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
⊢ AEMeasurable (fun x ↦ |(f' x).det|) (μ.restrict s)

Tactic:
refine continuous_abs.measurable.comp_aemeasurable ?_

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
⊢ AEMeasurable (fun x ↦ (f' x).det) (μ.restrict s)
Module: Mathlib.Geometry.Manifold.ChartedSpace

Goal state before:
pp: case refine_1
H : Type u
M : Type u_2
inst✝³ : TopologicalSpace H
inst✝² : TopologicalSpace M
inst✝¹ : ChartedSpace H M
inst✝ : LocallyConnectedSpace H
e : M → OpenPartialHomeomorph M H := chartAt H
⊢ ∀ (x : M),
    (𝓝 x).HasBasis (fun s ↦ (IsOpen[inst✝³] s ∧ ↑(e x) x ∈ s ∧ IsConnected s) ∧ s ⊆ (e x).target) fun s ↦
      ↑(e x).symm '' s

pp: case refine_2
H : Type u
M : Type u_2
inst✝³ : TopologicalSpace H
inst✝² : TopologicalSpace M
inst✝¹ : ChartedSpace H M
inst✝ : LocallyConnectedSpace H
e : M → OpenPartialHomeomorph M H := chartAt H
⊢ ∀ (x : M) (i : Set H),
    (IsOpen[inst✝³] i ∧ ↑(e x) x ∈ i ∧ IsConnected i) ∧ i ⊆ (e x).target → IsPreconnected (↑(e x).symm '' i)

Tactic:
· intro x
    simpa only [e, OpenPartialHomeomorph.symm_map_nhds_eq, mem_chart_source] using!
      ((LocallyConnectedSpace.open_connected_basis (e x x)).restrict_subset
        ((e x).open_target.mem_nhds (mem_chart_target H x))).map (e x).symm

Goal state after:
case refine_2
H : Type u
M : Type u_2
inst✝³ : TopologicalSpace H
inst✝² : TopologicalSpace M
inst✝¹ : ChartedSpace H M
inst✝ : LocallyConnectedSpace H
e : M → OpenPartialHomeomorph M H := chartAt H
⊢ ∀ (x : M) (i : Set H),
    (IsOpen[inst✝³] i ∧ ↑(e x) x ∈ i ∧ IsConnected i) ∧ i ⊆ (e x).target → IsPreconnected (↑(e x).symm '' i)
Module: Mathlib.Geometry.Manifold.ChartedSpace

Goal state before:
pp: H : Type u
M : Type u_2
inst✝³ : TopologicalSpace H
inst✝² : TopologicalSpace M
inst✝¹ : ChartedSpace H M
inst✝ : LocallyPathConnectedSpace H
x : M
s : Set M
hs : s ∈ 𝓝 x
e : OpenPartialHomeomorph M H := chartAt H x
⊢ ∃ i, (i ∈ 𝓝 x ∧ IsPathConnected i) ∧ id i ⊆ s

Tactic:
let t := s ∩ e.source

Goal state after:
H : Type u
M : Type u_2
inst✝³ : TopologicalSpace H
inst✝² : TopologicalSpace M
inst✝¹ : ChartedSpace H M
inst✝ : LocallyPathConnectedSpace H
x : M
s : Set M
hs : s ∈ 𝓝 x
e : OpenPartialHomeomorph M H := chartAt H x
t : Set M := s ∩ e.source
⊢ ∃ i, (i ∈ 𝓝 x ∧ IsPathConnected i) ∧ id i ⊆ s
Module: Mathlib.Geometry.Manifold.ChartedSpace

Goal state before:
pp: H : Type u_5
H' : Type u_6
α : Type u_7
f : H → α
⊢ (range fun p ↦ (f p.1, p.2)) = range f ×ˢ univ

Tactic:
rw [prod_range_univ_eq]

Goal state after:
H : Type u_5
H' : Type u_6
α : Type u_7
f : H → α
⊢ (range fun p ↦ (f p.1, p.2)) = range fun p ↦ (f p.1, p.2)
Module: Mathlib.Geometry.Manifold.LocalInvariantProperties

Goal state before:
pp: H : Type u_1
H' : Type u_3
inst✝¹ : TopologicalSpace H
inst✝ : TopologicalSpace H'
G : StructureGroupoid H
G' : StructureGroupoid H'
P : (H → H') → Set H → H → Prop
hG : G.LocalInvariantProp G' P
s : Set H
x : H
f : H → H'
e : OpenPartialHomeomorph H H
he : e ∈ G
hxe : x ∈ e.source
h : P (f ∘ ↑e.symm) (↑e.symm ⁻¹' s) (↑e x)
this : P ((f ∘ ↑e.symm) ∘ ↑e.symm.symm) (↑e.symm.symm ⁻¹' ↑e.symm ⁻¹' s) (↑e.symm (↑e x))
⊢ P f s x

Tactic:
e.symm_symm,

Goal state after:
H : Type u_1
H' : Type u_3
inst✝¹ : TopologicalSpace H
inst✝ : TopologicalSpace H'
G : StructureGroupoid H
G' : StructureGroupoid H'
P : (H → H') → Set H → H → Prop
hG : G.LocalInvariantProp G' P
s : Set H
x : H
f : H → H'
e : OpenPartialHomeomorph H H
he : e ∈ G
hxe : x ∈ e.source
h : P (f ∘ ↑e.symm) (↑e.symm ⁻¹' s) (↑e x)
this : P ((f ∘ ↑e.symm) ∘ ↑e) (↑e ⁻¹' ↑e.symm ⁻¹' s) (↑e.symm (↑e x))
⊢ P f s x
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'
f : OpenPartialHomeomorph M' H'
P : (H → H') → Set H → H → Prop
g : M → M'
s : Set M
x : M
hG : G.LocalInvariantProp G' P
hf : f ∈ maximalAtlas M' G'
xf : g x ∈ f.source
⊢ LiftPropWithinAt P g s x ↔ ContinuousWithinAt g s x ∧ LiftPropWithinAt P (↑f ∘ g) s x

Tactic:
rw [liftPropWithinAt_self_target, liftPropWithinAt_iff', and_congr_right_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'
f : OpenPartialHomeomorph M' H'
P : (H → H') → Set H → H → Prop
g : M → M'
s : Set M
x : M
hG : G.LocalInvariantProp G' P
hf : f ∈ maximalAtlas M' G'
xf : g x ∈ f.source
⊢ ContinuousWithinAt g s x →
    (P (↑(chartAt H' (g x)) ∘ g ∘ ↑(chartAt H x).symm) (↑(chartAt H x).symm ⁻¹' s) (↑(chartAt H x) x) ↔
      ContinuousWithinAt (↑f ∘ g) s x ∧
        P ((↑f ∘ g) ∘ ↑(chartAt H x).symm) (↑(chartAt H x).symm ⁻¹' s) (↑(chartAt H x) x))
Module: Mathlib.Geometry.Manifold.LocalInvariantProperties

Goal state before:
pp: H : Type u_1
inst✝¹ : TopologicalSpace H
G : StructureGroupoid H
inst✝ : ClosedUnderRestriction G
s : Set H
x : H
f : H → H
e' : OpenPartialHomeomorph H H
he'G : e' ∈ G
he'x : x ∈ e'.source
h : G.IsLocalStructomorphWithinAt f s x
hx : ↑e' x ∈ ↑e'.symm ⁻¹' s
hxs : x ∈ s
e : OpenPartialHomeomorph H H
heG : e ∈ G
hef : EqOn f (↑e.toPartialEquiv) (s ∩ e.source)
hex : x ∈ e.source
⊢ ∃ e ∈ G, EqOn (f ∘ ↑e'.symm) (↑e.toPartialEquiv) (↑e'.symm ⁻¹' s ∩ e.source) ∧ ↑e' x ∈ e.source

Tactic:
refine ⟨e'.symm.trans e, G.trans (G.symm he'G) heG, ?_, ?_⟩

Goal state after:
case refine_1
H : Type u_1
inst✝¹ : TopologicalSpace H
G : StructureGroupoid H
inst✝ : ClosedUnderRestriction G
s : Set H
x : H
f : H → H
e' : OpenPartialHomeomorph H H
he'G : e' ∈ G
he'x : x ∈ e'.source
h : G.IsLocalStructomorphWithinAt f s x
hx : ↑e' x ∈ ↑e'.symm ⁻¹' s
hxs : x ∈ s
e : OpenPartialHomeomorph H H
heG : e ∈ G
hef : EqOn f (↑e.toPartialEquiv) (s ∩ e.source)
hex : x ∈ e.source
⊢ EqOn (f ∘ ↑e'.symm) (↑(e'.symm ≫ₕ e).toPartialEquiv) (↑e'.symm ⁻¹' s ∩ (e'.symm ≫ₕ e).source)

case refine_2
H : Type u_1
inst✝¹ : TopologicalSpace H
G : StructureGroupoid H
inst✝ : ClosedUnderRestriction G
s : Set H
x : H
f : H → H
e' : OpenPartialHomeomorph H H
he'G : e' ∈ G
he'x : x ∈ e'.source
h : G.IsLocalStructomorphWithinAt f s x
hx : ↑e' x ∈ ↑e'.symm ⁻¹' s
hxs : x ∈ s
e : OpenPartialHomeomorph H H
heG : e ∈ G
hef : EqOn f (↑e.toPartialEquiv) (s ∩ e.source)
hex : x ∈ e.source
⊢ ↑e' x ∈ (e'.symm ≫ₕ e).source
Module: Mathlib.Geometry.Manifold.IsManifold.ExtChartAt

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
M : Type u_3
H : Type u_4
inst✝⁵ : NontriviallyNormedField 𝕜
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace 𝕜 E
inst✝² : TopologicalSpace H
inst✝¹ : TopologicalSpace M
I : ModelWithCorners 𝕜 E H
inst✝ : ChartedSpace H M
x : M
⊢ (interior (extChartAt I x).target).Nonempty

Tactic:
by_contra! H

Goal state after:
𝕜 : Type u_1
E : Type u_2
M : Type u_3
H✝ : Type u_4
inst✝⁵ : NontriviallyNormedField 𝕜
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace 𝕜 E
inst✝² : TopologicalSpace H✝
inst✝¹ : TopologicalSpace M
I : ModelWithCorners 𝕜 E H✝
inst✝ : ChartedSpace H✝ M
x : M
H : interior (extChartAt I x).target = ∅
⊢ False
Module: Mathlib.Geometry.Manifold.IsManifold.ExtChartAt

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
M : Type u_3
H : Type u_4
E' : Type u_5
M' : Type u_6
H' : Type u_7
inst✝²⁰ : NontriviallyNormedField 𝕜
inst✝¹⁹ : NormedAddCommGroup E
inst✝¹⁸ : NormedSpace 𝕜 E
inst✝¹⁷ : TopologicalSpace H
inst✝¹⁶ : TopologicalSpace M
I : ModelWithCorners 𝕜 E H
inst✝¹⁵ : NormedAddCommGroup E'
inst✝¹⁴ : NormedSpace 𝕜 E'
inst✝¹³ : TopologicalSpace H'
inst✝¹² : TopologicalSpace M'
I' : ModelWithCorners 𝕜 E' H'
inst✝¹¹ : ChartedSpace H M
inst✝¹⁰ : ChartedSpace H' M'
G : Type u_8
G' : Type u_9
F : Type u_10
F' : Type u_11
N : Type u_12
N' : Type u_13
inst✝⁹ : NormedAddCommGroup F
inst✝⁸ : NormedSpace 𝕜 F
inst✝⁷ : NormedAddCommGroup F'
inst✝⁶ : NormedSpace 𝕜 F'
inst✝⁵ : TopologicalSpace G
inst✝⁴ : TopologicalSpace N
inst✝³ : TopologicalSpace G'
inst✝² : TopologicalSpace N'
J : ModelWithCorners 𝕜 F G
J' : ModelWithCorners 𝕜 F' G'
inst✝¹ : ChartedSpace G N
inst✝ : ChartedSpace G' N'
f : M → N
g : M' → N'
x : M
x' : M'
⊢ writtenInExtChartAt (I.prod I') (J.prod J') (x, x') (Prod.map f g) =
    Prod.map (writtenInExtChartAt I J x f) (writtenInExtChartAt I' J' x' g)

Tactic:
by
  ext p <;>
  simp [writtenInExtChartAt, I.toPartialEquiv.prod_symm, (chartAt H x).toPartialEquiv.prod_symm]

Goal state after:
Module: Mathlib.Geometry.Manifold.ContMDiff.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'
n m : ℕ∞ω
hmn : m ≤ n
s : Set H
x : H
f : H → H'
e : OpenPartialHomeomorph H H
he : e ∈ contDiffGroupoid n I
hx : x ∈ e.source
h : ContDiffWithinAt 𝕜 m (↑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 n (↑I ∘ ↑e.symm ∘ ↑I.symm) (↑I.symm ⁻¹' e.target ∩ range ↑I) (↑I (↑e x))
⊢ ↑e x ∈ e.target

Tactic:
e.mapsTo hx

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

Goal state before:
pp: case h₁
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'
n m : ℕ∞ω
hmn : m ≤ n
s : Set H
x : H
f g : H → H'
h : ∀ y ∈ s, f y = g y
hx : f x = g x
hf : ContDiffWithinAtProp I I' m f s x
⊢ ∀ y ∈ ↑I.symm ⁻¹' s ∩ range ↑I, (↑I' ∘ g ∘ ↑I.symm) y = (↑I' ∘ f ∘ ↑I.symm) y

pp: case hx
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'
n m : ℕ∞ω
hmn : m ≤ n
s : Set H
x : H
f g : H → H'
h : ∀ y ∈ s, f y = g y
hx : f x = g x
hf : ContDiffWithinAtProp I I' m f s x
⊢ (↑I' ∘ g ∘ ↑I.symm) (↑I x) = (↑I' ∘ f ∘ ↑I.symm) (↑I x)

Tactic:
· intro y hy
      simp only [mfld_simps] at hy
      simp only [h, hy, mfld_simps]

Goal state after:
case hx
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'
n m : ℕ∞ω
hmn : m ≤ n
s : Set H
x : H
f g : H → H'
h : ∀ y ∈ s, f y = g y
hx : f x = g x
hf : ContDiffWithinAtProp I I' m f s x
⊢ (↑I' ∘ g ∘ ↑I.symm) (↑I x) = (↑I' ∘ f ∘ ↑I.symm) (↑I x)
Module: Mathlib.Geometry.Manifold.ContMDiff.Defs

Goal state before:
pp: case refine_2
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
M' : Type u_7
inst✝ : TopologicalSpace M'
f : M → M'
s : Set M
x : M
h : ContinuousWithinAt (f ∘ ↑(extChartAt I x).symm) (↑(extChartAt I x).symm ⁻¹' s ∩ range ↑I) (↑(extChartAt I x) x)
⊢ ContinuousWithinAt f s x

Tactic:
· rw [← continuousWithinAt_inter (extChartAt_source_mem_nhds (I := I) x)]
    have : ContinuousWithinAt ((f ∘ ↑(extChartAt I x).symm) ∘ ↑(extChartAt I x))
        (s ∩ (extChartAt I x).source) x := by
      apply h.comp (continuousAt_extChartAt x).continuousWithinAt
      intro y hy
      have : (chartAt H x).symm ((chartAt H x) y) = y :=
        OpenPartialHomeomorph.left_inv _ (by simpa using hy.2)
      simpa [this] using hy.1
    apply this.congr
    · intro y hy
      have : (chartAt H x).symm ((chartAt H x) y) = y :=
        OpenPartialHomeomorph.left_inv _ (by simpa using hy.2)
      simp [this]
    · simp

Goal state after:
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✝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))
⊢ ↑e x ∈ e.target

Tactic:
e.mapsTo hx

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

Goal state before:
pp: case mpr
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
n : ℕ∞ω
inst✝¹ : IsManifold I n M
inst✝ : IsManifold I' n M'
hcont : ContinuousOn f s
hdiff :
  ∀ (x : M) (y : M'),
    ContDiffOn 𝕜 n (↑(extChartAt I' y) ∘ f ∘ ↑(extChartAt I x).symm)
      ((extChartAt I x).target ∩ ↑(extChartAt I x).symm ⁻¹' (s ∩ f ⁻¹' (extChartAt I' y).source))
x : M
hx : x ∈ s
⊢ ContinuousWithinAt f s x ∧
    ContDiffWithinAtProp I I' n (↑(chartAt H' (f x)) ∘ f ∘ ↑(chartAt H x).symm)
      ((chartAt H x).target ∩ ↑(chartAt H x).symm ⁻¹' (s ∩ f ⁻¹' (chartAt H' (f x)).source)) (↑(chartAt H x) x)

Tactic:
refine ⟨hcont x hx, ?_⟩

Goal state after:
case mpr
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
n : ℕ∞ω
inst✝¹ : IsManifold I n M
inst✝ : IsManifold I' n M'
hcont : ContinuousOn f s
hdiff :
  ∀ (x : M) (y : M'),
    ContDiffOn 𝕜 n (↑(extChartAt I' y) ∘ f ∘ ↑(extChartAt I x).symm)
      ((extChartAt I x).target ∩ ↑(extChartAt I x).symm ⁻¹' (s ∩ f ⁻¹' (extChartAt I' y).source))
x : M
hx : x ∈ s
⊢ ContDiffWithinAtProp I I' n (↑(chartAt H' (f x)) ∘ f ∘ ↑(chartAt H x).symm)
    ((chartAt H x).target ∩ ↑(chartAt H x).symm ⁻¹' (s ∩ f ⁻¹' (chartAt H' (f x)).source)) (↑(chartAt H x) x)
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
ht : t ∈ 𝓝 x
⊢ MDiffAt[s ∩ t] f x ↔ MDiffAt[s] f x

Tactic:
by
  rw [MDifferentiableWithinAt, MDifferentiableWithinAt,
    differentiableWithinAt_localInvariantProp.liftPropWithinAt_inter ht]

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'
s : Set M
e : OpenPartialHomeomorph M H
e' : OpenPartialHomeomorph M' H'
x : M
he : e ∈ maximalAtlas I 1 M
he' : e' ∈ maximalAtlas I' 1 M'
hs : s ⊆ e.source
hx : x ∈ e.source
hy : f x ∈ e'.source
⊢ MDiffAt[s] f x ↔
    ContinuousWithinAt f s x ∧
      DifferentiableWithinAt 𝕜 (↑(e'.extend I') ∘ f ∘ ↑(e.extend I).symm) (↑(e.extend I) '' s) (↑(e.extend I) x)

Tactic:
rw [mdifferentiableWithinAt_iff_of_mem_maximalAtlas he he' hx hy, and_congr_right_iff]

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'
s : Set M
e : OpenPartialHomeomorph M H
e' : OpenPartialHomeomorph M' H'
x : M
he : e ∈ maximalAtlas I 1 M
he' : e' ∈ maximalAtlas I' 1 M'
hs : s ⊆ e.source
hx : x ∈ e.source
hy : f x ∈ e'.source
⊢ ContinuousWithinAt f s x →
    (DifferentiableWithinAt 𝕜 (↑(e'.extend I') ∘ f ∘ ↑(e.extend I).symm) (↑(e.extend I).symm ⁻¹' s ∩ range ↑I)
        (↑(e.extend I) x) ↔
      DifferentiableWithinAt 𝕜 (↑(e'.extend I') ∘ f ∘ ↑(e.extend I).symm) (↑(e.extend I) '' s) (↑(e.extend I) x))
Module: Mathlib.Geometry.Manifold.VectorBundle.FiberwiseLinear

Goal state before:
pp: 𝕜 : Type u_1
B : Type u_2
F : Type u_3
inst✝⁷ : TopologicalSpace B
inst✝⁶ : NontriviallyNormedField 𝕜
inst✝⁵ : NormedAddCommGroup F
inst✝⁴ : NormedSpace 𝕜 F
EB : Type u_4
inst✝³ : NormedAddCommGroup EB
inst✝² : NormedSpace 𝕜 EB
HB : Type u_5
inst✝¹ : TopologicalSpace HB
inst✝ : ChartedSpace HB B
IB : ModelWithCorners 𝕜 EB HB
n : ℕ∞ω
e : OpenPartialHomeomorph (B × F) (B × F)
U : Set B
hU : e.source = U ×ˢ univ
φ : ↑U → B → F ≃L[𝕜] F
u : ↑U → Set B
hu : ∀ (x : ↑U), IsOpen[inst✝⁷] (u x)
hUu : ∀ (x : ↑U), u x ⊆ U
hux : ∀ (x : ↑U), ↑x ∈ u x
hφ : ∀ (x : ↑U), ContMDiffOn IB 𝓘(𝕜, F →L[𝕜] F) n (fun x_1 ↦ ↑(φ x x_1)) (u x)
h2φ : ∀ (x : ↑U), ContMDiffOn IB 𝓘(𝕜, F →L[𝕜] F) n (fun x_1 ↦ ↑(φ x x_1).symm) (u x)
heφ : ∀ (x : ↑U), (e.restr (u x ×ˢ univ)).EqOnSource (FiberwiseLinear.openPartialHomeomorph (φ x) ⋯ ⋯ ⋯)
heuφ : ∀ (x : ↑U), EqOn (↑e) (fun q ↦ (q.1, (φ x q.1) q.2)) (u x ×ˢ univ)
huφ : ∀ (x x' : ↑U), ∀ y ∈ u x, y ∈ u x' → φ x y = φ x' y
hUu' : U = ⋃ i, u i
hU' : IsOpen[inst✝⁷] U
Φ₀ : ↑U → F ≃L[𝕜] F := iUnionLift u (fun x ↦ φ x ∘ Subtype.val) huφ U ⋯
Φ : B → F ≃L[𝕜] F := fun y ↦ if hy : y ∈ U then Φ₀ ⟨y, hy⟩ else ContinuousLinearEquiv.refl 𝕜 F
hΦ : ∀ (y : B) (hy : y ∈ U), Φ y = Φ₀ ⟨y, hy⟩
hΦφ : ∀ (x : ↑U), ∀ y ∈ u x, Φ y = φ x y
x : B
hx : x ∈ U
⊢ ContMDiffOn IB 𝓘(𝕜, F →L[𝕜] F) n (fun y ↦ ↑(Φ y)) (U ∩ u ⟨x, hx⟩)

Tactic:
refine (ContMDiffOn.congr (hφ ⟨x, hx⟩) ?_).mono inter_subset_right

Goal state after:
𝕜 : Type u_1
B : Type u_2
F : Type u_3
inst✝⁷ : TopologicalSpace B
inst✝⁶ : NontriviallyNormedField 𝕜
inst✝⁵ : NormedAddCommGroup F
inst✝⁴ : NormedSpace 𝕜 F
EB : Type u_4
inst✝³ : NormedAddCommGroup EB
inst✝² : NormedSpace 𝕜 EB
HB : Type u_5
inst✝¹ : TopologicalSpace HB
inst✝ : ChartedSpace HB B
IB : ModelWithCorners 𝕜 EB HB
n : ℕ∞ω
e : OpenPartialHomeomorph (B × F) (B × F)
U : Set B
hU : e.source = U ×ˢ univ
φ : ↑U → B → F ≃L[𝕜] F
u : ↑U → Set B
hu : ∀ (x : ↑U), IsOpen[inst✝⁷] (u x)
hUu : ∀ (x : ↑U), u x ⊆ U
hux : ∀ (x : ↑U), ↑x ∈ u x
hφ : ∀ (x : ↑U), ContMDiffOn IB 𝓘(𝕜, F →L[𝕜] F) n (fun x_1 ↦ ↑(φ x x_1)) (u x)
h2φ : ∀ (x : ↑U), ContMDiffOn IB 𝓘(𝕜, F →L[𝕜] F) n (fun x_1 ↦ ↑(φ x x_1).symm) (u x)
heφ : ∀ (x : ↑U), (e.restr (u x ×ˢ univ)).EqOnSource (FiberwiseLinear.openPartialHomeomorph (φ x) ⋯ ⋯ ⋯)
heuφ : ∀ (x : ↑U), EqOn (↑e) (fun q ↦ (q.1, (φ x q.1) q.2)) (u x ×ˢ univ)
huφ : ∀ (x x' : ↑U), ∀ y ∈ u x, y ∈ u x' → φ x y = φ x' y
hUu' : U = ⋃ i, u i
hU' : IsOpen[inst✝⁷] U
Φ₀ : ↑U → F ≃L[𝕜] F := iUnionLift u (fun x ↦ φ x ∘ Subtype.val) huφ U ⋯
Φ : B → F ≃L[𝕜] F := fun y ↦ if hy : y ∈ U then Φ₀ ⟨y, hy⟩ else ContinuousLinearEquiv.refl 𝕜 F
hΦ : ∀ (y : B) (hy : y ∈ U), Φ y = Φ₀ ⟨y, hy⟩
hΦφ : ∀ (x : ↑U), ∀ y ∈ u x, Φ y = φ x y
x : B
hx : x ∈ U
⊢ ∀ y ∈ u ⟨x, hx⟩, ↑(Φ y) = ↑(φ ⟨x, hx⟩ y)
Module: Mathlib.Geometry.Manifold.MFDeriv.Basic

Goal state before:
pp: case mpr
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
inst✝¹ : IsManifold I 1 M
inst✝ : IsManifold I' 1 M'
hcont : ContinuousOn f s
hdiff :
  ∀ (x : M) (y : M'),
    DifferentiableOn 𝕜 (↑(extChartAt I' y) ∘ f ∘ ↑(extChartAt I x).symm)
      ((extChartAt I x).target ∩ ↑(extChartAt I x).symm ⁻¹' (s ∩ f ⁻¹' (extChartAt I' y).source))
x : M
hx : x ∈ s
⊢ ContinuousWithinAt f s x ∧
    DifferentiableWithinAtProp I I' (↑(chartAt H' (f x)) ∘ f ∘ ↑(chartAt H x).symm)
      ((chartAt H x).target ∩ ↑(chartAt H x).symm ⁻¹' (s ∩ f ⁻¹' (chartAt H' (f x)).source)) (↑(chartAt H x) x)

Tactic:
refine ⟨hcont x hx, ?_⟩

Goal state after:
case mpr
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
inst✝¹ : IsManifold I 1 M
inst✝ : IsManifold I' 1 M'
hcont : ContinuousOn f s
hdiff :
  ∀ (x : M) (y : M'),
    DifferentiableOn 𝕜 (↑(extChartAt I' y) ∘ f ∘ ↑(extChartAt I x).symm)
      ((extChartAt I x).target ∩ ↑(extChartAt I x).symm ⁻¹' (s ∩ f ⁻¹' (extChartAt I' y).source))
x : M
hx : x ∈ s
⊢ DifferentiableWithinAtProp I I' (↑(chartAt H' (f x)) ∘ f ∘ ↑(chartAt H x).symm)
    ((chartAt H x).target ∩ ↑(chartAt H x).symm ⁻¹' (s ∩ f ⁻¹' (chartAt H' (f x)).source)) (↑(chartAt H x) x)
Module: Mathlib.Topology.FiberBundle.Constructions

Goal state before:
pp: B : Type u
F : Type v
E : B → Type w₁
B' : Type w₂
f✝ : B' → B
inst✝⁶ : TopologicalSpace B'
inst✝⁵ : TopologicalSpace (TotalSpace F E)
inst✝⁴ : TopologicalSpace F
inst✝³ : TopologicalSpace B
inst✝² : ∀ (_b : B), Nonempty (E _b)
K : Type U
inst✝¹ : FunLike K B' B
inst✝ : ContinuousMapClass K B' B
e : Trivialization F TotalSpace.proj
f : K
x : B' × F
h : f x.1 ∈ e.baseSet
⊢ ((TotalSpace.mk' F x.1 (e.symm (f x.1) x.2)).proj,
      (↑e (Pullback.lift (⇑f) (TotalSpace.mk' F x.1 (e.symm (f x.1) x.2)))).2) =
    x

Tactic:
simp_rw [Pullback.lift_mk, e.apply_mk_symm h]

Goal state after: