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

Goal state before:
pp: case inr
H : Type u
M : Type u_2
M' : Type u_3
inst✝² : TopologicalSpace H
inst✝¹ : TopologicalSpace M
inst✝ : TopologicalSpace M'
cm : ChartedSpace H M
cm' : ChartedSpace H M'
h : Nonempty H
e : OpenPartialHomeomorph (M ⊕ M') H
x : OpenPartialHomeomorph M' H
hx : x ∈ ChartedSpace.atlas
hxe : (fun e ↦ e.lift_openEmbedding ⋯) x = e
⊢ (∃ f ∈ atlas H M, e = f.lift_openEmbedding ⋯) ∨ ∃ f' ∈ atlas H M', e = f'.lift_openEmbedding ⋯

Tactic:
·

Goal state after:
case inr
H : Type u
M : Type u_2
M' : Type u_3
inst✝² : TopologicalSpace H
inst✝¹ : TopologicalSpace M
inst✝ : TopologicalSpace M'
cm : ChartedSpace H M
cm' : ChartedSpace H M'
h : Nonempty H
e : OpenPartialHomeomorph (M ⊕ M') H
x : OpenPartialHomeomorph M' H
hx : x ∈ ChartedSpace.atlas
hxe : (fun e ↦ e.lift_openEmbedding ⋯) x = e
⊢ (∃ f ∈ atlas H M, e = f.lift_openEmbedding ⋯) ∨ ∃ f' ∈ atlas H M', e = f'.lift_openEmbedding ⋯
Module: Mathlib.Geometry.Manifold.StructureGroupoid

Goal state before:
pp: case inr.inr
H✝ : Type u_1
inst✝¹ : TopologicalSpace H✝
H : Type u_2
inst✝ : TopologicalSpace H
e : OpenPartialHomeomorph H H
he : ∀ x ∈ e.source, ∃ s, IsOpen[inst✝] s ∧ x ∈ s ∧ e.restr s ∈ {OpenPartialHomeomorph.refl H} ∪ {e | e.source = ∅}
x : H
hx : x ∈ e.source
s : Set H
open_s : IsOpen[inst✝] s
xs : x ∈ s
x's : x ∈ (e.restr s).source
hs : e.restr s ∈ {e | e.source = ∅}
⊢ e ∈ {OpenPartialHomeomorph.refl H}

Tactic:
·

Goal state after:
case inr.inr
H✝ : Type u_1
inst✝¹ : TopologicalSpace H✝
H : Type u_2
inst✝ : TopologicalSpace H
e : OpenPartialHomeomorph H H
he : ∀ x ∈ e.source, ∃ s, IsOpen[inst✝] s ∧ x ∈ s ∧ e.restr s ∈ {OpenPartialHomeomorph.refl H} ∪ {e | e.source = ∅}
x : H
hx : x ∈ e.source
s : Set H
open_s : IsOpen[inst✝] s
xs : x ∈ s
x's : x ∈ (e.restr s).source
hs : e.restr s ∈ {e | e.source = ∅}
⊢ e ∈ {OpenPartialHomeomorph.refl H}
Module: Mathlib.Geometry.Manifold.StructureGroupoid

Goal state before:
pp: case inl
H✝ : Type u_1
inst✝¹ : TopologicalSpace H✝
H : Type u_2
inst✝ : TopologicalSpace H
e e' : OpenPartialHomeomorph H H
he'e : e' ≈ e
he : e ∈ {OpenPartialHomeomorph.refl H}
⊢ e' ∈ {OpenPartialHomeomorph.refl H} ∪ {e | e.source = ∅}

pp: case inr
H✝ : Type u_1
inst✝¹ : TopologicalSpace H✝
H : Type u_2
inst✝ : TopologicalSpace H
e e' : OpenPartialHomeomorph H H
he'e : e' ≈ e
he : e ∈ {e | e.source = ∅}
⊢ e' ∈ {OpenPartialHomeomorph.refl H} ∪ {e | e.source = ∅}

Tactic:
·

Goal state after:
case inl
H✝ : Type u_1
inst✝¹ : TopologicalSpace H✝
H : Type u_2
inst✝ : TopologicalSpace H
e e' : OpenPartialHomeomorph H H
he'e : e' ≈ e
he : e ∈ {OpenPartialHomeomorph.refl H}
⊢ e' ∈ {OpenPartialHomeomorph.refl H} ∪ {e | e.source = ∅}
Module: Mathlib.Geometry.Manifold.StructureGroupoid

Goal state before:
pp: case inr
H✝ : Type u_1
inst✝¹ : TopologicalSpace H✝
H : Type u_2
inst✝ : TopologicalSpace H
e e' : OpenPartialHomeomorph H H
he'e : e' ≈ e
he : e ∈ {e | e.source = ∅}
⊢ e' ∈ {OpenPartialHomeomorph.refl H} ∪ {e | e.source = ∅}

Tactic:
·

Goal state after:
case inr
H✝ : Type u_1
inst✝¹ : TopologicalSpace H✝
H : Type u_2
inst✝ : TopologicalSpace H
e e' : OpenPartialHomeomorph H H
he'e : e' ≈ e
he : e ∈ {e | e.source = ∅}
⊢ e' ∈ {OpenPartialHomeomorph.refl H} ∪ {e | e.source = ∅}
Module: Mathlib.Geometry.Manifold.HasGroupoid

Goal state before:
pp: case he
H : Type u
H' : Type u_1
M : Type u_2
M' : Type u_3
M'' : Type u_4
inst✝⁴ : TopologicalSpace H
inst✝³ : TopologicalSpace M
inst✝² : ChartedSpace H M
G : StructureGroupoid H
inst✝¹ : HasGroupoid M G
s : Opens M
inst✝ : ClosedUnderRestriction G
e e' : OpenPartialHomeomorph (↥s) H
w✝¹ : Set (OpenPartialHomeomorph (↥s) H)
x : ↥s
he : e = (chartAt H ↑x).subtypeRestr ⋯
hc : (fun x ↦ {(chartAt H ↑x).subtypeRestr ⋯}) x = w✝¹
w✝ : Set (OpenPartialHomeomorph (↥s) H)
x' : ↥s
he' : e' = (chartAt H ↑x').subtypeRestr ⋯
hc' : (fun x ↦ {(chartAt H ↑x).subtypeRestr ⋯}) x' = w✝
⊢ (chartAt H ↑x).symm ≫ₕ chartAt H ↑x' ∈ G

pp: case hs
H : Type u
H' : Type u_1
M : Type u_2
M' : Type u_3
M'' : Type u_4
inst✝⁴ : TopologicalSpace H
inst✝³ : TopologicalSpace M
inst✝² : ChartedSpace H M
G : StructureGroupoid H
inst✝¹ : HasGroupoid M G
s : Opens M
inst✝ : ClosedUnderRestriction G
e e' : OpenPartialHomeomorph (↥s) H
w✝¹ : Set (OpenPartialHomeomorph (↥s) H)
x : ↥s
he : e = (chartAt H ↑x).subtypeRestr ⋯
hc : (fun x ↦ {(chartAt H ↑x).subtypeRestr ⋯}) x = w✝¹
w✝ : Set (OpenPartialHomeomorph (↥s) H)
x' : ↥s
he' : e' = (chartAt H ↑x').subtypeRestr ⋯
hc' : (fun x ↦ {(chartAt H ↑x).subtypeRestr ⋯}) x' = w✝
⊢ IsOpen[inst✝⁴] ((chartAt H ↑x).target ∩ ↑(chartAt H ↑x).symm ⁻¹' ↑s)

Tactic:
·

Goal state after:
case he
H : Type u
H' : Type u_1
M : Type u_2
M' : Type u_3
M'' : Type u_4
inst✝⁴ : TopologicalSpace H
inst✝³ : TopologicalSpace M
inst✝² : ChartedSpace H M
G : StructureGroupoid H
inst✝¹ : HasGroupoid M G
s : Opens M
inst✝ : ClosedUnderRestriction G
e e' : OpenPartialHomeomorph (↥s) H
w✝¹ : Set (OpenPartialHomeomorph (↥s) H)
x : ↥s
he : e = (chartAt H ↑x).subtypeRestr ⋯
hc : (fun x ↦ {(chartAt H ↑x).subtypeRestr ⋯}) x = w✝¹
w✝ : Set (OpenPartialHomeomorph (↥s) H)
x' : ↥s
he' : e' = (chartAt H ↑x').subtypeRestr ⋯
hc' : (fun x ↦ {(chartAt H ↑x).subtypeRestr ⋯}) x' = w✝
⊢ (chartAt H ↑x).symm ≫ₕ chartAt H ↑x' ∈ G
Module: Mathlib.Geometry.Manifold.HasGroupoid

Goal state before:
pp: case hs
H : Type u
H' : Type u_1
M : Type u_2
M' : Type u_3
M'' : Type u_4
inst✝⁴ : TopologicalSpace H
inst✝³ : TopologicalSpace M
inst✝² : ChartedSpace H M
G : StructureGroupoid H
inst✝¹ : HasGroupoid M G
s : Opens M
inst✝ : ClosedUnderRestriction G
e e' : OpenPartialHomeomorph (↥s) H
w✝¹ : Set (OpenPartialHomeomorph (↥s) H)
x : ↥s
he : e = (chartAt H ↑x).subtypeRestr ⋯
hc : (fun x ↦ {(chartAt H ↑x).subtypeRestr ⋯}) x = w✝¹
w✝ : Set (OpenPartialHomeomorph (↥s) H)
x' : ↥s
he' : e' = (chartAt H ↑x').subtypeRestr ⋯
hc' : (fun x ↦ {(chartAt H ↑x).subtypeRestr ⋯}) x' = w✝
⊢ IsOpen[inst✝⁴] ((chartAt H ↑x).target ∩ ↑(chartAt H ↑x).symm ⁻¹' ↑s)

Tactic:
·

Goal state after:
case hs
H : Type u
H' : Type u_1
M : Type u_2
M' : Type u_3
M'' : Type u_4
inst✝⁴ : TopologicalSpace H
inst✝³ : TopologicalSpace M
inst✝² : ChartedSpace H M
G : StructureGroupoid H
inst✝¹ : HasGroupoid M G
s : Opens M
inst✝ : ClosedUnderRestriction G
e e' : OpenPartialHomeomorph (↥s) H
w✝¹ : Set (OpenPartialHomeomorph (↥s) H)
x : ↥s
he : e = (chartAt H ↑x).subtypeRestr ⋯
hc : (fun x ↦ {(chartAt H ↑x).subtypeRestr ⋯}) x = w✝¹
w✝ : Set (OpenPartialHomeomorph (↥s) H)
x' : ↥s
he' : e' = (chartAt H ↑x').subtypeRestr ⋯
hc' : (fun x ↦ {(chartAt H ↑x).subtypeRestr ⋯}) x' = w✝
⊢ IsOpen[inst✝⁴] ((chartAt H ↑x).target ∩ ↑(chartAt H ↑x).symm ⁻¹' ↑s)
Module: Mathlib.Geometry.Manifold.StructureGroupoid

Goal state before:
pp: case inl
H : Type u_1
inst✝ : TopologicalSpace H
u : StructureGroupoid H
f : OpenPartialHomeomorph H H
hf✝ : f ∈ (idGroupoid H).members
hf : f = OpenPartialHomeomorph.refl H
⊢ f ∈ u.members

pp: case inr
H : Type u_1
inst✝ : TopologicalSpace H
u : StructureGroupoid H
f : OpenPartialHomeomorph H H
hf✝ : f ∈ (idGroupoid H).members
hf : f.source = ∅
⊢ f ∈ u.members

Tactic:
·

Goal state after:
case inl
H : Type u_1
inst✝ : TopologicalSpace H
u : StructureGroupoid H
f : OpenPartialHomeomorph H H
hf✝ : f ∈ (idGroupoid H).members
hf : f = OpenPartialHomeomorph.refl H
⊢ f ∈ u.members
Module: Mathlib.Geometry.Manifold.StructureGroupoid

Goal state before:
pp: case inr
H : Type u_1
inst✝ : TopologicalSpace H
u : StructureGroupoid H
f : OpenPartialHomeomorph H H
hf✝ : f ∈ (idGroupoid H).members
hf : f.source = ∅
⊢ f ∈ u.members

Tactic:
·

Goal state after:
case inr
H : Type u_1
inst✝ : TopologicalSpace H
u : StructureGroupoid H
f : OpenPartialHomeomorph H H
hf✝ : f ∈ (idGroupoid H).members
hf : f.source = ∅
⊢ f ∈ u.members
Module: Mathlib.Geometry.Manifold.StructureGroupoid

Goal state before:
pp: case left
H : Type u_1
inst✝ : TopologicalSpace H
PG : Pregroupoid H
e e' : OpenPartialHomeomorph H H
he : e ∈ {e | PG.property (↑e) e.source ∧ PG.property (↑e.symm) e.target}
he' : e' ∈ {e | PG.property (↑e) e.source ∧ PG.property (↑e.symm) e.target}
⊢ PG.property (↑(e ≫ₕ e')) (e ≫ₕ e').source

pp: case right
H : Type u_1
inst✝ : TopologicalSpace H
PG : Pregroupoid H
e e' : OpenPartialHomeomorph H H
he : e ∈ {e | PG.property (↑e) e.source ∧ PG.property (↑e.symm) e.target}
he' : e' ∈ {e | PG.property (↑e) e.source ∧ PG.property (↑e.symm) e.target}
⊢ PG.property (↑(e ≫ₕ e').symm) (e ≫ₕ e').target

Tactic:
·

Goal state after:
case left
H : Type u_1
inst✝ : TopologicalSpace H
PG : Pregroupoid H
e e' : OpenPartialHomeomorph H H
he : e ∈ {e | PG.property (↑e) e.source ∧ PG.property (↑e.symm) e.target}
he' : e' ∈ {e | PG.property (↑e) e.source ∧ PG.property (↑e.symm) e.target}
⊢ PG.property (↑(e ≫ₕ e')) (e ≫ₕ e').source
Module: Mathlib.Geometry.Manifold.StructureGroupoid

Goal state before:
pp: case right
H : Type u_1
inst✝ : TopologicalSpace H
PG : Pregroupoid H
e e' : OpenPartialHomeomorph H H
he : e ∈ {e | PG.property (↑e) e.source ∧ PG.property (↑e.symm) e.target}
he' : e' ∈ {e | PG.property (↑e) e.source ∧ PG.property (↑e.symm) e.target}
⊢ PG.property (↑(e ≫ₕ e').symm) (e ≫ₕ e').target

Tactic:
·

Goal state after:
case right
H : Type u_1
inst✝ : TopologicalSpace H
PG : Pregroupoid H
e e' : OpenPartialHomeomorph H H
he : e ∈ {e | PG.property (↑e) e.source ∧ PG.property (↑e.symm) e.target}
he' : e' ∈ {e | PG.property (↑e) e.source ∧ PG.property (↑e.symm) e.target}
⊢ PG.property (↑(e ≫ₕ e').symm) (e ≫ₕ e').target
Module: Mathlib.Geometry.Manifold.StructureGroupoid

Goal state before:
pp: case left
H : Type u_1
inst✝ : TopologicalSpace H
PG : Pregroupoid H
e : OpenPartialHomeomorph H H
he :
  ∀ x ∈ e.source,
    ∃ s, IsOpen[inst✝] s ∧ x ∈ s ∧ e.restr s ∈ {e | PG.property (↑e) e.source ∧ PG.property (↑e.symm) e.target}
⊢ PG.property (↑e) e.source

pp: case right
H : Type u_1
inst✝ : TopologicalSpace H
PG : Pregroupoid H
e : OpenPartialHomeomorph H H
he :
  ∀ x ∈ e.source,
    ∃ s, IsOpen[inst✝] s ∧ x ∈ s ∧ e.restr s ∈ {e | PG.property (↑e) e.source ∧ PG.property (↑e.symm) e.target}
⊢ PG.property (↑e.symm) e.target

Tactic:
·

Goal state after:
case left
H : Type u_1
inst✝ : TopologicalSpace H
PG : Pregroupoid H
e : OpenPartialHomeomorph H H
he :
  ∀ x ∈ e.source,
    ∃ s, IsOpen[inst✝] s ∧ x ∈ s ∧ e.restr s ∈ {e | PG.property (↑e) e.source ∧ PG.property (↑e.symm) e.target}
⊢ PG.property (↑e) e.source
Module: Mathlib.Geometry.Manifold.StructureGroupoid

Goal state before:
pp: case right
H : Type u_1
inst✝ : TopologicalSpace H
PG : Pregroupoid H
e : OpenPartialHomeomorph H H
he :
  ∀ x ∈ e.source,
    ∃ s, IsOpen[inst✝] s ∧ x ∈ s ∧ e.restr s ∈ {e | PG.property (↑e) e.source ∧ PG.property (↑e.symm) e.target}
⊢ PG.property (↑e.symm) e.target

Tactic:
·

Goal state after:
case right
H : Type u_1
inst✝ : TopologicalSpace H
PG : Pregroupoid H
e : OpenPartialHomeomorph H H
he :
  ∀ x ∈ e.source,
    ∃ s, IsOpen[inst✝] s ∧ x ∈ s ∧ e.restr s ∈ {e | PG.property (↑e) e.source ∧ PG.property (↑e.symm) e.target}
⊢ PG.property (↑e.symm) e.target
Module: Mathlib.Geometry.Manifold.StructureGroupoid

Goal state before:
pp: case right.refine_1
H : Type u_1
inst✝ : TopologicalSpace H
PG : Pregroupoid H
e : OpenPartialHomeomorph H H
he :
  ∀ x ∈ e.source,
    ∃ s, IsOpen[inst✝] s ∧ x ∈ s ∧ e.restr s ∈ {e | PG.property (↑e) e.source ∧ PG.property (↑e.symm) e.target}
x : H
xu : x ∈ e.target
s : Set H
s_open : IsOpen[inst✝] s
xs : ↑e.symm x ∈ s
hs : e.restr s ∈ {e | PG.property (↑e) e.source ∧ PG.property (↑e.symm) e.target}
⊢ IsOpen[inst✝] (e.target ∩ ↑e.symm ⁻¹' s)

pp: case right.refine_2
H : Type u_1
inst✝ : TopologicalSpace H
PG : Pregroupoid H
e : OpenPartialHomeomorph H H
he :
  ∀ x ∈ e.source,
    ∃ s, IsOpen[inst✝] s ∧ x ∈ s ∧ e.restr s ∈ {e | PG.property (↑e) e.source ∧ PG.property (↑e.symm) e.target}
x : H
xu : x ∈ e.target
s : Set H
s_open : IsOpen[inst✝] s
xs : ↑e.symm x ∈ s
hs : e.restr s ∈ {e | PG.property (↑e) e.source ∧ PG.property (↑e.symm) e.target}
⊢ PG.property (↑e.symm) (e.target ∩ (e.target ∩ ↑e.symm ⁻¹' s))

Tactic:
·

Goal state after:
case right.refine_1
H : Type u_1
inst✝ : TopologicalSpace H
PG : Pregroupoid H
e : OpenPartialHomeomorph H H
he :
  ∀ x ∈ e.source,
    ∃ s, IsOpen[inst✝] s ∧ x ∈ s ∧ e.restr s ∈ {e | PG.property (↑e) e.source ∧ PG.property (↑e.symm) e.target}
x : H
xu : x ∈ e.target
s : Set H
s_open : IsOpen[inst✝] s
xs : ↑e.symm x ∈ s
hs : e.restr s ∈ {e | PG.property (↑e) e.source ∧ PG.property (↑e.symm) e.target}
⊢ IsOpen[inst✝] (e.target ∩ ↑e.symm ⁻¹' s)
Module: Mathlib.Geometry.Manifold.StructureGroupoid

Goal state before:
pp: case right.refine_2
H : Type u_1
inst✝ : TopologicalSpace H
PG : Pregroupoid H
e : OpenPartialHomeomorph H H
he :
  ∀ x ∈ e.source,
    ∃ s, IsOpen[inst✝] s ∧ x ∈ s ∧ e.restr s ∈ {e | PG.property (↑e) e.source ∧ PG.property (↑e.symm) e.target}
x : H
xu : x ∈ e.target
s : Set H
s_open : IsOpen[inst✝] s
xs : ↑e.symm x ∈ s
hs : e.restr s ∈ {e | PG.property (↑e) e.source ∧ PG.property (↑e.symm) e.target}
⊢ PG.property (↑e.symm) (e.target ∩ (e.target ∩ ↑e.symm ⁻¹' s))

Tactic:
·

Goal state after:
case right.refine_2
H : Type u_1
inst✝ : TopologicalSpace H
PG : Pregroupoid H
e : OpenPartialHomeomorph H H
he :
  ∀ x ∈ e.source,
    ∃ s, IsOpen[inst✝] s ∧ x ∈ s ∧ e.restr s ∈ {e | PG.property (↑e) e.source ∧ PG.property (↑e.symm) e.target}
x : H
xu : x ∈ e.target
s : Set H
s_open : IsOpen[inst✝] s
xs : ↑e.symm x ∈ s
hs : e.restr s ∈ {e | PG.property (↑e) e.source ∧ PG.property (↑e.symm) e.target}
⊢ PG.property (↑e.symm) (e.target ∩ (e.target ∩ ↑e.symm ⁻¹' s))
Module: Mathlib.Geometry.Manifold.HasGroupoid

Goal state before:
pp: case left
H : Type u
H' : Type u_1
M : Type u_2
M' : Type u_3
M'' : Type u_4
inst✝⁶ : TopologicalSpace H
inst✝⁵ : TopologicalSpace M
inst✝⁴ : ChartedSpace H M
inst✝³ : TopologicalSpace M'
inst✝² : TopologicalSpace M''
G : StructureGroupoid H
inst✝¹ : ChartedSpace H M'
inst✝ : ChartedSpace H M''
e : Structomorph G M M'
e' : Structomorph G M' M''
c : OpenPartialHomeomorph M H
c' : OpenPartialHomeomorph M'' H
hc : c ∈ atlas H M
hc' : c' ∈ atlas H M''
x : H
hx : x ∈ (c.symm ≫ₕ (e.trans e'.toHomeomorph).toOpenPartialHomeomorph ≫ₕ c').source
f₁ : OpenPartialHomeomorph M M' := ⋯
f₂ : OpenPartialHomeomorph M' M'' := ⋯
f : OpenPartialHomeomorph M M'' := ⋯
feq : f = f₁ ≫ₕ f₂
y : M' := ⋯
g : OpenPartialHomeomorph M' H := ⋯
hg₁ : chartAt H y ∈ atlas H M'
hg₂ : y ∈ (chartAt H y).source
s : Set H := ⋯
open_s : IsOpen[inst✝⁶] s
⊢ x ∈ (c.symm ≫ₕ f₁).source

pp: case right
H : Type u
H' : Type u_1
M : Type u_2
M' : Type u_3
M'' : Type u_4
inst✝⁶ : TopologicalSpace H
inst✝⁵ : TopologicalSpace M
inst✝⁴ : ChartedSpace H M
inst✝³ : TopologicalSpace M'
inst✝² : TopologicalSpace M''
G : StructureGroupoid H
inst✝¹ : ChartedSpace H M'
inst✝ : ChartedSpace H M''
e : Structomorph G M M'
e' : Structomorph G M' M''
c : OpenPartialHomeomorph M H
c' : OpenPartialHomeomorph M'' H
hc : c ∈ atlas H M
hc' : c' ∈ atlas H M''
x : H
hx : x ∈ (c.symm ≫ₕ (e.trans e'.toHomeomorph).toOpenPartialHomeomorph ≫ₕ c').source
f₁ : OpenPartialHomeomorph M M' := ⋯
f₂ : OpenPartialHomeomorph M' M'' := ⋯
f : OpenPartialHomeomorph M M'' := ⋯
feq : f = f₁ ≫ₕ f₂
y : M' := ⋯
g : OpenPartialHomeomorph M' H := ⋯
hg₁ : chartAt H y ∈ atlas H M'
hg₂ : y ∈ (chartAt H y).source
s : Set H := ⋯
open_s : IsOpen[inst✝⁶] s
⊢ x ∈ ↑(c.symm ≫ₕ f₁) ⁻¹' g.source

Tactic:
·

Goal state after:
case left
H : Type u
H' : Type u_1
M : Type u_2
M' : Type u_3
M'' : Type u_4
inst✝⁶ : TopologicalSpace H
inst✝⁵ : TopologicalSpace M
inst✝⁴ : ChartedSpace H M
inst✝³ : TopologicalSpace M'
inst✝² : TopologicalSpace M''
G : StructureGroupoid H
inst✝¹ : ChartedSpace H M'
inst✝ : ChartedSpace H M''
e : Structomorph G M M'
e' : Structomorph G M' M''
c : OpenPartialHomeomorph M H
c' : OpenPartialHomeomorph M'' H
hc : c ∈ atlas H M
hc' : c' ∈ atlas H M''
x : H
hx : x ∈ (c.symm ≫ₕ (e.trans e'.toHomeomorph).toOpenPartialHomeomorph ≫ₕ c').source
f₁ : OpenPartialHomeomorph M M' := ⋯
f₂ : OpenPartialHomeomorph M' M'' := ⋯
f : OpenPartialHomeomorph M M'' := ⋯
feq : f = f₁ ≫ₕ f₂
y : M' := ⋯
g : OpenPartialHomeomorph M' H := ⋯
hg₁ : chartAt H y ∈ atlas H M'
hg₂ : y ∈ (chartAt H y).source
s : Set H := ⋯
open_s : IsOpen[inst✝⁶] s
⊢ x ∈ (c.symm ≫ₕ f₁).source
Module: Mathlib.Geometry.Manifold.HasGroupoid

Goal state before:
pp: case right
H : Type u
H' : Type u_1
M : Type u_2
M' : Type u_3
M'' : Type u_4
inst✝⁶ : TopologicalSpace H
inst✝⁵ : TopologicalSpace M
inst✝⁴ : ChartedSpace H M
inst✝³ : TopologicalSpace M'
inst✝² : TopologicalSpace M''
G : StructureGroupoid H
inst✝¹ : ChartedSpace H M'
inst✝ : ChartedSpace H M''
e : Structomorph G M M'
e' : Structomorph G M' M''
c : OpenPartialHomeomorph M H
c' : OpenPartialHomeomorph M'' H
hc : c ∈ atlas H M
hc' : c' ∈ atlas H M''
x : H
hx : x ∈ (c.symm ≫ₕ (e.trans e'.toHomeomorph).toOpenPartialHomeomorph ≫ₕ c').source
f₁ : OpenPartialHomeomorph M M' := ⋯
f₂ : OpenPartialHomeomorph M' M'' := ⋯
f : OpenPartialHomeomorph M M'' := ⋯
feq : f = f₁ ≫ₕ f₂
y : M' := ⋯
g : OpenPartialHomeomorph M' H := ⋯
hg₁ : chartAt H y ∈ atlas H M'
hg₂ : y ∈ (chartAt H y).source
s : Set H := ⋯
open_s : IsOpen[inst✝⁶] s
⊢ x ∈ ↑(c.symm ≫ₕ f₁) ⁻¹' g.source

Tactic:
·

Goal state after:
case right
H : Type u
H' : Type u_1
M : Type u_2
M' : Type u_3
M'' : Type u_4
inst✝⁶ : TopologicalSpace H
inst✝⁵ : TopologicalSpace M
inst✝⁴ : ChartedSpace H M
inst✝³ : TopologicalSpace M'
inst✝² : TopologicalSpace M''
G : StructureGroupoid H
inst✝¹ : ChartedSpace H M'
inst✝ : ChartedSpace H M''
e : Structomorph G M M'
e' : Structomorph G M' M''
c : OpenPartialHomeomorph M H
c' : OpenPartialHomeomorph M'' H
hc : c ∈ atlas H M
hc' : c' ∈ atlas H M''
x : H
hx : x ∈ (c.symm ≫ₕ (e.trans e'.toHomeomorph).toOpenPartialHomeomorph ≫ₕ c').source
f₁ : OpenPartialHomeomorph M M' := ⋯
f₂ : OpenPartialHomeomorph M' M'' := ⋯
f : OpenPartialHomeomorph M M'' := ⋯
feq : f = f₁ ≫ₕ f₂
y : M' := ⋯
g : OpenPartialHomeomorph M' H := ⋯
hg₁ : chartAt H y ∈ atlas H M'
hg₂ : y ∈ (chartAt H y).source
s : Set H := ⋯
open_s : IsOpen[inst✝⁶] s
⊢ x ∈ ↑(c.symm ≫ₕ f₁) ⁻¹' g.source
Module: Mathlib.Geometry.Manifold.StructureGroupoid

Goal state before:
pp: case left
H : Type u_1
inst✝ : TopologicalSpace H
PG : Pregroupoid H
e e' : OpenPartialHomeomorph H H
he : e ∈ {e | PG.property (↑e) e.source ∧ PG.property (↑e.symm) e.target}
ee' : e' ≈ e
⊢ PG.property (↑e') e'.source

pp: case right
H : Type u_1
inst✝ : TopologicalSpace H
PG : Pregroupoid H
e e' : OpenPartialHomeomorph H H
he : e ∈ {e | PG.property (↑e) e.source ∧ PG.property (↑e.symm) e.target}
ee' : e' ≈ e
⊢ PG.property (↑e'.symm) e'.target

Tactic:
·

Goal state after:
case left
H : Type u_1
inst✝ : TopologicalSpace H
PG : Pregroupoid H
e e' : OpenPartialHomeomorph H H
he : e ∈ {e | PG.property (↑e) e.source ∧ PG.property (↑e.symm) e.target}
ee' : e' ≈ e
⊢ PG.property (↑e') e'.source
Module: Mathlib.Geometry.Manifold.StructureGroupoid

Goal state before:
pp: case right
H : Type u_1
inst✝ : TopologicalSpace H
PG : Pregroupoid H
e e' : OpenPartialHomeomorph H H
he : e ∈ {e | PG.property (↑e) e.source ∧ PG.property (↑e.symm) e.target}
ee' : e' ≈ e
⊢ PG.property (↑e'.symm) e'.target

Tactic:
·

Goal state after:
case right
H : Type u_1
inst✝ : TopologicalSpace H
PG : Pregroupoid H
e e' : OpenPartialHomeomorph H H
he : e ∈ {e | PG.property (↑e) e.source ∧ PG.property (↑e.symm) e.target}
ee' : e' ≈ e
⊢ PG.property (↑e'.symm) e'.target
Module: Mathlib.MeasureTheory.Function.Jacobian

Goal state before:
pp: case hm
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
u : ℕ → Set E := fun n ↦ disjointed (spanningSets μ) n
u_meas : ∀ (n : ℕ), MeasurableSet (u n)
A : s = ⋃ n, s ∩ u n
⊢ ∀ (i : ℕ), MeasurableSet (s ∩ u i)

pp: case hd
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
u : ℕ → Set E := fun n ↦ disjointed (spanningSets μ) n
u_meas : ∀ (n : ℕ), MeasurableSet (u n)
A : s = ⋃ n, s ∩ u n
⊢ Pairwise (Disjoint on fun n ↦ s ∩ u n)

Tactic:
·

Goal state after:
case hm
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
u : ℕ → Set E := fun n ↦ disjointed (spanningSets μ) n
u_meas : ∀ (n : ℕ), MeasurableSet (u n)
A : s = ⋃ n, s ∩ u n
⊢ ∀ (i : ℕ), MeasurableSet (s ∩ u i)
Module: Mathlib.MeasureTheory.Function.Jacobian

Goal state before:
pp: case hd
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
u : ℕ → Set E := fun n ↦ disjointed (spanningSets μ) n
u_meas : ∀ (n : ℕ), MeasurableSet (u n)
A : s = ⋃ n, s ∩ u n
⊢ Pairwise (Disjoint on fun n ↦ s ∩ u n)

Tactic:
·

Goal state after:
case hd
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
u : ℕ → Set E := fun n ↦ disjointed (spanningSets μ) n
u_meas : ∀ (n : ℕ), MeasurableSet (u n)
A : s = ⋃ n, s ∩ u n
⊢ Pairwise (Disjoint on fun n ↦ s ∩ u n)
Module: Mathlib.Geometry.Manifold.StructureGroupoid

Goal state before:
pp: case mp
H : Type u_1
inst✝ : TopologicalSpace H
G : StructureGroupoid H
⊢ ClosedUnderRestriction G → idRestrGroupoid ≤ G

pp: case mpr
H : Type u_1
inst✝ : TopologicalSpace H
G : StructureGroupoid H
⊢ idRestrGroupoid ≤ G → ClosedUnderRestriction G

Tactic:
·

Goal state after:
case mp
H : Type u_1
inst✝ : TopologicalSpace H
G : StructureGroupoid H
⊢ ClosedUnderRestriction G → idRestrGroupoid ≤ G
Module: Mathlib.Geometry.Manifold.StructureGroupoid

Goal state before:
pp: case mpr
H : Type u_1
inst✝ : TopologicalSpace H
G : StructureGroupoid H
⊢ idRestrGroupoid ≤ G → ClosedUnderRestriction G

Tactic:
·

Goal state after:
case mpr
H : Type u_1
inst✝ : TopologicalSpace H
G : StructureGroupoid H
⊢ idRestrGroupoid ≤ G → ClosedUnderRestriction G