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/MeasureZero.lean | measure_zero_image_iff_chart_domains | [163, 1] | [198, 41] | rw [iUnion_coe_set] | E✝ : Type u_1
inst✝²⁶ : NormedAddCommGroup E✝
inst✝²⁵ : NormedSpace ℝ E✝
H✝ : Type u_2
inst✝²⁴ : TopologicalSpace H✝
I✝ : ModelWithCorners ℝ E✝ H✝
M✝ : Type u_3
inst✝²³ : TopologicalSpace M✝
inst✝²² : ChartedSpace H✝ M✝
inst✝²¹ : I✝.Boundaryless
inst✝²⁰ : FiniteDimensional ℝ E✝
inst✝¹⁹ : MeasurableSpace E✝
inst✝¹⁸ : Bo... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
E✝ : Type u_1
inst✝²⁶ : NormedAddCommGroup E✝
inst✝²⁵ : NormedSpace ℝ E✝
H✝ : Type u_2
inst✝²⁴ : TopologicalSpace H✝
I✝ : ModelWithCorners ℝ E✝ H✝
M✝ : Type u_3
inst✝²³ : TopologicalSpace M✝
inst✝²² : ChartedSpace H✝ M✝
inst✝²¹ : I✝.Boundaryless
inst✝²⁰ : Fin... |
https://github.com/fpvandoorn/sard.git | f4a1bb550136e8591dcd20dca5bdcce766a958c7 | Sard/MeasureZero.lean | measure_zero_image_iff_chart_domains | [163, 1] | [198, 41] | rw [hTcover] | E✝ : Type u_1
inst✝²⁶ : NormedAddCommGroup E✝
inst✝²⁵ : NormedSpace ℝ E✝
H✝ : Type u_2
inst✝²⁴ : TopologicalSpace H✝
I✝ : ModelWithCorners ℝ E✝ H✝
M✝ : Type u_3
inst✝²³ : TopologicalSpace M✝
inst✝²² : ChartedSpace H✝ M✝
inst✝²¹ : I✝.Boundaryless
inst✝²⁰ : FiniteDimensional ℝ E✝
inst✝¹⁹ : MeasurableSpace E✝
inst✝¹⁸ : Bo... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
E✝ : Type u_1
inst✝²⁶ : NormedAddCommGroup E✝
inst✝²⁵ : NormedSpace ℝ E✝
H✝ : Type u_2
inst✝²⁴ : TopologicalSpace H✝
I✝ : ModelWithCorners ℝ E✝ H✝
M✝ : Type u_3
inst✝²³ : TopologicalSpace M✝
inst✝²² : ChartedSpace H✝ M✝
inst✝²¹ : I✝.Boundaryless
inst✝²⁰ : Fin... |
https://github.com/fpvandoorn/sard.git | f4a1bb550136e8591dcd20dca5bdcce766a958c7 | Sard/MeasureZero.lean | measure_zero_image_iff_chart_domains | [163, 1] | [198, 41] | rw [subset_antisymm (by simp) (hcovering)] | E✝ : Type u_1
inst✝²⁶ : NormedAddCommGroup E✝
inst✝²⁵ : NormedSpace ℝ E✝
H✝ : Type u_2
inst✝²⁴ : TopologicalSpace H✝
I✝ : ModelWithCorners ℝ E✝ H✝
M✝ : Type u_3
inst✝²³ : TopologicalSpace M✝
inst✝²² : ChartedSpace H✝ M✝
inst✝²¹ : I✝.Boundaryless
inst✝²⁰ : FiniteDimensional ℝ E✝
inst✝¹⁹ : MeasurableSpace E✝
inst✝¹⁸ : Bo... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
E✝ : Type u_1
inst✝²⁶ : NormedAddCommGroup E✝
inst✝²⁵ : NormedSpace ℝ E✝
H✝ : Type u_2
inst✝²⁴ : TopologicalSpace H✝
I✝ : ModelWithCorners ℝ E✝ H✝
M✝ : Type u_3
inst✝²³ : TopologicalSpace M✝
inst✝²² : ChartedSpace H✝ M✝
inst✝²¹ : I✝.Boundaryless
inst✝²⁰ : Fin... |
https://github.com/fpvandoorn/sard.git | f4a1bb550136e8591dcd20dca5bdcce766a958c7 | Sard/MeasureZero.lean | measure_zero_image_iff_chart_domains | [163, 1] | [198, 41] | simp | E✝ : Type u_1
inst✝²⁶ : NormedAddCommGroup E✝
inst✝²⁵ : NormedSpace ℝ E✝
H✝ : Type u_2
inst✝²⁴ : TopologicalSpace H✝
I✝ : ModelWithCorners ℝ E✝ H✝
M✝ : Type u_3
inst✝²³ : TopologicalSpace M✝
inst✝²² : ChartedSpace H✝ M✝
inst✝²¹ : I✝.Boundaryless
inst✝²⁰ : FiniteDimensional ℝ E✝
inst✝¹⁹ : MeasurableSpace E✝
inst✝¹⁸ : Bo... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
E✝ : Type u_1
inst✝²⁶ : NormedAddCommGroup E✝
inst✝²⁵ : NormedSpace ℝ E✝
H✝ : Type u_2
inst✝²⁴ : TopologicalSpace H✝
I✝ : ModelWithCorners ℝ E✝ H✝
M✝ : Type u_3
inst✝²³ : TopologicalSpace M✝
inst✝²² : ChartedSpace H✝ M✝
inst✝²¹ : I✝.Boundaryless
inst✝²⁰ : Fin... |
https://github.com/fpvandoorn/sard.git | f4a1bb550136e8591dcd20dca5bdcce766a958c7 | Sard/MeasureZero.lean | measure_zero_image_iff_chart_domains | [163, 1] | [198, 41] | rw [univ_inter] | E✝ : Type u_1
inst✝²⁶ : NormedAddCommGroup E✝
inst✝²⁵ : NormedSpace ℝ E✝
H✝ : Type u_2
inst✝²⁴ : TopologicalSpace H✝
I✝ : ModelWithCorners ℝ E✝ H✝
M✝ : Type u_3
inst✝²³ : TopologicalSpace M✝
inst✝²² : ChartedSpace H✝ M✝
inst✝²¹ : I✝.Boundaryless
inst✝²⁰ : FiniteDimensional ℝ E✝
inst✝¹⁹ : MeasurableSpace E✝
inst✝¹⁸ : Bo... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
E✝ : Type u_1
inst✝²⁶ : NormedAddCommGroup E✝
inst✝²⁵ : NormedSpace ℝ E✝
H✝ : Type u_2
inst✝²⁴ : TopologicalSpace H✝
I✝ : ModelWithCorners ℝ E✝ H✝
M✝ : Type u_3
inst✝²³ : TopologicalSpace M✝
inst✝²² : ChartedSpace H✝ M✝
inst✝²¹ : I✝.Boundaryless
inst✝²⁰ : Fin... |
https://github.com/fpvandoorn/sard.git | f4a1bb550136e8591dcd20dca5bdcce766a958c7 | Sard/MeasureZero.lean | measure_zero_image_iff_chart_domains | [163, 1] | [198, 41] | sorry | E✝ : Type u_1
inst✝²⁶ : NormedAddCommGroup E✝
inst✝²⁵ : NormedSpace ℝ E✝
H✝ : Type u_2
inst✝²⁴ : TopologicalSpace H✝
I✝ : ModelWithCorners ℝ E✝ H✝
M✝ : Type u_3
inst✝²³ : TopologicalSpace M✝
inst✝²² : ChartedSpace H✝ M✝
inst✝²¹ : I✝.Boundaryless
inst✝²⁰ : FiniteDimensional ℝ E✝
inst✝¹⁹ : MeasurableSpace E✝
inst✝¹⁸ : Bo... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
E✝ : Type u_1
inst✝²⁶ : NormedAddCommGroup E✝
inst✝²⁵ : NormedSpace ℝ E✝
H✝ : Type u_2
inst✝²⁴ : TopologicalSpace H✝
I✝ : ModelWithCorners ℝ E✝ H✝
M✝ : Type u_3
inst✝²³ : TopologicalSpace M✝
inst✝²² : ChartedSpace H✝ M✝
inst✝²¹ : I✝.Boundaryless
inst✝²⁰ : Fin... |
https://github.com/fpvandoorn/sard.git | f4a1bb550136e8591dcd20dca5bdcce766a958c7 | Sard/MeasureZero.lean | MeasureZero.isMeagre_of_isSigmaCompact | [205, 1] | [216, 85] | obtain ⟨K, ⟨hcompact, hcover⟩⟩ := h₂s | E✝ : Type u_1
inst✝²⁷ : NormedAddCommGroup E✝
inst✝²⁶ : NormedSpace ℝ E✝
H✝ : Type u_2
inst✝²⁵ : TopologicalSpace H✝
I✝ : ModelWithCorners ℝ E✝ H✝
M✝ : Type u_3
inst✝²⁴ : TopologicalSpace M✝
inst✝²³ : ChartedSpace H✝ M✝
inst✝²² : I✝.Boundaryless
inst✝²¹ : FiniteDimensional ℝ E✝
inst✝²⁰ : MeasurableSpace E✝
inst✝¹⁹ : Bo... | case intro.intro
E✝ : Type u_1
inst✝²⁷ : NormedAddCommGroup E✝
inst✝²⁶ : NormedSpace ℝ E✝
H✝ : Type u_2
inst✝²⁵ : TopologicalSpace H✝
I✝ : ModelWithCorners ℝ E✝ H✝
M✝ : Type u_3
inst✝²⁴ : TopologicalSpace M✝
inst✝²³ : ChartedSpace H✝ M✝
inst✝²² : I✝.Boundaryless
inst✝²¹ : FiniteDimensional ℝ E✝
inst✝²⁰ : MeasurableSpac... | Please generate a tactic in lean4 to solve the state.
STATE:
E✝ : Type u_1
inst✝²⁷ : NormedAddCommGroup E✝
inst✝²⁶ : NormedSpace ℝ E✝
H✝ : Type u_2
inst✝²⁵ : TopologicalSpace H✝
I✝ : ModelWithCorners ℝ E✝ H✝
M✝ : Type u_3
inst✝²⁴ : TopologicalSpace M✝
inst✝²³ : ChartedSpace H✝ M✝
inst✝²² : I✝.Boundaryless
inst✝²¹ : Fin... |
https://github.com/fpvandoorn/sard.git | f4a1bb550136e8591dcd20dca5bdcce766a958c7 | Sard/MeasureZero.lean | MeasureZero.isMeagre_of_isSigmaCompact | [205, 1] | [216, 85] | suffices ∀ n : ℕ, IsNowhereDense (K n) by
rw [isMeagre_iff_countable_union_isNowhereDense, ← hcover]
simp [IsMeagre]
exact ⟨range K, fun t ⟨n, hn⟩ ↦ hn ▸ this n, countable_range K, fun i ↦ subset_iUnion K i⟩ | case intro.intro
E✝ : Type u_1
inst✝²⁷ : NormedAddCommGroup E✝
inst✝²⁶ : NormedSpace ℝ E✝
H✝ : Type u_2
inst✝²⁵ : TopologicalSpace H✝
I✝ : ModelWithCorners ℝ E✝ H✝
M✝ : Type u_3
inst✝²⁴ : TopologicalSpace M✝
inst✝²³ : ChartedSpace H✝ M✝
inst✝²² : I✝.Boundaryless
inst✝²¹ : FiniteDimensional ℝ E✝
inst✝²⁰ : MeasurableSpac... | case intro.intro
E✝ : Type u_1
inst✝²⁷ : NormedAddCommGroup E✝
inst✝²⁶ : NormedSpace ℝ E✝
H✝ : Type u_2
inst✝²⁵ : TopologicalSpace H✝
I✝ : ModelWithCorners ℝ E✝ H✝
M✝ : Type u_3
inst✝²⁴ : TopologicalSpace M✝
inst✝²³ : ChartedSpace H✝ M✝
inst✝²² : I✝.Boundaryless
inst✝²¹ : FiniteDimensional ℝ E✝
inst✝²⁰ : MeasurableSpac... | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro
E✝ : Type u_1
inst✝²⁷ : NormedAddCommGroup E✝
inst✝²⁶ : NormedSpace ℝ E✝
H✝ : Type u_2
inst✝²⁵ : TopologicalSpace H✝
I✝ : ModelWithCorners ℝ E✝ H✝
M✝ : Type u_3
inst✝²⁴ : TopologicalSpace M✝
inst✝²³ : ChartedSpace H✝ M✝
inst✝²² : I✝.Boundaryl... |
https://github.com/fpvandoorn/sard.git | f4a1bb550136e8591dcd20dca5bdcce766a958c7 | Sard/MeasureZero.lean | MeasureZero.isMeagre_of_isSigmaCompact | [205, 1] | [216, 85] | intro n | case intro.intro
E✝ : Type u_1
inst✝²⁷ : NormedAddCommGroup E✝
inst✝²⁶ : NormedSpace ℝ E✝
H✝ : Type u_2
inst✝²⁵ : TopologicalSpace H✝
I✝ : ModelWithCorners ℝ E✝ H✝
M✝ : Type u_3
inst✝²⁴ : TopologicalSpace M✝
inst✝²³ : ChartedSpace H✝ M✝
inst✝²² : I✝.Boundaryless
inst✝²¹ : FiniteDimensional ℝ E✝
inst✝²⁰ : MeasurableSpac... | case intro.intro
E✝ : Type u_1
inst✝²⁷ : NormedAddCommGroup E✝
inst✝²⁶ : NormedSpace ℝ E✝
H✝ : Type u_2
inst✝²⁵ : TopologicalSpace H✝
I✝ : ModelWithCorners ℝ E✝ H✝
M✝ : Type u_3
inst✝²⁴ : TopologicalSpace M✝
inst✝²³ : ChartedSpace H✝ M✝
inst✝²² : I✝.Boundaryless
inst✝²¹ : FiniteDimensional ℝ E✝
inst✝²⁰ : MeasurableSpac... | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro
E✝ : Type u_1
inst✝²⁷ : NormedAddCommGroup E✝
inst✝²⁶ : NormedSpace ℝ E✝
H✝ : Type u_2
inst✝²⁵ : TopologicalSpace H✝
I✝ : ModelWithCorners ℝ E✝ H✝
M✝ : Type u_3
inst✝²⁴ : TopologicalSpace M✝
inst✝²³ : ChartedSpace H✝ M✝
inst✝²² : I✝.Boundaryl... |
https://github.com/fpvandoorn/sard.git | f4a1bb550136e8591dcd20dca5bdcce766a958c7 | Sard/MeasureZero.lean | MeasureZero.isMeagre_of_isSigmaCompact | [205, 1] | [216, 85] | have h : MeasureZero J (K n) := h₁s.mono (hcover ▸ subset_iUnion K n) | case intro.intro
E✝ : Type u_1
inst✝²⁷ : NormedAddCommGroup E✝
inst✝²⁶ : NormedSpace ℝ E✝
H✝ : Type u_2
inst✝²⁵ : TopologicalSpace H✝
I✝ : ModelWithCorners ℝ E✝ H✝
M✝ : Type u_3
inst✝²⁴ : TopologicalSpace M✝
inst✝²³ : ChartedSpace H✝ M✝
inst✝²² : I✝.Boundaryless
inst✝²¹ : FiniteDimensional ℝ E✝
inst✝²⁰ : MeasurableSpac... | case intro.intro
E✝ : Type u_1
inst✝²⁷ : NormedAddCommGroup E✝
inst✝²⁶ : NormedSpace ℝ E✝
H✝ : Type u_2
inst✝²⁵ : TopologicalSpace H✝
I✝ : ModelWithCorners ℝ E✝ H✝
M✝ : Type u_3
inst✝²⁴ : TopologicalSpace M✝
inst✝²³ : ChartedSpace H✝ M✝
inst✝²² : I✝.Boundaryless
inst✝²¹ : FiniteDimensional ℝ E✝
inst✝²⁰ : MeasurableSpac... | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro
E✝ : Type u_1
inst✝²⁷ : NormedAddCommGroup E✝
inst✝²⁶ : NormedSpace ℝ E✝
H✝ : Type u_2
inst✝²⁵ : TopologicalSpace H✝
I✝ : ModelWithCorners ℝ E✝ H✝
M✝ : Type u_3
inst✝²⁴ : TopologicalSpace M✝
inst✝²³ : ChartedSpace H✝ M✝
inst✝²² : I✝.Boundaryl... |
https://github.com/fpvandoorn/sard.git | f4a1bb550136e8591dcd20dca5bdcce766a958c7 | Sard/MeasureZero.lean | MeasureZero.isMeagre_of_isSigmaCompact | [205, 1] | [216, 85] | exact MeasureZero.isNowhereDense_of_isClosed J h (IsCompact.isClosed (hcompact n)) | case intro.intro
E✝ : Type u_1
inst✝²⁷ : NormedAddCommGroup E✝
inst✝²⁶ : NormedSpace ℝ E✝
H✝ : Type u_2
inst✝²⁵ : TopologicalSpace H✝
I✝ : ModelWithCorners ℝ E✝ H✝
M✝ : Type u_3
inst✝²⁴ : TopologicalSpace M✝
inst✝²³ : ChartedSpace H✝ M✝
inst✝²² : I✝.Boundaryless
inst✝²¹ : FiniteDimensional ℝ E✝
inst✝²⁰ : MeasurableSpac... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro
E✝ : Type u_1
inst✝²⁷ : NormedAddCommGroup E✝
inst✝²⁶ : NormedSpace ℝ E✝
H✝ : Type u_2
inst✝²⁵ : TopologicalSpace H✝
I✝ : ModelWithCorners ℝ E✝ H✝
M✝ : Type u_3
inst✝²⁴ : TopologicalSpace M✝
inst✝²³ : ChartedSpace H✝ M✝
inst✝²² : I✝.Boundaryl... |
https://github.com/fpvandoorn/sard.git | f4a1bb550136e8591dcd20dca5bdcce766a958c7 | Sard/MeasureZero.lean | MeasureZero.isMeagre_of_isSigmaCompact | [205, 1] | [216, 85] | rw [isMeagre_iff_countable_union_isNowhereDense, ← hcover] | E✝ : Type u_1
inst✝²⁷ : NormedAddCommGroup E✝
inst✝²⁶ : NormedSpace ℝ E✝
H✝ : Type u_2
inst✝²⁵ : TopologicalSpace H✝
I✝ : ModelWithCorners ℝ E✝ H✝
M✝ : Type u_3
inst✝²⁴ : TopologicalSpace M✝
inst✝²³ : ChartedSpace H✝ M✝
inst✝²² : I✝.Boundaryless
inst✝²¹ : FiniteDimensional ℝ E✝
inst✝²⁰ : MeasurableSpace E✝
inst✝¹⁹ : Bo... | E✝ : Type u_1
inst✝²⁷ : NormedAddCommGroup E✝
inst✝²⁶ : NormedSpace ℝ E✝
H✝ : Type u_2
inst✝²⁵ : TopologicalSpace H✝
I✝ : ModelWithCorners ℝ E✝ H✝
M✝ : Type u_3
inst✝²⁴ : TopologicalSpace M✝
inst✝²³ : ChartedSpace H✝ M✝
inst✝²² : I✝.Boundaryless
inst✝²¹ : FiniteDimensional ℝ E✝
inst✝²⁰ : MeasurableSpace E✝
inst✝¹⁹ : Bo... | Please generate a tactic in lean4 to solve the state.
STATE:
E✝ : Type u_1
inst✝²⁷ : NormedAddCommGroup E✝
inst✝²⁶ : NormedSpace ℝ E✝
H✝ : Type u_2
inst✝²⁵ : TopologicalSpace H✝
I✝ : ModelWithCorners ℝ E✝ H✝
M✝ : Type u_3
inst✝²⁴ : TopologicalSpace M✝
inst✝²³ : ChartedSpace H✝ M✝
inst✝²² : I✝.Boundaryless
inst✝²¹ : Fin... |
https://github.com/fpvandoorn/sard.git | f4a1bb550136e8591dcd20dca5bdcce766a958c7 | Sard/MeasureZero.lean | MeasureZero.isMeagre_of_isSigmaCompact | [205, 1] | [216, 85] | simp [IsMeagre] | E✝ : Type u_1
inst✝²⁷ : NormedAddCommGroup E✝
inst✝²⁶ : NormedSpace ℝ E✝
H✝ : Type u_2
inst✝²⁵ : TopologicalSpace H✝
I✝ : ModelWithCorners ℝ E✝ H✝
M✝ : Type u_3
inst✝²⁴ : TopologicalSpace M✝
inst✝²³ : ChartedSpace H✝ M✝
inst✝²² : I✝.Boundaryless
inst✝²¹ : FiniteDimensional ℝ E✝
inst✝²⁰ : MeasurableSpace E✝
inst✝¹⁹ : Bo... | E✝ : Type u_1
inst✝²⁷ : NormedAddCommGroup E✝
inst✝²⁶ : NormedSpace ℝ E✝
H✝ : Type u_2
inst✝²⁵ : TopologicalSpace H✝
I✝ : ModelWithCorners ℝ E✝ H✝
M✝ : Type u_3
inst✝²⁴ : TopologicalSpace M✝
inst✝²³ : ChartedSpace H✝ M✝
inst✝²² : I✝.Boundaryless
inst✝²¹ : FiniteDimensional ℝ E✝
inst✝²⁰ : MeasurableSpace E✝
inst✝¹⁹ : Bo... | Please generate a tactic in lean4 to solve the state.
STATE:
E✝ : Type u_1
inst✝²⁷ : NormedAddCommGroup E✝
inst✝²⁶ : NormedSpace ℝ E✝
H✝ : Type u_2
inst✝²⁵ : TopologicalSpace H✝
I✝ : ModelWithCorners ℝ E✝ H✝
M✝ : Type u_3
inst✝²⁴ : TopologicalSpace M✝
inst✝²³ : ChartedSpace H✝ M✝
inst✝²² : I✝.Boundaryless
inst✝²¹ : Fin... |
https://github.com/fpvandoorn/sard.git | f4a1bb550136e8591dcd20dca5bdcce766a958c7 | Sard/MeasureZero.lean | MeasureZero.isMeagre_of_isSigmaCompact | [205, 1] | [216, 85] | exact ⟨range K, fun t ⟨n, hn⟩ ↦ hn ▸ this n, countable_range K, fun i ↦ subset_iUnion K i⟩ | E✝ : Type u_1
inst✝²⁷ : NormedAddCommGroup E✝
inst✝²⁶ : NormedSpace ℝ E✝
H✝ : Type u_2
inst✝²⁵ : TopologicalSpace H✝
I✝ : ModelWithCorners ℝ E✝ H✝
M✝ : Type u_3
inst✝²⁴ : TopologicalSpace M✝
inst✝²³ : ChartedSpace H✝ M✝
inst✝²² : I✝.Boundaryless
inst✝²¹ : FiniteDimensional ℝ E✝
inst✝²⁰ : MeasurableSpace E✝
inst✝¹⁹ : Bo... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
E✝ : Type u_1
inst✝²⁷ : NormedAddCommGroup E✝
inst✝²⁶ : NormedSpace ℝ E✝
H✝ : Type u_2
inst✝²⁵ : TopologicalSpace H✝
I✝ : ModelWithCorners ℝ E✝ H✝
M✝ : Type u_3
inst✝²⁴ : TopologicalSpace M✝
inst✝²³ : ChartedSpace H✝ M✝
inst✝²² : I✝.Boundaryless
inst✝²¹ : Fin... |
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 ... |
https://github.com/fpvandoorn/sard.git | f4a1bb550136e8591dcd20dca5bdcce766a958c7 | Sard/LocallyLipschitz.lean | LipschitzOnWith.sum | [80, 1] | [96, 41] | ring | 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'
⊢ ↑Kf * ... | 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 | LocallyLipschitz.sum | [103, 1] | [112, 48] | intro x | 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
hf : LocallyLipschitz f
hg : LocallyLipschitz g
⊢ LocallyLipschitz (f + g) | 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
hf : LocallyLipschitz f
hg : LocallyLipschitz g
x : X
⊢ ∃ K, ∃ t ∈ 𝓝 x, LipschitzOnWith K (f + g) 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 : NormedAddCommGroup Y
inst✝ : NormedSpace ℝ Y
f g : X → Y
hf : LocallyLipschitz f
hg : LocallyLipschitz g
⊢ LocallyLipschitz (f + g)
TACTIC:
|
https://github.com/fpvandoorn/sard.git | f4a1bb550136e8591dcd20dca5bdcce766a958c7 | Sard/LocallyLipschitz.lean | LocallyLipschitz.sum | [103, 1] | [112, 48] | rcases hf x with ⟨Kf, t₁, h₁t, hfL⟩ | 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
hf : LocallyLipschitz f
hg : LocallyLipschitz g
x : X
⊢ ∃ K, ∃ t ∈ 𝓝 x, LipschitzOnWith K (f + g) t | case intro.intro.intro
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
hf : LocallyLipschitz f
hg : LocallyLipschitz g
x : X
Kf : ℝ≥0
t₁ : Set X
h₁t : t₁ ∈ 𝓝 x
hfL : LipschitzOnWith Kf f t₁
⊢ ∃ K, ∃ t ∈ 𝓝 x, LipschitzOnWith K (f + g) 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 : NormedAddCommGroup Y
inst✝ : NormedSpace ℝ Y
f g : X → Y
hf : LocallyLipschitz f
hg : LocallyLipschitz g
x : X
⊢ ∃ K, ∃ t ∈ 𝓝 x, LipschitzOnWith K (f + g) t
TACTIC:
|
https://github.com/fpvandoorn/sard.git | f4a1bb550136e8591dcd20dca5bdcce766a958c7 | Sard/LocallyLipschitz.lean | LocallyLipschitz.sum | [103, 1] | [112, 48] | rcases hg x with ⟨Kg, t₂, h₂t, hgL⟩ | case intro.intro.intro
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
hf : LocallyLipschitz f
hg : LocallyLipschitz g
x : X
Kf : ℝ≥0
t₁ : Set X
h₁t : t₁ ∈ 𝓝 x
hfL : LipschitzOnWith Kf f t₁
⊢ ∃ K, ∃ t ∈ 𝓝 x, LipschitzOnWith K (f + g) t | case intro.intro.intro.intro.intro.intro
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
hf : LocallyLipschitz f
hg : LocallyLipschitz g
x : X
Kf : ℝ≥0
t₁ : Set X
h₁t : t₁ ∈ 𝓝 x
hfL : LipschitzOnWith Kf f t₁
Kg : ℝ≥0
t₂ : Set X
h₂t : t₂ ∈ ... | 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 : NormedAddCommGroup Y
inst✝ : NormedSpace ℝ Y
f g : X → Y
hf : LocallyLipschitz f
hg : LocallyLipschitz g
x : X
Kf : ℝ≥0
t₁ : Set X
h₁t : t₁ ∈ 𝓝 x
hfL : LipschitzOnW... |
https://github.com/fpvandoorn/sard.git | f4a1bb550136e8591dcd20dca5bdcce766a958c7 | Sard/LocallyLipschitz.lean | LocallyLipschitz.sum | [103, 1] | [112, 48] | use Kf + Kg, t₁ ∩ t₂ | case intro.intro.intro.intro.intro.intro
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
hf : LocallyLipschitz f
hg : LocallyLipschitz g
x : X
Kf : ℝ≥0
t₁ : Set X
h₁t : t₁ ∈ 𝓝 x
hfL : LipschitzOnWith Kf f t₁
Kg : ℝ≥0
t₂ : Set X
h₂t : t₂ ∈ ... | case h
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
hf : LocallyLipschitz f
hg : LocallyLipschitz g
x : X
Kf : ℝ≥0
t₁ : Set X
h₁t : t₁ ∈ 𝓝 x
hfL : LipschitzOnWith Kf f t₁
Kg : ℝ≥0
t₂ : Set X
h₂t : t₂ ∈ 𝓝 x
hgL : LipschitzOnWith Kg g t₂... | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.intro.intro.intro.intro
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
hf : LocallyLipschitz f
hg : LocallyLipschitz g
x : X
Kf : ℝ≥0
t₁ : Set X
h₁t : t₁ ∈ 𝓝 x
... |
https://github.com/fpvandoorn/sard.git | f4a1bb550136e8591dcd20dca5bdcce766a958c7 | Sard/LocallyLipschitz.lean | LocallyLipschitz.sum | [103, 1] | [112, 48] | have hf' : LipschitzOnWith Kf f (t₁ ∩ t₂) := hfL.mono (Set.inter_subset_left t₁ t₂) | case h
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
hf : LocallyLipschitz f
hg : LocallyLipschitz g
x : X
Kf : ℝ≥0
t₁ : Set X
h₁t : t₁ ∈ 𝓝 x
hfL : LipschitzOnWith Kf f t₁
Kg : ℝ≥0
t₂ : Set X
h₂t : t₂ ∈ 𝓝 x
hgL : LipschitzOnWith Kg g t₂... | case h
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
hf : LocallyLipschitz f
hg : LocallyLipschitz g
x : X
Kf : ℝ≥0
t₁ : Set X
h₁t : t₁ ∈ 𝓝 x
hfL : LipschitzOnWith Kf f t₁
Kg : ℝ≥0
t₂ : Set X
h₂t : t₂ ∈ 𝓝 x
hgL : LipschitzOnWith Kg g t₂... | 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 : NormedAddCommGroup Y
inst✝ : NormedSpace ℝ Y
f g : X → Y
hf : LocallyLipschitz f
hg : LocallyLipschitz g
x : X
Kf : ℝ≥0
t₁ : Set X
h₁t : t₁ ∈ 𝓝 x
hfL : LipschitzOnWith Kf f t₁
Kg :... |
https://github.com/fpvandoorn/sard.git | f4a1bb550136e8591dcd20dca5bdcce766a958c7 | Sard/LocallyLipschitz.lean | LocallyLipschitz.sum | [103, 1] | [112, 48] | have hg' : LipschitzOnWith Kg g (t₁ ∩ t₂) := hgL.mono (Set.inter_subset_right t₁ t₂) | case h
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
hf : LocallyLipschitz f
hg : LocallyLipschitz g
x : X
Kf : ℝ≥0
t₁ : Set X
h₁t : t₁ ∈ 𝓝 x
hfL : LipschitzOnWith Kf f t₁
Kg : ℝ≥0
t₂ : Set X
h₂t : t₂ ∈ 𝓝 x
hgL : LipschitzOnWith Kg g t₂... | case h
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
hf : LocallyLipschitz f
hg : LocallyLipschitz g
x : X
Kf : ℝ≥0
t₁ : Set X
h₁t : t₁ ∈ 𝓝 x
hfL : LipschitzOnWith Kf f t₁
Kg : ℝ≥0
t₂ : Set X
h₂t : t₂ ∈ 𝓝 x
hgL : LipschitzOnWith Kg g t₂... | 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 : NormedAddCommGroup Y
inst✝ : NormedSpace ℝ Y
f g : X → Y
hf : LocallyLipschitz f
hg : LocallyLipschitz g
x : X
Kf : ℝ≥0
t₁ : Set X
h₁t : t₁ ∈ 𝓝 x
hfL : LipschitzOnWith Kf f t₁
Kg :... |
https://github.com/fpvandoorn/sard.git | f4a1bb550136e8591dcd20dca5bdcce766a958c7 | Sard/LocallyLipschitz.lean | LocallyLipschitz.sum | [103, 1] | [112, 48] | exact ⟨Filter.inter_mem h₁t h₂t, hf'.sum hg'⟩ | case h
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
hf : LocallyLipschitz f
hg : LocallyLipschitz g
x : X
Kf : ℝ≥0
t₁ : Set X
h₁t : t₁ ∈ 𝓝 x
hfL : LipschitzOnWith Kf f t₁
Kg : ℝ≥0
t₂ : Set X
h₂t : t₂ ∈ 𝓝 x
hgL : LipschitzOnWith Kg g t₂... | no goals | 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 : NormedAddCommGroup Y
inst✝ : NormedSpace ℝ Y
f g : X → Y
hf : LocallyLipschitz f
hg : LocallyLipschitz g
x : X
Kf : ℝ≥0
t₁ : Set X
h₁t : t₁ ∈ 𝓝 x
hfL : LipschitzOnWith Kf f t₁
Kg :... |
https://github.com/fpvandoorn/sard.git | f4a1bb550136e8591dcd20dca5bdcce766a958c7 | Sard/LocallyLipschitz.lean | helper | [115, 1] | [115, 98] | sorry | 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
a b : ℝ
⊢ ENNReal.ofReal (a * b) = ENNReal.ofReal a * ENNReal.ofReal b | 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
a b : ℝ
⊢ ENNReal.ofReal (a * b) = ENNReal.ofReal a * ENNReal.ofReal b
TACTIC:
|
https://github.com/fpvandoorn/sard.git | f4a1bb550136e8591dcd20dca5bdcce766a958c7 | Sard/LocallyLipschitz.lean | last' | [119, 1] | [120, 24] | exact mul_assoc a K c | 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
a K c : ℝ≥0
⊢ a * K * c = a * (K * c) | 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
a K c : ℝ≥0
⊢ a * K * c = a * (K * c)
TACTIC:
|
https://github.com/fpvandoorn/sard.git | f4a1bb550136e8591dcd20dca5bdcce766a958c7 | Sard/LocallyLipschitz.lean | last'' | [122, 1] | [124, 24] | exact mul_assoc _ K c | 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
a : ℝ
K c : ℝ≥0
⊢ ‖a‖.toNNReal * K * c = ‖a‖.toNNReal * (K * c) | 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
a : ℝ
K c : ℝ≥0
⊢ ‖a‖.toNNReal * K * c = ‖a‖.toNNReal * (K * c)
TACTIC:
|
https://github.com/fpvandoorn/sard.git | f4a1bb550136e8591dcd20dca5bdcce766a958c7 | Sard/LocallyLipschitz.lean | last | [126, 1] | [132, 8] | have : c ≠ ∞ := by sorry | 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
a : ℝ
K : ℝ≥0
c : ℝ≥0∞
⊢ ↑‖a‖.toNNReal * (↑K * c) = ↑(‖a‖.toNNReal * K) * c | 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
a : ℝ
K : ℝ≥0
c : ℝ≥0∞
this : c ≠ ⊤
⊢ ↑‖a‖.toNNReal * (↑K * c) = ↑(‖a‖.toNNReal * K) * c | 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
a : ℝ
K : ℝ≥0
c : ℝ≥0∞
⊢ ↑‖a‖.toNNReal * (↑K * c) = ↑(‖a‖.toNNReal * K) * c
TACTIC:
|
https://github.com/fpvandoorn/sard.git | f4a1bb550136e8591dcd20dca5bdcce766a958c7 | Sard/LocallyLipschitz.lean | last | [126, 1] | [132, 8] | lift c to ℝ≥0 using this | 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
a : ℝ
K : ℝ≥0
c : ℝ≥0∞
this : c ≠ ⊤
⊢ ↑‖a‖.toNNReal * (↑K * c) = ↑(‖a‖.toNNReal * K) * c | case intro
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
a : ℝ
K c : ℝ≥0
⊢ ↑‖a‖.toNNReal * (↑K * ↑c) = ↑(‖a‖.toNNReal * K) * ↑c | 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
a : ℝ
K : ℝ≥0
c : ℝ≥0∞
this : c ≠ ⊤
⊢ ↑‖a‖.toNNReal * (↑K * c) = ↑(‖a‖.toNNReal * K) * c
TACTIC:
|
https://github.com/fpvandoorn/sard.git | f4a1bb550136e8591dcd20dca5bdcce766a958c7 | Sard/LocallyLipschitz.lean | last | [126, 1] | [132, 8] | sorry | case intro
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
a : ℝ
K c : ℝ≥0
⊢ ↑‖a‖.toNNReal * (↑K * ↑c) = ↑(‖a‖.toNNReal * K) * ↑c | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case intro
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
a : ℝ
K c : ℝ≥0
⊢ ↑‖a‖.toNNReal * (↑K * ↑c) = ↑(‖a‖.toNNReal * K) * ↑c
TACTIC:
|
https://github.com/fpvandoorn/sard.git | f4a1bb550136e8591dcd20dca5bdcce766a958c7 | Sard/LocallyLipschitz.lean | last | [126, 1] | [132, 8] | sorry | 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
a : ℝ
K : ℝ≥0
c : ℝ≥0∞
⊢ c ≠ ⊤ | 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
a : ℝ
K : ℝ≥0
c : ℝ≥0∞
⊢ c ≠ ⊤
TACTIC:
|
https://github.com/fpvandoorn/sard.git | f4a1bb550136e8591dcd20dca5bdcce766a958c7 | Sard/LocallyLipschitz.lean | LipschitzOnWith.smul | [134, 1] | [150, 80] | intro x hx y hy | 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
K : ℝ≥0
s : Set X
hf : LipschitzOnWith K f s
a : ℝ
⊢ LipschitzOnWith ((ENNReal.ofReal ‖a‖).toNNReal * K) (fun x => a • f x) 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
K : ℝ≥0
s : Set X
hf : LipschitzOnWith K f s
a : ℝ
x : X
hx : x ∈ s
y : X
hy : y ∈ s
⊢ edist ((fun x => a • f x) x) ((fun x => a • f x) y) ≤ ↑((ENNReal.ofReal ‖a‖).toNNReal * K) * edist x y | 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
K : ℝ≥0
s : Set X
hf : LipschitzOnWith K f s
a : ℝ
⊢ LipschitzOnWith ((ENNReal.ofReal ‖a‖).toNNReal * K) (fun x => a • f x) s
TACTI... |
https://github.com/fpvandoorn/sard.git | f4a1bb550136e8591dcd20dca5bdcce766a958c7 | Sard/LocallyLipschitz.lean | LipschitzOnWith.smul | [134, 1] | [150, 80] | have : dist (a • f x) (a • f y) = ‖a‖ * dist (f x) (f y) := by
calc dist (a • f x) (a • f y)
_ = ‖(a • (f x)) - (a • (f y))‖ := by apply dist_eq_norm
_ = ‖a • ((f x) - (f y))‖ := by rw [smul_sub]
_ = ‖a‖ * ‖(f x) - (f y)‖ := by rw [norm_smul]
_ = ‖a‖ * dist (f x) (f y) := by rw [← dist_eq_no... | 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
K : ℝ≥0
s : Set X
hf : LipschitzOnWith K f s
a : ℝ
x : X
hx : x ∈ s
y : X
hy : y ∈ s
⊢ edist ((fun x => a • f x) x) ((fun x => a • f x) y) ≤ ↑((ENNReal.ofReal ‖a‖).toNNReal * K) * edist x y | 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
K : ℝ≥0
s : Set X
hf : LipschitzOnWith K f s
a : ℝ
x : X
hx : x ∈ s
y : X
hy : y ∈ s
this : dist (a • f x) (a • f y) = ‖a‖ * dist (f x) (f y)
⊢ edist ((fun x => a • f x) x) ((fun x => a • f 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 : NormedAddCommGroup Y
inst✝ : NormedSpace ℝ Y
f g : X → Y
K : ℝ≥0
s : Set X
hf : LipschitzOnWith K f s
a : ℝ
x : X
hx : x ∈ s
y : X
hy : y ∈ s
⊢ edist ((fun x => a • f x) x) ((fun x => a • ... |
https://github.com/fpvandoorn/sard.git | f4a1bb550136e8591dcd20dca5bdcce766a958c7 | Sard/LocallyLipschitz.lean | LipschitzOnWith.smul | [134, 1] | [150, 80] | calc edist (a • f x) (a • f y)
_ = ENNReal.ofReal (dist (a • f x) (a • f y)) := by rw [edist_dist]
_ = ENNReal.ofReal (‖a‖ * dist (f x) (f y)) := by rw [this]
_ = ENNReal.ofReal (‖a‖) * ENNReal.ofReal (dist (f x) (f y)) := by rw [← helper]
_ = ENNReal.ofReal ‖a‖ * edist (f x) (f y) := by rw [edist_dist]... | 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
K : ℝ≥0
s : Set X
hf : LipschitzOnWith K f s
a : ℝ
x : X
hx : x ∈ s
y : X
hy : y ∈ s
this : dist (a • f x) (a • f y) = ‖a‖ * dist (f x) (f y)
⊢ edist ((fun x => a • f x) x) ((fun x => a • f 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 : NormedAddCommGroup Y
inst✝ : NormedSpace ℝ Y
f g : X → Y
K : ℝ≥0
s : Set X
hf : LipschitzOnWith K f s
a : ℝ
x : X
hx : x ∈ s
y : X
hy : y ∈ s
this : dist (a • f x) (a • f y) = ‖a‖ * dist (... |
https://github.com/fpvandoorn/sard.git | f4a1bb550136e8591dcd20dca5bdcce766a958c7 | Sard/LocallyLipschitz.lean | LipschitzOnWith.smul | [134, 1] | [150, 80] | calc dist (a • f x) (a • f y)
_ = ‖(a • (f x)) - (a • (f y))‖ := by apply dist_eq_norm
_ = ‖a • ((f x) - (f y))‖ := by rw [smul_sub]
_ = ‖a‖ * ‖(f x) - (f y)‖ := by rw [norm_smul]
_ = ‖a‖ * dist (f x) (f y) := by rw [← dist_eq_norm]
_ = ‖a‖ * dist (f x) (f y) := by rw [← dist_smul₀] | 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
K : ℝ≥0
s : Set X
hf : LipschitzOnWith K f s
a : ℝ
x : X
hx : x ∈ s
y : X
hy : y ∈ s
⊢ dist (a • f x) (a • f y) = ‖a‖ * dist (f x) (f y) | 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
K : ℝ≥0
s : Set X
hf : LipschitzOnWith K f s
a : ℝ
x : X
hx : x ∈ s
y : X
hy : y ∈ s
⊢ dist (a • f x) (a • f y) = ‖a‖ * dist (f x) ... |
https://github.com/fpvandoorn/sard.git | f4a1bb550136e8591dcd20dca5bdcce766a958c7 | Sard/LocallyLipschitz.lean | LipschitzOnWith.smul | [134, 1] | [150, 80] | apply dist_eq_norm | 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
K : ℝ≥0
s : Set X
hf : LipschitzOnWith K f s
a : ℝ
x : X
hx : x ∈ s
y : X
hy : y ∈ s
⊢ dist (a • f x) (a • f y) = ‖a • f x - a • f y‖ | 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
K : ℝ≥0
s : Set X
hf : LipschitzOnWith K f s
a : ℝ
x : X
hx : x ∈ s
y : X
hy : y ∈ s
⊢ dist (a • f x) (a • f y) = ‖a • f x - a • f ... |
https://github.com/fpvandoorn/sard.git | f4a1bb550136e8591dcd20dca5bdcce766a958c7 | Sard/LocallyLipschitz.lean | LipschitzOnWith.smul | [134, 1] | [150, 80] | rw [smul_sub] | 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
K : ℝ≥0
s : Set X
hf : LipschitzOnWith K f s
a : ℝ
x : X
hx : x ∈ s
y : X
hy : y ∈ s
⊢ ‖a • f x - a • f y‖ = ‖a • (f x - f y)‖ | 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
K : ℝ≥0
s : Set X
hf : LipschitzOnWith K f s
a : ℝ
x : X
hx : x ∈ s
y : X
hy : y ∈ s
⊢ ‖a • f x - a • f y‖ = ‖a • (f x - f y)‖
TACT... |
https://github.com/fpvandoorn/sard.git | f4a1bb550136e8591dcd20dca5bdcce766a958c7 | Sard/LocallyLipschitz.lean | LipschitzOnWith.smul | [134, 1] | [150, 80] | rw [norm_smul] | 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
K : ℝ≥0
s : Set X
hf : LipschitzOnWith K f s
a : ℝ
x : X
hx : x ∈ s
y : X
hy : y ∈ s
⊢ ‖a • (f x - f y)‖ = ‖a‖ * ‖f x - f y‖ | 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
K : ℝ≥0
s : Set X
hf : LipschitzOnWith K f s
a : ℝ
x : X
hx : x ∈ s
y : X
hy : y ∈ s
⊢ ‖a • (f x - f y)‖ = ‖a‖ * ‖f x - f y‖
TACTIC... |
https://github.com/fpvandoorn/sard.git | f4a1bb550136e8591dcd20dca5bdcce766a958c7 | Sard/LocallyLipschitz.lean | LipschitzOnWith.smul | [134, 1] | [150, 80] | rw [← dist_eq_norm] | 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
K : ℝ≥0
s : Set X
hf : LipschitzOnWith K f s
a : ℝ
x : X
hx : x ∈ s
y : X
hy : y ∈ s
⊢ ‖a‖ * ‖f x - f y‖ = ‖a‖ * dist (f x) (f y) | 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
K : ℝ≥0
s : Set X
hf : LipschitzOnWith K f s
a : ℝ
x : X
hx : x ∈ s
y : X
hy : y ∈ s
⊢ ‖a‖ * ‖f x - f y‖ = ‖a‖ * dist (f x) (f y)
T... |
https://github.com/fpvandoorn/sard.git | f4a1bb550136e8591dcd20dca5bdcce766a958c7 | Sard/LocallyLipschitz.lean | LipschitzOnWith.smul | [134, 1] | [150, 80] | rw [← dist_smul₀] | 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
K : ℝ≥0
s : Set X
hf : LipschitzOnWith K f s
a : ℝ
x : X
hx : x ∈ s
y : X
hy : y ∈ s
⊢ ‖a‖ * dist (f x) (f y) = ‖a‖ * dist (f x) (f y) | 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
K : ℝ≥0
s : Set X
hf : LipschitzOnWith K f s
a : ℝ
x : X
hx : x ∈ s
y : X
hy : y ∈ s
⊢ ‖a‖ * dist (f x) (f y) = ‖a‖ * dist (f x) (f... |
https://github.com/fpvandoorn/sard.git | f4a1bb550136e8591dcd20dca5bdcce766a958c7 | Sard/LocallyLipschitz.lean | LipschitzOnWith.smul | [134, 1] | [150, 80] | rw [edist_dist] | 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
K : ℝ≥0
s : Set X
hf : LipschitzOnWith K f s
a : ℝ
x : X
hx : x ∈ s
y : X
hy : y ∈ s
this : dist (a • f x) (a • f y) = ‖a‖ * dist (f x) (f y)
⊢ edist (a • f x) (a • f y) = ENNReal.ofReal (dist... | 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
K : ℝ≥0
s : Set X
hf : LipschitzOnWith K f s
a : ℝ
x : X
hx : x ∈ s
y : X
hy : y ∈ s
this : dist (a • f x) (a • f y) = ‖a‖ * dist (... |
https://github.com/fpvandoorn/sard.git | f4a1bb550136e8591dcd20dca5bdcce766a958c7 | Sard/LocallyLipschitz.lean | LipschitzOnWith.smul | [134, 1] | [150, 80] | rw [this] | 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
K : ℝ≥0
s : Set X
hf : LipschitzOnWith K f s
a : ℝ
x : X
hx : x ∈ s
y : X
hy : y ∈ s
this : dist (a • f x) (a • f y) = ‖a‖ * dist (f x) (f y)
⊢ ENNReal.ofReal (dist (a • f x) (a • f y)) = ENNR... | 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
K : ℝ≥0
s : Set X
hf : LipschitzOnWith K f s
a : ℝ
x : X
hx : x ∈ s
y : X
hy : y ∈ s
this : dist (a • f x) (a • f y) = ‖a‖ * dist (... |
https://github.com/fpvandoorn/sard.git | f4a1bb550136e8591dcd20dca5bdcce766a958c7 | Sard/LocallyLipschitz.lean | LipschitzOnWith.smul | [134, 1] | [150, 80] | rw [← helper] | 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
K : ℝ≥0
s : Set X
hf : LipschitzOnWith K f s
a : ℝ
x : X
hx : x ∈ s
y : X
hy : y ∈ s
this : dist (a • f x) (a • f y) = ‖a‖ * dist (f x) (f y)
⊢ ENNReal.ofReal (‖a‖ * dist (f x) (f y)) = ENNRea... | 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
K : ℝ≥0
s : Set X
hf : LipschitzOnWith K f s
a : ℝ
x : X
hx : x ∈ s
y : X
hy : y ∈ s
this : dist (a • f x) (a • f y) = ‖a‖ * dist (... |
https://github.com/fpvandoorn/sard.git | f4a1bb550136e8591dcd20dca5bdcce766a958c7 | Sard/LocallyLipschitz.lean | LipschitzOnWith.smul | [134, 1] | [150, 80] | rw [edist_dist] | 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
K : ℝ≥0
s : Set X
hf : LipschitzOnWith K f s
a : ℝ
x : X
hx : x ∈ s
y : X
hy : y ∈ s
this : dist (a • f x) (a • f y) = ‖a‖ * dist (f x) (f y)
⊢ ENNReal.ofReal ‖a‖ * ENNReal.ofReal (dist (f 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 : NormedAddCommGroup Y
inst✝ : NormedSpace ℝ Y
f g : X → Y
K : ℝ≥0
s : Set X
hf : LipschitzOnWith K f s
a : ℝ
x : X
hx : x ∈ s
y : X
hy : y ∈ s
this : dist (a • f x) (a • f y) = ‖a‖ * dist (... |
https://github.com/fpvandoorn/sard.git | f4a1bb550136e8591dcd20dca5bdcce766a958c7 | Sard/LocallyLipschitz.lean | LipschitzOnWith.smul | [134, 1] | [150, 80] | exact mul_le_mul_left' (hf hx hy) _ | 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
K : ℝ≥0
s : Set X
hf : LipschitzOnWith K f s
a : ℝ
x : X
hx : x ∈ s
y : X
hy : y ∈ s
this : dist (a • f x) (a • f y) = ‖a‖ * dist (f x) (f y)
⊢ ENNReal.ofReal ‖a‖ * edist (f x) (f y) ≤ ↑‖a‖.to... | 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
K : ℝ≥0
s : Set X
hf : LipschitzOnWith K f s
a : ℝ
x : X
hx : x ∈ s
y : X
hy : y ∈ s
this : dist (a • f x) (a • f y) = ‖a‖ * dist (... |
https://github.com/fpvandoorn/sard.git | f4a1bb550136e8591dcd20dca5bdcce766a958c7 | Sard/LocallyLipschitz.lean | LipschitzOnWith.smul | [134, 1] | [150, 80] | exact last a K (edist x y) | 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
K : ℝ≥0
s : Set X
hf : LipschitzOnWith K f s
a : ℝ
x : X
hx : x ∈ s
y : X
hy : y ∈ s
this : dist (a • f x) (a • f y) = ‖a‖ * dist (f x) (f y)
⊢ ↑‖a‖.toNNReal * (↑K * edist x y) = ↑(‖a‖.toNNRea... | 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
K : ℝ≥0
s : Set X
hf : LipschitzOnWith K f s
a : ℝ
x : X
hx : x ∈ s
y : X
hy : y ∈ s
this : dist (a • f x) (a • f y) = ‖a‖ * dist (... |
https://github.com/fpvandoorn/sard.git | f4a1bb550136e8591dcd20dca5bdcce766a958c7 | Sard/LocallyLipschitz.lean | LocallyLipschitz.mysmul | [156, 1] | [160, 72] | intro x | 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
hf : LocallyLipschitz f
a : ℝ
⊢ LocallyLipschitz fun x => a • f x | 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
hf : LocallyLipschitz f
a : ℝ
x : X
⊢ ∃ K, ∃ t ∈ 𝓝 x, LipschitzOnWith K (fun x => a • f x) 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 : NormedAddCommGroup Y
inst✝ : NormedSpace ℝ Y
f g : X → Y
hf : LocallyLipschitz f
a : ℝ
⊢ LocallyLipschitz fun x => a • f x
TACTIC:
|
https://github.com/fpvandoorn/sard.git | f4a1bb550136e8591dcd20dca5bdcce766a958c7 | Sard/LocallyLipschitz.lean | LocallyLipschitz.mysmul | [156, 1] | [160, 72] | rcases hf x with ⟨Kf, t, ht, hfL⟩ | 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
hf : LocallyLipschitz f
a : ℝ
x : X
⊢ ∃ K, ∃ t ∈ 𝓝 x, LipschitzOnWith K (fun x => a • f x) t | case intro.intro.intro
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
hf : LocallyLipschitz f
a : ℝ
x : X
Kf : ℝ≥0
t : Set X
ht : t ∈ 𝓝 x
hfL : LipschitzOnWith Kf f t
⊢ ∃ K, ∃ t ∈ 𝓝 x, LipschitzOnWith K (fun x => a • f x) 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 : NormedAddCommGroup Y
inst✝ : NormedSpace ℝ Y
f g : X → Y
hf : LocallyLipschitz f
a : ℝ
x : X
⊢ ∃ K, ∃ t ∈ 𝓝 x, LipschitzOnWith K (fun x => a • f x) t
TACTIC:
|
https://github.com/fpvandoorn/sard.git | f4a1bb550136e8591dcd20dca5bdcce766a958c7 | Sard/LocallyLipschitz.lean | LocallyLipschitz.mysmul | [156, 1] | [160, 72] | exact ⟨ENNReal.toNNReal (ENNReal.ofReal ‖a‖) * Kf, t, ht, hfL.smul a⟩ | case intro.intro.intro
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
hf : LocallyLipschitz f
a : ℝ
x : X
Kf : ℝ≥0
t : Set X
ht : t ∈ 𝓝 x
hfL : LipschitzOnWith Kf f t
⊢ ∃ K, ∃ t ∈ 𝓝 x, LipschitzOnWith K (fun x => a • f x) 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 : NormedAddCommGroup Y
inst✝ : NormedSpace ℝ Y
f g : X → Y
hf : LocallyLipschitz f
a : ℝ
x : X
Kf : ℝ≥0
t : Set X
ht : t ∈ 𝓝 x
hfL : LipschitzOnWith Kf f t
⊢ ∃ K, ∃ t... |
https://github.com/fpvandoorn/sard.git | f4a1bb550136e8591dcd20dca5bdcce766a958c7 | Sard/Stuff.lean | image_null_of_C1_of_null | [22, 1] | [50, 13] | have h₁ : μ = μH[m] := by
have aux : μH[m] = volume := by
rw [← Fintype.card_fin m]
exact hausdorffMeasure_pi_real (ι := Fin m)
sorry | E : Type u_1
inst✝²² : NormedAddCommGroup E
inst✝²¹ : NormedSpace ℝ E
H : Type u_2
inst✝²⁰ : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type u_3
inst✝¹⁹ : TopologicalSpace M
inst✝¹⁸ : ChartedSpace H M
inst✝¹⁷ : I.Boundaryless
inst✝¹⁶ : SmoothManifoldWithCorners I M
inst✝¹⁵ : FiniteDimensional ℝ E
inst✝¹⁴ : Secon... | E : Type u_1
inst✝²² : NormedAddCommGroup E
inst✝²¹ : NormedSpace ℝ E
H : Type u_2
inst✝²⁰ : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type u_3
inst✝¹⁹ : TopologicalSpace M
inst✝¹⁸ : ChartedSpace H M
inst✝¹⁷ : I.Boundaryless
inst✝¹⁶ : SmoothManifoldWithCorners I M
inst✝¹⁵ : FiniteDimensional ℝ E
inst✝¹⁴ : Secon... | Please generate a tactic in lean4 to solve the state.
STATE:
E : Type u_1
inst✝²² : NormedAddCommGroup E
inst✝²¹ : NormedSpace ℝ E
H : Type u_2
inst✝²⁰ : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type u_3
inst✝¹⁹ : TopologicalSpace M
inst✝¹⁸ : ChartedSpace H M
inst✝¹⁷ : I.Boundaryless
inst✝¹⁶ : SmoothManifoldWi... |
https://github.com/fpvandoorn/sard.git | f4a1bb550136e8591dcd20dca5bdcce766a958c7 | Sard/Stuff.lean | image_null_of_C1_of_null | [22, 1] | [50, 13] | have h₂ : ν = μH[n] := by sorry | E : Type u_1
inst✝²² : NormedAddCommGroup E
inst✝²¹ : NormedSpace ℝ E
H : Type u_2
inst✝²⁰ : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type u_3
inst✝¹⁹ : TopologicalSpace M
inst✝¹⁸ : ChartedSpace H M
inst✝¹⁷ : I.Boundaryless
inst✝¹⁶ : SmoothManifoldWithCorners I M
inst✝¹⁵ : FiniteDimensional ℝ E
inst✝¹⁴ : Secon... | E : Type u_1
inst✝²² : NormedAddCommGroup E
inst✝²¹ : NormedSpace ℝ E
H : Type u_2
inst✝²⁰ : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type u_3
inst✝¹⁹ : TopologicalSpace M
inst✝¹⁸ : ChartedSpace H M
inst✝¹⁷ : I.Boundaryless
inst✝¹⁶ : SmoothManifoldWithCorners I M
inst✝¹⁵ : FiniteDimensional ℝ E
inst✝¹⁴ : Secon... | Please generate a tactic in lean4 to solve the state.
STATE:
E : Type u_1
inst✝²² : NormedAddCommGroup E
inst✝²¹ : NormedSpace ℝ E
H : Type u_2
inst✝²⁰ : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type u_3
inst✝¹⁹ : TopologicalSpace M
inst✝¹⁸ : ChartedSpace H M
inst✝¹⁷ : I.Boundaryless
inst✝¹⁶ : SmoothManifoldWi... |
https://github.com/fpvandoorn/sard.git | f4a1bb550136e8591dcd20dca5bdcce766a958c7 | Sard/Stuff.lean | image_null_of_C1_of_null | [22, 1] | [50, 13] | rw [h₁] at h₂s | E : Type u_1
inst✝²² : NormedAddCommGroup E
inst✝²¹ : NormedSpace ℝ E
H : Type u_2
inst✝²⁰ : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type u_3
inst✝¹⁹ : TopologicalSpace M
inst✝¹⁸ : ChartedSpace H M
inst✝¹⁷ : I.Boundaryless
inst✝¹⁶ : SmoothManifoldWithCorners I M
inst✝¹⁵ : FiniteDimensional ℝ E
inst✝¹⁴ : Secon... | E : Type u_1
inst✝²² : NormedAddCommGroup E
inst✝²¹ : NormedSpace ℝ E
H : Type u_2
inst✝²⁰ : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type u_3
inst✝¹⁹ : TopologicalSpace M
inst✝¹⁸ : ChartedSpace H M
inst✝¹⁷ : I.Boundaryless
inst✝¹⁶ : SmoothManifoldWithCorners I M
inst✝¹⁵ : FiniteDimensional ℝ E
inst✝¹⁴ : Secon... | Please generate a tactic in lean4 to solve the state.
STATE:
E : Type u_1
inst✝²² : NormedAddCommGroup E
inst✝²¹ : NormedSpace ℝ E
H : Type u_2
inst✝²⁰ : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type u_3
inst✝¹⁹ : TopologicalSpace M
inst✝¹⁸ : ChartedSpace H M
inst✝¹⁷ : I.Boundaryless
inst✝¹⁶ : SmoothManifoldWi... |
https://github.com/fpvandoorn/sard.git | f4a1bb550136e8591dcd20dca5bdcce766a958c7 | Sard/Stuff.lean | image_null_of_C1_of_null | [22, 1] | [50, 13] | have : μH[m] (f '' s) = 0 := by
have scifi : Convex ℝ U := sorry
apply locally_lipschitz_image_of_null_set_is_null_set_open (of_C1_on_open hU scifi hf) h₁s h₂s | E : Type u_1
inst✝²² : NormedAddCommGroup E
inst✝²¹ : NormedSpace ℝ E
H : Type u_2
inst✝²⁰ : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type u_3
inst✝¹⁹ : TopologicalSpace M
inst✝¹⁸ : ChartedSpace H M
inst✝¹⁷ : I.Boundaryless
inst✝¹⁶ : SmoothManifoldWithCorners I M
inst✝¹⁵ : FiniteDimensional ℝ E
inst✝¹⁴ : Secon... | E : Type u_1
inst✝²² : NormedAddCommGroup E
inst✝²¹ : NormedSpace ℝ E
H : Type u_2
inst✝²⁰ : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type u_3
inst✝¹⁹ : TopologicalSpace M
inst✝¹⁸ : ChartedSpace H M
inst✝¹⁷ : I.Boundaryless
inst✝¹⁶ : SmoothManifoldWithCorners I M
inst✝¹⁵ : FiniteDimensional ℝ E
inst✝¹⁴ : Secon... | Please generate a tactic in lean4 to solve the state.
STATE:
E : Type u_1
inst✝²² : NormedAddCommGroup E
inst✝²¹ : NormedSpace ℝ E
H : Type u_2
inst✝²⁰ : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type u_3
inst✝¹⁹ : TopologicalSpace M
inst✝¹⁸ : ChartedSpace H M
inst✝¹⁷ : I.Boundaryless
inst✝¹⁶ : SmoothManifoldWi... |
https://github.com/fpvandoorn/sard.git | f4a1bb550136e8591dcd20dca5bdcce766a958c7 | Sard/Stuff.lean | image_null_of_C1_of_null | [22, 1] | [50, 13] | rw [h₂, ← hd] | E : Type u_1
inst✝²² : NormedAddCommGroup E
inst✝²¹ : NormedSpace ℝ E
H : Type u_2
inst✝²⁰ : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type u_3
inst✝¹⁹ : TopologicalSpace M
inst✝¹⁸ : ChartedSpace H M
inst✝¹⁷ : I.Boundaryless
inst✝¹⁶ : SmoothManifoldWithCorners I M
inst✝¹⁵ : FiniteDimensional ℝ E
inst✝¹⁴ : Secon... | E : Type u_1
inst✝²² : NormedAddCommGroup E
inst✝²¹ : NormedSpace ℝ E
H : Type u_2
inst✝²⁰ : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type u_3
inst✝¹⁹ : TopologicalSpace M
inst✝¹⁸ : ChartedSpace H M
inst✝¹⁷ : I.Boundaryless
inst✝¹⁶ : SmoothManifoldWithCorners I M
inst✝¹⁵ : FiniteDimensional ℝ E
inst✝¹⁴ : Secon... | Please generate a tactic in lean4 to solve the state.
STATE:
E : Type u_1
inst✝²² : NormedAddCommGroup E
inst✝²¹ : NormedSpace ℝ E
H : Type u_2
inst✝²⁰ : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type u_3
inst✝¹⁹ : TopologicalSpace M
inst✝¹⁸ : ChartedSpace H M
inst✝¹⁷ : I.Boundaryless
inst✝¹⁶ : SmoothManifoldWi... |
https://github.com/fpvandoorn/sard.git | f4a1bb550136e8591dcd20dca5bdcce766a958c7 | Sard/Stuff.lean | image_null_of_C1_of_null | [22, 1] | [50, 13] | exact this | E : Type u_1
inst✝²² : NormedAddCommGroup E
inst✝²¹ : NormedSpace ℝ E
H : Type u_2
inst✝²⁰ : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type u_3
inst✝¹⁹ : TopologicalSpace M
inst✝¹⁸ : ChartedSpace H M
inst✝¹⁷ : I.Boundaryless
inst✝¹⁶ : SmoothManifoldWithCorners I M
inst✝¹⁵ : FiniteDimensional ℝ E
inst✝¹⁴ : Secon... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
E : Type u_1
inst✝²² : NormedAddCommGroup E
inst✝²¹ : NormedSpace ℝ E
H : Type u_2
inst✝²⁰ : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type u_3
inst✝¹⁹ : TopologicalSpace M
inst✝¹⁸ : ChartedSpace H M
inst✝¹⁷ : I.Boundaryless
inst✝¹⁶ : SmoothManifoldWi... |
https://github.com/fpvandoorn/sard.git | f4a1bb550136e8591dcd20dca5bdcce766a958c7 | Sard/Stuff.lean | image_null_of_C1_of_null | [22, 1] | [50, 13] | have aux : μH[m] = volume := by
rw [← Fintype.card_fin m]
exact hausdorffMeasure_pi_real (ι := Fin m) | E : Type u_1
inst✝²² : NormedAddCommGroup E
inst✝²¹ : NormedSpace ℝ E
H : Type u_2
inst✝²⁰ : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type u_3
inst✝¹⁹ : TopologicalSpace M
inst✝¹⁸ : ChartedSpace H M
inst✝¹⁷ : I.Boundaryless
inst✝¹⁶ : SmoothManifoldWithCorners I M
inst✝¹⁵ : FiniteDimensional ℝ E
inst✝¹⁴ : Secon... | E : Type u_1
inst✝²² : NormedAddCommGroup E
inst✝²¹ : NormedSpace ℝ E
H : Type u_2
inst✝²⁰ : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type u_3
inst✝¹⁹ : TopologicalSpace M
inst✝¹⁸ : ChartedSpace H M
inst✝¹⁷ : I.Boundaryless
inst✝¹⁶ : SmoothManifoldWithCorners I M
inst✝¹⁵ : FiniteDimensional ℝ E
inst✝¹⁴ : Secon... | Please generate a tactic in lean4 to solve the state.
STATE:
E : Type u_1
inst✝²² : NormedAddCommGroup E
inst✝²¹ : NormedSpace ℝ E
H : Type u_2
inst✝²⁰ : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type u_3
inst✝¹⁹ : TopologicalSpace M
inst✝¹⁸ : ChartedSpace H M
inst✝¹⁷ : I.Boundaryless
inst✝¹⁶ : SmoothManifoldWi... |
https://github.com/fpvandoorn/sard.git | f4a1bb550136e8591dcd20dca5bdcce766a958c7 | Sard/Stuff.lean | image_null_of_C1_of_null | [22, 1] | [50, 13] | sorry | E : Type u_1
inst✝²² : NormedAddCommGroup E
inst✝²¹ : NormedSpace ℝ E
H : Type u_2
inst✝²⁰ : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type u_3
inst✝¹⁹ : TopologicalSpace M
inst✝¹⁸ : ChartedSpace H M
inst✝¹⁷ : I.Boundaryless
inst✝¹⁶ : SmoothManifoldWithCorners I M
inst✝¹⁵ : FiniteDimensional ℝ E
inst✝¹⁴ : Secon... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
E : Type u_1
inst✝²² : NormedAddCommGroup E
inst✝²¹ : NormedSpace ℝ E
H : Type u_2
inst✝²⁰ : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type u_3
inst✝¹⁹ : TopologicalSpace M
inst✝¹⁸ : ChartedSpace H M
inst✝¹⁷ : I.Boundaryless
inst✝¹⁶ : SmoothManifoldWi... |
https://github.com/fpvandoorn/sard.git | f4a1bb550136e8591dcd20dca5bdcce766a958c7 | Sard/Stuff.lean | image_null_of_C1_of_null | [22, 1] | [50, 13] | rw [← Fintype.card_fin m] | E : Type u_1
inst✝²² : NormedAddCommGroup E
inst✝²¹ : NormedSpace ℝ E
H : Type u_2
inst✝²⁰ : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type u_3
inst✝¹⁹ : TopologicalSpace M
inst✝¹⁸ : ChartedSpace H M
inst✝¹⁷ : I.Boundaryless
inst✝¹⁶ : SmoothManifoldWithCorners I M
inst✝¹⁵ : FiniteDimensional ℝ E
inst✝¹⁴ : Secon... | E : Type u_1
inst✝²² : NormedAddCommGroup E
inst✝²¹ : NormedSpace ℝ E
H : Type u_2
inst✝²⁰ : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type u_3
inst✝¹⁹ : TopologicalSpace M
inst✝¹⁸ : ChartedSpace H M
inst✝¹⁷ : I.Boundaryless
inst✝¹⁶ : SmoothManifoldWithCorners I M
inst✝¹⁵ : FiniteDimensional ℝ E
inst✝¹⁴ : Secon... | Please generate a tactic in lean4 to solve the state.
STATE:
E : Type u_1
inst✝²² : NormedAddCommGroup E
inst✝²¹ : NormedSpace ℝ E
H : Type u_2
inst✝²⁰ : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type u_3
inst✝¹⁹ : TopologicalSpace M
inst✝¹⁸ : ChartedSpace H M
inst✝¹⁷ : I.Boundaryless
inst✝¹⁶ : SmoothManifoldWi... |
https://github.com/fpvandoorn/sard.git | f4a1bb550136e8591dcd20dca5bdcce766a958c7 | Sard/Stuff.lean | image_null_of_C1_of_null | [22, 1] | [50, 13] | exact hausdorffMeasure_pi_real (ι := Fin m) | E : Type u_1
inst✝²² : NormedAddCommGroup E
inst✝²¹ : NormedSpace ℝ E
H : Type u_2
inst✝²⁰ : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type u_3
inst✝¹⁹ : TopologicalSpace M
inst✝¹⁸ : ChartedSpace H M
inst✝¹⁷ : I.Boundaryless
inst✝¹⁶ : SmoothManifoldWithCorners I M
inst✝¹⁵ : FiniteDimensional ℝ E
inst✝¹⁴ : Secon... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
E : Type u_1
inst✝²² : NormedAddCommGroup E
inst✝²¹ : NormedSpace ℝ E
H : Type u_2
inst✝²⁰ : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type u_3
inst✝¹⁹ : TopologicalSpace M
inst✝¹⁸ : ChartedSpace H M
inst✝¹⁷ : I.Boundaryless
inst✝¹⁶ : SmoothManifoldWi... |
https://github.com/fpvandoorn/sard.git | f4a1bb550136e8591dcd20dca5bdcce766a958c7 | Sard/Stuff.lean | image_null_of_C1_of_null | [22, 1] | [50, 13] | sorry | E : Type u_1
inst✝²² : NormedAddCommGroup E
inst✝²¹ : NormedSpace ℝ E
H : Type u_2
inst✝²⁰ : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type u_3
inst✝¹⁹ : TopologicalSpace M
inst✝¹⁸ : ChartedSpace H M
inst✝¹⁷ : I.Boundaryless
inst✝¹⁶ : SmoothManifoldWithCorners I M
inst✝¹⁵ : FiniteDimensional ℝ E
inst✝¹⁴ : Secon... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
E : Type u_1
inst✝²² : NormedAddCommGroup E
inst✝²¹ : NormedSpace ℝ E
H : Type u_2
inst✝²⁰ : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type u_3
inst✝¹⁹ : TopologicalSpace M
inst✝¹⁸ : ChartedSpace H M
inst✝¹⁷ : I.Boundaryless
inst✝¹⁶ : SmoothManifoldWi... |
https://github.com/fpvandoorn/sard.git | f4a1bb550136e8591dcd20dca5bdcce766a958c7 | Sard/Stuff.lean | image_null_of_C1_of_null | [22, 1] | [50, 13] | have scifi : Convex ℝ U := sorry | E : Type u_1
inst✝²² : NormedAddCommGroup E
inst✝²¹ : NormedSpace ℝ E
H : Type u_2
inst✝²⁰ : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type u_3
inst✝¹⁹ : TopologicalSpace M
inst✝¹⁸ : ChartedSpace H M
inst✝¹⁷ : I.Boundaryless
inst✝¹⁶ : SmoothManifoldWithCorners I M
inst✝¹⁵ : FiniteDimensional ℝ E
inst✝¹⁴ : Secon... | E : Type u_1
inst✝²² : NormedAddCommGroup E
inst✝²¹ : NormedSpace ℝ E
H : Type u_2
inst✝²⁰ : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type u_3
inst✝¹⁹ : TopologicalSpace M
inst✝¹⁸ : ChartedSpace H M
inst✝¹⁷ : I.Boundaryless
inst✝¹⁶ : SmoothManifoldWithCorners I M
inst✝¹⁵ : FiniteDimensional ℝ E
inst✝¹⁴ : Secon... | Please generate a tactic in lean4 to solve the state.
STATE:
E : Type u_1
inst✝²² : NormedAddCommGroup E
inst✝²¹ : NormedSpace ℝ E
H : Type u_2
inst✝²⁰ : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type u_3
inst✝¹⁹ : TopologicalSpace M
inst✝¹⁸ : ChartedSpace H M
inst✝¹⁷ : I.Boundaryless
inst✝¹⁶ : SmoothManifoldWi... |
https://github.com/fpvandoorn/sard.git | f4a1bb550136e8591dcd20dca5bdcce766a958c7 | Sard/Stuff.lean | image_null_of_C1_of_null | [22, 1] | [50, 13] | apply locally_lipschitz_image_of_null_set_is_null_set_open (of_C1_on_open hU scifi hf) h₁s h₂s | E : Type u_1
inst✝²² : NormedAddCommGroup E
inst✝²¹ : NormedSpace ℝ E
H : Type u_2
inst✝²⁰ : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type u_3
inst✝¹⁹ : TopologicalSpace M
inst✝¹⁸ : ChartedSpace H M
inst✝¹⁷ : I.Boundaryless
inst✝¹⁶ : SmoothManifoldWithCorners I M
inst✝¹⁵ : FiniteDimensional ℝ E
inst✝¹⁴ : Secon... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
E : Type u_1
inst✝²² : NormedAddCommGroup E
inst✝²¹ : NormedSpace ℝ E
H : Type u_2
inst✝²⁰ : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type u_3
inst✝¹⁹ : TopologicalSpace M
inst✝¹⁸ : ChartedSpace H M
inst✝¹⁷ : I.Boundaryless
inst✝¹⁶ : SmoothManifoldWi... |
https://github.com/fpvandoorn/sard.git | f4a1bb550136e8591dcd20dca5bdcce766a958c7 | Sard/Stuff.lean | image_null_of_C1_of_null' | [52, 1] | [57, 77] | let hdiff := Iff.mpr contDiffOn_univ hf | E : Type u_1
inst✝²² : NormedAddCommGroup E
inst✝²¹ : NormedSpace ℝ E
H : Type u_2
inst✝²⁰ : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type u_3
inst✝¹⁹ : TopologicalSpace M
inst✝¹⁸ : ChartedSpace H M
inst✝¹⁷ : I.Boundaryless
inst✝¹⁶ : SmoothManifoldWithCorners I M
inst✝¹⁵ : FiniteDimensional ℝ E
inst✝¹⁴ : Secon... | E : Type u_1
inst✝²² : NormedAddCommGroup E
inst✝²¹ : NormedSpace ℝ E
H : Type u_2
inst✝²⁰ : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type u_3
inst✝¹⁹ : TopologicalSpace M
inst✝¹⁸ : ChartedSpace H M
inst✝¹⁷ : I.Boundaryless
inst✝¹⁶ : SmoothManifoldWithCorners I M
inst✝¹⁵ : FiniteDimensional ℝ E
inst✝¹⁴ : Secon... | Please generate a tactic in lean4 to solve the state.
STATE:
E : Type u_1
inst✝²² : NormedAddCommGroup E
inst✝²¹ : NormedSpace ℝ E
H : Type u_2
inst✝²⁰ : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type u_3
inst✝¹⁹ : TopologicalSpace M
inst✝¹⁸ : ChartedSpace H M
inst✝¹⁷ : I.Boundaryless
inst✝¹⁶ : SmoothManifoldWi... |
https://github.com/fpvandoorn/sard.git | f4a1bb550136e8591dcd20dca5bdcce766a958c7 | Sard/Stuff.lean | image_null_of_C1_of_null' | [52, 1] | [57, 77] | apply image_null_of_C1_of_null isOpen_univ hdiff μ ν hd (subset_univ s) hs | E : Type u_1
inst✝²² : NormedAddCommGroup E
inst✝²¹ : NormedSpace ℝ E
H : Type u_2
inst✝²⁰ : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type u_3
inst✝¹⁹ : TopologicalSpace M
inst✝¹⁸ : ChartedSpace H M
inst✝¹⁷ : I.Boundaryless
inst✝¹⁶ : SmoothManifoldWithCorners I M
inst✝¹⁵ : FiniteDimensional ℝ E
inst✝¹⁴ : Secon... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
E : Type u_1
inst✝²² : NormedAddCommGroup E
inst✝²¹ : NormedSpace ℝ E
H : Type u_2
inst✝²⁰ : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type u_3
inst✝¹⁹ : TopologicalSpace M
inst✝¹⁸ : ChartedSpace H M
inst✝¹⁷ : I.Boundaryless
inst✝¹⁶ : SmoothManifoldWi... |
https://github.com/fpvandoorn/sard.git | f4a1bb550136e8591dcd20dca5bdcce766a958c7 | Sard/Stuff.lean | image_measure_zero_of_C1_dimension_increase | [59, 1] | [90, 13] | let incl : E → E × (Fin (n-m) → ℝ) := fun x ↦ ⟨x, 0⟩ | E : Type u_1
inst✝²³ : NormedAddCommGroup E
inst✝²² : NormedSpace ℝ E
H : Type u_2
inst✝²¹ : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type u_3
inst✝²⁰ : TopologicalSpace M
inst✝¹⁹ : ChartedSpace H M
inst✝¹⁸ : I.Boundaryless
inst✝¹⁷ : SmoothManifoldWithCorners I M
inst✝¹⁶ : FiniteDimensional ℝ E
inst✝¹⁵ : Secon... | E : Type u_1
inst✝²³ : NormedAddCommGroup E
inst✝²² : NormedSpace ℝ E
H : Type u_2
inst✝²¹ : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type u_3
inst✝²⁰ : TopologicalSpace M
inst✝¹⁹ : ChartedSpace H M
inst✝¹⁸ : I.Boundaryless
inst✝¹⁷ : SmoothManifoldWithCorners I M
inst✝¹⁶ : FiniteDimensional ℝ E
inst✝¹⁵ : Secon... | Please generate a tactic in lean4 to solve the state.
STATE:
E : Type u_1
inst✝²³ : NormedAddCommGroup E
inst✝²² : NormedSpace ℝ E
H : Type u_2
inst✝²¹ : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type u_3
inst✝²⁰ : TopologicalSpace M
inst✝¹⁹ : ChartedSpace H M
inst✝¹⁸ : I.Boundaryless
inst✝¹⁷ : SmoothManifoldWi... |
https://github.com/fpvandoorn/sard.git | f4a1bb550136e8591dcd20dca5bdcce766a958c7 | Sard/Stuff.lean | image_measure_zero_of_C1_dimension_increase | [59, 1] | [90, 13] | let g' : E × (Fin (n-m) → ℝ) → F × (Fin (n-m) → ℝ) := fun ⟨y, _⟩ ↦ ⟨g y, 0⟩ | E : Type u_1
inst✝²³ : NormedAddCommGroup E
inst✝²² : NormedSpace ℝ E
H : Type u_2
inst✝²¹ : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type u_3
inst✝²⁰ : TopologicalSpace M
inst✝¹⁹ : ChartedSpace H M
inst✝¹⁸ : I.Boundaryless
inst✝¹⁷ : SmoothManifoldWithCorners I M
inst✝¹⁶ : FiniteDimensional ℝ E
inst✝¹⁵ : Secon... | E : Type u_1
inst✝²³ : NormedAddCommGroup E
inst✝²² : NormedSpace ℝ E
H : Type u_2
inst✝²¹ : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type u_3
inst✝²⁰ : TopologicalSpace M
inst✝¹⁹ : ChartedSpace H M
inst✝¹⁸ : I.Boundaryless
inst✝¹⁷ : SmoothManifoldWithCorners I M
inst✝¹⁶ : FiniteDimensional ℝ E
inst✝¹⁵ : Secon... | Please generate a tactic in lean4 to solve the state.
STATE:
E : Type u_1
inst✝²³ : NormedAddCommGroup E
inst✝²² : NormedSpace ℝ E
H : Type u_2
inst✝²¹ : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type u_3
inst✝²⁰ : TopologicalSpace M
inst✝¹⁹ : ChartedSpace H M
inst✝¹⁸ : I.Boundaryless
inst✝¹⁷ : SmoothManifoldWi... |
https://github.com/fpvandoorn/sard.git | f4a1bb550136e8591dcd20dca5bdcce766a958c7 | Sard/Stuff.lean | image_measure_zero_of_C1_dimension_increase | [59, 1] | [90, 13] | let pi : F × (Fin (n-m) → ℝ) → F := fun ⟨f, _⟩ ↦ f | E : Type u_1
inst✝²³ : NormedAddCommGroup E
inst✝²² : NormedSpace ℝ E
H : Type u_2
inst✝²¹ : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type u_3
inst✝²⁰ : TopologicalSpace M
inst✝¹⁹ : ChartedSpace H M
inst✝¹⁸ : I.Boundaryless
inst✝¹⁷ : SmoothManifoldWithCorners I M
inst✝¹⁶ : FiniteDimensional ℝ E
inst✝¹⁵ : Secon... | E : Type u_1
inst✝²³ : NormedAddCommGroup E
inst✝²² : NormedSpace ℝ E
H : Type u_2
inst✝²¹ : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type u_3
inst✝²⁰ : TopologicalSpace M
inst✝¹⁹ : ChartedSpace H M
inst✝¹⁸ : I.Boundaryless
inst✝¹⁷ : SmoothManifoldWithCorners I M
inst✝¹⁶ : FiniteDimensional ℝ E
inst✝¹⁵ : Secon... | Please generate a tactic in lean4 to solve the state.
STATE:
E : Type u_1
inst✝²³ : NormedAddCommGroup E
inst✝²² : NormedSpace ℝ E
H : Type u_2
inst✝²¹ : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type u_3
inst✝²⁰ : TopologicalSpace M
inst✝¹⁹ : ChartedSpace H M
inst✝¹⁸ : I.Boundaryless
inst✝¹⁷ : SmoothManifoldWi... |
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