mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.Geometry.Manifold.IsManifold.ExtChartAt

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
M : Type u_3
H : Type u_4
inst✝⁴ : NontriviallyNormedField 𝕜
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace 𝕜 E
inst✝¹ : TopologicalSpace H
inst✝ : TopologicalSpace M
f : OpenPartialHomeomorph M H
I : ModelWithCorners 𝕜 E H
s : Set M
x : M
hs : s ⊆ f.source
hx : x ∈ f.source
⊢ 𝓝[↑(f.extend I) '' s] ↑(f.extend I) x = 𝓝[↑(f.extend I) '' s] ↑(f.extend I) x

Tactic:
]

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

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
M : Type u_3
H : Type u_4
inst✝⁴ : NontriviallyNormedField 𝕜
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace 𝕜 E
inst✝¹ : TopologicalSpace H
inst✝ : TopologicalSpace M
I : ModelWithCorners 𝕜 E H
e e' : OpenPartialHomeomorph M H
⊢ (extendCoordChange e e').symm.source = ↑I '' (e.symm ≫ₕ e').symm.source

Tactic:
]

Goal state after:
𝕜 : Type u_1
E : Type u_2
M : Type u_3
H : Type u_4
inst✝⁴ : NontriviallyNormedField 𝕜
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace 𝕜 E
inst✝¹ : TopologicalSpace H
inst✝ : TopologicalSpace M
I : ModelWithCorners 𝕜 E H
e e' : OpenPartialHomeomorph M H
⊢ (extendCoordChange e e').symm.source = ↑I '' (e.symm ≫ₕ e').symm.source
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
f : OpenPartialHomeomorph M' H'
P : (H → H') → Set H → H → Prop
g : M → M'
s : Set M
hG : G.LocalInvariantProp G' P
he : e ∈ maximalAtlas M G
hf : f ∈ maximalAtlas M' G'
h : LiftPropOn P g s
y : H
hy : y ∈ e.target ∩ ↑e.symm ⁻¹' (s ∩ g ⁻¹' f.source)
⊢ y = y

Tactic:
]

Goal state after:
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 ∘ id) s x
⊢ ContinuousWithinAt f s x

Tactic:
]

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 ∘ id) s x
⊢ ContinuousWithinAt f s x
Module: Mathlib.Geometry.Manifold.IsManifold.ExtChartAt

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
M : Type u_3
H : Type u_4
inst✝⁴ : NontriviallyNormedField 𝕜
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace 𝕜 E
inst✝¹ : TopologicalSpace H
inst✝ : TopologicalSpace M
I : ModelWithCorners 𝕜 E H
e e' : OpenPartialHomeomorph M H
x : E
hx : x ∈ ↑I '' (e.symm ≫ₕ e').source
⊢ ↑I '' (e.symm ≫ₕ e').source ∈ 𝓝[range ↑I] x

Tactic:
]

Goal state after:
𝕜 : Type u_1
E : Type u_2
M : Type u_3
H : Type u_4
inst✝⁴ : NontriviallyNormedField 𝕜
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace 𝕜 E
inst✝¹ : TopologicalSpace H
inst✝ : TopologicalSpace M
I : ModelWithCorners 𝕜 E H
e e' : OpenPartialHomeomorph M H
x : E
hx : x ∈ ↑I '' (e.symm ≫ₕ e').source
⊢ ↑I '' (e.symm ≫ₕ e').source ∈ 𝓝[range ↑I] x
Module: Mathlib.Geometry.Manifold.IsManifold.Basic

Goal state before:
pp: case refine_2
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 s (↑I.symm (↑I x))
⊢ ContinuousWithinAt (f ∘ ↑I.symm) (↑I.symm ⁻¹' s ∩ range ↑I) (↑I x)

Tactic:
]

Goal state after:
case refine_2
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 s (↑I.symm (↑I x))
⊢ ContinuousWithinAt (f ∘ ↑I.symm) (↑I.symm ⁻¹' s ∩ range ↑I) (↑I x)
Module: Mathlib.Geometry.Manifold.IsManifold.ExtChartAt

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
M : Type u_3
H : Type u_4
inst✝⁴ : NontriviallyNormedField 𝕜
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace 𝕜 E
inst✝¹ : TopologicalSpace H
inst✝ : TopologicalSpace M
I : ModelWithCorners 𝕜 E H
e e' : OpenPartialHomeomorph M H
x : M
hxe : x ∈ e.source
hxe' : x ∈ e'.source
⊢ ↑(e.extend I) x ∈ ↑(e.extend I) '' (e.source ∩ e'.source)

Tactic:
]

Goal state after:
𝕜 : Type u_1
E : Type u_2
M : Type u_3
H : Type u_4
inst✝⁴ : NontriviallyNormedField 𝕜
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace 𝕜 E
inst✝¹ : TopologicalSpace H
inst✝ : TopologicalSpace M
I : ModelWithCorners 𝕜 E H
e e' : OpenPartialHomeomorph M H
x : M
hxe : x ∈ e.source
hxe' : x ∈ e'.source
⊢ ↑(e.extend I) x ∈ ↑(e.extend I) '' (e.source ∩ e'.source)
Module: Mathlib.Geometry.Manifold.IsManifold.Basic

Goal state before:
pp: 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
inst✝ : LocallyCompactSpace E
I : ModelWithCorners 𝕜 E H
x : H
⊢ (map (↑I.symm) (𝓝[range ↑I] ↑I x)).HasBasis (fun s ↦ s ∈ 𝓝 (↑I x) ∧ IsCompact s) fun s ↦ ↑I.symm '' (s ∩ range ↑I)

Tactic:
]

Goal state after:
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
inst✝ : LocallyCompactSpace E
I : ModelWithCorners 𝕜 E H
x : H
⊢ (map (↑I.symm) (𝓝[range ↑I] ↑I x)).HasBasis (fun s ↦ s ∈ 𝓝 (↑I x) ∧ IsCompact s) fun s ↦ ↑I.symm '' (s ∩ range ↑I)
Module: Mathlib.Geometry.Manifold.IsManifold.ExtChartAt

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
M : Type u_3
H : Type u_4
inst✝⁴ : NontriviallyNormedField 𝕜
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace 𝕜 E
inst✝¹ : TopologicalSpace H
inst✝ : TopologicalSpace M
I : ModelWithCorners 𝕜 E H
e e' : OpenPartialHomeomorph M H
⊢ UniqueDiffOn 𝕜 (↑I.symm ⁻¹' (e.symm ≫ₕ e').source ∩ range ↑I)

Tactic:
]

Goal state after:
𝕜 : Type u_1
E : Type u_2
M : Type u_3
H : Type u_4
inst✝⁴ : NontriviallyNormedField 𝕜
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace 𝕜 E
inst✝¹ : TopologicalSpace H
inst✝ : TopologicalSpace M
I : ModelWithCorners 𝕜 E H
e e' : OpenPartialHomeomorph M H
⊢ UniqueDiffOn 𝕜 (↑I.symm ⁻¹' (e.symm ≫ₕ e').source ∩ range ↑I)
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'
P : (H → H') → Set H → H → Prop
g : M → M'
s t : Set M
x : M
hG : G.LocalInvariantProp G' P
ht : t ∈ 𝓝[s] x
⊢ ContinuousWithinAt g s x ∧ P (↑(chartAt H' (g x)) ∘ g ∘ ↑(chartAt H x).symm) ?m.94 (↑(chartAt H x) x) ↔
    ContinuousWithinAt g s x ∧
      P (↑(chartAt H' (g x)) ∘ g ∘ ↑(chartAt H x).symm) (↑(chartAt H x).symm ⁻¹' s) (↑(chartAt H x) x)

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'
P : (H → H') → Set H → H → Prop
g : M → M'
s t : Set M
x : M
hG : G.LocalInvariantProp G' P
ht : t ∈ 𝓝[s] x
⊢ ↑(chartAt H x).symm ⁻¹' (s ∩ t) =ᶠ[𝓝 (↑(chartAt H x) x)] ?m.94

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'
P : (H → H') → Set H → H → Prop
g : M → M'
s t : Set M
x : M
hG : G.LocalInvariantProp G' P
ht : t ∈ 𝓝[s] x
⊢ Set H

Tactic:
]

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'
P : (H → H') → Set H → H → Prop
g : M → M'
s t : Set M
x : M
hG : G.LocalInvariantProp G' P
ht : t ∈ 𝓝[s] x
⊢ ↑(chartAt H x).symm ⁻¹' (s ∩ t) =ᶠ[𝓝 (↑(chartAt H x) x)] ↑(chartAt H x).symm ⁻¹' s
Module: Mathlib.Geometry.Manifold.IsManifold.ExtChartAt

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
M : Type u_3
H : Type u_4
inst✝⁴ : NontriviallyNormedField 𝕜
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace 𝕜 E
inst✝¹ : TopologicalSpace H
inst✝ : TopologicalSpace M
I : ModelWithCorners 𝕜 E H
e e' : OpenPartialHomeomorph M H
⊢ UniqueDiffOn 𝕜 (extendCoordChange e' e).source

Tactic:
]

Goal state after:
𝕜 : Type u_1
E : Type u_2
M : Type u_3
H : Type u_4
inst✝⁴ : NontriviallyNormedField 𝕜
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace 𝕜 E
inst✝¹ : TopologicalSpace H
inst✝ : TopologicalSpace M
I : ModelWithCorners 𝕜 E H
e e' : OpenPartialHomeomorph M H
⊢ UniqueDiffOn 𝕜 (extendCoordChange e' e).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
n : WithTop ℕ∞
I : ModelWithCorners 𝕜 E H
e e' : OpenPartialHomeomorph M H
inst✝ : ChartedSpace H M
he : e ∈ maximalAtlas I n M
he' : e' ∈ maximalAtlas I n M
⊢ ContDiffOn 𝕜 n (↑(extendCoordChange e e')) (↑I.symm ⁻¹' (e.symm ≫ₕ e').source ∩ range ↑I)

Tactic:
]

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
n : WithTop ℕ∞
I : ModelWithCorners 𝕜 E H
e e' : OpenPartialHomeomorph M H
inst✝ : ChartedSpace H M
he : e ∈ maximalAtlas I n M
he' : e' ∈ maximalAtlas I n M
⊢ ContDiffOn 𝕜 n (↑(extendCoordChange e e')) (↑I.symm ⁻¹' (e.symm ≫ₕ e').source ∩ range ↑I)
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'
P : (H → H') → Set H → H → Prop
g : M → M'
s t : Set M
x : M
hG : G.LocalInvariantProp G' P
hu : s =ᶠ[𝓝 x] t
⊢ LiftPropWithinAt P g (s ∩ {x | (fun x ↦ s x = t x) x}) x = LiftPropWithinAt P g (t ∩ {x | (fun x ↦ s x = t x) x}) x

Tactic:
]

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'
P : (H → H') → Set H → H → Prop
g : M → M'
s t : Set M
x : M
hG : G.LocalInvariantProp G' P
hu : s =ᶠ[𝓝 x] t
⊢ LiftPropWithinAt P g (s ∩ {x | (fun x ↦ s x = t x) x}) x = LiftPropWithinAt P g (t ∩ {x | (fun x ↦ s x = t x) x}) x
Module: Mathlib.Geometry.Manifold.IsManifold.Basic

Goal state before:
pp: 𝕜 : Type u
inst✝⁶ : NontriviallyNormedField 𝕜
E : Type v
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace 𝕜 E
H : Type w
inst✝³ : TopologicalSpace H
I : ModelWithCorners 𝕜 E H
E' : Type v'
inst✝² : NormedAddCommGroup E'
inst✝¹ : NormedSpace 𝕜 E'
H' : Type w'
inst✝ : TopologicalSpace H'
I' : ModelWithCorners 𝕜 E' H'
this : (range fun x ↦ (↑I x.1, ↑I' x.2)) = range (Prod.map ↑I ↑I')
⊢ if h : IsRCLikeNormedField 𝕜 then Convex ℝ (range ↑I ×ˢ range ↑I') else range ↑I ×ˢ range ↑I' = univ

Tactic:
]

Goal state after:
𝕜 : Type u
inst✝⁶ : NontriviallyNormedField 𝕜
E : Type v
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace 𝕜 E
H : Type w
inst✝³ : TopologicalSpace H
I : ModelWithCorners 𝕜 E H
E' : Type v'
inst✝² : NormedAddCommGroup E'
inst✝¹ : NormedSpace 𝕜 E'
H' : Type w'
inst✝ : TopologicalSpace H'
I' : ModelWithCorners 𝕜 E' H'
this : (range fun x ↦ (↑I x.1, ↑I' x.2)) = range (Prod.map ↑I ↑I')
⊢ if h : IsRCLikeNormedField 𝕜 then Convex ℝ (range ↑I ×ˢ range ↑I') else range ↑I ×ˢ range ↑I' = univ
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
n : WithTop ℕ∞
I : ModelWithCorners 𝕜 E H
e e' : OpenPartialHomeomorph M H
inst✝ : ChartedSpace H M
he : e ∈ maximalAtlas I n M
he' : e' ∈ maximalAtlas I n M
x : E
hx : x ∈ ↑I '' (e.symm ≫ₕ e').source
⊢ ↑I '' (e.symm ≫ₕ e').source ∈ 𝓝[range ↑I] x

Tactic:
]

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
n : WithTop ℕ∞
I : ModelWithCorners 𝕜 E H
e e' : OpenPartialHomeomorph M H
inst✝ : ChartedSpace H M
he : e ∈ maximalAtlas I n M
he' : e' ∈ maximalAtlas I n M
x : E
hx : x ∈ ↑I '' (e.symm ≫ₕ e').source
⊢ ↑I '' (e.symm ≫ₕ e').source ∈ 𝓝[range ↑I] x
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'
P : (H → H') → Set H → H → Prop
g : M → M'
s : Set M
x : M
hG : G.LocalInvariantProp G' P
h : LiftPropWithinAt P g univ x
hs : s ∈ 𝓝 x
⊢ LiftPropAt P g x

Tactic:
]

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'
P : (H → H') → Set H → H → Prop
g : M → M'
s : Set M
x : M
hG : G.LocalInvariantProp G' P
h : LiftPropWithinAt P g univ x
hs : s ∈ 𝓝 x
⊢ LiftPropAt P g x
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'
P : (H → H') → Set H → H → Prop
g : M → M'
s : Set M
x : M
hG : G.LocalInvariantProp G' P
h : LiftPropAt P g x
hs : s ∈ 𝓝 x
⊢ LiftPropWithinAt P g univ x

Tactic:
]

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'
P : (H → H') → Set H → H → Prop
g : M → M'
s : Set M
x : M
hG : G.LocalInvariantProp G' P
h : LiftPropAt P g x
hs : s ∈ 𝓝 x
⊢ LiftPropWithinAt P g univ x
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
n : WithTop ℕ∞
I : ModelWithCorners 𝕜 E H
e e' : OpenPartialHomeomorph M H
inst✝ : ChartedSpace H M
he : e ∈ maximalAtlas I n M
he' : e' ∈ maximalAtlas I n M
x : M
hxe : x ∈ e.source
hxe' : x ∈ e'.source
⊢ ↑(e.extend I) x ∈ ↑(e.extend I) '' (e.source ∩ e'.source)

Tactic:
]

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
n : WithTop ℕ∞
I : ModelWithCorners 𝕜 E H
e e' : OpenPartialHomeomorph M H
inst✝ : ChartedSpace H M
he : e ∈ maximalAtlas I n M
he' : e' ∈ maximalAtlas I n M
x : M
hxe : x ∈ e.source
hxe' : x ∈ e'.source
⊢ ↑(e.extend I) x ∈ ↑(e.extend I) '' (e.source ∩ 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
δ : (E →L[ℝ] E) → ℝ≥0
hδ :
  ∀ (A : E →L[ℝ] E),
    0 < δ A ∧
      (∀ (B : E →L[ℝ] E), ‖B - A‖ ≤ ↑(δ A) → |B.det - A.det| ≤ ↑ε) ∧
        ∀ (t : Set E) (g : E → E),
          ApproximatesLinearOn g A t (δ A) → ENNReal.ofReal |A.det| * μ t ≤ μ (g '' t) + ↑ε * μ t
t : ℕ → Set E
A : ℕ → E →L[ℝ] E
t_disj : Pairwise (Disjoint on t)
t_meas : ∀ (n : ℕ), MeasurableSet (t n)
t_cover : s ⊆ ⋃ n, t n
ht : ∀ (n : ℕ), ApproximatesLinearOn f (A n) (s ∩ t n) (δ (A n))
⊢ s = s ∩ ⋃ i, t i

Tactic:
]

Goal state after:
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
δ : (E →L[ℝ] E) → ℝ≥0
hδ :
  ∀ (A : E →L[ℝ] E),
    0 < δ A ∧
      (∀ (B : E →L[ℝ] E), ‖B - A‖ ≤ ↑(δ A) → |B.det - A.det| ≤ ↑ε) ∧
        ∀ (t : Set E) (g : E → E),
          ApproximatesLinearOn g A t (δ A) → ENNReal.ofReal |A.det| * μ t ≤ μ (g '' t) + ↑ε * μ t
t : ℕ → Set E
A : ℕ → E →L[ℝ] E
t_disj : Pairwise (Disjoint on t)
t_meas : ∀ (n : ℕ), MeasurableSet (t n)
t_cover : s ⊆ ⋃ n, t n
ht : ∀ (n : ℕ), ApproximatesLinearOn f (A n) (s ∩ t n) (δ (A n))
⊢ s = s ∩ ⋃ i, t i
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'
P : (H → H') → Set H → H → Prop
g : M → M'
s : Set M
hG : G.LocalInvariantProp G' P
h : ∀ x ∈ s, ∃ u, IsOpen[inst✝⁴] u ∧ x ∈ u ∧ LiftPropOn P g (s ∩ u)
x : M
hx : x ∈ s
u : Set M
u_open : IsOpen[inst✝⁴] u
xu : x ∈ u
hu : LiftPropOn P g (s ∩ u)
this : LiftPropWithinAt P g s x
⊢ LiftPropWithinAt P g s x

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'
P : (H → H') → Set H → H → Prop
g : M → M'
s : Set M
hG : G.LocalInvariantProp G' P
h : ∀ x ∈ s, ∃ u, IsOpen[inst✝⁴] u ∧ x ∈ u ∧ LiftPropOn P g (s ∩ u)
x : M
hx : x ∈ s
u : Set M
u_open : IsOpen[inst✝⁴] u
xu : x ∈ u
hu : LiftPropOn P g (s ∩ u)
this : LiftPropWithinAt P g (s ∩ u) x
⊢ u ∈ 𝓝 x

Tactic:
]

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'
P : (H → H') → Set H → H → Prop
g : M → M'
s : Set M
hG : G.LocalInvariantProp G' P
h : ∀ x ∈ s, ∃ u, IsOpen[inst✝⁴] u ∧ x ∈ u ∧ LiftPropOn P g (s ∩ u)
x : M
hx : x ∈ s
u : Set M
u_open : IsOpen[inst✝⁴] u
xu : x ∈ u
hu : LiftPropOn P g (s ∩ u)
this : LiftPropWithinAt P g s x
⊢ LiftPropWithinAt P g s x

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'
P : (H → H') → Set H → H → Prop
g : M → M'
s : Set M
hG : G.LocalInvariantProp G' P
h : ∀ x ∈ s, ∃ u, IsOpen[inst✝⁴] u ∧ x ∈ u ∧ LiftPropOn P g (s ∩ u)
x : M
hx : x ∈ s
u : Set M
u_open : IsOpen[inst✝⁴] u
xu : x ∈ u
hu : LiftPropOn P g (s ∩ u)
this : LiftPropWithinAt P g (s ∩ u) x
⊢ u ∈ 𝓝 x
Module: Mathlib.Geometry.Manifold.IsManifold.Basic

Goal state before:
pp: 𝕜 : Type u
inst✝⁴ : NontriviallyNormedField 𝕜
ι : Type v
inst✝³ : Fintype ι
E : ι → Type w
inst✝² : (i : ι) → NormedAddCommGroup (E i)
inst✝¹ : (i : ι) → NormedSpace 𝕜 (E i)
H : ι → Type u'
inst✝ : (i : ι) → TopologicalSpace (H i)
I : (i : ι) → ModelWithCorners 𝕜 (E i) (H i)
⊢ if h : IsRCLikeNormedField 𝕜 then Convex ℝ (univ.pi fun i ↦ range ↑(I i).toPartialEquiv)
  else (univ.pi fun i ↦ range ↑(I i).toPartialEquiv) = univ

Tactic:
]

Goal state after:
𝕜 : Type u
inst✝⁴ : NontriviallyNormedField 𝕜
ι : Type v
inst✝³ : Fintype ι
E : ι → Type w
inst✝² : (i : ι) → NormedAddCommGroup (E i)
inst✝¹ : (i : ι) → NormedSpace 𝕜 (E i)
H : ι → Type u'
inst✝ : (i : ι) → TopologicalSpace (H i)
I : (i : ι) → ModelWithCorners 𝕜 (E i) (H i)
⊢ if h : IsRCLikeNormedField 𝕜 then Convex ℝ (univ.pi fun i ↦ range ↑(I i).toPartialEquiv)
  else (univ.pi fun i ↦ range ↑(I i).toPartialEquiv) = univ
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'
P : (H → H') → Set H → H → Prop
g : M → M'
hG : G.LocalInvariantProp G' P
h : ∀ (x : M), ∃ u, IsOpen[inst✝⁴] u ∧ x ∈ u ∧ LiftPropOn P g u
⊢ LiftPropOn P g univ

Tactic:
]

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'
P : (H → H') → Set H → H → Prop
g : M → M'
hG : G.LocalInvariantProp G' P
h : ∀ (x : M), ∃ u, IsOpen[inst✝⁴] u ∧ x ∈ u ∧ LiftPropOn P g u
⊢ LiftPropOn P g univ
Module: Mathlib.Geometry.Manifold.IsManifold.Basic

Goal state before:
pp: 𝕜 : Type u
inst✝⁴ : NontriviallyNormedField 𝕜
ι : Type v
inst✝³ : Fintype ι
E : ι → Type w
inst✝² : (i : ι) → NormedAddCommGroup (E i)
inst✝¹ : (i : ι) → NormedSpace 𝕜 (E i)
H : ι → Type u'
inst✝ : (i : ι) → TopologicalSpace (H i)
I : (i : ι) → ModelWithCorners 𝕜 (E i) (H i)
⊢ (interior (univ.pi fun i ↦ range ↑(I i).toPartialEquiv)).Nonempty

Tactic:
]

Goal state after:
𝕜 : Type u
inst✝⁴ : NontriviallyNormedField 𝕜
ι : Type v
inst✝³ : Fintype ι
E : ι → Type w
inst✝² : (i : ι) → NormedAddCommGroup (E i)
inst✝¹ : (i : ι) → NormedSpace 𝕜 (E i)
H : ι → Type u'
inst✝ : (i : ι) → TopologicalSpace (H i)
I : (i : ι) → ModelWithCorners 𝕜 (E i) (H i)
⊢ (interior (univ.pi fun i ↦ range ↑(I i).toPartialEquiv)).Nonempty