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/ManifoldAux.lean | extendedChart_symm_smooth | [243, 1] | [265, 84] | exact this ▸ h | 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] | 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... | 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_differential_bijective | [267, 1] | [281, 59] | refine diffeoOn_differential_bijective 𝓘(ℝ, E) I (Eq.le rfl) ?_ e.open_source hx ?_ (mapsTo_image (e.extend I) e.source) ?_ ?_ ?_ ?_ | E : Type u_1
inst✝¹³ : NormedAddCommGroup E
inst✝¹² : NormedSpace ℝ E
H : Type u_2
inst✝¹¹ : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type u_3
inst✝¹⁰ : TopologicalSpace M
inst✝⁹ : ChartedSpace H M
inst✝⁸ : SmoothManifoldWithCorners I M
F : Type u_4
inst✝⁷ : NormedAddCommGroup F
inst✝⁶ : NormedSpace ℝ F
G : Ty... | case refine_1
E : Type u_1
inst✝¹³ : NormedAddCommGroup E
inst✝¹² : NormedSpace ℝ E
H : Type u_2
inst✝¹¹ : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type u_3
inst✝¹⁰ : TopologicalSpace M
inst✝⁹ : ChartedSpace H M
inst✝⁸ : SmoothManifoldWithCorners I M
F : Type u_4
inst✝⁷ : NormedAddCommGroup F
inst✝⁶ : NormedSp... | Please generate a tactic in lean4 to solve the state.
STATE:
E : Type u_1
inst✝¹³ : NormedAddCommGroup E
inst✝¹² : NormedSpace ℝ E
H : Type u_2
inst✝¹¹ : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type u_3
inst✝¹⁰ : TopologicalSpace M
inst✝⁹ : ChartedSpace H M
inst✝⁸ : SmoothManifoldWithCorners I M
F : Type u_4
... |
https://github.com/fpvandoorn/sard.git | f4a1bb550136e8591dcd20dca5bdcce766a958c7 | Sard/ManifoldAux.lean | extendedChart_symm_differential_bijective | [267, 1] | [281, 59] | exact e.extend_isOpenMapOn_source I e.open_source (Eq.subset rfl) | case refine_1
E : Type u_1
inst✝¹³ : NormedAddCommGroup E
inst✝¹² : NormedSpace ℝ E
H : Type u_2
inst✝¹¹ : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type u_3
inst✝¹⁰ : TopologicalSpace M
inst✝⁹ : ChartedSpace H M
inst✝⁸ : SmoothManifoldWithCorners I M
F : Type u_4
inst✝⁷ : NormedAddCommGroup F
inst✝⁶ : NormedSp... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case refine_1
E : Type u_1
inst✝¹³ : NormedAddCommGroup E
inst✝¹² : NormedSpace ℝ E
H : Type u_2
inst✝¹¹ : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type u_3
inst✝¹⁰ : TopologicalSpace M
inst✝⁹ : ChartedSpace H M
inst✝⁸ : SmoothManifoldWithCorners I M... |
https://github.com/fpvandoorn/sard.git | f4a1bb550136e8591dcd20dca5bdcce766a958c7 | Sard/ManifoldAux.lean | extendedChart_symm_differential_bijective | [267, 1] | [281, 59] | rintro x ⟨s, hs, rfl⟩ | case refine_2
E : Type u_1
inst✝¹³ : NormedAddCommGroup E
inst✝¹² : NormedSpace ℝ E
H : Type u_2
inst✝¹¹ : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type u_3
inst✝¹⁰ : TopologicalSpace M
inst✝⁹ : ChartedSpace H M
inst✝⁸ : SmoothManifoldWithCorners I M
F : Type u_4
inst✝⁷ : NormedAddCommGroup F
inst✝⁶ : NormedSp... | case refine_2.intro.intro
E : Type u_1
inst✝¹³ : NormedAddCommGroup E
inst✝¹² : NormedSpace ℝ E
H : Type u_2
inst✝¹¹ : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type u_3
inst✝¹⁰ : TopologicalSpace M
inst✝⁹ : ChartedSpace H M
inst✝⁸ : SmoothManifoldWithCorners I M
F : Type u_4
inst✝⁷ : NormedAddCommGroup F
inst✝... | Please generate a tactic in lean4 to solve the state.
STATE:
case refine_2
E : Type u_1
inst✝¹³ : NormedAddCommGroup E
inst✝¹² : NormedSpace ℝ E
H : Type u_2
inst✝¹¹ : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type u_3
inst✝¹⁰ : TopologicalSpace M
inst✝⁹ : ChartedSpace H M
inst✝⁸ : SmoothManifoldWithCorners I M... |
https://github.com/fpvandoorn/sard.git | f4a1bb550136e8591dcd20dca5bdcce766a958c7 | Sard/ManifoldAux.lean | extendedChart_symm_differential_bijective | [267, 1] | [281, 59] | have : (e.extend I).symm (e.extend I s) = s := e.extend_left_inv _ hs | case refine_2.intro.intro
E : Type u_1
inst✝¹³ : NormedAddCommGroup E
inst✝¹² : NormedSpace ℝ E
H : Type u_2
inst✝¹¹ : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type u_3
inst✝¹⁰ : TopologicalSpace M
inst✝⁹ : ChartedSpace H M
inst✝⁸ : SmoothManifoldWithCorners I M
F : Type u_4
inst✝⁷ : NormedAddCommGroup F
inst✝... | case refine_2.intro.intro
E : Type u_1
inst✝¹³ : NormedAddCommGroup E
inst✝¹² : NormedSpace ℝ E
H : Type u_2
inst✝¹¹ : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type u_3
inst✝¹⁰ : TopologicalSpace M
inst✝⁹ : ChartedSpace H M
inst✝⁸ : SmoothManifoldWithCorners I M
F : Type u_4
inst✝⁷ : NormedAddCommGroup F
inst✝... | Please generate a tactic in lean4 to solve the state.
STATE:
case refine_2.intro.intro
E : Type u_1
inst✝¹³ : NormedAddCommGroup E
inst✝¹² : NormedSpace ℝ E
H : Type u_2
inst✝¹¹ : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type u_3
inst✝¹⁰ : TopologicalSpace M
inst✝⁹ : ChartedSpace H M
inst✝⁸ : SmoothManifoldWit... |
https://github.com/fpvandoorn/sard.git | f4a1bb550136e8591dcd20dca5bdcce766a958c7 | Sard/ManifoldAux.lean | extendedChart_symm_differential_bijective | [267, 1] | [281, 59] | rw [this] | case refine_2.intro.intro
E : Type u_1
inst✝¹³ : NormedAddCommGroup E
inst✝¹² : NormedSpace ℝ E
H : Type u_2
inst✝¹¹ : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type u_3
inst✝¹⁰ : TopologicalSpace M
inst✝⁹ : ChartedSpace H M
inst✝⁸ : SmoothManifoldWithCorners I M
F : Type u_4
inst✝⁷ : NormedAddCommGroup F
inst✝... | case refine_2.intro.intro
E : Type u_1
inst✝¹³ : NormedAddCommGroup E
inst✝¹² : NormedSpace ℝ E
H : Type u_2
inst✝¹¹ : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type u_3
inst✝¹⁰ : TopologicalSpace M
inst✝⁹ : ChartedSpace H M
inst✝⁸ : SmoothManifoldWithCorners I M
F : Type u_4
inst✝⁷ : NormedAddCommGroup F
inst✝... | Please generate a tactic in lean4 to solve the state.
STATE:
case refine_2.intro.intro
E : Type u_1
inst✝¹³ : NormedAddCommGroup E
inst✝¹² : NormedSpace ℝ E
H : Type u_2
inst✝¹¹ : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type u_3
inst✝¹⁰ : TopologicalSpace M
inst✝⁹ : ChartedSpace H M
inst✝⁸ : SmoothManifoldWit... |
https://github.com/fpvandoorn/sard.git | f4a1bb550136e8591dcd20dca5bdcce766a958c7 | Sard/ManifoldAux.lean | extendedChart_symm_differential_bijective | [267, 1] | [281, 59] | exact hs | case refine_2.intro.intro
E : Type u_1
inst✝¹³ : NormedAddCommGroup E
inst✝¹² : NormedSpace ℝ E
H : Type u_2
inst✝¹¹ : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type u_3
inst✝¹⁰ : TopologicalSpace M
inst✝⁹ : ChartedSpace H M
inst✝⁸ : SmoothManifoldWithCorners I M
F : Type u_4
inst✝⁷ : NormedAddCommGroup F
inst✝... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case refine_2.intro.intro
E : Type u_1
inst✝¹³ : NormedAddCommGroup E
inst✝¹² : NormedSpace ℝ E
H : Type u_2
inst✝¹¹ : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type u_3
inst✝¹⁰ : TopologicalSpace M
inst✝⁹ : ChartedSpace H M
inst✝⁸ : SmoothManifoldWit... |
https://github.com/fpvandoorn/sard.git | f4a1bb550136e8591dcd20dca5bdcce766a958c7 | Sard/ManifoldAux.lean | extendedChart_symm_differential_bijective | [267, 1] | [281, 59] | exact fun x hx ↦ e.extend_right_inv _ hx | case refine_3
E : Type u_1
inst✝¹³ : NormedAddCommGroup E
inst✝¹² : NormedSpace ℝ E
H : Type u_2
inst✝¹¹ : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type u_3
inst✝¹⁰ : TopologicalSpace M
inst✝⁹ : ChartedSpace H M
inst✝⁸ : SmoothManifoldWithCorners I M
F : Type u_4
inst✝⁷ : NormedAddCommGroup F
inst✝⁶ : NormedSp... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case refine_3
E : Type u_1
inst✝¹³ : NormedAddCommGroup E
inst✝¹² : NormedSpace ℝ E
H : Type u_2
inst✝¹¹ : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type u_3
inst✝¹⁰ : TopologicalSpace M
inst✝⁹ : ChartedSpace H M
inst✝⁸ : SmoothManifoldWithCorners I M... |
https://github.com/fpvandoorn/sard.git | f4a1bb550136e8591dcd20dca5bdcce766a958c7 | Sard/ManifoldAux.lean | extendedChart_symm_differential_bijective | [267, 1] | [281, 59] | exact fun x hx ↦ e.extend_left_inv _ hx | case refine_4
E : Type u_1
inst✝¹³ : NormedAddCommGroup E
inst✝¹² : NormedSpace ℝ E
H : Type u_2
inst✝¹¹ : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type u_3
inst✝¹⁰ : TopologicalSpace M
inst✝⁹ : ChartedSpace H M
inst✝⁸ : SmoothManifoldWithCorners I M
F : Type u_4
inst✝⁷ : NormedAddCommGroup F
inst✝⁶ : NormedSp... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case refine_4
E : Type u_1
inst✝¹³ : NormedAddCommGroup E
inst✝¹² : NormedSpace ℝ E
H : Type u_2
inst✝¹¹ : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type u_3
inst✝¹⁰ : TopologicalSpace M
inst✝⁹ : ChartedSpace H M
inst✝⁸ : SmoothManifoldWithCorners I M... |
https://github.com/fpvandoorn/sard.git | f4a1bb550136e8591dcd20dca5bdcce766a958c7 | Sard/ManifoldAux.lean | extendedChart_symm_differential_bijective | [267, 1] | [281, 59] | exact SmoothOn.contMDiffOn (extendedChart_symm_smooth I he) | case refine_5
E : Type u_1
inst✝¹³ : NormedAddCommGroup E
inst✝¹² : NormedSpace ℝ E
H : Type u_2
inst✝¹¹ : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type u_3
inst✝¹⁰ : TopologicalSpace M
inst✝⁹ : ChartedSpace H M
inst✝⁸ : SmoothManifoldWithCorners I M
F : Type u_4
inst✝⁷ : NormedAddCommGroup F
inst✝⁶ : NormedSp... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case refine_5
E : Type u_1
inst✝¹³ : NormedAddCommGroup E
inst✝¹² : NormedSpace ℝ E
H : Type u_2
inst✝¹¹ : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type u_3
inst✝¹⁰ : TopologicalSpace M
inst✝⁹ : ChartedSpace H M
inst✝⁸ : SmoothManifoldWithCorners I M... |
https://github.com/fpvandoorn/sard.git | f4a1bb550136e8591dcd20dca5bdcce766a958c7 | Sard/ManifoldAux.lean | extendedChart_symm_differential_bijective | [267, 1] | [281, 59] | exact SmoothOn.contMDiffOn (extendedChart_smooth I he) | case refine_6
E : Type u_1
inst✝¹³ : NormedAddCommGroup E
inst✝¹² : NormedSpace ℝ E
H : Type u_2
inst✝¹¹ : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type u_3
inst✝¹⁰ : TopologicalSpace M
inst✝⁹ : ChartedSpace H M
inst✝⁸ : SmoothManifoldWithCorners I M
F : Type u_4
inst✝⁷ : NormedAddCommGroup F
inst✝⁶ : NormedSp... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case refine_6
E : Type u_1
inst✝¹³ : NormedAddCommGroup E
inst✝¹² : NormedSpace ℝ E
H : Type u_2
inst✝¹¹ : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type u_3
inst✝¹⁰ : TopologicalSpace M
inst✝⁹ : ChartedSpace H M
inst✝⁸ : SmoothManifoldWithCorners I M... |
https://github.com/fpvandoorn/sard.git | f4a1bb550136e8591dcd20dca5bdcce766a958c7 | Sard/ManifoldAux.lean | extendedChart_differential_bijective | [283, 1] | [298, 59] | have diff : ContMDiffOn 𝓘(ℝ, E) I 1 (e.extend I).symm (e.extend I '' e.source) :=
SmoothOn.contMDiffOn (extendedChart_symm_smooth I he) | E : Type u_1
inst✝¹³ : NormedAddCommGroup E
inst✝¹² : NormedSpace ℝ E
H : Type u_2
inst✝¹¹ : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type u_3
inst✝¹⁰ : TopologicalSpace M
inst✝⁹ : ChartedSpace H M
inst✝⁸ : SmoothManifoldWithCorners I M
F : Type u_4
inst✝⁷ : NormedAddCommGroup F
inst✝⁶ : NormedSpace ℝ F
G : Ty... | E : Type u_1
inst✝¹³ : NormedAddCommGroup E
inst✝¹² : NormedSpace ℝ E
H : Type u_2
inst✝¹¹ : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type u_3
inst✝¹⁰ : TopologicalSpace M
inst✝⁹ : ChartedSpace H M
inst✝⁸ : SmoothManifoldWithCorners I M
F : Type u_4
inst✝⁷ : NormedAddCommGroup F
inst✝⁶ : NormedSpace ℝ F
G : Ty... | Please generate a tactic in lean4 to solve the state.
STATE:
E : Type u_1
inst✝¹³ : NormedAddCommGroup E
inst✝¹² : NormedSpace ℝ E
H : Type u_2
inst✝¹¹ : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type u_3
inst✝¹⁰ : TopologicalSpace M
inst✝⁹ : ChartedSpace H M
inst✝⁸ : SmoothManifoldWithCorners I M
F : Type u_4
... |
https://github.com/fpvandoorn/sard.git | f4a1bb550136e8591dcd20dca5bdcce766a958c7 | Sard/ManifoldAux.lean | extendedChart_differential_bijective | [283, 1] | [298, 59] | refine diffeoOn_differential_bijective I 𝓘(ℝ, E) (Eq.le rfl) e.open_source ?_ hx (mapsTo_image (e.extend I) e.source) ?_ ?_ ?_ ?_ diff | E : Type u_1
inst✝¹³ : NormedAddCommGroup E
inst✝¹² : NormedSpace ℝ E
H : Type u_2
inst✝¹¹ : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type u_3
inst✝¹⁰ : TopologicalSpace M
inst✝⁹ : ChartedSpace H M
inst✝⁸ : SmoothManifoldWithCorners I M
F : Type u_4
inst✝⁷ : NormedAddCommGroup F
inst✝⁶ : NormedSpace ℝ F
G : Ty... | case refine_1
E : Type u_1
inst✝¹³ : NormedAddCommGroup E
inst✝¹² : NormedSpace ℝ E
H : Type u_2
inst✝¹¹ : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type u_3
inst✝¹⁰ : TopologicalSpace M
inst✝⁹ : ChartedSpace H M
inst✝⁸ : SmoothManifoldWithCorners I M
F : Type u_4
inst✝⁷ : NormedAddCommGroup F
inst✝⁶ : NormedSp... | Please generate a tactic in lean4 to solve the state.
STATE:
E : Type u_1
inst✝¹³ : NormedAddCommGroup E
inst✝¹² : NormedSpace ℝ E
H : Type u_2
inst✝¹¹ : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type u_3
inst✝¹⁰ : TopologicalSpace M
inst✝⁹ : ChartedSpace H M
inst✝⁸ : SmoothManifoldWithCorners I M
F : Type u_4
... |
https://github.com/fpvandoorn/sard.git | f4a1bb550136e8591dcd20dca5bdcce766a958c7 | Sard/ManifoldAux.lean | extendedChart_differential_bijective | [283, 1] | [298, 59] | exact e.extend_isOpenMapOn_source I e.open_source (Eq.subset rfl) | case refine_1
E : Type u_1
inst✝¹³ : NormedAddCommGroup E
inst✝¹² : NormedSpace ℝ E
H : Type u_2
inst✝¹¹ : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type u_3
inst✝¹⁰ : TopologicalSpace M
inst✝⁹ : ChartedSpace H M
inst✝⁸ : SmoothManifoldWithCorners I M
F : Type u_4
inst✝⁷ : NormedAddCommGroup F
inst✝⁶ : NormedSp... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case refine_1
E : Type u_1
inst✝¹³ : NormedAddCommGroup E
inst✝¹² : NormedSpace ℝ E
H : Type u_2
inst✝¹¹ : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type u_3
inst✝¹⁰ : TopologicalSpace M
inst✝⁹ : ChartedSpace H M
inst✝⁸ : SmoothManifoldWithCorners I M... |
https://github.com/fpvandoorn/sard.git | f4a1bb550136e8591dcd20dca5bdcce766a958c7 | Sard/ManifoldAux.lean | extendedChart_differential_bijective | [283, 1] | [298, 59] | rintro x ⟨s, hs, rfl⟩ | case refine_2
E : Type u_1
inst✝¹³ : NormedAddCommGroup E
inst✝¹² : NormedSpace ℝ E
H : Type u_2
inst✝¹¹ : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type u_3
inst✝¹⁰ : TopologicalSpace M
inst✝⁹ : ChartedSpace H M
inst✝⁸ : SmoothManifoldWithCorners I M
F : Type u_4
inst✝⁷ : NormedAddCommGroup F
inst✝⁶ : NormedSp... | case refine_2.intro.intro
E : Type u_1
inst✝¹³ : NormedAddCommGroup E
inst✝¹² : NormedSpace ℝ E
H : Type u_2
inst✝¹¹ : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type u_3
inst✝¹⁰ : TopologicalSpace M
inst✝⁹ : ChartedSpace H M
inst✝⁸ : SmoothManifoldWithCorners I M
F : Type u_4
inst✝⁷ : NormedAddCommGroup F
inst✝... | Please generate a tactic in lean4 to solve the state.
STATE:
case refine_2
E : Type u_1
inst✝¹³ : NormedAddCommGroup E
inst✝¹² : NormedSpace ℝ E
H : Type u_2
inst✝¹¹ : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type u_3
inst✝¹⁰ : TopologicalSpace M
inst✝⁹ : ChartedSpace H M
inst✝⁸ : SmoothManifoldWithCorners I M... |
https://github.com/fpvandoorn/sard.git | f4a1bb550136e8591dcd20dca5bdcce766a958c7 | Sard/ManifoldAux.lean | extendedChart_differential_bijective | [283, 1] | [298, 59] | have : (e.extend I).symm (e.extend I s) = s := e.extend_left_inv _ hs | case refine_2.intro.intro
E : Type u_1
inst✝¹³ : NormedAddCommGroup E
inst✝¹² : NormedSpace ℝ E
H : Type u_2
inst✝¹¹ : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type u_3
inst✝¹⁰ : TopologicalSpace M
inst✝⁹ : ChartedSpace H M
inst✝⁸ : SmoothManifoldWithCorners I M
F : Type u_4
inst✝⁷ : NormedAddCommGroup F
inst✝... | case refine_2.intro.intro
E : Type u_1
inst✝¹³ : NormedAddCommGroup E
inst✝¹² : NormedSpace ℝ E
H : Type u_2
inst✝¹¹ : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type u_3
inst✝¹⁰ : TopologicalSpace M
inst✝⁹ : ChartedSpace H M
inst✝⁸ : SmoothManifoldWithCorners I M
F : Type u_4
inst✝⁷ : NormedAddCommGroup F
inst✝... | Please generate a tactic in lean4 to solve the state.
STATE:
case refine_2.intro.intro
E : Type u_1
inst✝¹³ : NormedAddCommGroup E
inst✝¹² : NormedSpace ℝ E
H : Type u_2
inst✝¹¹ : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type u_3
inst✝¹⁰ : TopologicalSpace M
inst✝⁹ : ChartedSpace H M
inst✝⁸ : SmoothManifoldWit... |
https://github.com/fpvandoorn/sard.git | f4a1bb550136e8591dcd20dca5bdcce766a958c7 | Sard/ManifoldAux.lean | extendedChart_differential_bijective | [283, 1] | [298, 59] | rw [this] | case refine_2.intro.intro
E : Type u_1
inst✝¹³ : NormedAddCommGroup E
inst✝¹² : NormedSpace ℝ E
H : Type u_2
inst✝¹¹ : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type u_3
inst✝¹⁰ : TopologicalSpace M
inst✝⁹ : ChartedSpace H M
inst✝⁸ : SmoothManifoldWithCorners I M
F : Type u_4
inst✝⁷ : NormedAddCommGroup F
inst✝... | case refine_2.intro.intro
E : Type u_1
inst✝¹³ : NormedAddCommGroup E
inst✝¹² : NormedSpace ℝ E
H : Type u_2
inst✝¹¹ : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type u_3
inst✝¹⁰ : TopologicalSpace M
inst✝⁹ : ChartedSpace H M
inst✝⁸ : SmoothManifoldWithCorners I M
F : Type u_4
inst✝⁷ : NormedAddCommGroup F
inst✝... | Please generate a tactic in lean4 to solve the state.
STATE:
case refine_2.intro.intro
E : Type u_1
inst✝¹³ : NormedAddCommGroup E
inst✝¹² : NormedSpace ℝ E
H : Type u_2
inst✝¹¹ : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type u_3
inst✝¹⁰ : TopologicalSpace M
inst✝⁹ : ChartedSpace H M
inst✝⁸ : SmoothManifoldWit... |
https://github.com/fpvandoorn/sard.git | f4a1bb550136e8591dcd20dca5bdcce766a958c7 | Sard/ManifoldAux.lean | extendedChart_differential_bijective | [283, 1] | [298, 59] | exact hs | case refine_2.intro.intro
E : Type u_1
inst✝¹³ : NormedAddCommGroup E
inst✝¹² : NormedSpace ℝ E
H : Type u_2
inst✝¹¹ : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type u_3
inst✝¹⁰ : TopologicalSpace M
inst✝⁹ : ChartedSpace H M
inst✝⁸ : SmoothManifoldWithCorners I M
F : Type u_4
inst✝⁷ : NormedAddCommGroup F
inst✝... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case refine_2.intro.intro
E : Type u_1
inst✝¹³ : NormedAddCommGroup E
inst✝¹² : NormedSpace ℝ E
H : Type u_2
inst✝¹¹ : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type u_3
inst✝¹⁰ : TopologicalSpace M
inst✝⁹ : ChartedSpace H M
inst✝⁸ : SmoothManifoldWit... |
https://github.com/fpvandoorn/sard.git | f4a1bb550136e8591dcd20dca5bdcce766a958c7 | Sard/ManifoldAux.lean | extendedChart_differential_bijective | [283, 1] | [298, 59] | exact fun x hx ↦ e.extend_left_inv I hx | case refine_3
E : Type u_1
inst✝¹³ : NormedAddCommGroup E
inst✝¹² : NormedSpace ℝ E
H : Type u_2
inst✝¹¹ : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type u_3
inst✝¹⁰ : TopologicalSpace M
inst✝⁹ : ChartedSpace H M
inst✝⁸ : SmoothManifoldWithCorners I M
F : Type u_4
inst✝⁷ : NormedAddCommGroup F
inst✝⁶ : NormedSp... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case refine_3
E : Type u_1
inst✝¹³ : NormedAddCommGroup E
inst✝¹² : NormedSpace ℝ E
H : Type u_2
inst✝¹¹ : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type u_3
inst✝¹⁰ : TopologicalSpace M
inst✝⁹ : ChartedSpace H M
inst✝⁸ : SmoothManifoldWithCorners I M... |
https://github.com/fpvandoorn/sard.git | f4a1bb550136e8591dcd20dca5bdcce766a958c7 | Sard/ManifoldAux.lean | extendedChart_differential_bijective | [283, 1] | [298, 59] | exact fun x hx ↦ e.extend_right_inv _ hx | case refine_4
E : Type u_1
inst✝¹³ : NormedAddCommGroup E
inst✝¹² : NormedSpace ℝ E
H : Type u_2
inst✝¹¹ : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type u_3
inst✝¹⁰ : TopologicalSpace M
inst✝⁹ : ChartedSpace H M
inst✝⁸ : SmoothManifoldWithCorners I M
F : Type u_4
inst✝⁷ : NormedAddCommGroup F
inst✝⁶ : NormedSp... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case refine_4
E : Type u_1
inst✝¹³ : NormedAddCommGroup E
inst✝¹² : NormedSpace ℝ E
H : Type u_2
inst✝¹¹ : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type u_3
inst✝¹⁰ : TopologicalSpace M
inst✝⁹ : ChartedSpace H M
inst✝⁸ : SmoothManifoldWithCorners I M... |
https://github.com/fpvandoorn/sard.git | f4a1bb550136e8591dcd20dca5bdcce766a958c7 | Sard/ManifoldAux.lean | extendedChart_differential_bijective | [283, 1] | [298, 59] | exact SmoothOn.contMDiffOn (extendedChart_smooth I he) | case refine_5
E : Type u_1
inst✝¹³ : NormedAddCommGroup E
inst✝¹² : NormedSpace ℝ E
H : Type u_2
inst✝¹¹ : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type u_3
inst✝¹⁰ : TopologicalSpace M
inst✝⁹ : ChartedSpace H M
inst✝⁸ : SmoothManifoldWithCorners I M
F : Type u_4
inst✝⁷ : NormedAddCommGroup F
inst✝⁶ : NormedSp... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case refine_5
E : Type u_1
inst✝¹³ : NormedAddCommGroup E
inst✝¹² : NormedSpace ℝ E
H : Type u_2
inst✝¹¹ : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type u_3
inst✝¹⁰ : TopologicalSpace M
inst✝⁹ : ChartedSpace H M
inst✝⁸ : SmoothManifoldWithCorners I M... |
https://github.com/fpvandoorn/sard.git | f4a1bb550136e8591dcd20dca5bdcce766a958c7 | Sard/ObsoleteHelpers.lean | empty_iff_open_cover | [9, 1] | [13, 30] | have : ⋃ (α : I), s ∩ U α = s := by rw [←inter_iUnion, hcover, inter_univ s] | X I : Type
U : I → Set X
hcover : ⋃ α, U α = univ
s : Set X
⊢ s = ∅ ↔ ∀ (α : I), s ∩ U α = ∅ | X I : Type
U : I → Set X
hcover : ⋃ α, U α = univ
s : Set X
this : ⋃ α, s ∩ U α = s
⊢ s = ∅ ↔ ∀ (α : I), s ∩ U α = ∅ | Please generate a tactic in lean4 to solve the state.
STATE:
X I : Type
U : I → Set X
hcover : ⋃ α, U α = univ
s : Set X
⊢ s = ∅ ↔ ∀ (α : I), s ∩ U α = ∅
TACTIC:
|
https://github.com/fpvandoorn/sard.git | f4a1bb550136e8591dcd20dca5bdcce766a958c7 | Sard/ObsoleteHelpers.lean | empty_iff_open_cover | [9, 1] | [13, 30] | nth_rewrite 1 [← this] | X I : Type
U : I → Set X
hcover : ⋃ α, U α = univ
s : Set X
this : ⋃ α, s ∩ U α = s
⊢ s = ∅ ↔ ∀ (α : I), s ∩ U α = ∅ | X I : Type
U : I → Set X
hcover : ⋃ α, U α = univ
s : Set X
this : ⋃ α, s ∩ U α = s
⊢ ⋃ α, s ∩ U α = ∅ ↔ ∀ (α : I), s ∩ U α = ∅ | Please generate a tactic in lean4 to solve the state.
STATE:
X I : Type
U : I → Set X
hcover : ⋃ α, U α = univ
s : Set X
this : ⋃ α, s ∩ U α = s
⊢ s = ∅ ↔ ∀ (α : I), s ∩ U α = ∅
TACTIC:
|
https://github.com/fpvandoorn/sard.git | f4a1bb550136e8591dcd20dca5bdcce766a958c7 | Sard/ObsoleteHelpers.lean | empty_iff_open_cover | [9, 1] | [13, 30] | simp only [iUnion_eq_empty] | X I : Type
U : I → Set X
hcover : ⋃ α, U α = univ
s : Set X
this : ⋃ α, s ∩ U α = s
⊢ ⋃ α, s ∩ U α = ∅ ↔ ∀ (α : I), s ∩ U α = ∅ | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
X I : Type
U : I → Set X
hcover : ⋃ α, U α = univ
s : Set X
this : ⋃ α, s ∩ U α = s
⊢ ⋃ α, s ∩ U α = ∅ ↔ ∀ (α : I), s ∩ U α = ∅
TACTIC:
|
https://github.com/fpvandoorn/sard.git | f4a1bb550136e8591dcd20dca5bdcce766a958c7 | Sard/ObsoleteHelpers.lean | empty_iff_open_cover | [9, 1] | [13, 30] | rw [←inter_iUnion, hcover, inter_univ s] | X I : Type
U : I → Set X
hcover : ⋃ α, U α = univ
s : Set X
⊢ ⋃ α, s ∩ U α = s | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
X I : Type
U : I → Set X
hcover : ⋃ α, U α = univ
s : Set X
⊢ ⋃ α, s ∩ U α = s
TACTIC:
|
https://github.com/fpvandoorn/sard.git | f4a1bb550136e8591dcd20dca5bdcce766a958c7 | Sard/ObsoleteHelpers.lean | image_interior | [21, 1] | [26, 8] | sorry | α β : Type
inst✝¹ : TopologicalSpace α
inst✝ : TopologicalSpace β
e : PartialHomeomorph α β
s : Set α
⊢ e.target ∩ ↑e '' interior s = ↑e '' (e.source ∩ interior s) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
α β : Type
inst✝¹ : TopologicalSpace α
inst✝ : TopologicalSpace β
e : PartialHomeomorph α β
s : Set α
⊢ e.target ∩ ↑e '' interior s = ↑e '' (e.source ∩ interior s)
TACTIC:
|
https://github.com/fpvandoorn/sard.git | f4a1bb550136e8591dcd20dca5bdcce766a958c7 | Sard/ObsoleteHelpers.lean | homeo_preserves_empty_interior | [29, 1] | [32, 27] | rw [← Homeomorph.image_interior, h₂s] | α β : Type
inst✝¹ : TopologicalSpace α
inst✝ : TopologicalSpace β
f : α ≃ₜ β
s : Set α
h₂s : interior s = ∅
⊢ interior (⇑f '' s) = ∅ | α β : Type
inst✝¹ : TopologicalSpace α
inst✝ : TopologicalSpace β
f : α ≃ₜ β
s : Set α
h₂s : interior s = ∅
⊢ ⇑f '' ∅ = ∅ | Please generate a tactic in lean4 to solve the state.
STATE:
α β : Type
inst✝¹ : TopologicalSpace α
inst✝ : TopologicalSpace β
f : α ≃ₜ β
s : Set α
h₂s : interior s = ∅
⊢ interior (⇑f '' s) = ∅
TACTIC:
|
https://github.com/fpvandoorn/sard.git | f4a1bb550136e8591dcd20dca5bdcce766a958c7 | Sard/ObsoleteHelpers.lean | homeo_preserves_empty_interior | [29, 1] | [32, 27] | exact Set.image_empty ↑f | α β : Type
inst✝¹ : TopologicalSpace α
inst✝ : TopologicalSpace β
f : α ≃ₜ β
s : Set α
h₂s : interior s = ∅
⊢ ⇑f '' ∅ = ∅ | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
α β : Type
inst✝¹ : TopologicalSpace α
inst✝ : TopologicalSpace β
f : α ≃ₜ β
s : Set α
h₂s : interior s = ∅
⊢ ⇑f '' ∅ = ∅
TACTIC:
|
https://github.com/fpvandoorn/sard.git | f4a1bb550136e8591dcd20dca5bdcce766a958c7 | Sard/ObsoleteHelpers.lean | local_homeo_preserves_empty_interior | [35, 1] | [48, 8] | have h₁ : s = s ∩ f.source := by
rw [← @inter_eq_left α s f.source] at hs
symm
exact hs | α β : Type
inst✝¹ : TopologicalSpace α
inst✝ : TopologicalSpace β
f : PartialHomeomorph α β
s : Set α
hs : s ⊆ f.source
h₂s : interior s = ∅
⊢ interior (↑f '' s) = ∅ | α β : Type
inst✝¹ : TopologicalSpace α
inst✝ : TopologicalSpace β
f : PartialHomeomorph α β
s : Set α
hs : s ⊆ f.source
h₂s : interior s = ∅
h₁ : s = s ∩ f.source
⊢ interior (↑f '' s) = ∅ | Please generate a tactic in lean4 to solve the state.
STATE:
α β : Type
inst✝¹ : TopologicalSpace α
inst✝ : TopologicalSpace β
f : PartialHomeomorph α β
s : Set α
hs : s ⊆ f.source
h₂s : interior s = ∅
⊢ interior (↑f '' s) = ∅
TACTIC:
|
https://github.com/fpvandoorn/sard.git | f4a1bb550136e8591dcd20dca5bdcce766a958c7 | Sard/ObsoleteHelpers.lean | local_homeo_preserves_empty_interior | [35, 1] | [48, 8] | have h₂ : interior s = interior (s ∩ f.source) := by
sorry | α β : Type
inst✝¹ : TopologicalSpace α
inst✝ : TopologicalSpace β
f : PartialHomeomorph α β
s : Set α
hs : s ⊆ f.source
h₂s : interior s = ∅
h₁ : s = s ∩ f.source
⊢ interior (↑f '' s) = ∅ | α β : Type
inst✝¹ : TopologicalSpace α
inst✝ : TopologicalSpace β
f : PartialHomeomorph α β
s : Set α
hs : s ⊆ f.source
h₂s : interior s = ∅
h₁ : s = s ∩ f.source
h₂ : interior s = interior (s ∩ f.source)
⊢ interior (↑f '' s) = ∅ | Please generate a tactic in lean4 to solve the state.
STATE:
α β : Type
inst✝¹ : TopologicalSpace α
inst✝ : TopologicalSpace β
f : PartialHomeomorph α β
s : Set α
hs : s ⊆ f.source
h₂s : interior s = ∅
h₁ : s = s ∩ f.source
⊢ interior (↑f '' s) = ∅
TACTIC:
|
https://github.com/fpvandoorn/sard.git | f4a1bb550136e8591dcd20dca5bdcce766a958c7 | Sard/ObsoleteHelpers.lean | local_homeo_preserves_empty_interior | [35, 1] | [48, 8] | rw [h₁] at h₂s | α β : Type
inst✝¹ : TopologicalSpace α
inst✝ : TopologicalSpace β
f : PartialHomeomorph α β
s : Set α
hs : s ⊆ f.source
h₂s : interior s = ∅
h₁ : s = s ∩ f.source
h₂ : interior s = interior (s ∩ f.source)
⊢ interior (↑f '' s) = ∅ | α β : Type
inst✝¹ : TopologicalSpace α
inst✝ : TopologicalSpace β
f : PartialHomeomorph α β
s : Set α
hs : s ⊆ f.source
h₂s : interior (s ∩ f.source) = ∅
h₁ : s = s ∩ f.source
h₂ : interior s = interior (s ∩ f.source)
⊢ interior (↑f '' s) = ∅ | Please generate a tactic in lean4 to solve the state.
STATE:
α β : Type
inst✝¹ : TopologicalSpace α
inst✝ : TopologicalSpace β
f : PartialHomeomorph α β
s : Set α
hs : s ⊆ f.source
h₂s : interior s = ∅
h₁ : s = s ∩ f.source
h₂ : interior s = interior (s ∩ f.source)
⊢ interior (↑f '' s) = ∅
TACTIC:
|
https://github.com/fpvandoorn/sard.git | f4a1bb550136e8591dcd20dca5bdcce766a958c7 | Sard/ObsoleteHelpers.lean | local_homeo_preserves_empty_interior | [35, 1] | [48, 8] | have h₃ : f '' (interior s ∩ f.source) = ∅ := by sorry | α β : Type
inst✝¹ : TopologicalSpace α
inst✝ : TopologicalSpace β
f : PartialHomeomorph α β
s : Set α
hs : s ⊆ f.source
h₂s : interior (s ∩ f.source) = ∅
h₁ : s = s ∩ f.source
h₂ : interior s = interior (s ∩ f.source)
⊢ interior (↑f '' s) = ∅ | α β : Type
inst✝¹ : TopologicalSpace α
inst✝ : TopologicalSpace β
f : PartialHomeomorph α β
s : Set α
hs : s ⊆ f.source
h₂s : interior (s ∩ f.source) = ∅
h₁ : s = s ∩ f.source
h₂ : interior s = interior (s ∩ f.source)
h₃ : ↑f '' (interior s ∩ f.source) = ∅
⊢ interior (↑f '' s) = ∅ | Please generate a tactic in lean4 to solve the state.
STATE:
α β : Type
inst✝¹ : TopologicalSpace α
inst✝ : TopologicalSpace β
f : PartialHomeomorph α β
s : Set α
hs : s ⊆ f.source
h₂s : interior (s ∩ f.source) = ∅
h₁ : s = s ∩ f.source
h₂ : interior s = interior (s ∩ f.source)
⊢ interior (↑f '' s) = ∅
TACTIC:
|
https://github.com/fpvandoorn/sard.git | f4a1bb550136e8591dcd20dca5bdcce766a958c7 | Sard/ObsoleteHelpers.lean | local_homeo_preserves_empty_interior | [35, 1] | [48, 8] | sorry | α β : Type
inst✝¹ : TopologicalSpace α
inst✝ : TopologicalSpace β
f : PartialHomeomorph α β
s : Set α
hs : s ⊆ f.source
h₂s : interior (s ∩ f.source) = ∅
h₁ : s = s ∩ f.source
h₂ : interior s = interior (s ∩ f.source)
h₃ : ↑f '' (interior s ∩ f.source) = ∅
⊢ interior (↑f '' s) = ∅ | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
α β : Type
inst✝¹ : TopologicalSpace α
inst✝ : TopologicalSpace β
f : PartialHomeomorph α β
s : Set α
hs : s ⊆ f.source
h₂s : interior (s ∩ f.source) = ∅
h₁ : s = s ∩ f.source
h₂ : interior s = interior (s ∩ f.source)
h₃ : ↑f '' (interior s ∩ f.source) = ∅
⊢ ... |
https://github.com/fpvandoorn/sard.git | f4a1bb550136e8591dcd20dca5bdcce766a958c7 | Sard/ObsoleteHelpers.lean | local_homeo_preserves_empty_interior | [35, 1] | [48, 8] | rw [← @inter_eq_left α s f.source] at hs | α β : Type
inst✝¹ : TopologicalSpace α
inst✝ : TopologicalSpace β
f : PartialHomeomorph α β
s : Set α
hs : s ⊆ f.source
h₂s : interior s = ∅
⊢ s = s ∩ f.source | α β : Type
inst✝¹ : TopologicalSpace α
inst✝ : TopologicalSpace β
f : PartialHomeomorph α β
s : Set α
hs : s ∩ f.source = s
h₂s : interior s = ∅
⊢ s = s ∩ f.source | Please generate a tactic in lean4 to solve the state.
STATE:
α β : Type
inst✝¹ : TopologicalSpace α
inst✝ : TopologicalSpace β
f : PartialHomeomorph α β
s : Set α
hs : s ⊆ f.source
h₂s : interior s = ∅
⊢ s = s ∩ f.source
TACTIC:
|
https://github.com/fpvandoorn/sard.git | f4a1bb550136e8591dcd20dca5bdcce766a958c7 | Sard/ObsoleteHelpers.lean | local_homeo_preserves_empty_interior | [35, 1] | [48, 8] | symm | α β : Type
inst✝¹ : TopologicalSpace α
inst✝ : TopologicalSpace β
f : PartialHomeomorph α β
s : Set α
hs : s ∩ f.source = s
h₂s : interior s = ∅
⊢ s = s ∩ f.source | α β : Type
inst✝¹ : TopologicalSpace α
inst✝ : TopologicalSpace β
f : PartialHomeomorph α β
s : Set α
hs : s ∩ f.source = s
h₂s : interior s = ∅
⊢ s ∩ f.source = s | Please generate a tactic in lean4 to solve the state.
STATE:
α β : Type
inst✝¹ : TopologicalSpace α
inst✝ : TopologicalSpace β
f : PartialHomeomorph α β
s : Set α
hs : s ∩ f.source = s
h₂s : interior s = ∅
⊢ s = s ∩ f.source
TACTIC:
|
https://github.com/fpvandoorn/sard.git | f4a1bb550136e8591dcd20dca5bdcce766a958c7 | Sard/ObsoleteHelpers.lean | local_homeo_preserves_empty_interior | [35, 1] | [48, 8] | exact hs | α β : Type
inst✝¹ : TopologicalSpace α
inst✝ : TopologicalSpace β
f : PartialHomeomorph α β
s : Set α
hs : s ∩ f.source = s
h₂s : interior s = ∅
⊢ s ∩ f.source = s | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
α β : Type
inst✝¹ : TopologicalSpace α
inst✝ : TopologicalSpace β
f : PartialHomeomorph α β
s : Set α
hs : s ∩ f.source = s
h₂s : interior s = ∅
⊢ s ∩ f.source = s
TACTIC:
|
https://github.com/fpvandoorn/sard.git | f4a1bb550136e8591dcd20dca5bdcce766a958c7 | Sard/ObsoleteHelpers.lean | local_homeo_preserves_empty_interior | [35, 1] | [48, 8] | sorry | α β : Type
inst✝¹ : TopologicalSpace α
inst✝ : TopologicalSpace β
f : PartialHomeomorph α β
s : Set α
hs : s ⊆ f.source
h₂s : interior s = ∅
h₁ : s = s ∩ f.source
⊢ interior s = interior (s ∩ f.source) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
α β : Type
inst✝¹ : TopologicalSpace α
inst✝ : TopologicalSpace β
f : PartialHomeomorph α β
s : Set α
hs : s ⊆ f.source
h₂s : interior s = ∅
h₁ : s = s ∩ f.source
⊢ interior s = interior (s ∩ f.source)
TACTIC:
|
https://github.com/fpvandoorn/sard.git | f4a1bb550136e8591dcd20dca5bdcce766a958c7 | Sard/ObsoleteHelpers.lean | local_homeo_preserves_empty_interior | [35, 1] | [48, 8] | sorry | α β : Type
inst✝¹ : TopologicalSpace α
inst✝ : TopologicalSpace β
f : PartialHomeomorph α β
s : Set α
hs : s ⊆ f.source
h₂s : interior (s ∩ f.source) = ∅
h₁ : s = s ∩ f.source
h₂ : interior s = interior (s ∩ f.source)
⊢ ↑f '' (interior s ∩ f.source) = ∅ | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
α β : Type
inst✝¹ : TopologicalSpace α
inst✝ : TopologicalSpace β
f : PartialHomeomorph α β
s : Set α
hs : s ⊆ f.source
h₂s : interior (s ∩ f.source) = ∅
h₁ : s = s ∩ f.source
h₂ : interior s = interior (s ∩ f.source)
⊢ ↑f '' (interior s ∩ f.source) = ∅
TACTI... |
https://github.com/fpvandoorn/sard.git | f4a1bb550136e8591dcd20dca5bdcce766a958c7 | Sard/ObsoleteHelpers.lean | interior_zero_iff_open_cover | [53, 1] | [87, 10] | constructor | X : Type
inst✝ : TopologicalSpace X
I : Type
U : I → Set X
hU : ∀ (α : I), IsOpen (U α)
hcover : ↑(⋃ α, U α) = X
s : Set X
⊢ interior s = ∅ ↔ ∀ (α : I), interior (s ∩ U α) = ∅ | case mp
X : Type
inst✝ : TopologicalSpace X
I : Type
U : I → Set X
hU : ∀ (α : I), IsOpen (U α)
hcover : ↑(⋃ α, U α) = X
s : Set X
⊢ interior s = ∅ → ∀ (α : I), interior (s ∩ U α) = ∅
case mpr
X : Type
inst✝ : TopologicalSpace X
I : Type
U : I → Set X
hU : ∀ (α : I), IsOpen (U α)
hcover : ↑(⋃ α, U α) = X
s : Set X
⊢ (... | Please generate a tactic in lean4 to solve the state.
STATE:
X : Type
inst✝ : TopologicalSpace X
I : Type
U : I → Set X
hU : ∀ (α : I), IsOpen (U α)
hcover : ↑(⋃ α, U α) = X
s : Set X
⊢ interior s = ∅ ↔ ∀ (α : I), interior (s ∩ U α) = ∅
TACTIC:
|
https://github.com/fpvandoorn/sard.git | f4a1bb550136e8591dcd20dca5bdcce766a958c7 | Sard/ObsoleteHelpers.lean | interior_zero_iff_open_cover | [53, 1] | [87, 10] | intro hs α | case mp
X : Type
inst✝ : TopologicalSpace X
I : Type
U : I → Set X
hU : ∀ (α : I), IsOpen (U α)
hcover : ↑(⋃ α, U α) = X
s : Set X
⊢ interior s = ∅ → ∀ (α : I), interior (s ∩ U α) = ∅ | case mp
X : Type
inst✝ : TopologicalSpace X
I : Type
U : I → Set X
hU : ∀ (α : I), IsOpen (U α)
hcover : ↑(⋃ α, U α) = X
s : Set X
hs : interior s = ∅
α : I
⊢ interior (s ∩ U α) = ∅ | Please generate a tactic in lean4 to solve the state.
STATE:
case mp
X : Type
inst✝ : TopologicalSpace X
I : Type
U : I → Set X
hU : ∀ (α : I), IsOpen (U α)
hcover : ↑(⋃ α, U α) = X
s : Set X
⊢ interior s = ∅ → ∀ (α : I), interior (s ∩ U α) = ∅
TACTIC:
|
https://github.com/fpvandoorn/sard.git | f4a1bb550136e8591dcd20dca5bdcce766a958c7 | Sard/ObsoleteHelpers.lean | interior_zero_iff_open_cover | [53, 1] | [87, 10] | have aux: interior (s ∩ U α) ⊆ interior s := by
apply interior_mono
apply Set.inter_subset_left | case mp
X : Type
inst✝ : TopologicalSpace X
I : Type
U : I → Set X
hU : ∀ (α : I), IsOpen (U α)
hcover : ↑(⋃ α, U α) = X
s : Set X
hs : interior s = ∅
α : I
⊢ interior (s ∩ U α) = ∅ | case mp
X : Type
inst✝ : TopologicalSpace X
I : Type
U : I → Set X
hU : ∀ (α : I), IsOpen (U α)
hcover : ↑(⋃ α, U α) = X
s : Set X
hs : interior s = ∅
α : I
aux : interior (s ∩ U α) ⊆ interior s
⊢ interior (s ∩ U α) = ∅ | Please generate a tactic in lean4 to solve the state.
STATE:
case mp
X : Type
inst✝ : TopologicalSpace X
I : Type
U : I → Set X
hU : ∀ (α : I), IsOpen (U α)
hcover : ↑(⋃ α, U α) = X
s : Set X
hs : interior s = ∅
α : I
⊢ interior (s ∩ U α) = ∅
TACTIC:
|
https://github.com/fpvandoorn/sard.git | f4a1bb550136e8591dcd20dca5bdcce766a958c7 | Sard/ObsoleteHelpers.lean | interior_zero_iff_open_cover | [53, 1] | [87, 10] | exact Set.subset_eq_empty aux hs | case mp
X : Type
inst✝ : TopologicalSpace X
I : Type
U : I → Set X
hU : ∀ (α : I), IsOpen (U α)
hcover : ↑(⋃ α, U α) = X
s : Set X
hs : interior s = ∅
α : I
aux : interior (s ∩ U α) ⊆ interior s
⊢ interior (s ∩ U α) = ∅ | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case mp
X : Type
inst✝ : TopologicalSpace X
I : Type
U : I → Set X
hU : ∀ (α : I), IsOpen (U α)
hcover : ↑(⋃ α, U α) = X
s : Set X
hs : interior s = ∅
α : I
aux : interior (s ∩ U α) ⊆ interior s
⊢ interior (s ∩ U α) = ∅
TACTIC:
|
https://github.com/fpvandoorn/sard.git | f4a1bb550136e8591dcd20dca5bdcce766a958c7 | Sard/ObsoleteHelpers.lean | interior_zero_iff_open_cover | [53, 1] | [87, 10] | apply interior_mono | X : Type
inst✝ : TopologicalSpace X
I : Type
U : I → Set X
hU : ∀ (α : I), IsOpen (U α)
hcover : ↑(⋃ α, U α) = X
s : Set X
hs : interior s = ∅
α : I
⊢ interior (s ∩ U α) ⊆ interior s | case h
X : Type
inst✝ : TopologicalSpace X
I : Type
U : I → Set X
hU : ∀ (α : I), IsOpen (U α)
hcover : ↑(⋃ α, U α) = X
s : Set X
hs : interior s = ∅
α : I
⊢ s ∩ U α ⊆ s | Please generate a tactic in lean4 to solve the state.
STATE:
X : Type
inst✝ : TopologicalSpace X
I : Type
U : I → Set X
hU : ∀ (α : I), IsOpen (U α)
hcover : ↑(⋃ α, U α) = X
s : Set X
hs : interior s = ∅
α : I
⊢ interior (s ∩ U α) ⊆ interior s
TACTIC:
|
https://github.com/fpvandoorn/sard.git | f4a1bb550136e8591dcd20dca5bdcce766a958c7 | Sard/ObsoleteHelpers.lean | interior_zero_iff_open_cover | [53, 1] | [87, 10] | apply Set.inter_subset_left | case h
X : Type
inst✝ : TopologicalSpace X
I : Type
U : I → Set X
hU : ∀ (α : I), IsOpen (U α)
hcover : ↑(⋃ α, U α) = X
s : Set X
hs : interior s = ∅
α : I
⊢ s ∩ U α ⊆ s | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h
X : Type
inst✝ : TopologicalSpace X
I : Type
U : I → Set X
hU : ∀ (α : I), IsOpen (U α)
hcover : ↑(⋃ α, U α) = X
s : Set X
hs : interior s = ∅
α : I
⊢ s ∩ U α ⊆ s
TACTIC:
|
https://github.com/fpvandoorn/sard.git | f4a1bb550136e8591dcd20dca5bdcce766a958c7 | Sard/ObsoleteHelpers.lean | interior_zero_iff_open_cover | [53, 1] | [87, 10] | intro h | case mpr
X : Type
inst✝ : TopologicalSpace X
I : Type
U : I → Set X
hU : ∀ (α : I), IsOpen (U α)
hcover : ↑(⋃ α, U α) = X
s : Set X
⊢ (∀ (α : I), interior (s ∩ U α) = ∅) → interior s = ∅ | case mpr
X : Type
inst✝ : TopologicalSpace X
I : Type
U : I → Set X
hU : ∀ (α : I), IsOpen (U α)
hcover : ↑(⋃ α, U α) = X
s : Set X
h : ∀ (α : I), interior (s ∩ U α) = ∅
⊢ interior s = ∅ | Please generate a tactic in lean4 to solve the state.
STATE:
case mpr
X : Type
inst✝ : TopologicalSpace X
I : Type
U : I → Set X
hU : ∀ (α : I), IsOpen (U α)
hcover : ↑(⋃ α, U α) = X
s : Set X
⊢ (∀ (α : I), interior (s ∩ U α) = ∅) → interior s = ∅
TACTIC:
|
https://github.com/fpvandoorn/sard.git | f4a1bb550136e8591dcd20dca5bdcce766a958c7 | Sard/ObsoleteHelpers.lean | interior_zero_iff_open_cover | [53, 1] | [87, 10] | suffices ∀ V : Set X, (hV : V ⊆ s ∧ IsOpen V) → V = ∅ by sorry | case mpr
X : Type
inst✝ : TopologicalSpace X
I : Type
U : I → Set X
hU : ∀ (α : I), IsOpen (U α)
hcover : ↑(⋃ α, U α) = X
s : Set X
h : ∀ (α : I), interior (s ∩ U α) = ∅
⊢ interior s = ∅ | case mpr
X : Type
inst✝ : TopologicalSpace X
I : Type
U : I → Set X
hU : ∀ (α : I), IsOpen (U α)
hcover : ↑(⋃ α, U α) = X
s : Set X
h : ∀ (α : I), interior (s ∩ U α) = ∅
⊢ ∀ (V : Set X), V ⊆ s ∧ IsOpen V → V = ∅ | Please generate a tactic in lean4 to solve the state.
STATE:
case mpr
X : Type
inst✝ : TopologicalSpace X
I : Type
U : I → Set X
hU : ∀ (α : I), IsOpen (U α)
hcover : ↑(⋃ α, U α) = X
s : Set X
h : ∀ (α : I), interior (s ∩ U α) = ∅
⊢ interior s = ∅
TACTIC:
|
https://github.com/fpvandoorn/sard.git | f4a1bb550136e8591dcd20dca5bdcce766a958c7 | Sard/ObsoleteHelpers.lean | interior_zero_iff_open_cover | [53, 1] | [87, 10] | rintro V ⟨hVS, hV⟩ | case mpr
X : Type
inst✝ : TopologicalSpace X
I : Type
U : I → Set X
hU : ∀ (α : I), IsOpen (U α)
hcover : ↑(⋃ α, U α) = X
s : Set X
h : ∀ (α : I), interior (s ∩ U α) = ∅
⊢ ∀ (V : Set X), V ⊆ s ∧ IsOpen V → V = ∅ | case mpr.intro
X : Type
inst✝ : TopologicalSpace X
I : Type
U : I → Set X
hU : ∀ (α : I), IsOpen (U α)
hcover : ↑(⋃ α, U α) = X
s : Set X
h : ∀ (α : I), interior (s ∩ U α) = ∅
V : Set X
hVS : V ⊆ s
hV : IsOpen V
⊢ V = ∅ | Please generate a tactic in lean4 to solve the state.
STATE:
case mpr
X : Type
inst✝ : TopologicalSpace X
I : Type
U : I → Set X
hU : ∀ (α : I), IsOpen (U α)
hcover : ↑(⋃ α, U α) = X
s : Set X
h : ∀ (α : I), interior (s ∩ U α) = ∅
⊢ ∀ (V : Set X), V ⊆ s ∧ IsOpen V → V = ∅
TACTIC:
|
https://github.com/fpvandoorn/sard.git | f4a1bb550136e8591dcd20dca5bdcce766a958c7 | Sard/ObsoleteHelpers.lean | interior_zero_iff_open_cover | [53, 1] | [87, 10] | have h' : ∀ α : I, V ∩ U α = ∅ := by
intro α
have h₁ : IsOpen (V ∩ U α) := by exact IsOpen.inter hV (hU α)
have h₂ : V ∩ U α ⊆ s ∩ U α := by exact Set.inter_subset_inter_left (U α) hVS
have h₃ : V ∩ U α ⊆ interior (s ∩ U α) := by exact interior_maximal h₂ h₁
exact Set.subset_eq_empty h₃ (h α) | case mpr.intro
X : Type
inst✝ : TopologicalSpace X
I : Type
U : I → Set X
hU : ∀ (α : I), IsOpen (U α)
hcover : ↑(⋃ α, U α) = X
s : Set X
h : ∀ (α : I), interior (s ∩ U α) = ∅
V : Set X
hVS : V ⊆ s
hV : IsOpen V
⊢ V = ∅ | case mpr.intro
X : Type
inst✝ : TopologicalSpace X
I : Type
U : I → Set X
hU : ∀ (α : I), IsOpen (U α)
hcover : ↑(⋃ α, U α) = X
s : Set X
h : ∀ (α : I), interior (s ∩ U α) = ∅
V : Set X
hVS : V ⊆ s
hV : IsOpen V
h' : ∀ (α : I), V ∩ U α = ∅
⊢ V = ∅ | Please generate a tactic in lean4 to solve the state.
STATE:
case mpr.intro
X : Type
inst✝ : TopologicalSpace X
I : Type
U : I → Set X
hU : ∀ (α : I), IsOpen (U α)
hcover : ↑(⋃ α, U α) = X
s : Set X
h : ∀ (α : I), interior (s ∩ U α) = ∅
V : Set X
hVS : V ⊆ s
hV : IsOpen V
⊢ V = ∅
TACTIC:
|
https://github.com/fpvandoorn/sard.git | f4a1bb550136e8591dcd20dca5bdcce766a958c7 | Sard/ObsoleteHelpers.lean | interior_zero_iff_open_cover | [53, 1] | [87, 10] | have h'' : V = ⋃ (α : I), (V ∩ U α) := by
ext i
rw [Set.mem_iUnion]
sorry | case mpr.intro
X : Type
inst✝ : TopologicalSpace X
I : Type
U : I → Set X
hU : ∀ (α : I), IsOpen (U α)
hcover : ↑(⋃ α, U α) = X
s : Set X
h : ∀ (α : I), interior (s ∩ U α) = ∅
V : Set X
hVS : V ⊆ s
hV : IsOpen V
h' : ∀ (α : I), V ∩ U α = ∅
⊢ V = ∅ | case mpr.intro
X : Type
inst✝ : TopologicalSpace X
I : Type
U : I → Set X
hU : ∀ (α : I), IsOpen (U α)
hcover : ↑(⋃ α, U α) = X
s : Set X
h : ∀ (α : I), interior (s ∩ U α) = ∅
V : Set X
hVS : V ⊆ s
hV : IsOpen V
h' : ∀ (α : I), V ∩ U α = ∅
h'' : V = ⋃ α, V ∩ U α
⊢ V = ∅ | Please generate a tactic in lean4 to solve the state.
STATE:
case mpr.intro
X : Type
inst✝ : TopologicalSpace X
I : Type
U : I → Set X
hU : ∀ (α : I), IsOpen (U α)
hcover : ↑(⋃ α, U α) = X
s : Set X
h : ∀ (α : I), interior (s ∩ U α) = ∅
V : Set X
hVS : V ⊆ s
hV : IsOpen V
h' : ∀ (α : I), V ∩ U α = ∅
⊢ V = ∅
TACTIC:
|
https://github.com/fpvandoorn/sard.git | f4a1bb550136e8591dcd20dca5bdcce766a958c7 | Sard/ObsoleteHelpers.lean | interior_zero_iff_open_cover | [53, 1] | [87, 10] | rw [h''] | case mpr.intro
X : Type
inst✝ : TopologicalSpace X
I : Type
U : I → Set X
hU : ∀ (α : I), IsOpen (U α)
hcover : ↑(⋃ α, U α) = X
s : Set X
h : ∀ (α : I), interior (s ∩ U α) = ∅
V : Set X
hVS : V ⊆ s
hV : IsOpen V
h' : ∀ (α : I), V ∩ U α = ∅
h'' : V = ⋃ α, V ∩ U α
⊢ V = ∅ | case mpr.intro
X : Type
inst✝ : TopologicalSpace X
I : Type
U : I → Set X
hU : ∀ (α : I), IsOpen (U α)
hcover : ↑(⋃ α, U α) = X
s : Set X
h : ∀ (α : I), interior (s ∩ U α) = ∅
V : Set X
hVS : V ⊆ s
hV : IsOpen V
h' : ∀ (α : I), V ∩ U α = ∅
h'' : V = ⋃ α, V ∩ U α
⊢ ⋃ α, V ∩ U α = ∅ | Please generate a tactic in lean4 to solve the state.
STATE:
case mpr.intro
X : Type
inst✝ : TopologicalSpace X
I : Type
U : I → Set X
hU : ∀ (α : I), IsOpen (U α)
hcover : ↑(⋃ α, U α) = X
s : Set X
h : ∀ (α : I), interior (s ∩ U α) = ∅
V : Set X
hVS : V ⊆ s
hV : IsOpen V
h' : ∀ (α : I), V ∩ U α = ∅
h'' : V = ⋃ α, V ∩ ... |
https://github.com/fpvandoorn/sard.git | f4a1bb550136e8591dcd20dca5bdcce766a958c7 | Sard/ObsoleteHelpers.lean | interior_zero_iff_open_cover | [53, 1] | [87, 10] | simp only [Set.iUnion_eq_empty] | case mpr.intro
X : Type
inst✝ : TopologicalSpace X
I : Type
U : I → Set X
hU : ∀ (α : I), IsOpen (U α)
hcover : ↑(⋃ α, U α) = X
s : Set X
h : ∀ (α : I), interior (s ∩ U α) = ∅
V : Set X
hVS : V ⊆ s
hV : IsOpen V
h' : ∀ (α : I), V ∩ U α = ∅
h'' : V = ⋃ α, V ∩ U α
⊢ ⋃ α, V ∩ U α = ∅ | case mpr.intro
X : Type
inst✝ : TopologicalSpace X
I : Type
U : I → Set X
hU : ∀ (α : I), IsOpen (U α)
hcover : ↑(⋃ α, U α) = X
s : Set X
h : ∀ (α : I), interior (s ∩ U α) = ∅
V : Set X
hVS : V ⊆ s
hV : IsOpen V
h' : ∀ (α : I), V ∩ U α = ∅
h'' : V = ⋃ α, V ∩ U α
⊢ ∀ (i : I), V ∩ U i = ∅ | Please generate a tactic in lean4 to solve the state.
STATE:
case mpr.intro
X : Type
inst✝ : TopologicalSpace X
I : Type
U : I → Set X
hU : ∀ (α : I), IsOpen (U α)
hcover : ↑(⋃ α, U α) = X
s : Set X
h : ∀ (α : I), interior (s ∩ U α) = ∅
V : Set X
hVS : V ⊆ s
hV : IsOpen V
h' : ∀ (α : I), V ∩ U α = ∅
h'' : V = ⋃ α, V ∩ ... |
https://github.com/fpvandoorn/sard.git | f4a1bb550136e8591dcd20dca5bdcce766a958c7 | Sard/ObsoleteHelpers.lean | interior_zero_iff_open_cover | [53, 1] | [87, 10] | tauto | case mpr.intro
X : Type
inst✝ : TopologicalSpace X
I : Type
U : I → Set X
hU : ∀ (α : I), IsOpen (U α)
hcover : ↑(⋃ α, U α) = X
s : Set X
h : ∀ (α : I), interior (s ∩ U α) = ∅
V : Set X
hVS : V ⊆ s
hV : IsOpen V
h' : ∀ (α : I), V ∩ U α = ∅
h'' : V = ⋃ α, V ∩ U α
⊢ ∀ (i : I), V ∩ U i = ∅ | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case mpr.intro
X : Type
inst✝ : TopologicalSpace X
I : Type
U : I → Set X
hU : ∀ (α : I), IsOpen (U α)
hcover : ↑(⋃ α, U α) = X
s : Set X
h : ∀ (α : I), interior (s ∩ U α) = ∅
V : Set X
hVS : V ⊆ s
hV : IsOpen V
h' : ∀ (α : I), V ∩ U α = ∅
h'' : V = ⋃ α, V ∩ ... |
https://github.com/fpvandoorn/sard.git | f4a1bb550136e8591dcd20dca5bdcce766a958c7 | Sard/ObsoleteHelpers.lean | interior_zero_iff_open_cover | [53, 1] | [87, 10] | sorry | X : Type
inst✝ : TopologicalSpace X
I : Type
U : I → Set X
hU : ∀ (α : I), IsOpen (U α)
hcover : ↑(⋃ α, U α) = X
s : Set X
h : ∀ (α : I), interior (s ∩ U α) = ∅
this : ∀ (V : Set X), V ⊆ s ∧ IsOpen V → V = ∅
⊢ interior s = ∅ | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
X : Type
inst✝ : TopologicalSpace X
I : Type
U : I → Set X
hU : ∀ (α : I), IsOpen (U α)
hcover : ↑(⋃ α, U α) = X
s : Set X
h : ∀ (α : I), interior (s ∩ U α) = ∅
this : ∀ (V : Set X), V ⊆ s ∧ IsOpen V → V = ∅
⊢ interior s = ∅
TACTIC:
|
https://github.com/fpvandoorn/sard.git | f4a1bb550136e8591dcd20dca5bdcce766a958c7 | Sard/ObsoleteHelpers.lean | interior_zero_iff_open_cover | [53, 1] | [87, 10] | intro α | X : Type
inst✝ : TopologicalSpace X
I : Type
U : I → Set X
hU : ∀ (α : I), IsOpen (U α)
hcover : ↑(⋃ α, U α) = X
s : Set X
h : ∀ (α : I), interior (s ∩ U α) = ∅
V : Set X
hVS : V ⊆ s
hV : IsOpen V
⊢ ∀ (α : I), V ∩ U α = ∅ | X : Type
inst✝ : TopologicalSpace X
I : Type
U : I → Set X
hU : ∀ (α : I), IsOpen (U α)
hcover : ↑(⋃ α, U α) = X
s : Set X
h : ∀ (α : I), interior (s ∩ U α) = ∅
V : Set X
hVS : V ⊆ s
hV : IsOpen V
α : I
⊢ V ∩ U α = ∅ | Please generate a tactic in lean4 to solve the state.
STATE:
X : Type
inst✝ : TopologicalSpace X
I : Type
U : I → Set X
hU : ∀ (α : I), IsOpen (U α)
hcover : ↑(⋃ α, U α) = X
s : Set X
h : ∀ (α : I), interior (s ∩ U α) = ∅
V : Set X
hVS : V ⊆ s
hV : IsOpen V
⊢ ∀ (α : I), V ∩ U α = ∅
TACTIC:
|
https://github.com/fpvandoorn/sard.git | f4a1bb550136e8591dcd20dca5bdcce766a958c7 | Sard/ObsoleteHelpers.lean | interior_zero_iff_open_cover | [53, 1] | [87, 10] | have h₁ : IsOpen (V ∩ U α) := by exact IsOpen.inter hV (hU α) | X : Type
inst✝ : TopologicalSpace X
I : Type
U : I → Set X
hU : ∀ (α : I), IsOpen (U α)
hcover : ↑(⋃ α, U α) = X
s : Set X
h : ∀ (α : I), interior (s ∩ U α) = ∅
V : Set X
hVS : V ⊆ s
hV : IsOpen V
α : I
⊢ V ∩ U α = ∅ | X : Type
inst✝ : TopologicalSpace X
I : Type
U : I → Set X
hU : ∀ (α : I), IsOpen (U α)
hcover : ↑(⋃ α, U α) = X
s : Set X
h : ∀ (α : I), interior (s ∩ U α) = ∅
V : Set X
hVS : V ⊆ s
hV : IsOpen V
α : I
h₁ : IsOpen (V ∩ U α)
⊢ V ∩ U α = ∅ | Please generate a tactic in lean4 to solve the state.
STATE:
X : Type
inst✝ : TopologicalSpace X
I : Type
U : I → Set X
hU : ∀ (α : I), IsOpen (U α)
hcover : ↑(⋃ α, U α) = X
s : Set X
h : ∀ (α : I), interior (s ∩ U α) = ∅
V : Set X
hVS : V ⊆ s
hV : IsOpen V
α : I
⊢ V ∩ U α = ∅
TACTIC:
|
https://github.com/fpvandoorn/sard.git | f4a1bb550136e8591dcd20dca5bdcce766a958c7 | Sard/ObsoleteHelpers.lean | interior_zero_iff_open_cover | [53, 1] | [87, 10] | have h₂ : V ∩ U α ⊆ s ∩ U α := by exact Set.inter_subset_inter_left (U α) hVS | X : Type
inst✝ : TopologicalSpace X
I : Type
U : I → Set X
hU : ∀ (α : I), IsOpen (U α)
hcover : ↑(⋃ α, U α) = X
s : Set X
h : ∀ (α : I), interior (s ∩ U α) = ∅
V : Set X
hVS : V ⊆ s
hV : IsOpen V
α : I
h₁ : IsOpen (V ∩ U α)
⊢ V ∩ U α = ∅ | X : Type
inst✝ : TopologicalSpace X
I : Type
U : I → Set X
hU : ∀ (α : I), IsOpen (U α)
hcover : ↑(⋃ α, U α) = X
s : Set X
h : ∀ (α : I), interior (s ∩ U α) = ∅
V : Set X
hVS : V ⊆ s
hV : IsOpen V
α : I
h₁ : IsOpen (V ∩ U α)
h₂ : V ∩ U α ⊆ s ∩ U α
⊢ V ∩ U α = ∅ | Please generate a tactic in lean4 to solve the state.
STATE:
X : Type
inst✝ : TopologicalSpace X
I : Type
U : I → Set X
hU : ∀ (α : I), IsOpen (U α)
hcover : ↑(⋃ α, U α) = X
s : Set X
h : ∀ (α : I), interior (s ∩ U α) = ∅
V : Set X
hVS : V ⊆ s
hV : IsOpen V
α : I
h₁ : IsOpen (V ∩ U α)
⊢ V ∩ U α = ∅
TACTIC:
|
https://github.com/fpvandoorn/sard.git | f4a1bb550136e8591dcd20dca5bdcce766a958c7 | Sard/ObsoleteHelpers.lean | interior_zero_iff_open_cover | [53, 1] | [87, 10] | have h₃ : V ∩ U α ⊆ interior (s ∩ U α) := by exact interior_maximal h₂ h₁ | X : Type
inst✝ : TopologicalSpace X
I : Type
U : I → Set X
hU : ∀ (α : I), IsOpen (U α)
hcover : ↑(⋃ α, U α) = X
s : Set X
h : ∀ (α : I), interior (s ∩ U α) = ∅
V : Set X
hVS : V ⊆ s
hV : IsOpen V
α : I
h₁ : IsOpen (V ∩ U α)
h₂ : V ∩ U α ⊆ s ∩ U α
⊢ V ∩ U α = ∅ | X : Type
inst✝ : TopologicalSpace X
I : Type
U : I → Set X
hU : ∀ (α : I), IsOpen (U α)
hcover : ↑(⋃ α, U α) = X
s : Set X
h : ∀ (α : I), interior (s ∩ U α) = ∅
V : Set X
hVS : V ⊆ s
hV : IsOpen V
α : I
h₁ : IsOpen (V ∩ U α)
h₂ : V ∩ U α ⊆ s ∩ U α
h₃ : V ∩ U α ⊆ interior (s ∩ U α)
⊢ V ∩ U α = ∅ | Please generate a tactic in lean4 to solve the state.
STATE:
X : Type
inst✝ : TopologicalSpace X
I : Type
U : I → Set X
hU : ∀ (α : I), IsOpen (U α)
hcover : ↑(⋃ α, U α) = X
s : Set X
h : ∀ (α : I), interior (s ∩ U α) = ∅
V : Set X
hVS : V ⊆ s
hV : IsOpen V
α : I
h₁ : IsOpen (V ∩ U α)
h₂ : V ∩ U α ⊆ s ∩ U α
⊢ V ∩ U α =... |
https://github.com/fpvandoorn/sard.git | f4a1bb550136e8591dcd20dca5bdcce766a958c7 | Sard/ObsoleteHelpers.lean | interior_zero_iff_open_cover | [53, 1] | [87, 10] | exact Set.subset_eq_empty h₃ (h α) | X : Type
inst✝ : TopologicalSpace X
I : Type
U : I → Set X
hU : ∀ (α : I), IsOpen (U α)
hcover : ↑(⋃ α, U α) = X
s : Set X
h : ∀ (α : I), interior (s ∩ U α) = ∅
V : Set X
hVS : V ⊆ s
hV : IsOpen V
α : I
h₁ : IsOpen (V ∩ U α)
h₂ : V ∩ U α ⊆ s ∩ U α
h₃ : V ∩ U α ⊆ interior (s ∩ U α)
⊢ V ∩ U α = ∅ | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
X : Type
inst✝ : TopologicalSpace X
I : Type
U : I → Set X
hU : ∀ (α : I), IsOpen (U α)
hcover : ↑(⋃ α, U α) = X
s : Set X
h : ∀ (α : I), interior (s ∩ U α) = ∅
V : Set X
hVS : V ⊆ s
hV : IsOpen V
α : I
h₁ : IsOpen (V ∩ U α)
h₂ : V ∩ U α ⊆ s ∩ U α
h₃ : V ∩ U ... |
https://github.com/fpvandoorn/sard.git | f4a1bb550136e8591dcd20dca5bdcce766a958c7 | Sard/ObsoleteHelpers.lean | interior_zero_iff_open_cover | [53, 1] | [87, 10] | exact IsOpen.inter hV (hU α) | X : Type
inst✝ : TopologicalSpace X
I : Type
U : I → Set X
hU : ∀ (α : I), IsOpen (U α)
hcover : ↑(⋃ α, U α) = X
s : Set X
h : ∀ (α : I), interior (s ∩ U α) = ∅
V : Set X
hVS : V ⊆ s
hV : IsOpen V
α : I
⊢ IsOpen (V ∩ U α) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
X : Type
inst✝ : TopologicalSpace X
I : Type
U : I → Set X
hU : ∀ (α : I), IsOpen (U α)
hcover : ↑(⋃ α, U α) = X
s : Set X
h : ∀ (α : I), interior (s ∩ U α) = ∅
V : Set X
hVS : V ⊆ s
hV : IsOpen V
α : I
⊢ IsOpen (V ∩ U α)
TACTIC:
|
https://github.com/fpvandoorn/sard.git | f4a1bb550136e8591dcd20dca5bdcce766a958c7 | Sard/ObsoleteHelpers.lean | interior_zero_iff_open_cover | [53, 1] | [87, 10] | exact Set.inter_subset_inter_left (U α) hVS | X : Type
inst✝ : TopologicalSpace X
I : Type
U : I → Set X
hU : ∀ (α : I), IsOpen (U α)
hcover : ↑(⋃ α, U α) = X
s : Set X
h : ∀ (α : I), interior (s ∩ U α) = ∅
V : Set X
hVS : V ⊆ s
hV : IsOpen V
α : I
h₁ : IsOpen (V ∩ U α)
⊢ V ∩ U α ⊆ s ∩ U α | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
X : Type
inst✝ : TopologicalSpace X
I : Type
U : I → Set X
hU : ∀ (α : I), IsOpen (U α)
hcover : ↑(⋃ α, U α) = X
s : Set X
h : ∀ (α : I), interior (s ∩ U α) = ∅
V : Set X
hVS : V ⊆ s
hV : IsOpen V
α : I
h₁ : IsOpen (V ∩ U α)
⊢ V ∩ U α ⊆ s ∩ U α
TACTIC:
|
https://github.com/fpvandoorn/sard.git | f4a1bb550136e8591dcd20dca5bdcce766a958c7 | Sard/ObsoleteHelpers.lean | interior_zero_iff_open_cover | [53, 1] | [87, 10] | exact interior_maximal h₂ h₁ | X : Type
inst✝ : TopologicalSpace X
I : Type
U : I → Set X
hU : ∀ (α : I), IsOpen (U α)
hcover : ↑(⋃ α, U α) = X
s : Set X
h : ∀ (α : I), interior (s ∩ U α) = ∅
V : Set X
hVS : V ⊆ s
hV : IsOpen V
α : I
h₁ : IsOpen (V ∩ U α)
h₂ : V ∩ U α ⊆ s ∩ U α
⊢ V ∩ U α ⊆ interior (s ∩ U α) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
X : Type
inst✝ : TopologicalSpace X
I : Type
U : I → Set X
hU : ∀ (α : I), IsOpen (U α)
hcover : ↑(⋃ α, U α) = X
s : Set X
h : ∀ (α : I), interior (s ∩ U α) = ∅
V : Set X
hVS : V ⊆ s
hV : IsOpen V
α : I
h₁ : IsOpen (V ∩ U α)
h₂ : V ∩ U α ⊆ s ∩ U α
⊢ V ∩ U α ⊆... |
https://github.com/fpvandoorn/sard.git | f4a1bb550136e8591dcd20dca5bdcce766a958c7 | Sard/ObsoleteHelpers.lean | interior_zero_iff_open_cover | [53, 1] | [87, 10] | ext i | X : Type
inst✝ : TopologicalSpace X
I : Type
U : I → Set X
hU : ∀ (α : I), IsOpen (U α)
hcover : ↑(⋃ α, U α) = X
s : Set X
h : ∀ (α : I), interior (s ∩ U α) = ∅
V : Set X
hVS : V ⊆ s
hV : IsOpen V
h' : ∀ (α : I), V ∩ U α = ∅
⊢ V = ⋃ α, V ∩ U α | case h
X : Type
inst✝ : TopologicalSpace X
I : Type
U : I → Set X
hU : ∀ (α : I), IsOpen (U α)
hcover : ↑(⋃ α, U α) = X
s : Set X
h : ∀ (α : I), interior (s ∩ U α) = ∅
V : Set X
hVS : V ⊆ s
hV : IsOpen V
h' : ∀ (α : I), V ∩ U α = ∅
i : X
⊢ i ∈ V ↔ i ∈ ⋃ α, V ∩ U α | Please generate a tactic in lean4 to solve the state.
STATE:
X : Type
inst✝ : TopologicalSpace X
I : Type
U : I → Set X
hU : ∀ (α : I), IsOpen (U α)
hcover : ↑(⋃ α, U α) = X
s : Set X
h : ∀ (α : I), interior (s ∩ U α) = ∅
V : Set X
hVS : V ⊆ s
hV : IsOpen V
h' : ∀ (α : I), V ∩ U α = ∅
⊢ V = ⋃ α, V ∩ U α
TACTIC:
|
https://github.com/fpvandoorn/sard.git | f4a1bb550136e8591dcd20dca5bdcce766a958c7 | Sard/ObsoleteHelpers.lean | interior_zero_iff_open_cover | [53, 1] | [87, 10] | rw [Set.mem_iUnion] | case h
X : Type
inst✝ : TopologicalSpace X
I : Type
U : I → Set X
hU : ∀ (α : I), IsOpen (U α)
hcover : ↑(⋃ α, U α) = X
s : Set X
h : ∀ (α : I), interior (s ∩ U α) = ∅
V : Set X
hVS : V ⊆ s
hV : IsOpen V
h' : ∀ (α : I), V ∩ U α = ∅
i : X
⊢ i ∈ V ↔ i ∈ ⋃ α, V ∩ U α | case h
X : Type
inst✝ : TopologicalSpace X
I : Type
U : I → Set X
hU : ∀ (α : I), IsOpen (U α)
hcover : ↑(⋃ α, U α) = X
s : Set X
h : ∀ (α : I), interior (s ∩ U α) = ∅
V : Set X
hVS : V ⊆ s
hV : IsOpen V
h' : ∀ (α : I), V ∩ U α = ∅
i : X
⊢ i ∈ V ↔ ∃ i_1, i ∈ V ∩ U i_1 | Please generate a tactic in lean4 to solve the state.
STATE:
case h
X : Type
inst✝ : TopologicalSpace X
I : Type
U : I → Set X
hU : ∀ (α : I), IsOpen (U α)
hcover : ↑(⋃ α, U α) = X
s : Set X
h : ∀ (α : I), interior (s ∩ U α) = ∅
V : Set X
hVS : V ⊆ s
hV : IsOpen V
h' : ∀ (α : I), V ∩ U α = ∅
i : X
⊢ i ∈ V ↔ i ∈ ⋃ α, V ... |
https://github.com/fpvandoorn/sard.git | f4a1bb550136e8591dcd20dca5bdcce766a958c7 | Sard/ObsoleteHelpers.lean | interior_zero_iff_open_cover | [53, 1] | [87, 10] | sorry | case h
X : Type
inst✝ : TopologicalSpace X
I : Type
U : I → Set X
hU : ∀ (α : I), IsOpen (U α)
hcover : ↑(⋃ α, U α) = X
s : Set X
h : ∀ (α : I), interior (s ∩ U α) = ∅
V : Set X
hVS : V ⊆ s
hV : IsOpen V
h' : ∀ (α : I), V ∩ U α = ∅
i : X
⊢ i ∈ V ↔ ∃ i_1, i ∈ V ∩ U i_1 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h
X : Type
inst✝ : TopologicalSpace X
I : Type
U : I → Set X
hU : ∀ (α : I), IsOpen (U α)
hcover : ↑(⋃ α, U α) = X
s : Set X
h : ∀ (α : I), interior (s ∩ U α) = ∅
V : Set X
hVS : V ⊆ s
hV : IsOpen V
h' : ∀ (α : I), V ∩ U α = ∅
i : X
⊢ i ∈ V ↔ ∃ i_1, i ∈ ... |
https://github.com/fpvandoorn/sard.git | f4a1bb550136e8591dcd20dca5bdcce766a958c7 | Sard/LocallyLipschitz.lean | LocallyLipschitz.ToSubset.compatible_with_nhds | [16, 1] | [17, 117] | sorry | X : Type u_1
Y : Type u_2
Z : Type u_3
inst✝² : MetricSpace X
inst✝¹ : MetricSpace Y
inst✝ : MetricSpace Z
s t : Set X
x : ↑s
ht : t ∈ 𝓝 ↑x
⊢ t.toSubset s ∈ 𝓝 x | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
X : Type u_1
Y : Type u_2
Z : Type u_3
inst✝² : MetricSpace X
inst✝¹ : MetricSpace Y
inst✝ : MetricSpace Z
s t : Set X
x : ↑s
ht : t ∈ 𝓝 ↑x
⊢ t.toSubset s ∈ 𝓝 x
TACTIC:
|
https://github.com/fpvandoorn/sard.git | f4a1bb550136e8591dcd20dca5bdcce766a958c7 | Sard/LocallyLipschitz.lean | LocallyLipschitz.ToSubset.compatible_with_nhds_within | [19, 1] | [39, 60] | apply ToSubset.compatible_with_nhds | X : Type u_1
Y : Type u_2
Z : Type u_3
inst✝² : MetricSpace X
inst✝¹ : MetricSpace Y
inst✝ : MetricSpace Z
t U : Set X
x : ↑U
hU : IsOpen U
ht : t ∈ 𝓝[U] ↑x
this : t ∩ U ∈ 𝓝 ↑x
⊢ t.toSubset U ∈ 𝓝 x | case ht
X : Type u_1
Y : Type u_2
Z : Type u_3
inst✝² : MetricSpace X
inst✝¹ : MetricSpace Y
inst✝ : MetricSpace Z
t U : Set X
x : ↑U
hU : IsOpen U
ht : t ∈ 𝓝[U] ↑x
this : t ∩ U ∈ 𝓝 ↑x
⊢ t ∈ 𝓝 ↑x | Please generate a tactic in lean4 to solve the state.
STATE:
X : Type u_1
Y : Type u_2
Z : Type u_3
inst✝² : MetricSpace X
inst✝¹ : MetricSpace Y
inst✝ : MetricSpace Z
t U : Set X
x : ↑U
hU : IsOpen U
ht : t ∈ 𝓝[U] ↑x
this : t ∩ U ∈ 𝓝 ↑x
⊢ t.toSubset U ∈ 𝓝 x
TACTIC:
|
https://github.com/fpvandoorn/sard.git | f4a1bb550136e8591dcd20dca5bdcce766a958c7 | Sard/LocallyLipschitz.lean | LocallyLipschitz.ToSubset.compatible_with_nhds_within | [19, 1] | [39, 60] | exact Filter.mem_of_superset this (inter_subset_left t U) | case ht
X : Type u_1
Y : Type u_2
Z : Type u_3
inst✝² : MetricSpace X
inst✝¹ : MetricSpace Y
inst✝ : MetricSpace Z
t U : Set X
x : ↑U
hU : IsOpen U
ht : t ∈ 𝓝[U] ↑x
this : t ∩ U ∈ 𝓝 ↑x
⊢ t ∈ 𝓝 ↑x | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case ht
X : Type u_1
Y : Type u_2
Z : Type u_3
inst✝² : MetricSpace X
inst✝¹ : MetricSpace Y
inst✝ : MetricSpace Z
t U : Set X
x : ↑U
hU : IsOpen U
ht : t ∈ 𝓝[U] ↑x
this : t ∩ U ∈ 𝓝 ↑x
⊢ t ∈ 𝓝 ↑x
TACTIC:
|
https://github.com/fpvandoorn/sard.git | f4a1bb550136e8591dcd20dca5bdcce766a958c7 | Sard/LocallyLipschitz.lean | LocallyLipschitz.ToSubset.compatible_with_nhds_within | [19, 1] | [39, 60] | rcases ht with ⟨b, hb, U', hU', htaU⟩ | X : Type u_1
Y : Type u_2
Z : Type u_3
inst✝² : MetricSpace X
inst✝¹ : MetricSpace Y
inst✝ : MetricSpace Z
t U : Set X
x : ↑U
hU : IsOpen U
ht : t ∈ 𝓝[U] ↑x
⊢ t ∩ U ∈ 𝓝 ↑x | case intro.intro.intro.intro
X : Type u_1
Y : Type u_2
Z : Type u_3
inst✝² : MetricSpace X
inst✝¹ : MetricSpace Y
inst✝ : MetricSpace Z
t U : Set X
x : ↑U
hU : IsOpen U
b : Set X
hb : b ∈ 𝓝 ↑x
U' : Set X
hU' : U' ∈ Filter.principal U
htaU : t = b ∩ U'
⊢ t ∩ U ∈ 𝓝 ↑x | Please generate a tactic in lean4 to solve the state.
STATE:
X : Type u_1
Y : Type u_2
Z : Type u_3
inst✝² : MetricSpace X
inst✝¹ : MetricSpace Y
inst✝ : MetricSpace Z
t U : Set X
x : ↑U
hU : IsOpen U
ht : t ∈ 𝓝[U] ↑x
⊢ t ∩ U ∈ 𝓝 ↑x
TACTIC:
|
https://github.com/fpvandoorn/sard.git | f4a1bb550136e8591dcd20dca5bdcce766a958c7 | Sard/LocallyLipschitz.lean | LocallyLipschitz.ToSubset.compatible_with_nhds_within | [19, 1] | [39, 60] | rw [mem_nhds_iff] at hb | case intro.intro.intro.intro
X : Type u_1
Y : Type u_2
Z : Type u_3
inst✝² : MetricSpace X
inst✝¹ : MetricSpace Y
inst✝ : MetricSpace Z
t U : Set X
x : ↑U
hU : IsOpen U
b : Set X
hb : b ∈ 𝓝 ↑x
U' : Set X
hU' : U' ∈ Filter.principal U
htaU : t = b ∩ U'
⊢ t ∩ U ∈ 𝓝 ↑x | case intro.intro.intro.intro
X : Type u_1
Y : Type u_2
Z : Type u_3
inst✝² : MetricSpace X
inst✝¹ : MetricSpace Y
inst✝ : MetricSpace Z
t U : Set X
x : ↑U
hU : IsOpen U
b : Set X
hb : ∃ t ⊆ b, IsOpen t ∧ ↑x ∈ t
U' : Set X
hU' : U' ∈ Filter.principal U
htaU : t = b ∩ U'
⊢ t ∩ U ∈ 𝓝 ↑x | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.intro.intro
X : Type u_1
Y : Type u_2
Z : Type u_3
inst✝² : MetricSpace X
inst✝¹ : MetricSpace Y
inst✝ : MetricSpace Z
t U : Set X
x : ↑U
hU : IsOpen U
b : Set X
hb : b ∈ 𝓝 ↑x
U' : Set X
hU' : U' ∈ Filter.principal U
htaU : t = b ∩ U'
⊢ t ∩ ... |
https://github.com/fpvandoorn/sard.git | f4a1bb550136e8591dcd20dca5bdcce766a958c7 | Sard/LocallyLipschitz.lean | LocallyLipschitz.ToSubset.compatible_with_nhds_within | [19, 1] | [39, 60] | rcases hb with ⟨a, ha, haopen, hxa⟩ | case intro.intro.intro.intro
X : Type u_1
Y : Type u_2
Z : Type u_3
inst✝² : MetricSpace X
inst✝¹ : MetricSpace Y
inst✝ : MetricSpace Z
t U : Set X
x : ↑U
hU : IsOpen U
b : Set X
hb : ∃ t ⊆ b, IsOpen t ∧ ↑x ∈ t
U' : Set X
hU' : U' ∈ Filter.principal U
htaU : t = b ∩ U'
⊢ t ∩ U ∈ 𝓝 ↑x | case intro.intro.intro.intro.intro.intro.intro
X : Type u_1
Y : Type u_2
Z : Type u_3
inst✝² : MetricSpace X
inst✝¹ : MetricSpace Y
inst✝ : MetricSpace Z
t U : Set X
x : ↑U
hU : IsOpen U
b U' : Set X
hU' : U' ∈ Filter.principal U
htaU : t = b ∩ U'
a : Set X
ha : a ⊆ b
haopen : IsOpen a
hxa : ↑x ∈ a
⊢ t ∩ U ∈ 𝓝 ↑x | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.intro.intro
X : Type u_1
Y : Type u_2
Z : Type u_3
inst✝² : MetricSpace X
inst✝¹ : MetricSpace Y
inst✝ : MetricSpace Z
t U : Set X
x : ↑U
hU : IsOpen U
b : Set X
hb : ∃ t ⊆ b, IsOpen t ∧ ↑x ∈ t
U' : Set X
hU' : U' ∈ Filter.principal U
htaU : ... |
https://github.com/fpvandoorn/sard.git | f4a1bb550136e8591dcd20dca5bdcce766a958c7 | Sard/LocallyLipschitz.lean | LocallyLipschitz.ToSubset.compatible_with_nhds_within | [19, 1] | [39, 60] | rw [mem_nhds_iff] | case intro.intro.intro.intro.intro.intro.intro
X : Type u_1
Y : Type u_2
Z : Type u_3
inst✝² : MetricSpace X
inst✝¹ : MetricSpace Y
inst✝ : MetricSpace Z
t U : Set X
x : ↑U
hU : IsOpen U
b U' : Set X
hU' : U' ∈ Filter.principal U
htaU : t = b ∩ U'
a : Set X
ha : a ⊆ b
haopen : IsOpen a
hxa : ↑x ∈ a
⊢ t ∩ U ∈ 𝓝 ↑x | case intro.intro.intro.intro.intro.intro.intro
X : Type u_1
Y : Type u_2
Z : Type u_3
inst✝² : MetricSpace X
inst✝¹ : MetricSpace Y
inst✝ : MetricSpace Z
t U : Set X
x : ↑U
hU : IsOpen U
b U' : Set X
hU' : U' ∈ Filter.principal U
htaU : t = b ∩ U'
a : Set X
ha : a ⊆ b
haopen : IsOpen a
hxa : ↑x ∈ a
⊢ ∃ t_1 ⊆ t ∩ U, IsO... | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.intro.intro.intro.intro.intro
X : Type u_1
Y : Type u_2
Z : Type u_3
inst✝² : MetricSpace X
inst✝¹ : MetricSpace Y
inst✝ : MetricSpace Z
t U : Set X
x : ↑U
hU : IsOpen U
b U' : Set X
hU' : U' ∈ Filter.principal U
htaU : t = b ∩ U'
a : Set X
h... |
https://github.com/fpvandoorn/sard.git | f4a1bb550136e8591dcd20dca5bdcce766a958c7 | Sard/LocallyLipschitz.lean | LocallyLipschitz.ToSubset.compatible_with_nhds_within | [19, 1] | [39, 60] | use a ∩ U | case intro.intro.intro.intro.intro.intro.intro
X : Type u_1
Y : Type u_2
Z : Type u_3
inst✝² : MetricSpace X
inst✝¹ : MetricSpace Y
inst✝ : MetricSpace Z
t U : Set X
x : ↑U
hU : IsOpen U
b U' : Set X
hU' : U' ∈ Filter.principal U
htaU : t = b ∩ U'
a : Set X
ha : a ⊆ b
haopen : IsOpen a
hxa : ↑x ∈ a
⊢ ∃ t_1 ⊆ t ∩ U, IsO... | case h
X : Type u_1
Y : Type u_2
Z : Type u_3
inst✝² : MetricSpace X
inst✝¹ : MetricSpace Y
inst✝ : MetricSpace Z
t U : Set X
x : ↑U
hU : IsOpen U
b U' : Set X
hU' : U' ∈ Filter.principal U
htaU : t = b ∩ U'
a : Set X
ha : a ⊆ b
haopen : IsOpen a
hxa : ↑x ∈ a
⊢ a ∩ U ⊆ t ∩ U ∧ IsOpen (a ∩ U) ∧ ↑x ∈ a ∩ U | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.intro.intro.intro.intro.intro
X : Type u_1
Y : Type u_2
Z : Type u_3
inst✝² : MetricSpace X
inst✝¹ : MetricSpace Y
inst✝ : MetricSpace Z
t U : Set X
x : ↑U
hU : IsOpen U
b U' : Set X
hU' : U' ∈ Filter.principal U
htaU : t = b ∩ U'
a : Set X
h... |
https://github.com/fpvandoorn/sard.git | f4a1bb550136e8591dcd20dca5bdcce766a958c7 | Sard/LocallyLipschitz.lean | LocallyLipschitz.ToSubset.compatible_with_nhds_within | [19, 1] | [39, 60] | constructor | case h
X : Type u_1
Y : Type u_2
Z : Type u_3
inst✝² : MetricSpace X
inst✝¹ : MetricSpace Y
inst✝ : MetricSpace Z
t U : Set X
x : ↑U
hU : IsOpen U
b U' : Set X
hU' : U' ∈ Filter.principal U
htaU : t = b ∩ U'
a : Set X
ha : a ⊆ b
haopen : IsOpen a
hxa : ↑x ∈ a
⊢ a ∩ U ⊆ t ∩ U ∧ IsOpen (a ∩ U) ∧ ↑x ∈ a ∩ U | case h.left
X : Type u_1
Y : Type u_2
Z : Type u_3
inst✝² : MetricSpace X
inst✝¹ : MetricSpace Y
inst✝ : MetricSpace Z
t U : Set X
x : ↑U
hU : IsOpen U
b U' : Set X
hU' : U' ∈ Filter.principal U
htaU : t = b ∩ U'
a : Set X
ha : a ⊆ b
haopen : IsOpen a
hxa : ↑x ∈ a
⊢ a ∩ U ⊆ t ∩ U
case h.right
X : Type u_1
Y : Type u_2... | Please generate a tactic in lean4 to solve the state.
STATE:
case h
X : Type u_1
Y : Type u_2
Z : Type u_3
inst✝² : MetricSpace X
inst✝¹ : MetricSpace Y
inst✝ : MetricSpace Z
t U : Set X
x : ↑U
hU : IsOpen U
b U' : Set X
hU' : U' ∈ Filter.principal U
htaU : t = b ∩ U'
a : Set X
ha : a ⊆ b
haopen : IsOpen a
hxa : ↑x ∈ a... |
https://github.com/fpvandoorn/sard.git | f4a1bb550136e8591dcd20dca5bdcce766a958c7 | Sard/LocallyLipschitz.lean | LocallyLipschitz.ToSubset.compatible_with_nhds_within | [19, 1] | [39, 60] | calc a ∩ U
_ ⊆ b ∩ U := inter_subset_inter_left U ha
_ = b ∩ (U' ∩ U) := by congr; apply (Set.inter_eq_right.mpr hU').symm
_ ⊆ (b ∩ U') ∩ U := by rw [inter_assoc]
_ = t ∩ U := by rw [htaU] | case h.left
X : Type u_1
Y : Type u_2
Z : Type u_3
inst✝² : MetricSpace X
inst✝¹ : MetricSpace Y
inst✝ : MetricSpace Z
t U : Set X
x : ↑U
hU : IsOpen U
b U' : Set X
hU' : U' ∈ Filter.principal U
htaU : t = b ∩ U'
a : Set X
ha : a ⊆ b
haopen : IsOpen a
hxa : ↑x ∈ a
⊢ a ∩ U ⊆ t ∩ U | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h.left
X : Type u_1
Y : Type u_2
Z : Type u_3
inst✝² : MetricSpace X
inst✝¹ : MetricSpace Y
inst✝ : MetricSpace Z
t U : Set X
x : ↑U
hU : IsOpen U
b U' : Set X
hU' : U' ∈ Filter.principal U
htaU : t = b ∩ U'
a : Set X
ha : a ⊆ b
haopen : IsOpen a
hxa : ↑... |
https://github.com/fpvandoorn/sard.git | f4a1bb550136e8591dcd20dca5bdcce766a958c7 | Sard/LocallyLipschitz.lean | LocallyLipschitz.ToSubset.compatible_with_nhds_within | [19, 1] | [39, 60] | congr | X : Type u_1
Y : Type u_2
Z : Type u_3
inst✝² : MetricSpace X
inst✝¹ : MetricSpace Y
inst✝ : MetricSpace Z
t U : Set X
x : ↑U
hU : IsOpen U
b U' : Set X
hU' : U' ∈ Filter.principal U
htaU : t = b ∩ U'
a : Set X
ha : a ⊆ b
haopen : IsOpen a
hxa : ↑x ∈ a
⊢ b ∩ U = b ∩ (U' ∩ U) | case e_a
X : Type u_1
Y : Type u_2
Z : Type u_3
inst✝² : MetricSpace X
inst✝¹ : MetricSpace Y
inst✝ : MetricSpace Z
t U : Set X
x : ↑U
hU : IsOpen U
b U' : Set X
hU' : U' ∈ Filter.principal U
htaU : t = b ∩ U'
a : Set X
ha : a ⊆ b
haopen : IsOpen a
hxa : ↑x ∈ a
⊢ U = U' ∩ U | Please generate a tactic in lean4 to solve the state.
STATE:
X : Type u_1
Y : Type u_2
Z : Type u_3
inst✝² : MetricSpace X
inst✝¹ : MetricSpace Y
inst✝ : MetricSpace Z
t U : Set X
x : ↑U
hU : IsOpen U
b U' : Set X
hU' : U' ∈ Filter.principal U
htaU : t = b ∩ U'
a : Set X
ha : a ⊆ b
haopen : IsOpen a
hxa : ↑x ∈ a
⊢ b ∩ ... |
https://github.com/fpvandoorn/sard.git | f4a1bb550136e8591dcd20dca5bdcce766a958c7 | Sard/LocallyLipschitz.lean | LocallyLipschitz.ToSubset.compatible_with_nhds_within | [19, 1] | [39, 60] | apply (Set.inter_eq_right.mpr hU').symm | case e_a
X : Type u_1
Y : Type u_2
Z : Type u_3
inst✝² : MetricSpace X
inst✝¹ : MetricSpace Y
inst✝ : MetricSpace Z
t U : Set X
x : ↑U
hU : IsOpen U
b U' : Set X
hU' : U' ∈ Filter.principal U
htaU : t = b ∩ U'
a : Set X
ha : a ⊆ b
haopen : IsOpen a
hxa : ↑x ∈ a
⊢ U = U' ∩ U | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case e_a
X : Type u_1
Y : Type u_2
Z : Type u_3
inst✝² : MetricSpace X
inst✝¹ : MetricSpace Y
inst✝ : MetricSpace Z
t U : Set X
x : ↑U
hU : IsOpen U
b U' : Set X
hU' : U' ∈ Filter.principal U
htaU : t = b ∩ U'
a : Set X
ha : a ⊆ b
haopen : IsOpen a
hxa : ↑x ∈... |
https://github.com/fpvandoorn/sard.git | f4a1bb550136e8591dcd20dca5bdcce766a958c7 | Sard/LocallyLipschitz.lean | LocallyLipschitz.ToSubset.compatible_with_nhds_within | [19, 1] | [39, 60] | rw [inter_assoc] | X : Type u_1
Y : Type u_2
Z : Type u_3
inst✝² : MetricSpace X
inst✝¹ : MetricSpace Y
inst✝ : MetricSpace Z
t U : Set X
x : ↑U
hU : IsOpen U
b U' : Set X
hU' : U' ∈ Filter.principal U
htaU : t = b ∩ U'
a : Set X
ha : a ⊆ b
haopen : IsOpen a
hxa : ↑x ∈ a
⊢ b ∩ (U' ∩ U) ⊆ b ∩ U' ∩ U | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
X : Type u_1
Y : Type u_2
Z : Type u_3
inst✝² : MetricSpace X
inst✝¹ : MetricSpace Y
inst✝ : MetricSpace Z
t U : Set X
x : ↑U
hU : IsOpen U
b U' : Set X
hU' : U' ∈ Filter.principal U
htaU : t = b ∩ U'
a : Set X
ha : a ⊆ b
haopen : IsOpen a
hxa : ↑x ∈ a
⊢ b ∩ ... |
https://github.com/fpvandoorn/sard.git | f4a1bb550136e8591dcd20dca5bdcce766a958c7 | Sard/LocallyLipschitz.lean | LocallyLipschitz.ToSubset.compatible_with_nhds_within | [19, 1] | [39, 60] | rw [htaU] | X : Type u_1
Y : Type u_2
Z : Type u_3
inst✝² : MetricSpace X
inst✝¹ : MetricSpace Y
inst✝ : MetricSpace Z
t U : Set X
x : ↑U
hU : IsOpen U
b U' : Set X
hU' : U' ∈ Filter.principal U
htaU : t = b ∩ U'
a : Set X
ha : a ⊆ b
haopen : IsOpen a
hxa : ↑x ∈ a
⊢ b ∩ U' ∩ U = t ∩ U | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
X : Type u_1
Y : Type u_2
Z : Type u_3
inst✝² : MetricSpace X
inst✝¹ : MetricSpace Y
inst✝ : MetricSpace Z
t U : Set X
x : ↑U
hU : IsOpen U
b U' : Set X
hU' : U' ∈ Filter.principal U
htaU : t = b ∩ U'
a : Set X
ha : a ⊆ b
haopen : IsOpen a
hxa : ↑x ∈ a
⊢ b ∩ ... |
https://github.com/fpvandoorn/sard.git | f4a1bb550136e8591dcd20dca5bdcce766a958c7 | Sard/LocallyLipschitz.lean | LocallyLipschitz.ToSubset.compatible_with_nhds_within | [19, 1] | [39, 60] | exact ⟨IsOpen.inter haopen hU, ⟨hxa, Subtype.mem x⟩⟩ | case h.right
X : Type u_1
Y : Type u_2
Z : Type u_3
inst✝² : MetricSpace X
inst✝¹ : MetricSpace Y
inst✝ : MetricSpace Z
t U : Set X
x : ↑U
hU : IsOpen U
b U' : Set X
hU' : U' ∈ Filter.principal U
htaU : t = b ∩ U'
a : Set X
ha : a ⊆ b
haopen : IsOpen a
hxa : ↑x ∈ a
⊢ IsOpen (a ∩ U) ∧ ↑x ∈ a ∩ U | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h.right
X : Type u_1
Y : Type u_2
Z : Type u_3
inst✝² : MetricSpace X
inst✝¹ : MetricSpace Y
inst✝ : MetricSpace Z
t U : Set X
x : ↑U
hU : IsOpen U
b U' : Set X
hU' : U' ∈ Filter.principal U
htaU : t = b ∩ U'
a : Set X
ha : a ⊆ b
haopen : IsOpen a
hxa : ... |
https://github.com/fpvandoorn/sard.git | f4a1bb550136e8591dcd20dca5bdcce766a958c7 | Sard/LocallyLipschitz.lean | LocallyLipschitz.restrict | [46, 1] | [52, 106] | intro x | X : Type u_1
Y : Type u_2
Z : Type u_3
inst✝² : MetricSpace X
inst✝¹ : MetricSpace Y
inst✝ : MetricSpace Z
f : X → Y
hf : LocallyLipschitz f
s : Set X
⊢ LocallyLipschitz (s.restrict f) | X : Type u_1
Y : Type u_2
Z : Type u_3
inst✝² : MetricSpace X
inst✝¹ : MetricSpace Y
inst✝ : MetricSpace Z
f : X → Y
hf : LocallyLipschitz f
s : Set X
x : ↑s
⊢ ∃ K, ∃ t ∈ 𝓝 x, LipschitzOnWith K (s.restrict f) t | Please generate a tactic in lean4 to solve the state.
STATE:
X : Type u_1
Y : Type u_2
Z : Type u_3
inst✝² : MetricSpace X
inst✝¹ : MetricSpace Y
inst✝ : MetricSpace Z
f : X → Y
hf : LocallyLipschitz f
s : Set X
⊢ LocallyLipschitz (s.restrict f)
TACTIC:
|
https://github.com/fpvandoorn/sard.git | f4a1bb550136e8591dcd20dca5bdcce766a958c7 | Sard/LocallyLipschitz.lean | LocallyLipschitz.restrict | [46, 1] | [52, 106] | rcases hf x with ⟨K, t, ht, hfL⟩ | X : Type u_1
Y : Type u_2
Z : Type u_3
inst✝² : MetricSpace X
inst✝¹ : MetricSpace Y
inst✝ : MetricSpace Z
f : X → Y
hf : LocallyLipschitz f
s : Set X
x : ↑s
⊢ ∃ K, ∃ t ∈ 𝓝 x, LipschitzOnWith K (s.restrict f) t | case intro.intro.intro
X : Type u_1
Y : Type u_2
Z : Type u_3
inst✝² : MetricSpace X
inst✝¹ : MetricSpace Y
inst✝ : MetricSpace Z
f : X → Y
hf : LocallyLipschitz f
s : Set X
x : ↑s
K : ℝ≥0
t : Set X
ht : t ∈ 𝓝 ↑x
hfL : LipschitzOnWith K f t
⊢ ∃ K, ∃ t ∈ 𝓝 x, LipschitzOnWith K (s.restrict f) t | Please generate a tactic in lean4 to solve the state.
STATE:
X : Type u_1
Y : Type u_2
Z : Type u_3
inst✝² : MetricSpace X
inst✝¹ : MetricSpace Y
inst✝ : MetricSpace Z
f : X → Y
hf : LocallyLipschitz f
s : Set X
x : ↑s
⊢ ∃ K, ∃ t ∈ 𝓝 x, LipschitzOnWith K (s.restrict f) t
TACTIC:
|
https://github.com/fpvandoorn/sard.git | f4a1bb550136e8591dcd20dca5bdcce766a958c7 | Sard/LocallyLipschitz.lean | LocallyLipschitz.restrict | [46, 1] | [52, 106] | exact ⟨K, toSubset t s, ToSubset.compatible_with_nhds s t ht, LipschitzOnWith.restrict_subtype s t hfL⟩ | case intro.intro.intro
X : Type u_1
Y : Type u_2
Z : Type u_3
inst✝² : MetricSpace X
inst✝¹ : MetricSpace Y
inst✝ : MetricSpace Z
f : X → Y
hf : LocallyLipschitz f
s : Set X
x : ↑s
K : ℝ≥0
t : Set X
ht : t ∈ 𝓝 ↑x
hfL : LipschitzOnWith K f t
⊢ ∃ K, ∃ t ∈ 𝓝 x, LipschitzOnWith K (s.restrict f) t | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.intro
X : Type u_1
Y : Type u_2
Z : Type u_3
inst✝² : MetricSpace X
inst✝¹ : MetricSpace Y
inst✝ : MetricSpace Z
f : X → Y
hf : LocallyLipschitz f
s : Set X
x : ↑s
K : ℝ≥0
t : Set X
ht : t ∈ 𝓝 ↑x
hfL : LipschitzOnWith K f t
⊢ ∃ K, ∃ t ∈ 𝓝 x... |
https://github.com/fpvandoorn/sard.git | f4a1bb550136e8591dcd20dca5bdcce766a958c7 | Sard/LocallyLipschitz.lean | LocallyLipschitz.of_C1 | [54, 1] | [60, 11] | intro x | X : Type u_1
Y : Type u_2
Z : Type u_3
inst✝⁶ : MetricSpace X
inst✝⁵ : MetricSpace Y
inst✝⁴ : MetricSpace Z
E : Type u_4
F : Type u_5
f : E → F
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℝ E
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace ℝ F
hf : ContDiff ℝ 1 f
⊢ LocallyLipschitz f | X : Type u_1
Y : Type u_2
Z : Type u_3
inst✝⁶ : MetricSpace X
inst✝⁵ : MetricSpace Y
inst✝⁴ : MetricSpace Z
E : Type u_4
F : Type u_5
f : E → F
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℝ E
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace ℝ F
hf : ContDiff ℝ 1 f
x : E
⊢ ∃ K, ∃ t ∈ 𝓝 x, LipschitzOnWith K f t | Please generate a tactic in lean4 to solve the state.
STATE:
X : Type u_1
Y : Type u_2
Z : Type u_3
inst✝⁶ : MetricSpace X
inst✝⁵ : MetricSpace Y
inst✝⁴ : MetricSpace Z
E : Type u_4
F : Type u_5
f : E → F
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℝ E
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace ℝ F
hf : C... |
https://github.com/fpvandoorn/sard.git | f4a1bb550136e8591dcd20dca5bdcce766a958c7 | Sard/LocallyLipschitz.lean | LocallyLipschitz.of_C1 | [54, 1] | [60, 11] | rcases (ContDiffAt.exists_lipschitzOnWith (ContDiff.contDiffAt hf)) with ⟨K, t, ht, hf⟩ | X : Type u_1
Y : Type u_2
Z : Type u_3
inst✝⁶ : MetricSpace X
inst✝⁵ : MetricSpace Y
inst✝⁴ : MetricSpace Z
E : Type u_4
F : Type u_5
f : E → F
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℝ E
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace ℝ F
hf : ContDiff ℝ 1 f
x : E
⊢ ∃ K, ∃ t ∈ 𝓝 x, LipschitzOnWith K f t | case intro.intro.intro
X : Type u_1
Y : Type u_2
Z : Type u_3
inst✝⁶ : MetricSpace X
inst✝⁵ : MetricSpace Y
inst✝⁴ : MetricSpace Z
E : Type u_4
F : Type u_5
f : E → F
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℝ E
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace ℝ F
hf✝ : ContDiff ℝ 1 f
x : E
K : ℝ≥0
t : Set E... | Please generate a tactic in lean4 to solve the state.
STATE:
X : Type u_1
Y : Type u_2
Z : Type u_3
inst✝⁶ : MetricSpace X
inst✝⁵ : MetricSpace Y
inst✝⁴ : MetricSpace Z
E : Type u_4
F : Type u_5
f : E → F
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℝ E
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace ℝ F
hf : C... |
https://github.com/fpvandoorn/sard.git | f4a1bb550136e8591dcd20dca5bdcce766a958c7 | Sard/LocallyLipschitz.lean | LocallyLipschitz.of_C1 | [54, 1] | [60, 11] | use K, t | case intro.intro.intro
X : Type u_1
Y : Type u_2
Z : Type u_3
inst✝⁶ : MetricSpace X
inst✝⁵ : MetricSpace Y
inst✝⁴ : MetricSpace Z
E : Type u_4
F : Type u_5
f : E → F
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℝ E
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace ℝ F
hf✝ : ContDiff ℝ 1 f
x : E
K : ℝ≥0
t : Set E... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.intro
X : Type u_1
Y : Type u_2
Z : Type u_3
inst✝⁶ : MetricSpace X
inst✝⁵ : MetricSpace Y
inst✝⁴ : MetricSpace Z
E : Type u_4
F : Type u_5
f : E → F
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℝ E
inst✝¹ : NormedAddCommGroup F
inst✝ :... |
https://github.com/fpvandoorn/sard.git | f4a1bb550136e8591dcd20dca5bdcce766a958c7 | Sard/LocallyLipschitz.lean | LocallyLipschitz.of_C1_on_open | [62, 1] | [73, 116] | intro x | X : Type u_1
Y : Type u_2
Z : Type u_3
inst✝⁶ : MetricSpace X
inst✝⁵ : MetricSpace Y
inst✝⁴ : MetricSpace Z
E : Type u_4
F : Type u_5
f : E → F
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℝ E
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace ℝ F
U : Set E
h₁U : IsOpen U
h₂U : Convex ℝ U
hf : ContDiffOn ℝ 1 f U
⊢... | X : Type u_1
Y : Type u_2
Z : Type u_3
inst✝⁶ : MetricSpace X
inst✝⁵ : MetricSpace Y
inst✝⁴ : MetricSpace Z
E : Type u_4
F : Type u_5
f : E → F
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℝ E
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace ℝ F
U : Set E
h₁U : IsOpen U
h₂U : Convex ℝ U
hf : ContDiffOn ℝ 1 f U
x... | Please generate a tactic in lean4 to solve the state.
STATE:
X : Type u_1
Y : Type u_2
Z : Type u_3
inst✝⁶ : MetricSpace X
inst✝⁵ : MetricSpace Y
inst✝⁴ : MetricSpace Z
E : Type u_4
F : Type u_5
f : E → F
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℝ E
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace ℝ F
U : Se... |
https://github.com/fpvandoorn/sard.git | f4a1bb550136e8591dcd20dca5bdcce766a958c7 | Sard/LocallyLipschitz.lean | LocallyLipschitz.of_C1_on_open | [62, 1] | [73, 116] | have : ContDiffWithinAt ℝ 1 f U x := ContDiffOn.contDiffWithinAt hf (Subtype.mem x) | X : Type u_1
Y : Type u_2
Z : Type u_3
inst✝⁶ : MetricSpace X
inst✝⁵ : MetricSpace Y
inst✝⁴ : MetricSpace Z
E : Type u_4
F : Type u_5
f : E → F
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℝ E
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace ℝ F
U : Set E
h₁U : IsOpen U
h₂U : Convex ℝ U
hf : ContDiffOn ℝ 1 f U
x... | X : Type u_1
Y : Type u_2
Z : Type u_3
inst✝⁶ : MetricSpace X
inst✝⁵ : MetricSpace Y
inst✝⁴ : MetricSpace Z
E : Type u_4
F : Type u_5
f : E → F
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℝ E
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace ℝ F
U : Set E
h₁U : IsOpen U
h₂U : Convex ℝ U
hf : ContDiffOn ℝ 1 f U
x... | Please generate a tactic in lean4 to solve the state.
STATE:
X : Type u_1
Y : Type u_2
Z : Type u_3
inst✝⁶ : MetricSpace X
inst✝⁵ : MetricSpace Y
inst✝⁴ : MetricSpace Z
E : Type u_4
F : Type u_5
f : E → F
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℝ E
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace ℝ F
U : Se... |
https://github.com/fpvandoorn/sard.git | f4a1bb550136e8591dcd20dca5bdcce766a958c7 | Sard/LocallyLipschitz.lean | LocallyLipschitz.of_C1_on_open | [62, 1] | [73, 116] | let h := ContDiffWithinAt.exists_lipschitzOnWith this | X : Type u_1
Y : Type u_2
Z : Type u_3
inst✝⁶ : MetricSpace X
inst✝⁵ : MetricSpace Y
inst✝⁴ : MetricSpace Z
E : Type u_4
F : Type u_5
f : E → F
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℝ E
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace ℝ F
U : Set E
h₁U : IsOpen U
h₂U : Convex ℝ U
hf : ContDiffOn ℝ 1 f U
x... | X : Type u_1
Y : Type u_2
Z : Type u_3
inst✝⁶ : MetricSpace X
inst✝⁵ : MetricSpace Y
inst✝⁴ : MetricSpace Z
E : Type u_4
F : Type u_5
f : E → F
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℝ E
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace ℝ F
U : Set E
h₁U : IsOpen U
h₂U : Convex ℝ U
hf : ContDiffOn ℝ 1 f U
x... | Please generate a tactic in lean4 to solve the state.
STATE:
X : Type u_1
Y : Type u_2
Z : Type u_3
inst✝⁶ : MetricSpace X
inst✝⁵ : MetricSpace Y
inst✝⁴ : MetricSpace Z
E : Type u_4
F : Type u_5
f : E → F
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℝ E
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace ℝ F
U : Se... |
https://github.com/fpvandoorn/sard.git | f4a1bb550136e8591dcd20dca5bdcce766a958c7 | Sard/LocallyLipschitz.lean | LocallyLipschitz.of_C1_on_open | [62, 1] | [73, 116] | rcases (h h₂U) with ⟨K, t, ht, hf⟩ | X : Type u_1
Y : Type u_2
Z : Type u_3
inst✝⁶ : MetricSpace X
inst✝⁵ : MetricSpace Y
inst✝⁴ : MetricSpace Z
E : Type u_4
F : Type u_5
f : E → F
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℝ E
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace ℝ F
U : Set E
h₁U : IsOpen U
h₂U : Convex ℝ U
hf : ContDiffOn ℝ 1 f U
x... | case intro.intro.intro
X : Type u_1
Y : Type u_2
Z : Type u_3
inst✝⁶ : MetricSpace X
inst✝⁵ : MetricSpace Y
inst✝⁴ : MetricSpace Z
E : Type u_4
F : Type u_5
f : E → F
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℝ E
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace ℝ F
U : Set E
h₁U : IsOpen U
h₂U : Convex ℝ U
hf... | Please generate a tactic in lean4 to solve the state.
STATE:
X : Type u_1
Y : Type u_2
Z : Type u_3
inst✝⁶ : MetricSpace X
inst✝⁵ : MetricSpace Y
inst✝⁴ : MetricSpace Z
E : Type u_4
F : Type u_5
f : E → F
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℝ E
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace ℝ F
U : Se... |
https://github.com/fpvandoorn/sard.git | f4a1bb550136e8591dcd20dca5bdcce766a958c7 | Sard/LocallyLipschitz.lean | LocallyLipschitz.of_C1_on_open | [62, 1] | [73, 116] | exact ⟨K, toSubset t U, ToSubset.compatible_with_nhds_within t U h₁U ht, LipschitzOnWith.restrict_subtype U t hf⟩ | case intro.intro.intro
X : Type u_1
Y : Type u_2
Z : Type u_3
inst✝⁶ : MetricSpace X
inst✝⁵ : MetricSpace Y
inst✝⁴ : MetricSpace Z
E : Type u_4
F : Type u_5
f : E → F
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℝ E
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace ℝ F
U : Set E
h₁U : IsOpen U
h₂U : Convex ℝ U
hf... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.intro
X : Type u_1
Y : Type u_2
Z : Type u_3
inst✝⁶ : MetricSpace X
inst✝⁵ : MetricSpace Y
inst✝⁴ : MetricSpace Z
E : Type u_4
F : Type u_5
f : E → F
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℝ E
inst✝¹ : NormedAddCommGroup F
inst✝ :... |
https://github.com/fpvandoorn/sard.git | f4a1bb550136e8591dcd20dca5bdcce766a958c7 | Sard/LocallyLipschitz.lean | LipschitzOnWith.sum | [80, 1] | [96, 41] | intro y hy z hz | X : Type u_1
Y : Type u_2
Z : Type u_3
inst✝² : MetricSpace X
inst✝¹ : NormedAddCommGroup Y
inst✝ : NormedSpace ℝ Y
f g : X → Y
Kf Kg : ℝ≥0
s : Set X
hf : LipschitzOnWith Kf f s
hg : LipschitzOnWith Kg g s
⊢ LipschitzOnWith (Kf + Kg) (f + g) s | X : Type u_1
Y : Type u_2
Z : Type u_3
inst✝² : MetricSpace X
inst✝¹ : NormedAddCommGroup Y
inst✝ : NormedSpace ℝ Y
f g : X → Y
Kf Kg : ℝ≥0
s : Set X
hf : LipschitzOnWith Kf f s
hg : LipschitzOnWith Kg g s
y : X
hy : y ∈ s
z : X
hz : z ∈ s
⊢ edist ((f + g) y) ((f + g) z) ≤ ↑(Kf + Kg) * edist y z | Please generate a tactic in lean4 to solve the state.
STATE:
X : Type u_1
Y : Type u_2
Z : Type u_3
inst✝² : MetricSpace X
inst✝¹ : NormedAddCommGroup Y
inst✝ : NormedSpace ℝ Y
f g : X → Y
Kf Kg : ℝ≥0
s : Set X
hf : LipschitzOnWith Kf f s
hg : LipschitzOnWith Kg g s
⊢ LipschitzOnWith (Kf + Kg) (f + g) s
TACTIC:
|
https://github.com/fpvandoorn/sard.git | f4a1bb550136e8591dcd20dca5bdcce766a958c7 | Sard/LocallyLipschitz.lean | LipschitzOnWith.sum | [80, 1] | [96, 41] | have translation: ∀ w w' w'' : Y, edist (w + w'') (w' + w'') = edist w w' := by
intro w w' w''
simp only [edist_add_right] | X : Type u_1
Y : Type u_2
Z : Type u_3
inst✝² : MetricSpace X
inst✝¹ : NormedAddCommGroup Y
inst✝ : NormedSpace ℝ Y
f g : X → Y
Kf Kg : ℝ≥0
s : Set X
hf : LipschitzOnWith Kf f s
hg : LipschitzOnWith Kg g s
y : X
hy : y ∈ s
z : X
hz : z ∈ s
⊢ edist ((f + g) y) ((f + g) z) ≤ ↑(Kf + Kg) * edist y z | X : Type u_1
Y : Type u_2
Z : Type u_3
inst✝² : MetricSpace X
inst✝¹ : NormedAddCommGroup Y
inst✝ : NormedSpace ℝ Y
f g : X → Y
Kf Kg : ℝ≥0
s : Set X
hf : LipschitzOnWith Kf f s
hg : LipschitzOnWith Kg g s
y : X
hy : y ∈ s
z : X
hz : z ∈ s
translation : ∀ (w w' w'' : Y), edist (w + w'') (w' + w'') = edist w w'
⊢ edist ... | Please generate a tactic in lean4 to solve the state.
STATE:
X : Type u_1
Y : Type u_2
Z : Type u_3
inst✝² : MetricSpace X
inst✝¹ : NormedAddCommGroup Y
inst✝ : NormedSpace ℝ Y
f g : X → Y
Kf Kg : ℝ≥0
s : Set X
hf : LipschitzOnWith Kf f s
hg : LipschitzOnWith Kg g s
y : X
hy : y ∈ s
z : X
hz : z ∈ s
⊢ edist ((f + g) y)... |
https://github.com/fpvandoorn/sard.git | f4a1bb550136e8591dcd20dca5bdcce766a958c7 | Sard/LocallyLipschitz.lean | LipschitzOnWith.sum | [80, 1] | [96, 41] | simp only [Pi.add_apply, ENNReal.coe_add] | X : Type u_1
Y : Type u_2
Z : Type u_3
inst✝² : MetricSpace X
inst✝¹ : NormedAddCommGroup Y
inst✝ : NormedSpace ℝ Y
f g : X → Y
Kf Kg : ℝ≥0
s : Set X
hf : LipschitzOnWith Kf f s
hg : LipschitzOnWith Kg g s
y : X
hy : y ∈ s
z : X
hz : z ∈ s
translation : ∀ (w w' w'' : Y), edist (w + w'') (w' + w'') = edist w w'
⊢ edist ... | X : Type u_1
Y : Type u_2
Z : Type u_3
inst✝² : MetricSpace X
inst✝¹ : NormedAddCommGroup Y
inst✝ : NormedSpace ℝ Y
f g : X → Y
Kf Kg : ℝ≥0
s : Set X
hf : LipschitzOnWith Kf f s
hg : LipschitzOnWith Kg g s
y : X
hy : y ∈ s
z : X
hz : z ∈ s
translation : ∀ (w w' w'' : Y), edist (w + w'') (w' + w'') = edist w w'
⊢ edist ... | Please generate a tactic in lean4 to solve the state.
STATE:
X : Type u_1
Y : Type u_2
Z : Type u_3
inst✝² : MetricSpace X
inst✝¹ : NormedAddCommGroup Y
inst✝ : NormedSpace ℝ Y
f g : X → Y
Kf Kg : ℝ≥0
s : Set X
hf : LipschitzOnWith Kf f s
hg : LipschitzOnWith Kg g s
y : X
hy : y ∈ s
z : X
hz : z ∈ s
translation : ∀ (w ... |
https://github.com/fpvandoorn/sard.git | f4a1bb550136e8591dcd20dca5bdcce766a958c7 | Sard/LocallyLipschitz.lean | LipschitzOnWith.sum | [80, 1] | [96, 41] | calc edist (f y + g y) (f z + g z)
_ ≤ edist (f y + g y) (g y + f z) + edist (g y + f z) (f z + g z) := by apply edist_triangle
_ = edist (f y + g y) (f z + g y) + edist (g y + f z) (g z + f z) := by
simp only [add_comm, edist_add_right, edist_add_left]
_ ≤ edist (f y) (f z) + edist (g y) (g z) := by rw [tr... | X : Type u_1
Y : Type u_2
Z : Type u_3
inst✝² : MetricSpace X
inst✝¹ : NormedAddCommGroup Y
inst✝ : NormedSpace ℝ Y
f g : X → Y
Kf Kg : ℝ≥0
s : Set X
hf : LipschitzOnWith Kf f s
hg : LipschitzOnWith Kg g s
y : X
hy : y ∈ s
z : X
hz : z ∈ s
translation : ∀ (w w' w'' : Y), edist (w + w'') (w' + w'') = edist w w'
⊢ edist ... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
X : Type u_1
Y : Type u_2
Z : Type u_3
inst✝² : MetricSpace X
inst✝¹ : NormedAddCommGroup Y
inst✝ : NormedSpace ℝ Y
f g : X → Y
Kf Kg : ℝ≥0
s : Set X
hf : LipschitzOnWith Kf f s
hg : LipschitzOnWith Kg g s
y : X
hy : y ∈ s
z : X
hz : z ∈ s
translation : ∀ (w ... |
https://github.com/fpvandoorn/sard.git | f4a1bb550136e8591dcd20dca5bdcce766a958c7 | Sard/LocallyLipschitz.lean | LipschitzOnWith.sum | [80, 1] | [96, 41] | intro w w' w'' | X : Type u_1
Y : Type u_2
Z : Type u_3
inst✝² : MetricSpace X
inst✝¹ : NormedAddCommGroup Y
inst✝ : NormedSpace ℝ Y
f g : X → Y
Kf Kg : ℝ≥0
s : Set X
hf : LipschitzOnWith Kf f s
hg : LipschitzOnWith Kg g s
y : X
hy : y ∈ s
z : X
hz : z ∈ s
⊢ ∀ (w w' w'' : Y), edist (w + w'') (w' + w'') = edist w w' | X : Type u_1
Y : Type u_2
Z : Type u_3
inst✝² : MetricSpace X
inst✝¹ : NormedAddCommGroup Y
inst✝ : NormedSpace ℝ Y
f g : X → Y
Kf Kg : ℝ≥0
s : Set X
hf : LipschitzOnWith Kf f s
hg : LipschitzOnWith Kg g s
y : X
hy : y ∈ s
z : X
hz : z ∈ s
w w' w'' : Y
⊢ edist (w + w'') (w' + w'') = edist w w' | Please generate a tactic in lean4 to solve the state.
STATE:
X : Type u_1
Y : Type u_2
Z : Type u_3
inst✝² : MetricSpace X
inst✝¹ : NormedAddCommGroup Y
inst✝ : NormedSpace ℝ Y
f g : X → Y
Kf Kg : ℝ≥0
s : Set X
hf : LipschitzOnWith Kf f s
hg : LipschitzOnWith Kg g s
y : X
hy : y ∈ s
z : X
hz : z ∈ s
⊢ ∀ (w w' w'' : Y),... |
https://github.com/fpvandoorn/sard.git | f4a1bb550136e8591dcd20dca5bdcce766a958c7 | Sard/LocallyLipschitz.lean | LipschitzOnWith.sum | [80, 1] | [96, 41] | simp only [edist_add_right] | X : Type u_1
Y : Type u_2
Z : Type u_3
inst✝² : MetricSpace X
inst✝¹ : NormedAddCommGroup Y
inst✝ : NormedSpace ℝ Y
f g : X → Y
Kf Kg : ℝ≥0
s : Set X
hf : LipschitzOnWith Kf f s
hg : LipschitzOnWith Kg g s
y : X
hy : y ∈ s
z : X
hz : z ∈ s
w w' w'' : Y
⊢ edist (w + w'') (w' + w'') = edist w w' | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
X : Type u_1
Y : Type u_2
Z : Type u_3
inst✝² : MetricSpace X
inst✝¹ : NormedAddCommGroup Y
inst✝ : NormedSpace ℝ Y
f g : X → Y
Kf Kg : ℝ≥0
s : Set X
hf : LipschitzOnWith Kf f s
hg : LipschitzOnWith Kg g s
y : X
hy : y ∈ s
z : X
hz : z ∈ s
w w' w'' : Y
⊢ edis... |
https://github.com/fpvandoorn/sard.git | f4a1bb550136e8591dcd20dca5bdcce766a958c7 | Sard/LocallyLipschitz.lean | LipschitzOnWith.sum | [80, 1] | [96, 41] | apply edist_triangle | X : Type u_1
Y : Type u_2
Z : Type u_3
inst✝² : MetricSpace X
inst✝¹ : NormedAddCommGroup Y
inst✝ : NormedSpace ℝ Y
f g : X → Y
Kf Kg : ℝ≥0
s : Set X
hf : LipschitzOnWith Kf f s
hg : LipschitzOnWith Kg g s
y : X
hy : y ∈ s
z : X
hz : z ∈ s
translation : ∀ (w w' w'' : Y), edist (w + w'') (w' + w'') = edist w w'
⊢ edist ... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
X : Type u_1
Y : Type u_2
Z : Type u_3
inst✝² : MetricSpace X
inst✝¹ : NormedAddCommGroup Y
inst✝ : NormedSpace ℝ Y
f g : X → Y
Kf Kg : ℝ≥0
s : Set X
hf : LipschitzOnWith Kf f s
hg : LipschitzOnWith Kg g s
y : X
hy : y ∈ s
z : X
hz : z ∈ s
translation : ∀ (w ... |
https://github.com/fpvandoorn/sard.git | f4a1bb550136e8591dcd20dca5bdcce766a958c7 | Sard/LocallyLipschitz.lean | LipschitzOnWith.sum | [80, 1] | [96, 41] | simp only [add_comm, edist_add_right, edist_add_left] | X : Type u_1
Y : Type u_2
Z : Type u_3
inst✝² : MetricSpace X
inst✝¹ : NormedAddCommGroup Y
inst✝ : NormedSpace ℝ Y
f g : X → Y
Kf Kg : ℝ≥0
s : Set X
hf : LipschitzOnWith Kf f s
hg : LipschitzOnWith Kg g s
y : X
hy : y ∈ s
z : X
hz : z ∈ s
translation : ∀ (w w' w'' : Y), edist (w + w'') (w' + w'') = edist w w'
⊢ edist ... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
X : Type u_1
Y : Type u_2
Z : Type u_3
inst✝² : MetricSpace X
inst✝¹ : NormedAddCommGroup Y
inst✝ : NormedSpace ℝ Y
f g : X → Y
Kf Kg : ℝ≥0
s : Set X
hf : LipschitzOnWith Kf f s
hg : LipschitzOnWith Kg g s
y : X
hy : y ∈ s
z : X
hz : z ∈ s
translation : ∀ (w ... |
https://github.com/fpvandoorn/sard.git | f4a1bb550136e8591dcd20dca5bdcce766a958c7 | Sard/LocallyLipschitz.lean | LipschitzOnWith.sum | [80, 1] | [96, 41] | rw [translation, translation] | X : Type u_1
Y : Type u_2
Z : Type u_3
inst✝² : MetricSpace X
inst✝¹ : NormedAddCommGroup Y
inst✝ : NormedSpace ℝ Y
f g : X → Y
Kf Kg : ℝ≥0
s : Set X
hf : LipschitzOnWith Kf f s
hg : LipschitzOnWith Kg g s
y : X
hy : y ∈ s
z : X
hz : z ∈ s
translation : ∀ (w w' w'' : Y), edist (w + w'') (w' + w'') = edist w w'
⊢ edist ... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
X : Type u_1
Y : Type u_2
Z : Type u_3
inst✝² : MetricSpace X
inst✝¹ : NormedAddCommGroup Y
inst✝ : NormedSpace ℝ Y
f g : X → Y
Kf Kg : ℝ≥0
s : Set X
hf : LipschitzOnWith Kf f s
hg : LipschitzOnWith Kg g s
y : X
hy : y ∈ s
z : X
hz : z ∈ s
translation : ∀ (w ... |
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