url stringclasses 147
values | commit stringclasses 147
values | file_path stringlengths 7 101 | full_name stringlengths 1 94 | start stringlengths 6 10 | end stringlengths 6 11 | tactic stringlengths 1 11.2k | state_before stringlengths 3 2.09M | state_after stringlengths 6 2.09M | input stringlengths 73 2.09M |
|---|---|---|---|---|---|---|---|---|---|
https://github.com/fpvandoorn/sard.git | f4a1bb550136e8591dcd20dca5bdcce766a958c7 | Sard/LocallyLipschitzMeasureZero.lean | C1_preserves_null_sets_open | [104, 1] | [113, 55] | exact locally_lipschitz_image_of_null_set_is_null_set_open
(LocallyLipschitz.of_C1_on_open hU this hf) hsU hs | X : Type u_1
Y : Type u_2
inst✝¹⁴ : MetricSpace X
inst✝¹³ : MeasurableSpace X
inst✝¹² : BorelSpace X
inst✝¹¹ : MetricSpace Y
inst✝¹⁰ : MeasurableSpace Y
inst✝⁹ : BorelSpace Y
E : Type u_3
F : Type u_4
inst✝⁸ : NormedAddCommGroup E
inst✝⁷ : NormedSpace ℝ E
inst✝⁶ : MeasurableSpace E
inst✝⁵ : BorelSpace E
inst✝⁴ : SigmaC... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
X : Type u_1
Y : Type u_2
inst✝¹⁴ : MetricSpace X
inst✝¹³ : MeasurableSpace X
inst✝¹² : BorelSpace X
inst✝¹¹ : MetricSpace Y
inst✝¹⁰ : MeasurableSpace Y
inst✝⁹ : BorelSpace Y
E : Type u_3
F : Type u_4
inst✝⁸ : NormedAddCommGroup E
inst✝⁷ : NormedSpace ℝ E
ins... |
https://github.com/fpvandoorn/sard.git | f4a1bb550136e8591dcd20dca5bdcce766a958c7 | Sard/ManifoldAux.lean | PartialHomeomorph.isOpenMapOn_source | [35, 1] | [38, 72] | rw [(image_eq_target_inter_inv_preimage (e := e) hs)] | E : Type u_1
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace ℝ E
H : Type u_2
inst✝³ : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type u_3
inst✝² : TopologicalSpace M
inst✝¹ : ChartedSpace H M
inst✝ : SmoothManifoldWithCorners I M
e : PartialHomeomorph M H
s : Set M
hopen : IsOpen s
hs : s ⊆ e.source
⊢ IsOpen... | E : Type u_1
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace ℝ E
H : Type u_2
inst✝³ : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type u_3
inst✝² : TopologicalSpace M
inst✝¹ : ChartedSpace H M
inst✝ : SmoothManifoldWithCorners I M
e : PartialHomeomorph M H
s : Set M
hopen : IsOpen s
hs : s ⊆ e.source
⊢ IsOpen... | Please generate a tactic in lean4 to solve the state.
STATE:
E : Type u_1
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace ℝ E
H : Type u_2
inst✝³ : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type u_3
inst✝² : TopologicalSpace M
inst✝¹ : ChartedSpace H M
inst✝ : SmoothManifoldWithCorners I M
e : PartialHomeomo... |
https://github.com/fpvandoorn/sard.git | f4a1bb550136e8591dcd20dca5bdcce766a958c7 | Sard/ManifoldAux.lean | PartialHomeomorph.isOpenMapOn_source | [35, 1] | [38, 72] | exact e.continuousOn_invFun.isOpen_inter_preimage e.open_target hopen | E : Type u_1
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace ℝ E
H : Type u_2
inst✝³ : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type u_3
inst✝² : TopologicalSpace M
inst✝¹ : ChartedSpace H M
inst✝ : SmoothManifoldWithCorners I M
e : PartialHomeomorph M H
s : Set M
hopen : IsOpen s
hs : s ⊆ e.source
⊢ IsOpen... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
E : Type u_1
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace ℝ E
H : Type u_2
inst✝³ : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type u_3
inst✝² : TopologicalSpace M
inst✝¹ : ChartedSpace H M
inst✝ : SmoothManifoldWithCorners I M
e : PartialHomeomo... |
https://github.com/fpvandoorn/sard.git | f4a1bb550136e8591dcd20dca5bdcce766a958c7 | Sard/ManifoldAux.lean | PartialHomeomorph.symm_isOpenMapOn_target | [40, 1] | [43, 75] | have r : e.invFun '' t = e.source ∩ ↑e ⁻¹' t := symm_image_eq_source_inter_preimage (e := e) ht | E : Type u_1
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace ℝ E
H : Type u_2
inst✝³ : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type u_3
inst✝² : TopologicalSpace M
inst✝¹ : ChartedSpace H M
inst✝ : SmoothManifoldWithCorners I M
e : PartialHomeomorph M H
t : Set H
hopen : IsOpen t
ht : t ⊆ e.target
⊢ IsOpen... | E : Type u_1
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace ℝ E
H : Type u_2
inst✝³ : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type u_3
inst✝² : TopologicalSpace M
inst✝¹ : ChartedSpace H M
inst✝ : SmoothManifoldWithCorners I M
e : PartialHomeomorph M H
t : Set H
hopen : IsOpen t
ht : t ⊆ e.target
r : e.in... | Please generate a tactic in lean4 to solve the state.
STATE:
E : Type u_1
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace ℝ E
H : Type u_2
inst✝³ : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type u_3
inst✝² : TopologicalSpace M
inst✝¹ : ChartedSpace H M
inst✝ : SmoothManifoldWithCorners I M
e : PartialHomeomo... |
https://github.com/fpvandoorn/sard.git | f4a1bb550136e8591dcd20dca5bdcce766a958c7 | Sard/ManifoldAux.lean | PartialHomeomorph.symm_isOpenMapOn_target | [40, 1] | [43, 75] | exact r ▸ e.continuousOn_toFun.isOpen_inter_preimage e.open_source hopen | E : Type u_1
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace ℝ E
H : Type u_2
inst✝³ : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type u_3
inst✝² : TopologicalSpace M
inst✝¹ : ChartedSpace H M
inst✝ : SmoothManifoldWithCorners I M
e : PartialHomeomorph M H
t : Set H
hopen : IsOpen t
ht : t ⊆ e.target
r : e.in... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
E : Type u_1
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace ℝ E
H : Type u_2
inst✝³ : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type u_3
inst✝² : TopologicalSpace M
inst✝¹ : ChartedSpace H M
inst✝ : SmoothManifoldWithCorners I M
e : PartialHomeomo... |
https://github.com/fpvandoorn/sard.git | f4a1bb550136e8591dcd20dca5bdcce766a958c7 | Sard/ManifoldAux.lean | PartialHomeomorph.extend_isOpenMapOn_source | [56, 1] | [61, 83] | simp only [extend_coe, image_comp I e] | E : Type u_1
inst✝⁶ : NormedAddCommGroup E
inst✝⁵ : NormedSpace ℝ E
H : Type u_2
inst✝⁴ : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type u_3
inst✝³ : TopologicalSpace M
inst✝² : ChartedSpace H M
inst✝¹ : SmoothManifoldWithCorners I M
inst✝ : I.Boundaryless
e : PartialHomeomorph M H
s : Set M
hopen : IsOpen s
hs... | E : Type u_1
inst✝⁶ : NormedAddCommGroup E
inst✝⁵ : NormedSpace ℝ E
H : Type u_2
inst✝⁴ : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type u_3
inst✝³ : TopologicalSpace M
inst✝² : ChartedSpace H M
inst✝¹ : SmoothManifoldWithCorners I M
inst✝ : I.Boundaryless
e : PartialHomeomorph M H
s : Set M
hopen : IsOpen s
hs... | Please generate a tactic in lean4 to solve the state.
STATE:
E : Type u_1
inst✝⁶ : NormedAddCommGroup E
inst✝⁵ : NormedSpace ℝ E
H : Type u_2
inst✝⁴ : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type u_3
inst✝³ : TopologicalSpace M
inst✝² : ChartedSpace H M
inst✝¹ : SmoothManifoldWithCorners I M
inst✝ : I.Boundar... |
https://github.com/fpvandoorn/sard.git | f4a1bb550136e8591dcd20dca5bdcce766a958c7 | Sard/ManifoldAux.lean | PartialHomeomorph.extend_isOpenMapOn_source | [56, 1] | [61, 83] | apply (I.toOpenEmbedding.open_iff_image_open).mp (e.isOpenMapOn_source hopen hs) | E : Type u_1
inst✝⁶ : NormedAddCommGroup E
inst✝⁵ : NormedSpace ℝ E
H : Type u_2
inst✝⁴ : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type u_3
inst✝³ : TopologicalSpace M
inst✝² : ChartedSpace H M
inst✝¹ : SmoothManifoldWithCorners I M
inst✝ : I.Boundaryless
e : PartialHomeomorph M H
s : Set M
hopen : IsOpen s
hs... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
E : Type u_1
inst✝⁶ : NormedAddCommGroup E
inst✝⁵ : NormedSpace ℝ E
H : Type u_2
inst✝⁴ : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type u_3
inst✝³ : TopologicalSpace M
inst✝² : ChartedSpace H M
inst✝¹ : SmoothManifoldWithCorners I M
inst✝ : I.Boundar... |
https://github.com/fpvandoorn/sard.git | f4a1bb550136e8591dcd20dca5bdcce766a958c7 | Sard/ManifoldAux.lean | PartialHomeomorph.extend_symm_isOpenMapOn_target | [63, 1] | [76, 41] | have h : IsOpen (I.invFun '' t) := I.toOpenEmbedding_symm.open_iff_image_open.mp hopen | E : Type u_1
inst✝⁶ : NormedAddCommGroup E
inst✝⁵ : NormedSpace ℝ E
H : Type u_2
inst✝⁴ : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type u_3
inst✝³ : TopologicalSpace M
inst✝² : ChartedSpace H M
inst✝¹ : SmoothManifoldWithCorners I M
inst✝ : I.Boundaryless
e : PartialHomeomorph M H
t : Set E
hopen : IsOpen t
ht... | E : Type u_1
inst✝⁶ : NormedAddCommGroup E
inst✝⁵ : NormedSpace ℝ E
H : Type u_2
inst✝⁴ : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type u_3
inst✝³ : TopologicalSpace M
inst✝² : ChartedSpace H M
inst✝¹ : SmoothManifoldWithCorners I M
inst✝ : I.Boundaryless
e : PartialHomeomorph M H
t : Set E
hopen : IsOpen t
ht... | Please generate a tactic in lean4 to solve the state.
STATE:
E : Type u_1
inst✝⁶ : NormedAddCommGroup E
inst✝⁵ : NormedSpace ℝ E
H : Type u_2
inst✝⁴ : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type u_3
inst✝³ : TopologicalSpace M
inst✝² : ChartedSpace H M
inst✝¹ : SmoothManifoldWithCorners I M
inst✝ : I.Boundar... |
https://github.com/fpvandoorn/sard.git | f4a1bb550136e8591dcd20dca5bdcce766a958c7 | Sard/ManifoldAux.lean | PartialHomeomorph.extend_symm_isOpenMapOn_target | [63, 1] | [76, 41] | have : (e.extend I).target = I.symm ⁻¹' e.target := by
let r := e.extend_target I
rw [I.range_eq_univ, inter_univ] at r
exact r | E : Type u_1
inst✝⁶ : NormedAddCommGroup E
inst✝⁵ : NormedSpace ℝ E
H : Type u_2
inst✝⁴ : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type u_3
inst✝³ : TopologicalSpace M
inst✝² : ChartedSpace H M
inst✝¹ : SmoothManifoldWithCorners I M
inst✝ : I.Boundaryless
e : PartialHomeomorph M H
t : Set E
hopen : IsOpen t
ht... | E : Type u_1
inst✝⁶ : NormedAddCommGroup E
inst✝⁵ : NormedSpace ℝ E
H : Type u_2
inst✝⁴ : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type u_3
inst✝³ : TopologicalSpace M
inst✝² : ChartedSpace H M
inst✝¹ : SmoothManifoldWithCorners I M
inst✝ : I.Boundaryless
e : PartialHomeomorph M H
t : Set E
hopen : IsOpen t
ht... | Please generate a tactic in lean4 to solve the state.
STATE:
E : Type u_1
inst✝⁶ : NormedAddCommGroup E
inst✝⁵ : NormedSpace ℝ E
H : Type u_2
inst✝⁴ : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type u_3
inst✝³ : TopologicalSpace M
inst✝² : ChartedSpace H M
inst✝¹ : SmoothManifoldWithCorners I M
inst✝ : I.Boundar... |
https://github.com/fpvandoorn/sard.git | f4a1bb550136e8591dcd20dca5bdcce766a958c7 | Sard/ManifoldAux.lean | PartialHomeomorph.extend_symm_isOpenMapOn_target | [63, 1] | [76, 41] | have : I.symm '' t ⊆ e.target := calc I.symm '' t
_ ⊆ I.symm '' ((e.extend I).target) := image_subset _ ht
_ = I.symm '' (I.symm ⁻¹' e.target) := by rw [this]
_ ⊆ e.target := image_preimage_subset I.symm e.target | E : Type u_1
inst✝⁶ : NormedAddCommGroup E
inst✝⁵ : NormedSpace ℝ E
H : Type u_2
inst✝⁴ : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type u_3
inst✝³ : TopologicalSpace M
inst✝² : ChartedSpace H M
inst✝¹ : SmoothManifoldWithCorners I M
inst✝ : I.Boundaryless
e : PartialHomeomorph M H
t : Set E
hopen : IsOpen t
ht... | E : Type u_1
inst✝⁶ : NormedAddCommGroup E
inst✝⁵ : NormedSpace ℝ E
H : Type u_2
inst✝⁴ : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type u_3
inst✝³ : TopologicalSpace M
inst✝² : ChartedSpace H M
inst✝¹ : SmoothManifoldWithCorners I M
inst✝ : I.Boundaryless
e : PartialHomeomorph M H
t : Set E
hopen : IsOpen t
ht... | Please generate a tactic in lean4 to solve the state.
STATE:
E : Type u_1
inst✝⁶ : NormedAddCommGroup E
inst✝⁵ : NormedSpace ℝ E
H : Type u_2
inst✝⁴ : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type u_3
inst✝³ : TopologicalSpace M
inst✝² : ChartedSpace H M
inst✝¹ : SmoothManifoldWithCorners I M
inst✝ : I.Boundar... |
https://github.com/fpvandoorn/sard.git | f4a1bb550136e8591dcd20dca5bdcce766a958c7 | Sard/ManifoldAux.lean | PartialHomeomorph.extend_symm_isOpenMapOn_target | [63, 1] | [76, 41] | rw [extend_coe_symm, image_comp] | E : Type u_1
inst✝⁶ : NormedAddCommGroup E
inst✝⁵ : NormedSpace ℝ E
H : Type u_2
inst✝⁴ : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type u_3
inst✝³ : TopologicalSpace M
inst✝² : ChartedSpace H M
inst✝¹ : SmoothManifoldWithCorners I M
inst✝ : I.Boundaryless
e : PartialHomeomorph M H
t : Set E
hopen : IsOpen t
ht... | E : Type u_1
inst✝⁶ : NormedAddCommGroup E
inst✝⁵ : NormedSpace ℝ E
H : Type u_2
inst✝⁴ : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type u_3
inst✝³ : TopologicalSpace M
inst✝² : ChartedSpace H M
inst✝¹ : SmoothManifoldWithCorners I M
inst✝ : I.Boundaryless
e : PartialHomeomorph M H
t : Set E
hopen : IsOpen t
ht... | Please generate a tactic in lean4 to solve the state.
STATE:
E : Type u_1
inst✝⁶ : NormedAddCommGroup E
inst✝⁵ : NormedSpace ℝ E
H : Type u_2
inst✝⁴ : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type u_3
inst✝³ : TopologicalSpace M
inst✝² : ChartedSpace H M
inst✝¹ : SmoothManifoldWithCorners I M
inst✝ : I.Boundar... |
https://github.com/fpvandoorn/sard.git | f4a1bb550136e8591dcd20dca5bdcce766a958c7 | Sard/ManifoldAux.lean | PartialHomeomorph.extend_symm_isOpenMapOn_target | [63, 1] | [76, 41] | exact e.symm_isOpenMapOn_target h this | E : Type u_1
inst✝⁶ : NormedAddCommGroup E
inst✝⁵ : NormedSpace ℝ E
H : Type u_2
inst✝⁴ : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type u_3
inst✝³ : TopologicalSpace M
inst✝² : ChartedSpace H M
inst✝¹ : SmoothManifoldWithCorners I M
inst✝ : I.Boundaryless
e : PartialHomeomorph M H
t : Set E
hopen : IsOpen t
ht... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
E : Type u_1
inst✝⁶ : NormedAddCommGroup E
inst✝⁵ : NormedSpace ℝ E
H : Type u_2
inst✝⁴ : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type u_3
inst✝³ : TopologicalSpace M
inst✝² : ChartedSpace H M
inst✝¹ : SmoothManifoldWithCorners I M
inst✝ : I.Boundar... |
https://github.com/fpvandoorn/sard.git | f4a1bb550136e8591dcd20dca5bdcce766a958c7 | Sard/ManifoldAux.lean | PartialHomeomorph.extend_symm_isOpenMapOn_target | [63, 1] | [76, 41] | let r := e.extend_target I | E : Type u_1
inst✝⁶ : NormedAddCommGroup E
inst✝⁵ : NormedSpace ℝ E
H : Type u_2
inst✝⁴ : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type u_3
inst✝³ : TopologicalSpace M
inst✝² : ChartedSpace H M
inst✝¹ : SmoothManifoldWithCorners I M
inst✝ : I.Boundaryless
e : PartialHomeomorph M H
t : Set E
hopen : IsOpen t
ht... | E : Type u_1
inst✝⁶ : NormedAddCommGroup E
inst✝⁵ : NormedSpace ℝ E
H : Type u_2
inst✝⁴ : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type u_3
inst✝³ : TopologicalSpace M
inst✝² : ChartedSpace H M
inst✝¹ : SmoothManifoldWithCorners I M
inst✝ : I.Boundaryless
e : PartialHomeomorph M H
t : Set E
hopen : IsOpen t
ht... | Please generate a tactic in lean4 to solve the state.
STATE:
E : Type u_1
inst✝⁶ : NormedAddCommGroup E
inst✝⁵ : NormedSpace ℝ E
H : Type u_2
inst✝⁴ : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type u_3
inst✝³ : TopologicalSpace M
inst✝² : ChartedSpace H M
inst✝¹ : SmoothManifoldWithCorners I M
inst✝ : I.Boundar... |
https://github.com/fpvandoorn/sard.git | f4a1bb550136e8591dcd20dca5bdcce766a958c7 | Sard/ManifoldAux.lean | PartialHomeomorph.extend_symm_isOpenMapOn_target | [63, 1] | [76, 41] | rw [I.range_eq_univ, inter_univ] at r | E : Type u_1
inst✝⁶ : NormedAddCommGroup E
inst✝⁵ : NormedSpace ℝ E
H : Type u_2
inst✝⁴ : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type u_3
inst✝³ : TopologicalSpace M
inst✝² : ChartedSpace H M
inst✝¹ : SmoothManifoldWithCorners I M
inst✝ : I.Boundaryless
e : PartialHomeomorph M H
t : Set E
hopen : IsOpen t
ht... | E : Type u_1
inst✝⁶ : NormedAddCommGroup E
inst✝⁵ : NormedSpace ℝ E
H : Type u_2
inst✝⁴ : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type u_3
inst✝³ : TopologicalSpace M
inst✝² : ChartedSpace H M
inst✝¹ : SmoothManifoldWithCorners I M
inst✝ : I.Boundaryless
e : PartialHomeomorph M H
t : Set E
hopen : IsOpen t
ht... | Please generate a tactic in lean4 to solve the state.
STATE:
E : Type u_1
inst✝⁶ : NormedAddCommGroup E
inst✝⁵ : NormedSpace ℝ E
H : Type u_2
inst✝⁴ : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type u_3
inst✝³ : TopologicalSpace M
inst✝² : ChartedSpace H M
inst✝¹ : SmoothManifoldWithCorners I M
inst✝ : I.Boundar... |
https://github.com/fpvandoorn/sard.git | f4a1bb550136e8591dcd20dca5bdcce766a958c7 | Sard/ManifoldAux.lean | PartialHomeomorph.extend_symm_isOpenMapOn_target | [63, 1] | [76, 41] | exact r | E : Type u_1
inst✝⁶ : NormedAddCommGroup E
inst✝⁵ : NormedSpace ℝ E
H : Type u_2
inst✝⁴ : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type u_3
inst✝³ : TopologicalSpace M
inst✝² : ChartedSpace H M
inst✝¹ : SmoothManifoldWithCorners I M
inst✝ : I.Boundaryless
e : PartialHomeomorph M H
t : Set E
hopen : IsOpen t
ht... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
E : Type u_1
inst✝⁶ : NormedAddCommGroup E
inst✝⁵ : NormedSpace ℝ E
H : Type u_2
inst✝⁴ : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type u_3
inst✝³ : TopologicalSpace M
inst✝² : ChartedSpace H M
inst✝¹ : SmoothManifoldWithCorners I M
inst✝ : I.Boundar... |
https://github.com/fpvandoorn/sard.git | f4a1bb550136e8591dcd20dca5bdcce766a958c7 | Sard/ManifoldAux.lean | PartialHomeomorph.extend_symm_isOpenMapOn_target | [63, 1] | [76, 41] | rw [this] | E : Type u_1
inst✝⁶ : NormedAddCommGroup E
inst✝⁵ : NormedSpace ℝ E
H : Type u_2
inst✝⁴ : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type u_3
inst✝³ : TopologicalSpace M
inst✝² : ChartedSpace H M
inst✝¹ : SmoothManifoldWithCorners I M
inst✝ : I.Boundaryless
e : PartialHomeomorph M H
t : Set E
hopen : IsOpen t
ht... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
E : Type u_1
inst✝⁶ : NormedAddCommGroup E
inst✝⁵ : NormedSpace ℝ E
H : Type u_2
inst✝⁴ : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type u_3
inst✝³ : TopologicalSpace M
inst✝² : ChartedSpace H M
inst✝¹ : SmoothManifoldWithCorners I M
inst✝ : I.Boundar... |
https://github.com/fpvandoorn/sard.git | f4a1bb550136e8591dcd20dca5bdcce766a958c7 | Sard/ManifoldAux.lean | PartialHomeomorph.extend_image_mem_nhds_symm | [78, 1] | [87, 50] | rcases mem_nhds_iff.mp hn with ⟨t', ht's', ht'open, hxt'⟩ | E : Type u_1
inst✝⁶ : NormedAddCommGroup E
inst✝⁵ : NormedSpace ℝ E
H : Type u_2
inst✝⁴ : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type u_3
inst✝³ : TopologicalSpace M
inst✝² : ChartedSpace H M
inst✝¹ : SmoothManifoldWithCorners I M
inst✝ : I.Boundaryless
e : PartialHomeomorph M H
x : E
n : Set E
hn : n ∈ 𝓝 x... | case intro.intro.intro
E : Type u_1
inst✝⁶ : NormedAddCommGroup E
inst✝⁵ : NormedSpace ℝ E
H : Type u_2
inst✝⁴ : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type u_3
inst✝³ : TopologicalSpace M
inst✝² : ChartedSpace H M
inst✝¹ : SmoothManifoldWithCorners I M
inst✝ : I.Boundaryless
e : PartialHomeomorph M H
x : E
... | Please generate a tactic in lean4 to solve the state.
STATE:
E : Type u_1
inst✝⁶ : NormedAddCommGroup E
inst✝⁵ : NormedSpace ℝ E
H : Type u_2
inst✝⁴ : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type u_3
inst✝³ : TopologicalSpace M
inst✝² : ChartedSpace H M
inst✝¹ : SmoothManifoldWithCorners I M
inst✝ : I.Boundar... |
https://github.com/fpvandoorn/sard.git | f4a1bb550136e8591dcd20dca5bdcce766a958c7 | Sard/ManifoldAux.lean | PartialHomeomorph.extend_image_mem_nhds_symm | [78, 1] | [87, 50] | rw [mem_nhds_iff] | case intro.intro.intro
E : Type u_1
inst✝⁶ : NormedAddCommGroup E
inst✝⁵ : NormedSpace ℝ E
H : Type u_2
inst✝⁴ : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type u_3
inst✝³ : TopologicalSpace M
inst✝² : ChartedSpace H M
inst✝¹ : SmoothManifoldWithCorners I M
inst✝ : I.Boundaryless
e : PartialHomeomorph M H
x : E
... | case intro.intro.intro
E : Type u_1
inst✝⁶ : NormedAddCommGroup E
inst✝⁵ : NormedSpace ℝ E
H : Type u_2
inst✝⁴ : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type u_3
inst✝³ : TopologicalSpace M
inst✝² : ChartedSpace H M
inst✝¹ : SmoothManifoldWithCorners I M
inst✝ : I.Boundaryless
e : PartialHomeomorph M H
x : E
... | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.intro
E : Type u_1
inst✝⁶ : NormedAddCommGroup E
inst✝⁵ : NormedSpace ℝ E
H : Type u_2
inst✝⁴ : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type u_3
inst✝³ : TopologicalSpace M
inst✝² : ChartedSpace H M
inst✝¹ : SmoothManifoldWithCorner... |
https://github.com/fpvandoorn/sard.git | f4a1bb550136e8591dcd20dca5bdcce766a958c7 | Sard/ManifoldAux.lean | PartialHomeomorph.extend_image_mem_nhds_symm | [78, 1] | [87, 50] | refine ⟨(e.extend I).symm '' t', image_subset _ ht's', ?_, ?_⟩ | case intro.intro.intro
E : Type u_1
inst✝⁶ : NormedAddCommGroup E
inst✝⁵ : NormedSpace ℝ E
H : Type u_2
inst✝⁴ : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type u_3
inst✝³ : TopologicalSpace M
inst✝² : ChartedSpace H M
inst✝¹ : SmoothManifoldWithCorners I M
inst✝ : I.Boundaryless
e : PartialHomeomorph M H
x : E
... | case intro.intro.intro.refine_1
E : Type u_1
inst✝⁶ : NormedAddCommGroup E
inst✝⁵ : NormedSpace ℝ E
H : Type u_2
inst✝⁴ : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type u_3
inst✝³ : TopologicalSpace M
inst✝² : ChartedSpace H M
inst✝¹ : SmoothManifoldWithCorners I M
inst✝ : I.Boundaryless
e : PartialHomeomorph M... | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.intro
E : Type u_1
inst✝⁶ : NormedAddCommGroup E
inst✝⁵ : NormedSpace ℝ E
H : Type u_2
inst✝⁴ : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type u_3
inst✝³ : TopologicalSpace M
inst✝² : ChartedSpace H M
inst✝¹ : SmoothManifoldWithCorner... |
https://github.com/fpvandoorn/sard.git | f4a1bb550136e8591dcd20dca5bdcce766a958c7 | Sard/ManifoldAux.lean | PartialHomeomorph.extend_image_mem_nhds_symm | [78, 1] | [87, 50] | apply e.extend_symm_isOpenMapOn_target _ ht'open (Subset.trans ht's' hn') | case intro.intro.intro.refine_1
E : Type u_1
inst✝⁶ : NormedAddCommGroup E
inst✝⁵ : NormedSpace ℝ E
H : Type u_2
inst✝⁴ : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type u_3
inst✝³ : TopologicalSpace M
inst✝² : ChartedSpace H M
inst✝¹ : SmoothManifoldWithCorners I M
inst✝ : I.Boundaryless
e : PartialHomeomorph M... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.intro.refine_1
E : Type u_1
inst✝⁶ : NormedAddCommGroup E
inst✝⁵ : NormedSpace ℝ E
H : Type u_2
inst✝⁴ : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type u_3
inst✝³ : TopologicalSpace M
inst✝² : ChartedSpace H M
inst✝¹ : SmoothManifoldW... |
https://github.com/fpvandoorn/sard.git | f4a1bb550136e8591dcd20dca5bdcce766a958c7 | Sard/ManifoldAux.lean | PartialHomeomorph.extend_image_mem_nhds_symm | [78, 1] | [87, 50] | exact mem_image_of_mem (e.extend I).symm hxt' | case intro.intro.intro.refine_2
E : Type u_1
inst✝⁶ : NormedAddCommGroup E
inst✝⁵ : NormedSpace ℝ E
H : Type u_2
inst✝⁴ : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type u_3
inst✝³ : TopologicalSpace M
inst✝² : ChartedSpace H M
inst✝¹ : SmoothManifoldWithCorners I M
inst✝ : I.Boundaryless
e : PartialHomeomorph M... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.intro.refine_2
E : Type u_1
inst✝⁶ : NormedAddCommGroup E
inst✝⁵ : NormedSpace ℝ E
H : Type u_2
inst✝⁴ : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type u_3
inst✝³ : TopologicalSpace M
inst✝² : ChartedSpace H M
inst✝¹ : SmoothManifoldW... |
https://github.com/fpvandoorn/sard.git | f4a1bb550136e8591dcd20dca5bdcce766a958c7 | Sard/ManifoldAux.lean | unused_extendedChart_leftInverse' | [92, 1] | [93, 56] | sorry | E : Type u_1
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace ℝ E
H : Type u_2
inst✝³ : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type u_3
inst✝² : TopologicalSpace M
inst✝¹ : ChartedSpace H M
inst✝ : SmoothManifoldWithCorners I M
e : PartialHomeomorph M H
t : Set E
ht : t ⊆ ↑(e.extend I) '' e.source
⊢ ↑(e.ex... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
E : Type u_1
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace ℝ E
H : Type u_2
inst✝³ : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type u_3
inst✝² : TopologicalSpace M
inst✝¹ : ChartedSpace H M
inst✝ : SmoothManifoldWithCorners I M
e : PartialHomeomo... |
https://github.com/fpvandoorn/sard.git | f4a1bb550136e8591dcd20dca5bdcce766a958c7 | Sard/ManifoldAux.lean | extendedChart_image_nhds_on | [96, 1] | [99, 100] | rw [image_comp] | E : Type u_1
inst✝⁶ : NormedAddCommGroup E
inst✝⁵ : NormedSpace ℝ E
H : Type u_2
inst✝⁴ : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type u_3
inst✝³ : TopologicalSpace M
inst✝² : ChartedSpace H M
inst✝¹ : SmoothManifoldWithCorners I M
inst✝ : I.Boundaryless
e : PartialHomeomorph M H
x : M
n : Set M
hn : n ∈ 𝓝 x... | E : Type u_1
inst✝⁶ : NormedAddCommGroup E
inst✝⁵ : NormedSpace ℝ E
H : Type u_2
inst✝⁴ : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type u_3
inst✝³ : TopologicalSpace M
inst✝² : ChartedSpace H M
inst✝¹ : SmoothManifoldWithCorners I M
inst✝ : I.Boundaryless
e : PartialHomeomorph M H
x : M
n : Set M
hn : n ∈ 𝓝 x... | Please generate a tactic in lean4 to solve the state.
STATE:
E : Type u_1
inst✝⁶ : NormedAddCommGroup E
inst✝⁵ : NormedSpace ℝ E
H : Type u_2
inst✝⁴ : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type u_3
inst✝³ : TopologicalSpace M
inst✝² : ChartedSpace H M
inst✝¹ : SmoothManifoldWithCorners I M
inst✝ : I.Boundar... |
https://github.com/fpvandoorn/sard.git | f4a1bb550136e8591dcd20dca5bdcce766a958c7 | Sard/ManifoldAux.lean | extendedChart_image_nhds_on | [96, 1] | [99, 100] | apply I.toOpenEmbedding.isOpenMap.image_mem_nhds (e.image_mem_nhds (hn₂ (mem_of_mem_nhds hn)) hn) | E : Type u_1
inst✝⁶ : NormedAddCommGroup E
inst✝⁵ : NormedSpace ℝ E
H : Type u_2
inst✝⁴ : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type u_3
inst✝³ : TopologicalSpace M
inst✝² : ChartedSpace H M
inst✝¹ : SmoothManifoldWithCorners I M
inst✝ : I.Boundaryless
e : PartialHomeomorph M H
x : M
n : Set M
hn : n ∈ 𝓝 x... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
E : Type u_1
inst✝⁶ : NormedAddCommGroup E
inst✝⁵ : NormedSpace ℝ E
H : Type u_2
inst✝⁴ : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type u_3
inst✝³ : TopologicalSpace M
inst✝² : ChartedSpace H M
inst✝¹ : SmoothManifoldWithCorners I M
inst✝ : I.Boundar... |
https://github.com/fpvandoorn/sard.git | f4a1bb550136e8591dcd20dca5bdcce766a958c7 | Sard/ManifoldAux.lean | PartialHomeomorph.mapsTo_extend_symm | [101, 1] | [106, 11] | rintro x ⟨s, hs, rfl⟩ | E : Type u_1
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace ℝ E
H : Type u_2
inst✝³ : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type u_3
inst✝² : TopologicalSpace M
inst✝¹ : ChartedSpace H M
inst✝ : SmoothManifoldWithCorners I M
e : PartialHomeomorph M H
⊢ MapsTo (↑(e.extend I).symm) (↑(e.extend I) '' e.sou... | case intro.intro
E : Type u_1
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace ℝ E
H : Type u_2
inst✝³ : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type u_3
inst✝² : TopologicalSpace M
inst✝¹ : ChartedSpace H M
inst✝ : SmoothManifoldWithCorners I M
e : PartialHomeomorph M H
s : M
hs : s ∈ e.source
⊢ ↑(e.extend... | Please generate a tactic in lean4 to solve the state.
STATE:
E : Type u_1
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace ℝ E
H : Type u_2
inst✝³ : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type u_3
inst✝² : TopologicalSpace M
inst✝¹ : ChartedSpace H M
inst✝ : SmoothManifoldWithCorners I M
e : PartialHomeomo... |
https://github.com/fpvandoorn/sard.git | f4a1bb550136e8591dcd20dca5bdcce766a958c7 | Sard/ManifoldAux.lean | PartialHomeomorph.mapsTo_extend_symm | [101, 1] | [106, 11] | have : (e.extend I).symm (e.extend I s) = s := e.extend_left_inv _ hs | case intro.intro
E : Type u_1
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace ℝ E
H : Type u_2
inst✝³ : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type u_3
inst✝² : TopologicalSpace M
inst✝¹ : ChartedSpace H M
inst✝ : SmoothManifoldWithCorners I M
e : PartialHomeomorph M H
s : M
hs : s ∈ e.source
⊢ ↑(e.extend... | case intro.intro
E : Type u_1
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace ℝ E
H : Type u_2
inst✝³ : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type u_3
inst✝² : TopologicalSpace M
inst✝¹ : ChartedSpace H M
inst✝ : SmoothManifoldWithCorners I M
e : PartialHomeomorph M H
s : M
hs : s ∈ e.source
this : ↑(e.e... | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro
E : Type u_1
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace ℝ E
H : Type u_2
inst✝³ : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type u_3
inst✝² : TopologicalSpace M
inst✝¹ : ChartedSpace H M
inst✝ : SmoothManifoldWithCorners I M
e... |
https://github.com/fpvandoorn/sard.git | f4a1bb550136e8591dcd20dca5bdcce766a958c7 | Sard/ManifoldAux.lean | PartialHomeomorph.mapsTo_extend_symm | [101, 1] | [106, 11] | rw [this] | case intro.intro
E : Type u_1
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace ℝ E
H : Type u_2
inst✝³ : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type u_3
inst✝² : TopologicalSpace M
inst✝¹ : ChartedSpace H M
inst✝ : SmoothManifoldWithCorners I M
e : PartialHomeomorph M H
s : M
hs : s ∈ e.source
this : ↑(e.e... | case intro.intro
E : Type u_1
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace ℝ E
H : Type u_2
inst✝³ : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type u_3
inst✝² : TopologicalSpace M
inst✝¹ : ChartedSpace H M
inst✝ : SmoothManifoldWithCorners I M
e : PartialHomeomorph M H
s : M
hs : s ∈ e.source
this : ↑(e.e... | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro
E : Type u_1
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace ℝ E
H : Type u_2
inst✝³ : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type u_3
inst✝² : TopologicalSpace M
inst✝¹ : ChartedSpace H M
inst✝ : SmoothManifoldWithCorners I M
e... |
https://github.com/fpvandoorn/sard.git | f4a1bb550136e8591dcd20dca5bdcce766a958c7 | Sard/ManifoldAux.lean | PartialHomeomorph.mapsTo_extend_symm | [101, 1] | [106, 11] | exact hs | case intro.intro
E : Type u_1
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace ℝ E
H : Type u_2
inst✝³ : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type u_3
inst✝² : TopologicalSpace M
inst✝¹ : ChartedSpace H M
inst✝ : SmoothManifoldWithCorners I M
e : PartialHomeomorph M H
s : M
hs : s ∈ e.source
this : ↑(e.e... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro
E : Type u_1
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace ℝ E
H : Type u_2
inst✝³ : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type u_3
inst✝² : TopologicalSpace M
inst✝¹ : ChartedSpace H M
inst✝ : SmoothManifoldWithCorners I M
e... |
https://github.com/fpvandoorn/sard.git | f4a1bb550136e8591dcd20dca5bdcce766a958c7 | Sard/ManifoldAux.lean | ModelWithCorners.right_inv'' | [112, 1] | [114, 25] | have : x ∈ range I := by rw [I.range_eq_univ]; exact trivial | E : Type u_1
inst✝⁶ : NormedAddCommGroup E
inst✝⁵ : NormedSpace ℝ E
H : Type u_2
inst✝⁴ : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type u_3
inst✝³ : TopologicalSpace M
inst✝² : ChartedSpace H M
inst✝¹ : SmoothManifoldWithCorners I M
inst✝ : I.Boundaryless
x : E
⊢ (↑I ∘ I.invFun) x = x | E : Type u_1
inst✝⁶ : NormedAddCommGroup E
inst✝⁵ : NormedSpace ℝ E
H : Type u_2
inst✝⁴ : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type u_3
inst✝³ : TopologicalSpace M
inst✝² : ChartedSpace H M
inst✝¹ : SmoothManifoldWithCorners I M
inst✝ : I.Boundaryless
x : E
this : x ∈ range ↑I
⊢ (↑I ∘ I.invFun) x = x | Please generate a tactic in lean4 to solve the state.
STATE:
E : Type u_1
inst✝⁶ : NormedAddCommGroup E
inst✝⁵ : NormedSpace ℝ E
H : Type u_2
inst✝⁴ : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type u_3
inst✝³ : TopologicalSpace M
inst✝² : ChartedSpace H M
inst✝¹ : SmoothManifoldWithCorners I M
inst✝ : I.Boundar... |
https://github.com/fpvandoorn/sard.git | f4a1bb550136e8591dcd20dca5bdcce766a958c7 | Sard/ManifoldAux.lean | ModelWithCorners.right_inv'' | [112, 1] | [114, 25] | exact I.right_inv this | E : Type u_1
inst✝⁶ : NormedAddCommGroup E
inst✝⁵ : NormedSpace ℝ E
H : Type u_2
inst✝⁴ : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type u_3
inst✝³ : TopologicalSpace M
inst✝² : ChartedSpace H M
inst✝¹ : SmoothManifoldWithCorners I M
inst✝ : I.Boundaryless
x : E
this : x ∈ range ↑I
⊢ (↑I ∘ I.invFun) x = x | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
E : Type u_1
inst✝⁶ : NormedAddCommGroup E
inst✝⁵ : NormedSpace ℝ E
H : Type u_2
inst✝⁴ : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type u_3
inst✝³ : TopologicalSpace M
inst✝² : ChartedSpace H M
inst✝¹ : SmoothManifoldWithCorners I M
inst✝ : I.Boundar... |
https://github.com/fpvandoorn/sard.git | f4a1bb550136e8591dcd20dca5bdcce766a958c7 | Sard/ManifoldAux.lean | ModelWithCorners.right_inv'' | [112, 1] | [114, 25] | rw [I.range_eq_univ] | E : Type u_1
inst✝⁶ : NormedAddCommGroup E
inst✝⁵ : NormedSpace ℝ E
H : Type u_2
inst✝⁴ : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type u_3
inst✝³ : TopologicalSpace M
inst✝² : ChartedSpace H M
inst✝¹ : SmoothManifoldWithCorners I M
inst✝ : I.Boundaryless
x : E
⊢ x ∈ range ↑I | E : Type u_1
inst✝⁶ : NormedAddCommGroup E
inst✝⁵ : NormedSpace ℝ E
H : Type u_2
inst✝⁴ : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type u_3
inst✝³ : TopologicalSpace M
inst✝² : ChartedSpace H M
inst✝¹ : SmoothManifoldWithCorners I M
inst✝ : I.Boundaryless
x : E
⊢ x ∈ univ | Please generate a tactic in lean4 to solve the state.
STATE:
E : Type u_1
inst✝⁶ : NormedAddCommGroup E
inst✝⁵ : NormedSpace ℝ E
H : Type u_2
inst✝⁴ : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type u_3
inst✝³ : TopologicalSpace M
inst✝² : ChartedSpace H M
inst✝¹ : SmoothManifoldWithCorners I M
inst✝ : I.Boundar... |
https://github.com/fpvandoorn/sard.git | f4a1bb550136e8591dcd20dca5bdcce766a958c7 | Sard/ManifoldAux.lean | ModelWithCorners.right_inv'' | [112, 1] | [114, 25] | exact trivial | E : Type u_1
inst✝⁶ : NormedAddCommGroup E
inst✝⁵ : NormedSpace ℝ E
H : Type u_2
inst✝⁴ : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type u_3
inst✝³ : TopologicalSpace M
inst✝² : ChartedSpace H M
inst✝¹ : SmoothManifoldWithCorners I M
inst✝ : I.Boundaryless
x : E
⊢ x ∈ univ | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
E : Type u_1
inst✝⁶ : NormedAddCommGroup E
inst✝⁵ : NormedSpace ℝ E
H : Type u_2
inst✝⁴ : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type u_3
inst✝³ : TopologicalSpace M
inst✝² : ChartedSpace H M
inst✝¹ : SmoothManifoldWithCorners I M
inst✝ : I.Boundar... |
https://github.com/fpvandoorn/sard.git | f4a1bb550136e8591dcd20dca5bdcce766a958c7 | Sard/ManifoldAux.lean | PartialHomeomorph.extend_right_inv | [116, 1] | [123, 29] | have : I.invFun x ∈ e.target := by aesop | E : Type u_1
inst✝⁶ : NormedAddCommGroup E
inst✝⁵ : NormedSpace ℝ E
H : Type u_2
inst✝⁴ : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type u_3
inst✝³ : TopologicalSpace M
inst✝² : ChartedSpace H M
inst✝¹ : SmoothManifoldWithCorners I M
inst✝ : I.Boundaryless
e : PartialHomeomorph M H
x : E
hx : x ∈ ↑(e.extend I) ... | E : Type u_1
inst✝⁶ : NormedAddCommGroup E
inst✝⁵ : NormedSpace ℝ E
H : Type u_2
inst✝⁴ : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type u_3
inst✝³ : TopologicalSpace M
inst✝² : ChartedSpace H M
inst✝¹ : SmoothManifoldWithCorners I M
inst✝ : I.Boundaryless
e : PartialHomeomorph M H
x : E
hx : x ∈ ↑(e.extend I) ... | Please generate a tactic in lean4 to solve the state.
STATE:
E : Type u_1
inst✝⁶ : NormedAddCommGroup E
inst✝⁵ : NormedSpace ℝ E
H : Type u_2
inst✝⁴ : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type u_3
inst✝³ : TopologicalSpace M
inst✝² : ChartedSpace H M
inst✝¹ : SmoothManifoldWithCorners I M
inst✝ : I.Boundar... |
https://github.com/fpvandoorn/sard.git | f4a1bb550136e8591dcd20dca5bdcce766a958c7 | Sard/ManifoldAux.lean | PartialHomeomorph.extend_right_inv | [116, 1] | [123, 29] | have aux : ∀ y : H, y ∈ e.target → (e ∘ e.invFun) y = y := by intros; aesop | E : Type u_1
inst✝⁶ : NormedAddCommGroup E
inst✝⁵ : NormedSpace ℝ E
H : Type u_2
inst✝⁴ : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type u_3
inst✝³ : TopologicalSpace M
inst✝² : ChartedSpace H M
inst✝¹ : SmoothManifoldWithCorners I M
inst✝ : I.Boundaryless
e : PartialHomeomorph M H
x : E
hx : x ∈ ↑(e.extend I) ... | E : Type u_1
inst✝⁶ : NormedAddCommGroup E
inst✝⁵ : NormedSpace ℝ E
H : Type u_2
inst✝⁴ : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type u_3
inst✝³ : TopologicalSpace M
inst✝² : ChartedSpace H M
inst✝¹ : SmoothManifoldWithCorners I M
inst✝ : I.Boundaryless
e : PartialHomeomorph M H
x : E
hx : x ∈ ↑(e.extend I) ... | Please generate a tactic in lean4 to solve the state.
STATE:
E : Type u_1
inst✝⁶ : NormedAddCommGroup E
inst✝⁵ : NormedSpace ℝ E
H : Type u_2
inst✝⁴ : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type u_3
inst✝³ : TopologicalSpace M
inst✝² : ChartedSpace H M
inst✝¹ : SmoothManifoldWithCorners I M
inst✝ : I.Boundar... |
https://github.com/fpvandoorn/sard.git | f4a1bb550136e8591dcd20dca5bdcce766a958c7 | Sard/ManifoldAux.lean | PartialHomeomorph.extend_right_inv | [116, 1] | [123, 29] | calc ((e.extend I) ∘ (e.extend I).symm) x
_ = I ((e ∘ e.invFun) (I.invFun x)) := rfl
_ = I (I.invFun x) := by simp_rw [aux (I.invFun x) this]
_ = x := I.right_inv'' x | E : Type u_1
inst✝⁶ : NormedAddCommGroup E
inst✝⁵ : NormedSpace ℝ E
H : Type u_2
inst✝⁴ : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type u_3
inst✝³ : TopologicalSpace M
inst✝² : ChartedSpace H M
inst✝¹ : SmoothManifoldWithCorners I M
inst✝ : I.Boundaryless
e : PartialHomeomorph M H
x : E
hx : x ∈ ↑(e.extend I) ... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
E : Type u_1
inst✝⁶ : NormedAddCommGroup E
inst✝⁵ : NormedSpace ℝ E
H : Type u_2
inst✝⁴ : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type u_3
inst✝³ : TopologicalSpace M
inst✝² : ChartedSpace H M
inst✝¹ : SmoothManifoldWithCorners I M
inst✝ : I.Boundar... |
https://github.com/fpvandoorn/sard.git | f4a1bb550136e8591dcd20dca5bdcce766a958c7 | Sard/ManifoldAux.lean | PartialHomeomorph.extend_right_inv | [116, 1] | [123, 29] | aesop | E : Type u_1
inst✝⁶ : NormedAddCommGroup E
inst✝⁵ : NormedSpace ℝ E
H : Type u_2
inst✝⁴ : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type u_3
inst✝³ : TopologicalSpace M
inst✝² : ChartedSpace H M
inst✝¹ : SmoothManifoldWithCorners I M
inst✝ : I.Boundaryless
e : PartialHomeomorph M H
x : E
hx : x ∈ ↑(e.extend I) ... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
E : Type u_1
inst✝⁶ : NormedAddCommGroup E
inst✝⁵ : NormedSpace ℝ E
H : Type u_2
inst✝⁴ : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type u_3
inst✝³ : TopologicalSpace M
inst✝² : ChartedSpace H M
inst✝¹ : SmoothManifoldWithCorners I M
inst✝ : I.Boundar... |
https://github.com/fpvandoorn/sard.git | f4a1bb550136e8591dcd20dca5bdcce766a958c7 | Sard/ManifoldAux.lean | PartialHomeomorph.extend_right_inv | [116, 1] | [123, 29] | intros | E : Type u_1
inst✝⁶ : NormedAddCommGroup E
inst✝⁵ : NormedSpace ℝ E
H : Type u_2
inst✝⁴ : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type u_3
inst✝³ : TopologicalSpace M
inst✝² : ChartedSpace H M
inst✝¹ : SmoothManifoldWithCorners I M
inst✝ : I.Boundaryless
e : PartialHomeomorph M H
x : E
hx : x ∈ ↑(e.extend I) ... | E : Type u_1
inst✝⁶ : NormedAddCommGroup E
inst✝⁵ : NormedSpace ℝ E
H : Type u_2
inst✝⁴ : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type u_3
inst✝³ : TopologicalSpace M
inst✝² : ChartedSpace H M
inst✝¹ : SmoothManifoldWithCorners I M
inst✝ : I.Boundaryless
e : PartialHomeomorph M H
x : E
hx : x ∈ ↑(e.extend I) ... | Please generate a tactic in lean4 to solve the state.
STATE:
E : Type u_1
inst✝⁶ : NormedAddCommGroup E
inst✝⁵ : NormedSpace ℝ E
H : Type u_2
inst✝⁴ : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type u_3
inst✝³ : TopologicalSpace M
inst✝² : ChartedSpace H M
inst✝¹ : SmoothManifoldWithCorners I M
inst✝ : I.Boundar... |
https://github.com/fpvandoorn/sard.git | f4a1bb550136e8591dcd20dca5bdcce766a958c7 | Sard/ManifoldAux.lean | PartialHomeomorph.extend_right_inv | [116, 1] | [123, 29] | aesop | E : Type u_1
inst✝⁶ : NormedAddCommGroup E
inst✝⁵ : NormedSpace ℝ E
H : Type u_2
inst✝⁴ : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type u_3
inst✝³ : TopologicalSpace M
inst✝² : ChartedSpace H M
inst✝¹ : SmoothManifoldWithCorners I M
inst✝ : I.Boundaryless
e : PartialHomeomorph M H
x : E
hx : x ∈ ↑(e.extend I) ... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
E : Type u_1
inst✝⁶ : NormedAddCommGroup E
inst✝⁵ : NormedSpace ℝ E
H : Type u_2
inst✝⁴ : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type u_3
inst✝³ : TopologicalSpace M
inst✝² : ChartedSpace H M
inst✝¹ : SmoothManifoldWithCorners I M
inst✝ : I.Boundar... |
https://github.com/fpvandoorn/sard.git | f4a1bb550136e8591dcd20dca5bdcce766a958c7 | Sard/ManifoldAux.lean | PartialHomeomorph.extend_right_inv | [116, 1] | [123, 29] | simp_rw [aux (I.invFun x) this] | E : Type u_1
inst✝⁶ : NormedAddCommGroup E
inst✝⁵ : NormedSpace ℝ E
H : Type u_2
inst✝⁴ : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type u_3
inst✝³ : TopologicalSpace M
inst✝² : ChartedSpace H M
inst✝¹ : SmoothManifoldWithCorners I M
inst✝ : I.Boundaryless
e : PartialHomeomorph M H
x : E
hx : x ∈ ↑(e.extend I) ... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
E : Type u_1
inst✝⁶ : NormedAddCommGroup E
inst✝⁵ : NormedSpace ℝ E
H : Type u_2
inst✝⁴ : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type u_3
inst✝³ : TopologicalSpace M
inst✝² : ChartedSpace H M
inst✝¹ : SmoothManifoldWithCorners I M
inst✝ : I.Boundar... |
https://github.com/fpvandoorn/sard.git | f4a1bb550136e8591dcd20dca5bdcce766a958c7 | Sard/ManifoldAux.lean | bijective_iff_inverses' | [140, 1] | [151, 38] | have h : f.toFun ∘ g.toFun = id := calc f.toFun ∘ g.toFun
_ = (f.comp g).toFun := rfl
_ = (ContinuousLinearMap.id ℝ Y).toFun := by rw [h2]
_ = id := rfl | E : Type u_1
inst✝¹⁵ : NormedAddCommGroup E
inst✝¹⁴ : NormedSpace ℝ E
H : Type u_2
inst✝¹³ : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type u_3
inst✝¹² : TopologicalSpace M
inst✝¹¹ : ChartedSpace H M
inst✝¹⁰ : SmoothManifoldWithCorners I M
F : Type u_4
inst✝⁹ : NormedAddCommGroup F
inst✝⁸ : NormedSpace ℝ F
G : ... | E : Type u_1
inst✝¹⁵ : NormedAddCommGroup E
inst✝¹⁴ : NormedSpace ℝ E
H : Type u_2
inst✝¹³ : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type u_3
inst✝¹² : TopologicalSpace M
inst✝¹¹ : ChartedSpace H M
inst✝¹⁰ : SmoothManifoldWithCorners I M
F : Type u_4
inst✝⁹ : NormedAddCommGroup F
inst✝⁸ : NormedSpace ℝ F
G : ... | Please generate a tactic in lean4 to solve the state.
STATE:
E : Type u_1
inst✝¹⁵ : NormedAddCommGroup E
inst✝¹⁴ : NormedSpace ℝ E
H : Type u_2
inst✝¹³ : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type u_3
inst✝¹² : TopologicalSpace M
inst✝¹¹ : ChartedSpace H M
inst✝¹⁰ : SmoothManifoldWithCorners I M
F : Type u_... |
https://github.com/fpvandoorn/sard.git | f4a1bb550136e8591dcd20dca5bdcce766a958c7 | Sard/ManifoldAux.lean | bijective_iff_inverses' | [140, 1] | [151, 38] | have : g.toFun ∘ f.toFun = id := by calc g.toFun ∘ f.toFun
_ = (g.comp f).toFun := rfl
_ = (ContinuousLinearMap.id ℝ X).toFun := by rw [h1]
_ = id := rfl | E : Type u_1
inst✝¹⁵ : NormedAddCommGroup E
inst✝¹⁴ : NormedSpace ℝ E
H : Type u_2
inst✝¹³ : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type u_3
inst✝¹² : TopologicalSpace M
inst✝¹¹ : ChartedSpace H M
inst✝¹⁰ : SmoothManifoldWithCorners I M
F : Type u_4
inst✝⁹ : NormedAddCommGroup F
inst✝⁸ : NormedSpace ℝ F
G : ... | E : Type u_1
inst✝¹⁵ : NormedAddCommGroup E
inst✝¹⁴ : NormedSpace ℝ E
H : Type u_2
inst✝¹³ : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type u_3
inst✝¹² : TopologicalSpace M
inst✝¹¹ : ChartedSpace H M
inst✝¹⁰ : SmoothManifoldWithCorners I M
F : Type u_4
inst✝⁹ : NormedAddCommGroup F
inst✝⁸ : NormedSpace ℝ F
G : ... | Please generate a tactic in lean4 to solve the state.
STATE:
E : Type u_1
inst✝¹⁵ : NormedAddCommGroup E
inst✝¹⁴ : NormedSpace ℝ E
H : Type u_2
inst✝¹³ : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type u_3
inst✝¹² : TopologicalSpace M
inst✝¹¹ : ChartedSpace H M
inst✝¹⁰ : SmoothManifoldWithCorners I M
F : Type u_... |
https://github.com/fpvandoorn/sard.git | f4a1bb550136e8591dcd20dca5bdcce766a958c7 | Sard/ManifoldAux.lean | bijective_iff_inverses' | [140, 1] | [151, 38] | exact bijective_iff_inverses this h | E : Type u_1
inst✝¹⁵ : NormedAddCommGroup E
inst✝¹⁴ : NormedSpace ℝ E
H : Type u_2
inst✝¹³ : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type u_3
inst✝¹² : TopologicalSpace M
inst✝¹¹ : ChartedSpace H M
inst✝¹⁰ : SmoothManifoldWithCorners I M
F : Type u_4
inst✝⁹ : NormedAddCommGroup F
inst✝⁸ : NormedSpace ℝ F
G : ... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
E : Type u_1
inst✝¹⁵ : NormedAddCommGroup E
inst✝¹⁴ : NormedSpace ℝ E
H : Type u_2
inst✝¹³ : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type u_3
inst✝¹² : TopologicalSpace M
inst✝¹¹ : ChartedSpace H M
inst✝¹⁰ : SmoothManifoldWithCorners I M
F : Type u_... |
https://github.com/fpvandoorn/sard.git | f4a1bb550136e8591dcd20dca5bdcce766a958c7 | Sard/ManifoldAux.lean | bijective_iff_inverses' | [140, 1] | [151, 38] | rw [h2] | E : Type u_1
inst✝¹⁵ : NormedAddCommGroup E
inst✝¹⁴ : NormedSpace ℝ E
H : Type u_2
inst✝¹³ : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type u_3
inst✝¹² : TopologicalSpace M
inst✝¹¹ : ChartedSpace H M
inst✝¹⁰ : SmoothManifoldWithCorners I M
F : Type u_4
inst✝⁹ : NormedAddCommGroup F
inst✝⁸ : NormedSpace ℝ F
G : ... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
E : Type u_1
inst✝¹⁵ : NormedAddCommGroup E
inst✝¹⁴ : NormedSpace ℝ E
H : Type u_2
inst✝¹³ : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type u_3
inst✝¹² : TopologicalSpace M
inst✝¹¹ : ChartedSpace H M
inst✝¹⁰ : SmoothManifoldWithCorners I M
F : Type u_... |
https://github.com/fpvandoorn/sard.git | f4a1bb550136e8591dcd20dca5bdcce766a958c7 | Sard/ManifoldAux.lean | bijective_iff_inverses' | [140, 1] | [151, 38] | calc g.toFun ∘ f.toFun
_ = (g.comp f).toFun := rfl
_ = (ContinuousLinearMap.id ℝ X).toFun := by rw [h1]
_ = id := rfl | E : Type u_1
inst✝¹⁵ : NormedAddCommGroup E
inst✝¹⁴ : NormedSpace ℝ E
H : Type u_2
inst✝¹³ : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type u_3
inst✝¹² : TopologicalSpace M
inst✝¹¹ : ChartedSpace H M
inst✝¹⁰ : SmoothManifoldWithCorners I M
F : Type u_4
inst✝⁹ : NormedAddCommGroup F
inst✝⁸ : NormedSpace ℝ F
G : ... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
E : Type u_1
inst✝¹⁵ : NormedAddCommGroup E
inst✝¹⁴ : NormedSpace ℝ E
H : Type u_2
inst✝¹³ : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type u_3
inst✝¹² : TopologicalSpace M
inst✝¹¹ : ChartedSpace H M
inst✝¹⁰ : SmoothManifoldWithCorners I M
F : Type u_... |
https://github.com/fpvandoorn/sard.git | f4a1bb550136e8591dcd20dca5bdcce766a958c7 | Sard/ManifoldAux.lean | bijective_iff_inverses' | [140, 1] | [151, 38] | rw [h1] | E : Type u_1
inst✝¹⁵ : NormedAddCommGroup E
inst✝¹⁴ : NormedSpace ℝ E
H : Type u_2
inst✝¹³ : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type u_3
inst✝¹² : TopologicalSpace M
inst✝¹¹ : ChartedSpace H M
inst✝¹⁰ : SmoothManifoldWithCorners I M
F : Type u_4
inst✝⁹ : NormedAddCommGroup F
inst✝⁸ : NormedSpace ℝ F
G : ... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
E : Type u_1
inst✝¹⁵ : NormedAddCommGroup E
inst✝¹⁴ : NormedSpace ℝ E
H : Type u_2
inst✝¹³ : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type u_3
inst✝¹² : TopologicalSpace M
inst✝¹¹ : ChartedSpace H M
inst✝¹⁰ : SmoothManifoldWithCorners I M
F : Type u_... |
https://github.com/fpvandoorn/sard.git | f4a1bb550136e8591dcd20dca5bdcce766a958c7 | Sard/ManifoldAux.lean | diffeoOn_differential_bijective | [163, 1] | [206, 42] | set A := mfderiv I J f x | E : Type u_1
inst✝¹¹ : NormedAddCommGroup E
inst✝¹⁰ : NormedSpace ℝ E
H : Type u_2
inst✝⁹ : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type u_3
inst✝⁸ : TopologicalSpace M
inst✝⁷ : ChartedSpace H M
inst✝⁶ : SmoothManifoldWithCorners I M
F : Type u_4
inst✝⁵ : NormedAddCommGroup F
inst✝⁴ : NormedSpace ℝ F
G : Type... | E : Type u_1
inst✝¹¹ : NormedAddCommGroup E
inst✝¹⁰ : NormedSpace ℝ E
H : Type u_2
inst✝⁹ : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type u_3
inst✝⁸ : TopologicalSpace M
inst✝⁷ : ChartedSpace H M
inst✝⁶ : SmoothManifoldWithCorners I M
F : Type u_4
inst✝⁵ : NormedAddCommGroup F
inst✝⁴ : NormedSpace ℝ F
G : Type... | Please generate a tactic in lean4 to solve the state.
STATE:
E : Type u_1
inst✝¹¹ : NormedAddCommGroup E
inst✝¹⁰ : NormedSpace ℝ E
H : Type u_2
inst✝⁹ : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type u_3
inst✝⁸ : TopologicalSpace M
inst✝⁷ : ChartedSpace H M
inst✝⁶ : SmoothManifoldWithCorners I M
F : Type u_4
in... |
https://github.com/fpvandoorn/sard.git | f4a1bb550136e8591dcd20dca5bdcce766a958c7 | Sard/ManifoldAux.lean | diffeoOn_differential_bijective | [163, 1] | [206, 42] | let y := f x | E : Type u_1
inst✝¹¹ : NormedAddCommGroup E
inst✝¹⁰ : NormedSpace ℝ E
H : Type u_2
inst✝⁹ : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type u_3
inst✝⁸ : TopologicalSpace M
inst✝⁷ : ChartedSpace H M
inst✝⁶ : SmoothManifoldWithCorners I M
F : Type u_4
inst✝⁵ : NormedAddCommGroup F
inst✝⁴ : NormedSpace ℝ F
G : Type... | E : Type u_1
inst✝¹¹ : NormedAddCommGroup E
inst✝¹⁰ : NormedSpace ℝ E
H : Type u_2
inst✝⁹ : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type u_3
inst✝⁸ : TopologicalSpace M
inst✝⁷ : ChartedSpace H M
inst✝⁶ : SmoothManifoldWithCorners I M
F : Type u_4
inst✝⁵ : NormedAddCommGroup F
inst✝⁴ : NormedSpace ℝ F
G : Type... | Please generate a tactic in lean4 to solve the state.
STATE:
E : Type u_1
inst✝¹¹ : NormedAddCommGroup E
inst✝¹⁰ : NormedSpace ℝ E
H : Type u_2
inst✝⁹ : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type u_3
inst✝⁸ : TopologicalSpace M
inst✝⁷ : ChartedSpace H M
inst✝⁶ : SmoothManifoldWithCorners I M
F : Type u_4
in... |
https://github.com/fpvandoorn/sard.git | f4a1bb550136e8591dcd20dca5bdcce766a958c7 | Sard/ManifoldAux.lean | diffeoOn_differential_bijective | [163, 1] | [206, 42] | have hyx : g y = x := hleft_inv x hx | E : Type u_1
inst✝¹¹ : NormedAddCommGroup E
inst✝¹⁰ : NormedSpace ℝ E
H : Type u_2
inst✝⁹ : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type u_3
inst✝⁸ : TopologicalSpace M
inst✝⁷ : ChartedSpace H M
inst✝⁶ : SmoothManifoldWithCorners I M
F : Type u_4
inst✝⁵ : NormedAddCommGroup F
inst✝⁴ : NormedSpace ℝ F
G : Type... | E : Type u_1
inst✝¹¹ : NormedAddCommGroup E
inst✝¹⁰ : NormedSpace ℝ E
H : Type u_2
inst✝⁹ : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type u_3
inst✝⁸ : TopologicalSpace M
inst✝⁷ : ChartedSpace H M
inst✝⁶ : SmoothManifoldWithCorners I M
F : Type u_4
inst✝⁵ : NormedAddCommGroup F
inst✝⁴ : NormedSpace ℝ F
G : Type... | Please generate a tactic in lean4 to solve the state.
STATE:
E : Type u_1
inst✝¹¹ : NormedAddCommGroup E
inst✝¹⁰ : NormedSpace ℝ E
H : Type u_2
inst✝⁹ : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type u_3
inst✝⁸ : TopologicalSpace M
inst✝⁷ : ChartedSpace H M
inst✝⁶ : SmoothManifoldWithCorners I M
F : Type u_4
in... |
https://github.com/fpvandoorn/sard.git | f4a1bb550136e8591dcd20dca5bdcce766a958c7 | Sard/ManifoldAux.lean | diffeoOn_differential_bijective | [163, 1] | [206, 42] | have hysource : y ∈ t := hst hx | E : Type u_1
inst✝¹¹ : NormedAddCommGroup E
inst✝¹⁰ : NormedSpace ℝ E
H : Type u_2
inst✝⁹ : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type u_3
inst✝⁸ : TopologicalSpace M
inst✝⁷ : ChartedSpace H M
inst✝⁶ : SmoothManifoldWithCorners I M
F : Type u_4
inst✝⁵ : NormedAddCommGroup F
inst✝⁴ : NormedSpace ℝ F
G : Type... | E : Type u_1
inst✝¹¹ : NormedAddCommGroup E
inst✝¹⁰ : NormedSpace ℝ E
H : Type u_2
inst✝⁹ : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type u_3
inst✝⁸ : TopologicalSpace M
inst✝⁷ : ChartedSpace H M
inst✝⁶ : SmoothManifoldWithCorners I M
F : Type u_4
inst✝⁵ : NormedAddCommGroup F
inst✝⁴ : NormedSpace ℝ F
G : Type... | Please generate a tactic in lean4 to solve the state.
STATE:
E : Type u_1
inst✝¹¹ : NormedAddCommGroup E
inst✝¹⁰ : NormedSpace ℝ E
H : Type u_2
inst✝⁹ : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type u_3
inst✝⁸ : TopologicalSpace M
inst✝⁷ : ChartedSpace H M
inst✝⁶ : SmoothManifoldWithCorners I M
F : Type u_4
in... |
https://github.com/fpvandoorn/sard.git | f4a1bb550136e8591dcd20dca5bdcce766a958c7 | Sard/ManifoldAux.lean | diffeoOn_differential_bijective | [163, 1] | [206, 42] | have : f '' s = t := subset_antisymm (mapsTo'.mp hst) (fun y hy ↦ ⟨g y, hts hy, hright_inv y hy⟩) | E : Type u_1
inst✝¹¹ : NormedAddCommGroup E
inst✝¹⁰ : NormedSpace ℝ E
H : Type u_2
inst✝⁹ : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type u_3
inst✝⁸ : TopologicalSpace M
inst✝⁷ : ChartedSpace H M
inst✝⁶ : SmoothManifoldWithCorners I M
F : Type u_4
inst✝⁵ : NormedAddCommGroup F
inst✝⁴ : NormedSpace ℝ F
G : Type... | E : Type u_1
inst✝¹¹ : NormedAddCommGroup E
inst✝¹⁰ : NormedSpace ℝ E
H : Type u_2
inst✝⁹ : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type u_3
inst✝⁸ : TopologicalSpace M
inst✝⁷ : ChartedSpace H M
inst✝⁶ : SmoothManifoldWithCorners I M
F : Type u_4
inst✝⁵ : NormedAddCommGroup F
inst✝⁴ : NormedSpace ℝ F
G : Type... | Please generate a tactic in lean4 to solve the state.
STATE:
E : Type u_1
inst✝¹¹ : NormedAddCommGroup E
inst✝¹⁰ : NormedSpace ℝ E
H : Type u_2
inst✝⁹ : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type u_3
inst✝⁸ : TopologicalSpace M
inst✝⁷ : ChartedSpace H M
inst✝⁶ : SmoothManifoldWithCorners I M
F : Type u_4
in... |
https://github.com/fpvandoorn/sard.git | f4a1bb550136e8591dcd20dca5bdcce766a958c7 | Sard/ManifoldAux.lean | diffeoOn_differential_bijective | [163, 1] | [206, 42] | have : g '' t = s := by
rw [← this, ← image_comp]
exact EqOn.image_eq_self hleft_inv | E : Type u_1
inst✝¹¹ : NormedAddCommGroup E
inst✝¹⁰ : NormedSpace ℝ E
H : Type u_2
inst✝⁹ : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type u_3
inst✝⁸ : TopologicalSpace M
inst✝⁷ : ChartedSpace H M
inst✝⁶ : SmoothManifoldWithCorners I M
F : Type u_4
inst✝⁵ : NormedAddCommGroup F
inst✝⁴ : NormedSpace ℝ F
G : Type... | E : Type u_1
inst✝¹¹ : NormedAddCommGroup E
inst✝¹⁰ : NormedSpace ℝ E
H : Type u_2
inst✝⁹ : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type u_3
inst✝⁸ : TopologicalSpace M
inst✝⁷ : ChartedSpace H M
inst✝⁶ : SmoothManifoldWithCorners I M
F : Type u_4
inst✝⁵ : NormedAddCommGroup F
inst✝⁴ : NormedSpace ℝ F
G : Type... | Please generate a tactic in lean4 to solve the state.
STATE:
E : Type u_1
inst✝¹¹ : NormedAddCommGroup E
inst✝¹⁰ : NormedSpace ℝ E
H : Type u_2
inst✝⁹ : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type u_3
inst✝⁸ : TopologicalSpace M
inst✝⁷ : ChartedSpace H M
inst✝⁶ : SmoothManifoldWithCorners I M
F : Type u_4
in... |
https://github.com/fpvandoorn/sard.git | f4a1bb550136e8591dcd20dca5bdcce766a958c7 | Sard/ManifoldAux.lean | diffeoOn_differential_bijective | [163, 1] | [206, 42] | have hopen : IsOpen (g '' t) := by rw [this]; exact hs | E : Type u_1
inst✝¹¹ : NormedAddCommGroup E
inst✝¹⁰ : NormedSpace ℝ E
H : Type u_2
inst✝⁹ : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type u_3
inst✝⁸ : TopologicalSpace M
inst✝⁷ : ChartedSpace H M
inst✝⁶ : SmoothManifoldWithCorners I M
F : Type u_4
inst✝⁵ : NormedAddCommGroup F
inst✝⁴ : NormedSpace ℝ F
G : Type... | E : Type u_1
inst✝¹¹ : NormedAddCommGroup E
inst✝¹⁰ : NormedSpace ℝ E
H : Type u_2
inst✝⁹ : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type u_3
inst✝⁸ : TopologicalSpace M
inst✝⁷ : ChartedSpace H M
inst✝⁶ : SmoothManifoldWithCorners I M
F : Type u_4
inst✝⁵ : NormedAddCommGroup F
inst✝⁴ : NormedSpace ℝ F
G : Type... | Please generate a tactic in lean4 to solve the state.
STATE:
E : Type u_1
inst✝¹¹ : NormedAddCommGroup E
inst✝¹⁰ : NormedSpace ℝ E
H : Type u_2
inst✝⁹ : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type u_3
inst✝⁸ : TopologicalSpace M
inst✝⁷ : ChartedSpace H M
inst✝⁶ : SmoothManifoldWithCorners I M
F : Type u_4
in... |
https://github.com/fpvandoorn/sard.git | f4a1bb550136e8591dcd20dca5bdcce766a958c7 | Sard/ManifoldAux.lean | diffeoOn_differential_bijective | [163, 1] | [206, 42] | have hx2 : x ∈ g '' t := by simp_rw [this]; exact hx | E : Type u_1
inst✝¹¹ : NormedAddCommGroup E
inst✝¹⁰ : NormedSpace ℝ E
H : Type u_2
inst✝⁹ : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type u_3
inst✝⁸ : TopologicalSpace M
inst✝⁷ : ChartedSpace H M
inst✝⁶ : SmoothManifoldWithCorners I M
F : Type u_4
inst✝⁵ : NormedAddCommGroup F
inst✝⁴ : NormedSpace ℝ F
G : Type... | E : Type u_1
inst✝¹¹ : NormedAddCommGroup E
inst✝¹⁰ : NormedSpace ℝ E
H : Type u_2
inst✝⁹ : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type u_3
inst✝⁸ : TopologicalSpace M
inst✝⁷ : ChartedSpace H M
inst✝⁶ : SmoothManifoldWithCorners I M
F : Type u_4
inst✝⁵ : NormedAddCommGroup F
inst✝⁴ : NormedSpace ℝ F
G : Type... | Please generate a tactic in lean4 to solve the state.
STATE:
E : Type u_1
inst✝¹¹ : NormedAddCommGroup E
inst✝¹⁰ : NormedSpace ℝ E
H : Type u_2
inst✝⁹ : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type u_3
inst✝⁸ : TopologicalSpace M
inst✝⁷ : ChartedSpace H M
inst✝⁶ : SmoothManifoldWithCorners I M
F : Type u_4
in... |
https://github.com/fpvandoorn/sard.git | f4a1bb550136e8591dcd20dca5bdcce766a958c7 | Sard/ManifoldAux.lean | diffeoOn_differential_bijective | [163, 1] | [206, 42] | let A' := mfderiv J I g y | E : Type u_1
inst✝¹¹ : NormedAddCommGroup E
inst✝¹⁰ : NormedSpace ℝ E
H : Type u_2
inst✝⁹ : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type u_3
inst✝⁸ : TopologicalSpace M
inst✝⁷ : ChartedSpace H M
inst✝⁶ : SmoothManifoldWithCorners I M
F : Type u_4
inst✝⁵ : NormedAddCommGroup F
inst✝⁴ : NormedSpace ℝ F
G : Type... | E : Type u_1
inst✝¹¹ : NormedAddCommGroup E
inst✝¹⁰ : NormedSpace ℝ E
H : Type u_2
inst✝⁹ : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type u_3
inst✝⁸ : TopologicalSpace M
inst✝⁷ : ChartedSpace H M
inst✝⁶ : SmoothManifoldWithCorners I M
F : Type u_4
inst✝⁵ : NormedAddCommGroup F
inst✝⁴ : NormedSpace ℝ F
G : Type... | Please generate a tactic in lean4 to solve the state.
STATE:
E : Type u_1
inst✝¹¹ : NormedAddCommGroup E
inst✝¹⁰ : NormedSpace ℝ E
H : Type u_2
inst✝⁹ : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type u_3
inst✝⁸ : TopologicalSpace M
inst✝⁷ : ChartedSpace H M
inst✝⁶ : SmoothManifoldWithCorners I M
F : Type u_4
in... |
https://github.com/fpvandoorn/sard.git | f4a1bb550136e8591dcd20dca5bdcce766a958c7 | Sard/ManifoldAux.lean | diffeoOn_differential_bijective | [163, 1] | [206, 42] | have hr : 1 ≤ (r : ℕ∞) := Nat.one_le_cast.mpr (Nat.one_le_of_lt hr) | E : Type u_1
inst✝¹¹ : NormedAddCommGroup E
inst✝¹⁰ : NormedSpace ℝ E
H : Type u_2
inst✝⁹ : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type u_3
inst✝⁸ : TopologicalSpace M
inst✝⁷ : ChartedSpace H M
inst✝⁶ : SmoothManifoldWithCorners I M
F : Type u_4
inst✝⁵ : NormedAddCommGroup F
inst✝⁴ : NormedSpace ℝ F
G : Type... | E : Type u_1
inst✝¹¹ : NormedAddCommGroup E
inst✝¹⁰ : NormedSpace ℝ E
H : Type u_2
inst✝⁹ : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type u_3
inst✝⁸ : TopologicalSpace M
inst✝⁷ : ChartedSpace H M
inst✝⁶ : SmoothManifoldWithCorners I M
F : Type u_4
inst✝⁵ : NormedAddCommGroup F
inst✝⁴ : NormedSpace ℝ F
G : Type... | Please generate a tactic in lean4 to solve the state.
STATE:
E : Type u_1
inst✝¹¹ : NormedAddCommGroup E
inst✝¹⁰ : NormedSpace ℝ E
H : Type u_2
inst✝⁹ : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type u_3
inst✝⁸ : TopologicalSpace M
inst✝⁷ : ChartedSpace H M
inst✝⁶ : SmoothManifoldWithCorners I M
F : Type u_4
in... |
https://github.com/fpvandoorn/sard.git | f4a1bb550136e8591dcd20dca5bdcce766a958c7 | Sard/ManifoldAux.lean | diffeoOn_differential_bijective | [163, 1] | [206, 42] | have hgat : MDifferentiableAt J I g y :=
(hg.contMDiffAt (ht.mem_nhds (hst hx))).mdifferentiableAt hr | E : Type u_1
inst✝¹¹ : NormedAddCommGroup E
inst✝¹⁰ : NormedSpace ℝ E
H : Type u_2
inst✝⁹ : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type u_3
inst✝⁸ : TopologicalSpace M
inst✝⁷ : ChartedSpace H M
inst✝⁶ : SmoothManifoldWithCorners I M
F : Type u_4
inst✝⁵ : NormedAddCommGroup F
inst✝⁴ : NormedSpace ℝ F
G : Type... | E : Type u_1
inst✝¹¹ : NormedAddCommGroup E
inst✝¹⁰ : NormedSpace ℝ E
H : Type u_2
inst✝⁹ : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type u_3
inst✝⁸ : TopologicalSpace M
inst✝⁷ : ChartedSpace H M
inst✝⁶ : SmoothManifoldWithCorners I M
F : Type u_4
inst✝⁵ : NormedAddCommGroup F
inst✝⁴ : NormedSpace ℝ F
G : Type... | Please generate a tactic in lean4 to solve the state.
STATE:
E : Type u_1
inst✝¹¹ : NormedAddCommGroup E
inst✝¹⁰ : NormedSpace ℝ E
H : Type u_2
inst✝⁹ : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type u_3
inst✝⁸ : TopologicalSpace M
inst✝⁷ : ChartedSpace H M
inst✝⁶ : SmoothManifoldWithCorners I M
F : Type u_4
in... |
https://github.com/fpvandoorn/sard.git | f4a1bb550136e8591dcd20dca5bdcce766a958c7 | Sard/ManifoldAux.lean | diffeoOn_differential_bijective | [163, 1] | [206, 42] | have hfat : MDifferentiableAt I J f x :=
(hf.contMDiffAt (hs.mem_nhds hx)).mdifferentiableAt hr | E : Type u_1
inst✝¹¹ : NormedAddCommGroup E
inst✝¹⁰ : NormedSpace ℝ E
H : Type u_2
inst✝⁹ : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type u_3
inst✝⁸ : TopologicalSpace M
inst✝⁷ : ChartedSpace H M
inst✝⁶ : SmoothManifoldWithCorners I M
F : Type u_4
inst✝⁵ : NormedAddCommGroup F
inst✝⁴ : NormedSpace ℝ F
G : Type... | E : Type u_1
inst✝¹¹ : NormedAddCommGroup E
inst✝¹⁰ : NormedSpace ℝ E
H : Type u_2
inst✝⁹ : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type u_3
inst✝⁸ : TopologicalSpace M
inst✝⁷ : ChartedSpace H M
inst✝⁶ : SmoothManifoldWithCorners I M
F : Type u_4
inst✝⁵ : NormedAddCommGroup F
inst✝⁴ : NormedSpace ℝ F
G : Type... | Please generate a tactic in lean4 to solve the state.
STATE:
E : Type u_1
inst✝¹¹ : NormedAddCommGroup E
inst✝¹⁰ : NormedSpace ℝ E
H : Type u_2
inst✝⁹ : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type u_3
inst✝⁸ : TopologicalSpace M
inst✝⁷ : ChartedSpace H M
inst✝⁶ : SmoothManifoldWithCorners I M
F : Type u_4
in... |
https://github.com/fpvandoorn/sard.git | f4a1bb550136e8591dcd20dca5bdcce766a958c7 | Sard/ManifoldAux.lean | diffeoOn_differential_bijective | [163, 1] | [206, 42] | have inv1 := calc A'.comp A
_ = mfderiv I I (g ∘ f) x := (mfderiv_comp x hgat hfat).symm
_ = mfderivWithin I I (g ∘ f) (g '' t) x := (mfderivWithin_of_isOpen hopen hx2).symm
_ = mfderivWithin I I id (g '' t) x :=
mfderivWithin_congr (hopen.uniqueMDiffWithinAt hx2) (this ▸ hleft_inv) (hleft_inv x (this ▸ hx2))... | E : Type u_1
inst✝¹¹ : NormedAddCommGroup E
inst✝¹⁰ : NormedSpace ℝ E
H : Type u_2
inst✝⁹ : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type u_3
inst✝⁸ : TopologicalSpace M
inst✝⁷ : ChartedSpace H M
inst✝⁶ : SmoothManifoldWithCorners I M
F : Type u_4
inst✝⁵ : NormedAddCommGroup F
inst✝⁴ : NormedSpace ℝ F
G : Type... | E : Type u_1
inst✝¹¹ : NormedAddCommGroup E
inst✝¹⁰ : NormedSpace ℝ E
H : Type u_2
inst✝⁹ : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type u_3
inst✝⁸ : TopologicalSpace M
inst✝⁷ : ChartedSpace H M
inst✝⁶ : SmoothManifoldWithCorners I M
F : Type u_4
inst✝⁵ : NormedAddCommGroup F
inst✝⁴ : NormedSpace ℝ F
G : Type... | Please generate a tactic in lean4 to solve the state.
STATE:
E : Type u_1
inst✝¹¹ : NormedAddCommGroup E
inst✝¹⁰ : NormedSpace ℝ E
H : Type u_2
inst✝⁹ : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type u_3
inst✝⁸ : TopologicalSpace M
inst✝⁷ : ChartedSpace H M
inst✝⁶ : SmoothManifoldWithCorners I M
F : Type u_4
in... |
https://github.com/fpvandoorn/sard.git | f4a1bb550136e8591dcd20dca5bdcce766a958c7 | Sard/ManifoldAux.lean | diffeoOn_differential_bijective | [163, 1] | [206, 42] | have inv2 := calc A.comp A'
_ = mfderiv J J (f ∘ g) y := by
rw [← (hleft_inv x hx)] at hfat
exact (hyx ▸ (mfderiv_comp (f x) hfat hgat)).symm
_ = mfderivWithin J J (f ∘ g) t y := (mfderivWithin_of_isOpen ht hysource).symm
_ = mfderivWithin J J id t y :=
mfderivWithin_congr (ht.uniqueMDiffW... | E : Type u_1
inst✝¹¹ : NormedAddCommGroup E
inst✝¹⁰ : NormedSpace ℝ E
H : Type u_2
inst✝⁹ : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type u_3
inst✝⁸ : TopologicalSpace M
inst✝⁷ : ChartedSpace H M
inst✝⁶ : SmoothManifoldWithCorners I M
F : Type u_4
inst✝⁵ : NormedAddCommGroup F
inst✝⁴ : NormedSpace ℝ F
G : Type... | E : Type u_1
inst✝¹¹ : NormedAddCommGroup E
inst✝¹⁰ : NormedSpace ℝ E
H : Type u_2
inst✝⁹ : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type u_3
inst✝⁸ : TopologicalSpace M
inst✝⁷ : ChartedSpace H M
inst✝⁶ : SmoothManifoldWithCorners I M
F : Type u_4
inst✝⁵ : NormedAddCommGroup F
inst✝⁴ : NormedSpace ℝ F
G : Type... | Please generate a tactic in lean4 to solve the state.
STATE:
E : Type u_1
inst✝¹¹ : NormedAddCommGroup E
inst✝¹⁰ : NormedSpace ℝ E
H : Type u_2
inst✝⁹ : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type u_3
inst✝⁸ : TopologicalSpace M
inst✝⁷ : ChartedSpace H M
inst✝⁶ : SmoothManifoldWithCorners I M
F : Type u_4
in... |
https://github.com/fpvandoorn/sard.git | f4a1bb550136e8591dcd20dca5bdcce766a958c7 | Sard/ManifoldAux.lean | diffeoOn_differential_bijective | [163, 1] | [206, 42] | exact bijective_iff_inverses' inv1 inv2 | E : Type u_1
inst✝¹¹ : NormedAddCommGroup E
inst✝¹⁰ : NormedSpace ℝ E
H : Type u_2
inst✝⁹ : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type u_3
inst✝⁸ : TopologicalSpace M
inst✝⁷ : ChartedSpace H M
inst✝⁶ : SmoothManifoldWithCorners I M
F : Type u_4
inst✝⁵ : NormedAddCommGroup F
inst✝⁴ : NormedSpace ℝ F
G : Type... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
E : Type u_1
inst✝¹¹ : NormedAddCommGroup E
inst✝¹⁰ : NormedSpace ℝ E
H : Type u_2
inst✝⁹ : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type u_3
inst✝⁸ : TopologicalSpace M
inst✝⁷ : ChartedSpace H M
inst✝⁶ : SmoothManifoldWithCorners I M
F : Type u_4
in... |
https://github.com/fpvandoorn/sard.git | f4a1bb550136e8591dcd20dca5bdcce766a958c7 | Sard/ManifoldAux.lean | diffeoOn_differential_bijective | [163, 1] | [206, 42] | rw [← this, ← image_comp] | E : Type u_1
inst✝¹¹ : NormedAddCommGroup E
inst✝¹⁰ : NormedSpace ℝ E
H : Type u_2
inst✝⁹ : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type u_3
inst✝⁸ : TopologicalSpace M
inst✝⁷ : ChartedSpace H M
inst✝⁶ : SmoothManifoldWithCorners I M
F : Type u_4
inst✝⁵ : NormedAddCommGroup F
inst✝⁴ : NormedSpace ℝ F
G : Type... | E : Type u_1
inst✝¹¹ : NormedAddCommGroup E
inst✝¹⁰ : NormedSpace ℝ E
H : Type u_2
inst✝⁹ : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type u_3
inst✝⁸ : TopologicalSpace M
inst✝⁷ : ChartedSpace H M
inst✝⁶ : SmoothManifoldWithCorners I M
F : Type u_4
inst✝⁵ : NormedAddCommGroup F
inst✝⁴ : NormedSpace ℝ F
G : Type... | Please generate a tactic in lean4 to solve the state.
STATE:
E : Type u_1
inst✝¹¹ : NormedAddCommGroup E
inst✝¹⁰ : NormedSpace ℝ E
H : Type u_2
inst✝⁹ : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type u_3
inst✝⁸ : TopologicalSpace M
inst✝⁷ : ChartedSpace H M
inst✝⁶ : SmoothManifoldWithCorners I M
F : Type u_4
in... |
https://github.com/fpvandoorn/sard.git | f4a1bb550136e8591dcd20dca5bdcce766a958c7 | Sard/ManifoldAux.lean | diffeoOn_differential_bijective | [163, 1] | [206, 42] | exact EqOn.image_eq_self hleft_inv | E : Type u_1
inst✝¹¹ : NormedAddCommGroup E
inst✝¹⁰ : NormedSpace ℝ E
H : Type u_2
inst✝⁹ : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type u_3
inst✝⁸ : TopologicalSpace M
inst✝⁷ : ChartedSpace H M
inst✝⁶ : SmoothManifoldWithCorners I M
F : Type u_4
inst✝⁵ : NormedAddCommGroup F
inst✝⁴ : NormedSpace ℝ F
G : Type... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
E : Type u_1
inst✝¹¹ : NormedAddCommGroup E
inst✝¹⁰ : NormedSpace ℝ E
H : Type u_2
inst✝⁹ : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type u_3
inst✝⁸ : TopologicalSpace M
inst✝⁷ : ChartedSpace H M
inst✝⁶ : SmoothManifoldWithCorners I M
F : Type u_4
in... |
https://github.com/fpvandoorn/sard.git | f4a1bb550136e8591dcd20dca5bdcce766a958c7 | Sard/ManifoldAux.lean | diffeoOn_differential_bijective | [163, 1] | [206, 42] | rw [this] | E : Type u_1
inst✝¹¹ : NormedAddCommGroup E
inst✝¹⁰ : NormedSpace ℝ E
H : Type u_2
inst✝⁹ : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type u_3
inst✝⁸ : TopologicalSpace M
inst✝⁷ : ChartedSpace H M
inst✝⁶ : SmoothManifoldWithCorners I M
F : Type u_4
inst✝⁵ : NormedAddCommGroup F
inst✝⁴ : NormedSpace ℝ F
G : Type... | E : Type u_1
inst✝¹¹ : NormedAddCommGroup E
inst✝¹⁰ : NormedSpace ℝ E
H : Type u_2
inst✝⁹ : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type u_3
inst✝⁸ : TopologicalSpace M
inst✝⁷ : ChartedSpace H M
inst✝⁶ : SmoothManifoldWithCorners I M
F : Type u_4
inst✝⁵ : NormedAddCommGroup F
inst✝⁴ : NormedSpace ℝ F
G : Type... | Please generate a tactic in lean4 to solve the state.
STATE:
E : Type u_1
inst✝¹¹ : NormedAddCommGroup E
inst✝¹⁰ : NormedSpace ℝ E
H : Type u_2
inst✝⁹ : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type u_3
inst✝⁸ : TopologicalSpace M
inst✝⁷ : ChartedSpace H M
inst✝⁶ : SmoothManifoldWithCorners I M
F : Type u_4
in... |
https://github.com/fpvandoorn/sard.git | f4a1bb550136e8591dcd20dca5bdcce766a958c7 | Sard/ManifoldAux.lean | diffeoOn_differential_bijective | [163, 1] | [206, 42] | exact hs | E : Type u_1
inst✝¹¹ : NormedAddCommGroup E
inst✝¹⁰ : NormedSpace ℝ E
H : Type u_2
inst✝⁹ : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type u_3
inst✝⁸ : TopologicalSpace M
inst✝⁷ : ChartedSpace H M
inst✝⁶ : SmoothManifoldWithCorners I M
F : Type u_4
inst✝⁵ : NormedAddCommGroup F
inst✝⁴ : NormedSpace ℝ F
G : Type... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
E : Type u_1
inst✝¹¹ : NormedAddCommGroup E
inst✝¹⁰ : NormedSpace ℝ E
H : Type u_2
inst✝⁹ : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type u_3
inst✝⁸ : TopologicalSpace M
inst✝⁷ : ChartedSpace H M
inst✝⁶ : SmoothManifoldWithCorners I M
F : Type u_4
in... |
https://github.com/fpvandoorn/sard.git | f4a1bb550136e8591dcd20dca5bdcce766a958c7 | Sard/ManifoldAux.lean | diffeoOn_differential_bijective | [163, 1] | [206, 42] | simp_rw [this] | E : Type u_1
inst✝¹¹ : NormedAddCommGroup E
inst✝¹⁰ : NormedSpace ℝ E
H : Type u_2
inst✝⁹ : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type u_3
inst✝⁸ : TopologicalSpace M
inst✝⁷ : ChartedSpace H M
inst✝⁶ : SmoothManifoldWithCorners I M
F : Type u_4
inst✝⁵ : NormedAddCommGroup F
inst✝⁴ : NormedSpace ℝ F
G : Type... | E : Type u_1
inst✝¹¹ : NormedAddCommGroup E
inst✝¹⁰ : NormedSpace ℝ E
H : Type u_2
inst✝⁹ : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type u_3
inst✝⁸ : TopologicalSpace M
inst✝⁷ : ChartedSpace H M
inst✝⁶ : SmoothManifoldWithCorners I M
F : Type u_4
inst✝⁵ : NormedAddCommGroup F
inst✝⁴ : NormedSpace ℝ F
G : Type... | Please generate a tactic in lean4 to solve the state.
STATE:
E : Type u_1
inst✝¹¹ : NormedAddCommGroup E
inst✝¹⁰ : NormedSpace ℝ E
H : Type u_2
inst✝⁹ : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type u_3
inst✝⁸ : TopologicalSpace M
inst✝⁷ : ChartedSpace H M
inst✝⁶ : SmoothManifoldWithCorners I M
F : Type u_4
in... |
https://github.com/fpvandoorn/sard.git | f4a1bb550136e8591dcd20dca5bdcce766a958c7 | Sard/ManifoldAux.lean | diffeoOn_differential_bijective | [163, 1] | [206, 42] | exact hx | E : Type u_1
inst✝¹¹ : NormedAddCommGroup E
inst✝¹⁰ : NormedSpace ℝ E
H : Type u_2
inst✝⁹ : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type u_3
inst✝⁸ : TopologicalSpace M
inst✝⁷ : ChartedSpace H M
inst✝⁶ : SmoothManifoldWithCorners I M
F : Type u_4
inst✝⁵ : NormedAddCommGroup F
inst✝⁴ : NormedSpace ℝ F
G : Type... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
E : Type u_1
inst✝¹¹ : NormedAddCommGroup E
inst✝¹⁰ : NormedSpace ℝ E
H : Type u_2
inst✝⁹ : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type u_3
inst✝⁸ : TopologicalSpace M
inst✝⁷ : ChartedSpace H M
inst✝⁶ : SmoothManifoldWithCorners I M
F : Type u_4
in... |
https://github.com/fpvandoorn/sard.git | f4a1bb550136e8591dcd20dca5bdcce766a958c7 | Sard/ManifoldAux.lean | diffeoOn_differential_bijective | [163, 1] | [206, 42] | rw [← (hleft_inv x hx)] at hfat | E : Type u_1
inst✝¹¹ : NormedAddCommGroup E
inst✝¹⁰ : NormedSpace ℝ E
H : Type u_2
inst✝⁹ : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type u_3
inst✝⁸ : TopologicalSpace M
inst✝⁷ : ChartedSpace H M
inst✝⁶ : SmoothManifoldWithCorners I M
F : Type u_4
inst✝⁵ : NormedAddCommGroup F
inst✝⁴ : NormedSpace ℝ F
G : Type... | E : Type u_1
inst✝¹¹ : NormedAddCommGroup E
inst✝¹⁰ : NormedSpace ℝ E
H : Type u_2
inst✝⁹ : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type u_3
inst✝⁸ : TopologicalSpace M
inst✝⁷ : ChartedSpace H M
inst✝⁶ : SmoothManifoldWithCorners I M
F : Type u_4
inst✝⁵ : NormedAddCommGroup F
inst✝⁴ : NormedSpace ℝ F
G : Type... | Please generate a tactic in lean4 to solve the state.
STATE:
E : Type u_1
inst✝¹¹ : NormedAddCommGroup E
inst✝¹⁰ : NormedSpace ℝ E
H : Type u_2
inst✝⁹ : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type u_3
inst✝⁸ : TopologicalSpace M
inst✝⁷ : ChartedSpace H M
inst✝⁶ : SmoothManifoldWithCorners I M
F : Type u_4
in... |
https://github.com/fpvandoorn/sard.git | f4a1bb550136e8591dcd20dca5bdcce766a958c7 | Sard/ManifoldAux.lean | diffeoOn_differential_bijective | [163, 1] | [206, 42] | exact (hyx ▸ (mfderiv_comp (f x) hfat hgat)).symm | E : Type u_1
inst✝¹¹ : NormedAddCommGroup E
inst✝¹⁰ : NormedSpace ℝ E
H : Type u_2
inst✝⁹ : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type u_3
inst✝⁸ : TopologicalSpace M
inst✝⁷ : ChartedSpace H M
inst✝⁶ : SmoothManifoldWithCorners I M
F : Type u_4
inst✝⁵ : NormedAddCommGroup F
inst✝⁴ : NormedSpace ℝ F
G : Type... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
E : Type u_1
inst✝¹¹ : NormedAddCommGroup E
inst✝¹⁰ : NormedSpace ℝ E
H : Type u_2
inst✝⁹ : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type u_3
inst✝⁸ : TopologicalSpace M
inst✝⁷ : ChartedSpace H M
inst✝⁶ : SmoothManifoldWithCorners I M
F : Type u_4
in... |
https://github.com/fpvandoorn/sard.git | f4a1bb550136e8591dcd20dca5bdcce766a958c7 | Sard/ManifoldAux.lean | diffeo_differential_bijective | [208, 1] | [215, 50] | refine diffeoOn_differential_bijective I J hr isOpen_univ isOpen_univ trivial (mapsTo_univ f.toFun univ) (mapsTo_univ f.invFun univ) ?_ ?_ ?_ ?_ | E : Type u_1
inst✝¹¹ : NormedAddCommGroup E
inst✝¹⁰ : NormedSpace ℝ E
H : Type u_2
inst✝⁹ : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type u_3
inst✝⁸ : TopologicalSpace M
inst✝⁷ : ChartedSpace H M
inst✝⁶ : SmoothManifoldWithCorners I M
F : Type u_4
inst✝⁵ : NormedAddCommGroup F
inst✝⁴ : NormedSpace ℝ F
G : Type... | case refine_1
E : Type u_1
inst✝¹¹ : NormedAddCommGroup E
inst✝¹⁰ : NormedSpace ℝ E
H : Type u_2
inst✝⁹ : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type u_3
inst✝⁸ : TopologicalSpace M
inst✝⁷ : ChartedSpace H M
inst✝⁶ : SmoothManifoldWithCorners I M
F : Type u_4
inst✝⁵ : NormedAddCommGroup F
inst✝⁴ : NormedSpac... | Please generate a tactic in lean4 to solve the state.
STATE:
E : Type u_1
inst✝¹¹ : NormedAddCommGroup E
inst✝¹⁰ : NormedSpace ℝ E
H : Type u_2
inst✝⁹ : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type u_3
inst✝⁸ : TopologicalSpace M
inst✝⁷ : ChartedSpace H M
inst✝⁶ : SmoothManifoldWithCorners I M
F : Type u_4
in... |
https://github.com/fpvandoorn/sard.git | f4a1bb550136e8591dcd20dca5bdcce766a958c7 | Sard/ManifoldAux.lean | diffeo_differential_bijective | [208, 1] | [215, 50] | exact fun _ hx ↦ f.toPartialEquiv.left_inv' hx | case refine_1
E : Type u_1
inst✝¹¹ : NormedAddCommGroup E
inst✝¹⁰ : NormedSpace ℝ E
H : Type u_2
inst✝⁹ : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type u_3
inst✝⁸ : TopologicalSpace M
inst✝⁷ : ChartedSpace H M
inst✝⁶ : SmoothManifoldWithCorners I M
F : Type u_4
inst✝⁵ : NormedAddCommGroup F
inst✝⁴ : NormedSpac... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case refine_1
E : Type u_1
inst✝¹¹ : NormedAddCommGroup E
inst✝¹⁰ : NormedSpace ℝ E
H : Type u_2
inst✝⁹ : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type u_3
inst✝⁸ : TopologicalSpace M
inst✝⁷ : ChartedSpace H M
inst✝⁶ : SmoothManifoldWithCorners I M
F... |
https://github.com/fpvandoorn/sard.git | f4a1bb550136e8591dcd20dca5bdcce766a958c7 | Sard/ManifoldAux.lean | diffeo_differential_bijective | [208, 1] | [215, 50] | exact fun _ hy ↦ f.toPartialEquiv.right_inv' hy | case refine_2
E : Type u_1
inst✝¹¹ : NormedAddCommGroup E
inst✝¹⁰ : NormedSpace ℝ E
H : Type u_2
inst✝⁹ : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type u_3
inst✝⁸ : TopologicalSpace M
inst✝⁷ : ChartedSpace H M
inst✝⁶ : SmoothManifoldWithCorners I M
F : Type u_4
inst✝⁵ : NormedAddCommGroup F
inst✝⁴ : NormedSpac... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case refine_2
E : Type u_1
inst✝¹¹ : NormedAddCommGroup E
inst✝¹⁰ : NormedSpace ℝ E
H : Type u_2
inst✝⁹ : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type u_3
inst✝⁸ : TopologicalSpace M
inst✝⁷ : ChartedSpace H M
inst✝⁶ : SmoothManifoldWithCorners I M
F... |
https://github.com/fpvandoorn/sard.git | f4a1bb550136e8591dcd20dca5bdcce766a958c7 | Sard/ManifoldAux.lean | diffeo_differential_bijective | [208, 1] | [215, 50] | exact contMDiffOn_univ.mpr f.contMDiff_toFun | case refine_3
E : Type u_1
inst✝¹¹ : NormedAddCommGroup E
inst✝¹⁰ : NormedSpace ℝ E
H : Type u_2
inst✝⁹ : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type u_3
inst✝⁸ : TopologicalSpace M
inst✝⁷ : ChartedSpace H M
inst✝⁶ : SmoothManifoldWithCorners I M
F : Type u_4
inst✝⁵ : NormedAddCommGroup F
inst✝⁴ : NormedSpac... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case refine_3
E : Type u_1
inst✝¹¹ : NormedAddCommGroup E
inst✝¹⁰ : NormedSpace ℝ E
H : Type u_2
inst✝⁹ : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type u_3
inst✝⁸ : TopologicalSpace M
inst✝⁷ : ChartedSpace H M
inst✝⁶ : SmoothManifoldWithCorners I M
F... |
https://github.com/fpvandoorn/sard.git | f4a1bb550136e8591dcd20dca5bdcce766a958c7 | Sard/ManifoldAux.lean | diffeo_differential_bijective | [208, 1] | [215, 50] | exact contMDiffOn_univ.mpr f.contMDiff_invFun | case refine_4
E : Type u_1
inst✝¹¹ : NormedAddCommGroup E
inst✝¹⁰ : NormedSpace ℝ E
H : Type u_2
inst✝⁹ : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type u_3
inst✝⁸ : TopologicalSpace M
inst✝⁷ : ChartedSpace H M
inst✝⁶ : SmoothManifoldWithCorners I M
F : Type u_4
inst✝⁵ : NormedAddCommGroup F
inst✝⁴ : NormedSpac... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case refine_4
E : Type u_1
inst✝¹¹ : NormedAddCommGroup E
inst✝¹⁰ : NormedSpace ℝ E
H : Type u_2
inst✝⁹ : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type u_3
inst✝⁸ : TopologicalSpace M
inst✝⁷ : ChartedSpace H M
inst✝⁶ : SmoothManifoldWithCorners I M
F... |
https://github.com/fpvandoorn/sard.git | f4a1bb550136e8591dcd20dca5bdcce766a958c7 | Sard/ManifoldAux.lean | extendedChart_smooth | [218, 1] | [231, 97] | let e' : PartialHomeomorph E E := PartialHomeomorph.refl E | E : Type u_1
inst✝¹² : NormedAddCommGroup E
inst✝¹¹ : NormedSpace ℝ E
H : Type u_2
inst✝¹⁰ : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type u_3
inst✝⁹ : TopologicalSpace M
inst✝⁸ : ChartedSpace H M
inst✝⁷ : SmoothManifoldWithCorners I M
F : Type u_4
inst✝⁶ : NormedAddCommGroup F
inst✝⁵ : NormedSpace ℝ F
G : Typ... | E : Type u_1
inst✝¹² : NormedAddCommGroup E
inst✝¹¹ : NormedSpace ℝ E
H : Type u_2
inst✝¹⁰ : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type u_3
inst✝⁹ : TopologicalSpace M
inst✝⁸ : ChartedSpace H M
inst✝⁷ : SmoothManifoldWithCorners I M
F : Type u_4
inst✝⁶ : NormedAddCommGroup F
inst✝⁵ : NormedSpace ℝ F
G : Typ... | Please generate a tactic in lean4 to solve the state.
STATE:
E : Type u_1
inst✝¹² : NormedAddCommGroup E
inst✝¹¹ : NormedSpace ℝ E
H : Type u_2
inst✝¹⁰ : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type u_3
inst✝⁹ : TopologicalSpace M
inst✝⁸ : ChartedSpace H M
inst✝⁷ : SmoothManifoldWithCorners I M
F : Type u_4
i... |
https://github.com/fpvandoorn/sard.git | f4a1bb550136e8591dcd20dca5bdcce766a958c7 | Sard/ManifoldAux.lean | extendedChart_smooth | [218, 1] | [231, 97] | have h₁ : e ∈ maximalAtlas I M := subset_maximalAtlas _ he | E : Type u_1
inst✝¹² : NormedAddCommGroup E
inst✝¹¹ : NormedSpace ℝ E
H : Type u_2
inst✝¹⁰ : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type u_3
inst✝⁹ : TopologicalSpace M
inst✝⁸ : ChartedSpace H M
inst✝⁷ : SmoothManifoldWithCorners I M
F : Type u_4
inst✝⁶ : NormedAddCommGroup F
inst✝⁵ : NormedSpace ℝ F
G : Typ... | E : Type u_1
inst✝¹² : NormedAddCommGroup E
inst✝¹¹ : NormedSpace ℝ E
H : Type u_2
inst✝¹⁰ : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type u_3
inst✝⁹ : TopologicalSpace M
inst✝⁸ : ChartedSpace H M
inst✝⁷ : SmoothManifoldWithCorners I M
F : Type u_4
inst✝⁶ : NormedAddCommGroup F
inst✝⁵ : NormedSpace ℝ F
G : Typ... | Please generate a tactic in lean4 to solve the state.
STATE:
E : Type u_1
inst✝¹² : NormedAddCommGroup E
inst✝¹¹ : NormedSpace ℝ E
H : Type u_2
inst✝¹⁰ : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type u_3
inst✝⁹ : TopologicalSpace M
inst✝⁸ : ChartedSpace H M
inst✝⁷ : SmoothManifoldWithCorners I M
F : Type u_4
i... |
https://github.com/fpvandoorn/sard.git | f4a1bb550136e8591dcd20dca5bdcce766a958c7 | Sard/ManifoldAux.lean | extendedChart_smooth | [218, 1] | [231, 97] | have h₂ : e' ∈ maximalAtlas 𝓘(ℝ, E) E := subset_maximalAtlas _ (by rfl) | E : Type u_1
inst✝¹² : NormedAddCommGroup E
inst✝¹¹ : NormedSpace ℝ E
H : Type u_2
inst✝¹⁰ : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type u_3
inst✝⁹ : TopologicalSpace M
inst✝⁸ : ChartedSpace H M
inst✝⁷ : SmoothManifoldWithCorners I M
F : Type u_4
inst✝⁶ : NormedAddCommGroup F
inst✝⁵ : NormedSpace ℝ F
G : Typ... | E : Type u_1
inst✝¹² : NormedAddCommGroup E
inst✝¹¹ : NormedSpace ℝ E
H : Type u_2
inst✝¹⁰ : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type u_3
inst✝⁹ : TopologicalSpace M
inst✝⁸ : ChartedSpace H M
inst✝⁷ : SmoothManifoldWithCorners I M
F : Type u_4
inst✝⁶ : NormedAddCommGroup F
inst✝⁵ : NormedSpace ℝ F
G : Typ... | Please generate a tactic in lean4 to solve the state.
STATE:
E : Type u_1
inst✝¹² : NormedAddCommGroup E
inst✝¹¹ : NormedSpace ℝ E
H : Type u_2
inst✝¹⁰ : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type u_3
inst✝⁹ : TopologicalSpace M
inst✝⁸ : ChartedSpace H M
inst✝⁷ : SmoothManifoldWithCorners I M
F : Type u_4
i... |
https://github.com/fpvandoorn/sard.git | f4a1bb550136e8591dcd20dca5bdcce766a958c7 | Sard/ManifoldAux.lean | extendedChart_smooth | [218, 1] | [231, 97] | set f := e.extend I ∘ (e.extend I).symm | E : Type u_1
inst✝¹² : NormedAddCommGroup E
inst✝¹¹ : NormedSpace ℝ E
H : Type u_2
inst✝¹⁰ : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type u_3
inst✝⁹ : TopologicalSpace M
inst✝⁸ : ChartedSpace H M
inst✝⁷ : SmoothManifoldWithCorners I M
F : Type u_4
inst✝⁶ : NormedAddCommGroup F
inst✝⁵ : NormedSpace ℝ F
G : Typ... | E : Type u_1
inst✝¹² : NormedAddCommGroup E
inst✝¹¹ : NormedSpace ℝ E
H : Type u_2
inst✝¹⁰ : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type u_3
inst✝⁹ : TopologicalSpace M
inst✝⁸ : ChartedSpace H M
inst✝⁷ : SmoothManifoldWithCorners I M
F : Type u_4
inst✝⁶ : NormedAddCommGroup F
inst✝⁵ : NormedSpace ℝ F
G : Typ... | Please generate a tactic in lean4 to solve the state.
STATE:
E : Type u_1
inst✝¹² : NormedAddCommGroup E
inst✝¹¹ : NormedSpace ℝ E
H : Type u_2
inst✝¹⁰ : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type u_3
inst✝⁹ : TopologicalSpace M
inst✝⁸ : ChartedSpace H M
inst✝⁷ : SmoothManifoldWithCorners I M
F : Type u_4
i... |
https://github.com/fpvandoorn/sard.git | f4a1bb550136e8591dcd20dca5bdcce766a958c7 | Sard/ManifoldAux.lean | extendedChart_smooth | [218, 1] | [231, 97] | have h : ∀ x ∈ (e.extend I) '' e.source, f x = x := fun _ hx ↦ e.extend_right_inv I hx | E : Type u_1
inst✝¹² : NormedAddCommGroup E
inst✝¹¹ : NormedSpace ℝ E
H : Type u_2
inst✝¹⁰ : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type u_3
inst✝⁹ : TopologicalSpace M
inst✝⁸ : ChartedSpace H M
inst✝⁷ : SmoothManifoldWithCorners I M
F : Type u_4
inst✝⁶ : NormedAddCommGroup F
inst✝⁵ : NormedSpace ℝ F
G : Typ... | E : Type u_1
inst✝¹² : NormedAddCommGroup E
inst✝¹¹ : NormedSpace ℝ E
H : Type u_2
inst✝¹⁰ : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type u_3
inst✝⁹ : TopologicalSpace M
inst✝⁸ : ChartedSpace H M
inst✝⁷ : SmoothManifoldWithCorners I M
F : Type u_4
inst✝⁶ : NormedAddCommGroup F
inst✝⁵ : NormedSpace ℝ F
G : Typ... | Please generate a tactic in lean4 to solve the state.
STATE:
E : Type u_1
inst✝¹² : NormedAddCommGroup E
inst✝¹¹ : NormedSpace ℝ E
H : Type u_2
inst✝¹⁰ : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type u_3
inst✝⁹ : TopologicalSpace M
inst✝⁸ : ChartedSpace H M
inst✝⁷ : SmoothManifoldWithCorners I M
F : Type u_4
i... |
https://github.com/fpvandoorn/sard.git | f4a1bb550136e8591dcd20dca5bdcce766a958c7 | Sard/ManifoldAux.lean | extendedChart_smooth | [218, 1] | [231, 97] | apply (contMDiffOn_iff_of_mem_maximalAtlas' h₁ h₂ (Eq.subset rfl) (mapsTo_univ _ _)).mpr | E : Type u_1
inst✝¹² : NormedAddCommGroup E
inst✝¹¹ : NormedSpace ℝ E
H : Type u_2
inst✝¹⁰ : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type u_3
inst✝⁹ : TopologicalSpace M
inst✝⁸ : ChartedSpace H M
inst✝⁷ : SmoothManifoldWithCorners I M
F : Type u_4
inst✝⁶ : NormedAddCommGroup F
inst✝⁵ : NormedSpace ℝ F
G : Typ... | E : Type u_1
inst✝¹² : NormedAddCommGroup E
inst✝¹¹ : NormedSpace ℝ E
H : Type u_2
inst✝¹⁰ : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type u_3
inst✝⁹ : TopologicalSpace M
inst✝⁸ : ChartedSpace H M
inst✝⁷ : SmoothManifoldWithCorners I M
F : Type u_4
inst✝⁶ : NormedAddCommGroup F
inst✝⁵ : NormedSpace ℝ F
G : Typ... | Please generate a tactic in lean4 to solve the state.
STATE:
E : Type u_1
inst✝¹² : NormedAddCommGroup E
inst✝¹¹ : NormedSpace ℝ E
H : Type u_2
inst✝¹⁰ : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type u_3
inst✝⁹ : TopologicalSpace M
inst✝⁸ : ChartedSpace H M
inst✝⁷ : SmoothManifoldWithCorners I M
F : Type u_4
i... |
https://github.com/fpvandoorn/sard.git | f4a1bb550136e8591dcd20dca5bdcce766a958c7 | Sard/ManifoldAux.lean | extendedChart_smooth | [218, 1] | [231, 97] | exact ContMDiffOn.contDiffOn (fun x hx ↦ ContMDiffWithinAt.congr smoothWithinAt_id h (h x hx)) | E : Type u_1
inst✝¹² : NormedAddCommGroup E
inst✝¹¹ : NormedSpace ℝ E
H : Type u_2
inst✝¹⁰ : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type u_3
inst✝⁹ : TopologicalSpace M
inst✝⁸ : ChartedSpace H M
inst✝⁷ : SmoothManifoldWithCorners I M
F : Type u_4
inst✝⁶ : NormedAddCommGroup F
inst✝⁵ : NormedSpace ℝ F
G : Typ... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
E : Type u_1
inst✝¹² : NormedAddCommGroup E
inst✝¹¹ : NormedSpace ℝ E
H : Type u_2
inst✝¹⁰ : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type u_3
inst✝⁹ : TopologicalSpace M
inst✝⁸ : ChartedSpace H M
inst✝⁷ : SmoothManifoldWithCorners I M
F : Type u_4
i... |
https://github.com/fpvandoorn/sard.git | f4a1bb550136e8591dcd20dca5bdcce766a958c7 | Sard/ManifoldAux.lean | extendedChart_smooth | [218, 1] | [231, 97] | rfl | E : Type u_1
inst✝¹² : NormedAddCommGroup E
inst✝¹¹ : NormedSpace ℝ E
H : Type u_2
inst✝¹⁰ : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type u_3
inst✝⁹ : TopologicalSpace M
inst✝⁸ : ChartedSpace H M
inst✝⁷ : SmoothManifoldWithCorners I M
F : Type u_4
inst✝⁶ : NormedAddCommGroup F
inst✝⁵ : NormedSpace ℝ F
G : Typ... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
E : Type u_1
inst✝¹² : NormedAddCommGroup E
inst✝¹¹ : NormedSpace ℝ E
H : Type u_2
inst✝¹⁰ : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type u_3
inst✝⁹ : TopologicalSpace M
inst✝⁸ : ChartedSpace H M
inst✝⁷ : SmoothManifoldWithCorners I M
F : Type u_4
i... |
https://github.com/fpvandoorn/sard.git | f4a1bb550136e8591dcd20dca5bdcce766a958c7 | Sard/ManifoldAux.lean | ofSet_in_maximal_atlas | [234, 1] | [241, 35] | rw [maximalAtlas, mem_maximalAtlas_iff] | E : Type u_1
inst✝¹¹ : NormedAddCommGroup E
inst✝¹⁰ : NormedSpace ℝ E
H : Type u_2
inst✝⁹ : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type u_3
inst✝⁸ : TopologicalSpace M
inst✝⁷ : ChartedSpace H M
inst✝⁶ : SmoothManifoldWithCorners I M
F : Type u_4
inst✝⁵ : NormedAddCommGroup F
inst✝⁴ : NormedSpace ℝ F
G : Type... | E : Type u_1
inst✝¹¹ : NormedAddCommGroup E
inst✝¹⁰ : NormedSpace ℝ E
H : Type u_2
inst✝⁹ : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type u_3
inst✝⁸ : TopologicalSpace M
inst✝⁷ : ChartedSpace H M
inst✝⁶ : SmoothManifoldWithCorners I M
F : Type u_4
inst✝⁵ : NormedAddCommGroup F
inst✝⁴ : NormedSpace ℝ F
G : Type... | Please generate a tactic in lean4 to solve the state.
STATE:
E : Type u_1
inst✝¹¹ : NormedAddCommGroup E
inst✝¹⁰ : NormedSpace ℝ E
H : Type u_2
inst✝⁹ : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type u_3
inst✝⁸ : TopologicalSpace M
inst✝⁷ : ChartedSpace H M
inst✝⁶ : SmoothManifoldWithCorners I M
F : Type u_4
in... |
https://github.com/fpvandoorn/sard.git | f4a1bb550136e8591dcd20dca5bdcce766a958c7 | Sard/ManifoldAux.lean | ofSet_in_maximal_atlas | [234, 1] | [241, 35] | intro e' he' | E : Type u_1
inst✝¹¹ : NormedAddCommGroup E
inst✝¹⁰ : NormedSpace ℝ E
H : Type u_2
inst✝⁹ : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type u_3
inst✝⁸ : TopologicalSpace M
inst✝⁷ : ChartedSpace H M
inst✝⁶ : SmoothManifoldWithCorners I M
F : Type u_4
inst✝⁵ : NormedAddCommGroup F
inst✝⁴ : NormedSpace ℝ F
G : Type... | E : Type u_1
inst✝¹¹ : NormedAddCommGroup E
inst✝¹⁰ : NormedSpace ℝ E
H : Type u_2
inst✝⁹ : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type u_3
inst✝⁸ : TopologicalSpace M
inst✝⁷ : ChartedSpace H M
inst✝⁶ : SmoothManifoldWithCorners I M
F : Type u_4
inst✝⁵ : NormedAddCommGroup F
inst✝⁴ : NormedSpace ℝ F
G : Type... | Please generate a tactic in lean4 to solve the state.
STATE:
E : Type u_1
inst✝¹¹ : NormedAddCommGroup E
inst✝¹⁰ : NormedSpace ℝ E
H : Type u_2
inst✝⁹ : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type u_3
inst✝⁸ : TopologicalSpace M
inst✝⁷ : ChartedSpace H M
inst✝⁶ : SmoothManifoldWithCorners I M
F : Type u_4
in... |
https://github.com/fpvandoorn/sard.git | f4a1bb550136e8591dcd20dca5bdcce766a958c7 | Sard/ManifoldAux.lean | ofSet_in_maximal_atlas | [234, 1] | [241, 35] | rw [he'] | E : Type u_1
inst✝¹¹ : NormedAddCommGroup E
inst✝¹⁰ : NormedSpace ℝ E
H : Type u_2
inst✝⁹ : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type u_3
inst✝⁸ : TopologicalSpace M
inst✝⁷ : ChartedSpace H M
inst✝⁶ : SmoothManifoldWithCorners I M
F : Type u_4
inst✝⁵ : NormedAddCommGroup F
inst✝⁴ : NormedSpace ℝ F
G : Type... | E : Type u_1
inst✝¹¹ : NormedAddCommGroup E
inst✝¹⁰ : NormedSpace ℝ E
H : Type u_2
inst✝⁹ : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type u_3
inst✝⁸ : TopologicalSpace M
inst✝⁷ : ChartedSpace H M
inst✝⁶ : SmoothManifoldWithCorners I M
F : Type u_4
inst✝⁵ : NormedAddCommGroup F
inst✝⁴ : NormedSpace ℝ F
G : Type... | Please generate a tactic in lean4 to solve the state.
STATE:
E : Type u_1
inst✝¹¹ : NormedAddCommGroup E
inst✝¹⁰ : NormedSpace ℝ E
H : Type u_2
inst✝⁹ : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type u_3
inst✝⁸ : TopologicalSpace M
inst✝⁷ : ChartedSpace H M
inst✝⁶ : SmoothManifoldWithCorners I M
F : Type u_4
in... |
https://github.com/fpvandoorn/sard.git | f4a1bb550136e8591dcd20dca5bdcce766a958c7 | Sard/ManifoldAux.lean | ofSet_in_maximal_atlas | [234, 1] | [241, 35] | simp only [mfld_simps] | E : Type u_1
inst✝¹¹ : NormedAddCommGroup E
inst✝¹⁰ : NormedSpace ℝ E
H : Type u_2
inst✝⁹ : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type u_3
inst✝⁸ : TopologicalSpace M
inst✝⁷ : ChartedSpace H M
inst✝⁶ : SmoothManifoldWithCorners I M
F : Type u_4
inst✝⁵ : NormedAddCommGroup F
inst✝⁴ : NormedSpace ℝ F
G : Type... | E : Type u_1
inst✝¹¹ : NormedAddCommGroup E
inst✝¹⁰ : NormedSpace ℝ E
H : Type u_2
inst✝⁹ : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type u_3
inst✝⁸ : TopologicalSpace M
inst✝⁷ : ChartedSpace H M
inst✝⁶ : SmoothManifoldWithCorners I M
F : Type u_4
inst✝⁵ : NormedAddCommGroup F
inst✝⁴ : NormedSpace ℝ F
G : Type... | Please generate a tactic in lean4 to solve the state.
STATE:
E : Type u_1
inst✝¹¹ : NormedAddCommGroup E
inst✝¹⁰ : NormedSpace ℝ E
H : Type u_2
inst✝⁹ : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type u_3
inst✝⁸ : TopologicalSpace M
inst✝⁷ : ChartedSpace H M
inst✝⁶ : SmoothManifoldWithCorners I M
F : Type u_4
in... |
https://github.com/fpvandoorn/sard.git | f4a1bb550136e8591dcd20dca5bdcce766a958c7 | Sard/ManifoldAux.lean | ofSet_in_maximal_atlas | [234, 1] | [241, 35] | apply ofSet_mem_contDiffGroupoid | E : Type u_1
inst✝¹¹ : NormedAddCommGroup E
inst✝¹⁰ : NormedSpace ℝ E
H : Type u_2
inst✝⁹ : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type u_3
inst✝⁸ : TopologicalSpace M
inst✝⁷ : ChartedSpace H M
inst✝⁶ : SmoothManifoldWithCorners I M
F : Type u_4
inst✝⁵ : NormedAddCommGroup F
inst✝⁴ : NormedSpace ℝ F
G : Type... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
E : Type u_1
inst✝¹¹ : NormedAddCommGroup E
inst✝¹⁰ : NormedSpace ℝ E
H : Type u_2
inst✝⁹ : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type u_3
inst✝⁸ : TopologicalSpace M
inst✝⁷ : ChartedSpace H M
inst✝⁶ : SmoothManifoldWithCorners I M
F : Type u_4
in... |
https://github.com/fpvandoorn/sard.git | f4a1bb550136e8591dcd20dca5bdcce766a958c7 | Sard/ManifoldAux.lean | extendedChart_symm_smooth | [243, 1] | [265, 84] | have : IsOpen ((e.extend I) '' e.source) := e.extend_isOpenMapOn_source I e.open_source (Eq.subset rfl) | E : Type u_1
inst✝¹² : NormedAddCommGroup E
inst✝¹¹ : NormedSpace ℝ E
H : Type u_2
inst✝¹⁰ : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type u_3
inst✝⁹ : TopologicalSpace M
inst✝⁸ : ChartedSpace H M
inst✝⁷ : SmoothManifoldWithCorners I M
F : Type u_4
inst✝⁶ : NormedAddCommGroup F
inst✝⁵ : NormedSpace ℝ F
G : Typ... | E : Type u_1
inst✝¹² : NormedAddCommGroup E
inst✝¹¹ : NormedSpace ℝ E
H : Type u_2
inst✝¹⁰ : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type u_3
inst✝⁹ : TopologicalSpace M
inst✝⁸ : ChartedSpace H M
inst✝⁷ : SmoothManifoldWithCorners I M
F : Type u_4
inst✝⁶ : NormedAddCommGroup F
inst✝⁵ : NormedSpace ℝ F
G : Typ... | Please generate a tactic in lean4 to solve the state.
STATE:
E : Type u_1
inst✝¹² : NormedAddCommGroup E
inst✝¹¹ : NormedSpace ℝ E
H : Type u_2
inst✝¹⁰ : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type u_3
inst✝⁹ : TopologicalSpace M
inst✝⁸ : ChartedSpace H M
inst✝⁷ : SmoothManifoldWithCorners I M
F : Type u_4
i... |
https://github.com/fpvandoorn/sard.git | f4a1bb550136e8591dcd20dca5bdcce766a958c7 | Sard/ManifoldAux.lean | extendedChart_symm_smooth | [243, 1] | [265, 84] | let e' : PartialHomeomorph E E := PartialHomeomorph.ofSet (e.extend I '' e.source) this | E : Type u_1
inst✝¹² : NormedAddCommGroup E
inst✝¹¹ : NormedSpace ℝ E
H : Type u_2
inst✝¹⁰ : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type u_3
inst✝⁹ : TopologicalSpace M
inst✝⁸ : ChartedSpace H M
inst✝⁷ : SmoothManifoldWithCorners I M
F : Type u_4
inst✝⁶ : NormedAddCommGroup F
inst✝⁵ : NormedSpace ℝ F
G : Typ... | E : Type u_1
inst✝¹² : NormedAddCommGroup E
inst✝¹¹ : NormedSpace ℝ E
H : Type u_2
inst✝¹⁰ : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type u_3
inst✝⁹ : TopologicalSpace M
inst✝⁸ : ChartedSpace H M
inst✝⁷ : SmoothManifoldWithCorners I M
F : Type u_4
inst✝⁶ : NormedAddCommGroup F
inst✝⁵ : NormedSpace ℝ F
G : Typ... | Please generate a tactic in lean4 to solve the state.
STATE:
E : Type u_1
inst✝¹² : NormedAddCommGroup E
inst✝¹¹ : NormedSpace ℝ E
H : Type u_2
inst✝¹⁰ : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type u_3
inst✝⁹ : TopologicalSpace M
inst✝⁸ : ChartedSpace H M
inst✝⁷ : SmoothManifoldWithCorners I M
F : Type u_4
i... |
https://github.com/fpvandoorn/sard.git | f4a1bb550136e8591dcd20dca5bdcce766a958c7 | Sard/ManifoldAux.lean | extendedChart_symm_smooth | [243, 1] | [265, 84] | have h1 : e ∈ maximalAtlas I M := subset_maximalAtlas _ he | E : Type u_1
inst✝¹² : NormedAddCommGroup E
inst✝¹¹ : NormedSpace ℝ E
H : Type u_2
inst✝¹⁰ : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type u_3
inst✝⁹ : TopologicalSpace M
inst✝⁸ : ChartedSpace H M
inst✝⁷ : SmoothManifoldWithCorners I M
F : Type u_4
inst✝⁶ : NormedAddCommGroup F
inst✝⁵ : NormedSpace ℝ F
G : Typ... | E : Type u_1
inst✝¹² : NormedAddCommGroup E
inst✝¹¹ : NormedSpace ℝ E
H : Type u_2
inst✝¹⁰ : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type u_3
inst✝⁹ : TopologicalSpace M
inst✝⁸ : ChartedSpace H M
inst✝⁷ : SmoothManifoldWithCorners I M
F : Type u_4
inst✝⁶ : NormedAddCommGroup F
inst✝⁵ : NormedSpace ℝ F
G : Typ... | Please generate a tactic in lean4 to solve the state.
STATE:
E : Type u_1
inst✝¹² : NormedAddCommGroup E
inst✝¹¹ : NormedSpace ℝ E
H : Type u_2
inst✝¹⁰ : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type u_3
inst✝⁹ : TopologicalSpace M
inst✝⁸ : ChartedSpace H M
inst✝⁷ : SmoothManifoldWithCorners I M
F : Type u_4
i... |
https://github.com/fpvandoorn/sard.git | f4a1bb550136e8591dcd20dca5bdcce766a958c7 | Sard/ManifoldAux.lean | extendedChart_symm_smooth | [243, 1] | [265, 84] | have h2 : e' ∈ maximalAtlas 𝓘(ℝ, E) E := ofSet_in_maximal_atlas this | E : Type u_1
inst✝¹² : NormedAddCommGroup E
inst✝¹¹ : NormedSpace ℝ E
H : Type u_2
inst✝¹⁰ : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type u_3
inst✝⁹ : TopologicalSpace M
inst✝⁸ : ChartedSpace H M
inst✝⁷ : SmoothManifoldWithCorners I M
F : Type u_4
inst✝⁶ : NormedAddCommGroup F
inst✝⁵ : NormedSpace ℝ F
G : Typ... | E : Type u_1
inst✝¹² : NormedAddCommGroup E
inst✝¹¹ : NormedSpace ℝ E
H : Type u_2
inst✝¹⁰ : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type u_3
inst✝⁹ : TopologicalSpace M
inst✝⁸ : ChartedSpace H M
inst✝⁷ : SmoothManifoldWithCorners I M
F : Type u_4
inst✝⁶ : NormedAddCommGroup F
inst✝⁵ : NormedSpace ℝ F
G : Typ... | Please generate a tactic in lean4 to solve the state.
STATE:
E : Type u_1
inst✝¹² : NormedAddCommGroup E
inst✝¹¹ : NormedSpace ℝ E
H : Type u_2
inst✝¹⁰ : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type u_3
inst✝⁹ : TopologicalSpace M
inst✝⁸ : ChartedSpace H M
inst✝⁷ : SmoothManifoldWithCorners I M
F : Type u_4
i... |
https://github.com/fpvandoorn/sard.git | f4a1bb550136e8591dcd20dca5bdcce766a958c7 | Sard/ManifoldAux.lean | extendedChart_symm_smooth | [243, 1] | [265, 84] | apply (contMDiffOn_iff_of_mem_maximalAtlas' h2 h1 (Eq.subset rfl) (e.mapsTo_extend_symm I)).mpr | E : Type u_1
inst✝¹² : NormedAddCommGroup E
inst✝¹¹ : NormedSpace ℝ E
H : Type u_2
inst✝¹⁰ : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type u_3
inst✝⁹ : TopologicalSpace M
inst✝⁸ : ChartedSpace H M
inst✝⁷ : SmoothManifoldWithCorners I M
F : Type u_4
inst✝⁶ : NormedAddCommGroup F
inst✝⁵ : NormedSpace ℝ F
G : Typ... | E : Type u_1
inst✝¹² : NormedAddCommGroup E
inst✝¹¹ : NormedSpace ℝ E
H : Type u_2
inst✝¹⁰ : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type u_3
inst✝⁹ : TopologicalSpace M
inst✝⁸ : ChartedSpace H M
inst✝⁷ : SmoothManifoldWithCorners I M
F : Type u_4
inst✝⁶ : NormedAddCommGroup F
inst✝⁵ : NormedSpace ℝ F
G : Typ... | Please generate a tactic in lean4 to solve the state.
STATE:
E : Type u_1
inst✝¹² : NormedAddCommGroup E
inst✝¹¹ : NormedSpace ℝ E
H : Type u_2
inst✝¹⁰ : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type u_3
inst✝⁹ : TopologicalSpace M
inst✝⁸ : ChartedSpace H M
inst✝⁷ : SmoothManifoldWithCorners I M
F : Type u_4
i... |
https://github.com/fpvandoorn/sard.git | f4a1bb550136e8591dcd20dca5bdcce766a958c7 | Sard/ManifoldAux.lean | extendedChart_symm_smooth | [243, 1] | [265, 84] | apply ContMDiffOn.contDiffOn | E : Type u_1
inst✝¹² : NormedAddCommGroup E
inst✝¹¹ : NormedSpace ℝ E
H : Type u_2
inst✝¹⁰ : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type u_3
inst✝⁹ : TopologicalSpace M
inst✝⁸ : ChartedSpace H M
inst✝⁷ : SmoothManifoldWithCorners I M
F : Type u_4
inst✝⁶ : NormedAddCommGroup F
inst✝⁵ : NormedSpace ℝ F
G : Typ... | case a
E : Type u_1
inst✝¹² : NormedAddCommGroup E
inst✝¹¹ : NormedSpace ℝ E
H : Type u_2
inst✝¹⁰ : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type u_3
inst✝⁹ : TopologicalSpace M
inst✝⁸ : ChartedSpace H M
inst✝⁷ : SmoothManifoldWithCorners I M
F : Type u_4
inst✝⁶ : NormedAddCommGroup F
inst✝⁵ : NormedSpace ℝ F
... | Please generate a tactic in lean4 to solve the state.
STATE:
E : Type u_1
inst✝¹² : NormedAddCommGroup E
inst✝¹¹ : NormedSpace ℝ E
H : Type u_2
inst✝¹⁰ : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type u_3
inst✝⁹ : TopologicalSpace M
inst✝⁸ : ChartedSpace H M
inst✝⁷ : SmoothManifoldWithCorners I M
F : Type u_4
i... |
https://github.com/fpvandoorn/sard.git | f4a1bb550136e8591dcd20dca5bdcce766a958c7 | Sard/ManifoldAux.lean | extendedChart_symm_smooth | [243, 1] | [265, 84] | set f := e.extend I ∘ (e.extend I).symm ∘ (PartialEquiv.symm (PartialHomeomorph.extend e' 𝓘(ℝ, E))) | case a
E : Type u_1
inst✝¹² : NormedAddCommGroup E
inst✝¹¹ : NormedSpace ℝ E
H : Type u_2
inst✝¹⁰ : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type u_3
inst✝⁹ : TopologicalSpace M
inst✝⁸ : ChartedSpace H M
inst✝⁷ : SmoothManifoldWithCorners I M
F : Type u_4
inst✝⁶ : NormedAddCommGroup F
inst✝⁵ : NormedSpace ℝ F
... | case a
E : Type u_1
inst✝¹² : NormedAddCommGroup E
inst✝¹¹ : NormedSpace ℝ E
H : Type u_2
inst✝¹⁰ : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type u_3
inst✝⁹ : TopologicalSpace M
inst✝⁸ : ChartedSpace H M
inst✝⁷ : SmoothManifoldWithCorners I M
F : Type u_4
inst✝⁶ : NormedAddCommGroup F
inst✝⁵ : NormedSpace ℝ F
... | Please generate a tactic in lean4 to solve the state.
STATE:
case a
E : Type u_1
inst✝¹² : NormedAddCommGroup E
inst✝¹¹ : NormedSpace ℝ E
H : Type u_2
inst✝¹⁰ : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type u_3
inst✝⁹ : TopologicalSpace M
inst✝⁸ : ChartedSpace H M
inst✝⁷ : SmoothManifoldWithCorners I M
F : Typ... |
https://github.com/fpvandoorn/sard.git | f4a1bb550136e8591dcd20dca5bdcce766a958c7 | Sard/ManifoldAux.lean | extendedChart_symm_smooth | [243, 1] | [265, 84] | have cong : ∀ x ∈ e.extend I '' e.source, f x = x := fun x hx ↦ e.extend_right_inv I hx | case a
E : Type u_1
inst✝¹² : NormedAddCommGroup E
inst✝¹¹ : NormedSpace ℝ E
H : Type u_2
inst✝¹⁰ : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type u_3
inst✝⁹ : TopologicalSpace M
inst✝⁸ : ChartedSpace H M
inst✝⁷ : SmoothManifoldWithCorners I M
F : Type u_4
inst✝⁶ : NormedAddCommGroup F
inst✝⁵ : NormedSpace ℝ F
... | case a
E : Type u_1
inst✝¹² : NormedAddCommGroup E
inst✝¹¹ : NormedSpace ℝ E
H : Type u_2
inst✝¹⁰ : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type u_3
inst✝⁹ : TopologicalSpace M
inst✝⁸ : ChartedSpace H M
inst✝⁷ : SmoothManifoldWithCorners I M
F : Type u_4
inst✝⁶ : NormedAddCommGroup F
inst✝⁵ : NormedSpace ℝ F
... | Please generate a tactic in lean4 to solve the state.
STATE:
case a
E : Type u_1
inst✝¹² : NormedAddCommGroup E
inst✝¹¹ : NormedSpace ℝ E
H : Type u_2
inst✝¹⁰ : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type u_3
inst✝⁹ : TopologicalSpace M
inst✝⁸ : ChartedSpace H M
inst✝⁷ : SmoothManifoldWithCorners I M
F : Typ... |
https://github.com/fpvandoorn/sard.git | f4a1bb550136e8591dcd20dca5bdcce766a958c7 | Sard/ManifoldAux.lean | extendedChart_symm_smooth | [243, 1] | [265, 84] | have h : (e'.extend 𝓘(ℝ, E)) '' e'.source = e.extend I '' e.source := by
simp only [e', mfld_simps, image_id'] | case a
E : Type u_1
inst✝¹² : NormedAddCommGroup E
inst✝¹¹ : NormedSpace ℝ E
H : Type u_2
inst✝¹⁰ : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type u_3
inst✝⁹ : TopologicalSpace M
inst✝⁸ : ChartedSpace H M
inst✝⁷ : SmoothManifoldWithCorners I M
F : Type u_4
inst✝⁶ : NormedAddCommGroup F
inst✝⁵ : NormedSpace ℝ F
... | case a
E : Type u_1
inst✝¹² : NormedAddCommGroup E
inst✝¹¹ : NormedSpace ℝ E
H : Type u_2
inst✝¹⁰ : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type u_3
inst✝⁹ : TopologicalSpace M
inst✝⁸ : ChartedSpace H M
inst✝⁷ : SmoothManifoldWithCorners I M
F : Type u_4
inst✝⁶ : NormedAddCommGroup F
inst✝⁵ : NormedSpace ℝ F
... | Please generate a tactic in lean4 to solve the state.
STATE:
case a
E : Type u_1
inst✝¹² : NormedAddCommGroup E
inst✝¹¹ : NormedSpace ℝ E
H : Type u_2
inst✝¹⁰ : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type u_3
inst✝⁹ : TopologicalSpace M
inst✝⁸ : ChartedSpace H M
inst✝⁷ : SmoothManifoldWithCorners I M
F : Typ... |
https://github.com/fpvandoorn/sard.git | f4a1bb550136e8591dcd20dca5bdcce766a958c7 | Sard/ManifoldAux.lean | extendedChart_symm_smooth | [243, 1] | [265, 84] | have : ((e'.extend 𝓘(ℝ, E)) '' (e.extend I '' e.source)) = e.extend I '' e.source := by
have : e'.source = e.extend I '' e.source := by rw [PartialHomeomorph.ofSet_source]
exact this ▸ h | case a
E : Type u_1
inst✝¹² : NormedAddCommGroup E
inst✝¹¹ : NormedSpace ℝ E
H : Type u_2
inst✝¹⁰ : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type u_3
inst✝⁹ : TopologicalSpace M
inst✝⁸ : ChartedSpace H M
inst✝⁷ : SmoothManifoldWithCorners I M
F : Type u_4
inst✝⁶ : NormedAddCommGroup F
inst✝⁵ : NormedSpace ℝ F
... | case a
E : Type u_1
inst✝¹² : NormedAddCommGroup E
inst✝¹¹ : NormedSpace ℝ E
H : Type u_2
inst✝¹⁰ : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type u_3
inst✝⁹ : TopologicalSpace M
inst✝⁸ : ChartedSpace H M
inst✝⁷ : SmoothManifoldWithCorners I M
F : Type u_4
inst✝⁶ : NormedAddCommGroup F
inst✝⁵ : NormedSpace ℝ F
... | Please generate a tactic in lean4 to solve the state.
STATE:
case a
E : Type u_1
inst✝¹² : NormedAddCommGroup E
inst✝¹¹ : NormedSpace ℝ E
H : Type u_2
inst✝¹⁰ : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type u_3
inst✝⁹ : TopologicalSpace M
inst✝⁸ : ChartedSpace H M
inst✝⁷ : SmoothManifoldWithCorners I M
F : Typ... |
https://github.com/fpvandoorn/sard.git | f4a1bb550136e8591dcd20dca5bdcce766a958c7 | Sard/ManifoldAux.lean | extendedChart_symm_smooth | [243, 1] | [265, 84] | rw [PartialHomeomorph.ofSet_source, this] | case a
E : Type u_1
inst✝¹² : NormedAddCommGroup E
inst✝¹¹ : NormedSpace ℝ E
H : Type u_2
inst✝¹⁰ : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type u_3
inst✝⁹ : TopologicalSpace M
inst✝⁸ : ChartedSpace H M
inst✝⁷ : SmoothManifoldWithCorners I M
F : Type u_4
inst✝⁶ : NormedAddCommGroup F
inst✝⁵ : NormedSpace ℝ F
... | case a
E : Type u_1
inst✝¹² : NormedAddCommGroup E
inst✝¹¹ : NormedSpace ℝ E
H : Type u_2
inst✝¹⁰ : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type u_3
inst✝⁹ : TopologicalSpace M
inst✝⁸ : ChartedSpace H M
inst✝⁷ : SmoothManifoldWithCorners I M
F : Type u_4
inst✝⁶ : NormedAddCommGroup F
inst✝⁵ : NormedSpace ℝ F
... | Please generate a tactic in lean4 to solve the state.
STATE:
case a
E : Type u_1
inst✝¹² : NormedAddCommGroup E
inst✝¹¹ : NormedSpace ℝ E
H : Type u_2
inst✝¹⁰ : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type u_3
inst✝⁹ : TopologicalSpace M
inst✝⁸ : ChartedSpace H M
inst✝⁷ : SmoothManifoldWithCorners I M
F : Typ... |
https://github.com/fpvandoorn/sard.git | f4a1bb550136e8591dcd20dca5bdcce766a958c7 | Sard/ManifoldAux.lean | extendedChart_symm_smooth | [243, 1] | [265, 84] | exact fun x hx ↦ ContMDiffWithinAt.congr smoothWithinAt_id cong (cong x (h ▸ hx)) | case a
E : Type u_1
inst✝¹² : NormedAddCommGroup E
inst✝¹¹ : NormedSpace ℝ E
H : Type u_2
inst✝¹⁰ : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type u_3
inst✝⁹ : TopologicalSpace M
inst✝⁸ : ChartedSpace H M
inst✝⁷ : SmoothManifoldWithCorners I M
F : Type u_4
inst✝⁶ : NormedAddCommGroup F
inst✝⁵ : NormedSpace ℝ F
... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case a
E : Type u_1
inst✝¹² : NormedAddCommGroup E
inst✝¹¹ : NormedSpace ℝ E
H : Type u_2
inst✝¹⁰ : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type u_3
inst✝⁹ : TopologicalSpace M
inst✝⁸ : ChartedSpace H M
inst✝⁷ : SmoothManifoldWithCorners I M
F : Typ... |
https://github.com/fpvandoorn/sard.git | f4a1bb550136e8591dcd20dca5bdcce766a958c7 | Sard/ManifoldAux.lean | extendedChart_symm_smooth | [243, 1] | [265, 84] | simp only [e', mfld_simps, image_id'] | E : Type u_1
inst✝¹² : NormedAddCommGroup E
inst✝¹¹ : NormedSpace ℝ E
H : Type u_2
inst✝¹⁰ : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type u_3
inst✝⁹ : TopologicalSpace M
inst✝⁸ : ChartedSpace H M
inst✝⁷ : SmoothManifoldWithCorners I M
F : Type u_4
inst✝⁶ : NormedAddCommGroup F
inst✝⁵ : NormedSpace ℝ F
G : Typ... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
E : Type u_1
inst✝¹² : NormedAddCommGroup E
inst✝¹¹ : NormedSpace ℝ E
H : Type u_2
inst✝¹⁰ : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type u_3
inst✝⁹ : TopologicalSpace M
inst✝⁸ : ChartedSpace H M
inst✝⁷ : SmoothManifoldWithCorners I M
F : Type u_4
i... |
https://github.com/fpvandoorn/sard.git | f4a1bb550136e8591dcd20dca5bdcce766a958c7 | Sard/ManifoldAux.lean | extendedChart_symm_smooth | [243, 1] | [265, 84] | have : e'.source = e.extend I '' e.source := by rw [PartialHomeomorph.ofSet_source] | E : Type u_1
inst✝¹² : NormedAddCommGroup E
inst✝¹¹ : NormedSpace ℝ E
H : Type u_2
inst✝¹⁰ : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type u_3
inst✝⁹ : TopologicalSpace M
inst✝⁸ : ChartedSpace H M
inst✝⁷ : SmoothManifoldWithCorners I M
F : Type u_4
inst✝⁶ : NormedAddCommGroup F
inst✝⁵ : NormedSpace ℝ F
G : Typ... | E : Type u_1
inst✝¹² : NormedAddCommGroup E
inst✝¹¹ : NormedSpace ℝ E
H : Type u_2
inst✝¹⁰ : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type u_3
inst✝⁹ : TopologicalSpace M
inst✝⁸ : ChartedSpace H M
inst✝⁷ : SmoothManifoldWithCorners I M
F : Type u_4
inst✝⁶ : NormedAddCommGroup F
inst✝⁵ : NormedSpace ℝ F
G : Typ... | Please generate a tactic in lean4 to solve the state.
STATE:
E : Type u_1
inst✝¹² : NormedAddCommGroup E
inst✝¹¹ : NormedSpace ℝ E
H : Type u_2
inst✝¹⁰ : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type u_3
inst✝⁹ : TopologicalSpace M
inst✝⁸ : ChartedSpace H M
inst✝⁷ : SmoothManifoldWithCorners I M
F : Type u_4
i... |
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