mathlib-initiative/mathlib-tactics huggingface dataset:
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
b₀ b : M
hb : b ∈ (chartAt H b₀).source
⊢ Trivialization.symmL 𝕜 (trivializationAt E (TangentSpace I) b₀) b =
    (tangentBundleCore I M).coordChange (achart H b₀) (achart H b) b

Tactic:
by
  simp [hb]

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

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''
e : OpenPartialHomeomorph M M'
he✝ : MDifferentiable I I' e
e' : OpenPartialHomeomorph M' M''
he : MDifferentiable I I' e
x : M
hx : x ∈ e.source
y : TangentSpace I' (↑e x)
⊢ (↑(mfderiv% ↑e x)).toFun ((mfderiv% ↑e.symm (↑e x)) y) = y

Tactic:
have :
        (ContinuousLinearMap.id 𝕜 _ : TangentSpace I' (e x) →L[𝕜] TangentSpace I' (e x)) y = y :=
        rfl

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''
e : OpenPartialHomeomorph M M'
he✝ : MDifferentiable I I' e
e' : OpenPartialHomeomorph M' M''
he : MDifferentiable I I' e
x : M
hx : x ∈ e.source
y : TangentSpace I' (↑e x)
this : (ContinuousLinearMap.id 𝕜 (TangentSpace I' (↑e x))) y = y
⊢ (↑(mfderiv% ↑e x)).toFun ((mfderiv% ↑e.symm (↑e x)) y) = y
Module: Mathlib.Geometry.Manifold.MFDeriv.Atlas

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''
e : OpenPartialHomeomorph M M'
he✝ : MDifferentiable I I' e
e' : OpenPartialHomeomorph M' M''
he : MDifferentiable I I' e
x : M
hx : x ∈ e.source
y : TangentSpace I' (↑e x)
⊢ (↑(mfderiv% ↑e x)).toFun ((mfderiv% ↑e.symm (↑e x)) y) = y

Tactic:
have :
        (ContinuousLinearMap.id 𝕜 _ : TangentSpace I' (e x) →L[𝕜] TangentSpace I' (e x)) y = y :=
        rfl
      conv_rhs => rw [← this, ← he.comp_symm_deriv (e.map_source hx)]
      rw [e.left_inv hx]
      rfl

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

Goal state before:
pp: case e_a.hg
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
this : (fun z ↦ f (z.1, p.2)) = (fun z ↦ f (z, p.2)) ∘ Prod.fst
⊢ (MDiffAt fun z ↦ f (z, p.2)) p.1

pp: case e_a.hf
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
this : (fun z ↦ f (z.1, p.2)) = (fun z ↦ f (z, p.2)) ∘ Prod.fst
⊢ MDiffAt Prod.fst p

Tactic:
· exact hf.comp _ (mdifferentiableAt_id.prodMk mdifferentiableAt_const)

Goal state after:
case e_a.hf
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
this : (fun z ↦ f (z.1, p.2)) = (fun z ↦ f (z, p.2)) ∘ Prod.fst
⊢ MDiffAt Prod.fst p
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
e : Trivialization F TotalSpace.proj
inst✝ : MemTrivializationAtlas e
⊢ ContMDiffOn (IB.prod 𝓘(𝕜, F)) IB n (fun x ↦ (↑e.symm x).proj) e.target ∧
    ContMDiffOn (IB.prod 𝓘(𝕜, F)) 𝓘(𝕜, F) n (fun x ↦ (↑e (↑e.symm x)).2) e.target

Tactic:
refine ⟨contMDiffOn_fst.congr fun x hx ↦ e.proj_symm_apply hx,
    contMDiffOn_snd.congr fun x hx ↦ ?_⟩

Goal state after:
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
e : Trivialization F TotalSpace.proj
inst✝ : MemTrivializationAtlas e
x : B × F
hx : x ∈ e.target
⊢ (↑e (↑e.symm x)).2 = x.2
Module: Mathlib.Geometry.Manifold.VectorBundle.Tangent

Goal state before:
pp: 𝕜 : Type u_1
inst✝² : NontriviallyNormedField 𝕜
n : ℕ∞ω
E : Type u_2
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
V : (x : E) → TangentSpace 𝓘(𝕜, E) x
s : Set E
x : E
⊢ ContMDiffWithinAt 𝓘(𝕜, E) 𝓘(𝕜, E).tangent n (fun x ↦ ⟨x, V x⟩) s x ↔ ContDiffWithinAt 𝕜 n V s x

Tactic:
refine ⟨fun h ↦ ?_, fun h ↦ ?_⟩
  · exact ContMDiffWithinAt.contDiffWithinAt <|
      (contMDiff_snd_tangentBundle_modelSpace E 𝓘(𝕜, E)).contMDiffAt.comp_contMDiffWithinAt _ h
  · apply Bundle.contMDiffWithinAt_totalSpace.2
    refine ⟨contMDiffWithinAt_id, ?_⟩
    convert! h.contMDiffWithinAt with y
    simp

Goal state after:
Module: Mathlib.Topology.ShrinkingLemma

Goal state before:
pp: case refine_3.inr
ι : Type u_1
X : Type u_2
inst✝¹ : TopologicalSpace X
u : ι → Set X
s : Set X
inst✝ : NormalSpace X
v : PartialRefinement u s ⊤
hs : IsClosed s
i : ι
hi : i ∉ v.carrier
I : s ∩ ⋂ j, ⋂ (_ : j ≠ i), (v.toFun j)ᶜ ⊆ v.toFun i
C : IsClosed (s ∩ ⋂ j, ⋂ (_ : j ≠ i), (v.toFun j)ᶜ)
vi : Set X
ovi : IsOpen vi
hvi : s ∩ ⋂ j, ⋂ (_ : j ≠ i), (v.toFun j)ᶜ ⊆ vi
cvi : closure vi ⊆ v.toFun i
j : ι
hj : j ∈ v.carrier
⊢ closure (update v.toFun i vi j) ⊆ u j

Tactic:
· rw [update_of_ne (ne_of_mem_of_not_mem hj hi)]
      exact v.closure_subset hj

Goal state after:
Module: Mathlib.Topology.ShrinkingLemma

Goal state before:
pp: ι : Type u_1
X : Type u_2
inst✝² : TopologicalSpace X
u : ι → Set X
s : Set X
inst✝¹ : T2Space X
inst✝ : LocallyCompactSpace X
v : PartialRefinement u s fun w ↦ IsCompact (closure w)
hs : IsCompact s
i : ι
hi : i ∉ v.carrier
si : Set X := s ∩ (⋃ j, ⋃ (_ : j ≠ i), v.toFun j)ᶜ
hsi : si = s ∩ ⋂ i_1, ⋂ (_ : ¬i_1 = i), (v.toFun i_1)ᶜ
hsic : IsCompact si
x : X
hx : x ∈ si
j : ι
hj : j ≠ i
⊢ x ∉ v.toFun j

Tactic:
rw [hsi] at hx

Goal state after:
ι : Type u_1
X : Type u_2
inst✝² : TopologicalSpace X
u : ι → Set X
s : Set X
inst✝¹ : T2Space X
inst✝ : LocallyCompactSpace X
v : PartialRefinement u s fun w ↦ IsCompact (closure w)
hs : IsCompact s
i : ι
hi : i ∉ v.carrier
si : Set X := s ∩ (⋃ j, ⋃ (_ : j ≠ i), v.toFun j)ᶜ
hsi : si = s ∩ ⋂ i_1, ⋂ (_ : ¬i_1 = i), (v.toFun i_1)ᶜ
hsic : IsCompact si
x : X
hx : x ∈ s ∩ ⋂ i_1, ⋂ (_ : ¬i_1 = i), (v.toFun i_1)ᶜ
j : ι
hj : j ≠ i
⊢ x ∉ v.toFun j
Module: Mathlib.Topology.ShrinkingLemma

Goal state before:
pp: case refine_3.inr
ι : Type u_1
X : Type u_2
inst✝² : TopologicalSpace X
u : ι → Set X
s : Set X
inst✝¹ : T2Space X
inst✝ : LocallyCompactSpace X
v : PartialRefinement u s fun w ↦ IsCompact (closure w)
hs : IsCompact s
i : ι
hi : i ∉ v.carrier
si : Set X := s ∩ (⋃ j, ⋃ (_ : j ≠ i), v.toFun j)ᶜ
hsi : si = s ∩ ⋂ i_1, ⋂ (_ : ¬i_1 = i), (v.toFun i_1)ᶜ
hsic : IsCompact si
this : si ⊆ v.toFun i
vi : Set X
hvi : IsOpen vi ∧ si ⊆ vi ∧ closure vi ⊆ v.toFun i ∧ IsCompact (closure vi)
j : ι
hj : j ∈ v.carrier
⊢ closure (update v.toFun i vi j) ⊆ u j

Tactic:
· rw [update_of_ne (ne_of_mem_of_not_mem hj hi)]
      exact v.closure_subset hj

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

Goal state before:
pp: case neg
ι : Type uι
E : Type uE
inst✝⁶ : NormedAddCommGroup E
inst✝⁵ : NormedSpace ℝ E
F : Type uF
inst✝⁴ : NormedAddCommGroup F
inst✝³ : NormedSpace ℝ F
H : Type uH
inst✝² : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type uM
inst✝¹ : TopologicalSpace M
inst✝ : ChartedSpace H M
s : Set M
f : SmoothPartitionOfUnity ι I M s
n : ℕ∞
x₀ : M
g : ι → M → F
hφ : ∀ (i : ι), x₀ ∈ tsupport ⇑(f i) → ContMDiffAt I 𝓘(ℝ, F) (↑n) (g i) x₀
i : ι
hx : x₀ ∉ tsupport ⇑(f i)
⊢ ContMDiffAt I 𝓘(ℝ, F) (↑n) (fun x ↦ (f i) x • g i x) x₀

Tactic:
· exact contMDiffAt_of_notMem (compl_subset_compl.mpr
      (tsupport_smul_subset_left (f i) (g i)) hx) n

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

Goal state before:
pp: E : Type uE
inst✝¹⁶ : NormedAddCommGroup E
inst✝¹⁵ : NormedSpace ℝ E
H : Type uH
inst✝¹⁴ : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type uM
inst✝¹³ : TopologicalSpace M
inst✝¹² : ChartedSpace H M
inst✝¹¹ : FiniteDimensional ℝ E
inst✝¹⁰ : IsManifold I ∞ M
inst✝⁹ : SigmaCompactSpace M
inst✝⁸ : T2Space M
n : ℕ∞
F_fiber : Type u_1
inst✝⁷ : NormedAddCommGroup F_fiber
inst✝⁶ : NormedSpace ℝ F_fiber
V : M → Type u_2
inst✝⁵ : (x : M) → AddCommGroup (V x)
inst✝⁴ : (x : M) → TopologicalSpace (V x)
inst✝³ : (x : M) → Module ℝ (V x)
inst✝² : TopologicalSpace (TotalSpace F_fiber V)
inst✝¹ : FiberBundle F_fiber V
inst✝ : VectorBundle ℝ F_fiber V
t : (x : M) → Set (V x)
ht_conv : ∀ (x : M), Convex ℝ (t x)
W : M → Set M
h_nhds : ∀ (x₀ : M), W x₀ ∈ 𝓝 x₀
s_loc : M → (x : M) → V x
s_smooth : ∀ (x₀ : M), ContMDiffOn I (I.prod 𝓘(ℝ, F_fiber)) (↑n) (fun x ↦ ⟨x, s_loc x₀ x⟩) (W x₀)
h_mem_t : ∀ (x₀ y : M), y ∈ W x₀ → s_loc x₀ y ∈ t y
U : M → Set M := fun x ↦ interior (W x)
⊢ univ ⊆ ⋃ x, U x

Tactic:
intro x_pt _

Goal state after:
E : Type uE
inst✝¹⁶ : NormedAddCommGroup E
inst✝¹⁵ : NormedSpace ℝ E
H : Type uH
inst✝¹⁴ : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type uM
inst✝¹³ : TopologicalSpace M
inst✝¹² : ChartedSpace H M
inst✝¹¹ : FiniteDimensional ℝ E
inst✝¹⁰ : IsManifold I ∞ M
inst✝⁹ : SigmaCompactSpace M
inst✝⁸ : T2Space M
n : ℕ∞
F_fiber : Type u_1
inst✝⁷ : NormedAddCommGroup F_fiber
inst✝⁶ : NormedSpace ℝ F_fiber
V : M → Type u_2
inst✝⁵ : (x : M) → AddCommGroup (V x)
inst✝⁴ : (x : M) → TopologicalSpace (V x)
inst✝³ : (x : M) → Module ℝ (V x)
inst✝² : TopologicalSpace (TotalSpace F_fiber V)
inst✝¹ : FiberBundle F_fiber V
inst✝ : VectorBundle ℝ F_fiber V
t : (x : M) → Set (V x)
ht_conv : ∀ (x : M), Convex ℝ (t x)
W : M → Set M
h_nhds : ∀ (x₀ : M), W x₀ ∈ 𝓝 x₀
s_loc : M → (x : M) → V x
s_smooth : ∀ (x₀ : M), ContMDiffOn I (I.prod 𝓘(ℝ, F_fiber)) (↑n) (fun x ↦ ⟨x, s_loc x₀ x⟩) (W x₀)
h_mem_t : ∀ (x₀ y : M), y ∈ W x₀ → s_loc x₀ y ∈ t y
U : M → Set M := fun x ↦ interior (W x)
x_pt : M
a✝ : x_pt ∈ univ
⊢ x_pt ∈ ⋃ x, U x
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
ht : IsSeparable t
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
⊢ f =ᵐ[μ] 0

Tactic:
rcases ht with ⟨d, d_count, hd⟩

Goal state after:
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
⊢ f =ᵐ[μ] 0
Module: Mathlib.MeasureTheory.Function.AEEqOfIntegral

Goal state before:
pp: α : Type u_1
m0 : MeasurableSpace α
μ : Measure α
f : α → ℝ
hf_int_finite : ∀ (s : Set α), MeasurableSet s → μ s < ∞ → IntegrableOn f s μ
hf_zero : ∀ (s : Set α), MeasurableSet s → μ s < ∞ → 0 ≤ ∫ (x : α) in s, f x ∂μ
t : Set α
ht : MeasurableSet t
hμt : μ t ≠ ∞
s : Set α
hs : MeasurableSet s
x✝ : μ (s ∩ t) < ∞
⊢ 0 ≤ ∫ (x : α) in s ∩ t, f x ∂μ

Tactic:
refine hf_zero (s ∩ t) (hs.inter ht) ?_

Goal state after:
α : Type u_1
m0 : MeasurableSpace α
μ : Measure α
f : α → ℝ
hf_int_finite : ∀ (s : Set α), MeasurableSet s → μ s < ∞ → IntegrableOn f s μ
hf_zero : ∀ (s : Set α), MeasurableSet s → μ s < ∞ → 0 ≤ ∫ (x : α) in s, f x ∂μ
t : Set α
ht : MeasurableSet t
hμt : μ t ≠ ∞
s : Set α
hs : MeasurableSet s
x✝ : μ (s ∩ t) < ∞
⊢ μ (s ∩ t) < ∞
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 t : Set M
hs : IsOpen[inst✝⁵] s
ht : IsClosed[inst✝⁵] t
h : t ⊆ s
f : M → ℝ
f_supp : support f = s
f_diff : ContMDiff I 𝓘(ℝ, ℝ) (↑?m.68) f
f_pos : ∀ (x : M), 0 ≤ f x
g : M → ℝ
g_supp : support g = tᶜ
g_diff : ContMDiff I 𝓘(ℝ, ℝ) (↑?m.109) g
g_pos : ∀ (x : M), 0 ≤ g x
x : M
xs : x ∉ tᶜ
⊢ x ∈ t

Tactic:
simpa using xs

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

Goal state before:
pp: E : Type u_2
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace ℝ E
inst✝³ : CompleteSpace E
β : Type u_3
inst✝² : TopologicalSpace β
inst✝¹ : MeasurableSpace β
inst✝ : BorelSpace β
μ : Measure β
f : β → E
hf : Integrable f μ
h'f : ∀ (s : Set β), IsClosed[inst✝²] s → ∫ (x : β) in s, f x ∂μ = 0
⊢ f =ᵐ[μ] 0

Tactic:
suffices ∀ s, MeasurableSet s → ∫ x in s, f x ∂μ = 0 from
    hf.ae_eq_zero_of_forall_setIntegral_eq_zero (fun s hs _ ↦ this s hs)
  have A : ∀ (t : Set β), MeasurableSet t → ∫ (x : β) in t, f x ∂μ = 0
      → ∫ (x : β) in tᶜ, f x ∂μ = 0 := by
    intro t t_meas ht
    have I : ∫ x, f x ∂μ = 0 := by rw [← setIntegral_univ]; exact h'f _ isClosed_univ
    simpa [ht, I] using integral_add_compl t_meas hf
  intro s hs
  induction s, hs using MeasurableSet.induction_on_open with
  | isOpen U hU => exact compl_compl U ▸ A _ hU.measurableSet.compl (h'f _ hU.isClosed_compl)
  | compl s hs ihs => exact A s hs ihs
  | iUnion g g_disj g_meas hg => simp [integral_iUnion g_meas g_disj hf.integrableOn, hg]

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

Goal state before:
pp: case e'_5
X : Type u_1
inst✝ : PseudoMetricSpace X
ε : ℝ
E : ℕ × (ℕ → X × X)
⊢ E ∈ {E | ∑ i ∈ Finset.range E.1, dist (E.2 i).1 (E.2 i).2 < ε} ↔
    E ∈ (fun E ↦ ∑ i ∈ Finset.range E.1, dist (E.2 i).1 (E.2 i).2) ⁻¹' Ioo (0 - ε) (0 + ε)

Tactic:
simp only [mem_ofPred_eq, zero_sub, zero_add, mem_preimage, mem_Ioo, iff_and_self]

Goal state after:
case e'_5
X : Type u_1
inst✝ : PseudoMetricSpace X
ε : ℝ
E : ℕ × (ℕ → X × X)
⊢ ∑ i ∈ Finset.range E.1, dist (E.2 i).1 (E.2 i).2 < ε → -ε < ∑ i ∈ Finset.range E.1, dist (E.2 i).1 (E.2 i).2
Module: Mathlib.Analysis.Calculus.Rademacher

Goal state before:
pp: E : Type u_1
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace ℝ E
F : Type u_2
inst✝² : NormedAddCommGroup F
inst✝¹ : NormedSpace ℝ F
C : ℝ≥0
inst✝ : FiniteDimensional ℝ E
f : E → F
hf : LipschitzWith C f
s : Set E
hs : sphere 0 1 ⊆ closure[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] s
L : E →L[ℝ] F
x : E
hL : ∀ v ∈ s, HasLineDerivAt ℝ f (L v) x v
ε : ℝ
εpos : 0 < ε
δ : ℝ
δpos : 0 < δ
hδ : (↑C + ‖L‖ + 1) * δ = ε
q : Set E
hqs : q ⊆ s
q_fin : q.Finite
hq : sphere 0 1 ⊆ ⋃ y ∈ q, ball y δ
I : ∀ᶠ (t : ℝ) in 𝓝 0, ∀ v ∈ q, ‖f (x + t • v) - f x - t • L v‖ ≤ δ * ‖t‖
r : ℝ
r_pos : 0 < r
hr : ∀ (t : ℝ), ‖t‖ < r → ∀ v ∈ q, ‖f (x + t • v) - f x - t • L v‖ ≤ δ * ‖t‖
⊢ ∀ᶠ (x_1 : E) in 𝓝 0, ‖f (x + x_1) - f x - L x_1‖ ≤ ε * ‖x_1‖

Tactic:
apply Metric.mem_nhds_iff.2 ⟨r, r_pos, fun v hv ↦ ?_⟩

Goal state after:
E : Type u_1
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace ℝ E
F : Type u_2
inst✝² : NormedAddCommGroup F
inst✝¹ : NormedSpace ℝ F
C : ℝ≥0
inst✝ : FiniteDimensional ℝ E
f : E → F
hf : LipschitzWith C f
s : Set E
hs : sphere 0 1 ⊆ closure[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] s
L : E →L[ℝ] F
x : E
hL : ∀ v ∈ s, HasLineDerivAt ℝ f (L v) x v
ε : ℝ
εpos : 0 < ε
δ : ℝ
δpos : 0 < δ
hδ : (↑C + ‖L‖ + 1) * δ = ε
q : Set E
hqs : q ⊆ s
q_fin : q.Finite
hq : sphere 0 1 ⊆ ⋃ y ∈ q, ball y δ
I : ∀ᶠ (t : ℝ) in 𝓝 0, ∀ v ∈ q, ‖f (x + t • v) - f x - t • L v‖ ≤ δ * ‖t‖
r : ℝ
r_pos : 0 < r
hr : ∀ (t : ℝ), ‖t‖ < r → ∀ v ∈ q, ‖f (x + t • v) - f x - t • L v‖ ≤ δ * ‖t‖
v : E
hv : v ∈ ball 0 r
⊢ v ∈ {x_1 | (fun x_2 ↦ ‖f (x + x_2) - f x - L x_2‖ ≤ ε * ‖x_2‖) x_1}
Module: Mathlib.Analysis.Calculus.Taylor

Goal state before:
pp: t x : ℝ
n : ℕ
⊢ HasDerivAt (fun y ↦ (-y + x) ^ (n + 1)) (-(↑n + 1) * (-t + x) ^ n) t

Tactic:
rw [← neg_one_mul, mul_comm (-1 : ℝ), mul_assoc, mul_comm (-1 : ℝ), ← mul_assoc]

Goal state after:
t x : ℝ
n : ℕ
⊢ HasDerivAt (fun y ↦ (-y + x) ^ (n + 1)) ((↑n + 1) * (-t + x) ^ n * -1) t
Module: Mathlib.MeasureTheory.Function.AbsolutelyContinuous

Goal state before:
pp: a b : ℝ
E : Type u_3
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
f : ℝ → E
hf : ContDiffOn ℝ 1 f (uIcc a b)
⊢ AbsolutelyContinuousOnInterval f a b

Tactic:
by
  obtain ⟨K, hK⟩ := hf.exists_lipschitzOnWith (by decide) (convex_Icc _ _) isCompact_Icc
  exact hK.absolutelyContinuousOnInterval

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

Goal state before:
pp: s x y : ℝ
hx₀ : 0 < x
hx₁ : x < 1 / (1 + s ^ 2)
hy : |y| < s * x
H : √((1 - x) ^ 2 + y ^ 2) ≤ 1 - x / 2
⊢ 2 * √(1 + s ^ 2) * (x / 2) < (2 * √(1 + s ^ 2) + 1) * (x / 2)

Tactic:
by gcongr; exact lt_add_one _

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

Goal state before:
pp: case h.h.h.h
s : ℝ
hs : 0 < s
M ε : ℝ
hM : 0 < M
hε : 0 < ε
H : ∀ (x y : ℝ), 0 < x → x < ε → |y| < s * x → √(x ^ 2 + y ^ 2) < M * (1 - √((1 - x) ^ 2 + y ^ 2))
z : ℂ
hzl : 1 - z.re < ε
hzr : z ∈ stolzCone s
⊢ z ∈ stolzSet M

Tactic:
← one_re,

Goal state after:
case h.h.h.h
s : ℝ
hs : 0 < s
M ε : ℝ
hM : 0 < M
hε : 0 < ε
H : ∀ (x y : ℝ), 0 < x → x < ε → |y| < s * x → √(x ^ 2 + y ^ 2) < M * (1 - √((1 - x) ^ 2 + y ^ 2))
z : ℂ
hzl : re 1 - z.re < ε
hzr : z ∈ stolzCone s
⊢ z ∈ stolzSet M
Module: Mathlib.Analysis.Complex.AbelLimit

Goal state before:
pp: case h.h.h.h
s : ℝ
hs : 0 < s
M ε : ℝ
hM : 0 < M
hε : 0 < ε
H : ∀ (x y : ℝ), 0 < x → x < ε → |y| < s * x → √(x ^ 2 + y ^ 2) < M * (1 - √((1 - x) ^ 2 + y ^ 2))
z : ℂ
hzl : (1 - z).re < ε
hzr : |z.im| < s * (1 - z.re)
⊢ z ∈ stolzSet M

Tactic:
← one_re,

Goal state after:
case h.h.h.h
s : ℝ
hs : 0 < s
M ε : ℝ
hM : 0 < M
hε : 0 < ε
H : ∀ (x y : ℝ), 0 < x → x < ε → |y| < s * x → √(x ^ 2 + y ^ 2) < M * (1 - √((1 - x) ^ 2 + y ^ 2))
z : ℂ
hzl : (1 - z).re < ε
hzr : |z.im| < s * (re 1 - z.re)
⊢ z ∈ stolzSet M
Module: Mathlib.Analysis.Complex.AbelLimit

Goal state before:
pp: f : ℕ → ℂ
l : ℂ
h : Tendsto (fun n ↦ ∑ i ∈ range n, f i) atTop (𝓝 l)
M : ℝ
hM : 1 < M
s : ℕ → ℂ := fun n ↦ ∑ i ∈ range n, f i
g : ℂ → ℂ := fun z ↦ ∑' (n : ℕ), f n * z ^ n
ε : ℝ
εpos : ε > 0
B₁ : ℕ
hB₁ : ∀ n ≥ B₁, ‖∑ i ∈ range n, f i - l‖ < ε / 4 / M
F : ℝ := ∑ i ∈ range B₁, ‖l - s (i + 1)‖
z : ℂ
zn : ‖z‖ < 1
zm : ‖1 - z‖ < M * (1 - ‖z‖)
zd : ‖z - 1‖ < ε / 4 / (F + 1)
B₂ : ℕ
hB₂ : ‖l - ∑' (n : ℕ), f n * z ^ n - (1 - z) * ∑ i ∈ range (max B₁ B₂), (l - ∑ j ∈ range (i + 1), f j) * z ^ i‖ < ε / 2
S₁ : ‖1 - z‖ * ∑ i ∈ range B₁, ‖l - s (i + 1)‖ * ‖z‖ ^ i < ε / 4
⊢ ‖1 - z‖ * (ε / 4 / M) * ∑' (i : ℕ), ‖z‖ ^ i = ‖1 - z‖ * (ε / 4 / M) / (1 - ‖z‖)

Tactic:
tsum_geometric_of_lt_one (by positivity) zn,

Goal state after:
f : ℕ → ℂ
l : ℂ
h : Tendsto (fun n ↦ ∑ i ∈ range n, f i) atTop (𝓝 l)
M : ℝ
hM : 1 < M
s : ℕ → ℂ := fun n ↦ ∑ i ∈ range n, f i
g : ℂ → ℂ := fun z ↦ ∑' (n : ℕ), f n * z ^ n
ε : ℝ
εpos : ε > 0
B₁ : ℕ
hB₁ : ∀ n ≥ B₁, ‖∑ i ∈ range n, f i - l‖ < ε / 4 / M
F : ℝ := ∑ i ∈ range B₁, ‖l - s (i + 1)‖
z : ℂ
zn : ‖z‖ < 1
zm : ‖1 - z‖ < M * (1 - ‖z‖)
zd : ‖z - 1‖ < ε / 4 / (F + 1)
B₂ : ℕ
hB₂ : ‖l - ∑' (n : ℕ), f n * z ^ n - (1 - z) * ∑ i ∈ range (max B₁ B₂), (l - ∑ j ∈ range (i + 1), f j) * z ^ i‖ < ε / 2
S₁ : ‖1 - z‖ * ∑ i ∈ range B₁, ‖l - s (i + 1)‖ * ‖z‖ ^ i < ε / 4
⊢ ‖1 - z‖ * (ε / 4 / M) * (1 - ‖z‖)⁻¹ = ‖1 - z‖ * (ε / 4 / M) / (1 - ‖z‖)
Module: Mathlib.Analysis.SpecialFunctions.Trigonometric.Arctan

Goal state before:
pp: case inl
x : ℝ
hx_gt : -(π / 2) < x
hx_lt : ¬π / 2 ≤ x
r : ℤ
hxr_eq : x = (2 * ↑r + 1) * π / 2
h : 0 ≤ r
⊢ 1 * (π / 2) ≤ (2 * ↑r + 1) * (π / 2)

Tactic:
mul_le_mul_iff_left₀ (half_pos pi_pos)

Goal state after:
case inl
x : ℝ
hx_gt : -(π / 2) < x
hx_lt : ¬π / 2 ≤ x
r : ℤ
hxr_eq : x = (2 * ↑r + 1) * π / 2
h : 0 ≤ r
⊢ 1 ≤ 2 * ↑r + 1