mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.Topology.VectorBundle.Basic

Goal state before:
pp: case fst
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
hb : b ∈ e.baseSet
z : E b
⊢ (↑e { proj := b, snd := z }).1 = (b, (continuousLinearEquivAt R e b hb) z).1

pp: case snd
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
hb : b ∈ e.baseSet
z : E b
⊢ (↑e { proj := b, snd := z }).2 = (b, (continuousLinearEquivAt R e b hb) z).2

Tactic:
· refine e.coe_fst ?_
    rw [e.source_eq]
    exact hb

Goal state after:
case snd
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
hb : b ∈ e.baseSet
z : E b
⊢ (↑e { proj := b, snd := z }).2 = (b, (continuousLinearEquivAt R e b hb) z).2
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✝¹⁰ : NontriviallyNormedField 𝕜
inst✝⁹ : NormedAddCommGroup F
inst✝⁸ : NormedSpace 𝕜 F
inst✝⁷ : TopologicalSpace (TotalSpace F E)
inst✝⁶ : (x : B) → TopologicalSpace (E x)
EB : Type u_7
inst✝⁵ : NormedAddCommGroup EB
inst✝⁴ : NormedSpace 𝕜 EB
HB : Type u_8
inst✝³ : TopologicalSpace HB
IB : ModelWithCorners 𝕜 EB HB
inst✝² : TopologicalSpace B
inst✝¹ : ChartedSpace HB B
inst✝ : FiberBundle F E
s : (x : B) → E x
a : Set B
x₀ : B
⊢ ContMDiffWithinAt IB (IB.prod 𝓘(𝕜, F)) n (fun x ↦ TotalSpace.mk' F x (s x)) a x₀ ↔
    ContMDiffWithinAt IB 𝓘(𝕜, F) n (fun x ↦ (↑(trivializationAt F E x₀) { proj := x, snd := s x }).2) a x₀

Tactic:
simp_rw [contMDiffWithinAt_totalSpace, and_iff_right_iff_imp]

Goal state after:
n : ℕ∞ω
𝕜 : Type u_1
B : Type u_2
F : Type u_4
E : B → Type u_6
inst✝¹⁰ : NontriviallyNormedField 𝕜
inst✝⁹ : NormedAddCommGroup F
inst✝⁸ : NormedSpace 𝕜 F
inst✝⁷ : TopologicalSpace (TotalSpace F E)
inst✝⁶ : (x : B) → TopologicalSpace (E x)
EB : Type u_7
inst✝⁵ : NormedAddCommGroup EB
inst✝⁴ : NormedSpace 𝕜 EB
HB : Type u_8
inst✝³ : TopologicalSpace HB
IB : ModelWithCorners 𝕜 EB HB
inst✝² : TopologicalSpace B
inst✝¹ : ChartedSpace HB B
inst✝ : FiberBundle F E
s : (x : B) → E x
a : Set B
x₀ : B
⊢ ContMDiffWithinAt IB 𝓘(𝕜, F) n (fun x ↦ (↑(trivializationAt F E x₀) (TotalSpace.mk' F x (s x))).2) a x₀ →
    ContMDiffWithinAt IB IB n (fun x ↦ x) a x₀
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
⊢ MDiff[(extChartAt I x).target] ↑(extChartAt I x).symm

Tactic:
intro y 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
⊢ MDiffAt[(extChartAt I x).target] ↑(extChartAt I x).symm y
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'
x : M
t : Set E := ↑I.symm ⁻¹' (chartAt H x).target ∩ range ↑I
⊢ EqOn (writtenInExtChartAt I I (Sum.inl x) Sum.swap) id t

Tactic:
intro y hy

Goal state after:
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'
x : M
t : Set E := ↑I.symm ⁻¹' (chartAt H x).target ∩ range ↑I
y : E
hy : y ∈ t
⊢ writtenInExtChartAt I I (Sum.inl x) Sum.swap y = id y
Module: Mathlib.Geometry.Manifold.MFDeriv.SpecificFunctions

Goal state before:
pp: case inl
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'
x : M
t : Set E := ↑I.symm ⁻¹' (chartAt H x).target ∩ range ↑I
this : EqOn (writtenInExtChartAt I I (Sum.inl x) Sum.swap) id t
⊢ writtenInExtChartAt I I (Sum.inl x) Sum.swap =ᶠ[𝓝[range ↑I] ↑I (↑(chartAt H (Sum.inl x)) (Sum.inl x))] id

Tactic:
apply Filter.eventually_of_mem ?_ this

Goal state after:
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'
x : M
t : Set E := ↑I.symm ⁻¹' (chartAt H x).target ∩ range ↑I
this : EqOn (writtenInExtChartAt I I (Sum.inl x) Sum.swap) id t
⊢ t ∈ 𝓝[range ↑I] ↑I (↑(chartAt H (Sum.inl x)) (Sum.inl x))
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'
x : M'
t : Set E := ↑I.symm ⁻¹' (chartAt H x).target ∩ range ↑I
⊢ EqOn (writtenInExtChartAt I I (Sum.inr x) Sum.swap) id t

Tactic:
intro y hy

Goal state after:
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'
x : M'
t : Set E := ↑I.symm ⁻¹' (chartAt H x).target ∩ range ↑I
y : E
hy : y ∈ t
⊢ writtenInExtChartAt I I (Sum.inr x) Sum.swap y = id y
Module: Mathlib.Geometry.Manifold.MFDeriv.SpecificFunctions

Goal state before:
pp: case inr
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'
x : M'
t : Set E := ↑I.symm ⁻¹' (chartAt H x).target ∩ range ↑I
this : EqOn (writtenInExtChartAt I I (Sum.inr x) Sum.swap) id t
⊢ writtenInExtChartAt I I (Sum.inr x) Sum.swap =ᶠ[𝓝[range ↑I] ↑I (↑(chartAt H (Sum.inr x)) (Sum.inr x))] id

Tactic:
apply Filter.eventually_of_mem ?_ this

Goal state after:
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'
x : M'
t : Set E := ↑I.symm ⁻¹' (chartAt H x).target ∩ range ↑I
this : EqOn (writtenInExtChartAt I I (Sum.inr x) Sum.swap) id t
⊢ t ∈ 𝓝[range ↑I] ↑I (↑(chartAt H (Sum.inr x)) (Sum.inr 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
⊢ (extChartAt I c).source ∩ ↑(extChartAt I c) ⁻¹' support ↑f.toContDiffBump =
    (extChartAt I c).source ∩ ↑(extChartAt I c) ⁻¹' ball (↑(extChartAt I c) c) f.rOut

Tactic:
← (extChartAt I c).symm_image_target_inter_eq',

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
⊢ ↑(extChartAt I c).symm '' ((extChartAt I c).target ∩ support ↑f.toContDiffBump) =
    (extChartAt I c).source ∩ ↑(extChartAt I c) ⁻¹' ball (↑(extChartAt I c) c) f.rOut
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).symm '' ((extChartAt I c).target ∩ support ↑f.toContDiffBump) =
    (extChartAt I c).source ∩ ↑(extChartAt I c) ⁻¹' ball (↑(extChartAt I c) c) f.rOut

Tactic:
← (extChartAt I c).symm_image_target_inter_eq',

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
⊢ ↑(extChartAt I c).symm '' ((extChartAt I c).target ∩ support ↑f.toContDiffBump) =
    ↑(extChartAt I c).symm '' ((extChartAt I c).target ∩ ball (↑(extChartAt I c) c) f.rOut)
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).symm '' (ball (↑(extChartAt I c) c) f.rOut ∩ range ↑I)

Tactic:
← (extChartAt I c).symm_image_target_inter_eq',

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
⊢ ↑(extChartAt I c).symm '' ((extChartAt I c).target ∩ ball (↑(extChartAt I c) c) f.rOut) =
    ↑(extChartAt I c).symm '' (ball (↑(extChartAt I c) c) f.rOut ∩ range ↑I)
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
s : Set M
z : M
F' : Type u_21
inst✝¹ : NormedField F'
inst✝ : NormedAlgebra 𝕜 F'
p : M → F'
p' : TangentSpace I z →L[𝕜] F'
hp : HasMFDerivAt[s] p z p'
hp_ne : p z ≠ 0
⊢ HasMFDerivAt[s] p⁻¹ z (-(p z ^ 2)⁻¹ • p')

Tactic:
convert! hp.inv' hp_ne

Goal state after:
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
s : Set M
z : M
F' : Type u_21
inst✝¹ : NormedField F'
inst✝ : NormedAlgebra 𝕜 F'
p : M → F'
p' : TangentSpace I z →L[𝕜] F'
hp : HasMFDerivAt[s] p z p'
hp_ne : p z ≠ 0
⊢ -(p z ^ 2)⁻¹ • p' = -(MulOpposite.op (p z)⁻¹ • (p z)⁻¹ • p')
Module: Mathlib.Geometry.Manifold.VectorBundle.ContMDiffSection

Goal state before:
pp: case refine_1.h
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
f : M → 𝕜
s : (x : M) → V x
u : Set M
x₀ : M
hf : ContMDiffWithinAt I 𝓘(𝕜, 𝕜) n f u x₀
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
x : M
hx : x ∈ e.baseSet
⊢ (fun x ↦ (↑e ⟨x, (f • s) x⟩).2) x = (f • fun x ↦ (↑e ⟨x, s x⟩).2) x

Tactic:
apply (e.linear 𝕜 hx).2

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

Goal state before:
pp: 𝕜 : Type u_1
inst✝¹³ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝¹² : NormedAddCommGroup E
inst✝¹¹ : NormedSpace 𝕜 E
H : Type u_3
inst✝¹⁰ : TopologicalSpace H
I : ModelWithCorners 𝕜 E H
M : Type u_4
inst✝⁹ : TopologicalSpace M
inst✝⁸ : ChartedSpace H M
F : Type u_5
inst✝⁷ : NormedAddCommGroup F
inst✝⁶ : NormedSpace 𝕜 F
n : ℕ∞ω
V : M → Type u_6
inst✝⁵ : TopologicalSpace (TotalSpace F V)
inst✝⁴ : (x : M) → TopologicalSpace (V x)
inst✝³ : FiberBundle F V
inst✝² : (x : M) → AddCommGroup (V x)
inst✝¹ : (x : M) → Module 𝕜 (V x)
inst✝ : VectorBundle 𝕜 F V
u : Set M
s : (x : M) → V x
ψ : M → 𝕜
hψ : ContMDiffOn I 𝓘(𝕜, 𝕜) n ψ u
ht : IsOpen u
ht' : tsupport ψ ⊆ u
hs : ContMDiffOn I (I.prod 𝓘(𝕜, F)) n (fun x ↦ ⟨x, s x⟩) u
⊢ ∀ y ∈ (tsupport ψ)ᶜ, ⟨y, (ψ • s) y⟩ = zeroSection F V y

Tactic:
intro y hy

Goal state after:
𝕜 : Type u_1
inst✝¹³ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝¹² : NormedAddCommGroup E
inst✝¹¹ : NormedSpace 𝕜 E
H : Type u_3
inst✝¹⁰ : TopologicalSpace H
I : ModelWithCorners 𝕜 E H
M : Type u_4
inst✝⁹ : TopologicalSpace M
inst✝⁸ : ChartedSpace H M
F : Type u_5
inst✝⁷ : NormedAddCommGroup F
inst✝⁶ : NormedSpace 𝕜 F
n : ℕ∞ω
V : M → Type u_6
inst✝⁵ : TopologicalSpace (TotalSpace F V)
inst✝⁴ : (x : M) → TopologicalSpace (V x)
inst✝³ : FiberBundle F V
inst✝² : (x : M) → AddCommGroup (V x)
inst✝¹ : (x : M) → Module 𝕜 (V x)
inst✝ : VectorBundle 𝕜 F V
u : Set M
s : (x : M) → V x
ψ : M → 𝕜
hψ : ContMDiffOn I 𝓘(𝕜, 𝕜) n ψ u
ht : IsOpen u
ht' : tsupport ψ ⊆ u
hs : ContMDiffOn I (I.prod 𝓘(𝕜, F)) n (fun x ↦ ⟨x, s x⟩) u
y : M
hy : y ∈ (tsupport ψ)ᶜ
⊢ ⟨y, (ψ • s) y⟩ = zeroSection F V y
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)
Hloc :
  ∀ (x₀ : M),
    ∃ U_x₀ ∈ 𝓝 x₀,
      ∃ s_loc, ContMDiffOn I (I.prod 𝓘(ℝ, F_fiber)) (↑n) (fun x ↦ ⟨x, s_loc x⟩) U_x₀ ∧ ∀ y ∈ U_x₀, s_loc y ∈ t y
⊢ ∃ s, ∀ (x : M), s x ∈ t x

Tactic:
choose W h_nhds s_loc s_smooth h_mem_t using Hloc

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
⊢ ∃ s, ∀ (x : M), s x ∈ t x
Module: Mathlib.Analysis.Calculus.TangentCone.Pi

Goal state before:
pp: 𝕜 : Type u_1
inst✝⁶ : Semiring 𝕜
ι : Type u_2
E : ι → Type u_3
inst✝⁵ : (i : ι) → AddCommGroup (E i)
inst✝⁴ : (i : ι) → Module 𝕜 (E i)
inst✝³ : (i : ι) → TopologicalSpace (E i)
inst✝² : ∀ (i : ι), ContinuousAdd (E i)
inst✝¹ : ∀ (i : ι), ContinuousConstSMul 𝕜 (E i)
s : (i : ι) → Set (E i)
x : (i : ι) → E i
inst✝ : DecidableEq ι
i : ι
hi : ∀ (j : ι), j ≠ i → x j ∈ closure[inst✝³ j] (s j)
⊢ MapsTo (Pi.single i) (tangentConeAt 𝕜 (s i) (x i)) (tangentConeAt 𝕜 (closure[Pi.topologicalSpace] (univ.pi s)) x)

Tactic:
intro y hy

Goal state after:
𝕜 : Type u_1
inst✝⁶ : Semiring 𝕜
ι : Type u_2
E : ι → Type u_3
inst✝⁵ : (i : ι) → AddCommGroup (E i)
inst✝⁴ : (i : ι) → Module 𝕜 (E i)
inst✝³ : (i : ι) → TopologicalSpace (E i)
inst✝² : ∀ (i : ι), ContinuousAdd (E i)
inst✝¹ : ∀ (i : ι), ContinuousConstSMul 𝕜 (E i)
s : (i : ι) → Set (E i)
x : (i : ι) → E i
inst✝ : DecidableEq ι
i : ι
hi : ∀ (j : ι), j ≠ i → x j ∈ closure[inst✝³ j] (s j)
y : E i
hy : y ∈ tangentConeAt 𝕜 (s i) (x i)
⊢ Pi.single i y ∈ tangentConeAt 𝕜 (closure[Pi.topologicalSpace] (univ.pi s)) x
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 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:
obtain ⟨c, hc⟩ := NormedField.exists_lt_norm 𝕜 1

Goal state after:
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‖
⊢ ∃ 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)
Module: Mathlib.MeasureTheory.Function.AbsolutelyContinuous

Goal state before:
pp: case hh
a b : ℝ
⊢ Tendsto (fun E ↦ volume (⋃ i ∈ Finset.range E.1, uIoc (E.2 i).1 (E.2 i).2)) (totalLengthFilter ⊓ 𝓟 (disjWithin a b))
    (𝓝 0)

Tactic:
apply tendsto_volume_totalLengthFilter_nhds_zero.mono_left
    simp

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

Goal state before:
pp: case ht
F : Type u_2
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace ℝ F
f f' : ℝ → F
d b η : ℝ
hdb✝ : d ≤ b
hf : ∀ᵐ (x : ℝ), x ∈ Ioo d b → HasDerivAt f (f' x) x
hη : 0 < η
hdb : d < b
t : Set (ℝ × ℝ) := ⋯
s : Set ℝ := ⋯
⊢ ∀ a ∈ t, (interior (Icc a.1 a.2)).Nonempty

Tactic:
simp +contextual [t]

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

Goal state before:
pp: case succ
E : Type u_2
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
f : ℝ → E
x y : ℝ
s s' : Set ℝ
hs_unique : UniqueDiffOn ℝ s
hs' : s' ∈ 𝓝[s] y
hy : y ∈ s'
h : s' ⊆ s
hs'_unique : UniqueDiffWithinAt ℝ s' y
k : ℕ
hk :
  ContDiffOn ℝ (↑k) f s →
    DifferentiableWithinAt ℝ (iteratedDerivWithin k f s) s y →
      HasDerivWithinAt (fun t ↦ taylorWithinEval f k s t x)
        (((↑k !)⁻¹ * (x - y) ^ k) • iteratedDerivWithin (k + 1) f s y) s' y
hf : ContDiffOn ℝ (↑(k + 1)) f s
hf' : DifferentiableWithinAt ℝ (iteratedDerivWithin (k + 1) f s) s y
⊢ HasDerivWithinAt (fun t ↦ taylorWithinEval f (k + 1) s t x)
    (((↑(k + 1)!)⁻¹ * (x - y) ^ (k + 1)) • iteratedDerivWithin (k + 1 + 1) f s y) s' y

Tactic:
Nat.add_succ,

Goal state after:
case succ
E : Type u_2
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
f : ℝ → E
x y : ℝ
s s' : Set ℝ
hs_unique : UniqueDiffOn ℝ s
hs' : s' ∈ 𝓝[s] y
hy : y ∈ s'
h : s' ⊆ s
hs'_unique : UniqueDiffWithinAt ℝ s' y
k : ℕ
hk :
  ContDiffOn ℝ (↑k) f s →
    DifferentiableWithinAt ℝ (iteratedDerivWithin k f s) s y →
      HasDerivWithinAt (fun t ↦ taylorWithinEval f k s t x)
        (((↑k !)⁻¹ * (x - y) ^ k) • iteratedDerivWithin (k + 1) f s y) s' y
hf : ContDiffOn ℝ (↑(k + 1)) f s
hf' : DifferentiableWithinAt ℝ (iteratedDerivWithin (k + 1) f s) s y
⊢ HasDerivWithinAt (fun t ↦ taylorWithinEval f (k + 0).succ s t x)
    (((↑(k + 0).succ !)⁻¹ * (x - y) ^ (k + 0).succ) • iteratedDerivWithin ((k + 0).succ + 0).succ f s y) s' y