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

Goal state before:
pp: 𝕜 : Type u_1
inst✝³ : RCLike 𝕜
μ : Measure ℝ
E : Type u_2
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
inst✝ : NormedSpace 𝕜 E
a b : ℝ
bound : ℝ → ℝ
F F' : 𝕜 → ℝ → E
x₀ : 𝕜
s : Set 𝕜
hs : s ∈ 𝓝 x₀
hF_meas : ∀ᶠ (x : 𝕜) in 𝓝 x₀, AEStronglyMeasurable (F x) (μ.restrict (Ι a b))
hF'_meas : AEStronglyMeasurable (F' x₀) (μ.restrict (Ι a b))
h_bound : ∀ᵐ (x : ℝ) ∂μ.restrict (Ι a b), ∀ x_1 ∈ s, ‖F' x_1 x‖ ≤ bound x
h_diff : ∀ᵐ (x : ℝ) ∂μ.restrict (Ι a b), ∀ x_1 ∈ s, HasDerivAt (fun x_2 ↦ F x_2 x) (F' x_1 x) x_1
hF_int : IntegrableOn (F x₀) (Ι a b) μ
bound_integrable : IntegrableOn bound (Ι a b) μ
this :
  Integrable (F' x₀) (μ.restrict (Ι a b)) ∧
    HasDerivAt (fun n ↦ ∫ (a : ℝ) in Ι a b, F n a ∂μ) (∫ (a : ℝ) in Ι a b, F' x₀ a ∂μ) x₀
⊢ IntegrableOn (F' x₀) (Ι a b) μ ∧
    HasDerivAt (fun x ↦ (if a ≤ b then 1 else -1) • ∫ (t : ℝ) in Ι a b, F x t ∂μ)
      ((if a ≤ b then 1 else -1) • ∫ (t : ℝ) in Ι a b, F' x₀ t ∂μ) x₀

Tactic:
exact ⟨this.1, this.2.const_smul _⟩

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

Goal state before:
pp: H : Type u
M : Type u_2
inst✝² : TopologicalSpace H
inst✝¹ : TopologicalSpace M
inst✝ : ChartedSpace H M
s : Set M
⊢ (∀ (i : M), IsOpen[inst✝¹] ((chartAt H i).source ∩ s)) ↔
    ∀ (x : M), IsOpen[inst✝²] (↑(chartAt H x) '' ((chartAt H x).source ∩ s))

Tactic:
simp only [(chartAt H _).isOpen_image_iff_of_subset_source inter_subset_left]

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

Goal state before:
pp: H : Type u_1
inst✝ : TopologicalSpace H
S : Set (StructureGroupoid H)
e : OpenPartialHomeomorph H H
he : ∀ x ∈ e.source, ∃ s, IsOpen[inst✝] s ∧ x ∈ s ∧ ∀ i ∈ S, e.restr s ∈ i.members
i : StructureGroupoid H
hi : i ∈ S
x : H
hex : x ∈ e.source
⊢ ∃ s, IsOpen[inst✝] s ∧ x ∈ s ∧ e.restr s ∈ i.members

Tactic:
rcases he x hex with ⟨s, hs⟩

Goal state after:
H : Type u_1
inst✝ : TopologicalSpace H
S : Set (StructureGroupoid H)
e : OpenPartialHomeomorph H H
he : ∀ x ∈ e.source, ∃ s, IsOpen[inst✝] s ∧ x ∈ s ∧ ∀ i ∈ S, e.restr s ∈ i.members
i : StructureGroupoid H
hi : i ∈ S
x : H
hex : x ∈ e.source
s : Set H
hs : IsOpen[inst✝] s ∧ x ∈ s ∧ ∀ i ∈ S, e.restr s ∈ i.members
⊢ ∃ s, IsOpen[inst✝] s ∧ x ∈ s ∧ e.restr s ∈ i.members
Module: Mathlib.Geometry.Manifold.StructureGroupoid

Goal state before:
pp: H : Type u_1
inst✝ : TopologicalSpace H
S : Set (StructureGroupoid H)
e : OpenPartialHomeomorph H H
he : ∀ x ∈ e.source, ∃ s, IsOpen[inst✝] s ∧ x ∈ s ∧ ∀ i ∈ S, e.restr s ∈ i.members
i : StructureGroupoid H
hi : i ∈ S
x : H
hex : x ∈ e.source
s : Set H
hs : IsOpen[inst✝] s ∧ x ∈ s ∧ ∀ i ∈ S, e.restr s ∈ i.members
⊢ ∃ s, IsOpen[inst✝] s ∧ x ∈ s ∧ e.restr s ∈ i.members

Tactic:
exact ⟨s, ⟨hs.left, ⟨hs.right.left, hs.right.right i hi⟩⟩⟩

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

Goal state before:
pp: case inr
H : Type u
H' : Type u_1
M : Type u_2
M' : Type u_3
M'' : Type u_4
inst✝³ : TopologicalSpace H
inst✝² : TopologicalSpace M
inst✝¹ : TopologicalSpace M'
cm : ChartedSpace H M
cm' : ChartedSpace H M'
inst✝ : Nonempty H
x : M'
⊢ Sum.inr x ∈
    (Sum.elim (fun x ↦ (ChartedSpace.chartAt x).lift_openEmbedding ⋯)
        (fun x ↦ (ChartedSpace.chartAt x).lift_openEmbedding ⋯) (Sum.inr x)).source

Tactic:
rw [Sum.elim_inr, lift_openEmbedding_source,
        ← OpenPartialHomeomorph.lift_openEmbedding_source _ IsOpenEmbedding.inr]
      use x, cm'.mem_chart_source x

Goal state after:
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
⊢ IsOpen[c.toTopologicalSpace] e.source

Tactic:
by convert! c.open_source' he

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

Goal state before:
pp: H : Type u
M : Type u_2
inst✝² : TopologicalSpace H
inst✝¹ : TopologicalSpace M
inst✝ : ChartedSpace H M
s : Opens M
hs : Nonempty ↥s
e : OpenPartialHomeomorph (↥s) H
he : e ∈ atlas H ↥s
⊢ ∃ x, e = (chartAt H ↑x).subtypeRestr hs

Tactic:
by
  rcases he with ⟨xset, ⟨x, hx⟩, he⟩
  exact ⟨x, mem_singleton_iff.mp (by convert! he)⟩

Goal state after:
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
ε : ℝ≥0
εpos : 0 < ε
δ : (E →L[ℝ] E) → ℝ≥0
hδ :
  ∀ (A : E →L[ℝ] E),
    0 < δ A ∧
      (∀ (B : E →L[ℝ] E), ‖B - A‖ ≤ ↑(δ A) → |B.det - A.det| ≤ ↑ε) ∧
        ∀ (t : Set E) (g : E → E), ApproximatesLinearOn g A t (δ A) → μ (g '' t) ≤ (ENNReal.ofReal |A.det| + ↑ε) * μ t
t : ℕ → Set E
A : ℕ → E →L[ℝ] E
t_disj : Pairwise (Disjoint on t)
t_meas : ∀ (n : ℕ), MeasurableSet (t n)
t_cover : s ⊆ ⋃ n, t n
ht : ∀ (n : ℕ), ApproximatesLinearOn f (A n) (s ∩ t n) (δ (A n))
⊢ μ (f '' s) ≤ μ (⋃ n, f '' (s ∩ t n))

Tactic:
by
      apply measure_mono
      rw [← image_iUnion, ← inter_iUnion]
      exact Set.image_mono (subset_inter Subset.rfl t_cover)

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

Goal state before:
pp: H : Type u
H' : Type u_1
M : Type u_2
M' : Type u_3
M'' : Type u_4
inst✝⁸ : TopologicalSpace H
inst✝⁷ : TopologicalSpace M
inst✝⁶ : ChartedSpace H M
inst✝⁵ : TopologicalSpace M'
inst✝⁴ : TopologicalSpace M''
G : StructureGroupoid H
inst✝³ : ChartedSpace H M'
inst✝² : ChartedSpace H M''
e : OpenPartialHomeomorph M H
he : e ∈ atlas H M
inst✝¹ : HasGroupoid M G
inst✝ : ClosedUnderRestriction G
⊢ let s := { carrier := e.source, is_open' := ⋯ };
  let t := { carrier := e.target, is_open' := ⋯ };
  Structomorph G ↥s ↥t

Tactic:
by
  intro s t
  by_cases! h : Nonempty e.source
  · exact { e.toHomeomorphSourceTarget with
      mem_groupoid :=
        -- The atlas of H on itself has only one chart, hence c' is the inclusion.
        -- Then, compatibility of `G` *almost* yields our claim --- except that `e` is a chart
        -- on `M` and `c` is one on `s`: we need to show that restricting `e` to `s` and composing
        -- with `c'` yields a chart in the maximal atlas of `s`.
        fun c c' hc hc' ↦ G.compatible_of_mem_maximalAtlas (G.subset_maximalAtlas hc)
          (G.restriction_mem_maximalAtlas_subtype he h c' hc') }
  · have : IsEmpty t := isEmpty_coe_sort.mpr
      (by convert! e.image_source_eq_target ▸ image_eq_empty.mpr (isEmpty_coe_sort.mp h))
    exact { Homeomorph.empty with
      -- `c'` cannot exist: it would be the restriction of `chartAt H x` at some `x ∈ t`.
      mem_groupoid := fun _ c' _ ⟨_, ⟨x, _⟩, _⟩ ↦ (this.false x).elim }

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

Goal state before:
pp: case h
H : Type u_1
M : Type u_2
H' : Type u_3
inst✝³ : TopologicalSpace H
inst✝² : TopologicalSpace M
inst✝¹ : ChartedSpace H M
inst✝ : TopologicalSpace H'
G : StructureGroupoid H
G' : StructureGroupoid H'
e : OpenPartialHomeomorph M H
P : (H → H') → Set H → H → Prop
s : Set M
x : M
hG : G.LocalInvariantProp G' P
g : M → H'
he : e ∈ maximalAtlas M G
xe : x ∈ e.source
⊢ (g ∘ ↑(chartAt H x).symm) ∘ ↑((chartAt H x).symm ≫ₕ e).symm =ᶠ[𝓝 (↑e x)] g ∘ ↑e.symm

Tactic:
refine ((e.eventually_nhds' _ xe).mpr <| (chartAt H x).eventually_left_inverse
      (mem_chart_source H x)).mono fun y hy ↦ ?_

Goal state after:
case h
H : Type u_1
M : Type u_2
H' : Type u_3
inst✝³ : TopologicalSpace H
inst✝² : TopologicalSpace M
inst✝¹ : ChartedSpace H M
inst✝ : TopologicalSpace H'
G : StructureGroupoid H
G' : StructureGroupoid H'
e : OpenPartialHomeomorph M H
P : (H → H') → Set H → H → Prop
s : Set M
x : M
hG : G.LocalInvariantProp G' P
g : M → H'
he : e ∈ maximalAtlas M G
xe : x ∈ e.source
y : H
hy : ↑(chartAt H x).symm (↑(chartAt H x) (↑e.symm y)) = ↑e.symm y
⊢ ((g ∘ ↑(chartAt H x).symm) ∘ ↑((chartAt H x).symm ≫ₕ e).symm) y = (g ∘ ↑e.symm) y
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
I : ModelWithCorners 𝕜 E H
e e' : OpenPartialHomeomorph M H
⊢ UniqueDiffOn 𝕜 (extendCoordChange e e').target

Tactic:
rw [← extendCoordChange_symm, PartialEquiv.symm_target]
  exact uniqueDiffOn_extendCoordChange_source

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

Goal state before:
pp: 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
n : ℕ∞ω
M : Type u_4
inst✝³ : TopologicalSpace M
inst✝² : ChartedSpace H M
M' : Type u_5
inst✝¹ : TopologicalSpace M'
inst✝ : ChartedSpace H M'
hM : IsManifold I n M
hM' : IsManifold I n M'
e e' : OpenPartialHomeomorph (M ⊕ M') H
he : e ∈ atlas H (M ⊕ M')
he' : e' ∈ atlas H (M ⊕ M')
⊢ e.symm ≫ₕ e' ∈ contDiffGroupoid n I

Tactic:
by
    obtain (h | h) := isEmpty_or_nonempty H
    · exact ContDiffGroupoid.mem_of_source_eq_empty _ (eq_empty_of_isEmpty _)
    obtain (⟨f, hf, hef⟩ | ⟨f, hf, hef⟩) := ChartedSpace.mem_atlas_sum he
    · obtain (⟨f', hf', he'f'⟩ | ⟨f', hf', he'f'⟩) := ChartedSpace.mem_atlas_sum he'
      · rw [hef, he'f', f.lift_openEmbedding_trans f' IsOpenEmbedding.inl]
        exact hM.compatible hf hf'
      · rw [hef, he'f']
        apply ContDiffGroupoid.mem_of_source_eq_empty
        ext x
        exact ⟨fun ⟨hx₁, hx₂⟩ ↦ by simp_all, fun hx ↦ hx.elim⟩
    · -- Analogous argument to the first case: is there a way to deduplicate?
      obtain (⟨f', hf', he'f'⟩ | ⟨f', hf', he'f'⟩) := ChartedSpace.mem_atlas_sum he'
      · rw [hef, he'f']
        apply ContDiffGroupoid.mem_of_source_eq_empty
        ext x
        exact ⟨fun ⟨hx₁, hx₂⟩ ↦ by simp_all, fun hx ↦ hx.elim⟩
      · rw [hef, he'f', f.lift_openEmbedding_trans f' IsOpenEmbedding.inr]
        exact hM'.compatible hf hf'

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

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
n : ℕ∞ω
f : M → M
hf : ContMDiffOn I I n f s
hmaps : MapsTo f s s
k : ℕ
⊢ ContMDiffOn I I n f^[k] s

Tactic:
induction k with
  | zero => simpa using contMDiffOn_id
  | succ k h => simpa using h.comp hf hmaps

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

Goal state before:
pp: case refine_1
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'
hf : LiftPropOn (contDiffGroupoid n I).IsLocalStructomorphWithinAt (↑f) f.source
x : M
hx : x ∈ f.source
c : OpenPartialHomeomorph M H := chartAt H x
c' : OpenPartialHomeomorph M' H := chartAt H (↑f x)
hxf :
  (contDiffGroupoid n I).IsLocalStructomorphWithinAt (↑(chartAt H (↑f x)) ∘ ↑f ∘ ↑(chartAt H x).symm)
    (↑(chartAt H x).symm ⁻¹' f.source) (↑(chartAt H x) x)
e : OpenPartialHomeomorph H H
he : e ∈ contDiffGroupoid n I
he' : EqOn (↑c' ∘ ↑f ∘ ↑c.symm) (↑e) (↑c.symm ⁻¹' f.source ∩ e.source)
hex : ↑c x ∈ e.source
s : Set M := (f ≫ₕ c').source ∩ ((c ≫ₕ e) ≫ₕ c'.symm).source
⊢ (x ∈ f.source ∧ ↑f x ∈ c'.source) ∧ (x ∈ c.source ∧ ↑c x ∈ e.source) ∧ ↑e (↑c x) ∈ c'.target

Tactic:
rw [← he'] <;> simp only [c, c', hx, hex, mfld_simps]

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

Goal state before:
pp: 𝕜 : Type u_1
inst✝⁸ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : NormedSpace 𝕜 E
H : Type u_3
inst✝⁵ : TopologicalSpace H
I : ModelWithCorners 𝕜 E H
M : Type u_4
inst✝⁴ : TopologicalSpace M
inst✝³ : ChartedSpace H M
n : ℕ∞ω
M' : Type u_5
inst✝² : TopologicalSpace M'
inst✝¹ : IsManifold I n M
inst✝ : ChartedSpace H M'
IsM' : IsManifold I n M'
f : OpenPartialHomeomorph M M'
hf : LiftPropOn (contDiffGroupoid n I).IsLocalStructomorphWithinAt (↑f) f.source
⊢ ContMDiffOn I I n (↑f) f.source

Tactic:
apply contMDiffOn_of_locally_contMDiffOn
  intro x hx
  let c := chartAt H x
  let c' := chartAt H (f x)
  obtain ⟨-, hxf⟩ := hf x hx
  -- Since `f` is a local structomorph, it is locally equal to some transferred element `e` of
  -- the `contDiffGroupoid`.
  obtain
    ⟨e, he, he' : EqOn (c' ∘ f ∘ c.symm) e (c.symm ⁻¹' f.source ∩ e.source), hex :
      c x ∈ e.source⟩ :=
    hxf (by simp only [hx, mfld_simps])
  -- We choose a convenient set `s` in `M`.
  let s : Set M := (f.trans c').source ∩ ((c.trans e).trans c'.symm).source
  refine ⟨s, (f.trans c').open_source.inter ((c.trans e).trans c'.symm).open_source, ?_, ?_⟩
  · simp only [s, mfld_simps]
    rw [← he'] <;> simp only [c, c', hx, hex, mfld_simps]
  -- We need to show `f` is `ContMDiffOn` the domain `s ∩ f.source`.  We show this in two
  -- steps: `f` is equal to `c'.symm ∘ e ∘ c` on that domain and that function is
  -- `ContMDiffOn` it.
  have H₁ : ContMDiffOn I I n (c'.symm ∘ e ∘ c) s := by
    have hc' : ContMDiffOn I I n c'.symm _ := contMDiffOn_chart_symm
    have he'' : ContMDiffOn I I n e _ := contMDiffOn_of_mem_contDiffGroupoid he
    have hc : ContMDiffOn I I n c _ := contMDiffOn_chart
    refine (hc'.comp' (he''.comp' hc)).mono ?_
    dsimp [s, c, c']
    mfld_set_tac
  have H₂ : EqOn f (c'.symm ∘ e ∘ c) s := by
    intro y hy
    simp only [s, mfld_simps] at hy
    have hy₁ : f y ∈ c'.source := by simp only [hy, mfld_simps]
    have hy₂ : y ∈ c.source := by simp only [hy, mfld_simps]
    have hy₃ : c y ∈ c.symm ⁻¹' f.source ∩ e.source := by simp only [hy, mfld_simps]
    calc
      f y = c'.symm (c' (f y)) := by rw [c'.left_inv hy₁]
      _ = c'.symm (c' (f (c.symm (c y)))) := by rw [c.left_inv hy₂]
      _ = c'.symm (e (c y)) := by rw [← he' hy₃]; rfl
  refine (H₁.congr H₂).mono ?_
  mfld_set_tac

Goal state after:
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
x : B
⊢ (∃ i, x ∈ u i) → x ∈ U

Tactic:
rintro ⟨x, hx⟩

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
x✝ : B
x : ↑U
hx : x✝ ∈ u x
⊢ x✝ ∈ U
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✝⁷ : Semiring R
inst✝⁶ : TopologicalSpace F
inst✝⁵ : TopologicalSpace B
e✝ : Pretrivialization F TotalSpace.proj
x : TotalSpace F E
b✝ : B
y : E b✝
inst✝⁴ : AddCommMonoid F
inst✝³ : Module R F
inst✝² : (x : B) → AddCommMonoid (E x)
inst✝¹ : (x : B) → Module R (E x)
e : Pretrivialization F TotalSpace.proj
inst✝ : Pretrivialization.IsLinear R e
b : B
⊢ F →ₗ[R] E b

Tactic:
refine if hb : b ∈ e.baseSet then IsLinearMap.mk' (e.symm b) ?_ else 0

Goal state after:
R : Type u_1
B : Type u_2
F : Type u_3
E : B → Type u_4
inst✝⁷ : Semiring R
inst✝⁶ : TopologicalSpace F
inst✝⁵ : TopologicalSpace B
e✝ : Pretrivialization F TotalSpace.proj
x : TotalSpace F E
b✝ : B
y : E b✝
inst✝⁴ : AddCommMonoid F
inst✝³ : Module R F
inst✝² : (x : B) → AddCommMonoid (E x)
inst✝¹ : (x : B) → Module R (E x)
e : Pretrivialization F TotalSpace.proj
inst✝ : Pretrivialization.IsLinear R e
b : B
hb : b ∈ e.baseSet
⊢ IsLinearMap R (e.symm b)
Module: Mathlib.Geometry.Manifold.VectorBundle.Basic

Goal state before:
pp: B : Type u_2
F : Type u_4
E : B → Type u_6
inst✝⁶ : TopologicalSpace F
inst✝⁵ : TopologicalSpace (TotalSpace F E)
inst✝⁴ : (x : B) → TopologicalSpace (E x)
HB : Type u_7
inst✝³ : TopologicalSpace HB
inst✝² : TopologicalSpace B
inst✝¹ : ChartedSpace HB B
inst✝ : FiberBundle F E
x : TotalSpace F E
⊢ chartAt (ModelProd HB F) x =
    (trivializationAt F E x.proj).toOpenPartialHomeomorph ≫ₕ (chartAt HB x.proj).prod (OpenPartialHomeomorph.refl F)

Tactic:
by
  dsimp only [chartAt_comp, prodChartedSpace_chartAt, FiberBundle.chartedSpace'_chartAt,
    chartAt_self_eq]
  rw [Trivialization.coe_coe, Trivialization.coe_fst' _ (mem_baseSet_trivializationAt F E x.proj)]

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
M : Type u_5
E : B → Type u_6
inst✝²¹ : NontriviallyNormedField 𝕜
EB : Type u_7
inst✝²⁰ : NormedAddCommGroup EB
inst✝¹⁹ : NormedSpace 𝕜 EB
HB : Type u_8
inst✝¹⁸ : TopologicalSpace HB
IB : ModelWithCorners 𝕜 EB HB
inst✝¹⁷ : TopologicalSpace B
inst✝¹⁶ : ChartedSpace HB B
EM : Type u_9
inst✝¹⁵ : NormedAddCommGroup EM
inst✝¹⁴ : NormedSpace 𝕜 EM
HM : Type u_10
inst✝¹³ : TopologicalSpace HM
IM : ModelWithCorners 𝕜 EM HM
inst✝¹² : TopologicalSpace M
inst✝¹¹ : ChartedSpace HM M
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'
s : Set M
f : M → B
g : M → F
x : M
hf : ContMDiffWithinAt IM IB n f s x
hg : ContMDiffWithinAt IM 𝓘(𝕜, F) n g s x
he : f x ∈ e.baseSet
he' : f x ∈ e'.baseSet
⊢ ContMDiffWithinAt IM 𝓘(𝕜, F) n (fun y ↦ e.coordChange e' (f y) (g y)) s x

Tactic:
by
  refine ((hf.coordChangeL he he').clm_apply hg).congr_of_eventuallyEq ?_ ?_
  · have : e.baseSet ∩ e'.baseSet ∈ 𝓝 (f x) :=
     (e.open_baseSet.inter e'.open_baseSet).mem_nhds ⟨he, he'⟩
    filter_upwards [hf.continuousWithinAt this] with y hy
    exact (Trivialization.coordChangeL_apply' e e' hy (g y)).symm
  · exact (Trivialization.coordChangeL_apply' e e' ⟨he, he'⟩ (g x)).symm

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

Goal state before:
pp: case refine_1
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 j : ↑(atlas H M)
this : IsManifold I (0 + 1) M
⊢ (↑i).source ∩ (↑j).source ⊆ (↑i).source

Tactic:
exact inter_subset_left

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

Goal state before:
pp: n : ℕ∞ω
𝕜 : Type u_1
B : Type u_2
B' : Type u_3
F : Type u_4
M : Type u_5
E : B → Type u_6
inst✝²⁰ : NontriviallyNormedField 𝕜
EB : Type u_7
inst✝¹⁹ : NormedAddCommGroup EB
inst✝¹⁸ : NormedSpace 𝕜 EB
HB : Type u_8
inst✝¹⁷ : TopologicalSpace HB
IB : ModelWithCorners 𝕜 EB HB
inst✝¹⁶ : TopologicalSpace B
inst✝¹⁵ : ChartedSpace HB B
EM : Type u_9
inst✝¹⁴ : NormedAddCommGroup EM
inst✝¹³ : NormedSpace 𝕜 EM
HM : Type u_10
inst✝¹² : TopologicalSpace HM
IM : ModelWithCorners 𝕜 EM HM
inst✝¹¹ : TopologicalSpace M
inst✝¹⁰ : ChartedSpace HM M
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
ι : Type u_11
Z : VectorBundleCore 𝕜 B F ι
inst✝ : Z.IsContMDiff IB n
i i' : ι
⊢ ContMDiffOn IB 𝓘(𝕜, F →L[𝕜] F) n
    (fun b ↦ ↑(Trivialization.coordChangeL 𝕜 (Z.toFiberBundleCore.localTriv i) (Z.toFiberBundleCore.localTriv i') b))
    ((Z.toFiberBundleCore.localTriv i).baseSet ∩ (Z.toFiberBundleCore.localTriv i').baseSet)

Tactic:
refine (Z.contMDiffOn_coordChange IB i i').congr fun b hb ↦ ?_

Goal state after:
n : ℕ∞ω
𝕜 : Type u_1
B : Type u_2
B' : Type u_3
F : Type u_4
M : Type u_5
E : B → Type u_6
inst✝²⁰ : NontriviallyNormedField 𝕜
EB : Type u_7
inst✝¹⁹ : NormedAddCommGroup EB
inst✝¹⁸ : NormedSpace 𝕜 EB
HB : Type u_8
inst✝¹⁷ : TopologicalSpace HB
IB : ModelWithCorners 𝕜 EB HB
inst✝¹⁶ : TopologicalSpace B
inst✝¹⁵ : ChartedSpace HB B
EM : Type u_9
inst✝¹⁴ : NormedAddCommGroup EM
inst✝¹³ : NormedSpace 𝕜 EM
HM : Type u_10
inst✝¹² : TopologicalSpace HM
IM : ModelWithCorners 𝕜 EM HM
inst✝¹¹ : TopologicalSpace M
inst✝¹⁰ : ChartedSpace HM M
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
ι : Type u_11
Z : VectorBundleCore 𝕜 B F ι
inst✝ : Z.IsContMDiff IB n
i i' : ι
b : B
hb : b ∈ Z.baseSet i ∩ Z.baseSet i'
⊢ ↑(Trivialization.coordChangeL 𝕜 (Z.toFiberBundleCore.localTriv i) (Z.toFiberBundleCore.localTriv i') b) =
    Z.coordChange i i' b
Module: Mathlib.Geometry.Manifold.VectorBundle.Tangent

Goal state before:
pp: 𝕜 : Type u_1
inst✝³ : NontriviallyNormedField 𝕜
n : ℕ∞ω
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
⊢ ContMDiff I.tangent (I.prod 𝓘(𝕜, E)) n ⇑(tangentBundleModelSpaceHomeomorph I)

Tactic:
by
  apply contMDiff_iff.2 ⟨Homeomorph.continuous _, fun x y ↦ ?_⟩
  apply contDiffOn_id.congr
  simp only [mfld_simps, mem_range, TotalSpace.toProd, Equiv.coe_fn_symm_mk, forall_exists_index,
    Prod.forall, Prod.mk.injEq]
  rintro a b x rfl
  simp [PartialEquiv.prod]

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

Goal state before:
pp: E : Type uE
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace ℝ E
H : Type uH
inst✝³ : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type uM
inst✝² : TopologicalSpace M
inst✝¹ : ChartedSpace H M
c : M
f : SmoothBumpFunction I c
inst✝ : FiniteDimensional ℝ E
⊢ (extChartAt I c).source ∩ ↑(extChartAt I c) ⁻¹' ball (↑(extChartAt I c) c) f.rOut ⊆ (extChartAt I c).source

Tactic:
exact inter_subset_left

Goal state after: