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
hG : G.LocalInvariantProp G' P
U V : Opens M
hUV : U ≤ V
f : ↥V → M'
x : ↥U
⊢ LiftPropAt P f (inclusion hUV x) ↔ LiftPropAt P (f ∘ inclusion hUV) x

Tactic:
simp only [LiftPropAt, liftPropWithinAt_iff']

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
hG : G.LocalInvariantProp G' P
U V : Opens M
hUV : U ≤ V
f : ↥V → M'
x : ↥U
⊢ ContinuousWithinAt f univ (inclusion hUV x) ∧
      P (↑(chartAt H' (f (inclusion hUV x))) ∘ f ∘ ↑(chartAt H (inclusion hUV x)).symm)
        (↑(chartAt H (inclusion hUV x)).symm ⁻¹' univ) (↑(chartAt H (inclusion hUV x)) (inclusion hUV x)) ↔
    ContinuousWithinAt (f ∘ inclusion hUV) univ x ∧
      P (↑(chartAt H' ((f ∘ inclusion hUV) x)) ∘ (f ∘ inclusion hUV) ∘ ↑(chartAt H x).symm)
        (↑(chartAt H x).symm ⁻¹' univ) (↑(chartAt H x) x)
Module: Mathlib.Geometry.Manifold.ContMDiff.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
E' : Type u_5
inst✝⁵ : NormedAddCommGroup E'
inst✝⁴ : NormedSpace 𝕜 E'
H' : Type u_6
inst✝³ : TopologicalSpace H'
I' : ModelWithCorners 𝕜 E' H'
M' : Type u_7
inst✝² : TopologicalSpace M'
inst✝¹ : ChartedSpace H M
inst✝ : ChartedSpace H' M'
f : M → M'
n : ℕ∞ω
ι : Type u_11
s : ι → Set M
hf : ∀ (i : ι), ContMDiffOn I I' n f (s i)
hs : ∀ (i : ι), IsOpen[inst✝⁶] (s i)
hs' : ⋃ i, s i = univ
⊢ ContMDiffOn I I' n f univ

Tactic:
← hs'

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
E' : Type u_5
inst✝⁵ : NormedAddCommGroup E'
inst✝⁴ : NormedSpace 𝕜 E'
H' : Type u_6
inst✝³ : TopologicalSpace H'
I' : ModelWithCorners 𝕜 E' H'
M' : Type u_7
inst✝² : TopologicalSpace M'
inst✝¹ : ChartedSpace H M
inst✝ : ChartedSpace H' M'
f : M → M'
n : ℕ∞ω
ι : Type u_11
s : ι → Set M
hf : ∀ (i : ι), ContMDiffOn I I' n f (s i)
hs : ∀ (i : ι), IsOpen[inst✝⁶] (s i)
hs' : ⋃ i, s i = univ
⊢ ContMDiffOn I I' n f (⋃ i, s i)
Module: Mathlib.Geometry.Manifold.ContMDiff.Basic

Goal state before:
pp: case right
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
e : M → H
h : IsOpenEmbedding e
n : ℕ∞ω
inst✝ : Nonempty M
this✝¹ : IsManifold I ω M
x : M
y : H
z : E
hz : z ∈ (extChartAt I x).target ∩ ↑(extChartAt I x).symm ⁻¹' e ⁻¹' (extChartAt I y).source
this✝ : z ∈ ↑I.symm ⁻¹' (chartAt H x).target ∩ range ↑I
this : z ∈ ↑I.symm ⁻¹' (chartAt H x).target
⊢ ↑I.symm z ∈ range e

Tactic:
rw [mem_preimage, OpenPartialHomeomorph.singletonChartedSpace_chartAt_eq,
      h.toOpenPartialHomeomorph_target] at this

Goal state after:
case right
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
e : M → H
h : IsOpenEmbedding e
n : ℕ∞ω
inst✝ : Nonempty M
this✝¹ : IsManifold I ω M
x : M
y : H
z : E
hz : z ∈ (extChartAt I x).target ∩ ↑(extChartAt I x).symm ⁻¹' e ⁻¹' (extChartAt I y).source
this✝ : z ∈ ↑I.symm ⁻¹' (chartAt H x).target ∩ range ↑I
this : ↑I.symm z ∈ range e
⊢ ↑I.symm z ∈ range e
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 : NullMeasurableSet s μ
hf' : ∀ x ∈ s, HasFDerivWithinAt f (f' x) s x
hf : InjOn f s
⊢ AEMeasurable f (μ.restrict s)

Tactic:
apply ContinuousOn.aemeasurable₀ (fun x hx ↦ ?_) 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 : NullMeasurableSet s μ
hf' : ∀ x ∈ s, HasFDerivWithinAt f (f' x) s x
hf : InjOn f s
x : E
hx : x ∈ s
⊢ ContinuousWithinAt f s x
Module: Mathlib.Geometry.Manifold.ContMDiff.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
E' : Type u_5
inst✝⁵ : NormedAddCommGroup E'
inst✝⁴ : NormedSpace 𝕜 E'
H' : Type u_6
inst✝³ : TopologicalSpace H'
I' : ModelWithCorners 𝕜 E' H'
M' : Type u_7
inst✝² : TopologicalSpace M'
n : ℕ∞ω
inst✝¹ : ChartedSpace H M
inst✝ : Nonempty M'
e' : M' → H'
h' : IsOpenEmbedding e'
f : M → M'
hf : ContMDiff I I' n (e' ∘ f)
this : f = ↑(IsOpenEmbedding.toOpenPartialHomeomorph e' h').symm ∘ e' ∘ f
⊢ ContMDiff I I' n f

Tactic:
rw [this]

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
E' : Type u_5
inst✝⁵ : NormedAddCommGroup E'
inst✝⁴ : NormedSpace 𝕜 E'
H' : Type u_6
inst✝³ : TopologicalSpace H'
I' : ModelWithCorners 𝕜 E' H'
M' : Type u_7
inst✝² : TopologicalSpace M'
n : ℕ∞ω
inst✝¹ : ChartedSpace H M
inst✝ : Nonempty M'
e' : M' → H'
h' : IsOpenEmbedding e'
f : M → M'
hf : ContMDiff I I' n (e' ∘ f)
this : f = ↑(IsOpenEmbedding.toOpenPartialHomeomorph e' h').symm ∘ e' ∘ f
⊢ ContMDiff I I' n (↑(IsOpenEmbedding.toOpenPartialHomeomorph e' h').symm ∘ e' ∘ f)
Module: Mathlib.Geometry.Manifold.ContMDiff.Defs

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'
e : OpenPartialHomeomorph M H
e' : OpenPartialHomeomorph M' H'
f : M → M'
n : ℕ∞ω
x : M
he : e ∈ maximalAtlas I n M
he' : e' ∈ maximalAtlas I' n M'
hx : x ∈ e.source
hy : f x ∈ e'.source
⊢ ContMDiffAt I I' n f x ↔
    ContinuousAt f x ∧ ContDiffWithinAt 𝕜 n (↑(e'.extend I') ∘ f ∘ ↑(e.extend I).symm) (range ↑I) (↑(e.extend I) x)

Tactic:
rw [← contMDiffWithinAt_univ,
    contMDiffWithinAt_iff_of_mem_maximalAtlas he he' hx hy,
    continuousWithinAt_univ, preimage_univ, univ_inter]

Goal state after:
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 ∘ s.domRestrict f) (Measure.comap Subtype.val (μ.withDensity fun x ↦ ENNReal.ofReal |(f' x).det|)) ↔
    IntegrableOn (fun x ↦ |(f' x).det| • g (f x)) s μ

Tactic:
simp only [Set.domRestrict_eq, ← Function.comp_assoc, ENNReal.ofReal]

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) ∘ 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
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
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
⊢ ∃ e ∈ contDiffGroupoid n I,
    e.source ⊆ (c.symm ≫ₕ f ≫ₕ c').source ∧ EqOn (↑c' ∘ ↑f ∘ ↑c.symm) (↑e) e.source ∧ ↑c x ∈ e.source

Tactic:
rw [h2]

Goal state after:
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
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
⊢ ∃ e ∈ contDiffGroupoid n I,
    e.source ⊆ (c.symm ≫ₕ f ≫ₕ c').source ∧ EqOn (↑(c.symm ≫ₕ f ≫ₕ c')) (↑e) e.source ∧ ↑c x ∈ e.source
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 : Set M
h : MDiffAt[s] f x
hs : s ∈ 𝓝 x
⊢ s = univ ∩ s

Tactic:
rw [univ_inter]

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'
ι : Type u_11
s : ι → Set M
hf : ∀ (i : ι), MDiff[s i] f
hs : ∀ (i : ι), IsOpen[inst✝⁶] (s i)
hs' : ⋃ i, s i = univ
⊢ MDiff[univ] f

Tactic:
← hs'

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'
f : M → M'
ι : Type u_11
s : ι → Set M
hf : ∀ (i : ι), MDiff[s i] f
hs : ∀ (i : ι), IsOpen[inst✝⁶] (s i)
hs' : ⋃ i, s i = univ
⊢ MDiff[⋃ i, s i] f
Module: Mathlib.Geometry.Manifold.MFDeriv.Basic

Goal state before:
pp: 𝕜 : Type u_1
inst✝¹⁵ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝¹⁴ : NormedAddCommGroup E
inst✝¹³ : NormedSpace 𝕜 E
H : Type u_3
inst✝¹² : TopologicalSpace H
I : ModelWithCorners 𝕜 E H
M : Type u_4
inst✝¹¹ : TopologicalSpace M
inst✝¹⁰ : ChartedSpace H M
E' : Type u_5
inst✝⁹ : NormedAddCommGroup E'
inst✝⁸ : NormedSpace 𝕜 E'
H' : Type u_6
inst✝⁷ : TopologicalSpace H'
I' : ModelWithCorners 𝕜 E' H'
M' : Type u_7
inst✝⁶ : TopologicalSpace M'
inst✝⁵ : ChartedSpace H' M'
E'' : Type u_8
inst✝⁴ : NormedAddCommGroup E''
inst✝³ : NormedSpace 𝕜 E''
H'' : Type u_9
inst✝² : TopologicalSpace H''
I'' : ModelWithCorners 𝕜 E'' H''
M'' : Type u_10
inst✝¹ : TopologicalSpace M''
inst✝ : ChartedSpace H'' M''
f : M → M'
x : M
s : Set M
g : M' → M''
u : Set M'
f' : TangentSpace I x →L[𝕜] TangentSpace I' (f x)
g' : TangentSpace I' (f x) →L[𝕜] TangentSpace I'' (g (f x))
hg : HasMFDerivAt[u] g (f x) g'
hf : HasMFDerivAt[s] f x f'
hst : s ⊆ f ⁻¹' u
⊢ HasMFDerivAt[s] (g ∘ f) x (g' ∘SL f')

Tactic:
by
  refine ⟨ContinuousWithinAt.comp hg.1 hf.1 hst, ?_⟩
  have A :
    HasFDerivWithinAt (writtenInExtChartAt I' I'' (f x) g ∘ writtenInExtChartAt I I' x f)
      (ContinuousLinearMap.comp g' f' : E →L[𝕜] E'') ((extChartAt I x).symm ⁻¹' s ∩ range I)
      ((extChartAt I x) x) := by
    have :
      (extChartAt I x).symm ⁻¹' f ⁻¹' (extChartAt I' (f x)).source ∈
        𝓝[(extChartAt I x).symm ⁻¹' s ∩ range I] (extChartAt I x) x :=
      extChartAt_preimage_mem_nhdsWithin
        (hf.1.preimage_mem_nhdsWithin (extChartAt_source_mem_nhds _))
    unfold HasMFDerivWithinAt at *
    rw [← hasFDerivWithinAt_inter' this, ← extChartAt_preimage_inter_eq] at hf ⊢
    have : writtenInExtChartAt I I' x f ((extChartAt I x) x) = (extChartAt I' (f x)) (f x) := by
      simp only [mfld_simps]
    rw [← this] at hg
    apply HasFDerivWithinAt.comp ((extChartAt I x) x) hg.2 hf.2 _
    intro y hy
    simp only [mfld_simps] at hy
    have : f (((chartAt H x).symm : H → M) (I.symm y)) ∈ u := hst hy.1.1
    simp only [hy, this, mfld_simps]
  apply A.congr_of_eventuallyEq (writtenInExtChartAt_comp hf.1)
  simp only [mfld_simps]

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

Goal state before:
pp: 𝕜 : Type u_1
inst✝¹⁵ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝¹⁴ : NormedAddCommGroup E
inst✝¹³ : NormedSpace 𝕜 E
H : Type u_3
inst✝¹² : TopologicalSpace H
I : ModelWithCorners 𝕜 E H
M : Type u_4
inst✝¹¹ : TopologicalSpace M
inst✝¹⁰ : ChartedSpace H M
E' : Type u_5
inst✝⁹ : NormedAddCommGroup E'
inst✝⁸ : NormedSpace 𝕜 E'
H' : Type u_6
inst✝⁷ : TopologicalSpace H'
I' : ModelWithCorners 𝕜 E' H'
M' : Type u_7
inst✝⁶ : TopologicalSpace M'
inst✝⁵ : ChartedSpace H' M'
E'' : Type u_8
inst✝⁴ : NormedAddCommGroup E''
inst✝³ : NormedSpace 𝕜 E''
H'' : Type u_9
inst✝² : TopologicalSpace H''
I'' : ModelWithCorners 𝕜 E'' H''
M'' : Type u_10
inst✝¹ : TopologicalSpace M''
inst✝ : ChartedSpace H'' M''
f : M → M'
x : M
s : Set M
g : M' → M''
u : Set M'
y : M'
hg : MDiffAt[u] g y
hf : MDiffAt[s] f x
h : s ⊆ f ⁻¹' u
hy : f x = y
⊢ MDiffAt[s] (g ∘ f) x

Tactic:
by
  subst hy; exact hg.comp _ hf h

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

Goal state before:
pp: 𝕜 : Type u_1
inst✝¹⁵ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝¹⁴ : NormedAddCommGroup E
inst✝¹³ : NormedSpace 𝕜 E
H : Type u_3
inst✝¹² : TopologicalSpace H
I : ModelWithCorners 𝕜 E H
M : Type u_4
inst✝¹¹ : TopologicalSpace M
inst✝¹⁰ : ChartedSpace H M
E' : Type u_5
inst✝⁹ : NormedAddCommGroup E'
inst✝⁸ : NormedSpace 𝕜 E'
H' : Type u_6
inst✝⁷ : TopologicalSpace H'
I' : ModelWithCorners 𝕜 E' H'
M' : Type u_7
inst✝⁶ : TopologicalSpace M'
inst✝⁵ : ChartedSpace H' M'
E'' : Type u_8
inst✝⁴ : NormedAddCommGroup E''
inst✝³ : NormedSpace 𝕜 E''
H'' : Type u_9
inst✝² : TopologicalSpace H''
I'' : ModelWithCorners 𝕜 E'' H''
M'' : Type u_10
inst✝¹ : TopologicalSpace M''
inst✝ : ChartedSpace H'' M''
f : M → M'
x : M
s : Set M
g : M' → M''
u : Set M'
y : M'
hg : MDiffAt[u] g y
hf : MDiffAt[s] f x
h : f ⁻¹' u ∈ 𝓝[s] x
hy : f x = y
⊢ MDiffAt[s] (g ∘ f) x

Tactic:
by
  subst hy; exact MDifferentiableWithinAt.comp_of_preimage_mem_nhdsWithin _ hg hf h

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
x₀ x : M
hx : x ∈ (chartAt H x₀).source
⊢ Trivialization.symmL 𝕜 (trivializationAt E (TangentSpace I) x₀) x =
    mfderiv[range ↑I] ↑(extChartAt I x₀).symm (↑(extChartAt I x₀) x)

Tactic:
by
  have : MDiffAt[range I] ((chartAt H x₀).symm ∘ I.symm) (I (chartAt H x₀ x)) := by
    simpa using mdifferentiableWithinAt_extChartAt_symm (by simp [hx])
  simp [hx, mfderivWithin, this]

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

Goal state before:
pp: E : Type uE
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace ℝ E
H : Type uH
inst✝³ : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type uM
inst✝² : TopologicalSpace M
inst✝¹ : ChartedSpace H M
c : M
f : SmoothBumpFunction I c
x : M
inst✝ : FiniteDimensional ℝ E
hs : x ∈ (chartAt H c).source
hd : dist (↑(extChartAt I c) x) (↑(extChartAt I c) c) < f.rIn
⊢ ↑f =ᶠ[𝓝 x] 1

Tactic:
by
  filter_upwards [IsOpen.mem_nhds (isOpen_extChartAt_preimage c isOpen_ball) ⟨hs, hd⟩]
  rintro z ⟨hzs, hzd⟩
  exact f.one_of_dist_le hzs <| le_of_lt hzd

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

Goal state before:
pp: E : Type uE
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace ℝ E
H : Type uH
inst✝³ : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type uM
inst✝² : TopologicalSpace M
inst✝¹ : ChartedSpace H M
c : M
f : SmoothBumpFunction I c
inst✝ : FiniteDimensional ℝ E
s : Set M
hsc : IsClosed[inst✝²] s
hs : s ⊆ support ↑f
e : PartialEquiv M E := extChartAt I c
⊢ ∃ r ∈ Ioo 0 f.rOut, s ⊆ (chartAt H c).source ∩ ↑(extChartAt I c) ⁻¹' ball (↑(extChartAt I c) c) r

Tactic:
have : IsClosed (e '' s) := f.isClosed_image_of_isClosed hsc hs

Goal state after:
E : Type uE
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace ℝ E
H : Type uH
inst✝³ : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type uM
inst✝² : TopologicalSpace M
inst✝¹ : ChartedSpace H M
c : M
f : SmoothBumpFunction I c
inst✝ : FiniteDimensional ℝ E
s : Set M
hsc : IsClosed[inst✝²] s
hs : s ⊆ support ↑f
e : PartialEquiv M E := extChartAt I c
this : IsClosed[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] (↑e '' s)
⊢ ∃ r ∈ Ioo 0 f.rOut, s ⊆ (chartAt H c).source ∩ ↑(extChartAt I c) ⁻¹' ball (↑(extChartAt I c) c) r
Module: Mathlib.Topology.PartitionOfUnity

Goal state before:
pp: ι : Type u
X : Type v
inst✝ : TopologicalSpace X
s✝ : Set X
f : BumpCovering ι X s✝
i : ι
s : Set X
x✝ : X
j : ι
x : X
hx : j ∉ {i}
⊢ x ∉ support ⇑(Pi.single i 1 j)

Tactic:
rw [mem_singleton_iff] at hx

Goal state after:
ι : Type u
X : Type v
inst✝ : TopologicalSpace X
s✝ : Set X
f : BumpCovering ι X s✝
i : ι
s : Set X
x✝ : X
j : ι
x : X
hx : ¬j = i
⊢ x ∉ support ⇑(Pi.single i 1 j)
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:
· rw [HasFiniteMulSupport, mulSupport_one_sub]
      exact f.point_finite x

Goal state after:
Module: Mathlib.Topology.ShrinkingLemma

Goal state before:
pp: case neg.right
ι : Type u_1
X : Type u_2
inst✝² : TopologicalSpace X
u : ι → Set X
s : Set X
inst✝¹ : T2Space X
inst✝ : LocallyCompactSpace X
v : PartialRefinement u s fun w ↦ IsCompact (closure w)
hs : IsCompact s
i : ι
hi : i ∉ v.carrier
si : Set X := ⋯
hsi : si = s ∩ ⋂ i_1, ⋂ (_ : ¬i_1 = i), (v.toFun i_1)ᶜ
hsic : IsCompact si
this : si ⊆ v.toFun i
vi : Set X
hvi : IsOpen vi ∧ si ⊆ vi ∧ closure vi ⊆ v.toFun i ∧ IsCompact (closure vi)
j : ι
hj : j = i ∨ j ∈ v.carrier
h : ¬j = i
⊢ j ∈ v.carrier → IsCompact (closure (update v.toFun i vi j))

Tactic:
· intro hjmemv
        rw [update_of_ne h]
        exact v.pred_of_mem hjmemv

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

Goal state before:
pp: 𝕜 : Type u_1
inst✝⁵ : Semiring 𝕜
ι : Type u_2
E : ι → Type u_3
inst✝⁴ : (i : ι) → AddCommGroup (E i)
inst✝³ : (i : ι) → Module 𝕜 (E i)
inst✝² : (i : ι) → TopologicalSpace (E i)
inst✝¹ : ∀ (i : ι), ContinuousAdd (E i)
inst✝ : ∀ (i : ι), ContinuousConstSMul 𝕜 (E i)
s : (i : ι) → Set (E i)
x : (i : ι) → E i
h : ∀ (i : ι), Dense ↑(Submodule.span 𝕜 (tangentConeAt 𝕜 (s i) (x i))) ∧ x i ∈ closure[inst✝² i] (s i)
⊢ Dense ↑(Submodule.span 𝕜 (tangentConeAt 𝕜 (univ.pi s) x)) ∧ x ∈ univ.pi fun i ↦ closure[inst✝² i] (s i)

Tactic:
refine ⟨.of_closure <| (dense_pi univ fun i _ ↦ (h i).1).closure.mono ?_, fun i _ => (h i).2⟩

Goal state after:
𝕜 : Type u_1
inst✝⁵ : Semiring 𝕜
ι : Type u_2
E : ι → Type u_3
inst✝⁴ : (i : ι) → AddCommGroup (E i)
inst✝³ : (i : ι) → Module 𝕜 (E i)
inst✝² : (i : ι) → TopologicalSpace (E i)
inst✝¹ : ∀ (i : ι), ContinuousAdd (E i)
inst✝ : ∀ (i : ι), ContinuousConstSMul 𝕜 (E i)
s : (i : ι) → Set (E i)
x : (i : ι) → E i
h : ∀ (i : ι), Dense ↑(Submodule.span 𝕜 (tangentConeAt 𝕜 (s i) (x i))) ∧ x i ∈ closure[inst✝² i] (s i)
⊢ closure[Pi.topologicalSpace] (univ.pi fun i ↦ ↑(Submodule.span 𝕜 (tangentConeAt 𝕜 (s i) (x i)))) ⊆
    closure[Pi.topologicalSpace] ↑(Submodule.span 𝕜 (tangentConeAt 𝕜 (univ.pi s) x))
Module: Mathlib.MeasureTheory.Integral.IntervalIntegral.DerivIntegrable

Goal state before:
pp: case e'_3
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
f_deriv_nonneg : ∀ {x : ℝ}, x ∈ Ioo a b → 0 ≤ deriv f x
ebound : ∫ (x : ℝ) in a..b, ‖deriv f x‖ ≤ f b - f a
⊢ ∫ (x : ℝ) in a..b, deriv f x = ∫ (x : ℝ) in a..b, ‖deriv f x‖

Tactic:
refine intervalIntegral.integral_congr_uIoo ?_

Goal state after:
case e'_3
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
f_deriv_nonneg : ∀ {x : ℝ}, x ∈ Ioo a b → 0 ≤ deriv f x
ebound : ∫ (x : ℝ) in a..b, ‖deriv f x‖ ≤ f b - f a
⊢ EqOn (deriv f) (fun x ↦ ‖deriv f x‖) (uIoo a b)