mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.Analysis.Calculus.LineDeriv.Measurable

Goal state before:
pp: 𝕜 : Type u_1
inst✝⁹ : NontriviallyNormedField 𝕜
inst✝⁸ : LocallyCompactSpace 𝕜
E : Type u_2
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : NormedSpace 𝕜 E
inst✝⁵ : MeasurableSpace E
inst✝⁴ : OpensMeasurableSpace E
F : Type u_3
inst✝³ : NormedAddCommGroup F
inst✝² : NormedSpace 𝕜 F
inst✝¹ : CompleteSpace F
f : E → F
inst✝ : SecondCountableTopology E
hf : Continuous f
⊢ StronglyMeasurable fun p ↦ lineDeriv 𝕜 f p.1 p.2

Tactic:
borelize 𝕜

Goal state after:
𝕜 : Type u_1
inst✝⁹ : NontriviallyNormedField 𝕜
inst✝⁸ : LocallyCompactSpace 𝕜
E : Type u_2
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : NormedSpace 𝕜 E
inst✝⁵ : MeasurableSpace E
inst✝⁴ : OpensMeasurableSpace E
F : Type u_3
inst✝³ : NormedAddCommGroup F
inst✝² : NormedSpace 𝕜 F
inst✝¹ : CompleteSpace F
f : E → F
inst✝ : SecondCountableTopology E
hf : Continuous f
this✝¹ : MeasurableSpace 𝕜 := borel 𝕜
this✝ : BorelSpace 𝕜
⊢ StronglyMeasurable fun p ↦ lineDeriv 𝕜 f p.1 p.2
Module: Mathlib.Analysis.Complex.RemovableSingularity

Goal state before:
pp: case inl
E : Type u
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f : ℂ → E
s : Set ℂ
x : ℂ
hx : x ∈ s
hs : s ∈ 𝓝 x
hd : DifferentiableOn ℂ f (s \ {x})
hc : ContinuousAt f x
⊢ DifferentiableWithinAt ℂ f s x

Tactic:
refine (analyticAt_of_differentiable_on_punctured_nhds_of_continuousAt
      ?_ hc).differentiableAt.differentiableWithinAt
    refine eventually_nhdsWithin_iff.2 ((eventually_mem_nhds_iff.2 hs).mono fun z hz hzx => ?_)
    exact hd.differentiableAt (inter_mem hz (isOpen_ne.mem_nhds hzx))

Goal state after:
Module: Mathlib.Analysis.Complex.LocallyUniformLimit

Goal state before:
pp: E : Type u_1
ι : Type u_2
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
U : Set ℂ
φ : Filter ι
F : ι → ℂ → E
f : ℂ → E
inst✝¹ : CompleteSpace E
inst✝ : φ.NeBot
hf : TendstoLocallyUniformlyOn F f φ U
hF : ∀ᶠ (n : ι) in φ, DifferentiableOn ℂ (F n) U
hU : IsOpen U
x : ℂ
hx : x ∈ U
K : Set ℂ
hKU : K ⊆ U
hKx : K ∈ 𝓝 x
hK : IsCompact K
δ : ℝ
left✝¹ : δ > 0
left✝ : cthickening δ K ⊆ U
h1 : TendstoUniformlyOn (deriv ∘ F) (cderiv δ f) φ K
h2 : interior K ⊆ U
h3 : ∀ᶠ (n : ι) in φ, DifferentiableOn ℂ (F n) (interior K)
h4 : TendstoLocallyUniformlyOn F f φ (interior K)
h5 : TendstoLocallyUniformlyOn (deriv ∘ F) (cderiv δ f) φ (interior K)
h6 : ∀ x ∈ interior K, HasDerivAt f (cderiv δ f x) x
h7 : DifferentiableOn ℂ f (interior K)
⊢ DifferentiableWithinAt ℂ f U x

Tactic:
exact (h7.differentiableAt (interior_mem_nhds.mpr hKx)).differentiableWithinAt

Goal state after:
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✝ : LocallyConnectedSpace H
e : M → OpenPartialHomeomorph M H := chartAt H
⊢ LocallyConnectedSpace M

Tactic:
refine locallyConnectedSpace_of_connected_bases (fun x s ↦ (e x).symm '' s)
      (fun x s ↦ (IsOpen s ∧ e x x ∈ s ∧ IsConnected s) ∧ s ⊆ (e x).target) ?_ ?_

Goal state after:
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

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
H' : Type u_1
M : Type u_2
M' : Type u_3
M'' : Type u_4
inst✝ : TopologicalSpace H
c : ChartedSpaceCore H M
e✝ e : PartialEquiv M H
he : e ∈ c.atlas
this : TopologicalSpace M := c.toTopologicalSpace
⊢ ContinuousOn e.invFun e.target

Tactic:
apply continuousOn_isOpen_of_generateFrom

Goal state after:
H : Type u
H' : Type u_1
M : Type u_2
M' : Type u_3
M'' : Type u_4
inst✝ : TopologicalSpace H
c : ChartedSpaceCore H M
e✝ e : PartialEquiv M H
he : e ∈ c.atlas
this : TopologicalSpace M := ⋯
⊢ ∀ t ∈ ⋃ e ∈ c.atlas, ⋃ s, ⋃ (_ : IsOpen[inst✝] s), {↑e ⁻¹' s ∩ e.source}, IsOpen[inst✝] (e.target ∩ e.invFun ⁻¹' t)
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✝⁴ : NontriviallyNormedField 𝕜
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace 𝕜 E
inst✝¹ : TopologicalSpace H
inst✝ : TopologicalSpace M
f : OpenPartialHomeomorph M H
I : ModelWithCorners 𝕜 E H
⊢ (f.extend I).source = f.source

Tactic:
by
  rw [extend, PartialEquiv.trans_source, I.source_eq, preimage_univ, inter_univ]

Goal state after:
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✝⁴ : NontriviallyNormedField 𝕜
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace 𝕜 E
inst✝¹ : TopologicalSpace H
inst✝ : TopologicalSpace M
f : OpenPartialHomeomorph M H
I : ModelWithCorners 𝕜 E H
x : M
hx : x ∈ f.source
⊢ ↑I '' f.target ∈ map (↑(f.extend I)) (𝓝 x)

Tactic:
Filter.mem_map,

Goal state after:
𝕜 : Type u_1
E : Type u_2
M : Type u_3
H : Type u_4
inst✝⁴ : NontriviallyNormedField 𝕜
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace 𝕜 E
inst✝¹ : TopologicalSpace H
inst✝ : TopologicalSpace M
f : OpenPartialHomeomorph M H
I : ModelWithCorners 𝕜 E H
x : M
hx : x ∈ f.source
⊢ ↑(f.extend I) ⁻¹' ↑I '' f.target ∈ 𝓝 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'
e : OpenPartialHomeomorph M H
P : (H → H') → Set H → H → Prop
g : M → M'
s : Set M
x : M
hG : G.LocalInvariantProp G' P
he : e ∈ maximalAtlas M G
xe : x ∈ e.source
⊢ LiftPropWithinAt P g s x ↔ LiftPropWithinAt P (g ∘ ↑e.symm) (↑e.symm ⁻¹' s) (↑e x)

Tactic:
by
  rw [liftPropWithinAt_self_source, liftPropWithinAt_iff',
    e.symm.continuousWithinAt_iff_continuousWithinAt_comp_right xe, e.symm_symm]
  refine and_congr Iff.rfl ?_
  rw [Function.comp_apply, e.left_inv xe, ← Function.comp_assoc,
    hG.liftPropWithinAt_indep_chart_source_aux (chartAt _ (g x) ∘ g) he xe, Function.comp_assoc]

Goal state after:
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'
hG : G.LocalInvariantProp G' P
h : ∀ (x : M), ∃ u, IsOpen[inst✝⁴] u ∧ x ∈ u ∧ LiftPropOn P g u
⊢ LiftProp P g

Tactic:
by
  rw [← liftPropOn_univ]
  refine hG.liftPropOn_of_locally_liftPropOn fun x _ ↦ ?_
  simp [h x]

Goal state after:
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
y : E
hy : y ∈ (extChartAt I x).target
⊢ ↑(extChartAt I x).symm y ∈ (extChartAt I x).source

Tactic:
exact (extChartAt I x).map_target hy

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

Goal state before:
pp: 𝕜 : Type u_1
inst✝¹¹ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝¹⁰ : NormedAddCommGroup E
inst✝⁹ : NormedSpace 𝕜 E
H : Type u_3
inst✝⁸ : TopologicalSpace H
I : ModelWithCorners 𝕜 E H
M : Type u_4
inst✝⁷ : TopologicalSpace 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
inst✝¹ : ChartedSpace H' M'
n : ℕ∞ω
inst✝ : One M'
f : M → M'
hf : ∀ x ∈ mulTSupport f, ContMDiffAt I I' n f x
⊢ ContMDiff I I' n f

Tactic:
by
  intro x
  by_cases hx : x ∈ mulTSupport f
  · exact hf x hx
  · exact ContMDiffAt.congr_of_eventuallyEq contMDiffAt_const
      (notMem_mulTSupport_iff_eventuallyEq.1 hx)

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

Goal state before:
pp: case h₁
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
M' : Type u_16
inst✝¹ : TopologicalSpace M'
inst✝ : ChartedSpace H M'
n : WithTop ℕ∞
x : M
C : OpenPartialHomeomorph M H := chartAt H x
hC : C = chartAt H x
⊢ ↑(extChartAt I (Sum.inl x)) ∘ Sum.inl ∘ ↑(extChartAt I x).symm =ᶠ[𝓝[range ↑I] ↑(extChartAt I x) x] id

Tactic:
have : I.symm ⁻¹' C.target ∩ range I ∈ 𝓝[range I] (extChartAt I x) x := by
    rw [← I.image_eq (chartAt H x).target]
    exact (chartAt H x).extend_image_target_mem_nhds (mem_chart_source _ x)

Goal state after:
case h₁
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
M' : Type u_16
inst✝¹ : TopologicalSpace M'
inst✝ : ChartedSpace H M'
n : WithTop ℕ∞
x : M
C : OpenPartialHomeomorph M H := chartAt H x
hC : C = chartAt H x
this : ↑I.symm ⁻¹' C.target ∩ range ↑I ∈ 𝓝[range ↑I] ↑(extChartAt I x) x
⊢ ↑(extChartAt I (Sum.inl x)) ∘ Sum.inl ∘ ↑(extChartAt I x).symm =ᶠ[𝓝[range ↑I] ↑(extChartAt I x) x] id
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'
f : M → M'
s : Set M
x : M
n : ℕ∞ω
⊢ ContMDiffWithinAt I I' n f s x ↔
    ContinuousWithinAt f s x ∧
      ContDiffWithinAt 𝕜 n (↑(extChartAt I' (f x)) ∘ f ∘ ↑(extChartAt I x).symm)
        ((extChartAt I x).target ∩ ↑(extChartAt I x).symm ⁻¹' (s ∩ f ⁻¹' (extChartAt I' (f x)).source))
        (↑(extChartAt I x) x)

Tactic:
by
  simp only [ContMDiffWithinAt, liftPropWithinAt_iff']
  exact and_congr_right fun hc => contDiffWithinAt_congr_set <|
    hc.extChartAt_symm_preimage_inter_range_eventuallyEq

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
f' : E → E →L[ℝ] E
inst✝² : MeasurableSpace E
inst✝¹ : BorelSpace E
μ : Measure E
inst✝ : μ.IsAddHaarMeasure
f : OpenPartialHomeomorph E E
hf' : ∀ x ∈ f.source, HasFDerivAt (↑f) (f' x) x
g : E → F
⊢ ∫ (x : E) in f.target, g x ∂μ = ∫ (x : E) in f.source, |(f' x).det| • g (↑f x) ∂μ

Tactic:
have : f '' f.source = f.target := PartialEquiv.image_source_eq_target f.toPartialEquiv
  rw [← this]
  apply integral_image_eq_integral_abs_det_fderiv_smul μ f.open_source.measurableSet _ f.injOn
  intro x hx
  exact (hf' x hx).hasFDerivWithinAt

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

Goal state before:
pp: 𝕜 : Type u_1
inst✝⁴ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace 𝕜 E
E' : Type u_3
inst✝¹ : NormedAddCommGroup E'
inst✝ : NormedSpace 𝕜 E'
f : E → E'
s : Set E
x : E
⊢ MDiffAt[s] f x ↔ DifferentiableWithinAt 𝕜 f s x

Tactic:
simp only [mdifferentiableWithinAt_iff', mfld_simps]
  exact ⟨fun H => H.2, fun H => ⟨H.continuousWithinAt, H⟩⟩

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 : Set M
⊢ MDiffAt[s] f x ↔
    ContinuousWithinAt f s x ∧
      DifferentiableWithinAt 𝕜 (↑(extChartAt I' (f x)) ∘ f ∘ ↑(extChartAt I x).symm)
        ((extChartAt I x).target ∩ ↑(extChartAt I x).symm ⁻¹' (s ∩ f ⁻¹' (extChartAt I' (f x)).source))
        (↑(extChartAt I x) x)

Tactic:
by
  simp only [MDifferentiableWithinAt, liftPropWithinAt_iff']
  exact and_congr_right fun hc => differentiableWithinAt_congr_nhds <|
    hc.nhdsWithin_extChartAt_symm_preimage_inter_range

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
⊢ 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))

Tactic:
refine fun _ => differentiableWithinAt_congr_nhds ?_

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
x✝ : ContinuousWithinAt f s x
⊢ 𝓝[↑(e.extend I).symm ⁻¹' s ∩ range ↑I] ↑(e.extend I) x = 𝓝[↑(e.extend I) '' s] ↑(e.extend I) 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
f' : TangentSpace I x →L[𝕜] TangentSpace I' (f x)
h : t ∈ 𝓝 x
⊢ HasMFDerivAt[s ∩ t] f x f' ↔ HasMFDerivAt[s] f x f'

Tactic:
rw [HasMFDerivWithinAt, HasMFDerivWithinAt, extChartAt_preimage_inter_eq, hasFDerivWithinAt_inter,
    continuousWithinAt_inter h]
  exact extChartAt_preimage_mem_nhds h

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

Goal state before:
pp: 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
s : Set M
x : M
hxs : UniqueMDiffAt[s] x
⊢ mfderiv% id x = ContinuousLinearMap.id 𝕜 (TangentSpace I x)

Tactic:
exact mfderiv_id

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

Goal state before:
pp: case hx
𝕜 : Type u_1
inst✝¹⁴ : NontriviallyNormedField 𝕜
n : ℕ∞ω
E : Type u_2
inst✝¹³ : NormedAddCommGroup E
inst✝¹² : NormedSpace 𝕜 E
E' : Type u_3
inst✝¹¹ : NormedAddCommGroup E'
inst✝¹⁰ : NormedSpace 𝕜 E'
H : Type u_4
inst✝⁹ : TopologicalSpace H
I : ModelWithCorners 𝕜 E H
H' : Type u_5
inst✝⁸ : TopologicalSpace H'
I' : ModelWithCorners 𝕜 E' H'
M : Type u_6
inst✝⁷ : TopologicalSpace M
inst✝⁶ : ChartedSpace H M
M' : Type u_7
inst✝⁵ : TopologicalSpace M'
inst✝⁴ : ChartedSpace H' M'
F : Type u_8
inst✝³ : NormedAddCommGroup F
inst✝² : NormedSpace 𝕜 F
inst✝¹ : IsManifold I 1 M
inst✝ : IsManifold I' 1 M'
i : ↑(atlas H M)
x : M
hx : x ∈ (↑i).source
v : E
⊢ (↑((↑i).extend I) ∘ ↑((↑i).extend I).symm) (↑((↑i).extend I) x) = ↑((↑i).extend I) x

Tactic:
· simp_rw [Function.comp_apply, i.1.extend_left_inv hx]

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
g : M' → M''
hg : MDiffAt g (f x)
hf : MDiffAt f x
v : TangentSpace I x
⊢ (mfderiv% (g ∘ f) x) v = (mfderiv% g (f x)) ((mfderiv% f x) v)

Tactic:
rw [mfderiv_comp _ hg hf]

Goal state after:
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
g : M' → M''
hg : MDiffAt g (f x)
hf : MDiffAt f x
v : TangentSpace I x
⊢ (mfderiv% g (f x) ∘SL mfderiv% f x) v = (mfderiv% g (f x)) ((mfderiv% f x) v)
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

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_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
⊢ z ∈
    (tangentBundleCore I M).baseSet (achart H w) ∩ (tangentBundleCore I M).baseSet (achart H x) ∩
      (tangentBundleCore I M).baseSet (achart H y)
Module: Mathlib.Geometry.Manifold.VectorBundle.Tangent

Goal state before:
pp: 𝕜 : Type u_1
inst✝³ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace 𝕜 E
H : Type u_4
inst✝ : TopologicalSpace H
I : ModelWithCorners 𝕜 E H
p : TangentBundle I H
x : H
⊢ ↑(trivializationAt E (TangentSpace I) x) p = (p.proj, p.snd)

Tactic:
by
  simp only [TangentBundle.trivializationAt_apply]
  have : fderivWithin 𝕜 (↑I ∘ ↑I.symm) (range I) (I p.proj) =
      fderivWithin 𝕜 id (range I) (I p.proj) :=
    fderivWithin_congr' (fun y hy ↦ by simp [hy]) (mem_range_self p.proj)
  simp [this, fderivWithin_id (ModelWithCorners.uniqueDiffWithinAt_image I)]

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.transContinuousLinearEquiv e) x).symm (↑(extChartAt (I.transContinuousLinearEquiv e) x) x) ∈
    ⇑(Equiv.refl M).symm ⁻¹' (extChartAt I ((Equiv.refl M).symm x)).source

Tactic:
simp only [e.symm_apply_apply, Equiv.refl_symm, Equiv.coe_refl, mfld_simps]

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

Goal state before:
pp: 𝕜 : Type u_1
inst✝²⁶ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝²⁵ : NormedAddCommGroup E
inst✝²⁴ : NormedSpace 𝕜 E
E' : Type u_3
inst✝²³ : NormedAddCommGroup E'
inst✝²² : NormedSpace 𝕜 E'
F : Type u_4
inst✝²¹ : NormedAddCommGroup F
inst✝²⁰ : NormedSpace 𝕜 F
H : Type u_5
inst✝¹⁹ : TopologicalSpace H
H' : Type u_6
inst✝¹⁸ : TopologicalSpace H'
G : Type u_7
inst✝¹⁷ : TopologicalSpace G
G' : Type u_8
inst✝¹⁶ : TopologicalSpace G'
I : ModelWithCorners 𝕜 E H
I' : ModelWithCorners 𝕜 E' H'
J✝¹ : ModelWithCorners 𝕜 F G
J' : ModelWithCorners 𝕜 F G'
M : Type u_9
inst✝¹⁵ : TopologicalSpace M
inst✝¹⁴ : ChartedSpace H M
M'✝ : Type u_10
inst✝¹³ : TopologicalSpace M'✝
inst✝¹² : ChartedSpace H' M'✝
N✝ : Type u_11
inst✝¹¹ : TopologicalSpace N✝
inst✝¹⁰ : ChartedSpace G N✝
N'✝ : Type u_12
inst✝⁹ : TopologicalSpace N'✝
inst✝⁸ : ChartedSpace G' N'✝
n : ℕ∞ω
M' : Type u_13
inst✝⁷ : TopologicalSpace M'
inst✝⁶ : ChartedSpace H M'
M'' : Type u_14
inst✝⁵ : TopologicalSpace M''
inst✝⁴ : ChartedSpace H M''
N : Type u_15
J✝ : Type u_16
inst✝³ : TopologicalSpace N
inst✝² : ChartedSpace H N
J : ModelWithCorners 𝕜 E' H
N' : Type u_17
inst✝¹ : TopologicalSpace N'
inst✝ : ChartedSpace H N'
⊢ ContMDiff I I n ⇑(Equiv.sumAssoc M M' M'').symm

Tactic:
by
    apply ContMDiff.sumElim
    · exact ContMDiff.inl.comp ContMDiff.inl
    · exact ContMDiff.inr.sumMap contMDiff_id

Goal state after: