mathlib-initiative/mathlib-tactics huggingface dataset:
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
A : E →L[ℝ] E
m : ℝ≥0 := |A.det|.toNNReal + ε
I : ENNReal.ofReal |A.det| < ↑m
δ : ℝ≥0
h : ∀ (s : Set E) (f : E → E), ApproximatesLinearOn f A s δ → μ (f '' s) ≤ ↑m * μ s
δpos : 0 < δ
δ' : ℝ
δ'pos : 0 < δ'
hδ' : ∀ (B : E →L[ℝ] E), dist B A < δ' → dist B.det A.det < ↑ε
δ'' : ℝ≥0 := ⟨δ' / 2, ⋯⟩
B : E →L[ℝ] E
hB : ‖B - A‖ ≤ ↑(min δ δ'')
⊢ ↑(min δ δ'') ≤ ↑δ''

Tactic:
simp only [le_refl, NNReal.coe_min, min_le_iff, or_true]

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
this : TopologicalSpace M := c.toTopologicalSpace
t : Set M
e' : PartialEquiv M H
e'_atlas : e' ∈ c.atlas
s : Set H
s_open : IsOpen[inst✝] s
ts : t = ↑e' ⁻¹' s ∩ e'.source
⊢ IsOpen[inst✝] (e.target ∩ e.invFun ⁻¹' (↑e' ⁻¹' s ∩ e'.source))

Tactic:
let f := e.symm.trans e'

Goal state after:
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
this : TopologicalSpace M := c.toTopologicalSpace
t : Set M
e' : PartialEquiv M H
e'_atlas : e' ∈ c.atlas
s : Set H
s_open : IsOpen[inst✝] s
ts : t = ↑e' ⁻¹' s ∩ e'.source
f : PartialEquiv H H := e.symm ≫ e'
⊢ IsOpen[inst✝] (e.target ∩ e.invFun ⁻¹' (↑e' ⁻¹' s ∩ e'.source))
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
hf : InjOn f s
ε : ℝ≥0
εpos : 0 < ε
A : E →L[ℝ] E
δ' : ℝ
δ'pos : 0 < δ'
hδ' : ∀ (B : E →L[ℝ] E), dist B A < δ' → dist B.det A.det < ↑ε
δ'' : ℝ≥0 := ⟨δ' / 2, ⋯⟩
B : E →L[ℝ] E
hB : ‖B - A‖ ≤ ↑δ''
⊢ ‖B - A‖ < δ'

Tactic:
exact hB.trans_lt (half_lt_self δ'pos)

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
hf : InjOn f s
ε : ℝ≥0
εpos : 0 < ε
A : E →L[ℝ] E
δ' : ℝ
δ'pos : 0 < δ'
hδ' : ∀ (B : E →L[ℝ] E), dist B A < δ' → dist B.det A.det < ↑ε
δ'' : ℝ≥0 := ⋯
I'' : ∀ (B : E →L[ℝ] E), ‖B - A‖ ≤ ↑δ'' → |B.det - A.det| ≤ ↑ε
hA : A.det ≠ 0
m : ℝ≥0 := ⋯
I : ↑m < ENNReal.ofReal |A.det|
δ : ℝ≥0
h : ∀ (s : Set E) (f : E → E), ApproximatesLinearOn f A s δ → ↑m * μ s ≤ μ (f '' s)
δpos : 0 < δ
B : E →L[ℝ] E
hB : ‖B - A‖ ≤ ↑(min δ δ'')
⊢ ↑(min δ δ'') ≤ ↑δ''

Tactic:
simp only [le_refl, NNReal.coe_min, min_le_iff, or_true]

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

Goal state before:
pp: H : Type u_1
M : Type u_2
H' : Type u_3
M' : Type u_4
inst✝⁵ : TopologicalSpace H
inst✝⁴ : TopologicalSpace M
inst✝³ : ChartedSpace H M
inst✝² : TopologicalSpace H'
inst✝¹ : TopologicalSpace M'
inst✝ : ChartedSpace H' M'
G : StructureGroupoid H
G' : StructureGroupoid H'
e : OpenPartialHomeomorph M H
P : (H → H') → Set H → H → Prop
g : M → M'
s : Set M
x : M
hG : G.LocalInvariantProp G' P
he : e ∈ maximalAtlas M G
xe : x ∈ e.source
⊢ LiftPropWithinAt P g s x ↔ LiftPropWithinAt P (g ∘ ↑e.symm) (↑e.symm ⁻¹' s) (↑e x)

Tactic:
liftPropWithinAt_self_source,

Goal state after:
H : Type u_1
M : Type u_2
H' : Type u_3
M' : Type u_4
inst✝⁵ : TopologicalSpace H
inst✝⁴ : TopologicalSpace M
inst✝³ : ChartedSpace H M
inst✝² : TopologicalSpace H'
inst✝¹ : TopologicalSpace M'
inst✝ : ChartedSpace H' M'
G : StructureGroupoid H
G' : StructureGroupoid H'
e : OpenPartialHomeomorph M H
P : (H → H') → Set H → H → Prop
g : M → M'
s : Set M
x : M
hG : G.LocalInvariantProp G' P
he : e ∈ maximalAtlas M G
xe : x ∈ e.source
⊢ LiftPropWithinAt P g s x ↔
    ContinuousWithinAt (g ∘ ↑e.symm) (↑e.symm ⁻¹' s) (↑e x) ∧
      P (↑(chartAt H' ((g ∘ ↑e.symm) (↑e x))) ∘ g ∘ ↑e.symm) (↑e.symm ⁻¹' s) (↑e x)
Module: Mathlib.Geometry.Manifold.IsManifold.Basic

Goal state before:
pp: case refine_1
𝕜 : Type u_1
inst✝⁴ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace 𝕜 E
H : Type u_3
inst✝¹ : TopologicalSpace H
I : ModelWithCorners 𝕜 E H
X : Type u_4
inst✝ : TopologicalSpace X
f : H → X
s : Set H
x : H
h : ContinuousWithinAt (f ∘ ↑I.symm) (↑I.symm ⁻¹' s ∩ range ↑I) (↑I x)
this : ContinuousWithinAt ((f ∘ ↑I.symm) ∘ ↑I) ((fun x ↦ ↑I.symm (↑I x)) ⁻¹' s ∩ ↑I ⁻¹' range ↑I) x
⊢ ContinuousWithinAt f s x

Tactic:
I.left_inv,

Goal state after:
case refine_1
𝕜 : Type u_1
inst✝⁴ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace 𝕜 E
H : Type u_3
inst✝¹ : TopologicalSpace H
I : ModelWithCorners 𝕜 E H
X : Type u_4
inst✝ : TopologicalSpace X
f : H → X
s : Set H
x : H
h : ContinuousWithinAt (f ∘ ↑I.symm) (↑I.symm ⁻¹' s ∩ range ↑I) (↑I x)
this : ContinuousWithinAt ((f ∘ ↑I.symm) ∘ ↑I) ((fun x ↦ x) ⁻¹' s ∩ ↑I ⁻¹' range ↑I) x
⊢ ContinuousWithinAt f s x
Module: Mathlib.Geometry.Manifold.LocalInvariantProperties

Goal state before:
pp: H : Type u_1
inst✝¹ : TopologicalSpace H
G : StructureGroupoid H
inst✝ : ClosedUnderRestriction G
s : Set H
x : H
f : H → H
e' : OpenPartialHomeomorph H H
he'G : e' ∈ G
he'x : x ∈ e'.source
h : G.IsLocalStructomorphWithinAt f s x
hx : ↑e' x ∈ ↑e'.symm ⁻¹' s
hxs : x ∈ s
⊢ ∃ e ∈ G, EqOn (f ∘ ↑e'.symm) (↑e.toPartialEquiv) (↑e'.symm ⁻¹' s ∩ e.source) ∧ ↑e' x ∈ e.source

Tactic:
rcases h hxs with ⟨e, heG, hef, hex⟩

Goal state after:
H : Type u_1
inst✝¹ : TopologicalSpace H
G : StructureGroupoid H
inst✝ : ClosedUnderRestriction G
s : Set H
x : H
f : H → H
e' : OpenPartialHomeomorph H H
he'G : e' ∈ G
he'x : x ∈ e'.source
h : G.IsLocalStructomorphWithinAt f s x
hx : ↑e' x ∈ ↑e'.symm ⁻¹' s
hxs : x ∈ s
e : OpenPartialHomeomorph H H
heG : e ∈ G
hef : EqOn f (↑e.toPartialEquiv) (s ∩ e.source)
hex : x ∈ e.source
⊢ ∃ e ∈ G, EqOn (f ∘ ↑e'.symm) (↑e.toPartialEquiv) (↑e'.symm ⁻¹' s ∩ e.source) ∧ ↑e' x ∈ e.source
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✝⁵ : NontriviallyNormedField 𝕜
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace 𝕜 E
inst✝² : TopologicalSpace H
inst✝¹ : TopologicalSpace M
I : ModelWithCorners 𝕜 E H
inst✝ : ChartedSpace H M
y : E
x : M
hy : y ∈ (extChartAt I x).target
⊢ (extChartAt I x).target ∪ (range ↑I)ᶜ ∈ 𝓝 y

Tactic:
rw [← nhdsWithin_univ, ← union_compl_self (range I), nhdsWithin_union]

Goal state after:
𝕜 : Type u_1
E : Type u_2
M : Type u_3
H : Type u_4
inst✝⁵ : NontriviallyNormedField 𝕜
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace 𝕜 E
inst✝² : TopologicalSpace H
inst✝¹ : TopologicalSpace M
I : ModelWithCorners 𝕜 E H
inst✝ : ChartedSpace H M
y : E
x : M
hy : y ∈ (extChartAt I x).target
⊢ (extChartAt I x).target ∪ (range ↑I)ᶜ ∈ 𝓝[range ↑I] y ⊔ 𝓝[(range ↑I)ᶜ] y
Module: Mathlib.Geometry.Manifold.ContMDiff.Defs

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
E' : Type u_5
inst✝² : NormedAddCommGroup E'
inst✝¹ : NormedSpace 𝕜 E'
H' : Type u_6
inst✝ : TopologicalSpace H'
I' : ModelWithCorners 𝕜 E' H'
n m : ℕ∞ω
hmn : m ≤ n
s : Set H
x : H
f : H → H'
e : OpenPartialHomeomorph H H
he : e ∈ contDiffGroupoid n I
hx : x ∈ e.source
h : ContDiffWithinAt 𝕜 m (↑I' ∘ f ∘ ↑I.symm) (↑I.symm ⁻¹' s ∩ range ↑I) ((↑I ∘ ↑e.symm ∘ ↑I.symm) (↑I (↑e x)))
this✝¹ : ↑I x = (↑I ∘ ↑e.symm ∘ ↑I.symm) (↑I (↑e x))
this✝ : ↑I (↑e x) ∈ ↑I.symm ⁻¹' e.target ∩ range ↑I
this : ContDiffWithinAt 𝕜 n (↑I ∘ ↑e.symm ∘ ↑I.symm) (↑I.symm ⁻¹' e.target ∩ range ↑I) (↑I (↑e x))
⊢ ↑I.symm (↑I (↑e x)) ∈ e.target

Tactic:
I.left_inv,

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
E' : Type u_5
inst✝² : NormedAddCommGroup E'
inst✝¹ : NormedSpace 𝕜 E'
H' : Type u_6
inst✝ : TopologicalSpace H'
I' : ModelWithCorners 𝕜 E' H'
n m : ℕ∞ω
hmn : m ≤ n
s : Set H
x : H
f : H → H'
e : OpenPartialHomeomorph H H
he : e ∈ contDiffGroupoid n I
hx : x ∈ e.source
h : ContDiffWithinAt 𝕜 m (↑I' ∘ f ∘ ↑I.symm) (↑I.symm ⁻¹' s ∩ range ↑I) ((↑I ∘ ↑e.symm ∘ ↑I.symm) (↑I (↑e x)))
this✝¹ : ↑I x = (↑I ∘ ↑e.symm ∘ ↑I.symm) (↑I (↑e x))
this✝ : ↑I (↑e x) ∈ ↑I.symm ⁻¹' e.target ∩ range ↑I
this : ContDiffWithinAt 𝕜 n (↑I ∘ ↑e.symm ∘ ↑I.symm) (↑I.symm ⁻¹' e.target ∩ range ↑I) (↑I (↑e x))
⊢ ↑e x ∈ e.target
Module: Mathlib.Analysis.Normed.Operator.Prod

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
F : Type u_3
G : Type u_4
inst✝¹⁴ : NontriviallyNormedField 𝕜
inst✝¹³ : SeminormedAddCommGroup E
inst✝¹² : SeminormedAddCommGroup F
inst✝¹¹ : SeminormedAddCommGroup G
inst✝¹⁰ : NormedSpace 𝕜 E
inst✝⁹ : NormedSpace 𝕜 F
inst✝⁸ : NormedSpace 𝕜 G
M₁ : Type u_5
M₂ : Type u_6
M₃ : Type u_7
M₄ : Type u_8
inst✝⁷ : SeminormedAddCommGroup M₁
inst✝⁶ : NormedSpace 𝕜 M₁
inst✝⁵ : SeminormedAddCommGroup M₂
inst✝⁴ : NormedSpace 𝕜 M₂
inst✝³ : SeminormedAddCommGroup M₃
inst✝² : NormedSpace 𝕜 M₃
inst✝¹ : SeminormedAddCommGroup M₄
inst✝ : NormedSpace 𝕜 M₄
⊢ (fun p ↦ p.1.prodMap p.2) =
    ⇑(have Φ₁ := (compL 𝕜 M₁ M₂ (M₂ × M₄)) (inl 𝕜 M₂ M₄);
      have Φ₂ := (compL 𝕜 M₃ M₄ (M₂ × M₄)) (inr 𝕜 M₂ M₄);
      have Φ₁' := (compL 𝕜 (M₁ × M₃) M₁ (M₂ × M₄)).flip (fst 𝕜 M₁ M₃);
      have Φ₂' := (compL 𝕜 (M₁ × M₃) M₃ (M₂ × M₄)).flip (snd 𝕜 M₁ M₃);
      have Ψ₁ := fst 𝕜 (M₁ →L[𝕜] M₂) (M₃ →L[𝕜] M₄);
      have Ψ₂ := snd 𝕜 (M₁ →L[𝕜] M₂) (M₃ →L[𝕜] M₄);
      Φ₁' ∘SL Φ₁ ∘SL Ψ₁ + Φ₂' ∘SL Φ₂ ∘SL Ψ₂)

Tactic:
by
      apply funext
      rintro ⟨φ, ψ⟩
      refine ContinuousLinearMap.ext fun ⟨x₁, x₂⟩ => ?_
      simp

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

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
E' : Type u_5
inst✝² : NormedAddCommGroup E'
inst✝¹ : NormedSpace 𝕜 E'
H' : Type u_6
inst✝ : TopologicalSpace H'
I' : ModelWithCorners 𝕜 E' H'
s : Set H
x : H
f : H → H'
e : OpenPartialHomeomorph H H
he : e ∈ contDiffGroupoid 1 I
hx : x ∈ e.source
h : DifferentiableWithinAt 𝕜 (↑I' ∘ f ∘ ↑I.symm) (↑I.symm ⁻¹' s ∩ range ↑I) ((↑I ∘ ↑e.symm ∘ ↑I.symm) (↑I (↑e x)))
this✝¹ : ↑I x = (↑I ∘ ↑e.symm ∘ ↑I.symm) (↑I (↑e x))
this✝ : ↑I (↑e x) ∈ ↑I.symm ⁻¹' e.target ∩ range ↑I
this : ContDiffWithinAt 𝕜 1 (↑I ∘ ↑e.symm ∘ ↑I.symm) (↑I.symm ⁻¹' e.target ∩ range ↑I) (↑I (↑e x))
⊢ ↑I.symm (↑I (↑e x)) ∈ e.target

Tactic:
I.left_inv,

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
E' : Type u_5
inst✝² : NormedAddCommGroup E'
inst✝¹ : NormedSpace 𝕜 E'
H' : Type u_6
inst✝ : TopologicalSpace H'
I' : ModelWithCorners 𝕜 E' H'
s : Set H
x : H
f : H → H'
e : OpenPartialHomeomorph H H
he : e ∈ contDiffGroupoid 1 I
hx : x ∈ e.source
h : DifferentiableWithinAt 𝕜 (↑I' ∘ f ∘ ↑I.symm) (↑I.symm ⁻¹' s ∩ range ↑I) ((↑I ∘ ↑e.symm ∘ ↑I.symm) (↑I (↑e x)))
this✝¹ : ↑I x = (↑I ∘ ↑e.symm ∘ ↑I.symm) (↑I (↑e x))
this✝ : ↑I (↑e x) ∈ ↑I.symm ⁻¹' e.target ∩ range ↑I
this : ContDiffWithinAt 𝕜 1 (↑I ∘ ↑e.symm ∘ ↑I.symm) (↑I.symm ⁻¹' e.target ∩ range ↑I) (↑I (↑e x))
⊢ ↑e x ∈ e.target
Module: Mathlib.Geometry.Manifold.ContMDiff.Defs

Goal state before:
pp: 𝕜 : Type u_1
inst✝¹² : NontriviallyNormedField 𝕜
E : Type u_2
inst✝¹¹ : NormedAddCommGroup E
inst✝¹⁰ : NormedSpace 𝕜 E
H : Type u_3
inst✝⁹ : TopologicalSpace H
I : ModelWithCorners 𝕜 E H
M : Type u_4
inst✝⁸ : TopologicalSpace M
inst✝⁷ : ChartedSpace H M
E' : Type u_5
inst✝⁶ : NormedAddCommGroup E'
inst✝⁵ : NormedSpace 𝕜 E'
H' : Type u_6
inst✝⁴ : TopologicalSpace H'
I' : ModelWithCorners 𝕜 E' H'
M' : Type u_7
inst✝³ : TopologicalSpace M'
inst✝² : ChartedSpace H' M'
f : M → M'
s✝ : Set M
x : M
n : ℕ∞ω
inst✝¹ : IsManifold I n M
inst✝ : IsManifold I' n M'
hn : n ≠ ∞
s : Set M
hxs : x ∈ s
⊢ ContMDiffWithinAt I I' n f s x ↔ ∃ u ∈ 𝓝[s] x, ContMDiffOn I I' n f u

Tactic:
refine ⟨fun h ↦ ?_, fun ⟨u, hmem, hu⟩ ↦
    (hu _ (mem_of_mem_nhdsWithin hxs hmem)).mono_of_mem_nhdsWithin hmem⟩

Goal state after:
𝕜 : Type u_1
inst✝¹² : NontriviallyNormedField 𝕜
E : Type u_2
inst✝¹¹ : NormedAddCommGroup E
inst✝¹⁰ : NormedSpace 𝕜 E
H : Type u_3
inst✝⁹ : TopologicalSpace H
I : ModelWithCorners 𝕜 E H
M : Type u_4
inst✝⁸ : TopologicalSpace M
inst✝⁷ : ChartedSpace H M
E' : Type u_5
inst✝⁶ : NormedAddCommGroup E'
inst✝⁵ : NormedSpace 𝕜 E'
H' : Type u_6
inst✝⁴ : TopologicalSpace H'
I' : ModelWithCorners 𝕜 E' H'
M' : Type u_7
inst✝³ : TopologicalSpace M'
inst✝² : ChartedSpace H' M'
f : M → M'
s✝ : Set M
x : M
n : ℕ∞ω
inst✝¹ : IsManifold I n M
inst✝ : IsManifold I' n M'
hn : n ≠ ∞
s : Set M
hxs : x ∈ s
h : ContMDiffWithinAt I I' n f s x
⊢ ∃ u ∈ 𝓝[s] x, ContMDiffOn I I' n f u
Module: Mathlib.Geometry.Manifold.Algebra.Monoid

Goal state before:
pp: 𝕜 : Type u_1
inst✝¹² : NontriviallyNormedField 𝕜
H : Type u_2
inst✝¹¹ : TopologicalSpace H
E : Type u_3
inst✝¹⁰ : NormedAddCommGroup E
inst✝⁹ : NormedSpace 𝕜 E
I : ModelWithCorners 𝕜 E H
n : ℕ∞ω
G : Type u_4
inst✝⁸ : Mul G
inst✝⁷ : TopologicalSpace G
inst✝⁶ : ChartedSpace H G
E' : Type u_5
inst✝⁵ : NormedAddCommGroup E'
inst✝⁴ : NormedSpace 𝕜 E'
H' : Type u_6
inst✝³ : TopologicalSpace H'
I' : ModelWithCorners 𝕜 E' H'
M : Type u_7
inst✝² : TopologicalSpace M
inst✝¹ : ChartedSpace H' M
inst✝ : ContinuousMul G
⊢ Continuous fun p ↦ p.1 * p.2

Tactic:
exact continuous_mul

Goal state after:
Module: Mathlib.Geometry.Manifold.ContMDiff.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
n : ℕ∞ω
e : OpenPartialHomeomorph M H
he : e ∈ maximalAtlas I n M
⊢ e.target ⊆ (fun x ↦ ↑I.symm (↑I x)) ⁻¹' e.target

Tactic:
I.left_inv,

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
n : ℕ∞ω
e : OpenPartialHomeomorph M H
he : e ∈ maximalAtlas I n M
⊢ e.target ⊆ (fun x ↦ x) ⁻¹' e.target
Module: Mathlib.Geometry.Manifold.MFDeriv.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
x : M
s t : Set M
y : M
h : s =ᶠ[𝓝[{y}ᶜ] x] t
this : T1Space M
u : Set M
u_mem : u ∈ 𝓝 (↑(extChartAt I x).symm (↑(extChartAt I x) x))
hu : u ∩ {x}ᶜ ⊆ {y | (y ∈ s) = (y ∈ t)}
⊢ (extChartAt I x).target ∪ (range ↑I)ᶜ ∈ 𝓝 (↑(extChartAt I x) x)

Tactic:
rw [← nhdsWithin_univ, ← union_compl_self (range I), nhdsWithin_union]

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
s t : Set M
y : M
h : s =ᶠ[𝓝[{y}ᶜ] x] t
this : T1Space M
u : Set M
u_mem : u ∈ 𝓝 (↑(extChartAt I x).symm (↑(extChartAt I x) x))
hu : u ∩ {x}ᶜ ⊆ {y | (y ∈ s) = (y ∈ t)}
⊢ (extChartAt I x).target ∪ (range ↑I)ᶜ ∈ 𝓝[range ↑I] ↑(extChartAt I x) x ⊔ 𝓝[(range ↑I)ᶜ] ↑(extChartAt I x) x
Module: Mathlib.Topology.VectorBundle.Basic

Goal state before:
pp: R : Type u_1
B : Type u_2
F : Type u_3
E : B → Type u_4
inst✝⁸ : NontriviallyNormedField R
inst✝⁷ : (x : B) → AddCommMonoid (E x)
inst✝⁶ : (x : B) → Module R (E x)
inst✝⁵ : NormedAddCommGroup F
inst✝⁴ : NormedSpace R F
inst✝³ : TopologicalSpace B
inst✝² : TopologicalSpace (TotalSpace F E)
inst✝¹ : (x : B) → TopologicalSpace (E x)
inst✝ : FiberBundle F E
ι : Type u_5
Z : VectorBundleCore R B F ι
b : B
a : F
i : ι
x : B
x✝² : x ∈ (Z.localTriv i).baseSet
x✝¹ : R
x✝ : Z.Fiber x
⊢ (↑(Z.localTriv i) { proj := x, snd := x✝¹ • x✝ }).2 = x✝¹ • (↑(Z.localTriv i) { proj := x, snd := x✝ }).2

Tactic:
simp only [map_smul, localTriv_apply, mfld_simps]

Goal state after:
Module: Mathlib.Topology.VectorBundle.Basic

Goal state before:
pp: R : Type u_1
B : Type u_2
F : Type u_3
inst✝³ : NontriviallyNormedField R
inst✝² : NormedAddCommGroup F
inst✝¹ : NormedSpace R F
inst✝ : TopologicalSpace B
ι : Type u_5
Z : VectorBundleCore R B F ι
b : B
a : F
⊢ { proj := b, snd := a } ∈ (Z.localTrivAt b).source

Tactic:
rw [localTrivAt, mem_localTriv_source]

Goal state after:
R : Type u_1
B : Type u_2
F : Type u_3
inst✝³ : NontriviallyNormedField R
inst✝² : NormedAddCommGroup F
inst✝¹ : NormedSpace R F
inst✝ : TopologicalSpace B
ι : Type u_5
Z : VectorBundleCore R B F ι
b : B
a : F
⊢ { proj := b, snd := a }.proj ∈ Z.baseSet (Z.indexAt b)
Module: Mathlib.Geometry.Manifold.VectorBundle.Basic

Goal state before:
pp: case refine_1
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
⊢ (fun y ↦ e.coordChange e' (f y) (g y)) =ᶠ[𝓝[s] x] fun x ↦ ↑(Trivialization.coordChangeL 𝕜 e e' (f x)) (g x)

Tactic:
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

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

Goal state before:
pp: 𝕜 : Type u_1
inst✝¹⁰ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝⁹ : NormedAddCommGroup E
inst✝⁸ : NormedSpace 𝕜 E
H : Type u_3
inst✝⁷ : TopologicalSpace H
I : ModelWithCorners 𝕜 E H
M : Type u_4
inst✝⁶ : TopologicalSpace M
inst✝⁵ : ChartedSpace H M
E' : Type u_5
inst✝⁴ : NormedAddCommGroup E'
inst✝³ : NormedSpace 𝕜 E'
H' : Type u_6
inst✝² : TopologicalSpace H'
I' : ModelWithCorners 𝕜 E' H'
M' : Type u_7
inst✝¹ : TopologicalSpace M'
inst✝ : ChartedSpace H' M'
s : Set (M × M')
p : TangentBundle (I.prod I') (M × M')
hs : UniqueMDiffAt[s] p.proj
⊢ ⟨p.proj.2, (mfderiv[s] Prod.snd p.proj) p.snd⟩ = ⟨p.proj.2, p.snd.2⟩

Tactic:
mfderivWithin_snd hs

Goal state after:
𝕜 : Type u_1
inst✝¹⁰ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝⁹ : NormedAddCommGroup E
inst✝⁸ : NormedSpace 𝕜 E
H : Type u_3
inst✝⁷ : TopologicalSpace H
I : ModelWithCorners 𝕜 E H
M : Type u_4
inst✝⁶ : TopologicalSpace M
inst✝⁵ : ChartedSpace H M
E' : Type u_5
inst✝⁴ : NormedAddCommGroup E'
inst✝³ : NormedSpace 𝕜 E'
H' : Type u_6
inst✝² : TopologicalSpace H'
I' : ModelWithCorners 𝕜 E' H'
M' : Type u_7
inst✝¹ : TopologicalSpace M'
inst✝ : ChartedSpace H' M'
s : Set (M × M')
p : TangentBundle (I.prod I') (M × M')
hs : UniqueMDiffAt[s] p.proj
⊢ ⟨p.proj.2, (ContinuousLinearMap.snd 𝕜 (TangentSpace I p.proj.1) (TangentSpace I' p.proj.2)) p.snd⟩ =
    ⟨p.proj.2, p.snd.2⟩
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
⊢ support ↑f ⊆ (chartAt H c).source

Tactic:
rw [f.support_eq_inter_preimage, ← extChartAt_source I]; exact inter_subset_left

Goal state after:
Module: Mathlib.Topology.ShrinkingLemma

Goal state before:
pp: ι : Type u_1
X : Type u_2
inst✝ : TopologicalSpace X
u : ι → Set X
s : Set X
p : Set X → Prop
c : Set (PartialRefinement u s p)
i : ι
ne : c.Nonempty
⊢ i ∈ (find c ne i).carrier ↔ i ∈ chainSupCarrier c

Tactic:
rw [find]
  split_ifs with h
  · have := h.choose_spec
    exact iff_of_true this.2 (mem_iUnion₂.2 ⟨_, this.1, this.2⟩)
  · push Not at h
    refine iff_of_false (h _ ne.some_mem) ?_
    simpa only [chainSupCarrier, mem_iUnion₂, not_exists]

Goal state after:
Module: Mathlib.Topology.ShrinkingLemma

Goal state before:
pp: case inl
ι : Type u_1
X : Type u_2
inst✝ : TopologicalSpace X
u : ι → Set X
s : Set X
p : Set X → Prop
c : Set (PartialRefinement u s p)
hc : IsChain (fun x1 x2 ↦ x1 ≤ x2) c
ne : c.Nonempty
hfin : ∀ x ∈ s, {i | x ∈ u i}.Finite
hU : s ⊆ ⋃ i, u i
x : X
hxs : x ∈ s
i : ι
hi : i ∉ chainSupCarrier c
hxi : x ∈ u i
⊢ ∃ i, x ∈ (find c ne i).toFun i

Tactic:
use i
      simpa only [(find c ne i).apply_eq (mt (mem_find_carrier_iff _).1 hi)]

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

Goal state before:
pp: case insert
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
x₀ : M
ι : Type u_7
t : ι → (x : M) → V x
i : ι
s : Finset ι
hi : i ∉ s
h :
  (∀ i ∈ s, ContMDiffWithinAt I (I.prod 𝓘(𝕜, F)) n (fun x ↦ ⟨x, t i x⟩) u x₀) →
    ContMDiffWithinAt I (I.prod 𝓘(𝕜, F)) n (fun x ↦ ⟨x, ∑ i ∈ s, t i x⟩) u x₀
hs : ∀ i_1 ∈ insert i s, ContMDiffWithinAt I (I.prod 𝓘(𝕜, F)) n (fun x ↦ ⟨x, t i_1 x⟩) u x₀
⊢ ContMDiffWithinAt I (I.prod 𝓘(𝕜, F)) n (fun x ↦ ⟨x, ∑ i ∈ insert i s, t i x⟩) u x₀

Tactic:
simp only [Finset.sum_insert hi]

Goal state after:
case insert
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
x₀ : M
ι : Type u_7
t : ι → (x : M) → V x
i : ι
s : Finset ι
hi : i ∉ s
h :
  (∀ i ∈ s, ContMDiffWithinAt I (I.prod 𝓘(𝕜, F)) n (fun x ↦ ⟨x, t i x⟩) u x₀) →
    ContMDiffWithinAt I (I.prod 𝓘(𝕜, F)) n (fun x ↦ ⟨x, ∑ i ∈ s, t i x⟩) u x₀
hs : ∀ i_1 ∈ insert i s, ContMDiffWithinAt I (I.prod 𝓘(𝕜, F)) n (fun x ↦ ⟨x, t i_1 x⟩) u x₀
⊢ ContMDiffWithinAt I (I.prod 𝓘(𝕜, F)) n (fun x ↦ ⟨x, t i x + ∑ i ∈ s, t i x⟩) u x₀