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/MainTheorem.lean | sard_boundaryless | [84, 1] | [250, 15] | let A := mfderiv 𝓘(ℝ, E) I (e.extend I).symm x | case h'f'
E : Type u_1
inst✝²⁰ : NormedAddCommGroup E
inst✝¹⁹ : NormedSpace ℝ E
H : Type u_2
inst✝¹⁸ : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type u_3
inst✝¹⁷ : TopologicalSpace M
inst✝¹⁶ : ChartedSpace H M
inst✝¹⁵ : SmoothManifoldWithCorners I M
inst✝¹⁴ : FiniteDimensional ℝ E
inst✝¹³ : SecondCountableTopol... | case h'f'
E : Type u_1
inst✝²⁰ : NormedAddCommGroup E
inst✝¹⁹ : NormedSpace ℝ E
H : Type u_2
inst✝¹⁸ : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type u_3
inst✝¹⁷ : TopologicalSpace M
inst✝¹⁶ : ChartedSpace H M
inst✝¹⁵ : SmoothManifoldWithCorners I M
inst✝¹⁴ : FiniteDimensional ℝ E
inst✝¹³ : SecondCountableTopol... | Please generate a tactic in lean4 to solve the state.
STATE:
case h'f'
E : Type u_1
inst✝²⁰ : NormedAddCommGroup E
inst✝¹⁹ : NormedSpace ℝ E
H : Type u_2
inst✝¹⁸ : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type u_3
inst✝¹⁷ : TopologicalSpace M
inst✝¹⁶ : ChartedSpace H M
inst✝¹⁵ : SmoothManifoldWithCorners I M
i... |
https://github.com/fpvandoorn/sard.git | f4a1bb550136e8591dcd20dca5bdcce766a958c7 | Sard/MainTheorem.lean | sard_boundaryless | [84, 1] | [250, 15] | let B := mfderiv I J f ((e.extend I).symm x) | case h'f'
E : Type u_1
inst✝²⁰ : NormedAddCommGroup E
inst✝¹⁹ : NormedSpace ℝ E
H : Type u_2
inst✝¹⁸ : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type u_3
inst✝¹⁷ : TopologicalSpace M
inst✝¹⁶ : ChartedSpace H M
inst✝¹⁵ : SmoothManifoldWithCorners I M
inst✝¹⁴ : FiniteDimensional ℝ E
inst✝¹³ : SecondCountableTopol... | case h'f'
E : Type u_1
inst✝²⁰ : NormedAddCommGroup E
inst✝¹⁹ : NormedSpace ℝ E
H : Type u_2
inst✝¹⁸ : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type u_3
inst✝¹⁷ : TopologicalSpace M
inst✝¹⁶ : ChartedSpace H M
inst✝¹⁵ : SmoothManifoldWithCorners I M
inst✝¹⁴ : FiniteDimensional ℝ E
inst✝¹³ : SecondCountableTopol... | Please generate a tactic in lean4 to solve the state.
STATE:
case h'f'
E : Type u_1
inst✝²⁰ : NormedAddCommGroup E
inst✝¹⁹ : NormedSpace ℝ E
H : Type u_2
inst✝¹⁸ : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type u_3
inst✝¹⁷ : TopologicalSpace M
inst✝¹⁶ : ChartedSpace H M
inst✝¹⁵ : SmoothManifoldWithCorners I M
i... |
https://github.com/fpvandoorn/sard.git | f4a1bb550136e8591dcd20dca5bdcce766a958c7 | Sard/MainTheorem.lean | sard_boundaryless | [84, 1] | [250, 15] | let C := mfderiv J 𝓘(ℝ, F) (e'.extend J) ((f ∘ (e.extend I).symm) x) | case h'f'
E : Type u_1
inst✝²⁰ : NormedAddCommGroup E
inst✝¹⁹ : NormedSpace ℝ E
H : Type u_2
inst✝¹⁸ : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type u_3
inst✝¹⁷ : TopologicalSpace M
inst✝¹⁶ : ChartedSpace H M
inst✝¹⁵ : SmoothManifoldWithCorners I M
inst✝¹⁴ : FiniteDimensional ℝ E
inst✝¹³ : SecondCountableTopol... | case h'f'
E : Type u_1
inst✝²⁰ : NormedAddCommGroup E
inst✝¹⁹ : NormedSpace ℝ E
H : Type u_2
inst✝¹⁸ : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type u_3
inst✝¹⁷ : TopologicalSpace M
inst✝¹⁶ : ChartedSpace H M
inst✝¹⁵ : SmoothManifoldWithCorners I M
inst✝¹⁴ : FiniteDimensional ℝ E
inst✝¹³ : SecondCountableTopol... | Please generate a tactic in lean4 to solve the state.
STATE:
case h'f'
E : Type u_1
inst✝²⁰ : NormedAddCommGroup E
inst✝¹⁹ : NormedSpace ℝ E
H : Type u_2
inst✝¹⁸ : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type u_3
inst✝¹⁷ : TopologicalSpace M
inst✝¹⁶ : ChartedSpace H M
inst✝¹⁵ : SmoothManifoldWithCorners I M
i... |
https://github.com/fpvandoorn/sard.git | f4a1bb550136e8591dcd20dca5bdcce766a958c7 | Sard/MainTheorem.lean | sard_boundaryless | [84, 1] | [250, 15] | have hnA : e.extend I '' e.source ∈ 𝓝 x := by let x' := (e.extend I).symm x
have : e.extend I x' = x := e.extend_right_inv _ (hsbetter₀ hx)
rw [← this]
have hesource : e.source ∈ 𝓝 x' := by
have : (e.extend I).symm x = x' := rfl
have : x' ∈ e.source := by sorry exact e.open_source.mem_nhds this
exact ... | case h'f'
E : Type u_1
inst✝²⁰ : NormedAddCommGroup E
inst✝¹⁹ : NormedSpace ℝ E
H : Type u_2
inst✝¹⁸ : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type u_3
inst✝¹⁷ : TopologicalSpace M
inst✝¹⁶ : ChartedSpace H M
inst✝¹⁵ : SmoothManifoldWithCorners I M
inst✝¹⁴ : FiniteDimensional ℝ E
inst✝¹³ : SecondCountableTopol... | case h'f'
E : Type u_1
inst✝²⁰ : NormedAddCommGroup E
inst✝¹⁹ : NormedSpace ℝ E
H : Type u_2
inst✝¹⁸ : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type u_3
inst✝¹⁷ : TopologicalSpace M
inst✝¹⁶ : ChartedSpace H M
inst✝¹⁵ : SmoothManifoldWithCorners I M
inst✝¹⁴ : FiniteDimensional ℝ E
inst✝¹³ : SecondCountableTopol... | Please generate a tactic in lean4 to solve the state.
STATE:
case h'f'
E : Type u_1
inst✝²⁰ : NormedAddCommGroup E
inst✝¹⁹ : NormedSpace ℝ E
H : Type u_2
inst✝¹⁸ : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type u_3
inst✝¹⁷ : TopologicalSpace M
inst✝¹⁶ : ChartedSpace H M
inst✝¹⁵ : SmoothManifoldWithCorners I M
i... |
https://github.com/fpvandoorn/sard.git | f4a1bb550136e8591dcd20dca5bdcce766a958c7 | Sard/MainTheorem.lean | sard_boundaryless | [84, 1] | [250, 15] | have hr : 1 ≤ (r : ℕ∞) := Nat.one_le_cast.mpr (Nat.one_le_of_lt hr) | case h'f'
E : Type u_1
inst✝²⁰ : NormedAddCommGroup E
inst✝¹⁹ : NormedSpace ℝ E
H : Type u_2
inst✝¹⁸ : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type u_3
inst✝¹⁷ : TopologicalSpace M
inst✝¹⁶ : ChartedSpace H M
inst✝¹⁵ : SmoothManifoldWithCorners I M
inst✝¹⁴ : FiniteDimensional ℝ E
inst✝¹³ : SecondCountableTopol... | case h'f'
E : Type u_1
inst✝²⁰ : NormedAddCommGroup E
inst✝¹⁹ : NormedSpace ℝ E
H : Type u_2
inst✝¹⁸ : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type u_3
inst✝¹⁷ : TopologicalSpace M
inst✝¹⁶ : ChartedSpace H M
inst✝¹⁵ : SmoothManifoldWithCorners I M
inst✝¹⁴ : FiniteDimensional ℝ E
inst✝¹³ : SecondCountableTopol... | Please generate a tactic in lean4 to solve the state.
STATE:
case h'f'
E : Type u_1
inst✝²⁰ : NormedAddCommGroup E
inst✝¹⁹ : NormedSpace ℝ E
H : Type u_2
inst✝¹⁸ : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type u_3
inst✝¹⁷ : TopologicalSpace M
inst✝¹⁶ : ChartedSpace H M
inst✝¹⁵ : SmoothManifoldWithCorners I M
i... |
https://github.com/fpvandoorn/sard.git | f4a1bb550136e8591dcd20dca5bdcce766a958c7 | Sard/MainTheorem.lean | sard_boundaryless | [84, 1] | [250, 15] | have he : e ∈ maximalAtlas I M := by sorry | case h'f'
E : Type u_1
inst✝²⁰ : NormedAddCommGroup E
inst✝¹⁹ : NormedSpace ℝ E
H : Type u_2
inst✝¹⁸ : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type u_3
inst✝¹⁷ : TopologicalSpace M
inst✝¹⁶ : ChartedSpace H M
inst✝¹⁵ : SmoothManifoldWithCorners I M
inst✝¹⁴ : FiniteDimensional ℝ E
inst✝¹³ : SecondCountableTopol... | case h'f'
E : Type u_1
inst✝²⁰ : NormedAddCommGroup E
inst✝¹⁹ : NormedSpace ℝ E
H : Type u_2
inst✝¹⁸ : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type u_3
inst✝¹⁷ : TopologicalSpace M
inst✝¹⁶ : ChartedSpace H M
inst✝¹⁵ : SmoothManifoldWithCorners I M
inst✝¹⁴ : FiniteDimensional ℝ E
inst✝¹³ : SecondCountableTopol... | Please generate a tactic in lean4 to solve the state.
STATE:
case h'f'
E : Type u_1
inst✝²⁰ : NormedAddCommGroup E
inst✝¹⁹ : NormedSpace ℝ E
H : Type u_2
inst✝¹⁸ : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type u_3
inst✝¹⁷ : TopologicalSpace M
inst✝¹⁶ : ChartedSpace H M
inst✝¹⁵ : SmoothManifoldWithCorners I M
i... |
https://github.com/fpvandoorn/sard.git | f4a1bb550136e8591dcd20dca5bdcce766a958c7 | Sard/MainTheorem.lean | sard_boundaryless | [84, 1] | [250, 15] | have : I '' e.target ∈ 𝓝 x := sorry | case h'f'
E : Type u_1
inst✝²⁰ : NormedAddCommGroup E
inst✝¹⁹ : NormedSpace ℝ E
H : Type u_2
inst✝¹⁸ : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type u_3
inst✝¹⁷ : TopologicalSpace M
inst✝¹⁶ : ChartedSpace H M
inst✝¹⁵ : SmoothManifoldWithCorners I M
inst✝¹⁴ : FiniteDimensional ℝ E
inst✝¹³ : SecondCountableTopol... | case h'f'
E : Type u_1
inst✝²⁰ : NormedAddCommGroup E
inst✝¹⁹ : NormedSpace ℝ E
H : Type u_2
inst✝¹⁸ : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type u_3
inst✝¹⁷ : TopologicalSpace M
inst✝¹⁶ : ChartedSpace H M
inst✝¹⁵ : SmoothManifoldWithCorners I M
inst✝¹⁴ : FiniteDimensional ℝ E
inst✝¹³ : SecondCountableTopol... | Please generate a tactic in lean4 to solve the state.
STATE:
case h'f'
E : Type u_1
inst✝²⁰ : NormedAddCommGroup E
inst✝¹⁹ : NormedSpace ℝ E
H : Type u_2
inst✝¹⁸ : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type u_3
inst✝¹⁷ : TopologicalSpace M
inst✝¹⁶ : ChartedSpace H M
inst✝¹⁵ : SmoothManifoldWithCorners I M
i... |
https://github.com/fpvandoorn/sard.git | f4a1bb550136e8591dcd20dca5bdcce766a958c7 | Sard/MainTheorem.lean | sard_boundaryless | [84, 1] | [250, 15] | have hA : MDifferentiableAt 𝓘(ℝ, E) I (e.invFun ∘ I.invFun) x :=
((contMDiffOn_extend_symm (𝕜 := ℝ) he).contMDiffAt this).mdifferentiableAt hr | case h'f'
E : Type u_1
inst✝²⁰ : NormedAddCommGroup E
inst✝¹⁹ : NormedSpace ℝ E
H : Type u_2
inst✝¹⁸ : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type u_3
inst✝¹⁷ : TopologicalSpace M
inst✝¹⁶ : ChartedSpace H M
inst✝¹⁵ : SmoothManifoldWithCorners I M
inst✝¹⁴ : FiniteDimensional ℝ E
inst✝¹³ : SecondCountableTopol... | case h'f'
E : Type u_1
inst✝²⁰ : NormedAddCommGroup E
inst✝¹⁹ : NormedSpace ℝ E
H : Type u_2
inst✝¹⁸ : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type u_3
inst✝¹⁷ : TopologicalSpace M
inst✝¹⁶ : ChartedSpace H M
inst✝¹⁵ : SmoothManifoldWithCorners I M
inst✝¹⁴ : FiniteDimensional ℝ E
inst✝¹³ : SecondCountableTopol... | Please generate a tactic in lean4 to solve the state.
STATE:
case h'f'
E : Type u_1
inst✝²⁰ : NormedAddCommGroup E
inst✝¹⁹ : NormedSpace ℝ E
H : Type u_2
inst✝¹⁸ : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type u_3
inst✝¹⁷ : TopologicalSpace M
inst✝¹⁶ : ChartedSpace H M
inst✝¹⁵ : SmoothManifoldWithCorners I M
i... |
https://github.com/fpvandoorn/sard.git | f4a1bb550136e8591dcd20dca5bdcce766a958c7 | Sard/MainTheorem.lean | sard_boundaryless | [84, 1] | [250, 15] | have hB : MDifferentiableAt I J f ((e.invFun ∘ I.invFun) x) := hf.contMDiffAt.mdifferentiableAt hr | case h'f'
E : Type u_1
inst✝²⁰ : NormedAddCommGroup E
inst✝¹⁹ : NormedSpace ℝ E
H : Type u_2
inst✝¹⁸ : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type u_3
inst✝¹⁷ : TopologicalSpace M
inst✝¹⁶ : ChartedSpace H M
inst✝¹⁵ : SmoothManifoldWithCorners I M
inst✝¹⁴ : FiniteDimensional ℝ E
inst✝¹³ : SecondCountableTopol... | case h'f'
E : Type u_1
inst✝²⁰ : NormedAddCommGroup E
inst✝¹⁹ : NormedSpace ℝ E
H : Type u_2
inst✝¹⁸ : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type u_3
inst✝¹⁷ : TopologicalSpace M
inst✝¹⁶ : ChartedSpace H M
inst✝¹⁵ : SmoothManifoldWithCorners I M
inst✝¹⁴ : FiniteDimensional ℝ E
inst✝¹³ : SecondCountableTopol... | Please generate a tactic in lean4 to solve the state.
STATE:
case h'f'
E : Type u_1
inst✝²⁰ : NormedAddCommGroup E
inst✝¹⁹ : NormedSpace ℝ E
H : Type u_2
inst✝¹⁸ : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type u_3
inst✝¹⁷ : TopologicalSpace M
inst✝¹⁶ : ChartedSpace H M
inst✝¹⁵ : SmoothManifoldWithCorners I M
i... |
https://github.com/fpvandoorn/sard.git | f4a1bb550136e8591dcd20dca5bdcce766a958c7 | Sard/MainTheorem.lean | sard_boundaryless | [84, 1] | [250, 15] | have hBA : MDifferentiableAt 𝓘(ℝ, E) J (f ∘ (e.invFun ∘ I.invFun)) x := by
sorry | case h'f'
E : Type u_1
inst✝²⁰ : NormedAddCommGroup E
inst✝¹⁹ : NormedSpace ℝ E
H : Type u_2
inst✝¹⁸ : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type u_3
inst✝¹⁷ : TopologicalSpace M
inst✝¹⁶ : ChartedSpace H M
inst✝¹⁵ : SmoothManifoldWithCorners I M
inst✝¹⁴ : FiniteDimensional ℝ E
inst✝¹³ : SecondCountableTopol... | case h'f'
E : Type u_1
inst✝²⁰ : NormedAddCommGroup E
inst✝¹⁹ : NormedSpace ℝ E
H : Type u_2
inst✝¹⁸ : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type u_3
inst✝¹⁷ : TopologicalSpace M
inst✝¹⁶ : ChartedSpace H M
inst✝¹⁵ : SmoothManifoldWithCorners I M
inst✝¹⁴ : FiniteDimensional ℝ E
inst✝¹³ : SecondCountableTopol... | Please generate a tactic in lean4 to solve the state.
STATE:
case h'f'
E : Type u_1
inst✝²⁰ : NormedAddCommGroup E
inst✝¹⁹ : NormedSpace ℝ E
H : Type u_2
inst✝¹⁸ : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type u_3
inst✝¹⁷ : TopologicalSpace M
inst✝¹⁶ : ChartedSpace H M
inst✝¹⁵ : SmoothManifoldWithCorners I M
i... |
https://github.com/fpvandoorn/sard.git | f4a1bb550136e8591dcd20dca5bdcce766a958c7 | Sard/MainTheorem.lean | sard_boundaryless | [84, 1] | [250, 15] | have hnC : ((f ∘ e.invFun ∘ I.invFun) '' w) ∈ 𝓝 ((f ∘ e.invFun ∘ I.invFun) x) := sorry | case h'f'
E : Type u_1
inst✝²⁰ : NormedAddCommGroup E
inst✝¹⁹ : NormedSpace ℝ E
H : Type u_2
inst✝¹⁸ : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type u_3
inst✝¹⁷ : TopologicalSpace M
inst✝¹⁶ : ChartedSpace H M
inst✝¹⁵ : SmoothManifoldWithCorners I M
inst✝¹⁴ : FiniteDimensional ℝ E
inst✝¹³ : SecondCountableTopol... | case h'f'
E : Type u_1
inst✝²⁰ : NormedAddCommGroup E
inst✝¹⁹ : NormedSpace ℝ E
H : Type u_2
inst✝¹⁸ : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type u_3
inst✝¹⁷ : TopologicalSpace M
inst✝¹⁶ : ChartedSpace H M
inst✝¹⁵ : SmoothManifoldWithCorners I M
inst✝¹⁴ : FiniteDimensional ℝ E
inst✝¹³ : SecondCountableTopol... | Please generate a tactic in lean4 to solve the state.
STATE:
case h'f'
E : Type u_1
inst✝²⁰ : NormedAddCommGroup E
inst✝¹⁹ : NormedSpace ℝ E
H : Type u_2
inst✝¹⁸ : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type u_3
inst✝¹⁷ : TopologicalSpace M
inst✝¹⁶ : ChartedSpace H M
inst✝¹⁵ : SmoothManifoldWithCorners I M
i... |
https://github.com/fpvandoorn/sard.git | f4a1bb550136e8591dcd20dca5bdcce766a958c7 | Sard/MainTheorem.lean | sard_boundaryless | [84, 1] | [250, 15] | have hC : MDifferentiableAt J 𝓘(ℝ, F) (J ∘ e') ((f ∘ e.invFun ∘ I.invFun) x) := by
have : ContMDiffOn J 𝓘(ℝ, F) ∞ (J ∘ e') e'.source := extendedChart_smooth _ (chart_mem_atlas G _)
exact SmoothAt.mdifferentiableAt ((this.mono hw).contMDiffAt hnC) | case h'f'
E : Type u_1
inst✝²⁰ : NormedAddCommGroup E
inst✝¹⁹ : NormedSpace ℝ E
H : Type u_2
inst✝¹⁸ : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type u_3
inst✝¹⁷ : TopologicalSpace M
inst✝¹⁶ : ChartedSpace H M
inst✝¹⁵ : SmoothManifoldWithCorners I M
inst✝¹⁴ : FiniteDimensional ℝ E
inst✝¹³ : SecondCountableTopol... | case h'f'
E : Type u_1
inst✝²⁰ : NormedAddCommGroup E
inst✝¹⁹ : NormedSpace ℝ E
H : Type u_2
inst✝¹⁸ : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type u_3
inst✝¹⁷ : TopologicalSpace M
inst✝¹⁶ : ChartedSpace H M
inst✝¹⁵ : SmoothManifoldWithCorners I M
inst✝¹⁴ : FiniteDimensional ℝ E
inst✝¹³ : SecondCountableTopol... | Please generate a tactic in lean4 to solve the state.
STATE:
case h'f'
E : Type u_1
inst✝²⁰ : NormedAddCommGroup E
inst✝¹⁹ : NormedSpace ℝ E
H : Type u_2
inst✝¹⁸ : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type u_3
inst✝¹⁷ : TopologicalSpace M
inst✝¹⁶ : ChartedSpace H M
inst✝¹⁵ : SmoothManifoldWithCorners I M
i... |
https://github.com/fpvandoorn/sard.git | f4a1bb550136e8591dcd20dca5bdcce766a958c7 | Sard/MainTheorem.lean | sard_boundaryless | [84, 1] | [250, 15] | let comp := C.comp (B.comp A) | case h'f'
E : Type u_1
inst✝²⁰ : NormedAddCommGroup E
inst✝¹⁹ : NormedSpace ℝ E
H : Type u_2
inst✝¹⁸ : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type u_3
inst✝¹⁷ : TopologicalSpace M
inst✝¹⁶ : ChartedSpace H M
inst✝¹⁵ : SmoothManifoldWithCorners I M
inst✝¹⁴ : FiniteDimensional ℝ E
inst✝¹³ : SecondCountableTopol... | case h'f'
E : Type u_1
inst✝²⁰ : NormedAddCommGroup E
inst✝¹⁹ : NormedSpace ℝ E
H : Type u_2
inst✝¹⁸ : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type u_3
inst✝¹⁷ : TopologicalSpace M
inst✝¹⁶ : ChartedSpace H M
inst✝¹⁵ : SmoothManifoldWithCorners I M
inst✝¹⁴ : FiniteDimensional ℝ E
inst✝¹³ : SecondCountableTopol... | Please generate a tactic in lean4 to solve the state.
STATE:
case h'f'
E : Type u_1
inst✝²⁰ : NormedAddCommGroup E
inst✝¹⁹ : NormedSpace ℝ E
H : Type u_2
inst✝¹⁸ : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type u_3
inst✝¹⁷ : TopologicalSpace M
inst✝¹⁶ : ChartedSpace H M
inst✝¹⁵ : SmoothManifoldWithCorners I M
i... |
https://github.com/fpvandoorn/sard.git | f4a1bb550136e8591dcd20dca5bdcce766a958c7 | Sard/MainTheorem.lean | sard_boundaryless | [84, 1] | [250, 15] | let r := calc comp
_ = C.comp (B.comp A) := rfl
_ = C.comp ((mfderiv I J f ((e.invFun ∘ I.invFun) x)).comp (mfderiv 𝓘(ℝ, E) I (e.invFun ∘ I.invFun) x)) := rfl
_ = C.comp (mfderiv 𝓘(ℝ, E) J (f ∘ (e.invFun ∘ I.invFun)) x) := by rw [(mfderiv_comp x hB hA)]
_ = (mfderiv J 𝓘(ℝ, F) (J ∘ e') ((f ∘ e.invFun ∘ I.invF... | case h'f'
E : Type u_1
inst✝²⁰ : NormedAddCommGroup E
inst✝¹⁹ : NormedSpace ℝ E
H : Type u_2
inst✝¹⁸ : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type u_3
inst✝¹⁷ : TopologicalSpace M
inst✝¹⁶ : ChartedSpace H M
inst✝¹⁵ : SmoothManifoldWithCorners I M
inst✝¹⁴ : FiniteDimensional ℝ E
inst✝¹³ : SecondCountableTopol... | case h'f'
E : Type u_1
inst✝²⁰ : NormedAddCommGroup E
inst✝¹⁹ : NormedSpace ℝ E
H : Type u_2
inst✝¹⁸ : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type u_3
inst✝¹⁷ : TopologicalSpace M
inst✝¹⁶ : ChartedSpace H M
inst✝¹⁵ : SmoothManifoldWithCorners I M
inst✝¹⁴ : FiniteDimensional ℝ E
inst✝¹³ : SecondCountableTopol... | Please generate a tactic in lean4 to solve the state.
STATE:
case h'f'
E : Type u_1
inst✝²⁰ : NormedAddCommGroup E
inst✝¹⁹ : NormedSpace ℝ E
H : Type u_2
inst✝¹⁸ : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type u_3
inst✝¹⁷ : TopologicalSpace M
inst✝¹⁶ : ChartedSpace H M
inst✝¹⁵ : SmoothManifoldWithCorners I M
i... |
https://github.com/fpvandoorn/sard.git | f4a1bb550136e8591dcd20dca5bdcce766a958c7 | Sard/MainTheorem.lean | sard_boundaryless | [84, 1] | [250, 15] | let x' := (e.extend I).symm x | case h'f'
E : Type u_1
inst✝²⁰ : NormedAddCommGroup E
inst✝¹⁹ : NormedSpace ℝ E
H : Type u_2
inst✝¹⁸ : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type u_3
inst✝¹⁷ : TopologicalSpace M
inst✝¹⁶ : ChartedSpace H M
inst✝¹⁵ : SmoothManifoldWithCorners I M
inst✝¹⁴ : FiniteDimensional ℝ E
inst✝¹³ : SecondCountableTopol... | case h'f'
E : Type u_1
inst✝²⁰ : NormedAddCommGroup E
inst✝¹⁹ : NormedSpace ℝ E
H : Type u_2
inst✝¹⁸ : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type u_3
inst✝¹⁷ : TopologicalSpace M
inst✝¹⁶ : ChartedSpace H M
inst✝¹⁵ : SmoothManifoldWithCorners I M
inst✝¹⁴ : FiniteDimensional ℝ E
inst✝¹³ : SecondCountableTopol... | Please generate a tactic in lean4 to solve the state.
STATE:
case h'f'
E : Type u_1
inst✝²⁰ : NormedAddCommGroup E
inst✝¹⁹ : NormedSpace ℝ E
H : Type u_2
inst✝¹⁸ : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type u_3
inst✝¹⁷ : TopologicalSpace M
inst✝¹⁶ : ChartedSpace H M
inst✝¹⁵ : SmoothManifoldWithCorners I M
i... |
https://github.com/fpvandoorn/sard.git | f4a1bb550136e8591dcd20dca5bdcce766a958c7 | Sard/MainTheorem.lean | sard_boundaryless | [84, 1] | [250, 15] | have aux : x' ∈ s := hsbetter₁ (mem_image_of_mem _ hx) | case h'f'
E : Type u_1
inst✝²⁰ : NormedAddCommGroup E
inst✝¹⁹ : NormedSpace ℝ E
H : Type u_2
inst✝¹⁸ : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type u_3
inst✝¹⁷ : TopologicalSpace M
inst✝¹⁶ : ChartedSpace H M
inst✝¹⁵ : SmoothManifoldWithCorners I M
inst✝¹⁴ : FiniteDimensional ℝ E
inst✝¹³ : SecondCountableTopol... | case h'f'
E : Type u_1
inst✝²⁰ : NormedAddCommGroup E
inst✝¹⁹ : NormedSpace ℝ E
H : Type u_2
inst✝¹⁸ : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type u_3
inst✝¹⁷ : TopologicalSpace M
inst✝¹⁶ : ChartedSpace H M
inst✝¹⁵ : SmoothManifoldWithCorners I M
inst✝¹⁴ : FiniteDimensional ℝ E
inst✝¹³ : SecondCountableTopol... | Please generate a tactic in lean4 to solve the state.
STATE:
case h'f'
E : Type u_1
inst✝²⁰ : NormedAddCommGroup E
inst✝¹⁹ : NormedSpace ℝ E
H : Type u_2
inst✝¹⁸ : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type u_3
inst✝¹⁷ : TopologicalSpace M
inst✝¹⁶ : ChartedSpace H M
inst✝¹⁵ : SmoothManifoldWithCorners I M
i... |
https://github.com/fpvandoorn/sard.git | f4a1bb550136e8591dcd20dca5bdcce766a958c7 | Sard/MainTheorem.lean | sard_boundaryless | [84, 1] | [250, 15] | have : f' x' = B := by rw [← (hf' x' aux).mfderiv] | case h'f'
E : Type u_1
inst✝²⁰ : NormedAddCommGroup E
inst✝¹⁹ : NormedSpace ℝ E
H : Type u_2
inst✝¹⁸ : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type u_3
inst✝¹⁷ : TopologicalSpace M
inst✝¹⁶ : ChartedSpace H M
inst✝¹⁵ : SmoothManifoldWithCorners I M
inst✝¹⁴ : FiniteDimensional ℝ E
inst✝¹³ : SecondCountableTopol... | case h'f'
E : Type u_1
inst✝²⁰ : NormedAddCommGroup E
inst✝¹⁹ : NormedSpace ℝ E
H : Type u_2
inst✝¹⁸ : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type u_3
inst✝¹⁷ : TopologicalSpace M
inst✝¹⁶ : ChartedSpace H M
inst✝¹⁵ : SmoothManifoldWithCorners I M
inst✝¹⁴ : FiniteDimensional ℝ E
inst✝¹³ : SecondCountableTopol... | Please generate a tactic in lean4 to solve the state.
STATE:
case h'f'
E : Type u_1
inst✝²⁰ : NormedAddCommGroup E
inst✝¹⁹ : NormedSpace ℝ E
H : Type u_2
inst✝¹⁸ : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type u_3
inst✝¹⁷ : TopologicalSpace M
inst✝¹⁶ : ChartedSpace H M
inst✝¹⁵ : SmoothManifoldWithCorners I M
i... |
https://github.com/fpvandoorn/sard.git | f4a1bb550136e8591dcd20dca5bdcce766a958c7 | Sard/MainTheorem.lean | sard_boundaryless | [84, 1] | [250, 15] | have hBsurj : ¬ Surjective B := this ▸ h'f' _ aux | case h'f'
E : Type u_1
inst✝²⁰ : NormedAddCommGroup E
inst✝¹⁹ : NormedSpace ℝ E
H : Type u_2
inst✝¹⁸ : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type u_3
inst✝¹⁷ : TopologicalSpace M
inst✝¹⁶ : ChartedSpace H M
inst✝¹⁵ : SmoothManifoldWithCorners I M
inst✝¹⁴ : FiniteDimensional ℝ E
inst✝¹³ : SecondCountableTopol... | case h'f'
E : Type u_1
inst✝²⁰ : NormedAddCommGroup E
inst✝¹⁹ : NormedSpace ℝ E
H : Type u_2
inst✝¹⁸ : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type u_3
inst✝¹⁷ : TopologicalSpace M
inst✝¹⁶ : ChartedSpace H M
inst✝¹⁵ : SmoothManifoldWithCorners I M
inst✝¹⁴ : FiniteDimensional ℝ E
inst✝¹³ : SecondCountableTopol... | Please generate a tactic in lean4 to solve the state.
STATE:
case h'f'
E : Type u_1
inst✝²⁰ : NormedAddCommGroup E
inst✝¹⁹ : NormedSpace ℝ E
H : Type u_2
inst✝¹⁸ : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type u_3
inst✝¹⁷ : TopologicalSpace M
inst✝¹⁶ : ChartedSpace H M
inst✝¹⁵ : SmoothManifoldWithCorners I M
i... |
https://github.com/fpvandoorn/sard.git | f4a1bb550136e8591dcd20dca5bdcce766a958c7 | Sard/MainTheorem.lean | sard_boundaryless | [84, 1] | [250, 15] | have hA : Bijective A := extendedChart_symm_differential_bijective I (chart_mem_atlas H _) (hsbetter₀ hx) | case h'f'
E : Type u_1
inst✝²⁰ : NormedAddCommGroup E
inst✝¹⁹ : NormedSpace ℝ E
H : Type u_2
inst✝¹⁸ : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type u_3
inst✝¹⁷ : TopologicalSpace M
inst✝¹⁶ : ChartedSpace H M
inst✝¹⁵ : SmoothManifoldWithCorners I M
inst✝¹⁴ : FiniteDimensional ℝ E
inst✝¹³ : SecondCountableTopol... | case h'f'
E : Type u_1
inst✝²⁰ : NormedAddCommGroup E
inst✝¹⁹ : NormedSpace ℝ E
H : Type u_2
inst✝¹⁸ : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type u_3
inst✝¹⁷ : TopologicalSpace M
inst✝¹⁶ : ChartedSpace H M
inst✝¹⁵ : SmoothManifoldWithCorners I M
inst✝¹⁴ : FiniteDimensional ℝ E
inst✝¹³ : SecondCountableTopol... | Please generate a tactic in lean4 to solve the state.
STATE:
case h'f'
E : Type u_1
inst✝²⁰ : NormedAddCommGroup E
inst✝¹⁹ : NormedSpace ℝ E
H : Type u_2
inst✝¹⁸ : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type u_3
inst✝¹⁷ : TopologicalSpace M
inst✝¹⁶ : ChartedSpace H M
inst✝¹⁵ : SmoothManifoldWithCorners I M
i... |
https://github.com/fpvandoorn/sard.git | f4a1bb550136e8591dcd20dca5bdcce766a958c7 | Sard/MainTheorem.lean | sard_boundaryless | [84, 1] | [250, 15] | have hC : Bijective C :=
extendedChart_differential_bijective J (chart_mem_atlas G _) (hsbetter₃ (mem_image_of_mem _ hx)) | case h'f'
E : Type u_1
inst✝²⁰ : NormedAddCommGroup E
inst✝¹⁹ : NormedSpace ℝ E
H : Type u_2
inst✝¹⁸ : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type u_3
inst✝¹⁷ : TopologicalSpace M
inst✝¹⁶ : ChartedSpace H M
inst✝¹⁵ : SmoothManifoldWithCorners I M
inst✝¹⁴ : FiniteDimensional ℝ E
inst✝¹³ : SecondCountableTopol... | case h'f'
E : Type u_1
inst✝²⁰ : NormedAddCommGroup E
inst✝¹⁹ : NormedSpace ℝ E
H : Type u_2
inst✝¹⁸ : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type u_3
inst✝¹⁷ : TopologicalSpace M
inst✝¹⁶ : ChartedSpace H M
inst✝¹⁵ : SmoothManifoldWithCorners I M
inst✝¹⁴ : FiniteDimensional ℝ E
inst✝¹³ : SecondCountableTopol... | Please generate a tactic in lean4 to solve the state.
STATE:
case h'f'
E : Type u_1
inst✝²⁰ : NormedAddCommGroup E
inst✝¹⁹ : NormedSpace ℝ E
H : Type u_2
inst✝¹⁸ : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type u_3
inst✝¹⁷ : TopologicalSpace M
inst✝¹⁶ : ChartedSpace H M
inst✝¹⁵ : SmoothManifoldWithCorners I M
i... |
https://github.com/fpvandoorn/sard.git | f4a1bb550136e8591dcd20dca5bdcce766a958c7 | Sard/MainTheorem.lean | sard_boundaryless | [84, 1] | [250, 15] | have : Surjective B ↔ Surjective comp := by
have : comp = (C ∘ B) ∘ A := rfl
rw [this]
rw [hA.surjective.of_comp_iff (C ∘ B)]
rw [Surjective.of_comp_iff' hC B] | case h'f'
E : Type u_1
inst✝²⁰ : NormedAddCommGroup E
inst✝¹⁹ : NormedSpace ℝ E
H : Type u_2
inst✝¹⁸ : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type u_3
inst✝¹⁷ : TopologicalSpace M
inst✝¹⁶ : ChartedSpace H M
inst✝¹⁵ : SmoothManifoldWithCorners I M
inst✝¹⁴ : FiniteDimensional ℝ E
inst✝¹³ : SecondCountableTopol... | case h'f'
E : Type u_1
inst✝²⁰ : NormedAddCommGroup E
inst✝¹⁹ : NormedSpace ℝ E
H : Type u_2
inst✝¹⁸ : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type u_3
inst✝¹⁷ : TopologicalSpace M
inst✝¹⁶ : ChartedSpace H M
inst✝¹⁵ : SmoothManifoldWithCorners I M
inst✝¹⁴ : FiniteDimensional ℝ E
inst✝¹³ : SecondCountableTopol... | Please generate a tactic in lean4 to solve the state.
STATE:
case h'f'
E : Type u_1
inst✝²⁰ : NormedAddCommGroup E
inst✝¹⁹ : NormedSpace ℝ E
H : Type u_2
inst✝¹⁸ : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type u_3
inst✝¹⁷ : TopologicalSpace M
inst✝¹⁶ : ChartedSpace H M
inst✝¹⁵ : SmoothManifoldWithCorners I M
i... |
https://github.com/fpvandoorn/sard.git | f4a1bb550136e8591dcd20dca5bdcce766a958c7 | Sard/MainTheorem.lean | sard_boundaryless | [84, 1] | [250, 15] | rw [← r] | case h'f'
E : Type u_1
inst✝²⁰ : NormedAddCommGroup E
inst✝¹⁹ : NormedSpace ℝ E
H : Type u_2
inst✝¹⁸ : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type u_3
inst✝¹⁷ : TopologicalSpace M
inst✝¹⁶ : ChartedSpace H M
inst✝¹⁵ : SmoothManifoldWithCorners I M
inst✝¹⁴ : FiniteDimensional ℝ E
inst✝¹³ : SecondCountableTopol... | case h'f'
E : Type u_1
inst✝²⁰ : NormedAddCommGroup E
inst✝¹⁹ : NormedSpace ℝ E
H : Type u_2
inst✝¹⁸ : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type u_3
inst✝¹⁷ : TopologicalSpace M
inst✝¹⁶ : ChartedSpace H M
inst✝¹⁵ : SmoothManifoldWithCorners I M
inst✝¹⁴ : FiniteDimensional ℝ E
inst✝¹³ : SecondCountableTopol... | Please generate a tactic in lean4 to solve the state.
STATE:
case h'f'
E : Type u_1
inst✝²⁰ : NormedAddCommGroup E
inst✝¹⁹ : NormedSpace ℝ E
H : Type u_2
inst✝¹⁸ : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type u_3
inst✝¹⁷ : TopologicalSpace M
inst✝¹⁶ : ChartedSpace H M
inst✝¹⁵ : SmoothManifoldWithCorners I M
i... |
https://github.com/fpvandoorn/sard.git | f4a1bb550136e8591dcd20dca5bdcce766a958c7 | Sard/MainTheorem.lean | sard_boundaryless | [84, 1] | [250, 15] | have : ¬Surjective comp := by rw [← this]; exact hBsurj | case h'f'
E : Type u_1
inst✝²⁰ : NormedAddCommGroup E
inst✝¹⁹ : NormedSpace ℝ E
H : Type u_2
inst✝¹⁸ : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type u_3
inst✝¹⁷ : TopologicalSpace M
inst✝¹⁶ : ChartedSpace H M
inst✝¹⁵ : SmoothManifoldWithCorners I M
inst✝¹⁴ : FiniteDimensional ℝ E
inst✝¹³ : SecondCountableTopol... | case h'f'
E : Type u_1
inst✝²⁰ : NormedAddCommGroup E
inst✝¹⁹ : NormedSpace ℝ E
H : Type u_2
inst✝¹⁸ : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type u_3
inst✝¹⁷ : TopologicalSpace M
inst✝¹⁶ : ChartedSpace H M
inst✝¹⁵ : SmoothManifoldWithCorners I M
inst✝¹⁴ : FiniteDimensional ℝ E
inst✝¹³ : SecondCountableTopol... | Please generate a tactic in lean4 to solve the state.
STATE:
case h'f'
E : Type u_1
inst✝²⁰ : NormedAddCommGroup E
inst✝¹⁹ : NormedSpace ℝ E
H : Type u_2
inst✝¹⁸ : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type u_3
inst✝¹⁷ : TopologicalSpace M
inst✝¹⁶ : ChartedSpace H M
inst✝¹⁵ : SmoothManifoldWithCorners I M
i... |
https://github.com/fpvandoorn/sard.git | f4a1bb550136e8591dcd20dca5bdcce766a958c7 | Sard/MainTheorem.lean | sard_boundaryless | [84, 1] | [250, 15] | exact this | case h'f'
E : Type u_1
inst✝²⁰ : NormedAddCommGroup E
inst✝¹⁹ : NormedSpace ℝ E
H : Type u_2
inst✝¹⁸ : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type u_3
inst✝¹⁷ : TopologicalSpace M
inst✝¹⁶ : ChartedSpace H M
inst✝¹⁵ : SmoothManifoldWithCorners I M
inst✝¹⁴ : FiniteDimensional ℝ E
inst✝¹³ : SecondCountableTopol... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h'f'
E : Type u_1
inst✝²⁰ : NormedAddCommGroup E
inst✝¹⁹ : NormedSpace ℝ E
H : Type u_2
inst✝¹⁸ : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type u_3
inst✝¹⁷ : TopologicalSpace M
inst✝¹⁶ : ChartedSpace H M
inst✝¹⁵ : SmoothManifoldWithCorners I M
i... |
https://github.com/fpvandoorn/sard.git | f4a1bb550136e8591dcd20dca5bdcce766a958c7 | Sard/MainTheorem.lean | sard_boundaryless | [84, 1] | [250, 15] | let x' := (e.extend I).symm x | E : Type u_1
inst✝²⁰ : NormedAddCommGroup E
inst✝¹⁹ : NormedSpace ℝ E
H : Type u_2
inst✝¹⁸ : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type u_3
inst✝¹⁷ : TopologicalSpace M
inst✝¹⁶ : ChartedSpace H M
inst✝¹⁵ : SmoothManifoldWithCorners I M
inst✝¹⁴ : FiniteDimensional ℝ E
inst✝¹³ : SecondCountableTopology M
inst... | E : Type u_1
inst✝²⁰ : NormedAddCommGroup E
inst✝¹⁹ : NormedSpace ℝ E
H : Type u_2
inst✝¹⁸ : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type u_3
inst✝¹⁷ : TopologicalSpace M
inst✝¹⁶ : ChartedSpace H M
inst✝¹⁵ : SmoothManifoldWithCorners I M
inst✝¹⁴ : FiniteDimensional ℝ E
inst✝¹³ : SecondCountableTopology M
inst... | Please generate a tactic in lean4 to solve the state.
STATE:
E : Type u_1
inst✝²⁰ : NormedAddCommGroup E
inst✝¹⁹ : NormedSpace ℝ E
H : Type u_2
inst✝¹⁸ : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type u_3
inst✝¹⁷ : TopologicalSpace M
inst✝¹⁶ : ChartedSpace H M
inst✝¹⁵ : SmoothManifoldWithCorners I M
inst✝¹⁴ : F... |
https://github.com/fpvandoorn/sard.git | f4a1bb550136e8591dcd20dca5bdcce766a958c7 | Sard/MainTheorem.lean | sard_boundaryless | [84, 1] | [250, 15] | have : e.extend I x' = x := e.extend_right_inv _ (hsbetter₀ hx) | E : Type u_1
inst✝²⁰ : NormedAddCommGroup E
inst✝¹⁹ : NormedSpace ℝ E
H : Type u_2
inst✝¹⁸ : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type u_3
inst✝¹⁷ : TopologicalSpace M
inst✝¹⁶ : ChartedSpace H M
inst✝¹⁵ : SmoothManifoldWithCorners I M
inst✝¹⁴ : FiniteDimensional ℝ E
inst✝¹³ : SecondCountableTopology M
inst... | E : Type u_1
inst✝²⁰ : NormedAddCommGroup E
inst✝¹⁹ : NormedSpace ℝ E
H : Type u_2
inst✝¹⁸ : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type u_3
inst✝¹⁷ : TopologicalSpace M
inst✝¹⁶ : ChartedSpace H M
inst✝¹⁵ : SmoothManifoldWithCorners I M
inst✝¹⁴ : FiniteDimensional ℝ E
inst✝¹³ : SecondCountableTopology M
inst... | Please generate a tactic in lean4 to solve the state.
STATE:
E : Type u_1
inst✝²⁰ : NormedAddCommGroup E
inst✝¹⁹ : NormedSpace ℝ E
H : Type u_2
inst✝¹⁸ : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type u_3
inst✝¹⁷ : TopologicalSpace M
inst✝¹⁶ : ChartedSpace H M
inst✝¹⁵ : SmoothManifoldWithCorners I M
inst✝¹⁴ : F... |
https://github.com/fpvandoorn/sard.git | f4a1bb550136e8591dcd20dca5bdcce766a958c7 | Sard/MainTheorem.lean | sard_boundaryless | [84, 1] | [250, 15] | rw [← this] | E : Type u_1
inst✝²⁰ : NormedAddCommGroup E
inst✝¹⁹ : NormedSpace ℝ E
H : Type u_2
inst✝¹⁸ : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type u_3
inst✝¹⁷ : TopologicalSpace M
inst✝¹⁶ : ChartedSpace H M
inst✝¹⁵ : SmoothManifoldWithCorners I M
inst✝¹⁴ : FiniteDimensional ℝ E
inst✝¹³ : SecondCountableTopology M
inst... | E : Type u_1
inst✝²⁰ : NormedAddCommGroup E
inst✝¹⁹ : NormedSpace ℝ E
H : Type u_2
inst✝¹⁸ : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type u_3
inst✝¹⁷ : TopologicalSpace M
inst✝¹⁶ : ChartedSpace H M
inst✝¹⁵ : SmoothManifoldWithCorners I M
inst✝¹⁴ : FiniteDimensional ℝ E
inst✝¹³ : SecondCountableTopology M
inst... | Please generate a tactic in lean4 to solve the state.
STATE:
E : Type u_1
inst✝²⁰ : NormedAddCommGroup E
inst✝¹⁹ : NormedSpace ℝ E
H : Type u_2
inst✝¹⁸ : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type u_3
inst✝¹⁷ : TopologicalSpace M
inst✝¹⁶ : ChartedSpace H M
inst✝¹⁵ : SmoothManifoldWithCorners I M
inst✝¹⁴ : F... |
https://github.com/fpvandoorn/sard.git | f4a1bb550136e8591dcd20dca5bdcce766a958c7 | Sard/MainTheorem.lean | sard_boundaryless | [84, 1] | [250, 15] | have hesource : e.source ∈ 𝓝 x' := by
have : (e.extend I).symm x = x' := rfl
have : x' ∈ e.source := by sorry exact e.open_source.mem_nhds this | E : Type u_1
inst✝²⁰ : NormedAddCommGroup E
inst✝¹⁹ : NormedSpace ℝ E
H : Type u_2
inst✝¹⁸ : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type u_3
inst✝¹⁷ : TopologicalSpace M
inst✝¹⁶ : ChartedSpace H M
inst✝¹⁵ : SmoothManifoldWithCorners I M
inst✝¹⁴ : FiniteDimensional ℝ E
inst✝¹³ : SecondCountableTopology M
inst... | E : Type u_1
inst✝²⁰ : NormedAddCommGroup E
inst✝¹⁹ : NormedSpace ℝ E
H : Type u_2
inst✝¹⁸ : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type u_3
inst✝¹⁷ : TopologicalSpace M
inst✝¹⁶ : ChartedSpace H M
inst✝¹⁵ : SmoothManifoldWithCorners I M
inst✝¹⁴ : FiniteDimensional ℝ E
inst✝¹³ : SecondCountableTopology M
inst... | Please generate a tactic in lean4 to solve the state.
STATE:
E : Type u_1
inst✝²⁰ : NormedAddCommGroup E
inst✝¹⁹ : NormedSpace ℝ E
H : Type u_2
inst✝¹⁸ : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type u_3
inst✝¹⁷ : TopologicalSpace M
inst✝¹⁶ : ChartedSpace H M
inst✝¹⁵ : SmoothManifoldWithCorners I M
inst✝¹⁴ : F... |
https://github.com/fpvandoorn/sard.git | f4a1bb550136e8591dcd20dca5bdcce766a958c7 | Sard/MainTheorem.lean | sard_boundaryless | [84, 1] | [250, 15] | exact extendedChart_image_nhds_on I hesource (Eq.subset rfl) | E : Type u_1
inst✝²⁰ : NormedAddCommGroup E
inst✝¹⁹ : NormedSpace ℝ E
H : Type u_2
inst✝¹⁸ : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type u_3
inst✝¹⁷ : TopologicalSpace M
inst✝¹⁶ : ChartedSpace H M
inst✝¹⁵ : SmoothManifoldWithCorners I M
inst✝¹⁴ : FiniteDimensional ℝ E
inst✝¹³ : SecondCountableTopology M
inst... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
E : Type u_1
inst✝²⁰ : NormedAddCommGroup E
inst✝¹⁹ : NormedSpace ℝ E
H : Type u_2
inst✝¹⁸ : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type u_3
inst✝¹⁷ : TopologicalSpace M
inst✝¹⁶ : ChartedSpace H M
inst✝¹⁵ : SmoothManifoldWithCorners I M
inst✝¹⁴ : F... |
https://github.com/fpvandoorn/sard.git | f4a1bb550136e8591dcd20dca5bdcce766a958c7 | Sard/MainTheorem.lean | sard_boundaryless | [84, 1] | [250, 15] | have : (e.extend I).symm x = x' := rfl | E : Type u_1
inst✝²⁰ : NormedAddCommGroup E
inst✝¹⁹ : NormedSpace ℝ E
H : Type u_2
inst✝¹⁸ : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type u_3
inst✝¹⁷ : TopologicalSpace M
inst✝¹⁶ : ChartedSpace H M
inst✝¹⁵ : SmoothManifoldWithCorners I M
inst✝¹⁴ : FiniteDimensional ℝ E
inst✝¹³ : SecondCountableTopology M
inst... | E : Type u_1
inst✝²⁰ : NormedAddCommGroup E
inst✝¹⁹ : NormedSpace ℝ E
H : Type u_2
inst✝¹⁸ : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type u_3
inst✝¹⁷ : TopologicalSpace M
inst✝¹⁶ : ChartedSpace H M
inst✝¹⁵ : SmoothManifoldWithCorners I M
inst✝¹⁴ : FiniteDimensional ℝ E
inst✝¹³ : SecondCountableTopology M
inst... | Please generate a tactic in lean4 to solve the state.
STATE:
E : Type u_1
inst✝²⁰ : NormedAddCommGroup E
inst✝¹⁹ : NormedSpace ℝ E
H : Type u_2
inst✝¹⁸ : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type u_3
inst✝¹⁷ : TopologicalSpace M
inst✝¹⁶ : ChartedSpace H M
inst✝¹⁵ : SmoothManifoldWithCorners I M
inst✝¹⁴ : F... |
https://github.com/fpvandoorn/sard.git | f4a1bb550136e8591dcd20dca5bdcce766a958c7 | Sard/MainTheorem.lean | sard_boundaryless | [84, 1] | [250, 15] | have : x' ∈ e.source := by sorry | E : Type u_1
inst✝²⁰ : NormedAddCommGroup E
inst✝¹⁹ : NormedSpace ℝ E
H : Type u_2
inst✝¹⁸ : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type u_3
inst✝¹⁷ : TopologicalSpace M
inst✝¹⁶ : ChartedSpace H M
inst✝¹⁵ : SmoothManifoldWithCorners I M
inst✝¹⁴ : FiniteDimensional ℝ E
inst✝¹³ : SecondCountableTopology M
inst... | E : Type u_1
inst✝²⁰ : NormedAddCommGroup E
inst✝¹⁹ : NormedSpace ℝ E
H : Type u_2
inst✝¹⁸ : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type u_3
inst✝¹⁷ : TopologicalSpace M
inst✝¹⁶ : ChartedSpace H M
inst✝¹⁵ : SmoothManifoldWithCorners I M
inst✝¹⁴ : FiniteDimensional ℝ E
inst✝¹³ : SecondCountableTopology M
inst... | Please generate a tactic in lean4 to solve the state.
STATE:
E : Type u_1
inst✝²⁰ : NormedAddCommGroup E
inst✝¹⁹ : NormedSpace ℝ E
H : Type u_2
inst✝¹⁸ : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type u_3
inst✝¹⁷ : TopologicalSpace M
inst✝¹⁶ : ChartedSpace H M
inst✝¹⁵ : SmoothManifoldWithCorners I M
inst✝¹⁴ : F... |
https://github.com/fpvandoorn/sard.git | f4a1bb550136e8591dcd20dca5bdcce766a958c7 | Sard/MainTheorem.lean | sard_boundaryless | [84, 1] | [250, 15] | exact e.open_source.mem_nhds this | E : Type u_1
inst✝²⁰ : NormedAddCommGroup E
inst✝¹⁹ : NormedSpace ℝ E
H : Type u_2
inst✝¹⁸ : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type u_3
inst✝¹⁷ : TopologicalSpace M
inst✝¹⁶ : ChartedSpace H M
inst✝¹⁵ : SmoothManifoldWithCorners I M
inst✝¹⁴ : FiniteDimensional ℝ E
inst✝¹³ : SecondCountableTopology M
inst... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
E : Type u_1
inst✝²⁰ : NormedAddCommGroup E
inst✝¹⁹ : NormedSpace ℝ E
H : Type u_2
inst✝¹⁸ : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type u_3
inst✝¹⁷ : TopologicalSpace M
inst✝¹⁶ : ChartedSpace H M
inst✝¹⁵ : SmoothManifoldWithCorners I M
inst✝¹⁴ : F... |
https://github.com/fpvandoorn/sard.git | f4a1bb550136e8591dcd20dca5bdcce766a958c7 | Sard/MainTheorem.lean | sard_boundaryless | [84, 1] | [250, 15] | sorry | E : Type u_1
inst✝²⁰ : NormedAddCommGroup E
inst✝¹⁹ : NormedSpace ℝ E
H : Type u_2
inst✝¹⁸ : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type u_3
inst✝¹⁷ : TopologicalSpace M
inst✝¹⁶ : ChartedSpace H M
inst✝¹⁵ : SmoothManifoldWithCorners I M
inst✝¹⁴ : FiniteDimensional ℝ E
inst✝¹³ : SecondCountableTopology M
inst... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
E : Type u_1
inst✝²⁰ : NormedAddCommGroup E
inst✝¹⁹ : NormedSpace ℝ E
H : Type u_2
inst✝¹⁸ : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type u_3
inst✝¹⁷ : TopologicalSpace M
inst✝¹⁶ : ChartedSpace H M
inst✝¹⁵ : SmoothManifoldWithCorners I M
inst✝¹⁴ : F... |
https://github.com/fpvandoorn/sard.git | f4a1bb550136e8591dcd20dca5bdcce766a958c7 | Sard/MainTheorem.lean | sard_boundaryless | [84, 1] | [250, 15] | sorry | E : Type u_1
inst✝²⁰ : NormedAddCommGroup E
inst✝¹⁹ : NormedSpace ℝ E
H : Type u_2
inst✝¹⁸ : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type u_3
inst✝¹⁷ : TopologicalSpace M
inst✝¹⁶ : ChartedSpace H M
inst✝¹⁵ : SmoothManifoldWithCorners I M
inst✝¹⁴ : FiniteDimensional ℝ E
inst✝¹³ : SecondCountableTopology M
inst... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
E : Type u_1
inst✝²⁰ : NormedAddCommGroup E
inst✝¹⁹ : NormedSpace ℝ E
H : Type u_2
inst✝¹⁸ : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type u_3
inst✝¹⁷ : TopologicalSpace M
inst✝¹⁶ : ChartedSpace H M
inst✝¹⁵ : SmoothManifoldWithCorners I M
inst✝¹⁴ : F... |
https://github.com/fpvandoorn/sard.git | f4a1bb550136e8591dcd20dca5bdcce766a958c7 | Sard/MainTheorem.lean | sard_boundaryless | [84, 1] | [250, 15] | sorry | E : Type u_1
inst✝²⁰ : NormedAddCommGroup E
inst✝¹⁹ : NormedSpace ℝ E
H : Type u_2
inst✝¹⁸ : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type u_3
inst✝¹⁷ : TopologicalSpace M
inst✝¹⁶ : ChartedSpace H M
inst✝¹⁵ : SmoothManifoldWithCorners I M
inst✝¹⁴ : FiniteDimensional ℝ E
inst✝¹³ : SecondCountableTopology M
inst... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
E : Type u_1
inst✝²⁰ : NormedAddCommGroup E
inst✝¹⁹ : NormedSpace ℝ E
H : Type u_2
inst✝¹⁸ : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type u_3
inst✝¹⁷ : TopologicalSpace M
inst✝¹⁶ : ChartedSpace H M
inst✝¹⁵ : SmoothManifoldWithCorners I M
inst✝¹⁴ : F... |
https://github.com/fpvandoorn/sard.git | f4a1bb550136e8591dcd20dca5bdcce766a958c7 | Sard/MainTheorem.lean | sard_boundaryless | [84, 1] | [250, 15] | have : ContMDiffOn J 𝓘(ℝ, F) ∞ (J ∘ e') e'.source := extendedChart_smooth _ (chart_mem_atlas G _) | E : Type u_1
inst✝²⁰ : NormedAddCommGroup E
inst✝¹⁹ : NormedSpace ℝ E
H : Type u_2
inst✝¹⁸ : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type u_3
inst✝¹⁷ : TopologicalSpace M
inst✝¹⁶ : ChartedSpace H M
inst✝¹⁵ : SmoothManifoldWithCorners I M
inst✝¹⁴ : FiniteDimensional ℝ E
inst✝¹³ : SecondCountableTopology M
inst... | E : Type u_1
inst✝²⁰ : NormedAddCommGroup E
inst✝¹⁹ : NormedSpace ℝ E
H : Type u_2
inst✝¹⁸ : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type u_3
inst✝¹⁷ : TopologicalSpace M
inst✝¹⁶ : ChartedSpace H M
inst✝¹⁵ : SmoothManifoldWithCorners I M
inst✝¹⁴ : FiniteDimensional ℝ E
inst✝¹³ : SecondCountableTopology M
inst... | Please generate a tactic in lean4 to solve the state.
STATE:
E : Type u_1
inst✝²⁰ : NormedAddCommGroup E
inst✝¹⁹ : NormedSpace ℝ E
H : Type u_2
inst✝¹⁸ : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type u_3
inst✝¹⁷ : TopologicalSpace M
inst✝¹⁶ : ChartedSpace H M
inst✝¹⁵ : SmoothManifoldWithCorners I M
inst✝¹⁴ : F... |
https://github.com/fpvandoorn/sard.git | f4a1bb550136e8591dcd20dca5bdcce766a958c7 | Sard/MainTheorem.lean | sard_boundaryless | [84, 1] | [250, 15] | exact SmoothAt.mdifferentiableAt ((this.mono hw).contMDiffAt hnC) | E : Type u_1
inst✝²⁰ : NormedAddCommGroup E
inst✝¹⁹ : NormedSpace ℝ E
H : Type u_2
inst✝¹⁸ : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type u_3
inst✝¹⁷ : TopologicalSpace M
inst✝¹⁶ : ChartedSpace H M
inst✝¹⁵ : SmoothManifoldWithCorners I M
inst✝¹⁴ : FiniteDimensional ℝ E
inst✝¹³ : SecondCountableTopology M
inst... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
E : Type u_1
inst✝²⁰ : NormedAddCommGroup E
inst✝¹⁹ : NormedSpace ℝ E
H : Type u_2
inst✝¹⁸ : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type u_3
inst✝¹⁷ : TopologicalSpace M
inst✝¹⁶ : ChartedSpace H M
inst✝¹⁵ : SmoothManifoldWithCorners I M
inst✝¹⁴ : F... |
https://github.com/fpvandoorn/sard.git | f4a1bb550136e8591dcd20dca5bdcce766a958c7 | Sard/MainTheorem.lean | sard_boundaryless | [84, 1] | [250, 15] | rw [(mfderiv_comp x hB hA)] | E : Type u_1
inst✝²⁰ : NormedAddCommGroup E
inst✝¹⁹ : NormedSpace ℝ E
H : Type u_2
inst✝¹⁸ : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type u_3
inst✝¹⁷ : TopologicalSpace M
inst✝¹⁶ : ChartedSpace H M
inst✝¹⁵ : SmoothManifoldWithCorners I M
inst✝¹⁴ : FiniteDimensional ℝ E
inst✝¹³ : SecondCountableTopology M
inst... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
E : Type u_1
inst✝²⁰ : NormedAddCommGroup E
inst✝¹⁹ : NormedSpace ℝ E
H : Type u_2
inst✝¹⁸ : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type u_3
inst✝¹⁷ : TopologicalSpace M
inst✝¹⁶ : ChartedSpace H M
inst✝¹⁵ : SmoothManifoldWithCorners I M
inst✝¹⁴ : F... |
https://github.com/fpvandoorn/sard.git | f4a1bb550136e8591dcd20dca5bdcce766a958c7 | Sard/MainTheorem.lean | sard_boundaryless | [84, 1] | [250, 15] | rw [(mfderiv_comp x hC hBA)] | E : Type u_1
inst✝²⁰ : NormedAddCommGroup E
inst✝¹⁹ : NormedSpace ℝ E
H : Type u_2
inst✝¹⁸ : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type u_3
inst✝¹⁷ : TopologicalSpace M
inst✝¹⁶ : ChartedSpace H M
inst✝¹⁵ : SmoothManifoldWithCorners I M
inst✝¹⁴ : FiniteDimensional ℝ E
inst✝¹³ : SecondCountableTopology M
inst... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
E : Type u_1
inst✝²⁰ : NormedAddCommGroup E
inst✝¹⁹ : NormedSpace ℝ E
H : Type u_2
inst✝¹⁸ : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type u_3
inst✝¹⁷ : TopologicalSpace M
inst✝¹⁶ : ChartedSpace H M
inst✝¹⁵ : SmoothManifoldWithCorners I M
inst✝¹⁴ : F... |
https://github.com/fpvandoorn/sard.git | f4a1bb550136e8591dcd20dca5bdcce766a958c7 | Sard/MainTheorem.lean | sard_boundaryless | [84, 1] | [250, 15] | rw [← (hf' x' aux).mfderiv] | E : Type u_1
inst✝²⁰ : NormedAddCommGroup E
inst✝¹⁹ : NormedSpace ℝ E
H : Type u_2
inst✝¹⁸ : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type u_3
inst✝¹⁷ : TopologicalSpace M
inst✝¹⁶ : ChartedSpace H M
inst✝¹⁵ : SmoothManifoldWithCorners I M
inst✝¹⁴ : FiniteDimensional ℝ E
inst✝¹³ : SecondCountableTopology M
inst... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
E : Type u_1
inst✝²⁰ : NormedAddCommGroup E
inst✝¹⁹ : NormedSpace ℝ E
H : Type u_2
inst✝¹⁸ : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type u_3
inst✝¹⁷ : TopologicalSpace M
inst✝¹⁶ : ChartedSpace H M
inst✝¹⁵ : SmoothManifoldWithCorners I M
inst✝¹⁴ : F... |
https://github.com/fpvandoorn/sard.git | f4a1bb550136e8591dcd20dca5bdcce766a958c7 | Sard/MainTheorem.lean | sard_boundaryless | [84, 1] | [250, 15] | have : comp = (C ∘ B) ∘ A := rfl | E : Type u_1
inst✝²⁰ : NormedAddCommGroup E
inst✝¹⁹ : NormedSpace ℝ E
H : Type u_2
inst✝¹⁸ : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type u_3
inst✝¹⁷ : TopologicalSpace M
inst✝¹⁶ : ChartedSpace H M
inst✝¹⁵ : SmoothManifoldWithCorners I M
inst✝¹⁴ : FiniteDimensional ℝ E
inst✝¹³ : SecondCountableTopology M
inst... | E : Type u_1
inst✝²⁰ : NormedAddCommGroup E
inst✝¹⁹ : NormedSpace ℝ E
H : Type u_2
inst✝¹⁸ : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type u_3
inst✝¹⁷ : TopologicalSpace M
inst✝¹⁶ : ChartedSpace H M
inst✝¹⁵ : SmoothManifoldWithCorners I M
inst✝¹⁴ : FiniteDimensional ℝ E
inst✝¹³ : SecondCountableTopology M
inst... | Please generate a tactic in lean4 to solve the state.
STATE:
E : Type u_1
inst✝²⁰ : NormedAddCommGroup E
inst✝¹⁹ : NormedSpace ℝ E
H : Type u_2
inst✝¹⁸ : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type u_3
inst✝¹⁷ : TopologicalSpace M
inst✝¹⁶ : ChartedSpace H M
inst✝¹⁵ : SmoothManifoldWithCorners I M
inst✝¹⁴ : F... |
https://github.com/fpvandoorn/sard.git | f4a1bb550136e8591dcd20dca5bdcce766a958c7 | Sard/MainTheorem.lean | sard_boundaryless | [84, 1] | [250, 15] | rw [this] | E : Type u_1
inst✝²⁰ : NormedAddCommGroup E
inst✝¹⁹ : NormedSpace ℝ E
H : Type u_2
inst✝¹⁸ : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type u_3
inst✝¹⁷ : TopologicalSpace M
inst✝¹⁶ : ChartedSpace H M
inst✝¹⁵ : SmoothManifoldWithCorners I M
inst✝¹⁴ : FiniteDimensional ℝ E
inst✝¹³ : SecondCountableTopology M
inst... | E : Type u_1
inst✝²⁰ : NormedAddCommGroup E
inst✝¹⁹ : NormedSpace ℝ E
H : Type u_2
inst✝¹⁸ : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type u_3
inst✝¹⁷ : TopologicalSpace M
inst✝¹⁶ : ChartedSpace H M
inst✝¹⁵ : SmoothManifoldWithCorners I M
inst✝¹⁴ : FiniteDimensional ℝ E
inst✝¹³ : SecondCountableTopology M
inst... | Please generate a tactic in lean4 to solve the state.
STATE:
E : Type u_1
inst✝²⁰ : NormedAddCommGroup E
inst✝¹⁹ : NormedSpace ℝ E
H : Type u_2
inst✝¹⁸ : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type u_3
inst✝¹⁷ : TopologicalSpace M
inst✝¹⁶ : ChartedSpace H M
inst✝¹⁵ : SmoothManifoldWithCorners I M
inst✝¹⁴ : F... |
https://github.com/fpvandoorn/sard.git | f4a1bb550136e8591dcd20dca5bdcce766a958c7 | Sard/MainTheorem.lean | sard_boundaryless | [84, 1] | [250, 15] | rw [hA.surjective.of_comp_iff (C ∘ B)] | E : Type u_1
inst✝²⁰ : NormedAddCommGroup E
inst✝¹⁹ : NormedSpace ℝ E
H : Type u_2
inst✝¹⁸ : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type u_3
inst✝¹⁷ : TopologicalSpace M
inst✝¹⁶ : ChartedSpace H M
inst✝¹⁵ : SmoothManifoldWithCorners I M
inst✝¹⁴ : FiniteDimensional ℝ E
inst✝¹³ : SecondCountableTopology M
inst... | E : Type u_1
inst✝²⁰ : NormedAddCommGroup E
inst✝¹⁹ : NormedSpace ℝ E
H : Type u_2
inst✝¹⁸ : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type u_3
inst✝¹⁷ : TopologicalSpace M
inst✝¹⁶ : ChartedSpace H M
inst✝¹⁵ : SmoothManifoldWithCorners I M
inst✝¹⁴ : FiniteDimensional ℝ E
inst✝¹³ : SecondCountableTopology M
inst... | Please generate a tactic in lean4 to solve the state.
STATE:
E : Type u_1
inst✝²⁰ : NormedAddCommGroup E
inst✝¹⁹ : NormedSpace ℝ E
H : Type u_2
inst✝¹⁸ : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type u_3
inst✝¹⁷ : TopologicalSpace M
inst✝¹⁶ : ChartedSpace H M
inst✝¹⁵ : SmoothManifoldWithCorners I M
inst✝¹⁴ : F... |
https://github.com/fpvandoorn/sard.git | f4a1bb550136e8591dcd20dca5bdcce766a958c7 | Sard/MainTheorem.lean | sard_boundaryless | [84, 1] | [250, 15] | rw [Surjective.of_comp_iff' hC B] | E : Type u_1
inst✝²⁰ : NormedAddCommGroup E
inst✝¹⁹ : NormedSpace ℝ E
H : Type u_2
inst✝¹⁸ : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type u_3
inst✝¹⁷ : TopologicalSpace M
inst✝¹⁶ : ChartedSpace H M
inst✝¹⁵ : SmoothManifoldWithCorners I M
inst✝¹⁴ : FiniteDimensional ℝ E
inst✝¹³ : SecondCountableTopology M
inst... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
E : Type u_1
inst✝²⁰ : NormedAddCommGroup E
inst✝¹⁹ : NormedSpace ℝ E
H : Type u_2
inst✝¹⁸ : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type u_3
inst✝¹⁷ : TopologicalSpace M
inst✝¹⁶ : ChartedSpace H M
inst✝¹⁵ : SmoothManifoldWithCorners I M
inst✝¹⁴ : F... |
https://github.com/fpvandoorn/sard.git | f4a1bb550136e8591dcd20dca5bdcce766a958c7 | Sard/MainTheorem.lean | sard_boundaryless | [84, 1] | [250, 15] | rw [← this] | E : Type u_1
inst✝²⁰ : NormedAddCommGroup E
inst✝¹⁹ : NormedSpace ℝ E
H : Type u_2
inst✝¹⁸ : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type u_3
inst✝¹⁷ : TopologicalSpace M
inst✝¹⁶ : ChartedSpace H M
inst✝¹⁵ : SmoothManifoldWithCorners I M
inst✝¹⁴ : FiniteDimensional ℝ E
inst✝¹³ : SecondCountableTopology M
inst... | E : Type u_1
inst✝²⁰ : NormedAddCommGroup E
inst✝¹⁹ : NormedSpace ℝ E
H : Type u_2
inst✝¹⁸ : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type u_3
inst✝¹⁷ : TopologicalSpace M
inst✝¹⁶ : ChartedSpace H M
inst✝¹⁵ : SmoothManifoldWithCorners I M
inst✝¹⁴ : FiniteDimensional ℝ E
inst✝¹³ : SecondCountableTopology M
inst... | Please generate a tactic in lean4 to solve the state.
STATE:
E : Type u_1
inst✝²⁰ : NormedAddCommGroup E
inst✝¹⁹ : NormedSpace ℝ E
H : Type u_2
inst✝¹⁸ : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type u_3
inst✝¹⁷ : TopologicalSpace M
inst✝¹⁶ : ChartedSpace H M
inst✝¹⁵ : SmoothManifoldWithCorners I M
inst✝¹⁴ : F... |
https://github.com/fpvandoorn/sard.git | f4a1bb550136e8591dcd20dca5bdcce766a958c7 | Sard/MainTheorem.lean | sard_boundaryless | [84, 1] | [250, 15] | exact hBsurj | E : Type u_1
inst✝²⁰ : NormedAddCommGroup E
inst✝¹⁹ : NormedSpace ℝ E
H : Type u_2
inst✝¹⁸ : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type u_3
inst✝¹⁷ : TopologicalSpace M
inst✝¹⁶ : ChartedSpace H M
inst✝¹⁵ : SmoothManifoldWithCorners I M
inst✝¹⁴ : FiniteDimensional ℝ E
inst✝¹³ : SecondCountableTopology M
inst... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
E : Type u_1
inst✝²⁰ : NormedAddCommGroup E
inst✝¹⁹ : NormedSpace ℝ E
H : Type u_2
inst✝¹⁸ : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type u_3
inst✝¹⁷ : TopologicalSpace M
inst✝¹⁶ : ChartedSpace H M
inst✝¹⁵ : SmoothManifoldWithCorners I M
inst✝¹⁴ : F... |
https://github.com/fpvandoorn/sard.git | f4a1bb550136e8591dcd20dca5bdcce766a958c7 | Sard/MainTheorem.lean | sard | [252, 1] | [261, 8] | sorry | E : Type u_1
inst✝¹⁸ : NormedAddCommGroup E
inst✝¹⁷ : NormedSpace ℝ E
H : Type u_2
inst✝¹⁶ : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type u_3
inst✝¹⁵ : TopologicalSpace M
inst✝¹⁴ : ChartedSpace H M
inst✝¹³ : SmoothManifoldWithCorners I M
inst✝¹² : FiniteDimensional ℝ E
inst✝¹¹ : SecondCountableTopology M
inst... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
E : Type u_1
inst✝¹⁸ : NormedAddCommGroup E
inst✝¹⁷ : NormedSpace ℝ E
H : Type u_2
inst✝¹⁶ : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type u_3
inst✝¹⁵ : TopologicalSpace M
inst✝¹⁴ : ChartedSpace H M
inst✝¹³ : SmoothManifoldWithCorners I M
inst✝¹² : F... |
https://github.com/fpvandoorn/sard.git | f4a1bb550136e8591dcd20dca5bdcce766a958c7 | Sard/MainTheorem.lean | sard' | [267, 1] | [274, 99] | have : LocallyCompactSpace M := Manifold.locallyCompact_of_finiteDimensional I | E : Type u_1
inst✝²⁰ : NormedAddCommGroup E
inst✝¹⁹ : NormedSpace ℝ E
H : Type u_2
inst✝¹⁸ : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type u_3
inst✝¹⁷ : TopologicalSpace M
inst✝¹⁶ : ChartedSpace H M
inst✝¹⁵ : SmoothManifoldWithCorners I M
inst✝¹⁴ : FiniteDimensional ℝ E
inst✝¹³ : SecondCountableTopology M
inst... | E : Type u_1
inst✝²⁰ : NormedAddCommGroup E
inst✝¹⁹ : NormedSpace ℝ E
H : Type u_2
inst✝¹⁸ : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type u_3
inst✝¹⁷ : TopologicalSpace M
inst✝¹⁶ : ChartedSpace H M
inst✝¹⁵ : SmoothManifoldWithCorners I M
inst✝¹⁴ : FiniteDimensional ℝ E
inst✝¹³ : SecondCountableTopology M
inst... | Please generate a tactic in lean4 to solve the state.
STATE:
E : Type u_1
inst✝²⁰ : NormedAddCommGroup E
inst✝¹⁹ : NormedSpace ℝ E
H : Type u_2
inst✝¹⁸ : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type u_3
inst✝¹⁷ : TopologicalSpace M
inst✝¹⁶ : ChartedSpace H M
inst✝¹⁵ : SmoothManifoldWithCorners I M
inst✝¹⁴ : F... |
https://github.com/fpvandoorn/sard.git | f4a1bb550136e8591dcd20dca5bdcce766a958c7 | Sard/MainTheorem.lean | sard' | [267, 1] | [274, 99] | have : IsSigmaCompact s := isSigmaCompact_univ.of_isClosed_subset hs (subset_univ s) | E : Type u_1
inst✝²⁰ : NormedAddCommGroup E
inst✝¹⁹ : NormedSpace ℝ E
H : Type u_2
inst✝¹⁸ : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type u_3
inst✝¹⁷ : TopologicalSpace M
inst✝¹⁶ : ChartedSpace H M
inst✝¹⁵ : SmoothManifoldWithCorners I M
inst✝¹⁴ : FiniteDimensional ℝ E
inst✝¹³ : SecondCountableTopology M
inst... | E : Type u_1
inst✝²⁰ : NormedAddCommGroup E
inst✝¹⁹ : NormedSpace ℝ E
H : Type u_2
inst✝¹⁸ : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type u_3
inst✝¹⁷ : TopologicalSpace M
inst✝¹⁶ : ChartedSpace H M
inst✝¹⁵ : SmoothManifoldWithCorners I M
inst✝¹⁴ : FiniteDimensional ℝ E
inst✝¹³ : SecondCountableTopology M
inst... | Please generate a tactic in lean4 to solve the state.
STATE:
E : Type u_1
inst✝²⁰ : NormedAddCommGroup E
inst✝¹⁹ : NormedSpace ℝ E
H : Type u_2
inst✝¹⁸ : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type u_3
inst✝¹⁷ : TopologicalSpace M
inst✝¹⁶ : ChartedSpace H M
inst✝¹⁵ : SmoothManifoldWithCorners I M
inst✝¹⁴ : F... |
https://github.com/fpvandoorn/sard.git | f4a1bb550136e8591dcd20dca5bdcce766a958c7 | Sard/MainTheorem.lean | sard' | [267, 1] | [274, 99] | exact MeasureZero.isMeagre_of_isSigmaCompact J (sard I hf hf' h'f') (this.image (hf.continuous)) | E : Type u_1
inst✝²⁰ : NormedAddCommGroup E
inst✝¹⁹ : NormedSpace ℝ E
H : Type u_2
inst✝¹⁸ : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type u_3
inst✝¹⁷ : TopologicalSpace M
inst✝¹⁶ : ChartedSpace H M
inst✝¹⁵ : SmoothManifoldWithCorners I M
inst✝¹⁴ : FiniteDimensional ℝ E
inst✝¹³ : SecondCountableTopology M
inst... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
E : Type u_1
inst✝²⁰ : NormedAddCommGroup E
inst✝¹⁹ : NormedSpace ℝ E
H : Type u_2
inst✝¹⁸ : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type u_3
inst✝¹⁷ : TopologicalSpace M
inst✝¹⁶ : ChartedSpace H M
inst✝¹⁵ : SmoothManifoldWithCorners I M
inst✝¹⁴ : F... |
https://github.com/fpvandoorn/sard.git | f4a1bb550136e8591dcd20dca5bdcce766a958c7 | Sard/LocallyLipschitzMeasureZero.lean | null_set_from_countable_cover | [11, 1] | [23, 10] | have h : μ t ≤ 0 := by
have : ∀ n, μ (s n ∩ t) = 0 := fun n => measure_inter_null_of_null_left t (hnull n)
calc μ t
_ = μ ((⋃ (n : ℕ), s n) ∩ t) := by simp only [inter_eq_right.mpr hcov]
_ = μ (⋃ (n : ℕ), (s n ∩ t)) := by rw [iUnion_inter]
_ ≤ ∑' (n : ℕ), μ (s n ∩ t) := by apply OuterMeasure.iUnion_nat
... | X : Type u_1
Y : Type u_2
inst✝⁵ : MetricSpace X
inst✝⁴ : MeasurableSpace X
inst✝³ : BorelSpace X
inst✝² : MetricSpace Y
inst✝¹ : MeasurableSpace Y
inst✝ : BorelSpace Y
t : Set X
μ : Measure X
s : ℕ → Set X
hnull : ∀ (n : ℕ), ↑μ (s n) = 0
hcov : t ⊆ ⋃ n, s n
⊢ ↑μ t = 0 | X : Type u_1
Y : Type u_2
inst✝⁵ : MetricSpace X
inst✝⁴ : MeasurableSpace X
inst✝³ : BorelSpace X
inst✝² : MetricSpace Y
inst✝¹ : MeasurableSpace Y
inst✝ : BorelSpace Y
t : Set X
μ : Measure X
s : ℕ → Set X
hnull : ∀ (n : ℕ), ↑μ (s n) = 0
hcov : t ⊆ ⋃ n, s n
h : ↑μ t ≤ 0
⊢ ↑μ t = 0 | Please generate a tactic in lean4 to solve the state.
STATE:
X : Type u_1
Y : Type u_2
inst✝⁵ : MetricSpace X
inst✝⁴ : MeasurableSpace X
inst✝³ : BorelSpace X
inst✝² : MetricSpace Y
inst✝¹ : MeasurableSpace Y
inst✝ : BorelSpace Y
t : Set X
μ : Measure X
s : ℕ → Set X
hnull : ∀ (n : ℕ), ↑μ (s n) = 0
hcov : t ⊆ ⋃ n, s n
... |
https://github.com/fpvandoorn/sard.git | f4a1bb550136e8591dcd20dca5bdcce766a958c7 | Sard/LocallyLipschitzMeasureZero.lean | null_set_from_countable_cover | [11, 1] | [23, 10] | simp only [nonpos_iff_eq_zero, zero_le] at h ⊢ | X : Type u_1
Y : Type u_2
inst✝⁵ : MetricSpace X
inst✝⁴ : MeasurableSpace X
inst✝³ : BorelSpace X
inst✝² : MetricSpace Y
inst✝¹ : MeasurableSpace Y
inst✝ : BorelSpace Y
t : Set X
μ : Measure X
s : ℕ → Set X
hnull : ∀ (n : ℕ), ↑μ (s n) = 0
hcov : t ⊆ ⋃ n, s n
h : ↑μ t ≤ 0
⊢ ↑μ t = 0 | X : Type u_1
Y : Type u_2
inst✝⁵ : MetricSpace X
inst✝⁴ : MeasurableSpace X
inst✝³ : BorelSpace X
inst✝² : MetricSpace Y
inst✝¹ : MeasurableSpace Y
inst✝ : BorelSpace Y
t : Set X
μ : Measure X
s : ℕ → Set X
hnull : ∀ (n : ℕ), ↑μ (s n) = 0
hcov : t ⊆ ⋃ n, s n
h : ↑μ t = 0
⊢ ↑μ t = 0 | Please generate a tactic in lean4 to solve the state.
STATE:
X : Type u_1
Y : Type u_2
inst✝⁵ : MetricSpace X
inst✝⁴ : MeasurableSpace X
inst✝³ : BorelSpace X
inst✝² : MetricSpace Y
inst✝¹ : MeasurableSpace Y
inst✝ : BorelSpace Y
t : Set X
μ : Measure X
s : ℕ → Set X
hnull : ∀ (n : ℕ), ↑μ (s n) = 0
hcov : t ⊆ ⋃ n, s n
... |
https://github.com/fpvandoorn/sard.git | f4a1bb550136e8591dcd20dca5bdcce766a958c7 | Sard/LocallyLipschitzMeasureZero.lean | null_set_from_countable_cover | [11, 1] | [23, 10] | exact h | X : Type u_1
Y : Type u_2
inst✝⁵ : MetricSpace X
inst✝⁴ : MeasurableSpace X
inst✝³ : BorelSpace X
inst✝² : MetricSpace Y
inst✝¹ : MeasurableSpace Y
inst✝ : BorelSpace Y
t : Set X
μ : Measure X
s : ℕ → Set X
hnull : ∀ (n : ℕ), ↑μ (s n) = 0
hcov : t ⊆ ⋃ n, s n
h : ↑μ t = 0
⊢ ↑μ t = 0 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
X : Type u_1
Y : Type u_2
inst✝⁵ : MetricSpace X
inst✝⁴ : MeasurableSpace X
inst✝³ : BorelSpace X
inst✝² : MetricSpace Y
inst✝¹ : MeasurableSpace Y
inst✝ : BorelSpace Y
t : Set X
μ : Measure X
s : ℕ → Set X
hnull : ∀ (n : ℕ), ↑μ (s n) = 0
hcov : t ⊆ ⋃ n, s n
... |
https://github.com/fpvandoorn/sard.git | f4a1bb550136e8591dcd20dca5bdcce766a958c7 | Sard/LocallyLipschitzMeasureZero.lean | null_set_from_countable_cover | [11, 1] | [23, 10] | have : ∀ n, μ (s n ∩ t) = 0 := fun n => measure_inter_null_of_null_left t (hnull n) | X : Type u_1
Y : Type u_2
inst✝⁵ : MetricSpace X
inst✝⁴ : MeasurableSpace X
inst✝³ : BorelSpace X
inst✝² : MetricSpace Y
inst✝¹ : MeasurableSpace Y
inst✝ : BorelSpace Y
t : Set X
μ : Measure X
s : ℕ → Set X
hnull : ∀ (n : ℕ), ↑μ (s n) = 0
hcov : t ⊆ ⋃ n, s n
⊢ ↑μ t ≤ 0 | X : Type u_1
Y : Type u_2
inst✝⁵ : MetricSpace X
inst✝⁴ : MeasurableSpace X
inst✝³ : BorelSpace X
inst✝² : MetricSpace Y
inst✝¹ : MeasurableSpace Y
inst✝ : BorelSpace Y
t : Set X
μ : Measure X
s : ℕ → Set X
hnull : ∀ (n : ℕ), ↑μ (s n) = 0
hcov : t ⊆ ⋃ n, s n
this : ∀ (n : ℕ), ↑μ (s n ∩ t) = 0
⊢ ↑μ t ≤ 0 | Please generate a tactic in lean4 to solve the state.
STATE:
X : Type u_1
Y : Type u_2
inst✝⁵ : MetricSpace X
inst✝⁴ : MeasurableSpace X
inst✝³ : BorelSpace X
inst✝² : MetricSpace Y
inst✝¹ : MeasurableSpace Y
inst✝ : BorelSpace Y
t : Set X
μ : Measure X
s : ℕ → Set X
hnull : ∀ (n : ℕ), ↑μ (s n) = 0
hcov : t ⊆ ⋃ n, s n
... |
https://github.com/fpvandoorn/sard.git | f4a1bb550136e8591dcd20dca5bdcce766a958c7 | Sard/LocallyLipschitzMeasureZero.lean | null_set_from_countable_cover | [11, 1] | [23, 10] | calc μ t
_ = μ ((⋃ (n : ℕ), s n) ∩ t) := by simp only [inter_eq_right.mpr hcov]
_ = μ (⋃ (n : ℕ), (s n ∩ t)) := by rw [iUnion_inter]
_ ≤ ∑' (n : ℕ), μ (s n ∩ t) := by apply OuterMeasure.iUnion_nat
_ = ∑' (n : ℕ), 0 := by simp_rw [this]
_ = 0 := by rw [tsum_zero] | X : Type u_1
Y : Type u_2
inst✝⁵ : MetricSpace X
inst✝⁴ : MeasurableSpace X
inst✝³ : BorelSpace X
inst✝² : MetricSpace Y
inst✝¹ : MeasurableSpace Y
inst✝ : BorelSpace Y
t : Set X
μ : Measure X
s : ℕ → Set X
hnull : ∀ (n : ℕ), ↑μ (s n) = 0
hcov : t ⊆ ⋃ n, s n
this : ∀ (n : ℕ), ↑μ (s n ∩ t) = 0
⊢ ↑μ t ≤ 0 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
X : Type u_1
Y : Type u_2
inst✝⁵ : MetricSpace X
inst✝⁴ : MeasurableSpace X
inst✝³ : BorelSpace X
inst✝² : MetricSpace Y
inst✝¹ : MeasurableSpace Y
inst✝ : BorelSpace Y
t : Set X
μ : Measure X
s : ℕ → Set X
hnull : ∀ (n : ℕ), ↑μ (s n) = 0
hcov : t ⊆ ⋃ n, s n
... |
https://github.com/fpvandoorn/sard.git | f4a1bb550136e8591dcd20dca5bdcce766a958c7 | Sard/LocallyLipschitzMeasureZero.lean | null_set_from_countable_cover | [11, 1] | [23, 10] | simp only [inter_eq_right.mpr hcov] | X : Type u_1
Y : Type u_2
inst✝⁵ : MetricSpace X
inst✝⁴ : MeasurableSpace X
inst✝³ : BorelSpace X
inst✝² : MetricSpace Y
inst✝¹ : MeasurableSpace Y
inst✝ : BorelSpace Y
t : Set X
μ : Measure X
s : ℕ → Set X
hnull : ∀ (n : ℕ), ↑μ (s n) = 0
hcov : t ⊆ ⋃ n, s n
this : ∀ (n : ℕ), ↑μ (s n ∩ t) = 0
⊢ ↑μ t = ↑μ ((⋃ n, s n) ∩ ... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
X : Type u_1
Y : Type u_2
inst✝⁵ : MetricSpace X
inst✝⁴ : MeasurableSpace X
inst✝³ : BorelSpace X
inst✝² : MetricSpace Y
inst✝¹ : MeasurableSpace Y
inst✝ : BorelSpace Y
t : Set X
μ : Measure X
s : ℕ → Set X
hnull : ∀ (n : ℕ), ↑μ (s n) = 0
hcov : t ⊆ ⋃ n, s n
... |
https://github.com/fpvandoorn/sard.git | f4a1bb550136e8591dcd20dca5bdcce766a958c7 | Sard/LocallyLipschitzMeasureZero.lean | null_set_from_countable_cover | [11, 1] | [23, 10] | rw [iUnion_inter] | X : Type u_1
Y : Type u_2
inst✝⁵ : MetricSpace X
inst✝⁴ : MeasurableSpace X
inst✝³ : BorelSpace X
inst✝² : MetricSpace Y
inst✝¹ : MeasurableSpace Y
inst✝ : BorelSpace Y
t : Set X
μ : Measure X
s : ℕ → Set X
hnull : ∀ (n : ℕ), ↑μ (s n) = 0
hcov : t ⊆ ⋃ n, s n
this : ∀ (n : ℕ), ↑μ (s n ∩ t) = 0
⊢ ↑μ ((⋃ n, s n) ∩ t) = ↑μ... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
X : Type u_1
Y : Type u_2
inst✝⁵ : MetricSpace X
inst✝⁴ : MeasurableSpace X
inst✝³ : BorelSpace X
inst✝² : MetricSpace Y
inst✝¹ : MeasurableSpace Y
inst✝ : BorelSpace Y
t : Set X
μ : Measure X
s : ℕ → Set X
hnull : ∀ (n : ℕ), ↑μ (s n) = 0
hcov : t ⊆ ⋃ n, s n
... |
https://github.com/fpvandoorn/sard.git | f4a1bb550136e8591dcd20dca5bdcce766a958c7 | Sard/LocallyLipschitzMeasureZero.lean | null_set_from_countable_cover | [11, 1] | [23, 10] | apply OuterMeasure.iUnion_nat | X : Type u_1
Y : Type u_2
inst✝⁵ : MetricSpace X
inst✝⁴ : MeasurableSpace X
inst✝³ : BorelSpace X
inst✝² : MetricSpace Y
inst✝¹ : MeasurableSpace Y
inst✝ : BorelSpace Y
t : Set X
μ : Measure X
s : ℕ → Set X
hnull : ∀ (n : ℕ), ↑μ (s n) = 0
hcov : t ⊆ ⋃ n, s n
this : ∀ (n : ℕ), ↑μ (s n ∩ t) = 0
⊢ ↑μ (⋃ n, s n ∩ t) ≤ ∑' (... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
X : Type u_1
Y : Type u_2
inst✝⁵ : MetricSpace X
inst✝⁴ : MeasurableSpace X
inst✝³ : BorelSpace X
inst✝² : MetricSpace Y
inst✝¹ : MeasurableSpace Y
inst✝ : BorelSpace Y
t : Set X
μ : Measure X
s : ℕ → Set X
hnull : ∀ (n : ℕ), ↑μ (s n) = 0
hcov : t ⊆ ⋃ n, s n
... |
https://github.com/fpvandoorn/sard.git | f4a1bb550136e8591dcd20dca5bdcce766a958c7 | Sard/LocallyLipschitzMeasureZero.lean | null_set_from_countable_cover | [11, 1] | [23, 10] | simp_rw [this] | X : Type u_1
Y : Type u_2
inst✝⁵ : MetricSpace X
inst✝⁴ : MeasurableSpace X
inst✝³ : BorelSpace X
inst✝² : MetricSpace Y
inst✝¹ : MeasurableSpace Y
inst✝ : BorelSpace Y
t : Set X
μ : Measure X
s : ℕ → Set X
hnull : ∀ (n : ℕ), ↑μ (s n) = 0
hcov : t ⊆ ⋃ n, s n
this : ∀ (n : ℕ), ↑μ (s n ∩ t) = 0
⊢ ∑' (n : ℕ), ↑μ (s n ∩ t)... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
X : Type u_1
Y : Type u_2
inst✝⁵ : MetricSpace X
inst✝⁴ : MeasurableSpace X
inst✝³ : BorelSpace X
inst✝² : MetricSpace Y
inst✝¹ : MeasurableSpace Y
inst✝ : BorelSpace Y
t : Set X
μ : Measure X
s : ℕ → Set X
hnull : ∀ (n : ℕ), ↑μ (s n) = 0
hcov : t ⊆ ⋃ n, s n
... |
https://github.com/fpvandoorn/sard.git | f4a1bb550136e8591dcd20dca5bdcce766a958c7 | Sard/LocallyLipschitzMeasureZero.lean | null_set_from_countable_cover | [11, 1] | [23, 10] | rw [tsum_zero] | X : Type u_1
Y : Type u_2
inst✝⁵ : MetricSpace X
inst✝⁴ : MeasurableSpace X
inst✝³ : BorelSpace X
inst✝² : MetricSpace Y
inst✝¹ : MeasurableSpace Y
inst✝ : BorelSpace Y
t : Set X
μ : Measure X
s : ℕ → Set X
hnull : ∀ (n : ℕ), ↑μ (s n) = 0
hcov : t ⊆ ⋃ n, s n
this : ∀ (n : ℕ), ↑μ (s n ∩ t) = 0
⊢ ∑' (n : ℕ), 0 = 0 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
X : Type u_1
Y : Type u_2
inst✝⁵ : MetricSpace X
inst✝⁴ : MeasurableSpace X
inst✝³ : BorelSpace X
inst✝² : MetricSpace Y
inst✝¹ : MeasurableSpace Y
inst✝ : BorelSpace Y
t : Set X
μ : Measure X
s : ℕ → Set X
hnull : ∀ (n : ℕ), ↑μ (s n) = 0
hcov : t ⊆ ⋃ n, s n
... |
https://github.com/fpvandoorn/sard.git | f4a1bb550136e8591dcd20dca5bdcce766a958c7 | Sard/LocallyLipschitzMeasureZero.lean | lipschitz_image_null_set_is_null_set | [26, 1] | [33, 12] | have aux : μH[d] (f '' s) ≤ (K : ENNReal) ^ d * μH[d] s := hf.hausdorffMeasure_image_le hd | X : Type u_1
Y : Type u_2
inst✝⁵ : MetricSpace X
inst✝⁴ : MeasurableSpace X
inst✝³ : BorelSpace X
inst✝² : MetricSpace Y
inst✝¹ : MeasurableSpace Y
inst✝ : BorelSpace Y
d : ℝ
hd : d ≥ 0
s : Set X
f : X → Y
K : ℝ≥0
hf : LipschitzOnWith K f s
hs : ↑μH[d] s = 0
⊢ ↑μH[d] (f '' s) = 0 | X : Type u_1
Y : Type u_2
inst✝⁵ : MetricSpace X
inst✝⁴ : MeasurableSpace X
inst✝³ : BorelSpace X
inst✝² : MetricSpace Y
inst✝¹ : MeasurableSpace Y
inst✝ : BorelSpace Y
d : ℝ
hd : d ≥ 0
s : Set X
f : X → Y
K : ℝ≥0
hf : LipschitzOnWith K f s
hs : ↑μH[d] s = 0
aux : ↑μH[d] (f '' s) ≤ ↑K ^ d * ↑μH[d] s
⊢ ↑μH[d] (f '' s) =... | Please generate a tactic in lean4 to solve the state.
STATE:
X : Type u_1
Y : Type u_2
inst✝⁵ : MetricSpace X
inst✝⁴ : MeasurableSpace X
inst✝³ : BorelSpace X
inst✝² : MetricSpace Y
inst✝¹ : MeasurableSpace Y
inst✝ : BorelSpace Y
d : ℝ
hd : d ≥ 0
s : Set X
f : X → Y
K : ℝ≥0
hf : LipschitzOnWith K f s
hs : ↑μH[d] s = 0
... |
https://github.com/fpvandoorn/sard.git | f4a1bb550136e8591dcd20dca5bdcce766a958c7 | Sard/LocallyLipschitzMeasureZero.lean | lipschitz_image_null_set_is_null_set | [26, 1] | [33, 12] | rw [hs] at aux | X : Type u_1
Y : Type u_2
inst✝⁵ : MetricSpace X
inst✝⁴ : MeasurableSpace X
inst✝³ : BorelSpace X
inst✝² : MetricSpace Y
inst✝¹ : MeasurableSpace Y
inst✝ : BorelSpace Y
d : ℝ
hd : d ≥ 0
s : Set X
f : X → Y
K : ℝ≥0
hf : LipschitzOnWith K f s
hs : ↑μH[d] s = 0
aux : ↑μH[d] (f '' s) ≤ ↑K ^ d * ↑μH[d] s
⊢ ↑μH[d] (f '' s) =... | X : Type u_1
Y : Type u_2
inst✝⁵ : MetricSpace X
inst✝⁴ : MeasurableSpace X
inst✝³ : BorelSpace X
inst✝² : MetricSpace Y
inst✝¹ : MeasurableSpace Y
inst✝ : BorelSpace Y
d : ℝ
hd : d ≥ 0
s : Set X
f : X → Y
K : ℝ≥0
hf : LipschitzOnWith K f s
hs : ↑μH[d] s = 0
aux : ↑μH[d] (f '' s) ≤ ↑K ^ d * 0
⊢ ↑μH[d] (f '' s) = 0 | Please generate a tactic in lean4 to solve the state.
STATE:
X : Type u_1
Y : Type u_2
inst✝⁵ : MetricSpace X
inst✝⁴ : MeasurableSpace X
inst✝³ : BorelSpace X
inst✝² : MetricSpace Y
inst✝¹ : MeasurableSpace Y
inst✝ : BorelSpace Y
d : ℝ
hd : d ≥ 0
s : Set X
f : X → Y
K : ℝ≥0
hf : LipschitzOnWith K f s
hs : ↑μH[d] s = 0
... |
https://github.com/fpvandoorn/sard.git | f4a1bb550136e8591dcd20dca5bdcce766a958c7 | Sard/LocallyLipschitzMeasureZero.lean | lipschitz_image_null_set_is_null_set | [26, 1] | [33, 12] | simp only [mul_zero, nonpos_iff_eq_zero, hs] at aux ⊢ | X : Type u_1
Y : Type u_2
inst✝⁵ : MetricSpace X
inst✝⁴ : MeasurableSpace X
inst✝³ : BorelSpace X
inst✝² : MetricSpace Y
inst✝¹ : MeasurableSpace Y
inst✝ : BorelSpace Y
d : ℝ
hd : d ≥ 0
s : Set X
f : X → Y
K : ℝ≥0
hf : LipschitzOnWith K f s
hs : ↑μH[d] s = 0
aux : ↑μH[d] (f '' s) ≤ ↑K ^ d * 0
⊢ ↑μH[d] (f '' s) = 0 | X : Type u_1
Y : Type u_2
inst✝⁵ : MetricSpace X
inst✝⁴ : MeasurableSpace X
inst✝³ : BorelSpace X
inst✝² : MetricSpace Y
inst✝¹ : MeasurableSpace Y
inst✝ : BorelSpace Y
d : ℝ
hd : d ≥ 0
s : Set X
f : X → Y
K : ℝ≥0
hf : LipschitzOnWith K f s
hs : ↑μH[d] s = 0
aux : ↑μH[d] (f '' s) = 0
⊢ ↑μH[d] (f '' s) = 0 | Please generate a tactic in lean4 to solve the state.
STATE:
X : Type u_1
Y : Type u_2
inst✝⁵ : MetricSpace X
inst✝⁴ : MeasurableSpace X
inst✝³ : BorelSpace X
inst✝² : MetricSpace Y
inst✝¹ : MeasurableSpace Y
inst✝ : BorelSpace Y
d : ℝ
hd : d ≥ 0
s : Set X
f : X → Y
K : ℝ≥0
hf : LipschitzOnWith K f s
hs : ↑μH[d] s = 0
... |
https://github.com/fpvandoorn/sard.git | f4a1bb550136e8591dcd20dca5bdcce766a958c7 | Sard/LocallyLipschitzMeasureZero.lean | lipschitz_image_null_set_is_null_set | [26, 1] | [33, 12] | exact aux | X : Type u_1
Y : Type u_2
inst✝⁵ : MetricSpace X
inst✝⁴ : MeasurableSpace X
inst✝³ : BorelSpace X
inst✝² : MetricSpace Y
inst✝¹ : MeasurableSpace Y
inst✝ : BorelSpace Y
d : ℝ
hd : d ≥ 0
s : Set X
f : X → Y
K : ℝ≥0
hf : LipschitzOnWith K f s
hs : ↑μH[d] s = 0
aux : ↑μH[d] (f '' s) = 0
⊢ ↑μH[d] (f '' s) = 0 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
X : Type u_1
Y : Type u_2
inst✝⁵ : MetricSpace X
inst✝⁴ : MeasurableSpace X
inst✝³ : BorelSpace X
inst✝² : MetricSpace Y
inst✝¹ : MeasurableSpace Y
inst✝ : BorelSpace Y
d : ℝ
hd : d ≥ 0
s : Set X
f : X → Y
K : ℝ≥0
hf : LipschitzOnWith K f s
hs : ↑μH[d] s = 0
... |
https://github.com/fpvandoorn/sard.git | f4a1bb550136e8591dcd20dca5bdcce766a958c7 | Sard/LocallyLipschitzMeasureZero.lean | locally_lipschitz_image_of_null_set_is_null_set | [35, 1] | [87, 14] | let K : ℕ → Set X := compactCovering X | X : Type u_1
Y : Type u_2
inst✝⁶ : MetricSpace X
inst✝⁵ : MeasurableSpace X
inst✝⁴ : BorelSpace X
inst✝³ : MetricSpace Y
inst✝² : MeasurableSpace Y
inst✝¹ : BorelSpace Y
inst✝ : SigmaCompactSpace X
d : ℕ
f : X → Y
hf : LocallyLipschitz f
s : Set X
hs : ↑μH[↑d] s = 0
⊢ ↑μH[↑d] (f '' s) = 0 | X : Type u_1
Y : Type u_2
inst✝⁶ : MetricSpace X
inst✝⁵ : MeasurableSpace X
inst✝⁴ : BorelSpace X
inst✝³ : MetricSpace Y
inst✝² : MeasurableSpace Y
inst✝¹ : BorelSpace Y
inst✝ : SigmaCompactSpace X
d : ℕ
f : X → Y
hf : LocallyLipschitz f
s : Set X
hs : ↑μH[↑d] s = 0
K : ℕ → Set X := compactCovering X
⊢ ↑μH[↑d] (f '' s)... | Please generate a tactic in lean4 to solve the state.
STATE:
X : Type u_1
Y : Type u_2
inst✝⁶ : MetricSpace X
inst✝⁵ : MeasurableSpace X
inst✝⁴ : BorelSpace X
inst✝³ : MetricSpace Y
inst✝² : MeasurableSpace Y
inst✝¹ : BorelSpace Y
inst✝ : SigmaCompactSpace X
d : ℕ
f : X → Y
hf : LocallyLipschitz f
s : Set X
hs : ↑μH[↑d... |
https://github.com/fpvandoorn/sard.git | f4a1bb550136e8591dcd20dca5bdcce766a958c7 | Sard/LocallyLipschitzMeasureZero.lean | locally_lipschitz_image_of_null_set_is_null_set | [35, 1] | [87, 14] | have hcov : ⋃ (n : ℕ), K n = univ := iUnion_compactCovering X | X : Type u_1
Y : Type u_2
inst✝⁶ : MetricSpace X
inst✝⁵ : MeasurableSpace X
inst✝⁴ : BorelSpace X
inst✝³ : MetricSpace Y
inst✝² : MeasurableSpace Y
inst✝¹ : BorelSpace Y
inst✝ : SigmaCompactSpace X
d : ℕ
f : X → Y
hf : LocallyLipschitz f
s : Set X
hs : ↑μH[↑d] s = 0
K : ℕ → Set X := compactCovering X
⊢ ↑μH[↑d] (f '' s)... | X : Type u_1
Y : Type u_2
inst✝⁶ : MetricSpace X
inst✝⁵ : MeasurableSpace X
inst✝⁴ : BorelSpace X
inst✝³ : MetricSpace Y
inst✝² : MeasurableSpace Y
inst✝¹ : BorelSpace Y
inst✝ : SigmaCompactSpace X
d : ℕ
f : X → Y
hf : LocallyLipschitz f
s : Set X
hs : ↑μH[↑d] s = 0
K : ℕ → Set X := compactCovering X
hcov : ⋃ n, K n = ... | Please generate a tactic in lean4 to solve the state.
STATE:
X : Type u_1
Y : Type u_2
inst✝⁶ : MetricSpace X
inst✝⁵ : MeasurableSpace X
inst✝⁴ : BorelSpace X
inst✝³ : MetricSpace Y
inst✝² : MeasurableSpace Y
inst✝¹ : BorelSpace Y
inst✝ : SigmaCompactSpace X
d : ℕ
f : X → Y
hf : LocallyLipschitz f
s : Set X
hs : ↑μH[↑d... |
https://github.com/fpvandoorn/sard.git | f4a1bb550136e8591dcd20dca5bdcce766a958c7 | Sard/LocallyLipschitzMeasureZero.lean | locally_lipschitz_image_of_null_set_is_null_set | [35, 1] | [87, 14] | have hcompact : ∀ n : ℕ, IsCompact (K n) := isCompact_compactCovering X | X : Type u_1
Y : Type u_2
inst✝⁶ : MetricSpace X
inst✝⁵ : MeasurableSpace X
inst✝⁴ : BorelSpace X
inst✝³ : MetricSpace Y
inst✝² : MeasurableSpace Y
inst✝¹ : BorelSpace Y
inst✝ : SigmaCompactSpace X
d : ℕ
f : X → Y
hf : LocallyLipschitz f
s : Set X
hs : ↑μH[↑d] s = 0
K : ℕ → Set X := compactCovering X
hcov : ⋃ n, K n = ... | X : Type u_1
Y : Type u_2
inst✝⁶ : MetricSpace X
inst✝⁵ : MeasurableSpace X
inst✝⁴ : BorelSpace X
inst✝³ : MetricSpace Y
inst✝² : MeasurableSpace Y
inst✝¹ : BorelSpace Y
inst✝ : SigmaCompactSpace X
d : ℕ
f : X → Y
hf : LocallyLipschitz f
s : Set X
hs : ↑μH[↑d] s = 0
K : ℕ → Set X := compactCovering X
hcov : ⋃ n, K n = ... | Please generate a tactic in lean4 to solve the state.
STATE:
X : Type u_1
Y : Type u_2
inst✝⁶ : MetricSpace X
inst✝⁵ : MeasurableSpace X
inst✝⁴ : BorelSpace X
inst✝³ : MetricSpace Y
inst✝² : MeasurableSpace Y
inst✝¹ : BorelSpace Y
inst✝ : SigmaCompactSpace X
d : ℕ
f : X → Y
hf : LocallyLipschitz f
s : Set X
hs : ↑μH[↑d... |
https://github.com/fpvandoorn/sard.git | f4a1bb550136e8591dcd20dca5bdcce766a958c7 | Sard/LocallyLipschitzMeasureZero.lean | locally_lipschitz_image_of_null_set_is_null_set | [35, 1] | [87, 14] | suffices ass : ∀ n : ℕ, μH[d] (f '' (s ∩ K n)) = 0 by
apply null_set_from_countable_cover _ _ ass
rw [← image_iUnion]
have : s = ⋃ (n : ℕ), s ∩ K n := by calc s
_ = s ∩ univ := (inter_univ s).symm
_ = s ∩ ⋃ (n : ℕ), K n := by rw [hcov]
_ = ⋃ (n : ℕ), s ∩ K n := by apply inter_iUnion s
exact Eq... | X : Type u_1
Y : Type u_2
inst✝⁶ : MetricSpace X
inst✝⁵ : MeasurableSpace X
inst✝⁴ : BorelSpace X
inst✝³ : MetricSpace Y
inst✝² : MeasurableSpace Y
inst✝¹ : BorelSpace Y
inst✝ : SigmaCompactSpace X
d : ℕ
f : X → Y
hf : LocallyLipschitz f
s : Set X
hs : ↑μH[↑d] s = 0
K : ℕ → Set X := compactCovering X
hcov : ⋃ n, K n = ... | X : Type u_1
Y : Type u_2
inst✝⁶ : MetricSpace X
inst✝⁵ : MeasurableSpace X
inst✝⁴ : BorelSpace X
inst✝³ : MetricSpace Y
inst✝² : MeasurableSpace Y
inst✝¹ : BorelSpace Y
inst✝ : SigmaCompactSpace X
d : ℕ
f : X → Y
hf : LocallyLipschitz f
s : Set X
hs : ↑μH[↑d] s = 0
K : ℕ → Set X := compactCovering X
hcov : ⋃ n, K n = ... | Please generate a tactic in lean4 to solve the state.
STATE:
X : Type u_1
Y : Type u_2
inst✝⁶ : MetricSpace X
inst✝⁵ : MeasurableSpace X
inst✝⁴ : BorelSpace X
inst✝³ : MetricSpace Y
inst✝² : MeasurableSpace Y
inst✝¹ : BorelSpace Y
inst✝ : SigmaCompactSpace X
d : ℕ
f : X → Y
hf : LocallyLipschitz f
s : Set X
hs : ↑μH[↑d... |
https://github.com/fpvandoorn/sard.git | f4a1bb550136e8591dcd20dca5bdcce766a958c7 | Sard/LocallyLipschitzMeasureZero.lean | locally_lipschitz_image_of_null_set_is_null_set | [35, 1] | [87, 14] | intro n | X : Type u_1
Y : Type u_2
inst✝⁶ : MetricSpace X
inst✝⁵ : MeasurableSpace X
inst✝⁴ : BorelSpace X
inst✝³ : MetricSpace Y
inst✝² : MeasurableSpace Y
inst✝¹ : BorelSpace Y
inst✝ : SigmaCompactSpace X
d : ℕ
f : X → Y
hf : LocallyLipschitz f
s : Set X
hs : ↑μH[↑d] s = 0
K : ℕ → Set X := compactCovering X
hcov : ⋃ n, K n = ... | X : Type u_1
Y : Type u_2
inst✝⁶ : MetricSpace X
inst✝⁵ : MeasurableSpace X
inst✝⁴ : BorelSpace X
inst✝³ : MetricSpace Y
inst✝² : MeasurableSpace Y
inst✝¹ : BorelSpace Y
inst✝ : SigmaCompactSpace X
d : ℕ
f : X → Y
hf : LocallyLipschitz f
s : Set X
hs : ↑μH[↑d] s = 0
K : ℕ → Set X := compactCovering X
hcov : ⋃ n, K n = ... | Please generate a tactic in lean4 to solve the state.
STATE:
X : Type u_1
Y : Type u_2
inst✝⁶ : MetricSpace X
inst✝⁵ : MeasurableSpace X
inst✝⁴ : BorelSpace X
inst✝³ : MetricSpace Y
inst✝² : MeasurableSpace Y
inst✝¹ : BorelSpace Y
inst✝ : SigmaCompactSpace X
d : ℕ
f : X → Y
hf : LocallyLipschitz f
s : Set X
hs : ↑μH[↑d... |
https://github.com/fpvandoorn/sard.git | f4a1bb550136e8591dcd20dca5bdcce766a958c7 | Sard/LocallyLipschitzMeasureZero.lean | locally_lipschitz_image_of_null_set_is_null_set | [35, 1] | [87, 14] | choose Kx t ht hfL using fun x ↦ (hf x) | X : Type u_1
Y : Type u_2
inst✝⁶ : MetricSpace X
inst✝⁵ : MeasurableSpace X
inst✝⁴ : BorelSpace X
inst✝³ : MetricSpace Y
inst✝² : MeasurableSpace Y
inst✝¹ : BorelSpace Y
inst✝ : SigmaCompactSpace X
d : ℕ
f : X → Y
hf : LocallyLipschitz f
s : Set X
hs : ↑μH[↑d] s = 0
K : ℕ → Set X := compactCovering X
hcov : ⋃ n, K n = ... | X : Type u_1
Y : Type u_2
inst✝⁶ : MetricSpace X
inst✝⁵ : MeasurableSpace X
inst✝⁴ : BorelSpace X
inst✝³ : MetricSpace Y
inst✝² : MeasurableSpace Y
inst✝¹ : BorelSpace Y
inst✝ : SigmaCompactSpace X
d : ℕ
f : X → Y
hf : LocallyLipschitz f
s : Set X
hs : ↑μH[↑d] s = 0
K : ℕ → Set X := compactCovering X
hcov : ⋃ n, K n = ... | Please generate a tactic in lean4 to solve the state.
STATE:
X : Type u_1
Y : Type u_2
inst✝⁶ : MetricSpace X
inst✝⁵ : MeasurableSpace X
inst✝⁴ : BorelSpace X
inst✝³ : MetricSpace Y
inst✝² : MeasurableSpace Y
inst✝¹ : BorelSpace Y
inst✝ : SigmaCompactSpace X
d : ℕ
f : X → Y
hf : LocallyLipschitz f
s : Set X
hs : ↑μH[↑d... |
https://github.com/fpvandoorn/sard.git | f4a1bb550136e8591dcd20dca5bdcce766a958c7 | Sard/LocallyLipschitzMeasureZero.lean | locally_lipschitz_image_of_null_set_is_null_set | [35, 1] | [87, 14] | choose U hut hUopen hxU using fun x ↦ mem_nhds_iff.mp (ht x) | X : Type u_1
Y : Type u_2
inst✝⁶ : MetricSpace X
inst✝⁵ : MeasurableSpace X
inst✝⁴ : BorelSpace X
inst✝³ : MetricSpace Y
inst✝² : MeasurableSpace Y
inst✝¹ : BorelSpace Y
inst✝ : SigmaCompactSpace X
d : ℕ
f : X → Y
hf : LocallyLipschitz f
s : Set X
hs : ↑μH[↑d] s = 0
K : ℕ → Set X := compactCovering X
hcov : ⋃ n, K n = ... | X : Type u_1
Y : Type u_2
inst✝⁶ : MetricSpace X
inst✝⁵ : MeasurableSpace X
inst✝⁴ : BorelSpace X
inst✝³ : MetricSpace Y
inst✝² : MeasurableSpace Y
inst✝¹ : BorelSpace Y
inst✝ : SigmaCompactSpace X
d : ℕ
f : X → Y
hf : LocallyLipschitz f
s : Set X
hs : ↑μH[↑d] s = 0
K : ℕ → Set X := compactCovering X
hcov : ⋃ n, K n = ... | Please generate a tactic in lean4 to solve the state.
STATE:
X : Type u_1
Y : Type u_2
inst✝⁶ : MetricSpace X
inst✝⁵ : MeasurableSpace X
inst✝⁴ : BorelSpace X
inst✝³ : MetricSpace Y
inst✝² : MeasurableSpace Y
inst✝¹ : BorelSpace Y
inst✝ : SigmaCompactSpace X
d : ℕ
f : X → Y
hf : LocallyLipschitz f
s : Set X
hs : ↑μH[↑d... |
https://github.com/fpvandoorn/sard.git | f4a1bb550136e8591dcd20dca5bdcce766a958c7 | Sard/LocallyLipschitzMeasureZero.lean | locally_lipschitz_image_of_null_set_is_null_set | [35, 1] | [87, 14] | have hcovering : K n ⊆ ⋃ (x : X), U x := fun y _ ↦ mem_iUnion_of_mem y (hxU y) | X : Type u_1
Y : Type u_2
inst✝⁶ : MetricSpace X
inst✝⁵ : MeasurableSpace X
inst✝⁴ : BorelSpace X
inst✝³ : MetricSpace Y
inst✝² : MeasurableSpace Y
inst✝¹ : BorelSpace Y
inst✝ : SigmaCompactSpace X
d : ℕ
f : X → Y
hf : LocallyLipschitz f
s : Set X
hs : ↑μH[↑d] s = 0
K : ℕ → Set X := compactCovering X
hcov : ⋃ n, K n = ... | X : Type u_1
Y : Type u_2
inst✝⁶ : MetricSpace X
inst✝⁵ : MeasurableSpace X
inst✝⁴ : BorelSpace X
inst✝³ : MetricSpace Y
inst✝² : MeasurableSpace Y
inst✝¹ : BorelSpace Y
inst✝ : SigmaCompactSpace X
d : ℕ
f : X → Y
hf : LocallyLipschitz f
s : Set X
hs : ↑μH[↑d] s = 0
K : ℕ → Set X := compactCovering X
hcov : ⋃ n, K n = ... | Please generate a tactic in lean4 to solve the state.
STATE:
X : Type u_1
Y : Type u_2
inst✝⁶ : MetricSpace X
inst✝⁵ : MeasurableSpace X
inst✝⁴ : BorelSpace X
inst✝³ : MetricSpace Y
inst✝² : MeasurableSpace Y
inst✝¹ : BorelSpace Y
inst✝ : SigmaCompactSpace X
d : ℕ
f : X → Y
hf : LocallyLipschitz f
s : Set X
hs : ↑μH[↑d... |
https://github.com/fpvandoorn/sard.git | f4a1bb550136e8591dcd20dca5bdcce766a958c7 | Sard/LocallyLipschitzMeasureZero.lean | locally_lipschitz_image_of_null_set_is_null_set | [35, 1] | [87, 14] | let ⟨v, hv⟩ := IsCompact.elim_finite_subcover (hcompact n) U hUopen hcovering | X : Type u_1
Y : Type u_2
inst✝⁶ : MetricSpace X
inst✝⁵ : MeasurableSpace X
inst✝⁴ : BorelSpace X
inst✝³ : MetricSpace Y
inst✝² : MeasurableSpace Y
inst✝¹ : BorelSpace Y
inst✝ : SigmaCompactSpace X
d : ℕ
f : X → Y
hf : LocallyLipschitz f
s : Set X
hs : ↑μH[↑d] s = 0
K : ℕ → Set X := compactCovering X
hcov : ⋃ n, K n = ... | X : Type u_1
Y : Type u_2
inst✝⁶ : MetricSpace X
inst✝⁵ : MeasurableSpace X
inst✝⁴ : BorelSpace X
inst✝³ : MetricSpace Y
inst✝² : MeasurableSpace Y
inst✝¹ : BorelSpace Y
inst✝ : SigmaCompactSpace X
d : ℕ
f : X → Y
hf : LocallyLipschitz f
s : Set X
hs : ↑μH[↑d] s = 0
K : ℕ → Set X := compactCovering X
hcov : ⋃ n, K n = ... | Please generate a tactic in lean4 to solve the state.
STATE:
X : Type u_1
Y : Type u_2
inst✝⁶ : MetricSpace X
inst✝⁵ : MeasurableSpace X
inst✝⁴ : BorelSpace X
inst✝³ : MetricSpace Y
inst✝² : MeasurableSpace Y
inst✝¹ : BorelSpace Y
inst✝ : SigmaCompactSpace X
d : ℕ
f : X → Y
hf : LocallyLipschitz f
s : Set X
hs : ↑μH[↑d... |
https://github.com/fpvandoorn/sard.git | f4a1bb550136e8591dcd20dca5bdcce766a958c7 | Sard/LocallyLipschitzMeasureZero.lean | locally_lipschitz_image_of_null_set_is_null_set | [35, 1] | [87, 14] | have hnull: ∀ i : v, μH[d] (f '' (s ∩ U i)) = 0 := by
intro i
have h₂ : μH[d] (s ∩ U i) = 0 := measure_mono_null (inter_subset_left s (U ↑i)) hs
have h₁ := ((hfL i).mono (hut i)).mono (inter_subset_right s (U i))
refine lipschitz_image_null_set_is_null_set (Nat.cast_nonneg d) h₁ h₂ | X : Type u_1
Y : Type u_2
inst✝⁶ : MetricSpace X
inst✝⁵ : MeasurableSpace X
inst✝⁴ : BorelSpace X
inst✝³ : MetricSpace Y
inst✝² : MeasurableSpace Y
inst✝¹ : BorelSpace Y
inst✝ : SigmaCompactSpace X
d : ℕ
f : X → Y
hf : LocallyLipschitz f
s : Set X
hs : ↑μH[↑d] s = 0
K : ℕ → Set X := compactCovering X
hcov : ⋃ n, K n = ... | X : Type u_1
Y : Type u_2
inst✝⁶ : MetricSpace X
inst✝⁵ : MeasurableSpace X
inst✝⁴ : BorelSpace X
inst✝³ : MetricSpace Y
inst✝² : MeasurableSpace Y
inst✝¹ : BorelSpace Y
inst✝ : SigmaCompactSpace X
d : ℕ
f : X → Y
hf : LocallyLipschitz f
s : Set X
hs : ↑μH[↑d] s = 0
K : ℕ → Set X := compactCovering X
hcov : ⋃ n, K n = ... | Please generate a tactic in lean4 to solve the state.
STATE:
X : Type u_1
Y : Type u_2
inst✝⁶ : MetricSpace X
inst✝⁵ : MeasurableSpace X
inst✝⁴ : BorelSpace X
inst✝³ : MetricSpace Y
inst✝² : MeasurableSpace Y
inst✝¹ : BorelSpace Y
inst✝ : SigmaCompactSpace X
d : ℕ
f : X → Y
hf : LocallyLipschitz f
s : Set X
hs : ↑μH[↑d... |
https://github.com/fpvandoorn/sard.git | f4a1bb550136e8591dcd20dca5bdcce766a958c7 | Sard/LocallyLipschitzMeasureZero.lean | locally_lipschitz_image_of_null_set_is_null_set | [35, 1] | [87, 14] | have coe : ⋃ (i : v), U i = ⋃ (i : X) (_ : i ∈ v), U i := iUnion_coe_set _ _ | X : Type u_1
Y : Type u_2
inst✝⁶ : MetricSpace X
inst✝⁵ : MeasurableSpace X
inst✝⁴ : BorelSpace X
inst✝³ : MetricSpace Y
inst✝² : MeasurableSpace Y
inst✝¹ : BorelSpace Y
inst✝ : SigmaCompactSpace X
d : ℕ
f : X → Y
hf : LocallyLipschitz f
s : Set X
hs : ↑μH[↑d] s = 0
K : ℕ → Set X := compactCovering X
hcov : ⋃ n, K n = ... | X : Type u_1
Y : Type u_2
inst✝⁶ : MetricSpace X
inst✝⁵ : MeasurableSpace X
inst✝⁴ : BorelSpace X
inst✝³ : MetricSpace Y
inst✝² : MeasurableSpace Y
inst✝¹ : BorelSpace Y
inst✝ : SigmaCompactSpace X
d : ℕ
f : X → Y
hf : LocallyLipschitz f
s : Set X
hs : ↑μH[↑d] s = 0
K : ℕ → Set X := compactCovering X
hcov : ⋃ n, K n = ... | Please generate a tactic in lean4 to solve the state.
STATE:
X : Type u_1
Y : Type u_2
inst✝⁶ : MetricSpace X
inst✝⁵ : MeasurableSpace X
inst✝⁴ : BorelSpace X
inst✝³ : MetricSpace Y
inst✝² : MeasurableSpace Y
inst✝¹ : BorelSpace Y
inst✝ : SigmaCompactSpace X
d : ℕ
f : X → Y
hf : LocallyLipschitz f
s : Set X
hs : ↑μH[↑d... |
https://github.com/fpvandoorn/sard.git | f4a1bb550136e8591dcd20dca5bdcce766a958c7 | Sard/LocallyLipschitzMeasureZero.lean | locally_lipschitz_image_of_null_set_is_null_set | [35, 1] | [87, 14] | rw [← coe] at hv | X : Type u_1
Y : Type u_2
inst✝⁶ : MetricSpace X
inst✝⁵ : MeasurableSpace X
inst✝⁴ : BorelSpace X
inst✝³ : MetricSpace Y
inst✝² : MeasurableSpace Y
inst✝¹ : BorelSpace Y
inst✝ : SigmaCompactSpace X
d : ℕ
f : X → Y
hf : LocallyLipschitz f
s : Set X
hs : ↑μH[↑d] s = 0
K : ℕ → Set X := compactCovering X
hcov : ⋃ n, K n = ... | X : Type u_1
Y : Type u_2
inst✝⁶ : MetricSpace X
inst✝⁵ : MeasurableSpace X
inst✝⁴ : BorelSpace X
inst✝³ : MetricSpace Y
inst✝² : MeasurableSpace Y
inst✝¹ : BorelSpace Y
inst✝ : SigmaCompactSpace X
d : ℕ
f : X → Y
hf : LocallyLipschitz f
s : Set X
hs : ↑μH[↑d] s = 0
K : ℕ → Set X := compactCovering X
hcov : ⋃ n, K n = ... | Please generate a tactic in lean4 to solve the state.
STATE:
X : Type u_1
Y : Type u_2
inst✝⁶ : MetricSpace X
inst✝⁵ : MeasurableSpace X
inst✝⁴ : BorelSpace X
inst✝³ : MetricSpace Y
inst✝² : MeasurableSpace Y
inst✝¹ : BorelSpace Y
inst✝ : SigmaCompactSpace X
d : ℕ
f : X → Y
hf : LocallyLipschitz f
s : Set X
hs : ↑μH[↑d... |
https://github.com/fpvandoorn/sard.git | f4a1bb550136e8591dcd20dca5bdcce766a958c7 | Sard/LocallyLipschitzMeasureZero.lean | locally_lipschitz_image_of_null_set_is_null_set | [35, 1] | [87, 14] | have : f '' (s ∩ (K n)) ⊆ ⋃ (i : v), f '' (s ∩ (U i)) := by
calc f '' (s ∩ (K n))
_ ⊆ f '' (s ∩ (⋃ (i : v), U i)) := by
apply image_subset
apply inter_subset_inter_right s hv
_ = f '' ((⋃ (i : v), s ∩ (U i))) := by rw [inter_iUnion]
_ = ⋃ (i : v), f '' ( s ∩ (U i)) := image_iUnion | X : Type u_1
Y : Type u_2
inst✝⁶ : MetricSpace X
inst✝⁵ : MeasurableSpace X
inst✝⁴ : BorelSpace X
inst✝³ : MetricSpace Y
inst✝² : MeasurableSpace Y
inst✝¹ : BorelSpace Y
inst✝ : SigmaCompactSpace X
d : ℕ
f : X → Y
hf : LocallyLipschitz f
s : Set X
hs : ↑μH[↑d] s = 0
K : ℕ → Set X := compactCovering X
hcov : ⋃ n, K n = ... | X : Type u_1
Y : Type u_2
inst✝⁶ : MetricSpace X
inst✝⁵ : MeasurableSpace X
inst✝⁴ : BorelSpace X
inst✝³ : MetricSpace Y
inst✝² : MeasurableSpace Y
inst✝¹ : BorelSpace Y
inst✝ : SigmaCompactSpace X
d : ℕ
f : X → Y
hf : LocallyLipschitz f
s : Set X
hs : ↑μH[↑d] s = 0
K : ℕ → Set X := compactCovering X
hcov : ⋃ n, K n = ... | Please generate a tactic in lean4 to solve the state.
STATE:
X : Type u_1
Y : Type u_2
inst✝⁶ : MetricSpace X
inst✝⁵ : MeasurableSpace X
inst✝⁴ : BorelSpace X
inst✝³ : MetricSpace Y
inst✝² : MeasurableSpace Y
inst✝¹ : BorelSpace Y
inst✝ : SigmaCompactSpace X
d : ℕ
f : X → Y
hf : LocallyLipschitz f
s : Set X
hs : ↑μH[↑d... |
https://github.com/fpvandoorn/sard.git | f4a1bb550136e8591dcd20dca5bdcce766a958c7 | Sard/LocallyLipschitzMeasureZero.lean | locally_lipschitz_image_of_null_set_is_null_set | [35, 1] | [87, 14] | have hless : μH[d] (f '' (s ∩ (K n))) ≤ 0 := by
calc μH[d] (f '' (s ∩ (K n)))
_ ≤ μH[d] (⋃ (i : v), f '' (s ∩ (U i))) := measure_mono this
_ ≤ ∑' (i : v), (μH[d] (f '' (s ∩ (U i))) : ℝ≥0∞) := by apply measure_iUnion_le
_ = ∑' (_ : v), (0 : ℝ≥0∞) := tsum_congr hnull
_ = 0 := by simp | X : Type u_1
Y : Type u_2
inst✝⁶ : MetricSpace X
inst✝⁵ : MeasurableSpace X
inst✝⁴ : BorelSpace X
inst✝³ : MetricSpace Y
inst✝² : MeasurableSpace Y
inst✝¹ : BorelSpace Y
inst✝ : SigmaCompactSpace X
d : ℕ
f : X → Y
hf : LocallyLipschitz f
s : Set X
hs : ↑μH[↑d] s = 0
K : ℕ → Set X := compactCovering X
hcov : ⋃ n, K n = ... | X : Type u_1
Y : Type u_2
inst✝⁶ : MetricSpace X
inst✝⁵ : MeasurableSpace X
inst✝⁴ : BorelSpace X
inst✝³ : MetricSpace Y
inst✝² : MeasurableSpace Y
inst✝¹ : BorelSpace Y
inst✝ : SigmaCompactSpace X
d : ℕ
f : X → Y
hf : LocallyLipschitz f
s : Set X
hs : ↑μH[↑d] s = 0
K : ℕ → Set X := compactCovering X
hcov : ⋃ n, K n = ... | Please generate a tactic in lean4 to solve the state.
STATE:
X : Type u_1
Y : Type u_2
inst✝⁶ : MetricSpace X
inst✝⁵ : MeasurableSpace X
inst✝⁴ : BorelSpace X
inst✝³ : MetricSpace Y
inst✝² : MeasurableSpace Y
inst✝¹ : BorelSpace Y
inst✝ : SigmaCompactSpace X
d : ℕ
f : X → Y
hf : LocallyLipschitz f
s : Set X
hs : ↑μH[↑d... |
https://github.com/fpvandoorn/sard.git | f4a1bb550136e8591dcd20dca5bdcce766a958c7 | Sard/LocallyLipschitzMeasureZero.lean | locally_lipschitz_image_of_null_set_is_null_set | [35, 1] | [87, 14] | simp only [nonpos_iff_eq_zero, zero_le] at hless ⊢ | X : Type u_1
Y : Type u_2
inst✝⁶ : MetricSpace X
inst✝⁵ : MeasurableSpace X
inst✝⁴ : BorelSpace X
inst✝³ : MetricSpace Y
inst✝² : MeasurableSpace Y
inst✝¹ : BorelSpace Y
inst✝ : SigmaCompactSpace X
d : ℕ
f : X → Y
hf : LocallyLipschitz f
s : Set X
hs : ↑μH[↑d] s = 0
K : ℕ → Set X := compactCovering X
hcov : ⋃ n, K n = ... | X : Type u_1
Y : Type u_2
inst✝⁶ : MetricSpace X
inst✝⁵ : MeasurableSpace X
inst✝⁴ : BorelSpace X
inst✝³ : MetricSpace Y
inst✝² : MeasurableSpace Y
inst✝¹ : BorelSpace Y
inst✝ : SigmaCompactSpace X
d : ℕ
f : X → Y
hf : LocallyLipschitz f
s : Set X
hs : ↑μH[↑d] s = 0
K : ℕ → Set X := compactCovering X
hcov : ⋃ n, K n = ... | Please generate a tactic in lean4 to solve the state.
STATE:
X : Type u_1
Y : Type u_2
inst✝⁶ : MetricSpace X
inst✝⁵ : MeasurableSpace X
inst✝⁴ : BorelSpace X
inst✝³ : MetricSpace Y
inst✝² : MeasurableSpace Y
inst✝¹ : BorelSpace Y
inst✝ : SigmaCompactSpace X
d : ℕ
f : X → Y
hf : LocallyLipschitz f
s : Set X
hs : ↑μH[↑d... |
https://github.com/fpvandoorn/sard.git | f4a1bb550136e8591dcd20dca5bdcce766a958c7 | Sard/LocallyLipschitzMeasureZero.lean | locally_lipschitz_image_of_null_set_is_null_set | [35, 1] | [87, 14] | exact hless | X : Type u_1
Y : Type u_2
inst✝⁶ : MetricSpace X
inst✝⁵ : MeasurableSpace X
inst✝⁴ : BorelSpace X
inst✝³ : MetricSpace Y
inst✝² : MeasurableSpace Y
inst✝¹ : BorelSpace Y
inst✝ : SigmaCompactSpace X
d : ℕ
f : X → Y
hf : LocallyLipschitz f
s : Set X
hs : ↑μH[↑d] s = 0
K : ℕ → Set X := compactCovering X
hcov : ⋃ n, K n = ... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
X : Type u_1
Y : Type u_2
inst✝⁶ : MetricSpace X
inst✝⁵ : MeasurableSpace X
inst✝⁴ : BorelSpace X
inst✝³ : MetricSpace Y
inst✝² : MeasurableSpace Y
inst✝¹ : BorelSpace Y
inst✝ : SigmaCompactSpace X
d : ℕ
f : X → Y
hf : LocallyLipschitz f
s : Set X
hs : ↑μH[↑d... |
https://github.com/fpvandoorn/sard.git | f4a1bb550136e8591dcd20dca5bdcce766a958c7 | Sard/LocallyLipschitzMeasureZero.lean | locally_lipschitz_image_of_null_set_is_null_set | [35, 1] | [87, 14] | apply null_set_from_countable_cover _ _ ass | X : Type u_1
Y : Type u_2
inst✝⁶ : MetricSpace X
inst✝⁵ : MeasurableSpace X
inst✝⁴ : BorelSpace X
inst✝³ : MetricSpace Y
inst✝² : MeasurableSpace Y
inst✝¹ : BorelSpace Y
inst✝ : SigmaCompactSpace X
d : ℕ
f : X → Y
hf : LocallyLipschitz f
s : Set X
hs : ↑μH[↑d] s = 0
K : ℕ → Set X := compactCovering X
hcov : ⋃ n, K n = ... | X : Type u_1
Y : Type u_2
inst✝⁶ : MetricSpace X
inst✝⁵ : MeasurableSpace X
inst✝⁴ : BorelSpace X
inst✝³ : MetricSpace Y
inst✝² : MeasurableSpace Y
inst✝¹ : BorelSpace Y
inst✝ : SigmaCompactSpace X
d : ℕ
f : X → Y
hf : LocallyLipschitz f
s : Set X
hs : ↑μH[↑d] s = 0
K : ℕ → Set X := compactCovering X
hcov : ⋃ n, K n = ... | Please generate a tactic in lean4 to solve the state.
STATE:
X : Type u_1
Y : Type u_2
inst✝⁶ : MetricSpace X
inst✝⁵ : MeasurableSpace X
inst✝⁴ : BorelSpace X
inst✝³ : MetricSpace Y
inst✝² : MeasurableSpace Y
inst✝¹ : BorelSpace Y
inst✝ : SigmaCompactSpace X
d : ℕ
f : X → Y
hf : LocallyLipschitz f
s : Set X
hs : ↑μH[↑d... |
https://github.com/fpvandoorn/sard.git | f4a1bb550136e8591dcd20dca5bdcce766a958c7 | Sard/LocallyLipschitzMeasureZero.lean | locally_lipschitz_image_of_null_set_is_null_set | [35, 1] | [87, 14] | rw [← image_iUnion] | X : Type u_1
Y : Type u_2
inst✝⁶ : MetricSpace X
inst✝⁵ : MeasurableSpace X
inst✝⁴ : BorelSpace X
inst✝³ : MetricSpace Y
inst✝² : MeasurableSpace Y
inst✝¹ : BorelSpace Y
inst✝ : SigmaCompactSpace X
d : ℕ
f : X → Y
hf : LocallyLipschitz f
s : Set X
hs : ↑μH[↑d] s = 0
K : ℕ → Set X := compactCovering X
hcov : ⋃ n, K n = ... | X : Type u_1
Y : Type u_2
inst✝⁶ : MetricSpace X
inst✝⁵ : MeasurableSpace X
inst✝⁴ : BorelSpace X
inst✝³ : MetricSpace Y
inst✝² : MeasurableSpace Y
inst✝¹ : BorelSpace Y
inst✝ : SigmaCompactSpace X
d : ℕ
f : X → Y
hf : LocallyLipschitz f
s : Set X
hs : ↑μH[↑d] s = 0
K : ℕ → Set X := compactCovering X
hcov : ⋃ n, K n = ... | Please generate a tactic in lean4 to solve the state.
STATE:
X : Type u_1
Y : Type u_2
inst✝⁶ : MetricSpace X
inst✝⁵ : MeasurableSpace X
inst✝⁴ : BorelSpace X
inst✝³ : MetricSpace Y
inst✝² : MeasurableSpace Y
inst✝¹ : BorelSpace Y
inst✝ : SigmaCompactSpace X
d : ℕ
f : X → Y
hf : LocallyLipschitz f
s : Set X
hs : ↑μH[↑d... |
https://github.com/fpvandoorn/sard.git | f4a1bb550136e8591dcd20dca5bdcce766a958c7 | Sard/LocallyLipschitzMeasureZero.lean | locally_lipschitz_image_of_null_set_is_null_set | [35, 1] | [87, 14] | have : s = ⋃ (n : ℕ), s ∩ K n := by calc s
_ = s ∩ univ := (inter_univ s).symm
_ = s ∩ ⋃ (n : ℕ), K n := by rw [hcov]
_ = ⋃ (n : ℕ), s ∩ K n := by apply inter_iUnion s | X : Type u_1
Y : Type u_2
inst✝⁶ : MetricSpace X
inst✝⁵ : MeasurableSpace X
inst✝⁴ : BorelSpace X
inst✝³ : MetricSpace Y
inst✝² : MeasurableSpace Y
inst✝¹ : BorelSpace Y
inst✝ : SigmaCompactSpace X
d : ℕ
f : X → Y
hf : LocallyLipschitz f
s : Set X
hs : ↑μH[↑d] s = 0
K : ℕ → Set X := compactCovering X
hcov : ⋃ n, K n = ... | X : Type u_1
Y : Type u_2
inst✝⁶ : MetricSpace X
inst✝⁵ : MeasurableSpace X
inst✝⁴ : BorelSpace X
inst✝³ : MetricSpace Y
inst✝² : MeasurableSpace Y
inst✝¹ : BorelSpace Y
inst✝ : SigmaCompactSpace X
d : ℕ
f : X → Y
hf : LocallyLipschitz f
s : Set X
hs : ↑μH[↑d] s = 0
K : ℕ → Set X := compactCovering X
hcov : ⋃ n, K n = ... | Please generate a tactic in lean4 to solve the state.
STATE:
X : Type u_1
Y : Type u_2
inst✝⁶ : MetricSpace X
inst✝⁵ : MeasurableSpace X
inst✝⁴ : BorelSpace X
inst✝³ : MetricSpace Y
inst✝² : MeasurableSpace Y
inst✝¹ : BorelSpace Y
inst✝ : SigmaCompactSpace X
d : ℕ
f : X → Y
hf : LocallyLipschitz f
s : Set X
hs : ↑μH[↑d... |
https://github.com/fpvandoorn/sard.git | f4a1bb550136e8591dcd20dca5bdcce766a958c7 | Sard/LocallyLipschitzMeasureZero.lean | locally_lipschitz_image_of_null_set_is_null_set | [35, 1] | [87, 14] | exact Eq.subset (congrArg (image f) this) | X : Type u_1
Y : Type u_2
inst✝⁶ : MetricSpace X
inst✝⁵ : MeasurableSpace X
inst✝⁴ : BorelSpace X
inst✝³ : MetricSpace Y
inst✝² : MeasurableSpace Y
inst✝¹ : BorelSpace Y
inst✝ : SigmaCompactSpace X
d : ℕ
f : X → Y
hf : LocallyLipschitz f
s : Set X
hs : ↑μH[↑d] s = 0
K : ℕ → Set X := compactCovering X
hcov : ⋃ n, K n = ... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
X : Type u_1
Y : Type u_2
inst✝⁶ : MetricSpace X
inst✝⁵ : MeasurableSpace X
inst✝⁴ : BorelSpace X
inst✝³ : MetricSpace Y
inst✝² : MeasurableSpace Y
inst✝¹ : BorelSpace Y
inst✝ : SigmaCompactSpace X
d : ℕ
f : X → Y
hf : LocallyLipschitz f
s : Set X
hs : ↑μH[↑d... |
https://github.com/fpvandoorn/sard.git | f4a1bb550136e8591dcd20dca5bdcce766a958c7 | Sard/LocallyLipschitzMeasureZero.lean | locally_lipschitz_image_of_null_set_is_null_set | [35, 1] | [87, 14] | calc s
_ = s ∩ univ := (inter_univ s).symm
_ = s ∩ ⋃ (n : ℕ), K n := by rw [hcov]
_ = ⋃ (n : ℕ), s ∩ K n := by apply inter_iUnion s | X : Type u_1
Y : Type u_2
inst✝⁶ : MetricSpace X
inst✝⁵ : MeasurableSpace X
inst✝⁴ : BorelSpace X
inst✝³ : MetricSpace Y
inst✝² : MeasurableSpace Y
inst✝¹ : BorelSpace Y
inst✝ : SigmaCompactSpace X
d : ℕ
f : X → Y
hf : LocallyLipschitz f
s : Set X
hs : ↑μH[↑d] s = 0
K : ℕ → Set X := compactCovering X
hcov : ⋃ n, K n = ... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
X : Type u_1
Y : Type u_2
inst✝⁶ : MetricSpace X
inst✝⁵ : MeasurableSpace X
inst✝⁴ : BorelSpace X
inst✝³ : MetricSpace Y
inst✝² : MeasurableSpace Y
inst✝¹ : BorelSpace Y
inst✝ : SigmaCompactSpace X
d : ℕ
f : X → Y
hf : LocallyLipschitz f
s : Set X
hs : ↑μH[↑d... |
https://github.com/fpvandoorn/sard.git | f4a1bb550136e8591dcd20dca5bdcce766a958c7 | Sard/LocallyLipschitzMeasureZero.lean | locally_lipschitz_image_of_null_set_is_null_set | [35, 1] | [87, 14] | rw [hcov] | X : Type u_1
Y : Type u_2
inst✝⁶ : MetricSpace X
inst✝⁵ : MeasurableSpace X
inst✝⁴ : BorelSpace X
inst✝³ : MetricSpace Y
inst✝² : MeasurableSpace Y
inst✝¹ : BorelSpace Y
inst✝ : SigmaCompactSpace X
d : ℕ
f : X → Y
hf : LocallyLipschitz f
s : Set X
hs : ↑μH[↑d] s = 0
K : ℕ → Set X := compactCovering X
hcov : ⋃ n, K n = ... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
X : Type u_1
Y : Type u_2
inst✝⁶ : MetricSpace X
inst✝⁵ : MeasurableSpace X
inst✝⁴ : BorelSpace X
inst✝³ : MetricSpace Y
inst✝² : MeasurableSpace Y
inst✝¹ : BorelSpace Y
inst✝ : SigmaCompactSpace X
d : ℕ
f : X → Y
hf : LocallyLipschitz f
s : Set X
hs : ↑μH[↑d... |
https://github.com/fpvandoorn/sard.git | f4a1bb550136e8591dcd20dca5bdcce766a958c7 | Sard/LocallyLipschitzMeasureZero.lean | locally_lipschitz_image_of_null_set_is_null_set | [35, 1] | [87, 14] | apply inter_iUnion s | X : Type u_1
Y : Type u_2
inst✝⁶ : MetricSpace X
inst✝⁵ : MeasurableSpace X
inst✝⁴ : BorelSpace X
inst✝³ : MetricSpace Y
inst✝² : MeasurableSpace Y
inst✝¹ : BorelSpace Y
inst✝ : SigmaCompactSpace X
d : ℕ
f : X → Y
hf : LocallyLipschitz f
s : Set X
hs : ↑μH[↑d] s = 0
K : ℕ → Set X := compactCovering X
hcov : ⋃ n, K n = ... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
X : Type u_1
Y : Type u_2
inst✝⁶ : MetricSpace X
inst✝⁵ : MeasurableSpace X
inst✝⁴ : BorelSpace X
inst✝³ : MetricSpace Y
inst✝² : MeasurableSpace Y
inst✝¹ : BorelSpace Y
inst✝ : SigmaCompactSpace X
d : ℕ
f : X → Y
hf : LocallyLipschitz f
s : Set X
hs : ↑μH[↑d... |
https://github.com/fpvandoorn/sard.git | f4a1bb550136e8591dcd20dca5bdcce766a958c7 | Sard/LocallyLipschitzMeasureZero.lean | locally_lipschitz_image_of_null_set_is_null_set | [35, 1] | [87, 14] | intro i | X : Type u_1
Y : Type u_2
inst✝⁶ : MetricSpace X
inst✝⁵ : MeasurableSpace X
inst✝⁴ : BorelSpace X
inst✝³ : MetricSpace Y
inst✝² : MeasurableSpace Y
inst✝¹ : BorelSpace Y
inst✝ : SigmaCompactSpace X
d : ℕ
f : X → Y
hf : LocallyLipschitz f
s : Set X
hs : ↑μH[↑d] s = 0
K : ℕ → Set X := compactCovering X
hcov : ⋃ n, K n = ... | X : Type u_1
Y : Type u_2
inst✝⁶ : MetricSpace X
inst✝⁵ : MeasurableSpace X
inst✝⁴ : BorelSpace X
inst✝³ : MetricSpace Y
inst✝² : MeasurableSpace Y
inst✝¹ : BorelSpace Y
inst✝ : SigmaCompactSpace X
d : ℕ
f : X → Y
hf : LocallyLipschitz f
s : Set X
hs : ↑μH[↑d] s = 0
K : ℕ → Set X := compactCovering X
hcov : ⋃ n, K n = ... | Please generate a tactic in lean4 to solve the state.
STATE:
X : Type u_1
Y : Type u_2
inst✝⁶ : MetricSpace X
inst✝⁵ : MeasurableSpace X
inst✝⁴ : BorelSpace X
inst✝³ : MetricSpace Y
inst✝² : MeasurableSpace Y
inst✝¹ : BorelSpace Y
inst✝ : SigmaCompactSpace X
d : ℕ
f : X → Y
hf : LocallyLipschitz f
s : Set X
hs : ↑μH[↑d... |
https://github.com/fpvandoorn/sard.git | f4a1bb550136e8591dcd20dca5bdcce766a958c7 | Sard/LocallyLipschitzMeasureZero.lean | locally_lipschitz_image_of_null_set_is_null_set | [35, 1] | [87, 14] | have h₂ : μH[d] (s ∩ U i) = 0 := measure_mono_null (inter_subset_left s (U ↑i)) hs | X : Type u_1
Y : Type u_2
inst✝⁶ : MetricSpace X
inst✝⁵ : MeasurableSpace X
inst✝⁴ : BorelSpace X
inst✝³ : MetricSpace Y
inst✝² : MeasurableSpace Y
inst✝¹ : BorelSpace Y
inst✝ : SigmaCompactSpace X
d : ℕ
f : X → Y
hf : LocallyLipschitz f
s : Set X
hs : ↑μH[↑d] s = 0
K : ℕ → Set X := compactCovering X
hcov : ⋃ n, K n = ... | X : Type u_1
Y : Type u_2
inst✝⁶ : MetricSpace X
inst✝⁵ : MeasurableSpace X
inst✝⁴ : BorelSpace X
inst✝³ : MetricSpace Y
inst✝² : MeasurableSpace Y
inst✝¹ : BorelSpace Y
inst✝ : SigmaCompactSpace X
d : ℕ
f : X → Y
hf : LocallyLipschitz f
s : Set X
hs : ↑μH[↑d] s = 0
K : ℕ → Set X := compactCovering X
hcov : ⋃ n, K n = ... | Please generate a tactic in lean4 to solve the state.
STATE:
X : Type u_1
Y : Type u_2
inst✝⁶ : MetricSpace X
inst✝⁵ : MeasurableSpace X
inst✝⁴ : BorelSpace X
inst✝³ : MetricSpace Y
inst✝² : MeasurableSpace Y
inst✝¹ : BorelSpace Y
inst✝ : SigmaCompactSpace X
d : ℕ
f : X → Y
hf : LocallyLipschitz f
s : Set X
hs : ↑μH[↑d... |
https://github.com/fpvandoorn/sard.git | f4a1bb550136e8591dcd20dca5bdcce766a958c7 | Sard/LocallyLipschitzMeasureZero.lean | locally_lipschitz_image_of_null_set_is_null_set | [35, 1] | [87, 14] | have h₁ := ((hfL i).mono (hut i)).mono (inter_subset_right s (U i)) | X : Type u_1
Y : Type u_2
inst✝⁶ : MetricSpace X
inst✝⁵ : MeasurableSpace X
inst✝⁴ : BorelSpace X
inst✝³ : MetricSpace Y
inst✝² : MeasurableSpace Y
inst✝¹ : BorelSpace Y
inst✝ : SigmaCompactSpace X
d : ℕ
f : X → Y
hf : LocallyLipschitz f
s : Set X
hs : ↑μH[↑d] s = 0
K : ℕ → Set X := compactCovering X
hcov : ⋃ n, K n = ... | X : Type u_1
Y : Type u_2
inst✝⁶ : MetricSpace X
inst✝⁵ : MeasurableSpace X
inst✝⁴ : BorelSpace X
inst✝³ : MetricSpace Y
inst✝² : MeasurableSpace Y
inst✝¹ : BorelSpace Y
inst✝ : SigmaCompactSpace X
d : ℕ
f : X → Y
hf : LocallyLipschitz f
s : Set X
hs : ↑μH[↑d] s = 0
K : ℕ → Set X := compactCovering X
hcov : ⋃ n, K n = ... | Please generate a tactic in lean4 to solve the state.
STATE:
X : Type u_1
Y : Type u_2
inst✝⁶ : MetricSpace X
inst✝⁵ : MeasurableSpace X
inst✝⁴ : BorelSpace X
inst✝³ : MetricSpace Y
inst✝² : MeasurableSpace Y
inst✝¹ : BorelSpace Y
inst✝ : SigmaCompactSpace X
d : ℕ
f : X → Y
hf : LocallyLipschitz f
s : Set X
hs : ↑μH[↑d... |
https://github.com/fpvandoorn/sard.git | f4a1bb550136e8591dcd20dca5bdcce766a958c7 | Sard/LocallyLipschitzMeasureZero.lean | locally_lipschitz_image_of_null_set_is_null_set | [35, 1] | [87, 14] | refine lipschitz_image_null_set_is_null_set (Nat.cast_nonneg d) h₁ h₂ | X : Type u_1
Y : Type u_2
inst✝⁶ : MetricSpace X
inst✝⁵ : MeasurableSpace X
inst✝⁴ : BorelSpace X
inst✝³ : MetricSpace Y
inst✝² : MeasurableSpace Y
inst✝¹ : BorelSpace Y
inst✝ : SigmaCompactSpace X
d : ℕ
f : X → Y
hf : LocallyLipschitz f
s : Set X
hs : ↑μH[↑d] s = 0
K : ℕ → Set X := compactCovering X
hcov : ⋃ n, K n = ... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
X : Type u_1
Y : Type u_2
inst✝⁶ : MetricSpace X
inst✝⁵ : MeasurableSpace X
inst✝⁴ : BorelSpace X
inst✝³ : MetricSpace Y
inst✝² : MeasurableSpace Y
inst✝¹ : BorelSpace Y
inst✝ : SigmaCompactSpace X
d : ℕ
f : X → Y
hf : LocallyLipschitz f
s : Set X
hs : ↑μH[↑d... |
https://github.com/fpvandoorn/sard.git | f4a1bb550136e8591dcd20dca5bdcce766a958c7 | Sard/LocallyLipschitzMeasureZero.lean | locally_lipschitz_image_of_null_set_is_null_set | [35, 1] | [87, 14] | calc f '' (s ∩ (K n))
_ ⊆ f '' (s ∩ (⋃ (i : v), U i)) := by
apply image_subset
apply inter_subset_inter_right s hv
_ = f '' ((⋃ (i : v), s ∩ (U i))) := by rw [inter_iUnion]
_ = ⋃ (i : v), f '' ( s ∩ (U i)) := image_iUnion | X : Type u_1
Y : Type u_2
inst✝⁶ : MetricSpace X
inst✝⁵ : MeasurableSpace X
inst✝⁴ : BorelSpace X
inst✝³ : MetricSpace Y
inst✝² : MeasurableSpace Y
inst✝¹ : BorelSpace Y
inst✝ : SigmaCompactSpace X
d : ℕ
f : X → Y
hf : LocallyLipschitz f
s : Set X
hs : ↑μH[↑d] s = 0
K : ℕ → Set X := compactCovering X
hcov : ⋃ n, K n = ... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
X : Type u_1
Y : Type u_2
inst✝⁶ : MetricSpace X
inst✝⁵ : MeasurableSpace X
inst✝⁴ : BorelSpace X
inst✝³ : MetricSpace Y
inst✝² : MeasurableSpace Y
inst✝¹ : BorelSpace Y
inst✝ : SigmaCompactSpace X
d : ℕ
f : X → Y
hf : LocallyLipschitz f
s : Set X
hs : ↑μH[↑d... |
https://github.com/fpvandoorn/sard.git | f4a1bb550136e8591dcd20dca5bdcce766a958c7 | Sard/LocallyLipschitzMeasureZero.lean | locally_lipschitz_image_of_null_set_is_null_set | [35, 1] | [87, 14] | apply image_subset | X : Type u_1
Y : Type u_2
inst✝⁶ : MetricSpace X
inst✝⁵ : MeasurableSpace X
inst✝⁴ : BorelSpace X
inst✝³ : MetricSpace Y
inst✝² : MeasurableSpace Y
inst✝¹ : BorelSpace Y
inst✝ : SigmaCompactSpace X
d : ℕ
f : X → Y
hf : LocallyLipschitz f
s : Set X
hs : ↑μH[↑d] s = 0
K : ℕ → Set X := compactCovering X
hcov : ⋃ n, K n = ... | case h
X : Type u_1
Y : Type u_2
inst✝⁶ : MetricSpace X
inst✝⁵ : MeasurableSpace X
inst✝⁴ : BorelSpace X
inst✝³ : MetricSpace Y
inst✝² : MeasurableSpace Y
inst✝¹ : BorelSpace Y
inst✝ : SigmaCompactSpace X
d : ℕ
f : X → Y
hf : LocallyLipschitz f
s : Set X
hs : ↑μH[↑d] s = 0
K : ℕ → Set X := compactCovering X
hcov : ⋃ n,... | Please generate a tactic in lean4 to solve the state.
STATE:
X : Type u_1
Y : Type u_2
inst✝⁶ : MetricSpace X
inst✝⁵ : MeasurableSpace X
inst✝⁴ : BorelSpace X
inst✝³ : MetricSpace Y
inst✝² : MeasurableSpace Y
inst✝¹ : BorelSpace Y
inst✝ : SigmaCompactSpace X
d : ℕ
f : X → Y
hf : LocallyLipschitz f
s : Set X
hs : ↑μH[↑d... |
https://github.com/fpvandoorn/sard.git | f4a1bb550136e8591dcd20dca5bdcce766a958c7 | Sard/LocallyLipschitzMeasureZero.lean | locally_lipschitz_image_of_null_set_is_null_set | [35, 1] | [87, 14] | apply inter_subset_inter_right s hv | case h
X : Type u_1
Y : Type u_2
inst✝⁶ : MetricSpace X
inst✝⁵ : MeasurableSpace X
inst✝⁴ : BorelSpace X
inst✝³ : MetricSpace Y
inst✝² : MeasurableSpace Y
inst✝¹ : BorelSpace Y
inst✝ : SigmaCompactSpace X
d : ℕ
f : X → Y
hf : LocallyLipschitz f
s : Set X
hs : ↑μH[↑d] s = 0
K : ℕ → Set X := compactCovering X
hcov : ⋃ n,... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h
X : Type u_1
Y : Type u_2
inst✝⁶ : MetricSpace X
inst✝⁵ : MeasurableSpace X
inst✝⁴ : BorelSpace X
inst✝³ : MetricSpace Y
inst✝² : MeasurableSpace Y
inst✝¹ : BorelSpace Y
inst✝ : SigmaCompactSpace X
d : ℕ
f : X → Y
hf : LocallyLipschitz f
s : Set X
hs :... |
https://github.com/fpvandoorn/sard.git | f4a1bb550136e8591dcd20dca5bdcce766a958c7 | Sard/LocallyLipschitzMeasureZero.lean | locally_lipschitz_image_of_null_set_is_null_set | [35, 1] | [87, 14] | rw [inter_iUnion] | X : Type u_1
Y : Type u_2
inst✝⁶ : MetricSpace X
inst✝⁵ : MeasurableSpace X
inst✝⁴ : BorelSpace X
inst✝³ : MetricSpace Y
inst✝² : MeasurableSpace Y
inst✝¹ : BorelSpace Y
inst✝ : SigmaCompactSpace X
d : ℕ
f : X → Y
hf : LocallyLipschitz f
s : Set X
hs : ↑μH[↑d] s = 0
K : ℕ → Set X := compactCovering X
hcov : ⋃ n, K n = ... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
X : Type u_1
Y : Type u_2
inst✝⁶ : MetricSpace X
inst✝⁵ : MeasurableSpace X
inst✝⁴ : BorelSpace X
inst✝³ : MetricSpace Y
inst✝² : MeasurableSpace Y
inst✝¹ : BorelSpace Y
inst✝ : SigmaCompactSpace X
d : ℕ
f : X → Y
hf : LocallyLipschitz f
s : Set X
hs : ↑μH[↑d... |
https://github.com/fpvandoorn/sard.git | f4a1bb550136e8591dcd20dca5bdcce766a958c7 | Sard/LocallyLipschitzMeasureZero.lean | locally_lipschitz_image_of_null_set_is_null_set | [35, 1] | [87, 14] | calc μH[d] (f '' (s ∩ (K n)))
_ ≤ μH[d] (⋃ (i : v), f '' (s ∩ (U i))) := measure_mono this
_ ≤ ∑' (i : v), (μH[d] (f '' (s ∩ (U i))) : ℝ≥0∞) := by apply measure_iUnion_le
_ = ∑' (_ : v), (0 : ℝ≥0∞) := tsum_congr hnull
_ = 0 := by simp | X : Type u_1
Y : Type u_2
inst✝⁶ : MetricSpace X
inst✝⁵ : MeasurableSpace X
inst✝⁴ : BorelSpace X
inst✝³ : MetricSpace Y
inst✝² : MeasurableSpace Y
inst✝¹ : BorelSpace Y
inst✝ : SigmaCompactSpace X
d : ℕ
f : X → Y
hf : LocallyLipschitz f
s : Set X
hs : ↑μH[↑d] s = 0
K : ℕ → Set X := compactCovering X
hcov : ⋃ n, K n = ... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
X : Type u_1
Y : Type u_2
inst✝⁶ : MetricSpace X
inst✝⁵ : MeasurableSpace X
inst✝⁴ : BorelSpace X
inst✝³ : MetricSpace Y
inst✝² : MeasurableSpace Y
inst✝¹ : BorelSpace Y
inst✝ : SigmaCompactSpace X
d : ℕ
f : X → Y
hf : LocallyLipschitz f
s : Set X
hs : ↑μH[↑d... |
https://github.com/fpvandoorn/sard.git | f4a1bb550136e8591dcd20dca5bdcce766a958c7 | Sard/LocallyLipschitzMeasureZero.lean | locally_lipschitz_image_of_null_set_is_null_set | [35, 1] | [87, 14] | apply measure_iUnion_le | X : Type u_1
Y : Type u_2
inst✝⁶ : MetricSpace X
inst✝⁵ : MeasurableSpace X
inst✝⁴ : BorelSpace X
inst✝³ : MetricSpace Y
inst✝² : MeasurableSpace Y
inst✝¹ : BorelSpace Y
inst✝ : SigmaCompactSpace X
d : ℕ
f : X → Y
hf : LocallyLipschitz f
s : Set X
hs : ↑μH[↑d] s = 0
K : ℕ → Set X := compactCovering X
hcov : ⋃ n, K n = ... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
X : Type u_1
Y : Type u_2
inst✝⁶ : MetricSpace X
inst✝⁵ : MeasurableSpace X
inst✝⁴ : BorelSpace X
inst✝³ : MetricSpace Y
inst✝² : MeasurableSpace Y
inst✝¹ : BorelSpace Y
inst✝ : SigmaCompactSpace X
d : ℕ
f : X → Y
hf : LocallyLipschitz f
s : Set X
hs : ↑μH[↑d... |
https://github.com/fpvandoorn/sard.git | f4a1bb550136e8591dcd20dca5bdcce766a958c7 | Sard/LocallyLipschitzMeasureZero.lean | locally_lipschitz_image_of_null_set_is_null_set | [35, 1] | [87, 14] | simp | X : Type u_1
Y : Type u_2
inst✝⁶ : MetricSpace X
inst✝⁵ : MeasurableSpace X
inst✝⁴ : BorelSpace X
inst✝³ : MetricSpace Y
inst✝² : MeasurableSpace Y
inst✝¹ : BorelSpace Y
inst✝ : SigmaCompactSpace X
d : ℕ
f : X → Y
hf : LocallyLipschitz f
s : Set X
hs : ↑μH[↑d] s = 0
K : ℕ → Set X := compactCovering X
hcov : ⋃ n, K n = ... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
X : Type u_1
Y : Type u_2
inst✝⁶ : MetricSpace X
inst✝⁵ : MeasurableSpace X
inst✝⁴ : BorelSpace X
inst✝³ : MetricSpace Y
inst✝² : MeasurableSpace Y
inst✝¹ : BorelSpace Y
inst✝ : SigmaCompactSpace X
d : ℕ
f : X → Y
hf : LocallyLipschitz f
s : Set X
hs : ↑μH[↑d... |
https://github.com/fpvandoorn/sard.git | f4a1bb550136e8591dcd20dca5bdcce766a958c7 | Sard/LocallyLipschitzMeasureZero.lean | locally_lipschitz_image_of_null_set_is_null_set_open | [90, 1] | [92, 82] | sorry | X : Type u_1
Y : Type u_2
inst✝⁶ : MetricSpace X
inst✝⁵ : MeasurableSpace X
inst✝⁴ : BorelSpace X
inst✝³ : MetricSpace Y
inst✝² : MeasurableSpace Y
inst✝¹ : BorelSpace Y
inst✝ : SigmaCompactSpace X
d : ℕ
f : X → Y
U : Set X
hf : LocallyLipschitz (U.restrict f)
s : Set X
hsu : s ⊆ U
hs : ↑μH[↑d] s = 0
⊢ ↑μH[↑d] (f '' s)... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
X : Type u_1
Y : Type u_2
inst✝⁶ : MetricSpace X
inst✝⁵ : MeasurableSpace X
inst✝⁴ : BorelSpace X
inst✝³ : MetricSpace Y
inst✝² : MeasurableSpace Y
inst✝¹ : BorelSpace Y
inst✝ : SigmaCompactSpace X
d : ℕ
f : X → Y
U : Set X
hf : LocallyLipschitz (U.restrict f... |
https://github.com/fpvandoorn/sard.git | f4a1bb550136e8591dcd20dca5bdcce766a958c7 | Sard/LocallyLipschitzMeasureZero.lean | C1_preserves_null_sets_open | [104, 1] | [113, 55] | have : Convex ℝ U := sorry | X : Type u_1
Y : Type u_2
inst✝¹⁴ : MetricSpace X
inst✝¹³ : MeasurableSpace X
inst✝¹² : BorelSpace X
inst✝¹¹ : MetricSpace Y
inst✝¹⁰ : MeasurableSpace Y
inst✝⁹ : BorelSpace Y
E : Type u_3
F : Type u_4
inst✝⁸ : NormedAddCommGroup E
inst✝⁷ : NormedSpace ℝ E
inst✝⁶ : MeasurableSpace E
inst✝⁵ : BorelSpace E
inst✝⁴ : SigmaC... | X : Type u_1
Y : Type u_2
inst✝¹⁴ : MetricSpace X
inst✝¹³ : MeasurableSpace X
inst✝¹² : BorelSpace X
inst✝¹¹ : MetricSpace Y
inst✝¹⁰ : MeasurableSpace Y
inst✝⁹ : BorelSpace Y
E : Type u_3
F : Type u_4
inst✝⁸ : NormedAddCommGroup E
inst✝⁷ : NormedSpace ℝ E
inst✝⁶ : MeasurableSpace E
inst✝⁵ : BorelSpace E
inst✝⁴ : SigmaC... | Please generate a tactic in lean4 to solve the state.
STATE:
X : Type u_1
Y : Type u_2
inst✝¹⁴ : MetricSpace X
inst✝¹³ : MeasurableSpace X
inst✝¹² : BorelSpace X
inst✝¹¹ : MetricSpace Y
inst✝¹⁰ : MeasurableSpace Y
inst✝⁹ : BorelSpace Y
E : Type u_3
F : Type u_4
inst✝⁸ : NormedAddCommGroup E
inst✝⁷ : NormedSpace ℝ E
ins... |
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