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/LocallyLipschitz.lean | LipschitzOnWith.sum | [80, 1] | [96, 41] | ring | X : Type u_1
Y : Type u_2
Z : Type u_3
inst✝² : MetricSpace X
inst✝¹ : NormedAddCommGroup Y
inst✝ : NormedSpace ℝ Y
f g : X → Y
Kf Kg : ℝ≥0
s : Set X
hf : LipschitzOnWith Kf f s
hg : LipschitzOnWith Kg g s
y : X
hy : y ∈ s
z : X
hz : z ∈ s
translation : ∀ (w w' w'' : Y), edist (w + w'') (w' + w'') = edist w w'
⊢ ↑Kf * ... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
X : Type u_1
Y : Type u_2
Z : Type u_3
inst✝² : MetricSpace X
inst✝¹ : NormedAddCommGroup Y
inst✝ : NormedSpace ℝ Y
f g : X → Y
Kf Kg : ℝ≥0
s : Set X
hf : LipschitzOnWith Kf f s
hg : LipschitzOnWith Kg g s
y : X
hy : y ∈ s
z : X
hz : z ∈ s
translation : ∀ (w ... |
https://github.com/fpvandoorn/sard.git | f4a1bb550136e8591dcd20dca5bdcce766a958c7 | Sard/LocallyLipschitz.lean | LocallyLipschitz.sum | [103, 1] | [112, 48] | intro x | X : Type u_1
Y : Type u_2
Z : Type u_3
inst✝² : MetricSpace X
inst✝¹ : NormedAddCommGroup Y
inst✝ : NormedSpace ℝ Y
f g : X → Y
hf : LocallyLipschitz f
hg : LocallyLipschitz g
⊢ LocallyLipschitz (f + g) | X : Type u_1
Y : Type u_2
Z : Type u_3
inst✝² : MetricSpace X
inst✝¹ : NormedAddCommGroup Y
inst✝ : NormedSpace ℝ Y
f g : X → Y
hf : LocallyLipschitz f
hg : LocallyLipschitz g
x : X
⊢ ∃ K, ∃ t ∈ 𝓝 x, LipschitzOnWith K (f + g) t | Please generate a tactic in lean4 to solve the state.
STATE:
X : Type u_1
Y : Type u_2
Z : Type u_3
inst✝² : MetricSpace X
inst✝¹ : NormedAddCommGroup Y
inst✝ : NormedSpace ℝ Y
f g : X → Y
hf : LocallyLipschitz f
hg : LocallyLipschitz g
⊢ LocallyLipschitz (f + g)
TACTIC:
|
https://github.com/fpvandoorn/sard.git | f4a1bb550136e8591dcd20dca5bdcce766a958c7 | Sard/LocallyLipschitz.lean | LocallyLipschitz.sum | [103, 1] | [112, 48] | rcases hf x with ⟨Kf, t₁, h₁t, hfL⟩ | X : Type u_1
Y : Type u_2
Z : Type u_3
inst✝² : MetricSpace X
inst✝¹ : NormedAddCommGroup Y
inst✝ : NormedSpace ℝ Y
f g : X → Y
hf : LocallyLipschitz f
hg : LocallyLipschitz g
x : X
⊢ ∃ K, ∃ t ∈ 𝓝 x, LipschitzOnWith K (f + g) t | case intro.intro.intro
X : Type u_1
Y : Type u_2
Z : Type u_3
inst✝² : MetricSpace X
inst✝¹ : NormedAddCommGroup Y
inst✝ : NormedSpace ℝ Y
f g : X → Y
hf : LocallyLipschitz f
hg : LocallyLipschitz g
x : X
Kf : ℝ≥0
t₁ : Set X
h₁t : t₁ ∈ 𝓝 x
hfL : LipschitzOnWith Kf f t₁
⊢ ∃ K, ∃ t ∈ 𝓝 x, LipschitzOnWith K (f + g) t | Please generate a tactic in lean4 to solve the state.
STATE:
X : Type u_1
Y : Type u_2
Z : Type u_3
inst✝² : MetricSpace X
inst✝¹ : NormedAddCommGroup Y
inst✝ : NormedSpace ℝ Y
f g : X → Y
hf : LocallyLipschitz f
hg : LocallyLipschitz g
x : X
⊢ ∃ K, ∃ t ∈ 𝓝 x, LipschitzOnWith K (f + g) t
TACTIC:
|
https://github.com/fpvandoorn/sard.git | f4a1bb550136e8591dcd20dca5bdcce766a958c7 | Sard/LocallyLipschitz.lean | LocallyLipschitz.sum | [103, 1] | [112, 48] | rcases hg x with ⟨Kg, t₂, h₂t, hgL⟩ | case intro.intro.intro
X : Type u_1
Y : Type u_2
Z : Type u_3
inst✝² : MetricSpace X
inst✝¹ : NormedAddCommGroup Y
inst✝ : NormedSpace ℝ Y
f g : X → Y
hf : LocallyLipschitz f
hg : LocallyLipschitz g
x : X
Kf : ℝ≥0
t₁ : Set X
h₁t : t₁ ∈ 𝓝 x
hfL : LipschitzOnWith Kf f t₁
⊢ ∃ K, ∃ t ∈ 𝓝 x, LipschitzOnWith K (f + g) t | case intro.intro.intro.intro.intro.intro
X : Type u_1
Y : Type u_2
Z : Type u_3
inst✝² : MetricSpace X
inst✝¹ : NormedAddCommGroup Y
inst✝ : NormedSpace ℝ Y
f g : X → Y
hf : LocallyLipschitz f
hg : LocallyLipschitz g
x : X
Kf : ℝ≥0
t₁ : Set X
h₁t : t₁ ∈ 𝓝 x
hfL : LipschitzOnWith Kf f t₁
Kg : ℝ≥0
t₂ : Set X
h₂t : t₂ ∈ ... | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.intro
X : Type u_1
Y : Type u_2
Z : Type u_3
inst✝² : MetricSpace X
inst✝¹ : NormedAddCommGroup Y
inst✝ : NormedSpace ℝ Y
f g : X → Y
hf : LocallyLipschitz f
hg : LocallyLipschitz g
x : X
Kf : ℝ≥0
t₁ : Set X
h₁t : t₁ ∈ 𝓝 x
hfL : LipschitzOnW... |
https://github.com/fpvandoorn/sard.git | f4a1bb550136e8591dcd20dca5bdcce766a958c7 | Sard/LocallyLipschitz.lean | LocallyLipschitz.sum | [103, 1] | [112, 48] | use Kf + Kg, t₁ ∩ t₂ | case intro.intro.intro.intro.intro.intro
X : Type u_1
Y : Type u_2
Z : Type u_3
inst✝² : MetricSpace X
inst✝¹ : NormedAddCommGroup Y
inst✝ : NormedSpace ℝ Y
f g : X → Y
hf : LocallyLipschitz f
hg : LocallyLipschitz g
x : X
Kf : ℝ≥0
t₁ : Set X
h₁t : t₁ ∈ 𝓝 x
hfL : LipschitzOnWith Kf f t₁
Kg : ℝ≥0
t₂ : Set X
h₂t : t₂ ∈ ... | case h
X : Type u_1
Y : Type u_2
Z : Type u_3
inst✝² : MetricSpace X
inst✝¹ : NormedAddCommGroup Y
inst✝ : NormedSpace ℝ Y
f g : X → Y
hf : LocallyLipschitz f
hg : LocallyLipschitz g
x : X
Kf : ℝ≥0
t₁ : Set X
h₁t : t₁ ∈ 𝓝 x
hfL : LipschitzOnWith Kf f t₁
Kg : ℝ≥0
t₂ : Set X
h₂t : t₂ ∈ 𝓝 x
hgL : LipschitzOnWith Kg g t₂... | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.intro.intro.intro.intro
X : Type u_1
Y : Type u_2
Z : Type u_3
inst✝² : MetricSpace X
inst✝¹ : NormedAddCommGroup Y
inst✝ : NormedSpace ℝ Y
f g : X → Y
hf : LocallyLipschitz f
hg : LocallyLipschitz g
x : X
Kf : ℝ≥0
t₁ : Set X
h₁t : t₁ ∈ 𝓝 x
... |
https://github.com/fpvandoorn/sard.git | f4a1bb550136e8591dcd20dca5bdcce766a958c7 | Sard/LocallyLipschitz.lean | LocallyLipschitz.sum | [103, 1] | [112, 48] | have hf' : LipschitzOnWith Kf f (t₁ ∩ t₂) := hfL.mono (Set.inter_subset_left t₁ t₂) | case h
X : Type u_1
Y : Type u_2
Z : Type u_3
inst✝² : MetricSpace X
inst✝¹ : NormedAddCommGroup Y
inst✝ : NormedSpace ℝ Y
f g : X → Y
hf : LocallyLipschitz f
hg : LocallyLipschitz g
x : X
Kf : ℝ≥0
t₁ : Set X
h₁t : t₁ ∈ 𝓝 x
hfL : LipschitzOnWith Kf f t₁
Kg : ℝ≥0
t₂ : Set X
h₂t : t₂ ∈ 𝓝 x
hgL : LipschitzOnWith Kg g t₂... | case h
X : Type u_1
Y : Type u_2
Z : Type u_3
inst✝² : MetricSpace X
inst✝¹ : NormedAddCommGroup Y
inst✝ : NormedSpace ℝ Y
f g : X → Y
hf : LocallyLipschitz f
hg : LocallyLipschitz g
x : X
Kf : ℝ≥0
t₁ : Set X
h₁t : t₁ ∈ 𝓝 x
hfL : LipschitzOnWith Kf f t₁
Kg : ℝ≥0
t₂ : Set X
h₂t : t₂ ∈ 𝓝 x
hgL : LipschitzOnWith Kg g t₂... | Please generate a tactic in lean4 to solve the state.
STATE:
case h
X : Type u_1
Y : Type u_2
Z : Type u_3
inst✝² : MetricSpace X
inst✝¹ : NormedAddCommGroup Y
inst✝ : NormedSpace ℝ Y
f g : X → Y
hf : LocallyLipschitz f
hg : LocallyLipschitz g
x : X
Kf : ℝ≥0
t₁ : Set X
h₁t : t₁ ∈ 𝓝 x
hfL : LipschitzOnWith Kf f t₁
Kg :... |
https://github.com/fpvandoorn/sard.git | f4a1bb550136e8591dcd20dca5bdcce766a958c7 | Sard/LocallyLipschitz.lean | LocallyLipschitz.sum | [103, 1] | [112, 48] | have hg' : LipschitzOnWith Kg g (t₁ ∩ t₂) := hgL.mono (Set.inter_subset_right t₁ t₂) | case h
X : Type u_1
Y : Type u_2
Z : Type u_3
inst✝² : MetricSpace X
inst✝¹ : NormedAddCommGroup Y
inst✝ : NormedSpace ℝ Y
f g : X → Y
hf : LocallyLipschitz f
hg : LocallyLipschitz g
x : X
Kf : ℝ≥0
t₁ : Set X
h₁t : t₁ ∈ 𝓝 x
hfL : LipschitzOnWith Kf f t₁
Kg : ℝ≥0
t₂ : Set X
h₂t : t₂ ∈ 𝓝 x
hgL : LipschitzOnWith Kg g t₂... | case h
X : Type u_1
Y : Type u_2
Z : Type u_3
inst✝² : MetricSpace X
inst✝¹ : NormedAddCommGroup Y
inst✝ : NormedSpace ℝ Y
f g : X → Y
hf : LocallyLipschitz f
hg : LocallyLipschitz g
x : X
Kf : ℝ≥0
t₁ : Set X
h₁t : t₁ ∈ 𝓝 x
hfL : LipschitzOnWith Kf f t₁
Kg : ℝ≥0
t₂ : Set X
h₂t : t₂ ∈ 𝓝 x
hgL : LipschitzOnWith Kg g t₂... | Please generate a tactic in lean4 to solve the state.
STATE:
case h
X : Type u_1
Y : Type u_2
Z : Type u_3
inst✝² : MetricSpace X
inst✝¹ : NormedAddCommGroup Y
inst✝ : NormedSpace ℝ Y
f g : X → Y
hf : LocallyLipschitz f
hg : LocallyLipschitz g
x : X
Kf : ℝ≥0
t₁ : Set X
h₁t : t₁ ∈ 𝓝 x
hfL : LipschitzOnWith Kf f t₁
Kg :... |
https://github.com/fpvandoorn/sard.git | f4a1bb550136e8591dcd20dca5bdcce766a958c7 | Sard/LocallyLipschitz.lean | LocallyLipschitz.sum | [103, 1] | [112, 48] | exact ⟨Filter.inter_mem h₁t h₂t, hf'.sum hg'⟩ | case h
X : Type u_1
Y : Type u_2
Z : Type u_3
inst✝² : MetricSpace X
inst✝¹ : NormedAddCommGroup Y
inst✝ : NormedSpace ℝ Y
f g : X → Y
hf : LocallyLipschitz f
hg : LocallyLipschitz g
x : X
Kf : ℝ≥0
t₁ : Set X
h₁t : t₁ ∈ 𝓝 x
hfL : LipschitzOnWith Kf f t₁
Kg : ℝ≥0
t₂ : Set X
h₂t : t₂ ∈ 𝓝 x
hgL : LipschitzOnWith Kg g t₂... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h
X : Type u_1
Y : Type u_2
Z : Type u_3
inst✝² : MetricSpace X
inst✝¹ : NormedAddCommGroup Y
inst✝ : NormedSpace ℝ Y
f g : X → Y
hf : LocallyLipschitz f
hg : LocallyLipschitz g
x : X
Kf : ℝ≥0
t₁ : Set X
h₁t : t₁ ∈ 𝓝 x
hfL : LipschitzOnWith Kf f t₁
Kg :... |
https://github.com/fpvandoorn/sard.git | f4a1bb550136e8591dcd20dca5bdcce766a958c7 | Sard/LocallyLipschitz.lean | helper | [115, 1] | [115, 98] | sorry | X : Type u_1
Y : Type u_2
Z : Type u_3
inst✝² : MetricSpace X
inst✝¹ : NormedAddCommGroup Y
inst✝ : NormedSpace ℝ Y
f g : X → Y
a b : ℝ
⊢ ENNReal.ofReal (a * b) = ENNReal.ofReal a * ENNReal.ofReal b | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
X : Type u_1
Y : Type u_2
Z : Type u_3
inst✝² : MetricSpace X
inst✝¹ : NormedAddCommGroup Y
inst✝ : NormedSpace ℝ Y
f g : X → Y
a b : ℝ
⊢ ENNReal.ofReal (a * b) = ENNReal.ofReal a * ENNReal.ofReal b
TACTIC:
|
https://github.com/fpvandoorn/sard.git | f4a1bb550136e8591dcd20dca5bdcce766a958c7 | Sard/LocallyLipschitz.lean | last' | [119, 1] | [120, 24] | exact mul_assoc a K c | X : Type u_1
Y : Type u_2
Z : Type u_3
inst✝² : MetricSpace X
inst✝¹ : NormedAddCommGroup Y
inst✝ : NormedSpace ℝ Y
f g : X → Y
a K c : ℝ≥0
⊢ a * K * c = a * (K * c) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
X : Type u_1
Y : Type u_2
Z : Type u_3
inst✝² : MetricSpace X
inst✝¹ : NormedAddCommGroup Y
inst✝ : NormedSpace ℝ Y
f g : X → Y
a K c : ℝ≥0
⊢ a * K * c = a * (K * c)
TACTIC:
|
https://github.com/fpvandoorn/sard.git | f4a1bb550136e8591dcd20dca5bdcce766a958c7 | Sard/LocallyLipschitz.lean | last'' | [122, 1] | [124, 24] | exact mul_assoc _ K c | X : Type u_1
Y : Type u_2
Z : Type u_3
inst✝² : MetricSpace X
inst✝¹ : NormedAddCommGroup Y
inst✝ : NormedSpace ℝ Y
f g : X → Y
a : ℝ
K c : ℝ≥0
⊢ ‖a‖.toNNReal * K * c = ‖a‖.toNNReal * (K * c) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
X : Type u_1
Y : Type u_2
Z : Type u_3
inst✝² : MetricSpace X
inst✝¹ : NormedAddCommGroup Y
inst✝ : NormedSpace ℝ Y
f g : X → Y
a : ℝ
K c : ℝ≥0
⊢ ‖a‖.toNNReal * K * c = ‖a‖.toNNReal * (K * c)
TACTIC:
|
https://github.com/fpvandoorn/sard.git | f4a1bb550136e8591dcd20dca5bdcce766a958c7 | Sard/LocallyLipschitz.lean | last | [126, 1] | [132, 8] | have : c ≠ ∞ := by sorry | X : Type u_1
Y : Type u_2
Z : Type u_3
inst✝² : MetricSpace X
inst✝¹ : NormedAddCommGroup Y
inst✝ : NormedSpace ℝ Y
f g : X → Y
a : ℝ
K : ℝ≥0
c : ℝ≥0∞
⊢ ↑‖a‖.toNNReal * (↑K * c) = ↑(‖a‖.toNNReal * K) * c | X : Type u_1
Y : Type u_2
Z : Type u_3
inst✝² : MetricSpace X
inst✝¹ : NormedAddCommGroup Y
inst✝ : NormedSpace ℝ Y
f g : X → Y
a : ℝ
K : ℝ≥0
c : ℝ≥0∞
this : c ≠ ⊤
⊢ ↑‖a‖.toNNReal * (↑K * c) = ↑(‖a‖.toNNReal * K) * c | Please generate a tactic in lean4 to solve the state.
STATE:
X : Type u_1
Y : Type u_2
Z : Type u_3
inst✝² : MetricSpace X
inst✝¹ : NormedAddCommGroup Y
inst✝ : NormedSpace ℝ Y
f g : X → Y
a : ℝ
K : ℝ≥0
c : ℝ≥0∞
⊢ ↑‖a‖.toNNReal * (↑K * c) = ↑(‖a‖.toNNReal * K) * c
TACTIC:
|
https://github.com/fpvandoorn/sard.git | f4a1bb550136e8591dcd20dca5bdcce766a958c7 | Sard/LocallyLipschitz.lean | last | [126, 1] | [132, 8] | lift c to ℝ≥0 using this | X : Type u_1
Y : Type u_2
Z : Type u_3
inst✝² : MetricSpace X
inst✝¹ : NormedAddCommGroup Y
inst✝ : NormedSpace ℝ Y
f g : X → Y
a : ℝ
K : ℝ≥0
c : ℝ≥0∞
this : c ≠ ⊤
⊢ ↑‖a‖.toNNReal * (↑K * c) = ↑(‖a‖.toNNReal * K) * c | case intro
X : Type u_1
Y : Type u_2
Z : Type u_3
inst✝² : MetricSpace X
inst✝¹ : NormedAddCommGroup Y
inst✝ : NormedSpace ℝ Y
f g : X → Y
a : ℝ
K c : ℝ≥0
⊢ ↑‖a‖.toNNReal * (↑K * ↑c) = ↑(‖a‖.toNNReal * K) * ↑c | Please generate a tactic in lean4 to solve the state.
STATE:
X : Type u_1
Y : Type u_2
Z : Type u_3
inst✝² : MetricSpace X
inst✝¹ : NormedAddCommGroup Y
inst✝ : NormedSpace ℝ Y
f g : X → Y
a : ℝ
K : ℝ≥0
c : ℝ≥0∞
this : c ≠ ⊤
⊢ ↑‖a‖.toNNReal * (↑K * c) = ↑(‖a‖.toNNReal * K) * c
TACTIC:
|
https://github.com/fpvandoorn/sard.git | f4a1bb550136e8591dcd20dca5bdcce766a958c7 | Sard/LocallyLipschitz.lean | last | [126, 1] | [132, 8] | sorry | case intro
X : Type u_1
Y : Type u_2
Z : Type u_3
inst✝² : MetricSpace X
inst✝¹ : NormedAddCommGroup Y
inst✝ : NormedSpace ℝ Y
f g : X → Y
a : ℝ
K c : ℝ≥0
⊢ ↑‖a‖.toNNReal * (↑K * ↑c) = ↑(‖a‖.toNNReal * K) * ↑c | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case intro
X : Type u_1
Y : Type u_2
Z : Type u_3
inst✝² : MetricSpace X
inst✝¹ : NormedAddCommGroup Y
inst✝ : NormedSpace ℝ Y
f g : X → Y
a : ℝ
K c : ℝ≥0
⊢ ↑‖a‖.toNNReal * (↑K * ↑c) = ↑(‖a‖.toNNReal * K) * ↑c
TACTIC:
|
https://github.com/fpvandoorn/sard.git | f4a1bb550136e8591dcd20dca5bdcce766a958c7 | Sard/LocallyLipschitz.lean | last | [126, 1] | [132, 8] | sorry | X : Type u_1
Y : Type u_2
Z : Type u_3
inst✝² : MetricSpace X
inst✝¹ : NormedAddCommGroup Y
inst✝ : NormedSpace ℝ Y
f g : X → Y
a : ℝ
K : ℝ≥0
c : ℝ≥0∞
⊢ c ≠ ⊤ | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
X : Type u_1
Y : Type u_2
Z : Type u_3
inst✝² : MetricSpace X
inst✝¹ : NormedAddCommGroup Y
inst✝ : NormedSpace ℝ Y
f g : X → Y
a : ℝ
K : ℝ≥0
c : ℝ≥0∞
⊢ c ≠ ⊤
TACTIC:
|
https://github.com/fpvandoorn/sard.git | f4a1bb550136e8591dcd20dca5bdcce766a958c7 | Sard/LocallyLipschitz.lean | LipschitzOnWith.smul | [134, 1] | [150, 80] | intro x hx y hy | X : Type u_1
Y : Type u_2
Z : Type u_3
inst✝² : MetricSpace X
inst✝¹ : NormedAddCommGroup Y
inst✝ : NormedSpace ℝ Y
f g : X → Y
K : ℝ≥0
s : Set X
hf : LipschitzOnWith K f s
a : ℝ
⊢ LipschitzOnWith ((ENNReal.ofReal ‖a‖).toNNReal * K) (fun x => a • f x) s | X : Type u_1
Y : Type u_2
Z : Type u_3
inst✝² : MetricSpace X
inst✝¹ : NormedAddCommGroup Y
inst✝ : NormedSpace ℝ Y
f g : X → Y
K : ℝ≥0
s : Set X
hf : LipschitzOnWith K f s
a : ℝ
x : X
hx : x ∈ s
y : X
hy : y ∈ s
⊢ edist ((fun x => a • f x) x) ((fun x => a • f x) y) ≤ ↑((ENNReal.ofReal ‖a‖).toNNReal * K) * edist x y | Please generate a tactic in lean4 to solve the state.
STATE:
X : Type u_1
Y : Type u_2
Z : Type u_3
inst✝² : MetricSpace X
inst✝¹ : NormedAddCommGroup Y
inst✝ : NormedSpace ℝ Y
f g : X → Y
K : ℝ≥0
s : Set X
hf : LipschitzOnWith K f s
a : ℝ
⊢ LipschitzOnWith ((ENNReal.ofReal ‖a‖).toNNReal * K) (fun x => a • f x) s
TACTI... |
https://github.com/fpvandoorn/sard.git | f4a1bb550136e8591dcd20dca5bdcce766a958c7 | Sard/LocallyLipschitz.lean | LipschitzOnWith.smul | [134, 1] | [150, 80] | have : dist (a • f x) (a • f y) = ‖a‖ * dist (f x) (f y) := by
calc dist (a • f x) (a • f y)
_ = ‖(a • (f x)) - (a • (f y))‖ := by apply dist_eq_norm
_ = ‖a • ((f x) - (f y))‖ := by rw [smul_sub]
_ = ‖a‖ * ‖(f x) - (f y)‖ := by rw [norm_smul]
_ = ‖a‖ * dist (f x) (f y) := by rw [← dist_eq_no... | X : Type u_1
Y : Type u_2
Z : Type u_3
inst✝² : MetricSpace X
inst✝¹ : NormedAddCommGroup Y
inst✝ : NormedSpace ℝ Y
f g : X → Y
K : ℝ≥0
s : Set X
hf : LipschitzOnWith K f s
a : ℝ
x : X
hx : x ∈ s
y : X
hy : y ∈ s
⊢ edist ((fun x => a • f x) x) ((fun x => a • f x) y) ≤ ↑((ENNReal.ofReal ‖a‖).toNNReal * K) * edist x y | X : Type u_1
Y : Type u_2
Z : Type u_3
inst✝² : MetricSpace X
inst✝¹ : NormedAddCommGroup Y
inst✝ : NormedSpace ℝ Y
f g : X → Y
K : ℝ≥0
s : Set X
hf : LipschitzOnWith K f s
a : ℝ
x : X
hx : x ∈ s
y : X
hy : y ∈ s
this : dist (a • f x) (a • f y) = ‖a‖ * dist (f x) (f y)
⊢ edist ((fun x => a • f x) x) ((fun x => a • f x)... | Please generate a tactic in lean4 to solve the state.
STATE:
X : Type u_1
Y : Type u_2
Z : Type u_3
inst✝² : MetricSpace X
inst✝¹ : NormedAddCommGroup Y
inst✝ : NormedSpace ℝ Y
f g : X → Y
K : ℝ≥0
s : Set X
hf : LipschitzOnWith K f s
a : ℝ
x : X
hx : x ∈ s
y : X
hy : y ∈ s
⊢ edist ((fun x => a • f x) x) ((fun x => a • ... |
https://github.com/fpvandoorn/sard.git | f4a1bb550136e8591dcd20dca5bdcce766a958c7 | Sard/LocallyLipschitz.lean | LipschitzOnWith.smul | [134, 1] | [150, 80] | calc edist (a • f x) (a • f y)
_ = ENNReal.ofReal (dist (a • f x) (a • f y)) := by rw [edist_dist]
_ = ENNReal.ofReal (‖a‖ * dist (f x) (f y)) := by rw [this]
_ = ENNReal.ofReal (‖a‖) * ENNReal.ofReal (dist (f x) (f y)) := by rw [← helper]
_ = ENNReal.ofReal ‖a‖ * edist (f x) (f y) := by rw [edist_dist]... | X : Type u_1
Y : Type u_2
Z : Type u_3
inst✝² : MetricSpace X
inst✝¹ : NormedAddCommGroup Y
inst✝ : NormedSpace ℝ Y
f g : X → Y
K : ℝ≥0
s : Set X
hf : LipschitzOnWith K f s
a : ℝ
x : X
hx : x ∈ s
y : X
hy : y ∈ s
this : dist (a • f x) (a • f y) = ‖a‖ * dist (f x) (f y)
⊢ edist ((fun x => a • f x) x) ((fun x => a • f x)... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
X : Type u_1
Y : Type u_2
Z : Type u_3
inst✝² : MetricSpace X
inst✝¹ : NormedAddCommGroup Y
inst✝ : NormedSpace ℝ Y
f g : X → Y
K : ℝ≥0
s : Set X
hf : LipschitzOnWith K f s
a : ℝ
x : X
hx : x ∈ s
y : X
hy : y ∈ s
this : dist (a • f x) (a • f y) = ‖a‖ * dist (... |
https://github.com/fpvandoorn/sard.git | f4a1bb550136e8591dcd20dca5bdcce766a958c7 | Sard/LocallyLipschitz.lean | LipschitzOnWith.smul | [134, 1] | [150, 80] | calc dist (a • f x) (a • f y)
_ = ‖(a • (f x)) - (a • (f y))‖ := by apply dist_eq_norm
_ = ‖a • ((f x) - (f y))‖ := by rw [smul_sub]
_ = ‖a‖ * ‖(f x) - (f y)‖ := by rw [norm_smul]
_ = ‖a‖ * dist (f x) (f y) := by rw [← dist_eq_norm]
_ = ‖a‖ * dist (f x) (f y) := by rw [← dist_smul₀] | X : Type u_1
Y : Type u_2
Z : Type u_3
inst✝² : MetricSpace X
inst✝¹ : NormedAddCommGroup Y
inst✝ : NormedSpace ℝ Y
f g : X → Y
K : ℝ≥0
s : Set X
hf : LipschitzOnWith K f s
a : ℝ
x : X
hx : x ∈ s
y : X
hy : y ∈ s
⊢ dist (a • f x) (a • f y) = ‖a‖ * dist (f x) (f y) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
X : Type u_1
Y : Type u_2
Z : Type u_3
inst✝² : MetricSpace X
inst✝¹ : NormedAddCommGroup Y
inst✝ : NormedSpace ℝ Y
f g : X → Y
K : ℝ≥0
s : Set X
hf : LipschitzOnWith K f s
a : ℝ
x : X
hx : x ∈ s
y : X
hy : y ∈ s
⊢ dist (a • f x) (a • f y) = ‖a‖ * dist (f x) ... |
https://github.com/fpvandoorn/sard.git | f4a1bb550136e8591dcd20dca5bdcce766a958c7 | Sard/LocallyLipschitz.lean | LipschitzOnWith.smul | [134, 1] | [150, 80] | apply dist_eq_norm | X : Type u_1
Y : Type u_2
Z : Type u_3
inst✝² : MetricSpace X
inst✝¹ : NormedAddCommGroup Y
inst✝ : NormedSpace ℝ Y
f g : X → Y
K : ℝ≥0
s : Set X
hf : LipschitzOnWith K f s
a : ℝ
x : X
hx : x ∈ s
y : X
hy : y ∈ s
⊢ dist (a • f x) (a • f y) = ‖a • f x - a • f y‖ | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
X : Type u_1
Y : Type u_2
Z : Type u_3
inst✝² : MetricSpace X
inst✝¹ : NormedAddCommGroup Y
inst✝ : NormedSpace ℝ Y
f g : X → Y
K : ℝ≥0
s : Set X
hf : LipschitzOnWith K f s
a : ℝ
x : X
hx : x ∈ s
y : X
hy : y ∈ s
⊢ dist (a • f x) (a • f y) = ‖a • f x - a • f ... |
https://github.com/fpvandoorn/sard.git | f4a1bb550136e8591dcd20dca5bdcce766a958c7 | Sard/LocallyLipschitz.lean | LipschitzOnWith.smul | [134, 1] | [150, 80] | rw [smul_sub] | X : Type u_1
Y : Type u_2
Z : Type u_3
inst✝² : MetricSpace X
inst✝¹ : NormedAddCommGroup Y
inst✝ : NormedSpace ℝ Y
f g : X → Y
K : ℝ≥0
s : Set X
hf : LipschitzOnWith K f s
a : ℝ
x : X
hx : x ∈ s
y : X
hy : y ∈ s
⊢ ‖a • f x - a • f y‖ = ‖a • (f x - f y)‖ | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
X : Type u_1
Y : Type u_2
Z : Type u_3
inst✝² : MetricSpace X
inst✝¹ : NormedAddCommGroup Y
inst✝ : NormedSpace ℝ Y
f g : X → Y
K : ℝ≥0
s : Set X
hf : LipschitzOnWith K f s
a : ℝ
x : X
hx : x ∈ s
y : X
hy : y ∈ s
⊢ ‖a • f x - a • f y‖ = ‖a • (f x - f y)‖
TACT... |
https://github.com/fpvandoorn/sard.git | f4a1bb550136e8591dcd20dca5bdcce766a958c7 | Sard/LocallyLipschitz.lean | LipschitzOnWith.smul | [134, 1] | [150, 80] | rw [norm_smul] | X : Type u_1
Y : Type u_2
Z : Type u_3
inst✝² : MetricSpace X
inst✝¹ : NormedAddCommGroup Y
inst✝ : NormedSpace ℝ Y
f g : X → Y
K : ℝ≥0
s : Set X
hf : LipschitzOnWith K f s
a : ℝ
x : X
hx : x ∈ s
y : X
hy : y ∈ s
⊢ ‖a • (f x - f y)‖ = ‖a‖ * ‖f x - f y‖ | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
X : Type u_1
Y : Type u_2
Z : Type u_3
inst✝² : MetricSpace X
inst✝¹ : NormedAddCommGroup Y
inst✝ : NormedSpace ℝ Y
f g : X → Y
K : ℝ≥0
s : Set X
hf : LipschitzOnWith K f s
a : ℝ
x : X
hx : x ∈ s
y : X
hy : y ∈ s
⊢ ‖a • (f x - f y)‖ = ‖a‖ * ‖f x - f y‖
TACTIC... |
https://github.com/fpvandoorn/sard.git | f4a1bb550136e8591dcd20dca5bdcce766a958c7 | Sard/LocallyLipschitz.lean | LipschitzOnWith.smul | [134, 1] | [150, 80] | rw [← dist_eq_norm] | X : Type u_1
Y : Type u_2
Z : Type u_3
inst✝² : MetricSpace X
inst✝¹ : NormedAddCommGroup Y
inst✝ : NormedSpace ℝ Y
f g : X → Y
K : ℝ≥0
s : Set X
hf : LipschitzOnWith K f s
a : ℝ
x : X
hx : x ∈ s
y : X
hy : y ∈ s
⊢ ‖a‖ * ‖f x - f y‖ = ‖a‖ * dist (f x) (f y) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
X : Type u_1
Y : Type u_2
Z : Type u_3
inst✝² : MetricSpace X
inst✝¹ : NormedAddCommGroup Y
inst✝ : NormedSpace ℝ Y
f g : X → Y
K : ℝ≥0
s : Set X
hf : LipschitzOnWith K f s
a : ℝ
x : X
hx : x ∈ s
y : X
hy : y ∈ s
⊢ ‖a‖ * ‖f x - f y‖ = ‖a‖ * dist (f x) (f y)
T... |
https://github.com/fpvandoorn/sard.git | f4a1bb550136e8591dcd20dca5bdcce766a958c7 | Sard/LocallyLipschitz.lean | LipschitzOnWith.smul | [134, 1] | [150, 80] | rw [← dist_smul₀] | X : Type u_1
Y : Type u_2
Z : Type u_3
inst✝² : MetricSpace X
inst✝¹ : NormedAddCommGroup Y
inst✝ : NormedSpace ℝ Y
f g : X → Y
K : ℝ≥0
s : Set X
hf : LipschitzOnWith K f s
a : ℝ
x : X
hx : x ∈ s
y : X
hy : y ∈ s
⊢ ‖a‖ * dist (f x) (f y) = ‖a‖ * dist (f x) (f y) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
X : Type u_1
Y : Type u_2
Z : Type u_3
inst✝² : MetricSpace X
inst✝¹ : NormedAddCommGroup Y
inst✝ : NormedSpace ℝ Y
f g : X → Y
K : ℝ≥0
s : Set X
hf : LipschitzOnWith K f s
a : ℝ
x : X
hx : x ∈ s
y : X
hy : y ∈ s
⊢ ‖a‖ * dist (f x) (f y) = ‖a‖ * dist (f x) (f... |
https://github.com/fpvandoorn/sard.git | f4a1bb550136e8591dcd20dca5bdcce766a958c7 | Sard/LocallyLipschitz.lean | LipschitzOnWith.smul | [134, 1] | [150, 80] | rw [edist_dist] | X : Type u_1
Y : Type u_2
Z : Type u_3
inst✝² : MetricSpace X
inst✝¹ : NormedAddCommGroup Y
inst✝ : NormedSpace ℝ Y
f g : X → Y
K : ℝ≥0
s : Set X
hf : LipschitzOnWith K f s
a : ℝ
x : X
hx : x ∈ s
y : X
hy : y ∈ s
this : dist (a • f x) (a • f y) = ‖a‖ * dist (f x) (f y)
⊢ edist (a • f x) (a • f y) = ENNReal.ofReal (dist... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
X : Type u_1
Y : Type u_2
Z : Type u_3
inst✝² : MetricSpace X
inst✝¹ : NormedAddCommGroup Y
inst✝ : NormedSpace ℝ Y
f g : X → Y
K : ℝ≥0
s : Set X
hf : LipschitzOnWith K f s
a : ℝ
x : X
hx : x ∈ s
y : X
hy : y ∈ s
this : dist (a • f x) (a • f y) = ‖a‖ * dist (... |
https://github.com/fpvandoorn/sard.git | f4a1bb550136e8591dcd20dca5bdcce766a958c7 | Sard/LocallyLipschitz.lean | LipschitzOnWith.smul | [134, 1] | [150, 80] | rw [this] | X : Type u_1
Y : Type u_2
Z : Type u_3
inst✝² : MetricSpace X
inst✝¹ : NormedAddCommGroup Y
inst✝ : NormedSpace ℝ Y
f g : X → Y
K : ℝ≥0
s : Set X
hf : LipschitzOnWith K f s
a : ℝ
x : X
hx : x ∈ s
y : X
hy : y ∈ s
this : dist (a • f x) (a • f y) = ‖a‖ * dist (f x) (f y)
⊢ ENNReal.ofReal (dist (a • f x) (a • f y)) = ENNR... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
X : Type u_1
Y : Type u_2
Z : Type u_3
inst✝² : MetricSpace X
inst✝¹ : NormedAddCommGroup Y
inst✝ : NormedSpace ℝ Y
f g : X → Y
K : ℝ≥0
s : Set X
hf : LipschitzOnWith K f s
a : ℝ
x : X
hx : x ∈ s
y : X
hy : y ∈ s
this : dist (a • f x) (a • f y) = ‖a‖ * dist (... |
https://github.com/fpvandoorn/sard.git | f4a1bb550136e8591dcd20dca5bdcce766a958c7 | Sard/LocallyLipschitz.lean | LipschitzOnWith.smul | [134, 1] | [150, 80] | rw [← helper] | X : Type u_1
Y : Type u_2
Z : Type u_3
inst✝² : MetricSpace X
inst✝¹ : NormedAddCommGroup Y
inst✝ : NormedSpace ℝ Y
f g : X → Y
K : ℝ≥0
s : Set X
hf : LipschitzOnWith K f s
a : ℝ
x : X
hx : x ∈ s
y : X
hy : y ∈ s
this : dist (a • f x) (a • f y) = ‖a‖ * dist (f x) (f y)
⊢ ENNReal.ofReal (‖a‖ * dist (f x) (f y)) = ENNRea... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
X : Type u_1
Y : Type u_2
Z : Type u_3
inst✝² : MetricSpace X
inst✝¹ : NormedAddCommGroup Y
inst✝ : NormedSpace ℝ Y
f g : X → Y
K : ℝ≥0
s : Set X
hf : LipschitzOnWith K f s
a : ℝ
x : X
hx : x ∈ s
y : X
hy : y ∈ s
this : dist (a • f x) (a • f y) = ‖a‖ * dist (... |
https://github.com/fpvandoorn/sard.git | f4a1bb550136e8591dcd20dca5bdcce766a958c7 | Sard/LocallyLipschitz.lean | LipschitzOnWith.smul | [134, 1] | [150, 80] | rw [edist_dist] | X : Type u_1
Y : Type u_2
Z : Type u_3
inst✝² : MetricSpace X
inst✝¹ : NormedAddCommGroup Y
inst✝ : NormedSpace ℝ Y
f g : X → Y
K : ℝ≥0
s : Set X
hf : LipschitzOnWith K f s
a : ℝ
x : X
hx : x ∈ s
y : X
hy : y ∈ s
this : dist (a • f x) (a • f y) = ‖a‖ * dist (f x) (f y)
⊢ ENNReal.ofReal ‖a‖ * ENNReal.ofReal (dist (f x) ... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
X : Type u_1
Y : Type u_2
Z : Type u_3
inst✝² : MetricSpace X
inst✝¹ : NormedAddCommGroup Y
inst✝ : NormedSpace ℝ Y
f g : X → Y
K : ℝ≥0
s : Set X
hf : LipschitzOnWith K f s
a : ℝ
x : X
hx : x ∈ s
y : X
hy : y ∈ s
this : dist (a • f x) (a • f y) = ‖a‖ * dist (... |
https://github.com/fpvandoorn/sard.git | f4a1bb550136e8591dcd20dca5bdcce766a958c7 | Sard/LocallyLipschitz.lean | LipschitzOnWith.smul | [134, 1] | [150, 80] | exact mul_le_mul_left' (hf hx hy) _ | X : Type u_1
Y : Type u_2
Z : Type u_3
inst✝² : MetricSpace X
inst✝¹ : NormedAddCommGroup Y
inst✝ : NormedSpace ℝ Y
f g : X → Y
K : ℝ≥0
s : Set X
hf : LipschitzOnWith K f s
a : ℝ
x : X
hx : x ∈ s
y : X
hy : y ∈ s
this : dist (a • f x) (a • f y) = ‖a‖ * dist (f x) (f y)
⊢ ENNReal.ofReal ‖a‖ * edist (f x) (f y) ≤ ↑‖a‖.to... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
X : Type u_1
Y : Type u_2
Z : Type u_3
inst✝² : MetricSpace X
inst✝¹ : NormedAddCommGroup Y
inst✝ : NormedSpace ℝ Y
f g : X → Y
K : ℝ≥0
s : Set X
hf : LipschitzOnWith K f s
a : ℝ
x : X
hx : x ∈ s
y : X
hy : y ∈ s
this : dist (a • f x) (a • f y) = ‖a‖ * dist (... |
https://github.com/fpvandoorn/sard.git | f4a1bb550136e8591dcd20dca5bdcce766a958c7 | Sard/LocallyLipschitz.lean | LipschitzOnWith.smul | [134, 1] | [150, 80] | exact last a K (edist x y) | X : Type u_1
Y : Type u_2
Z : Type u_3
inst✝² : MetricSpace X
inst✝¹ : NormedAddCommGroup Y
inst✝ : NormedSpace ℝ Y
f g : X → Y
K : ℝ≥0
s : Set X
hf : LipschitzOnWith K f s
a : ℝ
x : X
hx : x ∈ s
y : X
hy : y ∈ s
this : dist (a • f x) (a • f y) = ‖a‖ * dist (f x) (f y)
⊢ ↑‖a‖.toNNReal * (↑K * edist x y) = ↑(‖a‖.toNNRea... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
X : Type u_1
Y : Type u_2
Z : Type u_3
inst✝² : MetricSpace X
inst✝¹ : NormedAddCommGroup Y
inst✝ : NormedSpace ℝ Y
f g : X → Y
K : ℝ≥0
s : Set X
hf : LipschitzOnWith K f s
a : ℝ
x : X
hx : x ∈ s
y : X
hy : y ∈ s
this : dist (a • f x) (a • f y) = ‖a‖ * dist (... |
https://github.com/fpvandoorn/sard.git | f4a1bb550136e8591dcd20dca5bdcce766a958c7 | Sard/LocallyLipschitz.lean | LocallyLipschitz.mysmul | [156, 1] | [160, 72] | intro x | X : Type u_1
Y : Type u_2
Z : Type u_3
inst✝² : MetricSpace X
inst✝¹ : NormedAddCommGroup Y
inst✝ : NormedSpace ℝ Y
f g : X → Y
hf : LocallyLipschitz f
a : ℝ
⊢ LocallyLipschitz fun x => a • f x | X : Type u_1
Y : Type u_2
Z : Type u_3
inst✝² : MetricSpace X
inst✝¹ : NormedAddCommGroup Y
inst✝ : NormedSpace ℝ Y
f g : X → Y
hf : LocallyLipschitz f
a : ℝ
x : X
⊢ ∃ K, ∃ t ∈ 𝓝 x, LipschitzOnWith K (fun x => a • f x) t | Please generate a tactic in lean4 to solve the state.
STATE:
X : Type u_1
Y : Type u_2
Z : Type u_3
inst✝² : MetricSpace X
inst✝¹ : NormedAddCommGroup Y
inst✝ : NormedSpace ℝ Y
f g : X → Y
hf : LocallyLipschitz f
a : ℝ
⊢ LocallyLipschitz fun x => a • f x
TACTIC:
|
https://github.com/fpvandoorn/sard.git | f4a1bb550136e8591dcd20dca5bdcce766a958c7 | Sard/LocallyLipschitz.lean | LocallyLipschitz.mysmul | [156, 1] | [160, 72] | rcases hf x with ⟨Kf, t, ht, hfL⟩ | X : Type u_1
Y : Type u_2
Z : Type u_3
inst✝² : MetricSpace X
inst✝¹ : NormedAddCommGroup Y
inst✝ : NormedSpace ℝ Y
f g : X → Y
hf : LocallyLipschitz f
a : ℝ
x : X
⊢ ∃ K, ∃ t ∈ 𝓝 x, LipschitzOnWith K (fun x => a • f x) t | case intro.intro.intro
X : Type u_1
Y : Type u_2
Z : Type u_3
inst✝² : MetricSpace X
inst✝¹ : NormedAddCommGroup Y
inst✝ : NormedSpace ℝ Y
f g : X → Y
hf : LocallyLipschitz f
a : ℝ
x : X
Kf : ℝ≥0
t : Set X
ht : t ∈ 𝓝 x
hfL : LipschitzOnWith Kf f t
⊢ ∃ K, ∃ t ∈ 𝓝 x, LipschitzOnWith K (fun x => a • f x) t | Please generate a tactic in lean4 to solve the state.
STATE:
X : Type u_1
Y : Type u_2
Z : Type u_3
inst✝² : MetricSpace X
inst✝¹ : NormedAddCommGroup Y
inst✝ : NormedSpace ℝ Y
f g : X → Y
hf : LocallyLipschitz f
a : ℝ
x : X
⊢ ∃ K, ∃ t ∈ 𝓝 x, LipschitzOnWith K (fun x => a • f x) t
TACTIC:
|
https://github.com/fpvandoorn/sard.git | f4a1bb550136e8591dcd20dca5bdcce766a958c7 | Sard/LocallyLipschitz.lean | LocallyLipschitz.mysmul | [156, 1] | [160, 72] | exact ⟨ENNReal.toNNReal (ENNReal.ofReal ‖a‖) * Kf, t, ht, hfL.smul a⟩ | case intro.intro.intro
X : Type u_1
Y : Type u_2
Z : Type u_3
inst✝² : MetricSpace X
inst✝¹ : NormedAddCommGroup Y
inst✝ : NormedSpace ℝ Y
f g : X → Y
hf : LocallyLipschitz f
a : ℝ
x : X
Kf : ℝ≥0
t : Set X
ht : t ∈ 𝓝 x
hfL : LipschitzOnWith Kf f t
⊢ ∃ K, ∃ t ∈ 𝓝 x, LipschitzOnWith K (fun x => a • f x) t | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.intro
X : Type u_1
Y : Type u_2
Z : Type u_3
inst✝² : MetricSpace X
inst✝¹ : NormedAddCommGroup Y
inst✝ : NormedSpace ℝ Y
f g : X → Y
hf : LocallyLipschitz f
a : ℝ
x : X
Kf : ℝ≥0
t : Set X
ht : t ∈ 𝓝 x
hfL : LipschitzOnWith Kf f t
⊢ ∃ K, ∃ t... |
https://github.com/fpvandoorn/sard.git | f4a1bb550136e8591dcd20dca5bdcce766a958c7 | Sard/MeasureZero.lean | MeasureZero.mono | [53, 1] | [61, 40] | intro μ hμ x | E : Type u_1
inst✝⁸ : NormedAddCommGroup E
inst✝⁷ : NormedSpace ℝ E
H : Type u_2
inst✝⁶ : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type u_3
inst✝⁵ : TopologicalSpace M
inst✝⁴ : ChartedSpace H M
inst✝³ : I.Boundaryless
inst✝² : FiniteDimensional ℝ E
inst✝¹ : MeasurableSpace E
inst✝ : BorelSpace E
s t : Set M
hs... | E : Type u_1
inst✝⁸ : NormedAddCommGroup E
inst✝⁷ : NormedSpace ℝ E
H : Type u_2
inst✝⁶ : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type u_3
inst✝⁵ : TopologicalSpace M
inst✝⁴ : ChartedSpace H M
inst✝³ : I.Boundaryless
inst✝² : FiniteDimensional ℝ E
inst✝¹ : MeasurableSpace E
inst✝ : BorelSpace E
s t : Set M
hs... | Please generate a tactic in lean4 to solve the state.
STATE:
E : Type u_1
inst✝⁸ : NormedAddCommGroup E
inst✝⁷ : NormedSpace ℝ E
H : Type u_2
inst✝⁶ : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type u_3
inst✝⁵ : TopologicalSpace M
inst✝⁴ : ChartedSpace H M
inst✝³ : I.Boundaryless
inst✝² : FiniteDimensional ℝ E
i... |
https://github.com/fpvandoorn/sard.git | f4a1bb550136e8591dcd20dca5bdcce766a958c7 | Sard/MeasureZero.lean | MeasureZero.mono | [53, 1] | [61, 40] | let e := chartAt H x | E : Type u_1
inst✝⁸ : NormedAddCommGroup E
inst✝⁷ : NormedSpace ℝ E
H : Type u_2
inst✝⁶ : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type u_3
inst✝⁵ : TopologicalSpace M
inst✝⁴ : ChartedSpace H M
inst✝³ : I.Boundaryless
inst✝² : FiniteDimensional ℝ E
inst✝¹ : MeasurableSpace E
inst✝ : BorelSpace E
s t : Set M
hs... | E : Type u_1
inst✝⁸ : NormedAddCommGroup E
inst✝⁷ : NormedSpace ℝ E
H : Type u_2
inst✝⁶ : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type u_3
inst✝⁵ : TopologicalSpace M
inst✝⁴ : ChartedSpace H M
inst✝³ : I.Boundaryless
inst✝² : FiniteDimensional ℝ E
inst✝¹ : MeasurableSpace E
inst✝ : BorelSpace E
s t : Set M
hs... | Please generate a tactic in lean4 to solve the state.
STATE:
E : Type u_1
inst✝⁸ : NormedAddCommGroup E
inst✝⁷ : NormedSpace ℝ E
H : Type u_2
inst✝⁶ : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type u_3
inst✝⁵ : TopologicalSpace M
inst✝⁴ : ChartedSpace H M
inst✝³ : I.Boundaryless
inst✝² : FiniteDimensional ℝ E
i... |
https://github.com/fpvandoorn/sard.git | f4a1bb550136e8591dcd20dca5bdcce766a958c7 | Sard/MeasureZero.lean | MeasureZero.mono | [53, 1] | [61, 40] | have : I ∘ e '' (e.source ∩ s) ⊆ I ∘ e '' (e.source ∩ t) := by
apply image_subset
exact inter_subset_inter_right e.source hst | E : Type u_1
inst✝⁸ : NormedAddCommGroup E
inst✝⁷ : NormedSpace ℝ E
H : Type u_2
inst✝⁶ : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type u_3
inst✝⁵ : TopologicalSpace M
inst✝⁴ : ChartedSpace H M
inst✝³ : I.Boundaryless
inst✝² : FiniteDimensional ℝ E
inst✝¹ : MeasurableSpace E
inst✝ : BorelSpace E
s t : Set M
hs... | E : Type u_1
inst✝⁸ : NormedAddCommGroup E
inst✝⁷ : NormedSpace ℝ E
H : Type u_2
inst✝⁶ : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type u_3
inst✝⁵ : TopologicalSpace M
inst✝⁴ : ChartedSpace H M
inst✝³ : I.Boundaryless
inst✝² : FiniteDimensional ℝ E
inst✝¹ : MeasurableSpace E
inst✝ : BorelSpace E
s t : Set M
hs... | Please generate a tactic in lean4 to solve the state.
STATE:
E : Type u_1
inst✝⁸ : NormedAddCommGroup E
inst✝⁷ : NormedSpace ℝ E
H : Type u_2
inst✝⁶ : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type u_3
inst✝⁵ : TopologicalSpace M
inst✝⁴ : ChartedSpace H M
inst✝³ : I.Boundaryless
inst✝² : FiniteDimensional ℝ E
i... |
https://github.com/fpvandoorn/sard.git | f4a1bb550136e8591dcd20dca5bdcce766a958c7 | Sard/MeasureZero.lean | MeasureZero.mono | [53, 1] | [61, 40] | exact measure_mono_null this (ht μ x) | E : Type u_1
inst✝⁸ : NormedAddCommGroup E
inst✝⁷ : NormedSpace ℝ E
H : Type u_2
inst✝⁶ : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type u_3
inst✝⁵ : TopologicalSpace M
inst✝⁴ : ChartedSpace H M
inst✝³ : I.Boundaryless
inst✝² : FiniteDimensional ℝ E
inst✝¹ : MeasurableSpace E
inst✝ : BorelSpace E
s t : Set M
hs... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
E : Type u_1
inst✝⁸ : NormedAddCommGroup E
inst✝⁷ : NormedSpace ℝ E
H : Type u_2
inst✝⁶ : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type u_3
inst✝⁵ : TopologicalSpace M
inst✝⁴ : ChartedSpace H M
inst✝³ : I.Boundaryless
inst✝² : FiniteDimensional ℝ E
i... |
https://github.com/fpvandoorn/sard.git | f4a1bb550136e8591dcd20dca5bdcce766a958c7 | Sard/MeasureZero.lean | MeasureZero.mono | [53, 1] | [61, 40] | apply image_subset | E : Type u_1
inst✝⁸ : NormedAddCommGroup E
inst✝⁷ : NormedSpace ℝ E
H : Type u_2
inst✝⁶ : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type u_3
inst✝⁵ : TopologicalSpace M
inst✝⁴ : ChartedSpace H M
inst✝³ : I.Boundaryless
inst✝² : FiniteDimensional ℝ E
inst✝¹ : MeasurableSpace E
inst✝ : BorelSpace E
s t : Set M
hs... | case h
E : Type u_1
inst✝⁸ : NormedAddCommGroup E
inst✝⁷ : NormedSpace ℝ E
H : Type u_2
inst✝⁶ : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type u_3
inst✝⁵ : TopologicalSpace M
inst✝⁴ : ChartedSpace H M
inst✝³ : I.Boundaryless
inst✝² : FiniteDimensional ℝ E
inst✝¹ : MeasurableSpace E
inst✝ : BorelSpace E
s t : S... | Please generate a tactic in lean4 to solve the state.
STATE:
E : Type u_1
inst✝⁸ : NormedAddCommGroup E
inst✝⁷ : NormedSpace ℝ E
H : Type u_2
inst✝⁶ : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type u_3
inst✝⁵ : TopologicalSpace M
inst✝⁴ : ChartedSpace H M
inst✝³ : I.Boundaryless
inst✝² : FiniteDimensional ℝ E
i... |
https://github.com/fpvandoorn/sard.git | f4a1bb550136e8591dcd20dca5bdcce766a958c7 | Sard/MeasureZero.lean | MeasureZero.mono | [53, 1] | [61, 40] | exact inter_subset_inter_right e.source hst | case h
E : Type u_1
inst✝⁸ : NormedAddCommGroup E
inst✝⁷ : NormedSpace ℝ E
H : Type u_2
inst✝⁶ : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type u_3
inst✝⁵ : TopologicalSpace M
inst✝⁴ : ChartedSpace H M
inst✝³ : I.Boundaryless
inst✝² : FiniteDimensional ℝ E
inst✝¹ : MeasurableSpace E
inst✝ : BorelSpace E
s t : S... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h
E : Type u_1
inst✝⁸ : NormedAddCommGroup E
inst✝⁷ : NormedSpace ℝ E
H : Type u_2
inst✝⁶ : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type u_3
inst✝⁵ : TopologicalSpace M
inst✝⁴ : ChartedSpace H M
inst✝³ : I.Boundaryless
inst✝² : FiniteDimensiona... |
https://github.com/fpvandoorn/sard.git | f4a1bb550136e8591dcd20dca5bdcce766a958c7 | Sard/MeasureZero.lean | MeasureZero.empty | [63, 1] | [66, 72] | intro μ _ x | E : Type u_1
inst✝⁸ : NormedAddCommGroup E
inst✝⁷ : NormedSpace ℝ E
H : Type u_2
inst✝⁶ : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type u_3
inst✝⁵ : TopologicalSpace M
inst✝⁴ : ChartedSpace H M
inst✝³ : I.Boundaryless
inst✝² : FiniteDimensional ℝ E
inst✝¹ : MeasurableSpace E
inst✝ : BorelSpace E
⊢ MeasureZero ... | E : Type u_1
inst✝⁹ : NormedAddCommGroup E
inst✝⁸ : NormedSpace ℝ E
H : Type u_2
inst✝⁷ : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type u_3
inst✝⁶ : TopologicalSpace M
inst✝⁵ : ChartedSpace H M
inst✝⁴ : I.Boundaryless
inst✝³ : FiniteDimensional ℝ E
inst✝² : MeasurableSpace E
inst✝¹ : BorelSpace E
μ : Measure E... | Please generate a tactic in lean4 to solve the state.
STATE:
E : Type u_1
inst✝⁸ : NormedAddCommGroup E
inst✝⁷ : NormedSpace ℝ E
H : Type u_2
inst✝⁶ : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type u_3
inst✝⁵ : TopologicalSpace M
inst✝⁴ : ChartedSpace H M
inst✝³ : I.Boundaryless
inst✝² : FiniteDimensional ℝ E
i... |
https://github.com/fpvandoorn/sard.git | f4a1bb550136e8591dcd20dca5bdcce766a958c7 | Sard/MeasureZero.lean | MeasureZero.empty | [63, 1] | [66, 72] | simp only [comp_apply, inter_empty, image_empty, OuterMeasure.empty'] | E : Type u_1
inst✝⁹ : NormedAddCommGroup E
inst✝⁸ : NormedSpace ℝ E
H : Type u_2
inst✝⁷ : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type u_3
inst✝⁶ : TopologicalSpace M
inst✝⁵ : ChartedSpace H M
inst✝⁴ : I.Boundaryless
inst✝³ : FiniteDimensional ℝ E
inst✝² : MeasurableSpace E
inst✝¹ : BorelSpace E
μ : Measure E... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
E : Type u_1
inst✝⁹ : NormedAddCommGroup E
inst✝⁸ : NormedSpace ℝ E
H : Type u_2
inst✝⁷ : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type u_3
inst✝⁶ : TopologicalSpace M
inst✝⁵ : ChartedSpace H M
inst✝⁴ : I.Boundaryless
inst✝³ : FiniteDimensional ℝ E
i... |
https://github.com/fpvandoorn/sard.git | f4a1bb550136e8591dcd20dca5bdcce766a958c7 | Sard/MeasureZero.lean | MeasureZero.iUnion | [68, 1] | [79, 11] | intro μ hμ x | E : Type u_1
inst✝⁹ : NormedAddCommGroup E
inst✝⁸ : NormedSpace ℝ E
H : Type u_2
inst✝⁷ : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type u_3
inst✝⁶ : TopologicalSpace M
inst✝⁵ : ChartedSpace H M
inst✝⁴ : I.Boundaryless
inst✝³ : FiniteDimensional ℝ E
inst✝² : MeasurableSpace E
inst✝¹ : BorelSpace E
ι : Type u_4
... | E : Type u_1
inst✝⁹ : NormedAddCommGroup E
inst✝⁸ : NormedSpace ℝ E
H : Type u_2
inst✝⁷ : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type u_3
inst✝⁶ : TopologicalSpace M
inst✝⁵ : ChartedSpace H M
inst✝⁴ : I.Boundaryless
inst✝³ : FiniteDimensional ℝ E
inst✝² : MeasurableSpace E
inst✝¹ : BorelSpace E
ι : Type u_4
... | Please generate a tactic in lean4 to solve the state.
STATE:
E : Type u_1
inst✝⁹ : NormedAddCommGroup E
inst✝⁸ : NormedSpace ℝ E
H : Type u_2
inst✝⁷ : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type u_3
inst✝⁶ : TopologicalSpace M
inst✝⁵ : ChartedSpace H M
inst✝⁴ : I.Boundaryless
inst✝³ : FiniteDimensional ℝ E
i... |
https://github.com/fpvandoorn/sard.git | f4a1bb550136e8591dcd20dca5bdcce766a958c7 | Sard/MeasureZero.lean | MeasureZero.iUnion | [68, 1] | [79, 11] | let e := chartAt H x | E : Type u_1
inst✝⁹ : NormedAddCommGroup E
inst✝⁸ : NormedSpace ℝ E
H : Type u_2
inst✝⁷ : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type u_3
inst✝⁶ : TopologicalSpace M
inst✝⁵ : ChartedSpace H M
inst✝⁴ : I.Boundaryless
inst✝³ : FiniteDimensional ℝ E
inst✝² : MeasurableSpace E
inst✝¹ : BorelSpace E
ι : Type u_4
... | E : Type u_1
inst✝⁹ : NormedAddCommGroup E
inst✝⁸ : NormedSpace ℝ E
H : Type u_2
inst✝⁷ : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type u_3
inst✝⁶ : TopologicalSpace M
inst✝⁵ : ChartedSpace H M
inst✝⁴ : I.Boundaryless
inst✝³ : FiniteDimensional ℝ E
inst✝² : MeasurableSpace E
inst✝¹ : BorelSpace E
ι : Type u_4
... | Please generate a tactic in lean4 to solve the state.
STATE:
E : Type u_1
inst✝⁹ : NormedAddCommGroup E
inst✝⁸ : NormedSpace ℝ E
H : Type u_2
inst✝⁷ : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type u_3
inst✝⁶ : TopologicalSpace M
inst✝⁵ : ChartedSpace H M
inst✝⁴ : I.Boundaryless
inst✝³ : FiniteDimensional ℝ E
i... |
https://github.com/fpvandoorn/sard.git | f4a1bb550136e8591dcd20dca5bdcce766a958c7 | Sard/MeasureZero.lean | MeasureZero.iUnion | [68, 1] | [79, 11] | have : I ∘ e '' (e.source ∩ (⋃ (n : ι), s n)) = ⋃ (n : ι), I ∘ e '' (e.source ∩ s n) := by
rw [inter_iUnion]
exact image_iUnion | E : Type u_1
inst✝⁹ : NormedAddCommGroup E
inst✝⁸ : NormedSpace ℝ E
H : Type u_2
inst✝⁷ : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type u_3
inst✝⁶ : TopologicalSpace M
inst✝⁵ : ChartedSpace H M
inst✝⁴ : I.Boundaryless
inst✝³ : FiniteDimensional ℝ E
inst✝² : MeasurableSpace E
inst✝¹ : BorelSpace E
ι : Type u_4
... | E : Type u_1
inst✝⁹ : NormedAddCommGroup E
inst✝⁸ : NormedSpace ℝ E
H : Type u_2
inst✝⁷ : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type u_3
inst✝⁶ : TopologicalSpace M
inst✝⁵ : ChartedSpace H M
inst✝⁴ : I.Boundaryless
inst✝³ : FiniteDimensional ℝ E
inst✝² : MeasurableSpace E
inst✝¹ : BorelSpace E
ι : Type u_4
... | Please generate a tactic in lean4 to solve the state.
STATE:
E : Type u_1
inst✝⁹ : NormedAddCommGroup E
inst✝⁸ : NormedSpace ℝ E
H : Type u_2
inst✝⁷ : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type u_3
inst✝⁶ : TopologicalSpace M
inst✝⁵ : ChartedSpace H M
inst✝⁴ : I.Boundaryless
inst✝³ : FiniteDimensional ℝ E
i... |
https://github.com/fpvandoorn/sard.git | f4a1bb550136e8591dcd20dca5bdcce766a958c7 | Sard/MeasureZero.lean | MeasureZero.iUnion | [68, 1] | [79, 11] | simp_all only [comp_apply, OuterMeasure.iUnion_null_iff] | E : Type u_1
inst✝⁹ : NormedAddCommGroup E
inst✝⁸ : NormedSpace ℝ E
H : Type u_2
inst✝⁷ : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type u_3
inst✝⁶ : TopologicalSpace M
inst✝⁵ : ChartedSpace H M
inst✝⁴ : I.Boundaryless
inst✝³ : FiniteDimensional ℝ E
inst✝² : MeasurableSpace E
inst✝¹ : BorelSpace E
ι : Type u_4
... | E : Type u_1
inst✝⁹ : NormedAddCommGroup E
inst✝⁸ : NormedSpace ℝ E
H : Type u_2
inst✝⁷ : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type u_3
inst✝⁶ : TopologicalSpace M
inst✝⁵ : ChartedSpace H M
inst✝⁴ : I.Boundaryless
inst✝³ : FiniteDimensional ℝ E
inst✝² : MeasurableSpace E
inst✝¹ : BorelSpace E
ι : Type u_4
... | Please generate a tactic in lean4 to solve the state.
STATE:
E : Type u_1
inst✝⁹ : NormedAddCommGroup E
inst✝⁸ : NormedSpace ℝ E
H : Type u_2
inst✝⁷ : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type u_3
inst✝⁶ : TopologicalSpace M
inst✝⁵ : ChartedSpace H M
inst✝⁴ : I.Boundaryless
inst✝³ : FiniteDimensional ℝ E
i... |
https://github.com/fpvandoorn/sard.git | f4a1bb550136e8591dcd20dca5bdcce766a958c7 | Sard/MeasureZero.lean | MeasureZero.iUnion | [68, 1] | [79, 11] | intro i | E : Type u_1
inst✝⁹ : NormedAddCommGroup E
inst✝⁸ : NormedSpace ℝ E
H : Type u_2
inst✝⁷ : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type u_3
inst✝⁶ : TopologicalSpace M
inst✝⁵ : ChartedSpace H M
inst✝⁴ : I.Boundaryless
inst✝³ : FiniteDimensional ℝ E
inst✝² : MeasurableSpace E
inst✝¹ : BorelSpace E
ι : Type u_4
... | E : Type u_1
inst✝⁹ : NormedAddCommGroup E
inst✝⁸ : NormedSpace ℝ E
H : Type u_2
inst✝⁷ : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type u_3
inst✝⁶ : TopologicalSpace M
inst✝⁵ : ChartedSpace H M
inst✝⁴ : I.Boundaryless
inst✝³ : FiniteDimensional ℝ E
inst✝² : MeasurableSpace E
inst✝¹ : BorelSpace E
ι : Type u_4
... | Please generate a tactic in lean4 to solve the state.
STATE:
E : Type u_1
inst✝⁹ : NormedAddCommGroup E
inst✝⁸ : NormedSpace ℝ E
H : Type u_2
inst✝⁷ : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type u_3
inst✝⁶ : TopologicalSpace M
inst✝⁵ : ChartedSpace H M
inst✝⁴ : I.Boundaryless
inst✝³ : FiniteDimensional ℝ E
i... |
https://github.com/fpvandoorn/sard.git | f4a1bb550136e8591dcd20dca5bdcce766a958c7 | Sard/MeasureZero.lean | MeasureZero.iUnion | [68, 1] | [79, 11] | apply hs | E : Type u_1
inst✝⁹ : NormedAddCommGroup E
inst✝⁸ : NormedSpace ℝ E
H : Type u_2
inst✝⁷ : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type u_3
inst✝⁶ : TopologicalSpace M
inst✝⁵ : ChartedSpace H M
inst✝⁴ : I.Boundaryless
inst✝³ : FiniteDimensional ℝ E
inst✝² : MeasurableSpace E
inst✝¹ : BorelSpace E
ι : Type u_4
... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
E : Type u_1
inst✝⁹ : NormedAddCommGroup E
inst✝⁸ : NormedSpace ℝ E
H : Type u_2
inst✝⁷ : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type u_3
inst✝⁶ : TopologicalSpace M
inst✝⁵ : ChartedSpace H M
inst✝⁴ : I.Boundaryless
inst✝³ : FiniteDimensional ℝ E
i... |
https://github.com/fpvandoorn/sard.git | f4a1bb550136e8591dcd20dca5bdcce766a958c7 | Sard/MeasureZero.lean | MeasureZero.iUnion | [68, 1] | [79, 11] | rw [inter_iUnion] | E : Type u_1
inst✝⁹ : NormedAddCommGroup E
inst✝⁸ : NormedSpace ℝ E
H : Type u_2
inst✝⁷ : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type u_3
inst✝⁶ : TopologicalSpace M
inst✝⁵ : ChartedSpace H M
inst✝⁴ : I.Boundaryless
inst✝³ : FiniteDimensional ℝ E
inst✝² : MeasurableSpace E
inst✝¹ : BorelSpace E
ι : Type u_4
... | E : Type u_1
inst✝⁹ : NormedAddCommGroup E
inst✝⁸ : NormedSpace ℝ E
H : Type u_2
inst✝⁷ : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type u_3
inst✝⁶ : TopologicalSpace M
inst✝⁵ : ChartedSpace H M
inst✝⁴ : I.Boundaryless
inst✝³ : FiniteDimensional ℝ E
inst✝² : MeasurableSpace E
inst✝¹ : BorelSpace E
ι : Type u_4
... | Please generate a tactic in lean4 to solve the state.
STATE:
E : Type u_1
inst✝⁹ : NormedAddCommGroup E
inst✝⁸ : NormedSpace ℝ E
H : Type u_2
inst✝⁷ : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type u_3
inst✝⁶ : TopologicalSpace M
inst✝⁵ : ChartedSpace H M
inst✝⁴ : I.Boundaryless
inst✝³ : FiniteDimensional ℝ E
i... |
https://github.com/fpvandoorn/sard.git | f4a1bb550136e8591dcd20dca5bdcce766a958c7 | Sard/MeasureZero.lean | MeasureZero.iUnion | [68, 1] | [79, 11] | exact image_iUnion | E : Type u_1
inst✝⁹ : NormedAddCommGroup E
inst✝⁸ : NormedSpace ℝ E
H : Type u_2
inst✝⁷ : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type u_3
inst✝⁶ : TopologicalSpace M
inst✝⁵ : ChartedSpace H M
inst✝⁴ : I.Boundaryless
inst✝³ : FiniteDimensional ℝ E
inst✝² : MeasurableSpace E
inst✝¹ : BorelSpace E
ι : Type u_4
... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
E : Type u_1
inst✝⁹ : NormedAddCommGroup E
inst✝⁸ : NormedSpace ℝ E
H : Type u_2
inst✝⁷ : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type u_3
inst✝⁶ : TopologicalSpace M
inst✝⁵ : ChartedSpace H M
inst✝⁴ : I.Boundaryless
inst✝³ : FiniteDimensional ℝ E
i... |
https://github.com/fpvandoorn/sard.git | f4a1bb550136e8591dcd20dca5bdcce766a958c7 | Sard/MeasureZero.lean | MeasureZero.union | [81, 1] | [90, 32] | let u : Bool → Set M := fun b ↦ cond b s t | E : Type u_1
inst✝⁸ : NormedAddCommGroup E
inst✝⁷ : NormedSpace ℝ E
H : Type u_2
inst✝⁶ : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type u_3
inst✝⁵ : TopologicalSpace M
inst✝⁴ : ChartedSpace H M
inst✝³ : I.Boundaryless
inst✝² : FiniteDimensional ℝ E
inst✝¹ : MeasurableSpace E
inst✝ : BorelSpace E
s t : Set M
hs... | E : Type u_1
inst✝⁸ : NormedAddCommGroup E
inst✝⁷ : NormedSpace ℝ E
H : Type u_2
inst✝⁶ : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type u_3
inst✝⁵ : TopologicalSpace M
inst✝⁴ : ChartedSpace H M
inst✝³ : I.Boundaryless
inst✝² : FiniteDimensional ℝ E
inst✝¹ : MeasurableSpace E
inst✝ : BorelSpace E
s t : Set M
hs... | Please generate a tactic in lean4 to solve the state.
STATE:
E : Type u_1
inst✝⁸ : NormedAddCommGroup E
inst✝⁷ : NormedSpace ℝ E
H : Type u_2
inst✝⁶ : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type u_3
inst✝⁵ : TopologicalSpace M
inst✝⁴ : ChartedSpace H M
inst✝³ : I.Boundaryless
inst✝² : FiniteDimensional ℝ E
i... |
https://github.com/fpvandoorn/sard.git | f4a1bb550136e8591dcd20dca5bdcce766a958c7 | Sard/MeasureZero.lean | MeasureZero.union | [81, 1] | [90, 32] | rw [union_eq_iUnion] | E : Type u_1
inst✝⁸ : NormedAddCommGroup E
inst✝⁷ : NormedSpace ℝ E
H : Type u_2
inst✝⁶ : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type u_3
inst✝⁵ : TopologicalSpace M
inst✝⁴ : ChartedSpace H M
inst✝³ : I.Boundaryless
inst✝² : FiniteDimensional ℝ E
inst✝¹ : MeasurableSpace E
inst✝ : BorelSpace E
s t : Set M
hs... | E : Type u_1
inst✝⁸ : NormedAddCommGroup E
inst✝⁷ : NormedSpace ℝ E
H : Type u_2
inst✝⁶ : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type u_3
inst✝⁵ : TopologicalSpace M
inst✝⁴ : ChartedSpace H M
inst✝³ : I.Boundaryless
inst✝² : FiniteDimensional ℝ E
inst✝¹ : MeasurableSpace E
inst✝ : BorelSpace E
s t : Set M
hs... | Please generate a tactic in lean4 to solve the state.
STATE:
E : Type u_1
inst✝⁸ : NormedAddCommGroup E
inst✝⁷ : NormedSpace ℝ E
H : Type u_2
inst✝⁶ : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type u_3
inst✝⁵ : TopologicalSpace M
inst✝⁴ : ChartedSpace H M
inst✝³ : I.Boundaryless
inst✝² : FiniteDimensional ℝ E
i... |
https://github.com/fpvandoorn/sard.git | f4a1bb550136e8591dcd20dca5bdcce766a958c7 | Sard/MeasureZero.lean | MeasureZero.union | [81, 1] | [90, 32] | exact MeasureZero.iUnion this | E : Type u_1
inst✝⁸ : NormedAddCommGroup E
inst✝⁷ : NormedSpace ℝ E
H : Type u_2
inst✝⁶ : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type u_3
inst✝⁵ : TopologicalSpace M
inst✝⁴ : ChartedSpace H M
inst✝³ : I.Boundaryless
inst✝² : FiniteDimensional ℝ E
inst✝¹ : MeasurableSpace E
inst✝ : BorelSpace E
s t : Set M
hs... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
E : Type u_1
inst✝⁸ : NormedAddCommGroup E
inst✝⁷ : NormedSpace ℝ E
H : Type u_2
inst✝⁶ : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type u_3
inst✝⁵ : TopologicalSpace M
inst✝⁴ : ChartedSpace H M
inst✝³ : I.Boundaryless
inst✝² : FiniteDimensional ℝ E
i... |
https://github.com/fpvandoorn/sard.git | f4a1bb550136e8591dcd20dca5bdcce766a958c7 | Sard/MeasureZero.lean | MeasureZero.union | [81, 1] | [90, 32] | intro i | E : Type u_1
inst✝⁸ : NormedAddCommGroup E
inst✝⁷ : NormedSpace ℝ E
H : Type u_2
inst✝⁶ : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type u_3
inst✝⁵ : TopologicalSpace M
inst✝⁴ : ChartedSpace H M
inst✝³ : I.Boundaryless
inst✝² : FiniteDimensional ℝ E
inst✝¹ : MeasurableSpace E
inst✝ : BorelSpace E
s t : Set M
hs... | E : Type u_1
inst✝⁸ : NormedAddCommGroup E
inst✝⁷ : NormedSpace ℝ E
H : Type u_2
inst✝⁶ : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type u_3
inst✝⁵ : TopologicalSpace M
inst✝⁴ : ChartedSpace H M
inst✝³ : I.Boundaryless
inst✝² : FiniteDimensional ℝ E
inst✝¹ : MeasurableSpace E
inst✝ : BorelSpace E
s t : Set M
hs... | Please generate a tactic in lean4 to solve the state.
STATE:
E : Type u_1
inst✝⁸ : NormedAddCommGroup E
inst✝⁷ : NormedSpace ℝ E
H : Type u_2
inst✝⁶ : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type u_3
inst✝⁵ : TopologicalSpace M
inst✝⁴ : ChartedSpace H M
inst✝³ : I.Boundaryless
inst✝² : FiniteDimensional ℝ E
i... |
https://github.com/fpvandoorn/sard.git | f4a1bb550136e8591dcd20dca5bdcce766a958c7 | Sard/MeasureZero.lean | MeasureZero.union | [81, 1] | [90, 32] | cases i | E : Type u_1
inst✝⁸ : NormedAddCommGroup E
inst✝⁷ : NormedSpace ℝ E
H : Type u_2
inst✝⁶ : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type u_3
inst✝⁵ : TopologicalSpace M
inst✝⁴ : ChartedSpace H M
inst✝³ : I.Boundaryless
inst✝² : FiniteDimensional ℝ E
inst✝¹ : MeasurableSpace E
inst✝ : BorelSpace E
s t : Set M
hs... | case false
E : Type u_1
inst✝⁸ : NormedAddCommGroup E
inst✝⁷ : NormedSpace ℝ E
H : Type u_2
inst✝⁶ : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type u_3
inst✝⁵ : TopologicalSpace M
inst✝⁴ : ChartedSpace H M
inst✝³ : I.Boundaryless
inst✝² : FiniteDimensional ℝ E
inst✝¹ : MeasurableSpace E
inst✝ : BorelSpace E
s t... | Please generate a tactic in lean4 to solve the state.
STATE:
E : Type u_1
inst✝⁸ : NormedAddCommGroup E
inst✝⁷ : NormedSpace ℝ E
H : Type u_2
inst✝⁶ : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type u_3
inst✝⁵ : TopologicalSpace M
inst✝⁴ : ChartedSpace H M
inst✝³ : I.Boundaryless
inst✝² : FiniteDimensional ℝ E
i... |
https://github.com/fpvandoorn/sard.git | f4a1bb550136e8591dcd20dca5bdcce766a958c7 | Sard/MeasureZero.lean | MeasureZero.union | [81, 1] | [90, 32] | exact ht | case false
E : Type u_1
inst✝⁸ : NormedAddCommGroup E
inst✝⁷ : NormedSpace ℝ E
H : Type u_2
inst✝⁶ : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type u_3
inst✝⁵ : TopologicalSpace M
inst✝⁴ : ChartedSpace H M
inst✝³ : I.Boundaryless
inst✝² : FiniteDimensional ℝ E
inst✝¹ : MeasurableSpace E
inst✝ : BorelSpace E
s t... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case false
E : Type u_1
inst✝⁸ : NormedAddCommGroup E
inst✝⁷ : NormedSpace ℝ E
H : Type u_2
inst✝⁶ : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type u_3
inst✝⁵ : TopologicalSpace M
inst✝⁴ : ChartedSpace H M
inst✝³ : I.Boundaryless
inst✝² : FiniteDimens... |
https://github.com/fpvandoorn/sard.git | f4a1bb550136e8591dcd20dca5bdcce766a958c7 | Sard/MeasureZero.lean | MeasureZero.union | [81, 1] | [90, 32] | exact hs | case true
E : Type u_1
inst✝⁸ : NormedAddCommGroup E
inst✝⁷ : NormedSpace ℝ E
H : Type u_2
inst✝⁶ : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type u_3
inst✝⁵ : TopologicalSpace M
inst✝⁴ : ChartedSpace H M
inst✝³ : I.Boundaryless
inst✝² : FiniteDimensional ℝ E
inst✝¹ : MeasurableSpace E
inst✝ : BorelSpace E
s t ... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case true
E : Type u_1
inst✝⁸ : NormedAddCommGroup E
inst✝⁷ : NormedSpace ℝ E
H : Type u_2
inst✝⁶ : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type u_3
inst✝⁵ : TopologicalSpace M
inst✝⁴ : ChartedSpace H M
inst✝³ : I.Boundaryless
inst✝² : FiniteDimensi... |
https://github.com/fpvandoorn/sard.git | f4a1bb550136e8591dcd20dca5bdcce766a958c7 | Sard/MeasureZero.lean | MeasureZero.open_implies_empty | [107, 1] | [134, 63] | suffices ∀ x : M, ((chartAt H x).source ∩ s) = ∅ by
by_contra h
obtain ⟨x, hx⟩ : Set.Nonempty s := Iff.mp nmem_singleton_empty h
specialize this x
have h₂: x ∈ (chartAt H x).toPartialEquiv.source ∩ s := by
simp
exact hx
rw [this] at h₂
contradiction | E : Type u_1
inst✝⁸ : NormedAddCommGroup E
inst✝⁷ : NormedSpace ℝ E
H : Type u_2
inst✝⁶ : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type u_3
inst✝⁵ : TopologicalSpace M
inst✝⁴ : ChartedSpace H M
inst✝³ : I.Boundaryless
inst✝² : FiniteDimensional ℝ E
inst✝¹ : MeasurableSpace E
inst✝ : BorelSpace E
s : Set M
h₁s ... | E : Type u_1
inst✝⁸ : NormedAddCommGroup E
inst✝⁷ : NormedSpace ℝ E
H : Type u_2
inst✝⁶ : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type u_3
inst✝⁵ : TopologicalSpace M
inst✝⁴ : ChartedSpace H M
inst✝³ : I.Boundaryless
inst✝² : FiniteDimensional ℝ E
inst✝¹ : MeasurableSpace E
inst✝ : BorelSpace E
s : Set M
h₁s ... | Please generate a tactic in lean4 to solve the state.
STATE:
E : Type u_1
inst✝⁸ : NormedAddCommGroup E
inst✝⁷ : NormedSpace ℝ E
H : Type u_2
inst✝⁶ : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type u_3
inst✝⁵ : TopologicalSpace M
inst✝⁴ : ChartedSpace H M
inst✝³ : I.Boundaryless
inst✝² : FiniteDimensional ℝ E
i... |
https://github.com/fpvandoorn/sard.git | f4a1bb550136e8591dcd20dca5bdcce766a958c7 | Sard/MeasureZero.lean | MeasureZero.open_implies_empty | [107, 1] | [134, 63] | intro x | E : Type u_1
inst✝⁸ : NormedAddCommGroup E
inst✝⁷ : NormedSpace ℝ E
H : Type u_2
inst✝⁶ : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type u_3
inst✝⁵ : TopologicalSpace M
inst✝⁴ : ChartedSpace H M
inst✝³ : I.Boundaryless
inst✝² : FiniteDimensional ℝ E
inst✝¹ : MeasurableSpace E
inst✝ : BorelSpace E
s : Set M
h₁s ... | E : Type u_1
inst✝⁸ : NormedAddCommGroup E
inst✝⁷ : NormedSpace ℝ E
H : Type u_2
inst✝⁶ : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type u_3
inst✝⁵ : TopologicalSpace M
inst✝⁴ : ChartedSpace H M
inst✝³ : I.Boundaryless
inst✝² : FiniteDimensional ℝ E
inst✝¹ : MeasurableSpace E
inst✝ : BorelSpace E
s : Set M
h₁s ... | Please generate a tactic in lean4 to solve the state.
STATE:
E : Type u_1
inst✝⁸ : NormedAddCommGroup E
inst✝⁷ : NormedSpace ℝ E
H : Type u_2
inst✝⁶ : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type u_3
inst✝⁵ : TopologicalSpace M
inst✝⁴ : ChartedSpace H M
inst✝³ : I.Boundaryless
inst✝² : FiniteDimensional ℝ E
i... |
https://github.com/fpvandoorn/sard.git | f4a1bb550136e8591dcd20dca5bdcce766a958c7 | Sard/MeasureZero.lean | MeasureZero.open_implies_empty | [107, 1] | [134, 63] | let e := chartAt H x | E : Type u_1
inst✝⁸ : NormedAddCommGroup E
inst✝⁷ : NormedSpace ℝ E
H : Type u_2
inst✝⁶ : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type u_3
inst✝⁵ : TopologicalSpace M
inst✝⁴ : ChartedSpace H M
inst✝³ : I.Boundaryless
inst✝² : FiniteDimensional ℝ E
inst✝¹ : MeasurableSpace E
inst✝ : BorelSpace E
s : Set M
h₁s ... | E : Type u_1
inst✝⁸ : NormedAddCommGroup E
inst✝⁷ : NormedSpace ℝ E
H : Type u_2
inst✝⁶ : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type u_3
inst✝⁵ : TopologicalSpace M
inst✝⁴ : ChartedSpace H M
inst✝³ : I.Boundaryless
inst✝² : FiniteDimensional ℝ E
inst✝¹ : MeasurableSpace E
inst✝ : BorelSpace E
s : Set M
h₁s ... | Please generate a tactic in lean4 to solve the state.
STATE:
E : Type u_1
inst✝⁸ : NormedAddCommGroup E
inst✝⁷ : NormedSpace ℝ E
H : Type u_2
inst✝⁶ : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type u_3
inst✝⁵ : TopologicalSpace M
inst✝⁴ : ChartedSpace H M
inst✝³ : I.Boundaryless
inst✝² : FiniteDimensional ℝ E
i... |
https://github.com/fpvandoorn/sard.git | f4a1bb550136e8591dcd20dca5bdcce766a958c7 | Sard/MeasureZero.lean | MeasureZero.open_implies_empty | [107, 1] | [134, 63] | obtain ⟨K''⟩ : Nonempty (PositiveCompacts E) := PositiveCompacts.nonempty' | E : Type u_1
inst✝⁸ : NormedAddCommGroup E
inst✝⁷ : NormedSpace ℝ E
H : Type u_2
inst✝⁶ : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type u_3
inst✝⁵ : TopologicalSpace M
inst✝⁴ : ChartedSpace H M
inst✝³ : I.Boundaryless
inst✝² : FiniteDimensional ℝ E
inst✝¹ : MeasurableSpace E
inst✝ : BorelSpace E
s : Set M
h₁s ... | case intro
E : Type u_1
inst✝⁸ : NormedAddCommGroup E
inst✝⁷ : NormedSpace ℝ E
H : Type u_2
inst✝⁶ : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type u_3
inst✝⁵ : TopologicalSpace M
inst✝⁴ : ChartedSpace H M
inst✝³ : I.Boundaryless
inst✝² : FiniteDimensional ℝ E
inst✝¹ : MeasurableSpace E
inst✝ : BorelSpace E
s :... | Please generate a tactic in lean4 to solve the state.
STATE:
E : Type u_1
inst✝⁸ : NormedAddCommGroup E
inst✝⁷ : NormedSpace ℝ E
H : Type u_2
inst✝⁶ : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type u_3
inst✝⁵ : TopologicalSpace M
inst✝⁴ : ChartedSpace H M
inst✝³ : I.Boundaryless
inst✝² : FiniteDimensional ℝ E
i... |
https://github.com/fpvandoorn/sard.git | f4a1bb550136e8591dcd20dca5bdcce766a958c7 | Sard/MeasureZero.lean | MeasureZero.open_implies_empty | [107, 1] | [134, 63] | let μ : Measure E := addHaarMeasure K'' | case intro
E : Type u_1
inst✝⁸ : NormedAddCommGroup E
inst✝⁷ : NormedSpace ℝ E
H : Type u_2
inst✝⁶ : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type u_3
inst✝⁵ : TopologicalSpace M
inst✝⁴ : ChartedSpace H M
inst✝³ : I.Boundaryless
inst✝² : FiniteDimensional ℝ E
inst✝¹ : MeasurableSpace E
inst✝ : BorelSpace E
s :... | case intro
E : Type u_1
inst✝⁸ : NormedAddCommGroup E
inst✝⁷ : NormedSpace ℝ E
H : Type u_2
inst✝⁶ : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type u_3
inst✝⁵ : TopologicalSpace M
inst✝⁴ : ChartedSpace H M
inst✝³ : I.Boundaryless
inst✝² : FiniteDimensional ℝ E
inst✝¹ : MeasurableSpace E
inst✝ : BorelSpace E
s :... | Please generate a tactic in lean4 to solve the state.
STATE:
case intro
E : Type u_1
inst✝⁸ : NormedAddCommGroup E
inst✝⁷ : NormedSpace ℝ E
H : Type u_2
inst✝⁶ : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type u_3
inst✝⁵ : TopologicalSpace M
inst✝⁴ : ChartedSpace H M
inst✝³ : I.Boundaryless
inst✝² : FiniteDimens... |
https://github.com/fpvandoorn/sard.git | f4a1bb550136e8591dcd20dca5bdcce766a958c7 | Sard/MeasureZero.lean | MeasureZero.open_implies_empty | [107, 1] | [134, 63] | specialize h₂s μ x | case intro
E : Type u_1
inst✝⁸ : NormedAddCommGroup E
inst✝⁷ : NormedSpace ℝ E
H : Type u_2
inst✝⁶ : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type u_3
inst✝⁵ : TopologicalSpace M
inst✝⁴ : ChartedSpace H M
inst✝³ : I.Boundaryless
inst✝² : FiniteDimensional ℝ E
inst✝¹ : MeasurableSpace E
inst✝ : BorelSpace E
s :... | case intro
E : Type u_1
inst✝⁸ : NormedAddCommGroup E
inst✝⁷ : NormedSpace ℝ E
H : Type u_2
inst✝⁶ : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type u_3
inst✝⁵ : TopologicalSpace M
inst✝⁴ : ChartedSpace H M
inst✝³ : I.Boundaryless
inst✝² : FiniteDimensional ℝ E
inst✝¹ : MeasurableSpace E
inst✝ : BorelSpace E
s :... | Please generate a tactic in lean4 to solve the state.
STATE:
case intro
E : Type u_1
inst✝⁸ : NormedAddCommGroup E
inst✝⁷ : NormedSpace ℝ E
H : Type u_2
inst✝⁶ : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type u_3
inst✝⁵ : TopologicalSpace M
inst✝⁴ : ChartedSpace H M
inst✝³ : I.Boundaryless
inst✝² : FiniteDimens... |
https://github.com/fpvandoorn/sard.git | f4a1bb550136e8591dcd20dca5bdcce766a958c7 | Sard/MeasureZero.lean | MeasureZero.open_implies_empty | [107, 1] | [134, 63] | by_contra h | case intro
E : Type u_1
inst✝⁸ : NormedAddCommGroup E
inst✝⁷ : NormedSpace ℝ E
H : Type u_2
inst✝⁶ : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type u_3
inst✝⁵ : TopologicalSpace M
inst✝⁴ : ChartedSpace H M
inst✝³ : I.Boundaryless
inst✝² : FiniteDimensional ℝ E
inst✝¹ : MeasurableSpace E
inst✝ : BorelSpace E
s :... | case intro
E : Type u_1
inst✝⁸ : NormedAddCommGroup E
inst✝⁷ : NormedSpace ℝ E
H : Type u_2
inst✝⁶ : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type u_3
inst✝⁵ : TopologicalSpace M
inst✝⁴ : ChartedSpace H M
inst✝³ : I.Boundaryless
inst✝² : FiniteDimensional ℝ E
inst✝¹ : MeasurableSpace E
inst✝ : BorelSpace E
s :... | Please generate a tactic in lean4 to solve the state.
STATE:
case intro
E : Type u_1
inst✝⁸ : NormedAddCommGroup E
inst✝⁷ : NormedSpace ℝ E
H : Type u_2
inst✝⁶ : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type u_3
inst✝⁵ : TopologicalSpace M
inst✝⁴ : ChartedSpace H M
inst✝³ : I.Boundaryless
inst✝² : FiniteDimens... |
https://github.com/fpvandoorn/sard.git | f4a1bb550136e8591dcd20dca5bdcce766a958c7 | Sard/MeasureZero.lean | MeasureZero.open_implies_empty | [107, 1] | [134, 63] | have h' : Set.Nonempty (interior (I ∘ e '' (e.source ∩ s))) := by
have : Set.Nonempty (I ∘ e '' (e.source ∩ s)) := (Iff.mp Set.nmem_singleton_empty h).image _
have : IsOpen (I ∘ e '' (e.source ∩ s)) := by
rw [Set.image_comp]
exact I.toHomeomorph.isOpenMap _ (e.isOpen_image_source_inter h₁s)
rwa [this.inte... | case intro
E : Type u_1
inst✝⁸ : NormedAddCommGroup E
inst✝⁷ : NormedSpace ℝ E
H : Type u_2
inst✝⁶ : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type u_3
inst✝⁵ : TopologicalSpace M
inst✝⁴ : ChartedSpace H M
inst✝³ : I.Boundaryless
inst✝² : FiniteDimensional ℝ E
inst✝¹ : MeasurableSpace E
inst✝ : BorelSpace E
s :... | case intro
E : Type u_1
inst✝⁸ : NormedAddCommGroup E
inst✝⁷ : NormedSpace ℝ E
H : Type u_2
inst✝⁶ : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type u_3
inst✝⁵ : TopologicalSpace M
inst✝⁴ : ChartedSpace H M
inst✝³ : I.Boundaryless
inst✝² : FiniteDimensional ℝ E
inst✝¹ : MeasurableSpace E
inst✝ : BorelSpace E
s :... | Please generate a tactic in lean4 to solve the state.
STATE:
case intro
E : Type u_1
inst✝⁸ : NormedAddCommGroup E
inst✝⁷ : NormedSpace ℝ E
H : Type u_2
inst✝⁶ : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type u_3
inst✝⁵ : TopologicalSpace M
inst✝⁴ : ChartedSpace H M
inst✝³ : I.Boundaryless
inst✝² : FiniteDimens... |
https://github.com/fpvandoorn/sard.git | f4a1bb550136e8591dcd20dca5bdcce766a958c7 | Sard/MeasureZero.lean | MeasureZero.open_implies_empty | [107, 1] | [134, 63] | exact (measure_pos_of_nonempty_interior (μ := μ) h').ne' h₂s | case intro
E : Type u_1
inst✝⁸ : NormedAddCommGroup E
inst✝⁷ : NormedSpace ℝ E
H : Type u_2
inst✝⁶ : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type u_3
inst✝⁵ : TopologicalSpace M
inst✝⁴ : ChartedSpace H M
inst✝³ : I.Boundaryless
inst✝² : FiniteDimensional ℝ E
inst✝¹ : MeasurableSpace E
inst✝ : BorelSpace E
s :... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case intro
E : Type u_1
inst✝⁸ : NormedAddCommGroup E
inst✝⁷ : NormedSpace ℝ E
H : Type u_2
inst✝⁶ : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type u_3
inst✝⁵ : TopologicalSpace M
inst✝⁴ : ChartedSpace H M
inst✝³ : I.Boundaryless
inst✝² : FiniteDimens... |
https://github.com/fpvandoorn/sard.git | f4a1bb550136e8591dcd20dca5bdcce766a958c7 | Sard/MeasureZero.lean | MeasureZero.open_implies_empty | [107, 1] | [134, 63] | by_contra h | E : Type u_1
inst✝⁸ : NormedAddCommGroup E
inst✝⁷ : NormedSpace ℝ E
H : Type u_2
inst✝⁶ : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type u_3
inst✝⁵ : TopologicalSpace M
inst✝⁴ : ChartedSpace H M
inst✝³ : I.Boundaryless
inst✝² : FiniteDimensional ℝ E
inst✝¹ : MeasurableSpace E
inst✝ : BorelSpace E
s : Set M
h₁s ... | E : Type u_1
inst✝⁸ : NormedAddCommGroup E
inst✝⁷ : NormedSpace ℝ E
H : Type u_2
inst✝⁶ : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type u_3
inst✝⁵ : TopologicalSpace M
inst✝⁴ : ChartedSpace H M
inst✝³ : I.Boundaryless
inst✝² : FiniteDimensional ℝ E
inst✝¹ : MeasurableSpace E
inst✝ : BorelSpace E
s : Set M
h₁s ... | Please generate a tactic in lean4 to solve the state.
STATE:
E : Type u_1
inst✝⁸ : NormedAddCommGroup E
inst✝⁷ : NormedSpace ℝ E
H : Type u_2
inst✝⁶ : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type u_3
inst✝⁵ : TopologicalSpace M
inst✝⁴ : ChartedSpace H M
inst✝³ : I.Boundaryless
inst✝² : FiniteDimensional ℝ E
i... |
https://github.com/fpvandoorn/sard.git | f4a1bb550136e8591dcd20dca5bdcce766a958c7 | Sard/MeasureZero.lean | MeasureZero.open_implies_empty | [107, 1] | [134, 63] | obtain ⟨x, hx⟩ : Set.Nonempty s := Iff.mp nmem_singleton_empty h | E : Type u_1
inst✝⁸ : NormedAddCommGroup E
inst✝⁷ : NormedSpace ℝ E
H : Type u_2
inst✝⁶ : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type u_3
inst✝⁵ : TopologicalSpace M
inst✝⁴ : ChartedSpace H M
inst✝³ : I.Boundaryless
inst✝² : FiniteDimensional ℝ E
inst✝¹ : MeasurableSpace E
inst✝ : BorelSpace E
s : Set M
h₁s ... | case intro
E : Type u_1
inst✝⁸ : NormedAddCommGroup E
inst✝⁷ : NormedSpace ℝ E
H : Type u_2
inst✝⁶ : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type u_3
inst✝⁵ : TopologicalSpace M
inst✝⁴ : ChartedSpace H M
inst✝³ : I.Boundaryless
inst✝² : FiniteDimensional ℝ E
inst✝¹ : MeasurableSpace E
inst✝ : BorelSpace E
s :... | Please generate a tactic in lean4 to solve the state.
STATE:
E : Type u_1
inst✝⁸ : NormedAddCommGroup E
inst✝⁷ : NormedSpace ℝ E
H : Type u_2
inst✝⁶ : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type u_3
inst✝⁵ : TopologicalSpace M
inst✝⁴ : ChartedSpace H M
inst✝³ : I.Boundaryless
inst✝² : FiniteDimensional ℝ E
i... |
https://github.com/fpvandoorn/sard.git | f4a1bb550136e8591dcd20dca5bdcce766a958c7 | Sard/MeasureZero.lean | MeasureZero.open_implies_empty | [107, 1] | [134, 63] | specialize this x | case intro
E : Type u_1
inst✝⁸ : NormedAddCommGroup E
inst✝⁷ : NormedSpace ℝ E
H : Type u_2
inst✝⁶ : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type u_3
inst✝⁵ : TopologicalSpace M
inst✝⁴ : ChartedSpace H M
inst✝³ : I.Boundaryless
inst✝² : FiniteDimensional ℝ E
inst✝¹ : MeasurableSpace E
inst✝ : BorelSpace E
s :... | case intro
E : Type u_1
inst✝⁸ : NormedAddCommGroup E
inst✝⁷ : NormedSpace ℝ E
H : Type u_2
inst✝⁶ : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type u_3
inst✝⁵ : TopologicalSpace M
inst✝⁴ : ChartedSpace H M
inst✝³ : I.Boundaryless
inst✝² : FiniteDimensional ℝ E
inst✝¹ : MeasurableSpace E
inst✝ : BorelSpace E
s :... | Please generate a tactic in lean4 to solve the state.
STATE:
case intro
E : Type u_1
inst✝⁸ : NormedAddCommGroup E
inst✝⁷ : NormedSpace ℝ E
H : Type u_2
inst✝⁶ : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type u_3
inst✝⁵ : TopologicalSpace M
inst✝⁴ : ChartedSpace H M
inst✝³ : I.Boundaryless
inst✝² : FiniteDimens... |
https://github.com/fpvandoorn/sard.git | f4a1bb550136e8591dcd20dca5bdcce766a958c7 | Sard/MeasureZero.lean | MeasureZero.open_implies_empty | [107, 1] | [134, 63] | have h₂: x ∈ (chartAt H x).toPartialEquiv.source ∩ s := by
simp
exact hx | case intro
E : Type u_1
inst✝⁸ : NormedAddCommGroup E
inst✝⁷ : NormedSpace ℝ E
H : Type u_2
inst✝⁶ : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type u_3
inst✝⁵ : TopologicalSpace M
inst✝⁴ : ChartedSpace H M
inst✝³ : I.Boundaryless
inst✝² : FiniteDimensional ℝ E
inst✝¹ : MeasurableSpace E
inst✝ : BorelSpace E
s :... | case intro
E : Type u_1
inst✝⁸ : NormedAddCommGroup E
inst✝⁷ : NormedSpace ℝ E
H : Type u_2
inst✝⁶ : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type u_3
inst✝⁵ : TopologicalSpace M
inst✝⁴ : ChartedSpace H M
inst✝³ : I.Boundaryless
inst✝² : FiniteDimensional ℝ E
inst✝¹ : MeasurableSpace E
inst✝ : BorelSpace E
s :... | Please generate a tactic in lean4 to solve the state.
STATE:
case intro
E : Type u_1
inst✝⁸ : NormedAddCommGroup E
inst✝⁷ : NormedSpace ℝ E
H : Type u_2
inst✝⁶ : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type u_3
inst✝⁵ : TopologicalSpace M
inst✝⁴ : ChartedSpace H M
inst✝³ : I.Boundaryless
inst✝² : FiniteDimens... |
https://github.com/fpvandoorn/sard.git | f4a1bb550136e8591dcd20dca5bdcce766a958c7 | Sard/MeasureZero.lean | MeasureZero.open_implies_empty | [107, 1] | [134, 63] | rw [this] at h₂ | case intro
E : Type u_1
inst✝⁸ : NormedAddCommGroup E
inst✝⁷ : NormedSpace ℝ E
H : Type u_2
inst✝⁶ : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type u_3
inst✝⁵ : TopologicalSpace M
inst✝⁴ : ChartedSpace H M
inst✝³ : I.Boundaryless
inst✝² : FiniteDimensional ℝ E
inst✝¹ : MeasurableSpace E
inst✝ : BorelSpace E
s :... | case intro
E : Type u_1
inst✝⁸ : NormedAddCommGroup E
inst✝⁷ : NormedSpace ℝ E
H : Type u_2
inst✝⁶ : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type u_3
inst✝⁵ : TopologicalSpace M
inst✝⁴ : ChartedSpace H M
inst✝³ : I.Boundaryless
inst✝² : FiniteDimensional ℝ E
inst✝¹ : MeasurableSpace E
inst✝ : BorelSpace E
s :... | Please generate a tactic in lean4 to solve the state.
STATE:
case intro
E : Type u_1
inst✝⁸ : NormedAddCommGroup E
inst✝⁷ : NormedSpace ℝ E
H : Type u_2
inst✝⁶ : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type u_3
inst✝⁵ : TopologicalSpace M
inst✝⁴ : ChartedSpace H M
inst✝³ : I.Boundaryless
inst✝² : FiniteDimens... |
https://github.com/fpvandoorn/sard.git | f4a1bb550136e8591dcd20dca5bdcce766a958c7 | Sard/MeasureZero.lean | MeasureZero.open_implies_empty | [107, 1] | [134, 63] | contradiction | case intro
E : Type u_1
inst✝⁸ : NormedAddCommGroup E
inst✝⁷ : NormedSpace ℝ E
H : Type u_2
inst✝⁶ : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type u_3
inst✝⁵ : TopologicalSpace M
inst✝⁴ : ChartedSpace H M
inst✝³ : I.Boundaryless
inst✝² : FiniteDimensional ℝ E
inst✝¹ : MeasurableSpace E
inst✝ : BorelSpace E
s :... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case intro
E : Type u_1
inst✝⁸ : NormedAddCommGroup E
inst✝⁷ : NormedSpace ℝ E
H : Type u_2
inst✝⁶ : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type u_3
inst✝⁵ : TopologicalSpace M
inst✝⁴ : ChartedSpace H M
inst✝³ : I.Boundaryless
inst✝² : FiniteDimens... |
https://github.com/fpvandoorn/sard.git | f4a1bb550136e8591dcd20dca5bdcce766a958c7 | Sard/MeasureZero.lean | MeasureZero.open_implies_empty | [107, 1] | [134, 63] | simp | E : Type u_1
inst✝⁸ : NormedAddCommGroup E
inst✝⁷ : NormedSpace ℝ E
H : Type u_2
inst✝⁶ : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type u_3
inst✝⁵ : TopologicalSpace M
inst✝⁴ : ChartedSpace H M
inst✝³ : I.Boundaryless
inst✝² : FiniteDimensional ℝ E
inst✝¹ : MeasurableSpace E
inst✝ : BorelSpace E
s : Set M
h₁s ... | E : Type u_1
inst✝⁸ : NormedAddCommGroup E
inst✝⁷ : NormedSpace ℝ E
H : Type u_2
inst✝⁶ : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type u_3
inst✝⁵ : TopologicalSpace M
inst✝⁴ : ChartedSpace H M
inst✝³ : I.Boundaryless
inst✝² : FiniteDimensional ℝ E
inst✝¹ : MeasurableSpace E
inst✝ : BorelSpace E
s : Set M
h₁s ... | Please generate a tactic in lean4 to solve the state.
STATE:
E : Type u_1
inst✝⁸ : NormedAddCommGroup E
inst✝⁷ : NormedSpace ℝ E
H : Type u_2
inst✝⁶ : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type u_3
inst✝⁵ : TopologicalSpace M
inst✝⁴ : ChartedSpace H M
inst✝³ : I.Boundaryless
inst✝² : FiniteDimensional ℝ E
i... |
https://github.com/fpvandoorn/sard.git | f4a1bb550136e8591dcd20dca5bdcce766a958c7 | Sard/MeasureZero.lean | MeasureZero.open_implies_empty | [107, 1] | [134, 63] | exact hx | E : Type u_1
inst✝⁸ : NormedAddCommGroup E
inst✝⁷ : NormedSpace ℝ E
H : Type u_2
inst✝⁶ : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type u_3
inst✝⁵ : TopologicalSpace M
inst✝⁴ : ChartedSpace H M
inst✝³ : I.Boundaryless
inst✝² : FiniteDimensional ℝ E
inst✝¹ : MeasurableSpace E
inst✝ : BorelSpace E
s : Set M
h₁s ... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
E : Type u_1
inst✝⁸ : NormedAddCommGroup E
inst✝⁷ : NormedSpace ℝ E
H : Type u_2
inst✝⁶ : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type u_3
inst✝⁵ : TopologicalSpace M
inst✝⁴ : ChartedSpace H M
inst✝³ : I.Boundaryless
inst✝² : FiniteDimensional ℝ E
i... |
https://github.com/fpvandoorn/sard.git | f4a1bb550136e8591dcd20dca5bdcce766a958c7 | Sard/MeasureZero.lean | MeasureZero.open_implies_empty | [107, 1] | [134, 63] | have : Set.Nonempty (I ∘ e '' (e.source ∩ s)) := (Iff.mp Set.nmem_singleton_empty h).image _ | E : Type u_1
inst✝⁸ : NormedAddCommGroup E
inst✝⁷ : NormedSpace ℝ E
H : Type u_2
inst✝⁶ : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type u_3
inst✝⁵ : TopologicalSpace M
inst✝⁴ : ChartedSpace H M
inst✝³ : I.Boundaryless
inst✝² : FiniteDimensional ℝ E
inst✝¹ : MeasurableSpace E
inst✝ : BorelSpace E
s : Set M
h₁s ... | E : Type u_1
inst✝⁸ : NormedAddCommGroup E
inst✝⁷ : NormedSpace ℝ E
H : Type u_2
inst✝⁶ : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type u_3
inst✝⁵ : TopologicalSpace M
inst✝⁴ : ChartedSpace H M
inst✝³ : I.Boundaryless
inst✝² : FiniteDimensional ℝ E
inst✝¹ : MeasurableSpace E
inst✝ : BorelSpace E
s : Set M
h₁s ... | Please generate a tactic in lean4 to solve the state.
STATE:
E : Type u_1
inst✝⁸ : NormedAddCommGroup E
inst✝⁷ : NormedSpace ℝ E
H : Type u_2
inst✝⁶ : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type u_3
inst✝⁵ : TopologicalSpace M
inst✝⁴ : ChartedSpace H M
inst✝³ : I.Boundaryless
inst✝² : FiniteDimensional ℝ E
i... |
https://github.com/fpvandoorn/sard.git | f4a1bb550136e8591dcd20dca5bdcce766a958c7 | Sard/MeasureZero.lean | MeasureZero.open_implies_empty | [107, 1] | [134, 63] | have : IsOpen (I ∘ e '' (e.source ∩ s)) := by
rw [Set.image_comp]
exact I.toHomeomorph.isOpenMap _ (e.isOpen_image_source_inter h₁s) | E : Type u_1
inst✝⁸ : NormedAddCommGroup E
inst✝⁷ : NormedSpace ℝ E
H : Type u_2
inst✝⁶ : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type u_3
inst✝⁵ : TopologicalSpace M
inst✝⁴ : ChartedSpace H M
inst✝³ : I.Boundaryless
inst✝² : FiniteDimensional ℝ E
inst✝¹ : MeasurableSpace E
inst✝ : BorelSpace E
s : Set M
h₁s ... | E : Type u_1
inst✝⁸ : NormedAddCommGroup E
inst✝⁷ : NormedSpace ℝ E
H : Type u_2
inst✝⁶ : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type u_3
inst✝⁵ : TopologicalSpace M
inst✝⁴ : ChartedSpace H M
inst✝³ : I.Boundaryless
inst✝² : FiniteDimensional ℝ E
inst✝¹ : MeasurableSpace E
inst✝ : BorelSpace E
s : Set M
h₁s ... | Please generate a tactic in lean4 to solve the state.
STATE:
E : Type u_1
inst✝⁸ : NormedAddCommGroup E
inst✝⁷ : NormedSpace ℝ E
H : Type u_2
inst✝⁶ : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type u_3
inst✝⁵ : TopologicalSpace M
inst✝⁴ : ChartedSpace H M
inst✝³ : I.Boundaryless
inst✝² : FiniteDimensional ℝ E
i... |
https://github.com/fpvandoorn/sard.git | f4a1bb550136e8591dcd20dca5bdcce766a958c7 | Sard/MeasureZero.lean | MeasureZero.open_implies_empty | [107, 1] | [134, 63] | rwa [this.interior_eq] | E : Type u_1
inst✝⁸ : NormedAddCommGroup E
inst✝⁷ : NormedSpace ℝ E
H : Type u_2
inst✝⁶ : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type u_3
inst✝⁵ : TopologicalSpace M
inst✝⁴ : ChartedSpace H M
inst✝³ : I.Boundaryless
inst✝² : FiniteDimensional ℝ E
inst✝¹ : MeasurableSpace E
inst✝ : BorelSpace E
s : Set M
h₁s ... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
E : Type u_1
inst✝⁸ : NormedAddCommGroup E
inst✝⁷ : NormedSpace ℝ E
H : Type u_2
inst✝⁶ : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type u_3
inst✝⁵ : TopologicalSpace M
inst✝⁴ : ChartedSpace H M
inst✝³ : I.Boundaryless
inst✝² : FiniteDimensional ℝ E
i... |
https://github.com/fpvandoorn/sard.git | f4a1bb550136e8591dcd20dca5bdcce766a958c7 | Sard/MeasureZero.lean | MeasureZero.open_implies_empty | [107, 1] | [134, 63] | rw [Set.image_comp] | E : Type u_1
inst✝⁸ : NormedAddCommGroup E
inst✝⁷ : NormedSpace ℝ E
H : Type u_2
inst✝⁶ : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type u_3
inst✝⁵ : TopologicalSpace M
inst✝⁴ : ChartedSpace H M
inst✝³ : I.Boundaryless
inst✝² : FiniteDimensional ℝ E
inst✝¹ : MeasurableSpace E
inst✝ : BorelSpace E
s : Set M
h₁s ... | E : Type u_1
inst✝⁸ : NormedAddCommGroup E
inst✝⁷ : NormedSpace ℝ E
H : Type u_2
inst✝⁶ : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type u_3
inst✝⁵ : TopologicalSpace M
inst✝⁴ : ChartedSpace H M
inst✝³ : I.Boundaryless
inst✝² : FiniteDimensional ℝ E
inst✝¹ : MeasurableSpace E
inst✝ : BorelSpace E
s : Set M
h₁s ... | Please generate a tactic in lean4 to solve the state.
STATE:
E : Type u_1
inst✝⁸ : NormedAddCommGroup E
inst✝⁷ : NormedSpace ℝ E
H : Type u_2
inst✝⁶ : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type u_3
inst✝⁵ : TopologicalSpace M
inst✝⁴ : ChartedSpace H M
inst✝³ : I.Boundaryless
inst✝² : FiniteDimensional ℝ E
i... |
https://github.com/fpvandoorn/sard.git | f4a1bb550136e8591dcd20dca5bdcce766a958c7 | Sard/MeasureZero.lean | MeasureZero.open_implies_empty | [107, 1] | [134, 63] | exact I.toHomeomorph.isOpenMap _ (e.isOpen_image_source_inter h₁s) | E : Type u_1
inst✝⁸ : NormedAddCommGroup E
inst✝⁷ : NormedSpace ℝ E
H : Type u_2
inst✝⁶ : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type u_3
inst✝⁵ : TopologicalSpace M
inst✝⁴ : ChartedSpace H M
inst✝³ : I.Boundaryless
inst✝² : FiniteDimensional ℝ E
inst✝¹ : MeasurableSpace E
inst✝ : BorelSpace E
s : Set M
h₁s ... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
E : Type u_1
inst✝⁸ : NormedAddCommGroup E
inst✝⁷ : NormedSpace ℝ E
H : Type u_2
inst✝⁶ : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type u_3
inst✝⁵ : TopologicalSpace M
inst✝⁴ : ChartedSpace H M
inst✝³ : I.Boundaryless
inst✝² : FiniteDimensional ℝ E
i... |
https://github.com/fpvandoorn/sard.git | f4a1bb550136e8591dcd20dca5bdcce766a958c7 | Sard/MeasureZero.lean | measure_zero_image_iff_chart_domains | [163, 1] | [198, 41] | let U : M → Set M := fun x ↦ (ChartedSpace.chartAt x : PartialHomeomorph M H).source | E✝ : Type u_1
inst✝²⁶ : NormedAddCommGroup E✝
inst✝²⁵ : NormedSpace ℝ E✝
H✝ : Type u_2
inst✝²⁴ : TopologicalSpace H✝
I✝ : ModelWithCorners ℝ E✝ H✝
M✝ : Type u_3
inst✝²³ : TopologicalSpace M✝
inst✝²² : ChartedSpace H✝ M✝
inst✝²¹ : I✝.Boundaryless
inst✝²⁰ : FiniteDimensional ℝ E✝
inst✝¹⁹ : MeasurableSpace E✝
inst✝¹⁸ : Bo... | E✝ : Type u_1
inst✝²⁶ : NormedAddCommGroup E✝
inst✝²⁵ : NormedSpace ℝ E✝
H✝ : Type u_2
inst✝²⁴ : TopologicalSpace H✝
I✝ : ModelWithCorners ℝ E✝ H✝
M✝ : Type u_3
inst✝²³ : TopologicalSpace M✝
inst✝²² : ChartedSpace H✝ M✝
inst✝²¹ : I✝.Boundaryless
inst✝²⁰ : FiniteDimensional ℝ E✝
inst✝¹⁹ : MeasurableSpace E✝
inst✝¹⁸ : Bo... | Please generate a tactic in lean4 to solve the state.
STATE:
E✝ : Type u_1
inst✝²⁶ : NormedAddCommGroup E✝
inst✝²⁵ : NormedSpace ℝ E✝
H✝ : Type u_2
inst✝²⁴ : TopologicalSpace H✝
I✝ : ModelWithCorners ℝ E✝ H✝
M✝ : Type u_3
inst✝²³ : TopologicalSpace M✝
inst✝²² : ChartedSpace H✝ M✝
inst✝²¹ : I✝.Boundaryless
inst✝²⁰ : Fin... |
https://github.com/fpvandoorn/sard.git | f4a1bb550136e8591dcd20dca5bdcce766a958c7 | Sard/MeasureZero.lean | measure_zero_image_iff_chart_domains | [163, 1] | [198, 41] | have hcovering : univ ⊆ ⋃ (x : M), U x := by
intro x _
rw [mem_iUnion]
exact ⟨x, mem_chart_source H x⟩ | E✝ : Type u_1
inst✝²⁶ : NormedAddCommGroup E✝
inst✝²⁵ : NormedSpace ℝ E✝
H✝ : Type u_2
inst✝²⁴ : TopologicalSpace H✝
I✝ : ModelWithCorners ℝ E✝ H✝
M✝ : Type u_3
inst✝²³ : TopologicalSpace M✝
inst✝²² : ChartedSpace H✝ M✝
inst✝²¹ : I✝.Boundaryless
inst✝²⁰ : FiniteDimensional ℝ E✝
inst✝¹⁹ : MeasurableSpace E✝
inst✝¹⁸ : Bo... | E✝ : Type u_1
inst✝²⁶ : NormedAddCommGroup E✝
inst✝²⁵ : NormedSpace ℝ E✝
H✝ : Type u_2
inst✝²⁴ : TopologicalSpace H✝
I✝ : ModelWithCorners ℝ E✝ H✝
M✝ : Type u_3
inst✝²³ : TopologicalSpace M✝
inst✝²² : ChartedSpace H✝ M✝
inst✝²¹ : I✝.Boundaryless
inst✝²⁰ : FiniteDimensional ℝ E✝
inst✝¹⁹ : MeasurableSpace E✝
inst✝¹⁸ : Bo... | Please generate a tactic in lean4 to solve the state.
STATE:
E✝ : Type u_1
inst✝²⁶ : NormedAddCommGroup E✝
inst✝²⁵ : NormedSpace ℝ E✝
H✝ : Type u_2
inst✝²⁴ : TopologicalSpace H✝
I✝ : ModelWithCorners ℝ E✝ H✝
M✝ : Type u_3
inst✝²³ : TopologicalSpace M✝
inst✝²² : ChartedSpace H✝ M✝
inst✝²¹ : I✝.Boundaryless
inst✝²⁰ : Fin... |
https://github.com/fpvandoorn/sard.git | f4a1bb550136e8591dcd20dca5bdcce766a958c7 | Sard/MeasureZero.lean | measure_zero_image_iff_chart_domains | [163, 1] | [198, 41] | have hopen : ∀ x : M, IsOpen (U x) := fun x => (ChartedSpace.chartAt x).open_source | E✝ : Type u_1
inst✝²⁶ : NormedAddCommGroup E✝
inst✝²⁵ : NormedSpace ℝ E✝
H✝ : Type u_2
inst✝²⁴ : TopologicalSpace H✝
I✝ : ModelWithCorners ℝ E✝ H✝
M✝ : Type u_3
inst✝²³ : TopologicalSpace M✝
inst✝²² : ChartedSpace H✝ M✝
inst✝²¹ : I✝.Boundaryless
inst✝²⁰ : FiniteDimensional ℝ E✝
inst✝¹⁹ : MeasurableSpace E✝
inst✝¹⁸ : Bo... | E✝ : Type u_1
inst✝²⁶ : NormedAddCommGroup E✝
inst✝²⁵ : NormedSpace ℝ E✝
H✝ : Type u_2
inst✝²⁴ : TopologicalSpace H✝
I✝ : ModelWithCorners ℝ E✝ H✝
M✝ : Type u_3
inst✝²³ : TopologicalSpace M✝
inst✝²² : ChartedSpace H✝ M✝
inst✝²¹ : I✝.Boundaryless
inst✝²⁰ : FiniteDimensional ℝ E✝
inst✝¹⁹ : MeasurableSpace E✝
inst✝¹⁸ : Bo... | Please generate a tactic in lean4 to solve the state.
STATE:
E✝ : Type u_1
inst✝²⁶ : NormedAddCommGroup E✝
inst✝²⁵ : NormedSpace ℝ E✝
H✝ : Type u_2
inst✝²⁴ : TopologicalSpace H✝
I✝ : ModelWithCorners ℝ E✝ H✝
M✝ : Type u_3
inst✝²³ : TopologicalSpace M✝
inst✝²² : ChartedSpace H✝ M✝
inst✝²¹ : I✝.Boundaryless
inst✝²⁰ : Fin... |
https://github.com/fpvandoorn/sard.git | f4a1bb550136e8591dcd20dca5bdcce766a958c7 | Sard/MeasureZero.lean | measure_zero_image_iff_chart_domains | [163, 1] | [198, 41] | let ⟨T, ⟨hTCountable, hTcover⟩⟩ := TopologicalSpace.isOpen_iUnion_countable U hopen | E✝ : Type u_1
inst✝²⁶ : NormedAddCommGroup E✝
inst✝²⁵ : NormedSpace ℝ E✝
H✝ : Type u_2
inst✝²⁴ : TopologicalSpace H✝
I✝ : ModelWithCorners ℝ E✝ H✝
M✝ : Type u_3
inst✝²³ : TopologicalSpace M✝
inst✝²² : ChartedSpace H✝ M✝
inst✝²¹ : I✝.Boundaryless
inst✝²⁰ : FiniteDimensional ℝ E✝
inst✝¹⁹ : MeasurableSpace E✝
inst✝¹⁸ : Bo... | E✝ : Type u_1
inst✝²⁶ : NormedAddCommGroup E✝
inst✝²⁵ : NormedSpace ℝ E✝
H✝ : Type u_2
inst✝²⁴ : TopologicalSpace H✝
I✝ : ModelWithCorners ℝ E✝ H✝
M✝ : Type u_3
inst✝²³ : TopologicalSpace M✝
inst✝²² : ChartedSpace H✝ M✝
inst✝²¹ : I✝.Boundaryless
inst✝²⁰ : FiniteDimensional ℝ E✝
inst✝¹⁹ : MeasurableSpace E✝
inst✝¹⁸ : Bo... | Please generate a tactic in lean4 to solve the state.
STATE:
E✝ : Type u_1
inst✝²⁶ : NormedAddCommGroup E✝
inst✝²⁵ : NormedSpace ℝ E✝
H✝ : Type u_2
inst✝²⁴ : TopologicalSpace H✝
I✝ : ModelWithCorners ℝ E✝ H✝
M✝ : Type u_3
inst✝²³ : TopologicalSpace M✝
inst✝²² : ChartedSpace H✝ M✝
inst✝²¹ : I✝.Boundaryless
inst✝²⁰ : Fin... |
https://github.com/fpvandoorn/sard.git | f4a1bb550136e8591dcd20dca5bdcce766a958c7 | Sard/MeasureZero.lean | measure_zero_image_iff_chart_domains | [163, 1] | [198, 41] | have : ∀ i : T, MeasureZero J (f '' ((U i) ∩ s)) := by
intro i
let e : PartialHomeomorph M H := ChartedSpace.chartAt i
have h : MeasureZero J (f '' (e.source ∩ s)) := hs i
have h₃ : U i = e.source := by rw [← Filter.principal_eq_iff_eq]
apply h.mono _
apply image_subset
rw [h₃] | E✝ : Type u_1
inst✝²⁶ : NormedAddCommGroup E✝
inst✝²⁵ : NormedSpace ℝ E✝
H✝ : Type u_2
inst✝²⁴ : TopologicalSpace H✝
I✝ : ModelWithCorners ℝ E✝ H✝
M✝ : Type u_3
inst✝²³ : TopologicalSpace M✝
inst✝²² : ChartedSpace H✝ M✝
inst✝²¹ : I✝.Boundaryless
inst✝²⁰ : FiniteDimensional ℝ E✝
inst✝¹⁹ : MeasurableSpace E✝
inst✝¹⁸ : Bo... | E✝ : Type u_1
inst✝²⁶ : NormedAddCommGroup E✝
inst✝²⁵ : NormedSpace ℝ E✝
H✝ : Type u_2
inst✝²⁴ : TopologicalSpace H✝
I✝ : ModelWithCorners ℝ E✝ H✝
M✝ : Type u_3
inst✝²³ : TopologicalSpace M✝
inst✝²² : ChartedSpace H✝ M✝
inst✝²¹ : I✝.Boundaryless
inst✝²⁰ : FiniteDimensional ℝ E✝
inst✝¹⁹ : MeasurableSpace E✝
inst✝¹⁸ : Bo... | Please generate a tactic in lean4 to solve the state.
STATE:
E✝ : Type u_1
inst✝²⁶ : NormedAddCommGroup E✝
inst✝²⁵ : NormedSpace ℝ E✝
H✝ : Type u_2
inst✝²⁴ : TopologicalSpace H✝
I✝ : ModelWithCorners ℝ E✝ H✝
M✝ : Type u_3
inst✝²³ : TopologicalSpace M✝
inst✝²² : ChartedSpace H✝ M✝
inst✝²¹ : I✝.Boundaryless
inst✝²⁰ : Fin... |
https://github.com/fpvandoorn/sard.git | f4a1bb550136e8591dcd20dca5bdcce766a958c7 | Sard/MeasureZero.lean | measure_zero_image_iff_chart_domains | [163, 1] | [198, 41] | have decomp : ⋃ (i : T), f '' ((U i) ∩ s) = f '' s :=
calc ⋃ (i : T), f '' ((U i) ∩ s)
_ = f '' (⋃ (i : T), (U i) ∩ s) := by rw [image_iUnion]
_ = f '' ((⋃ (i : T), (U i)) ∩ s) := by rw [iUnion_inter]
_ = f '' ((⋃ (i : M) (_ : i ∈ T), U i) ∩ s) := by rw [iUnion_coe_set]
_ = f '' ((⋃ (i : M), U i) ∩ s)... | E✝ : Type u_1
inst✝²⁶ : NormedAddCommGroup E✝
inst✝²⁵ : NormedSpace ℝ E✝
H✝ : Type u_2
inst✝²⁴ : TopologicalSpace H✝
I✝ : ModelWithCorners ℝ E✝ H✝
M✝ : Type u_3
inst✝²³ : TopologicalSpace M✝
inst✝²² : ChartedSpace H✝ M✝
inst✝²¹ : I✝.Boundaryless
inst✝²⁰ : FiniteDimensional ℝ E✝
inst✝¹⁹ : MeasurableSpace E✝
inst✝¹⁸ : Bo... | E✝ : Type u_1
inst✝²⁶ : NormedAddCommGroup E✝
inst✝²⁵ : NormedSpace ℝ E✝
H✝ : Type u_2
inst✝²⁴ : TopologicalSpace H✝
I✝ : ModelWithCorners ℝ E✝ H✝
M✝ : Type u_3
inst✝²³ : TopologicalSpace M✝
inst✝²² : ChartedSpace H✝ M✝
inst✝²¹ : I✝.Boundaryless
inst✝²⁰ : FiniteDimensional ℝ E✝
inst✝¹⁹ : MeasurableSpace E✝
inst✝¹⁸ : Bo... | Please generate a tactic in lean4 to solve the state.
STATE:
E✝ : Type u_1
inst✝²⁶ : NormedAddCommGroup E✝
inst✝²⁵ : NormedSpace ℝ E✝
H✝ : Type u_2
inst✝²⁴ : TopologicalSpace H✝
I✝ : ModelWithCorners ℝ E✝ H✝
M✝ : Type u_3
inst✝²³ : TopologicalSpace M✝
inst✝²² : ChartedSpace H✝ M✝
inst✝²¹ : I✝.Boundaryless
inst✝²⁰ : Fin... |
https://github.com/fpvandoorn/sard.git | f4a1bb550136e8591dcd20dca5bdcce766a958c7 | Sard/MeasureZero.lean | measure_zero_image_iff_chart_domains | [163, 1] | [198, 41] | rw [← decomp] | E✝ : Type u_1
inst✝²⁶ : NormedAddCommGroup E✝
inst✝²⁵ : NormedSpace ℝ E✝
H✝ : Type u_2
inst✝²⁴ : TopologicalSpace H✝
I✝ : ModelWithCorners ℝ E✝ H✝
M✝ : Type u_3
inst✝²³ : TopologicalSpace M✝
inst✝²² : ChartedSpace H✝ M✝
inst✝²¹ : I✝.Boundaryless
inst✝²⁰ : FiniteDimensional ℝ E✝
inst✝¹⁹ : MeasurableSpace E✝
inst✝¹⁸ : Bo... | E✝ : Type u_1
inst✝²⁶ : NormedAddCommGroup E✝
inst✝²⁵ : NormedSpace ℝ E✝
H✝ : Type u_2
inst✝²⁴ : TopologicalSpace H✝
I✝ : ModelWithCorners ℝ E✝ H✝
M✝ : Type u_3
inst✝²³ : TopologicalSpace M✝
inst✝²² : ChartedSpace H✝ M✝
inst✝²¹ : I✝.Boundaryless
inst✝²⁰ : FiniteDimensional ℝ E✝
inst✝¹⁹ : MeasurableSpace E✝
inst✝¹⁸ : Bo... | Please generate a tactic in lean4 to solve the state.
STATE:
E✝ : Type u_1
inst✝²⁶ : NormedAddCommGroup E✝
inst✝²⁵ : NormedSpace ℝ E✝
H✝ : Type u_2
inst✝²⁴ : TopologicalSpace H✝
I✝ : ModelWithCorners ℝ E✝ H✝
M✝ : Type u_3
inst✝²³ : TopologicalSpace M✝
inst✝²² : ChartedSpace H✝ M✝
inst✝²¹ : I✝.Boundaryless
inst✝²⁰ : Fin... |
https://github.com/fpvandoorn/sard.git | f4a1bb550136e8591dcd20dca5bdcce766a958c7 | Sard/MeasureZero.lean | measure_zero_image_iff_chart_domains | [163, 1] | [198, 41] | have todo : Encodable T := by sorry | E✝ : Type u_1
inst✝²⁶ : NormedAddCommGroup E✝
inst✝²⁵ : NormedSpace ℝ E✝
H✝ : Type u_2
inst✝²⁴ : TopologicalSpace H✝
I✝ : ModelWithCorners ℝ E✝ H✝
M✝ : Type u_3
inst✝²³ : TopologicalSpace M✝
inst✝²² : ChartedSpace H✝ M✝
inst✝²¹ : I✝.Boundaryless
inst✝²⁰ : FiniteDimensional ℝ E✝
inst✝¹⁹ : MeasurableSpace E✝
inst✝¹⁸ : Bo... | E✝ : Type u_1
inst✝²⁶ : NormedAddCommGroup E✝
inst✝²⁵ : NormedSpace ℝ E✝
H✝ : Type u_2
inst✝²⁴ : TopologicalSpace H✝
I✝ : ModelWithCorners ℝ E✝ H✝
M✝ : Type u_3
inst✝²³ : TopologicalSpace M✝
inst✝²² : ChartedSpace H✝ M✝
inst✝²¹ : I✝.Boundaryless
inst✝²⁰ : FiniteDimensional ℝ E✝
inst✝¹⁹ : MeasurableSpace E✝
inst✝¹⁸ : Bo... | Please generate a tactic in lean4 to solve the state.
STATE:
E✝ : Type u_1
inst✝²⁶ : NormedAddCommGroup E✝
inst✝²⁵ : NormedSpace ℝ E✝
H✝ : Type u_2
inst✝²⁴ : TopologicalSpace H✝
I✝ : ModelWithCorners ℝ E✝ H✝
M✝ : Type u_3
inst✝²³ : TopologicalSpace M✝
inst✝²² : ChartedSpace H✝ M✝
inst✝²¹ : I✝.Boundaryless
inst✝²⁰ : Fin... |
https://github.com/fpvandoorn/sard.git | f4a1bb550136e8591dcd20dca5bdcce766a958c7 | Sard/MeasureZero.lean | measure_zero_image_iff_chart_domains | [163, 1] | [198, 41] | apply MeasureZero.iUnion (ι := T) this | E✝ : Type u_1
inst✝²⁶ : NormedAddCommGroup E✝
inst✝²⁵ : NormedSpace ℝ E✝
H✝ : Type u_2
inst✝²⁴ : TopologicalSpace H✝
I✝ : ModelWithCorners ℝ E✝ H✝
M✝ : Type u_3
inst✝²³ : TopologicalSpace M✝
inst✝²² : ChartedSpace H✝ M✝
inst✝²¹ : I✝.Boundaryless
inst✝²⁰ : FiniteDimensional ℝ E✝
inst✝¹⁹ : MeasurableSpace E✝
inst✝¹⁸ : Bo... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
E✝ : Type u_1
inst✝²⁶ : NormedAddCommGroup E✝
inst✝²⁵ : NormedSpace ℝ E✝
H✝ : Type u_2
inst✝²⁴ : TopologicalSpace H✝
I✝ : ModelWithCorners ℝ E✝ H✝
M✝ : Type u_3
inst✝²³ : TopologicalSpace M✝
inst✝²² : ChartedSpace H✝ M✝
inst✝²¹ : I✝.Boundaryless
inst✝²⁰ : Fin... |
https://github.com/fpvandoorn/sard.git | f4a1bb550136e8591dcd20dca5bdcce766a958c7 | Sard/MeasureZero.lean | measure_zero_image_iff_chart_domains | [163, 1] | [198, 41] | intro x _ | E✝ : Type u_1
inst✝²⁶ : NormedAddCommGroup E✝
inst✝²⁵ : NormedSpace ℝ E✝
H✝ : Type u_2
inst✝²⁴ : TopologicalSpace H✝
I✝ : ModelWithCorners ℝ E✝ H✝
M✝ : Type u_3
inst✝²³ : TopologicalSpace M✝
inst✝²² : ChartedSpace H✝ M✝
inst✝²¹ : I✝.Boundaryless
inst✝²⁰ : FiniteDimensional ℝ E✝
inst✝¹⁹ : MeasurableSpace E✝
inst✝¹⁸ : Bo... | E✝ : Type u_1
inst✝²⁶ : NormedAddCommGroup E✝
inst✝²⁵ : NormedSpace ℝ E✝
H✝ : Type u_2
inst✝²⁴ : TopologicalSpace H✝
I✝ : ModelWithCorners ℝ E✝ H✝
M✝ : Type u_3
inst✝²³ : TopologicalSpace M✝
inst✝²² : ChartedSpace H✝ M✝
inst✝²¹ : I✝.Boundaryless
inst✝²⁰ : FiniteDimensional ℝ E✝
inst✝¹⁹ : MeasurableSpace E✝
inst✝¹⁸ : Bo... | Please generate a tactic in lean4 to solve the state.
STATE:
E✝ : Type u_1
inst✝²⁶ : NormedAddCommGroup E✝
inst✝²⁵ : NormedSpace ℝ E✝
H✝ : Type u_2
inst✝²⁴ : TopologicalSpace H✝
I✝ : ModelWithCorners ℝ E✝ H✝
M✝ : Type u_3
inst✝²³ : TopologicalSpace M✝
inst✝²² : ChartedSpace H✝ M✝
inst✝²¹ : I✝.Boundaryless
inst✝²⁰ : Fin... |
https://github.com/fpvandoorn/sard.git | f4a1bb550136e8591dcd20dca5bdcce766a958c7 | Sard/MeasureZero.lean | measure_zero_image_iff_chart_domains | [163, 1] | [198, 41] | rw [mem_iUnion] | E✝ : Type u_1
inst✝²⁶ : NormedAddCommGroup E✝
inst✝²⁵ : NormedSpace ℝ E✝
H✝ : Type u_2
inst✝²⁴ : TopologicalSpace H✝
I✝ : ModelWithCorners ℝ E✝ H✝
M✝ : Type u_3
inst✝²³ : TopologicalSpace M✝
inst✝²² : ChartedSpace H✝ M✝
inst✝²¹ : I✝.Boundaryless
inst✝²⁰ : FiniteDimensional ℝ E✝
inst✝¹⁹ : MeasurableSpace E✝
inst✝¹⁸ : Bo... | E✝ : Type u_1
inst✝²⁶ : NormedAddCommGroup E✝
inst✝²⁵ : NormedSpace ℝ E✝
H✝ : Type u_2
inst✝²⁴ : TopologicalSpace H✝
I✝ : ModelWithCorners ℝ E✝ H✝
M✝ : Type u_3
inst✝²³ : TopologicalSpace M✝
inst✝²² : ChartedSpace H✝ M✝
inst✝²¹ : I✝.Boundaryless
inst✝²⁰ : FiniteDimensional ℝ E✝
inst✝¹⁹ : MeasurableSpace E✝
inst✝¹⁸ : Bo... | Please generate a tactic in lean4 to solve the state.
STATE:
E✝ : Type u_1
inst✝²⁶ : NormedAddCommGroup E✝
inst✝²⁵ : NormedSpace ℝ E✝
H✝ : Type u_2
inst✝²⁴ : TopologicalSpace H✝
I✝ : ModelWithCorners ℝ E✝ H✝
M✝ : Type u_3
inst✝²³ : TopologicalSpace M✝
inst✝²² : ChartedSpace H✝ M✝
inst✝²¹ : I✝.Boundaryless
inst✝²⁰ : Fin... |
https://github.com/fpvandoorn/sard.git | f4a1bb550136e8591dcd20dca5bdcce766a958c7 | Sard/MeasureZero.lean | measure_zero_image_iff_chart_domains | [163, 1] | [198, 41] | exact ⟨x, mem_chart_source H x⟩ | E✝ : Type u_1
inst✝²⁶ : NormedAddCommGroup E✝
inst✝²⁵ : NormedSpace ℝ E✝
H✝ : Type u_2
inst✝²⁴ : TopologicalSpace H✝
I✝ : ModelWithCorners ℝ E✝ H✝
M✝ : Type u_3
inst✝²³ : TopologicalSpace M✝
inst✝²² : ChartedSpace H✝ M✝
inst✝²¹ : I✝.Boundaryless
inst✝²⁰ : FiniteDimensional ℝ E✝
inst✝¹⁹ : MeasurableSpace E✝
inst✝¹⁸ : Bo... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
E✝ : Type u_1
inst✝²⁶ : NormedAddCommGroup E✝
inst✝²⁵ : NormedSpace ℝ E✝
H✝ : Type u_2
inst✝²⁴ : TopologicalSpace H✝
I✝ : ModelWithCorners ℝ E✝ H✝
M✝ : Type u_3
inst✝²³ : TopologicalSpace M✝
inst✝²² : ChartedSpace H✝ M✝
inst✝²¹ : I✝.Boundaryless
inst✝²⁰ : Fin... |
https://github.com/fpvandoorn/sard.git | f4a1bb550136e8591dcd20dca5bdcce766a958c7 | Sard/MeasureZero.lean | measure_zero_image_iff_chart_domains | [163, 1] | [198, 41] | intro i | E✝ : Type u_1
inst✝²⁶ : NormedAddCommGroup E✝
inst✝²⁵ : NormedSpace ℝ E✝
H✝ : Type u_2
inst✝²⁴ : TopologicalSpace H✝
I✝ : ModelWithCorners ℝ E✝ H✝
M✝ : Type u_3
inst✝²³ : TopologicalSpace M✝
inst✝²² : ChartedSpace H✝ M✝
inst✝²¹ : I✝.Boundaryless
inst✝²⁰ : FiniteDimensional ℝ E✝
inst✝¹⁹ : MeasurableSpace E✝
inst✝¹⁸ : Bo... | E✝ : Type u_1
inst✝²⁶ : NormedAddCommGroup E✝
inst✝²⁵ : NormedSpace ℝ E✝
H✝ : Type u_2
inst✝²⁴ : TopologicalSpace H✝
I✝ : ModelWithCorners ℝ E✝ H✝
M✝ : Type u_3
inst✝²³ : TopologicalSpace M✝
inst✝²² : ChartedSpace H✝ M✝
inst✝²¹ : I✝.Boundaryless
inst✝²⁰ : FiniteDimensional ℝ E✝
inst✝¹⁹ : MeasurableSpace E✝
inst✝¹⁸ : Bo... | Please generate a tactic in lean4 to solve the state.
STATE:
E✝ : Type u_1
inst✝²⁶ : NormedAddCommGroup E✝
inst✝²⁵ : NormedSpace ℝ E✝
H✝ : Type u_2
inst✝²⁴ : TopologicalSpace H✝
I✝ : ModelWithCorners ℝ E✝ H✝
M✝ : Type u_3
inst✝²³ : TopologicalSpace M✝
inst✝²² : ChartedSpace H✝ M✝
inst✝²¹ : I✝.Boundaryless
inst✝²⁰ : Fin... |
https://github.com/fpvandoorn/sard.git | f4a1bb550136e8591dcd20dca5bdcce766a958c7 | Sard/MeasureZero.lean | measure_zero_image_iff_chart_domains | [163, 1] | [198, 41] | let e : PartialHomeomorph M H := ChartedSpace.chartAt i | E✝ : Type u_1
inst✝²⁶ : NormedAddCommGroup E✝
inst✝²⁵ : NormedSpace ℝ E✝
H✝ : Type u_2
inst✝²⁴ : TopologicalSpace H✝
I✝ : ModelWithCorners ℝ E✝ H✝
M✝ : Type u_3
inst✝²³ : TopologicalSpace M✝
inst✝²² : ChartedSpace H✝ M✝
inst✝²¹ : I✝.Boundaryless
inst✝²⁰ : FiniteDimensional ℝ E✝
inst✝¹⁹ : MeasurableSpace E✝
inst✝¹⁸ : Bo... | E✝ : Type u_1
inst✝²⁶ : NormedAddCommGroup E✝
inst✝²⁵ : NormedSpace ℝ E✝
H✝ : Type u_2
inst✝²⁴ : TopologicalSpace H✝
I✝ : ModelWithCorners ℝ E✝ H✝
M✝ : Type u_3
inst✝²³ : TopologicalSpace M✝
inst✝²² : ChartedSpace H✝ M✝
inst✝²¹ : I✝.Boundaryless
inst✝²⁰ : FiniteDimensional ℝ E✝
inst✝¹⁹ : MeasurableSpace E✝
inst✝¹⁸ : Bo... | Please generate a tactic in lean4 to solve the state.
STATE:
E✝ : Type u_1
inst✝²⁶ : NormedAddCommGroup E✝
inst✝²⁵ : NormedSpace ℝ E✝
H✝ : Type u_2
inst✝²⁴ : TopologicalSpace H✝
I✝ : ModelWithCorners ℝ E✝ H✝
M✝ : Type u_3
inst✝²³ : TopologicalSpace M✝
inst✝²² : ChartedSpace H✝ M✝
inst✝²¹ : I✝.Boundaryless
inst✝²⁰ : Fin... |
https://github.com/fpvandoorn/sard.git | f4a1bb550136e8591dcd20dca5bdcce766a958c7 | Sard/MeasureZero.lean | measure_zero_image_iff_chart_domains | [163, 1] | [198, 41] | have h : MeasureZero J (f '' (e.source ∩ s)) := hs i | E✝ : Type u_1
inst✝²⁶ : NormedAddCommGroup E✝
inst✝²⁵ : NormedSpace ℝ E✝
H✝ : Type u_2
inst✝²⁴ : TopologicalSpace H✝
I✝ : ModelWithCorners ℝ E✝ H✝
M✝ : Type u_3
inst✝²³ : TopologicalSpace M✝
inst✝²² : ChartedSpace H✝ M✝
inst✝²¹ : I✝.Boundaryless
inst✝²⁰ : FiniteDimensional ℝ E✝
inst✝¹⁹ : MeasurableSpace E✝
inst✝¹⁸ : Bo... | E✝ : Type u_1
inst✝²⁶ : NormedAddCommGroup E✝
inst✝²⁵ : NormedSpace ℝ E✝
H✝ : Type u_2
inst✝²⁴ : TopologicalSpace H✝
I✝ : ModelWithCorners ℝ E✝ H✝
M✝ : Type u_3
inst✝²³ : TopologicalSpace M✝
inst✝²² : ChartedSpace H✝ M✝
inst✝²¹ : I✝.Boundaryless
inst✝²⁰ : FiniteDimensional ℝ E✝
inst✝¹⁹ : MeasurableSpace E✝
inst✝¹⁸ : Bo... | Please generate a tactic in lean4 to solve the state.
STATE:
E✝ : Type u_1
inst✝²⁶ : NormedAddCommGroup E✝
inst✝²⁵ : NormedSpace ℝ E✝
H✝ : Type u_2
inst✝²⁴ : TopologicalSpace H✝
I✝ : ModelWithCorners ℝ E✝ H✝
M✝ : Type u_3
inst✝²³ : TopologicalSpace M✝
inst✝²² : ChartedSpace H✝ M✝
inst✝²¹ : I✝.Boundaryless
inst✝²⁰ : Fin... |
https://github.com/fpvandoorn/sard.git | f4a1bb550136e8591dcd20dca5bdcce766a958c7 | Sard/MeasureZero.lean | measure_zero_image_iff_chart_domains | [163, 1] | [198, 41] | have h₃ : U i = e.source := by rw [← Filter.principal_eq_iff_eq] | E✝ : Type u_1
inst✝²⁶ : NormedAddCommGroup E✝
inst✝²⁵ : NormedSpace ℝ E✝
H✝ : Type u_2
inst✝²⁴ : TopologicalSpace H✝
I✝ : ModelWithCorners ℝ E✝ H✝
M✝ : Type u_3
inst✝²³ : TopologicalSpace M✝
inst✝²² : ChartedSpace H✝ M✝
inst✝²¹ : I✝.Boundaryless
inst✝²⁰ : FiniteDimensional ℝ E✝
inst✝¹⁹ : MeasurableSpace E✝
inst✝¹⁸ : Bo... | E✝ : Type u_1
inst✝²⁶ : NormedAddCommGroup E✝
inst✝²⁵ : NormedSpace ℝ E✝
H✝ : Type u_2
inst✝²⁴ : TopologicalSpace H✝
I✝ : ModelWithCorners ℝ E✝ H✝
M✝ : Type u_3
inst✝²³ : TopologicalSpace M✝
inst✝²² : ChartedSpace H✝ M✝
inst✝²¹ : I✝.Boundaryless
inst✝²⁰ : FiniteDimensional ℝ E✝
inst✝¹⁹ : MeasurableSpace E✝
inst✝¹⁸ : Bo... | Please generate a tactic in lean4 to solve the state.
STATE:
E✝ : Type u_1
inst✝²⁶ : NormedAddCommGroup E✝
inst✝²⁵ : NormedSpace ℝ E✝
H✝ : Type u_2
inst✝²⁴ : TopologicalSpace H✝
I✝ : ModelWithCorners ℝ E✝ H✝
M✝ : Type u_3
inst✝²³ : TopologicalSpace M✝
inst✝²² : ChartedSpace H✝ M✝
inst✝²¹ : I✝.Boundaryless
inst✝²⁰ : Fin... |
https://github.com/fpvandoorn/sard.git | f4a1bb550136e8591dcd20dca5bdcce766a958c7 | Sard/MeasureZero.lean | measure_zero_image_iff_chart_domains | [163, 1] | [198, 41] | apply h.mono _ | E✝ : Type u_1
inst✝²⁶ : NormedAddCommGroup E✝
inst✝²⁵ : NormedSpace ℝ E✝
H✝ : Type u_2
inst✝²⁴ : TopologicalSpace H✝
I✝ : ModelWithCorners ℝ E✝ H✝
M✝ : Type u_3
inst✝²³ : TopologicalSpace M✝
inst✝²² : ChartedSpace H✝ M✝
inst✝²¹ : I✝.Boundaryless
inst✝²⁰ : FiniteDimensional ℝ E✝
inst✝¹⁹ : MeasurableSpace E✝
inst✝¹⁸ : Bo... | E✝ : Type u_1
inst✝²⁶ : NormedAddCommGroup E✝
inst✝²⁵ : NormedSpace ℝ E✝
H✝ : Type u_2
inst✝²⁴ : TopologicalSpace H✝
I✝ : ModelWithCorners ℝ E✝ H✝
M✝ : Type u_3
inst✝²³ : TopologicalSpace M✝
inst✝²² : ChartedSpace H✝ M✝
inst✝²¹ : I✝.Boundaryless
inst✝²⁰ : FiniteDimensional ℝ E✝
inst✝¹⁹ : MeasurableSpace E✝
inst✝¹⁸ : Bo... | Please generate a tactic in lean4 to solve the state.
STATE:
E✝ : Type u_1
inst✝²⁶ : NormedAddCommGroup E✝
inst✝²⁵ : NormedSpace ℝ E✝
H✝ : Type u_2
inst✝²⁴ : TopologicalSpace H✝
I✝ : ModelWithCorners ℝ E✝ H✝
M✝ : Type u_3
inst✝²³ : TopologicalSpace M✝
inst✝²² : ChartedSpace H✝ M✝
inst✝²¹ : I✝.Boundaryless
inst✝²⁰ : Fin... |
https://github.com/fpvandoorn/sard.git | f4a1bb550136e8591dcd20dca5bdcce766a958c7 | Sard/MeasureZero.lean | measure_zero_image_iff_chart_domains | [163, 1] | [198, 41] | apply image_subset | E✝ : Type u_1
inst✝²⁶ : NormedAddCommGroup E✝
inst✝²⁵ : NormedSpace ℝ E✝
H✝ : Type u_2
inst✝²⁴ : TopologicalSpace H✝
I✝ : ModelWithCorners ℝ E✝ H✝
M✝ : Type u_3
inst✝²³ : TopologicalSpace M✝
inst✝²² : ChartedSpace H✝ M✝
inst✝²¹ : I✝.Boundaryless
inst✝²⁰ : FiniteDimensional ℝ E✝
inst✝¹⁹ : MeasurableSpace E✝
inst✝¹⁸ : Bo... | case h
E✝ : Type u_1
inst✝²⁶ : NormedAddCommGroup E✝
inst✝²⁵ : NormedSpace ℝ E✝
H✝ : Type u_2
inst✝²⁴ : TopologicalSpace H✝
I✝ : ModelWithCorners ℝ E✝ H✝
M✝ : Type u_3
inst✝²³ : TopologicalSpace M✝
inst✝²² : ChartedSpace H✝ M✝
inst✝²¹ : I✝.Boundaryless
inst✝²⁰ : FiniteDimensional ℝ E✝
inst✝¹⁹ : MeasurableSpace E✝
inst✝... | Please generate a tactic in lean4 to solve the state.
STATE:
E✝ : Type u_1
inst✝²⁶ : NormedAddCommGroup E✝
inst✝²⁵ : NormedSpace ℝ E✝
H✝ : Type u_2
inst✝²⁴ : TopologicalSpace H✝
I✝ : ModelWithCorners ℝ E✝ H✝
M✝ : Type u_3
inst✝²³ : TopologicalSpace M✝
inst✝²² : ChartedSpace H✝ M✝
inst✝²¹ : I✝.Boundaryless
inst✝²⁰ : Fin... |
https://github.com/fpvandoorn/sard.git | f4a1bb550136e8591dcd20dca5bdcce766a958c7 | Sard/MeasureZero.lean | measure_zero_image_iff_chart_domains | [163, 1] | [198, 41] | rw [h₃] | case h
E✝ : Type u_1
inst✝²⁶ : NormedAddCommGroup E✝
inst✝²⁵ : NormedSpace ℝ E✝
H✝ : Type u_2
inst✝²⁴ : TopologicalSpace H✝
I✝ : ModelWithCorners ℝ E✝ H✝
M✝ : Type u_3
inst✝²³ : TopologicalSpace M✝
inst✝²² : ChartedSpace H✝ M✝
inst✝²¹ : I✝.Boundaryless
inst✝²⁰ : FiniteDimensional ℝ E✝
inst✝¹⁹ : MeasurableSpace E✝
inst✝... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h
E✝ : Type u_1
inst✝²⁶ : NormedAddCommGroup E✝
inst✝²⁵ : NormedSpace ℝ E✝
H✝ : Type u_2
inst✝²⁴ : TopologicalSpace H✝
I✝ : ModelWithCorners ℝ E✝ H✝
M✝ : Type u_3
inst✝²³ : TopologicalSpace M✝
inst✝²² : ChartedSpace H✝ M✝
inst✝²¹ : I✝.Boundaryless
inst✝²... |
https://github.com/fpvandoorn/sard.git | f4a1bb550136e8591dcd20dca5bdcce766a958c7 | Sard/MeasureZero.lean | measure_zero_image_iff_chart_domains | [163, 1] | [198, 41] | rw [← Filter.principal_eq_iff_eq] | E✝ : Type u_1
inst✝²⁶ : NormedAddCommGroup E✝
inst✝²⁵ : NormedSpace ℝ E✝
H✝ : Type u_2
inst✝²⁴ : TopologicalSpace H✝
I✝ : ModelWithCorners ℝ E✝ H✝
M✝ : Type u_3
inst✝²³ : TopologicalSpace M✝
inst✝²² : ChartedSpace H✝ M✝
inst✝²¹ : I✝.Boundaryless
inst✝²⁰ : FiniteDimensional ℝ E✝
inst✝¹⁹ : MeasurableSpace E✝
inst✝¹⁸ : Bo... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
E✝ : Type u_1
inst✝²⁶ : NormedAddCommGroup E✝
inst✝²⁵ : NormedSpace ℝ E✝
H✝ : Type u_2
inst✝²⁴ : TopologicalSpace H✝
I✝ : ModelWithCorners ℝ E✝ H✝
M✝ : Type u_3
inst✝²³ : TopologicalSpace M✝
inst✝²² : ChartedSpace H✝ M✝
inst✝²¹ : I✝.Boundaryless
inst✝²⁰ : Fin... |
https://github.com/fpvandoorn/sard.git | f4a1bb550136e8591dcd20dca5bdcce766a958c7 | Sard/MeasureZero.lean | measure_zero_image_iff_chart_domains | [163, 1] | [198, 41] | rw [image_iUnion] | E✝ : Type u_1
inst✝²⁶ : NormedAddCommGroup E✝
inst✝²⁵ : NormedSpace ℝ E✝
H✝ : Type u_2
inst✝²⁴ : TopologicalSpace H✝
I✝ : ModelWithCorners ℝ E✝ H✝
M✝ : Type u_3
inst✝²³ : TopologicalSpace M✝
inst✝²² : ChartedSpace H✝ M✝
inst✝²¹ : I✝.Boundaryless
inst✝²⁰ : FiniteDimensional ℝ E✝
inst✝¹⁹ : MeasurableSpace E✝
inst✝¹⁸ : Bo... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
E✝ : Type u_1
inst✝²⁶ : NormedAddCommGroup E✝
inst✝²⁵ : NormedSpace ℝ E✝
H✝ : Type u_2
inst✝²⁴ : TopologicalSpace H✝
I✝ : ModelWithCorners ℝ E✝ H✝
M✝ : Type u_3
inst✝²³ : TopologicalSpace M✝
inst✝²² : ChartedSpace H✝ M✝
inst✝²¹ : I✝.Boundaryless
inst✝²⁰ : Fin... |
https://github.com/fpvandoorn/sard.git | f4a1bb550136e8591dcd20dca5bdcce766a958c7 | Sard/MeasureZero.lean | measure_zero_image_iff_chart_domains | [163, 1] | [198, 41] | rw [iUnion_inter] | E✝ : Type u_1
inst✝²⁶ : NormedAddCommGroup E✝
inst✝²⁵ : NormedSpace ℝ E✝
H✝ : Type u_2
inst✝²⁴ : TopologicalSpace H✝
I✝ : ModelWithCorners ℝ E✝ H✝
M✝ : Type u_3
inst✝²³ : TopologicalSpace M✝
inst✝²² : ChartedSpace H✝ M✝
inst✝²¹ : I✝.Boundaryless
inst✝²⁰ : FiniteDimensional ℝ E✝
inst✝¹⁹ : MeasurableSpace E✝
inst✝¹⁸ : Bo... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
E✝ : Type u_1
inst✝²⁶ : NormedAddCommGroup E✝
inst✝²⁵ : NormedSpace ℝ E✝
H✝ : Type u_2
inst✝²⁴ : TopologicalSpace H✝
I✝ : ModelWithCorners ℝ E✝ H✝
M✝ : Type u_3
inst✝²³ : TopologicalSpace M✝
inst✝²² : ChartedSpace H✝ M✝
inst✝²¹ : I✝.Boundaryless
inst✝²⁰ : Fin... |
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