url stringclasses 147
values | commit stringclasses 147
values | file_path stringlengths 7 101 | full_name stringlengths 1 94 | start stringlengths 6 10 | end stringlengths 6 11 | tactic stringlengths 1 11.2k | state_before stringlengths 3 2.09M | state_after stringlengths 6 2.09M | input stringlengths 73 2.09M |
|---|---|---|---|---|---|---|---|---|---|
https://github.com/fpvandoorn/sard.git | f4a1bb550136e8591dcd20dca5bdcce766a958c7 | Sard/ManifoldAux.lean | extendedChart_differential_bijective | [283, 1] | [298, 59] | refine diffeoOn_differential_bijective I 𝓘(ℝ, E) (Eq.le rfl) e.open_source ?_ hx (mapsTo_image (e.extend I) e.source) ?_ ?_ ?_ ?_ diff | E : Type u_1
inst✝¹³ : NormedAddCommGroup E
inst✝¹² : NormedSpace ℝ E
H : Type u_2
inst✝¹¹ : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type u_3
inst✝¹⁰ : TopologicalSpace M
inst✝⁹ : ChartedSpace H M
inst✝⁸ : SmoothManifoldWithCorners I M
F : Type u_4
inst✝⁷ : NormedAddCommGroup F
inst✝⁶ : NormedSpace ℝ F
G : Ty... | case refine_1
E : Type u_1
inst✝¹³ : NormedAddCommGroup E
inst✝¹² : NormedSpace ℝ E
H : Type u_2
inst✝¹¹ : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type u_3
inst✝¹⁰ : TopologicalSpace M
inst✝⁹ : ChartedSpace H M
inst✝⁸ : SmoothManifoldWithCorners I M
F : Type u_4
inst✝⁷ : NormedAddCommGroup F
inst✝⁶ : NormedSp... | Please generate a tactic in lean4 to solve the state.
STATE:
E : Type u_1
inst✝¹³ : NormedAddCommGroup E
inst✝¹² : NormedSpace ℝ E
H : Type u_2
inst✝¹¹ : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type u_3
inst✝¹⁰ : TopologicalSpace M
inst✝⁹ : ChartedSpace H M
inst✝⁸ : SmoothManifoldWithCorners I M
F : Type u_4
... |
https://github.com/fpvandoorn/sard.git | f4a1bb550136e8591dcd20dca5bdcce766a958c7 | Sard/ManifoldAux.lean | extendedChart_differential_bijective | [283, 1] | [298, 59] | exact e.extend_isOpenMapOn_source I e.open_source (Eq.subset rfl) | case refine_1
E : Type u_1
inst✝¹³ : NormedAddCommGroup E
inst✝¹² : NormedSpace ℝ E
H : Type u_2
inst✝¹¹ : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type u_3
inst✝¹⁰ : TopologicalSpace M
inst✝⁹ : ChartedSpace H M
inst✝⁸ : SmoothManifoldWithCorners I M
F : Type u_4
inst✝⁷ : NormedAddCommGroup F
inst✝⁶ : NormedSp... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case refine_1
E : Type u_1
inst✝¹³ : NormedAddCommGroup E
inst✝¹² : NormedSpace ℝ E
H : Type u_2
inst✝¹¹ : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type u_3
inst✝¹⁰ : TopologicalSpace M
inst✝⁹ : ChartedSpace H M
inst✝⁸ : SmoothManifoldWithCorners I M... |
https://github.com/fpvandoorn/sard.git | f4a1bb550136e8591dcd20dca5bdcce766a958c7 | Sard/ManifoldAux.lean | extendedChart_differential_bijective | [283, 1] | [298, 59] | rintro x ⟨s, hs, rfl⟩ | case refine_2
E : Type u_1
inst✝¹³ : NormedAddCommGroup E
inst✝¹² : NormedSpace ℝ E
H : Type u_2
inst✝¹¹ : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type u_3
inst✝¹⁰ : TopologicalSpace M
inst✝⁹ : ChartedSpace H M
inst✝⁸ : SmoothManifoldWithCorners I M
F : Type u_4
inst✝⁷ : NormedAddCommGroup F
inst✝⁶ : NormedSp... | case refine_2.intro.intro
E : Type u_1
inst✝¹³ : NormedAddCommGroup E
inst✝¹² : NormedSpace ℝ E
H : Type u_2
inst✝¹¹ : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type u_3
inst✝¹⁰ : TopologicalSpace M
inst✝⁹ : ChartedSpace H M
inst✝⁸ : SmoothManifoldWithCorners I M
F : Type u_4
inst✝⁷ : NormedAddCommGroup F
inst✝... | Please generate a tactic in lean4 to solve the state.
STATE:
case refine_2
E : Type u_1
inst✝¹³ : NormedAddCommGroup E
inst✝¹² : NormedSpace ℝ E
H : Type u_2
inst✝¹¹ : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type u_3
inst✝¹⁰ : TopologicalSpace M
inst✝⁹ : ChartedSpace H M
inst✝⁸ : SmoothManifoldWithCorners I M... |
https://github.com/fpvandoorn/sard.git | f4a1bb550136e8591dcd20dca5bdcce766a958c7 | Sard/ManifoldAux.lean | extendedChart_differential_bijective | [283, 1] | [298, 59] | have : (e.extend I).symm (e.extend I s) = s := e.extend_left_inv _ hs | case refine_2.intro.intro
E : Type u_1
inst✝¹³ : NormedAddCommGroup E
inst✝¹² : NormedSpace ℝ E
H : Type u_2
inst✝¹¹ : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type u_3
inst✝¹⁰ : TopologicalSpace M
inst✝⁹ : ChartedSpace H M
inst✝⁸ : SmoothManifoldWithCorners I M
F : Type u_4
inst✝⁷ : NormedAddCommGroup F
inst✝... | case refine_2.intro.intro
E : Type u_1
inst✝¹³ : NormedAddCommGroup E
inst✝¹² : NormedSpace ℝ E
H : Type u_2
inst✝¹¹ : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type u_3
inst✝¹⁰ : TopologicalSpace M
inst✝⁹ : ChartedSpace H M
inst✝⁸ : SmoothManifoldWithCorners I M
F : Type u_4
inst✝⁷ : NormedAddCommGroup F
inst✝... | Please generate a tactic in lean4 to solve the state.
STATE:
case refine_2.intro.intro
E : Type u_1
inst✝¹³ : NormedAddCommGroup E
inst✝¹² : NormedSpace ℝ E
H : Type u_2
inst✝¹¹ : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type u_3
inst✝¹⁰ : TopologicalSpace M
inst✝⁹ : ChartedSpace H M
inst✝⁸ : SmoothManifoldWit... |
https://github.com/fpvandoorn/sard.git | f4a1bb550136e8591dcd20dca5bdcce766a958c7 | Sard/ManifoldAux.lean | extendedChart_differential_bijective | [283, 1] | [298, 59] | rw [this] | case refine_2.intro.intro
E : Type u_1
inst✝¹³ : NormedAddCommGroup E
inst✝¹² : NormedSpace ℝ E
H : Type u_2
inst✝¹¹ : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type u_3
inst✝¹⁰ : TopologicalSpace M
inst✝⁹ : ChartedSpace H M
inst✝⁸ : SmoothManifoldWithCorners I M
F : Type u_4
inst✝⁷ : NormedAddCommGroup F
inst✝... | case refine_2.intro.intro
E : Type u_1
inst✝¹³ : NormedAddCommGroup E
inst✝¹² : NormedSpace ℝ E
H : Type u_2
inst✝¹¹ : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type u_3
inst✝¹⁰ : TopologicalSpace M
inst✝⁹ : ChartedSpace H M
inst✝⁸ : SmoothManifoldWithCorners I M
F : Type u_4
inst✝⁷ : NormedAddCommGroup F
inst✝... | Please generate a tactic in lean4 to solve the state.
STATE:
case refine_2.intro.intro
E : Type u_1
inst✝¹³ : NormedAddCommGroup E
inst✝¹² : NormedSpace ℝ E
H : Type u_2
inst✝¹¹ : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type u_3
inst✝¹⁰ : TopologicalSpace M
inst✝⁹ : ChartedSpace H M
inst✝⁸ : SmoothManifoldWit... |
https://github.com/fpvandoorn/sard.git | f4a1bb550136e8591dcd20dca5bdcce766a958c7 | Sard/ManifoldAux.lean | extendedChart_differential_bijective | [283, 1] | [298, 59] | exact hs | case refine_2.intro.intro
E : Type u_1
inst✝¹³ : NormedAddCommGroup E
inst✝¹² : NormedSpace ℝ E
H : Type u_2
inst✝¹¹ : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type u_3
inst✝¹⁰ : TopologicalSpace M
inst✝⁹ : ChartedSpace H M
inst✝⁸ : SmoothManifoldWithCorners I M
F : Type u_4
inst✝⁷ : NormedAddCommGroup F
inst✝... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case refine_2.intro.intro
E : Type u_1
inst✝¹³ : NormedAddCommGroup E
inst✝¹² : NormedSpace ℝ E
H : Type u_2
inst✝¹¹ : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type u_3
inst✝¹⁰ : TopologicalSpace M
inst✝⁹ : ChartedSpace H M
inst✝⁸ : SmoothManifoldWit... |
https://github.com/fpvandoorn/sard.git | f4a1bb550136e8591dcd20dca5bdcce766a958c7 | Sard/ManifoldAux.lean | extendedChart_differential_bijective | [283, 1] | [298, 59] | exact fun x hx ↦ e.extend_left_inv I hx | case refine_3
E : Type u_1
inst✝¹³ : NormedAddCommGroup E
inst✝¹² : NormedSpace ℝ E
H : Type u_2
inst✝¹¹ : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type u_3
inst✝¹⁰ : TopologicalSpace M
inst✝⁹ : ChartedSpace H M
inst✝⁸ : SmoothManifoldWithCorners I M
F : Type u_4
inst✝⁷ : NormedAddCommGroup F
inst✝⁶ : NormedSp... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case refine_3
E : Type u_1
inst✝¹³ : NormedAddCommGroup E
inst✝¹² : NormedSpace ℝ E
H : Type u_2
inst✝¹¹ : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type u_3
inst✝¹⁰ : TopologicalSpace M
inst✝⁹ : ChartedSpace H M
inst✝⁸ : SmoothManifoldWithCorners I M... |
https://github.com/fpvandoorn/sard.git | f4a1bb550136e8591dcd20dca5bdcce766a958c7 | Sard/ManifoldAux.lean | extendedChart_differential_bijective | [283, 1] | [298, 59] | exact fun x hx ↦ e.extend_right_inv _ hx | case refine_4
E : Type u_1
inst✝¹³ : NormedAddCommGroup E
inst✝¹² : NormedSpace ℝ E
H : Type u_2
inst✝¹¹ : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type u_3
inst✝¹⁰ : TopologicalSpace M
inst✝⁹ : ChartedSpace H M
inst✝⁸ : SmoothManifoldWithCorners I M
F : Type u_4
inst✝⁷ : NormedAddCommGroup F
inst✝⁶ : NormedSp... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case refine_4
E : Type u_1
inst✝¹³ : NormedAddCommGroup E
inst✝¹² : NormedSpace ℝ E
H : Type u_2
inst✝¹¹ : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type u_3
inst✝¹⁰ : TopologicalSpace M
inst✝⁹ : ChartedSpace H M
inst✝⁸ : SmoothManifoldWithCorners I M... |
https://github.com/fpvandoorn/sard.git | f4a1bb550136e8591dcd20dca5bdcce766a958c7 | Sard/ManifoldAux.lean | extendedChart_differential_bijective | [283, 1] | [298, 59] | exact SmoothOn.contMDiffOn (extendedChart_smooth I he) | case refine_5
E : Type u_1
inst✝¹³ : NormedAddCommGroup E
inst✝¹² : NormedSpace ℝ E
H : Type u_2
inst✝¹¹ : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type u_3
inst✝¹⁰ : TopologicalSpace M
inst✝⁹ : ChartedSpace H M
inst✝⁸ : SmoothManifoldWithCorners I M
F : Type u_4
inst✝⁷ : NormedAddCommGroup F
inst✝⁶ : NormedSp... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case refine_5
E : Type u_1
inst✝¹³ : NormedAddCommGroup E
inst✝¹² : NormedSpace ℝ E
H : Type u_2
inst✝¹¹ : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type u_3
inst✝¹⁰ : TopologicalSpace M
inst✝⁹ : ChartedSpace H M
inst✝⁸ : SmoothManifoldWithCorners I M... |
https://github.com/fpvandoorn/sard.git | f4a1bb550136e8591dcd20dca5bdcce766a958c7 | Sard/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... |
https://github.com/fpvandoorn/sard.git | f4a1bb550136e8591dcd20dca5bdcce766a958c7 | Sard/LocallyLipschitzMeasureZero.lean | C1_preserves_null_sets_open | [104, 1] | [113, 55] | exact locally_lipschitz_image_of_null_set_is_null_set_open
(LocallyLipschitz.of_C1_on_open hU this hf) hsU hs | X : Type u_1
Y : Type u_2
inst✝¹⁴ : MetricSpace X
inst✝¹³ : MeasurableSpace X
inst✝¹² : BorelSpace X
inst✝¹¹ : MetricSpace Y
inst✝¹⁰ : MeasurableSpace Y
inst✝⁹ : BorelSpace Y
E : Type u_3
F : Type u_4
inst✝⁸ : NormedAddCommGroup E
inst✝⁷ : NormedSpace ℝ E
inst✝⁶ : MeasurableSpace E
inst✝⁵ : BorelSpace E
inst✝⁴ : SigmaC... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
X : Type u_1
Y : Type u_2
inst✝¹⁴ : MetricSpace X
inst✝¹³ : MeasurableSpace X
inst✝¹² : BorelSpace X
inst✝¹¹ : MetricSpace Y
inst✝¹⁰ : MeasurableSpace Y
inst✝⁹ : BorelSpace Y
E : Type u_3
F : Type u_4
inst✝⁸ : NormedAddCommGroup E
inst✝⁷ : NormedSpace ℝ E
ins... |
https://github.com/fpvandoorn/sard.git | f4a1bb550136e8591dcd20dca5bdcce766a958c7 | Sard/ObsoleteHelpers.lean | empty_iff_open_cover | [9, 1] | [13, 30] | have : ⋃ (α : I), s ∩ U α = s := by rw [←inter_iUnion, hcover, inter_univ s] | X I : Type
U : I → Set X
hcover : ⋃ α, U α = univ
s : Set X
⊢ s = ∅ ↔ ∀ (α : I), s ∩ U α = ∅ | X I : Type
U : I → Set X
hcover : ⋃ α, U α = univ
s : Set X
this : ⋃ α, s ∩ U α = s
⊢ s = ∅ ↔ ∀ (α : I), s ∩ U α = ∅ | Please generate a tactic in lean4 to solve the state.
STATE:
X I : Type
U : I → Set X
hcover : ⋃ α, U α = univ
s : Set X
⊢ s = ∅ ↔ ∀ (α : I), s ∩ U α = ∅
TACTIC:
|
https://github.com/fpvandoorn/sard.git | f4a1bb550136e8591dcd20dca5bdcce766a958c7 | Sard/ObsoleteHelpers.lean | empty_iff_open_cover | [9, 1] | [13, 30] | nth_rewrite 1 [← this] | X I : Type
U : I → Set X
hcover : ⋃ α, U α = univ
s : Set X
this : ⋃ α, s ∩ U α = s
⊢ s = ∅ ↔ ∀ (α : I), s ∩ U α = ∅ | X I : Type
U : I → Set X
hcover : ⋃ α, U α = univ
s : Set X
this : ⋃ α, s ∩ U α = s
⊢ ⋃ α, s ∩ U α = ∅ ↔ ∀ (α : I), s ∩ U α = ∅ | Please generate a tactic in lean4 to solve the state.
STATE:
X I : Type
U : I → Set X
hcover : ⋃ α, U α = univ
s : Set X
this : ⋃ α, s ∩ U α = s
⊢ s = ∅ ↔ ∀ (α : I), s ∩ U α = ∅
TACTIC:
|
https://github.com/fpvandoorn/sard.git | f4a1bb550136e8591dcd20dca5bdcce766a958c7 | Sard/ObsoleteHelpers.lean | empty_iff_open_cover | [9, 1] | [13, 30] | simp only [iUnion_eq_empty] | X I : Type
U : I → Set X
hcover : ⋃ α, U α = univ
s : Set X
this : ⋃ α, s ∩ U α = s
⊢ ⋃ α, s ∩ U α = ∅ ↔ ∀ (α : I), s ∩ U α = ∅ | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
X I : Type
U : I → Set X
hcover : ⋃ α, U α = univ
s : Set X
this : ⋃ α, s ∩ U α = s
⊢ ⋃ α, s ∩ U α = ∅ ↔ ∀ (α : I), s ∩ U α = ∅
TACTIC:
|
https://github.com/fpvandoorn/sard.git | f4a1bb550136e8591dcd20dca5bdcce766a958c7 | Sard/ObsoleteHelpers.lean | empty_iff_open_cover | [9, 1] | [13, 30] | rw [←inter_iUnion, hcover, inter_univ s] | X I : Type
U : I → Set X
hcover : ⋃ α, U α = univ
s : Set X
⊢ ⋃ α, s ∩ U α = s | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
X I : Type
U : I → Set X
hcover : ⋃ α, U α = univ
s : Set X
⊢ ⋃ α, s ∩ U α = s
TACTIC:
|
https://github.com/fpvandoorn/sard.git | f4a1bb550136e8591dcd20dca5bdcce766a958c7 | Sard/ObsoleteHelpers.lean | image_interior | [21, 1] | [26, 8] | sorry | α β : Type
inst✝¹ : TopologicalSpace α
inst✝ : TopologicalSpace β
e : PartialHomeomorph α β
s : Set α
⊢ e.target ∩ ↑e '' interior s = ↑e '' (e.source ∩ interior s) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
α β : Type
inst✝¹ : TopologicalSpace α
inst✝ : TopologicalSpace β
e : PartialHomeomorph α β
s : Set α
⊢ e.target ∩ ↑e '' interior s = ↑e '' (e.source ∩ interior s)
TACTIC:
|
https://github.com/fpvandoorn/sard.git | f4a1bb550136e8591dcd20dca5bdcce766a958c7 | Sard/ObsoleteHelpers.lean | homeo_preserves_empty_interior | [29, 1] | [32, 27] | rw [← Homeomorph.image_interior, h₂s] | α β : Type
inst✝¹ : TopologicalSpace α
inst✝ : TopologicalSpace β
f : α ≃ₜ β
s : Set α
h₂s : interior s = ∅
⊢ interior (⇑f '' s) = ∅ | α β : Type
inst✝¹ : TopologicalSpace α
inst✝ : TopologicalSpace β
f : α ≃ₜ β
s : Set α
h₂s : interior s = ∅
⊢ ⇑f '' ∅ = ∅ | Please generate a tactic in lean4 to solve the state.
STATE:
α β : Type
inst✝¹ : TopologicalSpace α
inst✝ : TopologicalSpace β
f : α ≃ₜ β
s : Set α
h₂s : interior s = ∅
⊢ interior (⇑f '' s) = ∅
TACTIC:
|
https://github.com/fpvandoorn/sard.git | f4a1bb550136e8591dcd20dca5bdcce766a958c7 | Sard/ObsoleteHelpers.lean | homeo_preserves_empty_interior | [29, 1] | [32, 27] | exact Set.image_empty ↑f | α β : Type
inst✝¹ : TopologicalSpace α
inst✝ : TopologicalSpace β
f : α ≃ₜ β
s : Set α
h₂s : interior s = ∅
⊢ ⇑f '' ∅ = ∅ | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
α β : Type
inst✝¹ : TopologicalSpace α
inst✝ : TopologicalSpace β
f : α ≃ₜ β
s : Set α
h₂s : interior s = ∅
⊢ ⇑f '' ∅ = ∅
TACTIC:
|
https://github.com/fpvandoorn/sard.git | f4a1bb550136e8591dcd20dca5bdcce766a958c7 | Sard/ObsoleteHelpers.lean | local_homeo_preserves_empty_interior | [35, 1] | [48, 8] | have h₁ : s = s ∩ f.source := by
rw [← @inter_eq_left α s f.source] at hs
symm
exact hs | α β : Type
inst✝¹ : TopologicalSpace α
inst✝ : TopologicalSpace β
f : PartialHomeomorph α β
s : Set α
hs : s ⊆ f.source
h₂s : interior s = ∅
⊢ interior (↑f '' s) = ∅ | α β : Type
inst✝¹ : TopologicalSpace α
inst✝ : TopologicalSpace β
f : PartialHomeomorph α β
s : Set α
hs : s ⊆ f.source
h₂s : interior s = ∅
h₁ : s = s ∩ f.source
⊢ interior (↑f '' s) = ∅ | Please generate a tactic in lean4 to solve the state.
STATE:
α β : Type
inst✝¹ : TopologicalSpace α
inst✝ : TopologicalSpace β
f : PartialHomeomorph α β
s : Set α
hs : s ⊆ f.source
h₂s : interior s = ∅
⊢ interior (↑f '' s) = ∅
TACTIC:
|
https://github.com/fpvandoorn/sard.git | f4a1bb550136e8591dcd20dca5bdcce766a958c7 | Sard/ObsoleteHelpers.lean | local_homeo_preserves_empty_interior | [35, 1] | [48, 8] | have h₂ : interior s = interior (s ∩ f.source) := by
sorry | α β : Type
inst✝¹ : TopologicalSpace α
inst✝ : TopologicalSpace β
f : PartialHomeomorph α β
s : Set α
hs : s ⊆ f.source
h₂s : interior s = ∅
h₁ : s = s ∩ f.source
⊢ interior (↑f '' s) = ∅ | α β : Type
inst✝¹ : TopologicalSpace α
inst✝ : TopologicalSpace β
f : PartialHomeomorph α β
s : Set α
hs : s ⊆ f.source
h₂s : interior s = ∅
h₁ : s = s ∩ f.source
h₂ : interior s = interior (s ∩ f.source)
⊢ interior (↑f '' s) = ∅ | Please generate a tactic in lean4 to solve the state.
STATE:
α β : Type
inst✝¹ : TopologicalSpace α
inst✝ : TopologicalSpace β
f : PartialHomeomorph α β
s : Set α
hs : s ⊆ f.source
h₂s : interior s = ∅
h₁ : s = s ∩ f.source
⊢ interior (↑f '' s) = ∅
TACTIC:
|
https://github.com/fpvandoorn/sard.git | f4a1bb550136e8591dcd20dca5bdcce766a958c7 | Sard/ObsoleteHelpers.lean | local_homeo_preserves_empty_interior | [35, 1] | [48, 8] | rw [h₁] at h₂s | α β : Type
inst✝¹ : TopologicalSpace α
inst✝ : TopologicalSpace β
f : PartialHomeomorph α β
s : Set α
hs : s ⊆ f.source
h₂s : interior s = ∅
h₁ : s = s ∩ f.source
h₂ : interior s = interior (s ∩ f.source)
⊢ interior (↑f '' s) = ∅ | α β : Type
inst✝¹ : TopologicalSpace α
inst✝ : TopologicalSpace β
f : PartialHomeomorph α β
s : Set α
hs : s ⊆ f.source
h₂s : interior (s ∩ f.source) = ∅
h₁ : s = s ∩ f.source
h₂ : interior s = interior (s ∩ f.source)
⊢ interior (↑f '' s) = ∅ | Please generate a tactic in lean4 to solve the state.
STATE:
α β : Type
inst✝¹ : TopologicalSpace α
inst✝ : TopologicalSpace β
f : PartialHomeomorph α β
s : Set α
hs : s ⊆ f.source
h₂s : interior s = ∅
h₁ : s = s ∩ f.source
h₂ : interior s = interior (s ∩ f.source)
⊢ interior (↑f '' s) = ∅
TACTIC:
|
https://github.com/fpvandoorn/sard.git | f4a1bb550136e8591dcd20dca5bdcce766a958c7 | Sard/ObsoleteHelpers.lean | local_homeo_preserves_empty_interior | [35, 1] | [48, 8] | have h₃ : f '' (interior s ∩ f.source) = ∅ := by sorry | α β : Type
inst✝¹ : TopologicalSpace α
inst✝ : TopologicalSpace β
f : PartialHomeomorph α β
s : Set α
hs : s ⊆ f.source
h₂s : interior (s ∩ f.source) = ∅
h₁ : s = s ∩ f.source
h₂ : interior s = interior (s ∩ f.source)
⊢ interior (↑f '' s) = ∅ | α β : Type
inst✝¹ : TopologicalSpace α
inst✝ : TopologicalSpace β
f : PartialHomeomorph α β
s : Set α
hs : s ⊆ f.source
h₂s : interior (s ∩ f.source) = ∅
h₁ : s = s ∩ f.source
h₂ : interior s = interior (s ∩ f.source)
h₃ : ↑f '' (interior s ∩ f.source) = ∅
⊢ interior (↑f '' s) = ∅ | Please generate a tactic in lean4 to solve the state.
STATE:
α β : Type
inst✝¹ : TopologicalSpace α
inst✝ : TopologicalSpace β
f : PartialHomeomorph α β
s : Set α
hs : s ⊆ f.source
h₂s : interior (s ∩ f.source) = ∅
h₁ : s = s ∩ f.source
h₂ : interior s = interior (s ∩ f.source)
⊢ interior (↑f '' s) = ∅
TACTIC:
|
https://github.com/fpvandoorn/sard.git | f4a1bb550136e8591dcd20dca5bdcce766a958c7 | Sard/ObsoleteHelpers.lean | local_homeo_preserves_empty_interior | [35, 1] | [48, 8] | sorry | α β : Type
inst✝¹ : TopologicalSpace α
inst✝ : TopologicalSpace β
f : PartialHomeomorph α β
s : Set α
hs : s ⊆ f.source
h₂s : interior (s ∩ f.source) = ∅
h₁ : s = s ∩ f.source
h₂ : interior s = interior (s ∩ f.source)
h₃ : ↑f '' (interior s ∩ f.source) = ∅
⊢ interior (↑f '' s) = ∅ | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
α β : Type
inst✝¹ : TopologicalSpace α
inst✝ : TopologicalSpace β
f : PartialHomeomorph α β
s : Set α
hs : s ⊆ f.source
h₂s : interior (s ∩ f.source) = ∅
h₁ : s = s ∩ f.source
h₂ : interior s = interior (s ∩ f.source)
h₃ : ↑f '' (interior s ∩ f.source) = ∅
⊢ ... |
https://github.com/fpvandoorn/sard.git | f4a1bb550136e8591dcd20dca5bdcce766a958c7 | Sard/ObsoleteHelpers.lean | local_homeo_preserves_empty_interior | [35, 1] | [48, 8] | rw [← @inter_eq_left α s f.source] at hs | α β : Type
inst✝¹ : TopologicalSpace α
inst✝ : TopologicalSpace β
f : PartialHomeomorph α β
s : Set α
hs : s ⊆ f.source
h₂s : interior s = ∅
⊢ s = s ∩ f.source | α β : Type
inst✝¹ : TopologicalSpace α
inst✝ : TopologicalSpace β
f : PartialHomeomorph α β
s : Set α
hs : s ∩ f.source = s
h₂s : interior s = ∅
⊢ s = s ∩ f.source | Please generate a tactic in lean4 to solve the state.
STATE:
α β : Type
inst✝¹ : TopologicalSpace α
inst✝ : TopologicalSpace β
f : PartialHomeomorph α β
s : Set α
hs : s ⊆ f.source
h₂s : interior s = ∅
⊢ s = s ∩ f.source
TACTIC:
|
https://github.com/fpvandoorn/sard.git | f4a1bb550136e8591dcd20dca5bdcce766a958c7 | Sard/ObsoleteHelpers.lean | local_homeo_preserves_empty_interior | [35, 1] | [48, 8] | symm | α β : Type
inst✝¹ : TopologicalSpace α
inst✝ : TopologicalSpace β
f : PartialHomeomorph α β
s : Set α
hs : s ∩ f.source = s
h₂s : interior s = ∅
⊢ s = s ∩ f.source | α β : Type
inst✝¹ : TopologicalSpace α
inst✝ : TopologicalSpace β
f : PartialHomeomorph α β
s : Set α
hs : s ∩ f.source = s
h₂s : interior s = ∅
⊢ s ∩ f.source = s | Please generate a tactic in lean4 to solve the state.
STATE:
α β : Type
inst✝¹ : TopologicalSpace α
inst✝ : TopologicalSpace β
f : PartialHomeomorph α β
s : Set α
hs : s ∩ f.source = s
h₂s : interior s = ∅
⊢ s = s ∩ f.source
TACTIC:
|
https://github.com/fpvandoorn/sard.git | f4a1bb550136e8591dcd20dca5bdcce766a958c7 | Sard/ObsoleteHelpers.lean | local_homeo_preserves_empty_interior | [35, 1] | [48, 8] | exact hs | α β : Type
inst✝¹ : TopologicalSpace α
inst✝ : TopologicalSpace β
f : PartialHomeomorph α β
s : Set α
hs : s ∩ f.source = s
h₂s : interior s = ∅
⊢ s ∩ f.source = s | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
α β : Type
inst✝¹ : TopologicalSpace α
inst✝ : TopologicalSpace β
f : PartialHomeomorph α β
s : Set α
hs : s ∩ f.source = s
h₂s : interior s = ∅
⊢ s ∩ f.source = s
TACTIC:
|
https://github.com/fpvandoorn/sard.git | f4a1bb550136e8591dcd20dca5bdcce766a958c7 | Sard/ObsoleteHelpers.lean | local_homeo_preserves_empty_interior | [35, 1] | [48, 8] | sorry | α β : Type
inst✝¹ : TopologicalSpace α
inst✝ : TopologicalSpace β
f : PartialHomeomorph α β
s : Set α
hs : s ⊆ f.source
h₂s : interior s = ∅
h₁ : s = s ∩ f.source
⊢ interior s = interior (s ∩ f.source) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
α β : Type
inst✝¹ : TopologicalSpace α
inst✝ : TopologicalSpace β
f : PartialHomeomorph α β
s : Set α
hs : s ⊆ f.source
h₂s : interior s = ∅
h₁ : s = s ∩ f.source
⊢ interior s = interior (s ∩ f.source)
TACTIC:
|
https://github.com/fpvandoorn/sard.git | f4a1bb550136e8591dcd20dca5bdcce766a958c7 | Sard/ObsoleteHelpers.lean | local_homeo_preserves_empty_interior | [35, 1] | [48, 8] | sorry | α β : Type
inst✝¹ : TopologicalSpace α
inst✝ : TopologicalSpace β
f : PartialHomeomorph α β
s : Set α
hs : s ⊆ f.source
h₂s : interior (s ∩ f.source) = ∅
h₁ : s = s ∩ f.source
h₂ : interior s = interior (s ∩ f.source)
⊢ ↑f '' (interior s ∩ f.source) = ∅ | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
α β : Type
inst✝¹ : TopologicalSpace α
inst✝ : TopologicalSpace β
f : PartialHomeomorph α β
s : Set α
hs : s ⊆ f.source
h₂s : interior (s ∩ f.source) = ∅
h₁ : s = s ∩ f.source
h₂ : interior s = interior (s ∩ f.source)
⊢ ↑f '' (interior s ∩ f.source) = ∅
TACTI... |
https://github.com/fpvandoorn/sard.git | f4a1bb550136e8591dcd20dca5bdcce766a958c7 | Sard/ObsoleteHelpers.lean | interior_zero_iff_open_cover | [53, 1] | [87, 10] | constructor | X : Type
inst✝ : TopologicalSpace X
I : Type
U : I → Set X
hU : ∀ (α : I), IsOpen (U α)
hcover : ↑(⋃ α, U α) = X
s : Set X
⊢ interior s = ∅ ↔ ∀ (α : I), interior (s ∩ U α) = ∅ | case mp
X : Type
inst✝ : TopologicalSpace X
I : Type
U : I → Set X
hU : ∀ (α : I), IsOpen (U α)
hcover : ↑(⋃ α, U α) = X
s : Set X
⊢ interior s = ∅ → ∀ (α : I), interior (s ∩ U α) = ∅
case mpr
X : Type
inst✝ : TopologicalSpace X
I : Type
U : I → Set X
hU : ∀ (α : I), IsOpen (U α)
hcover : ↑(⋃ α, U α) = X
s : Set X
⊢ (... | Please generate a tactic in lean4 to solve the state.
STATE:
X : Type
inst✝ : TopologicalSpace X
I : Type
U : I → Set X
hU : ∀ (α : I), IsOpen (U α)
hcover : ↑(⋃ α, U α) = X
s : Set X
⊢ interior s = ∅ ↔ ∀ (α : I), interior (s ∩ U α) = ∅
TACTIC:
|
https://github.com/fpvandoorn/sard.git | f4a1bb550136e8591dcd20dca5bdcce766a958c7 | Sard/ObsoleteHelpers.lean | interior_zero_iff_open_cover | [53, 1] | [87, 10] | intro hs α | case mp
X : Type
inst✝ : TopologicalSpace X
I : Type
U : I → Set X
hU : ∀ (α : I), IsOpen (U α)
hcover : ↑(⋃ α, U α) = X
s : Set X
⊢ interior s = ∅ → ∀ (α : I), interior (s ∩ U α) = ∅ | case mp
X : Type
inst✝ : TopologicalSpace X
I : Type
U : I → Set X
hU : ∀ (α : I), IsOpen (U α)
hcover : ↑(⋃ α, U α) = X
s : Set X
hs : interior s = ∅
α : I
⊢ interior (s ∩ U α) = ∅ | Please generate a tactic in lean4 to solve the state.
STATE:
case mp
X : Type
inst✝ : TopologicalSpace X
I : Type
U : I → Set X
hU : ∀ (α : I), IsOpen (U α)
hcover : ↑(⋃ α, U α) = X
s : Set X
⊢ interior s = ∅ → ∀ (α : I), interior (s ∩ U α) = ∅
TACTIC:
|
https://github.com/fpvandoorn/sard.git | f4a1bb550136e8591dcd20dca5bdcce766a958c7 | Sard/ObsoleteHelpers.lean | interior_zero_iff_open_cover | [53, 1] | [87, 10] | have aux: interior (s ∩ U α) ⊆ interior s := by
apply interior_mono
apply Set.inter_subset_left | case mp
X : Type
inst✝ : TopologicalSpace X
I : Type
U : I → Set X
hU : ∀ (α : I), IsOpen (U α)
hcover : ↑(⋃ α, U α) = X
s : Set X
hs : interior s = ∅
α : I
⊢ interior (s ∩ U α) = ∅ | case mp
X : Type
inst✝ : TopologicalSpace X
I : Type
U : I → Set X
hU : ∀ (α : I), IsOpen (U α)
hcover : ↑(⋃ α, U α) = X
s : Set X
hs : interior s = ∅
α : I
aux : interior (s ∩ U α) ⊆ interior s
⊢ interior (s ∩ U α) = ∅ | Please generate a tactic in lean4 to solve the state.
STATE:
case mp
X : Type
inst✝ : TopologicalSpace X
I : Type
U : I → Set X
hU : ∀ (α : I), IsOpen (U α)
hcover : ↑(⋃ α, U α) = X
s : Set X
hs : interior s = ∅
α : I
⊢ interior (s ∩ U α) = ∅
TACTIC:
|
https://github.com/fpvandoorn/sard.git | f4a1bb550136e8591dcd20dca5bdcce766a958c7 | Sard/ObsoleteHelpers.lean | interior_zero_iff_open_cover | [53, 1] | [87, 10] | exact Set.subset_eq_empty aux hs | case mp
X : Type
inst✝ : TopologicalSpace X
I : Type
U : I → Set X
hU : ∀ (α : I), IsOpen (U α)
hcover : ↑(⋃ α, U α) = X
s : Set X
hs : interior s = ∅
α : I
aux : interior (s ∩ U α) ⊆ interior s
⊢ interior (s ∩ U α) = ∅ | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case mp
X : Type
inst✝ : TopologicalSpace X
I : Type
U : I → Set X
hU : ∀ (α : I), IsOpen (U α)
hcover : ↑(⋃ α, U α) = X
s : Set X
hs : interior s = ∅
α : I
aux : interior (s ∩ U α) ⊆ interior s
⊢ interior (s ∩ U α) = ∅
TACTIC:
|
https://github.com/fpvandoorn/sard.git | f4a1bb550136e8591dcd20dca5bdcce766a958c7 | Sard/ObsoleteHelpers.lean | interior_zero_iff_open_cover | [53, 1] | [87, 10] | apply interior_mono | X : Type
inst✝ : TopologicalSpace X
I : Type
U : I → Set X
hU : ∀ (α : I), IsOpen (U α)
hcover : ↑(⋃ α, U α) = X
s : Set X
hs : interior s = ∅
α : I
⊢ interior (s ∩ U α) ⊆ interior s | case h
X : Type
inst✝ : TopologicalSpace X
I : Type
U : I → Set X
hU : ∀ (α : I), IsOpen (U α)
hcover : ↑(⋃ α, U α) = X
s : Set X
hs : interior s = ∅
α : I
⊢ s ∩ U α ⊆ s | Please generate a tactic in lean4 to solve the state.
STATE:
X : Type
inst✝ : TopologicalSpace X
I : Type
U : I → Set X
hU : ∀ (α : I), IsOpen (U α)
hcover : ↑(⋃ α, U α) = X
s : Set X
hs : interior s = ∅
α : I
⊢ interior (s ∩ U α) ⊆ interior s
TACTIC:
|
https://github.com/fpvandoorn/sard.git | f4a1bb550136e8591dcd20dca5bdcce766a958c7 | Sard/ObsoleteHelpers.lean | interior_zero_iff_open_cover | [53, 1] | [87, 10] | apply Set.inter_subset_left | case h
X : Type
inst✝ : TopologicalSpace X
I : Type
U : I → Set X
hU : ∀ (α : I), IsOpen (U α)
hcover : ↑(⋃ α, U α) = X
s : Set X
hs : interior s = ∅
α : I
⊢ s ∩ U α ⊆ s | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h
X : Type
inst✝ : TopologicalSpace X
I : Type
U : I → Set X
hU : ∀ (α : I), IsOpen (U α)
hcover : ↑(⋃ α, U α) = X
s : Set X
hs : interior s = ∅
α : I
⊢ s ∩ U α ⊆ s
TACTIC:
|
https://github.com/fpvandoorn/sard.git | f4a1bb550136e8591dcd20dca5bdcce766a958c7 | Sard/ObsoleteHelpers.lean | interior_zero_iff_open_cover | [53, 1] | [87, 10] | intro h | case mpr
X : Type
inst✝ : TopologicalSpace X
I : Type
U : I → Set X
hU : ∀ (α : I), IsOpen (U α)
hcover : ↑(⋃ α, U α) = X
s : Set X
⊢ (∀ (α : I), interior (s ∩ U α) = ∅) → interior s = ∅ | case mpr
X : Type
inst✝ : TopologicalSpace X
I : Type
U : I → Set X
hU : ∀ (α : I), IsOpen (U α)
hcover : ↑(⋃ α, U α) = X
s : Set X
h : ∀ (α : I), interior (s ∩ U α) = ∅
⊢ interior s = ∅ | Please generate a tactic in lean4 to solve the state.
STATE:
case mpr
X : Type
inst✝ : TopologicalSpace X
I : Type
U : I → Set X
hU : ∀ (α : I), IsOpen (U α)
hcover : ↑(⋃ α, U α) = X
s : Set X
⊢ (∀ (α : I), interior (s ∩ U α) = ∅) → interior s = ∅
TACTIC:
|
https://github.com/fpvandoorn/sard.git | f4a1bb550136e8591dcd20dca5bdcce766a958c7 | Sard/ObsoleteHelpers.lean | interior_zero_iff_open_cover | [53, 1] | [87, 10] | suffices ∀ V : Set X, (hV : V ⊆ s ∧ IsOpen V) → V = ∅ by sorry | case mpr
X : Type
inst✝ : TopologicalSpace X
I : Type
U : I → Set X
hU : ∀ (α : I), IsOpen (U α)
hcover : ↑(⋃ α, U α) = X
s : Set X
h : ∀ (α : I), interior (s ∩ U α) = ∅
⊢ interior s = ∅ | case mpr
X : Type
inst✝ : TopologicalSpace X
I : Type
U : I → Set X
hU : ∀ (α : I), IsOpen (U α)
hcover : ↑(⋃ α, U α) = X
s : Set X
h : ∀ (α : I), interior (s ∩ U α) = ∅
⊢ ∀ (V : Set X), V ⊆ s ∧ IsOpen V → V = ∅ | Please generate a tactic in lean4 to solve the state.
STATE:
case mpr
X : Type
inst✝ : TopologicalSpace X
I : Type
U : I → Set X
hU : ∀ (α : I), IsOpen (U α)
hcover : ↑(⋃ α, U α) = X
s : Set X
h : ∀ (α : I), interior (s ∩ U α) = ∅
⊢ interior s = ∅
TACTIC:
|
https://github.com/fpvandoorn/sard.git | f4a1bb550136e8591dcd20dca5bdcce766a958c7 | Sard/ObsoleteHelpers.lean | interior_zero_iff_open_cover | [53, 1] | [87, 10] | rintro V ⟨hVS, hV⟩ | case mpr
X : Type
inst✝ : TopologicalSpace X
I : Type
U : I → Set X
hU : ∀ (α : I), IsOpen (U α)
hcover : ↑(⋃ α, U α) = X
s : Set X
h : ∀ (α : I), interior (s ∩ U α) = ∅
⊢ ∀ (V : Set X), V ⊆ s ∧ IsOpen V → V = ∅ | case mpr.intro
X : Type
inst✝ : TopologicalSpace X
I : Type
U : I → Set X
hU : ∀ (α : I), IsOpen (U α)
hcover : ↑(⋃ α, U α) = X
s : Set X
h : ∀ (α : I), interior (s ∩ U α) = ∅
V : Set X
hVS : V ⊆ s
hV : IsOpen V
⊢ V = ∅ | Please generate a tactic in lean4 to solve the state.
STATE:
case mpr
X : Type
inst✝ : TopologicalSpace X
I : Type
U : I → Set X
hU : ∀ (α : I), IsOpen (U α)
hcover : ↑(⋃ α, U α) = X
s : Set X
h : ∀ (α : I), interior (s ∩ U α) = ∅
⊢ ∀ (V : Set X), V ⊆ s ∧ IsOpen V → V = ∅
TACTIC:
|
https://github.com/fpvandoorn/sard.git | f4a1bb550136e8591dcd20dca5bdcce766a958c7 | Sard/ObsoleteHelpers.lean | interior_zero_iff_open_cover | [53, 1] | [87, 10] | have h' : ∀ α : I, V ∩ U α = ∅ := by
intro α
have h₁ : IsOpen (V ∩ U α) := by exact IsOpen.inter hV (hU α)
have h₂ : V ∩ U α ⊆ s ∩ U α := by exact Set.inter_subset_inter_left (U α) hVS
have h₃ : V ∩ U α ⊆ interior (s ∩ U α) := by exact interior_maximal h₂ h₁
exact Set.subset_eq_empty h₃ (h α) | case mpr.intro
X : Type
inst✝ : TopologicalSpace X
I : Type
U : I → Set X
hU : ∀ (α : I), IsOpen (U α)
hcover : ↑(⋃ α, U α) = X
s : Set X
h : ∀ (α : I), interior (s ∩ U α) = ∅
V : Set X
hVS : V ⊆ s
hV : IsOpen V
⊢ V = ∅ | case mpr.intro
X : Type
inst✝ : TopologicalSpace X
I : Type
U : I → Set X
hU : ∀ (α : I), IsOpen (U α)
hcover : ↑(⋃ α, U α) = X
s : Set X
h : ∀ (α : I), interior (s ∩ U α) = ∅
V : Set X
hVS : V ⊆ s
hV : IsOpen V
h' : ∀ (α : I), V ∩ U α = ∅
⊢ V = ∅ | Please generate a tactic in lean4 to solve the state.
STATE:
case mpr.intro
X : Type
inst✝ : TopologicalSpace X
I : Type
U : I → Set X
hU : ∀ (α : I), IsOpen (U α)
hcover : ↑(⋃ α, U α) = X
s : Set X
h : ∀ (α : I), interior (s ∩ U α) = ∅
V : Set X
hVS : V ⊆ s
hV : IsOpen V
⊢ V = ∅
TACTIC:
|
https://github.com/fpvandoorn/sard.git | f4a1bb550136e8591dcd20dca5bdcce766a958c7 | Sard/ObsoleteHelpers.lean | interior_zero_iff_open_cover | [53, 1] | [87, 10] | have h'' : V = ⋃ (α : I), (V ∩ U α) := by
ext i
rw [Set.mem_iUnion]
sorry | case mpr.intro
X : Type
inst✝ : TopologicalSpace X
I : Type
U : I → Set X
hU : ∀ (α : I), IsOpen (U α)
hcover : ↑(⋃ α, U α) = X
s : Set X
h : ∀ (α : I), interior (s ∩ U α) = ∅
V : Set X
hVS : V ⊆ s
hV : IsOpen V
h' : ∀ (α : I), V ∩ U α = ∅
⊢ V = ∅ | case mpr.intro
X : Type
inst✝ : TopologicalSpace X
I : Type
U : I → Set X
hU : ∀ (α : I), IsOpen (U α)
hcover : ↑(⋃ α, U α) = X
s : Set X
h : ∀ (α : I), interior (s ∩ U α) = ∅
V : Set X
hVS : V ⊆ s
hV : IsOpen V
h' : ∀ (α : I), V ∩ U α = ∅
h'' : V = ⋃ α, V ∩ U α
⊢ V = ∅ | Please generate a tactic in lean4 to solve the state.
STATE:
case mpr.intro
X : Type
inst✝ : TopologicalSpace X
I : Type
U : I → Set X
hU : ∀ (α : I), IsOpen (U α)
hcover : ↑(⋃ α, U α) = X
s : Set X
h : ∀ (α : I), interior (s ∩ U α) = ∅
V : Set X
hVS : V ⊆ s
hV : IsOpen V
h' : ∀ (α : I), V ∩ U α = ∅
⊢ V = ∅
TACTIC:
|
https://github.com/fpvandoorn/sard.git | f4a1bb550136e8591dcd20dca5bdcce766a958c7 | Sard/ObsoleteHelpers.lean | interior_zero_iff_open_cover | [53, 1] | [87, 10] | rw [h''] | case mpr.intro
X : Type
inst✝ : TopologicalSpace X
I : Type
U : I → Set X
hU : ∀ (α : I), IsOpen (U α)
hcover : ↑(⋃ α, U α) = X
s : Set X
h : ∀ (α : I), interior (s ∩ U α) = ∅
V : Set X
hVS : V ⊆ s
hV : IsOpen V
h' : ∀ (α : I), V ∩ U α = ∅
h'' : V = ⋃ α, V ∩ U α
⊢ V = ∅ | case mpr.intro
X : Type
inst✝ : TopologicalSpace X
I : Type
U : I → Set X
hU : ∀ (α : I), IsOpen (U α)
hcover : ↑(⋃ α, U α) = X
s : Set X
h : ∀ (α : I), interior (s ∩ U α) = ∅
V : Set X
hVS : V ⊆ s
hV : IsOpen V
h' : ∀ (α : I), V ∩ U α = ∅
h'' : V = ⋃ α, V ∩ U α
⊢ ⋃ α, V ∩ U α = ∅ | Please generate a tactic in lean4 to solve the state.
STATE:
case mpr.intro
X : Type
inst✝ : TopologicalSpace X
I : Type
U : I → Set X
hU : ∀ (α : I), IsOpen (U α)
hcover : ↑(⋃ α, U α) = X
s : Set X
h : ∀ (α : I), interior (s ∩ U α) = ∅
V : Set X
hVS : V ⊆ s
hV : IsOpen V
h' : ∀ (α : I), V ∩ U α = ∅
h'' : V = ⋃ α, V ∩ ... |
https://github.com/fpvandoorn/sard.git | f4a1bb550136e8591dcd20dca5bdcce766a958c7 | Sard/ObsoleteHelpers.lean | interior_zero_iff_open_cover | [53, 1] | [87, 10] | simp only [Set.iUnion_eq_empty] | case mpr.intro
X : Type
inst✝ : TopologicalSpace X
I : Type
U : I → Set X
hU : ∀ (α : I), IsOpen (U α)
hcover : ↑(⋃ α, U α) = X
s : Set X
h : ∀ (α : I), interior (s ∩ U α) = ∅
V : Set X
hVS : V ⊆ s
hV : IsOpen V
h' : ∀ (α : I), V ∩ U α = ∅
h'' : V = ⋃ α, V ∩ U α
⊢ ⋃ α, V ∩ U α = ∅ | case mpr.intro
X : Type
inst✝ : TopologicalSpace X
I : Type
U : I → Set X
hU : ∀ (α : I), IsOpen (U α)
hcover : ↑(⋃ α, U α) = X
s : Set X
h : ∀ (α : I), interior (s ∩ U α) = ∅
V : Set X
hVS : V ⊆ s
hV : IsOpen V
h' : ∀ (α : I), V ∩ U α = ∅
h'' : V = ⋃ α, V ∩ U α
⊢ ∀ (i : I), V ∩ U i = ∅ | Please generate a tactic in lean4 to solve the state.
STATE:
case mpr.intro
X : Type
inst✝ : TopologicalSpace X
I : Type
U : I → Set X
hU : ∀ (α : I), IsOpen (U α)
hcover : ↑(⋃ α, U α) = X
s : Set X
h : ∀ (α : I), interior (s ∩ U α) = ∅
V : Set X
hVS : V ⊆ s
hV : IsOpen V
h' : ∀ (α : I), V ∩ U α = ∅
h'' : V = ⋃ α, V ∩ ... |
https://github.com/fpvandoorn/sard.git | f4a1bb550136e8591dcd20dca5bdcce766a958c7 | Sard/ObsoleteHelpers.lean | interior_zero_iff_open_cover | [53, 1] | [87, 10] | tauto | case mpr.intro
X : Type
inst✝ : TopologicalSpace X
I : Type
U : I → Set X
hU : ∀ (α : I), IsOpen (U α)
hcover : ↑(⋃ α, U α) = X
s : Set X
h : ∀ (α : I), interior (s ∩ U α) = ∅
V : Set X
hVS : V ⊆ s
hV : IsOpen V
h' : ∀ (α : I), V ∩ U α = ∅
h'' : V = ⋃ α, V ∩ U α
⊢ ∀ (i : I), V ∩ U i = ∅ | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case mpr.intro
X : Type
inst✝ : TopologicalSpace X
I : Type
U : I → Set X
hU : ∀ (α : I), IsOpen (U α)
hcover : ↑(⋃ α, U α) = X
s : Set X
h : ∀ (α : I), interior (s ∩ U α) = ∅
V : Set X
hVS : V ⊆ s
hV : IsOpen V
h' : ∀ (α : I), V ∩ U α = ∅
h'' : V = ⋃ α, V ∩ ... |
https://github.com/fpvandoorn/sard.git | f4a1bb550136e8591dcd20dca5bdcce766a958c7 | Sard/ObsoleteHelpers.lean | interior_zero_iff_open_cover | [53, 1] | [87, 10] | sorry | X : Type
inst✝ : TopologicalSpace X
I : Type
U : I → Set X
hU : ∀ (α : I), IsOpen (U α)
hcover : ↑(⋃ α, U α) = X
s : Set X
h : ∀ (α : I), interior (s ∩ U α) = ∅
this : ∀ (V : Set X), V ⊆ s ∧ IsOpen V → V = ∅
⊢ interior s = ∅ | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
X : Type
inst✝ : TopologicalSpace X
I : Type
U : I → Set X
hU : ∀ (α : I), IsOpen (U α)
hcover : ↑(⋃ α, U α) = X
s : Set X
h : ∀ (α : I), interior (s ∩ U α) = ∅
this : ∀ (V : Set X), V ⊆ s ∧ IsOpen V → V = ∅
⊢ interior s = ∅
TACTIC:
|
https://github.com/fpvandoorn/sard.git | f4a1bb550136e8591dcd20dca5bdcce766a958c7 | Sard/ObsoleteHelpers.lean | interior_zero_iff_open_cover | [53, 1] | [87, 10] | intro α | X : Type
inst✝ : TopologicalSpace X
I : Type
U : I → Set X
hU : ∀ (α : I), IsOpen (U α)
hcover : ↑(⋃ α, U α) = X
s : Set X
h : ∀ (α : I), interior (s ∩ U α) = ∅
V : Set X
hVS : V ⊆ s
hV : IsOpen V
⊢ ∀ (α : I), V ∩ U α = ∅ | X : Type
inst✝ : TopologicalSpace X
I : Type
U : I → Set X
hU : ∀ (α : I), IsOpen (U α)
hcover : ↑(⋃ α, U α) = X
s : Set X
h : ∀ (α : I), interior (s ∩ U α) = ∅
V : Set X
hVS : V ⊆ s
hV : IsOpen V
α : I
⊢ V ∩ U α = ∅ | Please generate a tactic in lean4 to solve the state.
STATE:
X : Type
inst✝ : TopologicalSpace X
I : Type
U : I → Set X
hU : ∀ (α : I), IsOpen (U α)
hcover : ↑(⋃ α, U α) = X
s : Set X
h : ∀ (α : I), interior (s ∩ U α) = ∅
V : Set X
hVS : V ⊆ s
hV : IsOpen V
⊢ ∀ (α : I), V ∩ U α = ∅
TACTIC:
|
https://github.com/fpvandoorn/sard.git | f4a1bb550136e8591dcd20dca5bdcce766a958c7 | Sard/ObsoleteHelpers.lean | interior_zero_iff_open_cover | [53, 1] | [87, 10] | have h₁ : IsOpen (V ∩ U α) := by exact IsOpen.inter hV (hU α) | X : Type
inst✝ : TopologicalSpace X
I : Type
U : I → Set X
hU : ∀ (α : I), IsOpen (U α)
hcover : ↑(⋃ α, U α) = X
s : Set X
h : ∀ (α : I), interior (s ∩ U α) = ∅
V : Set X
hVS : V ⊆ s
hV : IsOpen V
α : I
⊢ V ∩ U α = ∅ | X : Type
inst✝ : TopologicalSpace X
I : Type
U : I → Set X
hU : ∀ (α : I), IsOpen (U α)
hcover : ↑(⋃ α, U α) = X
s : Set X
h : ∀ (α : I), interior (s ∩ U α) = ∅
V : Set X
hVS : V ⊆ s
hV : IsOpen V
α : I
h₁ : IsOpen (V ∩ U α)
⊢ V ∩ U α = ∅ | Please generate a tactic in lean4 to solve the state.
STATE:
X : Type
inst✝ : TopologicalSpace X
I : Type
U : I → Set X
hU : ∀ (α : I), IsOpen (U α)
hcover : ↑(⋃ α, U α) = X
s : Set X
h : ∀ (α : I), interior (s ∩ U α) = ∅
V : Set X
hVS : V ⊆ s
hV : IsOpen V
α : I
⊢ V ∩ U α = ∅
TACTIC:
|
https://github.com/fpvandoorn/sard.git | f4a1bb550136e8591dcd20dca5bdcce766a958c7 | Sard/ObsoleteHelpers.lean | interior_zero_iff_open_cover | [53, 1] | [87, 10] | have h₂ : V ∩ U α ⊆ s ∩ U α := by exact Set.inter_subset_inter_left (U α) hVS | X : Type
inst✝ : TopologicalSpace X
I : Type
U : I → Set X
hU : ∀ (α : I), IsOpen (U α)
hcover : ↑(⋃ α, U α) = X
s : Set X
h : ∀ (α : I), interior (s ∩ U α) = ∅
V : Set X
hVS : V ⊆ s
hV : IsOpen V
α : I
h₁ : IsOpen (V ∩ U α)
⊢ V ∩ U α = ∅ | X : Type
inst✝ : TopologicalSpace X
I : Type
U : I → Set X
hU : ∀ (α : I), IsOpen (U α)
hcover : ↑(⋃ α, U α) = X
s : Set X
h : ∀ (α : I), interior (s ∩ U α) = ∅
V : Set X
hVS : V ⊆ s
hV : IsOpen V
α : I
h₁ : IsOpen (V ∩ U α)
h₂ : V ∩ U α ⊆ s ∩ U α
⊢ V ∩ U α = ∅ | Please generate a tactic in lean4 to solve the state.
STATE:
X : Type
inst✝ : TopologicalSpace X
I : Type
U : I → Set X
hU : ∀ (α : I), IsOpen (U α)
hcover : ↑(⋃ α, U α) = X
s : Set X
h : ∀ (α : I), interior (s ∩ U α) = ∅
V : Set X
hVS : V ⊆ s
hV : IsOpen V
α : I
h₁ : IsOpen (V ∩ U α)
⊢ V ∩ U α = ∅
TACTIC:
|
https://github.com/fpvandoorn/sard.git | f4a1bb550136e8591dcd20dca5bdcce766a958c7 | Sard/ObsoleteHelpers.lean | interior_zero_iff_open_cover | [53, 1] | [87, 10] | have h₃ : V ∩ U α ⊆ interior (s ∩ U α) := by exact interior_maximal h₂ h₁ | X : Type
inst✝ : TopologicalSpace X
I : Type
U : I → Set X
hU : ∀ (α : I), IsOpen (U α)
hcover : ↑(⋃ α, U α) = X
s : Set X
h : ∀ (α : I), interior (s ∩ U α) = ∅
V : Set X
hVS : V ⊆ s
hV : IsOpen V
α : I
h₁ : IsOpen (V ∩ U α)
h₂ : V ∩ U α ⊆ s ∩ U α
⊢ V ∩ U α = ∅ | X : Type
inst✝ : TopologicalSpace X
I : Type
U : I → Set X
hU : ∀ (α : I), IsOpen (U α)
hcover : ↑(⋃ α, U α) = X
s : Set X
h : ∀ (α : I), interior (s ∩ U α) = ∅
V : Set X
hVS : V ⊆ s
hV : IsOpen V
α : I
h₁ : IsOpen (V ∩ U α)
h₂ : V ∩ U α ⊆ s ∩ U α
h₃ : V ∩ U α ⊆ interior (s ∩ U α)
⊢ V ∩ U α = ∅ | Please generate a tactic in lean4 to solve the state.
STATE:
X : Type
inst✝ : TopologicalSpace X
I : Type
U : I → Set X
hU : ∀ (α : I), IsOpen (U α)
hcover : ↑(⋃ α, U α) = X
s : Set X
h : ∀ (α : I), interior (s ∩ U α) = ∅
V : Set X
hVS : V ⊆ s
hV : IsOpen V
α : I
h₁ : IsOpen (V ∩ U α)
h₂ : V ∩ U α ⊆ s ∩ U α
⊢ V ∩ U α =... |
https://github.com/fpvandoorn/sard.git | f4a1bb550136e8591dcd20dca5bdcce766a958c7 | Sard/ObsoleteHelpers.lean | interior_zero_iff_open_cover | [53, 1] | [87, 10] | exact Set.subset_eq_empty h₃ (h α) | X : Type
inst✝ : TopologicalSpace X
I : Type
U : I → Set X
hU : ∀ (α : I), IsOpen (U α)
hcover : ↑(⋃ α, U α) = X
s : Set X
h : ∀ (α : I), interior (s ∩ U α) = ∅
V : Set X
hVS : V ⊆ s
hV : IsOpen V
α : I
h₁ : IsOpen (V ∩ U α)
h₂ : V ∩ U α ⊆ s ∩ U α
h₃ : V ∩ U α ⊆ interior (s ∩ U α)
⊢ V ∩ U α = ∅ | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
X : Type
inst✝ : TopologicalSpace X
I : Type
U : I → Set X
hU : ∀ (α : I), IsOpen (U α)
hcover : ↑(⋃ α, U α) = X
s : Set X
h : ∀ (α : I), interior (s ∩ U α) = ∅
V : Set X
hVS : V ⊆ s
hV : IsOpen V
α : I
h₁ : IsOpen (V ∩ U α)
h₂ : V ∩ U α ⊆ s ∩ U α
h₃ : V ∩ U ... |
https://github.com/fpvandoorn/sard.git | f4a1bb550136e8591dcd20dca5bdcce766a958c7 | Sard/ObsoleteHelpers.lean | interior_zero_iff_open_cover | [53, 1] | [87, 10] | exact IsOpen.inter hV (hU α) | X : Type
inst✝ : TopologicalSpace X
I : Type
U : I → Set X
hU : ∀ (α : I), IsOpen (U α)
hcover : ↑(⋃ α, U α) = X
s : Set X
h : ∀ (α : I), interior (s ∩ U α) = ∅
V : Set X
hVS : V ⊆ s
hV : IsOpen V
α : I
⊢ IsOpen (V ∩ U α) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
X : Type
inst✝ : TopologicalSpace X
I : Type
U : I → Set X
hU : ∀ (α : I), IsOpen (U α)
hcover : ↑(⋃ α, U α) = X
s : Set X
h : ∀ (α : I), interior (s ∩ U α) = ∅
V : Set X
hVS : V ⊆ s
hV : IsOpen V
α : I
⊢ IsOpen (V ∩ U α)
TACTIC:
|
https://github.com/fpvandoorn/sard.git | f4a1bb550136e8591dcd20dca5bdcce766a958c7 | Sard/ObsoleteHelpers.lean | interior_zero_iff_open_cover | [53, 1] | [87, 10] | exact Set.inter_subset_inter_left (U α) hVS | X : Type
inst✝ : TopologicalSpace X
I : Type
U : I → Set X
hU : ∀ (α : I), IsOpen (U α)
hcover : ↑(⋃ α, U α) = X
s : Set X
h : ∀ (α : I), interior (s ∩ U α) = ∅
V : Set X
hVS : V ⊆ s
hV : IsOpen V
α : I
h₁ : IsOpen (V ∩ U α)
⊢ V ∩ U α ⊆ s ∩ U α | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
X : Type
inst✝ : TopologicalSpace X
I : Type
U : I → Set X
hU : ∀ (α : I), IsOpen (U α)
hcover : ↑(⋃ α, U α) = X
s : Set X
h : ∀ (α : I), interior (s ∩ U α) = ∅
V : Set X
hVS : V ⊆ s
hV : IsOpen V
α : I
h₁ : IsOpen (V ∩ U α)
⊢ V ∩ U α ⊆ s ∩ U α
TACTIC:
|
https://github.com/fpvandoorn/sard.git | f4a1bb550136e8591dcd20dca5bdcce766a958c7 | Sard/ObsoleteHelpers.lean | interior_zero_iff_open_cover | [53, 1] | [87, 10] | exact interior_maximal h₂ h₁ | X : Type
inst✝ : TopologicalSpace X
I : Type
U : I → Set X
hU : ∀ (α : I), IsOpen (U α)
hcover : ↑(⋃ α, U α) = X
s : Set X
h : ∀ (α : I), interior (s ∩ U α) = ∅
V : Set X
hVS : V ⊆ s
hV : IsOpen V
α : I
h₁ : IsOpen (V ∩ U α)
h₂ : V ∩ U α ⊆ s ∩ U α
⊢ V ∩ U α ⊆ interior (s ∩ U α) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
X : Type
inst✝ : TopologicalSpace X
I : Type
U : I → Set X
hU : ∀ (α : I), IsOpen (U α)
hcover : ↑(⋃ α, U α) = X
s : Set X
h : ∀ (α : I), interior (s ∩ U α) = ∅
V : Set X
hVS : V ⊆ s
hV : IsOpen V
α : I
h₁ : IsOpen (V ∩ U α)
h₂ : V ∩ U α ⊆ s ∩ U α
⊢ V ∩ U α ⊆... |
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