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

Goal state before:
pp: case mpr
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'
h₁ : ContMDiffOn I I n (↑f) f.source
h₂ : ContMDiffOn I I n (↑f.symm) f.target
x : M
hx : x ∈ f.source
⊢ (contDiffGroupoid n I).IsLocalStructomorphWithinAt (↑(chartAt H (↑f x)) ∘ ↑f ∘ ↑(chartAt H x).symm)
    (↑(chartAt H x).symm ⁻¹' f.source) (↑(chartAt H x) x)

Tactic:
let c := chartAt H x

Goal state after:
case mpr
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'
h₁ : ContMDiffOn I I n (↑f) f.source
h₂ : ContMDiffOn I I n (↑f.symm) f.target
x : M
hx : x ∈ f.source
c : OpenPartialHomeomorph M H := chartAt H x
⊢ (contDiffGroupoid n I).IsLocalStructomorphWithinAt (↑(chartAt H (↑f x)) ∘ ↑f ∘ ↑(chartAt H x).symm)
    (↑(chartAt H x).symm ⁻¹' f.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 : Set M
n : WithTop ℕ∞
hf : ContMDiffWithinAt I I' n f s x
hn : n ≠ 0
h : MDiffAt[s ∩ f ⁻¹' (extChartAt I' (f x)).source] f x
⊢ MDiffAt[s] f x

Tactic:
rwa [mdifferentiableWithinAt_inter'] at h

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'
x : M
s : Set M
n : WithTop ℕ∞
hf : ContMDiffWithinAt I I' n f s x
hn : n ≠ 0
h : MDiffAt[s ∩ f ⁻¹' (extChartAt I' (f x)).source] f x
⊢ f ⁻¹' (extChartAt I' (f x)).source ∈ 𝓝[s] 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 : Set M
hf : Injective ⇑(mfderiv[s] f x)
a✝ : Nontrivial E
⊢ MDiffAt[s] f x

Tactic:
have : Nontrivial (TangentSpace% x) := inferInstanceAs (Nontrivial E)

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'
x : M
s : Set M
hf : Injective ⇑(mfderiv[s] f x)
a✝ : Nontrivial E
this : Nontrivial (TangentSpace I x)
⊢ MDiffAt[s] f 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
y : B' × F
h : y ∈ (⇑f ⁻¹' e.baseSet) ×ˢ univ
⊢ TotalSpace.mk' F y.1 (e.symm (f y.1) y.2) ∈ Pullback.lift ⇑f ⁻¹' e.source

Tactic:
rw [mem_prod, mem_preimage] at h
    simp_rw [e.source_eq, mem_preimage, Pullback.lift_proj, h.1]

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

Goal state before:
pp: case inl
𝕜 : Type u_1
inst✝¹⁰ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝⁹ : NormedAddCommGroup E
inst✝⁸ : NormedSpace 𝕜 E
H : Type u_3
inst✝⁷ : TopologicalSpace H
I : ModelWithCorners 𝕜 E H
M : Type u_4
inst✝⁶ : TopologicalSpace M
inst✝⁵ : ChartedSpace H M
E' : Type u_5
inst✝⁴ : NormedAddCommGroup E'
inst✝³ : NormedSpace 𝕜 E'
H' : Type u_6
inst✝² : TopologicalSpace H'
I' : ModelWithCorners 𝕜 E' H'
M' : Type u_7
inst✝¹ : TopologicalSpace M'
inst✝ : ChartedSpace H' M'
f : M → M'
x : M
s : Set M
f' : TangentSpace I x →L[𝕜] TangentSpace I' (f x)
this : T1Space M
hf :
  ContinuousWithinAt f s x ∧
    HasFDerivWithinAt (writtenInExtChartAt I I' x f) f' (↑(extChartAt I x).symm ⁻¹' s ∩ range ↑I) (↑(extChartAt I x) x)
⊢ HasFDerivWithinAt (writtenInExtChartAt I I' x f) f' (↑(extChartAt I x).symm ⁻¹' insert x s ∩ range ↑I)
    (↑(extChartAt I x) x)

Tactic:
have : (extChartAt I x).target ∈
        𝓝[(extChartAt I x).symm ⁻¹' insert x s ∩ range I] (extChartAt I x) x :=
      nhdsWithin_mono _ inter_subset_right (extChartAt_target_mem_nhdsWithin x)

Goal state after:
case inl
𝕜 : Type u_1
inst✝¹⁰ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝⁹ : NormedAddCommGroup E
inst✝⁸ : NormedSpace 𝕜 E
H : Type u_3
inst✝⁷ : TopologicalSpace H
I : ModelWithCorners 𝕜 E H
M : Type u_4
inst✝⁶ : TopologicalSpace M
inst✝⁵ : ChartedSpace H M
E' : Type u_5
inst✝⁴ : NormedAddCommGroup E'
inst✝³ : NormedSpace 𝕜 E'
H' : Type u_6
inst✝² : TopologicalSpace H'
I' : ModelWithCorners 𝕜 E' H'
M' : Type u_7
inst✝¹ : TopologicalSpace M'
inst✝ : ChartedSpace H' M'
f : M → M'
x : M
s : Set M
f' : TangentSpace I x →L[𝕜] TangentSpace I' (f x)
this✝ : T1Space M
hf :
  ContinuousWithinAt f s x ∧
    HasFDerivWithinAt (writtenInExtChartAt I I' x f) f' (↑(extChartAt I x).symm ⁻¹' s ∩ range ↑I) (↑(extChartAt I x) x)
this : (extChartAt I x).target ∈ 𝓝[↑(extChartAt I x).symm ⁻¹' insert x s ∩ range ↑I] ↑(extChartAt I x) x
⊢ HasFDerivWithinAt (writtenInExtChartAt I I' x f) f' (↑(extChartAt I x).symm ⁻¹' insert x s ∩ range ↑I)
    (↑(extChartAt I x) x)
Module: Mathlib.Geometry.Manifold.MFDeriv.SpecificFunctions

Goal state before:
pp: 𝕜 : Type u_1
inst✝⁹ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝⁸ : NormedAddCommGroup E
inst✝⁷ : NormedSpace 𝕜 E
H : Type u_3
inst✝⁶ : TopologicalSpace H
I : ModelWithCorners 𝕜 E H
M : Type u_4
inst✝⁵ : TopologicalSpace M
inst✝⁴ : ChartedSpace H M
F₁ : Type u_17
inst✝³ : NormedAddCommGroup F₁
inst✝² : NormedSpace 𝕜 F₁
F₂ : Type u_18
inst✝¹ : NormedAddCommGroup F₂
inst✝ : NormedSpace 𝕜 F₂
x : M
f : M → F₁ × F₂
⊢ MDiffAt f x ↔ MDiffAt (Prod.fst ∘ f) x ∧ MDiffAt (Prod.snd ∘ f) x

Tactic:
rw [modelWithCornersSelf_prod, ← chartedSpaceSelf_prod]
  exact mdifferentiableAt_prod_iff f

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'
hg : MDiffAt[u] g (f x)
hf : MDiffAt[s] f x
h : s ⊆ f ⁻¹' u
hxs : UniqueMDiffAt[s] x
⊢ mfderiv[s] (g ∘ f) x = mfderiv[u] g (f x) ∘SL mfderiv[s] f x

Tactic:
apply HasMFDerivWithinAt.mfderivWithin _ hxs
  exact HasMFDerivWithinAt.comp x hg.hasMFDerivWithinAt hf.hasMFDerivWithinAt h

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

Goal state before:
pp: n : ℕ∞ω
𝕜 : Type u_1
B : Type u_2
F : Type u_4
E : B → Type u_6
inst✝¹⁵ : NontriviallyNormedField 𝕜
EB : Type u_7
inst✝¹⁴ : NormedAddCommGroup EB
inst✝¹³ : NormedSpace 𝕜 EB
HB : Type u_8
inst✝¹² : TopologicalSpace HB
IB : ModelWithCorners 𝕜 EB HB
inst✝¹¹ : TopologicalSpace B
inst✝¹⁰ : ChartedSpace HB B
inst✝⁹ : (x : B) → AddCommMonoid (E x)
inst✝⁸ : (x : B) → Module 𝕜 (E x)
inst✝⁷ : NormedAddCommGroup F
inst✝⁶ : NormedSpace 𝕜 F
inst✝⁵ : TopologicalSpace (TotalSpace F E)
inst✝⁴ : (x : B) → TopologicalSpace (E x)
inst✝³ : FiberBundle F E
inst✝² : VectorBundle 𝕜 F E
inst✝¹ : ContMDiffVectorBundle n F E IB
s : (x : B) → E x
x₀ : B
e : Trivialization F TotalSpace.proj
inst✝ : MemTrivializationAtlas e
hx₀ : x₀ ∈ e.baseSet
⊢ ContMDiffAt IB (IB.prod 𝓘(𝕜, F)) n (fun x ↦ TotalSpace.mk' F x (s x)) x₀ ↔
    ContMDiffAt IB 𝓘(𝕜, F) n (fun x ↦ (↑e { proj := x, snd := s x }).2) x₀

Tactic:
by
  simp_rw [← contMDiffWithinAt_univ]
  exact e.contMDiffWithinAt_section univ hx₀

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

Goal state before:
pp: case mpr
𝕜 : Type u_1
inst✝¹⁵ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝¹⁴ : NormedAddCommGroup E
inst✝¹³ : NormedSpace 𝕜 E
E' : Type u_3
inst✝¹² : NormedAddCommGroup E'
inst✝¹¹ : NormedSpace 𝕜 E'
F : Type u_4
inst✝¹⁰ : NormedAddCommGroup F
inst✝⁹ : NormedSpace 𝕜 F
H : Type u_5
inst✝⁸ : TopologicalSpace H
H' : Type u_6
inst✝⁷ : TopologicalSpace H'
G : Type u_7
inst✝⁶ : TopologicalSpace G
I : ModelWithCorners 𝕜 E H
I' : ModelWithCorners 𝕜 E' H'
J : ModelWithCorners 𝕜 F G
M : Type u_9
inst✝⁵ : TopologicalSpace M
inst✝⁴ : ChartedSpace H M
M' : Type u_10
inst✝³ : TopologicalSpace M'
inst✝² : ChartedSpace H' M'
N : Type u_11
inst✝¹ : TopologicalSpace N
inst✝ : ChartedSpace G N
n m : ℕ∞ω
h : M ≃ₘ^n⟮I, J⟯ N
f : N → M'
s : Set M
x : M
hm : m ≤ n
⊢ ContMDiffWithinAt J I' m f (⇑h '' s) (h x) → ContMDiffWithinAt I I' m (f ∘ ⇑h) s x

Tactic:
exact fun hf => hf.comp x (h.contMDiffWithinAt.of_le hm) (mapsTo_image _ _)

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

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
E' : Type u_3
inst✝¹² : NormedAddCommGroup E'
inst✝¹¹ : NormedSpace 𝕜 E'
F : Type u_4
inst✝¹⁰ : NormedAddCommGroup F
inst✝⁹ : NormedSpace 𝕜 F
H : Type u_5
inst✝⁸ : TopologicalSpace H
H' : Type u_6
inst✝⁷ : TopologicalSpace H'
G : Type u_7
inst✝⁶ : TopologicalSpace G
I : ModelWithCorners 𝕜 E H
I' : ModelWithCorners 𝕜 E' H'
J : ModelWithCorners 𝕜 F G
M : Type u_9
inst✝⁵ : TopologicalSpace M
inst✝⁴ : ChartedSpace H M
M' : Type u_10
inst✝³ : TopologicalSpace M'
inst✝² : ChartedSpace H' M'
N : Type u_11
inst✝¹ : TopologicalSpace N
inst✝ : ChartedSpace G N
n m : ℕ∞ω
h : M ≃ₘ^n⟮I, J⟯ N
f : M' → M
hm : m ≤ n
s : Set M'
x : M'
Hhf : ContMDiffWithinAt I' J m (⇑h ∘ f) s x
⊢ ContMDiffWithinAt I' I m f s x

Tactic:
by
    simpa only [Function.comp_def, h.symm_apply_apply] using
      (h.symm.contMDiffAt.of_le hm).comp_contMDiffWithinAt _ Hhf

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

Goal state before:
pp: case h
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
y : E
hy : y ∈ (extChartAt I x).target
h'y : ↑(extChartAt I x).symm y ∈ (extChartAt I x).source
h''y : ↑(extChartAt I x).symm y ∈ (chartAt H x).source
U' : UniqueMDiffAt[(extChartAt I x).source] (↑(extChartAt I x).symm y)
this :
  mfderiv% ↑(extChartAt I x) (↑(extChartAt I x).symm y) =
    mfderiv[(extChartAt I x).source] ↑(extChartAt I x) (↑(extChartAt I x).symm y)
⊢ (extChartAt I x).source ⊆ ↑(extChartAt I x) ⁻¹' range ↑I

Tactic:
intro z hz
    apply extChartAt_target_subset_range x
    exact PartialEquiv.map_source (extChartAt I x) hz

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 y : M
hy : y ∈ (extChartAt I x).source
this : y = ↑(extChartAt I x).symm (↑(extChartAt I x) y)
⊢ mfderiv[range ↑I] ↑(extChartAt I x).symm (↑(extChartAt I x) y) ∘SL mfderiv% ↑(extChartAt I x) y =
      ContinuousLinearMap.id 𝕜 (TangentSpace I y) ↔
    mfderiv[range ↑I] ↑(extChartAt I x).symm (↑(extChartAt I x) y) ∘SL
        mfderiv% ↑(extChartAt I x) (↑(extChartAt I x).symm (↑(extChartAt I x) y)) =
      ContinuousLinearMap.id 𝕜 (TangentSpace I (↑(extChartAt I x).symm (↑(extChartAt I x) y)))

Tactic:
rw [(extChartAt I x).left_inv (by simpa using hy)]

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 y : M
hy : y ∈ (extChartAt I x).source
h'y : ↑(extChartAt I x) y ∈ (extChartAt I x).target
⊢ (mfderiv% ↑(extChartAt I x) y).IsInvertible

Tactic:
have Z := ContinuousLinearMap.IsInvertible.of_inverse
    (mfderiv_extChartAt_comp_mfderivWithin_extChartAt_symm h'y)
    (mfderivWithin_extChartAt_symm_comp_mfderiv_extChartAt h'y)

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 y : M
hy : y ∈ (extChartAt I x).source
h'y : ↑(extChartAt I x) y ∈ (extChartAt I x).target
Z : (mfderiv% ↑(extChartAt I x) (↑(extChartAt I x).symm (↑(extChartAt I x) y))).IsInvertible
⊢ (mfderiv% ↑(extChartAt I x) y).IsInvertible
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:
filter_upwards [IsOpen.mem_nhds (isOpen_extChartAt_preimage c isOpen_ball) ⟨hs, hd⟩]

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
x : M
inst✝ : FiniteDimensional ℝ E
hs : x ∈ (chartAt H c).source
hd : dist (↑(extChartAt I c) x) (↑(extChartAt I c) c) < f.rIn
⊢ ∀ a ∈ (chartAt H c).source ∩ ↑(extChartAt I c) ⁻¹' ball (↑(extChartAt I c) c) f.rIn, ↑f a = 1 a
Module: Mathlib.Topology.Compactness.Paracompact

Goal state before:
pp: ι : Type u
X : Type v
Y : Type w
inst✝² : TopologicalSpace X
inst✝¹ : TopologicalSpace Y
inst✝ : CompactSpace X
⊢ ParacompactSpace X

Tactic:
refine ⟨fun ι s ho hu ↦ ?_⟩

Goal state after:
ι✝ : Type u
X : Type v
Y : Type w
inst✝² : TopologicalSpace X
inst✝¹ : TopologicalSpace Y
inst✝ : CompactSpace X
ι : Type v
s : ι → Set X
ho : ∀ (a : ι), IsOpen[inst✝²] (s a)
hu : ⋃ a, s a = univ
⊢ ∃ β t, (∀ (b : β), IsOpen[inst✝²] (t b)) ∧ ⋃ b, t b = univ ∧ LocallyFinite t ∧ ∀ (b : β), ∃ a, t b ⊆ s a
Module: Mathlib.Geometry.Manifold.VectorBundle.ContMDiffSection

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

Tactic:
induction s using Finset.induction_on with

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

Goal state before:
pp: ι : Type uι
E : Type uE
inst✝⁸ : NormedAddCommGroup E
inst✝⁷ : NormedSpace ℝ E
F : Type uF
inst✝⁶ : NormedAddCommGroup F
inst✝⁵ : NormedSpace ℝ F
H : Type uH
inst✝⁴ : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type uM
inst✝³ : TopologicalSpace M
inst✝² : ChartedSpace H M
inst✝¹ : FiniteDimensional ℝ E
inst✝ : IsManifold I ∞ M
n : ℕ∞
i : ι
s : Set M
j : ι
⊢ ContMDiff I 𝓘(ℝ, ℝ) ∞ ⇑((BumpCovering.single i s) j)

Tactic:
rcases eq_or_ne j i with (rfl | h)
    · simp only [contMDiff_one, ContinuousMap.coe_one, BumpCovering.coe_single, Pi.single_eq_same]
    · simp only [contMDiff_zero, BumpCovering.coe_single, Pi.single_eq_of_ne h,
        ContinuousMap.coe_zero]

Goal state after:
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)
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
⊢ f =ᵐ[μ.restrict t] 0

Tactic:
refine measure_eq_zero_of_trim_eq_zero hm ?_

Goal state after:
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
⊢ ((μ.restrict t).trim hm) {x | (fun x ↦ f x = 0 x) x}ᶜ = 0
Module: Mathlib.MeasureTheory.Function.AEEqOfIntegral

Goal state before:
pp: case hfi
E : Type u_2
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : NormedSpace ℝ E
inst✝⁵ : CompleteSpace E
β : Type u_3
inst✝⁴ : TopologicalSpace β
inst✝³ : MeasurableSpace β
inst✝² : BorelSpace β
inst✝¹ : SigmaCompactSpace β
inst✝ : R1Space β
μ : Measure β
f : β → E
hf : Integrable f μ
h'f : ∀ (s : Set β), IsCompact s → ∫ (x : β) in s, f x ∂μ = 0
s : Set β
hs : IsClosed[inst✝⁴] s
t : ℕ → Set β := ⋯
B : s = ⋃ n, t n
⊢ IntegrableOn f (⋃ n, t n) μ

Tactic:
exact hf.integrableOn

Goal state after:
Module: Mathlib.Analysis.Calculus.Taylor

Goal state before:
pp: E : Type u_2
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
f : ℝ → E
a b C x : ℝ
n : ℕ
hab : a ≤ b
hf : ContDiffOn ℝ (↑n + 1) f (Icc a b)
hx : x ∈ Icc a b
hC : ∀ y ∈ Icc a b, ‖iteratedDerivWithin (n + 1) f (Icc a b) y‖ ≤ C
h : a < b
hf' : DifferentiableOn ℝ (iteratedDerivWithin n f (Icc a b)) (Icc a b)
h' : ∀ y ∈ Ico a x, ‖((↑n !)⁻¹ * (x - y) ^ n) • iteratedDerivWithin (n + 1) f (Icc a b) y‖ ≤ (↑n !)⁻¹ * |x - a| ^ n * C
⊢ ∀ t ∈ Icc a x,
    HasDerivWithinAt (fun y ↦ taylorWithinEval f n (Icc a b) y x)
      (((↑n !)⁻¹ * (x - t) ^ n) • iteratedDerivWithin (n + 1) f (Icc a b) t) (Icc a x) t

Tactic:
intro t ht

Goal state after:
E : Type u_2
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
f : ℝ → E
a b C x : ℝ
n : ℕ
hab : a ≤ b
hf : ContDiffOn ℝ (↑n + 1) f (Icc a b)
hx : x ∈ Icc a b
hC : ∀ y ∈ Icc a b, ‖iteratedDerivWithin (n + 1) f (Icc a b) y‖ ≤ C
h : a < b
hf' : DifferentiableOn ℝ (iteratedDerivWithin n f (Icc a b)) (Icc a b)
h' : ∀ y ∈ Ico a x, ‖((↑n !)⁻¹ * (x - y) ^ n) • iteratedDerivWithin (n + 1) f (Icc a b) y‖ ≤ (↑n !)⁻¹ * |x - a| ^ n * C
t : ℝ
ht : t ∈ Icc a x
⊢ HasDerivWithinAt (fun y ↦ taylorWithinEval f n (Icc a b) y x)
    (((↑n !)⁻¹ * (x - t) ^ n) • iteratedDerivWithin (n + 1) f (Icc a b) t) (Icc a x) t
Module: Mathlib.Analysis.Calculus.TaylorIntegral

Goal state before:
pp: E : Type u_2
F : Type u_3
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedAddCommGroup F
inst✝² : NormedSpace ℝ E
inst✝¹ : NormedSpace ℝ F
f : E → F
x y : E
n : ℕ
inst✝ : CompleteSpace F
hf : ∀ t ∈ Set.uIcc 0 1, ContDiffAt ℝ (↑0 + 1) f (x + t • y)
⊢ Set.EqOn (fun t ↦ (fderiv ℝ f (x + t • y)) y) (deriv fun s ↦ f (x + s • y)) (Set.uIcc 0 1)

Tactic:
intro t ht

Goal state after:
E : Type u_2
F : Type u_3
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedAddCommGroup F
inst✝² : NormedSpace ℝ E
inst✝¹ : NormedSpace ℝ F
f : E → F
x y : E
n : ℕ
inst✝ : CompleteSpace F
hf : ∀ t ∈ Set.uIcc 0 1, ContDiffAt ℝ (↑0 + 1) f (x + t • y)
t : ℝ
ht : t ∈ Set.uIcc 0 1
⊢ (fun t ↦ (fderiv ℝ f (x + t • y)) y) t = deriv (fun s ↦ f (x + s • y)) t
Module: Mathlib.Analysis.Calculus.TaylorIntegral

Goal state before:
pp: E : Type u_2
F : Type u_3
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedAddCommGroup F
inst✝² : NormedSpace ℝ E
inst✝¹ : NormedSpace ℝ F
f : E → F
x y : E
n : ℕ
inst✝ : CompleteSpace F
hf : ∀ t ∈ Set.uIcc 0 1, ContDiffAt ℝ (↑0 + 1) f (x + t • y)
h_eq : Set.EqOn (fun t ↦ (fderiv ℝ f (x + t • y)) y) (deriv fun s ↦ f (x + s • y)) (Set.uIcc 0 1)
hf' : ∀ t ∈ Set.uIcc 0 1, DifferentiableAt ℝ (fun s ↦ f (x + s • y)) t
h₁ : ContinuousOn (fderiv ℝ f) ((fun t ↦ x + t • y) '' Set.uIcc 0 1)
⊢ Set.MapsTo (fun x_1 ↦ x + x_1 • y) (Set.uIcc 0 1) ((fun t ↦ x + t • y) '' Set.uIcc 0 1)

Tactic:
intro t ht

Goal state after:
E : Type u_2
F : Type u_3
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedAddCommGroup F
inst✝² : NormedSpace ℝ E
inst✝¹ : NormedSpace ℝ F
f : E → F
x y : E
n : ℕ
inst✝ : CompleteSpace F
hf : ∀ t ∈ Set.uIcc 0 1, ContDiffAt ℝ (↑0 + 1) f (x + t • y)
h_eq : Set.EqOn (fun t ↦ (fderiv ℝ f (x + t • y)) y) (deriv fun s ↦ f (x + s • y)) (Set.uIcc 0 1)
hf' : ∀ t ∈ Set.uIcc 0 1, DifferentiableAt ℝ (fun s ↦ f (x + s • y)) t
h₁ : ContinuousOn (fderiv ℝ f) ((fun t ↦ x + t • y) '' Set.uIcc 0 1)
t : ℝ
ht : t ∈ Set.uIcc 0 1
⊢ (fun x_1 ↦ x + x_1 • y) t ∈ (fun t ↦ x + t • y) '' Set.uIcc 0 1