mathlib-initiative/mathlib-tactics huggingface dataset:
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'
hf : MDiffAt f p
⊢ mfderiv% f p = (mfderiv% fun z ↦ f (z.1, p.2)) p + (mfderiv% fun z ↦ f (p.1, z.2)) p

Tactic:
erw [mfderiv_comp_of_eq hf (mdifferentiableAt_fst.prodMk mdifferentiableAt_const) rfl,
    mfderiv_comp_of_eq hf (mdifferentiableAt_const.prodMk mdifferentiableAt_snd) rfl,
    ← ContinuousLinearMap.comp_add,
    mdifferentiableAt_fst.mfderiv_prod mdifferentiableAt_const,
    mdifferentiableAt_const.mfderiv_prod mdifferentiableAt_snd, mfderiv_fst,
    mfderiv_snd, mfderiv_const, mfderiv_const]

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' → M''
p : M × M'
hf : MDiffAt f p
⊢ mfderiv% f p =
    mfderiv% f (p.1, p.2) ∘SL
      ((ContinuousLinearMap.fst 𝕜 (TangentSpace I p.1) (TangentSpace I' p.2)).prod 0 +
        ContinuousLinearMap.prod 0 (ContinuousLinearMap.snd 𝕜 (TangentSpace I p.1) (TangentSpace I' p.2)))
Module: Mathlib.Geometry.Manifold.VectorBundle.Tangent

Goal state before:
pp: 𝕜 : Type u_1
inst✝⁵ : NontriviallyNormedField 𝕜
n : ℕ∞ω
E : Type u_2
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace 𝕜 E
H : Type u_4
inst✝² : TopologicalSpace H
I : ModelWithCorners 𝕜 E H
M : Type u_6
inst✝¹ : TopologicalSpace M
inst✝ : ChartedSpace H M
h : IsManifold I (n + 1) M
⊢ (tangentBundleCore I M).IsContMDiff I n

Tactic:
have : IsManifold I n M := .of_le (n := n + 1) (le_self_add)
  refine ⟨fun i j => ?_⟩
  rw [contMDiffOn_iff_source_of_mem_maximalAtlas (subset_maximalAtlas i.2),
    contMDiffOn_iff_contDiffOn]
  · refine ((contDiffOn_fderiv_coord_change (I := I) i j).congr fun x hx => ?_).mono ?_
    · rw [PartialEquiv.trans_source'] at hx
      simp_rw [Function.comp_apply, tangentBundleCore_coordChange, (i.1.extend I).right_inv hx.1]
    · exact (i.1.extend_image_source_inter j.1).subset
  · apply inter_subset_left

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

Tactic:
have U : UniqueMDiffAt[range I] y := by
    apply I.uniqueMDiffOn
    exact extChartAt_target_subset_range x hy

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 : M
y : E
hy : y ∈ (extChartAt I x).target
U : UniqueMDiffAt[range ↑I] y
⊢ mfderiv% ↑(extChartAt I x) (↑(extChartAt I x).symm y) ∘SL mfderiv[range ↑I] ↑(extChartAt I x).symm y =
    ContinuousLinearMap.id 𝕜 (TangentSpace 𝓘(𝕜, E) y)
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
x : TangentBundle I H
y : ModelProd H E
⊢ ContDiffOn 𝕜 n
    (↑(extChartAt (I.prod 𝓘(𝕜, E)) y) ∘ ⇑(tangentBundleModelSpaceHomeomorph I) ∘ ↑(extChartAt I.tangent x).symm)
    ((extChartAt I.tangent x).target ∩
      ↑(extChartAt I.tangent x).symm ⁻¹'
        ⇑(tangentBundleModelSpaceHomeomorph I) ⁻¹' (extChartAt (I.prod 𝓘(𝕜, E)) y).source)

Tactic:
apply contDiffOn_id.congr

Goal state after:
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
x : TangentBundle I H
y : ModelProd H E
⊢ ∀
    x_1 ∈
      (extChartAt I.tangent x).target ∩
        ↑(extChartAt I.tangent x).symm ⁻¹'
          ⇑(tangentBundleModelSpaceHomeomorph I) ⁻¹' (extChartAt (I.prod 𝓘(𝕜, E)) y).source,
    (↑(extChartAt (I.prod 𝓘(𝕜, E)) y) ∘ ⇑(tangentBundleModelSpaceHomeomorph I) ∘ ↑(extChartAt I.tangent x).symm) x_1 =
      id x_1
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
x : ModelProd H E
y : TangentBundle I H
⊢ ContDiffOn 𝕜 n
    (↑(extChartAt I.tangent y) ∘ ⇑(tangentBundleModelSpaceHomeomorph I).symm ∘ ↑(extChartAt I.tangent x).symm)
    ((extChartAt I.tangent x).target ∩
      ↑(extChartAt I.tangent x).symm ⁻¹'
        ⇑(tangentBundleModelSpaceHomeomorph I).symm ⁻¹' (extChartAt I.tangent y).source)

Tactic:
apply contDiffOn_id.congr

Goal state after:
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
x : ModelProd H E
y : TangentBundle I H
⊢ ∀
    x_1 ∈
      (extChartAt I.tangent x).target ∩
        ↑(extChartAt I.tangent x).symm ⁻¹'
          ⇑(tangentBundleModelSpaceHomeomorph I).symm ⁻¹' (extChartAt I.tangent y).source,
    (↑(extChartAt I.tangent y) ∘ ⇑(tangentBundleModelSpaceHomeomorph I).symm ∘ ↑(extChartAt I.tangent x).symm) x_1 =
      id x_1
Module: Mathlib.Geometry.Manifold.MFDeriv.SpecificFunctions

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
M : Type u_4
inst✝³ : TopologicalSpace M
inst✝² : ChartedSpace H M
M' : Type u_21
inst✝¹ : TopologicalSpace M'
inst✝ : ChartedSpace H M'
p : M ⊕ M'
⊢ writtenInExtChartAt I I p Sum.swap =ᶠ[𝓝[range ↑I] ↑I (↑(chartAt H p) p)] id

Tactic:
by
  cases p with
    | inl x =>
      let t := I.symm ⁻¹' (chartAt H x).target ∩ range I
      have : EqOn (writtenInExtChartAt I I (Sum.inl x) (@Sum.swap M M')) id t := by
        intro y hy
        simp only [writtenInExtChartAt, extChartAt, Sum.swap_inl,
          ChartedSpace.sum_chartAt_inl, ChartedSpace.sum_chartAt_inr]
        dsimp
        rw [Sum.inr_injective.extend_apply, (chartAt H x).right_inv (by grind)]
        exact I.right_inv (by grind)
      apply Filter.eventually_of_mem ?_ this
      rw [Filter.inter_mem_iff]
      refine ⟨I.continuousWithinAt_symm.preimage_mem_nhdsWithin ?_, self_mem_nhdsWithin⟩
      exact (chartAt H x).open_target.mem_nhds (by simp)
    | inr x =>
      let t := I.symm ⁻¹' (chartAt H x).target ∩ range I
      have : EqOn (writtenInExtChartAt I I (Sum.inr x) (@Sum.swap M M')) id t := by
        intro y hy
        simp only [writtenInExtChartAt, extChartAt, Sum.swap_inr,
          ChartedSpace.sum_chartAt_inl, ChartedSpace.sum_chartAt_inr]
        dsimp
        rw [Sum.inl_injective.extend_apply, (chartAt H x).right_inv (by grind)]
        exact I.right_inv (by grind)
      apply Filter.eventually_of_mem ?_ this
      rw [Filter.inter_mem_iff]
      refine ⟨I.continuousWithinAt_symm.preimage_mem_nhdsWithin ?_, self_mem_nhdsWithin⟩
      exact (chartAt H x).open_target.mem_nhds (by simp)

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

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
x : M
φ : PartialEquiv M E := extChartAt I x
eq_nhd : ↑(extChartAt I x) ∘ ↑(extChartAt I x).symm =ᶠ[𝓝[range ↑I] ↑(extChartAt I x) x] id
⊢ fderivWithin 𝕜 (↑φ ∘ ↑φ.symm) (range ↑I) (↑φ x) = ContinuousLinearMap.id 𝕜 E

Tactic:
eq_nhd.fderivWithin_eq (by simp)

Goal state after:
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
x : M
φ : PartialEquiv M E := extChartAt I x
eq_nhd : ↑(extChartAt I x) ∘ ↑(extChartAt I x).symm =ᶠ[𝓝[range ↑I] ↑(extChartAt I x) x] id
⊢ fderivWithin 𝕜 id (range ↑I) (↑(extChartAt I x) x) = ContinuousLinearMap.id 𝕜 E
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
s : Set M
hsc : IsClosed[inst✝²] s
hs : s ⊆ (chartAt H c).source ∧ ↑(extChartAt I c) '' s ⊆ ball (↑(extChartAt I c) c) f.rOut
e : PartialEquiv M E := extChartAt I c
this : IsClosed[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] (↑e '' s)
⊢ ∃ r ∈ Ioo 0 f.rOut, s ⊆ (chartAt H c).source ∩ ↑(extChartAt I c) ⁻¹' ball (↑(extChartAt I c) c) r

Tactic:
rcases exists_pos_lt_subset_ball f.rOut_pos this hs.2 with ⟨r, hrR, hr⟩

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
inst✝ : FiniteDimensional ℝ E
s : Set M
hsc : IsClosed[inst✝²] s
hs : s ⊆ (chartAt H c).source ∧ ↑(extChartAt I c) '' s ⊆ ball (↑(extChartAt I c) c) f.rOut
e : PartialEquiv M E := extChartAt I c
this : IsClosed[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] (↑e '' s)
r : ℝ
hrR : r ∈ Ioo 0 f.rOut
hr : ↑e '' s ⊆ ball (↑(extChartAt I c) c) r
⊢ ∃ r ∈ Ioo 0 f.rOut, s ⊆ (chartAt H c).source ∩ ↑(extChartAt I c) ⁻¹' ball (↑(extChartAt I c) c) r
Module: Mathlib.Geometry.Manifold.VectorBundle.ContMDiffSection

Goal state before:
pp: case refine_2
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
s t : (x : M) → V x
u : Set M
x₀ : M
e : Trivialization F TotalSpace.proj := trivializationAt F V x₀
hs : ContMDiffWithinAt I 𝓘(𝕜, F) n (fun x ↦ (↑e ⟨x, s x⟩).2) u x₀
ht : ContMDiffWithinAt I 𝓘(𝕜, F) n (fun x ↦ (↑e ⟨x, t x⟩).2) u x₀
⊢ (↑e ⟨x₀, (s + t) x₀⟩).2 = ((fun x ↦ (↑e ⟨x, s x⟩).2) + fun x ↦ (↑e ⟨x, t x⟩).2) x₀

Tactic:
apply (e.linear 𝕜 (FiberBundle.mem_baseSet_trivializationAt' x₀)).1

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

Goal state before:
pp: case zero
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
s : Cₛ^n⟮I; F, V⟯
⊢ ⇑(0 • s) = 0 • ⇑s

Tactic:
zero_smul

Goal state after:
case zero
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
s : Cₛ^n⟮I; F, V⟯
⊢ ⇑(0 • s) = 0
Module: Mathlib.Geometry.Manifold.PartitionOfUnity

Goal state before:
pp: E : Type uE
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : NormedSpace ℝ E
H : Type uH
inst✝⁵ : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type uM
inst✝⁴ : TopologicalSpace M
inst✝³ : ChartedSpace H M
inst✝² : FiniteDimensional ℝ E
s : Set M
U : M → Set M
inst✝¹ : T2Space M
inst✝ : SigmaCompactSpace M
hs : IsClosed[inst✝⁴] s
hU : ∀ x ∈ s, U x ∈ 𝓝 x
⊢ ∃ ι f, f.IsSubordinate U

Tactic:
have : LocallyCompactSpace H := I.locallyCompactSpace
  have : LocallyCompactSpace M := ChartedSpace.locallyCompactSpace H M
  -- Next we choose a covering by supports of smooth bump functions
  have hB := fun x hx => SmoothBumpFunction.nhds_basis_support (I := I) (hU x hx)
  rcases refinement_of_locallyCompact_sigmaCompact_of_nhds_basis_set hs hB with
    ⟨ι, c, f, hf, hsub', hfin⟩
  choose hcs hfU using hf
  -- Then we use the shrinking lemma to get a covering by smaller open
  rcases exists_subset_iUnion_closed_subset hs (fun i => (f i).isOpen_support)
    (fun x _ => hfin.point_finite x) hsub' with ⟨V, hsV, hVc, hVf⟩
  choose r hrR hr using fun i => (f i).exists_r_pos_lt_subset_ball (hVc i) (hVf i)
  refine ⟨ι, ⟨c, fun i => (f i).updateRIn (r i) (hrR i), hcs, ?_, fun x hx => ?_⟩, fun i => ?_⟩
  · simpa only [SmoothBumpFunction.support_updateRIn]
  · refine (mem_iUnion.1 <| hsV hx).imp fun i hi => ?_
    exact ((f i).updateRIn _ _).eventuallyEq_one_of_dist_lt
      ((f i).support_subset_source <| hVf _ hi) (hr i hi).2
  · simpa only [SmoothBumpFunction.support_updateRIn, tsupport] using hfU i

Goal state after:
Module: Mathlib.MeasureTheory.Function.AEEqOfIntegral

Goal state before:
pp: α : Type u_1
E : Type u_2
𝕜 : Type u_3
m : MeasurableSpace α
μ : Measure α
inst✝² : RCLike 𝕜
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
t : Set E
f : α → E
hf : ∀ (c : StrongDual 𝕜 E), (fun x ↦ c (f x)) =ᵐ[μ] 0
h't : ∀ᵐ (x : α) ∂μ, f x ∈ t
d : Set E
d_count : d.Countable
hd : t ⊆ closure[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] d
this : Encodable ↑d
s : ↑d → StrongDual 𝕜 E
hs : ∀ (x : ↑d), ‖s x‖ ≤ 1 ∧ (s x) ↑x = ↑‖↑x‖
a : E
hat : a ∈ t
ha : a ≠ 0
a_pos : 0 < ‖a‖
a_mem : a ∈ closure[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] d
⊢ ∃ x, dist a ↑x < ‖a‖ / 2

Tactic:
rcases Metric.mem_closure_iff.1 a_mem (‖a‖ / 2) (half_pos a_pos) with ⟨x, h'x, hx⟩
      exact ⟨⟨x, h'x⟩, hx⟩

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

Goal state before:
pp: E : Type uE
inst✝⁸ : NormedAddCommGroup E
inst✝⁷ : NormedSpace ℝ E
H : Type uH
inst✝⁶ : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type uM
inst✝⁵ : TopologicalSpace M
inst✝⁴ : ChartedSpace H M
inst✝³ : FiniteDimensional ℝ E
inst✝² : IsManifold I ∞ M
inst✝¹ : SigmaCompactSpace M
inst✝ : T2Space M
n : ℕ∞
s : Set M
hs : IsOpen[inst✝⁵] s
f : SmoothPartitionOfUnity M I M
hf : f.IsSubordinate fun x ↦ (chartAt H x).source
g : M → H → ℝ
g_supp : ∀ (c : M), support (g c) = (chartAt H c).target ∩ ↑(chartAt H c).symm ⁻¹' s
g_diff : ∀ (c : M), ContMDiff I 𝓘(ℝ, ℝ) (↑n) (g c)
hg : ∀ (c : M), range (g c) ⊆ Icc 0 1
h'g : ∀ (c : M) (x : H), 0 ≤ g c x
h''g : ∀ (c x : M), 0 ≤ (f c) x * g c (↑(chartAt H c) x)
x : M
hx : x ∈ s
c : M
hc : 0 < (f c) x
Hx : x ∈ tsupport ⇑(f c)
⊢ x ∈ ↑(chartAt H c) ⁻¹' (chartAt H c).target ∩ ↑(chartAt H c) ⁻¹' ↑(chartAt H c).symm ⁻¹' s

Tactic:
simp [(chartAt H c).left_inv (hf c Hx), hx, (chartAt H c).map_source (hf c Hx)]

Goal state after:
Module: Mathlib.MeasureTheory.Function.AEEqOfIntegral

Goal state before:
pp: case refine_2
α : Type u_1
E : Type u_2
m0 : MeasurableSpace α
μ : Measure α
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
inst✝ : CompleteSpace E
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
t : Set α
ht : MeasurableSet t
hμt : μ t ≠ ∞
u : Set E
u_sep : IsSeparable u
hu : ∀ᵐ (x : α) ∂μ.restrict t, f x ∈ u
c : StrongDual ℝ E
s : Set α
hs : MeasurableSet s
hμs : μ s < ∞
⊢ c (∫ (x : α) in s, f x ∂μ) = 0

Tactic:
hf_zero s hs hμs

Goal state after:
case refine_2
α : Type u_1
E : Type u_2
m0 : MeasurableSpace α
μ : Measure α
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
inst✝ : CompleteSpace E
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
t : Set α
ht : MeasurableSet t
hμt : μ t ≠ ∞
u : Set E
u_sep : IsSeparable u
hu : ∀ᵐ (x : α) ∂μ.restrict t, f x ∈ u
c : StrongDual ℝ E
s : Set α
hs : MeasurableSet s
hμs : μ s < ∞
⊢ c 0 = 0
Module: Mathlib.Analysis.Calculus.Rademacher

Goal state before:
pp: case e'_11
E : Type u_1
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace ℝ E
inst✝³ : MeasurableSpace E
inst✝² : BorelSpace E
C : ℝ≥0
f : E → ℝ
μ : Measure E
inst✝¹ : FiniteDimensional ℝ E
inst✝ : μ.IsAddHaarMeasure
hf : LipschitzWith C f
v : E
L : ℝ →L[ℝ] E := ContinuousLinearMap.smulRight 1 v
p : E
this✝ : ∀ᵐ (s : ℝ), DifferentiableAt ℝ (fun t ↦ f (p + t • v)) s
s : ℝ
hs : DifferentiableAt ℝ (fun t ↦ f (p + t • v)) s
h's : DifferentiableAt ℝ (fun t ↦ f (p + t • v)) (s + 0)
this : DifferentiableAt ℝ (fun t ↦ s + t) 0
t : ℝ
⊢ f (p + s • v + t • v) = ((fun t ↦ f (p + t • v)) ∘ HAdd.hAdd s) t

Tactic:
simp only [add_assoc, Function.comp_apply, add_smul]

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

Goal state before:
pp: E : Type u_1
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace ℝ E
inst✝³ : MeasurableSpace E
inst✝² : BorelSpace E
C : ℝ≥0
f g : E → ℝ
μ : Measure E
inst✝¹ : FiniteDimensional ℝ E
inst✝ : μ.IsAddHaarMeasure
hf : LipschitzWith C f
h'f : HasCompactSupport f
hg : Continuous[PseudoMetricSpace.toUniformSpace.toTopologicalSpace, _] g
v : E
K : Set E := cthickening ‖v‖ (tsupport f)
K_compact : IsCompact K
t : ℝ
ht : t ∈ Ioc 0 1
t_pos : 0 < t
x : E
A : f x = 0
hx : x + t • v ∈ Function.support f
⊢ x ∈ K

Tactic:
apply mem_cthickening_of_dist_le _ _ (‖v‖) (tsupport f) (subset_tsupport _ hx)

Goal state after:
E : Type u_1
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace ℝ E
inst✝³ : MeasurableSpace E
inst✝² : BorelSpace E
C : ℝ≥0
f g : E → ℝ
μ : Measure E
inst✝¹ : FiniteDimensional ℝ E
inst✝ : μ.IsAddHaarMeasure
hf : LipschitzWith C f
h'f : HasCompactSupport f
hg : Continuous[PseudoMetricSpace.toUniformSpace.toTopologicalSpace, _] g
v : E
K : Set E := ⋯
K_compact : IsCompact K
t : ℝ
ht : t ∈ Ioc 0 1
t_pos : 0 < t
x : E
A : f x = 0
hx : x + t • v ∈ Function.support f
⊢ dist x (x + t • v) ≤ ‖v‖
Module: Mathlib.Analysis.Calculus.TangentCone.ProperSpace

Goal state before:
pp: case refine_1.hd₀
𝕜 : Type u_1
inst✝³ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace 𝕜 E
inst✝ : ProperSpace E
s : Set E
x : E
hx : AccPt x (𝓟 s)
u : ℕ → ℝ
u_pos : ∀ (n : ℕ), 0 < u n
u_lim : Tendsto u atTop (𝓝 0)
v : ℕ → E
hvx : ∀ (n : ℕ), v n ≠ x
hvu : ∀ (n : ℕ), v n ∈ closedBall x (u n)
hvs : ∀ (n : ℕ), v n ∈ s
d : ℕ → E := ⋯
M : ∀ (n : ℕ), x + d n ∈ s \ {x}
r : 𝕜
hr : 1 < ‖r‖
c : ℕ → 𝕜
c_ne : ∀ (n : ℕ), c n ≠ 0
c_le : ∀ (n : ℕ), ‖c n • d n‖ < 1
le_c : ∀ (n : ℕ), 1 / ‖r‖ ≤ ‖c n • d n‖
hc : ∀ (n : ℕ), ‖c n‖⁻¹ ≤ 1⁻¹ * ‖r‖ * ‖d n‖
l : E
l_mem : l ∈ closedBall 0 1 \ ball 0 (1 / ‖r‖)
φ : ℕ → ℕ
φ_strict : StrictMono φ
hφ : Tendsto ((fun n ↦ c n • d n) ∘ φ) atTop (𝓝 l)
⊢ Tendsto (d ∘ φ) atTop (𝓝 0)

Tactic:
refine Tendsto.comp ?_ φ_strict.tendsto_atTop
    exact squeeze_zero_norm (by simpa [dist_eq_norm] using hvu) u_lim

Goal state after:
Module: Mathlib.Analysis.Calculus.TangentCone.Seq

Goal state before:
pp: case mp.inl
𝕜 : Type u_1
E : Type u_2
inst✝² : NontriviallyNormedField 𝕜
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
s : Set E
x : E
hx : x ∈ closure[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] s
c : 𝕜
hc : 1 < ‖c‖
d : ℕ → E
hds : ∀ (n : ℕ), x + d n ∈ s
hd : ∀ (n : ℕ), ‖d n‖ < (1 / (2 * ‖c‖)) ^ n
⊢ ∃ c d,
    Tendsto (fun x ↦ ‖c x‖) atTop atTop ∧ (∀ᶠ (n : ℕ) in atTop, x + d n ∈ s) ∧ Tendsto (fun n ↦ c n • d n) atTop (𝓝 0)

Tactic:
refine ⟨(c ^ ·), d, ?tendsto_c, .of_forall hds, ?tendsto_cd⟩

Goal state after:
case tendsto_c
𝕜 : Type u_1
E : Type u_2
inst✝² : NontriviallyNormedField 𝕜
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
s : Set E
x : E
hx : x ∈ closure[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] s
c : 𝕜
hc : 1 < ‖c‖
d : ℕ → E
hds : ∀ (n : ℕ), x + d n ∈ s
hd : ∀ (n : ℕ), ‖d n‖ < (1 / (2 * ‖c‖)) ^ n
⊢ Tendsto (fun x ↦ ‖(fun x ↦ c ^ x) x‖) atTop atTop

case tendsto_cd
𝕜 : Type u_1
E : Type u_2
inst✝² : NontriviallyNormedField 𝕜
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
s : Set E
x : E
hx : x ∈ closure[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] s
c : 𝕜
hc : 1 < ‖c‖
d : ℕ → E
hds : ∀ (n : ℕ), x + d n ∈ s
hd : ∀ (n : ℕ), ‖d n‖ < (1 / (2 * ‖c‖)) ^ n
⊢ Tendsto (fun n ↦ (fun x ↦ c ^ x) n • d n) atTop (𝓝 0)
Module: Mathlib.MeasureTheory.Function.AbsolutelyContinuous

Goal state before:
pp: X : Type u_1
inst✝ : PseudoMetricSpace X
⊢ totalLengthFilter.HasBasis (fun ε ↦ 0 < ε) fun ε ↦ {E | ∑ i ∈ Finset.range E.1, dist (E.2 i).1 (E.2 i).2 < ε}

Tactic:
convert! Filter.HasBasis.comap (α := ℝ) _ (nhds_basis_Ioo_pos _) using 1
  ext ε E
  simp only [mem_ofPred_eq, zero_sub, zero_add, mem_preimage, mem_Ioo, iff_and_self]
  suffices 0 ≤ ∑ i ∈ Finset.range E.1, dist (E.2 i).1 (E.2 i).2 by grind
  exact Finset.sum_nonneg (fun _ _ ↦ dist_nonneg)

Goal state after:
Module: Mathlib.MeasureTheory.Integral.IntervalIntegral.DerivIntegrable

Goal state before:
pp: f : ℝ → ℝ
a b : ℝ
hf : MonotoneOn f (uIcc a b)
hab : a ≤ b
⊢ IntervalIntegrable (deriv f) volume a b

Tactic:
rw [uIcc_of_le hab] at hf

Goal state after:
f : ℝ → ℝ
a b : ℝ
hf : MonotoneOn f (Icc a b)
hab : a ≤ b
⊢ IntervalIntegrable (deriv f) volume a b
Module: Mathlib.MeasureTheory.Integral.IntervalIntegral.DerivIntegrable

Goal state before:
pp: f : ℝ → ℝ
a b : ℝ
hf : MonotoneOn f (uIcc a b)
hab : a ≤ b
⊢ ∫ (x : ℝ) in a..b, deriv f x ∈ uIcc 0 (f b - f a)

Tactic:
rw [uIcc_of_le hab] at hf

Goal state after:
f : ℝ → ℝ
a b : ℝ
hf : MonotoneOn f (Icc a b)
hab : a ≤ b
⊢ ∫ (x : ℝ) in a..b, deriv f x ∈ uIcc 0 (f b - f a)