDiff I 𝓘(ℝ, ℝ) ∞ g → HasCompactSupport g → ∫ (x : M), g x • f x ∂μ = 0
this✝² : LocallyCompactSpace H
this✝¹ : LocallyCompactSpace M
this✝ : SecondCountableTopology H
this : SecondCountableTopology M
x✝¹ : MetricSpace M := metrizableSpaceMetric M
s : Set M
hs : IsCompact s
δ : ℝ
δpos : 0 < δ
hδ : IsCompact (cthickening δ s)
u : ℕ → ℝ
u_pos : ∀ (n : ℕ), u n ∈ Ioo 0 δ
u_lim : Tendsto u atTop (𝓝 0)
v : ℕ → Set M := fun n ↦ thickening (u n) s
K : Set M
K_compact : IsCompact K
vK : ∀ (n : ℕ), v n ⊆ K
g : ℕ → M → ℝ
g_supp : ∀ (n : ℕ), support (g n) = v n
g_diff : ∀ (n : ℕ), ContMDiff I 𝓘(ℝ, ℝ) ∞ (g n)
g_range : ∀ (n : ℕ), range (g n) ⊆ Icc 0 1
hg : ∀ (n : ℕ), ∀ x ∈ s, g n x = 1
bound : M → ℝ := K.indicator fun x ↦ ‖f x‖
A : ∀ (n : ℕ), AEStronglyMeasurable (fun x ↦ g n x • f x) μ
B : Integrable bound μ
n : ℕ
x : M
x✝ : x ∈ K
⊢ ‖g n x‖ * ‖f x‖ ≤ ‖f x‖

Tactic:
have : ‖g n x‖ ≤ 1 := by
          have := g_range n (mem_range_self (f := g n) x)
          rw [Real.norm_of_nonneg this.1]
          exact this.2
        exact mul_le_of_le_one_left (norm_nonneg _) this

Goal state after:
Module: Mathlib.MeasureTheory.Function.AEEqOfIntegral

Goal state before:
pp: case refine_2
α : Type u_1
E : Type u_2
m m0 : MeasurableSpace α
μ : Measure α
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
inst✝ : CompleteSpace E
hm : m ≤ m0
f : α → E
hf_int_finite : ∀ (s : Set α), MeasurableSet s → μ s < ∞ → IntegrableOn f s μ
hf_zero : ∀ (s : Set α), MeasurableSet s → μ s < ∞ → ∫ (x : α) in s, f x ∂μ = 0
hf : FinStronglyMeasurable f (μ.trim hm)
t : Set α
ht_meas : MeasurableSet t
htf_zero✝ : ∀ x ∈ tᶜ, f x = 0
htμ : SigmaFinite ((μ.trim hm).restrict t)
this : SigmaFinite ((μ.restrict t).trim hm)
htf_zero : f =ᵐ[μ.restrict tᶜ] 0
hf_meas_m : StronglyMeasurable f
s : Set α
hs : MeasurableSet s
hμs : μ (s ∩ t) < ∞
⊢ ∫ (x : α), f x ∂(μ.restrict (s ∩ t)).trim hm = 0

Tactic:
← integral_trim hm hf_meas_m

Goal state after:
case refine_2
α : Type u_1
E : Type u_2
m m0 : MeasurableSpace α
μ : Measure α
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
inst✝ : CompleteSpace E
hm : m ≤ m0
f : α → E
hf_int_finite : ∀ (s : Set α), MeasurableSet s → μ s < ∞ → IntegrableOn f s μ
hf_zero : ∀ (s : Set α), MeasurableSet s → μ s < ∞ → ∫ (x : α) in s, f x ∂μ = 0
hf : FinStronglyMeasurable f (μ.trim hm)
t : Set α
ht_meas : MeasurableSet t
htf_zero✝ : ∀ x ∈ tᶜ, f x = 0
htμ : SigmaFinite ((μ.trim hm).restrict t)
this : SigmaFinite ((μ.restrict t).trim hm)
htf_zero : f =ᵐ[μ.restrict tᶜ] 0
hf_meas_m : StronglyMeasurable f
s : Set α
hs : MeasurableSet s
hμs : μ (s ∩ t) < ∞
⊢ ∫ (x : α) in s ∩ t, f x ∂μ = 0
Module: Mathlib.MeasureTheory.Function.AEEqOfIntegral

Goal state before:
pp: α : Type u_1
E : Type u_2
m m0 : MeasurableSpace α
μ : Measure α
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
inst✝ : CompleteSpace E
hm : m ≤ m0
f : α → E
hf_int_finite : ∀ (s : Set α), MeasurableSet s → μ s < ∞ → IntegrableOn f s μ
hf_zero : ∀ (s : Set α), MeasurableSet s → μ s < ∞ → ∫ (x : α) in s, f x ∂μ = 0
hf : FinStronglyMeasurable f (μ.trim hm)
⊢ f =ᵐ[μ] 0

Tactic:
obtain ⟨t, ht_meas, htf_zero, htμ⟩ := hf.exists_set_sigmaFinite
  have : SigmaFinite ((μ.restrict t).trim hm) := by rwa [restrict_trim hm μ ht_meas] at htμ
  have htf_zero : f =ᵐ[μ.restrict tᶜ] 0 := by
    rw [EventuallyEq, ae_restrict_iff' (MeasurableSet.compl (hm _ ht_meas))]
    exact Eventually.of_forall htf_zero
  have hf_meas_m : StronglyMeasurable[m] f := hf.stronglyMeasurable
  suffices f =ᵐ[μ.restrict t] 0 from
    ae_of_ae_restrict_of_ae_restrict_compl _ this htf_zero
  refine measure_eq_zero_of_trim_eq_zero hm ?_
  refine ae_eq_zero_of_forall_setIntegral_eq_of_sigmaFinite ?_ ?_
  · intro s hs hμs
    unfold IntegrableOn
    rw [restrict_trim hm (μ.restrict t) hs, Measure.restrict_restrict (hm s hs)]
    rw [← restrict_trim hm μ ht_meas, Measure.restrict_apply hs,
      trim_measurableSet_eq hm (hs.inter ht_meas)] at hμs
    refine Integrable.trim hm ?_ hf_meas_m
    exact hf_int_finite _ (hs.inter ht_meas) hμs
  · intro s hs hμs
    rw [restrict_trim hm (μ.restrict t) hs, Measure.restrict_restrict (hm s hs)]
    rw [← restrict_trim hm μ ht_meas, Measure.restrict_apply hs,
      trim_measurableSet_eq hm (hs.inter ht_meas)] at hμs
    rw [← integral_trim hm hf_meas_m]
    exact hf_zero _ (hs.inter ht_meas) hμs

Goal state after:
Module: Mathlib.Analysis.Calculus.TangentCone.Seq

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
inst✝⁴ : NormedDivisionRing 𝕜
inst✝³ : AddCommGroup E
inst✝² : Module 𝕜 E
inst✝¹ : TopologicalSpace E
inst✝ : ContinuousSMul 𝕜 E
s : Set E
x y : E
r : 𝕜
hr₀ : r ≠ 0
hr : ‖r‖ < 1
hs : ∀ᶠ (n : ℕ) in atTop, x + r ^ n • y ∈ s
⊢ ∀ᶠ (n : ℕ) in atTop, r ^ n ∈ {0}ᶜ

Tactic:
simp [hr₀]

Goal state after:
Module: Mathlib.MeasureTheory.Integral.IntervalIntegral.Slope

Goal state before:
pp: f : ℝ → ℝ
a b c : ℝ
hf : MonotoneOn f (Icc a (b + c))
hab : a ≤ b
hc✝ : 0 ≤ c
hc : 0 < c
⊢ a ≤ b + c

Tactic:
by linarith

Goal state after:
Module: Mathlib.MeasureTheory.Integral.IntervalIntegral.Slope

Goal state before:
pp: 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)
⊢ b ≤ b + c

Tactic:
by linarith

Goal state after:
Module: Mathlib.MeasureTheory.Integral.IntervalIntegral.DerivIntegrable

Goal state before:
pp: f : ℝ → ℝ
a b : ℝ
hf : MonotoneOn f (uIcc a b)
h : ∀ {a b : ℝ}, MonotoneOn f (uIcc a b) → a ≤ b → IntervalIntegrable (deriv f) volume a b
hab : ¬a ≤ b
⊢ b ≤ a

Tactic:
by linarith

Goal state after:
Module: Mathlib.MeasureTheory.Integral.IntervalIntegral.Slope

Goal state before:
pp: 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)
⊢ a ≤ a + c

Tactic:
by linarith

Goal state after: