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'
⊢ ContMDiffOn I I n (↑f) f.source ∧ ContMDiffOn I I n (↑f.symm) f.target →
    LiftPropOn (contDiffGroupoid n I).IsLocalStructomorphWithinAt (↑f) f.source

Tactic:
· -- We now show the converse: an open partial homeomorphism `f : M → M'` which is `C^n` in both
    -- directions is a local structomorphism.  We do this by proposing
    -- `((chart_at H x).symm.trans f).trans (chart_at H (f x))` as a candidate for a structomorphism
    -- of `H`.
    rintro ⟨h₁, h₂⟩ x hx
    refine ⟨(h₁ x hx).continuousWithinAt, ?_⟩
    let c := chartAt H x
    let c' := chartAt H (f x)
    rintro (hx' : c x ∈ c.symm ⁻¹' f.source)
    -- propose `(c.symm.trans f).trans c'` as a candidate for a local structomorphism of `H`
    refine ⟨(c.symm.trans f).trans c', ⟨?_, ?_⟩, (?_ : EqOn (c' ∘ f ∘ c.symm) _ _), ?_⟩
    · -- regularity of the candidate local structomorphism in the forward direction
      intro y hy
      simp only [mfld_simps] at hy
      have H : ContMDiffWithinAt I I n f (f ≫ₕ c').source ((extChartAt I x).symm y) := by
        refine (h₁ ((extChartAt I x).symm y) ?_).mono ?_
        · simp only [c, hy, mfld_simps]
        · mfld_set_tac
      have hy' : (extChartAt I x).symm y ∈ c.source := by simp only [c, hy, mfld_simps]
      have hy'' : f ((extChartAt I x).symm y) ∈ c'.source := by
        simp only [c, hy, mfld_simps]
      rw [contMDiffWithinAt_iff_of_mem_source hy' hy''] at H
      convert! H.2.mono _
      · simp only [c, hy, mfld_simps]
      · dsimp [c, c']; mfld_set_tac
    · -- regularity of the candidate local structomorphism in the reverse direction
      intro y hy
      simp only [mfld_simps] at hy
      have H : ContMDiffWithinAt I I n f.symm (f.symm ≫ₕ c).source
          ((extChartAt I (f x)).symm y) := by
        refine (h₂ ((extChartAt I (f x)).symm y) ?_).mono ?_
        · simp only [c', hy, mfld_simps]
        · mfld_set_tac
      have hy' : (extChartAt I (f x)).symm y ∈ c'.source := by simp only [c', hy, mfld_simps]
      have hy'' : f.symm ((extChartAt I (f x)).symm y) ∈ c.source := by
        simp only [c', hy, mfld_simps]
      rw [contMDiffWithinAt_iff_of_mem_source hy' hy''] at H
      convert! H.2.mono _
      · simp only [c', hy, mfld_simps]
      · dsimp [c, c']; mfld_set_tac
    -- now check the candidate local structomorphism agrees with `f` where it is supposed to
    · simp only [mfld_simps]; apply eqOn_refl
    · simp only [c, c', hx', mfld_simps]

Goal state after:
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'
⊢ (ContinuousOn f s ∧
      ∀ (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))) →
    MDiff[s] f

Tactic:
rintro ⟨hcont, hdiff⟩ x hx
    refine differentiableWithinAt_localInvariantProp.liftPropWithinAt_iff.mpr ?_
    refine ⟨hcont x hx, ?_⟩
    dsimp [DifferentiableWithinAtProp]
    convert! hdiff x (f x) (extChartAt I x x) (by simp only [hx, mfld_simps]) using 1
    mfld_set_tac

Goal state after:
Module: Mathlib.Topology.FiberBundle.Constructions

Goal state before:
pp: B : Type u
F : Type v
E : B → Type w₁
B' : Type w₂
inst✝¹ : TopologicalSpace B'
inst✝ : TopologicalSpace (TotalSpace F E)
f : B' → B
⊢ TotalSpace.topologicalSpace F E f ≤ induced TotalSpace.proj inst✝¹

Tactic:
Pullback.TotalSpace.topologicalSpace,

Goal state after:
B : Type u
F : Type v
E : B → Type w₁
B' : Type w₂
inst✝¹ : TopologicalSpace B'
inst✝ : TopologicalSpace (TotalSpace F E)
f : B' → B
⊢ pullbackTopology F E f ≤ induced TotalSpace.proj inst✝¹
Module: Mathlib.Topology.FiberBundle.Constructions

Goal state before:
pp: B : Type u
F : Type v
E : B → Type w₁
B' : Type w₂
inst✝¹ : TopologicalSpace B'
inst✝ : TopologicalSpace (TotalSpace F E)
f : B' → B
⊢ TotalSpace.topologicalSpace F E f ≤ induced (Pullback.lift f) inst✝

Tactic:
Pullback.TotalSpace.topologicalSpace,

Goal state after:
B : Type u
F : Type v
E : B → Type w₁
B' : Type w₂
inst✝¹ : TopologicalSpace B'
inst✝ : TopologicalSpace (TotalSpace F E)
f : B' → B
⊢ pullbackTopology F E f ≤ induced (Pullback.lift f) inst✝
Module: Mathlib.Topology.FiberBundle.Constructions

Goal state before:
pp: B : Type u
F : Type v
E : B → Type w₁
B' : Type w₂
inst✝¹ : TopologicalSpace B'
inst✝ : TopologicalSpace (TotalSpace F E)
f : B' → B
⊢ Pullback.TotalSpace.topologicalSpace F E f =
    induced (Prod.fst ∘ pullbackTotalSpaceEmbedding f) inst✝¹ ⊓ induced (Prod.snd ∘ pullbackTotalSpaceEmbedding f) inst✝

Tactic:
Pullback.TotalSpace.topologicalSpace,

Goal state after:
B : Type u
F : Type v
E : B → Type w₁
B' : Type w₂
inst✝¹ : TopologicalSpace B'
inst✝ : TopologicalSpace (TotalSpace F E)
f : B' → B
⊢ pullbackTopology F E f =
    induced (Prod.fst ∘ pullbackTotalSpaceEmbedding f) inst✝¹ ⊓ induced (Prod.snd ∘ pullbackTotalSpaceEmbedding f) inst✝
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 :
  ContinuousWithinAt f s x ∧
    HasFDerivWithinAt ((↑I' ∘ ↑(chartAt H' (f x))) ∘ f ∘ ↑(chartAt H x).symm ∘ ↑I.symm) 0
      (↑(chartAt H x).symm ∘ ↑I.symm ⁻¹' s ∩ range ↑I) (↑I (↑(chartAt H x) x))
⊢ fderivWithin 𝕜 ((↑I' ∘ ↑(chartAt H' (f x))) ∘ f ∘ ↑(chartAt H x).symm ∘ ↑I.symm)
      (↑(chartAt H x).symm ∘ ↑I.symm ⁻¹' s ∩ range ↑I) (↑I (↑(chartAt H x) x)) =
    0

Tactic:
rw [fderivWithin, if_pos]

Goal state after:
case hc
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 :
  ContinuousWithinAt f s x ∧
    HasFDerivWithinAt ((↑I' ∘ ↑(chartAt H' (f x))) ∘ f ∘ ↑(chartAt H x).symm ∘ ↑I.symm) 0
      (↑(chartAt H x).symm ∘ ↑I.symm ⁻¹' s ∩ range ↑I) (↑I (↑(chartAt H x) x))
⊢ HasFDerivWithinAt ((↑I' ∘ ↑(chartAt H' (f x))) ∘ f ∘ ↑(chartAt H x).symm ∘ ↑I.symm) 0
    (↑(chartAt H x).symm ∘ ↑I.symm ⁻¹' s ∩ range ↑I) (↑I (↑(chartAt H x) x))
Module: Mathlib.Topology.VectorBundle.Basic

Goal state before:
pp: R : Type u_1
B : Type u_2
F : Type u_3
E : B → Type u_4
inst✝⁹ : NontriviallyNormedField R
inst✝⁸ : (x : B) → AddCommMonoid (E x)
inst✝⁷ : (x : B) → Module R (E x)
inst✝⁶ : NormedAddCommGroup F
inst✝⁵ : NormedSpace R F
inst✝⁴ : TopologicalSpace B
inst✝³ : TopologicalSpace (TotalSpace F E)
inst✝² : (x : B) → TopologicalSpace (E x)
inst✝¹ : FiberBundle F E
e : Trivialization F TotalSpace.proj
inst✝ : Trivialization.IsLinear R e
b : B
⊢ Continuous[inst✝² b, PseudoMetricSpace.toUniformSpace.toTopologicalSpace] ⇑(Trivialization.linearMapAt R e b)

Tactic:
e.coe_linearMapAt b

Goal state after:
R : Type u_1
B : Type u_2
F : Type u_3
E : B → Type u_4
inst✝⁹ : NontriviallyNormedField R
inst✝⁸ : (x : B) → AddCommMonoid (E x)
inst✝⁷ : (x : B) → Module R (E x)
inst✝⁶ : NormedAddCommGroup F
inst✝⁵ : NormedSpace R F
inst✝⁴ : TopologicalSpace B
inst✝³ : TopologicalSpace (TotalSpace F E)
inst✝² : (x : B) → TopologicalSpace (E x)
inst✝¹ : FiberBundle F E
e : Trivialization F TotalSpace.proj
inst✝ : Trivialization.IsLinear R e
b : B
⊢ Continuous[inst✝² b, PseudoMetricSpace.toUniformSpace.toTopologicalSpace] fun y ↦
    if b ∈ e.baseSet then (↑e { proj := b, snd := y }).2 else 0
Module: Mathlib.Geometry.Manifold.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 f

Tactic:
rw [← mdifferentiableOn_univ, ← 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.Topology.VectorBundle.Basic

Goal state before:
pp: R : Type u_1
B : Type u_2
F : Type u_3
E : B → Type u_4
inst✝⁶ : NontriviallyNormedField R
inst✝⁵ : (x : B) → AddCommMonoid (E x)
inst✝⁴ : (x : B) → Module R (E x)
inst✝³ : NormedAddCommGroup F
inst✝² : NormedSpace R F
inst✝¹ : TopologicalSpace B
inst✝ : (x : B) → TopologicalSpace (E x)
a : VectorPrebundle R F E
e : Pretrivialization F TotalSpace.proj
he : e ∈ a.pretrivializationAtlas
e' : Pretrivialization F TotalSpace.proj
he' : e' ∈ a.pretrivializationAtlas
this : ContinuousOn (fun x ↦ (a.coordChange he' he x.1) x.2) ((e'.baseSet ∩ e.baseSet) ×ˢ univ)
⊢ ContinuousOn (↑e ∘ ↑e'.symm) ((e.baseSet ∩ e'.baseSet) ×ˢ univ)

Tactic:
inter_comm

Goal state after:
R : Type u_1
B : Type u_2
F : Type u_3
E : B → Type u_4
inst✝⁶ : NontriviallyNormedField R
inst✝⁵ : (x : B) → AddCommMonoid (E x)
inst✝⁴ : (x : B) → Module R (E x)
inst✝³ : NormedAddCommGroup F
inst✝² : NormedSpace R F
inst✝¹ : TopologicalSpace B
inst✝ : (x : B) → TopologicalSpace (E x)
a : VectorPrebundle R F E
e : Pretrivialization F TotalSpace.proj
he : e ∈ a.pretrivializationAtlas
e' : Pretrivialization F TotalSpace.proj
he' : e' ∈ a.pretrivializationAtlas
this : ContinuousOn (fun x ↦ (a.coordChange he' he x.1) x.2) ((e'.baseSet ∩ e.baseSet) ×ˢ univ)
⊢ ContinuousOn (↑e ∘ ↑e'.symm) ((e'.baseSet ∩ e.baseSet) ×ˢ univ)
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✝¹⁶ : NontriviallyNormedField 𝕜
EB : Type u_7
inst✝¹⁵ : NormedAddCommGroup EB
inst✝¹⁴ : NormedSpace 𝕜 EB
HB : Type u_8
inst✝¹³ : TopologicalSpace HB
IB : ModelWithCorners 𝕜 EB HB
inst✝¹² : TopologicalSpace B
inst✝¹¹ : ChartedSpace HB B
inst✝¹⁰ : (x : B) → AddCommMonoid (E x)
inst✝⁹ : (x : B) → Module 𝕜 (E x)
inst✝⁸ : NormedAddCommGroup F
inst✝⁷ : NormedSpace 𝕜 F
inst✝⁶ : TopologicalSpace (TotalSpace F E)
inst✝⁵ : (x : B) → TopologicalSpace (E x)
inst✝⁴ : FiberBundle F E
inst✝³ : VectorBundle 𝕜 F E
inst✝² : ContMDiffVectorBundle n F E IB
e e' : Trivialization F TotalSpace.proj
inst✝¹ : MemTrivializationAtlas e
inst✝ : MemTrivializationAtlas e'
⊢ ContMDiffOn IB 𝓘(𝕜, F →L[𝕜] F) n (fun b ↦ ↑(Trivialization.coordChangeL 𝕜 e e' b).symm) (e.baseSet ∩ e'.baseSet)

Tactic:
inter_comm

Goal state after:
n : ℕ∞ω
𝕜 : Type u_1
B : Type u_2
F : Type u_4
E : B → Type u_6
inst✝¹⁶ : NontriviallyNormedField 𝕜
EB : Type u_7
inst✝¹⁵ : NormedAddCommGroup EB
inst✝¹⁴ : NormedSpace 𝕜 EB
HB : Type u_8
inst✝¹³ : TopologicalSpace HB
IB : ModelWithCorners 𝕜 EB HB
inst✝¹² : TopologicalSpace B
inst✝¹¹ : ChartedSpace HB B
inst✝¹⁰ : (x : B) → AddCommMonoid (E x)
inst✝⁹ : (x : B) → Module 𝕜 (E x)
inst✝⁸ : NormedAddCommGroup F
inst✝⁷ : NormedSpace 𝕜 F
inst✝⁶ : TopologicalSpace (TotalSpace F E)
inst✝⁵ : (x : B) → TopologicalSpace (E x)
inst✝⁴ : FiberBundle F E
inst✝³ : VectorBundle 𝕜 F E
inst✝² : ContMDiffVectorBundle n F E IB
e e' : Trivialization F TotalSpace.proj
inst✝¹ : MemTrivializationAtlas e
inst✝ : MemTrivializationAtlas e'
⊢ ContMDiffOn IB 𝓘(𝕜, F →L[𝕜] F) n (fun b ↦ ↑(Trivialization.coordChangeL 𝕜 e e' b).symm) (e'.baseSet ∩ e.baseSet)
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'
hf : MDiffAt[s] f x
h : s ⊆ f ⁻¹' u
hg : MDiffAt[u] g (f x)
⊢ MDiffAt[s] (g ∘ f) x

Tactic:
exact hg.comp _ hf h

Goal state after:
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
x y z : M
h : z ∈ (extChartAt I x).source ∩ (extChartAt I y).source
⊢ ↑(extChartAt I x) z ∈ ((extChartAt I x).symm ≫ extChartAt I y).source

Tactic:
rw [PartialEquiv.trans_source'', PartialEquiv.symm_symm, PartialEquiv.symm_target]

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
x y z : M
h : z ∈ (extChartAt I x).source ∩ (extChartAt I y).source
⊢ ↑(extChartAt I x) z ∈ ↑(extChartAt I x) '' ((extChartAt I x).source ∩ (extChartAt I y).source)
Module: Mathlib.Geometry.Manifold.MFDeriv.SpecificFunctions

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'
p : TangentBundle I' M'
x₀ : M
⊢ tangentMap I' (I.prod I') (fun y ↦ (x₀, y)) p = ⟨(x₀, p.proj), (0, p.snd)⟩

Tactic:
by
  simp only [tangentMap, mfderiv_prod_right]
  rfl

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

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' → M''
p : M × M'
v : TangentSpace (I.prod I') p
hf : MDiffAt f p
⊢ (mfderiv% f p) v = ((mfderiv% fun z ↦ f (z, p.2)) p.1) v.1 + ((mfderiv% fun z ↦ f (p.1, z)) p.2) v.2

Tactic:
by
  rw [mfderiv_prod_eq_add_comp hf]
  rfl

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

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'
s : Set M
x : M
hs : UniqueMDiffAt[s] x
e : OpenPartialHomeomorph M M'
he : OpenPartialHomeomorph.MDifferentiable I I' e
hx : x ∈ e.source
⊢ UniqueMDiffAt[↑e '' (e.source ∩ s)] (↑e x)

Tactic:
inter_comm

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'
s : Set M
x : M
hs : UniqueMDiffAt[s] x
e : OpenPartialHomeomorph M M'
he : OpenPartialHomeomorph.MDifferentiable I I' e
hx : x ∈ e.source
⊢ UniqueMDiffAt[↑e '' (s ∩ e.source)] (↑e x)
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
r : ℝ
hr : r ∈ Ioo 0 f.rOut
⊢ support ↑(f.updateRIn r hr) = support ↑f

Tactic:
by
  simp only [support_eq_inter_preimage, updateRIn_rOut]

Goal state after:
Module: Mathlib.Topology.Compactness.Paracompact

Goal state before:
pp: case refine_1
ι : Type u
X : Type v
Y : Type w
inst✝³ : TopologicalSpace X
inst✝² : TopologicalSpace Y
inst✝¹ : CompactSpace X
inst✝ : ParacompactSpace Y
α : Type (max v w)
s : α → Set (X × Y)
ho : ∀ (a : α), IsOpen[instTopologicalSpaceProd] (s a)
hu : ⋃ a, s a = univ
a : X → Y → α
U : X → Y → Set X
V : X → Y → Set Y
hUo : ∀ (x : X) (y : Y), IsOpen[inst✝³] (U x y)
hVo : ∀ (x : X) (y : Y), IsOpen[inst✝²] (V x y)
hxU : ∀ (x : X) (y : Y), x ∈ U x y
hyV : ∀ (x : X) (y : Y), y ∈ V x y
hUV : ∀ (x : X) (y : Y), U x y ×ˢ V x y ⊆ s (a x y)
T : Y → Finset X
hT : ∀ (y : Y), ⋃ x ∈ T y, U x y = ⊤
W : Y → Set Y := fun y ↦ ⋂ x ∈ T y, V x y
hWo : ∀ (y : Y), IsOpen[inst✝²] (W y)
hW : ∀ (y : Y), y ∈ W y
E : Y → Set Y
hEo : ∀ (a : Y), IsOpen[inst✝²] (E a)
hE : ⋃ i, E i = univ
hEf : LocallyFinite E
hEA : ∀ (a : Y), E a ⊆ W a
x✝ : X × Y
x : X
y : Y
⊢ ∃ i, (x, y) ∈ U (↑i.snd) i.fst ×ˢ E i.fst

Tactic:
rcases iUnion_eq_univ_iff.1 hE y with ⟨b, hb⟩

Goal state after:
case refine_1
ι : Type u
X : Type v
Y : Type w
inst✝³ : TopologicalSpace X
inst✝² : TopologicalSpace Y
inst✝¹ : CompactSpace X
inst✝ : ParacompactSpace Y
α : Type (max v w)
s : α → Set (X × Y)
ho : ∀ (a : α), IsOpen[instTopologicalSpaceProd] (s a)
hu : ⋃ a, s a = univ
a : X → Y → α
U : X → Y → Set X
V : X → Y → Set Y
hUo : ∀ (x : X) (y : Y), IsOpen[inst✝³] (U x y)
hVo : ∀ (x : X) (y : Y), IsOpen[inst✝²] (V x y)
hxU : ∀ (x : X) (y : Y), x ∈ U x y
hyV : ∀ (x : X) (y : Y), y ∈ V x y
hUV : ∀ (x : X) (y : Y), U x y ×ˢ V x y ⊆ s (a x y)
T : Y → Finset X
hT : ∀ (y : Y), ⋃ x ∈ T y, U x y = ⊤
W : Y → Set Y := fun y ↦ ⋂ x ∈ T y, V x y
hWo : ∀ (y : Y), IsOpen[inst✝²] (W y)
hW : ∀ (y : Y), y ∈ W y
E : Y → Set Y
hEo : ∀ (a : Y), IsOpen[inst✝²] (E a)
hE : ⋃ i, E i = univ
hEf : LocallyFinite E
hEA : ∀ (a : Y), E a ⊆ W a
x✝ : X × Y
x : X
y b : Y
hb : y ∈ E b
⊢ ∃ i, (x, y) ∈ U (↑i.snd) i.fst ×ˢ E i.fst
Module: Mathlib.Topology.EMetricSpace.Paracompact

Goal state before:
pp: α : Type u_1
inst✝ : PseudoEMetricSpace α
pow_pos : ∀ (k : ℕ), 0 < 2⁻¹ ^ k
hpow_le : ∀ {m n : ℕ}, m ≤ n → 2⁻¹ ^ n ≤ 2⁻¹ ^ m
h2pow : ∀ (n : ℕ), 2 * 2⁻¹ ^ (n + 1) = 2⁻¹ ^ n
ι : Type u_1
s : ι → Set α
ho : ∀ (a : ι), IsOpen[PseudoEMetricSpace.toUniformSpace.toTopologicalSpace] (s a)
hcov : ∀ (x : α), ∃ i, x ∈ s i
w✝ : LinearOrder ι
wf : WellFoundedLT ι
ind : α → ι := fun x ↦ ⋯.min {i | x ∈ s i} ⋯
mem_ind : ∀ (x : α), x ∈ s (ind x)
notMem_of_lt_ind : ∀ {x : α} {i : ι}, i < ind x → x ∈ s i → False
D : ℕ → ι → Set α :=
  fun t ↦
    Nat.strongRec (motive := fun x ↦ ι → Set α)
      (fun n D' i ↦
        ⋃ x,
          ⋃ (_ : ind x = i),
            ⋃ (_ : eball x (3 * 2⁻¹ ^ n) ⊆ s i), ⋃ (_ : ∀ (m : ℕ) (H : m < n) (j : ι), x ∉ D' m H j), eball x (2⁻¹ ^ n))
      t
hD :
  D = fun t ↦
    Nat.strongRec (motive := fun x ↦ ι → Set α)
      (fun n D' i ↦
        ⋃ x,
          ⋃ (_ : ind x = i),
            ⋃ (_ : eball x (3 * 2⁻¹ ^ n) ⊆ s i), ⋃ (_ : ∀ (m : ℕ) (H : m < n) (j : ι), x ∉ D' m H j), eball x (2⁻¹ ^ n))
      t
Dn :
  ∀ (n : ℕ) (i : ι),
    D n i =
      ⋃ x,
        ⋃ (_ : ind x = i), ⋃ (_ : eball x (3 * 2⁻¹ ^ n) ⊆ s i), ⋃ (_ : ∀ m < n, ∀ (j : ι), x ∉ D m j), eball x (2⁻¹ ^ n)
n : ℕ
i : ι
y : α
⊢ y ∈ D n i ↔ ∃ x, ind x = i ∧ eball x (3 * 2⁻¹ ^ n) ⊆ s i ∧ (∀ m < n, ∀ (j : ι), x ∉ D m j) ∧ edist y x < 2⁻¹ ^ n

Tactic:
Dn n i

Goal state after:
α : Type u_1
inst✝ : PseudoEMetricSpace α
pow_pos : ∀ (k : ℕ), 0 < 2⁻¹ ^ k
hpow_le : ∀ {m n : ℕ}, m ≤ n → 2⁻¹ ^ n ≤ 2⁻¹ ^ m
h2pow : ∀ (n : ℕ), 2 * 2⁻¹ ^ (n + 1) = 2⁻¹ ^ n
ι : Type u_1
s : ι → Set α
ho : ∀ (a : ι), IsOpen[PseudoEMetricSpace.toUniformSpace.toTopologicalSpace] (s a)
hcov : ∀ (x : α), ∃ i, x ∈ s i
w✝ : LinearOrder ι
wf : WellFoundedLT ι
ind : α → ι := fun x ↦ ⋯.min {i | x ∈ s i} ⋯
mem_ind : ∀ (x : α), x ∈ s (ind x)
notMem_of_lt_ind : ∀ {x : α} {i : ι}, i < ind x → x ∈ s i → False
D : ℕ → ι → Set α :=
  fun t ↦
    Nat.strongRec (motive := fun x ↦ ι → Set α)
      (fun n D' i ↦
        ⋃ x,
          ⋃ (_ : ind x = i),
            ⋃ (_ : eball x (3 * 2⁻¹ ^ n) ⊆ s i), ⋃ (_ : ∀ (m : ℕ) (H : m < n) (j : ι), x ∉ D' m H j), eball x (2⁻¹ ^ n))
      t
hD :
  D = fun t ↦
    Nat.strongRec (motive := fun x ↦ ι → Set α)
      (fun n D' i ↦
        ⋃ x,
          ⋃ (_ : ind x = i),
            ⋃ (_ : eball x (3 * 2⁻¹ ^ n) ⊆ s i), ⋃ (_ : ∀ (m : ℕ) (H : m < n) (j : ι), x ∉ D' m H j), eball x (2⁻¹ ^ n))
      t
Dn :
  ∀ (n : ℕ) (i : ι),
    D n i =
      ⋃ x,
        ⋃ (_ : ind x = i), ⋃ (_ : eball x (3 * 2⁻¹ ^ n) ⊆ s i), ⋃ (_ : ∀ m < n, ∀ (j : ι), x ∉ D m j), eball x (2⁻¹ ^ n)
n : ℕ
i : ι
y : α
⊢ y ∈
      ⋃ x,
        ⋃ (_ : ind x = i),
          ⋃ (_ : eball x (3 * 2⁻¹ ^ n) ⊆ s i), ⋃ (_ : ∀ m < n, ∀ (j : ι), x ∉ D m j), eball x (2⁻¹ ^ n) ↔
    ∃ x, ind x = i ∧ eball x (3 * 2⁻¹ ^ n) ⊆ s i ∧ (∀ m < n, ∀ (j : ι), x ∉ D m j) ∧ edist y x < 2⁻¹ ^ n